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+{"text":"\\section{Introduction}\n\nThe stellarator concept \\citep{spitzerStellaratorConcept1958} offers a path to a steady-state and disruption-free fusion reactor with low recirculating power, but its complex three-dimensional geometry must be carefully designed to guarantee good plasma properties. In particular, the stellarator does not generally guarantee confinement of particles on collisionless trajectories due to its lack of continuous symmetry, leading to large neoclassical transport \\citep{helanderTheoryPlasmaConfinement2014}. However, the use of numerical optimisation techniques has led to advanced stellarator designs with good confinement properties, culminating in the design and construction of the HSX \\citep{andersonHelicallySymmetricExperiment1995} and W7-X stellarators \\citep{beidlerPhysicsEngineeringDesign1990}.\n\nAlthough gradient-based optimisation algorithms are generally more efficient than gradient-free algorithms, because of the large number of parameters (e.g. to represent the plasma boundary) they can be prohibitively expensive computationally if the gradients are evaluated via finite-differences. A more efficient way of obtaining gradient information is provided by adjoint methods, which were recently introduced in the stellarator optimisation field and have already found widespread application \\citep{landremanComputingLocalSensitivity2018, paulAdjointMethodGradientbased2018, antonsenAdjointApproachCalculating2019, paulAdjointMethodNeoclassical2019, paulAdjointApproachCalculating2020, paulAdjointMethodsStellarator2020, paulGradientbasedOptimization3D2021, geraldiniAdjointMethodDetermining2021, giulianiSinglestageGradientbasedStellarator2020}.\n\nPrevious work \\citep{antonsenAdjointApproachCalculating2019, paulAdjointApproachCalculating2020, paulGradientbasedOptimization3D2021} applied adjoint methods to ideal magnetohydrostatic (MHS) equilibria, building in the assumption of integrability, i.e. the existence of a set of nested flux surfaces. However, three-dimensional magnetic fields are generally not integrable due to the lack of continuous symmetry. Moreover, singularities arise at rational surfaces for linearised ideal MHS equilibria, making the computation of derivatives challenging \\citep{paulGradientbasedOptimization3D2021}. To overcome these challenges, different equilibrium models can be considered, such as vacuum or force-free fields. We herein apply adjoint methods to vacuum magnetic fields, relinquishing the assumption of global integrability, and avoiding the singular behaviour of MHS equilibria. Modeling the plasma magnetic field as a vacuum field is justified in the limit of vanishing plasma current and $\\beta$, the ratio of thermal pressure to magnetic pressure. Vacuum fields are thus broadly relevant for stellarators configurations, which tend to operate at low $\\beta$ and low plasma current, as non-axisymmetric shaping of the coils is used to generate rotational transform. Moreover, optimised vacuum solutions can serve as useful starting points for the optimisation of finite-pressure equilibria \\citep{boozerCurlfreeMagneticFields2019}.\n\nWe shall consider two objective functions, one targeting a rotational transform value on the boundary, and another targeting quasisymmetry on the boundary. As a subset of the larger class of omnigenous fields \\citep{hallThreedimensionalEquilibriumAnisotropic1975}, for which particles are confined on collisionless trajectories, quasisymmetric fields \\citep{nuhrenbergQuasihelicallySymmetricToroidal1988} have attracted strong interest, notably leading to the designs of the HSX \\citep{andersonHelicallySymmetricExperiment1995} and NCSX \\citep{zarnstorffPhysicsCompactAdvanced2001} stellarators. Multiple formulations of quasisymmetry exist \\citep{helanderTheoryPlasmaConfinement2014, rodriguezNecessarySufficientConditions2020, burbyMathematicsQuasisymmetry2020}, all of which employ flux coordinates and therefore require the existence of nested flux surfaces. We propose a method of constructing approximate flux coordinates on an isolated flux surface, on which quasisymmetry can then be defined and optimised for. The existence of at least one isolated flux surface will be guaranteed, by imposing the boundary condition that the magnetic field be tangential on a prescribed boundary. Note that we will not consider whether this boundary condition can actually be realised with a set of coils, a task pursued by codes like FOCUS \\citep{zhuNewMethodDesign2018}.\n\nWith the exception of \\cite{landremanMagneticFieldsPrecise2021}, previous optimisation studies \\citep{nuhrenbergQuasihelicallySymmetricToroidal1988, kuNewClassesQuasihelically2011, drevlakESTELLQuasiToroidallySymmetric2013, baderStellaratorEquilibriaReactor2019, hennebergPropertiesNewQuasiaxisymmetric2019, hennebergImprovingFastparticleConfinement2020, landremanStellaratorOptimizationGood2021} targeted quasisymmetry by minimising the symmetry-breaking components of the magnetic field strength in Boozer coordinates, often for vacuum magnetic fields. The most widely-used solver, whether for vacuum fields or plasmas with finite pressure, is the VMEC code \\citep{hirshmanThreedimensionalFreeBoundary1986}, which notably assumes the existence of nested flux surfaces. \nWe will employ the SPEC code \\citep{hudsonComputationMultiregionRelaxed2012}, which does not build in this assumption. Furthermore, in contrast to most previous studies, we use a formulation of quasisymmetry that does not rely on a Boozer coordinate transformation, although it still enables the specification of a desired helicity of the magnetic field strength. \n\nPrevious studies have sought to optimise for quasisymmetry either on a single flux surface \\citep[e.g.][]{hennebergImprovingFastparticleConfinement2020}, or on multiple flux surfaces \\citep[e.g.][]{landremanMagneticFieldsPrecise2021} with the aim of approximating quasisymmetry in a finite volume. We will herein consider quasisymmetry on a single flux surface only. Away from a surface with exact quasisymmetry, the deviation from quasisymmetry will generally increase linearly in the flux difference \\citep{senguptaVacuumMagneticFields2021}. It will thus be of interest to extend the present work on vacuum fields to multi-region relaxed magnetohydrodynamic (MRxMHD) equilibria. In this model, the interfaces between force-free regions are flux surfaces, such that quasisymmetry can be optimised for on multiple flux surfaces.\n\nThis paper is structured as follows. We begin with a brief introduction to adjoint methods in \\S\\ref{sec:basics_adjoint_methods}. A method of constructing approximate flux coordinates on a single flux surface is introduced in \\S\\ref{sec:approximate_flux_coordinates}. The derived adjoint equations for vacuum fields are presented in \\S\\ref{sec:adjoint_formalism_vacuum_fields}, first for a simpler objective function targeting a given rotational transform value on the boundary in \\S\\ref{sec:iota_fom_shape_derivative}, then for one targeting quasisymmetry on the boundary with a given helicity in \\S\\ref{sec:QS_fom_shape_derivative}. The resulting shape gradients are evaluated numerically and benchmarked against finite-difference calculations.\n\n\\section{Basics of adjoint methods}\n\\label{sec:basics_adjoint_methods}\n\nWe are interested in obtaining derivative information for a functional $f(\\mathcal{S}, u(\\mathcal{S}))$, called hereafter the objective function. This functional depends on the surface $\\mathcal{S}$ explicitly and also implicitly through the solution $u(\\mathcal{S})$ to a partial differential equation (PDE) $\\mathcal{P}(\\mathcal{S},u) = 0$. Here, $\\mathcal{P}$ is a general operator and $u$ is member of a Hilbert space with associated inner product $\\langle \\cdot , \\cdot \\rangle$, taken in our case to be the surface integral $\\int_\\mathcal{S} \\diff S \\;(\\cdot)(\\cdot)$. \n\nConsider a displacement of the surface $\\mathcal{S}$ in the direction $\\delta\\mathbf{x}$ with magnitude $\\epsilon$, resulting in a perturbed surface $\\mathcal{S}_\\epsilon = \\{ \\mathbf{x}_0 + \\epsilon \\delta\\mathbf{x}(\\mathbf{x}_0) : \\mathbf{x}_0 \\in \\mathcal{S}\\}$. The shape derivative of an arbitrary function $g(\\mathcal{S})$ in the direction $\\delta\\mathbf{x}$ is now defined as\n\\begin{equation}\n    \\delta g(\\mathcal{S})[\\delta\\mathbf{x}] = \\lim_{\\epsilon\\rightarrow 0} \\frac{g(\\mathcal{S_\\epsilon}) - g(\\mathcal{S})}{\\epsilon}.\n\\end{equation}\nIf $g$ depends only on the geometrical shape of the surface, the shape derivative $\\delta g[\\delta\\mathbf{x}]$ will be a function of only the normal component $\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec}$ of the displacement, as any tangential component of $\\delta\\mathbf{x}$ leaves the shape of $\\mathcal{S}$ unchanged to first order. Here, $\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}$ is a normal vector on $\\mathcal{S}$.\n\nTo compute derivatives of the objective function while enforcing the PDE constraint $\\mathcal{P}(\\mathcal{S},u) = 0$, the method of Lagrange multipliers is used. Consider the Lagrangian\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S},u,q) = f(\\mathcal{S},u) + \\int_\\mathcal{S} \\diff S \\; q \\; \\mathcal{P}(\\mathcal{S},u),\n\\end{equation}\nwith the Lagrange multiplier $q$. Its shape derivative $\\delta\\mathcal{L}[\\delta\\mathbf{x}]$ contains explicit contributions in the perturbation $\\delta\\mathbf{x}$, as well as implicit contributions through $\\delta q[\\delta\\mathbf{x}]$ and $\\delta u[\\delta\\mathbf{x}]$. The implicit dependencies are removed by making $\\mathcal{L}$ stationary with respect to $\\delta q[\\delta\\mathbf{x}]$, which is equivalent to enforcing the original PDE $\\mathcal{P}(\\mathcal{S},u) = 0$, and $\\delta u[\\delta\\mathbf{x}]$, which leads to an adjoint PDE for $q$. \n\nIf $\\mathcal{L}$ is stationary with respect to both $\\delta u[\\delta\\mathbf{x}]$ and $\\delta q[\\delta\\mathbf{x}]$, the remaining explicit dependence of its shape derivative $\\delta\\mathcal{L}(\\mathcal{S},u,q)[\\delta\\mathbf{x}]$ is equal to the shape derivative of the figure of merit $\\delta f(\\mathcal{S},u(\\mathcal{S}))[\\delta\\mathbf{x}]$ with $u=u(\\mathcal{S})$ satisfying the PDE constraint, as shown in e.g. \\citet{paulAdjointMethodsStellarator2020}. The Hadamard-Zol\\'esio structure theorem \\citep{delfourShapesGeometries2011} further states that the remaining contribution to the Lagrangian's shape derivative, provided $\\mathcal{L}$ is sufficiently smooth, can be expressed as\n\\begin{equation}\n    \\delta\\mathcal{L}(\\mathcal{S},u,q)[\\delta\\mathbf{x}] = \\int_\\mathcal{S} \\diff S \\;(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\; \\mathcal{G},\n\\end{equation}\nwhere $\\mathcal{G}$ is called the shape gradient, and can be interpreted as the local sensitivity of the objective function to perturbations of $\\mathcal{S}$.\n\nIn practice, the surface $\\mathcal{S}$ is typically represented by a finite set of parameters $\\Omega = \\{\\Omega_i,\\; i = 1, 2, \\dots N\\}$, e.g. Fourier coefficients $\\{R_{m,n}, Z_{m,n}\\}$, and the functional $f(\\mathcal{S},u(\\mathcal{S}))$ is approximated by a function $f(\\Omega,u(\\Omega))$. The derivative of $f(\\Omega,u(\\Omega))$ with respect to a parameter $\\Omega_i$ can be approximated as\n\\begin{equation}\n    \\frac{\\partial f(\\Omega, u(\\Omega))}{\\partial \\Omega_i} = \\int_\\mathcal{S} \\diff S\\; \\frac{\\partial \\mathbf{x}}{\\partial \\Omega_i}\\cdot \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} \\; \\mathcal{G}. \\label{eq:parameter_derivatives_shape_grad}\n\\end{equation}\nThe adjoint method of evaluating the parameter derivatives required for optimisation or sensitivity analysis \\citep{paulAdjointMethodsStellarator2020} thus consists in computing $\\delta\\mathcal{L}[\\delta\\mathbf{x}]$ to obtain the adjoint PDE for $q$ and the shape gradient $\\mathcal{G}$, which is then used to evaluate the right-hand-side of \\eqref{eq:parameter_derivatives_shape_grad}.\n\nWhen evaluating the parameter derivatives numerically through \\eqref{eq:parameter_derivatives_shape_grad}, errors are introduced from the inexact solutions to the original and adjoint PDEs. Indeed, these PDEs are assumed to be exactly satisfied in the preceding derivation to remove the implicit dependencies of $\\delta\\mathcal{L}[\\delta\\mathbf{x}]$ on $\\delta u[\\delta\\mathbf{x}]$ and $\\delta q[\\delta\\mathbf{x}]$, and also typically when deriving an expression for the shape gradient $\\mathcal{G}$.\n\nThe formalism presented above can easily be generalised to multiple PDE constraints, and, for a closed $\\mathcal{S}$, to PDEs satisfied not on $\\mathcal{S}$ but in the volume enclosed by it. This will be done in \\S\\ref{sec:adjoint_formalism_vacuum_fields}, where both the Laplace equation for the vacuum field and the straight field line equation, respectively valid in the volume and on the boundary, will be enforced as constraints, with two corresponding adjoint variables.\n\n\\section{Evaluating approximate flux coordinates on an isolated flux surface}\n\\label{sec:approximate_flux_coordinates}\n\nThe existence of nested flux surfaces is commonly assumed in theoretical studies of magnetically confined plasmas, e.g. to formulate quasisymmetry. In particular, many formulas involve $\\nabla\\psi$, where the toroidal flux $\\psi$ is a global flux surface label. However, three-dimensional magnetic fields lacking a continuous symmetry are not generally integrable. It is desirable to generalise $\\nabla\\psi$ to the case of an isolated flux surface, i.e. a flux surface in whose neighbourhood the field is generally non-integrable.\n\nOn a flux surface $\\mathcal{S}$, the magnetic field's normal component vanishes by definition, i.e.  $\\mathbf{B}\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0$ with $\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}$ the unit normal vector on $\\mathcal{S}$. The field line label $\\alpha$ on $\\mathcal{S}$ is defined through the straight field line equation $\\mathbf{B}\\cdot\\nabla_\\Upgamma\\alpha = 0$. Here, the tangential gradient $\\nabla_\\Upgamma$, defined in App.~\\ref{app:diff_operators_surfaces}, is the component of the 3D gradient tangential to the surface \\eqref{eq:def_tangential_gradient}. Note that in the integrable case, $\\nabla\\psi$ is normal to the flux surfaces, and the magnetic field satisfies $\\mathbf{B} = \\nabla\\psi\\times\\nabla\\alpha$.\n\nWe now define the generalisation $\\overline{\\nabla\\psi}$ on $\\mathcal{S}$ of the toroidal flux gradient $\\nabla\\psi$, by setting $\\overline{\\nabla\\psi}$ normal to $\\mathcal{S}$, and by requiring $\\mathbf{B} = \\overline{\\nabla\\psi}\\times\\nabla\\alpha$ to be satisfied on $\\mathcal{S}$. Squaring the latter equality and using $\\overline{\\nabla\\psi} = \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} |\\overline{\\nabla\\psi}|$ yields\n\\begin{equation}\n    \\overline{\\nabla\\psi} = \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} \\; \\frac{B}{\\abs{\\nabla_\\Upgamma \\alpha}}, \\label{eq:def_nabla_tilde_psi}\n\\end{equation}\nwhere $B = \\abs{\\mathbf{B}}$ is the magnetic field strength. Note that $\\overline{\\nabla\\psi}$ is defined through \\eqref{eq:def_nabla_tilde_psi}, and should not be misinterpreted as the gradient of a scalar function.\n\nThe defining expression for $\\overline{\\nabla\\psi}$ \\eqref{eq:def_nabla_tilde_psi} can be evaluated on any flux surface without requiring nested flux surfaces in its neighbourhood, and will revert to $\\overline{\\nabla\\psi} = \\nabla\\psi$ when the field is integrable in the neighbourhood of that flux surface. In practice, one might couple objective functions relying on \\eqref{eq:def_nabla_tilde_psi} with a figure of merit targeting integrability, aiming for a final plasma shape for which the field is integrable, such that $\\overline{\\nabla\\psi} = \\nabla\\psi$ and the minimised objective function represents the physical quantity of interest.\n\n\\begin{figure}\n\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/abs_nabla_psi_VMEC.pdf}\n  \\caption{}\n  \\label{fig:abs_nabla_psi_VMEC}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/abs_nabla_psi_SPEC.pdf}\n  \\caption{}\n  \\label{fig:abs_nabla_psi_SPEC}\n\\end{subfigure}\\\\\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/relative_diff_abs_nabla_psi.pdf}\n  \\caption{}\n  \\label{fig:relative_diff_abs_nabla_psi}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/poincare_surfaces_VMEC_SPEC.pdf}\n  \\caption{}\n  \\label{fig:poincare_surfaces_VMEC_SPEC}\n\\end{subfigure}%\n\\caption{Comparison of (a) the toroidal flux gradient $\\abs{\\nabla\\psi}$ evaluated with VMEC with (b) the generalised toroidal flux gradient $|\\overline{\\nabla\\psi}|$ \\eqref{eq:def_nabla_tilde_psi} obtained in SPEC, for a $5$-period rotating ellipse case with half a rotation per field period, major radius at the ellipse centre $R_0 = 5$\\;m, and ellipse major and minor axes values of $2$\\;m and $1$\\;m, respectively. The relative difference between the two quantities is below $1\\%$, as shown in (c). A small difference is to be expected in this case, where integrability is well satisfied, as attested in (d) by the Poincar\\'e plot at toroidal angle $\\phi=0$ from the SPEC calculation, which agrees well with the flux surfaces computed by VMEC. All data generated in this paper can be obtained from \\cite{niesDataCodesPaper2021}.}\n\\label{fig:abs_nabla_psi}\n\\end{figure}\n\nThe generalised toroidal flux gradient \\eqref{eq:def_nabla_tilde_psi} is evaluated in Fig.~\\ref{fig:abs_nabla_psi_SPEC} for a rotating ellipse configuration computed with the SPEC code. It agrees excellently with the toroidal flux gradient evaluated by VMEC, which can be calculated directly due to the imposed nestedness of flux surfaces, shown in Fig.~\\ref{fig:abs_nabla_psi_VMEC}. The relative difference between the two is below a percentage point in this case, as shown in Fig.~\\ref{fig:relative_diff_abs_nabla_psi}. The difference is expected to be small when integrability is satisfied, which indeed seems to hold here, as attested by the absence of islands and chaotic regions in the SPEC Poincar\\'e plot shown in Fig.~\\ref{fig:poincare_surfaces_VMEC_SPEC}. Note that SPEC solves for a vacuum magnetic field, while VMEC computes an ideal MHS equilibrium with vanishing thermal pressure, and with a plasma current that is small but finite due to the constraint of integrability.\n\nThe generalised toroidal flux gradient $\\overline{\\nabla\\psi}$ can be applied generally in any situation where local flux coordinates need to be evaluated, e.g. in calculations of perpendicular transport or magnetohydrodynamic stability. Isolated flux surfaces occur e.g. in fixed-boundary equilibrium calculations, where the plasma outer boundary is constrained to be a flux surface as a boundary condition on the magnetic field, or at the interfaces of MRxMHD equilibria computed by e.g. SPEC \\citep{hudsonComputationMultiregionRelaxed2012} or BIEST \\citep{malhotraTaylorStatesStellarators2019}. In the following, we will employ \\eqref{eq:def_nabla_tilde_psi} specifically for a fixed-boundary vacuum field to formulate quasisymmetry on the boundary.\n\n\\section{Application of adjoint formalism to vacuum fields}\n\\label{sec:adjoint_formalism_vacuum_fields}\n\nConsider a vacuum magnetic field $\\mathbf{B}$ in a toroidal domain $\\mathcal{V}$ bounded by the surface $\\mathcal{S} = \\partial \\mathcal{V}$. As the vacuum magnetic field is curl-free, it can be expressed as $\\mathbf{B}=\\nabla\\Phi$, with the scalar potential $\\Phi$. Because we consider a simple torus $\\mathcal{V}$, the most general form for the scalar potential is $\\Phi=G (\\omega+\\phi)$, where $G$ is a constant, $\\omega$ is a single-valued function on $\\mathcal{S}$, and $\\phi$ is an arbitrary toroidal angle. By integrating the magnetic field along a toroidal loop around the torus, the constant $G$ is found to be proportional to the net external current through the `hole' of the torus.\n\nAs the magnetic field is divergence-less, the magnetic scalar potential satisfies the Laplace equation. The field's normal component is constrained to vanish on $\\mathcal{S}$ by imposing a Neumann boundary condition on the magnetic scalar potential. Further prescribing e.g. $G$, or the toroidal flux, guarantees a unique solution to Laplace's equation. We herein opt to hold the toroidal flux fixed, although the shape derivative $\\delta G[\\delta\\mathbf{x}]$ will not appear in this study due to our normalisation of the figure of merit for quasisymmetry \\eqref{eq:definition_fQS}. A different choice of normalisation would lead to an additional contribution proportional to $\\delta G[\\delta\\mathbf{x}]$ in the shape derivative of the Lagrangian.\n\nFor convenience, we define the normalised magnetic field $\\mathbf{\\breve{B}}$ as\n\\begin{equation}\n    \\mathbf{\\breve{B}} \\equiv \\mathbf{B}\/G = \\nabla\\big(\\omega + \\phi\\big). \\label{eq:magnetic_field_scalar_pot}\n\\end{equation}\nLet us further assume the toroidal angle $\\phi$ to be the azimuthal angle in cylindrical coordinates, satisfying $\\Updelta\\phi=0$ in the domain of interest. We can thus write\n\\begin{subequations}\n\\begin{align}\n    \\nabla\\cdot\\mathbf{\\breve{B}} = \\Updelta\\omega = 0   \\qquad\\qquad\\qquad\\mathrm{in}\\;&\\mathcal{V}, \\label{eq:vacuum_field_Laplace}\\\\\n    \\mathbf{\\breve{B}}\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = \\nabla(\\omega+\\phi)\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0 \\qquad\\mathrm{on}\\; & \\mathcal{S} \\label{eq:vacuum_field_normal_BC},\n\\end{align}\n\\end{subequations}\nwith $\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}$ the normal unit vector on $\\mathcal{S}$. Furthermore, the shape derivative $\\delta\\omega[\\delta\\mathbf{x}]$ satisfies\n\\begin{subequations}\n\\begin{align}\n    \\Updelta(\\delta\\omega[\\delta\\mathbf{x}]) = 0   \\qquad\\qquad\\mathrm{in}\\;&\\mathcal{V}, \\label{eq:vacuum_field_pert_Laplace}\\\\\n    \\mathbf{\\breve{B}}\\cdot\\delta\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}[\\delta\\mathbf{x}] + \\nabla(\\delta\\omega[\\delta\\mathbf{x}])\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} + (\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\;\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla \\mathbf{\\breve{B}}\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0 \\quad\\mathrm{on}\\; & \\mathcal{S} \\label{eq:vacuum_field_pert_normal_BC},\n\\end{align}\n\\end{subequations}\nwhere the Laplace equation is obtained from noting the commutative property of shape and spatial derivatives, and the normal boundary condition on $\\delta\\omega$ was derived in e.g. \\citet[\\S 3.2]{sokolowskiIntroductionShapeOptimization1992}. The shape derivative of the normal vector $\\delta\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}[\\delta\\mathbf{x}]=-\\nabla_\\Upgamma(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})$ is derived in App.~\\ref{app:normal_extension_and_normal_vector_shape_derivative}.\n\nEvaluating the rotational transform and quasisymmetry figures of merit further requires the solution to the straight field line equation\n\\begin{equation}\n    0 = \\mathbf{B}\\cdot\\nabla_\\Upgamma\\alpha \\qquad\\mathrm{on}\\; \\mathcal{S}, \\label{eq:straight_field_line_eq}\n\\end{equation}\nwith the field line label\n\\begin{equation}\n   \\alpha \\equiv \\theta-\\iota\\phi+\\lambda(\\theta,\\phi), \\label{eq:def_field_line_label_alpha}\n\\end{equation}\nwhere $\\theta$ is a general poloidal angle, $\\lambda$ is a single-valued function of $\\theta$ and $\\phi$, and $\\iota$ is a scalar. Note that both $\\lambda$ and $\\iota$ are defined on the boundary $\\mathcal{S}$ only, through \\eqref{eq:def_field_line_label_alpha}.\n\nLet us define the Lagrangian corresponding to an arbitrary objective function $f(\\mathcal{S},\\omega,\\iota,\\lambda)$,\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}, \\omega, q_\\omega, \\iota, \\lambda, q_\\alpha ) = f(\\mathcal{S},\\omega,\\iota,\\lambda) + \\mathcal{M}(\\mathcal{S}, \\omega, q_\\omega) + \\mathcal{N}(\\mathcal{S}, \\omega, \\iota, \\lambda, q_\\alpha), \\label{eq:lagrangian_general}\n\\end{equation}\nwith the weak form of the Laplace equation \\eqref{eq:vacuum_field_Laplace}\n\\begin{equation}\n    \\mathcal{M}(\\mathcal{S}, \\omega, q_\\omega) = \\int_{\\mathcal{V}} \\diff V \\; q_\\omega \\Updelta \\omega, \\label{eq:weak_form_Laplace_vac}\n\\end{equation}\nand the weak form of the straight field line equation \\eqref{eq:straight_field_line_eq} normalised by $G$ \n\\begin{equation}\n    \\mathcal{N}(\\mathcal{S}, \\omega, \\iota, \\lambda, q_\\alpha) = \\int_{\\mathcal{S}} \\diff S \\; q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla_\\Upgamma \\alpha. \\label{eq:weak_form_straight_field_line_eq}\n\\end{equation}\nAs explained in \\S\\ref{sec:basics_adjoint_methods}, $q_\\omega$ and $q_\\alpha$ act as Lagrange multipliers: making the Lagrangian \\eqref{eq:lagrangian_general} stationary with respect to $\\delta q_\\omega[\\delta\\mathbf{x}]$ and $\\delta q_\\alpha[\\delta\\mathbf{x}]$ ensures that the Laplace \\eqref{eq:vacuum_field_Laplace} and straight field line \\eqref{eq:straight_field_line_eq} equations are satisfied, respectively. These trivial variations are omitted in the following under the assumption that \\eqref{eq:vacuum_field_Laplace} and \\eqref{eq:straight_field_line_eq} are satisfied, thus considering only the implicit dependencies of $\\delta\\mathcal{L}[\\delta\\mathbf{x}]$ on $\\delta\\omega[\\delta\\mathbf{x}]$, $\\delta\\iota[\\delta\\mathbf{x}]$, and $\\delta\\lambda[\\delta\\mathbf{x}]$ to obtain the adjoint equations for $q_\\omega$ and $q_\\alpha$.\n\nFirst, the shape derivative of $\\mathcal{M}$ \\eqref{eq:weak_form_Laplace_vac}, derived in App.~\\ref{app:derivation_laplace_eq_shape_derivative}, is\n\\begin{equation}\n    \\delta\\mathcal{M}[\\delta\\mathbf{x}] = \\int_{\\mathcal{V}} \\diff V \\; \\delta\\omega[\\delta\\mathbf{x}] \\Updelta q_\\omega - \\int_{\\mathcal{S}} \\diff S \\; \\bigg[ \\delta \\omega[\\delta\\mathbf{x}]  \\nabla q_\\omega \\cdot \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} - (\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\;  \\mathbf{\\breve{B}}\\cdot\\nabla q_\\omega  \\bigg]. \\label{eq:variation_M_tot}\n\\end{equation}\n\nSecond, the shape derivative of $\\mathcal{N}$ \\eqref{eq:weak_form_straight_field_line_eq}, derived in App.~\\ref{app:derivation_straight_field_line_eq_shape_derivative}, is\n\\begin{align}\n    \\delta\\mathcal{N}[\\delta\\mathbf{x}]  = \\int_{\\mathcal{S}} & \\diff S\\;  \\bigg[ - \\delta\\omega[\\delta\\mathbf{x}] \\;\\nabla_\\Upgamma \\cdot \\left(q_\\alpha \\nabla_\\Upgamma \\alpha\\right) - \\delta\\iota[\\delta\\mathbf{x}]\\; q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla\\phi   \\label{eq:variation_N_tot}\\\\\n    \\nonumber & - \\delta\\lambda[\\delta\\mathbf{x}]\\; \\nabla_\\Upgamma \\cdot \\left( q_\\alpha \\mathbf{\\breve{B}} \\right) + (\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec}) q_\\alpha \\left( \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{\\breve{B}}\\cdot\\nabla_\\Upgamma\\alpha - \\mathbf{\\breve{B}}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla_\\Upgamma\\alpha \\right) \\bigg].\n\\end{align}\nThe tangential gradient $\\nabla_\\Upgamma(\\cdot)$ and tangential divergence $\\nabla_\\Upgamma\\cdot(\\cdot)$ operators are defined in App.~\\ref{app:diff_operators_surfaces}.\n\nWe now proceed by computing the shape derivatives of two objective functions, first targeting a given rotational transform value on $\\mathcal{S}$ (\\S\\ref{sec:iota_fom_shape_derivative}), and second targeting quasi-symmetry on $\\mathcal{S}$ with a given helicity value (\\S\\ref{sec:QS_fom_shape_derivative}). We will then be able to evaluate the shape derivative of the Lagrangian \\eqref{eq:lagrangian_general}, yielding the adjoint equations and shape gradient formulas. Numerical verification and example shape gradients are shown for each figure of merit.\n\n\\subsection{Rotational transform objective function}\n\\label{sec:iota_fom_shape_derivative}\n\nBefore evaluating the more complicated shape gradient for the quasisymmetry figure of merit in \\S\\ref{sec:QS_fom_shape_derivative}, we consider a simple figure of merit targeting a given target rotational transform $\\iota_T$ on the surface $\\mathcal{S}$. We thus define\n\\begin{equation}\n    f_\\iota(\\iota) = \\frac{1}{2} ( \\iota - \\iota_T )^2  \\label{eq:f_iota},\n\\end{equation}\nwhere $\\iota$ is the rotational transform on $\\mathcal{S}$, obtained by solving the straight field line equation \\eqref{eq:straight_field_line_eq}. The shape derivative of $f_\\iota$ is simply\n\\begin{equation}\n    \\delta f_\\iota [\\delta\\mathbf{x}] = \\delta \\iota[\\delta\\mathbf{x}] \\; ( \\iota - \\iota_T ). \\label{eq:variation_f_iota_tot}\n\\end{equation}\nBy combining \\eqref{eq:variation_M_tot}, \\eqref{eq:variation_N_tot}, and \\eqref{eq:variation_f_iota_tot}, we obtain the shape derivative of the Lagrangian $\\mathcal{L}_\\iota$ [\\eqref{eq:lagrangian_general} with $f=f_\\iota$],\n\\begin{align}\n    \\delta\\mathcal{L}&_\\iota[\\delta\\mathbf{x}]  = \\int_{\\mathcal{V}} \\diff V \\; \\delta\\omega[\\delta\\mathbf{x}] \\; \\Updelta q_\\omega - \\int_{\\mathcal{S}} \\diff S \\;  \\delta \\omega [\\delta\\mathbf{x}] \\bigg[ \\nabla q_\\omega \\cdot \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} + \\nabla_\\Upgamma \\cdot \\Big(q_\\alpha \\nabla_\\Upgamma \\alpha \\Big) \\bigg] \\label{eq:variation_L_iota_tot}\\\\\n    \\nonumber & - \\delta\\iota[\\delta\\mathbf{x}]\\; \\bigg[ \\int_{\\mathcal{S}} \\diff S \\; q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla\\phi + ( \\iota - \\iota_T ) \\bigg] - \\int_{\\mathcal{S}} \\diff S \\;\\delta\\lambda[\\delta\\mathbf{x}]\\; \\nabla_\\Upgamma \\cdot \\Big( q_\\alpha \\mathbf{\\breve{B}} \\Big)  \\\\\n    \\nonumber & + \\int_{\\mathcal{S}} \\diff S \\;(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec}) \\bigg[\\nabla q_\\omega \\cdot \\mathbf{\\breve{B}} + q_\\alpha \\left( \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{\\breve{B}}\\cdot\\nabla_\\Upgamma\\alpha - \\mathbf{\\breve{B}}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla_\\Upgamma\\alpha \\right) \\bigg] \\Bigg].\n\\end{align}\n\nFirst, we obtain the adjoint equation for $q_\\alpha$ by requiring the second line of \\eqref{eq:variation_L_iota_tot} to vanish,\n\\begin{subequations}\n\\begin{align}\n   \\nabla_\\Upgamma \\cdot \\Big( q_\\alpha \\mathbf{\\breve{B}} \\Big) = 0 \\label{eq:iota_fom_adjoint_diff_eq_qsfl}, \\\\\n   \\int_{\\mathcal{S}} \\diff S \\; q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla\\phi + ( \\iota - \\iota_T ) = 0 \\label{eq:iota_fom_adjoint_integral_eq_qsfl},\n\\end{align}\n\\end{subequations}\nwith both equations defined on $\\mathcal{S}$. Using \\eqref{eq:surface_divergence_theorem}, the surface integral of \\eqref{eq:iota_fom_adjoint_diff_eq_qsfl} yields $0 = \\mathbf{\\breve{B}}\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}$, which is consistent with the boundary condition on the magnetic field \\eqref{eq:vacuum_field_normal_BC}. The first equation \\eqref{eq:iota_fom_adjoint_diff_eq_qsfl} can be recast in the form of a magnetic differential equation $\\mathbf{B}\\cdot\\nabla q_\\alpha = - q_\\alpha\\nabla_\\Upgamma\\cdot\\mathbf{B}$, while the second equation \\eqref{eq:iota_fom_adjoint_integral_eq_qsfl} is an integral condition on $q_\\alpha$ that ensures uniqueness of the solution.\n\nSecond, the adjoint equation for $q_\\omega$ is obtained by requiring the first line of \\eqref{eq:variation_L_iota_tot} to vanish,\n\\begin{subequations}\n\\begin{align}\n    \\Updelta q_\\omega = 0   \\qquad\\qquad\\qquad\\mathrm{in}\\;&\\mathcal{V}, \\label{eq:iota_fom_adjoint_eq_qomega}\\\\\n    \\nabla q_\\omega\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = -\\nabla_\\Upgamma \\cdot \\Big(q_\\alpha \\nabla_\\Upgamma \\alpha \\Big) \\qquad\\mathrm{on}\\; & \\mathcal{S} \\label{eq:iota_fom_adjoint_eq_qomega_normal_BC}.\n\\end{align}\n\\end{subequations}\nLike the magnetic potential $\\omega$, the adjoint variable $q_\\omega$ satisfies the Laplace equation in $\\mathcal{V}$ \\eqref{eq:iota_fom_adjoint_eq_qomega}. However, contrary to $\\omega$, $q_\\omega$ has a non-zero normal boundary condition on $\\mathcal{S}$ \\eqref{eq:iota_fom_adjoint_eq_qomega_normal_BC}, which notably depends on the straight field line adjoint variable $q_\\alpha$. Equations \\eqref{eq:iota_fom_adjoint_eq_qomega_normal_BC} and \\eqref{eq:iota_fom_adjoint_eq_qomega} are consistent, as $\\int_\\mathcal{V} \\diff V\\; \\Updelta q_\\omega = \\int_\\mathcal{S} \\diff S\\; \\nabla q_\\omega\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0$, by \\eqref{eq:surface_divergence_theorem}.\n\nFinally, the remaining contribution from the last line of \\eqref{eq:variation_L_iota_tot} yields the shape gradient\n\\begin{equation}\n   \\mathcal{G_\\iota} = \\frac{1}{G} \\Big[ \\mathbf{B}\\cdot \\nabla q_\\omega + q_\\alpha \\big( \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{B} - \\mathbf{B}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} \\big)\\cdot\\nabla_\\Upgamma \\alpha \\Big], \\label{eq:shape_gradient_iota_fom}\n\\end{equation}\nwith $\\delta\\mathcal{L}_\\iota[\\delta\\mathbf{x}]  = \\int_{\\mathcal{S}} \\diff S \\;(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\;\\mathcal{G_\\iota}$.\n\nWe now calculate the shape gradient numerically and verify it against a finite-difference evaluation. The solutions to Laplace's equation for the vacuum magnetic field \\eqref{eq:vacuum_field_Laplace} and adjoint equation for $q_\\omega$ \\eqref{eq:iota_fom_adjoint_eq_qomega} are calculated with the SPEC code \\citep{hudsonComputationMultiregionRelaxed2012}, employing the new Zernike polynomial implementation \\citep{quCoordinateParameterisationSpectral2020}. In all results shown, the radial resolution $L_\\mathrm{rad}$ in SPEC is tied to the poloidal Fourier resolution $M_\\mathrm{pol}$ through $L_\\mathrm{rad} = M_\\mathrm{pol} + 4$. The solutions to the straight field line and $q_\\alpha$ adjoint equations are obtained with a Fourier-Galerkin spectral solver.\n\n\\begin{figure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_analytic_iota_target.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_iota_fom_adjoint}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_findiff_iota_target.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_iota_fom_findiff}\n\\end{subfigure}\n\\\\\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_relative_error_iota_target.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_iota_fom_relative_error}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/dfdomega_convergence_epsilon_rel_error_iota_fom.pdf}\n  \\caption{}\n  \\label{fig:convergence_iota_fom}\n\\end{subfigure}\n\\caption{Shape gradient for the rotational transform objective function with $\\iota_T = 1$, evaluated through (a) adjoint methods and (b) a forward finite-difference scheme with step size $\\epsilon_\\mathrm{FD}=10^{-7}$, for the example rotating ellipse case introduced in Fig.~\\ref{fig:abs_nabla_psi}, with Fourier resolution $(N_\\mathrm{tor}, M_\\mathrm{pol}) = (16,16)$. The relative error, defined as the absolute error normalised by the maximal absolute value of the adjoint shape gradient, is shown in (c). The convergence of the relative error in the parameter derivative \\eqref{eq:parameter_derivatives_shape_grad} for a random direction in $\\Omega$ is shown in (d) as a function of the step-size $\\epsilon_\\mathrm{FD}$ and Fourier resolution $(N_\\mathrm{tor}, M_\\mathrm{pol})$. The black dashed line indicates the linear scaling in $\\epsilon_\\mathrm{FD}$ expected from the employed forward finite-difference scheme.}\n\\label{fig:iota_fom_numerical_eval_and_convergence}\n\\end{figure}\n\nThe shape gradient \\eqref{eq:shape_gradient_iota_fom} is shown in Fig.~\\ref{fig:shape_grad_iota_fom_adjoint} for the example rotating ellipse case introduced in Fig.~\\ref{fig:abs_nabla_psi}. The localisation at the ellipse tips is unsurprising, as near-axis expansions show that ellipticity of the flux surfaces generates rotational transform \\citep{mercierEquilibriumStabilityToroidal1964}. This shape gradient $\\mathcal{G}_\\mathrm{adjoint}$ can be verified against the direct finite-difference evaluation $\\mathcal{G}_\\mathrm{FD}$ shown in Fig.~\\ref{fig:shape_grad_iota_fom_findiff}, obtained by evaluating the parameter derivatives $\\partial f\/\\partial \\Omega_i$ through finite-differences and inverting \\eqref{eq:parameter_derivatives_shape_grad}, see \\citet{landremanComputingLocalSensitivity2018}. On the scale of the figure, the two shape gradients seem identical. The relative error is shown in Fig.~\\ref{fig:shape_grad_iota_fom_relative_error} to be small, limited to $\\sim 2\\%$ at the ellipse tips, and exhibits oscillations typical of a truncated Fourier resolution. The relative error is here defined as the absolute error normalised by the $L^\\infty$-norm of $\\mathcal{G}_\\mathrm{adjoint}$, i.e. its maximum absolute value. This choice is preferable to e.g. the $L^2$-norm, as the shape gradient and the error thereof have small average values on the boundary compared to their large values at the ellipse tips, such that unreasonably high relative errors would result at these locations if using the $L^2$-norm as normalisation. \n\nFurthermore, we test convergence of the shape gradient by evaluating a parameter derivative \\eqref{eq:parameter_derivatives_shape_grad} for a random direction in $\\Omega$. The parameter derivative is evaluated both through the adjoint shape gradient and by a forward finite-difference scheme. The relative error is shown in Fig.~\\ref{fig:convergence_iota_fom} as a function of the finite-difference step size $\\epsilon_\\mathrm{FD}$ and the Fourier resolution, which is used both in SPEC and the Fourier-Galerkin spectral solver. As $\\epsilon_\\mathrm{FD}$ is reduced, the error initially decreases linearly with $\\epsilon_\\mathrm{FD}$, as expected from the employed forward finite-difference scheme, until it plateaus at a value governed by the finite Fourier or radial resolution. As mentioned in \\S\\ref{sec:basics_adjoint_methods}, errors in the adjoint shape gradient are introduced by the assumption that the constraint and adjoint PDEs are exactly satisfied. In practice, these PDEs are solved only approximately, limited by the finite Fourier and radial resolution, such that a reduction of the error with increasing resolution is to be expected.\n\n\\subsection{Quasisymmetry objective function}\n\\label{sec:QS_fom_shape_derivative}\n\nFor a general (non-vacuum) magnetic field with nested flux surfaces, quasisymmetry can be expressed as\n\\begin{equation}\n    \\frac{\\mathbf{B}\\cdot\\nabla\\psi\\times\\nabla B}{\\mathbf{B}\\cdot\\nabla B} = -\\frac{MG + NI}{N-\\iota M}, \\label{eq:quasisymmetry_magnetic_field_condition}\n\\end{equation}\nwhere $I$ is the net toroidal plasma current and $N\/M$ is the helicity of the field strength in Boozer coordinates, see e.g. \\citet{helanderTheoryPlasmaConfinement2014}. For the vacuum field considered here, $I=0$. In the following, we will not consider quasi-poloidal symmetry, i.e. we will assume $M\\ne 0$. If desired, it would be straight-forward to extend the derived results to include the case $M=0$.\n\nFor magnetic fields with globally nested flux surfaces labelled by $\\psi$, \\eqref{eq:quasisymmetry_magnetic_field_condition} is defined globally. However, we are considering a generally non-integrable field, assuming only that the boundary $\\mathcal{S}$ is a flux surface. Using the generalised toroidal flux gradient defined in \\eqref{eq:def_nabla_tilde_psi}, we are able to define quasisymmetry on the isolated flux surface $\\mathcal{S}$, leading to the quasisymmetry (QS) objective function\n\\begin{equation}\n    f_\\mathrm{QS}(\\mathcal{S}, \\omega, \\iota, \\lambda) =  \\frac{1}{2} \\int_\\mathcal{S} \\diff S\\; v_\\mathrm{QS}^2(\\omega, \\iota, \\lambda), \\label{eq:definition_fQS}\n\\end{equation}\nwith\n\\begin{equation}\n    v_\\mathrm{QS} =  \\mathbf{\\breve{B}}\\cdot\\nabla \\breve{B}  -  \\mathbf{\\breve{B}}\\times\\frac{\\overline{\\nabla\\psi}}{G}\\cdot\\nabla \\breve{B}  \\left( \\iota - N\/M\\right). \\label{eq:definition_vQS}\n\\end{equation}\nIf $f_\\mathrm{QS}=0$ and the field is integrable in the neighbourhood of $\\mathcal{S}$, \\eqref{eq:quasisymmetry_magnetic_field_condition} will be satisfied on $\\mathcal{S}$, i.e. the field is quasisymmetric on the boundary.\n\nThe shape derivative of $f_\\mathrm{QS}$ is derived in App.~\\ref{app:derivation_QS_fom_shape_derivative}, with the final expression given in \\eqref{eq:delta_fQS_tot}. Combined with the shape derivatives of $\\mathcal{M}$ \\eqref{eq:variation_M_tot} and $\\mathcal{N}$ \\eqref{eq:variation_N_tot}, the shape derivative of the Lagrangian \\eqref{eq:lagrangian_general} with the quasisymmetric figure of merit follows \\eqref{eq:variation_L_QS_tot}.\n\nRequiring the Lagrangian to be stationary with respect to variations in $\\iota$ and $\\lambda$, the first two lines of \\eqref{eq:variation_L_QS_tot} yield the adjoint equations for $q_\\alpha$,\n\\begin{subequations}\n\\begin{align}\n   \\nabla_\\Upgamma \\cdot \\Big( q_\\alpha \\mathbf{\\breve{B}} \\Big) = -\\nabla_\\Upgamma\\cdot\\left[ \\nabla_\\Upgamma\\alpha \\left( v_\\mathrm{QS}\\; \\mathbf{\\breve{B}}\\times\\frac{\\overline{\\nabla\\psi}}{G}\\cdot\\nabla \\breve{B}\\; \\frac{\\iota - N\/M}{\\abs{\\nabla_\\Upgamma \\alpha}^2}\\right) \\right] \\label{eq:QS_fom_adjoint_diff_eq_qsfl}, \\\\\n   0=\\int_{\\mathcal{S}} \\diff S \\left\\{ q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla\\phi + v_\\mathrm{QS}\\; \\mathbf{\\breve{B}}\\times\\frac{\\overline{\\nabla\\psi}}{G}\\cdot\\nabla \\breve{B} \\left[ \\frac{\\nabla_\\Upgamma\\alpha\\cdot\\nabla_\\Upgamma \\phi}{\\abs{\\nabla_\\Upgamma \\alpha}^2} (\\iota-N\/M) + 1 \\right] \\right\\} \\label{eq:QS_fom_adjoint_integral_eq_qsfl}.\n\\end{align}\n\\end{subequations}\nSimilarly to the rotational transform objective function case, $q_\\alpha$ satisfies a magnetic differential equation \\eqref{eq:QS_fom_adjoint_diff_eq_qsfl} on $\\mathcal{S}$, with integral condition \\eqref{eq:QS_fom_adjoint_integral_eq_qsfl}. By \\eqref{eq:surface_divergence_theorem}, the surface integral of \\eqref{eq:QS_fom_adjoint_diff_eq_qsfl} is consistent with the magnetic field's normal component vanishing on the boundary \\eqref{eq:vacuum_field_normal_BC}.\n\nFurthermore, requiring the Lagrangian to be stationary with respect to variations in $\\omega$, we obtain the adjoint equations for $q_\\omega$ from the third and fourth lines of \\eqref{eq:variation_L_QS_tot}\n\\begin{subequations}\n\\begin{align}\n    \\Updelta q_\\omega & = 0   \\qquad\\qquad\\qquad\\mathrm{in}\\;\\mathcal{V}, \\label{eq:QS_fom_adjoint_eq_qomega}\\\\\n    \\nabla q_\\omega \\cdot \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} &= -\\nabla_\\Upgamma \\cdot \\Bigg\\{ q_\\alpha \\nabla_\\Upgamma \\alpha + v_\\mathrm{QS} \\;\\nabla_\\Upgamma \\breve{B} -  \\frac{\\mathbf{\\breve{B}}}{\\breve{B}} \\nabla_\\Upgamma \\cdot (v_\\mathrm{QS} \\; \\mathbf{\\breve{B}})\\label{eq:QS_fom_adjoint_eq_qomega_normal_BC} \\\\\n    \\nonumber & + \\left(\\iota-N\/M\\right) \\left[ v_\\mathrm{QS} \\; \\frac{\\overline{\\nabla\\psi}}{G}\\times\\nabla_\\Upgamma\\breve{B} - \\mathbf{\\breve{B}}\\; \\nabla_\\Upgamma\\cdot\\left(\\frac{1}{\\breve{B}}v_\\mathrm{QS} \\mathbf{\\breve{B}}\\times\\frac{\\overline{\\nabla\\psi}}{G}\\right)  \\right] \\Bigg\\} \\quad \\text{ on } \\mathcal{S}.\n\\end{align}\n\\end{subequations}\nAgain, $q_\\omega$ satisfies the Laplace equation in $\\mathcal{V}$ \\eqref{eq:QS_fom_adjoint_eq_qomega}, with a normal boundary condition on $\\mathcal{S}$ that is the tangential divergence of a vector tangential to the surface \\eqref{eq:QS_fom_adjoint_eq_qomega_normal_BC}. The boundary condition \\eqref{eq:QS_fom_adjoint_eq_qomega_normal_BC} is consistent with the Laplace equation, as $\\int_\\mathcal{V} \\diff V\\; \\Updelta q_\\omega = \\int_\\mathcal{S} \\diff S\\; \\nabla q_\\omega\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0$, by \\eqref{eq:surface_divergence_theorem}. \n\nFinally, we obtain the shape gradient from the last three lines of \\eqref{eq:variation_L_QS_tot},\n\\begin{align}\n    \\mathcal{G}&_\\mathrm{QS}  =  - (\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\breve{B}) \\nabla_\\Upgamma\\cdot\\left( v_\\mathrm{QS} \\mathbf{\\breve{B}} \\right) - v_\\mathrm{QS}\\;\\left(\\mathbf{\\breve{B}}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} - \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{\\breve{B}} \\right)\\cdot\\nabla_\\Upgamma\\breve{B} \\label{eq:QS_fom_shape_gradient} \\\\\n    \\nonumber & + \\left(\\iota-N\/M\\right)\\; \\frac{|\\overline{\\nabla\\psi}|}{G} \\; \\mathbf{\\breve{B}}\\times\\nabla\\breve{B}\\cdot  \\left[ \\abs{\\nabla_\\Upgamma\\alpha} \\nabla_\\Upgamma\\left( \\frac{v_\\mathrm{QS}}{\\abs{\\nabla_\\Upgamma\\alpha}} \\right) + \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\; v_\\mathrm{QS}\\; \\left( \\frac{\\nabla_\\Upgamma\\alpha\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla_\\Upgamma\\alpha}{\\abs{\\nabla_\\Upgamma \\alpha}^2} -h \\right) \\right]  \\\\\n    \\nonumber & + \\mathbf{\\breve{B}}\\cdot\\nabla q_\\omega + q_\\alpha \\left( \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{\\breve{B}} - \\mathbf{\\breve{B}}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} \\right)\\cdot\\nabla_\\Upgamma\\alpha + \\frac{h}{2} v_\\mathrm{QS}^2,\n\\end{align}\nwith $\\delta\\mathcal{L}_\\mathrm{QS}[\\delta\\mathbf{x}]  = \\int_{\\mathcal{S}} \\diff S \\;(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\;\\mathcal{G_\\mathrm{QS}}$, and $h$ the summed curvature.\n\n\\begin{figure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_analytic_quasisymm.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_QS_fom_adjoint}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_findiff_quasisymm.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_QS_fom_findiff}\n\\end{subfigure}\n\\\\\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_relative_error_quasisymm.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_QS_fom_relative error}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/dfdomega_convergence_epsilon_rel_error_QS_fom.pdf}\n  \\caption{}\n  \\label{fig:convergence_QS_fom}\n\\end{subfigure}\n\\caption{Shape gradient for the quasisymmetry objective function with helicity $N\/M=5$, evaluated through (a) adjoint methods and (b) a forward finite-difference scheme with step size $\\epsilon_\\mathrm{FD}=10^{-9}$, for the example rotating ellipse case introduced in Fig.~\\ref{fig:abs_nabla_psi}, with Fourier resolution $(N_\\mathrm{tor}, M_\\mathrm{pol}) = (16,16)$. The relative error, defined as the absolute error normalised by the maximal absolute value of the adjoint shape gradient, is shown in (c). The convergence of the relative error in the parameter derivative \\eqref{eq:parameter_derivatives_shape_grad} for a random direction in $\\Omega$ is shown in (d) as a function of the step-size $\\epsilon_\\mathrm{FD}$ and Fourier resolution $(N_\\mathrm{tor}, M_\\mathrm{pol})$. The black dashed line indicates the linear scaling in $\\epsilon_\\mathrm{FD}$ expected from the employed forward finite-difference scheme.}\n\\label{fig:QS_fom_numerical_eval_and_convergence}\n\\end{figure}\n\nThe shape gradient \\eqref{eq:QS_fom_shape_gradient} for targeted quasi-helical symmetry with helicity $N\/M = 5$ is shown in Fig.~\\ref{fig:shape_grad_QS_fom_adjoint} for the example rotating ellipse case introduced in Fig.~\\ref{fig:abs_nabla_psi}. The shape gradient obtained through adjoint methods is verified against a finite-difference evaluation in Fig.~\\ref{fig:shape_grad_QS_fom_findiff}. The error is visibly small, as is attested by the small relative error of the shape gradient shown in Fig.~\\ref{fig:shape_grad_QS_fom_relative error}. Convergence of the relative error for a parameter derivative in a random direction in $\\Omega$, evaluated with the adjoint method and with a centered finite-difference scheme, is shown in Fig.~\\ref{fig:convergence_QS_fom}. Akin to the rotational transform figure of merit convergence study in Fig.~\\ref{fig:convergence_iota_fom}, the error decreases linearly with $\\epsilon_\\mathrm{FD}$ until it plateaus due to finite Fourier or radial resolution. While the lowest resolution of $(N_\\mathrm{tor}, M_\\mathrm{pol}) = (8,8)$ seemed reasonable for the rotational transform figure of merit in Fig.~\\ref{fig:convergence_iota_fom}, a higher resolution is clearly required for the quasisymmetry figure of merit. This could be due to the fact that higher derivatives of the magnetic field are involved in the shape gradient for quasisymmetry \\eqref{eq:QS_fom_shape_gradient} than in the one for rotational transform \\eqref{eq:shape_gradient_iota_fom}, through derivatives of $v_\\mathrm{QS}$. The resulting fine-scale structure of $\\mathcal{G}$ is harder to resolve with a truncated Fourier series. However, the relative errors in Figs.~\\ref{fig:convergence_iota_fom}~and~\\ref{fig:convergence_QS_fom} are similarly small at the highest Fourier resolutions employed.\n\n\\section{Conclusions}\n\nIn this work, we derived the adjoint equations and shape gradient for the rotational transform and quasisymmetry of a vacuum field on a surface. The shape gradients allow fast computation of derivatives with respect to the parameters that describe the geometry of the surface, which are used in optimisation and sensitivity analyses. For a boundary represented by $N$ parameters, the speed-up from the adjoint method is $O(N)$ compared to a finite-difference evaluation. \n\nThis should enable future use of codes such as SPEC \\citep{hudsonComputationMultiregionRelaxed2012} in optimisation calculations, which was hitherto neglected in favour of the more widely-used VMEC code \\citep{hirshmanThreedimensionalFreeBoundary1986}. Contrary to VMEC, SPEC does not rely on the assumption of nested flux surfaces and can therefore model stochastic and island regions. In practice, employing adjoint methods and computing derivatives of quantities arising from ideal MHS equilibria is challenging, as the linearised MHS equilibrium equations possess regular singular points at every rational surface that resonates with the perturbation. These challenges can be avoided by the use of alternative equilibrium models, such as force-free magnetic fields, or the vacuum fields considered in this work. The generality of the results presented herein would also allow for their implementation in other solvers such as BIEST \\citep{malhotraTaylorStatesStellarators2019}. It is left for future work to extend the vacuum field results presented herein to the more general force-free fields modeled by SPEC.  Furthermore, the adjoint methods for vacuum fields introduced in this work could be fruitfully applied to other optimisation problems, e.g. in neoclassical transport calculations.\n\nIt is generally believed that exact quasisymmetry cannot be obtained exactly in a finite volume as near-axis expansions lead to an an overdetermined system of equations \\citep{garrenExistenceQuasihelicallySymmetric1991}, although that can be resolved by allowing for an anisotropic plasma pressure \\citep{rodriguezSolvingProblemOverdetermination2021a, rodriguezSolvingProblemOverdetermination2021}. Exact quasisymmetry on a surface is thought generally possible \\citep{garrenExistenceQuasihelicallySymmetric1991, plunkQuasiaxisymmetricMagneticFields2018}; and indeed, a vacuum solution near axisymmetry was recently found \\citep{senguptaVacuumMagneticFields2021}. The shape gradient for quasisymmetry derived in this work could be used to numerically probe the existence of quasisymmetric solutions on a surface that are not close to axisymmetry. For this purpose, the shape gradient for the rotational transform objective function \\eqref{eq:shape_gradient_iota_fom} could be used to avoid the axisymmetric solution at $\\iota = 0$, or also to avoid low order rationals. Furthermore, the shape gradients derived herein could be used to investigate if and how optimisation for quasisymmetry and for the rotational transform compete with each other. Finally, combining the derivatives of quasisymmetry and rotational transform with previously obtained derivatives of coil shapes \\citep{hudsonDifferentiatingShapeStellarator2018} and island size \\citep{geraldiniAdjointMethodDetermining2021} should, in principle, allow for the efficient search of a stellarator configuration with significant rotational transform, good integrability and neoclassical confinement at the boundary, realised by simple coils.\n\n\\begin{acknowledgments}\nThis work was supported by U.S. DOE DE-AC02-09CH11466, DE-SC0016072 and DE-AC02\u201376CH03073. A.B. acknowledges the generous support of the Simons Foundation.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nKontsevich has associated certain characteristic classes to\nfinite-dimensional $L_\\infty$- or $A_\\infty$-algebras equipped with an\ninvariant inner product, \\cite{Kncsg, KFd}. These are expressed in\nterms of the homology of certain complexes spanned by graphs with some\nadditional structures. This construction is by now well-understood\nboth from the point of view of Lie algebra homology and topological\nconformal field theory; see, for example, \\cite{HaLacc}.\n\nIn this note, we explore a natural generalization of this construction\nto the case of \\emph{curved} algebras, introduced by Positselski in\n\\cite{Pnqdc}. It turns out that a complete description of these\nclasses, and of the homology of the associated graph complexes, is\npossible. We show that these are all obtainable from\none-dimensional algebras, and that these classes are zero for algebras\nwith zero curvature. This contrasts with the corresponding problem for\nconventional graph complexes, which is still widely open.\n\nAs explained by Kontsevich, these graph complexes can be viewed as\ncomputing the stable homology of Lie algebras of symplectic vector\nfields on a vector space $W$ (in the $A_\\infty$ case, one should\ntake \\emph{noncommutative} symplectic vector fields).\nThis motivates us to consider the\nstability maps.  In this direction, we prove that the map from the Lie\nalgebra of symplectic vector fields on $W$ vanishing at the origin to\nthe homology of the Lie algebra of \\emph{all} vector fields on $W\n\\oplus \\CC \\cdot w$, where $w$ is an odd vector, is zero.  Similarly,\nwe show the same for the Lie algebra of noncommutative\nsymplectic vector fields.\n\nThe precise relation to the previous result is as follows. Any cyclic\n$L_\\infty$-algebra structure on $V$ defines an unstable characteristic\nclass in the homology of the Lie algebra of symplectic vector fields\non the shifted vector space $W = \\Pi V$. As $\\dim V \\rightarrow\n\\infty$, the homology of this Lie algebra converges to the graph\nhomology (at least if $V$ is only growing in the purely even or purely\nodd direction: see Theorem \\ref{GFgraph} below), and the image of the\nunstable characteristic class under the stability maps gives, in the\nlimit, the aforementioned (stable) characteristic class.  Hence, our\nresult above says that the unstable curved characteristic class of an\nalgebra with zero curvature already maps to zero under the first\nstability map $W:=\\Pi V \\into W \\oplus \\CC\\cdot w$.\n\n\nA related observation is the following: if $A$ is a curved\n(associative or Lie, or more generally $A_\\infty$- or $L_\\infty$-)\nalgebra with nonzero curvature, then $A$ is gauge equivalent (i.e.,\nhomotopy isomorphic) to the algebra with the same curvature and zero\nmultiplication, in a sense we will recall below.  In the case of\n\\emph{cyclic} curved algebras, we also compute the gauge equivalence\nclasses, which are less trivial: nontrivially curved algebras are\ngauge equivalent to the direct sum of a curved algebra of dimension at\nmost two of a certain form (but having nontrivial multiplications in\ngeneral), with a zero algebra.\n\nThis observation hints at a triviality of\ncurved infinity-algebras from a homological point of view, at least when the cyclic structure is not considered. A similar\nresult on the triviality of the corresponding derived categories was\nobtained recently in \\cite{KLNnv}; another manifestation of this triviality principle is briefly discussed in the last section of this paper.\n\nFinally, we generalize these results to the operadic setting, i.e., to\ntypes of algebras other than associative and Lie algebras. In\nparticular, we can apply it to Poisson, Gerstenhaber, BV, permutation,\nand pre-Lie algebras. For the most part, the generalization is\nstraightforward, and we restrict ourselves with giving only an outline\nof arguments in this section.  There is, however, one important aspect\nwhich is less visible in the special cases of commutative and ribbon\ngraphs: a curved graph complex associated with a cyclic (or even\nmodular) operad $\\mathcal O$ is quasi-isomorphic to a variant of the\n\\emph{deformation complex of a curved $\\mathcal O$-algebra on a\n  one-dimensional space}.  Therefore, this graph complex supports the\nstructure of a differential graded Lie algebra. This differential Lie\nalgebra, and its Chevalley-Eilenberg complex, appeared in various\nguises in the works of Zwiebach-Sen, Costello and\nHarrelson-Voronov-Zuniga on quantum master equation, \\cite{SZbias,\n  Cospftft, HVZocms}.\n\\subsection{Notation and conventions}\\label{notsec}\nIn this paper we work in the category of $\\mathbb Z\/2$-graded vector\nspaces (also known as (super)vector spaces) over $\\CC$ although all\nresults continue to hold in the $\\mathbb Z$-graded context and when\n$\\CC$ is replaced by any field of characteristic zero. We will usually\nrefer to these graded vector spaces simply as ``spaces.'' The parity of a homogeneous vector $v$ in a space will be denoted by $|v|$. The adjective\n`differential graded' will mean `differential $\\mathbb Z\/2$-graded'\nand will be abbreviated as `dg'. A (commutative) differential graded\n(Lie) algebra will be abbreviated as\n(c)dg(l)a.\nAll of\nour unmarked tensors are understood to be taken over $\\CC$. For a $\\mathbb Z\/2$-graded vector space $V=V_0\\oplus V_1$ the symbol $\\Pi\n V$ will denote the \\emph{parity reversion} of $V$; thus $(\\Pi\n V)_0=V_1$ while $(\\Pi V)_1=V_0$.\n\n\n We will make use of the language of \\emph{formal}\\footnote{Here the\n   word ``formal'' is understood in the sense of a formal\n   neighborhood, which differs from the notion of ``formality'' in\n   rational homotopy theory.}  spaces and algebras (which exist since\n \\cite{Lefat} under the name \"linearly compact\"; see, e.g.,\n \\cite{HaLactha} for a recent treatment relevant to the present work,\n under the present name).  A formal space is an inverse limit of\n finite-dimensional spaces. \n\n\n\n\n The functor of taking the linear dual establishes an\n anti-equivalence between the category of (discrete) vector spaces and that of\n formal vector spaces.\n\n It will always be clear from the context whether\n we work with formal or discrete vector spaces, and we will typically\n not mention this specifically later on; the tensor product of two formal spaces is understood to be their \\emph{completed} tensor product.  Furthermore, the symbol $V$\n will be reserved for a discrete space, with its dual $V^*$ therefore\n a formal space.\n\n  In particular, we will work with formal (c)dg(l)as.  The main\n examples will be completed tensor and symmetric algebras on formal\n spaces $W$; these will be denoted by $\\hat{T}W$ and\n $\\hat{S}W$ respectively. Note that we will \\emph{never} consider the\n uncompleted $TW$ and $SW$ when $W$ is formal, and similarly never\n consider the completed $\\hat{T}V$ or $\\hat{S}V$ when $V$ is discrete,\n so as to stay in either the category of formal spaces or that of\n discrete spaces.\n\n\n The Lie algebras of continuous derivations of $\\hat{T}W$ and\n of $\\hat{S}W$ will be denoted by $\\Der(\\hat{T}W)$ and\n $\\Der(\\hat{S}W)$ respectively; we will also consider their Lie\n subalgebras $\\Der^0(\\hat{T}W)$ and $\\Der^0(\\hat{S}W)$ consisting of\n derivations having no constant terms.\n\nWe mention here two potential pitfalls present in this framework. Firstly,\nthe categories of discrete and formal vector spaces are not disjoint: the\nspaces which are both discrete and formal are precisely finite-dimensional\nspaces. And secondly, not every space is either discrete or formal; moreover\nsuch spaces arise as a result of some natural operations with discrete or\nformal spaces. For example if $U$ and $W$ are infinite-dimensional formal or\ndiscrete spaces then the vector space $\\Hom(U,W)$ is neither formal nor\ndiscrete. Similarly, the Lie algebras\n$\\Der(\\hat{T}V^*)$ and $\\Der({T}V)$ will be neither\nformal nor discrete if $V$ is infinite-dimensional.\n\nTherefore, to avoid possible confusion, we make the blanket assumption\nthat the $\\ZZ\/2$-graded vector space $V$ which appears throughout the\npaper, in addition to being discrete as above, is in fact\n\\emph{finite-dimensional}.  This way all the objects we consider will\nlive in either the category of formal spaces or the category of\ndiscrete spaces (but not both: so each finite-dimensional space we\nconsider will be viewed in only one way). The price we pay is that\nsome of our results are not formulated in maximal generality; namely\nTheorem \\ref{curvtrivthm} and Claim \\ref{opclaim} do not need the\nspace $V$ to be finite-dimensional (although essentially none of the\nexposition needs to be modified to obtain this generalization).\n\n\n For a \\emph{formal} dgla $\\g$ its Chevalley-Eilenberg cohomological\n complex will be denoted by $\\CE^\\bullet(\\g)$.  This is defined as\n follows (note that the definition \\emph{differs} from the usual one\n in where completions are taken, since $\\g$ is formal rather than\n discrete):\nthe underlying graded vector space of $\\CE^\\bullet(\\g)$ is $S\\Pi{\\mathfrak g}^*$ and the differential is given as a sum of two maps $d_{\\operatorname I}$ and $d_{\\CE}$. Here $d_{\\operatorname I}$ and $d_{\\CE}$ are both specified by their restriction onto $\\Pi{\\mathfrak g}^*$ and extended to the whole $S\\Pi{\\mathfrak g}^*$ by the Leibniz rule; further $d_{\\operatorname I}:\\Pi{\\mathfrak g}^*\\to \\Pi{\\mathfrak g}^*$ is the shift of the dual of the internal differential on ${\\mathfrak g}$ whereas $d_{\\CE}:\\Pi{\\mathfrak g}^*\\to S^2(\\Pi{\\mathfrak g}^*)$ is induced by the commutator map $[,]:{\\mathfrak g}\\otimes {\\mathfrak g} \\to {\\mathfrak g}$.\n\nThis is in general a $\\ZZ\/2$-graded complex.  In the case that the\ndifferential $d$ is zero, it has an additional grading by\n\\emph{cohomological degree}, i.e., $S^i \\Pi\\mathfrak{ g}^*$ is in degree\n$i$.  The corresponding homological complex is the linear dual:\n$\\CE_\\bullet(\\g)=(\\CE^\\bullet(\\g))^*$, which has the underlying formal space $\\hat S \\Pi \\mathfrak{g}$.\nNote that in some papers (e.g. \\cite{HaLacc}) the Chevalley-Eilenberg complex of a graded Lie algebra ${\\mathfrak g}$ is defined using the (in our\ncase completed) \\emph{exterior} algebra $\\hat \\Lambda {\\mathfrak g}$; this definition is equivalent to ours under a canonical isomorphism $\\hat S(\\Pi{\\mathfrak g})\\cong \\hat \\Lambda {\\mathfrak g}$ where\n\\[\\Pi g_1\\ldots\\Pi g_n\\mapsto (-1)^{|g_{n-1}|+2|g_{n-1}|+\\ldots+(n-1)|g_1|}g_1\\wedge\\ldots\\wedge g_n.\\]\n\n\n\n \\section{Gauge equivalence classes of curved (cyclic) $A_\\infty$- and\n   $L_\\infty$-algebras}\n\\subsection{Curved $A_\\infty$- and $L_\\infty$- algebras}\nWe recall the definition of $A_\\infty$- and $L_\\infty$-algebras\nfollowing \\cite{HaLacc}, as well as their curved analogues; cf.~e.g.,\n\\cite{Nicbd} and references therein.\n\n\nA curved $A_\\infty$-algebra structure on a\n(finite-dimensional\\footnote{As in \\S \\ref{notsec}, $V$ is considered as\na discrete space throughout; in the present subsection\none could allow it to be infinite-dimensional and discrete.\nWe will not make further mention of this.}) space $V$ is\na continuous odd derivation $m$ of the formal dga $\\hat{T}\\Pi V^*$ and\na curved $L_\\infty$-algebra structure on $V$ is a continuous odd\nderivation $m$ of the formal cdga $\\hat{S}\\Pi V^*$; additionally $m$\nis required to square to zero in both cases. An ordinary (i.e. uncurved) $A_\\infty$-\nor $L_\\infty$-structure is specified by the requirement that $m$ have\nno constant term. The components $m_i:{T}^i\\Pi V\\to \\Pi V$ or\n${S}^i\\Pi V\\to \\Pi V$ of the dual of the restriction of $m$ to the\ntensor or symmetric powers $\\Pi V$ are the structure maps of the\ncorresponding $A_\\infty$- or $L_\\infty$-structure.\n\nWe note that sometimes it is convenient to use the more traditional\nway of writing the structure maps of an $A_\\infty$- or\n$L_\\infty$-algebra $V$ as ${T}^iV\\to V$ or ${\\Lambda}^iV\\to V$; these\nmaps will then be even or odd depending on whether $i$ is even or\nodd. To alleviate the notation we will still write $m_i$ for these\nmaps when the meaning is clear from the context.\n\nWe are interested in \\emph{gauge equivalence} classes of $A_\\infty$- or $L_\\infty$- structures on a fixed  space $V$. In particular, this equivalence relation implies other relations found in the literature under the names of homotopy or quasi-isomorphism.\n\nNamely, a gauge equivalence between $(V,m)$ and $(V,m')$ is defined as\na derivation $\\xi \\in \\Der^0(\\hat{T}\\Pi V^*)$ or $\\xi \\in\n\\Der^0(\\hat{S} \\Pi V^*)$ which is even and satisfies $m' = e^{\\ad \\xi}\nm$.\\footnote{In the case that our ground field is not $\\CC$, $e^\\xi$\n  still makes sense if we require additionally that $\\xi_1 = 0$ as\n  well, i.e., $\\xi$ has no linear term.  We can then modify the above\n  by saying that a gauge equivalence is a composition of such an\n  equivalence (for $\\xi_1=0$) with a linear isomorphism of $V$.} In\nparticular, such a gauge equivalence yields an isomorphism of dgas\n$e^\\xi: (\\hat{T} \\Pi V^*,m) \\iso (\\hat{T} \\Pi V^*, m')$ or cdgas\n$e^\\xi: (\\hat{S} \\Pi V^*,m) \\iso (\\hat{S} \\Pi V^*, m')$, and we usually\ndenote the gauge equivalence by $e^\\xi$.\n\\begin{theorem}\\label{curvtrivthm}\n  If $(V, m)$ is a curved $A_\\infty$- or $L_\\infty$- algebra for which\n  the curvature $m_0 = c \\in V_0$ is nonzero, then $m$ is gauge\n  equivalent to the structure $m' = c$ with all higher multiplications\n  $m'_i = 0$ for $i > 0$.\n\\end{theorem}\nRoughly, the above is saying that, when an (odd, noncommutative)\nformal vector field is nonzero evaluated at zero, then it is\nequivalent to a constant vector field up to (generally nonlinear) change of coordinates.\n\nAs a corollary of the theorem, it follows that any two\nnontrivially curved algebras with the same underlying graded vector space $V$\nare gauge equivalent.\n\\begin{proof}[Proof of Theorem \\ref{curvtrivthm}]\n  We consider the $A_\\infty$ case; the $L_\\infty$ case is similar.\n\n  Any $A_\\infty$-algebra structure $(V,m')$ with $m'_0=m_0=c$ can be\n  viewed as a deformation of $(V,m_0)$. Indeed, let us introduce a\n  formal parameter $\\hbar$; then $(V,m')$ is equivalent to the\n  deformed structure $(V,m'_\\hbar)$, where $m'_\\hbar = m'_0 + \\sum_{i\n    \\geq 1} \\hbar^i m'_i$.  This yields an equivalence with deformed\n  structures whose $i$-ary operations are homogeneous of degree $i$ in\n  $\\hbar$.\n\n  The structures of the form $(V,m')$ are governed by the dgla\n  $\\Der(\\hat T \\Pi V^*, [,c])$, where $c \\in V$ is viewed as an odd\n  constant derivation of $\\hat T \\Pi V^*$.  Formal deformations (such\n  as $(V, m'_\\hbar)$) are governed by the dgla $\\Der(\\hat T \\Pi V^*[[\\hbar]],\n  [,c])$.  Gauge equivalences $e^{\\xi}$ of Maurer-Cartan elements of\n  $\\Der(\\hat T \\Pi V^*, [,c])$, which we can assume satisfy $\\xi_0 = 0\n  = \\xi_1$ (so as to not change the curvature) are identified with\n  gauge equivalences $e^{\\frac{1}{\\hbar}\\xi_\\hbar}$ (for $\\xi_\\hbar :=\n  \\sum_{i} \\hbar^i \\xi_i$) of the corresponding Maurer-Cartan elements\n  of $\\Der(\\hat T \\Pi V^*[[\\hbar]], [,c])$, which preserve the grading\n  $|\\hbar|=1=|V^*|$.\n\n Since gauge equivalence classes of deformations of Maurer-Cartan\n elements are preserved under quasi-isomorphisms of dglas, it suffices\n to show that $\\Der(\\hat T \\Pi V^*, [,c])$ is acyclic. Let us write\n down this complex $C^\\bullet$ explicitly; note that $C^\\bullet$ is a version of\n the Hochschild complex in the curved setting.\n\n Set $C^i:=\\Hom((\\Pi V)^{\\otimes i}, \\Pi V)$ with the differential $d:C^i\\to\n C^{i-1}$  given by the formula, for $f\\in\n C^i$, and $x_1, \\ldots, x_{i-1} \\in \\Pi V$:\n\\[\ndf(x_1,\\ldots, x_{i-1})=\\sum_k(-1)^{|x_1|+\\cdots+|x_k|}f(x_1,\\ldots, x_{k},c,x_{k+1},\\ldots, x_i).\n\\]\nWe construct an explicit contracting homotopy.  Choose an odd linear\nmap $\\epsilon:\\Pi V\\to\\CC$ such that $\\epsilon(c)=1$ (here, odd means\nthat $\\epsilon|_{\\Pi (V_1) = (\\Pi V)_0} = 0$). Define maps $s_i:C^i\\to\nC^{i+1}$ by the formula, for $f\\in C^i$:\n\\begin{equation} \\label{siref}\ns_if(x_1,\\ldots,x_{i+1})=\\epsilon(x_1)f(x_2,\\ldots, x_{i+1}).\n\\end{equation}\nThen,\n\\begin{equation}\nd s_i f(x_1,\\ldots,x_{i}) + s_{i-1} d f(x_1, \\ldots, x_{i}) = \\epsilon(c) f(x_1, \\ldots, x_{i}) = f(x_1, \\ldots, x_{i}). \\qedhere\n\\end{equation}\n\\end{proof}\n\\subsection{Cyclic algebras}\nWe now extend the results of the previous subsection to the case of\nalgebras with a cyclic inner product.  By an \\emph{inner product} on a\n$\\ZZ\/2$-graded vector space $V$ we mean a nondegenerate symmetric\nbilinear form $( -, - ): V\\otimes V \\rightarrow \\CC$, where $V$ is\nrequired to be finite-dimensional. An \\emph{inner product space} is a\nspace equipped with an inner product.\n\\begin{definition} Let $(V,m)$ be a finite-dimensional $A_\\infty$- or\n  $L_\\infty$-algebra.  A cyclic inner product on $V$ is an inner\n  product $( -, - ): V\\otimes V \\rightarrow \\CC$ for which the tensors\n  $( m_i(v_1,\\ldots,v_i), v_{i+1})$ are invariant with respect to the\n  signed cyclic permutations of arguments:\n\\[\n(\n  m_i(v_1,\\ldots,v_i), v_{i+1})=\n (-1)^{i+|v_1|(|v_2|+\\ldots+|v_{i+1}|)}( m_i(v_{i+1},v_1\\ldots,v_{i-1}), v_{i})\n \\] We call an algebra $(V,m)$ a \\emph{cyclic} $A_\\infty$- or\n $L_\\infty$-algebra when it is equipped with such a cyclic inner\n product.\n\\end{definition}\nWe need to define a subspace of $\\Der(\\hat{T}\\Pi V^*)$ of\n\\emph{cyclic} derivations.  To do so, first note that $\\Der(\\hat{T}\n\\Pi V^*) \\cong \\Hom(\\Pi V^*, \\hat{T}\\Pi V^*)$ via the restriction map.\nUsing the inner product, we can identify the latter with\n$(\\Pi V^* \\otimes\n\\hat{T}\\Pi V^*)$.  Next, on any tensor power $(\\Pi V^*)^{\\otimes n}$, we\ncan define the (graded) cyclic permutation operator $\\sigma:\n(\\Pi V^*)^{\\otimes n} \\rightarrow (\\Pi V^*)^{\\otimes n}$. This extends to a\ncontinuous linear automorphism of $(\\Pi V^* \\otimes \\hat{T}\\Pi V^*)$, and\nyields a continuous linear automorphism of $\\Der(\\hat{T} \\Pi\nV^*)$.\n\\begin{definition}\\label{cycderdefn}\n  Let $\\CDer(\\hat{T} \\Pi V^*)$ denote the space of derivations\n  which are invariant under cyclic permutation: we call these\n  \\emph{cyclic} derivations.  Similarly, define $\\CDer^0(\\hat{T} \\Pi V^*)$ as those cyclic derivations with zero constant term (i.e., preserving the\naugmentation $\\hat{T}^{\\geq 1} \\Pi V^*$). Finally, define the spaces $\\CDer(\\hat{S} \\Pi V^*)$ and $\\CDer^0(\\hat{S} \\Pi V^*)$ analogously.\n\\end{definition}\nNote that $\\CDer(\\hat{S} \\Pi V^*)$ can be\nviewed as the subspace of $\\CDer(\\hat{T} \\Pi V^*)$ landing in symmetric\ntensors, and similarly for the version with zero constant term.\n\nGiven $V$ together with an inner product, cyclic curved\n$A_\\infty$-structures are the same as odd derivations $\\xi \\in\n\\CDer(\\hat{T} \\Pi V^*)$ which square to zero.  Similarly,\nuncurved structures correspond to odd square-zero $\\xi \\in\n\\CDer(\\hat{T} \\Pi V^*)$, and the $L_\\infty$-versions are\nobtained by replacing $\\hat{T}$ with $\\hat{S}$.  See, e.g.,\n\\cite{HaLactha} for details.\n\\begin{definition} A gauge equivalence of cyclic $A_\\infty$- or\n  $L_\\infty$- structures $(V,m)$ and $(V,m')$ on a fixed inner product\n  space $V$ is a map $e^\\xi$ where $\\xi \\in \\CDer^0(\\hat{T}\\Pi V^*)$\n  or $\\xi \\in \\CDer^0(\\hat{S} \\Pi V^*)$ satisfies $m' = e^{\\ad \\xi}\n  m$.\n\\end{definition}\n\\begin{remark}\n  In the literature, the term \\emph{symplectic} is sometimes used\n  instead of cyclic, the idea being that an inner product\n  on $V$ is equivalent to a (constant) symplectic structure on $\\Pi\n  V$, so that the cyclic derivations on $\\hat S \\Pi V^*$ or $\\hat T \\Pi V^*$ are the same\n  as formal (possibly noncommutative) symplectic vector fields on $\\Pi V$; a cyclic gauge equivalence could then be interpreted as a (formal, noncommutative) symplectomorphism (preserving the $\\ZZ\/2$-grading).\nNote that it follows from this interpretation that\nthe cyclic derivations of $V$ form a Lie superalgebra, and the cyclic\ngauge equivalences a Lie group.\n\\end{remark}\n\\begin{theorem}\\label{cyccurvtrivthm}\\\n \\begin{enumerate}\n\\item[(a)] If $(V, m)$ is a curved $A_\\infty$-algebra with a\n  cyclic inner product for which the curvature $m_0 = c \\in V_0$ is\n  nonzero, and $c' \\in V$ any even element for which $(c, c') = 1$, then $(V,m)$ is gauge equivalent to the\n  structure $m'$ with $m'_0 =c$, and higher multiplications\n\\begin{gather}\n  m_{2i-1}' = 0, i \\geq 1, \\\\\n  m_{2i}'(x_1, \\ldots, x_{2i}) = \\bigl(\\prod_{j=1}^{2i} (c', x_j) \\bigr) \\cdot (m_{2i}(c,c,\\ldots,c), c) \\cdot c'.\n\\end{gather}\n\\item[(b)] If $(V,m)$ is a curved $L_\\infty$-algebra with a cyclic inner product for which the curvature $m_0 = c \\in V_0$ is nonzero, then $(V,m)$ is gauge equivalent to the\n  structure $m'$ with $m'_0 =c$, and higher multiplications $m'_i = 0$ zero for all $i \\geq 1$.\n\\end{enumerate}\n\\end{theorem}\nOne immediately deduces\n\\begin{corollary}\\\n\\begin{enumerate}\n\\item[(a)]\nTwo curved cyclic $A_\\infty$-algebra structures $(V,m)$ and $(V,m')$ on the same underlying inner product space $V$ with nonzero curvature $m_0, m_0'$\nare gauge equivalent if and only if\n\\begin{equation}\n(m_0, m_0) = (m_0', m_0'), \\quad (m_{2i}(m_0, \\ldots, m_0), m_0) = (m_{2i}'(m_0', \\ldots, m_0'), m_0'), \\forall i \\geq 1.\n\\end{equation}\n\\item[(b)] Two curved cyclic $L_\\infty$-algebra structures $(V,m)$ and\n  $(V,m')$ on the same underlying inner product space $V$ with\n  nonzero curvature $m_0, m_0'$ are gauge equivalent if and only if\n  $(m_0, m_0) = (m_0', m_0')$.\n\\end{enumerate}\n\\end{corollary}\n\\begin{remark}\nAnother, perhaps intuitively more clear, way to understand this result is as follows. Consider the curved cyclic $A_\\infty$-algebra $V$ with curvature $c$ whose underlying inner product space is spanned by a single even vector $c$ with $(c,c)=1$ and higher products $m_{2i}(c,\\ldots,c)=t_ic$; $m_{2i+1}=0$ for $i=0,1,\\ldots$. Here $t_i$ are arbitrary numbers.\n\nConsider also  the curved cyclic $A_\\infty$-algebra $V^\\prime$ whose underlying space is spanned by two even vectors $c$ and $c^\\prime$ with $(c,c^\\prime)=(c^\\prime,c)=1, (c,c)=(c^\\prime,c^\\prime)=0$. The $A_\\infty$-structure is given as\n$m_0 = c$ and, for $i > 0$, $m_{2i}(c,\\ldots, c)=t_ic^\\prime$, where $t_i$ are arbitrary, and all other higher products are zero.\n\nThen any cyclic curved $A_\\infty$-algebra with nonzero curvature is gauge equivalent to the direct sum of either $V$ or $V^\\prime$ with an $A_\\infty$-algebra having zero $A_\\infty$-structure.\n\\end{remark}\n\\begin{proof}[Proof of Theorem \\ref{cyccurvtrivthm}]\n (a) We modify the previous\n  obstruction theory argument. Deformations of the algebra with $m_0=c$ and all\nhigher operations zero are governed by the subcomplex\n\\begin{equation}\\label{bbeq}\nB^\\bullet := \\CDer(\\hat{T}\\Pi V^*, [,c]) \\subset C^\\bullet\n\\end{equation}\nof the one\nconsidered in Theorem \\ref{curvtrivthm}.\n\nIn Lemma \\ref{qil} below, we show that $B^{\\bullet}$ is quasi-isomorphic to the subcomplex\n$B^{\\bullet}_0$ spanned by\n the cocycles\n\\begin{equation}\\label{ceq}\n\\epsilon^{i+1}(x_1, \\ldots, x_i) := \\epsilon(x_1) \\cdots \\epsilon(x_i) c', \\quad \\epsilon(v) := (c', v).\n\\end{equation}\nNote that, when $i$ is odd, $\\epsilon^{i+1} = 0$. Moreover, $B^{\\bullet}_0$\nis an abelian sub-dgla with zero differential.\n\nUsing the lemma, the result follows as in the proof of Theorem\n\\ref{curvtrivthm}. In more detail, the gauge equivalence classes of\nMaurer-Cartan elements of $B^\\bullet[[\\hbar]]$ and\n$B^\\bullet_0[[\\hbar]]$ which are zero modulo $\\hbar$ are\nidentified. Moreover, the formal Maurer-Cartan elements which are\nhomogeneous of the form $\\sum_{i \\geq 1} \\hbar^i m_i$ for $m_i \\in B^i$\nor $B^i_0$ are identified with actual Maurer-Cartan elements $\\sum_{i\n  \\geq 1} m_i$ with zero constant term.  Hence, the gauge equivalence\nclasses of cyclic curved $A_\\infty$- structures with $m_0 = c$ are\nidentified with gauge equivalence classes of Maurer-Cartan elements of\n$B^\\bullet_0$ with zero constant term. Since the latter is abelian\nwith zero differential, all elements are Maurer-Cartan, and define\ndistinct gauge equivalence classes.\n\nWe deduce that all curved $A_\\infty$-structures with $m_0 = c$ are gauge\nequivalent to one with all odd operations equal to zero, and even operations given by some multiple of the operation\n\\begin{equation}\n(x_1, \\ldots, x_{2i}) \\mapsto\n\\bigl(\\prod_{j=1}^{2i} (c', x_j) \\bigr) \\cdot c'.\n\\end{equation}\nIt remains to show that this multiple is $(m_{2i}(c,c,\\ldots,c), c)$.  This follows by taking the component of the original operation in the direct summand $B^i_0 \\subseteq B^i = (B^i_0 \\oplus B^i_+)$, using the definition of $B^i_+$ in\n\\eqref{bipdfn} below.\n\n(b) We can apply the same argument as above, except now with\n$B:=\\CDer(\\hat{S}\\Pi V^*, [,c])$.  The same argument as above applies\nand we deduce the same result, except that this time\n$m_{2i}(c,c,\\ldots,c) = 0$ for all $i \\geq 1$ by skew-symmetry of $m_{2i}$.\n\\end{proof}\n\n\\begin{lemma}\\label{qil} Keeping the assumptions and notation\n  of the theorem, the complex $B^\\bullet$ decomposes as $B^\\bullet =\nB^\\bullet_0 \\oplus B^\\bullet_+$, where $B^\\bullet_+$ is acyclic.  Moreover,\nthe inclusion $B^\\bullet_0 \\into B^\\bullet$ is a quasi-isomorphism of dglas.\n\\end{lemma}\n\\begin{proof}\nThe second statement follows from the first, since $B^\\bullet_0 \\into B^\\bullet$ is a dgla map.\n\nTo prove the first statement, we modify the contracting\n  homotopy $s_i$ from Theorem \\ref{curvtrivthm} to act on $B^\\bullet$. The result will \\emph{not}  be a contracting homotopy, but will instead accomplish the desired goal.\n\nDefine maps $s_i':B^i\\to B^{i+1}$ by\n\\begin{equation}\\label{sipfla1}\ns_i' f = \\sum_{j=0}^{i+1} \\sigma^j (s_i f),\n\\end{equation}\nwhere $\\sigma^j$ is the $j$-th power of the cyclic permutation defined above Definition \\ref{cycderdefn}.\nWe now compute $s_{i-1}' d + d s_i'$.  Assume that $f \\in B^i$.  In this\ncase, we may use the formula\n\\begin{multline}\\label{sipfla2}\ns_i' f(x_1,\\ldots,x_{i+1}) = \\sum_{j=1}^{i+1} (-1)^{|x_1|+\\ldots+|x_{j-1}|} \\epsilon(x_j) f(x_1, \\ldots, x_{j-1}, x_{j+1}, \\ldots, x_{i+1})\n\\\\\n+ (-1)^{|x_1| + \\ldots + |x_{i+1}|} (f(x_1, \\ldots, x_i), x_{i+1}) c'.\n\\end{multline}\nThen, we compute\n\\begin{multline}\\label{sipfla3}\n(s_{i-1}' d + d s_i')f(x_1, \\ldots, x_i) =  (i+1) \\epsilon(c) f(x_1, \\ldots, x_i) \\\\ - \\bigl( \\sum_{j=1}^{i+1} \\epsilon(x_j) f(x_1, \\ldots, x_{j-1}, c, x_{j+1}, \\ldots, x_{i}) + (f(x_1, \\ldots, x_{i}), c)c' \\bigr).\n\\end{multline}\nThus, the operator on the RHS is contractible.  Next, for $k \\leq i+1$, define subspaces\n\\begin{gather}\n  B^i_k \\subset B^i, \\quad B^i_k = B^i \\cap \\bigl(\\CC[S_{i+1}] \\cdot \\bigl( (\\CC \\cdot \\epsilon)^{\\otimes (i+1-k)} \\otimes ((\\CC \\cdot c)^\\perp)^{\\otimes k} \\bigr) \\bigr), \\label{bipdfn0}\\\\\n  B^i_+ := \\sum_{k =1}^{i+1} B^i_k. \\label{bipdfn}\n\\end{gather}\nNote that $B^\\bullet_+$ is a subcomplex, and $B^\\bullet = B^\\bullet_0\n\\oplus B^\\bullet_+$.\n\nWe claim that the RHS of \\eqref{sipfla3} acts as $k \\cdot \\Id$ on\n$B^i_k$ for all $k$. This follows directly.  As a result, $s_{i-1}' d + d\ns_i'$ restricts to zero on $B^\\bullet_0$ and to an automorphism on\n$B^\\bullet_+$.  This proves the lemma. \\qedhere\n\n\\end{proof}\n\\begin{remark}\n  Lemma \\ref{qil} is equivalent to the statement that the cyclic\n  (co)homology of any curved $A_\\infty$-algebra $V$ whose structure\n  maps are all zero, except for $m_0$ (which is not zero), is\n  isomorphic to the cyclic (co)homology of a one-dimensional\n  $A_\\infty$-algebra $V_1$ having the same property. See\n  \\cite{GJainfa, HaLactha} for the notion of cyclic (co)homology of\n  $A_\\infty$-algebras. These (co)homologies could be computed in\n  other ways from the above, for instance, with the help of Connes'\n  exact sequence connecting Hochschild (co)homology with cyclic\n  (co)homology (which appears in \\cite{GJainfa} in the curved\n  setting), since Theorem \\ref{curvtrivthm} shows that the Hochschild\n  (co)homologies of $V$ and $V_1$ are trivial.\n\\end{remark}\n\\section{Characteristic classes of curved algebras}\n\\subsection{The uncurved case}\\label{ccs}\nWe briefly recall Kontsevich's construction of characteristic classes\nof finite-dimensional $A_\\infty$- or $L_\\infty$-algebras with cyclic\ninner products.\nFirst, we recall the definition of certain graph complexes, for which\ncyclic $A_\\infty$- or $L_\\infty$-algebras will produce cycles.\n\\begin{definition}\n  A graph is a tuple $(H,V,E, \\varphi_V, \\varphi_E)$ of sets $H, V, E$\n  of \\emph{half-edges}, \\emph{vertices}, and \\emph{edges}, and\n  surjective maps $\\varphi_V: H \\rightarrow V, \\varphi_E: H\n  \\rightarrow E$ such that the fibers of $\\varphi_E$ all have\n  cardinality two.\n\\end{definition}\nGiven $\\Gamma = (H, V, E, \\varphi_V, \\varphi_E)$, we will also write $H_\\Gamma = H, V_\\Gamma = V, E_\\Gamma = E, \\varphi_V^\\Gamma =\n\\varphi_V$, and $\\varphi_E^\\Gamma = \\varphi_E$.\n\\begin{definition}\n  A ribbon graph is a graph together with a cyclic ordering on each\n  fiber $\\varphi_V^{-1}(t)$.\n\\end{definition}\nIntuitively, one may think of the edges of ribbon graphs as slightly\nfattened, which explains the cyclic ordering at vertices.\n\nKontsevich's graph complexes have a basis of graphs of a certain type,\nwith differential taking a graph to the sum over all edges of the\ncontracted graph obtained by shrinking that edge to a point, together\nwith a sign.  To make this precise requires the notion of\n\\emph{orientation}:\n\\begin{definition}\n  An orientation on a graph is a choice of ordering of all the\n  half-edges, ordering of all the vertices, and a sign $\\pm 1$, modulo\n  the relation that applying a transposition to the ordering of either\n  the half-edges or the vertices is the same as changing the sign.\n\\end{definition}\n\\begin{definition}\n  An oriented graph is a graph equipped with an orientation. An\n  isomorphism of oriented graphs is an isomorphism of graphs which\n  preserves orientation.  Similarly, the same definition applies replacing\n  ``graph'' with ``ribbon graph.''\n\\end{definition}\nNext, given a graph $\\Gamma = (H,V,E, \\varphi_V, \\varphi_E)$ and an\nedge $e \\in E$ with endpoints $v_1, v_2 \\in V$ meeting halves $h_1,\nh_2 \\in H$, one defines the contracted graph $d_e(\\Gamma) = (H, V \/\n\\{v_1 = v_2\\}, E, \\varphi_V', \\varphi_E)$ by identifying the endpoints\n$v_1$ and $v_2$. If $\\Gamma$ is moreover a ribbon graph, with the\ncyclically ordered sets $\\varphi_V^{-1}(v_1) = (a_1, a_2, \\ldots, a_i\n= h_1)$ and $\\varphi_V^{-1}(v_2) = (b_1, b_2, \\ldots, b_j = h_2)$,\nthen the cyclic ordering of the half edges at the new vertex $v = v_1\n= v_2$ is defined as $(a_1, a_2, \\ldots, a_{i-1}, b_1, b_2,\n\\ldots, b_{j-1})$. Finally, if $\\Gamma$ is equipped with an\norientation, where the half-edges are ordered as $h_1, h_2, p_1,\n\\ldots, p_m$ and with vertices ordered by $v_1, v_2, w_1, \\ldots,\nw_{k}$, the new orientation is given by the ordering $p_1, \\ldots,\np_m$ and $v, w_1, \\ldots, w_{k}$ of vertices, without changing the\nsign.\n\nConsider the graded vector space with basis the isomorphism classes of\noriented graphs \\emph{whose vertices have valence $\\geq 2$}, modulo\nthe relation that a graph is negative its opposite orientation. The\ngrading is given by the number of vertices.  Let $\\mathcal{G}$ be the\ncompletion of this graded vector space with respect to the number of\nedges (so \\emph{not} with respect to the defining grading on the\nvector space, which is by number of vertices).  Similarly, define\n$\\mathcal{G}_r$ using ribbon graphs rather than graphs.  In other\nwords, these are the spaces of possibly infinite linear combinations\nof isomorphism classes of oriented graphs which are not isomorphic to\nthe graph obtained by reversing the orientation.\n\nThen, it is a result of \\cite{Kncsg} that\n\\begin{equation}\nd(\\Gamma) := \\sum_{e \\in E_\\Gamma} d_e(\\Gamma)\n\\end{equation}\ndefines a differential on $\\mathcal{G}$ and $\\mathcal{G}_r$.\n\\begin{definition}\n  Kontsevich's graph complex is defined as $(\\mathcal{G}, d)$, and his\n  ribbon graph complex is defined as $(\\mathcal{G}_r, d)$.\n\\end{definition}\n\\begin{remark}\n  We could alternatively have used the uncompleted graph complex\n  above; however, the completed version is the one which naturally\n  contains characteristic classes of $L_\\infty$- or\n  $A_\\infty$-algebras.  In particular, taking homology commutes with\n  taking completion, for the following well-known reason: We can write the\n  uncompleted graph complex as a direct sum of the subcomplexes of\n  graphs of a fixed genus (i.e., first Betti number of the graph as a\n  topological space).  For each fixed genus, the completion with\n  respect to number of edges is the same as the completion with\n  respect to the grading, i.e., the number of vertices, since there\n  are only finitely many graphs with a fixed genus and number of\n  vertices.  Hence, the completed graph complex is the same as the\n  direct product of the completions of these subcomplexes with respect\n  to their usual grading.\n\\end{remark}\nFinally, given a cyclic $A_\\infty$-algebra $V$, one constructs an\nelement of $\\mathcal{G}_r$, given by a sum\n\\begin{equation}\n[V]:=\\sum_{\\Gamma} \\frac{1}{|\\Aut(\\Gamma)|} c_\\Gamma(V) \\cdot \\Gamma,\n\\end{equation}\nwhere we sum over isomorphism classes of ribbon graphs $\\Gamma$ (with\ngroup of automorphisms $\\Aut(\\Gamma)$), and $c_\\Gamma$ is given as\nfollows.  Equip $\\Gamma$ with an orientation. We will define\n$c_\\Gamma$ so that the opposite orientation would produce $-c_\\Gamma$.\nNamely, $c_\\Gamma$ is given by contracting the multiplications $m_i$\nof $V$ with the pairings $( -, -)$ according to the graph.\nIn more detail, consider\n\\begin{equation} \\label{multmaps} \\prod_{i=1}^n ( m(-), -\n  ): \\bigotimes_{i=1}^n V^{\\otimes |\\phi_V^{-1}(v_i)|}\n  \\rightarrow \\CC.\n\\end{equation}\nLet $h_1, \\ldots, h_{|H|}$ be the ordering of the half-edges and $v_1,\n\\ldots, v_{|V|}$ the ordering of the vertices defined by the\norientation, and assume that the sign is $1$. Let us pick\n\\emph{ciliations} of each of the vertices $v_1, \\ldots, v_{|V|}$,\nwhich means a linear ordering of the half-edges $\\varphi^{-1}_V(v_i)$\nmeeting each vertex $v_i$, compatible with the cyclic ordering given\nby the ribbon structure.  Up to changing the sign, let us assume that\nthe ordering of the half-edges is $\\varphi^{-1}_V(v_1),\n\\varphi^{-1}_V(v_2), \\ldots, \\varphi^{-1}_V(v_{|V|})$.  Let $f \\in V\n\\otimes V$ be the inverse to the pairing $(-,-): V \\otimes V\n\\rightarrow \\CC$. Finally, pick an arbitrary ordering of the edges\n$e_1, \\ldots, e_{|E|}$. Then, one applies \\eqref{multmaps} to the\nelement obtained by applying the signed permutation of components of\n$f^{\\otimes |E|} \\in V^{\\otimes |H|}$ which rearranges the half-edges\n$\\varphi_E^{-1}(e_1), \\ldots, \\varphi_E^{-1}(e_{|E|})$ into the\nordering $h_1, \\ldots, h_{|H|}$. One can check that the result does\nnot depend on the choices of orderings (but only depends on the\norientation of $\\Gamma$ by a sign, as mentioned above), and we let\n$c_\\Gamma(V)$ to be the result of this computation.\n\nIn a similar manner, one constructs from any cyclic $L_\\infty$-algebra $V$\nan element $[V]$ of $\\mathcal{G}$.  Then, the following result\nis due to Kontsevich:\n\\begin{proposition}\\cite{KFd} If $V$ is a cyclic $A_\\infty$- or $L_\\infty$-algebra then $[V]$ is a cycle on\n  $(\\mathcal{G}, d)$ or $(\\mathcal{G}_r, d)$ respectively.\n\\end{proposition}\n  A direct proof of the proposition in the $A_\\infty$-case\n  is contained, e.g., in \\cite{Igu}. More conceptually, one can view a\n  cyclic $A_\\infty$-algebra $V$ as an algebra over ${\\mathsf\n    F}\\underline{\\mathscr{A}\\textit{ss}}^{1}$, the Feynman transform of the $\\Det$-twisted naive\n  modular closure of the cyclic operad $\\mathscr{A}\\textit{ss}$; see \\cite{GKmo} and\n  \\cite{ChLadft} concerning these notions. Therefore, we obtain a map of\n  modular operads ${\\mathsf F}\\underline{\\mathscr{A}\\textit{ss}}^1\\to\n  \\mathscr{E}(V)$, where $\\mathscr{E}(V)$ is the modular endomorphism\n  operad of $V$. The map between the vacuum parts of the corresponding\n  operads\n\\[{\\mathsf F}\\underline{\\mathscr{A}\\textit{ss}}^1((0))\\to\n\\mathscr{E}(V)((0))\\cong \\CC\n\\]\nis precisely the characteristic class described above, and it follows that it does indeed give a cycle. One can prove the proposition similarly\nin the $L_\\infty$ case.\n\n\n\\subsection{Curved characteristic classes}\nThe preceding results have a natural generalization to the case of\ncurved algebras.  We need to remove the valence $\\geq 2$ condition,\nhowever, and study the graph complexes $(\\widetilde{\\mathcal{G}}, d),\n(\\widetilde{\\mathcal{G}_r}, d)$ of formal linear combinations of\ngraphs and ribbon graphs where vertices are allowed to have valence\n$1$ (we will not allow valence-zero vertices, since they don't add\nanything of value). We can consider this to be the complex of ``graphs\nwith stubs,'' where a stub is an edge incident to a valence-one\nvertex. Note that the conventional graph complex $(\\mathcal{G}, d)$ is\na subcomplex of $(\\widetilde{\\mathcal{G}}, d)$ and similarly in the\nribbon case. Then, everything else goes through exactly as above, and\nwe obtain the following result.\n\\begin{proposition} Any curved cyclic $A_\\infty$- or\n  $L_\\infty$-algebra $V$ induces a cycle $[V]$ on\n  $(\\widetilde{\\mathcal{G}_r}, d)$ or on $(\\widetilde{\\mathcal{G}},\n  d)$ respectively.\n\\end{proposition}\nThe curved graph homology could be expressed in terms of certain\nGelfand-Fuks type homology. Namely, let $W$ be a graded symplectic\nvector space and consider the graded formal Lie algebra $\\g(W)$ of formal\nsymplectic vector fields on $W$. Similarly consider the graded formal Lie\nalgebra $\\g_r(W)$ consisting of formal \\emph{noncommutative}\nsymplectic vector fields on $W$, i.e. the Lie algebra\n$\\CDer(\\Pi W)$. Taking the stable limit as the dimension\nof the even or odd part of $W$ goes to infinity, we arrive at\nfollowing result (which\nis a straightforward adaptation of \\cite[Theorem 1.1]{Kncsg}, up to technical\nproblems stemming from the lack of complete reducibility of finite-dimensional representations of simple Lie superalgebras).  Let\n$\\CC^{m}$ denote the even space of dimension $m$ and $\\Pi \\CC^m$ the odd\nspace of dimension $m$.  We equip $\\CC^{2m}$ with the standard symplectic\nform, and $\\Pi \\CC^m$ with the standard odd symplectic (i.e., orthogonal) form.\n\\begin{theorem}\\label{GFgraph}\n  Let $W$ be a fixed inner product space.\n  There are isomorphisms\n\\begin{equation}\n\\operatorname{H}_\\bullet\n  (\\widetilde{\\mathcal{G}_r}) \\cong \\lim_{m \\rightarrow \\infty}\n   \\HCE_\\bullet(\\g_r(W \\oplus \\CC^{2m})), \\quad \\operatorname{H}_\\bullet\n  (\\widetilde{\\mathcal{G}_r}) \\cong \\lim_{m \\rightarrow \\infty}\n   \\HCE_\\bullet(\\g_r(W \\oplus \\Pi \\CC^m))\n\\end{equation}\n between the stable Chevalley-Eilenberg\n  homology of the Lie algebra $\\g_r$ and of the\n  corresponding version of the curved graph complex.\nSimilarly, we have isomorphisms\n\\begin{equation}\nH_\\bullet(\\widetilde{\\mathcal{G}})\\cong \\lim_{m \\rightarrow \\infty}\n   \\HCE_\\bullet(\\g(W \\oplus \\CC^{2m})), \\quad H_\\bullet(\\widetilde{\\mathcal{G}})\\cong \\lim_{m \\rightarrow \\infty}\n   \\HCE_\\bullet(\\g(W \\oplus \\Pi \\CC^m)).\n\\end{equation}\n\\end{theorem}\n\\begin{remark}\nIt might be possible to further generalize this result\nto (certain) cases where both the even and the odd part have dimension going to\ninfinity, but that creates additional technical difficulties that we\nprefer to avoid (as we do not need such generality). In any case, they\nare the same difficulties that arise in the original uncurved setting\nof \\cite{Kncsg} (note that in \\emph{op.~cit.} only the even case is\nconsidered).\n\\end{remark}\n\\begin{proof}\n  The corresponding result for the uncurved graph complex and vector\n  fields vanishing at the origin was established by Kontsevich\n  \\cite{Kncsg} and his proof carries over to the present context, up\n  to some technical difficulties created by the fact that we are in\n  the super context, where $W_0$ and $W_1$ can both be nonzero. Define\n  a nonnegative grading on $\\CE^\\bullet(\\g_r(W))$ called\n  \\emph{weight}, which is the sum of the homological grading and the\n  degree of polynomial coefficients of the vector fields (this will\n  correspond, on graphs, to the number of half-edges).\n\n  The main tool is that the inclusion of subcomplexes\n\\begin{equation} \\label{invincleq}\n\\CE^\\bullet(\\g_r(W \\oplus \\CC^{2m}))^{\\mathfrak{osp}(W \\oplus\n    \\CC^{2m})} \\into \\CE^\\bullet(\\g_r(W \\oplus \\CC^{2m}))\n\\end{equation}\nis asymptotically a quasi-isomorphism, and similarly replacing\n$\\CC^{2m}$ with $\\Pi \\CC^m$.  By this, we mean that in weights $\\leq\nN$, there exists $M$ such that, if $m \\geq M$, then the inclusion is\nan isomorphism on homology in weights $\\leq N$.\n\nWe carry out the argument with $\\CC^{2m}$; replacing this by $\\Pi\n\\CC^m$ will not affect anything. Let $U_m := W \\oplus \\CC^{2m}$.  First note that $\\mathfrak{osp}(U_m)\n\\subseteq \\g_r(U_m)$ is the subspace of\nlinear vector fields and hence acts trivially on the cohomology of\n$\\CE^\\bullet(\\g_r(U_m))$.  Hence, the statement would\nfollow if it were true that $\\mathfrak{osp}(U_m)$ acted\ncompletely reducibly on $\\CE^\\bullet(\\g_r(U_m))$.  This\nis not, in general, true (when $W$ is not purely even); however, it\nfollows from Lemma\n\\ref{superredlem} below that,\nfor each $N \\geq 0$, there exists $M$ so that $m \\geq M$\nimplies that the weight $\\leq N$ part of $\\CE^\\bullet(\\g_r(U_m))$ is completely reducible as an $\\mathfrak{osp}(U_m)$-representation. This is sufficient\nto deduce that \\eqref{invincleq} is a quasi-isomorphism in weights $\\leq N$.\n\n\nTo complete the proof, it remains to relate the invariant subcomplex\nto graphs.  This part of the argument is nearly identical to\n\\emph{op.~cit.}, so we will be brief.  Associated to each graph is an\n$\\mathfrak{osp}(W)$-invariant element of $\\CE_\\bullet(\\g_r(W))$, as\ndescribed in the previous subsection. However, the resulting map\n$\\mathcal{G}_r \\to \\CE_\\bullet(\\g_r(W))$ does not linearly extend to a\nmap of complexes. Instead, if we attach to the dual of a graph in\n$\\mathcal{G}_r^*$ a canonical element of\n$\\CE^\\bullet(\\g_r(W))^{\\mathfrak{osp}(W)}$, as explained in\n\\emph{op.~cit.}, one obtains a canonical map of complexes\n$(\\mathcal{G}_r^*, d^*) \\to \\CE^\\bullet(\\g_r(W))^{\\mathfrak{osp}(W)}$.\nIn more detail, under the identification $\\mathcal{G}^*_r \\cong\n\\mathcal{G}_r$ using the basis of ribbon graphs (with fixed\norientations), this map sends each oriented ribbon graph $\\Gamma$ to\nthe corresponding functional which contracts elements of\n$\\CE_{|V_\\Gamma|}(\\g_r(W))$ using the symplectic form $W \\otimes\nW\\rightarrow \\CC$, similarly to the construction of \\S \\ref{ccs}.  That is,\nwe view $\\CE_{|V_\\Gamma|}(\\g_r(W)) \\subset \\Hom(W, \\hat{T}(W))$ as a subspace\nof $\\hat{T}(W)$ using the symplectic form, and contract with a\npermutation of $\\omega^{\\otimes |E_\\Gamma|}$, where $\\omega \\in W\n\\otimes W$ is the inverse to the symplectic form, so that the copy of\n$\\omega$ corresponding to each edge contracts the corresponding pair\nof half-edges.  For details on how to prove this indeed yields a map\nof complexes, see, e.g., \\cite[Theorem 4.10]{HaLacc}, and also\n\\cite{HamsaKt}.\n\nBy the fundamental theorems of invariant theory \\cite{Wftit} (see\nalso, e.g., \\cite{Howeftit}) and the super generalization found in\n\\cite{Seracit}, following the reasoning in the proof of Lemma\n\\ref{superredlem} below, it follows that this map of complexes is\nasymptotically an isomorphism: for every fixed $N \\geq 0$, there\nexists $M \\geq 0$ so that, when $\\dim W \\geq M$, this map is an\nisomorphism if we restrict to graphs with at most $N$ edges, and hence\nelements of $\\CE^\\bullet(\\g_r(W))^{\\mathfrak{osp}(W)}$ of weight $\\leq N$.\n\nThis construction goes through completely analogously for commutative\ngraphs and the Lie algebra $\\g$.\n\\end{proof}\nAs in the proof above, let $U_m := W \\oplus \\CC^{2m}$.  Also define $U_m' := W \\oplus \\Pi \\CC^m$.\n\\begin{lemma}\\label{superredlem}\nFor all $N \\geq 0$, there exists $M \\geq 0$ such that, when $m \\geq M$, $U_m^{\\otimes N}$ and $(U_m')^{\\otimes N}$ are completely reducible $\\mathfrak{osp}(U_m)$- and $\\mathfrak{osp}(U_m')$-modules, respectively.\n\\end{lemma}\n\\begin{proof}\n  We carry out the argument for $U_m$; the same applies for $U_m'$.\n  Suppose that one has subrepresentations $0 \\neq \\tau_1 \\subsetneq\n  \\tau_2 \\subseteq U_m^{\\otimes N}$.  We wish to show that the\n  inclusion $\\tau_1 \\into \\tau_2$ splits.  By adjunction, we\n  equivalently need to show that the composition $\\CC \\into \\tau_1^*\n  \\otimes \\tau_1 \\into \\tau_1^* \\otimes \\tau_2$ splits.  Since\n  $U_m^{\\otimes N}$ is self-dual using the pairing, $\\tau_1^* \\otimes\n  \\tau_2 \\subseteq U_m^{\\otimes 2N}$. This means that it suffices to\n  show that the inclusion of the invariant part, $(U_m^{\\otimes\n    2N})^{\\mathfrak{osp}(U_m)} \\into U_m^{\\otimes 2N}$, splits. Let us\n  replace $2N$ by $N$ for convenience.\n\n  This last fact follows using the fundamental theorems of invariant\n  theory.  Namely, by \\cite{Seracit}, all of the $\\mathfrak{osp}(U_m)$-invariants in the tensor algebra $T(U_m)$ are tensor products of the\n  pairing on $U_m$ with an invariant related to the determinant, whose\n  tensor weight (i.e., number of tensor components) is a function of\n  $m$ that goes to infinity when $m$ does.  Hence, for large enough\n  $m$, all of the invariants in weight $U_m^{\\otimes \\leq N}$ are\n  spanned by tensor products of the pairing on $W$. By the classical\n  second fundamental theorem of invariant theory \\cite{Wftit} (see\n  also \\cite{Howeftit}), for large enough $m$, there are no nontrivial\n  relations between these invariants.  The same argument\n  applied to $U_m^*$ shows that the coinvariants have the same\n  description. Moreover, for large enough $m$, the invariants of\n  $U_m^{\\otimes \\leq N}$ and invariants of $(U_m^*)^{\\otimes \\leq N}$\n  have a perfect pairing.  This implies that the composition\n  $(U_m^{\\otimes \\leq N})^{\\mathfrak{osp}(U_m)} \\into U_m^{\\otimes N}\n  \\onto (U_m^{\\otimes N})_{\\mathfrak{osp}(U_m)}$ is an isomorphism for\n  large enough $m$.\n\\end{proof}\n\nUnlike the case for the usual graph homology, it is possible to give a\ncomplete calculation of the homology of the curved graph complex and\ncurved ribbon graph complex:\n\\begin{theorem}\n\\label{acythm}\\\n\\begin{enumerate}\n\\item[(a)]\nThe homology of the complex  $(\\widetilde{\\mathcal{G}_r}, d)$ is identified with the space of\n  formal linear combinations of graphs all of whose connected\n  components are graphs whose vertices have valence one with the exception of at most a\n  single vertex, which has odd valence.\n\\item[(b)]\nThe homology of the complex $(\\widetilde{\\mathcal{G}}, d)$ is identified with the space of\n  formal linear combinations of graphs each of whose connected\n  components is the connected graph with one edge and two vertices.\n\\end{enumerate}\n\\end{theorem}\nWe can reinterpret this theorem as follows. First, it is enough to\ncompute the homology of the subcomplex of connected nonempty graphs,\n$\\widetilde{\\mathcal{G}_{r,c}} \\subset \\widetilde{\\mathcal{G}_r}$ and\n$\\widetilde{\\mathcal{G}_c} \\subset \\widetilde{\\mathcal{G}}$.  We call\nthese complexes the \\emph{connected} ribbon graph complex and the\nconnected graph complex.  Note that $\\widetilde{\\mathcal{G}_r} \\cong\n\\Sym \\widetilde{\\mathcal{G}_{r,c}}$ and $\\widetilde{\\mathcal{G}} \\cong\n\\Sym \\widetilde{\\mathcal{G}_c}$.  Then, the desired result is that the\nhomology of the former is identified with linear combinations of\nstar-shaped graphs with an odd number of edges and at most a single\nvertex of valence $\\geq 2$, and the latter is one-dimensional and\nspanned by the connected graph with two vertices and a single edge.\n\nIn other words, the theorem states that\n$\\widetilde{\\mathcal{G}_{r,c}}$ and $\\widetilde{\\mathcal{G}_c}$ are\nquasi-isomorphic to the subcomplexes spanned by connected graphs with\nat most a single vertex of valence $\\geq 2$.  These complexes are\nidentical with the deformation complexes of Theorem \\ref{cyccurvtrivthm}\nfor the one-dimensional curved $A_\\infty$ and $L_\\infty$ algebras with\ncurvature $c$ satisfying $(c,c) = 1$, and all higher operations zero.\nThat is, they are the graded vector spaces of noncommutative\nsymplectic vector fields and ordinary symplectic vector fields on the\nodd one-dimensional symplectic vector space $\\CC \\cdot c$, equipped\nwith the differential $\\ad(\\frac{\\partial}{\\partial c})$ (which turns out to be zero).\n\\begin{remark}\n  The stable homology of the Lie algebra of symplectic vector fields\n  has been computed by Guillemin and Shnider in \\cite{GSssrc}, and thus,\n  part (b) of the above theorem could be deduced from their\n  calculation, taking into account Theorem \\ref{GFgraph}. However we\n  include this result for completeness, and because the argument we\n  use in part (a) essentially extends without change to this case.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{acythm}]\n    Call a graph a \\emph{line segment} if it is topologically a line\n  segment, i.e., it is connected, and either it is a single vertex\nwith no edges, or all of its vertices have valence\n  two except for two vertices, which have valence one.  Given a\n  connected graph $\\Gamma$, let us call a vertex $v \\in V_\\Gamma$\n  \\emph{exterior} if it either has valence at most one, or one of the\n  connected components of $\\Gamma \\setminus v$ is a line segment. Call\n  all other vertices \\emph{interior}.\n\n  We will make use of a filtration on the connected (ribbon) graph\n  complex given by the number of interior vertices in the graph.  We\n  call this the \\emph{interior vertex filtration}.  The associated\n  graded complex is identified, as a vector space, with the (ribbon)\n  graph complex, and with the differential which is almost the same,\n  but only contracts edges incident to at least one exterior\n  vertex. Moreover, the associated graded complexes $\\gr\n  \\widetilde{\\mathcal{G}_{r,c}}$ and $\\gr \\widetilde{\\mathcal{G}_c}$\n  are graded not only by number of interior vertices, but by the\n  subgraph $\\Gamma_0 \\subseteq \\Gamma$ obtained by restricting to\n  interior vertices and edges which are incident only to interior\n  vertices (allowing here also the empty graph and the graph with a\n  single vertex and no edges).  It is clear that $\\Gamma_0$ is\n  connected (possibly empty).  Let $\\gr_{\\Gamma_0}\n  \\widetilde{\\mathcal{G}_{r,c}}$ or $\\gr_{\\Gamma_0}\n  \\widetilde{\\mathcal{G}_c}$ denote the resulting subcomplex graded by\n  $\\Gamma_0$.\n\n  (a) We claim that, when $\\Gamma_0$ contains at least one edge, then\n  $\\gr_{\\Gamma_0} \\widetilde{\\mathcal{G}_{r,c}}$ is acyclic.\n\n  Assume that $\\Gamma_0$ contains an edge.  Pick a half-edge $h$ of\n  $\\Gamma_0$, and let $v$ be the incident vertex.  We construct from\n  $v$ and $h$ a contracting homotopy $s$ on $\\gr_{\\Gamma_0}\n  \\widetilde{\\mathcal{G}_{r,c}}$.  Namely, for every oriented graph\n  $\\Gamma$ whose subgraph on internal vertices is $\\Gamma_0$, let\n  $s \\Gamma$ be the graph obtained from $\\Gamma$ by adding a new\n  univalent vertex together with an edge connecting it to $v$. The\n  resulting new half-edge incident to $v$ is, in the cyclic ordering\n  at $v$, one half-edge counterclockwise away from $h$.  Pick the\n  orientation on the new graph $s \\Gamma$ so that $\\Gamma$ is one of\n  the summands of $d(s \\Gamma)$.\n\n  We claim that $sd + ds = \\Id$.  It suffices to show that, for every\n  oriented graph $\\Gamma$ as above, $sd(\\Gamma) + ds(\\Gamma)=\\Gamma$.\n  In turn, it suffices to show that, for every edge $e \\in E_\\Gamma$,\n  $s d_e \\Gamma = - d_e (s \\Gamma)$.  This follows immediately from\n  our definition of $s$.\n\n\n\nHence, we deduce that $\\gr_{\\Gamma_0} \\widetilde{\\mathcal{G}_{r,c}}$\nis acyclic, as claimed.\n\nNext, we compute $\\gr_{\\Gamma_0} \\widetilde{\\mathcal{G}_{r,c}}$ where\n$\\Gamma_0$ is either empty or is the graph with a single vertex and no edges.\nCall the first graph ``$\\emptyset$'' and the second one ``$\\pt$.''\n\nFirst, $\\gr_{\\pt} \\widetilde{\\mathcal{G}_{r,c}}$ consists of star-shaped\ngraphs with a single vertex of valence $\\geq 3$, with the usual graph\ndifferential except that we do not allow to contract an edge that\nwould result in a graph without a vertex of valence $\\geq 3$, i.e., an\nedge which is incident to a vertex of valence $1$ and a vertex of\nvalence $3$. Consider the filtration by the valence of\nthe interior vertex. The associated graded complexes have homology\nwhich is one-dimensional, concentrated in the part where all vertices\nbut one have valence $1$.  Moreover, such graphs are actually zero\nwhen the node has even valence (because a cyclic symmetry can reverse\nthe orientation).  Hence, the associated spectral sequence computing\n$H_*(\\gr_{\\pt} \\widetilde{\\mathcal{G}_{r,c}})$ collapses at the first\npage, and the resulting homology is spanned by the star-shaped graphs\nwith a single vertex of odd valence $\\geq 3$ and with all other\nvertices of valence $1$.\n\nNext, $\\gr_\\emptyset \\widetilde{\\mathcal{G}_{r,c}}$ is the subcomplex of line\nsegments, whose homology is one-dimensional and spanned by the line segment\nwith two vertices.\n\nWe deduce that the first page of the spectral sequence of the interior\nvertex filtration on $\\widetilde{\\mathcal{G}_{r,c}}$ is concentrated\nin the part with $\\leq 1$ interior vertices.  The part in degree $1$\nis the span of star-shaped graphs with one vertex of odd valence $\\geq\n3$ and the other vertices of valence $1$, and the part in degree $0$\nis the span of the line segment with two vertices.  Since all of these\ngraphs have only odd numbers of edges, it is clear that the spectral\nsequence collapses at the first page, and the graphs above span the\nhomology of $\\widetilde{\\mathcal{G}_{r,c}}$, as desired.\n\n(b) The same argument as above applies in this case, except that,\nsince our graphs no longer have cyclic orderings of half edges at\nvertices, we modify the construction of the contracting homotopy $s$\n(used to show that $\\gr_{\\Gamma_0} \\widetilde{\\mathcal{G}_{r,c}}$ is\nacyclic when $\\Gamma_0$ contains an edge) accordingly. Namely, we\nremove the condition that the new edge be next to $h$ in the\ncounterclockwise cyclic ordering of half edges at the vertex $v$.\nEverything else goes through without change, except that now the\nstar-shaped graphs with a single vertex of valence $\\geq 1$ are all\nzero except for the one with only two vertices.  This implies the\ndesired result. \\qedhere\n\n\\end{proof}\nWe see that the complex of simply-connected graphs (which splits off\nas a direct summand) carries all of the homology of the complexes\n$(\\widetilde{\\mathcal{G}_r}, d)$ and $(\\widetilde{\\mathcal{G}}, d)$,\nand we obtain the following result.\n\\begin{corollary}\\label{curveduncurved}\n  The inclusions of complexes $(\\mathcal{G}_r,\n  d)\\subset(\\widetilde{\\mathcal{G}_r}, d)$ and $(\\mathcal{G},\n  d)\\subset(\\widetilde{\\mathcal{G}}, d)$ induce the zero maps on\n  homology.\n\\end{corollary}\n\\begin{remark}\n  It is natural to ask whether the nontrivial homology classes in the\n  curved graph complexes $\\widetilde{\\mathcal{G}_r}$ and\n  $\\widetilde{\\mathcal{G}}$ are detected by curved $A_\\infty$- and\n  $L_\\infty$-algebras. The answer is yes; in fact, they are detected by\n  one-dimensional algebras.  For the $A_\\infty$ case, let $\\Gamma(i)$\n  be the star-shaped graph whose central vertex has valence $2i+1$ and\n  $V(i)$ be the one-dimensional cyclic curved $A_\\infty$-algebra\n  spanned by an even vector $c$ such that $( c,c)=1$, $m_0=c$,\n  $m_{2i}(c,\\ldots,c)=c$, and all other $A_\\infty$-products are\n  zero. Then it is easy to see that the cycle\n  $[V(i)]\\in\\widetilde{\\mathcal{G}_r}$ is homologous to\n  $\\pm\\frac{1}{2i+1}\\Gamma(i)$ (the sign depends on the choice of an\n  orientation on $\\Gamma(i)$).  Next, let $\\Gamma(0)$ be the connected\n  graph with one edge and two vertices.  Then, we may let $V(0)$ be\n  the one-dimensional cyclic curved $A_\\infty$-algebra spanned by an\n  even vector $c$ such that $m_0=c$, $( c, c ) = 1$, and all the\n  higher $A_\\infty$-products are zero.  Then,\n  $[V(0)]\\in\\widetilde{\\mathcal{G}_r}$ is again homologous to\n  $\\pm\\frac{1}{2}\\Gamma(0)$.  The same statement holds for the\n  $L_\\infty$-setting, if we now let $\\Gamma(0)$ be an ordinary (not\n  ribbon) connected graph also with two vertices and one edge, and\n  consider the one-dimensional curved cyclic $L_\\infty$-algebra,\n  $V(0)$, with curvature $c$ satisfying $( c, c ) = 1$ and all higher\n  operations equal to zero: $[V(0)]\\in\\widetilde{\\mathcal{G}_r}$ is\n  homologous to $\\pm\\frac{1}{2}\\Gamma(0)$.\n\\end{remark}\n\\section{Homology of Lie algebras of vector fields and stability maps}\\label{unstabsec}\nIn order to refine Theorem \\ref{acythm}, we recall first a more\ngeneral way to view the construction of characteristic classes.\nLet $\\mathfrak{g}$ be a formal dgla.\nConsider a Maurer-Cartan element of $\\mathfrak{g}$, i.e., an element\n$x \\in \\Pi\\mathfrak{g}$ satisfying $dx + \\frac{1}{2} [x,x] = 0$.\nThen, $e^x = \\sum_{i \\geq 0}x^{i}\/i!$ defines a cycle in\n $\\CE_\\bullet(\\mathfrak{g})$, and hence a homology class of even\ntotal degree.  In the situation where the differential $d$ on\n$\\mathfrak{g}$ is zero, using the homological degree $|S^{i}\n\\Pi\\mathfrak{g}| = i$, each element $x^{ i}$ itself is a cycle, and we\nobtain \\emph{unstable} characteristic classes $[x^{ i}] \\in\n\\CE_i(\\mathfrak{g})$.\n\nMoreover, if $\\mathfrak{h} \\subseteq \\mathfrak g_0$ is a pronilpotent\nLie subalgebra of the even part of $\\mathfrak{g}$, then there is\ndefined a notion of \\emph{gauge equivalence} of Maurer-Cartan elements\ncorresponding to the adjoint action of the Lie group of the Lie\nalgebra $\\mathfrak{h}$; then it follows that if two Maurer-Cartan\nelements are gauge equivalent by a gauge in $\\mathfrak{h}$, then their\ncharacteristic classes are homologous. (One can more generally take\n$\\mathfrak{h} \\subseteq \\mathfrak{g}_0$ to be a Lie subalgebra which\nis the Lie algebra of a pro-Lie group).  This statement, as well as a\nmore detailed treatment of characteristic classes, can be found in,\ne.g., \\cite{Hamccms}.\n\nReturning to the situation of a (cyclic or curved) $L_\\infty$- or\n$A_\\infty$-algebra $V$, the corresponding element of the Lie algebra\n$\\Der^0(\\hat{S} (\\Pi V^*)), \\Der(\\hat{T}(\\Pi V^*))$, etc., defines a\ncanonical homology class.\n\nThe relationship to the aforementioned characteristic classes is\nKontsevich's result that the limit as the dimension of $V$ goes to\n$\\infty$ of the Lie homology of $\\CDer^0(\\hat{S} (\\Pi V^*))$ or\n$\\CDer^0(\\hat{T}(\\Pi V^*))$ is the completion (by number of edges) of\nthe homology of the graph complexes $(\\mathcal{G}, d)$. In the curved\nsituation the relevant result is Theorem \\ref{GFgraph}.\n\nFurthermore, given $V$, and any inner product space $W$,\n we can form the trivial extension $V \\oplus\nW$ where all multiplications with the second factor are zero. This\ninduces maps\n\\begin{gather} \\label{vplieeqn}\n\\varphi_{V,W}: \\CE_\\bullet(\\CDer(\\hat{S}(\\Pi V^*)))\n\\rightarrow \\CE_\\bullet(\\CDer(\\hat{S}(\\Pi (V \\oplus W)^*))), \\\\\n\\label{vpasseqn}\n\\varphi_{V,W}: \\CE_\\bullet(\\CDer(\\hat{T}(\\Pi V^*)))\n\\rightarrow \\CE_\\bullet(\\CDer(\\hat{T}(\\Pi (V \\oplus W)^*))),\n\\end{gather}\nand similarly the restrictions $\\varphi_{V,W}^0 :=\n\\varphi_{V,W}|_{\\CE_\\bullet(\\CDer^0(\\hat{S}(\\Pi V^*)))}$ or\n$\\varphi_{V,W}^0 := \\varphi_{V,W}|_{\\CE_\\bullet(\\CDer^0(\\hat{T}(\\Pi\n  V^*)))}$.  It is well known (and easy to check) that Kontsevich's\nconstruction (\\S \\ref{ccs}) is obtained from the above construction in\nthe limit: the image of the unstable characteristic cycle $\\sum_i\n\\xi^{i}\/i! \\in \\CE_\\bullet(\\CDer(\\hat{S}(\\Pi V^*)))$ under\n$\\varphi_{V,W}^0$ as $\\dim W_0 \\rightarrow \\infty$ (for fixed $W_1$)\nidentifies with the characteristic cycle on $(\\mathcal{G}, d)$ given\nin $\\S \\ref{ccs}$; similarly if we fix $W_0$ and let $\\dim W_1 \\to\n\\infty$.  In the curved setting, by Theorem \\ref{acythm}, for each\nfixed degree $i$, if we fix $W_1$, then for large enough $\\dim W_0$,\n$\\varphi_{V,W}$ induces a projection on homology,\n$$\\HCE_i(\\CDer(\\hat{S}(\\Pi V^*)))\n\\onto \\begin{cases} \\CC, & \\text{if $i$ is even}, \\\\ 0, & \\text{if $i$\n    is odd}.\\end{cases}$$\nThe same is true if we fix $W_0$ and consider\nlarge enough $\\dim W_1$.  A similar result is true in the associative\nversion, where now we project onto the span of graphs whose connected\ncomponents are stars with odd valence as in Theorem \\ref{acythm}.\n\n Using Theorem \\ref{cyccurvtrivthm}, we can prove an unstable\n analogue of Corollary \\ref{curveduncurved}, which gives information\n about the maps $\\varphi_{V,W}$ for all $W$ with $W_0 \\neq 0$:\n\\begin{theorem}\\label{unscurveduncurved}\n  If $V$ and $W$ are inner product spaces and $W_0 \\neq\n  0$, then the compositions\n\\begin{gather} \\label{lstabmaps}\n\\CE_i(\\CDer^0(\\hat S(\\Pi V^*))) \\to \\CE_i(\\CDer(\\hat S(\\Pi V^*))) \\mathop{\\to}^{\\varphi_{V,W}} \\CE_i(\\CDer(\\hat S(\\Pi (V \\oplus W)^*))), \\\\ \\label{astabmaps}\n\\CE_i(\\CDer^0(\\hat T(\\Pi V^*))) \\to \\CE_i(\\CDer(\\hat T(\\Pi V^*))) \\mathop{\\to}^{\\varphi_{V,W}} \\CE_i(\\CDer(\\hat T(\\Pi (V \\oplus W)^*)))\n\\end{gather}\nare zero on homology.\n\\end{theorem}\nBefore we prove the theorem, we first prove a lemma which may be\ninteresting in itself.  Given an element $c \\in V$ and a $L_\\infty$- or\n$A_\\infty$-structure given by a Maurer-Cartan element $\\xi \\in\n\\CDer^0(\\hat S(\\Pi V^*))$ or $\\xi \\in \\CDer^0(\\hat T(\\Pi V^*))$, we\nsay that $c$ is \\emph{central} if $[c, \\xi] = 0$. In particular, for\nan ordinary Lie algebra, $c$ is an element satisfying $\\{c, v\\}=0$ for\nall $v \\in V$, and for an ordinary associative algebra, $c$ satisfies\n$c \\cdot v = v \\cdot c$ for all $v \\in V$.\n\\begin{lemma}\nSuppose that $V$ is an (uncurved) $L_\\infty$- (or $A_\\infty$-) algebra with a nonzero even central element $c \\in V_0$.\nThen, the image of the resulting (unstable) characteristic class of $V$ under the appropriate map,\n\\begin{gather}\n\\CE_\\bullet(\\CDer^0(\\hat S(\\Pi V^*))) \\to \\CE_\\bullet(\\CDer(\\hat S(\\Pi V^*))) \\text{ or }\\\\\n\\CE_\\bullet(\\CDer^0(\\hat T(\\Pi V^*))) \\to \\CE_\\bullet(\\CDer(\\hat T(\\Pi V^*))),\n\\end{gather}\nis zero on homology.\n\\end{lemma}\n\\begin{proof}\nWe first consider the $L_\\infty$ case.  Let $\\mathfrak{g} :=\n  \\CDer(\\hat{S}(\\Pi V^*))$.  Let $\\xi\n  \\in \\CDer^0(\\hat{S}(\\Pi V^*)) \\subseteq\n  \\mathfrak{g}$ correspond to the $L_\\infty$-structure on $V$,\n  i.e., $\\xi$ satisfies $[\\xi, \\xi] = 0$, and viewed as an element\nof $\\mathfrak{g}$, $\\xi(v)$ has no\n  constant term for all $v \\in \\Pi V^*$.\n  Let $\\ell$ denote the structure maps for the algebra\n  $V$, with $\\ell_i: V^{\\otimes i} \\rightarrow V$ the $i$-th component.\n\n  Now, consider the $L_\\infty$-structure $\\{\\ell^\\lambda_i\\}$ on $V$\n  which is obtained by $\\ell^\\lambda_i := \\ell_i$ if $i \\geq 1$, and\n  $\\ell_0^\\lambda = \\lambda c$ for $\\lambda \\in \\CC$. Since $c$ is\n  central and even, these indeed define $L_\\infty$-structures.\n\n  Then, by Theorem \\ref{cyccurvtrivthm}, $V$ equipped with\n  $\\ell^\\lambda$ is gauge equivalent to an algebra with all higher\n  multiplications equal to zero, and curvature equal to $\\lambda\n  c$. This gauge equivalence is by vector fields with zero constant\n  and linear term, which form a pronilpotent dgla.  Hence, in the\n  limit as $\\lambda \\rightarrow 0$, we see that the characteristic\n  class of $(V, \\ell^\\lambda)$ becomes a boundary, i.e., the original\n  characteristic class of $(V, \\ell)$ is a boundary in\n  $\\CE_\\bullet(\\CDer(\\hat S(\\Pi V^*)))$, as desired.\n\n  In the $A_\\infty$ case, the same argument applies: as before, one\n  deforms the uncurved $A_\\infty$-structure $\\{m_i\\}$ to the curved\n  structure $\\{m^\\lambda_i\\}$ with $m^\\lambda_i = m_i$ for $i \\geq 1$\n  and $m^\\lambda_0 = \\lambda c$.  The difference is, by Theorem\n  \\ref{cyccurvtrivthm}, the resulting structure $(V, m^\\lambda)$ is\n  gauge equivalent to the algebra described there, which does not have\n  all higher operations zero.  Call this algebra structure\n  $\\{(m')^\\lambda_i\\}$.  Even though these are nonzero for infinitely\n  many $i$, it is still true that, as $\\lambda \\rightarrow 0$,\n  $(m')^\\lambda_i \\rightarrow 0$, and so we still deduce that the\n  original characteristic class was a boundary.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{unscurveduncurved}]\n  We treat only the $L_\\infty$ case, since the $A_\\infty$ case is\n  identical.  If $V$ is an uncurved $L_\\infty$-algebra, then $V \\oplus\n  W$ is an $L_\\infty$-algebra with a nonzero central element, namely,\n  any nonzero element of $W_0$.  Hence, the image of the resulting\n  characteristic class under \\eqref{lstabmaps} is zero.\n\nHowever, in general, not all homology classes of\n$\\CE_\\bullet(\\CDer^0(\\hat{S}(\\Pi V^*)))$ are obtainable in the above manner.\nTo fix this problem, we can consider nontrivial coefficients.  Given\nany cdga $\\mathfrak{a}$, we may consider\n$\\mathfrak{a}$-linear $L_\\infty$-algebra structures on $U := V\n\\otimes_\\CC \\mathfrak{a}$, for a fixed finite-dimensional\nspace $V$.  Denote $U^* := \\Hom_{\\mathfrak{a}}(U,\n\\mathfrak{a})$.  Then, these algebra structures on $U$ are, by\ndefinition, Maurer-Cartan elements of\n$\\Der^0(\\hat{S}_{\\mathfrak{a}}(\\Pi U^*))$, i.e., odd elements $f$\nsatisfying $df + \\frac{1}{2} [f, f] = 0$, where $d$ is the\ndifferential induced by the differential on $\\mathfrak{a}$. If we fix\nan inner product on $V$, we obtain an induced $\\mathfrak{a}$-linear\ninner product on $U$ (of the form $U \\otimes_{\\mathfrak{a}} U\n\\rightarrow \\mathfrak{a}$), and cyclic $L_\\infty$-algebras of this\nform are given by elements of $\\CDer^0(\\hat{S}_{\\mathfrak{a}}(\\Pi\nU^*))$.\n\nAs above, we obtain an unstable characteristic class of\n$\\CE_\\bullet(\\CDer^0(\\hat{S}_{\\mathfrak{a}}(\\Pi U^*)))$.  The above argument\nshows that the images of such classes under the tensor product of \\eqref{lstabmaps} with $\\Id_{\\mathfrak{a}}$ are boundaries.\n\nTo conclude the proof, we will use a universal example.\nQuite generally, if $\\mathfrak{h}$ is a formal dgla and\n$\\mathfrak{a}$ a cdga, then $\\CE^\\bullet(\\mathfrak{h}) =\nS\\Pi \\mathfrak{h}^*$ is naturally a cdga, and\nMaurer-Cartan elements of $\\mathfrak{h} \\otimes \\mathfrak{a}$\nidentify with cdga morphisms $\\CE^\\bullet(\\mathfrak{h})\n\\rightarrow \\mathfrak{a}$.\nIf we set $\\mathfrak{a}\n:= \\CE^\\bullet(\\mathfrak{h})$, then the identity map\n$\\Id_{\\CE^\\bullet(\\mathfrak{h})}$ yields a Maurer-Cartan element of\n$\\mathfrak{h} \\otimes \\mathfrak{a}$, and the resulting cycle $\\sum_i\n\\xi^{ i} \/ i! \\in \\widehat{\\CE_\\bullet(\\mathfrak{h}) \\otimes \\mathfrak{a}}\n\\cong \\End(\\CE^\\bullet(\\mathfrak{h}))$ is nothing but the identity map.\n\nApplying this construction to the case $\\mathfrak{h} =\n\\CDer^0(\\hat{S}(\\Pi V^*))$ and $\\mathfrak{a} =\n\\CE^\\bullet(\\CDer^0(\\hat{S}(\\Pi V^*)))$, we deduce from the above that\nthe $\\mathfrak{a}$-linear maps $\\widehat{\\varphi_{V,W} \\otimes\n  \\Id_{\\mathfrak{a}}}$ send the cycle $\\xi$ corresponding to the\nidentity element of $\\End(\\CE^\\bullet(\\mathfrak{h}))$ to a\nboundary. This implies that \\eqref{lstabmaps} itself is zero on\nhomology, as desired.\n\\end{proof}\n\n\n\n\\section{Operadic generalization}\nIn this section we sketch an operadic generalization of our main\nresults, from the associative and Lie cases to more general\nsettings. As we will show in the following section, these include\nPoisson, Gerstenhaber, BV, permutation, and pre-Lie algebras: see\nExamples \\ref{pgexam}, \\ref{bvexam}, and \\ref{permexam} in the\nfollowing section.\n\nWe may think of curved (cyclic) $A_\\infty$- and $L_\\infty$-algebras as\narising from the following construction, which we think of\nheuristically as a type of ``Koszul duality'' between operads\ngoverning curved algebras and those governing unital algebras (we do\nnot attempt to make this description precise).\n\\begin{remark}\n  For a somewhat related result, see \\cite{HiMickdt}, where resolutions\n  for operads of unital algebras are constructed by defining a Koszul\n  dual curved cooperad and performing a version of the cobar\n  construction. Here, we will not make use of the notion of curved\n  (co)operads defined in \\cite{HiMickdt}, and will only use ordinary (dg)\n  operads.\n\\end{remark}\nLet $\\mathcal{O}$ be a (cyclic) dg operad \\cite{GKcoch}, which we\nassume to be unital with unit $I \\in \\mathcal{O}(1)$. Moreover, we\nwill assume throughout that each $\\mathcal{O}(i)$ is a\nfinite-dimensional $\\ZZ\/2$-graded vector space (this isn't really\nessential, but it makes dualization less technical, and includes all\noperads we have in mind. Properly speaking, we view $\\mathcal{O}(i)$\nas a formal space, and everything generalizes to the\ninfinite-dimensional formal setting.)  Let $m_0 \\in \\mathcal{O}(0)$ be\nan element (corresponding to a ``0-ary'' operation).  Recall that\nevery (cyclic) operad is an $\\SS$ ($\\SS_+$)-module, where an\n$\\SS$-module is defined as a collection $\\{V_m\\}_{m \\geq 0}$ of\n$S_m$-modules for all $m \\geq 0$, and an $\\SS_+$-module is an\n$\\SS$-module where each $V_m$ is actually a module over $S_{m+1}$ (the\nunderlying $\\SS$-module is obtained using the inclusion $S_m \\subseteq\nS_{m+1}$ of permutations fixing $m+1$).\n\nWe will now consider $\\mathcal{O}$ as a nonunital operad and\nperform the cobar construction \\cite{GiKa}. Namely, let $\\mathcal{O}^* := \\{ \\mathcal{O}(m)^* \\}$ be the dual\n$\\SS$ (or $\\SS_+$)-module. Let $C(\\mathcal{O})$ be the free operad\ngenerated by $\\Pi \\mathcal{O}^*$, equipped with a differential\n$d_{C(\\mathcal{O})}$ obtained as follows.  For every $j \\geq 0$ and\nall $1 \\leq i \\leq k$, there is an operadic composition map\n\\begin{equation}\n\\circ^{k,j}_i: \\mathcal{O}(k) \\otimes \\mathcal{O}(j) \\rightarrow \\mathcal{O}(j+k-1),\n\\end{equation}\nwhich corresponds to plugging the element of $\\mathcal{O}(k)$ into the\n$i$-th input of the element of $\\mathcal{O}(j)$.  Let\n$(\\circ^{k,j}_i)^*$ be the linear dual to the above map.\nThen, we\ndefine the differential $d_{C(\\mathcal{O})}$ to be the\nunique extension to a derivation of the operation\n\\begin{equation}\nd_{C(\\mathcal{O})}|_{\\mathcal{O}(i)^*} = d_{\\mathcal{O}}^* +\n\\bigoplus_{j,k} (\\circ^{k,j}_i)^*.\n\\end{equation}\nNow, in the case that $\\mathcal{O}$ is the ordinary (non-dg) operad\ngoverning \\emph{unital} associative or commutative algebras, then\n$C(\\mathcal{O})$ is a dg operad with the property that\ngraded (dg) algebras $V$ over $C(\\mathcal{O})$, equipped\nwith zero differential, are the same as curved $A_\\infty$- or\n$L_\\infty$-algebras. In this case, $m_0 \\in \\mathcal{O}(0)$ is the unit of the\nmultiplication, and the curvature of a graded algebra $V$ over $C(\\mathcal{O})$ is the image of $m_0^* \\in \\mathcal{O}(0)^*$ in $V$.\n\n\nNext, suppose that\n$\\mathcal{O}'$ is the suboperad of $\\mathcal{O}$ in positive arity,\ni.e., $\\mathcal{O}'(0) = 0$ and $\\mathcal{O}'(i) = \\mathcal{O}(i)$ for $i \\geq 1$.\n Further\nsuppose that $\\mathcal{O}'$ is a Koszul operad with augmentation\n$\\overline{\\mathcal{O}'}$ (i.e., $\\overline{\\mathcal{O}'}$ is\na suboperad such that $\\mathcal{O}'(1) = \\overline{\\mathcal{O}'}(1) \\oplus \\CC \\cdot \\Id$, and\n$\\overline{\\mathcal{O}'}(i) = \\mathcal{O}'(i)$ for $i \\neq 1$).\nThen, $C(\\overline{\\mathcal{O}'})$ yields a resolution of the\nKoszul dual operad $(\\mathcal{O}')^!$ (this last fact is equivalent to\nKoszulity; see \\cite{GiKa}).  \\emph{Caution:} This operad $C(\\overline{\\mathcal{O}'})$ is sometimes called the cobar construction on the unital operad\n$\\mathcal{O}'$ itself, but\nhere we consider cobar constructions only on nonunital operads.\n\nWe think of an algebra over $C(\\mathcal{O})$ (with zero differential)\nas a certain type of \\emph{curved} version of an\n$((\\mathcal{O}')^!)_\\infty$-algebra, as opposed to an algebra over\n$C(\\overline{\\mathcal{O}'})$, which, in the case $\\mathcal{O}'$ is\nKoszul, is the same as an ordinary\n$((\\mathcal{O}')^!)_\\infty$-algebra.  (More generally, given a dg\nspace $V$ with possibly nontrivial differential, one could view a\n$C(\\mathcal{O})$-algebra structure on it as a dg generalization of a\ncurved $((\\mathcal{O}')^!)_\\infty$-algebra; thus one could speak, for\ninstance, about a curved dg $A_\\infty$- or $L_\\infty$-algebra. This\nwould be a Maurer-Cartan element in the appropriate differential\ngraded Lie algebras of (formal, noncommutative) vector fields rather\nthan simply square-zero odd derivations.)\n\n\nLet us return to the general setting of an arbitrary (cyclic) operad\n$\\mathcal{O}$. The analogues of Theorems \\ref{curvtrivthm} and\n\\ref{cyccurvtrivthm} are then the following.\n\\begin{definition}\n  A \\emph{weakly unital multiplication} $m_2 \\in \\mathcal{O}(2)$\n  is an element such that the image of the maps\n\\begin{equation}\\label{wtm2map}\nm_2(- \\otimes \\Id), m_2(\\Id \\otimes -): \\mathcal{O}(0) \\rightarrow \\mathcal{O}(1)\n\\end{equation}\nlie in the one-dimensional space $\\CC \\cdot \\Id \\subseteq \\mathcal{O}(1)$, and such that the maps are not both zero.\n\\end{definition}\nIf $m_2$ is weakly unital, then\n\\begin{equation} \\label{m2ideq}\nm_2(- \\otimes \\Id) = m_2(\\Id \\otimes -).\n\\end{equation}\nIndeed, if $m_0 \\in \\mathcal{O}(0)$ is any element such that\n$m_2(m_0 \\otimes \\Id) = \\lambda \\cdot \\Id$ for some nonzero $\\lambda\n\\in \\CC$, then it follows that $m_2(m_0 \\otimes m_0) = \\lambda\\cdot m_0$,\nand hence that $m_2(\\Id \\otimes m_0) = \\lambda \\cdot \\Id$.\n\nTherefore, when $m_2$ is weakly unital, we define $\\widetilde{m_2}\n\\in \\mathcal{O}(0)^*$ to be the resulting map \\eqref{m2ideq}.\n\n\nNext, given a $C(\\mathcal{O})$ algebra $V$, the $0$-ary operations form a map $\\gamma: \\mathcal{O}(0)^* \\rightarrow V$, which we call the \\emph{$0$-ary structure\nmap}.\n\\begin{definition}\nGiven an operad $\\mathcal{O}$ with weakly unital multiplication $m_2$,\na $C(\\mathcal{O})$ algebra $V$ with $0$-ary structure map $\\gamma$\nis said to be\n\\emph{nontrivially curved with respect to $m_2$}\nif the element $\\gamma(\\widetilde{m_2})$ is nonzero.\n\\end{definition}\nWe call the element $\\gamma(\\widetilde{m_2})$ the\n\\emph{curvature} of $V$ (for the element $m_2$).\n\nThen, Theorems \\ref{curvtrivthm} and \\ref{cyccurvtrivthm}\ngeneralize as follows.  Fix a graded vector space $V$ (with zero\ndifferential).  If $\\mathcal{O}$ is any dg operad, we denote a\n$\\mathcal{O}$-algebra structure on $V$ as a pair $(V, \\phi)$ for the\nalgebra structure, where given any $o \\in \\mathcal{O}(k)$, $\\phi(o) \\in\n\\Hom_\\CC(V^{\\otimes k}, V)$.\n\\begin{claim} Fix a graded vector space $V$. \\label{opclaim}\n\\begin{enumerate}\n\\item[(i)] Let $\\mathcal{O}$ be an operad with a weakly unital\n  multiplication $m_2$. Let $(V,\\phi)$ be a\n  $C(\\mathcal{O})$-algebra structure with nonzero curvature $c\n  \\in V$.\n  Then, $(V,\\phi)$ is gauge equivalent to the algebra $(V,\\phi')$ with\n  the same $0$-ary structure map $\\gamma$,\nbut with all\n  higher multiplications equal to zero.\n\\item[(ii)] Let $\\mathcal{O}$ be a cyclic operad with a weakly unital\n  multiplication $m_2$. Let $(V,\\phi)$ be a cyclic\n  $C(\\mathcal{O})$-algebra structure, with nonzero curvature $c\n  \\in V$.  Let $c' \\in V$ be an element such that $(c', c) = 1$.\n  Then, $(V,\\phi)$ is gauge equivalent to the algebra $(V,\\phi')$ with\n  the same $0$-ary structure map $\\gamma$, but with higher\n  multiplications of the form\n\\begin{equation}\\label{opclaim12eq}\n\\phi'(m)(v_1, \\ldots, v_i) = \\bigl(\\prod_{j=1}^i (c',v_j) \\bigr) \\cdot (\\phi(m)(c, c, \\ldots, c), c) \\cdot c'.\n\\end{equation}\n\\end{enumerate}\n\\end{claim}\nHere, gauge equivalence is defined as follows.  For any\n$C(\\mathcal{O})$-algebra structure on $V$, one can form the formal dg\n$\\mathcal{O}$-algebra $\\hat{C}_{\\mathcal{O}}(V)$, which is defined as the completed\nfree $\\mathcal{O}$-algebra generated by $\\Pi V^*$:\n\\[\n\\hat{C}_{\\mathcal{O}}(V)=\\prod_{n=0}^\\infty {\\mathcal\n  O}(n)\\otimes_{S_n}(\\Pi V^*)^{\\otimes n} ,\\] equipped with the\ndifferential $d_{\\hat{C}_{\\mathcal{O}}(V)}$ on\n$\\hat{C}_{\\mathcal{O}}(V)$ defined by the canonical linear map $\\Pi V^*\n\\rightarrow \\hat{C}_{\\mathcal{O}}(V)$, which is shifted dual to a\nrestriction of the structure maps $C(\\mathcal{O}) \\rightarrow\n\\Hom(V^{\\otimes n}, V)$.\n\nNext, assume for the moment that $V$ is equipped with the zero\n$C(\\mathcal{O})$-algebra structure, i.e.,\n$d_{\\hat{C}_{\\mathcal{O}}(V)} = 0$. We may then form the formal dgla\n$\\Der(\\hat{C}_{\\mathcal{O}}(V))$, where the differential is induced by\nthe differential on $V$ (and is zero in the case where $V$ was merely\na graded vector space, as in the setting of (non-dg) curved\n$A_\\infty$- or $L_\\infty$-algebras considered in earlier sections of\nthis paper).  It is then a standard fact (see, e.g., \\cite[Proposition\n2.15]{GJohaii}, Proposition 2.15 where, however, this result is\nformulated in the language of coalgebras) that the above yields a\nbijection between square-zero odd derivations of\n$\\hat{C}_{\\mathcal{O}}(V)$ and $C(\\mathcal{O})$-structures on $V$.\nFinally, we define two $C(\\mathcal{O})$-structures on $V$ to be gauge\nequivalent if the corresponding differentials\n$d_{\\hat{C}_{\\mathcal{O}}(V)}$ and $d_{\\hat{C}_{\\mathcal{O}}(V)}'$ are\ngauge equivalent, i.e., that there exists an even derivation $\\xi \\in\n\\Der^0(\\hat{C}_{\\mathcal{O}}(V))$ with zero constant term such that\n$d_{\\hat{C}_{\\mathcal{O}}(V)}' = e^{\\ad \\xi}\nd_{\\hat{C}_{\\mathcal{O}}(V)}$.\n\nSimilarly, if $\\mathcal{O}$ is a cyclic operad, and $V$ is a cyclic\nalgebra over $\\mathcal{O}$ (i.e., an algebra equipped with a\nnondegenerate inner product compatible with the cyclic structure on\n$\\mathcal{O}$), then there is a natural $\\ZZ\/(m+1)$ action on the\nsubspace $\\Pi V^* \\otimes \\mathcal{O}(m) \\otimes_{S_m} (\\Pi V^*)^{\\otimes m}\n\\cong \\Hom(\\Pi V^*, \\hat{C}_{\\mathcal{O}}(V)) = \\Der(\\hat{C}_{\\mathcal{O}}(V))$\ncoming from the $S_{m+1}$-structure on $\\mathcal{O}(m)$ and the\ncompatible inner product on $V$.  Then, we can define the \\emph{cyclic\n  derivations}, $\\CDer(\\hat{C}_{\\mathcal{O}}(V))$, to be the subspace of\n$\\ZZ\/(m+1)$-invariants in each degree $m$.  Then, a gauge equivalence\n$e^{\\ad\\xi}: d_{\\hat{C}_{\\mathcal{O}}(V)} \\iso d_{\\hat{C}_{\\mathcal{O}}(V)}'$ is\n\\emph{cyclic} if $\\xi$ is cyclic.\n\nFinally, we explain the main idea of the proof of the claim (without\ngoing into full detail).  We explain only the second part, since it is\nmore involved. We first form the cyclic deformation complex for the\n$C(\\mathcal{O})$-structure on $V$\nstructure with all higher operations $C(\\mathcal{O})(i), i > 0$ acting as\nzero, and with $0$-ary structure map $\\gamma: C(\\mathcal{O})(0) = \\Pi\\mathcal{O}(0)^* \\to V$.\nThis complex is $\\CDer(\\hat{C}_{\\mathcal{O}}(V))$, equipped with the\ndifferential $\\ad \\xi$, where $\\xi \\in\n\\CDer(\\hat{C}_{\\mathcal{O}}(V))$ is the derivation obtained from $\\Pi\\gamma^*: \\Pi V^* \\to \\mathcal{O}(0)$, by the formula (for $o \\in \\mathcal{O}(n)$ and $f_1, \\ldots, f_n \\in \\Pi V^*$):\n\\begin{equation}\n\\xi(o \\otimes_{S_n} (f_1 \\otimes \\cdots \\otimes f_n)) = \\sum_{i=1}^n (o \\circ_i \\Pi\\gamma^*(f_i)) \\otimes_{S_{n-1}} (f_1 \\otimes \\cdots \\otimes \\hat f_i \\otimes \\cdots \\otimes f_n),\n\\end{equation}\nwhere $\\hat f_i$ denotes omitting $f_i$ from the tensor product.\n\nThen, we again show that this complex is quasi-isomorphic\nto the subcomplex spanned by cochains such that $\\mathcal{O}(i)$ acts by\nmultiples of the element\n\\begin{equation}\\label{ceq2}\n\\epsilon^{i+1}(x_1, \\ldots, x_i) := \\epsilon(x_1) \\cdots \\epsilon(x_i) c', \\quad \\epsilon(v) := (c', v),\n\\end{equation}\nand hence that the original algebra structure is gauge equivalent to\none where all $\\geq 1$-ary operations are multiples of the operation\n\\begin{equation}\n\\phi'(m)(v_1, \\ldots, v_i) = \\bigl(\\prod_{j=1}^i (c',v_j) \\bigr) \\cdot c'.\n\\end{equation}\nFinally, as before, we can see that this multiple is $(\\phi(m)(c, c, \\ldots, c), c)$.\n\nThe main technical step in the above is showing that the complex is\nquasi-isomorphic to the subcomplex spanned by cochains as in\n\\eqref{ceq2}. This amounts to a generalization of Lemma \\ref{qil}.\nThe main point is that the map $s_i'$ defined in \\eqref{sipfla1} can\nbe generalized to this setting, as follows.  If $f = o\n\\otimes_{S_{n+1}} (f_1 \\otimes \\cdots \\otimes f_{n+1}) \\in\n\\mathcal{O}(n) \\otimes_{S_{n+1}} (\\Pi V^*)^{\\otimes (n+1)}$, viewed as\nan element of $\\CDer(\\hat{C}_{\\mathcal{O}}(V))$, then $s_n f \\in\n\\mathcal{O}(n+1) \\otimes_{S_{n+2}} (V^*)^{\\otimes(n+2)}$ is obtained\nas\n\\[\ns_n f = \\sum_{j=1}^{n+1} (o \\circ_j m_2) \\otimes_{S_{n+2}} (f_1 \\otimes \\cdots \\otimes f_{j-1} \\otimes \\epsilon \\otimes f_j \\otimes \\cdots \\otimes f_{n+1}).\n\\]\nThen, one obtains a similar result to \\eqref{sipfla3}, which gives the desired conclusion.\n\nNext, Theorem \\ref{acythm} generalizes as follows.  Let $\\cO$ be a\ncyclic operad, and let $\\mathcal{G}_{\\mathcal{O}}$ be the graph\ncomplex constructed from the operad $\\mathcal{O}$.  This complex is a\ngeneralization of Kontsevich's graph homology which has many\nequivalent definitions in the literature; one definition is via the\nFeynman transform construction of \\cite{GKmo}.  Namely, consider the\nnaive $\\Det$-twisted modular closure $\\underline{\\mathcal O}^1$ of\n$\\mathcal O$ by considering all contraction operations to be zero and\nall parts of $\\mathcal{O}$ of genus $\\geq 1$ to be zero; see, e.g.,\n\\cite[\\S 2]{ChLadft} for this notion. Then form the Feynman transform\noperad $\\mathsf{F}\\underline{\\mathcal O}^1$. The $0$-ary part\n$\\mathsf{F}\\underline{\\mathcal O}^1((0))$ is the desired graph\ncomplex.  As before, let $\\mathcal{G}_{\\mathcal{O},c} \\subseteq\n\\mathcal{G}_{\\mathcal{O}}$ be the subcomplex spanned by connected\nnonempty graphs; one has $\\mathcal{G}_{\\mathcal{O}} \\cong \\Sym\n\\mathcal{G}_{\\mathcal{O},c}$.\n\\begin{claim}\\label{opclaim2}\n  The complex $\\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic to\n  the quotient of the subcomplex spanned by graphs with at most one\n  vertex of valence $\\geq 2$ by the span of line segments with three\n  vertices whose central vertex is labeled by $\\Id$.\\footnote{Note\n    that, if $\\mathcal{O}(0)$ is even one-dimensional, as in the\n    preceding cases of $\\mathscr{C}\\!\\mathit{omm}$ or $\\mathscr{A}\\!\\mathit{ss}$, then these\n    line segments with central vertex labeled by $\\Id$ are already\n    zero. More generally, the span of these line segments is\n    isomorphic to $\\wedge^2 \\cO(0)$, by considering the labels at the\n    univalent vertices.}\n\\end{claim}\nIn other words, $\\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic to a\nquotient of the deformation complex of a certain canonical\ncyclic $C(\\mathcal{O})$-algebra (which we think of as a type of curved\n$((\\mathcal{O}')^!)_\\infty$-algebra), as follows.\\footnote{This\n  interpretation requires that $\\mathcal{O}(0)$ be finite-dimensional,\n  as we are assuming.  For infinite-dimensional formal\n  $\\mathcal{O}(0)$, while the claim above still holds, this\n  interpretation is technically not available.} Let $V :=\n\\mathcal{O}(0)^*$. View $V$ as a $\\mathcal{O}$-algebra with all $\\geq\n1$-ary operations trivial, and with $0$-ary structure $\\Id:\n\\mathcal{O}(0)^* \\to \\mathcal{O}(0)^*$. Since all higher operations\nare trivial, any inner product on $V$ is cyclic; we can fix one but it\nwill not really affect anything.  Then, the deformation complex of $V$\nas a $C(\\mathcal{O})$-algebra is\n$$\\CDer(\\hat{C}_{\\mathcal{O}}(V)) \\cong \\bigoplus_{m \\geq 0} \\mathcal{O}(m) \\otimes_{S_{m+1}} \\mathcal{O}(0),$$\nequipped with the differential $\\ad \\xi$ where $\\xi \\in\n\\CDer(\\hat{C}_{\\mathcal{O}}(V))$ is the element corresponding to $\\Id\n\\in V \\otimes \\mathcal{O}(0) \\cong \\Hom(V, V) \\subseteq\n\\Der(\\hat{C}_{\\mathcal{O}}(V))$.  By the above,\n$\\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic to the quotient\nof this deformation complex by the subcomplex in arity $2$, $\\Id\n\\otimes_{S_2} \\mathcal{O}(0) \\subseteq \\bigoplus_{m \\geq 0}\n\\mathcal{O}(m) \\otimes_{S_{m+1}} \\mathcal{O}(0)$.\n\nThe proof of Claim \\ref{opclaim2} is a direct generalization of the\nproof of Theorem \\ref{acythm}.  First, we generalize the notion of\ninterior vertex.  Note that $\\mathcal{G}_{\\mathcal{O},c}$ is\nspanned by oriented graphs of the following form: the vertices are\nordered, and each $m$-valent vertex is labeled by an element of\n$\\cO(m-1)$.  Next, the half-edges are also ordered.  Many of these\ngraphs are equivalent: applying a permutation of the half-edges\nincident to a given vertex is set equal to applying the corresponding\npermutation to the element of $\\cO$ labeling that vertex; also,\napplying any permutation of the half-edges is set equal to multiplying\nby the sign of that permutation. Finally, applying a permutation to\nthe vertices is the same as multiplying by the sign of that\npermutation.\n\nThen, an interior vertex of a graph as above is a vertex which has\nvalence $\\geq 2$ and whose removal does not result in a graph one of\nwhose connected components consists only of univalent vertices and\nbivalent vertices labeled by elements of $\\CC \\cdot \\Id \\subseteq\n\\cO(1)$.  As before, the number of interior vertices defines an\nincreasing filtration. Moreover, if we choose an arbitrary basis\n$\\{\\Gamma_i\\}$ of $\\mathcal{G}_{\\mathcal{O},c}$, then the\nassociated graded complex is graded by this basis, where\n$\\gr_{\\Gamma_i} \\mathcal{G}_{\\mathcal{O},c}$ is spanned by graphs\nwhose restriction to interior vertices yields $\\Gamma_i$ (note that\n$\\Gamma_i$ need not be a graph).\n\n\nThe main step is to generalize the construction of the contracting\nhomotopy which shows that $\\gr_{\\Gamma_0}\n\\mathcal{G}_{\\mathcal{O},c}$ is acyclic when $\\Gamma_0$ is a graph\ncontaining an edge. To do so, first fix an element $m_0 \\in \\cO(2)$\nsuch that $m_2 \\circ_1 m_0 = m_2 \\circ_2 m_0 = \\Id \\in \\cO(1)$.  We\nchoose the basis $\\{\\Gamma_i\\}$ to consist of graphs.  Let $\\Gamma_0$\nbe one such graph which contains an edge.  Fix a half edge $h$\nof $\\Gamma_0$\nbased at a vertex $v$.  Let us choose $v$ to be the last vertex in the\nordering of vertices, and $h$ to be the last half-edge in the ordering\nof half-edges based at $v$.  Suppose $v$ is $m$-valent, and let the\nlabel of $v$ be $o_v \\in \\cO(m-1)$. Label the half-edges of $v$, in\norder, $h_1, \\ldots, h_m$, with $h = h_m$.  The contracting homotopy\n$s$ then acts on any graph $\\Gamma \\in \\gr_{\\Gamma_0}\n\\mathcal{G}_{\\mathcal{O},c}$ by first adding a new half-edge\n$h_{m+1}$ to $v$ (last in the ordering at $v$).  Then, the label $o_v$\nis replaced by $o_v \\circ_m m_2$, where $m_2 \\in \\cO(2)$ is the weakly\nunital multiplication. Finally, one adds a new univalent vertex $y$ to\n$\\Gamma$ (which becomes the last vertex), labels it by $m_0$, and\nattaches it to $h_{m+1}$.  It is then straightforward to verify that\n$(sd + ds) \\Gamma = \\Gamma$, and hence $\\gr_{\\Gamma_0}\n\\mathcal{G}_{\\mathcal{O},c}$, is acyclic.\n\nFinally, we explain the appearance of the quotient by line segments\nwith three vertices whose central vertex is labeled by $\\Id$.  Namely,\nthese are exactly the graphs with a single vertex of valence $\\geq 2$\nwhich, nonetheless, have no interior vertices.  By a generalization of\nthe arguments in the proof of Theorem \\ref{acythm}, $\\gr_{\\pt}\n\\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic to the quotient of\nthe subcomplex spanned by star-shaped graphs with a single vertex of\nvalence $\\geq 2$ by the span of these graphs.  On the other hand,\n$\\gr_{\\emptyset} \\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic\nto the subcomplex spanned by graphs with two vertices, each univalent.\nThus, the second page of the spectral sequence for the interior vertex\nfiltration yields, in degree one, the claimed quotient\ncomplex modulo the graphs with only two vertices, and in degree zero,\nthe span of graphs with only two vertices and one edge.  The spectral\nsequence collapses at the third page to the homology of the whole\nquotient subcomplex stated in Claim \\ref{opclaim2}, which proves the\nresult. We omit further details.\n\nFinally, one can deduce an unstable version of Claim \\ref{opclaim2},\nanalogous to Theorem \\ref{unscurveduncurved}:\n\\begin{claim}\\label{opclaim3}\nThe composition\n\\begin{equation}\n\\CE_\\bullet(\\CDer^0(\\hat{C}_{\\mathcal{O}}(V))) \\to \\CE_\\bullet(\\CDer(\\hat{C}_{\\mathcal{O}}(V)))\n\\rightarrow \\CE_\\bullet(\\CDer(\\hat{C}_{\\mathcal{O}}(V \\oplus W)))\n\\end{equation}\nis zero on homology for any uncurved cyclic $C(\\mathcal{O})$-algebra $V$,\nwhere $W$ is an inner product space with $W_0 \\neq 0$.\n\\end{claim}\nWe omit the proof, which is obtained by combining the preceding\nmaterial with \\S \\ref{unstabsec}.\n\\section{Examples and further comments}\nIn this section we provide some remarks and examples regarding the\nmaterial of the previous section.\n\n\\subsection{Generalization to the modular case}\nHere we briefly sketch how the results of the previous section\ncan be extended from the\nsetting of cyclic operads to that of modular operads.\n\nAn analogous result to Claim \\ref{opclaim2} holds for the\n\\emph{twisted} version of the $\\mathcal O$-graph complex which\ncorresponds to the Feynman transform ${\\mathsf F}\\underline{\\mathcal\n  O}((0))$ (as opposed to ${\\mathsf F}\\underline{\\mathcal O}^1((0))$)\nwhich is the usual version of the graph complex.\\footnote{We note that\n  the operads ${\\mathsf F}\\underline{\\mathcal O}^d((n))$ are not,\n  strictly speaking, modular operads in the sense of \\cite{GKmo},\n  since the stability condition is not satisfied; however the\n  construction of the Feynman transform still makes sense and this\n  causes us no trouble. We will ignore this point henceforth.}  For\nexample, in the commutative and ribbon graph case the difference\nbetween two types of graph complex lies in a different notion of\norientation: a twisted orientation corresponds to ordering edges of a\ngraph, as opposed to ordering the half edges and vertices.\n\nMore generally, one can replace cyclic operads $\\cO$ by arbitrary\n(twisted) modular operads. Then, if $\\cO$ admits a weakly unital\nmultiplication, one still obtains in the same manner the result that\nthe graph complex for $\\cO$ is quasi-isomorphic to the subcomplex\ndefined analogously to the above.\n\nSimilarly, we can generalize all the constructions of the preceding\nsection from the cyclic to the modular setting. Let $\\cO$ be an\narbitrary (twisted) modular operad $\\cO$.  In this case, one replaces\n$C(\\cO)$ (used in the cyclic case above) by the Feynman transform\n(twisted) modular operad, ${\\mathsf F}\\mathcal O$, and modular\nalgebras over this operad are then thought of as curved algebras. The\nconstructions, results (Claims \\ref{opclaim}.(ii) and \\ref{opclaim3}),\nand proofs carry over to this setting.  In the special case where one\nhas a cyclic operad and considers it $\\Det$-twisted modular using the\nnaive $\\Det$-twisted modular closure, one recovers the above results.\nAnother example concerns so-called quantum $A_\\infty$-algebras: Let\n$\\mathcal{O}'$ be the $\\Det$-twisted modular closure of\n$\\mathscr{A}\\!ss$ such that ${\\mathsf F}\\mathcal O'$-algebras are\nso-called quantum $A_\\infty$-algebras (see for instance \\cite[Example\n5.2]{ChLafdmm} for an explanation of this notion). Then, if we let\n$\\mathcal{O} = \\mathcal{O}' \\oplus \\CC[0]$ be the unital version\n(adding a $0$-ary operation providing a unit for the associative\nmultiplication), then ${\\mathsf F}\\mathcal O$ defines a notion of\ncurved quantum $A_\\infty$-algebras.  As in the cyclic $A_\\infty$ case,\none sees that nontrivially curved quantum $A_\\infty$-algebra\nstructures on a fixed vector space are gauge equivalent to those where\nall the operations of positive arity are of the form\n\\eqref{opclaim12eq}, and in particular all land in a fixed\none-dimensional vector space. We refrain from making precise\nstatements.\n\n\\subsection{Examples}\nHere we provide examples of the preceding constructions for Poisson, Gerstenhaber, BV, and permutation (or pre-Lie) algebras.\n\\begin{example}\\label{pgexam} Consider the case of Poisson algebras.\n  It is natural to consider unital Poisson algebras, where here a unit\n  $f$ is an element satisfying $\\{1, f\\} = \\{f, 1\\} = 0$ for all $f$,\n  whereas $1 \\cdot f = f \\cdot 1 = f$. Let $\\mathcal{O}$ be the operad\n  governing unital Poisson algebras, i.e., $\\mathcal{O} =\n  u\\mathscr{P}\\!\\textit{oiss} = \\mathscr{P}\\!\\textit{oiss} \\oplus\n  \\CC[0]$ where $\\mathscr{P}\\!\\textit{oiss}$ is the Poisson operad and\n  $\\CC[0]$ is the one-dimensional vector space concentrated in degree\n  zero, which has zero compositions with the bracket of $\\mathcal{O}$,\n  and composition $m_2(m_0 \\otimes \\Id) = \\Id = m_2(\\Id \\otimes m_0)$\n  with the commutative multiplication, i.e.,\n  $\\mathscr{P}\\!\\textit{oiss} \\supset u\\mathscr{C}\\!\\textit{omm}$.\n  Then, by the above, one can think of $C(\\mathcal{O})$-algebras as\n  curved Poisson-infinity algebras (since Poisson is Koszul self-dual,\n  as in the associative case), and it follows from Claim\n  \\ref{opclaim}.(i) that nontrivially curved Poisson-infinity structures\n  on a vector space are all gauge equivalent.\n\n  Similarly, the Gerstenhaber operad $\\mathscr{G}\\!\\textit{erst}$ is\n  Koszul and dual to its suspension $\\mathfrak{s}\n  \\mathscr{G}\\!\\textit{erst}$; here the suspension is defined by\n  tensoring by the endomorphism operad of the one-dimensional odd\n  vector space (as a $\\ZZ\/2$-graded $\\SS$-module, this means that one\n  applies $\\Pi$ to the even-ary part).  As before, one can consider\n  unital $\\mathscr{G}\\!\\textit{erst}$ algebras,\n  $u\\mathscr{G}\\!\\textit{erst}$, and similarly its suspension $\\mathfrak{s}(u\n  \\mathscr{G} \\! \\textit{erst})$. Let $\\cO$ be this latter operad,\n  which we can also think of as $u\n  \\mathscr{G}\\!\\textit{erst}^!$. Here, the unit is odd.  We define\n  curved $\\mathscr{G}\\!\\textit{erst}_\\infty$ algebras as algebras over\n  $C(\\cO)$, where now the curvature is odd, rather than even.  From\n  Claim \\ref{opclaim}.(i), we deduce that any two nontrivially curved\n  Gerstenhaber-infinity algebra structures on the same graded vector\n  space are gauge equivalent.\n\n  In the case of the Poisson operad (but not for the Gerstenhaber operad),\n  in fact $\\mathscr{P}\\!\\textit{oiss}$ is cyclic, and hence one obtains\n  the notion of cyclic curved Poisson-infinity algebras. Claim\n  \\ref{opclaim}.(ii) then implies that all nontrivially curved cyclic\n  Poisson-infinity structures on a vector space are gauge equivalent\n  to those for which all operations of positive arity are of the form\n  \\eqref{opclaim12eq}, and in particular all land in a fixed\n  one-dimensional vector space.\n\n  Moreover, in fact one can compute the associated (unital Poisson)\n  graph homology. By Claim \\ref{opclaim2}, the associated graph\n  complex is quasi-isomorphic to the subcomplex spanned by graphs\n  whose vertices are all univalent except for at most one.  Since the\n  Poisson operad is the associated graded operad of the associative\n  operad, and in particular has the same $\\SS$-module structure, one\n  sees that this subcomplex has zero differential, and is spanned by\n  the star-shaped graphs with central vertex of odd valence.  That is,\n  the graph homology for the unital Poisson operad is isomorphic to\n  the homology of the graph complex $\\widetilde{\\mathcal{G}_r}$ for the unital\n  associative operad.\n\\end{example}\n\\begin{example}\\label{bvexam} The BV operad $\\BV$ \\cite{Getbva} is the homology operad of the operad of framed little discs; it is known to be cyclic. Its algebras, called $\\BV$-algebras, are Gerstenhaber algebras together with an odd operator $\\Delta$ which is a differential operator of second order with respect to the commutative multiplication and a derivation of the odd Lie bracket.  Let $\\overline{\\BV}$, as before, denote the augmentation ideal of the operad $\\BV$.\n  The $C(\\overline{\\BV})$-graph complex computes, according to\n  \\cite{Giafl2d}, the homology of the classifying space of diffeomorphism\n  groups of 3-dimensional oriented handlebodies.  Although the operad\n  $\\BV$ is not defined by (homogeneous) quadratic relations, in\n  \\cite{GTVhbva} it was shown to be Koszul in a more general sense,\n  and its Koszul dual, $\\BV^!$ was described, and shown to be\n  quasi-isomorphic to $C(\\overline{\\BV})$.\n\n  A unital $\\BV$-algebra is a $\\BV$-algebra with a unit with respect\n  to the commutative multiplication and such that the value of\n  $\\Delta$ on the unit is zero.\n\n  Consider the operad ${\\mathcal O} =u\\BV$ governing unital\n  $\\BV$-algebras; then we can view $C(\\mathcal O)$ as the operad\n  governing curved $\\BV^!_\\infty$-algebras. It follows that all\n  nontrivially curved $\\BV^!_\\infty$-algebras are gauge equivalent.\nFurthermore, according to Claim\n  \\ref{opclaim}.(ii) all nontrivially curved cyclic $\\BV^!_\\infty$-algebras are gauge equivalent\n  to those for which all operations of positive arity are of the form\n  \\eqref{opclaim12eq}.\n\n  Finally, by Claim \\ref{opclaim2}, the graph complex for $\\mathcal O$\n  is quasi-isomorphic to the subcomplex with at most one vertex of\n  valence $\\geq 2$.\n\\end{example}\n\\begin{example}\\label{permexam} Consider the case of (right) pre-Lie\n  algebras, i.e., those with a single operation $\\star$ satisfying the\n  relation\n\\begin{equation}\nx \\star (y \\star z) - (x \\star y) \\star z = x \\star (z \\star y) - (x \\star z) \\star y.\n\\end{equation}\nFor such algebras, it makes perfect sense to define the notion of a unit, $1$, such that\n\\begin{equation} \\label{prelieunit}\n1 \\star x = x = x \\star 1.\n\\end{equation}\nWe thus obtain an operad $\\mathcal{O} = u \\mathscr{P}\\!\\textit{re-Lie}\n= \\mathscr{P}\\!\\textit{re-Lie} \\oplus \\CC[0]$ governing \\emph{unital}\npre-Lie algebras, where now the compositions with $\\CC[0]$ are given\nby \\eqref{prelieunit}, or more precisely, $m_2(m_0 \\otimes \\Id) = \\Id\n= m_2(\\Id \\otimes m_0)$ where $m_0 = 1 \\in \\CC[0]$ and $m_2 \\in\n\\mathscr{P}\\!\\textit{re-Lie}[2]$ is the multiplication operation\n$\\star$.  By the above procedure, we then obtain a type of curved\nalgebra, namely algebras over $C(\\mathcal{O})$.  We will call these\n \\emph{curved (right)\n  $\\mathscr{P}\\!\\textit{erm}_{\\infty}$-algebras}, since the operad\n(right) $\\mathscr{P}\\!\\textit{erm}$ is Koszul dual to (right)\n$\\mathscr{P}\\!\\textit{re-Lie}$. In particular, curved\n  $\\mathscr{P}\\!\\textit{erm}_\\infty$ algebras with zero curvature\n  ($m_0=0$, i.e., the corresponding derivation of $C_{\\mathcal{O}}(V)$\n  has zero constant term) are the same as ordinary\n  $\\mathscr{P}\\!\\textit{erm}_\\infty$ algebras.  Let us recall here\n  that ordinary permutation algebras are algebras with an operation\n  $\\circ$ satisfying the relation\n\\begin{equation}\nx \\circ (z \\circ y) = x \\circ (y \\circ z) = (x \\circ y) \\circ z,\n\\end{equation}\ni.e., associative algebras additionally satisfying the first equality\nabove.  The operad $\\mathscr{P}\\!\\textit{erm}$ is the one whose algebras are\npermutation algebras.\n\nFrom the above results, we deduce that any two curved\n$\\mathscr{P}\\!\\textit{erm}_\\infty$-algebra structures on $V$ with nonzero curvature\nare gauge-equivalent.\n\nNote that, strictly speaking the operad $\\mathscr{P}\\!\\textit{re-Lie}$ is not cyclic, it is\nanticyclic \\cite[p. 9]{GKcoch}; however we can ignore this difference; any anticyclic operad gives rise to a cyclic one via the operadic suspension. Therefore, we obtain the corresponding result on the classification of nontrivially curved (anti)cyclic $\\mathscr{P}\\!\\textit{erm}_\\infty$-algebras, and on\n$u\\mathscr{P}\\!\\textit{re-Lie}$-graph homology.\n\\end{example}\n\\subsection{On the cobar construction of unital operads}\nFinally, we remark that there is a certain subtlety associated with\ntaking cobar-constructions of unital operads (such as an operad with a\nweakly unital multiplication), as we do. Let $\\mathcal O$ be\nsuch an operad. Then it is easy to see that for any $n\\geq 0$ the\ncomplex $C{\\mathcal O}(n)$ is contractible. However it does not follow\nthat any $C\\mathcal O$-algebra is gauge equivalent to a trivial\none. Indeed, one can take $\\mathcal O$ to be the operad $u{\\mathscr\n  A}\\!\\textit{ss}$ or $u{\\mathscr C}\\!\\textit{omm}$ governing unital\nassociative or unital commutative algebras; then $C\\mathcal\nO$-algebras (on a graded vector space $V$ with zero differential) will\nbe curved $A_\\infty$- or $L_\\infty$-algebras, respectively, and there is\nno reason for an arbitrary curved $A_\\infty$- or $L_\\infty$-algebra to\nbe gauge equivalent to a trivial one.  In fact, even if we let\n$\\mathcal O$ be the operad ${\\mathscr A}\\!\\textit{ss}$ or ${\\mathscr\n  C}\\!\\textit{omm}$, and take the bar construction of it (i.e., of the\nwhole unital operad, rather than just the augmentation ideal as one\nusually does in these cases), we again obtain that $C \\mathcal\nO$-algebras will be ordinary $A_\\infty$- or $L_\\infty$-algebras, with\nthe usual (nontrivial) gauge equivalence relation, even though $C\n\\mathcal{O}$ remains acyclic.\n\nThe explanation of this apparent paradox is that the operad $C\\mathcal\nO$ is \\emph{not cofibrant}; see \\cite{BMahto} for this notion. An algebra\nover $C\\mathcal O$ is a map from $C\\mathcal O$ to an endomorphism\noperad of a dg vector space and this map is not necessarily homotopic\nto zero even though $C\\mathcal O$ is acyclic. A similar phenomenon\noccurs when considering a cobar-construction $(T\\Pi V^*,d)$ for a\nunital associative algebra $V$; dg maps from $T\\Pi V^*$ to the field\n$\\CC$ are the Maurer-Cartan elements in $V$, i.e. the odd elements\n$v\\in V$ for which $dv+v^2=0$. Such elements need not be gauge\nequivalent to zero despite $(T\\Pi V^*,d)$ being acyclic; again,\nprecisely because $(T\\Pi V^*,d)$ is not a cofibrant dga.\n\nFurthermore, given a cyclic operad $\\mathcal O$ as above we can form its $\\Det^d$-modular closure $\\overline{\\mathcal O}^d$ and its naive $\\Det^d$-modular closure $\\underline{\\mathcal O}^d$. Then the same reasoning\nshows that the complexes ${\\mathsf F}\\overline{\\mathcal O}^d((n))$  and ${\\mathsf F}\\underline{\\mathcal O}^d((n))$ are acyclic\nfor $n>0$; here ${\\mathsf F}{\\mathcal O}$ is the Feynman transform of\n$\\mathcal O$.\n\nNext, the complexes  ${\\mathsf F}\\underline{u{\\mathscr C}\\!\\textit{omm}}^1((0))$ and ${\\mathsf\n  F}\\underline{u{\\mathscr A}\\!\\textit{ss}}^1((0))$ are\nnothing but our graph complexes $\\widetilde{\\mathcal G}$ and\n$\\widetilde{\\mathcal G}_r$. For $n > 0$, ${\\mathsf F}\\underline{u{\\mathscr C}\\!\\textit{omm}}^1((n))$ and ${\\mathsf\n  F}\\underline{u{\\mathscr A}\\!\\textit{ss}}^1((n))$ are similar except they are\nspanned by graphs which are additionally equipped with $n$ external\nlabeled edges (legs) which are not allowed to be contracted. When legs\nare present, these graph complexes are therefore acyclic.  However,\nthe vacuum (legless) part  ${\\mathsf F}\\overline{\\mathcal O}^d((n))$  and ${\\mathsf F}\\underline{\\mathcal O}^d((n))$ of the\nFeynman transform need not be acyclic, as our results demonstrate.\n\n\n\n\\vskip 10 pt\n\\noindent\n\\bibliographystyle{amsalpha}\n\\def\\cprime{$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tPerovskite photovoltaic solar cells (PPSC) have reached efficiency comparable to those of the best crystalline silicon (c-Si) devices within only a few years \\cite{nrel.gov}, thanks to the tremendous electrical and optical properties of the metal-halide hybrid perovskites materials \\cite{Snaith2013,jeon2015,Tress2017}. For single junction PPSC, the best performance yet demonstrated reach or even overpass a short-circuit current density $J_{sc}$ of 25.8 $mA.cm^{-2}$, an open circuit voltage $V_{oc}$ of 1.179 $V$, and a fill factor $FF$ of 0.846, leading to power conversion efficiency $PCE$ of 25.7\\% \\cite{nrel.gov}. These achievements are already quite close to the theoretically expected maximum values, especially for the $J_{sc}$, which is typically of 27.83 $mA.cm^{-2}$ under AM1.5G illumination, given that the band gap energy is estimated in their studies to 1.53 $eV$. Furthermore, tandem PPSC made of at least two different perovskite materials appears promising bust also challenging and less mature. In the case of the most common 2 terminal (2T) configuration, a net performance increase could have been recently demonstrated compared to single junction PPSC \\cite{lin_all-perovskite_2022}. \n\t\n\tAs for every optoelectronic device, light management (LM) in PPSC is of primary importance to reach the best performance. As schematized in Figure \\ref{fig:Interplays}, the LM is expected to impact three key performance parameters. First, thanks to the LM, impinging sunlight can be efficiently collected and then absorbed leading to a high $J_{sc}$. Then the LM can help to enhance the $V_{oc}$ thanks to the so-called photon recycling (PR)\\cite{miller2012strong}, provided a strong external luminescence \\cite{Ross1967} and considering the semiconductor exhibits weak non radiative recombinations. Finally, as recently proposed, the energy yield (EY) tends to substitute to the two previous criteria usually considered under standard test conditions (STC). For the latter, requested studies, however, remain complex since they intend to couple illumination models and angular-dependant optical, thermal and electrical models of the cell for one year \\cite{Peters2018}. However, simply considering the incidence angle in the evaluation of absorption and thus $J_{sc}$ is a first step towards EY improvement.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[scale=0.6]{Fig\/InterPlays.png}\n\t\t\\caption{The three main correlated aspects that governed Light Management.}\n\t\t\\label{fig:Interplays}\n\t\\end{figure}\n\t\n\t\n\tAs illustrated in Figure \\ref{fig:Interplays}, the LM in PPSC results from three interdependent considerations: the photonic engineering (PE) at the wavelength scale, the material choices, and the nanofabrication processes. Practically, the last two directly define the PPSC architecture.\n\t\n\tIn brief, still regardless of the envisaged semiconductor, the LM can invoke several kinds of PE. Most frequently, it results in anti-reflection (AR) over the whole absorbed spectral range, leading to a larger $J_{sc}$ as well as possibly larger EY if this the AR effect also occurs over a wide angular domain. Moreover, beyond the AR effects, light trapping (LT) mechanisms can be involved at the band edge of the perovskite material, also leading to a larger $J_{sc}$ \\cite{Bermel2007,Zhou2008}, whereas enhanced PR thanks to absorption suppression just above the band gap could enhance $V_{oc}$ \\cite{Sandhu2013}, still provided previously mentioned conditions. This well established PE is reminded in more details in section \\ref{light_management}. the LM could even be extended to the infrared domain to possibly limit the parasitic sub band gap absorption and enhance the thermal radiation, so as to decrease the operating temperature \\cite{dumoulin_radiative_2021}. Finally, for the specific case of 2T tandem PPSC, the LM is required to ensure at the same time absorption optimization and current matching between the two cells. \n\t\n\tIn the specific case of metal-halide perovskites, intrinsic properties make these materials particularly relevant for LT, compared to the standard case of c-Si solar cells. First, metal-halide perovskites are direct band gap semiconductors, the light is then more easily absorbed in a submicron thick layer, where LT and possibly PR can occur at their band edge, as detailed later. Second, their refractive indices are lower than that of c-Si, thus, AR effect is easier to obtain over the spectral range of interest. Lastly, the ability to recycle photons has been observed\/shown in the specific case of lead iodide perovskite material \\cite{Pazos-Outon2016}.\n\t\n\tOther key properties of metal-halide perovskites, mainly electrical and with regards to aging, have direct consequences on the typical architecture of the PPSC. The diffusion lengths of the carriers in the perovskites imply both contacts should be in the vicinity of the photocarrier generation, thereby covering both sides of the cell {\\cite{yang2019enhancing}}. Additionally, one of these contacts should obviously be transparent to the sunlight. The double heterojunction architecture is widespread, leading to the use of additional intermediate layers, the electron\/hole transport layers (ETL\/HTL). Then, the back metallic contact, ideally made of noble metal, can thus act as a back mirror. In any case, a careful optical design is required to keep low parasitic absorption in these surrounding layers. Moreover, an effective encapsulation should be added to prevent from the degradation of the perovskite. It should be carefully chosen considering the impact of its optical properties. \n\t\n\tLet us finally envisage the wide panel of low-cost elaboration techniques and strategies enabling the structuration of the perovskite thin film at various scales, from the sub-micron to the over-micron one. First PPSC mainly relied on a mesoporous architecture, in which the perovskite infiltrated a scaffold. Then, progress in the elaboration techniques of the perovskite in liquid or vapor phases basically results in a better microcrystallization, and thus larger diffusion lengths. Simple planar architectures have therefore replaced mesoporous ones \\cite{Liu2013,Liu2014b,Zhao2019}; thanks to these far larger diffusion lengths, current planar PPSC overpass the performances of the best mesoporous PPSC. \n\t\n\tFollowing these huge modifications, the LM has evolved: if the mesoporous architecture could have been optimized for efficient LM thanks to the scaffold (typically made of TiO$_2$) to enhance the $J_{sc}$, most of the planar cells demonstrated in the last few years (see e.g. supporting information of ref \\cite{Xu2020}) exhibit performances approaching the limits, up to about 90\\% of the maximum achievable $J_{sc}$, mostly by simply taking care of AR.\n\t\n\tHowever, refined the LM still enables additional improvements of the performances towards the limits. For example, reduced optical losses thanks to LT especially at the band edge helps to gain the last missing $mA\/cm^2$, as detailed in the following. In addition, the $V_{oc}$ could be improved thanks to an accurate treatment of the photocarrier recombination mechanisms and the associated rates.\n\t\n\tA few review papers discussed on various demonstrations of LM for PPSC at the time of their writing \\cite{Zhang2018,Zhang2019}. In this paper, we focus on PE mainly in corrugated dielectric, without any additional metallic material inside or at the vicinity of the perovskite material. Indeed, if using metal can induce LT due to plasmonics effect, other phenomena such parasitic absorption can frequently balance this effect \\cite{SiavashMoakhar2020}.\n\t\n\tAfter a brief presentation of the main PPSC architectures and material choices that impact LM, and summarizing the PE, we propose to further analyze various published structures relying on a wave optics approach. Thus, by unraveling the various mechanisms involved, we are able to compare the various proposed strategies to deduce guidelines for further optimization. These guidelines are finally illustrated thanks to simulations of the optical properties of a few case studies.\n\t\n\t\\section{Key PPSC materials and architectures}\n\t\n\tThe LM in the PPSC directly derive from their architecture, i.e. a stack of layers of various materials and thus optical indices, each layer having a thickness of the order of a few tenth of wavelengths, and with possible structurations from small to large scales compared to the wavelength. The main properties of the perovskite materials impacting LM have already been discussed. However, comparisons will appear difficult and need to be treated carefully, due to discrepancies between the structures investigated, as detailed hereafter. \n\t\n\t\\subsection{Patterning of the PPSC}\n\t\\label{possible_architecture}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/Architecture.png}\n\t\t\\caption{Various typical architectures of complete PPSC obtained using nanofabrication. a. Almost flat (possible nanoscale roughness) b. patterned perovskite using \"bottom-up\" approach, i.e. structuration of the ETL\/HTL, then overcoating with perovskite; c. patterned perovskite using \"top down\" approach, i.e. perovskite coating then structuration and overcoating. d. Flat perovskite but patterned surrounding layers and e. Patterned substrate coated with conformal stack. All envisaged patterned can be either periodic or aperiodic.}\n\t\t\\label{fig:Architecture}\n\t\\end{figure}\n\t\n\tLet us present an overview of the main patterning processes and deriving architectures that have been proposed in the literature.\n\tDue to the low diffusion lengths in the first fabricated metal-halide hybrid perovskites, first PPSC used mesoporous architectures to limit recombination. Indeed, TiO$_2$ scaffolds were infiltrated with the perovskite material, enabling a more efficient collection of the carriers, and, in some cases, LM \\cite{Lee2015,Lin2016,Ramos2016}. Nevertheless, as already mentioned in the introduction, most promising approaches relies nowadays in planar architectures. \n\t\n\tIn the following, layers will be considered as almost flat provided their roughness is at a sub-wavelength, nanometre scale, especially thanks to a good crystallinity in the case of the perovskite. \n\t\n\tHowever, a 2D structuration can be introduced in the PPSC stack, notably for single junction PPSC. In this frame, the perovskite layer itself can be one of the patterned layers, or only surrounding layers are structured, and the perovskite can be considered as flat. \n\tThe various envisaged technical solutions within each of the two strategies are sketched in Figure \\ref{fig:Architecture} b-e.\n\n\tPatterning of the perovskite layer can be achieved by various approaches. Such a structuration can first be generated by depositing this active layer on top of patterned layers, for example the ETL\/HTL standing underneath (\"bottom-up\" like approach, Figure \\ref{fig:Architecture} b.); the pattern can be either random \\cite{Xu2020,Huang2016,Pascoe2016}, or periodic \\cite{Paetzold2015,Abdelraouf2016,Wang2018,Kim2019}. Alternatively, the perovskite layer itself can be directly patterned \\textit{after its deposition} (\"top-down\" like approach, Figure \\ref{fig:Architecture} c.), again either in a random \\cite{Wang2018a,Wang2018b} or periodic way \\cite{Schmager2019,Schmager2019b}. \n\t\n\tAnother approach is to realize flat perovskite layers and to corrugate other adjacent layers (Figure \\ref{fig:Architecture} d.), such as the encapsulation layer \\cite{Nguyen2016,Li2019,kim_light_2021}, transport layers \\cite{kang2016b,Qarony2018,wang_coordinating_2021}, possibly associated with the metal back contact \\cite{Wei2017,Zhang2018b}, or only the front contacts \\cite{Hossain2020,Tockhorn2020}, and finally a glass substrate covering layer \\cite{Tavakoli2015,Dudem2016,Jost2017,Peer2017,doi:10.1002\/adom.201900018,Thangavel2020}.\n\t\n\tFinally, the front substrate or the back contact layer can be patterned within a broad range of sizes, from the hundreds of nanometers to almost the millimeter; then the other layers of the stack are conformally deposited, leading to a fully structured cell \\cite{Qarony2018,Wang2016b,Du2016,Shi2017,Soldera2018,Tooghi2020a,Wang2019,Soldera2020,Haque2020} (Figure \\ref{fig:Architecture} e.). It is noticeable that among these numerous papers, only two \\cite{Tockhorn2020,Soldera2020} are dealing with experimental results. Indeed, spin coating of perovskite on a patterned substrate remains challenging. The former mesoporous architecture has recently been revisited under the form of opal like structuring \\cite{Lobet2020}.\n\t\n\t\\subsection{Influence of the choice of Materials}\n\t\n\t\\subsubsection{Perovskite medium}\n\tThe choice of the absorbing material for PPSC results from multiple constraints. It should first exhibit an optimal band gap energy, in the range 1.1 - 1.4 $eV$\\cite{shockley1961detailed}, as well as optimal electrical properties and stability. Then, various deposition processes can be envisaged for a given metal halide perovskite. It may mainly lead to various grain sizes of the material during the crystallization process. For sizes of the order of the wavelength, this can affect LM. \n\t\n\tMethylammonium lead iodide perovskite material (CH$_3$NH$_3$PbI$_3$, MAPI) has been widely considered in the early stages of PPSC development, thanks to its rather simple chemical composition, and despite $E_g$ slightly above the optimal range. \n\tIt is even noticeable that different $E_g$ are reported for MAPI typically from 1.55 to 1.6 $eV$ (see e.g. \\cite{jiang2015}). This implies a more than 7\\% uncertainty on the maximum achievable $J_{sc}$, that is already of the order of the achievable gain using LM as reported in the following. Thus, the quantitative comparison between different studies, even implying the same material within the meaning of the chemical composition, is rather delicate. Differences of aging of the perovskite could also impact the comparison. Finally, various dispersion models can be used to fit the dielectric function \\cite{Li2020b}, leading to differences that also impact the optical indices that are used for simulations. \n\t\n\tFor all these reasons, it has to be emphasized that, in the following, only the relative impact of LM could be put into perspective, but the various given performances cannot be accurately compared. Finally, if MAPI can still be used as a typical case study, other materials have been developed in the past few years. They are more stable and exhibit a slightly lower $E_g$, mainly thanks to the substitution of MA by FA (Formamidinium, (NH$_2$)$_2$CH), possibly with Cs, as can be seen later.\n\t\n\t\\subsubsection{Transport and contact layer materials}\n\tFor the transport layers, mainly polymer materials are used, as well as thin layers of inorganic materials such as TiO$_2$. It is noticeable than materials such as those based on C$_{60}$ fullerene, acting as ETL, exhibit significant absorption at the shortest wavelengths of the solar spectrum, whereas polymers used as HTL mainly absorb at larger wavelengths, close to and above the band gap of the perovskite. Then, there is often a compromise to be found between suitable band levels, high electrical conductivity, and barrier to possible migrations, and low parasitic absorption, even if the thicknesses of these transport layers remain limited.\n\t\n\t\\section{Photonic engineering}\n\t\\label{Photonic concepts description}\n\t\n\t\\begin{figure}[b]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\linewidth]{Fig\/LightManagement.png}\n\t\t\t\\caption{Main Light Management strategies a. Without peculiar LM, reflectance can be important. b. Using a flat structure, or a patterned that can be considered as homogeneous (e.g. given its subwavelength dimensions), LM can lead to a rather broadband AR effect as well as to Fabry Perot modes that can enhance the absorption. c. Using a random texturation, LM can results from scattering, possibly Lambertian, that can also enhance the absorption. d. Using a periodic or strongly correlated patterned, in addition to AR and Fabry Perot effects, some Light Trapping can occur, drastically enhancing the absorption especially at band edge of perovskite.}\n\t\t\t\\label{fig:LightManagement}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\tLet us now summarize the various kinds of PE, mostly at the wavelength scale, that can be envisaged to improve LM in both single-junction and 2T multi-junction PPSC, and their impact on geometrical requirements in the PPSC. They are sketched in Figure \\ref{fig:LightManagement}. Most of the strategies derive from those already developed for other kinds of direct or even indirect band gap material used in thin film \\cite{massiot2020progress}.\n\t\n\tIn this frame, LM results from various light-structure interaction mechanisms. These mechanisms occur mainly either in the vertical direction (i.e. orthogonal to the stack) or in the directions of the patterns. Depending on the scale of the structure with respect to the wavelength, three main models can be used: first, at sub-wavelength scales, an effective medium; then, at wavelength scale, wave optics, or, at larger scales, geometrical optics.\n\t\n\t\\subsection{Photonic engineering in multilayer unpatterned stack}\n\t\\label{photonic concepts multilayer}\n\tIt is well known that geometrical optics are mainly suitable when the dimensions are far larger than the wavelength. Thus, it cannot accurately describe the interference effects that occur within the thickness of the numerous thin films that compose the cell. In contrast, the optical indices of medium textured at subwavelength scales can be homogenized in a approximated way using Effective Medium Theories such as the Bruggeman model. Then, descriptions involving wave optics are the most rigorous, and wave optics mechanisms might also be the most promising ones.\n\t\n\tLet us first briefly recall the two basic mechanisms able to enhance light harvesting in a flat stack, i.e. without the assistance of any patterning. In a rather high index, highly absorbing material, a properly designed stack enables to drastically reduce the impedance mismatch with the surrounding environment, whereas in a finite thickness absorbing layer, especially with a low absorption, it is rather a Fabry Perot like approach that can be used, leading to highly enhanced but narrow band absorption. \n\t\n\tWithin a typical PPSC stack, having a metallic contact that can act has a back mirror, these two approaches can be invoked. Schematically, at short wavelength, the large extinction coefficient of the perovskite enables the single pass absorption in the perovskite film, but AR effect is required. At larger wavelengths close to band gap, where the absorption drops, multiple passes are required, the second approach is well suited (Figure \\ref{fig:LightManagement} b.). \n\t\n\tHowever, the exact nature of the layers, and thus their optical indices and thicknesses are restricted by other constraints such as energy band levels, electrical resistance, risk of shunting, or even diffusion barriers for some species. Consequently, parasitic optical absorption has to be carefully studied. Moreover, broadband enhancement of the absorption under normal incidence is unlikely to be obtained using such simple architectures. \n\t\n\t\\subsection{Photonic engineering in multilayer, patterned stack}\n\t\\label{photonic_concepts_multilayer_patterned}\n\tIn addition to the continuum of propagative waves, light can be confined into the discrete set of in-plane, transverse guided modes existing in the stack. It is noticeable that the in-plane wave vector of any of these guided modes is larger than those of the free space modes. Thus these modes cannot be simply coupled from the free space in a perfectly flat stack, without any periodic or aperiodic corrugation. Then, the various kinds of in-plane structurations - having dimensions from the wavelength scale up to two orders larger - all induce diffraction. Whereas the impinging light lies around the normal incidence and has thus negligible in-plane wave vector $k_{in\\parallel}$, the light inside the device is diffracted. Indeed, the electromagnetic field can be expanded over a set of plane waves thanks to the in-plane spatial frequencies induced in the medium by the structuration. The reflected waves also experience diffraction accordingly. Considering the various kinds of structurations and thus of the resulting spatial frequencies, three main cases can be envisaged:\n\t\n\t\\begin{enumerate}[label=\\roman*)]\n\t\t\\item A random structuration tends to induce isotropic diffraction, so Lambertian scattering. Moreover, this scattering does not depend on the wavelength within the absorbed spectral range. It can typically lead to broadband AR effect (Figure \\ref{fig:LightManagement} c.). However the absorption enhancement remains limited. Indeed, given the already large absorption (except at the band edge) of the perovskite, and the back reflection induced by the metallic contact, it will be at most of the order of $4n^2$ with an index lower than other materials, in particular inorganic ones \\cite{Yablonovitch1982}.\n\t\t\\item A correlated disordered structuration that enhances a specific set of spatial frequencies, leads to a spectral dependent absorption enhancement. A careful choice of the sizes of the patterns is required to enhance the absorption in the desired spectral range \\cite{Vynck2012nmat}.\n\t\t\\item As a particular case from the previous case, periodic, so-called photonic crystal (PC) structurations can be envisaged. Most of them consists in a square or triangular lattice a simple pattern such as pillars or holes with a vertical profile, or smoother pyramidal, or even parabolic profiles. They mainly lead to discrete modal properties. A few complex patterns arranged in a periodic way have been explored \\cite{martins2013deterministic,Oskooi2014,Bozzola2014,ding2016design,Li2020}, enabling an absorption enhancement in a targeted spectral range thanks to a larger local density of modes. \n\t\\end{enumerate}\n\t\n\t\n\tIn any case, the diffraction efficiency, i.e. the amount of light that is effectively diffracted, and that is thus intended to be absorbed, strongly depends on the pattern of the structuration, including the optical index contrast, as well as on the diffraction order $p$. Whereas, as already mentioned, for scattering, the diffraction efficiency hardly depends on the direction, for diffraction by periodic structures, it is generally larger for the first diffraction order.\n\t\n\tIn this frame, it is noticeable that a LT phenomenon rigorously occurs when the impinging light is coupled thanks to, at least, strongly correlated structure, ideally periodic patterns, having respectively a correlation length or a period $\\Lambda$, within at least one guided mode of the stack (Figure \\ref{fig:LightManagement} d.). According to a perturbation approach \\cite{CamarilloAbad2020}, in a 1D case for sake of illustration without losing generality, a phase matching condition can be written under the form:\n\t\n\t\\begin{equation}\n\t\t\\beta_m=k_{in\\parallel}+p \\frac{2\\pi}{\\Lambda}\n\t\t\\label{QGM}\n\t\\end{equation}\n\tWhere $\\beta_m$ is the in-plane wave number of the $m^{th}$ guided mode of the stack. Due to the time reversal symmetry, the light coupled into the guided mode, if not fully absorbed, can be decoupled after a certain length. These modes are thus so-called \"quasi-guided mode\" (QGM).\n\t\n\tSuch modes can drastically enhance the light path inside the waveguide layer. Thus, if the in-coupled QGM is mainly confined in the perovskite layer rather than in the surrounding layers, the useful absorption is enhanced. Moreover, it also depends on the absorption; more precisely, the absorption is optimized at critical coupling, i.e. when an equilibrium is reached between diffraction efficiency and absorption. Consequently, the spectral bandwidth of the LT is set since the quality factor of the mode at the optimal absorption is of the order of $n\/2\\kappa$, $n,\\kappa$ being respectively the real and imaginary parts of the complex optical index \\cite{Park2009}.\n\t\n\tSuch a resonant LT appears promising to enhance the low absorption close to the band gap of the perovskite, provided it is not reduced elsewhere at shorter wavelengths. More precisely, within the width of the resonance, the absorption can be significantly larger than the previously mentioned broadband limit \\cite{Yu2010}. It could also be used in tandem PPSC mainly to optimize the $J_{sc}$, especially provided LT occurs in a guided mode specifically confined in one of the perovskite layers. As already discussed, for the other spatial frequencies that do not lead to LT, patterning can still result in a rather broadband AR effect, in addition to the one resulting from the design of the stack. Further analysis of all these effects can be found elsewhere \\cite{Zanotto2010,Brongersma2014}. \n\t\n\tIn addition to these PE approaches that contribute at first order to enhance the absorption and so the $J_{sc}$, other strategies could be used to enhance the $V_{oc}$, using a PR mechanism. In a simplified way, PR is supposed to induce at the $V_{oc}$ operating point a high density of photons into the absorbing materials, and so to give a chance to electrons-holes pairs to recombine radiatively rather than in a non radiative way. Typically, this requires at first order to inhibit the radiation, and consequently the absorption, in the luminescence spectral domain of the semiconductor. Thus, enhancing simultaneously the $J_{sc}$ using LT and the $V_{oc}$ using PR is a compromise \\cite{Sandhu2013}.\n\t\n\tIn any case, it has to be emphasized that all these effects should ideally target the perovskite layer only and not enhance the parasitic absorption in the other layers.\n\t\n\t\\subsection{Main geometrical requirements on patterns for Light Trapping}\n\t\n\tAs mentioned previously, 2D in-plane structurations are able to diffract the impinging sunlight under a quasi-normal incidence into the large number of modes of the PPSC, possibly including the guided modes.\n\tIndeed, the architectures of both single and multi-junction PPSC reviewed in the section \\ref{possible_architecture} can be schematized as one or several sub-micron thick high index perovskite layers lying between two lower indices ETL and HTL and possibly also lower index layers in the multi-junction case. The whole is lying on medium index TCO on a low index glass substrate and coated on top with metal.\n\tSuch structures exhibit several guided modes in both TE and TM polarization states.\n\tLet us first make the assumption of weak corrugation that doesn't change significantly the effective indices of the stack of active layers lying on a substrate. The largest effective index modes - typically 2.2 - are mainly confined in the perovskite, whereas the other modes, whose effective indices are below 1.9, can be confined in the whole stack of layers.\n\tIn this frame, for normal incident light ($k_{in\/\/}=0$) and for the highest efficient coupling at order $p=1$, equation (\\ref{QGM}) can be rewritten under the form:\n\t\\begin{equation}\n\t\t\\beta_m=\\frac{2\\pi N_{eff,m}}{\\lambda}=\\frac{2\\pi}{\\Lambda}\n\t\t\\label{phase_matching_equation}\n\t\\end{equation}\n\twhere $\\Lambda$ is the period, and could even be a characteristic length in correlated patterns. This 1D case can easily be extended to 2D patterns. \n\t\n\tThus, to be able to efficiently couple into the fundamental guided mode, which is mainly confined into the perovskite, at wavelengths corresponding to its band edge, e.g. 750 $nm$, $\\Lambda$ should be around 340 $nm$, whereas it should be larger than 400 $nm$ for higher order guided modes. This results in a pseudo-guided mode, as discussed at the end of section \\ref{photonic_concepts_multilayer_patterned}. At shorter wavelengths, such structurations can also couple light into the various low effective index modes (including Fabry Perot like modes) of the stack, that may also result in a broadband AR effect. \n\tThis typically implies that all the micron scale or even larger patterns in any of the layers of the stack are rather unlikely to couple the impinging light into guided modes in the low absorption domain, where such a coupling is of main interest. Such patterns only induce coupling into the Fabry Perot like modes of the stack and\/or diffraction at high orders with a limited efficiency.\n\t\n\t\\section{Examples of reported light management strategies in PPSC}\n\t\n\tWe report here on some studies to illustrate concrete LM strategies. For any case, the chosen perovskite material and its thickness are of primary importance. First of all, planar PPSC as envisaged in the section \\ref{possible_architecture} for their architecture and in section \\ref{photonic concepts multilayer} for the corresponding PE are discussed, putting mainly into evidence the importance of the thicknesses on LM. Then, results on in-plane patterned PPSC are reviewed, with focuses on the involved PE.\n\t\n\t\\subsection{Optimization of the thicknesses in planar PPSC}\n\t\n\tMost studies related to thickness optimization only focus on that of the perovskite layer, and report mainly on the derived useful absorption, as detailed just below. However, the whole stack has been considered in a few papers, with a view to investigate the parasitic absorption, as discussed later.\n\t\n\t\\subsubsection{Choice of the perovskite thickness in single junction PPSC}\n\t\n\tAs already mentioned, the optimal thickness of the sole perovskite layer is chosen as a compromise between maximizing optical absorption and keeping low bulk recombination of the carriers. In addition to the analysis of a few seminal cases of single junction PPSC, we present a synthesis of several studies, mainly experimental, that report on the effect of the perovskite layer thickness of planar single junction PPSC, first and mainly made of MAPI, then of other lead-halide perovskites. The detailed performances of the representative cells are summarized in Table \\ref{tableau 1} and, when available, the EQE are plotted in Figure \\ref{fig:EQE_tableau_1} in a spectral range limited from 400 to 800 $nm$ for sake of simple comparison.\\\\\n\t\n\t\\paragraph{MAPI}\n\t\n\tOne of the first high efficiency planar PPSC \\cite{Liu2013} which is made of an about 300 $nm$ thick MAPICl layer exhibited a $J_{sc}$ of 21.5 $mA.cm^{-2}$. In this seminal paper, M. Liu \\textit{et al.} already mentioned the possibility of an optimum thickness due to the balance between absorption and recombination. It was shown later by Y. Da \\textit{et al.} \\cite{Da2018} that this cell suffered from optical losses that represented 11.3\\% of the energy losses, and especially more than 20\\% of the achievable $J_{sc}$. This was mainly related to reflectance and parasitic absorption in the FTO. This thus emphasizes the importance of minimizing these effects by a careful design of the stack.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/T1_final.png}\n\t\t\\caption{EQE of various PPSC also reported in Table \\ref{tableau 1} \\cite{Liu2014b,Momblona2014,Liu2014,Liu2019b,Correa-Baena2016,jeong_pseudo-halide_2021,Ball2015,Lin2015}.}\n\t\t\\label{fig:EQE_tableau_1}\n\t\\end{figure}\n\t\n\tThen, the first high-efficiency PPSC, solution processed at room temperature, proposed by D. Liu \\textit{et al.} \\cite{Liu2014b} was made of an about 300 $nm$ thick MAPI layer. A $J_{sc}$ of 20.4 $mA.cm^{-2}$ was obtained, so about 5 $mA.cm^{-2}$ below the maximum achievable $J_{sc}$ given a typical $E_g$ of MAPI. The EQE (see Figure \\ref{fig:EQE_tableau_1}) was rather flat and remained lower than 0.8. The deficit could be due to both optical losses (too low absorption in the perovskite) and electrical losses (too high recombination). \n\t\n\tC. Momblona \\textit{et al.} \\cite{Momblona2014} conducted an experimental study on PPSC made of MAPI deposited by thermal evaporation, especially by increasing its thickness from 160 to 900 $nm$. If the $J_{sc}$ increased rather monotonously with the perovskite layer thickness, it never exceeded 20.4 $mA.cm^{-2}$. The main parameter that decreased with increasing perovskite layer thickness was the fill factor. As a result, the cell owning optimal performance was obtained for a perovskite layer thickness around 300 $nm$. More precisely, whereas the $J_{sc}$ slightly increases, both the $V_{oc}$ and the $FF$ significantly decreased at thicknesses larger than 300 $nm$. At these early ages of PPSC, the corresponding optimal PCE was 12.7\\%, with a $J_{sc}$ of 18.8 $mA.cm^{-2}$. Then, leading further investigations on the $J_{sc}$ deficit, it could be derived from the EQE (see Figure \\ref{fig:EQE_tableau_1}) that optical losses in almost 300 $nm$ thick perovskite were important at both short wavelengths and from 600 to 700 $nm$, compared to the 900 $nm$ thick perovskite layer that fully absorbs. The enhanced absorption in the range 700 to 750 $nm$ is likely due to a cavity effect. Thus, even if such a thickness of about 300 $nm$ appears to be optimal under the PCE point of view, it remains too thin to avoid optical losses. In addition, even with such thin absorbing layers, such cells still suffers from electrical losses. \n\t\n\tIn a similar way, D. Liu \\textit{et al.} \\cite{Liu2014} also then studied PPSC consisting in a ITO\/ZnO\/MAPI\/P3HT\/Ag stack, in which MAPI was obtained using thermal evaporation or solution processing. Again, in spite of a larger absorption and limited short-circuit for thicker films, a limited diffusion length led to an optimum thickness of the MAPI layer of about 330 $nm$ for the solution processed PPSC. It is noticeable that the corresponding EQE (see Figure \\ref{fig:EQE_tableau_1}) was rather low and flat. This could be related to scattering due to sub-micron crystallinity that mitigates cavity effects at larger wavelengths.\n\t\n\tMore recently, Y. Liu \\textit{et al.} \\cite{Liu2019b}, still using MAPI, similarly concluded that there were more power loss in PPSC when the thickness of the MAPI layer was either less or more than about 300 $nm$. They noticed that the grain size increased with the thickness, in favor of a larger thickness. However, it was noticeable that, at wavelengths larger than 650 $nm$, the EQE of the 303 $nm$ thick MAPI PPSC (see Figure \\ref{fig:EQE_tableau_1}) was lower than the one of the 564 $nm$ thick MAPI PPSC. This is related to too low absorption in this spectral range. \\\\\n\t\n\t\n\t\\paragraph{Other lead-halide perovskites}\n\t\n\tAs an alternative, J.P. Correa-Baena \\textit{et al.} \\cite{Correa-Baena2016} used multi-cations perovskites materials; these were recently under the spotlight, especially considering their better stability \\cite{jeon2015compositional}. Using FA$_{0.83}$MA$_{0.17}$Pb(I$_{0.83}$Br$_{0.17}$)$_3$ together with mesoporous TiO$_2$, the thickness that maximises the PCE was found to be at least 480 $nm$. The corresponding $J_{sc}$ was almost 24.4 $mA.cm^{-2}$. Considering the band gap of this material, reported to be about 1.63 eV \\cite{jacobsson2016exploration}, this is already the maximum achievable value. The EQE (see Figure \\ref{fig:EQE_tableau_1}) is noticeably high (also thanks to the an IQE of almost 100\\%) and flat at large wavelength. This could be related to the mesoporous TiO$_2$ that induces some scattering effect. \n\t\n\tUnlike the previous papers, M. Rai \\textit{et al.} \\cite{Rai2020} focused on the $V_{oc}$ deficit related to non-radiative recombination as a function of the thickness, expressed in terms of molar concentration of the precursor solution. This time, the studied perovskite was Cs$_{0.2}$FA$_{0.8}$Pb(I$_{0.85}$Br$_{0.15}$)$_3$, whose band gap has been found to be 1.62 $eV$, so close to the one of MAPI. They noticed as well that the $V_{oc}$ decreases with the thickness, whereas the $J_{sc}$ increases. As for the previously mentioned paper, the grain size increases with the thickness; the molar concentration of the precursor solution that maximizes PCE correspond to thickness of the perovskite layer of about 400 $nm$, so 30\\% larger than the one usually obtained using MAPI. This might be due to a larger diffusion length of the photocarriers for this perovskite.\n\t\n\tK.B. Nine \\textit{et al.} \\cite{nine_performance_2020} simulated optically and electrically the effect of the thickness of several FACsPI layers, leading to a larger optimal thickness, of almost 600 $nm$, so twice the thickness usually found using MAPI. Indeed, better electrical properties of this perovskite material, especially carrier mobility, allow such a thicker layer. \\\\\n\t\n\t\\paragraph{Synthesis}\n\t\n\t\n\t\n\t\\begin{table}[!h]\n\t\t\\begin{tabular}{@{}lllllll@{}}\n\t\t\t\\hline\n\t\t\tMaterial & Thickness ($nm$) & $J_{sc} (mA.cm^{-2})$ & $V_{oc} (V)$ & $FF$ & PCE (\\%) &  Ref \\\\\n\t\t\t\\hline\n\t\t\tMAPICl & 330 & 21.5 & 1.07 & 0.67 & 15.4 & \\cite{Liu2013} \\\\  \n\t\t\tMAPI & 300  & 20.4 & 1.03 & 0.749 & 15.7 & \\cite{Liu2014b} \\\\    \n\t\t\tMAPI & 285  & 18.8 & 1.07 & 0.63 & 12.7 & \\cite{Momblona2014} \\\\\n\t\t\tMAPI (solution process) & 330  & 17 & 0.94 & 0.62 & 11.8 & \\cite{Liu2014} \\\\\n\t\t\tMAPI & 303 & 21.3 & 1.07 & 0.715 & 18.4 & \\cite{Liu2019b} \\\\\n\t\t\tFAMAPBrI & 480 & 24 & 1.14 & 0.75 & 20.8 & \\cite{Correa-Baena2016} \\\\\n\t\t\tFAPI & 600 & 26.35 & 1.189 & 0.817 & 25.59 & \\cite{jeong_pseudo-halide_2021}\\\\\n\t\t\tMAPI & 492 & 21.56 & nr & nr & nr & \\cite{Ball2015}\\\\\n\t\t\tMAPI & 350 & 21.9 & 1.05 & 0.72 & 16.5 & \\cite{Lin2015} \\\\  \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{Experimental performances of planar PPSC with optimized perovskite thickness (nr: not reported)}\n\t\t\\label{tableau 1}\n\t\\end{table}\n\t\n\tIt is noticeable that these studies tend to an optimal thickness of about 300 $nm$ when using MAPI. This thickness could lead to state-of-the-art PCE of J. Li \\textit{et al.} \\cite{LI2018331}, with noticeably high $J_{sc}$ of 24.1 $mA.cm^{-2}$, mainly thanks to an ETL including graphdiyne, which improves electrical properties. Then, FAPI is the perovskite used for the best up-to-date single junction PPSC \\cite{jeong_pseudo-halide_2021} which performance were mentioned in the introduction, with a thickness of about 600 $nm$, even if, among numerous refinements of the architecture, the rough, about 50 $nm$ thick TiO$_2$ layer already may help to trap the light. \n\t\n\tIt appears that choosing a thickness smaller than the one maximizing the $J_{sc}$ could help to keep low resistance, so high FF and PCE. Then, with such a $J_{sc}$ around 90\\% of the maximum achievable $J_{sc}$, LM can optimize the absorption in the perovskite, especially at photon energies close to the band gap, where the absorption starts to decrease.\n\t\n\tIn any case, it is of high interest to understand the reasons why large variations of the internal perovskite absorption spectrum have been obtained for similar thicknesses of the same perovskite material (MAPI), as can be noticed through the cases 2 to 5 of Table \\ref{tableau 1} and the corresponding EQE plotted in Figure \\ref{fig:EQE_tableau_1}. They might be due to differences in the surroundings layers (thicknesses, indices) or in the perovskite itself, like microcrystallinity that can induce scattering. These phenomena will be described in the following, starting with the analysis of the effect of the surroundings layers in a unpatterned stack. \n\t\n\t\\subsubsection{Single junction unpatterned PPSC stack analysis}\n\tIf there can be an optimal thickness for the perovskite layer, the effect of the other layers, especially on the absorption and reflectance, should also be analyzed. Here, we review a selected set of publications focusing on this issue.\n\t\n\tJ.M. Ball \\textit{et al.} \\cite{Ball2015} studied planar PPSC using an optical model based on the transfer-matrix formalism with experimentally determined complex refractive index data. They focused on a typical stack made of FTO\/TiO$_2$\/MAPI\/SPIRO-OMeTAD\/Au. Under the assumption of $E_g = 1.56 eV$, the detailed analysis revealed that for a calculated $J_{sc}$ of 21.56 $mA.cm^{-2}$, parasitic absorption induces a lack of about 1.72 $mA.cm^{-2}$, and reflection losses corresponding to a $J_{sc}$ decrease of 1.36 $mA.cm^{-2}$, so a total of more than 10\\% of the collected current. On the other hand, the IQE losses can be estimated to another 10\\%. Moreover, most of the optical losses take place in the 420 $nm$ thick FTO. This underlines the importance of taking care of all the layers of the stack.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/lin_cavite.png}\n\t\t\t\\caption{Optical-field distribution for four wavelengths: for $\\lambda < 500 nm$ the optical-field distribution follows the Beer\u2013Lambert law and no optical field reaches to the back electrode as a result of the high absorption coefficient. In such cases the absorption is saturated and no optical interference occurs. For $\\lambda > 500 nm$ the optical field is governed by low-finesse cavity interference.}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]\n\t\t\t{Fig\/lin_eqe.png}\n\t\t\t\\caption{EQE spectra of devices with different MAPI-layer thicknesses. For $\\lambda < 500 nm$ there is minimal influence of the film thickness, but for $\\lambda < 500 nm$ the EQE is strongly thickness dependent because of the optical interference (Fabry Perot like modes)}\n\t\t\\end{subfigure}\n\t\t\\caption{PPSC properties as a function of the MAPI layer thickness \\cite{Lin2015}. Reproduced with permission}\n\t\t\\label{fig:lin eqe}\n\t\\end{figure}\n\t\n\tQ. Lin \\textit{et al.} \\cite{Lin2015} deeply investigated the properties of PPSC made of MAPI. Thanks to their measurement of the optical indices of the MAPI and of the other materials, they simulated the properties of PPSC, assuming an IQE of 100\\%. They identified the two absorption regimes of a flat PPSC, previously discussed in section \\ref{photonic concepts multilayer}, namely the single pass and the cavity regime. Thus, thanks to optical cavity effects at wavelengths larger than 500 $nm$, the noticeably high EQE of their cell (see Figure \\ref{fig:EQE_tableau_1}) appeared to be optimized for a thickness of about 370 $nm$ (see Figure \\ref{fig:lin eqe}). The experimental data noticeably confirmed the simulation study, since a cell having a 350 $nm$ thick MAPI layer led to the best properties, $J_{sc}=21.9 mA.cm^{-2}$, $V_{oc}=1.05 V$, $FF=0.72$ and $PCE=16.5\\%$. Given the resulting limited thickness of the MAPI, the obtained $J_{sc}$ was 87\\% of the maximum achievable value. These remarkable results were also obtained thanks to the optimization of the p and n type interlayers that were used to optimize the heterojunctions. It was especially shown that the PCDTBT p type interlayer has to be as thin as possible ( $\\sim 5 nm$), because of its absorption, even if it enabled a nice crystallinity of the MAPI. Otherwise, given the almost 100\\% IQE, most of the 4 to 5 $mA.cm^{-2}$ decrease of the photocurrent (depending on the exact value of the band gap energy of the considered MAPI) corresponds to optical losses, which could be either parasitic absorption or reflectance.\n\t\n\tM. Van Eerden \\textit{et al.} \\cite{VanEerden2017} analyzed the optical losses of 350 $nm$ thick MAPI layers PPSC. Within the considered cell, most of the losses are related to reflectance (equivalent to 4.2 $mA.cm^{-2}$), whereas the parasitic absorption remains limited to the half, and despite a subwavelength roughness (typically from 10 to 50 nm RMS) at both ITO\/TiO$_2$ and MAPI\/SPIRO-OMeTAD interfaces. \n\t\n\t\\subsubsection{Analysis of unpatterned, up-to-date record 2T tandem PPSC}\n\t\\label{tandem_plan}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/xiao_tandem_schema.png}\n\t\t\t\\caption{Stack of materials }\n\t\t\t\\label{fig:xiao tandem schema}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/xiao_tandem_jsc.png}\n\t\t\t\\caption{Simulated $J_{sc}$ as function of wide-band gap and narrow-band gap perovskite layer thicknesses.}\n\t\t\t\\label{fig:xiao tandem jsc }\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/xiao_tandem_eqe.png}\n\t\t\t\\caption{EQE of both subcells.}\n\t\t\t\\label{fig:xiao tandem eqe }\n\t\t\\end{subfigure}\n\t\t\\caption{Best up-to-date 2T tandem, all perovskite PPSC \\cite{xiao_all-perovskite_2020}. Reproduced with permission.}    \n\t\\end{figure}\n\t\n\tWhen compared to other tandem architectures, the 2T case leads to possibly limited optical losses, but such cells require an equilibrium of the current densities delivered by the two subcells. Among recent results \\cite{wang_prospects_2021}, K. Xiao \\textit{et al.} \\cite{xiao_all-perovskite_2020} have deeply investigated 2T tandem PPSC (see Figure \\ref{fig:xiao tandem schema}) both numerically and experimentally. They have shown using optical simulations that the reachable $J_{sc}$, which is indeed the smallest $J_{sc}$ delivered by each of the subcell, strongly depends on the thicknesses of each perovskite layer (see Figure \\ref{fig:xiao tandem jsc }), provided that thicknesses of the other layers have been set taking into account both electrical and optical properties (low parasitic absorption). Indeed, it appears that the top perovskite has to be thick enough to absorb enough light, but that a too thick layer also absorbs too much light, and finally that the overall $J_{sc}$ is limited by the bottom sub cell. The thickness of the perovskite bottom sub cell has to be large enough, more than 1.1 $\\mu m$.\n\tWhen fabricated on a small surface, such a cell exhibits even a slightly larger $J_{sc}$ than simulated, likely due to the too pessimistic predicted absorption. Even if this result remains a record at the time of its publication, its EQE (see Figure \\ref{fig:xiao tandem eqe }) reveals some non-idealities. Indeed, in addition to the relatively low values of the EQE plateaus, there is an overlap of the absorption domains of the two perovskite materials between 500 and 700 $nm$. This means that some undesirable thermalization still occurs below 700 $nm$ in the bottom sub cell. \n\t\n\t\\subsection{In-plane structuration for light management in single junction PPSC}\n\t\\label{light_management}\n\t\n\tAt this stage, it appears that even the best reported cells exhibit an EQE that can still be improved using PE at the wavelength scale, even if most of them already benefit from the roughness of the microcrystalline perovskite.\n\t\n\tIn this section, we will highlight important criteria that should be met to ensure a fair demonstration of LM. The report on selected studies is organized as a function of the various architectures as envisaged in section \\ref{possible_architecture}.\n\t\n\t\\subsubsection{General criteria for fair LM demonstration}\n\tThe impact of LM on the performance of a solar cell can be demonstrated by comparing a patterned device with an unpatterned reference. In case of superficial structurations or with the addition of specific LM layers, the absorption improvement appears generally obvious compared to the same unpatterned stack, or, even more favorable, to the unpatterned stack without the additional flat LM layer. Moreover, integrating LM layer could lead to a better charge collection, by decreasing the carrier path; the specific LM effect would then be difficult to distinguish. Therefore, great care should be taken in the definition of a reference structure or device. In particular, the volume of perovskite materials should be as close as possible in both structures. Additionally, the net enhancement of EQE, efficiency or even yield value can be accurately estimated only if the unpatterned reference has been optimized first. These conditions should be met in order to discuss LM for PPSC performance optimization. \n\t\n\t\\subsubsection{Periodic patterning for resonant LT}\n\t\\paragraph{Superficial structuration and light management layers}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/peer_bismwas_MLA_schema.png}\n\t\t\t\\caption{Schematic of the PPSC.}\n\t\t\t\\label{fig:peer bismwas_schema}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/peer_bismwas_absorption.png}\n\t\t\t\\caption{Simulated absorption spectrum for microlens array of period $~700 nm$ and height $800 nm$. The absorption of flat solar cell is overlaid for comparison.}\n\t\t\t\\label{fig:peer bismwas_absorption}\n\t\t\\end{subfigure}\n\t\t\\caption{PPSC with microlens array on air-glass side \\cite{Peer2017}. Reproduced with permission.}\n\t\\end{figure}\n\t\n\tA. Peer \\textit{et al.} \\cite{Peer2017} simulated planar PPSC with a stack including a 400 $nm$ thick MAPI layer. The cell lay on top of a 700 $\\mu m$ thick glass substrate. A micro lens array was then added on the top face of the glass substrate, by imprinting a polymer such as polystyrene (see Figure \\ref{fig:peer bismwas_schema}). Each micro lens of the triangular lattice had a smooth profile close to truncated pyramids. For an aspect ratio period\/height of the micro lenses close to 1, and a period of 700 $nm$, optimized gain of 6.3\\% of the $J_{sc}$ was reported compared to the flat reference, whose EQE noticeably shown dips likely due to cavity effects in the cell (see Figure \\ref{fig:peer bismwas_absorption}). Then, according to the EQE of the cell coupled to the micro lens array, the overall resulting broadband enhancement of the absorption is due at lower wavelength to AR effect, and to LT at the band edge. The optimal period of the pattern is rather large, in agreement with a low effective index of the coupled guided mode. It means that LT occurs in the mode that is hardly guided in the perovskite, but that has a nonnegligible overlap with the pattern. This overlap remains anyway limited, as revealed by the high quality factor of the resonance. Therefore, the absorption reaches almost 1, so is close to critical coupling conditions.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/wei_2017_adv_Energy_mat_afm.png}\n\t\t\t\\caption{SEM (left)  and  AFM (right) images of  the spin-coated layer (top) of grating patterned layer (middle) and moth-eye  patterned layer (bottom).}\n\t\t\t\\label{fig:wei 2017 adv Energy mat afm}    \n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/wei_2017_adv_Energy_mat_EQE2.png}\n\t\t\t\\caption{External quantum efficiency  (EQE) spectra of PPSC, and relative enhancement obtained by dividing the spectra of grating and moth-eye patterned devices by that of flat one.}\n\t\t\t\\label{fig:wei 2017 adv Energy mat perf}\n\t\t\\end{subfigure}\n\t\t\\caption{PPSC with bio inspired PCBM HTL \\cite{Wei2017}. Reproduced with permission.}\n\t\\end{figure}\n\t\n\tJ. Wei \\textit{et al.} \\cite{Wei2017} fabricated bioinspired back electrodes by imprinting the HTL made of PCBM before the conformal deposition of the Bphen\/Ag back contact. Patterns were either a periodic sinusoidal 1D grating or a uniform 2D moth-eye structure (see Figure \\ref{fig:wei 2017 adv Energy mat afm}), with a typical pitch of about 600 $nm$. It has then been experimentally shown (see Figure \\ref{fig:wei 2017 adv Energy mat perf}) that the EQE of the 240 $nm$ thick MAPICl PPSC in increased by 10\\% up to 40\\% at the band edge, mainly thanks to the 2D moth eye pattern. The overall $J_{sc}$ increased by 14.3\\%, mainly due to absorption increase, but a lower series resistance was also measured for patterned cells, which led to a slight increase of the IQE. As shown by there FDTD simulations of periodic structures, this is due light diffraction into the guided modes, rather than plasmonic effects. It can be confirmed by Fourier Analysis of the top view of the moth eye pattern that a significant part of the spatial frequencies lies in the optimal range for LT (see Figure SI-\\ref{fig:Supplemental Fourier Analysis Wei}).\n\t\n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/T2_final.png}\n\t\t\\caption{EQE of the patterned PPSC (solid line) and of their planar references (dashed line) (left) and EQE enhancement ((right) of reported simulated $J_{sc}$ enhancements of Table \\ref{tableau pattern mapi}, mainly resulting from Light Management, for MAPI single junction PPSC, with any kind of in-plane pattern (additional layers, structuration of the perovskite or conformal perovskite) and whatever the main identified photonic concept \\cite{Peer2017,Wei2017,kim_light_2021,Schmager2019,Du2016,Hossain2020,Qarony2018}}\n\t\t\\label{fig:EQE_table_2}\n\t\\end{figure}\n\t\n\t\n\tH. Kim \\textit{et al.} \\cite{kim_light_2021} simulated nanosphere arrays of TiO$_2$ conformally coated with silica on a standard MAPI PPSC stack, with a MAPI thickness limited to 100 $nm$. Provided suitable spacing and nanosphere diameter, the $J_{sc}$ could jump from 14 to 18.7 $mA.cm^{-2}$. Authors invoked the Mie scattering effect of the array to explain the enhancement, but it appears that the FEM simulation has likely been done using periodic boundary conditions, and thus that a LT effect occurs at several wavelengths, such as 705 or 790 $nm$ for those at the band edge of MAPI. This is in line with the larger EQE enhancement close to the band edge, as can be seen in Figure \\ref{fig:EQE_table_2}. It remains that the overall $J_{sc}$ is anyway limited due to the unusually low thickness. \\\\\n\t\n\t\\paragraph{Structuration of the perovskite} \n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/schmager_solmat_schema.png}\n\t\t\t\\caption{Layer stack of the simulated patterned PPSC, along with associated layer thicknesses. Three configurations of the active layer are simulated (1) a planar reference, (2) a cylindrical indention into the perovskite layer of variable depth; and (3) a hole geometry which corresponds to the maximum cylindrical indention.}\n\t\t\t\\label{fig:schmager solmat schema}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/schmager_solmat_simul.png}\n\t\t\t\\caption{Absorptance in the perovskite layer of the hole pattern and the 120 $nm$ deep indentation compared to the corresponding planar reference and the theoretical Yablonovitch limit for two different initial perovskite layer thicknesses of (left) 200 $nm$ and (middle) 300 $nm$. $J_{sc}$ for all data are compared (right). The displayed nanophotonic patterns employ a geometrical $ff=0.4$ and a period of 380 $nm$.}\n\t\t\t\\label{fig:schmager solmat simul}\n\t\t\\end{subfigure}\n\t\t\\caption{Various 2D in plane patterns of the MAPI layer in a PPSC \\cite{Schmager2019}. Reproduced with permission}\n\t\\end{figure}\n\t\n\tR. Schmager \\textit{et al.} \\cite{Schmager2019} simulated a classical MAPI PPSC (see Figure \\ref{fig:schmager solmat schema}). Compared to the planar configuration, etching of the MAPI layer was envisaged, thanks to a square lattice of holes, filled with Spiro-OMeTAD, with various etching depths. To avoid any short circuit, an optimized partial etching of 120 $nm$ led to a 5.6\\% increase of the $J_{sc}$, compared to the initially flat 300 $nm$ thick MAPI layer, provided an equivalent volume of perovskite. It results from a LT at the band edge, but without any sharp resonance, as can be observed on the spectral response of other patterns (see Figure \\ref{fig:schmager solmat simul}).\n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/schmager_solmat_carac.png}\n\t\t\\caption{External quantum efficiency and reflectance spectra of the planar and nanoimprinted perovskite solar cells. The nanoimprinted perovskite layer has a period of 480 $nm$. The nanoimprinted perovskite solar cell shows enhanced absorption and current generation close to the band gap. For energies below the band gap (wavelength larger than 790 $nm$) discrete peaks are visible in the reflectance measurement. \\cite{Schmager2019b}. Reproduced with permission.}\n\t\t\\label{fig:schmager solmat carac}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/T3_final.png}\n\t\t\\caption{EQE of the patterned PPSC (solid line) and of their planar references (dashed line) (left) and  EQE enhancement ((right) of Table \\ref{tableau pattern exp}, mainly resulting from Light Management, for  single junction PPSC, with any kind of in-plane pattern (additional layers, structuration of the perovskite or conformal perovskite) and whatever the main identified photonic concept \\cite{Schmager2019b,Tockhorn2020,Paetzold2015,Dudem2016,Jost2017,Zhang2018b,Pascoe2016}}\n\t\t\\label{fig:EQE_table_3}\n\t\\end{figure}\n\t\n\tThe same group \\cite{Schmager2019b} also realized nanoimprinted PPSC made of Cs$_{0.1}$(FA$_{0.83}$MA$_{0.17}$)$_{0.9}$Pb(I$_{0.83}$Br$_{0.17}$)$_3$. Their experimental results show a relative improvement of 2\\% of the PCE compared to the planar reference for the complete, gold coated solar cell. This was obtained thanks to an increase of the $J_{sc}$ from 19.1 to 19.4 $mA.cm^{-2}$ and noticeably identical $V_{oc}$ and $FF$.The EQE of the patterned cell was improved at wavelengths larger than 680 $nm$. A coupling to a quasi-guided mode is observed in the perovskite band edge (see Figure \\ref{fig:schmager solmat carac}), leading to a significant EQE enhancement (see Figure \\ref{fig:EQE_table_3}). \\\\\n\t\n\t\\paragraph{Textured substrate}\n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/du_john_schema.png}\n\t\t\t\\caption{Three-dimensional perspective  and two-dimensional side view of the inverted vertical cone perovskite unit cell.}\n\t\t\t\\label{fig:du john schema}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/du_john_simul.png}\n\t\t\t\\caption{Simulated absorption and reflection spectra of the optimized (period $a = 600 nm$, radius $R = 380 nm$) inverted cone PC and planar PPSC. The MAPI thickness of the planar cell is 180 $nm$, corresponding to the \"equivalent bulk thickness\" of MAPI in the inverted-cone PC device.}\n\t\t\t\\label{fig:du john simul}\n\t\t\\end{subfigure}\n\t\t\\caption{MAPI PPSC patterned with inverted cones\\cite{Du2016}. Reproduced with permission.}\n\t\\end{figure}\n\t\n\tQ. G. Du \\textit{et al.} \\cite{Du2016} simulated a strongly corrugated MAPI layer. It was obtained thanks to cones realized in ITO coated glass substrate with an intermediate conformal PEDOT layer lying on ITO (see Figure \\ref{fig:du john schema}). Thanks to optimized period and radius, a 15-17\\% improvement was expected in the PCE compared to the best flat cells of equivalent volume (see Figure \\ref{fig:du john simul}). The optimal period of about 400 $nm$ fits well the required period for LT into the guided modes, given the fact that the effective index of the modes is lower than our examples, due to the strong corrugation of the MAPI filled with low index material. AR and resonant light trapping are thus simultaneously obtained.\n\t\n\t\\subsubsection{Periodic structures for non resonant LM: anti-reflection and scattering}\n\t\n\t\\paragraph{Superficial structuration and light management layers}\n\t\n\tM. I. Hossain \\textit{et al.} \\cite{Hossain2020} proposed to coat the MAPBI like perovskite, having a thickness from 100 to 400 $nm$, with zinc oxide. More precisely, a first thick, lightly doped ZnO ensured the top contact, then it could be covered by texturations patterned either as pyramids or as non-resonant metasurfaces. The period was set to 800 $nm$. Accordingly to the mechanism typically occurring using such a period, no sharp resonance enhances the absorption at the band-edge, only an AR like effect occurs. Both kinds of patterns lead to simulated equivalent $J_{sc}$ enhancements of about 3 $mA.cm^{-2}$ whatever the thickness of the perovskite between 100 and 400 $nm$, so a relative enhancement roughly from 25\\% to 12\\% within the same range, but only 10\\% for the thickness 300 $nm$.\n\t\n\tP. Tockhorn \\textit{et al.} \\cite{Tockhorn2020} simulated and fabricated a 550 $nm$ thick PPSC made of a mixed cation, mixed halide Cs$_{0.05}$(FA$_{0.83}$MA$_{0.17}$)$_{0.95}$Pb(I$_{0.83}$Br$_{0.17}$) perovskite material on various periodically patterned glass substrates, in p-i-n configuration. The top side of the glass substrates was coated with low index, NaF thin film. Compared to the planar reference, patterned structures could exhibit a $J_{sc}$ up to 1 $mA\/cm^2$ larger. This results from a broadband enhancement of the EQE, including at the band edge, mainly attributed to AR effect, as verified by simulations where the volume of perovskite has been kept constant. This is in line with the flat EQE enhancement, as can be seen in Figure \\ref{fig:EQE_table_3} The resulting PCE reaches 19.7\\%, 1\\% absolute above the one of the planar reference.\\\\\n\t\n\t\\paragraph{Structuration of the perovskite} \n\t\n\tU. W. Paetzold \\textit{et al.} \\cite{Paetzold2015} proposed to pattern the front ITO electrode with a square lattice of pillars. The ETL was then corrugated, as well as the about 320 $nm$ thick MAPICl layer that planarized the stack, before the HTL and Al back contact deposition. They noticed increased absorption and EQE as the lattice period decreased, up to the smallest value envisaged, i.e. 500 $nm$. This led to an increase ofthe $J_{sc}$ by 5\\%. Broadband enhancements were observed, at wavelengths shorter than the band edge of the material, as well as a limited LT at the band edge, according to the EQE enhancement plotted in Figure \\ref{fig:EQE_table_3}. This is likely due to the lack of large enough spatial frequencies to couple the quasi-guided modes of the structure, given a period slightly above the optimal range. In addition, there might be a slight change in the volume of perovskite between the patterned cells compared to the flat reference.\\\\\n\t\n\t\\paragraph{Textured substrate}\n\t\n\tW. Qarony \\textit{et al.} \\cite{Qarony2018} calculated the EQE for three various configurations of PPSC using the same volume of perovskite and including moth eye periodical patterns with a period of about 150 $nm$, typically leading to scattering. The fist one had only a pattern at the top air \/ ZnO interface (thus is more comparable to former structures), but the two others considered a conformally patterned stack, either on a patterned Al substrate, or on a patterned NiO\/ITO layer. The patterned Al substrate clearly led to a lower EQE due to additional parasitic absorption. Moreover, a slightly lower $J_{sc}$ was obtained with the conformal stack on the flat Al substrate compared to the top patterned stack. Even this last case exhibits a low EQE enhancement (see Figure \\ref{fig:EQE_table_3}).\n\t\n\t\\subsubsection{Aperiodic patterning for LM}\n\t\n\t\\paragraph{Superficial structuration and light management layers}\n\tB. Dudem \\textit{et al.} \\cite{Dudem2016} proposed a multifunctional inverted micro-structured pyramidal Polydimethylsiloxane (PDMS) AR layer for enhancing the device efficiency, through AR and self cleaning. The MAPI layer was 340 $nm$ thick. The sizes of the pyramid were in the range from 1 to 10 $\\mu m$, therefore too large for efficient diffraction. Compared to flat PDMS, the $J_{sc}$ increases by 0.38 $mA.cm^{-2}$ up to 21.25 $mA.cm^{-2}$, corresponding to a limited AR effect, as confirmed by the flat EQE enhancement (see Figure \\ref{fig:EQE_table_3}).\n\t\n\tRather similarly, M. Jo\\v{s}t \\textit{et al.} \\cite{Jost2017} fabricated a so-called light management foil on the glass substrate of a planar 270 $nm$ thick MAPI cell, which led to a limited EQE enhancement (see Figure \\ref{fig:EQE_table_3}). At first, the thick glass substrate prevents from a strong overlap between any guided mode into the perovskite and the foil. Moreover, whatever the glass thickness, the lack of LT is also related to the far too low spatial frequencies resulting from the texturation. Indeed, thanks to available top view of the foil, a Fourier transform analysis reveals (see Figure SI-\\ref{fig:Supplemental Fourier Analysis Jost}) that most of Fourier's components lie below 7 $\\mu m^{-1}$, whereas the optimum $\\beta_m$ should be in the range 18 - 34 $\\mu m^{-1}$ at first order of diffraction, and even larger for other orders.\n\t\n\tOn the other side of stack, H. Zhang \\textit{et al.} \\cite{Zhang2018b} measured and modelled the impact of the roughness of the back mirror on the back scattering of MAPICl PPSC. The reference was made of a perovskite layer with a significant roughness after crystallization, coated with a thick Spiro layer that planarizes, then a flat gold mirror. With a thinner HTL, the gold mirror replicated the roughness of the perovskite layer leading to light back scattering. The PCE increased from 19.3\\% to 19.8\\%, mainly thanks to the $J_{sc}$ increase from 22.7 to 23.6 $mA.cm^{2}$. It is noticeable that the obtained EQE enhancement is the largest at the band edge. This could be related to the grain size that appears to be in the range 200 - 500 $nm$, so including the optimal range for efficient LT.\\\\\n\t\n\t\\paragraph{Structuration of the perovskite}\n\t\n\tA. R. Pascoe \\textit{et al.} \\cite{Pascoe2016} proposed textured MAPI at the scale of several hundreds of nanometers thanks to gas crystallization. It enhanced the EQE at wavelengths larger than 550 $nm$ (see Figure \\ref{fig:EQE_table_3}), thanks to an induced so-called scattering. Accordingly, the averaged $J_{sc}$ increased from 21.3 for planar references having an about 300 $nm$ thick MAPI layer to 22.1 $mA.cm^{2}$ for patterned samples of comparable volume. As previously, the noticeably large EQE enhancement close to the band edge can be related to the typical grain size of the order of 500 $nm$.\n\t\n\t\\subsubsection{Synthesis}\n\tIn the various previously described studies reporting on $J_{sc}$ enhancements thanks to LM, it can be noticed that most of the possible architectures envisaged in the section \\ref{possible_architecture} have been considered, using all the PE described in section \\ref{Photonic concepts description}. Tables \\ref{tableau pattern mapi} and \\ref{tableau pattern exp} summarize the $J_{sc}$ enhancements reported in the various previously described studies.\n\t\n\tFrom Table \\ref{tableau pattern mapi}, focusing on simulations of periodic patterning of MAPI based single junction PPSC, it comes out that LT can indeed significantly increase the $J_{sc}$ for very thin perovskite layers. The enhancement is more limited at a thickness of about 300 $nm$. It remains that an accurate comparison of the $J_{sc}$ enhancements is not possible, since performance enhancement strongly depends on the choice of the unpatterned reference, and especially its optimization in terms of LM as discussed previously. However, the EQE enhancements (Figure \\ref{fig:EQE_table_2}) can clearly reach larger values thanks to LT than for AR.\n\t\n\tFrom Table \\ref{tableau pattern exp}, focusing on experimental results, using various perovskite materials, the $J_{sc}$ enhancements are of the same order as the simulated ones, again with the same precautions as above. As for the simulated PPSC, the EQE enhancements (Figure \\ref{fig:EQE_table_3}) also reach larger values using patterns at the scale of the wavelength in the material. Moreover, the periodic case lead to the largest enhancement.\n\t\n\t\\begin{table}[h]\n\t\t\\caption{Reported simulated $J_{sc}$ enhancements mainly resulting from Light Management, for MAPI single junction PPSC, with any kind of in-plane pattern (additional layers, structuration of the perovskite or conformal perovskite) and whatever the main identified PE}\n\t\t\\label{tableau pattern mapi}\n\t\t\\begin{tabular}{@{}lllll@{}}\n\t\t\t\\hline\n\t\t\tPerov.thick. ($nm$ ) & $J_{sc}$ enhanc. (\\%) & Pattern. type & Photonic concept &  Ref \\\\\n\t\t\t\\hline\n\t\t\t400 & 6.3 & periodic add. patterned layer & Broadband AR and LT & \\cite{Peer2017} \\\\  \n\t\t\t120 & 31.7 & periodic add. patterned layer & Broadband AR and LT & \\cite{kim_light_2021} \\\\  \n\t\t\t300 & 5.6 & periodic struct. of the perovskite & Broadband AR and LT & \\cite{Schmager2019}  \\\\\n\t\t\t180 & 17 & periodic conformal perovskite & Broadband AR and LT & \\cite{Du2016} \\\\\n\t\t\t300 & 10 & periodic add. patterned layer & Broadband AR & \\cite{Hossain2020}  \\\\\n\t\t\t300 & 10 & periodic conformal perovskite & Broadband AR & \\cite{Qarony2018}\\\\     \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\begin{table}[!h]\n\t\t\\caption{Reported experimental $J_{sc}$ enhancements mainly resulting from Light Management, for single junction PPSC, with any kind of in-plane pattern, and whatever the main identified PE}\n\t\t\\begin{tabular}{@{}lllllll@{}}\n\t\t\n\t\t\t\\hline\n\t\t\tMaterial & Perov.thick. ($nm$) & $J_{sc}$ enhanc. (\\%) & Pattern. type & Photonic concept &  Ref \\\\\n\t\t\t\\hline\n\t\t\tMAPI & 240 & 14.3 & periodic add. patterned layer & Broadband AR and LT & \\cite{Wei2017} \\\\\n\t\t\tCsFAMAPBI & 370 & 2 & periodic struct. of the perovskite & Broadband AR and LT & \\cite{Schmager2019b}  \\\\    \n\t\t\tCsFAMAPBI & 550 & 6.3 & periodic, struct. of the ETL & Broadband AR & \\cite{Tockhorn2020} \\\\  \n\t\t\tMAPICl & 320 & 5 & periodic struct. of the perovskite & Broadband AR & \\cite{Paetzold2015}  \\\\\n\t\t\tMAPI & 340 & 1.8 & aperiodic add. patterned layer & Broadband AR & \\cite{Dudem2016} \\\\\n\t\t\tMAPI & 270 & 4.8 & aperiodic add. patterned layer & Broadband AR & \\cite{Jost2017} \\\\\n\t\t\tMAPICl & 300 & 4 & aperiodic add. patterned layer & Broadband AR & \\cite{Zhang2018b} \\\\\n\t\t\tMAPI & 300 & 3 & aperiodic substrate corrugation & Broadband AR & \\cite{Pascoe2016}\\\\\n\t\t\t\\hline\n\t\t\\end{tabular} \\label{tableau pattern exp}\n\t\\end{table}\n\t\n\t\n\t\\subsection{LM for PR}\n\t\n\tLM for $V_{oc}$ enhancement thanks to PR in PPSC is discussed in a limited number of publications.\n\t\n\tS. Nanz \\textit{et al.} \\cite{Nanz2019} investigated mainly theoretically the effect of various kinds of LM strategies on the PR, for multilayer stacks that are part of PPSC, mainly without the HTL and metallic contact. They were thus able to derive an upper limit $\\Delta V_{oc}$ for each case, under the assumption of pure radiative recombination. According to the summarized principle reminded previously, the $\\Delta V_{oc}$ resulting from PR in a patterned multilayer was in between the values obtained with the Lambertian multilayer and the rigorously planar multilayer. Indeed, the Lambertian multilayer, as it consists in a better absorber, also radiates the luminescence, whereas, for targeted thicknesses, luminescence can be partly guided, leading to recycling. Then, the quasi-guided mode of patterned multilayer led to enhanced PR compared to the Lambertian multilayer, and also an enhanced $J_{sc}$ compared to the flat multilayer.\n\tHowever, A. Bowman \\textit{et al.} \\cite{Bowman2020} showed that using a more realistic model including recombination, PR was rather unlikely to occur at maximum peak power even if the cell only interacts with a limited solid angle. In this context, it thus appears more promising to increase the absorption and thus the extraction, to the detriment of recycling.\n\t\n\t\\section{Simulations and Perspective}\n\t\n\tAs shown in the previous section, a limited number of studies, mainly focused on $J_{sc}$ enhancements, evidenced a LM effect in PPSC. This synthesis also illustrates that due to a lack of common references, different kinds of PE can hardly be compared, so as the most promising architectures for AR and light trapping. Therefore, we propose to simulate some of the promising patterns, to be able to compare their performances using typical materials and architecture of a PPSC.\n\t\n\t\\subsection{Methodology}\n\t\n\tThe Rigorous Coupled Wave Analysis \\cite{moharam1981rigorous} suits well for the simulation of the stacks periodically patterned under plane wave illumination. We have used the $S^4$ code \\cite{LIU20122233} available in the Solcore package \\cite{Alonso-Alvarez2018}. The derived $J_{sc}$ are obtained using AM1.5G spectrum \\cite{Gueymard1995} provided an IQE of 1.\n\t\n\tOptical indices used are from S. Manzoor \\textit{et al.} \\cite{Manzoor2018} for MAPI, from K.R. McIntosh \\textit{et al.} \\cite{McIntosh2009} for PMMA, and from J.M. Ball \\textit{et al.} \\cite{Ball2015} for all other materials: Sodalime Glass as a substrate, ITO, TiO$_2$, doped Spiro-OMeTAD (as HTL material), and gold. Thicknesses are set to 100 $nm$ for top contact, 20 $nm$ for ETL, and 300 $nm$ for Au.\n\t\n\tThe ETL is supposed to be a thin TiO$_2$ layer, dense and flat, to avoid scattering, and to limit parasitic absorption compared to other envisaged organic materials.\n\t\\subsection{Optimization of the key layers thicknesses in various planar single-junction PPSC}\n\t\n\tLet us first consider a planar PPSC on an infinitely thick substrate, illuminated through this substrate under normal incidence (see Figure \\ref{fig:perspective_single_junction}). The thicknesses of the MAPI layer $th_{MAPI}$ and of the HTL $th_{HTL}$ are supposed to vary in the ranges 300 - 700 $nm$ and 200 - 400 $nm$ respectively. As can be seen in Figure SI-\\ref{fig:jsc_thicknesses}, the $J_{sc}$ of such a cell increases with $th_{MAPI}$. Moreover, for a given $th_{MAPI}$, the $th_{HTL}$ has a non-negligible influence in the $J_{sc}$; e.g. for the smallest MAPI thickness of the considered range, the $J_{sc}$ can be increased by more than 0.5 $mA.cm^{-2}$, up to about 21.74 $mA.cm^{-2}$.\n\t\n\tIt remains that the low $J_{sc}$ for the thinnest considered MAPI is due to lower absorption at long wavelengths, as can be seen in Figure SI- \\ref{fig:absorption_spectra}, for a given thickness of HTL of 240 $nm$.\n\t\n\tGiven the possible identified advantages of using a thinner MAPI layer, mainly for electrical properties, its thickness will be set to 300 $nm$ in the following.\n\tAn AR PMMA layer is then coated on the glass substrate, which thickness is set to 1 $mm$ (see Figure \\ref{fig:perspective_single_junction}), with a negligible roughness. For the chosen $th_{MAPI}$, the coupled influence of PMMA thickness, $th_{PMMA}$, and $th_{HTL}$ is studied. According to Figure SI- \\ref{fig:jsc_thicknesses_pmma}, a significantly increased $J_{sc}$, up to about 21.92 $mA.cm^{-2}$, can be obtained for $th_{PMMA} = 360\\ nm$ together with $th_{HTL} = 250\\ nm$. This last structure will be then used as a planar but optimized reference for fair estimation of the impact of PE in the following. The corresponding spectrum is drawn in Figure \\ref{fig:A_tous}.\n\t\n\t\\subsection{Introduction of various 2D PC to enhance the current density}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/perspective_single_junction.png}\n\t\t\\caption{Various configurations of single junction PPSC simulated in the perspective. (a) planer PPSC on an infinite glass substrate, to study the effect of MAPI and Spiro-OmeTAD thicknesses (see results in Figure SI-\\ref{fig:jsc_thicknesses}), (b) Planar reference having a 300 $nm$ thick MAPI layer, and various PMMA and Spiro-Ometad thicknesses to maximise absorption (see results in Figure SI-\\ref{fig:jsc_thicknesses_pmma}), (c) Cross section of a 2D square lattice of cylindrical holes in MAPI, to study the effect of the corresponding PC parameters (see results in Figures SI-\\ref{fig:TOTAL_absorption_Jsc_Jo_fn_period_etch_depth40_PC_perovskite} and \\ref{fig:A_tous}), (d) Cross section of a 2D square lattice of cylindrical holes in PMMA, to study the effect of the corresponding PC parameters (see results in Figures SI-\\ref{fig:TOTAL_absorption_Jsc_Jo_fn_period_etch_depth40_PC_pmma} and \\ref{fig:A_tous}), (e) Cross section of two 2D distinct square lattices of cylindrical holes in PMMA and MAPI, to study the effect of PC parameters (see results in Figures SI-\\ref{fig:scan_N_Nprime} and \\ref{fig:A_tous})  }\n\t\t\\label{fig:perspective_single_junction}\n\t\\end{figure}\n\t\n\t\n\tAs already discussed, to further increase the absorption and thus the $J_{sc}$, the most efficient strategies should be to couple the impinging light into the guided modes thanks to properly designed patterns. In the following, we envisage 2D square lattices of cylindrical patterns. Each resulting 2D PC owns typically three parameters that can be optimized: i) its period $P$, ii) its filling fraction ($ff$) which is the ratio between the hole surface and the period surface, and iii) its thickness $t$. \n\tThese 2D PC can be located either (see Figure \\ref{fig:perspective_single_junction}):\n\t\\begin{itemize}\n\t\t\\item in the MAPI layer, made of holes in the MAPI layer filled with HTL material; for a fair comparison, the volume of MAPI material is the same as the planar PPSC, so its total thickness changes. Moreover, $t < th_{MAPI}$ to prevent from short circuits between HTL and ETL. A $t_{HTL}=250 nm$ thick slab of HTL is kept for planarization;    \n\t\t\\item in the top PMMA layer, patterned in a PC of air holes, with $t = th_{HTL}$, in favor of the diffraction efficiency, given the low index of the PMMA;\n\t\t\\item simultaneously at the two previous locations, but each PC has its own set of parameters.\n\t\\end{itemize}\n\t\n\tIn the following studies, all the PC parameters are scanned over realistic ranges. The step for $P$ and $t$ is 5 $nm$, whereas only 3 $ff$ have been envisaged: 0.3, 0.4 and 0.5; these appear to be the most realistic values compatible with a large area patterning at a reasonable cost. \n\t\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{Fig\/A_tous_cascade.png}\n\t\t\\caption{Simulated absorption spectra for various 1 mm thick glass substrate PPSC with maximized short circuit current density: the planar PPSC (Figure \\ref{fig:perspective_single_junction} (b)), the PPSC with a 2D PC into the MAPI (Figure \\ref{fig:perspective_single_junction} (c)), the PPSC with a 2D PC in the PMMA (Figure \\ref{fig:perspective_single_junction} (d)), and the PPSC with both 2D PC (Figure \\ref{fig:perspective_single_junction} (e)). Small amplitude, high spectral resolution Fabry Perot resonances take place in the $1 mm$ thick glass substrate.}\n\t\t\\label{fig:A_tous}\n\t\\end{figure}\n\t\n\t\n\t\\subsubsection{Single junction: 2D PC in the perovskite layer}\n\tWith respect to the best planar PPSC coated with PMMA, an increase of almost 1 $mA.cm^{-2}$, leading to a $J_{sc}$ about 22.88 $mA.cm^{-2}$ is obtained thanks to a PC into the MAPI (see Figure SI- \\ref{fig:TOTAL_absorption_Jsc_Jo_fn_period_etch_depth40_PC_perovskite}), with $P = 400\\ nm$, $ff= 0.4$ and $t= 110\\ nm$. It has been checked that $ff$ of 0.3 and 0.5 lead to lower current densities than the optimal one, but still larger than the reference. The corresponding spectra in Figure \\ref{fig:A_tous} reveal that the improvement is mainly due to a larger band edge absorption, so LT, as confirmed by the absorption enhancement in the same Figure.\n\t\n\tIt is noticeable that other periods, smaller than 550 $nm$, can lead to more limited $J_{sc}$ enhancements. However, it has been checked that using periods around 5 and 10 $\\mu m \\pm 0.5$ lead to a $J_{sc}$ lower than 21.85 $mA.cm^{-2}$. It confirms that such periods, far larger than the sub-micron optimal one, do not enable an efficient diffraction and thus do not lead to any $J_{sc}$ enhancement, when compared to an optimized planar reference.  \n\t\n\t\\subsubsection{Single junction: 2D PC in the PMMA covering layer}\n\t\n\tWith respect to the best planar PPSC coated with PMMA, an increase of almost 0.9 $mA.cm^{-2}$, leading to a $J_{sc}$ about 22.75 $mA.cm^{-2}$, is obtained thanks to a PC in the PMMA layer. The 2D PC parameters are $P = 665\\ nm$, $ff = 0.4$ and $t= 615\\ nm$ (see Figure SI-\\ref{fig:TOTAL_absorption_Jsc_Jo_fn_period_etch_depth40_PC_pmma}). It has been checked that $ff$ of 0.3 and 0.5 lead to lower current densities. It can be noticed on the corresponding spectra in Figure \\ref{fig:A_tous} that the improvement is due to both a AR effect and limited LT at band edge absorption, since it implies low effective index guided modes that are the only able to interact, weakly, with the pattern on top of the thick substrate. The enhancement remains lower than the one induced by the 2D PC in the perovskite layer. \n\tAgain, it has been checked that periods far larger than the optimal one, typically around around 5 and 10 $\\mu m \\pm 0.5$ lead to a $J_{sc}$ lower than 22 $mA.cm^{-2}$, so to a limited enhancement, because of a reduced diffraction efficiency.\n\t\n\t\\subsubsection{Single junction: Combination of the two 2D PC}\n\t\n\tGiven the previous enhancements, a structure that combines the two 2D PC, one in the PMMA and the second at the MAPI\/HTL interface, can be envisaged. It is noticeable that RCWA method implies that the period of such a combined architecture is a integer $N_{PMMA}$ times the period of the 2D PC in the PMMA ($P_{PMMA}$), and another integer $N_{MAPI}$ times the pitch of the 2D PC in the MAPI ($P_{MAPI}$); other PC parameters ($ff_{PMMA}$, $t_{PMMA}$ on the one hand, $ff_{MAPI}$, $t_{MAPI}$ on the other hand) can differ (see Figure \\ref{fig:perspective_single_junction}).\n\tFor the sake of illustration of a possible further $J_{sc}$ enhancement, it has been chosen to set the parameters of the 2D PC at the MAPI\/HTL as for the optimized cell with a flat PMMA layer, i.e. $P_{MAPI} = 400\\ nm$, $ff_{MAPI} = 0.4$, $t_{MAPI} = 110\\ nm$, as well as $ff_{PMMA} = 0.5$ and $t_{PMMA} = 405\\ nm$, among the thinnest most favorable values of the 2D PC in PMMA associated with planar MAPI. Then, $N_{PMMA}$ is scanned from 5 to 8 and $N_{MAPI}$ from 9 to 15, given the fact that $P_{PMMA}$ is typically larger $P_{MAPI}$ according to the previous studies. The simulated $J_{sc}$ displayed in Figure SI-\\ref{fig:scan_N_Nprime} shows that a limited increase, of about 0.26 $mA.cm^{-2}$, up to 23.14 $mA.cm^{-2}$, is possible provided $P_{PMMA} = 530\\ nm$. It can be seen in Figure \\ref{fig:A_tous} that such $J_{sc}$ enhancement results from both LT and AR effects compared to the planar reference. If the full space of various possible PC parameters has not been scanned, and thus the previous parameters not fully optimized, the interest of such a combination is yet demonstrated.\n\t\n\t\\subsection{2T tandem}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[scale=0.7]{Fig\/figures_perspective_tandem.png}\n\t\t\\caption{Various configurations of 2T tandem PPSC simulated in the perspective. (i) Planar ref on an infinite glass substrate with varying perovskite materials thicknesses, to reach and equilibrium of $J_{sc}$. (ii) Cross section of a 2D square lattice of ITO pillars in 1.77-eV perovskite, to enhance $J_{sc, 1.77eV-perovskite}$ (iii) Cross section of a 2D square lattice of ITO pillars in the 1.77-eV perovskite, and of dots of 1.22-eV perovskite in HTL. Inspired by the configuration proposed by K. Xiao \\textit{et al.} \\cite{xiao_all-perovskite_2020}}\n\t\t\\label{fig:figures perspective tandem}\n\t\\end{figure}\n\t\n\tA 2T tandem PPSC generally exhibits thermalization, as the one previously described in section \\ref{tandem_plan}. Thus, we will consider this example as a case study for possible improvement. In the following simulations, the optical indices they provided for the considered materials are used. The planar stack (see Figure \\ref{fig:figures perspective tandem}), considered as a reference, is close to the one shown in Figure \\ref{fig:xiao tandem schema}. Oppositely to our previous studies, the glass substrate is again supposed to be infinitely thick to avoid the additional interferences in the substrate. Within the stack, the layer thicknesses (except perovskites) have been set to realistic values such as $t_{ITO}=100\\ nm$, $t_{NiO}=t_{NVPB}= 10\\  nm$ (simplified version of a mixed material ETL), $t_{C60} = 10\\ nm$ for both HTL, as well as $t_{PEDOT-PSS} = 10\\ nm$ and $t_{Cu}=100\\ nm$. As justified later, the SnO$_2$ layer, with $t_{SnO_2}=100\\ nm$, acts as an optical spacer (the 1 $nm$ thick Au layer has thus been neglect).\n\t\n\tSetting the thicknesses of both perovskite layers to $t_{1.77\\ eV\\ PK} = 400\\ nm$ (also the maximum in the considered range) and $t_{1.22\\ eV\\ PK} = 880\\ nm$ (in the range 800 - 1200 $nm$) leads to the highest $J_{sc}$ of 16.25 $mA.cm^{-2}$, that appears to be limited by the 1.77 $eV$ perovskite subcell.\n\tThis value is slightly larger than the one obtained by K. Xiao \\textit{et al.}. Moreover, it is obtained in our case for different perovskite thicknesses, due to a mismatch between the thicknesses of the charge transport layers we have chosen and author's choices. However, our derived absorbance spectra for both sub cells (see Figure \\ref{fig:A_tous_tandem}) still exhibits a thermalization effect. Simply increasing $t_{1.77\\ eV\\ PK}$ could be at the expense of the charges collection. \n\t\n\tIn this frame, the possible enhancement of the $J_{sc,1.77\\ eV\\ PK}$ thanks to a 2D PC at the ITO \/ 1.77 $eV$ perovskite layer interface is studied. The 2D PC consists of a square lattice of ITO pillars, coated with conformal ETL and then with a 1.77 $eV$ perovskite layer that planarizes the corresponding subcell, while keeping an equivalent volume of perovskite as in the planar tandem cell. \n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.65\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/tandem.png}\n\t\t\t\\caption{Simulated absorption spectra of the various envisaged 2T tandem PPSC.}\n\t\t\t\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/tandem_zoom.png}\n\t\t\t\\caption{Zoom on LT in 1.77-eV perovskite material}\n\t\t\t\n\t\t\\end{subfigure}\n\t\t\\caption{2T tandem PPSC simulated in the perspective, Inspired by the configuration proposed by K. Xiao \\textit{et al.} \\cite{xiao_all-perovskite_2020}.}\n\t\t\\label{fig:A_tous_tandem}\n\t\\end{figure}\n\t\n\tA significant enhancement of $J_{sc,1.77\\ eV PK}$, up to 17.92 $mA.cm^{-2}$, has been found for $P=325\\ nm$, $ff=0.5$ (here defined as the ITO filling fraction) and $t=160\\ nm$.\n\tThe reason is twofold (see spectra in Figure \\ref{fig:A_tous_tandem}): a broadband AR as well as a limited LT at the band edge of the 1.77 $eV$ perovskite material that slightly reduces the thermalization. LT specifically occurs in one of the absorbing materials, to the detriment of the other one. It results from a guided mode mainly confined in the perovskite of interest, with a limited overlapping in the other perovskite, especially thanks to the rather large, 100 $nm$ SnO$_2$ optical spacer. Indeed, this layer can prevent evanescent coupling between quasi guided modes of the perovskites. Additionally, it could protect the 1.77 $eV$ perovskite material during the fabrication, but the junction between the two subcells might then also be less efficient. \n\t\n\tMoreover, this first step leads to a strong $J_{sc}$ disequilibrium between the two sub cells, since $J_{sc,1.22\\ eV\\ PK}$ even slightly decreases to 15.7 $mA.cm^{-2}$, compared to the planar reference. To increase in a second step $J_{sc,1.22\\ eV\\ PK}$, this low-Eg perovskite layer can also be patterned rather than simply increasing its already large thickness.\n\t\n\tStarting from the last structure, a second 2D PC is introduced at the 1.22 $eV$ perovskite material \/ HTL interface. To target a LT at larger wavelength, close to the band edge of this perovskite, its period has to be increased. However, given the constraint induced by the boundary conditions of the simulation, we simply chose a supercell having a period twice the one of the single PC device, with one pattern in the 1.22 $eV$ perovskite material, and two patterns in the 1.77 $eV$ perovskite material (see Figure \\ref{fig:figures perspective tandem}), while keeping $ff$ and thicknesses constant. Moreover, again, the volume of this perovskite is kept constant, HTL acting as a planariser, with a minimum thickness of 10 $nm$ (with a larger volume as in the planar alternative). \n\t\n\tIt appears that $J_{sc,1.77\\ eV PK}$ remains unchanged, whereas $J_{sc,1.22\\ eV\\ PK}$ is increased up to 16.2 $mA.cm^{-2}$, reducing, but not cancelling the disequilibrium. As expected, this results from LT at the band edge in the 1.22 $eV$ perovskite material (see Figure \\ref{fig:A_tous_tandem}). \n\tThis last result appears as a proof of concept to integrate two PC in order to induce LT, as well as possibly AR, in two different absorbing layers of a given stack, provided each one exhibits guided modes without overlapping. If not yet optimized, it is noticeable that, providing the first subcell is planarized, the two PC could practically have independent parameters, offering additional degrees of freedom to reduce disequilibrium. Given the required spacer, such a concept could be also applied to other kind of three or four terminal multijunctions cells. \n\t\n\t\\section{Conclusion - Perspectives}\n\t\n\tLight management for PV solar cells was mainly intended to increase the $J_{sc}$, simply thanks to a larger absorption. When using metal-halide perovskite materials, LM is shown to result from an interaction between the various material choices, the patterning processes and the PE. Moreover, demonstrating sole LM needs to meet some rigorous criteria to get rid of mixed electrical, material and photonic effects.\n\t\n\tThe investigations presented in this review have demonstrated that the easiest $J_{sc}$ enhancement is indeed the most frequently envisaged effect, especially for single junction cells, compared to the still highly challenging $V_{oc}$ enhancement, and the immature studies of the EY due to the lack of studies at the module scale. As demonstrated by several authors, even thicknesses optimization within flat PPSC can enhance absorption thanks to Fabry Perot effects while taking care of the electrical constraints. To further increase the absorption at the perovskite band edge, an in-plane pattern, at various scales, can be introduced in one of the layers. However, patterns at the wavelength scale appeared as the most efficient. In any case, the $J_{sc}$ enhancement compared to already optimized flat reference remained limited to a few percent. As regards the photonic regimes, the broadband AR effect was the most frequently observed, but only LT was able to significantly enhance the absorption at the band edge, as confirmed by our own simulations. In addition, we confirmed that direct patterning of the perovskite leads to more efficient LT than an additional layer on top of the substrate. Finally, in 2T tandem cells, we showed that one PC in each subcell can enhance each of the $J_{sc}$ separately.\n\t\n\tBeyond, some of the strategies developed for these opaque, single junction PPSC can be adjusted for other applications of hybrid perovskites for sun light harvesting. Indeed, accurately tuned spectral absorption is required in semitransparent single junction. Moreover, the same concepts can be tweaked to tailor reflectivity spectrum for perovskite-based color printing devices \\cite{Gholipour2017,Gao2018,Fan2019,Yoo2021} and engineering absorption management in full-color perovskite detectors \\cite{Hossain2020b,Qarony2020}.\n\t\n\tThis work is even part of far larger context. Indeed, other studies are ongoing concerning materials, which need to be more stable, and processes. In this frame, it can be noticed that record cells might not share the same encapsulation strategies as more realistic, large surface cells and modules \\cite{Wang2021b}, both for aging and safety reasons \\cite{wu_evolution_2021, wu_main_2021}. All perovskite tandem cells are even more challenging on the fabrication point of view \\cite{Zheng2020}, but are also very promising since they combine all the potential of perovskite based cells and modules with high yields.\n\t\n\tFinally, according to the reciprocity relation between absorption and emission \\cite{Rau2007}, light extraction strategies in perovskite LED \\cite{Gholipour2017,Wang:17,C7NR01631J}, as well as resonances of high quality factor in perovskite-based laser \\cite{Chen2016b,Pourdavoud2017,Qin2020} can use similar concepts to those derived in this work. Even LM in 2T tandem cells could be mimicked in white LEDs obtained by stacking several emitting materials. For LEDs, a high light extraction efficiency is a even more the key of good efficiency.\n\t\n\t\\textit{Acknowledgement:}\n\t\n\tRCWA simulations were performed on the Newton computer cluster facilities operated by PMCS2I at Ecole Centrale de Lyon and on the CNRS\/IN2P3 Computing Center in Lyon. R. M. L. acknowledges project EMIPERO (ANR-18-CE24-0016).\n\t\n\t\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nCubeSat space telescopes are a growing asset in astronomical research\\cite{Shkolnik_2018,douglas2019cubesats}. Their standardized format and compact volume enables the rapid development of high-impact mission concepts above the earth's atmosphere. Flight-tested commercial microelectromechanical systems (MEMS)  have further enhanced CubeSat telescopes by providing dynamic pointing and aberration control on-orbit (DeMi, NODE) \\cite{Yenchesky2019OptomechanicalDA,Clements2016}. The expanding capacity of CubeSats for astronomical research invites the development of novel technologies for their optical systems. However, the resolution in CubeSat optics is fundamentally limited by the clear aperture permitted by the CubeSat volume. The rayleigh criterion tells us that resolution ($\\Delta \\theta$) is inversely related to to the entrance pupil diameter ($D$) of the optical system.\n\n\\begin{equation}\n    \\Delta \\theta = 1.22 \\frac{\\lambda}{D}\n\\end{equation}\n\nReflective CubeSat objectives traditionally require mounting hardware that limits the available entrance pupil diameter. Single-point diamond turned (SPDT) mirrors can mitigate this limitation by enabling the manufacturer to machine the mounting hardware directly into the rear of the mirror substrate, eliminating the need for mounting hardware around the edge of the mirror and enhancing the nominal throughput and resolution. This also grants the objective a considerable degree of athermalization by eliminating the difference in the coefficient of thermal expansion between the mirror and mounting hardware\\cite{Zhang:17}. SPDT surfaces are typically used in longer wavelengths (MWIR, LWIR) due to the midspatial frequency errors left by tooling marks. However, recent developments in optical polishing\\cite{Jeon:17} have introduced Magnetorheological Finishing (MRF) to the fabrication of SPDT surfaces, reducing the dominant midspatial frequencies considerably and therefore extending their use into the optical. Athermal CubeSat objectives composed of low surface roughness mirrors with a large entrance pupil invites the design of high-performance optical CubeSat payloads to make the next generation of research and technology development in space more accessible. The Versatile CubeSat Telescope (VCT) concept is a prototypical fore-optic that boasts compatibility with a variety of research payloads. The goal of the VCT is to develop a flexible, large aperture telescope that can be replicated at low cost for easy adaptability to future research payloads and technology demonstrations. \n\n\\section{OPTICAL DESIGN - ON-AXIS V.S. OFF-AXIS IMPLEMENTATIONS}\nThe design of the VCT began with two realizations of a CubeSat telescope that could take advantage of a large primary mirror. The first, an on-axis telescope with a more classical Ritchey-Chretien objective outfitted with a plano-convex aspheric collimator. The second, an all-reflective off-axis Ritchey-Chretien solution with a freeform collimator. We baselined a 95mm entrance pupil diameter with a 20$\\%$ obscuration for the on-axis telescope, and scaled the entrance pupil diameter of the off-axis telescope to be equivalent in collecting area. A high pupil magnification is required for both designs to image the primary mirror onto the small MEMS FSM clear aperture (5mm). The specifications and system layouts are shown in table \\ref{tab:specs} and figure \\ref{fig:layout} respectively.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c}\n        \\hline\n        Specification & Value  \\\\\n        \\hline\n        Entrance Pupil Diameter & \\textbf{95 mm} \\\\\n        Exit Pupil Diameter & 5 mm \\\\\n        Half Field of View & 0.2$^{\\circ}$ \\\\ \n        Secondary Obscuration & \\textbf{20$\\%$} \\\\\n        Central Field Strehl & $>$0.99 \\\\\n        Average Field Strehl & $>$ 0.80 \\\\\n        OTA Packaging Volume & $<$ 2U \\\\\n        \\hline\n        \\\\\n    \\end{tabular}\n    \\qquad\n    \\begin{tabular}{c c}\n        \\hline\n        Specification & Value  \\\\\n        \\hline\n        Entrance Pupil Diameter & \\textbf{88 mm} \\\\\n        Exit Pupil Diameter & 5 mm \\\\\n        Half Fiel of View & 0.2$^{\\circ}$ \\\\ \n        Secondary Obscuration & \\textbf{0$\\%$} \\\\\n        Central Field Strehl & $>$0.99 \\\\\n        Average Field Strehl & $>$ 0.80 \\\\\n        OTA Packaging Volume & $<$ 2U \\\\\n        \\hline\n        \\\\\n    \\end{tabular}\n    \\caption{Optical design specifications for the (left) On-Axis and (right) Off-Axis telescope designs. All specifications are similar except for those in bold, which are set such that the apertures of each design are equal in throughput.}\n    \\label{tab:specs}\n\\end{table}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{vct_drawings.png}\n    \\caption{System layouts of the (left) On-Axis Telescope and the (right) Off-Axis Telescope designs. Each has three optical elements, two fast-steering mirrors (FSM) for fine pointing control, and a volume dedicated to a research payload.}\n    \\label{fig:layout}\n\\end{figure}\n\n Aspheric surfaces were necessary for both telescopes to achieve a high pupil magnification (19x and 17.5x for the on- and off-axis designs respectively) while maintaining a well-corrected field of view. While this slightly complicates the metrology of the mirrors it did not add any additional cost because the mirrors were already going to be diamond-turned. Both telescopes utilized a Ritchey-Chretien objective that forms an image near the primary mirror, which is collimated by an aspheric element to produce an external exit pupil. The size of the exit pupil was chosen to match the clear aperture of a MEMS fast-steering mirror (FSM)\\cite{memsfsm,Serra21} to correct for spacecraft pointing and jitter errors while in orbit. The external pupil also serves as a convenient interface for any instrument suite that would be included on the VCT (e.g. spectrographs, cameras, lasers).  The concept for the on-axis design was to make a well-established telescope format work given the pupil magnification and field of view specifications. The aspheric surfaces granted the design a well-corrected field of view in a compact format suitable for a CubeSat (2U). However, the design has a secondary obscuration which limits the throughput of the system and adds diffraction features to the image. The primary goal for the off-axis design was to offer a viable alternative to the classical on-axis design that would not suffer from undesirable diffraction effects brought upon by the secondary obscuration. Unfortuantely, shifting the primary parabolic mirror off-axis comes at the cost of more field-dependent aberration that limits the system's field of view. To accommodate this we selected the tertiary mirror of the off-axis design to be freeform. Before optimization of the freeform surface, the rotational symmetry of the system was broken by tilting the secondary and tertiary mirrors about their foci to make the system free of linear astigmatism. A linear-astigmatism free three mirror system (LAF-TMS) has been shown by Park et al\\cite{park_development_2020} to be an excellent starting point for freeform designs by using the mirror tilt angles and inter-mirror distances (Eq \\ref{eq:laf}) to mitigate a dominating aberration early in the design process.\n\n\\begin{equation}\n    \\frac{l_2^\\prime}{l_2}\\frac{l_3^\\prime}{l_3}tan{i_1}+\\left(1+\\frac{l_2^\\prime}{l_2}\\right)\\frac{l_3^\\prime}{l_3}tan{i_2+}\\left(1+\\frac{l_3^\\prime}{l_3}\\right)tan{i_3=0}\n    \\label{eq:laf}\n\\end{equation}\n\nIn this equation, parameters i$_{1,2,3}$ are the angles of incidence of the optical axis ray on surfaces 1,2, and 3, while $l_{2,3}$ and ${l^{\\prime}}_{2,3}$ are the front and rear focal lengths the secondary and tertiary mirror. Once the baseline LAF-TMS design had been implemented, the third mirror was changed to a freeform XY-Polynomial surface where coefficients were optimized that maintained the bilateral symmetry of the optical system. The mirrors were shifted off-axis in the $\\hat{y}$ direction, as was the field bias (+0.2$^{o}$). Consequently the freeform surface was constrained to be plane-symmetric about the $\\hat{y}-\\hat{z}$ plane by solely optimizing coefficients of even order in $\\hat{x}$. The resultant design is nearly diffraction-limited across the biased field of view.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{strhvsfov.png}\n    \\caption{Maps of strehl ratio v.s. field of view for the (left) On-Axis and (right) Off-Axis VCT. The symmetry of the optical system mirrors the performance quite readily, with the on-axis design having a rotationally symmetric field performance whereas the off-axis design exhibits plane-symmetric performance.}\n    \\label{fig:strvfov}\n\\end{figure}\n\n\\section{SENSITIVITY ANALYSIS}\n \nThe misalignment sensitivities of the two designs were explored in Zemax OpticStudio (ZOS) by iteratively perturbing each optic in six degrees of freedom using the Python ZOS-API. Our as-built performance goal for both systems was an average Strehl ratio $>$ 0.7 across the field of view to maintain reasonably diffraction-limited performance. The $0\\%$, $70\\%$, and $100\\%$ field of view in $\\pm \\hat{x}$ and $\\pm \\hat{y}$ were considered in the average calculation, considerably biasing the sensitivity analysis toward the edge of the field of view where performance is expected to degrade faster. This results in a more pessimistic analysis of misalignment sensitivity. The sensitivity analysis fixes the primary mirror while M2 and L3\/M3 were perturbed to better understand how misalignments with respect to the primary would impact the optical performance over all fields of view.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{qkdsens.png}\n    \\caption{Sensitivity to misalignment curves of the On-Axis (Solid Red) and Off-Axis (Dashed Blue) designs for their respective secondary (top) and tertiary (bottom) optic. The on-axis system is less sensitive for X\/Y decenter of both elements, X\/Y\/Z tilt of M2, and X\/Z tilt of M3.}\n    \\label{fig:my_label}\n\\end{figure}\n\nFigure 2 shows that in most cases the on-axis design was less sensitive to misalignment in six degrees of freedom. This result is consistent with the complexity of each system. The off-axis design employs off-axis conics and a freeform surface, both of which will be sensitive to misalignment due to the higher mirror slopes than the on-axis design. In applications where the secondary obscuration is tolerable, the on-axis design would be a better choice for applications where cost and misalignment risk are limiting factors.\n\n\\section{POLARIZATION ANALYSIS}\nMany spaceborne optical payloads have a degree of sensitivity to polarization effects. From laser experiments in support of quantum communications\\cite{Serra21} to polarimetry of atmospheric ice\\cite{Hart20}, there is a clear need to characterize the response of cubesat payloads to a vector electric field. To offer support for polarization-sensitive research payloads a comprehensive polarization ray trace (PRT) analysis was conducted for each realization of the VCT.  PRT is a ray-based approach to vector field calculations enabled by determining the local fresnel reflection coefficients that alter the transmission and phase of the electric field at each surface in the optical system. These effects are traced from the entrance pupil to the exit pupil of the optical system to determine the cumulative polarization behavior of the instrument. The proposed VCT designs use fast primary mirrors to achieve the high pupil magnification, so some polarization aberration is expected. However mirror substrates and coatings can be selected to mitigate the impact of polarization aberration for a given application. \n\nA PRT tool was written using the ZOS-API to create maps of diattenuation and retardance at the exit pupil of the system. The ZOS-API tool creates a batch raytrace and propagates it from the entrance pupil of the optical system to the surface under investigation. The output of the raytrace gives surface normal vectors in addition to exit ray vectors which are used to calculate the incident ray vectors and angle of incidence. Diattenuation ($D$) and retardance ($\\delta$) data is produced from the Zemax coating files and ray angles of incidence to generate the surface and pupil maps shown in figure \\ref{fig:polmaps_al}.\n\n\\begin{equation}\n    D = \\frac{|r_p|^{2} - |r_p|^{2}}{|r_s|^{2} + |r_p|^{2}};\\phantom{flapjackfacts}\\delta = |\\phi_{p} - \\phi_{s}|\n    \\label{eq:diat}\n\\end{equation}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{polplots.png}\n    \\caption{Polarization pupil maps assuming bare aluminum mirrors and uncoated N-BK7 lens surfaces. (Top) Diattenuation and (Bottom) retardance maps of the (left) On-Axis and (right) Off-axis versatile cubesat telescopes. These data were computed at $\\lambda$ = 780nm. At this wavelength the Off-Axis Aluminum VCT sees the greatest diattenuation, and the Off-Axis Gold sees the greatest retardance. This indicates that the off-axis design would introduce more challenges to polarization-sensitive payloads than the on-axis design would.}\n    \\label{fig:polmaps_al}\n\\end{figure}\n\nThe polarization pupil map permits analysis of the optimal material and configuration for a given polarization-dependent application. Figure \\ref{fig:polmaps_al} demonstrates that aluminum is more diattenuating than gold at $\\lambda=780nm$, and the off-axis configuration performs worse than the on-axis configuration due to the higher angles of incidence on the mirror surfaces. \n\n\\section{MECHANICAL DESIGN}\n\nThe VCT must be designed to deliver a high optical wavefront when operational on-orbit, yet also robust enough to survive the rigors of launch and orbital injection separation.  Achieving both goals in a compact package that affords straight-forward assembly and alignment is quite challenging.  Another important aspect of the design must also consider the mass contribution of the structural elements as this can be costly when considering launch requirements and on-orbit maneuvering.  With the prevalence of small satellite design ( $<$ 12U), development of a telescope that fits within a 2U volume with a mass less than 2 kg would present a low-cost solution that enables rapid prototyping of research payloads. For a prototype realization of the VCT, the optical system design called for SPDT Aluminum primary and secondary mirrors. All mechanical structural elements were also chosen to be aluminum to match the thermal behavior of the mounting structure to the optical elements. The collimating lens, which locates the instrument pupil, could also be easily mounted to the OTA using a common aluminum barrel structure with simple shims for precise alignment. \n\nThe first goal in the mechanical design of the telescope Optical Tube Assembly (OTA) is to support the optical elements with minimal effect due to expected thermal perturbations that would be encountered on-orbit.  These effects would include both rigid body misalignment and wavefront aberrations caused by subtle flexing of the optical elements.  In order to bound the problem, the first design decisions centered on overall sizing to fit within the small-sat form factor. The primary-secondary despace tolerance is very tight, so a truss structure was designed to properly space the two mirrors. The truss structure presents a good strength to weight ratio and is commonly used in ground-based telescope designs. For this space-based application, the truss structure would not allow for simple and easy shrouding of the optical path and thus would likely require additional structural pieces to support stray light baffling elements and thermal mitigation elements (i.e., mylar blankets).  Therefore, an early design decision was made to investigate a tube structure that would serve as both the metering structure that could maintain the mirror separation tolerance, transverse misalignment tolerance, and provide an in-place baffle that could be customized for the particular optical application.  A significant benefit of the tube design is the mass savings realized by the ability to select a very thin tube wall thickness that still retains the desired structural performance for all loading cases.  This tube would also allow for easy installation of thermal mitigation measures and targeted design of thermal conduction paths that further ensure optical performance stability while on-orbit.\n\nThe next major goal in the mechanical design considers the manufacturability of the system.  Considering its small size, the OTA manufacturability includes the fabrication processes for all structural components, as well as thoughtfully planned assembly and alignment processes that ensure the optical performance requirements can be met.  The aluminum mirrors provide a unique opportunity to meet the need for an assembly that is easy to integrate and align.  The primary mirror optical surface is thus applied directly to the structural component (primary mirror) which has appropriate design elements that consider thermal performance and mounting points for both the metering tube and the interface to the downstream optical system via a \\emph{hex plate} interface.  Clearly defining the rear of the primary mirror blank as the OTA support location allows for simple design of flexural components that will not propagate errors from other systems into the OTA, or vice-versa.  Similarly, the secondary mirror surface is applied directly to a head-ring structure that also considers thermal performance and mounting points to the metering tube, but also minimizes the size of support strut spiders that are common in the on-axis  design.  Considering the small size of the OTA, the one-piece secondary mirror\/head-ring simplifies its alignment features by moving them out to the tube diameter, rather than trying to squeeze them into the shadow of the secondary mirror on the entrance aperture.\n\nThe final goal of the mechanical design considers the interface of the OTA to the instrument and the spacecraft.  Most terrestrial optical system applications utilize a breadboard style support that allows for many mounting possibilities for both the OTA and any downstream instrument opto-mechanical elements, and can be purposefully designed to incorporate spacecraft mounting options that will isolate the payload from the spacecraft both thermally and dynamically as necessary.  The hex plate provides all of these interfaces in a single machined aluminum component.  The connection of the OTA to the hex plate is of particular importance for this discussion.  As mentioned earlier, the OTA must remain dynamically and thermally isolated from the other spacecraft or instrument components to preserve the telescope optical performance. A Finite Element Model (FEM) was created using the aluminum tube structure OTA, with both structural and optical components represented, and a Finite Element Analysis (FEA) was performed to determine the model behavior with both structurally dynamic boundary conditions and also varying thermal environmental conditions.  The FEM is shown in figure \\ref{fig:vct_fea}.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{vct_fea.png}\n    \\caption{The FEM of the VCT OTA highlighting the hex plate, flexure, and tube design.}\n    \\label{fig:vct_fea}\n\\end{figure}\n\nA simple three flexure design was selected to mount the OTA to the hex plate.  The flexures were positioned tangentially to the primary mirror blank to better accommodate thermal deformations that could arise at the hex plate. The flexure material properties were varied to determine the best trade of optical path stability, launch survivability, manufacturability, and cost. The materials considered were Aluminum (Al 6061-T6) and Titanium (Ti-6Al-4v).  For the structural analysis the hex plate was fixed at three points on its outer edge simulating spacecraft support, and MAC loads (100g accelerations in three orthogonal directions) were applied to the model.  Various flexure throat cross sectional areas were considered, and the best design was chosen based on the resulting maximum stresses revealed in the FEA.  The results for each material selection under this loading are shown in table \\ref{tab:macload}. \n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n     Load Case Direction & Max Flexure Stress [Al] & Max Flexure Stress [Ti] \\\\\n    \\hline\n        100g $\\hat{x}$ & 103ksi & 112ksi \\\\\n        100g $\\hat{y}$ & 98ksi & 103ksi \\\\ \n        100g $\\hat{z}$ & 42ksi & 46ksi \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{MAC loading in three orthogonal directions and the corresponding maximum flexure stress for Aluminum and Titanium flexures. The Ti flexures were chosen due to their higher maximum stress.}\n    \\label{tab:macload}\n\\end{table}\n\nAn additional analytical consideration for the structure is the modal behavior of the OTA.  The modal FEA reveals the lowest expected resonant frequency of the OTA under the prescribed mounting conditions for given material selections and flexure design choices.  The table below shows the first three modal FEA results for the same conditions discussed above.  These results indicate there would be little chance of exciting resonance in the OTA structure when considering expected launch loads.  And similarly, the telescope itself would not likely impart damaging resonance to any other spacecraft systems or partner satellites if launched as a secondary payload ride-share.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n        Al Flexures & Ti Flexures  \\\\\n    \\hline\n        130Hz & 156Hz \\\\\n        130Hz & 156Hz \\\\\n        363Hz & 434Hz \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{Modal behavior of the OTA for Al and Ti flexures for the cases described in \\ref{tab:macload}. Based on the results from the FEA for both materials, the Titanium flexure design was selected based on the extra margin provided on mass and stress.}\n    \\label{tab:my_label}\n\\end{table}\n\n Verification  of the selected flexure design needed to also consider the thermal behavior of the system.  For ease of creating the thermal FEM, the boundary conditions for the structure were preserved from the structural FEA and a 27.8\u00b0C differential in temperature was applied to the hex plate.  This set of conditions would bring into focus the thermal isolation of the OTA from the hex plate.  The results from this thermal analysis were normalized to show an expected mirror sag of 10nm P-V for each 1.0\u00b0C of temperature differential.  The results (shown below) also indicate an axisymmetric deformation of the OTA as would be expected.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{fea_results.PNG}\n    \\caption{Finite Element results of the hexplate-primary mirror temperature differential. The flexure design results in $\\approx$ 10nm of trefoil surface sag per degree Celsius.}\n    \\label{fig:fea_results}\n\\end{figure}\n\nOverall, an iterative analysis process was implemented to determine the final telescope structure geometry.  The results from dynamic survivability, modal, and thermal analyses were considered for each element.  Part of the FEM definition involves application of material properties that also allow for the consideration of mass as a secondary factor in selection of the final geometry.  Resulting optical performance effects are then analyzed in the Zemax OpticStudio STAR module for each iteration by exporting the three-dimensional nodal parameters from the FEA results. Other geometry selections made during the design process were more straightforward than the flexural interface between the OTA and the hex plate.  For instance, given the axisymmetric behavior for all analyses, the tube geometry selections were made without direct optical analyses as the major performance effect would be in focus change due to varying thermal conditions.  For this, common engineering knowledge can be applied to bias the focus alignment for all optical elements prior to integration with an instrument for the given expected on-orbit operational conditions (for this project +\/-1\u00b0C).  Similarly, the hex plate design only needed to preserve the axisymmetric geometry approximation when considering its overall shape and interface definition.  \nWhile not discussed in detail earlier, mass of each components was considered as an important mechanical design decision.  An overall maximum allowable mass budget of 2 kg was allotted to the design and did not present a problem during the iterative analyses.  The final telescope design mass is expected to be less than 1 kg once all hardware is integrated.  The only major design decision based on mass reduction involved light-weighting the primary mirror blank by cutting pockets into the structure and analyzing these pockets for their structural effect on mirror surface figure.  \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{3dprintlw.PNG}\n    \\caption{3D Prints of the primary mirror (left) and hex plate (right) demonstrating the lightweighting scheme used to achieve the mass requirement of $< 2kg$}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\section{STOP EVALUATION - MERGING FEA WITH SEQUENTIAL RAYTRACING}\n\nNumerous loading conditions were studied through FEA software to ensure the overall survival of the VCT payload, however, FEA analysis by itself could not determine how the deformed elements would impact the system's optical performance. Further studies of the wavefront deformation were necessary to confirm that the system would be able to function properly in its environment of operation. Typically, this kind of analysis would require the deformed surface to be decomposed into hundreds of polynomial terms to accurately model the shape of the surface, with no guarantee of the raytracing software being able to support the number of terms needed. However, through use of OpticStudio's STOP (Structural, Thermal, and Optical Performance) analysis feature \\emph{STAR} (Structural, Thermal, Analysis and Results) evaluation of the system's performance in various loading scenarios was made quick and efficient. STAR allows for the data from FEA software to be loaded directly into OpticStudio and onto select optical elements. By using the position and displacement of each node in a 6 column (x, y, z, dx, dy, dz) format STAR accurately deforms the surface shape, allowing for the deformed wavefront to be observed under various loading conditions. For the purposes of our design study we primarily looked at the surface deformation on the primary mirror caused by rotation about the optical axis.\n\nIn figure \\ref{fig:star} the change of the optical system's wavefront and PSF are shown after applying a -237 arcsecond rotation about the optical axis. This causes clear degradation in the system's overall optical performance. This optical analysis is critical for an iterative design process between optical and mechanical design teams. Quick optical performance analysis through STAR allows for the mechanical team to adjust the structures supporting the optical surfaces where necessary, ensuring performance specifications are met in realistic structural and thermal scenarios. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{starinv.png}\n    \\caption{Influence of STAR on the wavefront (left) and the PSF (right). A force tangent to the primary mirror was applied in FEA and the resultant surface deformation was loaded into the OpticStudio raytrace model. This permits evaluation of performance degradation as a function of structural and thermal deformation.}\n    \\label{fig:star}\n\\end{figure}\n\n\n\\section{Laboratory Prototype Status}\n\nThe next step for the development of the VCT is the construction of a laboratory prototype of the on-axis aluminum VCT. Mirrors for the on-axis VCT have been fabricated by Hanbat national university and are shown in figure \\ref{fig:fab_mirrors}. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{mirror_img.PNG}\n    \\caption{The fabricated (left) primary and (right) secondary of the on-axis VCT design manufactured by Hanbat National University in South Korea.}\n    \\label{fig:fab_mirrors}\n\\end{figure}\n\nThe primary and secondary mirror have arrived at the University of Arizona and are awaiting measurement and assembly so that they can be mounted into the OTA described in section 5 for system-level tests. The UArizona Space Astrophysics lab test facilities include a TVAC chamber and Hexapod that can be used for thermal vaccum and jitter tests respectively. Details of the comprehensive characterization and assembly of the Versatile Cubesat Telescope prototype will be published in the future.\n\n\\section{Conclusion}\n\nWe present diffraction-limited on- and off-axis designs for a Versatile CubeSat Telescope that fits within a 2U volume at low cost. Comprehensive analysis of the sensitivity to misalignment and polarization were considered, and a stable mechanical housing was created for the on-axis VCT. The VCT is a well-characterized high-performance design that can be adapted to a variety of space-borne research payloads. Future iterations of the VCT will expand upon the development of MEMS devices in space (e.g. DeMi\\cite{Morgan21}) by replacing the FSM with a Deformable Mirror for a higher degree of active wavefront control. Closed loop thermal simulations will be conducted to demonstrate the behavior of the VCT in response to a dynamic thermal environment.\n\n\\section{Acknowledgements}\nWe thank Zemax for the early access to their STAR analysis feature. This research made use of community-developed core Python packages, including: Numpy\\cite{numpy}, Matplotlib \\cite{matplotlib}, SciPy \\cite{jones_scipy_2001}, Astropy \\cite{the_astropy_collaboration_astropy_2013}, and Jupyter, IPython Interactive Computing architecture \\cite{perez_ipython_2007,kluyver_jupyter_2016}. Portions of this work were supported by the Arizona Board of Regents Technology Research Initiative Fund (TRIF).\n\n\n\\section{INTRODUCTION}\n\nCubeSat space telescopes are a growing asset in astronomical research\\cite{Shkolnik_2018,douglas2019cubesats}. Their standardized format and compact volume enables the rapid development of high-impact mission concepts above the earth's atmosphere. Flight-tested commercial microelectromechanical systems (MEMS)  have further enhanced CubeSat telescopes by providing dynamic pointing and aberration control on-orbit (DeMi, NODE) \\cite{Yenchesky2019OptomechanicalDA,Clements2016}. The expanding capacity of CubeSats for astronomical research invites the development of novel technologies for their optical systems. However, the resolution in CubeSat optics is fundamentally limited by the clear aperture permitted by the CubeSat volume. The rayleigh criterion tells us that resolution ($\\Delta \\theta$) is inversely related to to the entrance pupil diameter ($D$) of the optical system.\n\n\\begin{equation}\n    \\Delta \\theta = 1.22 \\frac{\\lambda}{D}\n\\end{equation}\n\nReflective CubeSat objectives traditionally require mounting hardware that limits the available entrance pupil diameter. Single-point diamond turned (SPDT) mirrors can mitigate this limitation by enabling the manufacturer to machine the mounting hardware directly into the rear of the mirror substrate, eliminating the need for mounting hardware around the edge of the mirror and enhancing the nominal throughput and resolution. This also grants the objective a considerable degree of athermalization by eliminating the difference in the coefficient of thermal expansion between the mirror and mounting hardware\\cite{Zhang:17}. SPDT surfaces are typically used in longer wavelengths (MWIR, LWIR) due to the midspatial frequency errors left by tooling marks. However, recent developments in optical polishing\\cite{Jeon:17} have introduced Magnetorheological Finishing (MRF) to the fabrication of SPDT surfaces, reducing the dominant midspatial frequencies considerably and therefore extending their use into the optical. Athermal CubeSat objectives composed of low surface roughness mirrors with a large entrance pupil invites the design of high-performance optical CubeSat payloads to make the next generation of research and technology development in space more accessible. The Versatile CubeSat Telescope (VCT) concept is a prototypical fore-optic that boasts compatibility with a variety of research payloads. The goal of the VCT is to develop a flexible, large aperture telescope that can be replicated at low cost for easy adaptability to future research payloads and technology demonstrations. \n\n\\section{OPTICAL DESIGN - ON-AXIS V.S. OFF-AXIS IMPLEMENTATIONS}\nThe design of the VCT began with two realizations of a CubeSat telescope that could take advantage of a large primary mirror. The first, an on-axis telescope with a more classical Ritchey-Chretien objective outfitted with a plano-convex aspheric collimator. The second, an all-reflective off-axis Ritchey-Chretien solution with a freeform collimator. We baselined a 95mm entrance pupil diameter with a 20$\\%$ obscuration for the on-axis telescope, and scaled the entrance pupil diameter of the off-axis telescope to be equivalent in collecting area. A high pupil magnification is required for both designs to image the primary mirror onto the small MEMS FSM clear aperture (5mm). The specifications and system layouts are shown in table \\ref{tab:specs} and figure \\ref{fig:layout} respectively.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c}\n        \\hline\n        Specification & Value  \\\\\n        \\hline\n        Entrance Pupil Diameter & \\textbf{95 mm} \\\\\n        Exit Pupil Diameter & 5 mm \\\\\n        Half Field of View & 0.2$^{\\circ}$ \\\\ \n        Secondary Obscuration & \\textbf{20$\\%$} \\\\\n        Central Field Strehl & $>$0.99 \\\\\n        Average Field Strehl & $>$ 0.80 \\\\\n        OTA Packaging Volume & $<$ 2U \\\\\n        \\hline\n        \\\\\n    \\end{tabular}\n    \\qquad\n    \\begin{tabular}{c c}\n        \\hline\n        Specification & Value  \\\\\n        \\hline\n        Entrance Pupil Diameter & \\textbf{88 mm} \\\\\n        Exit Pupil Diameter & 5 mm \\\\\n        Half Fiel of View & 0.2$^{\\circ}$ \\\\ \n        Secondary Obscuration & \\textbf{0$\\%$} \\\\\n        Central Field Strehl & $>$0.99 \\\\\n        Average Field Strehl & $>$ 0.80 \\\\\n        OTA Packaging Volume & $<$ 2U \\\\\n        \\hline\n        \\\\\n    \\end{tabular}\n    \\caption{Optical design specifications for the (left) On-Axis and (right) Off-Axis telescope designs. All specifications are similar except for those in bold, which are set such that the apertures of each design are equal in throughput.}\n    \\label{tab:specs}\n\\end{table}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{vct_drawings.png}\n    \\caption{System layouts of the (left) On-Axis Telescope and the (right) Off-Axis Telescope designs. Each has three optical elements, two fast-steering mirrors (FSM) for fine pointing control, and a volume dedicated to a research payload.}\n    \\label{fig:layout}\n\\end{figure}\n\n Aspheric surfaces were necessary for both telescopes to achieve a high pupil magnification (19x and 17.5x for the on- and off-axis designs respectively) while maintaining a well-corrected field of view. While this slightly complicates the metrology of the mirrors it did not add any additional cost because the mirrors were already going to be diamond-turned. Both telescopes utilized a Ritchey-Chretien objective that forms an image near the primary mirror, which is collimated by an aspheric element to produce an external exit pupil. The size of the exit pupil was chosen to match the clear aperture of a MEMS fast-steering mirror (FSM)\\cite{memsfsm,Serra21} to correct for spacecraft pointing and jitter errors while in orbit. The external pupil also serves as a convenient interface for any instrument suite that would be included on the VCT (e.g. spectrographs, cameras, lasers).  The concept for the on-axis design was to make a well-established telescope format work given the pupil magnification and field of view specifications. The aspheric surfaces granted the design a well-corrected field of view in a compact format suitable for a CubeSat (2U). However, the design has a secondary obscuration which limits the throughput of the system and adds diffraction features to the image. The primary goal for the off-axis design was to offer a viable alternative to the classical on-axis design that would not suffer from undesirable diffraction effects brought upon by the secondary obscuration. Unfortuantely, shifting the primary parabolic mirror off-axis comes at the cost of more field-dependent aberration that limits the system's field of view. To accommodate this we selected the tertiary mirror of the off-axis design to be freeform. Before optimization of the freeform surface, the rotational symmetry of the system was broken by tilting the secondary and tertiary mirrors about their foci to make the system free of linear astigmatism. A linear-astigmatism free three mirror system (LAF-TMS) has been shown by Park et al\\cite{park_development_2020} to be an excellent starting point for freeform designs by using the mirror tilt angles and inter-mirror distances (Eq \\ref{eq:laf}) to mitigate a dominating aberration early in the design process.\n\n\\begin{equation}\n    \\frac{l_2^\\prime}{l_2}\\frac{l_3^\\prime}{l_3}tan{i_1}+\\left(1+\\frac{l_2^\\prime}{l_2}\\right)\\frac{l_3^\\prime}{l_3}tan{i_2+}\\left(1+\\frac{l_3^\\prime}{l_3}\\right)tan{i_3=0}\n    \\label{eq:laf}\n\\end{equation}\n\nIn this equation, parameters i$_{1,2,3}$ are the angles of incidence of the optical axis ray on surfaces 1,2, and 3, while $l_{2,3}$ and ${l^{\\prime}}_{2,3}$ are the front and rear focal lengths the secondary and tertiary mirror. Once the baseline LAF-TMS design had been implemented, the third mirror was changed to a freeform XY-Polynomial surface where coefficients were optimized that maintained the bilateral symmetry of the optical system. The mirrors were shifted off-axis in the $\\hat{y}$ direction, as was the field bias (+0.2$^{o}$). Consequently the freeform surface was constrained to be plane-symmetric about the $\\hat{y}-\\hat{z}$ plane by solely optimizing coefficients of even order in $\\hat{x}$. The resultant design is nearly diffraction-limited across the biased field of view.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{strhvsfov.png}\n    \\caption{Maps of strehl ratio v.s. field of view for the (left) On-Axis and (right) Off-Axis VCT. The symmetry of the optical system mirrors the performance quite readily, with the on-axis design having a rotationally symmetric field performance whereas the off-axis design exhibits plane-symmetric performance.}\n    \\label{fig:strvfov}\n\\end{figure}\n\n\\section{SENSITIVITY ANALYSIS}\n \nThe misalignment sensitivities of the two designs were explored in Zemax OpticStudio (ZOS) by iteratively perturbing each optic in six degrees of freedom using the Python ZOS-API. Our as-built performance goal for both systems was an average Strehl ratio $>$ 0.7 across the field of view to maintain reasonably diffraction-limited performance. The $0\\%$, $70\\%$, and $100\\%$ field of view in $\\pm \\hat{x}$ and $\\pm \\hat{y}$ were considered in the average calculation, considerably biasing the sensitivity analysis toward the edge of the field of view where performance is expected to degrade faster. This results in a more pessimistic analysis of misalignment sensitivity. The sensitivity analysis fixes the primary mirror while M2 and L3\/M3 were perturbed to better understand how misalignments with respect to the primary would impact the optical performance over all fields of view.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{qkdsens.png}\n    \\caption{Sensitivity to misalignment curves of the On-Axis (Solid Red) and Off-Axis (Dashed Blue) designs for their respective secondary (top) and tertiary (bottom) optic. The on-axis system is less sensitive for X\/Y decenter of both elements, X\/Y\/Z tilt of M2, and X\/Z tilt of M3.}\n    \\label{fig:my_label}\n\\end{figure}\n\nFigure 2 shows that in most cases the on-axis design was less sensitive to misalignment in six degrees of freedom. This result is consistent with the complexity of each system. The off-axis design employs off-axis conics and a freeform surface, both of which will be sensitive to misalignment due to the higher mirror slopes than the on-axis design. In applications where the secondary obscuration is tolerable, the on-axis design would be a better choice for applications where cost and misalignment risk are limiting factors.\n\n\\section{POLARIZATION ANALYSIS}\nMany spaceborne optical payloads have a degree of sensitivity to polarization effects. From laser experiments in support of quantum communications\\cite{Serra21} to polarimetry of atmospheric ice\\cite{Hart20}, there is a clear need to characterize the response of cubesat payloads to a vector electric field. To offer support for polarization-sensitive research payloads a comprehensive polarization ray trace (PRT) analysis was conducted for each realization of the VCT.  PRT is a ray-based approach to vector field calculations enabled by determining the local fresnel reflection coefficients that alter the transmission and phase of the electric field at each surface in the optical system. These effects are traced from the entrance pupil to the exit pupil of the optical system to determine the cumulative polarization behavior of the instrument. The proposed VCT designs use fast primary mirrors to achieve the high pupil magnification, so some polarization aberration is expected. However mirror substrates and coatings can be selected to mitigate the impact of polarization aberration for a given application. \n\nA PRT tool was written using the ZOS-API to create maps of diattenuation and retardance at the exit pupil of the system. The ZOS-API tool creates a batch raytrace and propagates it from the entrance pupil of the optical system to the surface under investigation. The output of the raytrace gives surface normal vectors in addition to exit ray vectors which are used to calculate the incident ray vectors and angle of incidence. Diattenuation ($D$) and retardance ($\\delta$) data is produced from the Zemax coating files and ray angles of incidence to generate the surface and pupil maps shown in figure \\ref{fig:polmaps_al}.\n\n\\begin{equation}\n    D = \\frac{|r_p|^{2} - |r_p|^{2}}{|r_s|^{2} + |r_p|^{2}};\\phantom{flapjackfacts}\\delta = |\\phi_{p} - \\phi_{s}|\n    \\label{eq:diat}\n\\end{equation}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{polplots.png}\n    \\caption{Polarization pupil maps assuming bare aluminum mirrors and uncoated N-BK7 lens surfaces. (Top) Diattenuation and (Bottom) retardance maps of the (left) On-Axis and (right) Off-axis versatile cubesat telescopes. These data were computed at $\\lambda$ = 780nm. At this wavelength the Off-Axis Aluminum VCT sees the greatest diattenuation, and the Off-Axis Gold sees the greatest retardance. This indicates that the off-axis design would introduce more challenges to polarization-sensitive payloads than the on-axis design would.}\n    \\label{fig:polmaps_al}\n\\end{figure}\n\nThe polarization pupil map permits analysis of the optimal material and configuration for a given polarization-dependent application. Figure \\ref{fig:polmaps_al} demonstrates that aluminum is more diattenuating than gold at $\\lambda=780nm$, and the off-axis configuration performs worse than the on-axis configuration due to the higher angles of incidence on the mirror surfaces. \n\n\\section{MECHANICAL DESIGN}\n\nThe VCT must be designed to deliver a high optical wavefront when operational on-orbit, yet also robust enough to survive the rigors of launch and orbital injection separation.  Achieving both goals in a compact package that affords straight-forward assembly and alignment is quite challenging.  Another important aspect of the design must also consider the mass contribution of the structural elements as this can be costly when considering launch requirements and on-orbit maneuvering.  With the prevalence of small satellite design ( $<$ 12U), development of a telescope that fits within a 2U volume with a mass less than 2 kg would present a low-cost solution that enables rapid prototyping of research payloads. For a prototype realization of the VCT, the optical system design called for SPDT Aluminum primary and secondary mirrors. All mechanical structural elements were also chosen to be aluminum to match the thermal behavior of the mounting structure to the optical elements. The collimating lens, which locates the instrument pupil, could also be easily mounted to the OTA using a common aluminum barrel structure with simple shims for precise alignment. \n\nThe first goal in the mechanical design of the telescope Optical Tube Assembly (OTA) is to support the optical elements with minimal effect due to expected thermal perturbations that would be encountered on-orbit.  These effects would include both rigid body misalignment and wavefront aberrations caused by subtle flexing of the optical elements.  In order to bound the problem, the first design decisions centered on overall sizing to fit within the small-sat form factor. The primary-secondary despace tolerance is very tight, so a truss structure was designed to properly space the two mirrors. The truss structure presents a good strength to weight ratio and is commonly used in ground-based telescope designs. For this space-based application, the truss structure would not allow for simple and easy shrouding of the optical path and thus would likely require additional structural pieces to support stray light baffling elements and thermal mitigation elements (i.e., mylar blankets).  Therefore, an early design decision was made to investigate a tube structure that would serve as both the metering structure that could maintain the mirror separation tolerance, transverse misalignment tolerance, and provide an in-place baffle that could be customized for the particular optical application.  A significant benefit of the tube design is the mass savings realized by the ability to select a very thin tube wall thickness that still retains the desired structural performance for all loading cases.  This tube would also allow for easy installation of thermal mitigation measures and targeted design of thermal conduction paths that further ensure optical performance stability while on-orbit.\n\nThe next major goal in the mechanical design considers the manufacturability of the system.  Considering its small size, the OTA manufacturability includes the fabrication processes for all structural components, as well as thoughtfully planned assembly and alignment processes that ensure the optical performance requirements can be met.  The aluminum mirrors provide a unique opportunity to meet the need for an assembly that is easy to integrate and align.  The primary mirror optical surface is thus applied directly to the structural component (primary mirror) which has appropriate design elements that consider thermal performance and mounting points for both the metering tube and the interface to the downstream optical system via a \\emph{hex plate} interface.  Clearly defining the rear of the primary mirror blank as the OTA support location allows for simple design of flexural components that will not propagate errors from other systems into the OTA, or vice-versa.  Similarly, the secondary mirror surface is applied directly to a head-ring structure that also considers thermal performance and mounting points to the metering tube, but also minimizes the size of support strut spiders that are common in the on-axis  design.  Considering the small size of the OTA, the one-piece secondary mirror\/head-ring simplifies its alignment features by moving them out to the tube diameter, rather than trying to squeeze them into the shadow of the secondary mirror on the entrance aperture.\n\nThe final goal of the mechanical design considers the interface of the OTA to the instrument and the spacecraft.  Most terrestrial optical system applications utilize a breadboard style support that allows for many mounting possibilities for both the OTA and any downstream instrument opto-mechanical elements, and can be purposefully designed to incorporate spacecraft mounting options that will isolate the payload from the spacecraft both thermally and dynamically as necessary.  The hex plate provides all of these interfaces in a single machined aluminum component.  The connection of the OTA to the hex plate is of particular importance for this discussion.  As mentioned earlier, the OTA must remain dynamically and thermally isolated from the other spacecraft or instrument components to preserve the telescope optical performance. A Finite Element Model (FEM) was created using the aluminum tube structure OTA, with both structural and optical components represented, and a Finite Element Analysis (FEA) was performed to determine the model behavior with both structurally dynamic boundary conditions and also varying thermal environmental conditions.  The FEM is shown in figure \\ref{fig:vct_fea}.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{vct_fea.png}\n    \\caption{The FEM of the VCT OTA highlighting the hex plate, flexure, and tube design.}\n    \\label{fig:vct_fea}\n\\end{figure}\n\nA simple three flexure design was selected to mount the OTA to the hex plate.  The flexures were positioned tangentially to the primary mirror blank to better accommodate thermal deformations that could arise at the hex plate. The flexure material properties were varied to determine the best trade of optical path stability, launch survivability, manufacturability, and cost. The materials considered were Aluminum (Al 6061-T6) and Titanium (Ti-6Al-4v).  For the structural analysis the hex plate was fixed at three points on its outer edge simulating spacecraft support, and MAC loads (100g accelerations in three orthogonal directions) were applied to the model.  Various flexure throat cross sectional areas were considered, and the best design was chosen based on the resulting maximum stresses revealed in the FEA.  The results for each material selection under this loading are shown in table \\ref{tab:macload}. \n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n     Load Case Direction & Max Flexure Stress [Al] & Max Flexure Stress [Ti] \\\\\n    \\hline\n        100g $\\hat{x}$ & 103ksi & 112ksi \\\\\n        100g $\\hat{y}$ & 98ksi & 103ksi \\\\ \n        100g $\\hat{z}$ & 42ksi & 46ksi \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{MAC loading in three orthogonal directions and the corresponding maximum flexure stress for Aluminum and Titanium flexures. The Ti flexures were chosen due to their higher maximum stress.}\n    \\label{tab:macload}\n\\end{table}\n\nAn additional analytical consideration for the structure is the modal behavior of the OTA.  The modal FEA reveals the lowest expected resonant frequency of the OTA under the prescribed mounting conditions for given material selections and flexure design choices.  The table below shows the first three modal FEA results for the same conditions discussed above.  These results indicate there would be little chance of exciting resonance in the OTA structure when considering expected launch loads.  And similarly, the telescope itself would not likely impart damaging resonance to any other spacecraft systems or partner satellites if launched as a secondary payload ride-share.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n        Al Flexures & Ti Flexures  \\\\\n    \\hline\n        130Hz & 156Hz \\\\\n        130Hz & 156Hz \\\\\n        363Hz & 434Hz \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{Modal behavior of the OTA for Al and Ti flexures for the cases described in \\ref{tab:macload}. Based on the results from the FEA for both materials, the Titanium flexure design was selected based on the extra margin provided on mass and stress.}\n    \\label{tab:my_label}\n\\end{table}\n\n Verification  of the selected flexure design needed to also consider the thermal behavior of the system.  For ease of creating the thermal FEM, the boundary conditions for the structure were preserved from the structural FEA and a 27.8\u00b0C differential in temperature was applied to the hex plate.  This set of conditions would bring into focus the thermal isolation of the OTA from the hex plate.  The results from this thermal analysis were normalized to show an expected mirror sag of 10nm P-V for each 1.0\u00b0C of temperature differential.  The results (shown below) also indicate an axisymmetric deformation of the OTA as would be expected.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{fea_results.PNG}\n    \\caption{Finite Element results of the hexplate-primary mirror temperature differential. The flexure design results in $\\approx$ 10nm of trefoil surface sag per degree Celsius.}\n    \\label{fig:fea_results}\n\\end{figure}\n\nOverall, an iterative analysis process was implemented to determine the final telescope structure geometry.  The results from dynamic survivability, modal, and thermal analyses were considered for each element.  Part of the FEM definition involves application of material properties that also allow for the consideration of mass as a secondary factor in selection of the final geometry.  Resulting optical performance effects are then analyzed in the Zemax OpticStudio STAR module for each iteration by exporting the three-dimensional nodal parameters from the FEA results. Other geometry selections made during the design process were more straightforward than the flexural interface between the OTA and the hex plate.  For instance, given the axisymmetric behavior for all analyses, the tube geometry selections were made without direct optical analyses as the major performance effect would be in focus change due to varying thermal conditions.  For this, common engineering knowledge can be applied to bias the focus alignment for all optical elements prior to integration with an instrument for the given expected on-orbit operational conditions (for this project +\/-1\u00b0C).  Similarly, the hex plate design only needed to preserve the axisymmetric geometry approximation when considering its overall shape and interface definition.  \nWhile not discussed in detail earlier, mass of each components was considered as an important mechanical design decision.  An overall maximum allowable mass budget of 2 kg was allotted to the design and did not present a problem during the iterative analyses.  The final telescope design mass is expected to be less than 1 kg once all hardware is integrated.  The only major design decision based on mass reduction involved light-weighting the primary mirror blank by cutting pockets into the structure and analyzing these pockets for their structural effect on mirror surface figure.  \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{3dprintlw.PNG}\n    \\caption{3D Prints of the primary mirror (left) and hex plate (right) demonstrating the lightweighting scheme used to achieve the mass requirement of $< 2kg$}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\section{STOP EVALUATION - MERGING FEA WITH SEQUENTIAL RAYTRACING}\n\nNumerous loading conditions were studied through FEA software to ensure the overall survival of the VCT payload, however, FEA analysis by itself could not determine how the deformed elements would impact the system's optical performance. Further studies of the wavefront deformation were necessary to confirm that the system would be able to function properly in its environment of operation. Typically, this kind of analysis would require the deformed surface to be decomposed into hundreds of polynomial terms to accurately model the shape of the surface, with no guarantee of the raytracing software being able to support the number of terms needed. However, through use of OpticStudio's STOP (Structural, Thermal, and Optical Performance) analysis feature \\emph{STAR} (Structural, Thermal, Analysis and Results) evaluation of the system's performance in various loading scenarios was made quick and efficient. STAR allows for the data from FEA software to be loaded directly into OpticStudio and onto select optical elements. By using the position and displacement of each node in a 6 column (x, y, z, dx, dy, dz) format STAR accurately deforms the surface shape, allowing for the deformed wavefront to be observed under various loading conditions. For the purposes of our design study we primarily looked at the surface deformation on the primary mirror caused by rotation about the optical axis.\n\nIn figure \\ref{fig:star} the change of the optical system's wavefront and PSF are shown after applying a -237 arcsecond rotation about the optical axis. This causes clear degradation in the system's overall optical performance. This optical analysis is critical for an iterative design process between optical and mechanical design teams. Quick optical performance analysis through STAR allows for the mechanical team to adjust the structures supporting the optical surfaces where necessary, ensuring performance specifications are met in realistic structural and thermal scenarios. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{starinv.png}\n    \\caption{Influence of STAR on the wavefront (left) and the PSF (right). A force tangent to the primary mirror was applied in FEA and the resultant surface deformation was loaded into the OpticStudio raytrace model. This permits evaluation of performance degradation as a function of structural and thermal deformation.}\n    \\label{fig:star}\n\\end{figure}\n\n\n\\section{Laboratory Prototype Status}\n\nThe next step for the development of the VCT is the construction of a laboratory prototype of the on-axis aluminum VCT. Mirrors for the on-axis VCT have been fabricated by Hanbat national university and are shown in figure \\ref{fig:fab_mirrors}. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{mirror_img.PNG}\n    \\caption{The fabricated (left) primary and (right) secondary of the on-axis VCT design manufactured by Hanbat National University in South Korea.}\n    \\label{fig:fab_mirrors}\n\\end{figure}\n\nThe primary and secondary mirror have arrived at the University of Arizona and are awaiting measurement and assembly so that they can be mounted into the OTA described in section 5 for system-level tests. The UArizona Space Astrophysics lab test facilities include a TVAC chamber and Hexapod that can be used for thermal vaccum and jitter tests respectively. Details of the comprehensive characterization and assembly of the Versatile Cubesat Telescope prototype will be published in the future.\n\n\\section{Conclusion}\n\nWe present diffraction-limited on- and off-axis designs for a Versatile CubeSat Telescope that fits within a 2U volume at low cost. Comprehensive analysis of the sensitivity to misalignment and polarization were considered, and a stable mechanical housing was created for the on-axis VCT. The VCT is a well-characterized high-performance design that can be adapted to a variety of space-borne research payloads. Future iterations of the VCT will expand upon the development of MEMS devices in space (e.g. DeMi\\cite{Morgan21}) by replacing the FSM with a Deformable Mirror for a higher degree of active wavefront control. Closed loop thermal simulations will be conducted to demonstrate the behavior of the VCT in response to a dynamic thermal environment.\n\n\\section{Acknowledgements}\nWe thank Zemax for the early access to their STAR analysis feature. This research made use of community-developed core Python packages, including: Numpy\\cite{numpy}, Matplotlib \\cite{matplotlib}, SciPy \\cite{jones_scipy_2001}, Astropy \\cite{the_astropy_collaboration_astropy_2013}, and Jupyter, IPython Interactive Computing architecture \\cite{perez_ipython_2007,kluyver_jupyter_2016}. Portions of this work were supported by the Arizona Board of Regents Technology Research Initiative Fund (TRIF).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nCardiac magnetic resonance imaging is a valuable tool for myocardial structure, function, and tissue assessment, providing essential information for clinical diagnosis and treatment decisions in cardiovascular disease. Using standard segmented sequences in which data acquisition is segmented over multiple heart beats, good image quality can be obtained in patients with regular cardiac rhythm and good breath-holding ability; however, image quality can be degraded by motion artifacts when scanning patients with arrhythmia or poor breath-hold compliance. In comparison to segmented acquisitions, single-shot techniques can be applied for rapid image acquisition of a whole slice within a single shot, greatly reducing the scan time. Due to the short acquisition duration of single-shot techniques (typically < 200 ms), artifacts from intra-shot motion are negligible, therefore such methods tend to be robust against cardiac and breathing motion. However, this motion robustness comes at the expense of lower spatial resolution and signal to noise ratio (SNR). An example of the benefit of single-shot over segmented Late Gadolinium Enhanced (LGE) imaging in a patient who could not breath-hold is shown in \\ref{fig:fig1}. Recent techniques proposed to enhance the SNR of single-shot methods by motion correcting and then averaging multiple single-shot images acquired in free-breathing \\cite{Kellman2005}. While this technique shows good results with low acceleration factors, it may not provide optimal image quality for higher undersampling, introducing blurring and undersampling artifacts, mainly due to higher weight given to the regularization.  Batchelor et al. \\cite{Batchelor2005} proposed a first generalized reconstruction framework for motion compensation. The method allows arbitrary motion to be compensated by solving a general matrix inversion problem. This technique, however, requires an adequate knowledge of the displacement fields. The recent GRICS method \\cite{Odille2008a} extended this work by jointly estimating the motion and the recovered image, however, it relied on a motion model provided by external sensors (e.g. ECG, respiratory belt).\n\nIn this work, we sought to develop an efficient motion correction implementation suitable for reconstructing a high-resolution, high-SNR image from multiple accelerated single-shot images. The proposed method combines the benefits of using a hybrid self-navigated sampling scheme (see Fig. \\ref{fig:fig1}) with a joint reconstruction framework. In the image reconstruction step, a highly efficient feature-preserving regularization scheme (Beltrami) is proposed for recovering sharp details. We show that the proposed method is robust to high acceleration factors and yields results with efficient noise reduction and better overall image quality at a low computational cost.\n\n\n\\section{Theory}\n\n\\subsection{General Motion Compensation Framework}\n\nMotion compensation techniques aim to solve the following inverse problem: find an underlying image $\\rho$ free of motion artifacts, given derived measurements $s$ through the system $E$, affected by noise $\\nu: s = E\\rho + \\nu$. Where $E$ is the encoding matrix, generally composed of a sampling operator $\\xi$, a Fourier transform $F$, coil sensitivity maps $\\sigma$ (in case of parallel imaging reconstruction), and a motion warping operator $W$ describing a non-rigid deformation for each shot. Here $\\rho \\in \\mathbb{C}^{n_x \\times n_y}$ and $s \\in \\mathbb{C}^{n_x \\times n_y\/acc \\times n_r \\times n_c}$  are complex data with $n_c$ the number of receiver coils, $n_r$ the number of repetitions and $acc$ the acceleration factor. In this work, the acquired data s represents the k-space data from multiple single-shot images and is generally corrupted by noise. A typical approach to solve this problem is to minimize the squared difference as assessed by the Euclidean norm. However, this problem is generally ill-posed (e.g. due to undersampling and noise), leading to non-uniqueness of the solution, if it exists. Thus, regularity constraints on the unknown solution $\\rho$ have to be considered. Furthermore, motion should also be considered as unknown. The general joint optimization framework is then defined as\n\n\\begin{equation}\n\\label{eq:eq1}\n\\rho = \\argmin\\limits_{\\left( \\rho, \\vec{u}\\right)} \\lbrace \\Vert s - E\\left(\\vec{u}\\right) \\rho \\Vert_2^2 + \\lambda \\phi \\left( \\rho \\right) \\rbrace \\text{ where } E\\left( \\vec{u}\\right) = \\xi F \\sigma W \\left( \\vec{u}\\right)\n\\end{equation}\n\nHere $\\vec{u}$ represents the displacement fields, $\\phi$ is the chosen regularization function and $\\lambda > 0$ is the corresponding regularization parameter. The optimization problem in Equation \\ref{eq:eq1} is solved in four steps: (i) we first use the k-space center of the single-shot images to extract a self-navigation signal and to cluster the raw data into a reduced number of respiratory bins \\cite{Usman2013}; (ii) we reconstruct the images from each bin independently using a Beltrami-regularized SENSE (B-SENSE) reconstruction; (iii) then an estimate of the motion is obtained using a non-rigid registration (minimization of Equation \\ref{eq:eq1} with regards to $\\vec{u}$ \\cite{Odille2014}) and (iv) a high resolution\/SNR image is generated using the proposed motion-compensated reconstruction process (minimization with regards to $\\rho$). A general description of the method is shown in Fig. \\ref{fig:fig2}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.91\\textwidth]{Fig1.png}\n\\caption{(Left) Comparison between 2D segmented LGE (top) and 2D single-shot LGE (bottom) on a 77-year-old patient with breath-holding difficulties. The segmented LGE has higher resolution than the single-shot LGE but shows severe motion artefacts. (Right) Proposed hybrid k-space acquisition scheme including motion-calibration data (fully sampled center) and undersampled periphery, aimed to combine the resolution of segmented LGE with the motion-robustness of single-shot LGE.}\n\\label{fig:fig1}\n\\end{figure}\n\n\n\\subsection{Beltrami-Regularized SENSE}\n\nIn our framework, a respiratory signal is extracted from the motion calibration data itself. This pre-processing step is achieved by stacking the low resolution images along the time dimension. Singular value decomposition is then applied to the stack and the first left-singular vector is used as a good approximation of the true respiratory signal. A specific respiratory phase is then assigned to each acquired shot, as explained in \\cite{Usman2013}. This binning strategy would split the data into fewer motion states $n_b \\left( < n_r\\right)$ with negligible respiration motion and lower undersampling in each of them. Images from each respiratory bin $\\left( \\rho_i \\right)_{i = 1,\\dots,n_b}$ are then individually reconstructed by solving\n\n\\begin{equation}\n\\label{eq:eq2}\n\\rho_i = \\argmin\\limits_{\\rho} \\lbrace \\Vert s_i - E_i \\rho \\Vert_2^2 + \\lambda \\sqrt{1+\\beta^2 \\vert \\nabla \\rho \\vert^2} \\rbrace\n\\end{equation}\n\nThe first term in Equation \\ref{eq:eq2} is a data fidelity term that aims to minimize the difference between the reconstructed image and the acquired data. The Beltrami regularization $\\sqrt{1+\\beta^2 \\vert \\nabla \\rho \\vert^2}$ has been introduced in the field of string theory for physics and has shown high potential in several imaging problems, including image denoising and enhancement \\cite{Polyakov1981} and super-resolution reconstruction \\cite{Odille2015}. In particular, the metric can be chosen such that the Beltrami energy corresponds to an arbitrary interpolation between Gaussian diffusion $\\beta \\to 0$ and total variation (TV) \\cite{Rudin1992a} regularization $\\beta \\gg 1$. In \\cite{Zosso2014}, the authors showed that Beltrami regularization is able to maintain the advantage of TV (edges preserving, noise reduction) as well as reducing the effect of staircasing. B-SENSE is very similar to compressed sensing SENSE (CS-SENSE) methods presented by other authors \\cite{Liang2008}, where here Beltrami is making the image sparse in the gradient domain. Even though this suggests that B-SENSE has a close relationship with the compressed sensing (CS) theory, it is, however, not CS as defined by Cand\\`es et al. \\cite{Candes2006}, especially due to the pseudo-random undersampling pattern used here (i.e. a uniform random pattern is used in \\cite{Candes2006}). We propose to solve Equation \\ref{eq:eq2} by adopting a primal-dual projected gradient approach \\cite{Chan1999} with the potential to converge faster than the classic primal gradient-descent \\cite{Zosso2014}. Respiratory motion estimation is then accomplished using independent non-rigid registration of the images reconstructed from each respiratory bin. Here we use an iterative framework validated in a large patient database for myocardial $T_2$ mapping \\cite{Odille2014}, which is based on minimizing the sum-of-squared differences of the pixel intensities within a multi-resolution Gauss-Newton scheme.\n\n\n\\subsection{Motion Compensated Reconstruction with Preserved-Features}\n\nThis section presents the final step for solving the motion compensated problem in Equation \\ref{eq:eq1}. The aim of the method is to reconstruct the high resolution, high SNR image $\\rho$ from the acquired raw data $s = \\left( s_i \\right)_{i = 1,\\dots,n_b}$. For the motion compensated reconstruction, we solve the following optimization problem, with the acquisition model now including the estimated motion fields:\n\n\\begin{equation}\n\\label{eq:eq3}\n\\rho = \\argmin\\limits_{\\rho} \\lbrace \\Vert s - E\\left(\\vec{u}\\right) \\rho \\Vert_2^2 + \\lambda \\sqrt{1+\\beta^2 \\vert \\nabla \\rho \\vert^2} \\rbrace\n\\end{equation}\n\nAs in the previous section, we use a primal-dual projected gradient approach, employing the Beltrami energy as regularity prior \\cite{Zosso2014}. Note that regularization is always preferred in motion compensated reconstruction due to the ill-conditioning induced by the motion operators, as shown in \\cite{Atkinson2003}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\textwidth]{Fig2.png}\n\\caption{Schematic illustration of the proposed reconstruction, including the non-rigid motion extraction. Acquisition is performed using complementary trajectories, leading to uniform samplings in the phase encoding direction, which allows for an optimal combination of the k-spaces according to their positions in the breathing signal. The motion model, initialized by registering the images from each respiratory bin, is incorporated into the reconstruction process.}\n\\label{fig:fig2}\n\\end{figure}\n\n\\section{Material and Methods}\n\nThe proposed reconstruction algorithm was applied and validated with different experiments using Matlab (The MathWorks, Natick, MA) on a PC with Intel Xeon 3.3 GHz CPU and 64GB ram. The experiments were performed on 3T MR750w and 1.5T MR450w systems (GE Healthcare, WI, USA).\n\n\n\\begin{table}\n  \\centering\n  \\caption{Parameters used for the different experiments. The acquisition matrix size was 192 x 256.}\n  \\label{tab:table1}\n \\begin{tabular}{cccccccc}\n    \\toprule\n     & \\#repetition & \\#calib & \\#periphery & acc & acc shot & acc shot & NEX\\\\\n     & $n_r$ & lines & lines & peri & pre-bin & post-bin & sequence\\\\\n    \\midrule\n    Simulation 1 & 4 & 32 & 48 & 3.3 & 2.4 & - & 1.67\\\\\n    Simulation 2 & 4 & 32 & 32 & 5 & 3 & - & 1.33\\\\\n    Phantom      & 6 & 32 & 48 & 3.3 & 2.4 & - & 2.5\\\\\n    In vivo      & 15 & 17 & 43 & 4.1 & 3.2 & 1.1 (5 bins) & 4.7\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\n\\subsection{Offline Simulation on Synthetic Data}\n\nIn order to perform a realistic simulation, we first created a synthetic dataset based on actual LGE patient images. In one patient with suspected cardiovascular disease, four repetitions of a cardiac-gated, inversion recovery prepared, single-shot LGE scan were acquired in free-breathing 10 minutes after Gadolinium injection. Cardiac images were obtained with a spoiled fast gradient echo sequence and the following parameters: matrix size 192 x 256, in-plane spatial resolution 1.52 mm x 1.52 mm in short axis with slice thickness = 8 mm, readout flip angle = 20 degrees, echo time (TE) = 2.02 ms, mid-diastolic trigger delay, pulse repetition time (TR) = 4.43 ms and SENSE factor = 2 with partial Fourier. Synthetic k-space data were created by the application of synthetic coil sensitivity maps (with Gaussian profiles) to the LGE images, Fourier transformation and undersampling in the phase encoding direction. A full sampling of the central k-space area (17 lines) was used and the peripheral area was undersampled with a Golden Step Cartesian trajectory \\cite{Derbyshire2011} with an acceleration factor $R = 3.3$. Spacing between samples was proportional to the Golden ratio ($p = 0.618$). This trajectory enables an irregular but almost uniform distribution of the acquired data for any arbitrary number of repetitions, leading to incoherent aliasing (Fig. \\ref{fig:fig1}, right). The motion-free image was reconstructed using our reconstruction and compared to a motion correction method similar to that proposed by Kellman et al. \\cite{Kellman2005}, where the motion-free image is recovered by averaging the registered images obtained after the B-SENSE reconstruction step. We call this prior method reconstruction-registration-average (RRA).\n\n\\subsection{Phantom Imaging}\n\nSingle-shot pulse sequences were used to acquire phantom images with a 26-channel cardiac coil. The sequence was modified to take into account the same Golden ratio sampling as in our offline simulation experiment. The protocol was applied to acquire phantom images with a resolution of 1.48 mm x 1.48 mm. A translational motion was imposed to the table to mimic respiratory motion.\n\n\n\\subsection{In Vivo Validation Experiment with Self-Navigation}\n\nIn vivo cardiac datasets from two healthy adult volunteers were acquired on a 1.5T scanner using a 32-channel cardiac coil. A multi-shot slice (15 shots) of a free-breathing, cardiac-gated, spoiled fast gradient echo sequence (without inversion-recovery preparation) was collected with the following parameters: TE = 2.10 ms, TR = 4.52 ms, 8 mm slice thickness, FOV = 253 mm x 338 mm, matrix size 192 x 256 and 1.32 mm x 1.32 mm in-plane resolution, diastolic trigger delay. A fully sampled reference was acquired additionally in breath-hold for visual comparison. Each shot consists of 60 k-space lines: the central k-space was fully sampled with 17 lines and the periphery (43 lines) was undersampled, leading to a global acceleration factor of 3.2. An estimate of the respiratory signal was extracted using the proposed self-navigated technique, and was subsequently used to separate the acquired data into five respiratory bins. An overview of the parameters used in this study is given in Table \\ref{tab:table1}.\n\n\\section{Results}\n\nThe time needed to run the motion-compensated reconstruction for 15 shots of matrix size equal to 192 x 256 with 32-channel cardiac coils was about 1 min 35 s, including the time to compute the motion between shots.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Fig3.png}\n\\caption{Cardiac short-axis reconstruction of a synthetic dataset generated from 4 single-shot LGE acquisitions in free-breathing on a 37-year-old patient with acceleration factors $r = 3.3$ and $r = 5$. a) One reconstruction using a classic SENSE (192 x 256), b) Sum-of-Squares (all repetitions), c) Reconstruction-Registration-Average, d) proposed reconstruction.}\n\\label{fig:fig3}\n\\end{figure}\n\n\\subsection{Offline Simulation on Synthetic Data}\n\nExample reconstruction results on the simulated data generated from a patient with nonischemic cardiomyopathy are shown in Fig. \\ref{fig:fig3}. One can see a spatially blurred result with a standard reconstruction-registration-average (RRA) method. The proposed reconstruction exhibits significant quality improvements over each method with an acceleration factor $r = 3.3$ while reconstructing sharper edges (arrows) and small structures. For higher acceleration factors the performance of our method is much better compared to RRA, both in terms of reconstruction accuracy and image quality.\n\n\n\\subsection{Phantom Imaging}\n\nSimilar results can be observed in phantom experiments (Fig. \\ref{fig:fig4}). Comparisons with a classic Tikhonov reconstruction are shown in Fig. \\ref{fig:fig4}. The results present the reconstructed phantom motion experiments where here the motion has been applied with the table. The sum-of-squares reconstruction (Fig. \\ref{fig:fig4}, left) clearly exhibits the effect of motion. As in the previous experiment, the RRA method exhibits blurry results (due to the undersampling), although providing a motion-corrected denoised image. A visual improvement can be noticed when applying a motion compensated reconstruction with Tikhonov regularization. The latter method performs well but is, however, unable to recover sharp edges and some residual artifacts can still be seen on the recovered image. The use of a fast primal-dual algorithm combined with Beltrami regularization makes the proposed reconstruction robust with better performance in terms of image quality, reduced artifacts and sharpness (Fig. \\ref{fig:fig4}, Bel).\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=\\textwidth]{Fig4.png}\n\\caption{Reconstructions on a phantom using two different regularization methods with acceleration factor 3.3. Six single-shot repetitions have been acquired. From left to right: Sum-of-Squares (SoS), Reconstruction-Registration-Average (RRA), Tikhonov, Beltrami, Reference.}\n\\label{fig:fig4}\n\\end{figure}\n\n\\subsection{In Vivo Validation Experiment with Self-Navigation}\n\nFigure \\ref{fig:fig5} shows an example of the proposed fast and automatic self-navigated binning method on 150 consecutive slices of liver SPGR acquisition. The temporal rate was 400 ms, corresponding to a total acquisition duration of 1 min. The extracted respiratory signal (red) shows good agreement with the respiratory belt placed on the subject's thorax (blue), with a coefficient of determination $R^2 = 0.76$. Raw data acquired in similar motion states can be clustered into a reduced number of motion states, thereby improving the quality of images from which to extract motion in free-breathing without the need for navigators or external sensors.\nShort-axis images of the myocardium of a healthy subject without Gadolinium injection and without inversion recovery preparation are shown in Fig. \\ref{fig:fig6}. Both cardiac structures (myocardium wall, papillary muscles) and non-cardiac structures (blood vessels) are very well recovered with our reconstruction. The method yields significant sharpening of the myocardium wall and papillary muscles. However, due to the relatively high-undersampling, the RRA method is unable to recover a good quality image, exhibiting blurry structures and losing some of the details in the image such as blood vessels (arrows). This particularity is also seen in Fig. \\ref{fig:fig6} d where a specific intensity profile is plotted. The sharpness of the edges on our motion-corrected reconstruction is confirmed as well as the fidelity to the breath-hold acquisition.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Fig5.png}\n\\caption{Data binning step of the proposed self-navigated signal obtained from 150 consecutive 2D fast spoiled gradient echo acquisitions in free-breathing liver imaging.}\n\\label{fig:fig5}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\textwidth]{Fig6.png}\n\\caption{Data binning step of the proposed self-navigated signal obtained from 150 consecutive 2D fast spoiled gradient echo acquisitions in free-breathing liver imaging.}\n\\label{fig:fig6}\n\\end{figure}\n\n\\section{Discussion and Conclusion}\n\nWe introduced a new free-breathing single-shot LGE pipeline, including an optimized sampling and the associated joint reconstruction and motion correction algorithm designed for fast and robust cardiac imaging. By incorporating the estimated motion into the reconstruction process, we increased the robustness of the model and exhibited good quality images.\n\nIn this study, we used a fast and automatic self-navigated binning strategy that aims to cluster the acquired raw data into similar motion states. While preliminary results have shown improved image quality and better motion estimation, additional optimization of number of bins and number of repetitions is still required to maintain an optimal tradeoff between reconstruction quality, reconstruction time and accuracy of motion estimates. The motion corrected images show better visual quality than classic reconstructions but appear less sharp than corresponding breath-held acquisitions, especially for high accelerations. Possible explanations are the inaccuracies in motion estimates or other effects related to MR physics, such as spin history or changes in $B_0$ and $B_1$ inhomogeneities induced by breathing.\n\nOne interesting application of the proposed motion correction model is for high-resolution 3D isotropic LGE single-shot imaging of the heart, such as the one proposed recently in \\cite{Dzyubachyk2015} for myocardial scar assessment. This will allow for the reconstruction of 3D isotropic motion corrected volumes by keeping the advantages of a 2D acquisition (high tissue and vessel contrast, short acquisition time), e.g. using super-resolution techniques \\cite{Odille2015}. Other applications, such as abdominal imaging \\cite{Buerger2013} and coronary vessel imaging \\cite{Cruz2016}, are being investigated.\n\nA limitation to the method is that potential through-plane motion cannot be corrected, although it remains small compared to the slice thickness. To overcome this problem, one could consider weighting the images according to the motion amplitude compared to the target image or acquiring 3D slab instead of 2D slice data and applying motion compensation. The preliminary results presented in this work should be confirmed with further patient studies.\n\nThe feasibility of the proposed reconstruction has been evaluated in simulation, phantom and volunteer experiments. The method has been shown to allow non-rigid motion correction while efficiently recovering features, thanks to the Beltrami regularization scheme. The conventional segmented LGE acquisition is limited by the maximum breath-hold time, which limits the signal-to-noise ratio and\/or spatial resolution. This limitation is overcome by the presented free breathing approach. Ultimately, this method could enable accurate motion corrected reconstruction of single-shot images with higher spatial resolution and a higher signal-to-noise ratio compared to conventional segmented methods, with the potential to offer high-quality LGE imaging in challenging patients.\n\n\n\\subsection*{Acknowledgments}\nThe authors thank Mayo Clinic (Rochester, MN), Advanced Cardiovascular Imaging (New York, NY) and Morriston Hospital (Swansea, UK) for providing some of the imaging data. This publication was supported by the European Commission, through Grant Number 605162. The content is solely the responsibility of the authors and does not necessarily represent the official views of the EU.\n\n\n\\bibliographystyle{splncs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nConsider the linear transport equation in $1$D \n\\begin{equation}\\label{eq:transport}\n\\partial_t f + v \\partial_x f - \\partial_x \\Phi  \\partial_v f = 0,\n\\end{equation}\nfor an unknown function $f: [0,\\infty) \\times \\mathbb R_x \\times \\mathbb R_v \\to \\mathbb R_{\\geq 0}$\nwith a smooth external confining potential $\\Phi:\\mathbb R \\to \\mathbb R$. \n\nThe following is the main result of this note:\n\\begin{theorem}\\label{thm:main}\nLet $\\epsilon>0$ and $\\Phi(x) = \\frac {x^2}2 + \\frac{\\epsilon x^4}2$. Consider the unique solution $f$ to \\eqref{eq:transport} with initial data $f \\restriction_{t=0} = f_0$ such that \n\\begin{itemize}\n\\item $f_0: \\mathbb R_x \\times \\mathbb R_v \\to \\mathbb R_{\\geq 0}$ is smooth, and\n\\item there exists $c_s >0$ such that $\\mathrm{supp}(f_0) \\subseteq \\{ (x,v):  c_s \\leq \\frac{v^2}2 + \\Phi(x) \\leq c_s^{-1} \\}$.\n\\end{itemize}\n\nThen, for $\\epsilon$ sufficiently small, there exists $C>0$ depending on $\\epsilon$ and $c_s$ such that the following estimate holds:\n$$\\sup_{x\\in \\mathbb{R}} |\\partial_t \\varphi|(t,x) \\leq C \\langle t\\rangle^{-2} \\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} \\sum_{|\\alpha|+ |\\beta| \\leq 2} |\\partial_x^\\alpha \\partial_v^\\beta f_0|(x,v),$$\nwhere $\\varphi$ is defined by\n\\begin{equation}\\label{eq:phi}\n\\partial_{xx}^2\\varphi(t,x) =  \\int_{\\mathbb{R}} f(t,x,v)\\, \\mathrm{d} v,\\quad \\varphi(t,0) = \\partial_x \\varphi(t,0) = 0.\n\\end{equation}\n\\end{theorem}\n\nA few remarks of the theorem are in order.\n\n\\begin{remark}[Nonlinear Vlasov--Poisson system]\nThe reason that we are particularly concerned with $\\partial_t \\varphi$ is that it appears to be the quantity relevant for the stability of the\nzero solution for the nonlinear Vlasov--Poisson system in $1$D; see Section~\\ref{sec:VP}.\n\nIt should be noted that $\\varphi$ itself is not expect to decay to $0$ (since $\\int_{\\mathbb{R}} f\\, \\mathrm{d} v \\geq 0$). Thus the decay for $\\partial_t \\varphi$ can be viewed as a measure of the rate that $\\varphi$ approaches the limit $\\lim_{t\\to +\\infty} \\varphi(t,x)$.\n\n\\end{remark}\n\n\\begin{remark}[Derivatives of $\\partial_t\\varphi$]\nFor the applications on the Vlasov--Poisson system, one may also wish to obtain estimates for the derivatives of $\\partial_t \\varphi$. It is easy to extend our methods to obtain\n$$|\\partial_x \\partial_t\\varphi |\\lesssim \\langle t\\rangle^{-1},\\quad |\\partial_x^2 \\partial_t \\varphi|\\lesssim 1.$$\nNotice that these decay rates, at least by themselves, do not seem sufficient for a global nonlinear result.\n\\end{remark}\n\n\\begin{remark}[Phase mixing and the choice of $\\Phi$]\nThe result in Theorem~\\ref{thm:main} can be interpreted as a quantitative phase-mixing statement. It is well-known that for \n$$\\Phi(x) = \\frac {x^2}2,$$\nthe solution to \\eqref{eq:transport} does \\underline{not} undergo phase mixing (see chapter 3 in \\cite{BinTre_2011}). It is therefore important that we added the $\\frac{\\epsilon x^4}2$ term in the definition of the potential.\n\nOn the other hand, there are other choices of $\\Phi$ for which analogues of Theorem~\\ref{thm:main} hold. We expect that as long as $\\Phi$ is even and satisfies the non-degeneracy condition of \\cite{pRoS2020}, then a similar decay estimate holds. The particular example we used is only chosen for concreteness.\n\\end{remark}\n\n\\begin{remark}[Method of proof]\nIt is well-known that the linear transport equation \\eqref{eq:transport} can be written in action-angle variables, say $(Q, K)$, in which case \\eqref{eq:transport} takes the form\n\\begin{equation}\\label{eq:trivial}\n\\part_t f - c({K})\\part_{{Q}} f=0.\n\\end{equation}\n\nWhen $c'(K)$ is bounded away from $0$, phase mixing in the sense that $f$ converges \\emph{weakly} to a limit can be obtained after solving \\eqref{eq:trivial} with a Fourier series in $Q$; see \\cite{pRoS2020}. The point here is that $\\varphi$ is a (weighted) integral of $f$ over a region of phase space that is most conveniently defined with respect to the $(x,v)$ (as opposed to the action-angle) variables.\n\nWe quantify the strong convergence of $\\varphi_t\\to 0$ by finding an appropriate commuting vector field $Y$ that is adapted to the action-angle variables. The fact that $\\varphi$ is naturally defined as an integral over $v$ in $(x,v)$ coordinates makes it tricky to prove decay using this vector field. Furthermore, we are only able to prove $1\/\\jap{t}^2$ decay; this is for instance in contrast to the decay of the density for the free transport equation on a torus.\n\\end{remark}\n\n\\medskip\n\n\n\\subsection{Related result}\n\n\\subsection*{Linear phase mixing results}\nIn the particular context of Theorem~\\ref{thm:main}, decay of $\\partial_t\\varphi$, but without a quantitative rate, can be inferred from the work \\cite{pRoS2020}.\n\nThere are many linear phase mixing result, the simplest setting for this is the linear free transport equation. This is well-known; see for instance notes \\cite{villani2010landau} by Villani.\n\nOne of the most influential work on phase mixing is the groundbreaking paper \\cite{Landau1946} of Landau wherein he proposes a linear mechanism for damping for plasmas that does not involve dispersion or change in entropy. In the case of $\\mathbb T^d$, this is even understood in a nonlinear setting; see the section on nonlinear results below. The situation is more subtle in $\\mathbb R^d$, see \\cite{jBnMcM2020}, \\cite{rGjS1994}, \\cite{rGjS1995} and \\cite{dHKttNfR2020}.\n\nSee also \\cite{jBfW2020}, \\cite{faou2021linear} and \\cite{iT2017} for linear results on related models. In particular, we note that \\cite{faou2021linear} also rely on action-angle variables in their analysis.\n\n\\subsection*{Relation with other phase-mixing problems with integrable underlying dynamics}\nAs pointed out in \\cite{pRoS2020}, phase space mixing is relevant for the dynamics of kinetic models in many physical phenomena from stellar systems and dark matter halos to mixing of relativistic gas surrounding a black hole. See \\cite{dominguez2017description} for related discussions on dark matter halos. We also refer the interested reader to \\cite{BinTre_2011} for further background and discussions of phase mixing in other models, including the stability of galaxies.\n\nWe hope that the present work would also be a model problem and aid in understanding more complicated systems such as those described in \\cite{pRoS2020}. One particularly interesting problem is the stability of the Schwarzschild solution to the Einstein--Vlasov system in spherical symmetry. \n\n\\subsection*{Nonlinear phase mixing results}\nNonlinear Landau damping for Vlasov--Poisson on $\\mathbb{T}^d$ was first proven in analytic regularity by Mouhot--Villani in their landmark paper  \\cite{cMcV2011}. Since then their work has been extended and simplified in \\cite{jBnMcM2016} and \\cite{eGtNiR2020a}.\n\nSee also other nonlinear results, e.g.~in \\cite{jB2017}, \\cite{jBnMcM2018}, \\cite{chaturvedi2021vlasov}, \\cite{faou2016landau}, \\cite{dHKttNfR2019}, \\cite{bY2016}.\n\n\n\n\\subsection*{Collisional problems with confining potentials}\n\nConfining potentials for kinetic equations have been well-studied, particularly for collisional models. Linear stability results can be found in \\cite{carrapatoso2021special}, \\cite{dolbeault2009hypocoercivity}, \\cite{dolbeault2015hypocoercivity}, \\cite{duan2011hypocoercivity} and \\cite{duan2012hypocoercivity}.\n\nIn this connection, it would also be of interest to understand how phase mixing effects (studied in the present paper) interact with collisional effects (cf.~\\cite{jB2017}, \\cite{chaturvedi2021vlasov}, \\cite{iT2017}.)\n\n\n\\section{The Vlasov--Poisson system}\\label{sec:VP}\n\nThe motivation of our result is the Vlasov--Poisson system:\n\\begin{equation}\\label{eq:VP}\n\\begin{cases}\n\\part_t f+v\\part_x f-(\\part_x \\Phi+\\part_x \\varphi)\\part_v f=0, \\\\\n-\\part^2_x \\varphi=\\int_{\\mathbb{R}} f\\d v.\n\\end{cases}\n\\end{equation}\n\nNote that \\eqref{eq:VP} can be rewritten as \n\\begin{equation}\\label{e.VP}\n\\part_t f+\\{H,f\\}=0,\n\\end{equation}\nwhere $H$ is the Hamiltonian given by \n\\begin{equation}\\label{eq:nonlin-hamiltonian}\nH(x,v)=\\frac{v^2}{2}+\\Phi(x)+\\varphi(t,x).\n\\end{equation}\n\nNotice that $f \\equiv 0$ is a solution to \\eqref{eq:VP}, and the transport equation \\eqref{eq:transport} is \nthe linearization of \\eqref{eq:VP} near the zero solution.\n\nOne cannot hope that the term $\\partial_x \\varphi$ in the nonlinear term decays as $t\\to +\\infty$. \n(This can be seen by noting that $\\int_{\\mathbb{R}} f\\d v \\geq 0$ pointwise.) At best one can hope \nthat $\\partial_x \\varphi$ converges to some (non-trivial) limiting profile as $t\\to +\\infty$. For $f$ satisfying \nthe linear equation \\eqref{eq:transport}, such convergence (without a quantitative rate) has been shown\nin \\cite{pRoS2020}.\n\nIn anticipation of the nonlinear problem, it is important to understand the quantitative convergence. Since\n$\\partial_x\\varphi$ does not converge to $0$, it is natural to understand the decay rate of $\\partial_t \\partial_x \\varphi$.\n\nAs a first step to understand \\eqref{eq:VP}, we look at the linearized problem \\eqref{eq:transport} around the zero solution and prove that we get integrable decay for $\\varphi_t$ in the linearized dynamics.\n\n\\begin{remark}\nNote that the Poisson's equation above reads\n$$-\\part^2_x \\varphi=\\rho.$$\nIn particular, $\\varphi$ is only defined up to a harmonic function, i.e.~a linear function a $x$. \nIn Theorem~\\ref{thm:main}, we remove this ambiguity by setting $\\varphi(0) = (\\partial_x \\varphi)(0) = 0$. \nNotice that other normalization, e.g., $\\varphi(-\\infty) = (\\partial_x \\varphi)(-\\infty)=0$ would not change the function $\\varphi_t = \\partial_t \\varphi$.\n\\end{remark}\n\n\\section{The action-angle variables}\\label{sec:action}\n\n\\subsection{First change of variables}\nFrom now on we will consider the Hamiltonian $$H=\\frac{v^2}{2}+\\Phi(x).$$ This is the Hamiltonian for the equations \\eqref{eq:transport}, which is also \\eqref{eq:nonlin-hamiltonian} without the $\\varphi$ (the self-interaction term). As an intermediate step to getting the action-angle variables we use the change of coordinates \n\\begin{align*}\n(t,x,v)\\mapsto (t,\\chi,H) &\\hspace{5 em}\\text{when $x>0$}\\\\\n(t,x,v)\\mapsto (t,\\pi-\\chi,H) &\\hspace{5 em}\\text{when $x\\leq 0$},\n\\end{align*}\nwhere $\\chi:=\\arcsin\\left(\\frac{v}{\\sqrt{2H}}\\right).$\\\\\nFirst we check if the change of variables is well defined by calculating the Jacobian for $x>0$,\n\\begin{equation*}\nJ=\\begin{pmatrix}\n\\part_t t& \\part_x t&\\part_v t\\\\\n\\part_t H& \\part_x H&\\part_v H\\\\\n\\part_t \\chi& \\part_x \\chi&\\part_v \\chi\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1&0&0\\\\\n0&\\Phi_x&v\\\\\n0&-\\frac{v}{2H}\\cdot\\frac{\\Phi_x}{\\sqrt{\\Phi}}&\\frac{\\sqrt{\\Phi}}{H}\n\\end{pmatrix}\n\\end{equation*}\nNow \n\\begin{align*}\n\\det(J)&=\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\frac{(\\Phi+v^2\/2)}{H}\\\\\n&=\\frac{\\Phi_x}{\\sqrt{\\Phi}}.\n\\end{align*}\nSimilarly for $x\\leq 0$, \\begin{align*}\n\\det(J)&= - \\frac{\\Phi_x}{\\sqrt{\\Phi}}.\n\\end{align*}\nHence, \\begin{align*}\n\\det(J)\n&=\\sign{x}\\frac{\\Phi_x}{\\sqrt{\\Phi}}.\n\\end{align*}\nNext by chain rule and using that $H$ is independent of $t$, we get\n\n\\begin{align*}\n\\part_x&=\\sign{x}\\part_x \\chi\\part_{\\chi}+\\part_x H\\part_H\\\\\n&=-\\frac{v}{2H}\\cdot\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\part_{\\chi}+\\Phi_x\\part_H,\n\\end{align*}\n\\begin{align*}\n\\part_v&=\\sign{x}\\part_v \\chi\\part_{\\chi}+\\part_v H\\part_H\\\\\n&=\\frac{\\sqrt{\\Phi}}{H}\\part_{\\chi}+v\\part_H.\n\\end{align*}\nPluggin this in \\eref{VP}, we get the equation\n\\begin{equation}\\label{e.VP_chi_H_lin}\n\\part_t f-\\sign{x}\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\part_\\chi f=0.\n\\end{equation}\n\n\n\\subsection{Second change of variables} The coefficient in front of $\\part_\\chi f$ in \\eqref{e.VP_chi_H_lin} depends on both $\\chi$ and $H$. To take care of this, we reparametrize $\\chi$ (in a manner depending on $H$). More precisely, for a fixed $H$, we define $Q(\\chi,H)$ such that \n$$\\frac{\\d Q}{\\d \\chi}=\\frac{c(H)}{a(\\chi,H)},\\quad Q(0,H)=0,$$\nwhere $a(\\chi,H)=\\sign{x}\\frac{\\Phi_x(x)}{\\sqrt{\\Phi(x)}}$ such that $x=x(\\chi,H).$ To fix $c(H)$, we require that for every $H$, \n\\begin{equation}\\label{eq:C.def}\n2\\pi=\\int_0^{2\\pi}\\d Q=c(H)\\int_0^{2\\pi}\\frac{1}{a(\\chi,H)}\\d \\chi.\n\\end{equation}\nNow we define the change of variables, $(\\chi,H)\\mapsto (Q,K)$ where $K=H$. Then note,\n$$a(\\chi,H)\\partial_\\chi=c(H)\\part_Q$$\nand $$\\part_{H}=\\part_{K}+\\frac{\\part Q}{\\part H}\\part_Q.$$\nThus in these coordinates, we can rewrite \\eref{VP_chi_H_lin} as \n\\begin{equation}\\label{e.VP_Q_H_lin}\n\\part_t f-c({K})\\part_Q f=0.\n\\end{equation}\n\n\nFurther, the Jacobian is \n\\begin{equation*}\n\\begin{pmatrix}\n\\part_H K&\\part_\\chi K\\\\\n\\part_H Q& \\part_\\chi Q\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1&0\\\\\n\\part_H Q&\\frac{c(H)}{a(\\chi,H)}\n\\end{pmatrix}.\n\\end{equation*}\nNote that the determinant is $\\frac{c(H)}{a(\\chi,H)}$. Further, since \n\\begin{align*}\na(\\chi,H)&=\\sign{x}\\frac{\\Phi_x(x)}{\\sqrt{\\Phi(x)}}\\\\\n&=\\sqrt{2}\\frac{1+2\\varepsilon x^2}{\\sqrt{1+\\varepsilon x^2}},\n\\end{align*}\n\twe have that $a(\\chi,H) \\approx 1$ when $x$ is in a compact subset of $\\mathbb R$. As a result the determinant is bounded away from zero. For more details see Lemma~\\ref{lem:c-prime-bound}. \n\n\n\n\\section{The commuting vector field}\nWe first define the vector field $$Y=tc'(H)\\partial_Q-\\part_K.$$ In this section we prove that this vector field commutes with the transport operator as in \\eqref{e.VP_Q_H_lin} and that $|c'(H)|>0.$\n\\subsection{Commutation property}\n\nThe following commutation formula is an easy computation and thus we leave out the details.\n\\begin{lemma}\\label{lem:commutation}\nLet $Y = t c'(H) \\partial_Q - \\partial_{K}$. Then\n$$[\\partial_t - c(H) \\partial_Q, Y] = 0.$$\n\\end{lemma}\n\nThe following is an easy consequence of Lemma~\\ref{lem:commutation}:\n\\begin{lemma}\\label{lem:infinity-est}\nLet $f$ be a solution to \\eqref{e.VP_Q_H_lin} with initial data satisfying assumptions of Theorem~\\ref{thm:main}. Then\n$$\\sup_{(t,Q,K) \\in [0,\\infty)\\times \\mathbb T^1 \\times [c_s, c_s^{-1}]}\\hspace{.1em}\\sum_{\\ell\\leq 2} |Y^{\\ell} f|(t,Q,K) \\lesssim \\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} \\hspace{.1em}\\sum_{|\\alpha|+ |\\beta| \\leq 2} |\\partial_x^\\alpha \\partial_v^\\beta f_0|(x,v).$$\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{lem:commutation}, we have that $Y^\\ell f$ satisfies the transport equation \\eqref{e.VP_Q_H_lin} for any $\\ell \\in \\mathbb N \\cap \\{0\\}$. Hence we get the estimate\n$$\\sup_{(t,Q,K) \\in [0,\\infty)\\times \\mathbb T^1 \\times [c_s, c_s^{-1}]}\\hspace{.1em}\\sum_{\\ell\\leq 2} |Y^{\\ell} f|(t,Q,K)\\lesssim \\sup_{(Q,K) \\in \\mathbb T^1 \\times [c_s, c_s^{-1}]}\\hspace{.1em}\\sum_{\\ell \\leq 2} |\\part_K^{\\ell} f_0|(Q,K).$$\nSince the change of variables $(Q,K)\\to (x,v)$ is well-defined and bounded away from zero, we get the required result.\n\\end{proof}\n\\subsection{Positivity of $|c'(K)|$}\nWe prove that $|c'(K)|$ is uniformly bounded below on the support of $f$. This plays a key role in the next section ensuring phase mixing.\n\\begin{lemma}\\label{lem:c-prime-bound}\nFor every $c_s <+\\infty$, there exists $\\epsilon_0>0$ such that whenever $\\epsilon \\in (0,\\epsilon_0]$, there is a small constant $\\delta >0$ (depending on $c_s$ and $\\epsilon$) such that \n$$\\inf_{K \\in [c_s, c_s^{-1}]} |c'(K)| = \\inf_{H \\in [c_s, c_s^{-1}]} |c'(H)| \\geq \\delta.$$\n\\end{lemma}\n\\begin{proof}\nBy definition of $c(H)$, we have\n$$\\frac{2\\pi}{c(H)}=\\int_0^{2\\pi}\\left|\\frac{\\sqrt{\\Phi}}{\\Phi'}\\right|\\d \\chi,$$\nso that using $\\Phi=\\frac{x^2}{2}+\\frac{\\epsilon}{2} x^4$, we obtain\n$$\\frac{2\\pi}{c(H)}=\\int_0^{2\\pi}\\frac{\\sqrt{1+\\epsilon x^2}}{\\sqrt{2}(1+2\\epsilon x^2)}\\d \\chi.$$\n\nNotice that for $H \\in [c_s, c_s^{-1}]$, $|x|$ is bounded. It follows that $c(H) \\approx 1$. Therefore, to prove strict positivity of $|c'(H)|$, it suffices to prove positivity of $\\left|\\frac{c'(H)}{c^2(H)}\\right|.$ Note that \n\\begin{equation}\\label{eq:c'\/c}\n\\begin{split}\n\\frac{-2\\pi c'(H)}{c^2(H)}&=\\frac 1{\\sqrt 2}\\int_0^{2\\pi}\\part_H\\left(\\frac{\\sqrt{1+\\epsilon x^2}}{1+2\\epsilon x^2}\\right)\\d \\chi\\\\\n&=\\frac 1{\\sqrt 2}\\int_0^{2\\pi}\\part_H x\\left[\\frac{\\epsilon x}{\\sqrt{1+\\epsilon x^2}(1+2\\epsilon x^2)}-\\frac{4\\epsilon x\\sqrt{1+\\epsilon x^2}}{(1+2\\epsilon x^2)^2}\\right]\\d \\chi\\\\\n&=\\frac 1{\\sqrt 2}\\int_0^{2\\pi}\\part_H x\\left[\\frac{-3\\epsilon x-2\\epsilon^2x^3}{\\sqrt{1+\\epsilon x^2}(1+2\\epsilon x^2)^2}\\right]\\d \\chi.\n\\end{split}\n\\end{equation}\n\nNow we calculate $\\part_H x.$ First we use the equation, $H=\\frac{v^2}{2}+\\Phi(x).$ Precisely, we have \n$$1=v\\part_H v+\\Phi'(x) \\part_H x.$$ Thus \n\\begin{equation}\\label{eq:dxH}\n\\part_H x=\\frac{1-v\\part_H v}{\\Phi'}.\n\\end{equation}\nNext we use that $\\frac{v}{\\sqrt{2H}}=\\sin\\chi,$\n$$0=\\frac{\\part_H v}{\\sqrt{2H}}-\\frac{v}{(2H)^{\\frac{3}{2}}}.$$\nThus $\\part_H v=\\frac{v}{2H}.$ Plugging this into \\eqref{eq:dxH}, we get that \n\\begin{equation}\\label{eq:dxH.final}\n\\part_H x=\\frac{1-\\frac{v^2}{2H}}{\\Phi'}=\\frac{\\cos^2 \\chi}{\\Phi'(x)} = \\frac{\\cos^2\\chi}{x+2\\epsilon x^3},\n\\end{equation}\nwhere in the last equality we used $\\Phi=\\frac{x^2+\\epsilon x^4}{2}$.\n\nPlugging \\eqref{eq:dxH.final} back into \\eqref{eq:c'\/c}, we get that\n\\begin{align*}\n\\frac{-2\\pi c'(H)}{c^2(H)}&=\\int_0^{2\\pi}\\frac{\\cos^2\\chi}{\\sqrt{2} x(1+2\\epsilon x^2)}\\left[\\frac{-x(3\\epsilon +2\\epsilon^2x^2)}{\\sqrt{1+\\epsilon x^2}(1+2\\epsilon x^2)^2}\\right]\\d \\chi\\\\\n&=-\\int_0^{2\\pi}{\\cos^2\\chi}\\left[\\frac{(3\\epsilon +2\\epsilon^2x^2)}{\\sqrt{2(1+\\epsilon x^2)}(1+2\\epsilon x^2)^3}\\right]\\d \\chi.\n\\end{align*}\nFinally note that since $|x|$ is bounded on the region of interest, after choosing $\\epsilon_0$ sufficiently small, we have\n$$\\frac{(3\\epsilon +2\\epsilon^2x^2)}{\\sqrt{2(1+\\epsilon x^2)}(1+2\\epsilon x^2)^3}\\approx \\epsilon,$$ \nand thus $|c'(H)|>\\delta$. \\qedhere\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Decay for $\\varphi_t$}\nIn this section we finally prove the decay for $\\varphi_t$ (recall Theorem~\\ref{thm:main}). \n\nTo keep the notation lean, we will often suppress the explicit dependence on $t$.\n\\begin{lemma}\\label{lem:phi_t}\nFor $f$ satisfying the assumptions of Theorem~\\ref{thm:main}, and $\\varphi$ defined as in \\eqref{eq:phi}, we have the following formula\n$$\\varphi_t(x')=\\int_0^{x'}\\int_{\\mathbb{R}}v[f(y,v)-f(0,v)]\\d v\\d y.$$\n\\end{lemma}\n\\begin{proof}\nBy the continuity equation (following directly from \\eqref{eq:transport}), we have that $$\\rho_t=\\int_{\\mathbb{R}}v \\part_x f\\d v.$$\nThus $$-\\part^2_x \\varphi_t=\\rho_t=\\int_{\\mathbb{R}}v \\partial_x f\\d v.$$\nSolving the Laplace's equation (with boundary conditions \\eqref{eq:phi}), we get\n$$\\varphi_t(x')=\\int_0^{x'}\\int_0^y \\int_{\\mathbb{R}}v\\part_x f(z,v)\\d v\\d z\\d y.$$\nIntegrating by parts in $z$, we get\n$$\\varphi_t(x')=\\int_0^{x'}\\int_{\\mathbb{R}}v[f(y,v)-f(0,v)]\\d v\\d y.$$\n\\end{proof}\n\nIn view of Lemma~\\ref{lem:phi_t}, it suffices to bound $\\int_0^{x'}\\int_{\\mathbb{R}} v f(0,v) \\d v\\d y$ and $\\int_0^{x'}\\int_{\\mathbb{R}} v f(y,v) \\d v\\d y$, which will be achieved in the next two subsections respectively.\n\n\\subsection{Decay for the term involving $f(0,v)$}\nWe first prove decay for $\\int_{\\mathbb{R}} v f(0,v)\\d v.$ {Before proving the main estimate in Proposition~\\ref{prop:term-x=0}, we first prove a lemma.}\n\n\\begin{lemma}\\label{lem:Q.at.pi.2}\nThe level set $\\{x = 0\\}$ corresponds to the level sets $\\{ Q = \\frac \\pi 2 \\} \\cap \\{ Q = -\\frac \\pi 2\\} \\cup \\{(x,v) = (0,0)\\}$. \n\\end{lemma}\n\\begin{proof}\nFirst note that level set $\\{x = 0\\}$ corresponds to the level sets $\\{ \\chi = \\frac \\pi 2 \\} \\cap \\{ \\chi = -\\frac \\pi 2\\} \\cup \\{(x,v) = (0,0)\\}$. This is because when $x = 0$, $\\Phi(x)= 0 $, and thus by definition (when $v \\neq 0$) $\\chi:=\\arcsin\\left(\\frac{v}{\\sqrt{2H}}\\right) = \\arcsin (\\pm 1) = \\pm \\frac \\pi 2$.\n\nIt thus remains to show that\n\\begin{equation}\\label{eq:chi.Q.pi.2}\n\\chi = \\pm \\frac \\pi 2 \\iff Q = \\pm \\frac \\pi 2.\n\\end{equation}\n\nFix $H$, then since $a(\\chi,H)=\\lvert\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\rvert$ is independent of $v$, we have\n$$c(H)\\int_0^{\\pi} \\frac{1}{a(\\chi,H)}\\d \\chi=c(H)\\int_{\\pi}^{2\\pi} \\frac{1}{a(\\chi,H)}\\d \\chi.$$\nFurther, by the evenness of $\\Phi$, we have\n$$c(H)\\int_0^{\\pi\/2} \\frac{1}{a(\\chi,H)}\\d \\chi=c(H)\\int_{\\pi\/2}^{\\pi} \\frac{1}{a(\\chi,H)}\\d \\chi.$$\nFinally, since we have by construction, $$c(H)\\int_0^{2\\pi} \\frac{1}{a(\\chi,H)}\\d \\chi=2\\pi,$$ we have that $$Q(\\chi=\\pi\/2,H)=c(H)\\int_0^{\\pi\/2} \\frac{1}{a(\\chi,H)}\\d \\chi=\\pi\/2.$$\nSimilarly, $Q(\\chi=-\\pi\/2,H)=-\\pi\/2.$ Combining these, we obtain \\eqref{eq:chi.Q.pi.2}. \\qedhere \n\\end{proof}\n\n\\begin{proposition}\\label{prop:term-x=0}\nFor $f$ satisfying the assumptions of Theorem~\\ref{thm:main}, we have the following estimate:\n$$\\Big| \\int_{\\mathbb{R}} v f(0,v)\\d v \\Big| \\lesssim \\langle t \\rangle^{-2} \\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} \\hspace{.1em}\\sum_{|\\alpha|+ |\\beta| \\leq 2} |\\partial_x^\\alpha \\partial_v^\\beta f_0|(x,v).$$\n\\end{proposition}\n\\begin{proof}\nThe transport equation preserves $L^\\infty$ bounds so that by the support properties, we obviously have\n$$\\Big| \\int_{\\mathbb{R}} v f(0,v)\\d v \\Big| \\lesssim  \\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} \\hspace{.1em} | f_0|(x,v).$$\nIn other words, it suffices to prove the desired bound with $t^{-2}$ instead of $\\langle t \\rangle^{-2}$.\n\nNow note that $$\\int_{\\mathbb{R}} v f(0,v)\\d v=\\int_0^\\infty v [f(0,v)-f(0,-v)]\\d v.$$\nFor clarity of notation, we let $$\\bar{f}(t,Q,K) = f(t,x,v).$$ Now writing in the $(K,Q)$ variables, and using Lemma~\\ref{lem:Q.at.pi.2} together with the fact that $K = H = \\frac {v^2}2$ when $x=0$, we have\n$$\\int_{\\mathbb{R}} v f(0,v)\\d v=\\int_0^\\infty v [f(0,v)-f(0,-v)]\\d v=\\int_0^\\infty [\\bar f(\\pi\/2,K)-\\bar f(-\\pi\/2,K)]\\d { K}.$$\nBy the fundamental theorem of calculus, we have\n$$\\int_0^\\infty [\\bar f(\\pi\/2,K)- \\bar f(-\\pi\/2,K)]\\d { K}=\\int_{-\\pi\/2}^{\\pi\/2}\\int_0^\\infty \\part_Q \\bar f(Q,K)\\d {K}\\d Q.$$\nNext, the Cauchy--Schwarz inequality implies\n\\begin{align*}\n\\int_{-\\pi\/2}^{\\pi\/2}\\int_0^\\infty \\part_Q  \\bar f(Q,K)\\d {K}\\d Q&=\\sqrt{\\pi}\\left(\\int_{-\\pi\/2}^{\\pi\/2}\\left(\\int_0^\\infty \\part_Q  \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}\\\\\n&\\lesssim \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\part_Q  \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}.\n\\end{align*}\nNow using Poincare's inequality we get that for any $\\ell\\geq 2$\n\\begin{equation}\\label{eq:easy.term.after.Poincare}\n\\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\part_Q  \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}\\lesssim \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\part^{\\ell}_Q  \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}.\n\\end{equation}\n\nNow take $\\ell = 2$. We write $\\part_Q=\\frac{1}{c'(K)t}(Y+\\part_{K})$ so that \n\\begin{equation*}\n\\begin{split}\n&\\: \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\part^2_Q  \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}} \\\\\n= &\\: \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\frac{1}{|c'(K)|^2 t^2} ( Y^2 \\bar f+ 2\\part_{K} Y \\bar f+ \\partial^2_K \\bar f) (Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}} \\\\\n\\lesssim &\\: \\frac 1{t^2} \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty   (\\sum_{k=0}^2  |Y^k \\bar f|) (Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}.\n\\end{split}\n\\end{equation*}\nwhere in the last step we have integrated by parts in $K$ and bounded $\\frac 1{|c'(K)|}$, $\\frac{|c''(K)|}{|c'(K)|}$, etc.~using Lemma~\\ref{lem:c-prime-bound} and the smoothness of $c$.\n\nFinally, since $f(t,Q,K)$ is non-zero for $c_s\\leq K\\leq c_s^{-1}$ and $Q\\in [0,2\\pi]$, we can take supremum in $K$ and $Q$ followed by Lemma~\\ref{lem:infinity-est} to get the required result.\\qedhere\n\\end{proof}\n\n\n\\begin{remark}\nNotice that since we can take any $\\ell \\geq 2$ in \\eqref{eq:easy.term.after.Poincare}, we can write each $\\part_Q=\\frac{1}{c'(H)t}(Y+\\part_{K})$ and integrate by parts in $K$ many times to show that the term in Proposition~\\ref{prop:term-x=0} in fact decays faster than any inverse polynomial (depending on smoothness of $f$)! \n\nIn other words, the decay rate that we obtain in Theorem~\\ref{thm:main} is instead limited by the term treated in Proposition~\\ref{prop:bulk-term} below.\n\\end{remark}\n\n\\subsection{Decay for the term involving $f(y,v)$} \n\n{We now turn to the other term in Lemma~\\ref{lem:phi_t}. Before we obtain the main estimate in Proposition~\\ref{prop:bulk-term}, we first prove two simple lemmas.}\n\\begin{lemma}\\label{lem:Jacobian}\nUnder the change of variables $(x,v)\\mapsto (Q,K)$ as in Section~\\ref{sec:action}, the volume form transforms as follows:\n$$\\d v \\d x = c(K) \\,\\d Q\\d{K}.$$\n\\end{lemma}\n\\begin{proof}\nThe Jacobian determinant for the change of variables $(x,v)\\mapsto (\\chi,H)$ is $a(\\chi,H)=\\lvert\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\rvert$. Further the Jacobian determinant for the change of variables $(\\chi,H)\\to (Q,K)$ is $\\frac{c(H)}{a(\\chi,H)}$ and hence the Jacobian determinant for $(x,v)\\to (Q,K)$ is $c(H) = c(K)$. \\qedhere\n\\end{proof}\n\n\\begin{lemma}\\label{lem:f-to-g}\nLet $\\bar{f}(Q,K) = f(x,v)$ as above. There exists a function $\\bar{g}(Q,K)$ such that\n\\begin{equation}\\label{eq:f-to-g}\n\\partial_Q^2 \\bar{g} = \\partial_Q \\bar{f}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:g-bound}\n\\max_{\\ell\\leq 2} \\sup_{K} \\norm{Y^{\\ell} \\bar{g}}_{L^2_Q} \\lesssim \\max_{\\ell\\leq 2} \\sup_{Q,K} |Y^{\\ell} \\bar{f}|.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe use the Fourier series of $f$ in $Q$ to get that $$\\bar{f}(Q,K)=\\sum_{k=-\\infty}^{k=\\infty} \\widehat{\\bar{f}}_k(K)e^{ikQ}.$$\nNow we define $$\\bar{g}(Q,K):=\\sum_{k\\in\\mathbb{Z}\\backslash\\{0\\}} \\frac{1}{ik}\\widehat{\\bar{f}}_k(K)e^{ikQ}.$$ Then we see that $\\partial^2_Q \\bar{g}=\\partial_Q \\bar{f}.$\n\nUsing Plancheral's theorem we can easily see that\n$$\\max_{\\ell\\leq 2} \\sup_{K} \\norm{Y^{\\ell} \\bar{g}}_{L^2_Q} \\lesssim \\max_{\\ell\\leq 2} \\sup_{K} \\norm{Y^{\\ell} \\bar{f}}_{L^2_Q}.$$\nFinally, the result follows by taking supremum in $Q$. \\qedhere\n\\end{proof}\n\n\\begin{proposition}\\label{prop:bulk-term}\nFor $f$ satisfying the assumptions of Theorem~\\ref{thm:main}, we have the following estimate:\n$$\\int_0^{x'}\\int_{\\mathbb{R}}vf(t,y,v)\\d v\\d y\\lesssim \\jap{t}^{-2} {\\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} }\\hspace{.1em} \\sum_{|\\alpha|+ |\\beta| \\leq 2} |\\partial_x^\\alpha \\partial_v^\\beta f_0|(x,v).$$\n\\end{proposition}\n\\begin{proof}\n{As in the proof of Proposition~\\ref{prop:term-x=0}, boundedness is obvious and thus it suffices to prove an estimate with $\\langle t\\rangle^{-2}$ replaced by $t^{-2}$.}\n\nWe first note that\n$$\\int_0^{x'}\\int_{\\mathbb{R}}vf(y,v)\\d v\\d y=\\int_0^{x'}\\int_0^\\infty v[f(y,v)-f(y,-v)]\\d v\\d y.$$\nAgain let $$\\bar{f}(t,Q,K) = f(t,x,v).$$\nNext we use the change of variables $(x,v)\\mapsto (Q,K)$, that $v=\\sqrt{2H}\\sin \\chi$ and Lemma~\\ref{lem:Jacobian} followed by the fundamental theorem of calculus to obtain\n\\begin{align*}\n&\\: \\int_0^{x'}\\int_0^\\infty v[f(y,v)-f(y,-v)]\\d v\\d y\\\\\n= &\\: \\int_0^{\\Phi(x')}\\int_0^{\\pi\/2} c(K)\\sqrt{2K}S(Q,K)[\\bar f(Q,K)-\\bar f(-Q,K)]\\d Q\\d {K}\\\\\n&+\\int_{\\Phi(x')}^\\infty\\int_{\\mathfrak{Q}_K}^{\\pi\/2} c(K)\\sqrt{2K}S(Q,K)[\\bar f(Q,K)-\\bar f(-Q,K)]\\d Q\\d {K}\\\\\n=&\\:\\int_0^{\\Phi(x')}\\int_0^{\\pi\/2}\\int_{-Q}^{Q} c(K)\\sqrt{2K}S(Q,K)\\part_Q \\bar f(Q',K)\\d {Q'}\\d Q\\d {K}\\\\\n&+\\int_{\\Phi(x')}^\\infty\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2}\\int_{-Q}^{Q} c(K)\\sqrt{2K}S(Q,K)\\part_Q \\bar f(Q',K)\\d {Q'}\\d Q\\d {K}\\\\\n=: &\\: T_1+T_2,\n\\end{align*}\nwhere we have define\n\\begin{itemize}\n\\item $S(Q,K):=\\sin\\chi$, and  \n\\item $\\mathfrak{Q}_K$ to be the angle in $(Q,K)$ coordinates corresponding to angle $\\arccos\\left(\\frac{\\Phi(x')}{H}\\right)$ in $(\\chi,H)$ coordinates.\n\\end{itemize}\n\nNow using Fubini's theorem, we have \n$$T_1=\\int_{-\\pi\/2}^{\\pi\/2}\\int_0^{\\Phi(x')}\\left(\\int^{\\pi\/2}_{|Q'|} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'}$$\nand \n\\begin{align*}\nT_2&=\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\Phi(x')}^{\\mathfrak{H}_{Q'}}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'}\\\\\n&+\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'},\n\\end{align*}\nwhere $\\mathfrak{H}_{Q'}$ is such that $\\left(\\arccos\\left(\\frac{\\Phi(x')}{\\mathfrak{H}_{Q'}}\\right),\\mathfrak{H}_{Q'}\\right)$ in $(\\chi,H)$ coordinates gets mapped to $(|Q'|,\\mathfrak{H}_{Q'})$ in $(Q,K)$ coordinates  \n(such an $\\mathfrak{H}_{Q'}$ exists because $\\chi=\\arccos\\left(\\frac{\\Phi(x')}{H}\\right)$ increases as $H$ does and $Q$ is monotone\\footnote{Since $a(\\chi,H)>0$, we have that $Q$ is monotonically increasing as a function of $\\chi$ and vice-versa.} in $\\chi$.)\n\nPutting the above together we get,\n\\begin{align*}\nT_1+T_2=&\\int_{-\\pi\/2}^{\\pi\/2}\\int_{0}^{\\mathfrak{H}_{Q'}}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'}\\\\\n&+\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'}.\n\\end{align*}\nNow we use \\eqref{eq:f-to-g} from Lemma~\\ref{lem:f-to-g} and that $\\part_Q=\\frac{1}{c'({K})t}(Y+\\part_{K})$ to get that\n\\begin{align*}\nT_1+T_2=&t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{0}^{\\mathfrak{H}_{Q'}}\\frac{1}{c'({K})}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K} (Y+\\part_{K})\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}\\\\\n&+t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\frac{1}{c'({K})}\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}(Y+\\part_{K})\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}.\n\\end{align*}\n\nNext we integrate by parts in $K$. Since $\\mathfrak{Q}_{\\mathfrak{H}_{Q'}}=|Q'|$, we see that the boundary terms exactly cancel! Hence,\n\\begin{align*}\n&t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{0}^{\\mathfrak{H}_{Q'}}\\frac{1}{c'(K)}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K} \\part_{K}\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}\\\\\n&+t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\frac{1}{c'(K)}\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_{K}\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}\\\\\n&\\quad=-t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{0}^{\\mathfrak{H}_{Q'}}\\part_K\\left(\\frac{1}{c'(K)}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\right)\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}\\\\\n&\\quad -t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\part_K\\left(\\frac{1}{c'(K)}\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\right)\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}.\n\\end{align*}\n\nSince there is no boundary term we can integrate by parts after writing $\\part_Q=\\frac{1}{c'({K})t}(Y+\\part_{K})$ once more. Next note that that $\\bar{g}(Q,K)$ is nonzero only for $K \\in [c_s,c_{s}^{-1}]$ and that derivatives of $\\frac{c(K)}{c'(K)}$ is bounded as $|c'(K)|\\geq \\delta$ by Lemma~\\ref{lem:c-prime-bound}. Futher, $S(Q,K)=\\sin\\chi$ is smooth as a function of $K$. Thus $$\\sum_{\\ell\\leq 2} \\partial_K^\\ell \\left(\\frac{1}{c'(K)}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\right)\\lesssim 1.$$\n\nBy Cauchy--Schwarz in $Q'$ and $K$, we get that\n$$T_1+T_2\\lesssim \\sum_{\\ell\\leq 2}\\sup_{K}\\norm{Y^\\ell \\bar{g}}_{L^2_Q }.$$\nFinally, an application of \\eqref{eq:g-bound} from Lemma~\\ref{lem:f-to-g} followed by Lemma~\\ref{lem:infinity-est} gives us the required bound. \\qedhere\n\n\\end{proof}\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nThe proof follows by using Lemma~\\ref{lem:phi_t} and combining the estimates from Proposition~\\ref{prop:term-x=0} and Proposition~\\ref{prop:bulk-term}. \n\\end{proof}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and outline}\n\\label{sec:intro}\n\nThe goal of this paper is to understand the energy levels of near-extremal charged black holes in D dimensions from the perspective of the Euclidean gravitational path integral. The description of such black holes simplifies due to their  $AdS_2 \\times X_{D-2}$ near-horizon geometry, where $X_{D-2}$ is a compact space, specifying the geometry of the horizon. \n\n\nThe spectrum of near-extremal black holes has been a source of confusion over the years.\\footnote{By near-extremal we mean large charge black holes with a small but non-zero temperature ($T\\ll r_0^{-1}$, where $r_0$ is the horizon radius at extremality). } The first manifestation of this was pointed out by Preskill et al \\cite{Preskill:1991tb} (see also \\cite{Maldacena:1998uz} and \\cite{Page:2000dk}). The thermodynamics derived from a semiclassical evaluation of the gravitational path integral gives a temperature-dependent mass above extremality $E = M-M_0$ as $E \\sim 2\\pi^2 \\Phi_r T^2 $, where $M_0$ is the extremal mass. A similar analysis gives the semiclassical entropy $S=S_0 + 4\\pi^2 \\Phi_r T$, where $S_0=A_0\/(4G_N)$ is proportional to the extremal area $A_0$. This behavior is universal, but the precise value of the parameter $\\Phi_r$ depends on the details of the model. Because of this scaling,  it was noticed in \\cite{Preskill:1991tb} that the thermodynamic description of near-extremal black holes breaks down at low enough temperatures, $T\\lesssim 1\/\\Phi_r$. At such temperatures, the naive semiclassical analysis suggests that emitting even a single Hawking quanta is sufficient to change the black hole's temperature by a large amount.\\footnote{An equivalent issue was pointed out in \\cite{Maldacena:1998uz} due to the dependence of $\\Phi_r$ with Newton constant $G_N$ in particular models.} String theory microstate counting examples \\cite{Maldacena:1996ds} indicate the resolution of this issue is the presence of a mass gap of order $E_{gap}\\sim 1\/\\Phi_r$ in the black hole spectrum.\\footnote{In this paper, we will focus on scales of order $\\sim 1\/\\Phi_r$ also captured by the gravity description. Of course, at a completely different regime of even lower temperatures $T \\sim e^{-S_0}$, we expect to find a discrete spectrum with spacing of order $e^{-S_0}$. This scale requires a UV completion of the gravitational theory.} However, its origin from the gravitational description has, so far, been elusive.\n\nA second manifestation of the gap problem is the question of the extremal black hole degeneracy.  If the gravity description somehow breaks down at the energy scale $E \\sim 1\/\\Phi_r$ then it is possible that the black hole entropy $S_0$, obtained by a semiclassical computation, does not correctly capture the degeneracy of the extremal black hole; rather, $e^{S_0}$ could instead capture the number of extremal and near-extremal states within a $\\sim 1\/\\Phi_r$ energy interval. Nevertheless, while some authors claimed the extremal entropy vanishes \\cite{Hawking:1994ii}, string theory examples that preserve sufficient supersymmetry showed that the degeneracy at extremality (assumed to be captured by an index) matches with the Bekenstein-Hawking area $S_0$ \\cite{Strominger:1996sh}. \n\nIn \\cite{Iliesiu:2020qvm} these questions have been revisited using lessons from SYK \\cite{kitaevTalks, Maldacena:2016hyu} and Jackiw-Teitelboim (JT) gravity \\cite{Teitelboim:1983ux, Jackiw:1984je, Almheiri:2014cka, Jensen:2016pah, Maldacena:2016upp, Engelsoy:2016xyb,Iliesiu:2019lfc, Kapec:2019ecr}. The procedure used to compute the higher dimensional Euclidean path integral for non-supersymmetric black holes is the following. First, we separate the full near extremal geometry into the near-horizon $AdS_2 \\times X_{D-2}$ throat (where the interesting physics takes place) and the far away region, which we take to be \nflat. The second step is to recast the $D$-dimensional theory on the throat as a 2D theory on $AdS_2$. It was argued in \\cite{Iliesiu:2020qvm} that the temperature-dependence of the Euclidean path integral near-extremality is solely captured by a JT gravity theory (composed of fluctuations around the $AdS_2$ metric and a dilaton which captures the size of $X_{D-2}$), a dimensional reduction of $D$-dimensional gauge fields to 2D, and a gauge field originating from isometries of the horizon $X_{D-2}$. This reduction was considered classically in \\cite{Sachdev:2015efa, Almheiri:2016fws, Anninos:2017cnw, Turiaci:2017zwd, Nayak:2018qej, Moitra:2018jqs, Hadar:2018izi, Castro:2018ffi, Larsen:2018cts,  Moitra:2019bub, Sachdev:2019bjn, Hong:2019tsx, Castro:2019crn, Charles:2019tiu,Larsen:2020lhg} but the new feature of \\cite{Iliesiu:2020qvm} is the explanation that quantum effects at low temperatures are also captured by this simple theory. It can be shown that JT gravity exactly reduces to a boundary mode, the Schwarzian theory, which lives in the boundary of the throat (shown in figure \\ref{fig:4DnearBH}), and describes the breaking of the emergent $SL(2,\\mathbb{R})$ symmetry in the throat. In this theory, one can compute the path integral exactly and extract the near-extremal spectrum.\\footnote{Integrating out the massive KK modes and other fields have the only possible effect of introducing temperature-independent logarithmic corrections to $S_0$ previously computed in \\cite{Banerjee:2010qc, Banerjee:2011jp, Sen:2011ba, Sen:2012cj, Iliesiu:2020qvm} and thus such modes will not affect the computation of the density of states.}\n\n\\begin{figure}[t!]\n\\begin{center}\n \\begin{tikzpicture}[scale=1]\n \\pgftext{\\includegraphics[scale=0.6]{nearextBHWiggleCropped.png}} at (0,0);\n  \\draw (-5.5,3.5) node  {\\small Asymptotic flat region};\n    \\draw (-3.9,-1) node[text width=5cm,align=center]  {\\small Near-horizon boundary \\\\  $\\tau \\rightarrow f(\\tau)$ JT mode};\n  \\draw (-2.3,-3.5) node[text width=10cm,align=center]  {\\small Horizon};\n   \\draw (3,-2.3) node[text width=5cm,align=center]  {\\small Near AdS$_2$ region\\\\ Supported by a constant $U(1)$ flux};\n       \n    \\draw (4.5,1.5) node[text width=5cm,align=center]  {\\small Near extremal black hole\\\\ Charge $Q \\sim $ Mass};\n  \\end{tikzpicture}\n  \\caption{\\label{fig:4DnearBH} The near extremal 4D black hole. A similar picture applies to near-extremal rotating BTZ black hole in AdS$_3$.}\n    \\end{center}\n  \\end{figure}\n  \nWith this new perspective, the authors of \\cite{Iliesiu:2020qvm} addressed the puzzling questions about the thermodynamics of non-supersymmetric near-extremal Reissner-Nordstr\\\"om black holes. Regarding the first confusion pointed out above, the scale $T\\sim 1\/\\Phi_r$ identified by \\cite{Preskill:1991tb} was found to be the scale at which quantum effects in the gravitational path integral become large, and the Schwarzian description becomes strongly coupled. This effect signals that the breaking of the $SL(2, \\mathbb R)$ conformal symmetry in the throat becomes important \\cite{Almheiri:2016fws}. However, in contrast with string theory proposals \\cite{Maldacena:1996ds}, it was found that for non-supersymmetric black holes, there is no gap at the scale $E\\sim 1\/\\Phi_r$ and the gravitational path integral predicts a density of states that goes smoothly to zero as $E\\to0$. For convenience, this density of states is reproduced here in figure \\ref{fig:plot-of-rho} in the left column. Regarding the second confusion about the ground state degeneracy of the extremal black hole, \\cite{Iliesiu:2020qvm} also predicts the entropy of the extremal black hole to be much smaller than the naive prediction given by the area of the extremal horizon. A similar conclusion was obtained for near-extremal rotating black holes in 3D gravity \\cite{Ghosh:2019rcj,Maxfield:2020ale}.\n \n \n  \nThe results of \\cite{Iliesiu:2020qvm} thus show a very different spectrum of near-extremal black holes compared to results obtained from microscopic counting: string theory predicts that the ground state degeneracy of extremal black holes should agree with Bekenstein-Hawking extremal area \\cite{Strominger:1996sh} and that the value of the mass gap  $E_{gap}\\sim 1\/\\Phi_r$ can be extrapolated from the weak coupling regime  \\cite{Maldacena:1996ds}. The purpose of this paper is to resolve this puzzle. The solution is that most examples in string theory involve supergravity and the extremal black hole preserves some supersymmetries. In the cases studied in this paper, the temperature dependence of the Euclidean path integral is captured by a supersymmetric generalization of JT gravity, which describes the breaking of an emergent superconformal symmetry.\n\n\n In this paper, we will discuss the case of near-BPS near-extremal charged black holes in two setups. The first is in four dimensions ($D=4$), where such objects are described classically by the Reissner-Nordstr\\\"om solution with a $AdS_2 \\times S^2$ throat.   We will consider such black holes in 4D ungauged $\\cN=2$ supergravity in asymptotically flat space \\cite{Freedman:1976aw}.  The situation is depicted in Figure~\\ref{fig:4DnearBH}. In our conventions $M_0=Q\/\\sqrt{G_N}$, proportional to the charge of the black hole, and $\\Phi_r= \\sqrt{G_N}\\hspace{0.1cm} Q^3$. Such a 4D theory has the right ingredients for SUSY preserving extremal black holes \\cite{Gibbons:1982fy,Tod:1983pm,Ferrara:1995ih}. The BPS nature of such gravitational solutions allows one to identify them with corresponding string theory constructions, for example \\cite{Maldacena:1997de, Ooguri:2004zv,Simons:2004nm,Denef:2007vg}. The second setup we will consider is a near-extremal rotating black hole in $(4,4)$ supergravity in $AdS_3$, with a near-horizon geometry $AdS_2 \\times S^1$. This system is relevant for comparison with the D1\/D5 system \\cite{Strominger:1996sh}. Even though some questions about the spectrum can be addressed using solely the $(4,4)$ super-Virasoro algebra in this particular case thanks to the presence of an $AdS_3$ throat, we will be able to show in section \\ref{sec:ads3} that the spectrum matches with the one derived from JT gravity. The Virasoro analysis obviously does not generalize to other cases, for example the Reissner-Nordstr\\\"om black hole mentioned above, but the JT analysis does and is universal as we explain below.\n\n  \nThe main feature in 4D $\\cN=2$ supergravity and $(4,4)$ supergravity in $AdS_3$, compared to Einstein gravity, is the fact that the emerging symmetry in the throat becomes the superconformal group $PSU(1,1|2) \\supset SL(2,\\mathbb{R})\\times SU(2)$. In 4D, this includes the $AdS_2$ conformal symmetry and the $S^2$ isometries as bosonic subgroups \\cite{Kallosh:1997qw, Boonstra:1998yu,Claus:1998yw,Michelson:1999kn}. In $AdS_3$ the $SU(2)$ arises from a 3D gauge field. In the examples from string theory there is a second $SU(2)$ gauge field coming from the isometries of an extra $S^3$ factor. This second $SU(2)$ charge can be turned on without breaking supersymmetry, and corresponds to the black hole of \\cite{Breckenridge:1996is} which has rotation along the $S^3$. We find in both cases the relevant 2D mode in the throat controlling the temperature dependence of the partition function is given by $\\cN=4$ super-JT gravity, which describes the breaking of $PSU(1,1|2)$ and becomes strongly coupled at low temperatures. We will first define this theory and solve it exactly to extract the temperature dependence of the partition function, and from it the black hole spectrum. In order to do this, we will show that $\\cN=4$ super-JT gravity is equivalent to a $\\cN=4$ super-Schwarzian theory, which we introduce in this paper. We solve this theory using either path integral localization or canonical quantization.\\footnote{Previous attempts of defining the $\\mathcal{N}=4$ super-Schwarzian are \\cite{Aoyama:2018lfc, Aoyama:2018voj, Galajinsky:2020hsy} however, explicit formulae for all the components of the  $\\mathcal{N}=4$ super-Schwarzian were not presented. Furthermore, to our knowledge, no previous quantization attempt has been made. }\n  \n\\begin{figure}[t!]\n\\begin{center}\n \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.46]{dosem.pdf}} at (0,0);\n \\draw (-5.5,-0.4) node  {$e^{S_0}$};\n  \\draw (-5.5,2.75) node  {\\small $\\rho(E)$};\n  \\draw (-1.7,-3.2) node  {\\small $\\sim 1\/\\Phi_r$};\n   \\draw (4.55,-3) node  {\\small $E$};\n      \\draw (-1,-4.5) node  {\\small (a) Einstein gravity, $J=0$};\n  \\end{tikzpicture}\n  \\hspace{0.5cm}\n   \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.51]{dossugra.pdf}} at (0,0);\n \\draw (-5.7,-0.4) node  {$e^{S_0}$};\n   \\draw (-5.8,2.75) node  {\\small $\\rho(E)$};\n  \\draw (-1.7,-3.2) node  {\\small $E_{gap}= \\frac{1}{8 \\Phi_r}$};\n   \\draw (5,-3) node  {\\small $E$};\n     \\draw (-0.5,-4.5) node  {\\small (b) SUGRA, $J=0$};\n  \\end{tikzpicture}\n \\\\\n \\vspace{0.8cm}\n   \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.46]{dosj1.pdf}} at (0,0);\n \\draw (-5.5,-0.4) node  {$e^{S_0}$};\n  \\draw (-5.5,2.75) node  {\\small $\\rho(E)$};\n  \\draw (-1.7,-3.2) node  {\\small $J(J+1)\/2\\Phi_r$};\n   \\draw (4.55,-3) node  {\\small $E$};\n      \\draw (-1,-4.5) node  {\\small (c) Einstein gravity, $J\\neq0$};\n  \\end{tikzpicture}\n  \\hspace{0.5cm}\n   \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.51]{dosj1susy.pdf}} at (0,0);\n \\draw (-5.7,-0.4) node  {$e^{S_0}$};\n   \\draw (-5.8,2.75) node  {\\small $\\rho(E)$};\n  \\draw (-1.7,-3.2) node  {\\small $J^2\/2\\Phi_r$};\n   \\draw (5,-3) node  {\\small $E$};\n     \\draw (-0.5,-4.5) node  {\\small (d) SUGRA, $J\\neq0$};\n  \\end{tikzpicture}\n  \\caption{\\label{fig:plot-of-rho} Schematic shape of the black hole spectrum at fixed $SU(2)$ charge $J$ as a function of energy above extremality $E$. In 4D $J$ is angular momentum while in $AdS_3$ it is the $SU(2)$ charge (one of the two rotations along extra $S^3$ in string theory) that breaks supersymmetry. We show the semiclassical answer (red dashed) and the solution including quantum effects (purple). (a) Answer for Einstein gravity. We see there is no gap at scale $E\\sim 1\/\\Phi_r$ and the extremal entropy goes to zero. (b) Answer for supergravity (either $\\cN=2$ in 4D or $\\cN=(4,4)$ in 3D). We find a gap at the scale $E_{gap}=\\frac{1}{8\\Phi_r}$ and a number $e^{S_0}$ of extremal states, consistent with string theory expectations. (c) Einstein gravity spectrum for $J\\neq 0$. (d) Supergravity spectrum for $J\\neq 0$, the jumps indicate contributions from different supermultiplets $\\mathbf{J}, \\mathbf{J+1\/2}$ and $\\mathbf{J+1}$.}\n    \\end{center}\n  \\end{figure}\n\nTo contrast the results found in this paper, we show in Figure~\\ref{fig:plot-of-rho} the spectrum of near-extremal black holes derived from Einstein gravity in 4D coupled to a Maxwell field in the left panel. As previously mentioned, we see a small extremal entropy and a lack of a gap. In the right column, we show the main result of this paper, the density of states for near-extremal black holes in 4D $\\cN=2$ supergravity and $(4,4)$ supergravity in asymptotically $AdS_3$. Through a computation of the Euclidean path integral, independent of the UV completion of the theory, our analysis verifies and predicts the following results:\n\\begin{itemize}\n\\item We find that extremal BPS states exhibit an exact degeneracy. This degeneracy is given by the Bekenstein-Hawking horizon area, which is consistent with extremal microstate countings in string theory \\cite{Strominger:1996sh}. This is not true for extremal non-BPS states with $J\\neq 0$.\n\n\\item We observe the presence of a mass gap $E_{gap}=1\/(8\\Phi_r)$. In the context of $(4,4)$ supergravity in asymptotically $AdS_3$ and the D1\/D5 system, we verify the mass gap estimated at weak coupling from long strings \\cite{Maldacena:1996ds}. Here, this prediction is also expanded to black holes in 4D $\\cN=2$ supergravity. \n\n\\item We find that the extremal states are solely bosonic, implying the Witten index from a microscopic model coincides with the black hole degeneracy (For previous arguments see for example \\cite{Sen:2009vz, Mandal:2010cj, Benini:2015eyy}). This implies that the typical counting of microstates in string theory using an index, such as \\cite{Strominger:1996sh}, is correct.\n\n\\item A previous attempt to obtain the gap from a gravitational perspective was studied by Maldacena and Strominger \\cite{Maldacena:1997ih}. Their argument is only correct for black holes in AdS$_3$, and not for Reissner-Nordstr\\\"om for example. Our analysis that takes into account the breaking of $PSU(1,1|2)$ can in principle be applied to any system with this pattern of symmetry breaking and applies to situations without an $AdS_3$ factor in the throat.\n\n\\item In combination with the prior non-supersymmetric results from \\cite{Iliesiu:2020qvm}, we conclude that these features found in string theory examples heavily rely on supersymmetry and are special to supergravity. \n\\end{itemize}\n\n\n\nWithout jumping into all the details, we will give a summary of the technical results derived below. The $\\cN=4$ super-Schwarzian theory describing the spectrum of supersymmetric black holes, is given by the following action\n\\beq\n\\label{eq:intro-N=4-Schw-action}\nI_{\\cN=4} =-\\Phi_r \\int d\\tau \\left[ \\text{Sch}(f,\\tau) + {\\rm Tr} \\left( g^{-1} \\partial_\\tau g\\right)^2 + ({\\rm fermions}) \\right],\n\\eeq\nwhere the Schwarzian derivative is defined as \n\\beq\n\\text{Sch}(f,\\tau) \\equiv \\frac{\\partial_\\tau^3 f}{\\partial_\\tau f} - \\frac{3}2 \\left(\\frac{\\partial_\\tau^2 f}{\\partial_\\tau f} \\right)^2.\n\\eeq\nIn the action \\eqref{eq:intro-N=4-Schw-action},  $f(\\tau)$ is a reparametrization mode, $g(\\tau)$ is a $SU(2)$ element, and we ignore the terms involving fermionic fields $\\eta(\\tau)$ and $\\bar{\\eta}(\\tau)$ in the doublet of $SU(2)$. The field $g(\\tau)$ describes fluctuations in the angular momentum $J$ of the black hole in the 4D setup comes from the isometries of $S^2$. This theory has $\\cN=4$ Poincar\\'e supersymmetry but \\textbf{breaks} the emergent superconformal $PSU(1,1|2)$ space-time symmetry.\\footnote{Nonetheless, $PSU(1,1|2)$ is an important global symmetry for the action \\eqref{eq:intro-N=4-Schw-action}. We will clarify this in section \\ref{sec:space-time-and-global-symm}.} According to the unbroken $\\cN=4$ Poincar\\'e supersymmetry, supermultiplets organize into $\\mathbf{J}=(J)\\oplus2(J-\\frac{1}{2})\\oplus(J-1)$ for $E\\neq0$ and states $(J)$ for $E=0$. Solving this theory exactly gives the density of supermultiplet states labeled by their highest angular momentum $J$ as\n\\beq\n\\rho(J,E) = e^{S_0} \\delta_{J,0}\\delta(E)+ \n\\frac{e^{S_0}J}{4\\pi^2 \\Phi_r E^2}\\sinh \\left(2 \\pi \\sqrt{2\\Phi_r(E-E_0(J))} \\right)\\hspace{0.1cM} \\Theta\\Big(E-E_0(J)\\Big)\\,, \n\\eeq\nwhere we define $E_0(J)\\equiv J^2\/(2\\Phi_r)$. The details of this formula can be found in section \\ref{sec:N=4-super-Schw}. For example, states with zero angular momentum come from the contributions with $J=0$, $J=1\/2$ since $\\mathbf{1\/2}=(1\/2)\\oplus 2 (0)$ and $J=1$ since $\\mathbf{1}=(1)\\oplus 2(1\/2)\\oplus (0)$. This is the origin of the plot in the right panel of figure \\ref{fig:plot-of-rho}. In the continuous part states with angular momentum $J$ start at an energy $E_0(J)$. This is not necessarily surprising since the same feature appears in the non-supersymmetric case \\cite{Iliesiu:2020qvm}. From the gravity perspective this is the correction to the extremality bound of non-BPS extremal charged slightly-rotating black holes. The surprising feature in the supersymmetric theory is that there are no states with zero angular momentum in the range $0<E<1\/(8\\Phi_r)$. \n\n\n\nThe results of \\cite{Iliesiu:2020qvm} and the present work on the $\\cN=4$ super-Schwarzian theory can be seen as a low energy effective theory for black holes without and with supersymmetry, respectively. We are obtaining this effective theory from a detailed analysis of the gravitational path integral in specific examples, but we expect it to be universal and depend only on the pattern of symmetry breaking at low temperatures. We leave for future work the extensions of these ideas to cases such as near-BPS black holes in gauged supergravity in asymptotically $AdS_4$ or $AdS_5$, relevant within the AdS\/CFT correspondence. The symmetries suggest the near-extremal spectrum is described by the $\\cN=2$ super-Schwarzian theory introduced in \\cite{Fu:2016vas},\\footnote{To our knowledge, such a super-Schwarzian cannot describe the spectrum of near-BPS black holes in flatspace because the number of supercharges that are preserved by the extremal solution is a multiple of $4$ and 1D $\\cN=2$ super-Poincar\\'e only has two supercharges.} describing the breaking of $PSU(1,1|1) \\supset SL(2,\\mathbb{R})\\times U(1)$.\\footnote{We describe properties of the spectrum of the $\\cN=2$ super-Schwarzian in Appendix \\ref{app:N2}. Progress on relating the dynamics of near-BPS black holes in  $AdS_4$ to this super-Schwarzian theory was reported in \\cite{Forste:2020xwx}. Such a theory is, of course, also relevant when studying near-extremal black holes in $(2, 2)$ supergravity in AdS$_3$.} Such a super-Schwarzian theory sometimes exhibits a gap (depending on the value of the fundamental $U(1)$ R-symmetry charge and whether or not a certain mixed anomaly is present in the theory) but, in contrast to the $\\cN=4$ super-Schwarzian, does not generically have purely bosonic ground states.\n\nFinally, our investigations are relevant to the relation between 2D gravity and SYK models. While SYK models with $\\cN=1$ and $\\cN=2$ supersymmetry were constructed in \\cite{Fu:2016vas, Murugan:2017eto}, finding models with $\\cN=4$ has been elusive. We hope understanding $\\cN=4$ super-JT and super-Schwarzian can help find an SYK model with $\\cN=4$ supersymmetry, which is described at low temperatures by the $\\cN=4$ super-Schwarzian theory we described. We leave this for future work, but discuss difficulties in constructing such theories towards the end of the paper. \n\n\n\n\nThe rest of the paper is organized as follows. In section \\ref{sec:N=4superJT} we introduce $\\cN=4$ supersymmetric Jackiw-Teitelboim gravity, a 2D dilaton-gravity theory. We will use its BF description to argue that the disk partition function is computed by an $\\cN=4$ version of the Schwarzian theory living on the boundary. In section \\ref{sec:N=4-super-Schw}, we will study this boundary mode in its own right and compute its partition function and density of states as a function of energy and charge. We will explain in all details the features briefly described above. In section \\ref{sec:higherD}, we will perform the dimensional reduction in the throat of a 4D near extremal black holes in $\\cN=2$ ungauged supergravity and argue the resulting theory is $\\cN=4$ super-JT gravity. Using the results of section \\ref{sec:N=4-super-Schw}, we will present the final result for the spectrum of near-extremal black holes in supergravity. In section \\ref{sec:ads3} we analyze near-BPS near-extremal black holes in $(4,4)$ supergravity in asymptotically $AdS_3$ spaces, reaching similar conclusions regarding the black hole spectrum. This example helps to make a precise connection with the long string origin of the black hole gap in \\cite{Maldacena:1996ds}. In section \\ref{sec:conclusion} we conclude and discuss future directions.\n\n\n\\section{ $\\cN = 4$ Jackiw-Teitelboim gravity and the $\\cN = 4$ super-Schwarzian}\n\\label{sec:N=4superJT}\n\nIn order to be able to study the more complicated case of near-extremal black holes in 4D $\\cN=2$ supergravity, we first need to introduce 2D $\\cN = 4$ super-JT gravity and its relation to the $\\cN=4$ super-Schwarzian. As we shall explain, the latter can be studied using a conventional reformulation in terms of a BF theory. The difficulty will be in relating boundary conditions in this BF theory to SUSY preserving boundary conditions, typically imposed in the second formalism of gravity. The analysis of the $\\cN=4$ super-Schwarzian proves even more complicated. While there have been past papers defining the $\\cN=4$ super-Schwarzian derivative (in particular, we follow the definitions of \\cite{Matsuda:1988qf} and \\cite{Matsuda:1989kp}), extracting all the components of the super-symmetric action has, to our knowledge, not been previously done. \n\n\n\\subsection{Formulation as a $\\mathfrak{psu}(1,1|2)$ BF theory}\n\\label{sec:PSU(1,1|2)-BF-thy}\n\n To obtain the bulk action of $\\cN = 4$ super-JT gravity we start by considering a BF theory with a $\\mathfrak{psu}(1,1|2)$ superalgebra, \n\\be \n\\label{eq:psu(1,1|2)-superalgebra}\n[L_m, L_{n}] &=(m-n)L_{m-n}\\,, \\qquad \n[T_i, T_j] = i \\epsilon_{ijk} T_k\\,,\\nn \\\\\n[L_m, G_p{}^\\a] &=  \\left(\\frac{m}2 - \\a\\right) G_p{}^{\\a+m}\\,, \\qquad [L_m, \\bar G^p{}_{\\a}] = \\left(\\frac{m}2 - \\a\\right)\\bar  G^p{}_{\\a+m}\\,,\\nn \\\\\n[T_i,G_p{}^\\a] &= -\\frac{1}2 \\left(\\sigma^i\\right)_p{}^q G_{q}{}^\\a\\,, \\qquad [T_i,\\bar G^p{}_\\a] = \\frac{1}2 \\left(\\sigma^i\\right)^*{}^p{}_q G^{q}{}_\\a\\,,\\nn \\\\\n\\{G_p{}^\\a, \\bar G^q{}_\\b\\}&=  2 \\delta^q{}_{p} L_{\\a+\\b}-2(\\a-\\b)(\\sigma^i)^q{}_p T_i\\,,\n\\ee\nwith all other commutators\/anti-commutators between the generators vanishing. Here $L_0$ and $L_{\\pm 1}$ are bosonic generators forming an $\\mathfrak{sl}(2, \\mR)$ sub-algebra, $T_i$, $i = 1,\\,\\dots \\,3$, are bosonic generators forming an $\\mathfrak{su}(2)$ sub-algebra, and $G_p{}^\\a$ and $\\bar G^q{}_\\b$ are the fermionic generators with $p,q = 1, 2$ and $\\a, \\b = -\\frac{1}2, \\, \\frac{1}2$. \n\nThe action of $\\cN = 4$ super-JT gravity  is given by, \n\\be\nI_{BF} =- i \\int \\text{Str}\\, \\phi F\\,, ~~~~ F = dA - A\\wedge A\\,,\n\\ee\nwhere $A$ is a $\\mathfrak{psu}(1,1|2)$ gauge field with a field strength $F$ and $\\phi$ is a zero-form field transforming in the adjoint of $\\mathfrak{psu}(1,1|2)$. Above,  $\\text{Str}\\, \\phi F = \\< \\phi,\\,F\\>$ is a quadratic bilinear form invariant under adjoint transformations.  Such a form can be obtained from the quadratic Casimir of $\\mathfrak{psu}(1,1|2)$.\\footnote{The quadratic Casimir of the superalgebra can be written as, \\be\nC_2 = L_0^2 - \\frac{1}2 \\{L_1, L_{-1}\\} - T_i T^i +\\frac{1}4  (i \\sigma^2)^\\b{}_\\a [G_p{}^\\a ,\\bar G^{\\,p}{}_\\b],.\\nn\n\\ee\nRewriting $C_2 = g_{AB} X^A X^B$ with $X = \\{L_0, L_\\pm,\\,T_i,\\,G_p{}^\\a,\\, \\bar G^{\\,p}{}_\\a\\}$, we can define the invariant quadratic form by using the metric, $\n \\eta_{AB}^{\\mathfrak{psu(1,1|2)}} = (-1)^{[X_A]} (g_{AB}^{-1})\n$, \nwhere $[X_A] = 0,1$ for bosonic, and respectively, fermionic generators. In such a case, we can define the supertrace as, $\n\\text{Str} B F = \\< B, \\, F\\> = \\eta_{AB} B^A F^B$.\n} We define the gauge field in terms of the supermultiplet of the frame $e^a$ and spin connection $\\omega$, also consisting of the SU(2) gauge field $B^i$ and the gravitinos $\\psi_{p}{}^{ \\a}$ as:\\footnote{We choose such conventions for the $\\mathfrak{sl}(2, \\mR)$ components of the gauge field in order to agree with \\cite{Saad:2019lba}. }\n\\be \n\\label{eq:gauge-field-ansatz}\nA(x) &= \\sqrt{\\frac{\\Lambda}{2}} \\left[e^1(x) L_0 + \\frac{e^2(x)}2 \\left(L_{1} - L_{-1}\\right) \\right]- \\frac{\\omega(x)}2 \\left(L_1+L_{-1}\\right) + B^i(x) T_i + \\nonumber \\\\ &+   \\left(\\frac{\\Lambda}{2}\\right)^{\\frac{1}{4}}\\left(\\bar{\\psi}^{p}{}_{ \\a}(x) G_{p}{}^{ \\a} +  \\psi_{p}{}^{ \\a}(x) \\bar{G}^p{}_{ \\a}\\right)\\,.\n\\ee\nThe zero-form field $\\phi(x)$ is fixed in terms of the supermultiplet of the $SL(2, \\mR)$ Lagrange multipliers $\\phi^{1, 2}$ and $\\phi^0$, \n\\be\n\\label{eq:Lagr-multiplier}\n\\phi(x) &= 2\\phi^1(x) L_0 + {\\phi^2(x)} \\left(L_1 - L_{-1} \\right)- {\\phi^0(x)} \\left(L_1+L_{-1}\\right) + b^i (x) T_i + \\nn \\\\ &+ \\left(\\frac{\\Lambda}{2}\\right)^{-\\frac{1}{4}}\\left(\\bar{\\lambda}^{p}{}_{ \\a }(x) G_{p}{}^{ \\a}+   \\lambda_{p}{}^{ \\a }(x) \\bar G^{p}{}_{ \\a}\\right)\\,.\n\\ee\nHere, $\\lambda$ and $\\bar \\lambda$, and $\\psi$ and $\\bar \\psi$ should be understood as independent components of $A$ and are not related to the Hermitian conjugates of each other. In such a case, the action can be written as,\n\\be\n\\label{eq:JT-N=4-action}\nI_{JT}^{\\cN=4}& = {i} \\int \\bigg( \n\\underbrace{\\phi^0}_{\\sim\\text{ Dilaton}}\\bigg[\\underbrace{d\\omega + \\frac{\\Lambda}4 \\epsilon_{ab} e^a \\wedge e^b}_{\\frac{d^2 x}2 \\sqrt{g} (R+\\Lambda)} -\\underbrace{\\sqrt{2\\Lambda}\\bar{\\psi}^p{}_\\alpha \\wedge \\psi_p {}^\\alpha }_{\\substack{\\text{Gravitino $\\psi_p{}^\\a$ contribution}\\\\\\text{multiplying the dilaton}}}\\bigg]  \\nn \\\\ &-  \\sqrt{\\frac{\\Lambda}{2}}\\underbrace{\\phi^a}_{\\substack{\\text{Super-torsion}\\\\ \\text{Lagr.~multiplier}}}\\underbrace{\\left[de^a - \\epsilon_{ab}\\,\\omega\\wedge e^b +2\\left(\\bar \\gamma_a\\right)^{\\alpha}{}_{\\beta}\\bar{\\psi}^{p}{}_\\alpha\\wedge \\psi_p{}^{\\beta} \\right]}_{\\text{super-torsion component }\\tau^a} \\\\ \n&-  \\tr_{SU(2)} \\underbrace{\\left[b \\left(dB + B \\wedge B \\right)\\right]}_{SU(2)\\text{ BF theory}} \\,\\,+ \\,\\, \\sqrt{\\frac{\\Lambda}{2}}\\underbrace{b^q{}_p \\bar{\\psi}^p{} \\wedge \\bar{\\gamma}_3\\psi_q{} }_{SU(2)\\text{ BF super-partner}}\\nn \n+ \\, \\underbrace{2\\lambda\\mathcal{D} \\bar{\\psi} +2\\mathcal{D}^*\\psi \\bar{\\lambda}}_{\\substack{\\text{Gravitino $\\psi^p{}_\\a$ and }\\\\\\text{dilatino $\\lambda_p{}^\\a$ contribution}}}\\bigg)\\,, \n\\ee\nwhere we have outlined the important terms in the action which will ease the comparison with the near-horizon action which we shall obtain in section \\ref{ssec:4dSugra}.\\footnote{Above, our normalization of the $SU(2)$ trace is given such that $\\tr_{SU(2)}(T^i T^j) = -\\frac{1}2 \\delta^{ij}$.  The covariant derivative is given by $\\mathcal{D}=\\bar{\\gamma}_3d+ \\sqrt{\\frac{\\Lambda}{2}}\\bar{\\gamma}_ae^a+\\frac{1}{2}\\omega+\\frac{\\bar{\\gamma}_3}{2}B^i (\\sigma^i)$, and $\\cD^*$ is the conjugate of $\\cD$. We choose $\\bar{\\gamma}_1=\\sigma_1,\\,\\, \\bar{\\gamma}_2=-\\sigma_3$, and $\\bar \\gamma_3 = \\bar \\gamma_1 \\bar \\gamma_2 = i\\sigma_2$. }\n\n\nThe equations of motion for $\\phi^{1,2}$ act as Lagrange multipliers and imposes that the super-torsion vanishes. After integrating out $\\phi^{1,2}$, one can replace $e^1$, $e^2$ and $\\omega$ in terms of the metric $g_{\\mu \\nu}$ to obtain the supergravity action \\eqref{eq:JT-N=4-action} in the second order formalism. The JT gravity dilaton is given by $-i \\phi^0 \\equiv \\Phi$. When comparing the gravitational theory obtained from the dimensional reduction of the near-horizon region of near-BPS black holes to the $\\cN=4$ super-JT action we will use this latter form. For simplicity, in the remainder of this section we will fix the cosmological constant, $\\Lambda = 2$. Later, when discussing the effective action for the near-horizon region of the near-BP black holes we will revert to a general cosmological constant, determined by the radius of the black hole.\n\n\nWe can complete the dictionary between the BF theory \\eqref{eq:JT-N=4-action} and the second-order super-JT gravity by noting that  infinitesimal $PSU(1,1|2)$ gauge transformations in the former, map to infinitesimal diffeomorphisms in the latter. In particular, the infinitesimal supersymmetric transformations on all the fields in \\eqref{eq:JT-N=4-action} can be obtained by considering the infintesimal gauge transformation whose gauge parameter takes the form $\\Lambda = \\epsilon^{p\\a} G_{p \\a} + \\bar \\epsilon^{p \\a} \\bar G_{p \\a}$. \n\n\nWith this mapping between the BF theory \\eqref{eq:JT-N=4-action} and $\\cN=4$ super-JT gravity in mind, we can now analyze the supersymmetric boundary conditions applied to the theory  \\eqref{eq:JT-N=4-action} which will be necessary in section \\ref{sec:higherD} to understand the gluing of the near-horizon region of near-BPS black holes to asymptotic flatspace. As we will see, these boundary conditions will  reduce the gravitational path integral to that of the $\\cN=4$ super-Schwarzian. \n\n\n\n\\subsection{The super-JT  boundary conditions}\n\\label{sec:super-JT-bdy-cond}\n\nWe begin, just like in the non-supersymmetric case, by imposing that the boundary metric is fixed and proper boundary length is large, $L = \\beta\/\\varepsilon$, with $\\varepsilon$ small. Working in Fefferman-Graham gauge, where the metric can be written as\n\\be \n\\label{eq:FG-gauge-metric}\nds^2 = dr^2 + \\left(\\frac{1}4 e^{2r} - \\tilde \\cL(\\tau) + \\dots \\right) d\\tau^2 \\,,\n\\ee\nwe consider the boundary to be at fixed, but large, $r$. To satisfy the boundary conditions we identify $\\tau \\sim \\tau+\\beta$ and cut-off the geometry at $e^{-r} = \\varepsilon\/2$. We fix the leading component of the boundary metric and allow the sub-leading component, $\\tilde \\cL(\\tau)$, to vary. The $\\dots$ represent terms that are further sub-leading in $r$ which we do not need to fix.\n\nSimilarly, we require that the dilaton takes an asymptotically large and constant value at the boundary, $\\Phi|_{\\partial \\cM} \\equiv -i\\phi_0|_{\\partial \\cM} = \\Phi_r\/\\varepsilon$. Next, we discuss the boundary conditions for the super-partners of the frame and spin-connection. If working in the  Fefferman-Graham gauge \\eqref{eq:FG-gauge-metric}, then, in order to preserve supersymmetry, we impose that the leading component of the gravitino is fixed and vanishes $\\psi^{p \\alpha} = O(e^{-r\/2})$ and $\\bar \\psi^{q \\alpha} = O(e^{-r\/2})$. Similarly, to again preserve supersymmetry, we impose that the leading contribution of the dilatino vanishes at asymptotically large values $\\lambda  = O(e^{-r\/2})$. Finally, we describe the boundary conditions for the $SU(2)$ gauge field and its Lagrange multiplier. From the perspective of the higher dimensional  black holes which we will study in section \\ref{sec:higherD}, the $SU(2)$ gauge field appears from the isometry of $S^2$ along which we perform the dimensional reduction. We would therefore, like to fix Dirichlet boundary conditions at the boundary of the asymptotically flat region. As we will describe in detail in section \\ref{sec:higherD}, imposing these boundary conditions in the asymptotically flat region translates to fixing a linear combination of the zero-form field $b$ and the $SU(2)$ gauge field $B$ at the boundary of the near-horizon region. \n\n\nWe would now like to translate the boundary conditions discussed above in the second-order formalism to boundary conditions in the BF theory \\eqref{eq:JT-N=4-action}. We will follow the steps outlined in \\cite{Grumiller:2017qao} (explained also recently in \\cite{Saad:2019lba}) in non-supersymmetric JT gravity and will obtain results similar to \\cite{Cardenas:2018krd}, which studied boundary conditions in JT supergravity with an $OSp(2, \\cN)$ isometry group. Fixing the Fefferman-Graham gauge for the metric \\eqref{eq:FG-gauge-metric} yields the value of the frame $e^{1, \\, 2}$ and spin connection $\\omega$ \\cite{Saad:2019lba}\n\\be \n\\label{eq:FG-1st-order-formalism}\ne^1 = dr \\,, \\qquad e^2= \\left(\\frac{1}2 e^r - \\tilde \\cL(\\tau) e^{-r}\\right)d\\tau \\,, \\qquad \\omega  = -\\left(\\frac{1}2 e^r + \\tilde \\cL(\\tau) e^{-r}\\right)d\\tau \\,.\n\\ee \nFixing the decaying piece of the gravitino we can gauge fix all other components of the  $PSU(1,1|2)$ gauge field along the boundary to be\n\\be\n\\label{eq:PSU(1,1|2)-gauge-field-form}\nA_\\tau(\\tau) &= L_1 \\frac{e^r}2 + L_{-1} \\tilde \\cL(\\tau) e^{-r} + B^i_\\tau(\\tau) T_i \\nn \\\\ &+ e^{-\\frac{r}2} \\frac{\\bar\\psi^p(\\tau)}2  G_{p, \\, -\\frac{1}2} + e^{-\\frac{r}2} \\frac{\\psi^p(\\tau)}2 \\bar G_{p,\\, -\\frac{1}2} + O(e^{-2r})\\,,\n\\ee\nwhere we have used the shorthand notation $\\psi^{p}(\\tau) \\equiv \\psi^{p,\\, \\frac{1}2} $ and $\\bar \\psi^{p}(\\tau) \\equiv \\bar \\psi^{p,\\, \\frac{1}2} $ on the boundary $\\partial \\cM$. \nWe can now impose the equations of motion of $A_\\tau$ in the proximity of the boundary $D_\\tau \\phi = 0$, or rather due to only fixing the leading orders of $A_\\tau$ and $\\phi$, $D_\\tau \\phi = O(e^{-r})$, where $D$ denotes the $PSU(1,1|2)$ covariant derivative. This implies that the zero-form field $\\phi$ should be constrained to take the form \n\\be\n\\label{eq:b.c.-for-dilaton}\n\\phi(\\tau)  &= -{i\\Phi_r} L_1 e^r +  {2 i\\Phi_r'} L_0 +(\\bar \\psi \\lambda + \\psi \\bar \\lambda -  2i \\tilde\\cL \\Phi_r - 2i\\Phi_r'')L_{-1} e^{-r} + b^i(x) T_i \\nn \\\\&+ \\bar \\lambda^p G_{p,\\frac{1}2} e^{r\/2} - \\bigg(2(\\bar \\lambda^p)' - i\\Phi_r \\bar \\lambda^p  -  B_\\tau^i (\\sigma^i{}^*)^p{}_q\\, \\bar \\lambda^q\\bigg)  G_{p,-\\frac{1}2}e^{-r\/2} \\nn \\\\ &+ \\lambda^p \\bar G_{p,\\frac{1}2}e^{r\/2} - \\bigg(2 (\\lambda^p)' - i \\Phi_r  \\lambda^p  -  B_\\tau^i (\\sigma^i\\,)^p{}_q\\,  \\lambda^q\\bigg)  \\bar G_{p,-\\frac{1}2} e^{-r\/2} + O(e^{-2r})\\,,\n\\ee\nwhere, again, we have used the short-hand notation $\\lambda^{p}(\\tau) = \\lambda^{p, \\frac{1}2}$ and $\\bar \\lambda^{p}(\\tau) =\\bar \\lambda^{p, \\frac{1}2}$ on the boundary $\\partial \\cM$ and where $\\Phi'_r \\equiv \\partial_\\tau \\Phi_r(\\tau)$ denotes the derivative with respect to the boundary time $\\tau$.\\footnote{We will motivate why the parameter $\\Phi_r$ in \\eqref{eq:b.c.-for-dilaton} can be identified with the renormalized dilaton shortly.}\n\nIf we want to impose the boundary conditions for the dilaton ($\\Phi_r = \\text{const}$), the dilatinos ($\\lambda^{p, \\,\\frac{1}2} = 0$ and $\\bar \\lambda^{p,\\,\\frac{1}2} =0$) and the zero-form field $b^i(x)$ we can then relate the gauge field in \\eqref{eq:PSU(1,1|2)-gauge-field-form} to the zero-form field $\\phi(\\tau)$ as $-2i \\Phi_r \\hspace{0.1cm}A_\\tau(\\tau) = \\phi(\\tau)$, where $\\Phi_r$ is the renormalized value of the dilaton. Thus, in the first-order formalism and in the gauge in which $A_\\tau$ takes the form \\eqref{eq:PSU(1,1|2)-gauge-field-form}, the boundary condition that we want to impose is $\\delta(2i\\Phi_r A_\\tau(\\tau) + \\phi(\\tau))|_{\\partial \\cM}=0$. The boundary term necessary for such a boundary condition to have a well defined variational principle is \\cite{Cardenas:2018krd}:\n\\be \n\\label{eq:bdy-term-JT-grav-first-order}\nI_{BF,\\,\\text{bdy.}} = \\frac{i}2  \\int_{\\partial \\cM} \\text{Str}\\, \\phi A = { \\Phi_r} \\int_{\\partial \\cM} d\\tau\\, \\text{Str}\\,  A_\\tau^2 \\, . \n\\ee\nIntegrating out the field $\\Phi$ in the bulk we find that the bulk term yields no contribution. \nReplacing $A_\\tau$ in \\eqref{eq:bdy-term-JT-grav-first-order} we find that the action can then be rewritten as\n\\be \nI_{BF,\\,\\text{bdy.}} = -{\\Phi_r} \\int_{\\partial \\cM} d\\tau \\left[\\tilde \\cL(\\tau) + \\frac{1}2 \\left((B^1_\\tau)^2+(B^2_\\tau)^2+ (B^3_\\tau)^2\\right)\\right]\n\\ee\nThus, it is convenient to define \n\\be\n\\label{eq:def-cL(u)-and-bdy-Lagr}\n\\cL(\\tau)  = \\tilde \\cL(\\tau) + \\frac{1}2 \\left((B^1_\\tau)^2+(B^2_\\tau)^2+ (B^3_\\tau)^2\\right)\\,, \\,\\, \\text{ from which } \\,\\, I_{BF,\\,\\text{bdy.}} = -{\\Phi_r} \\int_{\\partial \\cM} d\\tau \\, \\cL(\\tau)\\,.\n\\ee\nWe will not determine $\\cL(\\tau)$ explicitly. Rather we will see how $\\cL(\\tau)$ (and all other variables in \\eqref{eq:PSU(1,1|2)-gauge-field-form}) transform under the gauge transformations that preserve the asymptotic form of \\eqref{eq:PSU(1,1|2)-gauge-field-form}. In general in a BF theory with gauge group $G$,  boundary gauge  transformations  are parametrized by functions $g$ in $\\text{loop}(G)\/G$. However, since we are preserving the asymptotic form \\eqref{eq:PSU(1,1|2)-gauge-field-form} which comes from working in the Fefferman-Graham gauge \\eqref{eq:FG-1st-order-formalism} the space of gauge transformations is instead parametrized by $\\text{Diff}(S^{1|4})\/PSU(1, 1|2)$. Therefore, the way in which $\\cL(\\tau)$ transforms under this special class of gauge transformations will yield how the boundary Lagrangian transforms under $\\text{Diff}(S^{1|4})\/PSU(1, 1|2)$. From here, we will show in section \\ref{sec:N=4-transf-law} that the boundary Lagrangian can be identified with the $\\cN=4$ super-Schwarzian derivative. Thus, to do this identification, we note that $\\cL(\\tau)$, $B_\\tau(\\tau)$, and $\\psi^p(\\tau)$ transform as \n\\bea \n\\label{eq:gauge-transf-preserving-asymp}\n\\delta_\\L \\cL &=&\\xi \\cL'+2 \\cL \\xi'+ \\xi'''- B^i_\\tau (t^i)'+\\frac{1}{2}\\left(3\\bar{\\psi} \\epsilon'+\\bar \\psi'\\epsilon -3\\bar{\\epsilon}' \\psi -\\bar{\\epsilon} \\psi'\\right), \\nn \\\\ \n\\delta_\\L \\psi^p &=& \\xi ( \\psi^p)'+\\frac{3}{2}\\psi^p \\xi'-\\epsilon^p \\cL -2(\\epsilon^p)''-\\frac{1}{2}t^i(\\sigma^i){}^p{}_q\\, \\psi^q + (B^i_\\tau)' (\\sigma^i){}^p_q \\epsilon^q +2 B^i_\\tau (\\sigma^i{})^p{}_q\\,(\\epsilon^q)',\\nn\\\\ \n\\delta_\\L B^i_\\tau&=& \\left( \\xi B^i_\\tau\\right)'-(t^i)'+ \\frac{1}2 \\bar \\psi \\sigma^i\\epsilon+ \\frac{1}2 \\bar{\\epsilon} \\sigma^i \\psi + i \\epsilon^{ijk} \\,t^j\\,B^k_\\tau\\,,\n\\eea\nunder the gauge transformation which preserves the form of $A_\\tau$ in \\eqref{eq:PSU(1,1|2)-gauge-field-form}:\n\\be\n\\Lambda(\\tau) &= \\frac{\\xi}2 L_1 - \\xi' L_{0} + \\left(\\frac{1}2 \\bar \\psi   \\epsilon +\\frac{1}2 \\psi \\bar \\epsilon - \\frac{1}2  \\xi (B^a)^2 +\\cL \\xi + \\xi'' \\right) L_{-1} - (t^i -\\xi B^i_\\tau) T_i \\nn \\\\ &+ \\frac{1}2  \\bar \\epsilon^p G_{p,\\frac{1}2} - \\left((\\bar \\epsilon^p)' - \\frac{1}2 \\xi \\bar \\psi^p  - \\frac{1}2 B_\\tau^i (\\sigma^i{}^*){}^p_q\\, \\bar \\epsilon^q\\right)  G_{p,-\\frac{1}2} + \\frac{1}2 \\epsilon^p \\bar G_{p,\\frac{1}2} \\nn \\\\ &- \\left( \\epsilon^p - \\frac{1}2 \\xi  \\psi^p  - \\frac{1}2 B_\\tau^i (\\sigma^i)^p_q\\,  \\epsilon^q\\right)  \\bar G_{p,-\\frac{1}2}\n\\ee\nIt is not a coincidence that $\\Lambda(\\tau)$ takes the same form (up to the redefinition \\eqref{eq:def-cL(u)-and-bdy-Lagr}) as $\\phi(\\tau)$ in \\eqref{eq:b.c.-for-dilaton}. This follows from requiring that the leading result in both $D_\\tau \\Lambda$ (since we require that the gauge transformation $\\Lambda$ not change the asymptotic form of $A_\\tau$ \\eqref{eq:PSU(1,1|2)-gauge-field-form})  and $D_\\tau \\phi$ (since we impose the equation of motion for $A$ by also using \\eqref{eq:PSU(1,1|2)-gauge-field-form}) both vanish. With these transformations in mind, we now proceed by introducing the necessary technology to define the $\\cN=4$ Schwarzian derivative. Following that, we will finally show that this Schwarzian derivative can be identified with the boundary Lagrangian  $\\cL(\\tau)$ from \\eqref{eq:def-cL(u)-and-bdy-Lagr}. \n\n\nFinally, we can extend this analysis to the case when the $SU(2)$ chemical potential, $\\a$, is turned on. In order to do this we can generalize the previous boundary condition from $ A_\\tau - \\frac{i \\phi}{2\\Phi_r} = 0$ to $ A_\\tau - \\frac{i\\phi}{2\\Phi_r} = \\frac{2\\pi i}{\\beta} \\alpha T^3 $, where $\\alpha\\sim \\alpha +1$. This does not modify the boundary conditions of gravity and the fermions but now the $SU(2)$ gauge field boundary condition is $B_\\tau - \\Phi^{-1}_r b = \\frac{2\\pi i}{\\beta} \\alpha T^3$, supporting the identification of $\\alpha$ with a chemical potential. In section \\ref{sec:subtleties} we explain how from the 4D near-extremal black hole perspective this boundary condition is equivalent to fixing the holonomy of the gauge field in the asymptotically flat region, which is related to fixing the boundary 4D angular velocity. \n\n\n\\subsection{The $\\cN=4$ supersymmetric boundary mode} \n\nSo far we have seen that $\\cN=4$ super-JT gravity can be reduced to a boundary theory. In this section, we will see this theory can be recasted as a $\\cN=4$ super-reparametrization mode with a super-Schwarzian action. We will first review in \\ref{sec:N=4-superDiffeos} the definition of super-diffeomorphisms. In \\ref{sec:super-Schw-action} we will define the super-Schwarzian derivative that will be the action and in \\ref{sec:N=4-transf-law} we will put everything together to write down the final boundary action. \n\n\\subsubsection{Super-diffeomorphisms}\n\\label{sec:N=4-superDiffeos}\n\nTo match with the $\\cN=4$ super-JT theory defined previously we will be interested in $SU(2)$ extended $\\cN=4$ supersymmetry.\\footnote{Other choices with also $\\cN=4$ are the $O(4)$ extended algebra studied for example in \\cite{Schoutens:1988ig}, we will not be interested in those for the purposes of this paper.} We will begin by giving a super-space description of $\\cN=4$ super-diffeomorphisms. \n\nConsider an $\\cN=4$ super-line parametrized by a bosonic variable $\\tau$ and fermionic variables $\\theta^p$ and $\\bar{\\theta}_q$, where $p,q=1,2$. The coordinates $\\theta^p$ and $\\bar{\\theta}_q$ transform respectively in the fundamental and antifundamental  representations of a local $SU(2)$ symmetry. We will omit the indices and simply write $\\theta$ and $\\bar{\\theta}$ when it is clear by context. We define the four super-derivatives as \n\\beq\nD_p = \\frac{\\partial}{\\partial \\theta^p} + \\frac{1}{2} \\bar{\\theta}_p \\partial_\\tau ,~~~~~\\bar{D}^q = \\frac{\\partial}{\\partial \\bar{\\theta}_q} + \\frac{1}{2} \\theta^q \\partial_\\tau.\n\\eeq\nThe $SU(2)$ indices will be raised and lowered by the antisymmetric tensors $\\varepsilon_{pq}$ and $\\varepsilon^{pq}$ with $\\varepsilon_{12}=\\varepsilon^{21}=1$. We will denote by $\\sigma^i$ with $i=1,2,3$ the Pauli matrices and indices will be contracted from `left-bottom' to `right-up', e.g. $\\bar{\\theta} \\sigma^i \\theta = \\bar{\\theta}_p (\\sigma^i)^p_q \\theta^q$. We will sometimes group the coordinates of the $\\cN=4$ super-line as $Z=(\\tau, \\theta^p, \\bar{\\theta}_p)$. \n\nWe will study general reparametrizations of the super-line, which will become the degrees of freedom in the path integral that defines the Schwarzian theory. These have the following form \n\\beq\\label{reparametrizations}\n\\tau \\to \\tau'(\\tau, \\theta, \\bar{\\theta}) ,~~~~\\theta^p \\to \\theta'{}^p(\\tau, \\theta, \\bar{\\theta}), ~~~~\\bar{\\theta}_q \\to \\bar{\\theta}'_q(\\tau, \\theta, \\bar{\\theta}),\n\\eeq\nand satisfy a set of constrains given by \n\\bea\\label{repa_constraints}\nD_p \\bar{\\theta}'_q =0&,&~~~\\bar{D}^p \\theta'{}^b=0,\\\\\nD_p \\tau' - \\frac{1}{2} (D_p \\theta'{}^q)\\bar{\\theta}'_q=0&,&~~~~\\bar{D}^p \\tau' - \\frac{1}{2} (\\bar{D}^p \\bar{\\theta}'_q)\\theta'{}^q=0\n\\eea\nanalyzed in \\cite{Matsuda:1988qf} and \\cite{Matsuda:1989kp}. They guarantee that the superderivatives transform homogeneously and preserve the global $SU(2)$ symmetry. We will refer to the space of solutions of these constrains as $\\cN=4$ super-diffeomorphisms and denote it as ${\\rm Diff}(S^{1|4})$, indicating one bosonic and four fermionic directions. \n\nNext we will look at some examples. The simplest super-reparametrizations are purely bosonic. They are given in terms of an arbitrary function $f(\\tau)$ and an arbitrary $SU(2)$ matrix $g(\\tau)$. The solutions of the constraints have the form \n\\bea\\label{eq:repbos}\n\\tau &\\to& f(\\tau) + \\frac{1}{8} f''(\\tau) (\\bar{\\theta}\\theta)^2,\\\\\n\\theta^p &\\to& g\\Big( \\tau -\\frac{1}{2}\\bar{\\theta}\\theta\\Big)^p_q  ~\\theta^q~ \\sqrt{f'\\Big(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\Big)} , \\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_q~ \\bar{g}\\Big(\\tau + \\frac{1}{2}\\bar{\\theta}\\theta\\Big)^q_p ~\\sqrt{f'\\Big(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\Big)}.\n\\eea\nAnother simple example are the `chiral' and `anti-chiral' fermionic transformations. The chiral ones are parametrized in terms of two fermionic functions $\\eta^p(\\tau)$ in the fundamental of $SU(2)$ and the reparametrization is given by\n\\bea\\label{eq:repfer}\n\\tau &\\to&\\tau + \\frac{1}{2} \\bar{\\theta} \\eta\\Big(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\Big) + \\frac{1}{4} \\partial_\\tau (\\bar{\\theta} \\eta(\\tau))^2,\\\\\n\\theta^p &\\to& \\theta^p + \\eta^p\\Big(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\Big)  , \\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_p\\Big( 1 + \\bar{\\theta} \\eta'\\Big(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\Big)  \\Big).\n\\eea\nThe anti-chiral are parametrized by $\\bar{\\eta}_p(\\tau)$ in the antifundamental, and the reparametrization is given by\n\\bea\\label{eq:repfer2}\n\\tau &\\to&\\tau + \\frac{1}{2} \\theta \\bar{\\eta}\\Big(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\Big) + \\frac{1}{4} \\partial_\\tau (\\theta \\bar{\\eta}(\\tau))^2,\\\\\n\\theta^p &\\to&\\theta^p\\Big( 1 +\\theta \\bar{\\eta}'\\Big(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\Big)  \\Big) , \\\\\n\\bar{\\theta}_p &\\to&  \\bar{\\theta}_p +\\bar{\\eta}_p\\Big(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\Big).\n\\eea\nThe most general element of ${\\rm Diff}(S^{1|4})$ is obtained by a general fermionic transformation followed by a bosonic one.\\footnote{ In writing the Schwarzian derivative, the order of the composition is important. In particular, to compute the $\\cN=4$ super-Schwarzian we compose the bosonic transformation with a fermionic one in this order. The order of compositions will however be unimportant when localizing the path integral.} We parametrize ${\\rm Diff}(S^{1|4})$ in terms of the degrees of freedom\n$\nf(\\tau),~~g(\\tau)\\in SU(2),~~\\eta^p(\\tau),~~\\bar{\\eta}_p(\\tau),\n$\nwhere $p=1,2$. It is hard to determine the finite form that has all parameters turned on since such answer contains a large number of terms. Therefore, we will not write it down explicitly although its straightforward to get from the results presented so far. Instead, it is useful to analyze the most general infinitesimal transformation. Up to $\\mathcal{O}(\\eta)$ is given by \n\\bea\n\\tau &\\to&f(\\tau-\\frac{1}{2}\\bar{\\eta}(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta)\\theta+\\frac{1}{2}\\bar{\\theta}\\eta(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta)) + \\frac{1}{8}f''(\\tau)\\left((\\bar{\\theta}+\\bar{\\eta})(\\theta+\\eta)\\right)^2\\\\\n\\theta^p&\\to& F^p{}_q\\left(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\right)\\left(\\theta^q+\\eta^q\\left(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\right)\\right)+\\partial_{\\tau}\\left(F^p{}_q\\left(\\tau\\right)\\theta^q \\theta \\bar{\\eta}\\left(\\tau\\right)\\right)\\\\ \n\\bar{\\theta}_p &\\to& \\left(\\bar{\\theta}_q+\\bar{\\eta}_q\\left(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\right)\\right)F^q{}_p\\left(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\right)+ \\partial_{\\tau}\\left(\\eta\\left(\\tau\\right)\\bar{\\theta}\\bar{\\theta}_qF^q{}_p\\left(\\tau\\right)\\right)\n\\eea\nwhere we use $F^p{}_q(\\tau)=\\sqrt{f'(\\tau)}g^p{}_q(\\tau) $.\n\nWe introduce now an important sub-group of the super-reparametrizations, the global superconformal group $PSU(1,1|2)$. It is generated by six bosonic variables (three from $SL(2)$ and three from $SU(2)$) and eight fermionic. The general case can be found in \\cite{Matsuda:1989kp}. Instead we will write down the bosonic generators \n\\bea\\label{infintesimalparameterb}\n\\tau &\\to& \\frac{a\\tau+b}{c\\tau+d} - \\frac{c}{4(c\\tau+d)^3}(\\bar{\\theta}\\theta)^2  ,\\\\\n\\theta^p &\\to&[e^{i \\vec{t} \\cdot \\vec{\\sigma}}]^p_q\\theta^q \\frac{1}{c(\\tau - \\frac{1}{2} \\bar{\\theta}\\theta)+d} ,\\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_q [e^{-i \\vec{t} \\cdot \\vec{\\sigma}}]^q_p \\frac{1}{c(\\tau + \\frac{1}{2} \\bar{\\theta}\\theta)+d}\\label{infintesimalparameterb-last} ,\n\\eea\nwith $a,b,c,d\\in \\mathbb{R}$ such that $a d- bc=1$ and $c>0$. This is precisely of the form \\eqref{eq:repbos} where $f(\\tau)$ is a global conformal transformation $PSL(2,\\mathbb{R})$ while $(t^1, t^2, t^3)$ parametrize a global $SU(2)$ transformation.  The fermionic generators can be parametrized by two constant spinor doublets $\\eta$ and $\\tilde{\\eta}$ and act as \n\\bea\n\\label{infintesimalparameterf-first}\n\\tau &\\to& \\tau + \\frac{1}{2} (\\bar{\\theta}-\\bar{\\tilde{\\eta}}) \\eta - \\frac{1}{2} \\bar{\\eta} (\\theta-\\tilde{\\eta})  ,\\\\\n\\theta^p &\\to& \\theta^p+ \\eta^p  - \\tilde{\\eta}^p,\\\\\n\\bar{\\theta}_p &\\to&\\bar{\\theta}_p + \\bar{\\eta}_p   - \\bar{\\tilde{\\eta}}_p,\n\\label{infintesimalparameterf}\n\\eea\n As anticipated there are eight fermionic generators. As explained above, the most general case is a composition of bosonic and fermionic. \n\n\\subsubsection{Super-Schwarzian action}\n\\label{sec:super-Schw-action}\n\nThe Schwarzian derivative associated to reparametrizations of the $\\cN=4$ super-circle was defined by Matsuda and Uematsu in \\cite{Matsuda:1989kp} (see also \\cite{Matsuda:1988qf}). The Schwarzian derivative is given in terms of superspace variables as \n\\beq\\label{Sderivative}\nS^i (Z; Z') = - 2 D \\sigma^i \\bar{D} \\log \\left(\\frac{1}{2} (D_p \\theta'{}^q)(\\bar{D}^p \\bar{\\theta}'_q)\\right).\n\\eeq\nIt satisfies the following chain rule \n\\beq\n\\label{chainrule}\nS^i (Z; Z') = \\frac{1}{ 2} (\\sigma^i)^p_q(\\sigma^j)^{q'}_{p'} (D_p \\theta''{}^{p'})(\\bar{D}^q \\bar{\\theta}''_{q'}) S^j (Z'';Z')+S^i (Z;Z'').\n\\eeq\nAnother defining property is the fact that $S^i (Z;Z')=0$ whenever the super-reparametrization is an element of the global $PSU(1,1|2)$ as in \\eqref{infintesimalparameterb}. This will prove consequential in section \\ref{sec:space-time-and-global-symm} when studying the global symmetries of the $\\cN=4$ super-Schwarzian action which we shall define shortly.  \n\nThe derivative $S^i$ is a superfield with several components. We can extract the bosonic piece we want to associate to the Schwarzian action. In the notation of \\cite{Matsuda:1989kp} it is \n\\beq\nS^i (Z; Z') \\supset - \\bar{\\theta} \\sigma^i \\theta \\hspace{0.1cm}S_b(\\tau, \\theta, \\bar{\\theta}; \\tau',\\theta',\\bar{\\theta}')\n\\eeq\nOne motivation of this choice is to look at purely bosonic transformations defined in equation \\eqref{eq:repbos}. For simplicity let's briefly consider $Z'=(\\tau',\\theta',\\bar{\\theta}')$ where $\\tau'$, $\\theta'$ and $\\bar \\theta'$ take the special form \\eqref{eq:repbos}, in terms of an arbitrary $f(\\tau)$ and set $g(\\tau)=1$. Then the definition above gives the super-Schwarzian \n\\beq\n\\label{eq:just-in-terms-of-Schw}\nS^i (Z,Z') = -{\\rm Sch}(f,\\tau) \\bar{\\theta} \\sigma^i \\theta.\n\\eeq\nAnother motivation is that when we interpret $S^i$ as a superconformal generator, that component generates bosonic translations along the circle and we want to identify this as the action of the Schwarzian theory.  \n\nThe super-Schwarzian satisfies the constrains $D_p D_q S^i = \\bar{D}^p \\bar{D}^q S^i=0$. Therefore, as a superfield it can be expanded in the following components \n\\bea\\label{sschwarzian}\nS^i &=& 2S_{T}^i + \\bar{\\theta} \\sigma^i S_f + \\bar{S}_f \\sigma^i \\theta - \\bar{\\theta} \\sigma^i \\theta S_b  + i \\epsilon^{ijk} \\bar{\\theta} \\sigma^j \\theta \\partial_\\tau S_{T}^k \\nonumber\\\\\n&&- \\frac{1}{2}(\\bar{\\theta}\\theta) \\bar{\\theta}\\sigma^i \\partial_\\tau S_f + \\frac{1}{2} (\\bar{\\theta} \\theta) \\partial_\\tau \\bar{S}_f \\sigma^i \\theta + \\frac{1}{4} (\\bar{\\theta} \\theta)^2 \\partial^2_\\tau S_T^i \n\\ea\nThen the super-Schwarzian action is \n\\beq\n\\label{Saction}\nI_{\\cN=4} = - \\Phi_r\\int d\\tau S_b[f(\\tau),g(\\tau),\\eta(\\tau)]\n\\eeq\nwhere $\\Phi_r$ can be viewed as a coupling constant whose role we shall soon discuss and the factor of $12$ above is chosen such that it simplifies the factor in \\eqref{eq:just-in-terms-of-Schw}.  \n\nWe can rewrite the action in super-field notation by defining $S = \\bar{\\theta} \\sigma^i \\theta S^i,$ where we sum over $i=1,2,3$. Then, using the expansion of $S^i$ gives,\\footnote{This is a simpler expression of $S^i$ for which more components are present. We can construct $S$ from an $S^i$ as long as $S^i$ satisfies  $D_p D_q S^i = \\bar{D}^p \\bar{D}^q S^i=0$. To derive this we use that \n\\bea\n&&(\\bar{\\theta} \\sigma^i \\theta)(\\bar{\\theta} \\sigma^i \\theta)=-3(\\bar{\\theta} \\theta)^2,~~~~~~\\varepsilon^{ijk}(\\bar{\\theta} \\sigma^i \\theta)(\\bar{\\theta} \\sigma^j \\theta)=0,\\nn\\\\\n&& (\\bar{\\theta} \\sigma^i \\theta) (\\bar{\\theta} \\sigma^i G) = -3 (\\bar{\\theta} \\theta) (\\bar{\\theta} G),~~~~(\\bar{\\theta} \\sigma^i \\theta) (\\bar{G} \\sigma^i \\theta) = 3 (\\bar{\\theta} \\theta) (\\bar{G} \\theta) \\nn\\,.\n\\eea\n}\n\\beq\nS =2 \\bar{\\theta} \\sigma^i S_T^i \\theta  - 3 (\\bar{\\theta} \\theta) \\bar{\\theta} S_f + 3 (\\bar{\\theta} \\theta) \\bar{S}_f \\theta + 3 S_b (\\bar{\\theta}\\theta)^2\\,.\n\\eeq\nThen the action \\eqref{Saction} can be rewritten as $I_{\\cN=4}\\sim \\Phi_r \\int d\\tau d^4\\theta S$.\nNote that in terms of the $ S[f(\\tau),g(\\tau),\\eta(\\tau)]$ there is no obvious chain rule analogous to \\eqref{chainrule}. For this reason it will oftentimes be easier to work with $S^i$ instead of the super-field $S$. To make things concrete, it is informative to write the Schwarzian action when focusing on purely bosonic components, when setting $\\eta(\\tau) = 0$ in \\eqref{Saction}:\n\\be\n\\label{eq:bosonic-compoents-of-the-action}\nI_{\\cN=4,\\text{ bosonic}}[f(\\tau), g(\\tau), \\eta(\\tau) = 0] =- \\int_0^\\b d\\tau \\Phi_r\\left[\\text{Sch}( f, \\tau ) + \\Tr(g^{-1} \\partial_\\tau g)^2\\right] \\,.\n\\ee\nSince $\\eta(\\tau)=0$ is a solution to the equations of motion we will soon use the above action to extract the classical saddle point when quantizing the super-Schwarzian action at the level of the path integral.\n\n\n\\subsubsection{Transformation law and a match with the JT boundary term } \n\\label{sec:N=4-transf-law}\n\n\nIn this section, we shall derive explicitly the infinitesimal transformation rules of the $\\cN=4$ super-Schwarzian (\\ref{sschwarzian}). We will expand $f(\\tau)\\approx\\tau+\\xi(\\tau)$, $g \\approx 1 + i t^{i}(\\tau) \\sigma^i$, $\\eta \\approx \\epsilon(\\tau)$ and $\\bar{\\eta}\\approx\\bar{\\epsilon}(\\tau)$ and work to linear order in $\\xi$, $t^i$, $\\epsilon$ and $\\bar{\\epsilon}$. A convenient way to encode the infinitesimal reparametrizations of the super line (\\ref{reparametrizations}) that automatically satisfies the constraints (\\ref{repa_constraints}) is to use a super-field as in \\cite{Matsuda:1988qf,Matsuda:1989kp}: \n\\bea\\label{repara_superfield}\nE(Z)=\\xi(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta)+\\xi(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta)+\\bar{\\theta} \\epsilon(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta)-\\bar{\\epsilon}(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta)\\theta+\\frac{1}{2}\\bar{\\theta}\\sigma^i\\theta t^i(\\tau),\n\\eea\nUnder such a reparametrization (\\ref{repara_superfield}), the super-Schwarzian (\\ref{sschwarzian}) transforms in the same way as the super-energy momentum tensor, which is given in \\cite{Matsuda:1989kp}: \n\\bea\\label{SStransformation}\n\\delta_E S^i= \\partial_{\\tau}\\left(E(Z)S^i\\right)+D E(Z)\\bar{D} S^i+ \\bar{D}E(Z)D S^i - i \\epsilon^{ijk}\\left(D\\sigma^j\\bar{D}E\\right) S^k-2D\\sigma^i\\bar{D}\\partial_{\\tau}E(z),\n\\eea\n Now substitute (\\ref{sschwarzian}) and (\\ref{repara_superfield}) into (\\ref{SStransformation}), and collect in components: \n\\bea\n\\delta_E S^i= 2\\delta_E S_T^i + \\bar{\\theta} \\sigma^i \\delta_ES_f + \\delta_E\\bar{S}_f \\sigma^i \\theta - \\bar{\\theta} \\sigma^i \\theta \\delta_E S_b  + i \\epsilon^{ijk} \\bar{\\theta} \\sigma^j \\theta \\partial_\\tau \\delta_E S_T^k+\\dots,\n\\eea\nwe obtain the infinitesimal transformation of $S_T^i, S_f, \\bar{S}_f, S_b$. Note that the terms in $\\dots$ are purely determined by lower components, and thus it is enough to focus on the terms up to $\\mathcal{O}(\\bar{\\theta}\\theta).$ As a result, the transformations of $S_T^i, S_f, \\bar{S}_f, S_b$ are given by:\n\\bea \n\\label{eq:Sb-transformation}\n\\delta_E S_b&=&\\xi S_b'+2S_b \\xi'+ \\xi'''- S_T^i (t^i)'+\\frac{1}{2}\\left(3\\bar{S}_f\\epsilon'+\\bar{S}_f'\\epsilon-3\\bar{\\epsilon}'S_f-\\bar{\\epsilon} S_f'\\right), \\nonumber\\\\ \n\\delta_E S_f^p&=& \\xi (S_f^{p})'+\\frac{3}{2}S_f^p \\xi'-\\epsilon^p S_b-2(\\epsilon^p)''-\\frac{1}{2}t^i (\\sigma^i){}^p_qS_f^q+(S_T^i)' (\\sigma^i{})^p_q\\epsilon^q+2 S_T^i (\\sigma^i{})^p_q(\\epsilon^q)',\\nonumber\\\\ \n\\delta_E S_T^i&=& \\left( \\xi S_T^i\\right)'-(t^i)'+\\frac{1}2 \\bar{S}_f\\sigma^i\\epsilon+ \\frac{1}2\\bar{\\epsilon}\\sigma^iS_f+i \\epsilon^{ijk}t^j S_T^k.\n\\eea\n\nWe note that they exactly agree with the infinitesimal transformation deduced from the boundary action of the BF theory \\eqref{eq:gauge-transf-preserving-asymp} with the field identification:\n\\be \n\\cL &\\leftrightarrow S_b\\,,\\nn \\\\ \n\\psi^p &\\leftrightarrow S_f^p\\,,\\nn \\\\ \nB^i_\\tau &\\leftrightarrow S_T^i\\,.\n\\ee\nSince the infinitesimal transformations and ${PSU}(1,1|2)$ invariance suffices to determine the global form of the action as the $\\cN=4$ super-Schwarzian, it then follows that the boundary action \\eqref{eq:def-cL(u)-and-bdy-Lagr} from BF theory agrees with our definition of the Schwarzian action \\eqref{Saction}. Therefore, the path integral in the $\\cN=4$ super-JT gravity with the boundary conditions discussed in section \\ref{sec:super-JT-bdy-cond} can be reduced to that for the $\\cN=4$ super-Schwarzian action defined by \\eqref{Saction}.  \n\n\\subsection{Spacetime and global symmetries}\n\\label{sec:space-time-and-global-symm}\nBefore analyzing the quantization of the super-Schwarzian theory, it is useful to study the space-time and global symmetries present in this action. \n\nA useful way to discuss symmetries of a quantum field theory is to cast it in terms of \\emph{internal symmetries} and \\emph{spacetime symmetries}.  The continuous internal symmetries of (\\ref{Saction}) form PSU$(1,1|2)$,\\footnote{It is sometimes confusing whether the group is SU$(1,1|2)$ or PSU$(1,1|2)$. We note that SU$(1,1|2)$ contains an extra $U(1)$ factor generated by identity, due to the cancellation of SL$(2)$ part and $SU(2)$ part of the super-trace in the algebra \\cite{BrittoPacumio:1999ax}. We do not have such an $U(1)$ factor in \\ref{infintesimalparameterb}, and, therefore, our symmetry group is PSU$(1,1|2).$ } generated by (\\ref{infintesimalparameterb})--(\\ref{infintesimalparameterf}), directly acting on $\\left(f(\\tau), g(\\tau), \\bar{\\eta}_p(\\tau), \\eta^p(\\tau)\\right).$ They are zero modes of the $\\cN=4$ super-Schwarzian derivative, and the action (\\ref{Saction}) is invariant due to the chain rule (\\ref{chainrule}). Specifically,\n\\be\n\\label{eq:invariance-under-PSU(1,1|2)}\nS^i(Z, h \\circ Z'(Z)) = S^i(Z,  Z'(Z)) \\,,\n\\ee\nwhere $h$ is a composition of the bosonic and fermionic transformations in (\\ref{infintesimalparameterb})--(\\ref{infintesimalparameterf}) which we apply to the super-reparametrization $Z'(Z)$.\nThese transformations are the supersymmetric generalization of the $SL(2,\\,\\mR):f(\\tau) \\to \\frac{a f(\\tau)+b}{c f(\\tau) +d}$, and we will have to quotient out such transformations as we proceed to compute the partition function \\cite{Jensen:2016pah,Maldacena:2016upp,Engelsoy:2016xyb}, in order to obtain a well-defined partition function. Furthermore, aside from the $PSU(1, 1|2)$ that has an $SU(2)$ subgroup which acts on the left of the field $g(\\tau)$ there is an additional $SU(2)$ symmetry which acts on the right of the field $g(\\tau)$. This additional symmetry does not act on $f(\\tau)$ or on the fermionic fields $\\bar \\eta$ and $\\eta$. Thus, to summarize the continuous internal symmetry is, up to discrete factors, $PSU(1,1|2) \\times SU(2)$.\\footnote{There is also an outer $SU(2)$ inherited from the $PSU(1,1|2)$ algebra. It acts on $(\\eta^1\\left(\\tau\\right), -\\bar{\\eta}_2\\left(\\tau\\right))$ as a doublet. In the $4d$ setup we later consider, this is the $R$ symmetry of the $\\mathcal{N}=2$ supergravity that is broken by stringy effects. }\n\nThe $\\cN=4$ super-Schwarzian theory also admits \\emph{spacetime} symmetries.  To see this consider the transformation \\eqref{eq:Sb-transformation}; for the transformation $\\xi(\\tau) = \\xi$ corresponding to $\\tau \\to \\tau + \\xi$ (time translations), $t^i(\\tau) =t^i$ corresponding to  constant infinitesimal rotation of $\\theta^p$ ($R$-symmetry rotations), and $\\epsilon(\\tau) = \\epsilon$ corresponding to $\\theta \\to \\theta+\\epsilon$ (super-translations), the action $S_b$ is invariant up to a total derivative. Together, all such infinitesimal transformations generate the $\\cN = 4$ super-Poincar\\'e spacetime symmetry. They satisfy the algebra\n\\bea\n\\{\\bar{Q}_p, Q^q\\}=2\\delta^q_p H, ~~~\\text{with }~~\\bar{Q}_p=i\\frac{\\partial}{\\partial \\theta^p}+\\bar{\\theta}_p\\partial_{\\tau},\\,\\,\\, Q^p=-i\\frac{\\partial}{\\partial \\bar{\\theta}_p}-\\theta^p \\partial_{\\tau},\\,\\,\\,H=\\partial_\\tau\n\\eea\nwhile all other commutators vanish. \nIt is straightforward to identify these $\\cN=4$ supercharges from the Noether procedure using the transformation \\eqref{eq:Sb-transformation}: $S_b$ is the charge generating time translations (and is thus the Hamiltonian of the theory), $S_f$ is the generator of super-translations, and $S_T^i$ is the generator of R-symmetry rotations.  Note that the Hamiltonian here is \\emph{not} one of the generators of PSU$(1,1|2),$ since it is in fact the super-Schwarzian itself, and thus proportional to the quadratic Casimir of the PSU$(1,1|2).$ This is quite analogous to the non-supersymmetric case \\cite{kitaevTalks, Mertens:2017mtv, Lin:2019qwu}. \n\nWe would like to stress that even though $PSU(1,1|2)$ plays a big role in constructing the Schwarzian action, the theory is not invariant under spacetime superconformal $PSU(1,1|2)$ symmetries. The spectrum is not organized according to $PSU(1,1|2)$ representations. Only the $\\cN=4$ super-Poincar\\'e sub-group is a spacetime symmetry. \n\nIt is also useful to briefly discuss some of the discrete internal and spacetime symmetries of the theory. The $\\cN=4$ super-Schwarzian has a time reversal symmetry $\\cT$ which acts on the fermionic fields as $\\cT \\eta(\\tau) = i \\eta(-\\tau)$ and $\\cT\\bar \\eta(\\tau) = i \\bar \\eta(-\\tau)$ and on the bosonic fields as $\\cT f(\\tau) = f(-\\tau)$ and $\\cT g(\\tau) = g(-\\tau)$; in such a case, $\\cT^2 = 1$ and the symmetry is $\\mathbb Z_2^\\cT$.  There is also a $\\mathbb Z_2^F$ fermionic symmetry $(-1)^F$ which solely acts on the fermionic fields.\\footnote{The same time-reversal properties are also true in the $\\cN=1$ theory \\cite{Stanford:2019vob}. } Thus, the symmetry is $\\mZ_2^F \\times \\mZ_2^\\cT$. We can now address discrete factors for the internal symmetry group of the theory. In (\\ref{infintesimalparameterb})--(\\ref{infintesimalparameterf}) we note that the transformation given by the center of $SL(2, \\mR)$ with $a=d=-1$ and $b=c=0$ (acting as $\\eta \\to -\\eta$ and $\\bar \\eta\\to -\\bar \\eta$) is redundant with the composition of the two center transformations for the two $SU(2)$, the first of which acts as $g \\to -g$, $\\eta \\to -\\eta$, $\\bar \\eta\\to -\\bar \\eta$, and the second of which acts solely on $g \\to -g$. Furthermore, this transformation is again redundant with the $(-1)^F$ symmetry mentioned previously. Thus, the bosonic subgroup of the symmetry group for the theory is given by \n\\be\n\\frac{SL(2, \\mR) \\times SU(2) \\times SU(2) \\times \\mZ_2^F}{\\mZ_2} \\times \\mZ_2^\\cT\\,.\n\\ee \nConsequently, we note that even-dimensional representations of $SU(2)$ (half-integer spins) need to be fermionic and odd-dimensional representations of $SU(2)$ (integer spins) need to be bosonic. Since we will be able to decompose our partition function as a sum over $SU(2)$ characters, this fact will play an important role in easily obtaining the supersymmetric index from the theory by using the result for the partition function. \n\n\\section{Quantizing the $\\cN=4$ Schwarzian theory}\n\\label{sec:N=4-super-Schw}\n\nIn this section, we will study the $\\cN=4$ super-Schwarzian theory in more detail. In particular we will compute the exact partition function and density of states. We will do the calculation in two ways, first using the localization method of Stanford and Witten \\cite{Stanford:2017thb} and second using the 2D CFT approach of \\cite{Mertens:2017mtv}, and find agreement. Then we will analyze the spectrum that we derive and point out some salient features like the large zero temperature degeneracy and the presence of a gap.\n\n\\subsection{The action}\n\n\nThe $\\cN=4$ super-line can be parametrized in superspace by $(\\tau, \\theta^p, \\bar{\\theta}_p)$, $p=1,2$, where $\\theta$ and $\\bar{\\theta}$ are Grassman variables transforming as fundamental and anti-fundamental of an $SU(2)$ symmetry.  $\\cN=4$ super-reparametrizations are parametrized by a bosonic field $f(\\tau)\\in {\\rm Diff}(S^1)$, a local transformation $g(\\tau) \\in SU(2)$ (or more precisely the loop group) and fermionic fields $\\eta^p(\\tau)$ and $\\bar{\\eta}_p(\\tau)$. In terms of a super-reparametrization these fields can be roughly written as \n\\bea\n\\tau &\\to& f(\\tau) + \\ldots,\\\\\n\\theta^p &\\to& g^p_{~q}(\\tau) \\theta^q \\sqrt{f'(\\tau)} + \\eta^p(\\tau) + \\ldots, \\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_q \\hspace{0.1cm}\\bar{g}^q_{~p}(\\tau) \\sqrt{f'(\\tau)} + \\bar{\\eta}_p(\\tau) + \\ldots.\n\\ea\nThe dots correspond to terms that are fixed by the super-reparametrization constrains and can be found in the previous section. We also defined a Schwarzian action $I_{\\cN=4} =-\\Phi_r \\int d\\tau S_b [ f, g, \\eta, \\bar{\\eta}]$ invariant under $PSU(1,1|2)$ transformations acting on the fields $(f,g,\\eta,\\bar{\\eta})$. The bosonic component of this action is \n\\beq\nI_{\\cN=4} =- \\Phi_r \\int_0^\\b d\\tau \\left[\\text{Sch}(f,\\tau) + \\Tr(g^{-1} \\partial_\\tau g)^2 + ({\\rm fermions})\\right]\n\\eeq\nwhich gives the usual Schwarzian action and a particle moving on $SU(2)$. The extra terms involve the fermions $\\eta$ and $\\bar{\\eta}$. \n\nIn this section, we will compute the Euclidean path integral giving the partition function,\\footnote{We leave the measure implicit in this formula. We take the measure to be the Pfaffian of the symplectic form over the integration space ${\\rm Diff}(S^{1|4})\/PSU(1,1|2)$, studied in \\cite{Aoyama:2018lfc}.}\n\\beq\nZ (\\beta,\\alpha) = \\int \\frac{\\mathcal{D} f \\mathcal{D} g \\mathcal{D} \\eta \\mathcal{D} \\bar{\\eta}}{PSU(1,1|2)} ~\\exp\\left( \\Phi_r \\int d\\tau S_b[f,g,\\eta,\\bar{\\eta}] \\right)\\,, \n\\eeq\nwhere $\\Phi_r$ is a dimensionful coupling constant of the theory. The inverse temperature $\\beta$ and chemical potentials $\\alpha$ appears in the path integral through the boundary conditions of the fields:\n\\beq\nf(\\tau+\\beta)=f(\\tau),~~~~g(\\tau+\\beta) = e^{2 \\pi i \\alpha \\sigma^3} g(\\tau),~~~  \\eta(\\tau+ \\beta) =- e^{2\\pi i \\alpha \\sigma^3} \\eta(\\tau),\n\\eeq\nand similarly for $\\bar{\\eta} $. In the rest of this section we will evaluate this path integral as a function of $\\beta$, $\\alpha$ and the coupling $\\Phi_r$, and rewrite it as a trace over a Hilbert space with a possibly continuous spectrum. \n \n \\subsection{The partition function}\n \n\\subsubsection{Method 1: Fermionic localization}\n\\label{sec:fermionic-localization}\n\nIn this section, we will solve the theory following \\cite{Stanford:2017thb}. The integration space is a coadjoint orbit and the super-Schwarzian action generates a $U(1)$ symmetry. Even though we will not work out the measure and symplectic form in detail, we will assume it is chosen such that we can apply the Duistermaat-Heckman theorem. Therefore, we will compute the classical saddles, the one-loop determinants, and put everything together into the final answer \\eqref{eq:localization-part-function}.\n\n\\paragraph{The $\\cN=4$ saddle-point:} As previously mentioned, the bosonic part of the $\\cN=4$ Schwarzian action is given by\n\\be\n\\label{eq:bosonic-compoents-of-the-action-2}\nI_{\\cN=4,\\text{ bosonic}} =- \\int_0^\\b d\\tau \\Phi_r\\left[\\text{Sch}(f,\\tau) + \\Tr(g^{-1} \\partial_\\tau g)^2\\right] \\,.\n\\ee\nThe equations of motion for $f(\\tau)$ and $g(\\tau)$ imply that:\n\\be\n\\partial_\\tau \\text{Sch}(f,\\tau) = 0\\,, \\qquad \\partial_\\tau \\Tr (g^{-1} \\partial_\\tau g)^2 =  0\n\\ee\nThe solution for the Schwarzian is well-known and is given by $f(\\tau) = \\tan(\\pi \\tau\/\\b)$. The solution for the $SU(2)$ adjoint field takes the form $g = \\exp(i t_i \\e^i \\tau)$, where $\\e^i$ is a constant that needs is set by the boundary conditions for the field $g(\\tau)$. To make the computation easier we note that all solutions can be transformed to the diagonal form ($g = \\exp(i \\sigma_3 \\e^3 \\tau)$) using an $SU(2)$ transformation. If  require that the field $g$ be periodic, than we have that  $\\e^{3} = 2\\pi n\/\\beta$, with $n \\in \\mZ$. More generally, the $SU(2)$ symmetry could have a fugacity   which would imply that the field $g$ is no longer periodic; rather, it has $g(\\beta) = z\\, g(0)$ where $z \\in SU(2)$ is the fugacity. Once again, since the partition function only depends on the conjugacy class of $z$, we can perform an $SU(2)$ transformation to diagonalize $z =  \\exp(2\\pi i \\a \\sigma_3 )$. The solution for $h$ is then given by $g = \\exp\\left(2\\pi i \\sigma_3 (n+\\a) \\frac{\\tau}\\beta\\right)$. \n\nIn such a case, the on-shell value of the action $I_{\\cN=4,\\text{bosonic}}$ is given by: \n\\be \n\\label{eq:on-shell-action-N=4}\nI_{\\cN=4,\\text{bosonic}}^{\\,\\text{on-shell}} = -\\frac{2\\pi^2 \\Phi_r} {\\b} \\left(1 -4  (n+\\a) ^2 \\right)\\,,\n\\ee\nNow that we have the  on-shell action we can proceed by computing the one-loop determinant which is sufficient for fully computing the partition function. \n \n\n\\paragraph{The one-loop determinant:} To compute the one-loop determinant, we have to account for all quadratic fluctuations in the theory. The quadratic fluctuations of the Schwarzian field have been analyzed in great detail \\cite{Maldacena:2016upp, Stanford:2017thb}, and its contributions to the one-loop determinant is given, up to an overall proportionality constant, by\n\\be \n\\label{eq:Schwarzian-one-loop-determinant}\n\\det_{\\text{Schw., one-loop}} = \\left(\\frac{\\Phi_r}{\\beta}\\right)^\\frac{3}2\\,.\n\\ee\nThe quadratic fluctuation around the saddle-point of the $SU(2)$-group element can be parametrized as $g(\\tau) = \\exp\\left( \\sigma_3 \\left[2\\pi i(n+\\a) \\frac{\\tau}\\beta + \\epsilon^3(\\tau) \\right]\\right)\\exp\\left( \\epsilon^2(\\tau) \\sigma_2\\right) \\exp\\left( \\epsilon^1(\\tau) \\sigma_1\\right)$, and yields a contribution to the action \\cite{Picken:190160}\n\\be \nI_{SU(2), \\text{ quad}} &=  \\frac{8\\pi^2 \\Phi_r} {\\b}  (n+\\a) ^2\\nn \\\\ &-2 \\Phi_r \\int_0^\\b d\\tau \\left((\\epsilon^1(\\tau)')^2 + (\\epsilon^2(\\tau)')^2 - (\\epsilon^3(\\tau)')^2 + \\frac{8\\pi (n+\\a)}{\\b} \\epsilon^2(\\tau) \\epsilon^1(\\tau)'  \\right)\\,,\n\\ee\nand the one-loop determinant obtained from integrating out these modes for each value of $n$ is given by \\cite{Picken:190160}:\n\\be\n\\label{eq:SO3-one-loop-determinant}\n \\det_{SU(2), \\text{ one-loop}}  = \\frac{\\Phi_r^{3\/2} (n+\\a) }{ \\beta^{3\/2} \\sin(2\\pi \\a)} \\,.\n\\ee\nFinally, we discuss the quadratic contribution of the fermionic fields. By using the saddle-point solution for $f(\\tau)$ and $g(\\tau)$ and by quadratically expanding the super-Schwarzian action:\n\\be\n I_{\\text{ferm., quad.}} =  \\Phi_r\\int_0^{\\beta} d\\tau\\, &\\bigg(\\eta^p \\left[\\frac{2\\pi^2}{\\beta^2}\\left(1+2(n+\\a)^2 \\right) \\partial_\\tau - \\partial_\\tau^3\\right]\\bar \\eta_p + \\nn \\\\&+ \\partial_\\tau \\eta^p \\left[\\frac{12\\pi^2(n+\\a)^2}{\\b^2} + \\frac{8i  \\pi (n+\\a)\\partial_\\tau}{\\beta} - 3\\partial_\\tau^2 \\right] \\bar \\eta_p\\bigg) \n \\ee\nExpanding  the fermionic fields in Fourier modes, \n\\bea \\eta^1(\\tau) &=&e^{i\\frac{2\\pi (n+\\a)\\tau}{\\b} }\\sum_{m_1 \\in \\dots, -\\frac{1}2, \\, \\frac{1}2, \\dots} \\sqrt{\\frac{\\b}{2\\pi}} \\,\\eta^1_{m_1} e^{-i\\frac{2\\pi m_1\\tau}{\\b} }\\\\\n\\eta^2(\\tau) &=&e^{-i\\frac{2\\pi (n+\\a)\\tau}{\\b} }\\sum_{m_2 \\in \\dots, -\\frac{1}2, \\, \\frac{1}2, \\dots} \\sqrt{\\frac{\\b}{2\\pi}}\\, \\eta^2_{m_2} e^{i\\frac{2\\pi m_2\\tau}{\\b}}\\,,\n\\ea\nwhere we impose anti-periodic boundary conditions for the fermionic fields when $\\alpha= 0$ and we impose boundary conditions consistent with the introduction of the fugacity $z$ when $\\alpha\\neq 0$. We can then rewrite the action as, \n\\be\n I_{\\text{ferm., quad.}} = \\frac{2\\pi^2 i \\Phi_r}{\\beta} \\left[\\sum_{p=1, 2}\\,\\sum_{m_p  \\in \\dots, -\\frac{1}2, \\, \\frac{1}2, \\dots} (m_p - n - \t\\a) (4m_p^2-1) \\eta^p_{m_p}  \\bar \\eta^p_{-m_p}\\right]\\,.\n\\ee\nWe are interested in computing the dependence of the one-loop determinant on $n$, $\\alpha$, $\\beta$ and $\\Phi_r$. The $\\beta$ and $\\Phi_r$ dependence is captured by the existence of the four-fermionic zero modes with $m_p = \\pm 1\/2$. As in \\cite{Stanford:2017thb}, to compute the rest of the one-loop determinant, we will regularize this result by dividing the result by the one-loop determinant with $n=0$ and $\\alpha = 0$. We thus find that the regularized one-loop determinant is given, again up to a proportionality constant, by \n\\be\n\\label{eq:ferm-one-loop}\n\\det_{\\text{ferm., one-loop}} = \\frac{\\beta^4}{\\Phi_r^4} \\prod_{p=1, 2}\\,\\,\\prod_{m_p \\in \\dots, -\\frac{5}2, -\\frac{3}2, \\frac{3}2, \\frac{5}2, \\dots} \\frac{m-n -\\a}{m} = \\frac{\\beta^4}{\\Phi_r^4} \\frac{\\cos(\\pi\\a)^2}{(1 - 4(n+\\a)^2)^2 }\\,. \n\\ee\n\n\n\\paragraph{Final answer:} Thus, accounting for the saddle-point value of the action \\eqref{eq:on-shell-action-N=4} together with the one-loop determinants \\eqref{eq:Schwarzian-one-loop-determinant}, \\eqref{eq:SO3-one-loop-determinant}, and \\eqref{eq:ferm-one-loop}, we find that the partition function of the $\\cN=4$ Schwarzian theory is given, up to an overall proportionality constant, by: \n\\be \n\\label{eq:localization-part-function}\nZ_{\\cN=4\\,\\,\\,\\text{Schw.}}  &= \\sum_{n \\in \\mathbb Z} \\det_{\\text{Schw., one-loop}} \\,\\,\\det_{SU(2), \\text{ one-loop}}\\,    \\det_{\\text{ferm., one-loop}} e^{-I_{\\cN=4,\\text{bosonic}}^{\\,\\text{on-shell}} } \\nn \\\\ &=\\sum_{n \\in \\mathbb Z} \\frac{ \\beta \\cot(\\pi \\a) (\\a+n)}{\\Phi_r (1- 4(n+\\a)^2)^2} e^{\\frac{2\\pi^2 \\Phi_r} {\\b} \\left(1 -4  (n+\\a) ^2 \\right)}\n\\ee \nWe will thus continue by matching this result using the completely distinct method of canonical quantization, after which we will come back to a detailed analysis of the spectrum associated to \\eqref{eq:localization-part-function} in section \\ref{sec:density-of-states}. \n\n\\subsubsection{Method 2: Canonical quantization} \\label{sec:meth2cq}\nIn this section, we will compute the partition function of the $\\cN=4$ super-Schwarzian theory using the canonical quantization approach of \\cite{Mertens:2017mtv} (for a very recent discussion explaining the connection to the localization approach see also \\cite{Alekseev:2020jja}). \n\nWe will illustrate briefly the idea first. The localization formula we used above can be applied to integrals that generally have the following form \n\\beq\nZ = \\int dqdp ~e^{-H(p,q)},\n\\eeq\nwhere the integral is over a symplectic space (a classical phase space) with coordinates $(q,p)$ and $H(q,p)$ generates via the Poisson brackets a $U(1)$ symmetry. In the case of the bosonic Schwarzian theory the integration manifold is ${\\rm Diff}(S^1)\/SL(2,\\mathbb{R})$ which a coadjoint orbit of the Virasoro group (and, therefore, symplectic), and $H(p,q)$ is the Schwarzian action. Instead of using localization, we can obtain this integral using the following identity \n\\beq\\label{eq:SKJDKD}\n\\lim_{\\hbar\\to 0} {\\rm Tr}\\left( e^{-H(p,q)}\\right) = \\int dqdp ~e^{-H(p,q)},\n\\eeq\nwhere the left-hand side is $\\hbar\\to 0$ limit of the trace evaluated over the Hilbert space obtained by quantizing the phase space. In the case of the Schwarzian, the quantization of ${\\rm Diff}(S^1)\/SL(2,\\mathbb{R})$ is the identity representation of the Virasoro algebra with central charge $c \\sim 1\/\\hbar$. The left hand side of \\eqref{eq:SKJDKD} can by very easily computed at finite $c$ as a Virasoro vacuum character by counting descendants, and a very simple calculation gives the Schwarzian path integral \\cite{Mertens:2017mtv}. In the bosonic case, the main advantage of this method is the possibility to compute correlation functions which are not available using localization. In this case, we will use it as a double check on our previous result.\n\n\nFor the case of the $\\cN=4$ super-Schwarzian the integration space is ${\\rm Diff}(S^{1|4})\/PSU(1,1|2)$ which is a coadjoint orbit of super-Virasoro and, therefore, also symplectic. We will assume that the quantization of this phase space, the Hilbert space in \\eqref{eq:SKJDKD}, is equivalent to the identity representation of the small $\\cN=4$ Virasoro algebra with central charge $c\\sim 1\/\\hbar$. The $\\cN=4$ super-Schwarzian partition function is then the semiclassical limit of the vacuum character.  \n\nLets begin then by recalling the super-Virasoro algebra involved in this problem. The bosonic generators are $L_n$ and $T_n^i$ where $n$ is an integer and $i=1,2,3$ label the generators of a Kac-Moody $SU(2)$ at level $k$. Their algebra is \n\\bea\n[ L_m,L_n ]&=& (m-n)L_{m+n}+\\frac{k}{2} m(m^2-1)\\delta_{n+m,0} \\label{eq:smalln4gen1}\\\\\n\\text{[} T^i_m,T^j_n \\text{]} &=& i \\epsilon^{ijk}T^k_{m+n} + \\frac{k}{2} m \\delta_{m+n,0} \\delta_{i,j} \\label{eq:smalln4gen2}\\\\\n\\text{[} L_m , T_n^i\\text{]}  & = & -n T_{m+n}^i.\\label{eq:smalln4gen3}\n\\ea\nThe central charge of the bosonic Virasoro sector is $c=6k$, which is fixed by a Jacobi identity. The fermionic generators are $G_r^p$ and $\\bar{G}_s^p$, $p=1,2$. They transform in the fundamental and antifundamental of the $SU(2)$. The Fourier mode parameter $r,s$ are integer in the Ramond sector or half-integer in the Neveu-Schwarz sector. The rest of the algebra, involving the fermionic generators, can be found for example in \\cite{Eguchi:1987sm}, and is given by\n\\bea\n&&\\{ G_r^p, \\bar{G}_s^q\\} = 2 \\delta^{pq} L_{r+s}-2(r-s)\\sigma^i_{pq} T^{i}_{r+s}+\\frac{k}{2}(4r^2-1)\\delta_{r+s,0},\\nn \\\\\n&& [T_m^i,G_r^p]=-\\frac{1}{2}\\sigma^i_{pq}G^q_{m+r},~~[T^i_m,\\bar{G}_r^p]=\\frac{1}{2}\\sigma^i_{pq}{}^\\star \\bar{G}_{m+r}^q,~~ \\{ G_r^p, G_s^q\\}=\\{\\bar{G}_r^p,\\bar{G}_s^q\\}=0\\nn  \\\\\n&&[L_m,G_r^p]=\\left(\\frac{m}{2}-r\\right)G_{m+r}^p,~~[L_m,\\bar{G}_r^p]=\\left(\\frac{m}{2}-r\\right)\\bar{G}_{m+r}^p,\\label{eq:smalln4gen4}\n\\ea\nwhere $\\sigma^i_{pq}$ are the Pauli matrices. In that reference, Eguchi and Taormina also construct the unitary representations of the algebra.  \n\nFor the application we have in mind in this paper, the Schwarzian path integral, we will only need the massless representations in the NS sector, due to the fact that we want the Schwarzian fermions to be antiperiodic as explained in \\cite{Mertens:2017mtv}.\\footnote{If we wanted the Schwarzian theory Witten index we would use the characters in the Ramond sector.} General representations are labeled by $h$, the eigenvalue of $L_0$, and $\\ell$, the spin of the $SU(2)$ representation, and the massless sector has $(h=\\ell, \\ell)$ with half-integer $\\ell=0, \\frac{1}{2}, \\ldots, \\frac{k}{2}$. For the Schwarzian path integral we will need the $\\ell=0$ representation. The characters are defined by\n\\beq\n\\chi_{\\ell} (k;q,z) \\equiv {\\rm Tr}_{NS}\\left[ (-1)^F q^{L_0-\\frac{c}{24}} z^{T^3_0}\\right],\n\\eeq\nover a representation $\\ell$ of the algebra. We need to insert $(-1)^F$ such that the fermions along the quantization direction are periodic and survive the semiclassical limit (see discussion in \\cite{Mertens:2017mtv}). \n\nThese characters were computed by Eguchi and Taormina \\cite{Eguchi:1987wf} by simply counting states. They are given by the following expression\\footnote{The origin of the first factor in the right hand side is explained in section 5 of \\cite{Kraus:2006nb}.}\n\\beq\n\\chi_{\\ell} (k;q,z=e^{2\\pi i y}) = e^{2\\pi i k \\frac{y^2}{\\tau}} q^{-\\frac{k}{4}}  q^{\\ell+\\frac{1}{4}} \\frac{i \\theta_3(q,-z) \\theta_3(q,-z^{-1})}{\\eta(q)^3 \\theta_1(q,z^2)} \\left[ \\mu(z,q) - \\mu(z^{-1},q) \\right],\n\\eeq\nwhere $\\theta_3(q,z)$ is the Jacobi theta function and we defined the function \n\\beq\n\\mu(z,q) \\equiv \\sum_{n\\in\\mathbb{Z}} \\frac{q^{(k+1)n^2+(2\\ell+1)n}z^{2(k+1)n + 2 \\ell + 1}}{(1- z q^{n+\\frac{1}{2}})(1- z q^{n+\\frac{1}{2}})}.\n\\eeq\n\nNow we have all the ingredients to extract the Schwarzian partition function from the $\\hbar \\sim 1\/k \\to 0$ ($c\\to\\infty$) limit applied to the above expression for the identity $\\ell=0$ representation. As explained in \\cite{Mertens:2017mtv} we need to consider the following scaling  \n\\beq\nz=e^{2 \\pi i \\alpha \\tau},~q=e^{2\\pi i \\tau}~~~~~{\\rm with}~~~~\\tau = \\frac{i}{k} \\frac{4\\pi \\Phi_r}{\\beta}.\n\\eeq\nWe then take $\\tau \\cdot k$ fixed in the limit and this constant is related to the ratio $\\Phi_r\/\\beta$ in the Schwarzian theory. This choice of $z$ and $q$ is written directly in terms of $\\alpha$ and $\\beta$ which will become the chemical potential and inverse temperature in the Schwarzian limit. \n\nWhen taking the Schwarzian limit we will only keep track of the dependence on $\\alpha$ and $\\beta$ since any prefactor can be absorbed in a redefinition of the zero-point entropy and energy. We will not go over all the details but some useful intermediate steps are \n\\beq\n\\mu(z,q)-\\mu(z^{-1},q) \\sim \\frac{8 e^{\\frac{2\\pi^2 \\Phi_r}{\\beta}4\\alpha^2}}{\\pi^2 |\\tau|^2} \\sum_{n\\in \\mathbb{Z}} \\frac{(\\alpha+n)}{(1-4(\\alpha+n)^2)^2} e^{-\\frac{2\\pi^2 \\Phi_r}{\\beta} 4(n+\\alpha)^2}.\n\\eeq\nUsing eq (3.15) of \\cite{Ahn:2003tt} gives the following limit\n\\beq\ni \\left( \\frac{\\theta_4(q,z)}{\\eta(q)^3}\\right)^2 \\frac{\\eta(q)^3}{\\theta_4(q,z^2q^{\\frac{1}{2}})} \\sim \\frac{\\tau}{\\tan \\pi \\alpha},\n\\eeq\nwhich is related in a simple way to the Jacobi theta functions appearing in the character. Including the rest of the terms the semiclassical $k\\to\\infty$ limit of the vacuum character is \n\\beq\n\\chi_{\\ell=0} (k\\to\\infty;q,z) \\sim  \\sum_{n\\in \\mathbb{Z}} \\frac{\\beta \\cot(\\pi \\alpha)(\\alpha+n)}{\\Phi_r(1-4(n+\\alpha)^2)^2} e^{\\frac{2 \\pi^2 \\Phi_r}{\\beta}(1-4(n+\\alpha)^2)}.\n\\eeq\nThis precisely reproduces the partition function computed by localization, given in equation \\eqref{eq:localization-part-function} (An analogous match was checked in \\cite{Mertens:2017mtv} for the case of $\\cN=1$ and $\\cN=2$ super-Schwarzian).\n\nWe can mention some interesting features of this expression. First of all the factor of $\\cot \\pi \\alpha$ is important for the formula to make sense. When $\\alpha \\to 0$ or $1$ it is crucial to include this factor for the final answer to be finite. The same happens when $\\alpha \\to 1\/2$ since otherwise the sum would be divergent. \n\nFrom the 2D CFT perspective the identity representation is invariant under the generators of the global $PSU(1,1|2)$ algebra. In terms of the Virasoro algebra those generators are  \n\\beq\n\\text{Bosonic:}~~~~L_{-1},L_0,L_1,~~~~T_0^1,T_0^2,T_0^3,~~~~\\text{Fermionic:}~~~~G_{\\pm \\frac{1}{2}}^a,~~~\\bar{G}^a_{\\pm \\frac{1}{2}},\n\\eeq\nwhich satisfy the same superalgebra as \\eqref{eq:psu(1,1|2)-superalgebra}. \nIt is important to take the fermionic generators in the NS sector. These produce the pre-factor of $\\beta^1$ in the character. In the localization calculation this factor basically counts the number of bosonic and fermionic zero modes $Z \\sim \\beta^{(\\#{\\rm fermion})\/2 - (\\#{\\rm bosons})\/2}$. In the case of the small $\\cN=4$ algebra there are $8$ fermionic zero modes and $6$ bosonic zero modes, giving a factor of $\\beta$. \n\n\\subsection{$\\cN=4$ supermultiplets}\n\nBefore extracting the spectrum from the exact partition function we first explain what properties we expect it to have. The super-Schwarzian theory we are studying captures the explicit breaking of the superconformal symmetry group $PSU(1,1|2)$. Still, as we have seen in section \\ref{sec:space-time-and-global-symm}, translations, super-translations and rigid $SU(2)$ rotations are symmetries. We can write the fermionic generators as $Q_p$ and $\\bar{Q}^p$ with $p=1,2$. Then a part of the algebra that we will use here is \n\\beq\n\\{ Q_p, \\bar{Q}^q\\} = 2 \\delta_a^b H,~~~\\{Q_p,Q_q\\}=\\{ \\bar{Q}_p, \\bar{Q}_q\\} =0\n\\eeq\nThese generators can be written in terms of the Schwarzian fields but we will not need it for the manipulations here. Imagine we first diagonalize $H$ and look at some states with energy $E$. Then as long as $E\\neq 0$ the operators $Q \\sim \\hat{a}$ act as a $SU(2)$ doublet of lowering fermionic operators and $\\bar{Q}\\sim \\hat{a}^\\dagger$ as a $SU(2)$ doublet of rising fermionic operators. To construct a representation we can begin with a state $|J\\rangle$ which transforms as a spin $J$ representation of $SU(2)$, constructed such that $Q_p|J\\rangle=0$. The supermultiplet will have states acting with a single charge $\\bar{Q}^q|J\\rangle$, which can be expanded into $(1\/2)\\otimes J = (J-1\/2) \\oplus (J+1\/2)$; and acting with two charges $\\bar{Q}_1 \\bar{Q}_2 |J\\rangle$ of spin $J$. Therefore, the supermultiplet with $E\\neq 0$, starting with $J\\neq 0$ is made of $(J-1\/2)\\oplus 2 (J) \\oplus (J+1\/2)$. When we construct a supermultiplet starting with a singlet $|0\\rangle$, the $\\bar{Q}^q|0\\rangle$ transforms as a doublet and $\\bar{Q}_1\\bar{Q}_2|0\\rangle$ as another singlet, giving $2(0)\\oplus(1\/2)$. Labeling the supermultiplet by the state with highest $SU(2)$ spin, the $E\\neq 0 $ part of the spectrum should organize as \n\\bea\n\\mathbf{J}&=&(J) \\oplus 2 (J-1\/2) \\oplus (J-1),~~~~J\\geq 1\\\\\n\\mathbf{1\/2}&=&(1\/2) \\oplus 2 (0).\n\\ea\nFinally we might also have states with $E=0$. Starting with a spin-$J$ representation $|J\\rangle$, having $H|J\\rangle =0$ implies that all supercharges annihilate the state and, therefore, that's the whole supermultiplet.\n\nTaking these considerations into account, we can expect the partition function of the $\\cN=4$ super-Schwarzian theory to be expanded in the following way\n\\bea\nZ(\\beta,\\alpha) &=& \\sum_J \\chi_J(\\alpha) \\rho_{\\rm ext}(J) + \\int dE ~e^{-\\beta E} \\left( \\chi_{1\/2}(\\alpha) +2\\chi_{0}(\\alpha)\\right) \\rho_{\\rm cont}(1\/2,E) \\nonumber\\\\\n&&+\\sum_{J\\geq 1} \\int dE ~e^{-\\beta E} \\left( \\chi_{J}(\\alpha) +2\\chi_{J-\\frac{1}{2}}(\\alpha)+ \\chi_{J-1}(\\alpha)\\right) \\rho_{\\rm cont}(J,E),\\label{sqwewq}\n\\ea\nwhere the sums are over half-integer $J$ and $\\chi_J(\\alpha)\\equiv \\sum_{m=-J}^J e^{4\\pi i \\alpha m} = \\frac{\\sin (2J+1)2\\pi \\alpha}{\\sin 2\\pi \\alpha}$ is the character of a spin-$J$ representation of $SU(2)$. In the first line, the first term corresponds to states with $E=0$ while the second term to the $E\\neq 0$ multiplet $\\mathbf{1\/2}$. The second line corresponds to the sum over all other $E\\neq 0$ supermultiplets. Therefore, $\\rho(J,E)$ is the density of supermultiplets with energy $E\\neq 0$ and highest spin $J$, while $\\rho_{\\rm ext}(J)$ is the density of $E=0$ states of spin $J$. \n\nWe will see in the next section that the spectrum of the $\\cN=4$ super-Schwarzian derived from the exact partition function we computed above has precisely this form (although with only singlet $J=0$ zero energy states). \n\n\n\n\n\\subsection{Exact density of states}\n\\label{sec:density-of-states}\nThe final answer for the exact $\\mathcal{N}=4$ super-Schwarzian theory partition function is given by the following function of inverse temperature $\\beta$ and $SU(2)$ chemical potential $\\alpha$ as\n\\beq\\label{exactPFN4d}\nZ(\\beta,\\alpha) =e^{S_0} \\sum_{n\\in \\mathbb{Z}} \\frac{\\beta}{\\Phi_r} \\frac{2 \\cot(\\pi \\alpha)(\\alpha+n)}{\\pi^3(1-4(n+\\alpha)^2)^2} e^{\\frac{2 \\pi^2 \\Phi_r}{\\beta}(1-4(n+\\alpha)^2)}\\,.\n\\eeq\nWe have fixed the overall normalization in a way that will be convenient later. We will write this answer as a trace over a Hilbert space (albeit with continuous spectrum) realizing it has precisely the form \\eqref{sqwewq}.\n\nTo understand the physics of this partition function we want to extract the density of states as a function of energy at fixed $SU(2)$ charge, which we will refer to as angular momentum (anticipating the application to near extremal black holes in 4D). To do that we begin by performing an inverse Laplace transform and define the fixed-chemical-potential density of states \n\\beq\nZ(\\beta,\\alpha) = \\int dE e^{-\\beta E} D(\\alpha, E).\n\\eeq\nApplying this to our result \\eqref{exactPFN4d} gives\n\\beq\nD(\\alpha,E) = D_{E=0}(\\alpha) \\delta(E) + D_{\\rm cont}(\\alpha,E),\n\\eeq\nwhere we separate the BPS and continuous part of the spectrum,\\footnote{The sum in $D_{E=0}(\\alpha)$ is at face value divergent. To regulate it we used the following prescription $\\lim_{N\\to\\infty} \\sum_{n=-N}^N\\frac{4(\\alpha+n)}{\\pi \\tan \\pi \\alpha (1-4(\\alpha+n)^2)} = 1$. We can verify that this is the correct prescription by checking that after integrating over energies, this gives back the original partition function.}\n\\bea\n\\label{toto}D_{E=0}(\\alpha) &=&e^{S_0} \\sum_{n\\in\\mathbb{Z}} \\frac{4(\\alpha+n)}{\\pi \\tan \\pi \\alpha (1-4(\\alpha+n)^2)} = e^{S_0}\\label{extD} \\\\\\label{contD}\nD_{\\rm cont}(\\alpha,E)&=&e^{S_0}\\sum_{n\\in \\mathbb{Z}} \\frac{4(\\alpha+n)}{\\pi \\tan \\pi \\alpha} \\frac{I_2\\left(2\\pi \\sqrt{2 \\Phi_r E(1-4(\\alpha+n)^2)}\\right)}{E(1-4(\\alpha+n)^2)}\n\\ea\nWe see the first line corresponding only to states with zero energy is independent of $\\alpha$. This means it only gets contributions from zero charge (angular momentum) states. We chose the normalization of the partition function such that this gives $\\exp \\left( S_0 \\right)$ and can be interpreted as the degeneracy of ground states.\n\nTo find the density of states we use the following identity to rewrite \\eqref{contD} as \n\\beq\\label{ble}\nD_{\\rm cont}(\\alpha,E) =e^{S_0} \\sum_{m=1}^\\infty \\frac{m \\sin 2\\pi m \\alpha}{ \\tan \\pi \\alpha} \\frac{\\sinh \\left(2 \\pi \\sqrt{2\\Phi_rE-\\frac{1}{4}m^2}\\right)}{2 \\pi^2 \\Phi_r E^2} \\Theta\\Big(E-\\frac{m^2}{8\\Phi_r}\\Big).\n\\eeq\nWe defined the Heaviside function $\\Theta(x)$ such that $\\Theta(x>0)=1$ and $\\Theta(x<0)=0$. The dependence with the chemical potential can be expanded in $SU(2)$ characters in the following simple way \n\\bea\\label{charactedID4}\n\\frac{2 \\sin 2\\pi m \\alpha}{ \\tan \\pi \\alpha} &=&  \\chi_{J}(\\alpha) +2\\chi_{J-\\frac{1}{2}}(\\alpha)+ \\chi_{J-1}(\\alpha),~~~J\\equiv m\/2,~{\\rm with}~m>1\\\\\n\\frac{2 \\sin 2\\pi \\alpha}{ \\tan \\pi \\alpha} &=&  \\chi_{1\/2}(\\alpha) +2\\chi_{0}(\\alpha),~~\\hspace{2.3cm}J\\equiv m\/2,~{\\rm with}~m=1\n\\eea\nwhere in the right hand side we defined the angular momentum $J$ in terms of the integer $m$. In principle we can use this formula to extract the density of states for each $SU(2)$ representation. Instead we will notice this is precisely the combination in equation \\eqref{sqwewq}\nwhere $J$ now labels the supermultiplet $\\mathbf{J}$. The second line with $m=1$ involves the special case $\\mathbf{1\/2}$. Comparing \\eqref{sqwewq} with \\eqref{ble} we can extract the density of supermultiplets $\\rho_{\\rm cont}(J,E)$ for $E\\neq0$ and using \\eqref{extD} we can write the density of $E=0$ states $\\rho_{\\rm ext}(J)$. The final answer is given by\n\\bea\n\\rho_{\\rm ext}(J) &=& e^{S_0} \\delta_{J,0}.\\label{sksks}\\\\\n\\rho_{\\rm cont}(J,E) &=& \n\\frac{e^{S_0}J}{4\\pi^2 \\Phi_r E^2}\\sinh \\left(2 \\pi \\sqrt{2\\Phi_r(E-E_0(J))} \\right)\\hspace{0.1cM} \\Theta\\Big(E-E_0(J)\\Big), \\hspace{0.1cm}\\text{for }J\\geq\\frac{1}{2},\\label{sksks2}\n\\ea\n where the gap for each supermultiplet labeled by $J$ is given by $E_0(J) \\equiv J^2\/(2\\Phi_r)$.\n\\begin{figure}\n    \\centering\n     \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.5]{superdos.pdf}} at (0,0);\n \\draw (-5.7,-0.4) node  {$e^{S_0}$};\n   \\draw (-5.8,2.75) node  {\\small $\\rho(E)$};\n   \\draw (-2,-3) node  {\\small $E_{\\rm gap}$};\n   \\draw (-0,-3) node  {\\small $E_0(1)$};\n   \\draw (-0,-1.5) node  {\\small $\\mathbf{1}$};\n   \\draw (-2.3,-1.5) node  {\\small $\\mathbf{1\/2}$};\n   \\draw (-4.6,-1.5) node  {\\small $\\mathbf{0}$};\n   \\draw (5,-3) node  {\\small $E$};\n \n  \\end{tikzpicture}\n  \\hspace{0.6cm}\n   \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.47]{dosj0.pdf}} at (0,0);\n \\draw (-5.4,-0.4) node  {$e^{S_0}$};\n   \\draw (-5.4,2.75) node  {\\small $\\rho(E)$};\n    \\draw (-2,-3) node  {\\small $E_{\\rm gap}$};\n   \\draw (0.2,-3) node  {\\small $E_0(1)$};\n   \\draw (5,-3) node  {\\small $E$};\n    \n  \\end{tikzpicture}\n    \\caption{\\textbf{Left:} Density of supermultiplets labeled by the highest spin $\\mathbf{J}$. We show $\\mathbf{0}$, which is simply a delta function at $E=0$; $\\mathbf{1\/2}$ which is continuous but starts at $E_{\\rm gap}\\equiv E_0(1\/2)$; and $\\mathbf{1}$ which is also continuous starting at $E_0(1)$. \\textbf{Right:} Degeneracy for all states with $J=0$. These come from $\\mathbf{0}$, the delta function at $E=0$, $\\mathbf{1\/2}$, starting at $E_{\\rm gap}$, and $\\mathbf{1}$, starting at $E_0(1)$. All other supermultiplets do not have a $J=0$ component.}\n    \\label{fig:my_label}\n\\end{figure}\n\n\n\nUsing this result we can get a simple picture of the shape of the spectrum. First we have a number $e^{S_0}$ of states at exactly $E=0$ which are all in the supermultiplet $\\mathbf{0}$, an $SU(2)$ singlet. These are the extremal BPS states of the black hole as we will see in the next section. For small energies there are no states until we reach the gap in the spectrum given by the threshold energy for the supermultiplet $\\mathbf{\\frac{1}{2}}=(\\frac{1}{2})\\oplus 2(0)$, given by \n\\beq\nE_{\\rm gap} = \\frac{1}{8 \\Phi_r},\n\\eeq\nand for $E>E_{\\rm gap}$ we have a continuum of states. Something similar is true for higher multiplets $J > 1\/2$, but now the continuum starts at a supermultiplet-dependent gap\n\\beq\nE_{0}(J) = \\frac{1}{2\\Phi_r} J^2.\n\\eeq\nIt is perhaps not surprising that states with spin $J$ start at $E_{0}(J)$. The surprising feature is that there are no states with $J=0$ at energies $0<E<E_{\\rm gap}$. \n\nIn figure \\ref{fig:my_label} we depict the shape of the spectrum of the $\\cN=4$ super-Schwarzian theory. In the left panel we show the degeneracy of supermultiplets, with the supermultiplet-dependent gap mentioned above. In the right we show the degeneracy of states (not supermultiplets) with $J=0$ as an illustration. We see the delta function from $\\mathbf{0}$ but also contributions from the continuum coming from $(0)\\subset\\mathbf{1\/2}$ and $(0)\\subset\\mathbf{1}$. \n\n\nFor each supermultiplet, the density of states has a spectral edge $\\rho_{\\rm cont}(J,E) \\sim (E-E_{0}(J))^{1\/2}$ characteristic of a hermitian random matrix model, although we will not follow this direction in this paper.\n\nWe can make some comments regarding the calculation of the $\\cN=4$ super-Schwarzian Witten index. First it is straightforward to show that the fermion number can be computed in the following way \n\\beq\n(-1)^F = e^{2 \\pi i T^3}\n\\eeq\nacting on a $SU(2)$ representation of spin $J$.\\footnote{Here $T^3 = T^3_L + T^3_R$ corresponding to the left and right $SU(2)$ symmetries of the $\\cN=4$ super-Schwarzian.} Using the density of states derived above it is easy to extract the index. First all states with $E=0$ have $J=0$ and, therefore, $(-1)^F=1$. States with $J>1\/2$ contribute in the following way \n\\beq\ne^{2\\pi i J} \\left[\\chi_J(\\alpha) - 2 \\chi_{J-1\/2}(\\alpha)+\\chi_{J-1}(\\alpha)\\right].\n\\eeq\nIt is easy to see that when $\\alpha=0$ this combination exactly vanishes since there is the same number of bosonic and fermionic states in the supermultiplet. The same is true for $\\mathbf{1\/2}$ which gives $\\chi_{1\/2}(\\alpha)-2\\chi_0(\\alpha)$ and also vanishes for $\\alpha=0$. Therefore, the Witten index of the $\\cN=4$ super-Schwarzian theory is given by $e^{S_0}$ and counts the number of ground states.\n\nAs a final comment, there are two different definitions of the $\\cN=2$ super-Schwarzian theory that differ on the presence of a 't Hooft anomaly, as we review in Appendix \\ref{app:N2}. Since the gauge group of the $\\cN=4$ super-Schwarzian is $SU(2)$ we think there cannot be such anomaly \\cite{Kapec:2019ecr} and the theory is unique, but this deserves further investigation.\n\n\n\\subsection{Comparison with a pure bosonic theory} \n\\label{sec:compbosth}\nTo finish this section we would like to compare this solution to a non-supersymmetric version of the theory, such as the one in \\cite{Iliesiu:2020qvm}. Imagine we have a bosonic Schwarzian theory coupled to an $SU(2)$ mode. The action is \n\\beq\\label{ejksjw}\nI =-\\Phi_r \\int d\\tau \\hspace{0.1cm} \\text{Sch}(f,\\tau) + K \\int d\\tau  {\\rm Tr} \\left( g^{-1} \\partial_\\tau g\\right)^2 ,\n\\eeq\nwhere $K$ and $\\Phi_r$ are independent parameters. This theory can be solved exactly \\cite{Mertens:2019tcm}. The density of states as a function of energy and angular momentum $J$ is given by\n\\beq\n\\rho_{bos.}(J,E) = e^{S_0} \\sinh \\left( 2\\pi \\sqrt{2\\Phi_r\\left( E- E_{bos.}(J)\\right)}\\right) \\Theta (E-E_{bos.}(J)),~~~E_{bos.}(J)\\equiv\\frac{J(J+1)}{2K}.\n\\eeq\nThe bosonic sector of the supersymmetric theory is special in two ways. First of all it necessarily has $K=\\Phi_r$. Therefore, at least semiclassically one can compute the gap scale by measuring the following quantity \n\\beq\n\\left( \\frac{\\partial J}{\\partial \\Omega} \\right)_{T=0,\\Omega=0}= K = \\Phi_r,\n\\eeq\nwhere $\\Omega=i\\alpha\/\\beta$ is the potential conjugated to $J$ (from 4D perspective, angular velocity). The second, and more important, feature is the fact that for the bosonic theory \\eqref{ejksjw}, even if $K=\\Phi_r$, there are states with $J=0$ for any energy $E>0$, namely $\\rho(J=0,E>0)\\neq 0$ and $\\rho(J=0,E=0)=0$. The $\\cN=4$ supersymmetric Schwarzian theory is completely different. We find a delta function at $E=0$ describing $e^{S_0}$ states. Moreover, even for $J=0$, there are no states in the range $0<E<\\frac{1}{8\\Phi_r}$. This is the surprising feature we are emphasizing in this paper.\n\n\n\\section{Near-BPS Black Holes in Flat Space}\n\\label{sec:higherD}\n\nAs discussed in the Introduction~\\ref{sec:intro}, extremal Reissner-Nordstr\\\"om black holes are a basic object of study in supergravity and string theory because in many examples they preserve supersymmetry and are exact solutions of the theory. The BPS nature of such gravitational solutions allows one to identify them with a corresponding D-brane configuration; such brane models allow for a precise microstate counting \\cite{Strominger:1996sh} which matches the result of the Bekenstein-Hawking entropy. While AdS$_2$ solutions of various supergravities are well studied, we focus on a minimalistic choice of black hole and AdS$_2 \\times S^2$ solutions of pure, ungauged, 4D $\\mathcal{N} = 2$ supergravity. As we will review, such a 4D theory already has the right ingredients for $\\frac{1}{2}$-BPS extremal black holes. Additionally, the pure $\\mathcal{N}=2$ supergravity theory will generically appear as a subsector of a more general matter coupled $\\mathcal{N}=2$ supergravity~\\cite{Freedman:2012zz}.\n\nIn this section, we will first recall some facts about 4D supergravity, then analyze the extremal AdS$_2 \\times S^2$ black hole solution. This solution has ordinary isometries corresponding to the AdS and sphere factors, and additionally permits covariantly constant spinors which are solutions of the Killing spinor equations. As is known in the literature \\cite{Kallosh:1997qw,Claus:1998yw,Boonstra:1998yu} the group of isometries for the near-horizon region enlarges, from the bosonic $SL(2, \\mathbb R)$, to the supergroup $PSU(1,1|2)$ and we will review this in some detail. The appearance of this supergroup corresponds to the symmetries of the $\\mathcal{N}=4$ JT and Schwarzian theories utilized in Section~\\ref{sec:N=4superJT}.\n\nIn this section, we are really interested in black holes which parametrically approach their BPS limits (i.e.~that have temperature of the same order or smaller than the conjectured $E_\\text{gap}$). In the effective AdS$_2$ near horizon dilaton supergravity, we allow fluctuations of the dilaton as well as Kaluza-Klein gauge fields and fermions. In the limit of a large black hole with small near BPS fluctuations, we will derive the effective 2D $\\cN=4$ JT supergravity. This will allow us to match the microscopic and thermodynamic properties of 4D black holes with the output of the $\\mathcal{N}=4$ Schwarzian from Section~\\ref{sec:density-of-states}.\n\n\n\n\n\\subsection{4D $\\mathcal{N}=2$ Supergravity}\n\\label{ssec:4dSugra}\nWe begin with some notions and notations for gravity in 4D. Some of these conventions will be different than those used in Section~\\ref{sec:N=4superJT} since we will be working in four-dimensional notation. In the end we will explain how to translate the results to be compatible with the two-dimensional $BF$ theory.\n\nWe use a mostly plus Lorentzian metric, and will eventually Wick rotate to Euclidean signature in 2D. We denote 4D curved space indices $M,N = 0 \\dots 3$ and the tangent space indices $A, B = 0, \\dots 3$. The 4D metric $G_{MN} = \\eta_{AB} E^A_M E^B_N$ is expressed in terms of the vierbein $E^A_M$. In 2D, the curved and tangent space indices are respectively $\\mu, \\nu = 0,1$ and $a,b = 0,1$, and we use the vielbein $e^a_\\mu$ and dualized spin connection $\\omega_\\mu$. We will occasionally use indices for the internal $S^2$ space, which will take values $m,n = 2,3$ for the coordinate indices and $a',b' = 2,3$ for the frame indices. When we are working in the near-horizon region, with the AdS$_2$ $\\times S^2$ product manifold, the vielbein will decompose as:\n\\begin{equation}\n    E^A_M = e^a_\\mu \\, , \\, \\, e^{a'}_m \\, .\n\\end{equation}\n We will generally suppress spinor indices when possible, but our spinor conventions are given in Appendix~(\\ref{app:spinorssugra}).\n\nPure, ungauged $\\mathcal{N}=2$ supergravity is a minimal theory containing only the graviton, $G_{MN}$; an $SU(2)_R$ doublet of gravitinos, $\\Psi_M^I$, $I=1,2$; and a $U(1)$ gauge field $A_M$ under which a black hole solution is electrically or magnetically charged. Without the addition of extra vector or hypermultiplets, this gravity theory does not have a precise embedding in string theory.\\footnote{As explained in \\cite{Becker:2007zj}, compactification of either Type II theory on a Calabi-Yau threefold will always produce at least one $\\mathcal{N}=2$ hypermultiplet along with the gravity multiplet discussed here. This follows from the reduction of the 10D graviton, dilaton, and 2-form.} Nevertheless, we argue that universal features of supersymmetric black holes are already present in this model, through the relationship with 2D Super-JT gravity.\n\nWe write the Lagrangian of pure 4D $\\mathcal{N}=2$ supergravity, following the conventions of \\cite{Freedman:2012zz}:\n\\begin{equation}\n    E^{-1} \\mathcal{L} = \\kappa^{-2}\\bigg(\\frac{1}{2}R - \\bar{\\Psi}_{I M} \\Gamma^{MNP}D_{N}\\Psi^I_P - \\frac{1}{4}F_{MN}F^{MN} + \\frac{\\varepsilon^{IJ}}{2\\sqrt{2}}\\bar{\\Psi}^M_I(F_{MN} + i\\, \\star  F_{MN}\\Gamma_5)\\Psi_J^{N}  \n    + \\textrm{4 gravitino} \\bigg)\\, ,\n    \\label{Eq:4DCompLag}\n\\end{equation}\nwith the standard definitions $D_M = \\partial_M + \\frac{1}{4}\\omega^{AB}_M \\Gamma_{AB} $ and $ F= dA$. We also write the 4D Newton constant as $  8 \\pi G_N = \\kappa^2$. In the above action and throughout this subsection, we follow the convention that a spinor $\\epsilon^I$ with an upper $SU(2)_R$ index has positive chirality, while a lower index has negative chirality:\n\\begin{equation}\n    \\Gamma_5 \\epsilon^I = + \\epsilon^I \\, , \\,\\,\\,\\,\\, \\Gamma_5 \\epsilon_I = - \\epsilon_I \\, .\n    \\label{Eq:chiralindex}\n\\end{equation}\nIn analyzing the supersymmetry and dimensional reduction later, we will often instead use Dirac spinors where the chirality will be implicit.\n\nWhile we have explicitly displayed the Pauli terms in Eq.~\\eqref{Eq:4DCompLag} proportional to the antisymmetric symbol $\\varepsilon^{IJ}$, we have omitted the more complicated 4-gravitino term. The form of this term is known in general,\\footnote{A general matter coupled $\\mathcal{N}=2$ supergravity theory including additional fermion terms is found, for instance, in \\cite{Andrianopoli:1996vr}.} but it is vanishing in the bosonic backgrounds corresponding to extremal black hole solutions. In the dimensional reduction to 2D, we will use other methods to fix the higher order fermion terms. With the understanding that some expressions are augmented by these terms higher order in fermions, Eq.~\\eqref{Eq:4DCompLag} is invariant under the following local supersymmetry transformations:\n\\begin{align}\n    \\delta_{\\epsilon} E^A_M &= \\frac{1}{2}\\bar{\\epsilon}^I \\Gamma^A \\Psi_{MI} + \\textrm{h.c.} \\, , \\\\\n    \\delta_{\\epsilon} A_M &= \\frac{1}{\\sqrt{2}} \\varepsilon^{IJ} \\bar{\\epsilon}_I\\Psi_{MJ} + \\textrm{h.c.} \\, , \\\\\n    \\delta_{\\epsilon}\\Psi^I_M &= (\\partial_M + \\frac{1}{4}\\omega^{AB}_M \\Gamma_{AB})\\epsilon^I  - \\frac{1}{4 \\sqrt{2}}\\Gamma^{AB}  F_{AB}\\Gamma_M \\varepsilon^{IJ}\\epsilon_J \\, . \\label{eq:gravsusytransform}\n\\end{align}\n\nThe Lagrangian and supersymmetry transformations above make manifest the $SU(2)_R$ symmetry of $\\mathcal{N}=2$ supergravity. In the application of this theory to nearly AdS$_2$ $\\times$ S$^2$ black holes described in Section~\\ref{sec:N=4superJT}, we considered a theory with $PSU(1,1|2)$ symmetry and bosonic subgroup $SL(2,\\mathbb{R})\\times SU(2)$. This latter $SU(2)$ is realized geometrically by the spatial rotations of the black hole, and is independent of the $SU(2)_R$ symmetry present in 4D. This feature is characteristic of the \\emph{small} $\\mathcal{N}=4$ superconformal algebra, which has the R-symmetry $SU(2) \\times SU(2)_{outer}$. In the context of this paper, it is possible to show that the outer R-symmetry corresponds to the 4D $SU(2)$ global symmetry.\\footnote{We thank E. Witten for emphasizing to us that the 4D $SU(2)$ global symmetry is only an approximate symmetry in supergravity; it should not be an exact symmetry of the near extremal black holes. Only the $SU(2)$ symmetry realized as rotations of the $S^2$ is a physical symmetry in a more complete model such as string theory.} Because the outer symmetry plays no role in our analysis, it will be convenient to use a formulation of $\\mathcal{N}=2$ supergravity with Dirac rather than Majorana gravitinos. This will hide the outer $SU(2)$ symmetry, but simplify many formulas in what follows. This version of the theory is presented in \\cite{Freedman:1976aw,Romans:1991nq}, and the passage to this formalism involves introduction of the Dirac spinors $\\Psi_M = \\Psi^1_M + i \\Psi_M^2$.\n\n\n\n\\subsection{The Black Hole and AdS$_2 \\times S^2$ Solutions}\n\\label{Sec:extremalBH}\n The extremal black hole of $\\mathcal{N} =2$ supergravity is a standard solution of Einstein-Maxwell theory which preserves 4 real supercharges of the 8 total. This is in contrast to the near horizon limit of this solution, AdS$_2 \\times S^2$, which preserves all supersymmetries. The supersymmetry enhancement~\\cite{Kallosh:1997qw, Claus:1998yw} corresponds to the addition of conformal supercharges, and the AdS background should have a full superconformal set of isometries. In this section, we briefly review these two solutions, which are by now somewhat standard.\n\nThe bosonic part of our action using our normalization conventions from \nEq.~\\eqref{Eq:4DCompLag} is the typical Einstein-Maxwell action:\n\\begin{equation}\n    S_{4D}^{\\text{(bosonic)}} =  \\frac{1}{\\kappa^2} \\int d^4x \\sqrt{-G}\\left (\\frac{1}{2}R - \\frac{1}{4}F_{MN}F^{MN} \\right ) \\, .\n    \\label{eq:4Dbosoniclag}\n\\end{equation}\n\nSolving the equations of motion, one can obtain the metric and gauge field of the celebrated extremal Reissner\u2013Nordstr\\\"om black hole solution, here given by: \n\\begin{align}\n    ds^2 &= -\\left(\\frac{r_0}{r} - 1\\right)^{2}dt^2 + \\left(\\frac{r_0}{r} - 1\\right)^{-2}dr^2 + r^2 d\\Omega_2^2 \\, , \\label{eq:extremalRNmetric}\\nn \\\\\n    F_{tr} &= -\\frac{\\kappa Q}{\\sqrt{4\\pi} r^2} \\, .\n\\end{align}\nWe also have the relationships between the extremal radius, mass, and charge, $r_0 = G_N M$ and $Q = \\sqrt{r_0 M}$. For simplicity, we have chosen an electrically charged black hole and set the magnetic charge $P=0$, but the results of this section can be easily generalized to this case.\nIn terms of the charge which we are keeping fixed, we have chosen units such that the extremal mass and extremal Bekenstein-Hawking entropy are:\\footnote{This choice is slightly non-standard, but will simplify some formulas later.}\n\\be \n\\label{eq:mass-and-entropy-BH-SUGRA}\nM_0 = Q\/\\sqrt{G_N}\\,, \\qquad \\qquad S_0 = \\frac{\\pi r_0^2}{G_N}= \\pi Q^2\\,.\n\\ee \n\n\n\nIn addition to the familiar bosonic symmetries of the solution~\\eqref{eq:extremalRNmetric} corresponding to translations and rotation, there are additional supersymmetries represented by covariantly constant spinors on the manifold. Since all fermions vanish on this background, we only need to analyze the gravitino supersymmetry transformation, Eq~\\eqref{eq:gravsusytransform}, adapted to Dirac spinors. Preserved Killing spinors are spacetime dependent $\\epsilon$ which are annihilated by the gravitino transformation rule:\\footnote{A discussion of the relationship between supersymmetry preserving black branes, Killing spinors, and Killing vectors may be found in \\cite{Gibbons:1982fy,Duff:1986hr,Lu:1998nu,Gauntlett:1998kc}.}\n\\begin{equation}\n    \\delta_{\\epsilon}\\Psi_M = \\hat{D}_M \\epsilon =  (\\partial_M + \\frac{1}{4}\\omega^{AB}_M \\Gamma_{AB})\\epsilon  + \\frac{i}{4 \\sqrt{2}}\\Gamma^{AB}  F_{AB}\\Gamma_M \\epsilon = 0\\, .\n    \\label{eq:KillingSpinor4D}\n\\end{equation}\nWe will typically use Dirac spinors and do not make use of the chiral components.\n\n\nThe extremal black hole of Eq.~\\eqref{eq:extremalRNmetric} can be thought of as a supergravity soliton which interpolates between two maximally supersymmetric vacuua; Minkowski space in the asymptotic far region, and AdS$_2 \\times S^2$ in the near horizon region. Because we are interested in an effective AdS$_2$ theory approximated by supersymmetric JT gravity, we now turn our attention to the AdS$_2 \\times S^2$ metric with the purely electric Bertotti-Robinson ansatz. \nStarting with Eq.~\\eqref{eq:extremalRNmetric}, we perform standard manipulations to take the near horizon limit. We shift the location of the event horizon then rescale the radial coordinate with to get the Poincare patch metric:\n\\begin{align}\n    ds^2 = \\frac{r_0^2}{z^2}(-dt^2 + dz^2) + r_0^2 d \\Omega_2^2 \\, , \\,\\,\\,\\, F = \\frac{1}{2} \\frac{\\kappa Q}{\\sqrt{4\\pi}}\\frac{1}{z^2} dt \\wedge dz\\, .\n    \\label{eq:poinAdSmetric}\n\\end{align}\n\nTo study this background, including the Killing spinors, we make the further change of coordinates $z = r_0 e^{-r\/r_0}$:\n\\begin{equation}\n    ds^2 = -e^{2r\/r_0}dt^2 + dr^2 + r_0^2 d \\Omega_2^2 \\, , \\, \\, \\, \\,\n    F = -\\frac{1}{2} \\frac{\\kappa Q}{\\sqrt{4\\pi} r_0^2}e^{r\/r_0} dt \\wedge dr\\, .\n\\end{equation}\nAt the risk of notational clutter, we denote frame indices with a hat, and in frame coordinates, the bosonic fields are:\n\\begin{align}\n    E^{\\hat t} &= e^{r\/r_0} dt \\, , \\,\\,\\,\\,\n    E^{\\hat r} = dr \\, , \\,\\,\\,\\,\n    E^{\\hat m} = (r_0 d \\theta \\, , \\, \\, r_0 \\sin{\\theta} d \\phi) \\, , \\,\\,\\,\\, \n    F_{\\hat{t} \\hat{r}} = - \\frac{\\kappa Q}{\\sqrt{4\\pi} r_0^2} = - \\frac{\\sqrt{2}}{r_0} \\label{eq:ads2s2frames}\\, . \\\\\n        \\omega^{\\hat{t}\\hat{r}} &= \\frac{1}{r_0}E^{\\hat t} = \\frac{1}{r_0}e^{r\/r_0} dt \\, , \\,\\,\\,\\,\n    \\omega^{\\hat{\\phi} \\hat{\\theta}} = \\omega^{\\phi \\theta}_{S^2} = \\cos \\theta d\\phi \\nn \\, .\n\\end{align}\nIn the expression for the flux, we used the definitions for $\\kappa$, $Q$, and $r_0$ in the extremal limit. The constant flux background is analogous to the Freund-Reubin ansatz~\\cite{Freund:1980xh} in that the spin connection also has a $1\/r_0$ behavior, and it is the cancellation between these two terms which will permit a maximally supersymmetric vacuum.\n\nBefore plugging the explicit background into Eq.~\\eqref{eq:KillingSpinor4D}, it is useful to first examine the integrability condition for the supercovariant derivative with a general background acting on a Killing spinor. A necessary condition for the existence of Killing spinors is\n\\begin{equation}\n    [\\hat{D}_M,\\hat{D}_{N}]\\epsilon = 0 \\, .\n\\end{equation}\nComputing this in the case of pure ungauged $\\mathcal{N}=2$ supergravity \\cite{Romans:1991nq} gives\n\\begin{equation}\n    \\left ((R_{MN}^{\\, \\, AB}\\Gamma_{AB} - \\frac{1}{8} \\slashed{F}\\Gamma_{[M}\\slashed{F}\\Gamma_{N]}) +\\frac{i}{\\sqrt{2}}\\Gamma_{AB}\\Gamma_{[N}(\\nabla_{M]}F^{AB}) \\right )\\epsilon = 0 \\, .\n\\end{equation}\nIt is possible to further simplify the Gamma matrix contractions for a general flux background, but the case of AdS$_2 \\times S^2$ has further simplifications as the third term vanishes identically. Allowing $a,b = 0,1$;  $a',b' = 2,3$ as frame indices and $\\mu,\\nu$;  $m,n$ as coordinate indices for AdS and the sphere respectively, we obtain the familiar conditions:\n\\begin{align}\n     \\big ( R_{\\mu \\nu}^{\\,\\, ab}\\gamma_{ab} &+ \\frac{1}{r_0^2}(e^a_\\mu e^b_\\nu - e^a_\\nu e^b_\\mu) \\gamma_{ab}  \\big )\\epsilon = 0 \\, , \\nn \\\\\n        \\big ( R_{m n}^{\\,\\, a'b'}\\gamma_{a'b'} &- \\frac{1}{r_0^2}(e^{a'}_m e^{b'}_n - e^{a'}_n e^{b'}_m) \\gamma_{a'b'}  \\big ) \\epsilon = 0 \\, .\n\\end{align}\nThese are identically satisfied by the Bertotti-Robinson metric as the AdS and sphere are maximally symmetric. We do not obtain any new algebraic conditions on the Killing spinors. These conditions, as well as the form of the spinors themselves, must be determined from the first order equation, Eq.~\\eqref{eq:KillingSpinor4D}. Inserting $F_{ab} = -\\frac{\\sqrt{2}}{r_0}\\epsilon_{ab}$ into this Killing spinor equation, we find that it splits into a pair of equations for the AdS and sphere parts:\n\\begin{align}\n(\\partial_\\mu + \\frac{1}{4}\\omega^{ab}_\\mu \\gamma_{ab}   - \\frac{i}{2 r_0}\\gamma_3 \\gamma_\\mu )&\\epsilon = 0 \\, , \\label{eq:adsKSequation} \\\\ \n(\\partial_m + \\frac{1}{4}\\omega^{a'b'}_m \\gamma_{a'b'}  - \\frac{i}{2 r_0}\\gamma_m  )&\\epsilon = 0 \\, , \\label{eq:sphereKSequation}\n\\end{align}\nwhere $\\gamma_3 = - \\gamma_0 \\gamma_1 = \\sigma_3 \\otimes \\mathbf{1}$ is the pseudo-chirality operator on the AdS space. These can be obtained by applying the conventions of Appendix \\ref{app:spinorssugra}. We observe that these equations are of the same form studied in \\cite{Lu:1998nu}, and we will give an explicit construction of the Killing spinors.\n\nTo find expressions for the Killing spinors, we make a slight abuse of notation in writing $\\epsilon_{4D} = \\epsilon \\otimes \\eta$. We will see that the complex 2 component $\\epsilon$ and $\\eta$ are Killing spinors on the AdS$_2$ and S$^2$ factors. Inserting the expressions of Eq.~\\eqref{eq:ads2s2frames}, we find the four Killing spinor equations,\n\\begin{align}\n&(\\partial_t - \\frac{1}{2r_0}e^{r\/r_0} \\gamma_{3} (1+ i \\gamma_0)) \\epsilon = 0 \\, ,\\nn \\\\ \n&(\\partial_r + \\frac{i}{2 r_0} \\gamma_0 ) \\epsilon = 0 \\, , \\nn\\\\ \n&(\\partial_\\theta - \\frac{i}{2}\\sigma_1 )  \\eta = 0 \\, ,\\nn\\\\\n&(\\partial_\\phi - \\frac{1}{2}\\cos\\theta \\sigma_{13} - \\frac{i}{2}\\sin \\theta \\sigma_3 ) \\eta = 0 \\, .\n\\end{align}\nTo solve these, we first introduce constant anticommuting Dirac spinors $\\epsilon^{0}_{\\pm}$ and $\\eta^\\alpha$. Here the $\\alpha$, $\\beta$ $= 1,2$ spinor indices are those for the internal $SU(2)$ on the sphere; this contrasts with the notation of Section~\\ref{sec:N=4superJT}, where these indices were denoted by $p,q$.\n\nThe $\\epsilon^0_\\pm$ are eigenstates under the supersymmetric projector:\n\\begin{equation}\n\\label{eq:bpsprojector}\n        \\epsilon^{0}_{\\pm} = \\mathcal{P}_\\pm \\epsilon^{0}_{\\pm} = \\frac{1}{2}(1 \\pm  i \\gamma_0 ) \\epsilon^{0}_{\\pm} \\, .\n\\end{equation}\nA convenient basis for the $\\eta^\\alpha$ is the one of definite chirality on the S$^2$. From Appendix~\\ref{app:spinorssugra}, we see that we may choose $\\eta^\\alpha = (\\eta^1, \\eta^2)$ with $\\sigma_2 \\eta^1 =  \\eta^1$ and $\\sigma_2 \\eta^2 = - \\eta^2$. For clarity, this explicit basis choice amounts to the familiar eigenstates: $\\epsilon^{0}_{+} = (1, i)$, $\\epsilon^{0}_{-} = (1, -i)\\,$, $\\eta^1 = (1, i)\\,$, $\\eta^2 = (1, -i)$.\n\nWith these projections, we find the AdS$_2$ $\\times$ S$^2$ solutions to the Killing spinor equations~\\eqref{eq:KillingSpinor4D} are:\n\\begin{equation}\n\\epsilon^\\alpha_- \\equiv e^{\\frac{r}{2r_0}} e^{\\frac{i \\theta}{2} \\sigma_1} e^{\\frac{\\phi}{2}  \\sigma_{13}} (\\epsilon^0_{-} \\otimes \\eta^\\alpha )\\, .\n\\label{eq:KS-spinor}\n\\end{equation}\n\\begin{equation}\n\\epsilon^\\alpha_+ \\equiv  \\left (1 + \\frac{t e^{r\/r_0}}{r_0} \\gamma_3 \\right ) e^{-\\frac{r}{2r_0}} e^{\\frac{i \\theta}{2} \\sigma_1} e^{\\frac{\\phi}{2}  \\sigma_{13}} (\\epsilon^0_{+} \\otimes \\eta^\\alpha )\\, .\n\\label{eq:KS+spinor}\n\\end{equation}\nSimilar expressions hold for the Dirac conjugate $\\bar{\\epsilon}^\\alpha_\\pm$, which we will need to form Killing vectors.\\footnote{Some of our conventions for Killing spinors may be different than elsewhere in the literature. For more, see \\cite{Fujii:1985bg,Lu:1998nu,Banerjee:2010qc,Banerjee:2011jp}.}\n\nThe super-isometry algebra of Killing vectors and spinors is typically determined by forming bilinears of spinors and checking that these correspond to the bosonic symmetries of the metric. Knowledge of the Killing spinors may be used to construct spacetime supercharges as integrals of the supercurrent~\\cite{Gauntlett:1998kc}; for our purposes it is sufficient to check that all bosonic isometries are generated by Killing spinor bilinears. This allows us to loosely interpret $\\epsilon^\\alpha_- \\sim G^\\alpha_{-1\/2}$ and $\\epsilon^\\alpha_+ \\sim G^\\alpha_{1\/2}$, and we will compute the Killing spinor analog of $\\{\\bar{G}^\\alpha_{\\pm}, G^\\beta_{\\pm'}\\}$ which were the anticommutators appearing in~\\eqref{eq:smalln4gen4}. Therefore, we consider objects of the form,\n\\begin{equation}\n    \\bar{\\epsilon}^\\alpha_{\\pm}\\Gamma^M \\epsilon^\\beta_{\\pm'} E_M \\, ,\n    \\label{Eq:KillvecSpin}\n\\end{equation}\nwhich transforms as a vector field. In fact, if we form bilinears from Killing spinors, this result will automatically be a Killing vector. A typical bilinear of the form Eq.~\\eqref{Eq:KillvecSpin} may vanish depending on the commutation of  $\\mathcal{P}_\\pm$ with the various gamma matrices. \n\nTo simplify the formulas, we will choose a unit normalization for the $\\eta^\\alpha$ and $\\epsilon^0_\\pm$ which solve Eq.~\\eqref{eq:bpsprojector}, but in general one would find constant spinor contractions appearing on the right hand side of the formulas to follow. As shown in Appendix~\\ref{ss:PSUblackhole}, the Killing vectors determined by \\eqref{Eq:KillvecSpin},\\eqref{eq:KS-spinor}, \\eqref{eq:KS+spinor} are:\n\\begin{align}\n    H &= \\partial_t \\, , \\,\\,\\,\\, D = t \\partial_t + z\\partial_z \\, , \\, \\, \\, \\, \\,K = (z^2 +t^2)\\partial_t + 2 tz \\partial_z \\, , \\\\\n    T_1 &= \\sin \\phi \\partial_\\theta + \\cot \\theta \\cos \\phi \\partial_\\phi \\, , \\,  T_2 =  \\cos \\phi \\partial_\\theta - \\cot \\theta \\sin \\phi \\partial_\\phi \\, , \\, \\, T_3 =  \\partial_\\phi \\, .\n    \\label{eq:4dS2Killing}\n\\end{align}\nwhich form the $SL(2)\\times SU(2)$ algebra, i.e.~the bosonic part of the supergroup $PSU(1,1|2)$. This motivates the conclusion that the near horizon theory should have this symmetry.\n\n\\subsection{Dimensional Reduction}\n\\label{ss:dimensionalreduction4d}\n\nOur goal is to now demonstrate that the effective 2D gravity theory describing the near extremal perturbations of Eq.~\\eqref{eq:poinAdSmetric} ultimately reduces to the supersymmetric completion of JT gravity described in Section~\\ref{sec:N=4superJT}. We will start with a convenient Kaluza-Klein ansatz for 4D black holes with a dilaton $\\chi$ capturing the fluctuating horizon area~\\cite{Duff:1986hr,Gibbons:2003gp}. Without the inclusion of extra massless fields, this ansatz does not generally satisfy all the higher dimensional Einstein equations~\\cite{Cvetic:2000dm}. However, as in the non-supersymmetric case~\\cite{Iliesiu:2020qvm}, for sufficiently small masses and spins above extremality, the reduced effective 2D theory does capture the higher dimensional near extremal black holes.\n\nIn our discussion of the Kaluza-Klein ansatz, distorted 4D quantities will be hatted. We begin with the distorted metric, written in terms of coordinates $x^M = (x^\\mu, y^m)$:\n\\begin{equation}\n    d\\hat{s}^2_{4D} = \\frac{r_0}{\\chi^{1\/2}}g_{\\mu \\nu}dx^\\mu dx^\\nu + \\chi \\, h_{mn}(dy^m + T^m_{i} B^{i}_{\\mu}dx^{\\mu})(dy^n + T^n_{j} B^{j}_{\\nu}dx^{\\nu}) \\, ,\n\\end{equation}\nwith an arbitrary 2D metric $g_{\\mu \\nu}$ and the unit $S^2$ metric:\n\\begin{equation}\n    h_{mn}dy^mdy^n = d\\theta^2 + \\sin^2 \\theta d\\phi^2 \\, .\n\\end{equation}\nWe parametrize the angular fluctuations with an $SU(2)$ gauge field $B^i_\\mu$, which transforms as a $\\mathbf{3}$ of $SO(3)$ and enters through the S$^2$ Killing vectors~\\eqref{eq:4dS2Killing}.\\footnote{In components, these obey the following identities in terms of the scalar harmonics $\\mu_{i}$~\\cite{Gibbons:2003gp}:\n\\begin{align}\n    \\mu_1 = \\sin\\theta \\cos\\phi \\, , \\, \\, \\, \\mu_2 = \\sin\\theta \\sin\\phi   \\, , \\, \\, \\, \\mu_3 = \\cos\\theta  \\, , \\, \\, \\, \n    T_{i}^{a'} = \\epsilon^{a'b'}\\partial_{b'} \\mu_{i} \\, , \\,\\,\\,\n    \\delta_{a'b'} T^{a'}_{i} T^{b'}_{j}&= \\delta_{i j} - \\mu_{i} \\mu_{j} \\, .\\nn\n\\end{align}\n}\n\nThe vierbein associated with this decomposition is\n\\begin{equation}\n\\label{Eq:Vielbeinreduction}\n    E_{M}^A dX^M \\rightarrow \\hat{e}^a = \\frac{r_0^{\\frac{1}2}}{\\chi^{\\frac{1}4}} e_{\\mu}^a dx^{\\mu} \\textrm{, }\\, \\,  \\hat{e}^{a'} = \\chi^{\\frac{1}2} e_{m}^{a'}(dy^m + T^{m}_{i} B^{i}_{\\mu}dx^{\\mu}) \\, ,\n\\end{equation}\nwith the metrics for the arbitrary 2D theory and the explicit metric of S$^2$:\n\\begin{equation}\n    g_{\\mu \\nu} = \\eta_{ab} e_{\\mu}^a e_{\\nu}^b\\, \\textrm{, } \\, \\, \\, \\,  h_{mn}= \\delta_{a'b'} e_{m}^{a'} e_{n}^{b'}\n\\end{equation}\n\nThe dimensional reduction of the Einstein-Hilbert term as well as the gravitino kinetic term both require the spin connection compatible with the vierbeins above. Even though the torsion tensor is not in general zero in supergravity due to fermions, for now we are free to use the torsion free first Cartan structure equations:\\footnote{Here, we use a hat notation to distinguish the modified vielbeins and connections from those corresponding to their 2D symmetric space versions, which satisfy:\n\\begin{align}\nd e^a + \\omega^{ab}\\wedge e_b = 0\\, , \\qquad\n d e^{a'} + \\omega^{a'b'}\\wedge e_{b'} =0 \\, .\\nn\n\\end{align}}\n\\begin{align}\n    d \\hat{E}^A &+ \\hat{\\omega}^{AB}\\wedge \\hat{E}_B = 0 \\, .\n\\end{align}\nWe will ultimately pass to the first order formulation of gravity in the effective AdS$_2$ theory. This first order theory does not automatically enforce the torsion free constraint, so we will only use the 4D distorted spin connection as an intermediate step. The expressions for the distorted spin connection, Ricci tensors and Ricci scalar are given in Appendix~\\ref{app:spinorssugra}. \n\n\n\nWe may now write the reduced gravitational action explicitly after dropping a total derivative on $\\chi$:\\footnote{Once we pass to the first order formalism in terms of the vielbein and spin connection, it is no longer generally true that integration by parts is valid due to the presence of torsion. In this dimensional reduction, all derivative terms on the dilaton $\\chi$ are true total derivatives, independent of torsion.}\n\\begin{align} \nS_{2D,\\text{ metric}} = \\frac{2 \\pi}{\\kappa^2} \\int \\! \\! d^2x \\sqrt{-g} \\left (\\chi R + \\frac{2r_0}{\\sqrt{\\chi}} -\\frac{1}{6} \\frac{\\chi^{\\frac{5}2}}{r_0} H^{i}_{ab}H_{i}^{ab} \\right ) \\, .\n\\end{align}\n\nOur ansatz for $F$ in the distorted case is a term which generalizes the constant flux background by the inclusion of the dilaton; in local coordinates it is \n\\begin{equation}\n    \\hat{F}_{ab} = - \\sqrt{2} \\frac{r_0}{\\chi} \\epsilon_{ab} \\, .\n\\end{equation}\nAs we will explain shortly, fixing the field strength in the near-horizon region is well motivated and we do not need to be concerned with quantum fluctuations of the $U(1)$ gauge field around this saddle. Thus, such a mode will not contribute to the Euclidean path integral when studying black holes in the canonical ensemble.\n\nOnce we expand around the near extremal background, this will have the effect of introducing a cosmological constant term. Additionally, we will perform the Wick rotation to Euclidean time, which will give the gravitational path integral an interpretation as a statistical partition function. In total, the reduction of the 4D bosonic Euclidean action becomes:\n\\begin{align}\nS_{2D}^{\\text{(bosonic)}} = -\\frac{2 \\pi}{\\kappa^2} \\int \\! \\! d^2x \\sqrt{g} \\left (\\chi R + \\frac{2r_0}{\\chi^{1\/2}} -\\frac{2r_0^3}{\\chi^{3\/2}} -\\frac{1}{3} \\frac{\\chi^{5\/2}}{r_0} \\tr_{SU(2)} H_{ab}H^{ab} \\right )  \\, .\n\\end{align}\nThis is the bosonic part of a more complicated 2d dilaton supergravity, see for example \\cite{Grumiller:2007ju}. This model is difficult to quantize directly, so we must make use of the near extremal condition in order to make contact with an extension of the JT model. To do this, we expand the dilaton to first order around the extremal squared radius, with a multiple of $G_N$ to obtain a convenient normalization:\n\\begin{equation}\n   \\frac{ \\chi(x)}{G_N} = \\frac{r_0^2}{G_N} + 2  \\Phi(x) + \\mathcal{O}\\left(\\frac{G_N \\Phi^2}{r_0^2}\\right)\\,.\n\\end{equation}\nTo this order, the action becomes\n\\begin{align}\nS_{2D}^{\\text{(bosonic)}} = -\\int \\! \\! d^2x \\sqrt{g} \\left[\\frac{r_0^2}{4G_N} R -\\frac{1}{12} \\frac{r_0^4}{G_N} \\tr_{SU(2)}H_{ab}H^{ab} + \\frac{\\Phi}2\\left(R + \\frac{2}{r_0^2} -\\frac{5}{6} r_0^2\\,\\tr_{SU(2)}H_{ab}H^{ab} \\right)\\right]  \\, .\n\\end{align}\nThe first term does not involve the dilaton, and is proportional to the Euler characteristic once we restore the Gibbons-Hawking-York boundary term containing the extrinsic curvature. This term thus sets the extremal entropy, and this will remain true independently of how we supersymmetrize the model. Thus, we find:\n\\begin{equation}\n    S_0 = \\frac{\\pi r_0 ^2}{ G_N}=\\pi Q^2 \\, ,\n\\end{equation}\nwhich is precisely the extremal entropy of Eq.~\\eqref{eq:mass-and-entropy-BH-SUGRA}.\n\n\n\nTo make contact with the bosonic part of the BF description presented in section \\ref{sec:PSU(1,1|2)-BF-thy}, we must introduce Lagrange multiplier fields for this gauge field. This is a scalar field $b_i$ transforming in the adjoint of $SU(2)$; we will additionally look for a form of the BF coupling $i b_i H^i$. In a large $r_0 $ expansion, this part of the action becomes:\n\\begin{equation}\n    S_{2D}^{\\text{(bosonic)}} \\supset -i \\int \\,   \\tr_{SU(2)}b H  + \\int \\! \\! d^2x \\sqrt{g}\\, \\frac{3G_N}{2r_0 ^4} \\,\\left (1 + \\mathcal{O}\\left(\\frac{G_N \\Phi}{r_0 ^2}\\right) \\right ) b_i b^i \\, .\n\\end{equation}\nWe see the quadratic terms in $b_i$ are suppressed by large powers of the extremal radius, so in the large mass near extremal limit ($r_0 \\gg 1\/G_N^{1\/2}$), the effective description is the unit normalized BF theory for $SU(2)$.\\footnote{One may worry that in integrating out the gauge field, the dilaton coupling leads to a divergent 1-loop determinant. There are various ways of addressing this, we refer the reader to \\cite{Iliesiu:2019lfc}.} Consequently, the dimensionally reduced bosonic part of the action is\n\\begin{align}\n\\label{eq:2D-bosonic-action}\nS_{2D}^{\\text{(bosonic)}} = S_0 - \\frac{1}{2} \\int \\! \\! d^2x \\sqrt{g} \\, \\Phi \\left (R + \\frac{2}{r_0 ^2}\\right ) -  i \\int   \\tr_{SU(2)}b H \\, + \\cO\\left(\\frac{G_N}{r_0^2}\\right).\n\\end{align}\nThis has the form of the standard JT bulk term plus a decoupled BF term. \n  This matches the fact that the extremal AdS$_2$ $\\times$ S$^2$ metric possesses $SL(2,\\mathbb{R})$ $\\times$ $SU(2)$ as bosonic symmetries. Of course, to the action \\eqref{eq:2D-bosonic-action}, one in principle has to add the GHY term and the boundary term for the $SU(2)$ gauge field. Nevertheless, since the correct boundary term was derived in directly in the first order formalism in section \\ref{sec:PSU(1,1|2)-BF-thy}, we will solely discuss the matching of bulk terms in this section. \n  \n  \n  Our next goal is to restore the appropriate fermionic terms to the 2D action. \n  Kaluza-Klein supergravity reductions on spheres are important for the AdS\/CFT correspondence, but performing this procedure for the full, consistent, nonlinear supergravity is somewhat more involved than what we will attempt here.\\footnote{One typically needs extra massless fields in higher dimensions, transgression terms in the KK ansatz, and higher order terms in fermions. For a review of some of the developments on this front, see \\cite{Duff:1986hr,deWit:1986oxb,Nastase:1999kf,Cvetic:2000dm}.} Rather than performing a full Kaluza-Klein reduction, we will instead derive only the terms that are linear (or less than linear) in the dilaton $\\Phi$. Higher powers of $\\Phi$ will come suppressed by factors of $G_N\/r_0^2$ (i.e.~similar to the higher powers in $\\Phi$ in \\eqref{eq:2D-bosonic-action}) and can thus be neglected for sufficiently large black holes. \n. \n\nThe basic fermion which must be included is the 2D gravitino. To ultimately match $PSU(1,1|2)$ BF theory, which naturally uses complex fermions, we will use the Dirac formulation of $\\mathcal{N}=2$ supergravity as discussed above. After dimensional reduction on $S^2$, the Dirac gravitinos $\\psi_\\mu^{\\alpha}$, $\\bar{\\psi}_{\\mu \\alpha}$ will transform in the $\\mathbf{2}$ and $\\bar{\\mathbf{2}}$ representations of the $SU(2)$ symmetry, which in this section, is parametrized by the spinor index $\\alpha=1,2$. The form of the gravitinos follows from the standard Kaluza Klein ansatz $\\Psi_\\mu \\sim \\psi_\\mu^{\\alpha}\\eta_\\alpha$, where $\\eta_\\alpha$ is a Dirac spinor which solves the angular equation \\eqref{eq:sphereKSequation}, i.e.~just the $S^2$ Killing spinor equation. \n\nThe angular solution is given by taking only the angular part of~\\eqref{eq:KS-spinor} and \\eqref{eq:KS+spinor} in terms of constant $\\eta^\\alpha$ spinors. A similar relation holds for the dilatino $\\lambda^\\alpha$, $\\bar{\\lambda}_\\alpha$ which come from the $S^2$ vector components of the gravitino.\\footnote{Some details of this are given for the asymptotically AdS$_4$ case in~\\cite{Forste:2020xwx}. }\n\nThe standard gravitino kinetic term $\\bar{\\psi}_{\\mu \\alpha} \\gamma^{\\mu \\nu \\rho}D_\\nu \\psi_\\rho^\\alpha$ vanishes in two dimensions. This is related to the Einstein-Hilbert term $R$ being a topological invariant, and the supersymmetric variation of this is automatically zero. The basic term arising from dimensional reduction after choosing the correct normalization for fermions and dropping higher order terms in $r_0 $ and the dilaton fields is:\n\\begin{equation}\n   S_{2D} \\supset 2i \\int d^2x \\, \\varepsilon^{\\mu \\nu} {\\lambda}^\\alpha \\mathcal{D}_\\mu \\bar \\psi_{\\nu\\alpha} \\, , + \\textrm{ h.c.}\n\\end{equation}\nfor $\\mathcal{D}_\\mu$ a suitable gauge supercovariant derivative whose form we will address directly in the first-order formalism.\\footnote{Such a term is natural in the generalization of JT theory to include a supercurvature~\\cite{Teitelboim:1983uy, Chamseddine:1991fg,Cardenas:2018krd}.} The next fermionic term, arising from the dimensional reduction, couples the dilaton $\\Phi$ to the 2D gravitinos $\\psi$ and $\\bar \\psi$. Such a term comes from the 4D terms $- \\bar{\\Psi}_{I M} \\Gamma^{MNP}D_{N}\\Psi^I_P + \\frac{\\varepsilon^{IJ}}{2\\sqrt{2}}\\bar{\\Psi}^M_I(F_{MN} + i \\star \\! F_{MN}\\Gamma_5)\\Psi_J^{N}$ in the full supergravity Lagrangian \\eqref{Eq:4DCompLag}. After passing to the Dirac spinors $\\Psi_M = \\Psi_M^1 + i \\Psi_M^2$, the resulting term in the 2D action is \n\\be \n S_{2D} \\supset \\int  \\frac{2}{r_0}\\Phi \\, (\\bar \\psi_\\a  \\wedge \\psi^\\alpha) \\, , \n\\ee\nwith all higher powers in $\\Phi$ suppressed by factors of $G_N^{1\/2}\/r_0$. The final term that involves quadratic fermionic terms couples $\\psi$ and $\\bar \\psi$ to the $SU(2)$ field strength $H$ arising from the KK reduction. Such a term arises from the dimensional reduction of  $- \\bar{\\Psi}_{I M} \\Gamma^{MNP}D_{N}\\Psi^I_P$ by using the dependence of the 4D spin connection on the field strength $H$ (see appendix \\ref{app:spinorssugra}), again dropping higher order terms. After integrating-in the zero form field $b$, the 2D gravitinos then couple to $b$ instead of to the field strength $H$, resulting in the term\n\\be \n S_{2D} \\supset i  \\int \\frac{b^i}{r_0} \\bar \\psi_\\a (\\gamma_3') (\\sigma^i)^\\a{}_\\b \\wedge \\psi^\\b \n\\ee\nIn principle integrating-out $b$ generates a four gravitino term, which is suppressed in $r_0$ and can be absorbed in a shift of the four gravitino term which we have only schematically written in the action \\eqref{Eq:4DCompLag}. Because such higher power fermionic terms are suppressed by factors of $G_N^{1\/2}\/r_0$ we will not be concerned with them when computing the linearized gravitational action.  \n\n\nWe can now put  together the bosonic and fermionic terms discussed above. In passing to the first order formulation, $\\omega^{ab}_\\mu$ will be promoted to a dynamical variable. Additionally, we will introduce standard 1-form notation for all fields, $e^a = e^a_\\mu dx^\\mu$, etc. We can now also now move to working in Euclidean signature, for which we will use the appropriate gamma matrices $\\gamma'^a$, as outlined in Appendix~\\ref{app:spinorssugra}. In terms of these, the first order bulk action (which will soon need to be modified) is: \n\\be\n\\label{eq:reducedbadaction}\nS_{2D} &\\supset - \\int \\Phi \\left ( d\\omega + \\frac{1}{2r_0 ^2} \\epsilon_{ab}e^a \\wedge e^b - \\frac{2}{r_0} \\bar \\psi_\\a  \\wedge \\psi^\\a \\right )+\\frac{i}{2} b_{i}\\left(H^{i} - \\frac{2}r_0 \\bar \\psi_\\a (\\gamma_3')(\\sigma^i)^\\a_\\b \\wedge \\psi^\\b \\right) \\nn \\\\ &- 2i {\\lambda} \\mathcal{D}  \\bar\\psi  + \\textrm{ h.c.} \\,,\n\\ee\nup to orders of $\\cO(G_N\/r_0^2)$ which we have neglected. Here, the supercovariant gauge exterior derivative is\n\\begin{equation}\n\\mathcal{D} \\psi^\\alpha=  d \\psi^\\alpha + \\frac{1}{2}\\omega \\gamma_3' \\wedge \\psi^\\alpha  - \\frac{1}{2 r_0 } \\gamma_3' \\gamma_a' e^a \\wedge \\psi^\\alpha + i B^i (\\sigma_i)^\\alpha{}_\\beta \\wedge \\psi^\\beta \\, .\n\\label{eq:2dsuperexterior}\n\\end{equation}\nThe problem with the action~\\eqref{eq:reducedbadaction} is that there is nothing to ensure the vanishing of the torsion tensor which has bosonic part $\\tau^a = de^a + \\varepsilon^{ab}\\omega \\wedge e_b $. In deriving Eq.~\\eqref{eq:4DRicci}, this was implicitly assumed. Additionally, in supergravity, it is only the supertorsion which vanishes. This will be remedied by introducing extra Lagrange multipliers to fix the supertorsion constraint explicitly. Therefore, we can add to the action an additional term $\\int \\phi_a \\tau^a$, where $\\phi_a$ are new scalars. \n\nThe various 1-forms and scalars can now be collected into super-multiplets as:\n\\begin{equation}\n\\label{eq:2d-SUGRA-supermultiplets}\n    (\\omega \\, , \\, e^a  \\, , \\, B^i \\, , \\, \\psi^{\\alpha} \\, , \\bar{\\psi}_{\\alpha})\\, \\, \\, , \\, \\, \\,  (\\Phi \\, , \\, \\phi^a \\, , \\, b^i \\, , \\, \\lambda^\\alpha \\, , \\, \\bar{\\lambda}_\\alpha) \\, ,\n\\end{equation}\nwhich we recognize as the matter content of equations~\\eqref{eq:gauge-field-ansatz} and~\\eqref{eq:Lagr-multiplier}, respectively. \n\nThus, after adding the super-torsion terms, the total linearized (in the dilaton) action arising from the dimensional reduction is:\\footnote{Alternatively, one can obtain \\eqref{eq:final-action-from-dim-red-nhr-2d} by restoring the $\\cN=4$ supersymmetry for the 2D bosonic action obtained in \\eqref{eq:2D-bosonic-action}. }\n\\begin{align}\n\\label{eq:final-action-from-dim-red-nhr-2d}\nS_{2D} = - \\int &\\Phi \\left ( d\\omega + \\frac{1}{2r_0 ^2} \\varepsilon_{ab} e^a \\wedge e^b -\\frac{2}{r_0 }\\bar{\\psi}_\\alpha  \\wedge \\psi^\\a \\right ) \\nonumber \\\\ &+i \\frac{\\phi_a}{r_0} \\left (d e^a + \\varepsilon^{ab} \\omega \\wedge e_b - 2\\,\\bar{\\psi}_\\alpha \\gamma'^a \\wedge  \\psi^{\\a} \\right )\\nonumber \\\\ & +\\frac{i}{2} b_{i}\\left(H^{i} -\\frac{2}{r_0}\\bar{\\psi}_\\alpha (\\gamma_3') (\\sigma^i)^{\\alpha}{}_\\beta \\wedge \\psi^\\beta \\right ) - \\left[2i \\lambda_\\a \\mathcal{D} \\bar \\psi^\\a  + \\textrm{ h.c.}\\right] \\,\n\\end{align}\nwhere the index $\\alpha$ is an $SU(2)$ index and the fermionic indices are contracted with the Euclidean gamma matrices $\\gamma_a'$ defined in Appendix~ \\ref{app:spinorssugra}. \n\nComparing  \\eqref{eq:final-action-from-dim-red-nhr-2d} to the $\\cN=4$ JT gravity action \\eqref{eq:JT-N=4-action} with $\\Lambda = 2\/r_0^2$, one finds an exact match after some simple field redefinitions having to do with the reality of dilaton and the choice of basis for the gamma matrices.\\footnote{In order to impose that the $PSU(1, 1|2)$ connection is flat the field $\\Phi$ should be integrated along an imaginary contour, thus identifying $\\Phi= i\\phi_0$. \n} \nBefore proceeding to obtain the full partition function of such black holes by using the results from section \\ref{sec:N=4-super-Schw}, we first address two subtleties about the dimensional reduction: how to obtain the boundary conditions presented in section \\ref{sec:PSU(1,1|2)-BF-thy}, and why other KK modes (besides the massless KK modes we have included thus far) do not contribute to the partition function in a significant way. \n\n \n \\subsection{Subtleties about the dimensional reduction}\n  \\label{sec:subtleties}\n  \n Before presenting the final results regarding the near-extremal spectrum of black holes in these supergravity theories, we will mention some subtleties and clarifications about the dimensional reduction. \n \n \\subsubsection*{Boundary conditions}\n \n  This paper aims to perform the path integral in the 4D geometry by separately analyzing the near-horizon contribution and the far away (asymptotically flat) region. The far away contribution is trivial, but importantly propagating the boundary conditions from infinity to the edge of the throat, we can derive the boundary conditions we should impose on the fields living in $AdS_2$ \\cite{Nayak:2018qej, Moitra:2019bub}. We will argue that the boundary conditions derived in this way are the same as the one used in section \\ref{sec:N=4superJT} to analyze $\\cN=4$ super-JT.\n \n This exercise has been done for the metric appearing in JT gravity and for the dilaton. We will follow the presentation of \\cite{Iliesiu:2020qvm}. The throat and far-away regions are separated by an arbitrary curve with fixed dilaton value $\\chi_b$, fixed intrinsic boundary metric $h_{\\tau\\tau}=1\/\\varepsilon^2$ and proper length $\\ell = \\int \\sqrt{h}= \\beta L_2 \/ \\varepsilon$, where $L_2$ is the AdS$_2$ radius. Curves of constant dilaton are fixed at some radial distance $r_0 + \\delta r_{\\rm bdy}$, with $r_{\\rm bdy}=L_2^2\/\\varepsilon$. Looking at the extremal solution in the far-away region we can derive the boundary condition for the dilaton $\\Phi_b = \\Phi_r\/\\varepsilon$ with $$\\Phi_r = \\frac{r_0 L_2^2}{G_N}.$$ In our case the AdS$_2$ radius is $L_2=r_0$. The definition of $\\Phi_r$ through the dilaton boundary condition agrees with the one used in section \\ref{sec:N=4superJT}, and this calculation relates it to 4D quantities.\n \n\n\nNext, we can analyze the $SU(2)$ gauge field boundary condition. We will derive this indirectly in the following way, ignoring for now its coupling to other fields. In \\cite{Iliesiu:2020qvm} we integrated out the 2D gauge field everywhere. We can see from the exact solution which boundary condition should be chosen in the throat in order to reproduce the angular-momentum dependence of the partition function. This gives the mixed boundary condition between the zero-form $b^i$ and the field strength $B^i$: $\\delta(2i\\Phi_r B_\\tau - b)\\big|_{\\rm NHR bdy} = 0 $. To prove this, we can consider the classical solution for the $SU(2)$ gauge field, in the region far-away from the horizon. We fix the holonomy of the gauge field $h = \\cP \\exp \\left(\\oint_{r\\to\\infty} B\\right) = \\exp(i 2\\pi \\alpha\\sigma^3) $ at the asymptotic boundary, with $\\alpha \\sim \\alpha +1$.   Following appendix A in \\cite{Iliesiu:2020qvm}, we will work in a gauge where $B_r = 0$, to obtain the general solution for the zero-form field $b$ and the gauge field $B$:\n\\be \nB= i\\frac{2\\pi \\alpha T^3}{\\b} \\left(1+\\frac{\\mathcal C}{r^3}\\right) d\\tau\\,, \\qquad H_{r\\tau} =-i\\frac{6\\pi \\alpha T^3}{\\b} \\frac{\\mathcal C}{r^4}\\,, \\qquad b = \\frac{4\\pi \\mathcal C \\,\\alpha\\, T^3}{\\b}\\,,\n\\ee\nwhere $\\mathcal C$ is an undetermined constant. Solving for $\\mathcal C$, at the boundary that separates the near-horizon region from the asymptotic region we find that $B_\\tau$ and $b$ are related as\n\\be\n\\label{eq:SU(2)-gauge-field-sol}\nB_\\tau  =\\frac{ 2\\pi  i\\alpha T^3}{\\b} + \\frac{i b \\,\\lpl^2\\,}{r^3}= \\frac{ 2\\pi  i \\alpha T^3}{\\b} + \\frac{i b }{2\\Phi_r}\n\\ee\nThus, the boundary condition which we want to choose for the  $SU(2)$ field is\n\\be\n\\label{eq:bc-SU(2)-gauge-field}\n\\delta\\left(2\\Phi_r i  B_\\tau - b \\right)= 0\\,,\n\\ee\nwhich is consistent with the boundary condition in $\\cN=4$ super-JT gravity discussed in section \\ref{sec:super-JT-bdy-cond}, given that $\\tau \\sim \\tau+\\beta$ in both the diffeomorphism gauges of \\eqref{eq:FG-gauge-metric} and for the Euclidean black hole metric. From a 4D perspective, the boundary holonomy of the $SU(2)$ gauge field is proportional to the boundary angular velocity $ \\Omega \\sim i \\alpha\/ \\beta$.\n\nNext, we mention why we could fix the flux of the $U(1)$ gauge field in the near-horizon region when working in the canonical ensemble. In such a case, the field strength (and not the gauge field itself) is fixed as $r\\to\\infty$. Once again, we can study how this happens by solving the equations of motion in the far-away region. The general solution takes the same form as \\eqref{eq:SU(2)-gauge-field-sol} for non-abelian $SU(2)$ gauge fields; however, since we now fix the field strength at $r \\to \\infty$ the constant $\\mathcal C$ in  \\eqref{eq:SU(2)-gauge-field-sol}  is also fixed. Therefore, the field strength is completely fixed at $r = r|_{\\rm NHR bdy}$, instead of fixing the linear combination \\eqref{eq:bc-SU(2)-gauge-field} as in the case of the $SU(2)$ gauge field. Since no massless field in the gravity, sector is charged under the $U(1)$ gauge field, one can integrate-out the 2D gauge field exactly and work in a sector of fixed charge. Since there is no temperature-dependent one-loop determinant when integrating out this field, the computation simply amounts to replacing the field strength in the near-horizon region with its classical values. Thus, this motivates our dimensional reduction in section  \\eqref{ss:dimensionalreduction4d}, where the $U(1)$ gauge field was absent from all supermultiplets. \n\n\nFinally, we will choose boundary conditions in the asymptotically flat space region for the fermions to vanish. Then it is reasonable to impose the same boundary condition in the boundary of the throat. This is also consistent with the choice made in section \\ref{sec:super-JT-bdy-cond}.\n\n\n\n\n\n\n  \\subsubsection*{Massive KK modes}\n\nPerforming the reduction in the previous sections, we have identified the massless Kaluza-Klein modes in the spectrum when reducing the supergravity sector of the 4D theory to 2D. We have ignored so far the contribution from towers of massive KK modes or other matter fields present in 4D. The reason is that they do not affect the temperature dependence of the partition function to leading order \\cite{Iliesiu:2020qvm}. \n\nThe first step to see this is to realize that in the throat, all interactions are suppressed by factors of $S_0$. This means that in $AdS_2$ besides having the JT mode, we have a tower of free fields to integrate out. Let's begin with the case of massless fields in 2D, which arise from an s-wave reduction of a massless field in 4D. We can couple it to JT and integrate it out. The answer has the following form\n\\beq\\label{eq:intoutout}\n\\log Z_{m=0} = \\delta E_0 \\beta + \\delta S_0 + \\mathcal{O}(\\varepsilon, \\beta^{-2}),\n\\eeq\nwhere $\\delta E_0 \\sim 1\/\\varepsilon$ and $\\delta S_0 \\sim \\log L_2$. The shift in energy is divergent and can be removed by a counterterm. The parameter $\\delta S_0$ simply shifts the overall prefactor of the partition function $e^{S_0}\\to e^{S_0 + \\delta S_0}$. Since $L_2 \\sim r_0$ this correction is logarithmic in the extremal area. The terms of order $\\varepsilon$ include couplings between matter and gravity, through the Schwarzian mode \\cite{Yang:2018gdb}. They are multiplied by a factor of the cut-off $\\varepsilon$ and therefore suppressed. Any other correction is further suppressed at low temperatures. The same conclusions are true for massive fields, and their contributions have the same structure as \\eqref{eq:intoutout}, producing only a shift of $S_0$. Finally, we can ask what happens when we integrate out a tower of massive KK modes coming from dimensionally reducing an individual field in 4D. This has been studied extensively by Sen and collaborators \\cite{Banerjee:2010qc,Banerjee:2011jp,Sen:2011ba,Sen:2012cj}. The answer always has the form \\eqref{eq:intoutout} and, in general, $\\delta S_0 \\sim \\log S_0$ with a prefactor depending on the 4D matter spectrum.    \n\nThe only exception to this rule are dimensional reductions of extra gauge fields in 4D. They reduce to gauge fields in 2D that can affect the temperature dependence in general. Nevertheless, one can focus on boundary condition of fixed charge for all these fields. This choice makes their contribution trivial, and temperature independent.\\footnote{An explanation of this is given in \\cite{Iliesiu:2020qvm}). }\n \n\n \n \\subsection{The black hole spectrum}\n \\label{sec:4D-BH-spectrum}\n \nWe can now put all results together. We argued the temperature dependence of the partition function of the near-extremal black hole is captured by $\\cN=4$ super-JT gravity, and in the previous sections, we precisely solved this theory. By matching boundary conditions, we have found the parameters of the JT theory in terms of their 4D origin \n\\beq\nS_0 = \\pi Q^2 ,~~~~~\\Phi_r = \\sqrt{G_N} \\hspace{0.1cm} Q^3.\n\\eeq\nWe are looking at the spectrum at fixed $U(1)$ charge and, therefore, these are fixed parameters. To extract the near-extremal black hole spectrum, we can use this identification in equations \\eqref{sksks} and \\eqref{sksks2}. In these expressions, the parameter $J$ is interpreted as the black hole angular momentum in 4D.  \n\nSince this spectrum was already analyzed in the introduction, so we will not repeat it here. The main features we want to point out are the following. If we look at states with zero angular momentum, we find a large extremal degeneracy and the presence of a mass gap in the black hole spectrum, given by\n\\beq\n\\text{Degeneracy of extremal BH}=e^{\\pi Q^2},~~~~E_{\\rm gap}=\\frac{1}{8 \\sqrt{G_N}\\hspace{0.1cm} Q^3}.\n\\eeq\nMoreover, our analysis in section \\ref{sec:density-of-states} shows that all extremal states are bosonic, and therefore a calculation of the index would match with the black hole degeneracy.\n \\begin{figure}\n     \\centering\n     \\includegraphics[scale=0.65]{SPlot.pdf}\\hspace{0.15cm}\n     \\includegraphics[scale=0.65]{EPlot.pdf}\n     \\caption{Plot of the entropy and energy dependence on temperature for near-BPS black holes in $\\cN=2$ ungauged supergravity. The black curves represent results obtained from the naive semiclassical computation while the red curves account for quantum corrections. The inset figures zoom in on the temperature range smaller than $E_\\text{gap}$ where quantum corrections become important. Both the entropy and energy approach zero temperature as $\\# + \\# e^{-E_\\text{gap}\/T}$ while in the semiclassical analysis, they approach zero temperature as $\\# T$ and $\\# T^2$, respectively.  }\n     \\label{fig:SPlot}\n \\end{figure}\n \nWe can also look at other angular momentum sectors. The correction to the extremal energy we find when we turn on rotation is given by $$E_{\\rm ext}(J)=\\frac{J^2}{2\\Phi_r}= \\frac{J^2}{2\\sqrt{G_N}\\hspace{0.1cm}Q^3} \\, ,$$ which is measured with respect to the $J=0$ extremal mass $M_0 = Q\/\\sqrt{G_N}$. This expression matches the expectation from 4D gravity: if we start with the Kerr-Newman metric, take the extremal limit with $Q\\neq 0$ $J\\neq 0$, we obtain precisely this correction when expanded to leading order in small $J$. Interestingly, we find the delta function giving the large extremal degeneracy at $J=0$ disappears at $J\\neq0$. This makes sense since extremal black holes with $J\\neq0$ do not preserve any supersymmetry, and they resemble the results in non-supersymmetry gravity in \\cite{Iliesiu:2020qvm}. \n\n\nAnother interesting aspect of the theory is that the supersymmetry in the throat relates the heat capacity $\\Phi_r$ of the black hole to a very different quantity $\\left(\\frac{\\partial J}{\\partial \\Omega} \\right)_{T=0,\\Omega=0}=\\Phi_r$, where $\\Omega$ is the black hole angular velocity. In non-supersymmetric theories, these are independent quantities. Moreover, by measuring the dependence of the angular momentum with the angular velocity at zero temperature, we can also determine the gap in the spectrum (since it is controlled by the heat capacity $\\Phi_r$). We comment on this in section \\ref{sec:compbosth} as well.\n\n\nFinally, if we decide to fix the metric in the asymptotically flat region and fix the angular velocity to vanish, it is equivalent to fixing the $SU(2)$ chemical potential to zero. With this choice we can use the exact results \\eqref{sksks} and \\eqref{sksks2} for the black hole spectrum to analyze how quantum corrections affect $S(T)$ and $E(T)$ when $\\alpha=0$. The results are shown in figure \\ref{fig:SPlot}. At large temperatures, the results above match the semiclassical result. At low temperatures, both the energy and the entropy behave exponentially as $E(T) \\sim e^{-1\/(8T  \\Phi_r)}$ and $S(T) \\sim S_0 + e^{-1\/(8 T  \\Phi_r)}$. This behavior is dictated by the fact that there is an actual gap in the spectrum. \n\n\n\n\n\n\n\n\\section{Near-BPS Black Holes in AdS$_3$}\\label{sec:ads3}\n\nIn this section, we will describe near-extremal near-BPS black holes in theories of supergravity in $AdS_3$. We show, in some cases, the theory again reduces to $\\cN=4$ super-JT near the horizon. In those cases, gravity predicts a large extremal entropy and a gap. We will mention concrete examples in string theory where this is relevant, and both the extremal states and the gap can be accounted for using D-brane techniques. \n\n\\subsection{$(4,4)$ Supergravity in $AdS_3$}\nWe will summarize features of supergravity in $AdS_3$ and compute the partition function with a torus boundary. We will focus on the case with $\\cN=(4,4)$ supersymmetry. These cases appear in string theory constructions that we will describe in the next section. We will also mention some features of theories with $\\cN=(4,0)$ in the end.\n\nWhen these theories are obtained from a dimensional reduction of six-dimensional supergravity on $AdS_3 \\times S^3$ we have a specific spectrum of matter and gauge fields coupled to gravity. In the near extremal limit, we already saw that as long as we pick boundary conditions with fixed charges for gauge fields, both them and the matter fields do not affect the partition function's temperature dependence \\cite{Iliesiu:2020qvm}. Therefore, in this section, we will focus on the pure supergravity sector on $AdS_3$. We will also give an argument for this fact near the end combining $AdS_3\/CFT_2$ with the modular bootstrap (which is a supersymmetric version of \\cite{Ghosh:2019rcj}).\n\nTheories of supergravity in $AdS_3$ appearing in dimensional reductions of string theory can be written as a difference between two Chern-Simons theories \\cite{Achucarro:1987vz}. We will follow the presentation in \\cite{deBoer:1998kjm}. Generally for a set of left and right moving symmetry groups $G_L \\times G_R $ we can define a gravity theory through the following action\n\\beq\nI_{G_L \\times G_R}= \\frac{k}{4\\pi}\\int \\left( \\mathcal{A} \\wedge d\\mathcal{A} + \\frac{2}{3} \\mathcal{A} \\wedge \\mathcal{A} \\wedge \\mathcal{A} \\right)-\\frac{k}{4\\pi} \\int  \\left(\\bar{\\mathcal{A}} \\wedge d\\bar{\\mathcal{A}} + \\frac{2}{3} \\bar{\\mathcal{A}} \\wedge \\bar{\\mathcal{A}} \\wedge \\bar{\\mathcal{A}} \\right)\n\\eeq\nwhere $\\mathcal{A}$ is a connection for $G_L$ and $\\bar{\\mathcal{A}}$ a connection for $G_R$. We will begin studying the $\\cN=(4,4)$ case, which corresponds to $G_L \\times G_R= PSU(1,1|2)_L \\times PSU(1,1|2)_R$. Expanding the gauge connection essentially involves two copies of the expansion performed in section \\ref{sec:PSU(1,1|2)-BF-thy} for BF theory. \n\nThe field content is a three-dimensional metric, a set of gravitini, and a $SU(2)_L \\times SU(2)_R$ Chern-Simons gauge field with level $k$. The action in terms of these fields can be found adapting the analysis of \\cite{Nishimura:1998ud}, for example. The metric part of the action in second-order formalism can be written as \n\\beq\nI_{G_L\\times G_R} \\supset \\frac{1}{16 \\pi G_3}\\int d^3 x \\sqrt{g}\\left( R+\\frac{2}{\\ell^2}\\right),\n\\eeq\nwhere $\\ell$ is the $AdS_3$ radius and $G_3$ the three-dimensional Newton constant. The Chern-Simons level is given by $k=\\ell\/(4G_3)$. The fact that this is also a level of $SU(2)$ implies $k$ is quantized.\n\nTo study this theory in asymptotically $AdS_3$ we consider a supersymmetric generalization of the Brown-Henneaux boundary conditions \\cite{Brown:1986nw}. The asymptotic symmetries in the $(4,4)$ case are given by two copies, left and right moving, of the small $\\cN=4$ Virasoro algebra. The generators of the bosonic Virasoro algebra are $L_n$, with integer $n$. The generators of the fermionic symmetries are $G_r^a$ where $a=1,2$ transforms in a double of $SU(2)$ and $r$ can be an integer (R-sector) or half-integer (NS-sector). Finally there is an $SU(2)$ current algebra at level $k$ generated by $T_n^i$, with integer $n$ and $i=1,2,3$ in the adjoint of $SU(2)$. The commutation relations were already written down above in equations \\eqref{eq:smalln4gen1}\\eqref{eq:smalln4gen2}\\eqref{eq:smalln4gen3}\\eqref{eq:smalln4gen4} and we will not repeat them here. The central charge of the algebra is $c$, and it is related through supersymmetry to the level $k$ of the $SU(2)$ algebra by $c=6k$. \n\nWe will be interested in computing the partition function on three dimensional surfaces with torus boundaries\n\\beq\nds^2 = d\\tau^2 + d\\varphi^2, ~~~~(\\tau,\\varphi) \\sim(\\tau,\\varphi+2\\pi)\\sim (\\tau+\\beta,\\varphi+\\theta). \n\\eeq\nWe take the moduli of the torus to be $q=e^{2\\pi i \\tau}$ and $\\bar{q}=e^{-2\\pi i \\bar{\\tau}}$, where\n\\beq\n\\tau=\\frac{\\theta+i\\beta}{2\\pi}= \\frac{i \\beta_L}{2\\pi}~~~~{\\rm and}~~~~\\bar{\\tau}=\\frac{\\theta-i\\beta}{2\\pi}= -\\frac{i \\beta_R}{2\\pi},\n\\eeq \nand it will be useful to think in terms of left and right moving temperatures. We also consider fixing the boundary chemical potential for the $SU(2)$ Chern-Simons field. We denote its fugacity by $z$ associated to the left-moving charge $T_0^3$ and $\\bar{z}$ for the right-moving charge $\\bar{T}_0^3$. Finally, we can pick the left- and right-moving fermions to be in the NS or R sector. In terms of $AdS\/CFT$, the partition function with these boundary conditions can be interpreted as a trace over a $CFT_2$ Hilbert space as\n\\beq\nZ_{NS\/R-NS\/R} = {\\rm Tr}_{NS\/R-NS\/R}\\left[ q^{L_0-\\frac{c}{24}}\\bar{q}^{\\bar{L}_0-\\frac{c}{24}} z^{T_0^3} \\bar{z}^{\\bar{T}^3_0} \\right],\n\\eeq\nand we have four possible partition functions.\n\n\\paragraph{Pure $\\mathbf{AdS_3}$:}  To begin, we start with considering the pure $AdS_3$ solution of this theory, and the fermions in the NS sector. In the bosonic case, the path integral of pure gravity including quantum fluctuations around this geometry can be exactly computed and it is one-loop exact \\cite{Maloney:2007ud}. An analogous argument applies for the supersymmetric generalizations considered here. The answer for the path integral around $AdS_3$ is given by the product of left- and right-moving vacuum super-Virasoro characters\n\\beq\\label{skekwe}\nZ_{NS-NS} = \\chi_{{\\rm id},NS}^{\\cN=4} (q,z) \\chi_{{\\rm id},NS}^{\\cN=4} (\\bar{q},\\bar{z}),\n\\eeq\nThese characters were computed by Eguchi and Taormina \\cite{Eguchi:1987wf} and are given by the following expression in the NS sector\n\\beq\n\\chi_{{\\rm id},NS}^{\\cN=4} (q,z) =  q^{-\\frac{k-1}{4}} \\frac{i \\theta_3(q,-z) \\theta_3(q,-z^{-1})}{\\eta(q)^3 \\theta_1(q,z^2)} \\left[ \\mu(z,q) - \\mu(z^{-1},q) \\right],\n\\eeq\nwhere $\\theta_3(q,z)$ is the Jacobi theta function and we defined the function \n\\beq\n\\mu(z,q) \\equiv \\sum_{n\\in\\mathbb{Z}} \\frac{q^{(k+1)n^2+n}z^{2(k+1)n + 1}}{(1- z q^{n+\\frac{1}{2}})(1- z q^{n+\\frac{1}{2}})}.\n\\eeq\nThe characters in the Ramond sector can be obtained by a simple replacement $z\\to z q^{1\/2}$. A similar expression applies for the right-mover sector.\n\n\\paragraph{BTZ:} Now we can consider quantum fluctuations around the BTZ solution. Classically, the metric is parametrized by an energy $E$ and angular momentum $P$ (for simplicity we turn off the $SU(2)$ fields now whose generators are denoted above by $T^i$). The metric is given by \n\\beq\nds^2 =  f d\\tau^2 + \\frac{\\ell^2 dr^2}{f} + r^2 \\left( d\\varphi - i\\frac{r_+r_-}{r^2 } d\\tau\\right)^2,~~~f=\\frac{(r^2-r_+^2)(r^2-r_-^2)}{r^2},\n\\eeq\nwhere some useful thermodynamics quantities given in terms of the inner and outer horizon radii $r_pm$ are given by \n\\beq\nE = \\frac{r_+^2 + r_-^2}{8 G_3 \\ell},~~~P=\\frac{r_+ r_-}{4 G_3 \\ell},~~~T =\\frac{r_+^2-r_-^2}{2\\pi \\ell r_+},~~~S = \\frac{2 \\pi r_+}{4 G_3}.\n\\eeq\nThe chemical potential for the momentum along the circle is $\\theta=i \\frac{1}{T}\\frac{r_-}{r_+}$. For simplicity we will work with a boundary torus with all chemical potentials fixed (both for rotation and $SU(2)$ charges). The partition function including quantum effects around the BTZ solution with these boundary conditions can be obtained from the pure AdS result \\eqref{skekwe} by a modular transformation. We will be interested in the R-R sector since the fermion becomes periodic in spatial $S^1$ and supersymmetry is preserved in the throat. The final answer is \n\\beq\nZ_{R-R} = e^{2\\pi i k \\frac{z^2}{\\tau}-2\\pi i k \\frac{\\bar{z}^2}{\\bar{\\tau}}}~ \\chi_{{\\rm id},NS}^{\\cN=4} (q',z') ~\\chi_{{\\rm id},NS}^{\\cN=4} (\\bar{q}',\\bar{z}'),\n\\eeq\nwhere we take into account that the modular transformation exchanges sectors and \n\\beq\nq'=e^{-2 \\pi i\/\\tau},~~z'=e^{2\\pi i \\alpha\/\\tau},~~~\\bar{q}'=e^{2 \\pi i\/\\bar{\\tau}},~~\\bar{z}'=e^{2\\pi i \\bar{\\alpha}\/\\bar{\\tau}},\n\\eeq\nwhere $\\alpha$ is the chemical potential for the $SU(2)$. In the next section we will use these expressions in the near extremal near-BPS limit. The origin of the first two factors is explained in section 5 of \\cite{Kraus:2006nb}. We can rewrite this answer as follows in a more transparent way\n\\beq\\label{eq:RRsumsaddle}\nZ_{R-R} = \\sum_{n,\\bar{n}\\in\\mathbb{Z}}\n  Z_{1-loop}^{n,\\bar{n}} e^{\\frac{i \\pi k}{2\\tau}(1-4(\\alpha+n)^2)-\\frac{i \\pi k}{2\\bar{\\tau}}(1-4(\\bar{\\alpha}+\\bar{n})^2)}\n\\eeq\nwhere the one-loop determinant is \n\\beq\\label{eq:RRoneloop}\nZ_{1-loop}^{n,\\bar{n}} = \\left|\\frac{(1-q')}{\\eta(q')} \\frac{z'{}^{2n}(q'{}^{(\\frac{1}{2}+n)^2} z' - q'{}^{(\\frac{1}{2}-n)^2}z'{}^{-1})}{\\theta_{1}(q,-z'{}^2)} \\frac{\\theta_{3}^2(q',-z')}{\\eta(q')^2 (1-z' q'{}^{n+\\frac{1}{2}})^2(1-z'{}^{-1} q'{}^{-n+\\frac{1}{2}})^2}\\right|^2.\n\\eeq\nThis way of rewritten the product of vacuum characters shows that the answer is given by a sum over saddles labeled by integers $n$ and $\\bar{n}$, with each contribution having a classical action written in \\eqref{eq:RRsumsaddle} and a one-loop determinant \\eqref{eq:RRoneloop} given by the product of the graviton, $SU(2)$ Chern-Simons and gravitino contributions.\n\n\\paragraph{Preserving only $\\mathbf{(4,0)}$ SUSY:}  This theory is also given by a combination of Chern-Simons actions with group $G_L \\times G_R= PSU(1,1|2)_L \\times \\left( SL(2,\\mathbb{R})\\times SU(2)\\right)_R$. In the $(4,0)$ case, we have as asymptotic a small $\\cN=4$ Virasoro algebra for left-movers and a bosonic Virasoro and an $SU(2)$ Kac-Moody algebra for the right-mover. The expressions for the partition functions look similar to the cases above, but the right-moving character has to be replaced by the bosonic ones.\n\n\n\\subsection{Near-BPS near-extremal limit}\nIn this section, we will consider a near-extremal rotating black hole state, for the theory considered in the previous section, at fixed $SU(2)$ charges.  \n\nWe will take the limit directly for the exact pure gravity result. If matter is present, we need to be more careful, divide the geometry into the throat and far-away region, and dimensionally reduce in the throat to $\\cN=4$ super-JT. To simplify, we also assume we pick fixed charge boundary conditions for any other gauge field that is not in the gravity sector. Then for the same reasons as in 4D \\cite{Iliesiu:2020qvm} the temperature dependence of the partition function is given by the $\\cN=4$ super-Schwarzian. \n\nUsing AdS$_3$\/CFT$_2$, we can give another perspective of this universality using the modular bootstrap: in the near-extremal limit, the vacuum block in the modular-dual channel dominates up to corrections suppressed by the twist gap \\cite{Ghosh:2019rcj}.\\footnote{If extra currents are present, they are easy to take into account and in the near-extremal limit looking at states where their charge is fixed, their fluctuations are frozen (this can be derived using results in \\cite{Mertens:2018fds})} This is a special case of the extended Cardy regime studied for example in \\cite{Benjamin:2019stq}.\n\nWe will be interested in the following limit. First, we take large $\\beta_R\\to0$. The ensemble is then dominated by black holes with a very large spin $P>0$.\\footnote{For the opposite orientation, we can take $\\beta_L\\to0$ then we have large $P<0$.} Second, we take large $k\\gg 1$ so that the gravity description in the bulk is accurate. Finally, we take $\\beta_L \\sim k \\gg 1$, which implies that we are looking at very low temperatures or states with $E \\sim P$. Since the state without left-moving excitations preserves supersymmetry, this is also a near-BPS limit \\cite{Coussaert:1993jp}. \n\nWe will follow the calculation first in the fixed $\\beta_L,\\beta_R$ ensemble. As explained in \\cite{Ghosh:2019rcj} when taking this near-extremal limit we can inverse Fourier transform to obtain fixed $P$ ensemble by basically replacing $\\beta_R \\to 2\\pi \\sqrt{c\/(24 P)}$ and $\\beta_L \\to 2 \\beta$ at the end of the calculation. We also keep the left-moving $SU(2)$ chemical potential $\\alpha$ fixed in this limit, and consider zero right-moving charge. Either taking the limit of the character or doing the reduction, the near-extremal near-BPS limit of the partition function is  \n\\beq\\label{eq:RReqnschwarzian}\nZ_{R-R}(\\beta,\\alpha) \\sim e^{2\\pi \\sqrt{k P}} \\sum_{n\\in \\mathbb{Z}} \\frac{\\beta}{k} \\frac{\\cot(\\pi \\alpha)(\\alpha+n)}{(1-4(n+\\alpha)^2)^2} e^{\\frac{ \\pi^2 k}{2\\beta}(1-4(n+\\alpha)^2)}\n\\eeq\nWe will not repeat the calculation here since it is completely analogous to section \\ref{sec:meth2cq}. The first term $e^{2\\pi \\sqrt{k P}}$ comes from the right-moving identity character which is basically evaluated in the usual Cardy limit, since $\\bar{q}' \\to 0$. The rest comes from the evaluation of the left-moving identity character, in the limit $q'\\to 1$. \n\nThe parameters describing the near-extremal near-BPS effective theory analyzed in section \\ref{sec:N=4-super-Schw} are given in terms of the level $k$ and the angular momentum $P$, by \n\\begin{equation}\n    S_0 = 2 \\pi \\sqrt{k P} ,~~~~\\Phi_r = \\frac{k}{4}.\n\\end{equation}\nTaking the inverse Laplace transform of this, we obtain the same density of states as the $\\cN=4$ super-Schwarzian theory with these parameters. In particular, we find a large degeneracy of BPS states given by $e^{2\\pi \\sqrt{ k P}}$, we find a gap to the first excited black hole state $E_{gap}=1\/(2k)$, and this predicts the index matches with the black hole degeneracy. \n\nThis can be easily generalized to cases with non-zero $SU(2)_R$ charge $\\bar{T}_0^3=J_R$. In this cases states with $T_0^3=0$ are still BPS and the contribution from the right-moving sector replaces $S_0 \\to 2\\pi \\sqrt{k P - J_R^2}$. With this modification, the spectrum as a function of temperature and $SU(2)_L$ chemical potential is still given by \\eqref{eq:RReqnschwarzian}, controlled by the $\\mathcal{N}=4$ super-Schwarzian.\n\n\\paragraph{Alternative construction  -- preserving $\\mathbf{(4,0)}$ SUSY:} In this case we obtain a similar conclusion for $\\beta_R \\to 0$ and large $\\beta_L$. The difference now is that we can take instead $\\beta_L \\to 0 $ and $\\beta_R$ large. This extremal limit breaks supersymmetry since the right-moving sector of the theory is purely bosonic. Therefore, we expect that in this case, the black hole spectrum has, to leading order, no extremal states and no gap, similar to \\cite{Ghosh:2019rcj} (or \\cite{Iliesiu:2020qvm}).\n\n\n\\subsection{Comparison to string theory constructions}\nIn this section, we will very briefly mention some string theory constructions using D-branes that can be dimensionally reduced to the $AdS_3$ supergravity theories studied above.\n\nFor example, take type IIB string theory compactified on $M^4$, being either $T^4$ or $K3$. We consider the D1-D5 system, which at low energies can be described by either a $(4,4)$ 2D superconformal field theory or supergravity on $AdS_3 \\times S^3$. Compactifying down to $AdS_3$ gives the $(4,4)$ supergravity theory we studied above. The bosonic part of the spectrum is a 3D metric on $AdS_3$, and a gauge symmetry coming from the $S^3$ factor in the metric $SO(4)\\sim SU(2)_L \\times SU(2)_R$ separated into left- and right-movers. For the reasons explained in the previous section, we expect the presence of other fields to leave the conclusions below unchanged. For this theory, we can derive the level of the $SU(2)$ current algebra by matching the chiral anomaly\n\\begin{equation}\n    k = Q_1 Q_5 ~~{\\rm for}~T^4,~~~~k=Q_1Q_5+1~~{\\rm for}~K3.\n\\end{equation}\nFor concreteness, we look at the $T^4$ case below. \n\nSo far we have vacuum $AdS_3$. We can add some momentum $P$ along the D1-string direction, which is identified in the BTZ gravitational description with the angular momentum $P$ defined above \\cite{Cvetic:1998xh}. The BPS extremal states correspond to no left-moving excitations of the string. Looking at low temperatures we have a near-extremal near-BPS black hole string with $AdS_2 \\times S^1 \\times S^3$ horizon. The parameters of the effective low-energy $AdS_2$ theory from the microscopic model is\n\\begin{equation}\n    S_0 = 2 \\pi \\sqrt{Q_1 Q_5 P},~~~~~\\Phi_r = \\frac{Q_1Q_5}{4}\n\\end{equation}\nUsing our solution we see we have $e^{S_0}$ states at extremality, consistent with \\cite{Strominger:1996sh}. Our analysis also explains why the index matches with the black hole degeneracy. This can be easily generalized to near BPS states with non-zero $SU(2)_R$ charges corresponding at zero temperature to BPS black holes with angular momentum in $S^3$ \\cite{Breckenridge:1996is}.\n\nFrom our gravitational analysis giving the low-temperature dependence of the partition function, we have also derived the gap to the first excited black hole, and it is $E_{gap}=1\/(8\\Phi_r)$. In terms of the microscopic model parameters, it is  \n\\begin{equation}\n    E_{gap} = \\frac{1}{2 Q_1 Q_5}.\n\\end{equation}\nThis answer matches with the string theory approach from \\cite{Maldacena:1996ds}. From this perspective, the extremal black hole states come from counting string configurations at the brane system with only left movers. The lowest energy excitation comes from the first excitation of long strings wound $Q_1Q_5$ times along the branes.\n\n\\paragraph{Alternative construction  -- Black holes in type I string theory:} We can also analyze a similar model in type I string theory instead of type II. We consider a D1-D5 brane system but now the supergravity theory emerging in $AdS_3$ has $(4,0)$ supersymmetry \\cite{Johnson:1998vd, Oz:1999it}. When we have an extremal black hole made out of right-movers, we expect the spectrum near-extremality to be analogous to the $\\cN=4$ super-Schwarzian. On the other hand, when the extremal black hole is made out of left-movers, supersymmetry is broken, and the near-extremal spectrum will look like the non-supersymmetric cases studied in \\cite{Ghosh:2019rcj} or \\cite{Iliesiu:2020qvm}.\n\nFinally, there are other interesting compactifications which we do not analyze in this paper, but whose role we briefly mention in the discussion section.\n\n\\section{Discussion}\n\\label{sec:conclusion}\n\nIn this paper, we have defined and solved $\\cN=4$ super-JT gravity. We show it reduces to a $\\cN=4$ generalization of the Schwarzian theory, which can be exactly solved. We argue that this theory captures the temperature-dependence of the gravitational path integral evaluated around near-extremal black holes in higher dimensions. We showed that both $\\cN=2$ ungauged supergravity in 4D flat space and $(4,4)$ supergravity in $AdS_3$ reduce to $\\cN=4$ super-JT in the near-horizon region of near-extremal black hole backgrounds. We found a gravitational explanation of the large extremal black hole degeneracy and for the presence of a gap in the spectrum. Thus our work addresses the strong tension between the non-supersymmetric results of \\cite{Iliesiu:2020qvm} and past micro-state countings in string theory \\cite{Strominger:1996sh}.\n\nWe finish here with some open questions and future directions:\n\n\\subsection*{Generalization to other black holes in AdS}\n\nWhile in this paper, we have focused on near-BPS black holes in 4D $\\cN=2$ supergravity in flatspace or in $(4,4)$ supergravity in $AdS_3$, there are numerous other near-extremal black hole solutions which are of interest in the AdS\/CFT correspondence.\\footnote{Similar considerations are also useful in computing quantum corrections to the Hartle-Hawking wavefunction in dS \\cite{Maldacena:2019cbz}.}\n\nThe first near-BPS solutions which we have not fully analyzed are those on $AdS_3 \\times S^3 \\times S^3 \\times S^1$ \\cite{Elitzur:1998mm}. The special feature about this compactification is that the extremal solution exhibits a large $\\cN=4$ symmetry, with $SU(2)_k\\times SU(2)_{k'}\\times U(1)$ current algebra. It would be interesting to analyze the 2D theory emerging in the throat for near-extremal near-BPS black holes. In this case, the symmetry of the boundary mode is now $D(2,1,\\alpha)$. We do not yet know how to study this version of the $\\cN=4$ Schwarzian theory, and we hope to address such a construction in future work.\n\nWe would also like to briefly mention the existence of near-BPS solutions in higher dimensional AdS. Extremal black holes in such theories typically preserve a smaller amount of supersymmetry; for instance, in AdS$_4$, such black holes exhibit an $OSp(2|2)$ isometry in the near-horizon region. The effective theory capturing the breaking of $OSp(2|2)$ was found to be $\\cN=2$ super-JT gravity \\cite{Forste:2020xwx} and the boundary dynamics is analogously given by the $\\cN=2$ super-Schwarzian (whose properties we have reviewed in appendix \\ref{app:N2}). As we explain in appendix \\ref{app:N2}, the $\\cN=2$ super-Schwarzian has a gap depending on the value of $\\hat q$ (which gives the periodicity of the identification of the $U(1)$ field $\\sigma$) and on whether or not the theory exhibits an anomaly (related to how we weigh the different saddles in the path integral). Thus, to conclude whether near-BPS black holes in such a theory exhibit a mass gap, we need to perform a rigorous analysis to account for all possible massless Kaluza-Klein modes that can appear in the near-horizon region, determine the analog of $\\hat q$ in supergravity and understand the situations in which the action of the boundary mode can exhibit an anomaly. \n\n\nOne purpose for studying the partition function of near-BPS black holes from the bulk perspective is to understand the gap in scaling dimensions between BPS and the near-BPS states in the dual CFT. If we find that the effective theory which captures the near-horizon dynamics exhibits a gap (as it did for the black holes in flatspace studied in this paper), then this translates to a scaling dimension gap, $\\Delta_{\\text{gap}}\\sim 1\/N^2$. It would be interesting to understand whether this gap in scaling dimensions is consistent with predictions from the large charge bootstrap \\cite{Jafferis:2017zna} in SCFTs. Finally, in comparing the partition function on the CFT side to the black hole partition function within a fixed large charge sector, there may be a mismatch coming from configurations with multiple black holes. Thus, it would be interesting to understand such corrections coming from multi-centered black hole solutions \\cite{Denef:2007vg, Denef:2000nb, Bates:2003vx, Sen:2007pg}.\\footnote{We thank G.~Moore for pointing out past works on this issue.} \n\n\n\n\n\n\n\\subsection*{On a possible $\\cN=4$ SYK model}\n\nAnother interesting possibility is whether there is a UV completion of the $\\mathcal{N}=4$ Super-Schwarzian theory in some quantum mechanical models, along the lines of \\cite{Fu:2016vas} for $\\mathcal{N}=1,2.$ To be more specific, we would like a random quantum mechanical model, with a stable unitary nearly conformal fixed point at low energies, with unbroken $\\mathcal{N}=4$ supersymmetry. A model involving dynamical bosons typically causes instability and exhibits supersymmetry breaking in the infrared (shown either as an operator with complex scaling dimension in the spectrum as in \\cite{Giombi:2017dtl,Klebanov:2018fzb}, or the absence of supersymmetric Dyson-Schwinger solutions as in \\cite{Anninos:2016szt,Chang:2018sve} ), and is rather inconvenient to study at finite $N$. In fact, in such a theory with a supermultiplet with $(b, \\psi, \\dots),$ and in a supersymmetric nearly conformal fixed point, $\\Delta_{\\psi}=\\Delta_b+\\frac{1}{2}.$ The Dyson-Schwinger equation of the dominant interaction in the infrared would constrain the dimensions of various fields so that the sum of scaling dimensions of the fields in the interaction is one, i.e.~$ n \\Delta_b+ 2m \\Delta_{\\psi}+ \\dots=1$, with $n,m\\in \\mathbb{Z}_{\\geq 0}$.  For the nearly conformal fixed point to be unitary, we require $\\Delta_{b},\\Delta_{\\psi}\\geq 0.$ Together with the supersymmetric constraint, we conclude the only non-trivial solutions possible is that \n\\begin{equation}\\label{bosondim}\n    \\Delta_b=0, \\Delta_{\\psi}=\\frac{1}{2}\\,.\n\\end{equation}\nHowever, such a solution indicates that the two-point function of $b$ must be logarithmic and typically causes a divergence in the Dyson-Schwinger equations (or \\eqref{bosondim} ceases to be a solution as in \\cite{Popov:2019nja}). \n\nOn the other hand, we may consider a theory with a fermionic super-multiplet. This scenario brings about yet another complication. Unlike the case of $\\mathcal{N}=1,2$, there is no relevant deformation of the $\\mathcal{N}=4$ that exists in the UV free theory. To see this, we can work in the $\\mathcal{N}=4$ superspace,\\footnote{Here, we note that this argument does not rule out possible theories without any kind of superspace realization. In particular, in one dimension one can consider a first derivative action in bosons as in \\cite{Tikhanovskaya:2020elb}, and this modifies the supersymmetry constraint to $\\Delta_b=\\Delta_{\\psi}$. Thus it allows a greater number of relevant interactions. } and the allowed action is \n\\begin{equation} \n    \\mathcal{L} \\sim \\int d^2\\theta W(\\Psi)+ h.c. + \\int d^4\\theta K(\\Psi, \\bar{\\Psi}),\n\\end{equation}\nwhere schematically \n\\begin{equation}\n    \\Psi= \\psi+ \\theta b+ \\dots,\n\\end{equation}\n where $\\Psi$ can sit in any representation of $SU(2)$ with half-integer $J$, and $\\psi$ is the lowest component. As a result, any local interaction must have dimension at least one due to $\\int d^2\\theta \\dots $, which implies that it cannot be relevant. This contrasts with the constructions of $\\mathcal{N}=1,2$ SYK-like models, where $\\int d\\theta W(\\Psi)$ is allowed and can produce relevant interactions. We can still ask if it's possible to have a marginally relevant deformation. Even if this were the case, in the infrared, a similar argument to (\\ref{bosondim}) would suggest \n \\begin{equation}\n     \\Delta_{\\psi}=0, \\Delta_b=\\frac{1}{2}, \\dots,\n \\end{equation}\nwhich coincides with the dimensions of the free theory. This analysis suggests that constructing an interacting IR fixed point is difficult when starting from a UV theory with the same amount of supersymmetry. Therefore, one might be tempted to consider a scenario in which the $\\cN=4$ supersymmetry solely emerges in the IR and is not present in the UV. We leave a more thorough investigation into these issues for future work. \n \n \n\\subsection*{Higher genus corrections to super-JT}\n\nMotivated by the existence of the gap in the leading density of states for the $\\cN=2$ and $\\cN=4$ super-Schwarzian, it would be interesting to understand whether the gap survives when accounting for corrections coming from higher genus geometries contributing to the 2D theory. Relatedly, due to the existence of the gap, it is interesting to note that the contribution from disk topologies to the spectral form factor $\\< Z(\\beta-i t) Z(\\beta + it)\\>$ dominates even at very late times. This result contrasts with non-supersymmetric or $\\cN=1$ JT gravity, where at late times, the cylindrical topology starts dominating, leading to a ``ramp'' in the spectral form factor, followed by a plateau at even later times. It would also be interesting to understand whether the genus expansion of the $\\cN=2$ and $\\cN=4$ super-JT gravity has an interpretation in terms of a matrix integral; this interpretation needs to go beyond the three Dyson\nensembles \\cite{Dyson:1962es} and the seven Altland-Zirnbauer ensembles \\cite{altland1997nonstandard}, whose gravitational interpretation was studied in \\cite{Stanford:2019vob}. \n\nThese non-perturbative corrections are relevant from the perspective of solving 2D $\\cN=4$ gravity exactly. It is not clear whether these corrections could be reliable from the higher dimensional picture, but it would be interesting if the presence of supersymmetry could help better understand issues of factorization in the D1\/D5 system, as one example. We leave this for future work.\n\n\n\n\n\n\\subsection*{Acknowledgements} \n\n\nWe thank R. Campos Delgado, A.~Castro, G.~Horowitz, I.~Klebanov, J.~Maldacena, S.~Pufu, D.~Stanford, H.~Verlinde and E.~Witten for valuable discussions and comments on the draft. MTH is supported in part by Department of Energy Grants DE-SC0007968, DE-SC0009988, and the Princeton Gravity Initiative. LVI was supported in part by the Simons Collaboration on the Conformal Bootstrap, a Simons Foundation Grant with No. 488653, and by the Simons Collaboration on Ultra-Quantum Matter, a Simons Foundation Grant with No. 651440. GJT is supported by a Fundamental Physics Fellowship. WZ was supported in part by the US NSF under Grants No. PHY-1620059 and PHY-1914860.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn order to tame the increasing data demands, small cell deployments are of key importance. To date, primary focus of small cell networks is to enhance the overall capacity by bringing users closer to serving base stations~\\cite{li2014throughput}. Small cells can be deployed as standalone self organizing networks, or operating in conjunction with the existing macro cellular network. Due to the ever-increasing data traffic demands, an ultra-dense deployment of small cells is not a cost-effective strategy due to CAPEX\/OPEX issues. At the same time, recent developments in unmanned aerial vehicles (UAVs), driven by Google and Facebook bring forward the idea of using UAVs for coverage extension and capacity enhancements. UAVs can be used as aerial access points acting as \\emph{pivot} between macro and small cell tiers. UAVs can autonomously provide a reliable multi-connectivity in areas prone to high demand or link failures. UAVs can be used as aerial access points, or relays between disconnected networks and enhanced connectivity~\\cite{guo2014performance}. A smart combination of all these networks can provide a vast range of applications in civilian networks~\\cite{merwaday2015uav}. One of the major issues faced by these networks is on-demand\/on-the-fly capacity provisioning. Capacity refers to data rate transmission towards ground users, whereas delays refer to latency of data transmission~\\cite{mozaffari2015unmanned}. Using drone small cells as aerial support to existing cellular network can handle these high traffic situations more cost-efficiently. While deploying a single UAV is relatively easy using a maximum coverage point over the demand area, deploying more UAVs operating in coordination is more challenging due to interference from other aerial nodes~\\cite{mozaffari2015drone}. Thus, an efficient approach is required that not only provides efficient topology for UAVs based on user demands, but also improved connectivity and enhanced coverage. Using multiple UAVs as relays between the existing macro cells and small cell networks is the primary focus of this letter, in which the goal is to increase capacity, and lower transmission delays.\n\nIn this letter, a cost function based multiple UAVs deployment model is presented. The proposed model uses user demand patterns to assign a cost and density function to each area and UAVs. These cost and density functions are then used to match each UAV to a particular demand zone via a reverse neural model based on user demand patterns~\\cite{chandrashekarappa2014forward}.\n\\begin{figure}[!hb]\n  \\centering\n \n\\includegraphics[width=200px]{fig1}\\\\\n  \\caption{Illustration of an UAV-overlaid network deployment, with UAVs acting as relays between MBS and UEs.}\\label{fig1}\n\\end{figure}\n\\section{System Model}\nThe proposed model aims at provisioning continuous data between macro and small cell user equipments (UEs). UAVs further enhance load balancing by forming multiple intermediate links between the macro cell and the small cell UEs. The proposed model focuses on UAV-to-UE links rather the MBS-to-UAV backhaul capacity links. A traditional small and macro cell network is shown in Fig.~\\ref{fig1}. This network provides a direct link between the UEs and the macro cell base station (MBS). However, with a continuous increase in number of users within the coverage of a particular small cell, it becomes almost impossible to sustain connectivity without any loss of data. This is a problematic issue for future generation 5G networks, whereby demand is set to be greater than the available capacity. This is the focus of this work harnessing the formation of UAVs for a reliable and load balanced network. The proposed approach leverages a cost based framework which uses neural demand patterns to identify areas with high demands. A predictive chart is formed at the MBS acting as a launch point for UAVs. This chart helps in finding appropriate positions and topology for UAVs. Network stability is attained by minimizing a cost function associated with both demand areas and deployed UAVs. A delay threshold is defined to analyze the network performance, defined as the network state with minimum delay in providing capacity coverage in high demand areas, minimum errors in mapping UAVs to particular demand area, and high reliability in terms of data rate delivery. The model consists of deploying $n$ UAVs each one capable of handling $S_{n}$ service requests in a zone governed by an MBS. If $S_{r}$ requests are made by active users in a small cell zone that the MBS is unable to handle, then the minimum number of UAVs required to handle this traffic is $\\frac{S_{r}}{S_{n}}$. Moreover, the optimal placement of multiple-UAVs in the required zone is a major issue. For this, the concept of \\emph{zone guider lines} is considered where \\emph{zone guider lines} divide a particular area into a set of small regular areas acting independently. For a low-complexity solution, the number of UAV is kept equal to the number of guider lines. The traditional hexagonal cell is divided into standard guider lines, then, the area with high user requests are marked. Next, the existing guider lines form the maximum and minimum limit for introducing new guider lines which will embark the area to be governed by a UAV. This procedure is presented in Fig~\\ref{fig3}.\n\\begin{figure}[!hb]\n  \\centering\n \n\\includegraphics[width=200px,height=180px]{fig3}\\\\\n  \\caption{Area distribution in the macro cell with an illustration of referral guider lines.}\\label{fig3}\n\\end{figure}\n\\subsection{Assumptions}\nIn order to validate the effectiveness of the proposed approach, some generic assumptions are as follows:\n\\begin{itemize}\n  \\item UAVs operate on same frequency spectrum.\n  \\item Each UAV is of the same make, whose configuration does not affect its positioning.\n\\end{itemize}\n\\section{Proposed Approach}\nUAVs are used as high altitude base stations to cover certain geographical areas. The proposed approach uses a cost based neural model to find the appropriate user demand zones where UAVs is placed. The cost function includes the cost of operation which has to be minimum, and the cost of handling UAVs. Thus, the proposed approach focuses on finding the appropriate cost function for demand areas and UAVs, allocating them to MBS, and defining a neural model to minimize the cost function. In the proposed model, UAVs operate at an altitude $h$ over an area $A$, with $x$ number of users. Users service request $S_{r}$ come with an arrival rate of $\\gamma$, and a mean packet size of 1\/$\\mu$. The load\/delay, denoted by $L$ (in seconds) for a user at location $y$, is computed as~\\cite{samarakoon2014opportunistic}:\n\\begin{equation}\\label{eq:d1}\nL (y) =\\frac{\\gamma}{W \\;\\log(1+SINR(y))\\times \\mu}.\n\\end{equation}\nThe channel modeling includes radio range and pathloss. Moreover, round robin approach is applied for scheduling. Further, the system model is defined with respect to UAVs rather than the MBS, assumed to operate on orthogonal band. The area load $L_{a}$ is given by~\\cite{samarakoon2014opportunistic}:\n\\begin{equation}\\label{eq:d2}\nL_{a}=\\int_{y \\in A} L(y) dy.\n\\end{equation}\nHere, $W$ is the system bandwidth, and assuming that UAVs operate on the same frequency spectrum, the signal-to-interference-plus-noise ratio $\\left(SINR\\right)$ from the $i^{th}$ UAV to a given UE at location $y$, considering UAV-to-UAV interference, is:\n\\begin{equation}\nSINR(y)= \\frac{\\frac{P \\; K  }{R_{iy}^{\\alpha}}}{\\sum_{j=1, j\\neq i}^{n} \\frac{P \\;K}{ R_{jy}^{\\alpha}} + N_{0}},\n\\end{equation}\nwhere $P$ is the UAV transmission power, $K$ is a factor that accounts for the geometrical parameters such as transmitter and receiver antenna heights, $R_{iy}$ is the distance between the $i^{th}$ UAV and the UE at location $y$, $\\alpha$ is the path loss exponent, and $N_{0}$ is the noise power spectral density. The spectral efficiency $N_{S}$ for a user at location $y$ following round robin scheduling is given by:\n\\begin{equation}\nN_{S}=W.\\;\\frac{\\log_{2} \\left( 1+SINR(y) \\right)}{x}\n\\end{equation}\nThe cost function is a function of capacity, delay, availability of line of sight (LOS), and coverage. Further, let $D_{f}$ denote the density function that quantifies the population of active\/non-active users based on users' request patterns. $D_{f}$ accounts for the number of active users $x$, packet loss (call drops) $C_{d}$, service requests $S_{r}$, and the total amount of users a cell can handle is $T_{r}$. For the considered network, two variants of the density function, one for a given area $D_{f}^{A}$, and the other for UAVs $D_{f}^{U}$ are computed as\n\\begin{equation}\\label{eq:1}\nD_{f}^{A}=\\min\\left(\\frac{\\left(\\frac{x}{T_{r}}\\right)^{S_{r}}\\;e^{- \\left(\\frac{x}{T_{r}}\\right)}}{S_{r}!}\\right),\n\\end{equation}\nand\n\\begin{equation}\\label{eq:2}\nD_{f}^{U}=\\min\\left(\\frac{\\left(\\frac{L_{a}}{n}\\right)^{S_{n}}\\;e^{- \\left(\\frac{L_{a}}{n}\\right)}}{S_{n}!}\\right),\n\\end{equation}\nrespectively. $D_{f}^{A}$ accounts for user distribution over the area, in which a higher value requires more UAVs, and a minimum value shows the efficient connectivity with no further requirement of intermediate relays. $D_{f}^{U}$ accounts for pending user requests in area A with respect to the number of UAVs deployed. A higher value denotes the requirement of more UAVs, and a minimum denotes efficient service handling using current deployment. Here, $\\frac{x}{T_{r}}$ denotes the ratio of active users to the total users a cell can handle. For 100\\% accuracy in mapping, $\\frac{x}{T_{r}}=1$. (\\ref{eq:1})-(\\ref{eq:2}) account for the cost function provided constraint (\\ref{eq:3}) holds.\n\\begin{equation}\\label{eq:3}\n\\sqrt{\\frac{1}{S_{r}} \\sum_{i=1}^{S_{r}} \\left( T_{r}^{i}-C_{d}^{i}\\right)} \\leq \\frac{x}{T_{r}}.\n\\end{equation}\nFor an efficient operation, the deviation in (\\ref{eq:3}), the number of users with unhandled service requests, should be kept minimum. Furthermore, the per area and UAV cost function $C_{f}^{A}$ and $C_{f}^{U}$ is given as:\n\\begin{equation}\\label{eq:4}\nC_{f}^{A}=\\min\\left(D_{f}^{A}\\; L_{a}\\; \\left(\\eta_{1}S_{r} + \\eta_{2}T_{r} \\right)\\right),\n\\end{equation}\nand\n\\begin{equation}\\label{eq:5}\nC_{f}^{U}=\\min\\left(D_{f}^{U}\\; R^{\\alpha}_{iy}\\; \\left( \\eta_{1}S_{r}+\\eta_{2} x\\right)\\right), LOS=true,\n\\end{equation}\nrespectively. Here, $\\eta_{1}$ and $\\eta_{2}$ are network balancing constants such that $\\left(\\eta_{1}, \\eta_{2} \\right) \\in \\left(0,1\\right)$. In general, $\\eta_{1}$ is driven by the network bandwidth and link speed, whereas $\\eta_{2}$ is driven by the number of active connections. For ideal state, $\\eta_{1}$ and $\\eta_{2}$ equals 1. In general, $0.5 \\leq \\eta_{1} \\leq 1$ and $\\eta_{1} \\leq \\eta_{2} \\leq 1$ which denotes that the network transfer rate must be higher than half of the initial configured rate. Both cost functions $C_{f}^{A}$ and $C_{f}^{U}$ are governed by the constraints of $D_{f}^{A}$ and $D_{f}^{U}$. The complete availability of LOS is one of the key driving factor for continuous connectivity. The overall cost function $C_{f}^{O}$ is computed at the MBS to maintain the overall network connectivity such that\n\\begin{equation}\\label{eq:6}\nC_{f}^{O}=\\min\\left( \\frac{1}{n} \\sum_{i=1}^{n} \\left(C_{f}^{U}\\right)_{i} + \\sum_{j=1}^{A_{T}} \\left(\\frac{C_{f}^{A}}{U_{T}} \\right)_{j}\\right).\n\\end{equation}\nHere, $U_{T}$ is the number of UAVs allocated to a particular area, $A_{T}$ is the number of total demand areas. With more UAVs, more resources are available in terms of transmission power, yielding high throughput and reduced delay. Further, the delay ($L_{d}$) at each node is computed as:\n\\begin{equation}\nL_{d}=L_{transmission}+L_{propagation}+L_{queue}+L_{processing}.\n\\end{equation}\nHere, $L_{transmission}$ is the transmission delay defined as the load\/delay of a particular user and is equal to $L$ (\\ref{eq:d1}), $L_{propagation}$ is the ratio of the distance between nodes to the propagation speed, $L_{queue}$ is the waiting time of the packet, and $L_{processing}$ is the network operational time.\n\\section{Neural Demand Patterns and Network Capacity}\nThe goal is to optimize the density functions and minimize the cost functions defined in (\\ref{eq:1}), (\\ref{eq:2}), (\\ref{eq:4}), (\\ref{eq:5}), and (\\ref{eq:6}). By controlling $D_{f}^{A}$ and $D_{f}^{U}$, the user distribution with pending requests are controlled, which in turn, minimizes $C_{f}^{A}$, $C_{f}^{U}$, and $C_{f}^{O}$. This minimization provides guaranteed service to UEs. Thus, the aim of the neural model is to accurately map UAVs to demand areas so as to minimize these cost functions.\n\\begin{figure}[!ht]\n  \\centering\n \n\\includegraphics[width=180px,height=170px]{fig4}\\\\\n  \\caption{Reverse neural model modeling the user demand pattern: $C_{f}^{A}$ refers to area cost function, $C_{f}^{U}$ refers to UAV cost function, $C_{f}^{O}$ is the overall cost function.}\\label{fig4}\n\\end{figure}\nDemand patterns are used to minimize the demand and cost functions by efficiently deploying aerial nodes as intermediate nodes between high demand areas and the MBS. These demand patterns are driven by a reverse multi-hierarchical neural model which is a combination of input, hidden, and output layer. Although neural models are slow and complex in operation, the reverse neural model accounts for accurate mapping of UAVs to demand areas with lesser iterations. This provides a low-complexity approach for UAV-to-Area mapping. The output for the proposed model is computed in forward direction, i.e. from Layer 3 to Layer 1, but nodes are deployed in reverse direction i.e. from Layer 1 to Layer 3, as shown in Fig.~\\ref{fig4}. Input involves demand areas that require services of UAVs. MBS is the driving factor of this reverse neural model, and it act as an output layer. UAVs are the intermediates, and are not having permanent connections with the demand areas, thus, acts as a hidden layer. These neural patterns are then topologically rearranged to form a stable network with minimized cost function. Layer 3 defines the cost function for underlying demand areas, layer 2 is for aerial nodes, and output layer 1 is mapped to the MBS. At first, a mesh like interface is initialized for the neural model (Fig.~\\ref{fig4}), then, the whole model is rearranged to allocate links that will lower the cost functions of all the zones. This procedure is governed by series of steps given in Algorithm~\\ref{algo1}.\n\\begin{algorithm}[!ht]\n\\fontsize{7}{9}\\selectfont\n\\caption{UAV to Area Mapping}\n\\label{algo1}\n\\begin{algorithmic}[1]\n\\State \\textbf{Input}: $U \\longleftarrow UAVs$, $A \\longleftarrow Demand \\;areas$\n\\State Initialize Network\n\\State Divide A into subdivisions $D_{M}$\n\\While{(U are not mapped to subarea $D_{M}$ \\&\\& ($C_{f}^{O}$=minimum))}\n\\State Store $D_{M}$ in area[] following descending order for $C_{f}^{A}$\n\\State i=1\n\\While{($i \\leq count(area[])$)}\n\\State allocate U to $D_{M}$, such that min($C_{f}^{U}$) is mapped to max($C_{f}^{A}$)\n\\State Compute $C_{f}^{A}$, $C_{f}^{U}$, $C_{f}^{O}$\n\\If {((U is mapped to $area[i]$) \\&\\& ($C_{f}^{A}(i)$=minimum))}\n\\State continue\n\\Else\n\\State re-initialize $U$, $C_{f}^{A}$\n\\State reset\n\\EndIf\n\\State \\textbf{end if}\n\\State $i=i+1$\n\\EndWhile\n\\State \\textbf{end while}\n\\EndWhile\n\\State \\textbf{end while}\n\\end{algorithmic}\n\\end{algorithm}\nInitially, the demand area $A$ is subdivided into small segments $D_{M}$, and the cost function is computed for each of the subdivided area. Each of the cost function is ranked in descending order, in which the UAV with a minimum cost function is mapped to the area with maximum cost function. This helps in balancing the overall load of the network. After initial allocation, all the cost functions are recomputed, and UAVs are mapped to the next high demand area based on the cost function. This procedure continues till a minimum is attained. With an efficient deployment, the cost function is minimized by sub-dividing the density function based on the area so that active users becomes equal to the total registered users i.e. $\\frac{x}{T_{r}}=1$ such that equation (\\ref{eq:1}) reduces to\n\\begin{equation}\nD_{f,avg}^{A}=\\frac{1}{e\\times S_{r}!}\n\\end{equation}\nwhere e=2.71828 approx. For verification at any stage, the average density function of each area must be less than equal to $D_{f,avg}^{A}$.\n\\begin{table}[!ht]\n\\fontsize{7}{9}\\selectfont\n\\centering\n\\caption{Parameter Configurations}\\label{self_conf\n\\begin{tabular}{l l l}\n\\hline\\\\\n\\textbf{Parameter} & \\textbf{Value} & \\textbf{Description}\\\\\n\\hline\n\\hline\\\\\n$A$ & 10000x10000 sq. m. & Simulation Area\\\\\nMBS & 10& Number of Macro Cell Base Station\\\\\n$T_{r}$ & 1200 (per MBS) & Max Users in a Cell\\\\\n$n$& 6 (per MBS)& Number of UAVs\\\\\n$S_{n}$ & 200 & Service Requests handled by each UAV\\\\\n$N_{0}$ & -170 dBm\/Hz & Noise Power Spectral Density\\\\\n$\\frac{1}{\\mu}$ & 1024 B&Packet Size \\\\\n$h$ &200-500 Feet& UAV altitude\\\\\n$\\frac{\\gamma}{\\mu}$ & 256 kbps& Offered Traffic\\\\\n$\\alpha$& 4&Path loss Exponent\\\\\n$K$ & -11 dB & Transmission Constant\\\\\n$P$ & 35 dBm & UAV Transmission Power \\\\\n$S_{r}$& 30-50 per zone& Service Requests\\\\\n$W$ & 10 MHz& System Bandwdith\\\\\n$x$ & 400 & Active Users\\\\\n\\hline\n\\end{tabular}\n\\end{table\n\\begin{figure}[!ht]\n\\begin{minipage}[t]{0.20\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g1}\n\\caption{\\fontsize{6}{6}\\selectfont Networks Delays vs. Extra Users }\n\\label{g1}\n\\end{minipage}\n\\hspace{\\fill}\n\\begin{minipage}[t]{0.26\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g7}\n\\caption{\\fontsize{6}{6}\\selectfont Throughput Coverage vs. Path loss Exponent}\n\\label{g3}\n\\end{minipage}\n\\end{figure}\n\\begin{figure}[!ht]\n\\begin{minipage}[t]{0.20\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g8}\n\\caption{\\fontsize{6}{6}\\selectfont Throughput Coverage vs. Extra Users}\n\\label{g4}\n\\end{minipage}\n\\hspace{\\fill}\n\\begin{minipage}[t]{0.26\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g9}\n\\caption{\\fontsize{6}{6}\\selectfont 5th Percentile Spectral Efficiency vs. Extra Users }\n\\label{g5}\n\\end{minipage}\n\\end{figure}\n\\begin{figure}[!ht]\n\\begin{minipage}[t]{0.20\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g12}\n\\caption{\\fontsize{6}{6}\\selectfont 5th Percentile Spectral Efficiency vs. Path loss Exponent }\n\\label{g6}\n\\end{minipage}\n\\hspace{\\fill}\n\\begin{minipage}[t]{0.26\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g5}\n\\caption{\\fontsize{6}{6}\\selectfont Probability of Guaranteed SINR vs. Extra Users }\n\\label{g7}\n\\end{minipage}\n\\end{figure}\n\\section{Performance Evaluation}\nThe proposed model is analyzed using network simulations, to efficiently allocates areas with higher demand patterns to UAVs based on their cost function. All parameters and configurations are presented in Table~\\ref{self_conf}. Active users refer to the number of users in a cell requesting services. Thus, at any time instance, more than the maximum supported user requests can be present in the cell. Results were recorded for delays, network capacity, reliability, and value of cost function with and without use of UAVs. Delays were traced for complete data sharing with 1000 iterations of user requests over 1000 seconds. Availability of LOS is the primary condition for UAVs to form a link with the UE. Finding appropriate position in the user demand areas causes UAVs to adjust their altitude to guarantee LOS towards the UE. High altitude provides less interference and appropriate LOS, but also induces more delays. Thus, an optimum altitude with availability of LOS is required for better coverage. For analysis, the altitude was varied between 200 ft. and 500 ft. with a multi-antenna relay support for communication and backhaul link capacity of 1.2 Gbps. Delay threshold is fixed at 200 ms, which defines the upper limit above which the packet drop increases abruptly. Results show that the proposed approach leveraging UAVs yields 37.7\\% lesser delays in comparison with a network comprising of small cell and macro cell. The performance delay for various UAV altitudes is given in Fig.~\\ref{g1}. Throughput coverage defined as the percentage of users whose SINR is above the threshold (0.03 bps\/Hz) is shown in Fig.~\\ref{g3}. Therein, the use of UAVs increases the overall 5th percentile throughput coverage by 15.5\\%, as shown in Fig.~\\ref{g3} and Fig.~\\ref{g4}. Moreover the optimal placement of UAVs according to user demand leveraging the reverse neural deployment model enhances the 5th percentile spectral efficiency. The accurate mapping optimally places UAVs according to demand patterns, thus, improving the 5th percentile spectral efficiency by 38\\% approx., as shown in Fig.~\\ref{g5} and Fig.~\\ref{g6}. Finally, Fig.~\\ref{g7} plots the probability of guaranteed SINR for a particular user in a given macro cell. Clearly, it can be noticed that the use of UAVs provides much guaranteed SINR above the threshold defined by $\\eta_{1}$ in (\\ref{eq:5}). Here, the SINR threshold is kept at 0.55 defining the value below which the network is unable to provide efficient connectivity to users. Results show that the proposed user demand based network model is capable of providing better capacity and prolonged connectivity than the existing cellular network.\n\\section{Conclusion}\nIn this letter, user demand based network model is proposed using multiple UAVs. The proposed model uses density and cost functions to compute areas with higher demands, whereby UAVs are deployed based on these cost functions. Analysis proved that the proposed model is capable of providing better capacity, reliability, and prolonged connectivity in comparison to existing ground-based wireless networks.\n\\bibliographystyle{ieeetr}\n\\nocite{*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\nEvergrowing concerns about user-privacy, censorship and central authority in popular social media have motivated both the development of federated social networks such as Mastodon and Diaspora~\\cite{rochko2018mastodon,Bielenberg:2012:TGD}, as well as research in academia~\\cite{Anderson:2009:PSN, Shakimov:2009:PCA}.\nThese networks aim to promote user control by decentralizing authority and relying on open-source software and open standards.\nAt the time of this writing, Mastodon has over 1 million users and 3500 instances which demonstrates the increasing acceptance of distributed social networks.\nAs with traditional social media, one key success factor of such a network is an active and engaged community.\n\nAs a community grows, overwhelming amounts of content make it increasingly difficult for a user to find interesting topics and other users to interact with.\nFor that reason, popular platforms such as Twitter, LinkedIn and Facebook introduce recommender systems that set out to solve a particular recommendation task.\nOne prominent example is the ``Who to Follow'' service by Twitter~\\cite{Gupta:2013:WFS}.\nDue to their recent emergence, those recommender systems do not exist for federated social networks, yet.\nHowever, they are needed to make distributed social media attractive to large user groups as well as competitive to centralized networks.\nAt the same time, recommender systems will contribute to develop, grow and sustain an active community.\n\nTo make federated social networks more attractive and feature complete, we implement and evaluate a topology-based user recommender based on personalized PageRank~\\cite{Page:1999:PCR}, a commonly used algorithm for link-prediction in social networks.\nWe compare this method against collaborative filtering based on link intersections~\\cite{Hannon:2010:RTU} and a random link predictor baseline~\\cite{Liben-Nowell:2003:LPP}.\nThe experiments are carried out on Mastodon, a federated social network for which user relations do not require reciprocation, and the network forms a directed graph.\nWe expect that the method and results are transferable to any other federated social network with similar characteristics.\n\nWe evaluate the systems in an offline and online scenario. For the offline evaluation, we collect an unbiased sample of the Mastodon user graph.\nThis sample is created by performing a Metropolis-Hastings Random Walk (MHRW) adapted for directed graphs~\\cite{Wang:2011:UGS,Wang:2010:USD}.\nThe collected data contains about 25\\% of the entire userbase of Mastodon.\nWe then evaluate the recommender systems according to standard performance metrics used in ranked retrieval systems, and deploy the two best performing methods to an online setting.\nBoth algorithms generate a list of personalized recommendations for 19 Mastodon users participating in the online trial and performance is measured with the balanced interleaving approach~\\cite{Joachims:2003:ERP}.\n\nThis paper is structured as follows.~\\cref{sec:dataset} explains how data are collected for the offline experiments and discusses the recommendation algorithms and their evaluation.\nIn~\\cref{sec:results} we present and discuss experimental results.~\\cref{sec:conclusion} concludes this paper and provides directions for future work.\n\n\\section{Dataset and Methods}\n\\label{sec:dataset}\n\n\\subsection{Recommendation Algorithms}\n\\label{sec:recommendation-algorithms}\nThe user recommendation problem for social networks can be formalized as follows.\nGiven a graph $G = (V,E)$ where $V$ and $E$ are vertices and edges, we seek to predict an interaction between a user $u \\in V$ and $v \\in V$ denoted by edge $(u,v)$.\nIn networks such as Mastodon and Twitter, a user interaction does not require reciprocation.\nThus, the graph is directed.\nWe consider two broad approaches to generate recommendations: (1) collaborative filtering-based recommendation and (2) topology-based recommendation.\n\nWith respect to the collaborative filtering, we use an approach inspired by~\\cite{Hannon:2010:RTU}. Each user $u \\in V$ is represented by a profile and recommendations are generated based on the similarity of profiles.\nWe distinguish between the three best performing strategies in~\\cite{Hannon:2010:RTU}:\n\\begin{description}[leftmargin=!,labelwidth=\\widthof{$following(u)$}]\n\t\\item[$\\mathit{following}(u)$] The set of user ID's $u$ follows\n\t\\item[$\\mathit{followers}(u)$] The set of user ID's that follow $u$\n\t\\item[$\\mathit{combined}(u)$] The combined set of following and follower ID's\n\\end{description}\nWe consider these profiles as documents to be indexed in a general purpose search engine. In order to generate recommendations for a user, the corresponding profile is extracted first.\nAfterwards, the retrieval system is queried with the profile and it ranks the indexed documents by their relevance to the query.\nEach ID in the user profile is a token of the query. If a  query consists of more than 10,000 tokens, we create a random subset of 10,000 tokens.\nUnlike~\\citet{Hannon:2010:RTU}, we use BM25 instead of TF-IDF to estimate the relevance score of each document and set parameters to common defaults ($k_1 = 1.2$, $b = 0.75$)~\\cite{Manning:2008:IIR:1394399}.\nThe final recommendation list contains the top-$k$ documents with highest retrieval score.\n\nThe collaborative filtering recommendations are compared to topology-based recommendations.\nSeveral methods have been proposed in literature which make use of link-based ranking algorithms such as HITS, PageRank and SALSA\\@.\nDue to the novelty of generating recommendations for federated social networks, we restrict our experiments to the personalized PageRank algorithm~\\cite{Page:1999:PCR} whose efficient computation is well-understood and which is used in the Twitter recommender system~\\citep{Gupta:2013:WFS}.\nWe apply the personalized PageRank for a seed node which is the user we want to generate recommendations for.\nAfter convergence, the list of user recommendations is constructed by taking the top-$k$ nodes with highest PageRank. Following~\\cite{Liben-Nowell:2003:LPP}, we set the damping factor $\\lambda = 0.85$.\n\n\\subsection{Data Collection}\nAcquiring the complete graph of a social network is always infeasible due to API limits and time constraints~\\cite{Wang:2011:UGS}.\nAn additional concern arises in a distributed social network.\nAs data is not stored at a central authority, there is no single API that provides access to all parts of the network.\nInstead, data is scattered around different sub-networks.\nBoth issues are addressed within this section.\n\nTo overcome the time constraint, we apply the Metropolis-Hastings Random Walk (MHRW) to acquire an unbiased sample that is still representative of the complete graph.\nMHRW is a Markov-Chain Monte Carlo algorithm that can be used to obtain node samples with a uniform probability distribution~\\cite{Wang:2011:UGS}.\nAs the MHRW is only applicable to undirected graphs, we apply a generalization that considers all directed edges as bidirectional edges~\\citep{Wang:2010:USD}.\nWe do not consider graph sampling methods such as Random Walk and Breadth-First Sampling as it has been shown that these methods yield samples biased towards high degree nodes~\\cite{Gjoka:2010:WFC}.\n\nDue to the fact that a distributed social network has no central API, one has to query the API of each individual sub-network referred to as \\textit{instance}.\nIn case of Mastodon, there are two public endpoints to acquire incoming and outgoing links: \\texttt{\/following} and \\texttt{\/followers}\\footnote{The following API URL pattern applies to any Mastodon instance:\\\\ \\texttt{https:\/\/<instance>\/users\/<user>\/<endpoint>.json}}.\nWhenever the MHRW visits an unexplored node, followers and followings of that node are fetched and stored in a document-oriented database.\nThis database is also used as a cache: if the random walk transitions to a node which it has already visited, we use the cached result rather than querying the API again.\nDuring the data collection, we apply fair crawling policies. Only instances that allow crawling as defined by the \\texttt{robots.txt} are considered.\nFurthermore, concurrent requests are throttled such that no more than 10 requests per second are issued (a rate which we believe any web server can sustain).\n\n\\subsection{Dataset Statistics}\n\\label{sec:dataset-statistics}\n\\cref{tab:dataset-statistics} summarizes the properties of the collected graph. The initial graph ($t_1$) has been crawled from the 16\/05\/18 until 17\/05\/18.\nThe MHRW was executed for 5500 iterations.\nDuring the crawl, 138 instances were disregarded either because of their \\texttt{robots.txt} or because they were no longer available.\nIn order to acquire a newer version of that graph ($t_2$), we visited the same users five days later and recorded new relationships.\nThe number of visited users in $t_2$ is slightly lower than in $t_1$, as some profiles were deleted or their instances became unavailable.\nThe updated graph is used as the ground-truth when evaluating our recommender systems.\n\nIt can be observed that the Network Average Clustering Coefficient (NCC) and the fraction of nodes in the largest Strongly Connected Component (SCC) is almost equal for the two given graphs.\nFurthermore, the graph is mildly disassortative.\nIt is important to mention that although the total number of nodes found $|V|$ is high (253,000), accounting for about 25\\% of the total Mastodon users, the number of visited nodes $|V^*|$ is much smaller (about 3400).\nIncoming and outgoing edges are only known for visited nodes.\n\n\\begin{table}[t]\n\\centering\n\\caption{Statistics of crawled graphs. The initial crawl at $t_1$ and the newer crawl of the same users at $t_2$.}\n\\label{tab:dataset-statistics}\n\\begin{tabular}{@{}llllllll@{}}\n\\toprule\nGraph & $|V|$ & $|V^*|$ & $|E|$ & Assort. & Deg. & NCC & SCC \\\\ \\midrule\n$t_1$ & 253,822 & 3437 & 754,037 & -0.015 & 5.94 & 0.31 & 0.175 \\\\\n$t_2$ & 255,638 & 3383 & 754,667 & -0.016 & 5.9 & 0.31 & 0.173 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsection{Evaluation}\n\\label{sec:evaluation}\nThe algorithms presented in~\\cref{sec:recommendation-algorithms} are evaluated in two phases: an offline evaluation and an online evaluation.\nFor the offline evaluation we measure precision at rank $k$ (p@k), Mean Average Precision (MAP) and success at rank $k$ (s@k), which are popular metrics for the evaluation of ranked retrieval systems~\\cite{Manning:2008:IIR:1394399}.\nThe newer graph at time $t_2$ serves as the ground-truth, whereas the graph at time $t_1$ can be seen as the training graph.\nIn information retrieval terms, the generated list of recommendations are the retrieved documents and the list of users a target user follows at time $t_2$ are the relevant documents. Significance is tested using a two-tailed paired t-test. We denote improvements with $^{\\blacktriangle}$ ($p<0.01$), deteriorations with $^{\\blacktriangledown}$ ($p<0.01$), and no significance by $^{\\circ}$.\n\nDuring the offline evaluation, all systems generate a list of 100 recommendations based on the training graph at time $t_1$.\nThis list is then compared with the actual links added to the graph in between time $t_1$ and $t_2$ (see~\\cref{sec:dataset-statistics}).\nIn case of the collected dataset, 329 of 3437 visited users started to follow another individual, and thus added a link to the graph.\nOnly for this set of users, recommendations are generated and evaluated.\n\nThe online evaluation is performed as follows.\nA recommendation bot is created on the Mastodon instance associated with the institute of the authors\\footnote{See \\url{https:\/\/mastodon.utwente.nl\/@Followdon}}.\nAfterwards, we ask users to follow this bot if they wish to receive personalized recommendations. For each participant, we generate a static web page consisting of a list of $N$ recommendations with the option to start to follow a suggested user.\nA link to this web page is then send to the user and we track the user interactions. A recommendation is considered relevant if the participant starts to follow a suggested user.\nThe recommendations of two algorithms are presented using balanced-interleaving, which is a relatively inexpensive evaluation method for online experiments compared to conventional A\/B testing.\nWe refer the reader to~\\cite{Joachims:2003:ERP} for a thorough discussion of this evaluation method.\n\nOne complication arises in the online evaluation. As an up-to-date graph is unavailable at recommendation time, such a graph has to be created.\nFor this, we explore the vicinity of a recommendation target $u \\in V$ by applying an egocentric random walk for a fixed amount of iterations.\nThis strategy resembles the ``circle-of-trust'' used in the Twitter recommender system~\\cite{Gupta:2013:WFS}.\nThe random walk is performed as follows. At each iteration, the algorithm either transitions to a random neighbor of the current user with probability $\\gamma$, or jumps back to $u$ with probability $1 - \\gamma$. In our experiments, we execute the random walk for 200 iterations and set $\\gamma = 0.8$.\nHere, we do not claim that this is the most efficient way of generating recommendations in an online setup. It is merely a way to deal with incomplete data in federated social networks.\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Offline Evaluation}\nThe collaborative filtering approach shows a consistently higher performance than a topology-based system using PageRank (see~\\cref{tab:results-offline}).\nWith respect to the success at rank $k$ metric, profile-based approaches (R2--R4) have up to two times higher retrieval scores than PageRank (R5).\nThe individual profiling strategies perform all rather similarly, which aligns with the findings in~\\cite{Hannon:2010:RTU}.\nAlso, a baseline system (R1) which generates recommendations by selecting 100 random users from the network topology is outperformed by a large margin.\nIn \\cref{fig:precision-at-k}, it can be observed that shorter recommendation lists have a higher precision for all recommendation strategies.\nPrecision at rank $k$ remains stable starting from a list length of $k = 50$ items.\nThis suggests that shorter lists are to be preferred in an online scenario.\n\n\\begin{table}[t]\n\\centering\n\\caption{Experimental results of offline evaluation. Significance for model in line $i > 1$ is tested against line $i - 1$.}\n\\label{tab:results-offline}\n\\begin{tabular}{@{}clllll@{}}\n\\toprule\n\\textbf{ID} & \\textbf{System}     & \\textbf{MAP} & \\textbf{s@1} & \\textbf{s@5} & \\textbf{s@10} \\\\ \\midrule\nR1          & Random              & 0.001       & 0.000            & 0.000            & 0.055         \\\\\nR2          & Profile (following) & \\textbf{0.019}$^{\\blacktriangle}$       & \\textbf{0.033}$^{\\blacktriangle}$        & 0.085$^{\\blacktriangle}$        & 0.152$^{\\blacktriangle}$         \\\\\nR3          & Profile (followers) & \\textbf{0.019}$^{\\circ}$       & 0.030$^{\\circ}$        & 0.100$^{\\circ}$        & 0.167$^{\\circ}$         \\\\\nR4          & Profile (combined)  & 0.018$^{\\circ}$       & \\textbf{0.033}$^{\\circ}$        & \\textbf{0.106}$^{\\circ}$        & \\textbf{0.173}$^{\\circ}$         \\\\\nR5          & Pers.\\ PageRank     & 0.014$^{\\circ}$       & 0.018$^{\\circ}$        & 0.061$^{\\blacktriangledown}$        & 0.082$^{\\blacktriangledown}$         \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.8\\columnwidth]{figures\/precisionatk-curve}\n\\caption{Precision for different recommendation list lengths ($k$) in offline evaluation.}\n\\label{fig:precision-at-k}\n\\end{figure}\n\nIt is important to mention that the list of possible suggestions from the profile-based recommender is smaller than the list from the PageRank recommender, which complicates the discussion.\nOnly visited nodes (see~\\cref{sec:dataset-statistics}) have been indexed in the document retrieval system.\nThis significantly reduces the pool size of possible users ($\\approx$3k).\nIn contrast, the PageRank recommender can suggest any user in the topology ($\\approx$255k).\nOne could overcome this issue as follows.\nEach user, regardless of whether or not it has been visited during the data collection, could be added to the search index.\nThen, incoming relationships can be inferred by inspecting the outgoing links of visited users.\nBy adding these relations as \\textit{followers} to the documents of unexplored nodes, the \\textit{following} strategy of the collaborative filtering can be applied.\nHowever, as no outgoing links are known for unexplored users, the \\textit{following} and \\textit{combined} strategies are not fully applicable.\nDue to time constraints, we did not further investigate this issue.\n\nFurthermore, it is worth to note that the chosen window of five days between $t_1$ and $t_2$ might not have been long enough to capture sufficient user activity.\nIn between the training snapshot at $t_1$ and the testing snapshot at $t_2$, six new connections were added to each user on average.\nThis gives rise to an interesting trade-off.\nFor a longer time span, one can capture larger amounts of activity within the network.\nIntuitively, more links will be added as users start to follow other users.\nHowever, the farther two snapshots are apart, the larger is the risk that the network deviates too much from the original structure.\nUsers might stop following other users or profiles could be deleted.\nMore severely, entire instances could become unavailable due to a temporary downtime, or they could even be discontinued.\nThis is a unique concern related to the distributed nature of federated social networks.\n\nFinally, we want to motivate which recommendation systems are evaluated in the online trial based on the results presented above.\nFrom~\\cref{tab:results-offline} it can be observed that the profile-based recommendation strategies perform rather similarly.\nHowever, the combined strategy (R4) performs best with respect to the success at rank 10 metric, which one seeks to maximize in an online system where 10 recommendations are presented to the user.\nTherefore, we pick R4 as the first recommendation system.\nAlthough the personalized PageRank recommender (R5) has a lower performance than the other profiling strategies, we expect that it produces valuable recommendations which are significantly different from the profile-based strategies.\nThis is due to the fact that it considers the network topology when generating recommendations.\nTherefore, we apply the balanced-interleaving evaluation to systems R4 and R5.\n\n\\subsection{Online Evaluation}\nThe online evaluation shows that neither the profile-based nor the topology-based system is superior (see~\\cref{tab:summary-online-evaluation}). Nineteen users participated in our online study. On average, they started to follow 1.8 users from our recommendations. For 5 users the profile-based approach performed best. For another 5 users, the topology-based approach performed best. For the remaining 9 users both system performed equally well, or no recommendation was followed.\nThe fact that valuable recommendations were generated that resulted in new followings shows that the two systems can be useful in practice.\nHowever, a larger group of participants is required to draw final conclusions on the recommender system performance.\n\n\\begin{table}[t]\n\\centering\n\\caption{Summary of online evaluation.}\n\\label{tab:summary-online-evaluation}\n\\begin{tabular}{@{}p{.7\\columnwidth}c@{}}\n\\toprule\n\\textbf{Characteristic}  & \\textbf{Value} \\\\ \\midrule\nNumber of participants   & 19 \\\\\nProfile-based recommender (R4) superior & 5\\\\\nPageRank recommender (R5) superior & 5\\\\\nDraw & 2 \\\\\nNo user interaction & 7 \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsection{Practical Considerations}\nThe generation of online recommendations turned out to be costly because the complete network data is not available.\nIn contrast to centralized social media, federated social networks do not have a single authority which stores data about the entire network graph.\nThe proposed method of crawling the vicinity of a target user at recommendation time (see~\\cref{sec:evaluation}) comes with a high overhead in network traffic and is not suitable for real-time systems that have to support large amounts of users.\nIn addition to that, the method is sensitive to the size of the vicinity.\nWe expect that a larger number of iterations yields a better picture of a user's vicinity, which in turn increases the quality of recommendations. However, an exploration of different parameter settings has been out of scope of this study.\nThe data collection issue is even more severe in the offline evaluation which requires large and representative samples of the entire network.\n\nTo reduce the overhead associated with crawling in an online setting, one might attempt to gradually construct a cached representation of the entire network graph.\nWhenever a recommendation is generated for a user, the vicinity is added to that graph. On subsequent recommendations, one might reuse parts of this network to avoid additional crawling.\nThis approach has two important issues that have to be considered. First, one has to address the question when parts of the network are considered to be out of date (i.e., when the cache expires). Second, and more importantly, such an approach seems to be in conflict with the intentions behind decentralization.\nBy constructing a database that aims to capture the entire network graph, one starts to centralize the data of a federated social network.\\looseness=-1\n\n\\section{Conclusion}\\label{sec:conclusion}\nUser recommendation algorithms commonly applied to centralized social media can be applied to incomplete data from federated social networks with the goal of developing an engaged community.\nWe showed that collaborative filtering-based recommenders outperform a topology-based recommender on a large unbiased sample of the federated social network Mastodon. The two recommenders outperform a random recommender by a large margin.\nA subsequent live user experiment on Mastodon using balanced interleaving shows that the two recommender approaches perform on par.\nAcquiring a sufficiently large snapshot of the network topology for offline recommendation proofed to be difficult and costly. Keeping the snapshot up-to-date needs constant re-sampling. Online recommendation was done by sampling the graph neighborhood for the current user.\n\nThere are several directions for future work. First, studying the extent to which incomplete data impacts the recommender performance may derive methods that are tailored towards federated social networks which operate with limited amounts of data.\nSecond, user recommendation algorithms in popular social media increasingly utilize user context information such as location data and interests.\nIt remains unclear how such data can be effectively acquired and utilized in federated social networks while preserving privacy.\nThird, BM25 might not be the best ranking function for the presented  recommender approach, and it should be compared to functions that also use popularity-based scoring.\nFinally, one may investigate how decentralized communication protocols such as ActivityPub can be extended to support community building algorithms while maintaining the notion of decentralized network data.\\looseness=-1\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s1:introduction}\n\nThe Advanced LIGO (aLIGO) observatories completed their first observing run ``O1''\nearly-2016, operating at a factor of $3-4$ higher gravitational-wave\n(GW) strain sensitivity than their first-generation \ncounterparts~\\cite{Shoemaker2009}.\nDuring O1, they made the first direct observation of gravitational \nwaves~\\cite{LIGOVirgo2016a}. Emitted by a pair of coalescing black holes, these\nwaves heralded an era of observational GW astrophysics as they traveled \nthrough Earth.\nTowards the end of this decade, we expect aLIGO to reach its design sensitivity.\nIn addition to the US-based efforts, we also expect the French-Italian detector Advanced\nVirgo~\\cite{aVIRGO,aVirgo2}, Japanese detctor KAGRA~\\cite{kagra,Somiya:2011np},\nand LIGO-India~\\cite{2013IJMPD..2241010U} to begin observing at comparable\nsensitivities within a few years. With a global network of sensitive GW\n observatories, we can expect GW astronomy to face significant developments over the\ncoming years.\n\n\nCoalescing compact binaries of stellar-mass black holes (BH) and\/or\nneutron stars (NS) are the primary targets for the second\ngeneration GW detectors~\\cite{Timmes:1995kp,Fryer:1999mi,RevModPhys.74.1015,\n2010ApJ...714.1217B,2010ApJ...715L.138B,Dominik:2014yma,Belczynski:2006zi,\n2012ApJ...749...91F,\nWex:1998wt,1991ApJ...379L..17N,Mandel:2015spa,Abbott:2016nhf}.\nA binary system of black holes was recently observed by \naLIGO~\\cite{LIGOVirgo2016a}. Previously, stellar-mass black holes had only \nbeen observed by inference in mixed binaries with stellar companion (through\nelectromagnetic observations of the companion)~\\cite{Lewin2010,\nRemillard:2006fc,Fragos:2010tm}.\nNeutron stars, on the other hand, have had numerous sightings. Thousands of\nelectromagnetically emitting neutron stars, or pulsars, have been \ndocumented~\\cite{Manchester:2004bp},\nin varied situations: as radio pulsars~\\cite{Lattimer:2012nd,Manchester:2004bp},\nin binary systems with a stellar companion~\\cite{1971ApJ...169L..23M,\nBond:2002eh,Lattimer:2012nd,Manchester:2004bp},\nand in binary neutron stars (BNS)~\\cite{Hulse:1975uf,Taylor:1982wi,\nWeisberg:2010zz,Lattimer:2012nd,Manchester:2004bp}.\nMixed binaries of black holes and neutron stars, is an astrophysically\ninteresting class of systems~\\cite{Wex:1998wt,\n1991ApJ...379L..17N,Janka1999,Fryer:2015jpa}, that has not yet been detected.\nWe expect to observe $\\mathcal{O}(10)$ mixed binaries per year with\naLIGO~\\cite{Abadie:2010cf}.\n\n\n\n\n\nNSBH binaries are of interest for multiple reasons. For instance,\nthey have been long associated with (as possible progenitors of) short\nGamma-ray Bursts (SGRBs)~\\cite{eichler:89,1992ApJ...395L..83N,moch:93,Barthelmy:2005bx,\n2005Natur.437..845F,2005Natur.437..851G,Shibata:2005mz,Paschalidis2014,\nTanvir:2013}. Depending on their equation of state (EoS), NSs can get disrupted by\nthe tidal field of their companion BHs. Once disrupted, most of the NS\nmaterial falls into the hole over an $\\mathcal{O}(1$ms$)$ time-scale,\nwith the rest partly getting ejected as unbound material\nand partly forming an accretion disk around the BH.\nThis short lived ($0.1-1s$) disk-BH system is hypothesized to drive SGRBs\nthrough the production of relativistic jets~\\cite{Foucart:2015a,\nLovelace:2013vma,Deaton2013,Foucart2012,Shibata:2005mz,Paschalidis2014}.\nHowever, whether or not such a system forms depends also on the nature of\nthe BH. Massive BHs (with $m_\\mathrm{BH}\\gtrsim 12M_\\odot$), as well as BHs with\nlarge retrograde spins, tend to swallow the NS whole without forming a\ndisk~\\cite{Foucart:2013psa}.\nOn the other hand, {\\it low-mass} BHs with $m_\\mathrm{BH}\\in[3M_\\odot, 12M_\\odot]$\\footnote{\nThe upper limit on BH mass that allows for NS disruption may very well be\nhigher, depending strongly on the magnitude of BH spin~\\cite{Foucart:2014nda}.},\ncan disrupt their companion NSs much before merger, forming long-sustained disks\nthat are required to sustain SGRBs~\\cite{Shibata:2007zm,2010PhRvD..81f4026F,\nLovelace:2013vma,Foucart:2014nda,Kawaguchi:2015}.\nA {\\it coincident} detection of both GWs and gamma-rays from an NSBH merger,\nwill provide us with a unique opportunity to confirm this hypothesized link\nbetween NSBH mergers and GRBs~\\cite{Abbott:2016wya}.\n\n\n\n\n\nAnother question that compact object mergers can help answer is `what is the \nnature of matter at nuclear densities supported by NSs'?\nA large fraction of past work aimed at measuring NS matter effects from GW\nsignals has consisted of inquiries about BNSs~\\cite{Lee1999a,Lee1999b,Lee2000,\noechslin:07,Read:2008iy,Markakis:2010mp,Markakis:2011vd,stergioulas:11,\nEast:2011xa,Lackey2014,Wade:2014vqa,Bauswein:2014qla}. In this paper, we will\ninstead focus on NSBHs.\nDuring the course of early inspiral, the tidal field of the BH produces a\ndeformation in its companion NS. The quadrupolar moment of the star associated\nwith this deformation also depends on its material properties, through an EoS-dependent\ntidal deformability parameter $\\Lambda_\\mathrm{NS}$. This induced quadrupolar moment\nchanges over the orbital time-scale, resulting in the emission of GWs in {\\it\ncoherence} with the orbital waves.\nThese waves draw more energy from the orbit and increase the inspiral rate (as\ncompared to an equivalent BBH)~\\cite{Flanagan2008}.\nCloser to merger, the strong tidal field of the BH can disrupt the NS. The\nquadrupolar moment of the disrupted binary system falls monotonically over a\nmillisecond time-scale~\\cite{Kyutoku:2010zd,Lackey:2013axa,Lovelace:2013vma,\nFoucart:2015a,Pannarale:2015jia}, resulting in the damping of GW amplitude.\nThis penultimate stage also depends strongly on the internal structure and energy\ntransport mechanism of the NS, and carries the strongest tidal signature in the\nGW spectrum~\\cite{Foucart:2014nda,Deaton2013}.\n\n\n\n\nGravitational waves emitted by coalescing NSBH binaries carry subtle hints of\nthe NS EoS from inspiral through to merger. During early inspiral, the tidal\ndephasing is relatively weak and has a frequency dependence equivalent to a\n$5^{th}$ Post-Newtonian (PN) order effect~\\cite{Vines2011}. Closer to merger,\na disruptive fate of the NS\ncan result in a strong suppression of GW emission above a cut-off \nfrequency~\\cite{Pannarale:2015jia}. Some past studies of tidal measurements\nwith NSBH binaries have used PN inspiral-only waveforms~\\cite{Maselli:2013rza}.\nIn doing so, however, they ignore (i) the merger signal which could contain significant\ninformation for NSBHs, and (ii) the errors due to unknown vaccum terms in PN \nwaveforms, which could dominate over the tidal terms themselves~\\cite{Barkett2015,\nYagi:2014}.\nSome other studies that account for merger effects via the use of complete\nnumerical simulations~\\cite{Foucart:2013psa}, are limited in the binary\nparameter space they sample.\nOthers, that do the same through the use of phenomenological waveform\nmodels~\\cite{Lackey2011,Lackey:2013axa} use the Fisher matrix to estimate\n$\\Lambda_\\mathrm{NS}$ measurement errors. Fisher matrix estimates may become\nunreliable at realistic signal-to-noise ratios (SNR)~\\cite{Vallisneri:2007ev},\nsuch as those as we might expect in the upcoming observing runs of GW\ndetectors~\\cite{Abadie:2010cf}, and we improve such studies with a\nfully Bayesian treatment of the problem here.\n\n\n\n\n\nIn this paper we study the measurability of neutron star's tidal deformability\nfrom realistic binaries of {\\it low}-mass BHs and NSs by aLIGO. We also probe\nhow tidal effects affect the estimation of other binary parameters for the same\nclass of systems. This study improves upon previous work in the following ways.\nFirst, we include tidal effects during inspiral and merger in a consistent\nway, by using the waveform model of Lackey {\\it et al.}~\\cite{Lackey:2013axa}\n(abbreviated henceforth to ``LEA'').\nSecond, we include the effect of black hole spin on tidal GW signals, in\naddition to the effect of BH mass, tidal deformability of the NS, and the SNR.\nThird, we perform a complete Bayesian analysis, instead of using the Fisher matrix\napproximation.\nAnd fourth, we explore how our measurement errors decrease as we gain information\nfrom multiple (realistic) events.\n\n\nWe now outline the main questions and results discussed in this paper.\nFirst, we probe the effect of ignoring tidal effects in\nthe recovery of non-tidal binary parameters, such as \ncomponent masses and spins. This is the case for current and planned aLIGO\nefforts.\nTo do so, we first use the enhanced-LEA (or ``LEA+'', see Sec.~\\ref{s2:waveforms})\nmodel to generate a set of realistic signals;\nand then use non-tidal (BBH) waveform filters to estimate the underlying\nbinary masses and spins with a Markov-chain Monte Carlo. Here and throughout,\nwe use the zero-detuning high-power design sensitivity curve~\\cite{Shoemaker2009}\nto characterize the expected detector noise.\nWe find that, for individual events, ignoring tidal effects will affect mass\nand spin-estimation only marginally; only for very loud signals (SNRs $\\gtrsim 30$)\nwill the systematic biases be large enough to exceed the underlying\nstatistical uncertainty. Furthermore, detection searches can ignore tidal\neffects without loss of sensitivity.\n\n\n\nSecond, we study the ability of aLIGO to constrain neutron star tidal \ndeformability with a single observation of an NSBH merger. For this, we\nuse the same setup for signal waveforms as before, but replace the filter\ntemplate model with one that includes tidal effects from inspiral\nthrough to merger (i.e. LEA+)~\\cite{Lackey:2013axa}. For most binaries with\nBH masses outside of the mass-gap $(2-5M_\\odot)$~\\cite{Bailyn:1997xt,\nKalogera:1996ci,Kreidberg:2012,Littenberg:2015tpa} and\/or realistic signal-to-noise\nratios (SNR), we find it difficult to put better than a factor of $2$ bound\non $\\Lambda_\\mathrm{NS}$ with a single observation. As we can see from\nFig.~\\ref{fig:TT_LambdaCIWidths90_0_Lambda_SNR}, it is only at SNRs \n$\\rho\\gtrsim 20-30$ (under otherwise favorable circumstances, such as a stiff\nequation of state) that we are able\nto bring this down to a $\\pm 75\\%$ bound on $\\Lambda_\\mathrm{NS}$. For signals louder\nthan $\\rho =30$, we can constrain $\\Lambda_\\mathrm{NS}$ to a much more meaningful degree\n(within $\\pm 50\\%$ of its true value).\nWhile this is discouraging at first, we turn to ask: what if we combine\ninformation from a population of low-SNR observations?\n\n\n\n\nThe EoS of matter at nuclear densities is believed to be universal among all\nneutron stars. The Tolman-Oppenheimer-Volkoff equation~\\cite{Tolman:1939jz,\nOppenheimer:1939ne,1934PNAS...20..169T}\nwould then predict that NS properties satisfy a universal relationship\nbetween $\\Lambda_\\mathrm{NS}$ and $m_\\mathrm{NS}$. As the final part of\nthis paper, we combine information from multiple observations of realistic NSBH\nsystems and perform a fully-Bayesian analysis of how our estimation of\n$\\Lambda_\\mathrm{NS}$ changes as we accumulate detections. This is similar to an earlier\nstudy~\\cite{DelPozzo:13} aimed at binary neutron stars.\nWe restrict ourselves to a population of NSs with masses clustered very tightly\naround $1.35M_\\odot$ (with a negligible variance), and negligible spins. We\nsample different nuclear EoSs by sampling entire populations fixing different\nvalues for the NS tidal deformability.\nFor all populations, we take source locations to be uniformly distributed in\nspatial volume, and source orientations to be uniform on the $2-$sphere. To\nsummarize, we find the following:\n(a) Our median estimate for $\\Lambda_\\mathrm{NS}$ starts out prior dominated, but \nconverges to within $10\\%$ of the true value within $10-20$ detections.\n(b) Measurement uncertainties for $\\Lambda_\\mathrm{NS}$, on the other hand, depend on\n$\\Lambda_\\mathrm{NS}$ itself. We find that for hard equations of state (with \n$\\Lambda_\\mathrm{NS}\\geq 1000$), $10-20$ observations are sufficient to constrain\n$\\Lambda_\\mathrm{NS}$ within $\\pm 50\\%$. For softer equations of state, the same level\nof certainty would require substantially more ($25-40$) observations.\n(c) Further, if the astrophysical ``mass-gap''~\\cite{Bailyn:1997xt,\nKalogera:1996ci,Kreidberg:2012,Littenberg:2015tpa} is real, we find that $20-50\\%$\nadditional observations would be required to attain the same measurement\naccuracy as above. (d) Putting tighter constraints on the $\\Lambda_\\mathrm{NS}$ of a\npopulation would require $50+$ NSBH observations, in any scenario.\nAnd, (e) it is the loudest $5-10$ events that will furnish the bulk\nof tidal information, and not the combination of a large number of \nlow-SNR events.\nAll of the above is possible within a few years of design\naLIGO operation~\\cite{Abadie:2010cfa}.\n\n\n\n\nIn this paper, we restrict our parameter space to span mass-ratios\n$q:=m_\\mathrm{BH}\/m_\\mathrm{NS}\\in[2,5]$, dimensionless BH spin (aligned with orbit)\n$\\chi_\\mathrm{BH}\\in[-0.5, +0.75]$, and dimensionless NS tidal deformability \n$\\Lambda_\\mathrm{NS}:= G\\left(\\frac{c^2}{G m_\\mathrm{NS}}\\right)^5\\lambda \\in[500, 2000]$.\nThese ranges are governed by the calibration of the LEA+ model which we use as\nfilters. \nMost of the disruptive NSBH simulations that LEA+ has been calibrated to\ninvolve $1.35M_\\odot$ NSs, and it is unclear how reliable the model is \nfor different NS masses~\\cite{Lackey:2013axa,Pannarale:2015jka}. This\nmotivates us to conservatively fix NS masses to $1.35M_\\odot$ in our \nsimulated signals (not templates). But, since the domain of calibration of\nLEA+ excludes NS spin completely, we fix $\\chi_\\mathrm{NS}=0$ in both signals as well as\nfilter templates. We expect the effect of ignoring NS mass and spin variations\nin our NSBH populations to be less severe than for\nBNSs~\\cite{Agathos:2015a}, considering the higher mass-ratios of NSBHs.\nThe accuracy of our quantitative results depends on the reliability of LEA+,\nwhich is the only model of its kind in current literature. A more recent \nwork~\\cite{Pannarale:2015jka} improves upon the amplitude description of LEA+,\nbut needs to be augmented with a compatible phase model. Overall, we expect\nour broad conclusions here to hold despite modeling inaccuracies (with errors\nnot exceeding $\\mathcal{O}(10\\%)$~\\cite{Pannarale:2015jka}).\nFinally, our results apply to LIGO instruments at design sensitivity,\nwhich they are projected to attain by $2019$~\\cite{Shoemaker2009,\nAbbott:2016wya}.\n\n\nThe remainder of the paper is organized as follows. \nSec.~\\ref{s1:techniques} discusses data analysis techniques and resources \nused in this paper, such as the waveform model, and parameter estimation \nalgorithm.\nSec.~\\ref{s1:PEwithnoNS} discusses the consequences of ignoring tidal \neffects in parameter estimation waveform models.\nSec.~\\ref{s1:PEwithNS} discusses the measurability for the leading order\ntidal parameter $\\Lambda_\\mathrm{NS}$ at plausible SNR values.\nSec.~\\ref{s1:multiple_observations} discusses the improvement in our\nmeasurement of $\\Lambda_\\mathrm{NS}$ with successive (multiple) observations of\nNSBH mergers.\nFinally, in Sec.~\\ref{s1:discussion} we summarize our results and discuss\nfuture prospects with Advanced LIGO.\n\n\n\n\n\n\\section{Techniques}\\label{s1:techniques}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim=20 18 18 18 0,clip=true,width=1.8\\columnwidth]{SingleSystemEta_q4_0_mc2_25_chi0_50}\\\\\n\\caption{{\\bf Illustrative posterior probability distributions for mass-ratio $\\eta$ at different SNR values:}\nWe show here probability distributions for mass ratio $\\eta$ as measured\nfor the same signal at different SNRs. The intrinsic parameters of the source\nare: $q = m_\\mathrm{BH}\/m_\\mathrm{NS} = 5.4M_\\odot\/1.35M_\\odot = 4$, $\\chi_\\mathrm{BH}=+0.5$, and $\\Lambda_\\mathrm{NS}=2000$;\nand the signal is injected at SNRs $\\rho=\\{20,30,50\\}$ (left to right). The templates\n{\\it ignore} tidal effects.\nIn each panel: the dashed red line marks the median value\n$\\eta^\\mathrm{Median}$, while the dashed green line show the true value\n$\\eta^\\mathrm{Injected}$. The darker shading shows\nthe recovered $90\\%$ credible interval for $\\eta$, $(\\Delta\\eta)^{90\\%}$.\nComparing systematic and statistical errors, we find that:\nat $\\rho=20$, $\\eta$ measurement is dominated by statistical\nerrors; at $\\rho=30$, the two become comparable; and \nfor louder signals ($\\rho\\simeq50$), the systematic errors dominate.\n}\n\\label{fig:SingleSystemEtaPDFvsSNR}\n\\end{figure*}\n\n\n\\subsection{Waveform Models}\\label{s2:waveforms}\n\n\n\nLackey {\\it et al.}~(LEA)~\\cite{Lackey:2013axa} developed a complete inspiral-merger\nwaveform model for disrupting NSBHs. Theirs is a frequency-domain\nphenomenological model that includes the effect of BH and NS masses and spins\n$\\{m_\\mathrm{BH}, \\chi_\\mathrm{BH}, m_\\mathrm{NS}\\}\\equiv\\vec{\\theta}$ and NS tidal deformability\n$\\Lambda_\\mathrm{NS}$. It was calibrated to a suite of $134$ numerical relativity (NR)\nsimulations of NSs inspiraling into spinning BHs, with\nNS masses ranging between $1.2M_\\odot\\leqm_\\mathrm{NS}\\leq 1.45M_\\odot$,\nmass-ratios $2\\leq q\\leq 5$, and BH spins $-0.5\\leq\\chi_\\mathrm{BH}\\leq+0.75$.\nThey also sample a total of $21$ two-parameter nuclear EoSs to cover the\nspectrum of NS deformability.\nThe GW strain $\\tilde{h}(f)$ per the LEA model can be written as\n\\begin{equation}\n \\tilde{h}_\\mathrm{NSBH}(f, \\vec{\\theta}, \\Lambda_\\mathrm{NS}) = \\tilde{h}_\\mathrm{BBH}(f, \\vec{\\theta})\\,A(f, \\vec{\\theta}, \\Lambda_\\mathrm{NS})\\,e^{\\mathrm{i} \\Delta\\Phi(f, \\vec{\\theta}, \\Lambda_\\mathrm{NS})},\n\\end{equation}\nwith NS spin $\\chi_\\mathrm{NS}=0$ identically. Here, $\\tilde{h}_\\mathrm{BBH}$ is\nan underlying BBH waveform model. In the original LEA model,\nthis was taken to be the SEOBNRv1\nmodel~\\cite{Taracchini:2012} of the Effective-one-body (EOB)\nfamily~\\cite{Buonanno99}. The factor $A(\\cdot)$ adjusts\nthe amplitude of the BBH model to match that of an NSBH merger of otherwise\nidentical parameters, with NS-matter effects parametrized by $\\Lambda_\\mathrm{NS}$.\nDuring early inspiral this term is set to\nunity, but is a sensitive function of $\\Lambda_\\mathrm{NS}$ close to merger. The term with\n$\\Delta\\Phi$ corrects the waveform phasing. During inspiral,\n$\\Delta\\Phi$ is set to the PN tidal phasing corrections,\nat the leading and next-to-leading orders~\\cite{Vines2011}; close to merger,\nadditional phenomenological terms are needed. Both $A$ and $\\Delta\\Phi$ are\ncalibrated to all $134$ available NR simulations.\n\n\nIn this paper we use LEA for our signal and template modeling, but switch the \nunderlying BBH model to SEOBNRv2 (and refer to it as enhanced-LEA or\n``LEA+'')~\\cite{Taracchini:2013rva}. We using the reduced-order\nfrequency-domain version of SEOBNRv2, which has the additional benefit of\nreducing computational cost~\\cite{Purrer:2015tud}. We expect this enhancement\nfrom LEA$\\rightarrow$LEA+ to make our conclusions more robust because: (a) the \nSEOBNRv2 model is more accurate~\\cite{Kumar:2015tha,Kumar:2016dhh}, and (b)\nthe differences between the two EOB models are caused by the\ninaccuracies of SEOBNRv1 during the {\\it inspiral} phase, many orbits before \nmerger~\\cite{Kumar:2015tha}.\nSince LEA only augments inspiral phasing with PN tidal terms, our\nchange in the underlying BBH model does not change LEA's construction, {\\it and}\nincreases the overall model accuracy during inspiral.\nFinally, we note that we approximate the full GW signal with its dominant\n$l=|m|=2$ modes, that are modeled by LEA+. For use in future LIGO science\nefforts, we have implemented the LEA+ model in the LIGO Algorithms\nLibrary~\\cite{LAL}.\n\n\\subsection{Bayesian methods}\\label{s2:bayesian}\n\n\n\nThe process of measuring systematic and statistical measurement errors\ninvolves simulating many artificial GW signals, and inferring source binary\nparameters from them using Bayesian statistics.\nWe start with generating a signal waveform, using the model LEA+, and injecting\nit in zero noise to obtain a stretch of data $d_n$. Source intrinsic parameters\n$\\vec{\\Theta}:=\\{m_\\mathrm{BH},m_\\mathrm{NS},\\chi_\\mathrm{BH},\\Lambda_\\mathrm{NS}\\}$ are reconstructed from this\ninjected signal. Extrinsic parameters $\\vec{\\theta}:=\\{t_c,\\phi_c\\}$ \nrepresenting the time of and phase at the arrival of signal are marginalized\nover numerically and analytically (respectively), while source location and\norientation parameters such as its luminosity distance, sky location, inclination\nand polarization angles are absorbed into a normalization, as describe later,\nand subsequently maximized over. This is justified because in this paper we\nconsider the {\\it single-detector case}. Using Bayes' theorem, the joint inferred\nprobability  distribution of $\\vec{\\Theta}$ can be evaluated as\n\\begin{equation}\\label{eq:postprob}\n p(\\vec{\\Theta} | d_n, H) = \\dfrac{p(d_n|\\vec{\\Theta}, H)\\,p(\\vec{\\Theta} | H)}{p(d_n|H)}.\n\\end{equation}\nHere, $p(\\vec{\\Theta} | H)$ is the {\\it a priori} probability of binary parameters\n$\\vec{\\Theta}$\ntaking particular values, given $H$ - which denotes all our collective knowledge,\nexcept for expectations on binary parameters that enter\nour calculations explicitly. Throughout this paper, we impose priors that are\nuniform in individual component masses, BH spin, and the tidal deformability of\nthe NS. In addition, we restrict mass-ratios to $q\\geq 2$, as LEA+ is not\ncalibrated for $1\\leq q\\leq 2$. $p(d_n|\\vec{\\Theta}, H)$ is the {\\it likelihood}\nof obtaining the given stretch of data $d_n$ if we assume that a\nsignal parameterized by $\\vec{\\Theta}$ is buried in it, and is given by\n\\begin{equation}\\label{eq:likelihood}\n p(d_n| \\vec{\\Theta}, H) \\equiv \\mathcal{L}(\\vec{\\Theta}) = \\mathcal{N}\\, \\mathrm{exp}[-\\frac{1}{2} \\langle d_n - h | d_n - h\\rangle ],\n\\end{equation}\nwhere $h\\equiv h(\\vec{\\Theta})$ is a filter template with parameters \n$\\vec{\\Theta}$, $\\langle\\cdot|\\cdot\\rangle$ is a suitably defined\ndetector-noise weighted inner-product\\footnote{The inner product\n$\\langle\\cdot|\\cdot\\rangle$  is defined as\n\\begin{equation}\n\\langle a|b\\rangle \\equiv 4\\,\\mathrm{Re}\\left[\\int_0^\\infty \\dfrac{\\tilde{a}(f) \\tilde{b}(f)^*}{S_n(|f|)}\\,\\mathrm{d} f\\right],\n\\end{equation}\nwhere $\\tilde{a}(f)$ is the Fourier transform of the finite time series $a(t)$,\nand $S_n(|f|)$ is the one-sided amplitude spectrum of detector noise. In this\nwork, we use the zero-detuning high-power design sensitivity curve~\\cite{\nShoemaker2009} for Advanced LIGO, with $15$Hz as the lower frequency cutoff.},\nand $\\mathcal{N}$ is the normalization constant that absorbs source distance,\norientation and sky location parameters. As in Ref.~\\cite{Purrer:2015nkh} we\nuse a likelihood that is maximized over the template\nnorm, allowing us to ignore the extrinsic parameters that only enter in the\ntemplate norm through $\\mathcal{N}$. As a result, we only need to sample over\n$\\vec{\\Theta}$ (or $\\vec{\\Theta} - \\{\\Lambda_\\mathrm{NS}\\}$ in the case of non-tidal\ntemplates).\nThe denominator in Eq.~\\ref{eq:postprob} is the {\\it a priori} probability of finding\nthe particular signal in $d_n$ and we assume that each injected signal is as\nlikely as any other. From the joint probability distribution\n$p(\\vec{\\Theta} | d_n, H)$ so constructed, extracting the measured probability distribution\nfor a single parameter (say $\\alpha$) involves integrating\n\\begin{equation}\\label{eq:marginalize}\n p(\\alpha | d_n, H) = \\int\\mathrm{d} \\vec{\\Theta}_\\alpha\\, p(\\vec{\\Theta} | d_n, H),\n\\end{equation}\nwhere $\\vec{\\Theta}_\\alpha$ is the set of remaining parameters, i.e.\n$\\vec{\\Theta}_\\alpha:=\\vec{\\Theta} - \\{\\alpha\\}$.\n\nWe use the ensemble sampler Markov-chain Monte-Carlo algorithm implemented in\nthe {\\tt emcee} package~\\cite{emcee}, to sample the probability distribution \n$p(\\vec{\\Theta} | d_n, H)$. We run 100 independent chains, each of which is\nallowed to collect 100, 000 samples and combine samples from chains that have\na Gelman-Rubin statistic~\\cite{gelman1992} close to unity. This procedure yields\nabout 10,000 independent samples.\nOne simplification we make to mitigate computational cost is to set the\nfrequency sampling interval to $\\Delta f=0.4$~Hz, which we find to be\nsufficient for robust likelihoods calculations in zero noise~\\cite{Purrer:2015nkh}.\nWe integrate\nEq.~\\ref{eq:marginalize} to obtain marginalized probability distributions\nfor the NS tidal deformability parameter: $p(\\Lambda_\\mathrm{NS}|d_n,H)$. We will quote\nthe median value of this distribution as our {\\it measured} value for\n$\\Lambda_\\mathrm{NS}$, and the $90\\%$ credible intervals associated with the distribution\nas the statistical error-bars.\n\n\n\n\n\\section{How is PE affected if we ignore NS matter effects?}\\label{s1:PEwithnoNS}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim=20 20 18 22 0,clip=true,width=1.95\\columnwidth]{TNMchirpBiasesOverCIWidths_CI90_0_Lambda_SNR30_70_linear}\n\\caption{{\\bf Ratio of systematic to statistical errors in measuring $\\mathcal{M}_c$, ignoring tidal effects:}\nWe show here the ratio of systematic and statistical\nmeasurement uncertainties for the binary chirp mass over the NSBH parameter \nspace. Each panel shows the same as a function of BH mass and spin. NS mass\nis fixed at $m_\\mathrm{NS}=1.35M_\\odot$, and its spin is set to zero. Down each column,\nwe can see the effect of the increasing tidal deformability of the NS at fixed\nSNR. Across each row, we can see the effect of increasing the signal strength\n(SNR), with the tidal deformability of the NS fixed. We show dashed contours\nfor $\\mathtt{R}_{\\mathcal{M}_c}=10\\%, 25\\%, 50\\%\\cdots$, with interleaving filled color\nlevels separated by $5\\%$.\nFor BBHs, the statistical errors dominate systematic ones for contemporary\nwaveform models~\\cite{Inprerp-LVC-WaveModels:2016,Kumar:2016dhh}. We find that\nits not much different for NSBH binaries, until we get to very high SNRs\n$\\rho\\gtrsim 70$.\n}\n\\label{fig:TN_chirpMassBias_vs_Lambda_SNR}\n\\end{figure*}\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[trim=20 20 18 21 0,clip=true,width=1.95\\columnwidth]{TNEtaBiasesOverCIWidths_CI90_0_Lambda_SNR20_50_linear}\n\\caption{{\\bf Ratio of systematic to statistical errors in measuring $\\eta$, ignoring tidal effects:}\nThis figure is similar to Fig.~\\ref{fig:TN_chirpMassBias_vs_Lambda_SNR}\nwith the difference that here we show the ratio of systematic and statistical\nerror sources for the symmetric mass-ratio $\\eta$ and not chirp mass. We find\nthat for fairly loud GW signals, at $\\rho\\simeq 50$, not including the\neffects of tidal deformation of the NS on GW emission can become the dominant\nsource of error for astrophysical searches with Advanced LIGO. However, for\nquieter signals with $\\rho\\leq 30$, it will have a negligible effect on the\nmeasurement of $\\eta$. We remind the reader that the SNRs here are always\nsingle detector values.\n}\n\\label{fig:TN_EtaBias_vs_Lambda_SNR}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[trim=20 20 18 20 0,clip=true,width=1.95\\columnwidth]{TNChiBHBiasesOverCIWidths_CI90_0_Lambda_SNR_linear}\n\\caption{{\\bf Ratio of systematic to statistical errors in measuring $\\chi_\\mathrm{BH}$, ignoring tidal effects:}\nThis figure shows the ratio of the systematic and statistical\nmeasurement errors for BH spins $\\mathtt{R}_{\\chi_\\mathrm{BH}}$. Information is arranged identically\nto Fig.~\\ref{fig:TN_chirpMassBias_vs_Lambda_SNR}, \nand~\\ref{fig:TN_EtaBias_vs_Lambda_SNR}, with the level spacing of filled contours\nincreased to $15\\%$.\nSimilar to the case of mass parameters, we find that below $\\rho\\approx 30$,\nignoring tidal effects in templates introduces minor systematic effects,\nwhich remain sub-dominant to the statistical measurement uncertainties.\n}\n\\label{fig:TN_BHspinBias_vs_Lambda_SNR}\n\\end{figure*}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim=10 20 8 18 0,clip=true,width=1.9\\columnwidth]{SingleSystemLambdaVary_q4_0_mc2_25_chi0_50_snr50}\n\\caption{{\\bf Illustrative posterior probability distributions for NS tidal\ndeformability $\\Lambda_\\mathrm{NS}$:}\nWe show here probability distributions recovered for the NS tidal\ndeformability parameter $\\Lambda_\\mathrm{NS}$ from three GW injections, with parameters:\n$q = m_\\mathrm{BH}\/m_\\mathrm{NS} = 5.4M_\\odot\/1.35M_\\odot = 4$, $\\chi_\\mathrm{BH}=+0.5$, and \n$\\Lambda_\\mathrm{NS}=\\{1000,1500,2000\\}$ from left to right. The injection SNR is fixed at\n$\\rho=50$. The templates {\\it include} tidal effects, with a prior $0\\leq\\Lambda_\\mathrm{NS}\\leq 4000$.\nIn each panel- the dashed red line marks the median value for\n$\\Lambda_\\mathrm{NS}$, and the dashed green line marks its {\\it true} value.\nThe darker shading shows the $90\\%$ credible interval, whose width\n$(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}$ is a direct measure of our statistical uncertainty.\nBy comparing the measurement uncertainty for these three injections, we see\nthat $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}$ grows very slowly with $\\Lambda_\\mathrm{NS}$. Therefore,\nthe fractional measurement error - $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}\/\\Lambda_\\mathrm{NS}$ -\ndecreases monotonically as $\\Lambda_\\mathrm{NS}$ increases (with signal strength fixed).\n}\n\\label{fig:SingleSystemLambdaPDFvsSNR}\n\\end{figure*}\nPast (and future) efforts with Advanced LIGO have used (or plan to use) BBH\nwaveform templates to search for and characterize NSBH sources. In doing so,\nthey ignore the signature of NS tidal effects on the emitted GWs. In this\nsection we present a fully Bayesian analysis of the effect of this\nsimplification on the recovery of non-tidal parameters from NSBH signals.\n\n\nWe inject LEA+ NSBH signals into zero noise, and run an MCMC sampler on\nthem using equivalent BBH templates (same model, tidal terms $\\rightarrow 0$).\nWe fix $m_\\mathrm{NS}=1.35M_\\odot$ and $\\chi_\\mathrm{NS}=0$, and explore a range of NS\nequations of state via the single tidal deformability parameter\n$\\Lambda_\\mathrm{NS}\\in\\{500, 800, 1000, 1500, 2000\\}$. Our injections also span a\nrectangular grid in the BH parameter space, with vertices at\n$q\\in\\{2,3,4,5\\}$, i.e.\n$m_\\mathrm{BH}\\in\\{2.7M_\\odot, 4.05M_\\odot, 5.4M_\\odot, 6.75M_\\odot\\}$, and BH spins\n$\\chi_\\mathrm{BH}\\in\\{-0.5, 0, +0.5, +0.75\\}$. \nFinally, we sample all other source-related parameters, that determine the\nsignal strength but not character\\footnote{For aligned-spin signals and\naligned-spin templates both, we only consider the contribution of the dominant\n$l=|m|=2$ waveform multipoles. This approximation has the additional benefit\nof combining the dependence of the waveforms on inclination, polarization\nand sky location angles, as well as on distance, into the luminosity\nor {\\it effective} distance. This quantity only appears as an overall scaling\nfactor, and therefore only affects signal strength~\\cite{Sathyaprakash:2009xs}.\n}, by sampling the SNR $\\rho\\in\\{20,30,50,70\\}$. Our choice of injection\nparameters here\nis motivated by two factors: (i) previous studies of the signatures of NS tidal\neffects on gravitational waves~\\cite{FoucartEtAl:2011,Foucart:2013psa,\nFoucart:2014nda} (which suggest that necessary conditions for the observation\nof tidal effects with aLIGO include high SNRs and a low-mass spinning companion\nBH); and (ii) technical constraints of our chosen LEA+ model~\\cite{Lackey:2013axa}.\nAt design sensitivity, if we expect $0.2-300$ NSBH detections a \nyear~\\cite{Abadie:2010cfa}, we can expect to see $0.02-25$ \n{\\it disruptive}\\footnote{We assume here that BH mass values are {\\it uniformly}\nlikely from $2M_\\odot$ to $\\sim 35M_\\odot$~\\cite{LIGOVirgo2016a}, but NSs are\ndisrupted in NSBH mergers only if $q\\leq 6$ and $\\chi_\\mathrm{BH}\\geq 0$~\\cite{Foucart:2014nda,\nFoucart:2013psa}.\n} NSBH mergers a year, of which we will have $0.005-7$ observations with \n$\\rho\\geq 20$, and $0.002-3$ a year with $\\rho\\geq 30$.\nTherefore, our injection parameters span a physically interesting subset of NSBH\nbinaries, that is {\\it also} likely observable in the near future. \nFor our Bayesian priors, we choose uniform distributions for both component\nmasses and black hole spin: $m_\\mathrm{BH}\\in[1.2,25]M_\\odot$; $m_\\mathrm{NS}\\in[1.2,3]M_\\odot$;\nand $-0.75\\leq \\chi_\\mathrm{BH}\\leq +0.75$.\n\n\n\nThe effect of ignoring\ntidal corrections in templates will manifest as a systematic shift\nof recovered median parameter values away from what they would be if we had used\ntidal templates with identical priors. \nIn zero noise, we expect the probability distributions recovered using tidal\ntemplates to be multi-dimensional Gaussians with the maximum likelihood\nparameter values approaching their true values. If the priors are not\nrestrictive, we expect the recovered median to also converge to the true value.\nHowever,\nthe LEA model imposes significantly more restrictive priors (both mass-ratio\nand spin) than SEOBNRv2~\\cite{Taracchini:2013rva,Lackey:2013axa}, which shifts\nthe median value of parameters recovered using {\\it our} tidal templates away\nfrom their true value. If we use LEA+ priors for our non-tidal templates, it\nwould add a caveat to our original question 'can we estimate non-tidal NSBH\nparameters with equivalent BBH templates'. Instead, we approximate the median\ntidally recovered parameters by their true injected values, as one would expect\nto recover with an ideal model for tidally disruptive NSBH mergers. With this\ncaveat, we estimate {\\it systematic} measurement bias\/errors as the\ndifferences between median and {\\it injected} parameter values. \nAs an illustration, in Fig.~\\ref{fig:SingleSystemEtaPDFvsSNR} we show the\nrecovered probability distributions for binary mass ratio $\\eta$ for\nthree NSBH injections, with $\\rho=20$ (left), $30$ (middle), and $\\rho=50$\n(right), and other parameters held fixed ($m_\\mathrm{NS}=1.35M_\\odot$, $\\chi_\\mathrm{NS}=0$,\n$m_\\mathrm{BH}=5.4M_\\odot$, $\\chi_\\mathrm{BH}=+0.5$ and $\\Lambda_\\mathrm{NS}=2000$). In each panel, both\nthe {\\it true} and median values of $\\eta$ are marked, and we use\nthe shift between the red and green vertical lines as our estimate of systematic\nmeasurement errors. Darker shading in all panels marks $90\\%$ credible intervals,\nwhose width $(\\Delta\\eta)^{90.0\\%}$ we use as a direct measure of our\n{\\it statistical} measurement uncertainty\/error\\footnote{We generalize the\nnotation $(\\Delta X)^{90.0\\%}$ to mean the $90\\%$ credible interval width\nfor any measured source parameter $X$.}.\nFor the illustrated binary,\nwe see clearly that even when the signal is moderately loud, with $\\rho=20$,\nstatistical errors dominate over systematics for $\\eta$. As we turn up the SNR\nfurther, the two error sources become comparable at $\\rho\\sim 30$,\nand systematic errors dominate finally when $\\rho\\simeq 50$.\n\n\n\n\n\nCredible intervals $(\\Delta X)^{90\\%}$ showing the precision with which\n$X=\\{\\mathcal{M}_c,\\eta,\\chi_\\mathrm{BH}\\}$ can be measured, are presented in\nAppendix~\\ref{as1:nontidalerrors}.\nWe remind ourselves that this {\\it precision} is only meaningful so long as the\nmeasurement is {\\it accurate} to begin with. Therefore, we define $\\mathtt{R}_X$ as\nthe ratio between systematic and statistical errors associated with the\nmeasurement of parameter $X$,\n\\begin{equation}\\label{eq:arr}\n\\mathtt{R}_X = \\dfrac{(X^\\mathrm{Median} - X^\\mathrm{Injected})}{(\\Delta X)^{90\\%}},\n\\end{equation}\nin order to compare the relative magnitude of both. Only when\n$|\\mathtt{R}_X| \\ll 1$ can we ignore tidal effects in our templates\nwithout hampering the measurement of non-tidal parameters from NSBH signals.\nWhen $\\mathtt{R}_X$ approaches a few tens of percent of unity, we can begin to favor\ntidal templates for NSBH studies.\n\n\nWe start with calculating $\\mathtt{R}_{\\mathcal{M}_c}$ as a function of various source\nparameters and show it in Fig.~\\ref{fig:TN_chirpMassBias_vs_Lambda_SNR}.\n$\\mathcal{M}_c$ is the leading order\nmass combination that affects the GW strain emitted by compact binaries as they\nspiral in, and is therefore determined the most precisely. We notice\nimmediately that for $\\rho\\leq 30$ the systematics are well under control\nand we can obtain reliable chirp mass estimates for NSBH signals using BBH\ntemplates.\nFor louder and less likely SNRs ($\\rho\\simeq 50$), we find that\n$\\mathtt{R}_{\\mathcal{M}_c}$ can become comparable to unity, but only if the BH has\nprograde spin $\\chi_\\mathrm{BH}\\gtrsim 0.4$, {\\it and} the true NS tidal deformability\nis large enough, s.t. $\\Lambda_\\mathrm{NS} \\gtrsim 1000$.\nWe therefore conclude that only for very loud signals, with\n$\\rho\\gtrsim 50-70$, will the inclusion of tidal terms in template\nmodels improve $\\mathcal{M}_c$ estimation. For lower SNRs, inclusion of new\nphysical content in templates will instead get washed out by detector noise.\nIn addition, we also note that $\\mathtt{R}_{\\mathcal{M}_c}\\geq 0$ always,\ni.e. $\\mathcal{M}_c$ is always being over-estimated. This is to be expected since\nthe tidal deformation of the NS drains energy faster from the orbit during\ninspiral (as compared to the BBH case), and its disruption close to merger\nreduces GW signal power at high frequencies. Both of these effects make the\nresulting signal resemble a BBH signal of higher chirp (or total) mass, although\nwe expect the latter effect to be dominant~\\cite{Pannarale:2011pk}.\n\n\n\nNext, in Fig.~\\ref{fig:TN_EtaBias_vs_Lambda_SNR}, we show the ratio of\nmeasurement errors $\\mathtt{R}_\\eta$ for the symmetric mass-ratio.\nGoing through the figure from left to right, we find that for realistic\nSNRs ($\\rho\\leq 30$) the systematics remain below statistical errors for\n$\\eta$ measurement. The worst case is of the most deformable NSs\n($\\Lambda_\\mathrm{NS} = 2000$), but even for them systematics in $\\eta$ are $2\\times$\nsmaller than the statistical measurement errors.\nMoving to louder signals with $\\rho\\simeq 50$, we find that for binaries of\nfairly deformable NSs ($\\Lambda_\\mathrm{NS}\\gtrsim 1500$) and low-mass BHs\n($m_\\mathrm{BH}\\leq 5M_\\odot$) that have prograde spins ($\\chi_\\mathrm{BH}\\gtrsim +0.4$), our\nmeasurement of mass-ratio can be seriously compromised by ignoring tidal\nphysics in template models. \nThis pattern is continued at even higher SNRs, as we can see from\nFig.~\\ref{fig:TN_EtaBias_vs_Lambda_SNR}. We therefore conclude that, even if\nunder moderate restrictions on BH and NS parameters, $\\rho=30-50$ is loud enough\nto motivate the use of tidal templates in aLIGO data analyses.\nIn addition, we also notice that, unlike for $\\mathcal{M}_c$, the median value of\n$\\eta$ is always {\\it lower} than its true value, which is what we expect if we\nwant BBH templates to fit NSBHs that disrupt and merge at lower frequencies.\n\n\n\nMoving on from mass to spin parameters, we now consider the measurement of BH\nspin angular momentum $\\chi_\\mathrm{BH}$. The ratio of systematic and statistical errors\nfor $\\chi_\\mathrm{BH}$ are shown in\nFig.~\\ref{fig:TN_BHspinBias_vs_Lambda_SNR}. The presentation of information in this \nfigure is identical to that of Fig.~\\ref{fig:TN_chirpMassBias_vs_Lambda_SNR}\nand~\\ref{fig:TN_EtaBias_vs_Lambda_SNR}. A diverging colormap is used so that both \nextremes of the colorbar range point to large systematic biases, while its zero (or\nsmall) value lies in the middle.\nFor the lowest SNR considered ($\\rho=30$), $\\chi_\\mathrm{BH}$ bias is about $2\\times$\nsmaller than its statistical measurement uncertainty, and is therefore mostly\nnegligible. Both do become somewhat comparable, but only when we have the most\ndeformable NSs in orbit around low-mass BHs. \nAt higher SNRs $(\\rho\\simeq50-70)$, we find that the systematics in $\\chi_\\mathrm{BH}$\nmeasurement can dominate completely, especially for binaries containing\nmass-gap violating BHs and\/or deformable NSs with $\\Lambda_\\mathrm{NS}\\geq1000$.\nFrom Fig.~\\ref{fig:TN_BHspinBias_vs_Lambda_SNR} we additionally note that when\nthe source spin magnitudes approach the highest allowed, i.e. at both extremes\nof the $x$-axes, $\\chi_\\mathrm{BH}\\times\\mathtt{R}_{\\chi_\\mathrm{BH}}<0$. This is to be expected because\nthe median of the recovered posterior distributions for $\\chi_\\mathrm{BH}$ can only get\npushed inwards from the boundaries.\n\n\nSummarizing these results, we find that irrespective of system parameters,\nbelow a signal-to-noise ratio of $30$, our measurements of mass and spin\nparameters of astrophysical NSBH binaries will remain limited by the intrinsic\nuncertainty due to instrument noise, and do not depend on whether we include\ntidal effects in template models. However, when the signal-to-noise ratio\nexceeds $30$ the systematic bias in binary mass and spin measurements become\ncomparable to and can exceed the uncertainty due to noise. Of the different\nnon-tidal parameters considered, we find that the measurement of $\\eta$\ndegrades worst (in a relative-error sense) due to the use of BBH templates in\ndeciphering an NSBH signal. Of all the sub-categories, we find that tidal\ntemplates could especially help with the parameter estimation of astrophysical\n{\\it mass-gap violating} NSBH binaries,\n\n\n\n\n\n\n\n\\section{What do we gain by using templates that include NS matter effects?}\\label{s1:PEwithNS}\n\\begin{figure*}\n\\centering    \n\\includegraphics[trim=35 21 15 21 0,clip=true,width=2.2\\columnwidth]{TTLambdaRawCIWidths90_0_Lambda_SNR}\n\\caption{{\\bf Statistical uncertainty in $\\Lambda_\\mathrm{NS}$ measurement:}\nHere we show the statistical uncertainty in the measurement of\n$\\Lambda_\\mathrm{NS}$. In each panel, the\nsame is shown as a function of the BH mass and spin, keeping $\\Lambda_\\mathrm{NS}$ and\ninjection's SNR $\\rho$ fixed (noted in the panel). Rows contain panels\nwith the same value of $\\Lambda_\\mathrm{NS}$, with $\\rho$ increasing from left to right.\nColumns contain panels with the same value of $\\rho$, with $\\Lambda_\\mathrm{NS}$ \nincreasing from top to bottom.\nContours at $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}=\\{50\\%, 75\\%, 100\\%, 150\\%, 200\\%\\}\\times\\Lambda_\\mathrm{NS}^\\mathrm{Injected}$ demarcate regions where we can constrain the\n$\\Lambda_\\mathrm{NS}$ parameter well (within a factor of two of the injected value).\nWe note that, as expected, the measurement accuracy for $\\Lambda_\\mathrm{NS}$ improves\nwith (i) increasing SNR, (ii) increasing $\\Lambda_\\mathrm{NS}$, (iii) increasing BH spin,\nand (iv) decreasing BH mass.\n}\n\\label{fig:TT_LambdaCIWidths90_0_Lambda_SNR}\n\\end{figure*}\n\\begin{figure}\n\\centering    \n\\includegraphics[trim=10 10 0 10 0,clip=true,width=1.05\\columnwidth]{TTSNRThresholdFor100LambdaMeasurement_BHspin_BHmass_Lambda2000_0_CI90_0}\n\\caption{\nWe show here, as a function of BH mass and spin, the {\\it minimum} signal\nstrength (SNR) required to constrain $\\Lambda_\\mathrm{NS}$ within an interval of width\nequal to $100\\%$ of its true value, i.e. with $\\pm 50\\%$ error-bars. The NS mass\nis fixed at $1.35M_\\odot$, spin at zero, and $\\Lambda_\\mathrm{NS}=2000$.\nWe can see that, even in the most conducive circumstances with large aligned \n$\\chi_\\mathrm{BH}$ and a comparable mass BH, we can only constrain $\\Lambda_\\mathrm{NS}$ to better\nthan $\\pm 50\\%$ {\\it if} the SNR is $\\gtrsim 29$. In the era of design\nsensitivity LIGO instruments, we expect this to happen approximately once in a\nyear of observation~\\cite{Abadie:2010cfa}.\n}\n\\label{fig:TT_SNRThresholds_BHspin_BHmass_CI90_0}\n\\end{figure}\n\n\nIn the previous section, we showed that the effects of the tidal deformation of\nNSs by their companion BHs become discernible in\nthe GW spectrum under certain favorable conditions, including (a) BH mass is\nsufficiently small, (b) BH spin is positive aligned, i.e. $\\chi_\\mathrm{BH}\\gtrsim +0.4$,\n(c) the NS is not very compact, with $\\Lambda_\\mathrm{NS}\\gtrsim 1000$, and (d) the\nsource location and orientation are such that its GW SNR $\\gtrsim 30$.\nBoth condition (a) and (b) enhance the tidal distortion of the star and increase\nthe number of orbits the system goes through at small separation, where the\ndifferences between NSBH and BBH signals are maximal.\nConditions (a)-(c) also reduce the onset frequency of the disruption of the NS,\nallowing for it to happen earlier in the orbit. \nWe expect that these conditions are also the ones which should maximize the\nlikelihood of {\\it measuring} tidal effects in NSBH signals. Here,\nwe turn the question around to ask: under similarly favorable circumstances,\ncan we gain insights about the internal structure of neutron stars from GW\nobservations?\n\n\n\nIn this section, we calculate the accuracy with which we measure $\\Lambda_\\mathrm{NS}$ from\n{\\it single} GW observations. We sample the same set of disruptive NSBH mergers\nas in the previous section, i.e. those with $q=\\{2,3,4,5\\}$,\n$\\chi_\\mathrm{BH}=\\{-0.5,0,+0.5,+0.75\\}$, and $\\Lambda_\\mathrm{NS}=\\{500, 800, 1000, 1500, 2000\\}$;\nfixing the NS mass $m_\\mathrm{NS}=1.35 M_\\odot$ and $\\chi_\\mathrm{NS}=0$. For each unique\ncombination of these parameters, we inject LEA+ signals into zero noise and\nperform a fully Bayesian parameter estimation analysis of each with LEA+\ntemplates. Our  priors on component masses and spins remain as in the\nprevious section, with mass-ratio additionally restricted to $2\\leq q\\leq 6$,\nand $\\Lambda_\\mathrm{NS}$ sampled uniformly from $[0,4000]$.\nAs an illustration of individual injections, we show the recovered probability\ndistribution for $\\Lambda_\\mathrm{NS}$ for three specific configurations in \nFig.~\\ref{fig:SingleSystemLambdaPDFvsSNR}. We fix\n$q = m_\\mathrm{BH} \/m_\\mathrm{NS} = 5.4M_\\odot\/1.35M_\\odot = 4$, with $\\chi_\\mathrm{BH}=+0.5$, and\nvary $\\Lambda_\\mathrm{NS}$ over $\\{1000, 1500, 2000\\}$ between the three panels.\nThe SNR is fixed at $\\rho=50$. The darker shaded regions mark the $90\\%$ credible\ninterval on $\\Lambda_\\mathrm{NS}$. We note that $\\Lambda_\\mathrm{NS}$ is estimated to within\n$\\pm 2000$ of its true value at this SNR. Another interesting thing to note \nis that while $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}$ slowly grows with $\\Lambda_\\mathrm{NS}$, the\nfractional uncertainty\n\\begin{equation}\n\\delta\\Lambda_\\mathrm{NS}^{90\\%}:= (\\Delta\\Lambda_\\mathrm{NS})^{90\\%}\/\\Lambda_\\mathrm{NS}\n\\end{equation}\ndecreases instead.\nFurther illustrations, showing the correlation between tidal and non-tidal\nparameters, are presented in Appendix~\\ref{as1:illustrations}.\nWe will continue here to focus on the measurement of $\\Lambda_\\mathrm{NS}$ itself.\n\n\n\n\nIn Fig.~\\ref{fig:TT_LambdaCIWidths90_0_Lambda_SNR} we show the main results of\nthis section. In each panel, as a function of black hole mass and spin, we show\nthe measured $90\\%$ credible interval widths $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}$. These\ncorrespond to the full width of the dark shaded regions in the illustrative\nFig.~\\ref{fig:SingleSystemLambdaPDFvsSNR}. The effect of increasing signal\nstrength can be seen as we go from left to right in each row. The effect of the\nNS tidal deformability parameter $\\Lambda_\\mathrm{NS}$ on its own measurability can be\nseen by comparing panels within each column, with the NS becoming more\ndeformable from top to bottom. \nA uniform pattern emerges in the left-most column, which corresponds to $\\rho=20$.\nWe find that at this signal strength, our measurement of $\\Lambda_\\mathrm{NS}$ is\ndominated by the width of our prior on it. The $90\\%$ credible intervals span\nthe entire allowed range for $\\Lambda_\\mathrm{NS}$, making a reasonable estimation of\n$\\Lambda_\\mathrm{NS}$ at $\\rho\\simeq20$ difficult.\nIncreasing the signal strength to $\\rho=30$ gives marginally better results,\nbringing down the statistical uncertainties to within $\\pm 75-100\\%$ of the\ntrue $\\Lambda_\\mathrm{NS}$ value~\\footnote{The symmetric error-bars of $\\pm\\mathrm{X}\\%$\ncorrespond to $\\delta\\lambdans^{90\\%} = 2\\mathrm{X}\\%$.}.\nIt is not until we reach an SNR as high as $\\rho\\gtrsim 50$, can we put\nmeaningful (i.e. $\\mathcal{O}(10\\%)$) constraints on $\\Lambda_\\mathrm{NS}$. For e.g.,\nwith a {\\it single} observation of a $q=4$ binary with $\\chi_\\mathrm{BH}\\geq 0.6$ and\n$\\rho = 50$~\\footnote{For an optimally oriented source with\n$q=4, m_\\mathrm{NS}=1.35M_\\odot, \\chi_\\mathrm{BH}=0.6$, an SNR of $\\rho = 50$ corresponds to\na luminosity distance of $\\approx 113$Mpc.}, we would be able to estimate \n$\\Lambda_\\mathrm{NS}$ to within $\\pm 40\\%$ of its true value (which is equivalent to\nmeasuring the ratio of NS radius to mass with an uncertainty of about\n$\\pm 10\\%$).\nThese results agree well with Sec.~\\ref{s1:PEwithnoNS}, and are consistent with\nFisher matrix estimates at high SNRs~\\cite{Lackey:2013axa}.\n\n\n\n\nAmongst other source parameters, BH mass and spin play a dominant role. A smaller\nBH with a larger spin always allows for a more precise measurement on $\\Lambda_\\mathrm{NS}$.\nWe can see this in the bottom right corner of each panel in\nFig.~\\ref{fig:TT_LambdaCIWidths90_0_Lambda_SNR}, which corresponds to low-mass BHs\nwith large spins, and is simultaneously the region of smallest measurement errors on $\\Lambda_\\mathrm{NS}$.\nThe actual deformability of the NS also plays an important role on its own\nmeasurability. For e.g., when $\\Lambda_\\mathrm{NS}\\leq 1000$, it is fairly difficult\nto meaningfully constrain $\\Lambda_\\mathrm{NS}$ without requiring the source to be\nclose ($\\approx 100$Mpc) with a GW SNR $\\rho\\gtrsim 50$. Quantifying this further,\nin Fig.~\\ref{fig:TT_SNRThresholds_BHspin_BHmass_CI90_0} we show the minimum\nsignal strength required to attain a certain level of credibility in our\n$\\Lambda_\\mathrm{NS}$ measurement, as a function of BH properties. The NS is allowed\nthe most favorable (hardest) EoS considered, with $\\Lambda_\\mathrm{NS}^\\mathrm{true}=2000$.\nWe first note that, even with the most favorable BH and NS properties, achieving\na $\\pm 50\\%$ measurement certainty on $\\Lambda_\\mathrm{NS}$ will require a GW SNR\n$\\rho\\gtrsim 30$. If we additionally restrict BH masses to lie outside of the so-called\nastrophysical mass-gap~\\cite{Bailyn:1997xt,Kalogera:1996ci,Kreidberg:2012,\nLittenberg:2015tpa}, we will simultaneously need to restrict BH spins\nto $\\chi_\\mathrm{BH}\\gtrsim +0.5$ to obtain the same measurement credibility at the same\nsource location.\n\n\nIt is interesting to note that the parameter ranges most favorable to\nthe measurability of $\\Lambda_\\mathrm{NS}$ are also those which produce\nrelatively more massive post-merger disks~\\cite{Foucart2012}. That is, \nthe subset of NSBHs that potentially produce SGRBs (using a sufficiently-large\ndisk mass as an indicator) would be the same subset most favorable\nfor measurement of tidal effects. Therefore the rate of SGRBs in the \nlocal universe (allowing for the fraction that are produced by NSBHs versus\nBNSs) would be an indicator of the rate of events most favorable for nuclear\nequation of state measurements.\n\n\nIn summary, with a single moderately loud ($\\rho\\lesssim 30$) GW signal from\na disruptive BHNS coalescence, we can constrain\nthe NS compactness parameter $\\Lambda_\\mathrm{NS}$ within $\\pm 100\\%$ of its true value.\nTo measure better with one observation, we will need a more fine-tuned source, with\n$\\rho\\geq 30$ and high BH spins, or $\\rho\\geq 50$.\nFinally, we note that these results are {\\it conservative}, and \nBHs with spins $\\chi_\\mathrm{BH} > 0.75$ will prove to be even more favorable laboratories\nfor $\\Lambda_\\mathrm{NS}$ measurement. However, we are presently unable to explore this case\nin quantitative detail due to waveform model restrictions~\\cite{Lackey:2013axa},\nwhich will also restrict our analyses of GW signals during the upcoming LIGO\nobserving runs.\n\n\n\n\\section{Combining observations: looking forward with Advanced LIGO}\\label{s1:multiple_observations}\n\\begin{figure}\n\\centering    \n\\includegraphics[trim=18 18 18 10 0,clip=true,width=\\columnwidth]{pdfLambda_vs_N_L800.pdf}\\\\\n\\includegraphics[trim=18 18 18 10 0,clip=true,width=\\columnwidth]{FillBetweenErrorBarsLambda_vs_N_L800.pdf}\n\\caption{{\\bf Recovery of $\\Lambda_\\mathrm{NS}$ for an increasingly large population of BH-NS signals.}\n{\\it Top}: Posterior probability distributions for $\\Lambda_\\mathrm{NS}$ (colored curves), and\nassociated $90\\%$ credible intervals (grey vertical lines), shown for different number\nof accumulated observations N. Distributions are normalized to unit area.  \n{\\it Bottom}: Measured median value of $\\Lambda_\\mathrm{NS}$ (as solid circles) and the\nassociated $90\\%$ credible intervals (as the vertical extent of filled region), shown as\na function of number of observations N. Solid horizontal line indicates the true value of\n$\\Lambda_\\mathrm{NS}=800$. Dashed and dotted horizontal lines (a pair for each line-style) demarcate\n$\\pm 25\\%$ and $\\pm 50\\%$ error bounds.\n}\n\\label{fig:TT_Lambda_vs_N_L800_CI90_0}\n\\end{figure}\n\\begin{figure}\n\\centering    \n\\includegraphics[width=1.05\\columnwidth,trim=1cm 0 0 0]{FillBetweenRelErrorBarsLambda_vs_NShifted_AllLambda.pdf}\n\\caption{{\\bf Improvement in $\\Lambda_\\mathrm{NS}$ measurement accuracy for different NS EoS:}\nIn this figure, the filled regions show how our measurement of $\\Lambda_\\mathrm{NS}$\nimproves as the number of observed events ($N$, shown on $x$-axis) increases.\nEach color corresponds to an independent population with its true value of\n$\\Lambda_\\mathrm{NS}$ given in the legend. For each population, we show the median \n$\\Lambda_\\mathrm{NS}$ value (as filled circles), as well as the associated\n$90\\%$ credible intervals for the measurement (as the vertical extent of the\nfilled region about the median), as functions of $N$.\n}\n\\label{fig:TT_Lambda_vs_N_CI90_0}\n\\end{figure}\n\\begin{figure*}\n\\centering\n\\includegraphics[trim=1cm 0 0 0, width=1.025\\columnwidth]{LambdaMedian_vs_N_AllPopulation}\n\\includegraphics[trim=0 0 1cm 0, width=1.025\\columnwidth]{LambdaMedian_vs_N_AstroPopulation}\\\\\n\\includegraphics[trim=1cm 0 0 0, width=1.025\\columnwidth]{LambdaMedian90pc_vs_N_AllPopulation}\n\\includegraphics[trim=0 0 1cm 0, width=1.025\\columnwidth]{LambdaMedian90pc_vs_N_AstroPopulation}\\\\\n\\caption{\n{\\bf No Mass-Gap}, {\\it top left}: The top figure shows the median value of the recovered\nprobability distribution for $\\Lambda_\\mathrm{NS}$, as a function of the number of observed \nevents $N$. There are four ensembles of curves,\ncorresponding to $\\Lambda_\\mathrm{NS}=\\{500,1000,1500,2000\\}$, with a hundred\nindependent population draws within each ensemble. One curve in each ensemble\nis highlighted in color, representing the realizations already plotted in\nFig.~\\ref{fig:TT_Lambda_vs_N_CI90_0}.\nIn the same color we show $\\pm 10\\%$ error-bounds on $\\Lambda_\\mathrm{NS}$ with\nhorizontal dash-dotted lines.\n{\\bf No Mass-Gap}, {\\it bottom left}: Here we show the interval of $\\Lambda_\\mathrm{NS}$ values within\nwhich the median $\\Lambda_\\mathrm{NS}$ lies for $90\\%$ of the populations in\neach ensemble shown in the top left panel.\nWe observe that within $10-25$ observations, the median of the measured \ncumulative probability distribution for $\\Lambda_\\mathrm{NS}$ converges to within $10\\%$\nof its true value.\n{\\bf Mass-Gap}, {\\it right column}: These panels are identical to their counterparts on the left,\nwith the only difference that the BH masses in each population are restricted\nto lie {\\it outside} the astrophysical mass-gap (i.e. paradigm B). The\ndifference that\nwe observe under this paradigm is that we need more ($30+$) events to achieve \nthe same ($10\\%$) measurement accuracy for populations with $\\Lambda_\\mathrm{NS}<1000$.\nFor more deformable neutron stars, $10-25$ events would suffice.\n}\n\\label{fig:TT_LambdaMedian_vs_N_AllInOne}\n\\end{figure*} \n\\begin{figure*}\n\\centering    \n\\includegraphics[width=1.025\\columnwidth,trim=1cm 0 0 0]{LambdaCIWidths90pc_vs_N_AllPopulation}\n\\includegraphics[width=1.025\\columnwidth,trim=0 0 1cm 0]{LambdaCIWidths90pc_vs_N_AstroPopulation}\n\\caption{{\\bf No Mass-Gap} ({\\it left}): This panel shows the width of $\\Lambda_\\mathrm{NS}$ interval within\nwhich the $90\\%$ credible intervals for $\\Lambda_\\mathrm{NS}$ lie, for $90\\%$ of \nthe populations in each ensemble, as a function of the number of observed events\n$N$. Details of how this is calculated are given in the text.\nThe populations are sampled under paradigm A, which allows BH masses to\nfall within the astrophysical mass-gap.\nEach panel corresponds to a unique value of populations' $\\Lambda_\\mathrm{NS}$,\ndecreasing from $2000\\rightarrow 500$ as we go from top to bottom.\nOne curve in each ensemble is highlighted in color (thin lines), representing the \nrealizations already plotted in Fig.~\\ref{fig:TT_Lambda_vs_N_CI90_0}.\n{\\bf Mass-Gap} ({\\it right}): This panel shows populations drawn under paradigm B, which\nrespects the mass-gap.\nWe find that with approximately $25$ or so events, we begin to put\nstatistically meaningful constraints on $\\Lambda_\\mathrm{NS}$, restricting it to within\n$\\pm 50\\%$ of the true value. We can expect to achieve this with a few years\nof design aLIGO operation~\\cite{Abadie:2010cfa}. Further tightening of \n$\\Lambda_\\mathrm{NS}$ credible intervals will require $40+$ events.\n}\n\\label{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne}\n\\end{figure*}\n\\begin{figure*}\n\\centering    \n\\includegraphics[width=1.025\\columnwidth,trim=1cm 0 0 0]{LambdaMedianCIWidths90pc_vs_N_AllPopulation_SNRSorted}\n\\includegraphics[width=1.025\\columnwidth,trim=0 0 1cm 0]{LambdaCIWidths90pc_vs_N_AstroPopulation_SNRSorted}\n\\caption{This figure is similar to Fig.~\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne},\nwith the only difference being that events in each population\nhave been sorted according to their signal strength (SNR), instead of\ntheir simulated chronology. We note that information about\nthe tidal deformability of neutron stars comes primarily \nfrom the loudest $5-10$ events, whether we allow BH masses\nin the mass gap (left panel) or restrict them to\n$m_\\mathrm{BH}\\geq 5M_\\odot$ (right panel). Left inset zooms\nin on the main figure for the first $15$ events. Right inset\nshows the actual (ensemble mean) SNR value for each event. We find that\nevents with $\\rho\\gtrsim 20-30$ provide the bulk of tidal information\nin our analysis.\n}\n\\label{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne_SNRSorted}\n\\end{figure*}\n\n\nIn the previous section, we showed that single observations of NSBH\ncoalescences at moderate SNRs have little information about the internal\nstructure of neutron stars that will be accessible to Advanced LIGO at its\ndesign sensitivity. We expect all neutron stars to share the same equation of\nstate, and hence the same $\\Lambda_\\mathrm{NS}(m_\\mathrm{NS})$. In addition, we know that the mass\ndistribution of (most) NSs that have not been spun up to millisecond periods\n(which are the ones we focus on in this paper, by setting $\\chi_\\mathrm{NS}\\approx 0$) is\nnarrowly peaked around $\\sim 1.35M_\\odot$~\\cite{Kiziltan2013}. Therefore,\ninformation from multiple NSBH observations can be combined to improve our\nestimation of $\\Lambda_\\mathrm{NS}$. We explore the same in this section within a fully\nBayesian framework. We refer the reader to Ref.~\\cite{Mandel:2009pc,Lackey2014,\nWade:2014vqa} for similar analyses of BNS inspirals.\n\n\n\n\nAn intuitive understanding of the problem is gained by considering first\nmultiple {\\it identical} sources with realistic but different SNRs. Let us consider the case\nof a population of optimally oriented binaries~\\footnote{An optimally oriented\nbinary is one which is located directly overhead the detector, with the \norbital angular momentum parallel to the line joining the detector to the\nsource. Such a configuration maximizes the observed GW signal strength in \nthe detector.}, distributed uniformly in spatial volume out\nto a maximum {\\it effective} distance~\\footnote{{\\it effective} distance $D$ \nis a combination of distance to the source, its orientation, and its sky\nlocation angles; and has a one-to-one correspondence with SNR for non-precessing\nsources. This is so because for such sources, their location and orientation\nremain constant over the timescales within which they sweep through\naLIGO's sensitive frequency band.}.\n$D^\\mathrm{max}$. $D^\\mathrm{max}$ is set by the minimum SNR \nthreshold $\\rho_\\mathrm{min}$ at which a source is considered\ndetectable~\\footnote{which\nwe take as $\\rho_\\mathrm{min}=10$ throughout.}. Next, we divide this volume into $I$\nconcentric shells, with radii $D_i$. If we have a measurement error\n$\\sigma_0$ for $\\Lambda_\\mathrm{NS}$, associated with a source located at $D=D_0$,\nthe same error for the same source located within the $i-$th shell would\nbe $\\sigma_i=\\sigma_0 \\dfrac{D_i}{D_0}$. Ref.~\\cite{Markakis:2010mp}\ncalculated that the combined error $\\sigma$ from $N$ independent\nmeasurements of $\\Lambda_\\mathrm{NS}$ in such a setting to be\n\\begin{align}\\label{eq:1oversigma}\n\\frac{1}{\\sigma^2} =& \\sum_{i=1}^I \\frac{N_i}{\\sigma_i^2} = \\left(\\frac{D_0}{\\sigma_0}\\right)^2 \\sum_{i=1}^I\\frac{N_i}{D_i^2}\\\\ \\nonumber =& \\left(\\frac{D_0}{\\sigma_0}\\right)^2 \\int_0^{D^\\mathrm{max}} \\dfrac{4\\pi D^2 n}{D^2}\\mathrm{d} D = \\left(\\frac{D_0}{\\sigma_0}\\right)^2 \\dfrac{3N}{(D^\\mathrm{max})^2},\n\\end{align}\nwhere $N_i$ is the number of sources within the $i-$th shell (s.t.\n$N:=\\sum N_i$), and $n$ is the number density of sources in volume.\nThe root-mean-square (RMS) averaged measurement error from $N$ sources is \nthen~\\cite{Markakis:2010mp}\n\\begin{equation}\\label{eq:rmsSigmaIdenticalSources}\n \\sigma_{avg} := \\frac{1}{\\sqrt{1\/\\sigma^{2}}} = \\frac{\\sigma_0}{D_0} D^\\mathrm{max} \\frac{1}{\\sqrt{3 N}},\n\\end{equation}\ngiven a fiducial pair $(\\sigma_0, D_0)$. It is straightforward to deduce from\nEq.~\\ref{eq:rmsSigmaIdenticalSources} that measurement uncertainty scales as \n$1\/\\sqrt{N}$, and the uncertainty afforded by a single observation with a high\nSNR $\\rho_c$ can be attained with $N = \\rho_c^2\/300$ realistic observations\nthat have $\\rho\\geq\\rho_\\mathrm{min}$. E.g., to get to the\nlevel of certainty afforded by a single observation with $\\rho=70$, we would\nneed $49\/3\\approx 16-17$ realistic (low SNR) detections.\n\nWhile we discussed Eq.~\\ref{eq:rmsSigmaIdenticalSources} for a population\nof optimally oriented sources, it is valid for a more general population\ndistributed uniformly in effective volume~\\cite{Markakis:2010mp}\n($\\propto D^3$).\nHowever, it \nstill only applies to sources with identical masses and spins, and we \novercome this limitation by performing a fully Bayesian analysis next.\n\n\n\\textbf{Astrophysical source population: }\\label{s2:astro_multiple}\nImagine that we have $N$ stretches of data, $d_1, d_2, \\cdots, d_N$, each \ncontaining a single signal emitted by an NSBH binary. Each of these signals can\nbe characterized by the non-tidal source parameters\n$\\vec{\\theta} := \\{m_\\mathrm{BH}, m_\\mathrm{NS}, \\chi_\\mathrm{BH}, \\chi_\\mathrm{NS}, \\vec{\\alpha}\\}$,\nand $\\{\\Lambda_\\mathrm{NS}\\}$, where $\\vec{\\alpha}$ contains extrinsic parameters,\nsuch as source distance, inclination, and sky location angles.\nAs before, let $H$ denote all of our collective prior knowledge; for instance,\n$H$ includes our assumption that all NSs in a single population have the same\ndeformability parameter $\\Lambda_\\mathrm{NS}$, and that its cumulative measurement is\ntherefore possible.\nThe probability distribution for $\\Lambda_\\mathrm{NS}$, given $N$ unique and\nindependent events, is\n\\begin{eqnarray}\n p(\\Lambda_\\mathrm{NS} |&& \\hspace{-4mm}d_1, d_2, \\cdots, d_N, H)\\hspace{50mm}\\nonumber\\\\ &=& \\dfrac{p(d_1,d_2,\\cdots,d_N |\\Lambda_\\mathrm{NS} , H)\\,p(\\Lambda_\\mathrm{NS}|H)}{\\int p(\\Lambda_\\mathrm{NS} |H) p(d_1,d_2,\\cdots,d_N |\\Lambda_\\mathrm{NS} , H)\\mathrm{d}\\Lambda_\\mathrm{NS}},\\label{eq:p11}\\\\\n  &=& \\dfrac{p(\\Lambda_\\mathrm{NS}|H) \\prod_{i} p(d_i|\\Lambda_\\mathrm{NS}, H)}{\\int p(\\Lambda_\\mathrm{NS} ) p(d_1,d_2,\\cdots,d_N |\\Lambda_\\mathrm{NS} , H)\\mathrm{d}\\Lambda_\\mathrm{NS}},\\label{eq:p12} \\\\\n  &=& \\dfrac{p(\\Lambda_\\mathrm{NS}|H) \\prod_{i} \\left( p(\\Lambda_\\mathrm{NS} |d_i, H)\\dfrac{p(d_i)}{p(\\Lambda_\\mathrm{NS}|H)} \\right)}{\\int\\, p(\\Lambda_\\mathrm{NS}|H )\\, p(d_1,d_2,\\cdots,d_N |\\Lambda_\\mathrm{NS} , H)\\mathrm{d}\\Lambda_\\mathrm{NS}}\\label{eq:p13};\n\\end{eqnarray}\nwhere Eq.~\\ref{eq:p11} and Eq.~\\ref{eq:p13} are application of Bayes' theorem,\nwhile Eq.~\\ref{eq:p12} comes from the mutual independence of all events.\nAssuming in addition that all events are {\\it equally likely}: \n$p(d_i) = p(d_j) = p(d)$, we get\n\\begin{eqnarray}\n p(&&\\hspace{-4mm}\\Lambda_\\mathrm{NS} | d_1, d_2, \\cdots, d_N, H)\\hspace{50mm}\\nonumber\\\\\n  &=& p(\\Lambda_\\mathrm{NS})^{1-N}\\times \\dfrac{p(d)^N}{\\int p(\\Lambda_\\mathrm{NS}) p(d_1, d_2, \\cdots, d_N |\\Lambda_\\mathrm{NS}, H)\\mathrm{d}\\Lambda_\\mathrm{NS}}\\nonumber\\\\ &&\\hspace{3mm}\\times\\prod_i p(\\Lambda_\\mathrm{NS} |d_i, H)\\label{eq:p22},\n\\end{eqnarray}\nwhere the prior probability $p(\\Lambda_\\mathrm{NS}|H)$ is written $p(\\Lambda_\\mathrm{NS})$ for\nbrevity. {\\it A priori}, we assume that no particular value of $\\Lambda_\\mathrm{NS}$ is\npreferred over another within the range $[0, 4000]$, i.e.\n\\begin{equation}\\label{eq:lprior}\n p(\\Lambda_\\mathrm{NS} | H) = \\dfrac{1}{4000}\\,\\mathrm{Rect}\\left(\\frac{\\Lambda_\\mathrm{NS}-2000}{4000}\\right).\n\\end{equation}\nWith a uniform prior, the first two factors in Eq.~\\ref{eq:p22} can be\nabsorbed into a normalization factor $\\mathcal{N}$, simplifying it to\n\\begin{equation}\\label{eq:lambdaMultiple}\n p(\\Lambda_\\mathrm{NS} | d_1, d_2, \\cdots, d_N; H) = \\mathcal{N}\\prod_{i=1}^N p(\\Lambda_\\mathrm{NS} | d_i, H).\n\\end{equation}\n\nIn the second set of terms in Eq.~\\ref{eq:lambdaMultiple} (of the form \n$p(\\Lambda_\\mathrm{NS} | d_i, H)$), each is the probability distribution for $\\Lambda_\\mathrm{NS}$\ninferred {\\it a posteriori} from the \\textit{i}-th observation by marginalizing\n\\begin{equation}\\label{eq:margpost}\n p(\\Lambda_\\mathrm{NS} | d_i, H) = \\int\\, p(\\vec{\\theta}, \\Lambda_\\mathrm{NS} | d_i, H)\\, \\mathrm{d} \\vec{\\theta},\n\\end{equation}\nwhere $p(\\vec{\\theta}, \\Lambda_\\mathrm{NS} | d_i, H)$ is the inferred joint probability \ndistribution of all source parameters $\\vec{\\theta}\\cup\\{\\Lambda_\\mathrm{NS}\\}$ for the \n$i$-th event, as given by Eq.~\\ref{eq:postprob}. We note that \nFig.~\\ref{fig:SingleSystemLambdaPDFvsSNR} illustrates $p(\\Lambda_\\mathrm{NS} | d_i, H)$\nfor three individual events. By substituting\nEq.~\\ref{eq:lprior}-\\ref{eq:margpost} into Eq.~\\ref{eq:lambdaMultiple}, we\ncalculate the probability distribution for $\\Lambda_\\mathrm{NS}$ as measured using $N$\nindependent events.\n\n\n\n\nOur goal is to understand the improvement in our measurement of $\\Lambda_\\mathrm{NS}$\nwith the number of recorded events. To do so, we simulate a population~\\footnote{%\nA population here is an ordered set of events, and an event itself is the \nset of parameters describing one astrophysical NSBH binary.}\nof $N$ events, and quantify what we learn from each successive observation \nusing Eq.~\\ref{eq:lambdaMultiple}. This allows us to quantify how\nrapidly our median estimate for $\\Lambda_\\mathrm{NS}$ converges to the true value,\nand how rapidly our credible intervals for the same shrink, with increasing\n$N$. Finally, we generate and analyze an ensemble of populations in order to\naverage over the stochastic process of population generation itself.\n\n\nIn order to generate each population, the first step is to fix\nthe NS properties: (i) NS mass $m_\\mathrm{NS}=1.35M_\\odot$, (ii) NS spin $\\chi_\\mathrm{NS}=0$\nand (iii) NS tidal deformability $\\Lambda_\\mathrm{NS}=$ fixed value chosen from\n$\\{500,800,1000,1500,2000\\}$. Next, we generate events, by sampling BH mass\n(uniformly) from $m_\\mathrm{BH}\\in[3M_\\odot,6.75M_\\odot]$, BH spin (uniformly)\nfrom $\\chi_\\mathrm{BH}\\in[0, 1]$, orbital inclination from $\\iota\\in[0,\\pi]$, and \nsource location uniform in spatial volume\\footnote{with a minimum SNR \n$\\rho_\\mathrm{min}=10$}.\nWe restrict ourselves to positive aligned BH spins, since binaries with\nanti-aligned spins have very little information to add at realistic SNRs,\nas demonstrated in Fig.~\\ref{fig:TT_LambdaCIWidths90_0_Lambda_SNR}. This is\nto be taken into account when the number of observations is related to detector\noperation time. \nWe repeat this process till we have an ordered set of $N$ events.\nSince we want to analyze not just a single realization of an astrophysical\npopulation, but an ensemble of them, we make an additional approximation to\nmitigate computational cost. Complete Bayesian parameter estimation is\nperformed for a set of simulated signals whose parameters are the vertices\nof a regular hyper-cubic grid (henceforth ``G'') in the space of\n$\\{q\\}\\times\\{\\chi_\\mathrm{BH}\\}\\times\\{\\rho\\}$, with each sampled at $q=\\{2,3,4,5\\}$,\n$\\chi_\\mathrm{BH}=\\{-0.5,0,0.5,0.75\\}$, and $\\rho=\\{10,20,30,50,70\\}$.\nAll events in each population draw are substituted by their respective nearest\nneighbours on the grid G.\nOur chosen signal parameter distribution is different from some other studies\nin literature, which often sample from more astrophysically motivated population\ndistribution functions~\\cite{Mandel:2009pc}. We chose one that is sufficiently\nagnostic in absence of actual known NSBHs, and pragmatic enough for generating\npopulation ensembles.\n\n\n\n\nIn Fig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0} we show illustrative results\nfor a single population with neutron star deformability $\\Lambda_\\mathrm{NS}=800$.\nIn the top panel, each curve shows the \nprobability distributions for $\\Lambda_\\mathrm{NS}$ as inferred from $N$ events, with $N$\nranging from $1-80$. We also mark the $90\\%$ credible intervals associated\nwith each of the probability distribution curves. The first few observations\ndo not have enough information to bound $\\Lambda_\\mathrm{NS}$ much more than\nour prior from Eq.~\\ref{eq:lprior} does. \nIn the bottom panel, we present information derived from the top panel.\nThe line-circle curve shows the measured median value from $N$ observations.\nThe pair of dashed (dotted) horizontal lines mark\n$\\pm25\\%$ ($\\pm50\\%$) error bars. At each $N$, the range spanned by the \nfilled region is the $90\\%$ credible interval deduced from the same \nevents. This figure somewhat quantifies the qualitative deductions we made\nfrom the left panel. We find that the median does track the true value quickly,\nreaching within its $10\\%$ with $10-15$ observations. This is as one expects of \ninjections in zero noise where random fluctuations are unable to shift the\nmedian away from the true value, so long as the measurement is not restricted\nby the prior. With the same information, our credible intervals also shrink to $\\pm 25\\%$.\nIn Fig.~\\ref{fig:TT_Lambda_vs_N_CI90_0} we show further results from four\nindependent populations for $\\Lambda_\\mathrm{NS}=\\{500,1000,1500,2000\\}$. As in the \nright panel of Fig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0}, the line-circle curves\ntrack the median $\\Lambda_\\mathrm{NS}$, while the filled regions show\nthe associated $90\\%$ credible intervals. From the figure, we observe\nthe following: (i) the shrinkage of credible interval widths with increasing\n$N$ happens in a similar manner for each $\\Lambda_\\mathrm{NS}$, \nand (ii) it takes\napproximately $20$ events to distinguish definitively (with $90\\%$ credibility)\nbetween deformable NSs with $\\Lambda_\\mathrm{NS}=2000$ and compact NSs with \n$\\Lambda_\\mathrm{NS}=500$, or equivalently to distinguish between hard, moderate and soft\nnuclear equations of state. This is comparable to what has been found for\nbinary neutron stars~\\cite{DelPozzo:13,Chatziioannou:2015uea,Agathos:2015a}.\n\n\n\nSo far we have discussed individual realizations of NSBH populations. The \nunderlying stochasticity of the population generation process makes it\ndifficult to draw generalized inferences (from a single realization of an\nNSBH population) about the measurability of $\\Lambda_\\mathrm{NS}$. In order to mitigate\nthis, we discuss ensembles of population draws next. In\nFig.~\\ref{fig:TT_LambdaMedian_vs_N_AllInOne} we show the median\n$\\Lambda_\\mathrm{NS}$ as a function of the number of observed\nevents, for four population ensembles, with a hundred population draws\nin each ensemble. Lets focus on the {\\it top left} panel first. In it, we\nshow the median $\\Lambda_\\mathrm{NS}$ for all populations in four ensembles,\nwith true $\\Lambda_\\mathrm{NS}=\\{2000,1500,1000,500\\}$ from top to bottom.\nPopulations highlighted in color are simply those that we discussed in\nFig.~\\ref{fig:TT_Lambda_vs_N_CI90_0}. Dash-dotted horizontal lines\ndemarcate $\\pm10\\%$ error intervals around the true $\\Lambda_\\mathrm{NS}$ values.\nThe panel just below it shows the range of $\\Lambda_\\mathrm{NS}$ that encloses the\nmedian $\\Lambda_\\mathrm{NS}$ for $90\\%$ of the populations in {\\it each}\nensemble. In other words, this panel shows the range of $\\Lambda_\\mathrm{NS}$ within which\nthe median $\\Lambda_\\mathrm{NS}$ value for $90\\%$ of NSBH populations is\nexpected to lie. From these panels, we observe that our median $\\Lambda_\\mathrm{NS}$\nvalues will be within $10\\%$ of the {\\it true} value after $\\sim 25$\ndetections of less deformable neutron stars ($\\Lambda_\\mathrm{NS}\\leq 1000$), or\nafter as few as $15$ detections of more deformable neutron stars\n($\\Lambda_\\mathrm{NS}\\geq 1500$). This is not surprising because we inject simulated\nsignals in {\\it zero} noise, which ensures that the median not be shifted away\nfrom the true value. That it takes $15+$ events for the median to approach\nthe true value is a manifestation of the fact that the measurement is limited\nby the prior on $\\Lambda_\\mathrm{NS}$ when we have fewer than $15$ events.\nThe results discussed in Fig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0},\n\\ref{fig:TT_Lambda_vs_N_CI90_0} and the left two panels\nof Fig.~\\ref{fig:TT_LambdaMedian_vs_N_AllInOne} apply to the parameter\ndistribution spanned by the grid G. This distribution allows for $m_\\mathrm{BH}$\nas low as $2.7M_\\odot$ (i.e. $q=2$).\nGiven that disruptive signatures are strongest for small $m_\\mathrm{BH}$, we now\ninvestigate an alternate paradigm in which no black hole masses fall within\nthe mass gap $2-5M_\\odot$ suggested by astronomical\nobservations~\\cite{Bailyn:1997xt,Kalogera:1996ci,Kreidberg:2012,\nLittenberg:2015tpa}. We will henceforth denote our standard paradigm, which\ndoes not respect the mass-gap, as paradigm A; with paradigm B being\nthis alternate scenario.\nBoth right panels of the figure are identical to their corresponding left\npanels, but drawn under population paradigm B. Under this paradigm, we\nexpectedly find that information accumulation is much slower. It would\ntake $25-40$ detections with $\\rho\\geq10$ under this paradigm, for our median\n$\\Lambda_\\mathrm{NS}$ to converge within $10\\%$ of its true value.\n\n\nFinally, we investigate the statistical uncertainties associated with\n$\\Lambda_\\mathrm{NS}$ measurements. We use $90\\%$ credible intervals as our measure of\nthe same. First, we draw an ensemble of a hundred populations each for\n$\\Lambda_\\mathrm{NS}=\\{500,1000,1500,2000\\}$. For each population $i$ in each ensemble,\nwe construct\nits $90\\%$ credible interval $[{\\Lambda_\\mathrm{NS}^{90\\%}}_{i-},{\\Lambda_\\mathrm{NS}^{90\\%}}_{i+}]$.\nNext, we construct the interval $[X^-,Y^-]$ that contains ${\\Lambda_\\mathrm{NS}^{90\\%}}_{i-}$\nfor $90\\%$ of the populations in each ensemble; and similarly $[X^+,Y^+]$\nfor ${\\Lambda_\\mathrm{NS}^{90\\%}}_{i+}$. Finally, in the left panel of \nFig.~\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne}, we show the\nconservative width $|Y^+ - X^-|$ that contains the $90\\%$ credible\nintervals for $90\\%$ of all populations in each ensemble~\\footnote{Drawn\nunder paradigm A (mass-gap not respected).}.\nFrom top to bottom, the population $\\Lambda_\\mathrm{NS}$\ndecreases from $\\Lambda_\\mathrm{NS}=2000\\rightarrow 500$, corresponding to decreasingly\ndeformable NSs with softer equations of state. We observe the following:\n(i) for moderately-hard to hard equations of state with $\\Lambda_\\mathrm{NS}\\geq 1000$,\nwe can constrain $\\Lambda_\\mathrm{NS}$ within $\\pm 50\\%$ using only $10-20$ events, and\nwithin $\\pm 25\\%$ (marked by black circles) with $25-40$ events; (ii) for softer \nequations of state with\n$\\Lambda_\\mathrm{NS}<1000$, we will achieve the same accuracy with $20-30$ and $50+$ \nevents, respectively; and (iii) for the first $5$ or so observations, our\nmeasurement spans the entire prior allowed range:\n$\\Lambda_\\mathrm{NS}\\in[0,4000]$, as shown by the plateauing of the $90\\%$ \ncredible intervals towards the left edge to $90\\%$ of $4000$, i.e. $3600$.\nThe right panel in Fig.~\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne}\nis identical to the left one, with the difference that populations are drawn\nunder paradigm B, which does {\\it not} allow for BH masses to fall within\nthe mass-gap. We find that\nfor NSs with $\\Lambda_\\mathrm{NS}\\leq 1000$, it would take $25-40$ events to\nconstrain $\\Lambda_\\mathrm{NS}$ within $\\pm 50\\%$ and $50+$ events to constrain it\nwithin $\\pm 25\\%$. This is somewhat slower than paradigm A, as is to be\nexpected since here we preclude the lowest mass-ratios, which correspond to\nsignals with largest tidal signatures. For $\\Lambda_\\mathrm{NS}>1000$ we find that we\ncan constrain $\\Lambda_\\mathrm{NS}$ within $\\pm 50\\%$ with a similar number of events as\nfor paradigm A, but will need more ($30-40$, as compared to $25-40$) events\nto further constrain it to within $\\pm 25\\%$ of the true value.\nUnder either paradigms, we find that measuring $\\Lambda_\\mathrm{NS}$ better than\n$25\\%$ will require $\\mathcal{O}(10^2)$ observations of disruptive NSBH\nmergers.\nThese results demonstrate that our probed set of disruptive NSBH mergers is as\ninformative of NS tidal properties as are BNS populations, if we assume a uniform mass\ndistribution for NSs and zero NS spins, and possibly more if NS masses are\nnot distributed uniformly~\\cite{Agathos:2015a}.\nHowever, being a subset of all NSBH binaries, the accumulation of information\nfrom NSBH signals in general may be slower than from binary neutron stars,\ndepending on the distribution of BH masses in coalescing NSBH binaries.\n\n\n{\\it Do all events matter?} In Ref.~\\cite{Lackey2014} the authors demonstrate\nthat the overwhelming majority of information about the NS equation of state\ncomes from the loudest $\\sim 5$ events in the case of binary neutron star\ndetections, and not from the combined effect of a large number of low-SNR\nsystems.\nThe question naturally arises if the same is true for NSBH sources as well.\nTherefore, in Fig.~\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne_SNRSorted}\nwe re-evaluate the accumulation of information with each successive NSBH \ndetection, sorting the events in each population by their SNR instead of \nsimulated chronology. We find the same qualitative behavior as in the case of \nBNSs~\\cite{Lackey2014}. \nWhether or not there is an astrophysical mass gap, we find\nthat the bulk of tidal information will be furnished by the loudest $5-10$\nNSBH detections of aLIGO detectors. {\\it This result is especially encouraging\nto NR follow-up efforts for GW detections, as we now know that only a handful\nof loudest NSBH events (with SNRs $\\rho\\gtrsim 20-30$) are the ones that may merit\nfull numerical-GR + magnetohydrodynamical follow-up simulations.}\n\n\n\nTo summarize, in this section we study the improvement in our measurement of \nNS deformability parameter $\\Lambda_\\mathrm{NS}$ with an increasing number of events. We\ndo so by simulating plausible populations of disrupting NSBH binaries (with\n$\\rho\\geq 10$). We find that:\n(i) for more deformable neutron stars (harder equation of states), the median\nvalue of $\\Lambda_\\mathrm{NS}$ comes within $10\\%$ of the true value with as \nfew as $10$ events, while achieving the same accuracy for softer equations of \nstate will take $15-20$ source detections; (ii) the statistical uncertainty\nassociated with $\\Lambda_\\mathrm{NS}$ measurement shrinks to within $\\pm50\\%$ with\n$10-20$ events, and to within $\\pm 25\\%$ with $50+$ events, when source \n$\\Lambda_\\mathrm{NS}\\geq 1000$; (iii) for softer equations of state, the same could take\n$25-40$ and $50+$ events, respectively for the two uncertainty thresholds;\nand (iv) if BHs really do observe the astrophysical mass-gap, the information\naccumulation is somewhat slower than if they do not. We conclude that within\n$20-30$ observations, aLIGO would begin to place very interesting bounds on \nthe NS deformability, which would allow us to rule out or rank different\nequations of state for neutron star matter. Within this population, we also\nfind that it will be the loudest $5-10$ events that will furnish most of the\ntidal information. Our key findings are summarized in \nFig.~\\ref{fig:TT_LambdaMedian_vs_N_AllInOne} -\n\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne_SNRSorted}.\n\n\n\n\n\n\\section{Discussion}\\label{s1:discussion}\n\n\nThe pioneering observation of gravitational waves by Advanced LIGO\nharbingers the dawn of an era of gravitational-wave astronomy where observations\nwould finally drive scientific discovery~\\cite{Abbott:2016blz}. As confirmed by\nthe first observations~\\cite{Abbott:2016blz,Abbott:2016nmj,Abbott:2016nhf},\nstellar-mass compact binary mergers emit GWs right in the sensitive frequency\nband of the LIGO observatories, and are their primary targets.\nNeutron star black hole binaries form a physically distinct sub-class of\ncompact binaries. We expect to detect the first of them in the upcoming\nobserving runs~\\cite{Abbott:2016ymx}, and subsequently at a healthy rate of\n$0.2-300$ mergers a year when aLIGO detectors reach design\nsensitivity~\\cite{Abadie:2010cf}.\n\nNSBH binaries are interesting for various reasons. Unlike BBHs, the presence of\nmatter allows for richer phenomena to occur alongside the strong-field\ngravitational dynamics. The quadrupolar moment of the NS changes during the\ncourse of inspiral, which increases the inspiral rate of the binary and alters the\nform of the emitted gravitational waves. Close to merger, under restricted but\nplausible conditions, the neutron star is disrupted by the tidal field of its \ncompanion black hole and forms an accretion disk around it. This disruption\nreduces the quadrupolar moment of the system, and decreases the amplitude of\nthe emitted GWs from the time of disruption through to the end of ringdown.\nBoth of these phenomena are discernible in their gravitational-wave signatures\nalone. In addition, if the neutron star matter is magnetized, the magnetic\nwinding above the remnant black hole poles can build up magnetic fields\nsufficiently to power short gamma-ray bursts (SGRB)~\\cite{Foucart:2015a,\nLovelace:2013vma,Deaton2013,Foucart2012,Shibata:2005mz,Paschalidis2014}.\nTherefore a coincident observation of gravitational waves from an NSBH merger\nand a SGRB can potentially confirm the hypothesis that the former is a\nprogenitor of the latter~\\cite{eichler:89,1992ApJ...395L..83N,moch:93,\nBarthelmy:2005bx,2005Natur.437..845F,2005Natur.437..851G,Shibata:2005mz,\nTanvir:2013,Paschalidis2014}.\n\n\nIn this paper we study the observability of tidal signatures in the\ngravitational-wave spectrum of NSBH binaries. More specifically, we investigate\nthree questions. First, what is the effect of not including tidal effects in \ntemplates while characterizing NSBH signals? Second, if we do include tidal \neffects, how well can we measure the tidal deformability of the NS\n(parameterized by $\\Lambda_\\mathrm{NS}$) from individual NSBH signals? And third, as we\nobserve more and more signals, how does our knowledge of $\\Lambda_\\mathrm{NS}$ improve?\nIn the following, we summarize our main findings.\n\n\n\n\n\nFirst, we study the effects of not including tidal terms in our search\ntemplates while characterizing NSBH signals. We expect that the waveform\ntemplate that best fits the signal would compensate for the reduced number of\ndegrees of freedom in the template model by moving away from the true\nparameters of the binary. This should result in a {\\it systematic} bias in \nthe recovered values of non-tidal source parameters, such as its masses \nand spins. In order to quantify it, we inject tidal signals into zero noise,\nand perform a Bayesian parameter estimation analysis on them using templates\n{\\it without} tidal terms.\nWe use the LEA+ model (c.f. Sec.~\\ref{s2:waveforms}) to produce tidal waveforms\nthat incorporate the effect\nof NS distortion during inspiral, and of its disruption close to merger. Our\ninjected signals sample the region of NSBH parameter space where NS disruption\nprior to binary merger is likely {\\it and} can be modeled using LEA+. Their\nparameters are given by combinations of $q=m_\\mathrm{BH}\/m_\\mathrm{NS}=\\{2,3,4,5\\},\n\\chi_\\mathrm{BH}=\\{-0.5, 0, 0.5, 0.75\\}$ and $\\Lambda_\\mathrm{NS}=\\{500,800,1000,1500,2000\\}$.\nOther parameters, such as source location and orientation, that factor out of\n$h(t)$ as amplitude scaling are co-sampled by varying $\\rho=\\{20,30,50,70\\}$.\n\n\n\nAt low to moderate SNRs ($\\rho\\lesssim 30$), we find that using BBH templates\ndoes not significantly hamper our estimation of non-tidal parameters for NSBH\nsignals. In the worst case, when the BH mass is within the astrophysical \nmass-gap~\\cite{Bailyn:1997xt,Kalogera:1996ci,Kreidberg:2012,Littenberg:2015tpa}\nand its spin is positive aligned, the systematic biases in $\\eta$ and $\\chi_\\mathrm{BH}$\nmeasurements do become somewhat comparable to statistical errors (ratio\n$\\sim 0.5-0.8$) under very restrictive conditions~\\footnote{requiring a\ncompanion BH with mass $m_\\mathrm{BH}\\lesssim 4.5M_\\odot$ (i.e. in the astrophysical\nmass-gap), and the\nhardest NS EoS considered (with $\\Lambda_\\mathrm{NS}\\simeq 2000$).}, but never exceed them.\nAt high SNRs ($\\rho\\gtrsim 50$), systematic biases in $\\mathcal{M}_c$ become larger\nthan the statistical uncertainties. For $\\eta$ and $\\chi_\\mathrm{BH}$ the difference\nis more drastic with the systematics reaching up to $4\\times$ the statistical\nerrors. We therefore conclude that $\\rho\\simeq 30-50$ is loud enough to\nmotivate the use of tidal templates for even the estimation of non-tidal\nparameters from NSBH signals.\nWe also conclude that low-latency parameter estimation algorithms, designed to\nclassify GW signals into electromagnetically active (NSBH and NSNS) and\ninactive (BBH) sources, can use BBH templates to trigger GRB \nalerts~\\cite{2012A&A...541A.155A,Singer:2014qca,Singer:2015ema,Pankow:2015cra,\nAbbott:2016wya,Abbott:2016gcq} for NSBH signals with low to moderate SNRs\n($\\rho\\lesssim 30$).\nThis is so because the primary requirement of identifying NS-X binaries (X =\n\\{NS, BH\\}) can be achieved just as easily with BBH templates, on the basis of\nthe smaller component's mass\\footnote{The smaller component mass is unlikely\nto be significantly biased by missing tidal effects in filter templates below\n$\\rho\\simeq 30$, as we show above.}.\nWe also speculate that NSBH detection searches are unlikely to be\naffected by the choice of ignoring tidal effects in matched-filtering\ntemplates, if these effects are too subtle to manifest in parameter estimation\nbelow $\\rho\\simeq 30$.\n\n\n\n\n\nSecond, we turn the question around to ask: can we measure the tidal effects if\nour template models did account for them? Tidal effects in our waveform model\nare parameterized using a single deformability parameter \n$\\Lambda_\\mathrm{NS}\\propto (R\/M)_\\mathrm{NS}^5$. In order to quantify the \nmeasurability of $\\Lambda_\\mathrm{NS}$, we inject the same tidal signals as before, and\nthis time perform a Bayesian analysis on them using {\\it tidal} templates. \nThe results are detailed in Sec.~\\ref{s1:PEwithNS}.\nAt low SNRs ($\\rho\\simeq 20$), we find that the best we can do is to constrain\n$\\Lambda_\\mathrm{NS}$ within $\\pm 75\\%$ of its true value at $90\\%$ credible level. This\ntoo only if the BH is spinning sufficiently rapidly, with $\\chi_\\mathrm{BH}\\gtrsim +0.7$,\nand the NS has $\\Lambda_\\mathrm{NS}\\gtrsim 1000$. At moderate SNRs ($\\rho\\simeq 30$), we\ncan constrain $\\Lambda_\\mathrm{NS}$ a little better, i.e. within $\\pm 50\\%$ of its true\nvalue. This level of accuracy, however, again requires that BH spin\n$\\chi_\\mathrm{BH}\\gtrsim+0.7$ and $\\Lambda_\\mathrm{NS}\\gtrsim 1000$. Binaries with smaller BH spins\nand\/or softer NS EoSs will furnish worse than $\\pm 75\\%-\\pm 100\\%$ errors for\n$\\Lambda_\\mathrm{NS}$. This trend continues as we increase the SNR from $\\rho=30-50$. It\nis not before we reach an SNRs as high as $\\rho\\simeq 70$ that we can shrink\n$\\Lambda_\\mathrm{NS}$ errors substantially with a single observation (i.e. within\n$\\pm 25\\%$ of its true value).\nIn summary, we find that with a single but moderately loud NSBH signal,\nAdvanced LIGO can begin to put a factor of $1-2\\times$ constraints on NS tidal\ndeformability parameter. These constraints can subsequently be used to assess\nthe likelihood of various candidate equations of state for nuclear matter, and\npossibly to narrow the range they span.\n\n\n\n\n\nThird, knowing that single observations can furnish only so much information\nabout the NS equation of state, we move on to investigate how well we do with\nmultiple signals. In order to quantify how $\\Lambda_\\mathrm{NS}$ measurement improves\nwith the number of observed events $N$, we generate populations of NSBH signals\nand combine the information extracted from each event.\nThe population generation procedure is as follows. The neutron star mass is\nheld fixed at $1.35M_\\odot$, its spin at $\\chi_\\mathrm{NS}=0$, and its tidal\ndeformability is fixed to each of $\\Lambda_\\mathrm{NS}=\\{500,1000,1500,2000\\}$. Black hole\nmass is sampled uniformly from the range $[2,5]\\times 1.35=[2.7, 6.75]M_\\odot$,\nand spin from $\\chi_\\mathrm{BH}\\in[0,1]$. As before, our parameter choice here is given\nby the intersection set of the mass range that allows for neutron star disruption\nand the range supported by LEA+~\\cite{Foucart2012,Foucart:2013a,Lackey:2013axa}.\nIn order to keep the computational cost reasonable, we make an additional\napproximation. For every population generated, we replace the parameters of each\nevent by their nearest neighbor on the uniform grid G, which has vertices\nat: $q=\\{2,3,4,5\\}\\times\\chi_\\mathrm{BH}=\\{-0.5,0,0.5,0.75\\}\\times\\Lambda_\\mathrm{NS}=\\{500,800,\n1000,1500,2000\\}\\times\\rho=\\{10,20,30,50,70\\}$.\nThis way, we only have to run full Bayesian parameter estimation analysis on\nthis fixed set of signals. \nThere are two sources of error that enter the deductions we make from\na single population generated in the manner described above. First, since the\ninjection parameters are pushed to their nearest neighbor on a grid, we\nfind discrete jumps in $\\Lambda_\\mathrm{NS}$ errors as a function of $N$. And second, an\nindividual population is one particular realization of a stochastic process and\ncould have excursions that may never be found in another population. To\naccount for both of these limitations, we generate an ensemble of populations,\nand conservatively combine information from all of them\\footnote{See \nSec.~\\ref{s1:multiple_observations} for further details.}.\n\n\n\n\nWe probe two astrophysical paradigms, one that allows for BH masses to lie\nwithin the astrophysical mass-gap (paradigm A), and one that does not (paradigm\nB).\n{\\it For paradigm A}, we find the following: (i) for the softer equations of\nstate that result in less deformable neutron stars, $15-20$ detections bring\nthe measured probability distribution for $\\Lambda_\\mathrm{NS}$ entirely within the prior,\nwhich ensures that the median $\\Lambda_\\mathrm{NS}$ tracks the true value to within $10\\%$.\n(ii) For NSBH populations with more deformable NSs ($\\Lambda_\\mathrm{NS}> 1000$),\nthe same is achievable within as few as $10$ (or $15$ at most) realistic\nobservations. (iii) The statistical uncertainty associated with $\\Lambda_\\mathrm{NS}$\nmeasurement can be restricted to be within $\\pm50\\%$ using $10-20$ observations\nwhen $\\Lambda_\\mathrm{NS}> 1000$), and using $25-40$ observations for softer equations\nof state. All of the above is possible within a few years of design\naLIGO operation~\\cite{Abadie:2010cfa}, if astrophysical BHs are allowed\nmasses $< 5M_\\odot$ (i.e. in the mass-gap). However, further\nrestricting $\\Lambda_\\mathrm{NS}$ will require $50+$ NSBH observations.\n{\\it For paradigm B}, we find the information accumulation to be somewhat slower.\nWhile the quantitative inferences for populations with $\\Lambda_\\mathrm{NS}>1000$ are\nnot affected significantly, we find that $\\Lambda_\\mathrm{NS}< 1000$ populations require \n$10-20\\%$ more events to attain the same measurement accuracy as under\nparadigm A. In either case, the accumulation of information from the general\nNSBH population will likely be slower than from BNS inspirals~\\cite{Mandel:2009pc,\nLackey2014,Wade:2014vqa,Agathos:2015a}, depending on the mass distribution of \nstellar-mass black holes. Though, template models for the latter may be more\nuncertain due to missing point-particle PN terms at orders comparable to\ntidal terms~\\cite{Lackey2014}.\nWe conclude that within as few as $20-30$ observations of disruptive NSBH\nmergers, aLIGO will begin to place interesting bounds on NS deformability.\nThis, amongst other things, will allow us to rank different equations of \nstate for neutron star matter from most to least likely, within a few years'\ndetector operation.\nWe also find that, within this population, the loudest $5-10$ events (with SNRs\n$\\rho\\gtrsim 20-30$) will provide us with most of the tidal information,\nand will therefore merit full NR follow-up.\nOur methods and results are detailed in Sec.~\\ref{s1:multiple_observations}.\n\n\n\n\n\n\n\n\n\nFinally, we note that the underlying numerical simulations used to calibrate\nthe waveform model used here have not been verified against\nindependent codes so far.\nIt is therefore difficult to assess the combined modeling error of LEA+ and its\neffect\non our results. Our results here are, therefore, limited by the limitations of\nour waveform model, and presented with this caveat. However, we do expect the\ncombined effect of modeling errors to {\\it not} affect our {\\it qualitative}\nconclusions, especially since the underlying point-particle component of LEA+\nincludes all high-order terms, unlike past BNS studies~\\cite{Lackey2014,\nWade:2014vqa}\nIn future, we plan to further the results presented here by using more recent\ntidal models~\\cite{Pannarale:2015jka,Hinderer:2016eia}, that\nmay improve upon LEA+\\footnote{One of them~\\cite{Pannarale:2015jka} is only an\namplitude model though, which has to be augmented with a compatible phase model\nfirst.}.\n\n\n\n\n\\begin{acknowledgments}\n  We thank Benjamin Lackey, Francesco Pannarale, Francois Foucart, and Duncan Brown\n  for helpful discussions. We gratefully acknowledge support\n  for this research at CITA from NSERC of Canada, the Ontario Early \n  Researcher Awards Program, the Canada Research\n  Chairs Program, and the Canadian Institute for Advanced Research.\n  Calculations were performed at the Vulcan\n  supercomputer at the Albert Einstein Institute;\n  H.P. and P.K. thank the Albert-Einstein Institute,\n  Potsdam, for hospitality during part of the time where this research\n  was completed. M.P. thanks CITA for hospitality where part of the work\n  was carried out.\n\\end{acknowledgments}\n\n\\begin{appendix}\n\n\\section{Statistical uncertainty in measuring non-tidal parameters}\\label{as1:nontidalerrors}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim = {2cm 0 0 0},width=2.\\columnwidth]{TNMchirpCIWidths90_0_Lambda_SNR}\\\\\n\\includegraphics[trim = {2cm 0 0 0},width=2.\\columnwidth]{TNEtaCIWidths90_0_Lambda_SNR}\\\\\n\\includegraphics[trim = {2cm 0 0 0},width=2.\\columnwidth]{TNChiBHCIWidths90_0_Lambda_SNR}\n\\caption{{\\bf Statistical measurement uncertainty for NSBH parameters,\nignoring tidal effects:}\nWe show here the statistical uncertainty associated with our measurement of\nnon-tidal parameters $\\mathcal{M}_c, \\eta,$ and $\\chi_\\mathrm{BH}$ (at $90\\%$ credibility),\nover the signal parameter space. Individual panels show the same as a function\nof BH mass and spin. Across each row, we see the effect of increasing signal\nstrength (i.e. SNR) with the tidal deformability of the NS $\\Lambda_\\mathrm{NS}$ fixed.\nDown each column, we see the effect of increasing $\\Lambda_\\mathrm{NS}$, at fixed SNR.\nTidal effects are ignored in templates.\n}\n\\label{fig:CIWidths90_Lambda_SNR}\n\\end{figure*}\nIn Fig.~\\ref{fig:CIWidths90_Lambda_SNR}, we show how {\\it precisely} can we\nmeasure non-tidal NSBH parameters $X=\\{\\mathcal{M}_c,\\eta,\\chi_\\mathrm{BH}\\}$ using BBH templates.\nThe three panels correspond to $\\mathcal{M}_c$ (top), $\\eta$ (middle), and $\\chi_\\mathrm{BH}$\n(bottom), and show the width of these credible intervals $(\\Delta X)^{90\\%}$\nas a function of BH mass\/spin (within each sub-panel), and NS properties, i.e.\n$\\Lambda_\\mathrm{NS}$ (downwards in each column)~\\footnote{We restrict NS mass to\n$1.35M_\\odot$ and its spin to zero. Varying its tidal deformability $\\Lambda_\\mathrm{NS}$\ndoes not significantly change the measurement uncertainties for non-tidal\nbinary parameters, as is evident from comparing the two rows in each panel of\nFig.~\\ref{fig:CIWidths90_Lambda_SNR}.}.\nFrom the left-most column, we find that: (i) at $\\rho=20$ the chirp mass is\nmeasured remarkably well - to a precision of $0.16\\%$ of its true value, and\n(ii) so is $\\chi_\\mathrm{BH}$. (iii) The dimensionless mass-ratio $\\eta$ is determined\nmore loosely, with $25+\\%$ uncertainty. If the signal is even louder\n($\\rho\\geq 30$), all three measurements gain further precision, especially\n$\\eta$, for which the relative errors shrink down to single-digit percents.\nWe remind ourselves that these results do not tell the full story since the\nprecision of a measurement is only meaningful if the measurement is accurate \nto begin with. In our case there are tidal effects that have not been\nincorporated into our search (BBH) templates, which can lead to a systematic\nbias in parameter recovery. We refer the reader to Sec.~\\ref{s1:PEwithnoNS} for\na comparative study of both systematic and statistical errors.\n\n\n\n\n\\section{Illustrations of Bayesian posteriors}\\label{as1:illustrations}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.05\\columnwidth,trim=2cm 0 0 0]{AllParamsMcEtPDF1D2D_q4_mc2_25_chi0_50_snr20\n\\includegraphics[width=1.05\\columnwidth,trim=0.7cm 0 1cm 0]{AllParamsMcEtPDF1D2D_q4_mc2_25_chi0_50_snr50\n\\caption{{\\bf Illustrative posterior probability distributions for NSBH parameters,\nfor signals at different SNRs:}\nWe illustrate here two sets of two-dimensional joint probability distributions,\ndiffering only in signal strength, with $\\rho=20$ in the left panel, and\n$\\rho=50$ in the right. The injected parameters are \n$q = m_\\mathrm{BH}\/m_\\mathrm{NS} = 5.4M_\\odot\/1.35M_\\odot = 4$, $\\chi_\\mathrm{BH}=+0.5$, and \n$\\Lambda_\\mathrm{NS}=2000$. Contours are shown for $\\{1-,2-,3-,\\cdots\\}\\sigma$ confidence levels.\nTemplates include tidal effects, as evident in the bottom rows\nof both panels which show the correlation of $\\Lambda_\\mathrm{NS}$ with non-tidal \nparameters. Contrasting the two panels illustrates the effect of increasing the\nSNR on various parameter measurements.\n}\n\\label{fig:SingleSystemLambda2DPDFs}\n\\end{figure*}\n\n\n\n\n\n\nIn Fig.~\\ref{fig:SingleSystemLambda2DPDFs} we show the correlation of\nmass, spin, and tidal parameter measurements. We keep the binary parameters\nas in Fig.~\\ref{fig:SingleSystemLambdaPDFvsSNR}, with $\\Lambda_\\mathrm{NS}=2000$, and\nset $\\rho=20$ (left panel) or $\\rho=50$ (right panel).\nWe find that the measurement of $\\Lambda_\\mathrm{NS}$ is weakly degenerate with\nother parameters, and at realistic SNRs it would improve by a few tens of \npercent if we knew non-tidal parameters to better accuracy. The predominant\nfactor that would enhance the measurement accuracy for $\\Lambda_\\mathrm{NS}$ is \nnevertheless the signal strength. Only when $\\rho\\gtrsim 50$ can we\nexpect $\\Lambda_\\mathrm{NS}$ measurement to be limited by its degeneracy with \nnon-tidal parameters (at a factor of few level), as also reported by previous\nstudies~\\cite{Lackey:2013axa}.\n\n\n\n\n\n\n\n\\section{Phenomenology of $\\Lambda_\\mathrm{NS}$ measurement errors}\n\\begin{figure}\n\\centering    \n\\includegraphics[width=1.05\\columnwidth]{PowerLawCoefficient_LambdaErrorvsN_vs_N.pdf}\n\\caption{%\nAssuming a power-law dependence of the measurement error on the number of\nevents: $\\delta\\Lambda_\\mathrm{NS}\\propto 1\/N^\\alpha$, we show $\\alpha$ in this figure\nas a function of the number of observed events $N$. Shown are five families\nof $100$ population draws each, with each family corresponding to one of\n$\\Lambda_\\mathrm{NS}=\\{500,800,1000,1500,2000\\}$. Each grey curve corresponds to one\nof these $100\\times5 = 500$ populations. The thicker curves, one from each\nfamily, shows the population we discussed in\nFig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0}-\\ref{fig:TT_Lambda_vs_N_CI90_0}.\nWe find that a power-law is a good approximation for the concerned dependence,\nand information accumulates {\\it faster} than $1\/\\sqrt{N}$. We estimate\n$\\alpha\\simeq 0.7^{+0.2}_{-0.2}$.\n}\n\\label{fig:TT_PowerLawLambdaErrorVsN}\n\\end{figure}\n\\begin{figure}\n\\centering    \n\\includegraphics[width=\\columnwidth]{PowerLawCoefficient_LambdaErrorvsLambda_vs_N_AllPopulations.pdf}\n\\caption{%\nIn this figure, which is similar to Fig.~\\ref{fig:TT_PowerLawLambdaErrorVsN},\nwe quantify the dependence of $\\delta\\Lambda_\\mathrm{NS}$ on $\\Lambda_\\mathrm{NS}$ itself. Of \nthe five families of simulated NSBH populations, we construct $100$\nindependent sets taking one population from each family. With each of \nthese $100$ sets, and assuming a power-law dependence:\n$\\delta\\Lambda_\\mathrm{NS}\\propto\\Lambda_\\mathrm{NS}^\\beta$, we estimate $\\beta$ and show it in\nthis figure as a function of the number of observed events $N$. The thicker\ncurve corresponds to the populations discussed in\nFig.~\\ref{fig:TT_Lambda_vs_N_CI90_0}.\nWe find that $\\beta$ can be estimated to lie within $[1\/6,5\/6]$ with a\nlikely value close to $1\/2$. Since $0<\\beta<1$, the relative error\n$\\delta\\Lambda_\\mathrm{NS}\/\\Lambda_\\mathrm{NS}$ {\\it decreases} as the star gets more \ndeformable, while the absolute error $\\delta\\Lambda_\\mathrm{NS}$ {\\it increases}.\n}\n\\label{fig:TT_PowerLawLambdaErrorVsLambda}\n\\end{figure}\nHere, we quantitatively explore the dependence of our statistical\nuncertainties for $\\Lambda_\\mathrm{NS}$ on the number of events, as well as on the true\nNS deformability itself. First, we will focus on the dependence on $N$. We\nassume a power-law dependence of the form\n$\\delta\\Lambda_\\mathrm{NS}\\propto\\ 1\/N^\\alpha$. For each of the $100$ populations \nfor each of $\\Lambda_\\mathrm{NS}=500-2000$, we compute the exponent $\\alpha$ as a\nfunction of the number of observed events $N$, and show it in \nFig.~\\ref{fig:TT_PowerLawLambdaErrorVsN}. There are $100\\times5=500$ curves\non the figure, with one highlighted for each value of population's $\\Lambda_\\mathrm{NS}$.\nThese highlighted values are only special in the sense that they correspond to\npopulations discussed earlier in this section (c.f.\nFig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0}-\\ref{fig:TT_Lambda_vs_N_CI90_0}).\nWe immediately observe two things, (i) there is a globally similar dependence\non $N$ for all populations, and (ii) information accumulates {\\it faster} than\n$1\/\\sqrt{N}$. In fact, we find that if\n$\\delta\\Lambda_\\mathrm{NS}\\propto\\frac{1}{N^\\alpha}$, $\\alpha$ lines in the range\n$0.7_{-0.2}^{+0.2}$.\nNext, we focus on the dependence of $\\delta\\Lambda_\\mathrm{NS}$ on $\\Lambda_\\mathrm{NS}$ of the\npopulation itself. As suggested by Fisher-matrix studies~\\cite{Lackey:2013axa},\nand as for $N$, we assume the form $\\delta\\Lambda_\\mathrm{NS}\\propto\\Lambda_\\mathrm{NS}^\\beta$.\nFrom each set of $100$ populations with a given $\\Lambda_\\mathrm{NS}$ value, we draw one\nat random, and form a set of $5$ similarly drawn populations, one for each of\n$\\Lambda_\\mathrm{NS}=\\{500,800,1000,1500,2000\\}$. With each set, we determine $\\beta$\nfor different number of observed events $N$. In all, we make $100$ independent\n$5-$population sets and show the value of $\\beta$ measured from each in \nFig.~\\ref{fig:TT_PowerLawLambdaErrorVsLambda}. We find that the assumed\nrelation $\\delta\\Lambda_\\mathrm{NS}\\propto\\Lambda_\\mathrm{NS}^\\beta$ gets fairly robust for \nlarger values of $N$, with $\\beta$ converging to $\\beta=0.5^{+0.33}_{-0.33}$.\nThe fact that $0<\\beta<1$ implies that the relative error\n$\\delta\\Lambda_\\mathrm{NS}\/\\Lambda_\\mathrm{NS}$ {\\it decreases} with increasing $\\Lambda_\\mathrm{NS}$, while\nthe absolute error {\\it increases}.\nFrom these results, we conclude that the measurement uncertainty for\n$\\Lambda_\\mathrm{NS}$ after $N$ observations is\n\\begin{equation}\n \\delta\\Lambda_\\mathrm{NS}\\propto \\dfrac{\\Lambda_\\mathrm{NS}^{0.5^{+0.33}_{-0.33}}}{N^{0.7_{-0.2}^{+0.2}}}.\n\\end{equation}\nWe also find that while these results are inferred from paradigm A populations,\nparadigm B gives very similar results.\n\n\n\n\n\\section{Choice of underlying BBH model in LEA}\n\\begin{figure}\n\\centering    \n\\includegraphics[width=1.05\\columnwidth]{match-q-spin1-PhenomC-mf0_01.pdf}\n\\caption{%\nWe compare two alternatives of the tidal NSBH model from Ref.~\\cite{\nLackey:2013axa}, which differ in their underlying BBH prescriptions. One which\nwe use in this study uses SEOBNRv2, while the other uses IMRPhenomC as its\nbase. In this figure, we show the normalized overlap (match) between the two for\n$2,000,000$ points sampled uniformly in the NSBH parameter space. We find that\nfor $q\\gtrsim 4$ the discrepancies between IMRPhenomC and SEOBNRv2 as reported\nin~\\cite{Kumar:2015tha} dominate over tidal terms.\n\\label{fig:PhenomC_vs_SEOBNRv2_LEA}\n}\n\\end{figure}\nThe waveform model used in this paper is a variant of those calibrated\nin Ref~\\cite{Lackey:2013axa}. In that work, the authors also calibrate\na tidal prescription with the phenomenological model IMRPhenomC~\\cite{\nSantamaria:2010yb}\nas the base BBH model. Previous work~\\cite{Kumar:2015tha,Kumar:2016dhh} has \nshown that IMRPhenomC can exhibit pathological behavior for mass-ratios\n$q\\gtrsim 4$ and\/or non-zero black hole spins. We compute noise-weighted\ninner-products between the two variants for $2,000,000$ points sampled\nover the NSBH parameter space, and show the results in \nFig.~\\ref{fig:PhenomC_vs_SEOBNRv2_LEA}. We restrict the comparison to frequencies\nthat are affected by the tidal disruption of the NS, by integrating\nthe inner-products from $f = \\mathrm{max}(15, 0.01\/M)$~Hz\n(where $M$ is expressed in seconds ($1M_\\odot \\simeq 4.925\\mu$S, see\nEq.~(32-34) of~\\cite{Lackey:2013axa}).\nWe find that the differences between the two variants of LEA+ have mismatches\nof a few percent, while the tidal corrections contribute at a sub-percent\nlevel. We conclude that the differences of the underlying BBH model in LEA+\ndominate over its tidal calibration, and since SEOBNRv2 has been shown to\nbe more reliable than IMRPhenomC~\\cite{Kumar:2015tha,Kumar:2016dhh}, we \nrecommend the use of SEOBNRv2-based LEA+ in upcoming LIGO-Virgo analyses.\n\n\n\\end{appendix}\n\n\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\noindent\nElectronic excitations have been established as an alternative to nuclear recoils in direct detection of sub-GeV dark matter (DM). Nuclear recoil searches lose sensitivity at lower DM masses due to kinematic mismatch between the DM and heavier nuclei, whereas electronic transitions can potentially extract all of the DM kinetic energy during a DM-electron scattering event by excitation across an energy gap. Proposed targets, including noble gas atoms with ${\\cal O}(10\\, \\rm eV)$ ionization energies~\\cite{Essig:2011nj,Graham:2012su,Lee:2015qva,Essig:2017kqs,Catena:2019gfa,Agnes:2018oej,Aprile:2019xxb,Aprile:2020tmw},\nsemiconductors with ${\\cal O}(\\rm eV)$ electronic band gaps~\\cite{Essig:2011nj,Graham:2012su,Essig:2012yx,Lee:2015qva,Essig:2015cda,Derenzo:2016fse,Hochberg:2016sqx,Bloch:2016sjj,Kurinsky:2019pgb,Trickle:2019nya, Griffin:2019mvc,Griffin:2020lgd,Du:2020ldo}, and superconductors and Dirac materials with ${\\cal O}(\\rm meV)$ band gaps~\\cite{Hochberg:2015pha,Hochberg:2015fth,Hochberg:2016ajh,Hochberg:2017wce,Coskuner:2019odd,Geilhufe:2019ndy,inzani2021prediction}, extend the reach on DM mass well below the limit of nuclear recoil. Experimental searches using dielectric crystal targets are currently underway, specifically with Si (DAMIC~\\cite{deMelloNeto:2015mca,Aguilar-Arevalo:2019wdi,Settimo:2020cbq}, SENSEI~\\cite{Tiffenberg:2017aac,Crisler:2018gci,Abramoff:2019dfb,Barak:2020fql}, SuperCDMS~\\cite{Agnese:2014aze, Agnese:2015nto, Agnese:2016cpb, Agnese:2017jvy, Agnese:2018col, Agnese:2018gze, Amaral:2020ryn}) and Ge (EDELWEISS~\\cite{Armengaud:2018cuy, Armengaud:2019kfj, Arnaud:2020svb}, as well as SuperCDMS) which have been predicted to have excellent sensitivity down to $\\mathcal{O}(\\text{MeV})$ DM masses based on their $\\mathcal{O}(\\text{eV})$ band gaps.\n\nReliable theoretical predictions of target-specific transition rates are important not only for current experiments, but also for planning the next generation of detectors. Compared to the DM-induced electron ionization rate in noble gases like xenon~\\cite{Essig:2011nj,Graham:2012su,Lee:2015qva,Essig:2017kqs,Catena:2019gfa,Agnes:2018oej,Aprile:2019xxb,Aprile:2020tmw}, calculations for the DM-electron scattering rate in a crystal are more complicated. Ionization rates for noble gases can be calculated by considering each noble gas atom as an individual target, where the calculation simplifies to finding the ionization rate from an isolated atom, for which the wave functions and energy levels are well tabulated~\\cite{Bunge:1993jsz}. However, for crystal targets the atoms are not isolated and more involved techniques are required to obtain an accurate characterization of DM-electron interactions in a many-body system.\n\nThere have been a variety of approaches taken to compute the DM-electron scattering rate in crystals. One of the first attempts, Ref.~\\cite{Graham:2012su}, computed the rate with semi-analytic approximations for the initial and final state wave functions, and used the density of states to incorporate the electronic band structure. Later, Ref.~\\cite{Lee:2015qva} continued in this direction and used improved semi-analytic approximations for the initial state wave functions. Meanwhile, a fully numerical approach was advanced in Refs.~\\cite{Essig:2011nj, Essig:2015cda, Derenzo:2016fse} where density functional theory (DFT) was employed to calculate the valence and conduction electronic band structures and wave functions. The latter approach, as implemented in the \\texttt{QEdark} program and embedded in the \\textsc{Quantum ESPRESSO} package~\\cite{QE-2009,QE-2017,doi:10.1063\/5.0005082}, has become the standard for first-principles calculations of DM detection rates. Recently, in Refs.~\\cite{Trickle:2019nya, Griffin:2019mvc} we used a similar DFT approach as implemented in our own program for a study of DM-electron scattering in a variety of target materials. More recently there has been work utilizing the relation between the dielectric function and the spin-independent scattering rate~\\cite{Hochberg:2021pkt,Knapen:2021run,Knapen:2021bwg}, which also properly incorporates screening effects. \n\nThe goal of this work is to further extend the DM-electron scattering calculation in several key aspects, and present state-of-the-art predictions for Si and Ge detectors using a combination of DFT and semi-analytic methods. \nAs we will elaborate on shortly, the time- and resource-consuming nature of DFT calculations presents an intrinsic difficulty that has limited the scope of previous work in this direction to a restricted region of phase space; typically only bands within a few tens of eV above and below the band gap were included and electronic wave functions were cut off at a finite momentum.\nWe overcome this difficulty by implementing all-electron (AE) reconstruction (whose importance was previously emphasized in Ref.~\\cite{Liang:2018bdb}) to recover higher momentum components of DFT-computed wave functions, and by extending the calculation to bands farther away from the band gap using semi-analytic approximations along the lines of Refs.~\\cite{Graham:2012su,Lee:2015qva}. \nAs we will see, the new contributions computed here have a significant impact on detection prospects in cases where higher energy and\/or momentum regions of phase space dominate the rate, including scattering via a heavy mediator, and experiments with $\\mathcal{O}(10\\,\\text{eV})$ or higher energy thresholds. \nWe also stress that in contrast to the recent work emphasizing the relation between spin-independent DM-electron scattering rates and the dielectric function~\\cite{Hochberg:2021pkt,Knapen:2021run,Knapen:2021bwg}, \nour calculation can be straightforwardly extended to DM models beyond the standard spin-independent coupling. \nFurthermore, we do not make assumptions about isotropy for the majority of our calculation, and our framework is capable of treating anisotropic targets which exhibit smoking-gun daily modulation signatures~\\cite{Coskuner:2019odd,Trickle:2019nya,Geilhufe:2019ndy} (see also Refs.~\\cite{Griffin:2018bjn,Coskuner:2021qxo} for discussions of daily modulation for phonon excitations).\n\nOur calculation is implemented in an open-source program \\texttt{EXCEED-DM} (EXtended Calculation of Electronic Excitations for Direct detection of Dark Matter), to be released in an upcoming publication. \nCurrently a beta version of the program is available \\href{https:\/\/github.com\/tanner-trickle\/EXCEED-DM}{here}~\\cite{tanner_trickle_2021_4747696}. \nWe also make available our DFT-computed wave functions~\\cite{Trickle2021} and the output of \\texttt{EXCEED-DM}~\\cite{Trickle2021a} for Si and Ge.\n\n\\subsection{Overview of the Calculation and Key Results}\n\\label{subsec:key_results}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{figs\/Si_band_schematic.pdf}\n\t\\hspace{0.05\\textwidth}\n\t\\includegraphics[width=0.4\\textwidth]{figs\/Ge_band_schematic.pdf}\n    \\caption{Schematic representation of electronic states in Si (left) and Ge (right), divided into core, valence (``val''), conduction (``cond'') and free. Shaded regions indicate the range of energies for each type of electronic states. In a scattering process, electrons transition from either core or valence states, below the Fermi surface at $E = 0$, to conduction or free states above the band gap $E_\\text{g}$. As outlined in Sec.~\\ref{subsec:key_results} and explained in detail in Sec.~\\ref{sec:elec_states}, we compute the valence and conduction states numerically using DFT (including all-electron reconstruction), model the core states semi-analytically with RHF wave functions, and treat the free states as plane waves.}\n\t\\label{fig:band_cartoon}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figs\/key_results_summary.pdf}\n\t\\caption{\\label{fig:summary_plot}\n\t\tSelection of results from Sec.~\\ref{sec:projected_constraints}, for DM-electron scattering via a heavy mediator in a Ge target. \n    \\textbf{Left:} Contribution from each of the four transition types, valence to conduction (v$\\to$c), valence to free (v$\\to$f), core to conduction (c$\\to$c), and core to free (c$\\to$f) to the scattering rate binned in energy deposition (with $\\Delta\\omega=1$\\,eV) for a 1\\,GeV DM at a given reference cross section $\\overline{\\sigma}_e = 10^{-40}\\; \\text{cm}^2$. \\textbf{Right:} 95$\\%$ C.L.\\ projected limit (3 events) on $\\overline{\\sigma}_e$ assuming 1\\,kg-year exposure, for energy thresholds corresponding to 1 and 5 electron-hole pairs. We compare our results with \\texttt{QEdark} calculations in Refs.~\\cite{Essig:2015cda,Derenzo:2016fse} and the semi-analytic model of Lee et al~\\cite{Lee:2015qva}; see text for details.}\n\\end{figure}\n\nBefore delving into the technical details, let us give a brief overview of the calculation and highlight some key results. \nWe divide the electronic states in a (pure) crystal into four categories: core, valence, conduction and free, as illustrated in Fig.~\\ref{fig:band_cartoon} for Si and Ge and discussed in more detail in Sec.~\\ref{sec:elec_states}. \nAt zero temperature, electrons occupy states up to the Fermi energy, defined as the maximum of the valence bands and denoted by $E=0$. \nThe band gap $E_\\text{g}$, {\\it i.e.}\\ the energy gap between the occupied valence bands and unoccupied conduction bands, is typically $\\mathcal{O}(\\text{eV})$ for semiconductors, {\\it e.g.}\\ 1.11\\,eV for Si and 0.67\\,eV for Ge; this sets a lower limit on the energy deposition needed for an electron transition to happen.\n\nThe electronic states near the band gap deviate significantly from atomic orbitals and need to be computed numerically. We apply DFT methods (including AE reconstruction) for this calculation, and refer to the DFT-computed states as valence and conduction. \nSpecifically, for both Si and Ge, we take the first four bands below the gap to be valence, which span an energy range of $-12\\,$eV to 0 and $-14\\,$eV to 0, respectively, and take bands above the gap up to $E_\\text{dft}=60\\,$eV to be conduction. \n\nWith more computing power we can in principle include more states, both below and above the band gap, in the DFT calculation. \nHowever, since the states far from the band gap can be modeled semi-analytically with reasonable accuracy, computing them with DFT is inefficient. \nBelow the valence bands, electrons are tightly bound to the atomic nuclei. We model them using semi-analytic atomic wave functions and refer to them as core states. \nThese include the 1s, 2s, 2p states in Si and 1s, 2s, 2p, 3s, 3p, 3d states in Ge (the 3d states in Ge are sometimes referred to as semi-core, and we will compare the DFT and semi-analytic treatment of them in Secs.~\\ref{subsec:atomic_wf} and \\ref{subsec:cc}). \nFinally, above $E_\\text{dft}=60\\,$eV, we model the states as free electrons as they are less perturbed by the crystal environment.\n\nWith the electronic states modeled this way, we compute the rate for valence to conduction (v$\\to$c), valence to free (v$\\to$f), core to conduction (c$\\to$c) and core to free (c$\\to$f) transitions induced by DM scattering, as discussed in detail in Sec.~\\ref{sec:general_calc}. \nThe total rate is the sum of all four contributions. \nWe then use our calculation to update the projected reach of direct detection experiments in Sec.~\\ref{sec:projected_constraints}, and compare our results with previous literature.\n\nFigure~\\ref{fig:summary_plot} gives a glimpse of some of our key results. \nHere we consider the case of DM scattering via a heavy mediator in a Ge target. \nThe impact of core (3d) to conduction contributions is clearly visible from both the differential rate (left panel, for $m_\\chi=1\\,$GeV) and the projected reach (right panel). \nThey dominate the total rate for $m_\\chi\\gtrsim 10\\,$MeV, and, as we can see from the right panel of Fig.~\\ref{fig:summary_plot}, lead to significantly more optimistic reach compared to previous DFT calculations implemented in \\texttt{QEdark}~\\cite{Essig:2015cda,Derenzo:2016fse}; this is especially true for higher detector thresholds (corresponding to higher $Q$ values). \nNote that while Refs.~\\cite{Essig:2015cda,Derenzo:2016fse} included the 3d states in their DFT calculation, their contributions were significantly underestimated due to the absence of AE reconstruction. \nThe importance of AE reconstruction is also seen from the valence to conduction differential rate in the left panel of Fig.~\\ref{fig:summary_plot}, where our calculation predicts a much higher rate at $\\omega \\gtrsim 15\\,$eV compared to the \\texttt{QEdark} calculation in Ref.~\\cite{Derenzo:2016fse}.\nMeanwhile, accounting for in-medium screening (see Sec.~\\ref{subsec:in-med}) we find, consistent with Ref.~\\cite{Knapen:2021run}, a lower rate at energy depositions just above the band gap, and as a result, weaker reach at low $m_\\chi$, compared to Refs.~\\cite{Essig:2015cda,Derenzo:2016fse}.\nOn the other hand, our modeling of the core (3d) states is similar to the semi-analytic approach of Ref.~\\cite{Lee:2015qva}, and indeed we find very similar reach at large $m_\\chi$; however, the approach of Ref.~\\cite{Lee:2015qva} overestimates the rate at smaller $m_\\chi$ due to reduced accuracy in the modeling of the valence and conduction states. \nWe reserve a more detailed comparison with the literature for Sec.~\\ref{subsec:compare_to_prev}.\n\n\\section{Electronic States}\n\\label{sec:elec_states}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/ae_no_ae_core_wfc.pdf}\n    \\caption{Comparison of the Bloch wave function magnitudes, defined in Eq.~\\eqref{eq:av_bloch_state}, computed with DFT with (red, ``AE'') and without (blue, ``no AE'') AE reconstruction, and the semi-analytic core approximation of Eq.~\\eqref{eq:u_jnlmk} (green, ``core''). Shaded bands indicate the maximum and minimum values across all the bands belonging to the state type indicated in the upper right corner of each panel. AE reconstruction, discussed in Sec.~\\ref{subsubsec:ae}, recovers the large momentum behavior of the electronic wave functions. Core electronic states, such as those shown in the right panels and discussed in Sec.~\\ref{subsec:atomic_wf}, can be well modeled semi-analytically with atomic wave functions, as seen by the good agreement between the ``core'' and ``AE'' curves. When applicable, the semi-analytic parameterization is advantageous since the electronic wave functions are then known to arbitrarily large momentum.}\n    \\label{fig:binned_core_wf}\n\\end{figure}\n\nTo compute the DM-electron scattering rate one must understand the electronic states of the target. In targets with a periodic potential, Bloch's theorem states that the energy eigenstates can be indexed by a momentum, $\\vect{k}$, which lies within the first Brillouin zone (1BZ). These Bloch states, $\\psi_{i,\\vect{k}}$, where $i$ represents additional quantum numbers, are eigenstates of the discrete translation operator such that $\\psi_{i,\\vect{k}}(\\vect{x} + \\vect{r}) = e^{i \\vect{k} \\cdot \\vect{r}} \\psi_{i,\\vect{k}}(\\vect{x})$, which means the electronic wave functions can be written as\n\\begin{align}\n    \\label{eq:bloch_form}\n    \\psi_{i,\\vect{k}}(\\vect{x}) = \\frac{1}{\\sqrt{V}}\\, e^{i \\vect{k} \\cdot \\vect{x}} \\,u_{i,\\vect{k}}(\\vect{x}) \\,,\n\\end{align}\nwhere $u_{i,\\vect{k}}(\\vect{x} + \\vect{r}) = u_{i,\\vect{k}}(\\vect{x})$ and $V$ is the target volume. For every $\\vect{k}$ there exists a tower of eigenstates (labeled by $i$) of the target Hamiltonian which constitutes the complete set of states in the target. Unfortunately this complete set is not known for a general material and therefore a combination of approximations must be used to calculate them. \nAs discussed in Sec.~\\ref{subsec:key_results} and illustrated in Fig.~\\ref{fig:band_cartoon}, we divide the states into core, valence, conduction and free, and use a combination of numerical calculations and semi-analytic modeling. \nIn this section, we expand on the treatment of each type of electronic states.  \n\nWe first discuss the DFT calculation for valence and conduction states in Sec.~\\ref{subsec:dft}, and then move on to explain the semi-analytic treatment of core states in Sec.~\\ref{subsec:atomic_wf}. Our main results are contained in Fig.~\\ref{fig:binned_core_wf} where we compare the average magnitude of electronic wave functions,\nbinned in momentum (see Eq.~\\eqref{eq:av_bloch_state}), computed with and without AE reconstruction, discussed further in Sec.~\\ref{subsubsec:ae}, and, for the highest energy core states (2p in Si and 3d in Ge), those computed using the core approximation discussed in Sec.~\\ref{subsec:atomic_wf}. \nWe find that the AE reconstruction includes a significant contribution from wave functions at large momentum as expected, and that for the core states, the semi-analytic approach reproduces the large momentum components of these AE reconstructed DFT wave functions. \nLastly we will discuss the analytic treatment of the free states in Sec.~\\ref{subsec:pw_approx}.\n\n\\subsection{DFT Wave Functions and Band Structures}\n\\label{subsec:dft}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{figs\/bandstructure_Si.pdf}\n\\includegraphics[width=0.48\\textwidth]{figs\/bandstructure_Ge.pdf}\n\\caption{Calculated band structures of Si (left) using a PBE xc-functional within DFT and Ge (right) using a hybrid functional HSE06. The band gaps have been scissor corrected to their measured values near zero temperature, 1.11\\,eV and 0.67\\,eV for Si and Ge, respectively. The Fermi level is set to 0\\,eV in both panels.}\n\\label{fig:bandstructures}\n\\end{figure}\n\nIn principle, DFT provides an exact solution to the many-electron Schr\\\"odinger equation by the Hohenberg-Kohn theorems that treat all properties of a quantum many-body system as unique functionals of the ground state density. \nThey further show that the exact ground state density and energy can be found by minimizing the total energy of the system~\\cite{Hohenberg:1964zz, martin2020electronic}. This becomes tractable by the Kohn-Sham (KS) equations that reduce the many-body problem to non-interacting electrons moving in an effective potential, $V_{\\mathrm{eff}}$, \n\\begin{equation}\n\\left( \\frac{p^2}{2m_e} + V_{\\mathrm{eff}} - \\epsilon_i \\right) \\psi_i = 0 \\,,\\qquad\\quad\nV_{\\mathrm{eff}} = V_{\\mathrm{ext}} + V_{\\mathrm{H}} + V_{\\mathrm{xc}}\\,,\n\\label{eq:dft_hamiltonian_eq}\n\\end{equation}\nwhere $\\epsilon_i$ is the orbital energy of the KS orbital $\\psi_i$~\\cite{Kohn:1965zzb}. The external potential $V_{\\mathrm{ext}}$ and Hartree potential $V_{\\mathrm{H}}$, are known, while the exchange-correlation (xc) potential $V_{\\mathrm{xc}}$, which contains the many-body interactions, must be approximated. Herein lies the deviation from the exact solution, and although various formulations of xc-energy functionals have been successful, the choice of xc-functional will affect the predicted electronic states and hence calculated transition rates. For Si, we use PBE~\\cite{Perdew1996}, a type of generalized gradient approximation (GGA) xc-functional which is one of the most popular and low-cost choices. Local and semi-local based xc-functionals, such as PBE, suffer from a self-interaction error and band gap underestimation, which we modify with a ``scissor correction'' where the bands are shifted to match the experimentally determined values of band gap. For Ge, this underestimation results in zero band gap with PBE, therefore we instead use a hybrid functional, which mixes a parameterized amount of exact exchange into the xc-functional, correcting band gaps and band widths by error cancellation at the cost of increased computation time. We use the range-separated hybrid functional HSE06~\\cite{Heyd2003a,Heyd2006}, which applies a screened Coulomb potential to correct the long-range behavior of the xc-potential, giving high accuracy at a mid-level computational cost. Our computed band structures for Si and Ge are shown in Fig.~\\ref{fig:bandstructures}.\n\nThe periodic Bloch wave functions, $u_{i,\\vect{k}}(\\vect{x})$, Eq.~\\eqref{eq:bloch_form}, for band $i$ and Bloch wave vector $\\vect{k}$ are computed by finding the Fourier coefficients, $\\widetilde{u}_{i, \\vect{k}, \\vect{G}}$ (which satisfy the normalization condition, $\\sum_\\vect{G} | \\widetilde{u}_{i, \\vect{k}, \\vect{G}}|^2 = 1$):\n\\begin{align}\n    u_{i, \\vect{k}}(\\vect{x}) = \\sum_\\vect{G} \\widetilde{u}_{i,\\vect{k},\\vect{G}} \\,e^{i \\vect{G} \\cdot \\vect{x}} \\, .\n    \\label{eq:bloch_fourier}\n\\end{align}\nThe number of reciprocal lattice vectors $\\vect{G}$ kept in the sum is conventionally set by an energy cutoff, $E_\\text{cut}$, such that $|\\vect{k} + \\vect{G}|^2 < 2 m_e E_\\text{cut}$. These Bloch wave function coefficients $\\widetilde{u}_{i,\\vect{k},\\vect{G}}$ for both Si and Ge are computed with the projector augmented wave (PAW) method~\\cite{Blo,Kresse1999} within \\textsc{vasp}~\\cite{Kresse1993,Kresse1994,Kresse1996,Kresse1996a} up to $E_\\text{cut} = 1$ keV on a $10\\times10\\times10$ uniform $\\vect{k}$ mesh over the 1BZ. We then include AE reconstruction effects up to a higher energy cutoff, $E_\\text{AE} = 2$ keV, which recovers higher momentum components of the wave functions up to $|\\vect{k} + \\vect{G}|^2 < 2 m_e E_\\text{AE}$, as discussed in more detail in Sec.~\\ref{subsubsec:ae}. The Bloch wave function coefficients, $\\widetilde{u}_{i,\\vect{k},\\vect{G}}$, for Si and Ge used for this work can be found here~\\cite{Trickle2021}.\n\nA final consideration of using DFT wave functions is that DFT is fundamentally a ground state method, and the KS conduction band states are only approximations to excited states. Excited state methodologies are much more computationally expensive than ground state KS-DFT. Furthermore, since excited state quasiparticles, such as excitons, have been argued to have a negligible effect on the calculation of DM scattering rates~\\cite{Derenzo:2016fse}, they are neglected in our calculations. \n\n\\subsubsection{All-electron Reconstruction}\n\\label{subsubsec:ae}\n\nThere are many different approaches to find the eigenstates of Eq.~\\eqref{eq:dft_hamiltonian_eq}. The PAW method~\\cite{Blo} is one such standard approach. The main idea of the PAW method is to split up the calculation of the eigenstates: near the ionic centers the wave functions resemble the eigenstates of an isolated atom, while further away they can be computed numerically with a pseudopotential. This greatly simplifies the numeric calculation since the focus is then on large distance (small momentum), and the small distance (high momentum) pieces can be self-consistently reintroduced after the main part of the DFT calculation. The large distance components of the wave function are known as ``pseudo wave functions'' (PS wave functions) and the total wave functions are known as the ``all-electron wave functions'' (AE wave functions), indicating that all of the wave function components are included. We will now give a brief overview of how the AE wave functions can be reconstructed from the PS wave functions, computed with PAW-based DFT codes, and refer the reader to Refs.~\\cite{Blo,Kresse1999,rostgaard2009projector,pawpy} for more detailed information.\\footnote{It is possible to calculate all electronic eigenstates, including the core, self-consistently by other more complex methods such as the full-potential linearized augmented plane wave (FP-LAPW) method or the relaxed-core PAW (RC-PAW) method.} \n\nThe AE wave functions, $| \\Psi^\\text{AE} \\rangle$ are built from two components. Near the ionic core, or inside an ``augmentation sphere,'' $| \\Psi^\\text{AE} \\rangle$ is expanded in a set of basis functions, $| \\phi^\\text{AE}_j \\rangle$, which are simply taken to be the wave functions of an isolated atom,\n\\begin{align}\n    | \\Psi^\\text{AE} \\rangle = \\sum_j c_j \\,| \\phi^\\text{AE}_j \\rangle \\, .\n\\end{align}\nOutside of the augmentation sphere, $| \\Psi^\\text{AE} \\rangle = | \\Psi^\\text{PS} \\rangle$. Near the ionic core the PS wave functions $| \\Psi^\\text{PS} \\rangle$ are expanded in a set of basis functions $| \\phi^\\text{PS}_j \\rangle$ that are computationally more convenient than the $| \\phi^\\text{AE}_j \\rangle$. Therefore,\n\\begin{align}\n    | \\Psi^\\text{AE} \\rangle = | \\Psi^\\text{PS} \\rangle - \\sum_j c_j' \\,| \\phi^\\text{PS}_j \\rangle + \\sum_j c_j \\,|\\phi^\\text{AE}_j \\rangle \\, ,\n\\end{align}\nwhich simply replaces the components in $| \\Psi^\\text{PS} \\rangle$ within the augmentation sphere with the AE wave function. To find the $c$ coefficients we insert an identity, $\\mathds{1} = \\sum_j | \\phi^\\text{AE}_j \\rangle \\langle p^\\text{AE}_j | = \\sum_j | \\phi^\\text{PS}_j \\rangle \\langle p^\\text{PS}_j |$, where $|p^{\\text{AE}\/\\text{PS}}_j \\rangle$ are projector functions, defined to satisfy this identity within the augmentation sphere. Therefore, $c_j = \\langle p^\\text{AE}_j | \\Psi^\\text{AE} \\rangle$, $c_j' = \\langle p^\\text{PS}_j | \\Psi^\\text{PS} \\rangle$. The last ingredient to compute $| \\Psi^\\text{AE} \\rangle$ from $| \\Psi^\\text{PS} \\rangle$ is to require that $| \\phi^\\text{AE}_j \\rangle$ is related to $| \\phi^\\text{PS}_j \\rangle$ via a transformation, $| \\phi^\\text{AE} \\rangle = \\mathcal{T} | \\phi^\\text{PS} \\rangle$. This implies that all the PS states are related to AE states by this transformation $\\mathcal{T}$, such that $c_j = c_j'$ and the AE reconstruction can be written as\n\\begin{align}\n    | \\Psi^\\text{AE} \\rangle = | \\Psi^\\text{PS} \\rangle + \\sum_j \\left( | \\phi^\\text{AE}_j \\rangle - | \\phi^\\text{PS}_j \\rangle \\right) \\langle p^\\text{PS}_j | \\Psi^\\text{PS} \\rangle \\, .\n\\end{align}\nIn practice, we implement the AE reconstruction with \\texttt{pawpyseed}~\\cite{pawpy}, and the plane wave expansion cutoff of $| \\Psi^\\text{AE} \\rangle$, $E_\\text{AE}$, can be increased from the initial $E_\\text{cut}$. We use $E_\\text{AE} = 2$ keV.\n\nTo visualize the effect of AE reconstruction, we plot in Fig.~\\ref{fig:binned_core_wf} the average magnitude of the Bloch wave functions, binned in $q$,\n\\begin{align}\n    \\left\\langle \\left| \\widetilde{u}_i \\right|^2 \\right\\rangle (q; \\Delta q) \\equiv \\frac{1}{N_q} \\sum_{\\vect{k}}\\sum_{\\vect{G}} |\\widetilde{u}_{i,\\vect{k},\\vect{G}}|^2 \\,\\theta(q + \\Delta q - |\\vect{k} + \\vect{G}|) \\,\\theta(|\\vect{k} + \\vect{G}| - q) \\,,\n    \\label{eq:av_bloch_state}\n\\end{align}\nwhere $\\widetilde{u}_{i,\\vect{k},\\vect{G}}$ are the Fourier components of the Bloch wave functions, defined in Eq.~\\eqref{eq:bloch_fourier}. Each bin in momentum space extends from $q$ to $q + \\Delta q$ with $\\Delta q=1\\,$keV, and $N_q$ is a normalization factor equal to the number of points in a bin, $N_q = \\sum_{\\vect{k}} \\sum_{\\vect{G}} \\theta(q + \\Delta q - |\\vect{k} + \\vect{G}|)\\,\\theta(|\\vect{k} + \\vect{G}| - q)$. \nWe see that AE reconstruction recovers the high momentum components, which as we will see can significantly affect the DM-induced transition rate for processes which favor large momentum transfers (such as processes mediated by heavy particles), or processes limited to larger $\\omega$ ({\\it e.g.}\\ higher experimental thresholds where large $q$ processes are the only kinematically allowed transitions).\nPrevious DFT calculations of DM-induced electron transition rates, with the exceptions of Ref.~\\cite{Liang:2018bdb, Trickle:2019nya, Griffin:2019mvc}, used only the pseudo wave functions, $| \\Psi^\\text{PS} \\rangle$ as opposed to the AE wave functions, $| \\Psi^\\text{AE} \\rangle $, and have therefore underestimated detection rates in several cases.\n\n\\subsection{Atomic Wave Functions}\n\\label{subsec:atomic_wf}\n\nIf one could reconstruct the AE wave functions arbitrarily deep into the band structure, and to arbitrarily high momentum, one could calculate an accurate representation of the complete set of electronic states with a DFT calculation. In practice, however, this is neither feasible nor necessary. States deep in the band structure are more isolated from the influence of the crystal environment, and so an isolated atomic approximation becomes valid. We refer to these inner, tightly bound electrons as core electrons. In Si, we will show that the 2p states and below can be treated as core, while in Ge, the 3d states and below can, as alluded to in Fig.~\\ref{fig:band_cartoon}. The purpose of this subsection is to expand on the atomic approximation for core electrons and discuss its accuracy. \n\nMore precisely, the initial states of a transition should be taken as a linear combination of isolated atomic wave functions that is in Bloch form (known as Wannier states):\n\\begin{align}\n    \\label{eq:core_elec_wf}\n    \\psi_{\\kappa nlm,\\vect{k}}(\\vect{x}) = \\frac{1}{\\sqrt{N}} \\sum_\\vect{r} e^{i \\vect{k} \\cdot (\\vect{r}+\\vect{x}_\\kappa)} \\,\\psi_{\\kappa nlm}^\\text{atom}(\\vect{x} - \\vect{r} - \\vect{x}_\\kappa) \\,,\n\\end{align}\nwhere $\\kappa$ labels the atom in the primitive cell, $n, l, m$ are the standard atomic quantum numbers, $\\vect{x}_\\kappa$ is the equilibrium position of the $\\kappa^\\text{th}$ atom, $\\sum_\\vect{r}$ sums over all primitive cells in the lattice, and $N$ is the total number of cells. \nIn contrast to the valence and conduction states discussed in the previous subsection, the core states are labeled by $(\\kappa nlm)$ rather than band index $i$. \nThe corresponding periodic (dimensionless) $u$ functions can be easily obtained via Eq.~\\eqref{eq:bloch_form}:\n\\begin{equation}\nu_{\\kappa nlm,\\vect{k}}(\\vect{x}) = \\sqrt{\\Omega}\\, \\sum_\\vect{r} e^{-i \\vect{k} \\cdot (\\vect{x}-\\vect{r}-\\vect{x}_\\kappa)} \\,\\psi_{\\kappa nlm}^\\text{atom}(\\vect{x} - \\vect{r} - \\vect{x}_\\kappa) \\,,\n\\label{eq:u_jnlmk}\n\\end{equation}\nwhere $\\Omega=V\/N$ is the primitive cell volume. \n\nIn general, the atomic wave functions $\\psi_{\\kappa nlm}^\\text{atom}$ are not known analytically, but are expanded in a basis of well-motivated analytic functions. The basis coefficients are then fit by solving the isolated atomic Hamiltonian, giving a semi-analytic expression for $\\psi_{\\kappa nlm}^\\text{atom}$. We use a basis of Slater type orbital (STO) wave functions whose radial component is\n\\begin{align}\n    R_\\text{STO}(r; Z, n) & = a_0^{-3\/2} \\frac{\\left(2 Z \\right)^{n + \\frac{1}{2}}}{\\sqrt{(2n)!}}  \\left( \\frac{r}{a_0} \\right)^{n - 1} e^{-Z r\/a_0} \\,, \n   \\label{eq:sto_wf}\n\\end{align}\nwhere $a_0 =0.53$\\,\\AA\\;$= (3.7\\,\\text{keV})^{-1}$ is the Bohr radius, and $Z$ is an effective charge of the ionic potential. Including the angular part, the atomic wave functions are then\n\\begin{align}\n    \\psi^{\\rm atom}_{\\kappa nlm}(\\vect{x}) = \\sum_j C_{jln,\\kappa} R_\\text{STO}(x; Z_{jl,\\kappa}, n_{jl,\\kappa}) Y_l^m(\\hat{\\vect{x}}) \\,,\n\\end{align}\nwhere $C_{jln,\\kappa}, Z_{jl,\\kappa}, n_{jl,\\kappa}$ are tabulated in Ref.~\\cite{Bunge:1993jsz}, and $Y_l^m(\\hat{\\vect{x}})$ are the spherical harmonics with the Condon-Shortley phase convention~\\cite{E.U.Condon1935}.\n\nTo assess the accuracy of the atomic wave function approximation, we temporarily push the DFT calculation beyond its default regime (valence and conduction), to the highest core states -- 2p states in Si and 3d states in Ge, where it is still computationally feasible -- and compare the numerical wave functions to the semi-analytic ones discussed above. \nThe results, in terms of the average magnitude of Bloch wave functions defined in Eq.~\\eqref{eq:av_bloch_state}, are shown in the right panels of Fig.~\\ref{fig:binned_core_wf}.\\footnote{The flatness of band structures offers a complementary check of the validity of the atomic approximation. We have verified that the DFT computed energy eigenvalues indeed have a small variance for the highest core states, as expected.}\nWe see that the atomic approximation accurately reproduces the numerical wave functions up to the momentum cutoff $\\sqrt{2m_e E_\\text{AE}}\\simeq 50\\,$keV for $E_\\text{AE}=2\\,$keV. \nThese plots also show the limitation of DFT calculations. \nWhile AE reconstruction recovers higher-momentum components of electronic wave functions, it is not feasible to expand the plane wave basis set to arbitrarily high cutoff. \nHowever, having verified the atomic approximation for the highest core states, we can use it for all core states with confidence, allowing us to more easily include the high momentum components beyond the DFT cutoff.\n\n\\subsection{Plane Wave Approximation}\n\\label{subsec:pw_approx}\n\nWith the inclusion of the semi-analytic core states, all of the states below the band gap have been modeled. States above the band gap can also be computed with DFT methods, as described in Sec.~\\ref{subsec:dft}. Similar to valence bands, there are practical limitations to how many conduction bands can be included. To remedy this in the simplest way possible, we model states far above the band gap as plane waves,\n\\begin{equation}\n    \\label{eq:free_elec_wf}\n    \\psi_{\\vect{G}, \\vect{k}}(\\vect{x})  = \\frac{1}{\\sqrt{V}}\\, e^{i \\left( \\vect{k} + \\vect{G} \\right) \\cdot \\vect{x}} \\,,\\qquad\\quad\n E_{\\vect{G}, \\vect{k}}   = \\frac{| \\vect{k} + \\vect{G} |^2}{2 m_e}\\,,\n\\end{equation}\nwhere $\\vect{G}$ is a reciprocal lattice vector, and plays the role of a band index. (To understand this, simply note that every momentum can be decomposed into a $\\vect{k}$ vector inside the 1BZ and a reciprocal lattice vector. Integrating over the momentum of plane wave states amounts to a $\\vect{k}$ integral within the 1BZ and a $\\vect{G}$ sum.)\nFrom Eq.~\\eqref{eq:bloch_form} we see that the corresponding periodic $u$ functions are simply\n\\begin{equation}\nu_{\\vect{G},\\vect{k}}(\\vect{x}) = e^{i\\vect{G}\\cdot\\vect{x}} \\,.\n\\label{eq:u_free}\n\\end{equation}\nThe plane wave approximation is often used in atomic ionization rate calculations, with the inclusion of a \\textit{Fermi factor}, $F(\\nu)$,\n\\begin{align}\n    \\label{eq:fermi_factor}\n    F(\\nu)  = \\frac{\\nu}{1 - e^{-\\nu}}\\,, \\qquad \\nu(Z_\\text{eff}, E)  = 2 \\pi Z_\\text{eff} \\frac{\\alpha m_e}{\\sqrt{2 m_e E}} \\, , \n\\end{align}\nwhere $E$ is the final state electron energy, and $Z_\\text{eff}$ is an effective charge parameter, which enhances the transition rate at low $E$ to account for the long range behavior of the Coulomb potential. See Refs.~\\cite{Essig:2011nj,Graham:2012su, Lee:2015qva, Agnes:2018oej} for more details. In atomic ionization calculations one usually takes $Z_\\text{eff}$ to be related to the binding energy of the initial state, $E_B$,\n\\begin{align}\n    \\label{eq:z_eff_binding_approx}\n    Z_\\text{eff} = n \\sqrt{\\frac{E_B}{13.6 \\text{ eV}}} \\, ,\n\\end{align}\nwhere $n$ is the principal quantum number. Since the rate is proportional to the Fermi factor, $Z_\\text{eff} = 1$ is seen as the conservative choice. Later in Secs.~\\ref{subsec:vf} and \\ref{subsec:cf} we quantify how much of an effect this has on the transition rate. This uncertainty is only important for very high experimental thresholds, and generally we find that $Z_\\text{eff} = 1$ leads to a smoother match (within an $\\mathcal{O}(1)$ factor) to conduction band contributions from DFT calculations.\n\n\\section{Electronic Transition Rates}\n\\label{sec:general_calc}\n\nWe now present the DM-induced electron transition rate calculation. We begin with a general discussion and then in Secs.~\\ref{subsec:vc}-\\ref{subsec:cf} consider the four different transition types in turn: valence to conduction ($\\mathrm{v} \\rightarrow \\mathrm{c}$), valence to free ($\\mathrm{v} \\rightarrow \\mathrm{f}$), core to conduction ($\\mathrm{c} \\rightarrow \\mathrm{c}$) and core to free ($\\mathrm{c}\\rightarrow\\mathrm{f}$). Finally, in Sec.~\\ref{subsec:in-med} we discuss the treatment of in-medium screening.\n\nThe general derivation has been discussed previously (see {\\it e.g.}\\ Refs.~\\cite{Graham:2012su, Essig:2015cda,Catena:2019gfa, Trickle:2019nya, Liang:2018bdb}), and we repeat it here for completeness and clarity, as a variety of conventions have been used. Beginning with Fermi's Golden Rule, the transition rate between electronic states $\\ket{i, s}$ and $\\ket{f, s'}$ due to scattering with an incoming non-relativistic DM particle, $\\chi$, with mass $m_\\chi$, velocity $\\vect{v}$, and spin $\\sigma$ is given by\n\\begin{align}\n    \\label{eq:fermi}\n    \\Gamma_{i, s, \\sigma \\rightarrow f, s', \\sigma'}(\\vect{v}) = 2 \\pi V \\int \\frac{d^3 q}{(2 \\pi)^3}\\, \\big|\\bra{\\vect{p'}, \\sigma'; f, s'} \\,\\delta\\hat{H} \\,\\ket{\\vect{p}, \\sigma; i, s} \\big|^2 \\,\\delta(E_{f, s'} - E_{i, s} - \\omega_\\vect{q})\\,,\n\\end{align}\nwhere $\\ket{\\vect{p}, \\sigma; i, s} = \\ket{\\vect{p}, \\sigma} \\otimes \\ket{i, s}$, $\\vect{q}$ is the momentum deposited onto the target, $\\vect{p} = m_\\chi \\vect{v}$, $\\vect{p}' = \\vect{p} - \\vect{q}$, $\\delta\\hat{H}$ is the interaction Hamiltonian, $V$ is total volume of the target, and $\\omega_\\vect{q}$ is the energy deposition:\n\\begin{equation}\n    \\omega_\\vect{q} = \\frac{1}{2} m_\\chi v^2 - \\frac{\\left( m_\\chi \\vect{v} - \\vect{q} \\right)^2}{2 m_\\chi} = \\vect{q} \\cdot \\vect{v} - \\frac{q^2}{2 m_\\chi} \\, .\n    \\label{eq:w_kin}\n\\end{equation}\nWe assume that all quantum states are unit normalized. Modulo in-medium screening effects, discussed below in Sec.~\\ref{subsec:in-med}, we can write Eq.~\\eqref{eq:fermi} in terms of the standard QFT matrix element, defined with plane wave incoming and outgoing states, by inserting $\\mathbbm{1} = V \\sum_s \\int \\frac{d^3k}{(2\\pi)^3} \\ket{\\vect{k},s} \\bra{\\vect{k},s}$ and using \n\\begin{align}\n    \\label{eq:qft_m}\n    \\langle \\vect{p}', \\sigma'; \\vect{k}', s' | \\,\\delta \\hat H\\, | \\vect{p}, \\sigma; \\vect{k}, s \\rangle \\equiv \\frac{(2\\pi)^3}{V^2} \\frac{\\mathcal{M}_{\\sigma' s'\\sigma s}(\\vect{p}', \\vect{k}', \\vect{p}, \\vect{k})}{4 m_e m_\\chi} \\,\\delta^{(3)}\\left( \\vect{p}' + \\vect{k}' - \\vect{p} - \\vect{k}\\right) \\, .\n\\end{align}\nWe find\\\\[-4pt]\n\\begin{minipage}{\\textwidth}\n\\begin{align}\n    \\label{eq:fermi_m}\n    \\Gamma_{i, s, \\sigma \\rightarrow f, s', \\sigma'}(\\vect{v}) = &\\;\\frac{2 \\pi}{16 V m_e^2 m_\\chi^2} \\int \\frac{d^3 q}{(2 \\pi)^3} \\,\\delta(E_{f, s'} - E_{i, s} - \\omega_\\vect{q}) \\nonumber\\\\\n    &\\times \\left| \\int \\frac{d^3k}{(2\\pi)^3} \\,\\mathcal{M}_{\\sigma' s'\\sigma s}(\\vect{p} -\\vect{q}, \\vect{k}+\\vect{q}, \\vect{p}, \\vect{k}) \\,\\widetilde{\\psi}_f^*(\\vect{k} + \\vect{q}) \\widetilde{\\psi}_i(\\vect{k}) \\right|^2  \\, ,\n\\end{align}\n\\end{minipage}\nwhere $\\widetilde{\\psi}_i(\\vect{k}) = \\sqrt{V} \\langle \\vect{k} | i \\rangle$. \n\nWe will limit our analysis to matrix elements which only depend on $\\vect{q}$, and assume that the electron energy levels are also spin independent, which allows the spin sums to be easily computed:\n\\begin{align}\n    \\label{eq:fermi_spin_av}\n    \\overline{\\Gamma}_{i \\rightarrow f} & \\equiv \\frac{1}{2} \\sum_{\\sigma, \\sigma'} \\sum_{s, s'} \\Gamma_{i, s, \\sigma \\rightarrow f, s', \\sigma'} \\nonumber\\\\\n    & = \\frac{4 \\pi}{16 V m_e^2 m_\\chi^2}\\int \\frac{d^3 q}{(2 \\pi)^3} \\,\\overline{|\\mathcal{M}(\\vect{q})|^2}\\, |f_{i \\rightarrow f}|^2 \\,\\delta \\left( E_f - E_i - \\omega_\\vect{q} \\right) , \\\\\n    f_{i \\rightarrow f} & \\equiv \\int \\frac{d^3k}{(2\\pi)^3} \\,\\widetilde{\\psi}_f^*(\\vect{k} + \\vect{q})\\, \\widetilde{\\psi}_i(\\vect{k}) = \\int d^3x \\, e^{i \\vect{q} \\cdot \\vect{x} }\\, \\psi_f^*( \\vect{x} )\\, \\psi_i(\\vect{x}) \\,,\n    \\label{eq:wf_form_fac}\n\\end{align}\nwhere $\\overline{\\left| \\mathcal{M} \\right|^2}$ is the spin averaged matrix element squared and we have defined a crystal form factor $f_{i\\rightarrow f}$, written in terms of both momentum and position space representations of the wave functions. \n\nThe transition rate per target mass, $R_{i \\rightarrow f}$, is then given by\n\\begin{align}\n    \\label{eq:rate}\n    R_{i \\rightarrow f} = \\frac{1}{\\rho_T} \\frac{\\rho_\\chi}{m_\\chi} \\int d^3v f_\\chi(\\vect{v}) \\,\\overline{\\Gamma}_{i \\rightarrow f} \\,,\n\\end{align}\nwhere $\\rho_T$ is the target density, $\\rho_\\chi = 0.4 \\text{ GeV}\/\\text{cm}^{3}$ is the local DM density, and $f_\\chi$ is taken to be a boosted Maxwell-Boltzmann distribution. The total rate, $R$, is then simply the sum over all possible transitions from initial to final states. Since the only $\\vect{v}$ dependence in Eq.~\\eqref{eq:rate} comes from the energy conserving delta function, we perform the $\\vect{v}$ integral first and define $g(\\vect{q}, \\omega) = 2 \\pi \\int d^3v f_\\chi(v) \\delta(\\omega - \\omega_\\vect{q})$. This integral can be evaluated analytically (see {\\it e.g.}\\ Refs.~\\cite{Coskuner:2019odd,Trickle:2019nya,Coskuner:2021qxo}):\n\\begin{align}\n    g(\\vect{q}, \\omega) &\n    = \\frac{2\\pi^2 v_0^2}{N_0}\\frac{1}{q} \\left( e^{-v_-^2\/v_0^2} - e^{-v_\\text{esc}^2\/v_0^2} \\right) ,\\\\\n    v_- & = \\text{min} \\left\\{ \\frac{1}{q} \\left| \\omega + \\frac{q^2}{2m_\\chi} + \\vect{q} \\cdot \\vect{v}_e  \\right| , v_{\\text{esc}} \\right\\} ,\\label{eq:v_minus}\n\\end{align}\nwhere $\\omega = E_f - E_i$ is the deposited energy, and $N_0$ is a normalization factor such that $\\int d^3\\vect{v} f_\\chi(\\vect{v}) = 1$. \nWe take the DM velocity distribution parameters to be $v_0 = 230$ km$\/$s , $v_\\text{esc} = 600$ km$\/$s, and $v_e = 240$ km$\/$s. \nThe total rate then becomes\n\\begin{align}\n    R = \\frac{2}{16 V m_e^2 m_\\chi^3} \\frac{\\rho_\\chi}{\\rho_T}\\sum_{i, f} \\int \\frac{d^3q}{(2\\pi)^3} \\overline{\\left| \\mathcal{M}(\\vect{q}) \\right|^2} \\,g(\\vect{q}, \\omega) \\left| f_{i \\rightarrow f}(\\vect{q}) \\right|^2 \\, .\n\\end{align}\nHere we focus on simple DM models, such as the kinetically mixed dark photon or leptophilic scalar mediator models. In these models $\\mathcal{M}(\\vect{q})$ can be factorized as $\\mathcal{M}(q) = \\mathcal{M}(q_0) \\mathcal{F}_\\text{med}(q_0\/q) \\left(f_e\/f_e^0 \\right)$, where $\\mathcal{F}_\\text{med}(q_0\/q) = 1$ for a heavy mediator and $\\mathcal{F}_\\text{med}(q_0\/q) = (q_0\/q)^2$ for a light mediator, and $f_e\/f_e^0$ is a screening factor discussed in more detail in Sec.~\\ref{subsec:in-med}. As in previous works, we choose the reference momentum transfer to be $q_0 = \\alpha m_e$. We can then finally write the rate in terms of a reference cross section,\n\\begin{align}\n    \\overline{\\sigma}_e = \\frac{\\mu_{\\chi e}^2}{16\\pi m_\\chi^2 m_e^2} \\overline{|\\mathcal{M}(q_0)|^2} \\, ,\n\\end{align}\nand find\n\\begin{align}\n    \\label{eq:rate_simple}\n    R = \\frac{2 \\pi \\overline{\\sigma}_e}{V \\mu_{\\chi e}^2 m_\\chi} \\frac{\\rho_\\chi}{\\rho_T} \\sum_{i, f} \\int \\frac{d^3q}{(2\\pi)^3} \\left( \\frac{f_e}{f_e^0} \\right)^2 \\mathcal{F}_\\text{med}^2 \\,g(\\vect{q}, \\omega) \\left| f_{i \\rightarrow f}(\\vect{q}) \\right|^2 \\, .\n\\end{align}\nAnother useful quantity is the \\textit{binned} rate (the rate for energy deposition between $\\omega$ and $\\omega + \\Delta \\omega$), $\\Delta R_\\omega$, defined as\n\\begin{align}\n    \\Delta R_\\omega = \\frac{2 \\pi \\overline{\\sigma}_e}{V \\mu_{\\chi e}^2 m_\\chi} \\frac{\\rho_\\chi}{\\rho_T}\\sum_{i, f}& \\,  \\theta(\\omega + \\Delta \\omega - E_f + E_i) \n    \\,\\theta(\\omega - E_f + E_i) \\nonumber \\\\\n& \\times \\int \\frac{d^3q}{(2\\pi)^3} \\left( \\frac{f_e}{f_e^0} \\right)^2\\,\\mathcal{F}_\\text{med}^2 \\,g(\\vect{q}, \\omega) \\left| f_{i \\rightarrow f}(\\vect{q}) \\right|^2 .\n    & \n    \\label{eq:binned_rate_E}\n\\end{align}\n\n\\subsection{Valence to Conduction}\n\\label{subsec:vc}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/vc_contribution_diff_rate.pdf}\n    \\caption{DM-electron scattering rate from valence to conduction bands binned in energy deposition (with $\\Delta\\omega=1$\\,eV) for 1\\,GeV DM, light (top row) and heavy (bottom row) mediators, assuming $\\overline{\\sigma}_e = 10^{-40}\\; \\text{cm}^2$, computed with vs.\\ without AE reconstruction. Valence states included are the first four bands below the band gap, and conduction states included are all bands up to $E_\\text{dft}=60$\\,eV.}\n    \\label{fig:vc_diff_rate}\n\\end{figure}\n\nWe begin with valence to conduction band transitions. The initial (final) states are indexed by band number, $i (f)$, and Bloch momentum, $\\vect{k}_i (\\vect{k}_f)$ inside the 1BZ. The wave functions in Eq.~\\eqref{eq:bloch_form} can be substituted into the crystal form factor in Eq.~\\eqref{eq:wf_form_fac},\n\\begin{align}\n    f_{i, \\vect{k}_i \\rightarrow f, \\vect{k}_f} & = \\frac{1}{V} \\int d^3x \\,e^{i \\left( \\vect{k}_i - \\vect{k}_f + \\vect{q} \\right) \\cdot \\vect{x}}\\, u^*_{f, \\vect{k}_f}(\\vect{x}) \\,u_{i, \\vect{k}_i}(\\vect{x}) \\nonumber \\\\\n                                              & = \\sum_\\vect{G} \\delta_{\\vect{q},\\, \\vect{k}_f - \\vect{k}_i + \\vect{G}} \\,\\frac{1}{\\Omega}\\, \\int_\\text{cell} d^3x \\,e^{i \\vect{G} \\cdot \\vect{x}} \\,u^*_{f, \\vect{k}_f}(\\vect{x}) \\,u_{i, \\vect{k}_i}(\\vect{x}) \\,,\n                                \\label{eq:f_val_to_cond}\n\\end{align}\nwhere the integral is over the primitive cell with volume $\\Omega$, and we have used the identity $\\sum_\\vect{r} e^{i \\vect{q} \\cdot \\vect{r}} = N \\sum_\\vect{G} \\delta_{\\vect{q}, \\vect{G}}$. The total rate in Eq.~\\eqref{eq:rate_simple} is then\n\\begin{align}\n    R = \\frac{2 \\pi \\overline{\\sigma}_e}{\\mu_{\\chi e}^2 m_\\chi} \\frac{\\rho_\\chi}{\\rho_T} \\sum_{i = 1}^{N_\\text{v}} \\sum_{f = 1}^{N_\\text{c}} & \\int_\\text{1BZ} \\frac{d^3k_i}{(2\\pi)^3} \\frac{d^3k_f}{(2\\pi)^3} \\sum_\\vect{G} \\left( \\frac{f_e}{f_e^0} \\right)^2 \\mathcal{F}_\\text{med}^2 \\, g(\\vect{q}, \\omega) \\nonumber \\\\\n                                                                                                                                                       & \\times \\left| \\frac{1}{\\Omega} \\int_\\text{cell} d^3x \\, e^{i \\vect{G} \\cdot \\vect{x}} \\,u^*_{f, \\vect{k}_f}(\\vect{x}) \\,u_{i,\\vect{k}_i}(\\vect{x}) \\right|^2 ,\n      \\label{eq:rate_val_to_cond}\n\\end{align}\nwhere $\\vect{q} = \\vect{k}_f - \\vect{k}_i + \\vect{G}$, $N_{\\text{v}(\\text{c})}$ is the number of valence (conduction) bands. This is identical to the rate formulae derived in~\\cite{Griffin:2019mvc, Trickle:2019nya, Essig:2015cda} but written in terms of the periodic Bloch functions, $u_{i,\\vect{k}}(\\vect{x})$, instead of their Fourier transformed components, $\\widetilde{u}_{i,\\vect{k},\\vect{G}}$, similar to Ref.~\\cite{Liang:2018bdb}. Numerically the position space form is superior since the integral over the primitive cell can be computed by Fast Fourier Transform. This reduces the computational complexity from $\\mathcal{O}(N_G^2)$ to $\\mathcal{O}(N_G \\log{N_G})$, where $N_G$ is the number of $\\vect{G}$ points, {\\it i.e.}\\ the number of Fourier components in the expansion of $\\widetilde{u}_{i,\\vect{k}}$ in Eq.~\\eqref{eq:bloch_fourier}.\n\nIn Fig.~\\ref{fig:vc_diff_rate} we show the scattering rate from valence to conduction transitions binned in energy deposition, defined in Eq.~\\eqref{eq:binned_rate_E}, for a 1\\,GeV DM. The main difference between the calculation performed here and in previous works is the effect of the AE reconstruction, as discussed in Sec.~\\ref{subsubsec:ae}. For the case of DM with a heavy mediator, the rate, even with experimental thresholds as low as $\\sim10$ eV, is significantly enhanced relative to previous work. The AE reconstruction plays less of a role in the light mediator case since the transition rate is dominated by small momentum transfers. However, at high thresholds, where only larger momentum components can contribute, the AE reconstruction can still significantly boost the scattering rate by fully including the contributions neglected in the pseudo wave functions. \n\nSince most earlier works computing DM-electron scattering include only valence to conduction transitions, it is useful to understand for which DM masses these are the only kinematically allowed transitions. If $\\omega < E_\\text{g} - E_\\text{max}^\\text{core}$, where $E_\\text{max}^\\text{core}$ is the maximum energy of the core states, then the core states cannot contribute; if $\\omega < E_\\text{dft}$ the free states are not available. Therefore if $\\omega < \\text{min}\\{ E_\\text{dft}, E_\\text{g} - E_\\text{max}^\\text{core} \\}$ only the valence to conduction transitions are allowed, which can be related to a DM mass via $\\omega_\\text{max}(m_\\chi) < \\text{min}\\{ E_\\text{dft}, E_\\text{g} - E_\\text{max}^\\text{core} \\}$, where\n\\begin{align}\n    \\omega_\\text{max}(m_\\chi) = \\frac{1}{2} m_\\chi v_\\text{max}^2 = 3.9 \\text{ eV} \\left( \\frac{m_\\chi}{\\text{MeV}} \\right) \\left( \\frac{v_\\text{max}}{840 \\text{ km}\/\\text{s}} \\right)^2 \\, ,\n    \\label{eq:w_max}\n\\end{align}\nwith $v_\\text{max} = v_e + v_\\text{esc}$, the maximum incoming DM velocity. For Si (Ge), $E_\\text{max}^\\text{core} = -116$\\,eV ($-28$\\,eV), this corresponds to \n\\begin{align}\n    m_\\chi <\n    \\begin{cases}\n        15.2 \\text{ MeV} & (\\text{Si}) \\,,\\\\\n        7.8 \\text{ MeV} & (\\text{Ge}) \\, .\n    \\end{cases}\n\\end{align}\nRequiring that $\\omega_\\text{max} > E_\\text{g}$, where $E_\\text{g}$ is the band gap, sets a lower bound on the minimum detectable mass, $m_\\chi^\\text{min}$,\n\\begin{align}\n    m_\\chi^\\text{min} = \\frac{2E_\\text{g}}{v_\\text{max}^2} = 0.25 \\text{ MeV} \\left( \\frac{E_\\text{g}}{\\text{eV}} \\right) \\left( \\frac{840 \\text{ km}\/\\text{s}}{v_\\text{max}} \\right)^2 \\, .\n\\end{align}\nFor Si (Ge), with a band gap of 1.11 (0.67) eV, $m_\\chi^\\text{min}$ is $0.28$ ($0.17$) MeV.\nLastly, we remark that for DM interactions characterized by higher-dimensional operators (not considered in this work), the scattering rate scales with higher powers of $q$ and therefore is even more sensitive to AE reconstruction (and also $\\mathrm{c} \\rightarrow \\mathrm{c}$ contributions discussed below in Sec.~\\ref{subsec:cc}), which must be included in the analysis.\n\n\\subsection{Valence to Free}\n\\label{subsec:vf}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/vc_vf_contribution_diff_rate.pdf}\n    \\caption{DM-electron scattering rate from valence to conduction (v$\\to$c) bands and from valence bands to free states (v$\\to$f) binned in energy deposition (with $\\Delta\\omega=1$\\,eV) for 1\\,GeV DM, light (top row) and heavy (bottom row) mediators, assuming $\\overline{\\sigma}_e = 10^{-40}\\; \\text{cm}^2$. The upper edge of the shaded region corresponds to using $Z_\\text{eff}$ from Eq.~\\eqref{eq:z_eff_binding_approx}, while the bottom edge corresponds to $Z_\\text{eff} = 1$.}\n    \\label{fig:vf_diff_rate}\n\\end{figure}\n\nFor valence to free transitions the initial states are identical to those from Sec.~\\ref{subsec:vc}, labeled by band number $i$ and Bloch momentum, $\\vect{k}_i$. The final state wave functions are simple plane waves given by Eq.~\\eqref{eq:free_elec_wf}, labeled by a momentum $\\vect{k}_f$ in the 1BZ with the bands labeled by $\\vect{G}$. \nWe can therefore directly substitute Eq.~\\eqref{eq:u_free} into Eq.~\\eqref{eq:f_val_to_cond} derived in the previous subsection, and obtain the crystal form factor:\n\\begin{align}\n f_{i, \\vect{k}_i \\rightarrow \\vect{G}_f,\\vect{k}_f } &=\n \\sum_\\vect{G} \\delta_{\\vect{q},\\, \\vect{k}_f - \\vect{k}_i + \\vect{G}} \\;\\frac{1}{\\Omega}\\, \\int_\\text{cell} d^3x \\,e^{i (\\vect{G}-\\vect{G}_f) \\cdot \\vect{x}}  \\,u_{i, \\vect{k}_i}(\\vect{x}) \\nonumber\\\\\n &= \\sum_\\vect{G} \\delta_{\\vect{q},\\, \\vect{k}_f - \\vect{k}_i + \\vect{G}} \\,\\widetilde{u}_{i,\\vect{k}_i, \\vect{G}_f - \\vect{G}} \\,,\n\\end{align}\nwhere $\\widetilde{u}_{i,\\vect{k}_i,\\vect{G}}$ are the Fourier components of the Bloch wave functions defined in Eq.~\\eqref{eq:bloch_fourier}. \nIncorporating the Fermi factor correction discussed in Sec.~\\ref{subsec:pw_approx}, we find the rate in Eq.~\\eqref{eq:rate_simple} is given by\n\\begin{equation}\n    R  = \\frac{2 \\pi \\overline{\\sigma}_e}{\\mu_{\\chi e}^2 m_\\chi} \\frac{\\rho_\\chi}{\\rho_T} \\sum_{i = 1}^{N_\\text{v}} \\sum_{\\vect{G}_f} \\int_\\text{1BZ} \\frac{d^3k_i}{(2\\pi)^3} \\frac{d^3k_f}{(2\\pi)^3} F(\\nu_{i,\\vect{k}_i}) \\sum_\\vect{G} \\left( \\frac{f_e}{f_e^0} \\right)^2  \\mathcal{F}_\\text{med}^2 \\, g(\\vect{q}, \\omega) \\left| \\widetilde{u}_{i,\\vect{k}_i, \\vect{G}_f - \\vect{G}}\\right|^2 ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\omega  \\equiv \\frac{\\left| \\vect{k}_f + \\vect{G}_f \\right|^2}{2 m_e} - E_{i,\\vect{k}_i} \\,,\\qquad\n\\nu_{i,\\vect{k}_i} = \\nu(Z_\\text{eff}^{i, \\vect{k}_i},\\, \\omega + E_{i,\\vect{k}_i}) \\,.\n\\end{equation}\nWith a change of variables, $\\vect{G}' = \\vect{G}_f - \\vect{G}$ and defining $\\vect{k}' \\equiv \\vect{k}_f + \\vect{G}_f$ (and then dropping the prime for simplicity), the rate becomes\n\\begin{align}\n    R & = \\frac{2 \\pi \\overline{\\sigma}_e}{\\mu_{\\chi e}^2 m_\\chi} \\frac{\\rho_\\chi}{\\rho_T} \\sum_{i = 1}^{N_\\text{v}} \\int_\\text{1BZ} \\frac{d^3k_i}{(2\\pi)^3} F(\\nu_{i,\\vect{k}_i}) \\sum_{\\vect{G}} \\left| \\widetilde{u}_{i,\\vect{k}_i,\\vect{G}}\\right|^2 \\int \\frac{d^3k}{(2\\pi)^3}  \\left( \\frac{f_e}{f_e^0} \\right)^2 \\mathcal{F}_\\text{med}^2 \\, g(\\vect{q}, \\omega) \\, .\n\\end{align}\nwhere $\\vect{q} = \\vect{k} - \\vect{k}_i - \\vect{G}$.\n\nIn Fig.~\\ref{fig:vf_diff_rate} we compare the binned rate from the valence to conduction (v$\\to$c) calculation in the previous subsection to the valence to free (v$\\to$f) one performed here, again for a 1\\,GeV DM. We see that for large $\\omega$, where the v$\\to$c calculation is limited by the number of conduction bands included, the v$\\to$f calculation extrapolates the results to higher $\\omega$ as expected. There is some uncertainty due to the choice of the effective charge parameters, which is why the results are shown in bands. The lower edge corresponds to the conservative choice of $Z_\\text{eff}^{i, \\vect{k}_i} = 1$ for all $i, \\vect{k}_i$, and the upper edge corresponds to the value set by the binding energy, Eq.~\\eqref{eq:z_eff_binding_approx} with $E_B=-E_{i,\\vect{k}_i}$. We find that the conservative choice $Z_\\text{eff}^{i, \\vect{k}_i} = 1$ is a better match to the edge for the v$\\to$c calculation, and will use this in our final projections in Sec.~\\ref{sec:projected_constraints}. Note that as the threshold increases, the effect of v$\\to$f transitions becomes more important, and for a heavy mediator non-negligible constraints can be placed even with $\\mathcal{O}(100)$\\,eV energy thresholds.\n\n\\subsection{Core to Conduction}\n\\label{subsec:cc}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/cc_contribution_diff_rate.pdf}\n    \\caption{DM-electron scattering rate from core states to conduction bands binned in energy deposition (with $\\Delta\\omega=5$\\,eV) for 1\\,GeV DM, light (top row) and heavy (bottom row) mediators, assuming $\\overline{\\sigma}_e = 10^{-40}\\; \\text{cm}^2$. The core states are labelled by the corresponding atomic orbitals, and the conduction states up to $E_\\text{dft}=60$\\,eV are included. For comparison we also show the v$\\to$c contribution (after AE reconstruction) from Fig.~\\ref{fig:vc_diff_rate} in gray.}\n    \\label{fig:cc_diff_rate}\n\\end{figure}\n\nWe now turn to core to conduction transitions. The initial core states are indexed by $\\kappa$, the atom in the primitive cell, the usual atomic quantum numbers, $n, l, m$, and the Bloch momentum, $\\vect{k}_i$. The final states are the DFT computed conduction states. The crystal form factor is simply obtained from Eq.~\\eqref{eq:f_val_to_cond} by substituting $u_{i,\\vect{k}_i}\\to u_{\\kappa nlm,\\vect{k}_i}$:\n\\begin{equation}\nf_{\\kappa nlm, \\vect{k}_i \\rightarrow f, \\vect{k}_f}  =\\sum_\\vect{G} \\delta_{\\vect{q},\\, \\vect{k}_f - \\vect{k}_i + \\vect{G}} \\,\\frac{1}{\\Omega}\\, \\int_\\text{cell} d^3x \\,e^{i \\vect{G} \\cdot \\vect{x}} \\,u^*_{f, \\vect{k}_f}(\\vect{x}) \\,u_{\\kappa nlm, \\vect{k}_i}(\\vect{x}) \\,,\n\\label{eq:f_c2c}\n\\end{equation}\nThe total scattering rate is then\n\\begin{align}\n    R = \\frac{2\\pi\\overline{\\sigma}_e}{\\mu_{\\chi e}^2 m_\\chi} \\frac{\\rho_\\chi}{\\rho_T} \\sum_{\\kappa = 1}^{N_\\text{a}} \\sum_{n = 1}^{N_\\text{p}^\\kappa} & \\sum_{l = 0}^{n-1} \\sum_{m = -l}^{l} \\sum_{f = 1}^{N_\\text{c}} \\int_\\text{1BZ} \\frac{d^3k_i}{(2\\pi)^3} \\frac{d^3k_f}{(2\\pi)^3} \\sum_\\vect{G} \\left( \\frac{f_e}{f_e^0} \\right)^2 \\mathcal{F}_\\text{med}^2\\, g(\\vect{q}, \\omega) \\nonumber \\\\  \n                                                                                                                                                       & \\times \\left| \\frac{1}{\\Omega} \\int_\\text{cell} d^3x \\, e^{i \\vect{G} \\cdot \\vect{x}} u_{f\\vect{k}_f}^*(\\vect{x}) \\,u_{\\kappa nlm\\vect{k}_i}(\\vect{x}) \\right|^2 ,\n\\end{align}\nwhere $N_\\text{a}$ is the number of atoms in the primitive cell, $N_\\text{p}^\\kappa$ is the largest principal quantum number for atom $\\kappa$, and $\\omega = E_{f,\\vect{k}_f} - E_{\\kappa nl}$. \nThe core wave functions $u_{\\kappa nlm,\\vect{k}_i}(\\vect{x})$ are given by Eq.~\\eqref{eq:u_jnlmk}, and involves a sum over primitive cells. \nSince the integral in Eq.~\\eqref{eq:f_c2c} is just over one primitive cell, only the atoms in this and neighboring cells can have a significant contribution. \nIn other words, the sum over $\\vect{r}$ converges very quickly due to the localized nature of atomic wave functions. We therefore restrict $\\vect{r}$ to be summed over only the $3\\times 3\\times 3$ nearest cells.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/Ge_3d_focus.pdf}\n    \\caption{Contribution to the DM-electron scattering rate binned in energy deposition (with $\\Delta\\omega=1$\\,eV) from 3d electrons to conduction bands in Ge, for 1\\,GeV DM, light (left) and heavy (right) mediators, assuming $\\overline{\\sigma}_e = 10^{-40}\\; \\text{cm}^2$. The three curves in each panel are computed using DFT with and without AE reconstruction, and using the semi-analytic core wave functions.}\n    \\label{fig:Ge_3d_focus}\n\\end{figure}\n\nThe contribution of core to conduction (c$\\to$c) transitions to the binned rate, for $m_\\chi = 1$\\,GeV, can be seen in Fig.~\\ref{fig:cc_diff_rate}. In most cases the v$\\to$c transitions are dominant compared to the c$\\to$c, but there are two main scenarios where this is not true. First, when the experimental threshold is raised; this excludes the v$\\to$c transitions and causes the c$\\to$c contribution to be dominant. For example, consider a Si detector and a DM model with a heavy mediator (bottom left panel of Fig.~\\ref{fig:cc_diff_rate}). If the experimental threshold is $\\sim 50$\\,eV the c$\\to$c contribution from the 2p states in Si gives the dominant contribution. \nSecond, for a Ge target, and a DM model with a heavy mediator, the 3d states dominate the rate even at the lowest experimental threshold. To understand this in more detail we present Fig.~\\ref{fig:Ge_3d_focus} which compares the binned rate taking different modeling approaches for the 3d states in Ge. We see that the large momentum components of the wave function, recovered only after AE reconstruction in the DFT calculation, dominate the rate, which explains why previous works have underestimated the importance of 3d electrons. \nMeanwhile, we see explicitly at the scattering rate level that the semi-analytic approach accurately reproduces the DFT calculation at low $\\omega$, and extends the latter beyond its cutoff at high $\\omega$, consistent with the observation at the wave function level in Fig.~\\ref{fig:binned_core_wf}.\n\n\\subsection{Core to Free}\n\\label{subsec:cf}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/cc_cf_contribution_diff_rate.pdf}\n    \\caption{DM-electron scattering rate from core states to conduction bands (c$\\to$c) and to free states (c$\\to$f) binned in energy deposition (with $\\Delta\\omega=10$\\,eV) for 1\\,GeV DM, light (top row) and heavy (bottom row) mediators, assuming $\\overline{\\sigma}_e = 10^{-40}\\; \\text{cm}^2$.  As in the v$\\to$f calculation in Fig.~\\ref{fig:vf_diff_rate}, the upper edge of the shaded bands corresponds to $Z_\\text{eff}$ from Eq.~\\eqref{eq:z_eff_binding_approx}, and the lower edge corresponds to $Z_\\text{eff} = 1$.}\n    \\label{fig:cf_diff_rate}\n\\end{figure}\n\nThe last transition type we consider involves a core electron initial state and a free electron final state.\nThe crystal form factor is most easily obtained by substituting Eqs.~\\eqref{eq:core_elec_wf} and \\eqref{eq:free_elec_wf} into its definition, Eq.~\\eqref{eq:wf_form_fac}:\n\\begin{align}\n    f_{\\kappa nlm,\\vect{k}_i \\rightarrow \\vect{G}_f, \\vect{k}_f} & = \\frac{1}{\\sqrt{N V}} \\sum_\\vect{r} e^{i\\vect{k}_i \\cdot (\\vect{r}+\\vect{x}_\\kappa)} \\int d^3x\\, e^{i (\\vect{q} - \\vect{k}_f - \\vect{G}_f) \\cdot \\vect{x}} \\,\\psi_{\\kappa nlm}^\\text{atom}(\\vect{x} - \\vect{r} - \\vect{x}_\\kappa) \\nonumber \\\\\n                                   & = \\frac{1}{\\sqrt{NV}} \\,e^{i(\\vect{k}_i+\\vect{q} - \\vect{k}_f - \\vect{G}_f) \\cdot \\vect{x}_\\kappa} \\sum_\\vect{r} e^{i (\\vect{q} - \\vect{k}_f + \\vect{k}_i) \\cdot \\vect{r}} \\int d^3x \\,e^{i (\\vect{q} - \\vect{k}_f - \\vect{G}_f) \\cdot \\vect{x}} \\,\\psi_{\\kappa nlm}^\\text{atom}(\\vect{x}) \\nonumber \\\\\n                                   & = \\frac{1}{\\sqrt{\\Omega}} \\,e^{i(\\vect{k}_i+\\vect{q} - \\vect{k}_f - \\vect{G}_f) \\cdot \\vect{x}_\\kappa} \\sum_\\vect{G} \\delta_{\\vect{q} - \\vect{k}_f + \\vect{k}_i, \\vect{G}}\\, \\widetilde{\\psi}_{\\kappa nlm}^\\text{atom}(-\\vect{k}_i + \\vect{G} - \\vect{G}_f) \\,,\n\\end{align}\nwhere the Fourier transform of the RHF Slater type orbital (STO) core wave functions, given in Eq.~\\eqref{eq:sto_wf}, are known analytically \\cite{Belkic1989}:\n\\begin{align}\n    \\widetilde{\\psi}_\\text{STO}(\\vect{q}; Z, n, l, m) & = \\int d^3x \\, e^{i \\vect{q} \\cdot \\vect{x}} R_\\text{STO}(x; Z, n) \\,Y_l^m(\\hat{\\vect{x}}) \\equiv \\chi_\\text{STO}(q; Z, n, l) \\,Y_l^m(\\hat{\\vect{q}}) \\,,\\\\\n    \\chi_\\text{STO}(q; Z, n) & = 4 \\pi N (n - l)! (2 Z)^{n} \\left( \\frac{i a_0 q}{Z} \\right)^l \\sum_{s = 0}^{ \\lfloor (n - l)\/2 \\rfloor} \\frac{\\omega_s^{nl}}{\\left( (a_0 q)^2 + Z^2 \\right)^{n - s + 1}} \\,,\\\\\n    \\omega_s^{nl} & = \\left(-4 Z^2\\right)^{-s} \\frac{(n - s)!}{s! (n - l - 2s)!} \\,.\n\\end{align}\nThe direct detection rate is then\n\\begin{align}\n    R = &\\; \\frac{2\\pi\\overline{\\sigma}_e}{\\mu_{\\chi e}^2 m_\\chi} \\frac{\\rho_\\chi}{\\rho_T \\Omega} \\sum_{\\kappa = 1}^{N_\\text{a}} \\sum_{n = 1}^{N_\\text{p}^\\kappa} \\sum_{l = 0}^{n-1} \\sum_{m = -l}^{l} \\int_\\text{1BZ} \\frac{d^3k_i}{(2\\pi)^3} \\frac{d^3k_f}{(2\\pi)^3}\\nonumber \\\\                                                                                                                                                               & \\times  \\sum_{\\vect{G}_f} \\sum_\\vect{G} F (\\nu_{\\kappa nl})  \\left( \\frac{f_e}{f_e^0} \\right)^2 \\mathcal{F}_\\text{med}^2\\, g(\\vect{q}, \\omega) \\left| \\widetilde{\\psi}_{\\kappa nlm}^\\text{atom}\\left( -\\vect{k}_i + \\vect{G} - \\vect{G}_f \\right) \\right|^2 ,\n\\end{align}\nwhere $\\vect{q} = \\vect{k}_f - \\vect{k}_i + \\vect{G}$, and $\\nu_{\\kappa nl}=\\nu(Z_\\text{eff}^{\\kappa nl},\\, \\omega + E_{\\kappa nl})$. We can now shift the $\\vect{G}_f$ variable, $\\vect{G'} \\equiv \\vect{G}_f - \\vect{G}$ and define $\\vect{k} = \\vect{k}_i + \\vect{G}'$ and $\\vect{k}' = \\vect{k}_f + \\vect{G}$. Therefore, $\\vect{q} = \\vect{k}' - \\vect{k}$ and\n\\begin{align}\n    R = \\frac{2\\pi\\overline{\\sigma}_e}{\\mu_{\\chi e}^2 m_\\chi} \\frac{\\rho_\\chi}{\\rho_T \\Omega} & \\sum_{\\kappa = 1}^{N_\\text{a}} \\sum_{n = 1}^{N_\\text{p}^\\kappa} \\sum_{l = 0}^{n - 1} \\sum_{m = -l}^{l} \\int \\frac{d^3k}{(2\\pi)^3} \\frac{d^3k'}{(2\\pi)^3} \\,F( \\nu_{\\kappa nl} ) \\left( \\frac{f_e}{f_e^0} \\right)^2 \\mathcal{F}_\\text{med}^2\\, g(\\vect{q}, \\omega) \\left| \\widetilde{\\psi}^\\text{atom}_{\\kappa nlm}\\left( \\vect{k} \\right) \\right|^2 ,\n\\end{align}\nwhich is the closest expression to the vacuum matrix element, with just the inclusion of the core wave functions acting as a form factor.\n\nIn Fig.~\\ref{fig:cf_diff_rate}, we compare the binned rate from the core to conduction (c$\\to$c) calculation to the core to free (c$\\to$f) calculation and see a reasonable extrapolation to higher $\\omega$. As with the transition region between v$\\to$c and v$\\to$f shown in Fig.~\\ref{fig:vf_diff_rate}, we find $Z_\\text{eff} = 1$ gives a better match between c$\\to$c and c$\\to$f. While the total number of electrons from these transitions is expected to be much less than lower energy transitions, this is the best available calculation for thresholds up to the kinematic limit of $\\omega_\\text{max}$. \n\n\\subsection{In-medium Screening}\n\\label{subsec:in-med}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{figs\/Si_dielectric.pdf}\n\t\\includegraphics[width=0.45\\textwidth]{figs\/Ge_dielectric.pdf}\n\t\\caption{Dielectric function $\\epsilon(q, \\omega)$, given by Eq.~\\eqref{eq:analytic_di} with the parameters in Table~\\ref{tab:in_med}, of Si (left) and Ge (right) used to incorporate screening effects. The solid line indicates the edge of the kinematically accessible region $\\omega \\lesssim q v$. The dashed line is the band gap of the target. While the static dielectric can be $\\mathcal{O}(10)$, in the kinematically allowed region $\\epsilon(q, \\omega)$ is an $\\mathcal{O}(1)$ number, leading to an $\\mathcal{O}(1)$ effect on the scattering rates when the latter are dominated by small $q, \\omega$ transitions.}\n\t\\label{fig:dielectrics}\n\\end{figure}\n\nDM-electron interactions mediated by a dark photon or scalar are screened due to the in-medium mixing between the mediator and the photon. \nThe relevance of screening has been recently emphasized in Ref.~\\cite{Knapen:2021run}. \nThe screening factor, $f_e\/f_e^0$, is related to the longitudinal dielectric, $f_e\/f_e^0 = (\\hat{\\vect{q}} \\cdot \\boldsymbol{\\epsilon} \\cdot \\hat{\\vect{q}})^{-1}$, where $\\boldsymbol{\\epsilon}$ is the dielectric tensor. \nIt can be computed from in-medium loop diagrams or extracted from optical data. \nHere we model the dielectric of Si and Ge following Ref.~\\cite{Cappellini1993}:\n\\begin{align}\n    \\epsilon(q, \\omega) = 1 + \\left[ \\frac{1}{\\epsilon_0 - 1} + \\alpha \\left( \\frac{q}{q_\\text{TF}} \\right)^2 + \\frac{q^4}{4m_e^2 \\omega_p^2} - \\left( \\frac{\\omega}{\\omega_p} \\right)^2  \\right]^{-1} \\, ,\n    \\label{eq:analytic_di}\n\\end{align}\nand $\\epsilon_{ij} = \\epsilon(q, \\omega) \\,\\delta_{ij}$. \nHere, $\\epsilon_0 \\equiv \\epsilon(0, 0)$ is the static dielectric, $\\alpha$ is a fitting parameter, $q_\\text{TF}$ is the Thomas-Fermi momentum, and $\\omega_p$ is the plasma frequency. The parameters used for Si and Ge are listed in Table~\\ref{tab:in_med}, and we plot the dielectric as a function of $q, \\omega$ in Fig.~\\ref{fig:dielectrics}. \n\n\\begin{table}[t]\n    \\begin{center}\n        \\begin{tabular}{c||c c c c}\n            \\hline\n            Target & $\\epsilon_0$ & $\\alpha$ & $\\omega_p$ $[\\text{eV}]$ & $q_\\text{TF}$ $[\\text{keV}]$ \\\\\n            \\hline\n            Si & 11.3 & 1.563 & 16.6 & 4.13 \\\\\n            Ge & 14 & 1.563 & 15.2 & 3.99 \\\\\n            \\hline\n        \\end{tabular}\n        \\caption{Parameters used in the model of dielectric function, Eq.~\\eqref{eq:analytic_di}, of Si and Ge from Ref.~\\cite{Cappellini1993}, which accounts for in-medium screening effects on the transition rate.}\n        \\label{tab:in_med}\n    \\end{center}\n\\end{table}\n\nNaively one might expect that the effect of the dielectric is to screen the rate by an $\\mathcal{O}(100)$ factor due to the fact that the static dielectric, $\\epsilon_0$, is $\\mathcal{O}(10)$. However, this is only the value of the dielectric function at $q = \\omega = 0$, while as $q \\rightarrow \\infty$ and $\\omega \\rightarrow \\infty$ the dielectric approaches unity. Therefore, the effect of the dielectric crucially depends on the region of the kinematic phase space being probed. For a given energy deposition, $\\omega$, the momentum transfer is limited to $q \\gtrsim \\omega \/ v$ where $v \\sim 10^{-3}$ is the DM velocity. Therefore, the absolute minimum momentum transfer is $q_\\text{min} \\sim E_\\text{g} \/ v \\sim \\mathcal{O}(\\text{keV})$, for $\\mathcal{O}(\\text{eV})$ band gap targets. This is parametrically the same size as the Thomas-Fermi momentum $q_\\text{TF}$, so the dielectric is expected to slightly deviate from one, which causes only an $\\mathcal{O}(1)$ shift to the scattering rate, as seen in Fig.~\\ref{fig:scr_effects}.\n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=\\textwidth]{figs\/screening_comparison.pdf}\n    \\caption{Effect of screening on the binned rate (top row, for 1\\,GeV DM) and total rate (bottom row, as a function of $m_\\chi$) from v$\\to$c transitions for DM models with a light (red) and heavy (blue) mediator. The unscreened rate $R^\\text{no scr}$ is obtained with $\\epsilon=1$, and the screened rate $R^\\text{scr}$ is obtained with the model of the dielectric function given in Eq.~\\eqref{eq:analytic_di}.}\n    \\label{fig:scr_effects}\n\\end{figure}\n\n\\section{Projected Sensitivity}\n\\label{sec:projected_constraints}\n\nWe now compile the results from the previous sections to compute the projected sensitivity. We also compare the relative importance of each transition type, and discuss differences between our results and previous calculations in the literature. When there are large differences, it is typically because of the inclusion of AE reconstruction and core states in the calculation. Since AE reconstruction and core states contribute predominantly at higher momentum transfer and energy deposition, we will find the largest differences typically occur for a massive mediator and higher detector threshold, where the effects in some cases can be more than an order of magnitude (especially for Ge). For the case of a massless mediator and lower detection threshold, the differences with previous literature are much smaller and mostly due to the inclusion of in-medium effects.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/total_diff_rate_compare.pdf}\n    \\caption{DM-electron scattering rate binned in energy deposition (with $\\Delta\\omega=1$\\,eV)  for 1\\,GeV DM, light (top row) and heavy (bottom row) mediators, from all four transition types: valence to conduction (v$\\to$c), valence to free (v$\\to$f), core to conduction (c$\\to$c), and core to free (c$\\to$f). We assume $\\overline{\\sigma}_e = 10^{-40}\\; \\text{cm}^2$, and take $Z_\\text{eff} = 1$ for all effective charges in the Fermi factor. Note that the c$\\to$c and c$\\to$f transitions involve semi-analytic treatment of 2p (3d) states and below in Si (Ge), which has been validated with DFT calculations including AE reconstruction; see Fig.~\\ref{fig:binned_core_wf}. We also overlay the binned rate from Ref.~\\cite{Derenzo:2016fse} which computed the v$\\to$c contribution using \\texttt{QEdark} (treating 3d states in Ge as valence, without including AE reconstruction effects).}\n    \\label{fig:compiled_diff_rate}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/compiled_rate.pdf}\n    \\caption{DM-electron scattering rate as a function of the DM mass, for light (top row) and heavy (bottom row) mediators, from all four transition types: valence to conduction (v$\\to$c), valence to free (v$\\to$f), core to conduction (c$\\to$c), and core to free (c$\\to$f). We assume $\\overline{\\sigma}_e = 10^{-40}\\; \\text{cm}^2$, take $Z_\\text{eff} = 1$ for all effective charges in the Fermi factor, and show results for several threshold $Q$ values which significantly impact the v$\\to$c contribution.}\n    \\label{fig:compiled_rate}\n\\end{figure}\n\nIn Fig.~\\ref{fig:compiled_diff_rate} we show the contribution to the binned rate from each of the four transition types, for a 1\\,GeV DM. \nWe see that valence to conduction (v$\\to$c) has a higher peak than the other three transition types, except for the Ge, heavy mediator case, where core to conduction (c$\\to$c) has the highest peak. \nFor comparison, Refs.~\\cite{Essig:2015cda, Derenzo:2016fse} compute the valence to conduction rates with DFT, including also the 3d states in Ge, but without AE reconstruction. \nAs expected, we find a lower rate at the lowest energy depositions due to the inclusion of in-medium screening, and a much higher rate at high $\\omega$ due to AE reconstruction and inclusion of core states. \n\nThe impact of these observations on the reach depends on the energy threshold. \nAssuming charge readout ({\\it e.g.}\\ via a CCD), the relevant quantity is the number of electron-hole pairs, $Q$, produced in an event. \nFor an energy deposition $\\omega$, this is given by\n\\begin{align}\nQ = 1 + \\left\\lfloor \\frac{\\omega - E_\\text{g}}{\\varepsilon} \\right\\rfloor ,\n\\label{eq:Q_bin}\n\\end{align}\nwhere the values for $\\varepsilon$ are $3.6$ eV and $2.9$ eV for Si and Ge respectively. \nIn Fig.~\\ref{fig:compiled_rate}, we show the total rate as a function of the DM mass, for $Q\\ge1, 5, 10$.\nThe threshold only affects the v$\\to$ c rate, as the other three transition types involve energy depositions corresponding to $Q>10$, and are therefore always fully included. \nWe see that for $Q\\ge1$, the valence to conduction (v$\\to$c) contribution dominates the total rate with the exception of the Ge, heavy mediator scenario, where core to conduction (c$\\to$c) is dominant for $m_\\chi\\gtrsim 30\\,$MeV. \nHigher thresholds significantly cut out v$\\to$c contributions in all cases, and render c$\\to$c more important for Ge, even in the light mediator scenario. \nFor Si, on the other hand, the total rate is still dominated by v$\\to$c because the core states are much deeper and contribute a lower rate. \nWe also see that v$\\to$f and c$\\to$f contributions are subdominant in all cases.\n\nFinally, we present the projected reach on the DM-electron reference cross section $\\overline\\sigma_e$ in Figs.~\\ref{fig:reach_compare_Q1} and \\ref{fig:reach_compare_Q10}, for $Q\\ge1$ and $Q\\ge10$, respectively. \nOur new calculation yields several important differences compared to the previous literature, and we discuss them in detail in the following subsection.\n\n\\subsection{Comparison With Previous Results}\n\\label{subsec:compare_to_prev}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/reach_compare_Q1.pdf}\n    \\caption{95\\% C.L.\\ exclusion reach (3 events) assuming 1 kg-year exposure, $Q \\ge 1$, for light (top row) and heavy (bottom row) mediators. The results shown are from this work, Griffin et al.~\\cite{Griffin:2019mvc}, Essig et al.~\\cite{Essig:2015cda}, Lee et al.~\\cite{Lee:2015qva}, and Knapen et al.~\\cite{Knapen:2021run} (with and without screening). See Sec.~\\ref{subsec:compare_to_prev} for detailed comparison.}\n    \\label{fig:reach_compare_Q1}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/reach_compare_Q10.pdf}\n    \\caption{95\\% C.L.\\ exclusion reach (3 events) assuming 1 kg-year exposure, $Q \\ge 10$, for light (top row) and heavy (bottom row) mediators. The results shown are from this work and Essig et al.~\\cite{Essig:2015cda}. See Sec.~\\ref{subsec:compare_to_prev} for detailed comparison.}\n    \\label{fig:reach_compare_Q10}\n\\end{figure}\n\nWe begin by comparing to our previous work, Ref.~\\cite{Griffin:2019mvc}, shown in brown in Fig.~\\ref{fig:reach_compare_Q1}. We previously restricted our analysis to the light mediator scenario, and $Q \\ge 1$, which is relatively unaffected by AE reconstruction effects since the rate is peaked at small energy\/momentum transfers, as seen in Fig.~\\ref{fig:vc_diff_rate}. The main reason the reach here is weaker is the inclusion of in-medium screening discussed in Sec.~\\ref{subsec:in-med}.  \n\nNext we compare to Ref.~\\cite{Essig:2015cda}, shown in red in Figs.~\\ref{fig:reach_compare_Q1} and \\ref{fig:reach_compare_Q10}. \nThose results were computed solely from valence to conduction (v$\\to$c) transitions. The largest discrepancy is in the high $m_\\chi$ regime scattering off a Ge target via a heavy mediator. This is due to high momentum contributions to the 3d wave functions in Ge. Ref.~\\cite{Essig:2015cda} computed the 3d states with DFT without AE reconstruction, which as we saw in Fig.~\\ref{fig:binned_core_wf} is crucial for recovering the dominant part of the 3d wave functions at high momentum. \nAs discussed in Sec.~\\ref{subsec:atomic_wf}, our modeling of 3d electrons in Ge as core states reproduces their DFT-computed wave functions up to the AE reconstruction cutoff, and provides a robust parameterization of higher momentum components.\nSince the valence states in Ge also contribute an appreciable amount, the $Q \\ge 1$ results in Fig.~\\ref{fig:reach_compare_Q1} only differ by about an order of magnitude. \nHowever, the difference is more stark when going to higher $Q$ thresholds in Fig.~\\ref{fig:reach_compare_Q10}, which essentially isolates the 3d electrons' contribution. \nIn the low mass regime the difference is less significant, and primarily due to the inclusion of screening effects. \nAnother difference that is important here is sampling of the 1BZ. Ref.~\\cite{Essig:2015cda} used a uniform $6\\times6\\times6$ mesh with extra 27 points chosen by hand close to the center of the 1BZ, whereas here (as well as in Ref.~\\cite{Griffin:2019mvc}) we use a uniform $10\\times10\\times10$ grid. While checking convergence we found our (unscreened) results using a $6\\times6\\times6$ uniform mesh were a closer match to Ref.~\\cite{Essig:2015cda}; generally, increasing the number of $\\vect{k}$ points reduces the rate toward convergence, {\\it i.e.}\\ $R_{10\\times10\\times10} < R_{9\\times9\\times9} < R_{8\\times8\\times8}$. This can be seen more directly in the difference between the brown and red lines in the light mediator scenario (as both are computed without screening), and it affects Ge more than Si, as is expected due to the smaller band gap and greater dispersions of nearby bands requiring denser $\\vect{k}$ point sampling for convergence.\n\nRef.~\\cite{Lee:2015qva} also computed DM-electron scattering rates in semiconductors, focusing on Ge. The approach taken in that paper was to semi-analytically model the Ge wave functions with the core wave functions (with the same set of RHF STO wave function coefficients tabulated in Ref.~\\cite{Bunge:1993jsz}) and treat the final states as free with a Fermi factor, analogous to the core to free calculation performed here. As we can see from Fig.~\\ref{fig:reach_compare_Q1}, while for most of the mass range and mediators the estimates are too optimistic due to incorrect modeling of the valence and conduction states, in the high mass region with a heavy mediator (bottom-right panel), where 3d states dominate, their estimates are in good agreement with ours presented here, as expected. \n\nFinally, we discuss the comparison with the most recent work, Ref.~\\cite{Knapen:2021run}, which was limited to valence to conduction transitions. To show the effect of screening, we show their projected reach with (purple) and without (green) screening in Fig.~\\ref{fig:reach_compare_Q1}. Again the largest discrepancy is in the heavy mediator scenario with a Ge target, primarily due to the neglect of the 3d states in Ref.~\\cite{Knapen:2021run}. When these are not important, {\\it i.e.}\\ the low mass regime or a light mediator, we generally find good agreement, with our reach being a bit stronger. Notably this does not seem due to a mis-model of the dielectric, since the effect of screening relative to our previous results, Ref.~\\cite{Griffin:2019mvc}, is consistent with their result. We also find that screening has a smaller effect at high masses in the heavy mediator scenario for Si. These small differences are harder to disentangle since they could be due to: 1) different xc-functionals used (PBE and HSE vs.\\ TB09); 2) local field effects which are only partially included here since we assume the screening factor is isotropic; 3) the plane wave expansion parameter, $E_\\text{cut}$, taken to be $500$\\,eV without AE reconstruction in Ref.~\\cite{Knapen:2021run}, vs.\\ 1\\,keV, AE corrected to 2 keV taken here; 4) DM velocity distribution parameters, studied in detail in Ref.~\\cite{Radick:2020qip}, for which Ref.~\\cite{Knapen:2021run} assumed $v_\\text{esc} = 500$ km$\/$s as opposed to $v_\\text{esc} = 600$ km$\/$s chosen here; and 5) Ref.~\\cite{Knapen:2021run} took a directionally averaged dielectric, whereas here we only assume isotropy in the screening factor but not the matrix element itself.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nDark matter-electron scattering in dielectric crystal targets, especially semiconductors like Si and Ge, are at the forefront of DM direct detection experiments. \nIt is therefore imperative to have accurate theoretical predictions for the excitation rates. In this work, we extended the scattering rate calculation in several key aspects. \nMuch of the focus of previous calculations has been on transitions from valence to conduction bands just across the band gap, which will be accessible to near-future experiments. \nWe performed state-of-the-art DFT calculations for these states, and highlighted the importance of all-electron reconstruction which has been neglected in most previous works. \nAlong with this, we extended the transition rate calculation by explicitly including the contributions from core electrons and additional states more than 60\\,eV above the band gap using analytic approximations. \n\nWe updated the projected reach with our new calculation and found important differences compared to previous results. \nIn particular, we found that in the heavy mediator scenario, 3d electrons in Ge give a dominant contribution to the detection rate for DM heavier than about 30\\,MeV. \nAlso, the rate can be significantly higher than predicted previously for higher experimental thresholds. \nThis is exciting because new DM parameter space will be within reach even before detectors reach the single electron ionization threshold. \n\nWe also release a beta version of \\texttt{EXCEED-DM} (available \\href{https:\/\/github.com\/tanner-trickle\/EXCEED-DM}{here}~\\cite{tanner_trickle_2021_4747696}) that implements our DM-electron scattering calculation for general crystal targets, and make the electronic wave function data for Si and Ge~\\cite{Trickle2021}, as well as the \\texttt{EXCEED-DM} output~\\cite{Trickle2021a}, publicly available so our present analysis can be reproduced.\nWe have previously used \\texttt{EXCEED-DM} for a target comparison study~\\cite{Griffin:2019mvc}, and to study the daily modulation signals that can arise in anisotropic materials~\\cite{Trickle:2019nya}. \nThe generality of \\texttt{EXCEED-DM} means that potential applications are vast. \nIt can be used to compute detection rates for other target materials (assuming DFT calculations of valence and conduction states are available), and can also be adapted to include additional DM interactions such as in an effective field theory framework similar to the study of atomic ionizations in Ref.~\\cite{Catena:2019gfa} (see Ref.~\\cite{Catena:2021qsr} for a recent effort in this direction).\nFor momentum-suppressed effective operators, a full calculation in our framework is even more important, as the effects of all-electron reconstruction and core states (overlooked in Ref.~\\cite{Catena:2021qsr}) are generally amplified. \nMoreover, the differential information that can be obtained from our program facilitates further studies including realistic backgrounds.\nDetails of \\texttt{EXCEED-DM} and additional example calculations will be presented in an upcoming publication.\n\n\\acknowledgments\nWe are grateful to Kyle Bystrom for assistance with \\texttt{pawpyseed}, and thank Alex Ganose, Thomas Harrelson, Andrea Mitridate, Michele Papucci and Tien-Tien Yu for helpful discussions. \nWe also thank Rouven Essig and Adrian Soto for early correspondence about \\texttt{QEdark}.\nThis material is based upon work supported by the U.S.\\ Department of Energy, Office of Science, Office of High Energy Physics, under Award No.~DE-SC0021431 (TT, ZZ, KZ), by a Simons Investigator Award (KZ) and the Quantum Information Science Enabled Discovery (QuantISED) for High Energy Physics (KA2401032).  \nThis research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S.\\ Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No.~DE-AC02-05CH11231.\nSome of the computations presented here were conducted on the Caltech High Performance Cluster, partially supported by a grant from the Gordon and Betty Moore Foundation. \nWork at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S.\\ Department of Energy under Contract No.~DE-AC02-05CH11231.\n\n\\bibliographystyle{utphys}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and summary}\n\\label{sec:intro}\n\nThis is an updated version of a Technical Report, \\cite{Hall-Wellner-79},\n     that, although never published, has been referenced repeatedly in the literature:   \n    e.g., \\cite{MR1416657}, \\cite{MR931630}, \\cite{MR856407}, \\cite{MR1927772}, \n    \\cite{MR1792793}, \\cite{MR2344641}.\n   \n   \n   \n   \n    \nLet $X_1, \\ldots , X_n$ be a random sample from a continuous d.f. $F$ \non ${\\mathbb  R}^+ = [0,\\infty)$ with finite mean $\\mu = E(X)$, \nvariance $\\sigma^2 \\le \\infty$, and density $f(x) > 0$.  Let $\\overline{F} = 1-F$\ndenote the survival function, let ${\\mathbb F}_n$ and $\\overline{{\\mathbb F}}_n$ denote the \nempirical distribution function and empirical survival function respectively, \nand let \n\\begin{eqnarray*}\ne(x) \\equiv e_F (x) \\equiv E(X-x | X>x) = \\int_x^\\infty \\overline{F} dI \/ \\overline{F} (x), \n\\ \\ \\ \\ 0 \\le x < \\infty\n\\end{eqnarray*}\ndenote the {\\sl mean residual life function} or {\\sl life expectancy function} \nat age $x$.\nWe use a subscript $F$ or $\\overline{F}$ on $e$ interchangeably, and $I$\ndenotes the identity function and Lebesgue measure on ${\\mathbb  R}^+$.\n\nA natural nonparametric or life table estimate of $e$ is the random function \n$\\hat{e}_n $ defined by \n\\begin{eqnarray*}\n\\hat{e}_n (x) = \\left \\{ \\int_x^\\infty \\overline{{\\mathbb F}}_n dI \/ \\overline{{\\mathbb F}}_n (x) \\right \\} \n1_{[0, X_{nn})} (x)\n\\end{eqnarray*}\nwhere $X_{nn} \\equiv \\max_{1 \\le i \\le n } X_i$; that is, the average, less $x$, of \nthe observations exceeding $x$.  \n\\cite*{MR0471233}\nstudied $\\hat{e}_n$ on a fixed finite interval $0 \\le x \\le T < \\infty$.\nShe proved that $\\hat{e}_n$ is a strongly uniformly consistent estimator of $e$ \non $[0,T]$, and that, when properly centered and normalized, it \nconverges weakly to a certain limiting Gaussian process on $[0,T]$.\n\nWe first extend\nYang's (1978) \nresults to all of ${\\mathbb  R}^+$ by introducing suitable \nmetrics.  Her consistency result is extended in Theorem~\\ref{ConsistencyTh1} by \nusing the techniques of \n\\cite{MR0651528,MR0651392};\nthen her weak convergence \nresult is extended in Theorem~\\ref{ProcessConvergenceThm}  using \n\\cite{MR0301846}\nand \\cite{MR0651392}. \n\nIt is intuitively clear that the variance of $\\hat{e}_n (x)$ is \napproximately $\\sigma^2 (x) \/ n(x)$ where \n\\begin{eqnarray*}\n\\sigma^2 (x) = Var[X-x| X> x]\n\\end{eqnarray*}\n is the residual variance and $n(x)$ is the number of observations exceeding $x$; \nthe formula would be justified if \nthese $n(x)$ observations were a random sample of fixed sized $n(x)$ from the \nconditional distribution $P( \\cdot | X>x)$.  Noting that \n$\\overline{{\\mathbb F}}_n (x) = n(x)\/n \\rightarrow \\overline{F}(x)$ a.s., we would then have\n\\begin{eqnarray*}\nn Var[ \\hat{e}_n (x)] = n \\sigma^2 (x) \/ n(x)  \\rightarrow \\sigma^2 (x) \/ \\overline{F} (x) .\n\\end{eqnarray*}\nProposition~\\ref{prop:VarianceCovarianceProp} and \nTheorem~\\ref{ProcessConvergenceThm}  validate this (see (2.4) below):  the variance of \nthe limiting distribution of $n^{1\/2} ( \\hat{e}_n (x) - e(x))$ is precisely \n$\\sigma^2 (x) \/ \\overline{F} (x)$.  \n\nIn Section~\\ref{sec:AlternativeSufficientConditions} \nsimpler sufficient conditions for Theorems~\\ref{ConsistencyTh1}  \nand \\ref{ProcessConvergenceThm}  are given and the \ngrowth rate of the variance of the limiting process for large $x$ is considered;\nthese results are related to those of \n\\cite{MR0359049}. \nExponential, Weibull, and Pareto examples are considered in Section 4.\n\nIn Section~\\ref{sec:confidenceBands}, by transforming (and reversing) the time scale and rescaling\nthe state space, we convert the limit process to standard Brownian motion on \nthe unit interval (Theorem~\\ref{ReversedProcBMThm}); this enables construction of nonparametric simultaneous\nconfidence bands for the function $e_F$ (Corollary~\\ref{UniformConfBandsCor}).\nApplication to survival data of guinea pigs subject to infection with \ntubercle bacilli as given by \n\\cite{Bjerkedal-60} \nappears in Section~\\ref{sec:illustrationOfBands}.\n\nWe conclude this section with a brief review of other previous work.\nEstimation of the function $e$, and especially the discretized life-table version,\nhas been considered by Chiang; see pages 630-633 of \n\\cite{Chiang-60}\nand page 214 of \n\\cite{Chiang-68}. \n(Also see \\cite{Chiang-68},\npage 189, for some \nearly history of the subject.)  The basis for {\\sl marginal} inference (i.e. at a specific\nage $x$) is that the estimate $\\hat{e}_n (x)$ is approximately normal with \nestimated standard error $S_k \/ \\sqrt{k}$, where $k = n \\overline{{\\mathbb F}}_n (x)$ \nis the observed number of observations beyond $x$ and $S_k$ is the \nsample standard deviation of those observations.  \nA partial justification of this is in \n\\cite{Chiang-60},\npage 630, (and is made precise in \nProposition~\\ref{prop:PointwiseAsympNormality} below).  \n\\cite{Chiang-68},\npage 214, gives the analogous marginal \nresult for grouped data in more detail, but again without proofs;  \nnote the solumn $S_{\\hat{e}_i}$ in his Table 8, page 213, which is based on a \nmodification and correction of a variance formula due to \n\\cite{Wilson-38}. \nWe know of no earlier work on simultaneous inference (confidence\nbands) for mean residual life.   \n   \nA plot of (a continuous version of) the estimated mean residual life function of  \n43 patients suffering from chronic gramulocytic leukemia is given \nby \n\\cite{MR0253494}. \n\\cite{Gross-Clark-75} \nbriefly mention the estimation of $e$ in a life - table setting, but \ndo not discuss the variability of the estimates (or estimates thereof).\nTests for exponentiality against decreasing mean residual life alternatives have\nbeen considered by \n\\cite{MR0395119}. \n\n\\section{Convergence on ${\\mathbb  R}^+$; covariance function of the limiting process}\n\\label{sec:convergenceOnRplus}\nLet $\\{ a_n \\}_{n\\ge1}$ be a sequence of nonnegative numbers with $a_n \\rightarrow 0$\nas $n\\rightarrow \\infty$.  For any such sequence and a d.f. $F$ as above, set \n$b_n = F^{-1} (1-a_n) \\rightarrow \\infty$ as $n\\rightarrow \\infty$.  \nThen, for any function $f$ on ${\\mathbb  R}^+$, define $f^*$ equal to $f$ for $x \\le b_n$ \nand $0$ for $x> b_n$:  $f^* (x) = f(x) 1_{[0,b_n ]} (x)$.  \nLet $\\| f \\|_a^b \\equiv \\sup_{a \\le x \\le b} | f(x)|$ and write $\\| f \\| $ \nif $a=0$ and $b=\\infty$.  \n\nLet ${\\cal H}(\\downarrow)$ denote the set of all nonnegative, decreasing \nfunctions $h$ on $[0,1]$ for which $\\int_0^1 (1\/h) dI< \\infty$.\n\\medskip\n\n\\par\\noindent\n{\\bf Condition 1a.}  There exists $h \\in {\\cal H} ( \\downarrow)$ such that\n\\begin{eqnarray*}\nM_1 \\equiv M_1 (h,F) \\equiv \\sup_x \\frac{\\int_x^\\infty h(F) dI\/ h(F(x))}{e(x)} < \\infty .\n\\end{eqnarray*}\nSince $0 < h(0) < \\infty$ and $e(0) = E(X) < \\infty$, \nCondition 1a implies that $\\int_0^\\infty h(F) dI < \\infty$.\nAlso note that $h(F)\/h(0)$ is a survival function on ${\\mathbb  R}^+$\nand that the numerator in Condition 1a is simply $e_{h(F)\/h(0)}$; \nhence Condition 1a may be rephrased as:  there exists $h \\in {\\cal H}(\\downarrow) $\nsuch that $M_1 \\equiv \\| e_{h(F)\/h(0)} \/ e_F \\| < \\infty$.\n\\medskip\n\n\\par\\noindent\n{\\bf Condition 1b.}  \nThere exists $h \\in {\\cal H}(\\downarrow)$ for which \n$\\int_0^\\infty h(F) dI< \\infty$ and $\\| e h(F) \\| < \\infty$.\n\\medskip\n\n\\par\\noindent\nBounded $e_F$ and existence of a moment of order greater than $1$ \nis more than sufficient for Condition 1b (see Section 3).\n\\medskip\n\n\\begin{thm}\n\\label{ConsistencyTh1}\nLet $a_n = \\alpha n^{-1} \\log \\log n$ with $\\alpha>1$.  \nIf Condition 1a holds for a particular $h \\in {\\cal H} (\\downarrow)$, then\n\\begin{eqnarray}\n&& \\rho_{h(F)e\/ \\overline{F}} ( \\hat{e}_n^* , e^* ) \\nonumber \\\\\n&& \\ \\ \\ \\equiv \\sup \\left \\{ \\frac{|\\hat{e}_n (x) - e(x) | \\overline{F} (x)}{h(F(x))e(x)} : \\ x \\le b_n \\right \\}\n\\rightarrow_{a.s.} 0 \\  \\ \\mbox{as} \\ \\ n \\rightarrow \\infty .\n\\label{WeightedConsistencyPart1}\n\\end{eqnarray}\nIf Condition 1b holds, then \n\\begin{eqnarray}\n&& \\rho_{1\/\\overline{F}} ( \\hat{e}_n^* , e^* )  \\nonumber \\\\\n&& \\ \\ \\ \\equiv \\sup \\{  |\\hat{e}_n (x) - e(x) | \\overline{F} (x) : \\ x \\le b_n \\}\n\\rightarrow_{a.s.} 0 \\  \\ \\mbox{as} \\ \\ n \\rightarrow \\infty .\n\\label{WeightedConsistencyPart2}\n\\end{eqnarray}\n\\end{thm}\n\nThe metric in (\\ref{WeightedConsistencyPart2}) turns out to be a natural one\n(see Section~\\ref{sec:confidenceBands}); that in (\\ref{WeightedConsistencyPart1}) is typically stronger.\n\n\\begin{proof}\nFirst note that for $x< X_{nn}$\n\\begin{eqnarray*}\n\\hat{e}_n (x) - e(x) \n= \\frac{\\overline{F}(x)}{\\overline{{\\mathbb F}}_n (x) }\n     \\left \\{ \\frac{-\\int_x^\\infty ( {\\mathbb F}_n - F)dI}{\\overline{F} (x)} \n     + \\frac{e(x)}{\\overline{F} (x)} ( {\\mathbb F}_n (x) - F(x) ) \\right \\} .\n\\end{eqnarray*}\nHence \n\\begin{eqnarray*}\n\\rho_{h(F)e\/ \\overline{F}} ( \\hat{e}_n^* , e^* ) \n& \\le & \\bigg \\| \\frac{\\overline{F}}{\\overline{{\\mathbb F}}_n } \\bigg \\|_0^{b_n} \n          \\left \\{ \\sup_x \\frac{| \\int_x^\\infty ({\\mathbb F}_n - F) dI |}{h(F(x))e(x)} \n           + \\sup_x \\frac{| {\\mathbb F}_n (x) - F(x) |}{h(F(x))} \\right \\} \\\\\n & \\le & \\bigg \\| \\frac{\\overline{F}}{\\overline{{\\mathbb F}}_n } \\bigg \\|_0^{b_n} \\cdot \n              \\rho_{h(F)} ( {\\mathbb F}_n , F ) (M_1 + 1 ) \\\\\n & \\rightarrow_{a.s.} & 0\n \\end{eqnarray*}\nusing Condition 1a, Theorem 1 of \n\\cite{MR0651528} \nto show $\\rho_{h(F)} ( {\\mathbb F}_n , F ) \\rightarrow_{a.s.} 0$ a.s., \nand Theorem 2 of \n\\cite{MR0651392} \nto show that \n$\\limsup_n \\| \\overline{F} \/ \\overline{{\\mathbb F}}_n \\|_0^{b_n} < \\infty$ a.s..\n\nSimilarly, using Condition 1b,\n\\begin{eqnarray*}\n\\rho_{1\/ \\overline{F}} ( \\hat{e}_n^* , e^* ) \n& \\le & \\bigg \\| \\frac{\\overline{F}}{\\overline{{\\mathbb F}}_n } \\bigg \\|_0^{b_n} \n          \\left \\{ \\sup_x | \\int_x^\\infty ({\\mathbb F}_n - F) dI |  \n           + \\sup_x  e(x) | {\\mathbb F}_n (x) - F(x) | \\right \\} \\\\\n & \\le & \\bigg \\| \\frac{\\overline{F}}{\\overline{{\\mathbb F}}_n } \\bigg \\|_0^{b_n} \\cdot \n              \\rho_{h(F)} ( {\\mathbb F}_n , F ) \\left ( \\int_0^\\infty h(F) dI + \\| e h(F) \\| \\right ) \\\\\n & \\rightarrow_{a.s.} & 0 .\n \\end{eqnarray*}\n\\end{proof} \n\nTo extend Yang's weak convergence results, we will use the \nspecial uniform empirical processes ${\\mathbb U}_n$ of the Appendix of \n\\cite{MR0301846} \nor \n\\cite{MR838963} \nwhich converge to a special Brownian bridge \nprocess ${\\mathbb U}$ in the strong sense that\n\\begin{eqnarray*}\n\\rho_q ({\\mathbb U}_n , {\\mathbb U}) \\rightarrow_p 0 \\ \\ \\mbox{as} \\ \\ n \\rightarrow \\infty \n\\end{eqnarray*}\nfor $q \\in {\\cal Q} (\\downarrow)$, the set of all continuous functions \non $[0,1]$ which are monotone decreasing on $[0,1]$ and \n$\\int_0^1 q^{-2} dI < \\infty$.  Thus ${\\mathbb U}_n = n^{1\/2} (\\Gamma_n - I)$\non $[0,1]$ where $\\Gamma_n$ is the empirical d.f. of special uniform \n$(0,1)$ random variables $\\xi_1 , \\ldots , \\xi_n$.\n\nDefine the {\\sl mean residual life process} on ${\\mathbb  R}^+$ by \n\\begin{eqnarray*}\nn^{1\/2} ( \\hat{e}_n (x) - e(x)) \n& = & \\frac{1}{\\overline{{\\mathbb F}}_n (x)} \\left \\{ - \\int_x^\\infty n^{1\/2} ( {\\mathbb F}_n - F) dI \n            + e(x) n^{1\/2} ( {\\mathbb F}_n (x) - F(x)) \\right \\} \\\\\n& \\stackrel{d}{=} & \\frac{1}{\\overline{\\Gamma}_n (F(x))} \\left \\{ \n              - \\int_x^\\infty {\\mathbb U}_n (F) dI + e(x) {\\mathbb U}_n (F(x)) \\right \\} \\\\\n& \\equiv & {\\mathbb Z}_n (x), \\ \\ \\ \\ 0\\le x < F^{-1} (\\xi_{nn} ) \n\\end{eqnarray*}\nwhere $\\xi_{nn} = \\max_{1 \\le i \\le n} \\xi_i$, and \n${\\mathbb Z}_n (x) \\equiv - n^{1\/2} e(x)$ for $x \\ge F^{-1} ( \\xi_{nn} )$.  \nThus ${\\mathbb Z}_n$ has the same law as $n^{1\/2} (\\hat{e}_n - e)$ \nand is a function of the special process ${\\mathbb U}_n$.  Define the corresponding \nlimiting process ${\\mathbb Z}$ by \n\\begin{eqnarray}\n{\\mathbb Z}(x) = \\frac{1}{\\overline{F} (x)} \\left \\{ - \\int_x^\\infty {\\mathbb U} (F) dI + e(x) {\\mathbb U}(F(x)) \\right \\}, \n\\ \\ \\ \\ 0 \\le x < \\infty .\n\\label{LimitProcess}\n\\end{eqnarray}\nIf $\\sigma^2 = Var(X) < \\infty$ (and hence under either Condition 2a or 2b below), \n${\\mathbb Z}$ is a mean zero Gaussian process on ${\\mathbb  R}^+$ with covariance function \ndescribed as follows:\n\n\\begin{prop}  \n\\label{prop:VarianceCovarianceProp}\nSuppose that $\\sigma^2 = Var(X) < \\infty$.\nFor $0 \\le x \\le y < \\infty$\n\\begin{eqnarray}\nCov [{\\mathbb Z}(x) , {\\mathbb Z}(y)] \n= \\frac{\\overline{F}(y)}{\\overline{F}(x)} Var [{\\mathbb Z}(y)] = \\frac{\\sigma^2 (y)}{\\overline{F}(y)} \n\\label{CovarianceFormulaOne}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n\\sigma^2 (t) \\equiv Var[X-t|X>t] \n= \\frac{\\int_t^\\infty (x-t)^2 F(x)}{\\overline{F} (t)} - e^2 (t) \n\\end{eqnarray*}\nis the residual variance function; also \n\\begin{eqnarray}\nCov[{\\mathbb Z}(x) \\overline{F}(x) , {\\mathbb Z}(y) \\overline{F}(y)]\n= Var[{\\mathbb Z}(y) \\overline{F} (y)] = \\overline{F}(y) \\sigma^2 (y) .\n\\label{CovarianceFormulaTwo}\n\\end{eqnarray}\n\\end{prop}\n\n\\begin{proof}  \nIt suffices to prove (\\ref{CovarianceFormulaTwo}).  \nLet ${\\mathbb Z}' \\equiv {\\mathbb Z} \\overline{F}$; from (\\ref{LimitProcess}) we find\n\\begin{eqnarray*}\nCov[{\\mathbb Z}' (x) , {\\mathbb Z}' (y)] \n& = & e(x) e(y) F(x) \\overline{F} (y) - e(x) \\int_y^\\infty F(x) \\overline{F} (z) dz \\\\\n&& \\ \\ \\ - \\ e(y) \\int_x^\\infty (F(y \\wedge z) - F(y) F(z) ) dz \\\\\n&& \\ \\ \\ + \\ \\int_x^\\infty \\int_y^\\infty (F(z \\wedge w) - F(z) F(w) ) dz dw .\n\\end{eqnarray*}\nExpressing integrals over $(x,\\infty)$ as the sum of integrals over $(x,y)$\nand $(y,\\infty)$, and recalling the defining formula for $e(y)$, we find that the \nright side reduces to \n\\begin{eqnarray*}\n\\lefteqn{\n\\int_y^\\infty \\int_y^\\infty (F(z \\wedge z) - F(z) F(w) dz dw - e^2 (y) F(y) \\overline{F} (y) }\\\\\n&= & \\int_y^\\infty (t-y)^2 dF(t) - \\overline{F} (y) e^2 (y) \\\\\n& = & \\overline{F} (y) \\sigma^2 (y) \n\\end{eqnarray*}\nwhich, being free of $x$, is also $Var[{\\mathbb Z}' (y)]$.  \n\\end{proof}\n\nAs in this proposition, the process ${\\mathbb Z}$ is often more easily studied \nthrough the process ${\\mathbb Z}' = {\\mathbb Z} \\overline{F}$;  such a study continues\nin Section 5.  \nStudy of the variance of ${\\mathbb Z}(x)$, namely $\\sigma^2 (x)\/ \\overline{F}(x)$,\nfor large $x$ appears in Section 3.\n\\medskip\n\n\\par\\noindent\n{\\bf Condition 2a}.  $\\sigma^2 < \\infty$ and there exists \n$q \\in {\\cal Q}(\\downarrow)$ such that\n\\begin{eqnarray*}\nM_2 \\equiv M_2 (q,F) \\equiv \\sup_x \\frac{\\int_x^\\infty q(F) dI\/q(F(x))}{e(x)} < \\infty .\n\\end{eqnarray*}\nSince $0 < q(0) < \\infty$ and $e(0) = E(X) < \\infty$, Condition 2a implies\nthat $\\int_0^\\infty q(F) dI < \\infty$; Condition 2a may be rephrased as:\n$M_2 \\equiv \\| e_{q(F)\/q(0)}\/ e_F \\| < \\infty$ where \n$e_{q(F)\/q(0)}$ denotes the mean residual life function for the \nsurvival function $q(F)\/q(0)$.  \n\\medskip\n\n\\par\\noindent\n{\\bf Condition 2b.}  $\\sigma^2 < \\infty$ and there exists \n$q \\in {\\cal Q}(\\downarrow)$ such that $\\int_0^\\infty q(F)dI< \\infty$. \n\\medskip\n\nBounded $e_F$ and existence of a moment of order greater than $2$ \nis more than sufficient for 2b (see Section~\\ref{sec:AlternativeSufficientConditions}).  \n\\medskip\n\n\\begin{thm}  (Process convergence). \n\\label{ProcessConvergenceThm} \nLet $a_n \\rightarrow 0$, $na_n \\rightarrow \\infty$.  \nIf Condition 2a holds for a particular $q \\in {\\cal Q}(\\downarrow)$, then\n\\begin{eqnarray}\n&& \\rho_{q(F) e \/ \\overline{F}} ( {\\mathbb Z}_n^* , {\\mathbb Z}^*) \\nonumber \\\\\n&& \\ \\ \\ \\ \\equiv \\sup \\left \\{ \\frac{| {\\mathbb Z}_n (x) - {\\mathbb Z}(x)| \\overline{F}(x)}{q(F(x))e(x)} : \\ \\ \nx \\le b_n \\right \\} \\rightarrow_p 0 \\ \\ \\ \\mbox{as} \\ \\ n \\rightarrow \\infty .\n\\label{ProcessConvergenceA}\n\\end{eqnarray}\nIf Condition 2b holds, then\n\\begin{eqnarray}\n\\label{ProcessConvergenceB}\n\\phantom{blablabla}\\rho_{1\/\\overline{F}} ( {\\mathbb Z}_n^*, {\\mathbb Z}^*) \n\\equiv \\sup \\{ | {\\mathbb Z}_n(x) - {\\mathbb Z}(x)| \\overline{F} (x): \\ x \\le b_n \\} \\rightarrow_p 0 \n\\ \\ \\ \\mbox{as} \\ \\ n \\rightarrow \\infty .\n\\end{eqnarray}\n\\end{thm}\n\n\\begin{proof}\nFirst write \n\\begin{eqnarray*}\n{\\mathbb Z}_n (x) - {\\mathbb Z}(x) \n= \\left \\{ \\frac{\\overline{F}(x)}{\\overline{\\Gamma}_n (F(x))} -1 \\right \\} {\\mathbb Z}_n^1 (x) \n     +  ({\\mathbb Z}_n^1 (x) - {\\mathbb Z}(x)) \n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n{\\mathbb Z}_n^1 (x) \\equiv \\frac{1}{\\overline{F}(x)} \\left \\{ - \n        \\int_x^\\infty {\\mathbb U}_n (F) dI + e(x) {\\mathbb U}_n (F(x)) \\right \\}, \\ \\ \\ \\ 0 \\le x < \\infty .\n\\end{eqnarray*}\nThen note that, using Condition 2a, \n\\begin{eqnarray*}\n\\rho_{q(F)e\/\\overline{F}} ({\\mathbb Z}_n^1 , 0 ) \n& \\le & \\sup_x \\frac{| \\int_x^\\infty {\\mathbb U}_n (F) dI }{q(F(x))e(x)} + \\rho_q ({\\mathbb U}_n , 0 ) \\\\\n& \\le & \\rho_q ({\\mathbb U}_n , 0)\\{ M_2 +1) \\} = O_p (1);\n\\end{eqnarray*}\nthat $\\| \\overline{I}\/ \\overline{\\Gamma}_n -1 \\|_0^{1-a_n} \\rightarrow_p 0$\nby Theorem 0 of Wellner (1978) since $na_n \\rightarrow \\infty$; and,\nagain using Condition 2a, that \n\\begin{eqnarray*}\n\\rho_{q(F) e\/ \\overline{F}} ({\\mathbb Z}_n^1 , {\\mathbb Z}) \n& \\le & \\sup_x \\frac{| \\int_x^\\infty ( {\\mathbb U}_n (F) - {\\mathbb U}(F))dI|}{q(F(x))e(x)} \n            + \\rho_q ({\\mathbb U}_n , {\\mathbb U})\\\\\n& \\le & \\rho_q ({\\mathbb U}_n , {\\mathbb U}) \\{ M_2 +1 \\} \\rightarrow_p 0 .\n\\end{eqnarray*}\nHence\n\\begin{eqnarray*}\n\\rho_{q(F)e\/ \\overline{F}} ({\\mathbb Z}_n^* , {\\mathbb Z}^* ) \n& \\le & \\bigg \\| \\frac{\\overline{I}}{\\overline{\\Gamma}_n} -1 \\bigg \\|_0^{1-a_n} \n            \\rho_{q(F)e\/\\overline{F}} ({\\mathbb Z}_n^1, 0) + \\rho_{q(F)e\/\\overline{F}} ({\\mathbb Z}_n^1, {\\mathbb Z})\\\\\n& = & o_p (1) O_p (1) + o_p (1) = o_p (1) .\n\\end{eqnarray*}\nSimilarly, using Condition 2b\n\\begin{eqnarray*}\n\\rho_{1\/\\overline{F}} ({\\mathbb Z}_n^1, 0) \n& \\le & \\sup_x \\bigg | \\int_x^\\infty {\\mathbb U}_n (F) dI \\bigg | + \\sup_x e(x)  | {\\mathbb U}_n (F(x)) | \\\\\n& \\le & \\rho_q ({\\mathbb U}_n , 0) \\left \\{ \\int_0^\\infty q(F) dI + \\| e q(F) \\| \\right \\} = O_p (1),\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\rho_{1\/\\overline{F}} ( {\\mathbb Z}_n^1, {\\mathbb Z}) \n& \\le & \\sup_x \\bigg | \\int_x^\\infty ({\\mathbb U}_n (F) - {\\mathbb U} (F) ) dI \\bigg | + \\sup_x e(x) | {\\mathbb U}_n (F(x)) - {\\mathbb U} (F(x)) |\\\\\n& \\le & \\rho_q ({\\mathbb U}_n , {\\mathbb U} ) \\left \\{ \\int_0^\\infty q(F) dI + \\| e q(F) \\| \\right \\} \\rightarrow_p 0,\n\\end{eqnarray*}\nand hence \n\\begin{eqnarray*}\n\\rho_{1\/\\overline{F}} ( {\\mathbb Z}_n^* , {\\mathbb Z}^*) \n& \\le & \\bigg \\| \\frac{\\overline{I}}{\\overline{\\Gamma}_n} -1 \\bigg \\|_0^{1-a_n} \n            \\rho_{1\/\\overline{F}} ({\\mathbb Z}_n^1 , 0 ) + \\rho_{1\/\\overline{F}} ({\\mathbb Z}_n^1, {\\mathbb Z}) \\\\\n& = & o_p (1) O_p (1) + o_p (1) = o_p (1) .\n\\end{eqnarray*}\n\\end{proof}\n\n\\section{Alternative sufficient conditions; $Var[{\\mathbb Z}(x)]$ as $x \\rightarrow \\infty$.}\n\\label{sec:AlternativeSufficientConditions}\nOur goal here is to provide easily checked conditions which will imply the \nsomewhat cumbersome Conditions 2a and 2b;  similar conditions also appear \nin the work of \n\\cite{MR0359049}, \nand we use their results to extend\ntheir formula for the residual coefficient of variation for large $x$ \n((\\ref{LimitingVarianceRelToMRLSquared}) below).  \nThis provides a simple description of the behavior of \n$Var[{\\mathbb Z}(x)]$, the asymptotic variance of $n^{1\/2} (\\hat{e}_n (x) - e(x))$\nas $x\\rightarrow \\infty$.\n\\medskip\n\n\\par\\noindent\n{\\bf Condition 3.}  $E(X^r) < \\infty$ for some $r>2$.\n\\smallskip\n\n\\par\\noindent\n{\\bf Condition 4a.}  Condition 3 and $\\lim_{x \\rightarrow \\infty}  \\frac{d}{dx} (1\/\\lambda (x))\n= c < \\infty$ where $\\lambda = f\/ \\overline{F}$, the hazard function.\n\\smallskip\n\n\\par\\noindent\n{\\bf Condition 4b.}  Condition 3 and $\\limsup_{x \\rightarrow \\infty} \\{\n\\overline{F}(x)^{1+\\gamma} \/ f(x) \\} < \\infty$ for some \n$r^{-1} < \\gamma < 1\/2$. \n\\medskip\n\n\\begin{prop}  \n\\label{RegVariationSuffCond}\nIf Condition 4a holds, then $0 \\le c \\le r^{-1}$, Condition 2a holds, \nand the squared residual coefficient of variation tends to $1\/(1-2c)$:\n\\begin{eqnarray}\n\\lim_{x \\rightarrow \\infty} \\frac{\\sigma^2 (x)}{e^2 (x)} = \\frac{1}{1-2c} .\n\\label{LimitingVarianceRelToMRLSquared}\n\\end{eqnarray}\nIf Condition 4b holds, then Condition 2b holds.\n\\end{prop}\n\n\\begin{cor}\nCondition 4a implies\n\\begin{eqnarray*}\nVar[ {\\mathbb Z}(x)] \\sim \\frac{e^2 (x)}{\\overline{F} (x)} (1- 2c)^{-1} \\ \\ \\ \n\\mbox{as} \\ \\ x \\rightarrow \\infty .\n\\end{eqnarray*}\n\\end{cor}\n\n\\begin{proof}\nAssume 4a.  Choose $\\gamma$ between $r^{-1}$ and $1\/2$; define\na d.f. $G$ on ${\\mathbb  R}^+$ by $\\overline{G} = \\overline{F}^{\\gamma}$ \nand note that $g\/\\overline{G} = \\gamma f \/ \\overline{F} = \\gamma \\lambda$.\nBy Condition 3 $x^r \\overline{F}(x) \\rightarrow 0$ as $x \\rightarrow \\infty$ \nand hence $x^{\\gamma r} \\overline{G}(x) \\rightarrow 0$ as $x \\rightarrow \\infty$. \nSince $\\gamma r > 1$, $G$ has a finite mean and therefore \n$e_G (x) = \\int_x^\\infty \\overline{G} dI \/ \\overline{G} (x)$ is well-defined.\n\nSet $\\eta = 1 \/ \\lambda  = \\overline{F}\/f$, and note that \n$\\eta (x) \\overline{G} (x) \\rightarrow 0$ as $x \\rightarrow \\infty$.  \n(If $\\limsup \\eta(x) < \\infty$, then it holds trivially; otherwise, $\\eta (x) \\rightarrow \\infty$\n(because of 4a) and $\\lim \\eta(x) \\overline{G} (x) = \\lim (\\eta(x)\/x) (x \\overline{G} (x)) \n= \\lim \\eta '' (x) x \\overline{G}(x) = 0$ by 4a and L'Hopital.\nThus by L'Hopital's rule\n\\begin{eqnarray*}\n0 & \\le & \\lim \\frac{\\eta(x)}{e_G (x)} \n               = \\lim \\frac{\\eta(x) \\overline{G} (x)}{\\int_x^\\infty \\overline{G} dI} \\\\\n& = & \\lim \\frac{\\eta (x) g(x) - \\overline{G} (x) \\eta' (x)}{\\overline{G} (x)} \\\\\n& = & \\gamma - \\lim \\eta' (x) = \\gamma - c \\ \\ \\ \\mbox{by 4a} .\n\\end{eqnarray*}\nThus $c \\le \\gamma$ for any $\\gamma > r^{-1}$ and it follows that \n$c \\le r^{-1}$.  It is elementary that $c \\ge 0$ since $\\eta = 1\/\\lambda$ \nis nonnegative.\n\nChoose $q(t) = (1-t)^{\\gamma}$.  Then $\\gamma -c> 0$, \n$q \\in {\\cal Q}(\\downarrow)$, and to verify 2a \nit now suffices to show that $\\lim (\\eta(x) \/ e_F(x) ) = 1-c < \\infty$\nsince it then follows that \n\\begin{eqnarray*}\n\\lim \\frac{e_G (x)}{e_F (x)} = \\lim \\frac{\\eta (x) \/ e_F (x)}{\\eta (x) \/ e_G (x)} \n= \\frac{1-c}{\\gamma-c} < \\infty .\n\\end{eqnarray*}\nBy continuity and $e_G (0) < \\infty$,  $0 < e_F (0) < \\infty$, \nthis implies Condition 2a.  But $r>2$ implies that $x \\overline{F}(x) \\rightarrow 0$ \nas $x \\rightarrow \\infty$ so $\\eta (x) \\overline{F} (x) \\rightarrow 0$ and hence\n\\begin{eqnarray*}\n\\lim \\frac{\\eta(x)}{e_F (x)} \n= \\lim \\frac{\\eta \\overline{F} (x)}{\\int_x^\\infty \\overline{F} dI} \n= \\lim ( 1 - \\eta' (x)) = 1-c .\n\\end{eqnarray*}\nThat (\\ref{LimitingVarianceRelToMRLSquared}) holds will now follows from results of \n\\cite{MR0359049}, \nas follows:\nTheir Corollary to Theorem 7 implies that \n$P( \\lambda (t) (X-t) > x | X>t) \\rightarrow e^{-x}$ if $c=0$ \nand $\\rightarrow (1+cx)^{-1\/c}$ if $c>0$.  \nThus, in the former case, $F$ is in the domain of attraction of the \nPareto residual life distribution and its related extreme value distribution.\nThen Theorem 8(a) implies convergence of the (conditional) mean \nand variance of $\\lambda (t) (X -t)$ to the mean and variance \nof the limiting Pareto distribution, namely $(1-c)^{-1}$ \nand $(1-c)^{-2} (1-2c)^{-1}$.  \nBut the conditional mean of $\\lambda (t) (X_t)$ is simply \n$\\lambda (t) e(t)$, so that $\\lambda (t) \\sim (1-c)^{-1} \/ e(t)$ \nand (\\ref{LimitingVarianceRelToMRLSquared}) now follows.\n\nIf Condition 4b holds, let $q(F) = \\overline{F}^{\\gamma}$ again.\nThen $\\int_0^{\\infty} q(F) dI< \\infty$, \nand it remains to show that \n$\\limsup\\{ e(x) \\overline{F} (x)^{\\gamma} \\} < \\infty$.\nThis follows from 4b by L'Hopital.  \n\\end{proof}\n\nSimilarly, sufficient conditions for Conditions 1a and 1b can be given:\nsimply replace ``2'' in Condition 3 and ``1\/2'' in Condition 4b with ``1'', \nand the same proof works.  Whether (\\ref{LimitingVarianceRelToMRLSquared}) \nholds when $r$ in Condition 3 is exactly $2$ is not known.\n\n\\section{Examples.}\n\\label{sec:examples}\nThe typical situation, when $e(x)$ has a finite limit and Condition 3 holds, \nis as follows:  $e\\sim \\overline{F}\/f \\sim f\/(-f')$ as $x \\rightarrow \\infty$ \n(by L'Hopital), and hence 4b, 2b, and 1b hold;  also \n$\\eta' \\equiv (\\overline{F}\/ f)' = [ (F\/f) (-f\/f')] -1 \\rightarrow 0$\n(4a with $c=0$, and hence 2a and 1a hold), $\\sigma (x) \\sim e(x) $ \nfrom (\\ref{LimitingVarianceRelToMRLSquared}), and \n$Var[{\\mathbb Z}] \\sim e^2 \/ \\overline{F} \\sim (\\overline{F}\/f)^2 \/\\overline{F}  \\sim 1\/(-f')$.  \nWe treat three examples, not all `typical', in more detail.\n\n\\begin{exmpl} \n(Exponential).  \nLet $\\overline{F} (x) = \\exp (-x\/\\theta)$, $x \\ge 0$, with \n$0 < \\theta < \\infty$.  Then $e(x) = \\theta $ for all $x \\ge 0$. \nConditions 4a and 4b hold (for all $r$, $\\gamma \\ge 0$) with $c=0$, \nso Conditions 2a and 2b hold by Proposition~\\ref{RegVariationSuffCond}\nwith $q(t) = (1-t)^{1\/2-\\delta}$,  $0 < \\delta < 1\/2$.  \nConditions 1a and 1b hold with $h(t) = (1-t)^{1-\\delta}$, $0 < \\delta < 1$.  \nHence Theorems~\\ref{ConsistencyTh1} and ~\\ref{ProcessConvergenceThm}  hold where now \n\\begin{eqnarray*}\n{\\mathbb Z}(x) = \\frac{{\\mathbb U} (F(x))}{1-(x)} - \\frac{1}{1-F(x)} \\int_{F(x)}^1 \\frac{{\\mathbb U}}{1-I} dI \n\\stackrel{d}{=} \\theta {\\mathbb B} (e^{x\/\\theta} ) , \\ \\ \\ 0 \\le x < \\infty\n\\end{eqnarray*}\nand ${\\mathbb B}$ is standard Brownian motion on $[0,\\infty)$.  \n(The process ${\\mathbb B}_1 (t) = {\\mathbb U} (1-t) - \\int_{1-t}^1 ( {\\mathbb U}\/ (1-I) ) dI$, \n$0 \\le t \\le 1$, is Brownian motion on $[0,1]$; and with \n${\\mathbb B}_2 (x) \\equiv x {\\mathbb B}_1 (1\/x)$ for $1 \\le x \\le \\infty$, \n${\\mathbb Z}(x) = \\theta {\\mathbb B}_2 (1\/\\overline{F} (x)) = \\theta {\\mathbb B}_2 (e^{x\/\\theta} )$.)\nThus, in agreement with (\\ref{CovarianceFormulaOne}), \n\\begin{eqnarray*}\nCov[ {\\mathbb Z}(x) , {\\mathbb Z}(y) ] = \\theta^2 e^{(x \\wedge y)\/\\theta}, \\ \\ \\ \n0 \\le x, y < \\infty .\n\\end{eqnarray*}\nAn immediate consequence is that \n$\\| {\\mathbb Z}_n^* \\overline{F} \\| \\rightarrow_d \\| \\overline{F} \\| \\stackrel{d}{=} \n\\theta \\sup_{0 \\le t \\le 1} | {\\mathbb B}_1 (t) |$; \ngeneralization of this to other $F$'s appears in Section 5. \n(Because of the ``memoryless'' property of exponential $F$, \nthe results for this example can undoubtedly be obtained by more\nelementary methods.)\n\\end{exmpl}\n \n\\begin{exmpl} (Weibull).  \nLet $\\overline{F} (x) = \\exp (-x^{\\theta} )$, $x \\ge 0$, with $0 < \\theta < \\infty$.\nConditions 4a and 4b hold (for all $r$, $\\gamma >0$) with $c=0$, so \nConditions 1 and 2 hold with $h$ and $q$ as in Example 1 by Proposition~\\ref{RegVariationSuffCond}. \nThus Theorems~\\ref{ConsistencyTh1}  and ~\\ref{ProcessConvergenceThm}  hold.  Also, $e(x) \\sim \\theta^{-1} x^{1-\\theta}$ as \n$x \\rightarrow \\infty$, and hence $Var[Z(x)] \\sim \\theta^{-2} x^{2(1-\\theta)} \\exp( x^{\\theta})$\nas $x \\rightarrow \\infty$.\n\\end{exmpl}\n\n\\begin{exmpl} (Pareto).\nLet $\\overline{F} (x) = (1+cx)^{-1\/c}$, $x \\ge 0$, with $0 < c < 1\/2$.\nThen $e(x) = (1-c)^{-1} (1+cx)$, and Conditions 4a and 4b hold for $r< c^{-1}$ \nand $\\gamma\\ge c$  (and $c$ of 4a is $c$).  Thus Proposition~\\ref{RegVariationSuffCond} holds with $r>2$\nand $c>0$ and $Var[{\\mathbb Z}(x)] \\sim c^{2+(1\/c)} (1-c)^{-2} (1-2c)^{-1} x^{2 +(1\/c)} $\nas $x \\rightarrow \\infty$.  \nConditions 1 and 2 hold with $h$ and $q$ as in Example 1, and Theorems~\\ref{ConsistencyTh1} \nand~\\ref{ProcessConvergenceThm}  hold.\n\nIf instead $1\/2\\le c < 1$, then $E(X) < \\infty$ but $E(X^2 ) = \\infty$, \nand 4a and 4b hold with $1 < r < 1\/c \\le 2$ and $\\gamma\\ge c$.  Hence Condition 1 and \nTheorem~\\ref{ConsistencyTh1} hold, but Condition 2 (and hence our proof of \nTheorem~\\ref{ProcessConvergenceThm}) fails.  \nIf $c \\ge 1$, then $E(X) = \\infty$ and $e(x) = \\infty$ for all $x \\ge 0$.\n\\end{exmpl}\n\nNot surprisingly, the limiting process ${\\mathbb Z}$ has a variance which grows quite rapidly, \nexponentially in the exponential and Weibull cases, and as a power $(>4)$ of $x$ \nin the Pareto case.\n\n\\section{Confidence bands for $e$.}  \n\\label{sec:confidenceBands}\nWe first consider the process ${\\mathbb Z}' \\equiv {\\mathbb Z} \\overline{F}$ on ${\\mathbb  R}^+$ \nwhich appeared in (\\ref{CovarianceFormulaTwo}) of Proposition~\\ref{prop:VarianceCovarianceProp}.\nIts sample analog ${\\mathbb Z}_n' \\equiv {\\mathbb Z}_n \\overline{{\\mathbb F}}_n$ is easily seen to be a \ncumulative sum (times $n^{-1\/2}$) of the observations exceeding $x$, \neach centered at $x + e(x)$;  as $x$ decreases the number of terms in the sum \nincreases.  Moreover, the corresponding increments \napparently act asymptotically independently so that ${\\mathbb Z}_n'$, in reverse time,\nis behaving as a cumulative sum of zero-mean independent increments.  \nAdjustment for the non-linear variance should lead to Brownian motion. \nLet us return to the limit version ${\\mathbb Z}'$.\n\nThe zero-mean Gaussian process ${\\mathbb Z}'$ has covariance function \n$Cov[{\\mathbb Z}' (x) , {\\mathbb Z}' (y) ] = Var[{\\mathbb Z}' (x \\vee y) ]$ (see (\\ref{CovarianceFormulaTwo})); \nhence, when viewed in reverse time, it has independent increments \n(and hence ${\\mathbb Z}'$ is a reverse martingale).  \nSpecifically, with ${\\mathbb Z}'' (s) \\equiv {\\mathbb Z}' (-\\log s)$, \n${\\mathbb Z}'' $ is a zero-mean Gaussian process on $[0,1]$ with independent \nincrements and $Var[{\\mathbb Z}'' (s)] = Var[{\\mathbb Z}' (-\\log s)] \\equiv \\tau^2 (s)$.  \nHence $\\tau^2$ is increasing in $s$, and, from (\\ref{CovarianceFormulaTwo}),\n\\begin{eqnarray}\n\\tau^2 (s) = \\overline{F} (-\\log s) \\sigma^2 (-\\log s) .\n\\label{FormulaForTau}\n\\end{eqnarray}\nNow $\\tau^2 (1) = \\sigma^2 (0) = \\sigma^2$, and \n$$\n\\tau^2 (0) = \\lim_{\\epsilon \\downarrow 0} \\overline{F} (-\\log \\epsilon) \\sigma^2 (-\\log \\epsilon)\n= \\lim_{x \\rightarrow \\infty} \\overline{F}(x) \\sigma^2 (x) = 0\n$$\nsince\n\\begin{eqnarray*}\n0 \\le \\overline{F} (x) \\sigma^2 (x) \\le \\overline{F}(x) E(X^2 | X> x) \n= \\int_x^\\infty y^2 dF(y) \\rightarrow 0.\n\\end{eqnarray*}\nSince $f(x)>0$ for all $x \\ge 0$,  $\\tau^2 $ is strictly increasing.  \n\nLet $g$ be the inverse of $\\tau^2$; then $\\tau^2 (g(t)) = t$, $g(0) = 0$,\nand $g(\\sigma^2 ) = 1$.  Define ${\\mathbb W}$ on $[0,1]$ by \n\\begin{eqnarray}\n{\\mathbb W}(t) \\equiv \\sigma^{-1} {\\mathbb Z}'' ( g(\\sigma^2 t)) = \\sigma^{-1} {\\mathbb Z}' (-\\log g(\\sigma^2 t)) .\n\\label{DefnOfW}\n\\end{eqnarray}\n\n\\begin{thm}  \n\\label{ReversedProcBMThm}\n${\\mathbb W}$ is standard Brownian motion on $[0,1]$.\n\\end{thm}\n\n\\begin{proof}\n${\\mathbb W}$ is Gaussian with independent increments and $Var[{\\mathbb W} (t)] =t$ \nby direct computation.  \n\\end{proof}\n\n\\begin{cor}\n\\label{FBarWeightedProcessConvergence}\nIf (\\ref{ProcessConvergenceB}) holds, then\n\\begin{eqnarray*}\n\\rho ({\\mathbb Z}_n^{'*}, {\\mathbb Z}^{'*} ) \\equiv \\sup_{x \\le b_n } | {\\mathbb Z}_n (x) \\overline{{\\mathbb F}}_n (x) - {\\mathbb Z}(x) \\overline{F} (x)|\n= o_p (1) \n\\end{eqnarray*}\nand hence $\\| {\\mathbb Z}_n \\overline{{\\mathbb F}}_n \\|_0^{b_n} \\rightarrow_d \\| {\\mathbb Z} \\overline{F} \\| \n= \\sigma \\| {\\mathbb W} \\|_0^1 \\ \\ \\ \\mbox{as} \\ \\ n \\rightarrow \\infty $.\n\\end{cor}\n\n\\begin{proof}\nBy Theorem 0 of \n\\cite{MR0651392} \n$\\| \\overline{{\\mathbb F}}_n \/ \\overline{F} -1 \\|_0^{b_n} \\rightarrow_p 0$ \nas $n\\rightarrow \\infty$, and this together with (\\ref{ProcessConvergenceB}) implies the \nfirst part of the statement.  The second part follows immediately from the \nfirst and (\\ref{DefnOfW}). \n\\end{proof}\n\nReplacement of $\\sigma^2$ by a consistent estimate $S_n^2$ (e.g. the sample variance based \non all observations), and of $b_n$  by $\\hat{b}_n = {\\mathbb F}_n^{-1} (1-a_n)$, the $(n-m)-$th order statistic \nwith $m = [ n a_n]$, leads to asymptotic confidence bands for $e = e_F$:  \n\n\\begin{cor}\n\\label{UniformConfBandsCor}\nLet $0 < a < \\infty$, and set \n$\\hat{d}_n (x) \\equiv n^{-1\/2} a S_n \/ \\overline{{\\mathbb F}_n (x)}$. \n If (\\ref{ProcessConvergenceB}) holds, $S_n^2 \\rightarrow_p \\sigma^2$, and \n$na_n \/ \\log \\log n \\uparrow \\infty$, then, as $n \\rightarrow \\infty$\n\\begin{eqnarray}\n&& P \\left (\\hat{e}_n (x)   - \\hat{d}_n (x) \\le e(x) \\le \\hat{e}_n (x) + \\hat{d}_n (x) \\ \\ \n\\mbox{for all \\ } 0 \\le x \\le \\hat{b}_n \\right ) \\nonumber \\\\\n&& \\ \\  \\rightarrow   Q(a)   \n\\label{ConfidenceProbConvergence}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray*}\nQ(a) & \\equiv & P( \\| {\\mathbb W} \\|_0^1 < a) = \\sum_{k=-\\infty}^\\infty (-1)^k \\{ \\Phi ((2k+1)a) - \\Phi( (2k-1)a) \\} \\\\\n& = & 1 - 4 \\{ \\overline{\\Phi} (a) - \\overline{\\Phi} (3a) + \\overline{\\Phi} (5a) - \\cdots \\}\n\\end{eqnarray*}\nand $\\Phi$ denotes the standard normal d.f. \n\\end{cor}\n\n\\begin{proof}\nIt follows immediately from Corollary~\\ref{FBarWeightedProcessConvergence} \nand $S_n \\rightarrow_p \\sigma > 0$ that \n\\begin{eqnarray*}\n\\| {\\mathbb Z}_n \\overline{{\\mathbb F}}_n \\|_0^{b_n} \/ S_n \\rightarrow_d \\| {\\mathbb Z} \\overline{F} \\|\/ \\sigma = \\| {\\mathbb W} \\|_0^1 .\n\\end{eqnarray*}\nFinally $b_n $ may be replaced by $\\hat{b}_n$ without harm:  letting $c_n = 2 \\log \\log \/(na_n) \\rightarrow 0$\nand using Theorem 4S of \n\\cite{MR0651392}, \nfor $\\tau>1$ and all $n \\ge N(\\omega , \\tau)$, \n$\\hat{b}_n \\equiv {\\mathbb F}_n^{-1} (1-a_n) \\stackrel{d}{=} F^{-1} (\\Gamma_n^{-1} (1-a_n)) \n\\le F^{-1} (\\{ 1 + \\tau c_n^{1\/2} \\} (1-a_n))$ w.p. 1.  This proves the convergence claimed in the corollary; \nthe expression for $Q(a)$ is well-known (e.g. see \n\\cite{MR0233396}, \npage 79).  \n\\end{proof}\n\nThe approximation $1- 4 \\overline{\\Phi} (a)$ for $Q(a)$ gives 3-place accuracy for $a>1.4$.  \nA short table appears below:\n\n\n\\begin{table}[h]\n\\caption{$Q(a)$ for selected $a$}\n\\label{smallQTable}\n\\begin{tabular}{| c c | c c |}\n\\hline\n$a$ & $Q(a)$ & $a$ & $Q(a)$ \\\\\n\\hline\n2.807 & .99 & 1.534 & .75 \\\\\n2.241 & .95 & 1.149 & .50 \\\\\n1.960 & .90 & 0.871 & .25  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThus, choosing $a$ so that $Q(a) = \\beta$, (\\ref{ConfidenceProbConvergence}) provides a two-sided simultaneous confidence\nband for the function $e$ with confidence coefficient asymptotically $\\beta$. \nIn applications we suggest taking $a_n = n^{-1\/2}$ so that $\\hat{b}_n$ is the \n$(n-m)-$th order statistic with $m = [n^{1\/2}]$; we also want $m $ large enough for an adequate \ncentral limit effect, remembering that the conditional life distribution may be quite skewed.  \n(In a similar fashion, one-sided asymptotic bands are possible, but they will be less trustworthy \nbecause of skewness.)  \n\nInstead of simultaneous bands for all real $x$ one may seek (tighter) bands on $e(x)$ for one or two specific\n$x-$values. \nFor this we can apply Theorem~\\ref{ProcessConvergenceThm}  and \nProposition~\\ref{prop:VarianceCovarianceProp} directly.  \nWe first require a consistent estimator of the asymptotic variance of \n$n^{1\/2} ( \\hat{e}_n (x) - e(x))$, namely $\\sigma^2 (x) \/ \\overline{F} (x)$.  \n\n\\begin{prop}  \n\\label{prop:VarianceEstConsistent}\nLet $0 \\le x < \\infty$ be fixed and let $S_n^2 (x)$ be the sample variance of those observations \nexceeding $x$.  If Condition 3 holds then \n$S_n^2 (x) \/ \\overline{{\\mathbb F}}_n (x) \\rightarrow_{a.s.} \\sigma^2 (x) \/ \\overline{F} (x) $. \n\\end{prop}\n\n\\begin{proof} \nSince $\\overline{{\\mathbb F}}_n (x) \\rightarrow_{a.s.} \\overline{F} (x) > 0$ and \n\\begin{eqnarray*}\nS_n^2 (x) = \\frac{2 \\int_x^\\infty (y-x) \\overline{{\\mathbb F}}_n (y) dy}{ \\overline{{\\mathbb F}}_n (x)} - \\hat{e}_n^2 (x), \n\\end{eqnarray*}\nit suffices to show that $\\int_x^\\infty y {\\mathbb F}_n (y) dy \\rightarrow_{a.s.} \\int_x^\\infty y \\overline{F} (y) dy$. \nLet $h(t) = (1-t)^{\\gamma+1\/2}$ and $q(t) = (1-t)^{\\gamma}$ with $r^{-1} < \\gamma < 1\/2$ \nso that $h \\in {\\cal H}(\\downarrow)$, $q \\in {\\cal Q}(\\downarrow)$, and \n$\\int_0^\\infty q(F) dI < \\infty$ by the proof of Proposition~\\ref{RegVariationSuffCond}.  \nThen,\n\\begin{eqnarray*}\n\\bigg | \\int_x^\\infty y \\overline{{\\mathbb F}}_n (y) dy - \\int_x^\\infty y \\overline{F} (y) dy \\bigg | \n\\le \\rho_{h(F)} ( {\\mathbb F}_n , F) \\int_0^\\infty I h(F) dI \\rightarrow_{a.s.} 0 \n\\end{eqnarray*}\nby Theorem 1 of Wellner (1977) since\n\\begin{eqnarray*}\n\\int_0^\\infty I h(F) dI = \\int_0^\\infty (I^2 \\overline{F} )^{1\/2} q(F) dI < \\infty .\n\\end{eqnarray*}\n\\end{proof}\n\nBy Theorem~\\ref{ProcessConvergenceThm}, \nPropositions~\\ref{prop:VarianceCovarianceProp} and \n~\\ref{prop:VarianceEstConsistent}, and Slutsky's theorem we have:\n\n\\begin{prop} \n\\label{prop:PointwiseAsympNormality}\nUnder the conditions of Proposition~\\ref{prop:VarianceEstConsistent},\n\\begin{eqnarray*}\nd_n (x) \\equiv n^{1\/2} ( \\hat{e}_n (x) - e(x)) \\overline{{\\mathbb F}}_n^{1\/2} (x) \/ S_n (x) \n\\rightarrow_d N(0,1) \\ \\ \\ \\mbox{as} \\ \\ n \\rightarrow \\infty .\n\\end{eqnarray*}\n\\end{prop}\n\nThis makes feasible an asymptotic confidence interval for $e(x)$ (at this particular fixed $x$).\nSimilarly, for $x< y$, using the joint asymptotic normality of $(d_n (x) , d_n (y))$ \nwith asymptotic correlation $ \\{ \\overline{F} (y) \\sigma^2 (y) \/ \\overline{F} (x) \\sigma^2 (x) \\}^{1\/2}$\nestimated by \n$$\n\\{ \\overline{{\\mathbb F}}_n (y) S_n^2 (y) \/ \\overline{{\\mathbb F}}_n (x) S_n^2 (x) \\} ^{1\/2},\n$$ \nan asymptotic confidence ellipse for $(e(x), e(y))$ may be obtained.  \n\n\\section{Illustration of the confidence bands.}  \n\\label{sec:illustrationOfBands}\nWe illustrate with two data sets presented by \n\\cite{Bjerkedal-60} \nand briefly mention one appearing in Barlow and Campo (1975).\n\nBjerkedal gave various doses of tubercle bacilli to groups of $72$ guinea pigs and recorded their \nsurvival times.  \nWe concentrate on Regimens 4.3 and 6.6 (and briefly mention 5.5, the only other complete data set \nin Bjerkedal's study M); see Figures 1 and 2 below. \n\nFirst consider the estimated mean residual life $\\hat{e}_n$, the center jagged line in each figure.\nFigure 1 has been terminated at day 200; the plot would continue approximately horizontally, \nbut application of asymptotic theory to this part of $\\hat{e}_n$, based only the last 23 survival\n times \n (the last at 555 days), seems unwise.  \n Figure 2 has likewise been terminated at 200 days, omitting only nine survival times \n (the last at 376 days);  the graph of $\\hat{e}_n$ would continue downward.  \n The dashed diagonal line is $\\overline{X} -x$;  if all survival times were equal, say $\\mu$, \n then the residual life function would be $(\\mu - x)^+$, a lower bound on $e(x)$ near the origin.  \n More specifically, a Maclaurin expansion yields\n \\begin{eqnarray*}\n e(x) = \\mu + (\\mu f_0 -1) x + (1\/2)\\{ (2 \\mu f_0 -1) f_0 + f_0' \\} x^2 + o(x^2)\n \\end{eqnarray*}\n where $f_0 = f(0)$, $f_0' = f' (0)$, if $f'$ is continuous at $0$, or \n \\begin{eqnarray*}\n e(x) = \\mu - x + \\frac{\\mu d}{r!} x^r + o(x^r) \n \\end{eqnarray*}\n if $f^{(s)} (0) = 0$ for $s< (r-1)$ ($\\ge 0$) and $= d$ for $s = r-1$ \n (if $f^{(r-1)}$ is continuous at $0$).  It thus seems likely from \n Figures 1 and 2 that in each of these cases either $f_0 = 0$ and $f_0' > 0$ \n or $f_0 $ is near $0$ (and $f_0'\\ge 0$).  \n \n Also, for large $x$,  $e(x) \\sim 1\/\\lambda (x)$, and Figure 1 suggests that the \n corresponding $\\lambda $ and $e$ have finite positive limits, whereas the $e$ \n of Figure 2 may eventually decrease ($\\lambda$ increase).  \n We know of no parametric $F$ that would exhibit behavior quite like these.  \n \n The upper and lower jagged lines in the figures provide $90\\%$ (asymptotic) \n confidence bands for the respective $e$'s, based on (\\ref{ConfidenceProbConvergence}).  \n At least for Regimen 4.3, a constant $e$ (exponential survival) can be rejected.\n \n The vertical bars at $x=0$, $x=100$, and $x=200$ in Figure 1, and at \n $0$, $50$, and $100$ in Figure 2, are $90\\%$ (asymptotic) pointwise confidence \n intervals on $e$ at the corresponding $x-$values (based on \n Proposition~\\ref{prop:PointwiseAsympNormality}). \n Notice that these intervals are not much narrower than the simultaneous bands early \n in the survival data, but are substantially narrower later on.  \n \n A similar graph for Regimen 5.5 (not presented) is somewhat similar to that \n in Figure 2, with the upward turn in $\\hat{e}_n$ occurring at 80 days \n instead of at $50$, and a possible downward turn at somewhere around 250 days\n (the final death occurring at 598 days).\n \n A similar graph was prepared for the failure data on 107 right rear tractor brakes presented \n by \n \\cite{MR0448771},\n page 462.  \n It suggests a quadratic decreasing $e$ for the first 1500 to 2000 hours (with $f(0) $ at or near $0$\n but $f'(0)$ definitely positive), with $\\overline{X} = 2024$, and with \n a possibly constant or slightly increasing $e$ from 1500 or so to 6000 hours.\n The $e$ for a gamma distribution with $\\lambda =2$ and $\\alpha = .001$ ($e(x) = \\alpha^{-1} (\\alpha x +2 )\/ (\\alpha x +1)$\n with $\\alpha = .001$) fits reasonably well -- i.e. is within the confidence bands, even for $25\\%$ confidence. \n Note that this is in excellent agreement with Figures 2.1(b) and 3.1(d) of \n \\cite{MR0448771}. \n\n Bryson and Siddiqui's (1969) \n data set was too small ($n=43$) for these asymptotic methods, except \n possibly early in the data set.)  \n \n \\begin{figure}\n\\centering\n\\includegraphics[width=1.00\\textwidth]{PLOTS\/GraphsMRL-1-ai.eps}\n\\caption{$90\\%$ confidence bands for mean residual life; Regimen 4.3}\n\\label{fig:Regimen4-3}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.00\\textwidth]{PLOTS\/GraphsMRL-2-ai.eps}\n\\caption{$90\\%$ confidence bands for mean residual life; Regimen 6.6}\n\\label{fig:Regimen6-6}\n\\end{figure}\n \n \\section{Further developments}\n \\label{sec:furtherDevelopments}\nThe original version of this paper, \\cite{Hall-Wellner-79}, ended with a one-sentence sketch of \ntwo remaining problems:  \n``Confidence bands on the difference between two mean residual life functions, and for the case of censored data,\n will be presented in subsequent papers.''   Although we never did address these questions ourselves, others\n took up these further problems.  \n \n Our aim in this final section is to briefly survey some of the developments since 1979 concerning \n mean residual life, including related studies of median residual life and other quantiles, as well as developments\n for  censored data, alternative inference strategies, semiparametric models involving mean or median residual life,\n and generalizations to higher dimensions.  \n For a review of further work up to 1988 see \\cite{Guess-Proschan:88}.\n\n \n\\subsection{Confidence bands and inference}\n\\cite{MR856407} gave a further detailed study of the asymptotic behavior \nof the mean residual life process as well as other related processes including the Lorenz curve.\n\\cite{MR931630} developed tests and confidence sets for comparing two mean residual life functions\nbased on independent samples from the respective populations.  These authors also gave a brief \ntreatment based on comparison of {\\sl median residual life}, to be discussed in Subsection~\\ref{subsec:MedianRL}\nbelow.\n \\cite{MR1416657} introduced weighted metrics into the study of the asymptotic behavior of the\n mean residual life process, thereby avoiding the intervals $[0,x_n]$ changing with $n$ involved in our\n Theorems 2.1 and 2.2, and thereby provided confidence \n bands and intervals for $e_F$ in the right tail.  \n \\cite{MR2256261} introduced empirical likelihood methods to the study of the mean residual life function.\n They obtained confidence intervals and confidence bands for compact sets $[0,\\tau]$ with \n $\\tau < \\tau_F \\equiv \\inf\\{ x : F(x) = 1\\}$.\n      \n\\subsection{Censored data}\n      \\label{subsec:CensoredMRL}\n      \n      \\cite{MR0458678} initiated the study of estimated mean residual life under random right censorship. \n      She used an estimator $\\hat{F}_n$ which is asymptotically equivalent to the Kaplan - Meier estimator \n      and considered, in particular,\n       the case when $X$ is bounded and stochastically smaller than the censoring variable $C$.\n      In this case she proved that $\\sqrt{n}(\\hat{e} (x) - e(x))$ converges weakly (as $n\\rightarrow \\infty$) \n      to a Gaussian process with mean zero. \n      \\cite{MR1416657} give a brief review of the challenges involved in this problem; see their page 1726.\n      \\cite{MR2344641} extended their earlier  study (\\cite{MR2256261}) of empirical likelihood methods to this case,\n      at least for the problem of obtaining pointwise confidence intervals.  The empirical likelihood methods\n      seem to have superior coverage probability properties in comparison to the Wald type intervals which follow from \n      our Proposition~\\ref{prop:PointwiseAsympNormality}.   \n        \\cite{MR1678973} introduced smooth estimates of mean residual life in the uncensored case.  In\n      \\cite{MR2459252} they introduce and study smooth estimators of $e_F$ based on corresponding smooth estimators \n         of $\\overline{F} = 1-F$ introduced by \\cite{MR1661441}.\n\n\\subsection{Median and quantile residual life functions} \n       \\label{subsec:MedianRL}\n  \n  Because mean residual life is frequently difficult, if not impossible, to estimate in the \n  presence of right-censoring, it is natural to consider surrogates for it which do not depend\n  on the entire right tail of $F$.  Natural replacements include median residual life and \n  corresponding {\\sl residual life quantiles}.  \n   The study of median residual life was apparently initiated in \\cite{MR628933}.  \n   Characterization issues and basic properties have been investigated by\n     \\cite{MR732677}, \n     \\cite{MR756012}, and \\cite{MR2131870}. \n     \\cite{MR764974}  proposed comparisons of two populations based on their corresponding \n      median (and other quantile) residual life functions.  \n      As noted by Joe and Proschan, \n      ``Some results differ notably from corresponding results for the mean residual life function''.\n    \n      \\cite{MR2422830} investigated estimation of median residual life with  right-censored data for \n      one-sample and two-sample problems.  They provided an interesting illustration of\n       their methods using a long-term follow-up \n      study (the National Surgical Adjuvant Breast and Bowel Project, NSABP) involving breast cancer patients. \n  \n     \n\\subsection{Semiparametric models for mean and median residual life}\n\n     \\cite{MR1064816}\n             investigated a characterization \n             related to  a {\\sl proportional mean residual life} model:  $e_G = \\psi e_F$ with $\\psi>0$.\n     \\cite{MR1278221}\n          studied several methods of estimation in a semiparametric \n          regression version of the proportional mean residual life model,  \n          $e(x|z) = \\exp (\\theta^T z) e_0 (x) $ where $e(x|z)$ denotes the conditional mean\n          residual life function given $Z = z$.   \n     \\cite{MR2135857}\n          provide a nice review of various models and study estimation \n          in the same semiparametric proportional mean residual life regression model\n          considered by   \\cite{MR1278221}, but in the presence of right censoring.  Their\n          proposed estimation method involves  inverse probability of censoring weighted (IPCW) \n          estimation methods  (\\cite{MR0053460}; \n          \\cite{Robins-Rotnitzky:92}). \n      \\cite{MR2158607} use counting process methods to develop alternative estimators \n           for the proportional mean residual life model in the presence of right censoring. \n          The methods of estimation considered by \\cite{MR1278221},  \\cite{MR2135857}, and  \\cite{MR2158607}\n          are apparently inefficient.\n     \\cite{MR2125050} \n          consider information calculations and likelihood based estimation in a two-sample version \n          of the proportional mean residual life model.  Their calculations suggest that certain weighted ratio-type\n          estimators may achieve asymptotic efficiency, but a definitive answer to the issue of efficient estimation\n          apparently remains unresolved. \n     \\cite{MR2278085} proposed an alternative additive semiparametric regression model involving mean \n            residual life.\n     \\cite{MR2753008}  considered a large family of semiparametric regression models which includes \n           both the additive model proposed by \\cite{MR2278085} and the proportional mean residual life \n           model considered by earlier authors, but advocated replacing mean residual life by median residual life.         \n     \n     \\cite{MR2155475} also developed a median residual life regression model with additive structure and took a \n           semiparametric Bayesian approach to inference. \n    \n\\subsection{Monotone and Ordered mean residual life functions}\n     \n     \\cite{MR1792793} consider estimation of $e_F$ subject to the shape restrictions that \n     $e_F$ is increasing or decreasing.  The main results concern ad-hoc estimators that are simple monotizations\n     of the basic nonparametric empirical estimators $\\hat{e}_n$ studied here.  These authors show that the \n     nonparametric maximum likelihood estimator does not exist in the increasing MRL case and that although \n     the nonparametric MLE exists in the decreasing MRL case, the estimator is difficult to compute. \n     \\cite{MR1238352} and \\cite{MR1927772} study estimation of two mean residual life functions $e_F$ and $e_G$ in \n      one- and two-sample settings \n      subject to the restriction $e_F (x) \\le e_G(x)$ for all $x$. \\cite{MR1927772} also develop large sample confidence \n      bands and intervals to accompany their estimators.\n      \n      \n\n\\subsection{Bivariate residual life} \n\n\\cite{MR663455} defined a multivariate mean residual life function and \nshowed that it uniquely determines the joint multivariate distribution, extending\nthe known univariate result of \\cite{MR0153061};\nsee \\cite{MR665274} for a review of univariate results of this type.\nSee \\cite{MR1412096, MR1649088} for further multivariate characterization results.\n\\cite{MR1941419} introduced a bivariate mean residual life function\nand propose natural estimators.  \n\\medskip\n\n\n\\noindent {\\bf Remarks:}\nThis revision was accomplished jointly by the authors in 2011 and 2012. \nThe first author passed away in October 2012.   Section 7 only covers further developments\nuntil 2012.  A MathSciNet search for ``mean residual life'' over the period 2011 - 2017 yielded \n148 hits on 10 July 2017.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe framework within which astronomers -- a term that will be used\nrather loosely in the rest of the document to indicate individuals\nperforming research in Astronomy, Astrophysics and Space Physics\n(AA\\&SP) -- have been functioning in Greece is not too different from\nother European countries.\n\nAs in most other countries in Europe, the educational and research\nactivity in Astronomy, Astrophysics and Space Physics in Greece has\nbeen fostered within public Universities and Research Institutes. Even\nthough this may change in the near future currently no private\nacademic or research institutions in AA\\&SP are operating in\nGreece. Thus the individuals who are employed full time to teach or do\nresearch in AA\\&SP are typically civil servants in permanent, tenure\ntrack, or fixed-term research associate positions. Currently the\nmajority of them (see Sect. 3, Fig. 1) are in Universities and only a\nsmall fraction is associated with Research Institutes or\nObservatories.\n\nUp until the early 1980s the structure of the University system in\nGreece followed the German style. It was based on ``Chairs'' of\nProfessors in specific research fields (i.e. Astronomy, Classical\nMechanics, etc.). The few astronomy Professors, typically as numerous\nas the corresponding number of university departments pursuing\nresearch in the various areas of AA\\&SP, held their position until the\nage of retirement. They made all major administrative decisions\nrelated to both teaching priorities and research directions in their\ninstitutions. Several junior staff members did support them in these\nactivities but those members had only marginal control and rather\nlimited independence to pursue their own research directions.\n\nThe University system presently in place was put forward in 1983,\n(with a few minor modifications over the past $\\sim$25 years) and has a\nstructure similar to the current academic system of the United\nStates. There are two ranks of tenure track positions: Lecturer and\nAssistant Professor, and two of tenured positions: Associate Professor\nand Full Professor. A minimum of three years is required on each rank\nbefore applying for promotion the next. Tenure is obtained upon\nsuccessful evaluation after spending three years at the level of\nAssistant Professor. A university faculty can, in principle pursue\nhis\/her own research direction, teach courses, and supervise graduate\nstudents.\n\nHowever, when this change in the academic system took place in the\nearly 1980's, there was no provision on the age distribution of the\nfaculty to be hired. As a result a large number of individuals who\nwere already affiliated with the universities at the time in junior\nlevel -- the so-called ``Assistant'' appointments, automatically\nobtained tenure at the rank of Lecturer upon the completion of their\nPhD. Others, who already had a PhD, were considered for tenure at\nhigher ranks. The evaluation for this process though was often not\nvery strict and with criteria based mostly on social reasons or giving\na disproportional emphasis to the teaching responsibilities of the\nfaculty, rather than mostly based on their research background and\npotential or relevance of the field to the future direction of modern\nastrophysics. In addition, since most of the individuals who obtained\nthese positions were past graduate students of the same universities,\nthere was a disproportional hiring from ``within''. This phenomenon,\nknown as ``academic inbreeding'', was more prevalent in the older\ninstitutions in Athens and Thessaloniki, which had the largest number\nof staff at the time. Even today there are institutions in Greece\nwhere well over 70\\% of their permanent staff members are past alumni\nwho did their dissertation in the same institute and did not spend\nmore than a couple of years away from their alma matter before\nobtaining permanent positions. It is beyond the scope of the present\ndocument to discuss this phenomenon and the serious negative\nconsequences it has on both the quality of research performed and on\nthe opening of new research horizons in academic institutions. We\nshould note though that this phenomenon is not unique to Greece, as it\nalso appears for example in the French academic system and in\nKorea\\footnote{See: Science 18 December 1998: Vol. 282. no. 5397} ,\nbut it is practically absent in the United States. Greek universities\nin the periphery of the country did not suffer much from this problem\nfor two reasons. Either they had not produced their own PhDs due to\ntheir youth as institutions, or their faculty made a conscious\ndecision to have a broader perspective in their hiring process. For\nexample, in the Department of Physics of the University of Crete,\nwhere the author is currently employed, only 1 out of the 33 faculty\nmembers obtained his PhD from this institution. All these political\ndecisions had implications that continue to affect the evolution of\nGreek astronomy, and academic system in general, well into the 21st\ncentury.\n\nResearch Institutes in Greece have a similar structure to the\nUniversities, with also four ranks, which are loosely indicated as\nResearcher-D, -C, -B and -A. Each researcher also has to remain in a\ngiven rank for a minimum of three years and tenure is obtained upon\npromotion from Researcher-C to Researcher-B.  Research Institutes can\nnot award academic degrees and as a result close collaboration with a\nUniversity is needed in order for a researcher to be able to\nco-supervise students.\n\nPublic funding for development of infrastructures, direct support of\nresearch in astronomy, or fellowships towards graduate studies in the\nfield, has been traditionally fairly limited. Such a low level support\nis not restricted to Greek astronomy but it is also the case in most\ndisciplines. In 2004 Greece spent only 0.58\\% of its Gross Domestic\nProduct (GDP) in R\\&D, which brings Greece as a nation in the last\nplace among the 15 EU countries in this category, a position it holds\nfor the past 5 years. At over the same period the European Union (EU)\naverage was 1.95\\%, more than 3 times higher\\footnote{Source EUROSTAT\nin http:\/\/europa.eu.int\/comm\/eurostat\/}. As a result the\npossibilities for Greek astronomers to join large international\ncollaborative projects, or just to obtain support to attend scientific\nmeetings outside Greece have been scarce. Even though the situation\nhas recently improved over the past decade, and the possibilities --\nmostly via the financial and organizational support of the European\nUnion -- are more numerous, the effects of this low level national\nfunding can be seen in most indices quantifying the overall astronomy\nscientific output from Greece. It is worth noting that Greece, which\njoined the European Union as the 10th member in 1981, is still not a\nmember state of the European Southern Observatory (ESO) and only\njoined the European Space Agency (ESA) in 2005.\n\n\\section{Governing bodies}\n\nThe policies that directly affect issues related to AA\\&SP in Greece\nare determined by the Ministry of Development, in particular the\nGeneral Secretariat for Research and Technology, and the Ministry of\nEducation. The first has administrative control over the research\ninstitutes and astronomy infrastructure and the latter controls the\nnational university system.\n\nGreek astronomers can express their opinion or shape policies on\nissues related AA\\&SP via the Greek National Committee for Astronomy\n(GNCA) or the Hellenic Astronomical Society (Hel.A.S.).\n\n\\subsection{The Greek National Committee for Astronomy (GNCA)}\n\nThe Greek National Committee for Astronomy (GNCA\\footnote{The web page\nof GNCA is: http:\/\/www.astro.noa.gr\/$\\sim$gnca}) was established, by Royal\nDecree, as the official advisory committee to the Greek Government for\nall matters relevant to Astronomical and Astrophysical research, in\n1957. It is the official body, responsible for the promotion and\ncoordination of Astronomy in Greece and for all matters related to\ninternational astronomical cooperation. The Minister of Development\nselects the members of GNCA and appoints them for a term of two\nyears. Its official seat is the National Observatory of Athens. Since\n1995, GNCA does not have its own budget, but obtains its funding from\nthe budget of the General Secretariat of Research and Technology\n(GSRT) of the Ministry of Development. The GNCA has the following\nprincipal objectives:\n\n\\begin{itemize}\n\n\\item To co-ordinate and promote the various astronomical activities\nin Greece, including research and education.\n\n\\item To act as the link between the Greek astronomical community and\nthe International Astronomical Union (IAU), officially representing\nGreece in the General Assembly of the IAU.\n\n\\item To facilitate the advancement of international collaboration\nbetween Greek and foreign astronomers and research groups.\n\n\\end{itemize}\n\nBesides the IAU, GNCA has taken responsibility for Greece's\nrepresentation to the Board of Directors of the journal \"Astronomy and\nAstrophysics\" (and its financial contributions), to the European Joint\nOrganization for Solar Observations (JOSO) and, recently, to the\nEuropean Union FP6, I3, Network OPTICON (see Sect. 8).\n\nThe board of GNCA consists of five ordinary and five substitute\nmembers. The current (2005-2007) ordinary members are\nProf. P. Laskarides (Chair - Univ. of Athens), Prof. T. Krimigis (Vice\nChair - Academy of Athens), Dr. I. Daglis (Nat. Obs. of Athens),\nProf. N. Kylafis (Univ. of Crete), and Prof. J.H. Seiradakis (Univ. of\nThessaloniki). The substitute members over the same period are:\nProf. S. Avgoloupis (Univ. of Thessaloniki), Dr. E. Dara (Academy of\nAthens), Prof. M. Kafatos (George Mason Univ., USA), Prof. A. Nindos\n(Univ. of Ioannina), and Prof. J. Papamastorakis (Univ. of Crete).\n\n\\begin{figure*} \n\\plotone{fig1.eps}\n\\figurenum{1} \n\n\\caption{Distribution of tenured and tenure track astronomers in the\nmajor research institutes in Greece. The total number of astronomers\nincluded in this study is 107.}\n\n\\end{figure*} \n\n\\subsection{The Hellenic Astronomical Society (Hel.A.S.)}\n\nThe Hellenic Astronomical Society (Hel.A.S.\\footnote{The web page of\nHel.A.S. is http:\/\/www.astro.auth.gr\/elaset}) exists for nearly 15\nyears and it is the major association of professional astronomers in\nGreece. Its overall structure and operation is similar to other\nnational societies such as the ``American Astronomical Society'' in\nthe US, or the ``Soci\\'{e}t\\'{e} Fran\\c{c}aise d' Astronomie et\nd'Astrophysique'' in France.\n\nHistorically, the first serious attempt to establish a Hellenic\nAstronomical Society was undertaken in 1982 during the XVIII General\nAssembly of the International Astronomical Union, which took place in\nPatras, Greece. There, during several meetings, a dozen astronomers\ngathered in order to put the foundations of the long sought\nSociety. The following years progress was slow even though material\nnecessary for setting up the framework for the Society was being\ncollected.  It was much later, in November 1991, when\nProf. P. Laskarides (Univ. of Athens) issued the first announcement of\nthe 1st Hellenic Astronomical Conference, that the idea of the\nestablishment of an Astronomical Society was formally put forward\nagain. With the help of several colleagues Prof. J.H. Seiradakis\n(Univ. of Thessaloniki) drafted the first Constitution for the\nSociety. The final version was presented to the participants of the\n1st Hellenic Astronomical Conference, which was held in Athens in\nSeptember 1992.\n\nDuring the Athens Conference, several astronomers became founding\nmembers of the Hellenic Astronomical Society. A few more founding\nmembers signed the Constitution during the next weeks bringing the\ntotal number of founding members to sixty six (66). Following the\nappropriate legal procedures, the Hellenic Astronomical Society\n(Hel.A.S.) was recognized by the Court of Justice in Athens on May 25\n1993. The appointed Council of Hel.A.S. became aware of the verdict of\nthe Court of Justice in June 1993. The President of the Council,\nProf. B. Barbanis (Univ. of Thessaloniki), assisted by the members\ninitiated the procedure for the first elections of Hel.A.S. In the\nelections, which took place on June 2nd 1994, participated 83\\% of the\nfounding members.\n\nAccording to its Constitution the Governing Council of\nHel.A.S. consists of a President, six (6) members and three (3)\nauditors. The Council is elected for a two-year term and an individual\ncannot serve on it for more than two consecutive terms. The candidates\nfor the Council must be members of Hel.A.S. who live and work\npermanently in Greece during the term of their candidacy and at least\n42\\% of them should be affiliated with institutions outside the Athens\nmetro area. The current Council, whose mandate ends in June 2006,\nconsists of Prof. P. Laskarides (Univ. of Athens) as the president and\nProf. D. Hatzidimitriou (Univ. of Crete), Prof. K. Tsinganos (Univ. of\nAthens), Prof. V. Geroyannis (Univ. of Patras), Prof. K. Kokkotas\n(Univ. of Thessaloniki) and Prof. X. Moussas (Univ. of Athens) as\nmembers. The auditors for the same period are Prof. E. Danezis\n(Univ. of Athens), Prof. E. Mavromichalaki (Univ. of Athens) and\nDr. E. Xilouris (National Obs. of Athens)\n\nThe Hellenic Astronomical Society has been very active and currently\nhas 272 members, 27\\% of which live and work outside Greece. It has\nbeen recognized as an Affiliated Member of the European Astronomical\nSociety (EAS) and has established links with other international\nastronomical societies. It has been organizing a major science meeting\nevery two years and in the summer of 1997 organized the Joint European\nand National Astronomical Meeting (JENAM-97).\n\n\\begin{figure*} \n\\plotone{fig2.eps}\n\\figurenum{2} \n\\caption{A histogram of the age distribution of astronomers in Greece\nin 2006. The vertical line indicates the 67th year of age which is the\ncurrent compulsory retirement age for civil servants.}\n\n\\end{figure*} \n\n\\section{Academic Institutions}\n\n\\subsection{Human Resources in Astronomy, Astrophysics \\& Space Physics}\n\nAs one would expect, since more than half of the population in Greece\nis concentrated in the Athens and Thessaloniki metro areas, most of\nthe astronomers in Greece are also associated with institutes located\nin these two cities. This is depicted in Figure 1 where the fraction\nof tenured and tenure track astronomy faculty in the major AA\\&SP\ninstitutions in Greece is presented. \n\nAn additional issue, which affects the current state and has direct\nimplications to the future of Greek astronomy, is related to the age\ndistribution of Greek professional astronomers. In Figure 2, a\nhistogram of 135 astronomers working in Greece is presented, using the\ndatabase of the members of the Hellenic Astronomical Society, as well\nas ancillary information collected by the author. The study was\nlimited to individuals over the age of 30, since this is typically the\nage when one is competitive for tenure track or long term research\nassociate positions. Some individuals over the retirement age of 67,\nwho are on an emeritus-type position and\/or still active, were\nincluded in the analysis. The error on a single 5-year bin is of the\norder of 5\\% but it is very likely that the values of bins at ages\ngreater than 55 are somewhat underestimated. This is due to the fact\nthat there are a number of individuals who formally have a tenured\nastronomy positions but as they are no longer active they were not\nincluded in the database of Hel.A.S, on which analysis was based.\n\n\nInspection of Figure 2 clearly reveals that almost 30\\% of Greek\nastronomers are near or over the age of 60. This was a direct\nconsequence of the legislative changes that took place in Greece in\nthe early 1980s mentioned in Sect. 1. Furthermore, statistics over the\nlast 10 years indicate that on average there were less than 3 new\ntenure track astronomy position openings per year in the country,\nincluding both universities and research institutions. The fraction of\nastronomers near the age of retirement is even larger if we were to\nconsider only the two older universities of Athens and\nThessaloniki. This implies that within the next 5 to 10 years a large\nnumber of their current faculty members will retire and they will have\nto be replaced in a very short time scale. This will be an interesting\nchallenge for Greek astronomy. Will it be possible for these\ninstitutions to find enough, well qualified, candidates from the\navailable pool of post-docs and research associates for their needs?\nWill they be forced to lower their hiring standards in order to hire\nfaculty for their teaching needs, or they will be able to hire with a\nlower pace, being selective and identifying the key scientific\nresearch areas they should be investing in? In 2016 we will know the\nanswer to these questions!\n\nAnother topic worth touching upon is gender diversity in Greek\nastronomy. At the time of writing this report 13\\% of the permanent or\ntenure track astronomy positions in Greece were held by women. This\npercentage is less than in France\\footnote{The percentages for the\nother countries mentioned are based on the 2003 report by Dr. Florence\nDurret (Institute d'Astrophysique de Paris, France) available at:\nhttp:\/\/www2.iap.fr\/sf2a\/courrier.html}, which leads the way with\n$\\sim$26\\%, or in Italy, Russia and Spain, all above 15\\%, but higher\nthan the fraction of female astronomers in the United States which is\n$\\sim$10\\%. We should note though, that only recently one female\nastronomer in Greece reached for the first time the highest possible\nacademic rank (Full Professor or Researcher A), a statistic that will\nhopefully improve very soon.\n\n\n\\subsection{Research in Astronomy, Astrophysics \\& Space Physics}\n\nThe latest organized effort to map the research activity in AA\\&SP in\nthe various institutes in Greece took place in 1998. Dr. E. Kontizas,\nas the president of GNCA at the time, appointed an international\nsix-member committee, chaired by Prof. Y. Terzian (Cornell Univ.,\nUSA), to report on the status of astronomy in Greece and propose\nrecommendations for the future. The report\\footnote{The complete\n``Terzian Report'' is available at:\nhttp:\/\/www.astro.noa.gr\/gnca\/NEWS\/ca-report2000.htm} was presented\nduring the workshop ``Astronomy 2000+: Greek Prospects for the 21st\nCentury'' which took place at the National Observatory of Athens on\nNovember 1998. The description presented in the following paragraphs\ndraws from material included in this report with some modifications\nmostly related to changes in the human resources of the institutes\ninvolved.\n\nThere are eight institutions in Greece with Departments or Sections\ndevoted to teaching and research in Astronomy and Astrophysics. Three\nare located in Athens: the largest is the Section within the\nDepartment of Physics of the National Kapodistrian University,\nfollowed by the National Observatory of Athens, and an astronomy\nSection of the national Academy of Athens. In Thessaloniki there is a\nvery small group within the Faculty of Engineering (Polytechnic\nSchool) and a considerably larger one within the Department of Physics\nof the Aristotle University. In Crete there is a Section of\nAstrophysics and Space Physics in the Department of Physics in\nHeraklion, while the Universities of Patras and Ioannina each have\nsmall Astronomy groups within either their Physics or Engineering\nDepartments. Some research activity in very specific areas\n(i.e. cosmology or general relativity) also exists in a few\nDepartments of Mathematics but the numbers of permanent staff are very\nsmall and there is no critical mass to be considered groups.\n\nThe principal institutions devoted to research and technical\ndevelopment in space sciences is the Department of Electrical and\nComputer Engineering at the ``Democritus\" University of Thrace (in\nparticular the Laboratory of Space Electrodynamics in the Section of\nTelecommunications and Space Science) and the Institute for Space\nApplications and Remote Sensing of the National Observatory of\nAthens. Significant research in ground based ionospheric and\natmospheric work is also a component of the overall Astrophysics and\nSpace Science Section at the University of Crete. Activity relating to\nspace science also exists in the Section of Astronomy, Astrophysics,\nand Mechanics of the University of Athens, and at the Research Center\nfor Astronomy in the Academy of Athens.\n\nIn the following subsections we present a brief description of the\nvarious institutes in Greece hosting research groups with active\nresearch in AA\\&SP. More detailed annual activity reports from most\ninstitutions and groups are being collected by the Greek National\nCommittee for Astronomy and they are made available from its web site\nmentioned in section 2.1.\n\n\\subsubsection{University of Athens}\n\nThe ``National \\& Kapodistrian'' University of Athens was founded in\n1837, soon after the independence of Greece. It was the first\nUniversity in Greece as well as in the Balkan Peninsula and the whole\neastern Mediterranean region. The Department of Physics was created in\n1904 and its current Section of Astronomy, Astrophysics, and Mechanics\nwas formed in the mid 1980s by merging the previously independent\nChairs indicated in its name. The Section is the largest in Greece and\nconsists of 24 tenured or tenure tract faculty. Most of them have\nresearch interests in the area of Astronomy and Astrophysics, while\nMechanics is a rather small constituent. The Department of Physics\nstarted a graduate school in 1994, within which the Section has its\nown Masters and PhD programs with 12 graduate courses. Since January\n2000 the Section also operates a 40cm Cassegrain telescope within a 5m\nrotating dome located on the top of the Physics building. The\ntelescope was constructed by DFM engineering (USA) has an f\/3 focal\nratio and it is mainly used for educational activities.\n\nIn addition to the pursuit of astronomy, it should also be mentioned\nthat the faculty of the Physics Department have been involved over\nmany years in building a deep sea High Energy Neutrino telescope,\nknown as NESTOR. Recently this effort has been put under the auspices\nof the National Observatory of Athens as an independent institute for\nAstroparticle Physics (see Sect. 3.2.2).\n\n\\subsubsection{National Observatory of Athens}\n\nThe National Observatory of Athens (NOA\\footnote{More information on\nthe National Observatory of Athens the can be found at:\nhttp:\/\/www.noa.gr}) was founded in 1842 and is the oldest research\ninstitute in Greece. It currently consists of five institutes three of\nwhich, the Institute of Astronomy and Astrophysics, the Institute for\nSpace Applications and Remote Sensing and the Institute of\nAstroparticle Physics -- Nestor, conduct research in AA\\&SP. The\ncurrent director of NOA is Prof. C. Zerefos.\n\nThe Institute of Astronomy and Astrophysics has 11 permanent staff\nscientists as well as research associates and support personnel. Their\nresearch interests include a variety of topics in extragalactic\nastronomy, observational cosmology, interstellar matter, X-ray\nastronomy, and binary stars. The Institute supports the Astronomical\nObservatory in Kryoneri as well as the new Chelmos Observatory where\nthe new 2.3m ``Aristarchos'' telescope, the largest in Greece, is\nlocated (see Sect. 4.1, 4.3). The institute is also very active in\npublic outreach activities, among which are the operation of a Visitor\nCenter and an annual summer school, which introduces basic concepts of\nmodern astrophysics to high-school students since 1996. The current\ndirector of the Institute is Prof. C. Goudis.\n\nThe Institute for Space Applications and Remote Sensing has 11 tenure\nor tenure track research staff. The activities of the Institute\nencompass a wide area in Space Research and Applications. Its main\nobjective is to carry out R\\&D projects in these fields, which include\nRemote Sensing, Telecommunications, Space and Ionospheric\nPhysics. Additional activities include the systematic collection and\nprocessing of data derived from observations made either from the\nearth or space as well as the performance of autonomous studies in\nother specific subjects of space research and applications. The\nInstitute is equipped with satellite and ionospheric ground stations,\nvarious RF and electronic test and measurement equipment, as well as\nan advanced computing center connected to international networks. The\ncurrent director of the Institute is Dr. I. Daglis.\n\nThe Institute of Astroparticle Physics -- NESTOR (Neutrino Extended\nSubmarine Telescope with Oceanographic Research) became the fifth\ninstitute of the national Observatory of Athens in 2003. The institute\nis leading the development of a deep-sea high energy neutrino\ntelescope approximately 14km off the shore from the town of Pylos in\nPeloponnese, at water depth of 4000m. NESTOR will detect the Cherenkov\nradiation produced by muons traversing the water when their parent\nneutrinos emitted from astrophysical objects, such as X-ray binaries,\nblack holes, or Active Galactic Nuclei, interact with water. The\ncurrent director of the Institute is Prof. L. Resvanis.\n\n\n\\subsubsection{Academy of Athens}\n\nThe Academy of Athens was formally founded in 1926. It currently has\namong its members two Academicians (Prof. G. Contopoulos and\nProf. T. Krimigis) with a background and research interests in\nastronomy. One of the centers of the Academy, the Research Center for\nAstronomy and Applied Mathematics, consists of 11 permanent research\nstaff, and conducts research in solar and space physics, cosmology,\nparticle physics and dynamical astronomy.\n\n\\subsubsection{University of Thessaloniki}\n\nThe ``Aristotle'' University of Thessaloniki was the second university\nin Greece and it was founded in 1925. There are two units in the\nUniversity with activity in Astronomy. The smallest, in the\nPolytechnic School, consists of two faculty members and their research\nis concentrated mainly on flare stars. The largest is the Section of\nAstrophysics, Astronomy and Mechanics (AAM\\footnote{The online\ndescription of the AAM Section in Thessaloniki can be found at:\nhttp:\/\/www.astro.auth.gr}) of the Department of Physics, with 17\nfaculty members, several research associates, graduate students, and\nsupport personnel. The Section was formed in the mid-80s when the\nadministrative structure of the Laboratories of Astronomy (founded in\n1943) and Mechanics changed (see Sect. 1). The staff is active in many\nareas of theoretical and observational astrophysics, including an\nactive theoretical group on gravitation and general relativity, as\nwell as in education and public outreach. In addition to the\nStephanion Observatory (see Sect. 4.4) the Section operates a 20cm\nrefracting telescope (made by Secretan, Paris) in a rotating\n6m-diameter dome, which is located within the University campus, and\nit is used for educational purposes.\n\n\n\n\\subsubsection{University of Crete}\n\nThe University of Crete was founded in 1973 but accepted its first\nstudents in 1978. Its Department of Physics was founded in 1978 and is\nthe youngest of similar Departments in Greece. The Section of\nAstrophysics and Space Physics\\footnote{The web page of the Astronomy\nSection in Crete can be found at: http:\/\/www.physics.uoc.gr\/en\/} has 7\nfaculty members as well as several research staff and graduate\nstudents. Two (2) more tenured track astronomers, from the Foundation\nfor Research and Technology-Hellas and the Technical Education\nInstitute of Heraklion, are actively collaborating with the members of\nthe Section. Research at the Univ. of Crete covers a broad range in\ntheoretical and observational problems related to both galactic and\nextragalactic astrophysics. Significant efforts are being devoted to\nthe operations of an Ionospheric Physics laboratory. Observations for\nseveral astronomical projects are also taken at the Skinakas\nObservatory (see Sect. 4.2) and others are performed using\ninternational ground and space born telescopes. The Department has a\ngraduate program through which students can pursue their graduate\nstudies in astrophysics.\n\n\\subsubsection{University of Thrace}\n\nThe Laboratory of Space Electrodynamics (LSE) at Department of\nElectrical and Computer Engineering of the ``Democritus\" University of\nThrace is the largest space physics group in Greece with extensive\nexperience in hardware development. It consists of 6 faculty, several\nresearch associates, support personnel, and many graduate and\nundergraduate students. The scientists are co investigators or\nassociated scientists on several international spacecraft missions\n(e.g. Ulysses, Geotail, Cluster II, and others), successfully funded\nthrough European programs and bilateral collaborations with other\ncountries, including the U.S. The LSE has designed, developed and\nsuccessfully flown particle experiments on a number of Russian\nspacecraft, as well as component systems to instruments involving data\nprocessing units and ASICs (Application Specific Integrated\nCircuits). Such high technology hardware capability in space\ninstrumentation is rather unique within Greece. The LSE group has\nexpanded their activities to antennae and propagation, satellite\ncommunications, and other related areas.\n\n\\subsubsection{University of Patras}\n\nThe University of Patras has a Laboratory of Astronomy and a Section\nof Astronomy in the Division of Theoretical and Mathematical Physics\nin the Department of Physics. A total of nine tenured and tenure track\nfaculty teach courses and conduct research in a few astronomy areas\nand there is an active theoretical group on celestial mechanics.\n\n\\subsubsection{University of Ioannina}\n\nThis is the smallest group of Astronomy in a Department of Physics in\nGreece. It has three faculty members in the Section of Astrogeophysics,\nwithin the Department of Physics. The staff performs research mostly\nin solar physics and in multi-wavelength observations of flare stars.\n\n \n\\begin{figure*} \n\\plotone{fig3.eps}\n\\figurenum{3} \n\n\\caption{An optical satellite image of Greece, obtained in 2004, in\nwhich the locations of the major observatories hosting functioning\noptical telescopes with a primary mirror diameter larger than 0.5m are\nindicated (Image courtesy of MODIS Rapid Response Project at\nNASA\/GSFC)}\n\n\\end{figure*} \n \n\\section{National Facilities}\n\nThe limited funding of the Greek government towards basic and applied\nresearch has had, as a result, the small investment in major\ninfrastructures for astronomical facilities in Greece. This affected\nthe oldest observatories in Greece, such as Penteli, Kryoneri and\nStephanion Observatory, which have difficulties keeping up-to-date\nwith the modern developments in telescope design, aperture size of the\ntelescope primary mirrors, as well as the instrumentation\navailable. More recent facilities, such as Skinakas Observatory, which\ncurrently hosts the largest operational telescope in Greece which is\n1.29m in diameter, are more modern and do provide high quality\ninstruments to the observers. However, they also suffer from the\nlimited national financial support and they cannot function as\nfacilities that can provide access to all Greek astronomers who may\nwish to use them. A major effort in improving the current situation\nhas been the ongoing construction of the 2.3m ``Aristarchos\" telescope\nby the National Observatory of Athens. The telescope had its first\nlight in the end of 2005 and when it becomes fully operational, before\nthe end of 2007, will be the largest in Greece.\n\n\n\n\\subsection{Chelmos Observatory}\n\nThe site selected for the new 2.3m telescope is located in Northern\nPeloponnese, on top of Chelmos mountain, near the small town of\nKalavrita approximately 150km from Athens, with longitude:\n22$^{o}$13', latitude: 37$^{o}$58' N and an elevation of 2340m. The\ntotal cost for the project is expected to be about 5 million Euros and\nit was financed mainly by the European Union, as well as by the\nGeneral Secretariat for Research and Technology of the Ministry of\nDevelopment. The telescope named ``Aristarchos\" is a\nRitchey-Chr\u017dtien with a focal ratio f\/8 and a 10\u00d5 field of view as\nwell an RC-corrected field of view of 1degree. The telescope and dome\nare constructed by Carl Zeiss (Germany).\n \n\\begin{figure*}[h] \n\\plotone{fig4.eps}\n\\figurenum{4} \n\\caption{Left: A picture of the dome of the 2.3m Aristarchos\ntelescope at Chelmos. Right: A photograph of the telescope inside the\ndome (Images courtesy of National Observatory of Athens).}\n\\end{figure*} \n\n\nThe image scale on the focal plane is 1''=85$\\mu$m and a 1024x1024 CCD\ncamera was the first light instrument. A medium resolution (2.5\\AA -\n6\\AA) spectrometer covering the range between 4270\\AA and 7730\\AA as\nwell as a 4096x4096 optical CCD will be the first generation\ninstruments of the telescope. These will be followed by an echelle\nspectrometer covering the range between 3900\\AA and 7500\\AA with a\nresolution of 6 km s$^{-1}$, as well as other instruments. The\nsupervision of the telescope construction as well as its operation are\nmanaged by the Institute of Astronomy and Astrophysics of the National\nObservatory of Athens\\footnote{Details on the Chelmos Observatory is\navailable at: http:\/\/www.astro.noa.gr\/ASC\\_2.3m\/ngt\\_main.htm}.\n\n\\subsection{Skinakas Observatory}\n\nThe Skinakas Observatory\\footnote{More information on Skinakas\nObservatory can be obtained from: http:\/\/skinakas.physics.uoc.gr}\noperates as part of a scientific research collaboration between the\nUniversity of Crete, the Foundation for Research and Technology-Hellas\n(FORTH) and the Max-Planck-Institut f\\\"{u}r Extraterrestrische (MPE)\nPhysik of Germany.\n\nThe site of the Observatory (Longitude: 24$^{o}$53'57''E, Latitude:\n35$^{o}$12'43''N), chosen on scientific and functional grounds, is the\nSkinakas summit of Mount Ida (also known as Psiloritis), at an\naltitude of 1750m and a distance of 60km from Heraklion. The\nObservatory has two telescopes: a Modified Ritchey-Chr\u017dtien\ntelescope with a 1.29m aperture (focal ratio f\/7.6), which became\noperational in 1995, and a 30cm telescope (focal ratio f\/3.2). The\nbuilding for the small telescope was constructed in 1986, and\nobservations started in 1987. The site is one of the best in Greece\nwith weather conditions often permitting photometric sub-arcsecond\nseeing. It includes a modern guest house powered with solar arrays and\nan Internet connection.\n \n \n\\begin{figure*} \n\\plotone{fig5.eps}\n\\figurenum{5} \n\n\\caption{An areal view of the Skinakas Observatory summit with the\nlarger dome of the 1.29m telescope seen on the left, along with the\nsmaller domes the guest house facilities. The 1.29m telescope inside\nits dome is seen in the right (Images courtesy of the Physics Dept.,\nUniv. of Crete).}\n\n\\end{figure*} \n\n\nThe optical system of the 1.29m telescope were manufactured by Carl\nZeiss (Germany). The mechanical parts were built by DFM Engineering\n(USA). The instrumentation available includes a focal reducer, a\nnumber of optical CCD cameras, and a low resolution long slit\nspectrograph. A 1024x1024 near-IR camera and an echelle spectrograph\nwill soon be available on site, along with OPTIMA, a fast\nphoto-polarimeter with microsecond time resolution intended for\nobservations of compact objects. Various research projects, both\ngalactic and extragalactic, mostly led by members of the Department of\nPhysics of the University of Crete or MPE astronomers have been\nongoing since the facilities became operational. The close\ncollaboration with the MPE group and the FORTH engineering support has\nhelped the astrophysics group in Crete in keeping the telescope and\nthe instruments in the forefront of technology, always taking into\naccount the limitations in the budget.\n\n\\subsection{Kryoneri Observatory}\n\nThe Astronomical Station of Kryoneri\\footnote{Additional information\non Kryoneri Observatory are at:\nhttp:\/\/www.astro.noa.gr\/ASK\\_1.2m\/ask\\_main.htm} was established in\n1972. It is located in the Northern Peloponnese, on top of mountain\nKilini at an elevation of 930m, near the small village Kryoneri 110km\nfrom Athens (longitude: 22$^{o}$37'E, latitude: 37$^{o}$58'N). The\n1.2m Cassegrain Coude telescope of the Astronomical Station Kryoneri,\nmade by Grubb Parsons Co., Newcastle, was installed in 1975. Its\noptical system consists of a paraboloidal primary mirror of 1.23m in\ndiameter and f\/3 focal ratio, and a hyperboloidal secondary mirror (31\ncm). Both mirrors are made of Zerodur. The telescope focal ratio is\nf\/13, its field of view is about 40' and the image scale is\n12.5\"\/mm. As with Chelmos Observatory, Kryoneri is operated by the\nInstitute of Astronomy and Astrophysics of the National Observatory of\nAthens.\n\n\n\\subsection{Stephanion Observatory}\n\nThe first observations at the Stephanion Observatory, in eastern\nPeloponnese, were undertaken in March 1967 with a guest 38cm reflector\nand a UBV photometer that belonged to the Bergedorf Observatory of the\nUniversity of Hamburg, Germany. Since then a large number of\ninstruments have been hosted at the 800-m altitude observatory, which\nis located at longitude: 22$^{o}$49'45''E, latitude: 37$^{o}$45'9''N,\nincluding French telescopes, for monitoring satellites, and a 40cm\nreflector from the Utrecht Observatory, Netherlands. In June 1971, the\n30-inch (76cm) Cassegrain reflector of the University of Thessaloniki\nwas installed at the Observatory. Until 1975, when the 1.23m\nCassegrain Coude reflector at Kryoneri became operational, this was\nthe largest telescope in Greece.\n\nThe 30-inch reflector is mounted asymmetrically and its focal ratio is\nf\/3 for the primary hyperbolic mirror and f\/13.5 for the Cassegrain\nfocus. It was constructed by Astro Mechanics, USA, a firm that has\nlong ago discontinued making astronomical instruments. The majority of\nobservations are carried out with a Johnson dual channel photoelectric\nphotometer with an offset guider unit mounted in the Cassegrain\nfocus. It includes an RCA 1P21 and an RCA 7102 photo-multipliers, both\nof which are refrigerated by dry ice. Key photometric observations of\nvariable stars (flare stars, Cepheid variables, RS CVns, etc) have\nbeen undertaken in co-operation with large ground or space\ninstruments. The international demand for co-operative and\nsimultaneous observations at the Stephanion Observatory stems from the\nstrict differential method used for obtaining absolute, above\natmosphere, stellar magnitudes in the international UBV system. The\nerror in the calibrated magnitudes obtained is usually better than\n0.02 magnitudes.\n\n\\subsection{Penteli Observatory}\n\nThe Astronomical Station on Penteli Mountain, just 15km from downtown\nAthens, was established in 1937 when it became apparent that it was\nnecessary to move the telescopes from the grounds of the old National\nObservatory in the center of Athens. In 1955 the National Observatory\nof Athens accepted the donation offered by the University of\nCambridge, for a 62.5cm telescope designed by R. S. Newall, and\nconstructed by the firm Thomas Cooke \\& Sons in 1868. Its big tube\n(about 9m in legth), the German-type equatorial mount and its weight\nof about nine tons, required careful dismounting, transportation and\ninstallation in a new dome that was built in Penteli. This telescope,\nno longer used for research, is still available on site today.\n\n\\subsection{Eudoxos Educational Observatory}\n\nThe ``Eudoxos\" observatory is a web-accessible complex of optical and\nradio telescopes, founded in 1999, whose facilities are located 16km\nfrom Argostoli, in the Ionian island of Kefallinia at a plateau 600m\nbelow the peak of mount Ainos (1628m). It operates a 0.6m Cassegrain\nrobotic telescope named after Dr. Andreas Michalitsianos, a Greek\nastrophysicist who was born in the island and had a successful career\nin NASA (USA) until his early passing away. The observatory was formed\nby a consortium of Greek institutes involving the National Research\nCenter of Physical Sciences ``Democritus\", the Hellenic Naval Academy,\nthe Ministry of Education and the Prefecture of Kefallinia and\nIthaki. It is being operated by the same consortium with the addition\nof the University of Athens and has already received substantial\nsupport from the Hellenic Air Force and the Ministry of Education. The\n0.6m telescope consists of a fully autonomous computerized optical\ntube assembly, automated enclosure, GPS smart antenna for time\nsynchronization, a full set of meteorological sensors, a large format\nimaging CCD camera and UBVRI wheeled photometric filters, as well as a\nfleet of peripheral instruments currently under construction or\ntesting. All equipment is completely controlled by two supervisory\ncomputers, which communicate via the Internet to the participating\nsecondary schools and institutions.\n\n\\section{High School and Undergraduate Studies in Astronomy}\n\nThe Greek secondary education system does provide substantial training\nin physics and mathematics to the students who wish to follow\nuniversity studies in sciences. Even though there is no compulsory\nastronomy course in high school (only an elective introductory\nastronomy course is available for high-school juniors) basic astronomy\nideas related to the solar system, stars, galaxies, and the formation\nof the universe are presented in other courses. Since 1996 the\n``Society for Space and Astronomy\" of Volos (see Sect. 7) has been\norganizing a very successful national astronomy competition in which\nstudents from all over Greece attending the last three years of high\nschool (``Lyceum'' in greek) can participate.  The top students are\nawarded various prizes while the first two are invited to attend an\nall-expenses-paid summer space-camp in the United States organized by\nNASA. This effort, mainly supported by private funds and volunteer\nwork, has helped substantially in popularizing astronomy among high\nschool students.\n\nAt the university level there is no Bachelors (BSc) degree in\nAstronomy or Space Science in Greece. Most individuals, who are now\nprofessionals in the field of AA\\&SP and did their undergraduate\nstudies in Greece, obtained their BSc degree in Physics following a\nfour-year program. Some, mostly theorists, have obtained their\nundergraduate degrees in Mathematics or Engineering. Even in the\nvarious Departments of Physics in Greece though, the curriculum of the\nastronomy courses varies depending on the number and research\nbackground of the astronomy faculty. Most Physics majors in Greece\nhave to follow at least one compulsory junior course in Astrophysics\nwhile some complementary topics on dynamical astronomy are typically\ncovered on compulsory sophomore and junior level courses in classical\nmechanics and modern physics. Most Departments of Physics offer the\npossibility of an astronomy specialization (or minor), even though\nthis is not formally awarded as a degree. Within this framework,\nstudents, who are interested in astronomy, have the opportunity to\nattend typically five to ten junior and senior level courses in\nastrophysics, space physics and celestial mechanics, thus obtaining a\nfairly solid background if they wish to continue for graduate studies.\n\nThe level of this University astronomy training is usually very good\nin the theoretical and encyclopedic part and the top students are\ncompetitive with international standards. What the students lack\nsometimes is the hands-on practical knowledge, which can only be\nobtained with access to engineering facilities or observatories. The\norganization of summer schools, such as the one taking place at the\nUniversity of Crete for the past 17 years, addressed to undergraduates\nat junior and senior level, can often fill this gap. There are also\nrecent efforts at various institutions, such as the Univ. of Athens,\nto enhance the observational astrophysics courses with a more\norganized usage of small telescopes and new instruments.\n\n\\section{Graduate Studies in Astronomy, Astrophysics \\& Space Physics}\n\nGraduate studies in Astronomy Astrophysics \\& Space Physics leading to\na Masters or a PhD degree can now be completed in most Greek\nUniversities. The first well-organized physics graduate program in\nGreece with coursework, qualifying exams, and at least partial\nfinancial support for students was developed in the University of\nCrete in 1984. This was soon to be followed by the University of\nAthens and other institutions.\n\nHowever, the system suffers from difficulties, which again stem from\nthe limited national funding. Less than a handful of state fellowships\nfor graduate studies in AA\\&SP are available each year. Providing\nfinancial support for graduate studies via European Union or national\nresearch proposals in astronomy is very challenging both due to\nlimited funds available in this field as well as due to various\nbureaucratic and organizational difficulties. As a result graduate\nstudents in Greece have to either work, or rely on other means to\nsupport themselves during their studies. This sometimes affects their\nability to invest the amount of time necessary for research in order\nto complete a very high quality PhD project.\n\nThese reasons have been pushing many of the Greek students to go\nabroad for their graduate studies. The most popular destinations are\nthe United States, the United Kingdom, Germany, France and The\nNetherlands. The improved facilities and competitive research\nenvironment in those countries do provide high quality training to the\nstudents but often decrease the likelihood of their return to work in\nGreece. Recently though, the opportunities made available by the\nEuropean Union, mostly via the Human Capital as well as Training and\nMobility programs, have ameliorated the situation. These new\npossibilities have provided the means to establish close links between\nGreek and other European institutions, which improves substantially\nthe training of local students thus bringing direct scientific return\nto the home institution.\n\n\\section{Amateur Astronomy in Greece}\n\nAmateur astronomy has been flourishing in Greece over the past\ndecade. The availability of high quality and low cost small telescopes\nand the use of Internet to organize and advertise the activities of\ngroups has greatly helped the development in the field. Many amateur\norganizations exist all over Greece. In particular one should mention\nthe ``Hellenic Astronomical Union\" which is the society of amateur\nastronomers in Athens, the ``Group of friends of Astronomy\" in\nThessaloniki, the ``Corfu Astronomical Society\", and the very active\n``Society for Space and Astronomy\" in the city of Volos.  Since 1999\nthe Greek amateur astronomers have been organizing a national meeting\nevery two years where they present their results and discuss issues of\ncommon interest.\n\n\\section{Greece and International Astronomy Organizations}\n\nGreece joined the International Astronomical Union as a funding member\nin 1920.  It also contributes to the support of the international\nrefereed journal of Astronomy \\& Astrophysics, which allows Greek\nastronomers to publish their scientific results without page charges.\nSince 2004 Greece also participates in OPTICON, a 19.2 million Euro\n5-year European FP6 Infrastructure Network, which provides access to a\nnumber of medium size telescope facilities around the world.\n\nIn early 2005 Greece joined the European Space Agency (ESA)\ncontributing to the annual budget of ESA with $\\sim$9 million\nEuros. This opens new opportunities for Astrophysics and Space Physics\nboth in terms of technology development as well as in science. Over\nthe past year significant organizational efforts have been taking\nplace in order to stimulate the Greek AA\\&SP community so that it will\nbe able to capitalize on this investment and join the rest of the\nwestern European countries in the forefront of space technology.\n\n\\section{Remarks}\n\nI believe that it is appropriate to end this article on the status of\nGreek astronomy with an optimistic note on the many improvements we\nhave all experienced over the past decade. As it can be seen from the\nmaterial presented in the previous sections, the environment, both\nresearch and academic, for the current and next generation of Greek\nastronomers is considerably better than what our predecessors had\nexperienced and worked through.\n\nI must also touch upon, a subject, which was mentioned earlier but\nonly briefly. Unfortunately Greece is still not a member of the\nEuropean Southern Observatory, the major astronomical organization in\nEurope. As a result it has no access to the current European\ninfrastructures of the Very Large Telescope (VLT), nor to the\ndevelopment of the Atacama Large Millimeter Array (ALMA) nor to the\ndesign of the future ESO projects, such as the 100m OverWhelmingly\nLarge telescope (OWL). I should stress that the report\\footnote{The\ncomplete ``Terzian Report'' is available at\nhttp:\/\/www.astro.noa.gr\/gnca\/NEWS\/ca-report2000.htm} of the\ninternational expert committee chaired by Prof. Terzian (Cornell\nUniv.) on the status of Greek Astronomy presented in 1998 during the\nworkshop ``Astronomy 2000+: Greek Prospects for the 21st Century''\nnoted that joining ESO should be the first astronomy priority for the\nnation. Current rough estimates indicate that the cost for Greece to\njoin ESO would be a one-time $\\sim$10 million Euros entrance fee,\nsimilar to our annual contribution to ESA, and an annual membership\nfee of only $\\sim$1 million Euros. Thus, if following the\nrecommendation of the international expert committee, ESO were to be\nour lofty astronomy goal for the present century, one can only hope\nthat the whole Greek community will embrace it and with a joined\neffort will convince the ``powers that be\" to turn the wheels and make\nit a reality before the end of the current decade.\n\n\\acknowledgments\n\nThis document used material from the online archives of the Hellenic\nAstronomical Society, the Greek National Committee for Astronomy, as\nwell as from the annual reports of the various institutes that were\navailable online. I would like to thank K. Kokkotas (Univ. of\nThessaloniki), N. Kylafis (Univ. of Crete), J.H. Seiradakis (Univ. of\nThessaloniki), K. Tsinganos (Univ. of Athens), and E. Xilouris\n(National Observatory of Athens) for making suggestions that improved\nthis article.\n\n\n\\end{document} \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}}\n\\renewcommand{\\vec}[1]{{\\bm #1}}\n\n\\usepackage{amsmath}\n\\usepackage{txfonts,wasysym,ulem,url}\n\\usepackage[usenames]{color}\n\n\\begin{document}\n\n\n\\title{\nChiral condensate with topological degeneracy in  graphene \nand its manifestation in edge states\n}\n\n\\author{Yuji Hamamoto}\n \\affiliation{%\nInstitute of Physics, University of Tsukuba, Tsukuba 305-8571, Japan%\n}%\n\n\\author{Hideo Aoki}\n \\affiliation{%\nDepartment of Physics, University of Tokyo,  Hongo, Tokyo 113-0033, Japan%\n}%\n\n\\author{Yasuhiro Hatsugai}%\n\\email{hatsugai@sakura.cc.tsukuba.ac.jp}\n \\affiliation{%\nInstitute of Physics, University of Tsukuba, Tsukuba 305-8571, Japan%\n}%\n \\affiliation{%\nTsukuba Research Center for Interdisciplinary Material Science, University of Tsukuba, Tsukuba 305-8571, Japan%\n}%\n\n\\date{\\today\n            \n\n\\begin{abstract}\nRole of chiral symmetry in many-body states\nof\ngraphene in strong magnetic fields is theoretically studied with the honeycomb lattice model.  \nFor a spin-split Landau level \nwhere the leading electron-electron interaction is \nthe nearest-neighbor repulsion,\na chiral condensate \nis shown to be, within the subspace of $n=0$ Landau level, an \nexact many-body ground state with \na finite gap, for which calculation of Chern numbers reveals that\nthe ground state is a Hall insulator with a {\\it topological degeneracy} \nof two.  \nThe topological nature of the ground state is shown to manifest itself as \na Kekul\\'ean bond order along armchair edges,\nwhile the pattern melts in the bulk due to\nquantum fluctuations.  \nThe whole story can be regarded as a realization of the \nbulk-edge correspondence peculiar to the chiral symmetry.\n\\end{abstract}\n\n\\pacs{73.22.Pr, 71.10.Fd, 73.43.-f\n                            \n                             \n\\maketitle\n\n\\mysection{Introduction}%\nWhile the physics of graphene started from the one-body \nelectronic structure as a Dirac fermion, \na possible relevance of electron correlation in graphene has \nbeen intensively studied after a gap opening in the $n=0$ \nLandau level (LL)\nwas experimentally observed in strong magnetic fields.~\\cite{PhysRevLett.96.136806,PhysRevLett.99.106802}\nSince it is difficult to explain the gap within a simple one-body problem,\nconsiderable theoretical efforts have ensued to clarify many-body effects \nin graphene quantum Hall (QH)\nsystems.~\\cite{PhysRevLett.96.256602,\nPhysRevB.74.075422,\nPhysRevB.74.161407,\nPhysRevB.74.195429,\nPhysRevB.75.165411,\n2007SSCom.143..504A,\nPhysRevLett.99.196802}\nHowever, little attention has been paid on how  \nmany-body effects should reflect the {\\it chiral symmetry} \nin the graphene QH regime, which is after all \na fundamental symmetry inherent in graphene's honeycomb lattice.\nOn the one-body level, the effects of chiral symmetry in graphene is well \nunderstood: \nTo start with, the symmetry guarantees the \nemergence of doubled Dirac cones in the Brillouin zone, \nwhich can be interpreted as a two-dimensional analogue\nof the Nielsen-Ninomiya theorem well-known \nin the four-dimensional lattice gauge theory.  \nWe can even examine the wave functions \nin terms of Aharonov-Casher argument, which states that\nchiral symmetry {\\it topologically protects}\nthe degeneracy of the $n=0$ LL against random gauge fields.~\\cite{PhysRevA.19.2461}\nA similar situation occurs for ripples in a graphene sheet, \nwhich can be modelled by random hopping amplitudes.  \nKawarabayashi {\\it et al.} have shown that  the $n=0$ LL exhibits\nan anomalously sharp (delta-function-like) density of \nstates (DOS) as soon as the spatial wavelength of the ripple \nexceed a few lattice constants.~\\cite{PhysRevLett.103.156804}\n\nIn the presence of electron-electron interactions, \non the other hand, \nthe role of chiral symmetry has primarily been investigated\nin zero magnetic fields \nin the context of spontaneous symmetry breaking.~\\cite{PhysRevB.74.195429,\nPhysRevLett.102.026802,\nPhysRevB.79.165425,\nPhysRevB.82.121403}\nWhile these studies mainly employ a Dirac field model \nin a continuum space to discuss many-body gap formation, \nsuch an effective treatment may well overlook \nthe essence of graphene's chiral symmetry,\nwhich is intimately related to the underlying honeycomb lattice.\n\nWith this background, we shed light in the present work on \nhow the chiral symmetry influences the many-body problem\nin graphene QH effect, by\nfully taking account of the lattice structure.  \nWe first examine \nthe many-body problem with exact diagonalization\nin a subspace projected onto the $n=0$ LL.\nWorking on the subspace enables us to classify \nmany-body states according to a notion of the {\\it total chirality} of \nthe filled zero modes.  In terms of this, \nfor a ``bipartite'' electron-electron interaction such as\nthe nearest neighbor repulsion,\nwhich is the dominant interaction for a spin-split LL, \nwe show that the many-body ground state is {\\it exactly} identified to be \na chiral condensate with a topological degeneracy of two.  \nWe confirm numerically that there exists a finite energy \ngap to the first-excited state, \nwhich makes the Chern number of the ground state  well-defined.\nThe total Chern number contributed by the filled zero modes\nalong with the negative energy states (``Dirac sea\") \nturns out to be zero, \nwhich implies the system is a Hall insulator with vanishing Hall conductance.\n\n\nDespite the cancellation of the Chern number in the bulk, however, \nwe move on to show that the topological nature of the chiral condensate \nis in fact made manifest as an emergence of a Kekul\\'ean bond order in the edge state \nalong armchair edges of the honeycomb lattice \n(in sharp contrast to zigzag edge states in one-body problem).  \nOn the mean-field (MF)  level \nthe Kekul\\'e pattern is shown to appear in the bulk \nwith a  Kekul\\'ean degeneracy of three, \nso that the present result amounts that the mean-field \norder, while dissolved in the topologically-degenerate \nchiral condensate in the bulk, resurfaces along an armchair edge.  \nThis can be interpreted as an example of bulk-edge correspondence,~\\cite{PhysRevLett.71.3697}\nwhich states that a topologically nontrivial bulk state\nshould always accompany a characteristic edge state.\nIn the case of graphene, the bulk-edge correspondence \nbecomes peculiar in that the chiral condensate \nmanifests itself as a freezing of a Kekul\\'ean bond order along a \nspecific (armchair) edges.\n\n\\mysection{Chiral symmetry}%\nTo model interacting electrons on a honeycomb lattice,\nwe assume that spin degeneracy is lifted by a Zeeman splitting, \nso that the leading Coulomb interaction reduces to the \nnearest-neighbor repulsion.\nThe\nHamiltonian then reads $\\mathcal{H}=\\mathcal{H}_{\\rm kin}+\\mathcal{H}_{\\rm int}$, where \nthe kinetic term\n$\\mathcal{H}_{\\rm kin}=-t\\sum_{\\langle ij\\rangle}\n(e^{i\\theta_{ij}}c^\\dagger_{i}c_{j}+{\\rm H.c.})\n\\equiv c^{\\dagger}H_{\\rm kin}c$\ndescribes electron hopping between adjacent sites\n$\\langle ij\\rangle$ with strength $t>0$.\nThe magnetic field is included as the Peierls phase\n$\\theta_{ij}$ such that magnetic flux per elementary hexagon equals\n$\\sum_{\\hexagon}\\theta_{ij}=2\\pi\\phi$ in units of a magnetic flux quantum $h\/e$.\nFor a honeycomb lattice with $N_{\\bullet(\\circ)}$\nsites in sublattice $\\bullet(\\circ)$,\n$c^{\\dagger}=(c^{\\dagger}_{\\bullet},c^{\\dagger}_{\\circ})$\nwith $c^{\\dagger}_{\\bullet(\\circ)}$ a row of creation operators\nfor sublattice $\\bullet(\\circ)$\nand $H_{\\rm kin}$ is a square matrix of dimension $N_{\\bullet}+N_{\\circ}.$\nIf we introduce $\\Gamma={\\rm diag}(I_{\\bullet},-I_{\\circ})$\nwith an identity matrix $I_{\\bullet(\\circ)}$ of dimension $N_{\\bullet(\\circ)}$,\nthe kinetic term satisfies an anticommutation relation\n$\\{H_{\\rm kin},\\Gamma\\}=0$, \nwhich defines the chiral symmetry.\nThe symmetry implies that, \nif $\\psi_{k}$ is the $k$-th eigenvector with energy $\\varepsilon_{k}$,\na chiral partner\n$\\Gamma\\psi_{k}$ exists with an energy $-\\varepsilon_{k}$.\nThis makes the $n=0$ LL special\nin that we can take\n$\\psi_{k}$ as an eigenstate of $\\Gamma$ \nas $\\Gamma\\psi_{k\\pm}=\\pm\\psi_{k\\pm}$.\nThe interaction between spin-polarized electrons \nis expressed in a particle-hole symmetric form as\n$\\mathcal{H}_{\\rm int}=\\sum_{i\\ne j}V_{ij}\n\\left(n_i-\\frac{1}{2}\\right)\\left(n_j-\\frac{1}{2}\\right)\n=\\frac{1}{2}\\sum_{i\\ne j}V_{ij}\n(c^\\dagger_ic^{\\dagger}_{j}c_jc_i\n+c_ic_{j}c^{\\dagger}_jc^{\\dagger}_i)+{\\rm const.}$,\nwhere $V_{ij}$ is the strength of electron-electron interaction,\n$n_i\\equiv c^\\dagger_ic_i$ the number operator at site $i$.\n\n\\mysection{Chiral condensate}%\nWe start with an investigation of \nthe many-body problem\nat half filling.\nSince it is difficult to treat all the many-body states exactly,\nwe shrink the Hilbert space\nby projecting onto the $n=0$ LL.\nSuch a treatment is valid as long as $|V_{ij}|$ is perturbatively small\ncompared with the Landau gaps around the $n=0$ LL.~\\cite{detail}\nIn the $n=0$ LL,\nwe take a zero mode multiplet $\\psi=(\\psi_{+},\\psi_{-})$\nwhere we have decomposed it into \neigenstates of the chiral operator, \n$\\psi_{\\pm}=(\\psi_{1\\pm},\\cdots,\\psi_{M_{\\pm}\\pm})$ with degeneracy $M_{\\pm}$.\nNote that the zero modes are localized on each of the sublattices\nas $\\psi_{+}=\\frac{1}{\\sqrt{2}}\\left(\n\\scriptsize{\n\\begin{array}{c}\n\\psi_{\\bullet}\\\\\n0\n\\end{array}\n}\n\\right)$ and $\\psi_{+}=\\frac{1}{\\sqrt{2}}\\left(\n\\scriptsize{\n\\begin{array}{c}\n0\\\\\n\\psi_{\\circ}\n\\end{array}\n}\n\\right)$.\nIf we introduce the negative-energy multiplet $\\varphi=(\\varphi_{1},\\varphi_{2},\\cdots)$\nsuch that $H_{\\rm kin}\\varphi_{k}=\\varepsilon_{k}\\varphi_{k}$\nwith $\\varepsilon_{k}<0$,\n$(\\psi_{+},\\psi_{-},\\varphi, \\Gamma\\varphi)$ form a complete set,\nso that the fermion operator is expanded as\n$c=\\psi d+\\varphi d_{<}+\\Gamma\\varphi d_{>}$,\nwhere\n$d=\\left(\n\\scriptsize{\n\\begin{array}{c}\nd_{+}\\\\\nd_{-}\n\\end{array}\n}\n\\right)$\nand $d_{\\lessgtr}$ are columns of fermion operators\nfor the zero modes and the $\\varepsilon\\lessgtr0$ states, respectively.\nIn the projected subspace,\nthe total Hamiltonian is written, up to a constant, as\n$\\tilde{\\mathcal{H}}=\\frac{1}{2}\\sum_{i\\ne j}V_{ij}\n(\\tilde{c}^{\\dagger}_{i}\\tilde{c}^{\\dagger}_{j}\\tilde{c}_{j}\\tilde{c}_{j}\n+\\tilde{c}_{i}\\tilde{c}_{j}\\tilde{c}^{\\dagger}_{j}\\tilde{c}^{\\dagger}_{i})$,\nwhere the projected fermion operator, $\\tilde{c}_{i}\\equiv\\psi d$,\nno longer satisfies the canonical anticommutation relations.\nStill, the generator of the total chirality of the filled zero modes\ncan be defined as $\\mathcal{G}=\\tilde{c}^{\\dagger}\\Gamma\\tilde{c}\n=d^{\\dagger}_{+}d_{+}-d^{\\dagger}_{-}d_{-}$.\nDue to the invariance of $\\tilde{\\mathcal{H}}$ for the chiral transformation\n$\\tilde{\\mathcal{H}}\\mapsto\ne^{i\\theta\\mathcal{G}}\\tilde{\\mathcal{H}}e^{-i\\theta\\mathcal{G}}\n=\\tilde{\\mathcal{H}}$,\nthe total chirality is conserved, $[\\tilde{\\mathcal{H}},\\mathcal{G}]=0$.\nThen the many-body states are classified according to the {\\it total chirality} \n$\\chi_{\\rm tot}$.\nFor repulsive interactions $V_{ij}>0$,\n$\\tilde{\\mathcal{H}}$ is semi-positive definite.\nFurthermore, when the interactions are bipartite (i.e., only \nact between different sublattices, $V_{i\\in\\bullet,j\\in\\bullet}=V_{i\\in\\circ,j\\in\\circ}=0$) as the case with the nearest-neighbor interaction, \nchiral condensates $|G_{\\pm}\\rangle=d^{\\dagger}_{1\\pm}\\cdots d^{\\dagger}_{M_\\pm\\pm}\n|D_{<}\\rangle$ (with $|D_<\\rangle\\equiv\\prod_{m}(c^{\\dagger}\\varphi)_{m}|0\\rangle$ denoting the Dirac sea) \nconstitute a {\\it ground state doublet} $\\Psi=(|G_{+}\\rangle,|G_{-}\\rangle)$,\nsince $c_{j}c_{i}|G_{\\pm}\\rangle=c^{\\dagger}_{j}c^{\\dagger}_{i}|G_{\\pm}\\rangle=0$\nfor $i\\in\\bullet$ and $j\\in\\circ$.~\\cite{detail,hamamoto}\nWe call this a topological degeneracy of two.\nUnlike a simple charge density wave (CDW),\none may mix $|G_{+}\\rangle$ and $|G_{-}\\rangle$\nthrough a unitary transformation $\\Psi\\mapsto\\Psi^{\\omega}\\omega$\nwith $\\omega\\in{\\rm U}(2)$ even in a finite system.\nOne can numerically confirm\nthat the chiral condensate remains the ground state\nunless the non-bipartite potential is large,~\\cite{PhysRevLett.99.196802}\neven though the ground-state energy is nonzero.\nIn the rest of the paper,\nwe focus on the leading interaction, i.e., the nearest-neighbor repulsion\nfor simplicity.\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[scale=1.15]{gap_Ne30.eps}\n\\caption{\\label{fig:phi-dependence}\n(color online)\nMagnetic-field dependence of the many-body gap $\\Delta$ in the $n=0$ LL.\nThe result is displayed for 30 electrons \nwith the consecutive values of the linear dimension of \nthe system, $L$, connected by a line with the results \nfor $L\\equiv 3$ (mod 3) forming a lower envelope \n($\\Delta\\propto\\phi^2$).  \nThe inset shows the gap at $L=3l$ against $1\/L$ \nfor the magnetic flux $\\phi=1\/12$ (circle), $\\phi=1\/27$ (triangle), \nand $\\phi=1\/48$ (square).\n}\n\\end{center}\n\\end{figure}\n\n\\mysection{Many-body gap}%\nWe next calculate excitation energies numerically\nwith the exact diagonalization method.\nIn the projected subspace,\nthe strength of the nearest-neighbor repulsion $V>0$ is the only \nenergy scale, which acts as the unit of energy.\nFull energy spectra of $\\tilde{\\mathcal{H}}$ for finite systems\nsuggest that the first excited state appears in the sector of\n$\\chi_{\\rm tot}=\\pm(M_\\pm-2)$,\nwhich is created from a chiral condensate $|G_{\\pm}\\rangle$\nby single chirality-flippings \nanalogous to the projected single-mode approximation.~\\cite{PhysRevLett.54.581,PhysRevB.33.2481,PhysRevLett.73.3568}\nNoticing this,\nwe further shrink the Hilbert space\nby focusing on the sector of $\\chi_{\\rm tot}=\\pm(M_\\pm-2)$.\nThis enables us to obtain the energy of the first excited state,\nor the energy gap $\\Delta$,\nwith calculation cost of the order of $\\mathcal{O}(M_\\pm^2)$.\nWe consider a system on a torus composed of $2L^2$ lattice sites\nwith a linear dimension $L$.\nFor investigating a weak-field regime, we adopt the string gauge,~\\cite{PhysRevLett.83.2246}\nwhere a magnetic flux is given by $\\phi=m\/L^2$ with an integer $m$ $(>0)$\nand $M_\\pm=m$ zero modes are obtained for each chirality.\n\nIn Fig.~\\ref{fig:phi-dependence},\nwe plot $\\Delta$ for $30$ electrons as a function of $\\phi$ with $L$ changed consecutively.  We immediately \nnotice that \nthe result exhibits a marked periodicity of three, where \nthe values for $L\\equiv 3$ (mod 3) form a clear lower \nenvelope with a scaling $\\Delta\\propto\\phi^2$,\nwhile those for $L\\ne3l$ deviates from this.  \nThe latter behavior is considered to be a finite-size effect, \nsince the deviation diminishes with the sample size.  \nTo confirm the scaling, the inset plots $\\Delta\/\\phi^2$ at $L=3l$ \nagainst $1\/L$ for various values of $\\phi$,\nwhich indicates the scaling law $\\Delta\\propto\\phi^2$ is \nvery accurately obeyed.\n\n\\mysection{Hall conductance}%\nLet us now consider Hall conductance of the chiral condensate.\nBy the Niu-Thouless-Wu formula,~\\cite{PhysRevB.31.3372}\nthe Hall conductance is written with the Chern number~\\cite{PhysRevLett.49.405} as\n\\[\n\\sigma_{xy}=\\frac{e^{2}}{h}\\frac{1}{N_{D}}C,\\qquad\nC=\\frac{1}{2\\pii}\\int{\\rm Tr}_{N_{D}}dA,\\qquad\nA=\\Psi^{\\dagger}d\\Psi\n\\]\nwhere $N_{D}$ is the degeneracy\nand $A$ is the non-Abelian Berry connection \nthat describes multiplets.~\\cite{2004JPSJ...73.2604H}\nIn terms of the basis that diagonalizes $\\mathcal{G}$, we have \n$C=C_{+}+C_{-}$ with\n$C_{\\pm}=\\frac{1}{2\\pii}\\int\\langle dG_{\\pm}|dG_{\\pm}\\rangle$.\nEach term is further decomposed as $C_{\\pm}=C_{\\psi_{\\pm}}+C_{D_{<}}$\nwith $C_{\\psi_{\\pm}}=\\frac{1}{2\\pii}\\int{\\rm Tr}_{M_{\\pm}}\nd\\psi^{\\dagger}_{\\pm}d\\psi_{\\pm}$\nand $C_{D_{<}}=\\frac{1}{2\\pii}\\int{\\rm Tr}d\\varphi^{\\dagger}d\\varphi$.\nBy the charge conjugation,\nwe have $C_{\\psi_{+}}+C_{D_{<}}=-(C_{\\psi_{-}}+C_{D_{>}})$\nwith $C_{D_{>}}\n=\\frac{1}{2\\pii}\\int{\\rm Tr}(\\Gamma d\\varphi)^{\\dagger}\\Gamma d\\varphi=C_{D_{<}}$.\nThus the total  Chern number of the ground-state doublet\nvanishes as $C=C_{\\psi_{+}}+C_{\\psi_{-}}+2C_{D_{<}}=0$,\nwhich may be called a topological cancellation.\nThis implies that the chiral condensate is a Hall insulator\nwith a nontrivial topological degeneracy $N_{D}=2$.\n\n\n\\begin{figure}[t]\n\\includegraphics[scale=1.15]{dos-obemf3_mf-bulk15x15p15v0.25e8i174s1323339888.eps}\n\\hfill\n\\includegraphics[scale=1.15]{wc-hbdmf3_mf-bulk15x15p15v0.25e8i174s1323339888.eps}\n\\caption{\\label{fig:bond-order}\n(color online) Mean-field results for the energy spectrum (a),\nits blowup around the $n=0$ LL (b),\nand the bond strength $|\\Delta_{ij}|$ plotted in a real-space (c).\nThe parameters are $L=15,\\phi=1\/15$, and $V\/t=0.25$.\nA dashed hexagon in (c) is an enlarged unit cell with arrows primitive vectors.}\n\\end{figure}\n\n\\mysection{Bond order}%\nAs have been confirmed in various systems,\nwhile topological phases are featureless in a bulk,\nthey show characteristic boundary states.~\\cite{PhysRevLett.71.3697}\nSo a natural question \nwe can pose here is: do the edge states in the present system \nexhibit special features despite the bulk Chern number \nbeing zero?  \nBefore presenting the result, however, let us first \nhave a look at the mean-field state in the present \nsystem in the bulk, which will turn out to be instructive.   \nOne virtue of a mean-field picture~\\cite{hatsugai08up} is that \nwe can introduce a bond order, \n$\\Delta_{ij}\\equiv V\\langle c^\\dagger_ic_j\\rangle$, \nfor adjacent sites $\\langle ij\\rangle$.~\\cite{PhysRevB.37.3774,\nPhysRevB.39.11413,\nPhysRevB.45.4027,\nPhysRevB.61.16377,\nPhysRevLett.98.146801}\nThe dominant part of the MF Hamiltonian is given by\n$\\mathcal{H}_{\\rm MF}=-\\sum_{\\langle ij\\rangle}[(te^{i\\theta_{ij}}\n+\\Delta_{ij}^\\ast)c^\\dagger_ic_j+{\\rm H.c.}]$,\nwhere $\\Delta_{ij}$ is determined self-consistently by diagonalizing\n$\\mathcal{H}_{\\rm MF}$.\nA spontaneous symmetry breaking is induced by the many-body effect for weak magnetic fields, \nwhere the density of states has a sharp peak at  \nthe Fermi energy.\nIn Fig.~\\ref{fig:bond-order},\nwe show a typical MF result for the ordered phase.\nThe energy spectrum is plotted in panel (a),\nwhere the qualitative structure of the LLs is preserved.\nThis comes from the fact that the convergent order parameters turn out \nto retain the initial Peierls phase\nas $\\Delta_{ij}=|\\Delta_{ij}|e^{-i\\theta_{ij}}$.\nThe influence of the electron-electron interaction appears most prominently in the $n=0$ LL,\nwhere a finite gap of the order of $\\phi$ opens as shown in the blowup Fig.~\\ref{fig:bond-order}(b).\nTo see how the symmetry is broken in the mean field, \nwe show in panel (c) a real-space image of the \nbond order $|\\Delta_{ij}|$, \nwhich is seen to exhibit a Kekul\\'e pattern.~\\cite{2008PhyE...40.1530H,hatsugai08up}\nThis makes the unit cell enlarged, which causes $K$ and $K'$ points \nto be coupled, and this in turn opens a finite gap.  In this sense \nwe can regard this a Peierls transition in the honeycomb lattice.\n\nOn the other hand,\nthe chiral condensate with its topological degeneracy of two\nexhibits in the bulk no bond order as we have seen in Fig.~\\ref{fig:bond-order}.\nDue to the quantum fluctuation, the bond order of the mean field\nis destroyed and the quantum liquid ground state is realized.\n\n\n\n\n\\mysection{Edge and defect states}%\nWe are now in a position to ask the question: what kind of \nedge states does the chiral condensate accommodate?\nBased on the bulk-edge correspondence,\nwe may expect a non-trivial behavior of the many-body states near edges.\nA prime example is the \nfractional\nQH states in a 2DEG, where \na CDW-like behavior emerges along edges\nof a\nribbon,~\\cite{PhysRevB.50.17199}\nwhile in the bulk it melts into the Laughlin liquid with no long-range order,\nwhich has a $q$-fold degeneracy of the fractional QH states at filling $\\nu=1\/q$.\nNote that a honeycomb lattice with edges\nhas QH edge states whose mode lies in a LL gap.  \nTo perform the projection onto the $n=0$ LL we set an energy cutoff,\nthe choice of which is shown to have little influence on the edge states shown below.\n\n\\begin{figure}[t]\n\\includegraphics[scale=1.15]{wc-hbd_arm12x192p1q192n12z24.eps}\n\\hfill\n\\includegraphics[scale=1.15]{wc-hbd_zig24x192p1q192n12z58.eps}\n\\caption{\\label{fig:bond-chiral}\n(color online)\nBond strength near an armchair (left) or zigzag (right) edge of the doubly-degenerate chiral condensate.\nMagnetic flux is $\\phi=1\/192$ for which the magnetic length\nis $l_{B}\\simeq 8.9a$.\nThe bond order decays in a bulk away from the edges.\n}\n\\end{figure}\n\nIn Fig.~\\ref{fig:bond-chiral} we show\n$|\\Delta_{ij}|$\nfor the chiral condensate\nplotted in a real space near armchair and zigzag edges.\nIn panel (a),\nwe can see that a Kekul\\'e-type bond order \nreminiscent of the mean-field result in Fig.~\\ref{fig:bond-order}\nemerges along the armchair edge.\nThis is the\nkey result in the present work.\nThe enhancement in bonds rapidly decays away from the edge\nin a few lattice constants,\nand $|\\Delta_{ij}|$ slightly oscillates\nwith a length scale of the order of the magnetic length\n$l_{B}\\sim a\/\\sqrt{\\phi}$\nwith $a$ being the interatomic spacing.\nThis may naively seem to be analogous to the\nfractional QH edge states in a 2DEG, \nbut here the honeycomb lattice structure is essential \nin the ground state.  Indeed,\nthe ring pattern is locked along the armchair edge \nin a Kekul\\'e pattern, \nwhile this is not the case with zigzag edges [see panel (b)].\nIn the latter case, \nthe ring pattern is blurred by the translational symmetry along a zigzag edge, \nand a very weak stripe pattern parallel to the edge appears.\nThese patterns related with the three-fold degeneracy of the \nKekul\\'e pattern  are washed out in the bulk chiral condensate.  \nAll these are a specific property of a honeycomb lattice model.\n\nWe can further endorse that the lattice structure is at the core \nby looking at the states around lattice defects.\nWhen a single atom is removed from the bulk honeycomb lattice,\none-body localized zero modes appear that are protected\nby the chiral symmetry.~\\cite{PhysRevLett.89.077002,2009SSCom.149.1061H}\nIn the presence of the electron-electron interactions, however,\nlocal chiral symmetry breaking occurs spontaneously to lower the energy\nby inducing effective hopping in the same sublattice.~\\cite{2004PhyE...22..679R,\n2009SSCom.149.1061H}\nThen what if two point defects come close to each other?\nSuch a divacancy\nconsists of two adjacent missing atoms, \nand is recently observed experimentally\nin ion-irradiated carbon samples.~\\cite{PhysRevB.85.121402}\nWe expect that the chiral symmetry may be partially recovered\nwith a reconfiguring of the two symmetry-breaking bonds.\nWe plot in Fig.~\\ref{fig:bond-defect} the \nbond order for the chiral condensate near the divacancy.\nWe do confirm that\nenhancement of the bond order near the divacancy\nwhich can be considered due to the revival of the chiral symmetry.\n\n\\begin{figure}[t]\n\\includegraphics[scale=1.15]{wc-hbd_z2def60x60p3q3600n3z6.eps}\n\\caption{\\label{fig:bond-defect}\n(color online)\nBond strength of the chiral condensate\nnear a divacancy composed of two adjacent sites missing.\nMagnetic flux is $\\phi=1\/1200$\nfor which the magnetic length is $l_B\\simeq 22.3a$.  \nThe bond order reflects twofold axial symmetry of the divacancy.\n}\n\\end{figure}\n\nWe have thus shown that the bond order emerges along edges \nand around vacancies, \ndespite the topological cancellation $C=0$ \nthat might first seem to wipe out any signature of the chiral condensate.\nIndeed, the charge density $\\langle n_i\\rangle$ itself is uniform\nfor the\nchiral condensate even along edges,\nwhich is due to the invariance of the chiral condensate\nfor the charge conjugation.\nThus it is the bond order $|\\Delta_{ij}|$ that we have to \nlook at as a probe for the chiral condensate.  \nThus the bond order provides a new probe for the many-body effect\nin half-filled graphene applied a magnetic field.\nThe bond order near\nedges\nshould be observable experimentally with some imaging techniques such as \nGreen's function scanning tunneling microscopes.~\\cite{PhysRevLett.74.306,PhysRevB.51.5502}\nSince the amplitude of\n$|\\Delta_{ij}|\/V=|\\langle c^\\dagger_ic_j\\rangle|$\nis of the order of the magnetic flux $\\phi$,\nthe magnetic field should have significant magnitudes.\n\n\n{\\it Summary.}---%\nThe many-body ground state\nat half filling\nin the honeycomb lattice is identified as a doubly-degenerate chiral condensate for a spin-split Landau level.  \nThe many-body effect opens a finite energy gap,\nwhich makes the chiral condensate a generic topological insulator.\nHowever, the system has a peculiar manifestation of \nthe bulk-edge correspondence in topological systems \nas an emergence of a bond order with a Kekul\\'e pattern \nalong armchair edges  in an exact ground state, \nwhile the pattern is dissolved in the bulk. \n\n{\\it Acknowledgement.}---%\nThe computation in this work has been done\nwith the facilities of the Supercomputer Center,\nInstitute for Solid State Physics, University of Tokyo.\nThis work was supported in part by Grants-in-Aid\nfor Scientific Research No. 23340112 and No. 23654128 from the JSPS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendices}\n\\subsection{Full List of Incompatibility Rules}\\label{appd:corr1a}\nThe 12 incompatibility rules of entity type-constrained biRoR introduced in Section~\\ref{sec:corr1a} are\n\\begin{enumerate}\n\\item Part-Whole (arg0) and Per-Soc (arg0)\n\\item Part-Whole (arg0) and Gen-Aff (arg0)\n\\item Part-Whole (arg1) and Per-Soc (arg0)\n\\item Part-Whole (arg1) and Gen-Aff (arg0)\n\\item Per-Soc (arg0) and Org-Aff (arg1)\n\\item Per-Soc (arg0) and Art (arg1)\n\\item Org-Aff (arg0) and Art (arg1)\n\\item Org-Aff (arg1) and Art (arg1)\n\\item Org-Aff (arg1) and Gen-Aff (arg0)\n\\item Art (arg0) and Art (arg1)\n\\item Art (arg1) and Gen-Aff (arg0)\n\\item Art (arg1) and Gen-Aff (arg1)\n\\end{enumerate}\n\n\n\n\\end{document}\n\n\\subsection{The Tradeoff between precision and recall}\n\\begin{table*}[!ht]\n  \n  \\centering\n  \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{l|ccc|ccc|ccc|ccc|ccc}\n    \\toprule\n     & \\multicolumn{3}{c|}{\\textbf{BC\\_dev}} & \\multicolumn{3}{c|}{\\textbf{BC\\_test}} & \\multicolumn{3}{c|}{\\textbf{CTS}} & \\multicolumn{3}{c|}{\\textbf{WL}} & \\multicolumn{3}{c}{\\textbf{Overall}} \\\\\n     & \\textbf{P} & \\textbf{R} & \\textbf{Micro} & \\textbf{P} & \\textbf{R} & \\textbf{Micro} & \\textbf{P} & \\textbf{R} & \\textbf{Micro} & \\textbf{P} & \\textbf{R} & \\textbf{Micro} & \\textbf{P} & \\textbf{R} & \\textbf{Micro} \\\\\n     \\hline\n     \n    \\textbf{$\\text{\\modelname{}}$}  & 76.76 & 64.25 & 69.8& 71.64 & \\textbf{62.08} & 67.17& 63.52 & \\textbf{60.25} & 59.5& 60.89 & \\textbf{54.77} & 58.4 & 65.35\n& \\textbf{59.03}\n& 61.69\\\\\n    \\textbf{$\\text{\\modelname{}}_\\text{local}$}   & 77.01 & 61.69 & 68.44& \\textbf{77.52} & 61.94 & 68.89& \\textbf{71.95} &  57.93 & 59.87& 65.51 & 54.61 & 60.6 & \\textbf{71.66}\n& 58.16\n& 63.12\\\\\n    \\textbf{$\\text{\\modelname{}}_\\text{non-local}$} & 79.57 & \\textbf{65.15} & \\textbf{70.97}& 72.64 & 61.78 & 67.88& 69.14 & 52.84 & 58.55& 63.17 & 52.96 & 58.52 & 68.32\n& 55.86\n& 61.65\\\\\n    \\textbf{$\\text{\\modelname{}}_\\text{glocal}$} & \\textbf{80.30} & 62.98 & 70.36& 77.03 & 61.88 & \\textbf{69.82}& 70.20 & 59.64 & \\textbf{61.17}& \\textbf{65.94} & 54.48 & \\textbf{61.7} & 71.06\n\n&58.67& \\textbf{64.23}\\\\\n    \\bottomrule\n    \\end{tabular}\n  }\n  \\caption{Precision and Recall on ACE05} \\label{tab:addlabel}\n\\end{table*}\n\n\n\n\\begin{table}[htbp]\n  \\centering\n    \\begin{tabular}{lccc}\n    \\toprule\n    \\textbf{Model}    & \\textbf{P} & \\textbf{R} & \\textbf{Micro} \\\\\n    \\midrule\n    \\multicolumn{4}{c}{\\textbf{\\emph{Non-ensemble models}}} \\\\\n           \\textbf{$\\text{\\modelname{}}$}  & 32.05 & \\textbf{49.37} & 37.9 \\\\\n           \\textbf{$\\text{\\modelname{}}_\\text{local}$}   & 39.31 & 45.3  & 39.24 \\\\\n           \\textbf{$\\text{\\modelname{}}_\\text{non-local}$}  & 39.08 & 47.68 & 39.63 \\\\\n           \\textbf{$\\text{\\modelname{}}_\\text{glocal}$}  & \\textbf{44.24} & 46.86 & \\textbf{39.74} \\\\\n    \\midrule\n    \\multicolumn{4}{c}{\\textbf{\\emph{Ensemble models}}} \\\\\n           \\textbf{E-$\\text{\\modelname{}}$}  & 40.77 & \\textbf{54.03} & 42.39 \\\\\n           \\textbf{E-$\\text{\\modelname{}}_\\text{local}$}    & 51.03 & 51.98 & 43.12 \\\\\n           \\textbf{E-$\\text{\\modelname{}}_\\text{non-local}$}  & 52.31 & 51.39 & \\textbf{43.98} \\\\\n           \\textbf{E-$\\text{\\modelname{}}_\\text{glocal}$}  & \\textbf{56.48} & 47.44 & 43.35 \\\\\n    \\bottomrule\n    \\end{tabular\n    \\caption{Results on SemEval 2018 Task 7.2}\n\n  \\label{tab:res_semeval}%\n\\end{table}%\n\n\\subsection{Which model captures which feature well?}\n\n2. example\n\\subsection{Ablation Study on ACE05}\n\n\n\\subsection{Two-Stage Training}\nWe study the performance of each stage in the two-stage learning on ACE05 validation set in Table~\\ref{tab:ablation}. The first stage is binary classification to distinguish whether two entities have ``no-relation'' or a valid relation. In this stage, the glocal model which combines both the local learner and non-local supervisor performs the best, scoring +2.55 F1 scores higher than the base model.\n\nWe observe that in both datasets, there are a dominant number of negative labels. The experiments show that the difficulty is more in identifying whether there exists a relation, than in classifying the specific relation. So it is also worth exploring a two-stage learning strategy for local and non-local correlations. Following \\cite{DBLP:journals\/corr\/abs-1909-11898}, in the first stage, we train an extraction model on identifying binary relations (where the ``no-relation'' label is treated as negative, and all other relations are positive). In the second stage, a classification model, which is trained to classify the different positive relations, will be applied on the samples which the extraction model in Stage 1 labels as positive. We denote the combination of a local extracting model and local classification model as \\textbf{$\\text{\\modelname{}}_{\\text{ex:local},  \\text{cl:local}}$}, and the combination of a glocal extracting model and local classification model as \\textbf{$\\text{\\modelname{}}_{\\text{ex:glocal}, \\text{cl:local}}$}.\n\n\n     \n\n\n\n\n\n\n\\section{Introduction}\\label{sec:intro}\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width= \\columnwidth]{img\/fig_intro.pdf}\n    \\caption{An example of the RE task.}\n    \\label{fig:intro-ex}\n\\end{figure}\n\nRelation Extraction (RE) is the task to identify the relation of given entities, based on the text that they appear in. As a fundamental task of knowledge extraction from text, RE has become an active area of research in the past several decades~\\cite{DBLP:conf\/tipster\/MillerCFRSSW98,DBLP:journals\/jmlr\/ZelenkoAR03,DBLP:conf\/naacl\/BunescuM05,DBLP:conf\/acl\/ZhouSZZ05,DBLP:conf\/semeval\/HendrickxKKNSPP10}. \n\nIn natural language, most text includes multiple entities. For example, the sentence in Figure~\\ref{fig:intro-ex} has seven entities. We find that 99.76\\% data instances of the widely used ACE 05 dataset \\cite{walker2006ace} have more than two entities, and there are 9.21 relations in each text on average. However, most previous research\nhas been confined to the simplified setting of only classifying the relation between every \\textit{two} entities at a time \\cite{DBLP:conf\/coling\/ZengLLZZ14,DBLP:conf\/semeval\/LuanOH18,DBLP:conf\/acl\/LiJ14,DBLP:conf\/emnlp\/GormleyYD15,DBLP:conf\/acl\/MiwaB16}. For the sentence in Figure~\\ref{fig:intro-ex} with 7 entities, most previous approaches would perform 49 independent relation classification tasks (if including self-reflexive relations). It is not feasible to reduce this number of classification tasks because existing methods require explicit annotation of the entities in the input. For example, to predict the relation between the entity pair (obstetricians, California), the input needs to be transformed into ``...$\\langle e1\\rangle$ obstetricians $\\langle \\backslash e1\\rangle$ in $\\langle e2\\rangle$ California $\\langle \\backslash e2\\rangle$ will pay \\$60,000 in Los Angeles ...''.\n\n\nThe problems exposed by such a previous paradigm is that it is not only inefficient, but also overlooks the interdependency among the multiple relations in one context. For example, in the 49 relations in Figure~\\ref{fig:intro-ex}, if we already know the relationship (Miami, is part of, south Florida), where ``is part of'' is a relation defined on two objects, then it is very unlikely for \\textit{Miami} to be in any other person-social relationship such as ``is the father of...''. \nRemember that, on average, there are 9.21 relations in each text in the ACE 05 dataset, for example, and each relation can provide information to other relations in the same text. We denote the frequently-appearing interdependency of the many relations in the same text as the ``relation of relations'' (RoR) phenomenon.\n\nTo capture RoR, we propose a new paradigm of RE by treating the predictions of all relations in the same text as a whole. Note that our work is distinct from \\cite{DBLP:conf\/acl\/WangTYCWXGP19}, which still treats the relation of each entity pair as independent classification tasks, but saves the computation power at the cost of accuracy by encoding all entities in one pass.\nInstead, our newly proposed paradigm is not about tradeoffs between computation cost and accuracy, but to increase the performance by capturing RoR.\n\nIn this paper, we first highlight the importance of RoR by identifying several types and their frequent occurrences in RE datasets in Section~\\ref{sec:ror_stats}. We then propose a data-driven approach {without} hand-crafted rules, using Graph Neural Networks (GNNs) to model each relation as a node to learn the pair-wise dependency of every two relations, and then a matrix transformer to learn the correlations involving multiple relations or numerical correlations of the count of relations. We evaluate the model on two benchmark datasets, ACE 2005 (ACE05)~\\cite{walker2006ace} and SemEval 2018 Task 7.2 (SemEval2018)~\\cite{DBLP:conf\/semeval\/GaborBSQZC18}. Our system outperforms the previous state-of-the-art (SOTA) models by +1.12\\% on ACE05 and +2.55\\% on SemEval2018. The contributions of our paper are as follows:\n\\begin{itemize}\n    \\item We propose a new paradigm of RE, according to the frequent RoR phenomenon. It provides a new perspective for future research in RE.\n    \\item We develop a model to learn the interdependency of all relations in the same text, based on a GNN and matrix transformer.\n    \\item We validate the effectiveness of our model, which outperforms SOTA models by a clear margin on two benchmark datasets.\n    \\item We open-source our model and evaluation codes.\n\\end{itemize}\n\n\\section{New formulation of RE}\nMost previous work formulates RE as multiple independent classification problems limited to two entities and the text: \n\\begin{quote}\n    Given two entity mentions $\\bm{e}_1$ and $\\bm{e}_2$, a text sequence $\\bm{t}=\\{w_1, w_2, \\ldots, w_N\\}$ involving $\\bm{e}_1$ and $\\bm{e}_2$, and a finite set relation types $\\mathcal{R}$, the task is to predict the relation type between the two entities.\n\\end{quote}\nUnder this setting, RE can be solved by the well-researched sentence classification task.\n\nHowever, based on the motivations in Section~\\ref{sec:intro}, we propose a new paradigm of RE:\n\\begin{quote}\nGiven~a~text~sequence~$\\bm{t}=\\{w_1, w_2, \\ldots, w_N\\}$ and all the entities $\\bm{e}_1, \\ldots, \\bm{e}_M$ mentioned in $\\bm{t}$, the model needs to predict the  relationship $r_{ij}$ between each two entities $(\\bm{e}_i, \\bm{e}_j)$, where $i, j \\in \\{1,\\ldots, M\\}$.\n\\end{quote}\nWe use a matrix $\\bm{R} = (r_{ij}) \\in \\mathbb{R}^{M \\times M}$ to represent all the relations of interest, as shown in Figure~\\ref{fig:problem_form}.\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width= \\columnwidth]{img\/fig_prob.pdf}\n    \\caption{New formulation of RE that considers all $M$ entities in the same text.}\n    \\label{fig:problem_form}\n\\end{figure}\n\\begin{table*}[!t]\n  \\centering\n    \\begin{tabular}{lll}\n    \\toprule\n    \\textbf{Relation} & \\textbf{Arg0} & \\textbf{Arg1} \\\\\n    \\midrule\n    Person-Social (Per-Soc)\n    (e.g., family) & PER & PER \\\\\n    Part-Whole (e.g., city-and-country) & FAC, LOC, GPE, ORG & FAC, LOC, GPE, ORG \\\\\n    Physical (Phys) (e.g., near) & PER, FAC, LOC, GPE & PER, FAC, LOC, GPE \\\\\n    Organization-Affiliation (Org-Aff) (e.g., employment) & PER, ORG, GPE & ORG, GPE \\\\\n    Agent-Artifact (Art) (e.g., owner-and-object) & PER, ORG, GPE & FAC \\\\\n    General-Affiliation (Gen-Aff) (e.g., citizen) & PER & PER, LOC, GPE, ORG \n    \\\\\n    \\bottomrule\n    \\end{tabular}%\n  \\caption{Valid entity types for each relation. The entity type abbreviations refer to Person (PER), Facility (FAC), Location (LOC), Geo-PoliticalEntity (GPE), and Organization (ORG).}\n  \\label{tab:ror_type1a}\n\\end{table*}\n\n\n\\section{Statistical analysis of RoR}\\label{sec:ror_stats}\nWe conduct our case study using the benchmark dataset ACE05 \\cite{walker2006ace}. We will introduce two forms of RoR: (1) biRoR, which is only between two relations, and (2) multiRoR, which involves three or more relations. \n\nNote that the purpose of the following analyses is to demonstrate the importance of RoR. Our model introduced in the subsequent section will \\textit{not} hand-craft such detailed rules but will learn RoR in a data-driven way.\n\n\\subsection{Data overview}\n\nACE05 \\cite{walker2006ace} is the most widely used dataset for RE. Its text is extracted from a variety of sources, including news programs, newspapers, newswire reports, and audio transcripts. The 6 relation types are shown in Table~\\ref{tab:ror_type1a}. There are 7 entity types valid for the relations: Facility (FAC), Geo-PoliticalEntity (GPE), Location (LOC), Organization (ORG), Person (PER), Vehicle (VEH), Weapon (WEA).\n\n\\subsection{BiRoR: Interdependency of two relations}\\label{sec:ror_local}\n\\subsubsection{Entity type-constrained biRoR }\\label{sec:corr1a}\nWe first introduce the simplest form of RoR, based on the entity types. Specifically, given the triple ($\\bm{e}_1$, $\\mathrm{RelType}_a$, $\\bm{e}_2$) of two entities and their relation, we can infer whether $\\bm{e}_1$ is unlikely to co-occur with a different relation type $\\mathrm{RelType}_b$. \n\nTo elucidate such constraint, we will make an example using the seven entity types in ACE05. As detailed in Table~\\ref{tab:ror_type1a}, only certain types are allowed to be the arguments of the relations. Therefore, we can deduct 12 rules of \\textit{incompatibility}. For example, \nthe same entity cannot be both the arg0 of Per-Soc and the arg0 of Part-Whole, because an entity must be a person (PER) in order to satisfy the Per-Soc relationship, but Part-Whole \\textit{cannot} involve PER. The full list of incompatibility rules are listed in Appendix~\\ref{appd:corr1a}.\n\n\n\\subsubsection{Semantic-constrained biRoR}\\label{sec:corr1b}\nAnother type of biRoR is constrained by the semantics of the relation. The intuition is that what a relation means can imply whether it can be shared or must be disjoint with another relation. For example, the Art relationship can describe a person (arg0) owning a facility (arg1), where the arg1 \\textit{must} be a facility. If such relation already exists, the same facility cannot be involved in the Part-Whole relation with a city, because cities (e.g., Boston) cannot be a part of a facility, semantically. This kind of incompatibility is not a hard constraint by the entity type, but it is implied by the semantics of relations.\n\nSemantics can also imply whether a relation should be symmetric. For example, Per-Soc is always symmetric because family and friends are commutative relations, whereas Org-Aff is always an asymmetric relation \\cite{ACE05eval}. Hence, if a relation $r_{ij}=$ Per-Soc, then $r_{ji}=$ Per-Soc. And if $r_{ij}=$ Org-Aff, then $r_{ji}\\neq$ Org-Aff.\n\n\n\n\n\n\\subsubsection{Empirical biRoR}\\label{sec:corr1c}\nTo form a more direct understanding of biRoR, we calculate the correlation of every two relations in Figure~\\ref{fig:corr2}. We can see that the incompatibility rules in Section~\\ref{sec:corr1a} is proven by the red negative color, the symmetric property of Per-Soc and Phys in Section~\\ref{sec:corr1b} is proven by the darker blue. \n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width= \\columnwidth]{img\/fig_corr2.pdf}\n    \\caption{Correlation of every two relations. Each cell $(\\mathrm{RelType}_i, \\mathrm{RelType}_j)$ represents the conditional probability that an entity will have $\\mathrm{RelType}_j$ given the existing $\\mathrm{RelType}_i$. For better visual effect, the zero probability is converted to $-1$ (dark red), signaling that these two relations have 0\\% co-occurrence.}\n    \\label{fig:corr2}\n\\end{figure}\nThere are also other correlations such as frequent \nco-occurrences of Part-Whole and Phys relations.\n\n\n\n\\subsection{MultiRoR: Correlation of 3+ relations}\\label{sec:ror_nonlocal}\n\n\\subsubsection{Entity type-constrained multiRoR}\nOther than biRoR, which only involves two relations, there are more complicated rules acting on multiple relations, namely multiRoR. The entity type-constrained multiRoR extends Section~\\ref{sec:corr1a} from the incompatibility between two relations to among 3+ relations. For example, if we already know two relations of an entity, which is both the arg1 of the Org-Aff relation and the Phys relation, then this entity cannot be the arg1 of any Art relations (because this entity must be GPE). As the number of relations centered on one entity increases in Figure~\\ref{fig:corr3_1}, the percentage of invalid combinations of multiple relations among all possible combinations will soon be over 50\\%, and even reached 83\\% when there are 7 relations.\n\\begin{figure}[ht!]\n  \\centering\n   \\resizebox{0.85 \\columnwidth}{!}{%\n            \\begin{tikzpicture}\n            \\pgfplotsset{\n            scale only axis,\n           \n            legend style={at={(0,0)},anchor=south west},\n        }\n        \n    \\pgfplotsset{\npercentage plot\/.style={\nnodes near coords align=vertical,\n    yticklabel=\\pgfmathprintnumber{\\tick}\\,$\\%$,\n    ymin=0,\n    ymax=100,\n    enlarge y limits={upper,value=0}\n}\n}\n\\begin{axis}[\n    legend style={cells={align=left},nodes={scale=0.7, transform shape},\n    legend cell align={left}},\n    title={Percentage of Invalid Combinations},\n    xlabel={Number of Relations Centered on One Entity},\n    ylabel={},\n    xmin=0.5, xmax=7.5,\n    ymin=0, ymax=100,\n    xtick={1,2,3,4,5,6,7},\n    ytick={0,20,40,60,80,100},\n    legend pos=north west,\n    ymajorgrids=true,\n    grid style=dashed,\n    width=7cm,\n    height=5cm,\n    every axis plot\/.append style={thick},\n    percentage plot\n]\n\n\\addplot[\n    color=blue,\n    mark=square,\n    ]\n    coordinates {\n    (1,0)(2,22)(3,41)(4,55)(5,65)(6,73)(7,83)\n    };\n    \\legend{}\n    \n\\node [above,font=\\small] at (axis cs:  1,  3) {\\pgfmathprintnumber{0}\\%};\n\\node [above,font=\\small] at (axis cs:  1.9,  24) {\\pgfmathprintnumber{22}\\%};\n\\node [above,font=\\small] at (axis cs:  2.9,  41.5) {\\pgfmathprintnumber{41}\\%};\n\\node [above,font=\\small] at (axis cs:  4,  55.5) {\\pgfmathprintnumber{55}\\%};\n\\node [above,font=\\small] at (axis cs:  5,  65.5) {\\pgfmathprintnumber{65}\\%};\n\\node [above,font=\\small] at (axis cs:  6,  73.5) {\\pgfmathprintnumber{73}\\%};\n\\node [above,font=\\small] at (axis cs:  7,  83.5) {\\pgfmathprintnumber{83}\\%};\n\n\\end{axis}\n\\end{tikzpicture}\n}\n\\vspace{0px}\n\\caption{Percentage of invalid combinations over all possible set of relations of an entity, as the total number of non-empty relations centered on the entity increases.}\n\\label{fig:corr3_1}\n\\end{figure}\n\n\n\\subsubsection{Numerically correlated multiRoR}\n\n\nWe can also discover numerical correlations of multiple relations. From all relation matrices $\\bm{R}$ in the dataset, we find that the number of occurrences of a specific type of relations can correlate with the number of another relation type. Note that it is counted as multiRoR, because the correlation is defined not between two single relations, but the total count of two relation types (each of which can include multiple occurrences). From the correlation plot in Figure~\\ref{fig:corr3_3}, Per-Soc and Gen-Aff show a strong positive linear dependency, whereas Art and Org-Aff are negatively related by numbers.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width= \\columnwidth]{img\/fig_corr3_3.png}\n    \\caption{Numerical correlation of every two relation types. Each cell $(\\mathrm{RelType}_i, \\mathrm{RelType}_j)$ represents the linear correlation between the number of occurrences of $\\mathrm{RelType}_i$ and that of $\\mathrm{RelType}_j$.\n    }\n    \\label{fig:corr3_3}\n\\end{figure}\n\n\\section{Method}\nBased on the rich RoR phenomenon analyzed in Section~\\ref{sec:ror_stats}, we aim to design a model that can mine these properties from data. A naive solution is to hand-craft many rules to impose every type of RoR, but it is not scalable when there are datasets of different features, or when there are some RoR that are difficult to be manually identified. Hence, we aim to design a model that has the capacity to learn RoR with no hand-crafting. \n\n\nOur overall training strategy is in Figure~\\ref{fig:archi}. In the following, we will elaborate on three key components: (1) initial embeddings of entities and relations, (2) the GNN-based biRoR learner, and (3) the matrix transformer that learns multiRoR.\n\n    \n    \n\n\\subsection{Initialization of entities and relations} \\label{sec:method_init}\nAs a preparation step, we first obtain the embedding of each entity. We pass the text through a pretrained BERT model \\cite{DBLP:conf\/naacl\/DevlinCLT19}, and obtain each entity representation by average pooling over its tokens' hidden states in the last layer of BERT. Note that our framework can easily adapt to other ways to retrieve pretrained embeddings, such as \\cite{DBLP:conf\/nips\/YangDYCSL19,DBLP:journals\/corr\/abs-1907-11692}. \n\nThen, we obtain the initial embedding of each relation by applying an feedforward layer on the concatenated embeddings of the two involved entities.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width= \\columnwidth]{img\/fig_archi.pdf}\n    \\caption{The model architecture of RoR.}\n    \\label{fig:archi}\n\\end{figure}\n\n\\subsection{BiRoR Learner}\\label{sec:method_local}\nWe use a GNN to learn the interdependency between every two relations, namely biRoR.\nFor each text sequence $\\bm{t}$ with $M$ entities, we formulate a\ngraph $\\mathcal{G} = (\\mathcal{V},\\mathcal{E})$, where all entities and relations form the nodes of the graph. Accordingly, we link every relation node to its two entity nodes.\n\nWe use a GNN to learn node-to-node interactions, and especially among the relation nodes. The first layer of GNN is initialized with the node embeddings (including all entity embeddings and relation embeddings) obtained from Section~\\ref{sec:method_init}. In each layer $\\ell$, it aggregates all hidden states passed from the neighbors to update its representation in the next layer $\\ell +1$. More specifically, we have\n\\begin{align}\n    h^{\\ell+1}_u = \\mathrm{FFN}\\left(W^O\\sum_{v \\in {\\mathcal{N}} (u)}\\alpha_{u,v}h^{\\ell}_v\\right)\n    ,\n\\end{align}\nwhere $h_i^{\\ell}$ is the hidden state of the $i$-th vertex in the $\\ell$-th layer, $\\mathrm{FFN}(\\cdot)$ is a feed-forward network, $W^O$ is the weight matrix, ${\\mathcal{N}} (u)$ is the set of neighbor nodes to the vertex $u$, and $\\alpha_{u,v}$ is the attention weight that $u$ has to $v$. This attention weight $\\alpha_{u,v}$ is obtained by\n\\begin{align*}\n    \\alpha_{u,v} = \\frac{ \\exp \\left[W^{Q}h^{\\ell}_v\\left(W^{K}h^{\\ell}_u\\right)^T\\right]}{ \\sum_{v' \\in {\\mathcal{N}} (u)}\\exp \\left[W^{Q}h^{\\ell}_{v'}\\left(W^{K}h^{\\ell}_u\\right)^T\\right]}\n    ,\n\\end{align*}\nwhere $W^{K}$ and $W^{Q}$ are the key and query weight matrices when calculating the attention.\n\n\n\n\\subsection{MultiRoR Learner} \\label{sec:method_global}\nThe GNN introduced in Section~\\ref{sec:method_local} has a strong ability to model node-to-node interaction, which corresponds to the biRoR, but it is not as strong when seizing the more complicated multiRoR as analyzed in Section~\\ref{sec:ror_nonlocal}. For example, if an entity has both the relation Org-Aff and Phys, then it cannot have the relation Art. But the GNN structure does not necessarily capture such complicated multiRoR that involve nested conditions. \n\nTherefore, we need another module to directly model the relation matrix $\\bm{R}\\in \\mathbb{R}^{M\\times M}$, which takes into consideration the dynamics among all relations as a whole, in order to capture multiRoR.\n\nTo this end, we propose a simple but effective module, a relation matrix transformer. As each relation $r_{ij}$ in the relation matrix $\\bm{R}$ needs to attend to all other relations, we build our relation matrix transformer by customizing the Transformer encoder architecture, which allows extensive mutual attention among all elements \\cite{DBLP:conf\/nips\/VaswaniSPUJGKP17}. Specifically, we customize the position encoding in the Transformer into two parts: row encoding and column encoding, each of which is a learnable mapping from the position index to a $d$-dimensional vector space. \n\nOur relation matrix transformer adds the position encoding, namely the sum of the row and column embedding, to the initial representations of relations obtained by procedures in Section~\\ref{sec:method_init}. The input is thus a tensor $\\bm{T} \\in \\mathbb{R}^{M \\times M \\times d}$, where $d$ is the dimension of features in the embedding.\nThe matrix transformer then learns the dynamics among all relations as a whole, and outputs new features of all relations by a transformed matrix $\\bm{T_{\\mathrm{Transformed}}} \\in \\mathbb{R}^{M \\times M \\times d'}$ which captures the multiRoR. \n\nFinally, we add together the relation embeddings learned by the GNN and the matrix transformer, and feed them into the final classification layer to obtain the type of each relation.\n\n\n\n\\section{Experiments}\n\\subsection{Datasets}\nWe use two benchmark RE datasets to evaluate the performance of our model.\\footnote{Since our focus is RE, we do not use a newly published RE dataset, DocRED \\cite{yao2019docred}. It contains entity linking annotations which can be a multi-tasking objective, and turns the task into a different setting.}\n\n\\paragraph{ACE05} \nThe ACE 2005 Multilingual Training Corpus (ACE05)~\\cite{walker2006ace} is a standard RE dataset. It has six relation categories, detailed in Section~\\ref{sec:ror_stats}. We process and split the dataset following the practice in \\cite{DBLP:conf\/emnlp\/GormleyYD15,DBLP:conf\/acl\/PlankM13}.\\footnote{\\url{https:\/\/github.com\/mgormley\/ace-data-prep}} There are six subdomains in the dataset: Broadcast Conversations (BC), Broadcast News (BN), Conversational Telephone Speech (CTS), Newswire (NW), Usenet Newsgroups (UN), and Weblogs (WL). The statistics of the resulted dataset is in Table~\\ref{tab:ace05_stats}.\nThe macro F1-score is the primary evaluation metric of the data, as used by most works. As ``no-relation'' is regarded as the negative label, all non-negative relations on wrong entities as treated as false positives. \n\n\\begin{table}[!h]\n  \\centering\n    \\begin{tabular}{clc}\n    \\toprule\n    & \\textbf{Domain} & \\textbf{\\# Relations} \\\\\n    \\midrule\n    \\multirow{1}{*}{\\textbf{Train}} &\n    NW+BN &  34,669  \\\\\n    \\midrule\n    \\multirow{1}{*}{\\textbf{Valid}} & BC\\_dev & 5,717 \\\\\n    \\midrule\n    \\multirow{3}{*}{\\textbf{Test}} & BC\\_test & 6,692 \\\\\n    & CTS & 11,683 \\\\\n    & WL &  11,022 \\\\\n    \\bottomrule\n    \\end{tabular}%\n  \\caption{Statistics of ACE05 dataset. \n  }\\label{tab:ace05_stats}\n\\end{table}\n\\paragraph{SemEval2018} \nWe use SemEval 2018 Task 7.2 \\cite{DBLP:conf\/semeval\/GaborBSQZC18} as the second RE dataset to evaluate our model.\\footnote{We choose Task 7.2 instead of the Task 7.1 used by \\citet{DBLP:conf\/acl\/WangTYCWXGP19} because Task 7.1 only tests the \\textit{classification} of positive relations, but Task 7.2 is the standard relation \\textit{extraction} which tests on both positive and negative relations.}\nThe corpus is collected from abstracts and introductions of\nscientific papers, and there are six types of semantic relations in total.\nNote that there are three subtasks of it: Subtask 1.1 and Subtask 1.2 are relation \\emph{classification} on clean and noisy data, respectively; Subtask 2 is the standard relation \\textit{extraction}, where the same training set as Subtask 1 is used, but the evaluation is to identify all relations including ``no-relation.''\nFollowing the main systems~\\cite{DBLP:conf\/semeval\/RotsztejnHZ18,DBLP:conf\/semeval\/NooralahzadehOL18}, we combine the training data of both Subtask 1.1 and 1.2 as the training data.\nThe dataset consists of 136,965 train, and 27,316 test examples. We count the relation of every entity pair as a sample, and if no specific relation is annotated for some entity pairs, we use ``no-relation'' as the label. The standard evaluation metric is also macro F1, which is used for the official ranking of submissions.\n\n\\subsection{Baselines for ACE05}\nFor ACE05, we compare our model with the following systems.\n\n\\paragraph{$\\text{BERT}_\\text{EntEmb}$}\n\nWe use the baseline from  \\cite{DBLP:conf\/acl\/WangTYCWXGP19} (called ``$\\text{BERT}_\\text{sp}$ with entity-indicator on input-layer'' in the original paper), which is essentially a BERT with entity embedding on the input layer.\n\n\\paragraph{$\\text{OnePass}_\\text{MRE}$ and $\\text{OnePass}_\\text{SRE}$} \nWe also compare with the state-of-the-art systems on ACE05 -- \\text{OnePass} by \\citet{DBLP:conf\/acl\/WangTYCWXGP19}. OnePass has two variations: single relation extraction ($\\text{OnePass}_\\text{SRE}$) that treats \nevery single relation extraction as an independent classification task including separate encoding and classification, and multiple relation extraction ($\\text{OnePass}_\\text{MRE}$) that encodes all entities in one pass, and then do the pair-wises classification per relation. \n\n\n\\paragraph{Other models} \nWe also compare our model with other previous RE models such as \\cite{DBLP:conf\/emnlp\/GormleyYD15,DBLP:conf\/naacl\/NguyenG15,DBLP:conf\/ijcnlp\/FuNMG17,DBLP:conf\/emnlp\/ShiFHZJLH18}.\n\n\n\\subsection{Baselines for SemEval 2018 Task 7.2}\nFor SemEval2018, we compare our results with the top 3 systems on the leaderboard. For a fair comparison, we compare our models against the non-ensemble and ensemble models separately. \n\n\\paragraph{Ensemble models} The top 1 system \\cite{DBLP:conf\/semeval\/RotsztejnHZ18} is an ensemble of CNNs and RNNs. Its training uses an ensemble size of 20, data augmentation, multi-task learning, and many other meticulous designs.\nNotably, when implementing our models, we only use a small ensemble size of 5, with no further tricks.\n\n\n\\begin{table*}[!ht]\n  \\centering\n    \\begin{tabular}{l|c|cccc}\n    \\toprule\n    \\textbf{} & \\multicolumn{1}{c|}{\\textbf{Dev}} & \\multicolumn{4}{c}{\\textbf{Test}} \\\\\n    \\textbf{} & BC\\_dev & BC\\_test & CTS   & WL & \\textbf{Overall} \\\\\n    \\hline\n    \\textbf{HybridFCM \\cite{DBLP:conf\/emnlp\/GormleyYD15}} & -- & 63.48 & 56.12 & 55.17 & 58.26\n    \\\\\n    \\textbf{DAN \\cite{DBLP:conf\/ijcnlp\/FuNMG17}} & -- & 65.16 & 55.55 & 57.19 & 59.30\n    \\\\\n    \\textbf{GSN \\cite{DBLP:conf\/emnlp\/ShiFHZJLH18}} & -- & 66.38 & 57.92 & 56.84 & 60.38\n    \\\\\n    \\textbf{$\\text{BERT}_\\text{EntEmb}$ \\cite{DBLP:conf\/acl\/WangTYCWXGP19}} & 65.32 & 66.86 & 57.65 & 53.56 & 59.36 \\\\\n    \\textbf{$\\text{OnePass}_\\text{MRE}$ \\cite{DBLP:conf\/acl\/WangTYCWXGP19}} & 67.46 & 69.25 & 61.7  & 58.48 & 63.14 \\\\\n    \\textbf{$\\text{OnePass}_\\text{SRE}$ \\cite{DBLP:conf\/acl\/WangTYCWXGP19}} & 68.90 & \\textbf{69.76} & 63.71  & 57.20 & 63.14 \\\\\n    \\hline\n    \\textbf{$\\text{\\modelname{}}_\\text{base}$} & 69.95 & 66.51 & 61.84 & 57.67 &  62.01 \\\\\n    \\textbf{$\\text{\\modelname{}}_\\text{bi-only}$} & 68.50 & \\textbf{68.86} & 64.18 & 59.57 & 64.20 \\\\\n    \\textbf{$\\text{\\modelname{}}_\\text{multi-only}$} & \\textbf{71.64}    & 66.77    & 59.90    & 57.62 & 61.43 \\\\\n    \\textbf{$\\text{\\modelname{}}_\\text{full}$}  & 70.59 & 68.63 & \\textbf{64.49} & \\textbf{59.67} & \\textbf{64.26} \\\\\n   \n   \n   \n   \n   \n   \n   \n   \n    \\bottomrule\n    \\end{tabular}%\n  \\caption{Macro F1 on ACE05. Following \\cite{DBLP:conf\/acl\/WangTYCWXGP19}, we reported the performances on the development set, and all subsets of the test set. ``Overall'' is the average of all three subsets in the test set. }\\label{tab:ace05}\n\\end{table*}\n\n\\paragraph{Non-ensemble models} The 2nd model  \\cite{DBLP:conf\/semeval\/LuanOH18} uses LSTMs to encode both word sequences and dependency\ntree structures, and perform relation extraction between concepts on top of them. And the 3rd model \\cite{DBLP:conf\/semeval\/NooralahzadehOL18} uses a CNN model over the shortest dependency path between two entities.\n\n\\subsection{Our Models}\nWe investigate the following four settings of our model.\n\n\\paragraph{$\\text{\\modelname{}}_\\text{base}$} For the base model in our architecture, we use a customized version of the $\\text{BERT}_\\text{EntEmb}$ baseline. Instead of the extra embedding for entities in $\\text{BERT}_\\text{EntEmb}$, we use the sentence index to indicate entities, so that the pretrained properties of BERT can be maintained as much as possible. We use this model as the base model that we build our biRoR and multiRoR learning modules on.\n\n\\paragraph{$\\modelname{}_\\text{bi-only}$ and $\\modelname{}_\\text{multi-only}$}\nBased on $\\text{\\modelname{}}_\\text{base}$, we implement {$\\modelname{}_\\text{bi-only}$}, which only the biRoR module based on GNN, and {$\\modelname{}_\\text{multi-only}$}, which uses only the multiRoR module based on a relation matrix transformer. \n\n\\paragraph{$\\modelname{}_\\text{full}$} Finally, we implement the full model $\\modelname{}_\\text{full}$ which adopts both the GNN and the relation matrix transformer to learn all types of RoR.\n\n\n\\subsection{Implementation details}\nWe follow \\cite{DBLP:conf\/acl\/WangTYCWXGP19}'s practice to use BERT-base (uncased) as the pretrained model.  \nThe GNN and matrix transformer has 4 layers, 8-head attention, and the hidden size of attention layers is 512. The feedforward layers of GNN have 1024 hidden units, and that of matrix transformer has 4096 hidden units. The batch size is 8.  The learning rate is set to 5e-5 on ACE05 and 1e-4 on SemEval2018, and the warmup is 0.1. We use the Adam optimizer and the cosine scheduler of the python package Transformers.\\footnote{\\url{https:\/\/github.com\/huggingface\/transformers}} We train the model for 30 epochs on both datasets. \nWe run all experiments with one Tesla V100 card on a Ubuntu system.\n\n\n\n\n\\section{Results and analysis}\n\\subsection{Main results}\n\\paragraph{ACE05}\nWe first analyze the experiments on the larger dataset, ACE05. From the main results in Table~\\ref{tab:ace05}, we can see that in the last column (overall performance), our full model $\\text{\\modelname{}}_\\text{full}$ achieves the strongest performance, outperforming all previous models. Specifically, our model improves over the strongest setting of OnePass by 1.12\\%, which is a large margin on the ACE05 dataset. Our full model Moreover, on the subsets of the test set, such as CTS and WL, which is different from the BC domain seen in the development set, our model also demonstrate consistent improvement over the previous best model.\n\\begin{table}[ht]\n  \\centering\n    \\begin{tabular}{lc}\n    \\toprule\n    \\textbf{Model}    & \\textbf{Macro} \\\\\n    \\midrule\n    \\multicolumn{2}{c}{\\textbf{\\emph{Non-ensemble models}}} \\\\\n           \\cite{DBLP:conf\/semeval\/LuanOH18} &  39.10  \\\\\n           \\cite{DBLP:conf\/semeval\/NooralahzadehOL18} &  33.60    \\\\\n           \\textbf{$\\text{\\modelname{}}_\\text{base}$} & 38.83  \\\\\n           \\textbf{$\\text{\\modelname{}}_\\text{bi-only}$}  & 41.99  \\\\\n           \\textbf{$\\text{\\modelname{}}_\\text{multi-only}$} & 42.87  \\\\\n           \\textbf{$\\text{\\modelname{}}_\\text{full}$} & \\textbf{45.46}  \\\\\n    \\midrule\n    \\multicolumn{2}{c}{\\textbf{\\emph{Ensemble models}}} \\\\\n           \\cite{DBLP:conf\/semeval\/RotsztejnHZ18} & 49.30   \\\\\n           \\textbf{E-$\\text{\\modelname{}}_\\text{base}$} & 46.47  \\\\\n           \\textbf{E-$\\text{\\modelname{}}_\\text{bi-only}$}  & 51.50   \\\\\n           \\textbf{E-$\\text{\\modelname{}}_\\text{multi-only}$} & \\textbf{51.85}  \\\\\n           \\textbf{E-$\\text{\\modelname{}}_\\text{full}$} & 51.56  \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Macro F1 scores on SemEval 2018 Task 7.2.}\n\n  \\label{tab:res_semeval}%\n\\end{table}%\n\n\\paragraph{SemEval2018}\n\nIn Table~\\ref{tab:res_semeval}, we compare our system with the top 3 systems \\cite{DBLP:conf\/semeval\/RotsztejnHZ18,DBLP:conf\/semeval\/LuanOH18,DBLP:conf\/semeval\/NooralahzadehOL18} on the SemEval 2018 Task 7.2 Leaderboard. For the non-ensemble setting, our model is +6.26\\% higher in performance than the most competitive models on the leaderboard. For the more difficult ensemble model leaderboard, we use a simpler setting (i.e., smaller ensemble size, and no data augmentation), but surpasses the top model by +2.55\\%. The improvement shows that our model with no complicated designs can demonstrate strong performance. \n\n\\paragraph{Summary of main experiments}\nExperiments on both datasets validate the effectiveness of our new perspective into RE. Because we model all the relation extraction tasks of the same text together, our method absorbs more knowledge from the same data than other models. Therefore, without data augmentation for SemEval2018, and without domain adaptation on ACE05 \\cite{DBLP:conf\/ijcnlp\/FuNMG17,DBLP:conf\/emnlp\/ShiFHZJLH18}, our model still surpasses all systems that utilizes extra data.\n\n\n\\subsection{Does the model learn multi-relation instances well?}\n\nWe want to further analyze our model's ability to learn complicated multiRoR relations, especially for some entities with a high number of non-empty relations. For the model outputs on the ACE05 test set, we look into the subset of relations centered on entities with $\\geq2$ relations. On this subset, the macro F1 of the previous model OnePass is 61.19\\%, which drops 2 percent under its overall reported performance. However, our method achieves 63.55\\%, keeping almost the same performance as the reported 64.26\\% on the whole test set. On the more challenging subset involving entities with $\\geq3$ relations, our model performs 56.63\\%, almost 4 percent over OnePass's F1 of 52.82\\%. This shows that the advantage of our model is stronger than others as the relation prediction gets more challenging.\n\n\n\n\\subsection{Does the model learn the conditional probability of relations well?}\n\nWe also compare our model's ability to learn biRoR versus OnePass. Remember the ground-truth conditional probability of every two relations mentioned before in Figure~\\ref{fig:corr2} of Section~\\ref{sec:corr1c}. For each row of the heatmap, it is normalized to probability 1, so each value in the row forms the probability distribution of any other relation conditioned on the observation of an existing relation. On the overall test set of ACE05, we analyze the output of our model and OnePass to obtain the corresponding heatmap, with each row representing a probability distribution. We calculate the average Jenson-Shannon (JS) distance of all the distributions. We find that OnePass is 0.0805 away from the gold distribution, and we are 0.0326 away from the gold distribution, $60\\%$ closer than OnePass, which is the strongest previous model.\n\n\n\n\n\\section{Related Work}\nRelation Extraction is one of the most important tasks in NLP. Conventional classification\napproaches have made use of contextual, lexical\nand syntactic features combined with richer linguistic and background knowledge such as WordNet and FrameNet \\cite{DBLP:conf\/semeval\/HendrickxKKNSPP10,DBLP:conf\/semeval\/RinkH10}, as well as kernel-based methods \\cite{DBLP:journals\/jmlr\/ZelenkoAR03,DBLP:conf\/naacl\/BunescuM05,DBLP:conf\/acl\/ZhouSZZ05}.\n\nThe recent advancement of deep neural networks result in a revolution in the methodology of RE. Many CNN-based \\cite{DBLP:conf\/coling\/ZengLLZZ14,DBLP:conf\/acl\/SantosXZ15,DBLP:conf\/naacl\/NguyenG15}, and RNN-based \\cite{DBLP:conf\/emnlp\/SocherHMN12,DBLP:journals\/corr\/ZhangW15a,DBLP:conf\/acl\/MiwaB16,DBLP:conf\/acl\/ZhouSTQLHX16} models achieve high performance in many datasets. The popularity of the field also gives birth to many shared tasks \\cite{DBLP:conf\/semeval\/HendrickxKKNSPP10} which turned into the cradle of many competitive, well-designed systems \\cite{DBLP:conf\/semeval\/LuanOH18,DBLP:conf\/semeval\/RotsztejnHZ18,DBLP:conf\/semeval\/JinDSMC18}.\nRecently, as the model innovation in single relation extraction gradually slow down, most work shifted to the direction of distant supervision \\cite{DBLP:conf\/acl\/MintzBSJ09,DBLP:conf\/emnlp\/ZengLC015,DBLP:conf\/acl\/LinSLLS16}.\n\nThese data-augmentation methods through distant supervision are orthogonal to the innovation in supervised models, and the focus of this paper is to innovate the supervised models to learn the same data with more thorough exploitation. To the best of our knowledge, we are the first paper in RE aiming to learn the correlation of relations in the given data.\n\nIn terms of our matrix formulation of the RoR, the most similar work is the table-filling approach of joint entity and relation extraction \\cite{miwa2014modeling,gupta2016table}. However, our basic unit is an entity as opposed to every word in the text. Moreover, our proposed GNN and Matrix Transformer are different from \\citeauthor{miwa2014modeling}'s \\citeyearpar{miwa2014modeling} history-based structured learning with complex features and heuristics, as well as \\citeauthor{gupta2016table}'s \\citeyearpar{gupta2016table} RNN sequence decoding model.\n\n\n\\section{Ethical Considerations}\nAs with any RE algorithms, there is the danger of using the model to analyze the massive amount of text online, and retrieve the relationship among users and mine their critical information. We are aware of this danger, and sought to minimize the risk. For this reason, we only work on anonymized data, and the set of relations only involves common knowledge graph relations such as part-whole. The algorithm provided in this paper should \\textit{not} be used to analyze any user-sensitive data, but as a tool to facilitate public knowledge graph construction.\n\n\\section{Conclusion}\nIn this paper, we proposed a new paradigm of RE, which is capable of modeling the interdependency among multiple relations in the same text. Our model uses a GNN and a matrix transformer to capture the RoR in data. Experiments validated that our model has a substantial improvement on the two benchmark datasets where most models compete for improvement on the decimal point. We also conducted analyses to reflect on our model's performance on learning multi-relation data and the proximity of distributions of correlation of the gold and predicted relations. \n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nWe appreciate Prof Rada Mihalcea and Di Jin for their constructive suggestions on the storyline of this paper. We also thank Qipeng Guo for advice on the design of GNNs. Zhijing Jin appreciates Yixuan Zhang and Zhutian Yang for their support.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn order to clearly state the question addressed here, it\nis important to recall some\npoints of the mathematical foundation of observables in quantum mechanics.\nThere will be two main\ncontributions, one related to some weak solutions of the Schr\\\"odinger\nequation and other to dimensional\ninterpretations.\n\nThe problem of finding the\ncorrect self-adjoint extension describing the quantum (Schr\\\"odinger)\noperator corresponding to a physical\nmodel can be subtle and difficult. Usually the physicist has a clear\nexpression for the operator, an\nunbounded one acting in a Hilbert space~$\\mathcal H$, but it is not\nobvious which\ndomain should be taken (some general references for what follows are\n\\cite{BEH,RS,Th}).\n\nLet $\\langle\\psi,\\phi\\rangle$ denote the inner product in~$\\mathcal\nH$; if $T$\nis a linear operator acting on its dense domain ${\\rm dom{\\,}} T\\subset \\mathcal\nH$, then to represent a physical\nobservable it is necessary that $T$ is hermitian, i.e.,\n\\[\n\\langle T\\psi,\\phi\\rangle = \\langle \\psi,T\\phi\\rangle, \\quad \\forall\n\\psi,\\phi\\in {\\rm dom{\\,}} T.\n\\]\nHowever this condition is not enough to guarantee that $T$ has real\nspectrum and the time evolution it generates is\nunitary; the right condition is self-adjointness. The domain of its\nadjoin $T^*$ is\n\\[\n{\\rm dom{\\,}} T^* = \\{\\xi\\in\\mathcal H: \\exists \\eta\\in{\\mathcal H} \\;{\\rm\nwith}\\; \\langle \\eta,\\phi\\rangle =\n\\langle \\xi,T\\phi\\rangle, \\; \\forall\\phi\\in{\\rm dom{\\,}} T \\},\n\\]\nand for $\\xi\\in{\\rm dom{\\,}} T^*,$ one has $T^*\\xi=\\eta.$ It follows that $T^*$\nis well defined if ${\\rm dom{\\,}} T$ is dense\nin $\\mathcal H$, and $T$ is hermitian if, and only if, $T^*$ is an\nextension of $T$. The operator $T$ is\nself-adjoint if $T=T^*$. Notice also that (often) for bounded operators the\ndistinction between hermitian and\nself-adjoint operators does not exist.\n\n\nAs already mentioned, usually $T$ is hermitian with dense domain, and\none asks if it is also self-adjoint or\nhas any self-adjoint extension; such extensions are the candidates for\nthe operator describing the related physical\nobservable. A nice situation that often occurs, in particular for the\nHamiltonian of the Hydrogen atom (and\nother atomic systems as well), is that $T$ is essentially self-adjoint,\ni.e., it has just one self-adjoint\nextension and the physical operator is well determined. However, there are\nsituations where there are\ninfinitely many self-adjoint extensions and each\none should correspond to a different physical circumstance; the choice\nis a physical one, not on mathematical bases. Even worse, some hermitian\noperators have no self-adjoint\nextensions!\n\nThe standard example of such framework is the momentum operator\n$P=-i\\frac{d}{dx}$ for a particle in a box\n$[0,1]$. In this case $\\mathcal H = L^2[0,1]$, it is natural to take\n${\\rm dom{\\,}} P$ as smooth functions\n$\\psi\\in\\mathcal H$ such that $\\psi(0)=0=\\psi(1)$ (so that the particle\nremains confined\nto the box); the self-adjoint extensions of this hermitian operator are\n$P_\\alpha$, where $\\alpha$ is a complex number with $|\\alpha|=1$, and\nall elements of ${\\rm dom{\\,}} P_\\alpha$ satisfy $\\psi(1)=\\alpha\\psi(0)$.\n\nIt is worth remarking that if $T$ is hermitian and ${\\rm dom{\\,}} T=\\mathcal H$,\nthen $T$ is bounded,\nso that in  general such domain questions\nare not avoidable. These interesting problems are well explored in the\nliterature, and as additional\nreferences see\n\\cite{AGHKH} and for applications to the one-dimensional hydrogen\natom see~\\cite{FLM,Gesz}.\n\nNevertheless, there are some delicate issues in the mathematical\nfoundations of quantum mechanics that seem\nnot yet exploited from the physical point of view. The main goal of\nthis work is to discuss one of\nsuch issues and relate it to a physical situation.\n\n\nRecall that a self-adjoint Hamiltonian operator\n$H$ generates a time evolution $\\psi(t)=U(t,0)\\psi=e^{-itH}\\psi$,\nwhich is a solution of the Schr\\\"odinger equation\n\\[\ni\\frac{d}{dt}\\psi(t) = H\\psi(t),\\quad \\psi=\\psi(0)\\in{\\rm dom{\\,}} {H}.\n\\]\nSince $U(t,0)$ is a family of unitary operators, for any time $t$ its\ndomain is\nthe\nwhole Hilbert space $\\mathcal H$, so\nthat it is meaningful to consider $U(t,0)\\varphi$ for $\\varphi\\in\\mathcal\nH$ but with $\\varphi\\notin {\\rm dom{\\,}} H$,\ni.e., the time evolution is not\nrestricted to the domain of $H$. Sometimes $U(t,0)\\varphi$, for $\\varphi\\notin {\\rm dom{\\,}} H$, is called a\nweak solution of the Schr\\\"odinger\nequation.\n\nAn unusual situation will be presented. A self-adjoint  operator $H$ with\ndense\n${\\rm dom{\\,}} H\\subset\\mathcal H=L^2(\\ensuremath{{\\mathrm{I\\!R}}}^3)$ will be considered,\n vectors\n$\\Xi\\in\\mathcal H$ not belonging to its domain will be given, although they are\npseudo-eigenvectors of $H$, that is,\n\\begin{equation}\\label{pseudoeigenvalue1}\nH\\Xi = \\lambda_\\Xi \\Xi,\n\\end{equation}\nfor $\\lambda_\\Xi\\in \\ensuremath{{\\mathrm{I\\!R}}}$. Some numerical calculations will indicate\nthat $\\Xi$ has a nonzero component in\nthe continuous subspace of $H$, so that the na\\\"{\\i}ve time evolution\nbuilt from (\\ref{pseudoeigenvalue1}) gives\nan incorrect answer. It will be argued that such solutions are related to\nthe same model in smaller\ndimensions. Furthermore, the physical system in question is one of the\nmost celebrated models in\nquantum mechanics, the three dimensional (3D) hydrogen atom.\n\n\n\\section{Pseudo-Eigenvectors as Weak Solutions}\n\nThe hermitian Hamiltonian of the 3D hydrogen atom is\n\\[\nH_0=-\\frac{\\hbar^2}{2\\mu}\\Delta - \\frac{e^2}{r},\\quad {\\rm dom{\\,}}\nH_0=C^\\infty_0(\\ensuremath{{\\mathrm{I\\!R}}}^3)\\subset L^2(\\ensuremath{{\\mathrm{I\\!R}}}^3),\n\\]\nwhere $\\mu$ is the electron mass, $e$ its electric charge and  $C^\\infty_0(\\ensuremath{{\\mathrm{I\\!R}}}^3)$ denotes the set of\nsmooth functions with compact\nsupport. This operator is essentially\nself-adjoint and its unique self-adjoint extension\n$H_H$, {\\it the} 3D hydrogen atom\nHamiltonian, reads\n\\begin{equation}\\label{HH}\nH_H=-\\frac{\\hbar^2}{2\\mu}\\Delta - \\frac{e^2}{r},\\quad {\\rm dom{\\,}}\nH_H=H^2(\\ensuremath{{\\mathrm{I\\!R}}}^3),\n\\end{equation}\nwith $H^2(\\ensuremath{{\\mathrm{I\\!R}}}^3)$ denoting an appropriate Sobolev space; in particular,\n$H^2(\\ensuremath{{\\mathrm{I\\!R}}}^3)$ is a subspace of $L^2(\\ensuremath{{\\mathrm{I\\!R}}}^3)$,\nit is also the natural domain of the free particle Hamiltonian and all\nits\nelements are continuous functions~\\cite{RS}.\n\nThe usual spectral analysis of $H_H$ can be performed and its well-known\neigenvalues\n\\[\n-\\frac{\\mu e^4}{2 \\hbar^2 n^2},\\quad n\\ge1,\n\\]\ncan be found. Recall that the closed subspace $\\mathcal H_p$ generated by its\neigenvectors is named the point\nsubspace of\n$H_H$ and its orthogonal complement $\\mathcal H_{ac}$ is a nontrivial subspace (i.e., it has nonzero elements) and\nnamed the absolutely continuous (or scattering) subspace of\n$H_H$. Physically, the members of $\\mathcal H_p$ are the bound states while\nthe\nelements of $\\mathcal H_{ac}$ describe the\nionizing atomic states (this interpretation follows, for instance, by the\nRAGE Theorem\n\\cite{CFKS,deOdoC}).\n\nThe eigenvalue equation for the 3D hydrogen atom Hamiltonian is separable\nin spherical\ncoordinates $r\\ge0,\\, 0\\le \\theta\\le\\pi,\\,0\\le \\phi\\le2\\pi$, and by taking the standard representation\n\\begin{equation}\\label{RTP}\n\\Psi(r,\\theta,\\phi)=R(r)\\Theta(\\theta)\\Phi(\\phi),\n\\end{equation}\n the equation for\n$\\Theta(\\theta)$ is given by~\\cite{pauling}\n\\begin{equation}\\label{ThetaEq}\n\\frac{1}{\\sin\\theta}\\frac{d}{d\\theta}\\left(\\sin\\theta\n\\frac{d\\Theta}{d\\theta}\\right) -\\left(\\frac{m^2}{\\sin^2\\theta}-\\ell(\\ell+1)\n\\right)\\Theta=0,\n\\end{equation}\nwith $m$ and $\\ell\\ge0$ being integer constants. For each $\\ell$ value one\nhas\n$-\\ell\\le m\\le \\ell$.\n\nConsider first  the particular\ncase  $\\ell=0$; it follows that $m=0$ and (\\ref{ThetaEq}) reduces to\n\\begin{equation}\\label{danadaEq}\n\\frac{1}{\\sin\\theta}\\frac{d}{d\\theta}\\left(\\sin\\theta\n\\frac{d\\Theta}{d\\theta}\\right) =0.\n\\end{equation}\nThe usual normalized solution of this equation is\n$\\Theta_{0,0}(\\theta)=\\Theta_{l=0,m=0}({\\theta})=1\/\\sqrt2$. However, there\nis also\nthe additional solution (that will play a major role here)\n\\begin{equation}\\label{qsi}\n\\xi_{0,0}(\\theta) = \\frac{\\sqrt6}{\\pi}\\ln\\left[ \\tan\\left(\n\\frac{\\theta}{2}\\right) \\right].\n\\end{equation}\nThis is just one instance of\nadditional solutions $\\xi_{\\ell,m}$ of\n(\\ref{ThetaEq}) for $\\ell,m$ as above; such\nsolutions are Legendre function of the second kind~\\cite{arfken,math}.\nThe $\\xi_{\\ell,m}$ solutions have been discarded in the mathematical\nliterature\nsince\nthey are not continuous at\n$\\theta=0$ and $\\theta=\\pi$, and so via (\\ref{RTP}) they do not generate\nelements in the domain of\n$H_H$; and discarded in the physical literature~\\cite{pauling} by\narguing they are not bounded functions.\n\nBy taking the usual radial $R_{n,\\ell}(r)$ and azimuthal $\\Phi_m(\\phi)=\\frac\n1{\\sqrt{2\\pi}}e^{im\\phi}$ solutions for the 3D hydrogen\natom, set ($\\ell<n$)\n\\[\nF_{n,\\ell,0}(r,\\theta,\\phi)= \\left[\\left(\\frac{2}{n a_0}\\right)^3\n\\frac{(n-\\ell-1)!}{4 \\pi n \\left[(n+\\ell)!\\right]^3} \\right]^\\frac12\n\\exp\\left(-\\frac{r}{n a_0}\\right)\\left(\\frac{2 r}{n a_0}\\right)^\\ell\nL^{2 \\ell+ 1}_{n+\\ell}\\left(\\frac{2 r}{n a_0}\n\\right)\n\\] so that one gets the\nstandard eigenfunctions (here restricted to $m=0$)\n\\[\n\\Psi_{n,\\ell,0}(r,\\theta,\\phi):= F_{n,\\ell,0}(r,\\theta,\\phi)\n\\Theta_{\\ell,0}(\\theta),\n\\]\nand now the additional ones\n\n\\[\n\\Xi_{n,\\ell,0}(r,\\theta,\\phi):=F_{n,\\ell,0}(r,\\theta,\\phi)\n\\xi_{\\ell,0}(\\theta),\n\\]\nwhere $a_0=\\hbar^2\/(\\mu e^2)$ denotes the Bohr radius and\n$L^{2\\ell+1}_{n+\\ell}$ are the\nLaguerre polynomials.\n\nFor $m\\ne0$ the probability density generated by $\\xi_{\\ell,m}$\n diverges (recall the Jacobian is\n$r^2 \\sin\\theta$), i.e.,\n\\[\n\\int_0^\\pi \\sin\\theta\\, |\\xi_{\\ell,m\\ne0}(\\theta)|^2\\, d\\theta =\\infty;\n\\]\nthus the corresponding functions $\\Xi_{n,\\ell,m\\ne0}(r,\\theta,\\phi)$  do not\nbelong to the Hilbert space\n$L^2(\\ensuremath{{\\mathrm{I\\!R}}}^3)$. Hence, it is meaningless to talk about their\ntime evolution even as weak\nsolutions of the 3D hydrogen atom Schr\\\"odinger equation.\n\nHowever, for $m=0$ the probability density generated by $\\xi_{\\ell,0}$\ndoes not diverge, i.e., by choosing\nappropriate constants it is found that\n\\[\n\\int_0^\\pi \\sin\\theta\\, |\\xi_{\\ell,0}(\\theta)|^2\\, d\\theta =1,\n\\]\nso that together with the  $R(r)$ and $\\Phi(\\phi)$ counterparts in\n(\\ref{RTP}),\n$\\xi_{\\ell,0}$ generates elements $\\Xi_{n,\\ell,0}$  of\n$L^2(\\ensuremath{{\\mathrm{I\\!R}}}^3)$, as given by the expression above.\n\n\n\nNotice that although $\\Xi_{n,\\ell,0}$ does not belong to the domain of\n$H_H$,\none formally finds\n\\begin{equation}\\label{pseudoeigenvalue}\nH_H\\Xi_{n,\\ell,0} = \\lambda_{n} \\Xi_{n,\\ell,0},\\quad \\lambda_{n} =\n-\\frac{\\hbar^2}{2\\mu a_0^2 n^2},\\quad \\forall n, 0\\le \\ell<n,\n\\end{equation}\nso that $\\Xi_{n,\\ell,0}$ and $\\lambda_{n}$ are pseudo-eigenvectors and\npseudo-eigenvalues of $H_H$,  respectively. Another\npoint supporting the use of the adjective ``pseudo'' is that\n$\\Xi_{n,\\ell,0}$\nare not orthogonal to\nevery $\\Psi_{n,\\ell',0}$ (for instance, $\\Xi_{n,0,0}$ is not orthogonal to\n $\\Psi_{n,\\ell',0}$ with odd~$\\ell'$).\n\nThe question to be addressed now is about the time evolution of\n$\\Xi_{n,\\ell,0}$, which is well defined since $\\Xi_{n,\\ell,0}\\in L^2(\\ensuremath{{\\mathrm{I\\!R}}}^3)$\nand so $U(t,0)\\Xi_{n,\\ell,0}$ is a weak solution of the 3D hydrogen atom\nSchr\\\"odinger equation\n\\[\ni\\hbar\\frac{\\partial}{\\partial t}\\psi = H_H \\psi.\n\\]\nBased on (\\ref{pseudoeigenvalue}) the na\\\"{\\i}ve expression for the solution $\\Xi_{n,\\ell,0}(t)=e^{-i H_H\nt\/\\hbar}\\Xi_{n,\\ell,0}$ is\n\\begin{equation}\\label{naivesolution}\n\\Xi_{n,\\ell,0}(t)=e^{-i \\lambda_{n} t\/\\hbar}\\Xi_{n,\\ell,0};\n\\end{equation}\nsuch solution is not correct for $\\Xi_{n,\\ell,0}$ is not in the\ndomain of $H_H$; if (\\ref{naivesolution})\nholds then $\\Xi_{n,\\ell,0}(t)$ would be strongly differentiable and so\none could conclude that\n$\\Xi_{n,\\ell,0}\\in{\\rm dom{\\,}} H_H$. Notice that (\\ref{naivesolution}) is correct\nif $\\Xi$ is replaced by\n$\\Psi$.\n\nFrom the dynamic point of view it is important to check if\n$\\Xi_{n,\\ell,0}$ belongs to the point subspace $\\mathcal H_p$\nassociated to $H_H$ or if it has a component in the absolutely continuous\nsubspace $\\mathcal H_{ac}$. In the latter case ionization can take place; in the former case these\ngeneralised eigenstates are written as superpositons of ordinary eigenstates, even if they do not belong to\nthe domain of the operator, and so such solutions would be, in some sense, between linear  combination of\nordinary eigenvectors and the continuous space (but with no ionization). \n\n\nIn order to check if $\\Xi_{n,\\ell,0}$\nis generated by $\\Psi_{n',\\ell',m}$, $n'\\ge1$,\n$\\ell'=0,\\cdots,n'-1,$\n$-\\ell'\\le m\\le \\ell'$, i.e., if $\\Xi_{n,\\ell,0}$ belongs to the point\nsubspace of\n$H_H$, consider\n\\[\n\\Xi_{n,\\ell,0} = \\sum_{n'=1}^\\infty \\sum_{l'=0}^{n'-1} C_{n',l'}^{(n)}\\,\n\\Psi_{n',l',0} + \\chi^c_{n,\\ell},\n\\]\nwith $\\chi^c_{n,\\ell}$ denoting the component of $\\Xi_{n,\\ell,0}$ in the\ncontinuous subspace\n$\\mathcal H_{ac}$. Notice that clearly\n$\\Xi_{n,\\ell,0}\\perp \\Psi_{n',\\ell',m}$ if $m\\ne0$. It was numerically found\nthat\n$\\|\\chi^c_{1,0}\\|>0.8$, as indicated in figure~\\ref{FigPN} for $n=1$ (in  figure~\\ref{FigPN} the\nvalues of $P(N)$  are exact, since symbolic calculus was used); the parameter $P(N)^2\\equiv\\sum_{n'=1}^N\n\\sum_{l=0}^{n'-1} |C_{n',l}^{(1)}|^2$ is an approximation for\n$1-\\|\\chi^c_{1,0}\\|^2$. Similar results were found for other values\nof~$\\ell$.\n\n\nTherefore one concludes that $\\Xi_{n,\\ell,0}$ have both nonzero point and\ncontinuous components, so that their time\nevolutions actually are not described by (\\ref{naivesolution}), but give\n nonzero\nprobabilities $\\|\\chi^c_{n,\\ell}\\|^2$ of ionization (and then far from being\nbound states).\n\n\\begin{figure}\n\\begin{center}\n\\vskip 10pt\n\\includegraphics[width=6cm, height=6cm]{Fig1proj.eps}\n\\caption{$P$ as function of $N$.}\n\\label{FigPN}\n\\end{center}\n\\end{figure}\n\n\n\\section{Lower Dimensional Hydrogen Atom}\nThe fact that all $\\Xi_{n,\\ell,0}$ belong the Hilbert space raises the possibility of finding physical\nmeanings  for them; this section aims at discussing possible physical contents of these pseudo-eigenvectors.\nThe first crucial remark is that formally $\\Xi_{n,\\ell,0}$ has null azimuthal angular\nmomentum ($m=0$). The  solution $\\Xi_{n,0,0}$ has also null total angular\nmomentum (both\n$\\ell=0=m$), but there is a lack of rotational symmetry (see~(\\ref{qsi}));\nthis particular solution gives\na clue on the physical interpretation. In fact, in comparision with ordinary eigenfunctions\n$\\Xi_{n,\\ell,0}(r,\\theta)$ \nare elongated over the $z$-axis with a logarithmic divergence at\n$\\theta=0$ and $\\pi$.\nFigure~\\ref{Figzerozero} shows the absolutely values of\n$\\xi_{0,0}(\\theta)$ and $\\Theta_{0,0}(\\theta)$ as a\nfunction of~$\\theta$, and Figure~\\ref{function3D} a boundary surface of the 3D wavefunction $\\Xi_{1,0,0}$,\nwhich is to be compared with $\\Psi_{0,0,0}$ that has complete radial symmetry (its\nboundary surfaces are spheres centrered at the origin). Hence, there is a strong indication that\n$\\Xi_{n,0,0}$ are reminiscent of classical\ntrajectories performing one-dimensional (1D) like motion, in agreement with\nits null angular momentum and lack\n of rotational symmetry.  So, it is natural to relate such\nwavefunctions to the 1D hydrogen atom, an interesting and\ncontroversial subject, popularized by the\nwork of Loudon \\cite{Lou} published in 1959.\n\n\\begin{figure}\n\\begin{center}\n\\vskip 10pt\n\\includegraphics[width=6cm, height=6cm]{FigQ00.eps}\n\\caption{$|\\xi_{0,0}(\\theta)|$ and  $|\\Theta_{0,0}(\\theta)|$ (dashed) as\nfunction of~$\\theta$.}\n\\label{Figzerozero}\n\\end{center}\n\\end{figure}\n\n\nLoudon stated that the 1D hydrogen atom was twofold degenerate, having\neven and odd eigenfunctions for each\neigenvalue, except for the (even) ground state having infinite binding\nenergy. Typically 1D systems have no\ndegenerate eigenvalues, and Loudon justified the double degeneracy as a\nconsequence of the singular atomic\npotential. Andrews \\cite{And1} questioned the existence of a ground\nstate with infinite binding energy. Ten\nyears later Haines and Roberts \\cite{HR} revised Loudon's work and\nobtained that the even wave functions,\nwith continuous eigenvalues, were complementary to odd functions, but\nsuch results were criticized by\nAndrews \\cite{And2}, who did not accepted the continuous eigenvalues.\nGomes and Zimerman \\cite{GZ} argued\nthat the even states with finite energy should be excluded. Spector and\nLee \\cite{SL} presented a\nrelativistic treatment that removed the problem of infinite binding\nenergy of the ground state. Several other\nworks \\cite{DPST,BKB,NVS,LC,OL,FLM,XDD,LL} (see also references therein)\nhave discussed this (apparent)\nsimple problem.\n\nThe 1D hydrogen atom has been used as a simplification of the 3D model\nin several theoretical and\nnumerical studies \\cite{JSS,DKS,LCO3}. It is then interesting that Cole\nand Cohen\n\\cite{CC} and Wong et al.\\ \\cite{W} have\nreported some experimental evidence for the 1D hydrogen atom. The\n``quasi-1D'' solutions $\\Xi_{n,0,0}$ are\nnatural candidates to describe such experimental observations and may\nbe relevant for an appropriate\njustification for the use of 1D simplifications. Lastly, the\n1D eigenvalues coincide with the\neigenvalues of the 3D hydrogen model.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm, height=8cm]{Fig03D.eps}\n\\caption{A boundary surface of $|\\Xi_{n=1,\\ell=0,0}(r,\\theta,\\phi)|^2$. }\n\\label{function3D}\n\\end{center}\n\\end{figure}\n\nNow the solutions $\\Xi_{n,\\ell\\ne0,0}$ have nonzero  total  angular\nmomentum while zero angular\nmomentum in  the $z$-direction, and the logarithm divergence for\n$\\theta=0$ and~$\\pi$ is also present for all\n$\\xi_{l,0}$, indicating that the $z$-axis plays a special role in the\nclassical trajectories analogy. So it\nis possible to interpret that $\\Xi_{n,\\ell\\ne0,0}$ is related to\ntwo-dimensional motions taking place in planes\ncontaining the $z$-axis, i.e., to the 2D hydrogen atom.\nFigure~\\ref{FigDoisDois} illustrates such\ninterpretation for\n$\\ell=2,m=0$. The\n2D hydrogen atom has also been considered in the literature (see\n\\cite{YGCWC,delCV,RR,VP,PP}\nand references therein), but its history is not as controversial as for\nthe 1D case.\n\n\nFinally, a word about $\\Xi_{n,\\ell,m\\ne0}$; since they do not belong to\nthe Hilbert space, based on the\nabove discussion and proceeding heuristically, it is tempting to interpret such solutions as the\n``contribution'' due to the classical\ntrajectories which come into collision with the nucleus, and the\nmathematical apparatus prudently avoids them\nexplicitly (maybe a mathematical consequence of the uncertainty principle).\n\n\n\\begin{figure}\n\\begin{center}\n\\vskip 10pt\n\\includegraphics[width=6cm, height=6cm]{FigQ22.eps}\n\\caption{$|\\xi_{2,0}(\\theta)|$ and $|\\Theta_{2,0}(\\theta)|$ (dashed) as\nfunction of $\\theta$.}\n\\label{FigDoisDois}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusions}\nOne is naturally inclined to presume that higher dimensional quantum\nmodels carry somehow lower\ndimensional dynamics, and the study of such simpler models could mimics\nimportant aspects of\nthe original one. Of course, in general the difficulties of performing\nsuch dimensional reductions are enormous, and usually carried out by\n``brute force.''\n\nThe case of the 3D hydrogen atom discussed in this\nwork has revealed a particular and\ninteresting framework: there are experimental evidence for the 1D\nhydrogen atom; the 3D hermitian model has just one self-adjoint extension,\nand its\n 3D eigenvalue equation\npresents formal solutions $\\Xi_{n,\\ell,0}$ that do not belong to the\ndomain of the corresponding Hamiltonian\noperator; in spite of being formal eigenvectors, these solutions live in\nthe underlying Hilbert space\nand present a component in the continuous subspace\nof the Hamiltonian so that, for an electron in such state, ionization can\ntake place;\n  these solutions have formally\nzero azimuthal angular momentum, with integrable probability\ndensities, and are concentrated around the $z$-axis,  indicating their 1D\nand 2D\ncharacter for $\\Xi_{n,0,0}$ and $\\Xi_{n,\\ell\\ne0,0}$, respectively.\nSumming up, such solutions are\nreminiscent of 1D and 2D classical trajectories and give a connection\nbetween the hydrogen atom in different\ndimensions.\n\n\nHow general is this framework? This is a fascinating open question, whose\nanswer could\neventually improve the interpretations.\n\nIn addition, notice that there is an attractive relation between the\ndimensional interpretations advocated in\nthis work and the mathematical formalism, which exhausts the possibilities for (pseudo-)eigenvectors. For\ngenuine 3D motion it presents eigenfunctions in the Hamiltonian\noperator domain; for 1D and 2D reminiscent trajectories it presents\neigenfunctions in the Hilbert space but not\nin the domain of the operator; and for those colliding trajectories (axial\ndivergence)  the\nformal eigenfunctions do not belong\nto the Hilbert space.\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection*{Acknowledgments} {\\small AL-C thanks FAPESP. CRdeO thanks\nthe partial\nsupport by CNPq}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec intro}\n\\thispagestyle{empty}\n\nThe local and global deformation theories of irreducible holomorphic symplectic manifolds enjoy many beautiful properties. For example, unobstructedness of deformations \\cite{Bo78,Ti,To89} and the local Torelli theorem \\cite{B} are at the origin of many results on symplectic manifolds. Among the highlights of the global theory are Huybrechts' surjectivity of the period map \\cite{Huy99} and Verbitsky's global Torelli theorem \\cite{V13} (see for example Markman's survey article \\cite{markmantor} and Huybrechts' Bourbaki talk \\cite{huybourbaki}). \nVerbitsky's result has since paved the way for many important developments, with applications to a wide variety of questons such as:  birational boundedness \\cite{Ch16}, lattice polarized mirror symmetry \\cite{Ca16}, algebraic cycles \\cite{CP}, hyperbolicity questions \\cite{KLV} and many more.  Recent progress in MMP and interest in singular symplectic varieties \\cite{GKP} has made it apparent that a global deformation theory of singular symplectic varieties would be equally valuable.\n\nIn the present article we initiate a systematic study of the moduli theory of singular symplectic varieties. \nWe prove general results concerning their deformation theory and building on this we develop a global moduli theory for locally trivial families of singular irreducible symplectic varieties. \nFor the purpose of this article it is convenient to define an \\emph{(irreducible) symplectic variety} $X$ to be a compact normal K\\\"ahler space admitting a resolution $\\pi:Y\\to X$ by an (irreducible) holomorphic symplectic manifold $Y$, see section \\ref{sec symplektisch}. In this situation we refer to $\\pi$ as an \\emph{(irreducible) symplectic resolution}.  In particular, if the varieties are projective, then our (irreducible) symplectic varieties are symplectic varieties in the sense of Beauville \\cite{Be} but our notion is of course much more restrictive.\n\nBy work of Namikawa \\cite{Na01,Na06}, a projective variety $X$ admitting a resolution $\\pi:Y\\to X$ by a symplectic manifold $Y$ also has unobstructed deformations, and the natural map ${\\operatorname{Def}}(Y)\\to{\\operatorname{Def}}(X)$ is finite.  However, $\\pi$ deforms generically to an isomorphism, and as a consequence the only natural period map is that of the resolution $Y$.\nFrom a Hodge-theoretic perspective it is therefore more natural to consider the \\emph{locally trivial} deformations of $X$ as in this case the pure weight two Hodge structure on $H^2(X,{\\mathbb Z})$ varies in a local system, and the resulting theory is very closely analogous to the smooth situation. \nOur first result is the following\n\n\\begin{theorem}[see Proposition \\ref{prop defo}]\\label{theorem def smooth} Let $\\pi:Y\\to X$ be an irreducible symplectic resolution and $N=N_1(Y\/X)\\subset H_2(Y,{\\mathbb Z})$ the group of 1-cycles contracted by $\\pi$.  The base space ${\\operatorname{Def}}^{\\rm{lt}}(X)$ of the universal locally trivial deformation  is smooth, and there is a diagram\n\\[\n\\xymatrix{\n{\\mathscr Y} \\ar[d]\\ar[r]& {\\mathscr X} \\ar[d]\\\\\n{\\operatorname{Def}}(Y,N) \\ar[r]^{\\cong} & {\\operatorname{Def}}^{\\rm{lt}}(X) \\\\\n}\n\\]\nwhere ${\\mathscr X}\\to {\\operatorname{Def}}^{\\rm{lt}}(X)$ is the universal locally trivial deformation of $X$, ${\\mathscr Y}\\to{\\operatorname{Def}}(Y,N)\\subset{\\operatorname{Def}}(Y)$ is the restriction of the universal deformation of $Y$ to the closed subspace along which $N$ remains algebraic, and ${\\mathscr Y}\\to{\\mathscr X}$ specializes to $\\pi$.\n\\end{theorem}\n The case of a divisorial contraction $\\pi:Y\\to X$ of a projective symplectic variety was treated by Pacienza and the second author in \\cite{LP}, where it was shown that locally trivial deformations of $X$ correspond to deformations of $Y$ such that all irreducible components of the exceptional divisor ${\\operatorname{Exc}}(\\pi)$ of $\\pi$ deform along. The space ${\\operatorname{Def}}^{\\rm{lt}}(X)$ of locally trivial deformations of $X$ is smooth of dimension $h^{1,1}(X)=h^{1,1}(Y)-m$ where $m$ is the number of irreducible components of ${\\operatorname{Exc}}(\\pi)$.  This description is equivalent to that of the theorem, since the Beauville--Bogomolov--Fujiki form $q$ on $H^2(Y,{\\mathbb Z})$ yields an isomorphism $\\tilde q: H^2(Y,{\\mathbb Q})\\xrightarrow{\\cong} H_2(Y,{\\mathbb Q})$ identifying the subspace of $H^2(Y,{\\mathbb Q})$ spanned by ${\\operatorname{Exc}}(\\pi)$ with $N_{\\mathbb Q}$.  In the case where $\\pi$ is a small contraction, $X$ is not ${\\mathbb Q}$-factorial, but we can still recast the description in the theorem in terms of line bundles:  under $\\tilde q$, the Hodge classes in the orthogonal $N^\\perp\\subset H^2(Y,{\\mathbb Q})$ are the ${\\mathbb Q}$-line bundles on $Y$ that vanish on $N$ and can therefore be pushed forward to ${\\mathbb Q}$-line bundles on $X$.\n\nAn important theorem of Huybrechts \\cite[Theorem 2.5]{Huy} shows that birational\\footnote{We will use the term birational instead of bimeromorphic also for K\\\"ahler varieties.} irreducible holomorphic symplectic manifolds are deformation equivalent.  Of course, birational irreducible symplectic varieties are not necessarily locally trivially deformation equivalent, and the correct analog of Huybrechts' theorem is the following:  \n\\begin{theorem}[See Theorem \\ref{theorem huybrechts}]\\label{intro theorem huybrechts}\nLet $\\pi:{Y}\\to X$ and $\\pi':Y'\\to X'$ be projective irreducible symplectic resolutions of projective irreducible symplectic varieties $X,X'$, and\nlet $\\phi:X\\dashrightarrow X'$ be a birational map.  Then the following are equivalent:\n\\begin{enumerate}\n \\item There is an isomorphism $\\varphi:{\\operatorname{Def}}^{\\rm{lt}}(X)\\to {\\operatorname{Def}}^{\\rm{lt}}(X')$ such that for each $t\\in {\\operatorname{Def}}^{\\rm{lt}}(X)$ we have a birational map $\\phi_t: {\\mathscr X}_t \\dashrightarrow {\\mathscr X}'_{\\varphi(t)}$ which is an isomorphism in codimension one. For general $t \\in {\\operatorname{Def}}^{\\rm{lt}}(X)$, the map $\\phi_{t}$ is an isomorphism. In particular, $X$ and $X'$ are locally trivial deformations of one another.\n\\item The map $\\tilde\\phi^*: H^2({Y}',{\\mathbb C}) \\to H^2({Y},{\\mathbb C})$ induced by the birational map $\\tilde\\phi:{Y} \\dashrightarrow {Y}'$ is an isomorphism that sends $H^2(X',{\\mathbb C})$ isomorphically to $H^2(X,{\\mathbb C})$. \n\\end{enumerate}\n\\end{theorem} \nFor a criterion that is more intrinsic to $X$, let ${\\operatorname{Cl}}(X)$ denote the group of Weil divisors modulo linear equivalence.\n\\begin{corollary}\\label{intro cor div}\nSuppose $\\phi:X\\dashrightarrow X'$ is a birational map of irreducible projective symplectic varieties which is an isomorphism in codimension one.  If $\\phi_*: {\\operatorname{Cl}}(X)_{\\mathbb Q} \\to {\\operatorname{Cl}}(X')_{\\mathbb Q}$ and its inverse send ${\\mathbb Q}$-line bundles to ${\\mathbb Q}$-line bundles, then $X$ and $X'$ are locally trivially deformation equivalent. \n\\end{corollary}\n\nFixing a lattice $\\Lambda$, there is a natural locally trivial $\\Lambda$-marked moduli space $\\mathfrak{M}^{\\rm{lt}}_\\Lambda$ obtained by gluing the universal locally trivial deformation spaces together, and we also have an associated notion of parallel transport operator.  Further, there is a period map $P:\\mathfrak{M}^{\\rm{lt}}_\\Lambda\\to\\Omega_\\Lambda$ to the associated period domain $\\Omega_\\Lambda$ that is a local isomorphism.  The two equivalent conditions in Theorem \\ref{intro theorem huybrechts} also imply -- as for smooth varieties -- that $X$ and $X'$ are inseparable in $\\mathfrak{M}^{\\rm{lt}}_\\Lambda$, see Theorem \\ref{theorem huybrechts strong}.  \nAs a converse, we have a version of the global Torelli theorem, see Theorem \\ref{theorem global torelli}. The following is an analog of Markman's Hodge theoretic version \\cite[Theorem 1.3]{markmantor} of the global Torelli theorem.\n\n\\begin{theorem}[Global Torelli Theorem]\\label{intro global}\nLet $X$, $X'$ be irreducible symplectic varieties and let $f:H^2(X,{\\mathbb Z}) \\to H^2(X',{\\mathbb Z})$ be a locally trivial parallel transport operator which is a Hodge isometry. Then there is a birational map $\\phi:X \\dashrightarrow X'$ which induces\\footnote{By this we mean that the induced birational map on irreducible symplectic resolutions induces an isomorphism on cohomology which identifies $H^2(X,{\\mathbb Z})$ with $H^2(X',{\\mathbb Z})$.} a Hodge isometry $\\phi_*:H^2(X,{\\mathbb Z})\\to H^2(X',{\\mathbb Z})$.\n\\end{theorem}\n\nAs in the smooth case, this has an equivalent formulation saying that every connected component of the marked moduli space maps injectively to the period domain, points in one fiber are birational, and this birational map induces an isomorphism on the second cohomology, see Corollary \\ref{corollary global torelli}. Moreover, we have analogs of Huybrechts' surjectivity of the period map \\cite{Huy99} as well as of a well-known theorem of Sullivan \\cite{Sullivan} in the smooth case, see Corollary \\ref{corollary monodromy group} and Theorem \\ref{theorem almost surjectivity}. Put together, we have\n\n\\begin{theorem}\\label{intro theorem torelli}Let $X$ be an irreducible symplectic variety with $b_2(X)>3$ admitting a symplectic resolution $\\pi:Y\\to X$ with $b_2(Y) > 5$.  Let $\\mathfrak{N}^{\\rm{lt}}_\\Lambda\\subset \\mathfrak{M}^{\\rm{lt}}_\\Lambda$ be a connected component containing $X$ (after choosing a marking).  Then\n\\begin{enumerate}\n\\item The period map $P:\\mathfrak{N}^{\\rm{lt}}_\\Lambda\\to\\Omega_\\Lambda$ is surjective, and the points in any fiber are pairwise nonseparated. Moreover, the nonseparated points precisely correspond to birational symplectic varieties such that the birational map induces a Hodge isometry on the second cohomology lattices.\n\\item   The locally trivial weight two monodromy group $\\operatorname{Mon}^2(X)^{\\rm{lt}}$ is a finite index subgroup of $\\O(H^2(X,{\\mathbb Z}))$, equal to the restriction of the subgroup of $\\operatorname{Mon}^2(Y)$ stabilizing $\\pi^*H^2(X,{\\mathbb Z})$. \n\\end{enumerate}\n\\end{theorem}\n\nThe analogous results in the smooth case heavily rely on the existence of a hyperk\\\"ahler metric, the theory of twistor lines, and deformations of complex structures. This presents a major difficulty for singular varieties as the aforementioned techniques are not available or much less understood as in the smooth case. \nWe therefore deduce the largeness of the image of the period map by density results built on Ratner's theorem (as first explored in this context by Verbitsky \\cite{V15}), and this is responsible for the numerical conditions on $b_2(X)$.  To the best of our knowledge this is the first general result in this direction which makes a statement about \\emph{large} deformations of singular symplectic varieties.  Note that $X$ with $b_2(X)=3$ are locally trivially rigid in the sense that their only locally trivial deformation is what should be the (unique) twistor deformation, and unlike in the smooth case we can give examples (see Remark \\ref{remark rigid}). On the other hand, all known smooth deformation types have $b_2(Y)>5$. \n\n \nThe locally trivial monodromy group interacts in an interesting way with the monodromy group of the resolution, permuting the different possible resolutions.  We deduce from the second part of Theorem \\ref{intro theorem torelli} the following classification of the locally trivial deformation types of contractions in terms of the monodromy group of the resolution (see Corollary \\ref{cor mon det}):\n  \\begin{corollary}\\label{intro theorem mono class}  With the above restrictions, the locally trivial deformation type of $X$ is uniquely determined by the $\\operatorname{Mon}^2(Y)$ orbit of $N_1(Y\/X)$ in $H_2(Y,{\\mathbb Z})$.  \n\\end{corollary}\n\nMuch more can be said when $X$ admits a resolution deformation equivalent to a Hilbert scheme of points on a $K3$ surface.  The possible projective contractions $M\\to X$ of a moduli space of sheaves $M$ on a $K3$ surface is completely described by wall-crossing in the space of Bridgeland stability conditions by work of Bayer--Macr\\`i \\cite{BM}; \nTheorem \\ref{theorem def smooth} combined with density results then implies that no new singularities occur in the general $X$ (see Proposition \\ref{proposition def to bridge}):\n\n\\begin{theorem}Let $\\pi:Y\\to X$ be an irreducible symplectic resolution where $Y$ is a $K3^{[n]}$-type manifold.  Provided $b_2(X)>3$, $X$ is locally trivially deformation equivalent to a wall-crossing contraction of a moduli space of Bridgeland stable objects on a K3 surface.  \n\\end{theorem}\n\nExtremal contractions are particularly amenable to the lattice theory involved, and we for example have the following generalization of a beautiful result of Arbarello and Sacc\\`a \\cite{AS}:\n\n\\begin{theorem}\\label{intro theorem nakajima} Let $\\pi:Y\\to X$ be an irreducible symplectic resolution where $Y$ is a $K3^{[n]}$-type manifold, and assume ${\\rm rk}\\,N_1(Y\/X)=1$.  Then for any closed point $x\\in X$ the analytic germ $(X,x)$ is isomorphic to that of a Nakajima quiver variety.\n\\end{theorem}\n\nAs another nice application, we are able to rule out the existence of certain contractions on $K3^{[n]}$-type varieties:\n\n\\begin{theorem} Let $\\pi:Y\\to X$ be a divisorial contraction of a $K3^{[n]}$-type manifold $Y$, and assume ${\\rm rk}\\,N_1(Y\/X)=1$. Then for a generic closed point $x\\in X^{\\operatorname{sing}}$ the analytic germ $(X,x)$ is isomorphic to $(S,0)\\times ({\\mathbb C}^{2n-2},0)$ where $(S,0)$ is an $A_1$-surface singularity.\n\\end{theorem}\n\nThe surprising fact is that relative Picard rank one contractions whose generic singularity is transversally an $A_2$-surface singularity \\emph{do not occur} on irreducible symplectic manifolds while on other arbitrary symplectic manifolds they do, see Corollary \\ref{corollary a2} and the discussion thereafter.\n\nWe expect most of our general results to apply to symplectic varieties not necessarily admitting a symplectic resolution by using a ${\\mathbb Q}$-factorial terminalization in place of the resolution but we do not pursue that level of generality here.  We do note however that this would fit very nicely with another result of Namikawa \\cite{Na06} that every flat deformation of a ${\\mathbb Q}$-factorial terminal projective symplectic variety is locally trivial.  As for the applications, we only restrict to $K3^{[n]}$-type varieties for simplicity, and our theory should yield similar results for the Kummer and O'Grady types.  These directions are the topic of a forthcoming article.\n\n\\subsection*{Outline}\nIn sections \\ref{sec hodge} and \\ref{sec symplektisch} we explain some basic facts about irreducible symplectic varieties and their Hodge structures. We recall that the Hodge structure on the second cohomology of an irreducible symplectic variety is always pure and prove the degeneration of Hodge-de Rham spectral sequence for singular symplectic varieties in a suitable range. We also examine the Mumford--Tate group of a general irreducible symplectic manifold. \n\nThe locally trivial deformation theory of irreducible symplectic varieties is studied in section \\ref{sec defo}. This is one of the two technical centerpieces of the present paper. All our moduli theory builds on this in an essential way. The main results are the proof of smoothness of the Kuranishi space, the comparison of deformations of a singular symplectic variety and its resolution, the local Torelli theorem, and the analog of Huybrechts' theorem. We also include some remarks and applications to the question of existence of algebraically coisotropic subvarieties. In section \\ref{sec monodromy} we introduce and investigate marked moduli spaces of locally trivial families of irreducible symplectic varieties. We study the relation to moduli spaces of the resolution through compatibly marked moduli spaces of irreducible symplectic resolutions as well as various associated monodromy groups. The analysis of how these spaces and their monodromy groups are connected is the second main ingredient of this work. In this section we make use of Verbitsky's ergodicity of complex structures and Amerik-Verbitsky's concept of MBM classes.\n\nIn section \\ref{sec k3type} we give applications to $K3^{[n]}$-type varieties. We essentially make use of the description of the birational geometry of Bridgeland moduli spaces of stable objects by Bayer and Macr\\`i. In principle, our methods allow to extend any result on singularities that can be proven for contractions of Bridgeland moduli spaces to arbitrary $K3^{[n]}$-type varieties. The above-mentioned generalization of Arbarello--Sacc\\`a's result is proven here.\n\n\\subsection*{Acknowledgments.} We would like to thank J\\'anos Koll\\'ar for helpful discussions on birational contractions. We benefited from discussions, remarks, emails of Chiara Camere, Daniel Greb, Klaus Hulek, Manfred Lehn, Giovanni Mongardi, Gianluca Pacienza, Jacob Tsimerman, Thomas Peternell, Stefan Kebekus, and Stefano Urbinati.\n\nChristian Lehn was supported by the DFG through the research grants Le 3093\/2-1 and  Le 3093\/3-1.\n\n\n\n\n\\section{Hodge structure for rational singularities}\\label{sec hodge}\n\nWe establish some basic facts about the Hodge structure on low degree cohomology groups of varieties with rational singularities. The following lemma is well-known, we include it for the reader's convenience.\n\n\\begin{lemma}\\label{lemma peternell}\nLet $\\pi:{Y}\\to X$ be a resolution of singularities of a compact normal variety $X$ with rational singularities. Then the sequence\n\\begin{equation}\\label{eq leray}\n 0 \\to H^2(X,{\\mathbb Z}) \\to[\\pi^*] H^2({Y},{\\mathbb Z}) \\to[\\varphi] H^0(R^2\\pi_*{\\mathbb Z})\n\\end{equation}\nis exact. In particular, if ${Y}$ is of class ${\\mathscr C}$, then $H^2(X,{\\mathbb Z})$ carries a pure Hodge structure. Moreover, the restriction of $\\pi^*$ to the transcendental lattice is an isomorphism over ${\\mathbb Q}$ and $\\pi^*\\operatorname{NS}(X)_{\\mathbb Q}$ is the subspace of $H^{1,1}({Y},{\\mathbb Q})$ of all ${\\mathbb Q}$-line bundles on ${Y}$ that vanish on the $\\pi$-exceptional curves.\n\\end{lemma}\n\\begin{proof}\nThe exponential sequence and rationality of singularities imply that $R^1\\pi_*{\\mathbb Z}=0$. Therefore, the sequence \\eqref{eq leray}, which comes from the Leray spectral sequence, is exact and by strictness of morphisms of Hodge structures, $H^2(X,{\\mathbb Z})$ carries a pure Hodge structure. \n\nThe last two statements follow from \\cite[(12.1.3) Theorem]{KM}.\n\\end{proof}\nThe proof shows that \\eqref{eq leray} is even defined and exact over ${\\mathbb Z}$ but we will not need this in the sequel.\n\nWe will investigate in how far the de Rham complex on a resolution or the one on the smooth part of a singular variety can be used to compute the singular cohomology and the Hodge decomposition on $X$. \n\\begin{lemma}\\label{lemma hodge}\nLet $X$ be a normal compact K\\\"ahler variety. Then the following hold:\n\\begin{enumerate}\n\t\\item\\label{lemma hodge item one} Let $\\pi:Y \\to X$ be a resolution of singularities by a compact K\\\"ahler manifold $Y$. If $X$ has rational singularities, then for all $k\\leq 2$ the canonical map $H^k(X,{\\mathbb C})\\to {\\mathbb H}^k(X,\\pi_*\\Omega_Y^\\bullet)$ is an isomorphism and the Hodge-de Rham spectral sequence\n\\begin{equation}\\label{eq spec seq}\n E^{p,q}_1=H^q(\\pi_*\\Omega_Y^p) \\Rightarrow {\\mathbb H}^{p+q}(X,\\pi_*\\Omega_Y^\\bullet)\n\\end{equation}\ndegenerates on $E_1$ in the region where $p+q \\leq 2$.\n\\item\\label{lemma hodge item two} Let us denote by $j:U=X^{\\operatorname{reg}}\\to X$ the inclusion of the regular part. If $X$ is projective with log terminal singularities, then for all $k\\leq 2$ the canonical map $H^k(X,{\\mathbb C})\\to {\\mathbb H}^k(X,j_*\\Omega_U^\\bullet)$ is an isomorphism and the Hodge-de Rham spectral sequence\n\\begin{equation}\\label{eq spec seq projective}\n E^{p,q}_1=H^q(j_*\\Omega_U^p) \\Rightarrow {\\mathbb H}^{p+q}(X,j_*\\Omega_U^\\bullet)\n\\end{equation}\ndegenerates on $E_1$ in the region where $p+q \\leq 2$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nWe may assume that $\\pi:{Y} \\to X$ is a log resolution  of singularities which is an isomorphism over $U$, i.e., $\\pi^{-1}(X^{\\operatorname{sing}})$ is a divisor with simple normal crossings. \nLog terminal singularities are rational by \\cite[Theorem 5.22]{KM}. As a consequence of \\cite[Theorem 1.4]{GKKP} we have $j_*\\Omega_U^\\bullet=\\pi_*\\Omega_{Y}^\\bullet$ so that \\eqref{lemma hodge item two} is a consequence of \\eqref{lemma hodge item one}, and it suffices to prove the first statement.\nWe obtain a commuting diagram\n\\begin{equation}\\label{eq hodge diag}\n\\xymatrix{\nH^k(X,{\\mathbb C}) \\ar[r]\\ar[d]^{\\pi^*} & {\\mathbb H}^k(\\pi_*\\Omega_Y^\\bullet)\\ar[d]^{\\psi}\\\\\nH^k({Y},{\\mathbb C}) \\ar[r] & {\\mathbb H}^k(\\Omega_{Y}^\\bullet)\\\\\n}\n\\end{equation}\nwhere the right vertical morphism comes from the canonical morphism of complexes $\\pi_*\\Omega_{Y}^\\bullet \\to R\\pi_*\\Omega_{Y}^\\bullet$. The lower horizontal map is an isomorphism by Grothendieck's theorem and $\\pi^*$ is injective by Lemma \\ref{lemma peternell}. \n We will show that for $k\\leq 2$ the map $\\psi$ is injective and its codimension is the same as for $\\pi^*$. We may and will assume that $X_0={Y}$ is the beginning of a semi-simplicial resolution $\\begin{xy}\\xymatrix@C=1em{\n\\ldots  \\ar[r]\\ar@<0.5ex>[r]\\ar@<-0.5ex>[r] & X_{1} \\ar@<0.5ex>[r]\\ar@<-0.5ex>[r] & X_0 \\ar[r]^\\varepsilon& X, \\\\}\\end{xy}$ which serves to calculate the Hodge structure on $H^k(X,{\\mathbb C})$, see \\cite[Chapter 5]{PS}. With this assumption, all morphisms in \\eqref{eq hodge diag} become filtered morphisms where on the left we speak about the Hodge filtrations and on the right about the filtrations on the cohomology of a complex $K^\\bullet$ given by \n$F^p{\\mathbb H}^k(K^\\bullet):=\\operatorname{im}({\\mathbb H}^k(K^{\\geq p})\\to {\\mathbb H}^k(K^\\bullet))$. All morphisms but $H^k(X,{\\mathbb C}) \\to {\\mathbb H}^k(j_*\\Omega_U^\\bullet)$ are clearly filtered and this last one is filtered because it comes from a morphism of filtered complexes\n\\[\n {\\mathbb C}_X \\to[\\textrm{qis}] {\\mathfrak s}\\left(R\\varepsilon_*\\Omega_{X_0}^\\bullet \\to R\\varepsilon_*\\Omega_{X_1}^\\bullet \\to \\ldots\\right) \\ot j_*\\Omega_U^\\bullet\n\\]\nwhere ${\\mathfrak s}(\\cdot)$ stands for the associated single complex.\n\nThe injectivity part of the claim will follow from the statement about degeneration of the spectral sequence.\nDegeneration on $E_1$ follows as the $E_1$-level of the spectral sequence \\eqref{eq spec seq} embeds into the $E_1$-level of the spectral sequence of the complex $\\Omega_{Y}^\\bullet$, which degenerates on $E_1$ by Hodge theory. \n\nWe have $H^k(X,{\\mathcal O}_X){\\ \\cong\\ } H^k({Y},{\\mathcal O}_{Y})$ by rationality of singularities for all $k \\in {\\mathbb N}_0$.\nThe inclusion $H^1(X,\\pi_*\\Omega_{Y}) \\subset H^1({Y},\\Omega_{Y})$ is deduced from the Leray spectral sequence. Thus, the spectral sequence for $\\pi_*\\Omega_Y^\\bullet$ degenerates in the realm $p+q \\leq 2$ and we obtain commutative diagrams\n\\begin{equation}\\label{eq hpq diag}\n\\xymatrix{\nH^{p,q}(X) \\ar@{^(->}[r]\\ar@{^(->}[d]^{\\pi^*} & H^q(\\pi_*\\Omega_Y^p)\\ar@{^(->}[d]^{\\psi^{p,q}}\\\\\nH^{p,q}({Y}) \\ar[r]^{\\ \\cong\\ } & H^q(\\Omega_{Y}^p)\\\\\n}\n\\end{equation}\nfor all such $p$ and $q$.\n\nIt remains to see that $H^1(X,\\pi_*\\Omega_{Y}) \\subset H^1({Y},\\Omega_{Y})$ has codimension equal to $m:=\\dim N_1(Y\/X)=\\dim H^{1,1}({Y})-\\dim H^{1,1}(X)$, where $N_1(Y\/X)$ is the kernel of the surjection $N_1(Y) \\to N_1(X)$, see \\cite[(12.1.5)]{KM} and note that log terminal singularities are rational.\nBut $\\operatorname{coker}\\psi^{1,1}$ is the image of $H^1(\\Omega_{Y}) \\to H^0(R^1\\pi_*\\Omega_{Y})$. So let $C_1, \\ldots, C_m$ be curves in ${Y}$ contracted to a point under $\\pi$ such that their classes form a basis of $N_1({Y}\/X)$ and let $L_1, \\ldots, L_m$ be line bundles on ${Y}$ such that their Chern classes $\\xi_i:={\\rm c}_1(L_i)\\in H^1(\\Omega_{Y})$ define linearly independent functionals on $N_1({Y}\/X)$. Choose irreducible components $F_1, \\ldots, F_m$ of fibers of $\\pi$ such that $C_i \\subset Y_i$ for all $i$. Take resolutions of singularities  $\\nu_i:\\tilde F_i \\to F_i$ and curves $\\tilde C_i \\subset \\tilde F_i$ such that ${\\nu_i}_*\\tilde C_i = C_i$ for all $i$. If we denote $F:=\\coprod{\\tilde F_i}$ and by $\\nu : F\\to {Y}$ the composition of the resolutions with the inclusion, then by the projection formula $\\nu^*\\xi_i.\\tilde C_j= \\xi_i.C_j$ so that the $\\nu^*\\xi_i$ are still linearly independent. This implies that the $\\xi_i$ are mapped to an $m$-dimensional subspace of $H^1(F,\\Omega_F)$ under the composition $H^1(\\Omega_{Y}) \\to H^0(R^1\\pi_*\\Omega_{Y})\\to H^1(F,\\Omega_F)$. In particular, ${\\rm rk} \\left(H^1(\\Omega_{Y}) \\to H^0(R^1\\pi_*\\Omega_{Y})\\right) \\geq m$, which completes the proof of the lemma. \n\\end{proof}\nWe immediately deduce\n\\begin{corollary}\\label{corollary hodge}\nLet $X$ be a normal compact K\\\"ahler variety, let $\\pi:Y \\to X$ be a resolution of singularities by a compact K\\\"ahler manifold $Y$, and denote by $j:U=X^{\\operatorname{reg}}\\to X$ the inclusion of the regular part. \n\\begin{enumerate}\n\t\\item If $X$ has rational singularities and $k,p+q \\leq 2$ then  \\begin{enumerate}\n \\item $H^{p,q}(X){\\ \\cong\\ } H^q(X,\\pi_*\\Omega_Y^p)$, \n \\item $H^k(X,{\\mathbb C}){\\ \\cong\\ } {\\mathbb H}^k(X,\\pi_*\\Omega_Y^\\bullet)$, and \n \\item $F^pH^k(X,{\\mathbb C}){\\ \\cong\\ } {\\mathbb H}^k(X,\\pi_*\\Omega_Y^{\\geq k})$.\n \\end{enumerate}\n\\item If $X$ is projective with log terminal singularities and $k,p+q \\leq 2$ then\n \\begin{enumerate}\n \\item $H^{p,q}(X){\\ \\cong\\ } H^q(X,j_*\\Omega_U^p)$, \n \\item $H^k(X,{\\mathbb C}){\\ \\cong\\ } {\\mathbb H}^k(X,j_*\\Omega_U^\\bullet)$, and \n \\item $F^pH^k(X,{\\mathbb C}){\\ \\cong\\ } {\\mathbb H}^k(X,j_*\\Omega_U^{\\geq k})$.\n \\end{enumerate}\n\\end{enumerate} \\qed\n\\end{corollary}\nNow we turn to the relative situation. The following result follows from the absolute case by homological algebra.\n\\begin{lemma}\\label{lemma hodge relativ}\nLet $X_0$ be a normal projective variety with log terminal singularities, let $f: X \\to S$ be a flat deformation of $X_0$ over a local Artinian base scheme $S$ of finite type over ${\\mathbb C}$, let $U_0 \\subset X_0$ be the regular locus and let $U\\to S$ be the induced deformation of $U_0$. Suppose that $j_*\\Omega^\\bullet_{U\/S}$ is flat over $S$ where $j:U{\\, \\hookrightarrow\\,} X$ is the inclusion. Then, for all $k\\leq 2$ the canonical map $H^k(X,f^{-1}{\\mathcal O}_S)\\to {\\mathbb H}^k(X,j_*\\Omega_{U\/S}^\\bullet)$ is an isomorphism and the Hodge-de Rham spectral sequence\n\\begin{equation}\\label{eq spec seq relativ}\n E^{p,q}_1=H^q(j_*\\Omega_{U\/S}^p) \\Rightarrow {\\mathbb H}^{p+q}(X,j_*\\Omega_{U\/S}^\\bullet)\n\\end{equation}\ndegenerates on $E_1$ in the region where $p+q \\leq 2$. Moreover, the ${\\mathcal O}_S$-modules $H^q(X,j_*\\Omega_{U\/S}^p)$ are free for $p+q \\leq 2$ and compatible with arbitrary base change.\n\nThe same holds true if $X \\to S$ is a locally trivial deformation of a normal K\\\"ahler variety $X_0$ with rational singularities such that for some resolution $\\pi:Y_0\\to X_0$ we have $\\pi_*\\Omega_{Y_0} = j_*\\Omega_{U_0}$.\n\\end{lemma}\n\\begin{proof}\nThe proof is the same in the projective and in the K\\\"ahler case.\nPut $R:=\\Gamma(S,{\\mathcal O}_S)$. The differentials on all $E_1^{p,q}$ with $p+q=n$ will be zero if and only if $\\sum_{p+q=n}\\lg_R E_1^{p,q}=\\lg_R {\\mathbb H}^n(X,j_*\\Omega_{U\/S}^\\bullet)$ where $\\lg_R$ denotes the length as an $R$-module. Note that both sides are finite.\n\nFlatness of $j_*\\Omega^\\bullet_{U\/S}$ entails that there is a bounded below complex $L^\\bullet$ of free $R$-modules such that there is an isomorphism $H^q(X,j_*\\Omega_{U\/S}^p \\otimes f^*M) {\\ \\cong\\ } H^q(L\\otimes_R M)$ which is functorial in the $R$-module $M$, see \\cite[Th\\'eor\\`eme (6.10.5)]{EGAIII2} in the projective case and \\cite[Ch~3, Th\\'eor\\`eme 4.1]{BS} in the K\\\"ahler case.\n By \\cite[(3.5.1)]{Deligne}, this implies that $\\lg_R H^q(X,j_*\\Omega_{U\/S}^p) \\leq \\lg R \\cdot \\lg_R H^q(X_0,j_*\\Omega_{U_0}^p)$ and equality holds if and only if $H^q(X,j_*\\Omega_{U\/S}^p)$ is $R$-free.\n \nFor $n\\leq 2$ we have \n\\begin{align*}\n\\lg_R {\\mathbb H}^n(X,j_*\\Omega_{U\/S}^\\bullet) &\\leq  \\sum_{p+q=n}\\lg_R H^q(X,j_*\\Omega_{U\/S}^p)\\\\\n&\\leq  \\lg R \\cdot \\sum_{p+q=n}\\dim_{\\mathbb C} H^q(X_0,j_*\\Omega_{U_0}^p) \\\\\n&= \\lg R \\cdot H^n(X_0,{\\mathbb C}) \n\\end{align*}\nwhere the first inequality is the existence of the spectral sequence, the second one was explained just before and the equality is the degeneracy of the spectral sequence for $X_0$, see Lemma \\ref{lemma hodge}.\nMoreover, ${\\mathbb H}^n(X,j_*\\Omega_{U\/S}^\\bullet) = H^n(X,f^{-1}{\\mathcal O}_S) =  H^n(X,\\ul{R}_X)$ for $n \\leq 2$ where $\\ul{R}_X$ denotes the constant sheaf $R$ as can be seen by induction over the length of $R$ and the corresponding statement for $X_0$ from Lemma \\ref{lemma hodge}.\nThe base change property follows from the local freeness by \\cite[(7.8.5)]{EGAIII2} in the projective case and by \\cite[Ch~3, Corollaire 3.10]{BS} in the K\\\"ahler case.\n\\end{proof}\n\\section{Symplectic varieties}\\label{sec symplektisch}\n\nRecall that an \\emph{irreducible symplectic manifold} is a simply connected compact K\\\"ahler manifold ${Y}$ such that $H^0({Y},\\Omega_{Y}^2)={\\mathbb C} \\sigma$ for a holomorphic symplectic $2$-form $\\sigma$. There is a nondegenerate quadratic form $q_{Y}:H^2({Y},{\\mathbb Z})\\to {\\mathbb Z}$, the \\emph{Beauville--Bogomolov--Fujiki form}, whose associated bilinear form gives an injection $\\tilde q:H^2({Y},{\\mathbb Z}){\\, \\hookrightarrow\\,} H_2({Y},{\\mathbb Z})$ which becomes an isomorphism over ${\\mathbb Q}$.\nThroughout this article, by \\emph{symplectic variety} we mean a normal compact K\\\"ahler\\footnote{K\\\"ahler spaces were first introduced by Grauert \\cite[\\S 3.3]{Gra}; their study has been continued by Moishezon \\cite{Moi}.} space $X$ such that its smooth part admits a holomorphic symplectic $2$-form that extends holomorphically to one (and hence to any) resolution of singularities \\emph{and} such that $X$ admits a resolution by a symplectic manifold. \nNote that the existence of a symplectic resolution is not required in \\cite{Be}.  We say that $X$ is an irreducible symplectic variety if the resolution is irreducible symplectic. An \\emph{irreducible symplectic resolution} is a resolution of a symplectic variety in our sense by an irreducible symplectic manifold.\n\nSuppose that $\\pi:{Y} \\to X$ is a proper birational morphism from an irreducible symplectic manifold ${Y}$ to a normal K\\\"ahler variety $X$ which is then automatically a symplectic variety. From \\cite[(12.1.3) Theorem]{KM} it follows that we have a short exact sequence \n\\[\n0\\to H_2(Y\/X,{\\mathbb Q}) \\to H_2(Y,{\\mathbb Q}) \\to H_2(X,{\\mathbb Q})\\to 0. \n\\]\n\nMoreover, by loc. cit. $H_2(Y\/X,{\\mathbb Q})$ is generated by algebraic cycles and therefore it coincides with $N_1(Y\/X)_{\\mathbb Q}$. Here, $N_1(Y\/X) \\subset N_1(Y)$ is defined as the kernel of the push forward map $\\pi_*:N_1(Y) \\to N_1(X)$.\n\n\\begin{lemma}\\label{lemma symplektisch}\nLet $\\pi:{Y} \\to X$ be an irreducible symplectic resolution. Then $\\pi^*:H^2(X,{\\mathbb C}) \\to H^2({Y},{\\mathbb C})$ is injective and the restriction of $q$ to $H^2(X,{\\mathbb C})$ is nondegenerate. \nThe $q$-orthogonal complement to  $H^2(X,{\\mathbb C})$ in $H^2({Y},{\\mathbb C})$ is $N:=\\tilde q^{-1}(N_1({Y}\/X))$. In particular, the inclusion $H^2(X,{\\mathbb C})\\subset H^2({Y},{\\mathbb C})$ is an equality on the transcendental part $H^2(X,{\\mathbb C})_{\\rm tr}=H^2({Y},{\\mathbb C})_{\\rm tr}$. Moreover, $N$ is negative definite with respect to the Beauville--Bogomolov--Fujiki form.\n\\end{lemma}\n\\begin{proof}\nInjectivity follows from Lemma \\ref{lemma peternell}. To see that $N$ is $q$-orthogonal to $H^2(X,{\\mathbb C})$ we argue as follows: as the bilinear form $q$ is a morphism of Hodge structures and $N \\subset H^{1,1}({Y})$ is orthogonal to the $(2,0)$-part of $H^2({Y},{\\mathbb C})$ and hence to $H^2({Y},{\\mathbb C})_{\\rm tr}$ (which is by the way isomorphic to $H^2(X,{\\mathbb C})_{\\rm tr}$ by Lemma \\ref{lemma hodge}). So it suffices to prove that $N$ is $q$-orthogonal to $N^1(X) \\subset H^2(X,{\\mathbb Q}) \\subset H^2({Y},{\\mathbb Q})$. For this we only have to unravel the definition of $N$: let $C$ be a curve in ${Y}$ that is contracted to a point under $\\pi$. Then the class $D_C=q(C)\\in \\operatorname{Pic}(Y)_{\\mathbb Q}$ is the unique class such that $C.D'=q(D_C,D')$ for all $D'\\in \\operatorname{Pic}(Y)_{\\mathbb Q}$. In particular, $q(D_C,D')=0$ for all ${\\mathbb Q}$-line bundles $D'$ on $X$. So $H^2(X,{\\mathbb C}) \\subset N^\\perp$ and to conclude it is sufficient to exhibit a positive class in $H^2(X,{\\mathbb R})$. As $X$ is supposed to be K\\\"ahler, a K\\\"ahler class $h \\in H^{1,1}(X,{\\mathbb R})$ will do.\nThen $q(h)>0$ as $h$ is big and nef on $X$, so the orthogonal to $h$ in $H^{1,1}({Y})$ is negative definite and thus we see that $H^2({Y},{\\mathbb C})=H^2(X,{\\mathbb C}) \\oplus N$.\n\\end{proof}\n\\begin{proposition}\\label{proposition gkkp for symplectic}\nLet $\\pi:Y\\to X$ be an irreducible symplectic resolution and let $j:U=X^{\\operatorname{reg}}\\to X$ be the inclusion. Then, we have \n\\begin{enumerate}\n\t\\item $j_*\\Omega_U=\\pi_*\\Omega_Y$.\n\t\\item $T_X = j_*\\Omega_U$.\n\\item $H^p(X,j_*\\Omega_U^q)=H^p(X,\\pi_*\\Omega_Y^q)$ for all $p+q \\leq 2$. \n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nLet $X_0=X^{\\operatorname{sing}}$ be the singular locus of $X$ with the reduced structure and let us consider $U_0=X {\\ \\setminus \\ } X_0^{\\operatorname{sing}}$. As a consequence of Kaledin's results \\cite{Kal}, see Remark \\ref{remark kaledin} and Corollary \\ref{corollary from kaledin} below, every point of $U_0$ is either smooth or a transversal ADE surface singularity. \n Let us denote $V = \\pi^{-1}(U)$, $V_0=\\pi^{-1}(U_0)$ and let us write for simplicity $j$ for all inclusions of $U,U_0,V,V_0$ to $X,Y$ and $j_0$ for the inclusions of $U,V$ to $U_0,V_0$. By semi-smallness \\cite[Lemma 2.11]{Kal}, we have $\\operatorname{codim}_Y (Y{\\ \\setminus \\ } V_0) \\geq 2$. Then the first two claims follow from\n\\[\nT_X=j_*{j_0}_*T_U= j_*{j_0}_*\\Omega_U=j_*\\pi_* \\Omega_{V_0}=\\pi_* j_*\\Omega_{V_0}=\\pi_* \\Omega_{Y},\n\\]\nwhere the first equality comes from the reflexivity of $T_X$, the second from the symplectic form on the regular part, the third is because the pushforward of the sheaf of holomorphic $p$-forms along the minimal resolution of ADE surface singularities is reflexive, the fourth is just functoriality of the push-forward, and the last equality holds for codimension reasons.\n\nLet us prove the third statement. As $X$ has rational singularities, we have that $R\\pi_* {\\mathcal O}_Y={\\mathcal O}_X=j_*{\\mathcal O}_U$ so that $H^k(j_*{\\mathcal O}_U)=H^k({\\mathcal O}_Y)$ for all $k$. The first statement implies that $H^k(j_*\\Omega_U)=H^k(\\pi_*\\Omega_Y)$ for all $k$ and the remaining equality $H^0(\\pi_*\\Omega^2_Y)=H^0(j_*\\Omega^2_U)$ \nfollows from $Y$ being an irreducible symplectic resolution. \n\\end{proof}\n\n\\begin{remark}\\label{remark kaledin}\nKaledin's article \\cite{Kal} is formulated for complex algebraic varieties, but we use his results in Proposition \\ref{proposition gkkp for symplectic} for arbitrary irreducible symplectic varieties.\nLet us comment on why they carry over to the compact K\\\"ahler setting as well. The crucial ingredient in Kaledin's proofs is the use of functorial mixed Hodge structures on cohomology groups of complex algebraic varieties and there is no such structure on the cohomology of arbitrary complex varieties. However, we have mixed a Hodge structure on the cohomology of compact K\\\"ahler varieties which is functorial for proper morphisms. It would even be sufficient to consider compact complex varieties of Fujiki class ${\\mathscr C}$, i.e., dominated by a K\\\"ahler manifold.\n\nWith this in mind, Kaledin's proofs work almost literally for compact K\\\"ahler varieties. More precisely, one first shows using mixed Hodge structures that Kaledin's proofs yield analogs of \\cite[Lemma 2.7]{Kal} and \\cite[Lemma 2.9]{Kal} in the compact K\\\"ahler setting. These are the key technical ingredients to prove the stratification and formal product decomposition \\cite[Theorem 2.3]{Kal} as well as \\cite[Theorem 2.5]{Kal} which relates the symplectic and Poisson structure. \nOther than mixed Hodge theory, Kaledin mainly uses Poisson structures, commutative algebra, or direct geometric arguments which all make sense also in our setting.\nFinally, also semi-smallness \\cite[Lemma 2.11]{Kal} is a consequence of geometric properties of the symplectic form and Lemma 2.9 of op. cit.\n\\end{remark}\nWe wish to be a bit more precise regarding the product decomposition on the level of germs, which has not been addressed by Kaledin. It follows easily using a well-known result of Artin.\n\\begin{corollary}\\label{corollary from kaledin}\nLet $X$ be a symplectic variety and let $X_1=X^{\\operatorname{sing}}$ be its singular locus with the reduced structure. Then $\\operatorname{codim}_X X_1^{\\operatorname{sing}} \\geq 4$ and for every point $p \\in U=X {\\ \\setminus \\ } X_1^{\\operatorname{sing}}$ we have an isomorphism of germs $(X,p) {\\ \\cong\\ } ({\\mathbb C}^{\\dim X -2},0) \\times (S,0)$ where $(S,0)$ is an ADE surface singularity.\n\\end{corollary}\n\\begin{proof}\nWith Remark \\ref{remark kaledin} in mind, we infer the claim on the codimension from \\cite[Theorem 2.3]{Kal}. The same theorem also gives the sought-for isomorphism on the level of formal complex spaces. Every local formal isomorphism between complex spaces is the formal completion of an analytic isomorphism by \\cite[Corollary 1.6]{Ar68}.\n\\end{proof}\n\n\nLater on we will need that the very general Hodge structure of a symplectic variety has no automorphisms different from $\\pm {\\rm id}$. As we were not able to find a reference, we include a short proof. In fact, we include a proof -- which is basically taken from \\cite{vGV} -- of a more general result about Mumford--Tate groups, which might be of independent interest, so let us recall this notion. Let $H$ be a finitely generated torsion free ${\\mathbb Z}$-module endowed with a pure Hodge structure of weight $k$ and consider the action $\\rho:\\SS^1\\to \\operatorname{End}(H_{\\mathbb C})$ of the unit circle $\\SS^1 \\subset {\\mathbb C}^\\times$ on $H_{\\mathbb C}=H\\otimes {\\mathbb C}$ defined by $\\rho(z).h:=z^{p-q}h$ for $h\\in H^{p,q}$. One easily checks that this action is defined over ${\\mathbb R}$.\n\n\n\\begin{definition}\nThe Mumford--Tate group of a pure Hodge structure $H$ is the smallest algebraic subgroup ${\\operatorname{MT}}(H) \\subset \\operatorname{GL}_H({\\mathbb R})$ which is defined over ${\\mathbb Q}$ and which contains the image of $\\rho$.\n\\end{definition}\n\nA \\emph{Hodge structure of hyperk\\\"ahler type} is a weight two Hodge structure $H$ with $h^{2,0}=h^{0,2}=1$ together with a nondegenerate bilinear form $q$ on $H$ of signature $(3,{\\rm rk} H -3)$ which is a morphism of Hodge structures and for which $(H^{1,1})^{\\perp_q} = H^{2,0}\\oplus H^{0,2}$ and the restriction of $q$ to $H^{2,0}\\oplus H^{0,2}$ is positive definite. Hodge structures of hyperk\\\"ahler type on $H$ are parametrized by the period domain $$\\Omega_H:=\\{\\omega \\in \\P(H_{\\mathbb C})\\mid (\\omega,\\omega)=0, (\\omega,\\bar\\omega)>0\\}.$$\n\nA Hodge structure of hyperk\\\"ahler type is called \\emph{Mumford--Tate general} if it is not contained in any Hodge locus $\\operatorname{Hdg}_\\alpha \\subset \\Omega_H$ of Hodge classes $\\alpha \\in H^s \\otimes (H^*)^r$ for some $r,s \\in {\\mathbb N}_0$ such that $\\operatorname{Hdg}_\\alpha \\subsetneq \\Omega_H$ is a proper subset.\n\n\\begin{proposition}\\label{proposition van geemen voisin}\nLet $(H,q)$ be a Mumford--Tate general Hodge structure of hyperk\\\"ahler type. Then ${\\operatorname{MT}}(H)=\\operatorname{SO}_H({\\mathbb R})$.\n\\end{proposition}\n\\begin{proof}The proof is basically identical to \\cite[Lemma 9]{vGV} so we only sketch it very briefly. It follows from the definition that ${\\operatorname{MT}}(H) \\subset \\operatorname{SO}_H({\\mathbb R})$. The main ingredient to show equality is that for every Hodge structure $H_\\omega$ corresponding to some $\\omega \\in \\Omega_H$ there is an inclusion of Mumford--Tate groups ${\\operatorname{MT}}(H_\\omega) \\subset {\\operatorname{MT}}(H)$. This is because the Mumford--Tate group may be characterized as the stabilizer group of all Hodge classes $\\alpha \\in H^s \\otimes (H^*)^r$ for all $r,s \\in {\\mathbb N}_0$. Then one proceeds inductively: for $\\alpha \\in H$ and $\\omega \\in \\alpha^\\perp \\subset \\Omega_H$ which is Mumford--Tate general with respect to the Hodge structure $\\alpha^\\perp$ we have that ${\\operatorname{MT}}(H_\\omega) = \\operatorname{SO}_{\\alpha^\\perp}({\\mathbb R})$. The result follows as $\\operatorname{SO}_H({\\mathbb R})$ is the smallest algebraic subgroup containing all $\\operatorname{SO}_{\\alpha^\\perp}({\\mathbb R})$.\n\\end{proof}\n\\begin{corollary}\\label{corollary mumford tate}\nA Mumford--Tate general Hodge structure of hyperk\\\"ahler type only has $\\pm {\\rm id}$ as automorphisms.\n\\end{corollary}\n\\begin{proof}\nThis is a consequence of Proposition \\ref{proposition van geemen voisin} together with the fact that $\\operatorname{End}_{{\\mathbb Q}-\\textrm{HS}}(H_{\\mathbb Q})=\\operatorname{End}(H_{\\mathbb Q})^{\\operatorname{SO}(H_{\\mathbb Q})}= {\\mathbb Q} {\\rm id}_H$.\n\\end{proof}\n\\section{Deformations}\\label{sec defo}\nAs usual in deformation theory, when we speak about the versal deformation ${\\mathscr Z} \\to {\\operatorname{Def}}(Z)$ of a complex space $Z$, then the complex space ${\\operatorname{Def}}(Z)$ has a distinguished point $0\\in{\\operatorname{Def}}(Z)$ such that the fiber of ${\\mathscr Z}\\to {\\operatorname{Def}}(Z)$ over $0$ is $Z$ and we should actually speak about the morphism of space germs $({\\mathscr Z},Z) \\to ({\\operatorname{Def}}(Z),0)$. All deformation theoretic statements have to be interpreted as statements about germs.\n\nLet $X$ be a normal compact complex variety with rational singularities and let $\\pi:Y \\to X$ be a resolution of singularities.\nRecall that by \\cite[Proposition 11.4]{KM} there is a morphism $p:{\\operatorname{Def}}(Y)\\to {\\operatorname{Def}}(X)$ between the Kuranishi spaces of $Y$ and $X$ and also between the versal families ${\\mathscr Y} \\to {\\operatorname{Def}}({Y})$ and ${\\mathscr X} \\to {\\operatorname{Def}}(X)$ fitting in a diagram\n\\begin{equation}\n\\label{eq defo diag}\n\\xymatrix{\n{\\mathscr Y} \\ar[d]\\ar[r]^{P}& {\\mathscr X} \\ar[d]\\\\\n{\\operatorname{Def}}({Y}) \\ar[r]^{p} & {\\operatorname{Def}}(X) \\\\\n}\n\\end{equation}\nRecall from \\cite{FK} that there exists a closed complex subspace ${\\operatorname{Def}}^{\\rm{lt}}(X)\\subset {\\operatorname{Def}}(X)$ parametrizing locally trivial deformations of $X$. More precisely, the restriction of the versal family to this subspace, which by abuse of notation we denote also by ${\\mathscr X} \\to {\\operatorname{Def}}^{\\rm{lt}}(X)$, is a locally trivial deformation of $X$ and is versal for locally trivial deformations of $X$.\n\n\nLet $\\pi:Y \\to X$ be an irreducible symplectic resolution of a compact singular symplectic variety $X$ of dimension $2n$. As $H^0(T_Y)=0$, every versal deformation of $Y$ is universal. We also have $H^0(T_X)=H^0(\\pi_*\\Omega_Y)=0$ by Proposition \\ref{proposition gkkp for symplectic} so that every versal deformation of $X$ is universal. \nLet us fix universal deformations of $X$ and $Y$ and a diagram as \\eqref{eq defo diag}. It is well-known that ${\\mathscr Y} \\to {\\operatorname{Def}}(Y)$ is a family of irreducible symplectic manifolds, at least in the sense of germs, i.e., possibly after shrinking the representative of ${\\operatorname{Def}}(Y)$. If $X$ is projective, then also ${\\mathscr X}\\to{\\operatorname{Def}}(X)$ is a family of irreducible symplectic varieties by \\cite[Theorem 2.2]{Na01}. \nThat this statement also holds without the projectivity assumption is highly probable, but it does not seem to be written somewhere in the literature. We will however not need this and even prove it for locally trivial deformations.\nOur first goal is to describe $p^{-1}({\\operatorname{Def}}^{\\rm{lt}}(X)) \\subset {\\operatorname{Def}}({Y})$. \n\nRecall from the introduction and Lemma \\ref{lemma symplektisch} that \nwe have an orthogonal decomposition\n\\begin{equation}\\label{eq n}\nH^2({Y},{\\mathbb Q})=H^2(X,{\\mathbb Q})\\oplus N \n\\end{equation}\nwhere $N$ corresponds under $\\tilde q$ to the curves contracted by $\\pi$ and put $m:=\\dim N$.\n\\begin{theorem}\\label{theorem deflt is smooth}\nLet $X$ be an irreducible symplectic variety. Then the space ${\\operatorname{Def}}^{\\rm{lt}}(X)$ of locally trivial deformations of $X$ is smooth of dimension $h^{1,1}(X) = h^{1,1}({Y}) - m$. \n\\end{theorem}\n\\begin{proof}\nSmoothness is shown using the $T^1$-lifting principle of Kawamata-Ran \\cite{Ran,Ka1,Ka2}, see also \\cite[\\S 14]{GHJ} or \\cite[VI.3.6]{mydiss} for more detailed introductions. The tangent space to ${\\operatorname{Def}}^{\\rm{lt}}(X)$ at the origin is $H^1(T_X)$ which thanks to the symplectic form can be identified with $H^1(j_*\\Omega_U)$ where $j:U=X^{\\operatorname{reg}} \\to X$ is the inclusion. For the $T^1$-lifting property one has to show that for every infinitesimal locally trivial deformation ${\\mathcal X} \\to S$ of $X$ over an Artinian base scheme $S$ the space $H^1(T_{{\\mathcal X}\/S})$ is locally ${\\mathcal O}_S$-free and compatible with arbitrary base change. We denote again by $j:{\\mathcal U} \\to {\\mathcal X}$ the inclusion of the smooth locus of ${\\mathcal X} \\to S$.  Take an extension $\\sigma \\in H^0({\\mathcal U},j_* \\Omega_{{\\mathcal X}\/S}^2)$ of the symplectic form on $U\\subset X$. It remains nondegenerate and hence yields an isomorphism $T_{{\\mathcal X}\/S}\\to j_*\\Omega_{{\\mathcal X}\/S}$, consequently also $H^1(T_{{\\mathcal X}\/S}){\\ \\cong\\ } H^1(\\Omega_{{\\mathcal X}\/S})$ which is free by Proposition \\ref{proposition gkkp for symplectic} and Lemma \\ref{lemma hodge relativ}. Thus, it satisfies the $T^1$-lifting property and thus the space ${\\operatorname{Def}}^{\\rm{lt}}(X)$ is smooth.\nIt follows from Corollary \\ref{corollary hodge} that $\\dim H^1(T_X) = h^{1,1}(X)$ which shows the dimension statement and completes the proof.\n\\end{proof}\n\\begin{corollary}\\label{corollary deformation remains isv}\nLet $\\pi:Y \\to X$ be an irreducible symplectic resolution and consider the induced morphism $p:{\\operatorname{Def}}(Y)\\to {\\operatorname{Def}}(X)$. Then ${\\operatorname{Def}}^{\\rm{lt}}(X)$ is contained in the image of $p$. Moreover, a small locally trivial deformation of an irreducible symplectic variety is again an irreducible symplectic variety.\n\\end{corollary}\n\\begin{proof}\nFor the first claim, it suffices to note that the tangent space $H^1(X,T_X)$ to ${\\operatorname{Def}}^{\\rm{lt}}(X)$ by Proposition \\ref{proposition gkkp for symplectic} is a subspace to the tangent space $H^1(Y,T_Y)$ to ${\\operatorname{Def}}(Y)$ and as both spaces are smooth, every locally trivial deformation of $X$ induces a deformation of $Y$. A small deformation of a K\\\"ahler space with rational singularities remains K\\\"ahler by \\cite[Proposition 5]{Na01a} and again by Proposition \\ref{proposition gkkp for symplectic} the symplectic form extends sideways.\n\\end{proof}\n\\begin{corollary}\\label{corollary projective}\nEvery irreducible symplectic resolution $\\pi:Y \\to X$ has a small locally trivial deformation to an irreducible symplectic resolution $\\pi:Y' \\to X'$ where $Y'$ and $X'$ are projective.\n\\end{corollary}\n\\begin{proof}\nThe subspace $N$ from \\eqref{eq n} is negative definite and of Hodge type (1,1) by Lemma \\ref{lemma symplektisch}. As $Y$ is an irreducible symplectic manifold, the signature of $q_Y$ on $H^{1,1}(Y,{\\mathbb R})$ is $(1,b_2(Y)-3)$ so that by Theorem \\ref{theorem deflt is smooth} the contraction $\\pi:Y\\to X$ deforms in a positive dimensional locally trivial family. By \\cite[Proposition 26.6]{GHJ} arbitrarily close to $\\pi:Y\\to X$ there is a small deformation $Y'\\to X'$ of $\\pi$ such that $Y'$ is projective. As $X'$ has rational singularities, see e.g. \\cite[Theorem 3.3.3]{Kirschner}, and is K\\\"ahler, it is projective by Namikawa's result \\cite[Corollary 1.7]{Na02}.\n\\end{proof}\n\nFor $N\\subset H^2(Y,{\\mathbb Q})$ as in \\eqref{eq n} let us denote by ${\\operatorname{Def}}({Y},N)\\subset {\\operatorname{Def}}({Y})$ the subspace of those deformations of ${Y}$ where all line bundles on $Y$ with first Chern class in $N$ deform along. This is also the subspace where $N$ remains of type $(1,1)$. It is a smooth submanifold of ${\\operatorname{Def}}({Y})$ of codimension $m=\\dim N$ by \\cite[1.14]{Huy99}.\n\\begin{proposition}\\label{prop defo}\nLet $\\pi: {Y}\\to X$ be an irreducible symplectic resolution. Let ${\\mathscr Y} \\to {\\operatorname{Def}}({Y},N)$ and ${\\mathscr X} \\to{\\operatorname{Def}}^{\\rm{lt}}(X)$ be the (restrictions of the) universal deformations. Then there is a diagram \n\\begin{equation}\\label{eq diag lt}\n\\xymatrix{\n{\\mathscr Y} \\ar[d]\\ar[r]^P& {\\mathscr X} \\ar[d]\\\\\n{\\operatorname{Def}}({Y},N) \\ar[r]^{p} & {\\operatorname{Def}}^{\\rm{lt}}(X) \\\\\n}\n\\end{equation}\nwhere $p$ is the restriction of the natural morphism $p:{\\operatorname{Def}}({Y})\\to{\\operatorname{Def}}(X)$. Moreover, $p$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nWe have to show that $p^{-1}({\\operatorname{Def}}^{\\rm{lt}}(X)) = {\\operatorname{Def}}({Y},N)$. We know by Corollary \\ref{corollary deformation remains isv} that for each $t\\in {\\operatorname{Def}}(Y)$ mapping to ${\\operatorname{Def}}^{\\rm{lt}}(X)$ the morphism $P_t:{\\mathscr Y}_t \\to {\\mathscr X}_{p(t)}$ is an irreducible symplectic resolution. By Lemma \\ref{lemma hodge relativ} and Proposition \\ref{proposition gkkp for symplectic} the second cohomology of locally trivial deformations of $X$ form a vector bundle on ${\\operatorname{Def}}^{\\rm{lt}}(X)$, in particular, $h^{1,1}({\\mathscr X}_{p(t)})=h^{1,1}(X)$. Thus, by the decomposition $H^2({Y},{\\mathbb C})=N\\oplus H^2(X,{\\mathbb C})$ from Lemma \\ref{lemma symplektisch} we see that the space $N_1({\\mathscr Y}_t\/{\\mathscr X}_{p(t)})$ of curves contracted by $P_t$ has dimension $m$ for all $t\\in p^{-1}({\\operatorname{Def}}^{\\rm{lt}}(X))$. As $N$ is the orthogonal complement of $H^2(X,{\\mathbb C})$, it also  varies in a local system. This shows the sought-for equality.\n\nOne shows as in \\cite{LP} that  $p$ is an isomorphism, we only sketch this: It suffices to show that the differential $T_{p,0}:T_{{\\operatorname{Def}}({Y},N),0}\\to T_{{\\operatorname{Def}}^{\\rm{lt}}(X),0}=H^1(T_X)$ is an isomorphism. \nWe know from \\cite[(1.8) and (1.14)]{Huy99} that $T_{{\\operatorname{Def}}({Y},N),0} \\subset H^1(T_{Y})$ can be identified with the orthogonal complement to $N \\subset H^{1,1}({Y})$ under the isomorphism $H^1(T_{Y}) {\\ \\cong\\ } H^{1,1}({Y})$ induced by the symplectic form. In other words, $T_{{\\operatorname{Def}}({Y},N),0} {\\ \\cong\\ } H^{1,1}(X) \\subset H^{1,1}({Y})$. That this is mapped to $H^1(T_X){\\ \\cong\\ } H^1(j_*\\Omega_U)$ under the restriction of $T_{p,0}:H^1(T_{Y})\\to \\operatorname{Ext}^1(\\Omega_X,{\\mathcal O}_X)$ is easily verified.\n\\end{proof}\nThe independence of the Beauville--Bogomolov--Fujiki pairing on the cohomology of a singular symplectic variety has been proven by Namikawa, see \\cite[Theorem 8]{Na01a}. In combination with the preceding result and the version for smooth varieties we immediately deduce the following variant of the local Torelli theorem for locally trivial deformations.\n\\begin{corollary}[Local Torelli Theorem]\\label{corollary local torelli}\nLet $X$ be an irreducible symplectic variety, let $q_X$ be its Beauville--Bogomolov--Fujiki form, and let \n\\begin{equation}\\label{eq local period domain}\n\\Omega(X):=\\{[\\sigma]\\in \\P(H^2(X,{\\mathbb C}))\\mid q_X(\\sigma)=0, q_X(\\sigma,\\bar\\sigma)>0\\} \\subset \\P(H^2(X,{\\mathbb C})) \n\\end{equation}\nbe the period domain for $X$. If $f:{\\mathscr X} \\to {\\operatorname{Def}}^{\\rm{lt}}(X)$ denotes the universal locally trivial deformation of $X$ and $X_t:=f^{-1}(t)$, then the period map \n\\[\n\\wp:{\\operatorname{Def}}^{\\rm{lt}}(X) \\to \\Omega(X), \\quad t \\mapsto H^{2,0}(X_t)\n\\]\nis a local isomorphism.\\qed\n\\end{corollary}\nIt should be mentioned that Namikawa has proven local Torelli theorem for certain singular projective symplectic varieties in loc. cit. and this has been generalized by Kirschner \\cite[Theorem 3.4.12]{Kirschner} to a larger class of varieties.  In particular, Kirschner has proven a local Torelli theorem in the context of symplectic compact K\\\"ahler spaces. Let us emphasize, however, that neither Namikawa's version nor Kirschner's is what we need as they do not make any statement about local triviality. Also observe that our version of local Torelli---unlike Namikawa's or Kirschner's---does not make any assumption on the codimension of the singular locus of the variety $X$.\n\\begin{remark}\\label{remark identify base}\nWe will from now on identify the spaces ${\\operatorname{Def}}({Y},N)$ and ${\\operatorname{Def}}^{\\rm{lt}}(X)$ via the morphism $p$ if we are given a birational contraction. \\end{remark}\n\nThe following result is needed in the proof of this section's main theorem. For a variety $X$ we denote by ${\\operatorname{Cl}}(X)$ the group of Weil divisors modulo linear equivalence on it and we write ${\\operatorname{Cl}}(X)_{\\mathbb Q}:={\\operatorname{Cl}}(X) \\otimes_{\\mathbb Z} {\\mathbb Q}$.\n\n\\begin{proposition}\\label{proposition qcartier}\nConsider a diagram $X \\ot[\\pi] Y \\to[\\pi'] X'$ of normal projective varieties where $Y$ is smooth, $\\pi, \\pi'$ are birational and $\\pi^* H^2(X,{\\mathbb Z})\\subset (\\pi')^*H^2(X',{\\mathbb Z})\\oplus M$ as subsets of $H^2(Y,{\\mathbb Z})$ where $M \\subset H^2(Y,{\\mathbb Z})$ is the subspace generated by $\\pi'$-exceptional divisors.\nSuppose that the birational map $\\phi:= \\pi' \\circ \\pi^{-1}: X \\dashrightarrow X'$ is an isomorphism in codimension one. \nThen the map $\\phi_*: {\\operatorname{Cl}}(X)_{\\mathbb Q} \\to {\\operatorname{Cl}}(X')_{\\mathbb Q}$ sends ${\\mathbb Q}$-Cartier divisors to ${\\mathbb Q}$-Cartier divisors.  \n\\end{proposition}\n\\begin{proof}\n Take a Cartier divisor $D$ on $X$ and write $\\pi^* D=\\tilde D + E$ where $\\tilde D$ is the strict transform and $E$ is $\\pi$-exceptional. By assumption, we have $\\pi^* D = (\\pi')^*D' + F$ for some Cartier divisor $D'$ on $X'$ and a $\\pi'$-exceptional divisor $F$ on $Z$. Then $\\pi'_*\\pi^* D = D'$ is Cartier. Now it suffices to note that $E$ is also $\\pi'$-exceptional so that $D'=\\pi'_* \\tilde D = \\phi_* D$ and the claim follows.\n\\end{proof}\n\nNote that is not necessary that $\\phi$ be an isomorphism in codimension one. It would be sufficient that every $\\pi$-exceptional divisor is also $\\pi'$-exceptional.\n\n\\begin{theorem}\\label{theorem huybrechts}\nLet $X$ and $X'$ be projective symplectic varieties, suppose that $X$ has a resolution of singularities $\\pi:{Y}\\to X$ by an irreducible symplectic manifold ${Y}$, and\nlet $\\phi:X\\dashrightarrow X'$ be a birational map which is an isomorphism in codimension one. Then, there exists a symplectic resolution $\\pi':{Y}' \\to  X'$ by an irreducible symplectic manifold ${Y}'$ and \nthe following are equivalent.\n\\begin{enumerate}\n \\item \\label{aeq eins} There is an isomorphism $\\varphi:{\\operatorname{Def}}^{\\rm{lt}}(X)\\to {\\operatorname{Def}}^{\\rm{lt}}(X')$ such that for every $t\\in {\\operatorname{Def}}^{\\rm{lt}}(X)$ we have a birational map $\\phi_t: {\\mathscr X}_t \\dashrightarrow {\\mathscr X}'_{\\varphi(t)}$ which is an isomorphism in codimension one. For general $t \\in {\\operatorname{Def}}^{\\rm{lt}}(X)$, the map $\\phi_{t}$ is an isomorphism. In particular, $X$ and $X'$ are locally trivial deformations of one another.\n\\item \\label{aeq drei} The map $\\tilde\\phi^*: H^2({Y}',{\\mathbb C}) \\to H^2({Y},{\\mathbb C})$ induced by the birational map $\\tilde\\phi:{Y} \\dashrightarrow {Y}'$ is an isomorphism that sends $H^2(X',{\\mathbb C})$ to $H^2(X,{\\mathbb C})$. \n\\end{enumerate}\nIf these conditions are satisfied and if we denote $\\wp:{\\operatorname{Def}}^{\\rm{lt}}(X)\\to \\Omega(X)$, $\\wp':{\\operatorname{Def}}^{\\rm{lt}}(X')\\to \\Omega(X')$  the associated period maps, then the isomorphism $\\varphi$ from \\eqref{aeq eins} coincides with $(\\wp')^{-1}\\circ \\wp$ where we identify $\\Omega(X)$ and $\\Omega(X')$ through the isomorphism induced by $\\tilde\\phi^*$.\n\\end{theorem}\n\\begin{proof}\nThe statement on the existence of a symplectic resolution is well known, we include the references for convenience. Let $\\pi':{Y}'\\to X'$ be a ${\\mathbb Q}$-factorial terminalization, which exists by \\cite[Corollary 1.4.3]{BCHM}.\nIt follows from \\cite[Theorem 1]{Kaw08} and \\cite[Example (flops) p.98]{Na06} that ${Y}'$ is an irreducible symplectic manifold as well.\n\nThe birational map $\\tilde \\phi:{Y} \\dashrightarrow {Y}'$ between irreducible symplectic manifolds induces an isomorphism between $H^2({Y},{\\mathbb Z})$ and $H^2({Y}',{\\mathbb Z})$ compatible with the Beauville--Bogomolov--Fujiki forms and the local Torelli theorem gives an isomorphism ${\\operatorname{Def}}({Y})\\to {\\operatorname{Def}}({Y}')$. \nLet us consider the orthogonal decompositions $H^2({Y},{\\mathbb Q})=H^2(X,{\\mathbb Q})\\oplus N$ and $H^2({Y}',{\\mathbb Q})=H^2(X',{\\mathbb Q})\\oplus N'$ as in \\eqref{eq n}.\nCondition \\eqref{aeq drei} is equivalent to saying that the isomorphism ${\\operatorname{Def}}({Y})\\to {\\operatorname{Def}}({Y}')$ restricts to an isomorphism ${\\operatorname{Def}}({Y},N) \\to {\\operatorname{Def}}({Y}',N')$ and therefore yields an isomorphism $\\varphi:{\\operatorname{Def}}^{\\rm{lt}}(X) \\to {\\operatorname{Def}}^{\\rm{lt}}(X')$ via Proposition~\\ref{prop defo}. \nIn order to deduce \\eqref{aeq eins} from \\eqref{aeq drei}, it remains to show the existence of a birational map $\\phi_t: {\\mathscr X}_t \\dashrightarrow {\\mathscr X}'_{\\varphi(t)}$ for $t\\in {\\operatorname{Def}}^{\\rm{lt}}(X)$ which is an isomorphism at the general point and an isomorphism in codimension one for every point. In this situation we will identify the spaces $$S:={\\operatorname{Def}}^{\\rm{lt}}(X){\\ \\cong\\ }{\\operatorname{Def}}^{\\rm{lt}}(X'){\\ \\cong\\ } {\\operatorname{Def}}({Y},N){\\ \\cong\\ } {\\operatorname{Def}}({Y}',N')$$ and consider the universal families\n\\[\n {\\mathscr Y} \\to {\\mathscr X} \\to S \\ot {{\\mathscr X}'} \\ot {{\\mathscr Y}'}\n\\]\nfrom Proposition \\ref{prop defo}. For a point $t\\in S$  the fibers ${\\mathscr Y}_t$ and ${\\mathscr Y}'_{t}$ are deformation equivalent (by Huybrechts' theorem, see \\cite[Theorem 2.5]{Huy}) and have the same periods, hence they are birational by Verbitsky's global Torelli theorem \\cite[Theorem 1.17]{V13}. \nWe obtain a birational map ${\\mathscr X}_{t}\\dashrightarrow {\\mathscr X}'_{t'}$ for all $t\\in S$. \nAs the deformations of $X$ and $X'$ are locally trivial, their exceptional divisors deform all over $S$ and are contracted under the morphisms ${\\mathscr Y}_t \\to {\\mathscr X}_{t}$ and ${\\mathscr Y}'_t \\to {\\mathscr X}'_{t}$ for all $t$.\nThis follows just as in the proof of \\cite[Proposition 2.3]{LP} if we use that the $H^2({\\mathscr X}'_{t},{\\mathbb Z})$ and $H^2({\\mathscr X}_{t},{\\mathbb Z})$ form local systems, more precisely, local subsystems of those formed by $H^2({\\mathscr Y}'_{t},{\\mathbb Z})$ and $H^2({\\mathscr Y}_{t},{\\mathbb Z})$. Hence, the $\\phi_t$ are isomorphisms in codimension one.\nIn order to show that $\\phi_t$ is an isomorphism for general $t$, it is sufficient to consider projective deformations of $X$. Projective deformations of $X$ are dense over every positive dimensional subvariety of $\\deflt$ as the same holds for ${Y}$, see \\cite[Proposition 26.6]{GHJ}. For a very general $t \\in S$ corresponding to projective ${\\mathscr X}_t$ and ${\\mathscr X}'_t$  both varieties have Picard number one. Moreover, by Proposition \\ref{proposition qcartier} the pullback of an ample line bundle on ${\\mathscr X}'_t$ under the rational map $\\phi_t$ is again an ample line bundle and thus $\\phi_t$ is an isomorphism.\n\nFor the converse, consider $\\phi: X \\dashrightarrow X'$. The induced birational map $\\tilde\\phi:Y \\dashrightarrow Y'$ induces a Hodge isometry $\\tilde\\phi^*: H^2({Y}',{\\mathbb C}) \\to H^2({Y},{\\mathbb C})$ which maps $H^2(X,{\\mathbb C})_{\\rm tr}$ to $H^2(X',{\\mathbb C})_{\\rm tr}$. By assumption $\\tilde\\phi^*$ is an isomorphism in codimension one, so it fulfills the hypotheses of Proposition \\ref{proposition qcartier}. Thus, $H^2(X,{\\mathbb C})$ is sent to $H^2(X',{\\mathbb C})$ and the claim follows.\n\nThe last statement is clear from the proof.\n\\end{proof}\n\nWe denote by $\\Delta=\\{z\\in {\\mathbb C}\\mid \\abs{z}\\leq 1\\}$ the complex unit disk and by $\\Delta^\\times:=\\Delta{\\ \\setminus \\{0\\} }$ the complement of the origin.\n\n\\begin{theorem}\\label{theorem huybrechts strong}\nLet $X$ and $X'$ be projective symplectic varieties, suppose that $X$ has resolution of singularities $\\pi:{Y}\\to X$ by an irreducible symplectic manifold ${Y}$, and\nlet $\\phi:X\\dashrightarrow X'$ be a birational map which is an isomorphism in codimension one such that $\\phi_*: {\\operatorname{Cl}}(X)_{\\mathbb Q} \\to {\\operatorname{Cl}}(X')_{\\mathbb Q}$ sends ${\\mathbb Q}$-line bundles to ${\\mathbb Q}$-line bundles. Then there are one parameter deformations $f:{\\mathscr X} \\to \\Delta$, $f':{\\mathscr X}' \\to \\Delta$ such that ${\\mathscr X}$ and ${\\mathscr X}'$ are birational over $\\Delta$ and such that ${\\mathscr X}^* = f^{-1}(\\Delta^\\times) {\\ \\cong\\ } (f')^{-1}(\\Delta^\\times) = ({\\mathscr X}')^* $. \n\\end{theorem}\n\\begin{proof}\n The argument of Huybrechts works in this context almost literally, see \\cite[Theorem 1.1]{LP} for the necessary changes.\n\\end{proof}\n\nFollowing Voisin \\cite[Definition 0.6]{Vo15}, we call a subvariety $P\\subset Y$ of an irreducible symplectic variety an \\emph{algebraically coisotropic subvariety} if it is coisotropic and admits a rational map $\\phi: P \\dashrightarrow B$ onto a variety of dimension $\\dim Y - 2\\cdot\\operatorname{codim} P$ such that the restriction of the symplectic form to $P$ satisfies $\\sigma\\vert_P=\\phi^*\\sigma_B$ for some 2-form $\\sigma_B$ on $B$. \n\n\\begin{proposition}\\label{proposition cmsb}\nEvery irreducible component $P$ of the exceptional locus of an irreducible symplectic resolution $Y\\to X$ is algebraically coisotropic and the coisotropic fibration of $P$ is given by $\\pi:P\\to B:=\\pi(P)$. In particular, it is  holomorphic. Moreover, the general fiber of $\\pi\\vert_P$ is rationally connected.\n\\end{proposition}\n\\begin{proof}\nBy Corollary \\ref{corollary projective} we may assume that $X$ and $Y$ are projective. Let $F$ denote a resolution of singularities of the general fiber of $P\\to B$. By \\cite[Lemma 2.9]{Kal} it follows that the pullback of the symplectic form $\\sigma$ of $X$ to $F$ vanishes identically so that $F$ is isotropic. It remains to show that $\\dim B = 2(n - \\dim F)$ where $\\dim X=2n$.\n This is a consequence of \\cite[Theorem 1.2]{Wierzba}. \n\nTo prove rational connectedness, we use that every rational curve $C$ on a symplectic variety the morphism $\\nu:\\P^1 \\to C \\subset X$ obtained by normalization and inclusion deforms in a family of dimension at least $2n-2$, see \\cite[Example 5.2]{Ran2} or \\cite[Proposition 3.1]{CP}. Being an exceptional locus $P$ is known to be uniruled. Let $H$ be an ample divisor on $Y$ and take an irreducible rational curve $C$ on $P$ contracted by $\\pi$ such that the intersection product $H.C$ is minimal among all rational curves on $P$. \nIn the Chow scheme of $Y$ we look at an irreducible component ${\\textrm{Ch}}$ containing $[C]$ of the locus parametrizing rational curves. \nBy what we noted above $\\dim {\\textrm{Ch}}\\geq 2n-2$ and by minimality of $H.C$ all points in $U$ correspond to irreducible and reduced rational curves.\nLet $U \\subset {\\textrm{Ch}} \\times Y$ be the graph of the universal family of cycles and denote by $p:U \\to {\\textrm{Ch}}$ and $q:U \\to Y$ the projections. As $C$ is contracted by $\\pi$, the curves in ${\\textrm{Ch}}$ cannot move out of the exceptional locus (this would e.g. change the intersection with a pullback of an ample divisor from $X$) and as $U$ is irreducible, we have in fact $U \\subset {\\textrm{Ch}} \\times P$. Moreover, by the Rigidity Lemma, $U\\to {\\textrm{Ch}}$ is a family of curves in the fibers of $P \\to B$. Now a simple dimension count together with the fact that a positive dimensional family of rational curves with two basepoints has to have reducible or nonreduced members (Bend and Break, see e.g. \\cite[Theorem (5.4.2)]{Ko}) imply that through the general point of a general fiber $F$ of $P\\to B$ there is a family of rational curves without further basepoints of dimension $i-1$. Consequently, $F$ is rationally connected.\n\\end{proof}\n\n\n\\begin{remark}\nRational connectedness of $F$ would follow from \\cite[Theorem 9.1]{CMSB} the proof of which however seems to be incomplete. Instead, it might also be possible to use \\cite[Theorem 2.8 (2)]{CMSB} and the well-known fact that a variety is rationally connected if and only if it contains a very free rational curve, see e.g. \\cite[Ch IV, 3.7 Theorem]{Ko}.\n\\end{remark}\n\n\\begin{remark}\n Rational chain connectedness follows from the much stronger result \\cite[Corollary 1.5]{HM}. This notion coincides for smooth varieties with rational connectedness, however, this is not the case for singular varieties. The cone over an elliptic curve is the easiest example of a variety which is rationally chain connected but not rationally connected. \n\\end{remark}\n\nRecall from \\cite[Theorem 1.1]{LP2} that given an algebraically coisotropic subvariety with almost holomorphic coisotropic fibration $\\phi$ whose generic fiber $F$ is smooth, then $F$ deforms all over its Hodge locus $\\operatorname{Hdg}_F \\subset {\\operatorname{Def}}(Y)$. Moreover, if for $t \\in \\operatorname{Hdg}_F$ we denote by $Y_t$ the corresponding deformation of $Y$, then the deformations of $F$ inside $Y_t$ cover an algebraically coisotropic subvariety $P_t \\subset Y_t$ with $F_t$ as a generic fiber of the coisotropic fibration. It seems, however, unclear how to relate the cycle class of $P_t$ with that of $P$ let alone to show that $P_t$ is a flat deformation. \n\nIn the context of birational contractions of symplectic varieties (in dimension $\\geq 4$ at least) it is rather common that the generic fiber of the exceptional locus over its image is smooth. Thus, it is worthwhile to mention that the main result of \\cite{LP2} can be strengthened in this special situation.\nBut first we need some notation. \n\nLet $F \\subset Y$ be a closed subvariety in an irreducible symplectic manifold. If ${\\mathscr Y} \\to {\\operatorname{Def}}(Y)$ denotes the universal deformation we let ${\\mathscr H} \\to {\\operatorname{Def}}(Y)$ be the union of all those components of the relative Hilbert scheme (or Douady space) of ${\\mathscr Y}$ over ${\\operatorname{Def}}(Y)$ which contain $[F]$. We define the closed subspace ${\\operatorname{Def}}(Y,F) \\subset {\\operatorname{Def}}(Y)$ to be the scheme theoretic image of ${\\mathscr H} \\to {\\operatorname{Def}}(Y)$; this is the space of deformations of $Y$ that contain a deformation of $F$. \n\n\\begin{theorem}\\label{theorem pacienza}\nLet $\\pi:Y\\to X$ be an irreducible symplectic resolution, let $P\\subset Y$ be the exceptional locus of $\\pi$, put $B:=\\pi(P)$, and let ${\\mathscr Y} \\to {\\mathscr X}$ be the restriction of the universal deformation of $Y\\to X$ over ${\\operatorname{Def}}(Y,N)$. Suppose that $P$ is irreducible and that a general fiber $F$ of $\\pi:P\\to B$ is smooth. Then we have ${\\operatorname{Def}}(Y,N) \\subset {\\operatorname{Def}}(Y,F)$ and the Hodge locus $\\operatorname{Hdg}_P$ of $P$ contains ${\\operatorname{Def}}(Y,N)$.\n\\end{theorem}\n\\begin{proof} \nLet ${\\mathscr F} \\to {\\mathscr H} \\to {\\operatorname{Def}}(F,Y)$ be the universal deformation of $F$ over the closed subspace ${\\mathscr H}$ of the relative Hilbert scheme of ${\\mathscr Y} \\to {\\operatorname{Def}}(Y,N)$. \nBy Proposition \\ref{proposition cmsb} the variety $P$ is algebraically coisotropic and the fibers of $\\pi:P\\to B$ are rationally connected so that $H^1(F,{\\mathcal O}_F)=0$ and \\cite[Theorem 1.1]{LP2} can be applied. We deduce that ${\\operatorname{Def}}(F,Y)$ and ${\\mathscr H}\\to{\\operatorname{Def}}(F,Y)$ are smooth at $0$ respectively $[F]$. In particular, ${\\mathscr H}$ is irreducible. Moreover, by \\cite[Corollary 1.2 and 1.3]{LP2} the period map identifies ${\\operatorname{Def}}(F,X)$ with $Q \\cap \\P(K)$ where $Q \\subset \\P(H^2(X,{\\mathbb C}))$ is the period domain of the irreducible symplectic manifold $Y$ and $K=\\ker(H^2(Y,{\\mathbb C})\\to H^2(F,{\\mathbb C}))$. If $p\\in B$ denotes the point with $F=\\pi^{-1}(b)$, then by commutativity of $$\\xymatrix{F \\ar[r]\\ar[d]& Y \\ar[d]\\\\ \\{p\\} \\ar[r] & X\\\\}$$  it follows that we have $H^2(X,{\\mathbb C}) \\subset K$ and hence using $H^2(X,{\\mathbb C})^\\perp = N$ and the period map once more we obtain ${\\operatorname{Def}}(Y,N) \\subset {\\operatorname{Def}}(F,Y)$.\n\nIn order to show that $P$ remains a Hodge class all over ${\\operatorname{Def}}(Y,N)$ we will construct flat families ${\\mathscr P}_\\Delta \\to \\Delta$ over curves $\\Delta \\subset {\\operatorname{Def}}(Y,N)$ passing through the origin such that the cycle underlying the central fiber ${\\mathscr P}_{\\Delta,0}$ is a multiple of $P$. To this end we replace ${\\mathscr F} \\to {\\mathscr H}$ as well as ${\\mathscr Y} \\to {\\mathscr X}$ by their restrictions to a given smooth curve germ $\\Delta \\subset {\\operatorname{Def}}(Y,N)$ and obtain morphisms\n$ \\xymatrix{\n{\\mathscr H} & \\ar[l] {\\mathscr F} \\ar[r]& {\\mathscr Y} \\ar[r] &{\\mathscr X}\\\\\n }$\nover $\\Delta$. The map in the middle is induced by the projection to the second factor of ${\\mathscr F} \\subset {\\mathscr H} \\times {\\mathscr Y}$. As $\\Delta$ is smooth and smoothness is stable under base change we may still assume that ${\\mathscr H}$, ${\\mathscr H} \\to \\Delta$ and ${\\mathscr F} \\to {\\mathscr H}$ are smooth at $[F]$ respectively in a neighborhood of $F\\subset {\\mathscr F}$. In particular, there is still a unique irreducible component of ${\\mathscr H}$ passing through $[F]$ and by shrinking the representative of ${\\operatorname{Def}}(F,Y)$ and throwing away components of ${\\mathscr H}$ we may assume that ${\\mathscr H}$ is irreducible.\n\nOn the other hand, as ${\\mathscr X} \\to \\Delta$ is a locally trivial deformation, it induces a flat (even locally trivial) deformation of all components of its singular locus (with the reduced structure). Let ${\\mathscr B} \\to {\\operatorname{Def}}(Y,N)$ be the so induced deformation of $B$. Then ${\\mathscr B} \\subset {\\mathscr X}$ is an irreducible (and reduced) subspace.\nIf we knew that also ${\\mathscr Y} \\to {\\mathscr X}$ were a locally trivial deformation of $Y\\to X$, the claim would follow immediately. As we cannot prove this so far, cf. Question \\ref{question contraction deforms locally trivially}, we have to argue differently.\n\nWe take the unique closed irreducible and reduced subspace ${\\mathscr F}' \\subset {\\mathscr F}$ that coincides with ${\\mathscr F}$ in a neighborhood of $F$. As $\\rho:{\\mathscr F}' \\to {\\mathscr H}$ is proper it therefore is surjective as well. If we take the Stein factorization, ${\\mathscr F}'\\to \\tilde{\\mathscr H} \\to {\\mathscr H}$, then\nby the Rigidity Lemma (see e.g. \\cite[Lemma 1.15]{Debarre}) there is a commutative diagram\n\\[\\xymatrix{\n{\\mathscr F} \\ar[d]\\ar[r]& {\\mathscr Y}\\ar[d]\\\\\n\\tilde{\\mathscr H} \\ar[r]&{\\mathscr X}\\\\\n}\\] \nNote that $\\tilde{\\mathscr H} \\to {\\mathscr H}$ is finite, birational, and an isomorphism over $[F]\\in {\\mathscr H}$ and that $\\tilde{\\mathscr H}$ is irreducible. Moreover, the image of $\\tilde{\\mathscr H} \\to {\\mathscr X}$ coincides in a neighborhood of $[f]\\in \\tilde{\\mathscr H}$ with the closed subvariety ${\\mathscr B} \\subset{\\mathscr X}$ thanks to the smoothness of ${\\mathscr F} \\to {\\mathscr H}$ in a neighborhood of $F\\subset {\\mathscr F}$. Invoking the irreducibility of $\\tilde{\\mathscr H}$ and ${\\mathscr B}$ we conclude that we have $\\tilde{\\mathscr H}{{\\twoheadrightarrow}} {\\mathscr B} \\subset {\\mathscr X}$. We define ${\\mathscr P}  = {\\mathscr P}_\\Delta \\subset {\\mathscr Y}$ to be the image of ${\\mathscr F}' \\to {\\mathscr Y}$. The variety ${\\mathscr F}'$ being irreducible the same holds true for ${\\mathscr P}$ and hence the induced map $\\rho:{\\mathscr P} \\to \\Delta$ is flat. It remains to show that $P\\subset {\\mathscr P}$ is the unique component of the central fiber ${\\mathscr P}_0$ of $\\rho$ of dimension $\\dim P$. This follows from the irreducibility of $P$ by invoking the Rigidity Lemma once more.\n\\end{proof}\n\nRecall from \\cite[Definition 1.5]{Vo15} that a cohomology class $p \\in H^{2i}(Y,{\\mathbb C})$ on a smooth symplectic variety is called \\emph{coisotropic} if it is a Hodge class and $[\\sigma]^{n-i+1}\\cup p = 0$ in $H^{2n+2}(Y,{\\mathbb C})$ where $\\sigma$ is the symplectic form on $Y$. We refer to Huybrechts' article \\cite{Hu14} for the notion of a constant cycle subvariety. This is roughly speaking a subvariety of a given variety all of whose points have the same cycle class in the ambient variety.\n\n\\begin{corollary}\nIn the situation of Theorem \\ref{theorem pacienza}, the class of $[P]$ remains an effective coisotropic Hodge class all over $S={\\operatorname{Def}}(Y,N)$. Moreover, there are varieties $P_t \\subset Y_t$ for each $t\\in S$ representing (a multiple of) $[P]$ which are algebraically coisotropic with rationally connected fibers. In particular, the fibers are constant cycle subvarieties of $Y_t$.\n\\end{corollary}\n\\begin{proof}\nIt follows from the preceding theorem that the class $[P_t]$ of the subvarieties $P_t\\subset Y_t$ for a deformation $Y_t$ of $Y$ with $t \\in {\\operatorname{Def}}(Y,N)$ is (a multiple of) $[P]$. The proof showed moreover that $P_t$ is covered by deformations of the general fiber $F$ of the coisotropic fibration of $P$. The claim follows as $F$ was rationally connected and rational connectedness is known to be invariant under deformations (for smooth varieties).\n\\end{proof}\n\n \n\n\\begin{question}\\label{question contraction deforms locally trivially}  Let $X$ be a symplectic variety, $Y\\to X$ a symplectic resolution, $N=\\tilde q^{-1}(N_1(Y\/X))$ (cf. Lemma \\ref{lemma symplektisch}), and\n${\\mathscr Y}\\to{\\mathscr X}$ be the morphism between universal families over $S:={\\operatorname{Def}}(Y,N)={\\operatorname{Def}}^{\\mathrm{lt}}(X)$ from diagram \\eqref{eq diag lt}. In this case we ask:\n\n\\emph{Is ${\\mathscr Y} \\to {\\mathscr X}$ a locally trivial deformation of $Y \\to X$? }\n\n\\noindent Note that if this were the case, the whole diagram \n\\[\\xymatrix{\n{\\mathscr E} \\ar[d]\\ar[r]& {\\mathscr Y}\\ar[d]\\\\\n{\\mathscr S} \\ar[r]&{\\mathscr X}\\\\\n}\\] \nof complex spaces over $S$ where ${\\mathscr S} \\to S$ is the singular locus of ${\\mathscr X} \\to S$ and ${\\mathscr E} \\to S$ is the exceptional locus of ${\\mathscr Y} \\to {\\mathscr X}$ were a locally trivial (in particular flat) deformation over $S$ of its central fiber.\n\\end{question}\n\n\n\n\n\n\n\n\n\\section{Period maps and monodromy groups}\\label{sec monodromy}\n\n\\subsection{Marked moduli spaces}  \n\nRecall that given a lattice $\\Lambda$ of signature $(3,{\\rm rk}\\,\\Lambda-3)$, we define an analytic coarse moduli space $\\mathfrak{M}_{\\Lambda}$ of $\\Lambda$-marked irreducible holomorphic symplectic manifolds $(Y,\\mu)$ where $\\mu:\\Lambda\\xrightarrow{\\cong}H^2(Y,{\\mathbb Z})$ is a marking by gluing together the Kuranishi families.  Likewise, we define $\\mathfrak{M}^{{\\rm{lt}}}_\\Lambda$ to be the analytic space obtained by gluing together the locally trivially Kuranishi spaces of $\\Lambda$-marked symplectic varieties. Note that $\\mathfrak{M}_\\Lambda$ itself is a disjoint union of connected components of $\\mathfrak{M}^{\\rm{lt}}_\\Lambda$.\n\nWe define the period domain\n\\[\\Omega_\\Lambda:=\\{{\\mathbb C}\\cdot\\sigma\\mid (\\sigma,\\sigma)=0, (\\sigma,\\bar\\sigma)>0\\}\\subset \\P(\\Lambda\\otimes{\\mathbb C}).\\]\nBy the local Torelli theorem, see Corollary \\ref{corollary local torelli}, there is a period map $P:\\mathfrak{M}^{\\rm{lt}}_\\Lambda\\to \\Omega_\\Lambda$ that is a local isomorphism. \n\n\nLet $X$ be a $\\Lambda$-marked symplectic variety.  By Namikawa's result \\cite[Theorem 2.2]{Na01}, the deformation type of a symplectic resolution is constant along each connected component of $\\mathfrak{M}^{\\rm{lt}}_\\Lambda$.  Given a signature $(3,{\\rm rk}\\,\\Lambda'-3)$ lattice $\\Lambda'$ and a primitive embedding $\\iota:\\Lambda{\\, \\hookrightarrow\\,}\\Lambda'$, we define a compatibly marked symplectic resolution $\\pi:(Y,\\mu)\\to (X,\\nu)$ to be a symplectic resolution $\\pi$ and a commutative diagram   \n\n\n\\[\\xymatrix{\n\\Lambda'\\ar[r]^{\\mu}_\\cong&H^2(Y,{\\mathbb Z})\\\\\n\\Lambda\\ar[u]^\\iota\\ar[r]_\\nu^\\cong&H^2(X,{\\mathbb Z})\\ar[u]_{\\pi^*}\n}\\]\n\nWe define $\\mathfrak{M}^\\mathrm{res}_{\\Lambda',\\Lambda}$ to be the analytic space obtained by gluing together the bases of locally trivial Kuranishi spaces ${\\mathscr X}\\to {\\operatorname{Def}}^{\\rm{lt}}(X)$ together with a choice of compatibly marked simultaneous resolution ${\\mathscr Y}\\to{\\mathscr X}$ modulo the following equivalence relation: we identify $\\pi:(Y,\\mu) \\to (X,\\nu)$ with $\\pi':(Y',\\mu') \\to (X',\\nu')$ provided there is an isomorphism \n\\[\\xymatrix{Y\\ar[r]^\\cong\\ar[d]_{\\pi}&Y'\\ar[d]^{\\pi'}\\\\\nX \\ar[r]_\\cong&X'}\\]\n compatible with the marking.  There are obvious forgetful maps that fit into a diagram\n \\[\\xymatrix{\n&\\mathfrak{M}^{\\rm{lt}}_\\Lambda\\ar[r]^P&\\Omega_{\\Lambda}\\ar[dd]\\\\ \n\\mathfrak{M}^\\mathrm{res}_{\\Lambda',\\Lambda}\\ar[ur]\\ar[dr]&&\\\\\n&\\mathfrak{M}_{\\Lambda'}\\ar[r]_P&\\Omega_{\\Lambda'}\\\\\n }\\]\n where the vertical arrow is the embedding of $\\Omega_\\Lambda$ into $\\Omega_{\\Lambda'}$ as the Noether-Lefschetz locus $\\Omega_{\\Lambda'}\\cap\\P(\\Lambda\\otimes{\\mathbb C})$.\n \n \\subsection{Monodromy groups}\n \n Each of the above moduli spaces $\\mathfrak{M}$ has a corresponding notion of parallel transport operator, which we call locally trivial parallel transport operators for $\\mathfrak{M}^{\\rm{lt}}_\\Lambda$ and simultaneously resolved parallel transport operators for $\\mathfrak{M}^\\mathrm{res}_{\\Lambda',\\Lambda}$.  A simultaneously resolved parallel transport operator from $\\pi:Y\\to X$ to $\\pi':Y'\\to X'$ yields a diagram\n \\[\\xymatrix{\n H^2(Y,{\\mathbb Z})\\ar[r]^f&H^2(Y',{\\mathbb Z})\\\\\n H^2(X,{\\mathbb Z})\\ar[u]^{\\pi^*}\\ar[r]_g&H^2(X',{\\mathbb Z})\\ar[u]_{\\pi'^*}\n }\\]\n and $f,g$ are evidently locally trivial parallel transport operators.\n    We adopt the following notation for the associated monodromy groups:  \n \\begin{enumerate}\n \\item \n $\\operatorname{Mon}(X)^{\\rm{lt}}\\subset \\operatorname{GL}(H^*(X,{\\mathbb Z}))$ will be the monodromy group associated to locally trivial families, and $\\operatorname{Mon}^2(X)^{\\rm{lt}}\\subset\\O(H^2(X,{\\mathbb Z}))$ its image in $\\operatorname{GL}(H^2(X,{\\mathbb Z}))$ under the restriction of the action to the weight 2 cohomology. Here, the orthogonal group is taken with respect to the restriction of the Beauville--Bogomolov--Fujiki form to $H^2(X,{\\mathbb C}) \\subset H^2(Y,{\\mathbb C})$ for any resolution of singularities. As mentioned before, by \\cite[Theorem 8]{Na01a} the restriction of the quadratic form does not depend on the choice of resolution.\n \\item \n Likewise for a symplectic resolution $\\pi:Y\\to X$ we define $\\operatorname{Mon}(\\pi)$ and $\\operatorname{Mon}^2(\\pi)$ to be the monodromy groups associated to deformations of $\\pi$ that are locally trivial on $X$.  \n \\item The monodromy groups associated to smooth deformations of $Y$ for which a given primitive sublattice $L\\subset H^2(Y,{\\mathbb Z})$ remains algebraic will be denoted $\\operatorname{Mon}(Y)_L$ and $\\operatorname{Mon}^2(Y)_L$.\n \n \\item  We define $\\operatorname{Iso}^2(\\pi)\\subset \\operatorname{Mon}^2(Y)$ to be the stabilizer of $\\pi^*H^2(X,{\\mathbb Z})$, and $\\operatorname{Iso}^2(X)$ to be the image of $\\operatorname{Iso}^2(\\pi)$ in $\\O(H^2(X,{\\mathbb Z}))$.\n \\end{enumerate}\n   Obviously $\\operatorname{Mon}^2(Y)_L\\subset\\operatorname{Iso}^2(\\pi)$ for $L=(\\pi^*H^2(X,{\\mathbb Z}))^\\perp$, the orthogonal complement of which is identified over ${\\mathbb Q}$ with $N_1(Y\/X)$ under the Beauville--Bogomolov--Fujiki pairing.  Finally, in the presence of a compatible marking $(\\mu,\\nu)$ with $\\mu(\\Pi)=L$, we replace $Y,X,\\pi,L$ by $\\Lambda',\\Lambda,\\iota,\\Pi$ in the notation to denote the groups obtained in $\\O(\\Lambda')$ and $\\O(\\Lambda)$ using the marking.\n   \nThe following is a corollary of a well-known theorem of Sullivan \\cite{Sullivan}, see Corollary \\ref{corollary monodromy group} for its analog for singular varieties.\n\\begin{corollary}\\label{corollary iso finite index} Let $\\pi:Y \\to X$ be an irreducible symplectic resolution. Then $\\operatorname{Iso}^2(X)$ is a finite index subgroup of $\\O(H^2(X,{\\mathbb Z}))$.\n\\end{corollary}\n\\begin{proof} Let $\\tilde\\O(H^2(X,{\\mathbb Z}))\\subset \\O(H^2(X,{\\mathbb Z}))$ be the subgroup acting as $\\pm 1$ on the discriminant group.  Then $\\tilde\\O(H^2(X,{\\mathbb Z}))$ embeds to $\\O(H^2(Y,{\\mathbb Z}))$ by acting as the identity on $N=(\\pi^*H^2(X,{\\mathbb Z}))^\\perp$, by standard lattice theory \\cite[Corollary 1.5.2]{Ni}. Thus, $\\operatorname{Iso}^2(\\pi)$ contains $\\tilde\\O(H^2(X,{\\mathbb Z}))\\cap \\operatorname{Mon}^2(Y)$. By \\cite{Sullivan} the group $\\operatorname{Mon}^2(Y)$ has finite index in $\\O(H^2(Y,{\\mathbb Z}))$, see also \\cite[Lemma 7.5]{markmantor}, so that the intersection $\\tilde\\O(H^2(X,{\\mathbb Z}))\\cap \\operatorname{Mon}^2(Y)$ has finite index in $\\tilde\\O(H^2(X,{\\mathbb Z}))$.  The inclusion $\\tilde\\O(H^2(X,{\\mathbb Z}))\\cap \\operatorname{Mon}^2(Y) \\subset \\operatorname{Iso}^2(\\pi)$ entails that $\\operatorname{Iso}^2(X)$ has finite index in $\\O(H^2(X,{\\mathbb Z}))$.\n\\end{proof}\n\n   \n Before stating the main proposition of this subsection, we make a brief detour to describe an assumption which allows for a complete description of $\\operatorname{Mon}^2(X)^{\\rm{lt}}$.  Recall that for $Y$ an irreducible holomorphic symplectic manifold, the positive cone $C^{1,1}_Y\\subset H^{1,1}(Y,{\\mathbb R})$ is the connected component of $\\{\\alpha \\in H^{1,1}(Y,{\\mathbb R})\\mid q(\\alpha)>0\\}$ containing the K\\\"ahler cone. \nIt has a wall-and-chamber decomposition such that the complement of the walls are the images of the K\\\"ahler cones of birational models \n(see \\cite{markmantor} for details).\n\\begin{property} \\label{property} For an irreducible holomorphic symplectic manifold $Y$ and a primitive negative definite sublattice let $L\\subset H^2(Y,{\\mathbb Z})$, we say that property $\\operatorname{Fin}_L(Y)$ holds if for the very general small deformation $Y'$ of $Y$ with $\\operatorname{Pic}(Y')=L$ there are only finitely many K\\\"ahler chambers.  For an irreducible symplectic variety $X$, we say property $\\operatorname{Fin}(X)$ holds if property $\\operatorname{Fin}_L(Y)$ holds, where $\\pi:Y\\to X$ is a symplectic resolution and $L=(\\pi^*H^2(X,{\\mathbb Z}))^\\perp$. \n\\end{property}\nProperty $\\operatorname{Fin}_L(Y)$ follows for all such lattices $L$ from a version of the Kawamata-Morrison conjecture for the K\\\"ahler cone, and is known to hold provided $b_2(Y)\\neq 5$ (see \\cite{AV}).  This in particular includes all known deformation types.  $\\operatorname{Fin}_L(Y)$ is also trivially true for any deformation class if ${\\rm rk}\\,L=1$, see Proposition \\ref{proposition rank one} below.\n\\begin{proposition}\\label{proposition parallel transport} Let $X_1,X_2$ be locally trivially deformation equivalent irreducible symplectic varieties.  \n\\begin{enumerate}\n\\item\\label{pto item one} Let $g:H^2(X_1,{\\mathbb Z})\\to H^2(X_2,{\\mathbb Z})$ be a locally trivial parallel transport operator. For any choice of symplectic resolutions $\\pi_i:Y_i\\to X_i$, there exists a parallel transport operator $f:H^2(Y_1,{\\mathbb Z})\\to H^2(Y_2,{\\mathbb Z})$ such that $f\\vert_{H^2(X_1,{\\mathbb Z})}=g$.\n\\item\\label{pto item two} Assume $b_2(X_1)>3$.  For any choice of symplectic resolutions $\\pi_i:Y_i\\to X_i$, and any parallel transport operator $f:H^2(Y_1,{\\mathbb Z})\\to H^2(Y_2,{\\mathbb Z})$ sending $\\pi^*H^2(X_1,{\\mathbb Z})$ to $\\pi^*H^2(X_2,{\\mathbb Z})$, the restriction \\linebreak $H^2(X_1,{\\mathbb Z})\\to~H^2(X_2,{\\mathbb Z})$ is a locally trivial parallel transport operator.\n\\item\\label{pto item three} Additionally assume property $\\operatorname{Fin}(X_1)$.  Then the locally trivial parallel transport operators $H^2(X_1,{\\mathbb Z})\\to H^2(X_2,{\\mathbb Z})$ are precisely those arising from restricting simultaneously resolved parallel transport operators from $\\pi_1$ to $\\pi_2$ for some choice of symplectic resolutions $\\pi_i:Y_i\\to X_i$.\n\\item\\label{pto item four} The simultaneously resolved parallel transport operators from $\\pi_1$ to $\\pi_2$ are precisely the parallel transport operators $H^2(Y_1,{\\mathbb Z})\\to H^2(Y_2,{\\mathbb Z})$ arising from families ${\\mathscr Y}\\to B$ along which $(\\pi_i^*H^2(X_i,{\\mathbb Z}))^\\perp$ remains algebraic.\n\\end{enumerate}\n\\end{proposition}\n \n \\begin{proof}\nFor the proof of \\eqref{pto item one}, let $\\pi_i:Y_i \\to X_i$ be resolutions and let ${\\mathscr X}\\to B$ be a locally trivial deformation of $X_1$ and $X_2$ over a connected base $B$. We can always choose the deformation in such a way that the very general fiber has Picard number zero. Let us call $b_1,b_2 \\in B$ the points corresponding to $X_1, X_2$. Let us choose a very general path $\\gamma$ from $b_1$ to $b_2$ that realizes the parallel transport operator $p$ and cover it with finitely many disks $D_1,\\ldots, D_n$ such that over each disk we have a simultaneous resolution ${\\mathscr Y}^{(i)}\\to {\\mathscr X}\\vert_{D_i} \\to D_i$, where for disks containing $b_1$ or $b_2$ the corresponding ${\\mathscr Y}^{(i)}$ should be a deformation of $Y_1$ respectively $Y_2$. We may assume that $D_i \\cap D_{i+1}$ is nonempty for each $i$ and that our path is obtained by concatenation of paths $\\gamma_i$ from $c_{i-1}$ to $c_i$ inside $D_i$ for some points $c_i \\in D_i \\cap D_{i+1}$ where $c_0=b_1, c_n=b_2$. We may furthermore assume that ${\\mathscr X}_{c_i}$ has Picard number zero for $i\\neq 0,n$. For each such $i$ the irreducible symplectic manifolds ${\\mathscr Y}^{(i)}_{c_i}$ and ${\\mathscr Y}^{(i+1)}_{c_i}$ are birational and therefore there exists a parallel transport operator $\\delta_i:H^2({\\mathscr Y}^{(i)}_{c_i},{\\mathbb Z}) \\to H^2({\\mathscr Y}^{(i+1)}_{c_i},{\\mathbb Z})$ which is a Hodge isometry. Combining with Lemma \\ref{lemma symplektisch} we see that $\\delta_i$ maps $H^2({\\mathscr X}_{c_i},{\\mathbb Z})$ to itself. If $f_i$ denotes the parallel transport operator induced by $\\gamma_i$ then $f:=f_n \\circ \\delta_n \\circ f_{n-1} \\circ \\ldots \\circ \\delta_1 \\circ f_1:H^2(Y_1,{\\mathbb Z})\\to H^2 (Y_2,{\\mathbb Z})$ is the sought-for parallel transport operator.\n\n  For part \\eqref{pto item two}, let $\\mathfrak{N}_{\\Lambda'}$ be a given connected component of $\\mathfrak{M}_{\\Lambda'}$, and denote by $\\mathfrak{N}_{\\Lambda}^{\\rm{lt}}\\subset \\mathfrak{M}_{\\Lambda}^{\\rm{lt}}$ the set of $(X,\\nu)$ admitting a compatibly marked symplectic resolution $\\pi:(Y,\n \\mu)\\to (X,\\nu)$ such that $(Y,\\mu)\\in \\mathfrak{N}_{\\Lambda'}$. The subset $\\mathfrak{N}_{\\Lambda}^{\\rm{lt}}\\subset \\mathfrak{M}_{\\Lambda}^{\\rm{lt}}$ is open and closed by Proposition \\ref{prop defo}, and therefore it is a union of path-components as $\\mathfrak{M}^{\\rm{lt}}_\\Lambda$ is locally path-connected.  It suffices to show $\\mathfrak{N}^{\\rm{lt}}_\\Lambda$ is path-connected, since if $(X_1,\\nu_1)\\in \\mathfrak{N}^{\\rm{lt}}_\\Lambda$ and $\\pi_1:(Y_1,\\mu_1)\\to (X_1,\\nu_1)$ is a compatibly marked resolution, then for $f$ as in the claim we have $(X_2,f\\nu_1)\\in \\mathfrak{N}_{\\Lambda}^{\\rm{lt}}$.  The path-connectedness will in turn follow from the following\n \n\\begin{lemma}\\label{lemma density}The image of the period map $P:\\mathfrak{N}_{\\Lambda}^{\\rm{lt}}\\to \\Omega_\\Lambda$ is an open $\\operatorname{Iso}^2(\\Lambda)$-invariant path-connected subset and the preimage over any Picard rank 0 point of $\\Omega_\\Lambda$ consists of a unique point.\n\\end{lemma}\n \\begin{proof}The fact that $\\mathfrak{N}^{\\rm{lt}}_\\Lambda$ is invariant under $\\operatorname{Iso}^2(\\Lambda)$ is just a restatement of the above observation, so the image $P(\\mathfrak{N}_\\Lambda^{\\rm{lt}})$ is clearly open and $\\operatorname{Iso}^2(\\Lambda)$-invariant. In fact, it is the complement of countably many real analytic sub-manifolds of codimension $\\geq 2$ by a result of Verbitsky \\cite[Theorem 1.16]{V15} (see Remark \\ref{remark verbitsky} below), and is therefore path-connected (see for example \\cite[Lemma 4.10]{V13}). Here we used that $\\operatorname{Iso}^2(\\Lambda)$ has finite index in $\\O(\\Lambda)$ by Corollary \\ref{corollary iso finite index}.\n\nIt remains to prove the final claim.  Suppose there are two points $(X_1,\\nu_1),$ $(X_2,\\nu_2)\\in \\mathfrak{N}^{\\rm{lt}}_\\Lambda$ with the same period and with Picard rank zero each.\n As for smooth varieties, neither $X_1$ nor $X_2$ contain any curves for otherwise the nondegeneracy of the intersection pairing would contradict the nonexistence of divisors.\n Note that $\\nu_2\\circ\\nu_1^{-1}:H^2(X_1,{\\mathbb Z})\\to H^2(X_2,{\\mathbb Z})$ is a Hodge isometry.  Let us choose compatibly marked resolutions $(Y_1,\\mu_1),(Y_2,\\mu_2)\\in\\mathfrak{N}_{\\Lambda'}$.  It follows that $\\mu_2\\circ\\mu_1^{-1}:H^2(Y_1,{\\mathbb Z})\\to H^2(Y_2,{\\mathbb Z})$ is a Hodge isometry as well.  Thus, $(Y_1,\\mu_1)$ and $(Y_2,\\mu_2)$ have the same period and are therefore birational, by the global Torelli theorem.  It follows that $X_1$ and $X_2$ are birational, and because neither has any effective curves, we have $X_1\\cong X_2$. This can be seen just as in the algebraic context, apply e.g. \\cite[Lemma 1.15 (b)]{Debarre} to a resolution of indeterminacy.\n\nLet us assume first, that the Hodge structure on $H^2(X_1,{\\mathbb Z})$ is Mumford--Tate general. Then the isomorphism induces $\\pm \\nu_2\\circ\\nu_1^{-1}$ on cohomology by Corollary \\ref{corollary mumford tate}. \nLet us recall from \\cite[Definition 3.5]{Markman} that Markman defined an orientation class in $H^2(C_Y,{\\mathbb Z}){\\ \\cong\\ } {\\mathbb Z}$ that is constant on the component $\\mathfrak{N}_{\\Lambda'}$, see also \\cite[\\S 4]{markmantor}, and we have a diagram\n \\[\\xymatrix{\n  H^2(C_{Y_2},{\\mathbb Z})\\ar[r]^{(\\mu_2\\circ\\mu_1^{-1})^*}& H^2(C_{Y_1},{\\mathbb Z})\\\\\n  H^2(C_{X_2}.{\\mathbb Z})\\ar[u]^{\\pi_2^*}_\\cong\\ar[r]_{(\\nu_2\\circ\\nu_1^{-1})^*}& H^2(C_{X_1},{\\mathbb Z})\\ar[u]_{\\pi_1^*}^\\cong\n }\\] \n so in fact $(X_1,\\nu_1)\\cong (X_2,\\nu_2)$.\n\n\tNow let $(X_1,\\nu_1),(X_2,\\nu_2)\\in \\mathfrak{N}^{\\rm{lt}}_\\Lambda$ be arbitrary Picard rank zero points with the same period. Also in this case, we have seen that $X_1,X_2$ are birational and having Picard rank zero, they are even biholomorphic. \n\t\nWe take a very general disk $\\Delta \\subset \\Omega_\\Lambda$ centered at $P(X_1,\\nu_1)=P(X_2,\\nu_2)$ and lift them to disks $\\Delta_1$ and $\\Delta_2$ in $\\mathfrak{N}^{\\rm{lt}}_\\Lambda$ centered at $(X_1,\\nu_1)$ respectively $(X_2,\\nu_2)$. Up to shrinking the disks, we may assume that there are locally trivial families ${\\mathscr X}^{(1)} \\to \\Delta_1$, ${\\mathscr X}^{(2)} \\to \\Delta_2$ of irreducible symplectic varieties with central fibers $X_1$ respectively $X_2$ such that $P(\\Delta_1)=\\Delta=P(\\Delta_2)$ where we endow these families with the markings induced by $\\nu_1$ and $\\nu_2$. \nAs $\\Delta$ was chosen very general, the above argument applies to the very general fiber of ${\\mathscr X}^{(1)}\\to\\Delta_1$, ${\\mathscr X}^{(2)}\\to\\Delta_2$ -- namely the Mumford--Tate general ones -- so that for the corresponding $t \\in \\Delta$ there is an isomorphism $\\phi_t:{\\mathscr X}^{(1)}_t \\to {\\mathscr X}^{(2)}_t$ such that $\\phi_t$ induces $\\nu_{2,t}\\circ \\nu_{1,t}^{-1}$ on cohomology.\nFor each such $t$ we consider the graph $\\Gamma_t' \\subset {\\mathscr X}^{(1)}_t \\times {\\mathscr X}^{(2)}_t$ and by the usual Hilbert scheme argument we obtain a family $\\Gamma \\subset {\\mathscr X}^{(1)}\\times_\\Delta {\\mathscr X}^{(2)}$ whose fibers $\\Gamma_t$ coincide with the above $\\Gamma_t'$ for uncountably many $t$.\nClearly, the locus of $t\\in \\Delta$ where the fiber $\\Gamma_t$ of $\\Gamma$ is an isomorphism is Zariski open and by shrinking our disk we may assume that $\\Gamma_t$ is the graph of an isomorphism everywhere outside $t=0$. \nUp to shrinking the disks once more we may assume that there are irreducible symplectic resolutions ${\\mathscr Y}^{(1)} \\to {\\mathscr X}^{(1)}$ and ${\\mathscr Y}^{(2)} \\to {\\mathscr X}^{(2)}$. We take the strict transform of $\\Gamma$ and obtain a cycle $\\wt\\Gamma \\subset {\\mathscr Y}^{(1)}\\times_\\Delta{\\mathscr Y}^{(2)}$. For each $t\\in \\Delta$ the induced morphism $[\\wt\\Gamma_t]_* : H^2({\\mathscr Y}^{(1)}_t,{\\mathbb Z}) \\to H^2({\\mathscr Y}^{(2)}_t,{\\mathbb Z})$ maps $H^2({\\mathscr X}^{(1)}_t,{\\mathbb Z})$ to $H^2({\\mathscr X}^{(2)}_t,{\\mathbb Z})$ and for $t\\neq 0$ the restriction to $H^2({\\mathscr X}^{(1)}_t,{\\mathbb Z})$ coincides with $\\nu_{2,t}\\circ \\nu_{1,t}^{-1}$. We conclude that it does so for $t=0$, too.\n\nIt remains to show that the restriction of $[\\wt\\Gamma_0]_*$ to $H^2(X_1,{\\mathbb Z})$ is induced by an isomorphism $X_1\\to[{\\ \\cong\\ }] X_2$. We argue as in the proof of \\cite[Theorem 4.3]{Huy99}: Let $Z$ be an irreducible component of $\\wt\\Gamma_0$. The restriction of $[Z]_*$ to $H^2(X_1,{\\mathbb Z})$ is a morphism of Hodge structures and hence either zero or an isomorphism as the Hodge structure on $H^2(X_1,{\\mathbb Z})$ is simple. Suppose that for the given $Z$ it is an isomorphism. Then there is $0\\neq \\lambda \\in {\\mathbb C}$ such that $[Z]_* \\sigma_1=\\lambda\\sigma_2$ where $\\sigma_1,\\sigma_2$ denote the symplectic forms on $Y_1={\\mathscr Y}^{(1)}_0,Y_2={\\mathscr Y}^{(2)}_0$. Observe that $\\sigma_i \\in H^2(X_i,{\\mathbb C}) \\subset H^2(Y_i,{\\mathbb C})$ by Lemma \\ref{lemma symplektisch} for $i=1,2$.\nDenoting $p_1$ and $p_2$ the projections from $Y_1\\times Y_2$ to its factors we have\n\\[\n\\begin{aligned}\n0 &\\neq \\lambda \\int_{Y_2} (\\sigma_2 \\bar\\sigma_2)^n = \\int_{Y_2} ({p_2}_*([Z] \\cup p_1^*\\sigma_1)) ({\\sigma_2}^{n-1} \\bar{\\sigma}_2^n)\\\\\n&= \\int_{Y_1\\times Y_2} [Z]\\cup p_1^*\\sigma_1 (p_2^*{\\sigma_2})^{n-1} (p_2^*\\bar{\\sigma}_2)^n = \\int_{Z} (p_1^*\\sigma_1) (p_2^*{\\sigma_2})^{n-1} (p_2^*\\bar{\\sigma}_2)^n.\\\\\n\\end{aligned}\n\\]\nReplacing $Z$ by a resolution of singularities we may assume it to be smooth. The integral can only be nonzero if $p_2:Z \\to Y_2$ (and hence also the composition $Z\\to X_2$) is dominant for otherwise the $(0,2n)$-form $(p_2^*\\bar{\\sigma}_2)^n$ would vanish. As $(p_1)_*[\\wt\\Gamma_t]=[({\\mathscr Y}_1)_t]$ and $(p_2)_*[\\wt\\Gamma_t]=[({\\mathscr Y}_2)_t]$ for $t\\neq 0$, the same is true for $t=0$ and therefore $Z$ has to map birationally onto $X_2$. We infer that $H^0(\\Omega_{Z}^2)= {\\mathbb C} \\cdot \\sigma_Z$ and $\\sigma_Z=p_2^*\t\\sigma_2$ is generically nondegenerate so that $Z\\to X_1$ is also birational (otherwise $p_1^*\\sigma_1=0$). Put together, the restriction of $[Z']_*$ to $H^2(X_1,{\\mathbb C})$ vanishes for all other irreducible components $Z'\\neq Z$ of $\\wt\\Gamma_0$ and hence $\\nu_2\\circ\\nu_1^{-1}=[Z]_*$ on $H^2(X_1,{\\mathbb C})$.\n The image of $Z$ in $X_1\\times X_2$ is the graph of a birational map which has to be an isomorphism $\\phi:X_2 \\to X_1$ as $X_1,X_2$ have Picard rank zero. The isomorphism $[Z]_*:H^2(Y_1,{\\mathbb C})\\to H^2(Y_2,{\\mathbb C})$ coincides with $p_{2*}\\circ p_1^*$ and we have that $\\phi \\circ \\pi_2 \\circ p_2 = \\pi_1 \\circ p_1$ so that\n\\[\n\\nu_2\\circ\\nu_1^{-1} = [Z]_* \\circ \\pi_1^* = \\pi_2^* \\circ \\phi^*\n\\]\nin cohomology which concludes the proof.\n  \\end{proof}\n \nThe proofs of parts \\eqref{pto item three} and \\eqref{pto item four} proceed as in \\cite[Corollary 5.11]{Ma13} (see also Section 7.1 of \\cite{markmantor}), so we only sketch the argument.  It will be enough if we can lift, for part \\eqref{pto item three}, families ${\\mathscr X}\\to B$ realizing locally trivial parallel transport operators and, for part \\eqref{pto item four}, families ${\\mathscr Y}\\to B$ realizing parallel transport operators, to simultaneously resolved families ${\\mathscr Y}'\\to{\\mathscr X}'\\to B'$.  It suffices to consider a disk $B$, and the claim follows after covering the family with liftable disks and gluing them along nodes if there is a dense set of points with finitely many lifts.  In both cases we use points corresponding to compatibly marked resolutions $\\pi:(Y,\\mu)\\to(X,\\nu)$ for which $\\operatorname{Pic}(X)=0$.  Indeed, for part \\eqref{pto item four}, given such a $(Y,\\mu)$, there is at most one compatibly marked contraction $(Y,\\mu)\\to(X,\\nu)$ by the lemma.  For part \\eqref{pto item three}, such a $(X,\\nu)$ admits finitely many compatibly marked resolutions assuming $\\operatorname{Fin}(X)$. \n \\end{proof}\n\n \\begin{remark}\nWith almost the same reasoning one can show that every $(X',\\nu'),(X'',\\nu'')\\in \\mathfrak{N}^{\\rm{lt}}_\\Lambda$ with the same period and with Picard group generated by an ample divisor are isomorphic as marked irreducible symplectic varieties.\n \\end{remark}\n\n \\begin{remark}\\label{remark verbitsky}\nWe make a remark about Verbitsky's result \\cite[Theorem 1.16]{V15}.  The period domain $\\Omega_\\Lambda$ can alternatively be thought of as the space of positively oriented positive-definite planes in $\\Lambda_{\\mathbb R}$.  Indeed, given a Hodge structure on $\\Lambda$, $(\\Lambda_ {\\mathbb C})^{2,0}\\oplus(\\Lambda_{\\mathbb C})^{0,2}$ is defined over ${\\mathbb R}$ and corresponds to a positively oriented positive definite plane $P\\subset \\Lambda_{\\mathbb R}$.  The orientation is given by $(\\Re\\sigma,\\Im\\sigma)$, for $\\sigma\\in (\\Lambda_{\\mathbb C})^{2,0}$.  Fixing such a plane $P_0$ as a basepoint, we obtain an isomorphism\n \\begin{equation} \\label{eq period homogeneous}\n\\Omega_\\Lambda\\cong \\operatorname{SO}(\\Lambda_{\\mathbb R})\/\\operatorname{SO}(P_0)\\times\\operatorname{SO}(P_0^\\perp)\n\\end{equation}\n Given an arithmetic lattice $\\Gamma\\subset \\operatorname{SO}(\\Lambda_{\\mathbb R})$, by an application of Ratner's theorem the orbit closure of a point $\\Gamma g\\in\\Gamma\\backslash\\operatorname{SO}(\\Lambda)$ under $\\operatorname{SO}(P_0)\\times\\operatorname{SO}(P_0^\\perp)$ is one of three possibilities, depending on $r:={\\rm rk}\\,(gP_0\\cap \\Lambda)$:\n \\begin{enumerate}\n \\item if $r=2$, the orbit is closed;\n \\item if $r=1$, the orbit closure is $\\Gamma g \\operatorname{SO}(gP_0^\\perp).\\operatorname{SO}(gP_0)$;\n \\item if $r=0$, the orbit is dense.\n \\end{enumerate} \n Correspondingly, there are three possibilities for the orbit closure of $\\omega\\in\\Omega_\\Lambda$ under $\\Gamma$ depending on the \\emph{transcendental rational rank} $r$ of the positive plane $P_\\omega$:  it is closed if $r=2$, dense if $r=0$, and if $r=1$ it is a union of $G_{\\lambda}$ for $\\lambda\\in\\Gamma.\\lambda_0$ in the orbit of some primitive $0\\neq \\lambda_0\\in\\Lambda$, where $G_\\lambda$ is the subset of positive planes $P$ containing some $\\lambda$.  Note that $G_\\lambda$ is a real-analytic submanifold of real codimension ${\\rm rk}\\, \\Lambda-2$ in $\\Omega_\\Lambda$.  The $r=1$ case was omitted in \\cite{V15}.\n \\end{remark}\nFor an irreducible symplectic resolution $\\pi:Y\\to X$ we denote by $N$ the negative definite lattice $(\\pi^*H^2(X,{\\mathbb Z}))^\\perp$, compare to \\eqref{eq n}. In terms of monodromy groups, Proposition \\ref{proposition parallel transport} means:\n  \\begin{corollary}\\label{corollary monodromy}  Let $\\pi:Y\\to X$ be an irreducible symplectic resolution.  \n \\begin{enumerate}\n\\item We have $\\operatorname{Mon}^2(X)^{\\rm{lt}}\\subset\\operatorname{Iso}^2(X)$ and if $b_2(X)>3$, then $\\operatorname{Mon}^2(X)^{\\rm{lt}} = \\operatorname{Iso}^2(X)$;\n\\item If in addition property $\\operatorname{Fin}(X)$ holds, then the image of $\\operatorname{Mon}^2(\\pi)$ in $\\operatorname{Mon}^2(X)^{\\rm{lt}}$ has finite index;\n\\item $\\operatorname{Mon}^2(\\pi)=\\operatorname{Mon}^2(Y)_N$.\n\n\\end{enumerate}\n \\end{corollary}\n In particular, we deduce the analog of Sullivan's theorem \\cite{Sullivan} in the smooth case:\n\n  \\begin{corollary}\\label{corollary monodromy group} Let $\\pi:Y \\to X$ be an irreducible symplectic resolution and suppose $b_2(X) >3$. Then the following hold:\n\\begin{enumerate}\n  \\item\\label{corollary monodromy group item one}  $\\operatorname{Mon}^2(X)^{\\rm{lt}}$ is a finite index subgroup of $\\O(H^2(X,{\\mathbb Z}))$;\n \\item\\label{corollary monodromy group item two}  If property $\\operatorname{Fin}(X)$ holds, then the forgetful map $\\operatorname{Mon}^2(\\pi)\\to\\operatorname{Mon}^2(X)^{\\rm{lt}}$ has finite kernel and finite index image.\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof}  By Corollary \\ref{corollary monodromy} it is sufficient to show that $\\operatorname{Iso}^2(X)$ has finite index in $\\O(H^2(X,{\\mathbb Z}))$, which has been done in Corollary \\ref{corollary iso finite index}. \nThe second claim follows from Corollary \\ref{corollary monodromy} and the fact that the kernel of $\\operatorname{Mon}^2(\\pi)\\to\\operatorname{Mon}^2(X)^{\\rm{lt}}$ is a subset of the finite group $O(N)$.\n \\end{proof}\n \n In fact, $\\operatorname{Mon}^2(X)^{\\rm{lt}}$ is a subgroup of the group $\\O^+(H^2(X,{\\mathbb Z}))$ of orthogonal transformations preserving the orientation of $H^2(C_X,{\\mathbb Z})$.  \n  \\begin{corollary}\\label{cor mon det}  Fix a deformation type of irreducible holomorphic symplectic manifolds.  The locally trivial deformation type of any irreducible symplectic variety $X$ with $b_2(X)>3$ admitting a symplectic resolution $\\pi: Y\\to X$ with $Y$ of that deformation type is determined by the $\\operatorname{Mon}^2(Y)$ orbit of $N_1(Y\/X)$ in $H_2(Y,{\\mathbb Z})$.  If property $\\operatorname{Fin}(X)$ holds, it is uniquely determined.\n\\end{corollary}\n\nWe define the relative Picard rank of $X$ to be ${\\rm rk}\\, N_1(Y\/X)$ for a symplectic resolution $Y\\to X$; it is independent of the choice of resolution.  In the special case that $X$ has relative Picard rank one, Property $\\operatorname{Fin}(X)$ trivially holds, and we can compute the relevant monodromy groups more precisely.\n\n\\begin{proposition}\\label{proposition rank one}  Let $\\pi:Y\\to X$ be an irreducible symplectic resolution of relative Picard rank one and assume $b_2(X)>3$.  Let $\\operatorname{Iso}^+(\\pi)\\subset \\operatorname{Iso}^2(\\pi)$ be the subgroup acting trivially on $(\\pi^* H^2(X,{\\mathbb Z}))^\\perp$.  Then we have $\\operatorname{Mon}^2(\\pi)=\\operatorname{Iso}^+(\\pi)$.\n\\end{proposition}\n\\begin{proof}  See \\cite{Ma13}.  In the notation of the proof of Proposition \\ref{proposition parallel transport}, consider the fiber product\n\\[\\xymatrix{\n\\mathfrak{N}_{\\Lambda',\\Lambda}^\\mathrm{res}\\ar[d]\\ar[r]&\\mathfrak{N}_{\\Lambda'}\\times_{\\Omega_{\\Lambda'}}\\Omega_\\Lambda\\ar[d]\\\\\n\\mathfrak{N}_\\Lambda^{\\rm{lt}}\\ar[r]&\\Omega_\\Lambda\n}\\]\nThus, $\\mathfrak{N}_{\\Lambda',\\Lambda}^\\mathrm{res} \\subset \\mathfrak{M}_{\\Lambda',\\Lambda}^\\mathrm{res}$ is the space of those tuples $(Y',\\mu',X',\\nu',\\pi')$ for which $(X',\\nu')\\in \\mathfrak{N}_{\\Lambda}^{\\rm{lt}}$ and $(Y',\\mu')\\in \\mathfrak{N}_{\\Lambda'}$.\nLet $\\lambda\\in \\Lambda^\\perp\\subset \\Lambda'$ be a generator.  The main point is that the sign of $(\\alpha,\\mu'(\\lambda))$ at a point $\\pi':(Y',\\mu')\\to (X',\\nu') \\in \\mathfrak{N}_{\\Lambda',\\Lambda}^\\mathrm{res}$ is locally constant on $\\mathfrak{N}^\\mathrm{res}_{\\Lambda',\\Lambda}$, $\\alpha\\in H^2(Y',{\\mathbb R})$ is a K\\\"ahler form, since at every point either $\\mu'(\\lambda)$ or $-\\mu'(\\lambda)$ as a class in $H_2(Y',{\\mathbb Z})\\supset H^2(Y',{\\mathbb Z})$ is a positive rational multiple of the class of a rational curve on $Y'$. Consequently, the space $\\mathfrak{N}_{\\Lambda',\\Lambda}^\\mathrm{res}$ has precisely two path-connected components, and $\\operatorname{Mon}^2(\\pi)\\subset \\operatorname{Iso}^+(\\pi)$.  On the other hand, certainly $\\operatorname{Iso}^+(\\pi)\\subset \\operatorname{Mon}^2(\\pi)$.  \n\\end{proof}\n\t\n\\begin{remark}\\label{remark rigid}\nIt should be noted that the assumption $b_2(X)>3$ that we used several times throughout this section is nontrivial. Unlike in the smooth case, we know that there are examples of irreducible symplectic varieties $X$ with $b_2(X)=3$:  it may well happen that a birational contraction $Y \\to X$ of an irreducible symplectic manifold $Y$ contracts a negative definite subspace of $H^2(Y,{\\mathbb R})$ of maximal dimension. In the case of $K3$ surfaces for example, one may take $X$ to be $S\/G$ where $S$ is a $K3$ surface and $G$ is a group of symplectic automorphisms with minimal invariant second cohomology lattice $H^2(S,{\\mathbb Z})^G$ (i.e. of rank $3$).\n Finite groups of symplectic automorphisms were classified by Nikulin \\cite{Nikulin} and Mukai \\cite{Mukai}, explicit examples of groups with the sought-for rank of $H^2(S,{\\mathbb Z})^G$ may be found in \\cite{Xi,Ha}. An example of a different kind may be found in \\cite{OZ}. The minimal resolution $Y \\to X$ then gives us an example of a contraction of relative Picard rank $19$. The induced contraction ${\\rm Hilb}^n(Y) \\to {\\rm Sym}^n(Y) \\to {\\rm Sym}^n(X)$ gives an example of a contraction of a $K3^{[n]}$-type variety of dimension $2n$ with relative Picard rank $20$.\n\\end{remark}\n\t\nWe now prove the analog of Markman's Hodge theoretic formulation of the global Torelli theorem, see \\cite[Theorem 1.3]{markmantor}.\n\\begin{theorem}[Global Torelli Theorem]\\label{theorem global torelli}\nLet $X$, $X'$ be irreducible symplectic varieties and let $g:H^2(X,{\\mathbb Z}) \\to H^2(X',{\\mathbb Z})$ be a locally trivial parallel transport operator which is a Hodge isometry. Then there is a birational map $\\phi:X \\dashrightarrow X'$ which induces an isomorphism $\\phi_*:H^2(X,{\\mathbb Z})\\to H^2(X',{\\mathbb Z})$.\n\\end{theorem}\n\\begin{proof}\nThe choice of a marking $\\nu$ on $X$ induces via the parallel transport operator a marking $\\nu'$ on $X'$ and so the period maps give isomorphisms $S:={\\operatorname{Def}}^{\\rm{lt}}(X) \\to[(P')^{-1}\\circ P] {\\operatorname{Def}}^{\\rm{lt}}(X')$ thanks to the local Torelli theorem \\ref{corollary local torelli}. \nLet $\\pi:Y\\to X$, $\\pi':Y'\\to X'$ be irreducible symplectic resolutions. We infer from the local Torelli theorem for smooth varieties and Proposition \\ref{prop defo} that the period map in the same manner gives an isomorphism $S{\\ \\cong\\ } {\\operatorname{Def}}(Y,N){\\ \\cong\\ } {\\operatorname{Def}}(Y',N')$ where $N=(\\pi^*H^2(X,{\\mathbb Z}))^\\perp,N'=({\\pi'}^*H^2(X',{\\mathbb Z}))^\\perp$. Proposition \\ref{proposition parallel transport} \\eqref{pto item one} guarantees that $Y$ and $Y'$ lie in the same component of the moduli space of marked irreducible symplectic manifolds so that by Verbitsky's global Torelli theorem \\cite[Theorem 1.17]{V13} for each $t\\in S$ there is a birational map $\\phi_t:Y_t\\dashrightarrow Y'_{\\psi(t)}$ where $Y_t$ and $Y'_{\\psi(t)}$ are the corresponding deformations of $Y$ and $Y'$. \nIt is easily seen that the induced isomorphism on cohomology maps $N$ to $N'$. The claim follows therefore from Theorem \\ref{theorem huybrechts}.\n\\end{proof}\n\nAs in the smooth case an equivalent formulation of the global Torelli theorem is the following analog of \\cite[Theorem 1.16]{V13}, see also \\cite[Theorem 2.2]{markmantor}. \n\n\\begin{corollary}\\label{corollary global torelli}\nLet ${\\mathfrak{N}}_\\Lambda^{\\rm{lt}} \\subset {\\mathfrak{M}}_\\Lambda^{\\rm{lt}}$ be a connected component of the marked moduli space of irreducible symplectic varieties. Then the period map $P: {\\mathfrak{N}}_\\Lambda^{\\rm{lt}}\\to \\Omega_\\Lambda$ is generically injective and if $(X,\\nu),(X',\\nu')$ are in the same fiber of $P$, then $X$ and $X'$ are birational, isomorphic in codimension one, and the birational map induces a Hodge isometry $H^2(X,{\\mathbb Z})\\to H^2(X',{\\mathbb Z})$ compatible with the markings.\\qed\n\\end{corollary}\n\n Recall that a period $\\omega \\in \\Omega_\\Lambda$ may also be described by a positively oriented, positive-definite two plane $P_\\omega \\subset \\Lambda_{\\mathbb R}$. We call ${\\rm rk}\\left(P_\\omega \\cap \\Lambda\\right)$ the transcendental rational rank of $\\omega$.\n\\begin{theorem}[Surjectivity of the Period map]\\label{theorem almost surjectivity}Let ${\\mathfrak{N}}_\\Lambda^{\\rm{lt}} \\subset {\\mathfrak{M}}_\\Lambda^{\\rm{lt}}$ be a non\\-empty connected component with ${\\rm rk}\\Lambda >3$ such that the deformation type of the symplectic resolution has $b_2> 5$.  Then the period map $P: {\\mathfrak{N}}_\\Lambda^{\\rm{lt}}\\to \\Omega_\\Lambda$ is surjective.\n\\end{theorem}\n\nBefore the proof we record an observation that is interesting in its own right.  Recall that Amerik--Verbitsky \\cite[Definition 1.13]{AV} have introduced the notion of a monodromy birationally minimal (MBM) class.  For a smooth irreducible holomorphic symplectic manifold $Y$, a nonzero class $\\alpha\\in H^{1,1}(Y,{\\mathbb Q})$ with $q_Y(\\alpha)<0$ is MBM if up to the action of the monodromy group $\\alpha^\\perp \\subset H^{1,1}(Y,{\\mathbb R})$ is a face of the K\\\"ahler cone of a birational model of $Y$. Note that here by a face of a convex cone $C$ in a vector space $V$ we mean a linear subspace $U$ whose intersection with the boundary $\\partial C$ has nonempty interior.\n  Via the isomorphism $\\tilde q_Y:H^2(Y,{\\mathbb Q})\\to H_2(Y,{\\mathbb Q})$ coming from $q_Y$, we will also refer to MBM homology classes.  MBM classes are deformation invariant along deformations for which they remain of Hodge type $(1,1)$ \\cite[Theorem 1.17]{AV}.  \n\\begin{lemma}\\label{MBM}Let $\\pi:Y\\to X$ be an irreducible symplectic resolution with $b_2(Y)> 5$.  Then $N=\\wt q^{-1}(N_1(Y\/X)) \\subset H^2(Y,{\\mathbb Z})$ is rationally generated by MBM classes.\n\\end{lemma}\n\\begin{proof}  By the deformation invariance and Corollary \\ref{corollary projective}, we may assume $X$ and $Y$ are projective.  Then $N^\\perp$ in $H^{1,1}(Y,{\\mathbb Q})_{\\mathbb R}$ is a face of the nef cone, since $N_1(Y\/X)$ is extremal in the effective cone of curves.  Let $C \\subset H^{1,1}(Y,{\\mathbb Q})_{\\mathbb R}$ be the positive cone and denote $W$ the union of hyperplanes $\\alpha^\\perp$ over all MBM classes $\\alpha$.\n By \\cite[Theorem 1.19]{AV}, the ample cone is a connected component of the complement of $W$ in $C$ and by \\cite[Corollary 1.6]{AV14} integral MBM classes have bounded square.  In particular, the collection $\\{\\alpha^\\perp\\}$ is locally finite (in the positive cone).  Since $N$ is negative definite by Lemma \\ref{lemma symplektisch}, we can take a small ball $B\\subset N^\\perp$ which is both contained in the boundary of the ample cone and in the positive cone, so $B$ must be open in an intersection of finitely many $\\alpha^\\perp$.  It follows that $N_{\\mathbb Q}$ is generated by MBM classes.  \n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{theorem almost surjectivity}]\nFor a point $(X',\\nu')\\in\\mathfrak{N}^{\\rm{lt}}_\\Lambda$, the marking $\\nu'$ induces markings on the universal locally trivial deformation ${\\mathscr X}' \\to S={\\operatorname{Def}}^{\\rm{lt}}(X')$ of $X'$ so that we may interpret $S$ as a small open neighborhood of $(X',\\nu')$ in ${\\gothN_\\Lambda^\\lt}$. We infer from the local Torelli theorem \\ref{corollary local torelli} that $P(S) \\subset \\Omega_\\Lambda$ is open as well.  Evidently, the period map $P:\\mathfrak{N}_\\Lambda^{\\rm{lt}}\\to \\Omega_\\Lambda$ is $\\operatorname{Mon}^2(\\Lambda)^{\\rm{lt}}$-equivariant, so the complement of the image is a closed $\\operatorname{Mon}^2(\\Lambda)^{\\rm{lt}}$-invariant subset.  By Remark \\ref{remark verbitsky}, and the fact that any transcendental rational rank one orbit closure contains a transcendental rational rank two orbit, it suffices to show that every maximal Picard rank period is in the image.\n\nAssume that $\\omega \\in  \\Omega_\\Lambda$ corresponds to a Hodge structure whose Picard rank equals ${\\rm rk} \\Lambda$.\nAs in Proposition \\ref{proposition parallel transport}, there is a unique component $\\mathfrak{N}_{\\Lambda'}\\subset\\mathfrak{M}_{\\Lambda'}$ of compatibly marked resolutions, and via the embedding $\\omega\\in \\Omega_\\Lambda\\subset \\Omega_{\\Lambda'}$ and Verbitsky's global Torelli theorem, there is a smooth marked irreducible holomorphic symplectic manifold $(Y,\\mu)$ with period $\\omega$ and $Y$ is certainly projective, by Huybrechts' criterion.  Furthermore, since $\\mu^{-1}(\\Lambda_{\\mathbb R}) \\subset H^2(Y,{\\mathbb R})$ is a wall in the decomposition of the positive cone in $H^{1,1}(Y,{\\mathbb R})$ into K\\\"ahler chambers by Lemma \\ref{MBM}, we can choose $(Y,\\mu)$ in its birational equivalence class so that $\\mu^{-1}(\\Lambda_{\\mathbb R})$ is a face of the ample cone of $Y$.  Choosing a line bundle $L$ with $\\mathrm{c}_1(L)$ in the intersection of the nef cone and the positive cone of $Y$, $L$ is big and nef, so by the basepoint-free theorem we obtain a birational contraction $\\pi:Y\\to X$ to a symplectic variety $X$, and $\\pi$ is naturally compatibly marked by $(\\mu,\\nu)$. By Corollary \\ref{cor mon det}, $X$ is a locally trivial deformation of $X'$. To finish, by Lemma \\ref{lemma density} the point $(X,\\nu)$ is in the same component $\\mathfrak{N}_\\Lambda^{\\rm{lt}}$, and the claim is proven.\n\n\\end{proof}\n\\begin{remark}The moduli space of quasi-polarized locally trivially deformations of a given projective symplectic variety is thus a locally symmetric variety of orthogonal type.  The geometry of such varieties, and in particular their Kodaira dimensions, have been studied using modular forms by Gritsenko--Hulek--Sankaran in a series of papers, see for example \\cite{GHS,GHS2,GHS3}. A rather general recent result in this direction has been obtained by Ma \\cite{Ma17}.\n\\end{remark}\n\n\\begin{remark}We needed to use Verbitsky's global Torelli theorem on the resolution, and this method of proof most likely cannot be used to give a different proof in the smooth case.  Note however the same argument does provide a simple proof of the surjectivity of the period map of K3 surfaces using three facts:  (1) any maximal Picard rank period is realized by a Kummer surface; (2) all K3 surfaces are deformation equivalent; and (3) the monodromy group is arithmetic.   \n\\end{remark}\n\\section{Applications to \\texorpdfstring{$K3^{[n]}$}{K3\\textasciicircum {[n]}}-type manifolds}\\label{sec k3type}\nRecall that a compact K\\\"ahler manifold $Y$ is said to be of $K3^{[n]}$-type if it is deformation equivalent to a Hilbert scheme of $n$ points on $K3\\;$ surface.  The $K3^{[n]}$-type manifolds form one of the two known infinite families of irreducible holomorphic symplectic manifolds.  We assume throughout that $n\\geq 2$ (i.e. that $\\dim Y\\geq 4$).   \n\nBy work of Markman \\cite[Corollary 9.5]{markmantor} there is a canonical extension of weight 2 integral Hodge structures\n\\begin{equation}\n0\\to H^2(Y,{\\mathbb Z})\\to \\tilde \\Lambda (Y,{\\mathbb Z})\\to Q\\to 0\\label{extperiod}\n\\end{equation}\nwhere $Q\\cong {\\mathbb Z}(-1)$.  The lattice underlying $\\tilde\\Lambda(Y,{\\mathbb Z})$ is the Mukai lattice $\\tilde\\Lambda_{K3}=E_8(-1)^2\\oplus U^4$.  We denote the primitive generator of the orthogonal to $H^2(Y,{\\mathbb Z})$ in $\\tilde\\Lambda(Y,{\\mathbb Z})$ by $v=v(Y)$, which is determined up to sign and satisfies $v^2=2-2n$.  Note that\n\\[H^2(Y,{\\mathbb Z})\\cong \\Lambda_{K3^{[n]}}:=E_8(-1)^2\\oplus U^3\\oplus (2-2n).\\]\n\nDenote by $\\operatorname{Mon}^2_{K3^{[n]}}\\subset O(\\Lambda_{K3^{[n]}})$ the image of the weight two monodromy representation, which has been computed by Markman \\cite{markmanmon} to be the subgroup $\\tilde O^+(\\Lambda_{K3^{[n]}})$ preserving the orientation class and acting as $\\pm 1$ on the discriminant group $D(\\Lambda_{K3^{[n]}}):=\\Lambda_{K3^{[n]}}^*\/\\Lambda_{K3^{[n]}}$.  We have the following well-known consequence of this computation and Verbitsky's global Torelli theorem:  the extension \\eqref{extperiod} determines the birational class of $Y$.  More precisely, for two symplectic manifolds $Y,Y'$, there is a Hodge isometry $\\phi:H^2(Y,{\\mathbb Z})\\to H^2(Y',{\\mathbb Z})$ lifting to a Hodge isometry $\\tilde\\phi:\\tilde\\Lambda(Y,{\\mathbb Z})\\to\\tilde\\Lambda(Y',{\\mathbb Z})$ of the Markman Hodge structures if and only if $Y$ is birational to $Y'$.  We therefore refer to \\eqref{extperiod} as the \\emph{extended period}.\n\n\n\n\\subsection{Bridgeland stability conditions}\\label{subsec bridgeland stability}\n\nRecall that for a $K3\\;$ surface $S$, the total cohomology $H^*(S,{\\mathbb Z})$ carries the so-called Mukai Hodge structure\n\\[\\tilde H(S,{\\mathbb Z}):= H^0(S,{\\mathbb Z})(-1)\\oplus H^2(S,{\\mathbb Z})\\oplus H^4(S,{\\mathbb Z})(1)\\]\nwhich comes equipped with the Mukai pairing defined by\n\\[(a_0+a_2+a_4,b_0+b_2+b_4):=(a_2,b_2)_S - (a_0,b_4) - (a_4,b_0)\n\\]\nfor $a_i, b_i \\in H^i(S,{\\mathbb Z})$.  For a generic Bridgeland stability condition $\\sigma$ on $S$, the moduli space $Y=M_\\sigma(v)$ of Bridgeland $\\sigma$-stable objects on $S$ of Mukai vector $v\\in \\tilde H(S,{\\mathbb Z})^{(1,1)}$ is a $K3$-type manifold, and we canonically have  $\\tilde\\Lambda(Y,{\\mathbb Z})=\\tilde H(S,{\\mathbb Z}) $ with $v(Y)=v$.  The identification $v^\\perp\\xrightarrow{\\cong} H^2(Y,{\\mathbb Z})$ is achieved by the Fourier--Mukai transform.\n\nNote that by work of Bayer-Macr\\`i \\cite{BM}, every symplectic birational model of a Bridgeland moduli space is a Bridgeland moduli space. We will need below the following Hodge-theoretic characterization of Bridgeland moduli spaces which follows from \\cite[Lemma 2.5]{Hu15} and \\cite[Proposition~4]{Addington}.  \n\\begin{proposition}\nA projective $K3^{[n]}$-type manifold $Y$ is isomorphic to a Bridgeland moduli space on a twisted projective $K3\\;$ surface if and only if one of the following equivalent conditions holds:\n\\begin{enumerate}\n\\item $\\tilde\\Lambda(Y,{\\mathbb Q})_{\\textrm{alg}}$ contains $U$ as a sublattice;\n\\item The rational transcendental lattice $H^2(Y,{\\mathbb Q})_{\\textrm{tr}}\\cong \\tilde\\Lambda(Y,{\\mathbb Q})_{\\textrm{tr}}$ is Hodge-isometric to the rational transcendental lattice of a projective $K3\\;$ surface.\n\\end{enumerate}\nFurthermore, the projective $K3\\;$ surface can be taken untwisted if and only if one of the following equivalent conditions holds: \n\\begin{enumerate}\n\\item[$(1')$] $\\tilde\\Lambda(Y,{\\mathbb Z})_{\\textrm{alg}}$ contains $U$ as a primitive sublattice;\n\\item[$(2')$] The transcendental lattice $H^2(Y,{\\mathbb Z})_{\\textrm{tr}}\\cong \\tilde\\Lambda(Y,{\\mathbb Z})_{\\textrm{tr}}$ is Hodge-isometric to the transcendental lattice of a projective $K3\\;$ surface.\n\\end{enumerate}\n\\end{proposition}\n\n\\subsection{$K3^{[n]}$-type contractions}\nWe now turn to the singular case.\n\\begin{definition}Let $X$ be a symplectic variety and $\\pi:Y\\to X$ a symplectic resolution.  We say $\\pi$ is a $K3^{[n]}$-type contraction if $Y$ is a $K3^{[n]}$-type manifold.  We will often abuse terminology and refer to $X$ itself as a $K3^{[n]}$-type contraction as well.\n\\end{definition}\nNote that if $X$ is a $K3^{[n]}$-type contraction, \\emph{every} symplectic resolution is a $K3^{[n]}$-type manifold by Huybrechts' theorem \\cite[Theorem 2.5]{Huy}.\n\\begin{example}Our main source of examples of $K3^{[n]}$-type contractions come from contractions of Bridgeland moduli spaces, which we call Bridgeland contractions.  Bridgeland moduli spaces of untwisted $K3$ surfaces will be called untwisted Bridgeland contractions for emphasis.  Their geometry is beautifully described via wall-crossing by Bayer--Macr\\`i theory \\cite{BM}.  Given a projective $K3$ surface $S$, a primitive Mukai vector $v\\in \\tilde H(S,{\\mathbb Z})_{\\mathrm{alg}}$, and an open chamber $\\mathcal{C}\\subset{\\operatorname{Stab}}^\\dagger(S)$ associated to $v$, then any $\\sigma_0\\in\\partial \\mathcal{C}$ yields a semiample class $\\ell_{\\sigma_0}$ on $M_\\mathcal{C}(v)$, and the associated morphism $\\pi:M_{\\mathcal{C}}(v)\\to M$ is a $K3^{[n]}$-type contraction.  The morphism $\\pi$ contracts a curve if and only if two generic stable objects in the corresponding family are $S$-equivalent with respect to $\\sigma_0$.\n\\end{example}\nMuch is known about the singularities of Bridgeland contractions, and Theorem \\ref{theorem huybrechts} roughly says that arbitrary $K3^{[n]}$-type contractions exhibit no new singularities:\n\\begin{proposition}\\label{proposition def to bridge}Let $X$ be a $K3^{[n]}$-type contraction with $b_2(X)>3$.  Then $X$ is locally trivially deformation-equivalent to a Bridgeland contraction $M$.  Furthermore, if $b_2(X)>4$, $M$ may be taken to be an untwisted Bridgeland contraction.\n\\end{proposition}\n\\begin{proof}  Since every smooth symplectic birational model of a smooth Bridgeland moduli space is a Bridgeland moduli space, and every contraction is a Bridgeland contraction, we need only argue that the periods of such moduli spaces are dense in $\\Omega_{\\Lambda}$ for any primitive signature $(3,b)$ sublattice $\\Lambda\\subset \\Lambda_{K3^{[n]}}$ with $b>0$.  We prove the second part first.  If we choose an arbitrary primitive embedding $U\\subset\\tilde\\Lambda_{K3}$ containing $\\Lambda_{K3^{[n]}}^\\perp$, then $U\\cap\\Lambda$ is of rank at most one and negative definite.  Any period $\\omega\\in\\Omega_\\Lambda$ orthogonal to $U\\cap \\Lambda$ is of the required form, and by Remark \\ref{remark verbitsky} the monodromy orbits of such periods are dense in $\\Omega_\\Lambda$ since ${\\rm rk}\\,\\Lambda\\cap U^\\perp\\geq 4$ provided $b_2(X)>4$.  \n\nFor the first claim, let $\\Pi$ be the orthogonal to $\\Lambda$ in $\\tilde\\Lambda$; we may assume ${\\rm rk}\\,\\Lambda=5$ by the last paragraph.  As ${\\rm rk}\\,\\Pi>4$, we can find $U\\subset \\Pi_{\\mathbb Q}$ since $\\Pi$ has (many) isotropic vectors by a classical theorem of Meyer.  The periods $\\omega\\in\\Omega_\\Lambda$ orthogonal to $U\\cap \\Lambda$ will now suffice.\n\\end{proof}\nThe work of Bayer-Macr\\`i \\cite{BM} in principle provides a complete description of the singularities of Bridgeland contractions as they are all realized by wall-crossing.  A Bridgeland stability condition $\\sigma_0$ on a $K3\\;$ surface $S$ comes with a central charge $Z_0:\\tilde H(S,{\\mathbb Z})\\to {\\mathbb C}$, and we denote by ${\\mathcal{H}}_{\\sigma_0}(v)\\subset \\tilde H(S,{\\mathbb Z})_{\\mathrm{alg}}$ the primitive sublattice of algebraic vectors $a$ such that $\\Im \\frac{Z(a)}{Z(v)}=0$, \\emph{i.e.}, those vectors for which $Z(a)$ and $Z(v)$ are ${\\mathbb R}$-linearly dependent.  A nearby generic stability condition $\\sigma$ yields a contraction $\\pi:M_\\sigma(v)\\to M$ which identifies $\\sigma$-stable sheaves which are $S$-equivalent with respect to $\\sigma_0$. The lattice $ {\\mathcal{H}}_{\\sigma_0}(v)$ is of signature $(1,\\rho)$, and \n\\[\nN_{\\sigma_0}(v):= {\\mathcal{H}}_{\\sigma_0}(v)\\cap H^2(M_{\\sigma}(v),{\\mathbb Z})={\\mathcal{H}}_{\\sigma_0}(v)\\cap v^\\perp\n\\]\n is negative definite.  Denoting by $R_{\\sigma_0}(v)\\subset H_2(M,{\\mathbb Z})$ the primitive sublattice corresponding to $N_{\\sigma_0}(v)$ under the isomorphism $H^2(M_\\sigma(v),{\\mathbb Q})\\cong H_2(M_\\sigma(v),{\\mathbb Q})$, we see that $N_{\\sigma_0}(v)_ {\\mathbb Q}$ is naturally identified with the orthogonal to $\\pi^*H^2(M,{\\mathbb Q})$ in $H^2(M_{\\sigma}(v),{\\mathbb Q})$.\n\nWalls corresponding to relative Picard rank one contractions are particularly easy to analyze, and for instance we have:\n\n\\begin{proposition}\\label{nakajima}  Let $X$ be a relative Picard rank one $K3^{[n]}$-type contraction and $x\\in X$ a (closed) point.  The analytic germ $(X,x)$ is isomorphic to that of a Nakajima quiver variety.\n\\end{proposition}\n\\begin{proof}By \\cite[Theorem 1.1]{AS}, the statement is known for Gieseker moduli spaces $X=M_{H_0}(v)$ where $v$ is a primitive Mukai vector of a pure 1-dimensional sheaf on a $K3$ surface $S$ with $v^2\\geq 2$ and $H_0$ is a nongeneric polarization.  Given a primitive sublattice ${\\mathcal{H}}\\subset\\tilde\\Lambda_{K3}$ associated to a relative Picard rank one contraction, we therefore only need to show that every such lattice arises from this construction up to the monodromy action.  By \\cite{BM}, every such lattice ${\\mathcal{H}}$ contains $v$ and a class $a\\in{\\mathcal{H}}$ with $0\\leq(v,a)\\leq (v,v)\/2$ and $(a,a)\\geq -2$.  Let $S$ be a $K3$ surface such that\n\\begin{enumerate}\n\\item $\\operatorname{Pic}(S)\\cong {\\mathcal{H}}$.  Let $D\\in\\operatorname{Pic}(S)$ correspond to $v$ and $A$ to $a$.\n\\item $D-\\epsilon A$ is ample.\n\\end{enumerate}\nSuch a surface $S$ exists since ${\\mathcal{H}}$ embeds primitively into $\\Lambda_{K3}$ \\cite{Ni}.  Then $D,A$ and $D-A$ are effective by the conditions on $a$, so $|D|$ contains reducible curves.  Choose an ample $H_0$ and $\\delta,\\alpha\\in{\\mathbb Z}$ nonzero such that\n\\[\\frac{\\delta}{H_0.D}=\\frac{\\alpha}{H_0.A}.\\]\nThen there are strictly $H_0$-semistable sheaves of Mukai vector $v_0=(0,D,\\delta)$.  Set $M_\\pm=M_{H^\\pm}(v_0)$ for $H^\\pm=H_0\\pm \\epsilon D$, and $M_0=M_{H_0}(v_0)$.  We conclude by noticing that the lattice ${\\mathcal{H}}$ in $\\tilde H(S,{\\mathbb Z})=\\tilde\\Lambda(M_+,{\\mathbb Z})$ associated to the wall crossing \n\\[\\xymatrix{\nM_+\\ar[dr]\\ar@{-->}[rr]&&M_-\\ar[ld]\\\\\n&M_0&\n}\\]\nis \n\\[\\left\\langle (0,D,\\delta), (0,A,\\alpha)\\right\\rangle\\cong \\operatorname{Pic}(S)\\cong {\\mathcal{H}}\\]\nwhere the isomorphism takes $v_0$ to $v$.\n\\end{proof}\n\nThe proof of Proposition \\ref{nakajima} gives explicit models for every relative Picard rank one $K3^{[n]}$-type contraction among compactified Jacobians of linear systems on $K3$ surfaces.   Knutsen, Lelli-Chiesa, and Mongardi \\cite{KLCM} have also used compactified Jacobians to construct contractible ruled subvarieties of $K3^{[n]}$-type manifolds, and analyze the geometry more closely.  Models for such contractions have further been treated by Hassett--Tschinkel \\cite{HT15}, where it is shown that every wall ${\\mathcal{H}}$ can be realized on the Hilbert scheme of points for a projective $K3$ surface of Picard rank one.  Note that we have the following simple consequence of Corollary \\ref{cor mon det}:   \n\n\\begin{corollary}\\label{cor monodromy class}  The locally trivial deformation type of a $K3^{[n]}$-type contraction $X$ is determined by $(\\lambda,\\lambda)$ and $\\div(\\lambda)$, for a primitive generator $\\lambda\\in (\\pi^*H^2(X,{\\mathbb Z}))^\\perp$ and a symplectic resolution $\\pi:Y\\to X$.\n\\end{corollary}\n\\begin{proof}Using Corollary \\ref{cor mon det}, this is a purely lattice-theoretic statement about the monodromy group of the $K3^{[n]}$-type deformation class, see \\cite[Section 10]{Eichler}.\n\\end{proof}\n\nThe Bayer--Macr\\`i picture strongly suggests that the answer to the following question is affirmative:\n\\begin{question}Let $\\pi:Y\\to X$ be a relative Picard rank one $K3^{[n]}$-type contraction, and let $E\\subset Y$ be an irreducible component of the exceptional locus.  Is the generic fiber of the map $E\\to X$ isomorphic to $\\P^{\\operatorname{codim} E}$?\n\n\\end{question}\nIndeed, for a Bridgeland moduli space $Y=M_{\\sigma_+}(v)$ and a contraction induced by a wall-crossing, the Harder--Narasimhan filtration of a generic point $[F]\\in E$ with respect to a generic nearby stability condition $\\sigma_-$ on the other side of the wall is often of the form\n\\begin{equation}0\\to A\\to F\\to B\\to 0\\label{HNfilt}\\end{equation}\nfor $A,B$ $\\sigma_0$-stable.  All such extensions are $\\sigma_+$-stable, and this yields a $\\P^k=\\P\\operatorname{Ext}^1(B,A)$ fiber that is contracted.  Moreover, setting $a=v(A)$ and $b=v(B)$,\n\\begin{align*}\\dim E&=k+\\dim M_{\\sigma_0}^{\\operatorname{1st}}(a)+\\dim M^{\\operatorname{1st}}_{\\sigma_0}(b)\\\\\n&=\\left((a,b)-1\\right)+\\left(a^2+2\\right)+\\left(b^2+2\\right)\\\\\n&=\\left(v^2+2\\right)-k\\end{align*}\nso $k=\\operatorname{codim} E$.  Thus, in this case, we are done if the Harder--Narasimhan filtration of the general point of $E$ has the form \\eqref{HNfilt} for fixed\\footnote{Note the constancy of the Mukai vectors is automatic if a universal family exists over $E$, by the existence of Harder--Narasimhan filtrations in families.} $a$ and $b$.  By Corollary \\ref{cor monodromy class} and \\cite{LP2}, it would be sufficient to consider one model in each monodromy orbit, and many special cases have been established previously, see \\cite{HT15,KLCM}.  It is also not difficult to prove the following special case:\n\n\\begin{proposition}\\label{proposition generic fiber}Let $\\pi:Y\\to X$ be a relative Picard rank one $K3^{[n]}$-type contraction, and let $E\\subset Y$ be an irreducible divisorial component of the exceptional locus.  Then the generic fiber of the map $E\\to X$ is $\\P^1$.\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{proposition def to bridge}, the contraction $\\pi$ is a locally trivial deformation of a Bridgeland contraction. Using \\cite[Theorem 1.1]{LP2} we may assume that $\\pi$ is itself a Bridgeland contraction. In our special case it also follows from uniqueness of relative minimal resolutions of ADE surface singularities.\nSo let $Y=M_{\\sigma_+}(v)$ be a Bridgeland moduli space on a $K3\\;$ surface $S$ and consider a wall-crossing contraction $\\pi:Y\\to X$ associated to the rank two lattice $v\\in{\\mathcal{H}}\\subset \\tilde H(S,{\\mathbb Z})_\\mathrm{alg}$.  Let $\\sigma_0$ be a generic stability condition on the wall.  Note that there is at most one divisorial component to the exceptional locus of a relative Picard rank one contraction.  It follows from the classification in \\cite{BM} that there are two cases:\n\\begin{enumerate}\n\\item ${\\mathcal{H}}$ contains no isotropic vectors.  For a spherical class $s\\in {\\mathcal{H}}$, the spherical reflection $\\rho_s:\\tilde H(S,{\\mathbb Z})\\to \\tilde H(S,{\\mathbb Z})$ gives a parallel transport operator from $v^\\perp$ to $v'^\\perp$ for $v'=\\rho_s(v)$.  As in \\cite[Section 7]{BM}, if the wall is totally semistable for $v$, we can produce $v'\\in {\\mathcal{H}}$ by a sequence of spherical reflections through spherical classes $s_i\\in{\\mathcal{H}}$ for which the wall is not totally semistable, so without loss of generality we may assume we are in this case.  By \\cite[Lemma 7.4]{BM}, the Harder--Narasimhan filtration of a generic point $[F]\\in U$ for some \\'etale open $U\\to E$ is given by \n\\[0\\to S\\to F\\to A\\to 0\\mbox{\\indent or\\indent}0\\to A\\to F\\to S\\to 0\\]\nfor $S$ and $A$ both $\\sigma_0$-stable and $S$ spherical.  Without loss of generality, assume we are in the first case (\\emph{i.e.} that $\\phi_{+}(S)\\leq \\phi_+(A)$).  There is a unique such $S$ (since necessarily $(v(S),v)=0$), and all such extensions are $\\sigma_+$-stable, so $\\P^1\\cong \\P\\operatorname{Ext}^1(A,S)$ gives the unique curve through the general point that is contracted by $\\pi$.\n\\item   ${\\mathcal{H}}$ contains an isotropic class.  The above proof still holds if there are no isotropic classes $w\\in{\\mathcal{H}}$ such that $(v,w)=1,2$ (see \\cite[Proposition 8.6]{BM}).  As in \\cite[Section 8]{BM}, there is an isotropic class $w_0\\in{\\mathcal{H}}$ such that $M_{\\sigma_0}(w_0)=M^{\\operatorname{1st}}_{\\sigma_0}(w_0)$, and we may assume $w=w_0$.  The case $(v,w_0)=1$ is the Hilbert--Chow contraction, and the remaining case is similarly treated by \\cite[Lemma 8.7]{BM}.\n\\end{enumerate}\n\\end{proof}\n\nIn \\cite{BB} the first author classified contractions at the other extreme, namely those for which the exceptional locus contains a Lagrangian $\\P^n$.\n\n\nAs an application, assume that ${Y}$ is a symplectic $2n$-fold which admits a divisorial contraction $\\pi:{Y}\\to X$ of relative Picard rank one such that $X$ has transversal $A_2$ singularities. We call this an $A_2$-contraction. Note that while $ADE$ singularities admit unique symplectic resolutions, in the relative Picard rank one setting the monodromy action on the set of components of the general fiber of the exceptional locus yields a group of automorphisms of the $ADE$ graph in question acting transitively on the nodes, so only $A_1$ and $A_2$ singularities remain as possibilities.  We would like to know whether an $A_2$ contraction exists if ${Y}$ is an irreducible symplectic manifold.\n\n\\begin{corollary}\\label{corollary a2}\nLet ${Y}$ be an irreducible symplectic manifold deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface. Then ${Y}$ does not admit any $A_2$-contraction of relative Picard rank one.\n\\end{corollary}\n\nNote that there are however examples of $A_2$-contractions of relative Picard rank one of smooth and projective symplectic varieties. See e.g. \\cite[\\S 1.4, Example 2]{Wierzba} for an explicit construction.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n\nMobile phone datasets allow for the analysis of human behavior on an unprecedented scale. The social network, temporal dynamics and mobile behavior of mobile phone users have often been analyzed independently from each other using mobile phone datasets. In this article, we explore the connections between various features of human behavior extracted from a large mobile phone dataset. We show that clustering and principal component analysis allows for a significant dimension reduction with limited loss of information. The most important features are related to geographical location. In particular, we observe that most people spend most of their time at only a few locations. With the help of clustering methods, we then robustly identify home and office locations and compare the results with official census data. Finally, we analyze the geographic spread of users' frequent locations and show that commuting distances can be reasonably well explained by a gravity model.\n\n\\section{Introduction}\nInformation and communication technologies have always been important sources of data and inspiration in sociology, especially in recent decades. These technologies influence the behavior of people, which is a subject of study in itself (e.g.\\cite{BALL1968,DEBAILLENCOURT-ET-AL2007,LICOPPESMOREDA2005}), but  they also provide massive amounts of data that can be used to analyze various aspects of human behavior. \n\nTelephone and mobile phone data have already been used to study social networks,\nsometimes in conjunction with features such as gender and age\n\\cite{SMOREDALICOPPE2000}. More recently, the mobile phone data available to\nresearchers have been enriched with geographical information. This allows to\nanalyze regularities, or even\nlaws\\cite{Brockmann2006,GONZALEZETAL-NATURE-2008,Candia2008}, governing the\nhighly predictable mobility~\\cite{Song2010} in everyday life. These insights can\nbe vital in emergency situations~\\cite{Bagrow2011a}, or in (preventing)\nspreading of diseases~\\cite{Bajardi2011,Balcan2009,Rocha2011} or mobile\nviruses~\\cite{Wang2009a}. Furthermore, users' mobility and their social network\nare intertwined: the one could be used to predict the other~\\cite{Wang2011,\nCrandall2010}, and the probability of two people calling over a distance follows\na gravity like model~\\cite{Lambiotte2008a,Krings2009,Calabrese2011,Levy2010}.\nResearch has also shown there are geographical clusters of highly connected\nantennas~\\cite{Expert2011} (e.g. resembling provinces) as well as clusters in\nthe social network consisting of groups of well connected\npeople\\cite{Palla2007,Blondel2008,Onnela2007}, although the connections between\nthe two are not yet fully understood~\\cite{Onnela2011}. Similar results have\nalso been obtained in a virtual mobility setting~\\cite{Szell2011}.\n\nIn this paper, we analyze anonymized communication data from a telecom operator in Portugal. The data cover a period of 15 months and the following information is available for each communication: the times of initiation and termination, the users involved, and the transmitting and receiving antennas (at the beginning of the communication). In addition, we also know the locations (longitude and latitude) of all antennas.\n\nWe first present a statistical analysis of the data. We define a set of 50 general features that we compute for each user, and using principal component analysis and clustering methods, we show that these features are highly redundant: they can all be recovered, with a loss of accuracy of less than  $5\\%$, using  a reduced set of only five meta-features. \n\nObserving that the most important features are geographical, we then pay specific attention to the most common locations of each user. By developing a procedure to extract these frequent positions, we observe that people spend most of their time in only a few locations. We then cluster the different calling patterns for each user and each location, and from this, we observe that only two types of locations are clearly identifiable, namely home and work. We compare our results to census data obtained from the Portuguese National Institute of Statistics.\n\nFinally, we analyze in more detail the behavior of users who have exactly one\nhome location and one office location. This allows us to predict the number of\ncommuters between different regions of the country using a gravity model. More\nprecisely, we observe that two different regimes exist, the first involving\ndistances smaller than \\unit{150}{\\kilo\\meter} (which is half the distance\nbetween the two largest cities) and the second involving larger distances. In\nthe latter case, only the number of offices in the destination region is\nstatistically significant.\n\nThe fundamental contribution of this paper is that we improve the understanding\nof frequent locations. Building on previous location inference\nwork\\cite{ZANG2010}, we construct a method for rigorously determining the type\nof locations. It had already been observed that people have only a few top\nlocations \\cite{GONZALEZETAL-NATURE-2008}, but it remained unclear what type of\nlocations they represent. Although it is often (tacitly) assumed they represent\nhome and office (\\cite{GONZALEZETAL-NATURE-2008, Song2010}), this had never been\nrigorously analyzed. We confirm this hypothesis, and also conclude that these\nare the only type of locations that are robustly detectable in the data.\n\n\\section{Data Mining and Feature Analysis}  \\label{sectanalysis}\nIn this section, we analyze the calling and geographic behaviors of mobile phone users based on features that summarize these behaviors. These features allow us to investigate interdependencies between characteristics such as call durations, the distances of calls, the distances of movements and the frequency of calls. This can be achieved, for example, by analyzing {correlations} between these features. In this section we also use principal component analysis and cluster analysis to better understand these interdependencies.\n\n\\subsection{Preprocessing} \\label{preprocsec}\nBefore proceeding with the analysis, some preprocessing of the raw data was necessary. The most important preprocessing step was the application of a {moving weighted average} filter on the calling positions of the users.\n\nThis filtering was crucial because the position of the antenna does not always accurately reflect the actual position of the user. Moreover, due to noise (such as that introduced by reflection and scattering in urban environments), the closest antenna is not always the one serving the call. Without proper filtering, these inaccuracies tend to accumulate, particularly for measures such as the total distance traveled.\n\nThe filtering was computed as follows. The positions were smoothed independently for all users. Assume that a user made calls at times $t(1), \\dots, t(n)$ and the coordinates of the antennas that served the calls are $x(1), \\dots, x(n)$. The smoothed positions of the user, denoted by $y(1), \\dots, y(n)$, can be calculated as\n\\begin{equation}\ny(i)= \\!\\! \\sum\\limits_{j \\in B_{\\delta} (i) } \\!\\!\\! w(j)\\, x(j)\n\\end{equation}\nwith $B_{\\delta}(i) = \\left\\{ \\, j : \\left| t(j) - t(i) \\right| \\leq \\delta \\,\n\\right\\}$ where $B_{\\delta}(i)$ denotes the indices of those calls that were initiated or received within a maximum interval $\\delta$ from the current time of the filtering. We used $\\delta=30$min for the dataset. Positions that were further distant from the current time of the decision had proportionally smaller weights:\n\n\\begin{equation}\nw(j) = 1 - \\frac{\\left|t(j) - t(i)\\right|}{\\delta},\n\\end{equation}\nwhere $i$ denotes the current index of the call that should be smoothed.\n\nIn addition to filtering, we took into account those customers who had made and\/or received at least $10$ calls during the period analyzed ($15$ months). Moreover, for compression and cluster analysis, we {normalized} (scaled) and {centered} the data. Most of the analysis was performed on $100\\,000$ randomly (uniformly) selected users. We performed Student's {t-tests} to examine the statistical significance of the results.\n\n\\subsection{Features}\nWe defined 50 features to summarize users' behavior. Each feature represents one particular aspect of users' behavior as a single number, such as the number of incoming or outgoing calls, the number of people who called or were being called by the user, the position (coordinates) of the user (mean and deviation), the coordinates of the two most frequently used antennas, the durations of incoming or outgoing calls (mean and deviation), the distances of the incoming or outgoing calls (mean and deviation), the directions of the incoming or outgoing calls (mean and deviation) and various movement measures.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics{1-features}\n\\end{center}\n\\caption{(a) The distribution of the average locations of the users. Brighter colors indicate areas in which higher numbers of users have their average locations. (b) Three methods of measuring customer movements: (1) gyration, (2) diameter of the convex hull and (3) total line segment length.}\n\\label{fig2}\n\\end{figure}\n\nIndividual features themselves can contain considerable information. For example, by analyzing the average locations of the users, we obtained information on the distribution of users across the country, and large cities can be recognized as bright spots in the left panel of Fig.~\\ref{fig2}.\n\nIn addition to a measure referred to as gyration~\\cite{GONZALEZETAL-NATURE-2008}, we propose two additional measures of customer movements: the {diameter of the convex hull} and the {total line segment length}. All three of these measures rely on the positions of the user during calls to give some indication of how much or how far he has traveled. We take into account both incoming (received) and outgoing (initiated) calls. The sequence of the positions (calls) is not important for the first two measures, but it is significant for determining line segment length. This is illustrated in the right panel of Fig.~\\ref{fig2}.\n\nGyration~\\cite{GONZALEZETAL-NATURE-2008} measures the deviation (mean square-error) of each of the user's positions from his average location. The diameter of the convex hull measures the maximum distance  between any two positions of the user during a given period. The {total line segment length} sums all of the distances between each pair of consecutive positions of the user. Note that the filtering procedure explained earlier can have a large impact on this final measure.\n\n\\subsection{Correlation Analysis}\n\nAfter computing the values of each feature for each user, we analyzed the {interdependencies} between these features using a {correlation analysis}. As mentioned earlier, we considered $100\\,000$ randomly selected users. We used t-statistics to confirm that our results are also valid for the complete dataset. In some cases, the correlations are better analyzed on a {logarithmic} scale, and we have therefore also analyzed the logarithmic correlations.\n\n\\begin{table}\n\\caption{Selected Results of Correlation Analysis}\n\\begin{center}\n\\begin{tabular}{|l   l| c c |} \\hline\n{\\em Feature A} & {\\em Feature B} & {\\em Cor.} & {\\em LogCor.}\\\\\n\\hline\n\\hline\nNo Calls         & No Callers       & .91 & .90\\\\\nDiam Conv Hull & No Antennas     & .55 & .20\\\\\nAvg Duration     & Avg Distance     & .31 & .64\\\\\nNo Antennas    & No Calls         & .60 & .68\\\\\nDiam Conv Hull & Avg Duration    & .05 & .18\\\\\nLine Segm Len & No Antennas     & .45 & .75\\\\\nGyration        & Std Dev Dist   & .60 & .40\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{Corr-table}\n\\end{table}\n\nTable \\ref{Corr-table} shows some of the correlations for the $100\\,000$\nrandomly selected users. The data in this table shows that movement-related features are correlated with some but not all of the other features. Some pairs of features there, such as the number of calls and the number of callers, are highly correlated as expected. Other pairs of features, such as the diameter of the convex hull and the average duration of a call, exhibit weaker correlations. Note that the correlation measures only the {linear} dependencies between two features, and more detailed relationships might be uncovered using more complex methods. We do not pursue this further here, but instead turn to an analysis of the redundancy of the data.\n\n\\subsection{Principal Component Analysis}\n\n\\begin{figure}\n\\begin{center}\n  \\includegraphics{2-clustering}\n\\end{center}\n\\caption{Illustration of the basic concept behind (a) principal component analysis and (b) cluster analysis.}\n\\label{fig4}\n\\end{figure}\n\nWe now analyze the interdependencies between features using another approach. To\nwhat extent are the analyzed features redundant? In other words, how much of the\ninformation represented by one feature can be expressed by (a combination of)\nother features? To address this question, we used {principal component analysis}\n(PCA), which is widely used in various disciplines.  The basic goal of PCA is to\nreduce the dimensions of the data. It can be proven that PCA provides an optimal\nlinear transformation for mean-square-based dimensionality reduction\n\\cite{Jolliffe}.\n\nThe core idea of PCA is as follow. Let $x_1, \\dots, x_n \\in \\mathbb{R}^{d}$ be\n(independent) realizations of a random vector $X$, where we assume that\n$\\mathbb{E}[X]=0$ (which can be guaranteed, for example, by substracting the\nsample mean from the measurements). In our case, the vectors $(x_i)_{i=1}^n$\nrepresent users, while each entry corresponds to a feature. \n\nWe aim at finding {\\em orthonormal} vectors $w_1, \\dots, w_d \\in\n\\mathbb{R}^{d}$, called the principal components, with the property that\nfor all $k \\in \\{1,\\dots, d\\}$, the linearly transformed vector $Y_k =\nW_k^\\mathrm{T} X$, where $W_k$ is $[w_1, \\dots, w_k]$, explains the maximum\npossible variance of $X$. In other words, if we transform our dataset $D = [x_1,\n\\dots, x_n]$ by $S_k = W_k^\\mathrm{T} D$, then we can reconstruct the matrix $D \\in\n\\mathbb{R}^{d \\times n}$ from the matrix $S_k \\in \\mathbb{R}^{k \\times n}$ (using\n$W_k$) with the smallest possible mean square error. Note that sometimes the\nrows of $S_k$, i.e., $s_i = w_i^\\mathrm{T} D$ are called the principal\ncomponents and the $w_i$'s are referred to as loadings or coefficients.\n\n\\newcommand{\\argmax}{\\mathop{\\vphantom{\\max}\\mathchoice\n  {\\hbox{arg\\,max}}\n  {\\hbox{arg\\,max}}{\\mathrm{A}}{\\mathrm{A}}}\\displaylimits}\n\nA recursive formulation of PCA can be given as follows. Let\n\\begin{align}\nw_1 &= \\argmax_{\\|w\\|=1} \\frac{1}{n}\\sum_{i=1}^n (w^\\mathrm{T} x_i)^2 \\nonumber \\\\\n    & \\approx \\argmax_{\\|w\\|=1} \\mathbb{E} \\left[(w^\\mathrm{T} X)^2\\right] =\n      \\argmax_{\\|w\\|=1} \\mbox{Var}\\left[w^\\mathrm{T} X \\right].\n\\end{align}\nThe vector $w_1$ points toward the direction in which the sample variance of the data is maximized. This is of course an approximation of what we would get using the full (unknown) distribution of $X$. Having defined the first \\(k-1\\) vectors, the \\(k\\)-th is determined as\n\\begin{align}\n  w_k &= \\argmax_{\\|w\\|=1} \\frac{1}{n}\\sum_{i=1}^n\n  \\Bigg(w^\\mathrm{T}\\Big(x_i-\\sum_{j=1}^{k-1}{w_j w_j^\\mathrm{T}\n  x_i}\\Big)\\Bigg)^{\\!\\!2} \\nonumber \\\\\n    &\\approx\n\\argmax_{\\|w\\|=1,\\, w \\perp (w_i)_{i=1}^{k-1}}\\! \\mbox{Var} \\left[ w^\\mathrm{T}X \\right],\n\\end{align}\nwhich is thus chosen to achieve the highest variance possible while being\northogonal ($\\perp$) to the previous choices.  The vectors $(w_i)_{i=1}^d$ can\nbe efficiently computed from the (estimate of the) covariance matrix \\(\\Sigma=\n\\mathbb{E}\\left[X X^\\mathrm{T}\\right]\\), since vector $w_i$, $i \\in \\{1,\\dots,\nd\\}$, is an eigenvector of the sample covariance matrix corresponding to its\n$i$-th largest eigenvalue. The basic concept behind PCA is illustrated in\nFig.~\\ref{fig4}.\n\n\\begin{table*}\n\\caption{Compression of Features by Principal Component Analysis}\n\\begin{center}\n\\begin{tabular}{|c | c || c | c |} \\hline\n{\\em Variance Kept} & {\\em Mean Square Err.} & {\\em Dimen.\\ Required} & {\\em Compress.\\ Rate}\\\\\n\\hline\n\\hline\n99\\% & 1\\% & 24 & 52\\%\\\\\n\\hline\n98\\% & 2\\% & 13 & 74\\%\\\\\n\\hline\n95\\% & 5\\% & 5 & 90\\%\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{PCA-table}\n\\end{table*}\n\nOur PCA analysis revealed high redundancy among the features analyzed. Table \\ref{PCA-table} shows the results of this analysis. It can be seen that if we allow a $1\\,\\%$ (mean square) error in the variance, the number of features can be reduced by more than $50\\,\\%$ (from $50$ to $24$). If the allowed error is raised to $5\\,\\%$, we can further reduce the number of features to $5$, which represents a compression rate of $90\\,\\%$. In other words, we can build five components using a linear combination of the original features, and using only the values of these five components, we can determine the values of any of the $50$ original features with a $5\\,\\%$ mean-square error. This implies that the features have many interdependencies and are highly redundant.\n\nIn order to identify which features are most relevant, we determined their\nimportance as follows. PCA produces a set of orthogonal vectors,\n$(w_i)_{i=1}^d$, which point toward the directions of maximum variance. As noted\nearlier they are eigenvectors corresponding to eigenvalues of the sample covariance\nmatrix. Furthermore, the $i$-th eigenvalue, $\\lambda_i$, equals to the (sample)\nvariance of $s_i = w_i^\\mathrm{T} D$. Then, each original feature can be\nidentified by an element of the canonical basis. For example, feature 1 can be\nidentified by $e_1 = \\left<1, 0, \\dots, 0 \\right>^\\mathrm{T}$. The {importance}\nof feature $i$ can then be defined as the max-norm of the projected vector $e_i$\non the basis defined by $(w_i\/{\\lambda_i})_{i=1}^{d}$. The basis was thus\nscaled to produce larger coordinates in directions of higher variances. Note\nthat we have also scaled the scores such that the most important feature has a\nrelative importance of $1$.\n\nFig.~\\ref{fig3} presents a list of the features in order of importance. The most\nimportant features, such as the average position of the user and the coordinates\nof the two most often used antennas, are geographic features. This indicates\nthat the locations of the users and their calls are among the most important\ncharacteristics.\n\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics{3-features-pca}\n\\end{center}\n\\caption{The relative importance of each feature according to PCA, defined as the max-norm of the projection of the feature basis on the principal components.}\n\\label{fig3}\n\\end{figure*}\n\n\\subsection{Cluster Analysis}\nAfter analyzing the data using PCA, we performed {cluster analysis}, to identify typical user classes based on calling behaviors. We used the {subtractive} clustering method~\\cite{Chiu} illustrated in Fig.~\\ref{fig4}, which is a variant of the classical mountain method. An advantage of the subtractive clustering method is that it can identify the number of clusters required.\n\nThe application of subtractive clustering to the {normalized} data for\n$100\\,000$ uniformly selected customers resulted in $5$ clusters. Each of these\nclusters is identified by its {central} element (a vector of feature values) and\nits range of influence. As with the PCA, we wished to identify the main\nconstituent features of these clusters. We therefore performed a similar\nanalysis as for the PCA, using the vectors of the cluster centers as the basis\nfor the dominant feature subspace. The results of this ordering, presented in\nFig.~\\ref{fig5}, indicate that location- and movement-related features are important characteristics, similar to PCA.\n\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics{4-features-clustering}\n\\end{center}\n\\caption{The relative importance of each feature according to cluster analysis.}\n\\label{fig5}\n\\end{figure*}\n\nNote that both PCA and cluster analysis reveal clear differences in importance\nbetween the $x$ and $y$ coordinates. This can be expected for an elongated\ncountry such as Portugal and is probably aggravated by the fact that most people\nlive along the coast.\n\nAlthough the features concerning the $y$ coordinates have a similar importance\nin both the PCA and cluster analysis, there are some clear differences as well.\nIn general, an explanation of this could be that PCA focuses on global\ncharacteristics, it tries to build components (by linear combination of feature\nvectors) which can explain the dataset with minimal mean square error.\nClustering, however, concentrates on local similarities, and tries to find\nclusters in which the feature vectors are ``close'' to each other.\nNevertheless, various geographical features have key importance according to\nboth orderings. The most notable difference concerns the diameter of the convex\nhull, which has a very high importance in clustering, while it has a relatively\nlow importance in PCA. From the PCA analysis this implies that the variance in\nthe diameter of the convex hull is not important for explaining a large part of\nthe data. From the cluster analysis, the differences in the diameter of the\nconvex hull are important, even though the variance might not contribute that\nmuch.  This suggests an interesting effect of the diameter of the convex hull.\nBesides the obvious importance of the $y$ position when clustering people, the\ndiameter of the convex hull separates people that share similar $y$ positions.\nIn conclusion, the features that are important for clustering people are: (1)\nfirst antenna; (2) second antenna; (3) diameter of convex hull; and (4) average\nposition.\n\n\\section{Frequent Locations} \\label{sectarnaud}\n\nIn the previous section, we analyzed several features and concluded that the\nmost important ones are related to geography. Additionally, we observed that\nmost people spend most of their time in only a few locations. In this section,\nwe focus on characterizing these frequent locations for each user by analyzing\nweekly calling patterns. Once these frequent locations are characterized, we\nanalyze them in greater depth. A related, although different, concept of\nhabitats~\\cite{Bagrow2012} was recently introduced, where habitats are clusters\nof the associated Markov mobility network. However, a single habitat might\ncontain several frequent locations.\n\nAs explained in Section \\ref{preprocsec}, the data are noisy, and often, any one of multiple antennas can be used to make a call from a given position. Because this can be true for frequent locations such as home and the office, we first develop a method for estimating which antennas are relevant for characterizing such locations.\n\nAfter extracting the frequent locations for each user, we estimate these positions more precisely using a maximum likelihood approach. We then present various statistics using these estimated positions. In particular, we estimate the amount of time people spend at work and home, characterize different combinations of frequent locations (multiple `homes' or `offices'), estimate the geographical density of homes and offices, compare our estimates to independent statistics and, finally, analyze distances between home and office (commuting distances).\n\n\\subsection{Detection of Frequent Locations}\n\nDetecting the most common locations of a user is only possible if enough calls involving that user are recorded\\footnote{In this section, we include both calls and text messages because we want to maximize the information on antenna usage; we refer to both as ``calls''.}. For users who make only a few calls, no locations can be called ``frequent'' with any certainty. We therefore selected only users who make at least one call a day on average and who make consecutive calls within $24$ hours 80\\% of the time. The latter constraint requires a certain regularity of users, and excludes users with highly bursty behavior\\cite{GONZALEZETAL-NATURE-2008}. From this selection, we selected a random sample of $100\\,000$ users.\n\nFor detecting frequent locations, it is appropriate to begin by identifying the most frequently used antenna (MFA). However, as stated earlier, the same antenna is not always used for calls made from a given position (due to load balancing or the effects of noise on the signal). Hence, other antennas located near the MFA may also be used to serve the frequent location. We must therefore consider sets of antennas that are relatively close together.\n\nWe first performed a Voronoi tessellation, which partitions the space into cells based on the distance between each point and the closest antenna. Each Voronoi cell includes the set of points that are closer to the antenna located in that cell than to any other antenna. Based on the Voronoi tessellation, a graph can be created in which nodes are neighbors if their associated Voronoi cells are adjacent. Each node corresponds to an antenna, and its neighbors are called the Delaunay neighbors.\n\nWe next grouped antennas around the MFA based on Delaunay neighborship. More precisely, we defined the Delaunay radius of each antenna to be the largest distance between an antenna and any of its Delaunay neighbors (this is later used in the estimation of the position; see Section~\\ref{sec:Neighborhood} for more information). We then merged all antennas around the MFA that are within twice this radius\\footnote{We observe that taking twice the Delaunay radius yields an error of less than $0.1$\\% for estimating positions. See Section~\\ref{sec:Neighborhood} for more information.} and assigned them one ``location'', the position of which will be defined later.\n\nAfter identifying the first MFA and merging the surrounding antennas, we moved on to the remaining antennas, selecting the most frequently used of those and repeating the procedure described above. We continued iterating until we identified a set of antennas that represented less than 5\\% of a  user's calls. We repeated this for each user in our selection and thus obtained a number of frequent locations for all users.\n\n\\begin{figure}\n  \\begin{center}\n     \\includegraphics{5-mfp_freq}\n  \\end{center}\n  \\caption{Histogram of the number of frequent locations per user}\n  \\label{fig:histFL}\n\\end{figure}\n\nThe results of this procedure are summarized in Fig.~\\ref{fig:histFL} and indicate that the average number of frequent locations per user is approximately $2.14$ and that $95\\%$ of the users have fewer than $4$ frequent locations. This implies that the 3 or 4 most common locations are sufficient to predict the position of user, most of the time~\\cite{Song2010}. A substantial number of users have only one single frequent location, which is usually an office or a home location (as we will see later on). This could reflect the possession of separate business and private phones, one of which is (almost) exclusively used at work and the other only at home.\n\n\n\\subsection{Clustering of Weekly Calling Patterns}\n\nThe data show two clearly identifiable periodic dynamics in mobile phone use: a daily cycle and a weekly cycle, as illustrated in Fig.~\\ref{fig:avg_pattern}. The daily cycle largely follows the human circadian rhythm, with a clear drop in activity during the night, a gradual increase in the morning and a decrease in the evening, with a small dip around lunch time. The weekly dynamic is related to the workweek, with different behavior on weekends as compared to work days.\n\nWe collected all of the calls made using antennas associated with each frequent location. Because we have the time stamps (beginning and end) of each call, we know the times at which each frequent location is used. The description of this usage at the weekly scale seems to be especially suitable for further analysis. We therefore divided the week into 168 hours and aggregated the usage pattern of the whole period. This resulted in a $168-$dimensional vector per frequent location with the calling frequency for one hour in each entry.\n\nBased on the aggregated call vectors for all frequent locations, we performed k-means clustering. We ran this clustering for $k=\\{2,\\ldots,10\\}$ to investigate what patterns of usage could be distinguished. We found that using $k=3$ yielded clear results, as displayed in Fig.~\\ref{fig:home_office}. The first cluster clearly represents a pattern related to work. During the weekdays, an increase in the usage of these antennas occurs during the morning, followed by a small dip around noon, and a decrease in usage from around 6 p.m. on. During the weekend, these antennas are used far less. This pattern is in excellent agreement with independent statistics from the Portuguese National Institute of Statistics (INE) in terms of time spent at work, as shown in Fig.~\\ref{fig:home_office}. The second cluster reflects a pattern of usage that appears to be more closely associated with a home position. The usage of these antennas is lower during the day, and the maximum usage occurs during the evening. These antennas are also used more during the weekend than are the antennas in the first cluster. Finally, the third cluster appears simply to contain locations that do not follow the dynamics of the previous two clusters. This cluster follows the more general dynamic displayed in Fig.~\\ref{fig:avg_pattern}.\n\n\\begin{figure}\n  \\begin{center}\n   \\includegraphics{6-call_dynamics}\n  \\end{center}\n  \\caption{Weekly dynamics on average (a) and for the three clusters (b) detected using k-means clustering: home, office and the remainder, shown along with independent time usage statistics from the Portuguese National Institute of Statistics (INE)~\\cite{INE1999}. Using more clusters yields similar results. The dotted lines indicate noon of each day.}\n  \\label{fig:home_office}\n  \\label{fig:avg_pattern}\n\\end{figure}\n\n\nWe observed that when more than three clusters are considered, they tend to yield results very similar to those shown here. We expected that we would be able to identify additional patterns of usage, such as those of calls made by students with a different rhythm from working people or calls made from weekend houses that show no activity during the week, but we did not observe these patterns. Such patterns certainly do exist, but they appear to be marginal when compared to the established home and office routine. Hence, there appears to be no identifiable patterns of usage other than the home and office patterns described above. However, using only two clusters obfuscates this result, and the separation between home and office positions is less clear in this case.\n\n\\begin{table}\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{cccc}\n    \\rowcolor{white}\n$\\#$ Home & $\\#$ Office & $\\#$ Unidentified & \\%\\\\\n\\hline \\hline\n &  1&   &  16.8\\\\ \n1&   &   &  16.5\\\\\n1&   &  1&  9.1\\\\\n1&  1&   &  6.6\\\\\n &  1&  1&  6.0\\\\\n1&   &  2&  5.0\\\\\n &   &  1&  3.5\\\\\n1&  1&  1&  3.5\\\\\n &   &  2&  3.4\\\\\n &  1&  2&  2.7\n\\end{tabular}\n\\caption{The $10$ most frequent combinations of frequent locations. Each\n  combination is composed of the number of homes, offices and unidentified\n  locations a user has. Each row indicates such a combination. The empty entries\n  indicate no such type of location is present in a combination. The last column\n  contains the percentage of how often such a combination occurs.}\n\\label{locPattern}\n\\end{center}\n\\end{table}\n\nThe top 10 most frequent combinations of frequent locations are displayed in Table~\\ref{locPattern}. Approximately 32\\% of the users have either a single home location or a single office location alone, whereas only 3.5\\% have only a single unidentified location. For users with two frequent locations, the most common combination is one home location and one unidentified location. Only 6.6\\% of all users have the combination of one home location, one office location and no unidentified locations. Approximately 85\\% of the users have at most one home and\/or one office location, and approximately 12\\% of the users have exactly one home and one office location (and possibly multiple unidentified locations).\n\nOf all frequent locations, approximately $60\\%$ ca be classified as ``home'' or\n``office'' (as in the first two columns of Table~\\ref{locPattern}). We observed\nthat users tend to have no more than two identifiable positions, as depicted in\nFig.~\\ref{fig:histFLID}. The majority of users have only one identifiable location, which is by definition either home or office. For users with two identifiable locations, over 50\\% have both a home and an office, and the rest has either two homes or two offices.\n\n\\begin{figure}\n  \\begin{center}\n     \\includegraphics{7-barIdentifiable}\n  \\end{center}\n  \\caption{Histogram of the number of frequent locations per user, with visual separation between ``home'' locations, ``office'' location and mixed locations (i.e., users with one of each)}\n  \\label{fig:histFLID}\n\\end{figure}\n\n\\subsection{Estimating the Position of Frequent Locations}\n\n  \\subsubsection{Basic model}\n\n  We propose a model to estimate the position of the home, the office and other frequent locations. We consider a simplified version of the model proposed in \\cite{ZANG2010}, which was also used in~\\cite{SocialEvent2011}. The underlying idea is that users connect to the antenna that has the highest signal strength, which is not necessarily the closest antenna. \n\n  We begin by estimating the total signal strength of an antenna $i$ at a\n  certain position $x$. We assume, similarly to~\\cite{ZANG2010}, that the total\n  signal strength consists of three components: the power of the antennas,\n  the loss of signal strength over distance and some stochastic fading of the signal due to scattering and reflection in the environment. Specifically, we use the following parameters.\n  \\begin{itemize}\n    \\item The position of antenna $i$ is denoted by $X_i$.\n    \\item The power is denoted by $p_i$, and we assume this to be constant and equal for all antennas because we have no information regarding the power of the antennas. Therefore, $p_i=p$ for all $i$.\n    \\item The loss of signal at position $x$ for antenna $i$ is modeled as\n      \\begin{equation}\n        L_i(x) = \\frac{1}{\\| x - X_i \\|^\\beta},\n      \\end{equation}\n      where $\\beta$ is a parameter indicating how quickly the signal decays.\n    \\item The so-called Rayleigh fading of the signal from antenna $i$ can be modeled by a unit mean exponential random variable $R_i$~\\cite{Tse:2005p10323}, for which the cumulative distribution function (cdf) is\n      \\begin{equation}\n        \\Pr(R_i \\leq r) = F(r) = 1 - e^{-r}.\n      \\end{equation}\n      Furthermore, we assume all $R_i$ to be independent.\n  \\end{itemize}\n  The total signal strength $S_i(x)$ of antenna $i$ at location $x$ is then modeled as \n  \\begin{equation}\n\t\tS_i(x) = p L_i(x) R_i,\n\t\\label{eq:signalstrength}\n\t\\end{equation}\n\tand we model the probability that a user at position $x$ connects to antenna $i$, $\\Pr(a=i | x)$, as the probability that the signal strength of antenna $i$ is larger than that of any other antenna:\n\t\\begin{align}\n\t\t\\Pr(a=i | x) &= \\Pr(S_i(x) > S_j(x), ~~ \\forall j) \\nonumber \\\\\n                 &= \\prod_j{\\Pr(S_i(x) > S_j(x))}.\n\t\t\\label{equ:probabilityAiX}\n\t\\end{align}\n  This probability density is displayed in Fig.~\\ref{fig:antenna_dens}.\n  \n\\begin{figure}\n  \\begin{center}\n     \\includegraphics{8-dens_plot}\n  \\end{center}\n  \\caption{Probability density $\\Pr(a=i | x)$ (represented by topographic curves) for a particular antenna $i$ (central black `X'), with neighboring antennas (red `X's) and the local Voronoi tessellation (dark lines) also shown. The probability density can be seen as a smoothed Voronoi tessellation in which there is a small probability of connection to antenna $i$ when the user is in another Voronoi cell.}\n  \\label{fig:antenna_dens}\n\\end{figure}\n\n\t\n\tThis probability can be seen as a smoothed Voronoi tessellation, in which a user will always connect to the closest antenna, by taking the limit of $\\beta \\to \\infty$. In that case, we are essentially considering the situation in which the path loss is dominant over the Rayleigh fading. Hence, little noise is involved, and whenever a closer antenna exists, it will be used.\n\n  \\subsubsection{Antenna neighborhoods}\\label{sec:Neighborhood}\n  \n  As mentioned in the previous section, the probability that a user will connect to a specific antenna depends on the position of other nearby antennas.  The relevant set of antennas $\\mathcal{X}$ can be rather large, which can slow down the computation of the probabilities. Using a local approximation might accelerate this process without affecting the results.\n  \n  The idea of using a local approximation is tied to the decreased probability that a call will be linked to an antenna that is far away. Only some of the antennas around a given position are in fact relevant. It therefore seems natural to construct local neighborhoods of antennas so as to make the method more efficient without introducing any significant error.\n  \n  We define the neighborhood $\\mathcal{X}_i$ and the domain $\\mathcal{D}_i$ of antenna $i$ to consist of the smallest circle enclosing at least all of the Delaunay neighbors (and possibly more). As mentioned previously, the Delaunay neighbors are those antennas located in adjacent Voronoi cells.\n  \\begin{itemize}\n    \\item For each antenna, we select all Delaunay neighbors and then select the\n      maximum distance between the focal antenna and any of these neighbors:\n      \\begin{equation}\n        \\rho_i = \\max \\{ d(X_i, X_j) | j \\text{~Delaunay neigh. of~} i\\},\n      \\end{equation}\n      where $d(X_i,X_j)$ is the distance between antenna $i$ and $j$.\n    \\item We then define the domain \n      \\begin{equation}\n        \\mathcal{D}_i = \\{x | \\|x - X_i\\| \\leq \\delta \\rho_i\\}\n        \\label{eq:domain}\n      \\end{equation}\n      as the region within radius $\\delta \\rho_i$, where $\\delta $ is a scaling factor. We observe that choosing $\\delta =2$ leads to an error of less than $0.1\\%$ in the computation of $\\Pr(a=i | x)$ compared\\footnote{average error based on $1000$ random points} to using the entire set $\\mathcal{X}$.\n    \\item Finally, the set of Delaunay neighbors\\footnote{To deal with antennas near the border of the country (for which the Delaunay neighbors can be far away), we take this border into account, and create a slightly different neighbor set.} is taken as all antennas within this region:\n      \\begin{equation} \\mathcal{X}_i = \\{j | X_j \\in \\mathcal{D}_i \\text{~for~} j \\in \\mathcal{X} \\}.\\end{equation}\n    Note that this set contains at least all of the Delaunay neighbors and may also contain other antennas.  \n  \\end{itemize}\n   \n  Finally, using equation (\\ref{equ:probabilityAiX}), we approximate the probability  as \n  \\begin{equation}\n    \\Pr(a=i | x) \\approx \\prod_{j \\in\\mathcal{X}_i}{Pr(S_i(x) > S_j(x))},\n    \\label{eq:probabilityAisubX}\n  \\end{equation}\n  leading to a large reduction in the computational time required\n\t\n  \\subsubsection{Maximum Likelihood Estimation}\n  \n  We use the model explained above to more accurately estimate the position of each frequent location. For each such location, we know the number of calls $k_i$ made using antenna $i$. The probability that $k_i$ calls were made using antenna $i$ given position $x$ is then $\\Pr(a=i | x)^{k_i}$.  Hence, the log likelihood of observing call frequencies $k$ for the antennas in $\\mathcal{X}_f$, where $f$ is the MFA of a frequent location, for a certain position $x$ is\n  \\begin{equation}\n    \\log \\mathcal{L}(x|k) = \\sum_{i \\in \\mathcal{X}_f} k_i \\log \\Pr(a=i | x).\n    \\label{equ:log_likelihood}\n  \\end{equation}\n  The maximum likelihood estimate (MLE) $\\hat{x}$ of the position of a frequent location is then given by\n  \\begin{equation}\n    \\hat{x} = \\arg \\max_{x} \\log \\mathcal{L}(x|k).\n    \\label{equ:mle}\n  \\end{equation}\n\nTo find the MLE, we employ a derivative-free optimization scheme because the\ngradient of the likelihood function is costly to evaluate. In particular, we use\nthe Nelder-Mead algorithm\\cite{NelderMead}, initialized with the weighted\naverage position of the antennas associated with the frequent location. The\ndistance between the average position of the antennas and the MLE is\n\\unit{1.7}{\\kilo\\meter} on average and reaches a maximum of approximately\n\\unit{35}{\\kilo\\meter}. This shows that although using the average position provides a reasonable approximation, it is not always accurate.\n\n\\subsection{Results}\nWe now analyze the results of the position estimation. First, we present our results concerning the geographical distribution of frequent locations around the country and compare these results to independent statistics. We then analyze commuting distances, i.e., the distances of travel between home and office, and develop a model of the number of commuters between each pair of counties\\footnote{We used the NUTS-3 data defined by Eurostat, which, in the case of Portugal, consists of groups of municipalities; we refer to these as ``counties'' for simplicity.}.\n\n\\subsubsection{Population density estimation}\n\n\\begin{figure*}\n  \\begin{center}\n   \\includegraphics{9-maps}\n  \\end{center}\n  \\caption{(a) Population sizes per county throughout the country (based on statistics from INE), (b) estimated number of homes per county, and (c) the distribution of all frequent locations. Lighter colors indicate higher values.}\n  \\label{fig:distr_FL}\n  \\label{fig:map_pop_dens}\n\\end{figure*}\n\nThe position estimates of all frequent locations can be used to analyze the population distribution throughout the country. Using the county level data, we counted the number of home locations for each county. We then compared these results to population density data obtained from the \\emph{Instituto Nacional de Estatistica\\footnote{http:\/\/www.ine.pt}} (INE). As shown in Fig.~\\ref{fig:map_pop_dens}, there is a strong correspondence between the INE population data for each county and our estimate. The correlation between the two is 0.92. This indicates that we can accurately estimate population size based on the mobile phone data. A more accurate density plot of the frequent locations is shown in Fig.~\\ref{fig:distr_FL}, which illustrate that these locations are concentrated in the cities. A comparison of Fig.~\\ref{fig:distr_FL} to the distribution of the average positions of users over the entire period (Fig.~\\ref{fig2}), shows that the distribution of frequent locations is more pronounced. Average positions are likely to be distorted by commutes and to interpolate between home and office.\n\n\\subsubsection{Commuting distances}\n\nThe home and office positions determined above can be used to estimate commuting distances. For individuals who have more than one home or one office, multiple commuting distances could be calculated, but it would be unclear which distance is the ``correct'' one. Therefore, for this analysis, we considered only the $12\\%$ of users who have exactly one home and one office (and possibly some unidentified frequent locations). This means that each user considered has exactly one commuting distance. These commutes are plotted in Fig.~\\ref{fig:commute_map}, with smaller distances indicated in brighter colors. Two things stand out on this map. First, the two largest cities in Portugal, Porto and Lisbon, are clearly discernible. Second, most of the cities appear to predominantly attract people living in the immediate surroundings.\n\n\\begin{figure}\n  \\begin{center}\n     \\includegraphics{10-commuting_map}\n  \\end{center}\n  \\caption{Commute map for our sample of users. Brighter colors indicate smaller commuting distances. Most of the commutes cover only small distances, although some commutes span half the country. The number of commutes decays approximately log-normally with distance.}\n  \\label{fig:commute_map}\n\\end{figure}\n\n\nThe distribution of commuting distances depicted in Fig.~\\ref{fig:commute_dist}\nappears to be affected by the location of Porto and Lisbon. Two different\nregimes can be discerned: one regime reflecting commuting distances of less than\n\\unit{150}{\\kilo\\meter} and the other reflecting larger distances. This\ncoincides with the distance between Lisbon and Porto, which is approximately\n\\unit{300}{\\kilo\\meter}. In fact, most of Portugal is within\n\\unit{150}{\\kilo\\meter} of one of these two cities. This suggests that most\npeople tend to work no further away than the closest largest city, i.e., it is\nunlikely that people living near Porto work in Lisbon. The set of commuting\ndistances that are less than \\unit{150}{\\kilo\\meter} can be reasonably well fitted using a log-normal distribution with parameters $\\mu=2.35$ and $\\sigma=0.94$, as displayed in Fig.~\\ref{fig:commute_dist}.\n\n\\begin{figure}\n    \\begin{center}\n       \\includegraphics{11-commuting_dist_distr}\n    \\end{center}\n    \\caption{The distribution of commuting distances revealed by the analysis of\n    mobile phone data. These distances exhibit a log-normal distribution for $d\n    < \\unit{150}{\\kilo\\meter}$ (blue line). The full distribution is shown, with\n    the line at \\unit{150}{\\kilo\\meter} separating the two different regimes.\n    These two distinct regimes probably arise because almost all of continental\n    Portugal is within \\unit{150}{\\kilo\\meter} of either Porto or Lisbon.}\n    \\label{fig:commute_dist}\n\\end{figure}\n\nA common model for analyzing commuting distance is the gravity model~\\cite{Balcan2009, GravityModel}, although recently another parameterless model has been suggested~\\cite{Simini2012}. This model formulates the number of trips $w_{ij}$ made between two locations $i$ and $j$ as proportional to the population sizes at the origin $P_i$ and at the destination $P_j$, with some decay, depending on the distance $d_{ij}$ between $i$ and $j$. More precisely, the model is formulated as\n\\begin{equation}\n \\hat{w}_{ij} \\sim \\frac{P^\\alpha_iP^\\beta_j}{f(d_{ij})},\n\\end{equation}\nwhere $f(d_{ij})$ is usually taken as either a power law $d^\\gamma_{ij}$ or an exponential decay $e^{\\gamma d_{ij}}$, with parameters $\\alpha$, $\\beta$ and $\\gamma$ to be estimated from the data.\n\nHere, we formulate the gravity model in terms of the number of trips (commutes) made between county $i$ and county $j$. Instead of simply considering the population size as $P_i$ and $P_j$, we can take into account our previous calculations of the distributions of both home positions and office positions. The probability of a trip from $i$ to $j$ can then be formulated in terms of the number of home locations at the origin $H_i$ and the number of office locations at the destination $O_j$. \n\n\\begin{table*}\n  \\begin{center}\n    \\begin{tabular}{llll}\n      Coefficient & Variable & $d_{ij} < \\unit{150}{\\kilo\\meter}$ & $d_{ij} \\geq \\unit{150}{\\kilo\\meter}$ \\\\\n    \\hline \\hline\n    $\\alpha$ & Number of homes at origin  & $0.17^{**} \\pm 0.013$  & $0.018 \\pm 0.013$ \\\\\n    $\\beta$  & Number of offices at destination & $0.21^{**} \\pm 0.013$  & $0.030^{*} \\pm 0.012$ \\\\\n    $\\gamma$ & Distance  & $0.37^{**} \\pm 0.018$  & $0.13 \\pm 0.11$ \\\\\n      & $R^2$ & $0.52$ & $0.26$ \\\\\n      & $R^2$ (exponential fit) & $0.46$ & $0.26$\n    \\end{tabular}\n  \\end{center}\n\n  \\caption{Fitted parameters and $R^2$ of the gravity model with $f(d_{ij}) = d_ij^\\gamma$, with standard errors reported. We also report $R^2$ for the exponential fit $f(d_{ij}) = e^{\\gamma d_{ij}}$, which is slightly worse. ${}^{**}$ $p < 0.001$, ${}^*$ $p < 0.05$.}\n  \\label{tab:gravity_fit}\n\\end{table*}\n\nAgain, we discern two regimes: a close-range regime with $d_{ij} < \\unit{150}{\\kilo\\meter}$ and a long-distance regime with $d_{ij} \\geq \\unit{150}{\\kilo\\meter}$. Fitting both the power law decay and the exponential decay, we find that the power law decay provides a slightly better fit. The results are displayed in Fig.~\\ref{fig:gravity_fit} and in Table~\\ref{tab:gravity_fit}. Interestingly, the decay distance parameter $\\gamma$ for large distances is not significant, suggesting that for distances $d_{ij} \\geq \\unit{150}{\\kilo\\meter}$, the number of trips no longer depends on the actual distance. In fact, the only coefficient that is significant for large distances is the coefficient of the number of offices at the destination. Thus, for larger distances, only the number of work opportunities at the destination appears to be important.\n\n\\begin{figure*}\n  \\begin{center}\n    \\subfloat[Power law decay]{\\includegraphics[width=0.5\\textwidth]{12-gravity_estimation_log_fit}}\n    \\subfloat[Exponential decay]{\\includegraphics[width=0.5\\textwidth]{13-gravity_estimation_exp_fit}}\n  \\end{center}\n  \\caption{Plots of the prediction ratio $\\hat{w}_{ij}\/w_{ij}$ for commuting\n  distance of $d < \\unit{150}{\\kilo\\meter}$ (left panels) and $d_{ij} \\geq\n  \\unit{150}{\\kilo\\meter}$ (right panels) for (a) the power law decay $f(d_{ij})\n  = d_{ij}^\\gamma$, and (b) the exponential decay $f(d_{ij}) = e^{\\gamma\n    d_{ij}}$.  Red squares indicate mean values, and blue circles indicate\n  medians.}\n  \\label{fig:gravity_fit}\n\\end{figure*}\n\nThe fit of the model is better when the numbers of home and office locations per county are used than when the population sizes are used. As shown in Table~\\ref{tab:gravity_fit}, the values of $R^2$ for the two regimes are $0.52$ and $0.26$, respectively, when the numbers of home and office locations are used, compared to $0.43$ and $0.24$, respectively, when population sizes are used. Hence, it is worth taking into account the numbers of offices and homes when modeling commuting distances instead of simply using population size as an approximation for both. \nIn the present case, the model slightly overestimates the number of shorter\ncommutes, indicating that there is room for improvement. This deviation might be\ndue to the aggregation of information at a small resolution. On the other hand,\nthis might also be due to a real effect: distances below some threshold have no\neffect. In this case, trips under about \\unit{2}{\\kilo\\meter} should be almost\nunaffected by distance. Higher resolution data is needed to investigate this in\nmore detail.\n\n\\section{Conclusion} \\label{sectpaul}\n\nIn this study, we analyzed the behavior of mobile phone customers based on their calling habits. We first sampled $100\\,000$ customers randomly and filtered their locations, as these are based on associated antenna locations, which are subject to disturbances. We then defined and computed $50$ features that describe the calling behaviors of the customers. We performed a correlation analysis on these features, which showed that movement- and location-related features are correlated with many other features. We then analyzed the data using principal component analysis (PCA). This showed that the original features are highly redundant and can be efficiently compressed if some reconstruction error (e.g., $5\\,\\%$) is allowed. We also performed a cluster analysis and that revealed a small number of typical user classes. We computed the relative importance of each feature in the PCA and the cluster analysis and found that location- and movement-related features are especially important in both cases. We therefore analyzed the users' most common locations.\n\nWe clustered these frequent locations based on weekly calling patterns and found that only home and office locations could be clearly identified. Other patterns of usage (such as use from weekend houses) are surely present in the data, but these are marginal when compared to the clear pattern of use from home and office locations. We characterized the number of frequent locations for each user and the most common combinations of frequent locations (e.g., multiple houses or offices). Finally, we estimated the positions of frequent locations based on a probabilistic inference framework. Using these positions, we derived a fairly accurate estimate of the distribution of the population, which showed a correlation of 0.92 with independent population statistics. These positions also allowed us to analyze commuting distances, and we found that the data are reasonably well explained by a gravity model. This model works better when the numbers of homes and offices are considered instead of population sizes. This indicates that when analyzing commuting distances, it is worth taking the distribution of home and office location into account.\n\nThe present study represents an exploratory analysis of the data. Further research into the frequent locations and associated user behavior should be undertaken. This data set contains both geographical data and social network data, and it would be interesting to further analyze the interaction between the two.\n\n\\section*{Acknowledgments}\nThe authors acknowledge support from the grant ``Actions de recherche concert\\'ees --- Large Graphs and Networks'' of the Communaut\\'e Fran\\c caise de Belgique and from the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian State Science Policy Office. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.\n\n\n\n\n\n\\bibliographystyle{model1-num-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe microscopic manifestation of temperature in lattice dynamics is generally understood through the \\textit{harmonic approximation}, where atomic motion is mapped onto a set of vibrational normal modes (i.e., phonons)~\\cite{Born1998}. \nModern studies into lattice dynamics of solids significantly expand on this view to account for anharmonic phenomena such as phase transitions~\\cite{Fultz2020,Bruce1981}, thermal expansion~\\cite{Barron1999} and thermal conductance~\\cite{Sun2010,Romero2015,Whalley2016}.\n\nIn contrast, studies that focus on the electronic properties of solids rarely go beyond a quasi-harmonic (QH) treatment, where the normal modes are re-normalized for each given temperature to account for thermal expansion~\\cite{Fultz2010}.\nThe effects of lattice dynamics are usually incorporated through the prism of electron-phonon interaction. \nThis can manifest in modified carrier lifetimes~\\cite{Eiguren2002,Eiguren2003}, corrections to the electronic band structure~\\cite{Gopalan1987}, mobility estimations~\\cite{Li2015,Fiorentini2016,Schweicher2019} and calculations for vibrational spectroscopy~\\cite{Strauch1995,Gillet2017,Cusco2007,Menedez1983}.\nFormally, almost any \\emph{ab initio} calculation of electron-phonon interactions relies on the harmonic approximation\\cite{Giustino2017}.\nA different manifestation of lattice dynamics in electronic properties is the temperature evolution of the material's dielectric response. \nThe need to extend anharmonic modeling to dielectric properties has been demonstrated perhaps most strikingly in halide perovskite semiconductors~\\cite{Zhu2016,Joshi2019,Martiradonna2018,Sender2016,Schilcher2021,Guo2019,Omer2017}.\nImportantly, by relying on molecular dynamics simulations, calculations of the dielectric response of a crystal are not limited to a perturbative treatment of anharmonicity.\n\nBecause inelastic light scattering originates in the vibrational modulation of a crystal's polarizability auto-correlation function~\\cite{Yu2010}, it is the ideal probe to study the effect of anharmonicity on dielectric properties.\nA main drawback of optical inelastic light scattering is that momentum conservation limits vibrational contributions to the $\\Gamma$, or zero crystal momentum point. \nThis limitation does not apply in 2\\textsuperscript{nd} order Raman scattering, which is driven by contributions from the entire Brillouin zone (BZ)~\\cite{Loudon1964}. \nSimply put, 2\\textsuperscript{nd} order Raman describes the inelastic scattering of a single photon with two phonons~\\cite{Cardona1982}. \nThe scattering intensity inside a given frequency interval $I\\left(\\Omega\\right)\\text{d}\\Omega$ emerges out of contributions from all possible phonon combinations with the appropriate energy and momentum conservation. \nIt is therefore sensitive to any anharmonic effect in either the phonon dispersion or the dielectric response.\n\nIn this study, we introduce an \\textit{ab initio} approach incorporating finite temperature effects into the dielectric response of a crystal. \nThe approach is based on the effective interatomic force constants used in the temperature dependent effective potential (TDEP) method~\\cite{Hellman2011,Hellman2013a,Hellman2013}. \nThe computational method is benchmarked with experimental measurements of dielectric response in terms of 2\\textsuperscript{nd} order Raman scattering.\nAs a model system we choose NaCl.\nIt is an ideal showcase since first order Raman is forbidden by symmetry, necessitating a computational methodology that goes beyond the lowest order.\nFurthermore, the central role of anharmonicity in NaBr was recently demonstrated~\\cite{Shen2020}, motivating further examination of the rock-salt structure.\nThe comparison between simulated and experimental results shows that the harmonic treatment is inadequate when considering dielectric response at finite temperatures.\nIn contrast, the method developed here not only reproduces experimental results more faithfully than conventional approaches, it also shows how higher order dielectric response is linked to other manifestations of anharmonicity.\nMore specifically, it accounts for phonon broadening, thermal transport, and demonstrates how a measurement of the 2\\textsuperscript{nd} order Raman spectrum can be used to conclude the dominant anharmonic expressions in a given material.\nImportantly, due to its \\emph{ab initio} nature, our method provides a full description of the linear dielectric response of the material and its dependence on atomic displacements. \n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{NaCl_measVSsim4.pdf}\n\\caption{Temperature dependence of the Raman spectrum for NaCl by measurement (black), by quasi-harmonic (red) and TDEP (blue) simulation. \\# symbols mark system artifacts.}\n\\label{fig:NaClTemp}\n\\end{figure}\n\n\n\nOur main results are presented in Fig.~\\ref{fig:NaClTemp}, where we compare the measured, temperature dependent, unpolarized \\footnote{measured\/calculated sums of the scattered intensities for a full rotation of incident polarization} Raman spectra of NaCl and corresponding simulations. \nThe technical details for the construction of an unpolarized spectrum, as well as the QH computation, are provided in Sec.~S\\rom{2},~S\\rom{7}, respectively, in the supplementary material (SM). \nOur experimental spectra are in good agreement with past measurements~\\cite{Welsh1949, Krauzman1969}. \nWe observe a continuous spectrum with some sharp non-Lorentzian features centered around 280 cm$^{-1}$.\n\nUnlike early analytical treatments\\cite{Birman1962,Birman1963,Lax1961}, our computation is able to predict the entire shape of the spectrum. \nIt is immediately evident that even at 300~K the QH calculation for the spectrum is imperfect.\nIt overestimates the high ($\\approx$300~cm$\\textsuperscript{-1}$) to low ($\\approx$100~cm$\\textsuperscript{-1}$) intensity ratio.\nIt also fails to capture the broad decaying intensity above 350~cm$\\textsuperscript{-1}$.\nThe discrepancies between experiment and QH calculation at 300~K are remarkable, as they demonstrate the significance of anharmonic effects in the dielectric response of an archetypal rock-salt crystal, even at room temperature. \nThese discrepancies get even larger at higher temperatures.\nAbove 500~K the main spectral feature around 300~cm$\\textsuperscript{-1}$ broadens and flattens but the QH calculation predicts an opposite trend. \nThese findings constitute direct evidence for the failure of a QH approach to explain the dielectric response of NaCl, due to finite temperature anharmonic effects.\n\nContrary to the QH approach, the TDEP-based computation shows much better agreement with experiment. \nAt 300~K the line-shape of the TDEP-based Raman spectrum (blue) follow more accurately the experimental spectrum (black). \nAgreement between experimental and TDEP spectra is even better above 500~K. \nThe main discrepancy remains the decaying intensity above 350~cm$\\textsuperscript{-1}$ which is not captured by either QH or TDEP. \nWe suspect this is the result of higher order ($>2$) scattering terms since, as discussed below, our calculation does not include them. \nHaving established that our computational method is both required and successful in predicting the Raman spectrum, we turn to discuss the TDEP-based calculation in more detail.\n\n\n\nWe start by briefly reiterating the main idea behind TDEP to describe the ionic motion. \nThe starting point is the lattice dynamical expansion for the crystal energy:\\cite{Born1998}\n\\begin{equation}\n\\begin{split}\n    H = & H_0 + \n    \\frac{1}{2}\n    \\sum_{ij\\alpha\\beta} \\Phi_{ij}^{\\alpha\\beta} \n    u_i^{\\alpha} u_j^{\\beta}\n     +\n    \\frac{1}{3!}\n    \\sum_{ikj\\alpha\\beta\\gamma} \\Phi_{ijk}^{\\alpha\\beta\\gamma} \n    u_i^{\\alpha} u_j^{\\beta} u_k^{\\gamma} +\n\\\\ &    \n    +\n    \\frac{1}{4!}\n    \\sum_{ikjl\\alpha\\beta\\gamma\\delta} \\Phi_{ijkl}^{\\alpha\\beta\\gamma\\delta} \n    u_i^{\\alpha} u_j^{\\beta} u_k^{\\gamma} u_l^{\\delta}\n    +\n    \\ldots\n\\end{split}\n\\end{equation}\nwhere $u_i^\\alpha$ denote the displacement of atom $i$ in direction $\\alpha$. The interatomic force constants $\\Phi$ are determined by minimizing the difference between the model system and \\emph{ab initio} calculated forces $\\vec{f}$.\n\\begin{equation}\n    \\vec{\\Phi} = \\arg\\min_{\\Phi} \\| \\vec{f}^{\\textrm{model}} - \\vec{f}^{\\textrm{ab initio}} \\|\n\\end{equation}\nThe \\emph{ab initio} calculations are performed in the canonical ensemble yielding an explicit temperature dependent model Hamiltonian -- either via molecular dynamics or a self-consistent stochastic sampling~\\cite{Shulumba2017}. \nIn this study we used the latter.\n\nSince the sampling is done at finite temperature, an explicit temperature dependence is built into the effective interaction parameters. The TDEP Hamiltonian is constructed to be the best possible fit at a certain temperature.\n\nTo asses the accuracy of our model Hamiltonian for the lattice dynamics we calculate the phonon spectral function and compare it with a neutron scattering experiment~\\cite{Raunio1969a}. \nThe calculated spectral functions agree well with experiments and previous non-harmonic calculations~\\cite{Ravichandran2018} (see Sec.~S\\rom{8} in the SM).\n\nThe need for a non-perturbative treatment when describing anharmonic lattice dynamics motivates an analogous treatment for the dielectric response. \nTo extend the formalism to dielectric properties we similarly expand the susceptibility $\\vec{P}$ and dipole moment $\\vec{M}$ in terms of atomic displacements $u$:\\cite{Born1998}\n\\begin{align}\n\\begin{split}\n    \\label{eq:pexpansion}\n    P^{\\mu\\nu} = & P_0^{\\mu\\nu} + \n    \\sum_{i\\alpha}\n    P^{\\mu\\nu\\alpha}_i u_i^{\\alpha} +\n   \n    \\frac{1}{2}\n    \\sum_{ij\\alpha\\beta}\n    P^{\\mu\\nu\\alpha\\beta}_{ij} u_i^{\\alpha}u_j^{\\beta} + \n    \\ldots\n\\end{split}\n\\\\\n\\begin{split}\n    \\label{eq:mexpansion}\n    M^{\\mu} = & M_0^{\\mu} + \n    \\sum_{i\\alpha}\n    M^{\\mu\\alpha}_i u_i^{\\alpha} + \n   \n    \\frac{1}{2}\n    \\sum_{ij\\alpha\\beta}\n    M^{\\mu\\alpha\\beta}_{ij} u_i^{\\alpha}u_j^{\\beta} + \n    \\ldots\n\\end{split}\n\\end{align}\nHere the indices $\\mu\\nu$ denote Cartesian components of electric field derivatives. \nIn principle this expansion is done for dynamic quantities -- including the electronic frequency dependence -- but in the present case for NaCl we are interested in a frequency range far from any resonances and can safely ignore the electronic frequency dependence.\n\nIn analogy with the process described for the force constants, the terms in the dipole moment and susceptibility expansions are determined by minimizing the difference between the model values and those obtained from simulation:\n\\begin{align}\n    \\vec{M} & = \\arg\\min_{\\vec{M}} \\left\\| \n        \\frac{\\partial \\vec{M}}{\\partial u}^{\\textrm{model}} - \n        \\frac{\\partial \\vec{M}}{\\partial u}^{\\textrm{ab initio}}\n    \\right\\|\n\\\\\n    \\vec{P} & = \\arg\\min_{\\vec{P}} \\left\\| \n        \\vec{P}^{\\textrm{model}} - \n        \\vec{P}^{\\textrm{ab initio}}\n    \\right\\|.\n\\end{align}\n\nThe end result is a set of interaction parameters that explicitly depend on temperature, incorporating all orders of non-harmonic effects in the linear dielectric response of a material. \nWe note that the inclusion of frequency dependence adds no conceptual difficulty, only the practical challenge of accurately determining the frequency dependent electronic dielectric response for a simulation with hundreds of atoms. \nThe algorithm for determining the interaction parameters is described in detail in Sec.~S\\rom{5} in the SM, and the specific computational details are given in Sec.~S\\rom{8}.\nHaving obtained the dielectric response we turn to compute the Raman scattering cross-section.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{NaCl_PO_maps_small.pdf}\n\\caption{NaCl Raman PO color map for measurement and TDEP simulation in 300~K. Frequency of spectral features in simulation slightly misses experimental values, so dashed color lines help guide comparison.\n}\n\\label{fig:NaClPO}\n\\end{figure*}\n\n\nTheoretical interpretations of 2\\textsuperscript{nd} order Raman spectra usually limit themselves to high symmetry areas of reciprocal space where most density of states singularities are expected~\\cite{Burstein1965,Agrawal1967,Karo1965,Weber1993,Wang1973,Smith2002}.\nThis approach adequately accounts for pronounced peaks, but severely limits the possibility of incorporating anharmonic effects.\nAlso, because of the symmetry breaking of moving away from the $\\Gamma$-point and generalization into multiple phonon states, selection rules in 2\\textsuperscript{nd} order Raman usually become too relaxed to place meaningful restrictions on the measured Raman tensor (see Sec.~S\\rom{4} in SM for more details).\nExisting \\textit{ab initio} approaches to 2\\textsuperscript{nd} order Raman all operated strictly within the harmonic approximation~\\cite{Gillet2017,Strauch1995}.\nA Shell-model based calculation by \\citet{Bruce1972} did incorporate phonon-phonon interactions, but his partial treatment lead him to conclude these are negligible in the Raman spectrum of NaCl.\n\nThe construction of the full 2\\textsuperscript{nd} order spectrum from first principles requires a few computational steps.\nThe starting point is to use the TDEP method to construct an anharmonic Hamiltonian that describes the dynamics of the ions.\nThis anharmonic Hamiltonian is then coupled to an expansion of the polarizability in terms of atomic displacements.\nThe coupled ionic-polarizability system can then be solved with many-body methods, providing the finite temperature spectrum.\nThe steps involved are discussed in detail below.\n\nFollowing \\citet{Cowley1963}, the Raman scattering cross-section is given by:\n\\begin{equation}\n    \\sigma(\\Omega) \\propto \n    \\sum_{\\mu\\nu\\xi\\rho}\n\tE^{\\textrm{out}}_{\\mu}\n\tE^{\\textrm{out}}_{\\xi}\n\tI_{\\mu\\nu,\\xi\\rho}(\\Omega)\n\tE^{\\textrm{in}}_{\\nu}\n\tE^{\\textrm{in}}_{\\rho}    \n\\end{equation}\nwhere $\\vec{E}^{\\textrm{in}}$ and $\\vec{E}^{\\textrm{out}}$ are the electric field vectors of the incoming and outgoing light, with Greek letters standing for Cartesian components.\nThe tensor $I$ depends on the probing frequency $\\Omega$ and constitutes the material property governing Raman scattering. \nIt is given by\n\\begin{equation}\n\\label{eq:Cowley_Beast}\n    I_{\\mu\\nu,\\xi\\rho}(\\Omega) = \\int \\text{d}t \\left\\langle P^{\\mu\\nu}(t)P^{\\xi\\rho}(0) \\right\\rangle e^{-i\\Omega t},\n\\end{equation}\nthe Fourier transformed thermal average of the polarizability-polarizability autocorrelation function.\nEq.~\\ref{eq:Cowley_Beast} says that the complete description for Raman scattering is given by a tensor detailing the vectorial relationship between incident and scattered fields.\nIt is therefore desirable to inspect polarization dependence when comparing theory and measurement.\n\nTo realize this, we performed a polarization-orientation (PO) Raman measurement of a NaCl single crystal. \nThe temperature dependent spectra presented in Fig.~\\ref{fig:NaClTemp} are actually calculated sums of the scattered intensities for a full rotation of incident polarization.\nWe probe separately the parallel and perpendicular scattered polarizations with respect to the linearly polarized incident beam orientation.\nDetails of the experimental setup are given in Sec.~S\\rom{1} in the SM.\n\nFigure~\\ref{fig:NaClPO} shows the full PO color map for measurement and simulation in 400~K (PO maps for all temperatures are given in Sec.~S\\rom{2} in the SM).\nOur theoretical calculation also fully accounts for the tensor nature of Raman scattering, making a detailed comparison possible.\nThe calculated model closely follows the PO dependence throughout the spectrum.\nDashed colored lines follow corresponding spectral features in simulation and experiment (some discrepancy in absolute frequency persists).\nOur calculation correctly predicts the periodicity with incident angle of each feature, as well as the relative phase between them, thereby faithfully reproducing the tensorial nature of the modulated dielectric response.\n\nThe full tensorial formalism for 2\\textsuperscript{nd} order Raman has already been rigorously worked out, but is rarely used and is repeated and extended here.\\cite{Cowley1964b,Wallis1971}\nFar from resonance, to lowest order, the tensor $I$ relating incident and outgoing light has four terms:\n\\begin{equation}\n    I_{\\mu\\nu,\\xi\\rho}(\\Omega) = \n    I^{(\\textrm{I})}_{\\mu\\nu,\\xi\\rho}+\n    I^{(\\textrm{II})}_{\\mu\\nu,\\xi\\rho}+\n    I^{(\\textrm{III})}_{\\mu\\nu,\\xi\\rho}+\n    I^{(\\textrm{IV})}_{\\mu\\nu,\\xi\\rho}\n\\end{equation}\nwith the different terms given by\n\\begin{align}\n    \\label{eq:ramanfirstorder}\n\tI^{(\\textrm{I})}_{\\mu\\nu,\\xi\\rho} = &\n\t(n(\\Omega)+1)\n\t\\sum_s P^{(\\textrm{I})}_{\\mu\\nu,\\xi\\rho}(s) \n\tJ_s(\\Omega)    \n\\\\\n\\label{eq:ramansecondorder}\n\\begin{split}\n\tI^{(\\textrm{II})}_{\\mu\\nu,\\xi\\rho} = &\n\t2\n\t\\sum_{ \\vec{q} s_1 s_2 }\n\tP^{(\\textrm{II})}_{\\mu\\nu,\\xi\\rho}(\\vec{q},s_1,s_2)\n\t\t\\int \n\tJ_{s_1}(\\Omega')(n(\\Omega')+1)\n \\times \\\\\n\t&\n\tJ_{s_2}(\\Omega-\\Omega')(n(\\Omega-\\Omega')+1)\n\td\\Omega'\n\\end{split}\n\\\\\n\\begin{split}\n\tI^{(\\textrm{III})}_{\\mu\\nu,\\xi\\rho} = & \n\t-6(n(\\Omega)+1)\n\t\\sum_{\\vec{q}s_1 s_2 s_3}\n\tJ_{s_3}(\\Omega)\n\t\\times \\\\\n\t& P^{(\\textrm{III})}_{\\mu\\nu,\\xi\\rho}(\\vec{q},s_1,s_2,s_3)\n\t\\,\\Im\\left\\{ S(s_1,s_2,\\Omega) \\right\\}\n\t\\vspace{5pt}\n\\end{split}\n\\\\\n\\begin{split}\n\tI^{(\\textrm{IV})}_{\\mu\\nu,\\xi\\rho} = & \n\t3(n(\\Omega)+1)\n\t\\sum_{\\vec{q} s_1 s_2}\n\tP^{(\\textrm{IV})}_{\\mu\\nu,\\xi\\rho}(\\vec{q},s_1,s_2) \\times\n\t\\\\ & \n\t(2n_{s_2}+1)\n\tJ_{s_1}(\\Omega)    \n\\end{split}\n\\end{align}\nwhere the matrix elements $\\vec{P}^{(\\textrm{I})}-\\vec{P}^{(\\textrm{IV})}$ are defined via the Fourier components of the terms in Eq.~\\ref{eq:pexpansion}, (see Sec.~S\\rom{7} in the SM for the explicit expressions and derivations of the above terms). \nThe phonon spectral function for mode $s$, $J_s = -\\Im\\left\\{ G_{s}(\\Omega) \\right\\}\/\\pi$ is obtained from\n\\begin{equation}\n    G_{s}(Z) = \\frac{2\\omega_s}{\\omega_s^2 - 2\\omega_s\\Sigma_s(Z) - Z^2}\n\\end{equation}    \nwhere\n\\begin{equation}\n\\begin{split}\n    \\Sigma_{\\vec{q}s}(Z) = & -18 \\sum_{\\vec{q}_1\\vec{q}_2 s_1 s_2}\n    \\left| \\Phi^{s s_1s_2}_{\\vec{q}\\vec{q}_1\\vec{q}_2} \\right|^2\n    S(s_1,s_2,Z) +\n\\\\ & +\n    12 \\sum_{\\vec{q}_1 s_1}\n    \\Phi^{s s s_1 s_1}_{\\vec{q}\\bar{\\vec{q}}\\vec{q}_1\\bar{\\vec{q}}_1}(2n_{\\vec{q}_1 s_1}+1)\n\\end{split}    \n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:Sfun}\n\\begin{split}\n    &S(s_a,s_b,Z) = \n    \\\\\n\t&(n_{a}+n_{b}+1)\n\t\\left[\n\t\\frac{1}{(\\omega_{a}+\\omega_{b}-Z)_p}-\n\t\\frac{1}{(\\omega_{a}+\\omega_{b}+Z)_p}\n\t\\right]\n\t\\\\\n\t&+\n\t(n_{a}-n_{b})\n\t\\left[\n\t\\frac{1}{(\\omega_{b}-\\omega_{a}+Z)_p}-\n\t\\frac{1}{(\\omega_{b}-\\omega_{a}-Z)_p}\n\t\\right]\n\\end{split}\t\n\\end{equation}\nThe first term, Eq.~\\eqref{eq:ramanfirstorder}, is the first order Raman scattering which comes down to the one-phonon spectral function weighted by the first order Raman matrix elements. \nThe spectral function contains any broadening and shifts due to anharmonicity, and also any deviation from a Lorentzian lineshape. \nThis term is what is commonly referred to as Raman scattering.\n\nThe second term, Eq.~\\eqref{eq:ramansecondorder}, is the second order Raman scattering. \nThe matrix elements contain the momentum conservation, and the convolution term contains the energy conservation. \nIf one were to set the matrix elements to unity it would yield a spectrum that follows the two-phonon density of states.\n\nThe third and fourth terms are more subtle, as they contain contributions from the whole BZ but are multiplied with the one-phonon lineshape. \nThey will thus decay rapidly away from the first order Raman peaks, with the net effect of slightly shifting and altering the shapes of the first order peaks. \nThese terms are one reason that neutron scattering and Raman scattering measurements might not coincide exactly. \n\nThe theoretical PO maps in Fig.~\\ref{fig:NaClPO} are calculated from Eq.~\\eqref{eq:ramansecondorder}. In NaCl first order Raman is forbidden by symmetry, which means $P^{(\\textrm{I})}$,$P^{(\\textrm{III})}$, and $P^{(\\textrm{IV})}$ are zero. We stress that the spectrum is calculated from the interacting phonons, i.e., a convolution of one-phonon spectral functions and includes explicit anharmonic effects.\n\n\nIt is worth dwelling on the temperature effects in 2\\textsuperscript{nd} order Raman, and their connection to anharmonic behavior.\nThe scattering is generated by phonons throughout the BZ. \nSince the TDEP method implicitly contains all orders of anharmonicity, this already imparts a non-trivial temperature dependence, but the effects of anharmonicity go beyond the temperature dependence of the bare phonon dispersion curve. \nInspecting Eq.~\\eqref{eq:ramansecondorder} one sees that it is a convolution of two one-phonon spectral functions. \nIf we replace the interacting spectral functions with the non-interacting ones we will recover the traditional result\\cite{Cowley1964b} (See SM, Eq.~S73).\nIn the present study we found it crucial to include the interacting spectral functions when calculating the 2\\textsuperscript{nd} order Raman spectra, without them agreement with experiment was markedly worse. \nIntuitively, if the one-phonon DOS is broadened, the two-phonon DOS must also broaden -- and since the two-phonon DOS is a convolution, non-Lorentzian lineshapes have a larger impact on two-phonon than on one-phonon spectra.\n\nBesides its own non-trivial temperature dependence, 2\\textsuperscript{nd} order Raman emerged as a uniquely sensitive probe for anharmonic lattice dynamics.\nThree-phonon scattering is ostensibly a straightforward perturbative anharmonic effect.\nOnly a small fraction of all possible three-phonon combinations satisfy energy and momentum conversion -- this is what we refer to as the scattering phase space. \nMathematically, this is defined via the imaginary part of Eq.~\\eqref{eq:Sfun}. \nThe temperature dependence of available phonon scattering phase space is, however, a non-perturbative anharmonic effect, and can lead to a host of surprising trends with temperature.\nThermal conductivity that does not follow $T^{-1}$\\cite{Romero2015}, phonon line widths that do not grow linearly with temperature\\cite{Delaire2011b}, or in our case, a Raman spectrum that does not follow the quasiharmonic approximation are all consequences of non-pertrubative anharmonicity.\n\nThe 2\\textsuperscript{nd} order Raman spectrum follows the same set of momentum and energy conservation rules as three-phonon scattering.\nIn fact, the spectra is directly proportional to the available scattering phase space. \nThis makes Raman scattering invaluable as a direct probe of anharmonicity: we can see how (a part) of the scattering phase space evolves with temperature, providing insight into the underlying mechanisms governing strongly anharmonic materials.\nIn this specific case we can verify prior theoretical studies. \\citet{Ravichandran2018} found it necessary to include temperature-dependent phonons to accurately describe the thermal transport in NaCl -- this can be verified experimentally by the change in the Raman spectra with temperature in Fig.~\\ref{fig:NaClTemp}. Moreover, four-phonon scattering was predicted to be important. The presence of higher order scattering is evident in the tail at large wavenumbers (second order Raman can only contribute up to wavenumbers equal to $2\\omega_{\\textrm{max}}$, third order Raman, which would involve four-phonon scattering, can contribute up to $3\\omega_{\\textrm{max}}$). \n\nThe methodology described in this letter is general with regard to the coupling between anharmonicity and dielectric response. \nIn Sec.~S\\rom{9} of the SM we present the infrared absorption spectra, as well as the temperature dependence of the dielectric tensor and Born charges. \nWe could not find any experimental results to verify these quantities with, so we leave them as predictions.\n\n\n\n\nIn conclusion, we have demonstrated the crucial role of anharmonic lattice dynamics effects in determining the phonon-modulated linear dielectric response of a crystal. \nThis was achieved by introducing an \\textit{ab initio} method to calculate the dielectric response of a crystal at finite temperatures and comparing it to the first continuous PO spectrum of an exclusively 2\\textsuperscript{nd} order Raman structure.\nOur generalized TDEP method incorporates all orders of non-harmonic effects through a sampling of temperature dependent lattice configurations, expressed as effective interatomic force constants.\nWe introduce analogous effective interaction tensors that govern the dielectric response, i.e., we expand the dipole moment and polarizability in terms of atomic displacements.\nUsing the new generalized TDEP method, other dielectric material properties such as ferroelectricity and capacitance may be calculated, along with their temperature dependence. \nBy isolating and evaluating the normal modes comprising the emergent temperature dependence of a material property, new design rules may be inferred for better functional materials operating at finite temperatures.\n\n\\section*{acknowledgments}\nThe authors would like to thank Dr. Lior Segev (WIS) for invaluable software development. O.Y. acknowledges funding from ISF(1861\/17), BSF (grant no. 2016650) and ERC (850041 -ANHARMONIC). O.H. acknowledges support from the Swedish Research Council (VR) program 2020-04630. Supercomputer resources were provided by the Swedish National Infrastructure for Computing (SNIC).  \n\n\n\\section{Experimental setup}  \\label{SM:Experiment}\nAll Raman measurements were performed on our home-built system. Figure \\ref{fig:SM_setup} shows a schematic of the system. \nThe beam path begins with a 488nm solid-state (Coherent Sapphire SF 488-100 CDRH) laser, which will excite the Raman scattering in the sample. \nThe laser is filtered for any amplified spontaneous emission by a volume holographic ASE (Ondax) filters. Next, the beam shape is optimized and culminated by a pinhole spatial filter.\nControl over incident and scattered polarization is realized by a set of two polarizers and two half-wave plates.\nFirst, linear polarization is insured by a (Thorlabds) calcite polarizer. A monochromatic half-wave plate (HW0 in Fig.~\\ref{fig:SM_setup}) is placed \\textit{before} the polarizer and rotated for optimal incident intensity, correcting for the original laser polarization.\nThe beam then passes through a (NoiseBlock 90\/10 Ondax) beam splitter (BS) with the experimental polarization orientation controlled by another monochromatic half-wave plate (HW1 in Fig.~\\ref{fig:SM_setup}). \nThe polarized beam then enters a microscope and focused on the sample by a 10X (Zeiss) objective.\nExcitation powers of 7-30 mW were used (depending on temperature), measured just before the microscope.\nThe sample (see \\ref{SM:Synthesis}) is placed inside a (TS1000 HiTemp) Linkam stage under Argon purge. \nPurging with a mono-atomic gas proved crucial, since under vacuum local laser heating would create defects, whisle nitrogen would dominate the Raman signal with its rotational excitations.  \nA mono-atomic gas is also the best option for effective thermal coupling to the Linkam hot plate.\nInside the Linkam the magic happens, and laser light is scattered off the thermally excited crystal. \nThe back-scattered light is collected by the same objective and passes back through HW1 and through the BS, where 90\\% of the Rayleigh is eliminated and Raman signal transmitted.\nNote that the beam's polarization at this point is not known, since we can not assume the scattered light, especially the Raman signal, scattered back into the same incident polarization, as is described in \\ref{fig:SM_setup} (\\textbf{b}).\nIndeed, there is no reason to assume the beam is still linearly polarized.  \nWe may, however, as always, describe the beam's polarization as the sum of two linear components, one parallel to the incident beam polarization, and the other perpendicular. \nAfter passing again through HW1, all light scattered in parallel to the incident beam polarization will now parallel the original polarization determined by the first polarizer. \nAll light scattered perpendicular to the incident beam, will now be also polarized perpendicular to the first polarizer.  \nAfter the BS, the scattered beam (orange line in \\ref{fig:SM_setup}) passes through an achromatic half-wave plate (HW2 in \\ref{fig:SM_setup}) which either leaves or rotates the signal polarization by 90\\textsuperscript{0}. \nAnother identical polarizer (designated \"analyzer\" in Fig.~\\ref{fig:SM_setup}) filters the polarization of the signal. \nWe thus separate the two scattered polarization components.\nWith HW2 at 0\\textsuperscript{0}, only the parallel signal will make it past the analyzer, with HW2 at 45\\textsuperscript{0}, only the perpendicular signal is transmitted.\nThe system achieved an extinction ratio $(I_{\\perp}\/I_{\\parallel})$ between 1\/500 and 1\/100 depending on the orientation of HW1.\nAfter the analyzer, the remaining Rayleigh component of the signal is attenuated by two (Ondax SureBlock ultra-narrowband) notch filters and the signal is directed into a one meter (Horiba FHR 1000) spectrometer and detected by a (Horiba Synapse) CCD.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{setupANDfaceon.png}\n    \\caption{({\\bfseries a}) Schematic of experimental setup. Laser light comes from top. Scattered signal goes right for detection. \n    ({\\bfseries b}) Diagram for the different polarizations involved in the measurement. $\\overset{\\rightarrow}{\\vec{E}_i}$ is the incident linear polarization. $\\overset{\\rightarrow}{\\vec{E}_s}$ is the polarization for the scattered signal and need not be linearly polarized.\n    Parallel measurement (HW2 at 0$\\textsuperscript{0}$) \n    probes the scattered field's projection on $\\hat{\\vec{e}}_{\\parallel}$. \n    The perpendicular measurement (HW2 at 45\\textsuperscript{0}) \n    probes the scattered field's projection on \n   \n    \\label{fig:SM_setup}}\n\\end{figure}\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{NaCl_400K_PO.pdf}\n\\caption{NaCl Raman PO color map for measurement and TDEP simulation in 400~K. Top Row is for experimental measurement, bottom row for calculation. Left column is for parallel configuration and right is for perpendicular.}\n\\label{fig:NaClPO400K}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{NaCl_500K_PO.pdf}\n\\caption{NaCl Raman PO color map for measurement and TDEP simulation in 500~K. Top Row is for experimental measurement, bottom row for calculation. Left column is for parallel configuration and right is for perpendicular.}\n\\label{fig:NaClPO500K}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{NaCl_600K_PO.pdf}\n\\caption{NaCl Raman PO color map for measurement and TDEP simulation in 600~K. Top Row is for experimental measurement, bottom row for calculation. Left column is for parallel configuration and right is for perpendicular.}\n\\label{fig:NaClPO600K}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{NaCl_700K_PO.pdf}\n\\caption{NaCl Raman PO color map for measurement and TDEP simulation in 700~K. Top Row is for experimental measurement, bottom row for calculation. Left column is for parallel configuration and right is for perpendicular.}\n\\label{fig:NaClPO700K}\n\\end{figure*}\n\n\\vspace{1.5cm}\n\\section{Measurement and analysis} \\label{SM:Measurement}\n\nHere We give the detailed description of all data acquisition processing and present all PO maps for all measured temperatures.\n\nTo accumulate a Raman crystallography data-set, we rotate the polarization orientation of the incident laser beam (polarization $\\overrightarrow{\\vec{E}}_i$\nin Fig.~\\ref{fig:SM_setup} (\\textbf{b}) in $10^0$ steps covering $\\theta=0-360^0$ in a plane parallel to the NaCl (001) crystal face. \nAt each $\\theta$ step, a Raman spectrum is collected for both HW2 configurations (see $\\hat{\\vec{e}}_{\\parallel}$ and $\\hat{\\vec{e}}_{\\perp}$ in Fig.~\\ref{fig:SM_setup}).\n\nThe signal of 2\\textsuperscript{nd} order Raman is very weak, so long exposure times and multiple accumulations are necessary. \nFor each measured spectrum, that is, for a given temperature, at a given incident polarization angle at given detection configuration, approximately 6 accumulations of 60 seconds were collected.\n\nSome residual baseline persisted in all measured spectra, resembling a Lorentzian tail towards lower frequencies. This was cancelled in all unpolarized figures via standard baseline reduction. \nAll PO maps are given in \\Cref{fig:NaClPO400K,fig:NaClPO500K,fig:NaClPO600K,fig:NaClPO700K}. These are presented with no augmentation other than a low-frequency cut-off, where the signal is governed by the notch filters and not the actual scattering. For each temperature, the corresponding TDEP calculation (see \\ref{SM:green}) is also presented.\n\n\\vspace{1.5cm}\n\\section{Single Crystal Synthesis} \\label{SM:Synthesis}\n\nThe NaCl single crystal used in Raman measurements was home grown.\nDouble-deionized water (DDW) was heated slowly to around $40-50^0C$.\nHigh purity $(\\geq99\\%)$ Sodium Chloride powder was added and stirred into the DDW until saturation.\nThe saturated solution was poured into another wide flask with a few fresh crystals for crystallization sites.\nThe flask was left covered with a filter paper (to avoid dust penetration and allow evaporation) in a chemical hood for several days until the crystals reached a size of $\\sim$1cm.\nFinally, the crystals were taken out of the dried flask and placed on a clean wipe to adsorb all residual water.\nA picture of the crystal used is given in figure \\ref{fig:Single_Crystal}.\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{Single_Crystal.jpg}\n\\caption{One of the NaCl single crystals synthesized for the experiment..}\n\\label{fig:Single_Crystal}\n\\end{figure*}\n\n\n\\vspace{1.5cm}\n\\section{Factor group analysis for 2\\textsuperscript{\\MakeLowercase{nd}} order Raman} \\label{SM:Group_theory}\n\nThe full factor group analysis for the rock-salt structure has already been worked out by Burnstein~\\cite{Burstein1965}, following methods established by Birman~\\cite{Birman1962} and Elliot~\\cite{Elliot1960}.\nThese ostensibly provide a complete tool-set to interpret 2\\textsuperscript{nd} order Raman spectra by assigning each spectral feature to a known combination of vibrational modes.  \nAs mentioned in the main text, this treatment may prove useful for mode assignment of a known phononic dispersion relations to an observed spectrum. \nHowever, even in the high symmetry case of the cubic $O_h$ point group, selection rules for 2\\textsuperscript{nd} order Raman are too relaxed to offer complete predictive power, even with reliable dispersion relations at hand. \n\nTo demonstrate this, we perform the factor group analysis for for one high symmetry point in the Birllouin zone, specifically, for the $\\mathrm{X'_5 (TO)},\\:\\mathrm{X'_5 (TA)}$ phonon pair in the reciprocal space X-point in a rock-salt structure.\n\nThe basic selection rule governing Raman scattering is~\\cite{Yu2010}:\n\\begin{equation} \\label{eq:selection_rule}\n    \\Gamma_f\\subseteq\\Gamma_{15}\\otimes\\Gamma_{15}\n\\end{equation}\nwith $\\Gamma_f$ the representation of the excited phonon state, and $\\Gamma_{15}$ the representation for the momentum operator in BSW notation (Raman scattering can be viewed as the product o two dipole transitions). The crux of the matter lies in the fact that in 2\\textsuperscript{nd} order Raman this final phonon state is generally composed of two off $\\Gamma$-point phonons, with its representation given by the Kronecker product of the two one phonon states. For example:\n\\begin{align}\n        \\nonumber\n       & \\textrm{Quantum state:}\\:\\:\n        \\left|\\mathrm{X}_{2phonon}\\right\\rangle =\n        \\left|\\mathrm{X'_5 (TO),\\:X'_5 (TA)}\\right\\rangle\\:\\to\\: \\\\  \\nonumber \n        \\\\ \n       & \\textrm{Representation:}\\:\\:\\:\n        \\mathrm{X}_{2phonon} =  \n        \\mathrm{X'_5 (TO)\\otimes X'_5 (TA) = X_1 \\oplus X_2 \\oplus X_3 \\oplus X_4}\n\\end{align}\nwith all the irreducible representations in the direct sum being Raman active according to Eq.~\\eqref{eq:selection_rule}. This leads to the symmetry allowed Raman tensor form~\\cite{Bilbao2006}:\n\\begin{equation}\n    R_{\\mathrm{X'}_5\\times\\mathrm{X'}_5} = R_{\\mathrm{X}_1}+R_{\\mathrm{X}_2}+R_{\\mathrm{X}_3}+R_{\\mathrm{X}_4}= \\\\\n    \\begin{pmatrix}\n    a & d & 0 \\\\\n    d & b & 0 \\\\\n    0 & 0 & c \n    \\end{pmatrix}\n\\end{equation}\nwhich is a very general Raman tensor indeed. Now, consider that this tensor is responsible for the contribution to the scattered intensity inside some frequency interval $I(\\Omega)\\mathrm{d}\\Omega$ from a single phonon combination.\nIn practice, equation $(11)$ allows a multitude of combinations to contribute to the same interval, such that accurately predicting the observed intensity, let alone the PO behavior of the signal, becomes intractable. \nThe situation becomes even worse for lower symmetry structures, where more phonon combinations will be Raman active and have more general Raman tensors.\n\n\n\n\\vspace{1.5cm}\n\\section{Numerical determination of interaction tensors} \\label{SM:Tensors}\n\nHere we briefly reiterate the TDEP procedure for consistency of notation and to clarify the analogy between how dipole and polarizibility interactions are determined and how interatomic force constants are determined. The starting point is a set of displacements $\\vec{u}$ and forces $\\vec{f}$ from a set of $N_c$ supercells sampled from a canonical ensemble at temperature $T$. Although omitted in the notation, all interaction tensors depend on temperature as well as the volume (or more generally strain). The interatomic force constants are determined as follows: consider a supercell with $N_a$ atoms and forces and displacements given by the $3N_{a} \\times 1$ vectors $\\vec{u}$ and $\\vec{f}$ (for clarity we will make note of the dimensions of the matrices in each step):\n\n\n\\begin{align}\n    \\underbrace{\\vec{f}}_{3N_a \\times 1} \n    & = - \n    \\underbrace{\\vec{\\Phi}}_{3N_a \\times 3 N_a} \n    \\underbrace{\\vec{u}}_{3N_a \\times 1} \n\\intertext{this can equivalently be written as}\n    \\underbrace{\\vec{f}}_{3N_a \\times 1} \n    & = - \n    \\underbrace{(\\vec{I} \\otimes \\vec{u}^T)}_{3N_a \\times (3N_a)^2}\n    \\underbrace{\\vec{\\Phi}_v}_{ (3N_a)^2 \\times 1}\n\\intertext{with the Kronecker product $\\otimes$. To express the forces from $N_c$ supercells the matrices are stacked on top of each other:}\n    \\underbrace{\n    \\begin{pmatrix}\n        \\vec{f}_1 \\\\\n        \\vdots \\\\\n        \\vec{f}_{N_c}\n    \\end{pmatrix}\n    }_{3N_a N_c \\times 1} \n    & = - \n    \\underbrace{\n    \\begin{pmatrix}\n        \\vec{I} \\otimes \\vec{u}_1^T \\\\\n        \\vdots \\\\\n        \\vec{I} \\otimes \\vec{u}_{N_c}^T \\\\\n    \\end{pmatrix}\n    }_{3N_aN_c \\times (3N_a)^2}\n    \\underbrace{\\vec{\\Phi}_v}_{ (3N_a)^2 \\times 1}\n    %\n\\intertext{I can without any loss of generality express the second order force constants as a linear combination of $N_x$ irreducible components $\\vec{x}^{\\Phi\\textrm{II}}$, where $N_x \\le (3N_a)^2 $, via a matrix $\\vec{C}^{\\Phi\\textrm{II}}$:}\n    \\underbrace{\n    \\begin{pmatrix}\n        \\vec{f}_1 \\\\\n        \\vdots \\\\\n        \\vec{f}_{N_c}\n    \\end{pmatrix}\n    }_{3N_a N_c \\times 1} \n    & = - \n    \\underbrace{\n    \\underbrace{\n    \\begin{pmatrix}\n        \\vec{I} \\otimes \\vec{u}_1^T \\\\\n        \\vdots \\\\\n        \\vec{I} \\otimes \\vec{u}_{N_c}^T \\\\\n    \\end{pmatrix}\n    }_{3N_aN_c \\times (3N_a)^2}\n    \\underbrace{\n    \\vec{C}^{\\Phi\\textrm{II}}\n    }_{(3N_a)^2 \\times N_x}\n    }_{=\\vec{A}^{\\Phi\\textrm{II}}\\,, 3N_a N_c \\times N_x }\n    \\underbrace{\\vec{x}^{\\Phi\\textrm{II}}}_{ N_x \\times 1}\n\\end{align}\nThe set of symmetry relations and invariances determines $\\vec{C}$ and $N_x$,\\cite{Leibfried1961,Maradudin1968,Born1998} and in general $N_x \\ll (3N_a)^2$ which considerably simplifies the problem of numerically determining $\\vec{x}$. In practice, only the matrix $\\vec{A}$ is determined and stored. The nominally very large matrices that go into the construction of $\\vec{A}$ are quite sparse and construction presents negligible computational cost. The second order force constants imply matrices that could possibly be treated naively, but the generalization to higher order quickly becomes intractable with dense matrix storage. The third order force constants, for example, comes down to\n\\begin{equation}\n    \\underbrace{\n    \\begin{pmatrix}\n        \\vec{f}_1 \\\\\n        \\vdots \\\\\n        \\vec{f}_{N_c}\n    \\end{pmatrix}\n    }_{3N_a N_c \\times 1} \n     = - \n    \\underbrace{\n    \\underbrace{\n    \\begin{pmatrix}\n        \\vec{I} \\otimes \\vec{u}_1^T \\otimes \\vec{u}_1^T\\\\\n        \\vdots \\\\\n        \\vec{I} \\otimes \\vec{u}_{N_c}^T \\otimes \\vec{u}_{N_c}^T \\\\\n    \\end{pmatrix}\n    }_{3N_aN_c \\times (3N_a)^3}\n    \\underbrace{\n    \\vec{C}^{\\Phi\\textrm{III}}\n    }_{(3N_a)^3 \\times N_x}\n    }_{=\\vec{A}^{\\Phi\\textrm{III}}\\,, 3N_a N_c \\times N_x }\n    \\underbrace{\\vec{x}^{\\Phi\\textrm{III}}}_{ N_x \\times 1}    \n\\end{equation}\nwhere the contracted matrix $\\vec{A}$ is many orders of magnitude smaller than the matrices it is built from. The interatomic force constants are determined in succession:\n\\begin{align}\n\\label{eq:f2}\n    \\vec{A}^{\\Phi\\textrm{II}}\n    \\vec{x}^{\\Phi\\textrm{II}} & = \\vec{f}\n\\\\\n\\label{eq:f3}\n    \\vec{A}^{\\Phi\\textrm{III}}\n    \\vec{x}^{\\Phi\\textrm{III}} & = \n    \\vec{f}-\n    \\vec{A}^{\\Phi\\textrm{II}}\n    \\vec{x}^{\\Phi\\textrm{II}}\n\\\\    \n\\label{eq:f4}\n    \\vec{A}^{\\Phi\\textrm{IV}}\n    \\vec{x}^{\\Phi\\textrm{IV}} & = \n    \\vec{f}\n    -\n    \\vec{A}^{\\Phi\\textrm{II}}\n    \\vec{x}^{\\Phi\\textrm{II}}\n    -\n    \\vec{A}^{\\Phi\\textrm{III}}\n    \\vec{x}^{\\Phi\\textrm{III}}\n\\end{align}\nWhere $\\vec{f}$ denotes the reference forces to be reproduced. Equations \\eqref{eq:f2}--\\eqref{eq:f4} are overdetermined and solved as a least squares problem. This ensures that the baseline harmonic part becomes as large as possible, and that the higher order terms become smaller and smaller.\n\nIn a similar manner we search for a way to determine $M$ and $P$ (the expansion of the dipole moment and polarizability with respect to position) to all orders. The input is again a supercell, where in addition to forces we also have determined $\\partial^2 U \/ \\partial E^2$ as well as $\\partial^2 U \/ \\partial E \\partial R$, that is the polarizability and Born charges. In a manner analogous to the force constants we determine the irreducible components of the interaction tensors and express the polarizability for a supercell as:\n\\begin{equation}\n\\begin{split}\n    \\underbrace{ \\partial_E \\partial_E U (\\vec{u}) }_{9 \\times 1}\n    &\n    =\n    P_0 + \n    \\underbrace{ \\vec{I} \\otimes \\vec{u}^T }_{9 \\times 27 N_a }\n    \\underbrace{ P }_{27 N_a \\times 1 }\n    +\n    \\underbrace{ \\vec{I} \\otimes \\vec{u}^T \\otimes \\vec{u}^T }_{9 \\times 9 (3N_a)^2 }\n    \\underbrace{ P }_{9 (3N_a)^2 \\times 1 }\n    +\n    \\ldots\n\\\\    \n    & = \n    P_0 + \n    \\vec{A}^{P\\textrm{I}}\n    \\vec{x}^{P\\textrm{I}}\n    +\n    \\vec{A}^{P\\textrm{II}}\n    \\vec{x}^{P\\textrm{II}}\n    +\n    \\ldots\n\\end{split}    \n\\end{equation}\nThe terms in the polarizability expansion are determined slightly differently than the interatomic force constants. For a set of $N_c$ supercells I build the set of differences in polarizability between supercell $i$ and $j$:\n\\begin{align}\n    \\Delta^{0} P_{ij} & = \n    \\partial_E \\partial_E U(\\vec{u}_i) -\n    \\partial_E \\partial_E U(\\vec{u}_j)\n\\intertext{and solve for $\\vec{x}^{P\\textrm{I}}$ from}\n    \\left(\n    \\vec{A}^{P\\textrm{I}}_{i} -\n    \\vec{A}^{P\\textrm{I}}_{j}\n    \\right)\\vec{x}^{P\\textrm{I}}\n    & = \\Delta^{0} P_{ij}\n\\intertext{where the set of all pairs of supercell $ij$ are used. The fluctuations from the first order are removed to construct}\n    \\Delta^{1} P_{ij} & = \n    \\left(\\partial_E \\partial_E U(\\vec{u}_i) - \\vec{A}^{P\\textrm{I}}_{i}\\vec{x}^{P\\textrm{I}}  \\right)  -\n    \\left(\n    \\partial_E \\partial_E U(\\vec{u}_j) -\n    \\vec{A}^{P\\textrm{I}}_{j}\\vec{x}^{P\\textrm{I}}\n    \\right)\n\\\\\n    \\left(\n    \\vec{A}^{P\\textrm{II}}_{i} -\n    \\vec{A}^{P\\textrm{II}}_{j}\n    \\right)\\vec{x}^{P\\textrm{II}}\n    & = \\Delta^{1} P_{ij}\n\\end{align}\nAnd so on for all orders considered. The baseline polarizability is then determined via\n\\begin{equation}\n    P_0 + P^{\\textrm{th}} = \\left\\langle \n    \\partial_E \\partial_E U -\n    \\vec{A}^{P\\textrm{I}}\n    \\vec{x}^{P\\textrm{I}} -\n    \\vec{A}^{P\\textrm{II}}\n    \\vec{x}^{P\\textrm{II}}\n    - \\ldots\n    \\right\\rangle\n\\end{equation}\nWhere $P^{\\textrm{th}}$ is the contribution to the susceptibility from the second order terms in the polarizability expansion:\\cite{Cowley1964b}\n\\begin{equation}\n    P^{\\textrm{th}} = \\sum_{\\lambda} P_{\\lambda\\lambda}(2n_{\\lambda}+1)\n\\end{equation}\nwhere the matrix elements are defined below.\nFor a perfectly harmonic system (or any harmonic sampling of phase space) the average of the first order term disappears.\n\nThe interaction parameters for the dipole moment are determined from the set of Born charges calculated in the supercell. We deliberately do not work with polarization directly. The quantized nature of polarization makes finding the polarization on the same branch for a large number of supercells tedious if not outright problematic. The starting point is again the idea of expressing the Born charges for a supercell in terms of the irreducible components:\n\\begin{equation}\n\\begin{split}\n    \\underbrace{ \\partial_E \\partial_R U }_{9N_a \\times 1}(\\vec{u})\n    &\n    =\n    \\underbrace{ \\vec{M}^{\\textrm{I}} }_{9N_a \\times 1}\n    +\n    \\underbrace{ \\vec{I} \\otimes \\vec{u}^T }_{9N_a \\times (9N_a)(3 N_a) }\n    \\underbrace{ M^{\\textrm{II}} }_{(9N_a) (3N_a) \\times 1 }\n    +\n    \\underbrace{ \\vec{I} \\otimes \\vec{u}^T \\otimes \\vec{u}^T }_{9N_a \\times (9N_a)(3 N_a)^2 }\n    \\underbrace{ M^{\\textrm{III}} }_{(9N_a) (3N_a)^2 \\times 1 }\n    +\n    \\ldots\n    \\\\\n    & = \n    \\vec{A}^{\\textrm{MI}}\\vec{x}^{\\textrm{MI}} + \n    \\vec{A}^{\\textrm{MII}}\\vec{x}^{\\textrm{MII}} + \n    \\vec{A}^{\\textrm{MIII}}\\vec{x}^{\\textrm{MIII}} + \n\\end{split}    \n\\end{equation}\nWhere the terms are determined in succesion:\n\\begin{align*}\n    \\vec{A}^{\\textrm{MI}}\\vec{x}^{\\textrm{MI}} & = \\partial_E \\partial_R U\n    \\\\\n    \\vec{A}^{\\textrm{MII}}\\vec{x}^{\\textrm{MII}} & = \\partial_E \\partial_R U -\\vec{A}^{\\textrm{MI}}\\vec{x}^{\\textrm{MI}}\n    \\\\\n    \\vec{A}^{\\textrm{MIII}}\\vec{x}^{\\textrm{MIII}} & = \\partial_E \\partial_R U -\\vec{A}^{\\textrm{MI}}\\vec{x}^{\\textrm{MI}}\n    -\\vec{A}^{\\textrm{MII}}\\vec{x}^{\\textrm{MII}}\n\\end{align*}\n\n\n\\vspace{1.5cm}\n\\section{Long-ranged electrostatics in polar materials}\n\nLong-ranged interactions are not captured appropriately by the interatomic force constants and have to be treated with care. The formalism to deal with this in quasiharmonic calculations is well established, see e.g. \\citet{Gonze1997}. For finite temperature calculations of interactions beyond pair interaction the formalism requires a slight adjustment.\n\nWe will first discuss the problem in quite general terms. Our starting point is a general long-ranged (polar) pair interaction $f(r)$ that we range-separate in some way:\n\\begin{equation}\n    f(r) = \n    f^{\\textrm{sr}}(r)+f^{\\textrm{lr}}(r).\n\\end{equation}\nIf we set the range-separation such that the short-ranged $f^{\\textrm{sr}}$ part fits snugly in the simulation cell, that part of the interaction is taken care of by the usual force constants and requires no additional consideration. The long-ranged tail $f^{\\textrm{lr}}$ is what remains to be treated. This is done by calculating the Hessian of the long-ranged part:\n\\begin{equation}\n    \\Phi^{\\textrm{lr}}_{ij\\alpha\\beta} = \\left. \\frac{\\partial^2 f^{\\textrm{lr}}(r)}{\\partial u_{i}^\\alpha \\partial u_{j}^\\beta} \\right|_{r=r_{ij}}\n\\end{equation}\nafter which the forces from the long-ranged interactions can be calculated\n\\begin{equation}\n    f_{i\\alpha}^{\\textrm{lr}} = -\\sum_{j\\beta} \\Phi^{\\textrm{lr}}_{ij\\alpha\\beta} u_{j\\beta}\n\\end{equation}\nthese long-ranged forces are removed from the calculated forces, such that the procedure for determining the interatomic force constants becomes:\n\\begin{align}\n\\label{eq:ff2}\n    \\vec{A}^{\\Phi\\textrm{II}}\n    \\vec{x}^{\\Phi\\textrm{II}} & = \\vec{f} - \\vec{f}^{\\textrm{lr}}\n\\\\\n\\label{eq:ff3}\n    \\vec{A}^{\\Phi\\textrm{III}}\n    \\vec{x}^{\\Phi\\textrm{III}} & = \n    \\vec{f}-\\vec{f}^{\\textrm{lr}}-\n    \\vec{A}^{\\Phi\\textrm{II}}\n    \\vec{x}^{\\Phi\\textrm{II}}\n\\\\    \n\\label{eq:ff4}\n    \\vec{A}^{\\Phi\\textrm{IV}}\n    \\vec{x}^{\\Phi\\textrm{IV}} & = \n    \\vec{f}- \\vec{f}^{\\textrm{lr}}\n    -\n    \\vec{A}^{\\Phi\\textrm{II}}\n    \\vec{x}^{\\Phi\\textrm{II}}\n    -\n    \\vec{A}^{\\Phi\\textrm{III}}\n    \\vec{x}^{\\Phi\\textrm{III}}\n\\end{align}\nWhen determining the dynamical matrix, the long-ranged pair interactions are added back again:\n\\begin{equation}\n    \\vec{\\Phi}(\\vec{q}) = \\vec{\\Phi}^{\\textrm{sr}}(\\vec{q})+\\vec{\\Phi}^{\\textrm{lr}}(\\vec{q})\n\\end{equation}\nThis procedure is beneficial in several ways. By subtracting forces (instead of dynamical matrices as in \\citet{Gonze1997}) numerical issues due to aliasing (finite size) errors are greatly reduced. More importantly, it also ensures there is no multiple counting: the long-ranged interactions are subtracted from interactions of all orders, ensuring the short-sightedness of higher order interactions. What we just described is the general procedure, for practical calculations one needs to assume an explicit form for the long-ranged interactions. We use the (now temperature-dependent) Born charges and dielectric constant, and the range-separatation is done using the standard Ewald technique. The resulting real-space force constants are then\\cite{Gonze1997}\n\\begin{align}\n\\Phi^{\\textrm{dd}}|^{\\alpha\\beta}_{ij} & = \\sum_{\\gamma\\delta} Z_{i}^{\\alpha\\gamma}Z_{j}^{\\beta\\delta} \\widetilde{\\Phi}^{\\gamma\\delta}_{ij} \\\\\n\\widetilde{\\Phi}^{\\alpha\\beta}_{ij} & =\n\\label{eq:SM_ddifc}\n\\frac{1}{4\\pi\\epsilon_0}\n\\frac{1}{\\sqrt{\\det \\epsilon}}\n\\left(\n\t\\frac\n\t{ \\epsilon^{-1}_{\\alpha\\beta} }\n\t{|\\Delta_{ij}|_{\\epsilon}^{3}}\n\t-3 \\frac\n\t{\\Delta_{ij}^{\\alpha}\\Delta_{ij}^{\\beta}}\n\t{|\\Delta_{ij}|_{\\epsilon}^{5}}\n\\right)\n\\end{align}\nHere $\\boldsymbol{\\epsilon}$ is the dielectric tensor and $\\Delta$ realspace distances using the dielectric tensor as a metric:\n\\begin{align}\n\\mathbf{r}_{ij} & = \\mathbf{R}_{j}+\\mathbf{\\tau}_j-\\mathbf{\\tau}_i \\\\\n\\mathbf{\\Delta}_{ij} & = \\boldsymbol{\\epsilon}^{-1}\\mathbf{r}_{ij} \\\\\n|\\Delta_{ij}|_{\\epsilon} & = \\sqrt{\\mathbf{\\Delta}_{ij} \\cdot \\mathbf{r}_{ij}}\n\\end{align}\nThis poses an issue when calculating the dynamical matrix: the interactions die off as $~1\/r^{3}$ which makes it necessary to extend the sum over lattice vectors to infinity. This is remedied with the usual Ewald technique:\n\\begin{equation}\n\\widetilde{\\mathbf{\\Phi}}_{ij} (\\mathbf{q})=\n\\widetilde{\\mathbf{\\Phi}}^\\textrm{r}+\\widetilde{\\mathbf{\\Phi}}^\\textrm{q}+\\widetilde{\\mathbf{\\Phi}}^\\textrm{c}\n\\end{equation}\nDividing the sum into a realspace part, a reciprocal part and a connecting part. The realspace part is given by\n\\begin{align}\n\\widetilde{\\mathbf{\\Phi}}^\\textrm{r}_{ij} & =\n-\\frac{\\Lambda^3}{4\\pi\\epsilon_0\\sqrt{\\det \\epsilon}}\n\\sum_{\\mathbf{R}}\n\\mathbf{H}(\\Lambda \\Delta_{ij},\\Lambda|\\Delta_{ij}|_{\\epsilon})\ne^{i \\mathbf{q} \\cdot \\mathbf{R}} \\\\\n\\frac{\\partial \\widetilde{\\mathbf{\\Phi}}^\\textrm{r}_{ij}}{\\partial q_\\alpha}  & =\n-\\frac{\\Lambda^3}{4\\pi\\epsilon_0\\sqrt{\\det \\epsilon}}\n\\sum_{\\mathbf{R}}\niR_\\alpha\n\\mathbf{H}(\\Lambda \\Delta_{ij},\\Lambda|\\Delta_{ij}|_{\\epsilon})\ne^{i \\mathbf{q} \\cdot \\mathbf{R}}\n\\end{align}\nwhere\n\\begin{equation}\nH_{\\alpha\\beta}(\\mathbf{x},y) = \\frac{x_{\\alpha}x_{\\beta} }{y^2}\n\\left[\n\\frac{3\\,\\textrm{erfc}\\,y}{y^3}\n+\n\\frac{2 e^{-y^2}}{\\sqrt{\\pi}}\\left(\\frac{3}{y^2}+2 \\right)\n\\right]\n-\\epsilon^{-1}_{\\alpha\\beta}\n\\left[\n\\frac{\\textrm{erfc}\\,y}{y^3} + \\frac{2 e^{-y}}{\\sqrt{\\pi} y^2 }\n\\right]\\,.\n\\end{equation}\nIn reciprocal space we have\n\\begin{align}\n\\label{eq:SM_philongrange}\n\t\\widetilde{\\Phi}_\\textrm{q}|_{ij}\n\t&=\n\t\\sum_{\\mathbf{K}=\\mathbf{G}+\\mathbf{q}}\n\t\\chi_{ij}(\\mathbf{K},\\Lambda)\n\t\\left( \\mathbf{K}\\otimes\\mathbf{K} \\right) \\\\\n\t\\frac{\\partial}{\\partial q_x} \\widetilde{\\Phi}_\\textrm{q}|_{ij}\n\t%\n\t&=\n\t\\sum_{\\mathbf{K}=\\mathbf{G}+\\mathbf{q}}\n\t\\chi_{ij}(\\mathbf{K},\\Lambda)\n\t\\left( \\mathbf{K}\\otimes\\mathbf{K} \\right)\n\t%\n\t\\left(\n\ti\\tau^x_{ij} - \\left[\\sum_{\\alpha} K_\\alpha \\epsilon_{\\alpha x}\n\t\\right]\n\t\\left[ \\frac{1}{\\|\\mathbf{K}\\|_{\\epsilon}}+\\frac{1}{4\\Lambda^2} \\right]\n\t\\right) +\n\t%\n\t\\chi_{ij}(\\mathbf{K},\\Lambda)\n\t\\begin{pmatrix}\n\t2K_x & K_y & K_z \\\\\n\tK_y & 0 & 0 \\\\\n\tK_z & 0 & 0\n\t\\end{pmatrix} \\\\\n\t\\frac{\\partial}{\\partial q_y} \\widetilde{\\Phi}_\\textrm{q}|_{ij}\n\t%\n\t&=\n\t\\sum_{\\mathbf{K}=\\mathbf{G}+\\mathbf{q}}\n\t\\chi_{ij}(\\mathbf{K},\\Lambda)\n\t\\left( \\mathbf{K}\\otimes\\mathbf{K} \\right)\n\t%\n\t\\left(\n\ti\\tau^y_{ij} - \\left[\\sum_{\\alpha} K_\\alpha \\epsilon_{\\alpha y}\n\t\\right]\n\t\\left[ \\frac{1}{\\|\\mathbf{K}\\|_{\\epsilon}}+\\frac{1}{4\\Lambda^2} \\right]\n\t\\right) +\n\t%\n\t\\chi_{ij}(\\mathbf{K},\\Lambda)\n\t\\begin{pmatrix}\n\t0 & K_x & 0 \\\\\n\tK_x & 2K_y & K_z \\\\\n\t0 & K_z & 0\n\t\\end{pmatrix}\\\\\n\t\\frac{\\partial}{\\partial q_z} \\widetilde{\\Phi}_\\textrm{q}|_{ij}\n\t%\n\t&=\n\t\\sum_{\\mathbf{K}=\\mathbf{G}+\\mathbf{q}}\n\t\\chi_{ij}(\\mathbf{K},\\Lambda)\n\t\\left( \\mathbf{K}\\otimes\\mathbf{K} \\right)\n\t%\n\t\\left(\n\ti\\tau^z_{ij} - \\left[\\sum_{\\alpha} K_\\alpha \\epsilon_{\\alpha z}\n\t\\right]\n\t\\left[ \\frac{1}{\\|\\mathbf{K}\\|_{\\epsilon}}+\\frac{1}{4\\Lambda^2} \\right]\n\t\\right) +\n\t%\n\t\\chi_{ij}(\\mathbf{K},\\Lambda)\n\t\\begin{pmatrix}\n\t0 & 0 & K_x \\\\\n\t0 & 0 & K_y \\\\\n\tK_x & K_y & 2K_z\n\t\\end{pmatrix}\n\\end{align}\nwhere\n\\begin{align}\n\\chi_{ij}(\\mathbf{K},\\Lambda) &  =\n\\frac{1}{\\Omega\\epsilon_0}\n\\frac{\n\\exp\\left(i \\mathbf{q} \\cdot \\mathbf{\\tau}_{ij} \\right)\n\\exp\\left( -\\frac{\\|\\mathbf{K}\\|_{\\epsilon}}{4\\Lambda^2}  \\right)\n}\n{\\|\\mathbf{K}\\|_{\\epsilon}} \\\\\n\\|\\mathbf{K}\\|_{\\epsilon} & =\\sum_{\\alpha\\beta}\\epsilon_{\\alpha\\beta}K_\\alpha K_\\beta\n\\end{align}\nand finally a connecting part given by\n\\begin{equation}\n\\widetilde{\\mathbf{\\Phi}}_\\textrm{c}|_{ij}=\n\\delta_{ij}\n\\frac{\\Lambda^3}{3 \\epsilon_0 \\pi^{3\/2} \\sqrt{ \\det \\boldsymbol{\\epsilon} } }\n\\widetilde{\\boldsymbol{\\epsilon}}\n\\end{equation}\nThis expression constitutes the dipole-dipole interactions at all distances. As explained in the beginning of the section we only want to subtract the long-ranged components from the forces. This separation is realized by choosing the Ewald parameter $\\Lambda$ such that $\\widetilde{\\Phi}^r$ disappears at the supercell boundary, and keeping only the reciprocal space part of the Ewald sum. So, to summarize, the procedure is as follows\n\\begin{itemize}\n    \\item Determine Ewald parameter $\\Lambda$ such that the short-ranged interactions die off at the supercell boundary\n    \\item Calculate $\\vec{\\Phi}^{\\textrm{lr}}(\\vec{q}=0)$ for the simulation supercell via Eq.~\\eqref{eq:SM_philongrange}\n    \\item Use $\\vec{\\Phi}^{\\textrm{lr}}(\\vec{q}=0)$ to calculate the forces $\\vec{f}^{\\textrm{lr}}$ originating from the long-ranged tails of the dipole-dipole interactions\n    \\item Use equations \\eqref{eq:ff2}--\\eqref{eq:ff4} to determine interatomic force constants.\n    \\item When determining the dynamical matrix at any $\\vec{q}$, add $\\vec{\\Phi}^{\\textrm{lr}}(\\vec{q})$ back again.\n\\end{itemize}\nFor reference we also specified the gradient of the dynamical matrix in case the group velocities are needed. \nTo conclude this section we note that there is no principal difference between this approach and what is regularly used, but in practice the presented approach is easier to implement, suffers from less numerical artifacts and makes avoiding double-counting easier. \nThe double-counting we refer to arises because if we only correct the dynamical matrices, some long-ranged polar interactions get picked up by the third and fourth order force constants. \nThe non-analytical behavior of Eq.~\\eqref{eq:SM_ddifc} is already included in the the long-ranged reciprocal space part and independent of Ewald coupling parameter.\n\n\\vspace{1.5cm}\n\\section{Determining Raman spectra}\n\\label{SM:green}\n\nIn this section we outline the procedure to determine Raman spectra of an anharmonic crystal. \nWe will begin by sketching the procedure to obtain the phonon-phonon correlation functions in the presence of three-phonon anharmonicity via the equation of motion approach, since intermediate results are needed to obtain the polarizability-polarizability correlation functions.\n\nIf we express the displacements in terms of creation and annihilation operators we can identify the reciprocal space coefficients needed for the perturbative expansion, repeated here for clarity and consistency of notation. \nStart with the standard creation and annihilation operators for a phonon with momentum $\\vec{q}$ and polarization $s$ (when there is no loss of clarity we use the compound index $\\lambda$ to denote $\\vec{q}s$):\n\\begin{align}\n\ta_{\\vec{q}s} = & \\frac{1}{\\sqrt{2N\\hbar}}\n\t\\sum_{i\\alpha} \\epsilon_{\\vec{q}s}^{i\\alpha}\n\t\\left( \\sqrt{m_i \\omega_{\\vec{q}s}} u_{i\\alpha}-i \\frac{p_{i\\alpha}}{ \\sqrt{ m_i \\omega_{\\vec{q}s}} } \\right) \n\te^{-i\\mathbf{q}\\cdot\\mathbf{r}_i} \n\\\\\n\ta_{\\vec{q}s}^\\dagger = & \\frac{1}{\\sqrt{2N\\hbar}}\n\t\\sum_{i\\alpha} \\epsilon_{\\vec{q}s}^{i\\alpha\\,\\dagger} \n\t\\left( \\sqrt{m_i \\omega_{\\vec{q}s}} u_{i\\alpha} + i \\frac{p_{i\\alpha}}{ \\sqrt{ m_i \\omega_{\\vec{q}s}} } \\right) \n\te^{i\\mathbf{q}\\cdot\\mathbf{r}_i}\n\\\\\n    A_{\\vec{q}s} = a_{\\vec{q}s} + a_{\\bar{\\vec{q}}s}^\\dagger = &\n\\frac{2}{\\sqrt{2N\\hbar}}\n\t\\sum_{i\\alpha} \\epsilon_{\\vec{q}s}^{i\\alpha}\n\t\\sqrt{m_i \\omega_{\\vec{q}s}} u_{i\\alpha}\n\te^{-i\\mathbf{q}\\cdot\\mathbf{r}_i} \n\\\\\n    B_{\\vec{q}s} = a_{\\vec{q}s} - a_{\\bar{\\vec{q}}s}^\\dagger = &\n-\\frac{2i}{\\sqrt{2N\\hbar}}\n\t\\sum_{i\\alpha} \\epsilon_{\\vec{q}s}^{i\\alpha}\n    \\frac{p_{i\\alpha}}{ \\sqrt{ m_i \\omega_{\\vec{q}s}} }\t\n\te^{-i\\mathbf{q}\\cdot\\mathbf{r}_i} \n\\end{align}\nHere $\\epsilon$ are phonon eigenvectors ($\\vec{\\epsilon}_i \\cdot \\vec{\\epsilon}_j = \\delta_{ij}$), $m$ atomic masses, $\\omega$ frequencies. The vector $\\vec{r}$ is a lattice vector, and $u$ and $p$ are the position and momentum operators. We use  the notation $\\bar{\\vec{q}}=-\\vec{q}$. \nI introduce scaled eigenvectors (to simplify notation) via\n\\begin{align}\n\t\\upsilon_{\\vec{q}s}^{i\\alpha} = & \\sqrt{ \\frac{\\hbar}{2 m_i \\omega_{\\vec{q}s} } } \\epsilon_{\\vec{q}s}^{i\\alpha}\n    %\n\\intertext{In this notation the matrix elements pertaining to anharmonicity become}\n    \\Phi_{\\lambda\\lambda'\\lambda''} & = \n    \\frac{1}{3!}\n\t\\sum_{ijk\\alpha\\beta\\gamma}\n\t\\upsilon_{\\vec{q}s}^{i\\alpha}\n\t\\upsilon_{\\vec{q}'s'}^{j\\beta}\n\t\\upsilon_{\\vec{q}''s''}^{k\\gamma}\n\t\\Phi_{ijk}^{\\alpha\\beta\\gamma}\n\te^{-i \n\t(\\vec{q}\\cdot\\vec{r}_i+ \\vec{q}'\\cdot\\vec{r}_j+\\vec{q}''\\cdot\\vec{r}_k)}\n\\\\\n    \\Phi_{\\lambda\\lambda'\\lambda''\\lambda'''} & = \n    \\frac{1}{4!}\n\t\\sum_{ijkl\\alpha\\beta\\gamma\\delta}\n\t\\upsilon_{\\vec{q}s}^{i\\alpha}\n\t\\upsilon_{\\vec{q}'s'}^{j\\beta}\n\t\\upsilon_{\\vec{q}''s''}^{k\\gamma}\n\t\\upsilon_{\\vec{q}'''s'''}^{l\\delta}\t\n\t\\Phi_{ijkl}^{\\alpha\\beta\\gamma\\delta}\n\te^{-i\n\t(\\vec{q}\\cdot\\vec{r}_i +\n\t\\vec{q}'\\cdot\\vec{r}_j +\n\t\\vec{q}''\\cdot\\vec{r}_k +\n\t\\vec{q}'''\\cdot\\vec{r}_l)\n\t} \n\\intertext{so that the anharmonic part of the Hamiltonian becomes}\n\tH_A & = \\Phi_{\\lambda\\lambda'\\lambda''}A_{\\lambda}A_{\\lambda'}A_{\\lambda''}+\n\t\\Phi_{\\lambda\\lambda'\\lambda''\\lambda'''}A_{\\lambda}A_{\\lambda'}A_{\\lambda''}A_{\\lambda'''} + \\ldots\t\n\\end{align}\nIn an analogous way we define the coefficients of the expansion of the polarizability as\n\\begin{align}\n    P^{\\mu\\nu}_{\\lambda} & =\n\t\\sum_{i\\alpha}\n\t\\upsilon_{\\vec{q}s}^{i\\alpha}\n\tP_{i}^{\\mu\\nu,\\alpha}\n\te^{-i\\vec{q}\\cdot\\vec{r}_i}\n\t\\Delta_{\\vec{q}} \n\t=\n\t\\sum_{i\\alpha}\n\t\\upsilon_{\\vec{\\Gamma}s}^{i\\alpha}\n\tP_{i}^{\\mu\\nu,\\alpha}\n\t=\n\tP^{\\mu\\nu}(s)\n\\\\\nP^{\\mu\\nu}_{\\lambda\\lambda'} & =\n    \\frac{1}{2!}\n\t\\sum_{ij\\alpha\\beta}\n\t\\upsilon_{\\vec{q}s}^{i\\alpha}\n\t\\upsilon_{\\vec{q}'s'}^{j\\beta}\n\tP_{ij}^{\\mu\\nu,\\alpha\\beta}\n\te^{-i\\vec{q}\\cdot\\vec{r}_i}\n\te^{-i\\vec{q}'\\cdot\\vec{r}_j}\n\t\\Delta_{\\vec{q}\\vec{q}'} =\n\\\\\n\t& =\t\n\t\\frac{1}{2!}\n\t\\sum_{ij\\alpha\\beta}\n\t\\upsilon_{\\vec{q}s}^{i\\alpha}\n\t(\\upsilon_{\\vec{q}s'}^{j\\beta})^{\\dagger}\n\tP_{ij}^{\\mu\\nu,\\alpha\\beta}\n\te^{i\\vec{q}\\cdot(\\vec{r}_j-\\vec{r}_i)}\n\\\\\n\t& =\n\t\\frac{1}{2!}\n\t\\sum_{ij\\alpha\\beta}\n\t(\\upsilon_{\\vec{q}s}^{i\\alpha})^{\\dagger}\n\t\\upsilon_{\\vec{q}s'}^{j\\beta}\n\tP_{ij}^{\\mu\\nu,\\alpha\\beta}\n\te^{-i\\vec{q}\\cdot(\\vec{r}_j-\\vec{r}_i)} = \n\tP^{\\mu\\nu}(\\vec{q},s,s') = \n\tP^{\\mu\\nu}(\\bar{\\vec{q}},s,s')^\\dagger =\n\tP^{\\mu\\nu}(\\bar{\\vec{q}},s',s)\n\\\\\nP^{\\mu\\nu}_{\\lambda\\lambda'\\lambda''} & =\n    \\frac{1}{3!}\n    \\sum_{ijk\\alpha\\beta\\gamma}\n\t\\upsilon_{\\vec{q}s}^{i\\alpha}\n\t\\upsilon_{\\vec{q}'s'}^{j\\beta}\n\t\\upsilon_{\\vec{q}'s'}^{k\\gamma}\n\tP_{ijk}^{\\mu\\nu,\\alpha\\beta\\gamma}\n\te^{-i\n\t(\\vec{q}\\cdot\\vec{r}_i+\n\t\\vec{q}'\\cdot\\vec{r}_j+\n\t\\vec{q}''\\cdot\\vec{r}_k)\n\t}\n\\intertext{and the dipole moment matrix elements become}\n    M^{\\mu}_{\\lambda} & =\n\t\\sum_{i\\alpha}\n\t\\upsilon_{\\vec{\\Gamma}s}^{i\\alpha}\n\tM_{i}^{\\mu,\\alpha}\n\t=\n\tM^{\\mu}(s)\n\\\\\nM^{\\mu\\nu}_{\\lambda\\lambda'} & =\n    \\frac{1}{2!}\n\t\\sum_{ij\\alpha\\beta}\n\t(\\upsilon_{\\vec{q}s}^{i\\alpha})^{\\dagger}\n\t\\upsilon_{\\vec{q}s'}^{j\\beta}\n\tM_{ij}^{\\mu,\\alpha\\beta}\n\te^{-i\\vec{q}\\cdot(\\vec{r}_j-\\vec{r}_i)} = \n\tM^{\\mu}(\\vec{q},s,s') = \n\tM^{\\mu}(\\bar{\\vec{q}},s,s')^\\dagger =\n\tM^{\\mu}(\\bar{\\vec{q}},s',s)\n\\\\\nM^{\\mu}_{\\lambda\\lambda'\\lambda''} & =\n    \\frac{1}{3!}\n    \\sum_{ijk\\alpha\\beta\\gamma}\n\t\\upsilon_{\\vec{q}s}^{i\\alpha}\n\t\\upsilon_{\\vec{q}'s'}^{j\\beta}\n\t\\upsilon_{\\vec{q}'s'}^{k\\gamma}\n\tM_{ijk}^{\\mu,\\alpha\\beta\\gamma}\n\te^{-i\n\t(\\vec{q}\\cdot\\vec{r}_i+\n\t\\vec{q}'\\cdot\\vec{r}_j+\n\t\\vec{q}''\\cdot\\vec{r}_k)\n\t}\n\\intertext{Such that}\n\\label{eq:polarizabilityexpansion}\nP^{\\mu\\nu} & = P^{\\mu\\nu}_0 + P^{\\mu\\nu}_{\\lambda} A_{\\lambda}\n+ P^{\\mu\\nu}_{\\lambda\\lambda'} \nA_{\\lambda}A_{\\lambda'}\n+ P^{\\mu\\nu}_{\\lambda\\lambda'\\lambda''} \nA_{\\lambda}A_{\\lambda'}A_{\\lambda''} + \\ldots\n\\\\\nM^{\\mu} & = M^{\\mu}_0 + M^{\\mu}_{\\lambda} A_{\\lambda}\n+ M^{\\mu}_{\\lambda\\lambda'} \nA_{\\lambda}A_{\\lambda'}\n+ M^{\\mu}_{\\lambda\\lambda'\\lambda''} \nA_{\\lambda}A_{\\lambda'}A_{\\lambda''} + \\ldots\n\\end{align}\n\n\\subsection{Phonon Green's functions}\n\nWe briefly sketch the derivation of the phonon self-energy in the presence of third order anharmonicity since intermediate results will be used to express the Raman and infrared spectra. Starting from the retarded Green's function\n\\begin{equation}\n    G^{X,Y} = -i\\theta(t)\\avg{[X(t),Y^\\dagger(0)]}\n\\end{equation}\nwe express the equations of motion of the phonon-phonon Green's function via\n\\begin{subequations}\n\\begin{align}\n\\label{eq:dr0}\n    \\partial_t A(t) & = -i[H,A]\n\\\\\n\t\\frac{d}{d t} G^{AA}_{\\lambda\\lambda'} \n\t& = - i \\omega_{\\lambda} G^{BA}_{\\lambda\\lambda'}\n\\\\\n\t\\frac{d}{d t} G^{BA}_{\\lambda\\lambda'}\n\t& = i 2 \\delta_{ \\lambda\\lambda' }\\delta( t )\n\t- i \\omega_{\\lambda} G^{AA}_{\\lambda\\lambda'}\n\t- i 6 \\sum_{\\mu_1 \\mu_2} \\Phi_{\\lambda \\mu_1 \\mu_2} G^{AA,A}_{\\mu_1 \\mu_2,\\lambda'}\n\\\\\n\\label{eq:dr1}\n\ti \\frac{d}{d t} G^{AA,A}_{\\mu_1 \\mu_2,\\lambda'} & =\n\t\\omega_{\\mu_1} G^{BA,A}_{\\mu_1 \\mu_2,\\lambda'} +\n\t\\omega_{\\mu_2} G^{AB,A}_{\\mu_1 \\mu_2,\\lambda'}\n\\\\\n\\label{eq:dr2}\t\n\ti \\frac{d}{d t} G^{AB,A}_{\\mu_1 \\mu_2,\\lambda'} & =\n\t\\omega_{\\mu_1} G^{BB,A}_{\\mu_1 \\mu_2,\\lambda'} +\n\t\\omega_{\\mu_2} G^{AA,A}_{\\mu_1 \\mu_2,\\lambda'} +\n\t\\sum_{\\nu_1 \\nu_2}\n\t6 \\Phi_{\\mu_2 \\nu_1 \\nu_2} \n\tG^{AAA,A}_{\\mu_1 \\nu_1 \\nu_2,\\lambda'}\n\\\\\n\\label{eq:dr3}\t\n\ti \\frac{d}{d t} G^{BA,A}_{\\mu_1 \\mu_2,\\lambda'} & =\n\t\\omega_{\\mu_1} G^{AA,A}_{\\mu_1 \\mu_2,\\lambda'} +\n\t\\omega_{\\mu_2} G^{BB,A}_{\\mu_1 \\mu_2,\\lambda'} +\n\t\\sum_{\\nu_1 \\nu_2}\n\t6 \\Phi_{\\nu_1 \\nu_2 \\mu_1} \n\tG^{AAA,A}_{\\nu_1 \\nu_2 \\mu_2,\\lambda'}\n\\\\\n\\label{eq:dr4}\n\ti \\frac{d}{d t} G^{BB,A}_{\\mu_1 \\mu_2,\\lambda'} & =\n\t\\omega_{\\mu_1} G^{AB,A}_{\\mu_1 \\mu_2,\\lambda'} +\n\t\\omega_{\\mu_2} G^{BA,A}_{\\mu_1 \\mu_2,\\lambda'}\n\\intertext{\nthe four-point Green's functions are decoupled\\cite{Semwal1972,Zubarev1960} via \n$\\avg{abcd}\\approx \\avg{ab}\\avg{cd} + \\avg{ac} \\avg{bd} + \\avg{ad} \\avg{bc}$ to yield\n}\n\t\\sum_{\\nu_1 \\nu_2}\n\t6 \\Phi_{\\mu_2 \\nu_1 \\nu_2} \n\tG^{AAA,A}_{\\mu_1 \\nu_1 \\nu_2,\\lambda'}\n\t& =\n\t12 (2n_{\\mu_1}+1) \\sum_{\\nu_1}\\Phi_{\\mu_1 \\mu_2 \\nu_1 } G^{AA}_{\\nu_1 \\lambda'}\n\\\\\n\\label{eq:dr5}\n\t\\sum_{\\nu_1 \\nu_2}\n\t6 \\Phi_{\\nu_1 \\nu_2 \\mu_1} \n\tG^{AAA,A}_{\\nu_1 \\nu_2 \\mu_2,\\lambda'}\n\t& =\n\t12 (2n_{\\mu_2}+1) \\sum_{\\nu_1}\\Phi_{\\mu_1 \\mu_2 \\nu_1 } G^{AA}_{\\nu_1 \\lambda'}\n\\end{align}\n\\end{subequations}\nAfter Fourier transforming equations \\eqref{eq:dr0}--\\eqref{eq:dr5} to the frequency domain we get a solvable set of equations that recover the usual definition of three-phonon anharmonicity:\\cite{Leibfried1961,Cowley1963,wallace1998thermodynamics,Semwal1972}\n\\begin{equation}\n    G_{\\lambda\\lambda'}(\\Omega)^{-1} = G^0_{\\lambda\\lambda'}(\\Omega)^{-1} + \\Sigma_{\\lambda\\lambda'}\n\\end{equation}    \nwhere\n\\begin{equation}\n    \\Sigma_{\\lambda\\lambda'}(Z) = -18 \\sum_{\\vec{q}_1\\vec{q}_2 s_1 s_2}\n    \\Phi^{\\lambda s_1s_2}_{\\vec{q}\\bar{\\vec{q}}_1\\bar{\\vec{q}}_2}\n    \\Phi^{\\lambda' s_1s_2}_{\\bar{\\vec{q}} \\vec{q}_1 \\vec{q}_2}\n    S(s_1,s_2,Z)\n\\end{equation}\nand\n\\begin{equation}\n    S(s_a,s_b,Z) = \n\t(n_{a}+n_{b}+1)\n\t\\left[\n\t\\frac{1}{(\\omega_{a}+\\omega_{b}-Z)_p}-\n\t\\frac{1}{(\\omega_{a}+\\omega_{b}+Z)_p}\n\t\\right]\n\t+\n\t(n_{a}-n_{b})\n\t\\left[\n\t\\frac{1}{(\\omega_{b}-\\omega_{a}+Z)_p}-\n\t\\frac{1}{(\\omega_{b}-\\omega_{a}-Z)_p}\n\t\\right]\n\\end{equation}\nNaturally if one considers the four-phonon interactions you recover the standard results, but the information we need right now is the intermediate result for the three-point Green's function expressed in terms of two-point Green's functions and three-phonon matrix elements:\n\\begin{equation}\n    \\label{eq:threepointGF}\n    G_{\\lambda\\lambda',\\lambda''} =\n    6 \\sum_{s} \\Phi_{\\bar{\\lambda}\\bar{\\lambda}'s} \n    S(\\lambda,\\lambda')\n    G_{ A_{s} A_{\\lambda''} }\n\\end{equation}\n\\subsection{Raman spectra}\nIf we start from the polarizability-polarizability correlation function and insert the expansion of the polarizability in terms of phonon coordinates, Eq.~\\eqref{eq:polarizabilityexpansion}, we get\n\\begin{equation}\n\\begin{split}\n    \\avg{P(t)P(0)} = i\\overset{>}{G}(P,P) = &\n    \\vec{P}_0 \\otimes \\vec{P}_0 + \n    %\n    \\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'} i\\overset{>}{G}(A_{\\lambda},A_{\\lambda'})\n+ \\\\\n    + &\n    \\vec{P}_{\\lambda\\lambda'}\n    \\otimes\n    \\vec{P}_{\\lambda''\\lambda'''}\n    i\\overset{>}{G}(A_{\\lambda}A_{\\lambda'},A_{\\lambda''}A_{\\lambda'''})\n+ \\\\    \n     + &\n    \\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'\\lambda''}\n    i\\overset{>}{G}(A_{\\lambda},A_{\\lambda'}A_{\\lambda''})\n+\n    \\vec{P}_{\\lambda\\lambda'}\n    \\otimes\n    \\vec{P}_{\\lambda''}\n    i\\overset{>}{G}(A_{\\lambda}A_{\\lambda'},A_{\\lambda''})\n+ \\\\\n    + &\n\\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'\\lambda''\\lambda'''}\n    i\\overset{>}{G}(A_{\\lambda},A_{\\lambda'}A_{\\lambda''}A_{\\lambda'''})\n+\n    \\vec{P}_{\\lambda\\lambda'\\lambda''}\n    \\otimes\n    \\vec{P}_{\\lambda'''}\n    i\\overset{>}{G}(A_{\\lambda}A_{\\lambda'}A_{\\lambda''},A_{\\lambda'''})\n\\end{split}\n\\end{equation}\nwhere we have omitted terms of $A^4$ or higher. We will deal with the terms in order. The constant term does not contribute to the Raman spectrum. The first term that gives a contribution is\n\\begin{equation}\n\\label{eq:SMramanI}\n    I^{\\textrm{I}} =\n    \\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'}\n    \\int\n    i \\overset{>}{G}(A_{\\lambda}(t),A_{\\lambda'}(0))\n    e^{-i\\Omega t} dt\n    =\n    \\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'}\n    (n(\\Omega)+1)\n    J_{\\lambda\\lambda'}(\\Omega)\n\\end{equation}\nwhere $J$ is the phonon spectral function\n\\begin{equation}\n\tJ(\\Omega) = \n\t- \\frac{1}{\\pi} \\Im\\left\\{ G(\\Omega) \\right\\} =\t\\frac{i}{(n(\\Omega,T)+1)}\\overset{>}{G}(\\Omega)\n\\end{equation}\nWe identify Eq.~\\eqref{eq:SMramanI} as the first order Raman spectra. \nThe next contribution comes from\n\\begin{equation} \\label{eq:2nd_ord_Raman}\n    I^{\\textrm{II}} =\n    \\vec{P}_{\\lambda\\lambda'} \\otimes \\vec{P}_{\\lambda''\\lambda'''}\n    \\int\n    \\overset{>}{G}(A_{\\lambda}(t)A_{\\lambda'}(t),A_{\\lambda''}(0)A_{\\lambda'''}(0))\n    e^{-i\\Omega t} dt\n\\end{equation}\nThe first step is to decouple the four-point Green's function:\\cite{Zubarev1960}\n\\begin{equation}\n\\begin{split}\n    \\overset{>}{G}(A_{\\lambda}(t)A_{\\lambda'}(t),A_{\\lambda''}(0)A_{\\lambda'''}(0))\n  & \\approx  \n    \\overset{>}{G}(A_{\\lambda}(t),A_{\\lambda'''}(0))\n    \\overset{>}{G}(A_{\\lambda'}(t),A_{\\lambda''}(0))\n    +\n\\\\  & +      \n    \\overset{>}{G}(A_{\\lambda}(t),A_{\\lambda''}(0))\n    \\overset{>}{G}(A_{\\lambda'}(t),A_{\\lambda'''}(0))\n    +\n\\\\  & +\n    \\underbrace{\\overset{>}{G}(A_{\\lambda}(t),A_{\\lambda'}(t))\n    \\overset{>}{G}(A_{\\lambda''}(0),A_{\\lambda'''}(0))}_{= \\textrm{constant}}\n\\end{split}    \n\\end{equation}\nand note that the Fourier transform of a product becomes a convolution:\n\\begin{equation}\n\\begin{split}\n    I^{\\textrm{II}} & =\n    \\vec{P}_{\\lambda\\lambda'} \\otimes \\vec{P}_{\\lambda''\\lambda'''}\n    \\frac{1}{\\pi^2}\n    \\bigg[    \n    \\int\n    (n(\\Omega')+1)\n    J_{\\lambda\\lambda''}(\\Omega')\n    (n(\\Omega-\\Omega')+1) J_{\\lambda'\\lambda'''}(\\Omega-\\Omega')\n    d\\Omega'\n    +\n\\\\\n    & +\n    \\int\n    (n(\\Omega')+1)\n    J_{\\lambda\\lambda'''}(\\Omega')\n    (n(\\Omega-\\Omega')+1) J_{\\lambda'\\lambda''}(\\Omega-\\Omega')\n    d\\Omega'\n    \\bigg]\n    =\n\\\\    \n& = 2\\Re\\left\\{ \n\\vec{P}_{\\lambda\\lambda'} \\otimes \\vec{P}_{\\lambda''\\lambda'''}\n \\right\\}\n     \\int\n    (n(\\Omega')+1)\n    J_{\\lambda\\lambda''}(\\Omega')\n    (n(\\Omega-\\Omega')+1) J_{\\lambda'\\lambda'''}(\\Omega-\\Omega')\n    d\\Omega' \n\\end{split}    \n\\end{equation}\nwhere we relabelled the summation indices in the second term. \nThis term is the second-order Raman spectra, slightly more general than it is usually presented. \nIf we assume a diagonal self-energy (and consequently a diagonal spectral function) and insert the non-interacting spectral function $J^0_{\\lambda} = \\delta(\\Omega-\\omega_{\\lambda}) - \\delta(\\Omega+\\omega_{\\lambda})$ the integral becomes\n\\begin{equation}\n\\begin{split}\n    & \\int\n    n(-\\Omega')\n    \\left[\n        \\delta(\\Omega'-\\omega_a)\n        -\n        \\delta(\\Omega'+\\omega_a)\n    \\right]\n    n(\\Omega'-\\Omega)\n    \\left[\n        \\delta(\\Omega-\\Omega'-\\omega_b)\n        -\n        \\delta(\\Omega-\\Omega'+\\omega_b)\n    \\right]\n    d\\Omega' =\n\\\\\n    = &\n    n(-\\omega_a)\n    n(\\omega_a-\\Omega)\n    \\left[\n        \\delta(\\Omega-\\omega_a-\\omega_b)\n        -\n        \\delta(\\Omega-\\omega_a+\\omega_b)\n    \\right] -\n\\\\    \n    & \n    n(\\omega_a)\n    n(-\\omega_a-\\Omega)\n    \\left[\n        \\delta(\\Omega+\\omega_a-\\omega_b)\n        -\n        \\delta(\\Omega+\\omega_a+\\omega_b)\n    \\right] =\n\\\\\n    = &\n    \\left[n(\\Omega)+1\\right]\n    \\left[\n    n(\\omega_a)+n(\\omega_b)+1\n    \\right]\n    \\left[\n    \\delta(\\Omega+\\omega_a+\\omega_b)\n    -\n    \\delta(\\Omega-\\omega_a-\\omega_b)\n    \\right]\n    +\n\\\\\n    & +\n    \\left[n(\\Omega)+1\\right]\n    \\left[\n    n(\\omega_a)-n(\\omega_b)\n    \\right]\n    \\left[\n    \\delta(\\Omega-\\omega_a+\\omega_b)\n    -\n    \\delta(\\Omega+\\omega_a-\\omega_b)\n    \\right] =\n\\\\\n    = & -\\frac{n(\\Omega)+1}{\\pi} \\Im\\left\\{ S(\\omega_a,\\omega_b,\\Omega) \\right\\}\n\\end{split}\n\\end{equation}\nwhere we made use of $n(a+b) = (1+n(a)+n(b))\/n(a)n(b)$ as well as $n(-a)=-n(a)-1$. \nWith this we recover the expression given by \\citet{Cowley1964b} as the limit of weak anharmonicity. \nIn this study, however, it was found necessary to keep the interacting spectral function when calculating the Raman spectra. \nIt is intuitive: the second order Raman is proportional to the two-phonon DOS, and if the phonons are broadened then the two-phonon DOS should also be broadened.\n\nNext we have a term given by\n\\begin{equation}\n    I^{\\textrm{III}} =\n    \\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'\\lambda''}\n    G^{>}(A_{\\lambda},A_{\\lambda'}A_{\\lambda''})\n+\n    \\vec{P}_{\\lambda\\lambda'}\n    \\otimes\n    \\vec{P}_{\\lambda''}\n    G^{>}(A_{\\lambda}A_{\\lambda'},A_{\\lambda''})\n\\end{equation}\nwhere we make use of the intermediate result for the three-point Green's functions in Eq.~\\eqref{eq:threepointGF} and arrive at\n\\begin{equation}\nI^{\\textrm{III}} =\n-6\n(n(\\Omega)+1)\n\\sum_{\\mu}\n\\left(\n\\vec{P}_{\\lambda''\\lambda'}\n\\otimes\n\\vec{P}_{\\lambda}\n\\Phi_{\\bar{\\lambda}''\\bar{\\lambda}'\\mu}\n+\n\\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'\\lambda''}\n\\Phi_{\\bar{\\lambda}'\\bar{\\lambda}''\\mu}\n\\right)\n\\Im\\left\\{ \nS_{\\lambda''\\lambda'}\n\\right\\}\nJ_{\\mu \\lambda}(\\Omega)\n\\end{equation}\nwhere, if we assume a diagonal self-energy, we recover previous results.\\cite{Cowley1964b} \nIt is worth noting is that -- as far as we have discovered -- the last two terms are in practice several orders of magnitude smaller than the usual first and second order terms. \nIn term III we have a product between the two-phonon DOS and the one-phonon spectral function, so to have any sizeable contribution these need to peak simultaneously. \nThe final term\n\\begin{equation}\n    I^{\\textrm{IV}} =\n\\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'\\lambda''\\lambda'''}\n    G(A_{\\lambda},A_{\\lambda'}A_{\\lambda''}A_{\\lambda'''})\n+\n    \\vec{P}_{\\lambda\\lambda'\\lambda''}\n    \\otimes\n    \\vec{P}_{\\lambda'''}\n    G(A_{\\lambda}A_{\\lambda'}A_{\\lambda''},A_{\\lambda'''})    \n\\end{equation}\nwe approach with the same decoupling procedure as before:\n\\begin{equation}\n\\begin{split}\n    G^{>}(A_{\\lambda}(t),A_{\\lambda'}(0),A_{\\lambda''}(0)A_{\\lambda'''}(0))\n  & \\approx  \n    G^{>}(A_{\\lambda}(t),A_{\\lambda'''}(0))\n    G^{>}(A_{\\lambda'}(0),A_{\\lambda''}(0))\n    +\n\\\\  & +      \n    G^{>}(A_{\\lambda}(t),A_{\\lambda''}(0))\n    G^{>}(A_{\\lambda'}(0),A_{\\lambda'''}(0))\n    +\n\\\\  & +\n    G^{>}(A_{\\lambda}(t),A_{\\lambda'}(0))\n    G^{>}(A_{\\lambda''}(0),A_{\\lambda'''}(0)) =\n\\\\\n    & = 3 G^{>}(A_{\\lambda}(t),A_{\\lambda'''}(0))\\delta_{\\lambda'\\lambda''}(2n_{\\lambda'}+1)\n\\end{split}    \n\\end{equation}\nwhere the factor 3 comes from relabelling indices. \nWe also used the equal-time thermal average of the phonon operators. This gives\n\\begin{equation}\n    I^{\\textrm{IV}} =\n    3\n    (n(\\Omega)+1)\n    \\left(\n    \\vec{P}_{\\lambda} \\otimes \\vec{P}_{\\lambda'\\lambda'\\lambda''}\n    +\n    \\vec{P}_{\\lambda''\\lambda'\\lambda'}\n    \\otimes\n    \\vec{P}_{\\lambda}\n    \\right)\n    (2n_{\\lambda'}+1)\n    J_{\\lambda\\lambda''}\n\\end{equation}\nWhat we derived in the section above holds for any operator that is expressed as a constant multiplied with the phonon operator, i.e. a quantity that only depends on the position of the atoms (it's frequency dependence can be ignored, or at least the cross-terms between the frequency dependence and the position of atoms). The derivation of the infrared absorption spectrum is identical, one only needs to replace the matrix elements. If the real part of the susceptibility if of interest it is conveniently calculated via a Kramers-Kronig transformation of the imaginary part.\n\nIn the main text I have defined compound matrix element for the polarizability to simplify the notation, they are given via:\n\\begin{align}\n\tP^{(\\textrm{I})}_{\\mu\\nu,\\xi\\rho}(s_1) & = P^{\\mu\\nu}(s_1) P^{\\xi\\rho}(s_1)^\\dagger\t\n\\\\\n\tP^{(\\textrm{II})}_{\\mu\\nu,\\xi\\rho}(\\vec{q},s_1,s_2) & =\n\tP^{\\mu\\nu} (\\vec{q},s_1,s_2)\n\tP^{\\xi\\rho} (\\vec{q},s_1,s_2)^\\dagger\t\n\\\\\n\tP^{(\\textrm{III})}_{\\mu\\nu,\\xi\\rho}(\\vec{q},s_1,s_2,s_3) & =\n\t\\Phi_{\\vec{q} \\bar{\\vec{q}} \\vec{\\Gamma}}^{s_1 s_2 s_3}\n\tP^{\\mu\\nu}( \\vec{q},s_1,s_2 )\t\n\tP^{\\xi\\rho}(s_3)\n\t+\n\t\\Phi_{\\vec{q} \\bar{\\vec{q}} \\vec{\\Gamma}}^{s_1 s_2 s_3}\n\tP^{\\xi\\rho}( \\vec{q},s_1,s_2 )\t\n\tP^{\\mu\\nu}(s_3)\n\\\\\n\tP^{(\\textrm{IV})}_{\\mu\\nu,\\xi\\rho}(\\vec{q},s_1,s_2) & =\n\tP^{\\mu\\nu}(\\vec{\\Gamma},\\bar{\\vec{q}},\\vec{q},s_1,s_2,s_2)\n\tP^{\\xi\\rho}(s_1)\n\t+\n\tP^{\\xi\\rho}(\\vec{\\Gamma},\\bar{\\vec{q}},\\vec{q},s_1,s_2,s_2)\n\tP^{\\mu\\nu}(s_1)\n\\intertext{In exactly the same way I can define the following dipole matrix elements:}\n\t\\label{eq:ircomp1}\n\tM^{(I)}_{\\alpha\\beta}(s_1) & = M^{\\alpha}_{s_1} (M^{\\beta}_{s_1})^\\dagger\n\t%\n\t\\\\\n\t%\n\t\\label{eq:ircomp2}\n\tM^{(II)}_{\\alpha\\beta}(s_1,s_2,\\vec{q}) & =\n\tM^\\alpha (^{s_1 s_2}_{\\vec{q} })\n\tM^\\beta (^{s_1 s_2}_{\\vec{q} })^\\dagger\n\t%\n\t\\\\\n\t%\n\t\\label{eq:ircomp3}\n\tM^{(III)}_{\\alpha\\beta}(s_1,s_2,s_3,\\vec{q}) & =\t\n\t\\Phi_{\\vec{q} \\bar{\\vec{q}} \\vec{0}}^{s_1 s_2 s_3}\n\tM^\\beta(^{s_1 s_2}_{\\vec{q}} )\t\n\tM^\\alpha_{s_3}\n\t+\n\t\\Phi_{\\vec{q} \\bar{\\vec{q}} \\vec{0}}^{s_1 s_2 s_3}\n\tM^\\alpha(^{s_1 s_2}_{\\vec{q}} )\t\n\tM^\\beta_{s_3}\n\t%\n\t\\\\\n\t%\n\t\\label{eq:ircomp4}\n\tM^{(IV)}_{\\alpha\\beta}(s_1,s_2,\\vec{q}) & =\t\t\n\tM^\\alpha(^{s_1 s_2 s_2}_{ \\vec{0}\\bar{\\vec{q}}\\vec{q} })\n\tM^\\beta_{s_1}\n\t+\n\tM^\\beta(^{s_1 s_2 s_2}_{ \\vec{0}\\bar{\\vec{q}}\\vec{q} })\n\tM^\\alpha_{s_1}\n\\end{align}\nHere I have explicitly used momentum conservation in the matrix elements. There is only a single q-vector that remains, which means that the computational cost of determining the spectras (given interaction tensors) are modest, at most a single sum over the Brillouin zone.\n\n\\vspace{1.5cm}\n\\section{Details of calculations}  \\label{SM:details}\n\n\\subsection{Self-consistency procedure}\n\nThe interatomic force constants are determined self-consistently as explained in \\citet{Shulumba2016b}, but it is worth explaining the practical procedure. \nThe basic idea is that a set of force constants gives us normal modes that can be thermally populated, generating thermally excited structures. \nThermally excited structures can, with the help of equation \\eqref{eq:ff2} yield a new set of force constants. \nThe procedure is terminated when the input and output force constants show no significant difference.\n\nThe self-consistency procedure is implemented via a geometric series:\n\\begin{itemize}\n    \\item Create an initial guess for the second order force constants (see \\citet{Shulumba2016b})  and generate 1 thermally excited supercell\n    \\item Generate new force constants and produce 2 thermally excited supercells.\n    \\item Use both the one old and two new supercells to generate a new set of force constants, and 4 new supercells\n    \\item Use 2+4 supercells to generate forceconstants and 8 supercells\n    \\item Use 4+8 supercells to generate forceconstants and 16 supercells\n    \\item Repeat until convergence\n\\end{itemize}\nThis procedure has the benefit of introducing some mixing (since you use supercells generated from different force constants) smoothing convergence. \nIn addition, the prescribed procedure leaves only 25\\% of calculations as waste. \nA straight implementation (generate N supercells each iteration, repeat until convergence) can waste a much larger fraction of calculations.\n\nFor this study all calculated spectra converged after 7 iterations, using 128+64 configurations. \nThe calculations were done on a grid of volumes and temperatures, and a few points were tested with one more iteration where no appreciable difference in calculated quantities could be seen.\n\n\\begin{figure*}[hbt]\n     \\centering\n     \\includegraphics[width=\\linewidth]{nacl_spectral_half.pdf}\n     \\caption{Phonon spectral functions for NaCl at different temperatures calculated using the TDEP method. The white circles in panel (a) are experimental neutron measurements from \\citet{Raunio1969a}. We show an overall good agreement with experiment bar a slight uniform underestimation of frequencies.}\n    \\label{fig:SMspectralfunction}\n\\end{figure*}\n\n\\subsection{Free energy minimization}\n\nTo determine the equilibrium volume at any given temperature we did calculations on a grid of volumes and temperatures. The grid is specified in Table \\ref{table:SM_simulationgrid}.\n\\begin{table*}\n\\begin{ruledtabular}\n\\begin{tabular}{r|llllll}\nTemperature (K) & \\multicolumn{6}{c}{Lattice constants (\\AA)} \\\\\n\\hline\n0 & 5.233848 & 5.324865 & 5.419088 & 5.510051 & 5.603468 & 5.717206 \\\\ \n150 & 5.255144 & 5.346531 & 5.441138 & 5.532472 & 5.626268 & 5.740469 \\\\ \n300 & 5.282018 & 5.373873 & 5.468963 & 5.560764 & 5.655040 & 5.769825 \\\\ \n500 & 5.326527 & 5.419155 & 5.515047 & 5.607621 & 5.702692 & 5.818444 \\\\ \n700 & 5.380951 & 5.474526 & 5.571398 & 5.664918 & 5.760960 & 5.877895 \\\\ \n1000 & 5.481180 & 5.576498 & 5.675174 & 5.770436 & 5.868267 & 5.987380 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption{\\label{table:SM_simulationgrid}\nList of temperatures and lattice parameters used in the simulations.}\n\\end{table*}\nOnce simulations are converged the free energy was determined via\n\\begin{equation}\n    F = U_0 + F_{\\textrm{ph}} + \\Delta F^{3\\textrm{ph}} + \\Delta F^{4\\textrm{ph}}\n\\end{equation}\nwhere $F_{\\textrm{ph}}$ is the usual phonon free energy and $U_0$ the renormalized baseline free energy given by\n\\begin{equation}\n    U_0 = \\avg{U -\n    \\frac{1}{2!}\\sum_{\\substack{ ij\\\\ \\alpha\\beta } }\\overset{\\textrm{lr}}{\\Phi}_{ij}^{\\alpha\\beta}\nu_i^\\alpha u_j^\\beta - \n    \\frac{1}{2!}\\sum_{\\substack{ ij\\\\ \\alpha\\beta } }\\Phi_{ij}^{\\alpha\\beta}\nu_i^\\alpha u_j^\\beta - \n \\frac{1}{3!}\n\\sum_{\\substack{ijk\\\\ \\alpha\\beta\\gamma}}\\Phi_{ijk}^{\\alpha\\beta\\gamma}\nu_i^\\alpha u_j^\\beta u_k^\\gamma - \n\\frac{1}{4!}\n\t\\sum_{\\substack{\n\tijkl\\\\\n\t\\alpha\\beta\\gamma\\delta\n\t}}\n\\Phi_{ijkl}^{\\alpha\\beta\\gamma\\delta}\nu_i^\\alpha u_j^\\beta u_k^\\gamma u_l^\\delta \n    }\n\\end{equation}\nHere it is important to note that we have to subtract the long-ranged polar interactions to avoid double-counting them. The explicit anharmonic contributions are given via\\cite{Leibfried1961,Cowley1963,wallace1998thermodynamics}\n\\begin{equation}\\label{eq:deltaF3}\n\t\\Delta F^{3\\textrm{ph}} =\n\t-6\n\t\\sum_{\\lambda\\lambda'\\lambda''}\n\t\\left|\n\t\t\\Phi_{\\lambda\\lambda'\\lambda''}\n\t\\right|^2 \n\t%\n\t\\left(\n\n\t\\frac{3n_{\\lambda} n_{\\lambda'} + 3n_{\\lambda} + 1}\n\t{(\\omega_{\\lambda}+\\omega_{\\lambda'}+\\omega_{\\lambda''})_p}\n\t+\n\t\\frac{ 6n_{\\lambda} n_{\\lambda''} - 3 n_{\\lambda} n_{\\lambda'} + 3n_{\\lambda''}}\n\t{(\\omega_{\\lambda}+\\omega_{\\lambda'}-\\omega_{\\lambda''})_p}\n\t\\right)\n\n\t%\n\t+9\\Phi_{\\lambda\\bar{\\lambda}\\lambda''}\\Phi_{\\lambda'\\bar{\\lambda}'\\bar{\\lambda}''}\n\t\\frac{4 n_{\\lambda}( n_{\\lambda'}+1)+1}\n\t{(\\omega_{\\lambda''})_p}\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{eq:deltaF4}\n\t\\Delta F^{4\\textrm{ph}} =\n\t3\\sum_{\\lambda\\lambda'}\n\t\\Phi_{\\lambda\\bar{\\lambda}\\lambda'\\bar{\\lambda}'}(2n_{\\lambda}+1)(2n_{\\lambda'}+1)\n\\end{equation}\nThe free energy was fitted to a Birch-Murnaghan equation of state, and pressure was determined via $P = -dF\/dV$.\n\n\\subsection{Parameters in calculations}\n\nAll density functional theory calculations were done using the projector augmented wave (PAW)\\cite{Blochl1994a} method as implemented in VASP.\\cite{Kresse1996c,Kresse1999,Kresse1996,Kresse1993b} \nWe treated exchange-correlation within the AM05 approximation\\cite{Armiento2005,Mattsson2009}. \nWe used a 288 atom supercell, and the Brillouim zone integrations used a $2 \\times 2 \\times 2$ Monkhorst-Pack mesh,\\cite{Monkhorst1976a} and a Gaussian smearing of 0.2 eV was applied. \nPhonon self-energies and Raman spectra were integrated on a $51 \\times 51 \\times 51$ q-point mesh. \nThe plane-wave energy cutoff was set to 350eV.\n\nThe statistics needed to determine the polarizability and Born charges on a large set of supercells used a slightly smaller simulation cell to offset the increased computational cost. \nHere we used a 64 atom supercell with a tighter $6 \\times 6 \\times 6$ k-point mesh.\n\n\\vspace{1.5cm}\n\\section{Additional quantities you can get from the expansion parameters} \\label{SM:dielectric_things_not_Raman}\n\nAs noted in the main manuscript, what we have developed in this paper is a general procedure for the interplay between anharmonicity and dielectric properties. \nExperimentally we have the Raman spectra to verify our results with, which is the focus in the main text. \nTo illustrate the general nature of the method we present a few other quantities that are accessible (but without any accessible experimental confirmation we leave them as predictions).\n\nIn Fig.~\\ref{fig:SMZandeps} we show the temperature dependence of the Born charges and static dielectric tensor. \nThe temperature dependence is modest as one would expect in a wide-bandgap material. \nEven so, the quasiharmonic treatment of Born charges overestimates the temperature dependence by a factor 2 as compared with the full temperature dependence. \nIt is easy to imagine that in a material with a much smaller bandgap this discrepancy could be significantly larger.\nIn \\ref{fig:nacl_Z_and_eps} we show the IR spectra of NaCl for a few temperatures.\nThe procedure to calculate these is identical to the Raman spectra, the only difference being the matrix elements.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{nacl_Z_and_eps.pdf}\n    \\caption{Temperature dependence of the Born effective charge and $\\epsilon_\\infty$ in NaCl treated three different ways. \n    The most realistic, at constant pressure (i.e. volume changing with temperature) with full temperature dependence of all parameters is denoted NPT. The same treatment without thermal expansion is denoted NVT. \n    For reference we also include the quasiharmonic results, denoted QH, where all interaction parameters are set to their values at 0K, and temperature is only included via thermal expansion.}\n    \\label{fig:SMZandeps}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{nacl_ir_spectra.pdf}\n    \\caption{Temperature dependence of the IR absorption spectra in NaCl. The spectra are calculated via equations (10)--(13) with the Raman matrix elements replaced with the IR matrix elements, equations \\eqref{eq:ircomp1}--\\eqref{eq:ircomp4}}\n    \\label{fig:nacl_Z_and_eps}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\chapter*{Abstract}\n\n\\begin{tabularx}{\\textwidth}{lX}\t\n{\\bf Keywords:} & Black holes, Proca fields, Hawking radiation, TeV gravity, scalar fields, Maxwell fields, asymptotically anti-de Sitter spacetimes, quasinormal modes, superradiance.\\\\\n\\\\\n{\\bf Abstract:} & This thesis presents recent studies on test scalar and vector fields around black holes in the classical theory of General Relativity. It is separated in two parts according to the asymptotic properties of the spacetime under study. \\\\\n& In the first part, we investigate scalar and Proca fields on an asymptotically flat background. For the Proca field, we obtain a complete set of equations of motion in higher dimensional spherically symmetric backgrounds. These equations are solved numerically, both to compute Hawking radiation spectra and quasi-bound states. In the former case, for the first time, we carry out a precise study of the longitudinal degrees of freedom induced by the mass of the field. This can be used to improve the modeling of evaporation of black holes coupled to massive vector fields, and black hole event generators currently used at the Large Hadron Collider to probe TeV gravity models with extra dimensions. Regarding quasi-bound states, we find arbitrarily long lived modes for a charged Proca field in a Reissner-Nordstr\\\"om black hole. As a comparison, we also find such long lived modes for a charged scalar field. \\\\\n& The second part of this thesis presents research on superradiant instabilities of scalar and Maxwell fields on an asymptotically anti-de Sitter background. For the scalar case, we introduce a charge coupling between the field and the background, and show that superradiant instabilities do exist for all values of the total angular momentum, $\\ell$, in higher dimensions. This result corrects a statement in the literature that such instabilities only appear in even dimensions. For the Maxwell case, we first propose a general prescription to impose boundary conditions on the Kerr-anti-de Sitter spacetime, and obtain two Robin boundary conditions which give two different quasinormal modes even in a simpler Schwarzschild-anti-de Sitter black hole. Then these\n\\end{tabularx}\n\n\\begin{tabularx}{\\textwidth}{lX}\t\n{\\color{white}{Abstract:}} & two boundary conditions are implemented to study superradiant unstable modes and vector clouds. In particular, we find that the new branch of quasinormal modes may be unstable in a larger parameter space. Furthermore, the existence of vector clouds indicates that one may find a vector hairy black hole solution for the Einstein-Maxwell-anti-de Sitter system at the nonlinear level, which implies, in such system, that the Kerr-Newman-anti-de Sitter black hole is not a unique solution.\n\\end{tabularx}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\cleardoublepage\n\n\n\n\n\n\n\n\n\\begin{tabularx}{\\textwidth}{lX}\t\n{\\bf Palavras-chave:} & Buracos negros, Campos de Proca, Radia\\c{c}\\~ao de Hawking, gravidade \\`a escala do TeV, campos escalares, campos de Maxwell, asimt\\'oticamente anti-de Sitter, modos quasi-normais, super-radi\\^ancia.\\\\\n\\\\\n{\\bf Resumo:} & Nesta tese apresentamos estudos recentes sobre campos escalares e vetoriais de teste, em torno de buracos negros na teoria cl\\'assica da relatividade geral. A tese encontra-se dividida em duas partes, de acordo com as propriedades asimt\\'oticas do espa\\c{c}o-tempo em estudo. \\\\\n& Na primeira parte, investigamos os campos escalar e de Proca num espa\\c{c}o asimt\\'oticamente plano. Para o campo de Proca, obtemos um conjunto completo de equa\\c{c}\\~oes do movimento em espa\\c{c}os esfericamente sim\\'etricos em dimens\\~oes elevadas. Estas equa\\c{c}\\~oes s\\~ao resolvidas numericamente, tanto para o c\\'alculo de radia\\c{c}\\~ao de Hawking como para o c\\'alculo de estados quasi-ligados. No primeiro c\\'alculo, pela primeira vez, efetuamos um estudo preciso dos graus de liberdade longitudinais que s\\~ao induzidos pelo termo de massa do campo. Este estudo pode ser usado para melhorar o modelo da evapora\\c{c}\\~ao de buracos negros acoplados a campos vetoriais massivos e geradores de eventos de buraco negro usados presentemente no Grande Colisor de H\\'adrons para testar modelos de gravidade com dimens\\~oes extra \\`a escala do TeV. Relativamente aos estados quasi-ligados, encontramos estados com tempos de vida arbitrariamente longos para o campo de Proca carregado, no buraco negro de Reissner-Nordstr\\\"om. Como compara\\c{c}\\~ao, obtemos estados com tempos de vida arbitrariamente longos tamb\\'em para o campo escalar.\\\\\n& Na segunda parte da tese, apresentamos investiga\\c{c}\\~ao sobre instabilidades super-radiantes para os campos escalar e de Maxwell em espa\\c{c}o asimt\\'oticamente anti-de Sitter. No caso escalar introduzimos um acoplamento de carga entre o campo e o background e mostramos que instabilidades super-radiantes existem para todos os valores do momento angular total, $\\ell$, em dimens\\~oes mais elevadas. Este resultado\n\\end{tabularx}\n\n\\begin{tabularx}{\\textwidth}{lX}\t\n{\\color{white}{Abstract:}} & corrige a afirma\\c{c}\\~ao na literatura de que estas instabilidades aparecem apenas em dimens\\~oes \\'impares. Para o caso do campo de Maxwell, propomos primeiro uma prescri\\c{c}\\~ao para impor condi\\c{c}\\~oes fronteira no espa\\c{c}o tempo de Kerr-anti-de Sitter obtendo duas condi\\c{c}\\~oes fronteira do tipo de Robin que originam dois tipos diferentes de modos quasi-normais, mesmo no caso mais simples do buraco negro de Schwarzschild-anti-de Sitter. Estas duas condi\\c{c}\\~oes fronteira s\\~ao implementadas no estudo de modos super-radiantes inst\\'aveis e nuvens vetoriais. Em particular, encontramos um novo ramo de modos quasi-normais que podem conter instabilidades mais fortes. Mostramos ainda que a exist\\^encia de nuvens vetoriais indica a poss\\'ivel exist\\^encia de solu\\c{c}\\~oes de buraco negro com cabelo vetorial para o sistema Einstein-Maxwell-anti-de Sitter a n\\'ivel n\\~ao linear, o que implica, nesse sistema, que o buraco negro de Kerr-Newman-anti-de Sitter poder\\'a n\\~ao ser \\'unico.\n\\end{tabularx}\n\n\n\n\n\n\n\n\n\n\n\n\\chapter*{Acknowledgements}\n\nConfucius said, ``\\textit{lost time is never around the clock}''. Along my way to study physics during these years, I have received enormous encouragement and assistance from my family, my friends and my colleagues. I think it is the time to express my gratitude to all of them.\n\nFirst of all, I am very thankful to my supervisor, Prof. Carlos Herdeiro, for his support and inspiration, for teaching me how to work on physics and for giving me the freedom to explore my own interests.\n\nSpecial thanks are reserved for Marco Sampaio, for our enjoyable collaboration on various projects. As my co-supervisor, he has taught me a lot of physics and numerical techniques, while as a friend, his positiveness always reminds me to be optimistic when I face problems. I would also like to thank the other colleagues from the gravitation and high energy physics group at University of Aveiro, for all the great discussions about physics and others, they are Carolina Benone, Fl\\'avio Coelho, Pedro Cunha, Juan Carlos Degollado, Jai Grover, Antonio Morais, Jo\\~ao Rosa, Eugen Radu, and Helgi R\\'unarsson. In particular, I am grateful to Juan Carlos Degollado and Jo\\~ao Rosa for their interesting courses, and various discussions on the topic of superradiance.\n\nI would like to acknowledge the financial support from Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia (FCT)--the International Doctorate Network in Particle Physics, Astrophysics and Cosmology (IDPASC) programme, with the grant SFRH\/BD\/51648\/2011 during the completion of this thesis.\n\nI owe my deepest gratitude to all my previous colleagues in China, especially Prof. Dezhi Huang, for encouraging me to work in a different field, and Prof. Jiliang Jing, for introducing me the black hole physics and supporting me to study aboard. I would also like to thank all my friends in China, who are too many to be listed individually, for their constant support.\n\nFinally, I would like to thank my parents, who teach me to be a virtuous man, and my sister, who is always on my side. Their endless love and support make me get better.\n\n\\chapter{Angular dependence of the news function}\n\\label{ch:analytics}\n\n\\epigraph{\\emph{Sym\\'etrie.} En ce qu'on voit d'une vue; fond\\'ee sur ce qu'il n'y a pas de raison de faire autrement.}{Blaise Pascal\\\\ \\emph{Pens\\'ees}}\n\nIn the previous chapter we showed that the radiation signal seen by an infinitely far away observer, which when squared and integrated gives the energy radiated, is encoded in a sole function of retarded time $\\tau$ and angular coordinate $\\theta$. Furthermore, this so called news function depends exclusively on the transverse metric components $h_{ij}$, whose formal solution was given in integral form in Chapter~\\ref{ch:dynamics}.\n\nIn this chapter we shall simplify that solution and study the angular dependence of the news function. We start in Section~\\ref{CL} by showing that the metric possesses a conformal symmetry at each order in perturbation theory which, at null infinity, implies a factorisation of the angular dependence of the news function at each order. By actually computing that angular factor, we shall write the inelasticity's angular distribution as a power series in $\\sin^2\\theta$. In particular, we show that a consistent truncation of this angular series at $O(n)$ requires knowledge of the metric perturbations up to $O(n+1)$. This clarifies the meaning of perturbation theory in this problem: it allows for an angular expansion of the news function off the collision axis. In Section~\\ref{asymp_waveforms} we simplify the integral solution and verify, explicitly, that it becomes effectively one-dimensional.\n\n\\section{A hidden symmetry}\n\\label{CL}\nD'Eath and Payne observed, in $D=4$, that if one starts with a shock wave and performs a boost followed by an appropriate scaling of all coordinates, the end result is just an overall conformal factor \\cite{D'Eath:1992hd}. This is easily understood in physical terms: since the only scale of the problem is the energy of the shock wave, the effect of the boost can be undone by rescaling the coordinates. In what follows we shall generalise this to $D\\geq4$.\n\nRecall the metric of one Aichelburg-Sexl shock wave in Rosen coordinates,\n\\begin{equation}\nds^2 = -d\\bar{u} d\\bar{v} + \\left(1+\\dfrac{\\kappa \\bar{u} \\theta(\\bar{u})}{2}\\Phi''\\right)^2d\\bar{\\rho}^{2} + \\bar{\\rho}^2\\left(1+ \\frac{\\kappa \\bar{u} \\, \\theta(\\bar{u})}{2 \\bar \\rho}\\Phi'\\right)^2 d\\bar{\\Omega}^2_{D-3} \\ .\n\\end{equation}\nNow let $L$ be the Lorentz transformation\n\\begin{equation}\n(\\bar{u},\\bar{v},\\bar{x}^i)\\xrightarrow{L}(e^{-\\beta}\\bar{u},e^\\beta\\bar{v},\\bar{x}^i)\\,,\n\\end{equation}\nand $C$ the conformal scaling\n\\begin{equation}\n(\\bar{u},\\bar{v},\\bar{x}^i)\\xrightarrow{C}e^{-\\frac{1}{D-3}\\beta}(\\bar{u},\\bar{v},\\bar{x}^i)\\,.\n\\end{equation}\nThen, under the combined action of $CL$,\n\\begin{eqnarray}\n(\\bar{u},\\bar{v},\\bar{x}^i)&\\xrightarrow{CL}&(e^{-\\frac{D-2}{D-3}\\beta}\\bar{u},e^{\\frac{D-4}{D-3}\\beta}\\bar{v},e^{-\\frac{1}{D-3}\\beta}\\bar{x}^i)\\,,\\\\\ng_{\\mu\\nu}(\\bar X)&\\xrightarrow{CL}&e^{\\frac{2}{D-3}\\beta}g_{\\mu\\nu}(\\bar X')\\; ,\n\\end{eqnarray}\nwhere $\\bar X$ and $\\bar X'$ denote the coordinates before and after the transformation respectively, i.e. $\\bar X\\xrightarrow{CL}\\bar X'$. The metric simply scales by an overall factor.\n\nSince this is a one parameter symmetry, one can find coordinates on $(D-1)$-dimensional sheets which are invariant under the transformation, and a normal coordinate parameterising inequivalent sheets. A suitable set of invariant coordinates on such sheets is $\\left\\{p,q,\\phi^a\\right\\}$, with\n\\begin{equation}\np\\equiv \\bar{v}\\bar{\\rho}^{D-4}\\,,\\qquad q\\equiv \\bar{u}\\bar{\\rho}^{-(D-2)}\\,, \n\\end{equation}\nand $\\phi^a$ the angles on the transverse plane. The normal coordinate along the orbits of the symmetry is simply $\\bar{\\rho}$ which transforms as\n\\begin{equation}\n\\bar{\\rho}\\rightarrow\\bar{\\rho}'=e^{-\\frac{1}{D-3}\\beta}\\bar{\\rho}\\,.\n\\end{equation} \n\nNow consider the superposition of two shocks. In the boosted frame with Rosen coordinates, the action of $CL$ is\n\\begin{equation}\ng_{\\mu\\nu}(\\nu,\\lambda;\\bar X)\\xrightarrow{CL}e^{\\frac{2}{D-3}\\beta}g_{\\mu\\nu}(\\nu,e^{-2\\beta}\\lambda;\\bar X')\\,.\n\\end{equation}\nThus the perturbative expansion remains the same except for $\\lambda\\rightarrow e^{-2\\beta}\\lambda$. In the centre-of-mass frame with Brinkmann coordinates, Eq.~\\eqref{metric_series} becomes\n\\begin{eqnarray}\\label{eq:CLonMetric}\ng_{\\mu\\nu}(X) \\xrightarrow{CL}g_{\\mu\\nu}(X')=e^{\\frac{2}{D-3}\\beta}\\left[\\eta_{ \\mu \\nu}+\\sum_{k=1}^\\infty e^{-2k\\beta}  h_{\\mu\\nu}^{(k)}(X')\\right]\\,.\n\\end{eqnarray}\nQuite remarkably, in the future of the collision the metric possesses a conformal symmetry at each order in perturbation theory.\n\nNote that the metric functions $h_{\\mu\\nu}^{(k)}$ in Eq.~\\eqref{eq:CLonMetric} are the same before and after the transformation, the only change being the coordinates they are evaluated on. On the other hand, for a generic coordinate transformation, the metric transforms as a rank-2 tensor,\n\\begin{equation}\ng_{\\mu \\nu}(X)=\\frac{\\partial x^{\\mu'}}{\\partial x^\\mu}\\frac{\\partial x^{\\nu'}}{\\partial x^\\nu}g_{\\mu' \\nu'}(X')\\,.\n\\end{equation}\nTogether with Eq.~\\eqref{eq:CLonMetric} this implies that\n\\begin{equation}\\label{CL_transf}\nh_{\\mu\\nu}^{(k)}(X')=e^{(2k+N_u-N_v)\\beta}h_{\\mu\\nu}^{(k)}(X)\\,,\n\\end{equation}\nwhere $N_u$ and $N_v$ are the number of $u$-indices and $v$-indices. Thus a given metric function evaluated on the transformed coordinates is the same function evaluated on the initial coordinates multiplied by an appropriate factor.\n\nIn these coordinates, the invariants $p,q$ read\n\\begin{equation}\np=(v-\\Phi(\\rho))\\rho^{D-4}\\,,\\qquad q=u\\rho^{-(D-2)}\\,,\\label{def_pq}\n\\end{equation}\nwhile $\\rho$ transforms as\n\\begin{equation}\n\\rho\\rightarrow \\rho'=e^{-\\frac{1}{D-3}\\beta}\\rho\\,.\n\\end{equation}\n\n\\subsection{Reduction to two dimensions}\n\nSince $\\rho$ is the only coordinate that transforms under the action of $CL$, Eq.~\\eqref{CL_transf} implies a separation of variables in the form\n\\begin{equation}\\label{eq:separation_rho}\nh_{\\mu\\nu}^{(k)}(p,q,\\rho,\\phi^a)=\\rho^{-(D-3)(2k+N_u-N_v)}f^{(k)}_{\\mu\\nu}(p,q,\\phi^a)\\,.\n\\end{equation}\nSince the angles $\\phi^a$ are ignorable, the problem becomes two dimensional at each order in perturbation theory. Let us see how.\n\nIn Section~\\ref{reduction_3D} we defined scalar functions of $(u,v,\\rho)$ by factoring out the trivial dependence of each metric component on $\\phi^a$, Eqs.~\\eqref{gen_perts}, and similarly for the sources $T^{(k-1)}_{\\mu\\nu}$. Then Eq.~\\eqref{eq:separation_rho} implies that, for each of those functions generically denoted by $F(u,v,\\rho)$ as before,\n\\begin{equation}\nF^{(k)}(u,v,\\rho)= \\dfrac{f^{(k)}(p,q)}{\\rho^{(D-3)(2k+N_u-N_v)}}\\,.\\label{f_pq}\n\\end{equation}\nFor its respective source as defined in Eq.~\\eqref{def_S}, since the d'Alembertian operator scales as $\\rho^{-2}$ under $CL$,\n\\begin{equation}\nS^{(k)}(u,v,\\rho)=\\dfrac{s^{(k)}(p,q)}{\\rho^{(D-3)(2k+N_u-N_v)+2}}\\,.\\label{def_s}\n\\end{equation}\n\nThus our problem is effectively two-dimensional in $(p,q)$ coordinates: at each order, $f^{(k)}(p,q)$ is the solution of a differential equation (inherited from the wave equation) with a source $s^{(k)}(p,q)$ and subject to initial conditions on $q=0$. This is an enormous computational advantage. In Section~\\ref{app:Green2D} of Appendix~\\ref{app:Green} we obtain the differential equation obeyed by $f^{(k)}(p,q)$ and the reduced Green's function $G^k_m(p,q;p',q')$. However, there is another consequence of this symmetry which only becomes manifest when one goes to null infinity.\n\n\\subsection{The $CL$ symmetry at null infinity}\nIn terms of $(r,\\tau,\\theta)$ the new coordinates $p$ and $q$ read\n\\begin{eqnarray}\np&=&(\\tau+r(1+\\cos\\theta)-\\Phi(r\\sin\\theta))(r\\sin\\theta)^{D-4}\\,,\\\\\nq&=&(\\tau+r(1-\\cos\\theta))(r\\sin\\theta)^{-(D-2)}\\,.\n\\end{eqnarray}\nOne can see they are not well-defined in the limit $r\\rightarrow\\infty$ since\n\\begin{equation}\np\\rightarrow\\infty\\,,\\qquad q\\rightarrow0\\,.\n\\end{equation}\n\nHowever, it is reasonable to expect that the finite, non-trivial dependence of the news function in $(\\tau,\\theta)$ should be given in terms of combinations of $p,q$ that remain finite and non-trivial at null infinity. An appropriate choice is\n\\begin{equation}\n\\hat{p}\\equiv\n\\left\\{\n\\begin{array}{ll}\n \\frac{pq-1}{q}+2\\log q\\ , &  D=4\\  \\vspace{2mm}\\\\\n\\displaystyle{ \\frac{pq-1}{q^{\\frac{1}{D-3}}}}\\ , & D>4\\ \\label{tau_transform}\\,\n\\end{array} \\right. \\ ,\n\\qquad \\hat{q}\\equiv q^{\\frac{1}{D-3}}\\,.\n\\end{equation}\n\nThis coordinate transformation, $(p,q)\\rightarrow(\\hat{p},\\hat{q})$, has a constant non-vanishing determinant,\n\\begin{equation}\n\\det \\left[\\frac{\\partial (p,q)}{\\partial(\\hat{p},\\hat{q})}\\right]=D-3\\,,\n\\end{equation}\nhence is well-defined everywhere.\n\nWhen $r\\rightarrow\\infty$,\n\\begin{eqnarray}\n\\hat{p}&\\rightarrow&2\\bar{\\tau}(\\tau,\\theta)+O\\left(r^{-1}\\right)\\,,\\\\\n\\hat{q}&\\rightarrow&(1-\\cos\\theta)^{\\frac{1}{D-3}}(\\sin\\theta)^{-\\frac{D-2}{D-3}}\\times r^{-1}+O\\left(r^{-2}\\right)\\,,\n\\end{eqnarray}\nwhere we have defined a new time coordinate\n\\begin{eqnarray}\n\\bar{\\tau}(\\tau,\\theta)=\\left\\{\n\\begin{array}{ll}\n \\tau\\times(1-\\cos\\theta)^{-1}+\\log\\left(\\frac{1-\\cos\\theta}{\\sin\\theta}\\right)\\ , &  D=4\\  \\vspace{2mm}\\\\\n\\displaystyle{ \\tau\\times(1-\\cos\\theta)^{-\\frac{1}{D-3}}(\\sin\\theta)^{-\\frac{D-4}{D-3}}}\\ , & D>4\\ \\label{tau_transform}\\,\n\\end{array} \\right. \\ .\n\\end{eqnarray}\n\nRecall that we had already factored out the $\\rho$ dependence in Eq.~\\eqref{f_pq}, leaving only a function of $(p,q)$ to compute. But since $\\bar\\tau$ is the only surviving quantity in the asymptotic limit, we expect the news function to become effectively one-dimensional at null infinity: besides the trivial dependence on $\\theta$ coming from the known powers of $r$ and $\\rho$ (remember that $\\rho=r\\sin\\theta$), it must be a function of $\\bar\\tau(\\tau,\\theta)$ only.\n\nFrom Eqs.~\\eqref{news2} and \\eqref{f_pq},\n\\begin{equation}\n\\dot{\\mathcal{E}}^{(k)}(\\tau,\\theta)=(\\sin\\theta)^{\\frac{D-4}{2}-2k(D-3)}\\times\\lim_{r\\rightarrow\\infty}\\frac{d}{d\\tau}\\left[r^{\\frac{D-2}{2}-2k(D-3)}\\hat{f}^{(k)}(\\hat{p},\\hat{q})\\right]\\,,\n\\end{equation}\nwhere $\\hat{f}^{(k)}$ contains the relevant contribution from $E^{(k)}$ and $H^{(k)}$, as a function of $(\\hat{p},\\hat{q})$. \n\nIf the quantity inside brackets is to remain finite at null infinity, then necessarily\n\\begin{equation}\n\\hat{f}^{(k)}(\\hat{p},\\hat{q})\\simeq\\alpha^{(k)}(\\hat{p})\\hat{q}^{\\frac{D-2}{2}-2k(D-3)}+\\dots\\,,\n\\end{equation}\nwhere $\\dots$ denotes higher powers of $\\hat{q}$ and $\\alpha^{(k)}(\\hat{p})$ some unknown function of $\\hat{p}$. \n\nThus, after taking the limit $r\\rightarrow\\infty$,\n\\begin{equation}\n\\dot{\\mathcal{E}}^{(k)}(\\tau,\\theta)=\\frac{1}{1-\\cos\\theta}\\left(\\frac{1+\\cos\\theta}{1-\\cos\\theta}\\right)^{k-1+\\frac{1}{4}\\frac{D-4}{D-3}}\\frac{d\\bar\\tau}{d\\tau}(\\theta)\\times\\frac{d}{d\\bar\\tau}\\alpha^{(k)}(2\\bar\\tau)\\,.\n\\end{equation}\n\nUsing Eq.~\\eqref{tau_transform}, and evaluating this expression at $\\theta=\\tfrac{\\pi}{2}$ noting that $\\bar\\tau\\left(\\tau,\\tfrac{\\pi}{2}\\right)=\\tau$, we conclude that\n\\begin{equation}\n\\dot{\\mathcal{E}}^{(k)}(\\tau,\\theta)=\\left(\\frac{1}{1-\\cos\\theta}\\right)^2\\left(\\frac{1+\\cos\\theta}{1-\\cos\\theta}\\right)^{k-1-\\frac{1}{4}\\frac{D-4}{D-3}}\\dot{\\mathcal{E}}^{(k)}\\left(\\bar\\tau(\\tau,\\theta),\\frac{\\pi}{2}\\right)\\,,\\label{Eq:angular_factorization}\n\\end{equation}\nwhich proves our proposition.\n\n\\subsection{The meaning of perturbation theory}\n\\label{meaning}\n\nThe perturbative expansion of the metric, Eq.~\\eqref{metric_series}, implies an analogous series for the inelasticity's angular distribution $\\epsilon(\\theta)$ defined in Eq.~\\eqref{e1},\n\\begin{equation}\n\\epsilon(\\theta)=\\sum_{N=1}^\\infty\\epsilon^{(N)}(\\theta)\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\epsilon^{(N)}(\\theta)=\\sum_{k=1}^{N}\\int d\\tau\\, \\dot{\\mathcal{E}}^{(k)}(\\tau,\\theta)\\dot{\\mathcal{E}}^{(N+1-k)}(\\tau,\\theta)\\,.\\label{N_N-k}\n\\end{equation}\n\nInserting the result of Eq.~\\eqref{Eq:angular_factorization} and changing the integration variable from $\\tau$ to $\\bar{\\tau}$, we conclude that\n\\begin{equation}\n\\epsilon^{(N)}(\\theta)=\\left(\\frac{1}{1-\\cos\\theta}\\right)^3 \\left(\\frac{1+\\cos\\theta}{1-\\cos\\theta}\\right)^{N-1}\\epsilon^{(N)}\\left(\\frac{\\pi}{2}\\right)\\, . \\label{seriesi2}\n\\end{equation}\nThus it suffices to compute the news function on the symmetry plane.\n\nThe whole series reads\n\\begin{eqnarray}\n\\epsilon(\\theta)&=&\\sum_{N=1}^\\infty \\epsilon^{(N)}(\\theta) =\\left(\\frac{1}{1-\\cos\\theta}\\right)^3\\sum_{N=1}^\\infty\\left(\\frac{1+\\cos\\theta}{1-\\cos\\theta}\\right)^{N-1}\\epsilon^{(N)}\\left(\\frac{\\pi}{2}\\right)\\,,\\\\\n&\\equiv&\\sum_{n=0}^\\infty\\alpha_n(\\theta) \\epsilon^{(n+1)}\\left(\\frac{\\pi}{2}\\right)\\,.\n\\label{series2}\n\\end{eqnarray}\n\nObserve that each individual $\\alpha_n(\\theta)$ is regular at $\\theta=\\pi$, where we saw that perturbation theory should be valid, but does not obey $\\alpha_n(\\theta)=\\alpha_n(\\pi-\\theta)$ since the transformation to Brinkmann coordinates broke the $\\mathbb{Z}_2$ symmetry. However, $\\epsilon(\\theta)$ should respect that symmetry. In particular, if it has a regular limit at the axis, it can be written as a power series in $\\sin^2\\theta$,\n\\begin{equation}\n\\epsilon(\\theta)=\\sum_{n=0}^\\infty \\epsilon_n(\\sin\\theta)^{2n}\\,.\\label{angular_series}\n\\end{equation}\n\nIndeed, writing $\\cos\\theta=-\\sqrt{1-\\sin^2\\theta}$, we see that near $\\theta=\\pi$,\n\\begin{equation}\n\\alpha_n(\\theta)\\sim (\\sin\\theta)^{2n}\\,.\n\\end{equation}\n\nFor example, the first three terms are given by\n\\begin{eqnarray}\n\\epsilon_0&=&\\frac{1}{8}\\epsilon^{(1)}\\left(\\frac{\\pi}{2}\\right)\\,,\\label{e0}\\\\\n\\epsilon_1&=&\\frac{1}{32}\\left[3\\epsilon^{(1)}\\left(\\frac{\\pi}{2}\\right)+\\epsilon^{(2)}\\left(\\frac{\\pi}{2}\\right)\\right]\\,,\\\\\n\\epsilon_2&=&\\frac{1}{128}\\left[9\\epsilon^{(1)}\\left(\\frac{\\pi}{2}\\right)+5\\epsilon^{(2)}\\left(\\frac{\\pi}{2}\\right)+\\epsilon^{(3)}\\left(\\frac{\\pi}{2}\\right)\\right] \\ .\n\\end{eqnarray}\n\nThus we conclude that a consistent truncation of the series in Eq.~\\eqref{angular_series}, i.e. the extraction of the coefficient $\\epsilon_n$, requires knowledge of $\\epsilon^{(n+1)}(\\tfrac{\\pi}{2})$ and hence, from Eq.~\\eqref{N_N-k}, of the metric up to $O(n+1)$. This clarifies the meaning of perturbation theory in this problem: it allows for the extraction of successive coefficients $\\epsilon_n$, thus amounting to an angular expansion off the collision axis.\n\nIt should be stressed that this result is a kinematical consequence of the $CL$ symmetry only: the dynamical (wave) equations have not yet been used (except for the background solution of course).\n\nFinally, the inelasticity is then given by\n\\begin{eqnarray}\n\\epsilon&=&\\sum_{n=0}^\\infty \\epsilon_n \\int_{-1}^1\\frac{d\\cos\\theta}{2}(\\sin\\theta)^{2n}\\,,\\\\\n&=&\\sum_{n=0}^\\infty \\frac{2^n n!}{(2n+1)!!}\\epsilon_n \\,,\\\\\n&=&\\epsilon_0+\\frac{2}{3}\\epsilon_1+\\ldots\\,.\\label{eq_second_order_e}\n\\end{eqnarray}\n\nIn the next section we shall obtain simplified expressions for the integrals giving the asymptotic metric functions that contribute to the news function. By working in Fourier space with respect to the retarded time $\\tau$, we shall also confirm, explicitly, that the coordinate transformation of Eq.~\\eqref{tau_transform} factorises the angular dependence out of the integrals.\n\n\n\\section{Asymptotic integral solution for the metric functions}\n\\label{asymp_waveforms}\nBack in Chapter~\\ref{ch:dynamics} we wrote the formal solution for each metric function $F(u,v,\\rho)$ in integral form, Eq.~\\eqref{sol_3D}. Since we are ultimately interested in the news function $\\dot{\\mathcal{E}}(\\tau,\\theta)$, we define the asymptotic waveform $\\dot{F}(\\tau,\\theta)$, obtained from $F(u,v,\\rho)$, according to Eq.~\\eqref{news2},\n\\begin{equation}\n\\dot{F}(\\tau,\\theta)\\equiv\\lim_{r\\rightarrow\\infty} r\\rho^{\\frac{D-4}{2}}\\frac{d}{d\\tau}F(u,v,\\rho)\\,.\n\\end{equation}\n\nActually, the Dirac delta in the Green's function is better dealt with in Fourier space, so we define\n\\begin{equation}\n\\hat{\\dot{F}}(\\omega,\\theta)\\equiv\\int d\\tau\\,\\dot{F}(\\tau,\\theta)e^{-i\\omega\\tau}\\,.\n\\end{equation}\nThis is equally useful to compute the inelasticity since, by the Parseval-Plancherel theorem,\n\\begin{equation}\n\\int d\\tau\\, \\dot{F}(\\tau,\\theta)^2=\\frac{1}{2\\pi}\\int d\\omega\\, |\\hat{\\dot{F}}(\\omega,\\theta)|^2\\,.\n\\end{equation}\n\nThe next steps are detailed in Appendix~\\ref{app:asympWF} due to their tedious and technical nature. \n\nIn short, one needs to take the asymptotic limit and integrate in $\\tau$ to obtain the Fourier transform. Dropping primes on integration variables for ease of notation, we get\n\\begin{equation}\n\\hat{\\dot{F}}(\\omega,\\theta)=-i^{\\frac{D-2}{2}+m}\\omega\\int_0^\\infty d\\rho\\,\\rho^{\\frac{D-2}{2}}J_{\\frac{D-4}{2}+m}(\\omega\\rho\\sin\\theta)\\hat{S}\\left(\\omega\\frac{1+\\cos\\theta}{2},\\omega\\frac{1-\\cos\\theta}{2};\\rho\\right)\\,,\\label{initial_volume}\n\\end{equation}\nwhere $J_\\nu(z)$ is the $\\nu$-th order Bessel function of the first kind and\n\\begin{equation}\n\\hat{S}(x,y;\\rho)=\\frac{1}{2}\\int du\\int dv\\, e^{-iux}e^{-ivy}\\, S(u,v,\\rho)\\,.\\label{S_hat}\n\\end{equation}\n\nNext, one finds it convenient to transform to a new frequency $\\Omega$,\n\\begin{equation}\n\\omega\\rightarrow\\Omega\\equiv\\omega^{D-3}\\left(\\frac{\\sin\\theta}{2}\\right)^{D-4}\\frac{1-\\cos\\theta}{2}\\,,\\label{omega_transformation}\n\\end{equation}\ntogether with a new waveform\n\\begin{equation}\n\\hat{\\mathcal{F}}(\\Omega,\\theta)\\equiv\\sqrt{\\omega'(\\Omega)}\\,\\hat{\\dot{F}}(\\omega(\\Omega),\\theta)\\,,\n\\end{equation}\nsuch that\n\\begin{equation}\n\\int d\\omega\\, |\\hat{\\dot{F}}(\\omega,\\theta)|^2=\\int d\\Omega\\, |\\hat{\\mathcal{F}}(\\Omega,\\theta)|^2\\,.\\label{oO_int}\n\\end{equation}\n\nThen, using the $CL$ symmetry, namely Eq.~\\eqref{def_s}, one shows that, at each order $k$ in perturbation theory,\n\\begin{equation}\n\\hat{\\mathcal{F}}^{(k)}(\\Omega,\\theta)=\\left(\\frac{1}{1-\\cos\\theta}\\right)^{\\frac{3}{2}}\\left(\\frac{1+\\cos\\theta}{1-\\cos\\theta}\\right)^{k-1}\\hat{\\mathcal{F}}^{(k)}\\left(\\Omega,\\frac{\\pi}{2}\\right)\\,, \\label{angular_factorization}\n\\end{equation}\nwhere the angular dependence is now completely factored out of the integral, and the (new) Fourier space waveform evaluated at $\\theta=\\tfrac{\\pi}{2}$, which we shall abbreviate to\n\\begin{equation}\n\\hat{\\mathcal{F}}(\\Omega)\\equiv\\hat{\\mathcal{F}}\\left(\\Omega,\\frac{\\pi}{2}\\right)\\,,\n\\end{equation}\nis given by\n\\begin{equation}\n\\hat{\\mathcal{F}}(\\Omega)\\equiv -\\sqrt{\\frac{8}{D-3}}i^{\\frac{D-2}{2}+m}\\,\\Omega^{2k-1}\\int_0^\\infty dR\\,R^{\\frac{D-2}{2}}J_{\\frac{D-4}{2}+m}(2R)\\hat{S}(\\Omega^{-1},\\Omega;R) \\; . \\label{Omega_form}\n\\end{equation}\n\nThis formulation in terms of the new frequency $\\Omega$ will prove to be useful later on in Chapter~\\ref{ch:surface} for the evaluation of surface terms. However, for the time being, we can invert the transformation in Eq.~\\eqref{omega_transformation} at $\\theta=\\tfrac{\\pi}{2}$, i.e.\n\\begin{equation}\n\\Omega\\rightarrow\\bar{\\omega}\\equiv2\\,\\Omega^{\\frac{1}{D-3}}\\,,\\qquad \\hat{\\mathcal{F}}(\\Omega)\\rightarrow\\sqrt{\\Omega'(\\bar{\\omega})}\\,\\hat{\\mathcal{F}}\\left(\\Omega(\\bar{\\omega})\\right)\\,.\n\\end{equation}\n\nThe net relationship between $\\omega$ and $\\bar{\\omega}$ is\n\\begin{equation}\n\\bar{\\omega}=\\omega\\times(1-\\cos\\theta)^{\\frac{1}{D-3}}(\\sin\\theta)^{\\frac{D-4}{D-3}}\\,,\n\\end{equation}\nwhich is equivalent, in real space, to a transformation of the time coordinate\n\\begin{equation}\n\\tau\\rightarrow\\bar{\\tau}(\\tau,\\theta)=\\tau\\times(1-\\cos\\theta)^{-\\frac{1}{D-3}}(\\sin\\theta)^{-\\frac{D-4}{D-3}}\\,.\n\\end{equation}\nApart from a $\\theta$-dependent shift in $D=4$ (indeed an example of a \\emph{supertranslation}~\\cite{Coelho:2012sy}) $\\bar{\\tau}$ is exactly the same as in \\eqref{tau_transform}.\n\nIn the next chapter we shall specialise this asymptotic solution to the surface case and compute all the terms that contribute to the inelasticity and depend linearly on the initial data. This will allows us to extract the isotropic coefficient $\\epsilon_0$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Angular momentum flux}\n\\label{app:angmomflux}\n\nThis appendix contains details on derivation of the angular momentum flux for the Maxwell field on the Kerr-AdS background. Our purpose is to show that the vanishing of the energy flux, a physical principle used in Chapter~\\ref{ch:KerrAdS} to impose boundary conditions for the Maxwell field on Kerr-AdS, leads to a vanishing angular momentum flux.\n\nFrom the definition of the energy-momentum tensor for the Maxwell field\n\\begin{equation}\nT_{\\mu \\nu}=F_{\\mu\\sigma}F^\\sigma_{\\;\\;\\;\\nu}+\\dfrac{1}{4}g_{\\mu\\nu}F^2\\;,\n\\end{equation}\nwe can calculate the angular momentum flux\n\\begin{equation}\n\\mathcal{J}=\\int_{S^2} \\sin\\theta d\\theta d\\varphi\\; r^2 \\left(T^r_{\\;\\;\\varphi,\\;\\uppercase\\expandafter{\\romannumeral1}}+T^r_{\\;\\;\\varphi,\\;\\uppercase\\expandafter{\\romannumeral2}}\\right)\\;,\\label{angmomf}\n\\end{equation}\nwith\n\\begin{align}\n&T^r_{\\;\\;\\varphi,\\;\\uppercase\\expandafter{\\romannumeral1}}=-a\\sin^2\\theta\\;T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral1}}\\;,\\label{relation1}\\\\\n&T^r_{\\;\\;\\varphi,\\;\\uppercase\\expandafter{\\romannumeral2}}=-\\dfrac{i\\sin\\theta\\sqrt{\\Delta_\\theta}(r^2+a^2)}{2\\Xi\\rho^4}\\Phi_1(\\Phi_2^\\ast+\\Delta_r\\Phi_0^\\ast)+c.c\\;,\\label{relation2}\n\\end{align}\nwhere $T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral1}}$ is given in Eq.~\\eqref{rtcom}, $c.c$ stands for the complex conjugate of the preceding terms, and\n\\begin{equation}\n\\Phi_0=\\phi_0\\;,\\;\\;\\;\\;\\;\\Phi_2=2\\bar{\\rho}^\\ast\\phi_2\\;,\\;\\;\\;\\;\\;\\bar{\\rho}=r+ia\\cos\\theta\\;.\n\\end{equation}\n\n\nFrom Eq.~\\eqref{relation1}, and considering the vanishing energy flux boundary conditions, i.e.\n\\begin{equation}\n\\int_{S^2} \\sin\\theta d\\theta d\\varphi\\; r^2 T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral1}}\\rightarrow0\\;,\n\\end{equation}\nasymptotically, one concludes that there is no contributions for the angular momentum flux from the first term $T^r_{\\;\\;\\varphi,\\;\\uppercase\\expandafter{\\romannumeral1}}$.\n\n\nFor the second term, from Eq.~\\eqref{relation2}, we notice that $\\Phi_1$ is involved, which has been derived in Eq.~\\eqref{Phi1eq}. For convenience, we list this solution and related expressions below. The solution for $\\Phi_1$ is\n\\begin{equation}\n\\bar{\\rho}^\\ast\\Phi_1=g_{+1}\\bar{\\mathscr{L}}_1S_{+1}-ia f_{-1}\\mathscr{D}_0P_{-1}\\;,\n\\end{equation}\nwith\n\\begin{align}\n&g_{+1}=\\frac{1}{B}(r\\mathscr{D}_0P_{-1}-P_{-1})\\;,\\\\\n&f_{-1}=\\frac{1}{B}(\\cos\\theta\\bar{\\mathscr{L}}_1S_{+1}+\\sin\\theta\\sqrt{\\Delta_\\theta}S_{+1})\\;,\\\\\n&\\bar{\\mathscr{L}}_1S_{+1}=\\dfrac{(2a\\omega\\Xi\\cos\\theta-\\lambda)S_{+1}-BS_{-1}}{2\\mathcal{Q}\\sqrt{\\Delta_\\theta}}\\;,\n\\end{align}\nwhere\n\\begin{equation}\n\\mathscr{D}_0=\\dfrac{\\partial}{\\partial r}-\\dfrac{iK_r}{\\Delta_r}\\;,\\;\\;\\;\\mathcal{Q}=\\dfrac{\\Xi(a\\omega\\sin^2\\theta-m)}{\\sin\\theta\\Delta_\\theta}\\;,\\;\\;\\;P_{-1}=BR_{-1}\\;,\n\\end{equation}\nand the constant $B$ is given by Eq.~\\eqref{Bvalue}, $S_{+1}$ and $S_{-1}$ are spin weighted AdS spheroidal harmonics. From the properties of these spheroidal harmonic functions\n\\begin{equation}\nS_{+1}(\\pi-\\theta)=S_{-1}(\\theta)\\;,\\;\\;\\;\\;\\;\\;S_{-1}(\\pi-\\theta)=S_{+1}(\\theta)\\;,\n\\end{equation}\nwhich are guaranteed by the angular equations~\\eqref{Spluseq} and~\\eqref{Sminuseq}, then we have the following properties\n\\begin{align}\n\\int_0^\\pi d\\theta\\sin\\theta f_{odd}(\\theta)S_{+1}(\\theta)S^\\ast_{+1}(\\theta)=-\\int_0^\\pi d\\theta\\sin\\theta f_{odd}(\\theta)S_{-1}(\\theta)S^\\ast_{-1}(\\theta)\\;,\\nonumber\\\\\n\\int_0^\\pi d\\theta\\sin\\theta f_{even}(\\theta)S_{+1}(\\theta)S^\\ast_{+1}(\\theta)=\\int_0^\\pi d\\theta\\sin\\theta f_{even}(\\theta)S_{-1}(\\theta)S^\\ast_{-1}(\\theta)\\;,\\nonumber\\\\\n\\int_0^\\pi d\\theta\\sin\\theta f_{odd}(\\theta)S_{-1}(\\theta)S^\\ast_{+1}(\\theta)=-\\int_0^\\pi d\\theta\\sin\\theta f_{odd}(\\theta)S_{+1}(\\theta)S^\\ast_{-1}(\\theta)\\;,\\nonumber\\\\\n\\int_0^\\pi d\\theta\\sin\\theta f_{even}(\\theta)S_{-1}(\\theta)S^\\ast_{+1}(\\theta)=\\int_0^\\pi d\\theta\\sin\\theta f_{even}(\\theta)S_{+1}(\\theta)S^\\ast_{-1}(\\theta)\\;,\\label{intepro}\n\\end{align}\nwhere\n\\begin{equation}\nf_{odd}(\\pi-\\theta)=-f_{odd}(\\theta)\\;,\\;\\;\\;\\;\\;\\;f_{even}(\\pi-\\theta)=f_{even}(\\theta)\\;.\\nonumber\n\\end{equation}\nWith all of these expressions at hand, and making use of the integration properties of the spin weighted AdS spheroidal harmonics given in~\\eqref{intepro}, Eq.~\\eqref{relation2} becomes\n\\begin{eqnarray}\n&&T^r_{\\;\\;\\varphi,\\;\\uppercase\\expandafter{\\romannumeral2}}=-\\dfrac{i\\sin\\theta\\sqrt{\\Delta_\\theta}(r^2+a^2)}{2\\Xi\\rho^4}(\\mathcal{C}_1S_{+1}S^\\ast_{-1}+\\mathcal{C}_2S_{-1}S^\\ast_{-1})\n+c.c\\;,\\label{Trphi2rep}\n\\end{eqnarray}\nwhere terms that vanish under the angular integration, due to the properties listed above, have been discarded. The expressions for $\\mathcal{C}_1$ and $\\mathcal{C}_2$ are messy in general, but they can be simplified asymptotically. The asymptotic expression for $\\mathcal{C}_1$ goes as\n\\begin{equation}\n\\mathcal{C}_1\\sim c_0+\\mathcal{O}(1\/r)\\;,\\label{c1expasy1}\n\\end{equation}\nwhere $c_0$ is proportional to $T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral1}}$ asymptotically, so that finally $\\mathcal{C}_1\\sim \\mathcal{O}(1\/r)$. Similar analysis can be done for $\\mathcal{C}_2$ as well. The asymptotic expression for $\\mathcal{C}_2$ is\n\\begin{equation}\n\\mathcal{C}_2\\sim \\hat{c}_0+\\mathcal{O}(1\/r)\\;,\\label{c1expasy2}\n\\end{equation}\nand, as in the former case, $\\hat{c}_0$ vanishes after the vanishing energy flux boundary conditions in Eq.~\\eqref{bc} are imposed. Then from Eq.~\\eqref{Trphi2rep}, together with Eqs~\\eqref{c1expasy1} and~\\eqref{c1expasy2}, we conclude that\n\\begin{eqnarray}\nr^2T^r_{\\;\\;\\varphi,\\;\\uppercase\\expandafter{\\romannumeral2}}\\sim \\mathcal{O}(1\/r)\\;,\n\\end{eqnarray}\nasymptotically, which leads to the vanishing of the angular momentum flux of Eq.~\\eqref{angmomf}.\n\n\n\n\n\n\n\\chapter{Functions and matrices for the charged Proca case}\n\\label{app:chargedP}\n\nThis appendix contains details on the functions and matrices used to rewrite the second order radial differential equations into a first order form, and recurrence relations used to initialize these radial differential equations close to the event horizon. This was used in Chapter~\\ref{ch:ChargedP}.\n\nThe functions that used in the text are (where $\\kappa_s^2=\\ell(\\ell+1)$):\n\\begin{align}\nA(r)&\\equiv \\sum^{2n+1}_{m=0}{a_m y^m}=r \\left[r^n-(1+Q^2)r+Q^2r^{n-2}\\right]^2 \\; ,\\nonumber\\\\\nB(r)&\\equiv \\sum^{2n}_{m=0}{b_m y^m}=2\\left[r^n-(1+Q^2)r+Q^2r^{n-2}\\right]^2\\; ,\\nonumber\\\\\nC(r)&\\equiv \\sum^{2n+1}_{m=0}{c_m y^m}= (\\omega r-qQ)^2r^{2n-1}\n-(\\kappa_s^2+\\mu_p^2r^2)r^{n-1}\\left[r^n-(1+Q^2)r+Q^2r^{n-2}\\right]  \\; ,\\nonumber\\\\\nE(r)&\\equiv \\sum^{2n}_{m=0}{e_m y^m}=iqQr^{n-1}\\left[(1+Q^2)(n-1)r-2Q^2r^{n-2}\\right]\\nonumber\\\\& \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;+i\\omega r^n\\left[2r^{n}-(1+Q^2)(n+1)r+4Q^2r^{n-2}\\right] \\, ,\\nonumber\\\\\n\\tilde A(r)&\\equiv \\sum^{2n}_{m=0}{\\tilde a_m y^m}=\\left[r^n-(1+Q^2)r+Q^2r^{n-2}\\right]^2  \\; ,\\nonumber\\\\\n\\tilde B(r)&\\equiv 0 \\;,\\nonumber\n\\end{align}\n\\begin{align}\n\\tilde C(r)\\equiv \\sum^{2n}_{m=0}{\\tilde c_m y^m}=& (\\omega r-qQ)^2r^{2n-2}\n -\\left(\\kappa_s^2+\\mu_p^2r^2\\right)r^{n-2}\\left[r^n-(1+Q^2)r+Q^2r^{n-2}\\right] \\;,\\nonumber\\\\\n\\tilde E(r)\\equiv \\sum^{2n-2}_{m=0}{\\tilde e_m y^m}=&iqQ r^{n-1}\\left[(n-3)(1+Q^2)+2r^{n-1}\\right]\\nonumber\\\\&-i\\omega r^{n-1}\\left[(1+Q^2)(n-1)r-2Q^2r^{n-2}\\right]\\;.\\nonumber\n\\end{align}\nThe recurrence relations are\n\\begin{align}\n\\mu_0&=\\nu_0\\;,\\nonumber\\\\\n\\mu_1&=-\\frac{\\left[a_3\\rho(\\rho-1)+b_2\\rho +c_1+e_1\\right]\\nu_0+ e_0 \\nu_1}{a_2\\rho(\\rho+1)+c_0}\\;,\\nonumber\\\\\n\\mu_j&=\\frac{\\tilde a_2(\\rho+j)(\\rho+j-1)+\\tilde c_0}{D_j}f_j-\\frac{e_0}{D_j}\\tilde\nf_j\\;,\\nonumber\\\\\n\\nu_j&=\\frac{a_2(\\rho+j)(\\rho+j-1)+c_0}{D_j}\\tilde\nf_j-\\frac{\\tilde e_0}{D_j}f_j\\;,\n\\label{CPrecurone}\n\\end{align}\nwith\n\\begin{align}\nD_j&=\\left[a_2(\\rho+j)(\\rho+j-1)+c_0\\right]\\left[\\tilde a_2(\\rho+j)(\\rho+j-1)+\\tilde c_0\\right]-\\tilde e_0 e_0\\;,\\nonumber\\\\\nf_j&=-\\sum^j_{m=1}\\left[(a_{m+2}(\\rho+j-m)(\\rho+j-m-1)+b_{m+1}(\\rho+j-m)+c_m\\big)\\mu_{j-m}+e_m\\nu_{j-m}\\right]\\;,\\nonumber\\\\\n\\tilde f_j&=-\\sum^j_{m=1}\\left[\\big(\\tilde a_{m+2}(\\rho+j-m)(\\rho+j-m-1)+\\tilde c_m\\big)\\nu_{j-m}+\\tilde e_m\\mu_{j-m}\\right]\\;.\\nonumber\n\\end{align}\n\\begin{text}\nThe coefficients used in the asymptotic expansion in the text are\n\\begin{align}\nc^\\pm=&\\dfrac{i}{2\\omega}\\Big[-\\kappa_s^2+Q^2(\\mu_p^2-2\\omega^2)-\\dfrac{q^2Q^2\\mu_p^2}{k^2}\\mp i\\dfrac{qQ\\omega}{k}+(2\\omega^2-\\mu_p^2)(1+Q^2)\\delta_{3,n}\\nonumber\\\\&+\\Big(\\pm i\\dfrac{\\mu_p^2(1+Q^2)}{2k}-\\Big(\\dfrac{\\mu_p^2(1+Q^2)}{2k}\\Big)^2+(1+Q^2)^2(2\\omega^2-\\mu_p^2)\\nonumber\\\\&+\\dfrac{qQ\\omega(1+Q^2)(\\mu_p^2-2k^2)}{k^2}\\Big)\\delta_{2,n}\\Big] \\; .\\nonumber\\label{CPcpm}\n\\end{align}\nThe relation between the new first order radial functions $\\mathbf{\\Psi}$ used in~\\eqref{eq:ODEcoupled}, and the 4-vector $\\mathbf{V}^T=(\\psi,d_r\\psi,\\chi,d_r\\chi)$ for the original fields and derivatives is found from Eqs.~\\eqref{asymptoticpsi},~\\eqref{asymptoticpchi} and their derivatives. The corresponding $r$-dependent matrix transformation $\\mathbf{T}$ is defined\n\\begin{equation}\n\\mathbf{V}= \\mathbf{T} \\mathbf{\\Psi} \\; ,\n\\end{equation}\nand its form is\n\\begin{equation}\\label{eq:CPT}\n\\mathbf{T}=\\left(\\begin{array}{cccc}\\frac{e^{i\\Phi}}{r} & \\frac{e^{-i\\Phi}}{r} & e^{i\\Phi} & e^{-i\\Phi} \\vspace{2mm}\\\\ \\frac{ike^{i\\Phi}}{r} & -\\frac{ike^{-i\\Phi}}{r} & \\left[ik+\\frac{i\\varphi}{r}\\right]e^{i\\Phi} & -\\left[ik+\\frac{i\\varphi}{r}\\right]e^{-i\\Phi} \\vspace{2mm}\\\\ 0 & 0 & \\left(-\\frac{k}{\\omega}+\\frac{c^+}{r}\\right)e^{i\\Phi}  & \\left(\\frac{k}{\\omega}+\\frac{c^-}{r}\\right)e^{-i\\Phi} \\vspace{2mm}\\\\ 0 & 0 & \\left[-\\frac{ik^2}{\\omega}+\\frac{ikc^+-\\frac{ik\\varphi}{\\omega}}{r}\\right]e^{i\\Phi}& -\\left[\\frac{ik^2}{\\omega}+\\frac{ikc^-+\\frac{ik\\varphi}{\\omega}}{r}\\right]e^{-i\\Phi} \\end{array}\\right) \\ ,\n\\end{equation}\nOn another hand, the original system~\\eqref{originalsys1} and\n~\\eqref{originalsys2}, can be written in a first order form\n\\begin{equation}\n\\dfrac{d\\mathbf{V}}{dr}=\\mathbf{X}\\mathbf{V} \\; ,\n\\end{equation}\nwhere the matrix $\\mathbf{X}$ is\n\\begin{equation}\\label{eq:CPX}\n\\mathbf{X}=\\left(\\begin{array}{cccc}0 & 1 & 0 & 0 \\vspace{2mm} \\\\ -\\dfrac{C}{A} & -\\dfrac{B}{A}& -\\dfrac{E}{A}  & 0 \\vspace{2mm}\\\\0 & 0 & 0 & 1 \\vspace{2mm} \\\\ -\\dfrac{\\tilde E}{\\tilde A}  & 0& -\\dfrac{\\tilde C}{\\tilde A}& 0 \\end{array}\\right) \\ ,\n\\end{equation}\nand the 2-vectors\n\\begin{equation}\\label{eq:CPyplus}\n\\mathbf{y}^{\\pm}=\\left(\\begin{array}{c}\\sqrt{\\dfrac{\\kappa_s^2 k}{\\omega^2}} a_0^{\\pm} \\\\ i\\sqrt{\\dfrac{k}{\\mu_p^2}}\\left[\\left(\\pm \\varphi+\\omega c^{\\pm}\\mp k(1+Q^2)\\delta_{n,2}\\pm\\dfrac{kqQ}{\\omega}\\right)a_0^{\\pm}\\pm ka_1^{\\pm}\\right] \\end{array}\\right) \\ ,\n\\end{equation}\n\\begin{equation}\\label{eq:hminus}\n\\mathbf{h}^-=\\left(\\begin{array}{c}\\sqrt{t} \\nu_0 \\\\ \\sqrt{\\dfrac{\\mu_p^2}{\\mu_p^4+q^2Q^2}}\\left(-i\\kappa_s^2\\nu_0+(\\omega-qQ)d \\right) \\end{array}\\right) \\; ,\n\\end{equation}\nwith\n\\begin{align}\nt&=\\dfrac{\\kappa_s^2}{\\mu_p^4+q^2Q^2}\\Big[(\\omega-qQ)\\left(qQa+b\\mu_p^2+2i\\rho \\mu_p^2\\alpha+\\dfrac{2qQ\\mu_p^2}{\\beta}\\right)-\\kappa_s^2\\mu_p^2\\Big]\\;,\\nonumber\\\\\nd&=\\left[\\left(1+\\rho-\\dfrac{iqQ}{\\mu_p^2}\\rho\\right)(a+ib)+\\left(1+\\dfrac{iqQ}{\\mu_p^2}\\right)\\left(\\dfrac{iqQ}{\\beta}-\\alpha\\rho\\right)\\right]\\nu_0+(1+2\\rho)\\left(\\dfrac{iqQ}{\\mu_p^2}-1\\right)\\nu_1 \\;,\\nonumber\\\\\na&=\\dfrac{qQ}{\\omega-qQ}\\;,\\;\\;\\;\\;\\;\\;\\;b=\\dfrac{qQ(2+n(n-3)(1+Q^2))}{\\beta^2}-\\dfrac{2\\omega\\alpha}{\\beta}+\\dfrac{\\kappa_s^2+\\mu_p^2}{\\omega-qQ}\\;,\\nonumber\\\\\n\\alpha&=\\dfrac{n(n-1)+(n-3)(n+2)Q^2}{2\\beta}\\;,\\;\\;\\;\\;\\;\\;\\;\\beta=(n-1)+(n-3)Q^2\\;.\\nonumber\n\\end{align}\n\\end{text}\n\n\n\n\n\n\\chapter{Functions and matrices for the neutral Proca case}\n\\label{app:neutralP}\n\nThis appendix contains details on the functions, recurrence relations and matrices, used in Chapter~\\ref{ch:NeutralP}.\n\nThe functions appeared in Eqs.~\\eqref{NPsysterm1} and~\\eqref{NPsysterm2} are $(\\kappa_s^2=\\ell(\\ell+n-1))$\n\\begin{align}\nM(r)&\\equiv \\sum^{2n-1}_{m=0}{\\alpha_m y^m}= r\\left(r^{n-1}-1\\right)^2 \\; ,\\nonumber\\\\\nN(r)&\\equiv \\sum^{2n-2}_{m=0}{\\beta_m y^m}=n\\left(r^{n-1}-1\\right)^2  \\; ,\\nonumber\\\\\nP(r)&\\equiv \\sum^{2n-1}_{m=0}{\\gamma_m y^m}=-(\\kappa_s^2+\\mu_p^2r^2)\\left(r^{2n-3}-r^{n-2}\\right) +\\omega^2r^{2n-1}  \\; ,\\nonumber\\\\\nQ(r)&\\equiv \\sum^{2n-2}_{m=0}{\\sigma_m y^m}= i\\omega r^{n-1}\\left(2r^{n-1}-n-1\\right) \\; ,\\nonumber\\\\\n\\tilde M(r)&\\equiv \\sum^{2n}_{m=0}{\\tilde \\alpha_m y^m}=r^2\\left(r^{n-1}-1\\right)^2  \\; ,\\nonumber\\\\\n\\tilde N(r)&\\equiv \\sum^{2n-1}_{m=0}{\\tilde \\beta_m y^m}=(n-2)r\\left(r^{n-1}-1\\right)^2\\;,\\nonumber\\\\\n\\tilde P(r)&\\equiv \\sum^{2n}_{m=0}{\\tilde \\gamma_m y^m}=-\\left(\\kappa_s^2+\\mu_p^2r^2\\right)\\left(r^{2n-2}-r^{n-1}\\right)\n+\\omega^2r^{2n}-(n-2)\\left(r^{n-1}-1\\right)^2\\;,\\nonumber\\\\\n\\tilde Q(r)&\\equiv \\sum^{n}_{m=0}{\\tilde \\sigma_m y^m}=-i\\omega(n-1)r^n\\;.\\nonumber\n\\end{align}\nThe recurrence relations are\n\\begin{align}\n\\mu_0&=\\nu_0\\;,\\nonumber\\\\\n\\mu_1&=-\\frac{\\left(\\rho(\\rho-1)\\alpha_3+\\rho\\beta_2+\\gamma_1+\\sigma_1\\right)\\nu_0+\\sigma_0\\nu_1}{\\rho(\\rho+1)\\alpha_2+\\gamma_0}\\;,\\nonumber\\\\\n\\mu_j&=\\frac{\\omega^2+(n-1)^2(\\rho+j)(\\rho+j-1)}{D_j}f_j+\\frac{i\\omega(n-1)}{D_j}\\tilde\nf_j\\;,\\nonumber\\\\\n\\nu_j&=\\frac{\\omega^2+(n-1)^2(\\rho+j)(\\rho+j-1)}{D_j}\\tilde\nf_j+\\frac{i\\omega(n-1)}{D_j}f_j\\;,\n\\label{NPrecurone}\n\\end{align}\nwith\n\\begin{align}\nD_j&=(n-1)^2\\omega^2+\\left(\\omega^2+(n-1)^2(\\rho+j)(\\rho+j-1)\\right)^2\\;,\\nonumber\\\\\nf_j&=-\\sum^j_{m=1}{\\Big[\\big(\\alpha_{m+2}(\\rho+j-m)(\\rho+j-m-1)+\\beta_{m+1}(\\rho+j-m)+\\gamma_m\\big)\\mu_{j-m}+\\sigma_m\\nu_{j-m}\\Big]}\\;,\\nonumber\\\\\n\\tilde\nf_j&=-\\sum^j_{m=1}{\\left[\\big(\\tilde\\alpha_{m+2}(\\rho+j-m)(\\rho+j-m-1)+\\tilde\\beta_{m+1}(\\rho+j-m)+\\tilde\\gamma_m\\big)\\nu_{j-m}+\\tilde\\sigma_m\\mu_{j-m}\\right]}\\;.\\nonumber\n\\end{align}\nThe coefficients used in the asymptotic expansion in the text are\n\\begin{equation}\\label{eq:NPcpm}\nc^\\pm=\\dfrac{i}{2\\omega}\\left[-\\kappa_s^2+\\dfrac{n(2-n)}{4}+(2\\omega^2-\\mu_p^2)\\left(\\delta_{2,n}+\\delta_{3,n}\\right)+\\left(\\pm i\\dfrac{\\mu_p^2}{2k}-\\left(\\dfrac{\\mu_p^2}{2k}\\right)^2\\right)\\delta_{2,n}\\right] \\; .\n\\end{equation}\nThe matrices used in the text are as follows\n\\begin{equation}\\label{eq:NPT}\n\\mathbf{T}=\\frac{1}{r^{\\frac{n-2}{2}}}\\left(\\begin{array}{cccc}\\frac{e^{i\\Phi}}{r} & \\frac{e^{-i\\Phi}}{r} & e^{i\\Phi} & e^{-i\\Phi} \\vspace{2mm}\\\\ \\frac{ike^{i\\Phi}}{r} & -\\frac{ike^{-i\\Phi}}{r} & \\left[ik+\\frac{i\\varphi-\\frac{n-2}{2}}{r}\\right]e^{i\\Phi} & -\\left[ik+\\frac{i\\varphi+\\frac{n-2}{2}}{r}\\right]e^{-i\\Phi} \\vspace{2mm}\\\\ 0 & 0 & \\left(-\\frac{k}{\\omega}+\\frac{c^+}{r}\\right)e^{i\\Phi}  & \\left(\\frac{k}{\\omega}+\\frac{c^-}{r}\\right)e^{-i\\Phi} \\vspace{2mm}\\\\ 0 & 0 & \\left[-\\frac{ik^2}{\\omega}+\\frac{ikc^+-\\frac{k}{\\omega}(i\\varphi-\\frac{n-2}{2})}{r}\\right]e^{i\\Phi}& -\\left[\\frac{ik^2}{\\omega}+\\frac{ikc^-+\\frac{k}{\\omega}(i\\varphi+\\frac{n-2}{2})}{r}\\right]e^{-i\\Phi} \\end{array}\\right) \\ ,\n\\end{equation}\n\\begin{equation}\\label{eq:X}\n\\mathbf{X}=\\left(\\begin{array}{cccc}0 & 1 & 0 & 0 \\vspace{2mm} \\\\ -\\dfrac{P}{M} & -\\dfrac{N}{M}& -\\dfrac{Q}{M}  & 0 \\vspace{2mm}\\\\0 & 0 & 0 & 1 \\vspace{2mm} \\\\ -\\dfrac{\\tilde Q}{\\tilde M}  & 0& -\\dfrac{\\tilde P}{\\tilde M}&-\\dfrac{\\tilde N}{\\tilde M} \\end{array}\\right) \\ ,\n\\end{equation}\nand the 2-vectors\n\\begin{equation}\\label{eq:NPyplus}\n\\mathbf{y}^{\\pm}=\\left(\\begin{array}{c}\\sqrt{\\dfrac{\\kappa_s^2 k}{\\omega}} a_0^{\\pm} \\\\ \\sqrt{\\dfrac{\\omega k}{\\mu_p^2}}\\left[(\\pm i\\varphi-\\frac{n-2}{2}+i\\omega c^{\\pm}\\mp ik\\delta_{n,2})a_0^{\\pm}\\pm ika_1^{\\pm}\\right] \\end{array}\\right) \\ ,\n\\end{equation}\n\\begin{equation}\\label{eq:NPhminus}\n\\mathbf{h}^-=\\left(\\begin{array}{c}\\sqrt{\\kappa_s^2} \\nu_0 \\\\ \\dfrac{i\\omega(\\mu_1-\\rho(\\frac{n}{2}\\nu_0+\\nu_1-\\mu_1))+\\kappa_s^2\\nu_0}{\\mu_p} \\end{array}\\right) \\ .\n\\end{equation} \n\\chapter{Proof of Theorem 1}\n\\label{app:ST1}\nIn this appendix, we prove Theorem~\\ref{thm1} given in Chapter~\\ref{ch:prelim}, by taking $\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}=BS_{-1}$, as an example. To do so, we start from Eq.~\\eqref{Spluseq}, by applying the operator $\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}$ on both sides,\n\\begin{align}\n-\\lambda\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}&=\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}(\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}-2a\\omega\\cos\\theta\\Xi)S_{+1}\\nonumber\\\\\n&=\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}S_{+1}-2a\\omega\\Xi\\bar{\\mathscr{L}_0}(\\cos\\theta\\bar{\\mathscr{L}_1}-\\sin\\theta\\sqrt{\\Delta_\\theta})S_{+1}\\nonumber\\\\\n&=\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}S_{+1}-2a\\omega\\Xi\\cos\\theta\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}+4a\\omega\\Xi\\sqrt{\\Delta_\\theta}\\sin\\theta \\bar{\\mathscr{L}_1}S_{+1}\\;,\\label{angularTeukol1}\n\\end{align}\nwhere relations\n\\begin{equation}\n\\bar{\\mathscr{L}_n}^\\dag\\cos\\theta=\\cos\\theta\\bar{\\mathscr{L}_n}^\\dag-\\sin\\theta\\sqrt{\\Delta_\\theta}\\;,\\;\\;\\;\\;\\;\\;\n\\sqrt{\\Delta_\\theta}\\sin\\theta\\bar{\\mathscr{L}}_{n+1}=\\bar{\\mathscr{L}_n}\\sqrt{\\Delta_\\theta}\\sin\\theta\\;,\n\\end{equation}\nhave been used. The first term in Eq.~\\eqref{angularTeukol1} can be further simplified\n\\begin{align}\n\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}S_{+1}&=\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}(\\bar{\\mathscr{L}_0}+2\\sqrt{\\Delta_\\theta}\\mathcal{Q})\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=\\bar{\\mathscr{L}_0}(\\bar{\\mathscr{L}_1}^\\dag-2\\sqrt{\\Delta_\\theta}\\mathcal{Q})\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}+2\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}\\sqrt{\\Delta_\\theta}\\mathcal{Q}\\bar{\\mathscr{L}_1}S_{+1}\\;,\n\\end{align}\nin which\n\\begin{align}\n\\bar{\\mathscr{L}_1}\\sqrt{\\Delta_\\theta}\\mathcal{Q}&=\\left(\\bar{\\mathscr{L}_0}+\\dfrac{1}{\\sin\\theta}\\dfrac{d}{d\\theta}(\\sqrt{\\Delta_\\theta}\\sin\\theta)\\right)\\sqrt{\\Delta_\\theta}\\mathcal{Q}\\nonumber\\\\\n&=\\sqrt{\\Delta_\\theta}\\mathcal{Q}\\bar{\\mathscr{L}_0}+\\sqrt{\\Delta_\\theta}\\dfrac{d}{d\\theta}(\\sqrt{\\Delta_\\theta}\\mathcal{Q})+\\dfrac{\\sqrt{\\Delta_\\theta}\\mathcal{Q}}{\\sin\\theta}\\dfrac{d}{d\\theta}(\\sqrt{\\Delta_\\theta}\\sin\\theta)\\nonumber\\\\\n&=\\sqrt{\\Delta_\\theta}\\mathcal{Q}\\bar{\\mathscr{L}_0}+\\dfrac{d}{d\\theta}(\\Xi H)+\\Xi H\\cot\\theta=\\sqrt{\\Delta_\\theta}\\mathcal{Q}\\bar{\\mathscr{L}_0}+2a\\omega\\Xi\\cos\\theta\\;,\n\\end{align}\nwhere $\\mathcal{Q}=\\tfrac{\\Xi H}{\\Delta_\\theta}$, as defined in Eq.~\\eqref{Qdef},\nand where the following relation is used\n\\begin{equation}\n\\dfrac{dH}{d\\theta}+H\\cot\\theta=2a\\omega\\cos\\theta\\;.\n\\end{equation}\nThen Eq.~\\eqref{angularTeukol1} becomes\n\\begin{align}\n-\\lambda\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}&=\\bar{\\mathscr{L}_0}(\\bar{\\mathscr{L}_1}^\\dag-2\\sqrt{\\Delta_\\theta}\\mathcal{Q})\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}+2\\bar{\\mathscr{L}_0}(\\sqrt{\\Delta_\\theta}\\mathcal{Q}\\bar{\\mathscr{L}_0}+2a\\omega\\Xi\\cos\\theta)\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&\\;\\;\\;\\;-2a\\omega\\Xi\\cos\\theta\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}+4a\\omega\\Xi\\sqrt{\\Delta_\\theta}\\sin\\theta \\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}^\\dag\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}+4a\\omega\\Xi\\bar{\\mathscr{L}_0}\\cos\\theta\\bar{\\mathscr{L}_1}S_{+1}-2a\\omega\\Xi\\cos\\theta\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\&\\;\\;\\;\\;+4a\\omega\\Xi\\sqrt{\\Delta_\\theta}\\sin\\theta \\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}^\\dag\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}+4a\\omega\\Xi(\\cos\\theta\\bar{\\mathscr{L}_0}-\\sin\\theta\\sqrt{\\Delta_\\theta})\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\&\\;\\;\\;\\;-2a\\omega\\Xi\\cos\\theta\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}\n+4a\\omega\\Xi\\sqrt{\\Delta_\\theta}\\sin\\theta \\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=(\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}^\\dag+2a\\omega\\Xi\\cos\\theta)\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}\\;.\\label{angularTeukol2}\n\\end{align}\nComparing Eq.~\\eqref{angularTeukol2} with Eq.~\\eqref{Sminuseq} and remembering the definition in Eq.~\\eqref{newangoperator}, one concludes that $\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}$ is proportional to $S_{-1}$. We identify the proportionality constant as $B$, such that\n\\begin{equation}\n\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}=BS_{-1}\\;.\\label{eqB1}\n\\end{equation}\nFollowing the same procedures as above, one can also prove that\n\\begin{equation}\n\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}^\\dag S_{-1}=BS_{+1}\\;,\\label{eqB2}\n\\end{equation}\nwhere the same proportionality constant $B$ is used. This property is guaranteed by the normalization conditions given in Eq.~\\eqref{angnorm}\n\\\\\nTherefore, Theorem~\\ref{thm1} is proved. \n\\chapter{Marginal charged clouds around charged black holes}\n\\label{ch:ChargedClouds}\n\n\n\n\n\n\n\\section{Introduction}\nScattering processes for the Proca field were studied in Chapters~\\ref{ch:NeutralP} and~\\ref{ch:ChargedP}. An interesting generalization is to study this field in the context of quasinormal modes and quasi-bound states. In fact, such studies have been performed for a neutral Proca field on a Schwarzschild BH~\\cite{Rosa:2011my}, and on a slowly rotating Kerr BH~\\cite{Pani:2012vp}. In this chapter, we are going to study quasi-bound states for a charged Proca field (as well as a charged scalar field) on a Reissner-N\\\"ordstrom (RN) BH. This study is particularly interesting since we would like to know if the charge couplings between the background and the Proca field could balance the gravitational attraction, similarly to what occurs to scalar clouds.\n\nScalar clouds~\\cite{Hod:2012px,Hod:2013zza,Herdeiro:2014goa} are equilibrium configurations of a complex, massive scalar field in the background of a Kerr BH. They are stationary bound states, with a real frequency. These configurations are possible due to the existence of two qualitatively different types of quasi-bound states -- i.e. bound field configurations with a \\textit{complex} frequency:\n\\begin{itemize}\n\\item[i)] Time decaying quasi-bound states; this is the generic behavior expected for matter around a BH, due to the purely ingoing boundary condition at the horizon. Indeed, this is the only kind of quasi-bound state that can be found around Schwarzschild BHs\n\\item[ii)] Time growing quasi-bound states; this occurs for Kerr BHs in the superradiant regime,\n    i.e. when the real part of the frequency, $\\omega$, of the quasi-bound state obeys $\\omega<m\\Omega_H$, where $m$ is the azimuthal harmonic index of the field mode and $\\Omega_H$ is the horizon angular velocity of the Kerr BH. This yields an instability of Kerr BHs in the presence of any field for which quasi-bound states can be found in the superradiant regime.\n\\end{itemize}\nScalar clouds exist at the boundary between these two regimes, i.e. when $\\omega=m\\Omega_H$, which is compatible with the bound state condition $\\omega<\\mu$ for scalar fields with mass $\\mu$ on the Kerr BH. Renewed interest concerning these clouds has been triggered by the recent observation that there are Kerr BHs with scalar hair~\\cite{Herdeiro:2014goa,Herdeiro:2014ima,Herdeiro:2014jaa}, found as exact solutions of the Einstein-Klein-Gordon system, and which correspond to the nonlinear realization of these clouds.\n\nIt has long been known that charged, i.e. RN BHs can amplify charged scalar fields through superradiant scattering~\\cite{Bekenstein:1973mi}. This is a process that has qualitative similarities with the superradiant scattering of (neutral) scalar fields by Kerr BHs. For the charged case, as explained in Section~\\ref{sc:SR}, there is no analogue of a charged superradiant instability for asymptotically flat charged BHs~\\cite{Furuhashi:2004jk,Hod:2013nn,Hod:2013eea}. As such, one concludes that there will be no charged scalar clouds around RN BHs analogue to the ones discussed above for Kerr BH, i.e. as true bound states. An open question is whether the same holds for higher spin massive fields, where an effective potential analysis (which indicates if there is a potential well to support quasi-bound states) is not always straightforward. Here we study the charged Proca field around a RN BH, and show that similar statements to those for a scalar field hold.\n\nAlthough no stationary bound states exist for charged scalar or Proca fields around RN BHs, we show in this chapter that stationary \\textit{marginally bound states} do exist. The first observation of such states for scalar case was reported in~\\cite{Degollado:2013eqa} in the double extremal limit wherein the BH charge tends to its mass $|Q|\\rightarrow M$ and the field charge tends to its mass $|q|\\rightarrow \\mu$. It was noticed that, in this limit, the imaginary part of the quasi-bound states vanishes and the real part tends to the mass of the field. Moreover, in this limit, the scalar field does not trivialize and it can be interpreted as a collection of scalar particles at rest, outside the horizon, in a no-force configuration with the BH, due to a balance of electromagnetic and gravitational forces. These particles are only marginally bound to the BH and we shall therefore dub the corresponding field configuration as \\textit{marginal (charged) clouds}.\n\nIn fact we show that one needs not take a double extremal limit (as in~\\cite{Degollado:2013eqa}) to get marginal (charged) scalar clouds around RN BHs. It is enough that the \\textit{threshold condition}\n %\n\\begin{equation}\nqQ=\\mu M \\ ,\n\\label{tc}\n\\end{equation}\nis obeyed, clearly indicating that the equilibrium is due to a force balance. Moreover, we show that a completely analogous behavior is found for a charged Proca field around RN BHs. It is worth emphasizing that although these clouds are only marginally bound, arbitrarily long lived quasi-bound states exist close to the limit where these clouds appear, which, for many practical purposes, may be faced as eternal clouds.\n\nThe structure of this chapter is organized as follows. In Section~\\ref{sec:qProca} we review the background geometry as well as the field equations for the charged Proca and scalar fields. In particular, in Section~\\ref{sec:eff_potential}, we analyze the effective potential for a decoupled mode of the Proca field, to illustrate some expected features of the result. In Section~\\ref{sec:num_strategy} we describe the numerical strategy to calculate the quasi-bound state frequencies for coupled and decoupled modes, summarize some known results in the small frequency limit and combine them to predict an approximation for the (hydrogen-like) real part of spectrum for the charged Proca field. In Section~\\ref{results} we present numerical results illustrating the variation of the quasi-bound state frequencies with particle charge and mass as to identify the condition for long lived states. In Section~\\ref{sec_analytic} we demonstrate that there are indeed solutions when the threshold condition holds, which means that the imaginary part of the frequency vanishes. In a double extremal limit we even find closed form solutions for both the scalar (already discussed in~\\cite{Degollado:2013eqa}) and also the Proca case. Finally we conclude with a summary of our results in Section~\\ref{Discussion}, in particular we discuss the possible existence of nonlinear realizations of these marginal clouds.\n\n\n\n\n\n\n\\section{Charged Proca and scalar fields on a charged black hole}\n\\label{sec:qProca}\n\\subsection{The Proca field}\n\nWe consider first a Proca field $W^{\\mu}$ with mass $\\mu$\\footnote{In this chapter, we use $\\mu$ in general to stand for the mass both for the Proca fields and scalar fields, which were denoted by $\\mu_p$ and $\\mu_s$, respectively, in previous chapters.}, which is charged under a $U(1)$ gauge symmetry associated with the gauge field $A_\\mu$.  The field $W^{\\mu}$ can describe $W$ particles in the SM coupled to gravity and its Lagrangian density is given in Eq.~\\eqref{Lagrangian}.\nThe background geometry for a $U(1)$ charged BH is given by the RN solution with line element\n\\begin{equation}\nds^2=-U(r)dt^2+\\dfrac{1}{U(r)}dr^2+r^2(d\\theta^2+\\sin^2\\theta d\\varphi^2) \\ ,\n\\end{equation}\nwhere\n\\begin{equation}\nU(r)=1-\\dfrac{2M}{r}+\\dfrac{Q^2}{r^2}\\;.\\nonumber\n\\end{equation}\n$M$ and $Q$ are the BH mass and charge parameters, respectively. For numerical convenience, we change to units such that the outer horizon radius is at $r_H=1$, i.e.\n\\begin{equation}\n2M=1+Q^2 \\ .\n\\end{equation}\nThe field equations for a Proca field on an Einstein-symmetric (and in particular spherically symmetric) background have been derived in Eqs.~\\eqref{originalsys1},~\\eqref{originalsys2},~\\eqref{Max} and~\\eqref{k0M}. In the following we set $n=2$ in those equations, and identify the metric function as $U(r)$. Then we obtain a system of two coupled modes $(\\psi,\\chi)$ obeying\n\\begin{align}\n&\\left[U^2\\dfrac{d}{dr}\\left(r^2\\dfrac{d}{dr}\\right)+\\left(\\omega r -qQ\\right)^2-U\\left(\\ell(\\ell+1)+\\mu^2r^2\\right)\\right]\\psi+\\left[2iU\\omega r-ir\\left(\\omega r-qQ\\right)\\dfrac{dU}{dr}\\right]\\chi=0 \\;, \\nonumber \\\\\n&\\left[U^2r^2\\dfrac{d^2}{dr^2}+\\left(\\omega r -qQ\\right)^2-U\\left(\\ell(\\ell+1)+\\mu^2r^2\\right)\\right]\\chi  +\\left[2iqQU-i r\\left(\\omega r-qQ\\right)\\dfrac{dU}{dr}\\right]\\psi=0 \\; \\;, \\; \\; \\; \\; \\label{coupledequations}\n\\end{align}\nand a decoupled transverse mode $\\Upsilon$\n\\begin{equation}\n\\left[r^2U\\dfrac{d}{dr}\\left(U\\dfrac{d}{dr}\\right)+\\left(\\omega r  -qQ\\right)^2-\\left(\\ell(\\ell+1)+\\mu^2r^2\\right)U\\right] \\Upsilon =0  \\label{TmodeEq}\\ ,\n\\end{equation}\nwhere $\\ell=1,2,\\ldots$. There is also an exceptional mode for $\\ell=0$ which is described by a decoupled radial equation\n\\begin{equation}\n\\left[\\dfrac{d}{dr}\\left(\\dfrac{Ur^4}{(\\omega r-qQ)^2-Ur^2\\mu^2}\\dfrac{d}{dr}\\right)+\\dfrac{r^2}{U}-\\dfrac{qQr^2\\left((\\omega r-qQ) r U'-2qQU\\right)}{\\left[(\\omega r-qQ)^2-Ur^2\\mu^2\\right]^2}\\right]\\psi^{(0)}=0 \\ .\n\\end{equation}\nThe asymptotics of the solutions of such systems has been detailed in Chapter~\\ref{ch:NeutralP} and Chapter~\\ref{ch:ChargedP}. Let us denote a generic field mode (coupled or not to others) by $\\psi_i$. Then the near horizon asymptotics for all modes takes the form\n\\begin{equation}\\label{Eq:NH_expansion}\n\\psi_i= (r-1)^{\\mp i\\frac{\\omega-qQ}{1-Q^2}}\\sum_{n=0}^{+\\infty} a_i^{(n)} (r-1)^n\\ ,\n\\end{equation}\nwhere the minus sign corresponds to an ingoing boundary condition at the horizon. On the other hand, in the asymptotic region $r\\rightarrow +\\infty$ the expansion is\n\\begin{equation}\n\\psi_i= e^{ i \\Phi}\\sum_{n=0}^{+\\infty}\\dfrac{c_{i,+}^{(n)}}{r^n}+ e^{-i \\Phi} \\sum_{n=0}^{+\\infty}\\dfrac{c_{i,-}^{(n)}}{r^n}\\label{eq:FFexpansion}\n\\end{equation}\nwith\n\\begin{align}\n \\Phi &\\equiv k r+\\varphi \\log r\\;,\\nonumber \\\\\n \\varphi&\\equiv \\dfrac{(\\omega^2+k^2)(1+Q^2)-2qQ\\omega}{2k} \\;,\\\\\n k&\\equiv \\sqrt{\\omega^2-\\mu^2} \\; .\n\\end{align}\nAll the necessary expansion coefficients were provided in Chapters~\\ref{ch:NeutralP} and~\\ref{ch:ChargedP}. As shown originally by Press and Teukolsky~\\cite{Press:1973zz}, if we consider initial data given on a compact support, the late time dynamics of generic linear perturbation is governed by a superposition of a discrete set of solutions in Fourier space. After imposing an ingoing boundary condition at the horizon, there are basically two types of solutions depending on the boundary conditions when $r\\rightarrow +\\infty$ (considering a mode oscillating as $e^{i\\Phi}$, i.e. $\\Re(k)>0$ in the square root and assuming, without loss of generality, $\\Re(\\omega)>0$):\n\\begin{itemize}\n\\item $\\Im(k)<0$: these are the quasinormal modes which describe time decaying oscillations. These modes are free to escape the BH potential and asymptotically grow exponentially~\\cite{Nollert:1999ji} (for large $r$).\n\\item $\\Im(k)>0$, these are quasi-bound states, i.e. they describe field configurations which are confined in the outside region of the BH and decay exponentially when $r\\rightarrow+\\infty$. They are possible if there is a confining potential well where the field can accumulate. The boundary condition for these states can thus be recast as\n\\begin{equation}\n\\lim_{r\\rightarrow+\\infty}\\psi_i=0\\;.\n\\end{equation}\n\n\\end{itemize}\nQuasi-bound states are very interesting in the presence of superradiance, since they provide the possibility of an instability as discussed in Chapter~\\ref{ch:intro}. If they exist within the superradiant regime, the field is able to extract energy from the BH which accumulates in the confining potential. This would be signaled by an exponential growth of the wave amplitude. The condition for the instability to appear is $\\omega_I\\equiv\\Im{(\\omega)}>0$, i.e. the time-dependence is\n\\begin{equation}\n\\psi_i\\sim e^{-i\\omega t}=e^{-i\\omega_Rt+\\omega_It} \\; .\n\\end{equation}\nIn the absence of instabilities (i.e. $\\omega_I<0$), the wave amplitude decays exponentially with a lifetime\n\\begin{equation}\n\\tau\\equiv |\\omega_I|^{-1}\\; .\n\\end{equation}\nIf $\\omega_I\\rightarrow 0$ is possible, then the state can be truly bound, or marginally bound if $\\omega_R\\rightarrow \\mu \\Leftrightarrow k_R\\rightarrow 0$. Observe that in this exact limit the wavelength diverges so the state becomes de-localized. Actually, $\\mu-\\omega_R$ is basically the binding energy of the state; since this goes to zero, in the limit, the state becomes marginally bound. Moreover, one can have arbitrarily long lived quasi-bound states near this limit which have a large amplitude in a compact domain (similarly to true bound states -- see the results in Section~\\ref{results}). If $\\tau$ is very large, then the state may be effectively considered bound (rather than quasi-bound) for many practical purposes.\n\n\\subsection{Effective potential for the transverse mode}\n\\label{sec:eff_potential}\nA natural strategy to investigate the possibility of quasi-bound states to appear is to recast the radial equations for the fields in a Schr\\\"odinger like form with an effective potential. In the case of the Proca field, this is not so straightforward for the coupled system. However, the transverse mode Eq.~\\eqref{TmodeEq}, can be easily recast in a Schr\\\"odinger like form ($dr_\\star \\equiv dr\/U $)\n\\begin{eqnarray}\n&&\\left[-\\dfrac{d^2}{dr_\\star^2}+V_{\\rm eff}\\right] \\Upsilon=0\\;,\\\\\n&&V_{\\rm eff}=\\left(\\tfrac{\\ell(\\ell+1)}{r^2}+\\mu^2\\right)U-\\left(\\omega -\\tfrac{qQ}{r}\\right)^2 \\; .\n\\end{eqnarray}\nTo classify the effective potential and investigate when a well forms, it is convenient to define a compactified coordinate $x\\equiv 1-1\/r\\in [0,1]$ such that\n\\begin{equation}\\label{eq:Veff}\nV_{\\rm eff}(x) = \\sum_{k=0}^4 b_k x^k \\ ,\n\\end{equation}\nwhere the coefficients $b_k$ are\n\\begin{align}\nb_0&=-(\\omega-qQ)^2 \\;,\\nonumber\\\\\nb_1&= (1-Q^2)(\\mu^2+\\ell(\\ell+1))-2qQ(\\omega-qQ)\\;,\\nonumber\\\\\nb_2&= -(2-3Q^2)\\ell(\\ell+1)+Q^2(\\mu^2-q^2)\\;,\\label{eq:bcoeffs}\\\\\nb_3&= \\ell(\\ell+1)(1-3Q^2)\\;,\\nonumber\\\\\nb_4&= \\ell(\\ell+1)Q^2\\; . \\nonumber\n\\end{align}\nThus we are dealing with a quartic polynomial. Taking into account its values at the end points a well can only form for two possible configurations with three roots (shown schematically in Fig.~\\ref{EffectiveVschematics}).\n\\begin{figure}\n\\begin{center}\n\\includegraphics[clip=true,trim = 40 450 50 50, width=0.44\\textwidth]{Veff\/Veff_schem.pdf}\n\\end{center}\n\\caption{\\label{EffectiveVschematics} Schematic representation of the effective potential (in the compactified coordinate $x$) for the two possible cases where a well may form.\n}\n\\end{figure}\nIn case~1 the derivative $V'_{\\rm eff}(x)$ must have three positive roots $x_1,x_2,x_3$. Then its form is $V'_{\\rm eff}(x)=A(x-x_1)(x-x_2)(x-x_3)$. But we also know that\n\\begin{eqnarray}\n  V'_{\\rm eff}(x)&=&\\sum_{k=0}^3 (k+1) b_{k+1} x^k\\ .\n\\end{eqnarray}\nEquating the two forms we conclude that $x_1x_2x_3=-b_1\/(4b_4)$. We are interested in finding a well in the superradiant regime, $\\omega<qQ$; but using Eqs.~\\eqref{eq:bcoeffs} we conclude that $-b_1\/b_4<0$, which is inconsistent with the positivity of the roots of the derivative. Thus case~1 is not possible in the superradiant regime.\n\nCase~2 on the other hand is defined by\\footnote{In this case we could not extract any other condition as for case~1.}\n\\begin{equation}\nV'_{\\rm eff}(1)>0 \\Leftrightarrow \\mu^2 >\\dfrac{2qQ\\omega}{1+Q^2} \\Leftrightarrow \\mu M >qQ\\dfrac{\\omega}{\\mu} \\Leftrightarrow \\dfrac{\\omega}{\\mu}<\\sqrt{\\dfrac{M}{r_H}}\\;,\\label{chargedclouds:case2relation}\n\\end{equation}\nwhere in the last step we have used the superradiance condition $\\omega r_H<qQ$. Except in the near-extremal case, Eq.~\\eqref{chargedclouds:case2relation} is not consistent with hydrogen-like spectrum. We have also made a numerical scan of the parameter space and found none of parameters satisfying the condition in Eq.~\\eqref{chargedclouds:case2relation}.\n\nThis analysis indicates that no potential well is possible in the superradiant regime for the transverse modes. For the coupled system it is not simple to make a similar analysis, so we proceed in the following sections investigating the quasi-bound state frequencies numerically and attempting to flow the parameters into the superradiant regime.\n\nWe should note at this point that Furuhashi and Nambu~\\cite{Furuhashi:2004jk} found a condition for bound states to exist for a charged scalar field, working in a small frequency and charge approximation, and using an analytic matching technique. They found the condition that $\\mu M\\gtrsim qQ$, which, from the applicability of their analysis, is not necessarily true for large parameters. We will find, however, a perfect agreement with this threshold condition for large parameters as well, therefore away from the applicability of the analytic matching approximation in~\\cite{Furuhashi:2004jk}.\n\n\\subsection{The scalar field}\n\\label{sec:scalar}\nIn this study we have also solved the problem for a charged massive scalar field. In addition to being interesting on its own this also serves as a comparison to check some of the features we have found for the Proca field.\n\nDiscussions of the charged massive scalar field can be found, for example, in~\\cite{Sampaio:2009ra,Sampaio:2009tp,sampaio2010production,Degollado:2013eqa} and in~\\cite{Furuhashi:2004jk} where the rotating case was also analyzed. Here we only present the potential since the structure is very similar to the $\\Upsilon$ mode of the Proca field. After decomposing the scalar field $\\varphi$ in spherical harmonics $\\varphi=e^{-i\\omega t}Y_\\ell^m(\\phi,\\theta)R(r)\/r$, the radial equation takes exactly the same Schr\\\"odinger like form as for $\\Upsilon$ with a modified effective potential\n\\begin{equation}\nV_{\\rm eff}^{\\mathrm{(scalar)}}=\\left(\\tfrac{\\ell(\\ell+1)}{r^2}+\\mu^2+\\dfrac{1}{r}\\dfrac{dU}{dr}\\right)U-\\left(\\omega -\\tfrac{qQ}{r}\\right)^2 \\; .\n\\end{equation}\nThen all the asymptotic analysis and boundary conditions follow our previous discussion, Eqs.~\\eqref{Eq:NH_expansion} and~\\eqref{eq:FFexpansion}, for the Proca field modes.\n\n\n\n\\section{Numerical strategy}\n\\label{sec:num_strategy}\nTo find the quasi-bound state frequencies, we scan over $\\omega$ on the complex plane. We first define a function of $\\omega$ which returns the value of the solutions $\\psi_i$ at a large $r_{far} \\sim (50\\sim100)\/k_R$ ($k_R=\\Re(k)$), i.e. at a very large distance in multiples of the typical wavelength of the wave. For each value of $\\omega$ this is done by integrating the radial equations outwards from the horizon, with initial conditions given by the power series expansions Eq.~\\eqref{Eq:NH_expansion} evaluated at $r=1.001$, with typically twenty terms.  This function is expected to be exponentially large for generic values of $\\omega$ except in the vicinity of a quasi-bound state where it should be very small. Numerically, the quasi-bound state frequency is never attained exactly since there is always some numerical contamination of the exponentially growing solution. For example for a decoupled mode\n\\begin{equation}\\label{eq:illustratedecay}\n\\psi_i\\sim c_{i,+}^{(0)}e^{ik_Rr-|k_I|r}+\\ldots+c_{i,-}^{(0)}e^{-ik_Rr+|k_I|r} \\; .\n\\end{equation}\nIn the quasi-bound state limit, $c_{i,-}^{(0)}\\rightarrow 0$. Numerically this corresponds to frequencies that minimize $|\\psi_i|_{r_{far}}$.\n\\begin{figure*}\n\\begin{center}\n\\hspace{-3mm}\\includegraphics[clip=true,trim = 90 400 50 30, width=0.51\\textwidth]{Figs\/ch6\/Surfaces\/SurfacePsi0Q0.pdf}\\includegraphics[clip=true,trim = 90 30 50 380, width=0.51\\textwidth]{Figs\/ch6\/Surfaces\/SurfacePsi0Q05.pdf}\n\\end{center}\n\\caption{\\label{SurfacesPsi0} Magnitude of the $\\psi^{(0)}$ mode at $r_{far}$ (i.e. in the asymptotic far region), as a function of the complex frequency $\\omega$ in the neutral limit (left) and for non-zero charges (right). The quasi-bound state frequencies occur at sharp dips indicated by the level numbers $n$.\n}\n\\end{figure*}\nIn Fig.~\\ref{SurfacesPsi0} we show this quantity for the $\\ell=0$ decoupled mode in the neutral case (left) and in the charged case (right). The quasi-bound state frequencies are clearly seen as sharp dips in the surface plots (note that in the vertical axis the logarithm of the scalar field value is represented). We observe that turning on the charge, with $qQ>0$, the real part of the frequency, $\\omega_R$, moves to values closer to the field mass, $\\mu$, and the (negative) imaginary part $\\omega_I$ moves to values closer to zero.\n\nFor the coupled sector, the procedure is analogous if one recalls the scattering matrices defined in Chapter~\\ref{ch:NeutralP}. First one notes that the general solution of a system of $n$ coupled fields $\\psi_i$, with second order linear equations, can be represented by $2n$ integration constants. Those constants can be defined either at the event horizon or at infinity. The linearity of the system implies that a linear transformation relates the integration constants at the horizon, with those at infinity. We denote the ingoing and outgoing wave coefficients at the horizon ($+\/-$ respectively)\n \\[\\vec{\\mathbf{h}}=({\\mathbf h}^+,{\\mathbf h}^-)=(h^+_{i},h^-_{i}) \\ , \\] where $i=1,2$ for the current coupled system, and, similarly, the coefficients at infinity are defined\n  \\[\\vec{\\mathbf{y}}=({\\mathbf y}^+,{\\mathbf y}^-)=(y^+_{i},y^-_{i}) \\ . \\]\nThe linear transformation is then represented as\n\\begin{equation}\n\\vec{\\mathbf{y}}=\\mathbf{S} \\vec{\\mathbf{h}} \\ \\ \\Leftrightarrow \\ \\ \\left(\\begin{array}{c} {\\mathbf y}^+ \\\\ {\\mathbf y}^- \\end{array} \\right)=\\left(\\begin{array}{c|c} {\\mathbf S}^{++} & {\\mathbf S}^{+-} \\\\ \\hline {\\mathbf S}^{-+}  & {\\mathbf S}^{--} \\end{array} \\right)\\left(\\begin{array}{c} {\\mathbf h}^+ \\\\ {\\mathbf h}^- \\end{array} \\right)  ,\n\\end{equation}\nwhere the scattering matrix $\\mathbf{S}$ depends on $\\omega$, $\\ell$, field couplings and the background. It encodes all the information on the scattering process and it can be constructed from specific combinations of modes with boundary conditions set at the horizon.\n\nSimilarly to the decoupled modes, we want to impose an ingoing boundary condition at the horizon i.e. ${\\mathbf h}^{+}=0$. Then the solution coefficients far away are\n\\begin{equation}\\label{scattering_ingoing}\n{\\mathbf y}^s={\\mathbf S}^{s-}{\\mathbf h}^{-} \\; .\n\\end{equation}\nWe then must impose that the coefficients of the solution which grows exponentially fast vanish (in analogy to Eq.~\\eqref{eq:illustratedecay}), i.e. we need that there is a particular initial condition $\\hat{\\mathbf h}$ at the horizon such that $\\mathbf{y}^{-}=0$, i.e.\n\\begin{equation}\n0={\\mathbf S}^{--}\\hat{\\mathbf h}^{-} \\; .\n\\end{equation}\nThus we need to choose the eigenvector with zero eigenvalue of ${\\mathbf S}^{--}$. The condition for this solution to be possible is then\n\\begin{equation}\n\\det{\\mathbf S}^{--}=0 \\; ,\n\\end{equation}\nwhich will occur at the quasi-bound state frequencies. Furthermore, one can check that (up to a normalization constant) the radial profile of the quasi-bound state solution is obtained using the following initial condition at the horizon\n\\begin{equation}\n\\left(\\begin{array}{c}\\hat{h}^{-}_1 \\\\ \\hat{h}^{-}_2\\end{array}\\right)\\propto\\left(\\begin{array}{c}-S^{--}_{12} \\\\ S^{--}_{11}\\end{array}\\right) \\; .\n\\end{equation}\nIn practice, there is numerical contamination of exponentially growing eigenmodes, which means we must again employ a minimization condition (similarly to the decoupled modes)\n\\begin{equation}\n  \\min_{\\omega}\\left[|{\\mathbf S}^{--}|_{r_{far}}\\right] \\; .\n\\end{equation}\nIn Fig.~\\ref{SurfacesCoupled} we show an example of this quantity for $\\ell=1$ coupled modes with a non-zero background charge and a zoom around the corner where the higher frequency levels pile up (right). We discuss in more detail the effect of the field charge in Sections~\\ref{results} and~\\ref{Discussion} so here we only highlight the differences compared with the decoupled cases.\n\\begin{figure*}\n\\begin{center}\n\\hspace{-3mm}\\includegraphics[clip=true,trim = 90 400 50 30, width=0.505\\textwidth]{Figs\/ch6\/Surfaces\/SurfaceCoupledQ0.pdf}\\includegraphics[clip=true,trim = 90 30 50 380, width=0.505\\textwidth]{Figs\/ch6\/Surfaces\/SurfaceCoupledQ0zoom.pdf}\n\\end{center}\n\\caption{\\label{SurfacesCoupled} Magnitude of the determinant of ${\\mathbf S}^{--}$ for the coupled system at $r_{far}$ (i.e. in the asymptotic far region), as a function of the complex frequency $\\omega$ in the neutral limit for $\\ell=1$. The various levels are indexed by $(n,S)$ as discussed in the text. The right panel shows a zoom to better display the positions of a few higher levels.\n}\n\\end{figure*}\nOnce again the procedure is to find the local minimum for each valley found in the figure.\n\nSince the search for the quasi-bound state frequencies reduces to a (local) minimization problem, one needs in general a good starting guess which is close enough to the valley. Several analytic estimates have been developed in the literature for small parameters. The general conclusion is that $\\omega_R$ follows a hydrogen like spectrum. Of particular relevance to our problem is the charged scalar spectrum approximation found in~\\cite{Furuhashi:2004jk}:\n\\begin{equation}\\label{AnalyticFN}\n\\omega_R\\simeq \\mu\\left[1-\\frac{1}{2}\\frac{(\\mu M-qQ)^2}{N^2}\\right]\\;,\\;N=\\ell+n+1\\;,\n\\end{equation}\nwith $n\\footnote{In the remaining part of this chapter, $n$ refers to the overtone number.}\\in \\mathbb{N}_0$.\nThis approximation suggests that, as the charge is turned on to positive values, there is a critical value at\\footnote{In this limit $\\omega_R=\\mu$ and in fact $\\omega_I=0$ (see~\\cite{Furuhashi:2004jk}).} $\\mu M =qQ$, after which we expect an exit from the quasi-bound state regime (since the exponentially decaying tail disappears at this threshold).\n\nAlso in the low energy limit, but in the neutral case ($q=Q=0$), a low energy approximation was found for the Proca field in~\\cite{Rosa:2011my}. Their analytic approximations can in principle be generalized to our case by introducing a charge dependence. Even though in this study we have not developed such analytic matching calculation, we found an excellent agreement (for small parameters) with the following ansatz\n\\begin{equation}\\label{eq:ProcaOmegaR}\n\\omega_R\\simeq \\mu\\left[1-\\frac{1}{2}\\frac{(\\mu M-qQ)^2}{N^2}\\right]\\; \\;, \\;  N=\\ell+S+n+1\n\\end{equation}\nwhich results from shifting $\\mu M\\rightarrow \\mu M-qQ$ in the formula of~\\cite{Rosa:2011my} as suggested by Eq.~\\eqref{AnalyticFN}. Here $S=0$ for the transverse modes and $S=1,-1$ for the coupled modes (for the exceptional mode $\\ell=0$ there is only $S=1$).\n\nIn Fig.~\\ref{SurfacesCoupled} we indicate the labels $(n,S)$ corresponding to this approximation, for the first few pairs of levels which arise in the coupled system. As expected, due to the degeneracy in $N$, i.e the same $N$ can be obtained by adding different combinations of $(n,S)$ for the same $\\ell$, the number of degrees of freedom is doubled. This can be observed in the existence of two distinct lines of valleys in the right panel. By continuity, this classification must hold also for larger parameters, so a natural strategy to obtain new frequencies is to perform a flow of the parameters in small steps from some reference quasi-bound state frequencies. This can be pictured as a flow of the valleys in the plots of Figs.~\\ref{SurfacesPsi0} and~\\ref{SurfacesCoupled}.\n\nThe two parameters we will want to vary continuously are the field mass $\\mu$ and charge $q$. Taking the mass $\\mu$ as an example, let us assume we have obtained a high precision quasi-bound state frequency $\\omega$ by minimization near a valley (with given mode labels $\\{\\ell,n,S\\}$ and fixed $\\mu,q,Q,M$). Then by continuity the frequency for this mode at a nearby mass $\\mu\\rightarrow \\mu+\\delta \\mu$ will shift by a small amount $\\omega\\rightarrow\\omega+\\delta \\omega$ (all other parameters are fixed). If the step is small enough, we can use the previous $\\omega$ as a first guess and refine it by minimization to obtain the new frequency. Thus, by using small steps, we can iterate this procedure to flow the frequencies as a function of a continuous parameter. The same procedure can be applied to flow the frequency with charge $q$.\n\nEven though analytic approximations give a very useful guide to the dependence of the frequencies, and help understanding the labels of the various levels, in practice (especially for larger parameters) we determined various reference initial estimates for the frequencies of each state graphically (using plots such as Fig.~\\ref{SurfacesPsi0}) and then refined them through minimization before using as seeds to the flows. In Table~\\ref{TabSeeds}, we provide a set of seed frequencies that were used as starting points to produce various of the plots in the results of Section~\\ref{results} for the Proca field.  Some scalar frequencies  that were used to obtain scalar profiles to compare with the Proca field in the region of parameters where long lived states were found (see Sections~\\ref{results} and~\\ref{Discussion}) are shown in Table~\\ref{TabSeedsScalar}.\n\\begin{table}\n\\begin{minipage}{\\textwidth}\n\\begin{center}\n$Qr_H=0$\\footnote{Note that the results shown in this table agree with the data used in~\\cite{Rosa:2011my}, to all of the significant figures quoted.}\\vspace{1mm}\\\\\n\\begin{tabular}{||c|c|c||}\n\\hline\n $N$ & $(\\ell,n,S)$ &$\\mu^{-1}(\\omega_R,\\omega_I)$ \\\\\n\\hline\n$1$ & $(1,0,-1)$ &$\\phantom{..}(0.95972157,-0.0037679893)\\phantom{.}$\\\\\n\\hline\n    & $(1,1,-1)$ & $(0.99032863,-0.00062350043)$\\\\\n$2$ & $(0,0,1)$  & $(0.99031646,-0.00051223870)$\\\\\n    & $(1,0,0)$  & $(0.99131023,-0.00001346974)$\\\\\n\\hline\n\\end{tabular}\\vspace{2mm}\\\\\n$Qr_H=0.3 \\; (Q\/M\\simeq 0.55\\;,\\;\\mu M \\simeq 0.27)$ \\vspace{1mm}\\\\\n\\begin{tabular}{||c|c|c||}\n\\hline\n $N$ & $(\\ell,n,S)$ &$\\mu^{-1}(\\omega_R,\\omega_I)$ \\\\\n\\hline\n$1$ & $(1,0,-1)$ & $(0.95270408,-0.0046419907)$\\\\\n\\hline\n     & $(1,1,-1)$ & $(0.98838598,-0.00084694923)$\\\\\n $2$ & $(0,0,1)$  & $(0.98812624,-0.00095739793)$\\\\\n     & $(1,0,0)$  & $(0.98953577,-0.00002375643)$\\\\\n\\hline\n\\end{tabular}\\vspace{2mm}\\\\\n$Qr_H=0.9\\;(Q\/M\\simeq 0.994\\;,\\;\\mu M \\simeq 0.45)$ \\vspace{1mm}\\\\\n\\begin{tabular}{||c|c|c||}\n\\hline\n $N$ & $(\\ell,n,S)$ &$\\mu^{-1}(\\omega_R,\\omega_I)$ \\\\\n\\hline\n$1$ & $(1,0,-1)$ & $\\phantom{..}(0.87274751,-0.031112471)\\phantom{...}$\\\\\n\\hline\n    & $(1,1,-1)$ & $(0.96469871,-0.011098631)$\\\\\n$2$ & $(0,0,1)$  & $(0.96824273,-0.016591447)$\\\\\n    & $(1,0,0)$  & $(0.96354486,-0.001705819)$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{TabSeeds} Some reference frequencies (found through minimization) used in the flow for the Proca field, with $\\mu r_H=0.5$ and $q=0$. We indicate the background charge both in inverse horizon radius units, and also in BH mass units for comparison.}\n\\begin{center}\n\\begin{tabular}{||c|c|c||}\n\\hline\n $N$ & $(\\ell,n)$ &$\\mu^{-1}(\\omega_R,\\omega_I)$ \\\\\n\\hline\n$1$ & $(0,0)$ & $\\phantom{..}(0.93713433,-0.083684629)\\phantom{...}$\\\\\n\\hline\n    & $(0,1)$ & $(0.97694630,-0.016936300)$\\\\\n$2$ & $(1,0)$ & $(0.96780454,-0.000797486)$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{TabSeedsScalar} Some reference frequencies (found through minimization) used in the flow for the scalar field, with $\\mu r_H=0.5$ and $q=0$. We have focused on $Q\/M=0.994$ to compare with the Proca field.}\n\\end{minipage}\n\\end{table}\nAll frequencies in these tables were obtained by refining initial estimates through minimization. The initial estimates were obtained graphically by zooming surfaces such as Figs.~\\ref{SurfacesPsi0} and~\\ref{SurfacesCoupled} around each valley. Once an initial estimate is found which is inside the valley, the minimization routines refine the result to the required precision. These high precision frequencies can be used to flow either the mass $\\mu$ or the charge $q$ in small steps, to other values as shown in the plots of Section~\\ref{results}.\n\n\n\\section{Results}\n\\label{results}\nIn this section, we present a selection of numerical results, focusing mostly on the positive charge coupling case. First, we note that we did not find any quasi-bound states in the superradiant region $\\omega<qQ$.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[clip=true, width=0.33\\textwidth]{Figs\/ch6\/BoundRealCoupllev0L1Q03qVar.pdf}\\includegraphics[clip=true, width=0.33\\textwidth]{Figs\/ch6\/BoundRealTVlev0L1Q03qVar.pdf}\\includegraphics[clip=true, width=0.33\\textwidth]{Figs\/ch6\/BoundRealCouplev0L1Q09qflow.pdf} \\vspace{2mm}\\\\\n\\includegraphics[clip=true, width=0.33\\textwidth]{Figs\/ch6\/BoundImagCouplev0L1Q03qVar.pdf}\\includegraphics[clip=true, width=0.33\\textwidth]{Figs\/ch6\/BoundImagTVlev0L1Q03qVar.pdf}\\includegraphics[clip=true, width=0.325\\textwidth]{Figs\/ch6\/BoundImagCouplev0L1Q09qflow.pdf}\n\\end{center}\n\\caption{\\label{Fundamental} Variation of the quasi-bound state frequencies with field mass $\\mu$ and charge $q$ for the Proca field. In the top panels we represent curves for the real part $\\omega_R$ whereas the corresponding curves for the imaginary part are in the bottom panels. The thick curves labeled in the key to the figures are obtained by numerical integration. The thin solid lines correspond to the analytic formula Eq.~\\eqref{eq:ProcaOmegaR}. The vertical lines give, for each case $\\mu M=qQ$.\n}\n\\end{figure*}\nTo understand why this is the case we analyze first Fig.~\\ref{Fundamental} where we represent the variation of the frequencies with charge and mass couplings on a fixed background. In the top row of panels we represent the real part of the frequency (or more precisely $1-\\omega_R\/\\mu$). The thick curves correspond to the numerical results that we have generated through direct integration. We have requested a minimum relative precision of six digits (i.e. $10^{-6}$ of relative error). The thin solid lines (matching the color of the solid curves) correspond to the analytic guess, Eq.~\\eqref{eq:ProcaOmegaR}. In the bottom panels we represent the imaginary part which is always negative. We also indicate with vertical lines the value where the  threshold condition  $\\mu M=qQ$ is attained.\n\nIn the left panels we represent the fundamental mode frequency of the Proca system ($N=1$) with various curves for different positive and negative field charge as a function of mass coupling $\\mu M$. The middle panels are similar except that we represent a first excited state ($N=2$). The main observations on these are:\n\\begin{itemize}\n\\item Generically, $\\omega_I$ is negative and its modulus decreases for smaller mass.\n\\item For negative field charge\\footnote{Throughout we assume the convention that the BH charge is positive.} $q$, as it goes to more negative values, $|\\omega_I|$ grows for fixed mass, which means that $\\tau$ decreases so the scalar mode should be quickly absorbed by the BH.\n\\item For positive $q$, however, as we flow to smaller mass, there is a sharp threshold in $\\omega_I$ precisely at $\\mu M= qQ$, i.e. when the mass coupling balances out exactly the electromagnetic coupling. At this threshold $\\tau\\rightarrow +\\infty$, which means that we can have arbitrarily long lived Proca quasi-bound states.\n\\item Furthermore, at the threshold $\\mu M= qQ$, the real part of the frequency also tends sharply to $\\mu$. This means that as we try to flow into the superradiant regime ($\\omega<qQ$), we leave the quasi-bound state regime into the scattering regime, before reaching it. This explains the fact that we did not find any quasi-bound states in that regime.\n\\item Finally, one can see a very good agreement between our guess, Eq.~\\eqref{eq:ProcaOmegaR}, (thin solid lines) not only for small mass, but also near the long lived state threshold even for larger mass and charge couplings.\n\\end{itemize}\nThese features are confirmed in the right panels of Fig.~\\ref{Fundamental}, where we have performed a flow of the charge $q$, with $Q$ close to the extremal BH background limit, for all the $N=1,2$ modes of the Proca system. Here it can be seen more clearly that as the charge $q$ is increased to larger positive values, there is a maximum value allowed which is given by $q_{max}=\\mu M\/Q$ (in the extremal limit this becomes $q=\\mu$).\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[clip=true, trim = 0 0 13 0, width=0.32\\textwidth]{Figs\/ch6\/WFRealCoupEigenlev0L1Q09qVar.pdf}\\hspace{1mm}\\includegraphics[clip=true, width=0.32\\textwidth]{Figs\/ch6\/WFRealTVlev0L1Q09qVar.pdf}\\hspace{1mm}\\includegraphics[clip=true, width=0.32\\textwidth]{Figs\/ch6\/WFReallev0L0Q09qVar.pdf} \\vspace{2mm}\\\\\n\\includegraphics[clip=true, trim = 0 0 13 0, width=0.436\\textwidth]{Figs\/ch6\/WFRealScalarlev0L0Q09qVar.pdf}\\hspace{3mm}\\includegraphics[clip=true, trim = 0 0 13 0, width=0.35\\textwidth]{Figs\/ch6\/WFRealScalarlev0L1Q09qVar.pdf}\n\\end{center}\n\\caption{\\label{WFuncs} Radial profiles for some quasi-bound state modes for the Proca field (top row) and scalar field (bottom row) for comparison (key in the bottom row applies to all plots). We also indicate for some cases $|r^2dV_{\\rm eff}\/dr|^{-1}$ whose poles indicate the local maximum and minimum of the effective potential (left and right peaks in each panel where it is represented in thin dashed lines). In the top left panel we also indicate the charge $q$ associated with each curve (the same ordering applies to all other panels).\n}\n\\end{figure*}\n\nIn Fig.~\\ref{WFuncs} we display some examples of quasi-bound state radial profiles. In all panels we vary the field charge from zero up to a value close to the maximum allowed by the threshold condition, to observe how the field binds around the BH close to that limit. The top row is for modes of the Proca field and we have also included, for comparison, some modes for the scalar field in the bottom row. Whenever an effective potential is known, we display the quantity $|r^2dV_{\\rm eff}\/dr|^{-1}$ (i.e. the inverse derivative). The latter has poles, one at a local maximum of the effective potential near the horizon and another at a minimum (further away), which define the location of the potential well.\n\nIn the top left panel we display the fundamental mode of the Proca field. This mode is in the coupled system and it is the eigenvector with zero eigenvalue of the $\\mathbf{S}^{--}$ matrix. It results from the linear combination of two exponentially growing solutions, so for large $r$ the fine cancellation that must occur in the linear combination becomes numerically more difficult. This can be seen in the brown ($q=0.8$) curve which starts oscillating due to numerical errors on the right tail region (in $r$). It is clear from the curves that as we approach the threshold charge the field spreads out to a larger radius and the oscillations close to the horizon (which are associated with the field's absorption by the BH) become less extended. This shows that the field configuration becomes more and more bound state like (rather than quasi-bound) since the field oscillations associated with absorption near the horizon become smaller in amplitude.\n\nIn the top middle panel we show an $N=2$ level obtained from the transverse $S=0$ equation. The features are similar to the left one except that the radial profile peaks more strongly. The peak is clearly correlated with the depth of the potential well where the field is more tightly bound, as seen in the dashed lines indicating the maximum and minimum of the potential.\n\nThe top right panel, shows the first $\\ell=0$ level. The main difference is that the curves in the bound region tend to be much more flat. This is due to the absence of orbital angular momentum for these states. In fact, in the bottom left panel we show the corresponding mode in the scalar case (for comparison) where exactly the same happens for $\\ell=0$.\n\nIn the scalar case (bottom row), similar features apply. In particular one can verify very clearly that the field profile in the long lived limit, extends inside the (increasingly wide) potential well.\n\n\n\\section{Analytic solutions}\n\\label{sec_analytic}\nWe will now show that in the limit in which the imaginary part of the frequency goes to zero, there are indeed non-trivial solutions, which are the limiting behavior of the modes found numerically in the previous section. This follows closely the analysis in~\\cite{Degollado:2013eqa} but extends it, not only because we also consider the Proca field but also because we do not restrict ourselves to the double extremal limit considered therein.\n\n\nFrom the results of Section~\\ref{results}, in the limit of the threshold condition\n \\begin{equation}\n\\mu M=qQ\\Leftrightarrow qQ=\\dfrac{(1+Q^2)}{2}\\mu\\;,\n\\label{forcebalance}\n\\end{equation}\nthe frequency becomes real and equal to the field mass\n\\begin{equation}\n\\omega = \\mu \\; .\n\\end{equation}\nWe replace these two conditions in the original equations, for both the Proca and the scalar field, and define $\\rho\\equiv r-1$ and\n\\begin{equation}\n\\epsilon\\equiv 1-Q^2 \\ .\n\\end{equation}\nIf $\\epsilon=0$, then $Q^2=1$; going back to natural units, this means $Q=M$ and $\\mu=q$. So this corresponds to a \\textit{double extremal limit}, i.e. an extremal limit for both the BH and the field. Thus $\\epsilon$ parameterizes the deviation from the double extremal limit, but keeping the force balance condition given by Eq.~\\eqref{forcebalance}. We will show that there are exact solutions for the Proca field when $\\epsilon=0$, an analogous situation to that observed in~\\cite{Degollado:2013eqa} for the scalar field case. But even for $\\epsilon\\neq 0$, as we will show later, non-trivial solutions of the Proca fields are existed.\n\nWith the definitions above, the equations for the Proca field simplify as follows. The transverse mode equation becomes\n\\begin{equation}\n\\left[\\dfrac{d^2}{d\\rho^2}+\\left(\\frac{1}{\\rho} - \\frac{2}{1+\\rho} + \\frac{1}{\\epsilon+\\rho}\\right)\\dfrac{d}{d\\rho}-\\dfrac{\\ell(\\ell+1)}{\\rho(\\epsilon+\\rho)}+\\left(\\dfrac{\\epsilon\\mu(1+\\rho)}{2\\rho(\\epsilon+\\rho)}\\right)^2\\right] \\Upsilon=0 \\;.\\label{TmodeEq2}\n\\end{equation}\nThe $\\ell=0$ mode equation is in fact very similar\n\\begin{equation}\n\\left[\\dfrac{d^2}{d\\rho^2}+\\left(\\frac{1}{\\rho} + \\frac{2}{1+\\rho} + \\frac{1}{\\epsilon+\\rho}\\right)\\dfrac{d}{d\\rho}-\\dfrac{2-\\epsilon}{\\rho(1+\\rho)(\\epsilon+\\rho)}+\\left(\\dfrac{\\epsilon\\mu(1+\\rho)}{2\\rho(\\epsilon+\\rho)}\\right)^2\\right] \\psi^{(0)}=0 \\;.\\label{Psi0_2}\n\\end{equation}\nThe coupled system is now\n\\begin{align}\n&\\left[\\dfrac{d^2}{d\\rho^2}+ \\frac{2}{1+\\rho}\\dfrac{d}{d\\rho}-\\dfrac{\\ell(\\ell+1)}{\\rho(\\epsilon+\\rho)}+\\left(\\dfrac{\\epsilon\\mu(1+\\rho)}{2\\rho(\\epsilon+\\rho)}\\right)^2\\right]\\psi+i\\mu\\dfrac{4\\rho^3+6\\epsilon \\rho^2+\\epsilon^2(\\rho-1)}{2\\rho^2(\\epsilon+\\rho)^2}\\chi=0 \\;, \\nonumber \\\\\n&\\left[\\dfrac{d^2}{d\\rho^2}-\\dfrac{\\ell(\\ell+1)}{\\rho(\\epsilon+\\rho)}+\\left(\\dfrac{\\epsilon\\mu(1+\\rho)}{2\\rho(\\epsilon+\\rho)}\\right)^2\\right]\\chi-\\dfrac{i\\epsilon^2\\mu(1+\\rho)}{2\\rho^2(\\epsilon+\\rho)^2}\\psi=0.\n\\label{coupledequations2}\n\\end{align}\nFinally, for the scalar field we have\n\\begin{equation}\n\\left[\\dfrac{d^2}{d\\rho^2}+\\left(\\frac{1}{\\rho} - \\frac{2}{1+\\rho} + \\frac{1}{\\epsilon+\\rho}\\right)\\dfrac{d}{d\\rho}-\\dfrac{\\ell(\\ell+1)}{\\rho(\\epsilon+\\rho)}+\\left(\\dfrac{\\epsilon\\mu(1+\\rho)}{2\\rho(\\epsilon+\\rho)}\\right)^2+\\dfrac{\\epsilon(\\rho-1)-2\\rho}{\\rho(1+\\rho)^2(\\epsilon+\\rho)}\\right] R=0  \\;.\\label{RscalarEq2}\n\\end{equation}\n\nWe first consider the scalar field case with $\\epsilon=0$, and obtain\n\\begin{equation}\nR_\\ell=(\\rho+1)\\left(A^{(R)}\\rho^{\\ell}+B^{(R)}\\rho^{-\\ell-1}\\right)\\; .\\label{scalarrecover}\n\\end{equation}\nThen the scalar field can be written as a linear combination of harmonic functions\\footnote{One may observe that the harmonic modes with $e^{-i\\mu t}$ can be transformed to be static by making a gauge transformation, $\\varphi\\rightarrow\\varphi e^{i\\xi}$ with $\\xi=\\mu t$, and $A_\\mu\\rightarrow A_\\mu+q^{-1}\\partial_\\mu\\xi$. This is also true for the Proca case, following the same transformation but $W_\\mu\\rightarrow W_\\mu e^{i\\xi}$. We would like to emphasize, however, that even without the above transformations, the energy momentum tensors for marginal scalar and Proca clouds are time independent, which imply that they are static.}\n\\begin{equation}\n\\varphi=e^{-i\\mu t}\\sum_{\\ell,m}Y_{\\ell m}(\\theta,\\phi)\\left(A^{(R)}\\rho^{\\ell}+B^{(R)}\\rho^{-\\ell-1}\\right)\\;.\\label{scalarsolnew}\n\\end{equation}\nTo interpret the meaning of the modes~\\eqref{scalarsolnew}, it is useful to rewrite the extremal RN BH, in isotropic coordinates\n\\begin{equation}\nds^2=-H^{-2}dt^2+H^2\\delta_{ij}dx^idx^j\\; , \\; A=(H^{-1}-1)dt \\ ,\\label{extremalRNnew}\n\\end{equation}\nwhere $H(x^i)$ is a harmonic function on Euclidean 3-space, i.e. $\\Delta_{\\mathbb{E}^3}H=0$. To describe a single extremal RN BH, $H$ is taken to have a single pole localized at the origin, $H=1+M\/|{\\bf x}|$. But multi-centered solutions, of Majumdar-Papapetrou type~\\cite{Majumdar:1947eu,Papapetrou1945}, are also possible if one chooses a harmonic function with multiple poles. Then one can solve the scalar field equation, by taking $\\varphi(t,x^i)=e^{-i\\mu t}U(x^i)$, with $\\mu=q$ on the background in Eq.~\\eqref{extremalRNnew}. The field equation thus becomes\n\\begin{equation}\n\\Delta U(\\mathbf{x})=-4\\pi\\mu\\delta^3(\\mathbf{x}-\\mathbf{x^\\prime})\\;,\n\\end{equation}\nwith the solution $U(\\mathbf{x})=\\mu\/|\\mathbf{x}-\\mathbf{x^\\prime}|$, and where the $\\delta$ function is introduced to describe that the field is localized at $\\mathbf{x^\\prime}$ (not coincide with the BH horizon). In terms of spherical coordinates, $(\\rho,\\theta,\\phi)$, we have\\footnote{This corrects a statement made in~\\cite{Sampaio:2014swa} and~\\cite{Degollado:2013eqa} that regular solutions at the horizon can be constructed from divergent partial waves. We would like to thank Sam Dolan for pointing out that only inhomogeneous solutions can be made regular at both boundaries.}\n\\begin{equation}\nU(\\mathbf{x})=\\dfrac{\\mu}{|\\mathbf{x}-\\mathbf{x^\\prime}|}=\\dfrac{4\\pi\\mu}{\\rho^\\prime}\\sum_{\\ell,m}\\dfrac{Y^\\ast_{\\ell m}(\\theta^\\prime,\\phi^\\prime)}{2\\ell+1}F_\\ell(\\rho)Y_{\\ell m}(\\theta,\\phi)\\;,\\label{Usol}\n\\end{equation}\nwhere the function $F_\\ell$ is constructed from two homogeneous solutions of Eq.~\\eqref{scalarsolnew}\n\\begin{equation}\nF_l(\\rho) \\equiv \\begin{cases} (\\rho \/ \\rho^\\prime)^\\ell , & \\rho \\le \\rho^\\prime , \\\\ (\\rho^\\prime \/ \\rho)^{\\ell+1}, & \\rho \\ge \\rho^\\prime . \\end{cases}\n\\end{equation}\nIt is obvious that the solution in Eq.~\\eqref{Usol} is regular both at the horizon and at infinity, and describe scalar particles at some spatial location in equilibrium with the BH (due to a force cancellation).\n\nConsidering the Proca field equations and $\\epsilon=0$, we find exact general solutions which are qualitatively similar to the scalar solution:\n\\begin{align}\n&\\Upsilon=((\\rho+1)\\ell+1)A^{(\\Upsilon)}\\rho^{\\ell}+((\\rho+1)\\ell+\\rho)B^{(\\Upsilon)}\\rho^{-\\ell-1}\\ , \\nonumber\\\\\n&\\psi^{(0)}=\\dfrac{1}{1+\\rho}\\left(A^{(0)}\\rho+B^{(0)}\\rho^{-2}\\right)\\ , \\nonumber\\\\\n&\\chi= A^{(\\chi)}\\rho^{\\ell+1}+B^{(\\chi)}\\rho^{-\\ell}\\ , \\nonumber\\\\\n&\\psi= \\dfrac{1}{1+\\rho}\\left[A^{(\\psi)}\\rho^{\\ell+1}+\\dfrac{B^{(\\psi)}}{\\rho^{\\ell}}-i\\mu\\left(A^{(\\chi)}\\dfrac{\\rho^{\\ell+2}(3+\\rho+\\ell(2+\\rho))}{(3+5\\ell+2\\ell^2)}-B^{(\\chi)}\\dfrac{\\rho^{1-\\ell}(\\ell(2+\\rho)-1)}{\\ell(2\\ell-1)}\\right)\\right] .\\label{extremalsol}\n\\end{align}\n\nWe now apply a scalar-vector decomposition, using invariant tensors on the Euclidean 3-space, of the form\n\\begin{equation}\nW_0=H^{-1}\\Psi\\;,\\;\\;\\;\\;\\;\\;\n\\vec{W}=\\nabla\\Phi +\\vec{V}\\;, \\;\\;\\;\\;\\;\\;\\nabla\\cdot \\vec{V}=0 \\ ,\n\\end{equation}\nwhere $W_\\mu=(W_0,\\vec{W})$. All vectors and differential operators are now in Euclidean 3-space. Then, $\\Phi$ becomes the non-dynamical mode which disappears due to the (gauged) Lorentz condition; this condition is a consequence of the Proca equations. As such, we have (as before) a longitudinal degree of freedom, $\\Psi$, and two transverse degrees of freedom, $\\vec{V}$. From the time and space components of the Proca equation, these obey a Laplace and a Maxwell-Faraday like equation, respectively:\n\\begin{equation}\n\\Delta \\Psi=0\\;,\\;\\;\\;\\;\\;\\;\n\\nabla\\times \\vec{\\beta}=-i\\mu \\nabla\\Psi\\;,\n\\end{equation}\nwhere we have defined\n\\begin{equation}\n\\vec{\\beta}=\\dfrac{1}{H^2}\\nabla \\times \\vec{V} \\; .\n\\end{equation}\nClearly, the longitudinal mode equation can be solved by a point like source centered at any point\n\\begin{equation}\n\\Psi=\\dfrac{A}{|\\mathbf{x}-\\mathbf{x}^\\prime|}\\; ,\n\\label{pc}\n\\end{equation}\nand it serves as a source to the transverse modes. Furthermore, due to the linear structure, we can superpose any linear combination of solutions.\n\nThe bottom line we wish to emphasize is that the marginal (charged) Proca clouds exist in the double extremal limit and, as the scalar clouds, can be made regular at both boundaries. This can be achieved, as shown for scalar case, by constructing an inhomogeneous solution of the form \\eqref{pc} with ${\\bf x^\\prime}$ not coinciding with the BH horizon, from two homogeneous solutions of each equation in~\\eqref{extremalsol}.\n\nWhen $\\epsilon \\neq 0$ we cannot find a closed form solution. Nevertheless, after asymptotically expanding the equations in the limit $\\rho \\gg 1$ we find\n\\begin{align}\n\\Upsilon&\\rightarrow \\rho\\left(A^{(\\Upsilon)}\\rho^{J_{(0)}}+B^{(\\Upsilon)}\\rho^{-1-J_{(0)}}\\right)\\;, \\nonumber\\\\\n\\psi^{(0)}&\\rightarrow \\dfrac{1}{\\rho}\\left(A^{(0)}\\rho^{J_{(+)}}+B^{(0)}\\rho^{-J_{(+)}-1}\\right)\\;, \\\\\n\\chi&\\rightarrow A_1\\rho^{J_{(-)}}+B_1\\rho^{-1-J_{(-)}}+A_2\\rho^{J_{(+)}}+B_2\\rho^{-1-J_{(+)}}\\;,\\nonumber \\\\\n\\rho\\,\\psi&\\rightarrow \\hat{A}_1\\rho^{2+J_{(-)}}+\\hat{B}_1\\rho^{1-J_{(-)}}+\\hat{A}_2\\rho^{2+J_{(+)}}+\\hat{B}_2\\rho^{1-J_{(+)}}\\;,\\nonumber\n\\end{align}\nwhere the hatted constants $\\hat{A}_i,\\;\\hat{B}_i$ are proportional to $A_i,\\;B_i$ (the constants are unimportant) and we have defined the effective total angular momentum\n\\begin{equation}\nJ_{(S)}\\equiv \\sqrt{\\left(\\frac{1}{2}+\\ell+S\\right)^2-\\left(\\frac{\\epsilon \\mu}{2}\\right)^2}-\\frac{1}{2} \\; .\n\\end{equation}\nFor the scalar field\n\\begin{equation}\nR\\rightarrow \\rho\\left(A^{(R)}\\rho^{J_{(0)}}+B^{(R)}\\rho^{-1-J_{(0)}}\\right)\\; .\n\\end{equation}\nThis result, justifies the existence of non-trivial solutions when the threshold condition $\\mu M=qQ$ is obeyed,  even away from double extremality.\n\n\\section{Summary}\n\\label{Discussion}\n\nIn this chapter we have studied quasi-bound state for test field configurations of charged Proca and scalar fields in the background of the RN BH. We have found some interesting properties which are present for both types of fields and are likely to be generic for massive charged bosonic fields.\n\nFirst, despite the existence of superradiant scattering of charged Proca fields by RN BHs as found in Chapter~\\ref{ch:ChargedP}, we did not find any quasi-bound states in the superradiant regime, hence no superradiant instabilities due to charged Proca fields. This agrees with what occurs for charged scalar fields~\\cite{Furuhashi:2004jk,Hod:2013nn,Hod:2013eea}.\n\nSecond, we have found, considering $\\mu M> q Q >0$, that either decreasing the mass $\\mu$  or increasing the charge $q$ as to achieve the threshold condition  $\\mu M=qQ$, one gets arbitrarily long lived states, since the imaginary part of the quasi-bound state frequency tends to zero. When taking this limit the frequency becomes equal to the field mass $\\omega\\rightarrow \\mu$, and so the states become marginally bound. Since they do not trivialize, we obtain configurations we have dubbed as marginal (charged) scalar and Proca clouds around RN BHs. But observe that these clouds are qualitatively different from those found in the Kerr case~\\cite{Hod:2012px,Hod:2013zza,Herdeiro:2014goa}, and recently extended to the Kerr-Newman case~\\cite{Hod:2014baa}, which are real bound states.\n\nBy analyzing the behavior of the field profiles when approaching the threshold condition, the region where the field has a large amplitude agrees well with the region where a potential well is present. The width of the well typically increases when approaching the marginally bound limit meaning that the field is large on a wider region away from the BH horizon. Since the quasi-bound state is localized away from the horizon, the gravitational and electromagnetic field interactions which are responsible for supporting the bound state should be dominated by their asymptotic Newtonian and Coulombian limit respectively. Thus the threshold  condition $\\mu M=qQ$ should in fact correspond to a force balance condition between the Newtonian force and the electrostatic force.\n\nScalar clouds around Kerr BHs can be promoted to nonlinear hair. Indeed Kerr BHs with scalar hair exist that connect precisely to the Kerr solutions that allow the existence of the corresponding clouds~\\cite{Herdeiro:2014goa,Herdeiro:2014ima,Herdeiro:2014jaa}. One may therefore ask if the existence of these marginal (charged) scalar and Proca clouds hints at the existence of nonlinear solutions of RN BHs with scalar or Proca hair. One possibility is along the lines of the discussion in~\\cite{Degollado:2013eqa}, for the scalar case. It was observed therein that the marginal scalar clouds (albeit this terminology was not used therein) can be seen as the partial waves of a distribution of charged scalar particles in a no-force balance with the RN BH. This suggests the existence of multi-object nonlinear solutions -- of Majumdar-Papapetrou type~\\cite{Majumdar:1947eu,Papapetrou1945}, but not necessarily multiple BHs. One other possibility, however, is that since the clouds that are regular at the horizon are partial waves with $\\ell\\neq 0$, the corresponding nonlinear configurations will have non-zero angular momentum and thus will be a Kerr-Newman BH with charged scalar (or Proca) hair, a possibility already anticipated in~\\cite{Herdeiro:2014goa,Hod:2014baa}. Of course, the existence (or nonexistence) of any of these possible solutions, can only be decided by a fully nonlinear analysis of the Einstein-(charged)-Klein-Gordon or Einstein-(charged)-Proca systems.\n\n\n\n\n\n\\chapter{Hawking radiation for a Proca field: charged case}\n\\label{ch:ChargedP}\n\n\n\n\\section{Introduction}\nIn this chapter, we are going to study Hawking radiation for a charged Proca field, by solving the charged Proca equations derived in Chapter~\\ref{ch:GeneralInFlat} numerically. This is a generalization of the study presented in Chapter~\\ref{ch:NeutralP}, by adding charge both to the field and the BH, on the 3+1 dimensional SM brane. This charged brane background is motivated by TeV gravity scenarios, in which the SM particles are confined on a 4-dimensional brane, while gravity propagates in extra dimensions, see Section~\\ref{sc:BM} for more details. As shown in~\\cite{Sampaio:2009ra}, the Schwinger emission alone does not suffice to discharge the BH, which makes the study of Hawking radiation for charged BHs essential. The effects of charge on the Hawking evaporation process, for scalar and fermion fields, have been performed in~\\cite{Sampaio:2009ra,Sampaio:2009tp}. As a first motivation for this study, we are going to complete this picture by exploring the charged Proca fields.\n\n\n\n\n\n\n\n\n\n\n\n\n\nCompared to the study for neutral Proca fields in Chapter~\\ref{ch:NeutralP}, a new feature due to the charge of the background and of the field is the existence of superradiant modes. These modes are amplified through the extraction of Coulomb energy, as well as charge, from the charged BH. Furthermore, in a rotating background, the Proca field equation variables are not known to separate, which presents an extra difficulty added to the non decoupling of the modes, making it difficult to study exactly the superradiance phenomenon -- see~\\cite{paniPRL,paniPRD} for a recent study in the slow rotation approximation. Thus, as a second motivation for this study, the charged BH background with spherical symmetry, yields a setup where superradiance of a massive spin 1 field can be explored without any approximation, albeit numerically. Such analysis will be performed herein.\n\nThe structure of this chapter is organized as follows. In Section~\\ref{sec:nearhorizon} we introduce the background geometry and study the near horizon and asymptotic behaviors of the coupled charged Proca equations. In Section~\\ref{fluxes} we discuss how to construct the scattering matrix from the electric current. The numerical results for the transmission factor and the associated Hawking fluxes are presented in Section \\ref{chargedhresults} and we summarize our results in the last section. To keep the main part of this chapter compact and clear, some technical relations are left to Appendix~\\ref{app:chargedP}.\n\n\n\\section{Boundary conditions and first order system}\\label{sec:nearhorizon}\nBefore starting to deal with the Proca equations, we first present the following brane BH geometry\n\\begin{equation}\nds^2=-V(r)dt^2+\\dfrac{1}{V(r)}dr^2+r^2(d\\theta^2+\\sin^2\\theta d\\varphi^2) \\; ,\\label{chargebrane}\n\\end{equation}\nwith metric function\n\\begin{equation}\nV(r)=1-\\dfrac{M}{r^{n-1}}+\\dfrac{Q^2}{r^2}\\;,\\label{metricchbrane}\n\\end{equation}\nwhere $M$ and $Q$ are the parameters related with BH bulk mass and brane charge. For numerical convenience, we choose units such that the outer horizon radius is $r_H=1$, i.e. $M=1+Q^2$ at the outer horizon.\n\nIt is easy to map the line element in Eq.~\\eqref{chargebrane}, to the general background geometry with the Einstein space given in Eq.~\\eqref{KI:metric}. Then the charged Proca field equations we are interested herein, can be obtained by setting $n=2$ in Eqs.~\\eqref{originalsys1},~\\eqref{originalsys2},~\\eqref{Max} and~\\eqref{k0M}\\footnote{Note that the explicit $n$ appearing in these wave equations only depends on the dimension of the Einstein space. Therefore, we set $n=2$ for the background in Eq.~\\eqref{chargebrane}. But we keep $n$ in general in the metric function~\\eqref{metricchbrane} since gravity may propagate in the extra dimensions.}.\n\nThe procedure to rewrite the second order wave equation for the coupled system into a first order form, to impose an ingoing boundary condition at the horizon, and to extract the asymptotic expansion coefficients, is similar to what was done in Section~\\ref{sec:NPnearhorizon}. For clarity, we summarize the main steps in the reminder of this section.\n\nTo determine the transmission factors, we need to integrate the radial equations from the horizon to the far away region with ingoing boundary conditions. The standard procedure is to find a series expansion of the solution near the horizon, which can be used to initialize the solution (we do so at $r=1.001$). Focusing on the coupled system $\\left\\{\\psi,\\chi\\right\\}$, if we define $y=r-1$, Eqs.~\\eqref{originalsys1} and\n~\\eqref{originalsys2} become\n\\begin{eqnarray}\n\\left[A(r)\\dfrac{d^2}{dy^2}+B(r)\\dfrac{d}{dy}+C(r)\\right]\\psi+ E(r)\\chi&=&0\\label{systerm1}\\ ,\\\\\n\\left[\\tilde A(r)\\dfrac{d^2}{dy^2}+\\tilde\nB(r)\\dfrac{d}{dy}+\\tilde C(r)\\right]\\chi+\\tilde E(r)\\psi&=&0\\label{systerm2}\\ ,\n\\end{eqnarray}\nwhere the polynomials $A,B,C,E,\\tilde{A},\\tilde{B},\\tilde{C},\\tilde{E}$ are defined in Appendix~\\ref{app:chargedP}. Making use of Frobenius' method to expand $\\psi$ and $\\chi$, we insert the following expansions into Eqs.~\\eqref{systerm1} and~\\eqref{systerm2}\n\\begin{equation}\n\\psi=y^\\rho\\sum^{\\infty}_{j=0}{\\mu_jy^j}\\label{defpsi} \\; , \\;\\;\\; \\chi=y^\\rho\\sum^{\\infty}_{j=0}{\\nu_jy^j} \\; , \\;\\;\\; \\rho=\\dfrac{-i(\\omega-qQ)}{(n-1)+(n-3)Q^2}\\;,\n\\end{equation}\nwhere the sign of $\\rho$ was chosen to impose an ingoing boundary condition. We then obtain the recurrence relations~\\eqref{CPrecurone} for the coefficients $\\mu_j$ and $\\nu_j$ found in the Appendix. A general solution close to the horizon can be parameterized by two free coefficients $\\nu_0$ and $\\nu_1$.\n\nSimilarly, to understand the asymptotic behavior of the waves at infinity we now expand $\\psi$ and $\\chi$ as\n\\begin{equation}\n\\psi=e^{\\beta r}r^{p}\\sum_{j=0}\\dfrac{a_j}{r^j}\\label{psifar} \\; , \\qquad \\chi=e^{\\beta r}r^{p}\\sum_{j=0}\\dfrac{b_j}{r^j} \\ ,\n\\end{equation}\nwhich, after insertion into Eqs.~\\eqref{originalsys1} and~\\eqref{originalsys2}, yield\n\\begin{equation}\n\\beta = \\pm ik\\; , \\;\\;\\;\\;\\;\\;\\;k=\\sqrt{\\omega^2-\\mu_p^2}\\;,\\qquad p = \\pm i\\varphi \\; ,\n\\end{equation}\nwhere $\\varphi=\\delta_{n,2}(\\omega^2+k^2)(1+Q^2)\/(2k)-qQ\\omega\/k$. Thus one can show that  asymptotically\n\\begin{equation}\n\\psi \\rightarrow \\left(a_0^++\\dfrac{a_1^+}{r}+\\ldots\\right)e^{i\\Phi}+\\left(a_0^-+\\dfrac{a_1^-}{r}+\\ldots\\right)e^{-i\\Phi} \\; , \\label{asymptoticpsi}\n\\end{equation}\n\\begin{eqnarray}\n\\chi \\rightarrow\n\\left[\\left(-\\frac{k}{\\omega}+\\dfrac{c^+}{r}\\right)a_0^++\\ldots\\right]e^{i\\Phi}\n+\\left[\\left(\\frac{k}{\\omega}+\\dfrac{c^-}{r}\\right)a_0^-+\\ldots\\right]e^{-i\\Phi} \\; ,\\label{asymptoticpchi}\n\\end{eqnarray}\nwhere $\\Phi\\equiv kr+\\varphi \\log r$ and $c^\\pm$ is defined in the Appendix, Eq.~\\eqref{CPcpm}.\nThus, as expected, each field is a combination of ingoing and outgoing waves at infinity.\nAsymptotically, the solution is parameterized by four independent coefficients $\\left\\{a_0^\\pm,a_1^\\pm\\right\\}$, two for each independent mode in the coupled system. In the same way as in Section~\\ref{sec:NPnearhorizon}, one can define a first order system of ODEs containing four radial functions $\\left\\{\\chi^\\pm,\\psi^\\pm\\right\\}$ which coincide with such coefficients at infinity, allowing for an easy extraction of the wave amplitudes. Our target system, which will be solved numerically in the remainder, is then\n\\begin{equation}\\label{eq:ODEcoupled}\n\\dfrac{d\\mathbf{\\Psi}}{dr}=\\mathbf{T}^{-1}\\left(\\mathbf{X}\\mathbf{T}-\\dfrac{d\\mathbf{T}}{dr}\\right) \\mathbf{\\Psi} \\ ,\n\\end{equation}\nwith $\\mathbf{\\Psi}^T=(\\psi_{+},\\psi_{-},\\chi_{+},\\chi_{-})$. The definition of the matrices $\\mathbf{X}$ and $\\mathbf{T}$, and how they relate with~\\eqref{systerm1} and~\\eqref{systerm2} can be found in Appendix~\\ref{app:chargedP}.\n\n\n\n\n\\section{Hawking fluxes}\n\\label{fluxes}\nWe shall now calculate the transmission factor for the coupled system as well as the Hawking fluxes generated from all the modes. In contrast to constructing a conserved flux from the energy-momentum tensor in Chapter~\\ref{ch:NeutralP} which was simple enough to extract the transmission factors, in the present case, we use the conserved electric current which is naturally defined for this charged field. One can show that such a current is given by\n\\begin{equation}\n\\mathcal{J}^{\\alpha}=W^{\\dagger\\alpha\\mu}W_{\\mu}+\\dfrac{1}{\\sqrt{-g}}\\partial_{\\beta}\\left(\\sqrt{-g}W^{\\dagger\\beta}W^{\\alpha}\\right)-c.c. \\;\\label{current}\n\\end{equation}\nThe radial flux at $r$ fixed is obtained by integrating the $\\alpha=r$ component on the sphere. We note that only the first term in Eq.~\\eqref{current} (denoted from now on $\\mathcal{J}^{\\alpha}_{\\uppercase\\expandafter{\\romannumeral1}}$) contributes, since the second term becomes a total derivative on the sphere.\n\nThe contribution for the flux of the coupled fields at infinity, is then found by simplifying the radial component of Eq.~\\eqref{current} using the equations of motion, and inserting the far away expansion at infinity\n\\begin{equation}\\label{eq:currentInf}\n\\mathcal{J}^{r, \\infty}_{\\uppercase\\expandafter{\\romannumeral1}-couple} =|y_0^-|^2-|y_0^+|^2+|y_1^-|^2-|y_1^+|^2\n\\equiv(\\mathbf{y}^-)^\\dagger \\mathbf{y}^--(\\mathbf{y}^+)^\\dagger \\mathbf{y}^+ \\; ,\n\\end{equation}\nwhere $y_i^s(s=\\pm; i=0,1)$ are linear combinations of the asymptotic coefficients $a_i^s$ given in the Appendix, Eq.~\\eqref{eq:CPyplus}.\nUsing the reflection matrix $(\\mathbf{R})$ defined in Chapter~\\ref{ch:NeutralP}, we obtain\n\\begin{equation}\\label{eq:TF1}\n\\mathcal{J}^{r, \\infty}_{\\uppercase\\expandafter{\\romannumeral1}-couple}=(\\mathbf{y}^-)^\\dagger\\left(\\mathbf{1}-\\mathbf{R}^\\dagger\\mathbf{R}\\right) \\mathbf{y}^-\\equiv (\\mathbf{y}^-)^\\dagger\\mathbf{T}\\, \\mathbf{y}^-  \\; ,\n\\end{equation}\nwhere we have defined a (hermitian) transmission matrix $\\mathbf{T}$, which can be diagonalised to find the decoupled asymptotic fields.\n\nFollowing the same procedure, one can also calculate the electric current at the horizon\n\\begin{equation}\\label{eq:currentH}\n\\mathcal{J}^{r, H}_{\\uppercase\\expandafter{\\romannumeral1}-couple}=\\dfrac{1}{\\omega-qQ}\\left(\\mathbf{h^-}\\right)^\\dagger\\mathbf{h^-} \\; ,\n\\end{equation}\nwhere the $h^-_i$ coefficients are linear combinations of the two independent $\\nu_i$ coefficients \\mbox{($i=0,1$)}, given in the Appendix, Eq.~\\eqref{eq:hminus}. It shows an important point in Eq.~\\eqref{eq:currentH} that the current can be positive or negative. This is expected, because for a bosonic field, the electric coupling can trigger superradiance.\n\nFurthermore, from the conservation law of the electric current, one can find an alternative expression for the transmission matrix. Using the scattering matrix $(\\mathbf{S}^{--})$ defined in Chapter~\\ref{ch:NeutralP}, we find\n\\begin{equation}\\label{eq:TF2}\n\\mathbf{T}=\\dfrac{1}{\\omega-qQ}(\\mathbf{S}^{--}\\mathbf{S}^{\\dagger--})^{-1} \\; .\n\\end{equation}\n\nOnce we have obtained the transmission factors, the number and energy fluxes are given by\n\\begin{equation}\\label{eq:HawkFlux}\n\\dfrac{d\\left\\{N,E\\right\\}}{dt d\\omega}=\\dfrac{1}{2\\pi}\\sum_{\\ell,\\zeta} \\dfrac{(2\\ell+1)\\left\\{1,\\omega \\right\\}}{\\exp((\\omega-qQ)\/T_H)-1} \\mathbb{T}_{\\ell,\\zeta} \\;,\n\\end{equation}\nwhere $\\zeta$ labels the mode and  $T_H$ is the Hawking temperature which, in our units, is\n\\begin{eqnarray}\nT_H=\\dfrac{(n-1)+(n-3)Q^2}{4\\pi}\\;.\\label{hawkingtemperature}\n\\end{eqnarray}\n\n\n\\section{Numerical Results}\n\\label{chargedhresults}\nWe now present a selection of numerical results for the transmission factor and the corresponding Hawking fluxes. In order to integrate the decoupled and coupled radial equations, we wrote independent codes in \\textsc{mathematica} and in \\textsc{c++}, finding agreement between the two codes. Using them we have generated a set of figures that we now describe.\n\nIn Fig.~\\ref{Superradiance01} transmission factors for different masses, spacetime dimensions and charges are shown to exhibit the superradiance phenomenon. As explained in Chapter~\\ref{ch:intro}, superradiant amplification of a bosonic field in a charged and\/or rotating BH occurs since there is Coulomb and\/or rotational energy that can be extracted without decreasing the BH area. The general condition of superradiance is $\\omega<m\\Omega_H+q\\Phi_H$; when this condition is verified, the transmission factor becomes negative and the scattered mode is amplified. In order to make the results clearer and more readable, here we just show the two coupled modes with $\\ell=1$, as an example. Other cases are qualitatively similar. In the left panel of Fig.~\\ref{Superradiance01}, we show the superradiance dependence on the field charge for $n=2$ (top row) and $n=4$ (bottom row). It is clear that superradiant amplification is enhanced with growing field charge except in the small energy regime; in this regime one observes, at least for the cases plotted, that the amplification decreases with increasing field charge (this effect is more noticeable for higher dimensions, see bottom row). The middle panel of Fig.~\\ref{Superradiance01} shows the dependence on the background charge -- the trend is the same as when the field charge is varied, i.e. a generic superradiance enhancement with the charge except for small energies. In the right panel of Fig~\\ref{Superradiance01}, we show the superradiance  dependence on the field mass; a suppression effect with increasing mass is observed, which is a generic behavior for the transmission factor (independently of the spin of the field or of being in the superradiant regime)~\\cite{Sampaio:2009ra,Sampaio:2009tp}.\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[clip=true,width=0.31\\textwidth]{Figs\/ch5\/coupledM01L1Q06n2varq-zoom.pdf}\n\\includegraphics[clip=true,width=0.292\\textwidth]{Figs\/ch5\/coupledM01n2L1q08varQ-zoom.pdf} \\hspace{2mm}\n\\includegraphics[clip=true,width=0.31\\textwidth]{Figs\/ch5\/coupledn2L1Q08q1varmass-zoom.pdf}\\\\\n\\includegraphics[clip=true,width=0.31\\textwidth]{Figs\/ch5\/coupledM01L1Q06n4varq-zoom.pdf}\n\\includegraphics[clip=true,width=0.292\\textwidth]{Figs\/ch5\/coupledM01n4L1q08varQ-zoom.pdf}\\hspace{2mm}\n\\includegraphics[clip=true,width=0.31\\textwidth]{Figs\/ch5\/coupledn4L1Q08q1varmass-zoom.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{Superradiance01} Negative transmission factors that show the superradiant amplification of the coupled modes with $\\ell= 1$ for different parameters. The two coupled modes are represented with the same color (solid and dashed lines). The top and bottom rows differ in the spacetime dimension. (Left panel)  Variation with the field charge $q$; (Middle panel) variation with the background charge $Q$; (Right panel) variation with the field mass.}\n\\end{figure*}\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[clip=true,width=0.338\\textwidth]{Figs\/ch5\/TotalFluxn4M001Q04Varq.pdf}\n\\includegraphics[clip=true,width=0.45\\textwidth]{Figs\/ch5\/FluxasymmetryQ03q03Mass001.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{asymmetryeffect} (Left panel) Variation of the number fluxes with field charge for fixed $n$ and BH parameters in the small field mass limit. (Right panel) Variation of the difference between positive charge and negative charge flux with $n$.}\n\\end{figure*}\n\nAfter we obtain the transmission factors, the Hawking fluxes (\\ref{eq:HawkFlux}) can be calculated directly. In the remainder we have chosen to show the flux of the number of particles. The flux of energy has similar features and can be simply calculated by multiplying each point with $\\omega$.\n\nIn Fig~\\ref{asymmetryeffect} we illustrate the number fluxes dependence on the field charge. On the left plots, we have kept the background charge parameter fixed (and positive), with $Q=0.4$ and $n=4$, varying the field charge $q$ from $-1$ to $1$. The plots show that there is a region at low energy, where the negative field charge flux is larger, whereas in the remaining part of the spectrum, positive charge emission is favored. It is also clear that if we integrate over the curves the emission of positive charge is always favored. This low energy behavior where negative charges are favored, only occurs for $n>3$ as can be seen on the right panel, where the difference between the positive charge and negative charge flux spectrum is presented for various $n$. This inverted charge splitting effect was also observed for scalars and fermions~\\cite{Sampaio:2009tp}, and it results from the interplay between the thermal factor and the transmission factor appearing in the expression for the number fluxes. Whereas the thermal factor always favors same charge emission, the transmission factor favors opposite charge emission and these factors dominate different parts of the spectrum. We have considered the same parameters as in~\\cite{Sampaio:2009tp}, $Q=|q|=0.3$, wherein scalars and fermions have been studied, to allow for an easy comparison.\n\nIn Fig.~\\ref{FluxVarQ}, we present the number flux dependence on the background charge for different field charge, in the same row, and different spacetime dimensions, in the same column. Consider first the $q=0$ case (middle column); it shows that the fluxes are suppressed\/maintained\/enhanced with the increase of background charge for $n=2$\/$n=3$\/$n=4$. To understand this behavior observe, from the definition of Hawking temperature, Eq.~\\eqref{hawkingtemperature},  that the Hawking temperature is decreased\/maintained\/increased with increasing background charge for $n=2$\/$n=3$\/$n=4$, in these units. Since we are using horizon radius units, as we vary the charge parameter $Q$, we are actually varying the mass of the BH as well as the charge while keeping $r_H=1$. Nevertheless, it is easy to see that, up to a stretching of the horizontal axis, if we fix the BH mass and vary the dimensionful charge, these conclusions for the variation of the height of the curves do not change since the number flux is dimensionless~\\footnote{The integrated flux however will not be the same for all background charges as expected, scaling as $r_H^{-1}$ for fixed BH mass and varying charge.}. For higher temperature, one expects a larger flux of particles, which is indeed the behavior shown in the second column of Fig.~\\ref{FluxVarQ}. Turning on the field charge we observe a more involved behavior. For $n=3$, in which the Hawking temperature does not vary with $Q$, we see in the first\/third column and for sufficiently large energies a monotonic suppression\/enhancement of the Hawking flux when the BH has the opposite\/same charge as the field. This is in agreement with the discussion of the left panel of Figure~\\ref{asymmetryeffect}. For $n=2,4$, varying $Q$ one also varies the Hawking temperature and more complex patterns are observed.  Another trend is that the number fluxes increase as the spacetime dimension increases, which may be understood from the existence of more modes that contribute to the transmission factor.\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-4.5mm}\\includegraphics[clip=true,width=0.32\\textwidth]{Figs\/ch5\/Fluxn2M04qm04VarQ.pdf} & \\hspace{-4.5mm}\n\\includegraphics[clip=true,width=0.32\\textwidth]{Figs\/ch5\/Fluxn2M04q0VarQ.pdf} & \\hspace{-5mm}\n\\includegraphics[clip=true,width=0.32\\textwidth]{Figs\/ch5\/Fluxn2M04q04VarQ.pdf}\n\\\\\n\\hspace{-2mm}\\includegraphics[clip=true,width=0.323\\textwidth]{Figs\/ch5\/Fluxn3M04qm04VarQ.pdf} &\n\\hspace{-1mm}\\includegraphics[clip=true,width=0.324\\textwidth]{Figs\/ch5\/Fluxn3M04q0VarQ.pdf} &\n\\hspace{-3mm}\\includegraphics[clip=true,width=0.318\\textwidth]{Figs\/ch5\/Fluxn3M04q04VarQ.pdf}\n\\\\\n\\includegraphics[clip=true,width=0.32\\textwidth]{Figs\/ch5\/Fluxn4M04qm04VarQ.pdf} &\n\\includegraphics[clip=false,width=0.32\\textwidth]{Figs\/ch5\/Fluxn4M04q0VarQ.pdf} &\n\\includegraphics[clip=true,width=0.32\\textwidth]{Figs\/ch5\/Fluxn4M04q04VarQ.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{FluxVarQ} Number fluxes dependence on the background charge for different spacetime dimensions and field charge, with fixed field mass $\\mu_p=0.4$. We vary the field charge in the same row and vary the spacetime dimension in the same column.\n}\n\\end{figure*}\n\n\\subsection{Mass effect and bulk\/brane emission}\nIn Figure~\\ref{Mass_spin}, we perform a comparison of the effect of introducing a mass term for the various spins which are relevant for the Standard Model brane degrees of freedom. We have used the data of~\\cite{Sampaio:2009tp} for scalars and fermions. Note that for fermions we have multiplied the data by a factor of two to take into account the two helicities of the Dirac field, since here we are also considering all the three modes for the Proca field. The dashed curves are for increasingly larger field mass (we have used the cases $\\mu\\footnote{In this subsection, the parameter $\\mu$, if not specified, referes to the mass for different spin fields, include scalar, fermion and Proca.}=0$, $0.5$ and $1$, as can be seen from the threshold points where the curves start). For $D=n+2=4$ (left panel), we observe a striking similarity between the Proca ($s=1$) flux with the scalar ($s=0$) flux for $\\mu=0$ (except at the high energy tail), which is due to the dominance of the $\\ell=0$ mode. We can see this feature is always true at small energies for larger $n$ (middle and right panels);  as we increase $n$, however, higher modes of the Proca field enhance the flux as compared to scalars. The extra modes of the Proca field contributing at higher energy, also explain the fact that the mass suppression is not as large as for Dirac fermions or scalars, as we see from the dashed curves corresponding to $\\mu=0.5$ for example. The high energy behavior contrasts with the low energy behavior, where scalar and Proca fields are dominated by the $s$-wave, whereas Dirac fermions are suppressed since they do not allow an $s$-wave. At high energy, we observe tails which are in the ratio $1:2:3$ following the number of degrees of freedom for the scalar, fermion and Proca fields respectively. This is in agreement with the fact that all transmission factors tend to one at high energy.\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[clip=true,width=0.33\\textwidth]{Figs\/ch5\/Spincomparisonn2.pdf}\n\\includegraphics[clip=true,width=0.331\\textwidth]{Figs\/ch5\/Spincomparisonn3.pdf}\n\\includegraphics[clip=true,width=0.324\\textwidth]{Figs\/ch5\/Spincomparisonn4.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{Mass_spin} Variation of the number fluxes for neutral particles with different spins (scalar, fermion and Proca) on the brane. For each type of particles we consider three different masses ($\\mu=0$, $0.5$ and $1$), which can be identified by the starting point of the curve. Note we have included the two helicities for the fermion field and all the three modes for the Proca field (we have used a small mass $\\mu_p=0.01$ for the latter, instead of $\\mu_p=0$).}\n\\end{figure*}\n\nIn the remainder, we combine our results for the Proca field on the brane with those in Chapter~\\ref{ch:NeutralP} for a Proca field in the bulk, to analyze the relative bulk-to-brane emissivity in the neutral case.\n\n\\begin{table}[h]\n\\begin{center}\n\\small\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n & $n=2$ & $n=3$ & $n=4$ & $n=5$ & $n=6$ & $n=7$\n\\\\\n\\hline\n$\\mu_p=0.1$ & 1 & 0.46 & 0.38 & 0.41 & 0.53 & 0.83\n\\\\\n\\hline\n$\\mu_p=0.3$ & 1 & 0.49 & 0.40 & 0.42 & 0.55 & 0.86\n\\\\\n\\hline\n$\\mu_p=0.6$ & 1 & 0.59 & 0.47 & 0.49 & 0.62 & 0.95\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Bulk-to-Brane relative energy emission rates for massive neutral vector fields for different mass $\\mu_p$ in terms of the spacetime dimension $D=2+n$.}\n\\label{bulkbranecomparison}\n\\end{table}\n\nThe total energy rate (or number rate) emitted in Hawking radiation for a given field, is obtained by integrating the fluxes of Eq.~\\eqref{eq:HawkFlux} (or their counterparts in the bulk) over $\\omega$. The bulk-to-brane energy emissivity ratio, for massive neutral Proca fields of different mass is shown in Table~\\ref{bulkbranecomparison} as a function of $n$. We have used the data in Chapter~\\ref{ch:NeutralP} to obtain the total emissivity for bulk fields, as well as the data presented here, for the brane emissivity. The entries of Table~\\ref{bulkbranecomparison}, show clearly that the emission of energy into brane-localized Proca particles is dominant, for $n$ larger than two, which is consistent with the argument that BHs radiate mainly on the brane~\\cite{EHMprl2000}. For fixed mass, as $n$ increases, the bulk-to-brane energy emission ratio initially decreases, reaching a minimum value for an intermediate $n$. However, if one increases $n$ further, the bulk-to-brane energy emission ratio increases again. Furthermore, we have also observed that the bulk-to-brane energy emission ratio increases with the field mass. A similar behavior  was observed for scalar fields~\\cite{Harris:2003eg,Kanti:2010mk}. Finally, we found that the bulk-to-brane energy emission ratio for the Proca field is larger than that for the scalar field for all $n$, being more noticeable for large $n$, say $n=6$ and $n=7$ (note the different notation for $D=2+n$ in this chapter as compared with~\\cite{Harris:2003eg,Kanti:2010mk}).\n\n\n\\section{Summary}\n\\label{discussion}\nIn this chapter we have completed the analysis started in Chapter~\\ref{ch:NeutralP}, by computing the transmission factor for a charged Proca field propagating in the background of a charged BH on a brane. Furthermore, this work completes the study of the effect of mass and charge for particles evaporating on the brane, for all spins relevant for the SM in brane world scenarios~\\cite{Sampaio:2009ra,Sampaio:2009tp}.\n\nOne of the main, and novel features arising from considering a charged Proca field on a charged background is the existence of negative transmission factors, a signature of superradiant scattering, which we have presented and described. Our results are consistent with the condition of superradiance as allowed by the area theorem, and we found that generically increasing the field or background charge amplifies the effect (though some inversions are possible at small energies). As we explained in Chapter~\\ref{ch:intro}, it is worth commenting that although superradiant scattering is observed for both rotating and charged BHs, rotating BHs exhibit superradiant instabilities against massive scalar fields, and in particular against the Proca field~\\cite{paniPRL,paniPRD}, whereas charged (non-rotating) BHs do not exhibit analogous instabilities against charged, massive bosonic fields. This will be explored for the Proca field in Chapter~\\ref{ch:ChargedClouds}, by computing the frequencies of quasi-bound states, a study that can also teach us how long lived Proca hair around charged BHs can be.\n\nAn effect observed herein, is that similarly to scalars and fermions, there is an inverted charge splitting effect at small energies for more than one extra spacetime dimension, where particles with charge opposite to the BH are emitted dominantly. Nevertheless, for larger energies, the normal splitting order, favoring the emission of same charge particles as to discharge the BH is restored and overall (by integrating out the flux), this channel tends to discharge the BH. This effect has been suggested to be another signature that could be found in the energy spectrum of charged fermions \\cite{Sampaio:2009tp}. Since fermions in the final state are also produced indirectly from the decay of vector bosons such as $W^{\\pm}$ or the $Z$ particles one may ask whether the effect survives. Here we have verified that the effect is present for $W^{\\pm}$, so at least for the case when the final state decay products are a charged lepton $\\ell^\\pm$ and a neutrino, these contributions will certainly enhance the effect.\n\nIn the neutral case, we have performed two comparisons. First we have compared the effect of the mass and spin on the Hawking fluxes, for all spins in the SM. Our main findings are that the Proca field spectrum departs from being similar to a scalar field in four dimensions and becomes increasingly dominant for larger number of dimensions, peaking and extending towards larger energies. This also means that the mass suppression effect is smaller for the Proca field than it is for scalars and fermions. Second, we have compared the bulk-to-brane emission ratio for Proca fields, confirming brane dominance in general, and the suppression of brane dominance with increasing mass.\n\nThese results can be used to improve the modelling of BH evaporation in TeV gravity scenarios, in the BH event generators~\\cite{Frost:2009cf,Dai:2007ki} that are in use at the ATLAS and CMS experiments to put bounds on extra dimensions in this channel~\\cite{CMS:2012yf,ATLAS-CONF-2011-065,ATLAS-CONF-2011-068,Gingrich:2012vs,Aad:2015mzg}.\n\n\\chapter{Conclusions and outlook}\n\\label{ch:conclusion}\nIn this final chapter we draw our conclusions, and address some open questions\n\nThis thesis covers studies of Hawking radiation and superradiance, for scalar and vector fields in the probe limit, using perturbative methods.\n\nIn Chapter~\\ref{ch:GeneralInFlat}, we have studied the wave equations of a Proca field on spherically symmetric higher dimensional spacetimes. Such background was chosen to avoid the non-separation of variables for the Proca equations in a rotating spacetime; while still being an interesting setup in TeV gravity models. Using the Kodama-Ishibashi formalism, we achieved separation of variables, and obtained a set of coupled equations, as well as some decoupled ones.\n\nThese equations were used to study Hawking radiation, for a neutral Proca field on a $D$-dimensional Schwarzschild black hole in Chapter~\\ref{ch:NeutralP} and a charged Proca field on a brane charged black hole in Chapter~\\ref{ch:ChargedP}. We have designed a numerical strategy to solve the coupled equations and showed that the coupled systems may be treated with an {\\bf S}-matrix type formalism which allows decoupling in the asymptotic regions. This {\\bf S}-matrix was used to define a transmission matrix the eigenvalues of which give us the transmission factors. Then the Hawking fluxes were calculated using the standard formulas. For a neutral Proca field, we found distinctive features by introducing the mass term, such as the lifting of the degeneracy of the two transverse modes in four dimensions, the appearance of longitudinal modes and in particular the $s$-wave. When both background and Proca field charges were included, we observed the existence of superradiant modes, and a charge splitting effect for small energies and for two or more extra dimensions. We also compared the Proca bulk-to-brane ratio of energy emission, showing that most of the energy is emitted on the brane.\n\nIn Chapter~\\ref{ch:ChargedClouds}, we have studied quasi-bound states for both the charged massive scalar field and the charged Proca field in Reissner-Nordstr\\\"om black holes. We established that no such states exist in the superradiant regime for the Proca field, a similar behavior to that known for the scalar field. For both fields, however, decaying quasi-bound states with an arbitrary small imaginary part of the frequency exist and thus which are arbitrarily long lived. In the limit of vanishing imaginary part of the frequency, the fields did not trivialize and we dubbed the corresponding configurations as marginal scalar or Proca clouds, since they were only marginal bound.\n\nA problem which is still open is the study of the Proca field in the Kerr black hole without any approximations. Indeed, this problem has been partially addressed for slow rotation~\\cite{paniPRL,paniPRD}. This is still interesting because of the following open questions: (1) is it possible to construct a modified Newman-Penrose formalism to deal with linear perturbations for massive fields in general? (2) Is it possible to establish a numerical method to solve the type of two dimensional partial differential equation arising in this problem\n\nIn Chapter~\\ref{ch:scalarHD}, we have studied superradiant instabilities for a charged scalar field in a $D$-dimensional Reissner-Nordstr\\\"om-AdS black hole. By employing an analytic matching method and a numerical method, we proved that superradiant instabilities do exist for all $\\ell$ modes in higher dimensions. Inspired by the large $D$ general relativity~\\cite{Emparan:2013moa}, it would also be interesting to develop another analytic treatment to look for superradiant modes.\n\nIn Chapter~\\ref{ch:KerrAdS}, we have studied Maxwell perturbations on Kerr-AdS black holes. From the viewpoint that the AdS boundary may be regarded as a perfectly reflecting mirror, we proposed \\textit{vanishing energy flux} boundary conditions which are physically motivated. Imposing such conditions, we obtained a set of two Robin boundary conditions even for a Schwarzschild-AdS black hole, where only one of them has been reported in the literature. Applying these two boundary conditions to Kerr-AdS black holes, we have studied superradiant instabilities, and observed that the new branch of quasinormal modes may be unstable in a larger parameter space. Our results also showed that superradiant instabilities for the Maxwell field may exist for (moderately) larger black hole size, when comparing with the scalar case.\n\nTo study stationary vector clouds, we have solved the Teukolsky equations at the onset of the superradiant instability. We found that both boundary conditions can yield vector clouds, which are characterized by existence lines in the parameter space. These lines are bounded by pure AdS spaces and the extremal black holes, which differs from that observed for gravitational perturbations, for which only one of the sets of clouds are bounded by the extremal black holes~\\cite{Cardoso:2013pza}. The existence of clouds at the linear level indicates nonlinear hairy black hole solutions, so the open question is to find the nonlinear realization of these vector clouds. There is already a well-known exact black hole family within the Einstein-Maxwell-AdS system: the Kerr-Newman-AdS family. It will then be very interesting to study properties of the new type of ``hairy'' black hole solutions, and understand the interplay between the Kerr-Newman-AdS black holes and the new family of ``hairy'' black holes.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Proca field equations\n\\label{ch:GeneralInFlat}\nIn the first part of this thesis, we are going to study Hawking radiation, for neutral and charged Proca fields; and quasi-bound states, for charged scalar and Proca fields. To explore these problems on a particular background, the first step is to obtain the wave equations for those fields. Two key properties that will allow the study of these wave equations are: separation of variables and decoupling of degrees of freedom. For scalar fields, which are governed by the Klein-Gordon equation, there is only one degree of freedom, and variables in general can be separated on Kerr-like spacetimes~\\cite{Carter:1968rr}. This is in part the reason why scalar fields have been explored so intensively in the literature\\footnote{It is now known that at least one fundamental scalar field exists in Nature with the discovery of the Higgs boson~\\cite{Chatrchyan:2012xdj,Aad:2012tfa}.}. The situation becomes complicated for massive bosonic fields, with spin.\n\nIn the standard model of particle physics (SM), massive spin-1 (Proca) fields describe the $Z$ and $W$ particles, where the former is neutral and the latter is charged. Perturbations of a neutral Proca field were first studied in~\\cite{Pawl:2004bx,Konoplya:2005hr} on spherically symmetric backgrounds. In these studies only the $\\ell=0$ mode was considered since the corresponding wave equation is directly decoupled and separated. Later on by using the Kodama-Ishibashi formalism, see Section~\\ref{sc:KI} for details, we were able to obtain a set of equations on $D$-dimensional spherically symmetric backgrounds, to cover all the possible modes for both neutral and charged Proca fields. This was done in the context of Hawking radiation and quasi-bound states~\\cite{Herdeiro:2011uu,Wang:2012tk,Sampaio:2014swa}. Similar equations for neutral Proca fields on Schwarzschild BHs were shortly after obtained using different perturbation variables, and these equations were applied to study quasinormal modes and quasi-bound states~\\cite{Rosa:2011my}. In the Kerr background, separation of variables for Proca fields can be only achieved in the slow rotation limit, and the corresponding coupled wave equations were obtained in~\\cite{paniPRL,paniPRD}, to study superradiant instabilities. A fully numerical study of the BH-Proca system was recently implemented~\\cite{Zilhao:2015tya}, in nonrotating spacetimes.\n\nIn this chapter, we present a complete set of Proca equations on spherically symmetric backgrounds, by using the Kodama-Ishibashi formalism introduced in Section~\\ref{sc:KI}.\n\n\nWe start by describing scalar and Proca fields, which may be complex and charged under a $U(1)$ electromagnetic field, with the Lagrangian\n\\begin{equation}\n\\mathcal{L}=-(\\mathcal{D}_\\mu\\Psi)^\\ast\\mathcal{D}^\\mu\\Psi-\\mu_s^2\\Psi^\\ast\\Psi -\\dfrac{1}{2}W^\\dagger_{\\mu\\nu}W^{\\mu\\nu}-\\mu_p^2W_\\mu^\\dagger W^\\mu-iqW_\\mu^\\dagger W_\\nu F^{\\mu\\nu} \\;,\\label{Lagrangian}\n\\end{equation}\nwhere $W_{\\mu\\nu}=\\mathcal{D}_\\mu W_\\nu-\\mathcal{D}_\\nu W_\\mu$, $\\mathcal{D}_\\mu\\equiv \\partial_\\mu-i q A_\\mu$ and the field charge is $q$\\footnote{In general, scalar and Proca fields may have different charge. Since they will be studied separately, for simplicity here we denote both charges by the same symbol $q$.}. Scalar and Proca fields are denoted by $\\Psi$ and $W_\\mu$, with mass $\\mu_s$ and $\\mu_p$, respectively. As one observes from the above Lagrangian, both scalar and Proca fields are coupled to the electromagnetic potential $A_\\mu$ through the gauge covariant derivative, while the latter one also couples to the electromagnetic field strength tensor $F_{\\mu\\nu}=\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu$, as determined by gauge invariance in the SM.\n\n\n\nThe equations of motion for Proca fields, when all the background fields are fixed, are\n\\begin{equation}\n\\dfrac{1}{\\sqrt{-g}}\\mathcal{D}_\\nu\\left(\\sqrt{-g}W^{\\mu\\nu}\\right)+\\mu_p^2W^\\mu+iqW_\\nu F^{\\mu\\nu}=0 \\;.\\label{Procaeq}\n\\end{equation}\nFor the gravitational background with an Einstein symmetric space shown in Eq.~\\eqref{KI:metric}, the Kodama-Ishibashi formalism~\\cite{Kodama:2000fa} can be applied.\nFor a Proca field $W_\\mu=\\{W_a,W_i\\}$, $W_a$ are $m$-scalars, with respect to the Einstein space $\\sigma_n$, so they must obey the scalar eigenvalue equations~\\eqref{intro:eigenscalar}. $W_i$ is a covector field, so it can be decomposed into a scalar $\\Phi$ obeying the scalar eigenvalue equation~\\eqref{intro:eigenscalar}, and a transverse covector $W^{T}_i$ obeying the vector eigenvalue equation~\\eqref{intro:eigenvector}.\n\n\nThis decomposition allows for an expansion of the various degrees of freedom $\\{W_a,\\Phi,W^T_i\\}$ in a basis of harmonics of the Einstein space.\nFurthermore, this decomposition allows for a decoupling of the field equations into an independent vector mode $W^T_i$ and $m+1$ coupled scalar fields for each set of quantum numbers labeling the basis of harmonic functions. Observe, however, that not all these modes correspond to physically independent degrees of freedom, as we will show.\n\\\\\nIn the following we shall consider, separately, two cases according to the scalar eigenvalue, i.e. $\\kappa_s$.\n\n\\section{Modes with $\\kappa_s\\neq 0$}\nExpanding the field equations~\\eqref{Procaeq}, with the decomposition~\\eqref{intro:decom} for $W_i$, and using conditions~\\eqref{intro:eigenscalar} for $\\{W_a,\\Phi\\}$ and~\\eqref{intro:eigenvector} for $W^T_i$, we obtain\n\\begin{align}\n&\\dfrac{\\kappa_s^2}{r^2}B_a+\\dfrac{h_{af}}{r^n}\\mathcal{D}_b\\left[h^{db}h^{cf}r^n\\left(\\mathcal{D}_cB_d-\\mathcal{D}_dB_c\\right)\\right]-\\mu_p^2B_a-\\mathcal{D}_a\\left[\\dfrac{1}{r^{n-2}}\\mathcal{D}_b\\left(r^{n-2}h^{bc}B_c\\right)\\right]+iqB_bF_a^{\\;\\;b}\\nonumber\\\\&=0 \\ , \\hspace{12mm}\\label{Ba}\\\\\n&\\dfrac{1}{r^{n-2}}\\mathcal{D}_a\\left(r^{n-2}h^{ab}B_b\\right)-\\mu_p^2\\Phi=0 \\ , \\label{Ge:PHIeq}\\\\\n&\\left[\\dfrac{1}{r^2}\\left(\\kappa_v^2+\\dfrac{\\hat{R}}{n}\\right)+\\mu_p^2+\\dfrac{1}{r^{n-2}}\\mathcal{D}_a\\left(r^{n-2}h^{ab}\\mathcal{D}_b\\right)\\right]\\hat\nW^{Tj}=0 \\label{WT}\\; ,\n\\end{align}\nwith definition\\footnote{Here we use the variables $B_a$, since $\\Phi$ can be determined by $B_a$ and becomes a non-dynamical degree of freedom, as shown in Eq.~\\eqref{Ge:PHIeq}.} $B_a\\equiv W_a-\\mathcal{D}_a\\Phi$, and $\\hat{W}^{Ti}=r(y)^2W^{Ti}$.\n\nWe consider the spherically symmetric case, by specifying the metric in Eq.~\\eqref{KI:metric} with $\\{y^a\\}=\\{t,r\\}$,\n$|h|=1$, $h_{ab}$ is diagonal,\n$h_{tt}=-1\/h_{rr}\\equiv -V$. Since $\\Phi$ is given by the second equation in terms of the other fields, it is a non-dynamical degree of freedom. In four dimensions, this agrees with the fact that a spin-1 massive field has three possible physical polarizations which in this case will be the two dynamical scalars and the transverse vector. In higher dimensions we will see that the transverse vector on the $n$-sphere contains more (degenerate) polarizations.\n\nWe can factor out the spherical harmonics through the decomposition\n\\begin{align}\nB_a&=\\beta_a^\\Lambda(y)\\mathcal{Y}_\\Lambda(x)\\ , \\nonumber \\\\\n\\hat{W}^{T i}&=q^\\Lambda(y)\\mathcal{Y}^i_{\\Lambda}(x) \\; ,\\label{Lexpansion}\n\\end{align}\nwhere $\\mathcal{Y}_\\Lambda$ is the scalar harmonic function with the definition~\\eqref{intro:ScalatrHarmonicFunction}, $\\mathcal{Y}^i_\\Lambda$ is the vector harmonic function with the definition~\\eqref{intro:HarmonicVector}, and $\\Lambda$ denotes the mode eigenvalues for the corresponding harmonic functions. Furthermore, making the ansatz\n\\[\n\\beta^{\\Lambda}_t=e^{-i\\omega t}\\psi(r) \\ , \\qquad \\beta^{\\Lambda}_r=e^{-i\\omega t}\\dfrac{\\chi(r)}{V} \\ , \\qquad q^{\\Lambda}=e^{-i\\omega t}\\Upsilon(r)\\ ,\n\\]\n and using Eqs.~\\eqref{Ba} and~\\eqref{WT}, we obtain\\footnote{Note that there is a symmetry for the coupled system, i.e. for real $\\omega$, if $(\\psi,\\chi)$ is a solution to the equations, $(-\\psi^\\ast,\\chi^\\ast)$ is also a solution. For complex $\\omega$, this statement still can be made as: if $(\\psi,\\chi)e^{-i\\omega t}$ is a solution to the equations, then $(-\\psi^\\ast,\\chi^\\ast)e^{-i\\omega^\\ast t}$ is also a solution.}\n\\begin{align}\n&\\left[V^2\\dfrac{d}{dr}\\left(\\dfrac{1}{r^{n-2}}\\dfrac{d}{dr}r^{n-2}\\right)+(\\omega+qA_t)^2-\\left(\\dfrac{\\kappa_s^2}{r^2}+\\mu_p^2\\right)V \\right]\\chi-i\\left((\\omega+qA_t) V^{\\prime}+2qA_t\\dfrac{V}{r}\\right)\\psi\\nonumber\\\\& =0\\ ,\\label{originalsys1}\\\\\n&\\left[\\dfrac{V^2}{r^n}\\dfrac{d}{dr}\\left(r^n\\dfrac{d}{dr}\\right)+(\\omega+qA_t)^2-\\left(\\dfrac{\\kappa_s^2}{r^2}+\\mu_p^2\\right)V\\right]\\psi+i\\left(\\dfrac{2\\omega V}{r}-(\\omega+qA_t)V^{\\prime}\\right)\\chi \\nonumber\\\\&=0 \\ ,\\label{originalsys2} \\\\\n&\\left[\\dfrac{V}{r^{n-2}}\\dfrac{d}{dr}\\left(r^{n-2}V\\dfrac{d}{dr}\\right)+(\\omega+qA_t)^2-\\left(\\dfrac{\\kappa_v^2+\\frac{\\hat{R}}{n}}{r^2}+\\mu_p^2\\right)V\\right] \\Upsilon=0 \\label{transverse}\n\\; ,\n\\end{align}\nwhere $A_t$ is the only nonvanishing component of electromagnetic potential due to the spherically symmetric background. Thus we obtain two second order coupled radial equations for $\\left\\{\\psi,\\chi\\right\\}$ and a decoupled equation for $\\Upsilon$. Note that $\\kappa_s^2=\\ell(\\ell+n-1)$ and $\\kappa_v^2=\\ell(\\ell+n-1)-1$ with $\\ell$ starting at zero and one respectively. The third combination is $\\kappa_v^2+\\frac{\\hat{R}}{n}=\\ell(\\ell+n-1)+n-2$.\n\n\nThe manipulations leading to the two coupled equations above are only valid for non-zero $\\mu_p$. In the exactly massless theory, a similar calculation leads to a single decoupled equation for one of the scalar modes which is\n\\begin{equation}\n\\left[V\\dfrac{d}{dr}\\left(\\dfrac{V}{r^{n-2}}\\dfrac{d}{dr}r^{n-2}\\right)-\\dfrac{2qd_rA_tV^2}{(\\omega+qA_t)r^{n-2}}\\dfrac{d}{dr}r^{n-2}\n+\\Big(\\omega+qA_t\\Big)^2-\\dfrac{\\kappa_s^2}{r^2}V\\right] \\chi=0 \\; , \\label{Max}\n\\end{equation}\nwhereas the other mode $\\psi=i V d_r(r^{n-2}\\chi)\/((\\omega+qA_t) r^{n-2})$ becomes non-dynamical\\footnote{Note that the coupled equations have only one dynamical degree of freedom in the massless limit. One can use either $\\psi$ or $\\chi$ to describe the dynamical mode, then the other one becomes non-dynamical and is determined in terms of the dynamical mode.}. Here $d_r\\equiv d\/dr$. The transverse mode -- described by equation (\\ref{transverse}) -- remains the same for any $\\mu_p$; in particular, for $\\mu_p=0$, and (only) $n=2$ it becomes equivalent to (\\ref{Max}). This will be manifest in the numerical results.\n\\section{Modes with $\\kappa_s= 0$}\nFor the exceptional modes with $\\kappa_s=0$, $\\Phi$ does not enter the wave equation so it is a free non-dynamical field. The corresponding equation for $W^{(0)}_a$ is (the superscript denotes it is the exceptional mode)\n\\begin{equation}\n\\dfrac{h_{af}}{r^n\\sqrt{|h|}}\\mathcal{D}_b\\left[h^{db}h^{cf}r^n\\sqrt{|h|}\\left(\\mathcal{D}_cW^{(0)}_d-\\mathcal{D}_dW^{(0)}_c\\right)\\right]+\\mu_p^2W^{(0)}_a+iqW_b^{(0)}F_a^{\\;\\;b}=0 \\; .\n\\end{equation}\nWhen $\\mu_p^2\\neq 0$ one uses an ansatz similar to the previous section to obtain a radial equation for a dynamical degree of freedom\n\\begin{align}\n&\\left[\\dfrac{\\mu_p^2}{r^n}\\dfrac{d}{dr}\\left(\\dfrac{r^nV}{(\\omega+qA_t)^2-\\mu_p^2V}\\dfrac{d}{dr}\\right)+\\dfrac{\\mu_p^2}{V}+\\dfrac{\\mu_p^2qd_rA_t}\n{((\\omega+qA_t)^2-\\mu_p^2V)^2}\\Big(2qVd_rA_t-(\\omega+qA_t)V'\\Big)\\right.\\nonumber\\\\\n&\\left.-\\dfrac{\\omega+qA_t}{r^n((\\omega+qA_t)^2-\\mu_p^2V)}\n\\dfrac{d}{dr}\\Big(qr^nd_rA_t\\Big)\\right]\\psi^{(0)}=0 \\; ,\\label{k0M}\n\\end{align}\nand a non-dynamical one,\n\\begin{equation}\n\\chi^{(0)}=\\dfrac{i V}{(\\omega+qA_t)^2-\\mu_p^2V}\\left((\\omega+qA_t)d_r\\psi^{(0)}-q\\psi^{(0)}d_rA_t\\right)\\;,\\nonumber\n\\end{equation}\nwhere $d_r\\equiv d\/dr$.\nOtherwise, for $\\mu_p^2=0$, we recover the well known result that all the exceptional modes are non-dynamical (see e.g. \\cite{Konoplya:2005hr}).\n\nNow that we have covered all possibilities, several comments are in order. First there is a discrete difference between the small mass limit and the exactly massless theory since we have different sets of equations for each case. This should not be surprising since there is an extra longitudinal mode for massive vector bosons. Second, the equations for the Maxwell theory case are all decoupled, in agreement with previous work \\cite{Page:1976df}.\n\nWith all of the Proca equations at hand, we are going to apply them to study Hawking radiation in Chapter~\\ref{ch:NeutralP} for a neutral Proca field, and in Chapter~\\ref{ch:ChargedP} for a charged Proca field. In Chapter~\\ref{ch:ChargedClouds}, we will then apply them to the computation of quasi--bound states in the Reissner-Nordstr\\\"om BH.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Introduction}\n\\label{ch:intro}\n\n\n\n\n\\section{Background and motivation}\n\\label{sc:BM}\nGeneral Relativity (GR), as one of the pillars of modern physics, was established one hundred years ago by Albert Einstein in 1915. It unifies space and time, and in particular gravity is described by the curvature of spacetime. Mathematically, the theory is formulated by the elegant Einstein field equations~\\cite{Einstein:1916vd}, where the geometry is related to the distribution of matter and radiation. These equations were interpreted by John Wheeler in his famous statement that~\\textit{matter tells spacetime how to curve, and spacetime tells matter how to move}. GR has been tested with high accuracy in the regime of weak gravity~\\cite{Will:2014kxa}, while in the strong gravity regime the first direct observation on gravitational waves has been reported recently~\\cite{Abbott:2016blz}. From an analysis of the waves, black holes are identified as the source for such an event~\\cite{Abbott:2016blz}.\n\n\nThe concept of black hole (BH) can be dated back to the end of the eighteenth century. At that time, John Michell~\\cite{Michell:1784xqa} and\nPierre-Simon Laplace~\\cite{Israel:1987ae} put forward an idea that the largest bodies in the universe may be invisible since they are so massive\nthat even light could not escape, which were dubbed as \\textit{dark stars}. This idea was revived by Robert Oppenheimer and his collaborators, more than one century later, in their studies of gravitational collapse, where they concluded that neutron stars above the Tolman-Oppenheimer-Volkoff limit (approximately 1.5 to 3 solar masses) would collapse~\\cite{Oppenheimer:1939ne}. Such collapsed objects were called \\textit{frozen stars}. In 1967, the term \\textit{black hole} was introduced by John Wheeler~\\cite{Wheeler:1998vs}, and since then it was quickly accepted for general use.\n\nFrom a modern viewpoint, BHs are the simplest macroscopic objects in nature, in the sense that they can be uniquely characterized by their mass, spin and charge. This is the well known~\\textit{no hair conjecture}~\\cite{Misner:1974qy}. This conjecture has been circumvented in different contexts, for example in the Einstein-Yang-Mills theory~\\cite{Bizon:1990sr,Kuenzle:1990is,Volkov:1989fi,volkov1990black}, in the Horndeski theory~\\cite{Sotiriou:2013qea,Sotiriou:2014pfa,Babichev:2013cya} and recently in the Einstein-Klein-Gordon system where a Kerr BH with scalar hair was found~\\cite{Herdeiro:2014goa}. Among them, scalar hairy BH solutions are supported by the phenomenon of superradiance~\\cite{Brito:2015oca}, which provides a mechanism to generate hairy BH solutions in general~\\cite{Herdeiro:2014goa,Herdeiro:2014ima}. Another interesting phenomenon in BH physics is the Hawking radiation~\\cite{Hawking:1974rv}. Hawking radiation has been attracting a lot of attention, not only because it relates gravity to quantum theory which may provide a connection to a \\textit{quantum theory} of gravity, but also because it might be visible in high energy processes.\n\n\n\n\n\n\nIn this thesis, we are going to study BHs interacting with scalar and vector fields at the linear level, in the context of \\textit{Hawking radiation}\nand \\textit{superradiance}. The motivation for these studies is as follows.\n\n{\\bf TeV gravity scenarios}\\\\\nThe study of gravitational theories in higher dimensions has been discussed for a century, at least since the works by Nordstr\\\"om~\\cite{Nordstrom:1988fi}, Kaluza~\\cite{Kaluza:1921tu} and Klein~\\cite{Klein:1926tv}. During the last four decades, moreover, the naturalness of extra dimensions within supergravity and string theory made it a topic of intense research within high energy theoretical physics. At the end of the last century, this research led to models that, aiming at solving the hierarchy problem\\footnote{The hierarchy problem refers to the relative weakness of gravity, around sixteen orders of magnitude, by comparing to the other fundamental interactions.}, predicted that the extra dimensions could be very large (or even infinite) in size, as compared to the traditional Planck scale.\n\nWithin such scenario, the true fundamental Planck scale could be as low as the TeV scale~\\cite{Antoniadis:1990ew,Arkani-Hamed:1998rs,Antoniadis:1998ig,Arkani-Hamed:1998nn} so that the formation and evaporation of microscopic BHs could be visible in realistic man-made particle accelerators\\footnote{See, for example, the latest reports from CMS~\\cite{CMS:2015iwr} and ATLAS~\\cite{Aad:2015mzg} in the search for microscopic BHs.}, such as the Large Hadron Collider (LHC). This motivates our study on Hawking radiation. In particular, the second run of the LHC is ongoing, with a center of mass energy of 13 TeV, where such scenarios will be properly tested. Therefore, any improvements on the phenomenology of these models are quite timely.\n\n{\\bf Asymptotically anti-de Sitter spacetimes}\n\\\\\nAnti-de Sitter (AdS) spacetime is the unique maximally symmetric solution of the vacuum Einstein equations with a negative cosmological constant. Asymptotically AdS spacetimes, referring to spacetimes which share the conformal boundary with AdS but may be different in the bulk, have attracted a lot of attention in theoretical physics. One reason is the AdS\/CFT correspondence~\\cite{Maldacena:1997re} which conjectures a duality between gravity in the $d$-dimensional AdS bulk and a quantum conformal field theory living in the $(d-1)$-dimensional conformal boundary. Another reason is the timelike property of the AdS boundary, which leads to interesting novel features, as compared to asymptotically flat spacetimes, such as the weak turbulent instability~\\cite{Bizon:2011gg} and the superradiant instability for massless fields~\\cite{Cardoso:2013pza,Uchikata:2009zz,Wang:2015fgp}.\n\nWith these motivations in mind, in the following we will briefly describe the physical phenomena of Hawking radiation and superradiance.\n\n\n\n\n\n\n\n\\section{Hawking radiation}\n\\label{sc:HR}\nHawking radiation~\\cite{Hawking:1974rv} describes black body radiation that is predicted to be released by BHs, due to quantum effects\nclose to the event horizon. It is one of the most important features arising from quantum field theory in curved spacetime, discovered by Hawking \\mbox{in 1974}. This effect was derived in a semiclassical framework, in the sense that the background geometry is classical (governed by classical gravitational theories) while the propagating fields are quantized. Since Hawking radiation connects classical gravity with quantum theory, it has inspired many works to re-derive Hawking\nradiation through alternative methods, see for example~\\cite{Hawking:1974sw,Hartle:1976tp,Damour:1976jd,Parikh:1999mf,Robinson:2005pd}, with an expectation to get a deeper understanding of gravity itself.\n\nPhysically, Hawking radiation can be understood through an intuitive picture by considering virtual particles generated from the vacuum\\footnote{Note that this interpretation may lead to a flawed intuition on where does Hawking radiation originate~\\cite{Giddings:2015uzr}.}. As it is well known since Dirac, the quantum vacuum is not completely empty but it contains fluctuations which produce particle-antiparticle pairs. Close to the event horizon of a BH, strong gravity effects may separate particle-antiparticle pairs, and if the antiparticle is attracted into the interior of the hole then the particle can escape to infinity thus generating Hawking radiation.\n\nThe phenomenon of Hawking radiation found recently an interesting application in TeV gravity models. In such models, scattering processes with center of mass\nenergy well above \\mbox{the fundamental} Planck scale, should be dominated by classical gravitational interactions~\\cite{'tHooft:1987rb}. Then, for sufficiently small impact parameter, miniature BHs should form in particle collisions, and in particular, Hawking radiation would be the main observable signature~\\cite{Banks:1999gd,Dimopoulos:2001hw}. This motivation led to an intensive study of Hawking radiation from higher-dimensional BHs--\nsee~\\cite{Kanti:2014vsa} for a recent review and reference therein.\n\n\nIf a microscopic BH is produced, it is expected that the decay process can be modeled by the following four phases~\\cite{Giddings:2001bu}, namely\n\\begin{itemize}\n\\item \\textit{Balding phase}: all original ``hair'' inherited from incoming particles (except mass and angular momentum) is lost through gravitational and Hawking\nradiation and, at the end of this stage, the BH is axisymmetric and rotating.\n\\item \\textit{Spin-down phase}: then the BH emits Hawking radiation, losing mass and angular momentum evolving towards the end, into a spherically symmetric BH.\n\\item \\textit{Schwarzschild phase}: the spherically symmetric BH continues to radiate losing its mass, until it reaches the Planck scale.\n\\item \\textit{Planck phase}: the semiclassical approximation of Hawking radiation becomes invalid at this stage, and quantum gravity starts to play a significant role in the BH emission process.\n\\end{itemize}\nIt is believed that the spin-down and Schwarzschild phases will dominate the lifetime of the BH, therefore they are the most promising stages to generate\nobservational signatures of Hawking radiation. Indeed these phases have been modeled in BH event generators, such as \\textsc{charybdis2}\n~\\cite{Frost:2009cf,sampaio2010production} and~\\textsc{blackmax}~\\cite{Dai:2007ki} currently in use at the LHC. These event generators, however, can still be improved.\n\n\n\n\nOne of the Hawking radiation channels that has not been properly addressed in the literature is that of massive vector bosons, both electrically neutral and electrically charged, to describe the emission of $Z$ and $W^{\\pm}$ particles of the Standard Model. As our first goal in this thesis, we are going to study Hawing radiation\nfor both a neutral and a charged Proca field in higher dimensions, to bridge this gap.\n\n\n\n\\section{Superradiance}\n\\label{sc:SR}\nSuperradiance is a phenomenon which refers to a radiation enhancement process in several physical contexts. This term was coined by Dicke, to describe an effect in quantum optics that radiation of a group of emitters could be amplified in a coherent fashion~\\cite{Dicke:1954zz}. In 1971, Zel'dovich~\\cite{zeldovich1,zeldovich2} pointed out that scalar and electromagnetic radiation, impinging on a rotating cylinder with absorbing surfaces\ncan be amplified if the condition\n\\begin{equation}\n\\omega<m\\Omega\\;, \\label{consup}\n\\end{equation}\nholds. Here $\\omega$ and $m$ are the waves' frequency and the azimuthal number, and $\\Omega$ is the angular velocity of the cylinder.\n\nSince the event horizon of a BH provides a natural absorbing surface at the classical level, Zel'dovich anticipated that a rotating BH with horizon\nangular velocity $\\Omega_H$ should display superradiant amplification within the regime given by Eq.~\\eqref{consup}. This was indeed observed by Misner in\nunpublished calculations, who found that certain modes of scalar fields are amplified by Kerr BHs, as a wave analogue of the Penrose process~\\cite{Penrose:1969pc}. This work was then generalized and verified by Teukolsky and Press~\\cite{Teukolsky:1974yv} who found that the amplification process also occurs for electromagnetic and gravitational waves on the same background. According to these observations, Bekenstein~\\cite{Bekenstein:1973mi} realized that in order to satisfy Hawking's area theorem~\\cite{Hawking:1971tu}, superradiant amplification is classically required when condition~\\eqref{consup} holds. From the same reasoning he also derived that superradiance of charged bosonic waves by a charged BH exists when~\\cite{Bekenstein:1973mi}\n\\begin{equation}\n\\omega<q\\Phi_H\\;,\\label{consup2}\n\\end{equation}\nis satisfied, where $q$ is the field charge and $\\Phi_H$ is the electrostatic potential at the horizon.\n\nThis story changes dramatically for fermion fields. As first shown by Unruh~\\cite{Unruh:1973bda}, superradiance is absent for neutral massless Dirac fields on the Kerr background. This conclusion was then generalized to massive Dirac fields~\\cite{Iyer:1978du} (by correcting a previous work~\\cite{Martellini:1977qf}) and on the Kerr-Newman background~\\cite{Lee:1977ti}. Absence of superradiance for fermions originates from the fact that the net number current flowing down the horizon is positive definite, which implies that it is impossible for fermion fields to extract energy and angular momentum from the hole~\\cite{Unruh:1973bda}. Moreover, the argument~\\cite{Bekenstein:1973mi} based on the area theorem does not apply to a fermion field since its energy-momentum tensor does not obey the weak energy condition~\\cite{Hawking:1974sw}. At the quantum level, it can also be understood as a consequence of the exclusion principle which does not allow for more than one particle in each outgoing wave and therefore the scattered wave can not be stronger than the incident wave~\\cite{Hawking:1974sw}.\n\nBy placing a reflecting mirror around a rotating BH, the system composed with bosonic fields may become unstable. This was first addressed by Press and\nTeukolsky~\\cite{Press:1972zz}, and was dubbed as \\textit{black hole bomb}. The role of the mirror is to feed back to the BH the amplified scattered wave,\nas to recurrently extract rotational energy. Then, the wave bounces back and forth between the mirror and the BH until radiation pressure destroys the mirror.\nIn fact, the reflecting mirror is not necessarily artificial, and it has several realizations in nature. One realization is the field's mass. For a massive\nbosonic field with mass $\\mu$ satisfying the bound state condition $\\omega < \\mu$, the mass term can provide a confining mechanism similar to a\nmirror~\\cite{Damour:1976kh,Detweiler:1980uk,Zouros:1979iw,Dolan:2007mj,Rosa:2009ei,Hod:2009cp,Witek:2012tr,Rosa:2012uz,Pani:2012vp,Hod:2012zza}. Another realization is AdS asymptotics, which may also bind superradiant modes~\\cite{Cardoso:2004hs,Cardoso:2006wa,Aliev:2008yk,Dias:2011at,Li:2012rx,Uchikata:2011zz,Cardoso:2013pza}.\nRecently a considerable interest has been devoted to study superradiant scattering and instabilities using numerical methods at the non-linear level\n\\cite{Cardoso:2012qm,Yoshino:2012kn,Witek:2012tr,Yoshino:2013ofa,East:2013mfa,Okawa:2014nda,Dolan:2015dha,Sanchis-Gual:2015lje,Bosch:2016vcp}.\n\nSince superradiance also exists for charged BHs under the condition in Eq.~\\eqref{consup2}, it is natural to ask if charged superradiant modes can be confined by the field's mass and the AdS asymptotics as in the rotating case, and generate instabilities by extracting Coulomb energy and BH charge. As proved by Hod~\\cite{Hod:2013nn,Hod:2013eea,Hod:2015hza} for the scalar case, the field's mass \\textit{cannot} bound the superradiant modes since the superradiance\ncondition and the bound state condition for a charged massive scalar field in an asymptotically flat charged BH cannot be satisfied simultaneously. This statement has been generalized to charged Proca fields recently, showing that Reissner-Nordstr\\\"om BHs are also stable against those massive vector fields~\\cite{Sampaio:2014swa}. There is still a possibility, however, to circumvent Hod's results by considering a non-minimal coupling between the field and the background since Hod's results only hold for minimally coupled scalar fields~\\cite{Hod:2013eea,Hod:2015hza}. At this moment it might be safe to state that a charged BH bomb may be only achieved with AdS asymptotics~\\cite{Uchikata:2011zz,Dias:2011tj,Wang:2014eha}\\footnote{Alternatively, a charged BH bomb in an asymptotically flat spacetime can be also built artificially by imposing mirror-like boundary conditions~\\cite{Herdeiro:2013pia,Hod:2013fvl,Degollado:2013bha,Sanchis-Gual:2015lje}.}.\n\nThe phenomenon of superradiance is interesting for various reasons\\footnote{To find more applications of superradiance, we refer to a recent review on this\ntopic~\\cite{Brito:2015oca}.}. Among others, we would like to mention that superradiance provides a systematic mechanism to construct \\mbox{(quasi-) stationary} BH-field systems at the linear level, and generates hairy BH solutions at the nonlinear level~\\cite{Herdeiro:2014goa,Herdeiro:2014ima}. As we mentioned before, in an asymptotically flat BH, the superradiant instability is absent for a charged massive bosonic field. But as shown in~\\cite{Degollado:2013eqa,Sampaio:2014swa}, one can still find quasi-stationary states for such a BH-field system, when gravity is balanced with the electric field. Adding a mirror, as recently shown in~\\cite{Dolan:2015dha}, a charged hairy BH solution was found. An even more interesting result was obtained on a rotating BH. As we explained above, the field mass can be used as a mirror and therefore the superradiant instability is present for a massive scalar field on a Kerr BH. At the linear level, stationary states can be found in such\na system~\\cite{Hod:2012px,Benone:2014ssa,Herdeiro:2014pka}, which are dubbed as~\\textit{clouds}. Considering the back reaction of a complex scalar field, scalar hairy Kerr solutions have been found in~\\cite{Herdeiro:2014goa,Herdeiro:2015tia,Herdeiro:2015gia}.\n\nBecause the AdS asymptotics is another realization of the mirror, one may expect similar effects in an asymptotically AdS spacetime. Indeed the superradiant\ninstability for scalar and gravitational fields has been studied on a Kerr-anti-de Sitter (Kerr-AdS) BH~\\cite{Uchikata:2009zz,Cardoso:2013pza}. In this thesis, we will complete this study by exploring the superradiant instability for the Maxwell field. Furthermore, since such an instability exists in AdS, one expects to find clouds for various fields on Kerr-AdS BHs. The only studied case before this thesis on a Kerr-AdS BH was the gravitational field~\\cite{Cardoso:2013pza}. To fill this gap, we will present studies of scalar and vector clouds on the same background.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Structure}\n\\label{sc:ST}\nThe structure of this thesis is as follows. In Chapter~\\ref{ch:prelim}, we review two important perturbative methods, the Kodama-Ishibashi formalism and the\nNewman-Penrose formalism, which set the foundations for the other chapters. In Chapter~\\ref{ch:GeneralInFlat}, we employ the Kodama-Ishibashi formalism to derive\nequations of motion for a Proca field (either neutral or charged), on a $D$-dimensional spherically symmetric background. Then these equations are used to study\na quantum semi-classical effect on BHs, i.e. Hawking radiation, for a neutral Proca field on a $D$-dimensional Schwarzschild BH in Chapter~\\ref{ch:NeutralP}.\nFollowing this line, in Chapter~\\ref{ch:ChargedP} we explore Hawking radiation for a charged Proca field in a brane charged BH. We turn to classical properties in Chapter~\\ref{ch:ChargedClouds}, to study quasi-bound states for a charged Proca field in a Reissner-Nordstr\\\"om BH. In Chapter~\\ref{ch:scalarHD}, we study\nsuperradiant instabilities for a charged scalar field in a $D$-dimensional Reissner-Nordstr\\\"om-AdS BH. This is then generalized to the Maxwell field on\na Kerr-AdS BH, leading to the study of superradiant instabilities and vector clouds, in Chapter~\\ref{ch:KerrAdS}. Conclusions and outlook are drawn in\nChapter~\\ref{ch:conclusion}.\n\n\n\\chapter{Maxwell perturbations on Kerr-AdS black holes}\n\\label{ch:KerrAdS}\n\n\n\\section{Introduction}\n\\label{Maxse:intro}\n\nThe global structure of asymptotically AdS spacetimes allows interesting novel features, as compared to asymptotically flat spacetimes. For instance, when a rotating BH exists in the bulk of an asymptotically AdS spacetime, superradiant instabilities can be triggered by a massless field, in contrast to the asymptotically flat case wherein such instabilities only arise for massive fields. Recently scalar~\\cite{Uchikata:2009zz} and gravitational perturbations~\\cite{Cardoso:2013pza} on Kerr-AdS BHs have been addressed, and have been shown to exhibit a rich physics.\n\nIn this chapter we complete the analysis of bosonic fields on this background by studying Maxwell perturbations. To do so, one has to assign physically relevant boundary conditions which depend on the specific problem. In the context of quasinormal modes in asymptotically AdS BHs, the most studied perturbations are those of scalar fields, for which \\textit{field vanishing} boundary conditions are usually imposed, see e.g.~\\cite{Uchikata:2009zz}. For other spin fields, the problem has only been partly addressed. The quasinormal modes for the Maxwell field and gravitational field on Schwarzschild-AdS BHs have been obtained using the Regge-Wheeler method~\\cite{Regge:1957td}, instead of the Teukolsky equation, in~\\cite{Cardoso:2001bb,Cardoso:2003cj}, exploring the spherical symmetry of the background. Furthermore, these works impose field vanishing boundary conditions. For non-spherically symmetric backgrounds, like in Kerr-AdS BHs, one must, however, use the Teukolsky formalism  and, since this formalism uses a different set of variables, it is not obvious how to impose boundary conditions for non-zero spin fields. Recently, superradiant instabilities of the gravitational field on Kerr-AdS BHs have been studied~\\cite{Cardoso:2013pza} with boundary conditions chosen as to preserve the asymptotic global AdS structure of the background~\\cite{Dias:2013sdc}. The general principle to impose boundary conditions for arbitrary spin fields on Kerr-AdS BHs, however, is still missing.\n\n\n\n\nThe AdS boundary may be regarded as a perfectly reflecting mirror, in the sense that no energy flux can cross it. We will take this viewpoint as our basic principle to impose boundary conditions for linear perturbations of asymptotically AdS spacetimes. It suggests taking \\textit{vanishing energy flux} (VEF) boundary conditions, which should be contrasted to the \\textit{field vanishing} boundary conditions we mentioned before. We will first illustrate how this simple physical principle, can lead to two different Robin boundary conditions, using the Maxwell field as an example.\n\nThese boundary conditions are then used to study quasinormal modes, superradiantly unstable modes, and vector clouds. To address superradiant instabilities triggered by the Maxwell field on Kerr-AdS BHs, we use both an analytical matching scheme as well as a numerical method to explore the problem. The former method provides an intuitive way to understand how these two boundary conditions produce different instabilities, in the small BH and slow rotation regime. The latter method provides a technique to understand the problem in a larger region of the parameter space. We find that Maxwell fields can trigger stronger instabilities than scalar fields~\\cite{Uchikata:2009zz} in the superradiant regime, for both boundary conditions.\n\nStationary clouds~\\cite{Hod:2012px} are bound \\mbox{state} solutions of test fields on a rotating background, at the linear level. They exist at the threshold of the superradiant instabilities triggered by that test field. Recently, a considerable number of studies of such clouds has \\mbox{appeared} in the literature, mostly in asymptotically flat spacetimes~\\cite{Hod:2012px,Hod:2013zza,Shlapentokh-Rothman:2013ysa,Herdeiro:2014goa,Hod:2014baa,Benone:2014ssa,Hod:2014sha,Hod:2014pza,Herdeiro:2014pka,Benone:2014nla,Hod:2015ota,Wilson-Gerow:2015esa,Herdeiro:2015gia,Hod:2015ynd}, but also in asymptotically AdS \\mbox{spacetimes}~\\cite{Dias:2011at,Cardoso:2013pza}. Most of these studies have addressed scalar field clouds\\footnote{Superradiance onset curves for gravitational perturbations on Kerr-AdS, which --  as the existence lines for stationary clouds -- identify the backgrounds supporting the zero-mode of the perturbation,  have been studied in~\\cite{Cardoso:2013pza}, and the corresponding ``hairy'' BH solutions have been constructed in~\\cite{Dias:2015rxy}.} (even though \\textit{marginal} clouds have been considered for a charged Proca field on a charged BH background in Chapter~\\ref{ch:ChargedClouds}). Here we perform a study of Maxwell clouds, which can exist around rotating BHs in asymptotically AdS spacetimes. As a comparison with the Maxwell clouds on Kerr-AdS BHs we also consider scalar clouds in the same background.\n\nIt was proposed in~\\cite{Herdeiro:2014goa,Herdeiro:2014ima} that the existence of stationary clouds of a given test field, as a \\textit{zero-mode} of the superradiant instability, indicates the existence of new families of ``hairy'' BH solutions, at fully nonlinear level, such as the Kerr BHs with scalar hair found (numerically) in ~\\cite{Herdeiro:2014goa}, whose existence was recently formally proved~\\cite{Chodosh:2015oma}. It is an open issue if these hairy BHs may be formed dynamically, as the end point of the instability. Interesting evidence in this direction was reported recently~\\cite{Dolan:2015dha} for the case of Reissner-Nordstr\\\"om BHs in a cavity, following the earlier discussion of superradiant instabilities in this setup~\\cite{Herdeiro:2013pia,Hod:2013fvl,Degollado:2013bha}. The existence of Maxwell clouds on Kerr-AdS BHs has, therefore, the interesting implication that new families of solutions of charged rotating BHs exist, within the Einstein-Maxwell-AdS system, besides the well known Kerr-Newman-AdS family, and branching off from the latter.\n\n\n\n\nThe structure of this chapter is as follows. In Section~\\ref{Maxsec:eqs} we present the Teukolsky equations for the Maxwell field on Kerr-AdS BHs. Then the corresponding boundary conditions are studied in Section~\\ref{Maxsec:BC}, by requiring VEF. In Section~\\ref{Maxsec:AM}, a standard analytical matching method is developed to study quasinormal modes in the small BH and slow rotation regime. In Section~\\ref{Maxsec:NM}, we introduce two numerical methods, to solve the radial and angular Teukolsky equations, respectively, in a larger parameter space. In Section~\\ref{Maxsec:SAdS}, we take the Schwarzschild-AdS BH as an example to show, even in this simpler case, that there is a new branch of quasinormal modes which has not been explored yet. In Section~\\ref{Maxsec:instability}, numerical results for quasinormal modes, superradiant unstable modes and vector clouds are presented. As a comparison, scalar clouds are studied as well. We summarize our results in the last section.\n\n\n\\section{Teukolsky equations of the Maxwell field}\n\\label{Maxsec:eqs}\nIn the Newman-Penrose formalism, the Maxwell fields are described in terms of three complex scalars, two of which are independent. These two scalars are denoted by $\\phi_0$ and $\\phi_2$, and can be expanded as\n\\begin{equation}\n\\phi_0=R_{+1}(r)S_{+1}(\\theta)\\;,\\;\\;\\;\\;\\;\\;\\phi_2=\\dfrac{1}{2(\\bar{\\rho}^\\ast)^2}R_{-1}(r)S_{-1}(\\theta)\\;,\\label{fielddeccom}\n\\end{equation}\nwhere $\\bar{\\rho}^\\ast=r-ia\\cos\\theta$, and the radial functions $R_{\\pm1}$ obey Eqs.~\\eqref{Rpluseq} and~\\eqref{Rminuseq} while the angular functions $S_{\\pm1}$ obey Eqs.~\\eqref{Spluseq} and~\\eqref{Sminuseq}.\n\nIn the following we present these equations (both the radial and angular equations) explicitly, governing a spin $s (s=\\pm1)$ perturbation.\n\\\\\nThe radial equation is\n\\begin{equation}\n\\Delta_r^{-s}\\dfrac{d}{dr}\\left(\\Delta_r^{s+1}\\dfrac{d R_{s}(r)}{dr}\\right)+H(r)R_{s}(r)=0\\;,\\label{radialeq}\n\\end{equation}\nwith\n\\begin{eqnarray}\nH(r)=\\dfrac{\\Xi^2K^2-is\\Xi K\\Delta_r^\\prime}{\\Delta_r}+2is\\Xi K^\\prime+\\dfrac{s+|s|}{2}\\Delta_r^{\\prime\\prime}\n+\\dfrac{a^2}{L^2}-\\lambda\\;,\\nonumber\n\\end{eqnarray}\nwhere $K$ is given in Eq.~\\eqref{KHeq}.\nThe angular equation is\n\\begin{equation}\n\\dfrac{d}{du}\\left(\\Delta_u\\dfrac{dS_{lm}}{du}\\right)+A(u)S_{lm}=0\\;,\\label{angulareq}\n\\end{equation}\nwith $u=\\cos\\theta$, and\n\\begin{equation}\nA(u)=-\\dfrac{K_u^2}{\\Delta_u}-4smu\\dfrac{\\Xi}{1-u^2}+\\lambda-|s|-2(1-u^2)\\dfrac{a^2}{L^2}\\;,\\nonumber\n\\end{equation}\nwhere\n\\begin{align}\nK_u&=\\Big(\\omega a (1-u^2)+(s u-m)\\Big)\\Xi\\;,\\nonumber\\\\\n\\Delta_u&=(1-u^2)\\left(1-\\dfrac{a^2}{L^2}u^2\\right)\\;.\\nonumber\n\\end{align}\n\n\n\n\\section{Boundary conditions}\n\\label{Maxsec:BC}\n\nTo solve a differential equation, such as the radial equation~\\eqref{radialeq} or the angular equation~\\eqref{angulareq}, one has to impose physically relevant boundary conditions. For the angular equation~\\eqref{angulareq}, one usually requires its solutions to be regular at the singular points $\\theta=0$ and $\\theta=\\pi$. This determines uniquely the set of angular functions labeled by $\\ell$ and $m$. For the radial equation~\\eqref{radialeq}, we have to impose conditions both at the horizon and at infinity. At the horizon, ingoing boundary conditions are imposed. At infinity, however, the boundary conditions are more subtle. For the often studied case of a scalar field on Kerr-AdS BHs, the boundary conditions typically imposed require the field itself to vanish~\\cite{Uchikata:2009zz,Cardoso:2003cj}, when looking for quasinormal modes. For the Maxwell field and in the Teukolsky formalism, the asymptotic boundary conditions have not been explored yet. In this section, we are going to discuss them for the general Kerr-AdS background.\n\nWe propose that in the Teukolsky formalism, when looking for quasinormal modes of the Maxwell field on Kerr-AdS BHs, VEF boundary conditions should be imposed, following the spirit that the AdS boundary is a perfectly reflecting mirror so that no energy flux can cross it. For the particular case of the electromagnetic field, these boundary conditions create an analogy between the AdS boundary and a perfect conductor. In fact the conductor condition for the Maxwell field has been considered in the Kerr-mirror system~\\cite{Brito:2015oca}. But the VEF boundary conditions, which for a scalar field can yield both standard Dirichlet and Neumann boundary conditions and for a Maxwell field can yield perfectly conducting boundary conditions, are a  general principle for any spin field, based on a sound physical rationale.\n\nThe energy-momentum tensor for the Maxwell field is\\footnote{Note that the energy-momentum tensor for the Maxwell field used in Eq.~\\eqref{EMTensor} has a minus sign difference with the usual definition.}\n\\begin{equation}\nT_{\\mu \\nu}=F_{\\mu\\sigma}F^\\sigma_{\\;\\;\\;\\nu}+\\dfrac{1}{4}g_{\\mu\\nu}F^2\\;,\\label{EMTensor}\n\\end{equation}\nwith the Maxwell tensor $F_{\\mu\\nu}$~\\cite{Teukolsky:1973ha}\n\\begin{equation}\nF_{\\mu\\nu}=2\\left(\\phi_1(n_{[\\mu} l_{\\nu]}+m_{[\\mu} \\bar{m}_{\\nu]})+\\phi_2 l_{[\\mu} m_{\\nu]}+\\phi_0 \\bar{m}_{[\\mu} n_{\\nu]}\\right)+{\\rm c.c}\\;,\\nonumber\n\\end{equation}\nwhere square brackets on subscripts stand for anti-symmetrization, the tetrad $\\{l_\\mu, n_\\mu, m_\\mu, \\bar{m}_\\mu\\}$ is given in Eq.~\\eqref{pre:tetrad}, and c.c stands for complex conjugate of the preceding terms.\n\nWith the Maxwell scalars defined in Eq.~\\eqref{Maxwellscalars}, we are now able to calculate the radial energy flux $T^r_{\\;\\;t}$, by substituting all of the above ingredients into Eq.~\\eqref{EMTensor}, which gives\n\\begin{equation}\nT^r_{\\;\\;t}=T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral1}}+T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral2}}\\;,\\nonumber\n\\end{equation}\nwhere\n\\begin{equation}\nT^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral1}}=\\dfrac{1}{2\\Xi}\\left(4|\\phi_2|^2-\\dfrac{\\Delta_r^2}{\\rho^4}|\\phi_0|^2\\right)\\;,\\label{rtcom}\n\\end{equation}\nwhile $T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral2}}$ becomes irrelevant at infinity, so we do not show its expression here.\\\\\nWith the fields decomposition in Eq.~\\eqref{fielddeccom}, integrating $T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral1}}$ over a sphere, we obtain the energy flux\n\\begin{equation}\n\\mathcal{F}|_r=\\int_{S^2} \\sin\\theta d\\theta d\\varphi\\; r^2 T^r_{\\;\\;t,\\;\\uppercase\\expandafter{\\romannumeral1}}\n=\\dfrac{r^2}{2\\;\\Xi\\;\\rho^4}(|R_{-1}|^2-\\Delta_r^2|R_{+1}|^2)\\;,\\label{flux}\n\\end{equation}\nup to an irrelevant normalization, and where the angular functions $S_{\\pm1}(\\theta)$ are normalized to unity.\nTo get the asymptotic boundary condition for $R_{-1}$, we expand Eq.~\\eqref{radialeq} with $s=-1$ asymptotically as\n\\begin{equation}\nR_{-1} \\sim \\;\\alpha^{-} r+\\beta^{-}+\\mathcal{O}(r^{-1})\\;,\\label{asysol}\n\\end{equation}\nwhere $\\alpha^{-}$ and $\\beta^{-}$ are two integration constants. In order to get the explicit form of the boundary conditions, the Starobinsky-Teukolsky identities are required. As we proved in Chapter~\\ref{ch:prelim}, these identities can be written explicitly as\n\\begin{align}\nBR_{+1}&=\\left(\\dfrac{d}{dr}-\\dfrac{i\\Xi K}{\\Delta_r}\\right)\\left(\\dfrac{d}{dr}-\\dfrac{i\\Xi K}{\\Delta_r}\\right)R_{-1}\\;,\\label{identity1}\\\\\nBR_{-1}&=\\Delta_r\\left(\\dfrac{d}{dr}+\\dfrac{i\\Xi K}{\\Delta_r}\\right)\\left(\\dfrac{d}{dr}+\\dfrac{i\\Xi K}{\\Delta_r}\\right)P_{+1}\\;.\\label{identity2}\n\\end{align}\n\nKeeping in mind the Starobinsky-Teukolsky identity~\\eqref{identity1}, making use of the radial equation~\\eqref{radialeq} with $s=-1$ and the asymptotic expansion in Eq.~\\eqref{asysol}, at the asymptotic boundary, the energy flux in Eq.~\\eqref{flux} becomes\n\\begin{eqnarray}\n\\mathcal{F}|_{r,\\infty}=B^2|\\alpha^{-}|^2-|(\\lambda-2\\omega^2\\Xi^2L^2)\\alpha^{-}+2i\\beta^{-}\\omega\\Xi|^2\\;,\\label{fluxinf}\n\\end{eqnarray}\nwhere an overall constant of proportionality has been ignored.\nIn order to impose the VEF boundary conditions, i.e. $\\mathcal{F}|_{r,\\infty}=0$, we have\n\\begin{equation}\nB^2|\\alpha^{-}|^2-|(\\lambda-2\\omega^2\\Xi^2L^2)\\alpha^{-}+2i\\beta^{-}\\omega\\Xi|^2=0\\;.\n\\end{equation}\nNote that $\\alpha^{-}$ and $\\beta^{-}$ are two independent integration constants; we can re-scale them so that the modulus in the above equation can be dropped\\footnote{Indeed there still might be a phase factor between these two constants, but this phase factor can be fixed by calculating the normal modes.}. Then it is easy to solve this quadratic equation and obtain the two solutions\n\\begin{equation}\n\\dfrac{\\alpha^{-}}{\\beta^{-}}=\\dfrac{2i\\omega\\Xi}{B-\\lambda+2\\omega^2\\Xi^2L^2}\\;,\\;\\;\\;\n\\dfrac{\\alpha^{-}}{\\beta^{-}}=\\dfrac{2i\\omega\\Xi}{-B-\\lambda+2\\omega^2\\Xi^2L^2}\\;.\\label{bc}\n\\end{equation}\nWe have also proved in Appendix~\\ref{app:angmomflux} that, the angular momentum flux of the Maxwell field vanishes asymptotically if the above boundary conditions are satisfied, similarly to the gravitational case~\\cite{Cardoso:2013pza}.\n\nFor Schwarzschild-AdS BHs, Eq.~\\eqref{bc} simplifies to\n\\begin{equation}\n\\dfrac{\\alpha^{-}}{\\beta^{-}}=\\dfrac{i}{\\omega L^2}\\;,\\;\\;\\;\\dfrac{\\alpha^{-}}{\\beta^{-}}=\\dfrac{i\\omega}{-\\ell(\\ell+1)+\\omega^2L^2}\\; .\\label{bcSAdS}\n\\end{equation}\nThese are, apparently, two distinct Robin boundary conditions, but at this moment it is unclear if they lead to physically different modes or if they are isospectral.\n\nWe can also follow the same procedure to calculate boundary conditions for the Teukolsky equation with $s=+1$. Instead of using $R_{+1}(r)$, we use $P_{+1}(r)$ for convenience, which relates to $R_{+1}(r)$ through $P_{+1}(r)=\\Delta_r R_{+1}(r)$. As before, we expand $P_{+1}(r)$ from Eq.~\\eqref{radialeq} with $s=+1$ asymptotically\n\\begin{equation}\nP_{+1} \\sim \\;\\alpha^{+} r+\\beta^{+}+\\mathcal{O}(r^{-1})\\;,\\label{asysol2}\n\\end{equation}\nwhere $\\alpha^{+}$ and $\\beta^{+}$ are two integration constants. Using the Starobinsky-Teukolsky identity in Eq.~\\eqref{identity2}, the asymptotic expansion in Eq.~\\eqref{asysol2}, the Teukolsky equation with $s=+1$ in Eq.~\\eqref{radialeq} and the transformation $P_{+1}(r)=\\Delta_r R_{+1}(r)$, then Eq.~\\eqref{flux} gives the  conditions\n\\begin{equation}\n\\dfrac{\\alpha^{+}}{\\beta^{+}}=-\\dfrac{2i\\omega\\Xi}{B-\\lambda+2\\omega^2\\Xi^2L^2}\\;,\\;\\;\\;\n\\dfrac{\\alpha^{+}}{\\beta^{+}}=-\\dfrac{2i\\omega\\Xi}{-B-\\lambda+2\\omega^2\\Xi^2L^2}\\;,\\label{bc2}\n\\end{equation}\nafter imposing the VEF boundary conditions. Comparing the two boundary conditions in Eq.~\\eqref{bc} and in Eq.~\\eqref{bc2}, we find that there is only a sign difference, or in other words, they are complex conjugate of each other. This is a consequence that $P_{+1}(r)$ and $R_{-1}(r)$ are proportional to complex conjugate functions of each other. We have checked that solving the radial equation~\\eqref{radialeq} for $s=-1$ and $s=+1$ with the corresponding boundary conditions~\\eqref{bc} and~\\eqref{bc2}, for Schwarzschild-AdS BHs, the same quasinormal frequencies are obtained, which is consistent with the argument that these two Teukolsky equations encode the same information. Thus, for concreteness and without loss of generality, in the following we specify $s=-1$, and consider the corresponding boundary conditions.\n\n\n\\section{Analytical matching}\n\\label{Maxsec:AM}\nIn this section, we present an analytical calculation of quasinormal frequencies for a Maxwell field on a Kerr-AdS BH, with the two Robin boundary conditions discussed in Section~\\ref{Maxsec:BC}. Such calculations can be used to illustrate how these Robin boundary conditions generate unstable modes.\n\nMaking use of the standard matching procedure, we shall first divide the space outside the event horizon into two regions: the \\textit{near region}, defined by the condition $r-r_+\\ll1\/\\omega$, and the \\textit{far region}, defined by the condition $r_+\\ll r-r_+$. Then we further require the condition $r_+\\ll1\/\\omega$ so that an overlapping region exists where the solutions obtained in the near region and in the far region are both valid. In the following analysis we focus on small AdS BHs $(r_+\\ll L)$ with slow rotation $(a\\ll r_+)$. The former condition allows treating the frequencies for the Kerr-AdS BH as a perturbation of the AdS normal frequencies; the latter condition together with $\\omega r_+\\ll1$, implying $\\omega a\\ll 1$ and $a\\ll L$, allow approximating the angular equation for the spin-weighted AdS-spheroidal harmonics by the spin-weighted spherical harmonics, so that the separation constant becomes\n\\begin{equation}\n \\lambda\\simeq\\ell(\\ell+1)\\;,\\;\\;\\;\\text{with}\\;\\;\\ell=1,2,3\\;,\\;\\cdot\\cdot\\cdot\\;,\\label{eigenvalues}\n\\end{equation}\nwhere $\\ell$ is the angular momentum quantum number.\n\n\n\\subsection{Near region solution}\nIn the near region, under the small BH, $r_+\\ll L$, and the slow rotation, $a\\ll r_+$, approximations, Eq.~\\eqref{radialeq} becomes\n\\begin{eqnarray}\n\\Delta_r R_{-1}''+\\left(\\dfrac{(r_+-r_-)^2\\hat{\\omega}}{\\Delta_r}-\\lambda\\right)R_{-1}=0 \\;,\\label{neareq1}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\hat{\\omega}=\\left(\\bar{\\omega}+\\dfrac{i}{2}\\right)^2+\\dfrac{1}{4}\\;,\\;\\;\\;\\bar{\\omega}=\\left(\\omega-m\\Omega_H\\right)\\Xi\\dfrac{r_+^2+a^2}{r_+-r_-}\\;,\\nonumber\n\\end{equation}\nwhere $\\Omega_H$ is the angular velocity of the event horizon, given by Eq.~\\eqref{RHorizonV}.\nIt is convenient to define a new dimensionless variable\n\\begin{equation}\nz\\equiv \\dfrac{r-r_+}{r-r_-}\\;,\\nonumber\n\\end{equation}\nto transform Eq.~\\eqref{neareq1} into\n\\begin{equation}\nz(1-z)\\frac{d^2R_{-1}}{dz^2}-2z\\frac{dR_{-1}}{dz}+\\left(\\dfrac{\\hat{\\omega}(1-z)}{z}-\\dfrac{\\lambda}{1-z}\\right)R_{-1}=0\\;.\\label{neareq2}\n\\end{equation}\nThe above equation can be solved in terms of the hypergeometric function\n\\begin{equation}\nR_{-1}\\sim z^{1-i\\bar{\\omega}}(1-z)^\\ell\\;F(\\ell+1,\\ell+2-2i\\bar{\\omega},2-2i\\bar{\\omega};z)\\;,\\label{nearsol}\n\\end{equation}\nwhere an ingoing boundary condition has been imposed.\n\nThe near region solution, Eq.~\\eqref{nearsol}, must be expanded for large $r$, in order to perform the matching with the far region solution below. To achieve this we take the $z\\rightarrow1$ limit, and obtain\n\\begin{equation}\nR_{-1} \\sim \\Gamma(2-2i\\bar{\\omega})\\left[\\dfrac{R^{\\rm near}_{-1,1\/r}}{r^{\\ell}} +R^{\\rm near}_{-1,r} r^{\\ell+1}\\right]\n\\;,\\label{nearsolfar}\n\\end{equation}\nby using the properties of the hypergeometric function~\\cite{abramowitz+stegun}, and where\n\\begin{align}\nR^{\\rm near}_{-1,1\/r} & \\equiv  \\dfrac{\\Gamma(-2\\ell-1) (r_+ - r_-)^\\ell}{\\Gamma(-\\ell)\\Gamma(1-\\ell-2i\\bar{\\omega})} \\ ,\\nonumber \\\\\nR^{\\rm near}_{-1,r} &\\equiv \\dfrac{\\Gamma(2\\ell+1)(r_+-r_-)^{-\\ell-1}}{\\Gamma(\\ell+1)\\Gamma(\\ell+2-2i\\bar{\\omega})} \\ .\n\\end{align}\n\n\n\\subsection{Far region solution}\nIn the far region, $r-r_+\\gg r_+$, the BH effects can be neglected ($M\\rightarrow0, a\\rightarrow0$) so that\n\\begin{equation}\n\\Delta_r\\simeq r^2 \\left(1+\\dfrac{r^2}{L^2}\\right)\\; .\\nonumber\n\\end{equation}\nThen Eq.~\\eqref{radialeq} becomes\n\\begin{equation}\n\\Delta_rR_{-1}''(r)+\\left(\\dfrac{K_r^2+i K_r \\Delta_r^\\prime}{\\Delta_r}-2iK_r^\\prime\n-\\ell(\\ell+1)\\right)R_{-1}(r)\n=0\\;,\\label{fareq1}\n\\end{equation}\nwith $K_r=\\omega r^2$.\n\\\\The general solution for Eq.~\\eqref{fareq1} is\n\\begin{align}\nR_{-1}=&\\;r^{\\ell+1}(r-iL)^{\\frac{\\omega L}{2}}(r+iL)^{-\\ell-\\frac{\\omega L}{2}}\\Big[C_1F\\Big(+\\omega L,2\\ell+2;\\dfrac{2r}{r+iL}\\Big)\\Big.\\nonumber \\\\ & \\Big.\\;-2^{-2\\ell-1}C_2\\left(1+\\dfrac{iL}{r}\\right)^{2\\ell+1}F\\Big(-\\ell-1,-\\ell+\\omega L,-2\\ell;\\dfrac{2r}{r+iL}\\Big)\\Big]\\;,\\label{farsol}\n\\end{align}\nwhere $C_1$, $C_2$ are two integration constants, and they will be constrained in the following to satisfy the boundary conditions.\\\\\nThe first boundary condition in Eq.~\\eqref{bc}, in the far region, becomes\n\\begin{equation}\n\\dfrac{\\alpha^{-}}{\\beta^{-}}=\\dfrac{i}{\\omega L^2}\\;.\\nonumber\n\\end{equation}\nIn order to impose this boundary condition, we first expand Eq.~\\eqref{farsol} at large $r$, in the form of Eq.~\\eqref{asysol}; then one obtains the first relation between $C_1$ and $C_2$\n\\begin{equation}\n\\dfrac{C_2}{C_1}=-2^{2\\ell+1}\\dfrac{\\ell}{\\ell+1}\\dfrac{F(\\ell+1,\\ell+1+\\omega L,2\\ell+2;2)}{F(-\\ell,-\\ell+\\omega L,-2\\ell;2)}\\;.\\label{c1c2rel1}\n\\end{equation}\nThe second boundary condition in Eq.~\\eqref{bc}, in the far region, turns into\n\\begin{equation}\n\\dfrac{\\alpha}{\\beta}=\\dfrac{i\\omega}{-\\ell(\\ell+1)+\\omega^2L^2}\\;.\\nonumber\n\\end{equation}\nTo impose the second boundary condition above, again expanding Eq.~\\eqref{farsol} at large $r$, to extract $\\alpha^{-}$ and $\\beta^{-}$, then one gets the second relation between $C_1$ and $C_2$\n\\begin{equation}\n\\dfrac{C_2}{C_1}=2^{2\\ell+1}\\left(\\dfrac{\\ell}{\\ell+1}\\right)^2\\dfrac{\\ell+1+\\omega L}{\\ell-\\omega L}\\dfrac{\\mathcal{A}_1}{\\mathcal{A}_2}\\;,\n\\end{equation}\nwhere\n\\begin{align}\n&\\mathcal{A}_1=(\\ell+1)F(\\ell,\\ell+1+\\omega L,2\\ell+2;2)+\\omega L F(\\ell+1,\\ell+2+\\omega L,2\\ell+3;2)\\;,\\nonumber\\\\\n&\\mathcal{A}_2=\\ell F(-\\ell-1,-\\ell+\\omega L,-2\\ell;2)-\\omega L F(-\\ell,-\\ell+1+\\omega L,1-2\\ell;2)\\;.\n\\end{align}\nIn order to match this solution to the near region solution, we expand Eq.~\\eqref{farsol} for small $r$, to obtain\n\\begin{equation}\nR_{-1}\\;\\sim\\;\\dfrac{R^{\\rm far}_{-1,1\/r}}{r^\\ell}+R^{\\rm far}_{-1,r}r^{\\ell+1}\\;,\\label{farsolnear}\n\\end{equation}\nwith\n\\begin{align}\n&R^{\\rm far}_{-1,1\/r}  \\equiv   -i L C_2 \\ ,\\nonumber \\\\\n&R^{\\rm far}_{-1,r} \\equiv (-1)^\\ell2^{2\\ell+1}L^{-2\\ell} C_1\\;.\\nonumber\n\\end{align}\n\n\n\\subsection{Overlap region}\nTo match the near region solution Eq.~\\eqref{nearsolfar} and the far region solution Eq.~\\eqref{farsolnear} in the intermediate region, we impose the matching condition $R^{\\rm near}_{-1,r}R^{\\rm far}_{-1,1\/r}=R^{\\rm far}_{-1,r}R^{\\rm near}_{-1,1\/r}$, then we obtain\n\\begin{align}\n&\\dfrac{\\Gamma(-2\\ell-1)}{\\Gamma(-\\ell)}\\dfrac{\\Gamma(\\ell+1)}{\\Gamma(2\\ell+1)}\\dfrac{\\Gamma(\\ell+2-2i\\bar{\\omega})}{\\Gamma(1-\\ell-2i\\bar{\\omega})}\\left(\\dfrac{r_+-r_-}{L}\\right)^{2\\ell+1}\n\\nonumber\\\\&=\\;i\\;(-1)^\\ell \\dfrac{\\ell}{\\ell+1}\\dfrac{F(\\ell+1,\\ell+1+\\omega L,2\\ell+2;2)}{F(-\\ell,-\\ell+\\omega L,-2\\ell;2)}\\;,\\label{match1}\n\\end{align}\nwith the first boundary condition given by Eq.~\\eqref{bc}, and\n\\begin{align}\n&\\dfrac{\\Gamma(-2\\ell-1)}{\\Gamma(-\\ell)}\\dfrac{\\Gamma(\\ell+1)}{\\Gamma(2\\ell+1)}\\dfrac{\\Gamma(\\ell+2-2i\\bar{\\omega})}{\\Gamma(1-\\ell-2i\\bar{\\omega})}\\left(\\dfrac{r_+-r_-}{L}\\right)^{2\\ell+1}\n\\nonumber\\\\\n&=\\;i\\;(-1)^{\\ell+1}\\left(\\dfrac{\\ell}{\\ell+1}\\right)^2\\dfrac{\\ell+1+\\omega L}{\\ell-\\omega L}\\;\\dfrac{\\mathcal{A}_1}{\\mathcal{A}_2}\\;,\\label{match2}\n\\end{align}\nwith the second boundary condition given by Eq.~\\eqref{bc}.\n\\\\\nBoth Eqs.~\\eqref{match1} and~\\eqref{match2} can be solved perturbatively to look for the imaginary part of the quasinormal frequencies, in the small BH $(r_+\\ll L)$ and slow rotation $(a\\ll r_+)$ approximations.\nIn order to do so, we first look for normal modes. For a small BH, the left term in Eqs.~\\eqref{match1} and~\\eqref{match2} vanish at the leading order, then we have to require the right hand side of both equations to vanish as well. These conditions give the normal modes for pure AdS\n\\begin{align}\n&F(\\ell+1,\\ell+1+\\omega L,2\\ell+2;2)=0\\;\\;\\Rightarrow\\;\\;\\omega_{1,N}L=2N+\\ell+2\\;,\\label{normalmode1}\\\\\n&\\mathcal{A}_1=0\\;\\;\\Rightarrow\\;\\;\\omega_{2,N}L=2N+\\ell+1\\;,\\label{normalmode2}\n\\end{align}\nwhere $N=0,1,2,\\cdot\\cdot\\cdot$, and $\\ell=1,2,3,\\cdot\\cdot\\cdot$. The two sets of modes are, in this case, isospectral up to one mode.\n\\\\\nWhen the BH effects are taken into account, a correction to the frequency will be introduced\n\\begin{equation}\n\\omega_j L=\\omega_{j,N} L+i\\delta_j\\;,\\label{normalmode}\n\\end{equation}\nwhere $j=1,2$ for the two different boundary conditions, and $\\delta$ is used to describe the damping (growth) of the quasinormal modes, and we replace $\\omega L$ appearing in the second line of Eqs.~\\eqref{match1} and~\\eqref{match2} by $\\omega_1 L$ and $\\omega_2 L$ in Eq.~\\eqref{normalmode}, respectively. Then, from each of these two equations, we obtain $\\delta_j$ perturbatively, at leading order in $a$.\n\nIt turns out that the general expression for $\\delta_j$ is quite complicated. As such, we only show here a few explicit examples. For $\\ell=1$ and $N=0$, from Eq.~\\eqref{match1}, we obtain\n\\begin{align}\n\\delta_1&=-\\dfrac{16}{\\pi}\\dfrac{r_+^4}{L^4}+m\\dfrac{16}{3\\pi}\\dfrac{ar_+^2}{L^3}+\\mathcal{O}\\left(\\frac{a}{L},\\frac{r_+^4}{L^4}\\right)=-\\dfrac{16}{3\\pi}\\dfrac{r_+^2}{L^2}\\left(3\\dfrac{r_+^2}{L^2}-m\\dfrac{a}{L}\\right)+\\cdot\\cdot\\cdot\\nonumber\\\\\n&\\simeq-\\dfrac{16}{3\\pi}\\dfrac{r_+^4}{L^3}\\left(\\omega_{1,0}-m\\Omega_H\\right)+\\cdot\\cdot\\cdot\\;,\\label{analy1}\n\\end{align}\nwhere the angular velocity has been approximated by $\\Omega_H\\sim a\/r_+^2$. It is manifest, from \\eqref{analy1}, that $\\delta_1<0$ when $\\omega_{1,0}>m\\Omega_H$, while $\\delta_1>0$ when $\\omega_{1,0}<m\\Omega_H$. Thus we find growing modes within the superradiant regime, as expected.\n\\\\\nKeeping the same parameters as in the previous paragraph, i.e. $\\ell=1$ and $N=0$, from Eq.~\\eqref{match2}, we obtain\n\\begin{align}\n\\delta_2&=-\\dfrac{8}{3\\pi}\\dfrac{r_+^4}{L^4}+m\\dfrac{4}{3\\pi}\\dfrac{ar_+^2}{L^3}+\\mathcal{O}\\left(\\frac{a}{L},\\frac{r_+^4}{L^4}\\right)=-\\dfrac{4}{3\\pi}\\dfrac{r_+^2}{L^2}\\left(2\\dfrac{r_+^2}{L^2}-m\\dfrac{a}{L}\\right)+\\cdot\\cdot\\cdot\\nonumber\\\\\n&\\simeq-\\dfrac{4}{3\\pi}\\dfrac{r_+^4}{L^3}\\left(\\omega_{2,0}-m\\Omega_H\\right)+\\cdot\\cdot\\cdot\\;,\\label{analy2}\n\\end{align}\nwhich also shows clearly that $\\delta_2<0$ when $\\omega_{2,0}>m\\Omega_H$, but $\\delta_2>0$ when $\\omega_{2,0}<m\\Omega_H$, signaling again superradiant instabilities.\n\nFurthermore, for both cases, $\\delta_1=0$ and $\\delta_2=0$ give $\\omega_{1,0}=m\\Omega_H$ and $\\omega_{2,0}=m\\Omega_H$, which are the conditions to form clouds, with the two different boundary conditions.\n\n\\section{Numerical method}\n\\label{Maxsec:NM}\nWhen the BH parameters lie beyond the small size and slow rotation approximations considered in the last section, the analytical method fails and we have to solve the problem numerically. Since the radial equation~\\eqref{radialeq} and the angular equation~\\eqref{angulareq} are coupled through their eigenvalues, we have to solve both equations simultaneously. In this section, we describe the numerical method used to solve both equations.\n\nThe radial equation~\\eqref{radialeq}, will be solved with a direct integration method, which has been used in Chapters~\\ref{ch:NeutralP} and~\\ref{ch:ChargedP}. To be complete, here we briefly outline the procedure.\n\\\\\nWe first use Frobenius' method to expand $R_{-1}$ close to the event horizon\n\\begin{equation}\nR_{-1}=(r-r_+)^\\rho \\sum_{j=0}^\\infty c_j\\;(r-r_+)^j\\;,\\nonumber\n\\end{equation}\nto initialize Eq.~\\eqref{radialeq}. The series expansion coefficients $c_j$ can be derived directly after inserting these expansions into Eq.~\\eqref{radialeq}.\nThe parameter $\\rho$ is chosen as\n\\begin{equation}\n\\rho=1-\\dfrac{i(\\omega-m\\Omega_H)}{4\\pi T_H}\\;,\\nonumber\n\\end{equation}\nso that an ingoing boundary condition is satisfied. The angular velocity $\\Omega_H$ and the Hawking temperature $T_H$ are given in Eq.~\\eqref{RHorizonV} and Eq.~\\eqref{RTemp}.\n\nAt infinity, the asymptotic behavior of $R_{-1}$ is given by Eq.~\\eqref{asysol}. The expansion coefficients, $\\alpha^{-}$ and $\\beta^{-}$, can be extracted from $R_{-1}$ and its first derivative. For that purpose, we define two new fields $\\left\\{\\chi,\\psi\\right\\}$, which asymptote respectively to $\\left\\{\\alpha^{-},\\beta^{-}\\right\\}$ at infinity. Such a transformation can be written in the matrix form\n\\begin{equation}\n\\mathbf{V}= \\left(\\begin{array}{cc} r & 1 \\vspace{2mm}\\\\ 1 & 0 \\end{array}\\right) \\mathbf{\\Psi} \\equiv \\mathbf{T} \\mathbf{\\Psi} \\;,\\nonumber\n\\end{equation}\nby defining the vector $\\mathbf{\\Psi}^T=(\\chi,\\psi)$ for the new fields, and another vector $\\mathbf{V}^T=(R_{-1},\\frac{d}{dr}R_{-1})$ for the original field and its derivative.\n\nTo obtain a first order system of ordinary differential equations for the new fields, we define another matrix $\\mathbf{X}$, through\n\\begin{equation}\n\\dfrac{d\\mathbf{V}}{dr}=\\mathbf{X}\\mathbf{V} \\; ,\n\\end{equation}\nwhich can be read off from the original radial equation~\\eqref{radialeq} directly.\nThen the radial equation~\\eqref{radialeq} becomes\n\\begin{equation}\n\\dfrac{d\\mathbf{\\Psi}}{dr}=\\mathbf{T}^{-1}\\left(\\mathbf{X}\\mathbf{T}-\\dfrac{d\\mathbf{T}}{dr}\\right) \\mathbf{\\Psi} \\;.\\label{radialmatrix}\n\\end{equation}\nThis is the final equation we are going to solve.\n\nThe angular equation~\\eqref{angulareq}, will be solved using a spectral method, to look for the separation constant $\\lambda$. By observing Eq.~\\eqref{angulareq} and considering the constraint on rotation $a<L$, one finds two regular singularities at $u=\\pm 1$. To impose regular boundary conditions at these regular singularities, we require\n\\begin{equation}\nS\\;\\sim\\;\n\\begin{cases}\n(1-u)^{\\frac{|m+s|}{2}}\\;\\;\\;\\;\\;\\text{when}\\;\\;\\;u\\rightarrow 1\\;,\\\\(1+u)^{\\frac{|m-s|}{2}}\\;\\;\\;\\;\\;\\text{when}\\;\\;\\;u\\rightarrow -1\\;,\n\\end{cases}\n\\end{equation}\nwhere as announced in Section~\\ref{Maxsec:BC}, $s=-1$. These asymptotic behaviors can be factored out by defining a new function $\\hat{S}$\n\\begin{equation}\nS=(1-u)^{\\frac{|m+s|}{2}}(1+u)^{\\frac{|m-s|}{2}}\\hat{S}\\;.\\label{trans}\n\\end{equation}\nThen the angular equation~\\eqref{angulareq} becomes\n\\begin{equation}\nY(u)\\hat{S}=\\lambda \\hat{S}\\;,\\label{eigenfunc}\n\\end{equation}\nwhere the operator $Y(u)$ can be obtained straightforwardly after inserting the transformation~\\eqref{trans} into the angular equation~\\eqref{angulareq}.\n\nWe choose a Chebyshev grid as the collocation points to discretize the operator $Y(u)$, which becomes a matrix. Then Eq~\\eqref{eigenfunc} becomes a linear algebraic equation, and $\\lambda$ is obtained by looking for the eigenvalues of the matrix $Y(u)$.\n\n\n\n\\section{Warm up for Schwarzschild-AdS black holes}\n\\label{Maxsec:SAdS}\nWe shall now apply the VEF boundary conditions to Maxwell perturbations on Schwarzschild-AdS BHs, in the Teukolsky formalism. We show that even in this simpler case, there is a new branch of quasinormal modes which has not been explored yet.\n\nAs we mentioned in Section~\\ref{Maxse:intro}, the quasinormal modes for the Maxwell field on Schwarzschild-AdS BHs have been studied using the Regge-Wheeler formalism~\\cite{Cardoso:2003cj}. Here we will tackle the same problem in the Teukolsky formalism, imposing the boundary conditions discussed in Section~\\ref{Maxsec:BC}. We find that:\n\\begin{itemize}\n\\item[$\\bullet$] when the first of the two boundary conditions in Eq.~\\eqref{bcSAdS} is imposed, we recover the results given in the literature~\\cite{Cardoso:2003cj,Berti:2003ud};\n\\item[$\\bullet$] when the second of the two boundary conditions in Eq.~\\eqref{bcSAdS} is imposed, there is one new branch of quasinormal modes.\n\\end{itemize}\nTo be complete and for comparison, we will show both results in the following. In the numerical calculations all physical quantities are normalized by the AdS radius $L$, so we set $L=1$. Also, observe that we use $\\omega_1$ ($\\omega_2$) to represent the quasinormal frequency corresponding to the first (second) boundary conditions.\n\nThe numerical method we are using here has already been illustrated in Section~\\ref{Maxsec:NM}. Since the angular function in this case becomes the standard spherical harmonics, we do not need to solve the angular equation numerically. The results are as follows.\n\nIn Table~\\ref{EM1}, we list a few fundamental $(N=0)$ quasinormal frequencies of $\\omega_1$ (with $\\ell=1$) and $\\omega_2$ (with $\\ell=2$), for different BH sizes. As observed in Section~\\ref{Maxsec:AM}, the normal modes displayed in Eqs.~\\eqref{normalmode1} and~\\eqref{normalmode2}, are isospectral under the mapping\n\\begin{equation}\n\\ell_2\\leftrightarrow \\ell_1+1 \\ ,\n\\end{equation}\nexcept for one mode for the $\\omega_2$ branch. Here $\\ell_1$ and $\\ell_2$ refer to the angular momentum quantum number in the spectrum of $\\omega_1$ and $\\omega_2$. The presence of a BH, however, breaks the isospectrality. To show this, we present in Table~\\ref{EM1}, the two sets of quasinormal frequencies, with $\\ell_1=1$ and $\\ell_2=2$, respectively.\nOne observes that the degeneracy between $\\omega_1$ and $\\omega_2$ gets broken, especially in the small BH and intermediate BH regimes. For large BHs, these two modes are, again, almost isospectral, which seems to be a general feature for any type of perturbation~\\cite{Berti:2003ud,Cardoso:2003cj}.\nFurthermore, for large BHs, the real part of the frequency for either of the sets vanishes, while the imaginary part scales linearly with the BH size $r_+$. This scaling can be equally stated in terms of the Hawking temperature, which relates to the BH size through $T_H=3r_+\/(4\\pi L^2)$ for large BHs, supporting the arguments given in~\\cite{Horowitz:1999jd}, where a similar linear relation was found for scalar fields.\nWe remark that the numerical data for $\\omega_1$ displayed in Table~\\ref{EM1} coincides with the numerical results presented in~\\cite{Cardoso:2001bb,Cardoso:2003cj}, at least within $4$ significant digits, which can be used as a check for our numerical method.\n\\begin{table}\n\\caption{\\label{EM1} Quasinormal frequencies of the Maxwell field on Schwarzschild-AdS. Some fundamental modes are shown, for different BH sizes $r_+$ and for the two sets of modes.}\n\\small\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l }\n\\hline\n\\hline\n$r_+$ & $\\omega_1 (\\ell=1)$ & $\\omega_2 (\\ell=2)$ \\\\\n\\hline\n0.2 & 2.6384 - 5.7947$\\times 10^{-2}$ i & 2.9403 - 1.0466$\\times 10^{-4}$ i\\\\\n0.5 & 2.2591 - 0.6573 i & 2.7804 - 0.07549 i\\\\\n0.8 & 2.1758 - 1.2870 i & 2.6923 - 0.2721 i\\\\\n1.0 & 2.1630 - 1.6991 i & 2.6647 - 0.4061 i\\\\\n5.0 & 0 - 8.7948 i & 0 - 5.0528 i\\\\\n10 & 0 - 15.5058 i & 0 - 13.8198 i\\\\\n50 & 0 - 75.0958 i & 0 - 74.7533 i\\\\\n100 & 0 - 150.048 i & 0 - 149.876 i\\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\nFor small BHs, it can be shown using a perturbative analytical matching method given in Section~\\ref{Maxsec:AM}, that the real part of the frequencies approaches the normal modes of empty AdS~\\cite{Konoplya:2002zu}, given by Eqs~\\eqref{normalmode1} and~\\eqref{normalmode2} for the two different boundary conditions. On the other hand, the imaginary part for both modes approaches zero as\n\\begin{equation}\n-\\omega_{j,I} \\propto r_+^{2\\ell+2}\\;,\\nonumber\n\\end{equation}\nwhich also seems to be a general feature for any type of perturbations~\\cite{Berti:2009wx}. In Fig.~\\ref{Mf}, left panel, we display the numerical data (thick lines) for the fundamental modes of each branch against the leading behavior obtained from the perturbative matching method in Section~\\ref{Maxsec:AM}. We find a good agreement for small $r_+$, which is another check for our numerical method.\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-5mm}\\includegraphics[width=0.95\\textwidth]{Figs\/ch8\/Fig1.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{Mf} {\\em Left:} Comparison of the imaginary part for quasinormal frequencies between the analytical approximation of small BHs (thin dashed lines) and the numerical data (thick lines) for the fundamental modes of each branch of solutions. {\\em Right:} Effect of the angular momentum quantum number $\\ell$ on the quasinormal frequencies for intermediate size BHs with $r_+=1$, and $N=0$. The red line is for $\\omega_1$ and the blue line is for $\\omega_2$.}\n\\end{figure*}\n\nIn Table~\\ref{EM2}, we consider intermediate size BHs to exemplify the effect of the angular momentum quantum number $\\ell$ on the frequencies\\footnote{We have checked that the effect of varying $\\ell$ is qualitatively similar for small BHs.}. As one can see, for both modes, the real (imaginary) part of the quasinormal frequencies increases (decreases) in magnitude as $\\ell$ increases. These behaviors are more clearly shown in the right panel of Fig.~\\ref{Mf}. Observe that the increase of the real part of the frequency with $\\ell$ is qualitatively similar to the one observed for empty AdS.\n\\begin{table}\n\\caption{\\label{EM2} Same as Table \\ref{EM1}, but fixing now the BH size to be $r_+=1$. Some fundamental modes are shown, considering different angular momentum quantum number $\\ell$.}\n\\small\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l }\n\\hline\n\\hline\n$\\ell$ & $\\omega_1$ & $\\omega_2$ \\\\\n\\hline\n1 & 2.16302 - 1.69909 i & 1.55360 - 0.541785 i\\\\\n2 & 3.22315 - 1.38415 i & 2.66469 - 0.406058 i\\\\\n3 & 4.23555 - 1.20130 i & 3.69923 - 0.334088 i\\\\\n4 & 5.23994 - 1.07445 i & 4.71659 - 0.286828 i\\\\\n5 & 6.24294 - 0.97775 i & 5.72784 - 0.252025 i\\\\\n6 & 7.24598 - 0.89976 i & 6.73632 - 0.224622 i\\\\\n7 & 8.24941 - 0.83447 i & 7.74335 - 0.202110 i\\\\\n8 & 9.25327 - 0.77838 i & 8.74952 - 0.183072 i\\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\nFinally, let us remark that, in the above, we have focused on fundamental modes because, on the one hand, our main interest has been to explore the new set of modes which arises even for $N=0$ and, on the other hand, these low lying modes are expected to dominate the late time behavior of time evolutions.\n\n\n\\section{Quasinormal modes and superradiant instabilities}\n\\label{Maxsec:instability}\nIn this section, we now turn on rotation, to study superradiant instabilities for the Maxwell field on a Kerr-AdS BH. With the numerical strategies described in Section~\\ref{Maxsec:NM}, and the boundary conditions given in Eq.~\\eqref{bc}, the eigenvalues $\\{\\omega,\\lambda\\}$ of the coupled system in Eqs.~\\eqref{radialmatrix} and~\\eqref{eigenfunc} can be obtained iteratively. This is achieved by starting with an initial guess for $\\omega$ or $\\lambda$, which are roots of a characteristic equation, and then polishing the solution until $\\{\\omega,\\lambda\\}$ become stable. The initial values for $\\omega$ or $\\lambda$ can be chosen from the results in Schwarzschild-AdS BHs~\\cite{Wang:2015goa} or $\\ell(\\ell+1)$, respectively.\n\nNote that we use $\\omega_1$ $(\\omega_2)$ to represent the quasinormal frequency and $\\lambda_1$ ($\\lambda_2$) to stand for the separation constant, corresponding to the first (second) boundary conditions.\n\nA few selected eigenvalues for $\\omega$ and $\\lambda$ are tabulated in Tables~\\ref{Table1-1}$-$\\ref{Table3}, with two boundary conditions for various BH sizes. Since it is a generic feature for superradiant instabilities that lower order modes exhibit a stronger instability, we focus on the lowest fundamental modes, characterized by $N=0,\\;\\ell=1$ and $m=0,\\;\\pm1$.\n\nIn Tables~\\ref{Table1-1} and~\\ref{Table1-2} we consider a small BH with size $r_+=0.1$. The first observation from these two tables is that superradiant instabilities exist for both boundary conditions, with positive $m$; this is because only positive $m$ modes can meet the superradiance condition, assuming positive frequencies.\n\n\\begin{table}\n\\caption{\\label{Table1-1} Quasinormal frequencies and separation constants of the Maxwell field with the first boundary condition, for $\\ell=1$ fundamental modes, on a Kerr-AdS BH with size $r_+=0.1$.}\n\\small\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l l }\n\\hline\n\\hline\n$(\\ell,m)$ & $a$ & $\\omega_1$ & $\\lambda_1$ \\\\\n\\hline\n(1,\\;0) & 0 & 2.8519 - 1.7050$\\times 10^{-3}$ i & 2\\\\\n  & 0.01 & 2.8505 - 1.7295$\\times 10^{-3}$ i & 2.0005 - 5.9147$\\times 10^{-7}$ i\\\\\n  & 0.05 & 2.8151 - 2.5818$\\times 10^{-3}$ i & 2.0133 - 2.1677$\\times 10^{-5}$ i\\\\\n  & 0.1 & 2.6740 - 2.2847$\\times 10^{-2}$ i & 2.0480 - 7.1650$\\times 10^{-4}$ i\\\\\n\\hline\n(1,\\;1) & 0.01 & 2.8436 - 9.5962$\\times 10^{-4}$ i & 1.9149 + 2.8541$\\times 10^{-5}$ i\\\\\n        & 0.05 & 2.7837 + 5.5800$\\times 10^{-4}$ i & 1.5879 - 8.0057$\\times 10^{-5}$ i\\\\\n        & 0.1 & 2.6493 + 2.0481$\\times 10^{-3}$ i & 1.2278 - 5.6148$\\times 10^{-4}$ i\\\\\n\\hline\n(1,\\;-1) & 0.01 & 2.8572 - 2.7984$\\times 10^{-3}$ i & 2.0859 - 8.4666$\\times 10^{-5}$ i\\\\\n         & 0.05 & 2.8422 - 1.7571$\\times 10^{-2}$ i & 2.4296 - 2.7419$\\times 10^{-3}$ i\\\\\n         & 0.1 & 2.7398 - 1.2611$\\times 10^{-1}$ i & 2.8269 - 4.0538$\\times 10^{-2}$ i\\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\n\\begin{table}\n\\caption{\\label{Table1-2} Quasinormal frequencies and separation constants of the Maxwell field with the second boundary condition, for $\\ell=1$ fundamental modes, on a Kerr-AdS BH with size $r_+=0.1$\n}\n\\small\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l l }\n\\hline\n\\hline\n$(\\ell,m)$ & $a$ & $\\omega_2$ & $\\lambda_2$ \\\\\n\\hline\n(1,\\;0) & 0 & 1.9533 - 1.8240$\\times 10^{-4}$ i & 2\\\\\n        & 0.01 & 1.9529 - 1.8329$\\times 10^{-4}$ i & 2.0003 - 4.2946$\\times 10^{-8}$ i\\\\\n        & 0.05 & 1.9452 - 2.0829$\\times 10^{-4}$ i & 2.0071 - 1.2092$\\times 10^{-6}$ i\\\\\n        & 0.1 & 1.9160 - 1.0036$\\times 10^{-3}$ i & 2.0276 - 2.2602$\\times 10^{-5}$ i\\\\\n\\hline\n(1,\\;1) & 0.01 & 1.9436 - 7.9809$\\times 10^{-5}$ i & 1.9417 + 2.3800$\\times 10^{-6}$ i\\\\\n        & 0.05 & 1.8989 + 1.9474$\\times 10^{-4}$ i & 1.7156 - 2.8302$\\times 10^{-5}$ i\\\\\n        & 0.1  & 1.8292 + 7.8282$\\times 10^{-4}$ i & 1.4552 - 2.1958$\\times 10^{-4}$ i\\\\\n\\hline\n  (1,\\;-1) & 0.01 & 1.9622 - 3.2541$\\times 10^{-4}$ i & 2.0589 - 9.8188$\\times 10^{-6}$ i\\\\\n           & 0.05 & 1.9900 - 2.1380$\\times 10^{-3}$ i & 2.2975 - 3.2933$\\times 10^{-4}$ i\\\\\n           & 0.1  & 1.9987 - 1.9186$\\times 10^{-2}$ i & 2.5914 - 6.0286$\\times 10^{-3}$ i\\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\nThe effect of varying the rotation parameter on both eigenvalues, for different values of $m$ with fixed $\\ell=1$, are shown in Figs.~\\ref{rp01refre}$-$\\ref{rp01imlam}. In Fig.~\\ref{rp01refre}, the real part of the frequency is shown. An immediate first impression from Fig.~\\ref{rp01refre}, is that it seems the rotation impacts differently on the $m=-1$ modes, for the two boundary conditions. Checking carefully the numerical data for Re($\\omega_2$), however, we find that its value decreases sightly when $a$ is approaching $0.1$. Thus,  for  $m=-1$ and for both boundary conditions, Re($\\omega$) starts by increasing with increasing rotation but then decreases. For the other two values of $m$, Re($\\omega$) always decreases with increasing rotation.\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp01Reomega1.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp01Reomega2.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{rp01refre} Variation of Re($\\omega$) with varying rotation parameter, for fixed $r_+=0.1$ and $\\ell=1$ but for different values of $m$. The left panel is for the first boundary condition while the right panel is for the second boundary condition.}\n\\end{figure*}\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp01Imomega1.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp01Imomega2.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{rp01imfre} Variation of Im($\\omega$) with varying rotation parameter, for fixed $r_+=0.1$ and $\\ell=1$ but for different values of $m$. The left panel is for the first boundary condition while the right panel is for the second boundary condition. The brown solid thin line corresponds to Im($\\omega$)=0, to exhibit more clearly superradiant instabilities.}\n\\end{figure*}\n\nIn Fig.~\\ref{rp01imfre}, the imaginary part of the frequency is shown, for both boundary conditions. Im($\\omega$) increases with increasing rotation when $m=1$, eventually becoming positive, signaling the presence of superradiant unstable modes.\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp01Relambda1.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp01Relambda2.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{rp01relam} Variation of Re($\\lambda$) with varying rotation parameter, for fixed $r_+=0.1$ and $\\ell=1$ but for different values of $m$. The left panel is for the first boundary condition while the right panel is for the second boundary condition.}\n\\end{figure*}\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp01Imlambda1.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp01Imlambda2.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{rp01imlam} Variation of Im($\\lambda$) with varying rotation parameter, for fixed $r_+=0.1$ and $\\ell=1$ but for different values of $m$. The left panel is for the first boundary condition while the right panel is for the second boundary condition. Again, the brown solid thin line corresponds to Im($\\lambda$)=0, to exhibit that the sign of Im($\\lambda$) changes when the superradiant instability occurs (seen in the insets).}\n\\end{figure*}\n\nIn Fig.~\\ref{rp01relam} and Fig.~\\ref{rp01imlam}, we show the real and imaginary part of the separation constant, respectively, for both boundary conditions. The real part increases with increasing rotation both when $m=-1$ and $m=0$, albeit only slightly in the latter case, and decreases with the rotation when $m=1$. As for the imaginary part, for $m=1$, it increases with the rotation initially in a small range; but it starts decreasing afterwards. For the other two values of $m$, the imaginary part of the separation constants always decrease with the rotation. We also note that Im($\\lambda$), for $m=0$, with the first boundary condition decays faster than its counterpart with the second boundary condition. From these first four figures, we conclude that the effect of varying the rotation on the eigenvalues is qualitatively similar for the two boundary conditions. In the following, therefore, we only show the rotation effect on the eigenvalues with the first boundary condition, when considering other BH sizes.\n\nWe continue our study by varying the BH size. In Tables~\\ref{Table2-1} and~\\ref{Table2-2} we list a few selected eigenvalues for $r_+=0.3$. The interesting feature that now emerges is that superradiant instabilities only occur for the second boundary condition. This implies that the second boundary condition may produce unstable modes in a larger parameter space. This feature will be shown more clearly in the parameter space for the vector clouds. The effect of varying the rotation on the eigenvalues is shown in Figs.~\\ref{rp03fre}$-$\\ref{rp03lam}, with the first boundary condition. In Fig.~\\ref{rp03fre}, it displays that, Re($\\omega$) increases with increasing rotation for the $m=-1$ mode but decreases for both $m=0$ and $m=1$ modes, while Im($\\omega$) increases with increasing rotation for the $m=1$ mode but decreases for both $m=0$ and $m=-1$ modes. The behaviors of the separation constant, shown in Fig.~\\ref{rp03lam}, are similar to the counterparts in the $r_+=0.1$ case\\footnote{Notice that for the $m=1$ mode, Im($\\lambda$) starts decreasing with rotation around $a\\simeq0.25$.}.\n\n\\begin{table}\n\\centering\n\\caption{\\label{Table2-1} Quasinormal frequencies and separation constants of the Maxwell field with the first boundary condition, for $\\ell=1$ fundamental modes, on a Kerr-AdS BH with size $r_+=0.3$.}\n\\small\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l l }\n\\hline\n\\hline\n$(\\ell,m)$ & $a$ & $\\omega_1$ & $\\lambda_1$ \\\\\n\\hline\n(1,\\;0) & 0 & 2.4481 - 0.2291 i & 2 \\\\\n  & 0.1 & 2.4093 - 0.2768 i & 2.0397 - 7.8286$\\times 10^{-3}$ i\\\\\n  & 0.2 & 2.3071 - 0.4480 i & 2.1373 - 4.5479$\\times 10^{-2}$ i \\\\\n  & 0.3 & 2.2136 - 0.7769 i & 2.2476 - 1.5300$\\times 10^{-1}$ i \\\\\n  \\hline\n(1,\\;1) & 0.1 & 2.3197 - 0.1512 i & 1.3185 + 4.1838$\\times 10^{-2}$ i \\\\\n  & 0.2 & 2.1325 - 0.1120 i & 0.7889 + 5.6598$\\times 10^{-2}$ i \\\\\n& 0.3 & 1.8707 - 7.4377$\\times 10^{-2}$ i & 0.4681 + 5.1126$\\times 10^{-2}$ i \\\\\n\\hline\n  (1,\\;-1) & 0.1 & 2.5468 - 0.3994 i & 2.7643 - 0.1276 i \\\\\n           & 0.2 & 2.6884 - 0.7162 i & 3.5705 - 0.4764 i \\\\\n           & 0.3 & 2.9588 - 1.2044 i & 4.4377 - 1.2239 i \\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{\\label{Table2-2} Quasinormal frequencies and separation constants of the Maxwell field with the second boundary condition, for $\\ell=1$ fundamental modes, on a Kerr-AdS BH with size $r_+=0.3$\n}\n\\small\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l l }\n\\hline\n\\hline\n$(\\ell,m)$ & $a$ & $\\omega_2$ & $\\lambda_2$ \\\\\n\\hline\n(1,\\;0) & 0 & 1.8152 - 3.8034$\\times 10^{-2}$ i & 2\\\\\n        & 0.1 & 1.8092 - 4.7641$\\times 10^{-2}$ i & 2.0252 - 1.0134$\\times 10^{-3}$ i\\\\\n        & 0.2 & 1.7921 - 8.8195$\\times 10^{-2}$ i & 2.0949 - 6.9871$\\times 10^{-3}$ i\\\\\n        & 0.3 & 1.7875 - 1.8395$\\times 10^{-1}$ i & 2.1957 - 2.9422$\\times 10^{-2}$ i\\\\\n\\hline\n(1,\\;1) & 0.1 & 1.7265 - 1.5809$\\times 10^{-2}$ i & 1.4841 + 4.4473$\\times 10^{-3}$ i\\\\\n        & 0.2 & 1.6328 - 2.8939$\\times 10^{-3}$ i & 1.0447 + 1.4994$\\times 10^{-3}$ i\\\\\n        & 0.3 & 1.5304 + 5.2612$\\times 10^{-3}$ i & 0.7048 - 3.6993$\\times 10^{-3}$ i\\\\\n\\hline\n(1,\\;-1) & 0.1 & 1.9063 - 8.8702$\\times 10^{-2}$ i & 2.5624 - 2.7793$\\times 10^{-2}$ i\\\\\n         & 0.2 & 2.0270 - 1.9387$\\times 10^{-1}$ i & 3.1487 - 1.2392$\\times 10^{-1}$ i\\\\\n         & 0.3 & 2.2192 - 3.6493$\\times 10^{-1}$ i & 3.7662 - 3.4830$\\times 10^{-1}$ i\\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp03Reomega1.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp03Imomega1.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{rp03fre} Variation of $\\omega$ with varying rotation parameter, for different values of $m$. The BH size is fixed as $r_+=0.3$ and the first boundary condition has been imposed. The left panel is for Re($\\omega$) while the right panel is for Im($\\omega$).}\n\\end{figure*}\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp03Relambda1.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp03Imlambda1.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{rp03lam} Variation of $\\lambda$ with varying rotation parameter, for different values of $m$. The BH size is fixed as $r_+=0.3$ and the first boundary condition has been imposed. The left panel is for Re($\\lambda$) while the right panel is for Im($\\lambda$).}\n\\end{figure*}\n\nThe results for $r_+=1$ are presented in Table~\\ref{Table3} and Figs.~\\ref{rp1fre}$-$\\ref{rp1lam}. From Table~\\ref{Table3} one observes that there is no superradiant instability for any of the boundary conditions.\n\n\\begin{table}\n\\centering\n\\caption{\\label{Table3} Quasinormal frequencies and separation constants of the Maxwell field with the two different Robin boundary conditions, for $\\ell=1$ fundamental modes, on a Kerr-AdS BH with size $r_+=1$.}\n\\small\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l l l l }\n\\hline\n\\hline\n$(\\ell,m)$ & $a$ & $\\omega_1$ & $\\lambda_1$ & $\\omega_2$ & $\\lambda_2$ \\\\\n\\hline\n(1,\\;0) & 0 & 2.1630 - 1.6991 i & 2 & 1.5536 - 0.5418 i & 2\\\\\n\n  & 0.1 & 2.1672 - 1.7274 i & 2.0162 - 4.4059$\\times 10^{-2}$ i & 1.5627 - 0.5510 i & 2.0186 - 1.0131$\\times 10^{-2}$ i\\\\\n  & 0.2 & 2.1805 - 1.8146 i & 2.0580 - 0.1757 i & 1.5909 - 0.5795 i & 2.0727 - 4.0857$\\times 10^{-2}$ i\\\\\n  & 0.3 & 2.2067 - 1.9686 i & 2.1067 - 0.3931 i &  1.6416 - 0.6297 i & 2.1574 - 9.2910$\\times 10^{-2}$ i\\\\\n  \\hline\n(1,\\;1)\n  & 0.1 & 1.9512 - 1.5858 i & 1.4111 + 0.4430 i & 1.4277 - 0.4948 i & 1.5675 + 0.1404 i\\\\\n  & 0.2 & 1.7679 - 1.5195 i & 0.9445 + 0.7797 i & 1.3209 - 0.4643 i & 1.2046 + 0.2445 i\\\\\n  & 0.3 & 1.6048 - 1.4929 i & 0.5998 + 1.0384 i & 1.2273 - 0.4463 i & 0.9151 + 0.3204 i\\\\\n  \\hline\n  (1,\\;-1)\n  & 0.1 & 2.4169 - 1.8746 i & 2.7065 - 0.5962 i & 1.7067 - 0.6116 i & 2.4983 - 0.1905 i\\\\\n  & 0.2 & 2.7349 - 2.1390 i & 3.5193 - 1.4235 i & 1.8996 - 0.7147 i & 3.0586 - 0.4532 i\\\\\n  & 0.3 & 3.1548 - 2.5377 i & 4.4164 - 2.6136 i & 2.1522 - 0.8684 i & 3.6770 - 0.8235 i\\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\nIn Fig.~\\ref{rp1fre}, we present the real and imaginary parts of the frequency for the first boundary condition and $r_+=1$. Re($\\omega$) increases with rotation for the $m=-1$ mode, decreases with rotation for the $m=1$ mode, and increases sightly with rotation for the $m=0$ mode. The behavior of Im($\\omega$) is almost the opposite, since Im($\\omega$) decreases with rotation for both $m=0$ and $m=-1$ modes, but for the $m=1$ mode, it increases first and then starts to decrease around $a\\simeq0.33$. Comparing Fig.~\\ref{rp1lam} with Fig.~\\ref{rp1fre}, shows that the effect of rotation on $\\lambda$ (both real part and imaginary part) mimics closely that on $\\omega$, except for Im($\\lambda$) of the $m=1$ mode, which always increases with the rotation parameter.\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp1Reomega1.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp1Imomega1.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{rp1fre} Variation of $\\omega$ with varying rotation parameter, for different values of $m$. The BH size is fixed as $r_+=1$ and the first boundary condition has been imposed. The left panel is for Re($\\omega$) while the right panel is for Im($\\omega$).}\n\\end{figure*}\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp1Relambda1.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.41\\textwidth]{Figs\/ch8\/rp1Imlambda1.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{rp1lam} Variation of $\\lambda$ with varying rotation parameter, for different values of $m$. The BH size is fixed as $r_+=1$ and the first boundary condition has been imposed. The left panel is for Re($\\lambda$) while the right panel is for Im($\\lambda$).}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\\section{Stationary vector and scalar clouds}\n\\label{Maxsec:clouds}\nStationary clouds are bound state solutions with real frequency of test fields around a rotating background, computed at the linear level. The existence of clouds indicates nonlinear hairy BH solutions~\\cite{Herdeiro:2014goa,Herdeiro:2014ima}, but the converse needs not be true~\\cite{Brihaye:2014nba,Herdeiro:2015kha}. In order to find such solutions, we demand $\\omega=m\\Omega_H$; in other words, stationary clouds are the zero modes of superradiance. Imposing this condition leads to a constraint on the BH parameters: BHs are quantized in the sense that only BHs with specific parameters can support a cloud with a given set of ``quantum'' numbers. This quantization defines \\textit{existence lines} in the BH parameter space. In the practical implementation of our numerical calculations, we use the same method as before, with the condition $\\omega=m\\Omega_H$, to look for the rotation parameter. Note that all the results presented in this subsection are for fundamental modes, characterized by $N=0$.\n\nThe vector clouds we have obtained shall be presented in a parameter space spanned by $R_+$ and $\\hat{\\Omega}_H$, where\n\\begin{eqnarray}\nR_+=\\sqrt{\\dfrac{r_+^2+a^2}{\\Xi}}\\;,\\;\\;\\;\\;\\;\\;\\hat{\\Omega}_H=\\Omega_H\\Xi+a\\;,\\label{NewDef}\n\\end{eqnarray}\nin which $\\hat{\\Omega}_H$ has been defined in Eq.~\\eqref{NRHAV}.\nThe reason to use this pair of parameters, instead of $r_+$ and $\\Omega_H$, is as follows. $\\Omega_H$, as defined in Eq.~\\eqref{RHorizonV}, is the horizon angular velocity measured relatively to a rotating frame at infinity, while $\\hat{\\Omega}_H$, defined in Eq.~\\eqref{NRHAV} (also in Eq.~\\eqref{NewDef}), is the horizon angular velocity measured with respect to a non-rotating observer at infinity. The latter one is more relevant in BH thermodynamics~\\cite{Gibbons:2004ai}. In the practical calculations, one can use either of them since they are simply related by Eq.~\\eqref{NewDef}. As one may check, $\\hat{\\Omega}_H$ is a monotonic function of $a$, in terms of $R_+$, but not of $r_+$. Also there is an intuitive geometric meaning for $R_+$, which is the areal horizon radius.\n\n\nIn Fig.~\\ref{vec}, the existence lines for some examples of vector clouds are displayed (left panel) together with the corresponding separation constants (right panel). In the left panel, the red solid line corresponds to extremal BHs, and regular BHs only exist below this extremality line. The first three existence lines (with $\\ell=m=1,2,3$) for the first boundary condition, and the first two existence lines (with $\\ell=m=1,2$) for the second boundary condition, are presented by dotted and dot dashed lines, respectively. These lines start from bound state solutions (normal modes), denoted by orange dots in Fig.~\\ref{vec}, of the Maxwell field on empty AdS, i.e.\n\\begin{eqnarray}\n\\hat{\\Omega}_{H,1\\;|R_+=0}=1+\\dfrac{2}{\\ell}\\;,\\;\\;\\;\\;\\;\\;\\hat{\\Omega}_{H,2\\;|R_+=0}=1+\\dfrac{1}{\\ell}\\;.\\label{bsAdS}\n\\end{eqnarray}\nThese are obtained by equating the superradiance condition\\footnote{$\\Omega_H$ is the same as $\\hat{\\Omega}_H$ in pure AdS.}, $\\omega=m\\Omega_H$, to the normal mode conditions in Eqs.~\\eqref{normalmode1} and~\\eqref{normalmode2}, together with $m=\\ell$, and $N=0$. Observe, in particular, that although the two sets of normal modes in AdS are isospectral, the existence lines for the two boundary conditions only coincide as $R_+\\rightarrow 0$, when taking $\\ell=1$ for the first boundary condition and $\\ell=2$ for the second.\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.45\\textwidth]{Figs\/ch8\/vectorcloudsf.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.45\\textwidth]{Figs\/ch8\/vectorcloudslambda.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{vec}Vector clouds (left panel) and the corresponding separation constants (right panel) in $R_+$ versus $\\hat{\\Omega}_H$ and $\\lambda$ versus $R_+$ plots, respectively. $\\ell_1 (\\ell_2)$ refer to the results obtained by imposing the first (second) boundary condition.}\n\\end{figure*}\n\nAn existence line with a particular $\\ell=m$, separates the superradiantly stable Kerr-AdS BHs (to the left side of the existence line) and the superradiantly unstable ones (to the right side of the existence line), against that particular mode. Therefore, as one may observe from the left panel of Fig.~\\ref{vec}, the stable region in the parameter space against the mode, say $\\ell=m=n$, where $n$ is some integer, with the second boundary condition is also stable against all the modes with the first boundary condition from $\\ell=m=1$ up to $\\ell=m=n+1$. From the data in this figure, together with the relation between $R_+$ and $r_+$, Eq.~(\\ref{NewDef}), it can be concluded that: with the first boundary condition, BHs with $r_+\\leq0.25$ are superradiantly unstable against the $\\ell=m=1$ fundamental mode; while with the second boundary condition, BHs with $r_+\\leq0.34$ are superradiantly unstable against the $\\ell=m=1$ fundamental mode. These observations also explain the fact that the superradiant instability only appears for modes with the second boundary condition in Table~\\ref{Table2-2}.\n\n\nAs a comparison with the Maxwell stationary clouds reported above, we have also computed stationary scalar clouds, by solving the massless Klein-Gordon equation on Kerr-AdS BHs, with VEF boundary condition which in this case is the same as the usual field vanishing boundary condition. For this case there is a single set of modes. The results for the scalar clouds are exhibited in Fig.~\\ref{SCs}, in terms of the same parameters $R_+$ and $\\hat{\\Omega}_H$.\n\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{c}\n\\hspace{-4mm}\\includegraphics[clip=true,width=0.45\\textwidth]{Figs\/ch8\/scalarcloudsfrequency.pdf}\\hspace{15mm}\\includegraphics[clip=true,width=0.45\\textwidth]{Figs\/ch8\/scalarcloudslambda.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{SCs}Scalar clouds (left panel) and the corresponding separation constants (right panel) in $R_+$ versus $\\hat{\\Omega}_H$ and $\\lambda$ versus $R_+$ plots, respectively.}\n\\end{figure*}\n\n\nIn the left panel of Fig.~\\ref{SCs}, the red solid line stands, as before, for extremal BHs, so that regular BHs only exist below this line. The first three existence lines, corresponding to the modes with $\\ell=m=1,2,3$, are described by dotted, dot dashed and dashed lines, respectively. The corresponding separation constants are also shown in the right panel of Fig.~\\ref{SCs}. The orange dots in both panels indicate again the normal modes and eigenvalues of the angular function in pure AdS, which are\n\\begin{eqnarray}\n\\hat{\\Omega}_{H,{\\rm scalar}}=1+\\dfrac{3}{\\ell}\\;,\\;\\;\\;\\;\\;\\lambda=\\ell(\\ell+1)\\;.\n\\end{eqnarray}\nAgain, the existence line with a particular $\\ell=m$, divides the parameter space into two regions: BHs in the left region are superradiantly stable while BHs in the right region are superradiantly unstable, against that particular mode.\n\nComparing Figs.~\\ref{vec} and~\\ref{SCs} it becomes clear that the existence lines for stationary vector clouds appear to the \\textit{left} of the existence line of a stationary scalar cloud with the same quantum numbers. Thus, there are BHs that are stable against the scalar mode but become unstable against the vector mode. In a sense, vector superradiance is stronger. Qualitatively, this conclusion agrees with the computation of the amplification factors for scalar and vector modes in superradiant scattering, in asymptotically flat spacetimes.\n\n\n\n\n\\section{Summary}\n\\label{Maxsec:con}\nThe behavior of test fields on an asymptotically AdS spacetime depends sensitively on the boundary conditions, since such spacetimes contain a timelike boundary. The AdS boundary is often regarded as a perfectly reflecting mirror in the sense that no flux (both energy flux and angular momentum flux) can cross it. Various types of boundary conditions in asymptotically AdS spacetimes have been explored, in particular consistent with this simple requirement. We took this requirement as the \\textit{guiding principle} to impose boundary conditions on test fields. Following this principle, two boundary conditions were found for Maxwell fields on asymptotically AdS spacetimes, of which only one had been previously discussed on the Schwarzschild-AdS background. These two boundary conditions were then used to study quasinormal modes, superradiant unstable modes and vector clouds for the Maxwell field on Kerr-AdS BHs.\n\n\nTo find quasinormal modes and superradiant modes, we have solved the Teukolsky equations both analytically and numerically. In the small BH and slow rotation regime, an analytical matching method was applied to exhibit how these two boundary conditions work and how they produce superradiant instabilities. A numerical method was then used to explore the parameter space away from the small BH and slow rotation approximations. We found that for small BHs characterized by $r_+=0.1$, unstable superradiant modes appear with both boundary conditions. Increasing the BH size, as exemplified for $r_+=0.3$, superradiant instabilities only appear with the second boundary condition, and eventually disappear for both boundary conditions, as exemplified when $r_+=1$. Our analysis also shows that superradiant instabilities for the Maxwell field may exist for (moderately) larger BH sizes, when comparing with the scalar case, for which superradiant instabilities appear in the regime $r_+\\leq0.16$~\\cite{Uchikata:2009zz}.\n\nTo study stationary vector clouds, which can occur for massless fields in AdS due to the box-like global structure, we have solved the Teukolsky equations at the onset of the superradiant instability, i.e. for $\\omega=m\\Omega_H$. We found that both boundary conditions can yield vector clouds, and that these clouds are bounded by the extremal BHs, as for the scalar clouds on asymptotically flat Kerr BHs~~\\cite{Hod:2012px,Hod:2013zza,Herdeiro:2014goa}. This behaviour differs from that observed for gravitational perturbations, for which only one of the sets of clouds are bounded by the extremal BHs~\\cite{Cardoso:2013pza}. The existence of clouds at the linear level indicates nonlinear hairy BH solutions~\\cite{Herdeiro:2014goa,Herdeiro:2014ima}, so it would be interesting to find the nonlinear realization of these vector clouds. There is already a well-known exact BH family within the Einstein-Maxwell-AdS system: the Kerr-Newman-AdS family. It will then be interesting to understand the interplay between this well known family and the new family of ``hairy'' BHs.\n\n\n\n\n\n\\chapter{Hawking radiation for a Proca field: neutral case}\n\\label{ch:NeutralP}\n\n\n\n\n\\section{Introduction}\n\n\n\nHawking radiation, as we described in Section~\\ref{sc:HR}, is an interesting quantum phenomenon. It has been widely studied on different backgrounds, for various types of emitting particles, and in diverse gravitational theories. In particular, the study of Hawking radiation from higher-dimensional BHs has gained a lot attention recently, motivated by the possibility of producing microscopic BHs at the LHC. This is a prediction of TeV gravity scenarios in which the fundamental Planck scale could be as low as the TeV scale. This motivation led to an intensive study of Hawking radiation from higher-dimensional BHs~\\cite{Ida:2002ez,Harris:2003eg,Harris:2005jx,Ida:2005ax,Duffy:2005ns,Casals:2005sa,Cardoso:2005vb,Cardoso:2005mh,Ida:2006tf,Casals:2006xp,Casals:2008pq,Sampaio:2009ra,Sampaio:2009tp,Herdeiro:2011uu,Wang:2012tk}.\n\nOne of the Hawking radiation channels that has not been properly addressed in the literature is that of massive vector bosons, both neutral and charged, to describe the emission of $Z$ and $W^{\\pm}$ particles of the SM. The basic difficulty is that the Proca equations do not decouple even in a spherically symmetric BH background\\footnote{In a rotating background, it appears that the Proca field can neither be decoupled nor separated.}. To bridge this gap, in this chapter, we study Hawking radiation for a neutral Proca field, by solving the Proca equations derived in Chapter~\\ref{ch:GeneralInFlat}, in the background of a $D$-dimensional spherically symmetric Schwarzschild BH~\\cite{Tangherlini:1963bw}.\n\nIn order to perform such a study, we have designed a numerical strategy to solve the coupled equations\\footnote{The decoupled equations have been solved simultaneously.} without decoupling and showed that the coupled system may be treated by an {\\bf S}-matrix type formalism which allows decoupling in the asymptotic regions. This {\\bf S}-matrix is used to define a transmission matrix which gives the transmission factors as its eigenvalues. We have computed transmission factors for various modes, masses and spacetime dimensions. The mass term lifts the degeneracy between transverse modes, in $D=4$, and excites the longitudinal modes, in particular the $s$-wave. Moreover, it increases the contribution of waves with larger $\\ell$, which can be dominant at intermediate energies. The transmission factors are then used to obtain the Hawking fluxes in this channel.\n\nThis chapter is organized as follows. In Section~\\ref{sec:NPnearhorizon} we study the near horizon and far region asymptotic behaviors of the coupled Proca equations, which can be used to extract relevant information to study Hawking radiation. In Section~\\ref{sec:scatteringM} we discuss how the scattering matrix is used to compute the transmission factor and the Hawking spectrum. In Section~\\ref{sec:resultsNP} we discuss the numerical method and results, and we summarize our results in Section~\\ref{sec:sumProneu}. Some technical relations are left to Appendix~\\ref{app:neutralP}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Boundary conditions and radial system}\n\\label{sec:NPnearhorizon}\nIn this section, we study the wave equation for a neutral Proca field by setting the field charge $q=0$ in all the field equations derived in Chapter~\\ref{ch:GeneralInFlat}, i.e. Eqs.~\\eqref{originalsys1},~\\eqref{originalsys2},~\\eqref{Max} and~\\eqref{k0M}, in the Schwarzschild-Tangherlini background with the metric function $V=1-\\frac{M}{r^{n-1}}$. In the numerics, we choose units such that the horizon radius is $r_H=1$, so then $M=r_H^{n-1}$. Since decoupled radial equations have been extensively studied in the literature we will not present the details of our analysis of such modes and refer to the method in~\\cite{Sampaio:2009ra,Sampaio:2009tp}. Thus in the following we will focus on the solutions of the coupled system for the massive theory, which will be used in conjunction with the decoupled modes to obtain the full Hawking spectrum in Section~\\ref{sec:resultsNP}.\n\nWe start by finding a series expansion of the solution near the horizon for the coupled system $\\left\\{\\psi,\\chi\\right\\}$. This will be used to initialize the corresponding fields for the radial integration at $r=1.001$. If we define $y=r-1$, Eqs.~\\eqref{originalsys1} and\n~\\eqref{originalsys2} become\n\\begin{eqnarray}\n\\left[M(r)\\dfrac{d^2}{dy^2}+N(r)\\dfrac{d}{dy}+P(r)\\right]\\psi+Q(r)\\chi=0\\label{NPsysterm1}\\ ,\\\\\n\\left[\\tilde M(r)\\dfrac{d^2}{dy^2}+\\tilde\nN(r)\\dfrac{d}{dy}+\\tilde P(r)\\right]\\chi+\\tilde Q(r)\\psi=0\\label{NPsysterm2}\\ ,\n\\end{eqnarray}\nwhere the polynomials are defined in Appendix~\\ref{app:neutralP}. Making use of Frobenius' method to expand $\\psi$ and $\\chi$ as\n\\begin{equation}\n\\psi=y^\\rho\\sum^{\\infty}_{j=0}{\\mu_jy^j}\\label{defpsi} \\; , \\qquad \\chi=y^\\rho\\sum^{\\infty}_{j=0}{\\nu_jy^j} \\ ,\n\\end{equation}\nand inserting the above two equations into Eqs.~\\eqref{NPsysterm1} and\n~\\eqref{NPsysterm2}, we obtain\n\\begin{equation}\n\\rho=\\pm\\frac{i\\omega}{n-1} \\; \\; \\mathrm{or} \\; \\; \\rho = 1\\pm\\frac{i\\omega}{n-1} \\; .\\label{rhoexp}\n\\end{equation}\nWe want to impose an ingoing boundary condition at the horizon, so we must choose the minus sign. Furthermore, after this sign choice, the right hand side solution produces a series expansion which is a special case of the left hand side (where the first coefficient is set to zero), so without loss of generality we choose $\\rho = -i\\omega\/(n-1)$. One then writes down the recurrence relations and concludes that a general solution, close to the horizon, can be parameterized by the two coefficients $\\nu_0$ and $\\nu_1$. The other coefficients are generated by the recurrence relations~\\eqref{NPrecurone}.\n\nTo understand the asymptotic behavior of the waves at infinity we now study a large $r$ asymptotic expansion in the form\n\\begin{equation}\n\\psi=e^{\\beta r}r^{p}\\sum_{j=0}\\dfrac{a_j}{r^j}\\label{psifar} \\; , \\qquad \\chi=e^{\\beta r}r^{p}\\sum_{j=0}\\dfrac{b_j}{r^j} \\ .\n\\end{equation}\nInserting this into Eqs.~\\eqref{originalsys1} and~\\eqref{originalsys2} we obtain, at leading order,\n\\begin{equation}\n\\beta = \\pm ik\\;,\\;\\;\\;\\;\\;\\;\\;k=\\sqrt{\\omega^2-\\mu_p^2}\\;,\\qquad p = 1-\\frac{n}{2}\\pm i\\varphi \\; \\; {\\rm or} \\; \\; p =-\\frac{n}{2}\\pm i\\varphi \\; ,\n\\end{equation}\nwhere $\\varphi=\\delta_{n,2}(\\omega^2+k^2)\/(2k)$. Thus one can show that  asymptotically\\footnote{We have used, without loss of generality, the leading power behavior for $p$ and discarded the second option which produces the same solution, similarly to the near horizon expansion.}\n\\begin{equation}\n\\psi \\rightarrow \\dfrac{1}{r^{\\frac{n}{2}-1}}\\left[\\left(a_0^++\\dfrac{a_1^+}{r}+\\ldots\\right)e^{i\\Phi}+\\left(a_0^-+\\dfrac{a_1^-}{r}+\\ldots\\right)e^{-i\\Phi}\\right] \\ , \\nonumber\n\\end{equation}\n\\begin{equation}\n\\chi \\rightarrow \\dfrac{1}{r^{\\frac{n}{2}-1}}\\left[\\left(\\left(-\\frac{k}{\\omega}+\\dfrac{c^+}{r}\\right)a_0^++\\ldots\\right)e^{i\\Phi}+\\left(\\left(\\frac{k}{\\omega}+\\dfrac{c^-}{r}\\right)a_0^-+\\ldots\\right)e^{-i\\Phi}\\right] \\; ,\n\\end{equation}\nwhere $\\Phi\\equiv kr+\\varphi \\log r$ and $c^\\pm$ is defined in the Appendix, Eq.~\\eqref{eq:NPcpm}.\nSo as expected, each field is a combination of ingoing and outgoing waves at infinity. This asymptotic expansion also shows that, for a generic wave at infinity, we can choose four independent constants $\\left\\{a_0^\\pm,a_1^\\pm\\right\\}$ to characterize the solution. This is expected, since we have two coupled scalar fields and for each scalar degree of freedom we must have an associated ingoing wave and outgoing wave. Thus we can define four new fields $\\left\\{\\chi^\\pm,\\psi^\\pm\\right\\}$ (which will asymptote respectively to $\\left\\{a_0^\\pm,a_1^\\pm\\right\\}$), by truncating the expansion for the fields and for their first derivatives at infinity. Such a transformation can be written in matrix form by defining the 4-vector $\\mathbf{\\Psi}^T=(\\psi_{+},\\psi_{-},\\chi_{+},\\chi_{-})$ for the new fields, and another 4-vector $\\mathbf{V}^T=(\\psi,d_r\\psi,\\chi,d_r\\chi)$ for the original fields and derivatives. Then the transformation is given in terms of an $r$-dependent matrix $\\mathbf{T}$ defined through\n\\begin{equation}\n\\mathbf{V}= \\mathbf{T} \\mathbf{\\Psi} \\; ,\n\\end{equation}\nwhich we provide in the Appendix, Eq.~\\eqref{eq:NPT}. In order to obtain a first order system of ordinary differential equations (ODEs) for the new fields, we first define a matrix $\\mathbf{X}$ through\n\\begin{equation}\n\\dfrac{d\\mathbf{V}}{dr}=\\mathbf{X}\\mathbf{V} \\; ,\n\\end{equation}\nwhich is read out from the original system in Eqs.~\\eqref{originalsys1} and~\\eqref{originalsys2}. Its explicit form is given in the Appendix, Eq.~\\eqref{eq:X}.\nThen we obtain\n\\begin{equation}\n\\dfrac{d\\mathbf{\\Psi}}{dr}=\\mathbf{T}^{-1}\\left(\\mathbf{X}\\mathbf{T}-\\dfrac{d\\mathbf{T}}{dr}\\right) \\mathbf{\\Psi} \\ .\n\\end{equation} We can write other equivalent systems using different $\\mathbf{T}$ matrices. In particular, we have also integrated a first order system using the fields $\\psi_{s}=k \\psi-isd_r\\psi$ and $\\chi_{s}=k \\chi-isd_r\\chi$ with $s=\\pm$, which produced numerically equivalent results. The only difference is that for such fields we need to extract $\\mathcal{O}(r^{-1})$ coefficients to obtain $a_1^s$.\n\n\\section{The Hawking spectrum}\n\\label{sec:scatteringM}\nThe boundary conditions we have chosen in Section~\\ref{sec:NPnearhorizon} are suitable for the computation of the Hawking spectrum of radiated quanta from the BH. The Hawking spectrum is generically given by a sum over a complete set of modes with labels $\\zeta$, of the transmission factor $\\mathbb{T}_\\zeta$ times a thermal average number of quanta produced at the horizon $\\left<n_\\zeta\\right>$. This is defined for a basis of decoupled modes. In our problem, we have a sub-set of modes, the transverse vector mode, and the $\\ell=0$ ($\\kappa_s=0$) mode, which are decoupled. But we also have a tower of modes which are coupled two by two for each $\\ell>0$, the two scalars $\\psi$ and $\\chi$. It is not obvious how to decouple them for all $r$ through an explicit transformation. Instead, let us try to understand how to extract the relevant information in the asymptotic regions.\n\n\nLet us denote the two coupled fields by a 2-vector $\\mathbf{U^T}=(\\psi,\\chi)$ and represent the coupled system of radial equations through a (linear) second order matrix differential operator $\\mathcal{D}^{(2)}$ acting on $\\mathbf{U}$, i.e. $\\mathcal{D}^{(2)}\\mathbf{U}=0$. The system is coupled because of the off diagonal elements of the $\\mathcal{D}^{(2)}$ operator. To decouple the system we would have to find a transformation of the fields $\\mathbf{U}=\\mathcal{A} \\mathbf{\\bar U}$, such that the new differential operator $\\bar{\\mathcal{D}}^{(2)}=\\mathcal{D}^{(2)}$\\textopenbullet$\\mathcal{A}$ is diagonal, i.e.\n\\begin{equation}\n\\bar{\\mathcal{D}}^{(2)}=\\left(\\begin{array}{cc}\n\\bar{\\mathcal{D}}^{(2)}_1 & 0 \\\\ 0 & \\bar{\\mathcal{D}}^{(2)}_2\n\\end{array}\\right) \\ .\n\\end{equation}\nEven without finding such a transformation explicitly, one can draw some conclusions by assuming its existence\\footnote{In fact, for example, if we consider $\\mathcal{A}$ to be a general $r$-dependent matrix, we can write down two conditions for the four arbitrary functions of such a matrix. Thus, in principle, there is enough freedom.}. In particular we may establish a map between our general solution of the coupled system and the actual decoupled solution, for each of the asymptotic regions (horizon and far field). To find such a map let us first summarize the information we have on the general solution of the coupled system.\n\nIn Section~\\ref{sec:NPnearhorizon} we have found that a general solution is parameterized by four independent coefficients in one of the asymptotic regions; either at the horizon or at infinity. Once we have chosen one set of coefficients, say at the horizon, due to the linearity of the equations, the four independent wave components at infinity are a linear combination of the four coefficients at the horizon. Let us formally denote the ingoing and outgoing wave coefficients at the horizon ($+\/-$ respectively) by\n \\[\\vec{\\mathbf{h}}=({\\mathbf h}^+,{\\mathbf h}^-)=(h^+_{i},h^-_{i}) \\ , \\] where $i=1,2$ since we have two fields.\nSimilarly, the coefficients at infinity are defined as the large $r$ limit of the $\\mathbf{\\Psi}$ field components (up to linear transformation which we will define next), i.e.\n  \\[\\vec{\\mathbf{y}}=({\\mathbf y}^+,{\\mathbf y}^-)=(y^+_{i},y^-_{i}) \\ , \\]\nwith $i=1,2$ for $\\psi$ and $\\chi$ respectively. Due to linearity, we can define a scattering matrix\n\\begin{eqnarray}\n\\vec{\\mathbf{y}}=\\mathbf{S} \\vec{\\mathbf{h}} &&\\ \\ \\Leftrightarrow \\ \\ \\left(\\begin{array}{c} {\\mathbf y}^+ \\\\ {\\mathbf y}^- \\end{array} \\right)=\\left(\\begin{array}{c|c} {\\mathbf S}^{++} & {\\mathbf S}^{+-} \\\\ \\hline {\\mathbf S}^{-+}  & {\\mathbf S}^{--} \\end{array} \\right)\\left(\\begin{array}{c} {\\mathbf h}^+ \\\\ {\\mathbf h}^- \\end{array} \\right) \\;,\\nonumber\\\\  &&\\ \\ \\Leftrightarrow \\ \\ \\left(\\begin{array}{c} {y}^+_i \\\\ { y}^-_i \\end{array} \\right)=\\sum_{j}\\left(\\begin{array}{c|c} S_{i j}^{++} & S_{i j}^{+-} \\\\ \\hline S_{i j}^{-+}  & S_{i j}^{--} \\end{array} \\right)\\left(\\begin{array}{c} h_{j}^+ \\\\ h_{j}^- \\end{array} \\right) \\; ,\n\\end{eqnarray}\nwhich is a set of numbers (depending on energy, angular momentum, etc\\ldots) containing all the information on the scattering process. It can be fully determined by considering specific modes at the horizon and integrating them outwards.\nIn our problem, we have imposed an ingoing boundary condition at the horizon which is simply ${\\mathbf h}^{+}=0$. Then\n\\begin{equation}\\label{NPscattering_ingoing}\n{\\mathbf y}^s={\\mathbf S}^{s-}{\\mathbf h}^{-} \\; .\n\\end{equation}\nTaking the $s=-$ component, and denoting the inverse matrix of ${\\mathbf S}^{--}$ by $({\\mathbf S}^{--})^{-1}$, we invert~\\eqref{NPscattering_ingoing} to obtain the wave at the horizon given the ingoing wave at infinity\n\\begin{equation}\n{\\mathbf h}^{-} =\\left({\\mathbf S}^{--}\\right)^{-1}{\\mathbf y}^{-}\\; .\n\\end{equation}\nInserting this relation back in the $s=+$ component of~\\eqref{NPscattering_ingoing}, we obtain the outgoing wave in terms of the ingoing wave, at infinity\n\\begin{equation}\\label{Reflection}\n{\\mathbf y}^+={\\mathbf S}^{+-}({\\mathbf S}^{--})^{-1}{\\mathbf y}^-\\equiv{\\mathbf R} \\, {\\mathbf y}^-\\; ,\n\\end{equation}\nwhere in the last line we have defined the reflection matrix $\\mathbf{R}$. Before proceeding, we note that there is still some freedom in the definition of the asymptotic coefficients since any (non-singular) linear combination is equally good from the point of view of satisfying the boundary condition. This freedom can be written in terms of 3 matrices $\\mathbf{M}^s$, $\\mathbf{M}_H^-$ relating some new fields (hatted) to the old fields\n\\begin{equation}\n{\\mathbf y}^s=\\mathbf{M}^s\\hat{\\mathbf y}^s \\; ,\\; \\; \\; \\; \\; \\; \\; \\; {\\mathbf h}^{-}=\\mathbf{M}_H^-\\hat{\\mathbf h}^{-} \\; .\n\\end{equation}\nSince this represents the most general parametrization of the solution in the asymptotic regions, there must be a choice which decouples the fields in those regions. To find the correct transformation we need a physical prescription.\n\nTo obtain the transmission factor for the  decoupled components, it is instructive to remind ourselves of the calculation of the transmission factor for a single decoupled field. It is defined as the fraction of the incident wave which is transmitted to the horizon. If we look at a wave with energy $\\omega$ (for an observer at infinity), with ingoing\/outgoing amplitudes $Y^{(\\infty)}_{\\mp}$, then~\\cite{Kanti:2004nr}\n\\begin{equation}\n\\mathbb{T}=\\dfrac{|Y^{(\\infty)}_{-}|^2-|Y^{(\\infty)}_{+}|^2}{|Y^{(\\infty)}_{-}|^2}=\\dfrac{\\omega\\left(|Y^{(\\infty)}_{-}|^2-|Y^{(\\infty)}_{+}|^2\\right)}{\\omega |Y^{(\\infty)}_{-}|^2}=\\dfrac{\\mathcal{F}^{in}_H}{\\mathcal{F}^{in}_\\infty} \\ ,\\label{deftrans}\n\\end{equation}\nwhere in the last step we note that $\\mathbb{T}$ can be re-expressed as a ratio between the total incident energy flux $\\mathcal{F}^{in}_H$ (which is the difference between the energy carried by the ingoing wave and the energy of the outgoing wave) and the incident energy flux associated with the ingoing wave at infinity ($\\mathcal{F}^{in}_\\infty$). The former is the flux of energy transmitted down to the horizon.\n\nWe now compute the energy fluxes through a sphere at radius $r$ using the energy-momentum tensor. This will allow us to identify the decoupled fields at infinity and at the horizon, and in particular, the ingoing and outgoing decoupled waves at infinity. Such a flux is shown to be conserved in our background, by using the conservation law for the energy-momentum tensor, combined with the fact that the spatial integral of $T_t^{\\phantom{t} t}$ for each energy eigen-mode is constant. It is defined, evaluated at $r$, as\n\\begin{equation}\\label{eq:FluxR}\n\\mathcal{F}|_r=-\\int_{S^n} d\\Sigma\\, T_t^{\\; r}\n\\end{equation}\nwhere $d\\Sigma$ is the volume element on a $t,r={\\rm constant}$ hyper-surface. The energy-momentum tensor for the complex neutral Proca field is\n\\begin{equation}\nT^{\\mu\\nu}=-\\dfrac{1}{2}\\left(W^{\\dagger \\mu \\alpha}W^{\\nu}_{\\; \\alpha}-\\mu_p^2W^{\\dagger \\mu}W^{\\nu}+c.c.\\right)-\\dfrac{g^{\\mu \\nu}}{2}\\mathcal{L} \\; ,\n\\end{equation}\nup to an irrelevant normalization. If we insert this in~\\eqref{eq:FluxR}, assume a field configuration with a well defined energy\/frequency $\\omega$, and make use of the equations of motion, then, for the non-trivial case of $\\mu_p^2\\neq 0 \\neq \\kappa_s^2$, we obtain\n\\begin{equation}\\label{eq:FluxR2}\n\\mathcal{F}|_r=\\sum_\\Lambda\\dfrac{i\\omega V\\Upsilon^{\\dagger}_\\Lambda}{2r^2}\\dfrac{d\\Upsilon_\\Lambda}{dr}+\\sum_\\Lambda\\left\\{\\dfrac{\\kappa_s^2}{2r^2}\\psi^\\dagger \\chi-\\dfrac{1}{2\\mu_p^2}\\left[\\dfrac{V}{r^n}\\dfrac{d(r^n\\xi^\\dagger)}{dr}-\\dfrac{\\kappa_s^2\\psi^\\dagger}{r^2}\\right]\\left[i\\omega\\xi+\\dfrac{\\kappa_s^2\\chi}{r^2}\\right]\\right\\}+c.c\n\\end{equation}\nwhere $\\Lambda$ denotes the mode eigenvalues for the corresponding harmonic functions, and for convenience we define $\\xi=d_r\\psi+i\\omega\\chi$. Modes with different angular momentum eigenvalues are clearly decoupled, as are the transverse vector mode contributions in the first sum. The terms in the second sum couple two fields for fixed $\\Lambda$. We can compute the flux at infinity and close to the horizon and express it in terms of the asymptotic coefficients in the corresponding region. Focusing on a specific mode and in the coupled part of the flux (second sum in~\\eqref{eq:FluxR2})\n\\begin{equation}\\label{eq:FluxInf}\n\\mathcal{F}^{\\mathrm{coupled}}_\\infty=|y_0^-|^2-|y_0^+|^2+|y_1^-|^2-|y_1^+|^2\\equiv(\\mathbf{y}^-)^\\dagger \\mathbf{y}^--(\\mathbf{y}^+)^\\dagger \\mathbf{y}^+ \\; ,\n\\end{equation}\nwhere $y_i^s$ are linear combinations of the asymptotic coefficients $a_i^s$ given in the Appendix, Eq.~\\eqref{eq:NPyplus}. This choice of $y_i^s$ is already in a form close to decoupled, since we have separated the modulus square of the incident contribution from the reflected contribution, without interference terms. This form is invariant under separate unitary transformation of $\\mathbf{y}^\\pm$. Using the reflection matrix we obtain\n\\begin{equation}\\label{eq:FluxInfT}\n\\mathcal{F}^{\\mathrm{coupled}}_\\infty=(\\mathbf{y}^-)^\\dagger\\left(\\mathbf{1}-\\mathbf{R}^\\dagger\\mathbf{R}\\right) \\mathbf{y}^-\\equiv (\\mathbf{y}^-)^\\dagger\\mathbf{T}\\, \\mathbf{y}^-  \\; ,\n\\end{equation}\nwhere we have defined a (hermitian) transmission matrix $\\mathbf{T}$. Note that this matrix is composed of the transmission matrix given in Eq.~\\eqref{Reflection}, and it is a generalization of the transmission factor, defined in Eq.~\\eqref{deftrans}, to the coupled system. The transmission matrix can be diagonalized through a unitary transformation which is the remaining freedom we have for $\\mathbf{y}^-$. In fact we can do even better, and diagonalize the reflection matrix $\\mathbf{R}$ with a bi-unitary transformation using the arbitrary unitary $\\mathbf{M}^{\\pm}$ transformations. Then the fields are manifestly decoupled at infinity, both at the level of the reflection matrix and the transmission matrix. As a consequence, in the decoupled basis, an incident wave is reflected back in the same decoupled mode without interference with the other mode. Finally, the transmission factors are simply the eigenvalues of $\\mathbf{T}$, since they are each associated with a decoupled component.\n\nFurthermore, one can use the conservation law for the flux, to find an alternative expression for the transmission matrix, at the horizon (this will be useful to control numerical errors). The total flux at the horizon is\n\\begin{equation}\\label{eq:FluxH}\n\\mathcal{F}^{\\mathrm{coupled}}_H=\\left(\\mathbf{h^-}\\right)^\\dagger\\mathbf{h^-} \\; ,\n\\end{equation}\nwhere the $h^-_i$ coefficients are linear combinations of the two independent $\\nu_i$ coefficients ($i=0,1$), given in the Appendix, Eqs.~\\eqref{eq:NPhminus}. Eq.~\\eqref{eq:FluxH} establishes the important point that the flux is positive definite, so the transmission factors must be positive definite (as expected since there is no superradiance in Schwarzschild spacetime). Finally, using the relation between $\\mathbf{y}^-$ and $\\mathbf{h}^-$ through\\footnote{Note that the relation between $\\mathbf{h}^-$ and $(\\mathbf{M}^-)^{-1}\\mathbf{y}^-$ can be made diagonal using $\\mathbf{M}^{-}_H$, so the problem is also decoupled at the horizon.} $\\mathbf{S}^{--}$, we find\n\\begin{equation}\\label{eq:FluxH2}\n\\mathbf{T}=(\\mathbf{S}^{--}\\mathbf{S}^{\\dagger--})^{-1} \\; .\n\\end{equation}\nOnce we have obtained the transmission factors, the number and energy fluxes are given by the standard result\n\\begin{equation}\\label{eq:HawkFlux}\n\\dfrac{d\\left\\{N,E\\right\\}}{dt d\\omega}=\\dfrac{1}{2\\pi}\\sum_\\ell\\sum_\\zeta \\dfrac{\\left\\{1,\\omega \\right\\}}{\\exp(\\omega\/T_H)-1}d_\\zeta \\mathbb{T}_{\\zeta} \\ ,\n\\end{equation}\nwhere $\\zeta$ is a label running over the final set of decoupled scalar modes and the transverse mode, and $d_\\zeta$ are the degeneracies of the corresponding spherical harmonics. Labeling the scalar and vector harmonic degeneracies by $d_S$ and $d_V$ respectively we have \\cite{Ishibashi:2011ws}\n\\begin{eqnarray}\\label{eq:degen}\nd_S&=& \\dfrac{(n+2\\ell-1)(n+\\ell-2)!}{(n-1)!\\ell!}\\ , \\\\\nd_V&=& \\dfrac{(n+2\\ell-1)(n+\\ell-1)(n+\\ell-3)!}{(\\ell+1)(\\ell-1)!(n-2)!} \\; .\n\\end{eqnarray}\nThe Hawking temperature in horizon radius units is\n\\begin{equation}\nT_H=\\dfrac{n-1}{4\\pi} \\; .\n\\end{equation}\n\n\n\\section{Results}\n\\label{sec:resultsNP}\nIn this section we present a selection of numerical results to illustrate the behavior of the transmission factors and the corresponding Hawking fluxes. To integrate the coupled and decoupled radial equations, we first wrote test codes in \\textsc{mathematica} and then a code in the \\textsc{c++} language, using the numerical integration routines of the Gnu Standard Library (GSL). Besides using different programming frameworks we have also tested different integration strategies which all agreed within relative numerical errors smaller than 0.1~\\%. In fact, most of our numerical points have a precision which is one order of magnitude better. To check numerical errors we have integrated the radial equations up to a large radius of typically $r=10^4r_H$ and varied this up to a factor of 3 to check the precision. Furthermore we have used the two expressions for the transmission factor from Eqs.~\\eqref{eq:FluxInfT} and~\\eqref{eq:FluxH2} which agree within the quoted precision for almost all energies. The exception is for small energy, where the first definition converges poorly. This can be explained by a simple analysis of propagation of errors combined with the fact that the $\\mathbf{y}^{\\pm}$ coefficients grow very fast as we decrease energy, thus requiring a very large precision for some fine cancellations to occur. The second expression is thus more natural in that limit since it does not need such cancellations and does not require such large precision.\n\nWe have generated several samples of transmission factors, some of which are displayed in Fig.~\\ref{fig:Tfacs}. Hereafter, we shall denominate the partial waves associated to the different modes of the Proca field by $\\ell_1,\\ell_2,\\ell_T$ and $\\ell=0$, where $\\ell_1,\\ell_2$ correspond to the two coupled modes described by Eqs.~\\eqref{originalsys1} and~\\eqref{originalsys2}, $\\ell_T$ to the decoupled mode described by Eq.~\\eqref{transverse} and $\\ell=0$ to the $\\kappa_s=0$ mode, described by Eq.~\\eqref{k0M}. Moreover, partial waves associated to the Maxwell field shall be denoted by $\\ell_E$, and are described by Eq.~\\eqref{Max}.\n\\afterpage{\n\\begin{figure}[t]\n\\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M001n2Lvar.pdf}\n\\hspace{-1.5mm}\n\\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M001n3Lvar.pdf}\n\\hspace{-2.5mm}\n\\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M001n4Lvar.pdf} \\vspace{3mm}\\\\\n\\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M03n2Lvar.pdf} \\hspace{-2.5mm} \\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M03n3Lvar.pdf} \\hspace{-2.5mm} \\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M03n4Lvar.pdf} \\vspace{3mm} \\\\\n\\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M1n2Lvar.pdf} \\hspace{-2.5mm} \\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M1n3Lvar.pdf} \\hspace{-2.5mm} \\includegraphics[scale=0.68,clip=true,trim= 0 0 0 0]{Figs\/ch4\/M1n4Lvar.pdf}\n\\caption{\\label{fig:Tfacs} {\\em Transmission factors:} The three rows of panels, show the first few partial waves contributing to the Hawking spectrum. Each row corresponds to a fixed mass and each column to a fixed dimension. In particular, the first row shows the small mass limit of the Proca theory in order to compare it with Maxwell's theory.\n}\n\\end{figure}\n\\clearpage\n}\n\nIn the top row panels of Fig.~\\ref{fig:Tfacs}, we show the partial wave contributions for $n=2,3,4$ in the zero mass limit. Some general properties are as follows. The $\\mathbb{T}_\\ell$ curve becomes shifted towards higher frequencies both as $\\ell$ is increased, for $n$ fixed, and as $n$ is increased, for $\\ell$ fixed. The former can be understood from standard geometrical optics arguments. Moreover, for this choice, there is always a numerical coincidence between one of the partial waves ($\\ell_1$) obtained from the two coupled fields and the electromagnetic partial wave $\\ell_E$. The $\\ell=0$ and $\\ell_2$ modes are always absent in the Maxwell theory, so they can be associated with the longitudinal polarization of the massive vector field. Similarly, the $\\ell_T$ and $\\ell_1$ partial waves are associated with the transverse polarizations of the field. A qualitative dependence on dimension is that for $n=2$, $\\ell_T$ and $\\ell_1$ (or $\\ell_E$) modes are all equal. Curiously, this is in agreement with the fact that they describe the same number of transverse degrees of freedom as can be seen from the degeneracies~\\eqref{eq:degen} specialized for $n=2$. This degeneracy is lifted for $n>2$.\n\nFor non-zero mass (middle and bottom row panels of Fig.~\\ref{fig:Tfacs}), the degeneracy observed for $n=2$ in the massless limit is lifted. Also, we observe, for all $n$, that modes with higher $\\ell$ partial waves  (especially $\\ell_1$ modes) become a more dominant contribution at lower energies, as compared to lower $\\ell$ partial waves of other modes. In particular for $\\mu_p=1$, the transmission factor for $\\ell_1$ becomes the largest for small energy. This effect of excitation of sub-dominant partial waves is well known to exist for example as we increase $n$ (and we can also observe such effect in our plots) as well as with the introduction of BH rotation \\cite{sampaio2010production}. If this effect persists cumulatively on a rotating background, then we may have enhanced angular correlations for massive Proca fields emitted from the BH, since higher $\\ell$ partial waves are less uniform.\n\nAnother outstanding point is that for large mass, when $n=2, 3$, it can be seen that the transmission factor starts from a constant non-zero value at the threshold $\\omega=\\mu_p$ ($k=0$), at least for small $\\ell$. We have checked that this does not happen for $n\\geq 4$ for masses as large as $\\mu_p=10\\sim 15$, where the curves always asymptote smoothly to zero at $k=0$. Note that the parameter in the radial equations is $\\mu_p^2$ so these are very large masses. A possible explanation for this phenomenon can be motivated from considerations about the range of the gravitational field in Rutherford scattering. In $n=2$, the total cross-section for Rutherford scattering diverges, so the Newtonian gravitational potential is long ranged. This means that the effective size of the gravitational potential is infinite. The same happens in $n=3$ but only at zero momentum $k=0$. This indicates that a possible reason is that an incident wave at infinity with a very small momentum will still be sufficiently attracted by the gravitational field so that a constant non-zero fraction is still absorbed by the potential. In particular we note that some of the radial equations are similar in form to those obeyed by massive scalar and massive fermion fields, so the same effect exists for such fields. To our knowledge, this feature has not been noted or discussed in the literature. The only exception is the paper by Nakamura and Sato~\\cite{Nakamura:1976nc} in four dimensions, where it is claimed that the reflection factor for a scalar field always goes to $1$ at $\\omega=\\mu_s$ (and thus the transmission factor goes to zero). Their result seems, however, inconsistent with Figs.~1,~2 and~3 of the paper by Page~\\cite{Page:1977um} (also in four dimensions), where the Hawking fluxes for massive fermions become constant at the $k=0$ threshold (in agreement with our result).\n\nOnce we obtain the transmission factors, the computation of the Hawking fluxes~\\eqref{eq:HawkFlux} follows straightforwardly by summing up partial waves with the appropriate degeneracy factors~\\eqref{eq:degen}. We have chosen to show the flux of number of particles. The flux of energy has similar features and is simply related by multiplying each point in the plots by $\\omega$.\n\nIn Fig.~\\ref{fig:NfluxM0} we compare the Hawking fluxes of the Maxwell theory with the small mass limit of the Proca theory. For the particular case of $n=2$ we have reproduced the results by Page~\\cite{Page:1976ki} for the electromagnetic field and found very good agreement.  All panels show a red solid curve corresponding to the total Hawking flux summed up over partial waves. The partial waves included in the sum are also represented, scaled up by the appropriate degeneracy factor. As claimed in the discussion of the transmission factors, as we increase $n$, partial waves with larger $\\ell$ become more important for both Maxwell and Proca fields. One can clearly see that there is a large contribution to the total flux from the longitudinal degrees of freedom, since the vertical scales are larger for the Proca field. In particular the $\\ell=0$ mode enhances the spectrum greatly at small energies. Note that these extra contributions associated with the longitudinal degrees of freedom cannot in general (for arbitrary mass) be described by a scalar field, since there is always a contribution from the coupled modes $\\ell_1,\\ell_2$. That is, however, the approximation done so far in BH event generators, where the $W$ and $Z$ fields Hawking spectra in use are those of the electromagnetic field (for transverse polarizations) and a scalar field (for the longitudinal polarization). Thus, our methods can be readily applied to improve this phenomenological modeling.\n\\begin{figure}[t]\n\\includegraphics[scale=0.65,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxMaxwelln2.pdf}\\hspace{0mm}  \\includegraphics[scale=0.65,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxMaxwelln3.pdf} \\hspace{-2.3mm} \\includegraphics[scale=0.64,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxMaxwelln4.pdf} \\vspace{0mm}\\\\\n\\includegraphics[scale=0.653,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxM001n2.pdf} \\hspace{-2.5mm} \\includegraphics[scale=0.653,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxM001n3.pdf} \\hspace{-2.5mm} \\includegraphics[scale=0.653,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxM001n4.pdf} \\vspace{0mm}\n\\caption{\\label{fig:NfluxM0} {\\em Number fluxes for $\\mu_p=0$ (top panels) and $\\mu_p\\rightarrow 0$ (bottom panels):} The red solid curve of the top panels shows the Hawking flux of particles summed over the dominant partial waves for the Maxwell theory. The different partial waves are multiplied by the corresponding degeneracies. In the bottom panels the small (but non zero) $\\mu_p$ limit of the Proca theory is shown for comparison. The $\\oplus$ symbol denotes the addition of modes which are numerically equal.}\n\\end{figure}\n\nIn Fig.~\\ref{fig:NfluxComparison} we show the variation of the total number flux with $n$ and $\\mu_p$. The left panel shows the expected variation with $\\mu_p$: that the flux not only gets cutoff at the energy threshold $\\omega=\\mu_p$, but it is also suppressed with $\\mu_p$ (the same holds for $n>2$). This is the same behavior as found in~\\cite{Sampaio:2009tp,Sampaio:2009ra}. As pointed out already, in event generators massive vector particles are modeled using the Hawking fluxes for the Maxwell field and a massless scalar, with a cutoff at the mass threshold. In \\cite{Sampaio:2009ra,Sampaio:2009tp} it was shown that simply imposing a sharp cut-off on the fluxes of massless scalars and fermions over-shoots the real amount of Hawking radiation emitted in the massive scalar and fermion channel. Qualitative inspection of our results suggests a similar effect for the $W$ and $Z$ channels in the evaporation. A quantitative comparison, however, requires a consideration of a Proca field confined to a thin brane, which will be studied in Chapter~\\ref{ch:ChargedP}. The middle and right panels show variation with $n$. In addition to the well known large scaling of the  area under the curve and the shift of the spectrum to larger energies, we can also see that more partial waves start contributing to the shape of the curve which becomes more wavy. This is particularly true because the degeneracy factors for fixed $\\ell$ increase rapidly with $n$, which is a consequence of the larger number of polarizations available for a vector boson in higher dimensions. Finally, regarding $n=2,3$ we confirm the feature that the flux becomes a constant at $k=0$. This can be seen more clearly in the right panel in a logarithmic scale where the lines for $n\\geq 4$ curve down very sharply around that point, whereas for $n=2,3$ they tend to a constant.\n\\begin{figure}[t]\n\\includegraphics[scale=0.652,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxMVarn2.pdf} \\hspace{-2.5mm} \\includegraphics[scale=0.652,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxM1nVar.pdf} \\hspace{-2.5mm} \\includegraphics[scale=0.652,clip=true,trim= 0 0 0 0]{Figs\/ch4\/NfluxM1nVarLog.pdf} \\vspace{0mm}\n\\caption{\\label{fig:NfluxComparison} {\\em Number fluxes for various $\\mu_p$ and $n$:} (Left panel) Variation of the flux of particles for fixed $n=2$ and variable mass. (Middle and right panels) Variation of the flux with $n$ in a linear and logarithmic scale respectively. The logarithmic scale shows more clearly that the limiting flux at $k=0$ is finite for $n=2,3$.}\n\\end{figure}\n\n\n\\section{Summary}\n\\label{sec:sumProneu}\n\n\nIn this chapter, we have studied Hawking radiation for a neutral Proca field, by solving the coupled wave equations as well as decoupled equations numerically, on a $D$-dimensional Schwarzschild BH. Our results exhibit distinctive features as we introduce the mass term, such as the lifting of the degeneracy of the two transverse modes in four dimensions, the appearance of longitudinal mode contributions (absent for Maxwell's theory) and in particular the $s$-wave. As we have shown, there is a large contribution from the longitudinal modes, to the Hawking fluxes. \n\nOne feature that appears not to have been discussed in the literature is that in four and five spacetime dimensions, the transmission factor has a non-vanishing value in the limit of zero spatial momentum. We also find the expected suppression with mass of the Proca field, but perhaps the most relevant feature is to notice the increasing importance of the longitudinal modes and larger $\\ell$ partial waves.\n\nOur results could be applied to improve the model used in the \\textsc{charybdis2} Monte Carlo event generator \\cite{Frost:2009cf}. This simulates the production and decay of higher dimensional BHs in parton-parton collisions, a scenario which is being constrained at the second run of the LHC. It is therefore quite timely to improve the phenomenology of these models. Indeed our knowledge of Hawking evaporation process can still be improved greatly through the numerical study of various wave equations in BH backgrounds, which approximate the ones that could be produced at the LHC. This is illustrated by our results in this study, which alert for the importance of modelling the longitudinal modes correctly, instead of treating them as decoupled scalars as in current BH event generators.\n\n\n\n\n\\chapter{Preliminaries}\n\\label{ch:prelim}\n\n\nWe start in this chapter by introducing the mathematical tools to deal with perturbations of test fields around BHs, as the foundation to perform the study for Proca and Maxwell fields. Two types of perturbation methods, the Kodama-Ishibashi formalism and the Newman-Penrose formalism, will be illustrated, respectively, in the following.\n\nThroughout this thesis we will use the signature $(-,+,...,+)$ and natural units $G=c=\\hbar=1$, unless explicitly stated otherwise.\n\n\n\\section{Kodama-Ishibashi formalism}\n\\label{sc:KI}\nThe Kodama-Ishibashi (KI) formalism~\\cite{Kodama:2000fa,Kodama:2003jz,Ishibashi:2003ap,Kodama:2003kk} (for a review see~\\cite{Kodama:2007ph,Ishibashi:2011ws}), is the generalization of the Regge-Wheeler-Zerilli formalism~\\cite{Regge:1957td,Zerilli:1970se} to higher dimensions. This method is applicable to any higher dimensional spacetime with maximal symmetry whose manifold structure can be locally written as a warped product between a Lorentzian manifold and an Einstein space. The basic idea of this method is to classify the perturbations into different types (scalar, vector and tensor types), based on their tensorial behavior in the Einstein space.\n\nTo be specific, let us consider the following gravitational background with the manifold structure $\\mathcal{M}=\\mathcal{N}\\times\\mathcal{K}$ in the form~\\cite{Kodama:2000fa,Kodama:2007ph,Ishibashi:2011ws}\n\\begin{equation}\ng_{MN}dx^Mdx^N=h_{ab}(y)dy^a dy^b+r(y)^2d\\sigma_n^2 \\;,\\label{KI:metric}\n\\end{equation}\nwhere $x^M=(y^a,z^j)$. Note that the Lorentzian manifold is denoted by $\\mathcal{N}$ with metric $h_{ab}$, and the Einstein space is denoted by $\\mathcal{K}$ with constant curvature $K (K=0,\\pm1)$ and metric $\\sigma_{ij}$,\n\\begin{equation}\nd\\sigma_n^2=\\sigma_{ij}(z)dz^idz^j \\; .\n\\end{equation}\nThen the Riemann tensor and Ricci tensor on an Einstein space are given by\n\\begin{equation}\n\\hat{R}_{ijkl}=K (\\sigma_{ik}\\sigma_{jl}-\\sigma_{il}\\sigma_{jk})\\;,\\;\\;\\;\\;\\;\\;\\hat{R}_{ij}=(n-1)K \\sigma_{ij}\\;.\n\\end{equation}\nWe use indices $\\{a,b,c,\\ldots\\}$ for the first set of coordinates, $\\{y^a\\}$, spanning on the $m$-dimensional space with metric $h_{ab}(y)$; and indices $\\{i,j,k,\\ldots\\}$ for the second set of coordinates, $\\{z^i \\}$, spanning on the $n$-dimensional Einstein space. Then the spacetime dimension is $d=m+n$. We denote the covariant derivatives, the Christoffel connection coefficients and the Riemann tensors on the manifolds $\\{\\mathcal{M}$, $\\mathcal{N}$, $\\mathcal{K}\\}$, by $\\{\\nabla_M$, $D_a$, $\\hat{D}_i\\}$, $\\{\\Gamma^M_{NL}, \\bar{\\Gamma}^a_{bc}, \\hat{\\Gamma}^i_{jk}\\}$, and $\\{R_{MNLS}, \\bar{R}_{abcd}, \\hat{R}_{ijkl}\\}$, respectively. We also define the Laplace operator on the Einstein space as $\\hat{\\Delta}=\\hat{D}_i\\hat{D}^i$.\nThe metric form in Eq.~\\eqref{KI:metric} covers several interesting cases such as $2+n$-dimensional spherically symmetric BHs or a singly rotating BH in $4+n$-dimensions.\n\nThe expressions of $\\Gamma^M_{NL}$ and $R^M_{\\;\\;NLS}$ can be written in terms of the corresponding quantities on the manifold $\\mathcal{N}$ with metric $h_{ab}(y)$ and on the Einstein space with metric $\\sigma_{ij}$~\\cite{Kodama:2000fa}, i.e.\n\\begin{equation}\n\\Gamma^a_{bc}=\\bar{\\Gamma}^a_{bc}\\;,\\;\\;\\;\\Gamma^a_{ij}=-rD^ar\\sigma_{ij}\\;,\\;\\;\\;\\Gamma^i_{aj}=\\dfrac{D_ar}{r}\\delta^i_j\\;,\\;\\;\\;\\Gamma^i_{jk}=\\hat{\\Gamma}^i_{jk}\\;,\n\\end{equation}\nwhere the other components of $\\Gamma^M_{NL}$ vanish, and\n\\begin{align}\n&R^a_{\\;\\;bcd}=\\bar{R}^a_{\\;\\;bcd}\\;,\\;\\;\\;R^i_{\\;\\;ajb}=-\\dfrac{D_aD_br}{r}\\delta^i_j\\;,\\;\\;\\;R^a_{\\;\\;ibj}=-\\dfrac{D^aD_br}{r}g_{ij}\\;,\\nonumber\\\\\n&R^i_{\\;\\;jkl}=(K-D_arD^ar)(\\delta^i_k\\sigma_{jl}-\\delta^i_l\\sigma_{jk})\\;.\n\\end{align}\nThen the Ricci tensors and Einstein tensors can be derived directly~\\cite{Kodama:2000fa}\n\\begin{equation}\nR_{ab}=\\bar{R}_{ab}-\\dfrac{n}{r}D_aD_br\\;,\\;\\;\\;R_{ai}=0\\;,\\;\\;\\;R_{ij}=\\left(-\\dfrac{\\bar{\\Box} r}{r}+(n-1)\\dfrac{K-D_arD^ar}{r^2}\\right)g_{ij}\\;,\n\\end{equation}\nwhere we have defined $\\bar{\\Box}\\equiv D^aD_a$, and\n\\begin{align}\n&G_{ab}=\\bar{G}_{ab}-\\dfrac{n}{r}D_aD_br-\\left(\\dfrac{n(n-1)}{2r^2}(K-D_arD^ar)-\\dfrac{n}{r}\\bar{\\Box} r\\right)g_{ab}\\;,\\\\\n&G_{ij}=\\left(-\\dfrac{\\bar{R}}{2}-\\dfrac{(n-1)(n-2)}{2r^2}(K-D_arD^ar)+\\dfrac{n-1}{r}\\bar{\\Box} r\\right)g_{ij}\\;,\\\\\n&G_{ai}=0\\;,\n\\end{align}\nwith the definition $G_{MN}=R_{MN}-\\tfrac{1}{2}g_{MN}R$.\n\nTo write down the equations of motion, we shall decompose the perturbations in terms of their tensorial harmonics on $\\mathcal{K}$~\\cite{Kodama:2007ph,Ishibashi:2011ws}.\\\\\nFor a vector field $v_i$, it can be uniquely decomposed into a scalar field $v^{(s)}$ and a transverse vector field $v^{(t)}_i$ as\n\\begin{equation}\nv_i=\\hat{D}_iv^{(s)}+v^{(t)}_i\\;,\\;\\;\\;\\;\\;\\;\\hat{D}_iv^{(t)i}=0\\;,\\label{intro:decom}\n\\end{equation}\nwhere $v^{(s)}$ and $v^{(t)}_i$ satisfy the corresponding scalar and vector eigenvalue equations\n\\begin{align}\n&(\\hat{\\Delta}+\\kappa_s^2)v^{(s)}=0\\;,\\label{intro:eigenscalar}\\\\\n&(\\hat{\\Delta}+\\kappa_v^2)v^{(t)}_i=0\\;,\\label{intro:eigenvector}\n\\end{align}\non an Einstein space with spherical topology, with $\\kappa_s^2=\\ell(\\ell+n-1)$ and \\mbox{$\\kappa_v^2=\\ell(\\ell+n-1)-1$}. Note that the angular momentum quantum number, $\\ell$, starts from zero in the scalar eigenvalue $\\kappa_s$ and one in the vector eigenvalue $\\kappa_v$, respectively. Then taking a derivative $\\hat{D}_i$ on Eq.~\\eqref{intro:decom}, we have the relation\n\\begin{equation}\n\\hat{\\Delta}v^{(s)}=\\hat{D}_iv^i\\;,\n\\end{equation}\nwhich determines $v^{(s)}$ from Eq.~\\eqref{intro:eigenscalar}, and $v^{(t)}_i$ from Eq.~\\eqref{intro:decom} after $v^{(s)}$ is obtained.\n\nThe scalar and vector harmonic functions on $n$-spheres, used in Eqs.~\\eqref{intro:eigenscalar} and~\\eqref{intro:eigenvector}, are defined as follows~\\cite{Ishibashi:2011ws}. Let us denote the homogeneous cartesian coordinates on $n$-spheres by $\\Omega^A$($A=1,\\cdots,n+1$), and define the function $Y_{{\\bf a}}$ by\n\\begin{equation}\nY_{{\\bf a}}(\\Omega)=a_{A_1\\cdots A_\\ell}\\Omega^{A_1}\\cdots \\Omega^{A_\\ell}\\;,\n\\label{intro:ScalatrHarmonicFunction}\n\\end{equation}\nin terms of a constant tensor ${\\bf a}=(a_{A_1\\cdots A_\\ell})$ ($A_1,\\cdots,A_\\ell=1,\\cdots,n+1$). Then $Y_{{\\bf a}}$ is a scalar harmonic function with the eigenvalue $\\kappa_s^2$ if and only if $\\bf{a}$ satisfies the conditions\n\\begin{equation}\na_{A_1\\cdots A_\\ell}=a_{(A_1\\cdots A_\\ell)}\\;,\\;\\;\\;a_{A_1\\cdots A_{\\ell-2}}{}^B{}_B=0\\quad (\\ell\\ge2)\\;.\n\\end{equation}\nSimilarly, we can define the vector field $V_{{\\bf b}}^i$ by\n\\begin{equation}\nV_{{\\bf b}}^i=b_{A_1\\cdots A_\\ell ;B}\\Omega^{A_1}\\cdots\\Omega^{A_\\ell }\n                   \\hat{D}^i\\Omega^B \\;,\n\\label{intro:HarmonicVector}\n\\end{equation}\nin terms of a constant tensor ${\\bf b}=(a_{A_1\\cdots A_\\ell ;B})$($A_1,\\cdots,A_\\ell,B=1,\\cdots,n+1$). Then, $V_{{\\bf b}}^i$ is a vector harmonic function on $n$-spheres with eigenvalue $\\kappa_v^2$, if and only if the constant tensor ${\\bf b}$ satisfies the conditions\n\\begin{equation}\nb_{A_1\\cdots A_\\ell ;B}=b_{(A_1\\cdots A_\\ell );B}\\;,\\;\\;\\;b_{A_1\\cdots A_{\\ell -2}i}{}^i{}_{;B}=0\\;,\\;\\;\\;b_{(A_1\\cdots A_\\ell ;A_{\\ell +1})}=0\\;.\n\\end{equation}\nFor more details on these harmonic functions on $n$-spheres and the corresponding properties, we refer readers to~\\cite{Ishibashi:2011ws}.\n\n\n\n\n\n\n\n\n\n\\section{Newman-Penrose formalism}\n\\label{sc:NP}\nThe Newman-Penrose formalism~\\cite{Newman:1961qr}, as the name indicates, was developed by Newman and Penrose in 1962, as an alternative way to formulate field equations, such as the Einstein equations and the Maxwell equations. This formalism is extremely useful in various contexts in GR, for example to construct exact solutions of the Einstein equations~\\cite{Stephani:2003tm}, to study perturbations of massless test fields on various BH backgrounds~\\cite{Chandrasekhar:1985kt,Frolov:1998wf}, and to extract gravitational radiation in numerical relativity~\\cite{baumgarte2010numerical}. For the problems we are interested in this thesis, we focus on the application of this formalism in the context of perturbation theory.\n\nAs first exhibited in the celebrated work of Teukolsky~\\cite{Teukolsky:1972my}, linear perturbations of gravitational and electromagnetic fields on the Kerr background both separate and decouple, in terms of the Newman-Penrose variables. This was subsequently generalized to rotating BHs with a cosmological constant~\\cite{Khanal:1983vb,Wu:2003qc,Yoshida:2010zzb,Dias:2012pp}. In this section, we review the Newman-Penrose formalism with application to the Maxwell field on Kerr-AdS BHs, and further present some new ingredients which have not been derived in the literature, in the presence of a cosmological constant, with details, including\n\\begin{itemize}\n\\item the derivation of Teukolsky-Starbinski identities, which was given in~\\cite{Wu:2003qc} without proof,\n\\item the derivation for a complete set of solutions for the Maxwell field, in particular for $\\Phi_1$, which is relevant for proving Appendix~\\ref{app:angmomflux}.\n   \n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Basics}\nIn order to introduce the Newman-Penrose formalism, we first define a complex null tetrad $\\{l^\\mu,n^\\mu,m^\\mu,\\bar{m}^\\mu\\}$, where the normalization conditions\n\\begin{equation}\nl_\\mu n^\\mu=-1\\;,\\;\\;\\;\\;\\;\\;m_\\mu \\bar{m}^\\mu=1\\;,\\label{tetrad1}\n\\end{equation}\nare satisfied, and all the other scalar products vanish. Note that here $\\bar{m}^\\mu$ is the complex conjugate of $m^\\mu$.\nThe tetrad is related with the metric\\footnote{Note that this relation depends on the conventions.}\n\\begin{equation}\ng_{\\mu\\nu}=-l_\\mu n_\\nu-l_\\nu n_\\mu+m_\\mu\\bar{m}_\\nu+m_\\nu\\bar{m}_\\mu\\;.\\label{teme1}\n\\end{equation}\n\nThen spin coefficients are defined in terms of the tetrad\n\\begin{align}\n-\\kappa&=l_{\\mu;\\nu}m^\\mu l^\\nu\\;,\\;\\;\\;-\\rho=l_{\\mu;\\nu}m^\\mu \\bar{m}^\\nu\\;,\\;\\;\\;-\\sigma=l_{\\mu;\\nu}m^\\mu m^\\nu\\;,\\;\\;\\;-\\tau=l_{\\mu;\\nu}m^\\mu n^\\nu\\;,\\nonumber\\\\\n\\mu&=n_{\\mu;\\nu}\\bar{m}^\\mu m^\\nu\\;,\\;\\;\\;\\nu=n_{\\mu;\\nu}\\bar{m}^\\mu n^\\nu\\;,\\;\\;\\;\\;\\;\\;\\lambda=n_{\\mu;\\nu}\\bar{m}^\\mu \\bar{n}^\\nu\\;,\\;\\;\\;\\;\\;\\;\\pi=n_{\\mu;\\nu}\\bar{m}^\\mu l^\\mu\\;,\\nonumber\\\\\n-\\epsilon&=\\frac{1}{2}(l_{\\mu;\\nu}n^\\mu l^\\nu-m_{\\mu;\\nu}\\bar{m}^\\mu l^\\nu)\\;,\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;-\\beta=\\frac{1}{2}(l_{\\mu;\\nu}n^\\mu m^\\nu-m_{\\mu;\\nu}\\bar{m}^\\mu m^\\nu)\\;,\\nonumber\\\\\n-\\gamma&=\\frac{1}{2}(l_{\\mu;\\nu}n^\\mu n^\\nu-m_{\\mu;\\nu}\\bar{m}^\\mu n^\\nu)\\;,\\;\\;\\;\\;\\;\\;\\;\\;-\\alpha=\\frac{1}{2}(l_{\\mu;\\nu}n^\\mu \\bar{m}^\\nu-m_{\\mu;\\nu}\\bar{m}^\\mu \\bar{m}^\\nu)\\;.\\label{spincoeffs}\n\\end{align}\n\nNext we introduce the projection of the covariant derivatives in the null tetrad vectors, with the following notations\n\\begin{equation}\nD=l^\\mu\\partial_\\mu\\;,\\;\\;\\;\\Delta=n^\\mu\\partial_\\mu\\;,\\;\\;\\;\\delta=m^\\mu\\partial_\\mu\\;,\\;\\;\\;\\bar{\\delta}=\\bar{m}^\\mu\\partial_\\mu\\;.\\label{deriv}\n\\end{equation}\n\nWith the above spin coefficients in Eq.~\\eqref{spincoeffs} and the directional derivatives in Eq.~\\eqref{deriv} at hand, one may rewrite the Maxwell field equations.\nFor the reader who is interested in the equations of motion for other spin fields in this formalism, such as the gravitational field and the Dirac field, a detailed account can be found in~\\cite{Chandrasekhar:1985kt,Frolov:1998wf}.\n\nIn the Newman-Penrose formalism, the Maxwell tensor is decomposed into three complex scalars\n\\begin{equation}\n\\phi_0=F_{\\mu\\nu}l^{\\mu}m^{\\nu}\\;,\\;\\;\\;\\;\\;\\;\\phi_1=\\frac{1}{2}F_{\\mu\\nu}(l^{\\mu}n^{\\nu}+\\bar{m}^{\\mu}m^{\\nu})\\;,\\;\\;\\;\\;\\;\\;\\phi_2=F_{\\mu\\nu}\\bar{m}^{\\mu}n^{\\nu}\\;.\\label{Maxwellscalars}\n\\end{equation}\nThen the Maxwell equations become\n\\begin{align}\nD\\phi_1-\\bar{\\delta}\\phi_0&=(\\pi-2\\alpha)\\phi_0+2\\rho\\phi_1-\\kappa\\phi_2\\;,\\nonumber\\\\\nD\\phi_2-\\bar{\\delta}\\phi_1&=-\\lambda\\phi_0+2\\pi\\phi_1+(\\rho-2\\epsilon)\\phi_2\\;,\\nonumber\\\\\n\\delta\\phi_1-\\Delta\\phi_0&=(\\mu-2\\gamma)\\phi_0+2\\tau\\phi_1-\\sigma\\phi_2\\;,\\nonumber\\\\\n\\delta\\phi_2-\\Delta\\phi_1&=-\\nu\\phi_0+2\\mu\\phi_1+(\\tau-2\\beta)\\phi_2\\;,\\label{Maxwelleq1}\n\\end{align}\nwhere the differential operators appearing on the left hand side are defined in Eq.~\\eqref{deriv} and the spin coefficients on the right hand side are given by~Eq.\\eqref{spincoeffs}.\n\nTo obtain the explicit form of the Maxwell equations, we shall specify a background geometry. For that purpose and for later application in the problems we are interested in, we first review the Kerr-AdS spacetimes in the next subsection.\n\\subsection{Kerr-AdS black holes}\n\\label{subsec:KerrAdS}\nIn an asymptotically AdS background, the most general stationary and axisymmetric BH solution of the four dimensional Einstein-AdS system, is the Kerr-AdS BH. It was found by Carter~\\cite{Carter:1968ks} firstly, a few years after the finding of the Kerr solution.\n\nThe line element for a Kerr-AdS BH, in Boyer-Lindquist coordinates, can be written as\n\\begin{equation}\nds^2=-\\dfrac{\\Delta_r}{\\rho^2\\Xi^2}\\Big(dt-a\\sin^2\\theta d\\varphi\\Big)^2+\\rho^2\\left(\\dfrac{dr^2}{\\Delta_r}+\\dfrac{d\\theta^2}{\\Delta_\\theta}\\right)+\\dfrac{\\Delta_\\theta \\sin^2\\theta}{\\rho^2\\Xi^2}\\Big(a\\,dt-(r^2+a^2)d\\varphi\\Big)^2\\;,\\label{RKerrAdS}\n\\end{equation}\nwith metric functions\n\\begin{align}\n&\\rho^2\\equiv\\bar{\\rho}\\bar{\\rho}^\\ast=r^2+a^2\\cos^2\\theta\\;,\\;\\;\\Delta_r=\\Big(r^2+a^2\\Big)\\left(1+\\frac{r^2}{L^2}\\right)-2Mr\\;,\\;\\;\\nonumber\\\\\n&\\Delta_\\theta=1-\\dfrac{a^2\\cos^2\\theta}{L^2}\\;,\\;\\;\\Xi=1-\\dfrac{a^2}{L^2}\\;,\\label{RKerrAdSmetric}\n\\end{align}\nwhere $\\bar{\\rho}^\\ast$ is the complex conjugate of $\\bar{\\rho}$, and $\\bar{\\rho}=r+ia\\cos\\theta$.\nThe other parameters shown in Eq.~\\eqref{RKerrAdSmetric} include, $L$, which is the AdS radius; $M$ and $a$, which are the mass and spin parameters and relate to the BH energy and angular momentum.\n\nIn this frame, the angular velocity of the event horizon and the Hawking temperature are given by\n\\begin{align}\n&\\Omega_H=\\dfrac{a}{r_+^2+a^2}\\;,\\label{RHorizonV}\\\\\n&T_H=\\dfrac{1}{\\Xi}\\left[\\dfrac{r_+}{2\\pi}\\left(1+\\dfrac{r_+^2}{L^2}\\right)\\dfrac{1}{r_+^2+a^2}-\\dfrac{1}{4\\pi r_+}\\left(1-\\dfrac{r_+^2}{L^2}\\right)\\right]\\;,\\label{RTemp}\n\\end{align}\nwhere the event horizon $r_+$ is determined as the largest root of $\\Delta_r(r_+)=0$. For a given $r_+$, the mass parameter $M$ can be expressed as\n\\begin{equation}\nM=\\dfrac{(r_+^2+a^2)(L^2+r_+^2)}{2r_+L^2}\\;.\\nonumber\n\\end{equation}\nTo require the existence of BHs and to avoid singularities, one may impose the following constraints on the rotation parameter $a$\n\\begin{align}\n&\\dfrac{a}{L}\\leq \\dfrac{r_+}{L} \\sqrt{\\dfrac{3r_+^2+L^2}{L^2-r_+^2}}\\;,\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{for}\\;\\;\\;\\;\\;  \\dfrac{r_+}{L}<\\dfrac{1}{\\sqrt{3}}\\;,\\nonumber\\\\\n&\\dfrac{a}{L}<1\\;,\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{for}\\;\\;\\;\\;\\;  \\dfrac{r_+}{L}\\geq\\dfrac{1}{\\sqrt{3}}\\;,\\label{intro_secKerrAdS_rotation}\n\\end{align}\nwhere the equality sign in the first line corresponds to an extremal BH.\n\nThe Boyer-Lindquist coordinates are convenient to solve the perturbation equations for test fields. These coordinates, however, obscure the structure of the geometry at infinity. In fact, the metric in Eq.~\\eqref{RKerrAdS} describes the Kerr-AdS spacetime in a rotating frame, which can be seen by calculating the angular velocity at infinity~\\cite{Dias:2013sdc}\n\\begin{equation}\n\\Omega_\\infty=\\dfrac{a}{a^2-L^2}\\;,\n\\end{equation}\nwhich is apparently non-zero. In order to obtain a non-rotating frame, which is relevant to BH thermodynamics~\\cite{Gibbons:2004ai}, we can make the following coordinate transformation~\\cite{Henneaux:1985tv,Winstanley:2001nx}\n\\begin{equation}\n\\hat{t}=\\dfrac{t}{\\Xi}\\;,\\;\\;\\;\\hat{\\varphi}=\\varphi+\\dfrac{a}{L^2}\\dfrac{t}{\\Xi}\\;,\\;\\;\\;\\hat{r}^2=\\dfrac{r^2\\Delta_\\theta+a^2\\sin^2\\theta}{\\Xi}\\;,\\;\\;\\;\\hat{r}\\cos\\hat{\\theta}=r\\cos\\theta\\;,\\label{RNTrans}\n\\end{equation}\nso that in these new frame the Kerr-AdS geometry~\\eqref{RKerrAdS} is simply AdS space in the usual spherical coordinates\n\\begin{equation}\nds^2=-\\left(1+\\dfrac{\\hat{r}^2}{L^2}\\right)d\\hat{t}^2+\\left(1+\\dfrac{\\hat{r}^2}{L^2}\\right)^{-1}d\\hat{r}^2+\\hat{r}^2(d\\hat{\\theta}^2+\\sin^2\\hat\\theta d\\hat{\\varphi}^2)\\;,\n\\end{equation}\nwhen $r\\rightarrow\\infty (\\hat{r}\\rightarrow\\infty)$.\nThe angular velocity of the event horizon in these coordinates then becomes\n\\begin{equation}\n\\hat{\\Omega}_H=\\Omega_H\\Xi+\\dfrac{a}{L^2}\\;,\\label{NRHAV}\n\\end{equation}\nwhere $\\Omega_H$, defined in Eq.~\\eqref{RHorizonV}, is measured relative to a rotating observer at infinity.\nSince the metric in coordinates $(\\hat{t},\\hat{r},\\hat{\\theta},\\hat{\\varphi})$ is complicated, and also because we have the coordinate transformations in Eq.~\\eqref{RNTrans}, in practice we always work in Boyer-Lindquist coordinates. Our goal is to solve the Maxwell equations in the frequency domain, therefore it is useful to relate the frequencies in these two different coordinates. This is given by\n\\begin{equation}\ne^{-i\\omega t}e^{im\\varphi}=e^{-i\\hat{\\omega} \\hat{t}}e^{im\\hat{\\varphi}}\\;,\\;\\;\\;\\;\\;\\;\\Rightarrow\\;\\;\\;\\;\\;\\;\\hat{\\omega}=\\omega\\Xi+m\\dfrac{a}{L^2}\\;.\\label{nonrotatingfre\n\\end{equation}\n\n\\subsection{Maxwell equations on Kerr-AdS}\nBy specializing the general Maxwell equations~\\eqref{Maxwelleq1} to the Kerr-AdS background~\\eqref{RKerrAdS}, and using a generalization of the Kinnersley tetrad~\\cite{Khanal:1983vb}, i.e.\n\\begin{align}\nl^\\mu&=\\left(\\dfrac{(r^2+a^2)\\Xi}{\\Delta_r},1,0,\\dfrac{a\\Xi}{\\Delta_r}\\right)\\;,\\;\\;\\;\\;\\;\\;\nn^\\mu=\\dfrac{1}{2\\rho^2}\\Big((r^2+a^2)\\Xi,-\\Delta_r,0,a\\Xi\\Big)\\;,\\nonumber\\\\\nm^\\mu&=\\dfrac{1}{\\sqrt{2\\Delta_\\theta}\\bar{\\rho}}\\left(ia\\Xi\\sin\\theta,0,\\Delta_\\theta,\\dfrac{i\\Xi}{\\sin\\theta}\\right)\\;,\\;\\;\\;\\;\\;\\;\\bar{m}^\\mu=(m^\\mu)^\\ast\\;,\\label{pre:tetrad}\n\\end{align}\nwhere $\\bar{\\rho}=r+ia\\cos\\theta$, one obtains\n\\begin{align}\n\\left(\\mathscr{D}_0+\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)\\Phi_1&=\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_1-\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)\\Phi_0\\;,\\label{Maxwelleq2_1}\\\\\n\\left(\\mathscr{D}_0-\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)\\Phi_2&=\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_0+\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)\\Phi_1\\;,\\label{Maxwelleq2_2}\\\\\n-\\Delta_r\\left(\\mathscr{D}_1^\\dag-\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)\\Phi_0&=\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_0^\\dag+\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)\\Phi_1\\;,\\label{Maxwelleq2_3}\\\\\n-\\Delta_r\\left(\\mathscr{D}_0^\\dag+\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)\\Phi_1&=\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_1^\\dag-\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)\\Phi_2\\;,\\label{Maxwelleq2_4}\n\\end{align}\nwhere\n\\begin{align}\n\\mathscr{D}_n&=\\dfrac{\\partial}{\\partial r}-\\dfrac{i\\Xi K}{\\Delta_r}+\\dfrac{n}{\\Delta_r}\\dfrac{d\\Delta_r}{dr}\\;,\\nonumber\\\\\n\\mathscr{D}_n^\\dag&=\\dfrac{\\partial}{\\partial r}+\\dfrac{i\\Xi K}{\\Delta_r}+\\dfrac{n}{\\Delta_r}\\dfrac{d\\Delta_r}{dr}\\;,\\nonumber\\\\\n\\mathscr{L}_n&=\\dfrac{\\partial}{\\partial\\theta}-\\frac{\\Xi H}{\\Delta_\\theta}+\\dfrac{n}{\\sqrt{\\Delta_\\theta}\\sin\\theta}\\dfrac{d}{d\\theta}\\left(\\sqrt{\\Delta_\\theta}\\sin\\theta\\right)\\;,\\nonumber\\\\\n\\mathscr{L}_n^\\dag&=\\dfrac{\\partial}{\\partial\\theta}+\\frac{\\Xi H}{\\Delta_\\theta}+\\dfrac{n}{\\sqrt{\\Delta_\\theta}\\sin\\theta}\\dfrac{d}{d\\theta}\\left(\\sqrt{\\Delta_\\theta}\\sin\\theta\\right)\\;,\\label{Defoperator}\n\\end{align}\nwith\n\\begin{equation}\nK=\\omega(r^2+a^2)-am\\;,\\;\\;\\;H=a\\omega\\sin\\theta-\\dfrac{m}{\\sin\\theta}\\;,\\label{KHeq}\n\\end{equation}\nand where the following transformations have been made\n\\begin{equation}\n\\phi_0=\\Phi_0\\;,\\;\\;\\;\\phi_1=\\dfrac{1}{\\sqrt{2}\\bar{\\rho}^\\ast}\\Phi_1\\;,\\;\\;\\;\\phi_2=\\dfrac{1}{2(\\bar{\\rho}^\\ast)^2}\\Phi_2\\;.\\label{phirelations}\n\\end{equation}\nNote that the time and azimuthal dependence, $e^{-i\\omega t+im\\varphi}$, has been factored out. By acting with the operator $\\sqrt{\\Delta_\\theta}(\\mathscr{L}_0^\\dag+\\tfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast})$ on Eq.~\\eqref{Maxwelleq2_1}, and with the operator $(\\mathscr{D}_0+\\tfrac{1}{\\bar{\\rho}^\\ast})$ on Eq.~\\eqref{Maxwelleq2_3}, one obtains a decoupled equation for $\\Phi_0$ by eliminating $\\Phi_1$, i.e.\n\\begin{equation}\n[\\sqrt{\\Delta_\\theta}\\mathscr{L}_0^\\dag\\sqrt{\\Delta_\\theta}\\mathscr{L}_1+\\Delta_r\\mathscr{D}_1\\mathscr{D}_1^\\dag+2i\\omega\\Xi\\bar{\\rho}]\\Phi_0=0\\;,\\label{Phi0eq}\n\\end{equation}\nwith aid of the identities\n\\[\n\\begin{cases}\n\\left(\\mathscr{D}_0+\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)\\Delta_r\\left(\\mathscr{D}_1^\\dag-\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)=\\Delta_r\\mathscr{D}_1\\mathscr{D}_1^\\dag+\\dfrac{2i\\Xi K}{\\bar{\\rho}^\\ast}\\;,\\\\\n\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_0^\\dag+\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_1-\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)=\\sqrt{\\Delta_\\theta}\\mathscr{L}_0^\\dag\\sqrt{\\Delta_\\theta}\\mathscr{L}_1\n-\\dfrac{2ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\Xi H\\;.\n\\end{cases}\n\\]\n\nFollowing the same procedure, by acting with the operator $\\sqrt{\\Delta_\\theta}(\\mathscr{L}_0+\\tfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast})$ on Eq.~\\eqref{Maxwelleq2_4} and with the operator $\\Delta_r(\\mathscr{D}_0^\\dag+\\tfrac{1}{\\bar{\\rho}^\\ast})$ on Eq.~\\eqref{Maxwelleq2_2}, $\\Phi_1$ is again eliminated so that we obtain a decoupled equation for $\\Phi_2$\n\\begin{equation}\n[\\sqrt{\\Delta_\\theta}\\mathscr{L}_0\\sqrt{\\Delta_\\theta}\\mathscr{L}_1^\\dag+\\Delta_r\\mathscr{D}_0^\\dag\\mathscr{D}_0-2i\\omega\\Xi\\bar{\\rho}]\\Phi_2=0\\;,\\label{Phi2eq}\n\\end{equation}\nwith aid of the identities\n\\[\n\\begin{cases}\n\\Delta_r\\left(\\mathscr{D}_0^\\dag+\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)\\left(\\mathscr{D}_0-\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)=\\Delta_r\\mathscr{D}_0^\\dag\\mathscr{D}_0-\\dfrac{2i\\Xi K}{\\bar{\\rho}^\\ast}\\;,\\\\\n\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_0+\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_1^\\dag-\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)=\\sqrt{\\Delta_\\theta}\\mathscr{L}_0\\sqrt{\\Delta_\\theta}\\mathscr{L}_1^\\dag\n+\\dfrac{2ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\Xi H\\;.\n\\end{cases}\n\\]\nNow taking\n\\begin{equation}\n\\Phi_0=R_{+1}(r)S_{+1}(\\theta)\\;\\;\\;\\text{and}\\;\\;\\;\\Phi_2=R_{-1}(r)S_{-1}(\\theta)\\;,\\label{decompfields}\n\\end{equation}\nwe finally obtain the Maxwell equations with separated variables\n\\begin{align}\n&\\left(\\Delta_r\\mathscr{D}_1\\mathscr{D}_1^\\dag+2i\\omega\\Xi r\\right)R_{+1}=\\lambda R_{+1}\\;,\\label{Rpluseq}\\\\\n&\\left(\\sqrt{\\Delta_\\theta}\\mathscr{L}_0^\\dag\\sqrt{\\Delta_\\theta}\\mathscr{L}_1-2a\\omega\\Xi\\cos\\theta\\right)S_{+1}=-\\lambda S_{+1}\\;,\\label{Spluseq}\n\\end{align}\nand\n\\begin{align}\n&\\left(\\Delta_r\\mathscr{D}_0^\\dag\\mathscr{D}_0-2i\\omega\\Xi r\\right)R_{-1}=\\lambda R_{-1}\\;,\\label{Rminuseq}\\\\\n&\\left(\\sqrt{\\Delta_\\theta}\\mathscr{L}_0\\sqrt{\\Delta_\\theta}\\mathscr{L}_1^\\dag+2a\\omega\\Xi\\cos\\theta\\right)S_{-1}=-\\lambda S_{-1}\\;,\\label{Sminuseq}\n\\end{align}\nfrom Eqs.~\\eqref{Phi0eq} and~\\eqref{Phi2eq}. Note that $\\lambda$ refers to the separation constant from now on, and should not be confused with the spin coefficient.\nUsing the commutative property $\\Delta_r\\mathscr{D}_{n+1}=\\mathscr{D}_{n}\\Delta_r$, Eq.~\\eqref{Rpluseq} can be rewritten as\n\\begin{equation}\n\\left(\\Delta_r\\mathscr{D}_0\\mathscr{D}_0^\\dag+2i\\omega\\Xi r\\right)\\left(\\Delta_rR_{+1}\\right)=\\lambda \\left(\\Delta_rR_{+1}\\right)\\;,\\label{Ppluseq}\n\\end{equation}\nwhich shows that $\\Delta_rR_{+1}$ and $R_{-1}$ satisfy complex conjugate equations, by comparing with Eq.~\\eqref{Rminuseq}.\n\nThe decoupled equations~\\eqref{Maxwelleq2_1}-~\\eqref{Maxwelleq2_4} provide solutions for $\\Phi_0$, $\\Phi_1$ and $\\Phi_2$. In particular, $\\Phi_0$ and $\\Phi_2$ can be separated into radial parts which satisfy Eqs.~\\eqref{Rpluseq} and~\\eqref{Rminuseq}, and angular parts which satisfy Eqs.~\\eqref{Spluseq} and~\\eqref{Sminuseq}. The solution of $\\Phi_1$ is ignored for now and we will be back to this problem later.\nThere is an additional issue to discuss, which is related to the solutions of $\\Phi_0$ and $\\Phi_2$.\n\nAs we have already shown in Eqs~\\eqref{Rminuseq} and~\\eqref{Ppluseq}, $\\Phi_0$ and $\\Phi_2$ satisfy equations which are complex conjugate of each other, but the relative normalization between these two solutions still remains to be determined. The answer to this problem is given by the famous Starobinsky-Teukolsky identities. In the following we will address this issue by proving the Starobinsky-Teukolsky identities for the Maxwell field on the Kerr-AdS background.\n\n\\underline{\\bf Starobinsky-Teukolsky identities}\n\\begin{thm}\\label{thm1\n$\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}$ is a constant multiple of $S_{-1}$,\\;i.e. $\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}=BS_{-1}$, \\\\\n{\\color{white}{and}}$\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}^\\dag S_{-1}$ is a constant multiple of $S_{+1}$, i.e. $\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}^\\dag S_{-1}=BS_{+1}$.\n\\end{thm}\nNote that the new angular operators are defined as\n\\begin{equation}\n\\bar{\\mathscr{L}_n}=\\sqrt{\\Delta_\\theta}\\mathscr{L}_n\\;,\\;\\;\\;\\;\\;\\;\\bar{\\mathscr{L}_n}^\\dag=\\sqrt{\\Delta_\\theta}\\mathscr{L}_n^\\dag\\;,\\label{newangoperator}\n\\end{equation}\nso that this theorem can be written in the same form as its counterpart for the Kerr background~\\cite{Chandrasekhar:1985kt}. Also note that the two proportionality constants in this theorem have been set to the same (denoted by $B$), which is guaranteed if we normalize both $S_{+1}$ and $S_{-1}$ to unity, i.e.\n\\begin{equation}\n\\int_0^\\pi S_{+1}^2\\sin\\theta d\\theta=\\int_0^\\pi S_{-1}^2\\sin\\theta d\\theta=1\\;.\\label{angnorm}\n\\end{equation}\nTheorem~\\ref{thm1} is proved in Appendix~\\ref{app:ST1} with details, by taking $\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}=BS_{-1}$, as an example.\n\nTo evaluate $B$, we start by applying the operator $\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}^\\dag$ to the first expression in Theorem~\\ref{thm1} and with aid of the second expression in this theorem, then we have\n\\begin{align}\nB^2S_{+1}&=\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}^\\dag\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}=\\bar{\\mathscr{L}_0}^\\dag(\\bar{\\mathscr{L}_1}+2\\mathcal{Q}\\sqrt{\\Delta_\\theta})(\\bar{\\mathscr{L}_0}^\\dag-2\\mathcal{Q}\\sqrt{\\Delta_\\theta})\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=\\bar{\\mathscr{L}_0}^\\dag\\left(\\bar{\\mathscr{L}_1}\\bar{\\mathscr{L}_0}^\\dag-\\dfrac{2}{\\sin\\theta}\\dfrac{d}{d\\theta}\\left(\\sin\\theta H\\right)\\right)\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}(2a\\omega\\Xi\\cos\\theta-\\lambda)S_{+1}-4a\\omega\\Xi\\bar{\\mathscr{L}_0}^\\dag\\cos\\theta\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=2a\\omega\\Xi\\bar{\\mathscr{L}_0}^\\dag(\\cos\\theta\\bar{\\mathscr{L}_1}-\\sin\\theta\\sqrt{\\Delta_\\theta})S_{+1}-\\lambda\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}S_{+1}-4a\\omega\\Xi\\bar{\\mathscr{L}_0}^\\dag\\cos\\theta\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=-2a\\omega\\Xi(\\cos\\theta\\bar{\\mathscr{L}_0}^\\dag-\\sin\\theta\\sqrt{\\Delta_\\theta})\\bar{\\mathscr{L}_1}S_{+1}-2a\\omega\\Xi\\sin\\theta\\sqrt{\\Delta_\\theta}\\bar{\\mathscr{L}_1}^\\dag S_{+1}-\\lambda\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}S_{+1}\\nonumber\\\\\n&=-(2a\\omega\\Xi\\cos\\theta+\\lambda)\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}S_{+1}+2a\\omega\\Xi\\sin\\theta\\sqrt{\\Delta_\\theta}(\\bar{\\mathscr{L}_1} -\\bar{\\mathscr{L}_1}^\\dag)S_{+1}\\nonumber\\\\\n&=-(2a\\omega\\Xi\\cos\\theta+\\lambda)(2a\\omega\\Xi\\cos\\theta-\\lambda)S_{+1}-4a\\omega\\sin\\theta\\Xi^2HS_{+1}\\nonumber\\\\\n&=\\left(\\lambda^2-4\\omega^2\\Xi^2(a^2-\\dfrac{am}{\\omega})\\right)S_{+1}\\;,\\label{derivaB}\n\\end{align}\nwhere the angular equation~\\eqref{Spluseq} was used in the above derivations, and $\\mathcal{Q}$ is defined as\n\\begin{equation}\n\\mathcal{Q}=\\dfrac{\\Xi H}{\\Delta_\\theta}\\;. \\label{Qdef\n\\end{equation}\nEq.~\\eqref{derivaB} finally gives the value of $B$, i.e.\n\\begin{equation}\nB^2=\\lambda^2-4\\omega\\Xi^2(\\omega a^2-ma)\\;.\\label{Bvalue}\n\\end{equation}\nThe sign of $B$ can be fixed by comparing with the spherical case $(a=0)$ when the angular functions reduce to the spin-weighted spherical harmonics~\\cite{goldberg1967spin}. This comparison requires us to choose the positive square root in Eq.~\\eqref{Bvalue}\n\\begin{thm}\\label{thm2}\n$\\Delta_r\\mathscr{D}_0\\mathscr{D}_0R_{-1}$ is a constant multiple of $\\Delta_rR_{+1}$,\\;i.e. $\\Delta_r\\mathscr{D}_0\\mathscr{D}_0R_{-1}=B\\Delta_rR_{+1}$, \\\\\n{\\color{white}{and}}$\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\Delta_r\\mathscr{D}_0^\\dag\\mathscr{D}_0^\\dag\\Delta_rR_{+1}$ is a constant multiple of $R_{-1}$, i.e. $\\Delta_r\\mathscr{D}_0^\\dag\\mathscr{D}_0^\\dag\\Delta_rR_{+1}=BR_{-1}$.\n\\end{thm}\nThe proportionality stated in this theorem can be proved as follows, by taking the first expression as an example. By applying the operator $\\mathscr{D}_0\\mathscr{D}_0$ to Eq.~\\eqref{Rminuseq}, we have\n\\begin{align}\n\\lambda\\mathscr{D}_0\\mathscr{D}_0R_{-1}&=\\mathscr{D}_0\\mathscr{D}_0(\\Delta_r\\mathscr{D}_0^\\dag\\mathscr{D}_0-2i\\omega\\Xi r)R_{-1}\\nonumber\\\\\n&=\\mathscr{D}_0\\mathscr{D}_0\\Delta_r\\mathscr{D}_0\\mathscr{D}_0R_{-1}+2i\\Xi\\mathscr{D}_0\\mathscr{D}_0K\\mathscr{D}_0R_{-1}-2i\\omega\\Xi\\mathscr{D}_0\\mathscr{D}_0(rR_{-1})\n\\nonumber\\\\\n&=\\mathscr{D}_0\\Delta_r\\mathscr{D}_1\\mathscr{D}_0\\mathscr{D}_0R_{-1}+2i\\Xi\\mathscr{D}_0(K\\mathscr{D}_0+2\\omega r)\\mathscr{D}_0R_{-1}\n-2i\\omega\\Xi\\mathscr{D}_0\\mathscr{D}_0(rR_{-1})\\nonumber\\\\\n&=\\mathscr{D}_0(\\Delta_r\\mathscr{D}_1+2i\\Xi K)\\mathscr{D}_0\\mathscr{D}_0R_{-1}+2i\\omega\\Xi(2\\mathscr{D}_0r\\mathscr{D}_0R_{-1}-r\\mathscr{D}_0\\mathscr{D}_0R_{-1}\n+2\\mathscr{D}_0R_{-1})\\nonumber\\\\\n&=\\mathscr{D}_0\\Delta_r\\mathscr{D}_1^\\dag\\mathscr{D}_0\\mathscr{D}_0R_{-1}+2i\\omega\\Xi r\\mathscr{D}_0\\mathscr{D}_0R_{-1}\\nonumber\\\\\n&=(\\Delta_r\\mathscr{D}_1\\mathscr{D}_1^\\dag+2i\\omega\\Xi r)(\\mathscr{D}_0\\mathscr{D}_0R_{-1})\\;.\n\\end{align}\nTherefore, $\\mathscr{D}_0\\mathscr{D}_0R_{-1}$ satisfies the same equation as $R_{+1}$, by comparing with Eq.~\\eqref{Rpluseq}. The second part in this theorem can be proved following the same logic. Notice that the constants of proportionality in this theorem have been fixed to $B$, the same constant used in Theorem~\\ref{thm1}. This fact, indeed, is a consequence of Theorem~\\ref{thm1}, and can be understood as follows. The Maxwell scalars $\\Phi_0$ and $\\Phi_2$ are governed by their equations \\eqref{Phi0eq} and \\eqref{Phi2eq}, but their relative normalization is still undetermined. To obtain this relative normalization constant, by applying the operator $\\sqrt{\\Delta_\\theta}(\\mathscr{L}_0+ia\\sin\\theta\/\\bar{\\rho}^\\ast)$ to Eq.~\\eqref{Maxwelleq2_1} and $(\\mathscr{D}_0+1\/\\bar{\\rho}^\\ast)$ to Eq.~\\eqref{Maxwelleq2_2}, and eliminating $\\Phi_1$, we obtain\n\\begin{equation}\n\\left(\\mathscr{D}_0+\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)\\left(\\mathscr{D}_0-\\dfrac{1}{\\bar{\\rho}^\\ast}\\right)\\Phi_2\n=\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_0+\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)\n\\sqrt{\\Delta_\\theta}\\left(\\mathscr{L}_1-\\dfrac{ia\\sin\\theta}{\\bar{\\rho}^\\ast}\\right)\\Phi_0\n\\end{equation}\nwhich can be further simplified\n\\begin{equation}\n\\mathscr{D}_0\\mathscr{D}_0\\Phi_2=\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}\\Phi_1\\;,\\label{complete1}\n\\end{equation}\nwith aid of the commutative property of the angular operator\n\\begin{equation}\n\\sqrt{\\Delta_\\theta}\\sin\\theta\\mathscr{L}_{n+1}=\\mathscr{L}_n\\sqrt{\\Delta_\\theta}\\sin\\theta\\;.\n\\end{equation}\nWith the field decompositions~\\eqref{decompfields} and the identity $\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}=BS_{-1}$ in Theorem~\\ref{thm1}, Eq.~\\eqref{complete1} becomes\n\\begin{equation}\n\\mathscr{D}_0\\mathscr{D}_0R_{-1}=BR_{+1}\\;.\n\\end{equation}\nThis gives the first identity in Theorem~\\ref{thm2}, by multiplying $\\Delta_r$ on both sides.\n\n\\underline{\\bf The solution for $\\Phi_1$}\n\nTo complete the solutions, now we step back to look for the solution for $\\Phi_1$.\nWe start from Eq.~\\eqref{Maxwelleq2_1}, by multiplying $\\bar{\\rho}^\\ast$ on both sides, then we have\n\\begin{equation}\n\\Big(\\bar{\\rho}^\\ast\\mathscr{D}_0+1\\Big)\\Phi_1=\\left(\\bar{\\rho}^\\ast\\bar{\\mathscr{L}_1}-ia\\sin\\theta\\sqrt{\\Delta_\\theta}\\right)\\Phi_0\\;,\n\\end{equation}\nwhich, from the definition of $\\mathscr{D}_0$, may be rewritten as\n\\begin{equation}\n\\mathscr{D}_0\\Big(\\bar{\\rho}^\\ast\\Phi_1\\Big)=\\left(\\bar{\\rho}^\\ast\\bar{\\mathscr{L}_1}-ia\\sin\\theta\\sqrt{\\Delta_\\theta}\\right)\\Phi_0\\;.\\label{Phi1sol1}\n\\end{equation}\nThen multiplying by $\\Delta_r$ and expanding $\\Phi_0$ as in Eq.~\\eqref{decompfields}, Eq.~\\eqref{Phi1sol1} becomes\n\\begin{align}\n\\Delta_r\\mathscr{D}_0\\Big(\\bar{\\rho}^\\ast\\Phi_1\\Big)&=\\left(\\bar{\\rho}^\\ast\\bar{\\mathscr{L}_1}-ia\\sin\\theta\\sqrt{\\Delta_\\theta}\\right)\\Big(\\Delta_r\\Phi_0\\Big)\\nonumber\\\\\n&=r\\Big(\\Delta_rR_{+1}\\Big)\\bar{\\mathscr{L}_1}S_{+1}-ia\\Big(\\Delta_rR_{+1}\\Big)\\left(\\cos\\theta\\bar{\\mathscr{L}_1}S_{+1}+\\sin\\theta\\sqrt{\\Delta_\\theta}S_{+1}\\right)\\nonumber\\\\\n&=\\Delta_r\\Big(\\mathscr{D}_0g_{+1}\\bar{\\mathscr{L}_1}S_{+1}-iaf_{-1}\\mathscr{D}_0\\mathscr{D}_0P_{-1}\\Big)\\;,\n\\end{align}\nwhich gives\n\\begin{equation}\n\\mathscr{D}_0\\Big(\\bar{\\rho}^\\ast\\Phi_1\\Big)=\\mathscr{D}_0\\Big(g_{+1}\\bar{\\mathscr{L}_1}S_{+1}-iaf_{-1}\\mathscr{D}_0P_{-1}\\Big)\\;,\\label{Phi11eq}\n\\end{equation}\nwhere we defined\n\\begin{align}\n&g_{+1}=\\dfrac{1}{B}\\Big(r\\mathscr{D}_0P_{-1}-P_{-1}\\Big)\\label{gpluseq}\\;,\\\\\n&f_{-1}=\\dfrac{1}{B}\\left(\\cos\\theta\\bar{\\mathscr{L}_1}S_{+1}+\\sin\\theta\\sqrt{\\Delta_\\theta}S_{+1}\\right)\\;,\\label{fminuseq}\n\\end{align}\nand\n\\begin{equation}\n\\Delta_r\\Phi_0\\equiv P_{+1}S_{+1}\\;,\\;\\;\\;\\Phi_2\\equiv P_{-1}S_{-1}\\;,\\;\\;\\;\\;\n\\end{equation}\nand where the Starobinski-Teukolsky identity in Theorem~\\ref{thm2}, i.e.\n\\begin{equation}\n\\Delta_r\\mathscr{D}_0\\mathscr{D}_0P_{-1}=BP_{+1}\\;,\n\\end{equation}\nhas been used.\\\\\nApplying a similar procedure to Eq.~\\eqref{Maxwelleq2_2}, we obtain\n\\begin{equation}\n\\bar{\\mathscr{L}_0}\\Big(\\bar{\\rho}^\\ast\\Phi_1\\Big)=\\bar{\\mathscr{L}_0}\\Big(g_{+1}\\bar{\\mathscr{L}_1}S_{+1}-iaf_{-1}\\mathscr{D}_0P_{-1}\\Big)\\;.\\label{Phi12eq}\n\\end{equation}\nBy comparing Eqs.~\\eqref{Phi11eq} and~\\eqref{Phi12eq}, and considering that $\\mathscr{D}_0$ is the differential operator only for the radial part while $\\bar{\\mathscr{L}_0}$ is the differential operator only for the angular part, we conclude that\\footnote{Strictly, we could add an extra function, say $F$, satisfying homogenous equations $\\mathscr{D}_0(F)=\\bar{\\mathscr{L}_0}(F)=0$, to the solution of $\\Phi_1$ in Eq.~\\eqref{Phi1eq}. The solution of $F$, however, is singular at $\\theta=0$ and $\\theta=\\pi\/2$, similar to the Kerr case~\\cite{Chandrasekhar:1985kt}. Therefore, we have not included $F$ in the solution for $\\Phi_1$.}\n\\begin{equation}\n\\bar{\\rho}^\\ast\\Phi_1=g_{+1}\\bar{\\mathscr{L}_1}S_{+1}-iaf_{-1}\\mathscr{D}_0P_{-1}\\;,\\label{Phi1eq}\n\\end{equation}\nwhich determines $\\Phi_1$ uniquely. This equation is relevant to derive the angular momentum flux for the Maxwell field on Kerr-AdS background, see Appendix~\\ref{app:angmomflux} for details.\n\nAlthough the scalar $\\Phi_1$ is obtained in Eq.~\\eqref{Phi1eq}, $\\bar{\\mathscr{L}_1}S_{+1}$ is still yet unknown. Indeed $\\bar{\\mathscr{L}_1}S_{+1}$ can be expressed in terms of $S_{+1}$ and $S_{-1}$, by\n\\begin{numcases}{}\n\\bar{\\mathscr{L}_1}S_{+1}=\\dfrac{(2a\\omega\\Xi\\cos\\theta-\\lambda)S_{+1}-BS_{-1}}{2\\sqrt{\\Delta_\\theta}\\mathcal{Q}}\\;,\\label{L1Sp1}\\\\\n\\bar{\\mathscr{L}_1}^\\dag S_{-1}=\\dfrac{(2a\\omega\\Xi\\cos\\theta+\\lambda)S_{-1}+BS_{+1}}{2\\sqrt{\\Delta_\\theta}\\mathcal{Q}}\\;,\\label{L1Sm1}\n\\end{numcases}\nwhere $B$ and $\\mathcal{Q}$ are given in Eqs.~\\eqref{Bvalue} and~\\eqref{Qdef}.\n\nTo prove Eq.~\\eqref{L1Sp1}, we start from Eq.~\\eqref{Spluseq},\n\\begin{align}\n\\left(\\bar{\\mathscr{L}_0}^\\dag\\bar{\\mathscr{L}_1}-2a\\omega\\Xi\\cos\\theta+\\lambda\\right)S_{+1}=0\\Rightarrow\n\\left(\\bar{\\mathscr{L}_0}+2\\mathcal{Q}\\sqrt{\\Delta_\\theta}\\right)\\bar{\\mathscr{L}_1}S_{+1}+\\left(\\lambda-2a\\omega\\Xi\\cos\\theta\\right)S_{+1}=0\\;,\n\\end{align}\nand considering $\\bar{\\mathscr{L}_0}\\bar{\\mathscr{L}_1}S_{+1}=BS_{-1}$ in Theorem~\\ref{thm2}, we obtain\n\\begin{equation}\nBS_{-1}+2\\mathcal{Q}\\sqrt{\\Delta_\\theta}\\bar{\\mathscr{L}_1}S_{+1}+\\left(\\lambda-2a\\omega\\Xi\\cos\\theta\\right)S_{+1}=0\\;,\n\\end{equation}\nwhich finally gives Eq.~\\eqref{L1Sp1}.\nFollowing a similar procedure, Eq.~\\eqref{L1Sm1} can be proved.\n\n\n\n\n\n\n\n\n\\chapter{List of publications}\nThis thesis is based on the following published papers by the author:\n\\begin{enumerate}\n\\item\n{\\textit{Hawking radiation for a Proca field in D-dimensions},\\\\\nCarlos~Herdeiro, Marco~O.~P.~Sampaio, Mengjie~Wang,\\\\\n{}\\href{http:\/\/journals.aps.org\/prd\/abstract\/10.1103\/PhysRevD.85.024005}{Phys.\\ Rev.\\ D {\\bf 85} (2012) 2, 024005}\\;$(${}\\href{http:\/\/arxiv.org\/abs\/1110.2485}{\\tt arXiv:1110.2485}$)$.\n}\n\n\\item\n{\\textit{Hawking radiation for a Proca field in D dimensions. II. charged field in a brane charged black hole},\\\\\nMengjie~Wang, Marco~O.~P.~Sampaio, Carlos~Herdeiro,\\\\\n{}\\href{http:\/\/journals.aps.org\/prd\/abstract\/10.1103\/PhysRevD.87.044011}{Phys.\\ Rev.\\ D {\\bf 87} (2013) 4, 044011}\\;$(${}\\href{http:\/\/arxiv.org\/abs\/1212.2197}{\\tt arXiv:1212.2197}$)$.\n}\n\n\\item\n{\\textit{Superradiant instabilities in a D-dimensional small Reissner-Nordstr\\\"om-anti-de Sitter black hole},\\\\\nMengjie~Wang, Carlos~Herdeiro,\\\\\n{}\\href{http:\/\/journals.aps.org\/prd\/abstract\/10.1103\/PhysRevD.89.084062}{Phys.\\ Rev.\\ D {\\bf 89} (2014) 8, 084062}\\;$(${}\\href{http:\/\/arxiv.org\/abs\/1403.5160}{\\tt arXiv:1403.5160}$)$.\n}\n\n\\item\n{\\textit{Marginal scalar and Proca clouds around Reissner-Nordstr\\\"om black holes},\\\\\nMarco~O.~P.~Sampaio, Carlos~Herdeiro, Mengjie~Wang,\\\\\n{}\\href{http:\/\/journals.aps.org\/prd\/abstract\/10.1103\/PhysRevD.90.064004}{Phys.\\ Rev.\\ D {\\bf 90} (2014) 6, 064004}\\;$(${}\\href{http:\/\/arxiv.org\/abs\/1406.3536}{\\tt arXiv:1406.3536}$)$.\n}\n\n\\item\n{\\textit{Maxwell perturbations on asymptotically anti-de Sitter spacetimes: generic boundary conditions and a new branch of quasinormal modes},\\\\\nMengjie~Wang, Carlos~Herdeiro, Marco O. P. Sampaio,\\\\\n{}\\href{http:\/\/journals.aps.org\/prd\/abstract\/10.1103\/PhysRevD.92.124006}{Phys.\\ Rev.\\ D {\\bf 92} (2015) 12, 124006}\\;$(${}\\href{http:\/\/arxiv.org\/abs\/1510.04713}{\\tt arXiv:1510.04713}$)$.\n}\n\n\\item\n{\\textit{Maxwell perturbations on Kerr-anti-de Sitter: quasinormal modes, superradiant instabilities and vector clouds},\\\\\nMengjie~Wang, Carlos~Herdeiro,\\\\\n{}\\href{https:\/\/journals.aps.org\/prd\/abstract\/10.1103\/PhysRevD.93.064066}\n{Phys.\\ Rev.\\ D {\\bf 93} (2016) 6, 064066}\\;$(${}\\href{http:\/\/arxiv.org\/abs\/1512.02262}{\\tt arXiv:1512.02262}$)$.\n}\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Superradiant instabilities in higher dimensions}\n\\label{ch:scalarHD}\n\n\n\\section{Introduction}\n\\label{intro}\n\n\nSuperradiant instabilities are interesting phenomena with various applications from high energy physics to astrophysics, see a recent review~\\cite{Brito:2015oca} and references therein. Most of these studies focus on four dimensional spacetimes\\footnote{Exceptionally a few works were carried out in five dimensional spacetimes, see~\\cite{Aliev:2008yk,Dias:2011tj}, for example.}, but a generic analysis for superradiant instabilities in higher dimensions is still missing. In order to perform such a study, we choose asymptotically AdS BHs, one of the confining mechanisms discussed in Section~\\ref{sc:SR}, in which bound states of perturbation fields can be supported.\n\nThe goal of this chapter is to present a complete study of superradiant instabilities triggered by a charged scalar field interacting with a $D$-dimensional Reissner-Nordstr\\\"om-anti-de Sitter (RN-AdS) BH. As a spin off we shall clarify a claim, made in~\\cite{Aliev:2008yk}, that, in $D=5$,  a subset of field modes -- those with odd angular momentum quantum number, $\\ell$ -- do not develop superradiant instabilities. We shall show otherwise: that in fact \\textit{all} $\\ell$ modes in \\textit{all} dimensions can develop superradiant instabilities.\n\nIn addition to the study of superradiant instabilities for a minimally coupled charged scalar field, other studies have been considered for charged AdS BHs. It was first observed in \\cite{Gubser:2000ec,Gubser:2000mm} that for BHs obtained within $\\mathcal{N}=8$ supergravity\\footnote{This is a theory containing four $U(1)$ gauge fields and three real scalar fields \\textit{non-minimally} coupled to the Maxwell fields.} in $D=4$, \\textit{large} RN-AdS BHs are dynamically  unstable due to the existence of a tachyonic mode in the scalar field perturbations. On the other hand, in purely Einstein-Maxwell theory in $D$ dimensions, RN-AdS BHs have been argued to be stable against gravitational perturbations~\\cite{Konoplya:2008rq}. Thus the existence of scalar fields with some coupling to Maxwell fields is central to the instability of~\\cite{Gubser:2000ec,Gubser:2000mm}.\n\nA qualitatively different instability of RN-AdS BHs has been discussed in the context of holographic superconductors. It occurs in the presence of a massive scalar field that may or may not be charged and leads to the formation of scalar hair around the BH \\cite{Hartnoll:2008kx}. Unlike the superradiant instability, this other instability occurs even if the scalar field is not charged. In that case the, say, $D=4$ RN-AdS BH should be nearly extremal, which means its near horizon geometry has a two dimensional AdS factor; moreover, the scalar field should have a tachyonic mass above the Breitenlohner-Freedman  bound \\cite{Breitenlohner:1982bm,Breitenlohner:1982jf} of the four dimensional AdS space, but below the Breitenlohner-Freedman bound of the two dimensional AdS factor in the near horizon geometry of the BH. This is why the scalar field provides an instability to the BH geometry, but not to four dimensional AdS space.\n\nIn order to investigate the superradiant instability of a charged, massive scalar field in a $D$-dimensional RN-AdS background we shall employ both an analytical and a numerical method. First, we solve the charged Klein-Gordon equation using a matching method, and obtain a solution near the BH region and another one far away from the BH. This is done for small BHs, i.e. BHs obeying $r_+\\ll L$, where $r_+$ and $L$ stand for the BH event horizon and the AdS radius, respectively. The analytical quasinormal frequency for small BHs is then obtained by matching the near and far solutions in an intermediate region. We find that the relation between $\\ell$ and $D$ plays a central role in determining the analytical quasinormal frequency formula. When $\\ell=(p+\\frac{1}{2})(D-3)$, where $p$ is a non-negative integer, the matching method fails. The reason is that the near region solution and the far region solution have different functional dependence in terms of the radial coordinate, which makes our matching impossible. Such difficulty in employing the matching method also occurs for extremal BHs as discussed in~\\cite{Rosa:2009ei}, where an alternative point matching method was used. For all other values of $\\ell$ and $D$, the matching method works and it may be used to show that a superradiance instability exists for all $\\ell$ modes, in a region of the parameters space.\n\nAfter the (approximate) analytical analysis, then we solve the charged Klein-Gordon equation numerically both to check the analytical results and to explore the special case where the analytical method fails. We find good agreement between the two methods in the regime where both are valid. For the special case $\\ell=(p+\\frac{1}{2})(D-3)$, the numerical results show that the superradiant instability does exist.\n\nThe structure of this chapter is organized as follows. In Section~\\ref{seceq} we introduce the background geometry and the scalar field equation. In Section~\\ref{secmatching} we solve the scalar field equation analytically using the matching method and obtain an analytical quasinormal frequency formula. We analyze this formula for different relations between $\\ell$ and $D$, and show the reason why the matching method fails for the special case $\\ell=(p+\\frac{1}{2})(D-3)$ in Section~\\ref{result analysis}. To confirm our analytical results and to be able to investigate if there is a superradiant instability for that special case, we resort to a numerical method to solve the Klein-Gordon equation in Section~\\ref{numerical}.\nConclusions are presented in the last section.\n\\section{Background and field equation}\n\\label{seceq}\nWe consider a $D$-dimensional RN-AdS BH with the line element\n\\begin{equation}\nds^2=-f(r)dt^2+\\dfrac{1}{f(r)}dr^2+r^2d\\Omega^2_n \\;,\\label{metric}\n\\end{equation}\nwhere $d\\Omega^2_n$ is the metric on the unit $n$-sphere. In the following it will be convenient to use $n=D-2$ rather than $D$ to parameterize the spacetime dimension. The metric function $f(r)$ takes the form\n\\begin{equation}\nf(r)=1-\\dfrac{M}{r^{n-1}}+\\dfrac{Q^2}{r^{2(n-1)}}+\\dfrac{r^2}{L^2} \\;,\\label{metricfunc}\n\\end{equation}\nwhere the parameters $M, Q$ and $L$ are related to the BH mass $\\mathcal{M}$, charge $\\mathbb{Q}$ and cosmological constant $\\Lambda$ through\n\\begin{equation}\nM=\\dfrac{16\\pi G\\mathcal{M}}{nS_n}\\;\\;,Q^2=\\dfrac{8\\pi G\\mathbb{Q}^2}{n(n-1)}\\;\\;,L^2=-\\dfrac{n(n+1)}{2\\Lambda}\\;,\\nonumber\n\\end{equation}\nand the area of a unit $n$-sphere is $S_n=\\frac{2\\pi^{\\frac{n+1}{2}}}{\\Gamma(\\frac{n+1}{2})}$.\nThe Hawking temperature is given by\n\\begin{equation}\nT=\\dfrac{1}{4\\pi}\\left[\\frac{(n-1)M}{r_+^n}-\\frac{2(n-1)Q^2}{r_+^{2n-1}}+\\frac{2r_+}{L^2}\\right]\\;,\\label{hawkingtemp}\n\\end{equation}\nwhere the event horizon $r_+$ is determined as the largest root of $f(r_+)=0$. For non-extremal BHs, we have $Q<Q_c$, where the critical charge $Q_c$ corresponds to the maximal charge (i.e the charge of an  extremal BH) and is given by\n\\begin{equation}\nQ_c\\equiv r_+^{n-1}\\sqrt{1+\\dfrac{n+1}{n-1}\\left(\\dfrac{r_+}{L}\\right)^2}\\;.\\label{critialq}\n\\end{equation}\nThe electromagnetic potential of the charged BH is\n\\begin{equation}\nA=\\left(-\\sqrt{\\dfrac{n}{2(n-1)}}\\dfrac{Q}{r^{n-1}}+C\\right)dt\\;,\\nonumber\n\\end{equation}\nwhere the choice of the constant $C$ is a gauge choice. For instance, we should fix $C$ as\n\\begin{equation}\nC=\\sqrt{\\dfrac{n}{2(n-1)}}\\dfrac{Q}{r_+^{n-1}}\\;,\\nonumber\n\\end{equation}\nin order to have a vanishing electromagnetic potential at the event horizon, a choice used in the context of the AdS\/CFT correspondence~\\cite{Horowitz:2010gk}. As argued in~\\cite{Uchikata:2011zz}, however, this constant just shifts the real part of quasinormal frequency as Re$(\\omega)\\rightarrow$ Re$(\\omega)+qC$ ($q$ is the field charge) without affecting the imaginary part of it. Therefore, as we are mostly interested in the superradiant instability, which is determined by the imaginary part of the quasinormal frequency, we take in the following $C=0$.\n\nFor a charged massive scalar field, the corresponding Klein-Gordon (K-G) equation can be written as\n\\begin{equation}\n\\dfrac{1}{\\sqrt{-g}}D_\\mu\\left[\\sqrt{-g}g^{\\mu\\nu}D_\\nu\\right]\\phi-\\mu^2\\phi=0 \\;,\\label{sacalarEq}\n\\end{equation}\nwhere $D_\\mu=\\partial_\\mu -iqA_\\mu$, $q$ and $\\mu$ are the field charge and mass, respectively. The scalar field $\\phi$ can be decomposed in terms of spherical scalar harmonics due to the spherical symmetry of the background\n\\begin{equation}\n\\phi={\\rm e}^{-i\\omega t}\\rm{R}(r)\\rm{Y}(\\theta_i) \\;,\\label{phicecompose}\n\\end{equation}\nwhere $\\rm{Y}(\\theta_i)$ is a scalar spherical harmonic on the $n$-sphere. Substituting the metric in Eq.~\\eqref{metric} and the field decomposition in Eq.~\\eqref{phicecompose} into the K-G equation~\\eqref{sacalarEq}, we have\n\\begin{equation}\n\\dfrac{\\Delta}{r^{n-2}}\\dfrac{d}{dr}\\left(\\dfrac{\\Delta}{r^{n-2}}\\dfrac{d{\\rm R}}{dr}\\right)-\\Big(\\lambda+\\mu^2r^2\\Big)\\Delta{\\rm R}\n+r^{2n}\\Big(\\omega+qA_t\\Big)^2 {\\rm R}=0 \\;,\\label{RadialEq}\n\\end{equation}\nwith\n\\begin{equation}\n\\Delta\\equiv r^{2(n-1)}-M r^{n-1}+Q^2+\\dfrac{r^{2n}}{L^2}\\;,\\nonumber\n\\end{equation}\nand the $n-$dimensional spherical harmonic eigenvalue $\\lambda$ is given by\n\\begin{equation}\n\\lambda\\equiv \\ell(\\ell+n-1)\\;,\\nonumber\n\\end{equation}\nwhere $\\ell$ is the angular momentum quantum number.\n\nFor numerical convenience, we may rewrite Eq.~\\eqref{RadialEq} in terms of a new function $\\rm{X}$\n\\begin{align}\n&f(r)^2\\dfrac{d^2\\rm X}{dr^2} + f(r)f'(r) \\dfrac{d \\rm X}{dr} + \\left[(\\omega+qA_t)^2 -f(r) \\left(\\dfrac{\\lambda}{r^2}+\\mu^2+\\dfrac{nf'(r)}{2r}+\\dfrac{n(n-2)f(r)}{4r^2}\\right)\\right] \\rm X\\nonumber\\\\&=0\\;,\\label{RadialEq2}\n\\end{align}\nwhere ${\\rm X}\\equiv r^{n\/2} \\rm R$.\n\nIn order to determine the quasinormal frequency, by solving Eq.~\\eqref{RadialEq2}, one has to impose boundary conditions. Near the event horizon, we impose an ingoing boundary condition\n\\begin{equation}\n{\\rm X} \\sim {\\rm e}^{-i\\left(\\omega-\\omega_0\\right)r_\\ast}\\;,\\label{boundaryingoing}\n\\end{equation}\nwhere $\\omega_0\\equiv -qA_t(r_+)$ and the tortoise coordinate $r_\\ast$ is defined by\n\\begin{equation}\n\\dfrac{dr_\\ast}{dr}=\\dfrac{1}{f(r)}\\;.\\nonumber\n\\end{equation}\nAt infinity, we impose a decaying boundary condition\n\\begin{equation}\n{\\rm X} \\sim r^{-\\frac{1}{2}(1+\\sqrt{4\\mu^2L^2+(n+1)^2})}\\; ,\\label{boundarydecaying}\n\\end{equation}\nthen\n\\begin{equation}\n{\\rm R} \\sim r^{-\\frac{1}{2}(n+1+\\sqrt{4\\mu^2L^2+(n+1)^2})}\\;.\\label{boundarydecaying2}\n\\end{equation}\nFor a massless field in $D$ dimensions, the radial function goes, asymptotically as ${\\rm R}\\sim 1\/r^{D-1}$, as expected for a normalizable massless scalar perturbation in $D$-dimensional AdS space.\n\n\\section{Analytic calculations}\n\\label{secmatching}\nIn this section, we present the analytic calculations of quasinormal frequencies for a charged massive scalar field in a higher dimensional RN-AdS BH. Using a standard procedure, we shall divide the space outside the event horizon into two regions: the \\textit{near region}, defined by the condition $r-r_+\\ll1\/\\omega$, and the \\textit{far region}, defined by the condition $r_+\\ll r-r_+$. Then, in order to perform a matching of the two solutions we consider a low frequency condition, i.e. $r_+\\ll1\/\\omega$, and match the two solutions in an  \\textit{intermediate region} defined by $r_+\\ll r-r_+\\ll1\/\\omega$. In the following analysis we focus on small AdS BH $(r_+\\ll L)$. This allows us to treat the frequencies for the RN-AdS BH as a perturbation of the AdS normal frequencies.\n\n\\subsection{Near region solution}\nFor the near region analysis, we rewrite Eq.~\\eqref{RadialEq} as\n\\begin{equation}\n\\left(n-1\\right)^2\\Delta\\dfrac{d}{dx}\\left(\\Delta\\dfrac{d {\\rm R}}{dx}\\right)-\\Big(\\lambda+\\mu^2r^2\\Big)\\Delta {\\rm R}\n+ r^{2n}\\Big(\\omega+qA_t\\Big)^2 {\\rm R}=0 \\;,\\label{NearEq1}\n\\end{equation}\nwhere $x\\equiv r^{n-1}$. In the following we shall neglect the mass term in the first line of Eq.~\\eqref{NearEq1}. This amounts to say that  $r\\ll\\frac{\\ell}{\\mu}$, which is obeyed if the Compton wave length of the scalar particles is much larger than the BH horizon size and indeed becomes the near region condition if, moreover, the scalar particle Compton wave length is much smaller than the AdS radius. Observe that the condition  $r\\ll\\frac{\\ell}{\\mu}$ fails for $\\ell=0$ modes, but it turns out that even in that case the analytical results we shall obtain are in good agreement with the numerical results for sufficiently small mass (cf. Section~\\ref{numerical}). Under this approximation Eq.~\\eqref{NearEq1} becomes\n\\begin{equation}\n(n-1)^2\\Delta\\dfrac{d}{dx}\\left(\\Delta\\dfrac{d{\\rm R}}{dx}\\right)-\\lambda\\Delta {\\rm R} + r_+^{2n}\\Big(\\omega+qA_t\\Big)^2 \\rm R=0 \\;.\\label{NearEq2}\n\\end{equation}\nFor convenience in the analytic calculations, one can define a new dimensionless variable\n\\begin{equation}\nz\\equiv \\dfrac{x-x_+}{x-x_-}\\;,\\nonumber\n\\end{equation}\nwith $x_+=r_+^{n-1}$ and $x_-=r_-^{n-1}$,\nwhere $r_+$ and $r_-$ refer to the event horizon and the Cauchy horizon, respectively.\nThen Eq.~\\eqref{NearEq2} can be transformed into\n\\begin{equation}\nz\\dfrac{d}{dz}\\left(z\\dfrac{d{\\rm R}}{dz}\\right)+\\left[\\bar{\\omega}^2-\\dfrac{\\lambda}{(n-1)^2}\\dfrac{z}{(1-z)^2}\\right] \\rm R=0\\;,\\label{NearEq}\n\\end{equation}\nwith\n\\begin{equation}\n\\bar{\\omega}\\equiv \\dfrac{x_+^{\\frac{n}{n-1}}}{(n-1)(x_+ - x_-)}\\left(\\omega-\\sqrt{\\dfrac{n}{2(n-1)}}\\dfrac{qQ}{x_+}\\right) \\;.\\label{superradiance}\n\\end{equation}\nObserve that $\\bar{\\omega}<0$ for $\\omega<\\sqrt{\\frac{n}{2(n-1)}}\\frac{qQ}{x_+}$. This will be shown below to correspond to the superradiant regime.\n\nOne may now obtain a solution for Eq.~\\eqref{NearEq} with ingoing boundary condition in terms of a hypergeometric function:\n\\begin{equation}\n{\\rm R} \\sim z^{-i\\bar{\\omega}}(1-z)^\\alpha F(\\alpha,\\alpha-2i\\bar{\\omega},1-2i\\bar{\\omega};z) \\;,\\label{NearSol}\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha\\equiv 1+\\dfrac{\\ell}{n-1}\\;.\\label{alpha}\n\\end{equation}\nIn order to match the far region solution, one must expand the near region solution, \\mbox{Eq.~\\eqref{NearSol}}, at large $r$. To achieve this we take the $z\\rightarrow1$ limit and use the properties of the hypergeometric function~\\cite{abramowitz+stegun}, then obtain\n\\begin{equation}\n{\\rm R} \\sim \\Gamma(1-2i\\bar{\\omega})\\left[\\dfrac{{\\rm R}^{\\rm near}_{1\/r}}{r^{n-1+\\ell}} +{\\rm R}^{\\rm near}_r r^{\\ell}\\right]\n\\;,\\label{NearsolFar}\n\\end{equation}\nwhere\n\\begin{align}\n{\\rm R}^{\\rm near}_{1\/r} & \\equiv  \\dfrac{\\Gamma(1-2\\alpha) (r_+^{n-1} - r_-^{n-1})^\\alpha}{\\Gamma(1-\\alpha)\\Gamma(1-\\alpha-2i\\bar{\\omega})} \\ ,\\nonumber \\\\\n{\\rm R}^{\\rm near}_r &\\equiv \\dfrac{\\Gamma(2\\alpha-1)(r_+^{n-1}-r_-^{n-1})^{1-\\alpha}}{\\Gamma(\\alpha)\\Gamma(\\alpha-2i\\bar{\\omega})} \\ .\n\\end{align}\nSince the Gamma function has poles at negative integers, one observes that special care must be taken with the factor $\\Gamma(1-2\\alpha)\/\\Gamma(1-\\alpha)$. Its analysis will play a role below.\n\n\n\\subsection{Far region solution}\nIn the far region, $r-r_+\\gg r_+$, the BH effects can be neglected ($M\\rightarrow0,\\;Q\\rightarrow0$) so that\n\\begin{equation}\n\\Delta\\simeq r^{2n} \\left(\\dfrac{1}{r^2}+\\dfrac{1}{L^2}\\right)\\; .\\nonumber\n\\end{equation}\nThen Eq.~\\eqref{RadialEq} becomes\n\\begin{equation}\n\\left(1+\\dfrac{r^2}{L^2}\\right)\\dfrac{d^2{\\rm R}}{dr^2}+\\left(\\dfrac{n}{r}+\\dfrac{(n+2)r}{L^2}\\right)\\dfrac{d {\\rm R}}{dr}\n+\\left[\\omega^2\\left(1+\\dfrac{r^2}{L^2}\\right)^{-1}-\\left(\\dfrac{\\lambda}{r^2}+\\mu^2\\right)\\right]\\rm R=0\\;.\\label{fareq1}\n\\end{equation}\nDefining a new variable\n\\begin{equation}\ny\\equiv 1+\\dfrac{r^2}{L^2}\\;,\\nonumber\n\\end{equation}\nEq.~\\eqref{fareq1} becomes\n\\begin{equation}\ny\\left(1-y\\right)\\dfrac{d^2{\\rm R}}{dy^2}+\\left(1-\\dfrac{n+3}{2}y\\right)\\dfrac{d {\\rm R}}{dy}-\\left[\\dfrac{\\omega^2L^2}{4y}-\\dfrac{\\mu^2L^2}{4}+\\dfrac{\\lambda}{4(1-y)}\\right]\\rm R=0\\;.\\label{fareq2}\n\\end{equation}\nThe above equation has a hypergeometric equation structure, which can be shown explicitly through the transformation\n\\begin{equation}\n{\\rm R}=y^{\\frac{\\omega L}{2}}(1-y)^{\\frac{\\ell}{2}}F(a,b;c;y) \\;,\\nonumber\n\\end{equation}\nwith parameters\n\\begin{align}\n&a\\equiv \\dfrac{n+1}{4}+\\dfrac{\\omega L}{2}+\\dfrac{\\ell}{2}+\\dfrac{1}{2}\\sqrt{\\mu^2L^2+\\left(\\frac{n+1}{2}\\right)^2}\\;,\\nonumber\\\\\n&b\\equiv \\dfrac{n+1}{4}+\\dfrac{\\omega L}{2}+\\dfrac{\\ell}{2}-\\dfrac{1}{2}\\sqrt{\\mu^2L^2+\\left(\\frac{n+1}{2}\\right)^2}\\;,\\nonumber\\\\\n&c\\equiv 1+\\omega L\\;.\\label{hyperparameter}\n\\end{align}\nThen considering a decaying boundary condition at infinity given in Eq.~\\eqref{boundarydecaying2}, one finds a solution for Eq.~\\eqref{fareq2} in the form\n\\begin{equation}\n{\\rm R}\\thicksim (1-y)^{\\frac{\\ell}{2}}y^{\\frac{\\omega L}{2}-a} F(a,1+a-c;1+a-b;\\frac{1}{y})\\;.\\label{farsolution}\n\\end{equation}\nTo achieve the small $r$ behavior of Eq.~\\eqref{farsolution}, making the transformation $\\frac{1}{y}\\rightarrow1-y$ and using properties of the hypergeometric function~\\cite{abramowitz+stegun}, we obtain\n\\begin{equation}\n{\\rm R} \\sim  \\Gamma(1+a-b)\\left[\\dfrac{{\\rm R}^{\\rm far}_{1\/r}}{r^{n-1+\\ell}} +{\\rm R}^{\\rm far}_r r^{\\ell}\\right]   \\;,\\label{farsolution@near}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\rm R}^{\\rm far}_{1\/r} \\equiv  \\dfrac{\\Gamma(\\ell+\\frac{n-1}{2})L^{n-1+\\ell}}{\\Gamma(a)\\Gamma(1+a-c)} \\;,\\;\\;\\;\\;\\;\\;\n{\\rm R}^{\\rm far}_r \\equiv \\dfrac{\\Gamma(-\\ell-\\frac{n-1}{2})L^{-\\ell}}{\\Gamma(1-b)\\Gamma(c-b)}\\ .\n\\label{far_branches}\n\\end{equation}\nThe solution in Eq.~\\eqref{farsolution@near} is for pure AdS. Regularity of the above solution at the origin $(r=0)$ of AdS requires\\footnote{We remark that, alternatively, one can also demand $\\Gamma(a)=0$, which gives the negative AdS spectrum. Without loss of generality, we only consider the positive spectrum here.}\n\\begin{equation}\n\\Gamma(1+a-c)=\\infty\\  \\Rightarrow  1+a-c=-N ,\\nonumber\n\\end{equation}\nwhich gives the discrete spectrum\n\\begin{equation}\n\\omega_{N} L=2N+\\frac{n+1}{2}+\\ell+\\sqrt{\\mu^2L^2+\\left(\\frac{n+1}{2}\\right)^2}\\;,\\label{AdSfrequency}\n\\end{equation}\nwhere $N$ is a non-negative integer.\nObserve that the AdS frequencies remain real even for tachyonic modes when $0>\\mu^2\\ge -(n+1)^2\/(4L^2)$. This is the well known Breitenlohner-Freedman bound already discussed in the introduction. In particular one may see that the bound is more negative for higher dimensional spaces. This is the reason why one may violate the bound for two dimensional AdS but obey it for four dimensional AdS.\n\nWhen the BH effects are taken into account, a correction to the frequency (which may be complex) will be generated\n\\begin{equation}\n\\omega=\\omega_N+i\\delta\\;,\n\\end{equation}\nwhere the real part of $\\delta$ is used to describe the damping of the quasinormal modes.\nThen, for small BHs, using the approximation $1\/\\Gamma(-N+\\epsilon)\\simeq (-1)^NN!\\epsilon$ for small $\\epsilon$, the first term inside the bracket of Eq.~\\eqref{farsolution@near} becomes\n\\begin{equation}\n{\\rm R}^{\\rm far}_{1\/r} = (-1)^{N+1} i\\delta N! \\dfrac{\\Gamma(\\ell+\\frac{n-1}{2})L^{n+\\ell}}{2\\Gamma(a)} \\;.\\nonumber\n\\end{equation}\nFinally, observe that there appears to be extra poles in ${\\rm R}_r^{\\rm far}$, Eq.~\\eqref{far_branches}, due to the Gamma function $\\Gamma(-\\ell-\\frac{n-1}{2})$ for odd $n$. In the ${\\rm R}_r^{\\rm far}$ expression, however, due to Eq.~\\eqref{AdSfrequency}, $\\Gamma(1-b)=\\Gamma(-\\ell-\\frac{n-1}{2}-N)$, which cancels the former poles.\n\n\n\\subsection{Overlap region}\nTo match the near region solution~(\\ref{NearsolFar}) and the far region solution~(\\ref{farsolution@near}) in the intermediate region, we impose the matching condition ${\\rm R}^{\\rm near}_r{\\rm R}^{\\rm far}_{1\/r}={\\rm R}^{\\rm far}_r{\\rm R}^{\\rm near}_{1\/r}$. Then $\\delta$ can be obtained perturbatively\n\\begin{align}\n\\delta=&(-1)^{N}2i\\dfrac{(r_+^{n-1}-r_-^{n-1})^{2\\alpha-1}}{N!L^{2\\ell+n}} \\dfrac{\\Gamma(1-2\\alpha)\\Gamma(\\alpha)}{\\Gamma(2\\alpha-1)\\Gamma(1-\\alpha)} \\dfrac{\\Gamma(a)}{\\Gamma(1-b)\\Gamma(c-b)} \\dfrac{\\Gamma(-\\ell-\\frac{n-1}{2})}{\\Gamma(\\ell+\\frac{n-1}{2})} \\nonumber\\\\ &\\times \\dfrac{\\Gamma(\\alpha-2i\\bar{\\omega})}{\\Gamma(1-\\alpha-2i\\bar{\\omega})}\\;.\\label{imaginarypart}\n\\end{align}\n\n\n\\section{Analytical result analysis}\n\\label{result analysis}\nTo analyze Eq.~\\eqref{imaginarypart}, we shall simplify the Gamma functions therein. Firstly, the following combination, which is independent of the relation between $\\ell$ and $n$, can be simplified as\n\\begin{align}\n\\dfrac{\\Gamma(a)}{\\Gamma(1-b)\\Gamma(c-b)}\\dfrac{\\Gamma\\left(-\\ell-\\frac{n-1}{2}\\right)}{\\Gamma\\left(\\ell+\\frac{n-1}{2}\\right)}\n=&\\dfrac{(-1)^N}{\\Gamma\\left(\\ell+\\frac{n-1}{2}\\right)} \\dfrac{\\Gamma\\left(N+\\frac{n+1}{2}+\\ell+\\sqrt{\\mu^2L^2+(\\frac{n+1}{2})^2}\\right)}{\\Gamma\\left(N+1+\\sqrt{\\mu^2L^2+(\\frac{n+1}{2})^2}\\right)}\\nonumber\\\\\n&\\times\\prod_{k=1}^{N}\\left(\\ell+\\frac{n-1}{2}+k\\right)\\;.\\nonumber\n\\end{align}\nThen one has to consider the following cases separately, because the simplification for the other Gamma functions in Eq.~\\eqref{imaginarypart} depends on the relation between $\\ell$ and $n$.\n\n\\subsection{$\\ell$ is an integer multiple of $(n-1)$}\nFor this case we can write $\\ell=p(n-1)$, where $p$ is a non-negative integer.\nThen, the corresponding Gamma functions in Eq.~\\eqref{imaginarypart} can be simplified to\n\\begin{equation}\n\\dfrac{\\Gamma(1-2\\alpha)\\Gamma(\\alpha)}{\\Gamma(2\\alpha-1)\\Gamma(1-\\alpha)}=\\dfrac{(-1)^{p+1}}{2}\\dfrac{(p!)^2}{(2p)!(2p+1)!}\\;,\\nonumber\n\\end{equation}\n\\begin{equation}\n\\dfrac{\\Gamma(\\alpha-2i\\bar{\\omega})}{\\Gamma(1-\\alpha-2i\\bar{\\omega})}=(-1)^{p+1} 2i\\bar{\\omega} \\prod_{k^{\\prime}=1}^p(k^{\\prime 2}+4\\bar{\\omega}^2)\\;.\\nonumber\n\\end{equation}\nTherefore, Eq.~\\eqref{imaginarypart} becomes\n\\begin{align}\n\\delta=&-2\\bar{\\omega}\\dfrac{(r_+^{n-1}-r_-^{n-1})^{1+\\frac{2\\ell}{n-1}}}{N! L^{2\\ell+n}\\Gamma\\left(\\ell+\\frac{n-1}{2}\\right)}\\dfrac{(p!)^2}{(2p)!(2p+1)!}\n\\dfrac{\\Gamma\\left(N+\\frac{n+1}{2}+\\ell+\\sqrt{\\mu^2L^2+(\\frac{n+1}{2})^2}\\right)}{\\Gamma\\left(N+1+\\sqrt{\\mu^2L^2+(\\frac{n+1}{2})^2}\\right)}\\nonumber\\\\ &\\times\\prod_{k=1}^N\\left(\\ell+\\frac{n-1}{2}+k\\right) \\prod_{k^{\\prime}=1}^p (k^{\\prime 2}+4\\bar{\\omega}^2)\\;.\\label{case1}\n\\end{align}\nThis equation shares a similar structure to the corresponding result in $D=4$. From the definition of $\\bar{\\omega}$ in Eq.~\\eqref{superradiance}, we find that in the superradiant regime, $\\bar{\\omega}<0$ which implies $\\delta>0$. In this superradiant regime the wave function of the scalar field will grow with time which means the BH is unstable. Moreover, from Eq.~\\eqref{superradiance}, one may get a condition for the onset of the superradiant instability, i.e.\n\\begin{equation}\n\\dfrac{Q}{Q_c} > \\sqrt{\\frac{2(n-1)}{n}} \\dfrac{\\omega_N}{q}\\;,\\label{onset1}\n\\end{equation}\nwhere $\\omega_N$ is given in Eq.~\\eqref{AdSfrequency}. For a massless field, Eq.~\\eqref{onset1} simplifies to\n\\begin{equation}\n\\dfrac{Q}{Q_c} > \\sqrt{\\frac{2(n-1)}{n}} \\dfrac{2N+n+1+\\ell}{q L}\\;.\\label{onset2}\n\\end{equation}\n\n\n\\subsection{$\\ell$ is not an integer multiple of $(n-1)$}\nFor this case, the corresponding Gamma function in Eq.~\\eqref{imaginarypart} can be simplified as\n\\begin{equation}\n\\dfrac{\\Gamma(1-2\\alpha)\\Gamma(\\alpha)}{\\Gamma(2\\alpha-1)\\Gamma(1-\\alpha)}=-\\dfrac{1}{2\\cos\\frac{\\pi \\ell}{n-1}}\\dfrac{\\Gamma^2(1+\\frac{\\ell}{n-1})}{\\Gamma(1+\\frac{2\\ell}{n-1})\\Gamma(2+\\frac{2\\ell}{n-1})}\\;.\\nonumber\n\\end{equation}\nIf $\\dfrac{\\ell}{n-1}\\neq p+\\frac{1}{2}$, then cos$\\frac{\\pi\\ell}{n-1}\\neq0$, and the parameter $\\delta$ becomes complex (not simply real as in the previous case).\nIn this case the real part of $\\delta$ reflects the instability, which is given by\n\\begin{align}\n\\mbox{Re} \\delta=&-2\\bar{\\omega}\\dfrac{(r_+^{n-1}-r_-^{n-1})^{1+\\frac{2\\ell}{n-1}}}{N! L^{2\\ell+n}} \\dfrac{\\Gamma^4(1+\\frac{\\ell}{n-1})}{\\Gamma(1+\\frac{2\\ell}{n-1})\\Gamma(2+\\frac{2\\ell}{n-1})\\Gamma(\\ell+\\frac{n-1}{2})}\\prod_{k=1}^N(\\ell+\\frac{n-1}{2}+k) \\nonumber\\\\\n&\\times\\dfrac{\\Gamma(N+\\frac{n+1}{2}+\\ell+\\sqrt{\\mu^2L^2+(\\frac{n+1}{2})^2})}{\\Gamma(N+1+\\sqrt{\\mu^2L^2+(\\frac{n+1}{2})^2})}\\;,\\label{case2}\n\\end{align}\nwhere we have expanded the terms $\\Gamma(x-2i\\bar{\\omega})$ around small $\\bar{\\omega}$ to clearly distinguish the superradiant regime. Thus, when $\\bar{\\omega}<0$, we obtain Re$\\delta$ $>0$ which implies that the BH is also unstable, and the corresponding onset of such instability is governed by Eq.~\\eqref{onset1} for a massive field and Eq.~\\eqref{onset2} for a massless field.\n\nIf $\\frac{\\ell}{n-1}=p+\\frac{1}{2}$, the matching method fails; a similar situation occurs for extremal Kerr BHs~\\cite{Rosa:2009ei}. In order to make this point clear, we can do the following analysis. First, from the definition of $\\alpha$ in Eq.~\\eqref{alpha} and the condition $\\frac{\\ell}{n-1}=p+\\frac{1}{2}$, one observes that the first expansion term inside the brackets of  Eq.~\\eqref{NearsolFar} is divergent, which means that we cannot expand Eq.~\\eqref{NearSol} into Eq.~\\eqref{NearsolFar} anymore when $\\frac{\\ell}{n-1}=p+\\frac{1}{2}$. Alternatively, using a property of the hypergeometric function~\\cite{abramowitz+stegun}, we shall expand Eq.~\\eqref{NearSol} as\n\\begin{equation}\n{\\rm R} \\sim \\Gamma(1-2i\\bar{\\omega})\\left[-\\dfrac{(r_+^{n-1} - r_-^{n-1})^\\alpha \\zeta}{\\Gamma(1-\\alpha)\\Gamma(1-\\alpha-2i\\bar{\\omega})\\Gamma(2\\alpha)}\\dfrac{1}{r^{n-1+\\ell}}+\\dfrac{\\Gamma(2\\alpha-1)(r_+^{n-1}-r_-^{n-1})^{1-\\alpha}}{\\Gamma(\\alpha)\\Gamma(\\alpha-2i\\bar{\\omega})}r^{\\ell}\\right]\\;,\n\\label{NearsolFar2}\n\\end{equation}\nwith\n\\begin{equation}\n\\zeta=\\log\\left(\\dfrac{r_+^{n-1}-r_-^{n-1}}{r^{n-1}}\\right)+\\gamma+\\psi(\\alpha)-\\psi(2\\alpha)+\\psi(\\alpha-2i\\bar{\\omega})\\;,\\nonumber\n\\end{equation}\nwhere $\\gamma$ is the Euler constant and $\\psi(x)$ denotes the digamma function.\nBecause the $\\log r$ term is associated with distinct powers of $r$, it is impossible to match Eqs.~\\eqref{farsolution@near} and \\eqref{NearsolFar2}. For this case, we have to resort to a numerical solution, which is discussed in the next section.\n\n\n\\section{Numerical results}\n\\label{numerical}\nIn order to confirm the above analytical results and to calculate the quasinormal frequencies for the special cases $\\frac{\\ell}{n-1}=p+\\frac{1}{2}$ where the analytical method fails, we shall solve, in this section, Eq.~\\eqref{RadialEq2} numerically. We use a direct numerical integration method to obtain the quasinormal frequency of the BH. To do so, taking the boundary conditions near the horizon in Eq.~\\eqref{boundaryingoing} and at infinity in Eq.~\\eqref{boundarydecaying}, we expand the radial function near the horizon as\n\\begin{equation}\n{\\rm X} \\sim e^{-i(\\omega-\\omega_0)r_\\ast} \\sum_{j=0}^{\\infty}\\alpha_j(r-r_+)^j\\;,\\label{nearHExp}\n\\end{equation}\nand at infinity as\n\\begin{equation}\n{\\rm X} \\sim r^{-\\frac{1}{2}(1+\\sqrt{4\\mu^2L^2+(n+1)^2})} \\sum_{j=0}^{\\infty}\\frac{\\beta_j}{r^j}\\;.\\label{infExp}\n\\end{equation}\nThe series expansion coefficients can be derived directly after inserting these expansions into Eq.~(\\ref{RadialEq2}).\nWe use the series expansion near the horizon Eq.~\\eqref{nearHExp} to initialize the radial system Eq.~\\eqref{RadialEq2} from a point $r_s$ which is close to $r_+$ through the relation $r_s=(1+0.01)r_+$, and integrate the radial system outwards up to a radial value $r_m$. Similarly we can also use Eq.~\\eqref{infExp} as initial condition to integrate the radial system inward from $r_l=1000r_+$ down to $r_m$. Then we have two solutions at an intermediate radius $r_m$, and these two solutions are linearly dependent if their Wronskian vanishes at $r_m$. Using a secant method one can solve $W(\\omega,r_m)=0$ iteratively to look for the quasinormal frequency of the BH. We also varied $r_s$, $r_m$ and $r_l$ to check the numerical accuracy.\n\nWe list some numerical results in Tables~\\ref{3DRNAdS}-\\ref{compmass2}. Note that all physical quantities are normalized by the AdS radius $L$ and we set $L=1$. In the first three tables, we focus on the fundamental modes of massless fields because they are typically the most unstable modes. To check the mass effect on the validity of the analytical formulas, we also consider $\\mu=0.5$ and $\\mu=3.0$ in the last two tables. As a check of our numerical method, we have calculated the quasinormal frequencies for small Schwarzschild-AdS BHs and we obtained results which are in good agreement with those reported in~\\cite{Konoplya:2002zu}.\n\nIn order to address the special case for which the analytical method fails, i.e when $\\ell=(p+\\frac{1}{2})(n-1)$, we chose $n=3$ (five dimensional spacetime) and $\\ell=1$, corresponding to $p=0$ in our condition. The results are shown in Table~\\ref{3DRNAdS}, with $r_+=0.1$, field mass $\\mu=0$ and field charge $q=8$. It shows clearly that a superradiant instability appears when $Q\/Q_c$ satisfies the condition in Eq.~\\eqref{onset2}. Moreover, we also list numerical results for the $\\ell=0$ mode in Table~\\ref{3DRNAdS}. It shows that the frequencies of the odd modes ($\\ell=1$) and even modes ($\\ell=0$) have a similar behavior; in other words, there is nothing special for odd modes.\n\\begin{table}\n\\caption{\\label{3DRNAdS} Frequencies of the fundamental modes with different $\\ell$ for a BH with $r_+=0.1$, $q=8$, $\\mu=0$ in $D=5$.}\n\\begin{center}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l }\n\\hline\n\\hline\n$Q\/Q_c$ & $\\ell$=0 & $\\ell$=1 \\\\\n\\hline\n0.1 & 3.958 - 1.335$\\times 10^{-2}$ i & 4.978 - 2.689$\\times 10^{-4}$ i\\\\\n0.3 & 3.997 - 6.435$\\times 10^{-3}$ i & 4.998 - 1.367$\\times 10^{-4}$ i\\\\\n0.5 & 4.030 - 1.522$\\times 10^{-3}$ i & 5.014 - 5.053$\\times 10^{-5}$ i\\\\\n0.7 & 4.058 + 1.996$\\times 10^{-3}$ i & 5.028 - 2.596$\\times 10^{-6}$ i\\\\\n0.8 & 4.070 + 3.198$\\times 10^{-3}$ i & 5.034 + 7.524$\\times 10^{-6}$ i\\\\\n0.9 & 4.081 + 3.954$\\times 10^{-3}$ i & 5.040 + 9.597$\\times 10^{-6}$ i\\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{center}\n\\end{table}\n\nTo confirm the validity of the analytical quasinormal frequency formulas in Eqs.~\\eqref{case1} and \\eqref{case2}, we also compare some analytical results with numerical data in Tables~\\ref{comp1}-\\ref{compmass2}. In Table~\\ref{comp1}, we present analytical results obtained from Eq.~\\eqref{case1} and numerical results with $r_+=0.01$, $q=6$ for the $\\ell=0$ massless fundamental mode in five dimensional spacetimes. They show good agreement; the difference is smaller than 1\\%. In Table~\\ref{comp2}, we present analytical results obtained from Eq.~\\eqref{case2} and numerical data for the $\\ell=1$ fundamental mode with $r_+=0.01$, $q=10$, $\\mu=0$ in $D=6$, and they show good agreement as well. From these two tables, we confirm the validity of the analytical matching method for \\mbox{$\\mu=0$}. Results for non-zero mass are reported in Tables~\\ref{compmass1} and~\\ref{compmass2}. Two conclusions may be drawn from these tables. First, as the mass increases the agreement between the analytical and numerical method becomes worse. This is expected in view of the discussion of the approximation employed in Section~\\ref{secmatching}. Second, as the mass increases, the mode with $Q\/Q_c=0.9$ becomes stable. This is in agreement with Eq.~\\eqref{onset1} since, for the parameters in Table~\\ref{compmass2}, superradiance is only expected for $Q\/Q_c \\gtrsim 1.1$.\n\\begin{table}[h]\n\\caption{\\label{comp1} Comparison of the frequencies for the $\\ell=0$ fundamental modes of a BH with $r_+=0.01$, $q=6$, $\\mu=0$ in $D=5$.}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l }\n\\hline\n\\hline\n$Q\/Q_c$ & Im$(\\omega)$ (numerical) & Im$(\\omega)$ (analytical) \\\\\n\\hline\n0.1 & -1.053$\\times 10^{-5}$  & -1.0441$\\times 10^{-5}$ \\\\\n0.3 & -7.369$\\times 10^{-6}$  & -7.3230$\\times 10^{-6}$ \\\\\n0.5 & -4.222$\\times 10^{-6}$  & -4.2050$\\times 10^{-6}$ \\\\\n0.7 & -1.088$\\times 10^{-6}$  & -1.0870$\\times 10^{-6}$ \\\\\n0.9 & 2.023$\\times 10^{-6}$  & 2.0310$\\times 10^{-6}$ \\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\\begin{table}[h]\n\\caption{\\label{comp2} Comparison of the frequencies for the $\\ell=1$ fundamental modes of a BH with $r_+=0.01$, $q=10$, $\\mu=0$ in $D=6$.}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l }\n\\hline\n\\hline\n$Q\/Q_c$ & Im$(\\omega)$ (numerical) & Im$(\\omega)$ (analytical) \\\\\n\\hline\n0.1 & -4.377$\\times 10^{-11}$  & -4.3678$\\times 10^{-11}$ \\\\\n0.3 & -2.830$\\times 10^{-11}$  & -2.8274$\\times 10^{-11}$ \\\\\n0.5 & -1.342$\\times 10^{-11}$  & -1.3418$\\times 10^{-11}$ \\\\\n0.7 & -1.538$\\times 10^{-12}$  & -1.5371$\\times 10^{-12}$ \\\\\n0.8 & 2.283$\\times 10^{-12}$  & 2.2846$\\times 10^{-12}$ \\\\\n0.9 & 3.778$\\times 10^{-12}$  & 3.7844$\\times 10^{-12}$ \\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\\begin{table}[h]\n\\caption{\\label{compmass1} Comparison of the frequencies for the $\\ell=0$ fundamental modes of a BH with $r_+=0.01$, $q=6$, $\\mu=0.5$ in $D=5$.}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l }\n\\hline\n\\hline\n$Q\/Q_c$ & Im$(\\omega)$ (numerical) & Im$(\\omega)$ (analytical) \\\\\n\\hline\n0.1 & -1.093$\\times 10^{-5}$  & -1.0844$\\times 10^{-5}$ \\\\\n0.3 & -7.711$\\times 10^{-6}$  & -7.6617$\\times 10^{-6}$ \\\\\n0.5 & -4.498$\\times 10^{-6}$  & -4.4797$\\times 10^{-6}$ \\\\\n0.7 & -1.300$\\times 10^{-6}$  & -1.2977$\\times 10^{-6}$ \\\\\n0.9 &  1.878$\\times 10^{-6}$  & 1.8842$\\times 10^{-6}$ \\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\\begin{table}\n\\caption{\\label{compmass2} Comparison of the frequencies for the $\\ell=0$ fundamental modes of a BH with $r_+=0.01$, $q=6$, $\\mu=3.0$ in $D=5$.}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}} l l l }\n\\hline\n\\hline\n$Q\/Q_c$ & Im$(\\omega)$ (numerical) & Im$(\\omega)$ (analytical) \\\\\n\\hline\n0.1 & -2.379$\\times 10^{-5}$  & -2.3423$\\times 10^{-5}$ \\\\\n0.3 & -1.886$\\times 10^{-5}$  & -1.8637$\\times 10^{-5}$ \\\\\n0.5 & -1.399$\\times 10^{-5}$  & -1.3850$\\times 10^{-5}$ \\\\\n0.7 & -9.155$\\times 10^{-6}$  & -9.0632$\\times 10^{-6}$ \\\\\n0.9 & -4.321$\\times 10^{-6}$  & -4.2765$\\times 10^{-6}$ \\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\n\n\n\\section{Summary}\n\\label{discussion}\nWe have studied  the superradiant instability of small charged AdS BHs in $D$ dimensions, in the presence of a charged scalar field. Very recently our result has been generalized to the $D$-dimensional singly rotating Myers-Perry BHs, where the same conclusion was drawn, i.e. the superradiant instability exist for all $\\ell$ modes in \\textit{all} dimensions~\\cite{Aliev:2015wla,Delice:2015zga}.\n\nFirst, we solved the Klein-Gordon equation for a charged scalar field in charged AdS BHs with a standard matching method. We found that the relation between the angular momentum quantum number $\\ell$ and the spacetime dimension $D$ plays an important role in determining the analytical quasinormal frequency formula. When $\\ell=p(D-3)$, for a non-negative integer $p$, we found that the quasinormal frequencies of the small RN-AdS BHs have only an imaginary correction to the AdS normal frequencies. This is the case for all modes (i.e. all $\\ell$) in $D=4$, even $\\ell$ in $D=5$, $\\ell=0,3,6,9,\\dots$ in $D=6$, $\\ell=0,4,8,12,\\dots$ in $D=7$ and so on.\n\nA more subtle case occurs when $\\ell=(p+\\frac{1}{2})(D-3)$. For this case the matching method fails because a $\\log r$ term appears in the near region solution --~Eq.~\\eqref{NearsolFar2} -- which cannot be matched to Eq.~\\eqref{farsolution@near}. Failure to observe this limitation has led to a claim that odd $\\ell$ modes in $D=5$ did not exhibit superradiance~\\cite{Aliev:2008yk}. Here we have shown otherwise that the superradiant instability indeed exists using a numerical method which is mandatory for analyzing this case, in view of the invalidity of the matching method. A similar conclusion applies to all cases defined by  $\\ell=(p+\\frac{1}{2})(D-3)$, i.e, odd $\\ell$ in $D=5$, $\\ell=2,6,10,14,\\dots$ in $D=7$ and so on. Observe that this case can only occur in odd dimensions.\n\nFinally, all other cases have a complex correction to the AdS normal frequencies, i.e. the real part of the frequency is also shifted.\n\nOur analytic results show good agreement with the numerical results in Section~\\ref{numerical}. In particular a central conclusion is that all $\\ell$ modes in all dimensions, for sufficiently large field charge $q$ display superradiance. Moreover, in $D=4$, the dependence of the instability on the various parameters seems to be in qualitative agreement with the study of cavity BHs in $D=4$~\\cite{Herdeiro:2013pia,Hod:2013fvl,Degollado:2013bha}, and it would be interesting to make a more detailed comparison between the two cases.\n\nLet us close this study with two questions. First, is there a simple pattern for the behaviour of the frequencies as $D\\rightarrow \\infty$? A preliminary analysis could not unveil a simple formula. Finding such behavior would be relevant in view of the recent interest on General Relativity in the large $D$ limit \\cite{Emparan:2013moa}. Second, can one follow this instability numerically into the non-linear regime? It has been recently shown that in four dimensions the end-point of the instability is a hairy charged AdS BH~\\cite{Bosch:2016vcp}. It would be interesting to generalize such study to higher dimensions, to display how the properties of hairy BHs depend on the spacetime dimension.\n\n\\part{Scalar and Proca fields on asymptotically flat spacetimes}\n\\label{main:p1}\n\\include{GeneralInFlat}\n\\include{NeutralP}\n\\include{ChargedP}\n\\include{ChargedClouds}\n\\part{Scalar and Maxwell fields on asymptotically AdS spacetimes}\n\\label{main:p2}\n\\include{scalarHD}\n\\include{KerrAdS}\n\\include{conclusion}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{intro}\nThe last years have seen a generalization of Riemannian geometry to sub-Rie\\-ma\\-nnian geometry \\cite{agrachev2012, gromov144, montgomery2006, strichartz1989, strichartz1986}.\nA  sub-Riemannian manifold is a connected manifold $G$ with a distribution $D\\subset TG$ such that successive Lie brackets fields in $D$ generate all the tangent space $TG$. In addition, a positive definite scalar product $\\langle\\cdot,\\cdot\\rangle$ is defined at $D$ such that it is possible to calculate the length of admissible curves, i.e., the tangent curves on $D$. As in Riemannian geometry, the distance $\\rho$ between two points $p$ and $q$ in $G$ is defined as the infimum length of admissible curves that connect the points $p$ and $q$. By means of this distance, the sub-Riemannian manifold becomes a metric space \\cite{Bellaiche1996} and can be endowed with a Hausdorff measure. Also, there is an equivalent of the Riemannian volume, the so called Popp's volume. Introduced in \\cite{montgomery2006}, it is a smooth volume which is canonically associated with the sub-Riemannian structure and was used in \\cite{agrachev2012} to define the sub-Laplacian in sub-Riemannian geometry.\n\nOn sub-Riemannian manifolds which are nilpotent Lie groups (stratified Lie gro\\-u\\-ps) the spherical Hausdorff measure, the Popp's measure and the Haar measure are mutually constant multiples one of another \\cite{Agrachev2012a, ghezzi2014, Magnani2005, mitchell1985, barilari2013}. Then, it is natural to ask what means a submanifold of\na stratified Lie group to be of minimal measure.\n\nThe aim of this paper is, using the measure proposed by Magnani and Vittone for non-horizontal submanifolds in stratified Lie groups \\cite{Magnani2008}, to calculate the first variation of these submanifolds and determine the necessary conditions for a non-horizontal submanifold to be of minimal measure. We will also give new examples of non-horizontal minimal submanifolds in the Heisenberg group, in particular minimal surfaces in 5-dimensional Heisenberg group . In order to present the main results, we briefly introduce some concepts which will be dealt with in detail in the subsequent sections.\n\nLet $\\mathbb G$ be a stratified Lie group with a graded  Lie algebra $\\mathfrak{g}=\\mathfrak{g}^1\\oplus\\cdots\\oplus\\mathfrak{g}^r$ with $r\\geq1$. If $\\langle\\cdot,\\cdot\\rangle$ is a scalar product on $\\mathfrak g^1$, we can extend it to $\\mathfrak g$ by induction (see Proposition \\ref{extend}).\nWe also consider the distribution $D\\subset T\\mathbb G$ generated by $\\mathfrak g^1$ and the scalar product in $D$ generated by $\\langle\\cdot,\\cdot\\rangle$. This way, $(\\mathbb G,D,\\langle\\cdot,\\cdot\\rangle)$ becomes a sub-Riemannian manifold, also called the Carnot group. Note that, traditionally the scalar product on $D$ is extended to $T\\mathbb G$ and the Riemannian connection on $T\\mathbb G$ is used to make calculations on $\\mathbb G$. This observation will be our starting point. We shall work with a covariant derivative $\\overline\\nabla$ defined by $\\overline\\nabla X=0$ for all $X\\in\\mathfrak g$. It has intrinsic torsion which is essentially the negative of the Lie bracket in $\\mathbb G$ and the zero curvature tensor. So, this covariant derivative permits us to establish an interesting parallel between the invariants of submanifolds in $\\mathbb R^n$ and the invariants of submanifolds in $\\mathbb G$.\n\nThe geometry of a submanifold $M$ of $\\mathbb G$ at each point depends on the relative position of\n$TM$ and $D$. The submanifolds with a ``high'' contact with $D$ at one point may have\nsingularities from the metric point of view, even though being a $C^\\infty$ submanifold. In this paper,\nwe avoid these situations by considering a non-horizontal submanifold $M$ transverse to D,\ni.e., $TM+D=T\\mathbb G$.  For this submanifold the horizontal normal subspace $TM^\\perp$ play the same role as the normal space does to a submanifold in $\\mathbb R^n$. We define $TM^\\perp$ as the orthogonal subspace to $TM\\cap D$ in $D$, i.e., the horizontal distribution $D=(TM\\cap D)\\oplus TM^\\perp$. Hence, $T\\mathbb G=TM\\oplus TM^\\perp$ and we can use this decomposition\n(in general not orthogonal!) to project $\\overline\\nabla$ to a connection $\\nabla$ over $TM$.\n\nLet $e_1,\\ldots, e_n$ be a orthonormal basis of $\\mathfrak{g}$ with its dual $e^1,\\ldots,e^n$. Let $f_1,\\ldots,f_n$ be a adapted frame in the $T\\mathbb{G}$ such that $f_1,\\ldots,f_p$ are orthogonal to $TM\\cap D$ and $f_{p+1},\\ldots,f_{d_1}$ is a orthonormal basis of $TM\\cap D$ in $D$. We complete $f_{p+1},\\ldots,f_{d_1}$ to a basis $f_{p+1},\\ldots,f_n$ of $TM$ taking\n$\nf_j=e_j-\\sum_{\\alpha=1}^pA_j^\\alpha f_{\\alpha},\n$\nfor $j=d_1+1,\\ldots,n$. If we denote by $f^1,\\ldots, f^n$ its dual basis, then the sub-Riemannian volume form on $\\mathbb G$ is defined as $\\dif V=e^1\\wedge\\cdots\\wedge e^n=f^1\\wedge\\cdots\\wedge f^n$. When $M$ is a hypersurface, the $H$-perimeter measure is traditionally used \\cite{Montefalcone2007, Montefalcone2012, Danielli2007, Hladky2013}. For the non-horizontal submanifolds of codimension $p\\geq 1$, the spherical Hausdorff measure has the following representation proved in \\cite{Magnani2008}:\n\\begin{equation}\\label{M}\n\\int_M\\theta(\\tau^d_M(x))\\dif S^d_\\rho(x)=\\int_M|\\tau^d_{M}(x)|\\dif\\mbox{vol}_{h}(x) \\ ,\n\\end{equation}\nwhere $h$ is a fixed Riemannian metric, $d$ is the Hausdorff dimension of $M$, $\\theta(\\tau^d_M(x))$ is the metric factor (see Section \\ref{hspace}), $S^d_\\rho$ is the\t $d$-spherical Hausdorff measure and $\\dif\\mbox{vol}_{h}$ is the Riemannian volume form on $M$ induced by $(\\mathbb G,h)$.\n\nWe will denote by $\\mu$ the measure in non-horizontal submanifolds defined by  $\\dif\\mu(x)=|\\tau^d_{M}(x)|\\dif\\mbox{vol}_{h}(x)$, which  is a nat\\-u\\-ral can\\-di\\-date to de\\-fine the vol\\-ume for non-horiz\\-ontal submanifolds.\nIf the metric factor $\\theta(\\tau^d_M(x))$ is constant on $M$ (the case of the Heisenberg group $\\mathbb{H}^n$, see Section \\ref{hspace}) the measure $\\dif\\mu(x)$ is a multiple of $(Q-p)$-spherical Hausdorff measure on $M$. Writing the density $\\dif\\mu$ in the  adapted frame  $f^1,\\ldots,f^n$ we obtain a  beautiful formula which we will use for the variational calculation, namely\n$\n\\dif \\mu=f^{p+1}\\wedge\\cdots\\wedge f^n\n$ (see Theorem \\ref{dS}).\n\nThe second result of this paper (Theorem \\ref{firstvar}) is a sufficient condition for minimality of non-horizontal submanifolds. We say that a non-horizontal submanifold is minimal if $H+\\sigma=0$ on $TM^\\perp$, where $H$ is the mean curvature (Definition \\ref{curvmedia}) and $\\sigma$ is the mean torsion (Definition \\ref{colchete}). In the case of hypersurfaces, the mean torsion is null and so the definition of minimality is the same as in \\cite{Montefalcone2007, Montefalcone2012, Danielli2007, Hladky2013, Hurtado2010, Ritore2008} and also of minimal submanifolds of Riemannian geometry.\n\nAs a direct application of Theorem \\ref{firstvar} we present the following example: if $M$ is minimal submanifold of $\\mathbb R^{2n}$, then $N=M\\times\\mathbb R$ is minimal submanifold of $\\mathbb{H}^n$.\nFurthermore, an interesting application of this theorem is discussed in section \\ref{H2}, where we find minimal non-horizontal surfaces in the $5$-dimensional Heisenberg group $\\mathbb H^2$. Observe that in the $3$-dimensional Heisenberg group $\\mathbb H^1$ the tangent horizontal curves to minimal surfaces are lines and hence this surfaces are ruled surfaces \\cite{Pansu1982}. Now, for $\\mathbb H^2$ the tangent horizontal curves to minimal surfaces can be more general. In section \\ref{H2}, we will present two cases: the curves are lines and we obtain ruled surfaces; the curves are circles and we obtain a family of circles, which we will call tubular surfaces.\n\nIn the last section, we prove for non-horizontal hypersurfaces the following: $-H_{f_1}=\\mbox{div}_{\\mathbb G}\\Big(\\frac{\\mbox{grad}_{D}\\phi}{|\\mbox{grad}_{D}\\phi|}\\Big)$, where $\\phi:\\mathbb G\\rightarrow\\mathbb R$ is smooth function, $\\mbox{div}_{\\mathbb G}$ is the divergence function on $\\mathbb G$ and $\\mbox{grad}_D$ is the horizontal gradient operator. With this formula we give a proof that the hyperboloid paraboloid is minimal.\n\n\\section{Stratified Lie groups}\\label{slg}\n\nA stratified Lie group $\\mathbb{G}$ is an $n$-dimensional connected, simply connected nilpotent Lie group whose Lie algebra $\\mathfrak{g}$ decomposes as $\\mathfrak{g}=\\mathfrak{g}^1\\oplus\\mathfrak{g}^2\\oplus\\cdots\\oplus\\mathfrak{g}^r$ and satisfies the condition $[\\mathfrak{g}^1,\\mathfrak{g}^j]=\\mathfrak{g}^{j+1}, j=1,\\ldots, r-1,\\quad [\\mathfrak{g}^j,\\mathfrak g^r]=0, j=1,\\ldots,r$. Write $d_i=\\dim\\mathfrak{g}^i$, choose a basis $e_1,\\ldots,e_n$ of $\\mathfrak{g}$ such that $e_{d_{j-1}+1},\\ldots,e_{d_j}$ is a basis of $\\mathfrak{g}^j$ and denote its dual basis by $e^1,\\ldots,e^n$. Then, we define the \\emph{degree} of $e_k$ as $\\deg k=j$ if $d_{j-1}<k\\leq d_j$.\n\nWe also define an covariant derivative $\\overline\\nabla$ on $T\\mathbb G$ by\n\\begin{equation*}\n\\overline\\nabla e_j=0\n\\end{equation*}\nfor every $j=1,\\ldots,n$. Clearly, the curvature $\\overline K$ of $\\mathbb G$ is null. If $\\overline T$ is the torsion of $\\overline\\nabla$, then\n$\n\\overline T(e_i,e_j)=-[e_i,e_j]=-\\sum_{k=1}^nc_{ij}^ke_k$\nor\n\\begin{equation*\n\\overline T=-\\frac 1 2\\sum_{i,j,k=1}^nc_{ij}^ke^i\\wedge e^j\\otimes e_k,\n\\end{equation*}\nwhere $c_{ij}^k$ are the structure constants of $\\mathfrak{g}$ associated with $e_1,\\ldots,e_n$.\n\nThe stratification hypothesis on $\\mathfrak{g}$ implies that $c_{ij}^k=0$ if $\\deg{k}\\neq \\deg{i}+\\deg{j}$. In particular, $c_{ij}^k=0$ for $k=1,\\ldots,d_1$. We can then write\n\\begin{equation*}\n\\overline T=\\sum_{k=d_1+1}^n\\overline T^k\\otimes e_k,\\,\\,\\, \\mbox{where}\\,\\,\\, \\overline T^k=-\\frac 1 2\\sum_{i,j=1}^nc^k_{ij}e^i\\wedge e^j\\,.\n\\end{equation*}\n\nLet $D\\subset T\\mathbb G$  be defined by $D_x=\\{X(x)\\,|\\, X\\in\\mathfrak{g}^1\\}$. Consider a positive definite scalar product $\\left\\langle \\,,\\right\\rangle$ on  $D$ such that $e_1,...,e_{d_1}$ are orthonormal. With this scalar product, $(\\mathbb G,D,\\left\\langle ,\\right\\rangle)$ becomes a \\emph{sub-Riemannian manifold}.\n\n\\begin{prop}\\label{extend}\nThere is a canonical extension of $\\left\\langle ,\\right\\rangle$ on $D$ to a scalar product  on $T\\mathbb{G}$.\n\\end{prop}\n\\begin{proof}\nThe proof goes by induction. Suppose that we have extended $\\left\\langle , \\right\\rangle$ to a scalar product of $\\mathfrak{g}^k$. Then we have a scalar product on $\\mathfrak{g}^1\\oplus \\cdots \\oplus\\mathfrak{g}^k$, and we can consider the map\n$B:\\mathfrak{g}^1\\otimes\\mathfrak{g}^k\\rightarrow \\mathfrak{g}^{k+1}$ defined by $B(X\\otimes Y)=[X,Y]$.\nThe scalar products on both $\\mathfrak{g}^1$ and $\\mathfrak{g}^k$ induce a scalar product on $\\mathfrak{g}^1\\otimes\\mathfrak{g}^k$. Thus, $B:(\\ker B)^\\perp\\rightarrow\\mathfrak{g}^{k+1}$ is an isomorphism, by which we can transport the scalar product of $(\\ker B)^\\perp$ to $\\mathfrak{g}^{k+1}$.\n\\end{proof}\nWe shall use the same symbol $\\langle,\\rangle$ for the extended scalar product. Therefore, from the definition of $\\overline\\nabla$ and Proposition \\ref{extend}, we get for every $X,Y\\in T\\mathbb G$ that\n\\begin{equation*}\n\\overline\\nabla\\left\\langle X,Y\\right\\rangle=\\left\\langle \\overline\\nabla X,Y\\right\\rangle+\\left\\langle X,\\overline\\nabla Y\\right\\rangle.\n\\end{equation*}\n\\begin{prop}\nSuppose that $f:\\mathfrak{g}\\rightarrow\\mathfrak{g}$ is a graded automorphism of Lie algebras such that\n$\\left.f\\right|_{\\mathfrak{g}^1}:\\mathfrak{g}^1\\rightarrow\\mathfrak{g}^1$\nis an isometry. Then $f$ is an isometry of $\\mathfrak{g}$.\n\\end{prop}\n\\begin{proof} The proof goes by induction. We will suppose that we have shown that  $f:\\mathfrak{g}^1\\oplus\\cdots\\oplus\\mathfrak{g}^k\\rightarrow\\mathfrak{g}^1\\oplus\\cdots\\oplus\\mathfrak{g}^k$ is an isometry. Consider $\\fof:\\mathfrak{g}^1\\otimes\\mathfrak{g}^k\\rightarrow\\mathfrak{g}^1\\otimes\\mathfrak{g}^k$ defined by $\\fof(x\\otimes Y)=f(x)\\otimes f(Y)$. Then, $\\fof$ is an isometry, i.e.,\n\\begin{align*}\n\\langle\\fof(x\\otimes Y),f\\otimes f(x'\\otimes Y')\\rangle=&\n\\langle fx,fx'\\rangle\\langle  fY, fY'\\rangle=\\langle x,x'\\rangle\\langle  Y, Y'\\rangle\\\\\n=&\\langle x\\otimes Y,x'\\otimes Y'\\rangle.\n\\end{align*}\nAs before, let $B:\\mathfrak{g}^1\\otimes\\mathfrak{g}^k\\rightarrow\\mathfrak{g}^{k+1}$ be defined by $B(x\\otimes Y)=[x,Y]$. Then\n\\begin{equation*}\nB(\\fof(x\\otimes Y))=B(f(x)\\otimes f(Y))=[fx,fY]=f[x,Y]=f(B(x\\otimes Y)).\n\\end{equation*}\nTherefore, $\\fof(\\ker B)=\\ker B$ and $\\fof((\\ker B)^\\perp)=(\\ker B)^\\perp$.\n\nTake $Z,Z'\\in\\mathfrak{g}^{k+1}$. Then,\n$Z=B(u),\\, Z'=B(u')$,\nwhere $u,u'\\in (\\ker B)^\\perp$. Note that $\\fof (u),f\\!\\otimes\\!f(u')\\in (\\ker B)^\\perp$  and $B$, restricted to this subspace, is an isometry. So, we have\n\\begin{align*}\n\\langle fZ,fZ'\\rangle=&\\langle B(\\fof (u)),B(\\fof(u'))=\\langle \\fof (u),f\\!\\otimes\\!f(u')\\rangle\\rangle=\\langle u,u'\\rangle\\\\\n=&\\langle B(u),B(u')\\rangle=\\langle Z,Z'\\rangle\\,.\n\\end{align*}\n\\end{proof}\n\n\\section{Non-horizontal submanifolds}\\label{nhs}\n\nConsider a stratified Lie group $\\mathbb G$. A submanifold $M\\subset\\mathbb G$ of codimension $p$ is \\emph{non-horizontal} if $TM+D=T\\mathbb G$. We then have that $TM\\cap D$ is a subvector bundle with $\\dim TM\\cap D=d_1-p$.\n\nLet $U$ be an open neighborhood of $\\mathbb G$ such that $M\\cap U\\neq \\emptyset$. An adapted basis to $M$ on $U$ is a basis $f_1,\\ldots,f_n$ on $U$ such that\n\\begin{equation*}\nf_j=\\sum_{k=1}^{d_1}a_j^ke_k,\\,  j=1,\\cdots,d_1,\n\\end{equation*}\nis an orthonormal basis of $D$  with\n$f_1,\\ldots,f_p$  orthogonal to $TM\\cap D$ on $D$ and $f_{p+1},\\ldots,f_{d_1}$ is a basis of $TM\\cap D$. It follows that the matrix $(a_j^k)_{1\\leq j,k\\leq d_1}$ is orthogonal.\nWe complete $f_{p+1},\\ldots,f_{d_1}$ to a basis $f_{p+1},\\ldots, f_n$ of $U$ by\n\\begin{equation}\\label{fjj}\nf_j=e_j-\\sum_{\\alpha=1}^pA_j^\\alpha f_{\\alpha},\\, j=d_1+1,\\ldots,n.\n\\end{equation}\nsuch that $f_j$ is tangent to $M$ on $U$.\n\nNow, let $i,j=1,\\ldots,n$ and let $\\overline\\omega_j^i$ be the connection $1$-forms defined in terms of the basis $f_i$. Then\n\\begin{equation}\\label{conex1}\n\\overline\\nabla f_j=\\sum_{i=1}^{d_1}\\overline\\omega _j^i\\otimes f_i,\n\\end{equation}\nwhere\n\\begin{align*}\n\\overline\\omega _j^i&=\\sum_{k=1}^{d_1}a^k_i\\dif a_j^k, i,j=1,\\ldots,d_1,\\\\\n\\overline\\omega_j^\\alpha&=-\\dif A_j^{\\alpha}-\\sum_{\\beta=1}^pA_j^{\\beta}\\overline\\omega _\\beta^\\alpha, \\alpha=1,\\cdots,p, j=d_1+1,\\ldots,n,\\\\\n\\overline\\omega_j^i&=-\\sum_{\\beta=1}^pA_j^{\\beta}\\overline\\omega _\\beta^i,i=p+1, \\ldots,d_1, j=d_1+1,\\ldots,n.\n\\end{align*}\n\n\\begin{prop}\\emph{({\\bf{Cartan structural equations}})}\\label{stru}\nThe $1$-forms $f^i$ and $\\overline\\omega_j^i$ satisfy\n\\begin{align*}\n\\dif f^\\alpha=&-\\sum_{j=1}^{n}\\overline\\omega^\\alpha_j\\wedge f^j+\\tilde T^\\alpha, \\ \\alpha=1,\\ldots,p,\\\\\n\\dif f^{i}=&-\\sum_{j=1}^n\\overline\\omega^i_j\\wedge f^j, \\ i=p+1,\\ldots,d_1,\\\\\n\\dif f^{j}=&\\,\\,\\overline T^j, \\ j=d_1+1,\\ldots,n,\\\\\n\\dif\\overline\\omega^k_i=&-\\sum_{j=1}^{d_1}\\overline\\omega^k_j\\wedge\\overline\\omega^j_i,\\,\nk=1,\\ldots,d_1,  \\ i=1,\\ldots,n,\n\\end{align*}\nwhere\n$\\widetilde T^\\alpha =\\sum_{k=d_1+1}^nA_k^\\alpha\\overline T^k , \\alpha=1,\\ldots,p$.\n\n\\end{prop}\n\\begin{proof}\nThese identities follow directly from the definition of the adapted basis to $M$.\n\\end{proof}\n\n\n\\subsection{The second fundamental form and the Weingarten operator}\\label{sff}\nWe denote by $TM^\\perp$ the subvector bundle of $T\\mathbb G$ on $M$ generated by $f_1, \\ldots,\\!f_p$. Then $TM^\\perp$  is orthogonal to $TM\\cap D$ on $D$ and  $T\\mathbb G=TM^\\perp\\oplus TM$ (in general non-orthogonal!). If $v\\in T_p\\mathbb G$, $p\\in M$, we  write $v=v^\\top + v^\\perp$, where $ v^\\top\\in TM$ and $ v^\\perp\\in TM^\\perp$. Therefore\n\\begin{equation*}\n\\overline\\nabla_{X}Y=(\\overline\\nabla_{X}Y)^\\top+(\\overline\\nabla_{X}Y)^\\perp\\,\\,\\, \\mbox{and}\\,\\,\\,\\overline T(X,Y)=\\overline T^\\top(X,Y)+\\overline T^\\perp(X,Y)\n\\end{equation*}\nfor every $X,Y\\in TM$.\n\nWe define the covariant derivative on $TM$ by\n$\n\\nabla_XY=(\\overline\\nabla_XY)^\\top,\\, \\forall\\, X,Y\\in TM\n$.\n\\begin{dfn}\\label{segforma}\nLet $S:TM\\times TM\\rightarrow TM^\\perp$ be the bilinear form defined by\n$$S(X, Y)=(\\overline\\nabla_XY)^\\perp.$$ We say that $S$\nis the \\emph{second fundamental form} associated to $M$.\n\\end{dfn}\nMoreover, given $\\xi\\in TM^\\perp$ and $X\\in TM$ we write $\n\\overline\\nabla_X\\xi=-A_\\xi(X)+\\nabla^\\perp_X\\xi,\n$\nwhere\n\\begin{equation*}\nA_\\xi(X)=-(\\overline\\nabla_X\\xi)^\\top\\in TM\\,\\,\\, \\mbox{and}\\,\\,\\,\\nabla^\\perp_X\\xi=(\\overline\\nabla_X\\xi)^\\perp\\in  TM^\\perp.\n\\end{equation*}\n\\begin{dfn} Let\n$A:TM\\times TM^ \\perp \\rightarrow TM$ be the bilinear form defined by $$A_\\xi(X)=-(\\overline\\nabla_X\\xi)^\\top.$$ We say that $A$ is the Weingarten operator associated to $M$.\n\\end{dfn}\nObserve that $A_\\xi:TM\\rightarrow TM\\cap D$, since $D$ is invariant under $\\overline\\nabla$. Also, we have that $\\nabla^\\perp:\\xi\\in TM^\\perp\\rightarrow \\nabla^\\perp\\xi\\in (TM)^\\ast\\otimes TM^\\perp$ is a linear connection on $TM^\\perp$, where $(TM)^\\ast$ is the dual space of $TM$.\n\n\\begin{rmk}\nThe second fundamental form is generally not symmetric, i.e., for every $X,Y\\in TM$,\n$\nS(X,Y)-S(Y,X)=(\\overline{\\nabla}_XY-\\overline{\\nabla}_YX-[X,Y])^\\perp=\\overline{T}^\\perp(X,Y).\n$\n\\end{rmk}\nLet us introduce\n$\nP:TM\\times TM^\\perp\\rightarrow\\mathbb{R}\n$\n as $P(X,\\xi)=\\langle X,\\xi\\rangle$ and\n\\begin{align*}\n\\widetilde\\nabla_X P(Y,\\xi)=&X(P(Y,\\xi))-P(\\nabla_XY,\\xi)-P(Y,\\nabla^\\perp_X\\xi)\\\\\n=&X\\langle Y,\\xi \\rangle-\\left\\langle\\nabla_XY,\\xi\\right\\rangle-\\langle Y,\\nabla^\\perp_X\\xi\\rangle.\n\\end{align*}\n\\begin{prop}For every $X,Y\\in TM$ and $\\xi\\in TM^\\perp$, we have that\n\\begin{equation*}\n\\left\\langle S(X,Y),\\xi\\right\\rangle=\\widetilde\\nabla_X P(Y,\\xi)+\\left\\langle A_\\xi(X),Y\\right\\rangle.\n\\end{equation*}\nIn particular, if $Y\\in TM\\cap D$, then $\\left\\langle S(X,Y),\\xi\\right\\rangle=\\left\\langle A_\\xi(X),Y\\right\\rangle$.\n\\end{prop}\n\\begin{proof}\nThe proof follows by Definition \\ref{segforma} and by compatibility of $\\overline\\nabla$ with the extended scalar product $\\left\\langle,\\right\\rangle$. Clearly, $\\widetilde\\nabla_X P(Y,\\xi)=0, \\forall\\,\\, Y\\in TM\\cap D$.\n\\end{proof}\nLet $K^\\perp$ be the curvature of $\\nabla^\\perp$, i.e.,\n$K^\\perp(X,Y)\\xi=\\!\\nabla^\\perp_X\\nabla^\\perp_Y\\xi-\\nabla^\\perp_Y\\nabla_X\\xi-\\nabla^\\perp_{[X,Y]}\\xi$\nand let us define\n\\begin{equation*}\n\\widetilde\\nabla_X S(Y,Z)=\\nabla^\\perp_X(S(Y,Z))-S(\\nabla_XY,Z)-S(Y,\\nabla_XZ).\n\\end{equation*}\nWe then obtain a sub-Riemannian version of the fundamental equations.\n\n\\begin{thm}The curvature $K$ of $\\nabla$, the curvature $K^\\perp$ of $\\nabla^\\perp$, the second fundamental form $S$ and the Weingarten operator $A$ satisfy\n\\begin{enumerate}\n\\item  (Gauss equation) $K(X,Y)Z=A_{S(Y,Z)}(X)-A_{S(X,Z)}(Y)$;\n\\item (Codazzi equation) $\\widetilde\\nabla_YS(X,Z)-\\widetilde\\nabla_XS(Y,Z)-S(T(X,Y),Z)=0$;\n\\item (Ricci equation) $K^\\perp(X,Y)\\xi=S(X,A_\\xi Y)-S(Y,A_\\xi X)$,\n\\end{enumerate}\nwhere $T$ is the torsion of $\\nabla$.\n\\end{thm}\nIn coordinates, we have the following. Taking the tangential component and the horizontal orthogonal component of (\\ref{conex1}) we obtain, respectively\n\\begin{equation}\\label{A}\nS(X,Y)=\\sum_{\\alpha=1}^p\\sum_{j=p+1}^nf^j(Y)\\overline\\omega^\\alpha_j(X)f_\\alpha\\,\\,\\, \\mbox{and}\\,\\,\\,  A_{f_\\alpha}(X)=-\\sum_{i=p+1}^{d_1}\\overline\\omega_\\alpha^i(X)f_i.\n\\end{equation}\nFurthermore, $\\nabla^\\perp_{X} f_\\alpha=\\sum_{\\beta=1}^p\\overline\\omega_\\alpha^\\beta(X)f_\\beta$.\n\nWe end this section with two definitions, the first of which being well known.\n\\begin{dfn}\\label{curvmedia}\nThe \\emph{mean curvature} of $M$ is the tensor $H:TM^\\perp\\rightarrow\\mathbb R$ defined by\n$H_\\xi=-\\mbox{\\emph{trace} } A_\\xi$.\n\\end{dfn}\n\\begin{dfn}\\label{colchete}\n The \\emph{mean torsion} of $M$ is the tensor $\\sigma:TM^\\perp\\rightarrow\\mathbb R$ defined by\n\\begin{equation*}\n\\sigma_{\\xi}=\\sum_{j=d_1+1}^{n}f^j(\\overline T( \\xi,f_j)).\n\\end{equation*}\n\\end{dfn}\n\n\\section{The $\\mu$-measure of non-horizontal submanifolds}\\label{Hmnhs}\n\nIn this section, using the adapted basis to non-horizontal submanifold $M$, we will give a new expression for the density of the measure $\\mu$ on $M$. When $\\mathbb G$ is the Heisenberg group, this measure is a constant multiple of the spherical Hausdorff measure. In the general case the density of the spherical Hausdorff measure on $M$ is a multiple of $\\mu$ by a function. This function depends on the position of $TM$ in relation to the distribution $D$.\n\nFollowing \\cite{Magnani2008}, let $\\tau_{M}(x)$ be a unit $(n-p)$-tangent vector of $M$ at $x$ with respect to the extended metric $h:=\\langle,\\rangle$ at $\\mathbb G$. We define $\\tau^{d}_{M}(x)$ as the part of $\\tau_{M}(x)$ with maximum degree $d=Q-p$, where $Q$ is the Hausdorff dimension of $\\mathbb G$.\nLet  $v_1,\\ldots,v_p$ be a basis of sections of the orthogonal space $TM^{\\perp_h}$ which is orthonormal with respect to $h$, i.e., $\\langle v_i,TM\\rangle=0$, and  $v=v_1\\wedge v_2\\wedge\\cdots\\wedge v_p$ is the unit $p$-form of the orthogonal space $TM^{\\perp_h}$. We consider the measure $\\mu$ on $M$ with density\n\\begin{equation}\n\\dif \\mu=|\\tau^d_{M}|\\mbox{vol}_{h}\\llcorner M,\n\\end{equation}\nwhere\n$\\mbox{vol}_{h}\\llcorner M:=(v \\ \\lrcorner \\ \\mbox{dvol}_{h})|_{M}$,\nthe symbol $\\lrcorner$ denotes the contraction (or interior product) of a differential form with a vector,\n$\n\\mbox{dvol}_h=e^1\\wedge\\cdots\\wedge e^n=f^1\\wedge\\cdots\\wedge f^n\n$\nis the density of the spherical Hausdorff measure on $\\mathbb G$ as a sub-Riemannian metric space,  and $\\left|\\cdot\\right|$ is the norm induced by the fixed Riemannian metric $h$.\nThen\n\\begin{equation*}\n\\dif \\mu=|\\tau^{d}_{M}|v \\ \\lrcorner \\ \\mbox{dvol}_{h}=|\\tau^{d}_{M}|v \\ \\lrcorner \\ f^{1}\\wedge\\cdots\\wedge f^{n}.\n\\end{equation*}\n\\begin{thm}\\label{dS}\n\\begin{equation}\n\\dif \\mu=f^{p+1}\\wedge\\cdots\\wedge f^n \\ .\\label{volume}\n\\end{equation}\n\\end{thm}\n\\begin{proof} First, we denote\n\\begin{equation*}\nb_{jk}=\\langle f_{j},f_{k}\\rangle=\\delta_{jk} + \\sum^{p}_{\\alpha=1}A^{\\alpha}_{j}A^{\\alpha}_{k}.\n\\end{equation*}\nObserve that $\\langle f_{\\alpha},f_{k}\\rangle=-A^{\\alpha}_{k}$ for $j,k=d_{1}+1,\\ldots ,n$ and $\\alpha=1,\\ldots,p$.\nIf $\\mathbf B$ is the $(n-d_1)$-square matrix $\\mathbf{B}=(b_{jk})$,  $\\mathbf{A}$ is the $(n-d_1)\\times p$ matrix\n\\begin{equation*}\n\\mathbf A=\\left(\\begin{array}{cccc}\nA^{1}_{d_{1}+1}&A^{2}_{d_{1}+1}&\\cdots&A^{p}_{d_{1}+1}\\\\\nA^{1}_{d_{1}+2}&A^{2}_{d_{1}+2}&\\cdots&A^{p}_{d_{1}+2}\\\\\n\\vdots &\\vdots & \\ddots &\\vdots\\\\\nA^{1}_{n}&A^{2}_{n}&\\cdots&A^{p}_{n}\n\\end{array}\\right)\n\\end{equation*}\nand $\\mathbf I_{n-d_1}$ is the $(n-d_1)$-identity matrix, we obtain\n\\begin{equation*}\n\\mathbf B=\\mathbf I_{n-d_1}+\\mathbf{A}\\mathbf{A}^{T}.\n\\end{equation*}\nLet us now calculate $v$. To do this, we project $f_1,\\ldots,f_p$ on $TM^{\\perp_h}$. These are\n\\begin{equation*}\nw_{\\alpha}:=f_{\\alpha}+\\sum^{n}_{j=d_{1}+1}x^{j}_{\\alpha}f_{j},\n\\end{equation*}\nwhere we choose $x^{j}_{\\alpha}$ such that  $\\langle w_{\\alpha},f_{j}\\rangle=0 $ for $\\alpha=1,\\ldots,p$, $j=p+1,\\ldots,n$. In particular,\n\\begin{equation}\\label{x}\n\\sum^{n}_{k=d_{1}+1}x^{k}_{\\alpha}b_{jk}=A_j^\\alpha.\n\\end{equation}\nfor $\\alpha=1,\\ldots,p$ and $j=d_1+1,\\ldots,n$.  Thus the vectors $w_{\\alpha}$, $\\alpha=1,\\ldots,p$, are linearly independent and orthogonal to $TM$. Hence, take\n\\begin{equation*}\nv=\\frac{ 1} {|w_{1}\\wedge\\cdots\\wedge w_{p}|}w_{1}\\wedge\\cdots\\wedge w_{p}.\n\\end{equation*}\nObserve that $|w_{1}\\wedge\\cdots\\wedge w_{p}|=|\\det\\mathbf{W}| $, where $\\mathbf{W}=(W_{\\alpha\\beta})$ is a square $p$-matrix with\n\\begin{equation*}\nW_{\\alpha\\beta}=\\langle w_\\alpha,w_\\beta\\rangle=\\delta_{\\alpha\\beta}-\\sum^{n}_{j=d_{1}+1}x^{j}_{\\beta}A^{\\alpha}_{j}.\n\\end{equation*}\nIf $c_{kj}$ are the entries of the inverse matrix of $\\mathbf B$, it follows from (\\ref{x}) that\n\\begin{equation}\nx^{j}_{\\alpha}=\\sum^{n}_{k=d_{1}+1}c_{kj}A^{\\alpha}_{k},\\label{prodC}\n\\end{equation}\nand therefore $W_{\\alpha\\beta}=\\delta_{\\alpha\\beta}-\\sum^{n}_{j=d_{1}+1}A^{\\alpha}_{j}c_{jk}A^{\\beta}_{k}$, or\n$\n\\mathbf W=\\mathbf I_p-\\mathbf{A}^T\\mathbf{B}^{-1}\\mathbf{A}\n$.\nSo,\n\\begin{equation*}\nv\\lrcorner\\mbox{dvol}_h=\\frac{w_{1}\\wedge\\cdots\\wedge w_{p}}{|w_{1}\\wedge\\cdots\\wedge w_{p}|}\\,\\lrcorner\\,f^{1}\\wedge\\cdots\\wedge f^{n}=\\frac{1}{|\\det\\mathbf W|}f^{p+1}\\wedge\\cdots\\wedge f^{n},\n\\end{equation*}\nbecause $f^\\alpha=0$ on $M$ for $\\alpha=1,\\ldots,p$.\n\nLet us now find $\\tau^{d}_{M}$. First\n\\begin{align*}\n\\tau_{M}&=|f_{p+1}\\wedge\\cdots\\wedge f_{n}|^{-1}f_{p+1}\\wedge\\cdots\\wedge f_{n}=|f_{d_1+1}\\wedge\\cdots\\wedge f_{n}|^{-1}f_{p+1}\\wedge\\cdots\\wedge f_{n}\\\\\n&=|\\det\\mathbf B|^{-1}f_{p+1}\\wedge\\cdots\\wedge f_{d_1}\\wedge(e_{d_1+1}-\\!\\sum_{\\alpha=1}^p\\!A_{d_1+1}^\\alpha f_{\\alpha})\\wedge\\cdots\\wedge (e_{n}-\\!\\sum_{\\omega=1}^p\\!A_{n}^\\omega f_{\\omega}).\n\\end{align*}\nTherefore\n\\begin{align}\\label{taudM}\n\\tau^{d}_{M}=|\\det\\mathbf B|^{-1}f_{p+1}\\wedge\\cdots\\wedge f_{d_1}\\wedge e_{d_1+1}\\wedge\\cdots\\wedge e_{n},\n\\end{align}\nwhere $d=d_1-p+2d_2+\\ldots+ld_l=Q-p$. It follows that\n$\n|\\tau^{d}_{M}|=|\\det\\mathbf B|^{-1}.\n$\nThen $$\\dif\\mu=|\\det\\mathbf B|^{-1}|\\det\\mathbf W|^{-1}f^{p+1}\\wedge\\cdots\\wedge f^{n}\\,.$$ To finish the proof, we must show that $|\\det\\mathbf B| |\\det\\mathbf W|=1$. To do this, we need only apply the matrix determinant lemma (see for instance \\cite{Harville1998}). Taking determinants of\n{\\small{\n\\begin{equation*}\n\\left(\\begin{array}{cc}\n\\mathbf I&\\mathbf A^T\\\\\n0&\\mathbf B\n\\end{array}\\right)\n\\left(\\begin{array}{cc}\n\\mathbf I-\\mathbf A^T\\mathbf B^{-1}\\mathbf A&0\\\\\n\\mathbf B^{-1}\\mathbf A&\\mathbf I\n\\end{array}\\right)=\\left(\\begin{array}{cc}\n\\mathbf I&\\mathbf A^{T}\\\\\n\\mathbf A& \\mathbf B\n\\end{array}\\right)=\n\\left(\\begin{array}{cc}\n\\mathbf I&0\\\\\n\\mathbf A&\\mathbf I\n\\end{array}\\right)\n\\left(\\begin{array}{cc}\n\\mathbf I&\\mathbf A^{T}\\\\\n0&\\mathbf B-\\mathbf A\\mathbf A^T\n\\end{array}\\right)\n\\end{equation*}}}\nwe have $\\det(\\mathbf B)\\det(\\mathbf W)=1$.\n\\end{proof}\n\n\\subsection{The Heisenberg group $\\mathbb H^n$}\\label{hspace}\n\nWe now will give a proof that the Heisenberg group has constant metric factor.\n\nAccording to \\cite{Magnani2008}, the metric factor $\\theta(\\tau)$ associated with a simple $s$-vector $\\tau$ of $\\wedge_s\\mathfrak{g}$ is defined by\n\\begin{align}\\label{teta}\n\\theta(\\tau)=\\mathcal{H}^s_{|\\cdot|}(F^{-1}(\\exp(\\mathcal{L}(\\tau))\\cap B_1)),\n\\end{align}\nwhere $F:\\mathbb R^n\\rightarrow\\mathbb G$ is a system of graded coordinates with respect to an adapted orthonormal basis $(X_1,\\ldots,X_n)$, $\\mathcal{H}^s_{|\\cdot|}$ is the $s$-dimensional Hausdorff measure with respect to the Euclidean norm of $\\mathbb R^n$, $\\mathcal L(\\tau)$ is the unique subspace of $\\mathfrak{g}$ associated to $\\tau$ and $B_1$ is the open unit ball centered at $e$ with respect to a fixed homogeneous distance.\n\nWe now consider $\\mathfrak h$ to be the Lie algebra of the Heisenberg group $\\mathbb H^n$ generated by $e_1,\\ldots, e_{2n}$, $e_{2n+1}$ with $[e_i,e_{i+n}]=e_{2n+1}$ for $i=1,\\ldots,n$ and all the other brackets equal to $0$. We have $\\mathfrak{h}=\\mathfrak{h}^1\\oplus\\mathfrak{h}^2$, where $\\mathfrak{h}^1$ is generated by $e_1,\\ldots,e_{2n}$ and $\\mathfrak{h}^2$ by $e_{2n+1}$. Take a scalar product on $\\mathfrak{h}^1$ such that $e_1,\\ldots,e_{2n}$ is an orthonormal basis. We extend the chosen scalar product to $\\mathfrak h$ so that $e_{2n+1}$ is unitary and orthogonal to $\\mathfrak{h}^1$.\n\nThe group operation of Heisenberg group $\\mathbb H^n$ is given in exponential coordinates by\n\\begin{align*}\n(x^1,\\ldots,x^{2n},x^{2n+1})(y^1,\\ldots,y^{2n},y^{2n+1})=&\\Big( x^1+y^1,\\ldots,x^{2n}+y^{2n},x^{2n+1}+y^{2n+1}\\\\\n&+\\frac{1}{2}\\sum_{i=1}^n(x^iy^{i+n}-y^ix^{i+n})\\Big)\n\\end{align*}\nand the left-invariant vector fields are\n\\begin{equation*}\ne_i=\\frac{\\partial}{\\partial x^i}-\\frac{1}{2}x^{i+n} \\frac{\\partial}{\\partial x^{2n+1}},\\,\ne_{i+n}=\\frac{\\partial}{\\partial x^{i+n}}+\\frac{1}{2}x^{i} \\frac{\\partial}{\\partial x^{2n+1}},\\,\ne_{2n+1}=\\frac{\\partial}{\\partial x^{2n+1}}.\n\\end{equation*}\n\nWe shall first find a parametrization for the open unit ball $B_1$ of $\\mathbb H^n$ whose expression also works for a parametrization of $\\exp ^{-1}(B_1) \\subset\\mathfrak{h}$.\n\nBy \\cite[Proposition~5.1]{Diniz2009}, the Carnot-Caratheodory geodesics  $c(t)$ passing by the identity are solutions of the following system\n\\begin{align*}\n\\dot{c}=\\sum_{r=1}^{2n}\\lambda_re_r,\\quad \\dot{\\lambda}_j=-\\lambda_0\\lambda_{j+n},\\quad \\dot{\\lambda}_{j+n}=\\lambda_0\\lambda_{j},\\quad\n\\dot{\\lambda}_0=0.\n\\end{align*}\nwith initial conditions $c(0)=(0,\\ldots,0)$ and $\\lambda_k(0)=\\mu_k$, $k=0,\\ldots,2n$. Therefore,\n\\begin{align*}\n\\lambda_0&=\\mu_0\\\\\n\\lambda_j&=\\mu_j\\cos\\left(\\mu_0 t\\right)-\\mu_{j+n}\\sin\\left(\\mu_0 t\\right)\\\\\n\\lambda_{j+n}&=\\mu_{j+n}\\cos\\left(\\mu_0 t\\right)+\\mu_{j}\\sin\\left(\\mu_0 t\\right)\n\\end{align*}\nand\n\\begin{align*}\n\\dot{x}^j&=\\mu_j\\cos\\left(\\mu_0 t\\right)-\\mu_{j+n}\\sin\\left(\\mu_0 t\\right)\\\\\n\\dot{x}^{j+n}&=\\mu_{j+n}\\cos\\left(\\mu_0 t\\right)+\\mu_{j}\\sin\\left(\\mu_0 t\\right)\\\\\n\\dot{x}^{2n+1}&=\\frac{1}{2}\\sum_{j=1}^n(\\dot{x}^{j+n}x^j-\\dot{x}^jx^{j+n}).\n\\end{align*}\nIntegrating from $0$ to $t$ we get\n\\begin{align*}\n{x}^j(\\mu_0,\\ldots,\\mu_{2n},t)&=\\frac{\\mu_j}{\\mu_0}\\sin\\left(\\mu_0 t\\right)-\\frac{\\mu_{j+n}}{\\mu_0}\\left(1-\\cos\\left(\\mu_0 t\\right)\\right)\\\\\n{x}^{j+n}(\\mu_0,\\ldots,\\mu_{2n},t)&=\\frac{\\mu_j}{\\mu_0}\\left(1-\\cos\\left(\\mu_0 t\\right)\\right)+\\frac{\\mu_{j+n}}{\\mu_0}\\sin\\left(\\mu_0 t\\right)\\\\\n{x}^{2n+1}(\\mu_0,\\ldots,\\mu_{2n},t)&=\\frac{1}{2\\mu_0^2}\\left(\\sum_{r=1}^{2n}\\mu_r^2\\right)\\left(\\mu_0 t-\\sin\\left(\\mu_0 t\\right)\\right).\n\\end{align*}\nHence, the unitary ball $B_1$ defined by the homogeneous distance $\\rho$ is parameterized by the Carnot-Caratheodory exponential as\n\\begin{equation*}\n\\exp_{CC}(\\mu_0,\\ldots,\\mu_{2n})=\\left({x}^1(\\mu_0,\\ldots,\\mu_{2n},1),\\ldots,{x}^{2n+1}(\\mu_0,\\ldots,\\mu_{2n},1)\\right)\n\\end{equation*}\non the cylinder $-2\\pi\\leq\\mu_0\\leq2\\pi$, $\\sum_{r=1}^{2n}\\mu_r^2\\leq 1$. Observe that\n\\begin{equation*}\n\\left[\\begin{array}{l}\n{x}^j(\\mu_0,\\ldots,\\mu_{2n},1)\\\\\n{x}^{j+n}(\\mu_0,\\ldots,\\mu_{2n},1)\n\\end{array}\\right]=\\frac{1}{\\mu_0}\n\\left[\n\\begin{array}{lr}\n\\sin(\\mu_0)&-(1-\\cos(\\mu_0))\\\\\n(1-\\cos(\\mu_0))&\\sin(\\mu_0)\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{l}\n\\mu_j\\\\\n\\mu_{j+n}\n\\end{array}\n\\right].\n\\end{equation*}\nWe transform the matrix on the right-hand side into an  orthogonal matrix as follows:\n\\begin{equation*}\n\\left[\\begin{array}{l}\n{x}^j(\\mu_0,\\ldots,\\mu_{2n},1)\\\\\n{x}^{j+n}(\\mu_0,\\ldots,\\mu_{2n},1)\n\\end{array}\\right]=\\frac{\\sqrt{2(1-\\cos(\\mu_0))}}{|\\mu_0|}\n\\left[\n\\begin{array}{lr}\n\\cos\\alpha&-\\sin\\alpha\\\\\n\\sin\\alpha&\\cos\\alpha\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{l}\n\\mu_j\\\\\n\\mu_{j+n}\n\\end{array}\n\\right]\n\\end{equation*}\nwhere $\\cos\\alpha=\\dfrac{|\\sin \\mu_0|}{\\sqrt{2(1-\\cos\\mu_0)}}$ and $\\sin\\alpha=\\dfrac{\\sqrt{2(1-\\cos\\mu_0)}}{2\\sgn(\\mu_0)}$. If we introduce the change of variables\n\\begin{equation*}\n\\left[\n\\begin{array}{l}\n\\overline\\mu_j\\\\\n\\overline\\mu_{j+n}\n\\end{array}\n\\right]\n=\n\\left[\n\\begin{array}{lr}\n\\cos\\alpha&-\\sin\\alpha\\\\\n\\sin\\alpha&\\cos\\alpha\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{l}\n\\mu_j\\\\\n\\mu_{j+n}\n\\end{array}\n\\right],\n\\end{equation*}\nthen we can parameterize $B_1$ by\n\\begin{align}\\label{par}\n\\mathcal{P}(\\mu_0,\\overline\\mu_1\\ldots,\\overline\\mu_{2n})=\\Big(\\varepsilon(\\mu_0)\\overline\\mu_1,\\ldots,\\varepsilon(\\mu_0)\\overline\\mu_{2n}, \\frac{\\mu_0 -\\sin\\mu_0}{2\\mu_0^2}\\sum_{r=1}^{2n}\\overline\\mu_r^2\\Big)\\,.\n\\end{align}\non the cylinder $-2\\pi\\leq\\mu_0\\leq2\\pi$, $\\sum_{r=1}^{2n}\\overline\\mu_r^2\\leq 1$, where $\\varepsilon(\\mu_0)= \\dfrac{\\sqrt{2(1-\\cos\\mu_0})}{|\\mu_0|}$.\n\nUsing \\eqref{par} we can prove the result following:\n\\begin{thm}\nIf $M$ is a non-horizontal submanifold of $\\mathbb H^n$, then  the $\\mu$-measure on $M$ is a constant multiple of the spherical Hausdorff measure.\n\\end{thm}\n\\begin{proof} According to formula \\eqref{M} and \\eqref{teta} we must show that\n\\begin{align*}\n\\theta(\\tau_M^d(x))=\\mathcal{H}^{2n+1-p}_{|\\cdot|}(F^{-1}(\\exp(\\mathcal{L}(\\tau_M^d(x)))\\cap B_1))\n\\end{align*}\nis constant on a non-horizontal submanifold $M$ of $\\mathbb{H}^n$.\n\nFrom \\eqref{taudM} we get\n\n\\begin{equation*}\n\\tau^{d}_{M}=|\\det\\mathbf B|^{-1}f_{p+1}\\wedge\\cdots\\wedge f_{2n}\\wedge e_{2n+1}.\n\\end{equation*}\nTherefore, ${\\mathcal L}(\\tau^d_M)$ is generated by $f_{p+1},\\ldots,f_{2n},e_{2n+1}$.\n\nLet $A:\\mathbb R^{2n+1}\\rightarrow \\mathfrak h$ be the linear transformation such that $F:\\mathbb R^{2n+1}\\rightarrow \\mathbb H^{2n+1}$ is given by $F=\\exp\\circ A$. Then,\n\\begin{align*}\nF^{-1}(\\exp ({\\mathcal L}(\\tau_M^d))\\cap B_1)=A^{-1}({\\mathcal L}(\\tau_M^d)\\cap \\exp^{-1}(B_1)).\n\\end{align*}\nLet $\\phi\\in O(2n+1,\\mathbb{R})$ such that $\\phi(0,\\ldots,0,1)=(0,\\ldots,0,1)$ and $\\tilde \\phi=A\\phi A^{-1}$.\nNote that $\\tilde\\phi$ acts in $\\mathfrak h$ as an isometry, fixes $e_{2n+1}$ and ${\\mathcal L}(\\tilde \\phi\\tau_M^d)=\\tilde \\phi{\\mathcal L}(\\tau_M^d)$.\n\nIt follows from formula (\\ref{par}) that $\\tilde \\phi(\\exp^{-1}(B_1))=\\exp^{-1}(B_1)$. Hence,\n\\begin{align*}\n\\theta(\\tilde \\phi\\tau_M^d)&={\\mathcal H}^{2n+1-p}_{|.|}\\Big(F^{-1}\\Big(\\exp({\\mathcal L}(\\tilde \\phi\\tau_M^d))\\cap B_1\\Big)\\Big)\\\\\n&={\\mathcal H}^{2n+1-p}_{|.|}\\Big(A^{-1}\\Big({\\mathcal L}(\\tilde \\phi\\tau_M^d)\\cap\\exp^{-1}(B_1)\\Big)\\Big)\\\\\n&={\\mathcal H}^{2n+1-p}_{|.|}\\Big(A^{-1}\\Big(\\tilde \\phi {\\mathcal L}(\\tau_M^d)\\cap \\tilde \\phi\\Big(\\exp^{-1}(B_1)\\Big)\\Big)\\Big)\\\\\n&={\\mathcal H}^{2n+1-p}_{|.|}\\Big(A^{-1}\\circ \\tilde \\phi\\Big({\\mathcal L}(\\tau_M^d)\\cap \\exp^{-1}(B_1)\\Big)\\Big)\\\\\n&={\\mathcal H}^{2n+1-p}_{|.|}\\Big(\\phi\\circ A^{-1}\\Big({\\mathcal L}(\\tau_M^d)\\cap \\exp^{-1}(B_1)\\Big)\\Big)\\\\\n&={\\mathcal H}^{2n+1-p}_{|.|}\\Big(A^{-1}\\Big({\\mathcal L}(\\tau_M^d)\\cap \\exp^{-1}(B_1)\\Big)\\Big)\\\\\n&=\\theta(\\tau_M^d).\n\\end{align*}\nSince, for any $y\\in M$, ${\\mathcal L}(\\tau^{d}_{M}(y))$ can be obtained by the action of some $\\tilde\\phi$ on ${\\mathcal L}(\\tau^{d}_{M}(x))$, the function $\\theta(\\tau^{d}_{M})$ is constant on $M$.\n\n\n\\end{proof}\n\n\\section{First variation of $\\mu$}\\label{vvf}\nConsider an immersion $i:M\\rightarrow\\mathbb G$ such that $i(M)$ in $\\mathbb G$ is a non-horizontal submanifold of codimension $p$.  On $\\mathbb G$ we have the volume element\n\\begin{equation}\\label{volumeG}\n\\dif V=e^1\\wedge\\cdots\\wedge e^n=f^1\\wedge\\cdots\\wedge f^n\n\\end{equation}\nand on $i(M)$ the volume form\n\\begin{equation*}\n\\tilde\\Gamma(0):=\\dif \\mu=f^{p+1}\\wedge\\cdots\\wedge f^n.\n\\end{equation*}\nLet $F:(-\\epsilon,\\epsilon)\\times M\\rightarrow\\mathbb{G}$ be a differentiable map such that $F_u:M\\rightarrow\\mathbb{G}$ is an immersion for all $u\\in(-\\epsilon,\\epsilon)$, where $F_u(p)=F(u,p)$, and $i=F_0$. The map $F$ is a variation of $i$.  On each submanifold $F_u(M)$ we have the volume form $\\tilde\\Gamma(u)$ as constructed in Section \\ref{Hmnhs}. Thus, we have a family of volume forms $\\Gamma(u)=F_u^*(\\tilde\\Gamma(u))$ on $M$ and by the arguments of \\cite[p.~286]{Spivak1979}  we have\n\\begin{align}\\label{leibniz}\n\\left.\\frac{\\dif}{\\dif u}\\right|_{u=u_0}\\int_M\\Gamma(u)=\\int_M\\left.\\frac{\\dif}{\\dif u}\\right|_{u=u_0}\\Gamma(u)\\,.\n\\end{align}\nWe now state the main result of this section.\n\\begin{thm} \\label{firstvar}\nLet $i:M\\rightarrow\\mathbb G$ be an immersion of an oriented $(n-p)$-dimensional manifold $M$ into a sub-Riemannian stratified Lie group $(\\mathbb G,D,\\langle\\, ,\\,\\rangle)$ as a non-horizontal submanifold and let $F:(-\\epsilon,\\epsilon)\\times M \\rightarrow\\mathbb{G}$ be a variation of $i$ through immersions with variation vector field $W$. If $\\Gamma(u)=F_u^*(\\tilde\\Gamma(u))$, where $\\tilde\\Gamma(u)$ is the volume form of the $\\mu$-measure in $F_u(M)$ in accordance with the given orientation of $M$, then\n\\begin{equation*}\n\\dot{\\Gamma}(0)=i^*\\left(\nH_{ W^\\perp}+\\sigma_{ W^\\perp}\\right)\\Gamma(0)+\\dif \\left(i_*^{-1}(W^\\top)\\lrcorner\\Gamma(0)\\right),\n\\end{equation*}\nwhere $H$ is the mean curvature and $\\sigma$ is the mean torsion of $i(M)$.\n\\end{thm}\n\\begin{proof} We follow the ideas of Spivak \\cite{Spivak1979}. Let $W=\\left.\\frac{\\dif}{\\dif u}F\\right|_{u=0}$ be the variation vector field. Choose a point $p_0\\in M$ such that $W_{p_0}=W(i(p_0))$ is not tangent to $i(M)$. Then, in a neighborhood $O$ of $p_0$ (and decreasing $\\epsilon$ if necessary) we can suppose that $F:(-\\epsilon,\\epsilon)\\times O\\rightarrow\\mathbb G$ is an embedding. Let us denote $F_u(O)=O_u$. On an open set $\\widetilde O$ of $\\mathbb G$ containing the image of $(-\\epsilon,\\epsilon)\\times O$ we can choose a basis of vector fields $f_1,\\ldots,f_n$ such that\n\\begin{enumerate}\n\\item [(i)] $f_1,\\ldots,f_p$ restricted to $O_u$ are in $TO_u^\\perp$;\n\\item [(ii)] $f_{p+1},\\ldots,f_n$ restricted to $O_u$ are in $TO_u$;\n\\item [(iii)] $f_1,\\ldots,f_{d_1}$ are orthonormal in $D$;\n\\item [(iv)] $f_i=e_i-\\sum_{\\alpha=1}^pA_i^\\alpha f_\\alpha$,  for $i=d_1+1,\\ldots,n$.\n\\end{enumerate}\nIf $f^1,\\ldots,f^n$ are the dual $1$-forms, then $F_u^*(f^\\alpha)=0$ for $\\alpha=1,\\ldots,p$. Furthermore, if\n$\n\\Phi=f^{p+1}\\wedge\\cdots\\wedge f^n,\n$\n we get $\\tilde \\Gamma(u)=\\Phi|_{O_u}$ and\n\\begin{equation*}\n\\Gamma(u)=F_u^*\\tilde \\Gamma(u)=F_u^*\\Phi=F_u^*(f^{p+1}\\wedge\\ldots\\wedge f^n)\n\\end{equation*}\n is a volume form on $M$.\n\nNow, the vector field $\\frac{\\dif}{\\dif u}F$ defined on $F((-\\epsilon,\\epsilon)\\times O)$ can be extended further to a vector field $\\widetilde W$ defined on the open set $\\widetilde O$. Associated to $\\widetilde W$ there is a local $1$-parameter group of diffeomorphisms $\\rho_u$ such that $\\rho_u(F_v(p))=F_{u+v}(p)$. Hence, if $X$ is a tangent vector field on $O$, then $(\\rho_u)_*(F_v)_*X=(F_{u+v})_*X$. Therefore,\n\n\\begin{equation}\\label{Fu}\n\\dot{\\Gamma}(u)=F_u^*L_{\\widetilde W}\\Phi=F_u^*\\left(\\widetilde W\\lrcorner\\dif\\Phi+\\dif \\widetilde W\\lrcorner\\Phi\\right),\n\\end{equation}\nwhere $L_{\\widetilde W}$ is Lie derivative with respect to $\\widetilde W$.\n\nOn the other hand, using the Proposition \\ref{stru} we obtain\n\\begin{align*}\n\\dif\\Phi=&\\sum_{j=p+1}^n(-1)^{j-(p+1)}f^{p+1}\\wedge\\cdots\\wedge\\dif f^j\\wedge\\cdots\\wedge f^n\\\\\n=&\\sum_{j=p+1}^{d_1}(-1)^{j-(p+1)}f^{p+1}\\wedge\\cdots\\wedge\\left(\\sum^n_{i=1}f^{i}\\wedge\\overline{\\omega}_i^{j}\\right)\\wedge\\cdots\\wedge f^n\\\\\n&+\\sum_{j=d_1+1}^n(-1)^{j-(p+1)}f^{p+1}\\wedge\\cdots\\wedge \\overline T^j\\wedge\\cdots\\wedge f^n\n\\end{align*}\nSince $\\overline T^j=\\frac{1}{2}\\sum_{k,l=1}^n\\overline T^j(f_k,f_l)f^k\\wedge f^l$, then we can write the above equality as follows.\n\\begin{align*}\n\\dif\\Phi=&\\sum_{\\alpha=1}^p\\left(\\sum^{d_1}_{i=p+1}\\overline{\\omega}^{i}_{\\alpha}(f_i)+\\sum^n_{j=d_1+1}\\overline T^j(f_\\alpha,f_j)\\right)f^\\alpha\\wedge\\Phi\\\\\n&+\\sum_{\\alpha,\\beta=1}^p\\left(\\sum^{d_1}_{i=p+1}\\overline{\\omega}^{i}_{\\alpha}(f_\\beta)+\\frac{1}{2}\\sum^n_{j=d_1+1}\\overline T^j(f_\\alpha,f_\\beta)\\right)f^\\alpha\\wedge\\Psi\\\\\n=&\\sum^p_{\\alpha=1}\\Big(H_{f_\\alpha}+\\sigma_{f_\\alpha}\\Big)f^\\alpha\\wedge\\Phi+\\sum_{\\alpha,\\beta=1}^p\\Omega_{\\alpha\\beta}f^\\alpha\\wedge\\Psi,\n\\end{align*}\nwhere\n\\begin{equation*}\n\\Psi=f^{p+1}\\wedge\\cdots\\wedge f^\\beta\\wedge\\cdots\\wedge f^n\\,\\, \\mbox{and}\\,\\, \\Omega_{\\alpha\\beta}=\\sum^{d_1}_{i=p+1}\\overline{\\omega}^{i}_{\\alpha}(f_\\beta)+\\frac{1}{2}\\sum^n_{j=d_1+1}\\overline T^j(f_\\alpha,f_\\beta).\n\\end{equation*}\nIt now follows that\n\\begin{align*}\n\\widetilde W\\lrcorner\\dif\\Phi=&\\sum^p_{\\alpha=1}\\Big(H_{f_\\alpha}+\\sigma_{f_\\alpha}\\Big)\\Big(\\widetilde W\\lrcorner(f^\\alpha\\wedge\\Phi)\\Big)+\\sum_{\\alpha,\\beta=1}^p\\!\\!\\Omega_{\\alpha\\beta}\\Big(\\widetilde W\\lrcorner(f^\\alpha\\wedge\\Psi)\\Big)\\\\\n=&\\sum^p_{\\alpha=1}\\Big(H_{f_\\alpha}+\\sigma_{f_\\alpha}\\Big)\\Big(f^\\alpha(\\widetilde W)\\Phi-f^\\alpha\\wedge(\\widetilde W\\lrcorner\\Phi)\\Big)\\\\\n&+\\sum_{\\alpha,\\beta=1}^p\\Omega_{\\alpha\\beta}\\Big(f^\\alpha(\\widetilde W)\\Psi-f^\\alpha\\wedge(\\widetilde W\\lrcorner\\Psi)\\Big).\n\\end{align*}\nSince $F_u^*f^\\alpha=0$ and $F_u^*\\Psi=0$, then\n\\begin{equation*}\nF_u^*(\\widetilde W\\lrcorner\\dif\\Phi)=F_{u}^*\\left(\\sum_{\\alpha=1}^pf^\\alpha(\\widetilde W)(H_{f_\\alpha}+\\sigma_{f_\\alpha})\\right)\\Gamma(u)=F_{u}^*\\left(H_{\\widetilde W^\\perp}+\\sigma_{\\widetilde W^\\perp}\\right)\\Gamma(u) \\ .\n\\end{equation*}\nFor the other term is proved easily that\n$\n\\dif (\\widetilde W\\lrcorner\\Phi)=\\dif(\\widetilde W^\\top\\lrcorner\\Phi).\n$\n\nNow, equation (\\ref{Fu}) implies\n\\begin{equation*}\nF_u^*L_{\\widetilde W}\\Phi=F_{u}^*\\left(H_{\\widetilde W^\\perp}+\\sigma_{\\widetilde W^\\perp}\\right)\\Gamma(u)+F_{u}^*(\\dif(\\widetilde W^\\top\\lrcorner\\Phi)) \\ .\n\\end{equation*}\nBy setting $u=0$ we obtain the proof of this theorem at any point $p_0$ for which $W(p_0)$ is not tangent to $i(M)$.\nIn the general case, we proceed as follows.\n\nLet ${\\bf{G}}=\\mathbb{G}\\times\\mathbb{R}$ be the Lie group whose Lie algebra ${\\bf{g}}=\\mathfrak{g}\\oplus {\\mathbb{R}}$ (where $\\mathfrak{g}$ is the Lie algebra of $\\mathbb{G}$) admits a stratification\n${\\bf{g}}={\\bf{g^1}\\oplus g^2\\oplus\\cdots\\oplus g^r}$ such that\n\\begin{equation*}\n{\\bf{g^1}}=\\mathfrak{g}^1\\oplus\\mathbb{R}, \\ {\\bf g^i}=\\mathfrak{g}^i, \\ \\mbox{for} \\ i=2,3,\\ldots,r \\ \\mbox{and} \\ [\\mathfrak{g}^i,\\mathbb{R}]=0 \\ \\forall \\ i .\n\\end{equation*}\nLet $\\bf{D}$ be the distribution on $\\bf{G}$ generated by $\\bf{g}^1$. Therefore, $(\\bf{G, D},\\langle,\\rangle)$ is a sub-Riemannian stratified Lie group, where  by $\\left\\langle,\\right\\rangle$ we denote the scalar product induced on $\\bf{G}$.\n\nDefine ${\\widehat{F}}:(-\\epsilon,\\epsilon)\\times M\\rightarrow{\\bf{G}}$ by\n$\n{\\widehat{F}}(u,p)=(F(u,p),u).\n$\nThe new variation vector field $\\bf{W}$ is\n$\n{\\bf{W}}(p)=(W(p),1),\n$\nwhere $1$ denotes the unit vector field on $\\mathbb{R}$. Observe that $\\bf{W}$ is not tangent to $\\widehat{F}_0(M)$, so the theorem holds for $\\widehat{F}$. On the other hand\n\\begin{equation*}\n{\\bf{A_{\\widetilde{W}^{\\perp}}}}({\\bf{X}})=(A_{\\widetilde{W}^{\\perp}}(X),0), \\ {\\bf\\overline{T}(\\widetilde{W}^\\perp,X)}=(\\overline{T}(\\widetilde{W}^\\perp,X),0)\n\\end{equation*}\nwhere ${\\bf{X}}=X+\\xi\\in T{\\bf{G}}=T\\mathbb G\\oplus\\mathbb{R}$, which proves the claim of theorem for $F$.\n\n\\end{proof}\n\n\\begin{cor}\\label{eqmin}\nLet $M$ be an oriented  compact manifold  of dimension $n-p$ with boundary, $\\mathbb G$ a stratified Lie group of dimension $n$ and $i:M\\rightarrow\\mathbb G$  an immersion  of $M$ in $\\mathbb G$ as a non-horizontal submanifold. Let $F:(-\\epsilon,\\epsilon)\\times M \\rightarrow\\mathbb G$ be  a variation of $i$ through immersions, with variation vector field $W$. If $\\Gamma(u)=F_u^*(\\tilde\\Gamma(u))$, where $\\tilde\\Gamma(u)$ is the volume form in $F_u(M)$ in accordance with the given orientation of $M$, and\n\\begin{equation*}\nV(u)=\\int_M\\Gamma(u)\n\\end{equation*}\nthen\n\\begin{equation*}\n\\left.\\frac{\\dif}{\\dif u}V(u)\\right|_{u=0}=\\int_M i^*\\left(\nH_{ W^\\perp}+\\sigma_{ W^\\perp}\\right)\\Gamma(0)+\\int_{\\partial M} i_*^{-1}(W^\\top)\\lrcorner\\Gamma(0),\n\\end{equation*}\nwhere $H$ is the mean curvature and $\\sigma$ the mean torsion of $i(M)$. In particular, if $F$ is a variation that keeps the boundary of $M$ fixed, then\n\\begin{equation*}\n\\left.\\frac{\\dif}{\\dif u}V(u)\\right|_{u=0}\n=\\int_M i^*\\left(\nH_{ W^\\perp}+\\sigma_{ W^\\perp}\\right)\\Gamma(0)\\,.\n\\end{equation*}\nThe immersion $i$ is a critical point for $V$ among all immersions $f:M\\rightarrow\\mathbb G$ with $f=i$ on $\\partial M$ if and only if\n\\begin{equation*}\nH+\\sigma=0\n\\end{equation*}\non $TM^\\perp$.\n\\end{cor}\n\\begin{proof} The first claim follows from Theorem \\ref{firstvar}, formula (\\ref{leibniz}) and the Stokes'  theorem. If $F$ keeps $\\partial M$ fixed, then $W=0$ on $M$, so $i_*^{-1}(W^\\top)\\lrcorner\\Gamma(0)$ on $\\partial M$, which proves the second statement. The third one follows from the choice of a vector field $v\\in TM^\\perp$ such that for every $w\\in TM^\\perp$, $H_w+\\sigma_w=\\langle v,w\\rangle$ and a positive function $\\phi$ on the interior of $M$ such that $\\left.\\phi\\right|_{\\partial M}=0$. The last statement follows by taking $W^\\perp=\\phi v$.\n\\end{proof}\n\\begin{cor}\\label{eqminhiper}\nLet the assumptions of Corollary \\ref{eqmin} hold. If $M$ is a hypersurface,\nthen\n\\begin{equation*}\n\\left.\\frac{\\dif}{\\dif u}V(u)\\right|_{u=0}\n=\\int_M i^*\\left(\nH_{ W^\\perp}\\right)\\Gamma(0)+\\int_{\\partial M} i_*^{-1}(W^\\top)\\lrcorner\\Gamma(0),\n\\end{equation*}\nwhere $H$ is the mean curvature of $i(M)$. In particular, if $F$ is a variation that keeps the boundary of $M$ fixed, then\n\\begin{equation*}\n\\left.\\frac{\\dif}{\\dif u}V(u)\\right|_{u=0}\n=\\int_M i^*\\left(\nH_{ W^\\perp}\\right)\\Gamma(0),\n\\end{equation*}\nThe immersion $i$ is a critical point for $V$, among all immersions $f:M\\rightarrow\\mathbb G$ with $f=i$ on $\\partial M$ if and only if\n$\nH=0\n$\non $TM^\\perp$.\n\\end{cor}\n\\begin{proof} Observe that\n\\begin{equation*}\n\\overline T^j(f_\\alpha,f_j)=\\overline T^j(f_\\alpha,e_j-\\sum_{\\beta=1}^p A_j^\\beta f_\\beta)=\\sum_{i=1}^{d_1} a_\\alpha^i\\overline T^j(e_i,e_j)-\\sum_{\\beta=1}^p A_j^\\beta\\overline T^j(f_\\alpha,f_\\beta)\n\\end{equation*}\nand as $c_{ij}^j=0$ we get\n\\begin{equation*}\n\\overline T^j(f_\\alpha,f_j)=-\\sum_{\\beta=1}^p A_j^\\beta\\overline T^j(f_\\alpha,f_\\beta)\n\\end{equation*}\n and $\\overline T^j(f_\\alpha,f_\\beta)=0$ if $\\deg j\\neq 2$. In particular, if $M$ is a hypersurface of $\\mathbb G$, then $p=1$ and we obtain $\\overline T^j(f_1,f_j)=0$ for every $j=d_1+1,\\ldots,n$.\n \\end{proof}\nA special  case is when $M$ has the same dimension of $\\mathbb G$, so that $M$ is a compact $n$-dimensional manifold with boundary in $\\mathbb G$. Then $T_pM=T_p\\mathbb G$ so $TM^\\perp=0$ for all $p\\in M$. Consequently, $H_{ W^\\perp}+\\sigma_{ W^\\perp}=0$ and\n\\begin{equation*}\n\\left.\\frac{\\dif}{\\dif u}V(u)\\right|_{u=0}=\\int_{\\partial M} i_*^{-1}(W)\\lrcorner\\Gamma(0),\n\\end{equation*}\nfor any $W\\in TM$.\nIf $\\partial M$ is non-horizontal in all of its points, we can choose an adapted  basis $f_1,\\ldots,f_n$ of $T\\mathbb G|_{\\partial M}$. Then\n$ i_*^{-1}(W)\\lrcorner\\Gamma(0)=f^1(W)f^2\\wedge\\cdots\\wedge f^n$. Since $f^1=0$ on $T\\partial M$ we have that\n\\begin{equation*}\n\\left.\\frac{\\dif}{\\dif u}V(u)\\right|_{u=0}=\\int_{\\partial M}f^1(W)\\dif\\mu,\n\\end{equation*}\nwhere $\\dif\\mu=f^2\\wedge\\cdots\\wedge f^n$.\n\nMotivated by Corollary \\ref{eqmin} we give the following definition.\n\\begin{dfn}\\label{Hmin}\nWe say that a non-horizontal submanifold is \\emph{minimal} if\n\\begin{equation*}\nH+\\sigma=0\\,\\, \\mbox{on}\\,\\, TM^\\perp.\n\\end{equation*}\n\\end{dfn}\n\n\\begin{ex}\\label{verticalH}\nLet $M\\subset\\mathbb{R}^{2n}$ be a minimal submanifold and $N\\subset\\mathbb H^n$ non-horizontal submanifold defined by  $N=\\{(x,t)\\in \\mathbb H^n: x\\in M, t\\in \\mathbb R\\}$. Then, $N$ is minimal.\n\\end{ex}\nLet  $\\pi:\n\\mathbb H^n\\rightarrow \\mathbb R^{2n}$ be the natural projection and let $\\nabla^{\\mathbb R}$ be the canonical Riemannian connection on $\\mathbb R^{2n}$.\n Let $g_1,\\ldots,g_p$ and $g_{p+1},\\ldots,g_{2n}$ be orthonormal basis of $TM^{\\perp}$ and $TM$, respectively. Let $f_1,\\ldots, f_{2n}$ be a basis of $D$ restricted to $N$ such that $\\pi_*(f_j)=g_j$ for $j=1,\\ldots, 2n$. Therefore, $f_{2n+1}=\\dfrac{\\partial}{\\partial x^{2n+1}}$ extends $f_{p+1},\\ldots, f_{2n}$ to a basis of $TN$. Clearly, $\\pi_*(\\overline\\nabla_{f_i} f_\\alpha)=\\nabla^{\\mathbb R}_{g_i}g_\\alpha$ for $i=p+1,\\ldots, 2n$ and $\\alpha=1,\\ldots,p$.\nAs $\n\\overline\\nabla f_\\alpha=\\sum_{i=p+1}^{2n}\\overline\\omega_\\alpha^i\\otimes f_i,\n$\nand if $\n\\nabla^{\\mathbb R} g_\\alpha=\\sum_{i=p+1}^{2n}\\psi_\\alpha^i\\otimes g_i\n$ for $j=p+1,\\ldots, 2n$, we have that $\\pi^*\\psi_\\alpha^i=\\overline{\\omega}_\\alpha^i.$ It follows from (\\ref{A}) that\n\\begin{equation*}\n\\pi_{*}(A_{f_\\alpha}(f_i))=-\\sum_{j=p+1}^{2n}\\pi_{*}(\\overline\\omega_\\alpha^j(f_i)f_j)=-\\sum_{j=p+1}^{2n}\\psi_\\alpha^j(g_i)g_j=A_{g_\\alpha}(g_i).\n\\end{equation*}\nBecause $M$ is minimal, then\n\\begin{align}\\label{AW}\n\\sum_{j=p+1}^{2n}\\langle A_{f_\\alpha}(f_j),f_j\\rangle=0.\n\\end{align}\nNow, for the mean torsion we have that\n\\begin{align}\\label{TW}\n \\overline T^{2n+1}(f_\\alpha,f_{2n+1})=\\sum_{k=1}^{2n}a_\\alpha^k \\overline T^{2n+1}(e_k,e_{2n+1})=0,\n\\end{align}\nbecause $ \\overline T(e_k,e_{2n+1})=-[e_k,e_{2n+1}]=0$ for $k=1,\\ldots,2n$. Hence, from (\\ref{AW}) and (\\ref{TW}), $N$ is minimal.\n\nThe next example is well known in the literature (see \\cite{Pansu1982}).\n\\begin{ex} Let $S$ be a non-horizontal surface in $\\mathbb H^1$ at all points. If $S$ is minimal surface, then $S$ is ruled.\n\\end{ex}\nBy Corollary \\ref{eqmin}, we have\n$\\langle A_{f_1}(f_2),f_2\\rangle=0$ or $\\langle \\overline\\nabla_{f_2}f_1,f_2\\rangle=0$.\nIt follows that\n$\\langle \\overline\\nabla_{f_2}f_2,f_1\\rangle=0$, and therefore $\\overline\\nabla_{f_2}f_2=0$.\nConsider $\\pi$ and $\\nabla^\\mathbb R$ as in the Example \\ref{verticalH}.  Let $\\gamma$ be a tangent horizontal curve to $S$, with $\\gamma'(t)=f_2(\\gamma(t))$ and $g(t)=\\pi(\\gamma(t))$ the projection of $\\gamma$ on $\\mathbb R^2$. Then, $|g'(t)|=|\\pi_*\\gamma'(t)|=1$ and\n\\begin{equation*}\n\\nabla^\\mathbb R_{g'(t)}g'(t)=\\pi_*(\\overline\\nabla_{f_2}f_2)=0.\n\\end{equation*}\nTherefore, the curvature of curve $g$ is $0$. It follows that $g$ is a line in $\\mathbb R^2$. If  $g(t)=(x_0+at,y_0+bt)$, then\n\\begin{equation*}\n\\gamma(t)=(x_0+at,y_0+bt,z_0+\\frac{1}{2}(bx_0-ay_0)t)\n\\end{equation*}\nis a line, and it follows that $S$ is ruled.\n\n\\section{Non-horizontal surfaces in $\\mathbb{H}^2$}\\label{H2}\n\t\nIn this section we give new examples of minimal non-horizontal surfaces in the 5-dimensional  Heisenberg group $\\mathbb{H}^2$.\n\nThe Lie algebra $\\mathfrak{h}=\\mathfrak{h}^1\\oplus\\mathfrak{h}^2$ of the Heisenberg group $\\mathbb{H}^2$ is generated by $e_1, e_2, e_3, e_4, e_5$ with $[e_1,e_3]=[e_2,e_4]=e_5$. Let us define two linear operators acting on $\\mathfrak{h}^1$. The first operator we define by $Je_1=e_3$, $Je_2=e_4$, $Je_3=-e_1$ and $Je_4=-e_3$.\nThe second we define by $Re_1=e_2$, $Re_2=-e_1$, $Re_3=-e_4$ and $Re_4=e_3$. Consider the scalar product in $\\mathfrak{h}^1$ such that $e_1, e_2, e_3, e_4$ is an orthonormal basis of $\\mathfrak{h}^1$. One can easily prove the following properties of $J$ and $R$.\n\\begin{prop}\\label{JR} The following relations for $J$ and $R$ are true:\n\\begin{enumerate}\n\\item $J^2=R^2=-\\mbox{Id}$;\n\\item $RJ+JR=0$.\n\\item $\\langle Rx,Ry\\rangle=\\langle Jx,Jy\\rangle=\\langle x,y\\rangle, \\ \\forall \\ x,y\\in\\mathfrak{h}^1$;\n\\item $\\langle Jx,Ry\\rangle+\\langle Rx,Jy\\rangle=0,\\forall \\ x,y\\in\\mathfrak{h}^1$;\n\\item $[x,Jy]=\\langle x,y\\rangle e_5,\\, [x,Ry]=\\langle Jx,Ry\\rangle e_5,\\, \\forall \\ x,y\\in\\mathfrak{h}^1$,\n\\end{enumerate}\nwhere $\\mbox{Id}$ is identity application of $\\mathfrak h^1$.\n\\end{prop}\n \\begin{prop}\\label{basissuph4}\nIf $v_4\\in\\mathfrak{h}_1$ is unitary vector and $v_3=Rv_4,\\, v_2=-Jv_4,\\, v_1=-Rv_2$, then $v_1, v_2, v_3, v_4$ is orthonormal basis of $\\mathfrak{h}^1$ such that the only non-null brackets are $\n[v_1,v_3]=[v_2,v_4]=e_5$.\n\\end{prop}\n\\begin{proof} It follows immediately from  Proposition \\ref{JR} that $v_1,v_4,v_3,v_4$ is an orthonormal basis of $\\mathfrak{h}^1$.\nMoreover,\n\\begin{align*}\n[v_1,v_3]&=-[Rv_2,Rv_4]=-\\langle JRv_2,Rv_4\\rangle e_5=\\langle RJv_2,Rv_4\\rangle e_5=\\langle Jv_2,v_4\\rangle e_5=e_5\\\\\n[v_2,v_4]&=[-Jv_4,v_4]=\\langle v_4,v_4\\rangle e_5=e_5,\\\\ [v_2,v_3]&=[-Jv_4,Rv_4]=\\langle Rv_4,v_4\\rangle e_5=0,\\\\ [v_1,v_4]&=-[Rv_2,v_4]=\\langle Jv_4,Rv_2\\rangle e_5=-\\langle v_2,Rv_2\\rangle e_5=0,\\\\\n[v_1,v_2]&=[Rv_2,Jv_4]=\\langle Rv_2,v_4\\rangle e_5 =-\\langle v_1,v_4\\rangle e_5=0,\\\\\n[v_3,v_4]&=[ Rv_4,v_4]=-\\langle Jv_4,Rv_4\\rangle=0\\,.\n\\end{align*}\n\\end{proof}\n Naturally, we can extend the operators $J$ and $R$ to $D$ and it satisfy $\\overline \\nabla J=0$ and $\\overline\\nabla R=0$. We then get the following consequence of Proposition \\ref{basissuph4}.\n\\begin{cor}\\label{baseD}\nIf $f_4\\in D$ is an unitary vector field and $f_3=Rf_4, f_2=-Jf_4, f_1=-Rf_2$, then $f_1,f_2,f_3,f_4$ is an orthonormal basis of $D$ such that $\\overline T^5(f_1,f_3)=\\overline T^5(f_2,f_4)=-1$ and others are null.\n\\end{cor}\n\n\nLet $M\\subset \\mathbb H^2$ be a non-horizontal submanifold of dimension two. Then $TM\\cap D$ is of dimension one. Choose a unitary vector field $f_4 $ in $TM\\cap D$. Using Corollary \\ref{baseD} we complete it to an orthonormal basis of $D$ by choosing $f_3=Rf_4$, $f_2=-Jf_4$, $f_1=-Rf_2$ in $TM^\\perp$. Moreover, we get for every $X\\in TM$\n\\begin{equation*}\n\\overline\\omega_4^1(X)=\\overline\\omega_3^2(X), \\overline\\omega_4^2(X)=-\\overline\\omega_3^1(X)\\,\\,\\, \\mbox{and}\\,\\,\\, \\overline\\omega_2^1(X)=-\\overline\\omega_3^4(X)\\,.\n\\end{equation*}\nIf $M$ is minimal non-horizontal surface, then\n$\n-\\langle A_{\\xi}(f_4),f_4\\rangle+\\overline T ^5(\\xi,f_5)=0,\\, \\forall \\xi\\in TM^\\perp\n$.\nSo,\n\\begin{equation*}\n-\\overline\\omega^4_1(f_4)=A_5^3,\\, -\\overline\\omega^4_2(f_4)=0,\\, \\overline\\omega^4_3(f_4)=-A_5^1\\,.\n\\end{equation*}\nTherefore,\n\\begin{equation*}\n\\overline\\nabla_{f_4}f_4=-A_5^3f_1+A_5^1f_3\\,.\n\\end{equation*}\nIf we set\n$\nf_4=a^1e_1+a^2e_2+a^3e_3+a^4e_4\n$, where $a^i:\\mathbb{H}^2\\rightarrow\\mathbb{R}$ are smooth functions, then\n\\begin{align*}\nf_2&=-Jf_4=a^3e_1+a^4e_2-a^1e_3-a^2e_4,\\\\\nf_3&=Rf_4=-a^2e_1+a^1e_2+a^4e_3-a^3e_4,\\\\\nf_1&=-Rf_2=a^4e_1-a^3e_2+a^2e_3-a^1e_4\\,.\n\\end{align*}\nLet $\\varphi(t,s)$ be a parametrization of $M$ such that $\\varphi_s(t,s)=f_4(\\varphi(t,s))$. Then,\n\\begin{eqnarray}\\label{system}\n\\frac{\\dif}{\\dif s}\\left[\\begin{array}{c}\na^1\\\\\na^2\\\\\na^3\\\\\na^4\n\\end{array}\\right]=\\left[\n\\begin{array}{cccc}\n0&-\\overline\\omega^3_4(f_4)&0&\\overline\\omega^1_4(f_4)\\\\\n\\overline\\omega^3_4(f_4)&0&-\\overline\\omega^1_4(f_4)&0\\\\\n0&\\overline\\omega^1_4(f_4)&0&\\overline\\omega^3_4(f_4)\\\\\n-\\overline\\omega^1_4(f_4)&0&-\\overline\\omega^3_4(f_4)&0\n\\end{array}\\right]\n\\left[\\begin{array}{c}\na^1\\\\\na^2\\\\\na^3\\\\\na^4\n\\end{array}\\right]\\,.\n\\end{eqnarray}\nWe shall denote by $O(4,\\mathbb{R})$ the set of orthogonal matrices of order $4$. So,\nif $\\Phi(t,s)\\in O(4,\\mathbb{R})$ is a fundamental solution of (\\ref{system}), then  it satisfies\n\\begin{eqnarray}\\label{systemsol}\n\\left[\\begin{array}{c}\na^1(t,s)\\\\\na^2(t,s)\\\\\na^3(t,s)\\\\\na^4(t,s)\n\\end{array}\\right]=\\Phi(t,s)\n\\left[\\begin{array}{c}\na^1(t,0)\\\\\na^2(t,0)\\\\\na^3(t,0)\\\\\na^4(t,0)\n\\end{array}\\right]\\,.\n\\end{eqnarray}\nTherefore, if we write $\\varphi:\\mathbb R^2\\rightarrow \\mathbb R^5$ as\n\\begin{equation*}\n\\varphi(t,s)=(x^1(t,s),x^2(t,s),x^3(t,s),x^4(t,s),x^5(t,s)),\n\\end{equation*}\nthen\n$\\dfrac{\\partial x^i}{\\partial s}=a^i$ for $i=1,2,3,4$ and $\\dfrac{\\partial x^5}{\\partial s}=\\dfrac 1 2 (a^3x^1+a^4x^2-a^1x^3-a^2x^4)$. From this we get\n$$x^i(t,s)=x^i_t(0)+\\int_0^sa^i_t(u)\\dif u$$\nfor $i=1,2,3,4$ and\n\\begin{equation*}\nx^5_t(s)=x^5_t(0)+\\frac 1 2\\int_0^s\\Big(a^3_{t}(u)x^1_t(u)+a^4_{t}(u)x^2_t(u)-a^1_{t}(u)x^3_t(u)-a^2_{t}(u)x^4_t(u)\\Big)\\dif u,\n\\end{equation*}\nwhere $a^i_{t}(s)=a^i(t,s)$ and $x^{i}_t(s)=x^i(t,s)$.\n\nIn addition, if we impose the following condition\n$\n\\left\\langle\\dfrac{\\partial \\varphi}{\\partial t}(t,0),\\dfrac{\\partial \\varphi}{\\partial s}(t,0)\\right\\rangle=0\n$\nthen $\\dfrac{\\partial \\varphi}{\\partial t}(t,0)=e^5\\left(\\dfrac{\\partial \\varphi}{\\partial t}(t,0)\\right)f_5(t,0)$ and\n\\begin{equation*}\n\\varphi(t,0)=\\!\\!\\int_0^t\\!\\!r(v)\\Big(e_5(v,0)-A_5^1(v,0)f_1(v,0)-A_5^2(v,0)f_2(v,0)-A_5^3(v,0)f_3(v,0)\\Big)\\dif v,\n\\end{equation*}\nwhere $r(t)=e^5\\Big(\\dfrac{\\partial \\varphi}{\\partial t}(t,0)\\Big)$.\n\nUsing all of the above we can study two models of minimal non-horizontal surfaces in $\\mathbb{H}^2$.\n\\begin{ex} Ruled surface\n\\end{ex}\n Consider the case where $\\overline\\omega_4^1(f_4)=\\overline\\omega_4^3(f_4)=0$. In this case $a^i,i=1,\\cdots,4$ are constant on the tangent curves on $TM\\cap D$. So,\n\\begin{equation}\\label{x1234}\nx^i(t,s)=x^i(t)+a^i(t)s, i=1,2,3,4\n\\end{equation}\nwhere $x^i(t)=x^i(t,0)$ and $a^i(t)=a^i(t,0)$. Also, we get\n\\begin{equation}\\label{x5}\nx^5(t,s)=x^5(t)+\\frac{s}{2}\\Big(a^3(t)x^1(t)+a^4(t)x^2(t)-a^1(t)x^3(t)-a^2(t)x^4(t)\\Big).\n\\end{equation}\nMoreover,\n\\begin{equation*}\n\\varphi(t,0)=\\int_0^tr(v)\\Big(e_5(v,0)-A_5^2(v,0)f_2(v,0)\\Big)\\dif v\\,.\n\\end{equation*}\nNow, we write\n\\begin{align*}\n\\varphi(t,s)-\\varphi(t,0)=&\\sum_{i=1}^5\\Big(x^i(t,s)-x^i(t,0)\\Big)\\frac{\\partial}{\\partial x^i}\\\\\n=&\\sum_{i=1}^4\\Big(x^i(t,s)-x^i(t,0)\\Big)e_i(t)+\\Big(x^5(t,s)-x^5(t,0)\\\\\n&+\\frac 1 2\\Big((x^1(t,s)-x^1(t,0))x^3(t,0)+(x^2(t,s)-x^2(t,0))x^4(t,0)\\\\\n&-(x^3(t,s)-x^3(t,0))x^1(t,0)-(x^4(t,s)-x^4(t,0))x^2(t,0)\\Big)\\Big)e_5(t)\\,.\n\\end{align*}\nIt follows from (\\ref{x1234}) and (\\ref{x5}) that\n\\begin{align*}\n\\varphi(t,s)=\\varphi(t,0)+s f_4(t,0)\\,.\n\\end{align*}\n\n\\begin{ex} Tubular surfaces\n\\end{ex}\nConsider $\\overline\\omega_4^1(f_4)=b_1(t)$ and $\\overline\\omega_4^3(f_4)=b_3(t)$ constants along tangent curves on $TM\\cap D$ such that $b_1(t)^2+b_3(t)^2\\neq0$.\nThe eigenvalues of the matrix (\\ref{system}) are $\\pm i\\sqrt{b_1(t)^2+b_3(t)^2}$, where $i=\\sqrt{-1}$. Resolving the differential equation (\\ref{system}) we obtain the fundamental matrix\n\\begin{equation*}\n\\Phi(t,s)=\\left[\n\\begin{array}{cccc}\n\\cos(sb(t))&-\\dfrac{b_3(t)\\sin(sb(t))}{b(t)}&0&\\dfrac{b_1(t)\\sin(sb(t))}{b(t)}\\\\\n\\dfrac{b_3(t)\\sin(sb(t))}{b(t)}&\\cos(sb(t))&-\\dfrac{b_1(t)\\sin(sb(t))}{b(t)}&0\\\\\n0&\\dfrac{b_1(t)\\sin(sb(t))}{b(t)}&\\cos(sb(t))&\\dfrac{b_3(t)\\sin(sb(t))}{b(t)}\\\\\n-\\dfrac{b_1(t)\\sin(sb(t))}{b(t)}&0&-\\dfrac{b_3(t)\\sin(sb(t))}{b(t)}&\\cos(sb(t))\n\\end{array}\\right],\n\\end{equation*} where $b(t)=\\sqrt{b_1(t)^2+b_3(t)^2}$. Then\n\\begin{align*}\na^1(t,s)&=\\cos(sb(t))a^1(t)-\\frac{b_3(t)\\sin(sb(t))}{b(t)}a^2(t)+\\frac{b_1(t)\\sin(sb(t))}{b(t)}a^4(t),\\\\\na^2(t,s)&=\\frac{b_3(t)\\sin(sb(t))}{b(t)}a^1(t)+\\cos(sb(t))a^2(t)-\\frac{b_1(t)\\sin(sb(t))}{b(t)}a^3(t),\\\\\na^3(t,s)&=\\frac{b_1(t)\\sin(sb(t))}{b(t)}a^2(t)+\\cos(sb(t))a^3(t)+\\frac{b_3(t)\\sin(sb(t))}{b(t)}a^4(t),\\\\\na^4(t,s)&=-\\frac{b_1(t)\\sin(sb(t))}{b(t)}a^1(t)-\\frac{b_3(t)\\sin(sb(t))}{b(t)}a^3(t)+\\cos(sb(t))a^4(t)\\,.\n\\end{align*}\nIt follows that\n{\\small{\\begin{align*}\nx^1(t,s)&=x^1(t)+\\frac{\\sin(sb(t))}{b(t)}a^1(t)+b_3(t)\\frac{\\cos(sb(t))-1}{b(t)^2}a^2(t)-b_1(t)\\frac{\\cos(sb(t))-1}{b(t)^2}a^4(t),\\\\\nx^2(t,s)&=x^2(t)-b_3(t)\\frac{\\cos(sb(t))-1}{b(t)^2}a^1(t)+\\frac{\\sin(sb(t))}{b(t)}a^2(t)+b_1(t)\\frac{\\cos(sb(t))-1}{b(t)^2}a^3(t),\\\\\nx^3(t,s)&=x^3(t)-b_1(t)\\frac{\\cos(sb(t))-1}{b(t)^2}a^2(t)+\\frac{\\sin(sb(t))}{b(t)}a^3(t)-b_3(t)\\frac{\\cos(sb(t))-1}{b(t)^2}a^4(t),\\\\\nx^4(t,s)&=x^4(t)+b_1(t)\\frac{\\cos(sb(t))-1}{b(t)^2}a^1(t)+b_3(t)\\frac{\\cos(sb(t))-1}{b(t)^2}a^3(t)+\\frac{\\sin(sb(t))}{b(t)}a^4(t)\\,.\n\\end{align*}}}\nPlugging these terms in (\\ref{x5}) we get\n\\begin{align*}\nx^5(t,s)=&\\,x^5(t)+\\frac{\\cos(sb(t))-1}{2b(t)^2}\\Big(-(b_1(t)a^2(t)+b_3(t)a^4(t))x^1(t)\\\\\n&+(b_1(t)a^1(t)+b_3(t)a^3(t))x^2(t)-(b_3(t)a^2(t)-b_1(t)a^4(t))x^3(t)\\\\\n&-(b_3(t)a^1(t)+b_1(t)a^3(t))x^4(t)\\Big)+\\frac{\\sin(sb(t))}{2b(t)}\\Big(a^3(t)x^1(t)+a^4(t)x^2(t)\\\\\n&-a^1(t)x^3(t)-a^2(t)x^4(t)\\Big).\n\\end{align*}\nThen, the parametrization of a minimal non-horizontal surface in $\\mathbb{H}^2$ is\n{\\small{\\begin{align*}\n\\varphi(t,s)-\\varphi(t,0)=&\\,\\frac{\\sin(sb(t))}{b(t)}\\sum_{i=1}^4a^i(t)e_i(t)\\\\\n&+\\frac{\\cos(sb(t))-1}{b(t)^2}b_3(t)\\Big(a^2(t)e_1(t)-a^1(t)e_2(t)-a^4(t)e_3(t)\\\\\n&+a^3(t)e_4(t)\\Big)\n+\\frac{\\cos(sb(t))-1}{b(t)^2}b_1(t)\\Big(-a^4(t)e_1(t)+a^3(t)e_2(t)\\\\\n&-a^2(t)e_3(t)+a^3(t)e_4(t)\\Big)+\\Big(x^5(t,s)-x^5(t,0)\\\\\n&+\\frac 1 2(x^1(t,s)-x^1(t,0))x^3(t,0)+\\frac 1 2(x^2(t,s)-x^2(t,0))x^4(t,0)\\\\\n&-\\frac 1 2(x^3(t,s)-x^3(t,0))x^1(t,0)-\\frac 1 2(x^4(t,s)-x^4(t,0))x^2(t,0)\\Big)e_5(t)\\, ,\n\\end{align*}}}\nHence,\n{\\small{\\begin{equation}\\label{tubular}\n\\varphi(t,s)=\\varphi(t,0)+\\frac{\\sin(sb(t))}{b(t)}f_4(t,0)-\\frac{\\cos(sb(t))-1}{b(t)^2}\\Big(b_3(t)f_3(t,0)+b_1(t)f_1(t,0)\\Big).\n\\end{equation}}}\n\nConversely, suppose that $\\gamma$ is a transverse curve in $\\mathbb{H}^2$ ($\\gamma$ is a one dimensional non-horizontal submanifod) and $f_4\\in D$ is a unitary vector field along  $\\gamma$. According to Corollary \\ref{baseD} we can choose $f_2=-Jf_4,f_3=Rf_4, f_1=-Rf_2$ which is an orthonormal basis of $D$ along $\\gamma$. We complete this basis to a basis of $T\\mathbb H^2$ with $f_5=e_5-\\sum^3_{\\alpha=1}A_5^\\alpha f_\\alpha$ such that $\\dfrac{\\dif\\gamma}{\\dif t}=\\lambda_1 f_4+\\lambda_2 f_5$, where $\\lambda_1, \\lambda_2$ are non-null smooth functions on $\\mathbb{H}^2$.\n\nNow, if we define a non-horizontal surface $M$ by (\\ref{tubular}) with\n$\\varphi(t,0)=\\gamma(t,0)$, $b_1(t)=-A_5^3(t,0)$ and $b_3(t)=A_5^1(t,0)$, it follows that $M$ is minimal.\n\\begin{ex}{\\emph{\nLet $\\gamma(t)=\\left(r\\cos\\Big(\\frac{t}{r}\\Big),0,r\\sin\\Big(\\frac{t}{r}\\Big),0,0\\right)$ be a transverse curve in $\\mathbb{H}^2$ and let $ f_4(t)=\\left(0,\\cos\\Big(\\frac{t}{r}\\Big),0,\\sin\\Big(\\frac{t}{r}\\Big),0\\right)$\nbe an unitary vector field along $\\gamma$. Then,\n\\begin{align*}\nf_1(t)&=\\sin\\Big(\\frac{t}{r}\\Big)e_1(t)+\\cos\\Big(\\frac{t}{r}\\Big)e_3(t),\\\\\nf_2(t)&=\\sin\\Big(\\frac{t}{r}\\Big)e_2(t)-\\cos\\Big(\\frac{t}{r}\\Big)e_4(t),\\\\\nf_3(t)&=-\\cos\\Big(\\frac{t}{r}\\Big)e_1(t)+\\sin\\Big(\\frac{t}{r}\\Big)e_3(t)\\,.\n\\end{align*}\nMoreover,\n\\begin{equation*}\n\\frac{\\dif\\gamma}{\\dif t}=\\left(-\\sin\\Big(\\frac{t}{r}\\Big),0,\\cos\\Big(\\frac{t}{r}\\Big),0,0\\right)=\\frac{r}{2}\\left(\\frac{2}{r}\\cos\\Big(\\frac{2t}{r}\\Big)f_1(t)\n+\\frac{2}{r}\\sin\\Big(\\frac{2t}{r}\\Big)f_3(t)+e_5(t)\\right)\n\\end{equation*}\nand\n\\begin{equation*}\nf_5(t)=e_5(t)+\\frac{2}{r}\\cos\\Big(\\frac{2t}{r}\\Big)f_1(t)+\\frac{2}{r}\\sin\\Big(\\frac{2t}{r}\\Big)f_3(t).\n\\end{equation*}\nSo,\n\\begin{equation*}\nb_1(t)=A_5^3(t,0)=-\\frac{2}{r}\\sin\\Big(\\frac{2t}{r}\\Big),\\quad \\mbox{and}\\quad b_3(t)=-A_5^1(t,0)=\\frac{2}{r}\\cos\\Big(\\frac{2t}{r}\\Big)\n\\end{equation*}\n and we obtain that\n $\n b(t)=\\sqrt{b_1(t)^2+b_3(t)^2}=\\dfrac{2}{r}.\n $\nTherefore, the parametrization $\\varphi(t,s)$ of minimal non-horizontal surface in $\\mathbb{H}^2$ generated by $\\gamma(t)$ and the unitary vector field $f_4(t)$ is\n\\begin{align*}\n\\varphi(t,s)=&\\frac{r}{2}\\Big(\\cos\\Big(\\frac{t}{r}\\Big)\\Big(1+\\cos\\Big(\\frac{2s}{r}\\Big)\\Big),\\sin\\Big(\\frac{2s}{r}\\Big)\\cos\\Big(\\frac{t}{r}\\Big),\\\\\n&\\sin\\Big(\\frac{t}{r}\\Big)\\Big(1+\\cos\\Big(\\frac{2s}{r}\\Big)\\Big),\\sin\\Big(\\frac{2s}{r}\\Big)\\sin\\Big(\\frac{t}{r}\\Big),0\\Big).\n\\end{align*}\nThis surface is non-horizontal in all points except $(0,0,0,0,0)$.}}\n\\end{ex}\n\\section{Minimal hypersufaces}\nLet $\nM=\\{x\\in \\mathbb G :\\phi(x)=0\\,\\, \\mbox{e}\\,\\, \\mbox{grad}_{D}\\phi(x)\\neq 0\\}\n$\n be a non-horizontal hypersurface of $\\mathbb G$,\nwhere $\\phi:\\mathbb G\\rightarrow\\mathbb R$ is a real function $C^\\infty$ and $\\mbox{grad}_{D}$ denotes the horizontal gradient operator defined by\n\\begin{equation*}\n\\mbox{grad}_D\\phi=\\sum_{i=1}^{d_1}\\phi_ie_i\\,\\,\\, \\mbox{or}\\,\\,\\, \\dif\\phi(X)=\\langle \\mbox{grad}_{D}\\phi,X\\rangle,\\, \\forall\\, X\\in D,\n \\end{equation*}\n where $\\phi_i=e_i(\\phi)$.\n\nTo introduce the notion of divergence,  observe that the volume form \\eqref{volumeG} is parallel with respect to $\\overline\\nabla$. According to \\cite[p.~282]{Kobayashi1963}, for every vector field $X$ on $\\mathbb{G}$ we have\n\\begin{equation*}\n\\mbox{div}_{\\mathbb G}X=\\sum_{k=1}^{n}e^k(\\overline\\nabla_{e_k}X).\n\\end{equation*}\nLet\n$f_1=\\frac{\\mbox{grad}_D\\phi}{N}\n$,\nwhere\n$\nN=\\sqrt{\\sum_{i=1}^{d_1}\\phi_i^2},\n$\nand let $f_2,\\ldots,f_n$ be the adapted basis on $TM$.\n\n\\begin{prop}\\label{divHiper}\n$$-H_{f_1}=\\emph{div}_{\\mathbb G}\\left(\\frac{\\emph{grad}_{D}\\phi}{|\\emph{grad}_{D}\\phi|}\\right)\\,.$$\n\\end{prop}\n\\begin{proof}  From the definition of the mean curvature it follows that\n\\begin{align*}\n-H_{f_1}=&-\\sum_{i=2}^{d_1}\\langle A_{f_1}(f_i),f_i\\rangle=\\sum_{i=2}^{d_1}\\langle \\overline\\nabla_{f_i}f_1,f_i\\rangle=\\sum_{i=2}^{d_1}\\sum_{j=1}^{d_1}\\left\\langle \\overline\\nabla_{f_i}\\Big(\\frac{\\phi_j}{N}e_j\\Big),f_i\\right\\rangle\\\\\n=&\\sum_{j=1}^{d_1}\\sum_{i=2}^{d_1}\\langle e_j,f_i\\rangle {f_i}\\Big(\\frac{\\phi_j}{N}\\Big)=\\sum_{j=1}^{d_1}\\Big(e_j-\\langle e_j,f_1 \\rangle f_1\\Big)\\Big(\\frac{\\phi_j}{N}\\Big)\\\\\n=&\\sum_{j=1}^{d_1}e_j\\Big(\\frac{\\phi_j}{N}\\Big)-\\sum_{j=1}^{d_1}\\langle e_j,f_1 \\rangle f_1\\Big(\\langle e_j,f_1 \\rangle\\Big)\\\\\n=&\\,\\mbox{div}_{\\mathbb G}\\Big(\\frac{\\mbox{grad}_{D}\\phi}{|\\mbox{grad}_{D}\\phi|}\\Big)-\\frac 1 2 f_1\\Big(\\sum_{j=1}^{d_1}\\langle e_j,f_1 \\rangle^2\\Big)=\\mbox{div}_{\\mathbb G}\\left(\\frac{\\mbox{grad}_{D}\\phi}{|\\mbox{grad}_{D}\\phi|}\\right),\n\\end{align*}\nsince $\\sum_{j=1}^{d_1}\\langle e_j,f_1 \\rangle^2=1$.\n\\end{proof}\n\\begin{cor}\nLet $M\\subset\\mathbb{H}^n$ be a  non-horizontal hypersurface defined by the function $\\phi=u(x_1,\\cdots,x_{2n})-x_{2n+1}=0$. Then, $M$ is minimal if and only if\n\\begin{equation}\\label{Apgrafico}\n\\sum_{i=1}^{2n}u_{i,i} -\\frac{1}{N^2}\\sum_{i,j=1}^{2n}\\phi_i\\phi_ju_{i,j}=0,\n\\end{equation}\nwhere $u_{i,j}=\\frac{\\partial u}{\\partial x_i\\partial x_j}, \\phi_j=e_j(\\phi)$ and $N=\\sqrt{\\phi_1^2+\\cdots+\\phi_{2n}^2}$.\n\\end{cor}\n\\begin{proof} For $j=1,\\cdots,n, \\phi_j=u_{x_j}-\\frac 1 2 x_{j+n},\\, \\phi_{j+n}=u_{x_{j+n}}+\\frac 1 2 x_j$,  where $u_{x_{k}}=\\frac{\\partial u}{\\partial x_k}$. So\n$e_i(\\phi_j)=u_{x_i,x_j},\\, e_i(\\phi_{j+n})=u_{x_i,x_{j+n}}+\\frac 1 2\\delta_{ij},\\,\ne_{i+n}(\\phi_j)=u_{x_{i+n},x_j}-\\frac 1 2 \\delta_{ij}$ and $e_{i+n}(\\phi_{j+n})=u_{x_{i+n},x_{j+n}}$, for $i,j=1,\\cdots,n$.\nTherefore,\n\\begin{align*}\n-H_{f_1}=&\\sum_{i=1}^{2n}e_i\\frac{\\phi_i}{N}=\\sum_{i=1}^{2n}\\left(\\frac{e_i(\\phi_i)}{N}-\\phi_i\\sum_{j=1}^{2n}\\frac{\\phi_je_i(\\phi_j)}{N^3}\\right)\\\\\n=&\\frac{1}{N}\\left(\\sum_{i=1}^{2n}u_{i,i} -\\frac{1}{N^2}\\sum_{i,j=1}^{2n}\\phi_i\\phi_ju_{i,j} \\right)\\,.\n\\end{align*}\n   \\end{proof}\nFrom this proposition it follows immediately that the \\emph{hyperbolic paraboloid} with\n\\begin{equation*}\nu(x_1,\\ldots,x_{2n})=\\frac 1 4\\sum_{i=1}^n(x_i^2-x_{i+n}^2)\n\\end{equation*} is minimal as in \\cite{Montefalcone2012}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMultiple sclerosis (MS) is a chronic, progressive, and incurable autoimmune disease. Inflammation damages the myelin sheath, the protective coating of nerve cells, and causes signal disruption in the brain and spinal cord. The deterioration of nerve cells eventually becomes irreversible and leads to the development of disabilities. At least 1.3 million people worldwide are afflicted with MS with an average onset age of 29 years~\\cite{Dua:2008uf}. The incidence and prevalence rates vary amongst countries but remains a global problem~\\cite{Dua:2008uf}. Currently no cure exists for MS, but medications can help manage the symptoms, modify the disease course, and enhance the lifestyle of MS patients. Clinical trials have provided evidence that early diagnosis and treatment can slow the progression of MS, delaying the development of disabilities~\\cite{Murray:2006ip,Ross:2010ia}. Thus accurate identification of patients with high risk of developing MS is crucial to limiting the disease activity and prolonging a `normal' patient lifestyle.\n\nEarly MS diagnosis is a difficult problem as it lacks a single diagnostic test and common clinical features are shared with other diseases. Neurologists rely primarily on either the Poser or McDonald diagnostic criteria to classify the disease. The Poser criteria separates MS into four groups based on attacks, clinical evidence, and paraclinical evidence~\\cite{Poser:1983uw}. The McDonald diagnostic criteria, developed in 2001, leverages advancements in magnetic resonance imagining (MRI) techniques to facilitate diagnosis in typical clinical presentations~\\cite{McDonald:2001tk}. Recent modifications to McDonald criteria improve the classification applicability to pediatric, Asian, and Latin American communities~\\cite{Polman:2011co}. Nonetheless, a neurologist still relies on performing an exclusion diagnosis in conjunction with the patient's symptoms and medical history.\n\nThe advent of electronic medical records (EMRs) has increased the availability of medical data. Consequently, data mining and machine learning techniques have been used to develop clinical decision support systems to aid medical professionals. The problem of identifying patients with high risk of MS is a prime candidate for using EMRs to develop a data-driven prediction model. This paper investigates the feasibility and performance of a predictive disease model based on existing EMRs. Although our work is limited to patient data over a 7-year period, we establish a sparse baseline risk prediction model and demonstrate reasonable classification accuracy. \n\n\\section{Background and Related Work}\n\\label{sec:prior-work}\n\nThe exact nature and cause of MS is still unknown. Epidemiology studies have focused on discovering the variables that influence the development of MS. Prior research has identified genetic, environmental, and comorbidity risk factors that affect the disease incidence rates. These variables have been used to build models to predict the diagnosis and progression of the disease.\n\n\\subsection{Risk factors}\nGenetic susceptibility to MS has been supported by the following risk factors: race, gender, and family history. Genetic epidemiology studies have demonstrated a rise in disease risk when a family member is affected with MS~\\cite{Kahana:2000vu, Sadovnick:2002us, Compston:2008js, Ramagopalan:2011co}. The increase in risk is correlated with the degree of kinship~\\cite{Kahana:2000vu, Sadovnick:2002us, Compston:2008js, Ramagopalan:2011co}. Furthermore, the familial implications may also pertain to other autoimmune diseases~\\cite{Compston:2008js, Lauer:2010ep, Ramagopalan:2011co}. An individual's race, which is genetically determined, also factors into the development of the disease. Certain races, such as Asian, Native American, and African American, are less susceptible~\\cite{Kahana:2000vu,Ramagopalan:2011co}. Additionally, MS predominantly afflicts females and exhibit an early onset of the disease than their male counterparts~\\cite{Tintore:2009bd, Ramagopalan:2011co}.\n\nThe place of residence during a patient's formative years is one of the environmental factors in the development of MS. Studies have shown that immigrants migrating before adolescence acquire the risk associated with their new residence while immigrants moving after adolescence retain their risk of their original residence~\\cite{Hutter:1996en, Ascherio:2007jz, Ramagopalan:2011co}. The effect of latitude and hygiene hypothesis may account for the geographic variations beyond genetic factors.\n\nThe latitude gradient, a rise in incidence and prevalance rates with an increase in latitude, was previously a prominent MS feature but has declined in recent years. Sunlight duration, sunlight intensity, and vitamin D levels have been proposed as potential explanations for this phenomenon~\\cite{Hutter:1996en, Wingerchuk:2001ve, Ascherio:2007cd, Lauer:2010ep, Ramagopalan:2011co}. The seasonal vulnerability also demonstrates the importance of sun exposure as a patient is most vulnerable during the winter~\\cite{Hutter:1996en}. However, there are notable exceptions to the latitude theory in costal regions of Norway and Japan where a high consumption of fish dampens the lack of sunlight exposure~\\cite{Hutter:1996en, Ramagopalan:2011co}.\n\nThe hygiene hypothesis postulated that early exposure to various infectious agents protects the patient against risk of MS~\\cite{Ascherio:2007cd, Lauer:2010ep, Ramagopalan:2011co}. One infection in particular, Epstein Barr Virus (EBV), has been heavily associated with MS. Individuals with high anti-EBV antibodies have an increased risk of MS~\\cite{Ascherio:2010hl}. Additionally, contracting EBV at a later age also increases the likelihood of developing MS~\\cite{Ascherio:2007cd, Lauer:2010ep, Ramagopalan:2011co}. Other strains of viruses and infections,  such as human herpesvirus 6 (HHV6) and \\textit{Chlamydia pneumoniae}, have been proposed but lack sufficient evidence to support a casual effect on disease risk~\\cite{Wingerchuk:2001ve, Lauer:2010ep}.\n\nOne consequence of the hygiene hypothesis is the relationship between vaccinations and the susceptibility to MS. Countries with higher hygiene standards generally mandate vaccine immunizations to reduce the number of infections. However, during the late 1990s, concerns grew over the hepatitis B vaccine increasing the risk for MS~\\cite{Marshall:1998jw}. Although subsequent studies~\\cite{Ascherio:2001bd, Confavreux:2001ff} failed to find a significant correlation between the vaccine and the development of MS, the hypothesis that vaccinations may influence the development of the disease should not be dismissed~\\cite{Ascherio:2007cd}.\n\nAn individual's lifestyle, through diet and smoking habits, also factors into the disease risk. Individuals who consume non-marine meat have higher risk of developing the disease~\\cite{Lauer:2010ep}. However, fish and seafood consumption protects against MS~\\cite{Hutter:1996en, Ascherio:2007cd, Lauer:2010ep, Ramagopalan:2011co}. Marine life has a higher concentration of polyunsaturated fatty acids and antioxidants, which has anti-inflammatory properties that help suppress the disease process~\\cite{Hutter:1996en, Ascherio:2007cd}. High consumption of saturated fatty acids during one's childhood may cause adolescent obesity, which is associated with an increased risk of MS~\\cite{Ramagopalan:2011co}. Additionally, cigarette smoking has been shown to influence the development of MS~\\cite{Ascherio:2007cd, Lauer:2010ep, Ramagopalan:2011co}. \n\nShifts in an individual's hormone levels have also been suggested as factors in the disease process. A decrease in the number of MS relapses during pregnancy suggests the transient benefits of higher levels of estrogen~\\cite{Ascherio:2007cd, Tintore:2009bd}. A study on British women showed that the recent use of oral contraceptives reduced the risk of MS~\\cite{Alonso:2005ce}. However, a subsequent US study~\\cite{Hernan:2000ul} was unable to obtain evidence that supported the benefits of oral contraceptives.\n\nOther autoimmune disorders and specific cancers have been proposed as potential comorbidities to MS. In a paper that summarized the environmental features researched in etiological research on MS~\\cite{Lauer:2010ep}, Lauer noted that inflammatory bowel disease (IBD), ulcerative colitis, and Type 1 diabetes have the strongest correlations to MS amongst the various autoimmune disorders. The paper also referenced potential associations with Hodgkin's, oral, and colon cancers with the caveat that there was insufficient evidence to support these connections.\n\n\\subsection{Predictive models}\nPredictive studies have primarily focused on the progression of the disease. Bergamaschi et. al~\\cite{Bergamaschi:2001ud} identified clinical features that could help predict the onset of secondary progression, defined by an increase in the Kurtzke's Expanded Disability Status Scale (EDSS), using patient data collected in the first year of the disease. The factors discovered in the study were then used to propose a Bayesian Risk Estimate for Multiple Sclerosis (BREMS) score to predict the risk of reaching the secondary progression~\\cite{Bergamaschi:2006jw}. A recent study suggested the use of EDSS ranking to identify patients at risk for high progression rates 5 years from the onset of the disease~\\cite{Hughes:2012fu}.\n\nScoring systems have also been developed to assess the risk of disability. A study showed that MS Functional Composite, originally proposed as a clinical outcome measure, could be used to determine risk of severe physical disability~\\cite{Rudick:2001wn}. The Magnetic Resonance Disease Severity Scale (MRDSS) combined MRI measures into a composite score to predict the progression of physical disabilities~\\cite{Bakshi:2008fx}. Bazelier~\\cite{Bazelier:2012cc} derived a score using Cox proportional hazard models to estimate the long-term risk of osteoporotic and hip fractures in MS patients. Another study conducted by Margaritella et. al~\\cite{Margaritella:2012ba} used Evoked Potentials score to predict the progression of disability and identify patients with benign MS. \n\nLimited research has been done with regards to predicting the risk of developing MS. One work predicted MS in patients with mono symptomatic optic neuritis using MRI examination findings, oligoclonal bands in cerebrospinal fluid (CSF), immunoglobulin (Ig) G index,  and the seasonal time of onset~\\cite{Jin:2003hb}. Thrower~\\cite{Thrower:2007js} suggested the use of clinical characteristics of optic neuritis and traverse myelitis to identify high-risk MS patients. More recently, De Jager et. al~\\cite{DeJager:2009el} proposed a weighted genetic risk score (wGRS) based on genetic susceptibility loci in the context of environmental risk factors. However, prior research relies on specialized measurements that are performed to confirm a MS diagnosis. The approaches suggested do not generalize to all patients and fail to allow for early diagnosis and intervention of high-risk MS patients.\n\n\\begin{table*}[tb]\n\\caption{The list of MS features and their associated categories. The order of feature introduction is denoted by the number next to the category name.}\n\\subtable{\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{Demographics (1) } \\\\\n\\hline\nGender \\\\\nEthnicity \\\\\nRace \\\\\nAge \\\\\n\\hline\n\\multicolumn{1}{c}{Family History (1) } \\\\ \n\\hline\nMS \\\\\nMental Illness \\\\\nColon Cancer \\\\\nBreast Cancer \\\\\nLupus \\\\\nThyroiditis \\\\\nDiabetes \\\\\nInflammatory Bowel \\\\\n\\hline\n\\end{tabular}\n}\n\\subtable{\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{Autoimmune (2) } \\\\ \n\\hline\nInflammatory Bowel \\\\\nCeliac \\\\\nUveitis \\\\\nThyroiditis \\\\\nLupus \\\\\nRheumatoid arthritis \\\\\nSjoren's syndrome \\\\\nBell's palsy \\\\\nGuillain Barre \\\\\nDiabetes \\\\\nVitamin D deficiency \\\\\n\\hline\n\\end{tabular}\n}\n\\subtable{\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{Microbial (3)} \\\\ \n\\hline\nMeasles, mumps, rubella\\\\\nEpstein Barr Virus \\\\\n\\hline\n\\end{tabular}\n}\n\\subtable{\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{Mental Illness (4)} \\\\ \n\\hline\nBipolar \\\\\nSchizophrenia \\\\\n\\hline\n\\end{tabular}\n}\n\\subtable{\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{Cancer (5)} \\\\ \n\\hline\nLymphoma \\\\\nOral \\\\\nBreast \\\\\nColon \\\\\n\\hline\n\\end{tabular}\n}\n\\subtable {\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{Vaccine (6)} \\\\ \n\\hline\nHepatitis (A+B) \\\\\nDiphteria, tetanus, pertussis \\\\\nPolio \\\\\nInfluenza \\\\\nMeasles, mumps, and rubella \\\\\nVaricella (chicken pox) \\\\\nMeningococcal \\\\\nPneumococcal \\\\\n\\textit{Haemophilus influenzae} type b\\\\\nHuman Papillomavirus \\\\\n\\hline\n\\end{tabular}\n}\n\\subtable{\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{Reproductive (7)} \\\\ \n\\hline\nHysterectomy \\\\\nOral contraceptive pills \\\\\nEstrogen replacement therapy \\\\\n\\hline\n\\end{tabular}\n}\n\\subtable{\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{MRI Scans + Obesity (8)} \\\\ \n\\hline\nObesity \\\\\nAbnormal brain MRI \\\\\nBrain MRI \\\\\nCervical spine MRI \\\\\nThoracic spine MRI \\\\\n\\hline\n\\end{tabular}\n}\n\\subtable{\n\\begin{tabular}{l}\n\\hline\n\\multicolumn{1}{c}{Blood Tests (9)} \\\\ \n\\hline\nErythrocyte sedimintation rate \\\\\nLyme \\\\\nB12 \\\\\nANA panel \\\\\nAnti-cardiolipin antibody \\\\\nZinc \\\\\nCerebrospinal fluid exam \\\\\n\\hline\n\\end{tabular}\n}\n\\label{tab:variables} \n\\end{table*}\n\n\\section{Materials and Methods}\n\n\\subsection{Data}\nOur retrospective study used de-identified patient data from the NorthShore Enterprise Data Warehouse (EDW). The data was collected from January 2006 to July 2012 and contained information pertaining to demographics, medications, medical encounters and procedures.\n\nThe study examined adults ($\\geq$ 18 years of age) with complete demographic data (age, gender, ethnicity, and race). Any individual diagnosed with an MS ICD-9 code (``340\") during a Neurology office visit was selected as a case patient. 1,456 case patients were identified in the NorthShore EDW. However, only 737 of the patients had recorded electronic medical encounters prior to the initial diagnosis date. For each of the 737 case patients, four control subjects with matching age and gender were selected from the general population for a total of 3,685 patients.\n\n\\subsection{Predictor Variables}\nA comprehensive list of potential features was curated from prior MS research, detailed in section \\ref{sec:prior-work}. The list was also expanded to include common vaccinations, cancers, mental illnesses, and autoimmune diseases. Unfortunately some of the variables, such as lifestyle factors (smoking and alcohol use) and diet were unrecorded in the EMR. In addition, some diseases were also excluded because none of the patients received the particular medical diagnosis during the study period. Table \\ref{tab:variables} enumerates the features used in our retrospective study.\n\nThe initial MS diagnosis date is used to define $t_0$ for case patients. Since control patients did not have a MS diagnosis, $t_0$ is a randomly selected from the patient's encounter date. Figure \\ref{fig:encounter} shows the frequency of the number of encounters with the same date and location prior to $t_0$. Case patients generally have a low number of previous encounters while there is a more even distribution amongst the control patients. An alternative option was to use the same $t_0$ for matching control patients. However, this exacerbated the discrepancy in the number of encounters prior to $t_0$ between case and control. Thus, a random encounter date was used to define $t_0$ for control patients.\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics[scale=0.40]{encounters-hist.png}\n\\end{center}\n\\caption{Histogram of the number of encounters before $t_0$}\n\\label{fig:encounter}\n\\end{figure}\n\nAll data, except those related to family history, obtained after $t_0$ were discarded to limit the potential effects of confounding factors. Family medical history spanned the entire study period because collection time is unimportant. A patient may fail to disclose all the family history in the first few medical encounters but reveals the information in later encounters after being diagnosed with a certain disease.\n\nBinary values were used to indicate the presence of a particular medical diagnosis (ICD-9 code) found in a patient's encounter data prior to $t_0$. The vaccination, reproductive, and family medical history was extracted in a similar fashion, denoting the existing of specific supplemental classification codes (ICD-9 V codes). MRI scan features, except for an abnormal MRI result which was extracted via an ICD-9 code, signified the presence of specific medical procedure requests. Given the sparsity of numeric blood tests results, the feature was converted to three levels: (1) Unobserved, (2) Observed-Normal, and (3) Observed-Abnormal based on the ranges provided by the EDW. The entire feature set was represented using a binary matrix, where categorical variables were converted to dummy variables.\n\n\\subsection{Model Development}\nMultivariate logistic regression models were used to predict a MS diagnosis at the next encounter. The selection of logistic regression model was motivated by the popularity of the model in the medical community, the simplicity of the model, and the interpretability of the results. To evaluate the effect on accuracy of specific predictor categories, new features were introduced in the order defined by Table \\ref{tab:variables}. The first feature set contained only demographic and family history data, and was designed to mimic the information available at the first office visit. The last feature set contains all the predictor variables listed in Table \\ref{tab:variables}. For each feature set, two sets of logistic regression models were trained: (1) forward stepwise model selection by Akaike information criterion (AIC) and (2) backward stepwise model selection by AIC. Stepwise model selection by AIC is used to minimize the model complexity, or encourage a sparser feature representation, without sacrificing predictive performance. 10-fold cross validation was used to estimate the accuracy of each model.\n\n\\section{Results}\n\n\\subsection{Feature Set Comparisons}\nFigure~\\ref{fig:auc} shows the area under the receiver operating characteristic curve (AUC) for each feature set and model selection. Both feature selection methods result in similar predictive performance. Given only data that is available at the first office visit (demographic and family history), the forward selection model with an AUC of $0.538 \\pm 0.016$ marginally outperforms random guess. Feature set 2, an expansion of the features to include autoimmune disorder diagnoses, increases the AUC by $0.072$. The performance then remains stagnant with the addition of the microbial, mental illness,  and cancer feature categories. However, vaccinations, MRI scans, and blood test results boosted the predictive performance. Using all the available features (feature set 9), the forward and backward feature selection models predict an MS diagnosis at the next visit with an AUC of $0.724 \\pm 0.033$ and $0.718 \\pm 0.030$ respectively.\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{AUC.pdf}\n\\end{center}\n\\caption{A comparison of the AUC as categories of variables are added to the feature set.}\n\\label{fig:auc}\n\\end{figure}\n\nStepwise feature selection using AIC produces a model using a sparse set of features. Figure \\ref{fig:features} displays the comparison for the number of selected variables per feature set. The number of features selected with backward stepwise regression remains fairly constant for each feature set. This suggests the later categories (blood test results and MRI scans) are more informative. In addition, for feature sets 7-9, the backward selection method results in a sparse set of features. Both selection models on feature set 9 select less than 15\\% of the potential features.\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Features.pdf}\n\\end{center}\n\\caption{A box plot of the number of variables selected per feature set and selection method.}\n\\label{fig:features}\n\\end{figure}\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Pos-FS.pdf}\n\\end{center}\n\\caption{A scatterplot comparison of the predicted probabilities between consecutive feature sets for 737 case patients. The dotted line signifies no change in predicted risk.}\n\\label{fig:positive-fs}\n\\end{figure*}\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Neg-FS.pdf}\n\\end{center}\n\\caption{A 2D kernel density estimate of the previous feature set and next feature set predicted probabilities for 2,948 control patients.}\n\\label{fig:negative-fs}\n\\end{figure*}\n\nFigure \\ref{fig:positive-fs} compares the joint predicted probabilities of consecutive feature sets for case patients and illustrates the effects of adding specific feature categories. Some of the transitions have been omitted since they are similar to the first feature set transition (1$\\rightarrow$2). The addition of autoimmune disease diagnoses (1$\\rightarrow$2) generally increases the predicted risk. The trend is most noticeable in the transition plot from feature 7 to 8, where the points lie predominately above the dotted line. Inclusion of blood test results (8$\\rightarrow$9) marginally improves the predicted risk but it also decreases a substantial portion of patients with high risk in the previous model.\n\nA joint density estimate of two consecutive feature sets for the control patients is demonstrated in Figure \\ref{fig:negative-fs}. For the first transition (1$\\rightarrow$2), the improvement to predictive performance can also be traced to the decrease in the predicted risk of low risk MS patients. In addition, the figure illustrates that as the feature is expanded, the density slowly shifts away from the top right corner to the bottom half of the plot. The transition from feature set 8 to 9 shows that the predicted risk is distributed more evenly for the control patients compared to the first feature set, where the probabilities are lie around 0.21.\n\nFigures \\ref{fig:positive-fs} and \\ref{fig:negative-fs} show the dispersion of risk probabilities as more features are introduced. The addition of features related to MRI scans, or feature set 8, improves the separation between case and control through higher predicted probabilities for high-risk patients. The AUC improvement obtained from the inclusion of blood test results, feature set 9, can be primarily attributed to the decrease in predicted risk of control patients. These figures provide a graphical analysis of the benefits of specific feature categories.\n\n\\subsection{Feature Set 9 Results}\nWe further focus on the model with the best predictive performance, the forward selection model trained on all the features (feature set 9). Table \\ref{tab:coeff} summarizes the variables for which the magnitude of the coefficient is larger than 1 in the majority of the folds. The table also displays the number of case and control patients with the feature, the odds ratio, and the p-value from a chi-square test to determine the significance of the variable. If we use p-value as an initial filter with $\\alpha$=0.05, the following features would be eliminated: EBV; Bell's Palsy; colon cancer; family history of mental illness, MS, and inflammatory bowel disease; varicella vaccine, schizophrenia, and the \\textit{Haemophilus influenzae} type b vaccine. However, the selection of some of these variables, such as history of mental illness and varicella vaccine, is surprising given the lack of sufficient evidence in prior work to support their effect on the development of MS.\n\n\\begin{table*}[htb]\n\\begin{center}\n\\begin{tabular}{l r r r r r}\n\\hline\nFeature & \\multicolumn{1}{c}{Beta}& \\multicolumn{1}{c}{Case} \t&  \\multicolumn{1}{c}{Control} &  \\multicolumn{1}{c}{Odds ratio} &  \\multicolumn{1}{c}{p-value}\\\\\n\\hline\nPresence CSF oligoclonal bands\t&16.255$\\pm$0.545 \t& 27\t\t& 0\t\t& $\\infty$\t& 0.000 \\\\\nMental Illness (FH)\t\t\t\t& 6.298$\\pm$3.101 \t\t& 3\t\t& 1\t\t& 3\t\t& 0.033 \\\\\nEBV\t\t\t\t\t\t\t& 3.974$\\pm$3.924\t\t& 3\t\t& 2\t\t& 1.5\t\t& 0.093 \\\\\nAbnormal brain MRI\t\t\t\t& 2.877$\\pm$0.313 \t\t& 10\t\t& 4\t\t& 2.5\t\t& 0.000 \\\\\nUnobserved B12 \t\t\t\t& 2.527$\\pm$0.149 \t\t& 674\t& 2619\t& 0.257\t& 0.047 \\\\\nObs-Normal B12 \t \t\t\t& 2.375$\\pm$0.175\t\t& 54\t\t& 162\t& 0.333\t& 0.071 \\\\\nObs-Normal ANA SSB \t\t\t& 1.141$\\pm$0.269 \t\t& 57\t\t& 95\t\t& 0.6\t\t& 0.000 \\\\\nBell's Palsy\t\t\t\t\t& 1.889$\\pm$0.295 \t\t& 2\t\t& 3\t\t& 0.667\t& 0.576 \\\\\nDiabetes\t\t\t\t\t\t& -1.036$\\pm$0.078\t\t& 19\t\t& 224\t& 0.085\t& 0.000 \\\\\nObs-Normal ANA DS\t\t\t&-1.066$\\pm$0.122\t\t& 44\t\t& 78\t\t& 0.564\t& 0.000 \\\\\nOral contraceptive\t\t\t\t&-1.244$\\pm$0.205\t\t& 2\t\t& 35\t\t& 0.057\t& 0.043 \\\\\nDTP vaccine \t\t\t\t\t& -1.829$\\pm$0.126\t\t& 12\t\t& 327\t& 0.037\t& 0.000 \\\\\nUnobserved Lyme Test \t\t\t& -1.980$\\pm$0.099 \t& 692\t& 2924\t& 0.237\t& 0.000 \\\\\nColon cancer\t\t\t\t\t&-2.927$\\pm$4.072 \t\t& 2\t\t& 15\t\t& 0.133\t& 0.584 \\\\\nAsian race\t\t\t\t\t&-2.925$\\pm$0.234 \t\t& 2\t\t& 93\t\t& 0.022\t& 0.000 \\\\\nMS (FH)\t\t\t\t\t\t&-3.356$\\pm$4.307 \t\t& 2\t\t& 19\t\t& 0.105\t& 0.023 \\\\\nUnobserved CSF IGG synthesis\t&-4.841$\\pm$0.296\t\t& 700\t& 2946\t& 0.238\t& 0.000 \\\\\nVaricella vaccine\t\t\t\t&-15.161$\\pm$0.087 \t& 0\t\t& 13\t\t& 0\t\t& 0.145 \\\\\nHPV vaccine\t\t\t\t\t&-15.728$\\pm$2.188 \t& 0\t\t& 82\t\t& 0\t\t& 0.000 \\\\\nSchizophrenia\t\t\t\t\t&-15.763$\\pm$0.369\t& 0\t\t& 10 \t\t& 0 \t\t& 0.235 \\\\\nEstrogen replacement\t\t\t&-15.823$\\pm$0.209\t& 0\t\t& 22\t\t& 0 \t\t& 0.037 \\\\\nIBD (FH)\t\t\t\t\t\t&-17.236$\\pm$0.420\t& 0 \t\t& 3 \t\t& 0 \t\t& 0.885 \\\\\nHIB vaccine\t\t\t\t\t&-18.156$\\pm$0.521\t& 0 \t\t& 2 \t\t& 0 \t\t& 1.000 \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Mean coefficient values, odds ratio, and p-value for variables picked in at least 5 of the 10 folds with $|\\beta| > 1$}\n\\label{tab:coeff}\n\\end{table*}\n\nFigure \\ref{fig:risk} shows the distribution of predicted risk values. The figure shows that even with all the features listed in Table \\ref{tab:variables}, there is still a considerable overlap of predicted probabilities between case and control patients. Better separation between these two classes can improve the risk prediction accuracy. The plot suggests incorporation of additional diagnoses or temporal aspects of existing diagnoses may be necessary to improve model performance.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Risk.pdf}\n\\end{center}\n\\caption{Box plot of the predicted probabilities using a forward selection model on all the features.}\n\\label{fig:risk}\n\\end{figure}\n\nFigure \\ref{fig:perf} contains the performance plots for the forward selection models trained on feature set 1 and feature set 9. Figure \\ref{fig:roc} demonstrates the noticeable improvement using all the available features. Additionally, the model trained on feature set 1, demographics and family history features, barely outperforms random chance. The tradeoff between sensitivity, specificity, and positive predictive value can be seen in Figure \\ref{fig:ppv-sens}. Feature set 9 has a higher intersection between the sensitivity and specificity curves, which is summarized in Table \\ref{tab:intersect}. In addition, the full-featured model generally achieves a better positive predictive value for all threshold values. However, the positive predictive value and sensitivity curves cross at the value $\\sim 0.40$. At this point, we can accurately diagnose 40\\% of the case patients, but only 2 out of every 5 patients predicted to have a high risk of MS will be diagnosed with MS at the next office visit, a high number of false positives.\n\n\\begin{figure*}[htb]\n\\subfigure[ROC curves compared to random assignment]{\n\\includegraphics[width=0.48\\textwidth]{ROCR.pdf}\n\\label{fig:roc}\n}\n\\subfigure[Sensitivity, specificity, and positive predictive value as function of threshold]{\n\\includegraphics[width=0.48\\textwidth]{Threshold.pdf}\n\\label{fig:ppv-sens}\n}\n\\caption{Model performance plots for feature sets 1 and 9.}\n\\label{fig:perf}\n\\end{figure*}\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{l c c c c}\n\\hline\nFeature Set & Cutoff & Sensitivity & Sensitivity & PPV \\\\\n\\hline\n1 & 0.212 & 0.528 & 0.528 & 0.218\\\\\n9 & 0.241 & 0.647 & 0.647 & 0.314 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The intersection of the sensitivity and specificity curve from Figure \\ref{fig:ppv-sens}.}\n\\label{tab:intersect}\n\\end{table}\n\\subsection{Discussion}\nThe results demonstrate reasonable predictive accuracy using all the available features. One potential hindrance lies in the current feature construction. As Figure \\ref{fig:encounter} shows, there are a limited number of encounters prior to $t_0$ for case patients. Thus, it is difficult to determine whether an unobserved diagnosis may be due to the lack of longitudinal data (the patient was diagnosed prior to the study period). Additionally, certain diagnoses, such as EBV, can only be verified through culture samples which are not performed for every patient.\n\nAnother limitation of our study is the reliance on ICD-9 and procedure codes. A patient may exhibit all the clinical symptoms for a specific disease but it is not present in the encounter data because the disorder has not been diagnosed. The ambiguity of ICD9-codes and diagnostic discrepancies between medical doctors can also impact our feature construction. Moreover, the blood test results' conversion to a categorical feature may be inaccurate as the testing protocol may have changed during the study window. Therefore, a patient's feature vector may not accurately reflect their medical history.\n\nOur study also suggests incorporating additional features. Given that some of the variables were unrecorded in the structured portion of the EMR, parsing through the clinical notes could result in information regarding lifestyle factors, diet, detailed family and medical history. In addition, temporal aspects of the medical diagnoses were not included in our feature set since the data was confined to medical encounters over a 6-year period. \n\n\\section{Conclusion}\nThis paper presented a risk prediction model from EMRs to help address the difficulty of early diagnosis in MS patients. A sparse set of features were selected to minimize model complexity while maintaining reasonable predictive performance. Our results show we are able to help identify patients at high-risk of developing MS, in spite of a limited sample of patient data. In addition, our models have the ability to generalize to other healthcare systems as we rely only on components commonly found in electronic patient data.\n\nThe work demonstrates the potential of leveraging EMRs to aid medical professionals with difficult tasks, especially with early disease diagnosis. Future work will focus on incorporating temporal components, such as time of diagnosis, into the model, decreasing the false positive rate, and integrating a larger control population.\n\n\\section{Acknowledgments}\nWe thank Afif Hentati and Demetrius \"Jim\" Maraganore for their guidance, advice and comments on this study. We acknowledge comments from and conversations with Kibaek Kim, Yubin Park, Xiang Zhong, and Sanjay Mehrotra. We are indebted to Justin Lakeman for extracting data from the NorthShore Enterprise Data Warehouse.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\n\nTraditional Density Functional Theory (DFT) \\cite{Dre90,Koh99,DFTLN} and its time-dependent generalization \\cite{Run84,Gro94}\nhave evolved into standard tools for the description of electronic properties in condensed-matter physics and quantum chemistry\nthrough the simple local density instead of the less tractable $N$-body wave function.\nStationary DFT is based on the Hohenberg-Kohn (HK) theorem \\cite{Hoh64}, which\nproves that, for any non-degenerate system of N Fermions\nor Bosons \\cite{Dre90} put into a local external potential, the $N$-body ground-state wave function can be written as a functional of the local ground-state density.\nA similar theorem exists for the time-dependent case \\cite{Run84,Gro94}, where a dependence on the initial state appears.\nThe Kohn-Sham (KS) scheme \\cite{Koh65} and its time-dependent generalization \\cite{Run84,Gro94} provide a straightforward method \nto compute self-consistently the density in a quantum framework,\ndefining the non-interacting system (i.e.\\ the local single-particle potential) which reproduces the exact density.\n\nTraditional DFT is particularly well suited to study the electronic properties in molecules \\cite{Kre01}.\nAs a molecule is a self-bound system, the corresponding Hamiltonian is translationally invariant \n(which ensures Galilean invariance of the wave function \\cite{foot1}),\nand one can apply the Jacobi coordinates method.\nThis permits to decouple the center-of-mass (c.m.) properties from the internal ones,\nand to treat correctly the redundant coordinate problem\n(i.e.\\ the fact that one coordinate is redundant for the description of the internal properties \\cite{Schm01a})\nand the c.m. correlations.\nBut as the nuclei are much heavier than the electrons,\nwe can apply the Jacobi coordinates method to the nuclei only,\nso that only the nuclei will carry the c.m. correlations,\nand use the clamped nuclei approximation.\nThen, one recovers the ``external'' potential of traditional DFT, of the form $\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i)$,\nwhich accounts for the nuclear background as seen by the electrons in the frame attached to the c.m. of the nuclei.\nThus, traditional DFT is particularly adapted to the study of the electronic properties in molecules \\cite{Kre01}.\nIt is implicitely formulated in the nuclear c.m. frame \\cite{foot2} and the energy functional \ndoes not contain any c.m. correlations.\nOf course, contrary to the whole molecule, the pure electronic system is not a self-bound system:\nthe $v_{ext}$ potential breaks translational invariance and is compulsory in order to reach bound states in the stationary case.\n\nFor other self-bound systems, as isolated atomic nuclei or He droplets,\nthe situation is intrinsically different because the masses of all the particles (Fermions or Bosons) are of the same order of magnitude.\nAs a consequence, to decouple the c.m. properties from the internal ones,\none has to apply the Jacobi coordinates method\nto \\textit{all} the particules.\nThe redundant coordinate problem (thus the c.m. correlations) will now concern all the particles and should be treated properly.\nIf a DFT exists, the c.m. correlations \nshould be taken into account in the functional.\n\nMoreover, no \"external\" potential of the form $\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i)$ can be justified in the corresponding self-bound Hamiltonians\n(we denote $\\mathbf{r}_i$ the $N$ \nparticules\ncoordinates related to any inertial frame as the laboratory).\nOne may be tempted to formulate a DFT using the traditional DFT conclusions in the limit $v_{ext} \\to 0$,\nbut this would lead to false and incoherent results because:\n\\begin{itemize}\n\\item in the stationary case, the Hohenberg-Kohn theorem is valid only for external potentials that lead to bound many-body states \\cite{Lie83},\nwhich is not the case anymore at the limit $v_{ext} \\to 0$ \nfor translational invariant particle-particle interactions \\cite{Mes09};\n\\item the form of $v_{ext}$ is not translationally invariant, but translational invariance is a key feature of self-bound systems \\cite{Schm01a,Pei62,Mes09};\n\\item traditional DFT concepts as formulated so far are not applicable in terms of a well-defined internal density $\\rho_{int}$, i.e.\\ the density relative \nto the system's c.m.\\ , which is of experimental interest \\cite{Kre01,Eng07,Mes09} (it is for example measured in nuclear scattering experiments).\n\\end{itemize}\n\nInstead of the traditional DFT potential $\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i)$, one might be tempted to introduce an arbitrary translational invariant potential of the form $\\sum_{i=1}^N v_{int}(\\mathbf{r}_i - \\mathbf{R})$, where $\\vec{R}=\\frac{1}{N}\\sum_{j=1}^{N}\\vec{r}_j$ is the total c.m.\\ of the particles.\nThis potential is an \"internal\" potential, i.e. is seen in the c.m. frame,\nand in \\cite{Mes09} we underlined that it is the only form which satisfies all the key formal properties.\nHowever, $v_{int}$ should be zero in the purely isolated self-bound case.\nThis is why in \\cite{Mes09} we presented it as a mathematical \"auxiliary\"\nto reach our goal and showed that it can be dropped properly at the end, conserving all the conclusions.\nThrough it (and using the Jacobi coordinates), we proved, by a different way than those found in \\cite{Eng07,Bar07},\nthe stationary \"Internal DFT\" theorem:\nthe internal many-body state can be written as a functional of $\\rho_{int}$.\nThen we formulated rigorously the corresponding \"Internal\" KS scheme (in the c.m.\\ frame).\nThe main interest of this work is\nto give a first step towards a fundamental justification to the use of internal density functionals\nfor stationary mean-field like calculations of nuclei \\cite{Ben03} or He droplets \\cite{Bar06} with effective interactions,\nshowing that there exists an ultimate functional which permits to reproduce the exact internal density, which was not clear up to now.\n\nIt is to be noted that the stationary Internal DFT \/ KS formalism \ngives a more fundamental justification than the Hartree-Fock (HF) framework\nto the stationary nuclear mean-field like calculations. Indeed, HF does not contain quantum correlations,\nnor treats correctly the redundant coordinate problem, which introduces a spurious coupling between the\ninternal properties and the c.m. motion \\cite{RS80,Schm01a}.\nA way to overcome this problem in the stationary case is to perform projected HF\n(projection before variation on c.m.\\ momentum), which permits to restore \nGalilean invariance, but at the price of abandoning the independent-particle \ndescription \\cite{Schm01a,Pei62,Ben03}.\nWithin the Internal DFT \/ KS formalism,\nwe proved that the c.m.\\ correlations can be included in the energy functional \/ the KS potential \\cite{Mes09},\nso that there would be no need for a c.m.\\ projection if the ultimate functional was known.\n\nIt is a question of interest to generalize the stationary Internal DFT \/ KS formalism to the time-dependent case.\nIt would provide a first step towards a fundamental justification to the use of density functionals in nuclear\ntime-dependent calculations with an effective mean-field \\cite{tdhf_nucl,Neg82},\nand would prove that the c.m.\\ correlations can be included in the functional.\nThis last point is even more interesting that the spurious c.m.\\ motion problem remains in time-dependent HF \\cite{Irv80,Uma09},\nbut that then the projected HF method becomes unmanageable and is not used in practise \\cite{Uma09}.\n\nIn this paper, we propose to set up the time-dependent Internal DFT \/ KS formalism.\nThe paper is organized as follows:\nwe first apply the Jacobi coordinates method to the\ntime-dependent full many-body Hamiltonian to decouple the internal properties from the c.m. ones, and define some useful ``internal'' observables, including the internal density (section II);\nthen we show that the internal many-body wave function (and thus the ``internal'' mean values of all the observables) can be written as a functional of the internal density (section III);\nfinally, we develop the associated time-dependent Internal KS scheme as a practical scheme to compute the internal density (section IV).\n\n\n\n\n\n\n\\section{Time-dependent N body formulation.}\n\n\n\n\\subsection{General formulation.}\n\nIn the time-dependent domain, the introduction of an \\textit{explicitely} time-dependent internal potential of the form\n\\begin{eqnarray}\n\\label{eq:v}\n\\sum_{i=1}^N v_{int}(\\mathbf{r}_i - \\mathbf{R};t)\n\\end{eqnarray}\ntakes a true meaning.\nThis is because self-bound systems are plagued by a c.m. problem.\nFor instance, in the stationary case, the c.m. will be delocalized in the whole space for \\textit{isolated} self-bound systems \\cite{Eng07,Kre01,Mes09}.\nThis does not occur in experiments because experimentally observed self-bound systems are not \\textit{isolated} anymore\n(they interact with the piece of matter they are inserted in which localizes the c.m.).\nIn the time domain, the c.m. motion remains uncomparable to the experimental one (this will be discussed in more detail later),\nso that it would not make sense to introduce a time-dependent potential\nwhich would act on the c.m. motion.\nIt are the internal properties which are of true experimental interest (experimentalists always deduce those properties \\cite{foot4}).\nThis justifies\nthe introduction of an \\textit{explicitely} time-dependent potential \nof the form (\\ref{eq:v}),\nwhich would act on the internal properties only,\nand models the internal effect (only) of time-dependent potentials used in experiments.\nSuch a potential does not appear any more simply as a mathematical auxiliary (as for the stationary Internal DFT \/ KS) and should not necessarily be dropped at the end.\n\n\nWe thus start from a general translationally invariant $N$-body Hamiltonian\ncomposed of the usual kinetic energy term, a \ntranslationally invariant \ntwo-body potential $u$, which describes the particle-particle \ninteraction,\nand an arbitrary translationally invariant \"internal\" potential $v_{int}$ which contains an explicit time dependence\n\\begin{equation}\n\\label{eq:H}\nH\n=   \\sum_{i=1}^{N} \\frac{\\vec{p}^2_i}{2m} \n  + \\sum_{\\stackrel{i,j=1}{i > j}}^{N} u (\\vec{r}_i-\\vec{r}_j) \n  + \\sum_{i=1}^{N} v_{\\text{int}} (\\vec{r}_i - \\vec{R} ; t)\n\\; .\n\\end{equation}\nFor the sake of simplicity we assume a 2-body interaction $u$ and $N$ identical Fermions or Bosons.\nThe generalization to 3-body etc interactions is straightforward;\nthe generalization to different types of particles is underway.\n\nWe rewrite the Hamiltonian (\\ref{eq:H}) using the ($N-1$) Jacobi coordinates $\\{\\xi_\\alpha;\\alpha=1,\\dots,N-1\\}$\nand the c.m.\\ coordinate $\\vec{R}$, defined as\n\\begin{eqnarray}\n&& \\mathbf{\\xi}_{1} = \\mathbf{r}_2-\\vec{r}_1, \n\\mathbf{\\xi}_2=\\mathbf{r}_3-\\frac{\\vec{r}_2+\\vec{r}_1}{2}, \\ldots,\n\\nonumber\\\\\n&& \\mathbf{\\xi}_{N-1} = \\frac{N}{N-1} \\, (\\vec{r}_N - \\vec{R}),\n\\nonumber\\\\\n&& \\vec{R}=\\frac{1}{N}\\sum_{j=1}^{N}\\vec{r}_j\n.\n\\label{eq:jacobi}\n\\end{eqnarray}\nThe $\\xi_\\alpha$ are relative to the c.m.\\ of the other \n$1, \\ldots, \\alpha-1$ particles and are independent from $\\vec{R}$. They are to be distinguished from the $N$ \n\"laboratory coordinates\" $\\vec{r}_i$, and the $N$ \"c.m. frame coordinates\" $(\\vec{r}_i-\\vec{R})$ relative to the total c.m. $\\vec{R}$.\nAs the $\\{\\vec{r}_i-\\vec{r}_{j\\ne i}\\}$ and the $\\{\\vec{r}_i - \\vec{R}\\}$ can be rewritten as functions of the $\\xi_\\alpha$\n(in Appendix \\ref{app:jacobi} is given the expression of the $\\{\\mathbf{r}_i-\\mathbf{R}\\}$ as a function of the $\\{\\xi_\\alpha\\}$ coordinates),\nthe interaction $u$ and the internal potential $v_{int}$\ncan be rewritten as functions of the $\\xi_\\alpha$. We denote $U$ and $V$\nthe interaction potential and the internal potential in the Jacobi coordinates representation:\n\\begin{eqnarray}\n\\sum_{\\stackrel{i,j=1}{i > j}}^{N} u (\\vec{r}_i-\\vec{r}_j)\n\\quad&\\rightarrow&\\quad\nU(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1}) \n\\nonumber\\\\\n\\sum_{i=1}^{N} v_{int} (\\vec{r}_i - \\vec{R} ; t)\n\\quad&\\rightarrow&\\quad\nV(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\n\\label{eq:V_int}\n.\n\\end{eqnarray}\nOf course we have $U[u]$ and $V[v_{int}]$.\nThe $V[v_{int}]$ potential is ($N-1$) body in the Jacobi coordinates representation and\n\\textit{cannot} be written in a simple form in this representation (see Appendix \\ref{app:jacobi}).\nMoreover, various $v_{int}$ can lead to the same $V$, which we will develop later.\n\nAfter having defined the conjugate momenta of $\\vec{R}$ and $\\xi_\\alpha$,\nwe can separate (\\ref{eq:H}) into $H = H_\\text{CM} + H_\\text{int}$, where ($M = Nm$ is the total mass)\n\\begin{equation}\nH_\\text{CM} = -\\frac{\\hbar^2 \\Delta_\\vec{R}}{2M} \n\\label{eq:H_cm}\n\\end{equation}\nis a one-body operator acting in  $\\vec{R}$ space only,\nand ($\\tau_\\alpha$ is the conjugate momentum of $\\xi_\\alpha$ and $\\mu_\\alpha = m\\frac{\\alpha}{\\alpha+1}$ the corresponding reduced mass)\n\\begin{eqnarray}\nH_\\text{int}=\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha} &+& U[u](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1}) \n\\nonumber\\\\\n&+& V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\n\\label{eq:H_int}\n\\end{eqnarray}\nis a $(N-1)$ body operator in the $\\{\\xi_\\alpha\\}$ space. It contains the \ninteraction and the internal potential.\n\nIn the time-dependent case, we can choose freely the initial state $\\psi(\\vec{r}_1, \\ldots , \\vec{r}_N ; t_0)$.\nWe start from an initial state which can be written \n\\begin{equation}\n\\label{eq:psi_init}\n\\psi(\\vec{r}_1, \\ldots , \\vec{r}_N ; t_0) \n= \\Gamma(\\vec{R} ; t_0) \\; \n  \\psi_{int} ({\\boldmath{\\xi}}_1, \\ldots , {\\boldmath{\\xi}}_{N-1} ; t_0)\n\\end{equation}\nin the Jacobi coordinates representation.\nThis form does not mix the c.m. motion with the internal one\n(mixing them would not make sense because the c.m. motion does anyway not correspond to the experimental one)\nand corresponds to the form of the stationary state  \\cite{Schm01a,Mes09}.\nAs $H_{CM}$ and $H_{int}$ act in two separate subspaces, the $\\mathbf{R}$ and $\\{\\xi_\\alpha\\}$ spaces\n(which implies $[H_{CM},H_{int}]=0$), it is easy to show that the state $|\\psi(t))$ can be built at all time $t\\geq t_0$ as a direct product of the form\n\\begin{equation}\n\\label{eq:psi}\n\\psi(\\vec{r}_1, \\ldots , \\vec{r}_N ; t) \n= \\Gamma(\\vec{R} ; t) \\; \n  \\psi_{int} ({\\boldmath{\\xi}}_1, \\ldots , {\\boldmath{\\xi}}_{N-1} ; t) ,\n\\end{equation}\nwith\n\\begin{eqnarray}\n&& H_{CM}|\\Gamma(t)) = i\\hbar\\partial_t |\\Gamma(t))\n\\label{eq:schro_int}\\\\\n&& H_{int}|\\psi_{int}(t)) = i\\hbar\\partial_t |\\psi_{int}(t))\n\\label{eq:schro}\n\\; .\n\\end{eqnarray}\nHence, the $N$-body wave function $\\psi$ can be separated into a one-body wave function \n$\\Gamma$ that depends on the position $\\mathbf{R}$ of the c.m. only, \nand an \"internal\" ($N-1$) body wave function \n$\\psi_{int}$ that depends on the remaining ($N-1$) Jacobi \ncoordinates ${\\boldmath{\\xi}}_\\alpha$.\nOf course, $\\psi_{int}$ could also be written as a function of the $N$ laboratory coordinates $\\mathbf{r}_i$,\nbut one of them would be redundant.\n$\\Gamma$ is solution of the free Schr\\\"odinger equation and describes the motion of the \\textit{isolated} system as \na whole in any chosen inertial frame of reference (as the laboratory).\nIf one starts from a normalizable initial state $|\\Gamma(t_0))$, $|\\Gamma(t))$\nis condemned to spread more and more.\nIn the stationary limit, the only solutions of Eq. (\\ref{eq:schro_int}) are plane waves, which are infinitely spread (thus not normalizable).\nThis does not correspond to experimental situations, where the system is not isolated anymore: interactions with other systems\nof the experimental apparatus localize the c.m.\nBut the formal decoupling between the c.m.\\ motion and the internal properties obtained when using the Jacobi coordinates method\npermits to let the c.m.\\ motion to the choice of experimental conditions,\nthe internal properties being comparable to the experimental ones.\n\n\n\n\n\n\n\\subsection{Some useful definitions.}\n\\label{par:def}\n\nWe define some quantities and relations that will be useful for the next considerations.\nIn \\cite{Mes09,Gir08b,Kaz86} is defined the internal one-body density\n\\begin{eqnarray}\n\\lefteqn{\\rho_{int}(\\vec{r},t)\/N}\n\\label{eq:rho_int0}\n\\\\\n& = & \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_N \\;\n      \\delta(\\mathbf{R})\n      |\\psi_{int}(\\vec{r}_1, \\ldots, \\vec{r}_{N};t)|^2\n      \\delta \\big( \\vec{r} - (\\vec{r}_i-\\mathbf{R}) \\big)\\, \n\\nonumber\\\\\n& = & \\Big(\\frac{N}{N-1}\\Big)^3 \n      \\int \\! d\\vec{\\xi}_1 \\cdots  d\\mathbf{\\xi}_{N-2} \\; \n      \\big| \\psi_{int} \\big(\\mathbf{\\xi}_1, \\ldots, \\mathbf{\\xi}_{N-2},\n                   \\tfrac{N\\vec{r}}{N-1} ;t \\big) \\big|^2\n\\nonumber .\n\\end{eqnarray}\nIt is is normalized to $N$.\nThe laboratory density $\\rho(\\mathbf{r},t)$ is obtained by convolution of $\\rho_{int}$ with the c.m.\\ wave \nfunction (following \\cite{Gir08b,Kaz86}):\n$\n\\rho(\\mathbf{r},t) = \\int d\\mathbf{R} |\\Gamma(\\mathbf{R},t)|^2 \\rho_{int}(\\mathbf{r} - \\mathbf{R},t) .\n$\n\nWe also introduced in \\cite{Mes09} the local part of the two-body internal density matrix\n\\begin{eqnarray}\n\\label{eq:gamint0}\n\\lefteqn{\\gamma_{int}(\\vec{r},\\vec{r'};t)}\n      \\\\\n& = & \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_N \\; \n      \\delta(\\mathbf{R}) |\\psi_{int}(\\vec{r}_1, \\ldots, \\vec{r}_{N};t)|^2 \\,\n      \\nonumber \\\\\n&   & \\hspace{1.cm} \\times \n      \\delta \\big( \\vec{r} - (\\vec{r}_i-\\mathbf{R}) \\big)\n      \\delta \\big( \\vec{r'} - (\\vec{r}_{j\\ne i}-\\mathbf{R}) \\big)\n      \\nonumber \\\\\n& = & \\frac{N(N-1)}{2} \\Big(\\frac{N-1}{N-2}\\Big)^3 \\Big(\\frac{N}{N-1}\\Big)^3\n      \\int \\! d\\mathbf{\\xi}_1 \\cdots  d\\mathbf{\\xi}_{N-3}\n      \\nonumber\\\\\n&   & \\hspace{1.cm} \\times \\Big| \\psi_{int} \\Big(\\mathbf{\\xi}_1, \\ldots, \\mathbf{\\xi}_{N-3}, \n      \\tfrac{\\vec{r'}+(N-1)\\vec{r} }{N-2},\\tfrac{N\\vec{r'}}{N-1} ;t \\Big) \\Big|^2\n      \\nonumber\n.\n\\end{eqnarray}\nIt has the required normalisation to $N(N-1)\/2$.\nFollowing similar steps than in \\cite{Gir08b,Kaz86},\nwe can show that the local part of the two-body laboratory density matrix \n$\\gamma(\\vec{r},\\vec{r'},t)$ is obtained by convolution of $\\gamma_{int}$ with the c.m.\\ wave \nfunction:\n$\n\\gamma(\\vec{r},\\vec{r'};t) = \\int d\\mathbf{R} |\\Gamma(\\mathbf{R},t)|^2 \\gamma_{int}(\\vec{r} - \\mathbf{R},\\vec{r'} - \\mathbf{R};t) .\n$\n\nThe definitions of $\\rho_{int}(\\vec{r},t)$ and $\\gamma_{int}(\\vec{r},\\vec{r'};t)$ show clearly that they are defined in the c.m. frame,\ni.e.\\ that the $\\vec{r}$, $\\vec{r'}$ coordinates are measured in the c.m. frame (see the $\\delta$ relations in (\\ref{eq:rho_int0}) and (\\ref{eq:gamint0})).\nCompared to the traditional definitions, a $\\delta(\\mathbf{R})$ appears in the definition of the internal densities calculated\nwith $\\psi_{int}$ in $\\{\\mathbf{r}_i\\}$ coordinates. As one of them is redundant, the $\\delta(\\mathbf{R})$ represents the dependence of the redundant coordinate on the others \\cite{foot3}.\n\nAnother quantity that will be very useful is the one-body internal probability current, defined in Appendix \\ref{app:int_current} ($c.c.$ denotes the complex conjugate)\n\\begin{eqnarray}\n\\lefteqn{\\mathbf{j}_{int}(\\mathbf{r},t)\/N}\n\\label{eq:j_int1}\\\\\n&=& \\frac{\\hbar}{2m i} \\Big(\\frac{N}{N-1}\\Big)^3 \\int d\\mathbf{\\xi}_1 ...  d\\mathbf{\\xi}_{N-2} \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2}, \\nu;t)\n\\nonumber\\\\\n&& \\hspace{1.5cm}\n\\times \\mathbf{\\nabla_{\\nu}} \\psi_{int}(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2}, \\nu;t) \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}} + c.c.\n\\nonumber\n\\end{eqnarray}\nwhich satisfies the ``internal'' continuity equation\n\\begin{eqnarray}\n\\partial_t \\rho_{int}(\\mathbf{r},t) + \\mathbf{\\nabla} . \\mathbf{j}_{int}(\\mathbf{r},t) =0 .\n\\label{eq:cont_rel}\n\\end{eqnarray}\nUsing (\\ref{eq:j_int1}), (\\ref{eq:H_int}) and (\\ref{eq:schro}), we obtain the relation\n\\begin{widetext}\n\\begin{eqnarray}\n\\lefteqn{i \\frac{\\partial}{\\partial t} \\mathbf{j}_{int}(\\mathbf{r},t)}\n\\label{eq:partial_j_int}\n\\\\\n&=& \\frac{N}{2m i} \\Big(\\frac{N}{N-1}\\Big)^3 \\int d\\mathbf{\\xi}_1 ...  d\\mathbf{\\xi}_{N-2} \\Big\\{ \n\\mathbf{\\nabla_{\\nu}} \\psi_{int}(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2},\\nu;t) i\\hbar\\partial_t \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2},\\nu;t) \\nonumber\\\\\n&& \\hspace{4.6cm} +\n\\mathbf{\\nabla_{\\nu}} \\Big( i\\hbar \\partial_t\\psi_{int}(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2},\\nu;t) \\Big) \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2},\\nu;t)  + c.c.\n\\Big\\}\\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}}\n\\nonumber\\\\\n&=& \\frac{N}{2m i} \\Big(\\frac{N}{N-1}\\Big)^3 \\int d\\mathbf{\\xi}_1 ...  d\\mathbf{\\xi}_{N-2} \\Big\\{ \n\\mathbf{\\nabla_{\\nu}} \\psi_{int}(\\mathbf{\\xi}_1, ..., \\nu;t)  \\frac{\\hbar^2\\Delta_{\\nu}}{2\\mu_{N-1}} \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\nu;t)\n-  \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\nu;t) \\mathbf{\\nabla_{\\nu}} \\frac{\\hbar^2\\Delta_{\\nu}}{2\\mu_{N-1}} \\psi_{int}(\\mathbf{\\xi}_1, ..., \\nu;t)\n\\nonumber\\\\\n&& \\hspace{4.5cm} +\n\\psi^*_{int}(\\mathbf{\\xi}_1, ..., \\nu;t) \\mathbf{\\nabla_{\\nu}} \\Big( U[u](\\mathbf{\\xi}_1, ..., \\nu) + V[v_{int}](\\mathbf{\\xi}_1, ..., \\nu;t) \\Big) \\psi_{int}(\\mathbf{\\xi}_1, ... ,\\nu;t) \n+c.c.\n\\Big\\} \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}},\n\\nonumber\n\\end{eqnarray}\n\\end{widetext}\nwhich will be a key equation for the next considerations.\n\n\n\n\n\n\n\\section{Time-dependent Internal DFT theorem.}\n\n\n\n\\subsection{Preliminaries.}\n\\label{sub:int}\n\nTo prove the time-dependent Internal DFT theorem, we adapt the considerations of \\cite{Run84,Gro94}\nto the internal Schr\\\"odinger equation (\\ref{eq:schro}).\nThe main differences lie in the definition of the corresponding internal density (\\ref{eq:rho_int0}) and probability current (\\ref{eq:j_int1}),\nand in the fact that the potential $V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$\ncannot be written as the sum of one-body potentials in the Jacobi coordinates representation\n(which introduces some subtleties due to the c.m. correlations and will bring us to use the integral mean value theorem to reach our goal).\n\nIn what follows, we consider a given type of Fermions or Bosons, i.e. a given particle-particle interaction $u$.\nSolving the ``internal'' Schr\\\"odinger equation (\\ref{eq:schro}) for a fixed initial state $|\\psi_{int}(t_0))$ and for various internal potentials $V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$ defines two maps \\cite{Run84,Gro94}\n\\begin{eqnarray}\n&& F: V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\rightarrow |\\psi_{int}(t))\n\\nonumber\\\\\n&& G: V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\rightarrow \\rho_{int}(\\mathbf{r},t).\n\\label{eq:map}\n\\end{eqnarray}\nWe first notice that two potentials $v_{int}$ and $v'_{int}$ which lead to two potentials\n$V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$ and $V[v'_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$ that differ by a scalar function of time only $c(t)$, \nwill give two wave functions that differ by a phase $e^{-i\\alpha(t)\/\\hbar}$ only \\cite{Run84,Gro94}:\n\\begin{eqnarray}\n&&V[v'_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t) - V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t) = c(t)\n\\nonumber\\\\\n&&\\quad\\quad\\Rightarrow\\hspace{1mm}\n|\\psi'_{int}(t))=e^{-i\\alpha(t)\/\\hbar}|\\psi_{int}(t)) ,\n\\nonumber\\\\\n&& \\hspace{2.5cm} \\rm{with} \\hspace{2mm} \\Dot{\\alpha}(t)=c(t)\n\\label{eq:v'}\n.\n\\end{eqnarray}\nThen, $|\\psi_{int}(t))$ and $|\\psi'_{int}(t))$ will give the same density $\\rho_{int}(\\mathbf{r},t)=\\rho'_{int}(\\mathbf{r},t)$.\nThe consequence is that the map $G$ is not fully invertible.\n\nLet us discuss a bit about the condition (\\ref{eq:v'}).\nThe form (\\ref{eq:V_int}) for $V[v_{int}]$ implies $V[v'_{int}]-V[v_{int}]=V[v'_{int}-v_{int}]$.\nWe define\n\\begin{eqnarray}\n\\Delta v_{int}(\\mathbf{r};t)=v'_{int}(\\mathbf{r};t)-v_{int}(\\mathbf{r};t).\n\\label{eq:delta_v}\n\\end{eqnarray}\nIt is to be noted that the condition $\\Delta v_{int}(\\mathbf{r};t)\\ne c(t)\/N$\nis necessary but not sufficient to ensure the condition (\\ref{eq:v'}),\nwhich can be rewritten $V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)$.\nIndeed, it is possible to have $\\Delta v_{int}(\\mathbf{r};t)\\ne c(t)\/N$ and nevertheless\n$V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t) = c(t)$,\nbecause compensations due to the c.m. correlations can happen.\n\nLet us reason on the two particules case, where only one Jacobi coordinate is sufficient to describe the internal properties.\nWe have (see Appendix \\ref{app:jacobi}):\n$V[\\Delta v_{int}](\\mathbf{\\xi}_1;t)=\\Delta v_{int} (-\\frac{1}{2}\\xi_1;t)+\\Delta v_{int} (\\frac{1}{2}\\xi_1;t) \\big( =\\sum_{i=1}^{2} v_{\\text{int}} (\\vec{r}_i - \\vec{R} ; t) \\big)$.\nWe see that if $\\Delta v_{int}(\\mathbf{r};t)$ is an odd function of $\\mathbf{r}$ at all $t$\n(up to an additional time-dependent function),\nwe have $V[\\Delta v_{int}](\\mathbf{\\xi}_1;t)=c(t)$\n$\\Rightarrow$ $\\rho_{int}=\\rho'_{int}$.\nThis is due to the c.m. correlations, that the non-trivial form of $V$ reflects.\nIf\n$\\Delta v_{int}$ tends to move the first particule in one direction,\nthe second particule will tend to move in the opposite direction because of the c.m. correlations.\nBut if this potential counter-acts perfectly the motion of the second particule (as does an odd potential in the c.m. frame), then the particules remain stuck and the density unchanged.\n\nThe same can occur for an arbitrary number of particules.\nFor instance, as $\\sum_{i=1}^N (\\mathbf{r}_i-\\mathbf{R})=0$,\nit is obvious with (\\ref{eq:V_int}) and (\\ref{eq:delta_v}) that every $\\Delta v_{int}(\\mathbf{r};t)=\\mathbf{b}(t).\\mathbf{r}+c(t)\/N$\nwill yield $V[\\Delta v_{int}] = c(t)$ (even if this form for $\\Delta v_{int}$ leads to internal potentials which are not null at infinity).\nAgain, this is because if a potential counter acts perfectly the motion due to the c.m. correlations, the particules remain stuck and the density unchanged.\nIn what follows, we consider only internal potentials $v_{int}$ and $v'_{int}$ that lead to $V[\\Delta v_{int}]\\ne c(t)$.\n\nWe come back to Eq. (\\ref{eq:v'}) and denote $|\\psi_{int}(t))=e^{-i\\alpha(t)\/\\hbar}|\\psi^0_{int}(t))$ where we define $\\psi^0_{int}$ as the wave function obtained for the choice $c(t)=0$,\ni.e. associated to a $V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$ where no additive time-dependent function can be split.\nIf we prove that the map $G$ is invertible up to an additive time-dependent function $c(t)$, then $\\psi^0_{int}$ is fixed by $\\rho_{int}$\nthrough the relation $|\\psi^0_{int}(t))=F G^{-1} \\rho_{int}(\\mathbf{r},t)$,\nwhich implies that $|\\psi^0_{int}(t))$ can be written as a functional of the internal density $\\rho_{int}$ defined in (\\ref{eq:rho_int0}).\nConsequently, any expectation value of an operator $\\hat{O}$ which does not contain a time derivative can be written as a functional of the internal density (as the phase cancels out):\n$(\\psi_{int}(t)|\\hat{O}|\\psi_{int}(t))=(\\psi^0_{int}[\\rho_{int}](t)|\\hat{O}|\\psi^0_{int}[\\rho_{int}](t))$.\n\nWe thus have to show that \na propagation of (\\ref{eq:schro}) with two potentials $v_{int}$ and $v_{int}'$ that yield\n$V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)$\nwill produce two different internal densities $\\rho_{int}$ and $\\rho_{int}'$.\n\n\n\n\n\\subsection{The proof.}\n\nWe start from a \\textit{fixed initial state} $|\\psi_{int}(t_0))$ and\npropagate it with two with two potentials $v_{int}$ and $v_{int}'$ that give\n$V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)$.\nWe deduce from Eq. (\\ref{eq:partial_j_int})\n\\begin{widetext}\n\\begin{eqnarray}\ni \\frac{\\partial}{\\partial t} \\Big( \\mathbf{j}_{int}(\\mathbf{r},t) - \\mathbf{j}'_{int}(\\mathbf{r},t) \\Big) \\Big|_{t=t_0}  =\n\\frac{N}{m i} \\Big(\\frac{N}{N-1}\\Big)^3\n\\int d\\mathbf{\\xi}_1 ...  d\\mathbf{\\xi}_{N-2} |\\psi_{int}(\\mathbf{\\xi}_1, ..., \\nu;t_0)|^2\n\\mathbf{\\nabla_{\\nu}} V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\nu;t_0) \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}} .\n\\end{eqnarray}\nUsing the ``internal'' continuity relation (\\ref{eq:cont_rel}) we obtain\n\\begin{eqnarray}\n\\lefteqn{\\frac{\\partial^2}{\\partial t^2} \\Big( \\rho_{int}(\\mathbf{r},t) - \\rho'_{int}(\\mathbf{r},t) \\Big) \\Big|_{t=t_0} =}\n\\label{eq:partial_j_int0}\\\\\n&& \\frac{N}{m} \\Big(\\frac{N}{N-1}\\Big)^3 \\mathbf{\\nabla_{\\mathbf{r}}} .\\int d\\mathbf{\\xi}_1 ...  d\\mathbf{\\xi}_{N-2}\n|\\psi_{int}(\\mathbf{\\xi}_1, ..., \\frac{N}{N-1}\\mathbf{r};t_0)|^2\n\\mathbf{\\nabla}_\\nu V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\nu;t_0) \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}}\n.\n\\nonumber\n\\end{eqnarray}\n\nWe now make the only hypothesis which is used in this derivation. Following \\cite{Run84,Gro94} we restrict the set of\npotentials $v_{int}$ to those that can be expanded into Taylor series with respect to the time at the initial time $t_0$\n(which is a reasonable hypothesis for physical potentials).\nAs we supposed that $V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)$, we have ($k$ is a positive integer)\n\\begin{eqnarray}\nV[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)\n\\quad\\Rightarrow\\quad\n\\exists k :\nw_k(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t_0) \\ne constant ,\n\\label{eq:Vint}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nw_k(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t_0)\n= \\frac{\\partial^k}{\\partial t^k} V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t) \\Big|_{t=t_0}\n\\label{eq:wk}\n.\n\\end{eqnarray}\nIt is to be noted that the condition $\\frac{\\partial^k}{\\partial t^k} \\Delta v_{int}(\\mathbf{r};t)\\Big|_{t=t_0} \\ne constant$\n$\\Rightarrow$ $\\mathbf{\\nabla_r} \\frac{\\partial^k}{\\partial t^k} \\Delta v_{int}(\\mathbf{r};t)\\Big|_{t=t_0}\\ne \\overrightarrow{0}$\nis necessary to ensure the condition (\\ref{eq:Vint}) (see (\\ref{eq:V_int}) and (\\ref{eq:wk})), but not sufficient.\n\nIn what follows, we consider $k$ as the \\textit{smallest} positive integer such that (\\ref{eq:Vint})\nis verified. Then, if we apply $k$ time derivatives to the Eq. (\\ref{eq:partial_j_int0}), we straightforwardly obtain\n\\begin{eqnarray}\n\\frac{\\partial^{k+2}}{\\partial t^{k+2}} \\Big( \\rho_{int}(\\mathbf{r},t) - \\rho_{int}'(\\mathbf{r},t) \\Big) \\Big|_{t=t_0} =\n\\frac{N}{m} \\Big(\\frac{N}{N-1}\\Big)^3 \\mathbf{\\nabla_{\\mathbf{r}}} . \\int d\\mathbf{\\xi}_1 ...  d\\mathbf{\\xi}_{N-2} |\\psi_{int}(\\mathbf{\\xi}_1, ..., \\nu;t_0)|^2 \\mathbf{\\nabla_\\nu} w_k(\\mathbf{\\xi}_1, ..., \\nu;t_0) \\Big) \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}}\n.\n\\end{eqnarray}\nAs, for every physical potential, $\\mathbf{\\nabla}_{\\mathbf{\\xi}_{N-1}} w_k(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t_0)$ is a real and continuous function in the whole position space, and as $|\\psi_{int}(\\mathbf{\\xi}_1, ...,\\mathbf{\\xi}_{N-1};t_0)|^2$ is a real and positive function in the whole position space, we can apply the integral mean value theorem generalized to many variables functions (demonstrated in Appendix \\ref{app:mean_val_th}) to the previous expression. We obtain\n\\begin{eqnarray}\n\\exists (\\beta_1,...,\\beta_{N-2}) \\hspace{1mm} :\n&& m \\frac{\\partial^{k+2}}{\\partial t^{k+2}} \\Big( \\rho_{int}(\\mathbf{r},t) - \\rho_{int}'(\\mathbf{r},t) \\Big) \\Big|_{t=t_0}\n\\nonumber\\\\\n&& \\hspace{5mm} = \\mathbf{\\nabla_{\\mathbf{r}}} . \\Big[\n\\mathbf{\\nabla}_\\frac{N\\mathbf{r}}{N-1} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0)\nN \\Big(\\frac{N}{N-1}\\Big)^3 \\int d\\mathbf{\\xi}_1 ...  d\\mathbf{\\xi}_{N-2}|\\psi_{int}(\\mathbf{\\xi}_1, ..., \\frac{N}{N-1}\\mathbf{r};t_0)|^2 \\Big]\n\\nonumber\\\\\n&& \\hspace{5mm} = \\mathbf{\\nabla_{\\mathbf{r}}} . \\Big[\n\\mathbf{\\nabla}_\\frac{N\\mathbf{r}}{N-1} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0)\n\\rho_{int}(\\mathbf{r},t_0) \\Big]\n.\n\\label{eq:partial_j_int2}\n\\end{eqnarray}\nTo prove the one-to-one correspondence $V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\leftrightarrow \\rho_{int}(\\mathbf{r},t)$ it remains to show that (\\ref{eq:partial_j_int2})\ncannot vanish for $v_{int}$ and $v_{int}'$ that lead to the relation (\\ref{eq:Vint}).\nThen the internal densities $\\rho_{int}(\\mathbf{r},t)$ and $\\rho_{int}'(\\mathbf{r},t)$ would become different infinitesimally later than $t_0$.\nWe use the \\textit{reductio ad absurdum} method, in the spirit of Refs. \\cite{Run84,Gro94}. We suppose that (\\ref{eq:partial_j_int2}) vanishes, which implies:\n\\begin{eqnarray}\n0&=&\\frac{N-1}{N}\\int d\\mathbf{r} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0) \\mathbf{\\nabla_{\\mathbf{r}}} . \\Big[\n\\mathbf{\\nabla}_\\frac{N\\mathbf{r}}{N-1} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0)\n\\rho_{int}(\\mathbf{r},t_0) \\Big]\n\\nonumber\\\\\n&=& \n-\\int d\\mathbf{r} \\Big[ \\mathbf{\\nabla}_\\frac{N\\mathbf{r}}{N-1} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0) \\Big]^2\n\\rho_{int}(\\mathbf{r},t_0)\n.\n\\label{eq:absurdum2}\n\\end{eqnarray}\nAs $w_k$ is a many-body function, the Eq. (\\ref{eq:Vint}) does not imply that\n$\\forall (\\beta_1,...,\\beta_{N-2}):\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)\\ne\\overrightarrow{0}$ in the general case.\nHowever, we check if this relation holds for the particular form (\\ref{eq:V_int}) we choose for $V$.\n\nInserting the results of Appendix \\ref{app:jacobi} in (\\ref{eq:V_int}) and (\\ref{eq:wk}), we obtain, if $N>2$ \n(the case $N=2$ will be discussed later on)\n\\begin{eqnarray}\n&& w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)=\n\\label{eq:w_k}\\\\\n&&\\quad\n\\frac{\\partial^{k}}{\\partial t^{k}} \\Delta v_{int} \\big(\\frac{N-1}{N}\\xi_{N-1};t\\big) \\Big|_{t=t_0}\n+ \\sum_{i=1}^{N-2}\\frac{\\partial^{k}}{\\partial t^{k}} \\Delta v_{int} \\big(\\gamma_i-\\frac{1}{N}\\xi_{N-1};t\\big) \\Big|_{t=t_0}\n+\\frac{\\partial^{k}}{\\partial t^{k}} \\Delta v_{int} \\big(-\\sum_{i=1}^{N-2}\\gamma_i-\\frac{1}{N}\\xi_{N-1};t\\big) \\Big|_{t=t_0} ,\n\\nonumber\n\\end{eqnarray}\nwhere we defined\n\\begin{eqnarray}\n\\gamma_{N-2}=\\frac{N-2}{N-1}\\beta_{N-2}\n\\quad\\quad \\text{and} \\quad\\quad\n\\forall i \\in [1,N-3]: \\gamma_{i}=\\frac{i}{i+1}\\beta_{i} - \\sum_{\\alpha=i+1}^{N-2} \\frac{1}{\\alpha+1}\\beta_{\\alpha}.\n\\label{eq:gamma}\n\\end{eqnarray}\nThe form of the third term of the right hand side of Eq. (\\ref{eq:w_k}) comes from the fact that $\\sum_{i=1}^N(\\mathbf{r}_i-\\mathbf{R})=0$, which implies, using the Appendix \\ref{app:jacobi}, that $-\\sum_{\\alpha=1}^{N-2} \\frac{1}{\\alpha+1}\\beta_{\\alpha}=-\\sum_{i=1}^{N-2}\\gamma_i$. \nWe see from Eq. (\\ref{eq:gamma}) that the set $(\\gamma_1,...,\\gamma_{N-2})$ is perfectly defined by the set $(\\beta_1,...,\\beta_{N-2})$ and vice versa.\nWe now can calculate\n\\begin{eqnarray}\n&&\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)=\n\\nonumber\\\\\n&&\\hspace{3cm}\n\\frac{N-1}{N} \\mathbf{D} \\big(\\frac{N-1}{N}\\xi_{N-1}\\big)-\\frac{1}{N}\\sum_{i=1}^{N-2} \\mathbf{D} \\big(\\gamma_i-\\frac{1}{N}\\xi_{N-1}\\big)\n-\\frac{1}{N} \\mathbf{D} \\big(-\\sum_{i=1}^{N-2}\\gamma_i-\\frac{1}{N}\\xi_{N-1}\\big)\n\\label{eq:w}\n,\n\\end{eqnarray}\nwhere we introduced\n\\begin{eqnarray}\n\\mathbf{D}(\\mathbf{r})=\\mathbf{\\nabla}_\\mathbf{r} \\frac{\\partial^{k}}{\\partial t^{k}} \\Delta v_{int} (\\mathbf{r};t) \\big|_{t=t_0}\n.\n\\label{eq:D}\n\\end{eqnarray}\nfor simplicity.\nWe now check if\n$\\exists (\\beta_1,...,\\beta_{N-2}):$ $\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)=\\overrightarrow{0}$,\nwhich is equivalent, according to (\\ref{eq:gamma}) and (\\ref{eq:w}), to check if\n\\begin{eqnarray}\n\\exists (\\gamma_1,...,\\gamma_{N-2}):\\quad\n(N-1) \\mathbf{D} \\big((N-1)\\mathbf{r}\\big)=\\sum_{i=1}^{N-2} \\mathbf{D} \\big(\\gamma_i-\\mathbf{r}\\big)\n+\\mathbf{D} \\big(-\\sum_{i=1}^{N-2}\\gamma_i-\\mathbf{r}\\big)\n.\n\\end{eqnarray}\n\\end{widetext}\nSome mathematical considerations show that this equation cannot be fulfilled for all $\\mathbf{r}$ when $N>2$, whatever the set of $(\\gamma_1,...,\\gamma_{N-2})$,\ninstead if $\\mathbf{D} (\\mathbf{r})=\\overrightarrow{const}$.\nBut if $\\mathbf{D} (\\mathbf{r})=\\overrightarrow{const.}$, then $\\Delta v_{int} (\\mathbf{r};t)$ should be \nequal to $\\mathbf{b}(t).\\mathbf{r}+c(t)\/N$, according to (\\ref{eq:D}),\nwhich is forbidden by the condition (\\ref{eq:Vint}), cf. discussion of the \\S \\ref{sub:int}.\n\nIt remains to discuss the case $N=2$. It is easy to show that then, we have\n$\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\xi_{N-1};t_0)=\n-\\frac{1}{2}\\mathbf{D}\\big(-\\frac{1}{2}\\xi_{N-1}\\big)+\\frac{1}{2}\\mathbf{D}\\big(\\frac{1}{2}\\xi_{N-1}\\big)$,\nwhich is null if $\\mathbf{D}(\\mathbf{r})$\nis any par function of $\\mathbf{r}$.\nBut if $\\mathbf{D} (\\mathbf{r})$ is par, then $\\frac{\\partial^{k}}{\\partial t^{k}}\\Delta v_{int} (\\mathbf{r};t)$ should be an odd function of $\\mathbf{r}$\n(up to an additional time-dependnt function), according to (\\ref{eq:D}),\nwhich is also forbidden by the condition (\\ref{eq:Vint}), cf. discussion of the \\S \\ref{sub:int}.\n\nThus, we can conclude that, in our case\n\\begin{eqnarray}\n\\forall (\\beta_1,...,\\beta_{N-2}):\n\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)\\ne\\overrightarrow{0}\n.\n\\nonumber\n\\label{eq:absurdum1}\n\\end{eqnarray}\nWe immediately deduce the incompatibility of\nthis relation,\nwhich is a consequence of (\\ref{eq:Vint}) and of the particular form (\\ref{eq:V_int}) of $V$, with (\\ref{eq:absurdum2}). Thus, the hypothesis we made is absurd: Eq. (\\ref{eq:partial_j_int2}) cannot vanish if $V[\\Delta v_{int}]\\ne c(t)$, so that the internal densities $\\rho_{int}(\\mathbf{r},t)$ and $\\rho_{int}'(\\mathbf{r},t)$ become different infinitesimally later than $t_0$. As a consequence, the map $G$, defined in (\\ref{eq:map}), is invertible (up to an additive time-dependent function) and $|\\psi^0_{int}(t))$ can be written as a functional of the internal density (we use the notation (\\ref{eq:v'})).\nThus, any expectation value of an operator $\\hat{O}$ which does not contain a time derivative can be written as a functional of $\\rho_{int}$ as the phase cancels out.\nThis achieves to prove the time-dependent Internal DFT theorem\n(which is a variant of the Runge-Gross theorem \\cite{Run84,Gro94} for self-bound systems and internal densities).\n\nMind that all the previous reasonings hold only for a fixed initial state $\\psi_{int}(t_0)$ (and a given type of particle), so that $\\psi^0_{int}$ is not only a functional of $\\rho_{int}$, but also depends on $\\psi_{int}(t_0)$. This will be discussed further.\n\n\n\n\n\\subsection{Link with traditional (time-dependent) DFT.}\n\nWe stress here the link and differences between the traditional DFT and internal DFT potentials.\nWe recall that the form of the potential $v_{ext}$ of traditional DFT can be fundamentally justified starting from the\nlaboratory Hamiltonian of\nan isolated molecule where the nuclei are treated explicitely.\nAs a molecule is a self-bound system, one can apply the Jacobi coordinates method.\nWe denote the N electronic coordinates related to \nthe laboratory frame\nas $\\mathbf{r}_i$, \nthe nuclear c.m. coordinate as $\\mathbf{R}^{nucl}$ and\nthe N electronic coordinates related to the c.m. of the nuclei as $\\mathbf{r}'_i=\\mathbf{r}_i - \\mathbf{R}^{nucl}$.\nA key point concerning the molecules is that, as the nuclei are much heavier than the electrons,\nthe c.m. of the whole molecule coincides with $\\mathbf{R}^{nucl}$,\nand it is an excellent approximation to apply the Jacobi coordinates to the nuclear coordinates only.\nAs a result, the c.m. motion will be described by a $\\Gamma(\\mathbf{R}^{nucl})$ wave function.\nThe redundant coordinate problem (thus the c.m.\\ correlations) will concern the nuclei only, and will be ``external'' to the electronic problem:\nthe N electrons are still described by N coordinates.\nThen, if one decouples the electronic motion from the nuclear one doing the clamped nuclei approximation,\nthe interaction of the electrons with the nuclear background is described by a potential of the form $\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i - \\mathbf{R}^{nucl})$,\nwhich becomes $\\sum_i v_{ext}(\\mathbf{r}'_i)$ when moving to the c.m. frame.\nWe then recover the form of the traditional DFT potential.\nThe potential $v_{ext}$, which is \\textit{internal} for the (self-bound) molecular problem, becomes \\textit{external} for the pure electronic problem.\nThose considerations also hold in the time domain, the difference being that the potential\n\\begin{eqnarray}\n\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i - \\mathbf{R}^{nucl};t)\n\\label{eq:v_ext}\n\\end{eqnarray}\ncan then contain an explicit time dependence in addition to the part which\ndescribes the interaction of the electrons with the nuclear background.\nWe recover the traditional time-dependent DFT potential \\cite{Run84,Gro94,Gro90,Mar04} when moving in the c.m. frame.\n\n\nThose reasonings explicit the link between the traditional DFT potential expressed with the laboratory coordinates, Eq. (\\ref{eq:v_ext}),\nand the Internal DFT potential expressed with the laboratory coordinates, Eq. (\\ref{eq:v}).\nThey both act only on the internal properties, and not on the c.m. motion\n(because it is anyway not comparable to the experimental one).\nThe difference is that as, in the molecular case, some particules are much heavier than the other,\nit is a very good approximation to assimilate the c.m. of the whole molecule with $\\mathbf{R}^{nucl}$,\nwhich permits to neglect the c.m. correlations for the electronic system,\nand to justify the clamped nuclei approximation.\nThis simplifies greatly the electronic problem and the traditional DFT can be used to study it.\nWhen the particules constituting the self-bound system have nearly the same masses, as it is the case for the nuclei or the He droplets, \nthe total c.m. ($\\mathbf{R}$) should be calculated with \\textit{all} the particules,\nso that the c.m. correlations will concern all the particules, and no clamped approximation can be justified.\nThen, we should use the formalism proposed here.\n\n\n\n\n\n\n\\section{Time-dependent Internal Kohn-Sham scheme.}\n\nWe now provide a practical scheme to calculate the internal density $\\rho_{int}$, which consists in the generalization of the stationary Internal KS scheme of \\cite{Mes09} to the time-dependent case.\nFirst, we note that for any normalizable initial state $|\\psi_{int}(t_0))$, which are the only allowed, the ``internal'' Schr\\\"odinger equation (\\ref{eq:schro}) stems\nfrom a variational principle\non the ``internal'' quantum action \\cite{ker76,Run84,Vig08}\n\\begin{eqnarray}\nA_{int} = \\int_{t_0}^{t_1} dt (\\psi_{int}(t)|i\\hbar\\partial_t-H_{int}|\\psi_{int}(t)) .\n\\label{eq:action}\n\\end{eqnarray}\nAs the function $c(t)$ possibly contained in the potential $V_{int}$ is perfectly canceled by the time derivative of the corresponding phase $e^{-i\\alpha(t)\/\\hbar}$ of $\\psi_{int}$, see (\\ref{eq:v'}),\nwe have $A_{int}=\\int_{t_0}^{t_1} dt (\\psi_{int}^0[\\rho_{int}](t)|i\\hbar\\partial_t-H_{int}|\\psi_{int}^0[\\rho_{int}](t))$\nif $V_{int}$ is chosen so that no additive time-dependent function can be split.\nThus, the internal quantum action can be considered as a functional of $\\rho_{int}$.\nIts\n$\\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|i\\hbar\\partial_t-\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha} - U[u]|\\psi_{int}^0(t))$\npart is a universal functional of $\\rho_{int}$ in the sense that, for a given type of particle (a given interaction $u$), the same dependence on $\\rho_{int}$ holds for every $V[v_{int}]$, thus $v_{int}$ (see (\\ref{eq:V_int})).\n\nUsing the Eq. (\\ref{eq:H_int}), we develop the ``internal'' quantum action as\n\\begin{eqnarray}\nA_{int}[\\rho_{int}]&=& \\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|i\\hbar\\partial_t-\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha}|\\psi_{int}^0(t))\n\\label{eq:action2}\\\\\n&& - \\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|U[u](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1})|\\psi_{int}^0(t))\n\\nonumber\\\\\n&& - \\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)|\\psi_{int}^0(t))\n\\nonumber .\n\\end{eqnarray}\nTo rewrite its last two terms in a more convenient way, we establish a useful relation.\nFor any function $f(\\vec{r}_1,...,\\vec{r}_N;t)$ of the laboratory coordinates,\nexpressible with the Jacobi coordinates [we denote $F(\\xi_1,...,\\xi_{N-1};t)$], we have\n\\begin{eqnarray}\n\\label{eq:rel}\n\\lefteqn{(\\psi_{int}^0(t)| F(\\xi_1,...,\\xi_{N-1};t) |\\psi_{int}^0(t))}\n\\\\\n& = & \\int \\! d\\mathbf{\\xi}_1 \\cdots  d\\mathbf{\\xi}_{N-1} F(\\xi_1,...,\\xi_{N-1};t) \n\\big| \\psi_{int}^0 (\\xi_1,...,\\xi_{N-1};t) \\big|^2\n\\nonumber\\\\\n& = & \\int \\! d\\mathbf{R} d\\mathbf{\\xi}_1 \\cdots  d\\mathbf{\\xi}_{N-1} \\delta(\\mathbf{R}) F(\\xi_1,...,\\xi_{N-1};t)\n\\nonumber\\\\\n&& \\hspace{4.5cm} \\times \\big| \\psi_{int}^0 (\\xi_1,...,\\xi_{N-1};t) \\big|^2\n\\nonumber\\\\\n& = & \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_{N} \\delta(\\mathbf{R}) f(\\vec{r}_1,...,\\vec{r}_N;t) \\big| \\psi_{int}^0(\\vec{r}_1,...,\\vec{r}_N;t) \\big|^2\n\\nonumber\n\\, .\n\\end{eqnarray}\nWe see that the \"internal mean values\" calculated with $\\psi_{int}$\nexpressed as a function of the ($N-1$) coordinates $\\xi_\\alpha$, can also be calculated with $\\psi_{int}$\nexpressed as a function of the $N$ coordinates $\\mathbf{r}_i$.\nAs one of them is redundant, a $\\delta(\\mathbf{R})$ which\nrepresents the dependence of the redundant coordinate on the others appears \\cite{foot3}.\n\nThe relation (\\ref{eq:rel}) leads to\n\\begin{eqnarray}\n\\label{eq:Eext}\n\\lefteqn{\n(\\psi_{int}^0(t)|V[v_{int}](\\xi_1,...,\\xi_{N-1};t)|\\psi_{int}^0(t)) \n} \\nonumber\\\\\n& = & \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_N \\; \n      \\delta(\\mathbf{R}) \\sum_{i=1}^N v_{int}(\\vec{r}_i - \\vec{R};t) |\\psi_{int}^0(\\vec{r}_1,...,\\vec{r}_N;t)|^2 \\, \n      \\nonumber\\\\\n& = & \\sum_{i=1}^N \n      \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \n      \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_N \\; \\delta(\\mathbf{R})\n      \\nonumber\\\\\n&   & \\hspace{2cm} \\times |\\psi_{int}^0(\\vec{r}_1, \\ldots, \\vec{r}_{N};t)|^2  \\delta \\big( \\vec{r}-(\\vec{r}_i-\\vec{R}) \\big) \n      \\nonumber\\\\\n& = & \\sum_{i=1}^N \n      \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \\, \n      \\, \\frac{\\rho_{int}(\\vec{r},t)}{N} \n      \\nonumber\\\\\n& = & \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \\, \\rho_{int}(\\vec{r},t)\n,\n\\end{eqnarray}\nwhere we used (\\ref{eq:rho_int0}) to obtain the penultimate equality.\nWe see that the potential $\\sum_{i=1}^N v_{int}(\\vec{r}_i - \\vec{R};t)$ that is $N$ body \nwith respect to the laboratory coordinates (and $(N-1)$ body when \nexpressed with Jacobi coordinates), becomes one body (and local) when\nexpressed with the c.m. frame coordinates\n(mind that $\\rho_{int}$ is defined in the c.m. frame, i.e. that $\\mathbf{r}$ is measured in the c.m. frame, cf. \\S \\ref{par:def}).\n\nApplying (\\ref{eq:rel}) to the second term of the action integral (\\ref{eq:action2}) gives\n$(\\psi_{int}^0(t)| U[u](\\xi_1,...,\\xi_{N-1}) |\\psi_{int}^0(t))=\\frac{1}{2} \\int \\! d\\vec{r} \\, d\\vec{r'} \\gamma_{int}(\\vec{r},\\vec{r'};t) u(\\vec{r}-\\vec{r'})$,\nwhere $\\gamma_{int}$ is defined in (\\ref{eq:gamint0}).\n\nThe action integral (\\ref{eq:action2}) can thus be rewritten\n\\begin{eqnarray}\nA_{int}[\\rho_{int}] &=& \\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|i\\hbar\\partial_t-\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha}|\\psi_{int}^0(t))\n\\nonumber\\\\\n&& - \\frac{1}{2} \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\, d\\vec{r'} \\gamma_{int}(\\vec{r},\\vec{r'};t) u(\\vec{r}-\\vec{r'})\n\\nonumber\\\\\n&& - \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \\rho_{int}(\\vec{r},t)\n.\n\\label{eq:action3}\n\\end{eqnarray}\n\nUp to now we did not do any hypothesis.\nTo recover the \\old{associated} Internal time-dependent KS scheme, we assume, as to obtain the\ntraditional time-dependent KS scheme \\cite{Run84,Gro94}, that there exists, \\textit{in the c.m.\\ frame}, a $N$-body non-interacting system\n(i.e. a local single-particle potential $v_S$)\n\\begin{eqnarray}\n\\label{eq:td_KS}\n\\Big( -\\frac{\\hbar^2\\Delta}{2m} + v_S(\\mathbf{r},t) \\Big)\\varphi^i_{int}(\\mathbf{r},t) = i\\hbar\\partial_t \\varphi^i_{int}(\\mathbf{r},t)\n\\end{eqnarray}\nwhich reproduces \\textit{exactly} the density $\\rho_{int}$ of the interacting system\n(mind that $\\rho_{int}$ is defined in the c.m. frame)\n\\begin{eqnarray}\n\\rho_{int}(\\mathbf{r},t) =\\sum_{i=1}^N |\\varphi^i_{int}(\\mathbf{r},t)|^2 .\n\\label{eq:rho_int}\n\\end{eqnarray}\nEven if only ($N-1$) coordinates are sufficient to describe the internal properties,\nthey still describe a system of $N$ particles. Thus, we have to introduce $N$ orbitals in the KS scheme (as we did)\nif we want them to be interpreted (to first order only) as single-particle orbitals\nand obtain a scheme comparable (but not equivalent) to mean-field like calculations with effective interactions.\n\nIn (\\ref{eq:td_KS}) we implicitely supposed that the particles are Fermions (a KS scheme to describe Boson condensates can be set similarly equalling all the $\\varphi^i_{int}$).\nUniqueness of the potential $v_S(\\mathbf{r},t)$ for a given density $\\rho_{int}(\\mathbf{r},t)$\n(and initial $|\\varphi^i_{int}(t_0))$ which yield the correct initial density $\\rho_{int}(\\mathbf{r},t_0)$)\nis ensured by a direct application of the traditional time-dependent DFT formalism \\cite{Run84,Gro94}.\nOf course, the question of the validity of the KS hypothesis, known as the \\textit{non-interacting v-representability} problem,\nremains, as in traditional (time-dependent) DFT \\cite{Dre90,Gro94}.\n\nTo use similar kinds of notations than the traditional DFT ones, we add and substract to the internal action integral (\\ref{eq:action3}) the internal Hartree term\n\\\\\n$\nA_{H}[\\rho_{int}] = \\frac{1}{2} \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\, d\\vec{r'} \\, \n\\rho_{int}(\\vec{r},t) \\, \\rho_{int}(\\vec{r'},t) \\, u(\\vec{r}-\\vec{r'})\n$,\nthe non-interacting kinetic energy term\\\\\n$\\int_{t_0}^{t_1} dt \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|\\frac{\\vec{p}^2}{2m}|\\varphi^i_{int}(t))$\nand the\n$\\int_{t_0}^{t_1} dt \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|i\\hbar\\partial_t|\\varphi^i_{int}(t))$\nterm.\nThis permits to rewrite the ``internal'' action integral (\\ref{eq:action3}) as\n\\begin{eqnarray}\nA_{int}&=& \\int_{t_0}^{t_1} dt \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|i\\hbar\\partial_t-\\frac{\\vec{p}^2}{2m}|\\varphi^i_{int}(t)) - A_{H}[\\rho_{int}]\n\\nonumber\\\\\n&& - A_{XC}[\\rho_{int}] - \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \\, \\rho_{int}(\\vec{r},t) \n\\label{eq:action4}\n\\end{eqnarray}\nwhere the internal exchange-correlation part is defined as\n\\begin{widetext}\n\\begin{eqnarray}\nA_{XC}[\\rho_{int}]\n&=& \\frac{1}{2} \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\, d\\vec{r'} \\, \n      \\Big( \\gamma_{int}(\\vec{r},\\vec{r'};t) - \\rho_{int}(\\vec{r},t) \\, \\rho_{int}(\\vec{r'},t) \\Big) \\, \n      u(\\vec{r}-\\vec{r'})\n\\nonumber\\\\\n&&    + \\int_{t_0}^{t_1} dt \\Big( \n      (\\psi_{int}^0(t)|\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha}|\\psi_{int}^0(t)) \n      - \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|\\frac{\\vec{p}^2}{2m}|\\varphi^i_{int}(t)) \\Big)\n\\nonumber\\\\\n&&    - \\int_{t_0}^{t_1} dt \\Big( (\\psi_{int}^0(t)|i\\hbar\\partial_t|\\psi_{int}^0(t))\n      - \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|i\\hbar\\partial_t|\\varphi^i_{int}(t)) \\Big)\n.\n\\label{eq:Axc}\n\\end{eqnarray}\n\\end{widetext}\nWe see that it contains the exchange-correlation which comes from the interaction $u$ (first line of (\\ref{eq:Axc})), but also the correlations contained in the interacting\nkinetic energy  (second line of (\\ref{eq:Axc})) and in the interacting ``$i\\hbar\\partial_t$'' term  (third line of (\\ref{eq:Axc})).\nA key point is that, as the KS assumption implies $\\varphi^i_{int}[\\rho_{int}]$ \\cite{Run84,Gro94,Dre90},\n$A_{XC}[\\rho_{int}](t)$ can be written as a functional of $\\rho_{int}$\n(for given $|\\psi_{int}^0(t_0))$ and $\\{|\\varphi^i_{int}(t_0))\\}$ which yield the same initial density $\\rho_{int}(\\mathbf{r},t_0)$).\n\nIt remains to vary the ``internal'' quantum action (\\ref{eq:action4}) to obtain the equations of motion (which define $\\rho_{int}$).\nVignale, see Ref. \\cite{Vig08}, showed recently that the correct formulation of the variational principle is not to stationarize the quantum action, i.e. $\\delta A_{int}[\\rho_{int}]=0$ as done so far \\cite{ker76,Run84,Gro94}, but\n\\begin{eqnarray}\n\\delta A_{int}[\\rho_{int}]=&&\ni\\big(\\psi_{int}[\\rho_{int}](t_1)\\big|\\delta \\psi_{int}[\\rho_{int}](t_1)\\big)\n\\nonumber\\\\\n&& - i \\big(\\psi^S_{int}[\\rho_{int}](t_1)\\big|\\delta \\psi^S_{int}[\\rho_{int}](t_1)\\big)\n\\label{eq:action}\n\\end{eqnarray}\n(where $\\psi^S_{int}$ is the Slater determinant constructed from the $\\varphi^i_{int}$).\nThe two formulations lead to identical final results for theorems derived form symmetries of the action functional because\ncompensations occur \\cite{Vig08}, but Vignales's formulation permits to solve the causality paradox of the previous formulation.\n\nVarying (\\ref{eq:action}) with respect to the $\\varphi^{i*}_{int}(\\mathbf{r},t)$, with $t\\in[t_0,t_1]$,\nleads straightforwardly to the Internal time-dependent KS equations for the $\\varphi^i_{int}$\n\\begin{equation}\n\\label{eq:varphi_i}\n\\Big(\n- \\frac{\\hbar^2}{2m}\\Delta \n+ U_H[\\rho_{int}] \n+ U_{XC}[\\rho_{int}] \n+ v_{int}\n\\Big) \\varphi^i_{int} = i\\hbar\\partial_t \\varphi^i_{int}\n\\end{equation}\nwith the potentials\n\\begin{widetext}\n\\begin{eqnarray}\n&& U_{H}[\\rho_{int}](\\vec{r},t) \n= \\frac{\\delta A_{H}[\\rho_{int}]}{\\delta \\rho_{int}(\\vec{r},t)}\n\\nonumber\\\\\n&& U_{XC}[\\rho_{int}](\\vec{r},t) \n= \\frac{\\delta A_{XC}[\\rho_{int}]}{\\delta \\rho_{int}(\\vec{r},t)}\n-i\\big(\\psi_{int}[\\rho_{int}](t_1)\\big|\\frac{\\delta \\psi_{int}[\\rho_{int}](t_1)}{\\delta \\rho_{int}(\\vec{r},t)}\\big)\n+ i \\big(\\psi^S_{int}[\\rho_{int}](t_1)\\big|\\frac{\\delta \\psi^S_{int}[\\rho_{int}](t_1)}{\\delta \\rho_{int}(\\vec{r},t)}\\big)\n\\label{eq:Uxc}\n\\end{eqnarray}\n\\end{widetext}\nwhich are local as expected ($v_S=U_H[\\rho_{int}]+ U_{XC}[\\rho_{int}] + v_{int}$ with the notations of Eq. (\\ref{eq:td_KS})).\nNote that the variational formulation of Vignale \\cite{Vig08} leads to the addition of the last\ntwo terms in the definition of $U_{XC}[\\rho_{int}](\\vec{r},t)$, see Eq. (\\ref{eq:Uxc}),\ncompared to the traditional result obtained by stationarization of the action. It are those terms which permit to solve the causality paradox \\cite{Vig08}.\n\nEquations~(\\ref{eq:varphi_i}) have the same form as the traditional time-dependent KS \nequations formulated \\old{in the laboratory frame} for non-translationally \ninvariant Hamiltonians \\cite{Koh65,Run84,Gro94}\nand permit to define $\\rho_{int}$ through (\\ref{eq:rho_int}).\nHere, we have justified their form \\textit{in the c.m. frame} for \nself-bound systems described with translationally invariant Hamiltonians.\n\nBut there is a major difference with the traditional DFT formalism.\nFollowing similar steps as in Eq.~(\\ref{eq:rel}), one can show that the\ninteracting kinetic energy term and the interacting ``$i\\hbar\\partial_t$'' term can be rewritten \\cite{foot3}\n\\begin{widetext}\n\\begin{eqnarray}\n&& (\\psi_{int}^0(t)|\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha}|\\psi_{int}^0(t))\n= \\int d\\vec{r}_1 \\cdots d\\vec{r}_N \\delta(\\mathbf{R}) \n\\psi_{int}^{0*}(\\vec{r}_1,...,\\vec{r}_N;t)\n\\sum_{i=1}^N \\frac{\\mathbf{p}_i^2}{2m}\\psi_{int}^0(\\vec{r}_1,...,\\vec{r}_N;t)\n\\nonumber\\\\\n&& (\\psi_{int}^0(t)|i\\hbar\\partial_t|\\psi_{int}^0(t)) =\n\\int d\\vec{r}_1 \\cdots d\\vec{r}_N \\delta(\\mathbf{R}) \\psi_{int}^{0*}(\\vec{r}_1,...,\\vec{r}_N;t)\ni\\hbar\\partial_t \\psi_{int}^0(\\vec{r}_1,...,\\vec{r}_N;t)\n,\n\\label{eq:cm_cor}\n\\end{eqnarray}\n\\end{widetext}\nwhich makes it clear that the differences with the non-interacting kinetic energy term\n$\\sum_{i=1}^{N} \\int d\\vec{r} \\varphi^{i*}_{int}(\\vec{r})\\frac{\\vec{p}^2}{2m}\\varphi^i_{int}(\\vec{r})$\nand the non-interacting ``$i\\hbar\\partial_t$ term'' $\\sum_{i=1}^{N} (\\varphi^i_{int}(t)|i\\hbar\\partial_t|\\varphi^i_{int}(t))$\n(found in the exchange-correlation functional (\\ref{eq:Axc}))\ncome, on the one hand, from the correlations neglected in the traditional independent-particle framework,\nbut also from the c.m. correlations described by the $\\delta(\\mathbf{R})$ term in (\\ref{eq:cm_cor}), which does not appear in traditional time-dependent DFT \\cite{Run84,Gro94}.\nThe inclusion of the c.m. correlations in the exchange-correlation functional (\\ref{eq:Axc}) and potential (\\ref{eq:Uxc})\nis the main difference with the traditional KS scheme, and is a key issue for self bound-systems as atomic nuclei.\n\nMind that all the previous considerations only hold for fixed initial states $|\\psi_{int}(t_0))$ and $\\{|\\varphi^i_{int}(t_0))\\}$ which should of course give the same initial density $\\rho_{int}(\\mathbf{r},t_0)$ (and also for a fixed type of particle).\nAs a consequence, $\\psi_{int}^0$ is not only a functional of $\\rho_{int}$, but also depends on the initial state $|\\psi_{int}(t_0))$, and $U_{XC}$, Eq. (\\ref{eq:Uxc}),\nalso depends on the initial orbitals $\\{|\\varphi^i_{int}(t_0))\\}$.\nAn important difference to the ground state Internal DFT formalism \/ KS scheme presented in \\cite{Mes09}\nis that $|\\psi_{int}(t_0))$ and the $\\{|\\varphi^i_{int}(t_0))\\}$\ncannot necessarily be written as functionals of $\\rho_{int}(\\mathbf{r},t_0)$.\nHowever, as underlined in \\cite{Run84,Gro94}, if one starts from initial states $|\\psi_{int}(t_0))$ and $\\{|\\varphi^i_{int}(t_0))\\}$\nthat are non-degenerate ground states, i.e. that can be written as functionals of $\\rho_{int}(\\mathbf{r},t_0)$ \\cite{Mes09},\n$\\psi_{int}$ and $U_{XC}$ become functionals of $\\rho_{int}(\\mathbf{r},t)$ alone.\nThen, in the limit of stationary ground states, the theory reduces to the stationary Internal DFT \/ KS.\n\nWe recall that, as in traditional DFT, the previously discussed functionals are defined only for internal densities $\\rho_{int}$ which correspond to some internal potential $v_{int}$, called \\textit{v-representable} internal densities \\cite{Run84,Gro94}.\nUp to now, we do not know exactly how large the set of v-representable densities is.\nThis has to be kept in mind when variations with arbitrary densities are done, as to obtain the time-dependent KS equations.\n\n\n\n\n\n\\section{Conclusion.}\n\nIn summary, we have shown that, for a fixed initial state, the internal wave function,\nwhich describes the internal properties of a time-dependent self-bound system, can be written (up to a trivial phase) as a functional of the internal density.\nThis implies that the \"internal\" expectation values of any observable (which does not contain a time derivative),\nthat are of experimental interest, can be regarded as functionals of the internal density.\nThen, we set up, in the c.m.\\ frame, a practical scheme which permits to calculate the internal density and whose form is similar to the traditional time-dependent KS equations,\nthe difference being that the exchange-correlation functional contains the c.m. correlations.\n\nThis work is a first step towards the justification to the use of density functionals\nfor time-dependent nuclear mean-field like calculations with effective interactions \\cite{tdhf_nucl,Neg82},\nproving that there exists an ultimate functional which permits to reproduce the exact internal density\n(up to the non-interacting v-representability question).\nIf this functional was known, there would be no need for a c.m.\\ correction.\n\nPractically speaking, the time-dependent Internal KS scheme can describe, for instance in the nuclear case,\nthe collision of two nuclei in the frame attached to the total c.m.\\ of the nuclei.\nThen, $v_{int}$ is zero but the dependency to the initial state allows to start from\na state which corresponds to two nuclei with different velocities, or ``boosts''\n(choosen such as the total kinetic momentum is zero because we are in the c.m. frame). According to the choice of the boosts, \nwe can describe a wide variety of physical phenomena, from nuclear fusion \\cite{tdhf_nucl} to\nCoulomb excitation \\cite{Ald56}. One of the nuclei can also simply consist in a particule as a proton,\nto describe the excitation of a nucleus by diffusion.\n\nA case where a non-zero $v_{int}$ would be interesting could be\nthe case of the laser irradiation ($v_{int}$ would then contain a laser potential switched on at $t>t_0$).\nThis is not of major interest in the nuclear case because, experimentally speaking,\nwe do not yet have lasers that are suited to the study of the laser irradiation of a nucleus.\nHowever, this could be interesting in view of a generalization of this work to the whole molecule\n(following from the generalization to different types of particules, which is underway).\n\nMany questions remain open.\nIn particular, the question of the form of the potential which describes the c.m.\\ correlations;\nin addition to its practical interest, this question would also give interesting arguments concerning the non-interacting v-representability question.\nGeneralization to different types of particles (Fermions or Bosons) appears desirable.\nFinally, the same reasoning should be applied\nto rotational invariance to formulate the theory in term of the so-called \"intrinsic\" one-body density \\cite{Gir08a}\n(which is not directly observable). This is more complicated because rotation\ndoes not decouple from internal motion, but it should be interesting concerning the symmetry breaking question.\n\n\n\n\n\n\n\\begin{acknowledgments}\n\nThe author is particularly grateful to M. Bender, E.K.U. Gross, and E. Suraud for enlightening discussions and\nreading of the manuscript,\nand thanks the referee for his pertinent remarks.\nThe author thanks the Centre d'Etudes Nucl\\'eaires de Bordeaux-Gradignan for warm hospitality,\nand the Institut Universitaire de France and the\nAgence Nationale de la Recherche (ANR-06-BLAN-0319-02) for financial support.  \n\n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHamel and Prahalad \\cite{2005Competing} point out that the rapid changes in technology and the non-stationary nature of market demand create dynamic behavioural characteristics of the environment, and that in today's society, where technological innovations are proliferating and transformations are taking place rapidly, the cumulative effects of technological innovation in firms are more likely to take on the characteristics of a complex, irregular and non-periodic non-linear dynamical system. This non-linear characteristic seems to be chaotic, but in fact it is in order. From the perspective of non-linear theory, the evolution of a system can be summarised as shown in Figure \\ref{fig:fig1} . McBride \\cite{2010Chaos} defines chaos as a qualitative study of the unstable, acyclic behaviour of a deterministic nonlinear dynamical system, and Sardar and Abrams \\cite{2004Introducing} argue that 'order and chaos coexist, order in chaos and chaos in order. The presence of chaos in a non-linear system should be categorised and discussed. There are situations where chaos should be avoided, times when it should be controlled, and times when it should be exploited. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.7]{figure1.png}\n    \\caption{System evolution illustration.}\n    \\label{fig:fig1}\n\\end{figure}\n\nSo as of now, research on models of the cumulative effects of technological innovation in the textile industry is still very limited and far from systematic. The textile industry, as an indispensable part of the population, plays an important role in clothing, food, housing and transport, and has experienced a long period of development. The cumulative effect of technological innovation in the industry is also prominent, so the textile industry is chosen as the subject of this study. This paper discusses the following: 1. the application of a model of the cumulative effect of technological innovation in the textile industry from the perspective of chaos theory; and 2. the provision of countermeasures for technological innovation in the textile industry to appropriately control and manage chaos and increase the effective innovation rate of the industry.\n\n\n\\section{A chaotic model of the cumulative effect of technological innovation}\n\\subsection{Chaos theory}\nThe basic characteristics of chaotic motion are as follows: deterministic, non-linear, sensitive dependence on initial conditions and non-periodic. The connotation of chaos is reflected in the fact that it is a seemingly random dynamic behaviour generated by a non-linear system \\cite{R1976Simple} . A deterministic system is said to be chaotic if, in the absence of external stochastic influences, 1. the state of motion of the system is irregular and complex, similar to the Brownian motion of small molecules; 2. the system has a sensitive dependence on the initial conditions; and 3. some characteristic of the system (e.g. positive Lyapunov exponent, positive topological entropy, attractor of fractional dimension, etc.) has little to do with the choice of initial conditions.\nIn this paper, on top of the basic framework of previous studies, a chaotic economic model of the cumulative effect of technological innovation in the textile industry is developed using the worm mouth model in chaos theory as an example, with appropriately given parameters so that it can be used to describe the dynamic evolution of the textile industry.\n\n\\subsection{The textile industry is chaotic in nature}\nThe evolution of the textile industry is irregular and complex. The reason is that it is impossible to predict which textile company will launch a new garment that will be liked by the people, or to judge success or failure by its industry innovations. The system of cumulative effects of technological innovation in the textile industry is sensitive to the initial conditions - the cumulative effects of the initial innovation - and is dependent on them. Moreover, Mensh \\cite{1979Stalemate}, Houstein et al. \\cite{1982Long}, Kleinknecht \\cite{1987Are} and Silverberg et al. \\cite{1993Long} have successively conducted empirical studies on the distribution function of the temporal occurrence pattern of innovation using relevant statistical data, and the results show that the realisation of innovation is a similar Poisson distribution and has an exponential growth trend of stochastic process. Therefore, both the textile industry itself and its innovation accumulation are chaotic in nature.\n\n\\subsection{A chaotic model of the cumulative effect of innovation in the textile industry}\nLet $X_t$ be the marginal contribution of the accumulation of innovative technology in the textile industry system to the growth rate of the total economic volume of the textile industry system at a certain moment $t$, i.e. when the accumulation of innovation in the textile industry increases by $1\\%$, the total economic volume of the textile industry increases by $X_t\\%$, and call $X_t$ the accumulation effect of innovation. It is easy to see that $X_t$ is a function about time, that is, the value of $X_t$ is time-dependent. Similar to the variables used to describe the changing state of economic growth, such as capital output rate, labour productivity and capital-labour ratio, $X_t$ should also be a state variable to describe the evolution of the textile system.\nThe contribution of innovation to economic growth in the textile industry is much more than a simple linear accumulation of individual innovation contributors, and therefore $X_t$ is a more complex variable, as are variables such as the capital output rate, labour productivity and the capital-labour ratio. According to the definition of $X_t$, its variation mainly reflects fluctuations in the structure of productivity within the textile industry due to the continuous generation and accumulation of innovation. Silverberg \\cite{1993Long} developed a model of technological progress and its evolutionary chaos, which not only explains the existence of the impact of innovation, but also shows that innovation makes productivity move in an irregular cycle, i.e. chaos. This is the theoretical basis for this paper's $X_t$.    \\\\\nIntroducing the model more commonly used in chaos economics for studying chaos in economic growth, as shown in equation \\ref{equ1}.\n\n\\begin{equation}\\label{equ1}\n    X_{t+1} = X_{t}^{\\beta} \\frac{\\sigma A}{1+\\lambda} ( s-X_t )^{\\gamma}\n\\end{equation}\n\nwhere $X_t$ is the capital-labour ratio, $\\sigma (\\sigma >0)$ is the savings rate, $\\lambda (0<\\lambda <1)$ is the natural growth rate of labour, $A(A>0)$ is the technological progress factor, $\\beta (\\beta >0)$ is the elasticity of the capital-labour ratio, $\\gamma$ is a constant greater than zero, and $s$ is the maximum capital-labour rate.    \\\\\nWhen $\\beta=\\gamma=s=1$, let $\\mu=\\sigma A\/(1+\\gamma) $, then \\ref{equ1} can be transformed into \\ref{equ2}.\n\n\\begin{equation}\\label{equ2}\n    X_{t+1} = \\mu X_{t}(1-X_{t})\n\\end{equation}\n\nequation \\ref{equ2} is the general form of the insect-population model. The purpose of introducing the transformation of equation \\ref{equ1} into equation \\ref{equ2} is to construct a chaotic economic model of the cumulative effect of technological innovation by analogy with a chaotic initialization \\cite{2018Improving}. Drawing on the insect-population model, the following assumptions are introduced: 1) Technological innovation does not arise out of thin air, and existing innovations are usually the basis for subsequent innovations, similar to the relationship between parent and offspring insects in nature. 2) Technological innovation is measured by its level of innovation, and usually there is no mixture of parent and offspring, i.e., there is a certain amount of substitution of the firm's new product for the old one. 3) The technological innovation of an enterprise is limited by its own economic resources, just like the survival environment of a worm in nature. Therefore, this paper considers that technological innovation in textile enterprises meets the prerequisites of the insect-population model. According to the previous paper, the cumulative effect of technological innovation is a chaotic economic variable, which evolves in much the same way as the capital-labour ratio, with a non-linear evolution mechanism. Therefore, a chaotic economic model of the cumulative effect of technological innovation can be defined similarly - equation \\ref{equ3}.\n\n\\begin{equation}\\label{equ3}\n    X_{t+1} = T \\epsilon X_{t}(1-X_{t})\n\\end{equation}\n\nwhere $X_t \\in (0,1)$, $\\epsilon \\in (0,10)$, $T\\epsilon \\in (0,4)$, $X_t$ represents the proportion of the cumulative effect of technological innovation at time $t$, as the state variable of the textile system; $\\epsilon$ denotes the government regulation parameter; $T$ is the specific coefficient of the textile firm (i.e. the combined coefficient of the growth rate of technological inputs $\\alpha$, the proportion of technological content of output $\\beta$ and the annual growth rate of labour force $n$ at a point in time, which The relationship between them is $T=\\frac{\\alpha + \\beta}{1+n}$, where $\\alpha, \\beta, n \\in (0,1)$, $T\\epsilon$ together constitute the innovation control parameters of the textile enterprises themselves.\n\n\\subsection{Chaos of the model}\nThe chaotic state of \\ref{equ3} can be determined using the Li-Yorke theorem. Constructing the function,\n\n\\begin{equation}\\label{equ4}\n    f(x)=T\\epsilon x(1-x)\n\\end{equation}\n\nIt is not difficult to determine where $f(0)=0$ and $f(x)$ is a single-peaked function. \\\\\nFirst, determine the point $x^*$ at which $f(x)$ reaches its maximum value. $x^*$ represents the maximum technological innovation accumulation effect value of $f(x)$. $x^*$ can be obtained by solving the following first order partial derivative equation.\n\n\\begin{equation}\\label{equ5}\n    \\frac{\\partial f(x^*)}{\\partial x^*}=0\n\\end{equation}\n\nThe mapping point $x^*$ corresponding to the value of the maximum reachable function is obtained,\n\n\\begin{equation}\\label{equ6}\n    x^*=1\/2\n\\end{equation}\n\nThe maximum innovation accumulation effect value $X_{max}$ is,\n\n\\begin{equation}\\label{equ7}\n    x_{max}=f(x^*)=\\frac{T\\epsilon}{2}\n\\end{equation}\n\nFrom the above definition, it follows that the value of the maximum innovation accumulation effect should satisfy,\n\n\\begin{equation}\\label{equ8}\n    x_{max} \\le s\n\\end{equation}\n\nNext, determine the initial image point $x_i$ , and from equations \\ref{equ4} and \\ref{equ6}, we have,\n\n\\begin{equation}\\label{equ9}\n    T\\epsilon x(1-x)=1\/2\n\\end{equation}\n\nThe left end of equation \\ref{equ9} is equation \\ref{equ4}, a single-peaked function, so the smaller root is taken to be $x_i$, \n\n\\begin{equation}\\label{equ10}\n    x_i = \\frac{T\\epsilon - \\sqrt{(T\\epsilon)^2-T\\epsilon}}{2T\\epsilon}\n\\end{equation}\n\nFinally, determine its third-order mapping $f(x_{max})$,\n\n\\begin{equation}\\label{equ11}\n    f(x_{max}) = \\frac{T^2\\epsilon ^2}{2}(1-T\\epsilon \/2)\n\\end{equation}\n\nprovided that the first-, second- and third-order mappings $x^*$,$x_{max}$, $f(x_{max})$ generated by the initial preimage $x_i$ are mapped from the interval $[0,s]$ to its own interval $[0,s]$ and satisfy the sufficient conditions for chaos in the Li-Yorke theorem, i.e.,\n\n\\begin{equation}\\label{equ12}\n    0 \\le f(x_{max}) \\le x_i \\le x^* \\le x_{max}\n\\end{equation}\n\nThen the textile industry system \\ref{equ3} appears chaotic, as shown in Figure \\ref{fig:fig2}, which is a chaotic phase diagram regarding the proportion of the cumulative effect of two adjacent generations of innovation in the textile industry innovation system. Where the curve is the image of equation (3) and the straight line is $X_{t+1} = X_t$, which represents the process of converting a preimage point to the next preimage point.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.7]{figure2.png}\n    \\caption{Chaotic phase diagrams in textile industry.}\n    \\label{fig:fig2}\n\\end{figure}\n\n\n\\section{Chaotic nature of the model}\n\\subsection{Local structural stability and cyclic bifurcation propertiest}\nFor model \\ref{equ3}, specifically, when $T\\epsilon$ has a determined value, if the corresponding $x_t$ also has only one value (indicating that the firm's innovation reaches a uniquely determined proportion, i.e. the level of intensification). That is, the technological innovation effect of a textile firm reaches a uniquely determined level. Then the period of $x_t$ is said to be $1$; if at this point $x_t$ has two values corresponding to it, then the period of $x_t$ is said to be $2$; if at this point $x_t$ has n values corresponding to it, then the period of $x_t$ is said to be $n$. In particular, the structure of $x_t$ is unstable when there is a bifurcation and the steady state can only be maintained when the period of $x_t$ is determined. This also means that a stable accumulation of technological innovation in textile companies can only be guaranteed when $T\\epsilon$ takes on a value within a particular interval. In other words, a good accumulation of technological innovation in the textile industry can be maintained to a certain extent when the role of policy innovation and industry-specific coefficients are well controlled. A diagram of the iterative process for varying values of the control parameter $T\\epsilon$ from $0$ to large is shown in Figure \\ref{fig:fig3}.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.7]{figure3.png}\n    \\caption{Bifurcations.}\n    \\label{fig:fig3}\n\\end{figure}\n\n\\subsection{Effect of parameter values on the initial innovation ratio in the textile industry}\nBased on its definition of firm intensification in a chaotic economic model, the effect of the $T\\epsilon$ parameter on the initial innovation ratio of textile firms is derived by analogy.  \n\\begin{itemize}\n\t\\item When $0<T\\epsilon \\le 1$, there exists a stable immovable point $x_t=0$, indicating that there is no set. That is, there does not exist a corresponding non-zero $x_t$ taking value.   \n\t\\item When $1<T\\epsilon \\le 3$, $x_t$ has a locally stable structure with period $1$. If $T\\epsilon$ takes any value in $(1, 3)$, $x_t$ takes the corresponding one between $(0, 0.664)$. This indicates that the innovation set $x_t$ is stable at that value, i.e. a $T\\epsilon$ leads to only one innovation set $x_t$.      \n\t\\item When $T\\epsilon = 3$, the first bifurcation value $T\\epsilon_1$ occurs and a forked bifurcation occurs. This will double the $x_t$ cycle. At this point the structure of $x_t$ goes from stable to unstable.    \n\t\\item When $3<T\\epsilon \\le 3.4495$, a new locally stable structure of $x_t$ emerges, but at this point the period becomes $2$, i.e. a $T\\epsilon$ parameter value will lead to two innovation sets $x_{t1}$ and $x_{t2}$. Over time, their innovation set bounces around $x_{t1}$ and $x_{t2}$, thus constituting two stable sequences of immobile point values that range from $0.664~0.85$ and $0.664~0.44$, respectively, as $T\\epsilon$ varies. \n\t\\item When $T\\epsilon = 0.4495$, a second bifurcation value $T\\epsilon_2$ occurs and the Hough bifurcation occurs. $x_t$ has an unstable structure and thereafter the $x_t$ period multiplies to $4$.  \n\t\\item When $3.4495<T\\epsilon \\le 3.5441$, the structure of $x_t$ changes from unstable to stable, and the period of $x_t$ is $4$. This indicates that one $T\\epsilon$ parameter value will lead to four innovation sets, at which time $x_t$ has four sets of stable sequences of immobile point values, which range from 0.85-0.883, 0.85-0.82, 0.44-0.52 and 0.44-0.3367, respectively, with different values of $T\\epsilon$.    \n\t\\item When $T\\epsilon=3.5441$ , a third bifurcation value $T\\epsilon_3$ occurs, and the structure of $x_t$ is unstable, after which the period multiplies to $8$. \n\t\\item When $T\\epsilon >3.5441$, the local structure of $x_t$ is stable and the period of $x_t$ is $8$. The rest of the process continues until $T\\epsilon=3.5699=T\\epsilon_{\\infty}$, when the period of $x_t$ is infinite, i.e. chaos emerges. In other words, there is no relationship between the firm's innovation accumulation and the control parameters at this point.\n\\end{itemize}\n\n\\subsection{Sexual analysis of different parameter values}\nFirstly, the values of $\\alpha$, $\\beta$, and $n$ are taken and defined as shown in Table \\ref{tab1}, and all data are obtained from the findings of the \"China Statistical Yearbook\".\n\n\\begin{table}[H]\n    \\caption{\\label{tab1} Variable definition and value}\n    \\centering\n    \\resizebox{\\linewidth}{32pt}{\n    \\begin{tabular}{ccc}\n        \\hline\n        Symbol & Description & Fetching method \\\\\n        \\hline\n        $\\alpha$ & Growth rate of investment in science and technology & Log difference of R\\&D inputs \\\\\n        $\\beta$  & Proportion of technological content of output & Log difference in the number of active patents in the textile industry \\\\\n        $n$     & Annual growth rate of the labour force & Log difference of the average number of workers employed \\\\\n        $\\epsilon$ & Government regulation parameters & By taking the product of $T\\epsilon$ backwards or assuming that given \\\\\n        \\hline\n    \\end{tabular}}\n\\end{table}\n\nFrom an econometric perspective, log-differencing helps to enhance the significance of the main regression to some extent. As a corollary, the significance of log-differencing the variables in this paper is to smooth out the effects of unreasonable outliers and reduce the \"misleading\" effect of episodic events on the overall judgment. In addition, as the average number of workers before 14 years could not be found directly, the calculation of business income\/per capita business income was used, and the use of logarithmic difference to calculate the growth rate can also reduce the subtle influence of different quantiles on the conclusion. The log-difference results are shown in Table \\ref{tab2} \\footnote{All data are obtained from the China Statistical Yearbook, and are rounded to four decimal places after calculation by stata. The values of $\\alpha$ range from 1.46\\% to 47.44\\%, $\\beta$ from 9.58\\% to 51.69\\% and $n$ from -16.46\\% to 4.79\\%, so the values of $\\alpha+\\beta$ and $1+n$ are inferred from this.}.\n\n\\begin{table}[H]\n\t\\caption{\\label{tab2} Log-differential results}\n\t\\centering\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\hline\n\t\t\tYear & $\\alpha$ & $\\beta$ & $n$ & $\\alpha+\\beta$ & $1+n$ \\\\\n\t\t\t\\hline\n\t\t\t2009&\/&\/&\/&\/&\/\t\\\\\n\t\t\t2010&.2017&.5088&.0479&.7105&1.0479\\\\\n\t\t\t2011&.4744&.1671&-.0947&.6416&.9053\\\\\n\t\t\t2012&.0146&.1347&-.0566&.1494&.9434\\\\\n\t\t\t2013&.1382&.1418&-.0614&.2800&.9386\\\\\n\t\t\t2014&.1144&.5169&-.0653&.6313&.9347\\\\\n\t\t\t2015&.1558&.0958&-0.540&.2516&.9460\\\\\n\t\t\t2016&.0574&.2867&-.0627&.3441&.9373\\\\\n\t\t\t2017&.0584&.2688&-.1090&.3273&.8910\\\\\n\t\t\t2018&.0912&.2629&-.1646&.3541&.8354\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\\end{table}\n\nTable \\ref{tab2} shows that the values of $\\alpha+\\beta$ range from 14.94\\% to 71.05\\%, and the values of $1+n$ range from 83.54\\% to 104.79\\%. Considering that the value of the parameter $T$ in equation \\ref{equ3} is proportional to $(\\alpha+\\beta)$ and inversely proportional to $(1+n)$, and that $T$ is related to the innovation accumulation $x_t$ of the textile enterprises in equation \\ref{equ3}, the sexual attitude of $x_t$ can be analysed through this. To facilitate the calculation, the three fixed cases of the lower, middle and upper limits of these elements are taken separately, and only $\\epsilon$ is adjusted to analyse the nature of $x_t$.\n\n\\begin{itemize}\n\t\\item Lower bound case. $\\alpha+\\beta=0.1494$ , $1+n=0.8354$ . When $\\epsilon=5.5917$, $T\\epsilon=1.0000$ . From equation \\ref{equ3} and Figure \\ref{fig:fig3}, $x_t=0$. when $\\epsilon=10$, $T\\epsilon=1.7884$ and $x_t=0.4408$ . This means that at $(\\alpha+\\beta=0.1494)$, the degree of innovation accumulation achieved by textile firms remains zero when the parameter of government regulation of textile firms is 5.5917. When adjusted to the maximum value $\\epsilon=10$, $T\\epsilon=1.7884$ (satisfying $1<T\\epsilon<3$), at which point the period of $x_t$ is $1$, the innovation intensification of textile enterprises is also only 44.08\\%. At the same time, considering the nature of the enterprise cycle of China's existing textile enterprises and the characteristics of the industry, it is easy to see that the government's R\\&D subsidies for the traditional textile industry are not significant enough. Therefore, it is difficult for the government's regulation parameter $\\epsilon$ for the textile industry to reach a maximum value of $10$. That is, in the actual process, the innovation intensity of textile enterprises will be much less than 44.08\\%.\n\t\\item Middle bound case. This is calculated for the case $\\alpha+\\beta<0.2754$ , $1+n=0.9417$. When the government regulation parameter $\\epsilon=10$, $T\\epsilon=2.9245 (T\\epsilon<3)$, it can be inferred that in this case, whatever the value of $\\epsilon$ in the above range, the bifurcation does not occur and the structure is stable. Therefore, in this case, the government can increase the regulation of the innovation intensity of the textile enterprises, and can increase the policies and subsidies for them.\n\t\\item Upper bound case. $\\alpha+\\beta=0.9913$ , $1+n=1.0479$ . When $\\epsilon=1.0571$, $T\\epsilon=1.0000$ , the innovation set is $0$; when $\\epsilon<3.1659$, $T\\epsilon<3=T\\epsilon_1$ , the period of $x_t$ is $1$ and $x_t<0.3339$ , i.e. the innovation set is less than 33.39\\%; when $3.1659<\\epsilon<3.6465$, $3<T\\epsilon<3.4495=T\\epsilon_2$ ,the period of $x_t$ is 2; when When $3.6465<\\epsilon<3.7465$, $3.4495<T\\epsilon<3.5441=T\\epsilon_3$ and the period of $x_t$ is 4. From the above, it can be seen that when $\\epsilon_1=3.1659$, $\\epsilon_2=3.6465$ and $\\epsilon_3=3.7465$, there will be three bifurcations of $T\\epsilon$, $T\\epsilon_1$, $T\\epsilon_2$ and $T\\epsilon_3$. This means that $\\epsilon_1$, $\\epsilon_2$ and $\\epsilon_3$ are dangerous values of strength that lead to a non-stationary state in the textile innovation accumulation model. As $\\epsilon$ increases further, $T\\epsilon$ also becomes larger until $\\epsilon = 3.7737 = \\epsilon_{\\infty}$ when $T\\epsilon = 3.5699 = T\\epsilon_{\\infty}$ and the period of $x_t$ is infinite, i.e. chaos will emerge. Clearly, $\\epsilon=3.7737$ is the more dangerous regulatory parameter. At the upper value, $x_t$ takes values of 0.5270-0.6847 when the regulation parameter $\\epsilon$ is kept between 2.1142-3.1713, which would make $T\\epsilon$ take values between 2-3. This suggests that under a good state of affairs for firms, an appropriate increase in government regulation parameters would be beneficial to their innovation accumulation. It is also important to note that the government regulation parameter needs to be controlled below $3.7737$ in this case, otherwise textile firms will not be able to present a state with a stable cycle.\t\n\\end{itemize}\n\n\\subsection{Further discussion}\nAccording to the above discussion, we can streamline the results.\t\\\\\n\\begin{itemize}\n\t\\item At $\\alpha+\\beta \\leq 0.1494$ and $n = -0.1646$, it is found that even with as much change as possible in the government regulation parameter $\\epsilon$ for the textile industry, $T\\epsilon$ cannot exceed 1.7884. Meanwhile, $x_t$ barely reaches 0.4408 and still does not reach 0.5, which means that increasing support for the textile industry at this point is very unreasonable and a waste of effective resources. This is because, with a declining textile workforce and limited development of its own, the textile industry is unable to use its resources effectively to generate the technological innovations it needs, and even greater support for the textile industry will not yield particularly effective results if the government does not manage to change the employment situation. In this case, the textile industry can only increase its $(\\alpha+\\beta)$ by investing in its own research and development in order to achieve an optimal value of $T\\epsilon$ above 0.5.\n\t\\item When $0.1494\\leq \\alpha+\\beta \\leq 0.2754$ and $-0.1646 \\leq n \\leq -0.0583$, it is found that: the reduction in the annual labour growth rate of the textile industry in such cases is not very serious, so that even if the government regulation parameter $\\epsilon$ achieves its maximum value, $T\\epsilon$ is only $2.9245 < 3 =T\\epsilon_1$, at which point there is still no bifurcation and the period of $x_t$ is still 1. Therefore, the Continued and increased support for the textile industry can still have some amount of positive effect.\n\t\\item When $0.2754 \\leq \\alpha+\\beta\\leq 0.9913$ and $-0.0583\\leq n\\leq 0.0479$, the situation becomes extremely complicated, but it is still possible to determine a reasonably appropriate parameter for the government's regulatory effort based on the values taken in different cases. It is worth noting that in such cases, the government's regulatory parameters should avoid the dangerous values of $\\epsilon_1 = 3.1659$, $\\epsilon_2 = 3.6465$, $\\epsilon_3 = 3.7465$ and $\\epsilon_{\\infty} = 3.7737$. This is then adjusted in order to find the optimal control parameter $\\epsilon$, which allows for a better accumulation of innovations in the textile industry.\n\\end{itemize}\n\n\n\n\\section{Conclusion}\nThrough the interpretation of chaos theory, the cumulative effect of technological innovation in the textile industry is explored from a non-linear perspective, and the level of innovation intensification is the level of estimation of innovation in the textile industry. It is easy to see that using the data from 2009-2018 as a base, the appropriate level of adjustment that the government should give under different circumstances can be well revealed. Specifically, this can be summarised as follows: 1). When an enterprise's workforce shows a more serious negative growth, the government should not directly support its technological innovation through a series of means such as technological subsidies, but should first start to address the situation of a declining workforce in the textile enterprise concerned. For example, the minimum wage for blue-collar workers should be increased. In order to effectively avoid a large loss of labour in the textile industry and thus maintain a good positive innovation accumulation effect in the textile industry. 2). When labour turnover in the textile industry is not obvious, the government should increase policy support for the textile industry to stimulate its potential power of technological innovation to ensure maximum technological capacity gains. 3). When the textile industry is essentially free of labour losses or even growth, the government, in judging the situation of the textile industry in order to provide appropriate regulation, should sufficiently avoid bifurcation points in order to ensure that the textile industry \"goes to the next level\" without losing its existing innovation accumulation and in order to stimulate its innovation surplus. The purpose of avoiding bifurcation points is to avoid the uncertainty of direction leading to a decrease rather than an increase in the innovation intensity of the textile industry, but care should always be taken to avoid chaos.\t\\\\\nAt the same time, there is a developable period of ongoing research in the article. The model in this paper only establishes the variable $\\epsilon$, the parameter of government regulation intensity, but does not find a very suitable quantitative criterion that can be used to measure this parameter. It is hoped that future research by scholars will further explore the quantitative indicators concerning the parameter of government regulation intensity affecting the textile industry. It is hoped that future research will further explore the quantitative indicators that affect the strength of government regulation in the textile industry and form a complete macro chaos model that will help the government to regulate the textile industry.\n\n\\section*{Statement}\n\\begin{itemize}\n\t\\item This paper was completed at the end of my undergraduate degree, so the overall idea of writing is underdeveloped.\n\t\\item This article is not sufficiently rigorous and provides only a brief analysis, which may not even be considered an analysis.\t\n\\end{itemize}\n\n\n\\bibliographystyle{unsrt}  \n\n\\section{Introduction}\nHamel and Prahalad \\cite{2005Competing} point out that the rapid changes in technology and the non-stationary nature of market demand create dynamic behavioural characteristics of the environment, and that in today's society, where technological innovations are proliferating and transformations are taking place rapidly, the cumulative effects of technological innovation in firms are more likely to take on the characteristics of a complex, irregular and non-periodic non-linear dynamical system. This non-linear characteristic seems to be chaotic, but in fact it is in order. From the perspective of non-linear theory, the evolution of a system can be summarised as shown in Figure \\ref{fig:fig1} . McBride \\cite{2010Chaos} defines chaos as a qualitative study of the unstable, acyclic behaviour of a deterministic nonlinear dynamical system, and Sardar and Abrams \\cite{2004Introducing} argue that 'order and chaos coexist, order in chaos and chaos in order. The presence of chaos in a non-linear system should be categorised and discussed. There are situations where chaos should be avoided, times when it should be controlled, and times when it should be exploited. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.7]{figure1.png}\n    \\caption{System evolution illustration.}\n    \\label{fig:fig1}\n\\end{figure}\n\nSo as of now, research on models of the cumulative effects of technological innovation in the textile industry is still very limited and far from systematic. The textile industry, as an indispensable part of the population, plays an important role in clothing, food, housing and transport, and has experienced a long period of development. The cumulative effect of technological innovation in the industry is also prominent, so the textile industry is chosen as the subject of this study. This paper discusses the following: 1. the application of a model of the cumulative effect of technological innovation in the textile industry from the perspective of chaos theory; and 2. the provision of countermeasures for technological innovation in the textile industry to appropriately control and manage chaos and increase the effective innovation rate of the industry.\n\n\n\\section{A chaotic model of the cumulative effect of technological innovation}\n\\subsection{Chaos theory}\nThe basic characteristics of chaotic motion are as follows: deterministic, non-linear, sensitive dependence on initial conditions and non-periodic. The connotation of chaos is reflected in the fact that it is a seemingly random dynamic behaviour generated by a non-linear system \\cite{R1976Simple} . A deterministic system is said to be chaotic if, in the absence of external stochastic influences, 1. the state of motion of the system is irregular and complex, similar to the Brownian motion of small molecules; 2. the system has a sensitive dependence on the initial conditions; and 3. some characteristic of the system (e.g. positive Lyapunov exponent, positive topological entropy, attractor of fractional dimension, etc.) has little to do with the choice of initial conditions.\nIn this paper, on top of the basic framework of previous studies, a chaotic economic model of the cumulative effect of technological innovation in the textile industry is developed using the worm mouth model in chaos theory as an example, with appropriately given parameters so that it can be used to describe the dynamic evolution of the textile industry.\n\n\\subsection{The textile industry is chaotic in nature}\nThe evolution of the textile industry is irregular and complex. The reason is that it is impossible to predict which textile company will launch a new garment that will be liked by the people, or to judge success or failure by its industry innovations. The system of cumulative effects of technological innovation in the textile industry is sensitive to the initial conditions - the cumulative effects of the initial innovation - and is dependent on them. Moreover, Mensh \\cite{1979Stalemate}, Houstein et al. \\cite{1982Long}, Kleinknecht \\cite{1987Are} and Silverberg et al. \\cite{1993Long} have successively conducted empirical studies on the distribution function of the temporal occurrence pattern of innovation using relevant statistical data, and the results show that the realisation of innovation is a similar Poisson distribution and has an exponential growth trend of stochastic process. Therefore, both the textile industry itself and its innovation accumulation are chaotic in nature.\n\n\\subsection{A chaotic model of the cumulative effect of innovation in the textile industry}\nLet $X_t$ be the marginal contribution of the accumulation of innovative technology in the textile industry system to the growth rate of the total economic volume of the textile industry system at a certain moment $t$, i.e. when the accumulation of innovation in the textile industry increases by $1\\%$, the total economic volume of the textile industry increases by $X_t\\%$, and call $X_t$ the accumulation effect of innovation. It is easy to see that $X_t$ is a function about time, that is, the value of $X_t$ is time-dependent. Similar to the variables used to describe the changing state of economic growth, such as capital output rate, labour productivity and capital-labour ratio, $X_t$ should also be a state variable to describe the evolution of the textile system.\nThe contribution of innovation to economic growth in the textile industry is much more than a simple linear accumulation of individual innovation contributors, and therefore $X_t$ is a more complex variable, as are variables such as the capital output rate, labour productivity and the capital-labour ratio. According to the definition of $X_t$, its variation mainly reflects fluctuations in the structure of productivity within the textile industry due to the continuous generation and accumulation of innovation. Silverberg \\cite{1993Long} developed a model of technological progress and its evolutionary chaos, which not only explains the existence of the impact of innovation, but also shows that innovation makes productivity move in an irregular cycle, i.e. chaos. This is the theoretical basis for this paper's $X_t$.    \\\\\nIntroducing the model more commonly used in chaos economics for studying chaos in economic growth, as shown in equation \\ref{equ1}.\n\n\\begin{equation}\\label{equ1}\n    X_{t+1} = X_{t}^{\\beta} \\frac{\\sigma A}{1+\\lambda} ( s-X_t )^{\\gamma}\n\\end{equation}\n\nwhere $X_t$ is the capital-labour ratio, $\\sigma (\\sigma >0)$ is the savings rate, $\\lambda (0<\\lambda <1)$ is the natural growth rate of labour, $A(A>0)$ is the technological progress factor, $\\beta (\\beta >0)$ is the elasticity of the capital-labour ratio, $\\gamma$ is a constant greater than zero, and $s$ is the maximum capital-labour rate.    \\\\\nWhen $\\beta=\\gamma=s=1$, let $\\mu=\\sigma A\/(1+\\gamma) $, then \\ref{equ1} can be transformed into \\ref{equ2}.\n\n\\begin{equation}\\label{equ2}\n    X_{t+1} = \\mu X_{t}(1-X_{t})\n\\end{equation}\n\nequation \\ref{equ2} is the general form of the insect-population model. The purpose of introducing the transformation of equation \\ref{equ1} into equation \\ref{equ2} is to construct a chaotic economic model of the cumulative effect of technological innovation by analogy with a chaotic initialization \\cite{2018Improving}. Drawing on the insect-population model, the following assumptions are introduced: 1) Technological innovation does not arise out of thin air, and existing innovations are usually the basis for subsequent innovations, similar to the relationship between parent and offspring insects in nature. 2) Technological innovation is measured by its level of innovation, and usually there is no mixture of parent and offspring, i.e., there is a certain amount of substitution of the firm's new product for the old one. 3) The technological innovation of an enterprise is limited by its own economic resources, just like the survival environment of a worm in nature. Therefore, this paper considers that technological innovation in textile enterprises meets the prerequisites of the insect-population model. According to the previous paper, the cumulative effect of technological innovation is a chaotic economic variable, which evolves in much the same way as the capital-labour ratio, with a non-linear evolution mechanism. Therefore, a chaotic economic model of the cumulative effect of technological innovation can be defined similarly - equation \\ref{equ3}.\n\n\\begin{equation}\\label{equ3}\n    X_{t+1} = T \\epsilon X_{t}(1-X_{t})\n\\end{equation}\n\nwhere $X_t \\in (0,1)$, $\\epsilon \\in (0,10)$, $T\\epsilon \\in (0,4)$, $X_t$ represents the proportion of the cumulative effect of technological innovation at time $t$, as the state variable of the textile system; $\\epsilon$ denotes the government regulation parameter; $T$ is the specific coefficient of the textile firm (i.e. the combined coefficient of the growth rate of technological inputs $\\alpha$, the proportion of technological content of output $\\beta$ and the annual growth rate of labour force $n$ at a point in time, which The relationship between them is $T=\\frac{\\alpha + \\beta}{1+n}$, where $\\alpha, \\beta, n \\in (0,1)$, $T\\epsilon$ together constitute the innovation control parameters of the textile enterprises themselves.\n\n\\subsection{Chaos of the model}\nThe chaotic state of \\ref{equ3} can be determined using the Li-Yorke theorem. Constructing the function,\n\n\\begin{equation}\\label{equ4}\n    f(x)=T\\epsilon x(1-x)\n\\end{equation}\n\nIt is not difficult to determine where $f(0)=0$ and $f(x)$ is a single-peaked function. \\\\\nFirst, determine the point $x^*$ at which $f(x)$ reaches its maximum value. $x^*$ represents the maximum technological innovation accumulation effect value of $f(x)$. $x^*$ can be obtained by solving the following first order partial derivative equation.\n\n\\begin{equation}\\label{equ5}\n    \\frac{\\partial f(x^*)}{\\partial x^*}=0\n\\end{equation}\n\nThe mapping point $x^*$ corresponding to the value of the maximum reachable function is obtained,\n\n\\begin{equation}\\label{equ6}\n    x^*=1\/2\n\\end{equation}\n\nThe maximum innovation accumulation effect value $X_{max}$ is,\n\n\\begin{equation}\\label{equ7}\n    x_{max}=f(x^*)=\\frac{T\\epsilon}{2}\n\\end{equation}\n\nFrom the above definition, it follows that the value of the maximum innovation accumulation effect should satisfy,\n\n\\begin{equation}\\label{equ8}\n    x_{max} \\le s\n\\end{equation}\n\nNext, determine the initial image point $x_i$ , and from equations \\ref{equ4} and \\ref{equ6}, we have,\n\n\\begin{equation}\\label{equ9}\n    T\\epsilon x(1-x)=1\/2\n\\end{equation}\n\nThe left end of equation \\ref{equ9} is equation \\ref{equ4}, a single-peaked function, so the smaller root is taken to be $x_i$, \n\n\\begin{equation}\\label{equ10}\n    x_i = \\frac{T\\epsilon - \\sqrt{(T\\epsilon)^2-T\\epsilon}}{2T\\epsilon}\n\\end{equation}\n\nFinally, determine its third-order mapping $f(x_{max})$,\n\n\\begin{equation}\\label{equ11}\n    f(x_{max}) = \\frac{T^2\\epsilon ^2}{2}(1-T\\epsilon \/2)\n\\end{equation}\n\nprovided that the first-, second- and third-order mappings $x^*$,$x_{max}$, $f(x_{max})$ generated by the initial preimage $x_i$ are mapped from the interval $[0,s]$ to its own interval $[0,s]$ and satisfy the sufficient conditions for chaos in the Li-Yorke theorem, i.e.,\n\n\\begin{equation}\\label{equ12}\n    0 \\le f(x_{max}) \\le x_i \\le x^* \\le x_{max}\n\\end{equation}\n\nThen the textile industry system \\ref{equ3} appears chaotic, as shown in Figure \\ref{fig:fig2}, which is a chaotic phase diagram regarding the proportion of the cumulative effect of two adjacent generations of innovation in the textile industry innovation system. Where the curve is the image of equation (3) and the straight line is $X_{t+1} = X_t$, which represents the process of converting a preimage point to the next preimage point.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.7]{figure2.png}\n    \\caption{Chaotic phase diagrams in textile industry.}\n    \\label{fig:fig2}\n\\end{figure}\n\n\n\\section{Chaotic nature of the model}\n\\subsection{Local structural stability and cyclic bifurcation propertiest}\nFor model \\ref{equ3}, specifically, when $T\\epsilon$ has a determined value, if the corresponding $x_t$ also has only one value (indicating that the firm's innovation reaches a uniquely determined proportion, i.e. the level of intensification). That is, the technological innovation effect of a textile firm reaches a uniquely determined level. Then the period of $x_t$ is said to be $1$; if at this point $x_t$ has two values corresponding to it, then the period of $x_t$ is said to be $2$; if at this point $x_t$ has n values corresponding to it, then the period of $x_t$ is said to be $n$. In particular, the structure of $x_t$ is unstable when there is a bifurcation and the steady state can only be maintained when the period of $x_t$ is determined. This also means that a stable accumulation of technological innovation in textile companies can only be guaranteed when $T\\epsilon$ takes on a value within a particular interval. In other words, a good accumulation of technological innovation in the textile industry can be maintained to a certain extent when the role of policy innovation and industry-specific coefficients are well controlled. A diagram of the iterative process for varying values of the control parameter $T\\epsilon$ from $0$ to large is shown in Figure \\ref{fig:fig3}.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.7]{figure3.png}\n    \\caption{Bifurcations.}\n    \\label{fig:fig3}\n\\end{figure}\n\n\\subsection{Effect of parameter values on the initial innovation ratio in the textile industry}\nBased on its definition of firm intensification in a chaotic economic model, the effect of the $T\\epsilon$ parameter on the initial innovation ratio of textile firms is derived by analogy.  \n\\begin{itemize}\n\t\\item When $0<T\\epsilon \\le 1$, there exists a stable immovable point $x_t=0$, indicating that there is no set. That is, there does not exist a corresponding non-zero $x_t$ taking value.   \n\t\\item When $1<T\\epsilon \\le 3$, $x_t$ has a locally stable structure with period $1$. If $T\\epsilon$ takes any value in $(1, 3)$, $x_t$ takes the corresponding one between $(0, 0.664)$. This indicates that the innovation set $x_t$ is stable at that value, i.e. a $T\\epsilon$ leads to only one innovation set $x_t$.      \n\t\\item When $T\\epsilon = 3$, the first bifurcation value $T\\epsilon_1$ occurs and a forked bifurcation occurs. This will double the $x_t$ cycle. At this point the structure of $x_t$ goes from stable to unstable.    \n\t\\item When $3<T\\epsilon \\le 3.4495$, a new locally stable structure of $x_t$ emerges, but at this point the period becomes $2$, i.e. a $T\\epsilon$ parameter value will lead to two innovation sets $x_{t1}$ and $x_{t2}$. Over time, their innovation set bounces around $x_{t1}$ and $x_{t2}$, thus constituting two stable sequences of immobile point values that range from $0.664~0.85$ and $0.664~0.44$, respectively, as $T\\epsilon$ varies. \n\t\\item When $T\\epsilon = 0.4495$, a second bifurcation value $T\\epsilon_2$ occurs and the Hough bifurcation occurs. $x_t$ has an unstable structure and thereafter the $x_t$ period multiplies to $4$.  \n\t\\item When $3.4495<T\\epsilon \\le 3.5441$, the structure of $x_t$ changes from unstable to stable, and the period of $x_t$ is $4$. This indicates that one $T\\epsilon$ parameter value will lead to four innovation sets, at which time $x_t$ has four sets of stable sequences of immobile point values, which range from 0.85-0.883, 0.85-0.82, 0.44-0.52 and 0.44-0.3367, respectively, with different values of $T\\epsilon$.    \n\t\\item When $T\\epsilon=3.5441$ , a third bifurcation value $T\\epsilon_3$ occurs, and the structure of $x_t$ is unstable, after which the period multiplies to $8$. \n\t\\item When $T\\epsilon >3.5441$, the local structure of $x_t$ is stable and the period of $x_t$ is $8$. The rest of the process continues until $T\\epsilon=3.5699=T\\epsilon_{\\infty}$, when the period of $x_t$ is infinite, i.e. chaos emerges. In other words, there is no relationship between the firm's innovation accumulation and the control parameters at this point.\n\\end{itemize}\n\n\\subsection{Sexual analysis of different parameter values}\nFirstly, the values of $\\alpha$, $\\beta$, and $n$ are taken and defined as shown in Table \\ref{tab1}, and all data are obtained from the findings of the \"China Statistical Yearbook\".\n\n\\begin{table}[H]\n    \\caption{\\label{tab1} Variable definition and value}\n    \\centering\n    \\resizebox{\\linewidth}{32pt}{\n    \\begin{tabular}{ccc}\n        \\hline\n        Symbol & Description & Fetching method \\\\\n        \\hline\n        $\\alpha$ & Growth rate of investment in science and technology & Log difference of R\\&D inputs \\\\\n        $\\beta$  & Proportion of technological content of output & Log difference in the number of active patents in the textile industry \\\\\n        $n$     & Annual growth rate of the labour force & Log difference of the average number of workers employed \\\\\n        $\\epsilon$ & Government regulation parameters & By taking the product of $T\\epsilon$ backwards or assuming that given \\\\\n        \\hline\n    \\end{tabular}}\n\\end{table}\n\nFrom an econometric perspective, log-differencing helps to enhance the significance of the main regression to some extent. As a corollary, the significance of log-differencing the variables in this paper is to smooth out the effects of unreasonable outliers and reduce the \"misleading\" effect of episodic events on the overall judgment. In addition, as the average number of workers before 14 years could not be found directly, the calculation of business income\/per capita business income was used, and the use of logarithmic difference to calculate the growth rate can also reduce the subtle influence of different quantiles on the conclusion. The log-difference results are shown in Table \\ref{tab2} \\footnote{All data are obtained from the China Statistical Yearbook, and are rounded to four decimal places after calculation by stata. The values of $\\alpha$ range from 1.46\\% to 47.44\\%, $\\beta$ from 9.58\\% to 51.69\\% and $n$ from -16.46\\% to 4.79\\%, so the values of $\\alpha+\\beta$ and $1+n$ are inferred from this.}.\n\n\\begin{table}[H]\n\t\\caption{\\label{tab2} Log-differential results}\n\t\\centering\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\hline\n\t\t\tYear & $\\alpha$ & $\\beta$ & $n$ & $\\alpha+\\beta$ & $1+n$ \\\\\n\t\t\t\\hline\n\t\t\t2009&\/&\/&\/&\/&\/\t\\\\\n\t\t\t2010&.2017&.5088&.0479&.7105&1.0479\\\\\n\t\t\t2011&.4744&.1671&-.0947&.6416&.9053\\\\\n\t\t\t2012&.0146&.1347&-.0566&.1494&.9434\\\\\n\t\t\t2013&.1382&.1418&-.0614&.2800&.9386\\\\\n\t\t\t2014&.1144&.5169&-.0653&.6313&.9347\\\\\n\t\t\t2015&.1558&.0958&-0.540&.2516&.9460\\\\\n\t\t\t2016&.0574&.2867&-.0627&.3441&.9373\\\\\n\t\t\t2017&.0584&.2688&-.1090&.3273&.8910\\\\\n\t\t\t2018&.0912&.2629&-.1646&.3541&.8354\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\\end{table}\n\nTable \\ref{tab2} shows that the values of $\\alpha+\\beta$ range from 14.94\\% to 71.05\\%, and the values of $1+n$ range from 83.54\\% to 104.79\\%. Considering that the value of the parameter $T$ in equation \\ref{equ3} is proportional to $(\\alpha+\\beta)$ and inversely proportional to $(1+n)$, and that $T$ is related to the innovation accumulation $x_t$ of the textile enterprises in equation \\ref{equ3}, the sexual attitude of $x_t$ can be analysed through this. To facilitate the calculation, the three fixed cases of the lower, middle and upper limits of these elements are taken separately, and only $\\epsilon$ is adjusted to analyse the nature of $x_t$.\n\n\\begin{itemize}\n\t\\item Lower bound case. $\\alpha+\\beta=0.1494$ , $1+n=0.8354$ . When $\\epsilon=5.5917$, $T\\epsilon=1.0000$ . From equation \\ref{equ3} and Figure \\ref{fig:fig3}, $x_t=0$. when $\\epsilon=10$, $T\\epsilon=1.7884$ and $x_t=0.4408$ . This means that at $(\\alpha+\\beta=0.1494)$, the degree of innovation accumulation achieved by textile firms remains zero when the parameter of government regulation of textile firms is 5.5917. When adjusted to the maximum value $\\epsilon=10$, $T\\epsilon=1.7884$ (satisfying $1<T\\epsilon<3$), at which point the period of $x_t$ is $1$, the innovation intensification of textile enterprises is also only 44.08\\%. At the same time, considering the nature of the enterprise cycle of China's existing textile enterprises and the characteristics of the industry, it is easy to see that the government's R\\&D subsidies for the traditional textile industry are not significant enough. Therefore, it is difficult for the government's regulation parameter $\\epsilon$ for the textile industry to reach a maximum value of $10$. That is, in the actual process, the innovation intensity of textile enterprises will be much less than 44.08\\%.\n\t\\item Middle bound case. This is calculated for the case $\\alpha+\\beta<0.2754$ , $1+n=0.9417$. When the government regulation parameter $\\epsilon=10$, $T\\epsilon=2.9245 (T\\epsilon<3)$, it can be inferred that in this case, whatever the value of $\\epsilon$ in the above range, the bifurcation does not occur and the structure is stable. Therefore, in this case, the government can increase the regulation of the innovation intensity of the textile enterprises, and can increase the policies and subsidies for them.\n\t\\item Upper bound case. $\\alpha+\\beta=0.9913$ , $1+n=1.0479$ . When $\\epsilon=1.0571$, $T\\epsilon=1.0000$ , the innovation set is $0$; when $\\epsilon<3.1659$, $T\\epsilon<3=T\\epsilon_1$ , the period of $x_t$ is $1$ and $x_t<0.3339$ , i.e. the innovation set is less than 33.39\\%; when $3.1659<\\epsilon<3.6465$, $3<T\\epsilon<3.4495=T\\epsilon_2$ ,the period of $x_t$ is 2; when When $3.6465<\\epsilon<3.7465$, $3.4495<T\\epsilon<3.5441=T\\epsilon_3$ and the period of $x_t$ is 4. From the above, it can be seen that when $\\epsilon_1=3.1659$, $\\epsilon_2=3.6465$ and $\\epsilon_3=3.7465$, there will be three bifurcations of $T\\epsilon$, $T\\epsilon_1$, $T\\epsilon_2$ and $T\\epsilon_3$. This means that $\\epsilon_1$, $\\epsilon_2$ and $\\epsilon_3$ are dangerous values of strength that lead to a non-stationary state in the textile innovation accumulation model. As $\\epsilon$ increases further, $T\\epsilon$ also becomes larger until $\\epsilon = 3.7737 = \\epsilon_{\\infty}$ when $T\\epsilon = 3.5699 = T\\epsilon_{\\infty}$ and the period of $x_t$ is infinite, i.e. chaos will emerge. Clearly, $\\epsilon=3.7737$ is the more dangerous regulatory parameter. At the upper value, $x_t$ takes values of 0.5270-0.6847 when the regulation parameter $\\epsilon$ is kept between 2.1142-3.1713, which would make $T\\epsilon$ take values between 2-3. This suggests that under a good state of affairs for firms, an appropriate increase in government regulation parameters would be beneficial to their innovation accumulation. It is also important to note that the government regulation parameter needs to be controlled below $3.7737$ in this case, otherwise textile firms will not be able to present a state with a stable cycle.\t\n\\end{itemize}\n\n\\subsection{Further discussion}\nAccording to the above discussion, we can streamline the results.\t\\\\\n\\begin{itemize}\n\t\\item At $\\alpha+\\beta \\leq 0.1494$ and $n = -0.1646$, it is found that even with as much change as possible in the government regulation parameter $\\epsilon$ for the textile industry, $T\\epsilon$ cannot exceed 1.7884. Meanwhile, $x_t$ barely reaches 0.4408 and still does not reach 0.5, which means that increasing support for the textile industry at this point is very unreasonable and a waste of effective resources. This is because, with a declining textile workforce and limited development of its own, the textile industry is unable to use its resources effectively to generate the technological innovations it needs, and even greater support for the textile industry will not yield particularly effective results if the government does not manage to change the employment situation. In this case, the textile industry can only increase its $(\\alpha+\\beta)$ by investing in its own research and development in order to achieve an optimal value of $T\\epsilon$ above 0.5.\n\t\\item When $0.1494\\leq \\alpha+\\beta \\leq 0.2754$ and $-0.1646 \\leq n \\leq -0.0583$, it is found that: the reduction in the annual labour growth rate of the textile industry in such cases is not very serious, so that even if the government regulation parameter $\\epsilon$ achieves its maximum value, $T\\epsilon$ is only $2.9245 < 3 =T\\epsilon_1$, at which point there is still no bifurcation and the period of $x_t$ is still 1. Therefore, the Continued and increased support for the textile industry can still have some amount of positive effect.\n\t\\item When $0.2754 \\leq \\alpha+\\beta\\leq 0.9913$ and $-0.0583\\leq n\\leq 0.0479$, the situation becomes extremely complicated, but it is still possible to determine a reasonably appropriate parameter for the government's regulatory effort based on the values taken in different cases. It is worth noting that in such cases, the government's regulatory parameters should avoid the dangerous values of $\\epsilon_1 = 3.1659$, $\\epsilon_2 = 3.6465$, $\\epsilon_3 = 3.7465$ and $\\epsilon_{\\infty} = 3.7737$. This is then adjusted in order to find the optimal control parameter $\\epsilon$, which allows for a better accumulation of innovations in the textile industry.\n\\end{itemize}\n\n\n\n\\section{Conclusion}\nThrough the interpretation of chaos theory, the cumulative effect of technological innovation in the textile industry is explored from a non-linear perspective, and the level of innovation intensification is the level of estimation of innovation in the textile industry. It is easy to see that using the data from 2009-2018 as a base, the appropriate level of adjustment that the government should give under different circumstances can be well revealed. Specifically, this can be summarised as follows: 1). When an enterprise's workforce shows a more serious negative growth, the government should not directly support its technological innovation through a series of means such as technological subsidies, but should first start to address the situation of a declining workforce in the textile enterprise concerned. For example, the minimum wage for blue-collar workers should be increased. In order to effectively avoid a large loss of labour in the textile industry and thus maintain a good positive innovation accumulation effect in the textile industry. 2). When labour turnover in the textile industry is not obvious, the government should increase policy support for the textile industry to stimulate its potential power of technological innovation to ensure maximum technological capacity gains. 3). When the textile industry is essentially free of labour losses or even growth, the government, in judging the situation of the textile industry in order to provide appropriate regulation, should sufficiently avoid bifurcation points in order to ensure that the textile industry \"goes to the next level\" without losing its existing innovation accumulation and in order to stimulate its innovation surplus. The purpose of avoiding bifurcation points is to avoid the uncertainty of direction leading to a decrease rather than an increase in the innovation intensity of the textile industry, but care should always be taken to avoid chaos.\t\\\\\nAt the same time, there is a developable period of ongoing research in the article. The model in this paper only establishes the variable $\\epsilon$, the parameter of government regulation intensity, but does not find a very suitable quantitative criterion that can be used to measure this parameter. It is hoped that future research by scholars will further explore the quantitative indicators concerning the parameter of government regulation intensity affecting the textile industry. It is hoped that future research will further explore the quantitative indicators that affect the strength of government regulation in the textile industry and form a complete macro chaos model that will help the government to regulate the textile industry.\n\n\\section*{Statement}\n\\begin{itemize}\n\t\\item This paper was completed at the end of my undergraduate degree, so the overall idea of writing is underdeveloped.\n\t\\item This article is not sufficiently rigorous and provides only a brief analysis, which may not even be considered an analysis.\t\n\\end{itemize}\n\n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince Kaluza and Klein proposed the idea that our fundamental spacetime may be more than four dimensions in 1920s \\cite{Kaluza1921,Klein1926}, extra dimension theories have received much attention from theoretical physicists. In Kaluza-Klein (KK) theory, the extra dimension is compactified as a circle with the radius of Planck length $l_{Pl} \\sim 10^{-35}$m, and this strategy of compactifying extra dimensions dominates higher-dimensional unified physics \\cite{Overduin1997}. Inspired by the string theory, the braneworld scenario provides another way to hide the extra dimensions.  In this scenario, our universe is a brane embedded in a higher-dimensional bulk. All the Standard Model particles are trapped on the brane and only the graviton can propagate into the extra dimensions. \n\nIn 1999,  Randall and Sundrum proposed a well-known braneworld model \\cite{Randall1999}, which is now dubbed RS1 model. In this model, the bulk is a slice of a five-dimensional anti-de Sitter (AdS) spacetime, and there is a brane at each boundary of the slice. In order to maintain a flat four-dimensional spacetime on the brane, the fine-tuning condition requires the tension of the ultraviolet (UV) brane to be positive while the tension of the infrared (IR) brane to be negative. Our universe lives on the IR brane, so owing to the exponentially warped extra dimension, a fundamental electroweak scale of the order of Plank scale ($M_{Pl}\\sim 10^{16}$TeV) is redshifted to the observed effective electroweak scale ($\\sim$246GeV). The RS1 model provides a natural way to solve the gauge hierarchy problem without introducing a new large hierarchy, so it draws lots of attention in the last two decades, see, e.g., Refs.~\\cite{Rubakov2001,Csaki2004,Maartens2010,Gherghetta2010} for the introduction. \n\nThe RS1 model is based on the 5D general relativity (GR), where the 5D graviton is massless and possesses 5 degrees of freedom (DOF), therefore, there is a massless 4D graviton, a massless scalar radion, and a tower of massive 4D graviton in the effective 4D theory on the brane \\cite{Randall1999}. In the braneworld scenario, the effective 4D theory on the brane is related to the DOF of the high-dimensional gravitational theory. So it is natural to ask what is the difference between the 4D effective theories on the brane if the higher-dimensional graviton possesses different numbers of DOF. One simple exploration is to replace GR with the massive gravity theory, where the higher-dimensional massive graviton possesses more numbers of DOF. \n\nMassive gravity is a generalization of GR by endowing the graviton with a nonzero mass. The first pioneering work was done by Fierz and Pauli in 1939 by adding non-derivative interaction terms, called Fierz-Pauli mass term, into the linearized level of GR \\cite{Fierz1939}. However, the theory does not reduce to GR in the massless limit, so it suffers from a discontinuity problem, known as van Dam-Veltman-Zakharov discontinuity \\cite{Dam1970,Zakharov1970,VanNieuwenhuizen1973}. This problem can be alleviated by generalizing the linear Fierz-Pauli mass term to nonlinear terms \\cite{Vainshtein1972}. However, the nonlinear terms will generate a higher derivative term and lead to the Boulware-Deser ghost instability \\cite{Boulware1972}. A breakthrough was achieved by de Rham, Gabadadze, and Tolley about a decade ago \\cite{Rham2010a,Rham2011}. They constructed a nonlinear massive gravity theory with the field equation at most second order in time derivatives by getting rid of ghosts for all nonlinear self-interactions of the longitudinal component in the decoupling limit \\cite{Hassan2012,Hassan2012a}. This massive gravity theory is now dubbed dRGT gravity. However, it is found that the dRGT gravity is suffering from other pathologies, such as the acausality problem \\cite{Deser2013,Deser2013a} and the lack of stable and realistic Friedmann-Lema\\text{\\^ \\i}tre-Robertson-Walker cosmological solutions \\cite{DAmico2011,Gumrukcuoglu2011,Gumrukcuoglu2012,DeFelice2012a,DeFelice2013}. \n\nOn the other hand, although the Lorentz invariance is widely regarded as a fundamental symmetry of nature, there are still some hints that the Lorentz invariance may be broken at high energies in quantum gravity \\cite{Kostelecky1989,AmelinoCamelia2003}.  Especially, the aforementioned standard problems of Lorentz-invariant massive gravity theories may be avoided by switching to the Lorentz-violating massive gravity theory, see Refs.~\\cite{Rubakov2004,Dubovsky2004,Dubovsky2005,Blas2009,Lin2013} for examples.  A simple example of Lorentz-violating massive gravities was proposed by Lin in Ref.~\\cite{Lin2014}, where the theory is built with the Einstein-Hilbert action plus 3 canonical massless scalar fields. The spatial condensation of the scalar fields breaks spatial diffeomorphisms, which gives rise to 3 Goldstone excitations. After ``eating\" the 3 Goldstone bosons in the unitary gauge, the graviton gets weight and possesses 5 well-behaved DOF. It was found that the theory could remove the infrared divergence of the inflationary loop diagram owing to the graviton mass \\cite{Lin2014}.  For a recent review on massive gravity theories see Refs.~\\cite{Hinterbichler2012,Rham2014,Rham2017} and the references therein.\n\nThe higher-dimensional scenario has a close relation with massive gravities, since the 4D graviton spectrum  naturally includes a tower of non-pathologic massive spin-2 gravitons. Nevertheless, they are usually not a purely massive gravity theories due to the presence of a massless graviton associated with the unbroken 4D diffeomorphisms. A more interesting theory is the Dvali-Gabadadze-Porrati model, whose graviton looks like a resonance and effectively acquires a soft mass \\cite{Dvali2000,Dvali2000a}. In Ref.~\\cite{Chacko2004}, the authors considered the RS1 model with a localized Fierz-Pauli mass term for the graviton on the IR brane, which reproduces GR for observers localized on the UV brane at distances smaller than the IR scale but realizes a massive gravity at longer distance with a ghost radion. The embedding of a 4D dRGT massive gravity into a 5D dRGT massive gravity by resorting to the single-brane RS2 model was studied by Gabadadze in Ref.~\\cite{Gabadadze2017}, where the value of strong scale $\\Lambda_3$ can be significantly increased. More relevant works see Refs.~\\cite{Gabadadze2004,Kakushadze2014,Gabadadze2018,Kaloper2019} for examples. \n\nIn this work, we are interested in generating the massive gravity with spatial condensation into a 5D spacetime and studying the braneworld scenario in the massive gravity. The background spacetime of the spatial condensation scenario is invariant under the translation and 3D rotations. In order to preserve the Poincar\\'e invariance on the flat brane, we require that the background spacetime is invariant under the 4D Poincar\\'e transformation in the higher dimensional generalization. The condensation of 4 background scalars spontaneously breaks the 4D diffeomorphisms, and generates 4 Goldstone excitations associated with the broken symmetries. Consequently, by ``eating\" the 4 Goldstone excitations in the unitary gauge, the 5D massless spin-2 graviton with 5 DOF becomes massive and possesses 9 DOF on the spectrum. \n\n The layout of the paper is as follows: In Sec.~\\ref{Model}, the toy model is built in the 5D massive theory. In Sec.~\\ref{Perturbation}, the full linear perturbations have been considered. In Sec.~\\ref{Hierarchy}, the mass spectra of KK particles have been calculated and solving the gauge hierarchy problem has been discussed as an application of the model. In Sec.~\\ref{Radius_stabilization}, the Goldberger-Wise (GW) mechanism has been used to stabilize the radius of the extra dimension. Finally, brief conclusions are presented. Throughout the paper, the small Latin letters $a, b, \\cdots=0,1,2,3,$ label the group indices of the internal metric of scalar fields, and the capital Latin letters $A, B, \\cdots=0,1,2,3,5,$ and Greek letters $\\mu,\\nu, \\cdots=0,1,2,3,$ label the 5D and 4D spacetime indices, respectively. \n\n\n\n\\section{Model building}\\label{Model}\n\nWe start from a 5D extension of the Lorentz-violating massive gravity \\cite{Lin2014}, and the action is given by a 5D Einstein-Hilbert term plus 4 canonical massless scalar fields, i.e.,\n\\beq\nS=M^3\\int{d^5x}\\sqrt{-g}\\lt[\\fc{R}{2}-\\fc{1}{2}m^2g^{MN}\\pt_M\\phi^a\\pt_N\\phi^b\\eta_{ab} -\\Lambda\\rt]-\\int d^4 x \\sqrt{-g_{\\text{I}}} V_{\\text{I}}-\\int d^4 x \\sqrt{-g_{\\text{II}}} V_{\\text{II}},\n\\label{Main_Action}\n\\eeq\nwhere $M$ is the 5D fundamental gravity scale, $\\Lambda$ is the 5D cosmological constant, $V_{I}$ and $V_{II}$ represent the brane tensions, $m$ is a parameter proportional to the mass of 5D graviton, and $\\eta_{ab}$ is the internal metric of scalar fields to maintain the 4D Poincar\\'e symmetry. The background solution spontaneously breaks the 5D Lorentz invariance when the scalar fields are fixed to their vacuum values \n\\beq\n\\langle\\phi^a\\rangle=\\delta_\\mu^a x^\\mu, \\label{Scalar_vacuum}\n\\eeq \nwith $x^\\mu$ the coordinates of 4D spacetime. Thus, the condensation spontaneously generates a preferred 4D frame.\n\n\nIn order to investigate a RS1 model-like braneworld solution, the metric ansatz  preserving the 4D Poincar\\'e invariance reads\n\\beq\nds^{2}_{5}=a^2(y){\\eta}_{\\mu\\nu}dx^{\\mu}dx^{\\nu}+dy^2,\n\\label{Brane_Metric}\n\\eeq\nwhere $a(y)$ is the warp factor and $y \\in [-y_\\pi, y_\\pi]$ denotes an $S^1\/Z_2$ orbifold extra dimension. The UV brane with tension $V_\\text{I}$ locates at $y=0$ and the IR brane with tension $V_{\\text{II}}$ locates at $y=y_\\pi$.\n\nThe field equations are obtained by varying the action (\\ref{Main_Action}) with respect to the metric $g^{MN}$, \n\\beqn\nG_{MN}&=&m^2 \\Big(\\partial_M \\phi ^a \\partial_N  \\phi ^a-\\frac{1}{2}  g_{MN} \\partial^K \\phi ^a \\partial_K \\phi ^a\\Big)-\\Lambda  g_{MN}\\nn\\\\\n&&-\\frac{\\sqrt{-g_{\\text{I}}} }{\\sqrt{-g}}\\fc{V_{\\text{I}}}{M^3 }g_{\\text{I\t\t\t$\\mu \\nu $}} \\delta ^{\\mu }_M \\delta ^{\\nu }_N \\delta  \\left(y\\right)-\\frac{\\sqrt{-g_{\\text{II}}} }{ \\sqrt{-g}}\\fc{V_{\\text{II}}}{M^3}g_{\\text{II}\\mu \\nu } \\delta ^{\\mu }_M \\delta ^{\\nu }_N \\delta  \\left(y-y_\\pi\\right).\n\\eeqn\nFurther, with the ansatzes of braneworld metric and scalar field condensation, the field equations are written explicitly as\n\\beqn\n3\\lt(H'+2H^2 \\rt)&=&-\\fc{m^2}{a^2}-\\Lambda-\\fc{1}{M^3}\\left[  V_\\text{I} \\delta\\left(y\\right)+ V_{\\text{II}} \\delta\\left(y-y_{\\text{b}}\\right)\\right],\\label{EoMII_1}\\\\\n6H^2&=&-\\fc{2m^2}{a^2}-\\Lambda, \\label{EoMII_2}\n\\eeqn\nwhere $H\\equiv a'\/a$ and the prime denotes the derivative with respect to the extra dimension coordinate $y$. \n\nThe field equations simply give the following solution\n\\beq\na(y)=e^{-k|y|}+\\epsilon^2e^{k|y|}, \n\\label{Warp_factor}\n\\eeq\nwhere $\\epsilon\\equiv\\fc{m}{2\\sqrt{3}k}$, $k^2\\equiv-\\Lambda\/6$, and $|y|$ represents the absolute value of $y$ in order to be consistent with the $Z_2$ symmetry. Since the minimum value of warp factor is $2\\epsilon$, the 5D graviton has to be light enough, i.e., $\\epsilon<a(y_\\pi)\\sim10^{-16}$, to generate enough redshift for the fundamental electroweak scale.  As illustrated in Fig.~\\ref{Fig_Warp_factor},  the warp factor is different from that of RS model only far from the center for a tiny graviton mass, $m\/k\\ll1$.\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=9cm,height=6cm]{Warpfactor.eps}\n\\end{center}\n\\caption{The profiles of the warp factor $a(y)$ with respect to different mass parameters.}\n\\label{Fig_Warp_factor}\n\\end{figure}\n\nAfter matching the delta functions in \\eqref{EoMII_1}, we have the fine-tuning conditions,\n\\beqn\nV_I&=&6 k M^3 \\left[1-\\frac{2 \\epsilon^2}{1+\\epsilon^2}\\right],\\\\\nV_{II}&=&6 k M^3 \\left[1-\\frac{2}{1+\\epsilon^2 e^{2 k y_\\pi}}\\right].\n\\eeqn\n\nIn the case of $e^{-k y_\\pi}>\\epsilon$, the exponential decay term of  warp factor \\eqref{Warp_factor} is dominant in the whole bulk. It is clear that $V_I>0$ and $V_{II}<0$ in this case, namely, there is a positive tension brane located at the origin and a negative tension brane at $y_\\pi$. This is the model we will focus on in the rest of the paper. Especially, in the limit $\\epsilon e^{k y_\\pi}\\ll  1$, the warp factor \\eqref{Warp_factor} approaches the exponential decay form of RS model. Correspondingly, the fine-turning conditions reduce to the case of RS model as well,\n\\beq\nV_I\\approx-V_{II}\\approx6 k M^3, ~~~ \\Lambda=-6k^2.\n\\eeq\n\nBesides, there is another interesting case that both brane tensions are positive in case of  $e^{-k|y_\\pi|}<\\epsilon<1$. It would generate a new brane configuration distinct from the RS1 model. Especially, the scenario that our universe is confined on the positive tension brane may potentially provide a possible model with better properties \\cite{Csaki1999,Shiromizu2000,Yang2012a}. This model will be studied in detail in our another work.\n\n\t\n\\section{Linear perturbations}\\label{Perturbation}\n\nIn order to investigate the stability of the model, we consider the full linear perturbations against the background. The perturbed metric is written as\n\\beq\nds^2=\\lt(g_{MN}+h_{MN}\\rt)dx^Mdx^N,\n\\label{Metric_NC}\n\\eeq\nwhere $h_{MN}$ represents the perturbations against the background metric $g_{MN}$ given in \\eqref{Brane_Metric}. Due to the 4D Lorentz symmetry of our background spacetime, it is convenient to decompose the perturbed metric into the scalar, transverse vector and transverse-traceless tensor modes, and to rewrite it by\n\\beqn\nh_{55}&=&-2\\xi,\\\\\nh_{\\mu 5}&=&-a \\lt(S_{\\mu}+\\pt_\\mu \\beta \\rt),\\\\\nh_{\\mu\\nu}&=&a^2\\lt[ D_{\\mu\\nu}+2\\eta_{\\mu\\nu}\\psi+\\fc{1}{2}\\lt(\\pt_{\\mu}F_{\\nu}+\\pt_{\\nu}F_{\\mu} \\rt)+2\\pt_\\mu\\pt_\\nu E \\rt],\n\\eeqn\nwhere the vector modes satisfy the transverse condition $\\pt^\\mu S_{\\mu}=\\pt^\\mu F_{\\mu}=0$, and the tensor mode satisfies the transverse-traceless (TT) condition $\\pt^\\mu D_{\\mu\\nu}=0$. \n\nCorrespondingly, the perturbed scalar fields are \n\\beq\n\\phi^a=x^a+\\pi^a,\n\\eeq \nwhere $\\pi^a=\\delta^a_\\mu\\pi^\\mu$ is the Goldstone excitation of the condensation. In order to maintain the scalar condensation \\eqref{Scalar_vacuum} under the general coordinate transformation $x^M \\to x^M+\\epsilon^M$, the Goldstone excitation has to transform opposite to the 4D coordinates simultaneously, i.e., $\n\\pi^\\mu\\to\\pi^\\mu-\\epsilon^\\mu$.  This St\\\"uckelberg trick non-linearly restores the general covariance of the theory. Therefore, it behaves like a vector field and can be decomposed as $\\pi^\\mu=\\eta^{\\mu\\nu}\\lt(\\pt_\\nu \\varphi+A_\\nu \\rt)$, with $\\varphi$ a scalar field and $A_\\mu$ a transverse vector field.\n\nTherefore, there are 1 TT tensor, 3 transverse vector and 5 scalar modes in total. By decomposing $\\epsilon_\\mu =a^2(\\epsilon^{V}_{\\mu}+\\pt_\\mu \\epsilon^s)$ with $\\epsilon^{V}_{\\mu}$ a transverse vector and $\\epsilon^s$ a scalar, these 9 perturbed modes transform as follows under the general coordinate transformation,\n\\beqn\nD_{\\mu\\nu} &\\to& D_{\\mu\\nu}, \\quad S_{\\mu } \\to S_{\\mu } + a  \\epsilon^{V}_{\\mu }{}' ,\\quad F_{\\mu } \\to  F_{\\mu }-2 \\epsilon^{V}_{\\mu } ,\\nn\\\\\nA_{\\mu } &\\to&  A_{\\mu }-{\\epsilon }^{V}_{\\mu }, \\quad \\psi \\to \\psi -H\\epsilon _5,  \\quad E \\to E-\\epsilon^s, \\nn\\\\\n\\beta &\\to& \\beta+a {\\epsilon }^s{}'+\\frac{\\epsilon _5}{a}, \\quad  \\xi \\to \\xi +\\epsilon '_5, \\quad \\varphi \\to \\varphi -{\\epsilon }^s.\n\\label{Gauge_transformation}\n\\eeqn\n\nNaively, we have 5 gauge freedoms to eliminate one vector and two scalar modes by fixing ${\\epsilon }^{V}_\\mu$, ${\\epsilon}^s$ and $\\epsilon _5$. A commonly used gauge choice is the so-called unitary gauge, in which the Goldstone excitations $\\pi^a$ of scalar fields are closed, i.e., $A_\\mu=\\varphi=0$. Moreover, we have another gauge freedom $\\epsilon _5$ to set $\\psi=0$. Then from the fact that $Z_\\mu-\\pi_\\mu$, with $Z_\\mu\\equiv a^2\\lt(F_\\mu\/2+\\pt_\\mu E \\rt)$, is a gauge invariant quantity, one observes that the 4 Goldstone excitations  $\\pi^\\mu$ are ``eaten\" by the 5D graviton in the unitary gauge. Consequently, the 5D massless spin-2 graviton with 5 DOF gets weight and becomes massive, with 9 DOF on the spectrum. After gauge fixing, we have  1 TT tensor, 2 transverse vector and 3 scalar modes left. However, not all modes among them are physical, since some of them can be eliminated by some constraint equations, which can be easily extracted by utilizing the Arnowitt-Deser-Misner (ADM) formalism. \n\nThe RS metric \\eqref{Brane_Metric} in ADM formalism reads \n\\beq\nds^2=N^2dy^2+\\gamma _{\\mu \\nu } \\left(dx^{\\mu }+N^{\\mu }dy\\right) \\left(dx^{\\nu }+N^{\\nu }dy \\right),\n\\eeq\nwith $N=1$, $N^\\mu=0$ and $\\gamma_{\\mu\\nu}=a^2(y)\\eta_{\\mu\\nu}$. Correspondingly, the bulk action of \\eqref{Main_Action} is rewritten as \n\\beqn\nS&=&\\frac{M^3}{2}\\int d^5x\\sqrt{-\\gamma }\\Big[N \\lt(R^{(4)}-2 \\Lambda -m^2 \\gamma ^{\\mu \\nu } \\partial_\\mu\\phi ^a \\partial_\\nu\\phi ^a \\rt) -N^{-1}\\lt(E_{\\mu \\nu } E^{\\mu \\nu }-E^2\\rt)\\nn\\\\\n&& -m^2 N^{-1}\\left(\\phi^{a}{}'-N^{\\mu }\\partial_{\\mu }\\phi ^a\\right)\\left(\\phi^{a}{}'-N^{\\nu }\\partial_{\\nu }\\phi ^a\\right)\\Big], \\label{ADM_action}\n\\eeqn\nwhere $E_{\\mu \\nu }=\\frac{1}{2} \\left(\\gamma '_{\\mu \\nu }-\\nabla _{\\nu }N_{\\mu }-\\nabla _{\\mu }N_{\\nu }\\right)$. The constraint equations are yielded by varying with respect to the lapse function $N$ and shift vector $N_\\mu$, i.e.,\n\\beqn\nR^{(4)}-2 \\Lambda-m^2 \\gamma ^{\\mu \\nu } \\partial_\\mu\\phi^a \\partial_\\nu\\phi^a +N^{-2}\\lt(E_{\\mu \\nu } E^{\\mu \\nu }-E^2\\rt)&&\\nn\\\\\n+m^2N^{-2}\\left(\\phi^{a}{}'-N^{\\mu }\\partial _{\\mu }\\phi ^a\\right)\\left(\\phi^{a}{}'-N^{\\nu }\\partial_{\\nu }\\phi ^a\\right)&=&0, \\label{Constraint_Eq_I}\\\\\n\\nabla _{\\nu }\\lt[N^{-1}\\lt(E^{\\mu}_{\\nu }-\\delta^{\\mu}_{\\nu}E \\rt)\\rt]-m^2N^{-1}\\partial_{\\mu }\\phi ^a\\lt(\\phi^a{}'-N^{\\alpha } \\partial_\\alpha\\phi^a\\rt)&=&0. \\label{Constraint_Eq_II}\n\\eeqn\nBy perturbing the constraint equations, one has a constraint equation for vector modes and two constraint equations for scalar modes. Thus, one vector mode and two scalar modes can be worked out algebraically, leaving us one vector and one scalar modes eventually.  \n\nAfter scalar-vector-tensor decomposition and expanding the action \\eqref{Main_Action} to the quadratic order  of fluctuations, the TT tensor, transverse vector and scalar modes are decoupled with each other, so they can be treated separately.\n\n\\subsection{Tensor mode} \nBy including only the tensor perturbation in the perturbed metric and dropping the boundary terms, the quadratic action for tensor perturbation is read as\n\\beq\nS^{(2)}_T=\\frac{M^3}{8}\\int d^4xdya^2\\left[-{a^{2}} D_{ \\alpha \\beta }' D{}'^{\\alpha \\beta } -{\\partial_\\lambda D_{\\alpha \\beta }}\\partial ^{\\lambda }D ^{\\alpha \\beta }-2 m^2 D_{\\alpha \\beta } D^{\\alpha \\beta } \\right],\n\\eeq\nwhere the indices are raised and lowered by the 4D Minkowski metric $\\eta_{\\mu\\nu}$. Further, after a coordinate transformation $dy=adz$ that turns the background metric into a conformal form and a rescaling $\\tilde{D }={D }\/{2}$, a canonical form is obtained as\n\\beq\nS^{(2)}_T=\\frac{M^3}{2}\\int d^4y dz a^3 \\left[-\\dot{\\tilde{D }}_{\\alpha \\beta }\\dot{\\tilde{D }}^{\\alpha \\beta }-\\partial_\\lambda \\tilde D_{\\alpha \\beta } \\partial ^{\\lambda }\\tilde D ^{\\alpha \\beta }-2 m^2\\tilde D_{\\alpha \\beta}  \\tilde D^{\\alpha \\beta } \\right],\n\\eeq\nwhere the dot denotes the derivative with respect to the extra dimension coordinate $z$. The tensor mode is free from the ghost instability due to the correct sign of kinetic term. Moreover, the action is expressed more concisely in the momentum space, where the d'Alembert operator $\\partial ^{\\alpha }\\partial_{\\alpha } $ is replaced by $ -k^2\\equiv -k^\\alpha k_\\alpha$, with $k^\\alpha$ the four-momentum of tensor mode. Then, it yields\n\\beq\nS^{(2)}_T=\\frac{M^3}{2}\\int d^4 k dz a^3 \\left[-\\dot{\\tilde{D }}_{\\alpha \\beta }\\dot{\\tilde{D }}^{\\alpha \\beta }-\\left({k^2}+{2 m^2}\\right)\\tilde D_{\\alpha \\beta }\\tilde D^{\\alpha \\beta } \\right].\n\\eeq\nBy variation with respect to $\\tilde{D }^{\\alpha \\beta }$, we have the equation of motion\n\\beq\n\\ddot{\\tilde{D} }_{\\alpha\\beta }+3H\\dot{\\tilde{D} }_{\\alpha\\beta }=(k^2+2m^2){\\tilde{D}_{\\alpha\\beta } }.\n\\label{eom_tensor}\n\\eeq\nAfter redefining $\\tilde{D}_{\\alpha\\beta } = a^{-\\fc{3}{2}}\\mathcal{D}_{\\alpha\\beta }$, a Schr\\\"odinger-like equation is obtained,\n\\beq\n-\\ddot{\\mathcal{D} }_{\\alpha\\beta } +\\left(\\frac{3}{2}\\dot H+\\frac{9}{4}H^2\\right) {\\mathcal{D} }_{\\alpha\\beta } =M_T^2{\\mathcal{D}}_{\\alpha\\beta } ,\n\\label{SE_Tensor}\n\\eeq\nwhere $M_T^2 \\equiv -k^2-2m^2$. The Hamiltonian can be further factorized as a supersymmetric quantum mechanics form, with $H_T=A_T^\\dag A_T=\\lt(\\pt_z+\\fc{3}{2}H\\rt)\\lt(-\\pt_z+\\fc{3}{2}H\\rt)$.\nThen with the boundary condition $\\pt_z \\tilde{D}_{ \\alpha \\beta } |_{z=0,z_b} =0$, the self-adjoint Hamiltonian gives non-negative eigenvalues \\cite{Yang2017}, i.e., $M_T^2\\geq 0$. Thus, with a positive 5D graviton mass $m$, the four-momentum of tensor mode $k^2=-M_T^2-2m^2<0$, i.e., the tensor excitations are all time-like particles. Thus, the model is also irrelevant to tachyonic instability. If $m$ is set to zero, then $M_T^2=-k^2\\geq 0$ gives us the well-known result that there is a massless graviton and a tower of massive gravitons in RS1 model.\n\n\\subsection{Vector modes} \n\nBy including only the vector perturbations in the perturbed metric and dropping the boundary terms, the quadratic action for vector perturbations is read\n\\beqn\nS_{V}^{(2)}&=&\\frac{M^3}{16}\\int d^4xdy a^2\\Big[-a^2\\pt_\\beta F_\\alpha{}'\\pt^\\beta F^\\alpha{}'-2m^2\\pt_\\beta F_\\alpha \\pt^\\beta F^\\alpha -8a^2m^2 A_\\alpha{}' A^\\alpha{}'-8m^2 \\pt_\\beta A_\\alpha \\pt^\\beta A^\\alpha \\nn\\\\\n&& -4\\partial_{\\beta }S_{\\alpha }\\partial^\\beta S^{\\alpha }-8m^2S_\\alpha S^\\alpha+8m^2 \\pt_\\beta F_\\alpha \\pt^\\beta A^\\alpha -4a\\pt_\\beta F_\\alpha{}'\\pt^\\beta S^\\alpha-16 a m^2 S_\\alpha A^\\alpha{}'\\Big].\n\\label{Action_SV}\n\\eeqn\nThe constraint equation can be obtained by counting the first order perturbations of \\eqref{Constraint_Eq_II}, or simplify by varying the above quadratic action with respect to $S^\\alpha$, i.e.,\n\\beq\n2\\partial _{\\beta }{\\partial^\\beta S_{\\alpha }}-4 m^2 S_{\\alpha }+a \\partial_{\\beta }{\\partial^\\beta  F_{\\alpha }'}-4am^2A_\\alpha'=0.\n\\label{Vector_Constraint_Eq}\n\\eeq\nThen, $S^{\\alpha }$ can be worked out in momentum space, namely,\n\\beq\nS_{\\alpha }=-\\frac{a k^2 F_{\\alpha }' +4 a m^2 A_\\alpha '}{2 (k^2+2 m^2)}. \n\\label{Vector_Constraint}\n\\eeq\nWorking in the unitary gauge, $A_\\alpha$ is gauged away. Then, substituting the relation into the action \\eqref{Action_SV} yields\n\\beq\nS_V^{(2)}= M^3\\int d^4 k dy  \\left[-\\frac{a^4 k^2 m^2 F_{\\alpha }{}' F^{\\alpha }{}'}{8 \\left(k^2+2 m^2\\right)}-\\frac{a^2 k^2 m^2}{8} F_{\\alpha } F^{\\alpha } \\right].\n\\eeq\nAfter a coordinate transformation into the coordinate $z$, a  canonical normalized form can be obtained by redefining the vector perturbation as $\\tilde{F}_{\\alpha }=\\frac{k m F_{\\alpha } }{2 \\sqrt{k^2+2 m^2}}$, i.e.,\n\\beq\nS^{(2)}_V=\\frac{M^3}{2}\\int d^4 k dz a^3  \\left[-\\dot{\\tilde{F }}_{\\alpha} \\dot{\\tilde{F }}^{\\alpha}-\\left({k^2}+{2 m^2}\\right)\\tilde{F}_{\\alpha } \\tilde{F}^{\\alpha }\\right].\n\\eeq\nThe correct sign of the kinetic term ensures that the vector perturbation is free from the ghost instability. Then the  equation of motion reads\n\\beq\n\\ddot{\\tilde{F} }_{\\alpha }+3H\\dot{\\tilde{F} }_{\\alpha }=(k^2+2m^2){\\tilde{F}_{\\alpha } }.\n\\eeq\nA Schr\\\"odinger-like equation is given by redefining $\\tilde{F}_{\\alpha }\\to a^{-\\frac{3}{2}} \\mathcal{F}_{\\alpha }$,\n\\beq\n-\\ddot{\\mathcal{F}}_\\alpha+\\left(\\frac{3}{2}\\dot H+\\frac{9}{4}H^2\\right) \\mathcal{F}_\\alpha=M_V^2\\mathcal{F}_\\alpha,\n\\label{SE_Vector}\n\\eeq\nwhere $M_V^2 \\equiv -k^2-2m^2$. The Hamiltonian can also be factorized as a supersymmetric quantum mechanics form, with $H_V=A_V^\\dag A_V=\\lt(\\pt_z+\\fc{3}{2}H\\rt)\\lt(-\\pt_z+\\fc{3}{2}H\\rt)$. With the boundary condition $\\pt_z \\tilde{F }_{\\alpha} |_{z=0,z_b} =0$, the eigenvalues are non-negative $M_V^2\\geq 0$. Thus, the vector excitations are also time-like particles and irrelevant to tachyonic instabilities for the cases of $M_V^2> 0$. \n\nHowever, for the case of $M_V^2=-k_0^2-2m^2=0$,  the formula \\eqref{Vector_Constraint} is invalid, and the constraint equation \\eqref{Vector_Constraint_Eq} leads to \n\\beq\nF_\\alpha'=2A_\\alpha'.\n\\eeq\nThis implies that $F_\\alpha-2A_\\alpha \\equiv f_\\alpha(x)$ is a purely 4D field. Especially, the field $f_\\alpha(x)$ is a gauge invariant quantity under the general coordinate transformation by the observation from \\eqref{Gauge_transformation}. Then, the quadratic action \\eqref{Action_SV} reduces to\n\\beqn\nS^{(2)}_{V0} &=& -\\frac{M^3}{8}\\int dz a^3 m^2 k_0^2 \\left(F_{\\alpha }-2 A_{\\alpha }\\right) \\left(F^{\\alpha }-2 A^{\\alpha }\\right)=-\\frac{M^3}{8}\\int dz a^3 m^2 k_0^2 f_\\alpha f^\\alpha .\n\\eeqn\nThe equation of motion reads $k_0^2f_\\alpha=0$. Since $k_0^2=-2m^2$, it leads to $f_\\alpha=0$. This implies that the lightest vector mode does not exist in the mass spectrum. This is curial to recover the mass spectrum of  RS1 model in the massless limit $m=0$, where no massless vector mode exists due to the lack of continuous isometries of the higher dimension in the presence of 3-branes \\cite{Randall1999}. Moreover, all the massive vector modes are gauge dependent in the massless limit $m=0$, therefore, they can be gauged away by gauge fixing.\n\n\\subsection{Scalar modes} \nBy including only the scalar perturbations in the perturbed metric and dropping the boundary terms, the quadratic action for scalar perturbations is read as\n\\beqn\nS^{(2)}_{S}&=&M^3\\int d^4xdy a^2\\Big(6 a^2 H^2 \\xi ^2+4 m^2 \\psi ^2-4 m^2 \\xi \\psi+6 a^2 \\psi '{}^2 +12 a^2 H \\xi  \\psi ' +3\\pt_\\alpha \\psi \\pt^\\alpha \\psi\\nn\\\\\n&& -\\fc{m^2}{2} \\pt_\\alpha \\beta \\pt^\\alpha \\beta +3\\xi \\pt_\\alpha \\pt^\\alpha \\psi+3 aH\\xi \\pt_\\alpha \\pt^\\alpha \\beta + m^2 \\xi \\pt_\\alpha \\pt^\\alpha\\varphi-m^2 \\xi \\pt_\\alpha \\pt^\\alpha E -2m^2 \\psi \\pt_\\alpha \\pt^\\alpha \\varphi \\nn\\\\\n&& +2m^2\\psi \\pt_\\alpha \\pt^\\alpha \\varphi +3 a^2 H \\xi \\pt_\\alpha \\pt^\\alpha E' +3 a\\beta \\pt_\\alpha \\pt^\\alpha\\psi' + m^2 a \\beta \\pt_\\alpha \\pt^\\alpha \\varphi' -\\fc{m^2}{2}  a^2 \\pt_\\alpha \\varphi' \\pt^\\alpha \\varphi'\\nn\\\\\n&&+3a^2 \\psi'\\pt_\\alpha \\pt^\\alpha E' -\\fc{m^2}{2}  \\pt_\\alpha \\pt^\\alpha E \\pt_\\lambda \\pt^\\lambda E -\\fc{m^2 }{2} \\pt_\\alpha \\pt^\\alpha \\varphi \\pt_\\lambda \\pt^\\lambda \\varphi +m^2 \\pt_\\alpha \\pt^\\alpha \\varphi \\pt_\\lambda \\pt^\\lambda E \\Big).\n\\label{Scalar_full_action}\n\\eeqn\nFrom the first order perturbation of Eqs.~\\eqref{Constraint_Eq_I} and \\eqref{Constraint_Eq_II} or simply varying the above action respect to the modes $\\beta$ and $\\xi$ respectively, the constraint equations are obtained as \n\\beqn\n12 a^2 H^2 \\xi-4m^2\\psi+12 a^2 H \\psi'+3 \\pt_\\alpha \\pt^\\alpha \\psi +3aH\\partial _{\\alpha }\\partial ^{\\alpha }\\beta  +m^2 \\partial _{\\alpha }\\partial ^{\\alpha }\\varphi\\nn\\\\\n-m^2\\partial _{\\alpha }\\partial ^{\\alpha }E +3a^2H\\partial _{\\alpha }\\partial ^{\\alpha }E'&=&0,\\\\\nm^2\\partial_\\alpha  \\beta + 3aH\\partial _{\\alpha }\\xi + 3a \\pt_\\alpha   \\psi' +m^2 a \\pt_\\alpha  \\varphi' &=&0.\n\\eeqn\nAfter closing the scalar perturbations $\\varphi$ and $\\psi$ in the unitary gauge, we only have to take the remaining perturbations $\\xi$, $\\beta$ and $E$ into account in the quadratic action, i.e.,  \n\\beqn\nS^{(2)}_{S}&=&M^3\\int d^4xdy a^2\\Big(6 a^2 H^2 \\xi ^2 -\\frac{m^2}{2}\\partial _\\alpha \\beta \\partial ^{\\alpha }\\beta +3 a H \\xi\\partial _{\\alpha }\\partial ^{\\alpha }\\beta-m^2\\xi \\partial _{\\alpha }\\partial ^{\\alpha }E\\nn\\\\\n&&+3a^2H\\xi \\partial _{\\alpha }\\partial ^{\\alpha }E'-\\frac{m^2}{2}\\partial _{\\alpha }\\partial ^{\\alpha }E\\partial _{\\lambda }\\partial ^{\\lambda }E\\Big).\n\\label{Scalar_action}\n\\eeqn\nCorrespondingly, the constraint equations are rewritten as \n\\beqn\n12 a^2 H^2 \\xi+3aH\\partial _{\\alpha }\\partial ^{\\alpha }\\beta -m^2\\partial _{\\alpha }\\partial ^{\\alpha }E +3a^2H\\partial _{\\alpha }\\partial ^{\\alpha }E'&=&0,\\\\\nm^2\\partial_\\alpha  \\beta+3aH\\partial _{\\alpha }\\xi  &=&0.\n\\eeqn\nIn momentum space,  $\\beta$ and $\\xi $ can be worked out from the constraint equations as \n\\beqn\n\\beta &=&-\\frac{3 aH}{m^2}\\xi ,\\\\\n\\xi &=&\\frac{k^2m^2(3 a^2 HE'- m^2 E)}{3 a^2 H^2 \\left(3 k^2+4 m^2\\right)}.\n\\eeqn\nThus, the perturbations $\\beta$ and $\\xi$ can be eliminated by substituting these relations into the action \\eqref{Scalar_action}, then a quadratic action for scalar perturbation $E$ is achieved,\n\\beq\nS^{(2)}_{S}\\supset M^3\\int dk^4 dy \\fc{3a^2k^4 m^2}{6 k^2+8 m^2}  \\left[-a^2E' E' -\\lt(k^2+2 m^2\\rt)E^2\\right].\n\\eeq\nThe correct sign of the kinetic term ensures that the scalar perturbation is free from the ghost instability as well. Further, after redefining the scalar perturbation as $\\tilde E= \\sqrt{\\frac{3 k^4 m^2}{3 k^2+4 m^2}}E$,  the above action can be rewritten as a canonical normalized form in the conformal coordinate $z$, i.e.,\n\\beq\nS^{(2)}_{S}\\supset\\frac{M^3}{2}\\int d^4 k dz a^3 \\left[-\\dot{\\tilde{E} }\\dot{\\tilde{E}} -\\left({k^2}+{2 m^2}\\right)\\tilde{E}^2\\right].\n\\eeq \nFurther, the equations of motion of scalar mode $\\tilde{E}$ is given by\n\\beq\n\\ddot{\\tilde{E} }+3H\\dot{\\tilde{E} }=(k^2+2m^2){\\tilde{E} }.\n\\eeq\nBy redefining $\\tilde{E}=a^{-\\fc{3}{2}}{\\varepsilon}$, it can be rewritten as  a Schr\\\"odinger-like equation,\n\\beq\n-\\ddot{{\\varepsilon }}+\\left(\\frac{9 }{4}H^2+\\frac{3 }{2}\\dot H\\right){\\varepsilon } = M_S^2{\\varepsilon},\n\\label{SE_Scalar}\n\\eeq\nwhere $M_S^2\\equiv-k^2-2m^2$. The Hamiltonian can be factorized as a supersymmetric quantum mechanics form as well, $H_S=A_S^\\dag A_S=\\lt(\\pt_z+\\fc{3}{2}H\\rt)\\lt(-\\pt_z+\\fc{3}{2}H\\rt)$, so with the boundary condition $\\pt_z \\tilde E |_{z=0,z_b} =0$, the eigenvalues are non-negative $M_S^2\\geq 0$. Since the excitations of scalar mode are all time-like particles, it is also irrelevant to the tachyonic instability in scalar perturbations. \n\nIn RS1 model, the presence of IR brane abruptly ends AdS space, so it spontaneously breaks the conformal invariance of AdS bulk in the IR. The massless radion is just the Goldstone boson associated with the broken dilatation invariance \\cite{Arkani-Hamed2001a,Rattazzi2001}. For the current model, the scalar curvature reads $R=-20k^2+\\fc{4m^2e^{2k y}}{1+\\epsilon^2 e^{2ky}}$ in the bulk. Thus, the scalar condensation deforms the bulk geometry a little bit, which is not a pure AdS bulk anymore. Consequently, the radion acquires a tiny mass due to this explicit symmetry breaking. In the massless limit $m=0$, the quadratic action \\eqref{Scalar_full_action} will reduce to that of RS model, with only a massless radion in the mass spectrum \\cite{Goldberger1999a,Charmousis2000,Callin2005}. \n\n\n\n\\section{Mass Spectra and Gauge Hierarchy}\\label{Hierarchy}\n\nFor the lightest tensor and scalar mode corresponding to $M_{T,S}=0$, their wave functions can be easily solved from Eqs.~\\eqref{SE_Tensor} and \\eqref{SE_Scalar}. After returning to the coordinate space,  and utilizing the KK decompositions, $\\mathcal{D}_{\\alpha\\beta }(x,z) =d_{\\alpha\\beta }(x)\\Psi(z)$ and $\\varepsilon(x,z) =e(x)\\Psi(z)$,  Eqs.~\\eqref{SE_Tensor} and \\eqref{SE_Scalar} can be further reduced to the 4D Klein-Gordon equations, $\\Box^{(4)}d_{\\mu\\nu}=m_T^2 d_{\\mu\\nu}$ and $\\Box^{(4)}e(x)=m_S^2 e(x)$, with $m_{T,S}$ the effective mass of KK particles in 4D point of view, and the Schr\\\"odinger-like equation,\n\\beq\n-\\ddot{\\Psi} +\\left(\\frac{3}{2} \\dot{H}+\\frac{9}{4}H^2\\right)\\Psi =M_{T,S}^2\\Psi,\n\\eeq\nwhere $M_{T,S}^2=m_{T,S}^2-2m^2$. The wave function of ground states is easily achieved by setting $M_{T,S}=0$ or $m_{T,S}=\\sqrt{2}m$ in above Schr\\\"odinger-like equation, \n\\beq\n\\Psi^{(0)}(z)=N_0a(z)^{\\fc{3}{2}},\n\\eeq \nwith $N_0$ a normalization factor.  The boundary conditions $\\pt_z \\tilde{D}_{ \\alpha \\beta } |_{z=0,z_b} =0$ and $\\pt_z \\tilde E |_{z=0,z_b} =0$ lead to $\\lt.\\lt( \\Psi-\\frac{3}{2}H \\rt)\\rt|_{z=0,z_b}=0$. Obviously, the wave function $\\Psi^{(0)}(z)$ satisfies the boundary condition. All KK particles are massive in this model, which is a significant difference from the RS1 model, where the lightest KK particles are massless spin-2 graviton and spin-0 radion. Especially, from the redefinitions $\\tilde{D}_{\\alpha\\beta } = a^{-\\fc{3}{2}}\\mathcal{D}_{\\alpha\\beta }$ and $\\tilde{E}=a^{-\\fc{3}{2}}{\\varepsilon}$, their canonical normalized field configurations are given by $\\tilde{D}^{(0)}_{\\alpha\\beta }=a^{-\\fc{3}{2}}\\mathcal{D}^{(0)}_{\\alpha\\beta }=d^{(0)}_{\\alpha\\beta }(x)$ and $\\tilde{E}^{(0)}=a^{-\\fc{3}{2}} {\\varepsilon}^{(0)} =e^{(0)}(x)$. Therefore, the lightest graviton and radion propagate only on the brane. \n\nHowever, the effective mass of 4D graviton is severely constrained by the gravitational experiments \\cite{Rham2017}, e.g., the bound of the graviton mass is $m_g\\leq 4.7\\times 10^{-23}$eV from the detection of gravitational waves \\cite{Abbott2019}, thus the parameter $m$ must be tinier than the experimental constraints, i.e., $m<3.3\\times 10^{-23}$eV. Since the lightweight radion has the same mass as the lightest graviton, exchanging such nearly massless scalar particle would cause a fifth force and violate experimental observations. Therefore, similar to the RS1 model, the radion must gain weight to meet the experimental expectations, which is realized through GW mechanism \\cite{Goldberger1999a} in the next section.\n\nWith the normalization condition $\\int^{z_b}_{-z_b}\\Psi_0^2dz=1$, the normalization factor is worked out as \n\\beqn\nN_0^{-2}=\\frac{1}{k}\\lt[1-e^{-2 k y_\\pi}+4\\epsilon^2 k y_\\pi -\\epsilon^4(1-e^{2 k y_\\pi})\\rt].\n\\label{Normalization_factor}\n\\eeqn\nBy shutting down the 5D graviton mass $m$, one recovers the result of RS1 model. However, if we remove the visible brane, i.e., $y_\\pi\\to\\infty$, the quasi-massless graviton is no longer normalizable. This means that the effective 4D gravity theory can not be recovered on the brane. So the RS2-like single brane model \\cite{Randall1999a} is not a physically available one in current massive gravity.\n\nThe braneworld scenario provides a natural way to solve the  gauge hierarchy problem, which is a crucial motivation of the well-known ADD model \\cite{Arkani-Hamed1998} and RS1 model \\cite{Randall1999}. In our toy model,  the low-energy effective  theory is obtained by including the nearly-massless gravitons, i.e.,\n\\beq\nds^2=a^2(y)\\bar g_{\\mu\\nu}(x)dx^\\mu dx^\\nu+dy^2=a^2(y)\\lt[\\eta_{\\mu\\nu}+\\gamma_{\\mu\\nu}(x)\\rt]dx^\\mu dx^\\nu+dy^2,\n\\label{Metric_Lowenergy}\n\\eeq\nthen the 4D effective gravitational mass scale $M_{\\text{eff}}$ is read from the curvature term in the action \\eqref{Main_Action},\n\\beq\nM_{\\text{eff}}^2=M^3\\int^{y_\\pi}_{-y_\\pi} a^2 dy={N_0^{-2}}{M^3}. \\label{Mass_scale_relation}\n\\eeq\n\nOn the other hand, in order to produce a large hierarchy between the Planck scale and the electroweak scale, our Universe should be embedded on the IR brane located at $y_\\pi$. Then the Higgs field action on the brane is non-canonically normalized, \n\\beq\nS_\\text{H}\\supset\\int{  d^4x \\sqrt{| g_\\text{II}|}\\lt[-{g}_\\text{II}^{\\mu\\nu}D_{\\mu}H^{\\dag}D_{\\nu}H-\\lambda(H^{\\dag}H-v_0^2)^2 \\rt]  },\n\\eeq\nwhere $g_{\\text{II}\\mu\\nu}=a(y_\\pi)\\tilde{g}_{\\mu\\nu}$ is the induced metric on the brane at $y_\\pi$, and $v_0$ the fundamental Higgs vacuum expectation value (VEV). Writting the warp factor explicitly, it is \n\\beq\nS_\\text{H}\\supset\\int{  d^4x \\sqrt{|\\tilde g|}\\lt[-a(y_\\pi)^{2}\\tilde{g}^{\\mu\\nu}D_{\\mu}H^{\\dag}D_{\\nu}H-a(y_\\pi)^4\\lambda(H^{\\dag}H-v_0^2)^2 \\rt]  },\n\\eeq\n With a field renormalization, $H \\to \\tilde{H}\/a(y_\\pi)$, the effective action of Higgs on the brane is \n \\beq\n S_\\text{H}\\supset\\int{  d^4x \\sqrt{|\\tilde g|}\\lt[-\\tilde{g}^{\\mu\\nu}D_{\\mu}\\tilde{H}^{\\dag}D_{\\nu}H-\\lambda(\\tilde{H}^{\\dag}H-v_\\text{eff}^2)^2 \\rt]  },\n \\eeq\n where the effective Higgs VEV, $v_\\text{eff}=a(y_\\pi) v_0$, sets the electroweak scale on the brane.  \n \n Therefore, in order to solve the gauge hierarchy problem, the fundamental parameters $M$, $k$, $v_0$ are all set to be the order of Plank scale $M_\\text{Pl}$. Then, the warp factor has to provide enough redshift to recover a TeV scale of Higgs VEV on the brane, namely, $a(y_\\pi)\\sim10^{-16}$. In the condition $e^{-k |y_\\pi|}>\\epsilon$ and $m<3.3\\times 10^{-23}$eV, it leads to $e^{-ky_\\pi} \\approx a(y_\\pi) -\\fc{\\epsilon^2}{a(y_\\pi)} \\sim10^{-16}$. Thus, it requires  $y_\\pi\\approx 37\/k$.    \n\n\nSince the mass splitting scale of massive KK modes is inversely proportional to the conformal size of extra dimension $z_b$, i.e., $\\Delta m_T \\propto 1\/z_b$. From the coordinate transformation $dy=adz$, one has\n\\beq\nz_\\pi=\\frac{1}{k \\epsilon} \\left[\\text{arctan}\\left(\\epsilon e^{k y_\\pi}\\right)-\\text{arctan} \\left(\\epsilon \\right)\\right],\n\\eeq\nwhere the integral constant has been chosen so that $z(y=0)=0$. Since $e^{-ky_\\pi}  \\sim10^{-16}$, $m<3.3\\times 10^{-23}$eV and $k\\sim M_\\text{Pl}$, one has $\\epsilon e^{k y_\\pi} \\ll 1$. Thus, the formula can be rewritten approximately as $z_\\pi \\approx \\fc{e^{ky_\\pi}}{k}\\lt[1+\\mathcal{O}(\\epsilon^2)\\rt]$. So the mass splitting scale is similar to the RS model, i.e., $\\Delta m_T \\sim  k e^{-ky_\\pi} \\sim \\mathcal{O}(\\text{TeV})$.\n\n\t\n\\section{Radius stabilization}\\label{Radius_stabilization}\n\nIn RS1 model, the radius is not dynamically fixed, so there is a massless radion in the effective theory, which corresponds to the fluctuations of the radius of compact extra dimensions. However, the massless radion is phenomenologically unacceptable since it would contribute to Newton's law and cause a fifth force. It is well-known that the GW mechanism \\cite{Goldberger1999a} can be introduced to stabilize the size of the extra dimension and to increase the mass of radion. In our model, the light weight radion and the lightest graviton have equal mass, which is unacceptable in phenomenology as well. Therefore, we also utilize the GW mechanism here to increase the mass of radion. \n\nBy adding a bulk scalar field $\\Phi$ into the model, which has interaction with the two branes, the action is given by\n\\beqn\nS_\\Phi&=&\\fc{1}{2}\\int d^4x\\int^\\pi_{-\\pi}d\\theta\\sqrt{-g}\\lt(-g^{MN}\\pt_M\\Phi\\pt_N\\Phi-m_\\Phi^2\\Phi^2 \\rt)-\\int{}d^4x\\sqrt{-g_\\text{I}}\\lambda_\\text{I}(\\Phi^2-v_\\text{I}^2)^2\\nn\\\\\n&&-\\int{}\\sqrt{-g_{\\text{II}}}\\lambda_{\\text{II}}(\\Phi^2-v_{\\text{II}}^2)^2,     \n\\eeqn\nwhere $g_{MN}$ is the background metric \\eqref{Brane_Metric} with the radius $r_c$ of compactified extra dimension given by $y=r_c \\theta$, $\\lambda_{\\text{I\/II}}$ is the coupling parameters, and $V_{\\text{I\/II}}=\\lambda_{\\text{I\/II}} v_{\\text{I\/II}}^4$.\n\nThe equation of motion of the scalar filed $\\Phi$ is achieved by varying with respect to  $\\Phi$, \n\\beqn\n\\frac{1}{r_c}{\\partial_\\theta }\\left(a^4 {\\partial_\\theta  \\Phi }\\right)-a^4 m_{\\Phi }^2\\Phi-{4 a^4  \\lambda_\\text{I} \\Phi  \\left(\\Phi ^2-v_\\text{I}^2\\right)}\\delta (\\theta)-4 a^4\\lambda _\\text{II} \\left(\\Phi ^2-v_\\text{II}^2\\right) \\Phi  \\delta (\\theta -\\pi ) =0.\n\\label{EoM_Phi}\n\\eeqn\nAway from the two branes at $\\theta=0,\\pi$, the general solution of this equation reads\n\\beq\n\\Phi(\\theta)=e^{2 \\sigma } \\left[A  {}_2\\text{F}_1\\left(2,\\nu +2,\\nu +1, -\\epsilon ^2e^{2 \\sigma} \\right) e^{ \\nu \\sigma }+B {}_2\\text{F}_1\\left(2,2-\\nu ,1-\\nu ,-\\epsilon ^2e^{2 \\sigma } \\right) e^{-\\nu \\sigma }\\right],\n\\eeq\nwhere $\\sigma(\\theta)=k r_c \\theta$, $\\nu=\\sqrt{4+m_\\Phi^2\/k^2}$, ${}_2\\text{F}_1$ is the hypergeometric function, $A$ and $B$ are integration constants. Especially, under the condition $\\epsilon e^{\\sigma} \\ll 1$, up to the first order of correction, the solution can be approximately written as\n\\beq\n\\Phi(\\theta)\\simeq e^{2 \\sigma } \\left[A  \\left(1-2 \\epsilon^2\\frac{\\nu +2}{\\nu +1}e^{2 \\sigma } \\right)e^{ \\nu \\sigma }+B \\left(1-2\\epsilon^2\\frac{\\nu -2}{\\nu -1} e^{2 \\sigma }\\right)e^{ -\\nu \\sigma }\\right].\n\\eeq\n If we close the mass of 5D graviton, i.e., $\\epsilon=0$, the solution reduces to the one in general relativity \\cite{Goldberger1999a}. The integration constants $A$ and $B$ can be fixed by the boundary conditions on the branes, which are obtained by inserting the approximate solution into the equation of motion \\eqref{EoM_Phi} and matching the delta functions,\n \\beqn\n k \\left[ (\\nu +2) \\left(1+2\\epsilon ^2\\frac{\\nu -2}{\\nu +1}\\right)A- (\\nu-2 ) \\left(1+2\\epsilon ^2\\frac{\\nu +2}{\\nu -1}\\right)B\\right]\\nn\\\\\n -2 \\lambda _\\text{I}   \\left(1+4 \\epsilon ^2\\right)  \\Phi(0)\\left( \\Phi(0)^2 -v_\\text{I}^2\\right)=0,~~~\\\\\n k e^{2  \\sigma(\\pi) } \\left[(\\nu +2) e^{\\nu  \\sigma(\\pi)} \\left(1+2\\epsilon ^2\\frac{\\nu-2}{\\nu +1} e^{2 \\sigma(\\pi)}\\right)A -(\\nu-2) e^{ - \\nu \\sigma(\\pi) } \\left(1+2\\epsilon ^2\\frac{\\nu +2}{\\nu -1} e^{2  \\sigma(\\pi) }\\right) B \\right]\\nn\\\\\n +2 \\lambda _v \\left(1+4 \\epsilon ^2 e^{2 \\sigma(\\pi)}\\right)\\Phi(\\pi )\\left(\\Phi(\\pi)^2 -v_v^2\\right) =0.~~~\n \\eeqn \n Rather than solving the above equations, we employ the trick of Ref.~\\cite{Goldberger1999a} to simplify the calculation. By substituting the approximate solution back into the action and integrating over $\\theta$, it yields the effective potential of compactified radius $r_c$, \n \\beqn\n V_\\Phi(r_c)&\\approx& k A^2  (\\nu +2) e^{2 \\nu k  r_c  \\pi } \\left(1-\\frac{8 \\epsilon^2 e^{2 \\pi  k r_c}}{\\nu +1}\\right)+\\lambda _\\text{I} \\left( \\Phi(0)^2 -v_\\text{I}^2\\right)^2\\nn\\\\\n&& +e^{-4 k r_c \\phi }\\lambda _\\text{II} \\left(1+4 \\epsilon ^2 e^{2 \\pi  k r_c}\\right)\\left( \\Phi(\\pi)^2 -v_\\text{II}^2\\right)^2,\n\\label{Potential_Phi}\n \\eeqn\n where the limit of $e^{k r_c \\pi}\\gg 1$ has been used in the calculation. Assuming that the interaction parameters $\\lambda_\\text{I}$ and $\\lambda_\\text{II}$ are very large  \\cite{Goldberger1999a}, this effective potential implies the solution  $\\Phi(0)=v_\\text{I}$ and $\\Phi(\\pi)=v_\\text{II}$. Then, the integration constants $A$ and $B$ can be solved approximately as \n \\beqn\n A&\\approx & v_\\text{II} e^{-(\\nu +2) \\sigma(\\pi) } \\left[1-2\\epsilon^2\\frac{\\nu +2}{\\nu +1} e^{2 \\sigma(\\pi) } \\right]-v_\\text{I} e^{-2 \\nu  \\sigma(\\pi) } \\left[1+4\\epsilon^2\\frac{ \\nu }{\\nu ^2-1} e^{2 \\sigma(\\pi) }\\right],\\\\\n B&\\approx & v_\\text{I} \\left[1+2\\epsilon^2\\frac{\\nu +2}{\\nu +1}  \\left(e^{\\nu  \\sigma(\\pi) }-e^{2 \\sigma(\\pi) }\\right)\\right]-v_\\text{II} e^{-(\\nu +2) \\sigma(\\pi) } \\left[1+2\\epsilon^2\\frac{\\nu +2}{\\nu +1}  e^{\\nu  \\sigma(\\pi) }\\right].\n \\eeqn\nFurther,  assuming that the mass of the scalar filed is a small quantity, i.e., ${m_\\Phi}\/{k}\\ll 1$, so that  $\\nu =\\sqrt{4+\\frac{m_\\Phi^2}{k^2}}\\simeq 2+\\delta$ with $\\delta=\\frac{m_\\Phi^2}{4 k^2}$ a tiny quantity. Then the effective potential can be rewritten as\n\\beqn\nV_\\Phi(r_c)&\\approx& (4+\\delta) k e^{-4 k r_c \\pi}\\left(1-\\frac{8 \\epsilon ^2e^{2 k r_c \\pi } }{\\delta +3}\\right) \\nn\\\\\n&&\\times \\left[\\left(1+\\frac{4 (\\delta +2) \\epsilon ^2e^{2 k r_c \\pi } }{\\delta ^2+4 \\delta+3 }\\right)e^{-\\delta  k r_c \\pi} v_\\text{I} -\\left(1-\\frac{2 (\\delta +4)  \\epsilon ^2e^{2k r_c \\pi }}{\\delta +3}\\right)v_\\text{II} \\right]^2.\n\\eeqn \nThus the effective potential has a minimum at \n\\beq\nr_c \\approx \\frac{1}{\\pi k\\delta }\\ln\\left(\\frac{v_h}{v_v}\\right)-\\frac{4\\epsilon ^2}{3\\pi k}  \\left(\\frac{v_h}{v_v}\\right)^{\\fc{2}{\\delta }}\\left(1+\\frac{v_v}{v_h}\\right).\n\\eeq\nThe first leading term is just the result of GW mechanism in general relativity \\cite{Goldberger1999a}, and the second term proportional to $\\epsilon^2$ is a tiny correction stemming from the 5D graviton mass. The mechanism provides a dynamical way to stabilize the compactified radius of the extra dimension without introducing another large hierarchy. For instance, if $m_\\Phi\/k=0.1$ and ${v_h}\/{v_v}=1.34$, then one obtains $ky_\\pi=k r_c\\pi\\approx 37$ to generate a proper hierarchy. After the radius stabilization, the radion acquires a mass roughly $\\mathcal{O}(\\delta^2)$ TeV \\cite{Goldberger2000}, which is somewhat smaller than the TeV scale. \n\n \n\\section{Conclusions}\\label{Conclusions}\n\n\nIn this work, we generalized the RS1 model in a 5D extension of the Lorentz-violating massive gravity. It is found that the theory supports two distinct brane configurations. The configuration possessing both positive and negative tension branes is similar to the RS1 model, while the other possessing only two positive tension branes is distinct from RS1 model. The full linear perturbations against the background metric were also analyzed. It is found that the models are free from the ghost and tachyonic instabilities, and all KK particles are massive. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=9cm]{Mass_spectra.eps}\n\\end{center}\n\\caption{The mass spectra of RS1 model and current model, where the blue lines refer to the tensor modes, red lines to vector modes, and green lines to scalar modes.}\n\\label{Mass_Spectra}\n\\end{figure}\n\nAs shown in Fig.~\\ref{Mass_Spectra},  the tensor and scalar modes have similar mass spectra which start from $\\sqrt{2}m$ with a mass splitting of TeV scale, nevertheless, the lightest vector mode does not exist in the mass spectrum. The ground states of tensor and scalar modes propagate only along the brane. The graviton mass $\\sqrt{2}m$ has to be tiny enough to fit the experimental constraints. However, the light weight radion would lead to a fifth force to violate experimental observations, so the GW mechanism was considered to stabilize the size of the extra dimension and to weight the mass of radion. After radius stabilization, the 4D effective theory on the brane includes a nearly massless graviton plus three towers of non-pathologic very massive spin-2,  spin-1 and  spin-0 particles.  \n\nThe gauge hierarchy problem was also solved as an application of the model. Furthermore, due to the very distinct KK mass spectra between current model and RS1 model, the new KK towers of vector and scalar modes introduce some new reaction channels and suggest some new signals in colliders. This is potentially an interesting property of current model. However, it is beyond the scope of current work and left for our future consideration. \n\n\n\n\n\\section*{ACKNOWLEDGMENTS}\n\nWe would like to especially thank Prof.~Yu-Xiao Liu for a very helpful discussion of our paper. This work was supported by the National Natural Science Foundation of China under Grant Nos. 12005174 and 12165013. K. Yang acknowledges the support of Natural Science Foundation of Chongqing, China under Grant No. cstc2020jcyj-msxmX0370.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec1}\n\n\\subsection{Background and motivation}\n\n\n\n\nThe study of the Boolean matrices \\cite{bapat, bapatbook, luce, rao} play an important role in linear algebra \\cite{belohlavek2015below, kim, rao72}, combinatorics \\cite{brual}, graph theory \\cite{berge}  and network theory \\cite{ledley, li2015logical}. However, this becomes particularly challenging to store huge volumes of multidimensional data.  This potential difficulty can be easily overcome, thanks to tensors,  which are natural multidimensional generalizations of matrices \\cite{kolda, loan}. Here the notion of tensors is different in physics and engineering (such as stress tensors) \\cite{Nar93}, which are generally referred to as tensor fields in mathematics \\cite{de2008tensor}.  However, it will be more appropriate if we study the Boolean tensors and the generalized inverses of Boolean tensors. Hence the generalized inverses of Boolean tensors will encounter in many branches of mathematics, including relations theory \\cite{plem}, logic, graph theory, lattice theory \\cite{birkhoff} and algebraic semigroup theory.\n\n\n\nRecently, there has been increasing interest in studying inverses \\cite{BraliNT13} and different generalized inverses of tensors based on the Einstein product \\cite{bm, wei18,  stan, sun}, and opened new perspectives for solving multilinear systems \\cite{kolda, Mao19}. In \\cite{wei18, stan}, the authors have introduced some basic properties of the range and null space of multidimensional arrays. Further, in \\cite{stan}, it was discussed the adequate definition of the tensor rank, termed as reshaping rank. Corresponding representations of the weighted Moore-Penrose inverse introduced in \\cite{BehMM19, we17} and investigated a few characterizations in \\cite{PanMi19}. Though this work is focusing on the binary case; i.e., concentrating some interesting results based on the Boolean tensors and generalized inverses of Boolean tensors via the Einstein product.  In many instances, the result in the general case does not immediately follow even though it is not difficult to conclude.\n\n\n\nOn the other hand, one of the most successful developments in the world of multilinear algebra is the concept of tensor decomposition \\cite{ Kolda01, kolda, LalmV00}. This concept gives a clear and convenient way to implement all basic operations efficiently.  Recently this concept is extended in Boolean tensors \\cite{, erdos2013discovering, khamis2017Boolean, rukat2018tensormachine}. Further,  the fast and scalable distributed algorithms for Boolean tensor decompositions were discussed in  \\cite{miettinen2011Boolean}. In addition to that, a few applications of these decompositions are discussed in \\cite{erdos2013discovering, metzler2015clustering} for information extraction and clustering. At that same time, Brazell, et al. in \\cite{BraliNT13} discussed decomposition of tensors from the isomorphic group structure on the influence of the Einstein Product and demonstrated that they are special cases of the canonical polyadic decomposition \\cite{carroll1970analysis}. The vast work on decomposition on the tensors and its several applications in different areas of mathematics in the literature, and the recent works in \\cite{BraliNT13, sun}, motivate us to study the generalized inverses and space decomposition in the framework of Boolean tensors. \n This study leads to introduce the rank and the weight for the Boolean tensor with its application to  generalized inverses.\n\n\n  \n\n\n\\subsection{Organization of the paper}\n\n\nThe rest of the paper is organized as follows. In Section 2 we present some definitions, notations, and preliminary results, which are essential in proving the main results.  The main results are discussed in Section 3. It has four subparts. In the first part, some identities are proved while the generalized inverses for Boolean tensor are discussed in the second part. The third part mainly focuses on weighted Moore-Penrose inverses. Space decomposition and its application to generalized inverses are discussed in the last part.  Finally, the results along with a few questions are concluded in Section 4.\n  \n \n\n\n\n\n\n\n\n\n\\section{Preliminaries}\nWe first introduce some basic definitions and notations which will be used throughout the article. \n\\subsection{Definitions and terminology}\nFor convenience, we first briefly explain some of the terminologies which will be used here onwards. The tensor notation and definitions are followed from the article \\cite{BraliNT13, sun}.  We refer  $\\mathbb{R}^{I_1\\times\\cdots\\times I_N}$  as the set of order $N$  real tensors. Indeed, a matrix is a second order tensor, and a vector is a first order tensor. Let $\\mathbb{R}^{I_1\\times\\cdots\\times I_N}$  be the set of  order $N$ and dimension $I_1 \\times \\cdots \\times I_N$ tensors over the real\nfield $\\mathbb{R}$.  $\\mc{A} \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N}$ is a tensor with  $N$-th order tensor, and each entry of $\\mc{A}$ is denoted by $a_{i_1...i_N}$.  Note that throughout the paper, tensors are represented in calligraphic letters like  $\\mc{A}$, and the notation $(\\mc{A})_{i_1...i_N}= a_{i_1...i_N}$ represents the scalars. The Einstein product (\\cite{ein}) $ \\mc{A}{*_N}\\mc{B} \\in \\mathbb{R}^{I_1\\times\\cdots\\times\nI_N \\times J_1 \\times\\cdots\\times J_M }$ of tensors $\\mc{A} \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times K_1\n\\times\\cdots\\times K_N }$ and $\\mc{B} \\in\n\\mathbb{R}^{K_1\\times\\cdots\\times K_N \\times J_1 \\times\\cdots\\times\nJ_M }$   is defined\nby the operation ${*_N}$ via\n\\begin{equation}\\label{Eins}\n(\\mc{A}{*_N}\\mc{B})_{i_1...i_Nj_1...j_M}\n=\\displaystyle\\sum_{k_1...k_N}a_{{i_1...i_N}{k_1...k_N}}b_{{k_1...k_N}{j_1...j_M}}.\n\\end{equation}\nSpecifically, if $\\mc{B} \\in \\mathbb{R}^{K_1\\times\\cdots\\times K_N}$, then $\\mc{A}{*_N}\\mc{B} \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N}$ and \n\\begin{equation*}\\label{Einsb}\n(\\mc{A}{*_N}\\mc{B})_{i_1...i_N} = \\displaystyle\\sum_{k_1...k_N}\na_{{i_1...i_N}{k_1...k_N}}b_{{k_1...k_N}}.\n\\end{equation*}\nThis product is discussed in the area of continuum mechanics  \\cite{ein} and the theory of relativity \\cite{lai}. Further, the addition of two tensors $\\mc{A}, ~\\mc{B}\\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times K_1 \\times\\cdots\\times K_N }$ is defined\nas \n\\begin{equation}\\label{Eins1}\n(\\mc{A} + \\mc{B})_{i_1...i_N k_1...k_N}\n=a_{{i_1...i_N}{k_1...k_N}} + b_{{i_1...i_N}{k_1...k_N}}.\n\\end{equation}\nFor a tensor $~\\mc{A}=(a_{{i_1}...{i_N}{j_1}...{j_M}})\n \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times J_1 \\times\\cdots\\times J_M},$  let  $\\mc{B} =(b_{{i_1}...{i_M}{j_1}...{j_N}}) \\in \\mathbb{R}^{J_1\\times\\cdots\\times J_M \\times I_1 \\times\\cdots\\times I_N}$, be the {\\it transpose} of $\\mc{A}$, where $b_{i_1\\cdots i_Mj_1\\cdots j_N} = a_{j_1\\cdots j_M i_1\\cdots i_N}.$ The tensor $\\mc{B}$ is denoted by $\\mc{A}^T$. Also, we denote  $\\mc{A}^T=\\left({a}_{{i_1}...{i_N}{j_1}...{j_M}}^t\\right).$ \n The trace of a tensor $\\mc{A}$ with entries $(\\mc{A})_{{i_1}...{i_N}{j_1}...{j_N}}$, denoted by $tr(\\mc{A})$,\n  is defined as the sum of the diagonal entries, i.e., \n$tr(\\mc{A}) = \\displaystyle\\sum_{i_1 \\cdots i_N}a_{{i_1...i_N}{i_1...i_N}}.$ Further, a tensor $\\mc{O}$ denotes the {\\it zero tensor} if  all the entries are zero. \n A tensor $\\mc{A}\\in\n\\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times I_1 \\times\\cdots\\times\nI_N}$ is {\\it symmetric}  if  $\\mc{A}=\\mc{A}^T,$  and {\\it orthogonal} if $\\mc{A}{*_M}\\mc{A}^T= \\mc{A}^T{*_N} \\mc{A}=\\mc{I}$. Further, a tensor\n$\\mc{A}\\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times I_1\n\\times\\cdots\\times I_N}$  is {\\it idempotent}  if $\\mc{A}\n{*_N} \\mc{A}= \\mc{A}$.  The definition of a diagonal tensor follows. \nFurther, a tensor with entries $(\\mc{D})_{{i_1}...{i_N}{j_1}...{j_N}}$ is\n   called a {\\it diagonal\n   tensor} if $d_{{i_1}...{i_N}{j_1}...{j_N}} = 0$ for $(i_1,\\cdots,i_N) \\neq (j_1,\\cdots,j_N).$\nA few more notations and definitions are discussed below for defining generalized inverses of Boolean tensors. We first recall the definition\nof an identity  tensor below.\n\n\n\n\n\n\\begin{definition} (Definition 3.13, \\cite{BraliNT13}) \\\\\nA tensor \n  with entries \n     $ (\\mc{I})_{i_1 \\cdots i_Nj_1\\cdots j_N} = \\prod_{k=1}^{N} \\delta_{i_k j_k}$,\n   where\n\\begin{numcases}\n{\\delta_{i_kj_k}=}\n  1, &  $i_k = j_k$,\\nonumber\n  \\\\\n  0, & $i_k \\neq j_k $.\\nonumber\n\\end{numcases}\n is  called a {\\it  unit tensor or identity tensor}.\n\\end{definition}\nThe permutation tensor is defined as follows.\n\n\\begin{definition}\\label{perm} \nLet $\\pi$ be a permutation map  on $(i_1,i_2,\\cdots, i_N,j_1,j_2,\\cdots , j_N)$ defined by \n$$\\pi:=\\begin{pmatrix}\ni_1&i_2&\\cdots &i_N&j_1&j_2&\\cdots &j_N \\\\\n\\pi(i_1)&\\pi(i_2)&\\cdots &\\pi(i_N)&\\pi(j_1)&\\pi(j_2)&\\cdots&\\pi(j_N)  \\\\\n\\end{pmatrix}.\n$$\nA  tensor $\\mc{P}$\n  with entries \n     $ (\\mc{P})_{i_1 \\cdots i_Nj_1\\cdots j_N} = \\prod_{k=1}^{N} \\epsilon_{i_k} \\epsilon_{j_k}$,\n   where\n\\begin{numcases}\n{\\epsilon_{i_k}\\epsilon_{j_k} = }\n  1, &  $\\pi(i_k) = j_k$,\\nonumber\n  \\\\\n  0, & otherwise.\\nonumber\n\\end{numcases}\n is  called a {\\it  permutation tensor}.\n\\end{definition}\n\n\nNow we recall the block tensor as follows. \n\\begin{definition}{\\cite{sun}}\nFor a tensor $\\mc{A} = (a_{i_1... i_Nj_1...j_M})\n \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times J_1 \\times\\cdots\\times\n J_M},\\\\\n  \\mc{A}_{(i_1...i_N|:)}= (a_{i_1...i_N:...:})\\in \\mathbb{R}^{J_1\\times\\cdots\\times J_M}$ is a\n  subblock of $\\mc{A}$. $Vec(\\mc{A})$ is obtained by lining up all the subtensors\n  in a column, and  $t$-th subblock of $Vec(\\mc{A})$ is  $\\mc{A}_{(i_1...i_N|:)}$,\n where $$t=i_N + \\displaystyle\\sum_{K=1}^{N-1} \\left[ (i_K - 1) \\displaystyle\\prod_{L=K+1}^{N} I_L \\right].$$\n\\end{definition}\n\\vspace{-0.5cm}\nLet $\\mc{A} = (a_{i_1\\cdots i_N j_1 \\cdots j_M}) \\in\n\\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times J_1 \\times\\cdots\\times\nJ_M}$ and $\\mc{B} = (b_{i_1\\cdots i_N k_1 \\cdots k_M}) \\in\n\\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times K_1 \\times\\cdots\\times\nK_M}$. The {\\it row block tensor} consisting  of $\\mc{A}$ and\n$\\mc{B}$ is denoted by\n$[\\mc{A} ~ \\mc{B}] \\in \\mathbb{R}^{\\alpha^N\\times\\beta_1\\times \\cdots \\times \\beta_M},$\nwhere $\\alpha^N = I_1\\times\\cdots\\times I_N, \\beta_i = J_i + K_i, i\n= 1, \\cdots, M$, and is defined by\n\\begin{equation*}\n[\\mc{A} ~ \\mc{B}]_{i_1 \\cdots i_N l_1 \\cdots l_M} =\n\\begin{cases}\na_{i_1 \\cdots i_N l_1 \\cdots l_M}, & i_1 \\cdots i_N \\in [I_1] \\times \\dots \\times [I_N], l_1 \\cdots l_M \\in [J_1] \\times \\cdots \\times [J_M];\n\\\\\nb_{i_1 \\cdots i_N l_1 \\cdots l_M}, & i_1 \\cdots i_N \\in [I_1] \\times \\dots \\times [I_N], l_1 \\cdots l_M \\in \\Gamma_1 \\times \\cdots \\times \\Gamma_M;\n\\\\\n0, & \\textnormal{otherwise}.\n\\end{cases}\n\\end{equation*}\nwhere $\\Gamma_i = \\{ J_i +1, \\cdots, J_i+K_i\\}, i=1,\\cdots, M.$\n\nLet $\\mc{C} = (c_{j_1 \\cdots j_M i_1 \\cdots i_N}) \\in\n\\mathbb{R}^{J_1\\times\\cdots\\times J_M \\times I_1 \\times\\cdots\\times\nI_N}$ and $\\mc{D} = (d_{k_1 \\cdots k_M i_1 \\cdots i_N})  \\in\n\\mathbb{R}^{K_1\\times\\cdots\\times K_M \\times I_1 \\times\\cdots\\times\nI_N}$. The {\\it column block tensor} consisting of $\\mc{C}$ and\n$\\mc{D}$ is\n\\begin{equation*}\\label{eq224}\n\\left[%\n\\begin{array}{c}\n  \\mc{C} \\\\\n  \\mc{D} \\\\\n\\end{array}%\n\\right]= [\\mc{C}^T ~ \\mc{D}^T]^T \\in \\mathbb{R}^{\\beta_1 \\times\n\\cdots \\times \\beta_M\\times\\alpha^N}.\n\\end{equation*}\nFor $\\mc{A}_1 \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times J_1\n\\times\\cdots\\times J_M}, \\mc{B}_1 \\in\n\\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times K_1 \\times\\cdots\\times\nK_M}, \\mc{A}_2 \\in \\mathbb{R}^{L_1\\times\\cdots\\times L_N \\times J_1\n\\times\\cdots\\times J_M}$ and $ \\mc{B}_2 \\in\n\\mathbb{R}^{L_1\\times\\cdots\\times L_N \\times K_1 \\times\\cdots\\times\nK_M}$, we denote $\\tau_1 = [\\mc{A}_1 ~ \\mc{B}_1]$ and $\\tau_2 =\n[\\mc{A}_2 ~ \\mc{B}_2]$ as the {\\it row block tensors}.\n The {\\it column block tensor}  $ \\left[\n\\begin{array}{c}\n  {\\tau}_1 \\\\\n  {\\tau}_2 \\\\\n\\end{array}\n\\right]\n$ can be written as\n\\begin{equation*}\\label{eq225}\n\\left[\n\\begin{array}{c}\n  \\mc{A}_1 ~~ \\mc{B}_1\\\\\n  \\mc{A}_2 ~~ \\mc{B}_2\\\\\n\\end{array}\n\\right] \\in \\mathbb{R}^{\\rho_1\\times\\cdots\\times \\rho_N \\times \\beta_1 \\times\\cdots\\times \\beta_M},\n\\end{equation*}\nwhere $\\rho_i = I_i +L_i, i=1,\\cdots,N; \\beta_j = J_j + K_j$ and $j=1,\\cdots , M.$\n\n\n\n\n\n\n\n\n\n\\begin{definition} (Definition 2.1, \\cite{stan}) \\\\\nThe range space and null space of a tensor $\\mc{A}\\in \\mathbb{R}^{{I_1}\\times \\cdots\\times {I_M}\\times {J_1}\\times\\cdots \\times {J_N}}$ are defined as per the following: \n$$\n\\mathfrak{R}(\\mc{A}) = \\left\\{\\mc{A}{*_N}\\mc{X}:~\\mc{X}\\in\\mathbb{R}^{{J_1}\\times\\cdots\\times {J_N}}\\right\\}\\mbox{ and } \\mc{N}(\\mc{A})=\\left\\{\\mc{X}:~\\mc{A}{*_N}\\mc{X}=\\mc{O}\\in\\mathbb{R}^{{I_1}\\times \\cdots \\times {I_M}}\\right\\}.\n$$\n\\end{definition}\nThe relation of range space for tensors is discussed in \\cite{stan} as follows.\n\\begin{lemma}[Lemma 2.2. \\cite{stan}]\\label{range-stan}\nLet  $\\mc{A}\\in \\mathbb{R}^{{I_1}\\times\\cdots\\times {I_M}\\times {J_1}\\times\\cdots\\times {J_N}}$,  $\\mc{B}\\in \\mathbb{R}^{{I_1}\\times\\cdots\\times {I_M}\\times {K_1}\\times\\cdots\\times {K_L}}.$ Then $\\mathfrak{R}(\\mc{B})\\subseteq\\mathfrak{R}(\\mc{A})$ if and only if there exists  $\\mc{U}\\in \\mathbb{R}^{{J_1}\\times\\cdots\\times {J_N}\\times {K_1}\\times\\cdots\\times {K_L}}$ such that \n$\\mc{B}=\\mc{A}{*_N}\\mc{U}.$\n\\end{lemma}\n\n\n\n\nThe next subsection is discussed the Boolean tensor and some useful definitions\n\\subsection{The Boolean tensor}\nThe binary Boolean algebra $\\mathfrak{B}$ consists of the set $\\{0,1\\}$ equipped with the operations of addition and multiplication defined as follows:\n\\begin{center}\n\\begin{tabular}{ c|c c } \n & 0 & 1 \\\\\n\\hline\n 0 & 0 & 1 \\\\ \n1 & 1 & 1 \\\\ \n\\end{tabular} \n\\hspace{2cm}\n\\begin{tabular}{ c|c c } \n. & 0 & 1 \\\\\n\\hline\n 0 & 0 & 0 \\\\ \n1 & 0 & 1 \\\\ \n\\end{tabular}\n\\end{center}\n\\begin{definition}\nLet $\\mc{A}=(a_{{i_1}...{i_M}{j_1}...{j_N}})\n \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_M \\times J_1 \\times\\cdots\\times J_N}.$ If $a_{{i_1}...{i_M}{j_1}...{j_N}}\\in\\{0,1\\},$ then the tensor $\\mc{A}$ is called Boolean tensor. \n \\end{definition}\n The addition and product of Boolean tensors  are defined as in Eqs. (\\ref{Eins}) and (\\ref{Eins1}) but addition and product of two entries will follow addition and product rule of Boolean algebra.\nThe order relation for tensors is defined as follows.\n\\begin{definition}\nLet $\\mc{A}=(a_{{i_1}...{i_M}{j_1}...{j_N}})\n \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_M \\times J_1 \\times\\cdots\\times J_N} \\text{ and }~\\mc{B}\n  =(b_{{i_1}...{i_M}{j_1}...{j_N}})~~ \\in  \\mathbb{R}^{I_1\\times\\cdots\\times I_M\n\\times J_1 \\times\\cdots\\times J_N}. $ Then $\\mc{A}\\leq \\mc{B}$ if and only if $a_{{i_1}...{i_M}{j_1}...{j_N}}\\leq b_{{i_1}...{i_M}{j_1}...{j_N}}$ for all $i_s$ and $j_t$ where $1\\leq s\\leq M$ and $1\\leq t\\leq N.$\n\\end{definition}\n\n\n\n We generalize the component-wise complement of the Boolean matrix \\cite{fitz} to Boolean tensors and defined below.\n\n\\begin{definition}\\label{CompDef} \nLet  $\\mc{A}=(a_{{i_1}...{i_N}{j_1}...{j_M}})\n \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times J_1 \\times\\cdots\\times J_M}$ be a Boolean tensor. A tensor $\\mc{B}=(b_{{i_1}...{i_N}{j_1}...{j_M}})\n \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_N \\times J_1 \\times\\cdots\\times J_M}$ is called component-wise complement of $\\mc{A}$ if \n\\begin{equation*}\nb_{{i_1}...{i_N}{j_1}...{j_M}}=\\left\\{\\begin{array}{cc}\n   1,   & \\mbox{  when  } a_{i_1\\cdots i\n  _Nj_1j_2\\cdots j\n  _M}=0. \\\\\n   0,   & \\mbox{  when } a_{i_1\\cdots i\n  _Nj_1j_2\\cdots j\n  _M}=1.\n \\end{array}\\right.\n \\end{equation*}\nThe tensor $\\mc{B}$ and its entries respectively, denoted by    $\\mc{A}^C$ and  $\\left(a_{i_1\\cdots i\n  _Nj_1\\cdots j\n  _M}^c\\right).$\n\\end{definition}\n\n\n\n\n \\section{Main Results} \nIn this section, we prove a few exciting results on tensors which are emphasized in the binary case. We divided this section into four folds.  In the first part of this section, we discuss some identities on the Boolean tensors.  Then, after having introduced some necessary ingredients, we study the generalized inverses of the Boolean tensor and some equivalence results to other generalized inverses in the second part.  The existence and uniqueness of weighted Moore-Penrose inverses are discussed in the third part. The space decomposition and its connection to generalized inverses are presented in the final part.\n\n\\subsection{Some identities on Boolean tensors}\n   \nBy the definition of  Boolean tensor $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_M\\times I_1\\times \\cdots \\times I_M},$ we always get $\\mc{A}+\\mc{A}=\\mc{A}.$ The infinite series of the Boolean tensor,  $\\displaystyle\\sum_{k=1}^\\infty\\mc{A}^k$, is convergent and reduces to a finite series, since there are only finite number of Boolean tensors of the same order. Now we denote  $\\overline{\\mc{A}}$ for the infinite series of the Boolean tensors, i.e.,  $$\\overline{\\mc{A}}=\\displaystyle\\sum_{k=1}^ \\infty \\mc{A}^k.$$ \n\n\n\nSince $\\mc{A}\\leq\\mc{A}+\\mc{B}$ for any two Boolean tensor  (suitable order for addition)  $\\mc{A}$ and $\\mc{B}$, likewise $\\mc{A}=\\mc{A}+\\mc{A}\\geq \\mc{A}+\\mc{B}$ for any two Boolean tensor $\\mc{A}\\geq \\mc{B}$. This is stated in the next result.\n\n\\begin{theorem}\\label{thm3.11}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots \\times I_M\\times J_1 \\times\\cdots \\times J_N}$ and $\\mc{B}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_M\\times J_1\\times\\cdots \\times J_N}.$ Then $\\mc{A}\\geq\\mc{B}$ if and only if $\\mc{A}+\\mc{B}=\\mc{A}$.\n\\end{theorem}\n\n\nIf we consider $\\mc{A}\\geq \\mc{I}$ in the above theorem, then it is easy to verify that   \n$\\mc{I}+\\mc{A}+\\cdots+\\mc{A}^n=\\mc{A}^n$ and hence we can have the following result as a corollary. \n\\begin{corollary}\\label{cor3.4}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times\\cdots \\times I_N\\times I_1 \\times\\cdots \\times I_N}$ and $\\overline{\\mc{A}}=\\sum_{k=1}^\\infty \\mc{A}^k.$ If $\\mc{A}\\geq\\mc{I},$ then there exist $n,$ such that\n\\begin{enumerate}\n    \\item[(a)] $\\overline{\\mc{A}}=\\mc{A}^{n};$\n    \\item[(b)] $\\left(\\overline{\\mc{A}}\\right)^2=\\overline{\\mc{A}};$\n    \\item[(c)] $\\overline{\\left(\\overline{\\mc{A}}\\right)}=\\overline{\\mc{A}}.$\n\\end{enumerate}\n \\end{corollary}\n\n\n   \nUsing the above theorem, we now prove another result on the Boolean tensor. As follows,\n\n   \n   \n\\begin{theorem}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots \\times  I_N\\times I_1 \\times \\cdots \\times I_N}$ and $\\mc{B}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_N\\times I_1\\times \\cdots \\times  I_N},$ with $\\mc{A}\\geq \\mc{I}$ and $\\mc{B}\\geq \\mc{I}.$ Then $$\\overline{(\\mc{A}+\\mc{B})}=\\overline{(\\overline{\\mc{A}}{*_N}\\overline{\\mc{B}})}=\\overline{(\\overline{\\mc{B}}{*_N}\\overline{\\mc{A}})}.$$ \n\\end{theorem}\n\n\n\n\\begin{proof}\nSince $\\mc{A}\\geq \\mc{I}$ and $\\mc{B}\\geq \\mc{I}.$ So $\\overline{\\mc{A}}\\geq \\mc{I}$ and  $\\overline{\\mc{B}}\\geq \\mc{I}.$  Also we have $\\overline{\\mc{A}}\\geq \\mc{A}$ and $\\overline{\\mc{B}}\\geq \\mc{B}.$ Combining these results, we get $\\overline{A}{*_N}\\overline{B}\\geq \\mc{A}$ and $\\overline{A}{*_N}\\overline{B}\\geq \\mc{B}.$ Thus $\\overline{A}{*_N}\\overline{B}\\geq \\mc{A}+\\mc{B}$ and hence\n\\begin{equation}\\label{eq3.61}\n  \\overline{\\left(\\overline{\\mc{A}}{*_N}\\overline{\\mc{B}}\\right)}\\geq \\overline{\\mc{A}+\\mc{B}}.  \n\\end{equation}\nNow $\\overline{\\mc{A}+\\mc{B}}\\geq \\overline{\\mc{A}}$ and $\\overline{\\mc{A}+\\mc{B}}\\geq \\overline{\\mc{B}}.$  By using Corollary \\ref{cor3.4} $(c)$, we get $\\overline{\\mc{A}}{*_N}\\overline{\\mc{B}}\\leq \\left(\\overline{\\mc{A}+\\mc{B}}\\right)^2=\\overline{\\mc{A}+\\mc{B}}.$ From Corollary \\ref{cor3.4} $(b)$, we have \n\\begin{equation}\\label{eq3.362}\n    \\overline{\\left(\\overline{\\mc{A}}{*_N}\\overline{\\mc{B}}\\right)}\\leq \\overline{\\left(\\overline{\\mc{A}+\\mc{B}}\\right)}=\\overline{\\mc{A}+\\mc{B}}.\n\\end{equation}\nFrom Eqs.(\\ref{eq3.61}) and (\\ref{eq3.362}), the proof is complete. \n\\end{proof}\n\nIf $\\mathfrak{R}(\\mc{B}^T)=\\mathfrak{R}(\\mc{B}^T{*_M}\\mc{A}^T),$ then there exist a tensor $\\mc{U}$ such that $\\mc{B}=\\mc{U}{*_M}\\mc{A}{*_N}\\mc{B}$ and hence, we obtain  $\\mc{B}{*_M}\\mc{C}=\\mc{U}{*_M}\\mc{A}{*_N}\\mc{B}{*_M}\\mc{C}=\\mc{U}{*_M}\\mc{A}{*_N}\\mc{B}{*_M}\\mc{D}=\\mc{B}{*_M}\\mc{D}.$ This leads the following result.\n\n\n\\begin{theorem}\\label{ltcan}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots \\times  I_M \\times J_1\\times \\cdots \\times  J_N}$, $\\mc{B}\\in\\mathbb{R}^{J_1\\times \\cdots \\times J_N\\times K_1\\times \\cdots \\times  K_M} $, \\\\ $\\mc{C}\\in\\mathbb{R}^{K_1\\times \\cdots \\times K_M\\times J_1\\times \\cdots\\times  J_N}$ and $\\mc{D}\\in\\mathbb{R}^{K_1\\times \\cdots \\times K_M \\times J_1\\times \\cdots \\times  J_N}$ be Boolean tensors with\\\\ $\\mc{A}{*_N}\\mc{B}{*_M}\\mc{C}=\\mc{A}{*_N}\\mc{B}{*_M}\\mc{D}.$ If $\\mathfrak{R}(\\mc{B}^T)=\\mathfrak{R}(\\mc{B}^T{*_N}\\mc{A}^T),$ then $\\mc{B}{*_M}\\mc{C}=\\mc{B}{*_M}\\mc{D}.$\n\\end{theorem}\n\nSimilar way, we can prove the following corollary.\n\n\n\\begin{corollary}\\label{rtcan}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M \\times J_1 \\times \\cdots \\times  J_N}$, $\\mc{B}\\in\\mathbb{R}^{J_1\\times \\cdots \\times J_N\\times K_1 \\times \\cdots \\times  K_M} $, \\\\ $\\mc{C}\\in\\mathbb{R}^{K_1\\times \\cdots \\times K_M\\times I_1\\times \\cdots\\times  I_M}$ and $\\mc{D}\\in\\mathbb{R}^{K_1 \\times \\cdots \\times  K_M \\times I_1\\times I_2\\times\\cdots\\times  I_M}$  be  Boolean tensors with $\\mc{C}{*_M}\\mc{A}{*_N}\\mc{B}=\\mc{D}{*_M}\\mc{A}{*_N}\\mc{B}.$ If $\\mathfrak{R}(\\mc{A})=\\mathfrak{R}(\\mc{A}{*_N}\\mc{B}),$ then $\\mc{C}{*_M}\\mc{A}=\\mc{D}{*_M}\\mc{A}.$\n\\end{corollary}\n\n\nWe now discuss the important result on a transpose of an arbitrary order Boolean tensor, as follows.\n\\begin{lemma}\\label{lemma1}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots \\times  I_M \\times J_1 \\times \\cdots \\times  J_N}$ be any Boolean tensor. Then $\\mc{A}\\leq\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{A}.$\n \\begin{proof}\n Let $\\mc{B} = \\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{A}.$ We need to show that\n \\begin{equation*}\n  {a}_{i_1\\cdots i_M j_1\\cdots j_N}\\leq {b}_{i_1\\cdots i_M j_1\\cdots j_N}.\n \\end{equation*}\n This inequality is trivial  if  ${a}_{i_1\\cdots i_M j_1\\cdots j_N}= 0.$ Let us assume ${a}_{i_1\\cdots i_M j_1\\cdots j_N}=1.$ Now\n\\begin{equation*}\n{b}_{i_1\\cdots i_M j_1\\cdots j_N} =\\sum_{k_1\\cdots k_N}\\sum_{l_1\\cdots l_M}a_{{i_1\\cdots i_M}{k_1\\cdots k_N}}a_{{l_1\\cdots l_M}{k_1\\cdots k_N}}a_{{l_1\\cdots l_M}{j_1\\cdots j_N}}.\n\\end{equation*}\nFor $1\\leq s\\leq N,$ if $k_s=j_s$ and $l_s=i_s,$ then\n\\begin{equation*}\n    {b}_{i_1\\cdots i_M j_1\\cdots j_N} \\geq ({a}_{i_1\\cdots i_M j_1\\cdots j_N})^3={a}_{i_1\\cdots i_M j_1\\cdots j_N}=1.\n\\end{equation*}\nHence the proof is complete.\n \\end{proof}\n\\end{lemma}   \n\n\n\\begin{theorem}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots \\times  I_N\\times J_1 \\times \\cdots \\times J_N}$ and $\\mc{B}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_N\\times J_1 \\times \\cdots \\times J_N}.$ Then the equation $\\mc{A}{*_N}\\mc{X}=\\mc{B}$ is solvable if and only if $\\mc{X}=\\mc{C},$ where \n$$ c_{i_1\\cdots i_N j_1\\cdots j_n}=\\left\\{\\begin{array}{cc}\n   1  &  \\mbox{ if } a_{i_1\\cdots i\n  _Ni_1\\cdots i\n  _N}=0 \\mbox{ or } b_{i_1\\cdots i\n  _Nj_1\\cdots j\n  _N}=1 \\mbox{ for all } i_k,~1\\leq k\\leq N,\\\\\n   0  & otherwise.\n\\end{array}\\right.\n$$\n\\end{theorem}\n\n\n\\begin{proof}\nLet $\\mc{A}{*_N}\\mc{X}=\\mc{B}$ is solvable and $\\mc{A}{*_N}\\mc{X}=D.$ To claim $\\mc{D}=\\mc{B},$ it is enough to show $d_{i_1\\cdots i\n  _Nj_1\\cdots j\n  _N}=1$ if and only if $b_{i_1\\cdots i\n  _Nj_1\\cdots j\n  _N}=1.$ Let $d_{i_1\\cdots i\n  _Nj_1\\cdots j\n  _N}=1.$ This implies $a_{i_1\\cdots i\n  _Np_1\\cdots p\n  _N}=1$ and $c_{p_1\\cdots p\n  _Nj_1\\cdots j\n  _N}=1$  for some $p_k~~1\\leq k\\leq N.$  The condition $c_{p_1\\cdots p\n  _Nj_1\\cdots j\n  _N}=1$ yields either $a_{i_1\\cdots i\n  _Np_1\\cdots p\n  _N}=0 $  or $b_{p_1\\cdots p\n  _Nj_1\\cdots j\n  _N}=1$ for all $p_k~~1\\leq k\\leq N.$  Since $a_{i_1\\cdots i\n  _Np_1\\cdots p\n  _N}=1$ which makes $b_{p_1\\cdots p\n  _Nj_1\\cdots j\n  _N}=1$ for all $p_k,~~1\\leq k\\leq N.$ Therefore $b_{i_1\\cdots i\n  _Nj_1\\cdots j\n  _N}=1.$ Now if $b_{i_1\\cdots i\n  _Nj_1\\cdots j\n  _N}=1,$ then $a_{i_1\\cdots i\n  _Nr_1\\cdots r\n  _N}=1$ and $x_{r_1\\cdots r\n  _Nj_1\\cdots j\n  _N}=1$  for some $r_k,~~1\\leq k\\leq N.$  Suppose $c_{r_1\\cdots r\n  _Nj_1\\cdots j\n  _N}=0.$ Then $a_{q_1\\cdots q\n  _Nr_1\\cdots r\n  _N}=1$ and $b_{q_1\\cdots q\n  _Nj_1\\cdots j\n  _N}=0$  for some $q_k,~~1\\leq k\\leq N.$ Combining $a_{q_1\\cdots q\n  _Nr_1\\cdots r\n  _N}=1$ and $x_{r_1\\cdots r\n  _Nj_1\\cdots j\n  _N}=1$, we get $b_{q_1\\cdots q\n  _Nj_1\\cdots j\n  _N}=1.$ Which is the contradiction. So $c_{r_1\\cdots r\n  _Nj_1\\cdots j\n  _N}=1$ and hence $d_{i_1\\cdots i\n  _Nj_1\\cdots j\n  _N}=1.$ The converse part is trivial. \n\\end{proof}\n\n\nIn view of the Definition \\ref{CompDef} the following theorem is true for Boolean tensors.\n\n\\begin{preposition}\\label{equitc}\nLet $\\mc{A} \\in\\mathbb{R}^{I_1\\times \\cdots \\times I_M\\times J_1\\times \\cdots \\times J_N}$ be a Boolean tensor,  then\n\\begin{enumerate}\n    \\item[(a)] $(\\mc{A}^C)^C=\\mc{A};$\n    \\item[(b)] $(\\mc{A}^C)^T=(\\mc{A}^T)^C = \\mc{A}^{CT}.$\n    \\end{enumerate}\n\\end{preposition}\n \n \n \n \\begin{remark}\nIn general $ \\mc{B}^C *_N \\mc{A}^C \\neq (\\mc{A}*_N\\mc{B})^C \\neq \\mc{A}^C *_N \\mc{B}^C$ for any two tensor $\\mc{A},~\\mc{B} \\in\\mathbb{R}^{I_1\\times\\cdots \\times I_M\\times I_1\\times\\cdots \\times I_M}$\n \\end{remark}\n \n\n\\begin{example}\nConsider two Boolean tensor\n$~\\mc{A}=(a_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ and $~\\mc{B}=(b_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ such that\n\\begin{eqnarray*}\na_{ij11} =\n    \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    0 & 0 &  1\n    \\end{pmatrix},\na_{ij12} =a_{ij13} =a_{ij21}=a_{ij22}=a_{ij23}=\n    \\begin{pmatrix}\n     0 & 0 & 0\\\\\n     0 & 0 & 1\n    \\end{pmatrix}, \\mbox{ and }\n\\end{eqnarray*}\n\\begin{eqnarray*}\nb_{ij11} =b_{ij12}=b_{ij13}=b_{ij21}=b_{ij22}=b_{ij23}=\n    \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    0 & 0 &  0\n    \\end{pmatrix}.\n\\end{eqnarray*}\nIt is easy to verify $ \\mc{B}^C {*_2} \\mc{A}^C \\neq (\\mc{A}{*_2}\\mc{B})^C \\neq \\mc{A}^C {*_2} \\mc{B}^C$, where\n$~(\\mc{A}*_2\\mc{B})^C=\\mc{X}=(x_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ $~\\mc{A}^C*_2\\mc{B}^C=\\mc{Y} =(y_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ and \n$~\\mc{B}^C*_2\\mc{A}^C=\\mc{Z} =(z_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}},$ where\n\\begin{eqnarray*}\nx_{ij11} =x_{ij12} =x_{ij13} =x_{ij21} =x_{ij22} =x_{ij23} =\n    \\begin{pmatrix}\n    0 & 1 & 1 \\\\\n    1 & 1 &  0\n    \\end{pmatrix},\n\\end{eqnarray*}\n\\begin{eqnarray*}\ny_{ij11} =y_{ij12} =y_{ij13} =y_{ij21} =y_{ij22} =y_{ij23} =\n    \\begin{pmatrix}\n    1 & 1 & 1 \\\\\n    1 & 1 & 0\n    \\end{pmatrix}, \\mbox{ and }\n\\end{eqnarray*}\n\\begin{eqnarray*}\nz_{ij11} =z_{ij12} =z_{ij13} =z_{ij21} = z_{ij22} =z_{ij23} =\n    \\begin{pmatrix}\n    0 & 1 & 1 \\\\\n    1 & 1 &  1\n    \\end{pmatrix}.\n\\end{eqnarray*}\n\\end{example}\n\n\nThe next result is the one of the important tool to prove trace of a Boolean tensor.\n\n\\begin{theorem}\n Let $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_N\\times J_1 \\times \\cdots \\times J_N}.$ Then $\\mc{A}{*_N}\\mc{A}^C=\\mc{O}$ if and only either $\\mc{A}=\\mc{O}$ or $\\mc{A}=\\mc{O}^C.$\n\\end{theorem}\n \\begin{proof}\n Since the converse part is trivial, it is enough to show the sufficient part only. Let $\\mc{A}{*_N}\\mc{A}^C=\\mc{O}.$ Thus \n$\\displaystyle\\sum_{k_1\\cdots k_N}a_{i_1\\cdots i_Nk_1\\cdots k_N}a_{k_1\\cdots k_Nj_1\\cdots j_N}^c=0$. This implies,  $ a_{i_1\\cdots i_Nk_1\\cdots k_N}a_{k_1\\cdots k_Nj_1\\cdots j_N}^c=0 $ for all $k_s,~1\\leq s\\leq N.$ Which again yields either $a_{i_1\\cdots i_Nk_1\\cdots k_N}=0 $  for all $i_s,~k_s$  or $a_{k_1\\cdots k_Nj_1\\cdots j_N}^c=0$  for all $j_s,~k_s,~1\\leq s\\leq N.$. Therefore either $\\mc{A}=\\mc{O}$ or $\\mc{A}^C=\\mc{O}.$ Hence completes the proof.\n \\end{proof}\n \nFurther, when $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_N\\times J_1 \\times \\cdots \\times J_N}$ is  symmetric Boolean tensor, one can write \n \\begin{eqnarray*}\n tr(\\mc{A}{*_N}\\mc{A}^C)&=&\\sum_{i_1\\cdots i_N}\\sum_{k_1\\cdots k_N}a_{i_1\\cdots i_Nk_1\\cdots k_N}a_{k_1\\cdots k_Ni_1\\cdots i_N}^c\\\\\n &=&\\sum_{i_1\\cdots i_N}\\sum_{k_1\\cdot, k_N}a_{k_1\\cdots k_Ni_1\\cdots i_N}^ca_{i_1\\cdots i_Nk_1\\cdots k_N}\\\\\n  &=&\\sum_{k_1\\cdots k_N}\\sum_{i_1\\cdots  i_N}a_{k_1\\cdots k_Ni_1\\cdots i_N}^ca_{i_1\\cdots i_Nk_1\\cdots k_N}\\\\\n&=&tr(\\mc{A}^C{*_N}\\mc{A}).\n  \\end{eqnarray*}\n\n Hence, the tensors in the trace of a product of symmetric tensor and its complement can be switched without changing the result. This is stated in the next result.\n \n \\begin{theorem}\\label{trresult}\n  Let $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_N\\times J_1 \\times \\cdots \\times J_N}.$ If $\\mc{A}$ is symmetric, then \n $$\n tr(\\mc{A}{*_N}\\mc{A}^C)=tr(\\mc{A}^C{*_N}\\mc{A}).\n $$\n \\end{theorem}\n \n\n \n \\begin{remark}\\label{rmk3.10}\n  In addition to the result of Theorem \\ref{trresult} one can write  $tr(\\mc{A}{*_N}\\mc{A}^C)=tr(\\mc{A}^C{*_N}\\mc{A}) = 0$ . \nFurther, the symmetricity condition in Theorem \\ref{trresult} is only sufficient  but not necessary.\n \\end{remark}\n  One can verify the Remark \\ref{rmk3.10} by the following example.\n \n \\begin{example}\n Let a Boolean tensor\n$~\\mc{A}=(a_{ijkl}) \\in \\mathbb{R}^{{2\\times2}\\times{2 \\rtimes 2}}$ such that\n\\begin{eqnarray*}\na_{ij11} =\n    \\begin{pmatrix}\n    1 & 0 \\\\\n    1 & 0\n    \\end{pmatrix},\na_{ij12} =\n    \\begin{pmatrix}\n     1 & 0 \\\\\n     1 & 0 \n    \\end{pmatrix},\na_{ij21} =\n    \\begin{pmatrix}\n     1 & 0 \\\\\n     1 & 0\n    \\end{pmatrix},\na_{ij22} =\n    \\begin{pmatrix}\n    1 & 0 \\\\\n    1 & 0\n    \\end{pmatrix}.\n\\end{eqnarray*}\nIt is clear that $\\mc{A}$ is not symmetric but $tr(\\mc{A}*_2\\mc{A}^C) = tr(\\mc{A}^C*_2\\mc{A}) = 2$, where $~\\mc{A}*_2\\mc{A}^C=(x_{ijkl}) \\in \\mathbb{R}^{{2\\times2}\\times{2 \\rtimes 2}}$ and $~\\mc{A}^C*_2\\mc{A}=(y_{ijkl}) \\in \\mathbb{R}^{{2\\times2}\\times{2 \\rtimes 2}}$ with entries\n\\begin{eqnarray*}\nx_{ij11} =\n    \\begin{pmatrix}\n    1 & 0 \\\\\n    1 & 0\n    \\end{pmatrix},\nx_{ij12} =\n    \\begin{pmatrix}\n     1 & 0 \\\\\n     1 & 0 \n    \\end{pmatrix},\nx_{ij21} =\n    \\begin{pmatrix}\n     1 & 0 \\\\\n     1 & 0\n    \\end{pmatrix},\nx_{ij22} =\n    \\begin{pmatrix}\n    1 & 0 \\\\\n    1 & 0\n    \\end{pmatrix},\n\\end{eqnarray*}\n\\begin{eqnarray*}\ny_{ij11} =\n    \\begin{pmatrix}\n    0 & 1 \\\\\n    0 & 1\n    \\end{pmatrix},\ny_{ij12} =\n    \\begin{pmatrix}\n     0 & 1 \\\\\n     0 & 1 \n    \\end{pmatrix},\ny_{ij21} =\n    \\begin{pmatrix}\n     0 & 1 \\\\\n     0 & 1\n    \\end{pmatrix},\ny_{ij22} =\n    \\begin{pmatrix}\n    0 & 1 \\\\\n    0 & 1\n    \\end{pmatrix}.\n\\end{eqnarray*}\n \\end{example}\n \n\n \n \n Using the complement of a tensor, we now prove the following result.\n \n \\begin{lemma}\\label{complem}\n Let $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_M\\times J_1 \\times \\cdots \\times  J_N},$ $\\mc{B}\\in\\mathbb{R}^{J_1 \\times \\cdots \\times  J_N\\times K_1 \\times \\cdots \\times  K_L}$ and $\\mc{C}\\in\\mathbb{R}^{K_1 \\times \\cdots \\times K_L\\times J_1 \\times \\cdots \\times J_N}$ be Boolean tensors. Then \n $$\\mc{A}{*_N}\\mc{B}{*_L}\\mc{C}\\leq \\mc{I}^C~~if ~and ~only ~if ~~\\mc{A}^C\\geq (\\mc{B}{*_L}\\mc{C})^T.$$\n \\end{lemma}\n \\begin{proof}\n $\\mc{A}{*_N}\\mc{B}{*_L}\\mc{C}\\leq \\mc{I}^C$  if and only $ \\sum_{j_1\\cdots j_N}\\sum_{k_1\\cdots k_L}a_{{i_1\\cdots i_M}{j_1\\cdots j_N}}b_{{j_1\\cdots j_N}{k_1\\cdots k_L}}c_{{k_1\\cdots k_L}{i_1\\cdots i_M}}=0$\nfor all $i_r,$ $1\\leq r\\leq M.$ This is equivalent to $a_{{i_1\\cdots i_M}{j_1\\cdots j_N}}b_{{j_1\\cdots j_N}{k_1\\cdots k_L}}c_{{k_1\\cdots k_L}{i_1\\cdots i_M}}=0$ \nfor all $i_r,$ $j_s$ and $k_t,$ where $1\\leq r\\leq M,$ $1\\leq s\\leq N,$ $1\\leq t\\leq L.$ This in turn is true if and only\n\\begin{eqnarray*}\n\\left(a_{{i_1\\cdots i_M}{j_1\\cdots j_N}}^c\\right)&\\geq& b_{{j_1\\cdots j_N}{k_1\\cdots k_L}}c_{{k_1\\cdots k_L}{i_1\\cdots i_M}}  \\mbox{ for all  $k_t$ }\\\\\n &=&\\left(c_{{i_1\\cdots i_M}{k_1\\cdots k_L}}^t\\right) \\left(b_{{k_1\\cdots k_L}{j_1\\cdots j_N}}^t\\right)~\\mbox{ for all $k_t$}\\\\\n  &\\geq& \\sum_{k_1\\cdots k_L}\\left(c_{{i_1\\cdots i_M}{k_1\\cdots k_L}}^t\\right) \\left(b_{{k_1\\cdots k_L}{j_1\\cdots j_N}}^t\\right)\\\\\n    &\\geq& \\left(\\mc{C}^T{*_N}\\mc{B}^T\\right)_{{i_1\\cdots i_M}{j_1\\cdots j_N}}=\\left(\\left(\\mc{B}{*_L}\\mc{C}\\right)^T\\right)_{{i_1\\cdots i_M}{j_1\\cdots j_N}}.\n \\end{eqnarray*}\n Thus the proof is complete.\n \\end{proof}\n Now we discuss the important result based on transpose and component-wise complement of an arbitrary order  Boolean tensor, as follows.\n\\begin{theorem}\\label{lucethm}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times  J_N}.$ Then $\\mc{X}{*_M}\\mc{A}\\leq \\mc{B}$ if and only if $\\mc{X}\\leq  \\left(\\mc{B}^C{*_N}\\mc{A}^T\\right)^C$, and $\\mc{A}{*_N}\\mc{X}\\leq \\mc{B}$ if and only if $\\mc{X}\\leq \\left(\\mc{A}^T {*_M} \\mc{B}^C\\right)^C$.\n\\end{theorem}\n\\begin{proof}\nLet $\\mc{X}{*_M}\\mc{A}\\leq \\mc{B}.$ This yields \n$\\sum_{k_1\\cdots k_M}x_{i_1\\cdots i_M k_1\\cdots k_M}a_{k_1\\cdots k_M j_1\\cdots j_N }\\leq b_{i_1\\cdots i_M j_1\\cdots j_N }$ for all $i_r,~(1\\leq r\\leq M)$ and $j_s,~(1\\leq s\\leq N).$ This is equivalent to $x_{i_1\\cdots i_M k_1\\cdots k_M}a_{k_1\\cdots k_M j_1\\cdots j_N }\\leq b_{i_1\\cdots i_M j_1\\cdots j_N }$  for all $i_r$ and $j_s$ and $k_t~(1\\leq t\\leq M).$ This in turns is true {\\it if and only if} $x_{i_1\\cdots i_M k_1\\cdots k_M}a_{k_1\\cdots k_M j_1\\cdots j_N } b_{i_1\\cdots i_M j_1\\cdots j_N }^c=0,$ for all $j_s$ and $k_t.$ Which is equivalent to\\\\ $x_{i_1\\cdots i_M k_1\\cdots k_M}a_{k_1\\cdots k_M j_1\\cdots j_N } \\{b_{j_1\\cdots j_N i_1\\cdots i_M }^t\\}^c=0$ for all $j_s$ and $k_t.$ Summing over all $j_s$ and $k_t,$ we get, $\\sum_{k_1\\cdots k_M}\\sum_{j_1\\cdots j_N} x_{i_1\\cdots i_M k_1\\cdots k_M}a_{k_1\\cdots k_M j_1\\cdots j_N } \\{b_{j_1\\cdots j_N i_1\\cdots i_M }^t\\}^c=0.$ This is true if and only $\\mc{X}{*_M}\\mc{A}{*_N}\\left(\\mc{B}^T\\right)^C\\leq \\mc{I}^C.$ By Preposition \\ref{equitc} $(a)$, this is equivalent to $\\mc{X}{*_M}\\mc{A}{*_N}\\left(\\mc{B}^C\\right)^T\\leq \\mc{I}^C.$ By Lemma \\ref{complem}, this in turns true if and only $\\mc{X}^C\\geq \\left(\\mc{A}{*_N}(\\mc{B}^C)^T\\right)^T,$ that is, {\\it if and only if} $ \\mc{X}\\leq \\left(\\mc{B}^C{*_M}\\mc{A}^T\\right)^C.$\n\nThis completes first part of the theorem. Similar way, we can show the second part of the theorem.\n\\end{proof}\n\\begin{corollary}\nLet $\\mc{E}=\\mc{O}^C,$ where $\\mc{O}$ is the zero tensor. Then the  following statements are equivalent:\n\\begin{enumerate}\n    \\item[(a)] $\\mc{X}{*_M}\\mc{A}=\\mc{O};$\n     \\item[(b)] $\\mc{X}\\leq \\left(\\left(\\mc{A}{*_N}\\mc{E}\\right)^T\\right)^C;$\n      \\item[(c)] $\\mc{E}{*_N}\\mc{X}\\leq \\left(\\left(\\mc{A}{*_N}\\mc{E}\\right)^T\\right)^C.$\n\\end{enumerate}\n\\end{corollary}\nThe same result is also true for $\\mc{A}{*_N}\\mc{X}=\\mc{O}.$  Also the following corollary easily follow from Theorem \\ref{lucethm}.\n\n\\begin{corollary}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_M\\times J_1 \\times \\cdots \\times  J_N}$ and  $\\mc{X}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_M\\times I_1 \\times \\cdots \\times  I_M}.$  Then $\\mc{X}{*_M}\\mc{A}=\\mc{B}$ has a solution if and only if $\\mc{B}\\leq \\left(\\mc{B}^C{*_N}\\mc{A}^T\\right)^C{*_M}\\mc{A}.$\n\\end{corollary}\n\n\n\n   \n   \n   \n   \n\\subsection{Generalized inverses of Boolean tensors}\nFor the generalization of the generalized inverses of Boolean matrix \\cite{rao}, we introduce the definition of $\\{i\\}$-inverses $(i = 1, 2, 3, 4)$ and the Moore-Penrose inverse of Boolean tensors via the Einstein product, as follows. \n\n \n \\begin{definition}\\label{defgi}\n For any Boolean tensor $\\mc{A} \\in \\mathbb{R}^{I_1\\times\\cdots\\times I_M \\times J_1 \\times\\cdots\\times J_N},$  consider the following equations in $\\mc{X} \\in\n\\mathbb{R}^{J_1\\times\\cdots\\times J_N \\times I_1 \\times\\cdots\\times\nI_M}:$\n\\vspace{-.4cm}\n\\begin{eqnarray*}\n&&(1)~\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A} = \\mc{A},\\\\\n&&(2)~\\mc{X}{*_M}\\mc{A}{*_N}\\mc{X} = \\mc{X},\\\\\n&&(3)~(\\mc{A}{*_N}\\mc{X})^T = \\mc{A}{*_N}\\mc{X},\\\\\n&&(4)~(\\mc{X}{*_M}\\mc{A})^T = \\mc{X}{*_M}\\mc{A}.\n\\end{eqnarray*}\n\\vspace{-.34cm}\nThen $\\mc{X}$ is called\n\\begin{enumerate}\n\\item[(a)] \na generalized inverse of $\\mc{A}$ if it satisfies $(1)$ and denoted by $\\mc{A}^{(1)}.$\n\\item[(b)]  a reflexive generalized inverse of $\\mc{A}$ if it satisfies $(1)$ and $(2)$, which is denoted by $\\mc{A}^{(1,2)}.$\n\\item[(c)] a $\\{1,3\\}$ inverse of $\\mc{A}$ if it satisfies $(1)$ and $(3)$, which is denoted by $\\mc{A}^{(1,3)}.$\n\\item[(d)]  a $\\{1,4\\}$ inverse of $\\mc{A}$ if it satisfies $(1)$ and $(4)$, which is  denoted by $\\mc{A}^{(1,4)}.$\n\\item[(e)] the Moore-Penrose inverse of $\\mc{A}$ if it satisfies all four conditions $[(1)-(4)]$, which is denoted by $\\mc{A}^{\\dagger}.$\n\\end{enumerate}\n\\end{definition}\n\nThe following remark and corollary are follows from the Definition \\ref{defgi}.\n\n\\begin{remark}\\label{rm11}\n If $\\mc{X}$ is the generalized inverse of a Boolean tensor $\\mc{A} \\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times J_N}$ then $\\mc{X}{*_M}\\mc{A}{*_N}\\mc{X}$ is the reflexive generalized inverse of $\\mc{A}.$\n\\end{remark}\n\n\\begin{corollary}\\label{corm1}\nIf $\\mc{X}$ is the generalized inverse of a Boolean tensor $\\mc{A} \\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times J_N}$ Then   \n\\begin{enumerate}\n    \\item[(a)] $\\mc{X}^T$ is the generalized inverse of $\\mc{A}^T;$\n    \\item[(b)] $(\\mc{X}_1+\\mc{X}_2)$ is the generalized inverse of of a Boolean tensor $\\mc{A}$ when $\\mc{X}_1$ and $\\mc{X}_2$ are two generalized inverse of $\\mc{A}.$\n\\end{enumerate}\n\\end{corollary}\n\n\nThus the existence of generalized inverse of a Boolean tensor guarantees the existence of a reflexive generalized inverse. In addition to that, the Remark \\ref{rm11} and Corollary \\ref{corm1} (b) ensures that the existence of one-generalized inverse implies the existence of finite number generalized inverses. In view of the fact, we define the maximum generalized inverse of a Boolean tensor, as follow:\n\n\\begin{definition}\n Let $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots \\times I_M\\times J_1 \\times \\cdots \\times J_N}.$ A tensor $\\mc{X}$ is called maximum generalized inverse of $\\mc{A}$ if $\\mc{G}\\leq \\mc{X}$ for every generalized inverse $\\mc{G}$ of $\\mc{A}.$\n\\end{definition}\n\nNote that, the generalized inverse of a Boolean tensor need not be unique which explained in the next example.\n\n\\begin{example}\\label{example18}\nConsider a Boolean tensor\n$~\\mc{A}=(a_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ with entries\n\\begin{eqnarray*}\na_{ij11} =a_{ij12} =a_{ij13} =a_{ij21} =a_{ij22} =a_{ij23} =\n    \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    1 & 0 &  0\n    \\end{pmatrix}.\n\\end{eqnarray*}\nThen it can be easily verified that both tensors\n$~\\mc{X}=(x_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$  and $~\\mc{Y}=(y_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$  with entries\n\\begin{eqnarray*}\nx_{ij11} =\n    \\begin{pmatrix}\n    0 & 1 & 1 \\\\\n    1 & 1 &  1\n    \\end{pmatrix},\nx_{ij12} =x_{ij13} =x_{ij21} = x_{ij22} =x_{ij23} =\n    \\begin{pmatrix}\n     0 & 0 & 0\\\\\n     0 & 0 & 0\n    \\end{pmatrix},\\mbox{ and }\n   \\end{eqnarray*}\n\\begin{eqnarray*}\ny_{ij11} =\n    \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    0 & 0 &  1\n    \\end{pmatrix},\ny_{ij12} =y_{ij13} =y_{ij21} =y_{ij22} =y_{ij23} =\n    \\begin{pmatrix}\n     0 & 0 & 0\\\\\n     0 & 0 & 0\n    \\end{pmatrix},\n\\end{eqnarray*}\nare satisfies the required condition of the Definition \\ref{defgi}.\n\\end{example}\n \n Foa a Boolean tensor $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_N \\times J_1 \\times \\cdots \\times   J_N},$ the number of generalized inverses are finite and the maximum number of generalized inverses is $2^{I_1 \\times \\cdots \\times  I_N\\times I_1 \\times \\cdots \\times I_N}.$\n The next result assures the uniqueness and is true only for invertiable  tensors.\n \\begin{lemma}\n Let $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_N\\times I_1 \\times \\cdots \\times  I_N}$ be any Boolean tensor. If $\\mc{A}$ is invertiable then $\\mc{A}^{-1}$ is the only generalized inverse of $\\mc{A}.$\n \\end{lemma}\n Next, we discus the equivalence condition for consistent system and generalized inverse. \n \n \\begin{theorem}\\label{eqgen}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times J_N}$ and $\\mc{X}\\in\\mathbb{R}^{J_1 \\times \\cdots \\times J_N\\times I_1 \\times \\cdots \\times  I_M}.$  Then the followings are equivalent:\n\\begin{enumerate}\n    \\item[(a)] $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}=\\mc{A}.$\n    \\item[(b)] $\\mc{X}{*_M}\\mc{Y}$ is a solution of the tensor equation $\\mc{A}{*_N}\\mc{Z}=\\mc{Y}$ whenever $\\mc{Y}\\in\\mathfrak{R}(\\mc{A}).$\n    \\item[(c)] $\\mc{A}{*_N}\\mc{X}$ is idempotent and $\\mathfrak{R}(\\mc{A})=\\mathfrak{R}(\\mc{A}{*_N}\\mc{X}).$\n     \\item[(d)] $\\mc{X}{*_M}\\mc{A}$ is idempotent and $\\mathfrak{R}(\\mc{A^T})=\\mathfrak{R}(\\mc{A}^T{*_N}\\mc{X}^T).$\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\n\\begin{proof}\nFirst we will claim $(a)$ {\\it if and only if} $(b).$ Let us assume $(a)$ holds and $\\mc{Y}\\in\\mathfrak{R}(\\mc{A}).$ Then there exists a Boolean tensor $\\mc{Z}\\in \\mathbb{R}^{J_1\\times J_2\\times\\cdots\\times  J_N}$ such that $\\mc{A}{*_N}\\mc{Z}=\\mc{Y}.$ Now \n$$\\mc{A}{*_N}\\mc{X}{*_M}\\mc{Y}=\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}{*_N}\\mc{Z}=\\mc{A}{*_N}\\mc{Z}=\\mc{Y}.$$ Therefore, $\\mc{X}{*_M}\\mc{Y}$ is a solution of $\\mc{A}{*_N}\\mc{Z}=\\mc{Y}.$ Conversely assume $(b)$ is true. That is $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{Y}=\\mc{Y}$ for all $\\mc{Y}\\in\\mathfrak{R}(\\mc{A}).$ Since $\\mc{Y}\\in\\mathfrak{R}(\\mc{A})$ which implies there exists $\\mc{U}\\in \\mathbb{R}^{J_1\\times\\cdots \\times  J_N}$ such that $\\mc{A}{*_N}\\mc{U}=\\mc{Y}.$ Thus $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}{*_N}\\mc{U}=\\mc{A}{*_N}\\mc{U}$ for all $\\mc{U}\\in \\mathbb{R}^{J_1\\times\\cdots\\times  J_N}.$ Therefore $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}=\\mc{A}.$ Next we show the equivalence between $(a)$ and $(c).$ Clearly $(a)$ implies $\\mc{A}{*_N}\\mc{X}$ idempotent. Since $\\mc{A}=\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}$ and $\\mc{A}{*_N}\\mc{X}=\\mc{A}{*_N} {\\mc{X}{*_M}\\mc{A}{*_N}\\mc{X}},$ so by Lemma \\ref{range-stan} $\\mathfrak{R}(\\mc{A})=\\mathfrak{R}(\\mc{A}{*_N}\\mc{X}).$ Using the same idea, we can easily show the equivalence between $(a)$ and $(d).$ Hence completes the proof. \n\\end{proof}\n\n\n\n\nSince $\\mc{A}^T{*_M}\\mc{A}{*_N}\\mc{X}_1{*_N}\\mc{A}^T{*_M}\\mc{A}=\\mc{A}^T{*_M}\\mc{A}$, so by Theorem \\ref{ltcan}, $\\mc{A}{*_N}\\mc{X}_1{*_N}\\mc{A}^T{*_M}\\mc{A}=\\mc{A}.$ Which leads the following corollary. \n\n\\begin{corollary}\nLet $\\mathfrak{R}(\\mc{A}^T)=\\mathfrak{R}(\\mc{A}^T{*_M}\\mc{A}).$ If $\\mc{X}_1$ and $\\mc{X}_2$ are generalized inverses of $\\mc{A}^T{*_M}\\mc{A}$ and $\\mc{A}{*_N}\\mc{A}^T$ respectively, then $\\mc{X}_1{*_N}\\mc{A}^T$ and $\\mc{A}^T{*_M}\\mc{X}_2$ are generalized inverse of $\\mc{A}.$\n\\end{corollary}\n\n\n\nFurther, from the range conditions, if $\\mathfrak{R}(\\mc{A}^T)\\subseteq \\mathfrak{R}(\\mc{B}^T)$ and $\\mathfrak{R}(\\mc{C})\\subseteq \\mathfrak{R}(\\mc{B}).$ Then  $\\mc{A}=\\mc{V}{*_M}\\mc{B}$ and $\\mc{C}=\\mc{B}{*_N}\\mc{U}$ for some tensors $\\mc{U}$ and $\\mc{V}.$ Now \n$\\mc{A}{*_N}\\mc{X}{*_M}\\mc{C}= \\mc{V}{*_M}\\mc{B}{*_N}\\mc{X}{*_M}\\mc{B}{*_N}\\mc{U}=\\mc{V}{*_M}\\mc{B}{*_N}\\mc{U}$ which does not rely on $\\mc{X}$. So it is invariant to the choice of $\\mc{X}.$ So, we conclude this observation in the following corollary.\n\n\n\n\n\n\n\n\n\\begin{corollary}\nLet $\\mc{A},$ $\\mc{B}$ and $\\mc{C}$ be suitable tensors such that $\\mathfrak{R}(\\mc{A}^T)\\subseteq \\mathfrak{R}(\\mc{B}^T)$ and $\\mathfrak{R}(\\mc{C})\\subseteq \\mathfrak{R}(\\mc{B}).$ If the generalized inverse of $\\mc{B}$ exists, then $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{C}$ is invariant to $\\mc{X},$ where $\\mc{X}$ is the generalized inverse  of $\\mc{B}.$\n\\end{corollary}\n\n\n\n\nTo prove the next result, we define regular and singular of tensors, i.e., \nA tensor $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots \\times I_M\\times J_1\\times \\cdots J_N},$ is called regular if the tensor equation $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}=\\mc{A}$ has a solution, otherwise called \\textit{singular}.\n\n\\begin{theorem}\\label{thm3.43}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times J_N},$  $\\mc{S}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times I_1 \\times \\cdots \\times I_M},$ and $\\mc{T}\\in\\mathbb{R}^{J_1 \\times \\cdots \\times  J_M\\times J_1 \\times \\cdots \\times  J_N}.$ If $\\mc{S}$ and $\\mc{T}$ are invertible, then the following are equivalent:\n\\begin{enumerate}\n    \\item[(a)] $\\mc{A}$ is regular.\n     \\item[(b)] $\\mc{S}{*_M}\\mc{A}{*_N}\\mc{T}$ is regular.\n     \\item[(c)] $\\mc{A}^T$ is regular.\n       \\item[(d)] $\\mc{T}{*_N}\\mc{A}^T{*_M}\\mc{S}$ is regular.\n\\end{enumerate}\n\\end{theorem}\n\n\n\nBased on the block tensor\\cite{sun} and their properties, we have the following lemma.\n\\begin{lemma}\\label{block}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_M\\times J_1 \\times \\cdots \\times  J_N}.$ Then $\\mc{A}$ is regular if and only if $\\begin{bmatrix}\n\\mc{A} &\\mc{O}\\\\\n\\mc{O} & \\mc{B}\n\\end{bmatrix}$ is regular for all regular tensors $\\mc{B}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times I_M\\times J_1 \\times \\cdots \\times  J_N}.$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mc{A}$ and $\\mc{B}$ be regular tensors. Then there exist tensors $\\mc{X}$ and $\\mc{Y}$ such that $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}=\\mc{A}$ and $\\mc{B}{*_N}\\mc{Y}{*_M}\\mc{B}=\\mc{B}.$ Let $\\mc{Z}=\\begin{bmatrix}\n\\mc{X} & \\mc{O}\\\\\n\\mc{O} & \\mc{Y}\\\\\n\\end{bmatrix}.$ Now \n\\begin{eqnarray*}\n\\begin{bmatrix}\n\\mc{A} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}\\\\\n\\end{bmatrix}{*_N}\\mc{Z}{*_M}\\begin{bmatrix}\n\\mc{A} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}\\\\\n\\end{bmatrix}&=&\\begin{bmatrix}\n\\mc{A} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}\\\\\n\\end{bmatrix}{*_N}\\begin{bmatrix}\n\\mc{X} & \\mc{O}\\\\\n\\mc{O} & \\mc{Y}\\\\\n\\end{bmatrix}{*_M}\\begin{bmatrix}\n\\mc{A} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}\\\\\n\\end{bmatrix}\\\\\n&=&\n\\begin{bmatrix}\n\\mc{A}{*_N}\\mc{X} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}{*_N}\\mc{Y}\\\\\n\\end{bmatrix}{*_M}\\begin{bmatrix}\n\\mc{A} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}\\\\\n\\end{bmatrix}\\\\\n&=&\\begin{bmatrix}\n\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}{*_N}\\mc{Y}{*_M}\\mc{B}\\\\\n\\end{bmatrix}=\\begin{bmatrix}\n\\mc{A} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}\\\\\n\\end{bmatrix}.\n\\end{eqnarray*}\nThus $\\begin{bmatrix}\n\\mc{A} & \\mc{O}\\\\\n\\mc{O} & \\mc{B}\\\\\n\\end{bmatrix}$ is regular. The converse part can be proved in the similar way. \n\\end{proof}\n\n   We now present another characterization of the generalized inverse of the Boolean tensor, as follows.\n\\begin{theorem}\\label{lemcomp}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times  J_N}.$ Then \n$$\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}\\leq \\mc{A}~~ if ~and~ only~ if ~~\\mc{X}\\leq \\left(\\mc{A}{*_N}\\mc{A}^{CT}{*_M}\\mc{A}\\right)^{CT}.$$\n\\end{theorem}\n\\begin{proof}\nApplying Theorem \\ref{lucethm} repetitively, we get \n$\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}\\leq \\mc{A}$ if and only if $\\mc{X}{*_M}\\mc{A}\\leq \\left(\\mc{A}^T{*_M}\\mc{A}^C\\right)^C$, which equivalently if and only if \n\\begin{equation*}\n\\mc{X}\\leq \\left(\\left(\\left(\\mc{A}^T{*_M}\\mc{A}^C\\right)^C\\right)^C{*_N}\\mc{A}^T\\right)^C = \\left(\\mc{A}^T{*_M}\\mc{A}^C{*_N}\\mc{A}^T\\right)^C =\\left(\\mc{A}{*_N}\\mc{A}^{CT}{*_M}\\mc{A}\\right)^{CT}.\n\\end{equation*}\n\\end{proof}\nUsing the Theorem \\ref{lemcomp},  and the fact of transpose and component-wise complement of a Boolean tensor, we obtain an important result for finding the maximum generalized inverse of a Boolean tensor. \n\\begin{corollary}\nLet  $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times  J_N}$ be  regular. Then the following are holds\n\\begin{enumerate}\n\\item[(a)] $\\mc{A}=\\mc{A}{*_N}\\left(\\mc{A}{*_N}\\mc{A}^{CT}{*_M}\\mc{A}\\right)^{CT}{*_M}\\mc{A};$\n    \\item[(b)] $\\left(\\mc{A}{*_N}\\mc{A}^{CT}{*_M}\\mc{A}\\right)^{CT}$ is the maximum generalized inverse of $\\mc{A};$ \n    \\item[(c)] $\\left(\\mc{A}{*_N}\\mc{A}^{CT}{*_M}\\mc{A}\\right)^{CT}{*_M}\\mc{A}{*_N}\\left(\\mc{A}{*_N}\\mc{A}^{CT}{*_M}\\mc{A}\\right)^{CT}$ is the maximum reflexive generalized inverse of $\\mc{A}.$\n\\end{enumerate} \n\\end{corollary}\n\n\n\n\nNext, we discuss some equivalence results between generalized and other inverses. \n\n\n\n\n\n\\begin{theorem}\\label{eqv14}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times   J_N}$ be any Boolean tensor, then the following statements are equivalent:\n\\begin{enumerate}\n\\item[(a)] $\\mc{A}^{(1,4)}$ exists.\n\\item[(b)] $\\mc{A}^{(1)}$ exists and $\\mathfrak{R}(\\mc{A}) = \\mathfrak{R}(\\mc{A}{*_N}\\mc{A}^T).$ \n\\item[(c)] $(\\mc{A}{*_N}\\mc{A}^T)^{(1)}$ exists and  $\\mc{X}{*_M}\\mc{A}{*_N}\\mc{A}^T = \\mc{A}^T$ for some tensor $\\mc{X}.$\n\\end{enumerate}\n\n\n\\begin{proof}\nConsider $(a)$ is true and $\\mc{A}^{(1,4)}=\\mc{X}.$ Existence of  $\\mc{A}^{(1)}$ is trivial and hence $\\mathfrak{R}(\\mc{A}) = \\mathfrak{R}(\\mc{A}{*_N}\\mc{A}^T).$ Now we claim $(b)\\Rightarrow (c).$ Let $\\mc{A}^{(1)}$ exists and  $\\mathfrak{R}(\\mc{A}) = \\mathfrak{R}(\\mc{A}{*_N}\\mc{A}^T).$  Then there exist a Boolean tensor  $\\mc{U}\\in\\mathbb{R}^{I_1\\times\\cdots \\times I_M\\times J_1\\times\\cdots\\times  J_N}$ such that $\\mc{A} = \\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{U}.$  Which implies $\\mc{A}{*_N}\\mc{A}^{T}=\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{U}{*_N}\\mc{U}^T{*_M}\\mc{A}{*_N}\\mc{A}^T.$ So generalized inverse of $\\mc{A}{*_N}\\mc{A}^T$ exists. If we take $\\mc{X}=\\mc{A}^T{*_N}(\\mc{A}{*_N}\\mc{A}^T)^{(1)},$ then \n\\begin{eqnarray*}\n\\mc{X}{*_M}\\mc{A}{*_N}\\mc{A}^T &=&  {\\mc{A}^T}{*_M}(\\mc{A}{*_N}\\mc{A}^T )^{(1)}{*_M}\\mc{A}{*_N} \\mc{A}^T=\\mc{U}^T{*_M}\\mc{A}{*_N}\\mc{A}^T{*_M}(\\mc{A}{*_N}\\mc{A}^T )^{(1)}{*_M}\\mc{A}{*_N}\\mc{A}^T\\\\\n&=&\\mc{U}^T{*_M}\\mc{A}{*_N}\\mc{A}^T=\\mc{A}^T.\n\\end{eqnarray*}\nFinally, we claim $(c)\\Rightarrow (a).$ Let $\\mc{X}{*_M}\\mc{A}{*_N}\\mc{A}^T = \\mc{A}^T$. Taking transpose on both sides, we get $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A} = \\mc{A} $, As\n\\begin{eqnarray*}\n(\\mc{X}{*_M}\\mc{A})^T &= &\\mc{A}^T{*_M}\\mc{X}^T = \\mc{X}{*_M}\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{X}^T\\\\\n&=& (\\mc{X}{*_M}\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{X}^T)^T = (\\mc{A}^T{*_M}\\mc{X}^T)^T = \\mc{X}{*_M}\\mc{A}. \n\\end{eqnarray*}\nThus $\\mc{X}=\\mc{A}^{(1,4)}.$ Hence the proof is complete.\n\\end{proof}\n\\end{theorem}\n\n\n\n\n\nUsing the similar way, we can show the following theorem.\n\n\n\n\n\\begin{theorem}\\label{eqv13}\nLet $\\mc{A}$ be any Boolean tensor, then the following statements are equivalent:\n\\begin{enumerate}\n\\item[(a)] $\\mc{A}^{(1,3)}$ exists.\n\\item[(b)] $\\mc{A}^{(1)}$ exists and $\\mathfrak{R}(\\mc{A}^T) = \\mathfrak{R}(\\mc{A}^T{*_M}\\mc{A}).$ \n\\item[(c)] There exists a Boolean tensor $\\mc{X}$ such that $\\mc{A}^T=\\mc{A}^T{*_M}\\mc{A}{*_N}\\mc{X}.$ \n\\end{enumerate}\n\\end{theorem}\n\n\n\n\nWe now discuss the characterization of Moore-Penrose inverse of Boolean tensors.  The similar proof of Theorem 3.2 in \\cite{sun}, we have the uniqueness of the Moore-Penrose inverse of a Boolean tensor in $\\mathbb{R}^{I_1\\times \\cdots \\times I_M\\times J_1\\times\\cdots\\times  J_N}$, as follows.\n\n\n\n\n\\begin{lemma}\\label{mpiu}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times  J_N}$ be any Boolean tensor. If the Moore-Penrose inverse of $\\mc{A}$ exists then it is unique.\n\\end{lemma}\n In the next lemma, we discuss an estimate of Moore-Penrose inverse a tensor, as follows.\n \\begin{lemma}\\label{lemma2}\n Let   $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times  J_N}$ be a Boolean tensor and suppose $\\mc{A}$ admits a Moore-Penrose inverse. Then  $\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{A}\\leq\\mc{A}$\n \\begin{proof}\n Let $\\mc{B}= \\mc{A}^T{*_M}\\mc{A}$. Since $\\mc{B}$ is a Boolean tensor of even order and there are finitely many  Boolean tensors of same order, so there must exist positive integers $s,t\\in\\mathbb{N}$ such that $\\mc{B}^s$ = $\\mc{B}^{s+t}.$ Without loss of generality, we can assume that $s$ is the smallest positive integer for which $\\mc{B}^s= \\mc{B}^{s+t}$ for some $t\\in \\mathbb{N}.$ Now we will show $s =1.$ Suppose $s\\geq 2$. Let $\\mc{X}$ be the Moore-Penrose inverse of $\\mc{A}$. Since $\\mc{B}= \\mc{A}^T{*_M}\\mc{A}$ and $\\mc{B}^s= \\mc{B}^{s+t}$ which implies $\\mc{A}^T{{*_M}}\\mc{A}{{*_N}}\\mc{B}^{s-1} = \\mc{A}^T{{*_M}}\\mc{A}{{*_N}}\\mc{B}^{s+t-1}.$  Pre-multiplying both side $\\mc{X}^T$ yields $\\mc{A}{{*_N}}\\mc{B}^{s-1} = \\mc{A}{{*_N}}\\mc{B}^{s+t-1},$  which implies $\\mc{A}{*_N}\\mc{A}^T{{*_M}}\\mc{A}{{*_N}}\\mc{B}^{s-2} = \\mc{A}{*_N}\\mc{A}^T{{*_M}}\\mc{A}{{*_N}}\\mc{B}^{s+t-2}.$ Further, pre-multiplying both side $\\mc{X}$ yields $\\mc{A}^T{*_M}\\mc{X}^T{*_N}\\mc{A}^T{{*_M}}\\mc{A}{{*_N}}\\mc{B}^{s-2} =\\mc{A}^T{*_M}\\mc{X}^T{*_N}\\mc{A}^T{{*_M}}\\mc{A}{{*_N}}\\mc{B}^{s+t-2},$  which implies $\\mc{B}^{s-1}= \\mc{B}^{s+t-1}.$\n \nThus, the minimality of $s$ is false and hence $s=1.$ Therefore $\\mc{B}=\\mc{B}^{t+1}$ for some $t\\in\\mathbb{N}.$ Again we have $\\mc{B}=\\mc{B}^{t+1}$, which implies $\\mc{A}^T{{*_M}}\\mc{A} = \\mc{A}^T{{*_M}}\\mc{A}{{*_N}}\\mc{B}^{t}$. Pre-multiplying $\\mc{X}^T$ both sides yields\n$\\mc{A}{*_N}\\mc{X}{{*_M}}\\mc{A} = \\mc{A}{*_N}\\mc{X}{{*_M}}\\mc{A}{{*_N}}\\mc{B}^{t}.$\n Thus \n \\begin{equation}\\label{eqfl}\n \\mc{A} = \\mc{A}{{*_N}}\\mc{B}^{t}=\\mc{A}{*_N}(\\mc{A}^T{*_M}\\mc{A})^t.\n \\end{equation}\nApplying Lemma \\ref{lemma1} to $\\mc{A}{*_N}\\mc{A}^T{*_N}\\mc{A}$ repetitively and combining Eq. (\\ref{eqfl}), we obtain\n$$\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{A}\\leq \\mc{A}{*_N}(\\mc{A}^T{*_M}\\mc{A})^2\\leq\\cdots\\leq \\mc{A}{*_N}(\\mc{A}^T{*_M}\\mc{A})^t=\\mc{A}.$$\n  \\end{proof}\n\\end{lemma}\n\n\nUsing the Lemma \\ref{lemma1} and \\ref{lemma2} one can obtain  an interesting result on invertibility of Boolean tensor as follows, \n\n\\begin{corollary}\nA Boolean tensor $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_N\\times I_1 \\times \\cdots \\times  I_N}$ is invertible if and only if \n $$\\mc{A}{*_N}\\mc{A}^T=\\mc{A}^T{*_M}\\mc{A}=\\mc{I}.$$ \n\\end{corollary}\nFrom the Definition \\ref{perm}, we obtain\n\\begin{eqnarray*}\n(\\mc{P}{*_N}\\mc{P}^T)_{i_1 \\cdots i_Nj_1\\cdots j_N}&=&\\sum_{k_1\\cdots k_N}(\\mc{P})_{i_1 \\cdots i_Nk_1\\cdots k_N}(\\mc{P}^T)_{k_1 \\cdots k_Nj_1\\cdots j_N}\\\\\n&=&\\sum_{k_1\\cdots k_N}(\\mc{P})_{i_1 \\cdots i_Nk_1\\cdots k_N}(\\mc{P})_{j_1 \\cdots j_Nk_1\\cdots k_N}\\\\\n&=&(\\mc{P})_{i_1 \\cdots i_N\\pi(j_1)\\cdots \\pi(j_N)}(\\mc{P})_{j_1 \\cdots j_N\\pi(j_1)\\cdots \\pi(j_N)}\\\\\n&=& \\left\\{\\begin{array}{cc}\n   1  & \\mbox{ if } i_s=j_s \\mbox{ for all } 1\\leq s\\leq N. \\\\\n    0 & \\mbox{ otherwise.}\n\\end{array}\\right.\\\\\n&=&(\\mc{I})_{i_1 \\cdots i_Nj_1\\cdots j_N}.\n\\end{eqnarray*}\nSimilar way, we can also show $\\mc{P}^T{*_N}\\mc{P}=\\mc{I}.$ Therefore, every permutation tensors are orthogonal and  invertible. Adopting this result, we now present a characterization of the permutation tensor, as follows.\n\\begin{preposition}\\label{permu}\nA Boolean tensor $\\mc{A}$ has an inverse {\\it if and only if} it is a permutation tensor.\n\\end{preposition}\nNext result contains five equivalent conditions involving the existence of Moore-Penrose inverse of a Boolean tensor.\n \\begin{theorem}\\label{thm1}\n Let $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M \\times J_1\\times \\cdots \\times  J_N.}$ be any tensor.  Then  the  following statements are equivalent:\n \\begin{enumerate}\n \\item[(i)] The  Moore-Penrose  inverse  of $ \\mc{A}$ exists and unique.\n \\item[(ii)] $\\mc{A}{{*_N}}\\mc{A}^T{{*_N}}\\mc{A} \\leq \\mc{A}.$\n \\item[(iii)] $\\mc{A}{{*_N}}\\mc{A}^T{{*_N}}\\mc{A} = \\mc{A}.$\n\\item[(iv)] The Moore-Penrose inverse of $\\mc{A}$ exists and equals $\\mc{A}^T$.\n\\item[(v)] There exist a tensor $\\mc{G}$ such that \n  $\\mc{G}{{*_N}}\\mc{A}{{*_N}}\\mc{A}^T=\\mc{A}^T$ and $\\mc{A}^T{{*_N}}\\mc{A}{{*_N}}\\mc{G}=\\mc{A}^T$.\n\\end{enumerate}\n \\end{theorem}\n \n \n \n \\begin{proof}\n If $(i)$ holds then by Lemma \\ref{lemma2} $(ii)$ holds. Also $(ii)\\Rightarrow (iii)$ by Lemma \\ref{lemma1}. The statements $(iii)\\Rightarrow (iv)$ and $(iv)\\Rightarrow (i)$ are trivial by definition. Now we will show equivalence between $(i)$ and $(v)$. Suppose $(i)$ holds. If we take $\\mc{G}=\\mc{A}^T$ then $(v)$ hold. Conversely assume $(v)$ is true. To prove Moore-Penrose inverse of $A$ exists, first we show the following results:\n \\begin{itemize}\n     \\item $\\mc{A}{*_N}\\mc{G}{*_M}\\mc{A} = \\mc{A}$\\\\\n     Since $\\mc{G}{*_M} \\mc{A}{*_N}\\mc{A}^T=\\mc{A}^T$ which implies $ \\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{G}^T = \\mc{A}. $ Pre multiplying $\\mc{G}$ and post multiplying $\\mc{A}^T$ both sides, we obtain $\\mc{G}{*_M}\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{G}^T{*_N}\\mc{A}^T=\\mc{G}{*_M}\\mc{A}{*_N}\\mc{A}^T. $ \n     Thus $\\mc{A}^T{*_M}\\mc{G}^T{*_N}\\mc{A}^T=\\mc{A}^T.$ \n     Hence $\\mc{A}{*_N}\\mc{G}{*_M}\\mc{A} = \\mc{A}.$\n      \\item $(\\mc{G}{*_M}\\mc{A})^T = \\mc{A}^T{*_M}\\mc{G}^T=\\mc{G}{{*_M}}\\mc{A}{{*_N}}\\mc{A}^T{*_M}\\mc{G}^T=\\mc{G}{*_M}\\mc{A}.$ Therefore $\\mc{G}{*_M}\\mc{A}$ is symmetric.\n     \\item $(\\mc{A}{*_N}\\mc{G})^T = \\mc{G}^T{*_N}\\mc{A}^T=\\mc{G}^T{*_N}\\mc{A}^T{*_M}\\mc{A}{*_N}\\mc{G}=\\mc{A}{*_N}\\mc{G}.$ Thus $\\mc{A}{*_N}\\mc{G}$ is symmetric.\n          \\end{itemize}\n Now we will show the tensor $\\mc{X}=\\mc{G}{*_M}\\mc{A}{*_N}\\mc{G}$ is the Moore-Penrose of $\\mc{A}.$ Since \n \\begin{enumerate}\n     \\item [$\\bullet$] \n  $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A} =\\mc{A}{*_N}\\mc{G}{*_M}\\mc{A}=\\mc{A}.$\n   \\item [$\\bullet$] \n  $\\mc{X}{*_M}\\mc{A}{*_N}\\mc{X}  =\\mc{G}{*_M}\\mc{A}{*_N}\\mc{G}{*_M}\\mc{A}{*_N}\\mc{G} =\\mc{X}.$\n   \\item [$\\bullet$] \n  $(\\mc{A}{*_N}\\mc{X})^T=(\\mc{A}{*_N}\\mc{G}{*_M}\\mc{A}{*_N}\\mc{G})^T=(\\mc{A}{*_N}\\mc{G})^T{*_M}(\\mc{A}{*_N}\\mc{G})^T\n =\\mc{A}{*_N}\\mc{G}{*_M}\\mc{A}{*_N}\\mc{G}=\\mc{A}{*_N}\\mc{X}.$\n   \\item [$\\bullet$] \n $(\\mc{X}{*_M}\\mc{A})^T=(\\mc{G}{*_M}\\mc{A}{*_N}\\mc{G}{*_M}\\mc{A})^T=(\\mc{G}{*_M}\\mc{A})^T{*_N}(\\mc{G}{*_M}\\mc{A})^T\n =\\mc{G}{*_M}\\mc{A}{*_N}\\mc{G}{*_M}\\mc{A}=\\mc{X}{*_M}\\mc{A}.$\n  \\end{enumerate}\n\n\n Therefore, $\\mc{X}$ is the Moore-Penrose inverse of $\\mc{A}$ and By Lemma \\ref{mpiu} it is unique.\n  \\end{proof}\n  \n  \n  The reverse order law for the Moore-Penrose inverses of tensors yields a class  of challenging problems that are fundamental research in the theory of generalized inverses. Research on reverse order law tensors has been very active recently \\cite{Mispa18, PanRad18} but as per the above theorem it is trivially true in case of Boolean tensors.  \n\n  \\begin{remark}\n   If Moore-Penrose inverses of $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times  J_1 \\times \\cdots \\times  J_N}$, $\\mc{B}\\in\\mathbb{R}^{J_1 \\times \\cdots \\times   J_N\\times K_1 \\times \\cdots \\times K_L},$ and $\\mc{A}{*_N}\\mc{B}$ exists, then the reverse-order law for the Moore-Penrose inverse is always exists, i.e., \n   $$(\\mc{A}{*_N}\\mc{B})^\\dagger=\\mc{B}^\\dagger{*_N}\\mc{A}^\\dagger.$$\n  \\end{remark}\n\n\n\n\n\\subsection{Weighted Moore-Penrose inverse}\n\nUtilizing the Einstein product, weighted Moore-Penrose inverse of even-order tensor and arbitrary-order tensor was introduced in \\cite{BehMM19, we17}, very recently. This work motivate us to study weighted Moore-Penrose inverse for Boolean tensors.\n\n\\begin{definition}\\label{wmpi}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times J_1 \\times \\cdots \\times J_N}$,  $\\mc{M}\\in\\mathbb{R}^{I_1 \\times \\cdots \\times  I_M\\times I_1 \\times \\cdots \\times  I_M}$ and $\\mc{N}\\in\\mathbb{R}^{J_1  \\times \\cdots \\times  J_N\\times J_1 \\times \\cdots \\times J_N}$ be three Boolean tensors. If a Boolean tensor $\\mc{Z}\\in\\mathbb{R}^{J_1 \\times \\cdots \\times J_N\\times I_1 \\times \\cdots \\times   I_M}$ satisfying\n\\vspace{-.5cm}\n\\begin{eqnarray*}\n&(1)&\\mc{A}{*_N}\\mc{Z}{*_M}\\mc{A} = \\mc{A},\\\\\n&(2)&\\mc{Z}{*_M}\\mc{A}{*_N}\\mc{Z} = \\mc{Z},\\\\\n&(3)&(\\mc{M}{*_M}\\mc{A}{*_N}\\mc{Z})^T = \\mc{M}{*_M}\\mc{A}{*_N}\\mc{Z},\\\\\n&(4)&(\\mc{Z}{*_M}\\mc{A}{*_N}\\mc{N})^T = \\mc{Z}{*_M}\\mc{A}{*_N}\\mc{N},\n\\end{eqnarray*}\nis called weighted Moore-Penrose inverse of $\\mc{A}$ and it is denoted by $A^{\\dagger}_{\\mc{M},\\mc{N}}.$\n\\end{definition}\nNote that, the weighted Moore-Penrose inverse need not be unique in general. This can be verified by the following example.\n\\begin{example}\nLet the Boolean tensor\n$~\\mc{A}=(a_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ be defined as in Example \\ref{example18} with $\\mc{N}=\\mc{O} \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ and  $\\mc{M}=(m_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ such that \n\\begin{eqnarray*}\nm_{ij11} =m_{ij21}=\n    \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    0 & 0 &  0\n    \\end{pmatrix},\nm_{ij12} = m_{ij22} =\n    \\begin{pmatrix}\n     1 & 0 & 0\\\\\n     0 & 0 & 1\n    \\end{pmatrix},\nm_{ij13} =m_{ij23}\n    \\begin{pmatrix}\n     0 & 0 & 0\\\\\n     0 & 0 & 1\n\\end{pmatrix}.\n\\end{eqnarray*}\nThen it can be easily verified that both $~\\mc{X}=(x_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$, $~\\mc{Y}=(y_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ defined in Example \\ref{example18} satisfies all conditions of Definition \\ref{wmpi}. \\end{example}\nThe uniqueness and existence of weighted Moore-Penrose inverse and some equivalent properties will be discussed in the next part of this subsection.\n\n \\begin{theorem}\\label{uwmpi}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots\\times I_M\\times  J_1 \\times \\cdots\\times J_N},~~\\mc{M}\\in\\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times I_1 \\times \\cdots\\times I_M},$\\\\ $\\mc{N}\\in\\mathbb{R}^{J_1 \\times \\cdots\\times J_N\\times  J_1 \\times \\cdots\\times J_N}$ be three Boolean tensors with $\\mathfrak{R}(\\mc{A})=\\mathfrak{R}(\\mc{A}{*_N}\\mc{N})$ and $\\mathfrak{R}(\\mc{A}^T)=\\mathfrak{R}(\\mc{A}^T{*_M}\\mc{M}^T).$  If $\\mc{A}_{\\mc{M},\\mc{N}}^{\\dagger}$ exists, then\n\\begin{enumerate}\n\\item[(a)] $\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T = \\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T;$\n\\item[(b)] $\\mc{A}^T{*_M}\\mc{M}^T{*_M}\\mc{A} = \\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A}; $\n\\item[(c)] $ \\mc{A}_{\\mc{M},\\mc{N}}^{\\dagger} $  is unique.\n\\end{enumerate}\n\\end{theorem}\n \\begin{proof}\n Let $\\mc{X}$ be a weighted Moore-Penrose inverse of $\\mc{A}.$ Now \n \\begin{eqnarray*}\n \\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T &=&\\mc{A}{*_N} {\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{X}^T}{*_N}\\mc{A}^T\n =\\mc{A}{*_N} {(\\mc{X}{*_M}\\mc{A}{*_N}\\mc{N})^T}{*_N}\\mc{A}^T\\\\\n &=& {\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}}{*_N}\\mc{N}{*_N}\\mc{A}^T\n = \\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T.\n \\end{eqnarray*}\n  This completes the proof of part $(a).$ Using the similar lines of part $(a)$ and relation $(3)$ of Definition \\ref{wmpi}, we can  prove part $(b).$ Next we will claim the uniqueness of $A^\\dagger_{\\mc{M},\\mc{N}}.$ \\\\\n Suppose there exists two weighted Moore-Penrose inverses (say $\\mc{X}_1$ and $\\mc{X}_2$)  for $\\mc{A}.$ Then  \n \\begin{eqnarray*}\n \\mc{X}_1{*_M}\\mc{A}{*_N}\\mc{N} &=&  \\mc{X}_1{*_N}\\mc{A}{*_N} {\\mc{X}_2{*_M}\\mc{A}{*_N}\\mc{N}}\n   = \\mc{X}_1{*_M} {\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T}{*_M}\\mc{X}_2^T\\\\\n   &= & {\\mc{X}_1{*_M}\\mc{A}{*_N}\\mc{N}}{*_N}\\mc{A}^T{*_M}\\mc{X}_2^T \n   = \\mc{N}^T{*_N} {\\mc{A}^T{*_M}\\mc{X}_1^T{*_N}\\mc{A}^T}{*_M}\\mc{X}_2^T\\\\\n  & =& \\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{X}_2^T\n   = \\mc{X}_2{*_M}\\mc{A}{*_N}\\mc{N}.\n   \\end{eqnarray*}\n   Since $\\mathfrak{R}(\\mc{A})=\\mathfrak{R}(\\mc{A}{*_N}\\mc{N}).$ Which implies there exists $\\mc{U}$ such that $\\mc{A}{*_N}\\mc{N}{*_N}\\mc{U} = \\mc{A}.$ Thus $\\mc{X}_1{*_M}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{U} = \\mc{X}_2{*_M}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{U}.$ Hence $\\mc{X}_1{*_M}\\mc{A} =\\mc{X}_2{*_M}\\mc{A}$. Therefore\n   \\begin{equation}\\label{eq4.51}\n      \\mc{X}_1=\\mc{X}_1{*_M}\\mc{A}{*_N}\\mc{X}_1=\\mc{X}_2{*_M}\\mc{A}{*_N}\\mc{X}_1.\n   \\end{equation}\n Now by using Eq. (\\ref{eq4.51}), we get\n \\begin{eqnarray*}\n   \\mc{M}{*_M}\\mc{A}{*_N} {\\mc{X}_1} &=&   {\\mc{M}{*_M}\\mc{A}{*_N}\\mc{X}_2}{*_M}\\mc{A}{*_N}\\mc{X}_1\n   = \\mc{X}_2^T{*_N} {\\mc{A}^T{*_M}\\mc{M}^T{*_M}\\mc{A}}{*_N}\\mc{X}_1\\\\\n   &=& \\mc{X}_2{*_N}\\mc{A}^T{*_M} {\\mc{M}{*_M}\\mc{A}{*_N}\\mc{X}_1}\n   = \\mc{X}_2^T{*_N} {\\mc{A}^T{*_M}\\mc{X}_1^T{*_N}\\mc{A}^T}{*_M}\\mc{M}^T\\\\\n   &=& \\mc{X}_2^T{*_N}\\mc{A}^T{*_N}\\mc{M}^T\n  = \\mc{M}{*_M}\\mc{A}{*_N}\\mc{X}_2.\n   \\end{eqnarray*}\n  Again as $\\mathfrak{R}(\\mc{A}^T)=\\mathfrak{R}(\\mc{A}^T{*_M}\\mc{M}^T).$ This implies there exists $\\mc{V}^T$ such that $\\mc{A}^T{*_M}\\mc{M}^T{*_M}\\mc{V}^T = \\mc{A}^T.$ It leads $\\mc{V}{*_M}\\mc{M}{*_M}\\mc{A}=\\mc{A}.$ Thus $ \\mc{V}{*_M}\\mc{M}{*_M}\\mc{A}{*_N}\\mc{X}_1 = \\mc{V}{*_M}\\mc{M}{*_M}\\mc{A}{*_N}\\mc{X}_2.$  Hence $\\mc{A}{*_N}\\mc{X}_1 =\\mc{A}{*_N}\\mc{X}_2$. Therefore\n   \\begin{equation}\\label{eq4.52}\n      \\mc{X}_2=\\mc{X}_2{*_M}\\mc{A}{*_N}\\mc{X}_2=\\mc{X}_2{*_M}\\mc{A}{*_N}\\mc{X}_1.\n   \\end{equation}\n  Combining Eq. (\\ref{eq4.51}) and (\\ref{eq4.52}), we obtain $\\mc{X}_1=\\mc{X}_2$ and hence the proof is complete. \n  \\end{proof}\n The existence of weighted Moore-Penrose inverse is not trivial like other generalized inverses. The next theorem discusses the existence of weighted Moore-Penrose inverse. \n \n \\begin{theorem}\\label{ewmpi}\n Let $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots\\times I_M\\times  J_1\\times \\cdots\\times J_N},~~\\mc{M}\\in\\mathbb{R}^{I_1\\times \\cdots \\times I_M\\times I_1\\times \\cdots\\times I_M},$\\\\ $\\mc{N}\\in\\mathbb{R}^{J_1 \\times \\cdots\\times  J_N\\times  J_1 \\times \\cdots\\times J_N}$ be three Boolean tensors with $\\mathfrak{R}(\\mc{A})=\\mathfrak{R}(\\mc{A}{*_N}\\mc{N})$ and $\\mathfrak{R}(\\mc{A}^T)=\\mathfrak{R}(\\mc{A}^T{*_M}\\mc{M}^T).$ If\n $ \\mc{M}\\geq\\mc{I}$ and $\\mc{N}\\geq\\mc{I},$ then $\\mc{A}_{\\mc{M},\\mc{N}}^\\dagger$ exists if and only if any one of the following conditions holds:\n \\begin{enumerate}\n \\item[(a)] $\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A} = \\mc{A}.$\n \\item[(b)] $\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A} = \\mc{A}.$\n \\item[(c)] $\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T{*_M}\\mc{A} = \\mc{A}.$\n \\item[(d)] $\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T{*_M}\\mc{A} = \\mc{A}.$\n \\end{enumerate}\n In particular, $\\mc{A}_{\\mc{M},\\mc{N}}^\\dagger = \\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T.$\n \\end{theorem}\n  \\begin{proof}\nAssume $\\mc{A}_{\\mc{M},\\mc{N}}^\\dagger$ exists and let $\\mc{X}=\\mc{A}_{\\mc{M},\\mc{N}}^\\dagger$. Let $\\mc{B}= \\mc{A}^T{*_M}\\mc{A}$. Since for every Boolean tensor, there are finitely many  Boolean tensors of same order, so there must exist positive integers $s,t\\in\\mathbb{N}$ such that \n\\begin{equation}\\label{eq4.6}\n    (\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^s =  (\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s+t}.\n\\end{equation}\nWithout loss of generality, we can assume that $s$ is the smallest positive integer for which Eq. (\\ref{eq4.6}) holds. Now we will claim $s=1.$ Suppose on contradiction, assume $s>1.$ Now using Eq. (\\ref{eq4.6}), and properties of weighted Mooore-Penrose inverse, we get \n\\begin{eqnarray}\\label{eq4.7}\n\\nonumber\n {\\mc{X}{*_M} \\mc{A}{*_N}\\mc{N}}{*_N}\\mc{A}^T{*_M}\\mc{M}^T{*_M}&&\\hspace*{-0.7cm}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1} .\\\\\n\\nonumber\n&&\\hspace*{-3.5cm}=   {\\mc{X}{*_M}\\mc{A}{*_N}\\mc{N}}{*_N}\\mc{A}^T{*_M}\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1+t}\\\\\n\\nonumber\\textnormal{This yield~~} \\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{X}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T&&\\hspace*{-0.7cm}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1} \\\\\n&&\\hspace*{-4.5cm}=\\mc{N}^T{*_N}\\mc{A}^T{*_N}\\mc{X}^T{*_M}\\mc{A}^T{*_M}\\mc{M}^T(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1+t}.\n\\end{eqnarray}\nSince $\\mathfrak{R}(\\mc{A})=\\mathfrak{R}(\\mc{A}{*_N}\\mc{N}),$ which implies there exists a  tensor $\\mc{U}$ such that $\\mc{A}{*_N}\\mc{N}{*_N}\\mc{U}=\\mc{A}.$ Now premultiplying $\\mc{U}^T$ to Eq. (\\ref{eq4.7}) and using the properties $\\mc{U}^T{*_N}\\mc{N}^T{*_N}\\mc{A}^T=\\mc{A}^T$ and $\\mc{A}^T {*_M}\\mc{X}^T{*_N}\\mc{A}^T=\\mc{A}^T,$ we get\n\\begin{equation}\\label{eq4.71}\n    \\mc{A}^T{*_M}\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1} = \\mc{A}^T{*_M}\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1+t}.\n\\end{equation}\n Again, premultiplying $\\mc{X}^T $ to Eq. (\\ref{eq4.71}) and using the symmetricity of  $\\mc{M}{*_M}\\mc{A}{*_N}\\mc{X}$, we get\n$\\mc{M}{*_M} {\\mc{A}{*_N}\\mc{X}{*_M}(\\mc{A}}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1}= \\mc{M}{*_M} {\\mc{A}{*_N}\\mc{X}{*_M}(\\mc{A}}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1+t}.$ This gives \n\\begin{equation}\\label{eq4.8}\n    \\mc{M}{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1} = \\mc{M}{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1+t}.\n\\end{equation}       \n Since $\\mathfrak{R}(\\mc{A}^T)=\\mathfrak{R}(\\mc{A}^T{*_N}\\mc{M}^T),$ which implies there exists a  tensor $\\mc{Z}$ such that $\\mc{Z}{*_M}\\mc{M}{*_M}\\mc{A}=\\mc{A}.$ Premultiplying $\\mc{Z}$ to Eq. (\\ref{eq4.8}) yields \n  $$(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1} = (\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{s-1+t}.$$\n  and contradicts the minimality of $s.$ Therefore \n  \\begin{equation}\\label{eq4.9}\n      \\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T = (\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{t+1},~\\mbox{for some }~t\\in\\mathbb{N}.\n  \\end{equation}\n Premultiplying Eq. (\\ref{eq4.9}) by $\\mc{X},$ and using $(\\mc{X}{*_M}\\mc{A}{*_N}\\mc{N})^T=\\mc{X}{*_M}\\mc{A}{*_N}\\mc{N},$ we obtain \n$$\n\\mc{N}^T{*_N} {\\mc{A}^T{*_M}\\mc{X}^T{*_N}\\mc{A}^T}{*_M}\\mc{M}^T = \\mc{N}^T{*_M} {\\mc{A}^T{*_N}\\mc{X}^T{*_N}\\mc{A}^T}{*_M}\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^t.$$\nSince $ \\mc{A}^T{*_M}\\mc{X}^T{*_N}\\mc{A}^T=\\mc{A}^T,$ we get\n\\begin{equation}\\label{eqn121}\n   \\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T = \\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^t. \n\\end{equation}\nPremultiplying Eq. (\\ref{eqn121}) by a tensor $\\mc{U}^T$ and using $\\mathfrak{R}(\\mc{A})=\\mathfrak{R}(\\mc{A}{*_N}\\mc{N}),$ we again obtain $ \\mc{A}^T{*_M}\\mc{M}^T = \\mc{A}^T{*_M}\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^t.$ Postmultiplying $\\mc{Z}^T$ and applying $\\mathfrak{R}(\\mc{A}^T)=\\mathfrak{R}(\\mc{A}^T{*_M}\\mc{M}^T),$  we have  \n$$\n \\mc{A}^T = \\mc{A}^T{*_M}\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^{t-1}{*_M} \\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T.$$\n\nNow \n\\begin{eqnarray*}\n \\mc{A}^T &=& \\mc{A}^T{*_M} {\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T}{*_M}\\mc{M}^T)^{t-1}{*_M}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T\\\\\n &=&\\mc{A}^T{*_M}(\\mc{M}^T{*_M}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T){*_M} {\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T}{*_M}\\mc{M}^T)^{t-2}{*_N}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T\\\\\n &=&\\mc{A}^T{*_M}(\\mc{M}^T{*_M}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T)^2{*_M} {\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T}{*_M}\\mc{M}^T)^{t-3}{*_N}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T\\\\\n &=&\\cdots~~~~~~~~~~~~~~\\cdots~~~~~~~~~~~~~\\cdots\\\\\n &=&\\mc{A}^T{*_M}(\\mc{M}^T{*_M}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T)^{t-2}{*_M} {\\mc{M}^T{*_M}(\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T}{*_M} {\\mc{M}^T){*_N}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T}\\\\\n &=&\\mc{A}^T{*_M}(\\mc{M}^T{*_M}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T)^{t}=\\mc{A}^T{*_M}\\left[(\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M})^{t}\\right]^T.\n\\end{eqnarray*}\nThus \n\\begin{equation}\\label{eq4.10}\n    \\mc{A}=(\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M})^{t}{*_M}\\mc{A}.\n\\end{equation}\nAs $\\mc{M}\\geq\\mc{I},\\mc{N}\\geq\\mc{I} $, so by Lemma \\ref{lemma1}\n\\begin{equation}\\label{eq4.11}\n \\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A}\\geq \\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{A}\\geq \\mc{A}.\n\\end{equation}\nPostmultiplying $\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A}$, we obtain \n \\begin{equation}\\label{eq4.12}\n   \\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}\\leq(\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M})^2{*_M}\\mc{A}.\n \\end{equation}\n Combining Eqs.(\\ref{eq4.10}), (\\ref{eq4.11}) and (\\ref{eq4.12}), we have\n \n\n \n\\begin{eqnarray*}\n\\mc{A}&\\leq&\\mc{A}{*_N}\\mc{N}^T {*_N} \\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A}\n\\leq  (\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T*_M\\mc{M})^2{*_M}\\mc{A} \\\\\n&\\leq & (\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M})^3{*_M}\\mc{A}\\leq \\cdots\\leq (\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M})^t{*_M}\\mc{A}=\\mc{A}.\n\\end{eqnarray*}\nTherefore \n\\begin{equation}\\label{eq4.13}\n     \\mc{A} = \\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A},\n\\end{equation}\nand hence completes the proof of the condition $(b).$ By using Theorem \\ref{uwmpi}, the other conditions are holds since\n\\begin{eqnarray}\\nonumber\n   \\mc{A} &=&  {\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T}{*_M}\\mc{M}{*_M}\\mc{A} = \\mc{A}{*_N}\\mc{N}{*_N} {\\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A}}\\\\\\label{eq199} &=& {\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T}{*_M}\\mc{M}^T{*_M}\\mc{A} = \\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T{*_M}\\mc{A}.\n\\end{eqnarray}\nFurther, we will claim not only the four conditions holds but also $\\mc{A}^\\dagger_{\\mc{M},\\mc{N}}=\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T.$ Let $\\mc{X}=\\mc{A}_{\\mc{M},\\mc{N}} ^\\dagger.$ From Eq. (\\ref{eq199}), $\\mc{A}=\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}$ and\n\\begin{equation*}\n    \\mc{X}{*_M}\\mc{A}{*_N}\\mc{X}=\\mc{N}^T{*_N} {\\mc{A}^T{*_M}\\mc{M}^T{*_M}\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T}{*_M}\\mc{M}^T=\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T=\\mc{X}.\n\\end{equation*}\nUsing Theorem \\ref{uwmpi}, we show \n\\begin{eqnarray*}\n\\mc{M}{*_M}\\mc{A}{*_N}\\mc{X} &=&\\mc{M}{*_M} {\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T}{*_M}\\mc{M}^T    =\\mc{M}{*_M}\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}^T\\\\\n     &=&(\\mc{M}{*_M}\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T)^T\n=(\\mc{M}{*_M}\\mc{A}{*_N}\\mc{X})^T.\n\\end{eqnarray*}\nTherefore, $\\mc{M}{*_M}\\mc{A}{*_N}\\mc{X}$ is symmetric. Similarly, we can show $\\mc{X}{*_M}\\mc{A}{*_N}\\mc{N}$ is symmetric. So $\\mc{A}_{\\mc{M},\\mc{N}} ^\\dagger=\\mc{N}^T{*_N}\\mc{A}^T{*_M}\\mc{M}^T $. Next we will show the converse part. Let $\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A} = \\mc{A}.$ Since $\\mc{M}\\geq \\mc{I}$ and $\\mc{N}\\geq\\mc{I}$, so by Lemma \\ref{lemma1}, \\begin{equation*}\n    \\mc{A}\\leq\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{A}\\leq\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{A}\\leq\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{M}{*_M}\\mc{A} = \\mc{A}\n\\end{equation*}\n and hence \n\\begin{equation}\\label{eq4.14}\n  \\mc{A} =\\mc{A}{*_N}\\mc{A}^T{*_M}\\mc{A} = \\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{A}.\n\\end{equation}\nUsing the Eq. (\\ref{eq4.14}) and symmetricity of $ \\mc{A}{*_N}\\mc{A}^T$, we obtain\n\\begin{equation}\\label{eq4.15}\n   \\mc{A}{*_N}\\mc{A}^T = \\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{A}{*_N}\\mc{A}^T\n=\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T\n=\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T .\n\\end{equation}\nSimilar argument yields,\n\\begin{eqnarray}\\label{eq4.16}\n\\nonumber\n    \\mc{A}^T{*_M}\\mc{A}&=&\\mc{A}^T{*_M}\\mc{M}^T{*_M} {\\mc{A}{*_N}\\mc{N}^T{*_N}\\mc{A}^T}{*_M} \\mc{A}=\\mc{A}^T{*_M}\\mc{M}^T{*_M} {\\mc{A}{*_N}\\mc{N}{*_N}\\mc{A}^T{*_M}\\mc{A}}\\\\&=&\\mc{A}^T{*_M}\\mc{M}{*_N}\\mc{A} = \\mc{A}^T{*_M}\\mc{M}^T{*_M}\\mc{A}.\n\\end{eqnarray}\nUsing Eqs. (\\ref{eq4.14})-(\\ref{eq4.16}), it can be easily verified that $\\mc{X}=\\mc{N}^T{*_N}\\mc{A}^T {*_M}\\mc{M}^T$ is satisfies all four conditions of the weighted Moore-Penrose inverse. Similarly, one can start from other conditions to verify the same. Thus the proof is complete.\n  \\end{proof}\n  \\begin{remark}\n The equality  condition  in Theorem \\ref{ewmpi} $(a)$ can be replaced by ${\\bf{`\\geq'}}.$\n \\end{remark}\n\n\n\\subsection{Space Decomposition}\nUsing the theory of Einstein product, we introduce the definition of the space decomposition for Boolean tensors, which generalizes the matrix space decomposition \\cite{rao}. \n\n\n\\begin{definition}\\label{FRD}\n Let  $\\mc{F}\\in\\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times K_1 \\times \\cdots\\times K_L}$ and $\\mc{R}\\in\\mathbb{R}^{K_1 \\times \\cdots\\times K_L\\times J_1 \\times \\cdots\\times J_N}$ be two tensors with\n \\vspace{-.5cm}\n\\begin{eqnarray*}\n     &&(a)~\\mc{A} = \\mc{F}{*_L}\\mc{R};\\\\\n     &&(b)~\\mathfrak{R}(\\mc{A}) = \\mathfrak{R}(\\mc{F});\\\\\n     &&(c)~\\mathfrak{R}(\\mc{A}^T) = \\mathfrak{R}(\\mc{R}^T),\n\\end{eqnarray*}\nthen the tensor $\\mc{A}$ is called space decomposable and this decomposition is called a space decomposition  of $\\mc{A}$.\n\\end{definition}\n\n\nIn connection with the fact of the  above Definition \\ref{FRD} and Lemma \\ref{range-stan}, one can conclude the existence of a generalized inverse, as follows. \n\n\n\n\\begin{theorem}\\label{exisgen}\nLet $\\mc{A}=\\mc{F}{*_L}\\mc{R}$ be a space decomposition of  $\\mc{A}\\in\\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times  J_1 \\times \\cdots\\times J_N},$ where $\\mc{F}\\in\\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times K_1 \\times \\cdots\\times K_L}$ and $\\mc{R}\\in\\mathbb{R}^{K_1 \\times \\cdots\\times K_L\\times J_1 \\times \\cdots\\times J_N}$ .   Then $\\mc{A}^{(1)}$ exists.\n\\end{theorem}\n\n\nWe now present one of our essential result which represents not only the existence of reflexive generalized inverse but also other inverses through this decomposition.\n\n\n\\begin{theorem}\\label{the336}\nLet $\\mc{X}$ be a generalized inverse of the Boolean tensor $\\mc{A}.$ If $\\mc{A}=\\mc{F}{*_L}\\mc{R}$ is a space decomposition of $\\mc{A},$ where $\\mc{F}\\in\\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times K_1 \\times \\cdots\\times K_L}$ and $\\mc{R}\\in\\mathbb{R}^{K_1 \\times \\cdots\\times K_L\\times J_1 \\times \\cdots\\times J_N}.$  Then the following are holds:\n\\begin{enumerate}\n\\label{eqvspace}\n    \\item[(a)] $\\mc{F}^{(1)}$ and $\\mc{R}^{(1)}$ exists.\n    \\item[(b)] $\\mc{F}^{(1)}{*_M}\\mc{F}=\\mc{R}{*_N}\\mc{R}^{(1)}.$\n    \\item[(c)] $\\mc{F}^{(1)}{*_M}\\mc{A}=\\mc{R}$ and $\\mc{A}{*_N}\\mc{R}^{(1)}=\\mc{F}.$\n    \\item[(d)] $\\mc{R}^{(1)}{*_M}\\mc{F}^{(1)}$ is a generalized inverse of $\\mc{A}.$\n    \\item[(e)] $\\mc{R}{*_N}\\mc{X}$ is a reflexive inverse of $\\mc{F}$ and $\\mc{X}{*_M}\\mc{F}$ is a reflexive inverse of $\\mc{R}.$ \n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nSince $\\mc{X}$ is the generalized inverse of $\\mc{A}.$ Then we have  $\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}=\\mc{A},$ which implies\\\\ \n$\\mc{F}{*_L}\\mc{R}{*_N}\\mc{X}{*_M}\\mc{F}{*_L}\\mc{R}=\\mc{F}{*_L}\\mc{R}=\\mc{I}{*_M}\\mc{F}{*_L}\\mc{R}.$ Further, using Corollary \\ref{rtcan}, we get $\\mc{F}{*_L}\\mc{R}{*_N}\\mc{X}{*_M}\\mc{F}=\\mc{F}.$ Thus $\\mc{R}{*_N}\\mc{X}$ is a generalized inverse of $\\mc{F}.$ Similarly, one can determine $\\mc{X}{*_M}\\mc{F}$ is a generalized inverse of $\\mc{R}$.  Hence $(a)$ is proved. Now using the result $(a),$ one can prove $(b)$ and $(c).$  To prove $(d)$ we use the fact $(a)$ and obtain. \n\\begin{equation*}\n    \\mc{A}{*_N}\\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}{*_M}\\mc{A}=  \\mc{A}{*_N}\\mc{X}{*_M}\\mc{F}{*_L}\\mc{R}{*_N}\\mc{X}{*_M}\\mc{A}=  \\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}{*_N}\\mc{X}{*_M}\\mc{A}=\\mc{A}.\n\\end{equation*}\nHence $\\mc{R}^{(1)}{*_M}\\mc{F}^{(1)}$ is a generalized inverse of $\\mc{A}.$ \nIn a similar manner, one can prove $(e)$ using the fact $\\mc{R}{*_N}\\mc{X}{*_M}\\mc{F}{*_L}\\mc{R}{*_N}\\mc{X}\n=\\mc{R}{*_N}\\mc{X}$ and $\\mc{X}{*_M}\\mc{F}{*_L}\\mc{R}{*_N}\\mc{X}{*_M}\\mc{F}\n=\\mc{X}{*_M}\\mc{F}.$ This completes the proof.\n\\end{proof}\nIn view of the above theorem one can draw a conclusion, as follows. \n\\begin{remark}\\label{rmk3.43}\n Every generalized inverse of $\\mc{A}$ need not of the form $\\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}.$ \n \\end{remark}\n We verify the Remark \\ref{rmk3.43} with the following example.\n\\begin{example}\\label{example3.45}\nLet\n$~\\mc{A}=(a_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ be a Boolean tensor with\n\\begin{eqnarray*}\na_{ij11} =\n    \\begin{pmatrix}\n    1 & 1 & 0 \\\\\n    1 & 0 &  0\n    \\end{pmatrix},\na_{ij12}=a_{ij13} =a_{ij21}=a_{ij22}=a_{ij23}=\n    \\begin{pmatrix}\n     0 & 0 & 0\\\\\n     0 & 0 & 0\n    \\end{pmatrix}.\n\\end{eqnarray*}\nConsider  $\\mc{A}^{(1)}=(x_{ijkl}) \\in \\mathbb{R}^{{2\\times3}\\times{2 \\rtimes 3}}$ is a generalized inverse of $\\mc{A}$ with \n\\begin{eqnarray*}\nx_{ij11} =\n    \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    0 & 0 &  0\n    \\end{pmatrix},\nx_{ij12} =\n    \\begin{pmatrix}\n     0 & 1 & 0\\\\\n     0 & 0 & 0\n    \\end{pmatrix},\nx_{ij13} =\n    \\begin{pmatrix}\n     0 & 0 & 0\\\\\n     0 & 0 & 0\n\\end{pmatrix},\\\\\n    %\n    %\n    %\nx_{ij21} =\n    \\begin{pmatrix}\n     0 & 0 & 0\\\\\n     1 & 0 & 0\n    \\end{pmatrix},\n    x_{ij22} =\n    \\begin{pmatrix}\n    0 & 0 & 0\\\\\n    0 & 0 &  0\n    \\end{pmatrix},\nx_{ij23} =\n    \\begin{pmatrix}\n     0 & 0 & 0\\\\\n     0 & 0 & 0\n    \\end{pmatrix}.\n\\end{eqnarray*}\n\nIn light of the Theorem \\ref{the336} (e) one can conclude\n$$\\mc{R}^{(1)}*_2\\mc{F}^{(1)}=\\mc{A}^{(1)}*_2\\mc{F}*_2\\mc{R}*_2\\mc{A}^{(1)}=\\mc{A}^{(1)}*_2\\mc{A}*_2\\mc{A}^{(1)} \\neq \\mc{A}^{(1)}.$$\nTherefore, every generalized inverse of $\\mc{A}$ need not of the form $\\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}.$ \n\\end{example}\nAt this point one may be interested to know when does the generalized inverse of a Boolean tensor of the form $\\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}$ ?  The answer to this question is explained in the following Remark. \n\\begin{remark}\\label{rmk3.46}\n If $\\mc{X}$ is a reflexive inverse of a Boolean tensor $\\mc{A}\\in \\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times J_1 \\times \\cdots\\times J_N}$ and $\\mc{A}=\\mc{F}{*_L}\\mc{R}$ has a space decomposition, where $\\mc{F}\\in\\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times K_1 \\times \\cdots\\times K_L}$ and $\\mc{R}\\in\\mathbb{R}^{K_1 \\times \\cdots\\times K_L\\times J_1 \\times \\cdots\\times J_N}.$ Then every generalized inverse is of the form $\\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}.$\n\\end{remark}\nNow considering the fact of Remark \\ref{rmk3.46} and the observation of the Example \\ref{example3.45},  one can get the desired result.\n\n\\begin{theorem}\\label{refeqv}\nLet  $\\mc{A} \\in \\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times J_1 \\times \\cdots\\times J_N}$ be a Boolean tensor and \n$\\mc{A}=\\mc{F}{*_L}\\mc{R}$ be a space decomposition of $\\mc{A}, $ where $\\mc{F}\\in\\mathbb{R}^{I_1 \\times \\cdots\\times I_M\\times K_1 \\times \\cdots\\times K_L},$  $\\mc{R}\\in\\mathbb{R}^{K_1 \\times \\cdots\\times K_L\\times J_1 \\times \\cdots\\times J_N}.$ Assume that generalized inverse of  either $\\mc{F}$ reflexive or $\\mc{R}$ reflexive. Then $\\mc{X}$  is a reflexive generalized inverse of $\\mc{A}$  if and only if $\\mc{X} = \\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}.$\n\\end{theorem}\n\\begin{proof}\nConsider the generalized inverse of $\\mc{F}$ is reflexive. Taking into account of Theorem \\ref{eqvspace} $(d)$, we obtain  $\\mc{X} = \\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}$, which is a generalized inverse of  $\\mc{A}$. Therefore, it is enough to show $\\mc{X}{*_M}\\mc{A}{*_N}\\mc{X}=\\mc{X}.$ Now using Theorem \\ref{eqvspace} $(c)$, we get\n\\begin{eqnarray*}\n\\mc{X}{*_M}\\mc{A}{*_N}\\mc{X} =\\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}{*_M} {\\mc{A}{*_N}\\mc{R}^{(1)}}{*_L}\\mc{F}^{(1)}\n=\\mc{R}^{(1)}{*_L} {\\mc{F}^{(1)}{*_M}\\mc{F}{*_L}\\mc{F}^{(1)}} = \\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}.\n\\end{eqnarray*}\nConversely, let $\\mc{X}$ be a reflexive inverse of $\\mc{A}.$ Then by Theorem \\ref{eqvspace} $(e)$,\n\\begin{eqnarray*}\n\\mc{X} = \\mc{X}{*_M}\\mc{A}{*_N}\\mc{X} = \\mc{X}{*_M}\\mc{F}{*_L}\\mc{R}{*_N}\\mc{X} = \\mc{R}^{(1,2)}{*_L}\\mc{F}^{(1,2)}=\\mc{R}^{(1)}{*_L}\\mc{F}^{(1)}.\n\\end{eqnarray*}\n\\vspace{-.3cm}\n\\end{proof}\n\n\n\n\n\\begin{remark}\\label{rk2.41}\nIf we drop the condition either $\\mc{F}$ or $\\mc{R}$ is reflexive generalized inverse of $\\mc{A}$ in Theorem \\ref{refeqv}, then the theorem will not true in general.\n \\end{remark}\n\n\n\nIn favour of the the Remark \\ref{rk2.41} we produce an example as follows. \n\n\n\n\n\\begin{example}\nLet $\\mc{A}$ be the Boolean tensor defined in Example \\ref{example3.45} and $\\mc{A}=\\mc{F}=\\mc{R}.$ Since $\\mc{A}{*_2}\\mc{I}{*_2}\\mc{A}=\\mc{I}$ and $\\mc{I}{*_2}\\mc{A}{*_2}\\mc{I}\\neq\\mc{I}$, it follows that $\\mc{I}$ is the generalized inverse for both $\\mc{F}$ and $\\mc{G}$ but not reflexive. In view of the Theorem \\ref{refeqv},   \none can conclude \n$\\mc{R}^{(1)}{*_2}\\mc{F}^{(1)}=\\mc{I}$\nis not a reflexive generalized inverse of $\\mc{A}.$\n\\end{example}\n\nIn \\cite{beasley} and \\cite{song}, the authors have defined the rank of a Boolean matrix through space decomposition. Next, we discuss the rank and weight of a Boolean tensors. \n\n\n\\begin{definition}\n Let $\\mc{A}\\in\\mathbb{R}^{I_1\\times\\cdots\\times I_M\\times J_1 \\times\\cdots\\times\n J_N}$ be a Boolean tensor. If there exist a least positive integer,   $r=K_1 \\times\\cdots\\times K_L$  such that the Boolean tensors $\\mc{B}\\in\\mathbb{R}^{I_1 \\times\\cdots\\times I_M\\times K_1\\times\\cdots\\times  K_L}$ and $\\mc{C}\\in\\mathbb{R}^{K_1 \\times\\cdots\\times K_L\\times J_1\\times\\cdots\\times J_N}$ satisfies  $\\mc{A}=\\mc{B}{*_L}\\mc{C}$.  Then  $r$ is called the Boolean rank of $\\mc{A}$ and denoted by $r_b(\\mc{A}).$\n\\end{definition}\n\n\\begin{example}\\label{exrank}\nConsider a Boolean tensor\n$~\\mc{A}=(a_{ijkl}) \\in \\mathbb{R}^{{2\\times2}\\times{2 \\rtimes 2}}$ with entries\n\\begin{eqnarray*}\na_{ij11} =\n    \\begin{pmatrix}\n    1 & 0  \\\\\n    0 &  0\n    \\end{pmatrix},~\na_{ij12} =\n    \\begin{pmatrix}\n     0 & 0\\\\\n     0 &  0\n    \\end{pmatrix},~\na_{ij21} =\n    \\begin{pmatrix}\n     0 & 0\\\\\n     1 &  0\n    \\end{pmatrix},~\na_{ij22} =\n    \\begin{pmatrix}\n    1 &  0\\\\\n    0 &  0\n    \\end{pmatrix}.\n    \\end{eqnarray*}\n    There exist a least positive integer $r=2$ and two tensor\n    $~\\mc{B}=(b_{ijk}) \\in \\mathbb{R}^{{2\\times2 \\times 2}}$ and $~\\mc{C}=(c_{ijk}) \\in \\mathbb{R}^{{2\\times2 \\times 2}}$ with entries\n    \\begin{eqnarray*}  \n    b_{ij1} =\n    \\begin{pmatrix}\n    1 &  0\\\\\n    0 &  0\n    \\end{pmatrix},\n       b_{ij2} =\n    \\begin{pmatrix}\n    0 &  0\\\\\n    1 &  0\n    \\end{pmatrix},\n       c_{ij1} =\n    \\begin{pmatrix}\n    1 &  0\\\\\n    0 &  1\n    \\end{pmatrix},\n       c_{ij2} =\n    \\begin{pmatrix}\n    0 &  1\\\\\n    0 &  0\n    \\end{pmatrix},\n    \\end{eqnarray*}\n    such that $\\mc{A}= \\mc{B}*_1\\mc{C}$. However, $r=1$ gives two matrices $B$ and $C$, which is impossible to get a tensor.  Thus rank of the tensor is $2$.\n\\end{example}\n\nOn the other hand, the rank of the Boolean tensor is zero if it is zero tensor. Further, we have $\\mc{A}=\\mc{I}_m{*_M}\\mc{A}=\\mc{A}{*_N}\\mc{I}_n$, where $\\mc{A}\\in\\mathbb{R}^{I_1\\times\\cdots\\times I_M\\times J_1 \\times\\cdots\\times\n J_N} $. It is quite apparent that\n$$0\\leq r_b(\\mc{A})\\leq \\min\\{I_1 \\times\\cdots\\times I_M,~J_1\\times\\cdots\\times  J_N\\}. $$\n\nTo prove the last result of this paper, we define weight of Boolean tensor as.\n\n\\begin{definition}\nThe weight of Boolean tensor is denoted by $w(\\mc{A})$ and defined as\n$$\nw(\\mc{A})=\\{\\mbox{ Total number of non zero elements of } \\mc{A}\\}.\n$$\n\\end{definition}\n\nThe existence of generalized inverse can be discussed through Boolean rank, as follows. \n\n\n\\begin{theorem}\\label{rankthm}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots \\times I_M\\times J_1 \\times \\cdots \\times  J_N}$ be any tensor with $r_b(\\mc{A})\\leq 1.$ Then $\\mc{A}$ is regular.\n\\end{theorem}\n\n\\begin{proof}\nIt is trivial for $r_b(\\mc{A})=0,$ as a consequence of the fact $\\mc{O}$ tensor is always regular.\n Further, consider $r_b(\\mc{A})=1$ and define  a tensor $\\mc{J},$ with no zero elements. Then there exist permutation tensors $P$ and $\\mc{Q}$ such that $\\mc{P}{*_M}\\mc{A}{*_N}\\mc{Q}=\\begin{bmatrix}\n\\mc{J} & \\mc{O}\\\\\n\\mc{O} & \\mc{O}\\\\\n\\end{bmatrix}.$ As $\\mc{J}$ is regular, it implies $\\mc{P}{*_M}\\mc{A}{*_N}\\mc{Q}$ is regular.\n In view of the Lemma \\ref{block} and Preposition \\ref{permu} one can conclude $\\mc{A}$ is regular \n\\end{proof}\nIt is clear, if the weight of a Boolean tensor is $1,$ then the rank is also $1$. In view of this we obtain the following result. \n\\begin{corollary}\\label{weightcoro}\nLet $\\mc{A}\\in\\mathbb{R}^{I_1\\times \\cdots I_M\\times J_1\\times\\cdots J_N}$ be any tensor with $w(\\mc{A})\\leq 1.$ Then $\\mc{A}$ is regular.\n\\end{corollary}\n\n\n\n\\section{Conclusion}\nIn this paper, we have introduced generalized inverses $(\\{i\\}$-inverses $(i = 1, 2, 3, 4))$  with the Moore-Penrose inverse and weighted Moore-Penrose inverse for Boolean tensors via the Einstein product, which is a generalization of the generalized inverses of Boolean matrices. In addition to this, we have discussed their existence and uniqueness. This paper also provides some characterization through complement and its application to generalized inverses. \nFurther, we explored the space decomposition for the Boolean  tensors, at the same time, we have studied rank and the weight for the Boolean tensor. \nIn particular, we limited our study for Boolean tensors with $r_b(\\mc{A}) \\leq 1$ and $w(\\mc{A}) \\leq 1$. Herewith left as open problems for future studies.\\\\\n{\\bf Problem:} If the Boolean rank or weight  of a tensor $\\mc{A}$ is greater than 1, then under which conditions the Boolean tensor $\\mc{A}$ is regular $?$\\\\\nAdditionally, it would be interesting to investigate more generalized inverses on the Boolean tensors; this work is currently underway.\n\n\n\n\n\n\n\n\\noindent {\\bf{Acknowledgments}}\\\\\nThis research work was supported by Science and Engineering Research Board (SERB), Department of Science and Technology, India, under the Grant No. EEQ\/2017\/000747.\n\n\\bibliographystyle{abbrv}\n\\bibliographystyle{vancouver}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\paragraph{Background and Aim} Choreographies are high-level descriptions of communicating \nsystems, inspired by the ``Alice \nand Bob'' notation for security protocols, where the behaviours of participants is defined from a \nglobal viewpoint. Over the last two decades, they have become popular and have been applied in \ndifferent contexts, including: the specification of communication protocols \nin web service standards \\cite{wscdl}, business process notations \\cite{bpmn}, and \ntheoretical models of communications \\cite{HYC16}; the synthesis of correct-by-construction \nconcurrent software \\cite{CM13,chor:website,DGGLM17}; the runtime verification of (object-oriented) \nactor systems \\cite{NY17}; and the static verification of concurrent process models \n\\cite{LTY15,CLM17}. A notable application to software engineering is the Testable Architecture \nmethodology \\cite{savara:website}, a development lifecycle that keeps service implementations \naligned with the choreographies specified by designers.\n\nThe promise of choreographies is that they will improve correctness in concurrent programming. \nUnfortunately, this promise remains unfulfilled, because the choreography models explored so far have the unrealistic assumption that communications never fail. The only exception \nis the work in \\cite{APN17} (the state of the art in the topic \nof failures in choreographies so far), which equips choreographies with optional blocks that can be \ncancelled non-deterministically at runtime. This is an interesting direction, but it still has \nlimitations that impede its applicability (\\eg, communications are \nsynchronous\/instantaneous; we discuss more in related work) and, just as important, does not allow \nchoreographies to specify how the system should recover from a failure.\n\nThe aim of this work is to develop a choreography model that brings choreographies all the way to \nbeing applicable to settings with realistic communication failures.\nReaching this objective is challenging, because we need to provide enough \nexpressivity to program the local recovery strategies of participants (which may \nbe different) and at the same time retain the global viewpoint on communications offered by \nchoreographies.\nTo this end, we split choreographic communications into their declarations and \nimplementations as shown in the snippet below.\n\\begin{snippet}\n\\cnewframe{k}{\\pid s}{\\pid r}{}{\n\t\\\n\t\\code{sendExpBackoff}(k,\\pid s)\\keyword{;}\\,\n\t\\ \n\t\\code{recvTimeout}(k,\\pid r)\n}\n\\end{snippet}\nHere, we use a choreographic notation---$\\cframe{k}{\\pid s}{\\pid r}{}$---to declare a \ncommunication from a sender process $\\pid s$ to a receiver process $\\pid r$, and \nname it $k$. Then, we \\emph{implement} the communication $k$ by invoking the \nprocedures \\code{sendExpBackoff} and \\code{recvTimeout}, which respectively handle the send and \nreceive part of the communication. Both procedures handle communication failures and may perform \ndifferent retries at sending or receiving in case of failures, but with different policies: the \nfirst procedure uses exponential backoff between send attempts, while the second is based on a \nfixed timeout.\nThus, implementation needs not be symmetric between sender(s) and receiver(s).\n\n\\paragraph{Contribution}\nWe develop Robust Choreographies (RC\\xspace for short), a new choreographic programming model \nwith support for communication failures. In RC\\xspace, the programmer declares which communications \nshould take place using an Alice and Bob notation, and then defines how processes will enact these \ncommunications through asynchronous send and receive actions. Code regarding different \ncommunications can be interleaved freely, allowing for the modelling of dependencies between \nthe implementations of different communications.\n\nDifferently from previous work on choreographies, all send and receive actions might fail, \nmodelling that there may be connection problems and\/or timeouts on both ends.\nWe formalise this behaviour by giving an operational semantics for RC\\xspace.\nWhen a process tries to perform an action, it can later check whether this action succeeded (as in \ntypical mainstream APIs for networking), and it is possible to program recovery (\\eg, by retrying \nto perform the action, or by executing alternative code).\nRC\\xspace supports further features that are studied in the setting of choreographies with communication \nfailures for the first time, like name mobility, dynamic creation of processes (networks in RC\\xspace can \ngrow at runtime), and branching. This allows us, for example, to program processes that offload \ncommunication attempts to parallel computations.\n\nRC\\xspace is intended as an implementation model that sits on a lower level than previous work \non choreographies. The abstraction of state-of-the-art choreographic models can be recovered by \nimplementing their language constructs as procedures and thus offer them ``as a library''---or \nDomain-Specific Languages (DSLs), if we see them as macros. For example, for previous models that \ntake communication robustness for granted, we can write parametric procedures for robust send and \nreceive actions that attempt at performing the desired action until it succeeds (or follow \nbest-effort strategies).\nWe illustrate this idea by implementing in RC\\xspace different language constructs from previous work,\nincluding procedural choreographies \\cite{CM17:forte}, \none-to-any\/any-to-one interactions \\cite{LNN16,LH17}, and scatter\/gather \\cite{LMMNSVY15,CMP18}.\nWe also exemplify how some of these constructs can be extended to allow for failure and specify \ncompensations.\nA pleasant consequence of sharing RC\\xspace as underlying model for these constructs is that we can now \ncombine these features from different works. (For example, the calculus with map\/reduce\nin \\cite{LNN16} does not support parametric procedures and \\cite{CM17:forte} \\viceversa.)\n\nThe realistic failure model in RC\\xspace allows us to identify more programming mistakes and \nprogram properties than in previous work, in particular some related to robustness. In general, we \nare interested in studying which guarantees we can provide for each communication that has been \n(choreographically) declared in a program, based on an analysis of its following implementation.\nExamples of relevant questions are: Can we check whether a communication will be eventually \nimplemented, even if failures occur? And, will it have the right (type of) payload?\nWe develop a novel typing discipline that can answer questions of this kind, and apply it to the\nstatic verification of at-most-once and exactly-once delivery guarantees for user-defined code.\n\nWe end our development by showing that the foundations given by RC\\xspace are applicable in practice. \nSpecifically, we define a formal translation (a compiler, if you like) from choreographies in RC\\xspace to \na more standard process model, \\ie, an asynchronous variant of the $\\pi$-calculus equipped with \nstandard I\/O actions that might fail. These asynchronous fallible I\/O actions are the only way \nprocesses may interact: there is no shared memory or agreement primitive. \nWe prove that, if the original choreography is well-typed, the synthesised\ncode is operationally equivalent and enjoys deadlock-freedom.\nFor space reasons, the synthesis procedure and process model are given in \\cref{sec:synthesis}.\n\n\\section{Failure model}\n\\label{sec:failure-model}\n\nWe discuss the failure model that we adopt in this work.\n\nCommunications are asynchronous and may fail.\nA successful send action implies that the sent message is\nnow handed over to the communication stack of the sender, which will attempt at transmitting the \nmessage to the receiver.\nIf transmission succeeds, the message reaches the receiver and is stored by the communication stack \nof the receiver in a dedicated memory.\nA successful receive action means that a message has been consumed by the intended receiver, \n\\ie, the message has been successfully \\emph{delivered}---this \nrequires that transmission was successful.\nA receive action fails if it is executed when there is no message that it can consume.\nThis models that there may be connection problems on the end of the receiver or that a timeout \noccurred on the receive action. We assume that communication and node failures are transient, \nmeaning that failing to interact with a node does not impede eventually doing it in later retries. \nWe leave persistent failures to future work.\n\nThere are two settings that we consider, depending on the kind of system that the programmer is dealing with.\n\n\\begin{setting}[Reliable Transmission]\n\\label{fm:reliable}\nSuccessfully executing a send action means that the message has been reliably stored in the \ncommunication medium between sender and receiver. By reliably stored, we mean that the message is \nnot going to be lost until its transmission from the sender's to the receiver's communication stack \neventually takes place.\nThis is the case, for example, of local Inter-Process Communication (IPC) mechanisms, like unnamed \npipes in POSIX systems, shared memory, or file-based communications. It is also the case of \ndistributed systems using reliable message delivery protocols like TCP, under the assumption that \nthere are no connection resets (or similar issues)---in these cases, middleware can be employed to \nre-establish connections.\nCommunication failures happen in this setting, \\eg, because send actions may \nfail to hand over messages to the sender's communication stack, and receive actions may fail \nbecause of timing issues (trying to receive messages before they reach the receiver's communication \nstack, or trying to receive something that is never sent).\n\\end{setting}\n\n\\begin{setting}[Unreliable Transmission]\n\\label{fm:unreliable}\nThis setting is more low-level than the previous one. Here, we assume no reliable middleware for \nmessage transport.\nThis is the case, for example, of distributed systems that use protocols with unreliable \nmessage delivery. It is also the case for systems that use protocols which, in theory, guarantee \nmessage delivery (like TCP) but, in practice, messages acknowledged on the protocol level may fail \nin reaching the application of the receiver due to connection resets and no middleware to deal with \nthese issues is employed. We assume absence of corruption (\\eg, through checksums).\n\nSuccessfully executing a send action in this setting still means that the local communication \nstack of the sender has accepted the task of eventually sending the message, but there is no \nguarantee that the message is actually going to be successfully passed on to the receiver's stack:\ncommunication media can lose messages.\nTherefore, a sender cannot know if a successfully sent message is going to be transmitted\nto the receiver, unless the programmer explicitly implements an acknowledgement mechanism on the \napplication level and such acknowledgement is received by the sender.\n\\end{setting}\n\nNaturally, the first setting allows for stronger guarantees. We will show that our typing can be \nused to guarantee at-most once message delivery (reception by the receiver's application) in both \nsettings. For the first setting, we will also show that typing can be used to guarantee \nexactly-once message delivery. For the second setting this is unrealistic, and as typically done in \npractice we have to switch to a best-effort strategy. Therefore, we demonstrate that our typing can \nbe used to guarantee best-effort delivery, in the sense that every time an application is \nprogrammed to receive a message, it is guaranteed at least the chance to receive it correctly.\n\n\\section{Related Work}\n\\label{sec:related}\n\nThe work nearest to ours is \\cite{APN17}, where choreographies are used to specify communication \nprotocols that abstract from data---a variant of Multiparty Session Types \\cite{HYC16}.\nUnreliability is modelled by allowing parts of a choreography declared in special optional blocks \nto become no-op non-deterministically. A static analysis guarantees that the network cannot get \nstuck even if all optional blocks are not executed. We see our work as complementary to\n\\cite{APN17}: while our initial motivation is similar, the aim is different. Our focus is \nproviding guarantees on implementations, and consequently choreographies in RC\\xspace are concrete \nprograms, in contrast with protocol specifications. There are also several major technical \ndifferences that make our choreography language more expressive. We mention the most \nrelevant ones.\n\nCommunications are synchronous in \\cite{APN17}. This means that if \na participant succeeds in sending a message, it knows that the receiver has also succeeded. In RC\\xspace, \nwe are interested in systems with asynchronous message passing. This requires \ndefining and analysing send and receive actions separately, since succeeding in one does not \nnecessarily mean succeeding with the other.\nSeparating between send and receive actions is also essential to the programming of recovery \nstrategies in RC\\xspace, which may be asymmetric for sender and receiver. For example, a sender may \nhave different conditions to check (\\eg, a number of retries) than those at the intended receiver \nfor deciding whether an action should be retried, which cannot be captured in \\cite{APN17}. \nRecovery strategies cannot be specified at all in the choreographies of \\cite{APN17}, which is \nanother key distinction with our work.\nThe modelling of recovery strategies in RC\\xspace is also what allows us to develop our type system for \nthe static verification of at-most-once and exactly-once delivery, which is not studied in \n\\cite{APN17}.\nThe choreography model in \\cite{APN17} does not include features equivalent to our \nprimitives for process spawning, name mobility, or parametric procedures. Parametric procedures are \nparticularly important for RC\\xspace: including error handling code in choreographies makes them \nnecessarily more complicated, and having procedures to modularise programs is useful, as we \nillustrate with our examples. Procedures are also key to our implementation\nof language constructs from previous choreography models ``as libraries'' in RC\\xspace.\n\nFrom a broader perspective, we think that merging our research direction with that of \n\\cite{APN17} (choreographic programs with protocol specifications) would be a very interesting \nfuture work, because it may yield a static analysis for RC\\xspace to check whether a given recovery \nstrategy guarantees the eventual execution of a high-level protocol (the latter might even \nabstract from failures, leaving their handling to the implementation).\n\nIn \\cite{CGY16}, choreographies for protocol specifications are augmented with controlled \nexceptions. These are different from communication failures, because they are controlled by the \nprogrammer and their propagation is ensured through communications that are assumed never to fail \n(thus, they are also different from our compensations for communication failures, where we react to \nunexpected failures and do not make this assumption). This approach has been refined in \n\\cite{CVBZE16}, by allowing for more fine-grained propagation of errors (but errors are still \nuser-defined, so similar comments apply).\n\nIn \\cite{LNN16}, the authors present a choreography model that considers potential failures of \nnodes (processes in our terminology). This approach is far from ours and that in \\cite{APN17}, \nsince the idea is that a system has redundant copies of a node type, and a choreography can \nspecify how many nodes of a type are needed to continue operating. No recovery can be programmed, \nand there is no presentation of how the approach can be adopted in realistic process models \n(compilation). Communications among functioning nodes are assumed infallible.\n\nOur work is also related to the research line on Choreographic Programming \n\\cite{M13:phd,CM13,DGGLM17}, a paradigm where choreographies are used to define implementations of \ncommunicating systems. This is the first work that studies how communication failures can be dealt \nwith in this paradigm. Our primitives for name mobility and parametric procedures are inspired \nfrom \\cite{CM17:forte}, but our methods are different, since we brought them into an asynchronous \nsetting with potentially-failing communications. Also, the fact that interactions are \nimplemented through separate send and receive actions in RC\\xspace is new to choreographic programming. \nPrevious work \\cite{CM17:ice} explored this distinction to achieve asynchronous communications in \nchoreographies, but these cannot fail and the distinction is used only in the runtime semantics (the \nseparate terms cannot be used programmatically).\n\nPrevious work explored a notion of bisimulation for a process calculus with explicit locations and \nlinks, where both nodes and links may fail at runtime \\cite{FH08}. Differently from our \nsetting, communications are synchronous, messages cannot be lost, and failures are permanent.\nExploring a similar notion of behavioural equivalence for RC\\xspace and our target process calculus is \ndefinitely an interesting future work, because it may lead to a substitutability principle for \ngenerated processes wrt choreographies. For example, we could replace a block of process code \nprojected from a choreography with an equivalent one without having to re-run compilation.\nAnother interesting application could be extending RC\\xspace to allow for ``undefined'' processes, whose \nbehaviour is left to be defined in other choreographies, or even (legacy) process code. Previous \nwork showed how to design such extensions for choreographies based on multiparty session \ntypes, obtaining choreographies with ``partial'' terms that refer to externally-defined code \n\\cite{MY13}. A notion of bisimulation for such partial terms could lead to relaxing the conditions \nfor well-typedness given in \\cite{MY13}.\n\nOur scatter construct in \\cref{sec:examples} recalls the unreliable \nbroadcast studied in \\cite{KGG14} for the setting of process calculi. \nA key difference is that, in \\cite{KGG14}, recovery cannot re-attempt failed communications \n(a process exits all current sessions when a failure occurs). Moreover, \ncommunications are synchronous in \\cite{KGG14}.\n\nOur formalisation of messages in transit partly recalls that in \\cite{FMN07}, which presents an \nagreement protocol that works under the assumption of quasi-reliable communications and node \ncrashes. While we do not consider (permanent) node crashes in this work, the programming of recovery \nstrategies is similar in our setting. This is to be expected, since in a distributed setting a node \nmay suspect that another node crashed by being unable to communicate with it (over some time).\nA detailed study of consensus in RC (and extentions to node failures) is definitely interesting \nfuture work.\n\nPrevious works on choreographies include choice operators that behave non-de\\-term\\-in\\-ist\\-ic\\-ally, \\eg,\n$C + C'$, read ``run either $C$ or $C'$'' \n\\cite{QZCY07,LGMZ08,CHY12} (and their labelled variant, in \n\\cite{HYC16}).\nThese operators do not capture the communication failures that we are interested in, for two \nreasons. First, they are programmed explicitly and are thus predictable.\nSecond, their formalisations assume that the propagation of choice \ninformation among processes is reliable (for compilation).\nThus, similar comments to those for the comparison with \\cite{CGY16} apply.\nObserve also that the soundness of these models is ensured under the assumption that any two \nprocesses involved in some interactions together perform exactly the same number of (dual) \ncommunication actions. This is unrealistic, since sender and receiver can have different policies in \npractice, as we already discussed.\n\nSome previous choreography models, like \\cite{CHY12}, include explicit parallel \ncomposition, $C \\mid C'$. RC\\xspace captures process parallelism using out-of-order \nexecution---a practice shared by other choreography models, see also\nthe paper that introduced it \\cite{CM13}. In general, actions performed by distinct processes can \nbe performed in any order. However, $C \\mid C'$ in \\cite{CHY12} (and other works, like \n\\cite{LTY15}) allows processes to have internal threads that share (and compete for) resources, \npossibly leading to races to due internal sharing. This is not allowed for in RC\\xspace; if you \nlike, this follows Go's slogan ``do not communicate by sharing memory; instead, share memory by \ncommunicating'' \\cite{effective-go}.\n\n\\section{Choreography Model}\n\\label{sec:chor-model}\n\n\\paragraph{Syntax}\nAn RC\\xspace program is a pair $\\langle \\mathcal{D},C\\rangle$, where $C$ is a choreography and \n$\\mathcal{D}$ is a set of (global) procedure definitions following the syntax displayed below.\nWe assume the Barendregt convention and work up to $\\alpha$-equivalence, renaming bound names \n(frame identifiers, process references, and procedure parameters) as needed.\n\\begin{align*}\n\t\\mathcal{D} \\Coloneqq {} &\n\t\t\\procdef{X}{\\vec{P}}{C}, \\mathcal{D} \\mid \\emptyset\n\t\\\\\n\tP  \\Coloneqq  {} &\n\t\t\\cframe{k}{\\pid p}{\\pid q}{T} \\mid \\pid p\\colon T \\mid f\\colon T\n\t\t\\mid l\n\t\\\\\n\tC \\Coloneqq  {} &\n\t\t\\cbindin{N}{C} \\mid \n\t\tI\\keyword{;}\\, C\\mid \\keyword{0} \n\t\\\\\n\tI \\Coloneqq  {} &\n\t\\clocal{\\pid p}{f} \\mid\n\t\t\\csend{k}{S} \\mid\n\t\t\\crecv{k}{R} \\mid\n\t\tX(\\vec{A})\\mid\n\t\t\\cond{E}{C}{C'} \\mid \n\t\\\\ \\mid {} & \\keyword{0} \n\t\\\\\n\tS  \\Coloneqq  {} &\n\t\t\\pid p.f \\mid \\pid p.\\pid r \\mid \\pid p.l\n\t\\\\\n\tR  \\Coloneqq  {} &\n\t\t\\pid q.f \\mid \\pid q\n\t\\\\\n\tE \\Coloneqq  {} &\n\t\t\\pid p.f \\mid \\pid p.\\csent{k} \\mid \\pid q.\\creceived{k} \\mid \\pid q.\\creceivedlbl{k}{l}\n\t\\\\\n\tA  \\Coloneqq  {} &\n\t\tk \\tteq k' \\mid \\pid p\\tteq \\pid p' \\mid f \\tteq f' \\mid l \\tteq l'\n\t\\\\\n\tN  \\Coloneqq  {} &\n\t\t\\cstart{\\pid p}{\\pid q}{f} \\mid \n\t\t\\cframe{k}{\\pid p}{\\pid q}{T}\n\\end{align*}\nProcess names ($\\pid p$,$\\pid q$,$\\pid r$,\\dots) identify processes that execute concurrently. Each process has exclusive access to a private memory cell for storing values of a fixed type $T$ from a fixed set $\\mathcal{V}$ of datatypes (\\eg $\\type{Nat}$, $\\type{Char}$, $\\type{Bool}$). Values are manipulated only via functions (terms $f$) specified in a \\emph{guest language} which is intentionally left as a parameter of the model. \nFollowing practices established in previous choreography models \\cite{M13:phd,CM13,DGGLM17,CM17:forte} we assume that evaluation of internal computations is local and terminates. \nThe only further assumptions about the guest language of internal computations \nare that it comes with a typing discipline and that it supports boolean values (or an equivalent mechanism). Typing judgements will have the form $\\vdash f\\colon T \\to S$ and the type of boolean values will be denoted as $\\type{Bool}$.\nBesides values used by internal computations, processes can communicate process names and label selections (terms $\\pid p.\\pid r$, $\\pid p.l$).\nThese payloads are inaccessible to the guest language and hence are assigned (disjoint) types $\\type{PID}$ and $\\type{LBL}$ not in $\\mathcal{V}$. For exposition convenience, we define $\\type{VAL}$ as the super type of all datatypes used by the guest language \\ie we define the subtyping relation $\\subtype$ as the smallest partial order such that $T \\subtype \\type{VAL}$ for any $T \\in \\mathcal{V}$. We assume a constructor $\\mid$ making (disjoint) union types based on $\\type{VAL}$, $\\type{PID}$, and $\\type{LBL}$ (\\eg $\\type{PID} \\mid \\type{LBL}$) and extend the relation $\\subtype$ accordingly ($\\type{PID} \\subtype \\type{PID} \\mid \\type{LBL}$, $\\type{LBL} \\subtype \\type{PID} \\mid \\type{LBL}$, \\etc). Note that we do not require the guest language to come with subtyping or unions.\nWe write $t \\in T$ to express that $t$ inhabits $T$.\n\nChoreography declarations ($N$) introduce new processes and frames in their continuation ($C$).\nTerm $\\cnewframe{k}{\\pid p}{\\pid q}{T}{C}$ declares a communication from $\\pid p$ to $\\pid q$ where $T$ is the payload type and $k$ is the frame identifier to be used by the implementation of the communication.\nTerm $\\cnewproc{\\pid p}{\\pid q}{f}{C}$ declares a new process $\\pid q$ where $\\pid p$ is the process that spawns $\\pid q$ and $f$ is a function used by $\\pid p$ to compute the initial value for the memory cell of $\\pid q$.\nChoreography statements ($I$) can be local computations, communication actions, conditionals, calls; all statements have continuations. Term \n$\\clocal{\\pid p}{f}$ represents an internal computation where process $\\pid p$ evaluates the \nfunction $f$ against its memory cell and updates its content.  Send and receive actions in \nthe implementation of $k$ are described by terms of the form $\\csend{k}{S}$ and $\\crecv{k}{R}$ \nwhere subterms $S$ and $R$ depend on the payload type.\nIn value exchanges, terms $\\csend{k}{\\pid p.f}$ and $\\crecv{k}{\\pid q.f'}$ read ``$\\pid p$ applies \n$f$ to the content of its memory cell and attempts to send the result on frame $k$'' and ``$\\pid q$ \nattempts to receive a value on frame $k$ and, if successful, applies $f'$ to its memory cell and \nthe received value'', respectively. (We assume functions in sends and receives to accept \nrespectively exactly one argument and exactly two arguments, where the first argument is the \nprocess memory content.)\nIn label selections, terms $\\csend{k}{\\pid p.l}$ and $\\crecv{k}{\\pid q}$ read\n``$\\pid p$ attempts to send the selection of label $l$ on frame $k$'' and ``$\\pid p$ attempts \nto receive a selection on frame $k$''. Selections are meant to propagate information regarding \ninternal choices and as such have no side effects on process memory or network knowledge. (As we \nwill discuss in \\cref{sec:synthesis}, this mechanism is crucial for synthesising correct \nimplementations of conditionals.)\nIn process exchanges, terms $\\csend{k}{\\pid p.\\pid r}$ and $\\crecv{k}{\\pid q.\\pid r}$ read ``$\\pid \np$ attempts to send $\\pid r$ on frame $k$'' and ``$\\pid q$ attempts to receive $\\pid r$ on frame \n$k$'', respectively. The only side effect of process exchanges is on network knowledge of the \nreceiver which may learn a new process reference. This is necessary since networks may grow during \nthe execution of choreographic programs as new processes are spawn. \nIn a conditional term $\\cond{E}{C_1}{C_2}$, a process evaluates the guard $E$ and chooses between \nthe possible continuations $C_1$ and $C_2$ accordingly. We explain the meaning of each kind of \nguard $E$ in the following.\n\\begin{itemize}\n\t\\item $\\pid p.f$: $\\pid p$ chooses $C_1$ if applying $f$ to its memory content yields \n$\\literal{true}$, and $C_2$ otherwise; \n\t\n\t\\item $\\pid p.\\csent{k}$: $\\pid p$ chooses $C_1$ if its last send attempt \nfor $k$ was successful, and $C_2$ otherwise; \n\t\n\t\\item $\\pid q.\\creceived{k}$: $\\pid q$ chooses $C_1$ if its last receive \nattempt for $k$ was successful, and $C_2$ otherwise;\n\t\n\t\\item $\\pid q.\\creceivedlbl{k}{l}$: $\\pid q$ chooses $C_1$ if it successfully\nreceived the label $l$ on $k$, and $C_2$ otherwise.\n\\end{itemize}\nTerm $X(\\vec{A})$ is a call of procedure $X$ with the set of named arguments $\\vec{A}$; these can \nbe frame identifiers, process names, or (names of) functions in the guest language.\nTerm $\\keyword{0}$ is the \\emph{no-op} statement, also used to represent terminated choreographies.\n\nProcedures are defined by terms $\\procdef{X}{\\vec{P}}{C}$ where $X$ is the procedure name, \n$\\vec{P}$ is a set of parameter declarations, and the program term $C$ is the procedure body. A term \n$f\\colon T\\to S$ in $\\vec{P}$ binds a function (name) in $C$ and specifies its type. A term \n$\\cframe{k}{\\pid p}{\\pid q}{T}$ in $\\vec{P}$ binds the frame identifier $k$ in $C$ and specifies its \ntype, sender and receiver. A term $\\pid p\\colon T$ in $\\vec{P}$ binds the process name $\\pid p$ in \n$C$ and specifies the type of its memory cell. \nA set of procedure definitions $\\mathcal{D}$ is well-formed provided that its procedure definitions \nhave unique names, all free names in their bodies are captured by their parameters, and all calls \nare to procedures in $\\mathcal{D}$.\n\nIn the sequel, we may omit $\\keyword{0}$, empty $\\keyword{else}$ branches, and use basic logical connectors in guards as syntactic sugar. \nFor instance, we write $\\scond{\\neg\\pid p.\\csent{k}}{C}$ as sugar for\n$\\cond{\\pid p.\\csent{k}}{\\keyword{0}}{C}$.\nIn procedure calls, we may omit assignments if formal and actual parameters have the same name and,\n\\eg, simply write $X(k)$ instead of $X(k \\tteq k)$.\n\n\\paragraph{Semantics}\nDynamics of RC\\xspace is specified by the reduction semantics defined in \\cref{fig:chor-semantics}. The \nsemantics is parameterised over global procedures $\\mathcal{D}$, and its states (hereafter \n\\emph{runtime configurations}) are quadruples $\\langle C,\\sigma,\\phi,G\\rangle$. We describe\nthe components of runtime configurations in the following.\n\\begin{description}\n\t\\item[$C$]\n\tThe first component is the current term program.\n\t\n\t\\item[$\\sigma$]\n\tThe second component, called \\emph{memory configuration}, keeps track of the memory cell of each process in the system, which is\n\taccessible to the guest language for performing internal computation.\n\tFormally, it is a partial map from process names to values, \\ie $\\sigma(\\pid p) = v$ denotes \n\tthat the memory cell of process $\\pid p$ stores value $v$.\n\t\t\n\t\\item[$\\phi$]\n\tThe third component, called (concrete) \\emph{frame dictionary}, is a partial map from frame \n\tnames to representations of their states.\n\tMore specifically, $\\phi(k) = \\phiframe{\\pid s}{u}{\\pid r}{u'}$ denotes that ``frame $k$ has \n\tstate $\\phiframe{\\pid s}{u}{\\pid r}{u'}$.\n\tThe processes $\\pid s$ and $\\pid r$ are, respectively, the (intended) sender and receiver of \n\t$k$. The elements $u$ and $u'$ represent the states of the communication stacks of sender and \n\treceiver for the specific frame, respectively. Formally, $u$ and $u'$ can be a payload (a \n\tvalue, a label, or a process name), $\\bot$ (the payload did not enter the stack), or a \n\tpayload flagged as ``removed from the stack'' (denoted by the decoration $\\checkmark$, as \n\tin $\\removed{v}$). Removal from the sender stack happens when the stack attempts at \n\ttransmitting the payload to the intended receiver, and \n\tremoval from the receiver stack means a successful delivery to the receiver's application.\n\t\n\t\\item[$G$]\n\tThe fourth component, called \\emph{connection graph}, is a directed graph with process names as \n\tnodes. A process neighbourhood represents the processes that it knows (and thus can communicate\n\twith).\n\\end{description}\nTogether, the last two components form a \\emph{network configuration}.\n\n\\begin{figure*}[t]\n\\begin{infrules}\n\t\\infrule[\\rname[C]{NP}][rule:c-new-proc]{\n\t\tf(\\sigma(\\pid p)) \\downarrow v\n\t\t\\and\n\t\tG' = G\\cup \\{\\pid p \\leftrightarrow \\pid q\\} \\cup \\{\\pid q \\rightarrow \\pid r \\mid G_1 \\vdash \\pid p \\rightarrow \\pid r \\}\n\t}{\n\t\t\\langle\\cnewproc{\\pid p}{\\pid q}{f}{C},\\sigma,\\phi, G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\n\t\t\\langle C,\\sigma(\\pid q)[v],\\phi, G'\\rangle\n\t}\n\t\\infrule[\\rname[C]{NF}][rule:c-new-frame]{\n\t}{\n\t\t\\langle\\cnewframe{k}{\\pid p}{\\pid q}{T}{C},\\sigma,\\phi, G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\n\t\t\\langle C,\\sigma,\\phi(k)[\\phiframe{\\pid p}{\\bot}{\\pid q}{\\bot}], G\\rangle\n\t}\n\n\t\\infrule[\\rname[C]{Int}][rule:c-int-comp]{\n\t\tf(\\sigma(\\pid p)) \\downarrow v\n\t}{\n\t\t\\langle\\clocal{\\pid p}{f}\\keyword{;}\\, C ,\\sigma,\\phi\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma(\\pid p)[v],\\phi, G\\rangle \n\t}\n\t\\infrule[\\rname[C]{Snd}][rule:c-send]{\n\t\ts(\\sigma(\\pid p)) \\downarrow u \\and\n\t\t\\phi(k) = \\phiframe{\\pid p}{\\_}{\\pid q}{\\_} \\and\n\t\tG \\vdash \\pid p \\to \\pid q \\and\n\t}{\n\t\t\\langle\\csend{k}{\\pid p.s}\\keyword{;}\\, C ,\\sigma,\\phi\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma,\\phi(k)_2[u], G\\rangle \n\t}\n\t\\infrule[\\rname[C]{SndFail}][rule:c-send-fail]{}{\n\t\t\\langle\\csend{k}{S}\\keyword{;}\\, C ,\\sigma,\\phi, G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma,\\phi, G\\rangle \n\t}\n\\\\\n\t\\infrule[\\rname[C]{Loss}][rule:c-loss]{\n\t\t\\phi(k) = \\phiframe{\\pid s}{u}{\\pid q}{\\_}\n\t\t\\and\n\t\tG \\vdash \\pid p \\leftrightarrow \\pid q\n\t}{\n\t\t\\langle C,\\sigma,\\phi\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma,\\phi(k)_2[\\removed{u}], G\\rangle \n\t}\n\t\\infrule[\\rname[C]{Comm}][rule:c-comm]{\n\t\t\\phi(k) = \\phiframe{\\pid p}{u}{\\pid q}{\\_}\n\t\t\\and\n\t\tu \\in \\mathcal{U}\n\t\t\\and\n\t\tG \\vdash \\pid p \\leftrightarrow \\pid q\n\t}{\n\t\t\\langle C,\\sigma,\\phi\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma,\\phi(k)[\\phiframe{\\pid p}{\\removed{u}}{\\pid q}{u}], G\\rangle \n\t}\n\t\\infrule[\\rname[C]{RcvV}][rule:c-recv-val]{\n\t\t\\phi(k) = \\phiframe{\\pid p}{\\_}{\\pid q}{u}\n\t\t\\and\n\t\tu \\in \\{v,\\removed{v}\\}\n\t\t\\and\n\t\tf(\\sigma(\\pid q),v) \\downarrow w\n\t}{\n\t\t\\langle \\crecv{k}{\\pid q.f}\\keyword{;}\\, C ,\\sigma,\\phi,G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma(\\pid q)[w],\\phi(k)_4[\\removed{w}], G\\rangle \n\t}\n\t\\infrule[\\rname[C]{RcvP}][rule:c-recv-pid]{\n\t\t\\phi(k) = \\phiframe{\\pid p}{\\_}{\\pid q}{u}\n\t\t\\and\n\t\tu \\in \\{\\pid r,\\removed{\\pid r}\\}\n\t}{\n\t\t\\langle \\crecv{k}{\\pid q}\\keyword{;}\\, C ,\\sigma,\\phi,G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma,\\phi(k)_4[\\removed{\\pid r}], G \\cup \\{\\pid q \\to \\pid r\\}\\rangle \n\t}\n\t\\infrule[\\rname[C]{RcvL}][rule:c-recv-lbl]{\n\t\t\\phi(k) = \\phiframe{\\pid p}{\\_}{\\pid q}{u}\n\t\t\\and\n\t\tu \\in \\{l,\\removed{l}\\}\n\t}{\n\t\t\\langle \\crecv{k}{\\pid q}\\keyword{;}\\, C ,\\sigma,\\phi,G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma,\\phi(k)_4[\\removed{l}], G\\rangle \n\t}\n\t\\infrule[\\rname[C]{RcvFail}][rule:c-recv-fail]{\n \t\t\\phi(k)_4 = \\bot\n\t}{\n\t\t\\langle\\crecv{k}{R}\\keyword{;}\\, C ,\\sigma,\\phi, G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C,\\sigma,\\phi, G\\rangle \n\t}\n\t\\infrule[\\rname[C]{IfSnt}][rule:c-if-sent]{\n\t\t\\phi(k)_1 = \\pid p \\and\n\t\t\\text{if } \\phi(k)_2 \\neq \\bot\n\t\t\\text{ then } j = 1\n\t\t\\text{ else } j = 2\n\t}{\n\t\t\\langle \\cond{\\pid p.\\csent{k}}{C_1}{C_2}\\keyword{;}\\, C ,\\sigma,\\phi,G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C_j\\keyword{;}\\, C,\\sigma,\\phi,G\\rangle \n\t}\n\t\\infrule[\\rname[C]{IfRcv}][rule:c-if-recv]{\n\t\t\\phi(k)_3 = \\pid q \\and\n\t\t\\text{if } \\phi(k)_4 \\in \\mathcal{U}^\\checkmark\n\t\t\\text{ then } j = 1\n\t\t\\text{ else } j = 2\n\t}{\n\t\t\\langle \\cond{\\pid q.\\creceived{k}}{C_1}{C_2}\\keyword{;}\\, C ,\\sigma,\\phi,G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C_j\\keyword{;}\\, C,\\sigma,\\phi,G\\rangle \n\t}\n\t\\infrule[\\rname[C]{IfLbl}][rule:c-if-lbl]{\n\t\t\t\\phi(k)_3 = \\pid q \\and\n\t\t\\text{if } \\sigma(k)_4 = \\removed{l}\n\t\t\\text{ then } j = 1\n\t\t\\text{ else } j = 2\n\t}{\n\t\t\\langle \\cond{\\pid q.\\creceivedlbl{k}{l}}{C_1}{C_2}\\keyword{;}\\, C ,\\sigma,\\phi,G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C_j\\keyword{;}\\, C,\\sigma,\\phi,G\\rangle \n\t}\n\t\\infrule[\\rname[C]{IfExp}][rule:c-if-exp]{\n\t\t\\text{if } f(\\sigma(\\pid p)) \\downarrow \\literal{true}\n\t\t\\text{ then } j = 1\n\t\t\\text{ else } j = 2\n\t}{\n\t\t\\langle \\cond{\\pid p.f}{C_1}{C_2}\\keyword{;}\\, C ,\\sigma,\\phi,G\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C_j\\keyword{;}\\, C,\\sigma,\\phi,G\\rangle \n\t}\n\t\n\t\\infrule[\\rname[C]{Str}][rule:c-struct]{\n\t\tC_1 \\precongr_{\\mathcal{D}} C_1'\n\t\t\\and\n\t\t\\langle C_1' ,\\sigma_1,\\phi_1,G_1\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C_2',\\sigma_2,\\phi_2,G_2\\rangle\n\t\t\\and\n\t\tC_2' \\precongr_{\\mathcal{D}} C_2\n\t}{\n\t\t\\langle C_1 ,\\sigma_1,\\phi_1,G_1\\rangle \n\t\t\\reducesto_{\\mathcal{D}}\t\t\n\t\t\\langle C_2,\\sigma_2,\\phi_2,G_2\\rangle \n\t}\n\n\t\\infrule[\\rname[C]{Unfold}][rule:c-unfold]{\n\t\tX(\\vec{P}) = C_2 \\in \\mathcal{D}\n\t\t\\qquad\n\t\t\\vec{P} = \\dom(\\vec{A})\n\t}{\n\t\tX(\\vec{A})\\keyword{;}\\, C_1\n\t\t\\precongr_{\\mathcal{D}}\t\n\t\tC_2[\\vec{A}]\\keyword{;}\\, C_1\n\t}\n\t\\infrule[\\rname[C]{Nil}][rule:c-nil]{}{\n\t\t\\keyword{0}\\keyword{;}\\, C \\precongr_{\\mathcal{D}}\tC\n\t}\n\t\\infrule[\\rname[C]{Swap}][rule:c-swap]{\n\t\tC_1\n\t\t\\dotrel{\\congr}\n\t\tC_2\n\t}{\n\t\tC_1\n\t\t\\congr_{\\mathcal{D}}\t\n\t\tC_2\n\t}\n\t\\infrule[\\rname[C]{I-I}][rule:c-i-i]{\n\t\t\\mathrm{pn}(I_1) \\cap \\mathrm{pn}(I_2) = \\emptyset\n\t}{\n\t\tI_1\\keyword{;}\\, I_2\n\t\t\\dotrel{\\congr}\n\t\tI_2\\keyword{;}\\, I_1\n\t}\n\t\\infrule[\\rname[C]{I-N}][rule:c-i-n]{\n\t\t\\mathrm{pn}(I) \\cap \\mathrm{pn}(N) = \\emptyset\t\n\t}{\n\t\tI\\keyword{;}\\, \\cbindin{N}{C}\n\t\t\\dotrel{\\congr}\n\t\t\\cbindin{N}{I \\keyword{;}\\, C}\n\t}\n\t\\infrule[\\rname[C]{N-N}][rule:c-n-n]{\n\t\t\\mathrm{pn}(N_1) \\cap \\mathrm{pn}(N_2) = \\emptyset\t\n\t}{\n\t\t\\cbindin{N_1}{\\cbindin{N_2}{C}}\n\t\t\\dotrel{\\congr}\n\t\t\\cbindin{N_2}{\\cbindin{N_1}{C}}\n\t}\n\t\\infrule[\\rname[C]{I-If}][rule:c-i-if]{\n\t\t\\mathrm{pn}(I) \\cap \\mathrm{pn}(E) = \\emptyset\n\t}{\n\t\t\\cond{E}{I\\keyword{;}\\, C_1}{I\\keyword{;}\\, C_2}\n\t\t\\dotrel{\\congr}\n\t\tI\\keyword{;}\\, \\cond{E}{C_1}{C_2}\n\t}\n\t\\infrule[\\rname[C]{If-I}][rule:c-if-i]{}{\n\t\t\\cond{E}\n\t\t{C_1\\keyword{;}\\, I}\n\t\t{C_2\\keyword{;}\\, I}\n\t\t\\dotrel{\\congr}\n\t\t\\cond{E}{C_1}{C_2}\n\t\t\\keyword{;}\\, I\n\t}\n\t\\infrule[\\rname[C]{N-If}][rule:c-n-if]{\n\t\t\t\\mathrm{pn}(N) \\cap \\mathrm{pn}(E) = \\emptyset\t\t\n\t}{\n\t\t\\cond{E}{\\cbindin{N}{C_1}}{\\cbindin{N}{C_2}}\n\t\t\\dotrel{\\congr}\n\t\t\\cbindin{N}{\\cond{E}{C_1}{C_2}}\n\t}\n\t\\infrule[\\rname[C]{If-If}][rule:c-if-if]{\n\t\t\\mathrm{pn}(E_1) \\cap \\mathrm{pn}(E_2) = \\emptyset\n\t}{\n\t\t\\begingroup\\def1.1{1.1}\n\t\t\\begin{array}{r}\n\t\t\t\\cond{E_1}\n\t\t\t\t{\\cond{E_2}{C_{1}^{1}}{C_{1}^{2}}\\\\\\!}\n\t\t\t\t{\\cond{E_2}{C_{2}^{1}}{C_{2}^{2}}}\n\t\t\\end{array}\n\t\t\\endgroup\n\t\t\\dotrel{\\congr}\n\t\t\\begingroup\\def1.1{1.1}\n\t\t\\begin{array}{r}\n\t\t\t\\cond{E_2}\n\t\t\t\t{\\cond{E_1}{C_{1}^{1}}{C_{2}^{1}}\\\\\\!}\n\t\t\t\t{\\cond{E_1}{C_{1}^{2}}{C_{2}^{2}}}\n\t\t\\end{array}\n\t\t\\endgroup\n\t}\n\\end{infrules}\n\t\\caption{Choreographic model, operational semantics}\n\t\\label{fig:chor-semantics}\n\\end{figure*}\n\n\nWe define some convenient notation for the definition of our semantics.\nLet $\\mathcal U$ be the set of all payloads, \\ie, $\\mathcal{U} = \\type{VAL} \\uplus \\type{LBL} \n\\uplus \\type{PID}$. A frame state $\\phiframe{\\pid s}{u}{\\pid r}{u'}$ is thus an element of\n$\\type{PID} \\times \\mathcal{U}{}_\\bot^\\checkmark\\times\\type{PID}\\times\\mathcal{U}{}_\\bot^\\checkmark$,\nwhere: \n$\\mathcal{U}{}_\\bot^\\checkmark = \n\\mathcal{U} \\uplus \\{\\bot\\} \\uplus \\mathcal{U}^\\checkmark$, with $\\mathcal{U}^\\checkmark \n= \\{\\removed{u} \\mid u \\in \\mathcal{U}\\}$ the set of payloads flagged as ``removed'' from the stack.\nAn expression $\\phi(k)_i$ denotes the $i$-th component of the frame state $\\phi(k)$. A judgement $G \n\\vdash \\pid p \\to \\pid q$ states that $G$ has the edge $\\pid p \\to \\pid q$.\nUpdates to $\\sigma$ and $\\phi$ are written using a square bracket notation, \nspecifically: $\\sigma(\\pid p)[v]$ is the function defined as $\\sigma$\neverywhere except $\\pid p$, which is mapped to $v$;\n$\\phi(k)[\\phiframe{\\pid s}{u}{\\pid r}{u'}]$ is as $\\phi$, but $k$ is now mapped to\n$\\phiframe{\\pid s}{u}{\\pid r}{u'}$; $\\phi(k)_2[u]$ changes the second element (the sender's \nside) of the frame state for $k$ to $u$; and, likewise, $\\phi(k)_4[u]$ changes the fourth element \n(the receiver's side) of the frame state for $k$ to $u$.\n\n\n\\begin{figure*}[t]\n\\centering\n\\begin{tikzpicture}[auto,yscale=1.4,xscale=3.5,\n\t\tstate\/.style={\n\t\t\trectangle split, \n\t\t\trectangle split horizontal, \n\t\t\trectangle split parts=2,\n\t\t\tdraw,\n\t\t\trounded corners=5pt,\n\t\t\touter sep=2pt,\n\t\t\tinner sep=2pt,\n\t\t\tfont=\\small,\n\t    text height=1.3ex,\n\t    text depth=.1ex,\n\t    text centered,\n\t    minimum height=1.8em,\n\t\t},\n\t\ttransition\/.style={\n\t\t\t>=open triangle 60,->,\n\t\t\trounded corners=5pt,\n\t\t},\n\t\ttransition loss\/.style={\n\t\t\ttransition,\n\t\t\tdashed,\n\t\t\tblue\n\t\t},\n\t\ttransition label\/.style={\n\t\t\tpos=.45,\n\t\t\tfont=\\footnotesize,\n\t\t}\n\t]\n\n\t\\newcounter{stc}\n\t\\def\\state(#1) at (#2) <#3|#4>{\n\t\t\\stepcounter{stc}\n\t\t\\node[state,\n\t\t\tlabel={[label distance=-3pt]110:\\color{white!40!black}\\footnotesize(\\alph{stc})}\n\t\t] (#1) at (#2) {\t\t\t\t\n\t\t\t\\code{#3}\n\t\t\t\\nodepart{two}\n\t\t\t\\ensuremath{#4}\n\t\t};\n\t}\n\n\t\\state (nA) at (1,3) <1,2,3,4|>\n\t\\state (nB) at (1,2) <  2,3,4|\\bot,\\bot>\n\t\\state (nC) at (2,2) <    3,4|v,\\bot>\n\t\\state (nD) at (2,3) <    3,4|\\removed{v},v>\n\t\\state (nE) at (3,3) <      4|\\removed{v},\\removed{v}>\n\t\\state (nF) at (1,1) <    2,4|\\bot,\\bot>\n\t\\state (nG) at (0,2) <    3,4|\\bot,\\bot>\n\t\\state (nH) at (0,1) <      4|\\bot,\\bot>\n\t\\state (nI) at (2,1) <      4|v,\\bot>\n\t\\state (nJ) at (3,2) <    3,4|\\removed{v},\\bot>\n\t\\state (nK) at (3,1) <      4|\\removed{v},\\bot>\n\t\\state (nL) at (2,0) <      4|\\removed{v},v>\n\t\t\t\t\t\n\t\\draw[transition] (nA) -- \n\t\tnode[transition label] {\\ref{rule:c-new-frame}} \n\t\t(nB);\n\t\\draw[transition] (nB) --\n\t\tnode[transition label] {\\ref{rule:c-send}} \n\t\t(nC);\n\t\\draw[transition] (nC) --\n\t\tnode[transition label,swap] {\\ref{rule:c-comm}} \n\t\t(nD);\n\t\\draw[transition] (nD) --\n\t\tnode[transition label] {\\ref{rule:c-recv-val}} \n\t\t(nE);\n\t\\draw[transition] (nB) -- \n\t\tnode[transition label] {\\ref{rule:c-recv-fail}} \n\t\t(nF);\n\t\\draw[transition] (nB) -- \n\t\tnode[transition label,swap] {\\ref{rule:c-send-fail}} \n\t\t(nG);\n\t\\draw[transition] (nF) -- \n\t\tnode[transition label,swap] {\\ref{rule:c-send-fail}} \n\t\t(nH);\n\t\\draw[transition] (nF) --\n\t\tnode[transition label] {\\ref{rule:c-send}} \n\t\t(nI);\n\t\\draw[transition] (nG) -- \n\t\tnode[transition label] {\\ref{rule:c-recv-fail}} \n\t\t(nH);\n\t\\draw[transition] (nC) -- \n\t\tnode[transition label] {\\ref{rule:c-recv-fail}} \n\t\t(nI);\n\t\\draw[transition loss] (nC) -- \n\t\tnode[transition label] {\\ref{rule:c-loss}} \n\t\t(nJ);\n\t\\draw[transition] (nI) --\n\t\tnode[transition label] {\\ref{rule:c-comm}} \n\t\t(nL);\n\t\\draw[transition loss] (nI) -- \n\t\tnode[transition label] {\\ref{rule:c-loss}} \n\t\t(nK);\n\t\\draw[transition] (nJ) -- \n\t\tnode[transition label] {\\ref{rule:c-recv-fail}} \n\t\t(nK);\n\\end{tikzpicture}\n\\caption{An end-to-end communication and its execution.}\n\\label{fig:com-execution}\n\\end{figure*}\n\n\nFor compactness, the presentation relies on the structural precongruence $\\precongr_\\mathcal{D}$ via the standard mechanism of \\cref{rule:c-struct}; the relation is defined as the smallest relation on choreographies closed under rules in \\cref{fig:chor-semantics} (discussed below) and under syntactic constructors of the language. \nHerein, $C \\congr_\\mathcal{D} C'$ is a shorthand \nfor $C \\precongr_\\mathcal{D} C'$ and $C' \\precongr_\\mathcal{D} C$.\nUnnecessary schematic variables are omitted and replaced by the wildcard $\\_$.\n\n\\Cref{rule:c-new-proc} describes the creation of a new process which inherits the network knowledge \nof its parent---as common in standard process models like, \\eg, the $\\pi$-calculus \\cite{MPW92}.\nThe expression $f(\\sigma(\\pid p)) \\downarrow v$ in the rule premises states that the evaluation of \n$f$ against the content of the memory of $\\pid p$ yields value $v$. \nIn the reactum the memory of $\\pid p$ is initialised to $v$ and the connection graph is updated to \ninclude: the mutual connection between $\\pid p$ and $\\pid q$; and a connection from $\\pid q$ to \neach process in the neighbourhood of $\\pid p$.\n\\Cref{rule:c-new-frame} models the creation of a new frame. In the reactum the frame is given \nstatus $\\phiframe{\\pid p}{\\bot}{\\pid q}{\\bot}$, meaning that neither the sender's or the \nreceiver's stacks contain the frame payload.\n\\Cref{rule:c-send,rule:c-send-fail} describe the execution of a send attempt for frame $k$. In the \nfirst case the sender computes a payload for $k$ and its stack accepts it, whereas in the second \ncase it rejects it (the send action failed). In both cases no information about the attempt is \npropagated to the receiver side. For conciseness, we extend the notation of function evaluation \n($f(\\sigma(\\pid p)) \\downarrow v$) to process names and labels by regarding them as constants---we \nsignal this abuse of the notation by writing $s$ and $u$ in place of $f$ and $v$, respectively.\nObserve that \\cref{rule:c-send} does not check the frame's state, meaning that a sender may \nperform multiple send actions resulting in the transmission of different payloads for the \nsame frame.\n\\Cref{rule:c-comm,rule:c-loss} model frame transmission and its non-deterministic outcome.\nOnly in the first case the payload reaches the receiver (successful transmission), but the sender \nhas no knowledge of this outcome since the effect on its stack is the same \n(both reacti set $\\phi(k)_2$ to $\\removed{u}$).\n\\Cref{rule:c-recv-val,rule:c-recv-pid,rule:c-recv-lbl,rule:c-recv-fail} define the execution of a \nreceive attempt for frame $k$. The first three rules model the delivery of different types of \npayloads (values, processes, and labels) and the fourth models failure due to any payload for $k$ \nnot having reached the receiver end yet. \\Cref{rule:c-unfold} unfolds procedure calls by replacing \nall occurrences of formal arguments in its body ($C_2$) with actual ones as prescribed by the \nsubstitution $\\vec{A}$ (provided its domain of definition coincides with the set $\\vec{P}$) and all \nnames bound in the procedure body $C_2$ with fresh ones as per Barendregt's convention.\n\\Cref{rule:c-i-i,rule:c-i-n,rule:c-n-n,rule:c-i-if,rule:c-if-i,rule:c-n-if,rule:c-if-if} model the dynamic \nrescheduling of non-interfering operations, like communications involving different processes, \nwhere $\\mathrm{pn}(C)$ is the set of process names in $C$. We omit the symmetric rules for \nconditionals, for swapping statements in\/from the continuation of conditionals.\n\n\\paragraph{Failure Model}\nWe can now formalise our two settings described in \\cref{sec:failure-model}.\nSpecifically, in the remainder: whenever we discuss \\cref{fm:reliable} (Reliable Transmission), we \nrefer to RC\\xspace without \\cref{rule:c-loss} (without that rule, messages cannot be lost in \nthe transport phase); whenever we discuss \\cref{fm:unreliable} (Unreliable Transmission), we use \nfull RC\\xspace (\\cref{rule:c-loss} is included).\nWe illustrate the (formal) difference between the two settings with an end-to-end communication \nexample.\nConsider the program term below.\n\\begin{snippet}\n\t\\cnewframe{k}{\\pid s}{\\pid r}{T}{\\\\\n\t\\indent \\csend{k}{\\pid s.f}\\keyword{;}\\, \\\\\n\t\\indent \\crecv{k}{\\pid r.g}\\keyword{;}\\, \\\\\n\t\\indent \\keyword{0}}\n\\end{snippet}\nAssume w.l.o.g.~that $v$ is the payload computed using $f$, that $f$, $g$, $v$, and memory cells of \nsender and receiver have the correct types, and that $\\pid s$ and $\\pid r$ are \nconnected.\nThe transition system in \\cref{fig:com-execution} depicts all possible executions for this program.\nIn \\cref{fm:reliable}, the dashed edges are not included since they require \\cref{rule:c-loss}. In \n\\cref{fm:unreliable}, instead, all transitions are possible (even the dashed ones).\nStates represent runtime configurations and edges reductions.\nFor exposition convenience states are assigned letters from (a) to (l) and only a subset of the data forming a runtime configuration is included:\nprogram terms are represented by the list of lines from the program above (left half of each state);\nprocess memory cells are omitted;\nonly stacks of $\\pid s$ and $\\pid r$ for $k$ are included (right half of each state);\nconnection graphs are omitted.\nEdges are labelled with the names of the reduction rules used to derive them; rules about structural precongruence are omitted---the reduction from (b) to (f) is the only one that requires also \\cref{rule:c-struct}.\n\nThe program execution begins in state (a) and every run reaches a configuration with program term \n$\\keyword{0}$, \\ie, every execution eventually terminates---there are configurations like (i) where $\\keyword{0}$ \nmay still admit some reductions but these are derivable only using \\cref{rule:c-comm,rule:c-loss} and eventually reach a configuration that cannot be further reduced.\nWe describe the different situations after executing all terms.\n\\begin{description}\n\\item[(e)] This is the only configuration where $v$ is marked as delivered and there is exactly one path from (a) to (e) that is the only chain of events without failures.\n\n\\item[(h)] This configuration is reached only if both the send and receive fail. There are two paths to (h), one passing through (g) and one through (f).\nIn the former the failure at the receiver side is consequential to the failure at the sender side \nwhereas in the second the two failures are independent. In fact, (f) may also reduce to (i).\n\n\\item[(i)] This configuration is reached only if the send succeeds and the receive fails regardless \nof the relative ordering of such events. Although the program has been reduced to $\\keyword{0}$, (i) \nadmits further reductions modelling a transmission attempt by the sender stack.\n\n\\item[(l)] Considerations made for (i) apply (l) too since it is reachable only from (i) and via a reduction modelling actions performed by the communication stack.\n\\end{description}\nOnly if we assume \\cref{fm:unreliable} (\\ie we include \\cref{rule:c-loss}) then, (j) and (k) become \nreachable. In particular, (k) configuration is reached only if the send succeeds, the receive fails, \nand the frame is lost during transmission. Similarly to (h), the last two events are consequential \nonly along the path through (j). Indeed (i) admits a reduction to a configuration unreachable from \n(j).\nIf we exclude any form of failure instead (\\ie \\cref{fm:reliable} and the additional assumption \nsend and receive operations may never fail) then, only configurations on the path to (e) remain \nreachable from (a)---as expected.\n\n\n\\section{Application Examples}\n\\label{sec:examples}\n\nConsider the procedure in the snippet below.\n\\begin{snippet}\n\t\\procdef{\\code{sendWhile}}{\n\t\t\\cframe{k}{\\pid s}{\\pid r}{T_k},\n\t\t\\pid s\\colon T_{\\pid s},\n\t\tf\\colon T_{\\pid s} \\to T_k,\n\t\tc\\colon T_{\\pid s} \\to T_{\\pid s},\n\t\tg\\colon T_{\\pid s} \\to \\type{Bool}\n\t}{\\\\\\indent\n\t\t\\scond{\\clocal{\\pid s}{g} \\land \\neg \\pid s.\\csent{k}}{\n\t\t\t\\\\\\indent\\indent\n\t\t\t\\csend{k}{\\clocal{\\pid s}{f}}\\keyword{;}\\,\n\t\t\t\\\\\\indent\\indent\n\t\t\t\\clocal{\\pid s}{c}\\keyword{;}\\,\n\t\t\t\\\\\\indent\\indent\n\t\t\t\\code{sendWhile}(k,\\pid s,f,c,g)\n\t\t\t\\\\\\indent\\mspace{-10.0mu}\n\t\t}\n\t}\n\\end{snippet}\nAbove, process $\\pid s$ tries to send a payload computed by $f$ for frame $k$ \nuntil its stack accepts it or the guard $g$ is falsified; $c$ is \n(locally) computed between attempts.\nWe can use \\code{sendWhile} to implement different recovery strategies, as other procedures:\n\\begin{itemize}\n\t\\item\n\ta procedure \\code{send} that never gives up sending until successful is implemented as a call \n\tto \\code{sendWhile} with a guard $g$ that is always true;\n\n\t\\item\n\ta procedure \\code{sendN} that gives up after $n$ attempts is implemented as a call to \n\t\\code{sendWhile} where $g$ and $c$ are used to test and increment a counter respectively;\n\n\t\\item\n\ta procedure \\code{sendT} that gives up after a timeout $t$ is implemented as a call to \n\t\\code{sendWhile} where $g$ tests a timer;\n\n\t\\item\n\tvariations of the above that use exponential backoff or (\\eg procedure\n\t\\code{sendExpBackoff} from \\cref{sec:intro}) are implemented by passing a delay computation \n\t(``sleep'') as $c$.\n\\end{itemize}\nProcedure \\code{recvWhile}, the analogue of \\code{sendWhile}, is likewise implemented: one just has \nto replace every frame operation with its dual.\n\n\\paragraph{Procedural Choreographies}\nAssume \\cref{fm:reliable}, and consider the procedure below.\n\\begin{snippet}\n\t\\procdef{\\code{com}}{\n\t\t\\pid s\\colon T_{\\pid s},\n\t\t\\pid r\\colon T_{\\pid r},\n\t\tf\\colon T_{\\pid s} \\to T_k,\n\t\tg\\colon T_{\\pid r}\\times T_{k} \\to T_{\\pid r}\n\t}{\\\\\\indent\n\t\t\\cnewframe{k}{\\pid s}{\\pid r}{T}{\n\t\t\\\\\\indent\n\t\t\\code{send}(k,\\pid s,f)\\keyword{;}\\,\n\t\t\\\\\\indent\n\t\t\\code{recv}(k,\\pid r,g)\n\t\t}\n\t}\n\\end{snippet}\nProcedure \\code{com} implements exactly-once delivery since \\code{send}  terminates only if the \npayload is accepted by the sender's stack, which in turn ensures transmission, and finally \n\\code{recv} terminates only after the payload is delivered.\n\nAs a consequence, we can recover the language of Procedural Choreographies (PC) \\cite{CM17:forte}, \nwhich abstracts from communication failures, ``as a library''. For more evocative notation we \ndefine ${\\pid s.f \\ttTo \\pid r.g}$, ${\\pid s \n\\ttTo \\pid r[l]}$, and ${\\pid s.\\pid p \\ttTo \\pid r}$ as syntactic sugar for\n$\\code{com}(\\pid s,\\pid r,f,g)$ and its equivalent versions for label and process name \ncommunications, respectively.\nThen, translating PC programs into RC\\xspace is a matter of rewriting a\nfew symbols, \\eg, $\\pid s.f \\ttto \\pid r.f'$ in PC becomes ${\\pid s.f \\ttTo \\pid r.g}$ in RC\\xspace.\n\n\\paragraph{A search engine}\nWe now present a more sophisticated scenario, where a search process $\\pid s$ queries\nproviders $\\pid p_1,\\dots,\\pid p_m$ making a limited number of attempts.\nProcedures allow us to hide the request-response implementation and write\n\\begin{snippet*}\n\t\\code{reqRes}(\\pid s,\\pid p_1,req,\\overline{req},\\overline{resp},resp_1)\\keyword{;}\\,\n\t\\\\\\dots\\keyword{;}\\,\\\\\n\t\\code{reqRes}(\\pid s,\\pid p_m,req,\\overline{req},\\overline{resp},resp_m)\\keyword{;}\\,\n\\end{snippet*}\\looseness=-1\nwhere handling of internal representations and computations is delegated to functions $req$, $\\overline{req}$, $\\overline{resp}$, $resp_i$ written in the guest language.\nSince all queries are independent we can offload them to worker processes spawned by $\\pid s$ as shown in procedure $\\code{reqRes}$ below.\n\\begin{snippet}[\n\t\t\\def\\alignedcomment#1{%\n\t\t\t\\hfill\\quad%\n\t\t\n\t\t\t\\commentline{#1}%\n\t\t\n\t\t}\n\t]\n\t\\procdef{\\code{reqRes}}{\n\t\t\\pid s\\colon T_\\pid{s},\n\t\t\\pid p\\colon T_\\pid{p},\n\t\t\\\\\\indent\\indent\n\t\treq\\colon T_\\pid{s} \\to \\type{Maybe(Str)},\n\t\t\\overline{req}\\colon \\type{Maybe(Str)} \\to T_\\pid{p},\n\t\t\\\\\\indent\\indent\n\t\t\\overline{resp}\\colon T_\\pid{p} \\to \\type{Str},\n\t\tresp\\colon \\type{Maybe(Str)} \\to T_{\\pid s}\n\t}{\n\t\\\\\\indent\n\t\\cnewproc{\\pid s}{\\pid w}{req}{}\n\t\\alignedcomment{start a worker initialised with $req$}\n\t\\\\\\indent\t\n\t\\code{comPID}(\\pid s,\\pid p,\\pid w)\\keyword{;}\\,\n\t\\alignedcomment{introduce worker and provider}\n\t\\\\\\indent\t\n\t\\cnewframe{k_1}{\\pid w}{\\pid p}{\\type{Str}}{}\n\t\\alignedcomment{declare a frame for the request}\n\t\\\\\\indent\n\t\\code{sendN}(k_1,\\pid w,req)\\keyword{;}\\,\n\t\\alignedcomment{sends the query; $n$ attempts}\n\t\\\\\\indent\n\t\\code{recvT}(k_1,\\pid p,\\overline{req})\\keyword{;}\\,\n\t\\alignedcomment{receive the query but set a timeout}\n\t\\\\\\indent\n\t\\cnewframe{k_2}{\\pid p}{\\pid w}{\\type{Maybe(Str)}}{}\n\t\\alignedcomment{a frame for the response}\n\t\\\\\\indent\t\n\t\\cond{\\pid p.\\creceived{k_2}}{\n\t\\\\\\indent\\indent\n\t\t\\code{comN}(k_2,\\pid p,\\pid w,\\code{some}(resp),id)\\keyword{;}\\,\n\t\t\\alignedcomment{send response}\n\t\\\\\\indent\\mspace{-10.0mu}\n\t}{\n\t\\\\\\indent\\indent\n\t\t\\code{comN}(k_2,\\pid p,\\pid w,\\code{none},id)\\keyword{;}\\,\n\t\t\\alignedcomment{send empty response}\n\t\\\\\\indent\\mspace{-10.0mu}\n\t}\n\t\\\\\\indent\n\t\\code{com}(\\pid w,\\pid s,id,\\overline{resp})\\keyword{;}\\,\n\t\\alignedcomment{rely response}\n\t}\n\\end{snippet}\n\n\\paragraph{Best-effort strategies}\nAssume \\cref{fm:unreliable}.\nIn this setting, procedure \\code{com} is not robust any more, for \npayloads may now be lost during transmission, leaving the receiver looping forever.\n\nThis is a common problem in practice, which is addressed by switching to ``best-effort'' strategies where delivery is possible (to varying degrees) but not certain.\nBelow is a procedure that implements a simple communication protocol with capped retries and acknowledgements to the sender. In this, the strategy implemented by \\code{comACK} can be regarded as a simplification of that of TCP; four-phase handshakes or other protocols are implementable in RC\\xspace as well.\n\\begin{snippet}\n\t\\procdef{\\code{comACK}}{\n\t\t\\cframe{k}{\\pid s}{\\pid r}{T_k},\n\t\t\\cframe{k_{ack}}{\\pid r}{\\pid s}{\\type{Unit}},\n\t\t\\pid s\\colon T_{\\pid s},\n\t\t\\pid r\\colon T_{\\pid r},\n\t\tf\\colon T_{\\pid s} \\to T_k,\n\t\tg\\colon T_{\\pid r}\\times T_{k} \\to T_{\\pid r}\n\t}{\n\t\t\\\\\\indent\n\t\t\\code{sendT}(k,\\pid s,f)\\keyword{;}\\,\n\t\t\\\\\\indent\n\t\t\\code{recvT}(k, \\pid r,g)\\keyword{;}\\, \n\t\t\\\\\\indent\n\t\t\\code{sendT}(k_{ack}, \\pid r, \\literal{unit})\\keyword{;}\\,\n\t\t\\\\\\indent\n\t\t\\code{sendUntilACK}(k,k_{ack},\\pid s,f)\\keyword{;}\\,\n\t}\n\t\\\\\n\t\\procdef{\\code{sendUntilACK}}{\n\t\t\\cframe{k}{\\pid s}{\\pid r}{T_k},\n\t\t\\cframe{k_{ack}}{\\pid r}{\\pid s}{\\type{Unit}},\n\t\t\\pid s\\colon T_{\\pid s},\\\\\\indent\\indent\n\t\tf\\colon T_{\\pid s} \\to T_k\n\t}{\n\t\t\\\\\\indent\n\t\t\\scond{\\pid s.\\code{n > 0} \\land \\neg \\pid s.\\creceived{k}}{\n\t\t\t\\\\\\indent\\indent\n\t\t\t\\pid s.\\code{n-{}-}\n\t\t\t\\\\\\indent\\indent\n\t\t\t\\code{send}(k,\\pid s,f)\\keyword{;}\\,\n\t\t\t\\\\\\indent\\indent\n\t\t\t\\code{recv}(k_{ack},\\pid s,\\code{noop})\\keyword{;}\\,\n\t\t\t\\\\\\indent\\indent\n\t\t\t\\code{sendUntilACK}(k,k_{ack},\\pid s, f)\\keyword{;}\\,\n\t\t\t\\\\\\indent\\mspace{-10.0mu}\n\t\t}\n\t}\n\\end{snippet}\n\n\\paragraph{Compensations}\nWith \\code{comACK} we can also use RC\\xspace to develop a new variant of PC that does not assume reliable transmission, \\ie, for \\cref{fm:unreliable}. \nIn this setting, a common patter to deal with failures of best-effort communications are \n\\emph{compensations}. Fault compensations can be defined in RC\\xspace (for both \n\\cref{fm:reliable,fm:unreliable}) using conditionals, \\code{comACK} (or variations thereof), and \nsome syntax sugar to improve readability.\nAn expression ${\\pid s.f \\ttTo^{BE} \\pid r.f'}\\{C_{\\pid s}\\}\\{C_{\\pid r}\\}$ is a communication as \nin $\\code{comACK}(\\pid s,\\pid r,f,g)$ where choreographies $C_{\\pid s}$ and $C_{\\pid r}$ are \nexecuted as compensations for faults detected by the sender $\\pid s$ (no ack) or the receiver $\\pid \nr$, respectively.\nAn example of communications with fault compensations is the communication construct defined in \n\\cite{APN17} where communication operations specify default values as compensations; this construct \nis recovered in RC\\xspace using local computations as, \\eg, in ${\\pid s.f \\ttTo^{BE} \\pid r.f'}\\{\\pid \ns.\\literal{foo}\\}\\{\\pid r.\\literal{42}\\}$.\n\n\\paragraph{Any\/Many communications}\nWe can also implement more complex communication primitives, like those in \\cite{LNN16,CMP18}.\nBelow are procedures that iteratively attempt\nat sending some frames until the sender \nstack accepts all or any of them, respectively, using a round-robin strategy.\n\\begin{snippet}\n\t\\procdef{\\code{sendAll}}{\n\t\t\\pid s\\colon T_{\\pid s},\n\t\t\\cframe{k_1}{\\pid s}{\\pid r_1}{T},\n\t\t\\dots,\n\t\t\\cframe{k_n}{\\pid s}{\\pid r_n}{T},\n\t\t\\\\\\indent\\indent\n\t\tf\\colon T_{\\pid s} \\to T\n\t}{\\\\\\indent\n\t\t\\csend{k_n}{\\pid s.f}\\keyword{;}\\, \n\t\t\\\\\\indent\n\t\t\\cond{\\pid s.\\csent{k_n}}{\n\t\t\\\\\\indent\\indent\n\t\t\t\t\\code{sendAll}(\n\t\t\t\t\t\\pid s,\n\t\t\t\t\tk_1,\n\t\t\t\t\t\\dots,\n\t\t\t\t\tk_{n-1},\n\t\t\t\t\tf)\n\t\t\\\\\\indent\\mspace{-10.0mu}\n\t\t}{\n\t\t\\\\\\indent\\indent\n\t\t\t\\code{sendAll}(\\pid s,\n\t\t\t\tk_1 \\tteq k_2,\n\t\t\t\t\\dots,\n\t\t\t\tk_{n-1} \\tteq k_n,\n\t\t\t\tk_n \\tteq k_1,\n\t\t\t\tf)\n\t\t\\\\\\indent\\mspace{-10.0mu}\n\t\t}\n\t}\n\\end{snippet}\n\\begin{snippet}\n\t\\procdef{\\code{sendAny}}{\n\t\t\\pid s\\colon T_{\\pid s},\n\t\t\\cframe{k_1}{\\pid s}{\\pid r_1}{T},\n\t\t\\dots,\n\t\t\\cframe{k_n}{\\pid s}{\\pid r_n}{T},\n\t\t\\\\\\indent\\indent\n\t\tf\\colon T_{\\pid s} \\to T\n\t}{\\\\\\indent\n\t\t\\csend{k_1}{\\pid s.f}\\keyword{;}\\, \n\t\t\\\\\\indent\n\t\t\\scond{\\pid s.\\neg\\csent{k_1}}{\n\t\t\\\\\\indent\\indent\n\t\t\t\\code{sendAny}(\\pid s,\n\t\t\t\tk_1 \\tteq k_2,\n\t\t\t\t\\dots,\n\t\t\t\tk_{n-1} \\tteq k_n,\n\t\t\t\tk_n \\tteq k_1,\n\t\t\t\tf)\n\t\t\\\\\\indent\\mspace{-10.0mu}\n\t\t}\n\t}\n\\end{snippet}\nWe omit the dual procedures for receiving all or some frames, which are similarly defined.\nCombining these it is possible to implement scatter\/gather communication primitives from \n\\cite{LNN16}. For instance, below is an implementation of scatter.\n\\begin{snippet}\n\t\\procdef{\\code{scatterAll}}{\n\t\t\\pid s\\colon T_{\\pid s},\n\t\t\\pid r_1\\colon T_{\\pid r},\n\t\t\\dots,\n\t\t\\pid r_n\\colon T_{\\pid r},\n\t\tf\\colon T_{\\pid s} \\to T,\n\t\tg\\colon T_{\\pid r}\\times T_{k} \\to T_{\\pid r}\n\t}{\\\\\\indent\n\t\t\\cbindin{\\cframe{k_1}{\\pid s}{\\pid r_1}{T}\n\t\t\\dots\n\t\t\\cframe{k_n}{\\pid s}{\\pid r_2}{T}}{\n\t\t\\\\\\indent\n\t\t\\code{sendAll}(k_1,\\dots,k_n,f)\\keyword{;}\\,\n\t\t\\\\\\indent\n\t\t\\code{recv}(k,\\pid r_1,g)\\keyword{;}\\,\n\t\t\\dots\n\t\t\\code{recv}(k,\\pid r_n,g)\n\t\t}\n\t}\n\\end{snippet}\n\n\\begin{remark}\nFor clarity, we remark that RC\\xspace would require a version of the above procedures (\\eg \n\\code{send<$T_k$,$T_{\\pid s}$,$T_{\\pid r}$>}) for each signature used by the program at hand, since \nwe do not support type variables. Extending RC\\xspace with \nparametric polymorphism for procedures or an erasure step seems straightforward. \n\\end{remark}\n\n\\section{Typing, Progress, and Robustness}\n\\label{sec:typing}\n\nIt is easy to write programs in RC\\xspace that get stuck or have inconsistent communication implementations: parties may not be connected, payload types may not be respected, communication attempts may be mismatched. \nTo address these issues we introduce a typing discipline for RC\\xspace that checks that:\n\\begin{enumerate}\n\\item types of processes, functions, and procedures are respected;\n\\item processes that need to communicate are properly connected;\n\\item the delivery of frames is guaranteed to be at-most-once and best-effort;\n\\item there are no unnecessary checks on network actions (to avoid dead branches).\n\\end{enumerate}\nAdditionally, in \\cref{fm:reliable} exactly-once delivery is also checkable.\n\nChoreography programs can be regarded as ``network transformers''. Under this perspective, typing judgements are naturally of form\n\\[\n\t\\Gamma \\vdash \\langle \\mathcal{D}, C\\rangle \\colon \\mathcal{N} \\to \\mathcal{N}'\n\\]\nand read ``under the environment $\\Gamma$, running $\\langle\\mathcal{D},C\\rangle$ on a network configuration of type $\\mathcal{N}$ yields one of type $\\mathcal{N}'$''. \n\nTyping environments specify labels, procedures, and process names that may be used as well as their type; they are collections of the following form\n\\[\\Gamma \\Coloneqq\n\t\t\\Gamma, \\pid p\\colon T \\mid \n\t\t\\Gamma, l \\mid\n\t\t\\Gamma, \\cframe{k}{\\pid p}{\\pid q}{T} \\mid \n\t\t\\Gamma, X(\\vec{P})\\colon \\mathcal{N} \\to \\mathcal{N}' \\mid\n\t\t\\varnothing\n\\]\nwhere labels are unique, processes are assigned unique types in $\\mathcal{V}$, and procedure may be assigned multiple types.\nA type of network configurations $\\mathcal{N}$ is a pair $\\hnet{F}{G}$ formed by an abstract frame dictionary $F$ and a connection graph $G$. Abstract frame dictionaries specify possible states of frames while abstracting payload of a value type and any information non accessible to a program: a frame sender and receiver may only test its status using conditionals: a sender may only know whether its component is $\\bot$ and the receiver only whether its component is in $\\removed{\\mathcal{U}}$. \nFormally, abstract frames are collections of the form\n\\[\n\tF \\Coloneqq F, \\hframe[k]{U}{U'} \\mid \\emptyset\n\\]\nwhere $U,U' \\subseteq \\mathcal{U}_\\hbot$ for $\\mathcal{U}_\\hbot \\triangleq \\{\\hbot,\\bullet\\} \\uplus \\type{PID} \\uplus \\type{LBL}$.\nThe sender and receiver components of a frame status are abstracted by the function $\\alpha^s\\colon \\mathcal{U}_\\bot^\\checkmark \\to \\mathcal{U}_\\hbot$ and $\\alpha^r\\colon \\mathcal{U}_\\bot^\\checkmark \\to \\mathcal{U}_\\hbot$, respectively.\nThe first is given by the assignments \n\\[\n\t\\bot \\mapsto \\hbot\n\t\\qquad\n\tv,\\removed{v} \\mapsto \\bullet\n\t\\qquad\n\tl,\\removed{l} \\mapsto l\n\t\\qquad\n\t\\pid p,\\removed{\\pid p} \\mapsto \\pid p\n\\]\nand the second by\n\\[\n\t\\bot,v,l,\\pid p \\mapsto \\hbot\n\t\\qquad\n\t\\removed{v} \\mapsto \\bullet\n\t\\qquad\n\t\\removed{l} \\mapsto l\n\t\\qquad\n\t\\removed{\\pid p} \\mapsto \\pid p\n\\]\nwhere  $v \\in \\type{VAL}$, $\\pid p \\in \\type{PID}$, and $l \\in \\type{LBL}$.\nConsider for instance a value exchange $k$, $\\hframe[k]{\\{\\bullet\\}}{\\{\\hbot,\\bullet\\}}$ is inhabited by any frame status of $k$, $\\hframe[k]{\\{\\hbot,\\bullet\\}}{\\{\\hbot\\}}$ by any frame status where the payload is not delivered to the receiver, and $\\hframe[k]{\\{\\bullet\\}}{\\{\\hbot,\\bullet\\}}$ by any frame status where the sender stack accepted the payload.\nA type of network $\\hnet{F}{G}$ is well-formed under $\\Gamma$ (written $\\Gamma \\vdash \\hnet{F}{G}$) if the following conditions are met:\n\\begin{enumerate}[label={\\em(\\alph{*})}]\n\t\\item \n\tif $\\Gamma \\vdash \\cframe{k}{\\pid p}{\\pid q}{T}$, then\n\t$G \\vdash \\pid p \\leftrightarrow \\pid q$,\n\t$\\hframe[k]{U}{U'} \\in F$, and payloads in $U$ and $U'$ are of type $T$;\n\t\n\t\\item \n\tif $\\Gamma \\vdash \\pid p\\colon T$, then $\\pid p \\in G$;\n\t\n\t\\item\n\tif $\\hframe[k]{U}{U'} \\in F$, then $\\Gamma \\vdash \\cframe{k}{\\pid p}{\\pid q}{T}$ for some $\\pid p$, $\\pid q$, $T$;\n\t\n\t\\item\n\tif $\\hframe[k]{U}{U'} \\in F$ and $\\pid r \\in U \\cup U'$ then $\\pid r \\in G$;\n\t\n\t\\item\n\tif $\\pid p \\in G$, then $\\Gamma \\vdash \\pid p\\colon T$ for some $T$.\n\\end{enumerate}\nHereafter network types are assumed well-formed whenever appearing in a judgement together with an environment.\nJudgements of (concrete) network configurations have form\n\\[\n\t\\Gamma \\vdash {\\phi,G} \\colon \\hnet{F}{G'}\n\\]\nand hold whenever the following conditions are met:\n\\begin{enumerate}[label={\\em(\\alph{*})}]\n\t\\item \n\tif $\\Gamma \\vdash \\cframe{k}{\\pid p}{\\pid q}{T}$ and $\\hframe[k]{U}{U'} \\in F$, then\n\t$\\phi(k) = \\phiframe{\\pid p}{u}{\\pid q}{u'}$,\n\t$\\alpha^s(u) \\in U$, $\\alpha^r(u') \\in U'$, and payloads are of type $T$;\n\n\t\\item\n\t$G \\subseteq G'$.\n\\end{enumerate}\n\n\\begin{figure*}\n\\begin{infrules}\n\t\\infrule[\\rname[T]{Weaken}][rule:t-weaken]{\n\t\t\\Gamma_1 \\vdash C \\colon \\hnet{F_1}{G_1} \\to \\hnet{F_2}{G_2}\n\t}{\n\t\t\\Gamma_0,\\Gamma_1 \\vdash C \\colon \n\t\t\\hnet{F_0,F_1}{G_0\\cup G_1}\t\\to\t\\hnet{F_0,F_2}{G_0\\cup G_2}\t\t\n\t}\n\n\t\\infrule[\\rname[T]{Tell}][rule:t-tell]{\n\t\t\\Gamma \\vdash \\cframe{k}{\\pid p}{\\pid q}{\\type{PID}} \\and\n\t\t\\Gamma \\vdash \\pid r\\colon T \\and\n\t\t\\Gamma \\vdash C \\colon \\hnet{F_0,\\hframe[k]{U}{\\{\\pid r\\}}}{\\{\\pid q \\to \\pid r\\} \\uplus G_0} \\to \\hnet{F_1}{G_1}\t\t\n\t}{\n\t\t\\Gamma \\vdash C \\colon \n\t\t\\hnet{F_0,\\hframe[k]{U}{\\{\\pid r\\}}}{G_0} \\to\t\\hnet{F_1}{G_1}\t\t\n\t}\n\n\t\\infrule[\\rname[T]{Swap}][rule:t-swap]{\n\t\tC_0 \\dotrel{\\congr} C_1 \\and\n\t\t\\Gamma \\vdash C_1 \\colon \\hnet{F_0}{G_0} \\to \\hnet{F_1}{G_1}\n\t}{\n\t\t\\Gamma \\vdash C_0 \\colon \n\t\t\\hnet{F_0}{G_0} \\to \\hnet{F_1}{G_1}\t\t\n\t}\n\n\t\n\t\\infrule[\\rname[T]{Int}][rule:t-local]{\n\t\t\\vdash f\\colon T \\to T \n\t}{\n\t\t\\pid p\\colon T \\vdash \\pid p.f \\colon\n\t\t\\hnet{\\varnothing}{\\varnothing} \\to \\hnet{\\varnothing}{\\varnothing}\n\t}\n\t\n\t\\infrule[\\rname[T]{Nil}][rule:t-nil]{\n\t}{\n\t\t\\varnothing \\vdash \\keyword{0} \\colon \\hnet{\\varnothing}{\\varnothing}\\to\\hnet{\\varnothing}{\\varnothing}\n\t}\n\n\t\\infrule[\\rname[T]{;}][rule:t-conc]{\n\t\t\\Gamma \\vdash\tC_0 \\colon \\hnet{F_0}{G_0} \\to \\hnet{F_1}{G_1}\n\t\t\\and\n\t\t\\Gamma \\vdash\tC_1 \\colon \\hnet{F_1}{G_1} \\to \\hnet{F_2}{G_2}\n\t}{\n\t\t\\Gamma \\vdash C_0\\keyword{;}\\, C_1 \\colon \\hnet{F_0}{G_0}\t\\to \\hnet{F_2}{G_2}\n\t}\n\t\t\n\t\\infrule[\\rname[T]{NP}][rule:t-new-proc]{\n\t\tG_{\\pid q} = \\{\\pid p \\leftrightarrow \\pid q\\} \\cup \\{\\pid q \\rightarrow \\pid r \\mid G_1 \\vdash \\pid p \\rightarrow \\pid r \\}\n\t\t\\and\n\t\t\\vdash f \\colon T \\to T' \\and\n\t\t\\Gamma \\vdash \\pid p \\colon T \\and\n\t\t\\Gamma, \\pid q \\colon T' \\vdash C \\colon \n\t\t\\hnet{F_0}{G_0 \\cup G_{\\pid q} } \\to \n\t\t\\hnet{F_1}{G_1}\n\t}{\n\t\t\\Gamma \\vdash \\cbindin{\\cstart{\\pid p}{\\pid q}{f}}{C} \\colon\n\t\t\\hnet{F_0}{G_0} \\to \\hnet{F_1}{G_1\\setminus \\{\\pid q\\}}\n\t}\n\t\n\t\\infrule[\\rname[T]{NF}][rule:t-new-frame]{\n\t\tG_0 \\vdash \\pid p \\leftrightarrow \\pid q\n\t\t\\and\n\t\t\\Gamma, \\cframe{k}{\\pid p}{\\pid q}{T} \\vdash C \\colon \n\t\t\\hnet{F_0, \\hframe[k]{\\{\\hbot\\}}{\\{\\hbot\\}}}{G_0} \\to \n\t\t\\hnet{F_1, \\hframe[k]{\\_}{\\_}}{G_1}\n\t}{\n\t\t\\Gamma \\vdash \\cbindin{\\cframe{k}{\\pid p}{\\pid q}{T}}{C} \\colon\n\t\t\\hnet{F_0}{G_0} \\to\\hnet {F_1}{G_1}\n\t}\n\t\t\n\t\\infrule[\\rname[T]{Call}][rule:t-call]{\n\t\t\\Gamma \\vdash X(P_1,\\dots,P_n)\\colon \\hnet{F_0}{G_0} \\to \\hnet{F_1}{G_1}\n\t\t\\and\n\t\t\\Gamma \\vdash P_i[\\vec{A}] \\text{ for any } 1 \\leq i \\leq n \n\t}{\n\t\t\\Gamma \\vdash\n\t\tX(\\vec{A}) \\colon \n\t\t\\hnet{F_0[\\vec{A}]}{G_0[\\vec{A}]} \\to\n\t\t\\hnet{F_1[\\vec{A}]}{G_1[\\vec{A}]}\n\t}\n\n\t\\infrule[\\rname[T]{IfExp}][rule:t-if-exp]{\n\t\t\\Gamma \\vdash \\pid p\\colon T \\and\n\t\t\\vdash f\\colon T \\to \\type{Bool}\n\t\t\\and\n\t\t\\Gamma \\vdash C_1 \\colon\n\t\t\t\t\\hnet{F_0}{G_0} \\to \\hnet{F_1}{G_1}\n\t\t\\and\n\t\t\\Gamma \\vdash C_2 \\colon\n\t\t\t\t\\hnet{F_0}{G_0} \\to \\hnet{F_2}{G_2}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\cond{\\pid p.f}{C_1}{C_2} \\colon\n\t\t\\hnet{F_0}{G_0} \\to \\hnet{F_1 \\Ydown F_2}{G_1\\cap G_2}\n\t}\n\n\t\\infrule[\\rname[T]{IfSnd}][rule:t-if-send]{\n\t\t\\Gamma \\vdash \\cframe{k}{\\pid p}{\\pid q}{T} \n\t\t\\and\n\t\t\\Gamma \\vdash C_1 \\colon\n\t\t\t\t\\hnet{F_0,\\hframe[k]{\\{u\\}}{U}}{G_0} \\to \\hnet{F_1}{G_1}\n\t\t\\and\n\t\t\\Gamma \\vdash C_2 \\colon\n\t\t\t\t\\hnet{F_0,\\hframe[k]{\\{\\hbot\\}}{U}}{G_0} \\to \\hnet{F_2}{G_2}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\cond{\\pid p.\\csent{k}}{C_1}{C_2} \\colon\n\t\t\\hnet{F_0,\\hframe[k]{\\{\\hbot,u\\}}{U}}{G_0} \\to \\hnet{F_1 \\Ydown F_}{G_2\\cap G_2}\n\t}\n\n\t\\infrule[\\rname[T]{IfRcv}][rule:t-if-recv]{\n\t\t\\Gamma \\vdash \\cframe{k}{\\pid p}{\\pid q}{T} \n\t\t\\and\n\t\t\\Gamma \\vdash C_1 \\colon\n\t\t\t\t\\hnet{F_0,\\hframe[k]{U}{\\{u\\}}}{G_0} \\to \\hnet{F_1}{G_1}\n\t\t\\and\n\t\t\\Gamma \\vdash C_2 \\colon\n\t\t\t\t\\hnet{F_0,\\hframe[k]{U}{\\{\\hbot\\}}}{G_0} \\to \\hnet{F_2}{G_2}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\cond{\\pid q.\\creceived{k}}{C_1}{C_2} \\colon\n\t\t\\hnet{F_0,\\hframe[k]{U}{\\{\\hbot,u\\}}}{G_0} \\to \\hnet{F_1 \\Ydown F_2}{G_1\\cap G_2}\n\t}\n\n\t\\infrule[\\rname[T]{IfLBL}][rule:t-if-recv-lbl]{\n\t\t\\Gamma \\vdash \\cframe{k}{\\pid p}{\\pid q}{\\type{LBL}} \\and\n\t\t\\Gamma \\vdash l \n\t\t\\and\n\t\t\\Gamma \\vdash C_1 \\colon\n\t\t\t\t\\hnet{F_0,\\hframe[k]{U}{\\{l\\}}}{G_0} \\to \\hnet{F_1}{G_1}\n\t\t\\and\n\t\t\\Gamma \\vdash C_2 \\colon\n\t\t\t\t\\hnet{F_0,\\hframe[k]{U}{\\{\\hbot\\}}}{G_0} \\to \\hnet{F_1}{G_2}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\cond{\\pid q.\\creceivedlbl{k}{l}}{C_1}{C_2} \\colon\n\t\t\\hnet{F_0,\\hframe[k]{U}{\\{\\hbot,l\\}}}{G_0} \\to \\hnet{F_1 \\Ydown F_2}{G_1\\cap G_2}\n\t}\n\t\n\t\\infrule[\\rname[T]{RcvV}][rule:t-recv-val]{\n\t\t\\Gamma = \\cframe{k}{\\pid p}{\\pid q}{T}, \\pid q \\colon T'\n\t\t\\and\n\t\t\\vdash f\\colon T \\times T' \\to T'\n\t\t\\and\n\t\tU \\subseteq \\{\\hbot\\}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\crecv{k}{\\pid q.f} \\colon \n\t\t\\hnet{\\hframe[k]{U \\cup \\{\\bullet\\}}{\\{\\hbot\\}}}{\\varnothing} \\to \n\t\t\\hnet{\\hframe[k]{U \\cup \\{\\bullet\\}}{\\{\\hbot,\\bullet\\}}}{\\varnothing}\n\t}\n\n\t\\infrule[\\rname[T]{RcvP}][rule:t-recv-pid]{\n\t\t\\Gamma = \\cframe{k}{\\pid p}{\\pid q}{\\type{PID}},\\pid r \\colon T\n\t\t\\and\n\t\tU \\subseteq \\{\\hbot\\}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\crecv{k}{\\pid q} \\colon \n\t\t\\hnet{\\hframe[k]{U \\cup \\{\\pid r\\}}{\\{\\hbot\\}}}{\\varnothing} \\to \n\t\t\\hnet{\\hframe[k]{U \\cup \\{\\pid r\\}}{\\{\\hbot,\\pid r\\}}}{\\varnothing}\n\t}\n\n\t\\infrule[\\rname[T]{RcvL}][rule:t-recv-lbl]{\n\t\t\\Gamma = \\cframe{k}{\\pid p}{\\pid q}{\\type{LBL}}, l\n\t\t\\and\n\t\tU \\subseteq \\{\\hbot\\}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\crecv{k}{\\pid q.f} \\colon \n\t\t\\hnet{\\hframe[k]{U \\cup \\{l\\}}{\\{\\hbot\\}}}{\\varnothing} \\to \n\t\t\\hnet{\\hframe[k]{U \\cup \\{l\\}}{\\{\\hbot,l\\}}}{\\varnothing}\n\t}\n\\end{infrules}\n\t\\caption{Typing choreographies, shared rules}\n\t\\label{fig:chor-typing-shared}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{infrules}\t\n\t\\infrule[\\rname[R]{SndV}][rule:r-send-val]{\n\t\t\\Gamma = \\cframe{k}{\\pid p}{\\pid q}{T}, \\pid p \\colon T',\tf\\colon T' \\to T\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\csend{k}{\\pid p.f} \\colon \n\t\t\\hnet{\\hframe[k]{\\{\\hbot\\}}{\\{\\hbot\\}}}{\\varnothing} \\to \n\t\t\\hnet{\\hframe[k]{\\{\\hbot,\\bullet\\}}{\\{\\hbot\\}}}{\\varnothing}\n\t}\n\n\t\\infrule[\\rname[R]{SndP}][rule:r-send-pid]{\n\t\t\\Gamma = \\pid r \\colon T,\t\\cframe{k}{\\pid p}{\\pid q}{\\type{PID}}\n\t\t\\and\n\t\tG = \\{\\pid p \\rightarrow \\pid r\\}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\csend{k}{\\pid p.\\pid r} \\colon \n\t\t\\hnet{\\hframe[k]{\\{\\hbot\\}}{\\{\\hbot\\}}}{G} \\to \n\t\t\\hnet{\\hframe[k]{\\{\\hbot,\\pid r\\}}{\\{\\hbot\\}}}{G}\n\t}\n\t\n\t\\infrule[\\rname[R]{SndL}][rule:r-send-lbl]{\n\t\t\\Gamma = l, \\cframe{k}{\\pid p}{\\pid q}{\\type{LBL}}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\csend{k}{\\pid p.\\pid r} \\colon \n\t\t\\hnet{\\hframe[k]{\\{\\hbot\\}}{\\{\\hbot\\}}}{\\varnothing} \\to \n\t\t\\hnet{\\hframe[k]{\\{\\hbot,l\\}}{\\{\\hbot\\}}}{\\varnothing}\n\t}\n\\end{infrules}\n\t\\caption{Typing choreographies, rules for \\cref{fm:reliable}.}\n\t\\label{fig:chor-typing-reliable}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{infrules}\n\t\\infrule[\\rname[U]{SndV}][rule:u-send-val]{\n\t\tU \\subseteq \\{\\hbot,\\bullet\\} \\and\n\t\t\\Gamma = \\cframe{k}{\\pid p}{\\pid q}{T}, \\pid p \\colon T',\tf\\colon T' \\to T\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\csend{k}{\\pid p.f} \\colon \n\t\t\\hnet{\\hframe[k]{U}{\\{\\hbot\\}}}{\\varnothing} \\to \n\t\t\\hnet{\\hframe[k]{\\{\\hbot,\\bullet\\}}{\\{\\hbot\\}}}{\\varnothing}\n\t}\n\t\n\t\\infrule[\\rname[U]{SndP}][rule:u-send-pid]{\n\t\tU \\subseteq \\{\\hbot,\\pid r\\}\n\t\t\\and\n\t\t\\Gamma = \\pid r \\colon T,\t\\cframe{k}{\\pid p}{\\pid q}{\\type{PID}}\n\t\t\\and\n\t\tG = \\{\\pid p \\rightarrow \\pid r\\}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\csend{k}{\\pid p.\\pid r} \\colon \n\t\t\\hnet{\\hframe[k]{U}{\\{\\hbot\\}}}{G} \\to \n\t\t\\hnet{\\hframe[k]{\\{\\hbot,\\pid r\\}}{\\{\\hbot\\}}}{G}\n\t}\n\t\n\t\\infrule[\\rname[U]{SndL}][rule:u-send-lbl]{\n\t\tU \\subseteq \\{\\hbot,l\\}\n\t\t\\and\n\t\t\\Gamma = l, \\cframe{k}{\\pid p}{\\pid q}{\\type{LBL}}\n\t}{\n\t\t\\Gamma \\vdash \n\t\t\\csend{k}{\\pid p.\\pid r} \\colon \n\t\t\\hnet{\\hframe[k]{U}{\\{\\hbot\\}}}{\\varnothing} \\to \n\t\t\\hnet{\\hframe[k]{\\{\\hbot,l\\}}{\\{\\hbot\\}}}{\\varnothing}\n\t}\n\\end{infrules}\n\t\\caption{Typing choreographies, rules for \\cref{fm:unreliable}.}\n\t\\label{fig:chor-typing-unreliable}\n\\end{figure*}\n\nJudgements of choreography terms have form \n\\[\n\t\\Gamma \\vdash C \\colon \\hnet{F}{G} \\to \\hnet{F'}{G'}\n\\]\nand are derived using rules in \\cref{fig:chor-typing-shared} together with either rules in \\cref{fig:chor-typing-reliable} or \\cref{fig:chor-typing-unreliable} according to whether \\cref{fm:reliable} or \\cref{fm:reliable} is assumed.\n\n\\Cref{rule:t-recv-val,rule:t-recv-pid,rule:t-recv-lbl} specify that a receive operation takes any network where the sender will eventually produce a payload of the expected type, the receiver knows the sender, and the payload has not been consumed yet, and yields a network where the payload may be consumed.\nIn particular, choreographies where receives cannot be matched to sends (\\eg $\\cnewframe{k}{\\pid p}{\\pid q}{T}{\\crecv{k}{\\pid q.f}}$) or have consecutive receive operations (\\eg $\\crecv{k}{\\pid q.f}\\keyword{;}\\,\\crecv{k}{\\pid q.f}$) are rejected since delivery is either impossible or inconsistencies may arise (\\eg  the second operation shadows a successful outcome of the first).\nGiven the same receive statement multiple typing can be derived for the same receive statement under the same environment, even once weakening is taken into account. Likewise for send operations (discussed below).\nFor instance, judgement \n$\\Gamma \\vdash \\crecv{k}{\\pid q.f} \\colon \\hnet{\\hframe[k]{\\{\\hbot,\\bullet\\}}{\\{\\hbot\\}}}{G} \\to \\hnet{\\hframe[k]{\\{\\hbot,\\bullet\\}}{\\{\\hbot,\\bullet\\}}}{G}$\nis derivable if and only if \n$\\Gamma \\vdash \\crecv{k}{\\pid q.f} \\colon \\hnet{\\hframe[k]{\\{\\bullet\\}}{\\{\\hbot\\}}}{G} \\to \\hnet{\\hframe[k]{\\{\\bullet\\}}{\\{\\hbot,\\bullet\\}}}{G}$ is derivable.\n\\Cref{rule:t-recv-pid} cannot update the connection graph since the outcome of a receive operation cannot be statically known; the update can only be done using \\cref{rule:t-tell} once the delivery is certain.\n\\Cref{rule:r-send-val,rule:r-send-pid,rule:r-send-lbl} and \n\\Cref{rule:u-send-val,rule:u-send-pid,rule:u-send-lbl} are intended for typing under the assumptions of \\cref{fm:reliable} and \\cref{fm:unreliable}, respectively.\nRules of the two groups are alike save for the requirements imposed on the sender stack: rules of both groups require that the frame is of the expected type, that the receiver is known, and that it may be so once the statement is executed, but only those from the former require that the stack has yet to accept a payload for the frame. \nThe more stringent set of rules forbids any send operation for a frame with a potentially accepted (hence transmitted) payload (\\eg $\\csend{k}{\\pid p.f}\\keyword{;}\\,\\csend{k}{\\pid p.f}$) since this is a programming error in \\cref{fm:reliable} but not in \\cref{fm:unreliable} where transmission is not guaranteed. In fact, when the stack does not guarantee transmission this has to be programmed at the application level \\eg by resending frames not acknowledged.\n\\Cref{rule:t-if-exp,rule:t-if-send,rule:t-if-recv,rule:t-if-recv-lbl} require branches to be live; graphs are intersected to remove connections created only in one branch; frame dictionaries are merged ($\\Ydown$) by pointwise union under the condition that whenever both branches specify a payload they agree on it \\ie whenever $U_1$ and $U_2$ are merged it must hold that the set $(U_1 \\cup U_2) \\cap (\\{\\bullet\\} \\uplus \\type{PID} \\uplus \\type{LBL})$ has at most one element.\n\\Cref{rule:t-call} requires that the substitution $\\vec{A}$ respects types of formal and actual parameters and that the call type is obtained applying $\\vec{A}$ to the selected procedure type---the discipline admits \\adhoc polymorphism.\n\\Cref{rule:t-swap} allow each step of a derivation to switch to any element in the (finite) equivalence class $[C]_{\\dotrel{\\congr}}$. As a direct consequence, $\\dotrel{\\congr}$ implies type equivalence \\ie types are unaffected by instruction scheduling---recursive calls are not unfold by $\\dotrel{\\congr}$.\nObserve that \\cref{rule:t-swap} is only required for typing choreographies that would be otherwise rejected like \\eg $\\crecv{k}{\\pid q.g}\\keyword{;}\\,\\csend{k}{\\pid p.f}$.\n\n\\begin{lemma}\nIf $\\Gamma \\vdash C\\colon \\mathcal{N} \\to \\mathcal{N'}$ has a derivation where \\ref{rule:t-swap} is used, then either:\n\\begin{itemize}\n\\item $\\Gamma \\vdash C\\colon \\mathcal{N} \\to \\mathcal{N'}$ has a derivation without \\cref{rule:t-swap} or \n\\item no judgement for $C$ has a derivation without \\cref{rule:t-swap}.\n\\end{itemize}\n\\end{lemma}\n\nTyping judgements for procedure definitions are derived using the rule\n\\begin{infrules}\n\t\\infrule[\\rname[T]{Proc}]{\n\t\t\\Gamma \\vdash X(\\vec{P}) \\colon \\hnet{F_1}{G_1} \\to \\hnet{F_2}{G_2}\n\t\t\\\\\n\t\t\\Gamma|_{\\mathcal{D}}, \\vec{P} \\vdash C \\colon \\hnet{F_1}{G_1} \\to \\hnet{F_2}{G_2}\n\t}{\n\t\t\\Gamma \\vdash \\procdef{X}{\\vec{P}}{C}\n\t}\n\\end{infrules}\nwhere $\\Gamma|_{\\mathcal{D}}$ is the restriction of $\\Gamma$ to procedures in $\\mathcal{D}$. This restriction guarantees every label, free process name, and free frame name in the procedure body $C$ is bound by a formal parameter.\nA procedure definition is is well-typed under $\\Gamma$ if $\\Gamma \\vdash \\procdef{X}{\\vec{P}}{C}$.\nA set of procedure definitions $\\mathcal{D}$ is well-typed under $\\Gamma$ (written $\\Gamma \\vdash \\mathcal{D}$) provided that all of its elements are well-typed.\n\nA memory configuration $\\sigma$ is well-typed under $\\Gamma$ (written $\\Gamma \\vdash \\sigma$) provided that $\\vdash \\sigma(\\pid p)\\colon T$ whenever $\\Gamma \\vdash \\pid p\\colon T$.\n\n\\begin{definition}[Well-typedness]\n\\label{def:well-typedness}\nFor $\\mathcal D$ a set of procedure definitions and $\\langle C, \\sigma, \\phi,G \\rangle$ a runtime \nconfiguration, $\\langle C, \\sigma, \\phi,G \\rangle$ is \\emph{well-typed} under $\\mathcal{D}$ if there exist $\\Gamma$, $F_1$, $G_1$, $F_2$, and $G_2$ such that \n$\\Gamma \\vdash \\mathcal D$,\n$\\Gamma \\vdash C\\colon \\hnet{F_1}{G_1} \\to \\hnet{F_2}{G_2}$,\n$\\Gamma \\vdash \\sigma$, and\n$\\Gamma \\vdash \\phi,G\\colon \\hnet{F_1}{G_1}$.\n\nA choreographic program $\\langle \\mathcal{D}, C\\rangle$ is \\emph{well-typed} if there exist $\\sigma$, \n$\\phi$ and $G$ s.t.~the runtime configuration $\\langle C,\\sigma,\\phi, G \\rangle$ is well-typed.\n\\end{definition}\n\nFor any configuration there is an environment and a finite set of pairs of network types that subsume any other typing judgement.\n\n\\begin{theorem}[Existence of minimal typing]\n\\label{thm:chor-minimal-type}\nLet $\\langle C, \\sigma, \\phi,G \\rangle$ be \\emph{well-typed} under $\\mathcal{D}$. There are $\\Gamma$, $(\\mathcal{N}_0 \\to \\mathcal{N}'_0),\\dots,(\\mathcal{N}_n \\to \\mathcal{N}'_n)$ with the property that\nwhenever $\\Gamma' \\vdash \\langle \\mathcal{D}, C, \\sigma, \\phi, G \\rangle \\colon {\\mathcal{N}} \\to {\\mathcal{N}'}$ there is $i \\leq n$ such that:\n\\[\n\t\\infer{\n\t\t\\Gamma' \\vdash \\langle \\mathcal{D}, C, \\sigma, \\phi, G \\rangle \\colon {\\mathcal{N}} \\to {\\mathcal{N}'}\n\t}{\n\t\t\\infer*{}{\n\t\t\\Gamma \\vdash \\langle \\mathcal{D}, C, \\sigma, \\phi, G \\rangle \\colon {\\mathcal{N}_i} \\to {\\mathcal{N}'_i}\n\t\t}\n\t}\n\t\\text{.}\n\\]\n\\end{theorem}\n\n\\begin{theorem}[Decidability of typing]\n\\label{thm:decidability}\nGiven a set of procedure definitions $\\mathcal D$ and a runtime configuration $\\langle C, \\sigma, \n\\phi,G \\rangle$, it is decidable whether $\\langle C, \\sigma, \\phi,G \\rangle$ is \\emph{well-typed} \nunder $\\mathcal{D}$.\nGiven a program $\\langle \\mathcal{D},C\\rangle$, it is decidable whether $\\langle \n\\mathcal{D},C\\rangle$ is well-typed.\n\\end{theorem}\n\n\\begin{proof}[Sketch]\nTyping for elements of the guest language is known by assumption.\nObserve that building derivations for typing judgements from \\cref{def:well-typedness} is completely mechanical:\nrule selection is deterministic save \\cref{rule:t-tell,rule:t-swap} which only introduce a finite \nnumber of cases and can be used a finite amount of times (\\ref{rule:t-tell} uses disjoint union and \n\\ref{rule:t-swap} cannot unfold calls). A heuristic is to delay uses of these rules until it is \nnecessary to infer new connections or consider scheduling alternatives in order to proceed. Hence, \nderivations can be built using straightforward non-deterministic case exploration. Furthermore, \nrules in \\cref{fig:chor-typing-shared,fig:chor-typing-reliable,fig:chor-typing-unreliable} can be \nused to construct network types.\nThe only nontrivial part is constructing typing environments but only a finite number of cases need \nto be checked since processes, frames, and labels that need to be part of the typing environment \nare inferred from free names in program terms and formal parameters.\nFinally, observe that the definition domain of memory configurations and concrete frame \ndictionaries is bounded by typing environments and network types, and that actual values in $\\sigma$ \nor $\\phi$ (the only possible source of infinity) are irrelevant provided that they are of the right \ntypes (which is checkable by the assumptions on the guest language).\n\\end{proof}\n\nTyping is preserved by all reductions.\n\\begin{theorem}[Type preservation]\n\t\\label{thm:type-preservation}\n\tIf $\\langle C,\\sigma,\\phi,G \\rangle$ is well-typed under $\\mathcal{D}$ and there is a reduction $\\langle C,\\sigma,\\phi,G\\rangle \\reducesto_{\\mathcal{D}} \\langle C',\\sigma',\\phi',G'\\rangle$, then the reductum $\\langle C',\\sigma',\\phi',G'\\rangle$ is well-typed under $\\mathcal{D}$.\n\\end{theorem}\n\nIn general, RC\\xspace programs may deadlock if \nthe types of functions, memory cells, or procedures are not respected.\nInstead, well-typed programs enjoy progress, \\ie, they either terminate or diverge.\n\\begin{theorem}[Progress]\n\t\\label{thm:progress}\n\tIf a runtime configuration $\\langle C,\\sigma,\\phi,G \\rangle$ is well-typed under $\\mathcal{D}$ then either\n\t\t $C \\preceq_{\\mathcal{D}} \\keyword{0}$ or\n\t\tthere are $C'$, $\\sigma'$, $\\phi'$, $G'$ such that $\\langle C,\\sigma,\\phi,G\\rangle \\reducesto_{\\mathcal{D}} \\langle C',\\sigma',\\phi',G'\\rangle$.\n\\end{theorem}\n\nFrames are never delivered (to the receiver application level) more than one time.\n\\begin{theorem}[At-most-once delivery]\n\\label{thm:chor-at-most-once}\nLet $\\langle C,\\sigma,\\phi,G \\rangle$ be well-typed under $\\mathcal{D}$.\nIf $C\\precongr_{\\mathcal{D}} \\crecv{k}{R}\\keyword{;}\\, C'$, then $\\phi(k)_4 \\notin \\removed{\\mathcal{U}}$.\n\\end{theorem}\n\nIn \\cref{fm:reliable}, typing identifies frames that are guaranteed to be delivered.\n\\begin{theorem}[At-least-once delivery]\n\\label{thm:chor-at-least-once}\nAssume \\cref{fm:reliable} and that\n$\\Gamma \\vdash \\langle \\mathcal{D}, C, \\sigma, \\phi, G \\rangle \\colon \\hnet{F_1}{G_1} \\to \\hnet{F_2}{G_2}$.\nFor $\\hframe[k]{U}{U'} \\in F_2$ such that $\\hbot \\notin U'$, \nif $\\langle C,\\sigma,\\phi,G \\rangle \\reducesto_{\\mathcal{D}}^\\ast \\langle C',\\sigma',\\phi',G' \\rangle$ and $k \\notin \\mathrm{fn}(C')$ then $\\phi'(k)_4 \\in \\removed{\\mathcal{U}}$.\n\\end{theorem}\n\nThere is always an execution where a given frame is delivered.\n\\begin{theorem}[Best-effort delivery]\n\\label{thm:chor-best-effort}\nAssume \\cref{fm:unreliable} and that $\\Gamma \\vdash \\langle \\mathcal{D}, C, \\sigma, \\phi, G \\rangle \\colon \\hnet{F_1}{G_1} \n\\to \\hnet{F_2}{G_2}$.\nFor any $\\hframe[k]{U}{U'} \\in F_2$ such that $\\hbot \\notin U$, \nif there is a sequence of reductions $\\langle C,\\sigma,\\phi,G \\rangle \\reducesto_{\\mathcal{D}}^\\ast \n\\langle \\crecv{k}{R}\\keyword{;}\\, C',\\sigma',\\phi',G' \\rangle$ such that $k \\notin \\mathrm{fn}(C')$, then \nthere exist $C''$, $\\sigma''$, $\\phi''$, $G''$ such that $\\langle \nC,\\sigma,\\phi,G \\rangle \\reducesto^\\ast \\langle \\crecv{k}{R}\\keyword{;}\\, C'',\\sigma'',\\phi'',G'' \\rangle$ and $\\phi''(k)_4 \\in \\mathcal{U}$.\n\\end{theorem}\n\nWe conclude this section pointing out a limitation of the type system and a possible future extension. Consider procedure \\code{sendAnyOfTwo} reported below.\n\\begin{snippet*}\n\t\\procdef{\\code{sendAnyOfwo}}{\n\t\t\\pid p \\colon T_{\\pid p},\n\t\t\\cframe{k_1}{\\pid p}{\\pid q_1}{T_k},\n\t\t\\cframe{k_2}{\\pid p}{\\pid q_2}{T_k},\n\t\t\t\t\\\\\\indent\\indent\n\t\tf \\colon T_{\\pid p} \\to T_K\n\t}{\\\\\\indent\n\t\t\\csend{k_1}{\\pid p.f}\\keyword{;}\\, \n\t\t\\\\\\indent\n\t\t\\scond{\\pid p.\\neg\\csent{k_1}}{\n\t\t\\\\\\indent\\indent\n\t\t\t\\code{sendAnyOfTwo}(\\pid p \\tteq \\pid p, k_1 \\tteq k_2,k_2 \\tteq k_1, f \\tteq f)\n\t\t\\\\\\indent\\mspace{-10.0mu}\n\t\t}\n\t}\n\\end{snippet*}\nThis procedure alternates attempts for $k_1$ and $k_2$ until exactly one is successful. However, \nthis property is not captured by its type:\n\\begin{multline*}\n\t\\code{sendAnyOfwo}(\n\t\t\\pid p \\colon T_{\\pid p},\n\t\t\\cframe{k_1}{\\pid p}{\\pid q_1}{T_k},\n\t\t\\cframe{k_2}{\\pid p}{\\pid q_2}{T_k}) \\colon\\\\\n\t\\hnet{\n\t\t\\hframe[k_1]{\\{\\hbot\\}}{\\{\\hbot\\}},\n\t\t\\hframe[k_2]{\\{\\hbot\\}}{\\{\\hbot\\}}\n\t}{G}\n\t\\to\\\\\n\t\\hnet{\n\t\t\\hframe[k_1]{\\{\\hbot,\\bullet\\}}{\\{\\hbot\\}},\n\t\t\\hframe[k_2]{\\{\\hbot,\\bullet\\}}{\\{\\hbot\\}}\n\t}{G}\n\\end{multline*}\nwhere $G = \\{\\pid p \\to \\pid q_1,\\pid p \\to \\pid q_2\\}$.\nIndeed, the type system is designed to verify single communications, not groups.\n\n\\section{Synthesis of implementations}\n\\label{sec:synthesis}\n\n\nIn this section we present an EndPoint Projection (EPP) procedure which compiles a choreography to a concurrent implementation represented in terms of a process calculus. This calculus assumes the same failure model assumed for the choreography model but foregoes global data like globally unique frame identifiers since these are unrealistic in distributed settings.\n\n\\subsection{Process model}\nThe target process model is an extension of Procedural Processes \\cite{CM17:forte} where send and receive operations may fail and exchanged messages are tagged with numeric identifiers. Differently from frame identifiers used at the choreography level, numeric ones are strictly local: \n\\begin{itemize}\n\\item each process maintains a counter for each known process (its neighbourhood in the choreography model);\n\\item frame declarations increment counters locally \\ie without synchronising with the other party (which may not even have a matching frame declaration);\n\\item frames are assigned the value held by the corresponding counter.\n\\end{itemize}\nNumeric frame identifiers may be regarded as sequence numbers. However, the model does not offer any mechanism for maintaining counters synchronised among connected processes nor can such mechanism be programmed since these counters are inaccessible. The only way to maintain synchrony is to write programs where frame declarations are carefully matched on each involved party.\n\n\\paragraph{Syntax} A network is a pair $\\langle \\mathcal{B}, N\\rangle$ where $\\mathcal{B}$ is a set of procedure definitions and $N$ is a parallel composition of processes and messages in transit. A process is written as $\\actor{\\pid p}{B}{\\sigma_\\pid p}{\\theta_\\pid p}$ where $\\pid p$ is its name, $\\sigma_{\\pid p}$ is its memory cell, and $\\theta_{\\pid p}$ is the memory reserved to the runtime for storing information about:\n\\begin{itemize}\n\t\\item open connections (known process, last frame index), \n\t\\item method requests (labels received), and\n\t\\item frame status (the last send\/receive operation succeeded).\n\\end{itemize}\nFormally, $\\theta$ is a function given as the combination of:\n\\begin{itemize}\n\\item \n$\\theta^{\\mathrm{fc}}\\colon \\type{PID} \\rightharpoonup \\type{FID}$, \n\\item\n$\\theta^{\\mathrm{lb}}\\colon \\type{PID} \\times \\type{FID} \\rightharpoonup \\type{LBL}$, and\n\\item\n$\\theta^{\\mathrm{fs}}\\colon \\type{PID} \\times  \\type{FID} \\to \\type{Bool}$.\n\\end{itemize}\nWe will omit superscripts $\\mathrm{fc}$, $\\mathrm{lb}$, and $\\mathrm{fs}$ provided that the intended component is clear from the context.\nThe full syntax of the language for programming in this model is defined by the grammar below.\n\\begin{align*}\n\t\\mathcal{B} \\Coloneqq {} & \n\t\t\\procdef{X}{\\vec{P}}{B}, \\mathcal{B} \\mid \\varnothing\n\t\\\\\n\tN\t\\Coloneqq {} & \n\t\t\\actor{\\pid p}{B}{\\sigma_\\pid p}{\\theta_\\pid p} \\mid\n\t\tN \\mathrel{\\keyword{\\bfseries |}} N' \\mid \n\t\t\\keyword{0}\n\t\\\\\n\tB \\Coloneqq {} & \n\t\t\\astart{\\pid q}{f}{B'} \\keyword{;}\\, B \\mid\n\t\t\\acomdecl{\\pid p}{k} \\keyword{;}\\, B \\mid\n\t\t\\arecv{\\pid p}{k}{}\\keyword{;}\\, B\\mid\n\t\t\\\\ \\mid {} & \n\t\t\\arecv{\\pid p}{k}{R}\\keyword{;}\\, B\\mid\n\t\t\\asend{\\pid q}{k}{S}\\keyword{;}\\, B\\mid\n\t\t\\abranch{\\pid p}{k}{l_i\\keyword{:} B_i}[i \\in I] \\mid\n\t\tX(\\vec{P})\\keyword{;}\\, B \\mid\n\t\t\\keyword{0} \\mid\n\t\t\\\\ \\mid {} &\n\t\t\\cond{E}{B}{B'}\\keyword{;}\\, B''\n\t\\\\\n\tS \\Coloneqq {} & \n\t\tf \\mid \\pid r \\mid l\n\t\\\\\n\tR \\Coloneqq {} & \n\t\tf \\mid (\\pid r)\n\t\\\\\n\tE \\Coloneqq {} &\n\t\te \\mid \\adelivered{\\pid p}{k}\n\t\\\\\n\tP \\Coloneqq {} &\n\t\tk \\mid \\pid p\n\\end{align*}\nTerm $\\acomdecl{\\pid p}{k}$ describes a behaviour that creates a new frame for its continuation. Send actions for $k$ are described by terms $\\asend{\\pid q}{k}{f}$, $\\asend{\\pid q}{k}{\\pid r}$, $\\asend{\\pid q}{k}{l}$, and receive ones by $\\arecv{\\pid p}{k}{f}$, $\\arecv{\\pid p}{k}{(\\pid r)}$, $\\arecv{\\pid p}{k}{}$, for values, process names, or label exchanges, respectively. We remark that terms for receiving process names bind them in their continuation.\nTerm $\\abranch{\\pid p}{k}{l_i\\keyword{:} B_i}[i \\in I]$ describes a selection based on a label communicated as frame $k$, if any label $l_i$ has been successfully received then, the process proceeds with the corresponding behaviour $B_i$ otherwise it proceeds with the one labelled with $\\lbl{\\color{keyword} default}$. This label is reserved exclusively for this purpose and cannot be sent. If $I = \\emptyset$, then the term is simply discarded. Guards $\\adelivered{\\pid p}{k}$ state that the last communication action for frame $k$ with $\\pid p$ has been successfully completed. Remaining terms are standard.\n\n\nPrograms are written using frame names which are replaced by numeric identifiers assigned at runtime when frames are created.\nTerms where frame names ($k$) are replaced by numeric identifiers ($n$) are reserved to the runtime. Messages in transit from $\\pid p$ to $\\pid q$ are represented by ``bags'' \\ie terms (of sort $N$) like $\\abag{n}{\\pid p}{\\pid q}{M}$ where the subterm $M$ stands for the payload. \n\n\\begin{figure*}[t]\n\\begin{infrules}\n\t\\infrule[\\rname[P]{NP}][rule:p-new-proc]{\n\t\tf(\\sigma) \\downarrow v\n\t\t\\and\n\t\t\\theta_{\\pid p}' = \\theta_{\\pid p}[\\pid q \\mapsto 0]\n\t\t\\and\n\t\t\\theta_{\\pid q} = \\{\\pid p \\mapsto 0\\}\n\t}{\n\t\t\\actor{\\pid p}{\\astart{\\pid q}{f}{B'}\\keyword{;}\\, B}{\\sigma}{\\theta_{\\pid p}}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid p}{B'}{\\sigma}{\\theta_{\\pid p}'}\n\t\t\\mathrel{\\keyword{\\bfseries |}}\n\t\t\\actor{\\pid q}{B}{v}{\\theta_{\\pid q}}\n\t}\n\t\\infrule[\\rname[P]{NF}][rule:p-new-frame]{\n\t\tn = \\mathrm{next}(\\theta(\\pid q)) \\and\n\t\t\\theta' = \\theta[\\pid q \\mapsto n, (\\pid q,n) \\mapsto \\bot]\n\t}{\n\t\t\\actor{\\pid p}{\\keyword{new}(\\pid q,k) \\keyword{;}\\, B[n\/k]}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid p}{B}{\\sigma}{\\theta'}\n\t}\n\t\\infrule[\\rname[P]{Int}][rule:p-comp]{\n\t\tf(\\sigma) \\downarrow v\n\t}{\n\t\t\\actor{\\pid p}{\\alocal{\\pid q}{f} \\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid p}{B}{v}{\\theta}\n\t}\n\t\\infrule[\\rname[P]{Snd}][rule:p-send]{\n\t\ts(\\sigma) \\downarrow u \\and\n\t\t\\and\n\t\t\\theta' = \\theta[(\\pid q,n) \\mapsto \\top]\n\t}{\n\t\t\\actor{\\pid p}{\\asend{\\pid q}{n}{s} \\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid p}{B}{\\sigma}{\\theta'}\n\t\t\\mathrel{\\keyword{\\bfseries |}}\n\t\t\\abag{n}{\\pid p}{\\pid q}{u}\n\t}\n\t\\infrule[\\rname[P]{SndFail}][rule:p-send-fail]{\n\t}{\n\t\t\\actor{\\pid p}{\\asend{\\pid q}{n}{s} \\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid p}{B}{\\sigma}{\\theta}\n\t}\n\t\\infrule[\\rname[P]{Loss}][rule:p-loss]{\n\t}{\n\t\t\\abag{n}{\\pid p}{\\pid q}{u}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\keyword{0}\n\t}\n\t\\infrule[\\rname[P]{RcvV}][rule:p-recv-val]{\n\t\tf(\\sigma,v) \\downarrow w\n\t\t\\and\n\t\t\\theta' = \\theta[(\\pid p,n) \\mapsto \\top]\n\t}{\n\t\t\\actor{\\pid q}{\\arecv{\\pid p}{n}{f} \\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\mathrel{\\keyword{\\bfseries |}}\n\t\t\\abag{n}{\\pid p}{\\pid q}{v}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid q}{B}{w}{\\theta'}\n\t}\n\t\\infrule[\\rname[P]{RcvP}][rule:p-recv-pid]{\n\t\t\\theta' = \\theta[\n\t\t\t(\\pid p,n) \\mapsto \\top,\n\t\t\t\\pid r \\mapsto 0\n\t\t]\n\t}{\n\t\t\\actor{\\pid q}{\\arecv{\\pid p}{n}{(\\pid s)} \\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\mathrel{\\keyword{\\bfseries |}}\n\t\t\\abag{n}{\\pid p}{\\pid q}{\\pid r}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid q}{B[\\pid r \/ \\pid s]}{\\sigma}{\\theta'}\n\t}\n\t\\infrule[\\rname[P]{RcvL}][rule:p-recv.lbl]{\n\t\t\\theta' = \\theta[(\\pid p,n) \\mapsto \\top,(\\pid p,n) \\mapsto l]\n\t}{\n\t\t\\actor{\\pid q}{\\arecv{\\pid p}{n} \\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\mathrel{\\keyword{\\bfseries |}}\n\t\t\\abag{n}{\\pid p}{\\pid q}{l}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid q}{B}{\\sigma}{\\theta'}\n\t}\n\t\\infrule[\\rname[P]{RcvFail}][rule:p-rcv-fail]{\n\t}{\n\t\t\\actor{\\pid q}{\\arecv{\\pid p}{n}{R} \\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\n\t\t\\actor{\\pid q}{B}{\\sigma}{\\theta}\n\t}\n\t\\infrule[\\rname[P]{IfFrame}][rule:p-if-frame]{\n\t\t\\text{if } \\theta(\\pid q,n) = \\top\n\t\t\\text{ then } i = 1\n\t\t\\text{ else } i = 2\n\t}{\n\t\t\\actor{\\pid p}{\\cond{\\adelivered{\\pid q}{n}}{B_1}{B_2}\\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\t\t\n\t\t\\actor{\\pid p}{B_i\\keyword{;}\\, B}{\\sigma}{\\theta}\n\t}\n\t\\infrule[\\rname[P]{IfExp}][rule:p-if-exp]{\n\t\t\\text{if } f(\\sigma) \\downarrow \\literal{true}\n\t\t\\text{ then } i = 1\n\t\t\\text{ else } i = 2\n\t}{\n\t\t\\actor{\\pid p}{\\cond{f}{B_1}{B_2}\\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\t\t\n\t\t\\actor{\\pid p}{B_i\\keyword{;}\\, B}{\\sigma}{\\theta}\n\t}\n\t\\infrule[\\rname[P]{Branch}][rule:p-branch]{\n\t\t\\theta(\\pid p,n) = l_i\t\n\t\t\\and\n\t\t\\lbl{default} \\neq l_i\n\t}{\n\t\t\\actor{\\pid q}{\\abranch{\\pid p}{n}{l_i\\keyword{:} B_i}[i \\in I]\\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\t\t\n\t\t\\actor{\\pid p}{B_i\\keyword{;}\\, B}{\\sigma}{\\theta}\n\t}\n\t\\infrule[\\rname[P]{BranchFail}][rule:p-branch-fail]{\n\t\t\\theta(\\pid p,n) = \\bot\n\t\t\\and\n\t\t\\lbl{default} = l_i\n\t}{\n\t\t\\actor{\\pid q}{\\abranch{\\pid p}{n}{l_i\\keyword{:} B_i}[i \\in I]\\keyword{;}\\, B}{\\sigma}{\\theta}\n\t\t\\reducesto_{\\mathcal{B}}\t\t\n\t\t\\actor{\\pid p}{B_i\\keyword{;}\\, B}{\\sigma}{\\theta}\n\t}\n\t\\infrule[\\rname[P]{Par}][rule:p-par]\n\t\t{N \\reducesto_{\\mathcal{B}} N'}\n\t\t{N \\mathrel{\\keyword{\\bfseries |}} M \\reducesto_{\\mathcal{B}} N' \\mathrel{\\keyword{\\bfseries |}} M}\t\t\n\t\\infrule[\\rname[P]{Str}][rule:p-str]\n\t\t{N \\precongr_{\\mathcal{B}} M \\qquad M \\reducesto_{\\mathcal{B}} M' \\qquad M' \\precongr_{\\mathcal{B}} N'}\n\t\t{N \\reducesto_{\\mathcal{B}} N'}\n\t\\infrule[\\rname[P]{NilBeh}][rule:p-nil-beh]{}{\n\t\t\\keyword{0}\\keyword{;}\\, B \\precongr_{\\mathcal{B}} B\n\t}\n\t\\infrule[\\rname[P]{NilProc}][rule:p-nil-proc]\n\t\t{}\n\t\t{\\actor{\\pid p}{\\keyword{0}}{\\sigma}{\\theta} \\precongr_{\\mathcal{B}} \\keyword{0}}\n\n\t\\infrule[\\rname[P]{NilRecv}][rule:p-nil-recv]\n\t\t{}\n\t\t{\\abag{n}{\\pid p}{\\pid q}{M} \\mathrel{\\keyword{\\bfseries |}} \\actor{\\pid q}{\\keyword{0}}{\\sigma}{\\theta} \\precongr_{\\mathcal{B}} \\actor{\\pid q}{\\keyword{0}}{\\sigma}{\\theta}}\n\t\\infrule[\\rname[P]{NilNet}][rule:p-nil-net]\n\t\t{}\n\t\t{\\keyword{0} \\mathrel{\\keyword{\\bfseries |}} N \\precongr_{\\mathcal{B}} N}\n\t\\infrule[\\rname[P]{Unfold}][rule:p-unfold]\n\t\t{\\procdef{X}{\\vec{P}}{B'} \\in D}\n\t\t{X(\\vec{A})\\keyword{;}\\, B \\precongr_{\\mathcal{B}} B'[\\vec{A}\/\\vec{P}]\\keyword{;}\\, B}\n\\end{infrules}\n\t\\caption{Process model, operational semantics}\n\t\\label{fig:proc-semantics}\n\\end{figure*}\n\n\\paragraph{Semantics} \nThe calculus semantics is given by the reduction relation on networks $\\reducesto_{\\mathcal{B}}$ in \\cref{fig:proc-semantics} and is parameterised in the set $\\mathcal{B}$ of procedure definitions. For compactness, the presentation relies on the structural precongruence $\\precongr_\\mathcal{B}$.\nAll rules follow the intuitive description of terms and their description will be omitted.\n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\\begin{autoflow}\t\n\t\t\\newcommand{\\afdbox}[1]{{$\\displaystyle#1$}\\endmath\\autoflow@AND\\math\\displaystyle}\n\n\t\t\\afdbox{\n\t\t\t\\epp{\\keyword{0}}[m][\\pid r] \n\t\t\t\\triangleq \\keyword{0}\n\t\t}\n\t\t\n\t\t\\afdbox{\n\t\t\t\\epp{\\keyword{0} \\keyword{;}\\, C}[m][\\pid r]\n\t\t\t\\triangleq \n\t\t\t\\keyword{0}\\keyword{;}\\,\\epp{C}[m][\\pid r]\n\t\t}\n\t\t\n\t\t\\afdbox{\n\t\t\t\\epp{\\cstart{\\pid p}{\\pid q}{f}{C}}[m][\\pid r] \n\t\t\t\\triangleq \n\t\t\t\\begin{cases}\n\t\t\t\t\\astart{\\pid q}{f}{\\epp{C}[m][\\pid q]}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \\text{if } \\pid r = \\pid p\\\\\n\t\t\t\t\\epp{C}[m][\\pid r] & \\text{otherwise}\n\t\t\t\\end{cases}\n\t\t}\n\t\t\n\t\t\\afdbox{\n\t\t\t\\epp{X(\\vec{A})\\keyword{;}\\, C}[m][\\pid r] \n\t\t\t\\triangleq \n\t\t\t\\begin{cases}\n\t\t\t\tX_{\\pid p}(m(\\vec{A}\\setminus \\pid p\\tteq \\pid r))\\keyword{;}\\, \\epp{C}[m][\\pid r] & \\text{if } \n\t\t\t\t\\pid p\\tteq \\pid r \\in \\vec{A} \\text{ for some } \\pid p\\\\\n\t\t\t\t\\epp{C}[m][\\pid r] & \\text{otherwise}\n\t\t\t\\end{cases}\n\t\t}\n\t\t\n\t\t\\afdbox{\n\t\t\t\\epp{\\cnewframe{k}{\\pid p}{\\pid q}{T}{C}}[m][\\pid r] \n\t\t\t\\triangleq  \n\t\t\t\\begin{cases}\n\t\t\t\t\\acomdecl{\\pid q}{k}\\keyword{;}\\, \\epp{C[k^{T,\\pid r,\\pid q}\/k]}[m[k \\mapsto k]][\\pid r] & \\text{if } \\pid r = \\pid p\\\\\n\t\t\t\t\\acomdecl{\\pid p}{k}\\keyword{;}\\, \\epp{C[k^{T,\\pid p,\\pid r}\/k]}[m[k \\mapsto k]][\\pid r] & \\text{if } \\pid r = \\pid q\\\\\n\t\t\t\t\\epp{C}[m][\\pid r] & \\text{otherwise}\n\t\t\t\\end{cases}\n\t\t}\n\t\t\n\t\t\\afdbox{\n\t\t\t\\epp{\\csend{k^{T,\\pid p,\\pid q}}{\\pid p.s}\\keyword{;}\\, C}[m][\\pid r] \n\t\t\t\\triangleq \n\t\t\t\\begin{cases}\n\t\t\t\t\\asend{\\pid q}{m(k)}{s}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \\text{if } \\pid r = \\pid p\\\\\n\t\t\t\t\\epp{C}[m][\\pid r] & \\text{otherwise}\n\t\t\t\\end{cases}\n\t\t}\n\t\t\n\t\t\\afdbox{\n\t\t\t\\epp{\\crecv{k^{T,\\pid p,\\pid q}}{R}\\keyword{;}\\, C}[m][\\pid r] \n\t\t\t\\triangleq \n\t\t\t\\begin{cases}\n\t\t\t\t\\arecv{\\pid p}{m(k)}{f}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \\text{if } R = \\pid r.f\\\\\n\t\t\t\t\\arecv{\\pid p}{m(k)}{(\\pid s)}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \\text{if } R = \\pid r \\text{ and } T\\eqtype\\type{PID}\\\\\n\t\t\t\t\\arecv{\\pid p}{m(k)}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \\text{if } R = \\pid r \\text{ and } T\\eqtype\\type{LBL}\\\\\n\t\t\t\t\\epp{C}[m][\\pid r] & \\text{otherwise}\n\t\t\t\\end{cases}\n\t\t}\n\t\t\t\t\n\t\t\\afdbox{\n\t\t\t\\epp{\\cond{E}{C_1}{C_2}\\keyword{;}\\, C}[m][\\pid r] \n\t\t\t\\triangleq \n\t\t\t\\begin{cases}\n\t\t\t\t\\cond{e}{\\epp{C_1}[m][\\pid r]}{\\epp{C_2}[m][\\pid r]}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \n\t\t\t\t\t\\text{if } E = \\pid r.e\\\\\n\t\t\t\t\\cond{\\adelivered{\\pid q}{m(k)}}{\\epp{C_1}[m][\\pid r]}{\\epp{C_2}[m][\\pid r]}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \n\t\t\t\t\t\\text{if } E = \\pid r.\\csent{k^{T,\\pid r,\\pid q}}\\\\\n\t\t\t\t\\cond{\\adelivered{\\pid p}{m(k)}}{\\epp{C_1}[m][\\pid r]}{\\epp{C_2}[m][\\pid r]}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \n\t\t\t\t\t\\text{if } E = \\pid r.\\creceived{k^{T,\\pid p,\\pid r}}\\\\\n\t\t\t\t\\abranch{\\pid p}{m(k)}{l\\colon \\epp{C_1}[m][\\pid r],\\lbl{\\color{keyword} default}\\keyword{:} \\epp{C_2}[m][\\pid r]}\\keyword{;}\\, \\epp{C}[m][\\pid r] & \n\t\t\t\t\t\\text{if } E = \\pid r.\\creceivedlbl{k^{T,\\pid p,\\pid r}}{l}\\\\\n\t\t\t\t(\\epp{C_1}[m][\\pid r] \\merge \\epp{C_2}[m][\\pid r])\\keyword{;}\\, \\epp{C}[m][\\pid r] & \t\n\t\t\t\t\t\\text{otherwise}\n\t\t\t\\end{cases}\n\t\t}\n\n\t\\end{autoflow}\n\t\\end{center}\n\t\\caption{Behaviour projection.}\n\t\\label{fig:behaviour-projection}\n\\end{figure*}\n\n\\subsection{EndPoint Projection}\n\\label{sec:epp}\n\nRecall that at process level, active frames have numeric identifiers that are locally and independently generated by each process as soon as a frame is declared whereas at the choreography level frames have unique global identifiers. As a consequence, a coherent mapping from the former to the latter is needed in order to project choreographies with free frame names as in the case of running ones. By \\emph{frame mapping} we mean any mapping taking frame names to frame numbers or to themselves---there is no reason for assigning a numeric identifier or a different name to a bound frame name. \nWrite $\\phi|_{\\pid p}$ for the set\n$\n\t\\{k \\mid \\phi(k) = (\\pid q,\\pid r,t,u) \\text{ and } \\pid p \\in \\{\\pid q,\\pid r\\}\\}\n$\nof all frames in $\\phi$ to or from $\\pid p$.\nA frame mapping $m$ is said to be \\emph{compatible with $\\phi$} if for any $\\pid p$ that occurs in $\\phi$, $m$ assigns to frames in $\\phi|_{\\pid p}$ unique and sequential numbers \\ie:\n\\[\n\t\\{m(k) \\mid k \\in \\phi|_{\\pid p}\\} = \\{1,2,\\dots,\\left|\\,\\phi|_{\\pid p}\\,\\right|\\}\\text{.}\n\\]\nAny $\\phi$ admits a compatible mapping under the mild assumption that frame names can be totally ordered.\nConsider the choreography $C = \\csend{k}{\\pid p.f}\\keyword{;}\\, \\csend{k}{\\pid q.f'}$.\nIf $C$ is part of an execution (of a well-typed program) then, $k$ must occur in $\\phi$ and hence the projections of $\\pid p$ and $\\pid q$ must refer to this frame via a numeric identifier (\\cf \\cref{rule:p-new-frame}) and they must agree on it. However, since the process model does not offer any mechanism for processes to negotiate an agreement on their internal frame counters this property must be derived from the choreography level, hence the necessity of $m$. We remark that this situation is limited to free names only: programs in RC\\xspace are projected with $m = id$.\n\n\nGiven a choreography $C$ and a frame mapping $m$, the projected behaviour of process $\\pid p$  in $C$ is defined as $\\epp{C}[m][\\pid p]$ where $\\epp{-}[m][\\pid p]$ is the partial function defined by structural recursion in \\cref{fig:behaviour-projection}---for conciseness, each frame occurring in the choreography $C$ is pre-annotated with its senders and receivers.\nEach case in the definition follows the intuition of projecting, for each choreographic term, the local actions performed by the given process.\nFor instance, $\\csend{k}{\\pid p.f}$ is skipped during the projection of any process but $\\pid p$ for which case the send action $\\asend{\\pid p}{m(k)}{f}$ is produced. Cases for frame reception, procedure calls, frame and process creation, are similar. \nThe case for conditionals is more involved but follows a standard approach (see \\eg \\cite{BCDLDY08,LGMZ08,CHY12,CM16:facs,CM17:forte}). The (partial) merging operator $\\merge$ from \\cite{CHY12} is used to merge the behaviour of a process that does not know (yet) which branch has been chosen by the the process evaluating the guard. Intuitively, $B \\merge B'$ is isomorphic to $B$ and $B'$ up to branching, where branches of $B$ or $B'$ with distinct labels are also included. One proceeds homomorphically (\\eg $\\csend{k}{\\pid p.f}\\keyword{;}\\, B \\merge \\csend{k}{\\pid p.f}\\keyword{;}\\, B'$ is $\\csend{k}{\\pid p.f}\\keyword{;}\\, (B \\merge B')$) on all terms but branches which are handled defining the merge of\n$\n\t\\abranch{\\pid p}{k}{l_i \\keyword{:} B_i}[i \\in I]\\keyword{;}\\, B \n$ and $\n\t\\abranch{\\pid p}{k}{l_j \\keyword{:} B'_j}[j \\in J]\\keyword{;}\\, B'\n$\nas\n$\\abranch{\\pid p}{k}{l_h \\keyword{:} B''_h}[h \\in H]\\keyword{;}\\, (B \\merge B')$\nwhere $\\{l_h\\colon B''_h\\}_{h \\in H}$ is the union of\n$\\{l_i \\keyword{:} B_i\\}_{i \\in I\\setminus J}$, $\\{l_j \\keyword{:} B'_j\\}_{j \\in J\\setminus I}$, and $\\{l_g \\keyword{:} B_g \\merge B'_g\\}_{g \\in I\\cap J}$.\n\nProjection of procedure definitions follows the approach introduced by Procedural Choreographies \\cite{CM17:forte}. For $\\mathcal{D}$ a set of procedure definitions, its projection is defined as follows:\n\\[\n\t\\epp{\\mathcal{D}} \\triangleq \\bigcup_{\\procdef{X}{\\vec{P}}{C} \\in \\mathcal{D}}\n\t\t\t\\left\\{\n\t\t\t\t\\procdef{X_{\\pid p}}{\\vec{P}\\setminus \\pid p}{\\epp{C}[id][\\pid p]}\n\t\t\t\\,\\middle|\\,\n\t\t\t\t\\pid p: T \\in \\vec{P}\n\t\t\t\\right\\}\n\t\\text{.}\n\\]\nObserve that since a procedure $X$ may be called multiple times on any combination of its arguments (hence assigning to a process different r\\^oles at each call) it is necessary to project the behaviour of each possible process parameter in $\\vec{P}$ as the procedure $X_{\\pid r}$. Here typing is crucial otherwise processes may be called to play r\\^oles for which they lack the necessary connections.\n\nTo designate a network as the projection of a configuration $\\langle C,\\sigma,\\phi,G\\rangle$ it remains only to distribute the information contained in the global state $\\sigma$, $\\phi$, $G$.\nReserved memory for process $\\pid p$ ($\\theta_{\\pid p}$) in the process model is completely determined (up to frame numbering) by $\\phi$ and $G$ from the choreography level as these contain all data regarding processes known to $\\pid p$ and frames exchanged by $\\pid p$. \nSpecifically, $\\epp{\\phi,G}[m][\\pid p]$ is defined as the function $\\theta$ where\n\\begin{align*}\n\t\\theta^{\\mathrm{fc}}(\\pid q) \\triangleq{}&\n\t\\begin{cases}\n\t\t\\left|\\phi|_{\\pid p}\\right|\n\t\t & \\text{if } G \\vdash \\pid p \\to \\pid q \\\\\n\t\t\\bot & \\text{otherwise}\n\t\\end{cases}\n\t\\\\\n\t\\theta^{\\mathrm{ln}}(\\pid q,n) \\triangleq{}&\n\t\\begin{cases}\n\t\tl & \\text{if } \\exists k \\in m^{-1}(n) \\text{ s.t.~}\\phi(k) = \\phiframe{\\pid p}{\\_}{\\pid q}{\\removed{l}} \\\\\n\t\t\\bot & \\text{otherwise}\n\t\\end{cases}\n\t\\\\\n\t\\theta^{\\mathrm{fn}}(\\pid q,n) \\triangleq{}&\n\t\\begin{cases}\n\t\t\\top & \n\t\t\t\\mspace{-10mu}\\array{l}\n\t\t\t\\text{if } \\exists k \\in m^{-1}(n) \\text{ s.t. either } \n\t\t\t\\phi(k) = \\phiframe{\\pid p}{u}{\\pid q}{\\_} \\\\\n\t\t\t\\text{and } u \\neq \\bot \\text{ or } \n\t\t\t\\phi(k) = \\phiframe{\\pid q}{\\_}{\\pid p}{u'} \\text{ and } u' \\in \\removed{\\mathcal{U}}\n\t\t\t\\endarray\\mspace{-15mu}\\\\\n\t\t\\bot & \\text{otherwise}\n\t\\end{cases}\n\\end{align*}\nThe only information of $\\phi$ and $G$ that cannot be reconstructed from the distributed state of the processes in a network is that of frames in transit. To this end, a term $\\abag{m(k)}{\\pid p}{\\pid q}{u}$ is added to the network for each $\\phi(k) = \\phiframe{\\pid p}{\\removed{u}}{\\pid q}{\\bot}$.\nThe projection $\\epp{C,\\sigma,\\phi,G}[m]$ of $\\langle C,\\sigma,\\phi,G\\rangle$ \nis defined as the network:\n\\[\n\t\\prod_{\\phi(k) = \\phiframe{\\pid p}{\\removed{u}}{\\pid q}{\\bot}}\n\t\\abag{m(k)}{\\pid p}{\\pid q}{u}\n\t\\mathrel{\\keyword{\\bfseries |}}\n\t\\prod_{\\pid r \\in \\mathrm{pn}(C)} \\actor{\\pid p}{\\epp{C}[m][\\pid p]}{\\sigma(\\pid p)}{\\epp{\\phi,G}[m][\\pid p]}\n\\]\nwhere $m$ is any mapping compatible with $\\phi$.\nObserve that mappings are all equivalent up to $\\alpha$-conversion and that if $\\epp{C,\\sigma,\\phi,G}[m]$ is defined for some $\\sigma$, $\\phi$, and $G$ then $\\epp{C,\\sigma',\\phi',G'}[m']$ is defined for any $\\sigma'$, $\\phi'$, and $G'$. We say that $C$ is projectable whenever $\\epp{C,\\sigma,\\phi,G}[m]$ is defined  for some $m$, $\\sigma$, $\\phi$, and $G$.\n\nThere is an operational correspondence between choreographies and their projections---up to the ``pruning'' relation $\\pruning$ (\\cite{CHY12,CM13}) that eliminates ``dead branches'' due to the merging operator $\\merge$ when they are no longer needed to follow the originating choreography. \n\n\\begin{theorem}[EPP]\n\t\\label{thm:epp}\n\tLet $\\langle \\mathcal{D},C\\rangle$ be a projectable program.\n\tFor any $\\langle C,\\sigma,\\phi,G \\rangle$ well-typed under $\\mathcal{D}$:\n\t\\begin{description}\n\t\t\\item[Compl.] \n\t\t\tIf $\\langle C,\\sigma,\\phi,G \\rangle \\reducesto_{\\mathcal{D}} \\langle C',\\sigma',\\phi',G' \\rangle$, \n\t\t\tthen there are mappings $m \\subseteq m'$ such that $|\\dom(m')\\setminus\\dom(m)| \\leq 1$ and\n\t\t\t$\\epp{C,\\sigma,\\phi,G}[m] \\reducesto_{\\epp{\\mathcal{D}}} N$\n\t\t\tfor some $N \\pruning \\epp{C',\\sigma',\\phi',G'}[m']$.\n\t\t\\item[Sound.]\n\t\t\tIf $\\epp{C,\\sigma,\\phi,G}[m] \\reducesto_{\\epp{\\mathcal{D}}} N$, \n\t\t\tthen there are $N'$,$C'$,$\\sigma'$,$\\phi'$,$G'$, and $m'$ s.t.~$N \\reducesto_{\\epp{\\mathcal{D}}}^\\ast N'$,\n\t\t\t$\\langle C,\\sigma,\\phi,G \\rangle \\reducesto_{\\mathcal{D}} \\langle C',\\sigma',\\phi',G' \\rangle$, \n\t\t\t$\\epp{C',\\sigma',\\phi',G'}[m'] \\pruning N'$,\n\t\t\t$|\\dom(m')\\setminus\\dom(m)| \\leq 1$, and\n\t\t\t$m \\subseteq m'$.\n\t\\end{description}\n\\end{theorem}\n\nIt follows from the operational correspondence in \\cref{thm:epp} that projected networks exhibit all relevant properties ensured by the typing disciplines introduced in \\cref{sec:typing}, namely: progress (\\cref{thm:progress}), at-most-once delivery (\\cref{thm:chor-at-least-once}), best-effort delivery (\\cref{thm:chor-best-effort}), and at-lest-once delivery (\\cref{thm:chor-at-least-once}).\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nProgramming methodologies based on structured communications, like choreographies, have been \ninvestigated for a long time now \\cite{Aetal16,Hetal16}. This is the first paper that investigates \nhow this research line can be applied to the programming of robust distributed systems in the \nsetting of communication failures, bringing us one step nearer to improved reliability in concurrent \ncomputing in the future.\n\nWe believe that the results achieved in this paper unlock a very promising research direction. A \nnatural continuation is to consider different failure models that take into account node failures \nand explore adversaries models (\\eg to include message loss, duplication, or forging).\nNode failures and adversaries are crucial for reasoning about agreement problems in distributed systems. These problems are as challenging as common in real-world distributed programming. Results in this direction may advance the development of correct-by-construction agreement protocols and their implementations.\n\nAnother interesting direction is to explore quantitative properties of programs in RC\\xspace. To this end we plan to develop quantitative semantics for the RC\\xspace model. For instance, in a probabilistic settings, failures are characterised by probability distributions and properties like progress, at-most-once, and exactly-once delivery are formalised as almost-certain properties (their complement event has null measure). Then it is possible to reason about reliability assumptions on communication links \\eg to understand how a certain failure probability impacts our program. Another interesting property is the expected number of retransmissions. Estimates of this value allow to optimise failure-recovery strategies. Likewise, stochastic or timed semantics will enable models with explicit timeouts.\n\nThe typing disciplines introduced in this work ensure that well-typed distributed programs have at-most-once or exactly-once delivery guarantees. As pointed out at the end of \\cref{sec:typing}, these guarantees are limited to single communications but our approach can be reasonably extended to communication groups. This extension has immediate applications \\eg to the statical verification of replication protocols where an update is deemed successful only if it accepted by enough replicas.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn this paper we study the following type of (markovian) backward\nstochastic differential equations with infinite horizon (that we\nshall call \\textit{ergodic} BSDEs or EBSDEs for short):\n\\begin{equation}\\label{EBSDE*}\nY^x_t=Y^x_T +\\int_t^T\\left[\\psi(X^x_\\sigma,Z^x_\\sigma)-\n\\lambda\\right]d\\sigma-\\int_T^T Z^x_\\sigma dW_\\sigma, \\quad 0\\le\nt\\le T <\\infty.\n\\end{equation}\nIn equation (\\ref{EBSDE*}) $X^x$ is the solution of a forward\nstochastic differential equation with\n values in a Banach space $E$\nstarting at $x$ and $(W_t)_{t\\geq 0}$ is a cylindrical Wiener\nprocess in a Hilbert space $\\Xi$.\n\n\nOur aim is to find a triple\n $(Y,Z,\\lambda)$, where $Y,Z$ are\n adapted processes taking values in\n $\\mathbb{R}$ and $\\Xi^*$ respectively and $\\lambda$\n  is a real number. $\\psi:E\\times \\Xi^*\\to \\mathbb R$ is\n  a given function.\nWe stress the fact that $\\lambda$ is part of the unknowns of\nequation (\\ref{EBSDE*}) and this is the reason  why the above is a\nnew class of BSDEs.\n\n\n$ $\n\n\nIt is by now well known that BSDEs provide an efficient\nalternative tool to study optimal control problems, see, e.g.\n\\cite{peng93}, \\cite{ElKaMaz} or, in an infinite dimensional\nframework, \\cite{{FuTe1}}, \\cite{masiero}. But up to our best\nknowledge, there exists no work in which  BSDE techniques are\napplied to\n optimal control problems with \\emph{ergodic} cost functionals that is\n functionals depending only on the asymptotic behavior of the state\n (see e.g. the cost defined in formula\n (\\ref{ergodic-cost*}) below).\n\n\n$ $\n\n\n\\noindent The purpose of the present paper is to show that\nbackward\n stochastic differential equations, in particular\n the class of EBSDEs mentioned above,  are a very useful tool in the\n treatment of ergodic control problems as\n well, especially in an infinite dimensional framework.\n\n\n $ $\n\n\n \\noindent There is a fairly large amount of literature dealing by\n analytic techniques with optimal ergodic control problems\n  for finite dimensional stochastic state\n equations.\n We just mention the basic papers by Bensoussan and Frehse\n  \\cite{BeFr} and by Arisawa and Lions \\cite{ArLi} where the\nproblem is treated through the study of  the corresponding\nHamilton-Jacobi-Bellman (HJB) equation (solutions  are understood\n in a classical sense and in a viscosity sense,\nrespectively).\n\n\nConcerning the infinite dimensional case it is known that both\nclassical and viscosity notions of solutions are not so suitable\nconcepts. Maslowski and Goldys in \\cite{GoMa} employ a mild\nformulation of the Hamilton-Jacobi-Bellman equation in a\n Hilbertian framework (see \\cite{C} and references within for the\ncorresponding mild formulations  in the standard cases). In\n\\cite{GoMa} the authors prove, by a fixed point argument that\nexploits the smoothing properties of the Ornstein-Uhlenbeck\nsemigroup corresponding to the state equation,  existence and\nuniqueness of the  solution of the stationary HJB equation for\ndiscounted infinite horizon costs. Then they pass to the limit, as\nthe discount goes to zero, to obtain a mild solution of the  HJB\nequation for the ergodic problem (see also \\cite{duncans}). Such\ntechniques need to assume, beside natural condition on  the\ndissipativity of the state equation, also non-degeneracy of the\nnoise and a limitation on the lipschitz constant (with respect to\nthe gradient variable)  of the hamiltonian function. This last\ncondition carries a bound on the size of the control domain (see\n\\cite{FuTe-ell} for similar conditions in the infinite horizon\ncase).\n\n\n$ $\n\n\n\n\nThe introduction of EBSDEs allow us to treat Banach valued state\nequations with general monotone nonlinear term and possibly\ndegenerate noise. Non-degeneracy is replaced by a structure\ncondition as it usually happens in BSDEs approach, see, for\ninstance, \\cite{ElKaMaz}, \\cite{FuTe1}. Moreover the use of\n$L^{\\infty}$ estimates specific to infinite horizon backward\nstochastic differential equations (see \\cite{bh}, \\cite{royer},\n\\cite{HuTe}) allow us to eliminate conditions on  the lipschitz\nconstant of the hamiltonian. On the other side we will only\nconsider bounded cost functionals.\n\n\n$ $\n\n\n To start being more precise we consider a forward equation\n$$dX_t^x=(AX_t^x+F(X_t^x))dt+G dW_t,\\qquad X_0=x$$\nwhere $X$ has values in a Banach space $E$, $F$ maps $E$ to $E $\nand $A$ generates a strongly continuous semigroup of contractions.\nAppropriate dissipativity assumptions on  $A+F$   ensure the\nexponential decay of the difference between the trajectories\nstarting from different points $x,x'\\in E$.\n\n\nThen we introduce the class of strictly monotonic backward\nstochastic differential equations\n\\begin{equation}\\label{bsderoyer*}\n{Y}^{x,\\alpha}_t={Y}^{x,\\alpha}_T +\\int_t^T(\\psi(X^{x}_\\sigma,\nZ^{x,\\alpha}_\\sigma)-\\alpha Y^{x,\\alpha}_\\sigma)d\\sigma-\\int_t^T\nZ^{x,\\alpha}_\\sigma dW_\\sigma, \\quad 0\\le t\\le T <\\infty.\n\\end{equation}\nfor all $\\alpha>0$ (see \\cite{bh}, \\cite{royer} or \\cite{HuTe})\nwhere $\\psi: E\\times\\Xi^*\\rightarrow \\mathbb{R}$ is bounded in the\nfirst variable and Lipschitz in the second. By estimates based on\na Girsanov argument introduced in \\cite{bh} we obtain uniform\nestimates on $\\alpha{Y}^{x,\\alpha}$ and\n${Y}^{x,\\alpha}-{Y}^{x',\\alpha}$ that allow us to prove that,\nroughly speaking, $({Y}^{x,\\alpha}-{Y}^{0,\\alpha}_0,\n{Z}^{x,\\alpha}, \\alpha {Y}^{0,\\alpha}_0)$ converge to a solution\n$(Y^x,Z^x,\\lambda)$ of the EBSDE (\\ref{EBSDE*}), for all $x\\in E$.\nWe also show that $\\lambda$ is unique under very general\nconditions. On the contrary, in general we can not  expect\nuniqueness of the solution to (\\ref{EBSDE*}), at least in the non\nmarkovian case. On the other side in the markovian case we show\nthat we can find a solution of (\\ref{EBSDE*}) with\n$Y^x_t=v(X^x_t)$ and $Z^x_t=\\zeta(X^x_t)$ where $v$ is Lipschitz\nand $v(0)=0$. Moreover $(v, \\zeta)$ are unique at least in a\nspecial case where $\\psi$ is the Hamiltonian of a control problem\nand the processes $X^x$ are recurrent (see Section \\ref{sec-uniq}\nwhere we adapt an argument from \\cite{GoMa}).\n\n\n$ $\n\n\nIf we further assume differentiability  of $F$ and\n $\\psi$ (in the Gateaux sense) then $v$ is differentiable,\n  moreover $\\zeta =\\nabla v G$\n and finally $(v,\\lambda)$ give a mild solution of the HJB equation\n \\begin{equation}\n\\mathcal{L}v(x)\n+\\psi\\left( x,\\nabla v(x)  G\\right)  = \\lambda, \\quad x\\in E,  \\label{hjb*}%\n\\end{equation}\nwhere  linear operator $\\mathcal{L}$ is formally defined by\n\\[\n\\mathcal{L}f\\left(  x\\right)  =\\frac{1}{2}Trace\\left(\nGG^{\\ast}\\nabla ^{2}f\\left(  x\\right)  \\right)  +\\langle Ax,\\nabla\nf\\left(  x\\right) \\rangle_{E,E^{\\ast}}+\\langle F\\left(  x\\right)\n,\\nabla f\\left(  x\\right) \\rangle_{E,E^{\\ast}}.\n\\]\nMoreover if the Kolmogorov semigroup satisfies the smoothing\nproperty in Definition \\ref{strongly-feller} and $F$ is genuinely\ndissipative (see Definition \\ref{gen-diss}) then $v$ is bounded.\n\n\n\n$ $\n\n\n\nThe above results are then applied to a control problem with cost\n\\begin{equation}\\label{ergodic-cost*}\nJ(x,u)=\\limsup_{T\\rightarrow\\infty}\\frac{1}{T}\\, \\mathbb E\\int_0^T\nL(X_s^x,u_s)ds,\n\\end{equation}\n where $u$ is an adapted process (an admissible control)\nwith values in a separable metric space $U$, and the state\nequation is a Banach valued evolution equation of the form\n$$dX_t^x=(AX_t^x+F(X_t^x))\\, dt+G(dW_t+R(u_t)\\,dt),$$\nwhere $R: U \\rightarrow \\Xi$ is bounded. It is clear that the\nabove functional depends only on the asymptotic behavior of the\ntrajectories of $X^x$. After appropriate formulation\n we prove that, setting $\\psi(x,z)= \\inf_{u\\in U} [L(x,u)+ zR(u)]$ in\n  (\\ref{EBSDE*}), then $\\lambda$ is optimal, that is\n  $$\\lambda=\\inf_{u}J(x,u)$$\n   where the infimum is over all admissible controls.\n   Moreover $Z$ allows to  construct on optimal feedback in the\n   sense that $$\\lambda=J(x,u) \\hbox{ if and only if } L(X_t^x,u_t)+Z_t\nR(u_t)=\\psi(X_t^x,Z_t).$$\n\n\n\nFinally, see Section \\ref{section-heat-eq}, we show that our\nassumptions allow us to treat ergodic optimal control problems for\na stochastic heat equation with polynomial nonlinearity and\nspace-time white noise. We notice that the Banach space setting is\nessential in order to treat nonlinear terms with superlinear\ngrowth  in the state equation.\n\n\n\n\n$ $\n\n\n\n\n\n\n\nThe paper is organized as follows.\nAfter a section on notation, we introduce the forward SDE; in section 4 we\nstudy the ergodic BSDEs; in section 5 we show in\naddition the differentiability of the\nsolution assuming that the coefficient is Gateaux differentiable.\nIn section 6 we study the ergodic Hamilton-Jacobi-Bellman\nequation and we apply our result to optimal\nergodic control in section 7. Section 8 is devoted to\nshow the uniqueness of Markovian solution and the last section\ncontains application to the ergodic control of a nonlinear stochastic\nheat equation.\n\n\n\\section{Notation}\n Let $E,F$ be Banach spaces, $H$ a Hilbert space, all\nassumed to be defined over the real field and to be separable. The\nnorms and the scalar product will be  denoted $|\\,\\cdot\\,|$,\n$\\langle\\,\\cdot\\,,\\,\\cdot\\,\\rangle$, with subscripts if needed.\nDuality between the dual space $E^*$ and\n $E$ is denoted $\\langle\\,\\cdot\\,,\\,\\cdot\\,\\rangle_{E^*,E}$.\n$L(E,F)$ is the space of linear bounded operators $E\\to F$, with\nthe operator norm.\nThe domain of a linear (unbounded) operator $A$ is denoted $D(A)$.\n\nGiven a bounded function\n$ \\phi: E\\rightarrow \\mathbb{R}$ we denote\n$\\Vert\\phi\\Vert_0=\\sup_{x\\in E}|\\phi(x)|$. If, in addition,\n$\\phi$ is also Lipschitz continuous then\n$\\Vert\\phi\\Vert_{\\hbox{lip}}=\\Vert\\phi\\Vert_0+\n\\sup_{x,x'\\in E,\\,x\\ne x'}|\\phi(x)-\\phi(x')||x-x'|^{-1}$.\n\n\nWe say that a function $F:E\\to F$ belongs to\nthe class ${\\cal G}^1(E,F)$ if it is continuous, has a Gateaux\ndifferential $\\nabla F(x)\\in L(E,F)$ at any point $x  \\in E$, and\nfor every $k\\in E$ the mapping $x\\to \\nabla F(x) k$  is continuous\nfrom $E$ to $F$ (i.e. $x\\to \\nabla F(x) $  is continuous from $E$\nto $L(E,F)$ if the latter space is endowed the strong operator\ntopology). In connection with stochastic equations,\nthe space ${\\cal G}^1$ has been introduced in \\cite{FuTe1},\nto which we refer the reader for further properties.\n\n\n Given a  probability space $\\left(\n\\Omega,\\mathcal{F},\\mathbb{P}\\right) $ with a filtration\n$({\\cal F}_t)_{t\\ge 0}$ we consider the following classes of\nstochastic processes with values in a real separable Banach space\n$K$.\n\n\\begin{enumerate}\n\\item\n$L^p_{\\mathcal{P}}(\\Omega,C([0,T],K))$, $p\\in [1,\\infty)$,\n$T>0$, is the space\nof predictable processes $Y$ with continuous paths\non $[0,T]$\nsuch that\n$$\n|Y|_{L^p_{\\mathcal{P}}(\\Omega,C([0,T],E))}^p\n= \\mathbb E\\, \\sup_{t\\in [0,T]}|Y_t|_K^p<\\infty.\n$$\n\\item\n$L^p_{\\mathcal{P}}(\\Omega,L^2([0,T];K))$, $p\\in [1,\\infty)$,\n$T>0$, is the space\nof predictable processes $Y$ on $[0,T]$ such that\n$$\n|Y|^p_{L^p_{\\mathcal{P}}(\\Omega,L^2([0,T];K))}=\n\\mathbb E\\,\\left( \\int_{0}^{T}|Y_t|_K^2\\,dt\\right)^{p\/2}<\\infty.\n$$\n\n\n\n\\item\n$L_{\\cal P, {\\rm loc}}^2(\\Omega;L^2(0,\\infty;K))$\nis the space\nof predictable processes $Y$ on $[0,\\infty)$ that belong\nto the space $L^2_{\\mathcal{P}}(\\Omega,L^2([0,T];K))$\nfor every $T>0$.\n\\end{enumerate}\n\n\n\n\\section{The forward equation}\nIn a complete probability space $\\left(\n\\Omega,\\mathcal{F},\\mathbb{P}\\right) ,$  we consider the following\nstochastic differential equation with values in a  Banach\nspace $E$:\n\\begin{equation}\n\\left\\{\n\\begin{array}[c]{l} dX_t  =AX_t  d t+F(X_t) dt +GdW_t ,\\text{ \\ \\ \\\n} t\\geq 0, \\\\\nX_0  =x\\in \\, E.\n\\end{array}\n\\right.  \\label{sde}\n\\end{equation}\nWe assume that $E$ is continuously and densely embedded in a\nHilbert space $H$, and that both spaces are real separable.\n\n\n We will work under the following\ngeneral assumptions:\n\n\n\\begin{hypothesis}\n\\label{general_hyp_forward}\n\\begin{enumerate}\n \\item The operator $A$ is the generator of  a strongly\n continuous semigroup of contractions in\n$E$. We assume that the semigroup\n $\\{e^{tA},\\, t\\geq0\\}$\n of bounded linear operators on $E$ generated by $A$\n admits an extension to a strongly\ncontinuous semigroup of bounded linear operators on $H$ that we\ndenote by $\\{S(t),\\, t\\geq 0\\}$.\n\n\n\n\\item $W$ is a cylindrical Wiener process in another real separable\nHilbert space $\\Xi$. Moreover by ${\\cal F}_{t}$ we denote the\n$\\sigma$-algebra generated by $\\{W_s,\\; s\\in [0,t]\\}$ and\n by the sets of ${\\cal F}$ with $\\mathbb P$-measure zero.\n\n\n\n\\item $F:E\\to E$ is continuous and has polynomial growth (that is\nthere exist $c>0, k\\ge 0$ such that $|F(x)|\\leq c (1+|x|^k)$,\n$x\\in E$). Moreover there exists $\\eta>0$\n such that $A+F+\\eta I$ is dissipative.\n\n\n\\item $G$ is a bounded linear operator from $\\Xi$ to $H$. The\nbounded linear, positive and symmetric  operators on $H$ defined\nby the formula\n\\[\nQ_{t}h=\\int_{0}^{t}S(s)GG^{\\ast}S^*(s)h\\,ds,\\qquad t\\geq 0,\\; h\\in\nH,\n\\]\nare assumed to be of trace class in $H$. Consequently we can\ndefine the stochastic convolution\n$$\nW^{A}_t  =\\int_{0}^{t}S(t-s) GdW_s,\\quad t\\geq 0,\n$$\nas a family of $H$-valued stochastic integrals. We assume that the\nprocess $\\{W^{A}_t,\\, t \\geq 0\\}$ admits an $E$-continuous\nversion.\n\\end{enumerate}\n\\end{hypothesis}\n\n\n\n\nWe recall that, for every $x\\in E$, with $x\\neq 0$, the\nsubdifferential of the norm at $x$, $\\partial\\left( |x| \\right) $,\nis the set of functionals $x^{\\ast}\\in E^{\\ast}$ such that\n$\\left\\langle x^{\\ast },x\\right\\rangle _{E^{\\ast},E}=| x| $ and $|\nx^{\\ast}|_{E^{\\ast}}=1$. If $x=0$ then $\\partial\\left( | x|\\right)\n$ is the set of functionals $x^{\\ast}\\in E^{\\ast}$ such that\n$|x^{\\ast}|_{E^{\\ast}}\\leq 1$. The dissipativity assumption on\n$A+F$ can be explicitly stated as follows: for $x,x'\\in\nD(A)\\subset E$ there exists $x^{\\ast } \\in\\partial\\left( \\left|\nx-x' \\right| \\right)  $ such that\n$$\\left\\langle\nx^{\\ast}  ,A( x-x'  )  +F\\left( x \\right) -F\\left( x' \\right)\n\\right\\rangle _{E^{\\ast},E}\n \\leq-\\eta\\left|\nx-x' \\right|.\n$$\n\n\nWe can state the following theorem, see e.g. \\cite{DP1}, theorem\n7.13 and \\cite{DP2}, theorem 5.5.13.\n\\begin{theorem}\n\\label{teo2 forward}Assume that Hypothesis\n\\ref{general_hyp_forward} holds true. Then for every $x\\in E$\nequation (\\ref{sde}) admits a unique mild solution, that is an\nadapted  $E$-valued process with continuous paths  satisfying\n$\\mathbb{P}$-a.s.\n\\[\nX_{t}=e^{  t  A}x+\\int_{0}^{t}e^{ (t-s ) A}F\\left(  X_{s}\\right)\nds+\\int_{0}^{t}e^{(t-s ) A}GdW_{s},\\text{ \\ \\ \\ }t\\geq 0 .\n\\]\n\\end{theorem}\n\n\nWe denote the solution by $X^x  $, $x\\in E$.\n\n\n Now we want to investigate the dependence of\nthe solution on the initial datum.\n\n\n\\begin{proposition}\n\\label{prop lip X} Under Hypothesis \\ref{general_hyp_forward}  it\nholds:\n\\[\n\\left| X_t^{x_1}  -X_t^{x_2} \\right| \\leq e^{-\\eta t  }\\left|\nx_{1}-x_{2}\\right| ,\\text{ }t\\ge 0, \\;\\; x_{1},x_{2}\\in E.\n\\]\n\n\n\\end{proposition}\n\n\n\\begin{proof}\nLet $X_{1}\\left(  t\\right)  =X^{x_1}_{t} $ and $X_{2}\\left(\nt\\right)  =X^{x_2}_{t} $, $x_{1},x_{2}\\in E$. For $i=1,2$ we set\n$X_{i}^{n}\\left(  t\\right) =J_n X_{i}\\left(  t\\right)  $, where\n$J_n\n =n\\left(  nI-A\\right) ^{-1}$. Since $X_{i}^{n}\\left(  t\\right)\n\\in {D}\\left(  A\\right)  $ for every $t\\geq 0 $, and\n\\[\nX_{i}^{n}\\left(  t\\right)  =e^{t A}J_n x_{i}+\\int\n_{0}^{t}e^{\\left(  t-s\\right)  A}J_n F\\left( X_{i}\\left(  s\\right)\n\\right) ds+\\int_{0}^{t}e^{\\left(  t-s\\right) A}J_n GdW_{s},\n\\]\nwe get\n\\[\n\\frac{d}{dt}\\left(  X_{1}^{n}\\left(  t\\right) -X_{2}^{n}\\left(\nt\\right)  \\right)  =A\\left(  X_{1}^{n}\\left(  t\\right)  -X_{2}\n^{n}\\left(  t\\right)  \\right)  +J_n \\left[ F\\left( X_{1}\\left(\nt\\right)  \\right)  -F\\left( X_{2}\\left( t\\right)  \\right)  \\right]\n.\n\\]\nSo, by proposition II.8.5 in \\cite{S}\\ also $\\left|\nX_{1}^{n}\\left( t\\right)  -X_{2}^{n}\\left(  t\\right) \\right| $\nadmits the left and right derivatives with respect to $t$ and\nthere exists $x_{n}^{\\ast }\\left(  t\\right) \\in\\partial\\left(\n\\left| X_{1}^{n}\\left( t\\right) -X_{2}^{n}\\left(  t\\right) \\right|\n\\right)  $ such that the left derivative of $\\left|\nX_{1}^{n}\\left( t\\right) -X_{2}^{n}\\left(  t\\right)  \\right| $\nsatisfies the following\n\\[\n\\frac{d^{-}}{dt}\\left| X_{1}^{n}\\left(  t\\right) -X_{2}^{n}\\left(\nt\\right)  \\right| =\\left\\langle x_{n}^{\\ast}\\left(  t\\right)\n,\\frac{d}{dt}\\left( X_{1}^{n}\\left(  t\\right)  -X_{2}^{n}\\left(\nt\\right)  \\right) \\right\\rangle _{E^{\\ast},E}.\n\\]\nSo we have\n$$ \\begin{array}{ll}\\displaystyle  \\frac{d^{-}}{dt}\\left| X_{1}^{n}\\left(\nt\\right) -X_{2}^{n}\\left( t\\right)  \\right|   & =\\left\\langle\nx_{n}^{\\ast}\\left(  t\\right)  ,A\\left( X_{1}^{n}\\left( t\\right)\n-X_{2}^{n}\\left(  t\\right)  \\right)  +F\\left(  X_{1} ^{n}\\left(\nt\\right)  \\right)  -F\\left(  X_{2}^{n}\\left( t\\right)\n\\right)  \\right\\rangle _{E^{\\ast},E}\\\\\n& \\quad +\\left\\langle x_{n}^{\\ast}\\left(  t\\right)  ,J_n F\\left(\nX_{1}\\left(  t\\right)  \\right)  -F\\left(  X_{1}\n^{n}\\left(  t\\right)  \\right)  \\right\\rangle _{E^{\\ast},E}\\\\\n&  \\quad -\\left\\langle x_{n}^{\\ast}\\left(  t\\right)  ,J_n F\\left(\nX_{2}\\left(  t\\right)  \\right)  -F\\left(  X_{2}\n^{n}\\left(  t\\right)  \\right)  \\right\\rangle _{E^{\\ast},E}\\\\\n&  \\leq-\\eta\\left|\n X_{1}^{n}\\left(  t\\right)  -X_{2}^{n}\\left(\nt\\right)  \\right| +\\left| \\delta_{1}^{n}\\left( t\\right)\n-\\delta_{2}^{n}\\left(  t\\right)  \\right| ,\n\\end{array}$$\nwhere for $i=1,2$ we have set $\\delta_{i}^{n}\\left( t\\right) =J_n\nF\\left(  X_{i}\\left( t\\right) \\right) -F\\left( X_{i}^{n}\\left(\nt\\right)  \\right)  $.\n\n\nMultiplying the above by $e^{\\eta t}$ we get\n$$\\frac{d^{-}}{dt}\\left( e^{\\eta t}\\left| X_{1}^{n}\\left( t\\right)\n-X_{2}^{n}\\left( t\\right) \\right|\\right)\\leq e^{\\eta t} \\left|\n\\delta_{1}^{n}\\left( t\\right) -\\delta_{2}^{n}\\left( t\\right)\n\\right|.$$ We note that $\\delta_{i} ^{n}\\left(  t\\right)  $ tends\nto $0$\\ uniformly in $t\\in \\left[0,T\\right]  $ for arbitrary\n$T>0$. Indeed,\n\\[\n\\delta_{i}^{n}\\left(  t\\right)  =nR\\left(  n,A\\right)  \\left[\nF\\left( X_{i}\\left(  t\\right)  \\right)  -F\\left( X_{i}^{n}\\left(\nt\\right)  \\right)  \\right]  +\\left( nR\\left( n,A\\right) -I\\right)\nF\\left(  X_{i}\\left( t\\right) \\right)  ,\n\\]\nand the convergence to $0$ follows by a classical argument, see\ne.g. the proof of theorem 7.10 in \\cite{DP1}, since\n$X_{i}^{n}\\left( t\\right)  $ tends to $X_{i}\\left(  t\\right)  $\nuniformly in $t\\in\\left[  0,T\\right]  $ and the maps $t\\mapsto\nX_{i}\\left(  t\\right)   $ and $t\\mapsto F\\left( X_{i}\\left(\nt\\right)  \\right)  $ are continuous with respect to $t$.\n\n\n Thus letting  $n\\rightarrow\\infty$ we can conclude\n\\[\n\\left| X_{1}\\left(  t\\right)  -X_{2}\\left(  t\\right) \\right| \\leq\ne^{-\\eta t } \\left| x_{1}-x_{2}\\right| .\n\\]\nand the claim is proved. \\end{proof}\n\n\n$ $\n\n\n\\noindent We will also need the following assumptions.\n\\begin{hypothesis}\n\\label{hyp_W_A F(W_A)} We have $\\sup_{t\\geq 0}\\,\n\\mathbb E\\,|W^A_t|^2<\\infty.$\n\\end{hypothesis}\n\\begin{hypothesis}\n\\label{hyp-convol-determ}  $e^{tA}G\\,(\\Xi)\\subset E$ for all $t>0$\nand $\\displaystyle \\int_0^{+\\infty} |e^{tA} G|_{L(\\Xi,E)} dt <\n\\infty$.\n\\end{hypothesis}\n\n\nWe recall that for arbitrary gaussian random variabile $Y$ with\nvalues in the Banach space $E$, the inequality\n$$\n\\mathbb E \\,\\phi (|Y|-\\mathbb E\\,|Y|)\\le \\mathbb E \\,\\phi (2\\sqrt{\\mathbb E\\,|Y|^2}\\,\\gamma)\n$$\nholds for any convex nonnegative continuous function $\\phi$\non $E$ and\nfor $\\gamma$ a real standard gaussian random variable, see e.g.\n\\cite{kw-woy}, Example 3.1.2. Upon taking $\\phi(x)=|x|^p$,  it\nfollows that for every $p\\ge 2$ there exists $c_p>0$ such that $\\mathbb E\n\\,|Y|^p\\le c_p(\\mathbb E \\,|Y|^2)^{p\/2}$. By the gaussian character of\n$W^A_t$ and the polynomial growth condition on $F$ stated in\nHypothesis \\ref{general_hyp_forward}, point 3, we see that\nHypothesis \\ref{hyp_W_A F(W_A)} entails that for every $p\\ge 2$\n\\begin{equation}\\label{stimegaussunif}\n\\sup_{t\\geq 0} \\mathbb E\\left[ |W^A_t|^p+ |F(W^A_t)|^p \\right] <\\infty.\n\\end{equation}\n\n\n\\begin{proposition}\\label{prop-X-L^p}\nUnder Hypothesis \\ref{general_hyp_forward} it holds, for arbitrary\n$T>0$ and arbitrary $p\\geq 1$\n\\begin{equation}\\label{prop-X-L^p-1}\n\\mathbb E\\sup_{t\\in [0,T]}  |X_t^x|^p \\leq C_{p,T}(1+|x|^p),\\qquad x\\in\nE.\n\\end{equation}\n If, in addition, Hypothesis\n\\ref{hyp_W_A F(W_A)} holds then, for a suitable constant C\n\\begin{equation}\\label{prop-X-L^p-2}\\sup_{t\\geq 0} \\mathbb E |X_t^x| \\leq C(1+|x|)\n,\\qquad x\\in E.\\end{equation} Moreover if, in addition, Hypothesis\n\\ref{hyp-convol-determ} holds, $\\gamma$ is a bounded, adapted,\n$\\Xi$-valued process and $X^{x,\\gamma}$ is the mild solution of\nequation\n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\ndX^{x,\\gamma}_t  =AX^{x,\\gamma}_{t} dt+F(\nX^{x,\\gamma}_{t} ) dt+GdW_{t}+G\\gamma_{t}\\,dt ,\\quad t\\geq 0, \\\\\nX^{x,\\gamma}_{0} =x\\in E.\n\\end{array} \\right.  \\label{sde-gamma}\n\\end{equation}\nthen it is still true that\n\\begin{equation}\\label{rel-estimate-Xgamma}\n \\sup_{t\\geq 0} \\mathbb E |X^{x,\\gamma}_t| \\leq C_{\\gamma}(1+|x|),\\qquad  x\\in E,\n\\end{equation}\nfor a suitable constant $C_{\\gamma}$ depending only on\n a uniform bound for $\\gamma$.\n\\end{proposition}\n\\begin{proof} We let $Z_t=X^x_t-W^A_t$,\n$Z^n_t=J_n Z_t $, then\n$$\\frac{d}{dt } Z^n_t =\nAZ^n_t +J_nF(X^x_t) = AZ^n_t +\\left[F(Z^n_t+J_n W^A_t) - F(J_n\nW^A_t)\\right]+F( W^A_t)+\\delta^n_t\n$$ where $$\\delta^n_t= J_n F(X^x_t)-F(J_n X^x_t)\n+F(J_n W^A_t)-F( W^A_t).$$ Proceeding as in the proof of\nProposition  \\ref{prop lip X} observing  that, for all $t>0$,\n $\\displaystyle \\int_0^{t}|\\delta^n_s| ds \\rightarrow 0$ as $n\\rightarrow\\infty$, we get:\n$$|Z_t|\\leq e^{-\\eta t}|x|+\\int_0^{t} e^{-\\eta (t-s)}\n|F(W^A_s)|ds,\\;\\;\\; \\mathbb{P}-\\hbox{a.s.}$$ and\n(\\ref{prop-X-L^p-2}) follows from (\\ref{stimegaussunif}).\n\n\nIn the case in which $X^x$ is replaced by $X^{x, \\gamma}$ the\nproof is exactly the same just replacing $W^A_t$ by\n$W^{A,\\gamma}_t=W^A_t+\\int_0^t e^{(t-s)A}G\\gamma_s ds$.\n\n\n\nFinally to prove (\\ref{prop-X-L^p-1}) we notice that (see the\ndiscussion in \\cite{masiero}) the process $W^A$ is a Gaussian\nrandom variable with values in $C([0,T],E)$. Therefore by the\npolynomial growth of $F$ we get\n$$ \\mathbb E\\sup_{t\\in [0,T]} \\left[|W^A_t|^p + |F(W^A_t)|^p\\right]\\leq\nC_{p,T}(1+|x|^p),$$ and the claim follows as above.\n\\end{proof}\n\n\n$ $\n\n\n Finally the following result  is proved exactly as\nTheorem 6.3.3. in \\cite{DP2}.\n\\begin{theorem}\\label{ergodicity}\nAssume that Hypotheses \\ref{general_hyp_forward} and\n \\ref{hyp_W_A F(W_A)} hold then equation (\\ref{sde}) has a unique\n invariant measure in $E$ that we will denote by $\\mu$. Moreover\n $\\mu$ is strongly\nmixing (that is, for all $x\\in E$, the law of $X_t^x$ converges\nweakly to $ \\mu$ as\n $t\\rightarrow \\infty$).\n Finally\nthere exists a constant $C>0$ such that for any bounded Lipschitz\nfunction $\\phi: E\\rightarrow \\mathbb{R}$,\n$$\\left|\\mathbb{E}\\phi(X^x_t)-\\int_E \\phi\\, d\\mu \\right|\\leq C(1+|x|)\ne^{-\\eta t \/2} \\Vert\\phi\\Vert_{\\hbox{\\em lip}}.$$\n\\end{theorem}\n\n\n\n\\section{Ergodic BSDEs (EBSDEs)}\n\n\nThis section is devoted to the following type of BSDEs with\ninfinite horizon\n\\begin{equation}\\label{EBSDE}\nY^x_t=Y^x_T +\\int_t^T\\left[\\psi(X^x_\\sigma,Z^x_\\sigma)-\n\\lambda\\right]d\\sigma-\\int_t^T Z^x_\\sigma\\, dW_\\sigma, \\quad 0\\le\nt\\le T <\\infty,\n\\end{equation}\nwhere $\\lambda$ is a real number and is part of the unknowns of\nthe problem; the equation is required to hold for every $t$ and\n$T$ as indicated. On the function $\\psi: E\\times  \\Xi^*\n\\rightarrow {\\mathbb R}$ and assume the following:\n\n\n\\begin{hypothesis}\\label{hypothesisroyer} $ $ There exists\n$K_x, K_z>0$ such that\n$$ |\\psi(x,z)\n-\\psi(x',z')|\\le K_x|x-x'|+ K_z |z-z'|, \\qquad\n x,x'\\in E,\\;\nz,z'\\in\\Xi^*.\n$$\nMoreover $\\psi(\\,\\cdot\\,,0)$ is bounded.  We denote $\\sup_{x\\in E\n}|\\psi(x,0)|$ by $M$.\n\\end{hypothesis}\nWe start by considering an infinite horizon equation with strictly\nmonotonic drift, namely, for $\\alpha>0$, the equation\n\\begin{equation}\\label{bsderoyer}\n{Y}^{x,\\alpha}_t={Y}^{x,\\alpha}_T +\\int_t^T(\\psi(X^{x}_\\sigma,\nZ^{x,\\alpha}_\\sigma)-\\alpha Y^{x,\\alpha}_\\sigma)d\\sigma-\\int_t^T\nZ^{x,\\alpha}_\\sigma dW_\\sigma, \\quad 0\\le t\\le T <\\infty.\n\\end{equation}\n\n\n\n\nThe existence and uniqueness of solution to (\\ref{bsderoyer})\nunder Hypothesis \\ref{hypothesisroyer} was first studied by Briand\nand Hu in \\cite{bh} and then generalized by Royer in \\cite{royer}.\n They have established the following result when $W$ is a finite dimensional\n  Wiener process but the extension to the case in which $W$ is a\n  Hilbert-valued Wiener process is immediate (see also  \\cite{HuTe}).\n\n\n\\begin{lemma}\\label{lemmaroyer} Let us suppose that Hypotheses\n\\ref{general_hyp_forward} and\n\\ref{hypothesisroyer} hold.\n Then\n there  exists a unique solution $(Y^{x,\\alpha},Z^{x,\\alpha})$\n to  BSDE (\\ref{bsderoyer})\nsuch that $Y^{x,\\alpha}$ is a bounded continuous process, and\n$Z^{x,\\alpha}$ belongs to $L_{\\cal P, {\\rm\nloc}}^2(\\Omega;L^2(0,\\infty;\\Xi^*))$.\n\n\nMoreover $|Y^{x,\\alpha}_t|\\leq {M}\/{\\alpha}$, $\\mathbb{P}$-a.s.\nfor all $t\\geq 0$.\n\\end{lemma}\nWe define $$v^{\\alpha}(x)=Y^{\\alpha,x}_0.\n$$\nWe notice that by the above $|v^{\\alpha}(x)|\\leq {M}\/{\\alpha}$ for\nall $x\\in E$. Moreover by the uniqueness of the solution of\nequation (\\ref{bsderoyer}) it follows that\n$Y^{\\alpha,x}_t=v^{\\alpha}(X^x_t)$\n\n\n\nTo establish Lipschitz continuity of $  v^{\\alpha}$ (uniformly in\n$\\alpha$)  we use a Girsanov argument due to P. Briand and Y. Hu,\nsee \\cite{bh}. Here and in the following we use an\ninfinite-dimensional version of the Girsanov formula that can be\nfound for instance in \\cite{DP1}.\n\\begin{lemma}\\label{lemma-lip-v}  Under Hypotheses \\ref{general_hyp_forward}\nand \\ref{hypothesisroyer}  the following\nholds for any $\\alpha>0$:\n$$|v^{\\alpha}(x) - v^{\\alpha}(x')| \\leq \\frac{K_x}{\\eta} |x-x'|,\n\\qquad  x,x'\\in E. $$\n\\end{lemma}\n\\begin{proof} We briefly report the argument for the reader's convenience.\n\n\nWe set $\\tilde{Y}=Y^{\\alpha,x}-Y^{\\alpha,x'}$,\n$\\tilde{Z}=Z^{\\alpha,x}-Z^{\\alpha,x'},$\n$$\\beta_t=\\begin{cases}\n\\frac{\\displaystyle \\psi(X^{x'}_t,Z^{\\alpha,\nx'}_t)-\\psi(X^{x'}_t,Z^{\\alpha,x}_t)} {\\displaystyle\n|Z^{\\alpha,x}_t - Z^{\\alpha,x'}_t|_{\\Xi^*}^2}\\left( Z^{\\alpha,x}_t\n- Z^{\\alpha,x'}_t\\right)^*,& \\hbox{ if } Z^{\\alpha,x}_t \\neq Z^{\\alpha,x'}_t \\\\\n0, & \\hbox{  elsewhere,   }\n \\end{cases}\n$$\n$$f_t=\\psi(X^{x}_t,\nZ^{x,\\alpha}_t)-\\psi(X^{x'}_t, Z^{x,\\alpha}_t).    $$ By\nHypothesis  \\ref{hypothesisroyer}, $\\beta$ is a bounded\n$\\Xi$-valued, adapted process thus there exists a probability\n$\\tilde{\\mathbb{P}}$ under which $\\tilde{W_{t}}=\\int_0^{t} \\beta_s\nds + W_{t}$ is a cylindrical $\\Xi$-valued Wiener process for\n${t}\\in [0,T]$. Then  $(\\tilde{Y},\\tilde{Z})$ verify, for all\n$0\\le t\\le T <\\infty$,\n\\begin{equation}\\label{bsderoyer-girsanov}\n\\tilde{Y}_t=\\tilde{Y}_T -\\alpha \\int_t^T \\tilde{Y}_\\sigma d\\sigma\n+\\int_t^T f_{\\sigma}d\\sigma- \\int_t^T \\tilde{Z}_\\sigma\nd\\tilde{W}_\\sigma.\n\\end{equation}\nComputing  $d (e^{-\\alpha t}\\tilde{Y}_t)$, integrating over\n$[0,T]$, estimating the absolute value and finally taking the\nconditional expectation\n $\\tilde{\\mathbb{E}}^{\\mathcal{F}_t}$ with respect to\n$\\tilde{\\mathbb{P}}$ and $\\mathcal{F}_t$ we get:\n$$  |\\tilde{Y}_t| \\leq  e^{-\\alpha(T-t)}  \\tilde{\\mathbb{E}}^{\\mathcal{F}_t}\n|   \\tilde{Y}_T |+\n \\tilde{\\mathbb{E}}^{\\mathcal{F}_t}\n  \\int_{t}^T  e^{-\\alpha(s-t)} |f_s| ds  $$\nNow we recall that $ \\tilde{Y}$ is bounded   and that $|f_t|\\leq\nK_x |X^{x}_t-X^{x'}_t|\\leq K_x e^{-\\eta t}|x-x'|$ by Proposition\n\\ref{prop lip X}. Thus if  $T\\rightarrow \\infty$ we get $\n|\\tilde{Y}_t| \\leq K_x (\\eta+\\alpha)^{-1}e^{\\alpha t} |x-x'|  $\nand the claim follows  setting $t=0$.\n\\end{proof}\n\n\n$ $\n\n\n\\noindent By the above Lemma if we set\n$$\\overline{v}^{\\alpha}(x)= {v}^{\\alpha}(x)- {v}^{\\alpha}(0),$$\nthen $   | \\overline{v}^{\\alpha}(x)|\\leq    K_x \\eta^{-1}|x|$ for\nall $x\\in E$ and all $\\alpha>0$. Moreover by Lemma\n\\ref{lemmaroyer} $\\alpha |{v}^{\\alpha}(0)|\\leq M$.\n\n\n \\noindent  Thus by a diagonal procedure   we can construct a\n sequence $\\alpha_n\\searrow 0$ such that for all $x$ in a\ncountable dense subset $D\\subset E$\n     \\begin{equation}\\label{def-of-lambda}\n    {\\overline{v}}^{\\alpha_n}(x)\\rightarrow \\overline{v}(x),\\qquad\n\\alpha_n v^{\\alpha_n}(0)\\rightarrow \\overline{\\lambda},\n    \\end{equation}\nfor a suitable  function $  \\overline{v}: D \\rightarrow\n\\mathbb{R}$ and for a suitable real number $\\overline{\\lambda}$.\n\n\n Moreover, by Lemma \\ref{lemma-lip-v}, $ | \\overline{v}^{\\alpha}(x)-\n\\overline{v}^{\\alpha}(x')|\\leq    K_x \\eta^{-1}|x-x'|$   for all\n$x,x'\\in E$ and all $\\alpha>0$. So $\\overline{v}$ can be extended\nto a Lipschitz function defined on the whole $E$ (with Lipschitz\nconstant $K_x \\eta^{-1}  $) and\n\\begin{equation}\\label{def-of-v} {\\overline{v}}^{\\alpha_n}(x)\\rightarrow\n\\overline{v}(x),\\qquad  x\\in E.\\end{equation}\n\n\n\\begin{theorem} \\label{main-EBSDE} Assume Hypotheses\n\\ref{general_hyp_forward}  and\n\\ref{hypothesisroyer} hold. Moreover let $\\bar \\lambda$ be the\nreal number in (\\ref{def-of-lambda}) and define $\\bar Y^x_t= \\bar\nv(X^x_t)$ (where $\\overline{v}$ is the Lipschitz function with\n$\\overline{v}(0)=0$ defined in (\\ref{def-of-v})). Then there\nexists a process $\\overline{Z}^{x}\\in L_{\\cal P, {\\rm\nloc}}^2(\\Omega;L^2(0,\\infty;\\Xi^*))$\n  such that $\\mathbb{P}$-a.s.  the EBSDE\n (\\ref{EBSDE}) is satisfied by\n $(\\bar Y^x,\\bar Z^x, \\bar \\lambda)$ for all $0\\leq t\\leq T$.\n\n\nMoreover  there exists a measurable function  $\\overline{\\zeta}:\nE\\rightarrow \\Xi^*$ such that\n$\\overline{Z}^{x}_t=\\overline{\\zeta}(X^x_t)$.\n\\end{theorem}\n\n\n\\begin{proof} Let $\\overline{Y}^{x,\\alpha}_t={Y}^{x,\\alpha}_t-v^{\\alpha}(0)=\n\\overline{v}^{\\alpha}({X}^{x}_t)$.  Clearly we have,\n$\\mathbb{P}$-a.s.,\n\\begin{equation}\\label{equation-proof-main-1}\n \\overline{Y}^{x,\\alpha}_t=\\overline{Y}^{x,\\alpha}_T +\\int_t^T(\\psi(X^{x}_\\sigma,\nZ^{x,\\alpha}_\\sigma)-\\alpha \\overline{Y}^{x,\\alpha}_\\sigma-\\alpha\n{v}^{\\alpha}(0))d\\sigma -\\int_t^T Z^{x,\\alpha}_\\sigma dW_\\sigma,\n\\quad 0\\le t\\le T <\\infty.\n\\end{equation}\nSince $|\\bar v^{\\alpha}(x)|\\leq K_x|x|\/\\eta $, inequality\n(\\ref{prop-X-L^p-1})   ensures that\n$\\mathbb{E}\\sup_{t\\in[0,T]}\\left[\\sup_{\\alpha>0}\n|\\overline{Y}^{x,\\alpha}_t|^2\\right]< +\\infty$  for any $T>0$.\nThus, if we define $\\overline{Y}^x=\\overline{v}(X^x)$, then by\ndominated convergence theorem\n$$\\mathbb{E} \\int_0^T |\\overline{Y}^{x,\\alpha_n}_t -\\overline{Y}^{x}_t|^2 dt\n \\rightarrow 0\\quad \\hbox{and}\\quad\n\\mathbb{E} |\\overline{Y}^{x,\\alpha_n}_T-\\overline{Y}^{x}_T|^2\n\\rightarrow 0\n$$\nas $n\\rightarrow \\infty$ (where $\\alpha_n \\searrow 0$ is a\nsequence for which (\\ref{def-of-lambda}) and (\\ref{def-of-v})\nhold).\n\n\n\nWe claim now that there exists $\\overline{Z}^{x}\\in L_{\\cal P,\n{\\rm loc}}^2(\\Omega;L^2(0,\\infty;\\Xi^*))$ such that\n $$\\mathbb{E} \\int_0^T |{Z}^{x,\\alpha_n}_t -\\overline{Z}^{x}_t|_{\\Xi^*}^2 dt\n  \\rightarrow 0$$\nLet  $\\tilde{Y}={\\bar Y}^{x,\\alpha_n}-{\\bar Y}^{x,\\alpha_m}$,\n$\\tilde{Z}={Z}^{x,\\alpha_n}-{Z}^{x,\\alpha_m}$. Applying It\\^o's\nrule to $\\tilde{Y}^2$ we get by standard computations\n$$\\tilde{Y}^2_0+\\mathbb{E}\\int_0^T |\\tilde{Z}_t|_{\\Xi^*}^2 dt\n=\\mathbb{E}{\\tilde Y}^2_T + 2\\mathbb{E}\\int_0^T \\tilde \\psi_t\n\\tilde Y_t dt -2 \\mathbb{E}\\int_0^T \\left[\\alpha_n\n{Y}^{x,\\alpha_n}_t - \\alpha_m {Y}^{x,\\alpha_m}_t\\right] \\tilde\nY_t\\,dt\n$$\nwhere  $\\tilde\n\\psi_t=\\psi(X^x_t,Z^{x,\\alpha_n}_t)-\\psi(X^x_t,Z^{x,\\alpha_m}_t)$.\nWe notice that $|\\tilde\\psi_t| \\leq  K_z|\\tilde Z _t|$ and\n$\\alpha_n |{Y}^{x,\\alpha_n}_t|\\leq M$. Thus\n$$\n\\mathbb{E}\\int_0^T |\\tilde{Z}_t|_{\\Xi^*}^2 dt \\leq c\\left[\n\\mathbb{E} (\\tilde Y^x_T)^2 +\\mathbb{E}\\int_0^T (\\tilde{Y}^x_t)^2\ndt +\\mathbb{E}\\int_0^T |\\tilde{Y}^x_t| dt \\right].$$ It follows\nthat the sequence $\\{{Z}^{x,\\alpha_m}\\}$ is Cauchy in\n$L^2(\\Omega;L^2(0,T;\\Xi^*))$ for all $T>0$ and our claim is\nproved.\n\n\nNow we can  pass to the limit as $n\\rightarrow \\infty$ in equation\n(\\ref{equation-proof-main-1}) to obtain\n\\begin{equation}\\label{equation-proof-main-2}\n \\overline{Y}^{x}_t=\\overline{Y}^{x}_T +\\int_t^T(\\psi(X^{x}_\\sigma,\n\\overline{Z}^{x}_\\sigma)-\\overline{\\lambda })d\\sigma-\\int_t^T\n\\overline{Z}^{x}_\\sigma dW_\\sigma, \\quad 0\\le t\\le T <\\infty.\n\\end{equation}\nWe notice  that the above equation also ensures continuity of the\ntrajectories of $\\overline{Y}$ It remains now to prove that we can\nfind a measurable function $\\bar \\zeta:E\\rightarrow \\Xi^*$ such\nthat\n $\\overline{Z}^{x}_t=\\bar \\zeta (X^x_t)$, $\\mathbb{P}$-a.s. for almost every $t\\geq 0$.\n\n\nBy a general argument, see for instance \\cite{Fu}, we know that\nfor all $\\alpha>0$ there exists  $\\zeta^{\\alpha}:E\\rightarrow\n\\Xi^*$ such that\n ${Z}^{x,\\alpha}_t=\\zeta^{\\alpha} (X^x_t)$, $\\mathbb{P}$-a.s.\n for almost every $t\\geq 0$.\n\n\nTo construct $\\zeta$ we need some more regularity of the processes\n${Z}^{x,\\alpha}$ with respect to $x$.\n\n\nIf we compute $d ({Y}^{x,\\alpha}_t-{Y}^{x',\\alpha}_t)^2$ we get by\nthe Lipschitz character of $\\psi$:\n$$ \\begin{array} {l}\n\\displaystyle \\mathbb{E}\\int_0^T\n|Z^{x,\\alpha}_t-Z^{x',\\alpha}_t|_{\\Xi^*}^2 dt \\leq \\mathbb{E}\n(v^{\\alpha}(X^x_T)- v^{\\alpha}(X^{x'}_T))^2\n  \\\\\n\\quad +  \\displaystyle \\mathbb{E}\\int_0^T\n\\left(K_x|X^x_s-X^{x'}_s|\n+K_z|Z^{x,\\alpha}_s-Z^{x',\\alpha}_s|\\right)\n\\left|v^{\\alpha}(X^x_s)- v^{\\alpha}(X^{x'}_s)\\right| ds\n  \\end{array}$$\nBy the Lipschitz continuity of $v^{\\alpha}$ (uniform in $\\alpha$)\nthat of $\\psi$  and Proposition \\ref{prop lip X}  we immediately\nget:\n\\begin{equation}\\label{lip-of-Z}\n    \\mathbb{E}\\int_0^T |Z^{x,\\alpha}_t-Z^{x',\\alpha}_t|_{\\Xi^*}^2 dt \\leq c |x-x'|^2.\n\\end{equation}\nfor a suitable constant $c$ (that may depend on $T$).\n\n\nNow we fix an arbitrary $T>0$ and, by a diagonal procedure  (using\nseparability of $E$) we  construct a subsequence\n$(\\alpha_n')\\subset (\\alpha_n)$ such that $\\alpha_n' \\searrow 0$\nand\n$$\\mathbb{E}\\int_0^T |Z^{x,\\alpha_n'}_t-Z^{x',\\alpha_m'}_t|_{\\Xi^*}^2 dt \\leq 2^{-n}\n$$ for all $m\\geq n$ and for all $x\\in E$.\nConsequently  $Z^{x,\\alpha_n'}_t\\rightarrow \\overline{Z}^x_t$,\n$\\mathbb{P}$-a.s. for a.e. $t\\in [0,T]$. Then we set:\n$$\\bar \\zeta(x)=\\left\\{\\begin{array}{ll} \\lim_n \\zeta^{\\alpha_n'}(x),\n& \\hbox{ if the limit exists in }\\Xi^*,\\\\\n0,  & \\hbox{ elsewhere.}\\end{array}\\right.$$ Since\n$Z^{x,\\alpha_n'}_t= \\zeta^{\\alpha_n'}(X^x_t)\\rightarrow\n\\overline{Z}^{x}_t$   $\\mathbb{P}$-a.s. for a.e. $t\\in [0,T]$ we\nimmediately get that, for all $x\\in E$, the process $X^x_t$\nbelongs   $\\mathbb{P}$-a.s. for a.e. $t\\in [0,T]$ to the set where\n$\\lim_n \\zeta^{\\alpha_n'}(x)$ exists  and consequently\n $\\overline{Z}^{x}_t=\\bar \\zeta(X^x_t)$.\n\\end{proof}\n\\begin{remark}\\begin{em} We notice that the solution we\nhave constructed above has the following ``linear growth''\nproperty with respect to $X$: there exists $c>0$ such that,\n$\\mathbb{P}$-a.s.,\n\\begin{equation}\\label{growt-of-Y}\n|\\overline{Y}^x_t|\\leq c |X^x_t|  \\hbox{ for all $t\\geq 0$}.\n\\end{equation}\n\\end{em}\n\\end{remark}\nIf we require similar conditions then we immediately obtain\nuniqueness of $\\lambda$.\n\\begin{theorem}\\label{th-uniq-lambda} Assume that,\nin addition to Hypotheses \\ref{general_hyp_forward}, \\ref{hyp_W_A\nF(W_A)} and \\ref{hypothesisroyer}, Hypothesis\n\\ref{hyp-convol-determ} holds as well.  Moreover suppose that, for\nsome $x\\in E$, the triple $(Y',Z',\\lambda')$ verifies\n$\\mathbb{P}$-a.s. equation\n (\\ref{EBSDE}) for all $0\\leq t\\leq T$,\nwhere\n $Y'$ is a progressively measurable continuous process, $Z'$ is a process\n in $L_{\\cal P, {\\rm loc}}^2(\\Omega;L^2(0,\\infty;\\Xi^*))$ and\n $\\lambda'\\in \\mathbb{R}$.\n Finally assume that there exists $c_x>0$ (that may depend\n on $x$) such that\n$\\mathbb{P}$-a.s.\n$$\n |Y'_t|\\leq c_x (|X^x_t|+1) , \\hbox{ for all $t\\geq 0$}.\n$$ Then $\\lambda'=\\bar \\lambda$.\n\\end{theorem}\n\\begin{proof}\nLet $\\tilde \\lambda=\\lambda'-\\lambda$, $\\tilde\nY=Y'-\\overline{Y}^x$, $\\tilde Z=Z'-\\overline{Z}^x$. By easy\ncomputations:\n$$\\tilde \\lambda=T^{-1}\\left[\\tilde Y_T-\\tilde Y_0\\right]+T^{-1}\\int_0^T  \\tilde Z_t \\gamma_t dt\n-T^{-1}\\int_0^T  \\tilde Z_t dW_t$$ where\n$$\\gamma_t:=\\begin{cases}   \\frac{\\displaystyle \\psi(X^{x}_t,Z'_t)-\\psi(X^{x}_t,\\overline{Z}^{x}_t)}{\\displaystyle |Z'_t - \\overline{Z}_t|_{\\Xi^*}^2}\\left(Z'_t - \\overline{Z}_t \\right)^*,& \\hbox{ if } Z'_t \\neq \\overline{Z}_t, \\\\\n0, & \\hbox{  elsewhere   },\n \\end{cases}\n$$\nis a bounded $\\Xi$-valued progressively measurable process. By the\nGirsanov Theorem  there exists a probability measure\n$\\mathbb{P}_{\\gamma}$ under which $W^{\\gamma}_t=-\\int_0^t \\gamma_s\nds+W_t$, $t\\in [0,T]$, is a cylindrical Wiener process in $\\Xi$.\nThus computing expectation with respect to $\\mathbb{P}_{\\gamma}$\nwe get\n$$\\tilde \\lambda=T^{-1}\\mathbb{E}^{\\mathbb{P}_{\\gamma}}\n\\left[\\tilde Y_T-\\tilde Y_0\\right].$$ Consequently, taking into\naccount (\\ref{growt-of-Y}),\n \\begin{equation}\\label{eq-proof-uniq-lambda}\n|\\tilde \\lambda|\\leq c T^{-1}\\mathbb{E}^{\\mathbb{P}_{\\gamma}}\n(|X^x_T|+1)+ c T^{-1}(|x|+1)\n   \\end{equation}\nWith respect to $W^{\\gamma}$, $X^x$ is the mild solution of\n$$\n\\left\\{\n\\begin{array}{l}\ndX^{x,\\gamma}_t  =AX^{x,\\gamma}_{t} dt+F( X^{x,\\gamma}_{t} )\ndt+GdW^{\\gamma}_{t}+G\\gamma_{t}\\,dt ,\n\\quad t\\geq 0 \\\\\nX^{x,\\gamma}_{0} =x\\in E.\n\\end{array} \\right.\n$$\nand by (\\ref{rel-estimate-Xgamma}) we get\n$\\sup_{T>0}\\mathbb{E}^{\\mathbb{P}_{\\gamma}}|X^x_T|<\\infty$. So if\nwe let $T\\rightarrow\\infty$ in  (\\ref{eq-proof-uniq-lambda}) we\nconclude that $\\tilde\\lambda=0$.\n\\end{proof}\n\n\n\\begin{remark} \\em\nThe solution to EBSDE (\\ref{EBSDE}) is, in general,  not unique.\nIt is evident that the equation is invariant with respect to\naddition of a constant to $Y$ but we can also construct an\narbitrary number of solutions that do not differ only by a\nconstant (even if we require them to be bounded). On the contrary\nthe solutions we construct are not Markovian.\n\n\n\nIndeed, consider the equation:\n\\begin{equation}\\label{eq:nouniqueness}\n-dY_t=[\\psi(Z_t)-\\lambda]dt-Z_tdW_t.\n\\end{equation}\nwhere $W$ is a standard brownian motion  and\n$\\psi:\\mathbb{R}\\rightarrow \\mathbb{R}$ is differentiable bounded\nand  has bounded derivative.\n\n\nOne solution is $Y=0;Z=0;\\lambda=\\psi(0)$ (without loss of\ngenerality we can suppose that $\\psi(0)=0$).\n\n\nLet now  $\\phi:\\mathbb{R}\\rightarrow \\mathbb{R}$ be an arbitrary\ndifferentiable function bounded and  with bounded derivative. The\nfollowing BSDE on $[t,T]$ admits a solution:\n$$\\left\\{\\begin{array}{rcl}\n-dY_s^{x,t}&=&\\psi(Z_s^{x,t})ds-Z_s^{x,t}dW_s,\\\\\nY_T^{x,t}&=&\\phi(x+W_T-W_t).\n\\end{array}\\right.$$\nIf we define $u(t,x)=Y_t^{x,t}$ then both $u$ and $\\nabla u$ are\nbounded.  Moreover if $\\tilde{Y}_t=Y_t^{0,0}=u(t,W_t),\\\n\\tilde{Z}_t=Z_t^{0,0}=\\nabla u(t,W_t)$ then\n$$\\left\\{\\begin{array}{rcl}\n-d\\tilde{Y}_t&=&\\psi(\\tilde{Z}_t)dt-\\tilde{Z}_tdW_t,\\quad\nt\\in [0,T],\\\\\n\\tilde{Y}_T&=&\\phi(W_T).\n\\end{array}\\right.$$\n Then it is enough to extend with\n$\\tilde{Y}_t=\\tilde{Y}_T,\\ \\tilde{Z}_t=0$ for $t>T$ to construct a\nbounded solution to (\\ref{eq:nouniqueness}).\n\\end{remark}\n\\begin{remark}\\em The existence result in Theorem  \\ref{main-EBSDE}\ncan be easily extended to the case of $\\psi$ only satisfying\nthe conditions\n$$ |\\psi(x,z)\n-\\psi(x',z)|\\le K_x|x-x'|,\\quad  |\\psi(x,0)|\\le M,\n\\quad |\\psi(x,z)|\n\\le K_z(1+|z|).\n$$\nIndeed we can construct a sequence $\\{\\psi_n : n\\in \\mathbb{N}\\}$\nof functions Lipschitz in $x$ and $z$ such that for all $x,x'\\in\nH$, $z \\in \\Xi^*$, $n\\in \\mathbb{N}$\n$$ |\\psi^n(x,z)\n-\\psi^n(x',z)|\\le K'_x|x-x'|;\\quad  |\\psi^n(x,0)|\\leq M';\\quad\n\\lim_{n\\rightarrow \\infty}|\\psi^n(x,z) -\\psi(x,z)|=0.\n$$\nThis can be done by projecting $x$ to the subspaces generated by\na basis in $\\Xi^*$ and then regularizing by the standard\nmollification techniques,  see \\cite{FuTeBE}.\nWe know that if $(\\bar Y^{x,n}, \\bar Z^{x,n},\\lambda_n)$ is the\nsolution of the EBSDE (\\ref{EBSDE}) with $\\psi$ replaced by\n$\\psi^n$ then $\\bar Y^{x,n}_t=\\bar v^n(X^x_t)$ with\n$$ |\\bar v^n(x)\n-\\bar v^n(x')|\\le \\dfrac{K'_x}{\\eta}|x-x'|;\\quad  \\bar v^n(0)=0\n;\\quad |\\lambda_n|\\leq M'\n$$\nThus we can assume (considering, if needed, a subsequence) that\n$\\bar v^n(x) \\rightarrow \\bar v(x)$ and $\\lambda_n \\rightarrow\n\\lambda$.\nThe rest of the proof is identical to the one of Theorem\n\\ref{main-EBSDE}.\n\\end{remark}\n\n\n\\section{Differentiability}\n\n\n\nWe are now interested in the differentiability of the\nsolution to the EBSDE (\\ref{EBSDE}) with respect to $x$.\n\n\\begin{theorem}\\label{th-diff} Assume that Hypotheses\n\\ref{general_hyp_forward} and\n\\ref{hypothesisroyer} hold. Moreover assume that $F$ is of class\n${\\cal G}^1(E,E)$ with $\\nabla F$ bounded on bounded sets of $E$.\nFinally assume that $\\psi$ is of class ${\\cal G}^1(E\\times\n\\Xi^*,E)$. Then the function $\\overline{v}$ defined in\n(\\ref{def-of-v}) is of class ${\\cal G}^1(E,\\mathbb{R})$.\n\\end{theorem}\n\\begin{proof} In \\cite{masiero} it is proved that for arbitrary $T>0$ the map\n$x\\rightarrow X^x$ is of class $\\mathcal{G}^1$ from $E$ to\n$L^p_{\\mathcal{P}}(\\Omega,C([0,T],E))$. Moreover Proposition\n\\ref{prop lip X} ensures that for all $h\\in E$,\n\\begin{equation}\\label{proof-diff-estim-nabla-X}\n |\\nabla X^x_t h|\\leq e^{-\\eta t}|h|,\\quad \\hbox{$\\mathbb{P}$-a.s.,\n for all $t\\in [0,T]$}.\n\\end{equation}\nUnder the previous conditions one can proceed exactly as\nin Theorem 3.1 of \\cite{HuTe} to\nprove that for all $\\alpha >0$ the map $v^{\\alpha}$ is of class\n$\\mathcal{G}^1$.\n\n\n$ $\n\n\nThen we consider again\nequation (\\ref{bsderoyer}):\n$$\n{Y}^{x,\\alpha}_t ={Y}^{x,\\alpha}_T\n+\\int_t ^T(\\psi(X^{x}_\\sigma, Z^{x,\\alpha}_\\sigma)-\\alpha\nY^{x,\\alpha}_\\sigma)d\\sigma-\\int_t ^T Z^{x,\\alpha}_\\sigma\ndW_\\sigma, \\quad 0\\le t \\le T <\\infty,\n$$\nwe recall that ${Y}^{x,\\alpha}_T={v}^{\\alpha}(X^{x}_T)$,\n and apply again \\cite{masiero} (see Proposition 4.2 there) and \\cite{FuTe1}\n (see Proposition 5.2 there) to obtain that for all $\\alpha >0 $\n the map $x\\rightarrow Y^{x,\\alpha}$ is of class $\\mathcal{G}^1$\n from $E$ to $L^2_{\\mathcal{P}}(\\Omega,C([0,T],\\mathbb{R}))$ and the map\n$x\\rightarrow Z^{x,\\alpha}$  is of class $\\mathcal{G}^1$ from $E$\nto $L^2_{\\mathcal{P}}(\\Omega,L^2([0,T],\\Xi^*))$. Moreover for all\n$h\\in E$ it holds (for all $t>0$ since $T$ was arbitrary)\n$$\n-d\\nabla Y^{\\alpha,x}_th=[\\nabla_x\\psi(X^x_t,Z_t^{\\alpha,x})\n\\nabla X_t^xh+\\nabla_z\\psi(X^x_t,Z_t^{\\alpha,x})\\nabla\nZ_t^{\\alpha,x}h-\\alpha\\nabla Y^{\\alpha,x}_th]dt\n-\\nabla Z^{\\alpha,x}h dW_t.\n$$\nWe also know that $|Y^{\\alpha,x}_t|\\le {M}\/{\\alpha}$. Now we set\n$$U^{\\alpha,x}_t=e^{\\eta t}\\nabla Y^{\\alpha,x}_t h,\n\\quad V^{\\alpha,x}=e^{\\eta t}\\nabla Z^{\\alpha,x}_t h.$$ Then\n$(U^{\\alpha,x},V^{\\alpha,x})$ satisfies the following BSDE:\n\\begin{eqnarray*}\n-dU^{\\alpha,x}_t&=&[e^{\\eta t}\\nabla_x\\psi(X^x_t,Z_t^{\\alpha,x})\n\\nabla X_t^x-(\\alpha+\\eta)U^{\\alpha,x}_t +\\nabla_z\n\\psi(X^x_t,Z_t^{\\alpha,x}) V^{\\alpha,x}_t]dt-V^{\\alpha,x}_tdW_t.\n\\end{eqnarray*}\nBy (\\ref{proof-diff-estim-nabla-X}) and the usual Girsanov\nargument (recall the  $\\nabla_x \\psi$ and $\\nabla_z \\psi$ are\nbounded),\n$$|U^{\\alpha,x}_t|\\le \\frac{c}{\\alpha+\\eta},\\;\n\\forall t\\geq 0,\\; \\hbox{$\\mathbb P-$a.s. $\\qquad$  i.e. } \\qquad\n|\\nabla Y_t^{x,\\alpha}|\\le e^{-\\eta t}\\frac{c}{\\alpha+\\eta}.$$\nMoreover, consider the limit equation, with unknown\n$(U^{x},V^{x})$,\n\\begin{equation}\\label{eq:limit}\n-dU^x_t=[e^{\\eta t}\\nabla_x\\psi(X^x_t,\\bar Z_t^{x}) \\nabla\nX_t^x-\\eta U^x_t+\\nabla_z\\psi(X^x_t,\\bar Z_t^{x}) V^x]dt-V^xdW_t,\n\\end{equation}\nwhich, since $|e^{\\eta t}\\nabla_x\\psi \\nabla_x X_t^x|$ is bounded,\nhas a unique  solution such that $U^x$ is bounded and $V^x$\nbelongs to $L_{\\cal P, {\\rm loc}}^2(\\Omega;L^2(0,\\infty;\\Xi^*))$\n(see \\cite{bh} and \\cite{royer}).\n\n\nWe know that for a suitable sequence $\\alpha_n \\searrow 0$,\n$$\\bar v^{\\alpha}(x)= Y^{x,\\alpha_n}_0-Y^{0,\\alpha_n}_0\\rightarrow \\bar{Y}^x_0,$$\nand we claim now that\n$$  \\nabla \\bar v^{\\alpha_n}(x)=\\nabla Y_0^{x,\\alpha_n}=U_0^{x,\\alpha_n}\n\\rightarrow U_0^x.$$ To prove this we introduce the finite horizon\nequations: for $t\\in [0,N]$,\n$$\\begin{cases}\n& -dU_t^{x,\\alpha,N}=[e^{\\eta t}\\nabla_x\\psi(X^x_t,Z_t^{x,\\alpha})\n \\nabla X_t^x-(\\alpha+\\eta)U_t^{x,\\alpha,N}\n +\\nabla_z \\psi (X^x_t,Z_t^{x,\\alpha}) V_t^{x,\\alpha,N}]dt\\\\\n& \\qquad\\qquad\\qquad - V^{x,\\alpha,N}_tdW_t,\\\\\n& U_N^{x,\\alpha,N}=0.\n\\end{cases}$$\n$$\\begin{cases}& -dU_t^{x,N}=[e^{\\eta t}\\nabla_x\\psi(X^x_t,\\bar Z_t^{x})\n\\nabla X_t^x-(\\alpha+\\eta)U_t^{x,N}\n+\\nabla_z \\psi (X^x_t,\\bar Z^{x}_t) V_t^{x,N}]dt-V^{x,N}_tdW_t,\\\\\n& U_N^{x,N}=0.\n\\end{cases}$$\nSince $\\displaystyle \\mathbb E\\int_0^N |Z^{x,\\alpha_n}_s-\\bar Z^{x}_s|^2\nds\\rightarrow 0$ it is easy to verify that, for all fixed $N>0$,\n$U_0^{x,\\alpha_n,N}\\rightarrow U_0^{x,N}$.\n\n\nOn the other side a standard application of Girsanov Lemma gives\n see   \\cite{HuTe},\n$$|U_0^{x,\\alpha_n,N}-U_0^{x,\\alpha_n}|\\le \\frac{c}{\\alpha_n+\\eta}e^{-\\eta N}, \\qquad |U_0^{x,N}-U_0^{x}|\\le \\frac{c}{\\eta}e^{-\\eta N}.$$\nfor a suitable constant $c$.\n\n\nThus a standard argument implies $U_0^{x,\\alpha_n}\\rightarrow\nU_0^{x}$. An identical argument also ensures continuity of\n$U_0^{x}$ with respect to $x$ (also taking into account\n\\ref{lip-of-Z}). The proof is therefore completed.\n\\end{proof}\n\n\n$ $\n\n\nAs  usual in the theory of markovian  BSDEs, the differentiability\nproperty allows to identify the process  $\\bar Z^x$ as a function\nof the process $X^x$. To deal with our Banach space setting we\nneed to make the following extra assumption:\n\n\\begin{hypothesis}\\label{Hyp-masiero}\nThere exists a Banach space $\\Xi_0$, densely and continuously\nembedded in $\\Xi$, such that $G\\, (\\Xi_0) \\subset \\Xi$  and $G\n:\\Xi_0 \\rightarrow E$ is continuous.\n\\end{hypothesis}\n\nWe note that this condition is satisfied in most applications. In\nparticular it is trivially true in the special case $E=H$ just by\ntaking $\\Xi_0=\\Xi$, since  $G$ is assumed to be a linear bounded\noperator from $\\Xi$ to $H$. The following is proved in\n\\cite[Theorem 3.17]{masiero}:\n\n\n\\begin{theorem}\n \\label{theorem-identif-Z}\nAssume that Hypotheses \\ref{general_hyp_forward},\n \\ref{hypothesisroyer} and \\ref{Hyp-masiero} hold.\nMoreover assume that $F$ is of class ${\\cal G}^1(E,E)$ with\n$\\nabla F$ bounded on bounded subsets of $E$ and $\\psi$ is of\nclass ${\\cal G}^1(E\\times \\Xi^*,E)$. Then $\\bar Z^x_t=\\nabla \\bar\nv(X^x_t)G$, $\\mathbb{P}$-a.s. for a.e. $t\\geq 0$.\n\\end{theorem}\n\\begin{remark} \\label{precision}\\begin{em}\nWe notice that $\\nabla \\bar v(x)G\\xi$ is only defined for $\\xi\\in\n\\Xi_0$ in general, and the conclusion of\nTheorem \\ref{theorem-identif-Z} should be stated more precisely\nas follows: for $\\xi\\in\n\\Xi_0$ the equality $Z^x_t\\xi=\\nabla \\bar v(X^x_t)G\\xi$\nholds $\\mathbb{P}$-a.s. for almost every $t\\geq 0$. However,\nsince $\\bar Z^x$\nis a process with values in $\\Xi^*$, and more specifically\na process in  $\nL^2_{\\mathcal{P}}(\\Omega,L^2([0,T],\\Xi^*))$, it follows that\n$\\mathbb P$-a.s. and\nfor almost every\n$t$  the\noperator $\\xi \\rightarrow \\nabla \\bar v(X^x_t)G\\xi$ can be\nextended to a bounded linear operator defined on the whole $\\Xi$.\nEquivalently,\nfor almost every\n$t$ and for almost all $x\\in E$  (with respect to the law of $X_t$)\nthe linear\noperator $\\xi \\rightarrow \\nabla \\bar v(x)G\\xi$ can be\nextended to a bounded linear operator defined on the whole $\\Xi$\n(see also Remark 3.18 in \\cite{masiero}).\n\\end{em}\n\\end{remark}\n\\begin{remark}\\label{boundedpsibar}\n\\begin{em} The above representation together with the fact that\n$\\bar v$ is Lipschitz with Lipschitz constant $K_x\\eta^{-1}$\nimmediately implies that, if $F$ is of class ${\\cal G}^1(E,E)$ and\n$\\psi$ is of class ${\\cal G}^1(E\\times \\Xi^*,E)$, then $|\\bar\n{Z}^x_t|_{\\Xi_0^*}\\leq K_x\\eta^{-1} |G|_{L(\\Xi_0,E)}$ for all $x\\in\nE$, $\\mathbb{P}$-a.s. for almost every $t\\geq 0$. Consequently we\ncan construct $\\bar \\zeta$ in Theorem \\ref{main-EBSDE}\n in such a way that it is bounded in the\n$\\Xi_0^*$ norm by $K_x\\eta^{-1} |G|_{L(\\Xi_0,E)}$.\n\n\nOnce this is proved we can extend the result to the case in which\n$\\psi$ is no longer differentiable but only Lipschitz, namely\nwe can prove than even in this case the process $\\bar\n{Z}^x$ is bounded. Indeed if we\nconsider a sequence $\\{\\psi_n : n\\in \\mathbb{N}\\}$ of functions of\nclass ${\\cal G}^1(E\\times \\Xi^*,E)$ such that for all $x,x'\\in H$,\n$z,z'\\in \\Xi^*$, $n\\in \\mathbb{N}$,\n$$ |\\psi_n(x,z)\n-\\psi_n(x',z')|\\le K_x|x-x'|+ K_z |z-z'|;\\quad \\lim_{n\\rightarrow\n\\infty}|\\psi_n(x,z) -\\psi(x,z)|=0.\n$$\nWe know that if $(\\bar Y^{x,n}, \\bar Z^{x,n},\\lambda_n)$ is the\nsolution of the EBSDE (\\ref{EBSDE}) with $\\psi$ replaced by\n$\\psi_n$ then $|\\bar {Z}^{x,n}_t|_{\\Xi_0^*}\\leq K_x\\eta^{-1}\n|G|_{L(\\Xi_0,E)}$. Then as we did above we can show (showing that the\ncorresponding equations with monotonic generator converge\nuniformly in $\\alpha$) that $\\mathbb{E}\\int_0^T|\\bar {Z}^{x,n}_t\n-\\bar {Z}^{x}_t|_{\\Xi_0^*}^2dt\\rightarrow 0$ and the claim\nfollows.\n\n\nWe also notice that by the same argument we also have $ |\\bar\n\\zeta^{\\alpha}(x)|_{\\Xi_0^*}\\leq K_x\\eta^{-1} |G|_{L(\\Xi_0,E)}$,\n$\\forall \\alpha>0$.\n\\end{em}\n\\end{remark}\nNow we introduce  the Kolmogorov semigroup corresponding to $X$:\n for  measurable and bounded  $\\phi:\nE\\rightarrow \\mathbb{R}$ we define\n\\begin{equation}\\label{def-of-p}\nP_t[\\phi](x)=\\mathbb{E}\\, \\phi(X^x_t)\\qquad t\\ge 0,\\, x\\in E.\n\\end{equation}\n\\begin{definition}\\label{strongly-feller}\nThe semigroup $(P_t)_{t\\geq 0}$ is called strongly Feller if for\nall $t>0$ there exists $k_t$ such that for all measurable and\nbounded $\\phi: E\\rightarrow \\mathbb{R}$,\n$$|\nP_t[\\phi](x)- P_t[\\phi](x')|\\leq k_t \\Vert\\phi\\Vert_0 |x-x'|,\n\\qquad x,x'\\in E,\n$$\nwhere $\\Vert\\phi\\Vert_0=\\sup_{x\\in E}|\\phi(x)|$.\n\\end{definition}\n\nThis terminology is somewhat different from the classical one\n(namely, that $P_t$ maps  measurable  bounded functions into\ncontinuous ones,\n for all\n$t>0$), but it will be convenient for us.\n\n\n\n\\begin{definition}\\label{gen-diss} We say that $F$\nis genuinely dissipative if there exist $\\epsilon>0$ and $c>0$\nsuch that, for all $x,x'\\in E$, there exists $z^*\\in \\partial\n|x-x'|$ such that $<z^*,F(x)-F(x')>_{E^*,E}\\leq c\n|x-x'|^{1+\\epsilon}$.\n\\end{definition}\n\n\n\n\\begin{lemma}\\label{lemma-SF-dissip}\nAssume that Hypotheses \\ref{general_hyp_forward} and \\ref{hyp_W_A\nF(W_A)} hold.\nIf the Kolmogorov\nsemigroup $(P_t)$ is strongly Feller then for all bounded\nmeasurable $\\phi: E\\rightarrow\\mathbb{R}$,\n$$\\left|P_t[\\phi](x)-\\int_E \\phi(x)\\mu (dx)\\right|\n\\leq c e^{-\\eta (t\/4)}(1+|x|)\\Vert\\phi\\Vert_0.$$\nIf in addition $F$ is genuinely dissipative then\n$$\\left|P_t[\\phi](x)-\\int_E \\phi(x)\\mu (dx)\\right|\n\\leq c e^{-\\eta (t\/4)}\\Vert\\phi\\Vert_0.$$\n\\end{lemma}\n\\begin{proof} We fix $\\epsilon >0$. For $t>2$ we have,\nby Theorem \\ref{ergodicity},\n$$\n\\begin{array}{r}\\displaystyle\\left|P_t[\\phi](x)-\\int_E \\phi(x)\\mu (dx)\\right|=\n\\left|P_{t-1}[P_1[\\phi]](x)-\\int_E P_{1}[\\phi](x)\\mu (dx)\\right|\n\\leq C(1+|x|)\ne^{-\\eta t \/4} \\Vert P_{1}[\\phi]\\Vert_{\\hbox{lip}}\\\\\n\\displaystyle \\leq C(1+|x|) e^{-\\eta t \/4} k_{1}\\Vert\\phi\\Vert_0,\n\\end{array}$$\nand the first claim follows since $\\left|P_t[\\phi](x)-\\int_E\n\\phi(x)\\mu (dx)\\right|\\leq 2  \\Vert\\phi\\Vert_0$.\n\n\nIf now $F$ is genuinely dissipative then in \\cite{DP2}, Theorem\n6.4.1 it is shown that\n$$\\left|\\mathbb{E}\\phi(X^x_t)-\\int_E \\phi\\, d\\mu \\right|\\leq\nC e^{-\\eta t \/2} \\Vert\\phi\\Vert_{\\hbox{lip}}$$ and the second\nclaim follows by the same argument.\n\\end{proof}\n\nWe are now able to state and prove two corollaries\nof Theorems  \\ref{th-diff}  and \\ref{theorem-identif-Z}.\n\n\n\\begin{corollary}\\label{characterization of lambda}\nAssume that Hypotheses \\ref{general_hyp_forward}, \\ref{hyp_W_A\nF(W_A)}, \\ref{hypothesisroyer} and \\ref{Hyp-masiero} hold.\nMoreover assume that $F$ is of class $\\mathcal{G}^1$ with $\\nabla\nF$ bounded on bounded subsets of $E$, and that $\\psi$ is bounded\non each set $E\\times B$, where $B$ is any ball of  $\\Xi_0^*$.\nFinally assume that the Kolmogorov semigroup $(P_t)$ is strongly\nFeller.\n\n\n\nThen the following holds:\n$$\\lambda=\\int_E \\psi(x,\\bar \\zeta(x))\\mu (dx),$$\nwhere $\\mu$ is the unique invariant measure of $X$.\n\\end{corollary}\n\\begin{proof} First notice that $\\overline{\\psi}:=\n\\psi(\\,\\cdot\\, , \\bar\\zeta(\\,\\cdot\\,))$ is bounded, by\nRemark \\ref{boundedpsibar}.\n Then\n$$T^{-1}\\mathbb{E}[\\bar Y ^x_0-\\bar Y ^x_T]=\nT^{-1}\\mathbb E \\int_0^T\\left  (\\psi(X^x_t,\\bar \\zeta( X^x_t))- \\int_E\n\\bar \\phi\\, d\\mu \\right)dt+ \\left(\\int_E \\bar \\phi\\, d\\mu\n-\\lambda\\right).$$ We know that $T^{-1}\\mathbb{E}[\\bar Y ^x_0-\\bar\nY ^x_T]\\rightarrow 0$, by the argument\nin Theorem \\ref{th-uniq-lambda}.\nMoreover by the first conclusion of Lemma\n\\ref{lemma-SF-dissip}\n$$ T^{-1}\\mathbb E \\int_0^T\\left  (\\psi(X^x_t,\\bar \\zeta( X^x_t))-\n\\int_E \\bar \\phi\\, d\\mu \\right)dt \\rightarrow 0,$$ and the claim\nfollows. \\end{proof}\n\n\\begin{corollary}\\label{boundedness of v}\nIn addition to the assumptions of Corollary \\ref{characterization\nof lambda} suppose that $F$ is genuinely dissipative. Then $\\bar\nv$ is bounded.\n\\end{corollary}\n\\begin{proof}\nLet $(Y^{x,\\alpha},Z^{x,\\alpha})$ be the solution of\n(\\ref{bsderoyer}). We know that $Y^{x,\\alpha}_t=v^{\\alpha}(X^x_t)$\nand $Z^{x,\\alpha}_t= \\zeta^{\\alpha}(X^x_t)$  with $v^{\\alpha}$\nLipschitz uniformly with respect to $\\alpha$ and $\\zeta^{\\alpha}$\nbounded in $\\Xi^*$ uniformly with respect to $\\alpha$. Let\n$\\psi^{\\alpha}=\\psi(\\,\\cdot\\,,\\bar \\zeta^{\\alpha}(\\,\\cdot\\,))$.\nUnder the present assumptions we conclude that also the maps\n$\\psi^{\\alpha}$ as well are bounded in $\\Xi^*$ uniformly with\nrespect to $\\alpha$.\n\n\nComputing $d (e^{-\\alpha t} \\bar Y^{x\\alpha}_t)$ we obtain,\n$$Y^{x,\\alpha}_0=\\mathbb{E} e^{-\\alpha T} Y^{x,\\alpha}_T+\n\\mathbb{E} \\int_0^T  e^{-\\alpha t} \\psi^{\\alpha} (X^x_t)dt,$$ and\nfor $T\\rightarrow\\infty$,\n$$Y^{x,\\alpha}_0=\n\\mathbb{E} \\int_0^\\infty  e^{-\\alpha t} \\psi^{\\alpha} (X^x_t)dt.$$\nSubtracting  to both sides $\\alpha^{-1}\\int_E\n\\psi^{\\alpha}(x)\\mu(dx)$ we obtain\n$$\\left|Y^{x,\\alpha}_0-\\alpha^{-1}\\int_E  \\psi^{\\alpha}(x)\\mu(dx)\\right|=\n\\left| \\int_0^\\infty  e^{-\\alpha t} \\left[P_t[\\psi^{\\alpha}]\n(x)-\\int_E  \\psi^{\\alpha}(x)\\mu(dx)\\right]dt\\right|\\leq 4c\n\\eta^{-1} \\Vert \\psi^\\alpha\\Vert_0 $$ where the last inequality\ncomes from the second conclusion of Lemma \\ref{lemma-SF-dissip}.\n\n\nThus  $\\left|Y^{x,\\alpha}_0-Y^{0,\\alpha}_0\\right| \\leq 8 c\n\\eta^{-1} \\Vert \\psi^\\alpha\\Vert_0 $ and the claim follows since\nby construction $Y^{x,\\alpha}_0-Y^{0,\\alpha}_0  \\rightarrow \\bar v\n(x)$.\n\\end{proof}\n\\section{Ergodic Hamilton-Jacobi-Bellman equations}\nWe briefly show here that if $\\bar Y_0^x=\\bar v(x)$ is of class\n${\\cal G}^1$ then the couple $(v,\\lambda)$  is a mild solution of\nthe following ``ergodic''  Hamilton-Jacobi-Bellman equation:\n\\begin{equation}\n\\mathcal{L}v(x)\n+\\psi\\left( x,\\nabla v(x)  G\\right)  = \\lambda, \\quad x\\in E,  \\label{hjb}%\n\\end{equation}\nWhere  linear operator $\\mathcal{L}$ is formally defined by\n\\[\n\\mathcal{L}f\\left(  x\\right)  =\\frac{1}{2}Trace\\left(\nGG^{\\ast}\\nabla ^{2}f\\left(  x\\right)  \\right)  +\\langle Ax,\\nabla\nf\\left(  x\\right) \\rangle_{E,E^{\\ast}}+\\langle F\\left(  x\\right)\n,\\nabla f\\left(  x\\right) \\rangle_{E,E^{\\ast}},\n\\]\nWe notice that we can define the transition semigroup\n $(P_t)_{t\\geq 0}$ corresponding to $X$ by the formula (\\ref{def-of-p})\nfor all measurable functions $\\phi:E\\to\\mathbb{ R}$ having\npolynomial growth, and we notice that $\\mathcal{L}$ is the formal\ngenerator of $(P_t)_{t\\geq 0}$.\n\n\n Since we are dealing with an elliptic equation it is natural to consider\n$(v,\\lambda)$ as a mild solution of equation (\\ref{hjb}) if and\nonly if, for arbitrary $T>0$, $v(x)$ coincides with the  mild\n solution $u(t,x)$ of the corresponding parabolic equation\n having $v$ as a terminal condition:\n\\begin{equation}\\left\\{\n\\begin{array}{l}\n \\dfrac{\\partial u(t,x)}{\\partial t}+\\mathcal{L}u\\left(  t,x\\right)\n+\\psi\\left(  x,\\nabla u\\left(  t,x\\right)  G\\right)\n -\\lambda=0, \\quad t\\in [0,T],\\; x\\in E,  \\\\ \\\\\nu(T,x)=v(x), \\quad  x\\in E.\n \\end{array}\\right. \\label{hjb-parab}\n\\end{equation}\nThus we are led to the following definition (see also\n\\cite{FuTe-ell}):\n\\begin{definition}\n\\label{defsolmildkolmo} A pair $(v,\\lambda)$ ($v: E\\rightarrow\n\\mathbb{R}$ and $\\lambda\\in \\mathbb{R}$) is a mild solution of the\nHamilton-Jacobi-Bellman equation (\\ref{hjb}) if the following are\nsatisfied:\n\n\n\\begin{enumerate}\n\\item $v\\in\\mathcal{G}^{1}\\left(  E,\\mathbb R \\right)  $;\n\n\n\\item  there exists $C>0$ such that $\\left|  \\nabla v\\left(\nx\\right)\nh\\right|  \\leq C\\left|  h\\right|  _{E}\\left(  1+\\left|  x\\right|  _{E}%\n^{k}\\right)  $ for every  $x,h\\in E$ and some positive integer\n$k$;\n\n\n\\item for $0\\le t\\le T$  and $x\\in E$,\n\\begin{equation}\nv(x)=P_{T-t}\\left[  v\\right]  \\left(  x\\right)\n+\\int_{t}^{T}\\left(P_{s -t }\\left[  \\psi(\\cdot,\\nabla v\\left(\n\\cdot\\right)  G)\\right]  \\left( x\\right) -\\lambda \\right)   \\,ds.\n\\label{mild sol hjb}\n\\end{equation}\n\n\n\\end{enumerate}\n\\end{definition}\n\n\n\n\nIn the right-hand side of (\\ref{mild sol hjb}) we notice\noccurrence of the term $\\nabla v\\left( \\cdot\\right)  G$, which is\nnot well defined as a function $E\\to\\Xi^*$, since $G$ is not\nrequired to map $\\Xi$ into $E$.\nThe situation is similar to Remark \\ref{precision}.\nIn general,\n for $x \\in E$, $\\nabla \\bar\nv(x)G\\xi$ is only defined for $\\xi\\in \\Xi_0$.\nIn (\\ref{mild sol hjb}) it is implicitly required\nthat, $\\mathbb P$-a.s. and\nfor almost every\n$t$,  the\noperator $\\xi \\rightarrow \\nabla \\bar v(X^x_t)G\\xi$ can be\nextended to a bounded linear operator defined on the whole $\\Xi$.\nNoting that\n$$\nP_{t }\\left[  \\psi(\\cdot,\\nabla v\\left( \\cdot\\right)  G)\\right]\n\\left( x\\right) = \\mathbb E \\, \\psi(X^x_{t},\\nabla v\\left( X^x_{t}\\right)\nG)\n$$\nthe equation (\\ref{mild sol hjb}) is now meaningful.\n\nUsing the results for the parabolic case,  see \\cite{masiero}, we\nget existence of the mild solution of equation (\\ref{hjb})\nwhenever we have proved that the function\n$\\bar v$ in Theorem \\ref{main-EBSDE} is differentiable.\n\n\n\\begin{theorem}\\label{th-EHJB}\nAssume that Hypotheses \\ref{general_hyp_forward},\n\\ref{hypothesisroyer} and \\ref{Hyp-masiero} hold.\nMoreover assume that $F$ is of class ${\\cal G}^1(E,E)$ with\n$\\nabla F$ bounded on bounded subsets of $E$ and $\\psi$ is of\nclass ${\\cal G}^1(E\\times \\Xi^*,E)$.\n\n\nThen $(\\bar v,  \\bar\\lambda)$ is a   mild solution of the\nHamilton-Jacobi-Bellman equation (\\ref{hjb}).\n\nConversely, if $(v,\\lambda)$ is a   mild solution of\n (\\ref{hjb}) then, setting  $ Y^x_t=\nv(X^x_t)$ and ${Z}^{x}_t=\n\\nabla v( X^x_t)  G$,\nthe triple\n $( Y^x, Z^x,  \\lambda)$ is a solution of\n  the EBSDE\n (\\ref{EBSDE}).\n\n\\end{theorem}\n\n\n\n\\section{Optimal ergodic control}\n\\label{optcontr}\n\nAssume that Hypothesis \\ref{general_hyp_forward} holds and let\n$X^x$ denote the solution to equation (\\ref{sde}).\n Let $U$ be a separable\n metric space. We define a control $u$ as an\n$({\\cal F}_t)$-progressively measurable $U$-valued process.  The cost\n corresponding to a given control\nis defined in the following way. We assume that the functions\n$R:U\\rightarrow \\Xi^*$ and $L:E\\times U \\rightarrow \\mathbb R$ are\nmeasurable and satisfy, for some constant $c>0$,\n\\begin{equation}\\label{condcosto}\n|R(u)|\\leq c,\\quad |L(x,u)|\\leq c, \\quad |L(x,u)-L(x',u)|\\leq\nc\\,|x-x'|,\\qquad u\\in U,\\,x,x'\\in E.\n\\end{equation}\nGiven an arbitrary control $u$ and $T>0$, we introduce the\nGirsanov density\n$$ \\rho_T^u=\\exp\\left(\\int_0^T R(u_s)dW_s\n-\\frac{1}{2}\\int_0^T |R(u_s)|_{\\Xi^*}^2 ds\\right)$$ and the\nprobability $\\mathbb P_T^u=\\rho_T^u\\mathbb P$ on ${\\cal F}_T$. The\nergodic cost  corresponding to $u$ and the starting point $x\\in E$\nis\n\\begin{equation}\\label{def-ergodic-cost}\n  J(x,u)=\\limsup_{T\\rightarrow\\infty}\\frac{1}{T} \\mathbb\nE^{u,T}\\int_0^T L(X_s^x,u_s)ds,\n\\end{equation}\nwhere $\\mathbb E^{u,T}$  denotes expectation with respect to\n$\\mathbb P_T^u$. We notice that $W_t^u=W_t-\\int_0^t R(u_s)ds$ is a\nWiener process on $[0,T]$ under $\\mathbb P^u$ and that\n$$dX_t^x=(AX_t^x+F(X_t^x))dt+G(dW_t^u+R(u_t)dt),\n\\quad t\\in [0,T]$$ and this justifies our formulation of the\ncontrol problem. Our purpose is to minimize the cost over all\ncontrols.\n\n\n To this purpose we first define the Hamiltonian in the\nusual way\n\\begin{equation}\\label{defhamiton}\n\\psi(x,z)=\\inf_{u\\in U}\\{L(x,u)+z R(u)\\},\\qquad x\\in E,\\,z\\in\n\\Xi^*,\n\\end{equation}\nand we remark that  if, for all $ x,z$, the  infimum is attained\nin (\\ref{defhamiton}) then there exists a measurable function\n$\\gamma:E\\times \\Xi^*\\rightarrow U$ such that\n$$\\psi(x,z)=l(x,\\gamma(x,z))+z R(\\gamma(x,z)).$$\nThis follows from an application of Theorem 4 of \\cite{McS-War}.\n\nWe notice that under  the present assumptions $\\psi$ is a\nLipschitz function and $\\psi(\\cdot,0)$ is bounded (here the fact\nthat $R$ depends only on $u$ is used). So if we assume Hypotheses\n\\ref{general_hyp_forward} and \\ref{hyp_W_A F(W_A)} then in Theorem\n\\ref{main-EBSDE} we have constructed, for every $x\\in E$, a triple\n\\begin{equation}\\label{richiamoebsde}\n(\\bar Y^x,\\bar Z^x, \\bar \\lambda)= (\\bar v (X^x),\\bar \\zeta(X^x),\n\\bar \\lambda)\n\\end{equation} solution to\n the EBSDE\n (\\ref{EBSDE}).\n\n\n\n\n\n\n\n\\begin{theorem}\\label{Th-main-control}\nAssume that Hypotheses \\ref{general_hyp_forward}, \\ref{hyp_W_A\nF(W_A)} and  \\ref{hyp-convol-determ} hold, and that\n(\\ref{condcosto}) holds as well.\n\n\nMoreover suppose that, for some $x\\in E$, a triple $(Y,Z,\\lambda)$\nverifies $\\mathbb{P}$-a.s. equation\n (\\ref{EBSDE}) for all $0\\leq t\\leq T$,\nwhere\n $Y$ is a progressively measurable continuous process, $Z$ is a process\n in $L_{\\cal P, {\\rm loc}}^2(\\Omega;L^2(0,\\infty;\\Xi^*))$ and\n $\\lambda\\in \\mathbb{R}$.\n Finally assume that there exists $c_x>0$ (that may depend\n on $x$) such that\n$\\mathbb{P}$-a.s.\n$$\n |Y_t|\\leq c_x (|X^x_t|+1) , \\hbox{ for all $t\\geq 0$}.\n$$\n\n\n\nThen the following holds:\n\\begin{enumerate}\n \\item[(i)] For arbitrary control\n $u$ we have $J(x,u)\\ge \\lambda=\\bar\\lambda,$\nand the equality holds if and only if $L(X_t^x,u_t)+Z_t\nR(u_t)=\\psi(X_t^x,Z_t)$, $\\mathbb P$-a.s. for almost every $t$.\n\n\n\\item[(ii)] If the  infimum is attained in (\\ref{defhamiton}) then\nthe control $\\bar u_t=\\gamma(X_t^x,Z_t)$ verifies $J(x,\\bar u)=\n\\bar\\lambda.$\n\\end{enumerate}\n\n\nIn particular, for the solution (\\ref{richiamoebsde}) mentioned\nabove, we have:\n\\begin{enumerate}\n \\item[(iii)] For arbitrary control\n $u$ we have $J(x,u)=\\bar\\lambda$ if and only if\n$L(X_t^x,u_t)+\\bar\\zeta (X_t^x) R(u_t)=\\psi(X_t^x,\\bar \\zeta\n(X_t^x))$, $\\mathbb P$-a.s. for almost every $t$. \\item[(iv)] If the\ninfimum is attained in (\\ref{defhamiton}) then the control $\\bar\nu_t=\\gamma(X_t^x,\\bar\\zeta (X_t^x))$ verifies $J(x,\\bar u)=\n\\bar\\lambda.$\n\\end{enumerate}\n\n\n\\end{theorem}\n\n\n\\begin{remark}\\em\n\\begin{enumerate}\n \\item\nThe equality $\\lambda=\\bar\\lambda$ clearly follows from Theorem\n\\ref{th-uniq-lambda}. \\item Points $(iii)$ and $(iv)$ are\nimmediate consequences of $(i)$ and $(ii)$. \\item The conclusion\nof point $(iv)$ is that there exists an optimal control in\nfeedback form, with the optimal feedback given by the function\n$x\\mapsto \\gamma(x,\\bar\\zeta (x))$. \\item Under the conditions of\nTheorem \\ref{th-EHJB}, the pair $(\\bar v, \\bar \\lambda)$ occurring\nin (\\ref{richiamoebsde}) is a mild solution of the\nHamilton-Jacobi-Bellman equation (\\ref{hjb}). \\item It follows\nfrom the proof below that if $\\limsup$ is changed into $\\liminf$\nin the definition (\\ref{def-ergodic-cost}) of the cost, then the\nsame conclusions hold, with the obvious modifications, and the\noptimal value is given by $\\bar\\lambda$ in both cases.\n\\end{enumerate}\n\\end{remark}\n\n\n\\begin{proof}\n As $(Y,{Z}, \\bar\\lambda)$ is a solution of the\nergodic BSDE, we have\n\\begin{eqnarray*}\n-d{Y}_t&=&[\\psi(X_t^x,{Z}_t)-\\bar\\lambda]dt-{Z}_tdW_t\\\\\n&=&[\\psi(X_t^x,{Z}_t)- \\bar\\lambda]dt-{Z}_tdW_t^u-{Z}_t R(u_t)dt,\n\\end{eqnarray*}\nfrom which we deduce that\n\\begin{eqnarray*}\n\\bar\\lambda&=&\\frac{1}{T}\\mathbb E^{u,T}[Y_T-Y_0]\n+\\mathbb E^{u,T}\\frac{1}{T}\\int_0^T[\\psi(X_t^x,{Z}_t)-{Z}_t r(u_t)-L(X_t^x,{Z}_t)]dt\\\\\n& &+\\frac{1}{T}\\mathbb E^{u,T}\\int_0^T L(X_t^x,{Z}_t)dt.\n\\end{eqnarray*}\n\n\nThus\n$$\\frac{1}{T}\\mathbb E^{u,T}\\int_0^T L(X_t^x,{Z}_t)dt\\ge\n \\frac{1}{T}\\mathbb E^{u,T}[Y_0-Y_T]+\\bar\\lambda.$$\nBut by (\\ref{rel-estimate-Xgamma}) we have\n$$|\\mathbb E^{u,T} Y_T|\\le c\\mathbb E^{u,T}(|X_T^x|+1)\\le c(1+|x|).$$\nConsequently $T^{-1}\\mathbb E^{u,T}[Y_0-Y_T]\\rightarrow 0,$ and\n$$\\limsup_{T\\rightarrow\\infty } \\frac{1}{T}\\mathbb E^{u,T}\\int_0^T L(X_t^x,{Z}_t)dt\n\\ge \\bar\\lambda.$$\n\n\nSimilarly, if $L(X_t^x,u_t)+ Z_t R(u_t)=\\psi(X_t^x,Z_t)$,\n$$\\frac{1}{T}\\mathbb E^{u,T}\\int_0^T L(X_t^x,{Z}_t)dt=\n\\frac{1}{T}\\mathbb E^{u,T}[Y_0-Y_T]+\\bar\\lambda,$$ and the claim\nholds.\n\\end{proof}\n\n\n\n\n\\section{Uniqueness}\\label{sec-uniq}\nWe wish now to adapt the argument in \\cite{GoMa} in order to\nobtain uniqueness of markovian solutions to the EBSDE. This will\nbe done by a control thoretic interpretation the requires that the\nMarkov process related to the state equation with continuous\nfeedback enjoys recurrence properties. In this section we assume\n\\begin{equation}\\label{addizionali}\nE=H \\qquad\\hbox{ and }\\qquad F  \\hbox{ is bounded.}\n\\end{equation}\n\n\\noindent We recall here a result due to \\cite{seid} on recurrence\nof solution to SDEs.\n\\begin{theorem}\\label{th-rec-seidler}\nConsider\n\\begin{equation}\\label{eq:u}\nd{X}_t=(A{X}_t+g({X}_t))dt+GdW_t.\n\\end{equation}\nwhere $g: H \\rightarrow H$ is bounded and weakly continuous (that\nif $x\\rightarrow\\<\\xi,g(x)\\>$ is continuous for all $\\xi\\in H$).\nLet\n$$Q_t=\\int_0^t e^{sA}GG^*e^{sA^*}ds.$$\nand assume the following\n\\begin{enumerate}\n \\item $\\sup_{t\\ge 0} \\hbox{Trace}\\,(Q_t)<\\infty$;\n\\item $Q_t$ is injective for $t>0$; \\item  $ e^{t A}(H)\\subset\n(Q_t)^{1\/2}(H)$ for $t>0$; \\item $\\int_0^t\n|Q_s^{-1\/2}e^{sA}|ds<\\infty$ for $t>0$; \\item there exists\n$\\beta>0$ such that $\\int_0^t s^{-\\beta}\\,\n\\hbox{Trace}\\,(S(s)S(s)^*)\\, ds<\\infty$\n  for $t>0$.\n\\end{enumerate}\nThen,  for all $T>0$, equation (\\ref{eq:u}) admits a martingale\nsolution on $[0,T]$, unique in law. The associated transition\nprobabilities $P(t,x,T,\\cdot)$ on $H$ ($0\\le t\\le T, x\\in H$)\nidentify a   recurrent  Markov process on $[0,\\infty)$.\n\\end{theorem}\n\n\nConsider now the ergodic control problem with state equation:\n$$d{X}^{x,u}_t=(A{X}^{x,u}_t+F({X}^{x,u}_t)+GR(u_t))dt+GdW_t, \\ X_0^{x,u}=x,$$\nand cost\n$$\\limsup_{T\\to\\infty}\\frac{1}{T}\\,\n\\mathbb E\\int_0^T l(X_s,u_s)ds$$ where  $R:U\\rightarrow\n\\Xi$ is continuous  and bounded.\n\n\nWe restrict ourselves to the class of controls given by continuous\nfeedbacks, i.e. given arbitrary\n continuous $u: H\\rightarrow U$ (called feedback) we define the\n corresponding trajectory as the solution of\n$$d{X}^{x,u}_t=(A{X}^{x,u}_t+F({X}^{x,u}_t))dt+G(R(u(X_t^{x,u}))dt+dW_t),\n\\ X_0^{u,x}=x.$$\nWe notice that for all $T>0$ there exists a weak solution $X^{x,u}$ of\nthis equation, and it is unique in law.\n\n\n\n$ $\n\n\n We set as usual\n$$\\psi(x,z)=\\inf_{u\\in U}\\{L(x,u)+zR(u)\\},$$\nand assume that $\\psi$ is continuous and there exists a continuous\n$\\gamma:H\\times\\Xi\\rightarrow U$ such that\n$$\\psi(x,z)=L(x,\\gamma(x,z))+zR(\\gamma(x,z)).$$\n\n\n\\begin{theorem}\\label{th-uniqueness}\nSuppose (\\ref{addizionali})\nand suppose that the assumptions of Theorem\n\\ref{th-rec-seidler} hold.\nLet $(v,\\zeta,\\lambda)$ with $v:H\\rightarrow \\mathbb{R}$  continuous,\n$\\zeta:H\\rightarrow \\mathbb{R}$  continuous, and $\\lambda$ a real number\nsatisfy the following conditions:\n\n\n\\begin{enumerate}\n \\item $|v(x)|\\le c|x|$;\n\\item\nfor an arbitrary  filtered probability space with\na Wiener process\n$(\\hat{\\Omega},\\hat{\\mathcal{F}},\n\\{\\hat{\\mathcal{F}}_t\\}_{t>0},\\hat{\\mathbb{P}},\\{\n\\hat{W}_t\\}_{t>0})$ and\nfor any solution of\n$$d\\hat{X}_t=(A\\hat{X}_t+F(\\hat{X}_t))dt+Gd\\hat{W}_t,\\qquad t\\in [0,T],$$\nsetting  $Y_t=v(\\hat{X}_t),\\\nZ_t=\\zeta(\\hat{X}_t)$, we have\n$$-dY_t=[\\psi(\\hat{X}_t,Z_t)-\\lambda]dt-Z_tdW_t\\quad t\\in [0,T].$$\n\\end{enumerate}\nLet\n$$\\tau_r^T=\\inf \\{s\\in [0,T]:|X_s^{u,x}|< r\\},$$\nwith the convention $\\tau_r^T=T$ if the indicated set\nis empty,\nand\n$$J(x,u)=\\limsup_{r\\rightarrow 0}\\limsup_{T\\rightarrow \\infty}\n\\mathbb E\\int_0^{\\tau_r^T} [\\psi(X_s^{x,u},u(X_s^{x,u}))-\\lambda]ds.$$\nThen\n$$v(x)=\\inf_u J(x,u),$$\nwhere the infimum (that is a minimum) is taken over all continuous\nfeedbacks $u$.\n\\end{theorem}\n\\begin{proof} Let $u:H\\to U$ be continuous.\nWe notice that $X^{x,u}$ solves on $[0,T]$:\n$$dX_t^{x,u}=(AX_t^{x,u}+F(X_t^{x,u}))ds+Gd\\tilde{W}_t^u,\\ t\\in [0,T],$$\nwhere $\\tilde{W}_t=\\int_0^t R(u(X_r^{x,u})dr+W_t$ is a Wiener\nprocess on $[0,T]$ under a suitable probability $\\hat{\\mathbb{P}}^{u,T}$.\n\n\nTherefore $Y_t=v(X_t^{x,u})$, $Z_t=\\zeta(X_t^{x,u})$ satisfy:\n$$\n-dY_t=[\\psi(X_t^{x,u},u(X_t^{x,u}))-\\lambda]dt-Z_t\nR(u(X_t^{x,u}))]dt-Z_tdW_t.$$ Integrating in $[0,\\tau_r^T]$ we get\n$$v(x)=\\mathbb E(v(X_{\\tau_r^T}^{x,u}))+\\mathbb E\\int_0^{\\tau_r^T}\n[\\psi(X_s^{u,x},u(X_s^{x,u}))-\\lambda-Z_s R(X_s^{x,u})]ds.$$ Thus,\n\\begin{equation}\\label{eq:x}\nv(x)\\le\\mathbb E(v(X_{\\tau_r^T}^{x,u}))+\\mathbb E\\int_0^{\\tau_r^T}\n[L(X_s^{u,x},u(X_s^{x,u}))-\\lambda]ds.\n\\end{equation}\nNow\n\\begin{eqnarray*}\n|\\mathbb E(v(X_{\\tau_r^T}^{x,u}))|\\le c\\mathbb\nE|X^{x,u}_{\\tau_r^T}|&\\le& c r+ (\\mathbb\nE(|X_T^{x,u}|^2))^{1\/2}(\\mathbb P(\\tau_r^T=r))^{1\/2}\\\\ &\\le & c r+\nc (\\mathbb P(\\tau_r^T=r))^{1\/2}\n\\end{eqnarray*}\nNotice that $\\mathbb P(\\tau_r^T=r)=\\tilde{\\mathbb P}(\\inf_{t\\in\n[0,T]}|\\tilde{X}_t|\\geq r),$ where $\\tilde{X}$ is the Markov\nprocess on the whole $[0,+\\infty)$ corresponding to the equation\n(\\ref{eq:u}) with $g=F(\\cdot)+GR(u(\\cdot))$.\n\n\n$ $\n\n\n\\noindent Since $\\tilde{X}$ is recurrent, for all $ r>0$ it holds\n$\\tilde{\\mathbb P}(\\inf_{t\\in [0,T]}|\\tilde{X}_t|>r)\\rightarrow 0$\nas $T\\rightarrow \\infty.$ Thus\n$$\\limsup_{r\\rightarrow 0}\\limsup_{T\\rightarrow \\infty}|\n\\mathbb E(v(X_{\\tau_r^T}^{x,u}))|\\rightarrow 0.$$\nHence,\n$$v(x)\\le \\limsup_{r\\rightarrow 0}\\limsup_{T\\rightarrow \\infty}\n\\mathbb E\\int_0^{\\tau_r^T}\n[l(X_s^{x,u},u(X_s^{x,u}))-\\lambda]ds.$$ The proof is completed\nnoticing that if  $u$ is chosen as ${u}(x)=\\gamma(x,\\zeta(x))$\nthen the  above\ninequality becomes an equality.\n\\end{proof}\n\nThis result combines with Theorems \\ref{th-uniq-lambda}\nand \\ref{th-EHJB}\nto give the following\n\n\\begin{corollary}\\label{HJB-uniqueness}\nSuppose  that all the assumptions of\nTheorems \\ref{th-uniq-lambda}, \\ref{th-EHJB} and \\ref{th-uniqueness} hold.\nThen  $(\\bar v,  \\bar\\lambda)$ is the unique   mild solution of the\nHamilton-Jacobi-Bellman equation (\\ref{hjb}) satisfying\n$|\\bar v (x)|\\le c|x|$.\n\\end{corollary}\n\n\n\\section{Application to ergodic control of a semilinear heat equation}\n\\label{section-heat-eq}\n\nIn this section we show how our results can be applied to perform\nthe synthesis of the ergodic optimal control when the state\nequation is a semilinear heat equation with additive noise.  More\nprecisely, we treat a stochastic heat equation in space dimension\none, with a dissipative nonlinear term and with control and noise\nacting on a subinterval. We consider homogeneous Dirichlet\nboundary conditions.\n\n\n\\noindent In $\\left(  \\Omega,\\mathcal{F},\\mathbb{P}\\right)  $ with\na filtration $\\left(  \\mathcal{F}_{t}\\right) _{t\\geq0}$ satisfying\nthe usual conditions, we consider, for $t \\in\\left[ 0,T\\right] $\nand $\\xi\\in\\left[  0,1\\right]  $, the following equation\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l} d_{t }X^{u}\\left(  t ,\\xi\\right)  =\\left[\n\\frac{\\partial^{2}}{\\partial \\xi^{2}}X^{u}\\left(  t ,\\xi\\right)\n+f\\left(  \\xi,X^{u}\\left(  t  ,\\xi\\right)  \\right)\n+\\chi_{[a,b]}(\\xi) u\\left( t ,\\xi\\right)  \\right] dt\n+\\chi_{[a,b]}(\\xi) \\dot{W}\\left(\nt ,\\xi\\right)  dt ,\\\\\nX^{u}\\left(  t ,0\\right)  =X^{u}\\left(  t ,1\\right)  =0,\\\\\nX^{u}\\left(  t,\\xi\\right)  =x_{0}\\left(  \\xi\\right)  ,\n\\end{array}\n\\right.  \\label{heat equation}\n\\end{equation}\nwhere $\\chi_{[a,b]}$ is the indicator function of $[a,b]$ with\n$0\\leq a\\leq b\\leq 1$; $\\dot{W}\\left( t ,\\xi\\right)  $ is a\nspace-time white noise on $\\left[ 0,T\\right] \\times\\left[\n0,1\\right] $.\n\n\n\\noindent We introduce the cost functional\n\\begin{equation}\nJ\\left(  x,u\\right)  = \\limsup_{T\\rightarrow\\infty}\\dfrac{1}{T}\n\\mathbb{E}\\int_{0}^{T}\\int_{0}^{1}l\\left(  \\xi ,X^{u}_s\\left(\n\\xi\\right)  ,u_s(\\xi)\\right)  \\mu\\left(  d\\xi\\right) \\, ds,\n \\label{heat costo diri}\n\\end{equation}\nwhere $\\mu$ is a finite Borel measure on $\\left[  0,1\\right]  $.\nAn admissible control $u\\left(  t  ,\\xi\\right)  $ is a predictable\nprocess such that for all $t \\geq 0$, and\n $\\mathbb{P}$-a.s.\n $u\\left(  t ,\\cdot\\right)\n\\in U:=\\{v\\in C\\left(  \\left[  0,1\\right]  \\right) :\\left\\vert\nv\\left( \\xi\\right)  \\right\\vert \\leq\\delta\\}$. We denote by\n$\\mathcal{U}$ the set of such admissible controls. We wish to\nminimize the cost  over $\\mathcal{U}$, adopting the formulation of\nSection \\ref{optcontr}, i.e. by a change of probability in the\nform of (\\ref{def-ergodic-cost}). The cost introduced in\n(\\ref{heat costo diri}) is well defined on the space of continuous\nfunctions on the interval $\\left[ 0,1\\right] $, but for an\narbitrary $\\mu$\\ it is not well defined on the Hilbert space of\nsquare integrable functions.\n\n\nWe  suppose the following:\n\n\n\\begin{hypothesis}\n\\label{heatipotesi}\n\\begin{enumerate}\n\\item $f:\\left[  0,1\\right]  \\times\\mathbb{R} \\to\\mathbb{R}$ is\ncontinuous and for every\n $\\xi\\in\\left[0,1\\right] $, $ f(\\xi,\\,\\cdot\\,)$ is decreasing.\nMoreover there exist $C>0$ and $m>0$ such that for every\n$\\xi\\in\\left[0,1\\right]  ,$ $x\\in\\mathbb{R}$,\n$$ |f\\left(\n\\xi,x\\right)|\\leq C(1+|x|)^m, \\qquad f\\left( 0,x\\right)= f\\left(\n1,x\\right)=0.\n$$\n\n\n\\item $l:\\left[  0,1\\right]  \\times\\mathbb{R} \\times\n[-\\delta,\\delta]\\rightarrow\\mathbb{R}$ is continuous and bounded,\nand $l(\\xi,\\cdot,u)$ is Lipschitz continuous uniformly with\nrespect to $\\xi \\in\\left[ 0,1\\right]$, $u\\in [-\\delta,\\delta]$.\n\n\n\n\n\\item $x_{0}\\in C\\left(  \\left[  0,1\\right]  \\right)  $,\n$x_{0}(0)=x_{0}(1)=0$.\n\\end{enumerate}\n\\end{hypothesis}\n\n\n\\noindent To rewrite the problem in an abstract way we set\n $H=\\Xi=L^{2}\\left(   0,1  \\right)  $\n and $E=C_0\\left(\\left[  0,1\\right]  \\right)\n =\\{y\\in C\\left(\\left[  0,1\\right]  \\right)\\,:\\, y(0)=y(1)=0\\} $.\n We define an operator $A$ in $E$\\ by\n\\[\nD\\left(  A\\right)  =\\{y\\in C^{2}\\left(  \\left[  0,1\\right]\n\\right)\\,:\\, y,y''\\in C_{0}\\left(  \\left[  0,1\\right] \\right)\\}\n,\\text{ \\ \\ \\ \\ }\\left( Ay\\right) \\left(  \\xi\\right)\n=\\frac{\\partial^{2}}{\\partial\\xi^{2}}y\\left(  \\xi\\right) \\text{\nfor  }y\\in D\\left(  A\\right).\n\\]\nWe notice that $A$ is the generator of a $C_0$ semigroup in $E$,\nadmitting and extension to $H$, and $\\left| e^{tA}\\right|\n_{L\\left( E,E\\right) }\\leq e^{- t}$ see, for instance,  Theorem\n11.3.1 in \\cite{DP2}. As a consequence, $A+ F+I$ is\n  dissipative in  $E$.\n\nWe set, for $x\\in E$, $\\xi \\in [0,1]$, $z\\in \\Xi$, $u\\in U$,\n\\begin{equation}\nF\\left(  x\\right)  \\left(  \\xi\\right)  =f\\left(  \\xi,x\\left(\n\\xi\\right)  \\right)  ,\\ \\ \\left(  Gz\\right)  \\left(  \\xi\\right)\n =\\chi_{[a,b]}\\left(  \\xi\\right)  z\\left(\n\\xi\\right) ,\\ \\ L\\left(  x,u\\right)\n=\\displaystyle\\int_{0}^{1}l\\left( \\xi,x\\left(  \\xi\\right) ,u\\left(\n\\xi\\right) \\right)  \\mu\\left(  d\\xi\\right)  ,\n\\label{heatnotazioni}\n\\end{equation}\nand let  $R$ denote the canonical imbedding of $C(  \\left[\n0,1\\right])$ in\n $L^2(   0,1)$.\n\n\n\\noindent Finally $\\left\\{  W_{t },t \\geq0\\right\\}  $ is a\ncylindrical Wiener process in $H$ with respect to the filtration\n$\\left( \\mathcal{F}_{t }\\right) _{t \\geq0}$\n\n\n$ $\n\n\n\\noindent It is easy to verify that Hypotheses\n\\ref{general_hyp_forward} and \\ref{hyp_W_A F(W_A)} are satisfied\n(for the proof of point $4$ in Hypothesis\n\\ref{general_hyp_forward} and of Hypothesis \\ref{hyp_W_A F(W_A)}\nsee again \\cite{DP2} Theorem 11.3.1.).\n\n\n\\noindent Moreover, see for instance \\cite{C}, for some $C>0$,\n\\[\n\\left| e^{tA}\\right| _{L\\left( H,E\\right)  }\\leq Ct^{-1\/4}, \\qquad\n t\\in(0,1] ,\n\\]\nthus Hypothesis \\ref{hyp-convol-determ} holds.\n\n\n\\noindent Also Hypothesis \\ref{Hyp-masiero} is satisfied by taking\n$\\Xi _{0}=\\left\\lbrace f\\in C_0\\left(  \\left[  0,1\\right]\n\\right):f(a)=f(b)=0 \\right\\rbrace $.\n\n\n$ $ \\noindent Clearly the controlled heat equation (\\ref{heat\nequation}) can now be written in abstract way in the Banach space\n$E$ as\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l}\ndX_{t }^{x_0,u}=\\left[  AX_{t }^{x_0,u}+F\\left( X_{t\n}^{x_0,u}\\right) \\right] dt +GRu_{t }dt +GdW_{t }\\text{\\ \\ \\ }t\n\\in\\left[\nt,T\\right] \\\\\nX^{x_0,u}_0=x_{0},\n\\end{array}\n\\right.  \\label{heat eq abstract}\n\\end{equation}\nand  the results of the previous sections can be applied to the\nergodic cost (\\ref{heat costo diri}) (reformulated by a change of\nprobability in the form of (\\ref{def-ergodic-cost})).\n\n\n\\noindent In particular if we define,\nfor all $x\\in C_0([0,1])$, $z\\in L^2(0,1)$, $u\\in U$\n(identifying $L^2(0,1)$ with its  dual)\n$$\\psi(x,z)=\\inf _{u\\in U}\\left\\{\\int_0^1 l (\\xi,x(\\xi),u(\\xi))\n \\mu (d\\xi)+ \\int_a^b z(\\xi) u(\\xi) d\\xi\\right\\}$$\nthen there exist $\\overline v: E \\rightarrow \\mathbb{R}$\nLipschitz continuous and with $\\overline v(0)=0$, $\\overline \\zeta : E\n\\rightarrow \\Xi^*$ measurable and $\\overline \\lambda \\in\n\\mathbb{R}$ such that if $X^{x_0}=X^{x_0,0}$ is the solution of\nequation (\\ref{heat eq abstract}) then $(\\overline v(X^{x_0}),\n\\overline \\zeta(X^{x_0}),\\overline \\lambda)$ is a solution of the\nEBSDE (\\ref{EBSDE}) and the characterization of the optimal\nergodic control stated in  Theorem \\ref{Th-main-control} holds (and\n $\\overline \\lambda$ is unique in the sense of Theorem\n\\ref{th-uniq-lambda}).\n\n $ $\n\n\n\\noindent Moreover if $ f$ is of class $C^1(\\mathbb{R})$\n (consequently $F$ will be of class ${\\cal G}^1(E,E)$) and $\\psi$\n  is of class ${\\cal G}^1(E\\times \\Xi^*,E)$ then by Theorem \\ref{th-diff}\n $ \\overline v$ is of class ${\\cal G}^1(E,E)$ and, by Theorem \\ref{th-EHJB},\nit is a mild solution of the ergodic HJB equation (\\ref{hjb}) and it holds\n $\\overline \\zeta=\\nabla \\overline v G$.\n\n\n$ $\n\n\n\\noindent Let us then consider the particular case in which $[a,b]\n=[0,1]$,  $f(x,\\xi)=f(x)$ is of class $C^1$ with derivative having polynomial\ngrowth, and satisfies $f(0)=0$,\n$[f(x+h)-f(x)]h\\leq - c |h|^{2+\\epsilon}$\nfor suitable $c,\\epsilon\n>0$ and all $x,h\\in \\mathbb{R}$ (for instance, $f(x)=-x^3$).\nIn that case the Kolmogorov semigroup corresponding to the process\n$X^{x_0}$ is strongly Feller, see\n \\cite{C} and  \\cite{masiero2},  and it is easy to verify that\n$F$ is genuinely dissipative (see Definition \\ref{gen-diss}).\nMoreover we can choose $\\Xi_0=C_0([0,1])$ and it turns out that\n$\\psi$\n is bounded\non each set $E\\times B$, where $B$ is any ball of  $\\Xi_0^*$. Thus\nthe claims of Corollaries \\ref{characterization of lambda} and\n\\ref{boundedness of v} hold true, and in particular $\\overline v$\nis bounded.\n\n\n$ $\n\n\n\n\\noindent Finally if we assume that $\\mu$ is Lebesgue measure and\n$f$ is bounded and Lipschitz we can choose\n$E=\\Xi=\\Xi_0=H=L^2(0,1)$. Then the assumptions of Theorem\n\\ref{th-rec-seidler} are satisfied and we can apply Theorem\n\\ref{th-uniqueness} to characterize the function $\\overline v$. In\nparticular if $f$ is of class $C^1(\\mathbb{R})$ and $\\psi$ is of\nclass ${\\cal G}^1(H\\times \\Xi^*,H)$ then $\\overline v$ is the\nunique mild solution of the ergodic HJB equation (\\ref{hjb}).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAutonomous trucks are expected to fundamentally transform the freight transportation industry, and the enabling technology is progressing rapidly.\nMorgan Stanley estimates the potential savings from automation at \\$168 billion annually for the US alone \\cite{Greene2013-AutonomousFreightVehicles}.\nAdditionally, autonomous transportation may improve on-road safety, and reduce emissions and traffic congestion \\cite{ShortMurray2016-IdentifyingAutonomousVehicle,SlowikSharpe2018-AutomationLongHaul}.\n\nSAE International defines different levels of driving automation, ranging from L0 to L5, corresponding to no-driving automation to full-driving automation \\cite{SAEInternational2018-TaxonomyDefinitionsTerms}.\nThe current focus is on L4 technology (high automation), which aims at delivering automated trucks that can drive without any need for human intervention in specific domains, e.g., on highways.\nThe automotive industry is actively involved in making L4 vehicles a reality.\nDaimler Trucks, one of the leading heavy-duty truck manufacturers in North America, is working with both Torc Robotics and Waymo, and will be testing the latest generation of L4 trucks in the Southwest in early 2021 \\cite{Engadget2020-WaymoDaimlerTeam}.\nIn 2020, truck and engine maker Navistar announced a strategic partnership with technology company TuSimple to develop L4 trucks, to go into production by 2024 \\cite{TransportTopics2020-NavistarTusimplePartner}.\nOther companies developing self-driving vehicles include Argo AI, Aurora, Cruise, Embark, Ford, Kodiak, Lyft, Motional, Nuro, and Volvo Cars \\cite{FleetOwner-TusimpleAutonomousTruck}.\n\nA study by Viscelli \\cite{Viscelli-Driverless?AutonomousTrucks} describes different scenarios for the adoption of autonomous trucks by the industry.\nThe most likely scenario, according to some of the major players, is the \\emph{transfer hub business model} \\cite{Viscelli-Driverless?AutonomousTrucks,RolandBerger2018-ShiftingGearAutomation,ShahandashtEtAl2019-AutonomousVehiclesFreight}.\nAn Autonomous Transfer Hub Network (ATHN) makes use of autonomous truck ports, or \\emph{transfer hubs}, to hand off trailers between human-driven trucks and driverless autonomous trucks.\nAutonomous trucks then carry out the transportation between the hubs, while conventional trucks serve the first and last miles.\nFigure~\\ref{fig:autonomous_example} presents an example of an autonomous network with transfer hubs.\nOrders are split into a first-mile leg, an autonomous leg, and a last-mile leg, each of which served by a different vehicle.\nA human-driven truck picks up the cargo at the customer location, and drops it off at the nearest transfer hub.\nA driverless autonomous truck moves the trailer to the transfer hub closest to the destination, and another human-driven truck performs the last leg.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[scale=0.6]{images\/autonomous_example_ryder.pdf}\n\t\\caption{An Example of an Autonomous Transfer Hub Network.}\n\t\\label{fig:autonomous_example}\n\\end{figure}\n\nATHNs apply automation where it counts: monotonous highway driving is automated, while more complex local driving and customer contact is left to humans.\nEspecially for long-haul transportation, the benefit of automation is expected to be high.\nGlobal consultancy firm Roland Berger \\cite{RolandBerger2018-ShiftingGearAutomation} estimates that operational cost savings may be between 22\\% and 40\\% in the transfer hub model, based on the cost difference between driverless trucks and conventional trucks.\nThese estimates are based on single trips and it is not clear that they can be realized in practice: in particular, they do not take into account the empty miles traveled by autonomous trucks to pick up their next orders.\n\nThis paper proposes a Constraint Programming (CP) model to schedule the ATHN operations for a given set of orders.\nThe resulting schedule details the autonomous operations and the first\/last-mile operations at each of the hubs, and specifies the movements of every load, vehicle, and driver.\n\\emph{The CP model is then used to provide, for the first time, a detailed quantitative study of the benefits of ATHNs by considering a real case study where actual operations are modeled and optimized with high fidelity.}\nIt examines whether the savings predicted by \\cite{RolandBerger2018-ShiftingGearAutomation} materialize when the network effects, e.g., empty miles for relocation, are taken into account.\nIt is found that it is computationally feasible to solve this large-scale optimization problem with more than 100,000 variables and 100,000 constraints, and that the benefits of ATHNs may indeed be realized in practice.\n\nThe remainder of this paper is organized as follows.\nSection~\\ref{sec:problem} defines the problem of scheduling freight operations on an ATHN, and Section~\\ref{sec:formulation} formulates a CP model to solve this problem.\nThe case study is presented in Section~\\ref{sec:casestudy}.\nThe final section of the paper summarizes the findings and provides the conclusions.\n\n\n\\section{Problem Statement}\n\\label{sec:problem}\n\nThis section defines the problem of scheduling freight operations on an ATHN to perform a given set orders with a given set of vehicles, with the objective to minimize the cost of driving empty.\nThe problem is defined on a directed graph $G=(V,A)$, with vertices $V$ and arcs $A$.\nThe vertices represent locations, and are partitioned into hub locations $V_H$ and customer locations $V_C$.\nArcs between the transfer hubs correspond to autonomous transportation, and the other arcs represent human-driven legs.\nEvery arc $(i,j) \\in A$ is associated with a non-negative travel time $\\tau_{ij}$ and a cost $c_{ij}$.\nFor convenience, define $\\tau_{ii} = 0$ and $c_{ii} = 0$ for all $i \\in V$.\n\nThe set of customer orders is given by $R$.\nOrder $r\\in R$ is supposed to be picked up at time $p(r)$ at the origin $o(r) \\in V_C$, and to be transported through the ATHN to the destination $d(r) \\in V_C$.\nBased on order $r\\in R$, three tasks are defined: the first-mile task $t_r^f$, the autonomous task $t_r^a$, and the last-mile task $t_r^l$.\nThe first-mile task consists of loading the trailer at the customer location, moving the freight to the closest transfer hub, and unloading the trailer.\nSimilarly, the autonomous task and the last-mile task consist of loading, driving (between the hubs and to the destination, respectively), and unloading.\n\nLet $T$ be the set of all tasks generated by the orders.\nEvery task $t\\in T$ corresponds to a single leg, and is defined by an origin $o(t) \\in V$, a destination $d(t) \\in V$, and a pickup time $p(t)$.\nThe duration of a task equals $\\tau_{o(t),d(t)} + 2S$, where $S \\ge 0$ is the fixed time for loading or unloading a trailer.\nThe pickup time $p(t_r^f)$ of the first-mile task is equal to the order pickup time $p(r)$, while subsequent pickup times are based on the time the freight is supposed to be available.\nThat is, $p(t_r^a) = p(t_r^f) + \\tau_{o(t_r^f),d(t_r^f)} + 2S$, and $p(t_r^l) = p(t_r^a) + \\tau_{o(t_r^a),d(t_r^a)} + 2S$.\n\nTo create a feasible schedule, every task must be given a starting time, and be assigned to one of the available trucks.\nIt is assumed that an appointment flexibility of $\\Delta \\ge 0$ minutes is permitted, which means that task $t\\in T$ may start anywhere in the interval $[p(t)-\\Delta, p(t)+\\Delta]$.\nThe set of trucks $K$ is partitioned into autonomous trucks $K_A$, and regular trucks $K_h$ at every hub $h \\in V_H$.\nTasks can only be assigned to the corresponding set of trucks, and tasks performed by the same vehicle must not overlap in time.\nIf $t\\in T$ and $t'\\in T$ are subsequent tasks for a single truck, and $d(t) \\neq o(t')$, then an empty relocation is necessary, which takes $\\tau_{d(t) o(t')}$ time units and has cost $c_{d(t) o(t')}$.\nThe objective is to assign the tasks such that the total relocation cost is minimized.\n\nNote that the problem described above can be decomposed and solved independently for the autonomous network and for the operations at each of the hubs.\nThis is possible because different trucks are used for each part of the ATHN, and because each task is given an independent pickup time, based on the expected time the freight is available.\nOne potential problem is that using the appointment flexibility in one part of the network may lead to an infeasibility in another part of the network, but the case study shows that this is not an issue in practice: The first and last-mile schedules are not very constrained, and shifting the schedule to accommodate flexibility in the autonomous network is straightforward.\nAlternatively, one may first optimize the autonomous network, and update the first and last-mile pickup times accordingly.\n\n\n\\section{Mathematical Formulation}\n\\label{sec:formulation}\n\nThis section presents a CP model to schedule orders on the ATHN.\nWithout loss of generality, the set of tasks and the set of trucks represent a single part of the network that can be optimized independently.\nThat is, either the autonomous operations, or the first\/last-mile operations at one of the hubs are considered.\n\n\\newsavebox{\\modelbox}\n\\begin{lrbox}{\\modelbox}\n\\begin{varwidth}{1.15\\textwidth}\n\\begin{lstlisting}\nrange Trucks = ...;\nrange Tasks = ...;\nrange Sites = ...;\nrange Horizon = ...;\nrange Types = Sites union { shipType };  \nint or[Tasks] = ...;                     \nint de[Tasks] = ...;                     \nint pickupTime[Tasks] = ...;\nint loadTime = ...;\nint flexibility = ...;\nint travelTime[Types,Types] = ...;\nint travelCost[Types,Types] = ...;       \n\ndvar interval task[t in Tasks] in Horizon\n              size travelTime[or[t],de[t]] + 2*loadTime;\ndvar interval ttask[k in Trucks,t in Tasks] optional in Horizon\n              size travelTime[or[t],de[t]] + 2*loadTime;\ndvar interval load[Trucks,Tasks] optional in Horizon size loadTime;\ndvar interval ship[k in Trucks,t in Tasks] optional in Horizon\n              size travelTime[ort],de[t]];\ndvar interval unload[Trucks,Tasks] optional in Horizon size loadTime;\ndvar sequence truckSeq[k in Trucks]\n  in append(all(t in Tasks)load[k,t],all(t in Tasks)ship[k,t],all(t in Tasks)unload[k,t])\n  types append(all(t in Tasks)or[t],all(t in Tasks)shipType,all(t in Tasks)de[t]);\ndvar int emptyMilesCost[Trucks,Tasks];\ndvar int truckEmptyMilesCost[Trucks];\n\nminimize sum(k in Trucks) truckEmptyMilesCost[k];\n\nconstraints {\n\n   forall(t in Tasks) \n      startOf(task[t]) >= pickupTime[t] - flexibility;\n      startOf(task[t]) <= pickupTime[t] + flexibility;\n\t\n   forall(k in Trucks,t in Tasks)\n      span(ttask[k,t],[load[k,t],ship[k,t],unload[k,t]]);\n      startOf(ship[k,t]) == endOf(load[k,t])\n      startOf(unload[k,t]) == endOf(ship[k,t])\t \n\t\n   forall(k in Trucks)\n      alternative(task[t],all(k in Trucks) ttask[k,t])\t\n\t\n   forall(k in Trucks,t in Tasks)\n      emptyMilesCost[k,t] = travelCost[destination[t],typeOfNext(truckSeq[k],ttask[k,t],destination[t],destination[t])];\n\t\n   forall(k in Trucks)\n      truckEmptyMilesCost[k] = sum(t in Tasks) emptyMilesCost[k,t];\n\t\n   forall(k in Trucks)\n      noOverlap(truckSeq,travelTime);\n\n}\n\\end{lstlisting}\n\\end{varwidth}\n\\end{lrbox}\n\n\\begin{figure}[!t]\n\\makebox[\\textwidth][c]{%\n\\fbox{\\begin{minipage}{1.13\\textwidth}\n\t\\usebox{\\modelbox}\n\\end{minipage}}\n}\n\\caption{Formulation for Scheduling Freight Operations on an ATHN.}\n\\label{fig:formulation}\n\\end{figure}\n\nThe model is depicted in Figure \\ref{fig:formulation} using OPL\nsyntax \\cite{VanHentenryck1999-OplOptimizationProgramming}. The data of the model is given in lines 1--12. It consists of\na number of ranges (line 1--5), information about the tasks (lines\n6--8) that include their origins, destinations, and pickup times, the\ntime to load\/unload a trailer (line 9), the flexibility around the\npickup times (line 10), and the matrices of travel times and travel\ncosts. These matrices are defined between the sites but also\ninclude a dummy location {\\tt shipType} for reasons that will become\nclear shortly.\n\nThe main decision variables are the interval variables {\\tt task[t]}\nthat specify the start and end times of task {\\tt t} when processed\nby the autonomous network, and the optional interval variables {\\tt\n\tttask[k,t]} that are present if task {\\tt t} is transported by\ntruck {\\tt k}. These optional variables consist of three subtasks that\nare captured by the interval variables {\\tt load[k,t]} for loading,\n{\\tt ship[k,t]} for transportation, and {\\tt unload[k,t]} for\nunloading. The other key decision variables are the sequence variables\n{\\tt truckSeq[k]} associated with every truck: these variables\nrepresent the sequence of tasks performed by every truck. They\ncontain the loading, shipping, and unloading interval variables\nassociated with the trucks, and their types. The type of a loading\ninterval variable is the origin of the task, the type of an unloading\ninterval variable is the destination of the task, and the type of the\nshipping interval variable is the specific type {\\tt shipType} that is\nused to represent the fact that there is no transition cost and\ntransition time between the load and shipping subtasks, and the\nshipping and destination subtasks. The model also contains two\nauxiliary decision variables to capture the empty mile cost between a\ntask and its successor, and the empty mile cost of the truck\nsequence.\n\nThe objective function (line 28) minimizes the total costs of empty\nmiles. The constraints in lines 32--34 specify the potential start\ntimes of the tasks, and are defined in terms of the pickup times and\nthe flexibility parameter. The {\\sc span} constraints (line 37) link\nthe task variables and their subtasks, while the constraints in lines\n38--39 link the subtasks together. The {\\sc alternative} constraints\non line 42 specify that each task is processed by a single truck. The\nempty mile costs between a task and its subsequent task (if it\nexists) is computed by the constraints in line 45: they use the {\\sc\n\ttypeOfNext} expression on the sequence variables. The total empty\nmile cost for a truck is computed in line 48. The {\\sc noOverlap}\nconstraints in line 51 impose the disjunctive constraints between the\ntasks and the transition times. \n\n\n\\section{Case Study}\n\\label{sec:casestudy}\n\nTo quantify the impact of autonomous trucking on a real transportation network, a case study is presented for the dedicated transportation business of Ryder System, Inc., commonly referred to as \\emph{Ryder}.\nRyder is one of the largest transportation and logistics companies in North America, and provides fleet management, supply chain, and dedicated transportation services to over 50,000 customers.\nIts dedicated business, \\emph{Ryder Dedicated Transportation Solutions}, offers supply-chain solutions in which Ryder provides both drivers and trucks, and handles all other aspects of managing the fleet.\nRyder's order data is used to design an ATHN, and to create a detailed plan for how it would operate.\nThis allows for a realistic evaluation of the benefits of autonomous trucking.\n\n\n\\subsection{Data Description}\n\\label{sec:inputdata}\n\nRyder prepared a representative dataset for its dedicated transportation business in the Southeast of the US, reducing the scope to orders that were strong candidates for automation.\nThe dataset consists of trips that start in the first week of October 2019, and stay completely within the following states: AL, FL, GA, MS, NC, SC, and TN.\nIt contains 11,264 rows, which corresponds to 2,090 orders, formatted as in Table~\\ref{tab:order_data}.\nEvery order has a unique \\emph{OrderNumber}, and every row corresponds to a stop for a particular order.\nStops have a unique identifier \\emph{StopNumber}, and the \\emph{Stop} column indicates the sequence within the order.\nThe columns \\emph{StopArrivalDate} and \\emph{StopDepartureDate} indicate the scheduled arrival and departure times, and \\emph{City} and \\emph{ZipCode} identify the location of the stop.\nThe \\emph{Status} column gives a code for the status of the vehicle on arrival, and the \\emph{Event} column indicates what happens at the stop.\n\n\\begin{adjustbox}{center,float={table}[!t]}\n\t\\centering\n\t\\scriptsize\n\t\\begin{threeparttable}\n\t\t\\caption{An Example of the Order Data.}\n\t\t\\label{tab:order_data}%\n\t\t\\begin{tabular}{rrrrrrrrrr}\n\t\t\t\\multicolumn{1}{l}{StopNum} & \\multicolumn{1}{l}{OrderNum} & \\multicolumn{1}{l}{StopArrivalDate} & \\multicolumn{1}{l}{StopDepartureDate} & \\multicolumn{1}{l}{Stop} & \\multicolumn{1}{l}{City} & ZipCode & \\multicolumn{1}{l}{Status} & Event &  \\\\\n\t\t\t\\toprule\n\t\t\t68315760 & 7366366 & 2-10-2019 09:01 & 2-10-2019 09:02 & 1     & Atlanta & 30303 & LD    & HPL \\\\\n\t\t\t68315761 & 7366366 & 2-10-2019 16:29 & 2-10-2019 18:33 & 2     & Tennessee & 37774 & LD    & LUL \\\\\n\t\t\t68315762 & 7366366 & 3-10-2019 11:00 & 3-10-2019 11:30 & 3     & Atlanta & 30303 & MT    & DMT \\\\\n\t\t\n\t\t\n\t\t\t\\dots & \\dots & \\dots & \\dots & \\dots & \\dots & \\dots & \\dots & \\dots\\\\\n\t\t\t46798427 & 5207334 & 7-10-2019 02:35 & 7-10-2019 02:50 & 1     & Alpharetta & 30009 & NaN   & LLD \\\\\n\t\t\t46798428 & 5207334 & 7-10-2019 08:10 & 7-10-2019 08:49 & 2     & Macon & 31201 & LD    & LUL \\\\\n\t\t\t46798429 & 5207334 & 7-10-2019 15:16 & 7-10-2019 15:45 & 3     & Alpharetta & 30009 & LD    & LUL \\\\\n\t\t\\end{tabular}%\n\t\\end{threeparttable}\n\\end{adjustbox}%\n\nThe example data in Table~\\ref{tab:order_data} displays two orders.\nThe first order is a trip from Atlanta to Tennessee and back.\nBased on the status code, the truck arrived in Tennessee loaded (LD) and returned to Atlanta empty (MT).\nThe event codes show that a preloaded trailer was hooked in Atlanta (HPL), followed by a live unload (LUL) in Tennessee, after which the truck dropped the empty trailer (DMT) in Atlanta.\nThe exact codes are not important for the purpose of this paper.\nWhat is important, is the ability to derive the parts of the trip when the truck is moving freight, and the parts when the truck is driving empty.\nIf a vehicle returns to the starting location after only making deliveries, it is assumed that the return trip is empty, and the data is corrected if needed.\n\nRoad system data was obtained from OpenStreetMap \\cite{OpenStreetMap2020}, and route distance and mileage were calculated with the GraphHopper library.\nThe provided orders are long-haul trips, with an average trip length of 431 miles.\nMost of this distance is driven on highways: 65\\% of the distance is driven on interstates and US highways, and this number goes up further to 87\\% if state highways are included.\nSignificant highway usage is typical for long-haul transportation, and indicates that a significant part of each trip can potentially be automated.\n\n\n\\subsection{ATHN Design}\n\nThe design of ATHNs needs to decide the locations of the transfer hubs, which are the gateways to the autonomous parts of the network.\nA natural choice is to locate the hubs in areas where many trucks currently enter or exit the highway system.\nHistorical order data is used to identify these common highway access points.\nFor a given order, the truck is routed through the existing road network, and the highway segments, and their access points, can easily be identified.\nThe transfer hubs are then placed in areas with many access points.\nThe case study considers two different sets of hubs: a \\emph{small network} with 17 transfer hubs in the areas where Ryder trucks most frequently access the highway system, and a \\emph{large network} that includes 13 additional hubs in locations with fewer highway access points.\nThe small and the large network are visualized in Figure~\\ref{fig:designsmall} and Figure~\\ref{fig:designlarge}, respectively.\nThe exact hub locations are masked, but the figures are accurate within a 50 mile range.\nThe large network extends further northeast into North-Carolina, and further south into Florida.\nIt also makes the network more dense in the center of the region.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\subfloat[Small Network (17 hubs).\\label{fig:designsmall}]{%\n\t\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth, trim=28cm 3cm 29cm 15cm, clip]{images\/obfuscated_design_small.png}\n\t}\n\t\\hfill\n\t\\subfloat[Large Network (30 hubs).\\label{fig:designlarge}]{%\n\t\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth, trim=28cm 3cm 29cm 15cm, clip]{images\/obfuscated_design_large.png}\n\t}\n\t\\caption{ATHN Network Designs for the Southeast.}\n\t\\label{fig:designs}\n\\end{figure}\n\n\n\\subsection{Order Selection}\n\nThe case study focuses on scheduling the 494 most \\emph{challenging orders}.\nThese orders consist of a single delivery, followed by an empty return trip.\nBecause they require empty travel for 50\\% of the trip, these challenging orders are an excellent target for cost savings.\nThey also make up 24\\% of the dataset, and account for 53\\% of the empty mileage.\n\nFor a given ATHN and given arc costs $c_{ij}$ for all $(i,j) \\in A$, it is first determined which of these orders may benefit from the autonomous network, and which are better served by human-driven trucks.\nFor example, a direct trip with a human-driven trucks may be preferred for trips that are short, or for trips between locations that are far from any hub.\nTwo options are compared for every order: The first option is to serve the order with a conventional truck, which necessitates an empty return leg.\nThe second option uses the autonomous network, which amounts to driving to the nearest hub with a human-driven truck (first mile), shipping the load over the autonomous network to the hub closest to the destination, and then serving the last mile with another human-driven truck.\nNo empty returns are added in this case, as every truck is immediately available for the next task.\nThe two options are compared in terms of arc costs, and the cheapest one is selected.\nDifferent costs are used for autonomous and non-autonomous arcs, as will be explained in the next section.\nThe CP model only considers the orders that may benefit from the ATHN, while the other orders are scheduled separately.\n\n\n\\subsection{Parameters and Settings}\n\nThe following scenario is defined as the \\emph{base case} for the upcoming experiments.\nThe base case uses the small network, presented by Figure~\\ref{fig:designsmall}.\nFor conventional trucks, the cost $c_{ij}$ for driving arc $(i,j)\\in A$ is equal to the road distance.\nFor autonomous trucks, this cost is reduced by the percentage $\\alpha$.\nThe value of $\\alpha$ is a parameter, and the case study uses values ranging from $\\alpha=25\\%$ to $\\alpha=40\\%$, with $\\alpha = 25\\%$ for the base case.\nThis results in a conservative estimate of the benefits of autonomous trucking, which is predicted to be 29\\% to 45\\% cheaper per mile \\cite{EngholmEtAl2020-CostAnalysisDriverless}.\n\nThe time for loading or unloading is estimated at $S = 30$ minutes, and the appointment time flexibility is set to one hour ($\\Delta=60$).\nThe number of autonomous trucks is set to $\\lvert K_A \\rvert=50$.\nThe CP model is used to schedule the orders for each independent part of the network: the autonomous operations, and the first\/last-mile operations at each of the hubs.\nEach model is solved with the CPLEX CP Optimizer version 12.8 \\cite{LaborieEtAl2018-IbmIlogCp}.\n\n\n\\subsection{Base Case Results}\n\nOut of the 494 challenging orders, 437 (88\\%) are found to potentially benefit from the autonomous network, and the CP model is used to schedule these orders.\nScheduling the autonomous part of the ATHN is the most challenging: the model has close to 110,000 decision variables and more than 110,000 constraints.\nThe model is given an hour of CPU time and returns the best found solution, which is visualized in Figure~\\ref{fig:basecase_schedule}.\nThis figure shows both the autonomous tasks and the and the relocation tasks for the first week of October.\n\n\\begin{figure}[p]\n\t\\centering\n\t\\includegraphics[trim=20 50 10 40,clip,width=\\linewidth]{images\/small_25_gantt.pdf}\n    \\caption{Autonomous Truck Schedule for the Base Case.}\n\t\\label{fig:basecase_schedule}\n\\end{figure}\n\n\\begin{figure}[p]\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{images\/obfuscated_vehicle_20.png}\n\t\\caption{Single Autonomous Truck Route in the Base Case (blue is loaded, red is empty).}\n\t\\label{fig:singleroute}\n\\end{figure}\n\nFigure~\\ref{fig:basecase_schedule} shows that the transportation tasks are close together, and only a relatively small amount of relocation is necessary.\nIt is interesting to see that only a small number of autonomous trucks is driving during the weekend.\nThis is because the appointment times are still based on the current agreements with the customers, and having drivers work during the weekends is typically avoided.\nFor autonomous trucks, this would not be a problem, which again underlines that making full use of autonomous transportation requires adapting the business model and current practices.\nIn Section~\\ref{sec:flexibility}, the importance of time flexibility is considered in more detail.\n\nThe routes that are driven by the autonomous trucks consist of serving autonomous legs of different orders, with relocations in between.\nFigure~\\ref{fig:singleroute} shows a representative single truck route from the schedule.\nBlue arrows indicate that freight is being moved, and red arrows indicate that the truck is driving empty.\nThe truck starts at the west coast of Florida, where it picks up a load that has been delivered to the hub by a first\/last-mile truck driver.\nThe freight is then transported to the east coast, where it is unloaded so that a regular truck with driver can complete the last-mile.\nThe autonomous truck immediately starts serving the autonomous leg of the next order, returning to the Tampa area.\nAfter that, legs are served that are going north.\nThe first time a relocation is needed, is when the truck makes a delivery near Valdosta, close to the Georgia and Florida border.\nNo freight is immediately available, and the vehicle drives empty to the next location to pick up a load there.\nThe routes are clearly complex, which emphasizes the power of optimization: It is unlikely that this solution can be found by manual planners, but it is possible with optimization techniques.\n\nCompared to the autonomous trucks, the total distance driven by regular trucks is relatively short.\nThe optimization model was used to schedule the first and last-mile operations at selected hubs, and it was found that the amount of work is often insufficient to keep drivers occupied throughout the week.\nTo prevent driver idle time, it may be beneficial to outsource these legs, or to consolidate them with other operations in the area.\nIn terms of mileage, the percentage of empty miles at the hubs is typically under 25\\%, which is used as an estimate for the first\/last-mile efficiency in the remainder.\n\nTable~\\ref{tab:basecase_costs} quantifies the impact of autonomous trucking on the operating costs for the 437 selected orders.\nThe \\emph{Mileage} column indicates the total miles driven for both the current network and for the ATHN.\nThe numbers are separated based on whether the distance was driven while loaded or empty, and percentages are shown in the \\emph{\\% of total} column.\nThe \\emph{Cost without autonomous trucks} converts the miles into dollars, using \\$2 per mile as an approximation for the cost of human-driven trucks.\nRecall that driving autonomously is assumed to be $\\alpha=25\\%$ cheaper, which is reflected by the \\emph{Cost adjustment} column.\nThe \\emph{Cost} column presents the cost when autonomous trucks are available, and is obtained by multiplying the cost without autonomous trucks by the cost adjustment factor.\n\n\\begin{adjustbox}{center,float={table}[!t]}\n\t\\centering\n\t\\footnotesize\n\t\\begin{threeparttable}\n\t\t\\caption{Cost Table for the Base Case (437 orders).}\n\t\t\\label{tab:basecase_costs}%\n\t\t\\begin{tabular}{ccrrrrrrr}\n\t\t\t\\toprule\n\t\t\t&\t\t  &\t\t  &\t\t    &\t\t\t  & Cost without  &           & \\\\\n\t\t\t&       &       & \\quad Mileage & \\quad \\% of total & \\quad auton. trucks & \\quad Cost adj. & \\quad\\quad\\quad Cost \\\\\n\t\t\t\\midrule\n\t\t\t\\multirow{3}[0]{*}{Current network} &       & \\multicolumn{1}{l}{Loaded} & 96,669 & 50\\%  & \\$ 193,338 & 1.00  & \\$ 193,338 \\\\\n\t\t\t&       & \\multicolumn{1}{l}{Empty} & 96,698 & 50\\%  & \\$ 193,396 & 1.00  & \\$ 193,396 \\\\\n\t\t\t&       & \\multicolumn{1}{l}{Total} & 193,367 & 100\\% & \\$ 386,734 & 1.00  & \\$ 386,734 \\\\\n\t\t\t\\midrule\n\t\t\t\\multirow{9}[0]{*}{\\parbox{2.5 cm}{Autonomous transfer\\\\hub network}} & \\multirow{3}[0]{*}{Autonomous} & \\multicolumn{1}{l}{Loaded} & 91,618 & 67\\%  & \\$ 183,235 & 0.75  & \\$ 137,426 \\\\\n\t\t\t&       & \\multicolumn{1}{l}{Empty} & 44,217 & 33\\%  & \\$ 88,433 & 0.75  & \\$ 66,325 \\\\\n\t\t\t&       & \\multicolumn{1}{l}{Total} & 135,834 & 100\\% & \\$ 271,668 & 0.75  & \\$ 203,751 \\\\\n\t\t\t\\cmidrule{2-8}\n\t\t\t& \\multirow{3}[0]{*}{First\/last mile} & \\multicolumn{1}{l}{Loaded} & 29,286 & 75\\%  & \\$ 58,573 & 1.00  & \\$ 58,573 \\\\\n\t\t\t&       & \\multicolumn{1}{l}{Empty\\tnote{*}} & 9,762 & 25\\%  & \\$ 19,524 & 1.00  & \\$ 19,524 \\\\\n\t\t\t&       & \\multicolumn{1}{l}{Total} & 39,049 & 100\\% & \\$ 78,097 & 1.00  & \\$ 78,097 \\\\\n\t\t\t\\cmidrule{2-8}\n\t\t\t& Total &       & 174,883 &       & \\$ 349,766 &       & \\$ 281,848 \\\\\n\t\t\t\\cmidrule{2-8}\n\t\t\t& Savings &       & 18,484 &       & \\$ 36,969 &       & \\$ 104,886 \\\\\n\t\t\t& Savings (\\%) &       & 10\\% &       & 10\\% &       & 27\\% \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}%\n\t\t\\begin{tablenotes}\n\t\t\t\\item[*] estimated\n\t\t\\end{tablenotes}\n\t\\end{threeparttable}\n\\end{adjustbox}%\n\nCompared to the current network, the ATHN allows for significant savings for the selected orders: Table~\\ref{tab:basecase_costs} shows that the total cost goes down by 27\\%.\nAt a cost of \\$2 per mile, this corresponds to \\$104,886 per week, or \\$5.5M per year.\nThe \\emph{Mileage} column shows that almost 80\\% of the mileage in the ATHN can be automated, which partly explains the large savings.\nWhat is very interesting to observe is that the total mileage for the ATHN is actually \\emph{less} than the total mileage for the direct trips in the current network.\nIn the transfer hub network, there is no need to return back empty after a delivery, and there is no need to limit working hours or to return to a domicile at the end of the day.\nAs a result, only 33\\% of the automated distance is driven empty, compared to 50\\% for the current system.\nThis means that even if autonomous trucks would be as expensive as trucks with drivers, costs would still go down by 10\\% due to the additional flexibility that automation brings.\n\n\n\\subsection{Impact of the Size of the Network}\n\nA larger autonomous network results in shorter first\/last-mile trips, and may have a larger area of coverage.\nTo evaluate the impact of the size of the network, the calculations for the base case are repeated using the large network (Figure~\\ref{fig:designlarge}) with 30 hubs, instead of the small network (Figure~\\ref{fig:designsmall}) with 17 hubs.\nFor the large network, 468 of the 494 orders (95\\%) may benefit from the autonomous network, compared to only 88\\% for the base case.\nThis immediately implies that there is more potential for savings.\nIt also means a higher utilization of the autonomous trucks, as more legs are served by the same 50 vehicles.\n\nTable~\\ref{tab:large_25_costs} shows that the relative cost savings for the large network (29\\%) are similar to those for the small network (27\\%).\nThis means that the average benefit of automation is similar for both designs, \\emph{for the orders that are automated}.\nHowever, the large network allows more trips to benefit from automation, which is why the cost savings of \\$ 116,582 are 11\\% higher than the savings for the small network (\\$ 104,886).\nThe average benefit of automation is similar for the two designs due to two effects that cancel out.\nFirst, the same autonomous trucks have to serve more orders on the large network.\nThis increases the utilization of the vehicles, but also increases the percentage of empty miles from 33\\% to 35\\%.\nThe reason for this increase is that there is less time available to wait around at a hub for the next order, as the trucks are needed to perform other orders in the meantime.\nOn the other hand, the first and last-mile trips are shorter due to the additional hubs, which saves costs.\n\n\\begin{adjustbox}{center,float={table}[!t]}\n\t\\centering\n\t\\footnotesize\n\t\\begin{threeparttable}\n\t\t\\caption{Cost Table for the Large Network (468 orders).}\n\t\t\\label{tab:large_25_costs}%\n\t\t\\begin{tabular}{ccrrrrrr}\n\t    \\toprule\n\t\t&\t\t  &\t\t  &\t\t    &\t\t\t  & Cost without  &           & \\\\\n\t\t&       &       & \\quad Mileage & \\quad \\% of total & \\quad auton. trucks & \\quad Cost adj. & \\quad\\quad\\quad Cost \\\\\n\t\t\\midrule\n\t\t\\multirow{3}[0]{*}{Current network} &       & \\multicolumn{1}{l}{Loaded} & 101,213 & 50\\%  & \\$ 202,425 & 1.00  & \\$ 202,425 \\\\\n\t\t&       & \\multicolumn{1}{l}{Empty} & 96,698 & 50\\%  & \\$ 193,396 & 1.00  & \\$ 193,396 \\\\\n\t\t&       & \\multicolumn{1}{l}{Total} & 202,476 & 100\\% & \\$ 404,953 & 1.00  & \\$ 404,953 \\\\\n\t\t\\midrule\n\t\t\\multirow{8}[0]{*}{\\parbox{2.5 cm}{Autonomous transfer\\\\hub network}} & \\multirow{3}[0]{*}{Autonomous} & \\multicolumn{1}{l}{Loaded} & 97,326 & 65\\%  & \\$ 194,653 & 0.75  & \\$ 145,990 \\\\\n\t\t&       & \\multicolumn{1}{l}{Empty} & 53,247 & 35\\%  & \\$ 106,493 & 0.75  & \\$ 79,870 \\\\\n\t\t&       & \\multicolumn{1}{l}{Total} & 150,573 & 100\\% & \\$ 301,146 & 0.75  & \\$ 225,860 \\\\\n\t\t\\cmidrule{2-8}\n\t\t& \\multirow{3}[0]{*}{First\/last mile} & \\multicolumn{1}{l}{Loaded} & 23,442 & 75\\%  & \\$ 46,883 & 1.00  & \\$ 46,883 \\\\\n\t\t&       & \\multicolumn{1}{l}{Empty \\tnote{*}} & 7,814 & 25\\%  & \\$ 15,628 & 1.00  & \\$ 15,628 \\\\\n\t\t&       & \\multicolumn{1}{l}{Total} & 31,256 & 100\\% & \\$ 62,511 & 1.00  & \\$ 62,511 \\\\\n\t\t\\cmidrule{2-8}\n\t\t& Total &       & 181,829 &       & \\$ 363,657 &       & \\$ 288,371 \\\\\n\t\t\\cmidrule{2-8}\n\t\t& Savings &       & 20,648 &       & \\$ 41,296 &       & \\$ 116,582 \\\\\n\t\t& Savings (\\%) &       & 10\\%  &       & 10\\%  &       & 29\\% \\\\\n\t\t\\bottomrule\n\t\t\\end{tabular}%\n\t\t\\begin{tablenotes}\n\t\t\t\\item[*] estimated\n\t\t\\end{tablenotes}\n\t\\end{threeparttable}\n\\end{adjustbox}%\n\n\n\\subsection{Impact of the Cost of Autonomous Trucking}\n\nFor the base case, it was assumed that autonomous trucks are $\\alpha= 25\\%$ cheaper per mile than trucks with a driver.\nHowever, this number is yet far from certain, and higher cost reductions have also been reported in the literature.\nTo investigate the impact of the cost of autonomous trucking, Table~\\ref{tab:overview_costs} presents results for $\\alpha$ ranging from $25\\%$ to $40\\%$, for both the small and the large network.\nThe column \\emph{Autom.\n\torders} gives the number of orders that may benefit from automation, and are considered in the ATHN.\nThe relative cost savings (\\emph{Rel.\n\tsavings}) state the cost reduction compared to serving these orders with conventional trucks.\nThe \\emph{Cost savings} column gives the absolute cost savings in dollars.\nThe final column compares the absolute savings to the savings obtained for the baseline (small network, $\\alpha=25\\%$).\n\nTable~\\ref{tab:overview_costs} shows that, as autonomous trucking gets cheaper, and as more hubs are added to the network, the savings compared to the current system go up.\nAdditionally, more orders start using the ATHN, which increases the absolute savings further.\nEven though the autonomous trucks only perform the transportation between the hubs, the relative cost savings for the complete system often exceed the mileage cost reduction for autonomous trucks ($\\alpha$).\nThis again shows that the benefit of autonomous trucks is not only the lower cost per mile, but also the additional flexibility.\nCompared to the base case, cheaper autonomous transportation results in significantly larger savings.\nSimilar as in the previous section, increasing the size of the network does not strongly impact the average cost benefit per order, but does increase the total amount of orders that can be automated, which leads to more profits.\nIn the best case (large network, $\\alpha=40\\%$), the total benefit of the ATHN is \\$ 161,762 per week for the challenging orders, which corresponds to \\$8.4M savings per year.\n\n\\begin{table}[!t]\n\t\\centering\n\t\\footnotesize\n\t\\caption{Overview of Cost Savings under Different Assumptions.}\n\t\\label{tab:overview_costs}%\n\t\\begin{tabular}{crrrrr}\n\t\t\\toprule\n\t\t&       &       &       &       & Additional savings \\\\\n\t\t\\multicolumn{1}{l}{Network} & $\\alpha$ & \\quad Autom. orders & \\quad Rel. savings & \\quad Cost savings & \\quad comp. to base case \\\\\n\t\t\\midrule\n\t\t\\multirow{4}[0]{*}{Small} & 25\\%  & 437   & 27\\%  & \\$ 104,886 & +0\\% \\\\\n\t\t& 30\\%  & 439   & 32\\%  & \\$ 122,396 & +17\\% \\\\\n\t\t& 35\\%  & 443   & 35\\%  & \\$ 135,572 & +29\\% \\\\\n\t\t& 40\\%  & 443   & 38\\%  & \\$ 148,984 & +42\\% \\\\\n\t\t\\midrule\n\t\t\\multirow{4}[0]{*}{Large} & 25\\%  & 468   & 29\\%  & \\$ 116,582 & +11\\% \\\\\n\t\t& 30\\%  & 469   & 33\\%  & \\$ 131,798 & +26\\% \\\\\n\t\t& 35\\%  & 472   & 37\\%  & \\$ 149,586 & +43\\% \\\\\n\t\t& 40\\%  & 472   & 40\\%  & \\$ 161,762 & +54\\% \\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\\end{table}%\n\n\\subsection{Impact of Appointment Flexibility}\n\\label{sec:flexibility}\n\nFor the base case, the appointment flexibility was assumed to be $\\Delta = 60$ minutes.\nDeviating from a previously agreed appointment must be negotiated with the customer, but if there are significant benefits in terms of efficiency, this may be worth the effort.\nTo determine the impact of appointment flexibility, Table~\\ref{tab:time_flexibility} presents results for the base case (small network, $\\Delta=60$), in which the value of $\\Delta$ is varied.\nThe model is given four hours of CPU time for each setting.\nThe columns are similar to the previous table, and show the number of automated orders, the relative and absolute savings, and the additional savings compared to the base case.\nNote that the appointment flexibility does not affect the amount of orders that may benefit from automation, which is 437 for all four experiments.\n\nTable~\\ref{tab:time_flexibility} reveals that, if the appointment flexibility is already limited to one hour, limiting it further to 30 minutes to increase the service level is relatively inexpensive: the cost savings would only go down by 0.8\\%.\nIncreasing the flexibility by 30 minutes, on the other hand, goes a long way.\nUsing $\\Delta=90$ instead of $\\Delta=60$ results in 5\\% additional savings.\nThis indicates that the impact of appointment flexibility can be substantial.\nAlso note that the additional benefit is almost half that of the additional benefit for extending the network (+11\\%).\n\n\\begin{table}[!t]\n\t\\centering\n\t\\footnotesize\n\t\\caption{Overview of Cost Savings under Different Values of $\\Delta$.}\n\t\\label{tab:time_flexibility}%\n\t\\begin{tabular}{crrrrr}\n\t\t\\toprule\n\t\t&       &       &       &       & Additional savings \\\\\n\t\t\\multicolumn{1}{l}{Network} & $\\Delta$ & Autom. Orders & Rel. savings & Cost savings & comp. to base case \\\\\n\t\t\\midrule\n\t\t\\multirow{4}[0]{*}{Small} & 30    & 437   & 29\\%  & \\$ 110,887 & -0.8\\% \\\\\n\t\t& 60    & 437   & 29\\%  & \\$ 111,802 & 0.0\\% \\\\\n\t\t& 90    & 437   & 30\\%  & \\$ 117,354 & 5.0\\% \\\\\n\t\t& 120   & 437   & 30\\%  & \\$ 116,865 & 4.5\\% \\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\\end{table}%\n\nIt is surprising to see that increasing $\\Delta$ from 90 to 120 actually leads to a schedule that is less efficient, while more flexibility is available.\nThis is due to the CP model not finding the optimal solution.\nFinding the best schedule is a very challenging task, and increasing the flexibility increases the search space, which makes this task even more challenging.\nFurther investigation is needed to determine the actual additional savings that can be realized for $\\Delta=120$.\nThe results do suggest that no schedule could easily be found that was significantly better than the schedule for $\\Delta=90$, which hints that the advantage of additional flexibility is leveling off after $\\Delta=90$.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nAutonomous freight transportation is expected to completely transform the industry, and the technology is advancing rapidly, with different players developing and testing high automation L4 trucks.\nA crucial factor for the adoption of autonomous trucks is its return on investment, which is still uncertain.\nThis study contributes to the discussion by quantifying the benefits of the ATHN model, which is one of the most likely future scenarios.\nThe benefits are estimated based on a real transportation network, taking into account the detailed operations of the autonomous network.\n\nA CP model was presented to schedule the orders on an ATHN, and the model was used to conduct a case study on a real transportation network.\nIt was found that solving this large-scale optimization problem with CP is computationally feasible.\nFurthermore, ATHN may lead to substantial cost savings.\nFor some of the most challenging orders in the Southeast (orders that make a single delivery and return empty), operational cost may be reduced by 27\\% to 40\\%, which could save an estimated \\$5.5M to \\$8.4M per year on these orders.\nThis shows that the cost savings estimated by \\cite{RolandBerger2018-ShiftingGearAutomation} (22\\%-40\\%) may indeed be realized.\nThe savings are mainly attributed to a reduction in labor cost, but the increased flexibility of autonomous trucks also plays a significant role: Even if autonomous trucks would have the same cost per mile as human-driven trucks, cost savings would still be possible.\n\nIt was also explored how different assumptions impact the ATHN.\nIncreasing the size of the autonomous network mainly increases the number of orders that can benefit from automation, while the average benefit per automated order remains similar.\nThe impact of the cost per mile for autonomous trucking was also studied.\nAs autonomous trucks become cheaper, it is cost-efficient to automate more orders, and existing trips become cheaper as well.\nDue to the additional flexibility, it was found that the system benefit of automation often exceeds the benefit of the lower cost per mile.\nFinally, it was analyzed how appointment flexibility impacts the efficiency, and it was found that allowing deviations to be even 30 minutes larger can go a long way.\n\nThis paper quantified the impact of autonomous trucking on a real transportation network, and substantial benefits were found in terms of labor costs and flexibility.\nThese results strengthen the business case for autonomous trucking, and major opportunities may arise in the coming years.\nTo seize these opportunities, transport operators will have to update their business models, and use optimization technology to operate the more complex systems.\nDeveloping more detailed models and solution methods to support this transition is an interesting direction for future research.\n\n\n\\subsection*{Acknowledgements}\n\nThis research was funded through a gift from Ryder. Special thanks to the Ryder team for their invaluable support, expertise, and insights.\n\n\\clearpage\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:1}\n\nElectromagnetic waves propagating in disordered media are progressively scrambled by refractive index fluctuations and,\nthanks to interference, result into mesoscopic phenomena, such as speckle correlations and weak\nlocalization~\\cite{Akkermans2011, Sheng2010}. Polarization is an essential characteristic of electromagnetic waves that,\nconsidering the ubiquity of scattering processes in science, prompted the development of research in statistical\noptics~\\cite{Goodman2015, Brosseau1998} and impacted many applications, from optical imaging in biological\ntissues~\\cite{Tuchin2006} to material spectroscopy (e.g.\\@\\xspace, rough surfaces)~\\cite{Maradudin2007}, and radiation transport\nin turbulent atmospheres~\\cite{Andrews2005, Shirai2003}. Although the topic has experienced numerous developments and\noutcomes in the past decades, recent studies have revealed that much remains to be explored and understood on the\nrelation between the microscopic structure of scattering media and the polarization properties of the scattered field.\nIn particular, it was found that important information about the morphology of a disordered medium is contained in\nthe three-dimensional (3D) polarized speckles produced in the near-field above its surface~\\cite{Apostol2003,\nCarminati2010, Parigi2016} and in the spontaneous emission properties of a light source in the bulk~\\cite{Caze2010,Sapienza2011a}.\nSimilarly, the light scattered by random ensembles of large spheres was shown to exhibit unusual polarization features\ndue to the interplay between the various multipolar scatterer resonances~\\cite{Schmidt2015}.\n\nThe fact that light transport is affected by the microscopic structural properties of disordered media is well known.\nStructural correlations, coming from the finite scatterer size or from the specific morphology of porous\nmaterials~\\cite{Torquato2005, RojasOchoa2004, Garcia2007}, typically translate into an anisotropic phase function,\n$p(\\cos \\theta)$, which describes the angular response of a single scattering event with the scattering angle $\\theta$. The\naverage cosine of the phase function, known as the anisotropic scattering factor, $g=\\left\\langle \\cos \\theta \\right\\rangle$ (with\n$-1 \\leq g \\leq 1$), then leads to the standard definition of the transport mean free path (the average distance after\nwhich the direction of light propagation is completely randomized) as $\\ell^*=\\ell\/(1-g)$, where $\\ell$ is the\nscattering mean free path (the average distance between two scattering events). Single scattering anisotropy naturally\naffects how the polarization diffuses in disordered media, one of the most notable findings being that circularly\npolarized light propagates on longer distances compared to linearly polarized light in disordered media exhibiting\nforward single scattering ($g>0$) ---the so-called ``circular polarization memory effect''~\\cite{MacKintosh1989a,\nXu2005a, Gorodnichev2007}.\n\nRecent observations in mesoscopic optics also motivate deeper investigations on polarized light transport in correlated\ndisordered media. Indeed, numerical simulations revealed that uncorrelated ensembles of point scatterers cannot exhibit\n3D Anderson localization due to the vector nature of light~\\cite{Skipetrov2014, Bellando2014}. By contrast, it was found that the\ninterplay between short-range structural correlations and scatterer resonances could yield the opening of a 3D photonic\ngap in disordered systems~\\cite{Edagawa2008, Liew2011} and promote localization phenomena at its\nedges~\\cite{Imagawa2010}. To date, the respective role of polarization and structural correlations on mesoscopic optical\nphenomena remains largely to be clarified.\n\nTheoretically describing the propagation of polarized light in disordered media exhibiting structural correlations is a\ndifficult task. A first approach consists in using the vector radiative transfer equation~\\cite{Chandrasekhar1960,\nPapanicolaou1975, Mishchenko2006}, in which electromagnetic waves are described via the Stokes parameters and the\nscattering and absorption processes are related via energy conservation arguments. The various incident polarizations\n(linear, circular) and the single scattering anisotropy are explicitly implemented, thereby allowing for the\ninvestigation of a wide range of problems~\\cite{Amic1997, Gorodnichev2014}. A second approach relies on a transfer\nmatrix formalism based on a scattering sequence picture, where each scattering event (possibly anisotropic) yields a partial\nredistribution of the light polarization along various directions~\\cite{Akkermans1988, Xu2005, Rojas-Ochoa2004a}. The\napproach is phenomenological, yet very intuitive, making it possible to gain important physical insight into mesoscopic\nphenomena such as coherent backscattering~\\cite{Akkermans1988}.\n\nThe most \\textit{ab-initio} approach to wave propagation and mesoscopic phenomena in disordered systems is the so-called\nmultiple scattering theory, which directly stems from Maxwell's equations and relies on perturbative expansions on the\nscattering potential~\\cite{Sheng2010, Akkermans2011}. The formalism is often used to investigate mesoscopic\nphenomena, such as short and long-range (field and intensity) correlations or coherent backscattering, in a large\nvariety of complex (linear or nonlinear) media, including disordered dielectrics and atomic clouds. Unfortunately, it\nalso rapidly gains in complexity when the vector nature of light is considered. In fact, multiple scattering theory for\npolarized light has so far been restricted to uncorrelated disordered media only~\\cite{Stephen1986, MacKintosh1988,\nOzrin1992, VanTiggelen1996, VanTiggelen1999, Muller2002, Vynck2014}.\n\nIn this article, we present a model based on multiple scattering theory that describes how the diffusion of polarized\nlight is affected by short-range structural correlations, thereby generalizing previous models limited to uncorrelated\ndisorder. We do not aim at developing a complete theory for polarization-related mesoscopic phenomena in correlated\ndisordered media but at showing that, by a series of well-controlled approximations, important steps towards this\nobjective can be made. Starting from the (exact) Dyson and the Bethe-Salpeter equations for the average field and the\nfield correlation function, we derive a radiative transfer equation for the polarization-resolved specific\nintensity in the limit of short-range structural correlations and weak scattering. To analyze the impact of short-range\nstructural correlations on the diffusion of polarization, we then apply a $P_1$ approximation and decompose the\npolarization-resolved energy density into ``polarization eigenmodes'', as was done previously for uncorrelated\ndisordered media~\\cite{Ozrin1992, Muller2002, Vynck2014}. An interesting outcome of this decomposition is the\nobservation that each polarization eigenmode is affected independently and differently by short-range structural\ncorrelations. More precisely, each mode is characterized by a specific transport mean free path, and thus a specific\nattenuation length (describing the depolarization process) for its intensity. The transport mean free path of each\neigenmode depends non-trivially on the anisotropy factor $g$, and differently from the $(1-g)^{-1}$ rescaling\nwell known for the diffusion of scalar waves.\n\nThe paper is organized as follows. The radiative transfer equation for polarized light is derived\n\\textit{ab-initio} in Sect.~\\ref{sec:2}. The diffusion limit and the eigenmode decomposition are applied in\nSect.~\\ref{sec:3}. In Sect.~\\ref{sec:4}, we discuss the model and the results deduced from it, paying special attention to \nthe consistency of the approximations that have been made. Our conclusions are given in Sect.~\\ref{sec:5}. \nTechnical details about the average Green's function, the range of validity of the short-range structural correlation approximation, \nand the particular case of uncorrelated disorder, are presented in Appendices~\\ref{sec:A1}--\\ref{sec:A3}, respectively.\n\n\\section{Radiative transfer for polarized light}\\label{sec:2}\n\n\\subsection{Spatial field correlation}\n\nWe consider a disordered medium described by a real dielectric function of the form\n$\\epsilon(\\bm{r})=1+\\delta\\epsilon(\\bm{r})$, where $\\delta\\epsilon(\\bm{r})$ is the fluctuating part with the\nstatistical properties\n\\begin{equation}\\label{eq:disorder}\n   \\left\\langle \\delta\\epsilon(\\bm{r}) \\right\\rangle = 0,\n   \\qquad \\left\\langle \\delta\\epsilon(\\bm{r}) \\delta\\epsilon(\\bm{r}') \\right\\rangle = u f(\\bm{r}-\\bm{r}')\n\\end{equation}\nwhere $\\left\\langle\\ldots\\right\\rangle$ indicates ensemble averaging. The function\n$f(\\bm{r}-\\bm{r}')$ describes the structural correlation of the medium and $u$ is an amplitude whose expression will be\nderived below. We assume that the medium is statistically isotropic and invariant by translation. Considering a\nmonochromatic wave with free-space wavevector $k_0=\\omega\/c=2\\pi\/\\lambda$, $\\omega$ being the frequency,\n$\\lambda$ the wavelength and $c$ the speed of light in vacuum, the electric field $\\textbf{E}$ satisfies the vector \npropagation equation\n\\begin{equation}\n   \\nabla \\times \\nabla \\times \\bm{E}(\\bm{r})-k_0^2 \\epsilon(\\bm{r}) \\bm{E}(\\bm{r})=i \\mu_0 \\omega \\bm{j}(\\bm{r}),\n\\end{equation}\nwhere the current density $\\bm{j}(\\bm{r})$ describes a source distribution in the disordered medium. \nIntroducting the dyadic Green's function $G_{ik}$, the $i$th component of the electric field reads\n\\begin{equation}\\label{eq:Efield-green}\nE_i(\\bm{r})=i \\mu_0  \\omega \\int G_{ik}(\\bm{r},\\bm{r}') j_k(\\bm{r}') d\\bm{r}',\n\\end{equation} \nwhere implicit summation of repeated indices is assumed.\nThe spatial correlation function of the electric field $\\left\\langle E_i(\\bm{r}) E_j^\\star(\\bm{r}') \\right\\rangle$ obeys the Bethe-Salpeter equation\n\\begin{multline}\\label{eq:BS_field}\n   \\left\\langle E_i(\\bm{r}) E_j^\\star(\\bm{r}') \\right\\rangle = \\left\\langle E_i(\\bm{r}) \\right\\rangle \\left\\langle E_j^\\star (\\bm{r}') \\right\\rangle\n\\\\\n   + k_0^4 \\int \\left\\langle G_{im}(\\bm{r}-\\bm{r}_1) \\right\\rangle \\left\\langle G_{jn}^\\star(\\bm{r}'-\\bm{r}_1') \\right\\rangle\n\\\\\n   \\times \\Gamma_{mnrs} (\\bm{r}_1,\\bm{r}_1',\\bm{r}_2,\\bm{r}_2') \\left\\langle E_r (\\bm{r}_2) E_s^\\star (\\bm{r}_2') \\right\\rangle d\\bm{r}_1 d\\bm{r}_1' d\\bm{r}_2 d\\bm{r}_2'\n\\end{multline}\nthat can be derived from diagrammatic calculations~\\cite{Akkermans2011,Sheng2010}. In this expression\nthe superscript $\\star$ denotes complex conjugation, and\n$\\Gamma_{mnrs}$ is the four-point irreducible vertex that describes all possible scattering sequences\nbetween four points. In Eq.~(\\ref{eq:BS_field}), the first term in the right-hand side corresponds to the ballistic \nintensity, that is attenuated due to scattering at the scale of the scattering mean free path $\\ell$, and\nthe second term describes the multiple-scattering process. Note that at this level, Eq.~(\\ref{eq:BS_field}) \nis an exact closed-form equation.\n\nIt is also interesting to remark that the field correlation function $\\left\\langle E_i(\\bm{r}) E_j^\\star(\\bm{r}') \\right\\rangle$ is\none of the key quantities in statistical optics (where it is usually denoted by cross-spectral density matrix), since it\nencompasses the polarization and coherence properties of fluctuating fields in the frequency domain~\\cite{Goodman2015, Brosseau1998}. \nThe study of light fluctuations in 3D multiple scattering media has stimulated a revisiting of the concepts of degree of polarization \nand coherence~\\cite{Setala2002, Dennis2007, Refregier2014, Gil2014,Dogariu2015}, initially defined for 2D paraxial fields.\n\nTo proceed further, we assume weak disorder, such that the scattering mean free path $\\ell$\n is much larger than the wavelength ($k_0\\ell \\gg 1$). In this regime, only the two diagrams\nfor which the field and its complex conjugate follow the same trajectories (the so-called ladder and most-crossed\ndiagrams) contribute to the average intensity. The ladder diagrams are the root of radiative transport theory, that describes\nthe transport of intensity as an incoherent process. The most-crossed diagrams are responsible for weak localization and\ncoherent backscattering. In the ladder approximation and assuming independent scattering, the four-point irreducible\nvertex reduces to\n\\begin{multline}\n   \\Gamma_{mnrs}(\\bm{r}_1,\\bm{r}_1',\\bm{r}_2,\\bm{r}_2')\n\\\\\n   \\begin{split}\n      & = \\left\\langle \\delta\\epsilon(\\bm{r}_1) \\delta\\epsilon(\\bm{r}_1') \\right\\rangle \\delta(\\bm{r}_1-\\bm{r}_2) \\delta(\\bm{r}_1'-\\bm{r}_2') \\delta_{mr} \\delta_{ns}\n   \\\\\n      & = u f(\\bm{r}_1-\\bm{r}_1') \\delta(\\bm{r}_1-\\bm{r}_2) \\delta(\\bm{r}_1'-\\bm{r}_2') \\delta_{mr} \\delta_{ns},\n   \\end{split}\n\\end{multline}\nyielding\n\\begin{multline}\\label{eq:BS_field2}\n   \\left\\langle E_i(\\bm{r}) E_j^\\star(\\bm{r}') \\right\\rangle = \\left\\langle E_i(\\bm{r}) \\right\\rangle \\left\\langle E_j^\\star (\\bm{r}') \\right\\rangle\n\\\\\n   + uk_0^4 \\int \\left\\langle G_{im}(\\bm{r}-\\bm{r}_1) \\right\\rangle \\left\\langle G_{jn}^\\star(\\bm{r}'-\\bm{r}_1') \\right\\rangle\n\\\\\n   \\times f(\\bm{r}_1-\\bm{r}_1') \\left\\langle E_m (\\bm{r}_1) E_n^\\star (\\bm{r}_1') \\right\\rangle d\\bm{r}_1 d\\bm{r}_1'.\n\\end{multline}\nWe consider the source to be a point electric dipole located at $\\bm{r}_0$, such that\n\\begin{equation}\n   j_k(\\bm{r})=-i\\omega p_k \\delta(\\bm{r}-\\bm{r}_0),\n\\end{equation}\nwhere $p_k$ is the dipole moment along direction $k$. \nEquation~(\\ref{eq:Efield-green}) simplifies into $E_i(\\bm{r})= \\mu_0 \\omega^2 \\, G_{ik}(\\bm{r}-\\bm{r}_0) p_k$ and the\nBethe-Salpeter equation (\\ref{eq:BS_field2}) can be rewritten in terms of the dyadic Green's function in the form\n\\begin{multline}\\label{eq:BS_green}\n   \\left\\langle G_{ik}(\\bm{r}-\\bm{r}_0) G_{jl}^\\star(\\bm{r}'-\\bm{r}_0) \\right\\rangle = \\left\\langle G_{ik}(\\bm{r}-\\bm{r}_0) \\right\\rangle \\left\\langle G_{jl}^\\star (\\bm{r}'-\\bm{r}_0) \\right\\rangle\n\\\\\n   + uk_0^4 \\int \\left\\langle G_{im}(\\bm{r}-\\bm{r}_1) \\right\\rangle \\left\\langle G_{jn}^\\star(\\bm{r}'-\\bm{r}_1') \\right\\rangle f(\\bm{r}_1-\\bm{r}_1')\n\\\\\n   \\times \\left\\langle G_{mk} (\\bm{r}_1-\\bm{r_0}) G_{nl}^\\star (\\bm{r}_1'-\\bm{r}_0) \\right\\rangle d\\bm{r}_1 d\\bm{r}_1'.\n\\end{multline}\nUsing the change of variables $\\bm{r}-\\bm{r}_0=\\bm{R}+\\bm{X}\/2$ and $\\bm{r}'-\\bm{r}_0=\\bm{R}-\\bm{X}\/2$, and \ntransforming Eq.~(\\ref{eq:BS_green}) into reciprocal space, with $\\bm{K}$ and $\\bm{q}$ the reciprocal variables of\n$\\bm{R}$ and $\\bm{X}$ respectively, we finally obtain\n\\begin{widetext}\n   \\begin{multline}\\label{eq:BS_green_fourier}\n      \\left\\langle G_{ik}\\left(\\bm{q}+\\frac{\\bm{K}}{2}\\right) G_{jl}^\\star\\left(\\bm{q}-\\frac{\\bm{K}}{2}\\right) \\right\\rangle\n         = \\left\\langle G_{ik}\\left(\\bm{q}+\\frac{\\bm{K}}{2}\\right) \\right\\rangle \\left\\langle G_{jl}^\\star\\left(\\bm{q}-\\frac{\\bm{K}}{2}\\right) \\right\\rangle\n      + uk_0^4 \\left\\langle G_{im}\\left(\\bm{q}+\\frac{\\bm{K}}{2}\\right) \\right\\rangle \\left\\langle G_{jn}^\\star\\left(\\bm{q}-\\frac{\\bm{K}}{2}\\right) \\right\\rangle\n   \\\\\n      \\times \\int f(\\bm{q}-\\bm{q}') \\left\\langle G_{mk} \\left(\\bm{q}'+\\frac{\\bm{K}}{2}\\right) G_{nl}^\\star\n         \\left(\\bm{q}'-\\frac{\\bm{K}}{2}\\right) \\right\\rangle \\frac{d\\bm{q}'}{8\\pi^3}.\n   \\end{multline}\n\\end{widetext}\nA direct resolution of Eq.~(\\ref{eq:BS_green_fourier}) is possible for $f(\\bm{q}-\\bm{q}')=1$,\nand this approach was used in Ref.~\\cite{Vynck2014} to study the coherence and polarization properties of light in\nan uncorrelated disordered medium. In the case of a medium with structural correlations, a direct resolution is out of reach\nand we need to follow a different strategy.\n\n\\subsection{From field correlation to radiative transfer}\n\nIn this section we derive a radiative transfer equation for polarized light.\nWe proceed by evaluating the average Green's tensor $\\left\\langle \\bm{G} \\right\\rangle$, that obeys the Dyson\nequation~\\cite{Akkermans2011}. In its most general form, it reads~\\cite{Tai1993}\n\\begin{equation}\\label{eq:averageG}\n   \\left\\langle \\bm{G} (\\bm{q}) \\right\\rangle = \\left[ k_0^2 \\bm{I} - q^2 \\bm{P}(\\hat{\\bm{q}}) - \\bm{\\Sigma}(\\bm{q}) \\right]^{-1},\n\\end{equation}\nwith $\\bm{I}$ the unit tensor, $\\bm{P}(\\hat{\\bm{q}})=\\bm{I}-\\hat{\\bm{q}} \\otimes \\hat{\\bm{q}}$ the transverse\nprojection operator, $\\hat{\\bm{q}}=\\bm{q}\/q$ and $q=|\\bm{q}|$. $\\bm{\\Sigma}(\\bm{q})$ is the self-energy, which contains\nthe sum over all multiple scattering events that cannot be factorized in the averaging process. As shown in\nAppendix~\\ref{sec:A1}, for arbitrary structural correlations, $\\bm{\\Sigma}(\\bm{q})$ is non-scalar. \nThe problem can be simplified by assuming short-range structural correlations,\nin which case $\\bm{\\Sigma}(\\bm{q})=\\Sigma(\\bm{q})\\bm{I}$.  The average Green's tensor can then be written as\n\\begin{equation}\\label{eq:Green-mulet}\n   \\left\\langle \\bm{G} (\\bm{q}) \\right\\rangle = \\left\\langle G(\\bm{q}) \\right\\rangle \\left( \\bm{I} - \\frac{\\bm{q} \\otimes \\bm{q}}{k_0^2-\\Sigma(\\bm{q})} \\right),\n\\end{equation}\nwith $\\left\\langle G(\\bm{q}) \\right\\rangle =[k_0^2-q^2-\\Sigma(\\bm{q})]^{-1}$ the scalar Green's function. In a dilute medium,\nthe scattering events are assumed to take place on large distances compared to the wavelength (near-field interactions between \nscatterers can be neglected). In this case, the average Green's tensor $\\left\\langle \\bm{G} (\\bm{q}) \\right\\rangle$ can be reduced to its transverse\ncomponent~\\cite{Arnoldus2003}, yielding\n\\begin{equation}\n   \\left\\langle \\bm{G} (\\bm{q}) \\right\\rangle \\simeq \\left\\langle G(\\bm{q}) \\right\\rangle \\bm{P}(\\hat{\\bm{q}}).\n\\end{equation}\nAfter some simple algebra, the first term in the right-hand side in Eq.~(\\ref{eq:BS_green_fourier}) can be written as\n\\begin{multline}\n   \\left\\langle G_{ik}\\left(\\bm{q}+\\frac{\\bm{K}}{2}\\right) \\right\\rangle \\left\\langle G_{jl}^\\star\\left(\\bm{q}-\\frac{\\bm{K}}{2}\\right) \\right\\rangle\n\\\\\n   = M_{ik} M'_{jl}\\frac{\\left\\langle G(\\bm{q}+\\bm{K}\/2) \\right\\rangle - \\left\\langle G^\\star(\\bm{q}-\\bm{K}\/2) \\right\\rangle}\n      {2 \\bm{q} \\cdot \\bm{K} + \\Sigma(\\bm{q}+\\bm{K}\/2) - \\Sigma^\\star(\\bm{q}-\\bm{K}\/2)},\n\\end{multline}\nwhere we have defined the polarization factors $M_{ik}=\\delta_{ik} - (q_i + K_i\/2) (q_k + K_k\/2)\/|\\bm{q}+\\bm{K}\/2|^2$ and\n$M'_{jl}=\\delta_{jl} - (q_j - K_j\/2) (q_l - K_l\/2)\/|\\bm{q}-\\bm{K}\/2|^2$. In a dilute medium, we can assume that $|\\bm{K}| \\ll\n|\\bm{q}|$. This means that there are two different space scales in the correlation function of Green's tensor: A short scale associated to $\\bm{q}$ and corresponding to the dependence on direction of the specific intensity that we will introduce in Eq.~(\\ref{eq:specific_intensity}), and a large scale associated to $\\bm{K}$ and corresponding to the dependence of the specific intensity on position. This leads to\n\\begin{multline}\\label{eq:Green1}\n   \\left\\langle G_{ik}\\left(\\bm{q}+\\frac{\\bm{K}}{2}\\right) \\right\\rangle \\left\\langle G_{jl}^\\star\\left(\\bm{q}-\\frac{\\bm{K}}{2}\\right) \\right\\rangle\n\\\\\n   = (\\delta_{ik} - \\hat{q}_i \\hat{q}_k) (\\delta_{jl} - \\hat{q}_j \\hat{q}_l)\n      \\frac{\\left\\langle G(\\bm{q}) \\right\\rangle - \\left\\langle G^\\star(\\bm{q}) \\right\\rangle}{2 \\bm{q} \\cdot \\bm{K} + 2i \\operatorname{Im}[\\Sigma(\\bm{q})]}.\n\\end{multline}\nThe self-energy $\\Sigma(\\bm{q})$ renormalizes the propagation constant in the medium by defining a complex effective\npermittivity $\\epsilon_\\text{eff}=1-\\Sigma(\\bm{q})\/k_0^2$. The real part of $\\Sigma$ yields a change in the phase velocity,\nand the imaginary an attenuation of the field amplitude due to scattering. Hence, we can write\n\\begin{equation}\\label{eq:avGreen}\n   \\left\\langle G(\\bm{q}) \\right\\rangle = \\frac{1}{k_0^2 \\operatorname{Re}[\\epsilon_\\text{eff}] - q^2 + i k_0^2 \\operatorname{Im}[\\epsilon_\\text{eff}]}.\n\\end{equation}\nSince $\\operatorname{Im}[\\epsilon_\\text{eff}] \\ll \\operatorname{Re}[\\epsilon_\\text{eff}]$ in a dilute medium, we can rewrite\nEq.~(\\ref{eq:avGreen}) using the identity\n\\begin{equation}\n   \\lim_{\\varepsilon \\rightarrow 0} \\frac{1}{x-x_0-i \\varepsilon} = \\operatorname{PV}\\left[ \\frac{1}{x-x_0} \\right] - i \\pi \\delta (x-x_0),\n\\end{equation}\nwhere $\\operatorname{PV}$ stands for principal value.\nDefining $q_e=k_0\\sqrt{\\operatorname{Re}[\\epsilon_\\text{eff}]}$ as an effective wavevector, Eq.~(\\ref{eq:Green1}) becomes\n\\begin{multline}\\label{eq:Green2}\n   \\left\\langle G_{ik}\\left(\\bm{q}+\\frac{\\bm{K}}{2}\\right) \\right\\rangle \\left\\langle G_{jl}^\\star\\left(\\bm{q}-\\frac{\\bm{K}}{2}\\right) \\right\\rangle\n\\\\\n   = (\\delta_{ik} - \\hat{q}_i \\hat{q}_k) (\\delta_{jl} - \\hat{q}_j \\hat{q}_l) \\frac{\\pi \\delta(q_e^2-q^2)}{ i \\bm{q} \\cdot \\bm{K} - \\operatorname{Im}[\\Sigma(\\bm{q})]}.\n\\end{multline}\nIn order to derive a radiative transfer equation, we then introduce the quantity $L_{ijkl}$  by the relation\n\\begin{multline}\\label{eq:specific_intensity}\n   \\left\\langle G_{ik}\\left(\\bm{q}+\\frac{\\bm{K}}{2}\\right) G_{jl}^\\star\\left(\\bm{q}-\\frac{\\bm{K}}{2}\\right) \\right\\rangle\n\\\\\n      = \\frac{4\\pi^2}{q_e} \\delta(q_e^2 - q^2) L_{ijkl}(\\bm{K},q_e \\hat{\\bm{q}}).\n\\end{multline}\nHere, we assume that the correlation function of Green's tensor propagates on shell, {\\it i.e.} with a wavevector $q=q_e$. The impact of the on-shell approximation, which is the key step to solve the Bethe-Salpeter equation in the presence of  structural correlations, will be discussed in Sec.~\\ref{sec:4}. From Eqs.~(\\ref{eq:Green2}) and (\\ref{eq:specific_intensity}), we can rewrite the Bethe-Salpeter equation (\\ref{eq:BS_green_fourier}) in the form\n\\begin{multline}\n   \\frac{4\\pi^2}{q_e} \\delta(q_e^2 - q^2) L_{ijkl}(\\bm{K},q_e \\hat{\\bm{q}})\n\\\\\n   = \\frac{\\pi \\delta(q_e^2 - q^2)}{ i \\bm{q} \\cdot \\bm{K} - \\operatorname{Im}[\\Sigma(\\bm{q})]}\n      \\left[\\vphantom{\\int} (\\delta_{ik} - \\hat{q}_i \\hat{q}_k) (\\delta_{jl} - \\hat{q}_j \\hat{q}_l)\\right.\n\\\\\n   + u k_0^4 (\\delta_{im} - \\hat{q}_i \\hat{q}_m) (\\delta_{jn} - \\hat{q}_j \\hat{q}_n)\n\\\\\n   \\left. \\times  \\frac{4\\pi^2}{q_e} \\int f(\\bm{q}-\\bm{q}') \\delta(q_e^2 - q'^2) L_{mnkl}(\\bm{K},q_e \\hat{\\bm{q}}')\n   \\frac{d\\bm{q}'}{8\\pi^3} \\right].\n\\end{multline}\nIntegrating both sides of the equation over $q$, performing the integral on the right-hand side over $q'$, and using the relation $\\int_0^\\infty f(\\bm{r}) \\delta(r^2-r_0^2) r^2 dr=r_0f(\\bm{r}=r_0 \\hat{\\bm{r}})\/2$, we obtain\n\\begin{multline}\\label{eq:preRTE}\n   L_{ijkl}(\\bm{K},q_e \\hat{\\bm{q}})\n\\\\\n   =\\frac{q_e}{4\\pi}\\frac{1}{ i q_e \\hat{\\bm{q}} \\cdot \\bm{K} - \\operatorname{Im}[\\Sigma(q_e \\hat{\\bm{q}})]}\n      \\left[\\vphantom{\\int} (\\delta_{ik} - \\hat{q}_i \\hat{q}_k) (\\delta_{jl} - \\hat{q}_j \\hat{q}_l) \\right.\n\\\\\n   + \\frac{u k_0^4}{4\\pi} (\\delta_{im} - \\hat{q}_i \\hat{q}_m) (\\delta_{jn} - \\hat{q}_j \\hat{q}_n)\n\\\\\n   \\left.\\times \\int f(q_e (\\hat{\\bm{q}}-\\hat{\\bm{q}}')) L_{mnkl}(\\bm{K}, q_e \\hat{\\bm{q}}') d\\hat{\\bm{q}}' \\right].\n\\end{multline}\nThe quantity $L_{ijkl}(\\bm{K},q_e \\hat{\\bm{q}})$ is proportional to the specific intensity introduced in radiative transfer theory~\\cite{Chandrasekhar1960}, and has the meaning of a local and directional radiative flux. Actually, Eq.~(\\ref{eq:preRTE}) can be cast in the form of a radiative transfer equation, as we will now show.\n\nSince the disordered medium is statistically isotropic and translational-invariant, the correlation function $f$ only depends on $|\\hat{\\bm{q}}-\\hat{\\bm{q}}'|$, or equivalently on $\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}'$. It is directly related to the classical phase function $p(\\hat{\\bm{q}}\\cdot\\hat{\\bm{q}}')$ of radiative transfer theory as\n\\begin{equation}\\label{eq:correlation-phase}\n   f(q_e |\\hat{\\bm{q}}-\\hat{\\bm{q}}'|) = A \\, p(\\hat{\\bm{q}}\\cdot\\hat{\\bm{q}}'),\n\\end{equation}\nwhere $A$ is a constant whose value is determined by energy conservation, and $\\int p(\\hat{\\bm{q}}\\cdot\\hat{\\bm{q}}') d\\hat{\\bm{q}} = 4\\pi$.\nTo order $(k_0\\ell)^{-1}$ and for short-range structural correlations, one has $\\operatorname{Im}[\\Sigma(q_e \\hat{\\bm{q}})] =\n-q_e\/\\ell$ and $u=6\\pi\/k_0^4\\ell$ (these results are derived in Appendix~\\ref{sec:A1}). This allows us to rewrite Eq.~(\\ref{eq:preRTE})\nin its final form \n\\begin{widetext}\n   \\begin{equation}\\label{eq:RTE}\n      \\left[ i \\hat{\\bm{q}} \\cdot \\bm{K} + \\frac{1}{\\ell} \\right] L_{ijkl}(\\bm{K},\\hat{\\bm{q}})\n         = \\frac{1}{4\\pi} (\\delta_{ik} - \\hat{q}_i \\hat{q}_k) (\\delta_{jl} - \\hat{q}_j \\hat{q}_l)\n         + \\frac{3 A}{8 \\pi \\ell} (\\delta_{im} - \\hat{q}_i \\hat{q}_m) (\\delta_{jn} - \\hat{q}_j \\hat{q}_n)\n         \\int p(\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}') L_{mnkl}(\\bm{K}, \\hat{\\bm{q}}') d\\hat{\\bm{q}}'\n   \\end{equation}\n\\end{widetext}\nwhere an implicit summation over $m$ and $n$ is assumed.\nThis expression takes the form of a radiative transfer equation (RTE) for the polarization-resolved specific intensity.\nIt differs from the standard vector radiative transfer equation~\\cite{Chandrasekhar1960} in the sense that it is not written\nin terms of Stokes vector, but using a fourth-order tensor representing the specific intensity for polarized light, and relating two\ninput and two output polarization components. Nevertheless, the various terms in Eq.~(\\ref{eq:RTE}) have a very clear\nphysical meaning. The first and second terms on the left-hand-side respectively describe the total variation of specific\nintensity along direction $\\hat{\\bm{q}}$ and the extinction of the ballistic light due to scattering (i.e.\\@\\xspace, Beer-Lambert's\nlaw). The first and second terms on the right-hand-side describe the increase of specific intensity along direction\n$\\hat{\\bm{q}}$ due to the presence of a source, and to the light originally propagating along direction $\\hat{\\bm{q}}'$\nand being scattered along $\\hat{\\bm{q}}$, respectively.\n\nConservation of energy requires the scattering losses to be compensated by the gain due to scattering after\nintegration over all angles. The energy conservation relation has to be written on the intensity, i.e.\\@\\xspace by setting $i=j$ and\nsumming over polarization components in Eq.~(\\ref{eq:RTE}), in the form\n\\begin{multline}\n   \\frac{1}{\\ell}\\sum_i \\int L_{iikl}(\\bm{K},\\hat{\\bm{q}}) d\\hat{\\bm{q}}  = \\frac{3 A}{8\\pi\\ell}\n\\\\\n   \\times \\sum_{i,m} \\int (\\delta_{im} - \\hat{q}_i \\hat{q}_m)^2 p(\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}') L_{mmkl}(\\bm{K}, \\hat{\\bm{q}}') d\\hat{\\bm{q}}' d\\hat{\\bm{q}}.\n\\end{multline}\nThis leads to the following relation on the coefficient $A$\n\\begin{equation}\\label{eq:energy_conservation_constant}\n   \\frac{3}{8\\pi} \\sum_m \\int (\\delta_{im} - \\hat{q}_i \\hat{q}_m)^2 p(\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}') d\\hat{\\bm{q}} = \\frac{1_i}{A},\n\\end{equation}\nwhere $1_i$ is the unit vector. At this stage, we have obtained a transport equation for polarized light\n[Eq.~(\\ref{eq:RTE})] that takes the form of a RTE. This equation stems directly from the Dyson and Bethe-Salpeter equations, \nfulfills energy conservation, and is valid for dilute media and short-range correlated disorder.\n\n\\section{Diffusion of polarization}\\label{sec:3}\n\n\\subsection{$P_1$ approximation}\n\nIn short-range correlated media, the phase function $p(\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}')$ is expected to be\nquasi-isotropic. It can therefore be expanded into a Legendre series, which, to order $\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}'$,\nreads\n\\begin{equation}\\label{eq:phasefunction}\n   p(\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}') = 1 + 3g (\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}'),\n\\end{equation}\nwhere $g$ is the anisotropic scattering factor, defined as\n\\begin{equation}\n   g =\\frac{1}{4\\pi} \\int p({\\bm{q}} \\cdot \\hat{\\bm{q}}') \\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}' d\\hat{\\bm{q}},\n\\end{equation}\nand satisfying\n\\begin{equation}\n   g \\hat{\\bm{q}}' =\\frac{1}{4\\pi} \\int p({\\bm{q}} \\cdot \\hat{\\bm{q}}') \\hat{\\bm{q}} d\\hat{\\bm{q}}.\n \\end{equation}\nInserting Eq.~(\\ref{eq:phasefunction}) into Eq.~(\\ref{eq:RTE}), the RTE can be rewritten as\n\\begin{multline}\\label{eq:newRTE}\n   \\left[ i \\hat{\\bm{q}} \\cdot \\bm{K} + \\frac{1}{\\ell} \\right] L_{ijkl}(\\bm{K},\\hat{\\bm{q}})\n   = \\frac{1}{4\\pi} (\\delta_{ik} - \\hat{q}_i \\hat{q}_k) (\\delta_{jl} - \\hat{q}_j \\hat{q}_l)\n\\\\\n   + \\frac{3 A}{2 \\ell} (\\delta_{im} - \\hat{q}_i \\hat{q}_m) (\\delta_{jn} - \\hat{q}_j \\hat{q}_n)\n   \\\\\n   \\times \\left[\n      L^{(0)}_{mnkl}(\\bm{K})+ \\frac{3g}{4 \\pi } \\bm{j}_{mnkl}(\\bm{K}) \\cdot \\hat{\\bm{q}}\\right],\n\\end{multline}\nwhere $L^{(0)}_{ijkl}$ and $\\bm{j}_{ijkl}$ are the (polarization-resolved)\nirradiance and radiative flux vector, respectively, defined as\n\\begin{align}\n   L^{(0)}_{ijkl}(\\bm{K}) & = \\frac{1}{4\\pi} \\int L_{ijkl}(\\bm{K},\\hat{\\bm{q}}) d\\hat{\\bm{q}},\n\\\\\n   \\bm{j}_{ijkl}(\\bm{K}) & = \\int \\hat{\\bm{q}} L_{ijkl}(\\bm{K},\\hat{\\bm{q}}) d\\hat{\\bm{q}}.\n\\end{align}\n\nTo gain insight into the effect of short-range correlations on the propagation of polarized light, it is convenient to\ninvestigate the diffusion limit, which is reached after propagation on distances much larger than the \nscattering mean free path $\\ell$. In this limit, the specific intensity becomes quasi-isotropic. \nExpanding $L_{ijkl}$ into Legendre polynomials $P_n$ to first order in $\\hat{\\bm{q}}$, we have \n\\begin{equation}\\label{eq:P1_approx}\n   L_{ijkl}(\\bm{K},\\hat{\\bm{q}}) = L^{(0)}_{ijkl}(\\bm{K}) + \\frac{3}{4\\pi} \\bm{j}_{ijkl}(\\bm{K}) \\cdot \\hat{\\bm{q}}\n\\end{equation}\nwhich is the so-called $P_1$ approximation.\nInserting Eq.~(\\ref{eq:P1_approx}) into Eq.~(\\ref{eq:newRTE}) and calculating the zeroth and first moments of the\nresulting equation (which amounts to performing the integrations $\\int - d\\hat{\\bm{q}}$ and $\\int - \\hat{\\bm{q}} d\\hat{\\bm{q}}$,\nrespectively), we eventually arrive to a pair of equations relating $L^{(0)}_{ijkl}$ and $\\bm{j}_{ijkl}$:\n\\begin{widetext}\n   \\begin{align}\n      i \\bm{K} \\cdot \\bm{j}_{ijkl}(\\bm{K}) + \\frac{4\\pi}{\\ell} L^{(0)}_{ijkl}(\\bm{K})\n         & = \\frac{2}{3} S_{ijkl} + \\frac{4\\pi}{\\ell} A S_{ijmn} L^{(0)}_{mnkl}(\\bm{K}), \\label{eq:zeroth_moment}\n   \\\\\n      -\\frac{4\\pi}{3} K^2 \\ell L^{(0)}_{ijkl}(\\bm{K}) + i \\bm{K} \\cdot \\bm{j}_{ijkl}(\\bm{K})\n         & = i g \\frac{9A}{8\\pi} \\int (\\delta_{im} - \\hat{q}_i \\hat{q}_m) (\\delta_{jn} - \\hat{q}_j \\hat{q}_n)\n         \\left( \\bm{j}_{mnkl}(\\bm{K}) \\cdot \\hat{\\bm{q}} \\right) \\left( \\bm{K} \\cdot \\hat{\\bm{q}} \\right)\n         d\\hat{\\bm{q}}.\\label{eq:first_moment}\n   \\end{align}\n\\end{widetext}\nHere, we have defined\n\\begin{equation}\n   S_{ijkl}=\\frac{3}{8\\pi} \\int (\\delta_{ik} - \\hat{q}_i \\hat{q}_k) (\\delta_{jl} - \\hat{q}_j \\hat{q}_l) d\\hat{\\bm{q}},\n\\end{equation}\nand used the relations $\\int (\\delta_{im} - \\hat{q}_i \\hat{q}_m) (\\delta_{jn} - \\hat{q}_j \\hat{q}_n) \\hat{\\bm{q}}\nd\\hat{\\bm{q}}=0$, $\\int \\hat{q}_i \\hat{q}_j d\\hat{\\bm{q}}= 4\\pi \/ 3\\delta_{ij}$ and $\\int \\hat{q}_i \\hat{q}_j\n\\hat{q}_k d\\hat{\\bm{q}}= 0$. The additional complexity of the polarization mixing due to structural correlations can be\napprehended from Eq.~(\\ref{eq:first_moment}), where the relation between $L^{(0)}_{ijkl}$ and $\\bm{j}_{ijkl}$ in terms\nof input and output polarization components becomes particularly intricate as soon as $g \\neq 0$. Much deeper insight\ninto the diffusion of polarized light can be gained via an eigenmode decomposition, as shown below.\n\n\\subsection{Polarization eigenmodes}\n\nAnalytical expressions for all terms in the $L_{ijkl}^{(0)}(\\bm{K})$ and $\\bm{j}_{ijkl}(\\bm{K})$ tensors can be\nobtained by solving Eqs.~(\\ref{eq:zeroth_moment}) and (\\ref{eq:first_moment}), which we have done imposing $\\bm{K}$ to\nbe along one of the main spatial directions, without loss of generality, and using the software Mathematica~\\cite{Mathematica}. The\nobtained expressions at this stage are long and complicated, containing in particular high-order terms in powers of $K$\nand $g$ (that are not physical and will be neglected below). We now introduce a polarization-resolved energy density\n$U_{ijkl}=6\\pi \/ cL_{ijkl}^{(0)}$ and decompose it in terms of ``polarization eigenmodes''\nas in Refs.~\\cite{Ozrin1992, Muller2002, Vynck2014}:\n\\begin{equation}\\label{eq:eigenmode_decomposition}\n   U_{ijkl}(\\bm{K}) = \\sum_p U^{(p)}(\\bm{K}) \\left|ij\\right\\rangle_p \\left\\langle kl \\right|_p.\n\\end{equation}\nThe eigenvalues $U^{(p)}$ provide the characteristic length and time scales of the diffusion of each eigenmode and the projectors\n$\\left|ij\\right\\rangle_p \\left\\langle kl \\right|_p$, which will be denoted by ``polarization eigenchannels'', relate input polarization pairs\n$(k,l)$ to output polarization pairs $(i,j)$. The $U_{ijkl}$ is represented as a $9 \\times 9$ matrix (9 pairs of\npolarization components in input and output) and is diagonalized using Mathematica, leading again to full analytical expressions.\n\nAt this stage, the obtained expressions still depend on the coefficient $A$, originally defined in\nEq.~(\\ref{eq:correlation-phase}) and used to ensure energy conservation in the RTE, Eq.~(\\ref{eq:RTE}). To predict how\n$A$ depends on structural correlations, we rely on the particular case of the Henyey-Greenstein (HG) phase function~\\cite{Henyey1941}\n\\begin{equation}\\label{eq:HG-phasefunction}\n   p_\\text{HG}(\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}') = \\frac{1-g}{\\left[1+g^2-2g (\\hat{\\bm{q}} \\cdot \\hat{\\bm{q}}')\\right]^{3\/2}}.\n\\end{equation}\nThe HG phase function is very convenient since it provides a closed-form expression with $g$ as a single parameter, and\napproximates the phase functions of a wide range of disordered media (e.g.\\@\\xspace, interstellar dust clouds, biological\ntissues). The energy conservation equation, Eq.~(\\ref{eq:energy_conservation_constant}), can be solved analytically in\nthis case, yielding the surprisingly simple relation\n\\begin{equation}\\label{eq:A_HG}\n   A_\\text{HG} = \\frac{1}{1+g^2\/2}.\n\\end{equation}\nNote that the modification in energy conservation due to structural correlations appears at order $g^2$.\n\nWe can finally insert Eq.~(\\ref{eq:A_HG}) into the eigenvalues and eigenvectors found from\nEq.~(\\ref{eq:eigenmode_decomposition}) and develop analytical expressions valid to orders $K^2$ (diffusion\napproximation) and $g$ (weakly correlated disorder). The eigenvectors take the expressions already obtained for\nuncorrelated disorder~\\cite{Ozrin1992, Muller2002, Vynck2014}\n\\begin{align}\\label{eq:eigenvectors}\n   \\left|kl\\right\\rangle_{1} & = \\frac{1}{\\sqrt{3}} \\delta_{kl}, \\nonumber \\\\\n   \\left|kl\\right\\rangle_{2,3,4} & = \\frac{1}{\\sqrt{2}} (\\delta_{ka} \\delta_{lb} - \\delta_{kb} \\delta_{la}), \\nonumber \\\\\n   \\left|kl\\right\\rangle_{5} & = \\frac{1}{\\sqrt{2}} (\\delta_{ka} \\delta_{la} - \\delta_{kb} \\delta_{lb}), \\nonumber \\\\\n   \\left|kl\\right\\rangle_{6,7,8} & = \\frac{1}{\\sqrt{2}} (\\delta_{ka} \\delta_{lb} + \\delta_{kb} \\delta_{la}), \\nonumber \\\\\n   \\left|kl\\right\\rangle_{9} & = \\frac{1}{\\sqrt{6}} (\\delta_{ka} \\delta_{la} + \\delta_{kb} \\delta_{lb}- 2 \\delta_{kc} \\delta_{lc}).\n\\end{align}\nThe first eigenchannel is the scalar mode, relating uniformly pairs of identical polarization components ($xx$, $yy$ and\n$zz$), which describe the classical intensity, between themselves. The other eigenchannels either redistribute\nnonuniformly the energy between pairs of identical polarization ($p=5$ and $9$), thereby participating as well in the\npropagation of the classical intensity, or are concerned with pairs of orthogonal polarizations ($xy$, $xz$, etc), which\ncan participate, for instance, in magneto-optical media in which light polarization can rotate~\\cite{MacKintosh1988,\nVanTiggelen1996, VanTiggelen1999}.\n\nThe eigenvalues take the form of the solution of the diffusion equation in reciprocal space\n\\begin{equation}\\label{eq:diffusionsolution}\n   U^{(p)}(\\bm{K}) = \\frac{1}{\\mathcal{D}^{(p)} K^2 + \\mu_a^{(p)} c},\n\\end{equation}\nwhere $\\mathcal{D}^{(p)}$ and $\\mu_a^{(p)}$ are the diffusion constant and attenuation coefficient of the $p$th\npolarization mode. The eigenmode energy densities in real space therefore read\n\\begin{equation}\n   U^{(p)}(\\bm{R}) = \\frac{1}{4\\pi \\mathcal{D}^{(p)} R} \\exp\\left[- \\frac{R}{\\ell_\\text{eff}^{(p)}} \\right],\n\\end{equation}\nwith $R=|\\bm{R}|$ and $\\ell_\\text{eff}^{(p)}=\\sqrt{\\mathcal{D}^{(p)}\/\\mu_a^{(p)} c}$, which is an effective attenuation\nlength, describing the depolarization process.\n\nTable~\\ref{tab:diffcorr} summarizes the diffusion constants, attenuation coefficients and effective attenuation lengths\nof the different polarization eigenchannels. As in the case of uncorrelated\ndisorder previously studied in Ref.~\\cite{Vynck2014}, all modes exhibit different diffusion constants, thereby spreading at different speeds, and\nonly the scalar mode persists at large distances ($\\ell_\\text{eff}^{(1)}=\\infty$), all other modes being attenuated on a\nlength scale on the order of a mean free path.\n\n\\begin{table*}\n   \\caption{Summary of the diffusion constants $\\mathcal{D}^{(p)}$, attenuation coefficients $\\mu_a^{(p)}$ and effective\n   attenuation lengths $\\ell_\\text{eff}^{(p)}$ characterizing the diffusion properties of the energy density through the\n   individual polarization eigenchannels and the depolarization process. Note that all quantities are given to order\n   $g$. Quite remarkably, structural correlations, via the scattering asymmetry factor $g$, are found to affect\n   differently and independently each mode.}\\label{tab:diffcorr}\n   \\begin{ruledtabular}\n      \\def1.5{1.5}\n      \\begin{tabular}{c|c|c|c|c|c|c}\n         $p$ & 1 & 2 & 3,4 & 5,6 & 7,8 & 9 \\\\ \\hline\n         $\\mathcal{D}^{(p)}$ & $ \\left(1-g\\right)^{-1} \\frac{c\\ell}{3}$ & $\\left(\\frac{1}{2}-\\frac{9}{20}g\\right)^{-1} \\frac{c\\ell}{3}$ & $\\left(\\frac{1}{2}-\\frac{3}{20}g\\right)^{-1} \\frac{c\\ell}{3}$ & $\\left(\\frac{7}{10}-\\frac{69}{100}g\\right)^{-1} \\frac{c\\ell}{3}$ & $\\left(\\frac{7}{10}-\\frac{39}{100}g\\right)^{-1} \\frac{c\\ell}{3}$ & $\\left(\\frac{7}{10}-\\frac{29}{100}g\\right)^{-1} \\frac{c\\ell}{3}$ \\\\ \\hline\n        \n         $\\mu_a^{(p)}$ & $0$ & $\\frac{1}{\\ell}$ & $\\frac{1}{\\ell}$ & $\\frac{3}{7\\ell}$ & $\\frac{3}{7\\ell}$ & $\\frac{3}{7\\ell}$ \\\\ \\hline\n         $\\ell_\\text{eff}^{(p)}$ & $\\infty$ & $\\left( 1-\\frac{9}{20}g \\right)^{-1} \\sqrt{\\frac{2}{3}}\\ell$ & $\\left( 1-\\frac{3}{20}g \\right)^{-1} \\sqrt{\\frac{2}{3}}\\ell$ & $\\left( 1-\\frac{69}{140}g \\right)^{-1} \\frac{\\sqrt{10}}{3}\\ell$ & $\\left( 1-\\frac{39}{140}g \\right)^{-1} \\frac{\\sqrt{10}}{3}\\ell$ & $\\left( 1-\\frac{29}{140}g \\right)^{-1} \\frac{\\sqrt{10}}{3}\\ell$\n        \n      \\end{tabular}\n   \\end{ruledtabular}\n\\end{table*}\n\nMore interestingly, our study brings new information on the influence of short-range structural correlations on transport and depolarization.\nLet us first remark that we properly recover the diffusion constant of the scalar mode, $\\mathcal{D}=c\\ell^*\/3$ with $\\ell^*=\\ell\/(1-g)$ the\ntransport mean free path, which is a good indication of the validity of the model. The second and\nmore interesting finding in this study is the fact that the propagation characteristics of each polarization mode is\naffected independently and differently by short-range structural correlations. One may have anticipated that the\ndiffusion constant of each polarization mode would be simply rescaled by the $(1-g)^{-1}$ factor relating scattering and\ntransport mean free paths. Instead, we show that a transport mean free path can be defined for each polarization mode,\n$\\ell^{*(p)}=3\\mathcal{D}^{(p)}\/c$ and its dependence on the anisotropy factor $g$ can change significantly, as shown in Fig.~\\ref{fig1}(a). This, in\nturn, implies that the spatial attenuation of each polarization mode (due to depolarization) is affected differently by\nstructural correlations, as shown in Fig.~\\ref{fig1}(b).\n\n\\begin{figure}\n   \\centering\n\t\\includegraphics[width=0.5\\textwidth]{fig1}\n   \\caption{(Color online only) Evolution of (a) the\n   transport coefficient, $1\/\\ell^{*(p)}$, and (b) the attenuation coefficient, $1\/\\ell_\\text{eff}^{(p)}$, of\n   polarization eigenmodes with short-range structural correlations. The coefficients are given in units of $1\/\\ell$ and\n   shown on a restricted range of $g$ since the model is expected to remain valid to first order near $g=0$. The scalar\n   mode ($p=1$, cyan solid curve) has a transport coefficient scaling as $(1-g)$ and an attenuation coefficient equal to\n   zero (not shown). The polarization modes $p=2$--$4$ (gray dashed curves) and $p=5$--$9$ (orange dot-dashed curves)\n   exhibit different slopes, indicating that both their transport properties are affected differently by short-range\n   structural correlations.}\n   \\label{fig1}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:4}\n\nPrevious studies based on the multiple scattering theory for the propagation of polarized light relied on the\n\\textit{direct} resolution of the Bethe-Salpeter equation, Eq.~(\\ref{eq:BS_green_fourier}), using an expansion of the average Green's\ntensors and its correlation function to order $K^2$ (diffusion approximation). This strategy is however\npossible only for uncorrelated disorder, for which $f(\\bm{q}-\\bm{q}')=1$. Here, we proposed an alternative strategy based on\nthe derivation of a transport equation taking the form of an RTE, which allowed us to reach the same final goal (eigenmode decomposition) including\nshort-range structural correlations. This strategy, however, involves an additionnal approximation that has some implications. To\nclarify this point, let us consider our predictions in the limit of an uncorrelated disorder. Setting $g=0$ in the\npredictions of Table~\\ref{tab:diffcorr} yields the values reported in Table~\\ref{tab:diffuncorr}. An alternative\nstraightforward derivation from Eqs.~(\\ref{eq:zeroth_moment}) and (\\ref{eq:first_moment}), which yields the same\nresults, is proposed in Appendix~\\ref{sec:A3}. Compared to previous results (see, e.g.\\@\\xspace, Ref.~\\onlinecite{Vynck2014}), we\nobserve that the eigenvectors, or polarization eigenchannels, remain unchanged, but the eigenvalues are now 1, 3 and\n5-fold degenerate, yielding the same attenuation coefficients $\\mu_a^{(p)}$ but different diffusion constants\n$\\mathcal{D}^{(p)}$. This apparent discrepancy can be explained by the on-shell approximation, which ``smoothes out''\nthe polarization dependence in the correlation function of Green's tensor. Nevertheless, it is important to note that the\n\\textit{average} diffusion constants for the various degenerate modes are strictly identical:\n\\begin{equation}\n\\frac{1}{3} \\left(\\frac{6}{5}c\\ell + 2\\frac{2}{5}c\\ell \\right) = 2 \\frac{c\\ell}{3},\n\\end{equation}\nand\n\\begin{equation}\n\\frac{1}{5} \\left(2\\frac{230}{343}c\\ell + 2\\frac{130}{343}c\\ell + \\frac{290}{1029}c\\ell \\right) = \\frac{10}{7}\\frac{c\\ell}{3}\n\\end{equation}\nThis brings us to the conclusion that the model is consistent with the approximations that have been made.\n\n\\begin{table}[H]\n   \\caption{Summary of the diffusion constants $\\mathcal{D}^{(p)}$, attenuation coefficients $\\mu_a^{(p)}$ and effective\n   attenuation lengths $\\ell_\\text{eff}^{(p)}$ characterizing the diffusion properties of the energy density through the\n   individual polarization eigenchannels for an uncorrelated disorder ($g=0$).}\\label{tab:diffuncorr}\n   \\begin{ruledtabular}\n      \\def1.5{1.5}\n      \\begin{tabular}{l|l|l|l}\n         $p$ & 1 & 2-4 & 5-9 \\\\ \\hline\n         $\\mathcal{D}^{(p)}$ & $\\frac{c\\ell}{3}$ & $2 \\frac{c\\ell}{3}$ & $\\frac{10}{7}\\frac{c\\ell}{3}$ \\\\ \\hline\n         $\\mu_a^{(p)}$ & $0$ & $\\frac{1}{\\ell}$ & $\\frac{3}{7\\ell}$ \\\\ \\hline\n         $\\ell_\\text{eff}^{(p)}$ & $\\infty$ & $\\sqrt{\\frac{2}{3}}\\ell$ & $\\frac{\\sqrt{10}}{3}\\ell$\n      \\end{tabular}\n   \\end{ruledtabular}\n\\end{table}\n\nA second point deserving a comment is the fact that the attenuation length $1\/\\mu_a^{(p)}$ of the polarization eigenmodes\ndoes not depend on $g$ to first order, the effect of short-range structural correlations on the spatial\ndecay of polarization away from the source being implemented via the definition of mode-specific transport mean free\npaths. This picture contrasts with previous studies based on the phenomenological transfer matrix approach~\\cite{Xu2005,\nRojas-Ochoa2004a}, which relate the depolarization length $\\ell_p$ for linearly polarized light to the \\textit{scalar}\ntransport mean free path via a linear relation with $g$. In this sense, our model provides a different perspective on\nthis basic problem of light transport in disordered media. Intuitively, this picture also appears more physically sound,\nsince it is known that the relation between depolarization and transport mean free path varies with the incident\npolarization (linear, circular) or in presence of magneto-optical effects~\\cite{MacKintosh1988, VanTiggelen1996}.\n\nRelated to this point, it is also important to discuss the validity of the diffusion limit to retrieve depolarization\ncoefficients. Reaching the regime of diffusive transport typically requires light to experience several multiple\nscattering events. However, as pointed out previously (see, e.g.\\@\\xspace, Ref.~\\onlinecite{Gorodnichev2014}), this limit can hardly be\nachieved for the polarization modes, for which the depolarization occurs on the scale of a mean free path. It is then\nlegitimate to question the accuracy of the expressions reported in Table~\\ref{tab:diffcorr}. Nevertheless,\nwe do not expect this question to impact our claim that different polarization modes are individually and differently\naffected by short-range structural correlations. Actually, the established RTE for the polarization-resolved specific\nintensity, Eq.~(\\ref{eq:RTE}), like the standard vector radiative transfer equation, does not assume diffusive\ntransport. On this aspect, our study constitutes a very good starting point to investigate the validity of the diffusion\napproximation, which may be done either numerically by solving the RTE by Monte-Carlo methods, or analytically by adding\nhigher-order Legendre polynomials $P_n$ in the following steps.\n\nFinally, let us remark that the results of our model, in which disorder is described by a continuous and randomly fluctuating function of position [Eq.~(\\ref{eq:disorder})], should apply not only to heterogeneous materials with complex textit{connected} morphologies (e.g., porous media) but also to random ensembles of finite-size scatterers. Indeed, the Fourier transform of the structural correlation $f(\\bm{r}-\\bm{r}')$ directly leads to the definition of the phase function $p(\\hat{\\bm{q}}\\cdot\\hat{\\bm{q}}')$ [Eq.~(\\ref{eq:correlation-phase})], which is the same function to which one arrives when investigating light scattering by finite-size scatterers (it is, in this case, defined from the differential scattering cross-section). For the sake of broadness of applications and convenience, the final results here have been given for the HG phase function [Eq.~(\\ref{eq:HG-phasefunction})] but other phase functions (e.g., Mie for spherical scatterers) may be used to describe specific disordered media. Note that for ensembles of finite-size scatterers, the short-range correlation approximation restricts the validity range of the model to small scatterers.\n\n\\section{Conclusion}\\label{sec:5}\n\nTo conclude, we have proposed a model based on multiple scattering theory to describe the propagation of polarized light\nin disordered media exhibiting short-range structural correlations. Our results assume weak disorder ($k_0\\ell \\gg 1$),\nshort-range structural correlations (first order in $g$), and are obtained in the ladder approximation. Starting from the exact\nDyson and Bethe-Salpeter equations for the average field and the field correlation, we have derived a RTE for the\npolarization-resolved specific intensity [Eq.~(\\ref{eq:RTE})] and applied the $P_1$ approximation to investigate the\npropagation of polarized light in the diffusion limit. Interestingly, we have found that the polarization modes, described\nso far for uncorrelated disorder only, are independently and differently affected by short-range structural\ncorrelations. In practice, each mode is described by its own transport mean free path, which does not trivially depend\non $g$ (see Table~\\ref{tab:diffcorr}). \n\nIn essence, our study partly unveils the intricate relation between the complex morphology of disordered media and the\npolarization properties of the scattered intensity. The road towards a possible description of polarization-related\nmesoscopic phenomena in correlated disorder is long, yet we hope that the present work, which highlights several\ntheoretical challenges when dealing with polarized light and structural correlations, will motivate future\ninvestigations. The model may be generalized, for instance, by including the most-crossed diagrams in the derivation to\nenable the study of phenomena such as weak localization, or frequency dependence to investigate ---via a generalized\nRTE--- the temporal response to incident light pulses. Another line of research could be to study the impact of\nshort-range structural correlations on spatial coherence properties, which appears extremely relevant to the optical\ncharacterization of complex nanostructured media~\\cite{Dogariu2015}.\n\n\\section*{Acknowledgements}\n\nThe authors acknowledge John Schotland for stimulating discussions. This work is supported by LABEX WIFI (Laboratory of\nExcellence within the French Program ``Investments for the Future'') under references ANR-10-LABX-24 and\nANR-10-IDEX-0001-02 PSL$^*$, by INSIS-CNRS via the LILAS project and the CNRS ``Mission for Interdisciplinarity''\nvia the NanoCG project.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe work of Seiberg and Witten \\cite{sewi,sewi2} on the exact solution \nto the low-energy effective action of $N=2$ supersymmetric Yang-Mills theory \nhas afforded not only a renewed insight into the charge confinement \n\\cite{sewi2} and the chiral symmetry breaking \\cite{sewi2}, \nbut also a marvelous insight into topological \ninvariants \\cite{wi} and conformal field theories in four \ndimensions \\cite{scft}. The key ingredients in obtaining these exact results\nare duality and the appearance of massless monopoles\/dyons in the strong\ncoupling regions of the theory. \nIn the weak coupling region, the exact solution enables us to\ndetermine full non-peturbative instanton corrections to the \neffective coupling constant, whose evaluation is otherwise quite cumbersome\nin the standard framework of quantum field theory.\n\nThe low-energy effective action is described in terms of a prepotential\nwhich is a single holomorphic function of superfields of  $N=2$ \n$U(1)$ vector multiplets. \nThe exact solution for the prepotential may be \ncharacterized by the period integrals of \nthe special type of one-form on a hyper-elliptic curve.\nThe curves associated with a variety of $N=2$ \nsupersymmetric Yang-Mills theories and QCD \nhave been studied extensively \\cite{KlLeThYa}$-$\\cite{itsa}.\nThe moduli space of the curve contains singularities, at which \nsome solitons become massless. \nIn order to investigate the strong coupling physics, one needs to \nevaluate the period integrals near the singular locus.\nAn efficient approach to study this problem is to use the Picard-Fuchs \nequation, the differential \nequation which the periods obey \\cite{Cer,KlLeTh,Ma,ItYa}.\n\nIn the present article, we shall investigate the quantum moduli space\nof $N=2$ supersymmetric Yang-Mills theories with massless hypermultiplets\nand gauge groups $SU(2)$ and $SU(3)$.\nWe shall derive the Picard-Fuchs equation for the scalar part of $N=2$ \n$U(1)$ vector multiplets and their duals.\nBy solving the non-linear differential equation obeyed by the prepotential\nfor $G=SU(2)$, we will explicitly evaluate\nthe non-perturvative contributions in the prepotential \nin both weak and strong coupling regions.\n\n\n\\section{$N=2$ Supersymmetric Yang-Mills Theory with Massless Hypermultiplets}\nWe begin with reviewing some basic properties of the low-energy \neffective action of the $N=2$ supersymmetric $G=SU(N_{c})$ QCD\n\\cite{sewi2,arplsh}. \nIn $N=1$ superfield formulation, \nthe theory contains chiral \nmultiplets $\\Phi^{a}$ and  chiral field strength $W^{a}$ \n($a=1,\\ldots, {\\rm dim}G$) both in the adjoint representation of $G$, \nand chiral superfields $Q^{i}$ in the $N_{c}$ and \n$\\tilde{Q}^{i}$ ($i=1,\\cdots, N_{f}$)\nin the $\\bar{N}_{c}$ representation of the gauge group.\nThe superpotential is given by \n${\\cal W}=\\tilde{Q}T^{a}\\Phi^{a}Q+M^{i}_{j}\\tilde{Q}_{i} Q^{j}$\nwhere $T^{a}$ is the generator of $G$ and $M^{i}_{j}$ is the mass matrix.\nAlong the flat direction the scalar fields $\\phi$ of $\\Phi$ get\nvacuum expectation values, which break the gauge group to the Cartan subgroup \n$U(1)^{r}$ where $r=N_{c}-1$ is the rank of $G=SU(N_{c})$. \nWhen the squark fields do not have vacuum expectation values, the low-energy\neffective theory is in \nthe Coulomb branch and contains $r$ $U(1)$ vector multiplets \n$(A^{i}, W_{\\alpha}^{i})$ $(i=1,\\cdots,r)$, where $A^{i}$ are $N=1$\nchiral superfields and $W_{\\alpha}^{i}$ are $N=1$ vector superfields. \nThe quantum moduli space may be characterized by \nthe low-energy effective Lagrangian ${\\cal L}$ \nwith the prepotential ${\\cal F}(A)$\n\\begin{equation}\n{\\cal L}={1\\over 4\\pi} {\\rm Im }\n\\left(\\int d^2\\theta d^2\\bar{\\theta}A_{D i}\\bar{A}^{i}+{1\\over2}\n\\int d^2\\theta \\tau^{i j}W_{\\alpha}^{i} W^{j \\alpha}\\right), \n\\end{equation}\nwhere $A_{D i}={\\partial {\\cal F}\\over \\partial A^{i}}$ is a field dual to $A^{i}$ \nand \n$\\tau^{i j}={\\partial^{2}{\\cal F}\\over \\partial A^{i} \\partial A^{j}}$ the effective\ncoupling constants.\nWe denote the scalar component of $A^{i}$, $A_{D i}$ by $a^{i}$, $a_{D i}$.\nThe pairs $(a_{D i},a_{i})$ are the $Sp(2r,{\\bf Z})$ section over the \nspace of gauge invariant parameters $s_{i}$ $(i=2,\\cdots, N_{c})$\ndefined by ${\\rm det}(x-\\phi)=x^{N_{c}}-\\sum_{i=2}^{N_{c}} s_{i} x^{N_{c}-i}$.\nThe quantum moduli space of the Coulomb branch is parametrized by the \ngauge invariants $s_{i}$ and the eigenvalues $m_{1},\\ldots, m_{N_{f}}$ \nof the mass matrix.\n\nThe sections $(a_{D i},a_{i})$ are obtained as\nthe period integrals of a meromorphic differential\n$\\lambda$ over the hyper-elliptic curve ${\\cal C}$ \\cite{haoz}:\n\\begin{eqnarray}\ny^{2}&=&C(x)^2-G(x), \\nonumber \\\\\nC(x)&=& x^{N_{c}}-\\sum_{i=2}^{N_{c}} s_{i} x^{N_{c}-i}\n       +{\\Lambda^{2N_{c}-N_{f}}\\over 4} \\sum_{i=0}^{N_{f}-N_{c}}\n t_{i}(m) x^{N_{f}-N_{c}-i},\n\\nonumber \\\\\nG(x)&=& \\Lambda^{2N_{c}-N_{f}}\\prod_{i=1}^{N_{f}}(x+m_{i}),\n\\end{eqnarray}\nwhere $\\Lambda$ stands for the QCD scale parameter and $t_{i}(m)$ is defined by\n$\\prod_{i=1}^{N_{f}}(x+m_{i})=\\sum_{i=0}^{N_{f}} t_{i}(m) x^{N_{f}-i}$ with\n$t_{0}(m)=1$.\nThe terms proportional to $\\Lambda^{2N_{c}-N_{f}}$ in $C(x)$ are absent \nin the case of $N_{f}<N_{c}$. Notice that for $N_c=2$, $N_f=2$ theory\n$C(x)$ should read\n\\begin{equation}\nC(x)=x^2-s_2+{\\Lambda^2\\over 8}.\n\\end{equation}\n\n\nThe meromorphic differential $\\lambda$ is given by\n\\begin{equation}\n\\lambda={x\\over 2\\pi i} d\\log {C-y\\over C+y}.\n\\end{equation}\nThis differential satisfies the relations\n\\begin{equation}\n{\\partial \\lambda \\over \\partial s_{i}}=-\\omega_{N_{c}+1-i}+d(*),\n\\end{equation}\nwhere $\\omega_{i}={x^{i-1}dx \\over y}$ is the basis of the holomorphic \none-forms of the hyper-elliptic curve ${\\cal C}$.\nChoosing the basis of homology cycles $(A_{i},B_{i})$ on ${\\cal C}$ with\nthe canonical intersection form $A_{i}\\cdot B_{j}=\\delta_{i j}$, \n$A_{i}\\cdot A_{j}=B_{i}\\cdot B_{j}=0$, one can express \nthe pairs $(a_{D i},a_{i})$ as\n\\begin{eqnarray}\n\\alpha_{i}\\cdot a_{D} &= & \\int_{B_{i}}\\lambda, \\nonumber \\\\\n(\\lambda_{i}-\\lambda_{i-1})\\cdot a &=& \\int_{A_{i}}\\lambda,  \\hskip7mm\ni=1,\\cdots,r,\n\\end{eqnarray}\nwhere $\\alpha_{i}$ are the simple roots of $G$ and $\\lambda_{i}$ \nthe fundamental weights of $G$ ($\\lambda_{0}\\equiv0$).\n\nThe BPS mass formula $M\\geq |q_{i}a_{i}+h_{i}a_{D i}|$\nshows that the massless soliton may be characterized by the vanishing cycle\n$q_{i}A_{i}+h_{i}B_{i}\\sim 0$.\nTherefore the singularities of the quantum moduli space are determined by the \ndiscriminant $\\Delta$ of the curve.\nLet us enumerate explicit forms of the discriminants $\\Delta^{N_{c}}_{N_{f}}$ \nfor the $SU(N_c)$ theories ($N_c =2,\\, 3$) with $N_{f}(<2 N_{c})$ matter\nhypermultiplets. In the following \n$\\Lambda_{N_{f}}$ denotes the scale parameter corresponding to \nthe $N_{f}$-flavor theory.\n\\begin{itemize}\n\\item $N_{c}=2$ ($u\\equiv s_{2}$)\n\\begin{eqnarray}\n\\Delta^{2}_{0}&=& 256 \\Lambda_{0}^{8} (u-\\Lambda_{0}^2)(u+\\Lambda_{0}^2)\n\\nonumber \\\\\n\\Delta^{2}_{1}&=& -\\Lambda_{1}^{6} (27 \\Lambda_{1}^{6}+256 u^3) \\nonumber \\\\\n\\Delta^{2}_{2}&=& \\Lambda_{2}^4 (8 u-\\Lambda_{2}^2)^2 (8 u+\\Lambda_{2}^2)^2\/16\n\\nonumber \\\\\n\\Delta^{2}_{3}&=& \\Lambda_{3}^{2} (\\Lambda_{3}^{2}-256 u) u^4 \\nonumber \\\\\n\\end{eqnarray}\n\\item $N_{c}=3$ ($u\\equiv s_{2}$, $v\\equiv s_{3}$)\n\\begin{eqnarray}\n\\Delta^{3}_{0}&=& 64 \\Lambda_{0}^{18}\n       (-27 (v-\\Lambda_{0}^3)^2+4 u^3)\n                    (-4 u^3+27 (v+\\Lambda_{0}^3)^2)\n\\nonumber \\\\\n\\Delta^{3}_{1}&=&\\Lambda_{1}^{15}\n (-3125 \\Lambda_{1}^{15}+ 256 \\Lambda_{1}^{5}  u^{5}\n  + 22500 \\Lambda_{1}^{10}   u v - 1024 u^{6}  v \\nonumber \\\\\n& & -\n 43200 \\Lambda_{1}^{5}  u^2  v^2  + 13824 u^3  v^3  - 46656 v^5 ) \\nonumber \\\\\n\\Delta^{3}_{2}&=&  64 \\Lambda_{2}^{12} v^2 (-4 (\\Lambda_{2}^2-u)^3 - 27 v^2) \n(-4 (\\Lambda_{2}^2+u)^3+ 27 v^2)\n\\nonumber \\\\\n\\Delta^{3}_{3}&=& \\Lambda_{3}^9  (-\\Lambda_{3}^3  + 4 v)^3\n   (729 \\Lambda_{3}^6  u^3  - 256 u^6  - 3888 \\Lambda_{3}^3  u^3  v \\nonumber \\\\\n  & & + 3456 u^3  v^2  +\n      2916 \\Lambda_{3}^3  v^3  - 11664 v^4 )) \/ 16 \\nonumber \\\\\n\\Delta^{3}_{4}&=& 4 \\Lambda_{4}^6 v^4 (\\Lambda_{4}^4 u - 8 \\Lambda_{4}^2 u^2 +\n 16 u^3 + 2 \\Lambda_{4}^3 v - 72 \\Lambda_{4} u v - 108 v^2)\\nonumber  \\\\\n& &    (-\\Lambda_{4}^4 u + 8 \\Lambda_{4}^2 u^2 - 16 u^3 + 2 \\Lambda_{4}^3 v\n- 72 \\Lambda_{4} u v + 108 v^2) \\nonumber \\\\\n\\Delta^{3}_{5}&=& 4 \\Lambda_{5}^3  v^5  (\\Lambda_{5}^2  u^5  - 256 u^6  +\n2 \\Lambda_{5}^3  u^3  v - 528 \\Lambda_{5} u^4  v +\n \\Lambda_{5}^4  u v^2 \\nonumber \\\\\n& &  -\n300 \\Lambda_{5}^2  u^2  v^2  + 3456 u^3  v^2  + 4 \\Lambda_{5}^3  v^3  -\n4212 \\Lambda_{5} u v^3  - 11664 v^4 ). \\nonumber \\\\\n\\end{eqnarray}\n\\end{itemize}\nNote that for the gauge group $SU(2)$ the zeros of the discriminant realize \nthe discrete\n${\\bf Z}_{4-N_{f}}$ symmetry which arises from the $U(1)_{\\cal R}$ \nanomaly and the parity transformation in the flavor symmetry $O(2 N_{f})$.\nOrder of the zeros corresponds to the number\nof massless hypermultiplets appearing at the singularities. \nMassless monopoles obey the spinor representation of the flavor \nsymmetry $SO(2 N_{f})$. \nFor $SU(3)$ symmetries are less obvious  particularly for odd flavors.\n\n\n\\section{The Picard-Fuchs equations}\nLet us study the Picard-Fuchs equations for \n$(a_{D i},a_{i})$ and examine their solutions.\nWe first discuss the $SU(2)$ case in detail for simplicity. \nOne can change the curve of fourth order to the third one by \nusing the M\\\"obius transformation. \nThe curves are given by\n\\begin{equation}\ny^2=x^2(x-u)+{1 \\over 4} \\Lambda_0^4 \\, x\n\\end{equation}\nfor $N_f=0$ and\n\\begin{equation}\ny^2=x^2(x-u)-{1 \\over 64} \\Lambda_{N_f}^{2(4-N_f)} (x-u)^{N_f-1}\n\\end{equation}\nfor $N_f=1,\\, 2,\\, 3$.\nThe meromorphic differential $\\lambda$ for this curve\nis given by \n\\begin{equation}\n\\lambda ={\\sqrt{2} \\over 8\\pi}\\, {2u-(4-N_f)x \\over y}\\, dx\n\\end{equation}\nwhich satisfies\n\\begin{equation}\n{d\\lambda \\over du}={\\sqrt{2} \\over 8\\pi} {d x \\over y}+d(*).\n\\end{equation}\nIt is not difficult to see that the period $\\Pi=\\oint \\lambda$ satisfies the \ndifferential equation \n\\begin{equation}\np(u){d^2 \\Pi \\over du^2}+ \\Pi =0,\n\\label{pfsu2}\n\\end{equation}\nwhere \\cite{Cer,KlLeTh,ItYa}\n\\begin{equation}\np(u)=\\left\\{ \\begin{array}{ll}\n\\displaystyle{4 (u^2-\\Lambda_{0}^4)} & \\qquad N_{f}=0 \\\\\n\\displaystyle{4u^2+{27 \\Lambda_1^6\\over 64u}} & \\qquad N_{f}=1 \\\\\n\\displaystyle{\\ 4\\Big(u^2-{\\Lambda_2^4\\over 64}\\Big)} & \\qquad N_{f}=2 \\\\\n\\displaystyle{u\\Big(4u-{\\Lambda_3^2\\over 64}\\Big)} & \\qquad N_{f}=3.\n\\end{array}\n\\right.\n\\end{equation}\nNote that the regular singularities of (\\ref{pfsu2}) correspond to the\nzeros of the discriminant. Introducing a  new variable\n\\begin{equation}\nz=\\left\\{\\begin{array}{ll}\n\\displaystyle{\\left({u\\over \\Lambda_{0}^2}\\right)^2} & \\qquad N_{f}=0 \\\\\n\\ \\displaystyle{-{256\\over 27} \\left({u\\over \\Lambda_{1}^2}\\right)^3}\n & \\qquad N_{f}=1 \\\\\n\\displaystyle{64 \\left({u\\over \\Lambda_{2}^2}\\right)^2 }& \\qquad N_{f}=2 \\\\\n\\displaystyle{256 \\left({u\\over \\Lambda_{3}^2}\\right)} & \\qquad N_{f}=3, \\\\\n\\end{array}\n \\right.\n\\end{equation}\nit is shown that the Picard-Fuchs equation (\\ref{pfsu2}) turns out to be the\nGauss's hypergeometric system\n\\begin{equation}\nz(1-z) {d^{2} \\Pi \\over d z^{2}}\n+(\\gamma-(\\alpha+\\beta+1)z){d \\Pi \\over d z}-\\alpha\\beta \\Pi=0,\n\\label{hyperg}\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha=\\beta={-1\\over 2(4-N_{f})}, \\hskip10mm  \n\\gamma={3-N_{f} \\over 4-N_{f}}.\n\\end{equation}\nUsing the fundamental solutions of (\\ref{hyperg}), we may evaluate \nthe power series solutions for $(a_{D},a)$ near the singularities, \nwhich has been  explicitly done in our previous work \\cite{ItYa}.\n\nNext we write down the Picard-Fuchs equations for $SU(3)$ theory \nwith flavors $N_{f}\\leq 5$.\nThe periods $\\Pi=\\oint \\lambda$ now depend on two parameters $u$ and $v$.\nThe Picard-Fuchs equations take the form\n\\begin{equation}\n{\\cal L}_{1}\\Pi=0, \\quad {\\cal L}_{2}\\Pi=0,\n\\end{equation}\nwhere ${\\cal L}_{1}$ and ${\\cal L}_{2}$ are given by\n\\begin{itemize}\n\\item $N_{f}=0$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {1\\over 3} (-27\\Lambda_{0}^{6}+4 u^3+27 v^2)\\partial_{v}^{2}\n          +12 u v \\partial_{u}\\partial_{v}\n          +3 v \\partial_{v}+1 \\nonumber \\\\\n{\\cal L}_{2}&=& {1\\over u } (-27\\Lambda_{0}^{6}+4 u^3+27 v^2)\\partial_{u}^{2}\n          +12 u v \\partial_{u}\\partial_{v}\n          +3 v \\partial_{v}+1\n\\end{eqnarray}\n\\item $N_{f}=1$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {-25\\Lambda_{1}^{5}u^2+84 u^3 v +405 v^3 \\over 45 v}\\partial_{v}^{2}\n          +{64 u^4-1125 \\Lambda_{1}^{5} v+2160 u v^{2}\\over 180 v}\n \\partial_{u}\\partial_{v} \\nonumber \\\\\n       & &+ {16 u^3+135 v^2 \\over 45 v} \\partial_{v}+1  \\nonumber \\\\\n{\\cal L}_{2}&=& {4 (-25\\Lambda_{1}^{5}u^2+84 u^3 v +405 v^3)\\over\n            -25 \\Lambda_{1}^5 +84 u v} \\partial_{u}^{2} \\nonumber \\\\\n& &\n+{(-25 \\Lambda_{1}^5 +36 u v) (-25 \\Lambda_{1}^5+96 u v)\\over -25 \\Lambda_{1}^{5} +84 u v}\n \\partial_{u}\\partial_{v} \\nonumber \\\\\n & &    +  {3 v (-25 \\Lambda_{1}^5 +36 u v)\\over -25 \\Lambda_{1}^5 +84 u v} \\partial_{v}\n          +1\n\\end{eqnarray}\n\\item  $N_{f}=2$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {-8\\Lambda_{2}^{4}u+8 u^3  +27 v^2\\over 3}\\partial_{v}^{2}\n+{4 (2 \\Lambda_{2}^8 -4 \\Lambda_{2}^4 u^2 +2 u^4+27 u v^2)\\over 9 v}\n \\partial_{u}\\partial_{v} \\nonumber \\\\\n  & &     + { -8 \\Lambda_{2}^4 u +8 u^3 +27 v^2 \\over 9 v} \\partial_{v}+1  \\nonumber \\\\\n{\\cal L}_{2}&=& {-8\\Lambda_{2}^{4}u+8 u^3  +27 v^2\\over 2 u} \\partial_{u}^{2}\n          + {3v (\\Lambda_{2}^4+3 u^2)\\over u} \\partial_{u}\\partial_{v}     +  1\n\\end{eqnarray}\n\\item  $N_{f}=3$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {( 4 v-\\Lambda_{3}^3) (4 u^3+9 v^2)\\over 4 v} \\partial_{v}^{2}\n+{u (64 u^3-81 \\Lambda_{3}^3 v +432  v^2) \\over 36 v}  \\partial_{u}\\partial_{v} \\nonumber \\\\\n       & &     + { 16 u^3 +27 v^2 \\over 9 v} \\partial_{v}+1  \\nonumber \\\\\n{\\cal L}_{2}&=& {4 u^3+9 v^2\\over u} \\partial_{u}^{2}\n          +{u (-9 \\Lambda_{3}^3 +32  v)\\over 4} \\partial_{u}\\partial_{v}\n          - v  \\partial_{v}\n                +1\n\\end{eqnarray}\n\n\\item  $N_{f}=4$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& { -\\Lambda_{4}^4 u -56 \\Lambda_{4}^2 u^2+240 u^3+324 v^2\\over 36} \\partial_{v}^{2}\n+ {-8 \\Lambda_{4}^2 u^2 +32 u^3+27 v^2 \\over 9 v} \\partial_{v} \\nonumber \\\\\n& &\n +{2 \\Lambda_{4}^4 u^2 -16 \\Lambda_{4}^2 u^3 +32 u^4\n  -9 \\Lambda_{4}^2 v^2  +108 u v^2\\over 9 v} \\partial_{u}\\partial_{v}\n         +1  \\nonumber \\\\\n{\\cal L}_{2}&=&\n{ -\\Lambda_{4}^4 u -56 \\Lambda_{4}^2 u^2+240 u^3+324 v^2 \\over \\Lambda_{4}^2+60 u}\n \\partial_{u}^{2} \\nonumber \\\\\n& & +{v(\\Lambda_{4}^2+12 u) (-\\Lambda_{4}^2+36 u) \\over \\Lambda_{4}^2+60 u}\\partial_{u}\\partial_{v}\n     +{3 v (\\Lambda_{4}^{2}-36 u)\\over \\Lambda_{4}^2+60 u} \\partial_{v}+1 \\nonumber \\\\\n\\end{eqnarray}\n\n\\item\n $N_{f}=5$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {v (-2\\Lambda_{5}^2 u^2 +528 u^3+57\\Lambda_{5} u v +324 v^2)\\over\n 5 \\Lambda_{5} u + 36 v}\\partial_{v}^{2} \\nonumber \\\\\n& &     +{ -5 \\Lambda_{5}^2 u^3 +1280 u^4 -\\Lambda_{5}^3 u v +332 \\Lambda_{5} u^2 v\n-4 \\Lambda_{5}^2 v^2+1728 u v^2 \\over 4 (5 \\Lambda_{5} u\n + 36 v)} \\partial_{u}\\partial_{v} \\nonumber \\\\\n & & +\n{-5 \\Lambda_{5}^2 u^2 +1280 u^3+88 \\Lambda_{5} u v \n+432 v^2 \\over 4(5 \\Lambda_{5} u + 36 v)} \\partial_{v}+\n1  \\nonumber \\\\\n{\\cal L}_{2}&=& \n{-2\\Lambda_{5}^2 u^2 +528 u^3+57\\Lambda_{5} u v +324 v^2\\over 132 u}\n \\partial_{u}^2 \\nonumber \\\\\n& & +{-\\Lambda_{5} u^3-5 \\Lambda_{5}^2 u v+864 u^2 v-36 \\Lambda_{5} v^{2}\n\\over 132 u}\\partial_{u}\\partial_{v}\n -{\\Lambda_{5} u +324 v\\over 132} \\partial_{v}+1. \\nonumber \\\\\n\\end{eqnarray}\n\\end{itemize}\nThe Picard-Fuchs equations  for $N_{c}=3$, $N_{f}=0$ theory\nhave been investigated in detail by Klemm, Lerche and Theisen \\cite{KlLeTh}.\nThey show that the system reduces to the Appell's hypergeometric system of \ntype $F_{4}$. Although other cases are not classified as the Appell's system, \nthe solutions of\nthese equations may be obtained in the form of the power series expansion \nin $u$ and $v$. \n\n\n\\section{Calculation of the Prepotentials}\nSolving the system of the Picard-Fuchs equations, we evaluate the \nnon-perturbative  corrections to the prepotentials, which can be \nregarded as the multi-instanton contributions in the weak coupling \nregion and the threshold corrections in the strong coupling regions.\nIn the previous work \\cite{ItYa}, we have calculated some \nnon-trivial  corrections by using the solutions of the hypergeometric system. \nIn this article, we shall instead take more direct approach by solving the \nnon-linear differential equation for the prepotential in $SU(2)$ theory. \nThe present approach was proposed originally by Matone for $SU(2)$ pure Yang-\nMills theory \\cite{Ma}.\n\nWe start with the Picard-Fuchs equation (\\ref{pfsu2}). \nNotice that the equation carries no first derivative term. \nThis implies that the Wronskian of the system\n\\begin{equation}\nW=\\left|\n\\begin{array}{cc}\na & a_{D} \\\\\n\\partial_{u}a & \\partial_{u}a_{D}\n\\end{array} \\right|\n\\end{equation}\nis independent of $u$; $\\partial_{u}W=0$. When we integrate this with respect to\n$u$ there appears a constant, the explicit value of which \nis evaluated at any singularity in the moduli space. One finds that\n\\begin{equation}\nW={i b \\over 4\\pi},\n\\label{wrons}\n\\end{equation}\nwhere $b=4-N_{f}$ is the coefficient of the beta-function of $SU(2)$ $N=2$ \nQCD.  \nIn the present case, we can further simplify the equation (\\ref{wrons}) since\nthe Wronskian is written in the form of the total derivative\n\\begin{equation}\nW=\\partial_{u} \\left( a {\\partial {\\cal F} \\over \\partial a} -2 {\\cal F}\\right).\n\\label{totald}\n\\end{equation}\nThus we get the following relation between the prepotential and the \ngauge invariant parameter $u$\n\\begin{equation}\na {\\partial {\\cal F} \\over \\partial a} -2 {\\cal F}={i b \\over 4\\pi}u + {\\rm const.}\n\\label{scaling}\n\\end{equation}\nThe constant term may be absorbed in the definition of ${\\cal F}$ by shifts,\nwhich does not affect the form of the low-energy effective action.\nThis identity has been generalized for any $N=2$ Yang-Mills theories with \nor without matter hypermultiplets \\cite{sothya,egya} \nand is important in view of \nits relation to the soliton theory \\cite{whto}.\n\nThe relation (\\ref{scaling}) allows us to express $u$ in terms of $a$\n\\begin{equation}\nu={\\cal G}(a)\\equiv {4\\pi \\over i b}\n\\left(a {\\partial {\\cal F} \\over \\partial a} -2 {\\cal F}\\right).\n\\end{equation}\nThe Picard-Fuchs equation becomes\n\\begin{equation}\n-p({\\cal G}(a)){d^2 {\\cal G} \\over d a^2}+\na \\left({d {\\cal G}\\over d a}\\right)^3=0.\n\\label{nonl}\n\\end{equation}\nThis non-linear equation is utilized to  determine the instanton coefficients \nrecursively.\nNear the singularity at $u=\\infty$, the prepotential takes the form\n\\begin{equation}\n{\\cal F}(a)=\n\\left\\{\\begin{array}{ll}\n{i a^{2}\\over 2\\pi}\n\\left\\{ 2 \\log {a^{2}\\over \\Lambda_{0}^{2}}\n+\\sum_{k=0}^{\\infty} {\\cal F}_{k} (0)\n\\left( {\\Lambda_{0}\\over a} \\right)^{4k}\\right\\}\n & N_{f}=0 \\\\\n{i a^{2}\\over 4\\pi}\n\\left\\{ (4-N_{f}) \\ln {a^{2}\\over \\Lambda_{N_{f}}^{2}}\n+\\sum_{k=0}^{\\infty} {\\cal F}_{k} (N_{f})\n\\left( {\\Lambda_{N_{f}}^{2} \\over a^{2}} \\right)^{k(4-N_{f})}\\right\\}\n& N_{f}=1,2,3.\n\\end{array}\\right.\n\\end{equation}\nThe first four coefficients are listed in the table\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$N_{f}$ & ${\\cal F}_{1}$ & ${\\cal F}_{2}$ & ${\\cal F}_{3}$ & ${\\cal F}_{4}$ \\\\\n\\hline\n        &                              & & & \\\\\n0       &  $-{1\\over 2^5}$              & $-{5\\over 2^{14}}$    &\n        $-{3\\over 2^{18}}$ & $-{1469\\over 2^{18}}$  \\\\\n        &                              & & & \\\\\n1       &  ${3\\over 2^{12}}$             & -${3\\cdot 17 \\over 2^{288}}$  &\n           ${5\\cdot 7 \\cdot 11\\over 3 \\cdot 2^{38}}$ &\n           ${3^2 \\cdot  5 \\cdot 7\\cdot 9679 \\over 2^{59}}$  \\\\\n        &                              & & & \\\\\n2       &  $-{1\\over 2^{11}}$\n        &  $ -{5\\over 2^{26}}$\n        &  $-{3\\over 2^{36}}$\n        &  $ -{13\\cdot 113 \\over 2^{55}}$ \\\\\n        &                              & & & \\\\\n3       &  $-{1\\over 2^{5}}$\n        &  $-{1\\over 2^{24}}$\n        &  $0$\n       &   $-{5\\over 2^{51}}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nand summarized by the formulas ($N_{f}=1,2,3$)\n\\footnote{We have corrected an error of  ${\\cal F}_{2}(N_{f})$ in the \nprevious work \\cite{ItYa}.}:\n\\begin{eqnarray}\n{\\cal F}_{1} (N_{f})&=&-{1  \\over 2^{b} c(N_{f})}\n                         {1\\over 2\\, b^2}, \\nonumber \\\\\n{\\cal F}_{2} (N_{f})&=&-{1\\over 2^{2b} c(N_{f})^{2}}\n                       {5-8b+4b^{2} \\over\n                         8^{2}\\, b^{4}}, \\nonumber \\\\\n{\\cal F}_{3} (N_{f})&=&{1\\over 2^{3b}c(N_{f})^{3}}\n                      {   (1-b)(1-2b)\n                         (-23+30b-16b^{2}) \\over\n                       1728\\, b^{6}}, \\nonumber \\\\\n{\\cal F}_{4} (N_{f})&=&{1\\over 2^{4b} c(N_{f})^{4}}\n{(1 - 2b )(-677 + 3910b - 8124b^2 +\n      8456b^3 - 4672b^4 + 1152b^5) \\over\n294912\\, b^8}, \\nonumber \\\\\n\\label{weakcorre}\n\\end{eqnarray}\nwhere \n$c(N_{f})=\\eta_{N_{f}} 2^{8} (4-N_{f})^{N_{f}-4}$ ($N_{f}=1,2,3$) and\n$\\eta_{1}=-1,\\, \\eta_{2}=\\eta_{3}=1$.\nThe $N_{f}=0$ result \\cite{KlLeTh,Ma} may be \nobtained by replacing $c(N_{f})$ with 1 in the $N_{f}=2$ case.\nThe results (\\ref{weakcorre}) indeed agree with those obtained previously\n\\cite{ItYa}.\n\nSimilarly we may solve the non-liner differential equation \nin the strong coupling region where monopoles become massless.\nIn this case the dual field $a_{D}$ is a good coordinate around the \nsingularity $u=u_{0}$ where\n\\begin{equation}\nu_{0}=\\left\\{\\begin{array}{ll}\n\\Lambda_{0}^2 & \\qquad N_{f}=0 \\\\\n\\displaystyle{-3\\cdot 2^{-{8\\over 3}}\\Lambda_{1}^2 }\n & \\qquad N_{f}=1 \\\\\n\\displaystyle{\\Lambda_{2}^2\/8} & \\qquad N_{f}=2 \\\\\n0 & \\qquad N_{f}=3. \\\\\n\\end{array}\n \\right.\n\\end{equation}\nIntroduce the dual prepotential ${\\cal F}_{D}(a_{D})$ satisfying\n$a={d {\\cal F}_{D}\\over d a_{D}}$. Then, instead of (\\ref{totald}), the\nWronskian is expressed in terms of ${\\cal F}_{D}$. Hence\naround $u=u_{0}$ we get\n\\begin{equation}\nu=u_{0}+{4\\pi\\over b} \\left( \n2  {\\cal F}_{D} -a_{D}{d  {\\cal F}_{D} \\over d a_{D}} \\right).\n\\end{equation}\nSimilarly to the weak-coupling computation we now obtain the non-linear\ndifferential equation which makes it possible to calculate ${\\cal F}_{D}$\nrecursively. Using the expansion\n\\begin{equation}\n {\\cal F}_{D}={i \\over 8\\pi} a_{D}^2\\left\\{ k \\log \n\\left({a_{D}^2\\over \\Lambda_{N_{f}}^2}\\right)\n+\\sum_{n\\geq-1} {\\cal F}_{D n}(N_{f}) \n\\left({a_{D}\\over \\Lambda_{N_{f}}}\\right)^n \\right\\}\n\\end{equation}\nwith $k=1$ ($N_{f}=0$) and $k=2^{N_{f}-1}$ ($N_{f}=1,2,3$), \nwe find the coefficients as follows\n\\begin{eqnarray}\n{\\cal F}_{D\\ -1}(N_{f})&=& 16 k b^2 \\tilde{c}(N_{f}), \\nonumber \\\\\n{\\cal F}_{D \\ 1}(N_{f})&=&{2\\cdot 2 k\n \\over 3 \\tilde{c}(N_{f})}\n                         {(1-2b)(-5+2b)\\over 16 b^{2}}, \\nonumber \\\\\n{\\cal F}_{D \\ 2}(N_{f})&=&{-2 k\\over 2 \\tilde{c}(N_{f})^{2}}\n{(1-2b)(-61+190b-92 b^{2}+8b^{3})\n\\over 1152 b^{4}}, \\nonumber \\\\\n{\\cal F}_{D \\ 3}(N_{f})&=&{2\\cdot 2k\\over 5 \\tilde{c}(N_{f})^{3}}\n{(1-2b)(1379-8162b+14596 b^{2}-7288b^{3}+960b^{4})\n\\over 110592 b^{6}}, \\nonumber \\\\\n\\end{eqnarray}\nwhere\n$\\tilde{c}(1)=-3^{-1\/2}\\cdot 2^{-17\/6}$,$\\tilde{c}(2)=-i 2^{-9\/2}$ and\n$\\tilde{c}(3)=2^{-13\/2}$.\nThe $N_{f}=0$ result is obtained by putting $b=2$ and \n$\\tilde{c}(0)=-i\/4$.\nHere we have corrected some misprints in the previous work \\cite{ItYa}.\n\nFor $SU(3)$ theory the prepotential ${\\cal F}$ obeys the basic identity\nanalogous to ({\\ref{scaling}) \\cite{sothya,egya}. \nIn contrast to the $SU(2)$ case, combining\nthis with the $SU(3)$ Picard-Fuchs equations does not yield a simple\nrecursion relation. It is not clear at present if the existence of such\nrecursion relation is peculiar to $SU(2)$ theory. It will be interesting\nif one could find an additional constraint equation for ${\\cal F}$ which\neventually leads to the direct evaluation of instanton corrections.\n\n\\vskip5mm\n\\section*{Acknowledgments}\nThe work of K.I. is supported in part by University of Tsukuba\nResearch Projects and the Grant-in-Aid for Scientific Research from\nthe Ministry of Education (No. 07210210).\nThe work of S.-K.Y. is supported in part by Grant-in-Aid for Scientific\nResearch on Priority Area 231 ``Infinite Analysis'',\nthe Ministry of Education, Science and Culture, Japan.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nPhase transitions occurring in the bulk, but driven by specific conditions at\nits boundaries, are called {\\it boundary-induced phase\n  transitions}~\\cite{HenkelSchuetz94}.  Examples include diffusive\ntransport~\\cite{Krug91,Schuetz00} and traffic flow~\\cite{PopkovEtAl01}\nmodels. A simple example for this is provided by the one-dimensional totally\nasymmetric simple exclusion process \\cite{Derrida93}, where particles enter\nthe system at the left boundary, jump to the right in the bulk, and exit at\nthe right boundary. Depending on the entering and exiting rate values, the\nsystem exhibits qualitatively different phenomenologies (maximal current,\nlarge current and low density, or small current and high density), with\nstraightforward applications to traffic flow problems.\n\nIn the present work, we are interested in boundary-induced phase transitions in\nsystems with absorbing states.  An absorbing state is a dynamical trap which\ncan be accessed but cannot be left~\\cite{Hinrichsen00,Lubeck04,Odor04}.\nSystems with absorbing phase transitions are controlled by a parameter,\ndepending on which the system either enters the absorbing state with certainty\nor survives in a stationary fluctuating\/active state. The most prominent\nfamily of phase transitions into an absorbing state is the very robust direct\npercolation (DP) universality class.  A recent breakthrough has been the\nexperimental observation of DP critical behavior for the first\ntime~\\cite{TakeuchiEtAl07}.\n\nA paradigmatic model in the DP universality class is the contact process (CP)\n\\cite{MarroDickman99}. It can be viewed as a simple model for the propagation\nof a disease where sick individuals can infect healthy neighbors or become\nhealthy spontaneously. More precisely, in the CP in $d$ spatial dimensions, a\nparticle (infected individual) can be created at an ``empty'' site with a rate\n$\\lambda n\/2d$, where $n$ is the number of nearest neighbors occupied by a\nparticle, and an occupied site can become empty at rate $1$.  The empty\nconfiguration is an absorbing state.  For $\\lambda$ larger than a certain\ncritical threshold, $\\lambda_c$, the process is able to sustain (in an\ninfinite lattice) a non-vanishing density of particles, while for\n$\\lambda<\\lambda_c$ the dynamics ends up, ineluctably, in the absorbing state.\n\nAs continuous phase transitions involve long-range correlations, boundary\neffects may play an important role. In the context of absorbing phase\ntransitions, previous studies focused primarily on DP confined to\nparabolas~\\cite{KaiserTurban94,KaiserTurban95}, active\nwalls~\\cite{HinrichsenKoduvely98}, as well as absorbing walls and\nedges~\\cite{Froedh98a,Froedh01a}. Although such boundaries influence the\ndynamics deep into the bulk, the universality class of the bulk transition is\nnot inherently changed, rather it is extended by an additional independent\nexponent describing the order parameter near the boundary.  Therefore, the\nquestion arises whether it is possible to find boundary-induced absorbing\nphase transitions, absent in the corresponding systems without boundaries,\nconstituting independent universality classes.\n\nIn this paper we present a detailed discussion of two slightly different\nmodels introduced in Ref.~\\cite{BaratoHinrichsen08} and\n~\\cite{DeloubriereWijland02}, respectively.  Both of them exhibit a\nboundary-induced nonequilibrium phase transition into an absorbing\nstate. These are one-dimensional particle systems consisting of a single site\n(and, at most, its nearest neighbor), where creation and annihilation of\nparticles occur, and a bulk, where particles move diffusively. While in the\nfirst reference \\cite{BaratoHinrichsen08}, the dynamics at the boundary is a\ncontact process, in the second one~\\cite{DeloubriereWijland02} particles at\nthe origin annihilate only pairwise ($A+A \\rightarrow 0$).  We study different\nversions of these models to compare them and scrutinize the relevance of\nrelaxing the fermionic constraint (i.e. occupation number not restricted to\n$0$ or $1$) both at the bulk and at the boundary.  Our study includes mean\nfield approximations, numerical analysis, some field theoretical arguments, as\nwell as the relation with a (0+1)-dimensional non-Markovian\nmodel~\\cite{BaratoHinrichsen09}.\n\nThe paper is organized as follows: In the next section we define the first\nmodel and present numerical results. In Sec.~\\ref{mean_field} we discuss\nvarious types of mean field approximations and show that this model has indeed\na non-trivial behavior. Sec.~\\ref{related_models} is concerned with two\nbosonic versions of the first model and the study of models with pair\nannihilation at the boundary. One of the bosonic versions is solved exactly\nand it is shown to have trivial critical behavior. Instead, the other bosonic\nversion and models with pair annihilation are shown to share the same critical\nbehavior as the first model. In Sec.~\\ref{non-Markovian} we discuss the\nrelation with a (0+1)-dimensional non-Markovian\nmodel~\\cite{BaratoHinrichsen09} and, finally, we present our main conclusions.\n\n\\section{Basic model definition and simulations}\n\\label{simulations}\n\\subsection{Definition of the model}\n\\label{definition}\n\\begin{figure}\n\\includegraphics[width=70mm]{.\/fig1.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Model on a semi-infinite lattice with symmetric\n  diffusion in the bulk and special dynamical rules at the left boundary (see\n  main text).\n \\label{model}}\n\\end{figure}\nThe model presented in \\cite{BaratoHinrichsen08} is defined on a\none-dimensional semi-infinite discrete lattice where each site is either\noccupied by a particle ($s_i=1$) or empty ($s_i=0$). All lattice sites have\ntwo neighbors, except for the boundary ($i=0$) with a single one. The dynamics\nis a combination of an unbiased random walk in the bulk and a contact\nprocess-like dynamics at the left boundary. It is implemented as follows:\n\\begin{enumerate}\n\\item[(a)] A particle is randomly selected.\n\\item[(b)] If it is located at the leftmost site, it generates another particle\n  at site $1$ with probability $p$, provided that it is empty ($s_1=0$), or it\n  dies ($s_0=0$) with probability $1-p$.\n\\item[(c)] Particles in the bulk perform a symmetric exclusion process,\n  moving to any of their two neighbors with equal probability, provided that\n  the destination site is empty (otherwise nothing happens).\n\\end{enumerate}\nStarting with a single particle at the leftmost site in an otherwise absorbing\n(i.e. empty) configuration, the process evolves as follows: the initial\nparticle at site $0$ either dies or generates another particle at the\nneighboring site $1$. This last performs a random walk in the bulk until,\neventually, it returns to the origin to create another offspring or\ndisappear. A critical point, located at $p_c= 0.74435(15)$ has been reported\nto separate the absorbing phase, in which the total number of particles\nvanishes, from another with indefinitely sustained activity\n\\cite{BaratoHinrichsen08}.\n\\subsection{Order parameters}\n\\label{order_parameters}\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig2.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Density of particles at the leftmost site\n  $\\rho_0$ (left) and the average total number of particles $N$\n  (right) as functions of time for different values of $\\Delta$,\n  below, above, and at criticality. At the critical point,\n  $\\rho_0(t)$ decays as $t^{-1\/2}$ while and $\\langle N(t)\\rangle$ is\n  essentially constant.\n  \\label{fig:decay}}\n\\end{figure}\nA possible order parameter for this model is the average density of\nparticles at the leftmost site:\n\\begin{equation}\n  \\rho_0= \\langle s_0 \\rangle\\,,\n\\end{equation}\nwhere $\\langle \\rangle$ stands for ensemble averages. This quantity is plotted\nin Fig.~\\ref{fig:decay} as a function of $\\Delta:=p-p_c$. At the critical\npoint, $\\rho_0(t)$ decays algebraically in time, as:\n\\begin{equation} \n\\label{rhodecay}\n\\rho_0(t)\\;\\sim\\; t^{-\\alpha}\n\\end{equation}\nwith an exponent $\\alpha= 0.50(1)$, compatible with a rational value $\\alpha=\n1\/2$.\n\nAnother possibility is to choose as an order parameter the average total\nnumber of particles, $\\langle N(t) \\rangle$, which, as shown in\nFig.~\\ref{fig:decay}, goes to zero for $p<p_c$ and increases steadily for\n$p>p_c$ (actually, it is limited only by the system size).  At criticality,\n$\\langle N (t) \\rangle$ is found to be constant in the large time limit.\n\nIn the usual scaling picture of absorbing phase transitions, the critical\nexponent $\\beta$ is related to the probability that a given site belongs to an\ninfinite cluster generated from a fully occupied lattice at $t=-\\infty$. This\nquantity tends to zero as the control parameter approaches the critical value\nfrom above. Similarly, the exponent $\\beta'$ is related to the probability\nthat a localized seed generates an infinite cluster extending to\n$t=+\\infty$. Therefore, in the supercritical phase $(\\Delta >0)$, the averaged\nactivity of the site at the origin for $t\\to\\infty$ \\textit{measured in seed\n  simulations} averaging over all runs, scales as $\\rho^s \\sim\n\\Delta^{\\beta+\\beta'}$, where the superscript `s' stands for `stationary'.  At\ncriticality, this function is expected to decay as $\\rho(t) \\sim\nt^{-(\\beta+\\beta')\/\\nu_\\parallel}$, where $\\nu_\\parallel$ is the correlation\ntime exponent.  Moreover, in the DP class a special {\\it time reversal\n  symmetry} implies that $\\beta=\\beta'$ \\cite{Hinrichsen00}.\n\nAs shown in~\\cite{DeloubriereWijland02}, time reversal symmetry also holds in\nthe present type of models. This implies that, in supercritical seed\nsimulations, the density of active sites at the boundary is expected to\nsaturate as:\n\\begin{equation}\n  \\rho_0^s\\sim \\Delta^{2\\beta}\\,,\n\\end{equation}\nwhile, at criticality:\n\\begin{equation}\n  \\rho_0(t) \\sim t^{-2\\beta\/\\nu_\\parallel},\n\\end{equation}\nimplying that $\\alpha$ in Eq.~(\\ref{rhodecay}) is\n\\begin{equation}\n\\alpha=2\\beta\/\\nu_\\parallel\\,.\n\\label{alpha}\n\\end{equation}\nAssuming that $\\alpha=1\/2$, then $\\beta\/\\nu_\\parallel = 1\/4$.\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig3.eps}\n\\vspace{2mm}\n\\caption{(Color online) Left panel: Density of particles at the leftmost site,\n  at the time when it reaches its minimum value, as a function of the distance\n  from criticality $\\Delta$. This gives the exponent $\\beta= 0.68(5)$.  Right\n  panel: The same quantity, at criticality and in the stationary state, as a\n  function of the external field $h$, giving $\\delta_h^{-1}= 0.29(5)$, compatible\n  with the conjectured value $1\/3$.\n\\label{fig:off}}\n\\end{figure}\n\n\\subsection{Stationary properties}\n\\label{stationary}\nIn numerical simulations in the active phase, it takes a very long time,\nspecially for small values of $\\Delta$, to reach the steady state.  Moreover,\nwe observed the unusual fact that, for $\\Delta>0$, the density $\\rho_0$ goes\nthrough a minimum before reaching the stationary state (see\nFig.~\\ref{fig:decay} and also \\cite{FES}, where similar non-monotonous curves\nwere reported). However, it turns out that the value $\\rho_0^m$ at the minimum\nand the saturation value $\\rho_0^s$ differ by a constant factor, entailing\nthat both quantities scale in the same way, i.e.:\n\\begin{equation}\n\\rho_0^m\\sim \\Delta^{2\\beta} \\,.\n\\end{equation}\nNote that this can be true only if the density $\\rho_0(t)$ in seed simulations\nobeys the scaling relation:\n\\begin{equation}\n\\rho_0(t) = \\Delta^{2\\beta}\\, R(t\\Delta^{\\nu_\\parallel})\n\\end{equation}\ni.e. if it is possible to collapse the data by plotting\n$\\rho_0\\Delta^{-2\\beta}$ versus $t\\Delta^{4\\beta}$. Indeed, this will be shown\nto be the case in Sec.~\\ref{non-Markovian} for a $0$-dimensional non-Markovian\nprocess argued to be in the same universality class.\n\nRelying on this observation, one can determine the value of the exponent\n$\\beta$ by measuring the density $\\rho_0^m$ at the minimum, which is reached\nmuch earlier than the stationary state. In Fig.~\\ref{fig:off} we plot $\\rho_m$\nas a function of $\\Delta$, inferring $\\beta= 0.68(5)$.\n\n\\subsection{External field}\n\\label{external}\nIn ordinary directed percolation, an external field, conjugate to the order\nparameter, can be implemented by creating active sites at some constant rate\n$h$, thereby destroying the absorbing nature of the empty configuration. At\ncriticality, the external field is known to drive a $d$+1-dimensional DP\nprocess towards a stationary state with $\\rho^s \\sim h^{1\/\\delta_h}$ where\n$\\delta_h^{-1}=\\beta\/(\\nu_\\parallel+d\\nu_\\perp-\\beta')$, and $\\nu_\\perp$ is the\ncorrelation length critical exponent.\n\nIn the present model, the external field, conjugate to the order parameter\n$\\rho_0$, corresponds to spontaneous creation of activity at the leftmost site\nat rate $h$. The above hyperscaling relation for $\\delta_{h}$ is thus expected\nto be fulfilled by taking $d=0$:\n\\begin{equation}\n  \\rho_0^s\\sim h^{1\/\\delta_h}.\n\\end{equation}\nwith\n\\begin{equation}\n  \\delta_h^{-1}= \\beta\/(\\nu_\\parallel-\\beta').\n\\end{equation}\nFrom this expression, exploiting the fact that $\\beta=\\beta'$ and using\nEq.(\\ref{alpha}) as well as the conjectured rational value $\\alpha= 1\/2$, a\nprediction $\\delta_h^{-1}=1\/3$ is obtained. Our numerical estimate, $\\delta_h^{-1}=\n0.29(5)$ (see Fig.~\\ref{fig:off}) is compatible with this result.\n\n\\subsection{Survival probability}\n\\label{survival}\n\n\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig4.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Survival probability $P_s(t)$ as function of time\n  below, above, and at criticality.  At the critical point, it decays with the\n  exponent $\\delta=0.15(2)$, different from $\\beta\/\\nu_\\parallel$, and in\n  agreement with the conjectured value $1\/6$.\n\\label{fig:survival}}\n\\end{figure}\nThe survival probability $P_s(t)$ is defined as the fraction of runs that,\nstarting with a single seed at the boundary, survive \\textit{at least} until\ntime $t$. At criticality, this quantity is expected to decay algebraically:\n\\begin{equation}\nP_s(t)\\sim t^{-\\delta},\n\\end{equation}\nwith the so-called survival exponent $\\delta$, while in the super-critical\nregime it saturates in the long time limit. Since $P_s(\\infty)$ coincides with\nthe probability for a seed to generate an infinite cluster, the saturation\nvalue of the survival probability as a function of the distance from\ncriticality gives the exponent $\\beta'$. As in DP, one expects $P_s(t)$ to\ndecay in time with an exponent $\\delta=\\beta'\/\\nu_\\parallel =1\/4$. However, as\nshown in Fig.~\\ref{fig:survival}, one finds a much smaller exponent $\\delta=\n0.15(2)$. Therefore, the usual relation $\\delta= \\beta'\/\\nu_\\parallel$ does\nnot hold. We also observed that it is not possible to collapse different\ncurves of $P_s(t)$ for different values of $\\Delta$, i.e. the survival\nprobability seems to exhibit an anomalous type of scaling behavior. We expect\nthat off-critical simulations of the survival probability give the exponent\n$\\beta'$ but the simulation times needed to reach steady state are\nprohibitively long.\n\nAn explanation for the value $\\delta= 0.15(2)$, differing from\n$\\beta\/\\nu_\\parallel$, is given in the following subsection.\n\n\n\n\\subsection{Time reversal symmetry}\n\n\\label{time_reversal}\n\nIn ordinary bond DP, the statistical weight of a configuration of percolating\npaths does not depend on the direction of time. More specifically, the\nprobability to find an open path from at least one site at time $t=0$ to a\nparticular site at time $t$ coincides with the probability to find an open\npath from a particular site at time $t=0$ to at least one site at time\n$t$. This implies that, in bond DP, {\\it i)} the density $\\rho(t)$ in simulations with\nfully occupied initial state and {\\it ii)} the survival probability $P_s(t)$ in seed\nsimulations coincide; hence $\\beta=\\beta'$. In other realizations of DP\n(e.g. site DP), this time reversal symmetry is not exact but only\nasymptotically realized.\n\nApplying the same arguments to the present model, the survival probability\n$P_s(t)$ in seed simulations should scale in the same way as the density of\nactive sites at the boundary $\\rho_0(t)$ in a process starting with a\n\\textit{fully occupied lattice} in the bulk. A numerical test, which\napproximates such a situation, confirms this conjecture, i.e. one has\n$\\rho_0(t)\\sim t^{-\\delta}$ with $\\delta \\approx 0.15$ for a fully occupied\ninitial state.\n\nFollowing the arguments of~\\cite{BaratoHinrichsen09} in a related model, this\nobservation can be used to provide an heuristic explanation for the fact that\n$\\delta\\neq\\beta\/\\nu_\\parallel$.\n\nIt is known that, if the boundary acts as a sink or perfect trap (e.g.  if\n$p=0$), then, in a process starting with a fully occupied lattice, one\nobserves a growing depletion zone around the boundary whose linear size\n$l'(t)$ increases as $l'\\sim t^{\\alpha_{l}}$, with $\\alpha_{l}=1\/2$ (see\n\\cite{rabbits} and the next subsection). Thus, the density of active sites\ndecays as $t^{-1\/2}$. Hence, the influx of particles from the bulk to the\nleftmost site may be considered as an effective time-dependent external field\n$h(t)\\sim t^{-1\/2}$. Making the assumption that this field varies so slowly\nthat the response of the process (i.e. the actual average activity at the\nboundary) behaves adiabatically, as if the field was constant, then in a\ncritical process starting from an initially fully occupied state:\n\\begin{equation}\n  \\rho_0(t) \\sim t^{-\\frac{1}{2\\delta_h}} \\sim t^{-1\/6}.\n\\end{equation}\nOwing to the time reversal property, this quantity should decay as the\nsurvival probability. This chain of heuristic arguments leads to the\nconjecture that the survival exponent is given by $ \\delta = 1\/6$, in\nagreement with the numerical estimate $\\delta=0.15(2)$.\n\nThis unusual value of the exponent $\\delta$ is clearly related to the fact\nthat the present problem is inhomogeneous. The argumentation presented above\ndoes not work for the CP, for example, since there is no special site and,\ntherefore, a fully occupied lattice cannot be interpreted as a time dependent\nfield acting on a special site.\n\n\n\\subsection{Density profile}\n\\label{density}\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig5.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Left: Data collapse of the rescaled profiles of the\n  particle density at criticality for $t_0=64,128,\\ldots, 8192$ (blue) compared\n  to a Gaussian distribution (red). Inset: The same data collapse in a\n  double-logarithmic representation. Right: Density of particles (blue) and\n  pairs (green) at $t_0=10^6$, showing the presence of correlations which\n  decay in space as $x^{-1\/2}$, indicating that $\\beta\/\\nu_\\perp=1\/2$.\n  \\label{fig:profile}}\n\\end{figure}\nNow, we consider the density profile $\\rho(x,t)$ in the bulk, where $x \\in\n\\mathbb{N}$ is the spatial coordinate (distance to the boundary), computed at\nthe critical point. In the left panel of Fig.~\\ref{fig:profile}, we compare\nthe data collapse of the curves $\\rho(x,t)t^{1\/2}$ as a function of\n$x\/t^{1\/2}$ with a Gaussian and observe an excellent agreement, indicating\nrandom-walk like behavior with a dynamical exponent $z=2$.  However, in\ncontrast to a simple random walk, particles are mutually correlated. This is\nillustrated in the right panel of Fig.~\\ref{fig:profile}, where the connected\ncorrelation function between two nearest neighbors:\n\\begin{equation}\n  \\rho^{\\rm pair}(x,t)= \\langle\\rho(x+1,t)\\rho(x,t)\\rangle- \n  \\langle\\rho(x+1,t)\\rangle\\langle\\rho(x,t)\\rangle\n\\end{equation}\nin a system at the critical point is plotted against time. One observes an\nalgebraic decay, $x^{-1\/2}$, with distance. According to the standard scaling\ntheory this implies that $\\beta\/\\nu_\\perp=1\/2$ , confirming that\n$z=\\nu_\\parallel\/\\nu_\\perp=2$.  Moreover, these results are in full agreement\nwith field theoretical calculations presented in\nRef.~\\cite{DeloubriereWijland02} (see section \\ref{vanWijland}), which predict\n$z=2$ and $\\alpha=1\/2$.\n\n\\section{Mean field approximation}\n\\label{mean_field}\nHere, we  study mean field approximations at different levels. Let us denote by\n$\\eta_i$ the probability to find a particle at site~$i$; the temporal\nevolution within a simple (one-site) mean field approximation is given by:\n\\begin{eqnarray}\n\\frac{d\\eta_0}{dt}&=& -(1-p)\\eta_0+ \\frac{1}{2}\\eta_1(1-\\eta_0),\n\\label{eqmf1}\\\\\n\\frac{d\\eta_1}{dt}&=& p\\eta_0(1-\\eta_1)+\n\\frac{1}{2}\\left(\\eta_{2}+\\eta_{0}\\eta_1-2\\eta_{1}\\right),\n\\label{eqmf2}\\\\\n\\frac{d\\eta_i}{dt}&=& \\frac{1}{2}\\left(\\eta_{i+1}+\n\\eta_{i-1}-2\\eta_{i}\\right),~~ \\mbox{ for } i=2,3,\\hdots\\,\n\\label{eqmf3}\n\\end{eqnarray}\nNote that the equations for the boundary site and its neighbor,\nEq.~(\\ref{eqmf1}) and Eq.~(\\ref{eqmf2}), include quadratic terms due to the\nexclusion constraint, while the equation for sites at the bulk,\nEq.~(\\ref{eqmf3}), describes in this approximation a symmetric random walk,\n{\\it i.e.} it is a diffusion equation. The critical point within simple mean\nfield theory (where the equation for $\\eta_1$ also becomes a diffusion\nequation) is $p_c= 1\/2$.\n\nConsidering a localized initial condition at the boundary,\n$\\eta_i=\\delta_{i,0}$, after a transient time the densities at sites $0$ and\n$1$ should, approximately, coincide.  Therefore, from Eq.~(\\ref{eqmf1}) with\n$\\eta_0 \\approx \\eta_1$, it follows that, at criticality, $\\eta_0\\sim\nt^{-1\/2}$.\n\nIn the stationary regime, Eq.~(\\ref{eqmf1}) leads to $\\eta_0\\sim (p-1\/2)$ for\n$p\\ge 1\/2$. From these results we have:\n\\begin{equation}\n\\alpha^{MF}=1\/2\\,,\\qquad \\beta^{MF}=1\\,.\n\\end{equation}\nTo obtain the survival exponent, $\\delta$, we follow the arguments of the\npreceding section and study the decay of activity from a fully occupied\nlattice, $\\eta_i=1$ for all $i$.  Integrating Eqs.~(\\ref{eqmf1}),\nEq.~(\\ref{eqmf2}) and Eq.~(\\ref{eqmf3}) numerically with this initial\ncondition, we obtain an exponent in agreement with\n\\begin{equation}\n\\delta^{MF}= 1\/4\\,.\n\\end{equation}\nA more accurate approximation can be obtained by keeping the correlation\nbetween the first two sites, which is expected to be more relevant than the\ncorrelation between other neighboring sites.  Such a pair-approximation was\nused recently in a model where a boundary site also plays a special\nrole~\\cite{Sugden07}. In this approximation, the master equation reads:\n\\begin{eqnarray}\n\\frac{d\\sigma_{00}}{dt}&=& (1-p)\\sigma_{10}+\n\\frac{1}{2}[\\sigma_{01}(1-\\eta_2)- \\sigma_{00}\\eta_2],\n\\\\ \\frac{d\\sigma_{01}}{dt}&=& (1-p)\\sigma_{11}+\n\\frac{1}{2}[\\sigma_{00}\\eta_2-\\sigma_{01}(2-\\eta_2)],\\nonumber\n\\\\ \\frac{d\\sigma_{10}}{dt}&=& -\\sigma_{10}+\n\\frac{1}{2}[\\sigma_{01}-\\sigma_{10}\\eta_2+\\sigma_{11}(1-\\eta_2)],\\nonumber\n\\\\ \\frac{d\\sigma_{11}}{dt}&=& p\\sigma_{10}- (1-p)\\sigma_{11}+\n\\frac{1}{2}[\\sigma_{10}\\eta_2-\\sigma_{11}(1-\\eta_2)],\\nonumber\n\\\\ \\frac{d\\eta_2}{dt}&=& \\frac{1}{2}\\left(\\eta_{3}+ \\sigma_{11}+ \\sigma_{01}\n-2\\eta_{2}\\right),\\nonumber \\\\ \\frac{d\\eta_i}{dt}&=&\n\\frac{1}{2}\\left(\\eta_{i+1}+ \\eta_{i-1}-2\\eta_{i}\\right) \\mbox{ for }\ni=3,4,\\hdots\\,\\nonumber\n\\end{eqnarray}\nwhere $\\sigma_{s_0s_1}$ is the probability that the occupation numbers of the\nfirst two sites are $s_0$ and $s_1$. Numerical integration of these equations\nleads to an improved critical point estimation, $p_c\\approx 0.634$, but to the\nsame mean-field exponents as above.\n\\section{Related models and field theoretical approaches}\n\\label{related_models}\n\n\\subsection{Bosonic variant}\n\\label{bosonic}\nThe model defined above is fermionic in the sense that each site can be\noccupied by, at most, one particle. We now consider a bosonic variant without\nsuch a constraint.  This means that diffusion is independent of the\nconfiguration of particles and that particles can be created at the boundary\nsite without restriction.  More specifically, the update rules are:\n\\begin{enumerate}\n\\item[(a)] A particle is chosen randomly.\n\\item[(b)] If the particle is located at the leftmost site it can:\ncreate another particle at the leftmost site ($s_0=s_0+1$) at rate $\\lambda$,\ndie ($s_0=s_0-1$) at rate $\\sigma$, or\ndiffuse to the next neighbor at rate $D$.\n      \\item[(c)] If the particle is located in the bulk, it diffuses to the\n        right or to the left at equal rates $D$.\n\\end{enumerate}\nThe corresponding master equation is:\n\\begin{eqnarray}\n\\frac{dP(\\{n\\},t)}{dt} &=& \\lambda\\bigl[(n_0-1)P(n_0-1,...,t)-\n  n_0P(\\{n\\},t)\\bigr]\\nonumber \\\\\n& +&\\sigma\\bigl[(n_0+1)P(n_0+1,...,t)- n_0P(\\{n\\},t)\\bigr]\\nonumber \\\\\n&+& D \\Bigl[ \\sum_{\\langle ij\\rangle}P(...,n_i-1,n_j+1,...,t) \\\\\n&+&P(...,n_i+1,n_j-1,...,t)-2P(\\{n\\},t) \\Bigr] \\nonumber\n\\label{eqboson1}\n\\end{eqnarray} \nwhere $P(\\{n\\},t)$ is the probability to find a given configuration $\\{n\\}=\nn_0, n_1, n_2\\ldots$ and the sum runs over all nearest neighbors, $j$, of site\n$i$ (recall that site $0$ has only one neighbor). Defining the state vector:\n\\begin{equation}\n|\\psi(t)\\rangle= \\sum_{\\{n\\}}P(\\{n\\},t)|\\{n\\}\\rangle,\n\\end{equation}\nwhere $|\\{n\\}\\rangle=\\otimes_i|n_i\\rangle$ denotes the usual configuration\nbasis, the master equation can be expressed in the form\n\\begin{equation}\n\\frac{d}{dt} |\\psi(t)\\rangle= -\\hat{H}|\\psi(t)\\rangle\\,,\n\\end{equation}\nwhere $\\hat{H}$ is the time evolution operator. Using bosonic creation and\nannihilation operators, defined by $\\hat{a}_i|n_i\\rangle= n_i|n_i-1\\rangle$\nand $\\hat{a}_i^\\dagger|n_i\\rangle= |n_i+1\\rangle$, the master equation\nEq.~(\\ref{eqboson1}) can be shown to correspond to the time evolution\noperator:\n\\begin{eqnarray}\n\\hat{H}&=& D\\sum_{\\langle\n  ij\\rangle}(\\hat{a}^\\dagger_i-\\hat{a}^\\dagger_j)(\\hat{a}_i-\\hat{a}_j)\n\\label{eqbosonH}\n\\\\ &&+\\sigma(\\hat{a}_0^\\dagger-1)\\hat{a}_0 +\\lambda\n\\hat{a}_0^\\dagger(1-\\hat{a}_0^\\dagger)\\hat{a}_0.\\nonumber\n\\end{eqnarray}\nIn this formalism, the expectation value of an operator $\\hat{B}$ is given by\n$\\langle\\hat{B}\\rangle= \\langle 1|\\hat{B} |\\psi(t)\\rangle$ where $\\langle 1|=\n\\sum_{\\{n\\}}\\langle \\{n\\}|$. As is the case for the bosonic contact process\n\\cite{Baumann05}, the equations for the time evolution of the density of\nparticles close. From the Heisenberg equation of motion, $\\frac{d\\hat{B}}{dt}=\n[\\hat{H},\\hat{B}]$ and Eq.~(\\ref{eqbosonH}), one obtains:\n\\begin{eqnarray} \n\\frac{d\\rho_0}{dt} &=& D(\\rho_1-\\rho_0)+ \\Delta\\rho_0 \\label{eqboson1b}\n\\\\ \\frac{d\\rho_i}{dt} &=& D(\\rho_{i+1}+\\rho_{i-1}-2\\rho_i)\\qquad\ni=1,2,3\\ldots\\nonumber\n\\end{eqnarray}\nwhere $\\rho_i(t)= \\langle a^\\dagger_i(t)a_i(t)\\rangle= \\langle a_i(t)\\rangle$\nand $\\Delta= \\lambda-\\sigma$.  Alternatively, one could have written a\nLangevin equation equivalent to Eq.(\\ref{eqbosonH}), and from it, averaging\nover the resulting noise, one readily arrives at the same set of equations\nEq.(\\ref{eqboson1b}).\n\nFrom these equations, we can see that the critical point is $\\Delta= 0$, where\nEq.(\\ref{eqboson1b}) is a diffusion equation.  In the continuum limit,\nEq.~(\\ref{eqboson1b}) reads:\n\\begin{equation}\n\\frac{\\partial\\rho(x,t)}{\\partial t}= \\frac{\\partial^2\\rho(x,t)} {\\partial\n  x^2}+ \\Delta\\delta(x)\\rho(x,t)\\,\n\\label{eqboson2}\n\\end{equation}\nwhere $x$ is the spatial coordinate and, without loss of generality, we have\nset $D=1$. We note that in order to take the continuum limit in equation (\\ref{eqboson1b}), a site $-1$, with $\\rho_{-1}= \\rho_{0}$, has to be introduced, so that appropriate boundary conditions are satisfied. The solution of this inhomogeneous diffusion equation is:\n\\begin{equation}\n\\rho(x,t)= \\int_0^\\infty \\delta(\\zeta)G(x,\\zeta,t)d\\zeta+\\nonumber\n\\end{equation}\n\\begin{equation}\n\\int_0^t\\int_0^\\infty\\Delta\\delta(\\zeta)\\rho(\\zeta,\\tau)G(x,\\zeta,t-\\tau),\nd\\zeta d\\tau \\,\n\\label{eqboson3}\n\\end{equation}\nwhere $G(x,\\zeta,t)= (e^{-(x+\\zeta)^2\/(4t)}+e^{-(x-\\zeta)^2\/(4t)})\/(\\sqrt{\\pi t})$ is the Green\nfunction and the first term in the right hand side comes from the initial\ncondition $\\rho(x,0)= \\delta(x)$. From Eq.~(\\ref{eqboson3}) we have\n\\begin{equation}\n\\rho_0(t)= \\frac{2}{\\sqrt{\\pi t}}+2\\Delta\\frac{d^{-1\/2}}{dt^{-1\/2}}\\rho_0(t) \\,\n\\label{eqboson4}\n\\end{equation}\nwhere $\\rho_0(t)= \\rho(0,t)$, and the operator $\\frac{d^{-1\/2}}{dt^{-1\/2}}$,\ndefined by\n\\begin{equation}\n\\frac{d^{-1\/2}}{dt^{-1\/2}}f(t)= \\int_0^t\\frac{f(\\tau)}{\\sqrt{\\pi (t-\\tau)}}d\\tau,\n\\end{equation}\nis a half integral operator \\cite{Oldham}.  Equation~(\\ref{eqboson4}) involves\n(owing to the delta function in the interaction term in Eq.~(\\ref{eqboson2}))\nonly the density at the leftmost site. This justifies the mapping of this\nmodel onto an effective one-site non-Markovian process (see next\nsection). Using some rules for half integration \\cite{Oldham} to solve\nEq.~(\\ref{eqboson4}), we find:\n\\begin{equation}   \n\\rho_0(t)= \\frac{2}{\\sqrt{\\pi t}}+ 4\\Delta\\exp(4\\Delta^2t)\\mbox{erf}(-2\\Delta\\sqrt{t}),\n\\label{eqboson5}\n\\end{equation}\nwhere $\\mbox{erf(x)}$ is the error function. This implies that, above the\ncritical point, $\\rho_0$ grows exponentially in the long time limit, and does\nnot reach a stationary value, {\\it i.e.} there is a first order transition\nand, hence, $\\beta= 0$ in this bosonic model.  From equation\nEq.~(\\ref{eqboson5}), we deduce $\\beta'=1$ and $\\nu_\\parallel=2$. We have not\nbeen able to calculate the survival-probability exponent exactly, but\nnumerical simulations suggest $\\delta= 1\/4$, in agreement with the mean field\nexponent.\n\n\\subsection{Partially bosonic variant}\n\\label{partially_bosonic}\n\nLet us now introduce a {\\it partially bosonic} variant of the previous model\nby retaining the exclusion constraint only at the boundary, but not in the\nbulk.  The rules, in this case, are:\n\\begin{enumerate}\n\\item[(a)] A particle is randomly chosen.\n\\item[(b)] If it is at the leftmost site, it can generate a particle at site\n  $1$ (provided that $s_1=0$) with probability $p$ or die ($s_0:=0$) with\n  probability $1-p$.\n\\item[(c)] Particles in the bulk diffuse to the right or to the left with the\n  same probability, $1\/2$.\n\\end{enumerate}\nNumerical simulations show that this variant exhibits the same critical\nbehavior as the original model, even if the critical point is shifted to $p_c=\n0.6973(1)$. This shows that the fermionic constraint is relevant only at the\nboundary, where it induces a saturation of the particle density and leads the\ntransition to become continuous.\n\n\\subsection{Models with pair annihilation at the boundary}\n\\label{vanWijland}\nIn the models discussed so far, particles at the boundary either create an\noffspring or die spontaneously at some rate. Instead, a very similar model was\nintroduced in Ref.~\\cite{DeloubriereWijland02}, for which particles at the\nboundary annihilate only in \\textit{pairs}.  In its fermionic variant,\nparticles at sites $0$ and $1$ annihilate with each other (provided that both\nsites are occupied) at some rate, while isolated particles at the boundary\ncannot disappear:\n\\begin{eqnarray*}\n\\mbox{present models:} & A\\to2A \\,,\\quad A\\to\\emptyset\\,, \\\\ \\mbox{models of\n  Ref.~\\cite{DeloubriereWijland02}:} & A\\to2A \\,,\\quad 2A\\to\\emptyset\\,.\n\\end{eqnarray*}\nAnalogously, one can define a bosonic version, in which two particles at the\nboundary can annihilate. In the following discussion we consider these two\nvariants in $d$ spatial dimensions where, as is the case $d=1$, only a single\nsite has ``special\" dynamics.\n \nA detailed field theoretical analysis of these pair-annihilating models was\npresented in \\cite{DeloubriereWijland02}. In the bosonic case, proceeding as\nabove (see Eq.(\\ref{eqbosonH})) one obtains the following time evolution\noperator:\n\\begin{eqnarray}\n\\hat{H}&=& D\\sum_{\\langle ij\\rangle}(\\hat{a}^\\dagger_i-\\hat{a}^\\dagger_j)\n(\\hat{a}_i-\\hat{a}_j) \\nonumber\n\\\\ &&+\\sigma[(\\hat{a}_0^\\dagger)^2-1]\\hat{a}_0^2 +\\lambda\n\\hat{a}_0^\\dagger(1-\\hat{a}_0^\\dagger)\\hat{a}_0.\n\\label{H2}\n\\end{eqnarray}  \nwhich, after eliminating higher order terms and taking the continuum limit, is\nequivalent to a Langevin equation identical to the one for DP except for the\nfact that\nall terms, except for the Laplacian, are multiplied by a $\\delta$ function at\nthe boundary; {\\it i.e.} the non-diffusive part of the dynamics operates only\nat the boundary.  An $\\epsilon$-expansion analysis of Eq.(\\ref{H2}) (see\n\\cite{DeloubriereWijland02}) leads to $\\alpha=1\/2$ and $z=2$ as exact results\nin all orders of perturbation theory, and to $\\beta=1-3(4-3d)\/8$, up to first\norder in $\\epsilon =4\/3-d$ around the critical dimension $d_c=4\/3$.  Also, it\nwas shown that the time reversal symmetry is preserved.\n\nWe have verified all these predictions in computer simulations of the bosonic\nannihilation model. For instance, from the time decay of $\\rho_0(t)$, as shown\nin Fig.~\\ref{FSS}, we determine $\\delta = 0.21(3)$, while from a finite size\nscaling analysis of the saturation values of the order parameter at\ncriticality we measure $\\beta\/\\nu_{\\perp}=0.51(2)$ (see Fig.~\\ref{FSS}), in\nreasonable agreement with the expected results, $\\delta= 1\/6$ and\n$\\beta\/\\nu_{\\perp}=1\/2$, respectively. Moreover, from spreading simulations\n(not shown) we estimate $\\alpha \\approx 1\/2 $ and $z \\approx 2$. All the\nexponents are in agreement with the ones presented in the previous section for\nsingle particle annihilation models.\n\nActually, a simple argument explains why the model of section\n\\ref{simulations} and the pair-annihilation model share the same\ncritical behavior. This is plausible because the chain reaction $A \\to\n2A \\to \\emptyset$ in the model with pair annihilation generates\neffectively the reaction $A\\to\\emptyset$ of the model considered with\nCP-like dynamics.\n\nHence, the field theoretical predictions discussed above\n\\cite{DeloubriereWijland02,BaratoHinrichsen08} apply also to the CP-like\nmodel. In $d=1$, the one-loop prediction $\\beta= 5\/8= 0.625$\n\\cite{DeloubriereWijland02}, is not far from the exponent measured in section\n\\ref{simulations}, $\\beta=0.68(5)$.\n\nOn the other hand, the fermionic version of the pair-annihilating model has\nbeen conjectured to yield in a different universality class, and a prediction\nfor its critical exponents is made in \\cite{DeloubriereWijland02} (for\ninstance, $\\beta=1$).  Our numerical simulations disprove such a claim; all\nthe measured critical exponents for the fermionic variant of the\npair-annihilation model are numerically indistinguishable from their bosonic\ncounterparts (see Fig.~\\ref{FSS}).\n\\begin{figure}[t]\n\\begin{center}\n  \\includegraphics[height=5cm]{fig6.eps}\n\\end{center}\n\\caption{\\footnotesize{(Color online) Temporal behavior of $\\rho_{0}$ for the bosonic\n    pair-annihilating model, starting from a homogeneous initial condition for\n    different system sizes (from $L=64$ to $L=2048$).  The exponent\n    $\\beta\/\\nu_{\\perp}$ can be measured from the scaling of the different\n    saturation values as a function of system size (see inset; yellow\n    line). Also, in the inset (dashed green line), we show the scaling of saturation\n    values for the fermionic version of the same model, showing the same type\n    of scaling. }}\n\\label{FSS}\n\\end{figure}\n\nIn summary, all the defined models, either with single particle annihilation\nor with pair-annihilation, fermionic or bosonic, exhibit a boundary induced\nphase transitions and, except for one of them, they all are continuous and\nshare the same critical behavior. The exception to this rule is the CP-like\nmodel without a fermionic constraint at the boundary, which lacks of a\nsaturation mechanism in the active phase, leading to unbounded growth of \nparticle density at the leftmost site above the critical point and to a discontinuous transition.\n\\section{Relation to a $(0+1)$-dimensional non-Markovian process}\n\\label{non-Markovian}\n\nIn Ref.~\\cite{DeloubriereWijland02}, by integrating out the fields related to\ndiffusion in the bulk from the corresponding action, it was shown that the\nclass of boundary-induced phase transitions into an absorbing state considered\nhere can be related to a non-Markovian single site process. The properties of\nsuch a spreading process on a time line has been studied in further detail in\nRef.~\\cite{BaratoHinrichsen09}.\n\nOn an heuristic basis, the relation can be explained as follows: consider the\nCP-like model only from the perspective of the leftmost site. A particle at\nthe origin may die or create a new particle that will go for a random walk\ncoming back to the origin after a time $\\tau$. What happens during this random\nwalk is irrelevant from the perspective of the leftmost site; the only\nrelevant aspect is the time needed for a created particle to come back to the\nboundary.  Once it returns it may die or create new offsprings which, on their\nturn, will undergo random walks in the bulk.\n\nOur simulations above show that the fermionic constraint is irrelevant in the\nbulk. Therefore, we can consider without lost of generality the bulk-bosonic\nversion in which there is no effective interaction among diffusing\nparticles. In this case, the probability distribution of the returning time to\nthe origin has the well-known asymptotic form~\\cite{Redner01}:\n\\begin{equation}\n\\label{WaitingTimeDistribution}\nP(\\tau)\\sim \\tau^{-3\/2}\\,.\n\\end{equation}\nTaking all these elements into account we define the following\nnon-Markovian model on a single site~\\cite{BaratoHinrichsen08}:\n\\begin{enumerate}\n\\item[(a)] Set initially $s(t):= \\delta_{t,0}$ for all times, $t$.\n\\item[(b)] Select the lowest $t$ for which $s(t)=1$.\n\\item[(c)] With probability $\\mu$, generate a waiting time $\\tau$ according to\n  the distribution~Eq.~(\\ref{WaitingTimeDistribution}), truncate it to an\n  integer, and set $s(t+\\tau):=1$; otherwise (with probability $1-\\mu$) set\n  $s(t):=0$.\n\\item[(d)] Go back to (b).\n\\end{enumerate}\nThe process runs until the system enters the absorbing state ($s(t')=0$ for\nall $t'>t$) or a predetermined maximum time is exceeded.\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig7.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Off-critical data collapse with the one-site model:\n  $\\langle s(t)\\rangle\\Delta^{-2\\beta}$ as a function of $t\\Delta^{4\\beta}$\n  for different values of $\\Delta$, with $\\beta= 0.71(2)$.\n  \\label{fig:datacollapse}}\n\\end{figure}\nThe density of particles at the leftmost site of the original model is related\nto $\\langle s(t)\\rangle$ in the single-site model, the survival probability at\ntime $t$ is given by the fraction of runs surviving at least up to $t$, and\nthe initial condition $s(t):= \\delta_{t,0}$ corresponds to start with a single\nparticle at the boundary in the full model.  Critical exponents can be defined\nas in the original model. However, the simulation results for the single-site\nnon-Markovian model are more reliable because it is possible to perform much\nlonger runs and, in the case of off-critical simulations, one can work with\nsmaller values of $\\Delta$. With time-dependent simulations at the critical\npoint $\\mu_c=0.574262(2)$, we obtained $\\alpha=0.500(5)$ and\n$\\delta=0.165(3)$, in good agreement with the conjectured values $\\alpha= 1\/2$\nand $\\delta= 1\/6$.  As an example, we show the results of supercritical\nsimulations in Fig.~\\ref{fig:datacollapse}, where we obtained a convincing\ndata collapse by plotting $\\langle s(t)\\rangle\\Delta^{-2\\beta}$ as a function\nof $t\\Delta^{4\\beta}$ for different values of $\\Delta$ with $\\beta=\n0.71(2)$. The latter estimate is in agreement with $\\beta= 0.68(5)$, coming\nfrom the original model.\n\nAs shown in previous studies (see e.g.~\\cite{Hinrichsen07} and references\ntherein), a non-Markovian time evolution with algebraically distributed\nwaiting times $P(\\tau) \\sim \\tau^{-1-\\kappa}$ is generated by so-called\nfractional derivatives $\\partial_t^\\kappa$ which are defined by:\n\\begin{equation}\n\\label{eq:IntegralTime}\n\\partial_t^\\kappa \\, \\rho(t)  \\;=\\;  \\frac{1}{\\mathcal{N}_\\parallel(\\kappa)}\n\\int_0^{\\infty} {\\rm d}t' \\, {t'}^{-1-\\kappa} [\\rho(t)-\\rho(t-t')]\\,,\n\\end{equation}\nwhere $\\kappa\\in[0,1]$ and $\\mathcal{N}_\\parallel(\\kappa)=-\\Gamma(-\\kappa)$ is\na normalization constant. Hence, we expect this model to be described by a\nDP-like $0$-dimensional Langevin equation with a half-time derivative, instead\nof the usual one, to account for the non-Markovian character of the model:\n\\begin{equation}\n\\label{Langevin}\n\\partial_t^{1\/2} \\rho(t) = a \\rho(t) - \\rho(t)^2 + \\xi(t)\\,\n\\end{equation}\nwhere $a$ is proportional to the distance from criticality and $\\xi$ is a\nmultiplicative noise with correlations $\\langle \\xi(t)\n\\xi(t')\\rangle=\\rho(t)\\delta(t-t')$. This equation can be obtained from the\neffective action that arises when the fields related to diffusion in the bulk\nare integrated out, and the relation of the order of the fractional derivative\nin a generalized one-site model with the dimension in the full model is\n$\\kappa= (2-d)\/2$ \\cite{DeloubriereWijland02}. An analysis of this one-site\nmodel with general $\\kappa$ and a comparison with the results coming from\nfield theory is presented in \\cite{BaratoHinrichsen09}.\n\\section{Conclusion}\n\\label{conclusions}\n\nWe have studied boundary-induced phase transitions into an absorbing\nstate in one-dimensional systems with creation\/annihilation dynamics\nat the boundary and simple diffusive dynamics in the bulk.  The\nnon-trivial dynamics at the boundary induces a phase transition in the\nbulk. We have analyzed such a transition for different though similar\nmodels, including different ingredients: either single-particle\nannihilation or pairwise annihilation, fermionic constraint or lack of\nit, etc.\n\nA particular bosonic version can be exactly solved; owing to the lack of any\nsaturation mechanism, the density of particles grows unboundedly in the active\nphase, leading to a discontinuous transition with trivial critical exponents.\n\nThe rest of the analyzed models exhibit a continuous transition and define a\nunique universality class. At the bulk, the dynamics is governed by\nrandom-walks, entailing the exponent values $z=2$ and $\\alpha=1\/2$. On the\nother hand, some critical exponents take non-trivial values:\n{\\it i)} the survival probability from a localized seed at\nthe boundary exponent, which from an heuristic argument supported by\nsimulations results, turns out to be $\\delta=1\/6$, as well as {\\it ii)} the\norder parameter exponent, $\\beta= 0.71(2)$.  The remaining exponents can be\nobtained from these ones using scaling relations.\n\nFinally, it has been shown that the class of boundary induced phase\ntransitions studied here can be related to a single-site non-Markovian\nprocess.  This process is particularly suitable for numerical\nsimulations and it is also of conceptual interest in the sense that it\nshows that nonequilibrium phase transitions can occur even in $0+1$\ndimensions by choosing an adequate non-Markovian dynamics. It is also\nconvenient for the comparison of the results obtained form the\n$\\epsilon$-expansion and simulations\n\\cite{BaratoHinrichsen09,DeloubriereWijland02}.\n\nThe models studied here possibly constitute the simplest universality class of\nnonequilibrium phase transition into an absorbing state, in the sense that the transition occurs \nbecause of the special dynamics of just one site and, in contrast to DP, some critical exponents\ncan be obtained exactly from the field theory.\n\n\\begin{acknowledgments}\n  We thank X. Durang and M. Henkel for helpful discussions. \n  Financial support by the Deutsche Forschungsgemeinschaft (HI\n  744\/3-1), by the Spanish MEyC-FEDER, project FIS2005-00791, and from\n  Junta de Andaluc{\\'\\i}a as group FQM-165 is gratefully acknowledged.\n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAt the eve of data taking at the LHC, the electroweak standard\nmodel~(SM) with a fundamental scalar Higgs doublet remains an\nextremely successful effective description of all data collected in\nparticle physics experiments at colliders.  Nevertheless, the\nmicroscopic dynamics of the electroweak symmetry breaking~(EWSB)\nsector has not yet been tested directly.  Therefore, detailed studies\nof the SM and realistic alternative scenarios for EWSB are an\nessential part of the LHC experimental program.\n\nIn the past decade, additional dimensions of space-time at the\nTeV-scale have become an important paradigm for electroweak~(EW) model\nbuilding.  Planck-scale extra dimensions have long been a solid\nprediction of superstring theory, but they are outside of the\nexperimental range of collider experiments.  In contrast, TeV-scale\nextra dimensions will be tested at the LHC.\n\nModels with just one additional space dimension that have the geometry of\na five dimensional Anti-de Sitter space~($\\text{AdS}_5$)~\\cite{Randall:1999ee},\nplay a special role, because the conjectured AdS\/CFT\ncorrespondence~\\cite{Maldacena:1997re} reveals such models as dual to conformal\nfield theories~(CFT) on the four dimensional boundary branes.  In\nparticular, a weakly interacting~$\\text{AdS}_5$ model turns out to be\ndual to a strongly interacting Technicolor~(TC) like model for EWSB.\nIf the conjectured AdS\/CFT correspondence is exact, the extra\ndimension can be viewed as a technically convenient description\nof a strongly interacting dynamics.\n\nModel building for EWSB with extra dimensions does not require them to\nbe continuous, instead they can be\ndeconstructed~\\cite{ArkaniHamed:2001ca} as a discrete lattice with a\nfinite number of sites.  In this approach, the extra dimensions play a\nmetaphorical role as a organization principle for gauge theories with\nlarge non-simple gauge groups and complicated matter representations,\nsimilar to moose models.  It turns out that a minimal version of\nwarped higgsless models can be (de)constructed on just three lattice\nsites in the extra dimension and is known as the Three Site Higgsless\nModel~(3SHLM)~\\cite{SekharChivukula:2006cg}.\n\nIn order to be compatible with the EW precision tests~(EWPT), any\nadditional heavy gauge bosons should couple weakly to the SM fermions.\nThe 3SHLM ensures this by ``ideal fermion\ndelocalization''~\\cite{SekharChivukula:2006cg,SekharChivukula:2005xm}\nand the predominant production mechanism at the LHC will be in vector\nboson fusion~\\cite{He:2007ge}.  However, it has been pointed out\nrecently~\\cite{Abe:2008hb}, that the EWPT actually require a small,\nbut nonvanishing coupling of the heavy gauge bosons to SM fermions.\nThis allows their production in the $s$-channel at the LHC.  In fact,\na measurement of the relative strengths of the production mechanisms\nfor heavy vector bosons will be required to constrain higgsless models\nof EWSB.\n\nIn this paper, we complement the existing phenomenological\nstudies~\\cite{He:2007ge} of the 3SHLM by allowing for non ideal\ndelocalization and the production of~$W'$ and $Z'$~bosons in the\n$s$-channel at LHC.  We perform parton level Monte Carlo studies to\nidentify the regions of parameter space where the coupling of the\n$W'$~boson to SM fermions can be measured at the LHC.  We show how the\ncontributions from the nearly degenerate ~$W'$ and $Z'$~bosons can be\nseparated for this purpose.\n\nThis paper is organized as follows: in section~\\ref{sec:3shl} we\nreview the features of the 3SHLM that are relevant for our\ninvestigation.  In section~\\ref{sec-cpl} we discuss the relevant\ncouplings, masses and widths that are used in our Monte Carlo\neventgenerator described in section~\\ref{sec:whizard}.\nIn sections~\\ref{sec-hzprod} and~\\ref{sec-hwprod} we discuss our\nresults for the production of heavy~$Z'$ and $W'$~bosons, respectively.\nLimitations arising from finite jet mass resolutions are described\nin~\\ref{sec-jetres}.  We conclude in section~\\ref{sec:concl}.\n\n\\section{The Three Site Higgsless Model}\n\\label{sec:3shl}\n\nThe Three Site Higgsless Model~\\cite{SekharChivukula:2006cg} can be\nviewed as a warped 5D model of EWSB,\ndimensionally deconstructed~\\cite{ArkaniHamed:2001ca}\nto three lattice sites. The structure and field content of the model\nis shown in moose notation in figure~\\ref{fig-moose}.\n\\begin{figure}\n\\centerline{\\includegraphics{moose}}\n\\caption{The field content and structure of the 3SHLM in moose notation. The dashed lines\nconnecting fermions represent Yukawa couplings, the dotted blob illustrates the nontrivial\n$\\mathbf{U}(1)_2$ charge carried by all fermions.}\n\\label{fig-moose}\n\\end{figure}\nThe gauge group consists of two $\\mathbf{SU}(2)$ group factors located at the\nlattice sites~$0$ and~$1$ with gauge fields $A^\\mu_{0\/1}$ and gauge couplings $g_{0\/1}$ and a\n$\\mathbf{U}(1)$ gauge group located at the third lattice site with the gauge field\n$A^\\mu_2$ and gauge coupling $g_2$.  Note that the continuous 5D\nanalogue of this is a bulk $\\mathbf{SU}(2)$\nbroken to $\\mathbf{U}(1)$ on one brane by boundary conditions. The lattice sites are\nlinked by $\\mathbf{SU}(2)$ valued Wilson line fields $\\Sigma_{0\/1}$\nthat transform bi-unitarily under\ngauge transformations as\n\\begin{equation*}\n \\Sigma_0 \\longrightarrow U_0\\Sigma_0 U_1^\\dagger \\quad,\\quad\n \\Sigma_1 \\longrightarrow U_1\\Sigma_0 e^{-i\\theta\\frac{\\sigma_3}{2}}\\,.\n\\end{equation*}\nIf the potential for the Wilson line fields is arranged such that these acquire a vacuum\nexpectation value\n\\begin{equation*}\n  \\left<\\Sigma_{0\/1}\\right> = \\sqrt{2}v\\,,\n\\end{equation*}\nthe symmetry group is broken spontaneously\nto the electromagnetic~$\\mathbf{U}(1)_\\text{em}$. The kinetic terms for~$\\Sigma_{0\/1}$\ncontain covariant derivatives which produce mass terms for the gauge bosons;\nafter diagonalization we find a massless photon, two massive charged\ngauge bosons~$W$ and~$W^\\prime$ and two neutral massive gauge bosons~$Z$ and~$Z^\\prime$. \n\nChoosing~$g_1\\gg g_0,g_2$, the mass gap between the massive\ngauge bosons becomes large and the lighter ones can be identified\nwith the SM~$W$  and $Z$~bosons. These are mostly localized at the\nbrane sites, while the heavy modes\nare strongly localized at the bulk site. This symmetry breaking setup is similar to the\nBESS model~\\cite{Casalbuoni:1985kq}. After fixing the electric charge and\nthe~$W$ and $Z$~masses from the observed values,\nthe only remaining free parameter in the gauge sector is the $W^\\prime$ mass~$m_{W'}$.\n\nFermions are incorporated into the model by putting left-handed $\\mathbf{SU}(2)$\ndoublets $\\Psi_{0\/1,L}$ on the sites~$0$ and~$1$, a right-handed doublet $\\Psi_{1,R}$ on\nsite~$1$ and singlets $\\Psi^{u\/d}_{2,R}$ on site~$2$ for every SM\nfermion (cf.~figure~\\ref{fig-moose}).  The $\\mathbf{U}(1)_2$~charges\nof the $\\Psi^{u\/d}_{2,R}$ fermions are taken from the SM hypercharge\nassignments for the corresponding righthanded singlets, whereas the\n$\\mathbf{U}(1)_2$~charges of all other left- and righthanded fermions\nare taken from the SM hypercharge assignments for the corresponding\n\\emph{left}handed doublets.\n\nIn addition to the kinetic terms,\nYukawa couplings are added to the fermion Lagrangian\n\\begin{multline}\n\\label{equ-lgr-fermion}\n\\mathcal{L}_\\text{Yukawa} = \\\\\n\\lambda\\sum_i\n\\left[\\epsilon_L\\overline{\\Psi}_{0,L}^i\\Sigma_0\\Psi_{1,R}^i +\n\\sqrt{2}v\\overline{\\Psi}_{1,L}^i\\Psi_{1,R}^i +\n\\overline{\\Psi}_{1,L}^i\\Sigma_1\n  \\begin{pmatrix}\n     \\epsilon_{u,R}^i & 0 \\\\\n     0                & \\epsilon_{d,R}^i\n  \\end{pmatrix}\n\\Psi_{2,R}^i\\right]\n\\end{multline}\nwith the index~$i$ running over all SM fermions.\nThe parameter $\\epsilon_L$ is chosen universally for all fermions and\nsuch that the tree-level corrections to the EWPT\nvanish.  This will be referred to as ``ideal fermion\ndelocalization''~\\cite{SekharChivukula:2006cg,SekharChivukula:2005xm}.\nThe parameter $\\lambda$\nis also chosen universally for all fermions; only the~$\\epsilon_{u\/d,R}$ have a nontrivial\nflavor structure and are used to implement the mass splitting of\nquarks and leptons, as well as CKM flavor mixing. The\nvacuum expectation value $v$ breaks the symmetry and the mass\neigenstates are the SM\nfermions (localized mostly at the branes) and heavy partner fermions (localized\nmostly in the bulk).\n\nThe only remaining free parameter in the fermion sector after fixing the SM fermion\nmasses and $\\epsilon_L$ is the heavy fermion mass scale $m_\\text{bulk}=\\sqrt{2}\\lambda v$.\nTherefore the model is fixed uniquely by setting the SM parameters,\n$m_{W'}$ and $m_\\text{bulk}$ and \nby the requirement of ideal delocalization.\nLoop corrections to the EWPT\nand other phenomenological bounds limit the minimal values for these two parameters,\nrequiring $m_{W'}>\\unit[380]{GeV}$ and\n$m_\\text{bulk}>\\unit[2]{TeV}$~\\cite{SekharChivukula:2006cg}.\n\nThe spectrum of the model consists of the SM gauge bosons and\nfermions, the $W^\\prime$ and $Z^\\prime$ and a heavy partner fermion\nfor each SM fermion. The masses of the two new heavy gauge bosons are\nquasi-degenerate ($|m_{W'}-m_{Z'}|=\\mathcal{O}(\\unit[1]{GeV})$), and the\nmasses of the partner fermions are of the order $m_\\text{bulk}$ with\nthe~$t^\\prime$ being slightly heavier than the rest.\n\n\\section{Couplings, widths and branching ratios}\n\\label{sec-cpl}\n\nIdeal fermion delocalization implies that the couplings of the light SM fermions\nto the $W^\\prime$ vanish and that those to the $Z^\\prime$ are small\n($\\mathcal{O}(10^{-2}))$. The both heavy gauge bosons also\ncouple to the SM $Z$ and $W$~bosons with couplings of order $\\mathcal{O}(10^{-2})$.\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{widths}}\n\\caption{\\label{plot-widths}%\nThe relative width~$\\Gamma_V\/m_V$ of the\n$W^\\prime$ and $Z^\\prime$ bosons as a function of $m_{W'}$ with ideal\ndelocalization and $m_\\text{bulk}=\\unit[5]{TeV}$.}\n\\end{figure}\nTherefore, the only decay channel for the $W^\\prime$ in the ideally delocalized scenario is\nthe decay into a $W$ and a $Z$. The $Z^\\prime$ can in principle also decay into SM\nfermions; however, the decay of the longitudinal mode enhances the $WZ$ decay\nchannel by a factor of\n\\begin{equation*}\n  \\frac{m_{Z^\\prime}^4}{16m_W^2m_Z^2}\n\\end{equation*}\nover the decay into a fermion pair causing the latter decay to be highly\nsuppressed by a factor of the order of $\\mathcal{O}(10^{-2})$ (cf.~\\cite{He:2007ge}).\nLooking at figure~\\ref{plot-widths} we find that the resonances are rather narrow\n($\\Gamma_V\/m_V\\approx1-3\\%$) improving the prospects for observing these particles at the LHC.\n\nThe new heavy fermions decay into their light partner and a gauge boson, the resulting\nwidths being of the order $\\Gamma_f\/m_f\\approx0.1$, which, combined with their large mass\n($>\\unit[2]{TeV}$), will make the direct detection as a resonance at a collider rather\nchallenging.\n\nFor a massless SM fermion, the Yukawa coupling between the sites~$1$ and~$2$ vanishes.\nFrom~$\\mathcal{L}_{\\text{Yukawa}}$ we find that the wave function is\ncompletely fixed by the delocalization parameter $\\epsilon_L$. Therefore, the influence of\n$m_\\text{bulk}$ on the wave functions of the light SM fermions and their couplings is very\nsmall and the dependence of the cross section on $m_\\text{bulk}$ is\nalmost negligible at LHC energies.\n\nAlthough ideal delocalization guarantees compatibility with the\nconstraints from EWPT at tree level~\\cite{SekharChivukula:2005xm},\na recent 1-loop analysis~\\cite{Abe:2008hb} has shown that a deviation from\nideal delocalization is necessary to comply with the EWPT constraints at loop\nlevel. According to the authors of~\\cite{Abe:2008hb}, this deviation corresponds to an\non-shell coupling between the $W^\\prime$ and the light fermions as large as $1-2\\%$ of\nthe isospin gauge coupling $g_W\\approx g_0$.\n\n\\begin{figure}\n\\centerline{\\includegraphics{operator1}}\n\\caption{The tree-level diagram generating the operator~$O_1$~(cf.~(\\ref{equ-op1})) after\nintegrating out the bulk fermions.}\n\\label{fig-op1}\n\\end{figure}\nThe coupling $g_{W'ff}$ to which the bounds derived in~\\cite{Abe:2008hb} apply is defined\nin the effective theory obtained by integrating out the bulk fermions and is renormalized\nat the $W^\\prime$ mass shell. There are two operators contributing to this coupling in the\none loop analysis in addition to the coupling of the left-handed\nfermions to the component of\nthe $W^\\prime$ sitting at site $0$. The first one\n\\begin{equation} \n\\label{equ-op1}\n  O_1 = \\overline{\\Psi}_{0,L}\\Sigma_0\\fmslash{A_1}\\Sigma_0^\\dagger\\Psi_{0,L}\n\\end{equation}\nencodes a coupling between the component of the $W^\\prime$ sitting at site $1$ and the\nleft-handed SM fermion and is generated by integrating out the bulk fermion\nfrom the diagram in figure~\\ref{fig-op1} (see\nalso~\\cite{SekharChivukula:2006cg}). The second operator\n\\begin{equation}\n\\label{equ-op2}\n   O_2 = \n    \\overline{\\Psi}_{0,L}\\left(D_\\nu\\left(\\Sigma_0 F_1^{\\mu\\nu}\\Sigma_0^\\dagger\\right)\n      \\right)\\gamma_\\mu\\Psi_{0,L}\n\\end{equation}\narises from loop corrections. Although this operator also contains a contribution to the\ncoupling between the left-handed fermion and the gauge bosons at site $1$, it has a\nnontrivial momentum structure. However, using a non-linear field redefinition in the\nspirit of on-shell effective field theory~\\cite{Georgi:1991ch}, the\ncorresponding part of~$O_2$ \ncan be converted to the same form as~$O_1$ at the price of introducing additional\nhigher dimensional operators coupling at least two gauge bosons to two fermions whose\ncontributions are suppressed by another power of the gauge couplings. This allows the\noperator~$O_2$ to be included into $g_{W^\\prime ff}$ where it contributes to\nthe bounds derived by the authors of~\\cite{Abe:2008hb}.\n\nTherefore, the contributions of both these operators can be accounted for by adjusting the\ndelocalization parameter $\\epsilon_L$ in the tree level\nLagrangian $\\mathcal{L}_{\\text{Yukawa}}$\nto generate the coupling $g_{W^\\prime ff}$. The model parameters then should be understood\nto be renormalized at the $W^\\prime$~mass.\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{gWff_500}}\n\\caption{$g_{W'ff}$, $g_{Z'uu,L}$ and $g_{Z'dd,L}$ normalized to the site $0$ gauge\ncoupling as a function of the delocalization parameter $\\epsilon_L$.\nThe gray rectangle marks the range for $g_{W^\\prime ff}$ allowed by the EWPT as\nderived by the authors of~\\cite{Abe:2008hb}.}\n\\label{fig-gwff-500}\n\\end{figure}\nIn the case of light SM fermions and their partners, only the wave functions of the\nleft-handed fermions depend on the delocalization parameter $\\epsilon_L$. Therefore, the\nright-handed couplings between the new gauge bosons and the light SM fermions are not\naffected by the departure from ideal delocalization. Denoting the wave functions\nby~$\\phi_{f,L,i}$ and~$\\phi_{Z',i}$ and using the normalization of the fermion\nwave functions, the left-handed coupling of a fermion\nto the~$Z^\\prime$ can be written as\n\\begin{equation}\n  \\sum_{i=0}^{1}\\phi_{f,L,i}^2\\left(\\pm\\frac{1}{2} g_i\\phi_{Z',i}\n      + Y g_2\\phi_{Z',2}\\right)\n    = \\pm\\frac{1}{2}\\sum_{i=0}^1 g_i\\phi_{Z',i}\\phi_{f,L,i}^2 + Yg_2\\phi_{Z',2}\\,,\n\\label{equ-cpl-zff}\\end{equation}\nwith the sign depending on the isospin of the fermion and $Y$ denoting the hypercharge.\nAs tuning away from ideal delocalization shifts the light mode of the\nfermion towards the heavy $Z^\\prime$ sitting at site $1$, the isospin dependent part\nin~(\\ref{equ-cpl-zff}) grows, while the correction to~$g_{Z'uu}$ differs only in sign\nfrom the correction to~$g_{Z'dd}$.\n\nFigure~\\ref{fig-gwff-500} shows the dependence of\n$g_{W'ff}$, $g_{Z'dd}$ and $g_{Z'uu}$ on the delocalization parameter and clearly\ndemonstrates this behavior. Considering that we have both $u\\bar u$\nand $d\\bar d$ initial states at\nthe LHC, that both couplings start with positive values of the same order of magnitude at\nthe point of ideal delocalization and that we also have right-handed couplings of the same\norder of magnitude which don't depend on $\\epsilon_L$ at all, we don't\nexpect a large impact from changing $\\epsilon_L$ on $Z^\\prime$~production in the $s$-channel.\nOn the other hand, the effect on the $s$-channel $W^\\prime$~production\nshould be sizable, because $\\epsilon_L$ interpolates between this channel\nbeing forbidden and being about the same order of magnitude as\n$Z^\\prime$ production.\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{gWff_380} &\n\\includegraphics[angle=270,width=6.5cm]{gWff_600}\n\\end{tabular}}\n\\caption{\\label{fig-gwff-380-600}%\nThe same plots as figure~\\ref{fig-gwff-500}, but for the other parts of parameter\nspace probed by our Monte Carlo simulations.}\n\\end{figure}\n\nFigure~\\ref{fig-gwff-380-600} shows the same plot as\nfigure~\\ref{fig-gwff-500} for the other\nregions of parameter space probed in our Monte Carlo simulations. In all three plots,\nchanging $m_\\text{bulk}$ doesn't generate a visible change of the actual couplings, but\ndoes move the gray band of acceptable $g_{W'ff}$ values.\n\n\\section{Implementation}\n\\label{sec:whizard}\n\nWe have coded a FORTRAN 90 module which diagonalizes the lagrangian of the model\nand calculates all masses and couplings. Furthermore, the module calculates the tree\nlevel widths of all new particles. Non ideal delocalization is implemented by tuning the\nparameter $\\epsilon_L$ away from the value required for vanishing $g_{W'ff}$.\nFor the automatized generation of tree level matrix\nelements, we encoded the model in unitarity gauge into the optimizing\nmatrix element generator O'Mega~\\cite{Moretti:2001zz,Kilian:2007gr}\nwhich is part of the Monte Carlo eventgenerator generator WHIZARD~\\cite{Kilian:2007gr}.\nThe results presented below are based on Monte\nCarlo simulations using WHIZARD~\\cite{Kilian:2007gr}.\n\nWe checked the couplings calculated by our FORTRAN code against all the couplings for\nwhich analytic expressions are given in~\\cite{SekharChivukula:2006cg}. To check the\nvalidity of our implementation of the model, we compared the cross sections for a number of\n$2\\rightarrow 2$ processes to the SM, taking $m_{W^\\prime}$,\n$m_\\text{bulk}$ and $m_\\text{Higgs}$ to be huge. The widths calculated by the FORTRAN\nmodule using analytic formulae were checked against numeric results obtained from\namplitudes generated by O'Mega.\n\nWe also checked gauge invariance by numerically checking the Ward Identities in the model \nobtained by taking the limit\n\\begin{equation*}\n  \\sqrt{2}v=\\left<\\Sigma_{0\/1}\\right>\\rightarrow 0\\,,\n\\end{equation*}\nwhere the exact $\\mathbf{SU}(2)_0\\times\\mathbf{SU}(2)_1\\times\\mathbf{U}(1)_2$\ngauge symmetry is restored.\n\nIn addition, we compared several $2\\rightarrow 2$ cross sections\nto the CalcHep~\\cite{Pukhov:2004ca} implementation of the model used by the authors\nof~\\cite{He:2007ge}. After plugging in the correct $W^\\prime$ and $Z^\\prime$ widths,\nthe results turn out to be in perfect agreement.\n\n\\section{$Z^\\prime$ production in the $s$-channel}\n\\label{sec-hzprod}\n\nIn the ideally delocalized scenario, only the $Z^\\prime$ has nonvanishing tree level\ncouplings to the SM fermions, while the $W^\\prime$ is perfectly\nfermiophobic. As explained above, the $Z^\\prime$ decays with a branching ratio of over\n$95\\%$ into a $W^+W^-$ pair, rendering the resulting four fermion final state highly\nfavored over the two lepton one. This is in sharp contrast to many new heavy neutral\ngauge bosons predicted by other extensions of the SM (Little Higgs, GUTs\netc.) which usually have larger fermion couplings but small or vanishing couplings to the\nSM gauge bosons, because they typically originate from different gauge group factors and have little\nor no mixing with the SM gauge bosons~\\cite{Rizzo:2006nw,Langacker:2008yv}.\n\nThe most interesting final states for $Z^\\prime$ production are thus $jjjj$, $l\\nu jj$ and\n$l\\nu l\\nu$. The four jet final state however is highly contaminated from\nbackgrounds containing gluon jets, and the two neutrino final state suffers from\nthe momentum information missing for the two neutrinos, leaving $l\\nu jj$ as\nthe most promising candidate assuming one can cope with the missing neutrino momentum.\n\\begin{figure}\n\\centerline{\\includegraphics{hzprod}}\n\\caption{Representative of the class of diagrams contributing to the $Z^\\prime$\nproduction signal in $pp\\rightarrow l\\nu jj$.}\n\\label{fig-diag-hzprod}\n\\end{figure}\nFigure~\\ref{fig-diag-hzprod} shows a representative of the class of diagrams contributing to\nthe signal in this process.\nIn addition to the signal, there are also reducible backgrounds from events with\nneutral jet pairs and an irreducible background from diagrams not of the type\nfigure~\\ref{fig-diag-hzprod} contributing to the same final state.  In\nthis and the next section, we assume that a veto on forward tagging\njets is effective in suppressing the background from vector boson fusion.\n\nFor the construction of an observable that can deal with the missing longitudinal neutrino\nmomentum, consider the decay of an on-shell $W$ into a lepton with momentum $p_l=q$ and a\nneutrino with momentum $p_\\nu=p$. The mass shell conditions of neutrino and $W$ boson then give\ntwo equations involving the neutrino energy $p_0$ and longitudinal\nmomentum~$p_L$\n\\begin{subequations}\n\\begin{align}\n\\label{equ-nurec1}\n  p_0^2 - p_L^2 - \\left|\\vec p_\\perp\\right|^2 &= 0 \\\\\n\\label{equ-nurec2}\n  p_0 q_0 - p_L q_L - \\vec p_\\perp \\vec q_\\perp &= \\frac{m_W^2}{2}\n\\end{align}\n\\end{subequations}\n(assuming the lepton to be massless), with~$\\vec p_\\perp$ and~$\\vec\nq_\\perp$ the projections of the momenta onto the transverse plane.\n(\\ref{equ-nurec1}) describes a hyperbola in the $p_L-p_0$\nplane and~(\\ref{equ-nurec2}) describes a straight line with the modulus of the slope\nsmaller than $1$. These curves are parametrized by $\\vec p_\\perp$, $q$ and $m_W$ and one\nof their (two in general) intersections gives the neutrino energy and longitudinal\nmomentum as a function of these quantities. This geometrical situation is depicted in\nfigure~\\ref{fig-nurec}.\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{mnurec}}\n\\caption{The two curves generated by the mass shell conditions for $W$ and neutrino in the\ncase of a $W$ decaying to $l\\nu_l$.}\n\\label{fig-nurec}\n\\end{figure}\n\nThis construction allows us to reconstruct the full neutrino momentum from the lepton\nmomentum and the missing $p_T$ for the events coming from the decay of a quasi-on-shell\n$W$. However, owing to the modulus of the slope of the straight line being smaller than\none, we always have two solutions, none of which is preferred on kinematical\ngrounds. We have elected to deal with this by counting \\emph{both} solutions in the histograms,\neffectively doubling the amount of background events while preserving the size of the signal.\nThe two points of intersection can be obtained analytically by\n\\begin{equation}\np_0 = \\frac{q_0^2\\left(m_W^2 + 2\\vec{p}_\\perp\\vec{q}_\\perp\\right) \\pm q_L A}\n\t{2q_0\\left(q_0^2  - q_L^2\\right)} \\quad,\\quad\np_L = \\frac{q_L\\left(m_W^2 + 2\\vec{p}_\\perp\\vec{q}_\\perp\\right) \\pm A}\n\t{2\\left(q_0^2 - q_L^2\\right)}\\,,\n\\label{equ-nurec3}\n\\end{equation}\nwith the abbreviation\n\\[\nA = q_0\\sqrt{\\left(m_W^2 + 2\\vec{p}_\\perp\\vec{q}_\\perp\\right)^2 +\n\t4\\vec{p}_\\perp^2\\left(q_L^2 - q_0^2\\right)} \\,.\n\\]\n\nTo investigate the possibility of discovering the $Z^\\prime$ in\n$pp\\rightarrow jjl+p_{T,\\text{miss}}$ at the LHC we have performed full parton-level Monte\nCarlo simulations for an integrated luminosity of $\\int\\mathcal{L}=\\unit[100]{fb^{-1}}$,\nthe lepton being either an electron or a\nmuon and each jet being either a quark (excluding the top) or a gluon. To suppress the\nbackgrounds, we have applied $p_T$-cuts to all visible particles and to\n$p_{T,\\text{miss}}$\n\\[ p_T \\ge \\unit[50]{GeV}\\,. \\]\nIn addition, we have required the polar and intermediary angles of all visible particles\nto lie within\n\\[ -0.95 \\le \\cos\\theta \\le 0.95 \\]\nand also applied a small-$x$ cut to the ingoing partons\n\\[ x \\ge 1.4\\cdot 10^{-3} \\]\nto avoid infrared singularities in the amplitude. For identifying the intermediary $W$ we\napplied a cut to the invariant mass of the two jets\\footnote{See\n  section \\ref{sec-jetres} for a discussion of the effects of finite jet \n  resolution on the identification of the $W$.}\n\\[ \\unit[75]{GeV} \\le m_{jj} \\le \\unit[85]{GeV}\\,. \\]\nWe used~(\\ref{equ-nurec3}) to reconstruct the neutrino momentum, counting both solutions\ninto the histograms and discarding those with negative neutrino energy.\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{hist_hz_nurec} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hz_deloc}\n\\end{tabular}}\n\\caption{\\emph{Left:} Invariant mass distribution in $pp\\rightarrow l\\nu_ljj$ obtained\nfrom the reconstructed neutrino momenta vs. the distribution obtained from $p_\\nu$\ntaken from Monte Carlo data. \\emph{Right:} The effect of tuning $\\epsilon_L$ away from\nideal delocalization (cf.~figure~\\ref{fig-gwff-500}).}\n\\label{hist-hzprod-nurec}\n\\end{figure}\nThe plot on the left of figure~\\ref{hist-hzprod-nurec} compares  the\ninvariant mass distribution obtained\nfrom the reconstructed neutrino momenta to that obtained from\nthe unobservable neutrino momenta taken from Monte Carlo\ndata for $m_{W^\\prime}=\\unit[500]{GeV}$ and $m_\\text{bulk}=\\unit[3.5]{TeV}$.\nIn both cases the peak from the $Z^\\prime$ is clearly visible. As expected, counting\nboth solutions obtained from the reconstruction doubles the amount of background events\nwhile the number of events contained in the peak stays roughly the same. However, the peak is\nbroadened by the reconstruction, which can been seen when comparing to a\nSM simulation (dotted line). The broadening at the center of the peak is mainly\ncaused by the mismatch between reconstructed and true neutrino momentum of the signal events\ndue to the $W$ not being exactly on-shell; the sidebands of the peak are caused by the\nsecond solutions for $p_\\nu$ of events at the center of the peak.\n\nThe plot on the right of figure~\\ref{hist-hzprod-nurec} shows the\neffect of changing the delocalization parameter\n$\\epsilon_L$ in the range allowed by the EWPT at one loop\n(cf.~section \\ref{sec-cpl}),\nagain for $m_{W^\\prime}=\\unit[500]{GeV}$ and $m_\\text{bulk}=\\unit[3.5]{TeV}$. As argued\nbefore, the impact on the invariant mass distribution is not strong, the peak\nstaying clearly visible over the whole range of allowed values of $\\epsilon_L$.\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{hist_hz_fullrange}}\n\\caption{Invariant mass distribution in $pp\\rightarrow l\\nu_ljj$ for different values of\n$m_{W^\\prime}$ and $m_\\text{bulk}$.}\n\\label{hist-hzprod-fullrange}\n\\end{figure}\nFigure~\\ref{hist-hzprod-fullrange} shows the invariant mass distributions obtained for\n\\[ m_{W^\\prime}\\in\\left\\{\\unit[380]{GeV},\\unit[500]{GeV},\\unit[600]{GeV}\\right\\}\\,,\\]\nwhich covers the whole range of values allowed by the EWPT\nat one loop\\footnote\n{We also changed $m_\\text{bulk}$ as shown in figure~\\ref{hist-hzprod-fullrange} to\ncomply with the EWPT; however, as explained in section \\ref{sec-cpl}, this has no\nnoticeable effect on the cross section.}~\\cite{SekharChivukula:2006cg,Abe:2008hb}. As\nthe masses of the $Z^\\prime$ and $W^\\prime$ are quasi-degenerate,\nthe $Z^\\prime$ peak moves with\nchanging $m_{W^\\prime}$. The histogram shows that the peak stays clearly observable,\nalthough it decreases in size as $m_{W^\\prime}$ becomes larger owing\nto the smaller parton\ndistribution functions for the sea quarks at larger values of~$x$.\n\nTo get a quantitative handle on the significance of the signal and to estimate the minimal\nluminosity necessary for discovering the $Z^\\prime$, we define the raw signal $N$ to be the\nnumber of events in the $\\pm\\unit[20]{GeV}$ region around the peak. To estimate the\nbackground we have generated SM events for an integrated luminosity of\n$\\int\\mathcal{L}=\\unit[400]{fb^{-1}}$, analyzed this data the in the same way\nas the Monte Carlo data for the three site model and then downscaled the resulting\ndistributions by a factor of $4$ to reduce the error coming\nfrom fluctuations in the background. We denote the number of background events in the\n$\\pm\\unit[20]{GeV}$ region around the peak obtained this way by $N_b$.\n\nWe define the signal $N_s$ as\n\\begin{equation}\n   N_s = N - N_b\\,.\n\\end{equation}\nThe number of background events in the original Monte Carlo data $N^\\prime_b$\nis roughly doubled by our momentum reconstruction\n\\begin{equation*}\n  N_b = 2N^\\prime_b\n\\end{equation*}\nand the standard deviation of $\\sigma_{N_b}$ of $N_b$ must scale\naccordingly, resulting in\n\\begin{equation*}\n \\sigma_{N_b} = 2\\sigma_{N^\\prime_b} = 2\\sqrt{N^\\prime_b} = \\sqrt{2N_b}\\,.\n\\end{equation*}\nWe then define the significance in the usual way:\n\\begin{equation}\ns = \\frac{N_s}{\\sigma_{N_b}} = \\frac{N - N_b}{\\sqrt{2N_b}}\\,.\n\\label{equ-sgn-rec}\\end{equation}\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{significance_hz}}\n\\caption{The significance as defined in the text as a function of the integrated\nluminosity. The dotted lines mark the $3\\sigma$ resp. $5\\sigma$ discovery thresholds.}\n\\label{fig-sig-hz}\n\\end{figure}\nThe significance of the signal in the ideally delocalized scenario thus calculated is\nshown in figure~\\ref{fig-sig-hz} together with the $5\\sigma$ and $3\\sigma$ discovery\nthresholds. The $5\\sigma$ thresholds are approx.~$\\unit[1]{fb^{-1}}$,\n$\\unit[2]{fb^{-1}}$, $\\unit[5]{fb^{-1}}$ for\n$m_{W^\\prime}=\\unit[380]{GeV}$, $\\unit[500]{GeV}$, $\\unit[600]{GeV}$, respectively.\nConsidering the fact that tuning\n$\\epsilon_L$ into the region allowed by the EWPT does not\nsignificantly change the signal,\nthe three-site $Z^\\prime$ may be discovered as early as in the first\n$\\unit[1-2]{fb^{-1}}$ and even in the worst case can be expected to\nmanifest itself in the first\n$\\unit[10-20]{fb^{-1}}$ of data.\n\n\\section{$W^\\prime$ production in the $s$-channel without ideal delocalization}\n\\label{sec-hwprod}\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics{hwjj} &\n\\includegraphics{hwll}\n\\end{tabular}}\n\\caption{\\emph{Left: }Representative of the class of diagrams contributing to the\n$W^\\prime$ production signal in $pp\\rightarrow l\\nu_ljj$. \\emph{Right: }One of the signal\ndiagrams in the $lljj$ decay channel of the $W^\\prime$.}\n\\label{fig-diag-hwprod}\n\\end{figure}\nAs discussed in section \\ref{sec-cpl}, the deviation from ideal delocalization required by\nthe EWPT at one loop leads to non-vanishing couplings of the\n$W^\\prime$ to the SM fermions of the same order of magnitude as the $Z^\\prime ff$\ncouplings. This allows for the possibility of producing the $W^\\prime$ in the $s$-channel\nat the LHC.\n\nThere are two possible decay channels for the $W^\\prime$ that are promising candidates for\ndiscovering this resonance. The first possibility is the decay $W^\\prime\\rightarrow\nWZ\\rightarrow l\\nu_ljj$ (cf.~the left plot in figure~\\ref{fig-diag-hwprod}),\nwhich is the final state already discussed in the last section\nand which can be treated the same way (replacing the cut on the $W$ mass with a cut on the\n$Z$ mass). The second possibility is the decay of the $ZW$ pair into two leptons and two\njets (cf.~the right plot in figure~\\ref{fig-diag-hwprod}). The absence\nof missing $p_T$ is a clear advantage\nof this decay mode allowing for background suppression by cutting on the invariant mass of\nthe lepton pair; unfortunately, the branching ratio is smaller than that for the $l\\nu_ljj$\nmode.\n\nTo probe the $l\\nu_ljj$ final state we have used the same Monte Carlo data and cuts as in\nsection \\ref{sec-hzprod} replacing the cut on the invariant mass\\footnote{See\n  section~\\ref{sec-jetres} for a discussion of the effects of finite jet \n  resolution on the separation of $W$ and $Z$.}\nof the jet pair with\n\\[ \\unit[86]{GeV} \\le m_{jj} \\le \\unit[96]{GeV}\\,. \\]\nFor probing the $lljj$ final state we again performed Monte Carlo simulations\nfor an integrated luminosity of $\\int\\mathcal{L}=\\unit[100]{fb^{-1}}$. We applied the\nsame $p_T$, $x$ and angular cuts as in the last section together with the\nidentification cuts\n\\begin{equation}\n\\unit[75]{GeV} \\le m_{jj} \\le \\unit[85]{GeV} \\quad,\\quad\n\\unit[86]{GeV} \\le m_{ll} \\le \\unit[96]{GeV}\n\\label{equ-cut-wzident}\\end{equation}\non the invariant mass of the jet pair and on that of the dilepton system.\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{hist_hwjj_fullrange} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hwll_fullrange}\n\\end{tabular}}\n\\caption{\\emph{Left: }Invariant mass distribution for $W^\\prime$ production in\n$pp\\rightarrow l\\nu_ljj$ for different $W^\\prime$ masses and large $g_{W'ff}$.\n\\emph{Right: }The same distribution for the $lljj$ final state.}\n\\label{hist-hw-fullrange}\n\\end{figure}\nFigure~\\ref{hist-hw-fullrange} shows the invariant mass distributions obtained for both final\nstates for $m_{W^\\prime}=\\unit[380]{GeV}$, $\\unit[500]{GeV}$, $\\unit[600]{GeV}$ and\n$\\epsilon_L$ chosen from the allowed range such as to give large\nvalues\\footnote{Even larger values of $g_{W'ff}$ are allowed by\n  increasing $m_\\text{bulk}$, but we are more interested in the lowest\n  possible value for which the $W^\\prime$ might still be detected in\n  this channel at the LHC.}\nof~$g_{W'ff}$ (cf.~figures~\\ref{fig-gwff-500} and~\\ref{fig-gwff-380-600}). For both final\nstates, the resonance peaks can be clearly seen for all three values of $m_{W^\\prime}$.\nThe total number of events for $lljj$ is much smaller compared to $l\\nu_l jj$\nowing to the smaller branching\nratio, but the cuts on both $m_Z$ and $m_W$ and the absence of the double counting\nintroduced by the neutrino reconstruction significantly improve the\nsignal to background ratio.\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{hist_hwjj_38_deloc} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hwll_38_deloc} \\\\\n\\includegraphics[angle=270,width=6.5cm]{hist_hwjj_50_deloc} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hwll_50_deloc} \\\\\n\\includegraphics[angle=270,width=6.5cm]{hist_hwjj_60_deloc} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hwll_60_deloc}\n\\end{tabular}}\n\\caption{\\emph{Left column: }The $W^\\prime$ resonance peak in the invariant mass\ndistribution for $pp\\rightarrow l\\nu_ljj$ for different values of the delocalization\nparameter. \\emph{Right column: }The same distributions in the case of the $lljj$ final\nstate.}\n\\label{hist-hw-deloc}\n\\end{figure}\nThe dependence of the resonance peak on the delocalization parameter $\\epsilon_L$ is shown\nin figure~\\ref{hist-hw-deloc}. The left column shows the $\\pm\\unit[50]{GeV}$ region around\nthe peak for the $l\\nu_ljj$ final state for different values of $\\epsilon_L$ \nand for the case of ideal delocalization. For\n$m_{W^\\prime}=\\unit[500]{GeV}$ and $m_{W^\\prime}=\\unit[600]{GeV}$ the peak vanishes in the\ncase of ideal delocalization which demonstrates that the cut~(\\ref{equ-cut-wzident}) is\nsufficient to discriminate between jets coming from the decay of $W$ and those coming from\na $Z$. In the case of\n$m_{W^\\prime}=\\unit[380]{GeV}$, a small peak remains even in the case of ideal\ndelocalization which stems from jets coming from\n$pp\\rightarrow Z^\\prime\\rightarrow l\\nu_ljj$ misidentified as a $Z$ (we will discuss the\npossibility of unfolding these two contribution in the next section).\n\nThe histograms show\nthat tuning $\\epsilon_L$ towards the point of ideal delocalization quickly decreases the\nsize of the peak making it invisible for the lowest chosen values of~$\\epsilon_L$.\nThe right column shows the same region around the peak for the final state\n$lljj$ and the same values of $\\epsilon_L$. As should be expected, the same decrease of\nthe peak size is visible.\n\nTo obtain a numerical estimate for the integrated luminosity required\nfor a $s=5\\sigma$ or~$3\\sigma$\ndiscovery of the $W^\\prime$ at some given value of the delocalization\nparameter $\\epsilon_L$ we exploit the fact that the significance of\nthe signal scales as~$g_{W'ff}^2$\nwith the coupling of $W^\\prime$ to left-handed SM fermions. This allows\nus to estimate the integrated luminosity required for\nobtaining a signal with significance $s_0$ in terms of the significance of the signal\nfor other values of coupling and integrated luminosity.\n\nFor the actual determination of $s$ from Monte Carlo data we define the signal as in section\n\\ref{sec-hzprod}. In case of the $l\\nu_ljj$ final state we calculate~$s$\nvia~(\\ref{equ-sgn-rec}), while for the case of $lljj$ it can be calculated simply as\n\\begin{equation}\n  s = \\frac{N_s}{\\sqrt{N_b}}\\,,\n\\end{equation}\nbecause we don't have the additional doubling of the\nbackground events by the neutrino momentum reconstruction in this case\\footnote\n{Because of the lower number of events in the final state $lljj$, the background for this\ncase was calculated for an integrated luminosity of $\\int\\mathcal{L}=\\unit[1000]{fb^{-1}}$\nand scaled down.}.\n\n\\begin{table}\n\\centerline{\n\\begin{tabular}{|c|c|c||c|}\n\\hline\\multicolumn{4}{|c|}{$W^\\prime\\rightarrow l\\nu_ljj$} \\\\\\hline\\hline\n$m_{W^\\prime}\\:\\left[\\unit{GeV}\\right]$ & $m_\\text{bulk}\\:\\left[\\unit{TeV}\\right]$\n& $\\epsilon_L$ & $s$ \\\\\n\\hline 380 & 3.5 & 0.338 & 5.6 \\\\\n\\hline 500 & 3.5 & 0.254 & 8.6 \\\\\n\\hline 600 & 4.3 & 0.211 & 6.4 \\\\ \\hline\n\\end{tabular}\\hspace{1cm}\n\\begin{tabular}{|c|c|c||c|}\n\\hline\\multicolumn{4}{|c|}{$W^\\prime\\rightarrow lljj$} \\\\\\hline\\hline\n$m_{W^\\prime}\\:\\left[\\unit{GeV}\\right]$ & $m_\\text{bulk}\\:\\left[\\unit{TeV}\\right]$\n& $\\epsilon_L$ & $s$ \\\\\n\\hline 380 & 3.5 & 0.338 & 5.1 \\\\\n\\hline 500 & 3.5 & 0.254 & 8.5 \\\\\n\\hline 600 & 4.3 & 0.211 & 8.3 \\\\ \\hline\n\\end{tabular}}\n\\caption{The significance of the signal calculated at different points in parameter space\nfor both final states.}\n\\label{tab-sgn-hw}\n\\end{table}\nThe significances calculated this way at different points in parameter\nspace are shown in\ntable~\\ref{tab-sgn-hw}. For $m_{W^\\prime}=\\unit[380]{GeV}$ and\n$m_{W^\\prime}=\\unit[500]{GeV}$, both final states seem to do equally well at revealing the\nfermionic couplings of the $W^\\prime$; however, for $m_{W^\\prime}=\\unit[600]{GeV}$ the\ndilepton final state appears to give a slightly better signal owing to the\nbetter ratio of signal to background.\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\multicolumn{2}{c}{\\includegraphics[angle=270,width=6.5cm]{sign_hw_380}} \\\\\n\\includegraphics[angle=270,width=6.5cm]{sign_hw_500} &\n\\includegraphics[angle=270,width=6.5cm]{sign_hw_600}\n\\end{tabular}}\n\\caption{The integrated luminosity required for a $5\\sigma$ resp. $3\\sigma$ discovery of the\n$W^\\prime$ in the $s$-channel.}\n\\label{fig-lumi-hw}\n\\end{figure}\nThe integrated luminosity necessary for a $5\\sigma$ resp. $3\\sigma$ discovery of the\n$W^\\prime$ in the $s$-channel is shown in figure~\\ref{fig-lumi-hw} together with the range of the\ndelocalization parameter $\\epsilon_L$ allowed for the different choices of $m_{W^\\prime}$\nand $m_\\text{bulk}$. Taking the integrated luminosity collected over the full LHC running\ntime to be around $\\unit[400]{fb^{-1}}$ and considering the fact that the band of allowed\n$\\epsilon_L$ (and $g_{W^\\prime ff}$) can be moved further towards smaller values by\nlowering $m_\\text{bulk}$, it is evident from figure~\\ref{fig-lumi-hw} that there is a part of\nthe allowed parameter space in which the $W^\\prime$ would appear perfectly fermiophobic\nat the LHC. However, there also is a big region of parameter space in which the coupling of\nthe $W^\\prime$ to the SM fermions eventually should be discovered, although\nthis still would take several years of running time as the lowest integrated luminosity\nrequired for $3\\sigma$ is around $\\unit[10]{fb^{-1}}$ even at the point in parameter space\nmost easily accessible.\n\n\\section{Finite jet resolution and $W$\/$Z$ identification}\n\\label{sec-jetres}\n\nSince flavor tagging is impossible for light quark flavors, we have to\nrely on invariant mass cuts for the jet pairs to be able to separate\nthe case of the two jets in $l\\nu_ljj$ coming from the decay of a~$W$\nin $Z^\\prime$~production from that of the jets being produced by a\ndecaying $Z$ in $W^\\prime$~production.\nHowever, it may very well be impossible to obtain a resolution of order\n$\\pm\\unit[5]{GeV}$ in the jet invariant mass from experimental data.\nIn this section, we discuss the effect of a gaussian smearing of\nthe invariant mass of the jets on our analysis.\n\nIn the ideal case of exact $m_{jj}$ measurement, events coming from the decay of a\nintermediary $W$\/$Z$ are distributed according to a Breit-Wigner distribution\n\\[ p_b(x,m,\\Gamma)\\:dx =\n\\frac{n_b(m,\\Gamma)^{-1}}{\\left(x^2-m^2\\right)^2+\\Gamma^2 m^2}\\:dx\\,, \\]\nwith the normalization factor\n\\[ n_b(m,\\Gamma) = \\frac{\\pi}{4m^3}\\left(1+\\frac{\\Gamma^2}{m^2}\\right)^{-\\frac{3}{4}}\n\\sin^{-1}\\left(\\frac{1}{2}\\atan\\frac{\\Gamma}{m}\\right)\\,. \\]\nEmulating the measurement error in the jet mass by convoluting $p_\\text{bw}$ with a\ngaussian of standard deviation $\\sigma$\n\\[ \np_\\text{g}(x,\\sigma)\\:dx = \\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{x^2}{2\\sigma^2}}\\:dx\n\\]\nwe obtain the smeared distribution\n\\[ p_\\text{sm}(x,m,\\Gamma,\\sigma)\\: dx = \\int_0^\\infty dy\\:p_b(y,m,\\Gamma)\np_\\text{g}(x-y,\\sigma)\\,.\n\\]\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{smearedpeaks}}\n\\caption{The effect of a gaussian smearing on the Breit-Wigner shape of the $W$ and $Z$\nresonances for various widths $\\sigma$ of the gaussian.}\n\\label{fig-smearedpeaks}\n\\end{figure}\n\nFigure~\\ref{fig-smearedpeaks} shows the effect of this smearing on the Breit-Wigner peaks of\nthe $Z$ and the $W$. Turning on the smearing and increasing $\\sigma$ causes the sharp\nBreit-Wigner peaks to decay rapidly, and for $\\sigma=\\unit[10]{GeV}$, only two very broad\nbumps are left. The result is that, if a cross section has one contribution which stems\nfrom the decays of a virtual $Z$ and one coming from a virtual $W$, any attempt to isolate\nthe $Z$ contribution by cutting on the resonance will inevitably also select events coming\nfrom the $W$ decay contaminating the sample (and vice versa). Therefore, our analysis of\nthe $l\\nu_ljj$ final state will show a $W^\\prime$ peak even in the case of ideal\ndelocalization which is caused by jet pairs from a decaying $W$ misidentified as a $Z$.\n\nIf we try to isolate the $W$ peak with a cut on the invariant mass $m_{jj}$\n\\[ L_W \\le m_{jj} \\le U_W \\]\nand the $Z$ peak with a cut\n\\[ L_Z \\le m_{jj} \\le U_Z\\,, \\]\nthen the resulting event counts $\\widetilde{N}_W, \\widetilde{N}_Z$ can be\ncalculated from the true event counts $N_W, N_Z$  coming from a decaying $W$ or $Z$\nvia a matrix $T$ as\n\\[ \\begin{pmatrix} \\widetilde{N}_W \\\\ \\widetilde{N}_Z \\end{pmatrix} =\n\\begin{pmatrix} T_{WW} & T_{WZ} \\\\ T_{ZW} & T_{ZZ} \\end{pmatrix}\n\\begin{pmatrix} N_W \\\\ N_Z \\end{pmatrix} \\]\nwith entries\n\\[ T_{ij} = \\int_{L_i}^{U_i}dm\\:p_\\text{sm}(m,m_j,\\Gamma_j,\\sigma)\\,. \\]\nInverting $T$ we can calculate the event counts $N_W$ and $N_Z$\n\\begin{equation} \\begin{pmatrix} N_W \\\\ N_Z \\end{pmatrix} = T^{-1}\n\\begin{pmatrix} \\widetilde{N}_W \\\\ \\widetilde{N}_Z \\end{pmatrix}\\,.\n\\label{equ-trans-mat}\\end{equation}\nThe entries of $T$ give the probability of misidentifying an event and can be readily\ncalculated numerically; for example, choosing cuts\n\\[ L_W=\\unit[60]{GeV} \\quad,\\quad U_W=\\unit[85]{GeV} \\quad,\\quad\nL_Z=\\unit[86]{GeV} \\quad,\\quad U_Z=\\unit[111]{GeV} \\]\nyields\n\\[ T \\approx \\begin{pmatrix} 0.64 & 0.27 \\\\ 0.29 & 0.62 \\end{pmatrix}\n\\quad,\\quad\nT^{-1} \\approx \\begin{pmatrix} 1.9 & -0.85 \\\\ -0.89 & 2.0 \\end{pmatrix}\\,. \\]\nThis way, we can in principle use $T$ to disentangle the\ncontributions from $W$ and $Z$ resonances\nto the signal in the presence of a measurement error which causes the Breit-Wigner\npeaks to lose their shape. However, to apply this to actual data, it is vital\nto separate the signal from both the reducible and the irreducible\nbackgrounds, because they don't follow a Breit-Wigner distribution.\n\nIn order to estimate the significance of a signal obtained this way, we calculate the\nstandard deviation $\\sigma_{N_i}$ of $N_i$ according to\n\\[ \\sigma_{N_i} = \\sqrt{\\sum_{j\\in W,Z}\\left(T^{-1}_{ij}\\right)^2\\sigma_{\\widetilde{N}_j}^2}\\,. \\]\nIn our analysis, we obtain the signal events inside the smeared Breit-Wigner peaks\n$\\widetilde{N}_i$ by subtracting the background $N_{b,i}$ from the total number of events\n$N_{t,i}$. The error on $N_{t,i}$ is\n\\[ \\sigma_{N_{t,i}} = \\sqrt{N_i + 2N_{b,i}} = \\sqrt{N_{t,i} + N_{b,i}}\\,, \\]\nbecause of the neutrino momentum reconstruction doubling the amount of background events\n(cf.~section~\\ref{sec-hzprod}), and we finally arrive at\n\\begin{equation}\n\\sigma_{N_i} = \\sqrt{\\sum_{j\\in W,Z}\\left(T^{-1}_{ij}\\right)^2\\left(N_{t,j} + N_{b,j}\\right)}\n\\label{equ-sigma-after-transfer}\\end{equation}\nFor a simulation of the effect of the measurement error our analysis we have\nrandomly distributed the invariant mass of the jet pairs within a gaussian with width\n$\\sigma=\\unit[10]{GeV}$ centered around the correct value calculated from Monte Carlo data. We\nthen did the same analysis as in sections \\ref{sec-hzprod} and \\ref{sec-hwprod}\nwith $m_{W^\\prime}=\\unit[500]{GeV}$ and $m_\\text{bulk}=\\unit[3.5]{TeV}$ both for\n$\\epsilon_L=0.254$ and for the ideally delocalized scenario. The only difference to the\nprevious analysis are the cuts on $m_{jj}$ which we enlarged to\n\\[ \\unit[60]{GeV}\\le m_{jj}\\le\\unit[85]{GeV} \\quad\\text{resp.}\\quad\n\\unit[86]{GeV}\\le m_{jj} \\le\\unit[111]{GeV}\\,. \\]\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{hist_w_smear_ideloc} &\n\\includegraphics[angle=270,width=6.5cm]{hist_w_smear_254}\n\\end{tabular}}\n\\caption{\\emph{Left: }Signal in the $W^\\prime$ detection channel for the case of ideal\ndelocalization smeared with a gaussian error. \\emph{Right: }The same for the case\nof nonzero $g_{W'ff}$}\n\\label{hist-wsmear}\n\\end{figure}\nFigure~\\ref{hist-wsmear} shows the resulting effect on the $W^\\prime$ peak for the cases\nof ideal delocalization (left) and for $\\epsilon_L=0.254$ (right). In both cases a peak is\nclearly visible, which in the ideally delocalized scenario is only composed of events\nwith jets coming from a decaying $W$ misidentified as a $Z$.\n\nThe number of signal events\n$\\widetilde{N}_{W\/Z}$ after smearing, the significance $s_{W\/Z}$ of these\ncalculated via~(\\ref{equ-sgn-rec}), $N_{W\/Z}$ obtained from applying the transfer\nmatrix $T^{-1}$ (\\ref{equ-trans-mat}) and the resulting significance\n$N_i\/\\sigma_{N_i}$ obtained from~(\\ref{equ-sigma-after-transfer}) are shown in\ntable \\ref{tab-sgn-wzsep}. All peaks are significant with $s>5\\sigma$; however, after\napplying the transfer matrix, the $W^\\prime$ peak vanishes within one standard deviation\nfor ideal delocalization,\nwhile in the case of $\\epsilon_L=0.254$ a residue as big as $2\\sigma$ remains. The\n$Z^\\prime$ peak remains significant after applying the transfer matrix, however, the\nsignificance is reduced because the transfer matrix enlarges the error.\n\\begin{table}\n\\centerline{\n\\begin{tabular}{|c||c|c|c|c|}\n\\hline\\multicolumn{5}{|c|}{ideal delocalization}\\\\\\hline\\hline\n & $\\widetilde{N}_i$ & $s_i$ & $N_i$ & $\\frac{N_i}{\\sigma_{N_i}}$ \\\\\\hline\\hline\n$i=W$ & $3193$ & $17$ & $5126$ & $13$ \\\\\\hline\n$i=Z$ & $1371$ & $7.5$ & $-96.10$ & $0.24$ \\\\\\hline\n\\end{tabular}\n\\hspace{1cm}\n\\begin{tabular}{|c||c|c|c|c|}\n\\hline\\multicolumn{5}{|c|}{$\\epsilon_L=0.254$}\\\\\\hline\\hline\n & $\\widetilde{N}_i$ & $s_i$ & $N_i$ & $\\frac{N_i}{\\sigma_{N_i}}$ \\\\\\hline\\hline\n$i=W$ & $3767$ & $21$ & $5628$ & $14$ \\\\\\hline\n$i=Z$ & $2083$ & $11$ & $811.6$ & $2.0$ \\\\\\hline\n\\end{tabular}\n}\n\\caption{Comparison of the signals $\\widetilde{N}_{W\/Z}$ obtained with an gaussian\nsmearing of the invariant mass of the jets with $\\sigma=\\unit[10]{GeV}$ to the ``true''\nsignals $N_{W\/Z}$ calculated from the measured ones via the transfer matrix $T^{-1}$.}\n\\label{tab-sgn-wzsep}\n\\end{table}\n\nWhat are the consequences for the detection of $Z^\\prime$ and $W^\\prime$ in the $l\\nu_ljj$\nfinal state? The detection of the $Z^\\prime$ is not affected by inaccuracies in the\njet mass resolution as the peak is always present with little variations of its size over\nthe whole parameter space, and we can always compensate for the smearing of the jet\nmass by enlarging the cut window on $m_{jj}$. However, the separation of a possible $W^\\prime$\ncontribution to the peak (which depends heavily on the point in parameter space) by\ncutting on $m_{jj}$ alone is spoiled by the error in $m_{jj}$; we have to apply additional\ntricks like the transfer matrix~(\\ref{equ-trans-mat}) to disentangle the two contributions.\nWhile this seems to work in principle, the significance of the $W^\\prime$ signal is\nreduced by this analysis, rendering this final state much less suitable for detecting a\ncoupling between $W^\\prime$ and SM fermions than the decay into $lljj$ which\nis not contaminated by a contribution of the $Z^\\prime$.\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nWe have studied the production of the heavy~$W'$ and $Z'$ bosons of\nthe three site higgsless model in the $s$-channel at the LHC.  Unlike\nvector boson fusion, this production mode allows to directly measure\nthe couplings of the new bosons to standard model fermions.  These\ncouplings are constrained by electroweak precision tests and their\nmeasurement is therefore crucial for consistency checks of models of\nelectroweak symmetry breaking with extended gauge sectors.\n\nWe have found a method that will allow the separation of~$W'$ from\n$Z'$~processes at the parton level.  Our results show that the\nobservation of $s$-channel production of $Z'$ bosons will not require\na lot of integrated luminosity for all of the allowed parameter space.\nIn contrast, $W'$ production in the $s$-channel is much more sensitive\nto the model parameters and there are regions of parameter space where\nan observation will be very challenging, if not impossible.  A more\ndetailed experimental analysis should investigate the effects of\nhadronization and detector response on our results.\n\n\n\n\\section*{Acknowledgments}\nThis research is supported by Deutsche Forschungsgemeinschaft through\nthe Research Training Group 1147 \\textit{Theoretical Astrophysics and\nParticle Physics}, by Bundesministerium f\\\"ur Bildung und Forschung\nGermany, grant 05HT6WWA and by the Helmholtz Alliance \\textit{Physics\nat the Terascale}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\indent Throughout this paper, the term ``polyhedron''  refers only to convex polyhedron in Euclidean 3-space $\\mathbb{E}^3$ (convex 3-polytope). For any polyhedron $P$, we denote the set of all vertices, edges and faces of $P$ by $F_0(P)$, $F_1(P)$ and $F_2(P)$ respectively. These sets together with the empty set $\\emptyset$ and $P$ itself, form a lattice $F(P)$ under inclusion, with $P$ as maximum and $\\emptyset$ as minimum. The lattice $F(P)$ of $P$ is determined by the {\\it incidence matrix} $M_P$ of $P$, where we label the vertices by $v_1,\\ldots, v_r$ and the polygonal faces by $A_1,\\ldots,A_s$, say, and define $M_P$ to be the $r \\times s$ matrix whose $(i,j)th$ element $m_{i,j}$ is 1 if $v_i \\in A_j$ and 0 otherwise. The number $\\mu=\\mu(M_P)=\\mu(P)$, of nonzero elements of $M_P$ is called the {\\it multiplicity} of $P$. Since each edge determines four incidences of the adjacent faces and vertices and each incidence corresponds to two edges, we have $\\mu=\\mu(P)=2e$, where $e=e(P)$ is the number of edges of $P$.\n\n\\indent Two polyhedra $P$ and $Q$ are face equivalent (also called combinatorially equivalent), written $P \\approx Q$, if there is a lattice isomorphism from $F(P)$ to $F(Q)$. The set $[P]$ of all $Q \\approx P$ is the \\emph{face type} or, as often called, the \\emph{isomorphism type} of $P$. Since $P$ is the convex hull of the set $F_0(P)$ of its vertices and since $P$ determines $\\F$, we can identify $P$ with the point  $(v_1,\\ldots,v_r)$ in $\\mathbb{E}^{3r}=(\\mathbb{E}^3)^r$, where $\\F = \\{ v_1,\\ldots, v_r \\} $. Thus we may identify a neighborhood of $P$ in $[P]$ with a neighborhood of $(v_1,\\ldots, v_r)$ in $\\mathbb{E}^{3r}$, since any $P'\\approx P$ sufficiently close to $P$ may have its vertices labeled $v'_1,\\ldots v'_r$ in a unique way so that $v'_i$ is close to $v_i$, $i=1,\\ldots,r$.\n\nA graph  $\\mathscr{G}=(V,E)$   with vertex set  $V$  and edge set $E$  is said to be $3$-connected if any two vertices are connected by three internally disjoint paths (each pair of paths have only the two vertices in common); $\\mathscr{G}$  is planar if it can be embedded in the plane such that no two edges intersect internally.\n\nNow, for  any polyhedron $P$ each  edge has exactly two vertices. Therefore we can define the  $1$-skeleton, or edge graph, $\\mathscr{G}_P$ of any  polyhedron  $P$ as  follows. The vertices of $\\mathscr{G}_P$, are vertices of   $P$, and two vertices are connected by an edge in graph $\\mathscr{G}_P$ if they are vertices of the same edge of $P$. This planar graph which is called ``\\emph{Schlegel diagram}'' of $P$, can be obtained by projecting $P$ on one of its faces (outside face).\n\nDue to Steinitz's famous Theorem called ``Fundamental theorem of convex types'' a graph $\\mathscr{G}_P=(V,E)$  is the edge graph of  a polyhedron $P$ if and only if it is planar and $3$-connected. Beyond this remarkable result, it is interesting that, Steinitz discovered some important facts about the topological structure of $[P]$ which is called the realization  space  of $P$ (see Section 2 for the relevant definitions).\n\nIndeed, a careful inspection of the proof of Steinitz's Theorem shows that $[P]$  is smooth manifold with dimension $\\dim[P]=e-1$, up to similarities, where $e=e(P)$ is the number of edges of $P$ .\n\nPerhaps the naturality of the above result of Steinitz was the main reason for other mathematicians to believe that the realization  space of polytopes, in dimensions higher than three, might also be a smooth manifolds. Theorem, on page 18 of \\cite{Rob}, asserts that $[P]$ is manifold in general case, where $P$ is an $n$-dimensional convex polytope.  But a striking result of Mnev \\cite{RG,Mnev} shows that, from the topological point of view, the realization  space of a convex polytope may be arbitrarily complicated. The argument given in \\cite{Rob} does not take full account of coplanarity conditions on the intersections of the faces, but works for $n=3$ (see Section 3).\n\nThe structure of this note is as follows. In Section 2 it is introduced the explicit definition of the {\\it realization space} of a polyhedron in terms of a relevant metric. Then, we give a proof that the realization space is in fact a smooth manifold with a specific dimension. The proof, which is independent of implicit proof of Steinitz himself, is illustrated with  an example adopted from (\\cite{Rob}, p. 21).\n\n\nIn Section 3, for any polyhedron $P$ we define $\\langle P \\rangle$ the {\\it symmetry type} of $P$, together with giving some basic facts about the actions of transformation groups on manifolds. Then, we recall the {\\it stratification} of $\\mathbb{E}^3$  and the decomposition of the manifold  of the space of all polyhedra into {\\it strata} of symmetry types under the action of the point groups.\n\nIn Section 4, we first consider finite point groups generated by reflections. Then, with the help of the familiar notation of {\\it fundamental region} of a reflection group, we prove our main theorem  which relates the dimension of the symmetry type of polyhedron $P$ to the number of its edge orbits, under the action of $G(P)$, the symmetry group of $P$ on its face lattice $F(P)$. This is the proof of Deicke's conjecture in  \\cite{Rob} as well, for polyhedra having reflection symmetry groups. For polyhedra with rotation groups we refer to \\cite{Ros1}, and for the basic properties of polyhedra with symmetry we refer to Robertson's book \\cite{Rob}.\n\n\n\n\\section{The realization space of a polyhedron}\n\n\\indent Our purpose in this Section is to define realization space and determine its topological structure. But first we present some basic definitions.\n\nLet $\\left( \\mathbb{E}^n , \\, d \\right)$ be the $n$-dimensional Euclidean space with induced norm\n$$\nd(x,y) \\, = \\, \\|x-y \\|, \\, x,y\\in \\mathbb{E}^n \\, .\n$$\n\nDenote by $\\mathbb{K}^n$ hyperspace of all nonempty compact convex subsets (convex bodies) of $\\mathbb{E}^n$. Then, $\\mathbb{K}^n$ is endowed with the following familiar metric, called \\emph{Hausdorff metric}.\n\n\\begin{defi}\nFor $P\\in \\mathbb{K}^n$ and $\\varepsilon >0$, let\n$$\n\\mathcal{U}_{\\varepsilon}(P) \\, = \\, \\left\\{ x\\in \\mathbb{E}^n \\, \\mid \\, d(x,P) < \\varepsilon \\right\\} \\, ,\n$$\n\n\\noindent where $d(x,P) \\, = \\, \\stackrel[p\\in P]{}{\\inf} \\left\\{ d(x,p)=\\|x-p \\| \\right\\}$ .\\\\\n\nNow, for $P, \\, P' \\in \\mathbb{K}^n$, let $\\rho(P,P') \\, = \\, \\inf  \\left\\{ \\varepsilon \\, \\mid \\, P' \\subseteq \\, \\, \\mathcal{U}_{\\varepsilon}(P) \\right\\}$. Then,\n$$\\,\n\\,\n\\begin{array}{rcl}\n  d_{\\mathcal{H}}(P,P') & = & \\max  \\left\\{ \\rho(P,P') \\, , \\, \\rho(P',P) \\right\\} \\\\[4pt]\n                        & = & \\inf \\left\\{ \\varepsilon >0 \\, \\mid \\, P \\subseteq \\, \\, \\mathcal{U}_{\\varepsilon}(P') \\text{ and } P' \\subseteq \\, \\, \\mathcal{U}_{\\varepsilon}(P) \\right\\} \\, ,\n\\end{array}\n$$\n\n\\noindent is a metric on $\\mathbb{K}^n$ which is called Hausdorff metric (distance).\n\\end{defi}\n\n\nWe denote by $\\mathscr{P}$ the set of all polyhedra in $\\mathbb{E}^3$ as a topological subspace of the metric space $\\left( \\mathbb{K}^3, \\, d_{\\mathcal{H}} \\right)$.\n\n\n\\begin{defi}\nLet $P$ be a polyhedron (convex 3-polytope). Denote by $[P]$ the face type of $P$, the set of all polyhedra $Q$ face equivalent to $P$.\n$$\n[P] \\, = \\, \\left\\{ Q \\, \\mid \\, Q {\\text{ is polyhedron and }} Q\\approx P \\right\\} \\, .\n$$\n\n\\noindent Then, $[P]$ together with its natural subspace topology induced by Hausdorff metric $d_{\\mathcal{H}}$ is called the realization space of $P$.\n\\end{defi}\n\nAlternatively, any polyhedron $P$ determines, and it is determined by, the set $F_0(P) = \\{ v_1,\\ldots,v_r \\}$ of its vertices. Since we can identify $P$ with the point $(v_1,\\ldots,v_r)\\in \\left( \\mathbb{E}^3 \\right)^r$, $[P]$ can be interpreted as a topological subspace of $\\mathbb{E}^{3r}$ with its topology induced by the vertices.\n\nGiven $\\varepsilon >0$, sufficiently small, there is an open neighborhood $V_{\\varepsilon}(P)$ of $P$ in $[P]$ such that, for all $P' \\in V_{\\varepsilon}(P)$ we have $F_0(P') = \\{ v_1^{'},\\ldots,v_r^{'} \\}$, where $\\| v_i - v_i^{'} \\| < \\varepsilon$, $i=1, \\ldots , r$.\n\nTherefore, $[P]$ can be topologized locally by $V_{\\varepsilon}(P)$ neighborhood of $P$. Thus $V_{\\varepsilon}(P)$, in fact, is the open space of the small perturbations of the vertices of $P$.\n\n\\vspace*{0,2cm}\nNow, let $\\Pi_j$ be the plane containing $A_j$. Then, we may suppose without loss of generality that for all $j=1,\\ldots,s$, the origin $O$ does not lie in $\\Pi_j$. Otherwise translate  $P$ to ensure this condition. Then, for all $j$ there is a unique $a_j \\in \\mathbb{E}^3$ such that $\\Pi_j$ is given by the equation $\\langle x,a_j\\rangle=1, x\\in \\mathbb{E}^3$. Thus $\\Pi_j$ is given by $a_j$ and $P$ itself by the point\n$$(v_1,\\ldots,v_r,a_1,\\ldots,a_s)\\in \\mathbb{E}^{3r}\\times \\mathbb{E}^{3s}=\\mathbb{E}^{3(r+s)} \\, ,$$\n\n\\noindent where $\\langle v_i,a_j\\rangle=1$ for all $i,j$ such that $m_{i,j}=1$. Now let $P'$ be the polyhedron with $P\\approx P'$, having vertices $v'_1,\\ldots,v'_r$ and faces given by $a'_1,\\ldots,a'_s$ where the plane $\\Pi'_j$ of $A'_j$ has equation $\\langle x,a'_j\\rangle= 1$ and again $\\langle v'_i,a'_j \\rangle=1$ if $m_{ij}^{'}=1$ (with obvious notation).\nSuppose that $P'$ is close to $P$, so that\n$$v'_i=v_i+\\xi_i,\\,\\,\\,\\,a'_j=a_j+\\eta_j \\, ,$$\n\n\\noindent for some $\\xi_j,\\eta_j \\in \\mathbb{E}^3$ with $||\\xi_i||$ and $||\\eta_j||$ small ($i=1,\\ldots,r;j=1,\\ldots,s$). Then,\n\\begin{equation}\\label{eq1}\n\\bf{\\Phi_{[i,j]}(\\xi,\\eta):=\\langle v_i,\\eta_j\\rangle+\\langle \\xi_i,a_j\\rangle+\\langle\\xi_i,\\eta_j\\rangle=0},\n\\end{equation}\n\n\\noindent if $m_{ij}=1$ (and hence $m_{ij}^{'}=1$).\n\n\\vspace*{0,2cm}\nConversely, for any sufficiently small $\\xi \\in \\mathbb{E}^{3r}$ and $\\eta \\in \\mathbb{E}^{3s}$, the point\n$$(\\nu_1+\\xi_1,\\ldots,\\nu_r+\\xi_r,a_1+\\eta_1,\\ldots,a_s+\\eta_s)$$\n\n\\noindent represents a unique polyhedron $P'\\approx P$, provided equations (\\ref{eq1}) hold. Let us now consider the polynomial map $\\Phi: \\mathbb{E}^{3r}\\times \\mathbb{E}^{3s}\\rightarrow \\mathbb{E}^\\mu$ given by  $\\Phi(\\xi,\\eta)=(y_1,\\ldots,y_\\mu)$ where the pairs $(i,j)$ with $m_{ij}=1$ are arranged in lexicographical order, and if $(i,j)$ is the $[i,j]th$ such pair then, $y_{[i,j]}=\\Phi_{[i,j]}(\\xi,\\eta)$ is defined as in (\\ref{eq1}). We may for convenience suppress component suffices of $\\xi_i$ and $\\eta_j$, and write\n$$\\frac{\\partial y_{[i,j]}}{\\partial \\xi_i}=a_j+\\eta_j,\\,\\,\\,\\,\\frac{\\partial y_{[i,j]}}{\\partial \\eta_j}=\\nu_i+\\xi_i \\, .$$\n\nThus the Jacobian matrix $J_{\\Phi}(0,0)$, has order  $\\mu\\times(3(r+s))$ with $a_j$ in the $[i,j]$th row and the $i$th triplet of columns, and $\\nu_i$ in the $[i,j]$th row and the $(r+j)$th triplet of columns. All the other elements of $J_{\\Phi}(0,0)$ are $0$.\n\nA simple example may helps to clarify these remarks. Let $P$ be the polyhedron shown, with its accompanying Schlegel diagram in the following figure.\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.75cm} \\includegraphics[scale=.2]{FIG_1.eps}\n\\vspace*{-0,4cm}\\caption{{\\small{Schlegel diagram of $P$ with ``outside face'' $A_1$}}}\n{\\label{im1}}\n\\end{figure}}\n\n\\newpage\nIn this example, $r=7,s=8$ and $\\mu=26$. Notice that $3(r+s)=45>26=\\mu$. In fact, for any polyhedron, $3(r+s)=3(e+2)=3e+6=\\mu+e+2>\\mu$, by Euler's Theorem. Thus $\\mu$ is the largest value of the rank of $J_{\\Phi}(0,0)$. In the example, $J_{\\Phi}(0,0)$ has 26 rows and 45 columns, shown below in the truncated $26\\times 15$ form.\n\n\\begin{center} {\\scriptsize{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n  \\hline\n\n    & 1 & 2 & 3 & 4 & 5 & 6 & 7  & 1 & 2 & 3 & 4 & 5 & 6 & 7   & 8  \\\\\n    \\hline\n 15 & $a_5$ &   &   &   &   &   &     &   &   &   & $\\nu_1$   &  &  &  & \\\\\n  \\hline\n  16 & $a_6$ &   &   &   &   &   &     &   &   &   &   & $\\nu_1$ &   & & \\\\\n  \\hline\n  17 & $a_7$ &   &   &   &   &   &     &   &   &   &   &   &  & $\\nu_1$ & \\\\\n  \\hline\n  18 & $a_8$ &   &   &   &   &   &     &   &   &   &   &   &   &   &$\\nu_1$ \\\\\n  \\hline\n  22 &   & $a_2$ &   &   &   &   &     &  & $\\nu_2$  &   &   &   &   &   & \\\\\n  \\hline\n  23 &  & $a_3$ &   &   &   &   &      &         &  &  $\\nu_2$ &   &   &   &   & \\\\\n  \\hline\n  25 &   & $a_5$ &   &   &   &   &     &   &   &  &  & $\\nu_2$  &   &   & \\\\\n  \\hline\n  26 &   & $a_6$ &   &   &   &   &     &   &   &   &   &  & $\\nu_2$  &   & \\\\\n  \\hline\n  32 &   &   & $a_2$ &   &   &   &     &  & $\\nu_3$  &   &   &   &   &   & \\\\\n  \\hline\n  34 &   &   & $a_4$ &   &   &   &     &   &  &   & $\\nu_3$ &   &   &   & \\\\\n  \\hline\n  35 &   &   & $a_5$ &   &   &   &     &   &   &   &  & $\\nu_3$  &   &   & \\\\\n  \\hline\n  38 &   &   & $a_8$ &   &   &   &    &   &   &   &   &   &   &   & $\\nu_3$\\\\\n  \\hline\n  41 &   &   &   & $a_1$ &   &   &   & $\\nu_4$   &   &   &   &   &   &  & \\\\\n  \\hline\n  44 &   &   &   & $a_4$ &   &   &   &     &   &  & $\\nu_4$  &   &   &   & \\\\\n  \\hline\n  47 &   &   &   & $a_7$ &   &   &   &     &   &   &   &   &  & $\\nu_4$ & \\\\\n  \\hline\n  48 &   &   &   & $a_8$ &   &   &   &     &   &   &   &   &   &   &$\\nu_4$\\\\\n  \\hline\n  51 &   &   &   &   & $a_1$ &   &   & $\\nu_5$   &   &   &   &   &   &   & \\\\\n  \\hline\n  53 &   &   &   &  &  $a_3$ &   &   &     &  & $\\nu_5$  &   &   &   &   & \\\\\n  \\hline\n  56 &   &   &   &   & $a_6$ &   &   &      &   &   &   &  &  $\\nu_5$ &   & \\\\\n  \\hline\n  57 &   &   &   &   & $a_7$ &   &   &     &   &   &   &   &  & $\\nu_5$  & \\\\\n  \\hline\n  61 &   &   &   &   &   & $a_1$ &   &  $\\nu_6$  &   &   &   &   &   &   &\\\\\n  \\hline\n  62 &   &   &   &   &   & $a_2$ &     &  &  $\\nu_6$ &   &   &   &   &   &\\\\\n  \\hline\n  63 &   &   &   &   &   &$ a_3$ &     &   &  & $\\nu_6$  &   &   &   &   &\\\\\n  \\hline\n  71 &   &   &   &   &   &   & $a_1$ &  $\\nu_7$ &   &   &   &   &   &   & \\\\\n  \\hline\n  72 &   &   &   &   &   &   & $a_2$ &   & $\\nu_7 $  &   &   &   &   &   & \\\\\n  \\hline\n  74 &   &   &   &   &   &   & $a_4$ &    &   &  &   $\\nu_7$ &   &   &   &\\\\\n  \\hline\n\\end{tabular}}} \\end{center}\n\n\\begin{center}\n[truncated $J_{\\Phi}$]\n\\end{center}\n\n\nRecall that a similarity is a map\n\\vspace*{-.3cm}\n$$ s:\\mathbb{E}^3 \\rightarrow \\mathbb{E}^3$$\n\n\\vspace*{-.3cm}\n\\noindent such that for some real $r>0$, and all $x,y\\in \\mathbb{E}^3$\n\\vspace*{-.3cm}\n$$\nrd(x,y)=d(s(x), s(y)) \\, .\n$$\n\n\\vspace*{-.3cm}\nThe set of all similarities is a group which is denoted by $Sim(3)$, and called similarity group of $\\mathbb{E}^3$.\n\nThe map $\\varphi: Sim(3)\\rightarrow \\mathbb{R}_\\ast$ defined by $\\varphi(f)=r$, where $r$ is as above and $\\mathbb{R}_\\ast$ is multiplicative group of positive real numbers, is a group homomorphism.\nIt is well known that the kernel of this homomorphism is the isometry group or the Euclidian group $\\mathbb{E}(3)$. We may identify $Sim(3)$ as $\\mathbb{E}(3) \\times \\mathbb{R}_\\ast$. Since the manifold $\\mathbb{E}(3)$ has dimension $6$, the dimension of $Sim(3)$ is equal to $7$.\n\n\\vspace*{.2cm}\nNow we are ready to prove our main theorem in this Section:\n\\begin{teo} Let $P$ be a convex 3-polyhedron. Then, $[P]\/Sim(3)$, the realization space of $P$ modulo similarities, is a manifold of dimension $\\dim \\left( [P]\/Sim(3) \\right) \\, = \\, e-1$, where $e=e(P)$ the number of edges of $P$.\n\\end{teo}\n\\vspace*{-.1cm} {\\bf Proof:} To show, equivalently, that for any $P$, the face type $[P]$ is a manifold of dimension $e+6$, it is enough to show that $J_{\\Phi}(0,0)$ has rank $\\mu$. The result follows by the implicit function Theorem, since $e-1=3(r+s)-\\mu-7$.\n\nWe want to show that the rows of $J_{\\Phi}(0,0)$ are linearly independent. Suppose then, that for some real number $\\alpha_{[i,j]}$, we have\n\\vspace*{-.1cm}\n\\begin{equation}\\label{eq2}\n\\sum_j \\alpha_{[i,j]}a_j=\\sum_i \\alpha_{[i,j]}\\nu_i=0 \\, ,\n\\end{equation}\n\n\\vspace*{-.3cm}\n\\noindent for each $i=1,\\ldots,r$ and each $j=1,\\ldots,s$. We know that for any $i$ and any three values of $j_1,j_2,j_3$ of $j$ with $m_{ij}=1$, $a_{j_1},a_{j_2},a_{j_3}$ are linearly independent. Hence for each $i$, the equations $\\sum  \\alpha_{[i,j]}a_j=0$ have solution space of dimension $s_i-3$ where $s_i$  is the number of the values of $j$ with $m_{i,j}=1$, and we can express any three of the  numbers  $\\alpha_{[i,j]}$ as linear functions  of the remaining $s_i-3$. Hence the $\\mu$ numbers $\\alpha_{[i,j]}$ are expressible as linear functions of $\\sum_{i=1}^r(s_i-3)=\\mu-3r$ independent variables. In particular, for each $i$ for which $s_i=3$, all the numbers $\\alpha_{[i,j]}$ are 0.\n\nNow consider the equations $\\sum \\alpha_{[i,j]}\\nu_i=0$. Again, for each $i$, any three of the vertices $v_i$ are linearly independent. So again we can express any three of the numbers $\\alpha_{[i,j]}$ as linear combinations of the remaining $r_j-3$. So the numbers $\\alpha_{[i,j]}$ are expressible as linear functions of $\\sum_{i=1}^s(v_j-3)=\\mu-3s$ independent variables.\n\nBut the labels $\\alpha_{[i,j]}$ for each $j$ are already expressed as linear functions of $\\mu-3r$ independent variables, as described above. Hence the numbers $\\alpha_{[i,j]}$ are over determined by the two sets of equations in (\\ref{eq2}), since the solution space of the equations in (\\ref{eq2}) has dimension $\\mu-3(r+s)$. But  $\\mu-3(r+s)<0$ by Euler formula. It follows that $\\alpha_{[i,j]}=0$ for all $i,j$ where $m_{i,j}=1$.\\hfill$\\square$\n\n\n\\vspace*{.5cm}\nReferring to the above example, we may express each of, say  $\\alpha_{[1,6]}$, $\\alpha_{[1,7]}$ and $\\alpha_{[1,8]}$ as linear function of $\\alpha_{[15]}$ that is as a multiple of $\\alpha_{[15]}$. Thus $\\alpha_{[16]}=\\lambda_{16}\\alpha_{[15]}$, $\\alpha_{[17]}=\\lambda_{17}\\alpha_{[15]}$, $\\alpha_{[18]}=\\lambda_{18}\\alpha_{[15]}$. Likewise, taking the first number $\\alpha_{[i,j]}$ in each block parameter where possible we have  $\\alpha_{[23]}=\\lambda_{23}\\alpha_{[22]}$, $\\alpha_{[25]}=\\lambda_{25}\\alpha_{[22]}$ and $\\alpha_{[26]}=\\lambda_{26}\\alpha_{[22]}$, and so for each of the values $i=1,2,3,4,5$. For $i=6$ and for $i=7$, $s_i=3$ and immediately we find that $\\alpha_{[6i]}=\\alpha_{[7j]}=0$ for each $j$ with $m_{6j}=1$ or $m_{7j}=1$.\n\nNow repeat this procedure with numbers $\\alpha_{[i,j]}$ with $j$ fixed. Thus  $\\alpha_{[51]}=\\mu_{51}\\alpha_{[41]}$, $\\alpha_{[61]}=\\mu_{61}\\alpha_{[41]}$, and $\\alpha_{[71]}=\\mu_{71}\\alpha_{[41]}$, and likewise, for $j=2$. For $j=3,4,5,6,7$ and $8$, only three values of $i$  corresponds to each values of $j$, because the associated faces $A_j$ are triangles. So all $\\alpha_{[ij]}=0$ for all $i,j$ with $m_{i,j}=1$, as in general case.\n\n\n\\begin{obs}\n{\\label{obs1}}\n{\\rm It is worth mentioning that Richter-Gebert in (\\cite{RG}, Section 13.3) by fixing a suitable affine basis, proves that the realization space of a polyhedron $P$ (denoted there by $\\mathcal{R}(P)$), that is the space of coordinatization for the combinatorial type of $P$ with $e(P)=e$ edges is a smooth open manifold of dimension $e-6$, module the natural action of $12$-dimensional affine transformation group.\n\nThe realization space $\\mathcal{R}(P)$ is understood as a subspace of $\\mathbb{E}^{3n}$ by identifying the $3n$ coordinates of the $n$ vertices of $P$ with points in $\\mathbb{E}^{3n}$. It is described by the set of all solutions of a collection of polynomial equations and inequalities with integer coefficients. Such sets are called a simple semi-algebraic variety.\n\nFurthermore, by fixing an affine basis in the definition of the realization space, one makes sure that the ``reflection'' (mirror images in Steinitz's proof) do not create a second component of the realization space. Therefore, $\\mathcal{R}(P)$ is indeed (path connected) \\emph{contractible} and has \\emph{Steinitz's isotopy property} (\\cite{Stei}, Section 69), i. e., any two realizations $P_1$ and $P_2$ of $P$ can be continuously deformed into each other while maintaining the same structure throughout.\n\nHowever, our approach to study the face type manifold $[P]$ of a polyhedron $P$, is different. Although the realization spaces such as $[P]$ are usually (for example as above) defined modulo affine or Euclidian groups, in this work we consider $[P]\/Sim(3)$, $[P]$ modulo similarity group. The topology of the realization space $[P]$ of polyhedron $P$ is induced by Hausdorff metric, and since the action $Sim(3)$ on $[P]$ is not (fixed-point) free (see Section 3 for definition), the manifold $[P]\/Sim(3)$ is not contractible, and clearly does not satisfy Steinitz's isotopy property.}\n\\end{obs}\n\n\\begin{obs}\n{\\rm Let $P$ be a polyhedron with $f_0(P)=r$ vertices, $f_1(P)=e$ edges and $f_2(P)=s$ faces. Consider the face type $[P]$ of $P$. In (\\cite{Rob}, p. 75) the dimension of this manifold is given by\n$$\\dim [P]=3(r+s)-\\mu(P) \\, .$$\n\nAn intuitive derivation of the above formula may be given as follows. If the vertices and faces of $P$ were allowed to move independently in $\\mathbb{E}^3$ then, they would have $3(r+s)$ \\emph{degrees of freedom} (see page 11 for definition). But they are not independent. In fact for each incidence relation $(v,F)\\in F_0(P)\\times F_2(P)$ with $v \\in F$, the whole space loses one degree of freedom. Hence we have\n$$\\dim[P] \\, = \\, 3(r+s)-\\mu(P)\\,.$$\n\n\nNote that, since the similarity Lie group $Sim(3)$ has dimension 7, by Euler's formula\n\\vspace*{-.1cm}\n$$n-e+f=2 \\: , $$\n\n\\vspace*{-.1cm}\n\\noindent we have\n\\vspace*{-.1cm}\n$$\n\\begin{array}{rcl}\n  \\dim \\left( [P]\/Sim(3) \\right) & = & 3(r+s)-\\mu(P)-7 \\\\\n  \\, & = &  3(r+s)-2e-7 \\, = \\, e-1\\, (\\text{\\small{\\emph{module similarities}}}) \\, .\n\\end{array}\n$$}\n\\end{obs}\n\n\n\\section{Symmetry type of a polyhedron and stratifications}\n\n{\\bf Transformation groups}\n\n\\vspace*{0,2cm}\n\\indent We start this Section by introducing some basic definitions and standard results from the theory of Lie groups acting on smooth manifolds and refer the reader to \\cite{Dei} and \\cite{Kaw} for more details.\n\n\\vspace*{0,2cm}\nLet $G$ be a Lie group, with identity element $e$, and $M$ a smooth manifold. A smooth action of $G$ on $M$ is a $C^{\\infty}$ mapping:\n\\vspace*{-0,2cm}\n$$\\phi: G\\times M \\rightarrow M \\: , \\:\\: \\phi(g,x)=\\phi_g(x)=g(x) \\, ,$$\nsuch that\n$$\n\\begin{array}{rcll}\n  e(x) & = & x, & x\\in M \\, ,\\\\\n  (g_1g_2)(x) & = & g_1(g_2(x)), & g_1, g_2 \\in G \\, , \\: x\\in M \\, .\n\\end{array}\n$$\n\n\nIn this case we say that $M$ is a $G$-manifold. For any $x\\in M$ the subgroup\n\\vspace*{-.2cm}\n$$G_x=\\{g \\in G \\mid g(x)=x\\} \\, ,$$\nof $G$ is called the \\emph{stabilizer} or \\emph{isotropy} subgroup at $x$.\n\nThe set $ \\, G(x)=\\{ g(x) \\mid g\\in G \\} \\,$ is called the \\emph{$G$-orbit} of $x$. The \\emph{orbit space} of the action of $G$ on $M$ is the space $M\/G$, the space of all $G$-orbits endowed with the quotient topology given by canonical projection\n\\vspace*{-.2cm}\n$$\n\\begin{array}{ccccl}\n  \\pi & : & M & \\rightarrow & M\/G\\\\\n   &  & x & \\mapsto & G(x)\n\\end{array} \\, ,\n$$\n\\noindent and the differentiable structure of $M\/G$ is induced by the same structure of $M$.\n\n\nThe action is called \\emph{free} if, for each $x\\in M$, $ \\: G_x=\\{ e \\} \\, $ .\n\nIf a Lie group $G$ acts on a smooth manifold $M$ via $\\phi$, we call $(M,G):=(M,G,\\phi)$ a transformation group, and $M$ is said to be a $G$-manifold.\n\nNow, for each polyhedron $P$, a symmetry of  $P$ is a rigid  transformation  (or self isometry) $f:\\mathbb{E}^3\\rightarrow \\mathbb{E}^3$  such that $f(P)=P$. Any such symmetry maps vertices to vertices, edges to edges and faces to faces and preserves inclusions (incidences). Hence any symmetry induces an  automorphism on $F(P)$. The set $G(P)$  of all symmetries of $P$ is a finite subgroup of the Euclidean group  $\\mathbb{E}(3)$ acting on $F(P)$ as a group of automorphisms. We may assume that the centroid is $O$, so $G(P)$ is a finite subgroup of the orthogonal group $\\mathcal{O}(3)$. If a finite subgroup $G$ of $\\mathcal{O}(3)$ is the symmetry group of a convex polyhedron $P$, we also call $P$ a $G$-polyhedron.\n\n\n\n\\begin{defi}\nTwo polyhedra $P$ and $Q$ are \\emph{symmetry equivalent}, and write $P\\cong Q$, if there is an isomorphism\n$$\\lambda: F(P)\\rightarrow F(Q)$$\n\n\\noindent of the face lattices and some isometry $f:\\mathbb{E}^3 \\rightarrow \\mathbb{E}^3$ such that for all\n$g \\in G(P)$ and all $x \\in F(P)$,\n$$\\lambda(gx)=(f g f^{-1})(\\lambda(x)) \\, .$$\n\\end{defi}\n\nIf we further assume that $P$ and $Q$ are both $G$-polyhedra, that is having the same (rather than conjugate) subgroups then, with the above condition, we say that $P$ and $Q$ are $G$-equivalent. Hence, in this case\n$$\\lambda(gx)=g\\lambda(x) \\, .$$\n\n\\begin{defi}\nLet $P$ be a $G$-polyhedron. The symmetry type $\\langle P \\rangle$ of $P$ is defined by\n$$\\langle P \\rangle = \\{ Q \\, | \\, Q \\text{ is $G$-polyhedron and $Q$ is $G$-equivalent to $P$}, \\,\\, Q\\cong P \\} \\, . $$\n\\end{defi}\n\nNow consider $\\mathscr{P}$ the space of all convex polyhedra in $\\mathbb{E}^3$. Since the subdivision of $\\mathscr{P}$ into face types and symmetry types both respect the Euclidian similarities, it is convenient to look at the action of the similarity group $Sim(3)$ on the space $\\mathscr{P}$ of all polyhedra in $\\mathbb{E}^3$. This action partitions the quotient space\n$$ \\, \\mathscr{S}:=\\mathscr{P}\/Sim(3) \\, ,$$\n\n\\vspace*{-.2cm}\n\\noindent of similarity classes or ``shapes'' of polyhedra, into orbit types, where each orbit type consists of all those orbits on which the isotropy subgroups at any polyhedron in the orbit are conjugate.\n\nThus the symmetry types partitions each $[P]$ into mutually disjoint subsets refining the partitions of $\\mathscr{S}$ into face types.\n\nBut the isotropy subgroup at $P$ is just the symmetry group $G(P)$ itself. It follows that the symmetry types are composed of components of the orbit types. The principal orbit type corresponds to the trivial isotropy subgroup, that is to say to the symmetry type of any polyhedron $Q$ in $[P]$ with trivial symmetry group $G(Q)=\\{ e \\}$. This type is open in $[P]\/Sim(3)$ of dimension $e(P)-1$. All other symmetry types have lower dimensions.\n\nAs an example, let $P$ be a polyhedron combinatorially equivalent to cube. Then, $[P]\/Sim(3)$ has dimension 11. The principal orbit type corresponds to the realization space of polyhedron $Q\\approx P$ with trivial group, is an open and dense submanifold of $[P]\/Sim(3)$ of dimension $e(Q)-1=11$.\n\nNow, since each orbit type is a submanifold of $[P]$, by the Slice Theorem of transformation groups (see \\cite{Kaw}, Th. 4.11, and \\cite{Rob}, p.42), we have the following well known theorem.\n\n\n\n\\begin{teo} {\\rm (in \\cite{Rob}, p. 42 and \\cite{Ros2})}  Let $P$ be a polyhedron in $\\mathbb{E}^3$. Then, $\\langle P \\rangle$, the  symmetry type of $P$, is a smooth manifold.\\end{teo}\n\nIn order to study the dimension of the symmetry type of $P$, $\\dim \\langle P \\rangle$, we simply study those polyhedra $Q$ that lie in some neighborhood of $P$ in $\\mathscr{S}$ and are symmetry equivalent to $P$. We can simplify our discussion by restricting our attention to those $Q$ whose symmetry group is not merely conjugate to $G(P)$ but $G(P)$ itself. Along this restriction we also factor out components that came from similarities. In this way, we may determine the value of  $\\dim \\langle P \\rangle$. We first look at a simple example again.\n\nLet $P$ be the right pyramid over a square (Figure \\ref{image1}) with symmetry group $G(P)$ which is dihedral reflection group.\n\n\nSuppose we fix the group $G = G(P)$. Then, vertex $v_1$ can be chosen only on the axis of $G$. Therefore it has one degree of freedom (see page 11). Likewise $v_2$  must lie on reflection plane, hence has only two degrees of freedom. Having chosen  $v_2$, the vertices $v_3$,$v_4$ and $v_5$  which are on the same orbit of $v_2$ , have no degree of freedom at all, since they are determined by our choice of G and $v_2$ . Hence the vertices have a total of $1 + 2 = 3$ degrees of freedom. Similarly for the faces, each triangular face or equivalently the plane that contains it has only two degrees of freedom in the space of affine plane in $\\mathbb{E}^3$, since each plane is invariant under a reflection element of $G$. But the square face has only one degree freedom, because it is orthogonal to the axis of rotation of $G$. Therefore the faces have just $2 + 1 = 3$ degrees of freedom. Of course the faces and vertices can not be chosen independently of one another. The incidence of $v_1$ with respect to any of the four triangular faces adjacent to it, determines the incidence of that vertex to the other three faces under the action of $G$. Hence $v_1$  has only one ``independent'' incidence. The vertices $v_2$, $v_3$, $v_4$ and $v_5$ are in the same $G$-orbit and each one is incident with two triangles and one square faces. Take one of them say $v_2$. There is a reflection which fixes $v_2$ and sends adjacent triangular faces each one to the other. Thus the number of independent multiplicity (to be defined later) of $P$ is $1 + 2 = 3$. Each such incidence relation in the form of the condition that a vertex lies in a particular face, reduces the dimension of the symmetry type by one. Now, if we take into account the fact that the center of $P$ can be chosen only on the fixed point set of $G$, which is one dimensional and considering also the dilation of $P$ which in each case reduces $\\dim\\langle P\\rangle $ by one, we get\n$$\\dim\\langle P\\rangle \\, = \\, (1 + 2) + (1 + 2) - (1 + 2) - 2 \\, = \\, 1 \\, .$$\n\n\\vspace*{.5cm}\n{ \\begin{figure}[!htb] \\centering\n\\vspace*{-.5cm} \\hspace*{.75cm} \\includegraphics[scale=.25]{figura1.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{image1}\n\\end{figure}}\n\n\\vspace*{.5cm}\nIn fact, in Figure \\ref{image1} or in any right pyramid with a regular polygon as base, if we denote the height and radius of the base of P by $h$ and $r$, respectively and consider the ratio $\\zeta=\\frac{h}{r}$  the two such pyramids are similar if they have the same ratio $\\zeta$. Therefore we can parameterize the symmetry type of $P$ by $\\zeta$ with $0<\\zeta$. Hence $\\langle P \\rangle$ has the structure of the open interval and $\\dim \\langle P\\rangle  = 1$. Indeed the action of $G(P)$ on $F(P)$ has $\\epsilon=2$ edge orbits. Hence\n$\\dim \\langle P\\rangle =\\epsilon -1 = 1$.  Note that in the right pyramid with regular base the ratio $\\zeta$ is similarity invariant and the symmetry type is a connected 1-manifold with boundary 0-dimensional symmetry types one for regular base and the other a segment (1-polytope). The following figure illustrates this idea.\n\n\\vspace*{.5cm}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.5]\n\\filldraw [black]  (0,0) circle (2pt)\n            (15,0) circle (2pt);\n\\draw (15,0) -- (0,0) node[above] {\\small{$h\\rightarrow 0$}};\n\\draw (15,0) -- (0,0) node[below] {\\footnotesize{regular base}};\n\\draw (0,0) -- (15,0) node[above] {\\small{$r\\rightarrow 0$}};\n\\draw (0,0) -- (15,0) node[below] {\\footnotesize{1-polytope}};\n\\end{tikzpicture}\n\\end{center}\n\n\\begin{center}\n\\vspace*{-1.1cm} {\\footnotesize{right pyramid}}\n\\end{center}\n\n\n\\begin{center}\n{ \\begin{figure}[!htb] \\centering\n\\vspace*{-.55cm} \\includegraphics[scale=.075]{extra.eps}\n\\caption{$\\,$}\n\\label{extra}\n\\end{figure}}\n\\end{center}\n\n\\vspace*{-1cm}\nThe idea of this example can be applied in general to find the dimension of the symmetry type of any polyhedron $P$.\n\nLet us denote by $F_0(P), F_1(P)$ and $F_2(P)$ the set of all vertices, edges and faces of $P$, respectively, with symmetry group $G=G(P)$.\n\n\\begin{defi}\\label{TwoP}\nTwo ordered pairs $(v, F)$   and   $(v', F')$   in   $F_0(P)\\times F_2(P)$   are  called  $G$-independent incidences or simply independent incidences, if and only if, there exists no $g \\in G$   such   that  $g(v)=v'$    and $g(F)=F'$.  By  $\\mu_*(P)$   we  mean  the  number  of  $G$-independent incidences $(v, F)$ where $v \\in F_0(P)$,  $F \\in F_2(P)$  and $v \\in F$. Therefore $G(P)$ acts on the set of all such incident pairs $(v, F)$ with  $\\mu_*(P)$  orbits of independent incidences or ``incident orbits''.\n\\end{defi}\n\nFor example let $P$ be a rhombic dodecahedron (Figure \\ref{f2}) then, $\\mu(P) = 2e = 48$ but  $\\mu_*(P)=2$.\n\\vspace*{-.25cm}\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.5cm} \\includegraphics[scale=0.15]{figura2.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{f2}\n\\end{figure}}\n\n\\vspace*{.5cm}\n{\\bf Stratifications}\n\n\n\nFor a transformation group $(M,G)$ the structure of the orbit space $M\/G$ usually is complicated, for example it is not necessarily a manifold.\n\nHowever, when the Lie group $G$ is compact and the manifold $M$ is without boundary it can be shown that they are stratified into smooth manifolds.\n\nWe now describe briefly a stratification of  $\\mathbb{E}^3$ associated with a finite subgroup $G$ of the orthogonal group $O(3)$ that will help us to understand the relationship between the action of $G$ and the number $\\dim \\langle P\\rangle $.\n\n\\begin{defi} ({\\rm Stratification}) Let $X$ be a topological subspace of some Euclidian space  $\\mathbb{E}^3$.\n\nA partition  $\\sum=\\{M_i \\, | \\, i=1,\\ldots,k\\}$ of (pairwise disjoint) subsets of $X$ is called a stratification of $X$ if $\\sum$ satisfies the followings:\n\n\\begin{enumerate}\n\\item Each $M_i,\\,i=1,\\ldots,k$ is a connected smooth submanifold of $\\mathbb{E}^3$, called a \\newline $\\sum$-stratum.\n\n\\item For each $i$, the closure $\\overline{M_i}$ is the union of $M_i$ and the $M_j$'s with lower dimensions than the dimension of $M_i$, that is, the relative closure $X \\cap \\overline{M_i}$ is the union of elements of $\\sum$, one being $M_i$ itself and the others being of dimension less than the $\\dim M_i$.\n\n\\end{enumerate}\n\nThis condition is called {\\it frontier condition}. The dimension $\\dim(X)$ is\n\\vspace*{-0.4cm}\n$$\\max \\{\\dim(M_i) \\mid i=1,\\ldots,k\\} \\ .$$\n\\end{defi}\n\nThe stratification mainly is done by the help of the theorem so called Slice Theorem which is fundamental in studding the structure of the transformation groups (see \\cite{Kaw}, Th. 4.11).\n\n\\vspace*{.25cm}\nLet $X=\\mathbb{E}^3$ and $G$ a compact subgroup of $O(3)$, acting via $\\phi$ on $\\mathbb{E}^3$ as above.\nFor each  $x \\in \\mathbb{E}^3$ let $\\: \\: G_x = \\{g \\in G \\mid g(x)=x\\} \\: \\:$ be the isotropy subgroup of $G$ at $x$ and $F_x=Fix \\left( G_x \\right)$ be the set of all fixed points of $G_x$. Thus\n\\vspace*{-0,2cm}\n$$F_x=\\{y \\in \\mathbb{E}^3 \\mid \\text{ for all } g \\in G_x, \\, g(y)=y\\} \\, .$$\n\n\nSince $x \\in F_x$ and for all $g \\in G$, $g(0)=0$  and for $y,z$ in $F_x$  and  $\\lambda,\\mu\\in \\mathbb{R}$ we have $g(\\lambda y+\\mu z)=\\lambda y+\\mu z$, $F_x$  is a linear subspace of $\\mathbb{E}^3$. Define an equivalence relation $\\sim_{G}$  on $\\mathbb{E}^3$  as follows. Put $x\\sim_{G} y$  if $F_x=F_y$.\n\nNow let $x \\in \\mathbb{E}^3$ and $y \\in F_x$. If $F_x=F_y$ then, $y \\in [x]$, the equivalence class of $x$ in  $\\sim_{G}$.  However $y \\in F_x$  implies that $F_y \\subseteq F_x$, since  $G_x \\subseteq G_y$. Thus $ F_y$  is a linear subspace of  $F_x$. For $z\\in \\mathbb{E}^3-F_x$, we cannot have $z \\sim_{G} x$. Therefore $[x]=\\{y\\in F_x: F_y=F_x\\}$. So $[x]$ is complement in  $F_x$ of finitely many subspaces of $F_x$. Hence the equivalence classes $[x]$, $x \\in \\mathbb{E}^3$ stratify $\\mathbb{E}^3$ with finitely many such strata (orbit types) (see \\cite{Dei}, Th. 5.11). The dimension of $F_x$ denoted by $\\delta(x)$ is called the \\emph{degree of freedom} of $x$.\n\nFor example, let $G$ be a group generated by rotation matrix\n$$A=\\left(\n    \\begin{array}{ccc}\n      \\cos(\\theta) & -\\sin(\\theta) & 0 \\\\\n      \\sin(\\theta)& \\cos(\\theta) & 0 \\\\\n      0 & 0 & 1 \\\\\n    \\end{array}\n  \\right),\\,\\,\\,\\,\\,\\,\\,\\theta=\\frac{2\\pi}{n},\n$$\n\n\\noindent about $z$-axis through $\\theta$.\n\nThen, there are just four strata under this group action on $\\mathbb{E}^3$, namely the $0$-stratum $\\{0\\}$, the open rays $x=y=0, z>0$ and $x=y=0, z<0$, and the complement of the $z$-axis.\n\nNow let $P$ be a polyhedron with $G=G(P)$ some finite subgroup of $\\mathcal{O}(3)$ and $Q\\in \\langle P \\rangle$ as above.\nUnder the restriction imposed on $Q$ within the symmetry type of $P$, each vertex of $Q$ may be moved along a line, or within a plane, or in any direction in  $\\mathbb{E}^3$ near (without changing the symmetry type) its initial position in $P$ itself, having one, two or three degrees of freedom. Likewise, each face $F$ of $Q$ may have one, two or three degrees of freedom close to the corresponding face of $P$, according as F intersects a 1-stratum in an interior point of $F$ (necessarily at right angles), or intersects a 2-stratum in the interior of $F$ (again at right angles), or neither of these.\n\nConsidering the action of $G=G(P)$ on $F(P)$, let\n$$\n\\overline{v} \\, = \\, G(v) \\, = \\, \\left\\{ g(v) \\, \\mid \\, g\\in G \\right\\} \\: , \\: \\: \\overline{F} \\, = \\, G(F) \\, = \\, \\left\\{ g(F) \\, \\mid \\, g\\in G \\right\\} \\, , \\, \\,\\, \\text{ and }\n$$\n\\vspace*{-.6cm}\n$$\nF_0(P)\/ G \\, = \\, \\left\\{ \\overline{v} \\, \\mid \\, v\\in F_0(P) \\right\\} \\: , \\: \\: F_2(P)\/ G \\, = \\, \\left\\{ \\overline{F} \\, \\mid \\, F\\in F_2(P) \\right\\}\n$$\n\n\\noindent be the collection of orbits under the action of $G$ on face lattice $F(P)$.\n\nDefine $\\mathcal{M}_P$ to be the set of all pairs $(v,F)\\in F_0(P) \\times F_2(P)$ for which $v\\in F$ and\n\\vspace*{-.2cm}\n$$\n\\mathcal{M}_P\/ G \\, = \\, \\left\\{ \\left( g(v), g(F) \\right) \\, \\mid \\, g\\in G \\text{ and } (v,F)\\in \\mathcal{M}_P \\right\\} \\, .\n$$\n\nClearly $\\mu (P)$ and $\\mu _{*}(P)$ are the cardinalities of $\\mathcal{M}_P$ and $\\mathcal{M}_P\/ G$, respectively.\n\n\\vspace*{.2cm}\nWe observe that if $v,u\\in F_0(P)$ and $\\overline{v}=\\overline{u}$ then,\n\\vspace*{-.2cm}\n$$\n\\dim(v) \\, = \\, \\dim(u) \\, .\n$$\n\n\\vspace*{-.2cm}\nTherefore we can define the fixed dimensions $\\delta (\\xi)$, $\\xi \\in F_0(P)\/ G$, $\\delta (\\xi)=\\dim(v)$, the degree of freedom of $v$ for an arbitrary $\\overline{v}\\in \\xi$.\nThe same holds for $\\delta (\\zeta)$, $\\zeta \\in F_2(P)\/ G$.\n\nOur aim is to count the number of the vertices and the faces with dimensions $k=1,2,3$ and then, by subtracting the independent incidences $\\mu _{*}(P)$, express the $\\dim \\langle P \\rangle$ in terms of edge orbits alone.\n\n\n\n\n\n\n\\section{Fundamental regions and main theorem}\n\nIn this Section we consider finite subgroups of isometries which are generated by reflections namely $[q], [2, q], [3, 3], [3, 4]$ and $[3,5]$ (Table 2).\n\n\\vspace*{.2cm}\nThe finite subgroups of $\\mathbb{E}(3)$ which are generated by reflections in the plane, are given in the following table. They are called reflection groups, for obvious reason.\n\\begin{center}\n\\vspace*{-.2cm} {\\small{\\begin{tabular}{|c|l|c|}\n  \\hline\n \n  Symbol & description & order \\\\\n  \\hline\n                  & $q=1$: One plane of reflection; $q\\geq 2$: $q$ equally inclined planes  &   \\\\\n    $[q],q\\geq 1$ & of reflection passing through a $q$-fold axis of rotation, dihedral \\hspace{11cm}& $2q$ \\\\\n                  & reflection group. &   \\\\\n \\hline\n                  & $q$ equally inclined planes of reflection passing through a $q$-fold &   \\\\\n    $[2,q]$       & axis of rotation and reflection in a equatorial plane.   & 4q \\\\\n                  & $q$ 2-fold axes of rotation. The group of $q$-prism.&   \\\\\n\\hline\n     $[3,3]$      & Four 3-fold and three 2-fold axes. Six planes of reflection. &  \\\\\n                  & Symmetry group of the regular tetrahedron. & 24  \\\\\n\\hline\n                  & Three 4-fold and four 3-fold and six 2-fold axes of rotation. &   \\\\\n     $[3,4]$      & Nine planes of reflection. Symmetry group of the cube. & 48 \\\\\n\\hline\n                  & Six 5-fold, ten 3-fold and fifteen 2-fold axes of rotation.  &   \\\\\n     $[3,5]$      & Fifteen planes of reflection. Symmetry group of the icosahedron. & 120 \\\\\n  \\hline\n\\end{tabular}}}\n\\end{center}\n\n\\begin{center} Table 2: Reflection groups \\end{center}\n\nIt is well known that the \\emph{fundamental region} $\\Delta$ for the action of $[3, 3], [3, 4], [3, 5]$ and $[2, q]$ on the sphere $S^2$  are spherical triangles [1]. For $[q]$ the dihedral reflection group generated by two reflections, the fundamental region is a ``lune'' of angle $\\frac{\\pi}{q}$.\nWe may use the fundamental region of a reflection group to construct a stratification of $\\mathbb{E}^3$. For instance consider the tetrahedron $OABC$ (or its spherical projection) as a fundamental region of [3, 4] in Figure \\ref{f3} (\\cite{Rob}, p. 81).\nWe take the origin $O$ as a $0{\\text{-stratum}}$. By removing the origin from the rays $OA$, $OB$ and $OC$ we get three $1{\\text{-strata}}$, the interiors of the region $AOB$, $AOC$ and $BOC$ are $0{\\text{-strata}}$.\n\nFinally, the interior points of $\\mathbb{E}^3$  bounded by sectors $AOB$, $AOC$ and $BOC$ is $3{\\text{-strata}}$. By transferring these strata under the action of [3, 4] we obtain the required stratification of $\\mathbb{E}^3$.\n\nFor other reflection groups a stratification of $\\mathbb{E}^3$ is constructed in analogues fashion.\nReturning to our main problem, we now consider the following notion. Suppose that a finite reflection group $G$ in $\\mathcal{O}(3)$ has its fundamental region a spherical triangle $\\Delta$ and, let $P$ be a polyhedron with $G(P)=G$.\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.5cm} \\includegraphics[scale=.2]{figura3.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{f3}\n\\end{figure}}\n\nWe denote by  $\\Delta_p$ that portion of the surface of $P$ (namely those vertices, edges and subpolygonal faces) which lie within $\\Delta$, and call $\\Delta_p$ a \\emph{basic region} of $P$ (Figure \\ref{f4}).\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.75cm} \\includegraphics[scale=.225]{figura4.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{f4}\n\\end{figure}}\n\n\\vspace*{-0.1cm}\nHence $\\Delta_p$ is a simple closed planar polygonal region. If $n_{\\Delta_p}, e_{\\Delta_p}$ and $f_{\\Delta_p}$ denote the total number of distinct vertices, edges and subpolygonal faces of  $\\Delta_p$ respectively then, from Euler's formula by stereographic projection we get\n\\vspace*{-0,2cm}\n$$n_{\\Delta_p} - e_{\\Delta_p} + f_{\\Delta_p}=1 \\, .$$\n\nAs illustrated example, let $P$ be truncated cuboctahedron (Figure \\ref{f4}) with symmetry group $G(P)=[3,4]$ of cube, with order 48. Thus the basic region $\\Delta_p$ has $n_{\\Delta_p}=7$, $e_{\\Delta_p}=9$, $f_{\\Delta_p}=3$, and $n_{\\Delta_p} - e_{\\Delta_p} + f_{\\Delta_p}=7-9+3=1$.\n\n\\vspace*{0.1cm}\nNow having our necessary tools, we are in the position to state and prove our main theorem in this Section.\n\n\\begin{teo}Let $G$ be a finite reflection group in $\\mathbb{E}(3)$ and $P$ a polyhedron with $G(P)=G$. Then, $\\dim \\langle P\\rangle= \\epsilon-1$ , where $\\epsilon$ is the number of edge orbits of the action of $G$ on the set of edges of $P$.\n\\end{teo}\n\nFirst we prove the following lemma.\n\n\\begin{lema} Assuming the hypothesis of the theorem, let  $\\Delta_p$  be a basic region for $P$ such that the corners of fundamental region of $\\Delta$ of $G$ are vertices of $P$ (see Figure \\ref{f5}). Then, the number of incident pairs of vertices and faces of  $\\Delta_p$, the multiplicity $\\mu(\\Delta_p)$  of  $\\Delta_p$  is given by  $\\mu(\\Delta_p)=2e-\\beta$ where $e$ is the total number of edges of   $\\Delta_p$ and $\\beta$ the number of vertices on the boundary of the fundamental region.\n\\end{lema}\n{\\bf Proof:} By adjoining an extra face, say $K$, to  $\\Delta_p$, namely the complement of   $\\Delta_p$ itself with respect to the sphere we get a map $M_P$ on sphere. But the number of edges (and vertices) of $M_P$ is equal to the number of edges (and vertices) of  $\\Delta_p$. Hence  $\\mu(\\Delta_p)=2e$. Since there are $\\beta$ vertices on boundary of  $\\Delta$, with respect to that extra face $K$, we have  $\\mu(\\Delta_p)=\\mu(P)-\\beta=2e - \\beta$.\n\\vspace*{-0.3cm}\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.75cm} \\includegraphics[scale=.15]{figura5.eps}\n\\vspace*{-0,1cm}\\caption{$\\,$}\n\\label{f5}\n\\end{figure}}\n\n\\vspace*{-0,5cm}\n$\\:$ \\hfill$\\square$\n\n\\vspace*{-0,1cm}\n \\begin{obs}\n  {\\rm Let $P$ be a polyhedron and $v$ a vertex of $P$. According to stratification of  $\\mathbb{E}^3$ with reflection group $G(P)$ of $P$, $\\delta(v)$, the degree of freedom of $v$,  is one, two or three if $v$ is on a corner or side or within the interior of   $\\Delta_p$ respectively. Similarly if a face $F$ of $P$ has as its interior point a corner of $\\Delta$ then, $\\delta(F)=1$. If $F$ is orthogonal to a side of  $\\Delta_p$  or lies inside  $\\Delta_p$  then, $\\delta(F)=2 $ or 3, respectively.}\n \\end{obs}\n\n \\noindent {\\bf Proof of theorem:} Let  $\\eta(1),\\eta(2)$ and $\\eta(3)$ be the number of vertices of $\\Delta_p$   and   $\\phi(1),\\phi(2)$ and $\\phi(3)$ the number of faces of $\\Delta_p$ with one, two and three degrees of freedom respectively.\n\n First we assume that $P$ has no face $F$ with $\\delta(F)$ equal to one or two. Then, $\\mu_*(P)=\\mu(\\Delta_p)=2e-\\beta$. After factoring out the effect of dilation we get\n $$\n \\begin{array}{rcl}\n   \\dim\\langle P \\rangle & = & 1\\eta(1)+2\\eta(2)+3\\eta(3)+3\\phi(3)-\\mu(\\Delta_p)-1 \\\\[8pt]\n     & = & 3(n_{\\Delta_p}+\\phi_{\\Delta_p})-2e_{\\Delta_p}-4 \\, ,\n \\end{array}$$\n\\noindent since $\\eta(1)=3$.\n\nBut $n_{\\Delta_p}+\\phi_{\\Delta_p}=e_{\\Delta_p}+1$. Hence,\n$$\\dim\\langle P \\rangle=3(e_{\\Delta_p}+1)-2e_{\\Delta_p}-4= e_{\\Delta_p}-1 \\, .$$\n\\noindent Now clearly the number of edges of  $\\Delta_p$  is exactly the number of edge orbits of $P$. Therefore the theorem follows in this case.\n\nNext suppose $P$ has one face $F$ with $\\delta(F)=1$. This means that there is a face $F$ such that one corner say $v$ of $\\Delta_p$   is an interior point of $F$.\n\nLet $P'$ be a polyhedron which we get, by changing ``fake'' edges in $\\Delta_p$ into real ones (see Figure \\ref{f6}). This is done as follows.\n\nWe remove the constraint that the plane of  $F \\cap \\Delta_p$  is perpendicular to the ray $ov$. The edges of  $F \\cap \\Delta_p$  that lie in the boundary of  $\\Delta_p$  are also edges of $P'$ where $P'$ has a basic region  $\\Delta_{p'}$ , say, with the same combinatorial structure as $\\Delta_p$, and with vertices arbitrarily close to those of $\\Delta_p$.\nSince we have substituted a face with one degree of freedom by a face with three degrees of freedom and since the new vertex and its incidence cancel each other and hence do not effect our calculation for  $ \\dim\\langle P \\rangle $, we have $ \\dim\\langle P' \\rangle = \\dim\\langle P \\rangle  + 2$. But $\\dim\\langle P' \\rangle = \\epsilon'-1$ and $\\epsilon'=\\epsilon+2$ with obvious notations. Hence\n$$\\dim\\langle P \\rangle = \\dim\\langle P' \\rangle -2= \\epsilon'-1-2=\\epsilon-1 \\, .$$\n\nThe process will continue if $P$ has two or three (on other corners of $\\Delta$) faces of degree one.\n{ \\begin{figure}[!htb] \\centering\n\\vspace*{-.5cm} \\hspace*{.25cm} \\includegraphics[scale=.175]{figura6.eps}\n\\vspace*{-0,4cm}\\caption{\\small{\\emph{Broken lines represent ``fake'' edges and ``o'' a fake vertex.}}}\n\\label{f6}\n\\end{figure}}\n\nHere we remark that, in the process of changing ``fake'' edges in $\\Delta_p$  into real ones, since the transforms of $\\Delta_{p'}$ under corresponding group action is a polyhedral graph (planar and 3-connected), by the Theorem of Steinitz \\cite{Stei}, there exists a polyhedron P' which geometrically realizes $\\Delta_{P'}$.\n\n\\vspace*{.25cm}\nFinally, suppose $P$ has a face $F$ with $\\delta(F)=2$ (Figure \\ref{f7}). We construct $P'$ by adjoining the fake edge to $\\Delta_p$. Then, $\\dim\\langle P' \\rangle $ differs by one from\n$\\dim\\langle P \\rangle$ for replacement of a face with two degrees of freedom, by a face of three degrees of freedom. Hence\n$\\dim\\langle P' \\rangle = \\dim\\langle P \\rangle +1$. Because of $\\dim\\langle P' \\rangle =\\epsilon'-1$ and $\\epsilon=\\epsilon'-1$, we have\n\\vspace*{-0,3cm}\n$$\\dim\\langle P \\rangle +1=\\epsilon'-1=\\epsilon\\,\\,\\,\\,\\text{ and }\\,\\,\\,\\, \\dim\\langle P \\rangle =\\epsilon-1 \\, .$$\n\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.25cm} \\includegraphics[scale=.175]{figura7.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{f7}\n\\end{figure}}\n\n\\newpage\nNow this inductive process can be continued if $P$ has any number of faces with two degrees of freedom. In each step of construction, $\\dim\\langle P \\rangle$ and $\\epsilon$ each increase by one, while the operation leaves every other quantity in our calculation fixed. For the case of the group $[q]$ where $\\Delta$ is a ``lune'', the proof proceeds in similar way and is omitted here to avoid repetition. The proof of the theorem now is complete. \\hfill$\\square$\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn Sun-as-a-star helioseismology, it is common practice to fix the amplitude ratios between the $m$-components of the $l = 2$ and 3 multiplets (the so-called $m$-amplitude ratio) during the peak-fitting procedure when estimating the p-mode characteristics (e.g., Salabert et al. [1]), while the amplitudes of the $l=1$, 2, and 3 modes relative to the $l=0$ modes are left free (the so-called mode visibility). However, in asteroseismology, the mode visibilities are fixed to theoretical values due to lower signal-to-noise ratio (SNR) and shorter time series, and the $m$-amplitude ratios are expressed as a function of the inclination of the rotation axis only (Appourchaux et al. [2]; Garc\\'\\i a et al. [3]). In both cases, they are supposed to not depend on the star magnetic activity. However, in the near future, this situation could change when stellar activity cycles will be measured in asteroseismic targets (e.g., Garc\\'\\i a et al. [4]). After several years of observations collected by the Kepler mission, the SNR will be high enough to measure these parameters in a wide range of solar-like stars in the HR diagram at different evolution stages (Bedding et al. [5]; Chaplin et al. [6]). Moreover, simultaneous observations from SONG (Doppler velocity) and Kepler (intensity) will be extremely useful to better understand stellar atmospheres. \nAlthough, variations with the height in the solar atmosphere at which the measurements are obtained have been observed in the intensity VIRGO\/SPM data at the beginning of the SoHO mission (Fr{\\\"o}hlich et al. [7]), it has never been verified if these values change in the radial velocity GOLF measurements between the blue- and red-wing observing periods. \n\n\\section{Observations and analysis}\nWe used observations collected by the space-based instruments Global Oscillations at Low Frequency (GOLF) and Variability of Solar Irradiance and Gravity Oscillations (VIRGO) onboard the {\\it Solar and Heliospheric Observatory} (SoHO) spacecraft. GOLF (Gabriel et al. [8]) measures the Doppler velocity at different heights in the solar atmosphere depending on the wing -- Blue or Red -- of the sodium doublet  -- D1 and D2 -- in which the observations were performed (Garc\\'\\i a et al. [9]). VIRGO (Fr{\\\"o}hlich et al. [10]) is composed of three Sun photometers (SPM) at 402~nm (blue), 500~nm (green) and 862~nm (red). A total of 5021 days of GOLF and VIRGO observations starting on 1996 April 11 and ending on 2010 January 8 were analyzed, with respective duty cycles of  95.4\\% and 94.7\\%. The power spectra of the time series were fitted to extract the mode parameters (Salabert et al. [11]) using a standard likelihood maximization function (power spectrum with a $\\chi^2$ with 2 d.o.f. statistics). Each mode component was parameterized using an asymmetric Lorentzian profile. Since SoHO observes the Sun equatorwards, only the $l+|m|$ even components are visible in Sun-as-a-star observations of GOLF and VIRGO. In order to obtain observational estimates of the $m$-amplitude ratios, the $m = \\pm2$ and $m = 0$ components of the $l = 2$ multiplet, and the $m = \\pm3$ and $m = \\pm1$ components of the $l = 3$ multiplet were fitted using independent amplitudes, assuming that components with opposite azimuthal order $m$ have the same amplitudes ($H_{l,n,-m}=H_{l,n,+m}$). Note that the blue and red periods of GOLF were also analyzed separately, as well as the mean power spectrum of the three VIRGO  SPMs.\n\n\n\n\\section{Mode visibilities and $m$-amplitude ratios}\nThe amplitude of a given multiplet $(l,n)$ is defined as the sum of the amplitudes of its $m$-components, as $H_{l,n} = \\sum_{m=-l}^{m=+l} H_{l,n,m}$. \nThen, the visibilities of the $l=1$, 2, and 3 modes relative to the $l=0$ mode are respectively defined as the ratios $H_{l=1,n} \/ H_{l=0,n}$, $H_{l=2,n-1} \/ H_{l=0,n}$, and $H_{l=3,n-1}$. The left panel of Fig.~\\ref{fig:visi} shows these visibilities for both radial velocity (GOLF) and intensity (VIRGO) measurements as a function of frequency. These raw mode visibilities present a variation with frequency -- especially for the $l=1$ mode -- that is due to the large variation of the mode amplitudes with frequency, even over half a large frequency separation.\nThus, the visibilities are biased and in order to correct them we interpolated (using a spline interpolation) the amplitudes of the $l=1$, 2, and 3 modes to the frequencies of the $l=0$ mode (right panel of Fig.~\\ref{fig:visi}). \nFigure~\\ref{fig:meanvisi} shows the mode visibilities averaged over frequency for both GOLF and VIRGO observations as a function of $l$  (see Table~\\ref{tab:visigolf}). The amplitude ratios between the $m$-components of the $l = 2$ and $l = 3$ multiplets, defined as $H_{l=2,m=0}\/H_{l=2,m=\\pm2}$ and $H_{l=3,m=\\pm1}\/H_{l=3,m=\\pm3}$ respectively, are represented on Fig.~\\ref{fig:mratio} in the case of the radial velocity GOLF measurements and are also reported in Table \\ref{tab:mratiogolf}.\n\n\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.31]{GOLF_visi_nocorrection_forpaper.eps}  \\includegraphics[scale=0.31]{GOLF_visi_withcorrection_forpaper.eps}\n\\end{center} \n\\caption{\\label{fig:visi} Raw (left) and corrected (right) mode visibilities of $l = 1$ ($\\opencircle$), $l=2$ ($\\opensquare$), and $l=3$ ($\\opendiamond$) relative to $l = 0$ as a function of frequency in GOLF ($\\full$) and VIRGO ($\\dashed$) observations.} \n\\end{figure*} \n\n\n\\begin{figure}\n\\includegraphics[width=2.5in]{visib_vs_degree_forpaper.eps}\\hspace{0.5pc}%\n\\begin{minipage}[b]{22pc}\\caption{\\label{fig:meanvisi} Mode visibilities as a function of angular degree $l$ in GOLF ($\\full$, $\\fullcircle$) and VIRGO ($\\dashed$, $\\opensquare$) measurements. For comparison, the mode visibilities of the CoRoT target HD49385 measured by Deheuvels et al. [12] are also represented ($\\dotted$, $\\opentriangledown$).}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.25]{GOLF_mratio_forpaper.eps}\\hspace{0.5pc}%\n\\begin{minipage}[b]{20.5pc}\\caption{\\label{fig:mratio} $m$-amplitude ratios of the $l = 2$ (top) and $l = 3$ (bottom) modes as a function of frequency in the GOLF measurements.}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{table}\n\\caption{\\label{tab:visigolf} Mode visibilities in radial velocity GOLF and intensity VIRGO measurements.}\n\\begin{center}\n\\begin{tabular}{lllll}\n\\br\nMode visibility & GOLF & GOLF & GOLF& \\\\\nRadial velocity &         & Blue wing & Red wing&\\\\\n\\mr\n$H_{l=1}\/H_{l=0}$ & 1.69$\\pm$0.04 & 1.60$\\pm$0.05 & 1.85$\\pm$0.06&\\\\\n$H_{l=2}\/H_{l=0}$ & 0.81$\\pm$0.03 & 0.74$\\pm$0.04 & 0.98$\\pm$0.05&\\\\\n$H_{l=3}\/H_{l=0}$ & 0.17$\\pm$0.01 & 0.14$\\pm$0.02 & 0.28$\\pm$0.03&\\\\\n\\br\nMode visibility & VIRGO & VIRGO & VIRGO & VIRGO \\\\\nIntensity        &               & Blue      & Green   & Red \\\\\n\\mr\n$H_{l=1}\/H_{l=0}$ & 1.53$\\pm$0.05 & 1.55$\\pm$0.05 & 1.52$\\pm$0.05 & 1.39$\\pm$0.05\\\\\n$H_{l=2}\/H_{l=0}$ & 0.59$\\pm$0.03 & 0.63$\\pm$0.03 & 0.57$\\pm$0.03 & 0.42$\\pm$0.03\\\\\n$H_{l=3}\/H_{l=0}$ & 0.09$\\pm$0.02 & 0.10$\\pm$0.02 & 0.09$\\pm$0.02 & 0.05$\\pm$0.02\\\\\n\\br\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\caption{\\label{tab:mratiogolf} $m$-amplitude ratios in radial velocity GOLF and intensity VIRGO measurements.}\n\\begin{center}\n\\begin{tabular}{lll}\n\\br\n$m$-amplitude ratio & GOLF & VIRGO\\\\\n\\mr\n$H_{l=2,m=0}\/H_{l=2,m=\\pm2}$ & 0.63$\\pm$0.03 & 0.75$\\pm$0.06\\\\\n$H_{l=3,m=\\pm1}\/H_{l=3,m=\\pm3}$ & 0.40$\\pm$0.02 & 0.63$\\pm$0.06\\\\\n\\br\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{Models}\nWhen the contribution of a solar-disk element to the total flux depends only on its distance to the limb, the mode visibility and the $m$-amplitude ratio are decoupled (e.g. Gizon \\& Solanki [13]; Ballot et al. [14]). For VIRGO observations, this is verified since the contribution depends mainly on the limb-darkening. However, for GOLF, this is no more the case and we have performed complete computation taking into account the instrumental response, which differ for the blue and red wings. Results of these computations are listed in Table~\\ref{tab:visimodel} . The limb-darkening law of Neckel \\& Labs [15] has been used. Indeed, visibility values for GOLF vary with frequency by a few percents due to the horizontal motions of modes that increase at low frequency.\nIn general, these predictions agree with the observations. There is nevertheless some shortcomings: (i) even if the trend is correct, the difference between blue and red wings for GOLF is larger than expected; (ii) the visibility of the $l = 3$ modes in VIRGO are sensitively higher than expected. That could be explained by stronger effects of limb-darkening.\n\n\n\\begin{table}\n\\caption{\\label{tab:visimodel} Modeled visibilities and $m$-amplitude ratios in intensity VIRGO and radial velocity GOLF measurements.}\n\\begin{center}\n\\begin{tabular}{llll}\n\\br\nMode visibility \\& & VIRGO & GOLF            & GOLF\\\\\n $m$-amplitude ratio       &               & Blue  wing   &   Red wing\\\\\n\\mr\n$H_{l=1}\/H_{l=0}$ & 1.51 & 1.84 & 1.86\\\\\n$H_{l=2}\/H_{l=0}$ & 0.53 & 1.09 & 1.14\\\\\n$H_{l=3}\/H_{l=0}$ & 0.025 & 0.27 & 0.31\\\\\n\\mr\n$H_{l=2,m=0}\/H_{l=2,m=\\pm2}$ & 0.67 & 0.59 & 0.58\\\\\n$H_{l=3,m=\\pm1}\/H_{l=3,m=\\pm3}$ & 0.60 & 0.43 & 0.40\\\\\n\\br\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\ack\nThe authors want to thank Catherine Renaud and Antonio Jim\\'enez for the calibration and preparation of the GOLF and VIRGO datasets. The GOLF and VIRGO instruments onboard SoHO are a cooperative effort of many individuals, to whom we are indebted. SoHO is a project of international collaboration between ESA and NASA. DS acknowledges the support of the grant PNAyA2007-62650 from the Spanish National Research Plan. This work has been partially supported by the CNES\/GOLF grant at the SAp\/CEA-Saclay.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Supplementary information}\n\\subsection{Extraction of the exciton population from measured data}\n\\label{population}\n\nThe exciton population created by a pump pulse tuned into the phonon sideband\ncan be extracted from our two-pulse spectra (see Fig.~2) in three different ways\nby analyzing the peak heights of the three transitions: \n$|0\\rangle \\rightarrow |X\\rangle$, $|0\\rangle \\rightarrow |\\bar X\\rangle$\nand $|X\\rangle \\rightarrow |2X\\rangle$. The relation\nbetween the photocurrent ($PC$) signal and the exciton population is as follows. The $PC$  measured in our experiment is determined by the number of\nelectron-hole pairs in the sample, including pairs in the quantum dot (QD) and pairs\nexcited in the surrounding material by the pump and probe pulses, according to:\n\\begin{equation}\nPC=\\alpha(C^{'}_X+C^{'}_{\\bar{X}})+2\\beta C^{'}_{2X}+\\gamma N^{'}_\\text{s},\n\\label{Eq:PC}\n\\end{equation}\nwhere $C^{'}_X$ and $C^{'}_{\\bar{X}}$ are the populations of the two exciton states \nwith opposite spins that are created in the QD \nafter circularly polarized pump and probe pulses, while $C^{'}_{2X}$ is the biexciton population. \n$N^{'}_\\text{s}$ is the number of electron-hole pairs created \nin the surrounding material which scales linearly with the laser intensity.\n$\\alpha$, $\\beta$ and $\\gamma$ are the detection efficiencies of our \nphotocurrent measurement for each type of charge complexes. \nSpecifically, for our pump-probe measurement, $PC$ can be further subdivided into two parts: \na reference level $PC_\\text{R}$ and the change of the photocurrent signal originating \nfrom the QD induced by the probe pulse $\\Delta PC$ according to:\n\n\\begin{subequations}\n\\begin{align}\nPC&=PC_\\text{R} + \\Delta PC,\\\\\nPC_\\text{R} &= (\\alpha(C_X+C_{\\bar{X}})+ 2\\beta C_{2X})e^{-\\tau_\\text{delay}\/T_1}\n+ \\gamma N^{'}_\\text{s},\\\\\n\\Delta PC &= \\alpha (\\Delta C_X +\\Delta C_{\\bar{X}})+2\\beta \\Delta C_{2X},\n\\end{align}\n\\label{Eq:C}\n\\end{subequations}\nwhere $C_{X,\\bar{X},2X}$ are the exciton and biexciton\npopulations created immediately after the pump pulse\nand $\\Delta C_{X,\\bar{X},2X}$ is the change of the exciton population induced\nby the probe pulse in the QD. Since the electron can\ntunnel out from the QD, the exciton and biexciton populations decay exponentially with\ntime. Thus, $C_{X,\\bar{X},2X}e^{(-\\tau_\\text{delay}\/T_1)}=C^{'}_{X,\\bar{X},2X}$ represent the pump-created exciton or biexciton\npopulations that have remained in the QD  until the arrival of the probe pulse. \nHere, $\\tau_\\text{delay}$ is the delay time between the pump and probe\npulses and $T_1$ is the electron tunnelling time determined using inversion recovery measurement \\cite{Kolodka2007}. In the above discussion, we have neglected \nthe loss of the exciton population due to the radiative decay, since the radiative decay time\n($\\sim 600$~ps) is much longer than our typical delay times $\\tau_\\text{delay}$ (10 - 30ps). \n$PC_\\text{R}$ is determined by measuring the photocurrent signal from \nan off-resonant measurement where  the pump pulse is tuned into the phonon \nsideband and the probe frequency is far from any of the resonances of the dot or the surrounding material\nsuch that $\\Delta PC$ becomes negligible and the photocurrent signal coincides with $PC_\\text{R}$\nin this case. \n$\\Delta PC$ is the differential photocurrent signal that is discussed in the paper, \nwhich is determined from our experiment by subtracting the $PC_\\text{R}$ obtained from the\noff-resonant measurement from the total photocurrent signal $PC$. \nThe detection efficiency $\\alpha$ can be extracted from the single $\\pi$ pulse experiment. \nDenoting by $\\Delta PC(\\pi)$ the maximum of the differential photocurrent signal reached \nwith a single $\\pi$ pulse we find:\n\\begin{equation}\n\\Delta PC(\\pi)=\\alpha C_X(\\pi)=\\alpha.\n\\label{Eq:alpha}\n\\end{equation}\nHere, we have used that according to our path-integral simulations,\nthe phonon-induced deviation of $C_X(\\pi)$ from the ideal value of 1 \nis negligible at low temperatures.\n\n\nNow let us derive the relation between the differential PC signal and the exciton population for \nthe case when pump and probe pulse are co-circularly polarized and the probe pulse\nis resonant to the $|0\\rangle \\rightarrow |X\\rangle$  transition.\nIn this case, firstly, the $\\sigma^{+}$ polarized pump pulse creates a certain exciton population $C_X$\nand consequently the ground-state occupation after the pump pulse is given by $C_{0}=1-C_{X}$.\nThen, in the time interval until the probe pulse arrives, the exciton population\nis reduced to $C_Xe^{-\\tau_\\text{delay}\/T_1}$ due to the tunneling. The ground-state occupation, on the other hand,\nis not affected by the tunneling and thus stays at the value of $C_{0}=1-C_{X}$ until the arrival of the probe.\nFinally, the $\\pi$-pulse $\\sigma^{+}$ polarized probe exchanges the populations of the\nstates $|0\\rangle$ and $|X\\rangle$ resulting in an exciton  population after the probe of $C^{'}_{X}=1-C_{X}$.\nTherefore, the change of the $X$-exciton occupation induced by the probe pulse with the energy of $\\hbar \\omega_X$ \nis $\\Delta C_X = 1-(1+ e^{-\\tau_\\text{delay}\/T_1})\\,C_{X}$. \nUsing this result \ntogether with the fact, that for co-polarized $\\sigma^{+}$-pulses the populations of the $\\bar X$-exciton \nand the biexciton never build up, we find from Eq.~2(c):\n\\begin{equation}\n\\Delta PC_{0-X}=\\alpha[1-(1+ e^{-\\tau_\\text{delay}\/T_1})\\,C_{X}]\n\\label{Eq:Hph}\n\\end{equation}\nand thus with the help of Eq.~(\\ref{Eq:alpha}) we eventually end up with:\n\\begin{equation}\nC_X=\\dfrac{1}{1+e^{-\\tau_\\text{delay}\/T_1}}\\left(1-\\dfrac{\\Delta PC_{0-X}}{\\Delta PC(\\pi)}\\right).\n\\end{equation}\n\nThe situation is different when cross-circularly polarized pulses are used.  Let\nus first discuss the case where the $\\bar X$ polarized probe pulse is resonant\nto the $|0\\rangle \\rightarrow |\\bar X\\rangle$  transition.  After the action of\nthe circularly $\\sigma^{+}$ polarized pump pulse, the QD again has a certain probability $C_X$\nto be in the $|X\\rangle$ state and the probability to find the dot in the ground-state is $C_{0}=1-C_{X}$. The $\\sigma^{-}$ polarized probe pulse induces transitions\nfrom the ground-state to the $\\bar X$ exciton. Since the probe pulse has a pulse\narea of $\\pi$ and the occupation of $|\\bar X\\rangle$ is zero before the arrival\nof the probe, the probe pulse fully converts the occupation that was left in \nthe ground-state after the pump pulse \ninto an occupation of the $\\bar X$ exciton, i.e., $C^{'}_{\\bar X}=C_{0}=1-C_{X}$.\nAgain, the ground-state occupation is not affected by the electron tunnelling\nand therefore no correction involving the tunnelling time $T_{1}$ should be applied.\nSince the probe pulse is off-resonant to the $X-2X$ transition we can neglect the probe induced change\nof the $|X\\rangle$ and $|2X\\rangle$ occupations.\nRecalling that  $|\\bar X\\rangle$ is unoccupied before the probe, we find for\nthe resulting differential photocurrent signal $\\Delta PC_{0-\\bar{X}}$ with the \nprobe pulse being in resonance to the exciton transition:\n\\begin{equation}\n\\Delta PC_{0-\\bar{X}}=\\alpha (1-C_X),\n\\label{Eq:Hph}\n\\end{equation}\nwhich yields:\n\\begin{equation}\nC_X=1-\\Delta PC_{0-\\bar{X}}\/ \\Delta PC(\\pi).\n\\label{Eq:Hph}\n\\end{equation}\n\nBesides from the data measured at the $|0\\rangle \\rightarrow |X\\rangle$ \nand $|0\\rangle \\rightarrow |\\bar X\\rangle$ transitions, the exciton population\ncreated by the pump can also be extracted from the exciton to biexciton\ntransition. An $\\sigma^{+}$ polarized pump pulse tuned into the high-energy phonon\nsideband of the neutral exciton transition again creates a certain exciton population\n$C_X$ which evolves into $C_Xe^{-\\tau_\\text{delay}\/T_1}$ until the arrival of the probe.\nBiexcitons are not created, i.e., we have $C_{2X}=0$. A cross-polarized\n$\\pi$-power probe pulse resonant to the \n$|X\\rangle \\rightarrow |2X\\rangle$ transition\nconverts the $|X\\rangle$ population completely into\nan $|2X\\rangle$ population, which gives $C^{'}_X=0$ and\n$C^{'}_{2X}=C_Xe^{-\\tau_\\text{delay}\/T_1}$. Since the probe is now off-resonant to the\n$|0\\rangle \\rightarrow |\\bar X\\rangle$ transition, the occupation of the\n$|\\bar X\\rangle$ exciton induced by the probe is negligible \nand the ground-state occupation is not affected. \nThus, the $\\Delta PC$ signal resulting from a probe pulse \nin resonance to  the $|X\\rangle \\rightarrow |2X\\rangle$\ntransition is given by:\n\\begin{equation}\n\\Delta PC_{X-2X}=2\\beta \\Delta C_{2X}+\\alpha \\Delta C_X=(2\\beta - \\alpha)C_Xe^{-\\tau_\\text{delay}\/T_1}.\n\\label{Eq:PCbiexciton}\n\\end{equation}\n$\\beta$ can be determined from a separate experiment in whihc the pump is a $\\pi$ pulse resonant with $X$ and the probe is a $\\pi$ pulse resonant with $2X$. According to Eqs.~\\eqref{Eq:alpha} and \\eqref{Eq:PCbiexciton}, we have:\n\\begin{equation}\n\\beta=0.5(e^{\\tau_\\text{delay}\/T_1}\\Delta PC_{X-2X}(\\pi)+\\Delta PC(\\pi)).\n\\label{Eq:beta}\n\\end{equation}\n\nInserting Eqs.~\\eqref{Eq:alpha} and \\eqref{Eq:beta} into Eq.~\\eqref{Eq:PCbiexciton} \nwe can extract the exciton population after the pump from:\n\\begin{equation}\nC_X= \\dfrac{\\Delta PC_{X-2X}}{\\Delta PC_{X-2X}(\\pi)}.\n\\end{equation}\n\n\\subsection{Model}\n\\label{Model}\n\nFor our calculations we used the same model for an optically driven strongly \nconfined quantum dot as in Ref.~\\cite{Glassl2013},\nwhich is based on the Hamiltonian\n\\begin{align}\n \\label{eq:Hamiltonian}\n H = H_{\\rm{QD-light}} + H_{\\rm{QD-phonon}},\n\\end{align}\nwhere  \n\\begin{align}\n H_{\\rm{QD-light}} = \\hbar\\omega^{0}_{X}| X\\rangle\\langle X|\n+\\frac{\\hbar\\Omega(t)}{2} \\left[ | 0\\rangle\\langle X| + |X\\rangle \\langle 0| \\right],\n\\end{align}\nand\n\\begin{align}\n H_{\\rm{QD-phonon}} \\!=\\! \\sum_{\\bf q} \\hbar\\omega_{\\bf q}\\,b^\\dag_{\\bf q} b_{\\bf q} \n\\!+\\! \\sum_{\\bf q} \\hbar \\big( \\gamma_{\\bf q} b_{\\bf q} \\!+\\! \\gamma^{\\ast}_{\\bf q} b^\\dag_{\\bf q}\n                          \\big) |X \\rangle\\langle X|.\n\\label{dot-ph}\n\\end{align}  \nThe ground-state $|0\\rangle$ is chosen as the zero of the energy and\nthe phonon-free energy of the transition to the single exciton state $|X\\rangle$  is denoted\nby $\\hbar\\omega^{0}_{X}$.  The Rabi frequency $\\Omega(t)$ is proportional to the\nelectric field envelope of a circularly polarized Gaussian laser pulse with\nfrequency $\\omega_{L}$, which is detuned  from the ground-state to exciton\ntransition by $\\Delta = \\omega_{L}-\\omega_{X}$, where $\\omega_{X}$ is the\nfrequency of the single exciton resonance which deviates from $\\omega^{0}_{X}$\nby the polaron shift that results from the dot-phonon coupling\nin Eq.~(\\ref{dot-ph}). The coupling to the laser field\nis treated in the common rotating wave and dipole approximations.  The operator\n$b^\\dag_{\\bf q}$ creates a longitudinal acoustic (LA) bulk phonon with wave\nvector $\\bf{q}$ and energy $\\hbar \\omega_{\\bf{q}}$.  We assume a linear\ndispersion relation $\\omega_{\\bf{q}} = c_{s} |\\bf{q}|$, where $c_{s}$ denotes\nthe speed of sound.  The phonons are coupled via the deformation potential only\nto the exciton state. This coupling is expressed by the exciton-phonon coupling\n$\\gamma_{\\bf{q}}=\\frac{|\\bf{q}|}{\\sqrt{2V\\rho \\hbar \\omega_{\\bf{q}}}}\n\\left(D_{\\rm{e}} \\Psi^{\\rm{e}}({\\bf q}) - D_{\\rm{h}} \\Psi^{\\rm{h}}({\\bf\nq})\\right)$, where $\\rho$ denotes the mass density of the crystal, $V$ the mode\nvolume, $D_{\\rm{e\/h}}$ the deformation potential constants, and\n$\\Psi^{\\rm{e\/h}}(\\bf{q})$ the formfactors of electron and hole, respectively. As\nexplained in the main article, we calculate the formfactors from the\nground-state wavefunctions of a spherical symmetric, parabolic confinement\npotential.  It should be noted that, in the pure dephasing model for the\ndot-phonon coupling, no transitions between the bare electronic states can be\ninduced by the continuum of LA phonons, which can change the electronic\noccupations only in the presence of the laser field. We assume the system to be\ninitially in a product state of a thermal phonon-distribution at the temperature\nof the cryostat and a pure ground-state of the electronic subsystem.  We use the\nmaterial parameters given in Ref.~\\cite{Krummheuer2002} for GaAs, which are: $\\rho =\n5370 \\; \\rm{kg}\/\\rm{m}^3$, $c_{s} = 5110 \\; \\rm{m}\/\\rm{s}$, $D_{\\rm{e}} = 7.0 \\;\n\\rm{eV}$, and $D_{\\rm{h}} = -3.5 \\; \\rm{eV}$.\n\nTo obtain the time evolution of the electronic density matrix elements predicted \nby this model, we make use of a numerically exact real-time path-integral\napproach, described in detail in Ref.~\\cite{Vagov2011}.  This gives us the\nopportunity to calculate the dynamics of the quantum dot with a high and\ncontrollable numerical precision and without further approximations to the\ngiven Hamiltonian. This includes taking into account all multi-phonon processes\nand non-Markovian effects.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion and Future Work}\n\\label{sec:conclusion}\n\nWe present Flexible BFT, a protocol that supports diverse clients with different assumptions to use the same ledger. Flexible BFT allows the clients to tolerate combined (Byzantine plus alive-but-corrupt\\xspace) faults exceeding 1\/2 and 1\/3 for synchrony and partial synchrony respectively. At a technical level, under synchrony, we show a synchronous protocol where the replicas execute a network speed protocol and only the commit rule uses the synchrony assumption. For partial synchrony, we introduce the notion of Flexible Byzantine Quorums by deconstructing existing BFT protocols to understand the role played by the different quorums. We combine the two to form Flexible BFT which obtains the best of both worlds.\n\nOur liveness proof in Section~\\ref{sec:proof}\nemploys a strong assumption that all clients have correct commit rules.\nThis is because our alive-but-corrupt\\xspace fault model did not specify \nwhat these replicas would do if they can violate safety for some clients.\nIn particular, they may stop helping liveness. \nHowever, we believe this will not be a concern \nonce we move to a more realistic rational model.\nIn that case, the best strategy for alive-but-corrupt\\xspace replicas\nis to attack the safety of clients with unsafe commit rules \nwhile preserving liveness for clients with correct commit rules. \nSuch an analysis in the rational fault model\nremains interesting future work.\nOur protocol also assumes that all replicas have clocks that advance at the same rate. It is interesting to explore whether our protocol can be modified to work with clock drifts.\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nAs we have seen, three parameters \n${q_{r}}$, ${q_c}$, and $\\Delta$ determine the protocol. \n${q_{r}}$ is the only parameter for the replicas\nand is picked by the service administrator.\nThe choice of ${q_{r}}$\ndetermines a set of client assumptions that can be supported. \n${q_c}$ and $\\Delta$ are chosen by clients to commit blocks.\nIn this section, we first discuss the client assumptions supported by a\ngiven ${q_{r}}$ and then discuss the trade-offs \nbetween different choices of ${q_{r}}$.\n\n\\subsection{Client Assumptions Supported by ${q_{r}}$}\n\\label{sec:client-beliefs-given}\n\n\\begin{figure}[tbp]\n  \\centering\n    \\includegraphics[width=0.45\\textwidth]{single-curve.pdf}\n    \\caption{\\textbf{Clients supported for ${q_{r}} = 2\/3$.}}\n    \\label{fig:clients-supported}\n\\end{figure}\n\nFigure~\\ref{fig:clients-supported} represents\nthe clients supported at ${q_{r}} = 2\/3$. \nThe x-axis represents Byzantine faults\nand the y-axis represents total faults (Byzantine plus a-b-c\\xspace). \nEach point on this graph \nrepresents a client fault assumption as a pair: (Byzantine faults, total faults).\nThe shaded gray area indicates an ``invalid area'' since we\ncannot have fewer total faults than Byzantine faults.\nA missing dimension\nin this figure is the choice of $\\Delta$. \nThus, the synchrony guarantee shown in this figure is for clients\nthat choose a correct synchrony bound. \n\nClients with partial-synchrony assumptions\ncan get fault tolerance on (or below) the starred orange line. \nThe right most point on the line is $(1\/3, 1\/3)$, i.e., we\ntolerate less than a third of Byzantine replicas and no additional\na-b-c\\xspace replicas.  \nThis is the setting of existing partially synchronous consensus\n protocols~\\cite{DLS88,castro1999practical,yin2018hotstuff}. \nFlexible BFT\\xspace generalizes these protocols by giving clients the option of\nmoving up-left along the line, \ni.e., tolerating fewer Byzantine and more total faults.\nBy choosing ${q_c}>{q_{r}}$, a client tolerates \n$< {q_c}+{q_{r}}-1$ total faults for safety \nand $\\leq 1-{q_c}$ Byzantine faults for liveness. \nIn other words, as a client moves left,  \nfor every additional vote it requires,\nit tolerates one fewer Byzantine fault and gains overall\none higher total number of faults (i.e., two more a-b-c\\xspace faults).\nThe left most point on this line $(0, 2\/3)$ tolerating no \nByzantine replicas and the highest fraction of a-b-c\\xspace replicas. \n\nMoreover, for clients who believe in synchrony,\nif their $\\Delta$ assumption is correct,\nthey enjoy 1\/3 Byzantine tolerance and 2\/3 total tolerance\nrepresented by the green diamond.\nThis is because synchronous commit rules are not parameterized by\nthe number of votes received. \n\n\\paragraph{How do clients pick their commit rules?}\nIn Figure~\\ref{fig:clients-supported}, the shaded starred orange\nportion of the plot represent fault tolerance provided \nby the partially synchronous commit rule (CR1). \nSpecifically, setting ${q_c}$ to the total fault \nfraction yields the necessary commit rule. On the other hand, if\na client's required fault tolerance lies in the circled green portion\nof the plot, then the synchronous commit rule (CR2) with an\nappropriate $\\Delta$ picked by the client yields the necessary\ncommit rule. Finally, if a client's target fault tolerance corresponds to the\nwhite region of the plot, then \nit is not achievable with this ${q_{r}}$.\n\n\\paragraph{Clients with incorrect assumptions and recovery.}\nIf a client has incorrect assumption with respect to the fault\nthreshold or synchrony parameter $\\Delta$, then it can lose\nsafety or liveness. If a client believing in synchrony\npicks too small a $\\Delta$ and commits a value $b$, \nit is possible that a conflicting\nvalue $b'$ may also be certified. Replicas may\nchoose to extend the branch containing $b'$, effectively\nreverting $b$ and causing a safety violation. Whenever a client\ndetects such a safety violation, it may need to revert\nsome of its commits and increase $\\Delta$ to recover.\n\nFor a client with partial-synchrony assumption, if it loses safety, \nit can update its fault model to move left along\nthe orange starred line, i.e., tolerate higher total faults but fewer\nByzantine. On the other hand, if it observes no progress as its\nthreshold ${q_c}$ is not met, then it moves towards the\nright. However, if the true fault model is in the circled green\nregion in Figure~\\ref{fig:clients-supported}, then the client\ncannot find a partially synchronous commit rule that is both safe and live\nand eventually has to switch to using a synchronous commit rule.\n\nRecall that the goal of a-b-c\\xspace replicas is to attack \nsafety. Thus, clients with\nincorrect assumptions may be exploited by\na-b-c\\xspace replicas for their own gain (e.g., by double-spending). \nWhen a client updates to a correct assumption \nand recovers from unsafe commits, their subsequent commits\nwould be safe and final. This is remotely analogous to Bitcoin \n-- if a client commits to a transaction when it is a few blocks deep and a\npowerful adversary succeeds in creating an alternative longer fork, the\ncommit is reverted. \n\n\\subsection{Comparing Different ${q_{r}}$ Choices}\n\\label{sec:comp-diff-qmins}\n\\begin{figure}[tbp]\n  \\centering\n    \\includegraphics[width=0.45\\textwidth]{pbft.pdf}\n    \\caption{Clients supported by Flexible BFT\\xspace at different\n      ${q_{r}}$'s.  The legend represents the different ${q_{r}}$ values.}\n    \\label{fig:varying-qmin}\n\\end{figure}\nWe now look at the service administrator's choice at picking\n${q_{r}}$. \nIn general, the service administrator's goal is to tolerate a large number of Byzantine\nand a-b-c\\xspace faults, i.e., move towards top and\/or right of the figure.\nFigure~\\ref{fig:varying-qmin} shows the trade-offs\nin terms of clients supported by different ${q_{r}}$ values in Flexible BFT\\xspace.\n\nFirst, it can be observed that for clients with partial-synchrony assumptions, \n${q_{r}} \\geq 2\/3$ dominates ${q_{r}} < 2\/3$. \nObserve that the fraction of Byzantine\nreplicas $(B)$ are bounded by $B < {q_c}+{q_{r}}-1$ and $B \\leq\n1-{q_c}$, so $B \\leq {q_{r}}\/2$. \nThus, as ${q_{r}}$ decreases, Byzantine fault tolerance decreases.\nMoreover, since the total fault tolerance is \n${q_c} + {q_{r}} - 1$, a lower ${q_{r}}$ also tolerates a\nsmaller fraction of total faults for a fixed ${q_c}$.\n\nFor ${q_{r}} \\geq 2\/3$ or for clients believing in synchrony, \nno value of ${q_{r}}$ is Pareto optimal. \nFor clients with partial-synchrony assumptions, as ${q_{r}}$ increases, \nthe total fault tolerance for safety increases.\nBut since ${q_c} \\geq {q_{r}}$, we have $B \\leq 1 - {q_{r}}$, and\nhence the Byzantine tolerance for liveness decreases.\nFor clients believing in synchrony, the total fault tolerance\nfor safety is $< {q_{r}}$ and the Byzantine fault tolerance for\nliveness is $\\geq 1-{q_{r}}$. \nIn both cases, the choice of ${q_{r}}$ represents a safety-liveness\ntrade-off.\n\n\n\\subsection{Separating Alive-but-corrupt\\xspace Resilience from Diversity}\n\nSo far, we presented the Flexible BFT\\xspace techniques and protocols \nto simultaneously support diverse client support and stronger a-b-c\\xspace fault tolerance.\nIndeed, we believe both properties are desirable and they strengthen each other.\nBut we remark that these two properties can be provided separately.\n\nIt is relatively straightforward to provide stronger fault tolerance in the a-b-c\\xspace model in a classic uniform setting. \nFor example, under partial-synchrony, one can simply use a larger quorum in PBFT (without the ${q_{r}}$\/$q$ replica\/client quorum separation).\nBut we note that a higher total (a-b-c\\xspace plus Byzantine) tolerance comes at the price of a lower Byzantine tolerance.\nIn a uniform setting, this means \\emph{all} clients have to sacrifice some Byzantine tolerance.\nIn the diverse setting, Flexible BFT\\xspace gives clients the freedom to choose the fault assumption they believe in,\nand a client can choose the classic Byzantine fault model.\n\nOn the flip side, if one hopes to support diverse clients in the classic Byzantine fault (no a-b-c\\xspace faults),\nthe ``dimension of diversity'' reduces.\nOne example is the network speed replica protocol in Section~\\ref{sec:overv-synchr-flex},\nwhich supports clients that believe in different synchrony bounds.\nThat protocol can be further extended to support clients with a (uniform) partial-synchrony assumption.\nClients with partial-synchrony assumption are uniform since\nwe have not identified any type of ``diversity'' outside a-b-c\\xspace faults for them. \n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nByzantine fault tolerant (BFT) protocols are used to build replicated services~\\cite{PSL80,LSP82,schneider1990implementing}.\nRecently, they have received revived interest\nas the algorithmic foundation of what is known as decentralized ledgers, or blockchains. \n\nIn the classic approach to BFT protocol designs,\na protocol designer or a service administrator first picks a set of assumptions \n(e.g., the fraction of Byzantine faults and certain timing assumptions)\nand then devises a protocol (or chooses an existing one) tailored for that particular setting.\nThe assumptions made by the protocol designer are imposed upon all parties involved --- \nevery replica maintaining the service as well as every client (also known as the\n\"learner\" role) using the service.\nSuch a protocol collapses if deployed under settings that differ from the one it is designed for. \nIn particular, optimal-resilience partially synchronous solutions~\\cite{DLS88,castro1999practical} \nbreak (lose safety and liveness) if the fraction of Byzantine faults exceeds\n$1\/3$. \nSimilarly, optimal-resilience synchronous solutions~\\cite{abraham2018synchronous,hanke2018dfinity}\ndo not obtain safety or liveness if the fraction of Byzantine faults exceeds\n$1\/2$ or if the synchrony bound is violated.\n\nIn this work, we introduce a new approach for BFT protocol design called \\emph{Flexible BFT\\xspace}. \nOur approach offers advantages in the two aspects above.\nFirst, the Flexible BFT\\xspace approach enables protocols \nthat tolerate more than $1\/3$ (resp.\\ $1\/2$) corruption faults in the\npartial-synchrony (resp.\\ synchrony) model\n--- provided that the number of Byzantine faults do not exceed the respective resilience bounds. \nSecond, the Flexible BFT\\xspace approach allows a certain degree of separation between \nthe fault model and the protocol design.\nAs a result, Flexible BFT\\xspace allows diverse clients with different \nfault assumptions and timing assumptions (synchrony or not)\nto participate in the same protocol.\nWe elaborate on these two aspects below.\n\n\\paragraph{Stronger resilience.}\nWe introduce a mixed fault model with a new type of fault called\n\\emph{alive-but-corrupt\\xspace} (a-b-c\\xspace for short) faults. \nAlive-but-corrupt\\xspace replicas actively try to disrupt the system from maintaining a safe consensus decision \nand they might arbitrarily deviate from the protocol for this purpose. \nHowever, if they cannot break safety,\nthey will not try to prevent the system from reaching a (safe) decision.\nThe rationale for this new type of fault is that \nviolating safety may provide the attacker gains (e.g., a double spend attack)\nbut preventing liveness usually does not.\nIn fact, a-b-c\\xspace replicas may gain rewards from keeping the replicated service\nlive, e.g., by collecting service fees.\nWe show a family of protocols \nthat tolerate a combination of Byzantine and a-b-c\\xspace faults \nthat exceeds $1\/3$ in the partially synchronous model\nand exceeds $1\/2$ in the synchronous model.\nOur results do not violate existing resilience bounds \nbecause the fraction of Byzantine faults is always smaller than the respective bounds.\n\n\n\\paragraph{Diversity.}\nThe Flexible BFT\\xspace approach further provides certain separation between the fault model\nand the protocol. The design approach builds a protocol whose transcript can be\ninterpreted by external clients with diverse beliefs, who draw different consensus\ncommit decisions based on their beliefs.\nFlexible BFT\\xspace guarantees safety and liveness so far as the\nclients' beliefs are correct; thus two clients with correct\nassumptions agree with each other.\nClients specify (i) the fault threshold they need to tolerate, \nand (ii) the message delay bound, if any, they believe in.  \nFor example, one instance of Flexible BFT\\xspace can support a client that requires tolerance against $1\/5$ Byzantine faults plus $3\/10$ a-b-c\\xspace faults, \nwhile simultaneously supporting another client who requires tolerance against $1\/10$ Byzantine faults plus $1\/2$ a-b-c\\xspace faults,\nand a third client who believes in synchrony and requires \n$3\/10$ Byzantine plus $2\/5$ a-b-c\\xspace tolerance.\n\nThis novel separation of fault model from protocol design\ncan be useful in practice in several ways.\nFirst, different clients may naturally hold different assumptions about the system.\nSome clients may be more cautious and require a higher resilience than others;\nsome clients may believe in synchrony while others do not.\nMoreover, even the same client may assume a larger fraction of faults \nwhen dealing with a \\$1M transaction compared to a \\$5 one. \nThe rationale is that more replicas may be willing to collude\nto double spend a high-value transaction.\nIn this case, the client can wait for more votes before committing the \\$1M transaction.\nLast but not least, a client may update its assumptions based on certain events it observes.\nFor example, if a client receives votes for conflicting values,\nwhich may indicate an attempt at attacking safety,\nit can start requiring more votes than usual;\nif a client who believes in synchrony notices abnormally long message delays,\nwhich may indicate an attack on network infrastructure,\nit can update its synchrony bound to be more conservative or switch to a partial-synchrony assumption.\n\n\nThe notion of ``commit'' needs to be clarified in our new model.\nClients in Flexible BFT\\xspace have different assumptions and hence different commit rules.\nIt is then possible and common that a value is committed by one client but not another.\nFlexible BFT\\xspace guarantees that any two clients whose assumptions are correct \n(but possibly different) commit to the same value.\nIf a client's assumption is incorrect, however,\nit may commit inconsistent values which may later be reverted.\nWhile this new notion of commit may sound radical at first, \nit is the implicit behavior of existing BFT protocols.\nIf the assumption made by the service administrator is violated in a classic BFT protocol\n(e.g., there are more Byzantine faults than provisioned),\nclients may commit to different values and they have no recourse.\nIn this sense, Flexible BFT\\xspace is a robust generalization of classic BFT protocols. \nIn Flexible BFT\\xspace, if a client performs conflicting commits,\nit should update its assumption to be more cautious\nand re-interpret what values are committed under its new assumption.\nIn fact, this ``recovery'' behavior is somewhat akin to Bitcoin.\nA client in Bitcoin decides how many confirmations are needed \n(i.e., how ``deeply buried'') to commit a block.\nIf the client commits but subsequently an alternative longer fork appears, its commit is reverted.\nGoing forward, the client may increase the number of confirmations it requires.\n\n\n\\paragraph{Key techniques.}\nFlexible BFT\\xspace centers around two new techniques.\nThe first one is a novel synchronous BFT protocol with replicas\n executing at \\emph{network speed}; \nthat is, the protocol run by the replicas does not assume synchrony.\nThis allows clients in the same protocol \nto assume different message delay bounds and commit at their own\npace. \nThe protocol thus \n separates timing assumptions of replicas from timing assumptions of clients.\nNote that this is only possible via Flexible BFT\\xspace's separation of\nprotocol from the fault model:\nthe action of committing is only carried out by clients, not by replicas.\nThe other technique involves a breakdown of the different roles that \nquorums play in different steps of partially synchronous BFT protocols.\nOnce again, made possible by the separation in Flexible BFT\\xspace,\nwe will use one quorum size for replicas to run a protocol,\nand let clients choose their own quorum sizes for committing in the protocol.\n\n\\paragraph{Contributions.}\nTo summarize, our work has the following contributions.\n\n\\begin{enumerate}[topsep=8pt,itemsep=8pt]\n\t\t\t\t\n\\item \n\\textbf{Alive-but-corrupt faults.} We introduce a new type of fault, called\nalive-but-corrupt\\xspace fault, which attack safety but not liveness.\n\n\\item \\textbf{Synchronous BFT with network speed replicas.} \n\tWe present a synchronous protocol in which \n\tonly the commit step requires synchrony.\n\tSince replicas no longer perform commits in our approach,\n\tthe protocol simultaneously supports clients assuming different synchrony bounds.\n\t\t\n\\item \\textbf{Flexible Byzantine Quorums.} \n\tWe deconstruct existing BFT protocols to understand the role played by different quorums \n\tand introduce the notion of Flexible Byzantine Quorums.\n\tA protocol based on Flexible Byzantine Quorums simultaneously supports clients assuming different fault models.\n\n\\item \\textbf{One BFT Consensus Solution for the Populace.} \n\tPutting the above together, we present a new approach for BFT design, Flexible BFT\\xspace.\n\tOur approach has stronger resilience and diversity: Flexible BFT\\xspace tolerates a fraction of combined (Byzantine plus a-b-c\\xspace) faults \n\tbeyond existing resilience bounds. And clients with diverse fault\nand timing beliefs are supported  in the same protocol. \n\n\\end{enumerate}\n\n\\paragraph{Organization.}\nThe rest of the paper is organized as follows.\nSection~\\ref{sec:model} defines the Flexible BFT\\xspace model where replicas and clients are separated.\nWe will describe in more detail our key techniques \nfor synchrony and partial-synchrony in\nSections~\\ref{sec:overv-synchr-flex}~and~\\ref{sec:overview-async}, respectively. \nSection~\\ref{sec:protocol} puts these techniques together and presents the final protocol.\nSection~\\ref{sec:discussion} discusses the result obtained by the\nFlexible BFT design and \nSection~\\ref{sec:related-work} describes related work.\n\n\n\n\\section*{Acknowledgement}\nWe thank Ittai Abraham and Ben Maurer for many useful discussions\non Flexible BFT. We thank Marcos Aguilera for many insightful\ncomments on an earlier draft of this work\n\\bibliographystyle{plain}\n\n\\section{Modeling Flexible BFT\\xspace}\n\\label{sec:model}\n\nThe goal of Flexible BFT\\xspace is to build a replicated service that takes requests from clients\nand provides clients an interface of a single non-faulty server,\ni.e., it provides clients with the same totally ordered sequence of values. \nInternally, the replicated service uses multiple servers,\nalso called replicas, to tolerate some number of faulty servers. \nThe total number of replicas is denoted by $n$. In this paper, whenever we\nspeak about a set of replicas or messages, we denote the set size as its fraction over $n$.\nFor example, we refer to a set of $m$ replicas as ``$q$ replicas'' where $q=m\/n$.\n\nBorrowing notation from Lamport~\\cite{lamport2006fast}, \nsuch a replicated service has three logical actors: \n\\emph{proposers} capable of sending new values,\n\\emph{acceptors} who add these values to a totally ordered sequence (called a blockchain), \nand \\emph{learners} who decide on a sequence of values based on the transcript of the protocol and execute them on a state machine. \nExisting  replication protocols provide the following two properties:\n\n\\begin{description}[topsep=8pt,itemsep=4pt]\n\\item[-] \\textbf{Safety.} Any two learners learn the same sequence of values.\n\\item[-] \\textbf{Liveness.} A value proposed by a proposer will\n  eventually be executed by every learner.\n\\end{description}\n\nIn existing replication protocols, \nthe learners are assumed to be \\emph{uniform}, \ni.e., they interpret a transcript using the same rules \nand hence decide on the same sequence of values.\nIn Flexible BFT\\xspace, we consider diverse learners with different assumptions. \nBased on their own assumptions, \nthey may interpret the transcript of the protocol differently. \nWe show that so far as the assumptions \nof two different learners are both correct, \nthey will eventually learn the same sequence of values.\nA replication protocol in the Flexible BFT\\xspace approach\nsatisfies the following properties:\n\n\\begin{description}[topsep=8pt,itemsep=4pt]\n\\item[-] \\textbf{Safety for diverse learners.} Any two\n  learners with correct but potentially different assumptions\n  learn the same sequence of values.\n\\item[-] \\textbf{Liveness for diverse learners.} A value\n  proposed by a proposer will eventually be executed by every learner\n  with a correct assumption.\n\\end{description}\n\nIn a replicated service, clients act as proposers and learners, \nwhereas the replicas (replicated servers) are acceptors.\nThus, safety and liveness guarantees are defined with respect to clients.\n\n\\paragraph{Fault model.} \nWe assume two types of faults within the replicas: Byzantine and \\emph{alive-but-corrupt\\xspace} (a-b-c\\xspace for short). \nByzantine replicas behave arbitrarily. \nOn the other hand, the goal of a-b-c\\xspace replicas\nis to attack safety but to preserve liveness.\nThese replicas will take any actions that help them break safety of the protocol.\nHowever, if they cannot succeed in breaking safety, they will help provide liveness.\nConsequently, in this new fault model, \nthe safety proof should treat a-b-c\\xspace replicas similarly to Byzantine.\nThen, \\emph{once safety is proved},\nthe liveness proof can treat a-b-c\\xspace replicas similarly to honest.\nWe assume that the adversary is static, i.e., the adversary determines which\nreplicas are Byzantine and a-b-c\\xspace before the start of the protocol. \n\n\\paragraph{Other assumptions.} \nWe assume hash functions, digital signatures and a public-key infrastructure (PKI).\nWe use $\\sig{x}_R$ to denote a message $x$ signed by a replica $R$. \nWe assume pair-wise communication channels between replicas. \nWe assume that all replicas have clocks that advance at the same\nrate.\n\n\n\\section{Flexible Byzantine Quorums for Partial Synchrony - Overview}\n\\label{sec:overview-async}\n\nIn this section, we explain the high-level insights of Flexible\nByzantine Quorums in Flexible BFT\\xspace. \nAgain, for ease of exposition, we\nfocus on a single-shot consensus and do not consider termination.  \nWe start by reviewing the Byzantine Quorum\nSystems~\\cite{Malkhi:1997:BQS:258533.258650} that underlie\nexisting partially synchronous protocols  \nthat tolerate 1\/3 Byzantine faults (Section~\\ref{sec:byz-quorums}). \nWe will illustrate that multiple uses of 2\/3-quorums \nactually serve different purposes in these protocols.\nWe then generalize these protocols to use  \n\\emph{Flexible Byzantine Quorums}~(Section~\\ref{sec:flexible-byz-quorum}),\nthe key idea that enables more than 1\/3 fault tolerance \nand allows diverse clients with varying assumptions to co-exist. \n\n\\subsection{Background: Quorums in PBFT}\n\\label{sec:byz-quorums}\n\nExisting protocols for solving consensus in the partially synchronous setting \nwith optimal $1\/3$-resilience\nrevolve around voting by \\emph{Byzantine quorums} of replicas. \nTwo properties of Byzantine quorums are utilized for achieving safety and liveness.\nFirst, any two quorums intersect at one honest replica -- quorum intersection.\nSecond, there exists a quorum that contains no Byzantine faulty replicas -- quorum availability.\nConcretely, when less than $1\/3$ the replicas are Byzantine,\nquorums are set to size ${q_{r}}=2\/3$.\n(To be precise, ${q_{r}}$ is slightly larger than 2\/3, \ni.e., $2f+1$ out of $3f+1$ where $f$ is the number of faults,\nbut we will use ${q_{r}}=2\/3$ for ease of exposition.)\nThis guarantees an intersection of size at least $2{q_{r}}-1=1\/3$, \nhence at least one honest replica in the intersection.\nAs for availability, there exist ${q_{r}}=2\/3$ honest replicas to form\na quorum.\n\nTo dissect the use of quorums in BFT protocols, consider their use in\nPBFT~\\cite{castro1999practical} for providing safety and liveness. \nPBFT operates in a view-by-view manner.\nEach view has a unique leader and consists of the following steps: \n\n\\begin{itemize}\n\n\\item[-] \\textbf{Propose.} \n  A leader $L$ proposes a value $b$.\n\n\\item[-] \\textbf{Vote 1.} \n  On receiving the first value $b$ for a view $v$, a replica votes for $b$ \n  if it is \\emph{safe}, as determined by a locking mechanism described below.\n  A set of ${q_{r}}$ votes form a certificate $\\mathcal{C}^{{q_{r}}}(b)$.\n\n\\item[-] \\textbf{Vote 2.} \n  On collecting $\\mathcal{C}^{{q_{r}}}(b)$,\n  a replica ``locks'' on $b$ and votes for $\\mathcal{C}^{{q_{r}}}(b)$.\n\n\\item[-] \\textbf{Commit.} \n  On collecting ${q_{r}}$ votes for $\\mathcal{C}^{{q_{r}}}(b)$, a client learns that proposal\n$b$ becomes a committed decision. \n\n\\end{itemize}\nIf a replica locks on a value $b$ in a view, then it votes only for $b$ in subsequent views \nunless it ``unlocks'' from $b$.\nA replica ``unlocks'' from $b$ if it learns that ${q_{r}}$ replicas are \\emph{not}\nlocked on $b$ in that view or higher \n(they may be locked on other values or they may not be locked at all).\n\nThe properties of Byzantine quorums are harnessed in PBFT for safety and liveness as follows:\n\n\\begin{description}[topsep=8pt,itemsep=4pt]\n\n\\item[Quorum intersection within a view.]\nSafety within a view is ensured by the first round of votes.\nA replica votes only once per view.\nFor two distinct values to both obtain certificates, \none honest replica needs to vote for both, which cannot happen.\n\n\\item[Quorum intersection across views.]\nSafety across views is ensured by the locking mechanism.\nIf $b$ becomes a committed decision in a view, \nthen a quorum of replicas lock on $b$ in that view.\nFor an honest replica among them to unlock from $b$,\na quorum of replicas need to claim they are not locked on $b$.\nAt least one replica in the intersection is honest \nand would need to falsely claim it is not locked, which cannot happen.\n\n\\item[Quorum availability within a view.]\nLiveness within each view is guaranteed by having an honest quorum respond to a\nnon-faulty leader.\n\\end{description}\n\n\n\\subsection{Flexible Byzantine Quorums}\n\\label{sec:flexible-byz-quorum}\n\nOur Flexible BFT\\xspace approach separates the quorums used in \nBFT protocols for the replicas (acceptors) from the quorums used for learning when a decision becomes\ncommitted. \nMore specifically,\nwe denote the quorum used for forming certificates (locking) by ${q_\\text{lck}}$\nand the quorum used for unlocking by ${q_\\text{ulck}}$. \nWe denote the quorum employed by clients for learning certificate uniqueness by\n${q_\\text{unq}}$, and the quorum used for learning commit safety by ${q_\\text{cmt}}$. \nIn other words, clients mandate ${q_\\text{unq}}$ first-round votes and ${q_\\text{cmt}}$ second-round votes in order\nto commit a decision. \nBelow, we outline a modified PBFT-like protocol that uses these different quorum sizes instead of a single quorum size $q$.\nWe then introduce a new definition, Flexible Byzantine Quorums, that\ncapture the requirements needed for these quorums to provide\nsafety and liveness.\n\n\\begin{figure}[h]\n\\centering\n\\begin{boxedminipage}{\\columnwidth}\n\\begin{itemize}[itemsep=4pt,leftmargin=*]\n\n\\item[-] \\textbf{Propose.} \n  A leader $L$ proposes a value $b$.\n\n\\item[-] \\textbf{Vote 1.} \n  On receiving the first value $b$ for a view $v$, a replica votes for $b$ \n  if it is \\emph{safe}, as determined by a locking mechanism described below.\n  A set of ${q_\\text{lck}}$ votes forms a certificate $\\mathcal{C}^{{q_\\text{lck}}}(b)$.\n\n\\item[-] \\textbf{Vote 2.} \n  On collecting $\\mathcal{C}^{{q_\\text{lck}}}(b)$,\n  a replica ``locks'' on $b$ and votes for $\\mathcal{C}^{{q_\\text{lck}}}(b)$.\n\n\\item[-] \\textbf{Commit.} \n  On collecting ${q_\\text{unq}}$ votes for $b$ and ${q_\\text{cmt}}$ votes for $\\mathcal{C}^{{q_\\text{lck}}}(b)$, \n  a client learns that proposal $b$ becomes a committed decision. \n\n\\end{itemize}\nIf a replica locks on a value $b$ in a view, then it votes only for $b$ in subsequent views \nunless it ``unlocks'' from $b$ by learning that ${q_\\text{ulck}}$\n replicas are not locked on $b$.\n\\end{boxedminipage}\n\\end{figure}\n\n\\begin{description}[topsep=8pt,itemsep=4pt]\n\\item[Flexible quorum intersection (a) within a view.]\nContrary to PBFT, in Flexible BFT\\xspace, a pair of ${q_\\text{lck}}$ certificates need not necessarily\nintersect in an honest replica. \nIndeed, locking on a value does not preclude conflicting locks.\nIt only mandates that every ${q_\\text{lck}}$ quorum\nintersects with every ${q_\\text{unq}}$ quorum at at least one honest replica. \nFor safety, it is essential\nthat the fraction of faulty replicas is less than ${q_\\text{lck}}+{q_\\text{unq}}-1$.\n\n\\item[Flexible quorum intersection (b) across views.]\nIf a client commits a value $b$ in a view, \n${q_\\text{cmt}}$ replicas lock on $b$ in that view.\nFor an honest replica among them to unlock from $b$,\n${q_\\text{ulck}}$ replicas need to claim they are not locked on $b$.\nThis property mandates that every ${q_\\text{ulck}}$ quorum intersects\nwith every ${q_\\text{cmt}}$ quorum at at least one honest replica.\nThus, for safety, it is essential that the fraction of faulty\nreplicas is less than ${q_\\text{ulck}} + {q_\\text{cmt}} - 1$. \n\n\\item[Flexible quorum availability within a view.]\nFor liveness, Byzantine replicas cannot exceed \n{$1-\\max({q_\\text{unq}}, {q_\\text{cmt}}, {q_\\text{lck}}, {q_\\text{ulck}})$} \nso that the aforementioned quorums can be formed at different stages of the protocol. \n\\end{description}\n\nGiven the above analysis, Flexible BFT\\xspace ensures safety  \nif the fraction of faulty replicas is less than\n$\\min({q_\\text{unq}} + {q_\\text{lck}} - 1, {q_\\text{cmt}} + {q_\\text{ulck}} - 1)$,\nand provides liveness if the fraction of Byzantine replicas \nis at most $1-\\max({q_\\text{unq}}, {q_\\text{cmt}}, {q_\\text{lck}}, {q_\\text{ulck}})$.\nIt is optimal to use \\emph{balanced quorum sizes}\nwhere ${q_\\text{lck}} = {q_\\text{ulck}}$ and ${q_\\text{unq}} = {q_\\text{cmt}}$.\nTo see this, first note that we should make sure \n${q_\\text{unq}} + {q_\\text{lck}} = {q_\\text{cmt}} + {q_\\text{ulck}}$;\notherwise, suppose the right-hand side is smaller, \nthen setting $({q_\\text{cmt}},{q_\\text{ulck}})$ to equal $({q_\\text{unq}},{q_\\text{lck}})$\nimproves safety tolerance without affecting liveness tolerance.\nNext, observe that if we have ${q_\\text{unq}} + {q_\\text{lck}} = {q_\\text{cmt}} + {q_\\text{ulck}}$\nbut ${q_\\text{lck}} > {q_\\text{ulck}}$ (and hence ${q_\\text{unq}} < {q_\\text{cmt}}$),\nthen once again setting $({q_\\text{cmt}},{q_\\text{ulck}})$ to equal $({q_\\text{unq}},{q_\\text{lck}})$\nimproves safety tolerance without affecting liveness tolerance.\n\nThus, in this paper, we set ${q_\\text{lck}} = {q_{r}}$ and ${q_\\text{unq}} =\n{q_\\text{cmt}} = {q_c}$. Since replicas use ${q_{r}}$ votes to lock,\nthese votes can always be used by the clients to commit\n${q_\\text{cmt}}$ quorums. \nThus, ${q_c} \\geq {q_{r}}$.\nThe Flexible Byzantine Quorum requirements collapse\ninto the following two conditions.\n\\begin{description}\n\\item[Flexible quorum intersection.]\nThe fraction of faulty replicas is $< {q_c} + {q_{r}} - 1$.\n\n\\item[Flexible quorum availability.]\nThe fraction of Byzantine replicas is $\\leq 1-{q_c}$.\n\\end{description}\n\n\\paragraph{Tolerating a-b-c\\xspace faults.}\nIf all faults in the system are Byzantine faults, \nthen the best parameter choice is ${q_c}={q_{r}} \\geq 2\/3$\nfor $<1\/3$ fault tolerance,\nand Flexible Byzantine Quorums degenerate to basic Byzantine quorums.\nHowever, in our model, a-b-c\\xspace replicas are only interested in attacking safety but not liveness.\nThis allows us to tolerate  \n${q_c} + {q_{r}} - 1$ total faults (Byzantine plus a-b-c\\xspace), which can be more than $1\/3$. \nFor example, if we set ${q_{r}} = 0.7$ and ${q_c} = 0.8$, \nthen such a protocol can tolerate $0.2$ Byzantine faults plus\n$0.3$ a-b-c\\xspace faults.\nWe discuss the choice for ${q_{r}}$ and ${q_c}$ and their rationale in Section~\\ref{sec:discussion}.\n\n\\paragraph{Separating client commit rules from the replica protocol.}\n\\label{sec:client-coexist-partial}\nA key property of the Flexible Byzantine Quorum approach is that\nit decouples the BFT protocol from \nclient commit rules. \nThe decoupling allows \nclients assuming different fault models\nto utilize the same protocol.\nIn the above protocol, the propose and two voting steps \nare executed by the replicas and they are only parameterized by ${q_{r}}$.\nThe commit step can be carried by different clients using different commit thresholds ${q_c}$. \nThus, a fixed ${q_{r}}$ determines a possible set of clients with\nvarying commit rules (in terms of Byzantine and a-b-c\\xspace adversaries).\nRecall that a Byzantine adversary can behave arbitrarily and thus may not\nprovide liveness whereas an a-b-c\\xspace adversary only\nintends to attack safety but not liveness.\nThus, a client who believes that a large fraction of\nthe adversary may attempt to break safety, not progress, can choose a larger ${q_c}$.\nBy doing so, it seeks stronger safety against dishonest replicas, while trading \n liveness.\nConversely, a client that assumes that a large fraction of the adversary\nattacks liveness must choose a smaller ${q_c}$. \n\n\n\n\\subsection{Safety and Liveness}\n\\label{sec:proof}\n\nWe introduce the notion of \\emph{direct} and \\emph{indirect} commit to aid the proofs.\nWe say a block is committed \\emph{directly} under \\textbf{CR1} if\nthe block and its  immediate successor both get ${q_c}$ votes in the same view.\nWe say a block is committed \\emph{directly} under \\textbf{CR2} if some honest replica\nreports an undisturbed-$2\\Delta$ period after its successor block\nwas obtained.\nWe say a block is committed \\emph{indirectly} if neither condition applies to it but\nit is committed as a result of a block extending it being committed directly.\nWe remark that the direct commit notion, especially for \\textbf{CR2}, is merely a proof technique.\nA client cannot tell whether a replica is honest,\nand thus has no way of knowing whether a block is directly committed under \\textbf{CR2}. \n\n\\begin{lemma}\nIf a client directly commits a block $B_l$ in view $v$ using a correct commit rule,\nthen a certified block that ranks no lower than $\\CommitCert_{v}(B_{l})$ must equal or extend $B_l$. \n\\label{lemma:unique-cert}\n\\end{lemma}\n\n\\begin{proof}\nTo elaborate on the lemma, \na certified block $\\CommitCert_{v'}(B'_{l'})$ ranks no lower than $\\CommitCert_{v}(B_{l})$\nif either (i) $v'=v$ and $l' \\geq l$, or (ii) $v'>v$.\nWe need to show that if $B_l$ is directly committed, \nthen any certified block that ranks no lower either equals or extends $B_l$.\nWe consider the two commit rules separately. \nFor both commit rules, we will use induction on $v'$ to prove the lemma.\n\n\\bigskip\nFor $\\textbf{CR1}$ with parameter ${q_c}$ to be correct, \nflexible quorum intersection needs to hold, i.e.,\nthe fraction of faulty replicas must be less than ${q_c}+{q_{r}}-1$.\n$B_l$ being directly committed under $\\textbf{CR1}$ with parameter ${q_c}$ implies that \nthere are ${q_c}$ votes in view $v$ for $B_l$ and $B_{l+1}$ where $B_{l+1}$ extends $B_l$. \n\nFor the base case, a block $B'_{l'}$ with $l'\\geq l$ that does not extend $B_l$ cannot get certified in view $v$,\nbecause that would require ${q_c}+{q_{r}}-1$ replicas to vote for two equivocating blocks in view $v$.\n\nNext, we show the inductive step.\nNote that ${q_c}$ replicas voted for $B_{l+1}$ in view $v$,\nwhich contains $\\CommitCert_{v}(B_{l})$.\nThus, they lock $B_l$ or a block extending $B_l$ by the end of view $v$.\nDue to the inductive hypothesis,\nany certified block that ranks equally or higher from view $v$ up to view $v'$ \neither equals or extends $B_l$. \nThus, by the end of view $v'$,\nthose ${q_c}$ replicas still lock $B_l$ or a block extending $B_l$.\nSince the total fraction of faults is less than ${q_c}+{q_{r}}-1$,\nthe status $\\mathcal{S}$ shown by the leader of view $v'+1$ \nmust include a certificate for $B_l$ or a block extending it;\nmoreover, any certificate that ranks equal to or higher than $\\CommitCert_{v}(B_{l})$\nis for a block that equals or extends $B_l$. \nThus, only a block that equals or extends $B_l$ can gather votes from those ${q_c}$ replicas in view $v'+1$\nand only a block that equals or extends $B_l$ can get certified in view $v'+1$.\n\n\\bigskip\nFor $\\textbf{CR2}$ with synchrony bound $\\Delta$ to be correct, \n$\\Delta$ must be an upper bound on worst case message delay \nand the fraction of faulty replicas is less than ${q_{r}}$. \n$B_l$ being directly committed under $\\textbf{CR2}$ with $\\Delta$-synchrony implies that\nat least one honest replica voted for $B_{l+1}$ extending $B_l$ in view $v$,\nand did not hear an equivocating block or view change within $2\\Delta$ time after that.\nCall this replica $h$.\nSuppose $h$ voted for $B_{l+1}$ extending $B_l$ in view $v$ at time $t$,\nand did not hear an equivocating block or view change by time $t+2\\Delta$. \n\nWe first show the base case: a block $B'_{l'}$ with $l'\\geq l$ certified in view $v$ must equal or extend $B_l$.\nObserve that if $B'_{l'}$ with $l'\\geq l$ does not equal or extend $B_l$,\nthen it equivocates $B_l$.\nNo honest replica voted for $B'_{l'}$ before time $t+\\Delta$,\nbecause otherwise $h$ would have received the vote for $B'_{l'}$ by time $t+2\\Delta$,\nNo honest replica would vote for $B'_{l'}$ after time $t+\\Delta$ either,\nbecause by then they would have received (from $h$) and voted for $B_l$.\nThus, $B'_{l'}$ cannot get certified in view $v$.\n\nWe then show the inductive step.\nBecause $h$ did not hear view change by time $t+2\\Delta$,\nall honest replicas are still in view $v$ by time $t+\\Delta$,\nwhich means they all receive $B_{l+1}$ from $h$ by the end of view $v$.\nThus, they lock $B_l$ or a block extending $B_l$ by the end of view $v$.\nDue to the inductive hypothesis,\nany certified block that ranks equally or higher from view $v$ up to view $v'$ \neither equals or extends $B_l$. \nThus, by the end of view $v'$,\nall honest replicas still lock $B_l$ or a block extending $B_l$.\nSince the total fraction of faults is less than ${q_{r}}$,\nthe status $\\mathcal{S}$ shown by the leader of view $v'+1$ \nmust include a certificate for $B_l$ or a block extending it;\nmoreover, any certificate that ranks equal to or higher than $\\CommitCert_{v}(B_{l})$\nis for a block that equals or extends $B_l$. \nThus, only a block that equals or extends $B_l$ can gather honest votes in view $v'+1$\nand only a block that equals or extends $B_l$ can get certified in view $v'+1$.\n\\end{proof}\n\n\n\\begin{theorem}[Safety]\nTwo clients with correct commit rules commit the same block $B_k$ for each height $k$.\n\\label{thm:safety}\n\\end{theorem}\n\n\\begin{proof}\nSuppose for contradiction that two distinct blocks \n$B_k$ and $B'_k$ are committed at height $k$.\nSuppose $B_k$ is committed as a result of $B_{l}$ being directly committed in view $v$\nand $B'_k$ is committed as a result of $B'_{l'}$ being directly committed in view $v'$.\nThis implies $B_l$ is or extends $B_k$;\nsimilarly, $B'_{l'}$ is or extends $B'_k$.\nWithout loss of generality, assume $v \\leq v'$.\nIf $v=v'$, further assume $l \\leq l'$ without loss of generality.\nBy Lemma~\\ref{lemma:unique-cert},\nthe certified block $\\CommitCert_{v'}(B'_{l'})$ must equal or extend $B_l$.\nThus, $B'_k=B_k$.\n\\end{proof}\n\n\n\n\\begin{theorem}[Liveness]\nIf all clients have correct commit rules, \nthey all keep committing new blocks. \n\\label{thm:liveness}\n\\end{theorem}\n\n\\begin{proof}\nBy the definition of a-b-c\\xspace faults,\nif they cannot violate safety, they will preserve liveness.\nTheorem~\\ref{thm:safety} shows that if all clients have correct commit rules,\nthen safety is guaranteed \\emph{even if a-b-c\\xspace replicas behave arbitrarily}.\nThus, once we proved safety, we can treat a-b-c\\xspace replicas \nas honest when proving liveness. \n\nObserve that a correct commit rule tolerates at most $1-{q_{r}}$ Byzantine faults.\nIf a Byzantine leader prevents liveness, \nthere will be ${q_{r}}$ blame messages against it,\nand a view change will ensue to replace the leader. \nEventually, a non-Byzantine (honest or a-b-c\\xspace) replica becomes the leader \nand drives consensus in new heights.\nIf replicas use increasing timeouts,\neventually, all non-Byzantine replicas stay in the same view for sufficiently long.\nWhen both conditions occur, \nif a client's commit rule is correct (either \\textbf{CR1} and \\textbf{CR2}),\ndue to quorum availability,\nit will receive enough votes in the same view to commit. \n\\end{proof}\n\n\n\\section{Flexible BFT Protocol}\n\\label{sec:protocol}\n\nIn this section, we combine the ideas presented in\nSections~\\ref{sec:overv-synchr-flex} and \\ref{sec:overview-async}\nto obtain a final protocol that supports both types of clients.\nA client can either assume partial synchrony, with freedom to choose ${q_c}$ as described\nin the previous section, or assume synchrony with its own choice of\n$\\Delta$, as described in Section~\\ref{sec:overv-synchr-flex}. \nReplicas execute a protocol at the network speed\nwith a parameter ${q_{r}}$.\nWe first give the protocol executed by the replicas \nand then discuss how clients commit depending on their assumptions. \nMoreover, inspired by Casper~\\cite{DBLP:journals\/corr\/abs-1710-09437} and\nHotStuff~\\cite{yin2018hotstuff}, we show a protocol where the\nrounds of voting can be pipelined. \n\n\n\\subsection{Notation}\nBefore describing the protocol, we will first define some\ndata structures and terminologies that will aid presentation.\n\n\\paragraph{Block format.} \nThe pipelined protocol forms a chain of values.\nWe use the term \\emph{block} to refer to each value in the chain.\nWe refer to a block's position in the chain as its \\emph{height}. \nA block $B_k$ at height $k$ has the following format\n$$B_k := (b_k, h_{k-1})$$ \nwhere $b_k$ denotes a proposed value at height $k$ \nand $h_{k-1} := H(B_{k-1})$ is a hash digest of the predecessor block.\nThe first block $B_1=(b_1, \\bot)$ has no predecessor.\nEvery subsequent block $B_k$ must specify a predecessor block $B_{k-1}$ by including a hash of it.\nWe say a block is \\emph{valid} if \n(i) its predecessor is valid or $\\bot$, and \n(ii) its proposed value meets application-level validity conditions and is consistent with its chain of ancestors (e.g., does not double spend a transaction in one of its ancestor blocks). \n\n\\paragraph{Block extension and equivocation.}\nWe say $B_l$ \\emph{extends} $B_k$, \nif $B_k$ is an ancestor of $B_l$ ($l>k$). \nWe say two blocks $B_l$ and $B'_{l'}$ \\emph{equivocate} one another \nif they are not equal and do not extend one another.\n\n\n\\paragraph{Certificates and certified blocks.} \nIn the protocol, replicas vote for blocks by signing them.\nWe use $\\CommitCert_{v}{(B_{k})}$ to denote a set of signatures \non $h_{k} = H(B_{k})$ by ${q_{r}}$ replicas in view $v$. \n${q_{r}}$ is a parameter fixed for the protocol instance. \nWe call $\\CommitCert_{v}{(B_{k})}$ a certificate for $B_{k}$ from view $v$.\nCertified blocks are ranked \nfirst by the views in which they are certified and then by their heights.\nIn other words, a block $B_k$ certified in view $v$ \nis ranked \\emph{higher} than a block $B_{k'}$ certified in view $v'$\nif either (i) $v > v'$ or (ii) $v = v'$ and $k>k'$. \n\n\\paragraph{Locked blocks.} \nAt any time, a replica locks the highest certified block to its knowledge.\nDuring the protocol execution, each replica keeps track of all signatures for all blocks\nand keeps updating its locked block.\nLooking ahead, the notion of locked block will be\nused to guard the safety of a client commit.\n\n\\subsection{Replica Protocol}\n\nThe replica protocol progresses in a view-by-view fashion. Each view has a\ndesignated leader who is responsible for driving consensus on a\nsequence of blocks. Leaders can be chosen\nstatically, e.g., round robin, \nor randomly using more sophisticated techniques~\\cite{CKS05,Algorand}. \nIn our description, we assume a round robin selection of leaders,\ni.e.,  ($v$ {\\sf mod} $n$) is the leader of view $v$. \n\nAt a high level, the protocol does the following: \nThe leader proposes a block to all replicas. \nThe replicas vote on it if safe to do so. \nThe block becomes certified once ${q_{r}}$ replicas vote on it. \nThe leader will then propose another block extending the previous one, \nchaining blocks one after another at increasing heights.\nUnlike regular consensus protocols\nwhere replicas determine when a block is committed, in\nFlexible BFT\\xspace, replicas only certify blocks while committing is\noffloaded to the clients.\nIf at any time replicas detect malicious leader behavior or lack of\nprogress in a view, they blame the leader \nand engage in a view change protocol \nto replace the leader and move to the next view. \nThe new leader collects a status from different replicas and continues to propose\nblocks based on this status.\nWe explain the steady state and view change protocols in more detail below.\n\n\\begin{figure*}[tb]\n\\centering\n\\begin{boxedminipage}{\\textwidth}\n\nLet $v$ be the current view number and \nreplica $L$ be the leader in this view. \nPerform the following steps in an iteration.\n\n\\begin{enumerate}[topsep=8pt,itemsep=8pt,leftmargin=*]\n\\setlength\\itemsep{0.5em}\n\\item \\textbf{Propose. } \\label{step:propose} \n\\Comment{Executed by the leader of view $v$}\n\nThe leader $L$ broadcasts $\\sig{\\mathsf{propose}, B_k, v, \\CommitCert_{v'}(B_{k-1}), \\mathcal{S}}_L$.\nHere, $B_k := (b_k, h_{k-1})$ is the newly proposed block\nand it should extend the highest certified block known to $L$.\nIn the steady state, an honest leader $L$ would extend the\nprevious block it proposed,\nin which case $v'=v$ and $\\mathcal{S} = \\bot$. \nImmediately after a view change, \n$L$ determines the highest certified block from\nthe status $\\mathcal{S}$ received during the view change. \n\n\\item \\label{step:vote} \\textbf{Vote.} \n\\Comment{Executed by all replicas}\n\nWhen a replica $R$ receives a valid proposal \n$\\sig{\\mathsf{propose}, B_k, v, \\CommitCert_{v'}(B_{k-1}), \\mathcal{S}}_L$\nfrom the leader $L$, $R$ broadcasts the proposal and a vote $\\sig{\\mathsf{vote}, B_k, v}_R$ \nif (i) the proposal is the first one in view $v$,\nand it extends the highest certified block in $\\mathcal{S}$, \nor (ii) the proposal extends the last proposed block in the view. \n\n\nIn addition, replica $R$ records the following based on the messages it receives.\n\\begin{itemize}[topsep=8pt,itemsep=4pt]\n\\item[-] $R$ keeps track of the number of\n  votes received for this block in this view as $q_{B_{k},v}$. \n\\item[-] If block $B_{k-1}$ has been proposed in view\n  $v$, $R$ marks $B_{k-1}$ as a locked block and\n  records the locked time as ${\\mathsf{t\\text{-}{lock}}}_{k-1,v}$.\n\\item[-] If a block equivocating $B_{k-1}$\n  is proposed by $L$ in view $v$ (possibly received through a vote),\n  $R$ records the time ${\\mathsf{t}\\text{-}\\mathsf{equiv}}_{k-1,v}$ at\n  which the equivocating block is received. \n\\end{itemize}\n\nThe replica then enters the next iteration.\nIf the replica observes no progress or equivocating blocks in the same view $v$, \nit stops voting in view $v$ \nand sends $\\sig{\\mathsf{view\\text{-}change},v}_r$ message to all replicas. \n\\end{enumerate}\n\n\\end{boxedminipage}\n\\caption{Flexible BFT steady state protocol.}\n\\label{fig:steady}\n\\end{figure*}\n\n\\paragraph{Steady state protocol.}\nThe steady state protocol is described in Figure~\\ref{fig:steady}.\nIn the steady state, there is a unique leader who, in an iteration,\nproposes a block, waits for votes from ${q_{r}}$\nreplicas and moves to the next iteration.\nIn the steady state, an honest leader always extends the previous block it proposed.\nImmediately after a view change, \nsince the previous leaders could have been\nByzantine and may have proposed equivocating blocks, \nthe new leader needs to determine a safe block to propose. \nIt does so by collecting a status of\nlocked blocks from ${q_{r}}$ replicas denoted by $\\mathcal{S}$\n(described in the view change protocol).\n\nFor a replica $R$ in the steady state, on receiving a proposal for block\n$B_{k}$, a replica votes for it if \nit extends the previous proposed block in the view or \nif it extends the highest certified block in $\\mathcal{S}$.\nReplica $R$ can potentially receive blocks out of\norder and thus receive $B_{k}$ before its ancestor blocks. \nIn this case, replica $R$ waits until it receives the ancestor blocks, \nverifies the validity of those blocks and $B_k$ before voting for $B_k$.\nIn addition, replica $R$ records the following to aid a client commit:\n\\begin{itemize}\n\\item[-] \\textbf{Number of votes.} It records the number of votes received\n  for $B_{k}$ in view $v$ as $q_{B_{k},\n    v}$. Observe that votes are broadcast by \n  all replicas and the number of votes for a block can be greater\n  than ${q_{r}}$. $q_{B_{k}, v}$ will be updated each time the\n  replica hears about a new vote in view $v$.\n\\item[-] \\textbf{Lock time.} If $B_{k-1}$ was proposed in the\n  same view $v$, it locks $B_{k-1}$ and records the locked time as\n  ${\\mathsf{t\\text{-}{lock}}}_{k-1, v}$.\n\\item[-] \\textbf{Equivocation time.} If the replica ever observes an\n  equivocating block at height $k$ in view $v$ through a\n  proposal or vote, \n  it stores the time of equivocation as ${\\mathsf{t}\\text{-}\\mathsf{equiv}}_{k,v}$.\n\\end{itemize}\nLooking ahead, the locked time ${\\mathsf{t\\text{-}{lock}}}_{k-1,v}$ and\nequivocation time ${\\mathsf{t}\\text{-}\\mathsf{equiv}}_{k-1,v}$ will\nbe used by clients with synchrony assumptions to commit,\nand the number of votes $q_{B_{k}, v}$\nwill be used by clients with partial-synchrony assumptions to commit.\n\n\\paragraph{Leader monitoring.}\nIf a replica detects a lack of progress in view $v$ or\nobserves malicious leader behavior\nsuch as more than one height-$k$ blocks in the same view,\nit blames the leader of view $v$\nby broadcasting a $\\sig{\\mathsf{view\\text{-}change}, v}$ message. It quits\nview $v$ and stops voting and broadcasting blocks in\nview $v$. \nTo determine lack of progress, the replicas may simply guess a time bound for\nmessage arrival or use increasing timeouts for each view~\\cite{castro1999practical}.\n\n\\paragraph{View change.}\nThe view change protocol is described in Figure~\\ref{fig:vc}.\nIf a replica gathers ${q_{r}}$ $\\sig{\\mathsf{view\\text{-}change}, v}$\nmessages from distinct replicas,\nit forwards them to all other replicas and enters a new view\n$v + 1$ (Step~\\ref{step:new_view}). It records the time\nat which it received the blame certificate as ${\\mathsf{t\\text{-}{viewchange}}}_{v}$.\nUpon entering a new view,\na replica reports to the leader of the new view $L'$ its locked block \nand transitions to the steady state (Step~\\ref{step:status}). \n${q_{r}}$ status messages form the status $\\mathcal{S}$.\nThe first block $L'$ proposes in the new view \nshould extend the highest certified block among these ${q_{r}}$ status messages.\n\n\\begin{figure*}[htbp]\n\\centering\n\\begin{boxedminipage}{\\textwidth}\n\nLet $L$ and $L'$ be the leaders of views $v$ and $v+1$, respectively.\n\\begin{enumerate}[topsep=8pt,itemsep=8pt,leftmargin=*,label=(\\roman*)]\n\\setlength\\itemsep{0.5em}\n\\item \\label{step:new_view} \\textbf{New-view.}\nUpon gathering ${q_{r}}$ $\\sig{\\mathsf{view\\text{-}change}, v}$ messages,\nbroadcast them and enter view $v+1$. Record the time as ${\\mathsf{t\\text{-}{viewchange}}}_{v}$.\n\n\\item \\label{step:status}  \\textbf{Status.}\nSuppose $B_j$ is the block locked by the replica. Send a\nstatus of  its locked block to the leader $L'$ using\n$\\sig{\\mathsf{status}, v, B_j, \\CommitCert_{v'}(B_j)}$ and transition\nto the steady state. Here, $v'$ is the view in which $B_j$\nwas certified.\n\n\\end{enumerate}\n\n\\end{boxedminipage}\n\\caption{Flexible BFT view change protocol.}\n\\label{fig:vc}\n\\end{figure*}\n\n\n\n\\subsection{Client Commit Rules}\n\\label{sec:commit-rules}\n\n\n\\begin{figure*}[htbp]\n  \\centering\n  \\begin{boxedminipage}{\\textwidth}\n    \\begin{enumerate}[itemsep=8pt,leftmargin=0.2cm,label=]\n      \n      \\item \\textbf{(CR1) Partially-synchronous commit.} A\n        block $B_k$ is committed under the partially synchronous rule with parameter ${q_c}$  \n        iff there exist $l \\geq k$ and $v$ such that\n        \\begin{enumerate}[topsep=8pt,itemsep=4pt]\n\t\t\t\t\\item $\\CommitCert_{v}(B_l)$ and $\\CommitCert_{v}(B_{l+1})$ \n\t\t\t\t\t\texist where $B_{l+1}$ extends $B_l$ and $B_k$ (if $l = k$, $B_l = B_k$).\n        \\item $q_{B_{l}, v} \\geq {q_c}$ and $q_{B_{l+1}, v} \\geq {q_c}$.\n        \\end{enumerate}\n        \n\t\t\\item \\textbf{(CR2) Synchronous commit.}\n        A block $B_k$ is committed assuming $\\Delta-$synchrony \n\t\tiff the following holds for ${q_{r}}$ replicas.\n\t\tThere exist $l \\geq k$ and $v$ (possibly\n                different across replicas) such that,\n        \\begin{enumerate}[topsep=8pt,itemsep=4pt]\n        \\item $\\CommitCert_{v}(B_l)$ exists where $B_l$ extends $B_k$ (if $l = k$, $B_l = B_k$).\n        \\item An undisturbed-$2\\Delta$ period is observed after\n          $B_{l+1}$ is obtained, i.e., no equivocating block\n          or view change of view $v$ were\n           observed before $2\\Delta$ time after $B_{l+1}$\n           was obtained, i.e.,\n        $$\\min({\\mathsf{current}\\text{-}\\mathsf{time}}, {\\mathsf{t}\\text{-}\\mathsf{equiv}}_{l,v},\n        {\\mathsf{t\\text{-}{viewchange}}}_v) - {\\mathsf{t\\text{-}{lock}}}_{l,v}\n        \\geq 2\\Delta$$\n        \\end{enumerate}\n    \\end{enumerate}\n  \\end{boxedminipage}\n  \\caption{Flexible BFT commit rules}\n  \\label{fig:commit-rules}\n\\end{figure*}\n\nAs mentioned in the introduction, Flexible BFT\\xspace supports \nclients with different assumptions. \nClients in Flexible BFT\\xspace learn the\nstate of the protocol from the replicas and based on their\nown assumptions determine whether a block has been committed.\nBroadly, we supports two types of clients:\nthose who believe in synchrony and those who believe in partial synchrony.\n\n\\subsubsection{Clients with Partial-Synchrony Assumptions (CR1)}\nA client with partial-synchrony assumptions deduces whether a block has been committed\nby based on the number of votes received by a block. \nA block $B_l$ (together with its ancestors) is committed with parameter ${q_c}$\niff $B_l$ and its immediate\nsuccessor both receive $\\geq {q_c}$ votes in the same view. \n\n\\paragraph{Safety of CR1.}\nA CR1 commit based on ${q_c}$ votes is safe \nagainst $<{q_c}+{q_{r}}-1$ faulty replicas (Byzantine plus a-b-c\\xspace).\nObserve that if $B_l$ gets ${q_c}$ votes in view $v$,  \ndue to flexible quorum intersection, \na conflicting block cannot be certified in view $v$, \nunless $\\geq {q_c}+{q_{r}}-1$ replicas are faulty. \nMoreover, $B_{l+1}$ extending $B_l$ \nhas also received ${q_c}$ votes in view $v$. \nThus, ${q_c}$ replicas lock block $B_l$ in view $v$.\nIn subsequent views, honest replicas that have locked \n$B_l$ will only vote for a block \nthat equals or extends $B_l$ unless they unlock. \nHowever, due to flexible quorum intersection, \nthey will not unlock \nunless $\\geq {q_c}+{q_{r}}-1$ replicas are faulty. \nProof of Lemma~\\ref{lemma:unique-cert} formalizes this argument. \n\n\\subsubsection{Client with Synchrony Assumptions (CR2)}\nIntuitively, a CR2 commit involves ${q_{r}}$ replicas collectively stating that \nno ``bad event'' happens within ``sufficient time'' in a view.\nHere, a bad event refers to either leader equivocation or view change\n(the latter indicates sufficient replicas believe leader is\nfaulty) and the ``sufficient time'' is $2\\Delta$; \nwhere $\\Delta$ is a synchrony bound chosen by the client. \nMore formally, a replica states that a synchronous commit for block $B_k$\nfor a given parameter $\\Delta$ (set by a client) is satisfied iff the following holds.\nThere exists $B_{l+1}$ that extends $B_l$ and $B_k$,\nand the replica observes an undisturbed-$2\\Delta$ period after\nobtaining $B_{l+1}$ during which (i) no equivocating block\nis observed, and (ii) no blame certificate\/view\nchange certificate for view $v$ was obtained, i.e., \n$$\\min({\\mathsf{current}\\text{-}\\mathsf{time}}, {\\mathsf{t}\\text{-}\\mathsf{equiv}}_{l,v},\n        {\\mathsf{t\\text{-}{viewchange}}}_v) - {\\mathsf{t\\text{-}{lock}}}_{l,v}\n        \\geq 2\\Delta$$\nwhere ${\\mathsf{t}\\text{-}\\mathsf{equiv}}_{l,v}$ denotes the time \nequivocation for $B_l$ in view $v$ was\nobserved ($\\infty$ if no equivocation), ${\\mathsf{t\\text{-}{viewchange}}}_v$\ndenotes the time at which view change happened from view $v$\nto $v + 1$ ($\\infty$ if no view change has happened\nyet), and ${\\mathsf{t\\text{-}{lock}}}_{l,v}$ denotes the time at which \n$B_l$ was locked (or $B_{l+1}$ was proposed) in\nview $v$.\nNote that the client does not require the ${q_{r}}$ fraction of replicas \nto report the same height $l$ or view $v$.\n\n\\paragraph{Safety of CR2.}\nA client believing in synchrony assumes that all messages between replicas\narrive within $\\Delta$ time after they were sent. \nIf the client's chosen $\\Delta$ is a correct upper bound on message delay, \nthen a CR2 commit is safe against ${q_{r}}$ faulty replicas (Byzantine plus a-b-c\\xspace),\nas we explain below.\nIf less than ${q_{r}}$ replicas are faulty, at least one honest replica reported \nan \\emph{undisturbed-$2\\Delta$} period.\nLet us call this honest replica $h$ and \nanalyze the situation from $h$'s perspective\nto explain why an undisturbed $2\\Delta$ period ensures safety.\nObserve that replicas in Flexible BFT\\xspace forward the proposal when voting. \nIf $\\Delta$-synchrony holds, every other honest replica \nlearns about the proposal $B_l$ at most $\\Delta$ time after $h$ learns about it.\nIf any honest replica voted for a conflicting block \nor quit view $v$, $h$ would have known within $2\\Delta$ time. \n\n\\input{proof}\n\n\\subsection{Efficiency}\n\n\\paragraph{Latency.}\nClients with a synchrony assumption incur a latency of\n$2\\Delta$ plus a few network speed rounds. In terms of the\nmaximum network delay $\\Delta$, this matches the \nstate-of-the-art synchronous\nprotocols~\\cite{abraham2019sync}. The distinction though is that\n$\\Delta$ now depends on the client assumption and \nhence different clients may commit with different latencies\nClients with partial-synchrony assumptions incur a latency of two\nrounds of voting; this matches PBFT~\\cite{castro1999practical}.\n\n\\paragraph{Communication.}\nEvery vote and new-view messages are broadcast to all replicas,\nincurring $O(n^2)$ communication messages. This is the same\ncomplexity of PBFT~\\cite{castro1999practical} and Sync\nHotStuff~\\cite{abraham2019sync}.\n\n\n\\section{Related Work}\n\\label{sec:related-work}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\begin{subfigure}{0.45\\textwidth}\n    \\includegraphics[width=1\\linewidth]{compare.pdf}\n  \\end{subfigure}\n  \n  \\begin{subfigure}{0.45\\textwidth}\n   \n     \n      \\begin{tabular}{c l}\n        $+$ & Partially Synchronous protocols~\\cite{castro1999practical,Yin03,Martin06,Zyzzyva07,yin2018hotstuff,buchman2016tendermint}\\\\\n        $\\times$ & Synchronous Protocols~\\cite{pass2018thunderella,hanke2018dfinity,abraham2019sync,abraham2018synchronous}\\\\\n        $\\blacktriangle$ & Thunderella, Sync HotStuff ($\\vartriangle$: optimistic)~\\cite{pass2018thunderella,abraham2019sync}\\\\\n        $\\blacklozenge$ & Zyzzyva, SBFT ($\\lozenge$: optimistic)~\\cite{Zyzzyva07}\n      \\end{tabular}\n   \n   \n  \\end{subfigure}\n  \\caption{\\textbf{Comparing Flexible BFT\\xspace to existing consensus\n      protocols.} The legend represent different ${q_{r}}$ values.}\n  \\label{fig:compare}\n\\end{figure}\n\nMost BFT protocols are designed with a uniform assumption about the\nsystem. The literature on BFT consensus is vast and is largely beyond scope for\nreview here; we refer the reader to the standard\ntextbooks in distributed computing~\\cite{lynch1996distributed,attiya2004distributed}.\n\n\n\\paragraph{Resilience.} Figure~\\ref{fig:compare} compares resilience in Flexible BFT\\xspace\nwith some existing consensus protocols.  The x axis represents a Byzantine\nresilience threshold, the y axis the total resilience against\ncorruption under the a-b-c\\xspace fault mode.  The three\ndifferent colors (red, green, blue) represent three possible instantiations of\nFlexible BFT\\xspace at different ${q_{r}}$'s. \n\nEach point in the figure represents an abstract ``client'' belief. For the partial\nsynchrony model, client beliefs form lines, and for synchronous settings,\nclients beliefs are individual circles.\nThe locus of points on a given color represents all client\nassumptions supported for a corresponding ${q_{r}}$, representing the diversity of\nclients supported. \nThe figure depicts state-of-art resilience combinations by existing consensus solutions via \nuncolored shapes, $+, \\times, \\vartriangle, \\blacktriangle, \\lozenge, \\blacklozenge$.\nPartially synchronous protocols~\\cite{castro1999practical, yin2018hotstuff,buchman2016tendermint} that tolerate one-third Byzantine faults can all be represented  by the `+' symbol at $(1\/3, 1\/3)$. \nSimilarly, synchronous\nprotocols~\\cite{hanke2018dfinity,abraham2018dfinity,abraham2018synchronous}\nthat tolerate one-half Byzantine faults are represented by the\n`$\\times$' symbol at $(1\/2, 1\/2)$.\nIt is worth noting that some of these works \nemploy two commit rules that differ in number of votes \nor synchrony~\\cite{Martin06,Zyzzyva07,pass2018thunderella,abraham2019sync}.\nFor instance, Thunderella and Sync HotStuff optimistically commit \nin an asynchronous fashion based on quorums of size $\\geq 3\/4$, \nas represented by a hollow triangle at $(1\/4, 1\/2)$.\nSimilarly, FaB~\\cite{Martin06}, Zyzzyva~\\cite{Zyzzyva07} and SBFT~\\cite{gueta2018sbft}\noptimistically commit when they receive all votes but wait for two rounds of votes otherwise.\nThese are represented by two points in the figure. \nDespite the two commit rules, \nthese protocols do not have client diversity, \nall parties involved (replicas and clients) make the same assumptions\nand reach the same commit decisions.\n\n\\paragraph{Diverse client beliefs.}\nA simple notion of client diversity exists in Bitcoin's probabilistic commit rule.\nOne client may consider a transaction committed after six confirmations while another may require only one confirmation.\nGenerally, the notion of client diversity has been discussed informally at public blockchain forums. \n\nAnother example of diversity is considered in the XFT protocol~\\cite{XFT}.\nThe protocol supports two types of clients: clients that assume crash faults under partial synchrony,\nor clients that assume Byzantine faults but believe in synchrony.\nYet another notion of diversity is considered \nby the federated Byzantine consensus model and the Stellar\nprotocol~\\cite{mazieres2015stellar}.\nThe Stellar protocol allows nodes to pick their own quorums.\nOur Flexible BFT\\xspace approach instead considers diverse clients \nin terms of a-b-c\\xspace adversaries and synchrony.\nThe model and techniques in~\\cite{mazieres2015stellar} and our paper\nare largely orthogonal and complementary.\n\n\\paragraph{Flexible Paxos.} Flexible Paxos by Howard et\nal.~\\cite{DBLP:conf\/opodis\/HowardMS16} observes that \nPaxos may use non-intersecting quorums within a view \nbut an intersection is required across views.\nOur Flexible Quorum Intersection (b) can be viewed as\nits counterpart in the Byzantine and a-b-c\\xspace setting.\nIn addition, Flexible BFT\\xspace applies the flexible quorum idea\nto support diverse clients with different fault model and timing assumptions.\n\n\\paragraph{Mixed fault model.}\nFault models that mix Byzantine and crash faults have been considered in various\nworks, e.g., FaB~\\cite{Martin06} and SBFT~\\cite{abraham2019sync}. The a-b-c\\xspace\nfaults are in a sense the opposite of crash faults, mixing Byzantine with\n``anti-crashes''. \nOur a-b-c\\xspace adversary bears similarity to a rational adversary in the BAR\nmodel~\\cite{aiyer2005bar}, with several important differences.\nBAR assumes no collusion exists among rational replicas themselves and between\nrational and Byzantine replicas, whereas a-b-c\\xspace replicas have no such\nconstraint.\nBAR solutions are designed to expose cheating behavior and thus deter rational\nreplicas from cheating. The Flexible BFT\\xspace approach does not rely on deterrence for good\nbehavior, and breaks beyond the $1\/3$ ($1\/2$) corruption tolerance threshold in\nasynchronous (synchronous) systems. \nLast, BAR solutions address only the partial synchrony settings. \nAt the same time, BAR provides a game theoretic proof of rationality. \nMore generally, game theoretical modeling and analysis with collusion have been performed to other problems \nsuch as secret sharing and multiparty computation~\\cite{abraham2006distributed,lysyanskaya2006rationality,gordon2006rational,kol2008cryptography}. \nAnalyzing incentives for the a-b-c\\xspace model remains an open challenge.\n\n\n\n\\section{Synchronous  BFT with Network Speed Replicas - Overview}\n\\label{sec:overv-synchr-flex}\n\\begin{figure*}[ht]\n\\begin{boxedminipage}{\\textwidth}\n\\paragraph{Protocol executed by the replicas.}\n\\begin{enumerate}\n\\setlength\\itemsep{0.5em}\n\n\\item \\textbf{Propose. } \\label{step:sync-propose} \n  The leader $L$ of view $v$ \n  proposes a value $b$.\n\\item \\textbf{Vote.} \\label{step:sync-vote}\n  On receiving the first value $b$ in a view $v$, a replica broadcasts $b$ and votes for $b$ if it is \\emph{safe} to do so, as determined by a locking mechanism described later. \nThe replica records the following.\n\\begin{itemize}\n\\item[-] If the replica collects ${q_{r}}$ votes on $b$, \n\t\tdenoted as $\\mathcal{C}^{q_{r}}_v(b)$ and called a certificate of $b$ from view $v$, \n\t\tthen it ``locks'' on $b$ and records the lock time as ${\\mathsf{t\\text{-}{lock}}}_v$.\n\\item[-] If the replica observes an equivocating value signed by $L$ at any time after entering view $v$, it records the time of equivocation as ${\\mathsf{t}\\text{-}\\mathsf{equiv}}_v$. It blames the leader by broadcasting $\\sig{\\mathsf{view\\text{-}change}, v}$ and the equivocating values.\n\\item[-] If the replica does not receive a proposal for sufficient time in view $v$, it times out and broadcasts $\\sig{\\mathsf{view\\text{-}change}, v}$. \n\\item[-] If the replica collects a set of ${q_{r}}$ $\\sig{\\mathsf{view\\text{-}change}, v}$ messages, \nit records the time as ${\\mathsf{t\\text{-}{viewchange}}}_v$, broadcasts them and enters view $v+1$. \n\\end{itemize}\n\\end{enumerate}\n\nIf a replica locks on a value $b$ in a view, then it votes only for $b$ in subsequent views \nunless it ``unlocks'' from $b$ by learning that ${q_{r}}$ replicas\nare not locked on $b$ in that view or higher views (they may be locked on other values or they may not be locked at all).\n\n\\paragraph{Commit rules for clients.}\nA value $b$ is said to be committed by a client assuming $\\Delta$-synchrony \niff ${q_{r}}$ replicas\neach report that there exists a view $v$ such that, \n\\begin{enumerate}\n\\item $b$ is certified, i.e., $\\mathcal{C}^{q_{r}}_v(b)$ exists.\n\\item the replica observed an undisturbed-$2\\Delta$ period after certification, i.e., \n\t\tno equivocating value or view change was observed at a time before $2\\Delta$ after it was certified, \n\t\tor more formally, $\\min({\\mathsf{current}\\text{-}\\mathsf{time}}, {\\mathsf{t}\\text{-}\\mathsf{equiv}}_v, {\\mathsf{t\\text{-}{viewchange}}}_v) - {\\mathsf{t\\text{-}{lock}}}_v \\geq 2\\Delta$\n\\end{enumerate}\n\\end{boxedminipage}\n\\caption{Synchronous BFT with network speed replicas.}\n\\label{fig:sync-bft}\n\\end{figure*}\n\nEarly synchronous protocols~\\cite{DS83,katz2009expected,micali2017optimal} have relied on synchrony in two ways.\nFirst, the replicas assume a maximum network delay $\\Delta$ for communication between them. Second, they require a lock step execution, i.e., all replicas are in the same round at the same time.\nHanke et al. showed a synchronous protocol without lock step execution~\\cite{hanke2018dfinity}. \nTheir protocol still contains a synchronous step in which all replicas perform a blocking wait of $2\\Delta$ time before proceeding to subsequent steps. \nSync HotStuff~\\cite{abraham2019sync} improves on it further to remove replicas' blocking waits during good periods (when the leader is honest),\nbut blocking waits are still required by replicas during bad situations (view changes).\n\nIn this section, we show a synchronous protocol where the\nreplicas do not ever have blocking waits and execute at the network speed.\nIn other words, replicas run a partially synchronous protocol and do not rely on synchrony at any point. \nClients, on the other hand, rely on synchrony bounds to commit.\nThis separation is what allows our protocol to support clients with different assumptions on the value of $\\Delta$. \nTo the best of our knowledge, this is the first synchronous protocol to achieve such a separation.\nIn addition, the protocol tolerates a combined Byzantine plus a-b-c\\xspace fault ratio greater than a half (Byzantine fault tolerance is still less than half). \n\nFor simplicity, in this overview, we show a protocol for single shot consensus.\nIn our final protocol in Section~\\ref{sec:protocol}, we will consider a pipelined version of the protocol for consensus on a sequence of values. \nWe do not consider termination for the single-shot consensus protocol in this overview \nbecause our final replication protocol is supposed to run forever. \n\nThe protocol is shown in Figure~\\ref{fig:sync-bft}. It runs in a sequence of views. Each view has a designated leader who may be selected in a round robin order. The leader drives consensus in that view.\nIn each view, the protocol runs in two steps -- propose and vote. In the propose step, the leader proposes a value $b$. In the vote step, replicas vote for the value if it is \\emph{safe} to do so.\nThe vote also acts as a \\emph{re-proposal} of the value.\nIf a replica observes a set of ${q_{r}}$ votes on $b$, called a\ncertificate $\\CommitCert(b)$, it ``locks'' on $b$. \nFor now, we assume ${q_{r}} = 1\/2$.\n(To be precise, ${q_{r}}$ is slight larger than 1\/2, e.g., $f+1$ out of $2f+1$.)\nWe will revisit the choice\nof ${q_{r}}$ in Section~\\ref{sec:discussion}.\nIn subsequent views, a replica will not vote for a value other than $b$ unless it learns that ${q_{r}}$ replicas are not locked on $b$. In addition, the replicas switch views (i.e., change leader) if they either observe an equivocation or if they do not receive a proposal from the leader within some timeout. \nA client commits $b$ if ${q_{r}}$ replicas state that there exists a view in which $b$ is certified \nand no equivocating value or view change was observed at a time before $2\\Delta$ after it was certified. \nHere, $\\Delta$ is the maximum network delay the client believes in. \n\nThe protocol ensures safety if there are fewer than ${q_{r}}$ faulty replicas.\nThe key argument for safety is the following: \nIf an honest replica $h$ satisfies the commit condition for some value $b$ in a view, then \n(a) no other value can be certified and \n(b) all honest replicas are locked on $b$ at the end of that view.\nTo elaborate, satisfying the commit condition implies that some honest replica $h$ has observed an undisturbed-$2\\Delta$ period after it locked on $b$, i.e., it did not observe an equivocation or a view change. \nSuppose the condition is satisfied at time $t$. \nThis implies that other replicas did not observe an equivocation or a view change before $t-\\Delta$. \nThe two properties above hold if the quorum honesty conditions described below hold.\nFor liveness, if Byzantine leaders equivocate or do not propose a safe value, they will be blamed by both honest and a-b-c\\xspace replicas and a view change will ensue. \nEventually there will be an honest or a-b-c\\xspace leader to drive consensus if quorum availability holds.\n\n\\begin{description}[topsep=8pt,itemsep=4pt]\n\\item[Quorum honesty (a) within a view.] \nSince the undisturbed period starts after $b$ is certified, \n$h$ must have voted (and re-proposed) $b$ at a time earlier than $t-2\\Delta$. \nEvery honest replica must have received $b$ before $t - \\Delta$.\nSince they had not voted for an equivocating value by then, they must have voted for $b$.\nSince the number of faults is less than ${q_{r}}$,\nevery certificate needs to contain an honest replica's vote.\nThus, no certificate for any other value can be formed in this view. \n\t\t\n\\item[Quorum honesty (b) across views.] \n$h$ sends $\\mathcal{C}^{q_{r}}_v(b)$ at time $t-2\\Delta$. \nAll honest receive $\\mathcal{C}^{q_{r}}_v(b)$ by time $t-\\Delta$ and become locked on $b$. \nFor an honest replica to unlock from $b$ in subsequent views, \n${q_{r}}$ replicas need to claim that they are not locked on $b$. \nAt least one of them is honest \nand would need to falsely claim it is not locked, which cannot happen.\n\n\\item[Quorum availability.]\tByzantine replicas do not exceed $1-{q_{r}}$ so that ${q_{r}}$ replicas respond to the leader. \n\\end{description}\n\n\n\\paragraph{Tolerating a-b-c\\xspace faults.}\nIf we have only honest and Byzantine replicas (and no a-b-c\\xspace replicas), quorum honesty requires the fraction of Byzantine replicas $B < {q_{r}}$. Quorum availability requires $B \\leq 1-{q_{r}}$. If we optimize for maximizing $B$, we obtain ${q_{r}} \\geq 1\/2$.\nNow, suppose $P$ represents the fraction of a-b-c\\xspace replicas. Quorum honesty requires $B + P < {q_{r}}$, and quorum availability requires $B \\leq 1-{q_{r}}$. Thus, the protocol supports varying values of $B$ and $P$ at different values of ${q_{r}} > 1\/2$ such that safety and liveness are both preserved.\n\n\\paragraph{Separating client synchrony assumption from the replica protocol.}\nThe most interesting aspect of this protocol is the separation of the client commit rule from\nthe protocol design. In particular, although this is a synchronous protocol, the replica protocol does not rely on any synchrony bound. This allows clients to choose their own message delay bounds. \nAny client that uses a correct message delay bound enjoys safety. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConvolutional neural network (CNN) has gained success in many fields and become the main method in pattern recognition and computer vision since AlexNet won the 2012 ImageNet competition \\cite{krizhevsky2012imagenet,deng2009imagenet}. Nowadays, the technology has also been used in real-world applications, such as applications in smartphones, intelligent recommendations on products or music, target recognition and face detection \\cite{lecun2015deep}.\n\nHowever, designing a high performance architecture is still a challenging problem. This usually needs expertise and takes long time of research by trail and error \\cite{hanxiao}. Because of numerous configurations with respect to the number of layers and details in each layer, it is impossible to explore the whole architecture space. Hence, how to automatically generate a competitive architecture becomes an important problem \\cite{bergstra2012random,miller1989designing,negrinho2017deeparchitect}.\n\nReinforcement Learning and Evolutionary Algorithm are two main architecture learning methods used for CNN structure learning \\cite{Baker,zhao,Zoph,Barret,zhao,Esteban}. Reinforcement learning uses models' performances on validation dataset as rewards and uses an agent to maximize the cumulative rewards when interacting with an environment. Evolutionary algorithm simulates evolution processes to iteratively improve models' performances.\n\nHowever, performances of the generated networks are limited by architecture search space which is determined by the algorithm's encoding system. If the encoding system can not represent one architecture, that architecture can never be learned. For example, \\cite{Lingxi} used fixed-number stages to represent network structures. Different stages are separated by pooling layer. Each stage is a directed acyclic graph (DAG). In a DAG, nodes represent layers and edges represent operations such as 1$\\times$1 convolution and 3$\\times$3 convolution. Therefore, the architecture space is directly limited by the fixed-number stages and pooling layers. Because of this, it is impossible to generate many network architectures. For example, it can't generate one network where there is a skip connection between one layer in first stage and another layer in last stage. Some other encoding systems just represent plain architectures which are composed of stacking layers. But plain networks usually suffer from learning problems such as vanishing gradient problem and exploding gradient problem. In this paper, we propose a new encoding system which have little limitations on search space. For example, the order of layers are randomly set. In our encoding system, we use short DNA strands which are composed of a series of DNA molecules (A,G,C,T) to represent layers and long DNA strands composed of short strands to represent architectures.\n\nIt is well known that network depth is of crucial importance because it can raise the ``levels'' of features. However, simply stacking layers causes the degradation problem. Skip-connections can solve the degradation problem when deepening networks \\cite{srivastava2015highway,srivastava2015training,he2016deep}. ResNet \\cite{he2016deep} uses skip-connections via residual blocks. Skip-connections containing in ResNet provide a shorter path between bottom layers and target output layer compared with normal plain architectures. Parameters of bottom layers are thus easier to be trained. DenseNet \\cite{huang2017densely} also adopts skip-connections by reusing feature maps. Because skip-connections existing in ResNet and DenseNet ease learning problem and improve model's capacity, we explore neural network architectures with skip-connections.\n\nDNA Computing is a method of computation using molecular biology techniques \\cite{braich2000solution,adleman1994molecular,boneh1996computational,kari2000using}. DNA molecules (A, G, C, T) can be used to encode information just like binary strings. DNA strands are composed of DNA molecules and can be regarded as a piece of information. DNA computing uses DNA strands to carry information. It generates large numbers of short DNA strands and then put them into DNA soup. In DNA soup, short DNA strands connect to each other to form longer DNA strands based on base pairing principle if provided suitable reaction environment. After a period of time, the soup contains a sets of candidate DNA strands that represent desired results. Then we can pick DNA strands representing desired results from the soup. For example, ``travelling salesman problem\" can be solved by DNA computing \\cite{adleman1994molecular}. We can generate different DNA strands and use them to represent a city that have to be visited. Each strand has a linkage with other strands. Within seconds, the strands form bigger ones that represent different travel routes. Then the DNA strands representing longer routes can be eliminated through chemical reaction. The remains are the solutions. In DNA soup, all connecting processes happen at the same time so that DNA computing can reduce reaction time.\n\nIn this paper, we use DNA computing algorithm to generate neural network architectures. In DNA computing algorithm, short DNA strand denoted as Layer Strand encodes one layer architecture and a piece of skip-connection information which determines whether the layer has a skip-connection with one of its previous layers. Long strands denoted as Architecture Strands are composed of short Layer Strands via base pairing. Because each layer in network have at most one skip connection with one its previous layers, DNA computing algorithm aims to explore networks with skip-connections. We have little limitations on search space. We don't limit the number of pooling layers and the depth of the architecture. The skip-connections are also randomly set for that any two layers can have a skip-connection. During DNA computing algorithm, we use Layer Strands (representing layers) as our reaction units and learn Architecture Strands (representing architectures) via base paring. After getting models (Architecture Strands) via DNA computing algorithm, we train those models on training data set and select one model according to their performance on the validation data set. We achieve 0.27\\% test error on MINST data set and 4.9\\% test error on CIFAR-10 data set.\n\n\\section{Related Work}\nIn this section, we introduce convolutional neural networks firstly. Then we introduce reinforcement learning and evolutionary algorithms for structure learning of deep networks.\n\\subsection{Convolutional Neural Networks}\nConvolutional neural networks (CNN) \\cite{Alex,Simonyan} have achieved great success in various computer tasks \\cite{Backpropagation}. Convolution neural networks are usually composed of convolution layers, pooling layers and fully connected layers. By stacking convolution layers, pooling layers and fully connected layers, we can get plain architectures.\n\nSpecialists have tried a lot to improve neural network's capacity and find that increased depth can help a lot. For example, deep models, from depth of sixteen \\cite{simonyan2014very} to thirty \\cite{ioffe2015batch}, perform well on ImageNet dataset. However, vanishing gradients and exploding gradients prevent models from being deeper \\cite{he2016deep}. By normalized initialization \\cite{lecun1998efficient,glorot2010understanding,saxe2013exact,he2015delving} and intermediate normalization layers, the problem has been solved a lot and networks can extend to tens of layers. But degradation problem happens when the models become deeper and deeper. Degradation problems mean that models' performance degrades with increased depth.\n\nResNet  \\cite{he2016deep} and DenseNet  \\cite{huang2017densely} can solve degradation problems well via skip-connections. Both have gained good results in ImageNet and CIFAR-10. They can also be generalized to many other data sets. Skip-connections make bottom layers have shorter pathes to output layer which makes learning easily and enriches the features. ResNet uses residual blocks to form whole architecture. In each residual block, input layer is added to output layer which is called a skip-connection. So bottom layers in ResNet have a very short path to output layer. The gradients can thus easily and effectively flow to the bottom layers via skip-connections. DenseNet reuses feature maps and increases width of each layer with little increased parameters. The input layer becomes one part of the output layer which can also be called a skip-connection. \\\\\nAs the neural networks containing skip-connections perform well in many tasks, we explore neural network architectures with skip-connections. In our encoding system, we don't limit search space. we don't limit the number of pooling layer compared with \\cite{Lingxi}. Locations of convolution layers and pooling layers are all randomly set and skip-connections are also randomly set for that one layer can have a skip-connection with any one of its previous layers.\n\\subsection {Reinforcement Learning and Evolutionary Algorithm}\nEven though Resnet and DenseNet perform well in many data sets, network architectures still need to be carefully designed for specific data set. Therefore, how to design a convolutional neural network is a very worthwhile issue. The traditional neural network is designed based on a large number of experimental experience. Recently, more and more researchers focus their research on automatically generating networks and networks have been automatically generated through Reinforcement Learning \\cite{Baker} and genetic algorithms \\cite{zhao}.\n\nReinforcement Learning usually uses a meta-controller to determine neural network architectures. It uses architectures' performance on validation data set as reward to update the meta-controller. Thus, a neural network architecture can be treated as a training sample. Because deep learning is a data driving technology, Reinforcement Learning needs a lot of samples to learn a high performance meta-controller. However, training a model spends huge computation resources. Evolutionary Algorithm simulates the process of evolution. It iterates improve its performance by operators such as mutation, crossover and selection. Evolutionary algorithm selects models according to their performances on validation data set. Thus, Evolutionary Algorithm also spends huge computation resources and time.\n\nAs the Reinforcement Learning and Evolutionary Algorithm are all data hungry methods and need huge computation sources, we aim to address that we can get a good model via DNA computing algorithm from high quality search space and only train a few of models.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=3in,height=3in]{DNA_strands.jpg} \n\\caption{ Layer Strand 9 in one Architecture Strand whose maximum depth is 24 (not including fully connected layers). \\textcircled{1}: Left: Head fragment AAACG (9) of Layer Strand 9; Right: Tail of Layer Strand 8. The tail pairs with the head to form a longer strand. \\textcircled{2}: Layer type fragment AAGTC (30). Because the result of 30 mod 24 is 6 and 6 is more than 4, the layer belongs to a convolution layer. \\textcircled{3}: kernel size fragement AAAGC (6). 6 mod 4 is 2. Because the layer is a convolution layer, thus the number 2 represents 5*5 kernel size. \\textcircled{4}: Channel number fragment AAGAC (18). The result of 18 mod 6 is 0, so the channel number is 32. \\textcircled{5}: Skip-connection fragment AAAGC (6). 6 mod 9 is 6. So there is a skip-connection between layer 6 and layer 9. \\textcircled{6}: Left: The head fragment of layer 10. Right: The tail fragment of layer 9.}\n\\label{f1}\n\\end{figure*}\n\\section{Our Approach}\nIn this section, we introduce how DNA strands encode neural network architectures. And then, we introduce how to generate architectures using DNA Computing Algorithm. After introduction of model generation, we introduce how to train the learned models.\n\\subsection{Coding System}\\label{codesys}\nDNA computing algorithm uses DNA strands to encode neural network architectures and use DNA computing algorithm to generate the strands that represent architectures. In our DNA computing algorithm, we use short DNA strands denoted as Layer Strands to represent layer architectures (convolution layers and pooling layers) and long strands denoted as Architecture Strands to represent overall neural network architectures. In DNA computing algorithm, Layer Strands can form Architecture Strands via base paring between the exposed heads and tails of different Layer Strands, which is like the process that stacking layers to form architectures. The skip-connection information is encoded in each Layer Strand. And one layer can have at most one skip-connection with one of its previous layer. Those architectures with skip-connections form architecture space learned by DNA computing algorithm. We must emphasize we don't limit the search space for that we only set the maximum depth of the model. The number of pooling layers, location of convolution layers and skip-connection in each layer are all random. The detail of encoding method are described below.\n\nBecause the similarity between pooling layer and convolution layer, pooling layer and convolution layer are represented by Layer Strands that have same constructions. Each Layer Strand is composed of 6 DNA-fragments representing specific parameters of pooling layer or convolution layer, such as kernel size or channel number. Those 6 DNA-fragments include head fragment, layer type fragment, kernel size fragment, channel number fragment, skip-connection fragment and tail fragment. The head fragment represents layer number. For example, some Layer Strands represent first layer while some Layer strands represent second layer and so on. If one Layer Strand represent ith layer, it is called Layer Strand i-1. The tail fragment are specially designed to pair the head fragment of its previous Layer strand so that two Layer Strands can form a longer strand by base paring method. The layer type fragment determines which kinds of layer this strand represents, convolution layer or pooling layer. The channel number fragment determines the number of channels. The number of channels is chosen from \\{32, 64, 96, 128, 160, 192\\}. The channel number fragment of pooling layer is useless for that the channel number of pooling layer is same as its previous layer. The kernel size fragment determines the kernel size. For convolution layer, the kernel size is chosen from \\{1$\\times$1, {3$\\times$3}, {5$\\times$5}, {7$\\times7$}\\}. For pooling layer, the kernel size is chosen from \\{2$\\times$2, 3$\\times$3\\}. As for skip-connection fragment, it determines whether this layer has a skip-connection with one of its previous layer. Each layer has at most one skip-connection. Six DNA fragments are arranged according to head, layer type, kernel size, channel number, skip-connection, and tail order. In reality, DNA is a double-stranded structure with exposed head and tail. So in addition to the head and tail fragments, the other four fragments belong to double-stranded structures. The head and tail fragments are exposed with single layer structure. Thus different Layer Strands can be connected between their exposed head and tail fragments' base paring. Six fragments compose a Layer strand which is a basic unit in DNA computing.\n\nJust as Figure \\ref{f1} shows, each fragment of Layer Strand is composed of five pairs of molecules or five single molecules (head and tail). For double-stranded structure fragments, only the strand along the same side as the tail and head is useful during decoding. Therefore, the length of a Layer Strand is 30. We use molecules A, G, C, T to represent 0, 1, 2, 3 respectively. According to the quaternary decoding method, five molecules represent integers from 0 to 1023. Thus one fragment can encode 1024 (4$^5$) kinds of information. There are redundant integers in each strands and we use different groups of integers to represent different parameter values in the four double-structured fragments so that all the permutation of five molecules can be utilized. The detail will be described below.\n\nHyper-parameter N specifies the maximum number of layers in each neural network. Each fragment can be translated into a real number and the specific parameter are determined by the real number. The head fragment can be translated into integer n and the integer n represents layer n. In generation, only Layer Strands represent layer 0 (first layer) to layer N-1 can be generated. Thus, only head fragments represent layer 0 to layer N-1 can be generated. The Layer Strand representing layer i can be denoted as Layer Strand i. In this way, we can define maximum depth of neural network. Because the tail fragment representing layer L needs to pair the head fragment of layer L+1 to form a longer strand, the tail fragment of layer L is thus determined by the tail fragment of layer L+1. For example, the head fragment representing layer one is AAAAG (can be translated into number 1), then the tail fragment of the layer 0 must be TTTTC (A pairs with T and G pairs with C). Thus, the two Layer Strands can be connected by base paring between exposed AAAAG and TTTTC single-layer fragments to form a long strand. So, only N kinds of tail fragments corresponding to head tails can be generated. As for the other four kinds of fragment, they are not limited in generation. All kinds of permutation of five molecules can be generated.\n\nDuring decoding, one fragment is translated into real number n and the concrete meanings of fragments are determined by integer groups which the integers belong to. Layer type fragment can be translated into n$_t$. If the result of n$_t$ mod N is less than 4, the layer belong to pooling layer. Otherwise, it is a convolution layer. That means we may get about 4 pooling layers in the N layers. The kernel size fragment can be translated into number n$_k$. If the layer is a convolution layer, the result of n$_k$ mod 4 determines the kernel size and 0, 1, 2, 3 represent 1$\\times$1, {3$\\times$3}, {5$\\times$5}, {7$\\times7$} kernel sizes respectively. If the layer is a pooling layer, the result of n$_k$ mod 2 determines the kernel size and 0, 1 represent 2$\\times$2, 3$\\times$3 kernel sizes respectively. The channel number fragment can be translated into real number n$_c$. The results of n$_c$ mod 6 determine the channel numbers and 0, 1, 2, 3, 4, 5 represent 32, 64, 96, 128, 160, 192 respectively. The skip-connection fragment can be translated into real number n$_s$. If the layer number is L and the result of n$_c$ mod L is l, there is a skip-connection between the layer L and layer l. l should be less than L-1. Otherwise, the skip-connection fragments are useless.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3in,height=5.8in]{1234.jpg} \n\\caption{One of our models on CIFAR-10 that gains high accuracy. }\n\\label{f3}\n\\end{figure}\n\\subsection{Generation via DNA Computing Algorithm}\nLayer Strands are our basic reaction units of DNA computing algorithm. Hyper-parameter P defines the number of architectures need to be generated. In generation, we generate P Layer Strands i (0$\\leq$i$\\le$N-1). We can get P$\\times$N Layer Strands. In each Layer Strand, only head fragments and tail fragments are generated specially while other fragments are randomly generated. \n\nWe then put all the Layer Strands into DNA soup. In DNA soup, Layer Strand i and Layer Strand i+1 are connected via base paring between their exposed head and tail. The head fragment of Layer Strand i pairs with the tail of Layer Strand i+1 to form a double-structure fragment. Thus, the two strands form a longer strand. All the connection processes can happen at the same time.\n\nThe process finish within seconds. Then we can select DNA strands from DNA soup and eliminate the strands whose start part is not Layer Strand 0. After we get Architecture Strands, we translate them into real architectures. In the translating, we add a fully connected layer at the end of each architecture. We train them on training data set and select one model based on their performances on validation data set. We simulate DNA computing algorithm via computer. Algorithm \\ref{alg:1} illustrates the process of DNA computing algorithm.\n\\begin{algorithm}[ht]\n\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\t\\caption{DNA Computing Algorithm}\n\t\\label{alg:1}\n\t\\begin{algorithmic}[1]\n\t\t\\REQUIRE {Dataset, maximum number of layers in one architecture (N), number of Layer Strands representing one layer (P);\\\\}\n\t\t\\ENSURE   The network structure with the highest accuracy on test data set.\n        \\FOR { i=1 to N }\n        \\FOR { j=1 to P }\n\t\t \\STATE {Generate one Layer Strand i randomly. }\n        \\ENDFOR\n        \\ENDFOR\n\t\t\\STATE{Put all the Layer Strands into DNA soup and provide proper reaction environment.}\\\\\n        \\STATE{Select Architecture Strands from DNA soup. Count the number of Architecture Strands and get number Num. }\n\t\t\\FOR{ i=1 to Number}\n\t\t\\STATE {{S}=Generate real CNN network.}\n        \\ENDFOR\n        \\STATE {{G} = Randomly select 100 models.}\n\t\t\\FOR { i=1 to 100 }\n\t\t \\STATE { Training network i in {G} on training data set and record its network structure and final accuracy on validation data set}\n        \\ENDFOR\n\t\t\\STATE {Select the model R from G that has highest validation accuracy.}\n        \\STATE {Train model R on the whole training data set and validation data set and get its accuracy A on test data set.}\n\t\t\\STATE \\textbf{return} R, A\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Model Training}\nJust as algorithm \\ref{alg:1}, we train all the models on the training data set and get their accuracies on validation data set after getting model via DNA Computing Algorithm. We select the best model according to their performances on validation data set. After getting the best model, we merge the training data set and validation data set and train the model again. we then use the model's performance on test data set as our algorithm output. The training details are described below.\n\nTo compress the search space, we used a pre-activated convolution unit (PCC). That's to say, we use batch normalization(BN) \\cite{ioffe2015batch} and ReLU activation \\cite{Alex} before convolution operation. The stride of convolution layer is set as 1 while the stride of pooling layer is set as 2. As for skip-connections, if the layer i and layer j (i$\\leq$j) has a skip-connection. We then add layer i and layer j as output of layer j. If the two layers' channels don't map, we use 1$\\times$1 convolution with stride 1 to change channels. If the feature map size don't map, we use 1$\\times$1 convolution with stride 2 to down sample.\n\nAs for optimizer, we use momentum optimizer with momentum set to 0.9. The initial learning rate is 0.1 and the weight decay is 0.0001. The total training epochs is 60. In the tenth epoch, the learning rates is set to 0.01. In the thirtieth epoch, the learning rates is set to 0.001. \n\nThe models are trained for at most 60 epochs on training sets (CIFAR-10). As for MNIST, 10 epochs is enough for the models to be converged. Carefully designing total epochs reduces huge time. At training time, we find that with a fixed learning rate, the algorithm is converged around some epochs and had a little progress on the further epochs. If we do not decrease the learning rate, the accuracy increase little. So it's necessary to immediately decrease the learning rate after the accuracy increase slowly. If the learning rate is set properly, it reduce huge time. During the training process, we used L2 regularization.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3in]{6.png} \n\\caption{The learning curve of MNIST and CIFAR10 is very similar and proves the early stop strategy can also be used on MNIST.}\n\\label{f6}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3in]{5.png} \n\\caption{  We randomly chose eight of all our generated models without early stop strategy. As the figure shows, the competitive models perform well all the time, but the poor models are contrary. That demonstrates we can eliminate the poor model according their learning curve. }\n\\label{f5}\n\\end{figure}\n\\subsection{Early Stop Strategy}\nIn order to further reduce running time, we used early stop strategy. Just as shown in Figure \\ref{f5} and Figure \\ref{f6}, the epochs can be reduced by carefully design. We have two main discoveries. Firstly, all the models have similar learning curves. That indicates us we can eliminate models without finishing all the training epochs. Secondly, most of models in our search space perform similarly showing that we can get a high performance model with fewer times of training. We can stop training if the model can not perform well until the specified epoch. The early stop strategy eliminates poor performance models with high probability for that good models usually perform well on an early stage. This strategy reduces the huge time and has little impact on accuracy. We set three model performance thresholds on CIFAR-10 data set that are trained for at most 60 epochs. (1) 10th epoch, 80\\% test accuracy. (2) 20th epoch, 85\\% accuracy. (3) 45th epoch, 90\\% accuracy. If the model does not reach the specified accuracy after the there epochs respectively, the models will be eliminated.\n\n\n\n\\section{Results}\nIn this section, we introduce our results on CIFAR-10 and MNIST data sets. We simulate DNA computing algorithm by computer.\n\\begin{table}[h]\n  \\centering\n  \\setlength{\\tabcolsep}{0.8mm}\n    \\scalebox{0.9}[0.9]{\n  \\begin{tabular} {|l|l|l|l|}\n  \\hline\n  Model&MNIST\\\\\n  \\hline\n  Lecun\\emph{et.al}\\cite{lecun1998gradient}&0.7\\\\\n  \\hline\n  Lauer\\emph{et.al}\\cite{lauer2007trainable}&0.54\\\\\n  \\hline\n  Jarrett\\emph{et.al}\\cite{jarrett2009best}&0.53\\\\\n  \\hline\n  Ranzato\\emph{et.al}\\cite{poultney2007efficient}&0.39\\\\\n  \\hline\n  Cirecsan\\emph{et.al}\\cite{cirecsan2012multi}&0.23\\\\\n\n  \\hline\n  {Our Method}&{0.27}\\\\\n  \\hline\n  \\end{tabular}\n  }\n  \\caption{Comparison of the recognition error rate (\\%) on MNIST.}\n\\end{table}\n\n\\subsection{Results on the MNIST Data Set}\nThe MNIST database of handwritten digits has a training set of 60,000 examples, and a test set of 10,000 examples. We test our algorithm on MNIST data set in order to reduce time. We divide the training data set into two parts. 55,000 images are used as training set while 5,000 images are used as validation data set. After we get the best model according their performances on validation data set, the whole 60,000 images are used for training the model. We then test the model on test data set and use the accuracy as our method's final output.\n\nWe use DNA Computing algorithm to generate model architectures. Many of our models have gained test accuracies higher than 99.60\\% and the highest test accuracy is 99.73\\%. We show one hundred models' learning in Figure \\ref{f7}. Only few of them perform poorly indicating that our neural network architectures with random skip-connection composes a high performance search space.\n\\begin{table}[h]\n  \\centering\n  \\setlength{\\tabcolsep}{1.0mm}\n\n   \\scalebox{0.9}[0.9]{\n  \\begin{tabular}{p{1.2cm}|p{6cm}|p{1.4cm}}\n  \\hline\n  &Model&CIFAR-10\\\\\n  \\hline\n  &Maxout \\cite{goodfellow2013maxout}&9.38\\\\\n  human&ResNet(depth=110) \\cite{he2016deep}&6.61\\\\\n  designed&ResNet(pre-activation)\\cite{he2016identity}&4.62\\\\\n  &DenseNet(L=40,k=12)\\cite{huang2017densely}&5.24\\\\\n\n  \\hline\n  &GeNet\\#2(G-50) \\cite{Lingxi}&7.10\\\\\n  auto&Large-Scale Evolution\\cite{Esteban}&5.40\\\\\n  designed&NAS (depth=39)\\cite{Lee}&6.05\\\\\n  &MetaQNN\\cite{Baker}&6.92\\\\\n  &EAS\\cite{cai2018efficient}&4.23\\\\\n  \\hline\n  &{Our Method(depth=49)}& {4.9}\\\\\n  \\hline\n  \\end{tabular}\n  }\n  \\caption{Comparison of the recognition error rate (\\%) on CIFAR-10.}\n\n\n\\end{table}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3in]{7.png} \n\\caption{One hundred models test accuracy on MNIST. Most of models have gained similar performances.}\n\\label{f7}\n\\end{figure}\n\n\\subsection{Results on the CIFAR-10 Data Set}\nThe CIFAR-10 dataset consists of 60000 32x32 colour images in 10 classes, with 6000 images per class. There are 50,000 training images and 10,000 test images. The training data set are divided into two parts, new training data set(55,000 images) and validation data set(5,000 images).\n\n\\begin{table}[h]\n  \\centering\n  \\setlength{\\tabcolsep}{0.8mm}\n    \\scalebox{0.9}[0.9]{\n  \\begin{tabular} {p{5.5cm}|p{2cm}}\n  \\hline\n  Model&CIFAR-10\\\\\n  \\hline\n  DNA computing algorithm (depth=13)&7.34\\\\\n  \\hline\n  DNA computing algorithm (depth=17)&6.7\\\\\n  \\hline\n  DNA computing algorithm (depth=21)&6.1\\\\\n  \\hline\n  DNA computing algorithm (depth=25)&5.65\\\\\n  \\hline\n  DNA computing algorithm (depth=49)&4.9\\\\\n  \\hline\n  \\end{tabular}\n  }\n  \\caption{Comparison of our models' recognition error rate (\\%) with different depth of models on CIFAR-10 data set.}\n\\end{table}\nOur best model gain 95.10\\% test accuracy with 49 layers architectures composed of convolution layers, pooling layers and a fully connected layer. Note that we use data augmentation (flip, crop, and data normalization). With few fully trained models, we still gain high test accuracy. It proves that learning models from carefully designed search space via DNA computing algorithm can gain high accuracy. It is possible for non-experts without expertise to get a high accuracy models in specific task. One of our high performance models are showed in Figure \\ref{f3}.\n\n\\section{Conclusion}\nWe propose DNA computing algorithm to learn neural networks from a well defined architecture search space. Our search space is defined by architectures with skip-connections. During training models, we use early stop strategy which saves time and computation sources. We find most models perform similarly in our search space and have similar learning curves. We prove that learning neural networks via DNA computing algorithm is feasible and gain high accuracy. And we find that local minimal is not of importance during training models and using early stop strategy can eliminate models just after several epochs of training. We conduct the algorithm in two data sets (CIFAR-10 and MNIST) and we get competitive results in comparison with evolutionary algorithm and Reinforcement Learning but training fewer models.\nWe simulate DNA computing algorithm by computer. In future work, We consider doing biochemical experiments to verify the feasibility of the method. \n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThis document serves as an example submission. It illustrates the format\nwe expect authors to follow when submitting a paper to ECCV. \nAt the same time, it gives details on various aspects of paper submission,\nincluding preservation of anonymity and how to deal with dual submissions,\nso we advise authors to read this document carefully.\n\n\\section{Initial Submission}\n\n\\subsection{Language}\n\nAll manuscripts must be in English.\n\n\\subsection{Paper length}\nPapers submitted for review should be complete. \nThe length should match that intended for final publication. \nPapers accepted for the conference will be allocated 14 pages (plus additional pages for references) in the proceedings. \nNote that the allocated 14 pages do not include the references. 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This makes reviewing more efficient, because reviewers can refer to a line on a page. Line numbering is removed in the camera-ready.\n\n\n\\subsection{Mathematics}\n\nPlease number all of your sections and displayed equations.  Again,\nthis makes reviewing more efficient, because reviewers can refer to a\nline on a page.  Also, it is important for readers to be able to refer\nto any particular equation.  Just because you didn't refer to it in\nthe text doesn't mean some future reader might not need to refer to\nit.  It is cumbersome to have to use circumlocutions like ``the\nequation second from the top of page 3 column 1''.  (Note that the\nline numbering will not be present in the final copy, so is not an\nalternative to equation numbers).  Some authors might benefit from\nreading Mermin's description of how to write mathematics:\n\\url{www.pamitc.org\/documents\/mermin.pdf}.\n\\section{Policies}\nTo avoid confusion, in case of discrepancies between policies mentioned here and those in the ECCV 2022 webpage, the web page is the one that is updated regularly and its policies shall overrule those appearing here. \n\n\\subsection{Review Process}\nBy submitting a paper to ECCV, the authors agree to the review process and understand that papers are processed by the Toronto system to match each manuscript to the best possible chairs and reviewers.\n\\subsection{Confidentiality}\nThe review process of ECCV is confidential. Reviewers are volunteers not part of the ECCV organisation and their efforts are greatly appreciated. The standard practice of keeping all information confidential during the review is part of the standard communication to all reviewers. Misuse of confidential information is a severe professional failure and  appropriate measures will be taken when brought to the attention of ECCV organizers. It should be noted, however, that the organisation of ECCV is not and cannot be held responsible for the consequences when reviewers break confidentiality.\n\nAccepted papers will be published by Springer (with appropriate copyrights) electronically up to three weeks prior to the main conference. Please make sure to discuss this issue with your legal advisors as it pertains to public disclosure of the contents of the papers submitted.\n\\subsection{Dual and Double Submissions}\nBy submitting a manuscript to ECCV 2022, authors acknowledge that it has not been previously published or accepted for publication in substantially similar form in any peer-reviewed venue including journal, conference, or workshop. Furthermore, no paper substantially similar in content has been or will be submitted to a journal, another conference or workshop during the review period (March 07, 2022 \u2013 July 3, 2022). The authors also attest that they did not submit substantially similar submissions to ECCV 2022. Violation of any of these conditions will lead to rejection and the violation will be reported to the other venue or journal, which will typically lead to rejection there as well. \n\nThe goals of the dual submission policy are (i) to have exciting new work be published for the first time at ECCV 2022, and (ii) to avoid duplicating the efforts of the reviewers.\nTherefore, all papers under review are checked for dual submissions and this is not allowed, independent of the page size of submissions. \n\nFor already published papers, our policy is based upon the following particular definition of ``publication''. A publication, for the purposes of the dual submission policy, is defined to be a written work longer than four pages that was submitted for review by peers for either acceptance or rejection, and, after review, was accepted. In particular, this definition of publication does not depend upon whether such an accepted written work appears in a formal proceedings or whether the organizers declare that such work ``counts as a publication''. \n\nAn arXiv.org paper does not count as a publication because it was not peer-reviewed for acceptance. The same is true for university technical reports. However, this definition of publication does include peer-reviewed workshop papers, even if they do not appear in a proceedings, if their length is more than 4 pages including citations. Given this definition, any submission to ECCV 2022 should not have substantial overlap with prior publications or other concurrent submissions. As a rule of thumb, the ECCV 2022 submission should contain no more than 20 percent of material from previous publications. \n\n\\subsection{Requirements for publication}\nPublication of the paper in the ECCV 2022 proceedings of Springer requires that at least one of the authors registers for the conference and present the paper there. It also requires that a camera-ready version that satisfies all formatting requirements is submitted before the camera-ready deadline. \n\\subsection{Double blind review}\n\\label{sec:blind}\nECCV reviewing is double blind, in that authors do not know the names of the area chair\/reviewers of their papers, and the area chairs\/reviewers cannot, beyond reasonable doubt, infer the names of the authors from the submission and the additional material. Avoid providing links to websites that identify the authors. Violation of any of these guidelines may lead to rejection without review. If you need to cite a different paper of yours that is being submitted concurrently to ECCV, the authors should (1) cite these papers, (2) argue in the body of your paper why your ECCV paper is non trivially different from these concurrent submissions, and (3) include anonymized versions of those papers in the supplemental material.\n\nMany authors misunderstand the concept of anonymizing for blind\nreview. Blind review does not mean that one must remove\ncitations to one's own work. In fact it is often impossible to\nreview a paper unless the previous citations are known and\navailable.\n\nBlind review means that you do not use the words ``my'' or ``our''\nwhen citing previous work.  That is all.  (But see below for\ntechnical reports).\n\nSaying ``this builds on the work of Lucy Smith [1]'' does not say\nthat you are Lucy Smith, it says that you are building on her\nwork.  If you are Smith and Jones, do not say ``as we show in\n[7]'', say ``as Smith and Jones show in [7]'' and at the end of the\npaper, include reference 7 as you would any other cited work.\n\nAn example of a bad paper:\n\\begin{quote}\n\\begin{center}\n    An analysis of the frobnicatable foo filter.\n\\end{center}\n\n   In this paper we present a performance analysis of our\n   previous paper [1], and show it to be inferior to all\n   previously known methods.  Why the previous paper was\n   accepted without this analysis is beyond me.\n\n   [1] Removed for blind review\n\\end{quote}\n\n\nAn example of an excellent paper:\n\n\\begin{quote}\n\\begin{center}\n     An analysis of the frobnicatable foo filter.\n\\end{center}\n\n   In this paper we present a performance analysis of the\n   paper of Smith [1], and show it to be inferior to\n   all previously known methods.  Why the previous paper\n   was accepted without this analysis is beyond me.\n\n   [1] Smith, L. and Jones, C. ``The frobnicatable foo\n   filter, a fundamental contribution to human knowledge''.\n   Nature 381(12), 1-213.\n\\end{quote}\n\nIf you are making a submission to another conference at the same\ntime, which covers similar or overlapping material, you may need\nto refer to that submission in order to explain the differences,\njust as you would if you had previously published related work. In\nsuch cases, include the anonymized parallel\nsubmission~\\cite{Authors14} as additional material and cite it as\n\\begin{quote}\n1. Authors. ``The frobnicatable foo filter'', BMVC 2014 Submission\nID 324, Supplied as additional material {\\tt bmvc14.pdf}.\n\\end{quote}\n\nFinally, you may feel you need to tell the reader that more\ndetails can be found elsewhere, and refer them to a technical\nreport.  For conference submissions, the paper must stand on its\nown, and not {\\em require} the reviewer to go to a techreport for\nfurther details.  Thus, you may say in the body of the paper\n``further details may be found in~\\cite{Authors14b}''.  Then\nsubmit the techreport as additional material. Again, you may not\nassume the reviewers will read this material.\n\nSometimes your paper is about a problem which you tested using a tool which\nis widely known to be restricted to a single institution.  For example,\nlet's say it's 1969, you have solved a key problem on the Apollo lander,\nand you believe that the ECCV audience would like to hear about your\nsolution.  The work is a development of your celebrated 1968 paper entitled\n``Zero-g frobnication: How being the only people in the world with access to\nthe Apollo lander source code makes us a wow at parties'', by Zeus.\n\nYou can handle this paper like any other.  Don't write ``We show how to\nimprove our previous work [Anonymous, 1968].  This time we tested the\nalgorithm on a lunar lander [name of lander removed for blind review]''.\nThat would be silly, and would immediately identify the authors. Instead\nwrite the following:\n\\begin{quotation}\n\\noindent\n   We describe a system for zero-g frobnication.  This\n   system is new because it handles the following cases:\n   A, B.  Previous systems [Zeus et al. 1968] didn't\n   handle case B properly.  Ours handles it by including\n   a foo term in the bar integral.\n\n   ...\n\n   The proposed system was integrated with the Apollo\n   lunar lander, and went all the way to the moon, don't\n   you know.  It displayed the following behaviours\n   which show how well we solved cases A and B: ...\n\\end{quotation}\nAs you can see, the above text follows standard scientific convention,\nreads better than the first version, and does not explicitly name you as\nthe authors.  A reviewer might think it likely that the new paper was\nwritten by Zeus, but cannot make any decision based on that guess.\nHe or she would have to be sure that no other authors could have been\ncontracted to solve problem B. \\\\\n\nFor sake of anonymity, it's recommended to omit acknowledgements\nin your review copy. They can be added later when you prepare the final copy.\n\n\\section{Manuscript Preparation}\n\nThis is an edited version of Springer LNCS instructions adapted\nfor ECCV 2022 first paper submission.\nYou are strongly encouraged to use \\LaTeX2$_\\varepsilon$ for the\npreparation of your\ncamera-ready manuscript together with the corresponding Springer\nclass file \\verb+llncs.cls+.\n\nWe would like to stress that the class\/style files and the template\nshould not be manipulated and that the guidelines regarding font sizes\nand format should be adhered to. This is to ensure that the end product\nis as homogeneous as possible.\n\n\\subsection{Printing Area}\nThe printing area is $122  \\; \\mbox{mm} \\times 193 \\;\n\\mbox{mm}$.\nThe text should be justified to occupy the full line width,\nso that the right margin is not ragged, with words hyphenated as\nappropriate. Please fill pages so that the length of the text\nis no less than 180~mm.\n\n\\subsection{Layout, Typeface, Font Sizes, and Numbering}\nUse 10-point type for the name(s) of the author(s) and 9-point type for\nthe address(es) and the abstract. For the main text, please use 10-point\ntype and single-line spacing.\nWe recommend using Computer Modern Roman (CM) fonts, which is the default font in this template.\nItalic type may be used to emphasize words in running text. Bold\ntype and underlining should be avoided.\nWith these sizes, the interline distance should be set so that some 45\nlines occur on a full-text page.\n\n\\subsubsection{Headings.}\n\nHeadings should be capitalized\n(i.e., nouns, verbs, and all other words\nexcept articles, prepositions, and conjunctions should be set with an\ninitial capital) and should,\nwith the exception of the title, be aligned to the left.\nWords joined by a hyphen are subject to a special rule. If the first\nword can stand alone, the second word should be capitalized.\nThe font sizes\nare given in Table~\\ref{table:headings}.\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}\n\\begin{center}\n\\caption{Font sizes of headings. Table captions should always be\npositioned {\\it above} the tables. The final sentence of a table\ncaption should end without a full stop}\n\\label{table:headings}\n\\begin{tabular}{lll}\n\\hline\\noalign{\\smallskip}\nHeading level & Example & Font size and style\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nTitle (centered)  & {\\Large \\bf Lecture Notes \\dots} & 14 point, bold\\\\\n1st-level heading & {\\large \\bf 1 Introduction} & 12 point, bold\\\\\n2nd-level heading & {\\bf 2.1 Printing Area} & 10 point, bold\\\\\n3rd-level heading & {\\bf Headings.} Text follows \\dots & 10 point, bold\n\\\\\n4th-level heading & {\\it Remark.} Text follows \\dots & 10 point,\nitalic\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\setlength{\\tabcolsep}{1.4pt}\n\nHere are some examples of headings: ``Criteria to Disprove Context-Freeness of\nCollage Languages'', ``On Correcting the Intrusion of Tracing\nNon-deterministic Programs by Software'', ``A User-Friendly and\nExtendable Data Distribution System'', ``Multi-flip Networks:\nParallelizing GenSAT'', ``Self-determinations of Man''.\n\n\\subsubsection{Lemmas, Propositions, and Theorems.}\n\nThe numbers accorded to lemmas, propositions, and theorems etc. should\nappear in consecutive order, starting with the number 1, and not, for\nexample, with the number 11.\n\n\\subsection{Figures and Photographs}\n\\label{sect:figures}\n\nPlease produce your figures electronically and integrate\nthem into your text file. For \\LaTeX\\ users we recommend using package\n\\verb+graphicx+ or the style files \\verb+psfig+ or \\verb+epsf+.\n\nCheck that in line drawings, lines are not\ninterrupted and have constant width. Grids and details within the\nfigures must be clearly readable and may not be written one on top of\nthe other. Line drawings should have a resolution of at least 800 dpi\n(preferably 1200 dpi).\nFor digital halftones 300 dpi is usually sufficient.\nThe lettering in figures should have a height of 2~mm (10-point type).\nFigures should be scaled up or down accordingly.\nPlease do not use any absolute coordinates in figures.\n\nFigures should be numbered and should have a caption which should\nalways be positioned {\\it under} the figures, in contrast to the caption\nbelonging to a table, which should always appear {\\it above} the table.\nPlease center the captions between the margins and set them in\n9-point type\n(Fig.~\\ref{fig:example} shows an example).\nThe distance between text and figure should be about 8~mm, the\ndistance between figure and caption about 5~mm.\n\\begin{figure}\n\\centering\n\\includegraphics[height=6.5cm]{eijkel2}\n\\caption{One kernel at $x_s$ ({\\it dotted kernel}) or two kernels at\n$x_i$ and $x_j$ ({\\it left and right}) lead to the same summed estimate\nat $x_s$. This shows a figure consisting of different types of\nlines. Elements of the figure described in the caption should be set in\nitalics,\nin parentheses, as shown in this sample caption. The last\nsentence of a figure caption should generally end without a full stop}\n\\label{fig:example}\n\\end{figure}\n\nIf possible (e.g. if you use \\LaTeX) please define figures as floating\nobjects. \\LaTeX\\ users, please avoid using the location\nparameter ``h'' for ``here''. If you have to insert a pagebreak before a\nfigure, please ensure that the previous page is completely filled.\n\n\n\\subsection{Formulas}\n\nDisplayed equations or formulas are centered and set on a separate\nline (with an extra line or halfline space above and below). Displayed\nexpressions should be numbered for reference. The numbers should be\nconsecutive within the contribution,\nwith numbers enclosed in parentheses and set on the right margin.\nFor example,\n\\begin{align}\n  \\psi (u) & = \\int_{0}^{T} \\left[\\frac{1}{2}\n  \\left(\\Lambda_{0}^{-1} u,u\\right) + N^{\\ast} (-u)\\right] dt \\; \\\\\n& = 0 ?\n\\end{align}\n\nPlease punctuate a displayed equation in the same way as ordinary\ntext but with a small space before the end punctuation.\n\n\\subsection{Footnotes}\n\nThe superscript numeral used to refer to a footnote appears in the text\neither directly after the word to be discussed or, in relation to a\nphrase or a sentence, following the punctuation sign (comma,\nsemicolon, or full stop). Footnotes should appear at the bottom of\nthe\nnormal text area, with a line of about 2~cm in \\TeX\\ and about 5~cm in\nWord set\nimmediately above them.\\footnote{The footnote numeral is set flush left\nand the text follows with the usual word spacing. Second and subsequent\nlines are indented. Footnotes should end with a full stop.}\n\n\n\\subsection{Program Code}\n\nProgram listings or program commands in the text are normally set in\ntypewriter font, e.g., CMTT10 or Courier.\n\n\\noindent\n{\\it Example of a Computer Program}\n\\begin{verbatim}\nprogram Inflation (Output)\n  {Assuming annual inflation rates of \n   years};\n   const\n     MaxYears = 10;\n   var\n     Year: 0..MaxYears;\n     Factor1, Factor2, Factor3: Real;\n   begin\n     Year := 0;\n     Factor1 := 1.0; Factor2 := 1.0; Factor3 := 1.0;\n     WriteLn('Year  \n     repeat\n       Year := Year + 1;\n       Factor1 := Factor1 * 1.07;\n       Factor2 := Factor2 * 1.08;\n       Factor3 := Factor3 * 1.10;\n       WriteLn(Year:5,Factor1:7:3,Factor2:7:3,Factor3:7:3)\n     until Year = MaxYears\nend.\n\\end{verbatim}\n\\noindent\n{\\small (Example from Jensen K., Wirth N. (1991) Pascal user manual and\nreport. Springer, New York)}\n\n\n\n\\subsection{Citations}\n\nThe list of references is headed ``References\" and is not assigned a\nnumber\nin the decimal system of headings. The list should be set in small print\nand placed at the end of your contribution, in front of the appendix,\nif one exists.\nPlease do not insert a pagebreak before the list of references if the\npage is not completely filled.\nAn example is given at the\nend of this information sheet. For citations in the text please use\nsquare brackets and consecutive numbers: \\cite{Alpher02},\n\\cite{Alpher03}, \\cite{Alpher04} \\dots\n\n\\section{Submitting a Camera-Ready for an Accepted Paper}\n\\subsection{Converting Initial Submission to Camera-Ready}\nTo convert a submission file into a camera-ready for an accepted paper:\n\\begin{enumerate}\n    \\item  First comment out \\begin{verbatim}\n        \\usepackage{ruler}\n    \\end{verbatim} and the line that follows it.\n    \\item  The anonymous title part should be removed or commented out, and a proper author block should be inserted, for which a skeleton is provided in a commented-out version. These are marked in the source file as \\begin{verbatim}\n       \n    \\end{verbatim} and \\begin{verbatim}\n       \n    \\end{verbatim}\n    \\item Please write out author names in full in the paper, i.e. full given and family names. If any authors have names that can be parsed into FirstName LastName in multiple ways, please include the correct parsing in a comment to the editors, below the \\begin{verbatim}\\author{}\\end{verbatim} field.\n    \\item Make sure you have inserted the proper Acknowledgments.\n  \\end{enumerate}  \n \n\\subsection{Preparing the Submission Package}\nWe need all the source files (LaTeX files, style files, special fonts, figures, bib-files) that are required to compile papers, as well as the camera ready PDF. For each paper, one ZIP-file called XXXX.ZIP (where XXXX is the zero-padded, four-digit paper ID) has to be prepared and submitted via the ECCV 2022 Submission Website, using the password you received with your initial registration on that site. The size of the ZIP-file may not exceed the limit of 60 MByte. The ZIP-file has to contain the following:\n  \\begin{enumerate}\n \\item  All source files, e.g. LaTeX2e files for the text, PS\/EPS or PDF\/JPG files for all figures.\n \\item PDF file named ``XXXX.pdf\" that has been produced by the submitted source, where XXXX is the four-digit paper ID (zero-padded if necessary). For example, if your paper ID is 24, the filename must be 0024.pdf. This PDF will be used as a reference and has to exactly match the output of the compilation.\n \\item PDF file named ``XXXX-copyright.PDF\": a scanned version of the signed copyright form (see ECCV 2022 Website, Camera Ready Guidelines for the correct form to use). \n \\item If you wish to provide supplementary material, the file name must be in the form XXXX-supp.pdf or XXXX-supp.zip, where XXXX is the zero-padded, four-digit paper ID as used in the previous step. Upload your supplemental file on the ``File Upload\" page as a single PDF or ZIP file of 100 MB in size or less. Only PDF and ZIP files are allowed for supplementary material. You can put anything in this file \u2013 movies, code, additional results, accompanying technical reports\u2013anything that may make your paper more useful to readers.  If your supplementary material includes video or image data, you are advised to use common codecs and file formats.  This will make the material viewable by the largest number of readers (a desirable outcome). ECCV encourages authors to submit videos using an MP4 codec such as DivX contained in an AVI. Also, please submit a README text file with each video specifying the exact codec used and a URL where the codec can be downloaded. Authors should refer to the contents of the supplementary material appropriately in the paper.\n \\end{enumerate}\n\nCheck that the upload of your file (or files) was successful either by matching the file length to that on your computer, or by using the download options that will appear after you have uploaded. Please ensure that you upload the correct camera-ready PDF\u2013renamed to XXXX.pdf as described in the previous step as your camera-ready submission. Every year there is at least one author who accidentally submits the wrong PDF as their camera-ready submission.\n\nFurther considerations for preparing the camera-ready package:\n  \\begin{enumerate}\n    \\item Make sure to include any further style files and fonts you may have used.\n    \\item References are to be supplied as BBL files to avoid omission of data while conversion from BIB to BBL.\n    \\item Please do not send any older versions of papers. There should be one set of source files and one XXXX.pdf file per paper. Our typesetters require the author-created pdfs in order to check the proper representation of symbols, figures, etc.\n    \\item  Please remove unnecessary files (such as eijkel2.pdf and eijkel2.eps) from the source folder. \n    \\item  You may use sub-directories.\n    \\item  Make sure to use relative paths for referencing files.\n    \\item  Make sure the source you submit compiles.\n\\end{enumerate}\n\nSpringer is the first publisher to implement the ORCID identifier for proceedings, ultimately providing authors with a digital identifier that distinguishes them from every other researcher. ORCID (Open Researcher and Contributor ID) hosts a registry of unique researcher identifiers and a transparent method of linking research activities to these identifiers. This is achieved through embedding ORCID identifiers in key workflows, such as research profile maintenance, manuscript submissions, grant applications and patent applications.\n\\subsection{Most Frequently Encountered Issues}\nPlease kindly use the checklist below to deal with some of the most frequently encountered issues in ECCV submissions.\n\n{\\bf FILES:}\n\\begin{itemize}\n    \\item My submission package contains ONE compiled pdf file for the camera-ready version to go on Springerlink.\n\\item I have ensured that the submission package has all the additional files necessary for compiling the pdf on a standard LaTeX distribution.\n\\item I have used the correct copyright form (with editor names pre-printed), and a signed pdf is included in the zip file with the correct file name.\n\\end{itemize}\n\n{\\bf CONTENT:}\n\\begin{itemize}\n\\item I have removed all \\verb| \\vspace| and \\verb|\\hspace|  commands from my paper.\n\\item I have not used \\verb|\\thanks|  or \\verb|\\footnote|  commands and symbols for corresponding authors in the title (which is processed with scripts) and (optionally) used an Acknowledgement section for all the acknowledgments, at the end of the paper.\n\\item I have not used \\verb|\\cite| command in the abstract.\n\\item I have read the Springer author guidelines, and complied with them, including the point on providing full information on editors and publishers for each reference in the paper (Author Guidelines \u2013 Section 2.8).\n\\item I have entered a correct \\verb|\\titlerunning{}| command and selected a meaningful short name for the paper.\n\\item I have entered \\verb|\\index{Lastname,Firstname}| commands for names that are longer than two words.\n\\item I have used the same name spelling in all my papers accepted to ECCV and ECCV Workshops.\n\\item I have inserted the ORCID identifiers of the authors in the paper header (see http:\/\/bit.ly\/2H5xBpN for more information).\n\\item I have not decreased the font size of any part of the paper (except tables) to fit into 14 pages, I understand Springer editors will remove such commands.\n\\end{itemize}\n{\\bf SUBMISSION:}\n\\begin{itemize}\n\\item All author names, titles, and contact author information are correctly entered in the submission site.\n\\item The corresponding author e-mail is given.\n\\item At least one author has registered by the camera ready deadline.\n\\end{itemize}\n\n\n\\section{Conclusions}\n\nThe paper ends with a conclusion. \n\n\n\\clearpage\\mbox{}Page \\thepage\\ of the manuscript.\n\\clearpage\\mbox{}Page \\thepage\\ of the manuscript.\n\nThis is the last page of the manuscript.\n\\par\\vfill\\par\nNow we have reached the maximum size of the ECCV 2022 submission (excluding references).\nReferences should start immediately after the main text, but can continue on p.15 if needed.\n\n\\clearpage\n\\bibliographystyle{splncs04}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, we addressed the problem of long-horizon exploration and planning by introducing a novel Long-HOT benchmark. Further, we proposed a modular hierarchical transport policy (HTP) that builds a topological graph of the scene to perform exploration with the help of weighted frontiers and simplify navigation in long-horizons through a combination of motion planning and RL policy robust to imperfect hand-offs.\nOur sub-task policies are connected in novel ways with different levels of hierarchical control requiring different state representations to perform object transport. We show how our approach leads to large improvements in performance on the transport task, it's ability to generalize to harder long-horizon task settings while only training on simpler versions, and also achieve state-of-the-art numbers on MultiON. \n\n\n\\section{Experiments}\n\\input{figures\/results}\n\nWe evaluate all approaches on two tasks set in photo-realistic Matterport3D scenes (MP3d)\\cite{Matterport3D} in Habitat~\\cite{habitat_iccv2019}: Long-HOT transport, and Multi-ON~\\cite{wani2020multion} object navigation. \n\n\n\\vspace{0.4cm}\n\\noindent \\textbf{Long-HOT:}\nWe split MP3D\\cite{Matterport3D} scenes into disjoint train, validation, and test scenes~\\cite{anderson2018evaluation} each with 61\/14\/15 scenes respectively. We generate 10,000 training task configurations among the training scenes, and 3000 each of validation and test configurations. \nEach task configuration consists of a specific configuration of objects, container, goal location, and agent starting location and pose. \nFirst, we sample a goal $(x,y)$ location in the map, then sample the four object locations. These object locations (1) lie in a specified range of distances from the goal (``goal-range''), (2) at a specified minimum distance  (``obj-dist-min'') away from other objects, and at (3) within a specified maximum distance (``obj-dist-max'') from at least one other object. Next, container and agent starting locations are also sampled to lie within the same goal-range as objects. All distances are geodesic. These settings permit modulating complexity: for example, large goal distances lead to harder tasks that are more exploration-intensive and need a longer task horizon. \n\n\nTable \\ref{tab:dataset} shows settings for different task levels used in our experiments. We train all methods on \\emph{default}-level tasks on 61 scenes.\nAfter training, we first evaluate them on 15 disjoint test scenes in the \\textit{default} setting and call this as ``Standard Long-HOT Task''.\nWe then perform a more focused ``Large Long-HOT Task'' evaluation on \\textit{large} scenes that have at least one dimension $>40m$ and sample \\emph{default}, \\emph{hard} and \\emph{harder} level tasks from Table \\ref{tab:dataset}.\n\n\n\\input{tables\/dataset_table}\n\n\n\n\n\n\n\\vspace{0.4cm}\n\\noindent \\textbf{MultiOn Dataset:}\nMultiOn\\cite{wani2020multion} is a sequential multi-object navigation task where the agent is required to visit objects in a predefined sequence.\nWe adapt our proposed transport policy to test it on the challenging MultiOn (3-On) task using the object goal vector as input to the  high-level controller (more details in supplementary). \n\n\\input{tables\/results}\n\n\\vspace{0.4cm}\n\\noindent \\textbf{Baselines:} We compare our proposed transport policy with several baseline methods and ablations. \n\\begin{itemize}[leftmargin=*]\n\\item \\textbf{NoMap:} This baseline policy, trained using PPO\\cite{ppo}, maps RGBD image $I$, hand state $O_h$ and goal state $O_g$ directly to low-level robot actions.\n\\item \\textbf{OracleMap:} This method improves NoMap by assuming additional access to a ground truth \n2D occupancy map of the $10m\\times 10m$ area centered on the agent in the overhead view. Similar to \\cite{wani2020multion}, we evaluate two versions of OracleMap: with occupancy alone (``Occ''), and with extra annotated true locations of the task-relevant objects and container (``Occ+Obj''). \n\\item \\textbf{OracleMap-Waypoints:} This baseline represents the a popular hierarchical approach in embodied navigation~\\cite{krantz2021waypoint,xia2021relmogen}: setting navigation waypoints for a motion planner, such as A\\text{*}~\\cite{astar}. It trains an RL policy to select discretized $(x, y)$ waypoints on the map. We provide access to OracleMap (Occ+Obj) for this baseline. See supplementary for details.\n\n\\item \\textbf{MultiOn Baselines:} For Long-HOT object transport, we adapt the ProjNeuralMap baseline from \\cite{wani2020multion} that projects perspective features in top view to our task. To this, we add additional hand-state and goal state embeddings instead of object goal embeddings as in MultiOn.\nFor MultiOn, we compare against the authors' baselines~\\cite{wani2020multion}, as well as the best-performing methods from the public leaderboard.\n\n\\item \\textbf{Exploration Ablations:} We study three variants of our method with different exploration strategies: NearestFrontier, CNN, GCN. HTP-NearestFrontier uses vanilla frontier exploration\\cite{frontier} and picks the closest frontier from the agent location as the next exploration subgoal. HTP-CNN and HTP-GCN use our proposed CNN and GCN-based exploration scores for weighted frontier exploration, explained in Sec~\\ref{sec:score_pred}.\n\n\n\\end{itemize}\n\n\n\n\\subsection{Metrics}\nWe use standard evaluation metrics following previous works \\cite{Weihs_2021_CVPR,wani2020multion,gupta2019cognitive,jain2020cordial,chen2020soundspaces,anderson2018evaluation,habitat_iccv2019} and adapt a few other metrics relevant to our task setting. \n\\vspace{-0.3cm}\n\\paragraph{\\bf \\%Success:} It measures the percentage of successful episodes across the test set. An episode is successful if the agent moves all $K$ objects to the goal location. \n\n\\vspace{-0.3cm}\n\\paragraph{\\bf \\%Progress:} It measures the percentage of target objects successfully transported to the goal location.\n\n\\vspace{-0.3cm}\n\\paragraph{\\bf SPL \\& PPL:} SPL is Success weighted-by Path Length, and PPL is Progress weighted by Path Length. Since there multiple ways in which one can complete this task we substitute optimal path length in SPL and PPL calculations with a reference path length $G_{ref}$ (details in supplemetary). Any execution with path length $G_{pl} \\leq G_{ref}$ weights the success and progress values by $1.0$. Hence  $\\texttt{SPL} = 1_{\\texttt{success}}\\times \\min(G_{pl}\/G_{ref},1.0)$ and $\\texttt{PPL} = \\texttt{Progress}\\times \\min(G_{pl}\/G_{ref},1.0)$. \n\n\\vspace{-0.3cm}\n\\paragraph{\\bf Episode Energy:} We adapt a similar metric from \\cite{Weihs_2021_CVPR} to our task setting. \nIt measures the amount of remaining energy to complete the episode and gives partial credit if the agent successfully moves the object closer to goal. It is defined as $E = \\sum_{k=1}^K d_{g2t^k}\/ \\sum_{k=1}^K D_{g2t^k}$ where numerator and denominator represent sum of geodesic distance of target objects to goal location at the ending and starting of the episode respectively. \n\n\\vspace{-0.3cm}\n\\paragraph{\\bf \\% Picked:} This metric measures the percentage of target objects that are successfully picked\n\\input{tables\/largeLHOT_multion}\n\n\\subsection{Results} \\label{ssec:quant_compare}\n\n\n\n\\paragraph{Standard Long-HOT Task:}  \n\nTable \\ref{tab:standard} shows the results of evaluations on Standard Long-HOT task for 1000 test episodes generated using the \\emph{default} task level.\nAll variants of HTP clearly outperform NoMap and ProjNeuralMap on all six  metrics. \nFig. \\ref{fig:example_results} visualizes an episode of HTP-CNN. We show video results of our work in supplementary.\n\n\n\n\\vspace{0.2cm}\n\\noindent\\emph{Are hierarchies good?} HTP-NearestFrontier already outperforms the non-hierarchical flat baselines by a large margin, showing the importance of our modular hierarchical approach involving separate policies for different task phases, coupled with a topological map. Interestingly, not all hierarchies are good: in particular, OracleMap-Waypoints, which sets waypoint subgoals for a motion planner, performs clearly worse than flat OracleMap (``Occ+Obj''). Note that OracleMap methods have access to ground truth map information and are not directly comparable with HTP, but can be meaningfully compared among themselves. \n\n\\vspace{0.2cm}\n\\noindent\\emph{Does weighted frontier exploration work?} Among HTP variants, both HTP-GCN and HTP-CNN, which use predicted scores for weighted frontier exploration, clearly outperform HTP-NearestFrontier. Between them, GCN and CNN are roughly equivalent in this setting.\n\n\\vspace{0.2cm}\n\\noindent\\emph{How important is good agent-centered occupancy and object location information?} On this test set, access to the ground truth occupancy and object maps centered around the agent significantly improves performance, with OracleMap (Occ+Obj) performing the best out of all methods.\n\n\n\n\n\n\n\\vspace{-0.2cm}\n\\paragraph{Large Long-HOT Task:} We now evaluate these same trained policies on more challenging settings in large scenes, with more difficult transport task levels. This evaluates generalization and highlights the benefits of effective hierarchy and modularity. Note that scenes used for testing in Large Long-HOT have some overlap with training scenes but not with the same episodes. This is due to limited large scenes available in the disjoint test set. But our observation of better generalization stands since all methods have the same advantage, but others suffer severe drops compared to ours.\n\nTable \\ref{tab:deep_transport} (left) shows the results. Flat end-to-end approaches like NoMap and OracleMap deteriorate catastrophically on \\textit{hard} and \\textit{harder} level tasks. \nNoMap's \\%Success drops from 39\\% on \\emph{default} to\na mere 5.2\\% and 2.8\\% on \\emph{hard} and \\emph{harder} levels. \nHTP  methods degrade more gracefully, achieving up to 33\\% and 22\\% on \\textit{hard} and \\textit{harder}. In fact, as task difficulty increases, HTP significantly closes the performance gap to OracleMap-Waypoints despite the oracle  method's access to ground truth map information. We believe this is because our modular approach with weighted frontier exploration leads to better generalization compared to waypoint setting as in OracleMap-Waypoints. Further, OracleMap-Waypoints performs better than OracleMap (Occ+Obj) confirming the benefits of subgoals in long-horizon settings. \nTo further study the dependence of HTP's model components on the task performance and its generalization to increasing number of object goals we conduct several ablations, results for which are available in the supplementary. \n\n\n\nOverall, this large effect of small increments in the spatial task scale at \\textit{hard} and \\textit{harder} levels  (Tab~\\ref{tab:dataset}) shows how Long-HOT stress-tests planning, exploration and reasoning over long spatial and temporal horizons. This is different from prior efforts~\\cite{wani2020multion} that extend a task by adding new objects, but pre-specify a sequence of single-object sub-tasks.\nOur HTP approach, which leverages hierarchical policies and topological maps, is a first step towards addressing these unique challenges induced by Long-HOT. \nNote that, difficulty of Long-HOT is expected to be a lot higher with the end criterion of MultiON, analysis for which will be in the supplementary.\n\n\n\n\n\n\n\n\n\n\n\\vspace{-0.2cm}\n\\paragraph{Results on MultiOn:} Finally, we also evaluate our proposed HTP framework on the MultiOn\\cite{wani2020multion} challenge. Table \\ref{tab:deep_transport} (right) shows that our method significantly outperforms other baselines from \\cite{wani2020multion}. Moreover, its performance is nearly on par or better with the CVPR 2021 Embodied AI workshop challenge winner \\cite{marza2021teaching}. Note that the techniques proposed in \\cite{marza2021teaching} are complementary to ours. \n\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\\input{figures\/teaser}\n\nA robot tasked with finding an object in a large environment or executing a complex maneuver must reason over a long horizon. Existing end-to-end reinforcement learning (RL) approaches often suffer in long-horizon embodied navigation tasks due to a combination of challenges: (a) inability to provide exploration guarantees when the point or object of interest is not visible, (b) difficulty in backtracking previously seen locations and (c) difficulty in planning over long horizons. In this work, we address these issues by proposing a novel long-horizon embodied transport task, as well as modular hierarchical methods for embodied transport and navigation.\n\nOur proposed long-horizon object transport task, Long-HOT, is designed to study modular approaches in the Habitat environment \\cite{habitat_iccv2019}. It requires an embodied agent to pick up objects placed at unknown locations in a large environment and drop them at a known goal location, while satisfying load constraints, which may be relaxed by picking up a special container object (Fig.~\\ref{fig:teaser}). While tasks like MultiON for sequential navigation also benefit from long-range planning \\cite{wani2020multion}, the proposed transport task requires more complex decision-making, such as order of pick-up and exploration-exploitation trade-off with respect to searching for the container. We abstract away the physical reasoning for pickup and drop actions, since unlike TDW-Transport \\cite{threedworld_transport}, our focus is on deeper exploration and long-horizon planning to find and transport objects in large scenes.\n\nWe argue that modularity is a crucial choice for tackling the above challenge, whereby navigation and interaction policies can be decoupled through temporal and state abstractions that significantly reduce training cost and enhance semantic interpretability compared to end-to-end approaches. This is distinct from existing hierarchical methods for subgoal generation \\cite{krantz2021waypoint,xia2021relmogen} in long horizon tasks, where the expressivity of subgoals is largely limited to goal reaching for embodied navigation and which still face scalability challenges when the task requires long trajectory demonstrations. \n\nOur modular approach for long-horizon embodied tasks constitutes a topological graph based exploration framework and atomic policies to execute individual sub-tasks (Fig.~\\ref{fig:teaser}). The higher level planner is a finite state machine that decides on the next sub-routine to execute from one of \\texttt{\\small \\{Explore, Pickup, Drop\\}} actions. The topological map representation consists of nodes connected in the form of a graph that serves to infuse geometric and odometry information to aid deeper exploration and facilitate backtracking. Unlike methods that utilize 360-degree panoramic images as input \\cite{chaplot2020neural,VGM,krantz2021waypoint}, we divide every node to aggregate representations from several directions in its vicinity. The representation within a specific node and direction consists of latent features $\\mathcal{F}_A$ from a pre-trained encoder, an exploration score $\\mathcal{F}_E$ that captures the likelihood of the agent finding an object if it explores a frontier in that direction and object closeness score $\\mathcal{F}_O$ that indicates distance to the object within the agent's field of view. We also propose a novel weighted improvement of frontier exploration \\cite{frontier} using the predicted exploration scores.\n\nUnlike methods \\cite{xia2021relmogen,krantz2021waypoint,VGM,habitat2o} that completely rely on either motion planning algorithms \\cite{planningalgo,roboticsbook} or use pure RL for low-level actions \\cite{VGM,wani2020multion}, our approach uses the best of both worlds with motion planning for point goals within explored regions and RL policies to travel the last-mile towards semantic targets at unknown locations. Indeed, on both Long-HOT and MultiON , we show that our proposed modular hierarchical approach performs significantly better, especially on longer horizons, compared to agents operating without map or other hierarchical approaches that sample navigation subgoals for task completion. Moreover, it realizes a key benefit of modularity, namely, adaptability to harder settings when trained on a simpler version of the task.\n\nIn summary, our contributions are the following:\n\\begin{tight_itemize}\n    \\item A novel object transport task, Long-HOT, for evaluating embodied methods in complex long-horizon settings.\n    \\item A modular transport framework that builds a topological graph and explores an unknown environment using weighted frontiers.\n    \\item Strong empirical performance on object transport and navigation tasks, with adaptability to longer horizons . \n\\end{tight_itemize}\n\n\n\\if 0,\n\nMany real world tasks require agent to be robust towards varying task complexities. Imagine a robot trying to find a car key within a room and within the whole building.\nExisting end-to-end RL approaches fail in adapting to such long-horizon settings when the complexity of the problem scales. These methods face three main challenges in embodied navigation, a) their inability to provide exploration guarantees when the point or object of interest is not visible, b) difficulty in backtracking previously seen locations and c) difficulty in planning over long horizons. \nThus far, hierarchical approaches \\cite{gupta2019relay,Barto03recentadvances} have been better at handling such temporally extended problems. In this work, we aim to address these issues by proposing modular hierarchical methods for embodied transport and navigation.\n\\JD{Feels a little disjointed. What constitutes task complexity? Are we studying specifically navigation settings?}\n\nAs our first contribution, to study the benefits of modular approaches in we build a long horizon object transport task called Long-HOT in Habitat\\cite{habitat_iccv2019} environment. Our task requires agents to transport objects to a goal locations with access to a container where these objects are scattered randomly at different locations across the floor on Matterport\\cite{Matterport3D} scenes. While there exists previous approaches\\cite{threedworld_transport} with similar task definition requiring additional physics based reasoning for object pickup\nwe abstract such physics in our setup and focus on defining these tasks in large environments where agent requires deeper exploration and long-horizon planning to find and transport objects more efficiently Fig. \\ref{fig:teaser}.\n\nFurther, we study hierarchical control in the context of object transport and navigation tasks. Existing hierarchical approaches have looked into the problem of subgoal generation \\cite{krantz2021waypoint,xia2021relmogen} for solving complex long horizon tasks but the expressivity of these subgoals have mostly been limited to goal reaching for embodied navigation and still suffer from the problem of scalability when the task requires long trajectory demonstrations. Modular approaches are arguably more a natural choice to this problem where navigation and interaction policies can be decoupled through means of temporal and state abstractions that significantly reduce the training cost and provide means for more semantic interpretability compared to end-to-end approaches.\n\nIn this work we present a novel modular approach for long-horizon transport problems by building a topological graph based exploration framework and atomic policies to execute individual sub-tasks. \nOur higher level planner is a finite state machine that decides on next sub-routine to execute from one of \\texttt{\\small \\{Explore, Pickup, Drop\\}} actions.\nWe build a topological map representation which consists of nodes that are connected in the form of a graph. Unlike methods \\cite{chaplot2020neural,VGM,krantz2021waypoint} that utilize 360-degree panoramic images as input we divide every node to aggregate representations from $\\theta = 12$ directions \nwithin its vicinity. A representation within a specific node and direction consists of latent features from a pre-trained encoder ($\\mathcal{F}_A$),\nan exploration ($\\mathcal{F}_E$) score that captures how likely will the agent find an object if it explores a frontier in that direction and object closeness ($\\mathcal{F}_O$) scores that indicates distances to object within the agent's field of view. Further, we propose a novel weighted improvement of the frontier exploration\\cite{frontier} \nusing the predicted exploration scores ($\\mathcal{F}_E$). Unlike methods\\cite{xia2021relmogen,krantz2021waypoint,VGM,habitat2o} that completely rely on either motion planning algorithms \\cite{planningalgo,roboticsbook} or use pure RL for low-level actions \\cite{VGM,wani2020multion} we propose to use the best of both worlds where we use motion planning algorithms for point goals within explored regions and make use of RL policies to travel the last-mile towards semantic targets at unknown locations. \nWe show that our proposed modular hierarchical approach performs significantly better especially in long-horizon compared to agents operating without map or other hierarchical approaches that samples navigation subgoals for task completion. \n\nIn summary, our contributions are the following:\n\\begin{itemize}\\itemsep-2pt\n    \\item A novel object transport task called Long-HOT, focused on evaluating the performance of methods in complex long-horizon settings.\n    \\item A hierarchical transport framework that builds a topological graph and explores the environment using weighted frontiers.\n    \\item Strong empirical performance on the proposed object transport and navigation tasks \\cite{wani2020multion}. Further, we show better adaptability of our method to much harder settings when trained on a simple version of the same task. \n   \n\\end{itemize}\n\n\\fi\n\n\n\n\n\n\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{Habitat Transport Policy}\n\n\n\n\n\n\\section{Hierarchical Transport Policy (HTP)}\nWe now describe a modular policy (Fig. \\ref{fig:framework}) that builds a topological map of the environment and operates at different levels of temporal abstraction. Our framework consists of three modules: {\\it a high-level controller}, {\\it an exploration module} and {\\it a pick-drop module}. The {\\it high-level controller} decides on the next high-level action to execute from one of $\\mathcal{A}_H =$\\texttt{\\small \\{Explore, Pickup[Object], Drop\\}} actions. The appropriate sub-task module then takes over to perform the given high-level action. At any point, if the high-level controller decides to execute a different sub-task, the current execution is interrupted and control is passed to the module executing the next one. The modules in our framework are made of several functions which we describe briefly in Sec. \\ref{ssec:components} and then provide details on how those components are connected to our overall framework.\n\\input{figures\/hierarchy}\n\n\\subsection{HTP Model Components}\\label{ssec:components}\nAn overview of our model components along with their connections in the HTP framework is shown in Fig. \\ref{fig:pipeline}. Our sub-task modules consist of the following components: a) a topological graph builder, b) exploration score predictor c) object-closeness predictor, d) object navigation policy and e) a low-level controller. We now describe each of them in detail. \n\\input{figures\/pipeline}\n\n\\subsubsection{Topological Graph Builder}\\label{ssec:graphupdate}\nThis function is responsible for creating a topological map of the environment as a graph ${G} = ({V}, {E})$, where $V \\in \\mathbb{R}^{N_t\\times f_d}$ and $E$ represent spatial nodes and connecting edges respectively. Here, $N_t$ represents number of nodes at timestep $t$ and $f_d$ represents the length of node features. Node features for a node $V_i \\in {V}$ consist of concatenated ``node-direction'' features $V_{i, \\theta}$ corresponding to $D=12$ directions $\\theta \\in \\{1,...,D\\}$ spanning 360 degrees, centered on the node.\nThese node-direction features $V_{i, \\theta}$ in turn are computed by encoding perspective RGB images through an encoder $\\mathcal{F}_A$, pretrained in an autoencoder (details in supplementary).\n\n\n\n\n\n\nAt every timestep $t$, the graph builder updates the map as follows. \nIt takes the pose information ($P_{\\phi xy}$)\nand encoded egocentric image features from $\\mathcal{F}_A$ as input.\nThe agent's location $P_{xy}$ is mapped to the nearest existing graph node $V_i$, and its heading angle $P_{\\phi}$ is mapped onto the nearest node-direction $\\theta$. Then that corresponding node-direction representation \n$V_{i,\\theta}$ is updated to the image feature vector.\nWe add a new node $V_{i+1}$ if the agent is not within a distance threshold $l_{th} = 2m$ of existing nodes, and store the corresponding node coordinate $P_{xy}$. \nWhen the agent transitions between two nodes in the graph, the graph is updated to add an edge between them.\nSimilar to \\cite{VGM}, we also keep track of the last visited time-step for every node to provide additional context for downstream processing.\n\n\n\n\n\n\\subsubsection{Exploration and Object Closenes Scores}\\label{sec:score_pred}\\label{ssec:scorepred}\nAt every node-direction, indexed by $(i, \\theta)$, aside from the feature vector above, we store two score predictions: (a) an {\\it exploration score} for frontiers\\cite{frontier} that predicts the likelihood of finding an object by exploring that frontier and (b) an {\\it object closeness score} that indicates distance to various objects in the current field of view.\n\n\\vspace{-0.2cm}\n\\paragraph{Exploration Score Predictor ($\\mathcal{F}_E$):}\\label{para:exp_score_pred} \nWe explore two variants of this function: (1) a Q learning based graph convolutional network (GCN)\\cite{kipf2017semi} that reasons for frontiers over the entire topological graph and (2) a convolutional network (CNN) predicts frontier scores on a per-frame basis.\n\n\\vspace{0.2cm}\n\\noindent{\\it GCN Exploration Score:} This function operates on (1) \nthe current topological map\n$G = (V,E)$ with associated features as computed above, and (2) a binary mask $M^{N_t \\times \\theta_d}$ indicating the availability of a frontier at every node-direction, computed using the method described in Sec. \\ref{ssec:expmodule}.\nProvided these inputs, we train a graph convolutional network (GCN)~\\cite{kipf2017semi} to produce reinforcement learning Q-values for each node-direction, representing future rewards for finding objects after visiting each frontier associated with that node-direction.\nThe object-finding reward function $r_t^e$ at every timestep $t$ is:\n\\begin{equation}\n    r_t^e = \\mathbb{I}_{success} \\cdot r^e_{success} +  r^e_{slack} + \\sum_o \\mathbb{I}^o_{found} \\cdot r^e_{found},\n\\end{equation}\nwhere $\\mathbb{I}^o_{found}$ is the indicator if object $o$ was found at timestep $t$, $r^e_{found}$ is the reward for finding a new object, $r^e_{slack}$ is the time penalty for every step that encourages finding the objects faster, $\\mathbb{I}_{success}$ is an indicator if \\textit{all} objects were found and $r^e_{success}$ is the associated success bonus. We consider the object to be found if the object is in the agent's field of view with distance less than maximum pre-defined distance. \nOur GCN architecture involves three layers of graph convolution layers and a fully connected layer. \n\n\n\n\\vspace{0.2cm}\n\\noindent{\\it CNN Exploration Score:} This variant of the exploration score is computed directly from the agent's current RGBD view.\nGiven this view, a CNN predicts three exploration scores:\neach score represents the chances of finding an object if the agent explores the farthest frontier available within a corresponding range of angles centered on the current agent heading: $-45^\\circ$ to $-15^\\circ$, $-15^\\circ$ to $+15^\\circ$, and $+15^\\circ$ to $+45^\\circ$ respectively for the three scores. This CNN is trained with labels set to $\\max(\\max_o(d_{a,o} - d_{f,o})\/5, 0)$ where $d_{a,o}, d_{f,o}$ represent geodesic distance to object $o$ from the agent and frontier respectively. If a frontier is not available then the score is set to 0. These three scores are then stored respectively to three consecutive node-directions $\\theta-1, \\theta, \\theta+1$, centered on the current direction $\\theta$, and at the current node $i$. \n\n\n\n\n\n\n\\vspace{-0.2cm}\n\\paragraph{Object Closeness Predictor ($\\mathcal{F}_O$):} This CNN maps the current RGBD observation $I$ to a ``closeness score'' for every object. It is trained with supervised learning to predict target closeness labels for each object, which are set to $\\max(1-d\/5, 0)$ where $d$ is the true distance to the object in m. So, objects farther than $5m$ away (or invisible) have labels $0$, and very close objects have labels $\\approx 1$. Each node-direction has an associated closeness score for each object.\n\n\\subsubsection{Object Navigation Policy}\\label{ssec:objectnav}\nNext, we discuss our object navigation policy. Given the RGBD observation $I$ and a one-hot encoding $k_o$ of a target object, the policy must select navigation actions from \\texttt{\\small \\{FORWARD, TURN-LEFT, TURN-RIGHT\\}} that take it closer towards the target. This policy is trained with the following reward $r^n_t$ \\cite{wani2020multion} at each timestep $t$:\n\\begin{equation}\nr^n_t = \\mathbb{I}_{[reached-obj]} \\cdot r^n_{obj} + r^n_{slack} + r^n_{d2o} + r^n_{collision},\n   \n       \n   \n\\end{equation}\nwhere $r^n_{obj}$ is the success reward if it reaches closer than a threshold distance $d_{th}$ with the target object, $r_{slack}$ is a constant time penalty for every step, $r^n_{d2o} = (d_{t-1} - d_t)$ is the decrease in geodesic distance with the target object and $r^n_{collision}$ is the penalty for collision with the environment. \n\nWe train this policy using the proximal policy optimization (PPO) \\cite{ppo} reinforcement learning algorithm, for approximately 40M iterations using 24 simulator instances. We use mini-batch size of 4 and perform 2 epochs in each PPO update. We use other hyper-parameters similar to \\cite{wani2020multion}.\n\n\\subsubsection{Low-Level Navigation Controller}\\label{ssec:lc}\nOur final module is a low-level controller that takes a goal location (from within the explored regions)\nto be reached as input. \nIt then plans a path towards the specified goal location using the classical A*\\cite{astar} planning algorithm using a pre-built occupancy map.\n\n\\subsection{HTP Control Flow}\\label{ssec:method} \nWe are now ready to describe how  HTP manages the flow of control between these components to perform long-horizon transport tasks.\nNote that while we describe the HTP algorithm for object transport, we show in Sec.~\\ref{ssec:quant_compare} that HTP also works for other embodied navigation tasks.\n\n\n\n\\subsubsection{High-Level Controller}\\label{ssec:highlevel}\nThe high-level controller ($\\pi^H$) is a finite state machine. Based on object closeness scores $\\mathcal{F}_O$ (Sec. \\ref{sec:score_pred}), hand state $O_h$, and goal state $O_g$,\nit selects one subtask from among $\\mathcal{A}_H =$\\texttt{\\small \\{Explore, Pickup[Object], Drop\\}}. \nAt timestep $t$, if the next high level action predicted by the controller is different from the current sub-task that is being executed, the controller interrupts the execution, and agent performs the updated high-level action. For example, during exploration if the agent finds an object with closeness score higher than a some threshold it then switches control from exploration to picking the object if the hand is not full or if it holds a container. \n\n\n\n\\subsubsection{Weighted Frontier  Exploration}\\label{ssec:expmodule}\n\nIf $\\pi^H$ selects the $<$\\texttt{\\small Explore}$>$ sub-task, the exploration module is executed. For exploration, we introduce a weighted frontier technique based on the predicted exploration score function $\\mathcal{F}_E$ (Sec \\ref{sec:score_pred}). For every timestep $t$, we calculate the set of frontiers ${\\bf S}$ over the explored and unexplored regions using occupancy information \\cite{frontier}.\nWhen a new frontier is identified, we assign a parent node-direction $Y_r = (i,\\theta^n)_r$ for the $r^{th}$ frontier, where $(i,\\theta)$ is the current localized node-direction and $\\theta^n$ is calculated based on the angle made by the frontier with the agent. Here the agent's field of view is $90^\\circ$, so $\\theta^n$ for the newly found frontiers can assume one of $\\{\\theta-1,\\theta,\\theta+1\\}$ directions.\nFor all existing frontiers from timestep $t-1$, we copy the same parent node-direction from the previous timestep. Finally, we calculate a representative frontier $ S^{(i,\\theta)}$ for node-direction $(i,\\theta)$, as: $S^{(i,\\theta)} = \\{s_k: \\argmin_k \\|s_k - X_c\\|\\ \\forall\\ Y_k = (i,\\theta)\\}$ where $s_k \\in {\\bf S}$ and $X_c$ is the center of frontiers associated with $Y_k=(i,\\theta)$. \n\n\nAt each timestep during its execution, the exploration module selects a node-direction $(i,\\theta)$ from the topological graph $G$ that has the highest exploration score $\\mathcal{F}_E$. Its corresponding frontier $S^{(i,\\theta)}$ is then set as the goal location for the agent's low-level controller, which begins to move towards this goal.  \nThe highest-score goal frontier is recomputed at every timestep, and may switch as new views are observed during exploration. \n\n\n\\subsubsection{Pick-Drop Module}\\label{ssec:pickdrop}\nThis module performs the pick or drop actions in the object transport task when the controller $\\pi^H$ selects an action $a_H \\in$\\ \\{\\texttt{\\small Pickup[Object], DropAtGoal}\\}. \nWhen called, this module first selects a node $(i,\\theta)$ from graph $G$ with the highest object closeness score $\\mathcal{F}_O$ for the target object.\nIf the agent is not already in the selected $i^{th}$ node, then its location $P_{xy}(i)$ is set as the goal for the low-level controller. Once the agent is localized to the $i^{th}$ node, it orients in the direction of $\\theta$. At this point, control is passed to the Object Navigation policy, targeting the object selected by $\\pi_H$. \nThe module then selects the pickup or drop action whenever the object closeness score $\\mathcal{F}_O$ for the target object, based on the current view, exceeds a threshold. \nThe sub-task is successful when the hand state or goal state is changed accordingly and the controller $\\pi^H$ predicts the next high level action to execute.  We execute this module till it performs the pick\/drop or for a maximum of $T_p$ steps after which the control is given back to the high-level controller $\\pi^H$.\n\n\n\n\n\n\\section{Related Work}\n\\paragraph{\\bf Embodied intelligence.} \n\nThe community has developed several simulation environments \\cite{habitat_iccv2019,ai2thor,shen2021igibson,threedworld_transport,chen2020soundspaces,xiazamirhe2018gibsonenv,RoboTHOR} and associated tasks to study embodied agents in tasks like object goal navigation \\cite{batra2020objectnav,chaplot2020object,wortsman2019learning,yang2018visual,wani2020multion}, point goal navigation \\cite{anderson2018evaluation,habitat_iccv2019,wijmans2020ddppo,ramakrishnan2020occupancy}, rearrangement \\cite{threedworld_transport,shridhar2020alfred,Weihs_2021_CVPR,habitat2o}, instruction following \\cite{shridhar2020alfred,anderson2018visionandlanguage} and several others in this regard.\nWhile there are handful of previous works designed for navigation \\cite{batra2020objectnav,chaplot2020object,wani2020multion} or rearrangement \\cite{threedworld_transport,Weihs_2021_CVPR}, they do not extensively stress tests methods with increasing task complexities. We find the typically used flat policy architectures \\cite{wani2020multion} in embodied AI tasks fail completely when executing over longer horizons. Hence, we propose a new benchmark called Long-HOT that has potential to serve as a testbed for, and accelerate the development of novel architectures for planning, exploration, and reasoning over long spatial and temporal horizons. \n\n\nOur task builds on previous transport tasks defined in embodied intelligence \\cite{Weihs_2021_CVPR,habitat2o,threedworld_transport,wani2020multion} but differs in ways that it requires deeper exploration and long horizon planning. While previous work like \\cite{Weihs_2021_CVPR} focus on identifying state changes using visual inputs to perform rearrangement or \\cite{habitat2o} use geometrically specified goals in single apartment environments these works operate in minimal exploration scenarios where the focus is shifted more towards perception or interaction with objects. \nOur task is closest to \\cite{threedworld_transport}, while \\cite{threedworld_transport} focuses on performing transport including physics based simulations, we abstract our interactions and focus more on complex long-horizon planning.\nOur work extends \\cite{wani2020multion} but rather than focusing on navigation in a predefined sequential fashion, our task requires more complex decision making to determine the order of picking and decide whether to perform a greedy transport if it sees the goal or to explore more in hopes of finding the container for efficient transport.\n\n\n\n\n    \n\n\n\\paragraph{\\bf Modular-Hierarchical Frameworks.} \nSolutions to long-horizon tasks typically involved hierarchical\\cite{NIPS1997_hierarchy,subgoal_discovery,Sutton:1999,bacon2016optioncritic} policies in reinforcement learning. They provide better exploration behavior through long-term commitment towards a particular subtask. \\cite{xia2021relmogen,krantz2021waypoint} present one such approach where they sample navigation subgoals to be executed by the low-level controller. While these methods can temporally abstract navigation to an extent we find their performance to drop significantly in longer horizon settings. In HTP we show that modularity enables generalization while only training on the simpler versions of the task. \n\n\nCloser to our work are modular approaches\\cite{NMCdas2019,chaplot2020neural,krantz2021waypoint,xia2021relmogen} that provide an intuitive way to divide complex long horizon tasks as a combination of individual sub-tasks that can be solved using existing approaches. Das et al. \\cite{NMCdas2019} present a  modular approach to solve embodied question answering \\cite{das2017embodied} through a combination of several navigation policies each for finding an object, to find a room or to exit one. \nThis can blow-up with number of sub-routines required to navigate across a building or inability of the agent to find a room of particular type in large environments. Rather than navigating to individual rooms our method proposes a weighted frontier technique that provides exploration guarantees.\nChaplot et al. \\cite{chaplot2020neural} propose a method for image goal navigation by generating a topological map of the scene using 360-degree panoramic images. Our approach operates on perspective images and divides a node representation into segments across different directions.\n\nOur work also closely relates to task and motion planning (TAMP) literature\\cite{garrett2020integratedtamp,ffrob} where the closest work in this domain is \\cite{yamada2020motion} which proposes a MP augmented RL approach in manipulation settings where they realize large displacements of a manipulator through a MP. \nWhile \\cite{yamada2020motion} tackles a simple manipulation domain for 2D block pushing where target objects are fully observable, we tackle a more complex navigation setting and propose to use a combination of MP and object navigation policies where a motion planner first moves to a region with high likelihood of object presence and gives control to the navigation policy that takes it closer to the goal object.\n\n\n\n\n\n\n\\section{Habitat Transport Task}\nWe propose a novel transport task for embodied agents that simulates object search, interaction, and transport in large indoor environments. A robot assistant might be expected to perform such tasks in a warehouse or a hospital.\nIn each episode, the agent must transport $K$ target objects to a specified object goal location in a large partially observed Habitat~\\cite{habitat_iccv2019} 3D indoor environment with many rooms and obstacles.\nThe environment also contains a special ``container'' object (in yellow) that can be used to transport other objects, another special goal object (in green) whose position is the goal location. In our setting, all objects are cylinders of various colors, placed into the environment.\nUnlike previously studied tasks such as~\\cite{habitat2o}, the agent needs to explore the environment to find all objects, and does not have access to their geometric coordinates. \nAt each step, the agent can turn by angle $\\alpha=30^{\\circ}$ to the left or right, move forward by $x=0.25m$, or execute object ``Pickup'' or ``Drop'' actions.  \n\n\nThe agent has access to standard egocentric perspective RGB and depth views. Fig.~\\ref{fig:pipeline} (left) shows an example of the agent's view of a scene with a prominent red object. Aside from this, the agent has access to odometry ($P_{\\phi xy}$),\nas well as the hand state $O_h$ and goal state $O_g$, which indicate if a target\/container object is either held by the agent or already at the goal respectively. \nFollowing \\cite{wani2020multion,Weihs_2021_CVPR}, if the agent is within $R = 1.5m$ of any pick-able object and a Pickup action is called, the closest object is removed from the scene and the hand state $O_h$ is updated to include it. For the Drop action, any objects in the agent's hand are dropped near the agent's location. If the goal is within distance $R$, the goal state is updated to include the object.\nThe agent can hold limited items in its hands at once, and is therefore constrained to carry at most two objects at a time unless it picks up the container, in which case any number of objects may be carried. Picking up the container requires the agent's hands to be empty. Each episode runs for a maximum of $T = 2500$ timesteps.\n\n\nThis transport task naturally entails additional complexity compared to previously proposed navigation settings, and has properties that are not emphasized in previous benchmark tasks for embodied agents~\\cite{wani2020multion,habitat2o,habitat_iccv2019,xiazamirhe2018gibsonenv}. It includes multiple task phases (searching for, navigating to, and interacting with objects), reasoning about the environment at various scales (such as coarse room connectivity charts over the explored map for planning long trajectories, and fine-grained local occupancy maps for object interaction), accounting for carrying capacity constraints. It also involves dynamically selecting among various sub-task sequences under uncertainty: for example, having found an object far from the goal, should an agent immediately return to drop it off at the goal, or should it look for another object before returning for efficiency?\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Further Analysis for Long-HOT and HTP}\n\n\\subsection{Ablation Study}\nHere, we study the dependence of various components in HTP to the overall task performance and also their generalizability to increasing number of target objects for the transport task. For creating a test episode, we sample additional targets with object colors that were used during the training. \nNote that we only test our methods with increasing number of object goals by training them on a \\emph{default} task level with 4 objects. The ablations were conducted using the GCN variant in HTP on 250 test episodes with 15 scenes.\n\n\\vspace{-0.3cm}\n\\paragraph{\\bf HTP w\/o graph:}\nWe perform an ablation of HTP without a graph memory for closeness scores. When the high-level action is one of \\texttt{\\small\\{Pickup, Drop\\}} action, the Pick-Drop module provides direct control to object-nav policy bypassing the low-level controller. Here, instead of reaching the node with highest closeness score we directly use object-nav policy to move closer towards target objects. \n\n\\input{figures\/ablation}\n\n\\vspace{-0.3cm}\n\\paragraph{\\bf HTP - random closeness scores:} To study how the values of closeness scores affect the task performance, we replace closeness values to be random numbers. Intuitively, this should affect agent the most as random closeness values will take agent to a very different node location compared to the one closest to the target object. Further, it also makes agent execute pick-drop operations at unintended locations thereby affecting the task performance. \n\n\\vspace{-0.3cm}\n\\paragraph{\\bf HTP - random exploration scores:} In this ablation, we replace the frontier exploration scores to random values which affects the frontier selection and the exploration strategy. \n\n\\vspace{-0.3cm}\n\\paragraph{\\bf HTP - random object navigation (50\\%):} Here we replace the actions from the object navigation policy with random actions 50\\% of the time. \n\n\nFig. \\ref{fig:ablation} reports the success rate and SPL of various HTP variants for increasing number of target objects. As shown, the performance of HTP is considerably higher than the all other ablations indicating the purpose of each of these model components. \nHTP w\/o graph provides significant drop in performance compared to HTP due to the long-horizon exploration and navigation required by object-nav for finding target objects. With increasing number of target objects the performance of HTP w\/o graph deteriorates even further relatively indicating the importance of topological graph. HTP with random closeness score affect agents the most due to its influence on pick drop operations as incorrect values take agent to a completely different node compared to the closest one and the performance even reduces close to 0\\% with increase in number of target objects. \n\nHTP with random exploration score provides low success rates as incorrect frontier values makes agent switch between frontiers that are farther away making exploration less efficient as the agent travels within the explored regions for significant portion of its time while switching. \nHTP is not affected much even when perturbed with random actions for object navigation 50\\% of the time, indicating the robustness of HTP. This could be due to HTP simplifying the navigation process through a combination of motion planning and RL where, we use motion planning for navigation within explored regions and RL policies for navigating towards semantic targets at unknown locations. \n\nWhile the performance of all the HTP variants decrease with increase in number of target objects, HTP still provides good enough performance for 8 object transport while only training on transport task involving 4 objects. This also shows HTP's ability to generalize towards increasing number of target objects.\n\n\n\n\n\\input{tables\/standard_wtermination}\n\\subsection{Episode termination for wrong pickup}\nTable \\ref{tab:standard_wterminate} compares the performance of HTP and baseline methods with early termination criteria for a wrong pickup action. An episode is terminated if there are no objects within agent's vicinity when a pickup action is called. The numbers indicate Long-HOT's difficulty to be a lot higher with an end criterion similar to MultiOn\\cite{wani2020multion}. We relax those hard constraints as training becomes more difficult with sparse rewards for long-horizon tasks and rather focus on task completion for our agents. While the performance of RL methods drop significantly in Table \\ref{tab:standard_wterminate}, the proposed HTP is nearly unaffected. It can be attributed to the modular hierarchical design of HTP where pick-drop actions are executed only when the object closeness score is higher than a threshold.\n\n\n\n\n\\subsection{Long-HOT episode statistics}\nFig. \\ref{fig:dataset_statistics} shows a histogram of geodesic distances with fraction of total episodes in the corresponding histogram bin for various datasets used in Long-HOT experiments. Note the range of reference path length increases with different task configurations in Large Long-HOT indicating a increasing task complexity.\n\\input{figures\/data_statistics}\n\n\\vspace{-0.4cm}\n\\subsection{Discussion and limitations}\nThe accompanying video shows some failure cases of HTP which includes situations where an agent is unable to move around an obstacle or has small frontier regions which are ignored by the exploration module and situations where the closeness scores for some object visible only from some particular node direction is overwritten by values obtained when the perspective image does not contain the object viewed from a different location localized to the same node direction.\n\nThe work assumes noiseless odometry and depth for task completion, but earlier works like \\cite{chaplot2020object} have shown that semantic mapping and navigation work well in the real world even with noisy pose and depth. Future works can relax these assumptions to build methods that work more robustly with different forms of noisy inputs.\n\n\n\\section{Todo}\n    \n\n\\section{Further Details on Implementation, Training and Metrics}\n\nIn this section, we provide additional implementation details of baseline architectures and the proposed HTP.\n\n\\subsection{No Map baseline}\n\nWe adapt an architecture similar to \\cite{wani2020multion} for the No Map policy. \n\n\\vspace{-0.4cm}\n\\paragraph{Inputs and Outputs:} No Map takes an RGBD image of size $256\\times256$ along with the hand state $O_h$ (size $5\\times1$), goal state $O_g$ (size $4\\times1$) and previous action as inputs to the policy. It then predicts one of \\texttt{\\{\\small FORWARD, TURN LEFT, TURN RIGHT, PICKUP, DROP\\}} actions at every timestep.\n\n\\vspace{-0.4cm}\n\\paragraph{Architecture:} The RGBD image is passed through a sequence of three convolutional layers + ReLU\\cite{relu} and a linear layer + ReLU that transforms the input into a feature vector of length $512$. The convolutional layers consist of kernels with size $\\{8,4,3\\}$, strides $\\{4,2,1\\}$ and output channels $\\{32,64,32\\}$ respectively. The hand state $O_h$ and goal state $O_g$ are passed through dense layers to get respective feature vectors (dim $32$). The previous action is embedded through an embedding layer of length $32$. Finally, image features, hand-goal features, and previous action embedding are concatenated and passed through a recurrent unit to output features that are used to predict actions and the approximate value function.\n\n\n\\vspace{-0.4cm}\n\\paragraph{Rewards:} The following rewards $r_t$ is provided at every timestep $t$ to train the No Map agent:\n\\begin{equation}\n    \\begin{split}\n        r_t = & \\mathbb{I}_{success} \\cdot r_{success} + \\mathbb{I}_{pick} \\cdot r_{pick} + r_{d2o} +r_{d2g} \\\\ & + \\sum_o \\mathbb{I}^o_{goal} \\cdot r_{goal} + r_{collision} +  r_{fpd} + r_{slack}\n    \\end{split}\n\\end{equation}\nwhere, $\\mathbb{I}_{success}, \\mathbb{I}_{pick}, \\mathbb{I}^o_{goal}$ are functions that indicate successful completion of the episode, any target object picked for the first time and target objects that are transported to the goal respectively. $r_{success}, r_{pick}, r_{goal}$ are the rewards associated with $\\mathbb{I}_{success}, \\mathbb{I}_{pick}, \\mathbb{I}^o_{goal}$. $r_{d2o} = (d^o_{t-1} - d^o_t)$ is the decrease in geodesic distance from the agent's position to the closest object. $r_{d2g} = \\max_o (d^o_{t-1} - d^o_t)$ is the maximum decrease in geodesic distance of target objects with the goal. $r_{collision}$ is the collision penalty for agents and $r_{slack}$ is the slack reward for every timestep the agent delays in completing the episode.\n\n\n\\vspace{-0.4cm}\n\\paragraph{Training:} The policy is trained using proximal policy optimization (PPO) \\cite{ppo} technique, for approximately 40M iterations using 24 simulator instances. The hyper-parameters used are similar to \\cite{wani2020multion}.\n\n\\subsection{OracleMap Baselines}\nWe first describe the OracleMap (Occ) and OracleMap (Occ+Obj) baselines and then provide details on OracleMap-Waypoints policy.\n\n\\subsubsection{OracleMap (Occ \/ Occ+Obj)}\nThe policy architecture for OracleMap (Occ \/ Occ + Obj) is similar to No Map agent with an additional map input that covers an area of $10m\\times10m$. First top view map embeddings (dim. 16) are generated and then passed through a map encoder. The encoder consists of convolutions with kernels $\\{4,3,2\\}$, stride $\\{3,1,1\\}$ and output channels $\\{32,64,32\\}$. The map encoder produces a feature vector of length $256$ and is concatenated as one of the inputs to the recurrent unit. The output action space and rewards used to train OracleMap (Occ \/ Occ + Obj) is similar to the No Map baseline. \n\n\n\n\n\n\\subsubsection{OracleMap-Waypoints}\nThe inputs to the baseline are OracleMap (Occ+Obj), hand state $O_h$ and goal state $O_g$. It then predicts waypoints to be reached as $(x,y)$ locations on the map. The prediction is discretized into $M=100$ bins within a $5m\\times5m$ range centered on the agent. The agent then selects a bin as one of its action and uses A*\\cite{astar} to reach its location. The predicted subgoal is also associated with one of \\texttt{\\small \\{Pickup, Drop\\}} high-level actions. The action space in Waypoints policy contains $M\\times2$ in total. Once the agent reaches the predicted subgoal, the corresponding high level action \\texttt{\\small \\{Pickup, Drop\\}} is executed. The subgoals are updated for every $t_k$ steps irrespective of agent reaching previously assigned subgoal. Agent's prediction range is kept higher than maximum traversable distance in $t_k$ steps for agents to provide subgoals that avoid taking pickup or drop actions within $t_k$ timesteps.  The policy architecture and rewards used are similar to OracleMap (Occ+Obj) baselines.\n\n\n\n\n\n\\subsection{Hierarchical Transport Policy}\n\n\\paragraph{High-level Controller:} \nThe controller takes hand state $O_h$, goal state $O_g$ and closeness scores $\\mathcal{F}_O$ of objects with respect to the nodes as inputs. The high-level action is assigned based on the following conditions, if the agent already holds maximum number of objects in its hand and goal object was discovered (closeness score of goal $> F_O^{th}$ in one of the nodes) a \\texttt{\\{Drop\\}} action is executed else the agent executes \\texttt{\\{Explore\\}} to find the goal object. If the agent has capacity to carry more objects, and some closeness scores for objects are greater than $F_O^{th}$ in the nodes, a \\texttt{\\{Pickup[Object]\\}} action is executed for objects that are not either held by the agent or transported already. The agent executes \\texttt{\\{Explore\\}} if none of the objects satisfy the closeness score criteria. \\texttt{\\{Pickup[Container]\\}} action is only executed when the container is discovered and the agent does not hold any object in its hand.\n\n\n\n\n\n\n\\paragraph{Pre-trained Encoder $\\mathcal{F}_A$ :}\nLatent features from a pre-trained auto-encoder is used as node features in HTP. It consists of a ResNet-18\\cite{resnet18} style encoder-decoder architecture with latent features of $32$ dimensions. The auto-encoder is trained with a weighted MSE loss that weighs pixels of target objects with a weight $\\lambda=2.0$.\n\n\n\\subsection{Reference Trajectory Calculation}\nA reference demonstration in the transport task first picks up the container and then picks up consecutive closest objects from its previous location to finally drop them at goal. The sum of geodesic distances in executing this reference trajectory in an episode from agent's starting location is used as the reference path length $G_{ref}$.\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHelioseismology -- the study of solar oscillations -- is a powerful probe of the\nstructure and dynamics of the Sun which has provided great improvements in our\nunderstanding of stellar evolution and structure \\citep[][ and references\ntherein]{TurckChieze1993, JCD2002}. Those successes push the community to apply\nseismic techniques to other stars, opening the doors to asteroseismology, the\nstudy of stellar oscillations. These oscillations have already been observed\nfrom ground and space. The ground-based observations are limited by the\nday-night cycle, which introduces aliases in the observations, but allow to use\nDoppler velocity measurements. They have provided data with {sufficient\nquality} to detect solar-like oscillations \\citep[see][ and references\ntherein]{BouchyCarrier2003, BeddingKjeldsen2003}. To reduce the aliases,\nmulti-site campaigns have been carried out but they are too short to have a good\nfrequency resolution. Space photometry missions and ground-based velocity\nnetworks must be used to provide observations of stellar oscillations without\nthese limitations. With the current MOST\\footnote{Microvariability and\nOscillations of STars \\citep{Matthews1998}} and WIRE\\footnote{Wide-field Infra\nRed Explorer \\citep{Buzasi2000}} satellites and the future\nCOROT\\footnote{Convection Rotation and planetary Transits \\citep{Baglin2001}}\nmission asteroseismology is blooming. However, we still have to deal -- in the\ncase of solar-like oscillations -- with very small signal-to-noise ratio\n(hereafter $S\/N$) observations as a consequence of the weakness of the\nluminosity variations. Moreover, stars cannot be spatially resolved yet. Only\nglobal oscillation modes can be observed. In addition, we cannot have access  to\nthe rotation rates and the rotation-axis inclination separately. Without knowing\nthese two key stellar properties, the tagging of the modes in terms of their\nproperties ($\\ell, m$) and successive $n$ may be extremely difficult. In fact,\nthe main problem to face will not be to fit the peaks (``peak-bagging'') but to\nprovide a good description of the model to be fitted after having put the\ncorrect labels on the modes (``peak tagging''). To do this, it has been proposed\nto use the echelle diagram where the modes follow ridges depending on the\nstellar properties. To improve the S\/N ratio \\citet{Bedding2004} proposed to\nfilter this diagram by a vertical {smoothing}. However the\n{smoothing} works well only when the ridges are quasi-vertical which\nmeans a very good \\textit{a priori} knowledge of the large difference and is\nrestricted to the asymptotic part of the spectrum. We propose here to follow a\nsimilar approach but using new mathematical denoising techniques better suited\nto the study of curved ridges.\n\nAt the end of the last decade, the application of mathematical transforms based\non wavelets to analyze astronomical images has been widely developed. The first\nwavelet algorithms were well adapted to treat images with isotropic elements.\nHowever, this description presented a limitation in the context of astrophysics,\nwhere objects such as filaments or spiral structures exhibit a highly\nanisotropic character (in shape and scale). New transforms, the ridgelet\n\\citep{Candes1998} and curvelet transforms \\citep{CandesDonoho1999, Starck2002},\nwere then developed to deal efficiently with such objects. Astrophysical\napplications (image denoising) of this technique have been presented in\n\\citet{Starck2003, Starck2004} to analyze images of gravitational arcs, the\nSaturn rings or the CMB (Cosmic Microwave Background) map.\n\nIn this paper we suggest to use the curvelet transform to analyze asteroseismic\nobservations (more precisely the stellar echelle diagrams), in order to improve\nthe ``peak tagging'' of the oscillation modes and even the resultant ``peak\nbagging''. To illustrate the application of this denoising technique in the\nasteroseismic case, we have performed Monte Carlo simulations of ideal\nasteroseismic data contaminated by different levels of stochastic noise. We\nstart in Sect.~2 by a quick reminder of the properties of stellar oscillation\nmodes in the solar-like case and the construction of the echelle diagram. In\nSect.~3 we introduce multiscale transforms, in particular the ridgelet and the\ncurvelet transforms. In Sect.~4, the simulated data of a star with an\noscillation spectrum similar to the Sun but with different rotation axis\ninclinations and rotation rates, are presented. In Sect.~5 we discuss the\nresults obtained in the simulations.\n\n\\section{Properties of solar-like oscillations}\n\n\\begin{figure}\n\t\\includegraphics[scale=0.35,angle=0]{Lambert2006_fig1a.eps}\n\t\\includegraphics[scale=0.35,angle=90]{Lambert2006_fig1b.eps}\n\t\\caption{Portion of the theoretical spectrum (top) and echelle diagram (bottom)\nfor a sun spinning ten times faster than the Sun and seen under an angle of\n50$\\degr$. This is the ideal power spectrum used in the simulations described in\nSect.~5.}\n\t\\label{theorique}\n\\end{figure}\n\nOnly low-degree stellar oscillation modes can be detected and observed with the\npresent generation of instruments. The asymptotic theory of oscillation modes\n($n\\gg \\ell$) is then adequate and can be used to study them. First order\n\\citep{Tassoul1980} and second order developments \\citep{Vorontsov1991,\nLopes1994, Roxburgh2000a, Roxburgh2000b} have been made to describe solar and\nstellar oscillations. In the case of solar-like stars, where p-modes\npredominate, the frequencies can be developed as:\n\\begin{equation}\\label{secondordre}\n\t\\nu_{n,\\ell} \\approx \\Delta\\nu_0 \\big(  n+\\frac{\\ell}{2}+\\frac{1}{4}\n+\\alphaup(\\nu) \\big) + \\frac{\\Delta\\nu_0}{4\\pi^2\\nu_{n,\\ell}}\\big((\\ell +\n1\/2)^2A + \\psi\\big) \n\\end{equation}\nin this expression $\\ell$ and $n$ are respectively the degree and the radial\norder of the modes and  \n\\begin{eqnarray*}\n\\tau_c\t\t\t&=&\t\\int_{r_{in}}^{r_{out}}\\frac{dr}{c_s} \\\\\n\\Delta\\nu_0\t&=&\t\\frac{1}{2\\tau_c} \\\\\nA\t\t\t\t&=&\t\\frac{1}{4\\pi^2\\nu_{n,\\ell}}\\big(\\frac{c_s(R_\\star)}{R_\\star} -\n\\int_{r_{in}}^{r_{out}}\\frac{dc_s}{dr}\\frac{dr}{r}\\big) \n\\end{eqnarray*}\n$c_s$ is the internal stellar sound speed, $\\alphaup$ is a phase-shift term and\n$\\psi$  is a function which allows to take into account the gravitational\npotential in the central region \\citep{Lopes1994}. From the asymptotic approach,\nwe can extract general properties of modes and better understand the physics\nhidden in the frequencies behavior. The large frequency spacing, defined as\n$\\Delta\\nu_{n,\\ell}=\\nu_{n+1,\\ell} - \\nu_{n,\\ell}$, tends asymptotically to\n$\\Delta\\nu_0$, related to the mass and radius of the star; the small frequency\nspacing, $\\delta_{\\ell,\\ell+2}\\nu=\\nu_{n,\\ell}-\\nu_{n-1,\\ell+2}$, can be\napproximated to first order by\n$(4\\ell+6)\\Delta\\nu_0\/(4\\pi^2\\nu_{n,\\ell})\\int_0^{R_\\star}\\frac{dc_s}{dr}\\frac{dr}{r}$.\nThis variable is related to the derivative of the sound speed and enhances the\neffect coming from the central regions, providing constraints on the age of the\nstar. Finally the second difference is defined as\n$\\delta_2\\nu=\\nu_{n+1,\\ell}-2\\nu_{n,\\ell}+\\nu_{n-1,\\ell}$. Its variations\nprovide information about the extent of the convective zone \\citep{Monteiro2000,\nBallot2004} or the helium abundance in the stellar envelope \\citep{Basu2004}.\n\nUnder the rotation effects the azimuthal order $m$ ($-\\ell \\leqslant m \\leqslant\n\\ell$) is needed to characterize the oscillation spectrum. If the angular\nvelocity $\\Omega$ is uniform \\citep{Ledoux1951}, the mode frequencies are\nasymptotically approximated by:\n\\begin{equation}\\label{unifangvel}\n\t\\nu_{n,\\ell,m}\\approx\\nu_{n,\\ell}+m\\Omega\/2\\pi = \\nu_{n,\\ell}+m\\delta\\nu\n\\end{equation}\nwhere $\\delta\\nu$ is the rotational splitting. Equation~\\ref{unifangvel} shows\nthat modes are ($2\\ell+1$)-times degenerated among the azimuthal order: a single\npeak in the spectrum becomes a multiplet. Its corresponding structure depends on\nthe rotation rate, the inclination axis of the star and its stochastic\nexcitation. The solar-like mode lifetimes (a few days) are expected to be much\nshorter than the length of the future space observations (a few months). In\nconsequence, the relative amplitude ratios inside a multiplet will only depend,\nin average, on the inclination angle and the spacing between these different\nm-components \\citep{GizonSolanki2003}. Thus if the different m-components of a\nmultiplet can be identified and tagged with the correct $(\\ell,m)$, they can\nprovide a good estimation of both the rotation-axis inclination $i$ and the\nrotational splitting $\\delta\\nu$, allowing a better mode parameter extraction\nthrough the fitting of the spectra. The effect of the stochastic excitation on\nan isolated mode could be minimized by computing the average of these parameters\non several modes \\citep[see for example the\nn-collapsogramme;][]{BallotYale2004}.\n \nEquation~\\ref{secondordre} shows that the even ($\\ell=0,2$) and odd ($\\ell=1,3$)\nmodes have respectively almost the same frequency, only separated by the small\nspacing $\\delta_{\\ell,\\ell+2}\\nu$. In addition, they are separated regularly in\nfrequency by the large spacing $\\Delta \\nu_{n,\\ell}$. This property allows us to\nbuild the so-called echelle diagram \\citep{Grec1983}, which is currently used to\nidentify modes for solar-like oscillations. It is a 2D representation of the\npower spectrum where this one is folded onto itself in units of the large\nspacing. In such representation the modes appear as almost locally vertical\nridges (see Fig.~\\ref{theorique}). The echelle diagram is a powerful tool for\nthe ``peak tagging'' since assigning the correct $(\\ell,m)$ values to the peaks\nis easier when the multiplet structure is well identified in this diagram. The\nsuccessive $n$ values are obtained from each individual horizontal line. \n\n    \\section{Multiscale Transforms}\n        \\subsection{The Wavelet Transform}\n\nThe wavelet transform provides a framework for decomposing images into their\nelementary constituents across scales by reducing the number of significant\ncoefficients necessary to represent an image. The continuous wavelet transform\nof a 2D signal is defined as:\n\n\\begin{eqnarray}\n   W(a,b_i, b_j) = \\frac{1}{\\sqrt{a}}\\int\\!\\!\\!\\int\nf(x,y)\\psi^*\\left(\\frac{x-b_i}{a},\\frac{y-b_j}{a}\\right)dxdy\n\\end{eqnarray}\nwhere $W(a,b)$ are the wavelet coefficients of the function $f(x)$, $\\psi(x)^*$\nis the conjugate of the analyzing wavelet, $a>0$ is the scale parameter and $b$\nis the position parameter. The continuous wavelet transform is the sum over all\nthe positions of the signal $f(x,y)$ multiplied by the scaled and shifted\nversions of the wavelet $\\psi((x-b_i) \/ a,(y-b_j) \/ a)$ (cf.\nFig.~\\ref{examples}, top panels). This process produces wavelet coefficients\nthat are a function of scale and position.\n\nHowever, the classical wavelet transform only address a portion of the whole\nrange of interesting phenomena: isotropic features at all scales and locations.\nOne of the drawbacks of the two-dimensional wavelet transform is that it does\nnot achieve an efficient analysis of images which present high anisotropy. For\ninstance, the wavelet transform does not efficiently approximate 2D edges, since\na large number of large wavelet coefficients, scale after scale, are required,\nmaking difficult its analysis. In order to solve this problem two new\nmathematical transforms, namely the ridgelet transform and the curvelet\ntransform, were introduced.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.245]{Lambert2006_fig2a.ps}\n    \\includegraphics[scale=0.245]{Lambert2006_fig2b.ps}\n    \\includegraphics[scale=0.245]{Lambert2006_fig2c.ps}\n    \\includegraphics[scale=0.245]{Lambert2006_fig2d.ps}\n    \\caption{Examples of 2D wavelets (top panels) and ridgelets (bottom panels).\nThe top right wavelet has a greater scale parameter than this on the left. The\nbottom right ridgelet has different orientation and width than the left one.}\n    \\label{examples}\n\\end{figure}\n\n        \\subsection{The Ridgelet transform}\n\nThe ridgelet transform was developed to process images including ridges elements\n\\citep{Candes1998}. It provides a representation of perfectly straight edges.\nGiven a function $f(x_1,x_2)$, the representation of this latter is the\nsuperposition of elements of the form\n$a^{-1\/2}\\psi((x_1\\cos\\theta+x_2\\sin\\theta-b)\/a)$, where $\\psi$ is a wavelet,\n$a>0$ a scale parameter, $b$ a location parameter and $\\theta$ an orientation\nparameter. The ridgelet is constant along lines\n$x_1\\cos\\theta+x_2\\sin\\theta=\\mathrm{const}$, and transverse to these ridges it\nis a wavelet. Thus, contrary to a unique wavelet transform, the ridgelet has two\nsupplementary characteristics: a length, equal to this of the image and an\norientation, allowing the analysis of an image in every direction and so\nexhibiting the edge structure. Fig.~\\ref{examples} (bottom panels) shows two\nexamples of ridgelets. The problem is that in the nature edges are typically\ncurved rather than straight so ridgelets alone cannot yield an efficient\nrepresentation.\n\n\t\\subsection{The Curvelet transform}\n\t\t\\subsubsection{Description}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.32]{Lambert2006_fig3.eps}                        \n    \\caption{Sketch illustrating the curvelet transform applied to an image. The\nimage is decomposed into subbands followed by a spatial partitioning of each\nsubband. The ridgelet transform is applied to each block. The finest details\ncorrespond to the highest frequencies.}\n    \\label{curveletgraphe}\n\\end{figure}\n\nRidgelets can be adapted to represent objects with curved edges using an\nappropriate multiscale localization: at a sufficiently fine scale a curved edges\ncan be considered as almost straight. \\citet{CandesDonoho1999} developed the\ncurvelet transform using ridgelets in this localized manner.\nFig.~\\ref{curveletgraphe} shows the different steps of the curvelet analysis of\nan image:\n\n\\begin{enumerate}\n\t\\item Image decomposition into subbands: as a set of wavelets bands through a\n2D isotropic wavelet transform. Each band corresponds to a different scale.\n\t\\item Smooth partitioning: each subband is partitioned into squares -- blocks\n--, whose size is appropriate to each scale. The finest is the scale, the\nsmaller are the blocks.\n\t\\item Ridgelet analysis: it's applied to each square.\n\\end{enumerate}\n\nThe implementation of the curvelet transform offers an exact reconstruction and\na low computational complexity. Like ridgelets, curvelets occur at all scales,\nlocations and orientations. Moreover contrary to ridgelets, which have a given\nlength (the image size) and a variable width, the curvelets have also a variable\nlength (the block size) and consequently a variable anisotropy. The finest the\nscale is, the more sensitive to the curvature the analysis is. As a consequence,\ncurved singularities can be well approximated with very few coefficients.\n\n\t\t\\subsubsection{Denoising images: filtering curvelet coefficients}\n\nTo remove noise a simple thresholding of the curvelet coefficients has been\napplied to select only significant coefficients. One possible thresholding of a\nnoisy image consists in setting to $0$ all non-significant curvelet coefficients\n$\\tilde c_{i,j,l}$, $i$, $j$ and $l$ respectively the indexes of the line, row\nand scale: it is the so-called hard-thresholding:\n\\begin{eqnarray}\n\t\\tilde c_{i,j,l} = \\left\\{ \\begin{array}{ll} \t\\mbox{1} & \\mbox{if }  \tc_{i,j,l}\n\\mbox{ is significant} \\\\ \n\t\t\t\t\t\t\t\t\t\t\t\\mbox{0} & \\mbox{if } \tc_{i,j,l} \\mbox{ is not significant}\n\\end{array} \\right.\n\\end{eqnarray}\nCommonly, $c_{i,j,l}$ is significant if the probability that the curvelet\ncoefficient is due to noise is small, i.e., if the curvelet coefficient is\ngreater than a given threshold. A basic problem remains: the choice of the\nthreshold. Usually, this threshold is taken equal to $k\\sigma_j$, where\n$\\sigma_j$ is the noise standard deviation at the scale $j$ and $k$ is a\nconstant taken equal to 5 in our filterings.\n\nSimple thresholding of the curvelet coefficients is very competitive\n\\citep{Starck2002} with ``state of the art'' techniques based on wavelets,\nincluding thresholding of decimated or undecimated wavelet transforms.\n\n\\section{Simulation of data}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.65]{Lambert2006_fig4_lowres.eps}\n    \\caption{Effect of the curvelet denoising on the mode visibility for\n$S\/N=5$. Each picture shows 120 realizations out of the 500 done in our Monte\nCarlo simulation. Each horizontal line corresponds to a single realization. The\ntop panel is the raw spectra and the bottom is the curvelet filtered one. \n\\label{montecarlo}}\n\\end{figure*}\n\nTo characterize the curvelet denoising technique applied to the asteroseismic\ndata, we have simulated typical solar-like observations varying different\nparameters: S\/N ratios, observational lengths, rotation-axis inclinations,\nrotation rates... With this approach we know the input parameters in advance and\nwe can evaluate the quality of the results given by the curvelet analysis and\nits limits.\n\nIn the simulations shown in this paper, we use the oscillation spectrum of a\nstar similar to the Sun but seen under different conditions. Different rotation-axis inclinations ($i=50\\degr$ and $90\\degr$) and rotation\nrates ($\\Omega= \\Omega_{\\sun}$, $5\\Omega_{\\sun}$, and $10\\Omega_{\\sun}$) have\nbeen considered. An ideal power spectrum were constructed first. Only the modes\n$\\ell\\le3$, $n=12$--$25$ were simulated. The mode parameters -- frequencies\n($\\nu$), amplitudes ($A$) and widths ($\\Gamma$) -- were obtained from the\nanalysis of GOLF (Global Oscillations at Low Frequency) data \\citep{Garcia2004}.\nThe amplitudes were corrected to take into account the difference between\nintensity and velocity observations. Modes were simulated with symmetrical\nLorentzian profiles as the asymmetry is expected to be at the level of the\nnoise. Following the method described in \\citet{FierryFraillon1998}, a\nmultiplicative noise, a $\\chi^2$ with 2 d.o.f. statistics, has been introduced\nto reproduce the stochastic excitation of such modes \\citep[see\nalso][]{Anderson1990}. The $S\/N$ ratio of the ``resultant'' raw power spectrum\nwas defined as the maximum of the bell-shaped p-mode power (i.e. the highest\nsimulated p mode) divided by the noise dispersion. The simulated background is\nflat assuming that it has been previously fitted and removed as it is usually\ndone for the Sun \\citep{Harvey1985}.\n\nSeveral Monte Carlo simulations have been performed for each ideal spectrum.\nRealistic $S\/N$, with values ranging from 5 to 15, have been used to cover a\nwide range of situations (compatible with what it is expected, \\citep[see\n][]{Baglin2001}). In each realization of the Monte Carlo simulation the same\nlevel of noise has been randomly added to the corresponding ideal spectra.\nTherefore all the realizations, in a given Monte Carlo simulation, have the same\n$S\/N$ ratio. The simulated spectra have been computed for two resolutions,\n$\\approx0.38$ and $\\approx0.077~\\mu$Hz, corresponding respectively to 30-day and\n150-day observations. The first are representative of MOST observations and the\nshort CoRoT runs while the latter are of the same length than the long CoRoT\nruns.\n\nSimulations of other stars, like some potential main CoRoT targets, with\ndifferent masses, ages and, in consequence, internal structures have been made.\nThe results have already been presented and discussed during the CoROT workshops\n\\#8 and \\#9 obtaining the same qualitative results. For the sake of clarity,\nthey are not shown here.\n\n\\section{Discussion}\n\\begin{figure*}\n    \t\\includegraphics[scale=0.35,angle=90.]{Lambert2006_fig5a.eps}\n    \t\\includegraphics[scale=0.35,angle=90.]{Lambert2006_fig5b.eps}\n    \t\\includegraphics[scale=0.35,angle=90.]{Lambert2006_fig5c.eps}\n    \t\\includegraphics[scale=0.35,angle=90.]{Lambert2006_fig5d.eps}\n    \t\\includegraphics[scale=0.35,angle=90.]{Lambert2006_fig5e.eps}\n\t\\includegraphics[scale=0.35,angle=90.]{Lambert2006_fig5f.eps}\n\t\\caption{Raw (left) and filtered (right) power spectra (top and middle panels)\nand echelle diagrams (bottom panels) for a $S\/N=5$ realization. The short dashed\nlines in the power spectra represent the position of the theoretical\nfrequencies. From left to right, the three first equidistant lines indicate the\ncomponents $m=-1,0,1$ of $\\ell=1$ modes, the two next indicate the strongest\ncomponents of $\\ell=2$ ($m=-1$ and $1$), and the the last indicates $\\ell=0$. In\nthis case only two components of the $\\ell=1$ and the $\\ell=0$ mode are slightly\nvisible in the raw diagram. On the curvelet filtered one, the three $\\ell=1$\ncomponents appear as well as the $\\ell=0$ and the components $m=\\pm 1$ of the\n$\\ell=2$ modes.\\label{diagechellesnr5}}\n\\end{figure*}\n\nOnce the spectra have been computed, the echelle diagrams can be built with a\nfixed folding frequency. This one corresponds to the mean large frequency\nspacing $\\Delta\\nu_0$, identified either by computing the FFT, the\nautocorrelation of the spectra or any other technique \\citep[see for\nexample][]{Regulo2002}. The denoising based on the curvelet transform is then\napplied to this echelle diagrams. It is important to note that artifacts may\nappear in the filtered spectra at frequencies $\\nu^*$=$\\nu_0$+$k\\Delta\\nu_0$,\nwith $k$ an integer, when random small structures appear in the echelle\ndiagrams. However, their appearance and position strongly depend on the folding\nfrequency and are very sensitive to its value. Therefore they can be easily\nidentified. The artifacts can be reduced (in contrast to the regions containing\nsignal) by building echelle diagrams with slightly different folding frequencies\nand averaging the resultant filtered spectra.\n\nIn order to present the results of data analysis using the curvelet denoising\nmethod, we have selected the case of a sun-like star seen with an inclination\nangle $i=50\\degr$ and with a rotation $\\Omega=10 \\Omega_{\\sun}$. A portion of\nthe ideal spectra constructed for this star can be seen in Fig.~\\ref{theorique}\n(top panel). Monte Carlo simulations were then performed, giving rise to\ndifferent sets (each one with 500 realizations) of raw spectra with different\n$S\/N$ ratios. The echelle diagrams were constructed using a folding frequency of\n135.18$~\\mu$Hz, obtained by computing the FFT of the raw spectrum.\n\n    \\subsection{Peak tagging}\n\nIn those cases, with a high $S\/N$ (typically 15), the mode structure is clearly\nvisible in each raw spectrum and also on the echelle diagram. The different\nridges can be easily identified and tagged. Although the filtering gives\nenhanced denoised diagrams and unfolded spectra, it does not contribute\nsignificantly to the mode identification. \n\nIn the lower $S\/N$ cases, however, the situation is different.\nFigure~\\ref{montecarlo} shows some of the results of the Monte Carlo simulation\nfor $S\/N$=5. \nThe upper panel corresponds to 120 realizations among the 500 computed for the\nraw spectra in the frequency range 2450--2920$~\\mu$Hz. Each horizontal line\ncorresponds to a single realization. Some patterns can hardly be seen. The lower\npanel represents the same spectra after applying the curvelet filtering. A\nseries of vertical ridges clearly appears. From the left to the right on the\npanels, they can be identified as the ($\\ell=2$; $m=\\pm 1$), the $\\ell=0$\n(blended with the $\\ell=2$; $m=+2$ ) and the ($\\ell=1$; $m=-1,0,+1$). The\nimprovement of the contrast is important in all the realizations and allows to\ndistinguish the different components of a mode, making easier the identification\nand the tagging. \n\nThe identification is harder when looking at each individual spectrum and\nrequires the use of the echelle diagram. Fig.~\\ref{diagechellesnr5} shows an\nexample of raw (left) and filtered (right) 150-day observation power spectra\n(top and middle panels) and the corresponding echelle diagrams (bottom panels)\nfor a $S\/N=5$ realization. Input frequencies are indicated by the short dashed\nlines above the spectra. The mode peaks can hardly be distinguished in the raw\nspectrum and can easily be confused with noise. For the range 2780-2920$~\\mu$Hz,\nonly a strong peak at 2900$~\\mu$Hz can be considered not to be noise. In the\nregion 3060--3180$~\\mu$Hz the peaks are visible and we can attempt to identify\nthe $\\ell$=1 and $\\ell$=0 modes but it is still unclear. On the contrary, on the\ncorresponding parts of the filtered spectrum, the structures of the $\\ell$=1\nmode with three components, the $\\ell$=0 mode and even the strongest components\nof the $\\ell$=2 mode are visible. \nThe raw echelle diagram gives no extra information because of the very weak\nridges and low contrast with the background. The weakest components can hardly\nbe detected and no tagging can be done. The curvelet filtering provides a\ncontrast enhancement of the ridges on the echelle diagram. Thus three almost\nequidistant strong ridges appear on the left of the diagram and one strong ridge\nwith two weaker ones on the right. The corresponding patterns can be seen on the\nfiltered spectrum corresponding well to the theoretical frequencies. Since the\nmodes $\\ell=3$ are not visible, and according to the amplitude of the strongest\npeak on the left, we can suggest that the three strongest peaks correspond to a\n$\\ell=1$ multiplet and the other ones to the $\\ell=2$ and $\\ell=0$ modes. \n\nConsequently,  when the tagging is done it is  also easier to have a first\nestimation of both the mean rotational splitting and the rotation-axis\ninclination, since the visibility of the multiplet is increased. From the\nspacing between the components of the mode $\\ell=1$, a first estimation of the\nmean rotational splitting of the star can be done, as well as an estimation of\nthe inclination angle, according to their relative amplitude ratios. We have\nselected the extraction of one parameter: the mean rotational splitting of the\n$\\ell$=1 mode at low frequency (2540--2550$~\\mu$Hz), to quantify the improvement\nof the curvelet filtering. This region is particularly interesting because the\nline width is still small and the modes, when they are visible, can be easily\nidentified. Thus, in a sample of 100 realizations of the Monte Carlo simulation,\nwe have obtained in 90 of them a better estimation of this parameter in the\nfiltered spectra. In fact, in the raw spectra it was very exceptional to obtain\na good result. With the filtered spectra a mean rotational splitting of $\\langle\n\\delta\\nu \\rangle=4.05\\pm0.30~\\mu$Hz was found, which is very close to the\nactual splitting included in the ideal spectra $\\langle \\delta\\nu\n\\rangle=4.0~\\mu$Hz. In addition,  specific methods can be applied to improve the\nextraction of these parameters by using different strategies of spectra fitting\nas the ones developed by \\citet{GizonSolanki2003} or \\citet{Ballot2006}. \nIn the case of the 30-day observations, the curvelet filtered echelle diagram is\nstill very noisy and it does not help in recognizing the ridges. However the\ncorresponding denoised power spectrum is much better despite the lower\nresolution (5 times less than in the long runs), even for small $S\/N$ ratios\n($\\sim5$). The modes $\\ell=0,2$ and $\\ell=1$ can be distinguished, at the\nmaximum power, while it is not obvious to do so in the raw spectra. Therefore,\nwe consider that a \n30-day run is the minimum length needed to have reliable results with the\ncurvelet denoising technique.\n\n\\citet{Garcia2005} analyzed the first available MOST public Procyon A data\n(32-day observation) using the curvelet technique. Previous analysis by\n\\citet{Matthews2004} did not reveal the presence of any p-mode structure in this\nstar. Therefore, due to its tiny S\/N ratio the results of the curvelet denoising\nshould be taken with care. Nevertheless, an excess of power seems to appear in\nthe region where it is expected and taking the 15 most prominent peaks in this\nregion, many are in agreement, inside the error bars, with previous tagged modes\nusing ground-based velocity observations.\n\n\\subsection{Extraction of p-mode parameters}\n\nOnce the mode identification and tagging are done, the extraction of the mode\nparameters can be performed. To illustrate how this extraction can be improved\nby using the denoised spectrum we have extracted the central frequency of the\nmodes in both the raw and the filtered spectra. To determine this parameter,\nmodes have been fitted by Lorentzian profiles using a maximum-likelihood\nestimator in the classical way: adjacent pairs of even ($\\ell=0$ and $\\ell=2$)\nmodes are fitted together,  while $\\ell=1$ is fitted alone, due to the small\namplitudes of $\\ell=3$ modes. For each multiplet, the fitted parameters are the\ncentral frequency $\\tilde\\nu_{n,\\ell}$, the amplitude $\\tilde A_{n,\\ell}$, the\nlinewidth $\\tilde\\Gamma_{n,\\ell}$ and the background $b$. The amplitude ratios\ninside the multiplets and the rotational splittings have been fixed thanks to\nthe preliminary estimation done in the previous section (cf. 5.1). The fitting\nprocedure provides for each adjusted parameter $\\tilde{X}$ an associated error\n$\\sigma(\\tilde{X})$ {computed by Hessian-matrix inversion}.\n \nThe raw spectra follow a $\\chi^2$ with 2 d.o.f. statistics, whereas the filtered\nspectra have a $\\chi^2$ with a higher d.o.f. statistics (close to a Gaussian\ndistribution depending on the number of filtered coefficients). {According to \\citet{Appourchaux2003}, it is possible to fit spectra following a\n$\\chi^2$ with more than\n2 d.o.f. statistics with a classical procedure developed for a $\\chi^2$ with 2\nd.o.f. statistics: parameters\nare correctly fitted, but computed errors have to be adapted \\textit{a\nposteriori}. However in our case, \ndue to filtering, points of filtred spectra are correlated (we have estimated\nthat one point is correlated with $\\sim$10 neighbouring points). This\ncorrelation should have to be considered, but we have neglected its \neffect on the fitting procedure in the present study. This assumption is\nvalidated by the Monte Carlo simulations.\nSuch a global filtering induces also correlations between the different lines\nof the echelle diagram. Thus the errors on parameters of different modes\n(typically $(n,\\ell)$ and $(n+1,\\ell)$)\ncan be correlated. These correlations will have to be taken into account\nespecially during the comparison of frequencies extracted by this way to stellar\nmodels.}\n\nFrom the 500 realizations of the Monte Carlo simulation, we derived for each\nmode and for both the raw and the filtered spectra the mean value of the\nextracted frequencies $\\langle \\tilde\\nu_{n,\\ell} \\rangle$, their mean computed\nerrors $\\langle\\sigma(\\tilde\\nu_{n,\\ell})\\rangle$ and the dispersion of\nfrequency distribution $\\sigma^{*}(\\tilde\\nu_{n,\\ell})$ (the real error). We\nhave verified that $\\sigma^{*}(\\tilde\\nu_{n,\\ell}) \\approx\n\\langle\\sigma(\\tilde\\nu_{n,\\ell})\\rangle$ for fits performed on the raw spectra\nand {we have $\\sigma^{*}(\\tilde\\nu_{n,\\ell}) <\n\\langle\\sigma(\\tilde\\nu_{n,\\ell})\\rangle$ for fits performed on the filtered\nones. As expected, the error bars on the fitted frequencies, computed by Hessian-matrix inversion, are overestimated.\n}\n\n\nFigure~\\ref{ecart} shows the difference between the mean fitted frequencies\n$\\langle \\tilde\\nu_{n,\\ell} \\rangle$ and the theoretical frequencies $\\nu_{in}$\nof the simulated star discussed in the previous section ($S\/N=5$). The error\nbars correspond to the dispersion $\\sigma^{*}(\\tilde\\nu_{n,\\ell})$. For each\n$\\ell$, the error bars of the filtered spectra are smaller than those of the raw\nspectra. In addition, the range where modes can be detected, tagged and fitted\nis extended. While the difference $\\langle \\tilde\\nu_{n,\\ell} \\rangle -\n\\nu_{in}$ is only flat in the central region of the raw power spectrum (e.g. for\n$\\ell=0$, in the range $n=18$--$22$), it extends at higher and lower frequencies\n(e.g. for $\\ell=0$, the range is extended to $n=16$--$23$) in the filtered one. \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.5,angle=90]{Lambert2006_fig6.eps}\n    \\caption{Differences between the mean fitted frequencies $\\langle \\tilde\n\\nu_{n,\\ell} \\rangle$ and the input frequencies $\\nu_{in}$, for $\\ell=0,1,2$,\nfor the raw (dashed line with triangles) and filtered (full line with diamonds)\nspectra ($S\/N=5$, 150-day observation). The error bars correspond to the\ndispersion $\\sigma^*(\\tilde\\nu_{n,\\ell})$ of the frequency distribution. For\nclarity the values for the raw case are shifted by $20~\\mu$Hz towards the\nright.\\label{ecart}}\n\\end{figure*}\n\n\\section{Conclusions}\n\nThe application of a noise reduction technique based on the curvelet transform\nto echelle diagrams improves the identification -- ``peak tagging'' -- of\nstellar acoustic modes. In observations with a $S\/N$ ratio as small as 5 we are\nstill able to recover the mode pattern and extract reliable asteroseismic\ninformation in both small and long runs (30-day and 150-day observations\nrespectively). Below this S\/N and with shorter observations, the method\nefficiency is reduced drastically. The rotational splittings and the\nrotation-axis inclination can be better estimated using the filtered spectrum.\nIn particular, Monte Carlo simulations showed that a better extraction of the\nmean rotational splitting from modes at low frequency can be done in 90 out of\n100 realizations using the filtered spectra. The uncertainty on the extracted\nrotational splitting of a typical sun-like star seen with an inclination angle\n$i=50\\degr$ and with a rotation $\\Omega=10 \\Omega_{\\sun}$ is very small,\n$\\sim$0.30 $\\mu$Hz. These parameters can then be used to have a set of guesses\nor \\textit{a priori} values to perform individual fits of the spectra. We have\nalso shown that the range of the frequency extraction can be extended at higher\nand lower frequencies using the filtered spectra. Finally, simulations of the\nshort run observations have demonstrated that this method can also be applied to\nlower resolution spectra with good results.\n\n\\begin{acknowledgements}\nP. Lambert thanks Dr. D. Neuman for useful discussions. \n\\end{acknowledgements}\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe binding energy of the nucleus, or its mass, contains\ninformation about interactions between its constituent protons and\nneutrons. Precision mass data as well as separation energies\nextracted from them can unveil much about the underlying nuclear\nstructure~\\cite{Blaum,Blaum2}. For instance, magic numbers can easily be seen from\nseparation energies which reveal a huge drop (depending on the\nsize of the closed shell), or a jump or a gap (depending on the way the\nseparation energy is constructed) after a magic number~\\cite{Novikov}. \nIn addition, separation energies can reflect\ncollective effects as we will discuss below. Such data are\ntherefore valuable in understanding the underlying shell structure\nin nuclei and the evolution of collective effects, and are\ntherefore essential to provide the basis for the development and\ntesting of a comprehensive theory of nuclei. In turn, a reliable\nnuclear theory is of utmost importance for calculating the\nproperties of unknown nuclei, which are needed, for example, in\nthe modelling of the rapid neutron capture process (r-process) of\nnucleosynthesis in stars~\\cite{Bertulani}. \nTherefore, new data on neutron-rich heavy\nnuclei are essential.\n\nHowever, such data are scarce mainly due to the complexity of\nproducing these exotic nuclei. This becomes evident if one\nglances at the chart of nuclides (see e.g. Ref.~\\cite{nchart}) where\nfor elements $Z\\sim70-80$ the number of observed neutron-rich\nnuclides is very limited. Very recently, several tens of new\nisotopes were discovered in projectile fragmentation of uranium\nbeams, though no spectroscopic information could yet be obtained for\nthese nuclei~\\cite{kurcewicz}. Their production rates are tiny, \nwhich requires very efficient measurement\ntechniques. One such technique for mass measurements is\nstorage-ring mass spectrometry~\\cite{FGM}.\n\nIn this paper we report on direct mass measurements of\nneutron-rich nuclei in the element range from lutetium to osmium.\nMasses for nine nuclei were measured for the first time, and for\nthree nuclei the mass uncertainty was improved. \nIt is known that nuclear collective effects can be seen in the behavior of nucleon separation energies~\\cite{Cakirli2}.\nHere, we investigate the relation between rather subtle effects in two-neutron separation energies, $S_{2n}$,\nand changes in both collectivity and neutron number.\nObserved irregularities in the smooth two-neutron separation\nenergies for Hf and W isotopes are linked to changes in\ncollective observables. The importance of the number of valence\nnucleons is discussed in the context of collective\ncontributions to binding energies calculated with the IBA\nmodel~\\cite{IBA}.\n\n\\section{Experiment}\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=\\textwidth]{fig1m.pdf}\n\\caption{Example of a Schottky frequency spectrum with the\ncorresponding isotope identification. This spectrum is\ncombined from eight independent injections acquired for the electron\ncooler voltage $U_c=209$~kV. Nuclides with known and previously\nunknown masses are indicated by different fonts (see legend).}\n\\label{id}\n\\end{figure*}\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\linewidth]{fig2_u}\n\\caption{A zoom of the Schottky frequency spectrum illustrated in\nFig.~\\ref{id} on a quadruplet of $A=190$ isobars. The peaks\ncorresponding to nuclides in isomeric states are labeled with\n$m$.} \\label{spzoom}\n\\end{figure}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig3}\n\\caption{(Color online) A part of the chart of nuclides indicating the\nnuclides measured in this work as well as the nuclides in the\nground and isomeric states identified in the other part of this\nexperiment devoted to the search for new K-isomers in this\nregion~\\cite{Matt,Matt2}.} \\label{nchart}\n\\end{figure*}\nThe experiment was conducted at GSI Helmholtzzentrum f\\\"ur\nSchwerionenforschung in Darmstadt. Heavy neutron-rich nuclei of\ninterest were produced in projectile fragmentation of $^{197}$Au\nprimary beams. The experiment described here was part of a\nlarger experimental campaign, some results of which are described\nin Refs.~\\cite{Matt,Matt2,Matt25,Matt3}. The $^{197}$Au beams were accelerated\nto the energy of 11.4~MeV\/u in the linear accelerator UNILAC and\nthen injected into the heavy ion synchrotron SIS-18~\\cite{sis},\nwhere they were further accelerated to an energy of 469.35~MeV\/u.\nThe $^{197}$Au$^{65+}$ beams were fast extracted (within about\n1~$\\mu$s) and focused on a production target located at the\nentrance of the fragment separator FRS~\\cite{frs,Geissel1992NIM}.\nAs target we used 1036~mg\/cm$^2$ thick $^9$Be with a 221~mg\/cm$^2$\nNb backing for more efficient electron stripping. The reaction\nproducts emerged from the target as highly-charged ions having\nmostly 0, 1, or 2 bound electrons. The nuclides of interest were\ntransported through the FRS, being operated as a pure magnetic\nrigidity ($B\\rho$) analyzer~\\cite{Geissel1992NIM}, and injected\ninto the cooler-storage ring ESR~\\cite{esr}. The transmission\nthrough the FRS and the injection into the ESR were optimized with\nthe primary beam, and the magnetic setting of FRS-ESR was fixed at\n$B\\rho=7.9$~Tm throughout the entire experiment. All ion species\nwithin the acceptance of the FRS-ESR of about $\\pm0.2$\\% were\ninjected and stored. Only $25$\\% of the ESR acceptance is filled\nat the injection. We note, that in contrast to the settings\ndescribed in Refs.~\\cite{Matt,Matt2}, in this experiment no\nenergy-loss degraders were employed in the FRS.\n\nThe relationship between relative revolution frequencies ($f$),\nrelative mass-over-charge ratios ($m\/q$) and velocities ($v$) of\nthe particles stored in a ring is given\nby~\\cite{FGM,Ra-PRL,Ra-NPA,LiBo}:\n\\begin{equation}\n\\label{sms1}\n\\frac{\\Delta f}{f}=-\\alpha_p\\frac{\\Delta \\frac{m}{q}}{\\frac{m}{q}}+(1-\\alpha_p\\gamma^2)\\frac{\\Delta v}{v},\n\\end{equation}\nwhere $\\gamma$ is the relativistic Lorentz factor, $\\alpha_p$\nis the momentum compaction factor, which characterizes\nthe relative variation of the orbit length of stored particles per\nrelative variation of their magnetic rigidity (for more details\nsee Refs.~\\cite{FGM,Ra-PRL,Ra-NPA,LiBo}). For the ESR, $\\alpha_p$\nis nearly constant for the entire revolution frequency acceptance\nand is $\\alpha_p\\approx0.179$. From Eq.~\\eqref{sms1} it becomes\nobvious that the revolution frequency is a measure of the\nmass-over-charge ratios of the stored ions provided that the\nsecond term on the right hand side, containing the velocity spread\n($\\Delta v\/v$), can be eliminated. The latter is achieved by\napplying electron cooling~\\cite{elcool}. For this purpose the\nstored ions are merged over a length of about 2.5~m with a\ncontinuous beam of electrons in the electron cooler device. The\nmean circumference of the ESR is 108.4~m and at our energies the\nions circulate with a frequency of about 2~MHz passing the\nelectron cooler at each revolution. The energy of the electrons is\nvery accurately defined by the applied acceleration potential.\nWithin a few seconds the mean velocity of the ions becomes equal\nto the mean velocity of the electrons. The velocity spread of the\nstored ions, which is $\\Delta v\/v\\approx4\\cdot10^{-3}$ at the injection,\nis thereby reduced to $\\Delta v\/v\\approx10^{-7}$~\\cite{elcool}.\n\nThe Schottky mass spectrometry (SMS) technique has been applied to\nthe electron cooled ions~\\cite{Ra-NPA,FGM}. In this technique,\nevery stored highly-charged ion at each revolution in the ESR\ninduces mirror charges on a couple of parallel electrostatic\ncopper plates, the Schottky pick-up installed inside the ring aperture.\nThe noise from the pick-up, which is dominated by the\nthermal noise, is amplified by a broad-band\nlow-noise amplifier~\\cite{Schaaf}. In the present experiment, we analyzed the\nnoise power at about 60~MHz, corresponding to the 30$^{th}$\nharmonic of the revolution frequency of the stored ions. The\npick-up signal was down-mixed using a $\\sim$60~MHz reference\nfrequency from an external frequency generator. The acceptance of\nthe ESR at the 30$^{th}$ harmonic corresponds to about\n320~kHz~\\cite{Li-2004,Li-NPA}. Therefore, to cover the entire ESR\nacceptance we digitized the signal with a sampling frequency of\n640~kHz using a commercial 16-bit ADC~\\cite{Kaza}. A Fourier\ntransform of the digitized data yielded the noise-power density\nspectrum, or the Schottky frequency spectrum~\\cite{FGM,Li-NPA}.\n\nNew ions were injected every few minutes. At the injection into\nthe ESR, the previously stored ions were removed. Several Schottky\nfrequency spectra were created for each injection. The parameters\nof the Fourier transform algorithm were optimized offline. A\nfrequency resolution of 4.77~Hz\/channel was chosen, which\ncorresponds to the time resolution of 0.21~s per spectrum.\nFurthermore, every 50 consecutive Schottky spectra were averaged\nto enhance the signal-to-noise ratio. Thus, Schottky spectra\nintegrated over 10~s were produced. The latter means that several\nindependent subsequent frequency spectra were obtained for each\ninjection of the ions into the ESR.\n\nThe electron cooling forces the ions to the same mean velocity\nthus filling the entire acceptance of the ESR of $\\Delta\nB\\rho\/B\\rho\\sim\\pm1.5\\%$ (see Ref.~\\cite{Ra-NPA}). Since $B\\rho=m\nv \\gamma \/ q$, by changing the velocity of the electrons in\ndifferent injections, ions with different $m\/q$ can be studied. In\nthe present experiment we varied the electron cooler voltage in\nthe range from 204~kV to 218~kV. On average eight injections were\nrecorded for each cooler setting. In order to facilitate the\nassignment of the revolution frequencies with the corresponding\nisotope identification, all spectra within each cooler setting\nwere combined together. In this case the maximum number of ion\nspecies present in each setting can be used for the\nidentification. The latter is done based on Eq.~\\eqref{sms1}.\nAs a starting point for the identification we used\nthe frequency of the stored $^{197}$Au$^{76+}$ primary ions. An\nexample of the combined Schottky frequency spectrum for an electron\ncooler voltage of $U_c=209$~kV is illustrated in Fig.~\\ref{id}.\nFig.~\\ref{spzoom} shows a zoom on a quadruplet of lines of\n$A=190$ isobars present in ground and\/or isomeric states. The\nlatter are indicated with a label $m$. The peak finding and the\nisotope identification were done automatically with a dedicated\nROOT-based~\\cite{ROOT} software~\\cite{Dasha}. The nuclides\nobserved in this experiment are illustrated on the chart of nuclides\nin Fig.~\\ref{nchart} together with the nuclides in the ground\nand isomeric states identified in the other part of this\nexperiment (see Refs.~\\cite{Matt,Matt2,Matt25,Matt3}).\n\n\\section{Data Analysis and Results}\n\\begin{table}[t!]\n\\caption{Nuclides with accurately known masses used as references\nto calibrate Schottky frequency spectra. Listed are the proton\n($Z$) and mass ($A$) numbers, the number of experimental settings \n($N_{set}$) in which this reference mass was observed, literature\nmass excess values from the Atomic-Mass Evaluation ~\\cite{AME} \n($ME_{AME}$) as well as the re-determined mass excess values\n($ME$) (see text) with the corresponding $\\sigma_{stat}$ uncertainty \nand its difference to the literature value\n($\\delta=ME-ME_{AME}$). \nNote that the systematic uncertainty of $\\sigma_{syst}=38$~keV (see text) is \nnot added here.\n\\label{references}}\n\\begin{center}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\nZ & A &  $N_{set}$ & $ME_{\\rm AME}$ & $ME$& $\\delta$ \\\\\n   &     &                     &  (keV)                      & (keV)                      &         (keV)     \\\\\n\\hline\n72 & 181 & 1 & -47412(2) & -47412(40) & 0(40)  \\\\\n\\hline\n73 & 181 & 1 & -48442(2) & -48383(40)& 59(40)  \\\\\n73 & 182 & 3 & -46433(2) & -46466(29)& -32(29)  \\\\\n73 & 183 & 3 & -45296(2) & -45276(16)& 20(16)  \\\\\n73 & 185 & 6 & -41396(14)& -41350(14)& 46(20)  \\\\\n\\hline\n74 & 184 & 5 & -45707(1) & -45663(17)& 44(17)  \\\\\n74 & 186 & 7 & -42510(2) & -42493(12)& 17(13)  \\\\\n74 & 187 & 8 & -39905(2) & -39863(8)& 41(8)  \\\\\n\\hline\n75 & 189 & 9 & -37978(8) & -38063(10)& -85(13)  \\\\\n75 & 191 & 9 & -34349(10)& -34364(3)& -15(11)  \\\\\n\\hline\n76 & 188 & 7 & -41136(1) & -41115(12)& 21(12)  \\\\\n76 & 190 & 7 & -38706(2) & -38637(15)& 69(15)  \\\\\n76 & 192 & 7 & -35881(3) & -35833(8)& 48(8)  \\\\\n76 & 193 & 7 & -33393(3) & -33329(8)& 63(8)  \\\\\n\\hline\n77 & 191 & 1 & -36706(2) & -36650(71)& 56(71) \\\\\n\\hline\n78 & 194 & 4 & -34763(1) & -34779(24)& -16(24)  \\\\\n78 & 196 & 9 & -32647(1) & -32655(4)& -7(4)  \\\\\n\\hline\n79 & 196 & 5 & -31140(3) & -31126(4)& 14(5)  \\\\\n\\hline\n\\hline\n\\end{tabular}\\end{center}\n\\end{table}\n\nIn order to determine the unknown mass-over-charge ratios, the\nSchottky frequency spectra have to be\ncalibrated~\\cite{Ra-NPA,Li-NPA}. For this purpose we selected the\nnuclides which were identified in our spectra and for which masses are\nknown experimentally according to the Atomic-Mass Evaluation 2003\n(AME)~\\cite{AME}. We note that an update of the AME was made\navailable in 2011~\\cite{AME11}, which however contains no new\ninformation in the mass region studied here. \nThe data of the present work were already included in the latest AME published very recently~\\cite{AME12}.\nFurthermore we\nrequired that the reference masses were obtained by more than one\nindependent measurement technique and that there must exist no\nother ionic species with a close mass-to-charge ratio which could simultaneously be stored in the ESR.\nAlso the peaks corresponding to long-lived isomeric states (we observed\n$^{182}$Hf$^{\\rm m1}$, $^{186}$W$^{\\rm m2}$, $^{190}$Os$^{\\rm m1}$ and $^{190}$Ir$^{\\rm m2}$ isomers)\nwere not used for calibration. The list of reference masses is\ngiven in Table~\\ref{references}.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\linewidth]{fig4} \n\\caption{Top: Mass-over-charge ratio $m\/q$ as a function of revolution\nfrequency. The solid line illustrates a straight line fit through\nthe calibration $m\/q$-values. Bottom: The same as top but with\nthe subtracted linear fit.} \\label{mqf}\n\\end{figure}\n\nThe momentum compaction factor $\\alpha_p$, although nearly\nconstant, is a complicated function of the revolution frequency\n$f$. If $\\alpha_p$ were exactly constant, then the $m\/q$ would linearly\ndepend on $f$. Fig.~\\ref{mqf} (top) shows an example of a linear\nfit through the calibration $m\/q$-values for one of the measured spectra. The\nresiduals of the fit (bottom panel of the same figure) clearly\nshow that the calibration function is more complicated than a\nlow-order polynomial function. Polynomials of up to 4th order (5\nfree coefficients) were employed, but different compared to the analyses\nperformed in Refs.~\\cite{Ra-NPA, Li-NPA,Chen2012}, the quality of\nthe fits was found unacceptable. \nA possible reason for the latter is the small number of reference masses in individual 10-s Schottky spectra.\nFurthermore, due to time variations of the storage ring and electron cooler parameters, \nsuch as, e.g., their magnetic fields, it is not possible to establish a universal calibration curve. \nTherefore, we employed an analysis procedure \nin which we used linear splines to approximate the calibration curve in each individual spectrum. \n\nChanges of the electron cooler voltage were done in steps of 0.5~kV\nso that for adjacent cooler settings the measured frequency spectra have a significant overlap.\nFurthermore, the same nuclide can be present in different charge states, which allows for a\nredundant analysis. The nuclides whose masses have been measured\nfor the first time or which mass accuracy was improved in this work are listed in Table~\\ref{newm}.\n\n\\begin{table}[t!]\n\\caption{Nuclides whose masses were determined for\nthe first time (in boldface) or whose mass uncertainty was improved in this work.\nListed are the proton ($Z$) and mass ($A$) numbers, the number of\nexperimental settings ($N_{set}$) in which this nuclide was\nobserved, and the obtained mass excess value ($ME$) with the\ncorresponding $1\\sigma$ total ($\\sqrt{\\sigma^2_{stat}+\\sigma^2_{syst}}$) uncertainty ($\\sigma(ME)$).\\label{newm}}\n\\begin{center}\n\\begin{tabular}{rrrrr}\n\\hline \\hline\nZ & A & $N_{set}$ &$ME$& $\\sigma(ME)$ \\\\\n   &     &                    & (keV) & (keV)                 \\\\\n\\hline\n{\\bf 71}&   {\\bf 181}& {\\bf 1} & {\\bf -44797}& {\\bf 126}\\\\\n{\\bf 71}&   {\\bf 183}&  {\\bf 1} & {\\bf -39716}& {\\bf 80}\\\\\n\\hline\n{\\bf 72}& {\\bf 185}& {\\bf 1} & {\\bf -38320}& {\\bf 64}\\\\\n{\\bf 72}& {\\bf 186}& {\\bf 1} & {\\bf -36424}&{\\bf 51}\\\\\n\\hline\n{\\bf 73}& {\\bf 187}& {\\bf 2} & {\\bf -36896}&{\\bf 56}\\\\\n{\\bf 73}&{\\bf 188}& {\\bf 2} & {\\bf -33612}&{\\bf 55}\\\\\n\\hline\n74& 189& 5 &    -35618& 40\\\\\n74& 190& 7 &    -34388& 41\\\\\n{\\bf 74}&{\\bf 191}&{\\bf 1} & {\\bf -31176}&{\\bf 42}\\\\\n\\hline\n{\\bf 75}&{\\bf 192}&{\\bf 1} & {\\bf -31589}&{\\bf 71}\\\\\n{\\bf 75}&{\\bf 193}&{\\bf 7} & {\\bf -30232}&{\\bf 39}\\\\\n\\hline\n76& 195& 1 &    -29512& 56\\\\\n\\hline\n\\hline\n\\end{tabular}\\end{center}\n\\end{table}\n\nSince the calibration curve is not known exactly and is approximated with linear splines, and\nsince the number of calibration points in each spectrum is small,\nthere is inevitably a systematic error introduced by the analysis method.\nWays to estimate systematic uncertainty have been described in our previous works~\\cite{Ra-NPA,Li-NPA,Chen2012}.\nFor this purpose in the present work, we re-determined the mass of each reference nuclide. \nThis was done consecutively by setting each of the references as ``no''-reference and\nobtaining its mass from the remaining 17 references. \nThe re-determined mass excess values are listed in Table~\\ref{references} along with their literature values~\\cite{AME}. \nThe systematic error $\\sigma_{syst}$ has been obtained from solving the following equation:\n\\begin{equation}\n\\sum_{i=1}^{N_{ref}}\\frac{(ME^{(i)}_{AME}-ME^{(i)})^2}{\\sigma_{AME(i)}^2+\\sigma_{(i)}^2+\\sigma_{syst}^2}=N_{ref},\n\\end{equation}\nwhere $N_{ref}=18$ is the number of reference nuclides, $ME^{(i)}$ ($\\sigma_{(i)}$) \nand $ME_{AME}^{(i)}$ ($\\sigma_{AME(i)}$) are the re-calculated and literature mass excess\nvalues (statistical uncertainties) of the $i$-th reference nuclide, respectively.\nThe systematic uncertainty of the present analysis amounts to $\\sigma_{syst}=38 $~keV. \nThe final uncertainties listed in Table~\\ref{newm} were obtained from a quadratic sum of the systematic and statistical uncertainties.\n\nWe note, that in contrast to Ref.~\\cite{Chen2012} we do not observe any significant systematic dependence \nof the re-calculated mass values versus their proton number and correspondingly do not reduce the systematic errors.\nA dedicated study should be performed to investigate the origin of this inconsistency.\n\n\\section{Discussion}\n\n\nThe new masses allow us to obtain interesting information on\nnuclear structure. Fig.~\\ref{s2n_e2} (left, middle) shows\ntwo-neutron separation energies ($S_{2n}$) as a function of\nneutron number for $Z=66-78$ in the $A\\sim180$ region.\nFig.~\\ref{s2n_e2} (left) is for even proton numbers while\nFig.~\\ref{s2n_e2} (middle) is for odd proton numbers. The new\n$S_{2n}$ values, for $^{181,183}$Lu, $^{185,186}$Hf,\n$^{187,188}$Ta, $^{191}$W, $^{192,193}$Re, calculated from the\nmasses measured in this study, are marked in red color. For known\nmasses whose values were improved in this experiment,\n$^{189,190}$W and $^{195}$Os, the literature $S_{2n}$ values are\nillustrated with black color.\n\nBy inspecting Fig.~\\ref{s2n_e2} (left), one can notice, that in\nthe $S_{2n}$ values of the even-$Z$ nuclei a flattening in Yb, Hf\nand W is seen at almost the last neutron numbers experimentally\nknown. The flattening in $S_{2n}$(W), using the improved W masses,\nis confirmed and $S_{2n}$ at $N=117$ continues with the same\nbehavior. In contrast to Hf and W, the new $S_{2n}$(Os) point,\nwhich is a bit lower at $N=119$ than in previous measurements,\nshows no flattening.\n\n\\begin{figure*}\n\\includegraphics[height=5.4cm]{fig5a}\\hspace{8mm}\n\\includegraphics[height=5.3cm]{fig5b}\n\\includegraphics[height=5.3cm]{fig5c}\n\\caption{(Color online) Data for the $A\\sim180$ region,\ntwo-neutron separation energies \\cite{AME} as a function of\nneutron number from $N=100$ to $N=124$ for even-$Z$ (left) from Dy\nto Pt and odd-$Z$ (middle) from Ho to Re. The new $S_{2n}$ values\nobtained from this work are shown in red color while the\nliterature $S_{2n}$ values are shown in black color. Right :\nEnergies of the first excited 2$^+$ states \\cite{NDC} for the same\neven-even nuclei as in the left panel.\\label{s2n_e2}}\n\\end{figure*}\n\nFig.~\\ref{s2n_e2} (middle) with the new measured masses does not\nshow similar effects in $S_{2n}$ as in Fig.~\\ref{s2n_e2} (left).\nHowever, there is a small change in slope (a more rapid fall-off)\nat $N=110$ compared to the lower-$N$ trend in $S_{2n}$($_{71}$Lu),\n$S_{2n}$($_{73}$Ta) and $S_{2n}$($_{75}$Re) (we also see this drop\nfor $_{72}$Hf and $_{74}$W in Fig.~\\ref{s2n_e2} (left)), and maybe\nat $N=115$ in $S_{2n}$($_{73}$Ta) as well. It is highly desirable\nto have more odd-$Z$ mass measurements in this region for a\ncomparison with even-$Z$ where more data are available. Thus, we\nconcentrate below on discussing even-$Z$ nuclei.\n\n\nFig.~\\ref{s2n_e2} (right) shows the energy of the first excited\n2$^+$ states against neutron number for $Z=70-78$ in the\n$A\\sim180$ region. This important, simple, observable has a high\nenergy (can be a few MeV) at magic numbers where nuclei are\nspherical and very low energies (less than 100 keV for well\ndeformed heavy nuclei) near mid-shell. The $E$(2$_1^+$) values\nusually decrease smoothly between the beginning and middle of a\nshell except when there is a sudden change in structure. Since\n$N\\sim104$ is mid-shell for the nuclei illustrated in\nFig.~\\ref{s2n_e2}, $E$(2$_1^+$) has a minimum at or close to\n$N=104$. After the mid-shell, the energy increases towards the\n$N=126$ magic number.\n\nLet us now focus on the W-Pt nuclei in Fig.~\\ref{s2n_e2} and\ncompare the behavior of $E$(2$_1^+$) and $S_{2n}$. In particular,\nwe look at $S_{2n}$ in isotopes where $E$(2$_1^+$) changes\nrapidly, indicating a sudden change in structure. In W,\n$E$(2$_1^+$) increases from $N=114$ to 116 by a considerably\nlarger amount compared to the other W isotopes (see\nRef.~\\cite{podolyak2000}). This jump signals a structural change\nfrom approximately constant deformation to decreasing deformation\nat $N=116$ (after $N=114$). Note that this neutron number is\nexactly where $S_{2n}$ exhibits flattening.\n\nSimilar to W, Os at $N=120$ (after $N=118$) has a jump in\n$E$(2$_1^+$). However, $S_{2n}$($^{196}$Os) does not reveal an\nobvious change at the same neutron number. At first glance, this\nseems inconsistent with the interpretation explained for W above\nbut, in fact, there might be an explanation for this different\nbehavior which would provide additional insight into the relation\nof binding to collectivity.\n\nReference~\\cite{Cakirli2} showed the structural sensitivity of\ncalculated collective contributions to binding. In addition\nRef.~\\cite{Cakirli2} stressed that collective binding is very\nsensitive to the number of valence nucleons.\nCalculated collective contributions to binding using the IBA-1\nmodel for boson numbers $N_{B}=5$ (left) and $N_{B}=16$ (right)\nare illustrated in the symmetry triangle \\cite{Cakirli2} in\nFig.~\\ref{triangle}. The three corners of the triangle describe\nthree dynamical symmetries, U(5) (vibrator), SU(3) (rotor) and\nO(6) ($\\gamma$-soft) (for more details, see Ref.~\\cite{Iachello}).\nThe color code in Fig.~\\ref{triangle} changes from yellow to red\nwhen the collective effects increase. Needless to say, nuclei have\nmore valence particles (so boson numbers) around the SU(3) corner\nthan the U(5) (and also O(6)) corner. One sees that the collective\nbinding energy (B.E.) rapidly increases for nuclei with axial\ndeformation, that is, near SU(3). Note that the triangles are\npresented for fixed boson numbers. In both, the color scale is\nkept the same to point out that the collective B.E.s are larger in\nN$_B$=16 than 5. As shown in Fig.~4 of Ref.~\\cite{Cakirli2}, the\ncollective binding energies vary approximately as the square of\nthe number of valence nucleons in the context of IBA calculations.\nTherefore, for a lower number of valence nucleons,\nFig.~\\ref{triangle} shows similar trends for both  $N_{B}=5$ and\n$N_{B}=16$, but the overall binding is considerably less (compare\n(left) and (right) of Fig.~\\ref{triangle}). We now suggest that\nthe behavior of $E(2_1^+)$ and $S_{2n}$ in Fig.~\\ref{s2n_e2} can\nbe understood in terms of this dual dependence of binding on\ncollectivity and valence nucleon number.\n\nIf $^{190}$W and $^{196}$Os are mapped in the symmetry triangle,\n$^{190}$W will likely be closer to the SU(3) corner than\n$^{196}$Os \\cite{Cakirli-pri}. One of the ways to understand this\nis from the $P$-factor \\cite{RCasten2}, defined as $P= N_p\\cdot N_n \/ (N_p +\nN_n)$, where $N_p$ denotes the number of valence protons (proton\nholes) and $N_n$ the number of valence neutrons (neutron holes).\nThus $P$ is a quantity that can provide a guide to structure. For\nexample, the onset of deformation in heavy nuclei corresponds to\nthe $P$-factor around 4 and 5. Generally, if $P$ is larger than 3,\ncollective effects increase. That is, nuclei become deformed and\napproach closer to the SU(3) corner.\n\nFig.~\\ref{Pfactor} shows color-coded values for the $P$-factor\nfor the $Z=50-82$, $N=82-126$ region, and indicates the\n$P$-factors for the nuclei relevant to this discussion. $^{190}$W\nhas 8 valence protons and 10 valence neutrons while $^{196}$Os has\n$N_p=6$ and $N_n=6$. Correspondingly, these nuclei have\n$P$-factors of 4.4 and 3, respectively. The greater collectivity\nof $^{190}$W compared to $^{196}$Os suggested by their $P$-factors\nis reflected in its lower 2$_1^+$ energies as seen in\nFig.~\\ref{s2n_e2} (right). Thus, for two reasons -- both greater\ncollectivity and more valence nucleons -- the collective binding\nshould be much greater in $^{190}$W than in $^{196}$Os and changes\nin binding energies ($S_{2n-coll}$ values) should be on a larger\nscale. We suggest that this accounts for the fact that we see a\nflattening in $^{190}$W clearly but not in $^{196}$Os in\nFig.~\\ref{s2n_e2} (left). Obviously, it is very important to have\nnew data, both masses and spectroscopic information, on even more\nneutron rich W isotopes although such experiments are difficult.\nEven the mass of $^{192}$W alone would be telling since the trend\nin $E$(2$_1^+$) is quite clear already.\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig6}\n\\caption{(Color online) The symmetry triangle of the IBA showing\nthe three dynamical symmetries at the vertices. The colors\nindicate calculated collective contributions in MeV to binding\nenergies for $N_B = 5$ (left) and $N_B=16$ (right). A similar\ntriangle for $N_B=16$ was presented in\nRef.~\\cite{Cakirli2}.\\label{triangle}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig7}\n\\caption{(Color online) $P$-factor values illustrated with a color\ncode for even-even nuclei in the $Z=50-82$ and $N=82-126$ shells.\nBlack points marked are for the key nuclei discussed, namely,\n$^{190}$W, $^{196}$Os, $^{188}$Pt,  $^{198}$Pt, and\n$^{152}$Sm.\\label{Pfactor}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig8}\n\\caption{Calculated $E$(2$_1^+$) (top), $\\delta E$(2$_1^+$) (middle)\nand $\\delta S_{2n-coll}$ (bottom) values from IBA calculations as\na function of boson number $N_B$. The points for\n$\\delta E$(2$_1^+)$ and $\\delta S_{2n-coll}$  correspond to a set\nof schematic IBA calculations in which $\\kappa$, and $\\chi$ are\nconstant (at 0.02 and -1.32, respectively) while $\\epsilon$, and\n$N_B$ vary in a smooth way to simulate a spherical-to-deformed\ntransition region. The following equations are used for\n$\\delta E$(2$_1^+$) and $\\delta S_{2n-coll}$:\n$\\delta E(2_1^+)(Z,N)=[E(2_1^+)(Z,N) - E(2_1^+)(Z,N+2)] \/\nE(2_1^+)(Z,N)$ and\n$\\delta S_{2n-coll}=-[S_{2n-coll}(Z,N) -\nS_{2n-coll}(Z,N+2)$], respectively.\\label{dS2n-E2}}\n\\end{figure}\n\nExisting data on Pt nicely illustrate and support these ideas.\nFig.~\\ref{s2n_e2} (right) shows two jumps in $E$(2$_1^+$) for\nPt, around $N\\sim110$ and $N\\sim118-120$. Looking at $S_{2n}$ for\nPt, there is a kink near $N\\sim110$ but a smooth behavior near\n$N\\sim118$. For $^{188}$Pt, $N_p$ is 4 and $N_n$ is 16 so the\n$P$-factor is 3.2. This isotope, with 20 valence nucleons, is\nrelatively collective and once again one sees an anomaly in\n$S_{2n}$ as well. In contrast, for $^{198}$Pt$_{120}$ with only 10\nvalence nucleons, and a $P$-factor of only 2.4, the lower\ncollectivity (seen in the much higher 2$^+$ energy) and the lower\nnumber of valence nucleons are such that $S_{2n}$ shows no\nanomaly, but rather a nearly straight behavior.\n\nThis qualitative interpretation is supported by collective model\ncalculations. A thorough and detailed study of this or any\ntransition region requires a very careful and systematic\nassessment of all the data on energies, transition rates, and\nbinding energies, the choice for the specific terms to include in\nthe Hamiltonian and the optimum approach to fitting the data. We\nare undertaking such a study and will present the results in a\nfuture publication \\cite{Cakirli3}. Nevertheless, it is useful to\npresent an example of the model results here to validate the ideas\npresented above. To this end, we have carried out a schematic set\nof IBA calculations using the Hamiltonian \\cite{7,8}\n\n\\begin{equation}\n\\label{eqH}\nH = \\epsilon \\hat{n}_d - \\kappa{Q} \\cdot {Q}\n\\end{equation}\n\n\\noindent where $Q$ is a quadrupolar operator\n\n\\noindent\n\n\\begin{equation}\n\\label{eqQ} \n{Q}= (s^{\\dagger}\\tilde{d} +\nd^{\\dagger}s) + \\chi(d^{\\dagger}\\tilde{d})^{(2)}.\n\\end{equation}\n\n\\noindent The first term in Eq.~\\eqref{eqH} drives nuclei spherical while\nthe $Q\\cdot Q$ term induces collectivity and deformation. Therefore a\nspherical-deformed transition region involves a systematic change\nin the ratio of $\\epsilon$ to $\\kappa$. No generality is lost by\nkeeping $\\kappa$ constant (at 0.02 MeV). We follow a trajectory\nalong the bottom axis of the triangle corresponding to $\\chi\n=-1.3228$. Fig.~\\ref{dS2n-E2} illustrates the results, for $N_B\n= 6-16$ showing $E$(2$_1^+$) and the differentials of $E$(2$_1^+$) (for\n$N_B = 6-15$) and for the collective contributions to $S_{2n}$,\n$S_{2n-coll}$, (for $N_B = 6-14$). There is a clear change in\nstructure  at $N_B\\sim$10 which is seen in a change in trend of\n$E$(2$_1^+$). Between $N_B = 10$ and 11, R$_{4\/2}$ changes from 2.60\nto 3.13. This corresponds to a maximum in the normalized\ndifferential of $E$(2$_1^+$). Confirming our association of\nstructural changes with kinks in $S_{2n}$, the differential of the\ncollective part of $S_{2n}$ also shows an extremum at exactly the\nsame point. These ideas will be expanded in our future publication\n\\cite{Cakirli3}.\n\nBesides the experimental examples of a correlation of $E$(2$_1^+$) energies \nand $S_{2n}$ values discussed in the context of Fig.~\\ref{s2n_e2},  \nour interpretation can\neasily be illustrated with the Sm isotopes around $N=90$. As is well\nknown, there is a sudden onset of deformation for the rare earth\nnuclei from $N=88$ to 90. This effect is clear from various\nobservables. One example is seen in Fig.~\\ref{Sm-S2n-E2} which\nshows the experimental $E$(2$_1^+$) energies (top) and $S_{2n}$\n(bottom) as a function of neutron number. Note that we plot these\nagainst decreasing neutron number so that the deformed nuclei are\non the left and spherical ones on the right to make the comparison\nwith Fig.~\\ref{s2n_e2} easier. Note also that the overall trend in\n$S_{2n}$ is opposite from that in Fig.~\\ref{s2n_e2} since $S_{2n}$\nvalues decrease with increasing neutron number (going to the left in Fig.~\\ref{Sm-S2n-E2} (bottom) which simply \nreflects the filling of the shell model orbits. The noticeable\ndeviation occurs near $N\\sim90$ where there is\na distinct flattening. To correlate the trends in these two\nobservables in the $N=90$ region, one can use the same\ninterpretation as above for W at $N=116$, namely, if there is a\nvisible change at neutron number $N$ in $E$(2$_1^+$) and there are\nmany valence nucleons, we expect to see a change in the behavior\nof $S_{2n}$. In Fig.~\\ref{Sm-S2n-E2} (top), the $E$(2$_1^+$) change\noccurs at $N\\sim90$. The isotope $^{62}$Sm at $N=90$ has $N_p=12$ and\n$N_n=8$ so it has 10 bosons and its $P$-factor is 4.8 (see\nFig.~\\ref{Pfactor}). One therefore expects to see a change in\n$S_{2n}$. Fig.~\\ref{Sm-S2n-E2} (bottom) confirms this\nexpectation. The clear structural change at $N=90$ shown in\n$E$(2$_1^+$) is correlated with a larger binding compared to the\ngeneral trend in $S_{2n}$ as a function of $N$.\n\nSimilar correlations can be seen in some other nuclei as well.\nFurther details will be discussed in Ref.~\\cite{Cakirli3}.\nHowever, here, it is worth mentioning two more examples marked in\nFig.~\\ref{s2n_e2}. The case of Yb-isotopes is interesting. Yb at\n$N=107$ starts to change slope in $S_{2n}$ and a flattening occurs\nat $N=108$. The $P$-factor is $\\sim7$. With the interpretation\nabove, one would expect to see a change in $E$(2$_1^+$) after\n$N=106$, at $N=108$. However, there is no sudden change in\n$E$(2$_1^+$) in $^{178}$Yb. To understand Yb around $N\\sim108$\nbetter, it might therefore be useful to have additional $S_{2n}$\nvalues (mass measurements) and also more spectroscopic results for\nthe neutron-rich Yb isotopes.\n\n\nHf at $N=114$ has a $P$-factor 5.4 and one sees a flattening in\n$S_{2n}$. The corresponding 2$^+$ energies, however, are not\nknown. Thus, similarly as in Yb, we need more spectroscopic\nresults for Hf.\n\nTo summarize, we observed a correlation between the behavior of\n$S_{2n}$ obtained from our measured masses with the spectroscopic\ndata for $E(2_1^+)$, which could be related to nuclear\ncollectivity and valence nucleon number.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig9}\n\\caption{Experimental $E$(2$_1^+$) (top) and\n$S_{2n}$ (bottom) values against neutron number for $_{62}$Sm\n\\cite{AME11, NDC}. \\label{Sm-S2n-E2}}\n\\end{figure}\n\n\\section{Conclusion}\n\nDirect mass measurements of neutron-rich $^{197}$Au projectile\nfragments at the cooler-storage ring ESR yielded new mass data.\nMasses of nine nuclides were obtained for the first time and for\nthree nuclei the mass uncertainty was improved.\n\nWith the new masses, two-neutron separation energies, $S_{2n}$,\nare investigated. We showed that changes in structure, as\nindicated by changes in the collective observable $E$(2$^+_1$),\nare reflected in $S_{2n}$ values in nuclei such as $^{190}$W and\n$^{188}$Pt, which have large $P$-factors, are collective,  and\nhave large valence nucleon numbers. For nuclei with similar\nchanges in $E$(2$_1^+$), such as $^{196}$Os, and $^{198}$Pt, which\nhave lower collectivity and $P$-factors, and fewer valence\nnucleons, the sensitivity of collective binding to structure is\ngreatly reduced and smooth trends in $S_{2n}$ are observed. In Hf,\nthere are new $S_{2n}$ values at $N=113$, 114 where we see a\nflattening but there is no spectroscopic data at $N=114$. To\nconfirm the ideas discussed in this paper and also in\nRef.~\\cite{Cakirli3}, it would be useful to measure the\n$E$(2$_1^+$) for Hf at $N=114$. Similarly, mass and spectroscopic\nmeasurements are suggested for nuclei such as Yb with $N\\sim108$.\nTo conclude, these new data illustrate subtle changes in structure\nand the correlation with $E$(2$_1^+$) reveals a valuable way to\ncorrelate changes in structure in terms of both masses and\nspectroscopic observables. Of course, to quantitatively test these\nideas requires a systematic collective model study of the\nmass-structure relationship in this region. Such a project has\nbeen initiated~\\cite{Cakirli3} and we illustrated some of the\nresults here.\n\n\n\n\n\\section{Acknowledgments}\n\n\nThe authors would like to thank the GSI accelerator team for the excellent technical support. \nThis work was supported by the BMBF Grant in the framework of the Internationale Zusammenarbeit in Bildung und Forschung Projekt-FKZ 01DO12012,\nby the Alliance Program of the Helmholtz Association (HA216\/EMMI), by the Max-Planck Society and the US DOE under Grant No. DE-FG02-91ER-40609.\nD.S. is supported by the International Max Planck Research School for Precision Tests of Fundamental Symmetries at MPIK. \nR.B.C. thanks the Humboldt Foundation for support.\nK.B. and Y.A.L. thank ESF for support within the EuroGENESIS program.\nK.B. acknowledge support by the Nuclear Astrophysics Virtual Institute (NAVI) of the Helmholtz Association. \nZ.P. would like to acknowledge the financial support by Narodowe Centrum Nauki (Poland) grant No. 2011\/01\/B\/ST2\/05131. \nM.S.S. acknowledges the support by the Helmholtz International Centre for FAIR within the framework of the LOEWE program launched by the State of Hesse. \nB.S. is partially supported by NCET, NSFC (Grants No. 10975008, 11105010 and 11035007).\nP.M.W. acknowledges the support by the UK STFC and AWE plc.\nT.Y. is grateful for a grant-in-aid for a scientific research No. A19204023 by the Japanese Ministry of Education, Science and Culture. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nDespite the fantastic successes of the Standard Model (SM)\nof electroweak interactions,\nits scalar sector remains largely untested~\\cite{hhg}.\nAn alternative to the single Higgs doublet of the SM\nis provided by the two-Higgs-doublet model (THDM),\nwhich can be supplemented by symmetry requirements\non the Higgs fields $\\Phi_1$ and $\\Phi_2$.\nSymmetries leaving the kinetic terms unchanged\\footnote{It has\nbeen argued by Ginsburg~\\cite{Gin} and by\nIvanov~\\cite{Ivanov1,Ivanov2} that one should also consider\nthe effect of non-unitary global symmetry transformations\nof the two Higgs fields, as the most general renormalizable\nHiggs Lagrangian allows for kinetic mixing of the two Higgs fields.\nIn this work, we study the possible\nglobal symmetries of the effective low-energy Higgs theory that arise\n\\textit{after} diagonalization of the Higgs kinetic energy terms.\nThe non-unitary transformations that diagonalize the Higgs kinetic\nmixing terms also transform the parameters of the Higgs potential,\nand thus can determine the structure of the remnant Higgs flavor symmetries\nof effective low-energy Higgs scalar potential.  It is the latter \nthat constitutes the main focus of this work.}\nmay be of two types.  On the one hand,\none may relate $\\Phi_a$ with some unitary transformation\nof $\\Phi_b$.\nThese are known as Higgs Family symmetries, or HF symmetries.\nOn the other hand,\none may relate $\\Phi_a$ with some unitary transformation\nof $\\Phi_b^\\ast$.\nThese are known as generalized CP symmetries,\nor GCP symmetries.\nIn this article we consider all such symmetries that are\npossible in the THDM,\naccording to their impact on the Higgs potential.\nWe identify three classes of GCP symmetries.\n\nThe study is complicated by the fact that one may perform a basis\ntransformation on the Higgs fields, thus hiding what might otherwise\nbe an easily identifiable symmetry. The need to seek basis invariant\nobservables in models with many Higgs was pointed out by Lavoura and\nSilva \\cite{LS}, and by Botella and Silva \\cite{BS}, stressing\napplications to CP violation. Refs.~\\cite{BS,BLS} indicate how to\nconstruct basis invariant quantities in a systematic fashion for any\nmodel, including multi-Higgs-doublet models. Work on basis\ninvariance in the THDM was much expanded upon\nby Davidson and Haber \\cite{DavHab},\nby Gunion and Haber \\cite{GunHab,Gun},\nby Haber and O'Neil \\cite{HabONe},\nand by other authors \\cite{others}.\nThe previous approaches highlight the role\nplayed by the Higgs fields. An alternative approach, spearheaded by\nNishi \\cite{Nishi1,Nishi2}, by Ivanov \\cite{Ivanov1,Ivanov2} and by\nManiatis {\\em et al}~\\cite{mani}, highlights the role played by\nfield bilinears, which is very useful for studies of the vacuum\nstructure of the model \\cite{Barroso,earlier_bilinears}. In this\npaper, we describe all classes of HF and GCP symmetries in both\nlanguages.\nOne problem with two classes of GCP identified here is that\nthey lead to an exceptional region of parameter space\n(ERPS) previously identified as problematic\nby Gunion and Haber \\cite{GunHab} and\nby Davidson and Haber \\cite{DavHab}.\nIndeed, no basis invariant quantity exists in the literature that\ndistinguishes between the $Z_2$ and $U(1)$ HF symmetries in\nthe ERPS.\n\nIf evidence for THDM physics is revealed in future experiments, then\nit will be critical to employ analysis techniques that are free\nfrom model-dependent assumptions.  It is for this reason that\na basis-independent formalism for the THDM is so powerful.\nNevertheless, current experimental data already impose significant\nconstraints on the most general THDM.   In particular, we know that\ncustodial symmetry breaking effects, flavor changing neutral\ncurrent (FCNC) constraints, and (to a lesser extent) CP-violating phenomena \nimpose some significant restrictions on the structure of the THDM\n(including the Higgs-fermion interactions).  For example,\nthe observed suppression of FCNCs implies that either the two heaviest\nneutral Higgs bosons of the THDM have masses above 1 TeV,\nor certain Higgs-fermion Yukawa couplings must be absent~\\cite{Pas}.\nThe latter can be achieved by imposing certain discrete symmetries on\nthe THDM.  Likewise, in the most general THDM, mass splittings between\ncharged and neutral Higgs bosons can yield custodial-symmetry breaking\neffects at one-loop that could be large enough to be in \nconflict with the precision electroweak data~\\cite{precision}.  \nOnce again, symmetries\ncan be imposed on the THDM to alleviate any potential disagreement\nwith data.  The implications of such symmetries for THDM phenomenology\nhas recently been explored by Gerard and collaborators~\\cite{Gerard}\nand by Haber and O'Neil~\\cite{custodial}.\n\nThus, if THDM physics is discovered, it will be important to \ndevelop experimental methods that can reveal the presence or absence\nof underlying symmetries of the most general THDM.  This requires two\nessential pieces of input.  First, one must identify all possible\nHiggs symmetries of interest.  Second, one must relate these\nsymmetries to basis-independent observables that can be probed by\nexperiment.  In this paper, we primarily address the first step,\nalthough we also provide basis-independent characterizations of these\nsymmetries.  Our analysis focuses the symmetries of the THDM scalar\npotential.  In principle, one can extend our study of these symmetries to the\nHiggs-fermion Yukawa interactions, although this lies beyond the scope\nof the present work.\n\n\nThis paper is organized as follows.\nIn section~\\ref{sec:notation} we\nintroduce our notation and define an invariant that does\ndistinguish the $Z_2$ and $U(1)$ HF symmetries in\nthe ERPS.\nIn section~\\ref{sec:vacuum} we explain the role\nplayed by the vacuum expectation values in\npreserving or breaking the $U(1)$ symmetry,\nand we comment briefly on renormalization.\nIn section~\\ref{sec:GCP} we introduce the GCP\ntransformations and explain why they are organized\ninto three classes.\nWe summarize our results and set them in the\ncontext of the existing literature in section~\\ref{sec:summary},\nand\nin section~\\ref{sec:allisCP} we prove a surprising result:\nmultiple applications of\nthe standard CP symmetry can be used to\nbuild all the models we identify,\nincluding those based on  HF symmetries.\nWe draw our conclusions in\nsection~\\ref{sec:conclusions}.\n\n\n\\section{\\label{sec:notation}The scalar sector of the THDM}\n\n\\subsection{Three common notations for the scalar potential}\n\nLet us consider a $SU(2) \\otimes U(1)$ gauge theory with\ntwo Higgs-doublets $\\Phi_a$,\nwith the same hypercharge $1\/2$,\nand with vacuum expectation values (vevs)\n\\begin{equation}\n\\langle \\Phi_a \\rangle\n=\n\\left(\n\\begin{array}{c}\n0\\\\\nv_a\/\\sqrt{2}\n\\end{array}\n\\right).\n\\label{vev}\n\\end{equation}\nThe index $a$ runs from $1$ to $2$,\nand we use the standard definition for the electric\ncharge,\nwhereby the upper components of the $SU(2)$ doublets are\ncharged and the lower components neutral.\n\nThe scalar potential may be written as\n\\begin{eqnarray}\nV_H\n&=&\nm_{11}^2 \\Phi_1^\\dagger \\Phi_1 + m_{22}^2 \\Phi_2^\\dagger \\Phi_2\n- \\left[ m_{12}^2 \\Phi_1^\\dagger \\Phi_2 + \\textrm{H.c.} \\right]\n\\nonumber\\\\[6pt]\n&&\n+ \\tfrac{1}{2} \\lambda_1 (\\Phi_1^\\dagger\\Phi_1)^2\n+ \\tfrac{1}{2} \\lambda_2 (\\Phi_2^\\dagger\\Phi_2)^2\n+ \\lambda_3 (\\Phi_1^\\dagger\\Phi_1) (\\Phi_2^\\dagger\\Phi_2)\n+ \\lambda_4 (\\Phi_1^\\dagger\\Phi_2) (\\Phi_2^\\dagger\\Phi_1)\n\\nonumber\\\\[6pt]\n&&\n+ \\left[\n\\tfrac{1}{2} \\lambda_5 (\\Phi_1^\\dagger\\Phi_2)^2\n+ \\lambda_6 (\\Phi_1^\\dagger\\Phi_1) (\\Phi_1^\\dagger\\Phi_2)\n+ \\lambda_7 (\\Phi_2^\\dagger\\Phi_2) (\\Phi_1^\\dagger\\Phi_2)\n+ \\textrm{H.c.}\n\\right],\n\\label{VH1}\n\\end{eqnarray}\nwhere $m_{11}^2$, $m_{22}^2$, and $\\lambda_1,\\cdots,\\lambda_4$\nare real parameters.\nIn general,\n$m_{12}^2$, $\\lambda_5$, $\\lambda_6$ and $\\lambda_7$\nare complex. ``H.c.''~stands for Hermitian conjugation.\n\nAn alternative notation,\nuseful for the construction of invariants\nand championed by Botella and Silva \\cite{BS} is\n\\begin{eqnarray}\nV_H\n&=&\nY_{ab} (\\Phi_a^\\dagger \\Phi_b) +\n\\tfrac{1}{2}\nZ_{ab,cd} (\\Phi_a^\\dagger \\Phi_b) (\\Phi_c^\\dagger \\Phi_d),\n\\label{VH2}\n\\end{eqnarray}\nwhere Hermiticity implies\n\\begin{eqnarray}\nY_{ab} &=& Y_{ba}^\\ast,\n\\nonumber\\\\\nZ_{ab,cd} \\equiv Z_{cd,ab} &=& Z_{ba,dc}^\\ast.\n\\label{hermiticity_coefficients}\n\\end{eqnarray}\nThe extremum conditions are\n\\begin{equation}\n\\left[ Y_{ab}\n+ Z_{ab,cd}\\, v_d^\\ast v_c \\right]\\ v_b = 0\n\\hspace{3cm}(\\textrm{for\\ } a = 1,2).\n\\label{stationarity_conditions}\n\\end{equation}\nMultiplying by $v_a^\\ast$ leads to\n\\begin{equation}\nY_{ab} (v_a^\\ast v_b) = - Z_{ab,cd}\\,\n(v_a^\\ast v_b)\\, (v_d^\\ast v_c).\n\\label{aux_1}\n\\end{equation}\n\nOne should be very careful when comparing Eqs.~(\\ref{VH1})\nand (\\ref{VH2}) among different authors,\nsince the same symbol may be used for quantities\nwhich differ by signs, factors of two, or complex conjugation.\nHere we follow the definitions of Davidson and Haber\n\\cite{DavHab}.\nWith these definitions:\n\\begin{eqnarray}\nY_{11}=m_{11}^2, &&\nY_{12}=-m_{12}^2,\n\\nonumber \\\\\nY_{21}=-(m_{12}^2)^\\ast && Y_{22}=m_{22}^2,\n\\label{ynum}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nZ_{11,11}=\\lambda_1, && Z_{22,22}=\\lambda_2,\n\\nonumber\\\\\nZ_{11,22}=Z_{22,11}=\\lambda_3, && Z_{12,21}=Z_{21,12}=\\lambda_4,\n\\nonumber \\\\\nZ_{12,12}=\\lambda_5, && Z_{21,21}=\\lambda_5^\\ast,\n\\nonumber\\\\\nZ_{11,12}=Z_{12,11}=\\lambda_6, && Z_{11,21}=Z_{21,11}=\\lambda_6^\\ast,\n\\nonumber \\\\\nZ_{22,12}=Z_{12,22}=\\lambda_7, && Z_{22,21}=Z_{21,22}=\\lambda_7^\\ast.\n\\label{znum}\n\\end{eqnarray}\n\nThe previous two notations look at the Higgs fields $\\Phi_a$ individually.\nA third notation is used by Nishi \\cite{Nishi1,Nishi2} and\nIvanov \\cite{Ivanov1,Ivanov2},\nwho emphasize\nthe presence of field bilinears $(\\Phi_a^\\dagger \\Phi_b)$\n\\cite{earlier_bilinears}.\nFollowing Nishi \\cite{Nishi1} we write:\n\\begin{equation}\nV_H = M_\\mu r_\\mu + \\Lambda_{\\mu \\nu} r_\\mu r_\\nu,\n\\label{VH3}\n\\end{equation}\nwhere $\\mu = 0,1,2,3$ and\n\\begin{eqnarray}\nr_0 &=&\n\\frac{1}{2}\n\\left[(\\Phi_1^\\dagger \\Phi_1) + (\\Phi_2^\\dagger \\Phi_2) \\right],\n\\nonumber\\\\\nr_1 &=&\n\\frac{1}{2}\n\\left[(\\Phi_1^\\dagger \\Phi_2) + (\\Phi_2^\\dagger \\Phi_1) \\right]\n= \\textrm{Re}\\, (\\Phi_1^\\dagger \\Phi_2),\n\\nonumber\\\\\nr_2 &=&\n- \\frac{i}{2}\n\\left[(\\Phi_1^\\dagger \\Phi_2) - (\\Phi_2^\\dagger \\Phi_1) \\right]\n= \\textrm{Im}\\, (\\Phi_1^\\dagger \\Phi_2),\n\\nonumber\\\\\nr_3 &=&\n\\frac{1}{2}\n\\left[(\\Phi_1^\\dagger \\Phi_1) - (\\Phi_2^\\dagger \\Phi_2) \\right].\n\\label{r_Ivanov}\n\\end{eqnarray}\nIn Eq.~(\\ref{VH3}),\nsummation of repeated indices is\nadopted with Euclidean metric.\nThis differs from Ivanov's notation \\cite{Ivanov1,Ivanov2},\nwho pointed out that $r_\\mu$ parametrizes the gauge orbits\nof the Higgs fields,\nin a space equipped with a Minkowski metric.\n\nIn terms of the parameters of Eq.~(\\ref{VH1}),\nthe $4$-vector $M_\\mu$ and $4 \\times 4$ matrix $\\Lambda_{\\mu\\nu}$ are written\nrespectively as:\n\\begin{equation}\nM_\\mu =\n\\left(\n\\begin{array}{cccc}\nm_{11}^2 + m_{22}^2,\n&\n-2\\, \\textrm{Re}\\, m_{12}^2,\n&\n2\\, \\textrm{Im}\\, m_{12}^2,\n&\nm_{11}^2 - m_{22}^2\n\\end{array}\n\\right),\n\\label{M_mu}\n\\end{equation}\nand\n\\begin{equation}\n\\Lambda_{\\mu \\nu} =\n\\left(\n\\begin{array}{cccc}\n(\\lambda_1+\\lambda_2)\/2 + \\lambda_3\\\n&\\,\\,\\,\n\\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\n&\\,\\,\\,\n- \\textrm{Im}\\, (\\lambda_6 + \\lambda_7)\n&\\,\\,\\,\n(\\lambda_1 - \\lambda_2)\/2\n\\\\\n \\phantom{-}\\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\\\n&\\,\\,\\,\n\\lambda_4 + \\textrm{Re}\\, \\lambda_5\n&\\,\\,\\,\n- \\textrm{Im}\\, \\lambda_5\n&\\,\\,\\,\n \\phantom{-}\\textrm{Re}\\, (\\lambda_6 - \\lambda_7)\\\n\\\\\n- \\textrm{Im}\\, (\\lambda_6 + \\lambda_7)\\\n&\\,\\,\\,\n- \\textrm{Im}\\, \\lambda_5\n&\\,\\,\\,\n\\lambda_4 - \\textrm{Re}\\, \\lambda_5\n&\\,\\,\\,\n- \\textrm{Im}\\, (\\lambda_6 - \\lambda_7)\\\n\\\\\n(\\lambda_1 - \\lambda_2)\/2\n&\\,\\,\\,\n\\textrm{Re}\\, (\\lambda_6 - \\lambda_7)\n&\\,\\,\\,\n- \\textrm{Im}\\, (\\lambda_6 - \\lambda_7)\n&\\,\\,\\,\n\\ (\\lambda_1+\\lambda_2)\/2 - \\lambda_3\n\\end{array}\n\\right).\n\\label{Lambda_munu}\n\\end{equation}\nEq.~(\\ref{VH3}) is related to Eq.~(\\ref{VH2}) through\n\\begin{eqnarray}\nM^\\mu &=&\n\\sigma^\\mu_{ab}\\, Y_{ba},\n\\label{M_vs_Y}\n\\\\\n\\Lambda^{\\mu \\nu} &=&\n\\tfrac{1}{2} Z_{ab,cd}\\, \\sigma^\\mu_{ba} \\sigma^\\nu_{dc},\n\\label{Lambda_vs_Z}\n\\end{eqnarray}\nwhere the matrices $\\sigma^i$ are the three Pauli matrices,\nand $\\sigma^0$ is the $2 \\times 2$ identity matrix.\n\n\n\n\\subsection{Basis transformations}\n\nWe may rewrite the potential in terms of new fields $\\Phi^\\prime_a$,\nobtained from the original ones by a simple \n(global) basis transformation\n\\begin{equation}\n\\Phi_a \\rightarrow \\Phi_a^\\prime = U_{ab} \\Phi_b,\n\\label{basis-transf}\n\\end{equation}\nwhere $U\\in U(2)$ is a $2 \\times 2$ unitary matrix.\nUnder this unitary basis transformation,\nthe gauge-kinetic terms are unchanged,\nbut the coefficients $Y_{ab}$ and $Z_{ab,cd}$ are transformed as\n\\begin{eqnarray}\nY_{ab} & \\rightarrow &\nY^\\prime_{ab} =\nU_{a \\alpha}\\, Y_{\\alpha \\beta}\\, U_{b \\beta}^\\ast ,\n\\label{Y-transf}\n\\\\\nZ_{ab,cd} & \\rightarrow &\nZ^\\prime_{ab,cd} =\nU_{a\\alpha}\\, U_{c \\gamma}\\,\nZ_{\\alpha \\beta,\\gamma \\delta}\\, U_{b \\beta}^\\ast \\, U_{d \\delta}^\\ast ,\n\\label{Z-transf}\n\\end{eqnarray}\nand the vevs are transformed as\n\\begin{equation}\nv_a \\rightarrow v_a^\\prime = U_{a b} v_b.\n\\label{vev-transf}\n\\end{equation}\nThus,\nthe basis transformations $U$ may be utilized in order to absorb\nsome of the degrees of freedom of $Y$ and\/or $Z$,\nwhich implies that not all parameters of Eq.~(\\ref{VH2})\nhave physical significance.\n\n\n\n\\subsection{\\label{subsec:HFsymmetry}Higgs Family symmetries}\n\nLet us assume that the scalar potential in\nEq.~(\\ref{VH2}) has some explicit internal symmetry.\nThat is,\nwe assume that the coefficients of $V_H$ stay\n\\textit{exactly the same} under a transformation\n\\begin{equation}\n\\Phi_a \\rightarrow \\Phi_a^S = S_{ab} \\Phi_b.\n\\label{S-transf-symmetry}\n\\end{equation}\n$S$ is a unitary matrix,\nso that the gauge-kinetic couplings\nare also left invariant by this Higgs Family symmetry\n(HF symmetry).\nAs a result of this symmetry,\n\\begin{eqnarray}\nY_{a b} & = &\nY^S_{a b} =\nS_{a \\alpha}\\, Y_{\\alpha \\beta}\\, S_{b \\beta}^\\ast ,\n\\label{Y-S}\n\\\\\nZ_{ab,cd} & = &\nZ^S_{ab,cd} =\nS_{a \\alpha}\\, S_{c \\gamma}\\,\nZ_{\\alpha \\beta, \\gamma \\delta}\\, S_{b \\beta}^\\ast \\, S_{d \\delta}^\\ast .\n\\label{Z-S}\n\\end{eqnarray}\nNotice that this is \\textit{not} the situation considered\nin Eqs.~(\\ref{basis-transf})--(\\ref{Z-transf}).\nThere,\nthe coefficients of the Lagrangian\n\\textit{do change}\n(although the quantities that are physically\nmeasurable are invariant with respect to any change of basis).\nIn contrast, Eqs.~(\\ref{S-transf-symmetry})--(\\ref{Z-S})\nimply the existence of a HF symmetry $S$ of the scalar potential\nthat leaves the coefficients of $V_H$ unchanged.\n\nThe Higgs Family symmetry group must be a subgroup of full $U(2)$\ntransformation group of $2\\times 2$ unitary \nmatrices employed in Eq.~(\\ref{basis-transf}).  Given the most\ngeneral THDM scalar potential, there is always a $U(1)$ subgroup\nof $U(2)$ under which the scalar potential is invariant.\nThis is the global hypercharge $U(1)_Y$ symmetry group:\n\\begin{equation}\nU(1)_Y:\n\\hspace{4ex}\n\\Phi_1 \\rightarrow e^{i \\theta} \\Phi_1,\n\\hspace{4ex}\n\\Phi_2 \\rightarrow e^{i \\theta} \\Phi_2,\n\\label{U1Y}\n\\end{equation}\nwhere $\\theta$ is an arbitrary angle (mod $2\\pi$).  The invariance\nunder the global $U(1)_Y$ is trivially guaranteed by the \ninvariance under the $SU(2)\\otimes U(1)$ electroweak gauge symmetry.\n\\textit{Since the  global hypercharge $U(1)_Y$ is always present, we shall \nhenceforth define the \nHF symmetries as those Higgs Family symmetries that are \northogonal to $U(1)_Y$.}\n\nWe now turn to the interplay between\nHF symmetries and basis transformations.\nLet us imagine that,\nwhen written in the basis of fields $\\Phi_a$,\n$V_H$ has a symmetry $S$.\nWe then perform a basis transformation from\nthe basis $\\Phi_a$ to the basis $\\Phi^\\prime_a$,\nas given by Eq.~(\\ref{basis-transf}).\nClearly,\nwhen written in the new basis,\n$V_H$ does \\textit{not} remain invariant under $S$.\nRather, it will be invariant under\n\\begin{equation}\nS^\\prime = U S U^\\dagger .\n\\label{S-prime}\n\\end{equation}\nAs we change basis,\nthe form of the potential changes\nin a way that may obscure the presence\nof a HF symmetry.   In particular, two HF symmetries\nthat naively look distinct\nwill actually yield precisely the same physical predictions\nif a unitary matrix $U$ exists such that Eq.~\\eqref{S-prime} is satisfied.\n\nHF symmetries in the two-Higgs-doublet model (THDM) have a long\nhistory.\nIn papers by Glashow and Weinberg and by Paschos~\\cite{Pas}, the discrete\n$Z_2$ symmetry was introduced,\n\\begin{equation}\nZ_2:\n\\hspace{4ex}\n\\Phi_1 \\rightarrow \\Phi_1,\n\\hspace{4ex}\n\\Phi_2 \\rightarrow - \\Phi_2,\n\\label{Z2}\n\\end{equation}\nin order to preclude flavour-changing neutral currents \\cite{Pas}.\nThis is just the interchange\n\\begin{equation}\n\\Pi_2:\n\\hspace{4ex}\n\\Phi_1 \\leftrightarrow \\Phi_2,\n\\label{Pi2}\n\\end{equation}\nseen in a different basis,\nas shown by applying Eq.~(\\ref{S-prime}) in the form\n\\begin{equation}\n\\left(\n\\begin{array}{cc}\n0 & 1 \\\\\n1 & 0 \\\\\n\\end{array}\n\\right)\n=\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{cc}\n1 & 1 \\\\\n1 & -1 \\\\\n\\end{array}\n\\right)\n\\\n\\left(\n\\begin{array}{cc}\n1 & 0 \\\\\n0 & -1 \\\\\n\\end{array}\n\\right)\n\\\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{cc}\n1 & 1 \\\\\n1 & -1 \\\\\n\\end{array}\n\\right).\n\\label{Z2ToPi2}\n\\end{equation}\nPeccei and Quinn~\\cite{PQ} introduced the continuous $U(1)$ symmetry\n\\begin{equation}\nU(1):\n\\hspace{4ex}\n\\Phi_1 \\rightarrow e^{-i \\theta} \\Phi_1,\n\\hspace{4ex}\n\\Phi_2 \\rightarrow e^{i \\theta} \\Phi_2,\n\\label{U1}\n\\end{equation}\ntrue for any value of $\\theta$,\nin connection with the strong CP problem.\nOf course,\na potential invariant under $U(1)$ is also invariant\nunder $Z_2$.\n\nFinally, we examine the largest possible Higgs Family symmetry group\nof the THDM, namely $U(2)$.  In this case, a basis transformation\nwould have no effect on the Higgs potential parameters.  Since\n$\\delta_{ab}$ is the only $U(2)$-invariant tensor, it follows that\n\\begin{eqnarray}\nY_{ab}&=& c_1 \\delta_{ab}\\,,\\label{u2y}\\\\\nZ_{ab,cd} &=& c_2 \\delta_{ab} \\delta_{cd} + c_3 \\delta_{ad} \\delta_{bc}\\,,\n\\label{u2z}\n\\end{eqnarray}\nwhere $c_1$, $c_2$ and $c_3$ are arbitrary real \nnumbers.~\\footnote{Note that there is no $\\delta_{ac} \\delta_{bd}$ term\ncontributing to $Z_{ab,cd}$, as such a term is not invariant under\nthe transformation of Eq.~(\\ref{Z-transf}).}\nOne can easily check from Eqs.~(\\ref{Y-transf}) and (\\ref{Z-transf})\nthat the unitarity of $U$ implies that $Y'=Y$ and $Z'=Z$ \nfor any choice of basis, as required\nby the $U(2)$-invariance of the scalar potential.\nEqs.~(\\ref{u2y}) and (\\ref{u2z}) impose the following constraints on\nthe parameters of the THDM scalar potential (independently of the\nchoice of basis):\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_1 = \\lambda_2=\\lambda_3+\\lambda_4\\,,\n& \\hspace{4ex} &\n\\lambda_5 = \\lambda_6 = \\lambda_7= 0\\,.\n\\label{u2pot}\n\\end{eqnarray}\nAs there are no non-zero potentially complex scalar potential parameters,\nthe $U(2)$-invariant THDM is clearly CP-invariant.\n\nAs previously noted, the\n$U(2)$ symmetry contains the global hypercharge\n$U(1)_Y$ as a subgroup.  Thus, in order to identify\nthe corresponding HF symmetry that is orthogonal to $U(1)_Y$,\nwe first observe that\n\\begin{equation}\nU(2)\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} SU(2)\\otimes U(1)_Y\/Z_2 \\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} SO(3)\\otimes U(1)_Y\\,.\n\\end{equation}\nTo prove the above isomorphism, simply note that any $U(2)$ matrix can\nbe written as $U=e^{i\\theta}\\hat{U}$, where $\\hat U\\in SU(2)$.\nTo cover the full $U(1)_Y$ group, we must take $0\\leq\\theta<2\\pi$.\nBut since both $\\hat U$ and $-\\hat U$ are elements of $SU(2)$\nwhereas $+1$ and $-1=e^{i\\pi}$ are elements of $U(1)_Y$, we must\nidentify $\\hat U$ and $-\\hat U$ as the same group element in order not to\ndouble cover the full $U(2)$ group.  The identification of $\\hat U$\nwith $-\\hat U$ in $SU(2)$ is isomorphic to $SO(3)$, using the well known\nisomorphism $SO(3)\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} SU(2)\/Z_2$.  Consequently, we have identified\nSO(3) as the HF symmetry that constrains the scalar potential\nparameters as indicated in Eq.(\\ref{u2pot}).\n\n\n\nThe impact of these symmetries on the potential parameters\nin Eq.~(\\ref{VH1}) is shown in section~\\ref{sec:summary}.\nAs mentioned above,\nif one makes a basis change, the potential parameters\nchange and so does the explicit form of the symmetry and\nof its implications.\nFor example,\nEq.~(\\ref{Z2ToPi2}) shows that the symmetries $Z_2$ and $\\Pi_2$\nare related by a basis change.\nHowever,\nthey have a different impact on the parameters in their\nrespective basis.\nThis can be seen explicitly in Table~\\ref{master1}\nof section~\\ref{sec:summary}. One can also easily prove that\nthe existence of either the $Z_2$, $\\Pi_2$ or Peccei-Quinn $U(1)$\nsymmetry is sufficient to guarantee the existence of a basis choice in\nwhich all scalar potential parameters are real.  That is, \nthe corresponding scalar Higgs sectors are explicitly CP-conserving.\n\nBasis invariant signs of HF symmetries were discussed\nextensively in Ref.~\\cite{DavHab}.\nRecently,\nFerreira and Silva~\\cite{FS2} extended these methods to include\nHiggs models with more than two Higgs doublets.\n\nConsider first the THDM scalar potentials that are invariant under \nthe so-called \\textit{simple} HF symmetries of Ref.~\\cite{FS2}.\nWe define a simple HF symmetry to be a symmetry\ngroup $G$ with the following property: the requirement that the\nTHDM scalar potential is invariant under a particular element \n$g\\in G$ (where $g\\neq e$ and $e$ is the identity element)\nis sufficient to guarantee invariance under the entire\ngroup $G$.  The discrete cyclic group \n$Z_n=\\{e\\,,\\,g\\,,\\,g^2\\,,\\,\\ldots\\,,\\,g^{n-1}\\}$,\nwhere $g^n = e$,\nis an example of a possible simple HF symmetry group.  \nIf we restrict the TDHM scalar potential to include terms of\ndimension-four or less (e.g., the tree-level scalar potential of the THDM),\nthen one can show that the Peccei-Quinn $U(1)$ symmetry is also\na simple HF symmetry.  For example, consider the matrix\n\\begin{equation}\nS=\\left( \\begin{array}{cc}\ne^{- 2 i \\pi\/3} & 0\\\\\n0 & e^{2 i \\pi\/3}\n\\end{array} \\right)\\,.\n\\end{equation}\nNote that $S$ is an element of the cyclic sub-group\n$Z_3=\\{S\\,,\\,S^2\\,,\\,S^3=1\\}$ of the\nPeccei-Quinn $U(1)$ group.\nAs shown in Ref.~\\cite{FS2}, the \ninvariance of the tree-level THDM scalar potential\nunder $\\Phi_a\\to S_{ab}\\Phi_b$ automatically implies the\ninvariance of the scalar potential under the full Peccei-Quinn $U(1)$\ngroup.  In contrast, the maximal HF symmetry, $SO(3)$, introduced\nabove is not a simple HF symmetry, as there is no single element of\n$S\\in SO(3)$ such that invariance under  $\\Phi_a\\to S_{ab}\\Phi_b$ \nguarantees invariance of the tree-level THDM \nscalar potential under the\nfull SO(3) group of transformations.\n\n\nTypically, the simple HF\nsymmetries take on a simple form for a particular choice of basis\nfor the Higgs fields.\nWe summarize here a few of the results of Ref.~\\cite{FS2}:\n\\begin{enumerate}\n\\item In the THDM, there are only two\n\\textit{independent} classes of \\textit{simple} symmetries:\na discrete $Z_2$ flavor symmetry, and a continuous Peccei-Quinn $U(1)$\nflavor symmetry.\n\\item Other discrete flavor symmetry groups $G$ that are subgroups of $U(1)$\nare not considered independent.  That is, if $S\\in G$ (where\n$S\\neq e$), then invariance under the\nthe discrete symmetry $\\Phi\\to S\\Phi$\nmakes the scalar potential automatically invariant under the full\nPeccei-Quinn $U(1)$ group;\n\\item In most regions of parameter space,\none can build quantities invariant under basis transformations\nthat detect these symmetries;\n\\item There exists a so-called exceptional region of parameter space (ERPS)\ncharacterized by\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = - \\lambda_6.\n\\label{ERPS}\n\\end{eqnarray}\nAs shown by Davidson and Haber \\cite{DavHab},\na theory obeying these constraints does have a $Z_2$ symmetry,\nbut it may or not have a $U(1)$ symmetry.\nWithin the ERPS, the invariants in the literature cannot be\nused to distinguish the two cases.\n\\end{enumerate}\n\nThe last statement above is a result of the following considerations.\nIn order to distinguish between $Z_2$ and $U(1)$,\nDavidson and Haber \\cite{DavHab} construct two\ninvariant quantities given by Eqs.~(46) and (50) of Ref.~\\cite{DavHab}.\nOutside the ERPS,\nthese quantities are zero if and only if $U(1)$ holds.\nUnfortunately,\nin the ERPS these quantities vanish automatically\nindependently of whether or not $U(1)$ holds.\nSimilarly,\nFerreira and Silva \\cite{FS2} have\nconstructed invariants detecting HF symmetries.\nBut their use requires the existence of a matrix, obtained\nby combining $Y_{ab}$ and $Z_{ab,cd}$,\nthat has two distinct eigenvalues.\nThis does not occur when the ERPS is due to a symmetry.\nFinally, in the ERPS,\nIvanov \\cite{Ivanov1} states that the symmetry might be\n``$(Z_2)^2$ or $O(2)$''\n[our $Z_2$ \\textit{or} our $U(1)$]\nand does not provide a way to distinguish the\ntwo possible flavor symmetries \\cite{oversight}.\n\nGunion and Haber \\cite{GunHab} have shown that\nthe ERPS conditions of Eq.~(\\ref{ERPS})\nare basis independent;\nif they hold in one basis, then they hold in any basis.\nMoreover, for a model in the ERPS,\na basis may be chosen such that all parameters are \nreal.\\footnote{Given a scalar potential whose parameters satisfy\nthe ERPS conditions\nwith ${\\rm Im}(\\lambda_5^* \\lambda_6^2)\\neq 0$, the unitary matrix\nrequired to transform into a basis in which all the scalar potential\nparameters are real can be determined only by numerical means.}\nHaving achieved such a basis,\nDavidson and Haber \\cite{DavHab} demonstrate that\none may make one additional basis transformation\nsuch that\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = \\lambda_6 = 0,\n\\hspace{4ex} \\textrm{Im}\\, \\lambda_5 = 0.\n\\label{ERPS2}\n\\end{eqnarray}\nThese conditions express the ERPS for a specific basis choice.\n\nOne might think that this is such a special region of parameter\nspace that it lacks any relevance.\nHowever,\nthe fact that the conditions in Eq.~(\\ref{ERPS}) hold in\n\\textit{any} basis is a good indication that a\nsymmetry may lie behind this condition.\nIndeed,\nas pointed out by Davidson and Haber \\cite{DavHab},\ncombining the two symmetries $Z_2$ and $\\Pi_2$\n\\textit{in the same basis} one is lead immediately to\nthe ERPS in the basis of Eq.~(\\ref{ERPS2}).\nUp to now,\nwe considered the impact of imposing\non the Higgs potential only one symmetry.\nThis was dubbed a simple symmetry.\nNow we are considering the possibility that the\npotential must remain invariant under one symmetry\nand \\textit{also} under a second symmetry;\nthis implies further constraints on the parameters\nof the Higgs potential.\nWe refer to this possibility as a multiple symmetry.\nAs seen from Table~\\ref{master1} of section~\\ref{sec:summary},\nimposing $Z_2$ and $\\Pi_2$ in the same basis leads\nto the conditions in Eq.~(\\ref{ERPS2}).\nIncidentally,\nthis example shows that a model which lies in the\nERPS,\nis automatically invariant under $Z_2$.\n\nIn section~{\\ref{sec:GCP}} we will show that\nall classes of non-trivial CP transformations lead\ndirectly to the ERPS,\nreinforcing the importance of this particular region of\nparameter space.\n\n\n\\subsection{Requirements for $U(1)$ invariance}\n\nIn the basis in which the $U(1)$ symmetry takes the form\nof Eq.~(\\ref{U1}),\nthe coefficients of the potential must obey\n\\begin{equation}\nm^{\\prime\\,2}_{12}= 0,\n\\hspace{4ex}\n\\lambda^\\prime_5 = \\lambda^\\prime_6 = \\lambda^\\prime_7 = 0.\n\\label{U1_conditions}\n\\end{equation}\nImagine that we have a potential of Eq.~(\\ref{VH1})\nin the ERPS:\n$m_{11}^2 = m_{22}^2$,\n$m_{12}^2=0$,\n$\\lambda_2=\\lambda_1$,\nand $\\lambda_7 = - \\lambda_6$.\nWe now wish to know whether a transformation $U$ may be chosen\nsuch that the potential coefficients in the new basis\nsatisfy the $U(1)$ conditions in Eq.~(\\ref{U1_conditions}).\nUsing the transformation rules in Eqs.~(A13)-(A23) of Davidson and\nHaber \\cite{DavHab},\nwe find that such a choice of $U$ is possible if and only if the\ncoefficients in the original basis satisfy\n\\begin{equation}\n2 \\lambda_6^3 - \\lambda_5 \\lambda_6(\\lambda_1 - \\lambda_3 - \\lambda_4)\n- \\lambda_5^2 \\lambda_6^\\ast = 0,\n\\label{can_change_to_usual_U1}\n\\end{equation}\nsubject to the condition that $\\lambda_5^\\ast \\lambda_6^2$ is real.\n\n\n\\subsection{\\label{subsec:D}The D invariant}\n\nHaving established the importance of the ERPS\n(as it can arise from a symmetry),\nwe will now build a basis invariant quantity that\ncan be used to detect the presence of a U(1)\nsymmetry in this special case.\n\nThe quadratic terms of the Higgs potential are always\ninsensitive to the difference between $Z_2$ and $U(1)$.\nMoreover,\nthe matrix $Y$ is proportional to the unit matrix in the ERPS.\nOne must thus look at the quartic terms.\nWe were inspired by the expression of $\\Lambda_{\\mu \\nu}$\nin Eq.~(\\ref{Lambda_munu}),\nwhich appears in the works of Nishi \\cite{Nishi1,Nishi2} and\nIvanov \\cite{Ivanov1,Ivanov2}.\nIn the ERPS of Eq.~(\\ref{ERPS}),\n$\\Lambda_{\\mu \\nu}$ breaks into a $1 \\times 1$ block\n($\\Lambda_{00}$),\nand a $3 \\times 3$ block\n($\\tilde{\\Lambda} = \\left\\{\\Lambda_{ij}\\right\\}$; $i,j=1,2,3$).\nA basis transformation $U$ belonging to $SU(2)$ on the $\\Phi_a$ fields\ncorresponds to an orthogonal $SO(3)$ transformation\nin the $r_i$ bilinears,\ngiven by\n\\begin{equation}\nO_{ij} = \\hbox{$\\frac{1}{2}$}\\,\\textrm{Tr} \n\\left[ U^\\dagger \\sigma_i U \\sigma_j \\right].\n\\label{O}\n\\end{equation}\nAny matrix $O$ of $SO(3)$ can be obtained by considering an\nappropriate matrix $U$ of $SU(2)$\n(unfortunately this property does not generalize for\nmodels with more than two Higgs doublets).\nA suitable choice of $O$ can be made that diagonalizes\nthe $3 \\times 3$ matrix $\\tilde{\\Lambda}$,\nthus explaining Eq.~(\\ref{ERPS2}).\nIn this basis,\nthe difference between the usual choices for $U(1)$ and $Z_2$ corresponds\nto the possibility that $\\textrm{Re} \\lambda_5$ might\nvanish or not, respectively.\n\nWe will now show that,\nonce in the ERPS,\nthe condition for the existence of $U(1)$ is that\n$\\tilde{\\Lambda}$ has two eigenvalues which are equal.\nThe eigenvalues of a $3 \\times 3$ matrix are the\nsolutions to the secular equation\n\\begin{equation}\nx^3 + a_2 x^2 + a_1 x + a_0 = 0,\n\\label{secular}\n\\end{equation}\nwhere\n\\begin{eqnarray}\na_0 &=&\n\\det \\tilde{\\Lambda}\n= - \\tfrac{1}{3}  \\textrm{Tr\\,} (\\tilde{\\Lambda}^3)\n- \\tfrac{1}{6} ( \\textrm{Tr\\,} \\tilde{\\Lambda} )^3\n+ \\tfrac{1}{2} ( \\textrm{Tr\\,} \\tilde{\\Lambda} )\n\\textrm{Tr\\,} (\\tilde{\\Lambda}^2)\n\\nonumber\\\\\n&=&\n- \\tfrac{1}{3} Z_{ab,cd}\n\\left( Z_{dc,gh} Z_{hg,ba} - \\tfrac{3}{2} Z^{(2)}_{dc} Z^{(2)}_{ba} \\right)\n+ \\tfrac{1}{2} Z_{ab,cd} Z_{dc,ba}\n\\textrm{Tr\\,} \\left( Z^{(1)} - \\tfrac{1}{2} Z^{(2)} \\right)\n\\nonumber\\\\\n& &\n-\\tfrac{1}{6} \\left( \\textrm{Tr\\,} Z^{(1)}\\right)^3\n+\\tfrac{1}{4} \\left( \\textrm{Tr\\,} Z^{(1)}\\right)^2 \\textrm{Tr\\,} Z^{(2)}\n- \\tfrac{1}{2} \\textrm{Tr\\,} Z^{(1)} \\left( \\textrm{Tr\\,} Z^{(2)}\\right)^2,\n\\\\\na_1 &=&\n\\tfrac{1}{2}  ( \\textrm{Tr\\,} \\tilde{\\Lambda} )^2\n- \\tfrac{1}{2} \\textrm{Tr\\,} (\\tilde{\\Lambda}^2)\n\\nonumber\\\\\n&=&\n\\tfrac{1}{2}\n\\left[\n\\left( \\textrm{Tr\\,} Z^{(1)}\\right)^2\n- \\textrm{Tr\\,} Z^{(1)} \\textrm{Tr\\,} Z^{(2)}\n+ \\left( \\textrm{Tr\\,} Z^{(2)}\\right)^2\n- Z_{ab,cd} Z_{dc,ba}\n\\right],\n\\\\\na_2 &=&\n- \\textrm{Tr\\,} \\tilde{\\Lambda}\n\\nonumber\\\\\n&=&\n\\tfrac{1}{2} \\textrm{Tr\\,} Z^{(2)} - \\textrm{Tr\\,} Z^{(1)},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nZ_{ab}^{(1)}\n&\\equiv&\nZ_{a \\alpha,\\alpha b} =\n\\left( \\begin{array}{cc}\n  \\lambda_1 + \\lambda_4 & \\quad \\lambda_6 + \\lambda_7 \\\\\n  \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad \\lambda_2 + \\lambda_4\n\\end{array} \\right),\n\\\\\nZ_{ab}^{(2)}\n&\\equiv&\nZ_{\\alpha \\alpha,a b} =\n\\left( \\begin{array}{cc}\n  \\lambda_1 + \\lambda_3 & \\quad \\lambda_6 + \\lambda_7 \\\\\n  \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad \\lambda_2 + \\lambda_3\n\\end{array} \\right).\n\\end{eqnarray}\nThe cubic equation, Eq.~(\\ref{secular}), has at least two\ndegenerate solutions if \\cite{AbrSte}\n\\begin{equation}\nD \\equiv\n\\left[ \\tfrac{1}{3} a_1 - \\tfrac{1}{9} a_2^2 \\right]^3\n+  \\left[ \\tfrac{1}{6} (a_1 a_2 - 3 a_0) - \\tfrac{1}{27} a_2^3 \\right]^2\n\\end{equation}\nvanishes.\n\nThe expression of $D$ in terms of the parameters in Eq.~(\\ref{VH1})\nis rather complicated,\neven in the ERPS.\nBut one can show by direct computation that if the $U(1)$-symmetry\ncondition of Eq.~(\\ref{can_change_to_usual_U1}) holds\n(subject to $\\lambda_5^\\ast \\lambda_6^2$ being real),\nthen $D=0$.\nWe can simplify the expression for $D$ by changing to a\nbasis where all parameters are real \\cite{GunHab},\nwhere we get\n\\begin{equation}\nD =\n- \\tfrac{1}{27}\n\\left[ \\lambda_5 (\\lambda_1 - \\lambda_3 - \\lambda_4 + \\lambda_ 5)\n       - 2 \\lambda_6^2 \\right]^2\n\\left[ (\\lambda_1 - \\lambda_3 - \\lambda_4 - \\lambda_ 5)^2\n       + 16 \\lambda_6^2 \\right].\n       \\label{eq:D}\n\\end{equation}\nIf $\\lambda_6 \\neq 0$, then $D=0$ means\n\\begin{equation}\n2 \\lambda_6^2 = \\lambda_5 (\\lambda_1 - \\lambda_3 - \\lambda_4 + \\lambda_ 5).\n\\label{L6NEQ0}\n\\end{equation}\nIf $\\lambda_6 = 0$,\nthen $D=0$ corresponds to one of three possible conditions:\n\\begin{equation}\n\\lambda_5 = 0,\n\\hspace{4ex}\n\\lambda_5 = \\pm (\\lambda_1 - \\lambda_3 - \\lambda_4).\n\\label{L6EQ0}\n\\end{equation}\nNotice that Eqs.~(\\ref{L6NEQ0}) and (\\ref{L6EQ0}) are\nequivalent to Eq.~(\\ref{can_change_to_usual_U1})\nin any basis where the coefficients are real.\n\nAlthough $D$ can be defined outside the ERPS,\nthe condition $D=0$ only guarantees that the model is invariant under\n$U(1)$ inside the ERPS of Eq.~(\\ref{ERPS}).\nOutside this region one can detect the presence of a $U(1)$ symmetry\nwith the invariants proposed by Davidson and Haber \\cite{DavHab}.\nThis closes the last breach in the literature concerning basis-invariant\nsignals of discrete symmetries in the THDM.\nThus, in the ERPS $D=0$ is a necessary and sufficient condition for\nthe presence of a $U(1)$ symmetry.\n\n\n\\section{\\label{sec:vacuum}Vacuum structure and renormalization}\n\nThe presence of a $U(1)$ symmetry in the Higgs potential\nmay (or not) imply the existence of a massless scalar, the axion,\ndepending on whether (or not) the $U(1)$ is broken by the vevs.\nIn the previous section we related the basis-invariant condition\n$D=0$ in the ERPS with the presence of a $U(1)$ symmetry.\nIn this section we will show that,\nwhenever the basis-invariant condition $D=0$ is\nsatisfied in the ERPS,\nthere is always a stationary point for which a massless scalar,\nother than the usual Goldstone bosons, exists.\n\nWe start by writing the extremum conditions for the THDM in the\nERPS.\nFor simplicity, we will be working in a basis where all the\nparameters are real~\\cite{GunHab}.\nFrom Eqs.~\\eqref{stationarity_conditions} and~\\eqref{znum},\nwe obtain\n\\begin{align}\n0 = & \\;\\;\nY_{11}\\,v_1\\,+\\,\\frac{1}{2}\\,\\left[\\lambda_1\\,v_1^3\\,+\n\\,\\lambda_{345}\\,v_1\\,v_2^2\\,+\\,\n\\lambda_6\\,(3\\,v_1^2\\,v_2\\,-\\,v_2^3)\\right],\n\\nonumber\\\\\n0 = & \\;\\;\nY_{11}\\,v_2\\,+\\,\\frac{1}{2}\\,\\left[\\lambda_1\\,v_2^3\\,+\n\\,\\lambda_{345}\\,v_2\\,v_1^2\\,+\\,\n\\lambda_6\\,(v_1^3\\,-\\,3\\,v_2^2\\,v_1)\\right],\n\\label{eq:stat}\n\\end{align}\nwhere we have defined\n$\\lambda_{345}\\equiv\\lambda_3 + \\lambda_4 + \\lambda_5$.\nWe now compute the mass matrices.\nAs we will be considering only vacua with real vevs,\nthere will be no mixing between the real and imaginary\nparts of the doublets.\nAs such, we can define the mass matrix of the CP-even scalars as given by\n\\begin{equation}\n\\left[M^2_h\\right]_{ij}\\;=\\;\\frac{1}{2}\\,\\frac{\\partial^2\nV}{\\partial \\mbox{Re}(\\Phi_i^0)\\,\n\\partial \\mbox{Re}(\\Phi_j^0)}\n\\end{equation}\nwhere $\\Phi_i^0$ is the neutral (lower) component of the $\\Phi_i$\ndoublet. Thus, we obtain, for the entries of this matrix, the\nfollowing expressions:\n\\begin{align}\n\\left[M^2_h\\right]_{11}\\;=& \\;\nY_{11}\\,+\\,\\frac{1}{2}\\,\\left(3\\,\\lambda_1\\,v_1^2\\,+\n\\,\\lambda_{345}\\,v_2^2\n\\,\n+\\,6\\,\\lambda_6\\,v_1\\,v_2\\right)\\nonumber \\vspace{0.2cm} \\\\\n\\left[M^2_h\\right]_{22}\\;=&\\;\nY_{11}\\,+\\,\\frac{1}{2}\\,\\left(3\\,\\lambda_1\\,v_2^2\\,+\n\\,\\lambda_{345}\\,v_1^2\n\\,\n+\\,6\\,\\lambda_6\\,v_1\\,v_2\\right)\\nonumber \\vspace{0.2cm} \\\\\n\\left[M^2_h\\right]_{12}\\;=& \\;\\lambda_{345}\\,v_1\\,v_2\\,\n\\,+\\,\\frac{3}{2}\\,\\lambda_6\\,(v_1^2\\,-\\,v_2^2)\\;\\;\\; .\n\\label{eq:mh}\n\\end{align}\nLikewise, the pseudoscalar mass matrix is defined as\n\\begin{equation}\n\\left[M^2_A\\right]_{ij}\\;=\\;\\frac{1}{2}\\,\\frac{\\partial^2\nV}{\\partial \\mbox{Im}(\\Phi_i^0)\n\\partial \\mbox{Im}(\\Phi_j^0)}\n\\end{equation}\nwhose entries are given by\n\\begin{align}\n\\left[M^2_A\\right]_{11}\\;=&\\;\nY_{11}\\,+\\,\\frac{1}{2}\\,\\left[\\lambda_1\\,v_1^2\\,+\\,\n\\left(\\lambda_3\\,+\\,\\lambda_4\\,-\\,\\lambda_5\\right)\\,v_2^2\n\\,+\\,2\\,\\lambda_6\\,v_1\\,v_2 \\right]\n\\nonumber \\vspace{0.2cm} \\\\\n\\left[M^2_A\\right]_{22}\\;=&\\;\nY_{11}\\,+\\,\\frac{1}{2}\\,\\left[\\lambda_1\\,v_2^2\\,+\\,\n\\left(\\lambda_3\\,+\\,\\lambda_4\\,-\\,\\lambda_5\\right)\\,v_1^2\n\\,-\\,2\\,\\lambda_6\\,v_1\\,v_2 \\right]\n\\nonumber \\vspace{0.2cm} \\\\\n\\left[M^2_A\\right]_{12}\\;=& \\;\\lambda_5\\,v_1\\,v_2 \\,+\\,\n\\frac{1}{2}\\,\\lambda_6\\,(v_1^2\\,-\\,v_2^2)\\;\\;\\; .\n\\label{eq:mA}\n\\end{align}\nThe expressions ~\\eqref{eq:mh} and~\\eqref{eq:mA} are valid for all\nthe particular cases we will now consider.\n\n\n\\subsection{Case $\\lambda_6\\,=\\,0$, $\\{v_1\\,,\\,v_2\\}\\,\\neq\\,0$}\n\nLet us first study the case $\\lambda_6\\,=\\,0$, wherein we may solve\nthe extremum conditions in an analytical manner. It is trivial\nto see that Eqs.~\\eqref{eq:stat} have three types of solutions: both\nvevs different from zero, one vev equal to zero (say, $v_2$) and\nboth vevs zero (trivial non-interesting solution). For a solution\nwith $\\{v_1\\,,\\,v_2\\}\\,\\neq\\,0$, a necessary condition must be\nobeyed so that there is a solution to Eqs.~\\eqref{eq:stat}:\n\\begin{equation}\n\\lambda_1^2\\,-\\,\\lambda_{345}^2 \\;\\neq\\;0\\;\\;\\; . \\label{eq:det}\n\\end{equation}\nIf we use the extremum conditions to evaluate\n$\\left[M^2_h\\right]$, we obtain\n\\begin{equation}\n\\left[M^2_h\\right]\\;=\\;\\begin{pmatrix} \\lambda_1\\,v_1^2 & \\quad\n\\lambda_{345}\\,v_1\\,v_2 \\\\ \\lambda_{345}\\,v_1\\,v_2 & \\quad\n\\lambda_1\\,v_2^2\n\\end{pmatrix}\n\\end{equation}\nwhich only has a zero eigenvalue if Eq.~\\eqref{eq:det} is broken.\nThus, there is no axion in this matrix in this case. As for\n$\\left[M^2_A\\right]$, we get\n\\begin{equation}\n\\left[M^2_A\\right]\\;=\\;-\\,\\lambda_5\\,\\begin{pmatrix}\nv_1^2 &\\quad  v_1\\,v_2 \\\\\nv_1\\,v_2 &\\quad v_2^2 \\end{pmatrix}\n\\end{equation}\nwhich clearly has a zero eigenvalue corresponding to the $Z$\nGoldstone boson. Further, this matrix will have an axion if\n$\\lambda_5\\,=\\,0$, which is the first condition of\nEq.~\\eqref{L6EQ0}.\n\n\\subsection{Case $\\lambda_6\\,=\\,0$, $\\{v_1\\,\\neq\\,0,\\,v_2\\,=\\,0\\}\\,$}\n\nReturning to Eq.~\\eqref{eq:stat}, this case gives us\n\\begin{equation}\nY_{11}\\,=\\,-\\,\\frac{1}{2}\\,\\lambda_1\\,v_1^2\\;\\;\\; ,\n\\end{equation}\nwhich implies $Y_{11}\\,<\\,0$. With this condition, the mass matrices\nbecome considerably simpler:\n\\begin{equation}\n\\left[M^2_h\\right]\\;=\\;\\begin{pmatrix} \\lambda_1\\,v_1^2 & \\quad 0 \\\\\n 0 & \\quad \\frac{1}{2}\\,(\\lambda_{345}\\,-\\,\\lambda_1)\\,v_1^2\n\\end{pmatrix}\n\\label{eq:mhll}\n\\end{equation}\nand\n\\begin{equation}\n\\left[M^2_A\\right]\\;=\\;\\frac{1}{2}\\,\\begin{pmatrix} 0 & \\quad 0 \\\\\n 0 & \\quad (\\lambda_3\\,+\\,\\lambda_4\\,-\\lambda_5\\,-\\,\\lambda_1)\\,v_1^2\n\\end{pmatrix} \\;\\;\\; .\n\\label{eq:mAll}\n\\end{equation}\nSo, we can have an axion in the matrix~\\eqref{eq:mhll} if\n\\begin{equation}\n\\lambda_{345}\\,-\\,\\lambda_1\\,=\\,0\\;\\;\\Leftrightarrow\\;\\;\\lambda_5\\,=\\,\n\\lambda_1\\,-\\,\\lambda_3\\,-\\,\\lambda_4 \\label{eq:c1}\n\\end{equation}\nor an axion in matrix~\\eqref{eq:mAll} if\n\\begin{equation}\n\\lambda_5\\,=\\, -\\lambda_1\\,+\\,\\lambda_3\\,+\\,\\lambda_4 \\;\\;\\; .\n\\label{eq:c2}\n\\end{equation}\nThat is, we have an axion if the second or third conditions of\nEq.~\\eqref{L6EQ0} are satisfied. The other possible case,\n$\\{v_1\\,=\\,0,\\,v_2\\,\\neq\\,0\\}\\,$, produces exactly the same\nconclusions.\n\n\\subsection{Case $\\lambda_6\\,\\neq\\,0$}\n\nThis is the hardest case to treat, since we cannot obtain analytical\nexpressions for the vevs. Nevertheless a full analytical treatment is\nstill possible. First, notice that with $\\lambda_6\\,\\neq\\,0$\nEqs.~\\eqref{eq:stat} imply that both vevs have to be non-zero. At\nthe stationary point of Eqs.~\\eqref{eq:stat}, the pseudoscalar mass\nmatrix has a Goldstone boson and an eigenvalue given by\n\\begin{equation}\n-\\lambda_5\\,(v_1^2\\,+\\,v_2^2)\\,-\n\\,\\lambda_6\\,\\frac{v_1^4\\,-v_2^4}{2\\,v_1\\,v_2}\n\\;\\;\\; .\n\\end{equation}\nSo, an axion exists if we have\n\\begin{equation}\n\\frac{v_1^2\\,-\\,v_2^2}{v_1\\,v_2}\\;=\\;-\n\\,\\frac{2\\,\\lambda_5}{\\lambda_6}\\;\\;\\;.\n\\label{eq:ves}\n\\end{equation}\nOn the other hand, after some algebraic manipulation, it is simple\nto obtain from~\\eqref{eq:stat} the following condition:\n\\begin{equation}\n\\lambda_1\\,-\\,\\lambda_{345}\\;=\\;\n\\lambda_6\\,\\left(\\frac{v_1^2\\,-\\,v_2^2}{v_1\\,v_2}\\,-\n\\,\\frac{4\\,v_1\\,v_2}{v_1^2\\,-\\,v_2^2}\\right)\n\\label{eq:les}\n\\end{equation}\nSubstituting Eq.~\\eqref{eq:ves} into~\\eqref{eq:les}, we obtain\n\\begin{equation}\n\\lambda_1\\,-\\,\\lambda_{345}\\,=\\,\n\\lambda_6\\,\\left(-\\,\\frac{2\\,\\lambda_5}{\\lambda_6}\\,+\n\\,\\frac{2\\,\\lambda_6}{\\lambda_5}\\right)\n\\;\\Longleftrightarrow\\; 2 \\lambda_6^2 \\,=\n\\, \\lambda_5 (\\lambda_1 -\n\\lambda_3 - \\lambda_4 + \\lambda_ 5).\n\\end{equation}\n\nThus, we have shown that all of the conditions stemming from the\nbasis-invariant condition $D=0$\nguarantee the existence of some stationary point for\nwhich the scalar potential yields an axion.\nNotice that, however,\nthis stationary point need not coincide with the global minimum\nof the potential.\n\n\n\\subsection{Renormalization group invariance}\n\nWe now briefly examine the renormalization group (RG) behavior of our\nbasis-invariant condition $D=0$. It would be meaningless to say that\n$D=0$ implies a $U(1)$ symmetry if that condition were only valid at\na given renormalization scale.  That is, it could well be that a numerical\naccident forces $D=0$ at only a given scale. To avoid such a\nconclusion, we must verify if\n$D=0$ is a RG-invariant condition (in addition to being\nbasis-invariant). For a given renormalization scale $\\mu$, the\n$\\beta$-function of a given parameter $x$ is defined as\n$\\beta_x\\,=\\,\\mu\\,\\partial x\/\\partial \\mu$. For simplicity, let us\nrewrite $D$ in Eq.~\\eqref{eq:D} as\n\\begin{equation}\nD\\;=\\;-\\,\\frac{1}{27}\\,D_1^2\\,D_2\\;\\;\\; ,\n\\end{equation}\nwith\n\\begin{align}\nD_1 &=\\; \\lambda_5 (\\lambda_1 - \\lambda_3 - \\lambda_4 + \\lambda_ 5)\n       - 2 \\lambda_6^2 \\nonumber \\\\\nD_2 &=\\; (\\lambda_1 - \\lambda_3 - \\lambda_4 - \\lambda_ 5)^2\n       + 16 \\lambda_6^2 \\;\\;\\;.\n\\end{align}\nIf we apply the operator $\\mu\\,\\partial \/\\partial \\mu$ to $D$, we\nobtain\n\\begin{equation}\n\\beta_D\\,=\\,-\\,\\frac{1}{27}\\,\n\\left(2\\,D_1\\,D_2\\,\\beta_{D_1}\\,+\\,D_1^2\\,\\beta_{D_2}\\right)\n\\;\\;\\;.\n\\end{equation}\n\nIf $D_1=0$ (which corresponds to three of the conditions presented\nin Eqs.~\\eqref{L6NEQ0} and~\\eqref{L6EQ0}) then we immediately have\n$\\beta_D\\,=\\,0$.  That is, if $D=0$ at a given scale, it is zero at\nall scales.\n\nIf $D_2=0$ and $D_1\\neq 0$ we will only have $\\beta_D\\,=\\,0$ if\n$\\beta_{D_2}\\,=\\,0$, or equivalently,\n\\begin{equation}\n2\\,(\\lambda_1 - \\lambda_3 - \\lambda_4 - \\lambda_ 5)\\,\n(\\beta_{\\lambda_1} - \\beta_{\\lambda_3} - \\beta_{\\lambda_4} -\n\\beta_{\\lambda_ 5}) \\,+\\,32\\,\\beta_{\\lambda_6}\\,\\lambda_6\\;=\\;0\n\\;\\;\\; .\n\\end{equation}\nGiven that $D_2=0$ implies that $\\lambda_6\\,=\\,0$ and\n$\\lambda_5\\,=\\,\\lambda_1 - \\lambda_3 - \\lambda_4$, we once\nagain obtain $\\beta_D\\,=\\,0$.\n\nThus, the condition $D=0$ is RG-invariant. A direct verification of\nthe RG invariance of Eqs.~\\eqref{L6NEQ0} and~\\eqref{L6EQ0}, and of\nthe conditions that define the ERPS itself, would require the\nexplicit form of the $\\beta$ functions of the THDM involving the\n$\\lambda_6$ coupling. That verification will be made\nelsewhere~\\cite{drtj}.\n\n\\section{\\label{sec:GCP}Generalized CP symmetries}\n\nIt is common to consider the standard CP transformation\nof the scalar fields as\n\\begin{equation}\n\\Phi_a (t, \\vec{x}) \\rightarrow\n\\Phi^{\\textrm{CP}}_a (t, \\vec{x}) = \\Phi_a^\\ast (t, - \\vec{x}),\n\\label{StandardCP}\n\\end{equation}\nwhere the reference to the time ($t$) and space ($\\vec{x}$)\ncoordinates will henceforth be suppressed.\nHowever,\nin the presence of several scalars with the same quantum numbers,\nbasis transformations can be included in the definition of the\nCP transformation.\nThis yields generalized CP transformations (GCP),\n\\begin{eqnarray}\n\\Phi^{\\textrm{GCP}}_a\n&=& X_{a \\alpha} \\Phi_\\alpha^\\ast\n\\equiv X_{a \\alpha} (\\Phi_\\alpha^\\dagger)^\\top,\n\\nonumber\\\\\n\\Phi^{\\dagger \\textrm{GCP}}_a\n&=& X_{a \\alpha}^\\ast \\Phi_\\alpha^\\top\n\\equiv X_{a \\alpha}^\\ast (\\Phi_\\alpha^\\dagger)^\\ast,\n\\label{GCP}\n\\end{eqnarray}\nwhere $X$ is an arbitrary unitary\nmatrix~\\cite{GCP1,GCP2}.\\footnote{Equivalently, one can \nconsider a generalized time-reversal\ntransformation proposed in Ref.~\\cite{Branco:1983tn}\nand considered further in Appendix A of Ref.~\\cite{GunHab}.}\n\n\nNote that the transformation\n$\\Phi_a\\to\\Phi^{\\rm GCP}_a$, where $\\Phi^{\\rm GCP}_a$ is given\nby Eq.~\\eqref{GCP},\nleaves the kinetic terms invariant.\nThe GCP transformation of a field bilinear yields\n\\begin{equation}\n\\Phi^{\\dagger \\textrm{GCP}}_a\n\\Phi^{\\textrm{GCP}}_b\n=\nX_{a \\alpha}^\\ast X_{b \\beta}\n(\\Phi_\\alpha \\Phi_\\beta^\\dagger)^\\top.\n\\end{equation}\nUnder this GCP transformation,\nthe quadratic terms of the potential may be written as\n\\begin{eqnarray}\nY_{ab} \\Phi^{\\dagger \\textrm{GCP}}_a\n\\Phi^{\\textrm{GCP}}_b\n&=&\nY_{ab} X_{a \\alpha}^\\ast X_{b \\beta}\n\\Phi_\\beta^\\dagger \\Phi_\\alpha\n\\nonumber\\\\\n&=&\nX_{b \\beta} Y_{ba}^\\ast X_{a \\alpha}^\\ast\n\\Phi_\\beta^\\dagger \\Phi_\\alpha\n\\nonumber\\\\\n&=&\nX_{\\alpha a} Y_{\\alpha \\beta}^\\ast X_{\\beta b}^\\ast\n\\Phi_a^\\dagger \\Phi_b\n=\n( X^\\dagger\\, Y\\, X )^\\ast_{ab}\n\\Phi_a^\\dagger \\Phi_b.\n\\end{eqnarray}\nWe have used the Hermiticity condition\n$Y_{ab}=Y_{ba}^\\ast$ in going to the second line;\nand changed the dummy indices $a \\leftrightarrow \\beta$\nand $b \\leftrightarrow \\alpha$ in going to the third line.\nA similar argument can be made for the quartic terms.\nWe conclude that the potential is invariant\nunder the GCP transformation\nof Eq.~\\eqref{GCP} if and only if the coefficients obey\n\\begin{eqnarray}\nY_{ab}^\\ast\n&=&\nX_{\\alpha a}^\\ast Y_{\\alpha \\beta} X_{\\beta b}\n= ( X^\\dagger\\, Y\\, X )_{ab},\n\\nonumber\\\\\nZ_{ab,cd}^\\ast\n&=&\nX_{\\alpha a}^\\ast X_{\\gamma c}^\\ast\nZ_{\\alpha \\beta, \\gamma \\delta} X_{\\beta b} X_{\\delta d}.\n\\label{YZ-CPtransf}\n\\end{eqnarray}\n\nIntroducing\n\\begin{eqnarray}\n\\Delta Y_{ab}\n&=&\nY_{ab} -\nX_{\\alpha a} Y_{\\alpha \\beta}^\\ast X_{\\beta b}^\\ast\n= \\left[Y - ( X^\\dagger\\, Y\\, X )^\\ast \\right]_{ab},\n\\nonumber\\\\\n\\Delta Z_{ab,cd}\n&=&\nZ_{ab,cd} -\nX_{\\alpha a} X_{\\gamma c}\nZ_{\\alpha \\beta, \\gamma \\delta}^\\ast X_{\\beta b}^\\ast X_{\\delta d}^\\ast.\n\\label{DY-DZ}\n\\end{eqnarray}\nwe may write the conditions for invariance under GCP as\n\\begin{eqnarray}\n\\Delta Y_{ab}\n&=&\n0,\n\\label{DY-GCP}\n\\\\\n\\Delta Z_{ab,cd}\n&=&\n0.\n\\label{DZ-GCP}\n\\end{eqnarray}\nGiven Eqs.~\\eqref{hermiticity_coefficients},\nit is easy to show that\n\\begin{eqnarray}\n\\Delta Y_{ab} &=& \\Delta Y_{ba}^\\ast,\n\\nonumber\\\\\n\\Delta Z_{ab,cd} \\equiv \\Delta Z_{cd,ab} &=& \\Delta Z_{ba,dc}^\\ast.\n\\label{DY-DZ-hermiticity}\n\\end{eqnarray}\nThus, we need only consider the real coefficients\n$\\Delta Y_{11}$, $\\Delta Y_{22}$,\n$\\Delta Z_{11,11}$, $\\Delta Z_{22,22}$,\n$\\Delta Z_{11,22}$, $\\Delta Z_{12,21}$,\nand the complex coefficients\n$\\Delta Y_{12}$, $\\Delta Z_{11,12}$,\n$\\Delta Z_{22,12}$, and $\\Delta Z_{12,12}$.\n\n\n\n\n\n\\subsection{GCP and basis transformations}\n\nWe now turn to the interplay between GCP transformations and basis\ntransformations.\nConsider the potential of Eq.~\\eqref{VH2} and call it\n$V(\\Phi)$.\nNow consider the potential obtained from $V(\\Phi)$\nby the basis transformation\n$\\Phi_a \\rightarrow \\Phi^\\prime_a = U_{ab} \\Phi_b$:\n\\begin{equation}\nV (\\Phi^\\prime) =\nY_{ab}^\\prime (\\Phi_a^{\\prime \\dagger} \\Phi^\\prime_b) +\n\\tfrac{1}{2}\nZ^\\prime_{ab,cd} (\\Phi_a^{\\prime \\dagger} \\Phi^\\prime_b)\n(\\Phi_c^{\\prime \\dagger} \\Phi^\\prime_d),\n\\end{equation}\nwhere the coefficients in the new basis are given by\nEqs.~(\\ref{Y-transf}) and (\\ref{Z-transf}).\nWe will now prove the following theorem: If $V(\\Phi)$ is invariant under\nthe GCP transformation of Eq.~(\\ref{GCP}) with the matrix $X$,\nthen $V (\\Phi^\\prime)$ is invariant under a new GCP transformation\nwith matrix\n\\begin{equation}\nX^\\prime = U X U^\\top.\n\\label{X-prime}\n\\end{equation}\nBy hypothesis $V(\\Phi)$ is invariant under\nthe GCP transformation of Eq.~(\\ref{GCP}) with the matrix $X$.\nEq.~(\\ref{YZ-CPtransf}) guarantees that $Y^\\ast = X^\\dagger Y X$.\nNow,\nEq.~(\\ref{Y-transf}) relates the coefficients in the two\nbasis through $Y = U^\\dagger Y^\\prime U$.\nSubstituting gives\n\\begin{equation}\nU^\\top Y^{\\prime \\ast} U^\\ast\n= X^\\dagger (U^\\dagger Y^\\prime U) X,\n\\end{equation}\nor\n\\begin{equation}\nY^{\\prime \\ast}\n= (U^\\ast X^\\dagger U^\\dagger) Y^\\prime (U X U^\\top)\n= X^{\\prime \\dagger} Y^\\prime X^\\prime,\n\\end{equation}\nas required.\nA similar argument holds for the quartic terms and the proof is complete.\n\nThe fact that the transpose $U^\\top$ appears in Eq.~(\\ref{X-prime})\nrather than $U^\\dagger$ is crucial.\nIn Eq.~(\\ref{S-prime}),\napplicable to HF symmetries,\n$U^\\dagger$ appears.\nConsequently,\na basis may be chosen where the HF symmetry is represented by\na diagonal matrix $S$.\nThe presence of $U^\\top$ in Eq.~(\\ref{X-prime}) implies\nthat, \ncontrary to popular belief,\n\\textit{it is not possible to reduce all GCP transformations\nto the standard CP transformation} of Eq.~(\\ref{StandardCP})\nby a basis transformation.\nWhat is possible,\nas we shall see below,\nis to reduce an invariance of the THDM potential under any GCP\ntransformation,\nto an invariance under the standard CP transformation\nplus some extra constraints.\n\nTo be more specific, the following result is easily established. If\nthe unitary matrix $X$ is symmetric, then it follows\nthat\\footnote{Here, we make use of a theorem in linear algebra that\nstates that for any unitary symmetric matrix $X$, a unitary matrix\n$V$ exists such that $X=VV^\\top$.  A proof of this result can be\nfound, e.g., in Appendix B of Ref.~\\cite{GunHab}.}\na unitary matrix\n$U$ exists such that $X'=UXU^\\top=1$, in which case\n$Y^{\\prime\\,*}=Y^\\prime$.  In this case, a basis exists in which the\nGCP is a standard CP transformation.\nIn contrast, if the unitary matrix $X$ is not symmetric, then no\nbasis exists in which $Y$ and $Z$ are real for generic values of the\nscalar potential parameters.\nNevertheless, as we shall demonstrate below, by \\textit{imposing}\nthe GCP symmetry on the scalar potential, the parameters of the\nscalar potential are constrained in such a way that for an\nappropriately chosen basis change, $Y^{\\prime\\,*}\n=X^{\\prime\\,\\dagger}Y'X'=Y'$ (with a similar result for $Z'$).\n\n\nGCP transformations were studied in \nRefs.~\\cite{GCP1,GCP2}. In\nparticular, Ecker, Grimus, and Neufeld \\cite{GCP2} proved that for\nevery matrix $X$ there exists a unitary matrix $U$ such that\n$X^\\prime$ can be reduced to the form\n\\begin{equation}\nX^\\prime = UXU^\\top=\n\\left(\n\\begin{array}{cc}\n   \\phantom{-}\\cos{\\theta} & \\quad \\sin{\\theta}\\\\\n   - \\sin{\\theta} & \\quad \\cos{\\theta}\n\\end{array}\n\\right),\n\\label{GCP-reduced}\n\\end{equation}\nwhere $0 \\leq \\theta \\leq \\pi\/2$. Notice the restricted range for\n$\\theta$. The value of $\\theta$ can be determined in either of two\nways: (i) the eigenvalues of $(X+X^\\top)^\\dagger(X+X^\\top)\/2$ are\n$\\cos{\\theta}$, each of which is twice degenerate; or (ii) $X\nX^\\ast$ has the eigenvalues $e^{\\pm 2 i \\theta}$.\n\n\n\\subsection{The three classes of GCP symmetries}\n\nHaving reached the special form of $X^\\prime$ in\nEq.~(\\ref{GCP-reduced}),\nwe will now follow the strategy adopted by Ferreira and\nSilva \\cite{FS2} in connection with HF symmetries.\nWe substitute Eq.~(\\ref{GCP-reduced}) for $X$ in\nEq.~(\\ref{YZ-CPtransf}),\nin order to identify the constraints imposed by this\nreduced form of the GCP transformations on\nthe quadratic and quartic couplings.\nFor each value of $\\theta$,\ncertain constraints will be forced upon the couplings.\nIf two different values of $\\theta$ enforce the same constraints,\nwe will say that they are in the same class\n(since no experimental distinction between the two will then be\npossible).\nWe will start by considering the special cases of $\\theta=0$\nand $\\theta=\\pi\/2$,\nand then turn our attention to $0 < \\theta < \\pi\/2$.\n\n\n\\subsubsection{CP1: $\\theta=0$}\n\nWhen $\\theta=0$,\n$X^\\prime$ is the unit matrix,\nand we obtain the standard CP transformation,\n\\begin{eqnarray}\n\\Phi_1 &\\rightarrow& \\Phi_1^\\ast,\n\\nonumber\\\\\n\\Phi_2 &\\rightarrow& \\Phi_2^\\ast, \\label{eq:cp1}\n\\end{eqnarray}\nunder which Eqs.~(\\ref{YZ-CPtransf}) take the very simple\nform\n\\begin{eqnarray}\nY_{ab}^\\ast\n&=&\n Y_{ab} ,\n\\nonumber\\\\\nZ_{ab,cd}^\\ast\n&=&\nZ_{ab,cd}.\n\\end{eqnarray}\nWe denote this CP transformation by CP1.\nIt forces all couplings to be real.\nSince most couplings are real by the Hermiticity of\nthe Higgs potential,\nthe only relevant constraints are\n$\\textrm{Im}\\, m_{12}^2 = \\textrm{Im}\\, \\lambda_5 =\n\\textrm{Im}\\, \\lambda_6 = \\textrm{Im}\\, \\lambda_7 = 0$.\n\n\n\\subsubsection{CP2: $\\theta=\\pi\/2$}\n\nWhen $\\theta=\\pi\/2$,\n\\begin{equation}\nX^\\prime =\n\\left(\n\\begin{array}{cc}\n   \\phantom{-} 0 & \\quad 1\\\\\n   - 1 & \\quad 0\n\\end{array}\n\\right), \\label{eq:cp2}\n\\end{equation}\nand we obtain the CP transformation,\n\\begin{eqnarray}\n\\Phi_1 &\\rightarrow& \\Phi_2^\\ast,\n\\nonumber\\\\\n\\Phi_2 &\\rightarrow& - \\Phi_1^\\ast,\n\\end{eqnarray}\nwhich we denote by CP2.\nThis was considered by Davidson and Haber \\cite{DavHab}\nin their Eq.~(37),\nwho noted that if this symmetry holds in one basis,\nit holds in \\textit{all} basis choices.\nUnder this transformation,\nEq.~(\\ref{DY-GCP}) forces the matrix of quadratic\ncouplings to obey\n\\begin{equation}\n0 =\n\\Delta Y =\n\\left(\n\\begin{array}{cc}\n   m_{11}^2 - m_{22}^2 & \\quad -2 m_{12}^2\\\\\n   - 2 m_{12}^{2 \\ast} & \\quad m_{22}^2 - m_{11}^2,\n\\end{array}\n\\right)\n\\end{equation}\nleading to $m_{22}^2 = m_{11}^2$ and $m_{12}^2=0$.\nSimilarly,\nwe may construct a matrix of matrices containing\nall coefficients $\\Delta Z_{ab,cd}$.\nThe uppermost-leftmost matrix corresponds to $\\Delta Z_{11,cd}$.\nThe next matrix along the same line corresponds\nto $\\Delta Z_{12,cd}$, and so on.\nTo enforce invariance under CP2,\nwe equate it to zero,\n\\begin{equation}\n0 =\n\\left(\n\\begin{array}{cc}\n   \\left(\n   \\begin{array}{cc}\n      \\lambda_1 - \\lambda_2 & \\quad\n      \\lambda_6 + \\lambda_7 \\\\\n      \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad\n      0\n   \\end{array}\n   \\right)\n   &\n   \\left(\n   \\begin{array}{cc}\n      \\lambda_6 + \\lambda_7 & \\quad\n      0 \\\\\n      0 & \\quad\n      \\lambda_6 + \\lambda_7\n   \\end{array}\n   \\right)\n   \\\\*[7mm]\n   \\left(\n   \\begin{array}{cc}\n   \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad\n   0 \\\\\n   0 & \\quad\n   \\lambda_6^\\ast + \\lambda_7^\\ast\n   \\end{array}\n   \\right)\n   &\n   \\left(\n   \\begin{array}{cc}\n   0 & \\quad\n   \\lambda_6 + \\lambda_7 \\\\\n   \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad\n   \\lambda_2 - \\lambda_1\n   \\end{array}\n   \\right)\n\\end{array}\n\\right).\n\\end{equation}\nWe learn that invariance under CP2 forces\n$m_{22}^2 = m_{11}^2$ and $m_{12}^2=0$,\n$\\lambda_2=\\lambda_1$, and $\\lambda_7 = - \\lambda_6$,\nleading precisely to the ERPS of Eq.~(\\ref{ERPS}).\nRecall that Gunion and Haber \\cite{GunHab} found that,\nunder these conditions we can always find a basis where\nall parameters are real.\nAs a result,\nif the potential is invariant under CP2,\nthere is a basis where CP2 still holds and in which\nthe potential is also invariant under CP1.\n\n\n\n\\subsubsection{CP3: $0 < \\theta < \\pi\/2$}\n\nFinally we turn to the cases where $0 < \\theta < \\pi\/2$.\nImposing Eq.~(\\ref{DY-GCP}) yields\n\\begin{eqnarray}\n0 = \\Delta Y_{11} &=&\n\\left[ (m_{11}^2 - m_{22}^2)\\ s - 2\\ \\textrm{Re}\\, m_{12}^2\\ c \\right] s,\n\\nonumber\\\\\n0 = \\Delta Y_{22} &=&\n- \\Delta Y_{11},\n\\nonumber\\\\\n0 = \\Delta Y_{12} &=&\n\\textrm{Re}\\, m_{12}^2\\ ( c_2 - 1) - 2 i\\ \\textrm{Im}\\, m_{12}^2\n       + \\tfrac{1}{2} (m_{22}^2 - m_{11}^2)\\ s_2,\n\\end{eqnarray}\nwhere we have used $c=\\cos{\\theta}$, $s=\\sin{\\theta}$,\n$c_2=\\cos{2 \\theta}$, and $s_2=\\sin{2 \\theta}$.\nSince $\\theta \\neq 0, \\pi\/2$,\nthe conditions $m_{22}^2 = m_{11}^2$ and $m_{12}^2=0$ are imposed,\nas in CP2.\nSimilarly, Eq.~(\\ref{DZ-GCP}) yields\n\\begin{eqnarray}\n0 = \\Delta Z_{11,11} &=&\n\\lambda_1 (1-c^4) - \\lambda_2 s^4\n- \\tfrac{1}{2} \\lambda_{345} s_2^2\n+ 4\\ \\textrm{Re}\\, \\lambda_6 c^3 s + 4\\ \\textrm{Re}\\, \\lambda_7 c s^3,\n\\nonumber\\\\\n0 = \\Delta Z_{22,22} &=&\n\\lambda_2 (1-c^4) - \\lambda_1 s^4\n- \\tfrac{1}{2} \\lambda_{345} s_2^2\n- 4\\ \\textrm{Re}\\, \\lambda_7 c^3 s - 4\\ \\textrm{Re}\\, \\lambda_6 c s^3,\n\\nonumber\\\\\n0 = \\Delta Z_{11,22}\n&=&\n- \\tfrac{1}{4} s_2\n\\left[\n4 \\textrm{Re}\\, (\\lambda_6 - \\lambda_7) c_2\n+ (\\lambda_1 + \\lambda_2 - 2 \\lambda_{345})s_ 2\n\\right],\n\\nonumber\\\\\n0 = \\Delta Z_{12,21} &=&\n\\Delta Z_{11,22}\n\\nonumber\\\\\n0 = \\textrm{Re}\\, \\Delta Z_{11,12} &=&\n\\tfrac{1}{4} s \\left[\n(-3 \\lambda_1 + \\lambda_2 + 2 \\lambda_{345}) c\n- (\\lambda_1 + \\lambda_2 - 2 \\lambda_{345}) c_3\n\\right.\n\\nonumber\\\\\n& & \\hspace{7mm}\n\\left.\n+ 4 \\textrm{Re}\\, \\lambda_6 (2 s + s_3)\n- 4 \\textrm{Re}\\, \\lambda_7 s_3\n\\right],\n\\nonumber\\\\\n0 = \\textrm{Re}\\, \\Delta Z_{22,12} &=&\n\\tfrac{1}{4} s \\left[\n(- \\lambda_1 + 3 \\lambda_2 - 2 \\lambda_{345}) c\n+ (\\lambda_1 + \\lambda_2 - 2 \\lambda_{345}) c_3\n\\right.\n\\nonumber\\\\\n& & \\hspace{7mm}\n\\left.\n- 4 \\textrm{Re}\\, \\lambda_6 s_3\n+ 4 \\textrm{Re}\\, \\lambda_7 (2 s + s_3)\n\\right],\n\\nonumber\\\\\n0 = \\textrm{Re}\\, \\Delta Z_{12,12} &=&\n\\Delta Z_{11,22}\n\\label{Real-5}\n\\\\\n0 = \\textrm{Im}\\, \\Delta Z_{11,12} &=&\n\\tfrac{1}{2}\n\\left[\n\\textrm{Im}\\, \\lambda_6 (3+c_2)\n+ \\textrm{Im}\\, \\lambda_7 (1-c_2)\n- \\textrm{Im}\\, \\lambda_5 s_2\n\\right],\n\\nonumber\\\\\n0 = \\textrm{Im}\\, \\Delta Z_{22,12} &=&\n\\tfrac{1}{2}\n\\left[\n\\textrm{Im}\\, \\lambda_6 (1-c_2)\n+ \\textrm{Im}\\, \\lambda_7 (3+c_2)\n+ \\textrm{Im}\\, \\lambda_5 s_2\n\\right],\n\\nonumber\\\\\n0 = \\textrm{Im}\\, \\Delta Z_{12,12} &=&\n2 c\n\\left[\n\\textrm{Im}\\, \\lambda_5 c + \\textrm{Im}\\,(\\lambda_6-\\lambda_7)s\n\\right],\n\\label{Im-3}\n\\end{eqnarray}\nwhere $\\lambda_{345}=\\lambda_3 + \\lambda_4 + \\textrm{Re}\\, \\lambda_5$,\n$c_3 = \\cos{3 \\theta}$, and $s_3 = \\sin{3 \\theta}$.\n\nThe last three equations may be written as\n\\begin{equation}\n0=\n\\left[\n\\begin{array}{ccc}\n-s_2 &\\quad (3+c_2) & \\quad (1-c_2)\\\\\n\\phantom{-}s_2 &\\quad  (1-c_2) &\\quad  (3+c_2)\\\\\n(1+c_2) &\\quad  s_2 &\\quad  -s_2\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{c}\n\\textrm{Im}\\, \\lambda_5 \\\\\n\\textrm{Im}\\, \\lambda_6 \\\\\n\\textrm{Im}\\, \\lambda_7\n\\end{array}\n\\right].\n\\end{equation}\nThe determinant of this homogeneous system of three equations\nin three unknowns is $32 c^2$,\nwhich can never be zero since we are assuming that $\\theta \\neq \\pi\/2$.\nAs a result,\n$\\lambda_5$, $\\lambda_6$, and $\\lambda_7$ are real,\nwhatever the value of $0 < \\theta < \\pi\/2$ chosen for the GCP\ntransformation.\nSince $m_{12}^2=0$,\nall potentially complex parameters must be real.\nWe conclude that a potential invariant\nunder any GCP with $0 < \\theta < \\pi\/2$\nis automatically invariant under CP1.\nCombining this with what we learned from CP2,\nwe conclude the following:\nif a potential is invariant under some GCP transformation,\nthen a basis may be found in which it is also invariant\nunder the standard CP transformation,\nwith some added constraints on the parameters.\n\nThe other set of five independent homogeneous equations in\nfive unknowns has a determinant equal to zero,\nmeaning that not all parameters must vanish.\nWe find that\n\\begin{eqnarray}\n0=\n\\Delta Z_{11,11} - \\Delta Z_{22,22}\n&=&\n2s \\left[\ns\\ (\\lambda_1-\\lambda_2) + c\\ 2 \\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\n\\right],\n\\nonumber\\\\\n0=\n\\textrm{Re}\\, \\Delta Z_{11,12}\n- \\textrm{Re}\\, \\Delta Z_{22,12}\n&=&\ns \\left[\n-c\\ (\\lambda_1-\\lambda_2) + s\\ 2 \\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\n\\right].\n\\end{eqnarray}\nSince $s \\neq 0$,\nwe obtain the homogeneous system\n\\begin{equation}\n0=\n\\left[\n\\begin{array}{cc}\n\\phantom{-}s & \\quad c\\\\\n-c & \\quad s\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{c}\n\\lambda_1 - \\lambda_2 \\\\\n2 \\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\n\\end{array}\n\\right],\n\\end{equation}\nwhose determinant is unity.\nWe conclude that $\\lambda_2 = \\lambda_1$ and $\\lambda_7 = - \\lambda_6$.\nThus,\nGCP invariance with any value of $0 < \\theta \\leq \\pi\/2$ leads\nto the ERPS of Eq.~(\\ref{ERPS}).\nSubstituting back we obtain\n$\\Delta Z_{11,11} = \\Delta Z_{22,22} = - \\Delta Z_{11,22}$\nand\n$\\textrm{Re}\\, \\Delta Z_{11,12} = - \\textrm{Re}\\, \\Delta Z_{22,12}$,\nleaving only two independent equations:\n\\begin{eqnarray}\n0=\n\\Delta Z_{11,11}\n&=&\n\\tfrac{1}{2} s_2 \\left[\n(\\lambda_1 - \\lambda_{345}) s_2 + 4 \\lambda_6 c_2 \\right],\n\\nonumber\\\\\n0=\n\\textrm{Re}\\, \\Delta Z_{22,12}\n&=&\n\\tfrac{1}{2} s_2 \\left[\n(\\lambda_1 - \\lambda_{345}) c_2 - 4 \\lambda_6 s_2 \\right],\n\\end{eqnarray}\nwhere we have used $c + c_3 = 2 c c_2$ and $s + s_3 = 2 c s_2$.\nSince $s_2 \\neq 0$,\nthe determinant of the system does not vanish,\nforcing $\\lambda_1=\\lambda_{345}$ and $\\lambda_6=0$.\n\nNotice that our results do not depend on which exact\nvalue of $ 0 < \\theta < \\pi\/2$ in Eq.~(\\ref{GCP-reduced}) we have chosen.\nIf we require invariance of the potential under GCP with some\nparticular value of $ 0 < \\theta < \\pi\/2$,\nthen the potential is immediately invariant under GCP\nwith any other value of $ 0 < \\theta < \\pi\/2$.\nWe name this class of CP invariances, CP3.\nCombining everything,\nwe conclude that invariance under CP3 implies\n\\begin{eqnarray}\nm_{11}^2 = m_{22}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = \\lambda_6 = 0,\n\\nonumber\\\\\n\\textrm{Im}\\, \\lambda_5 = 0,\n& \\hspace{4ex} &\n\\textrm{Re}\\, \\lambda_5 = \\lambda_1 - \\lambda_3 -\\lambda_4.\n\\label{Region-CP3}\n\\end{eqnarray}\nThe results of this section are all summarized in Table~\\ref{master1}\nof section~\\ref{sec:summary}.\n\n\\subsection{The square of the GCP transformation}\n\nIf we apply a GCP transformation twice to the scalar fields, we will\nhave, from Eq.~\\eqref{GCP}, that\n\\begin{equation}\n\\left(\\Phi^{\\textrm{GCP}}_a\\right)^{\\textrm{GCP}}\n\\;=\\;\nX_{a \\alpha} \\left(\\Phi^{\\textrm{GCP}}_\\alpha\\right)^\\ast\n\\;=\\;\nX_{a \\alpha}\\, X_{\\alpha b}^\\ast\\ \\Phi_b \\;\\;\\; ,\n\\end{equation}\nso that the square of a GCP transformation is given by\n\\begin{equation}\n(GCP)^2 \\;=\\; XX^\\ast \\;\\;\\; .\n\\label{eq:cpq}\n\\end{equation}\nIn particular, for a generic unitary matrix $X$, $(GCP)^2$ is a Higgs Family \nsymmetry transformation.\n\nUsually, only GCP transformations with $(GCP)^2 = \\boldsymbol{1}$\n(where $\\boldsymbol{1}$ is the unit matrix)\nare considered in the literature.\nFor such a situation,\n$X=X^\\dagger=X^*$,\nand one can always find\na basis in which $X=\\boldsymbol{1}$.\nIn this case,\na GCP transformation is equivalent to a standard CP\ntransformation in the latter basis choice.\nFor example,\nthe restriction that $(GCP)^2 = \\boldsymbol{1}$\n(or equivalently, requiring the squared of the corresponding generalized \ntime-reversal transformation to equal the unit matrix)\nwas imposed in \nRef.~\\cite{GunHab} and more recently in Ref.~\\cite{mani}. \nHowever, as we have illustrated in this section, the invariance\nunder a GCP transformation, in which  $(GCP)^2 \\neq \\boldsymbol{1}$ \n(corresponding to a unitary matrix $X$ that is not symmetric)\nis a \\textit{stronger} restriction on the parameters of the \nscalar potential than the invariance under a standard CP transformation.   \n\nAs we see from the results in the previous sections,\n$X$ is {\\em not} symmetric for the symmetries CP2 and CP3.\nIn fact, this feature provides a strong distinction among the\nthree GCP symmetries previously introduced.  \nLet us briefly examine $(GCP)^2$ for the three possible\ncases $CP1$, $CP2$ and $CP3$.\n\n\\subsubsection{$(CP1)^2$}\n\nComparing Eqs.~\\eqref{GCP} and~\\eqref{eq:cp1}, we come to the\nimmediate conclusion that $X_{CP1}\\,=\\,\\boldsymbol{1}$, so that\nEq.~\\eqref{eq:cpq} yields\n\\begin{equation}\n(CP1)^2\\;=\\;\\boldsymbol{1}\\,.\n\\end{equation}\nThis implies that a CP1-invariant scalar potential \nis invariant under the symmetry group\n$Z_2=\\{\\boldsymbol{1}\\,,\\,CP1\\}$.\n\n\n\\subsubsection{$(CP2)^2$}\n\nThe matrix $X_{\\textrm{CP2}}$ is shown in Eq.~\\eqref{eq:cp2} so that, by\nEq.~\\eqref{eq:cpq}, we obtain\n\\begin{equation}\n(CP2)^2\\;=\\;-\\,\\boldsymbol{1}\\,.\n\\end{equation}\nAlthough this result significantly distinguished CP2 from CP1, \nthe authors of Ref.~\\cite{mani} noted (in considering their\n$CP_g^{(i)}$ symmetries) that the transformation law for $\\Phi_a$\nunder (CP2)$^2$ can be reduced to the identity by a global\nhypercharge transformation.  That is,\nif we start with the symmetry group $Z_4=\\{\\boldsymbol{1}\\,,\\,CP2\\,,\\,\n-\\boldsymbol{1}\\,,\\,-CP2\\}$, we can impose an equivalence relation\nby identifying two elements of $Z_4$ related by multiplication  \nby $-\\boldsymbol{1}$.  If we denote $(Z_2)_Y=\\{\\boldsymbol{1}\\,,\n-\\boldsymbol{1}\\}$ as the two-element\ndiscrete subgroup of the global hypercharge\n$U(1)_Y$, then the discrete symmetry group that is orthogonal to $U(1)_Y$\nis given by $Z_4\/(Z_2)_Y\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} Z_2$.   Hence,\nthe CP2-invariant scalar potential exhibits\na $Z_2$ symmetry orthogonal to the Higgs flavor symmetries\nof the potential. \n\n\n\\subsubsection{$(CP3)^2$}\n\nThe matrix $X_{\\textrm{CP3}}$ is given in Eq.~\\eqref{GCP-reduced}, with\n$0<\\theta<\\pi\/2$, so that, by Eq.~\\eqref{eq:cpq}, we obtain\n\\begin{equation}\n(CP3)^2\\;=\\;\\left(\n\\begin{array}{cc}\n  \\phantom{-} \\cos{2\\theta} & \\quad \\sin{2\\theta}\\\\\n   - \\sin{2\\theta} & \\quad \\cos{2\\theta}\n\\end{array}\n\\right)\\,,\n\\end{equation}\nwhich once again is {\\em not} the unit matrix.\nHowever, the transformation law for $\\Phi_a$ under (CP3)$^2$ \n\\textit{cannot} be reduced to the identity by a global\nhypercharge transformation. \nThis is the reason why Ref.~\\cite{mani} did not consider CP3.\nHowever, $(CP3)^2$ is a non-trivial HF symmetry of the CP3-invariant\nscalar potential.\\footnote{In Section~\\ref{sec:summary}B, we shall\nidentify $(CP3)^2$ with the Peccei Quinn U(1) symmetry defined as\nin Eq.~(\\ref{U1}) and then transformed to a new basis\naccording to the unitary matrix defined in Eq.~(\\ref{UPQ}).}\nThus, one can always reduce the square of \nCP3 to the identity by applying a suitable HF symmetry transformation.\nIn particular, a CP3-invariant scalar potential also exhibits a $Z_2$ symmetry\nthat is orthogonal to the Higgs flavor symmetries of the potential.\n\n\nIn this paper, we prove that there are three and only three\nclasses of GCP transformations.\nOf course, within each class,\none may change the explicit form of the\nscalar potential by a suitable basis transformation;\nbut that will not alter its physical consequences.\nSimilarly,\none can set some parameters to zero in some ad-hoc fashion,\nnot rooted in a symmetry requirement. \nBut, as we have shown, the constraints imposed on the scalar potential\nby a single GCP symmetry can be grouped into three classes:\nCP1, CP2, and CP3.\n\n\n\\section{\\label{sec:summary}Classification of the HF and\nGCP transformation classes in the THDM}\n\n\\subsection{Constraints on scalar potential parameters}\n\nSuppose that one is allowed one single symmetry requirement\nfor the potential in the THDM.\nOne can choose an invariance under one particular Higgs Family\nsymmetry.\nWe know that there are only two independent classes\nof such simple symmetries: $Z_2$ and Peccei-Quinn $U(1)$.\nOne can also choose an invariance under a particular\nGCP symmetry.  \nWe have proved that there are three classes of\nGCP symmetries, named CP1, CP2, and CP3.\nIf any of the above symmetries is imposed on the THDM scalar\npotential (in a specified basis), then the coefficients\nof the scalar potential are constrained, as summarized in\nTable~\\ref{master1}.  For completeness, we also exhibit\nthe constraints imposed by $SO(3)$,\nthe largest possible continuous HF symmetry that is orthogonal to\nthe global hypercharge $U(1)_Y$ transformation.\n\\begin{table}[ht!]\n\\caption{Impact of the symmetries on the coefficients\nof the Higgs potential in a specified basis.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccccc}\nsymmetry & $m_{11}^2$ & $m_{22}^2$ & $m_{12}^2$ &\n$\\lambda_1$ & $\\lambda_2$ & $\\lambda_3$ & $\\lambda_4$ &\n$\\lambda_5$ & $\\lambda_6$ & $\\lambda_7$ \\\\\n\\hline\n$Z_2$ &   &   & 0 &\n   &  &  &  &\n   & 0 & 0 \\\\\n$U(1)$ &  &  & 0 &\n &  & &  &\n0 & 0 & 0 \\\\\n$SO(3)$ &  & $ m_{11}^2$ & 0 &\n   & $\\lambda_1$ &  & $\\lambda_1 - \\lambda_3$ &\n0 & 0 & 0 \\\\\n\\hline\n$\\Pi_2$ &  & $ m_{11}^2$ & real &\n   & $ \\lambda_1$ &  &  &\nreal &  & $\\lambda_6^\\ast$\n\\\\\n\\hline\nCP1 &  &  & real &\n & &  &  &\nreal & real & real \\\\\nCP2 &  & $m_{11}^2$ & 0 &\n  & $\\lambda_1$ &  &  &\n   &  & $- \\lambda_6$ \\\\\nCP3 &  & $m_{11}^2$ & 0 &\n   & $\\lambda_1$ &  &  &\n$\\lambda_1 - \\lambda_3 - \\lambda_4$ (real) & 0 & 0 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{master1}\n\\end{table}\n\nEmpty entries in Table~\\ref{master1} correspond to a lack of constraints on the\ncorresponding parameters.\nTable~\\ref{master1}has been constructed for those basis choices in which\n$Z_2$ and $U(1)$ have the specific forms in Eqs.~(\\ref{Z2}) and\n(\\ref{U1}), respectively.\nIf, for example,\nthe basis is changed and $Z_2$ acquires the form $\\Pi_2$ in\nEqs.~(\\ref{Pi2}),\nthen the constraints on the coefficients are altered,\nas shown explicitly on the\nfourth line of Table~\\ref{master1}.\nHowever,\nthis does not correspond to a new model.\nAll physical predictions are the same since\nthe specific forms of $Z_2$ and $\\Pi_2$ differ only by\nthe basis change in Eq.~(\\ref{Z2ToPi2}).\nThe constraints for CP1, CP2, and CP3 shown in Table I\napply to the basis in which the GCP transformation\nof Eq.~(\\ref{GCP}) is used where $X$ has been transformed into $X^\\prime$\ngiven by Eq.~(\\ref{GCP-reduced}),\nwith $\\theta=0$, $\\theta = \\pi\/2$,\nand $0 < \\theta < \\pi\/2$, respectively.\n\n\n\n\n\\subsection{Multiple symmetries and GCP}\n\nWe now wish to consider the possibility of simultaneously imposing\nmore than one symmetry requirement on the Higgs potential.\nFor example, one can require that $Z_2$ and $\\Pi_2$ be enforced\n\\textit{within the same basis}.  In what follows, we shall indicate\nthat the two symmetries are enforced simultaneously by writing\n$Z_2\\oplus\\Pi_2$.\nCombining the constraints from the appropriate\nrows of Table~\\ref{master1},\nwe conclude that,\nunder these two simultaneous requirements\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = \\lambda_6 = 0,\n\\hspace{4ex} \\textrm{Im}\\, \\lambda_5 = 0.\n\\label{Z2+Pi2}\n\\end{eqnarray}\nThis coincides exactly with the conditions of the ERPS\nin a very special basis,\nas shown in Eq.~(\\ref{ERPS2}).\nSince CP2 leads to the ERPS of Eq.~(\\ref{ERPS}),\nwe conclude that\n\\begin{equation}\nZ_2 \\oplus \\Pi_2 \\equiv \\textrm{CP2 in some specific basis}.\n\\label{equiv-CP2}\n\\end{equation}\nThis was noted previously by Davidson and Haber \\cite{DavHab}.\nNow that we know what all classes of HF and CP symmetries can\nlook like,\nwe can ask whether all GCP symmetries can be written as\nthe result of some multiple HF symmetry.\n\nThis is clearly not possible for CP1 because of parameter counting.\nTable~\\ref{master1} shows that CP1 reduces the scalar potential to ten\nreal parameters.\nWe can still perform an orthogonal basis change while keeping\nall parameters real.\nThis freedom can be used to remove one further parameter;\nfor example, setting $m_{12}^2=0$ by diagonalizing the $Y$\nmatrix.\nNo further simplification is allowed.\nAs a result, CP1 leaves nine independent parameters.\nThe smallest HF symmetry is $Z_2$.\nTable~\\ref{master1} shows that $Z_2$ reduces the potential to six\nreal and one complex parameter.\nThe resulting eight parameters could never account for the\nnine needed to fully describe the most general model\nwith the standard CP invariance CP1.\\footnote{In\nIvanov's language, this is clear since CP1 corresponds to\na $Z_2$ transformation of the vector $\\vec{r}$,\nwhich is the simplest transformation on $\\vec{r}$ one\ncould possibly make.\nSee section~\\ref{more_multiple}.}\n\n\nBut one can utilize two HF symmetries in order\nto obtain the same constraints obtained by invariance under CP3.\nLet us impose \\textit{both} $U(1)$ and $\\Pi_2$\n\\textit{in the same basis}.\nFrom Table~\\ref{master1},\nwe conclude that,\nunder these two simultaneous requirements\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = \\lambda_6 = 0,\n\\hspace{4ex} \\lambda_5 = 0.\n\\label{U1+Pi2}\n\\end{eqnarray}\nThis does not coincide with the\nconditions for invariance under CP3 shown in Eq.~(\\ref{Region-CP3}).\nHowever,\none can use the transformation rules in Eqs.~(A13)-(A23)\nof Davidson and Haber \\cite{DavHab},\nin order to show that a basis transformation, \n\\begin{equation} \\label{UPQ}\nU=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{cc} \\phantom{-}1 & \\quad -i \\\\\n-i & \\quad\\phantom{-}1\\end{array}\\right)\\,, \n\\end{equation}\nmay be chosen which takes us from Eqs.~(\\ref{Region-CP3}),\nwhere $\\textrm{Re}\\, \\lambda_5 = \\lambda_1 -\\lambda_3 -\\lambda_4$,\nto Eqs.~(\\ref{U1+Pi2}),\nwhere $\\lambda_5=0$ (while maintaining the other relations among\nthe scalar potential parameters).\nWe conclude that\n\\begin{equation}\nU(1) \\oplus \\Pi_2 \\equiv \\textrm{CP3 in some specific basis}.\n\\label{equiv-CP3}\n\\end{equation}\nNote that in the basis in which the CP3 relations of  Eq.~(\\ref{Region-CP3})\nare satisfied with $\\lambda_5\\neq 0$, the discrete HF symmetry\n$\\Pi_2$ is still respected.\nHowever, using Eq.~(\\ref{UPQ}), it follows that the U(1)-Peccei Quinn\nsymmetry corresponds to the invariance of the scalar potential under\n$\\Phi_a\\to \\mathcal{O}_{ab}\\Phi_b$, where $\\mathcal{O}$ is an arbitrary\n$SO(2)$ matrix.\n\nThe above results suggest that it should be possible to distinguish CP1,\nCP2, and CP3 in a basis invariant fashion.\nBotella and Silva \\cite{BS} have built three so-called $J$-invariants\nthat detect any signal of CP violation (either explicit or\nspontaneous) after the minimization\nof the scalar potential.  However, in this paper we are concerned\nabout the symmetries of the scalar potential independently of the\nchoice of vacuum.  Thus, we shall consider\nthe four so-called $I$-invariants built by\nGunion and Haber \\cite{GunHab} in order to detect any\nsignal of \\textit{explicit} CP violation present (before the vacuum state\nis determined).\nIf any of these invariants is nonzero, then CP is explicitly violated,\nand neither CP1, nor CP2, nor CP3 hold.\nConversely,\nif all $I$-invariants are zero, then CP is explicitly conserved, but we cannot\ntell a priori which GCP applies.\nEqs.~(\\ref{equiv-CP2}) and (\\ref{equiv-CP3}) provide the crucial hint.\nIf we have CP conservation, $Z_2\\oplus\\Pi_2$ holds,\nand $U(1)$ does not,\nthen we have CP2.\nAlternatively,\nif we have CP conservation, and $U(1)\\oplus\\Pi_2$ also holds,\nthen we have CP3.\nWe recall that both CP2 and CP3 lead to the ERPS,\nand that the general conditions for the ERPS in Eq.~(\\ref{ERPS})\nare basis independent.\nThis allows us to distinguish CP2 and CP3 from CP1.\nBut, prior to the present work,\nno basis-independent quantity had been identified in the literature\nthat could distinguish $Z_2$ and $U(1)$ in the ERPS.\nThe basis-independent quantity $D$ introduced\nin subsection~\\ref{subsec:D} is precisely the invariant required for\nthis task.  That is,\nin the ERPS $D\\neq0$ implies CP2, whereas $D=0$ implies CP3.\n\nOne further consequence of the results of Table~\\ref{master1}\ncan be seen by simultaneously imposing the U(1) Peccei-Quinn symmetry\nand the CP3 symmetry \\textit{in the same basis}.  The resulting\nconstraints on the scalar potential parameters are precisely those of\nthe SO(3) HF symmetry.  Thus, we conclude that\n\\begin{equation}\nU(1) \\oplus \\textrm{CP3} \\equiv SO(3).\n\\label{equiv-O3}\n\\end{equation}\nIn particular, $SO(3)$ is not a simple HF symmetry, as the invariance\nof the scalar potential under a single element of SO(3) is not\nsufficient to guarantee invariance under the full SO(3) group of\ntransformations. \n\n\n\n\\subsection{Maximal symmetry group of the scalar potential\northogonal to $U(1)_Y$}\n\nThe standard CP symmetry, CP1,\nis a discrete $Z_2$ symmetry that transforms the scalar\nfields into their complex conjugates, and hence \nis not a subgroup of the $U(2)$ transformation group\nof Eq.~\\ref{basis-transf}.  We have previously noted that\nTHDM scalar potentials that exhibit \\textit{any} non-trivial\nHF symmetry $G$ is automatically CP-conserving.  Thus, the actual\nsymmetry group of the scalar potential is in fact \nthe semidirect product\\footnote{In general, the non-trivial element of\n$Z_2$ will not commute with all elements of $G$, in which case the\nrelevant mathematical structure is that of a semidirect product.  In\ncases where the non-trivial element of $Z_2$ commutes with all\nelements of $G$, we denote the corresponding direct product as\n$G\\otimes Z_2$.}\nof $G$ and $Z_2$, which we write as $G\\rtimes Z_2$.\nNoting that $U(1)\\rtimes Z_2\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} SO(2)\\rtimes Z_2\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} O(2)$, and\n$SO(3)\\otimes Z_2\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} O(3)$, we conclude that the maximal \nsymmetry groups of the scalar potential orthogonal to $U(1)_Y$\nfor the possible choices of HF symmetries are given in \nTable~\\ref{maximal}.\\footnote{For ease of notation, we denote\n$Z_2\\otimes Z_2$ by $(Z_2)^2$ and $Z_2\\otimes Z_2\\otimes Z_2$\nby $(Z_2)^3$.}\n\n\n\n\\begin{table}[ht]\n\\caption{Maximal symmetry groups [orthogonal to global $U(1)_Y$\nhypercharge] of the scalar sector of the THDM.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\ndesignation &  HF symmetry group  & maximal symmetry group\\\\\n\\hline\n$Z_2$ & $Z_2$ & $(Z_2)^2$ \\\\\nPeccei-Quinn & $U(1)$ & $O(2)$ \\\\\n$SO(3)$ & $SO(3)$ & $O(3)$ \\\\\nCP1  & --- & $Z_2$ \\\\\nCP2  & $(Z_2)^2$ & $(Z_2)^3$ \\\\\nCP3 & $O(2)$ & $O(2)\\otimes Z_2$\n\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{maximal}\n\\end{table}\n\nFinally, we reconsider CP2 and CP3.  Eq.~(\\ref{equiv-CP2})\nimplies that the CP2 symmetry is equivalent to a $(Z_2)^2$ HF\nsymmetry.  To prove this statement, we note that in the\ntwo-dimensional flavor space of Higgs fields, the $Z_2$ and $\\Pi_2$\ndiscrete symmetries defined by Eqs.~(\\ref{Z2}) and (\\ref{Pi2})\nare given by:\n\\begin{equation}\nZ_2=\\{S_0\\,,\\,S_1\\}\\,,\\qquad\\qquad \\Pi_2=\\{S_0\\,,\\,S_2\\}\\,,\n\\end{equation}\nwhere $S_0\\equiv \\boldsymbol{1}$ is the $2\\times 2$ identity matrix and\n\\begin{equation}\nS_1=\\left(\n\\begin{array}{cc}\n1 & \\quad\\phantom{-} 0 \\\\\n0 & \\quad -1 \\\\\n\\end{array}\n\\right)\\,,\\qquad\\qquad\nS_2=\\left(\n\\begin{array}{cc}\n0 & \\quad 1 \\\\\n1 & \\quad 0 \\\\\n\\end{array}\n\\right)\\,.\n\\end{equation}\nIf we impose the $Z_2$ and $\\Pi_2$ symmetry in the same basis, then the\nscalar potential is invariant under the dihedral group of eight elements,\n\\begin{equation}\nD_4=\\{S_0\\,,\\,S_1\\,,\\,S_2\\,,\\,S_3\\,,\\,-S_0\\,,\\,-S_1\\,,\\,-S_2\\,,\\,-S_3\\}\\,,\n\\end{equation}\nwhere $S_3=S_1 S_2=-S_2 S_1$.  \nAs before, we identify $(Z_2)_Y\\equiv\\{S_0\\,,\\,-S_0\\}$ as the\ntwo-element discrete subgroup of the global hypercharge $U(1)_Y$.\nHowever, we have defined the HF symmetries to be orthogonal to $U(1)_Y$.\nThus, to determine the HF symmetry group of CP2, we identify as\nequivalent those elements of $D_4$ that are related by multiplication\nby $-S_0$.  Group theoretically, we identify the HF symmetry group\nof CP2 as\n\\begin{equation}\nD_4\/(Z_2)_Y\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} Z_2\\otimes Z_2\\,.\n\\end{equation}\n\nThe HF symmetry group of CP2 is not the maximally allowed symmetry\ngroup.  In particular, the constraints of CP2 on the scalar potential\nimply the existence of a basis in which all scalar potential\nparameters are real.  Thus, the scalar potential is explicitly\nCP-conserving.  The $Z_2$ symmetry associated with this CP\ntransformation is orthogonal to the HF symmetry as previously noted.  \n(This is easily checked explicitly by employing a four-dimensional real\nrepresentation of the two complex scalar fields.)  Thus,\nthe maximal symmetry group of the CP2-symmetric scalar potential\nis $(Z_2)^3$.  Similarly, Eq.~(\\ref{equiv-CP3})\nimplies that the CP3 symmetry is equivalent to a $U(1)\\rtimes Z_2$ HF\nsymmetry. This is isomorphic to an $O(2)$ HF symmetry, which is\na subgroup of the maximally allowed $SO(3)$ HF symmetry group. \nHowever, the constraints of CP3 on the scalar potential\nimply the existence of a basis in which all scalar potential\nparameters are real.  Thus, the scalar potential is explicitly\nCP-conserving.  Once again, the $Z_2$ symmetry associated with this CP\ntransformation is orthogonal to the HF symmetry noted above.  Thus,\nthe maximal symmetry group of the CP3-symmetric scalar potential\nis $O(2)\\otimes Z_2$.  \n\nThe above results are also summarized in \nTable~\\ref{maximal}.  In all cases, the maximal symmetry group is\na direct product of the HF symmetry group and the $Z_2$ corresponding\nto the standard CP-transformation, whose square is the identity operator.\n\nOne may now ask whether Table~\\ref{maximal} exhausts all\npossible independent symmetry constraints that one\nmay place on the Higgs potential.\nPerhaps one can choose other combinations,\nor maybe one can combine three, four, or more\nsymmetries.\nWe know of no way to answer this problem\nbased only on the transformations of the scalar\nfields $\\Phi_a$.\nFortunately,\nIvanov has solved this problem~\\cite{Ivanov1} by looking\nat the transformation properties of field bilinears,\nthus obtaining for the first time the list of symmetries given in the last\ncolumn of Table~\\ref{maximal}.\n\n\n\\subsection{\\label{more_multiple}More on multiple symmetries}\n\n\nWe start by looking at the implications of the symmetries we have\nstudied so far on the vector $\\vec{r} = \\{ r_1, r_2, r_3\\}$,\nwhose components were introduced in Eq.~(\\ref{r_Ivanov}).\nNotice that a unitary transformation $U$ on\nthe fields $\\Phi_a$ induces an orthogonal\ntransformation $O$ on the vector of bilinears\n$\\vec{r}$,\ngiven by Eq.~(\\ref{O}).\nFor every pair of unitary transformations $\\pm U$\nof $SU(2)$,\none can find some corresponding transformation $O$\nof $SO(3)$,\nin a two-to-one correspondence.\nWe then see what these symmetries imply\nfor the coefficients of Eq.~(\\ref{VH3})\n(recall the $\\Lambda_{\\mu \\nu}$ is a symmetric matrix).\nBelow, we list the transformation of $\\vec{r}$ under which the\nscalar potential is invariant, followed by the \ncorresponding constraints on the\nquadratic and quartic scalar potential parameters, $M_\\mu$ and\n$\\Lambda_{\\mu\\nu}$.  \n\nUsing the results of Table I, we find that $Z_2$ implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{c}\n-r_1\\\\\n-r_2\\\\\n\\phantom{-}r_3\n\\end{array}\n\\right],\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\nM_3\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 &\\,\\,\\, 0 &\\,\\,\\, \\Lambda_{03}\\\\\n 0 &\\,\\,\\, \\Lambda_{11} &\\,\\,\\, \\Lambda_{12} &\\,\\,\\, 0\\\\\n 0 & \\,\\,\\,\\Lambda_{12} &\\,\\,\\, \\Lambda_{22} &\\,\\,\\, 0\\\\\n\\Lambda_{03} &\\,\\,\\, 0 &\\,\\,\\, 0 &\\,\\,\\, \\Lambda_{33}\n\\end{array}\n\\right],\n\\label{Iv-Z2}\n\\end{equation}\n$U(1)$ implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{ccc}\nc_2 & -s_2 & \\phantom{-}0\\\\\ns_2 &\\phantom{-}c_2 & \\phantom{-}0\\\\\n0 & \\phantom{-}0 & \\phantom{-}1\n\\end{array}\n\\right]\\ \\vec{r},\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\nM_3\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 & \\,\\,\\,0 & \\,\\,\\,\\Lambda_{03}\\\\\n 0 &\\,\\,\\, \\Lambda_{11} &\\,\\,\\, 0 &\\,\\,\\, 0\\\\\n 0 & \\,\\,\\,0 & \\,\\,\\,\\Lambda_{11} & \\,\\,\\,0\\\\\n\\Lambda_{03} & \\,\\,\\,0 & \\,\\,\\,0 &\\,\\,\\, \\Lambda_{33}\n\\end{array}\n\\right],\n\\label{Iv-U1}\n\\end{equation}\nand SO(3) implies \n\\begin{equation}\n\\vec{r} \\rightarrow\n\\mathcal{O} \\vec{r},\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\n0\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 & \\,\\,\\,0 & \\,\\,\\,0\\\\\n 0 &\\,\\,\\, \\Lambda_{11} &\\,\\,\\, 0 &\\,\\,\\, 0\\\\\n 0 & \\,\\,\\,0 & \\,\\,\\,\\Lambda_{11} & \\,\\,\\,0\\\\\n0 & \\,\\,\\,0 & \\,\\,\\,0 &\\,\\,\\, \\Lambda_{11}\n\\end{array}\n\\right],\n\\label{Iv-SO3}\n\\end{equation}\nwhere $\\mathcal{O}$ is an arbitrary $3\\times 3$ orthogonal \nmatrix of unit determinant.\nIn the language of bilinears, a basis invariant condition for the presence of\n$SO(3)$ is that the three eigenvalues of $\\tilde{\\Lambda}$ are equal.\n(Recall that $\\tilde{\\Lambda} = \\left\\{\\Lambda_{ij}\\right\\}$; $i,j=1,2,3$).\n\n\nAs for the GCP symmetries,\nCP1 implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{c}\n\\phantom{-}r_1\\\\\n-r_2\\\\\n\\phantom{-}r_3\n\\end{array}\n\\right],\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\nM_1\\\\\n0\\\\\nM_3\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} & \\,\\,\\,\\Lambda_{01}  & \\,\\,\\, 0 & \\,\\,\\, \\Lambda_{03}\\\\\n\\Lambda_{01} & \\,\\,\\, \\Lambda_{11} & \\,\\,\\, 0 & \\,\\,\\, \\Lambda_{13}\\\\\n 0 & \\,\\,\\, 0 & \\,\\,\\, \\Lambda_{22} &  \\,\\,\\,0\\\\\n\\Lambda_{03} &  \\,\\,\\,\\Lambda_{13} &  \\,\\,\\,0 & \\,\\,\\, \\Lambda_{33}\n\\end{array}\n\\right],\n\\label{Iv-CP1}\n\\end{equation}\nCP2 implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{c}\n-r_1\\\\\n-r_2\\\\\n-r_3\n\\end{array}\n\\right],\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\n0\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 &\\,\\,\\, 0 &\\,\\,\\, 0\\\\\n0 &\\,\\,\\, \\Lambda_{11}\\,\\,\\, &\\,\\,\\, \\Lambda_{12} & \\,\\,\\,\\Lambda_{13}\\\\\n0 &\\,\\,\\, \\Lambda_{12} &\\,\\,\\, \\Lambda_{22} &\\,\\,\\, \\Lambda_{23}\\\\\n0 &\\,\\,\\, \\Lambda_{13} & \\,\\,\\,\\Lambda_{23} & \\,\\,\\,\\Lambda_{33}\n\\end{array}\n\\right],\n\\label{Iv-CP2}\n\\end{equation}\nand CP3 implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{ccc}\n\\phantom{-}c_2 & \\phantom{-}0 & \\phantom{-}s_2\\\\\n\\phantom{-}0 & -1 & \\phantom{-}0\\\\\n-s_2 & \\phantom{-}0 & \\phantom{-}c_2\n\\end{array}\n\\right]\\ \\vec{r},\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\n0\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 & \\,\\,\\,0 & 0\\,\\,\\,\\\\\n0 &\\,\\,\\, \\Lambda_{11} &\\,\\,\\, 0 &\\,\\,\\, 0\\\\\n0 &\\,\\,\\, 0 &\\,\\,\\, \\Lambda_{22} &\\,\\,\\, 0\\\\\n0 &\\,\\,\\, 0 &\\,\\,\\, 0 & \\,\\,\\,\\Lambda_{11}\n\\end{array}\n\\right].\n\\label{Iv-CP3}\n\\end{equation}\nNotice that in CP3 two of the eigenvalues of $\\Lambda$ are equal,\nin accordance with our observation that $D$ can be used\nto distinguish between CP2 and CP3.\n\nBecause each unitary transformation on the fields $\\Phi_a$\ninduces an $SO(3)$ transformation on the vector\nof bilinears $\\vec{r}$,\nand because the standard CP transformation\ncorresponds to an inversion of $r_2$\n(a $Z_2$ transformation on the vector $\\vec{r}$),\nIvanov \\cite{Ivanov1}\nconsiders all possible proper and improper transformations\nof $O(3)$ acting on $\\vec{r}$.  \nHe identifies the following six classes of transformations:\n(i) $Z_2$; (ii) $(Z_2)^2$; (iii) $(Z_2)^3$;\n(iv) $O(2)$; (v) $O(2) \\otimes Z_2$; and (vi) $O(3)$.\nNote that these symmetries are all orthogonal to the global $U(1)_Y$\nhypercharge symmetry, as the bilinears $r_0$ and $\\vec{r}$\nare all singlets under a $U(1)_Y$ transformation.\nThe six classes above identified by Ivanov\ncorrespond precisely to the six possible maximal\nsymmetry groups identified in Table~\\ref{maximal}.\nNo other independent symmetry transformations are possible.\n\n\nOur work permits one to identify the abstract transformation\nof field bilinears utilized by Ivanov in terms of\ntransformations on the scalar fields themselves,\nas needed for model building.\nCombining our work with Ivanov's,\nwe conclude that there is only one new type\nof symmetry requirement which one can place on\nthe Higgs potential via multiple symmetries.\nCombining this with our earlier results,\nwe conclude that all possible symmetries on the scalar\nsector of the THDM can be reduced to multiple HF symmetries,\nwith the exception of the standard CP transformation (CP1).\n\n\n\n\n\\section{\\label{sec:allisCP}Building all symmetries with the standard CP}\n\nWe have seen that there are only six independent\nsymmetry requirements, listed in Table~\\ref{maximal},\nthat  one can impose on the Higgs potential.\nWe have shown that all possible symmetries of the scalar\nsector of the THDM can be reduced to multiple HF symmetries,\nwith the exception of the standard CP transformation (CP1).\nNow we wish to show a dramatic result:\n\\textit{all possible symmetries on the scalar\nsector of the THDM can be reduced to multiple applications of\nthe standard CP symmetry.}\n\nUsing Eq.~(\\ref{X-prime}),\nwe see that the basis transformation of Eq.~(\\ref{basis-transf}),\nchanges the standard CP symmetry of Eq.~(\\ref{StandardCP})\ninto the GCP symmetry of Eq.~(\\ref{GCP}),\nwith\n\\begin{equation}\nX=U U^\\top.\n\\end{equation}\nIn particular,\nan orthogonal basis transformation does not affect the\nform of the standard CP transformation.\nSince we wish to generate $X \\neq 1$,\nwe will need complex matrices $U$.\n\nNow we wish to consider the following situation.\nWe have a basis (call it the original basis) and\nimpose the standard CP symmetry CP1 on that original basis.\nNext we consider the same model in a different basis\n(call it $M$) and impose the standard CP symmetry on that basis $M$.\nIn general, this procedure of imposing\nthe standard CP symmetry in the original basis \\textit{and also}\nin the rotated basis $M$ leads to two independent impositions.\nThe first imposition makes all parameters real in\nthe original basis.\nOne way to combine the second imposition with the first\nis to consider the basis transformation $U_M$ taking us\nfrom basis $M$ into the original basis.\nAs we have seen,\nthe standard CP symmetry in basis $M$ turns,\nwhen written in the original basis,\ninto a symmetry under\n\\begin{eqnarray}\n\\Phi^{\\textrm{CP}}_a\n&=& (X_M)_{a \\alpha} \\Phi_\\alpha^\\ast,\n\\nonumber\\\\\n\\Phi^{\\dagger \\textrm{CP}}_a\n&=&\n(X_M)^\\ast_{a \\alpha} (\\Phi_\\alpha^\\dagger)^\\ast,\n\\label{CGP-M}\n\\end{eqnarray}\nwith $X_M=U_M U_M^\\top$.\nNext we consider several such possibilities.\n\nWe start with\n\\begin{equation}\nU_{A} =\n\\left(\n\\begin{array}{cc}\n\\phantom{-}c_{\\pi\/4} & \\quad -i s_{\\pi\/4} \\\\\n- i s_{\\pi\/4} & \\quad \\phantom{-}c_{\\pi\/4}\n\\end{array}\n\\right),\n\\hspace{3ex}\nX_{A} =\n\\left(\n\\begin{array}{cc}\n\\phantom{-}0 & \\quad -i  \\\\\n- i & \\quad \\phantom{-}0\n\\end{array}\n\\right).\n\\end{equation}\nHere and henceforth $c$ ($s$) with a subindex indicates the\ncosine (sine) of the angle given in the subindex.\nWe denote by CP1$_A$ the imposition of the CP symmetry\nin Eq.~(\\ref{CGP-M}) with $X_M=X_A$\n(which coincides with the imposition of the standard CP\nsymmetry in the basis $M=A$).\n\nNext we consider\n\\begin{equation}\nU_{B} =\n\\left(\n\\begin{array}{cc}\ne^{-i \\pi\/4} & \\quad 0 \\\\\n0 & \\quad e^{i \\pi\/4}\n\\end{array}\n\\right),\n\\hspace{3ex}\nX_{B} =\n\\left(\n\\begin{array}{cc}\n-i & \\quad 0  \\\\\n\\phantom{-}0 & \\quad i\n\\end{array}\n\\right).\n\\end{equation}\nWe denote by CP1$_B$ the imposition of the CP symmetry\nin Eq.~(\\ref{CGP-M}) with $X_M=X_B$\n(which coincides with the imposition of the standard CP\nsymmetry in the basis $M=B$).\n\nA third possible choice is\n\\begin{equation}\nU_{C} =\n\\left(\n\\begin{array}{cc}\ne^{i \\delta\/2} & \\quad 0 \\\\\n0 & \\quad e^{-i \\delta\/2}\n\\end{array}\n\\right),\n\\hspace{3ex}\nX_{C} =\n\\left(\n\\begin{array}{cc}\ne^{i \\delta} & \\quad 0 \\\\\n0 & \\quad e^{-i \\delta}\n\\end{array}\n\\right),\n\\end{equation}\nwhere $\\delta \\neq n \\pi\/2$ with $n$ integer.\nWe denote by CP1$_C$ the imposition of the CP symmetry\nin Eq.~(\\ref{CGP-M}) with $X_M=X_C$\n(which coincides with the imposition of the standard CP\nsymmetry in the basis $M=C$).\n\nFinally, we consider\n\\begin{equation}\nU_{D} =\n\\left(\n\\begin{array}{cc}\n\\phantom{i}c_{\\delta\/2} & \\quad i s_{\\delta\/2} \\\\\ni s_{\\delta\/2} & \\phantom{i}\\quad c_{\\delta\/2}\n\\end{array}\n\\right),\n\\hspace{3ex}\nX_{D} =\n\\left(\n\\begin{array}{cc}\n\\phantom{i}c_\\delta & \\quad i s_\\delta  \\\\\ni s_\\delta & \\quad \\phantom{i}c_\\delta\n\\end{array}\n\\right),\n\\end{equation}\nwhere $\\delta \\neq n \\pi\/2$ with $n$ integer.\nWe denote by CP1$_D$ the imposition of the CP symmetry\nin Eq.~(\\ref{CGP-M}) with $X_M=X_D$\n(which coincides with the imposition of the standard CP\nsymmetry in the basis $M=D$).\n\nThe impact of the first three symmetries on the coefficients of the\nHiggs potential are summarized in\nTable~\\ref{master3}.\n\\begin{table}[ht!]\n\\caption{Impact of the CP1$_M$ symmetries on the coefficients\nof the Higgs potential.\nThe notation ``imag'' means that the\ncorresponding entry is purely imaginary.\nCP1 in the original basis has been included for reference.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccccc}\nsymmetry & $m_{11}^2$ & $m_{22}^2$ & $m_{12}^2$ &\n$\\lambda_1$ & $\\lambda_2$ & $\\lambda_3$ & $\\lambda_4$ &\n$\\lambda_5$ & $\\lambda_6$ & $\\lambda_7$ \\\\\n\\hline\nCP1 &  &  & real &\n &  & &  &\nreal & real & real\\\\\n\\hline\nCP1$_A$ &   & $m_{11}^2$  &  &\n   & $\\lambda_1$ &  &  &\n   &  & $\\lambda_6$ \\\\\nCP1$_B$ &  &  & imag &\n &  & &  &\nreal & imag & imag \\\\\nCP1$_C$ &  &  & $|m_{12}^2| e^{i \\delta}$ &\n & & & & $|\\lambda_5| e^{2 i \\delta}$  &\n$|\\lambda_6| e^{i \\delta}$ & $|\\lambda_7| e^{i \\delta}$\n\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{master3}\n\\end{table}\n\n\\noindent\nImposing CP1$_D$ on the Higgs potential leads to the more complicated\nset of equations:\n\\begin{eqnarray}\n2 \\textrm{Im} \\left( m_{12}^2 \\right)\\, c_\\delta\n+ (m_{22}^2 - m_{11}^2)\\, s_\\delta\n&=& 0,\n\\nonumber\\\\\n2 \\textrm{Im} \\left( \\lambda_6 - \\lambda_ 7 \\right)\\, c_{2 \\delta}\n+ \\lambda_{12345}\\, s_{2 \\delta}\n&=& 0,\n\\nonumber\\\\\n2 \\textrm{Im} \\left( \\lambda_6 + \\lambda_ 7 \\right)\\, c_\\delta\n+ \\left( \\lambda_1 - \\lambda_2 \\right)\\, s_\\delta\n&=& 0,\n\\nonumber\\\\\n\\textrm{Im} \\lambda_5\\, c_\\delta\n+ \\textrm{Re} \\left( \\lambda_6 - \\lambda_ 7 \\right)\\, s_\\delta\n&=& 0,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\lambda_{12345} = \\tfrac{1}{2} \\left( \\lambda_1 + \\lambda_ 2 \\right)\n- \\lambda_3 - \\lambda_4 + \\textrm{Re} \\lambda_5.\n\\end{equation}\n\nCombining these results with those in Table~\\ref{master1},\nwe have shown that\n\\begin{eqnarray}\n\\textrm{CP1} \\oplus \\textrm{CP1}_B\n&=& Z_2\\ \\  \\textrm{in some specific basis},\n\\nonumber\\\\\n\\textrm{CP1} \\oplus \\textrm{CP1}_C\n&=& U(1),\n\\nonumber\\\\\n\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_B\n&=& \\textrm{CP2}\\ \\  \\textrm{in some specific basis},\n\\nonumber\\\\\n\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_C\n&=& \\textrm{CP3}\\ \\  \\textrm{in some specific basis},\n\\nonumber\\\\\n\\textrm{CP1} \\oplus \\textrm{CP1}_C \\oplus \\textrm{CP1}_D\n&=& SO(3).\n\\label{incredible}\n\\end{eqnarray}\nLet us comment on the ``specific basis choices'' needed.\nImposing $\\textrm{CP1} \\oplus \\textrm{CP1}_B$ leads to\n$ m_{12}^2=\\lambda_6=\\lambda_7=0$ and $\\textrm{Im} \\lambda_5=0$,\nwhile imposing $Z_2$ leads to $ m_{12}^2=\\lambda_6=\\lambda_7=0$\nwith no restriction on $\\lambda_5$.\nHowever, when $Z_2$ holds one may rephase $\\Phi_2$\nby the exponential of $-i \\arg(\\lambda_5)\/2$,\nthus making $\\lambda_5$ real.\nIn this basis,\nthe restrictions of $Z_2$ coincide with the restrictions\nof $\\textrm{CP1} \\oplus \\textrm{CP1}_B$.\nSimilarly,\nimposing $\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_C$\nleads to $m_{12}^2=\\lambda_5=\\lambda_6=\\lambda_7=0$,\n$m_{22}^2=m_{11}^2$ and $\\lambda_2=\\lambda_1$.\nWe see from Table~\\ref{master1} that CP3 has these features,\nexcept that $\\lambda_5$ need not vanish; it is real and\n$\\textrm{Re} \\lambda_5 = \\lambda_1-\\lambda_3-\\lambda_4$.\nStarting from the CP3 conditions and\nusing the transformation rules in Eqs.~(A13)-(A23) of Davidson and\nHaber \\cite{DavHab},\nwe find that a basis choice is possible such that\n$\\textrm{Re} \\lambda_5=0$.\\footnote{Notice that, in the new basis,\n$\\lambda_1$ differs in general from $\\lambda_3 + \\lambda_4$;\notherwise the larger $SO(3)$ Higgs Family symmetry would hold.}\nPerhaps it is easier to prove the equality\n\\begin{equation}\n\\textrm{CP1} \\oplus \\textrm{CP1}_B \\oplus \\textrm{CP1}_D\n= \\textrm{CP3}\\ \\  \\textrm{in some specific basis}.\n\\end{equation}\nIn this case,\nthe only difference between the impositions from the\ntwo sides of the equality come from the sign of $\\textrm{Re} \\lambda_5$,\nwhich is trivial to flip through the basis change\n$\\Phi_2 \\rightarrow - \\Phi_2$.\nFinally,\nimposing $\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_B$\nwe obtain $m_{12}^2=\\textrm{Im} \\lambda_5=\\lambda_6=\\lambda_7=0$,\n$m_{22}^2=m_{11}^2$ and $\\lambda_2=\\lambda_1$.\nThis does not coincide with the conditions of CP2 which\nlead to the ERPS of Eq.~(\\ref{ERPS}).\nFortunately,\nand as we mentioned before,\nDavidson and Haber \\cite{DavHab} proved that\none may make a further basis transformation\nsuch that Eq.~(\\ref{ERPS2}) holds,\nthus coinciding with the conditions imposed by\n$\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_B$.\n\nNotice that our description of CP2 in terms of several\nCP1 symmetries is in agreement with the results found by the\nauthors of Ref.~\\cite{mani}.\nThese authors also showed a very interesting\nresult, concerning  spontaneous symmetry breaking in 2HDM models\npossessing a CP2 symmetry.\nNamely, they prove (their Theorem 4)\nthat electroweak symmetry breaking will {\\em necessarily}\nspontaneously break CP2.\nHowever, they also show that the vacuum\nwill respect at least one of the CP1 symmetries which compose\nCP2.\nWhich is to say, in a model which has a CP2 symmetry,\nspontaneous symmetry breaking necessarily respect the CP1\nsymmetry.\n\nIn summary,\nwe have proved that all possible symmetries on the scalar\nsector of the THDM,\nincluding Higgs Family symmetries,\ncan be reduced to multiple applications of\nthe standard CP symmetry.\n\n\n\\section{\\label{sec:conclusions}Conclusions}\n\nWe have studied the application of generalized CP symmetries\nto the THDM,\nand found that there are only two independent classes\n(CP2 and CP3),\nin addition to the standard CP symmetry (CP1).\nThese two classes lead to an exceptional region of parameter,\nwhich exhibits either a $Z_2$ discrete symmetry or\na larger $U(1)$ Peccei-Quinn symmetry.\nWe have succeeded in\nidentifying a basis-independent invariant quantity that can\ndistinguish between the $Z_2$ and $U(1)$ symmetries.\nIn particular, such an invariant is required\nin order to distinguish between CP2 and CP3,\nand completes the description of all symmetries in the THDM\nin terms of basis-invariant quantities.\nMoreover, CP2 and CP3 can be obtained by combining\ntwo Higgs Family symmetries and that this is not possible\nfor CP1.  \n\nWe have shown that all symmetries of the THDM previously identified\nby Ivanov \\cite{Ivanov1} can be achieved through simple symmetries.\nwith the exception of $SO(3)$.\nHowever, the $SO(3)$ Higgs Family symmetry\ncan be achieved by imposing a $U(1)$ Peccei-Quinn  \nsymmetry and the CP3-symmetry in the same basis.\nFinally, we have demonstrated that\nall possible symmetries of the scalar\nsector of the THDM can be reduced to multiple applications of\nthe standard CP symmetry.\nOur complete description of the symmetries on the scalar fields\ncan be combined with symmetries in the quark and lepton sectors,\nto aid in model building.\n\n\n\\begin{acknowledgments}\nWe would like to thank Igor Ivanov and Celso Nishi for their\nhelpful comments on the first version of this manuscript.\nThe work of P.M.F. is supported in part by the Portuguese\n\\textit{Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e a Tecnologia} (FCT)\nunder contract PTDC\/FIS\/70156\/2006. The work of H.E.H. is\nsupported in part by the U.S. Department of Energy, under grant\nnumber DE-FG02-04ER41268. The work of J.P.S. is supported in\npart by FCT under contract CFTP-Plurianual (U777).\n\nH.E.H. is most grateful for the kind hospitality and support of the\nCentro de F\\'{\\i}sica Te\\'orica e Computacional at Universidade de\nLisboa\n(sponsored by the Portuguese FCT and\nFunda\\c{c}\\~{a}o Luso-Americana para o Desenvolvimento)\nand the Centro de F\\'{\\i}sica Te\\'orica de Part\\'{\\i}culas at\nInstituto Superior T\\'ecnico during his visit to Lisbon. This work\nwas initiated during a conference in honor of Prof. Augusto Barroso,\nto whom we dedicate this article.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nYoung high-mass stars (M $ \\geq \\,$ 8 M$_\\odot$) probed in cm-wavelength interferometric studies typically appear as fairly bright (flux densities of $\\sim$ few mJy to Jy)  regions of ionized gas  that are classified according to their size and emission measure, e.g., compact, ultracompact (UC), and hypercompact (HC) H{\\small II} regions \\citep[e.g.,][]{2005IAUS..227..111K}. It is generally thought that once  nuclear burning  has  begun the star  produces enough UV radiation to photoionize the surrounding gas. However, theories of the earliest stages remain poorly constrained by observations mainly due to the characteristics of the regions where they are born, which are highly dust-obscured, distant ($\\gtrsim$ 1 kpc) regions that undergo rapid evolution, and they reach the zero-age main sequence (ZAMS) while still heavily accreting. In fact, an evolutionary sequence for high-mass stars has not yet been established \\citep[e.g.,][]{2013A&A...550A..21S, 2014prpl.conf..149T}, although significant progress has been achieved on both observational and theoretical fronts \\citep[e.g.,][]{2018ARA&A..56...41M}. The identification and study of\nobjects in the early stages of their evolution will help us to\n  discriminate among proposed mechanisms for their formation; the two main\nscenarios being core accretion (i.e., scaled-up version of low-mass\nstar formation) and competitive accretion (i.e., in which stars in a cluster attract each other while they accrete from a shared reservoir of gas; see \\citealp{2014prpl.conf..149T}). The low-mass star formation process is modeled by\naccretion via a circumstellar disk and a collimated jet\/outflow that removes angular momentum and allows accretion to proceed \\citep[e.g.,][]{1988ApJ...328L..19S}. The jet\/outflow system is powered  magnetohydrodynamically by rotating magnetic fields coupled to either the disk (disk winds: e.g., \\citealp{2000prpl.conf..759K}) and\/or the protostar (X-winds: e.g., \\citealp{1987ARA&A..25...23S}).  Additionally, protostellar collisions have been  proposed as an alternative mechanism for the formation of high-mass stars \\citep{1998MNRAS.298...93B, 2005AJ....129.2281B}. \\\\\n\nMassive molecular outflows are a common phenomenon in high-mass star forming regions (e.g., \\citealp{1996ApJ...457..267S, 2002A&A...383..892B}); hence accretion disks and ionized jets similar to those found towards low-mass protostars are also expected. \n In addition, several surveys toward high-mass star forming regions in the NIR spectral lines of H$_2$ have detected a large number of molecular jets \\citep[e.g.,][]{2017ApJ...844...38W, 2015MNRAS.450.4364N}.\nHowever, the current sample of known high-mass protostars associated with disks \\citep[see review by][]{2016A&ARv..24....6B} and collimated jets \\citep[e.g.,][]{1995ApJ...449..184M, 1998ApJ...502..337M, 2006ApJ...638..878C, 2008AJ....135.2370R} is inadequate to draw conclusions about the entire population.\nThe detection of sources at the onset of high-mass star formation and the measurement of their physical properties is essential  to test theoretical models of high-mass star formation \\citep[e.g.,][]{2014prpl.conf..149T}. Furthermore, the  most sensitive instruments are necessary  to place significant constraints on the occurrence rate and parameters of these detections.\\\\\n\nIn  \\citet[][hereafter Paper I]{2016ApJS..227...25R} we described our high sensitivity ($\\sim$3 -- 10 $\\mu$Jy beam$^{-1}$) continuum survey, which aimed to identify candidates in early evolutionary phases of high-mass star formation and to study their centimeter continuum emission. We observed  58 high-mass star forming region candidates using the Karl G. Jansky Very Large Array (VLA)\\footnote{The National Radio Astronomy \nObservatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.} at 1.3 and 6 cm wavelengths at an angular resolution ${\\scriptstyle <}\\,$0\\rlap.$^{\\prime \\prime}$6.  The 58 targets were grouped into three categories based on their mid and far-IR luminosity as well as the temperature of the cores: 25 hot molecular cores (HMCs), 15 cold molecular cores with mid-IR point source association (CMC--IRs), and 18 cold molecular cores (CMCs) devoid of IR point source associations. The cores in our sample cover a wide range of parameters such as bolometric luminosity and distance. They have similar masses and densities, however, the latter two types of cores---mainly found within infrared dark clouds (IRDCs)---have lower temperatures (T$\\sim$ 10--20 K) than HMCs (T${\\scriptstyle >}\\,$50 K; depending on the probe and scale). In \\citetalias{2016ApJS..227...25R} we reported  detection rates of 1\/18 (6$\\%$) CMCs, 8\/15 (53$\\%$) CMC-IRs and 25\/25 (100$\\%$) HMCs.\nIn several cases, we detected multiple sources within a region, which resulted in a total detection of 70 radio sources associated with 1.2 mm dust clumps.\nThe 100$\\%$ detection rate of centimeter emission in the HMCs  is a higher fraction than previously reported. This suggests that radio continuum may be present, albeit weak, in {\\it all} HMCs although in many cases it is only detectable with the superior sensitivity now available with the upgraded VLA.   Our results show further evidence for an evolutionary sequence in the formation of high-mass stars, from  very early stage cold cores (i.e., CMCs) to relatively more evolved ones (i.e., HMCs).\\\\\n  \n\n\nA number of physical processes can cause centimeter continuum emission associated with high-mass star forming regions (see \\citealt{2012ApJ...755..152R} and \\citealt{2013ApJ...766..114S} summaries of thermal and non-thermal emission detected at centimeter wavelengths from YSOs). Recently, \\citet*[][hereafter TTZ16]{2016ApJ...818...52T}  developed a model to predict the radio emission from high-mass stars forming via core accretion. The \\citetalias{2016ApJ...818...52T} model predicts that during the first stages of ionization the H{\\small II} region is initially confined to the vertical (or outflow) axis and produces free-free emission with  similar features and parameters as observed towards ionized jets. Ionized jets are detected as weak and  compact centimeter continuum sources. At subarcsecond resolutions, they usually show a string-like morphology, often aligned  with a large-scale molecular outflow  of size up to a few parsecs. Ionized jets trace outflows on  smaller scales, providing the location of the driving protostar, that otherwise are deeply embedded in the natal clump and generally remain undetected at other wavelengths  due to the high extinction in the region \\citep{1998AJ....116.2953A}.  However, less extincted sources may\nhave molecular jet counterparts visible in H$_2$ line emission from shocked gas.\nThese sources are also called `thermal radio jets' due to their characteristic  rising spectrum which is consistent with free-free radiation from ionized gas.  \nThe  ionization mechanism of these jets has been proposed to be \\emph{shock-induced ionization} when the wind from the central protostar ionizes itself through  shocks due to variations in velocity of the flow or variations of the mass loss rate  (\\citealt*{1987RMxAA..14..595C}; \\citealt{1989ApL&C..27..299C}).  Unlike the simple model of a  uniform electron density H{\\small II} region,  ionized jets and winds have a radial density gradient and thus  are partially optically thick.  \\citet{1986ApJ...304..713R} discussed the behavior of  collimated jets and the dependency  of their physical parameters (such as temperature, velocity, density and ionization fraction) on  morphology, independently of the mechanism of ionization, and showed that the spectral index of a partially ionized jet  ranges  between $-0.1 \\leq \\alpha \\leq 1.1$.\\\\\n\nThe detection of ionized jets toward  high-mass stars at their early stages, as predicted by \\citetalias{2016ApJ...818...52T}, can help to distinguish between accretion scenarios (highly organized outflows are expected from core accretion but not from competitive accretion scenarios; \\citealt{2016ApJ...821L...3T}),  and ultimately will give us insight about accretion disks around high-mass stars. Several systematic studies searching for ionized jets have been reported in the literature. \\citet{2012ApJ...753...51G},  from a sample of 33 IR luminous objects, detected 2 ionized jets using the Australia Telescope Compact Array (ATCA) with a 4$\\sigma$ detection limit and an image rms ($\\sigma$) of $\\sim$0.1-0.2 mJy beam$^{-1}$  at 4.8 and 8.6 GHz.  \\citet{2016A&A...585A..71M} observed  11 high-mass YSOs using the Jansky VLA and detected 5 collimated ionized jets and 6 ionized wind candidates with a 3$\\sigma$ detection limit and an rms $\\sim$ 11 $\\mu$Jy beam$^{-1}$ at $\\sim$ 6.2 GHz. \\citet{2016MNRAS.460.1039P} observed 49 high-mass YSOs using the ATCA and detected 16 ionized jets and 12 jet candidates  with a 3$\\sigma$ detection limit and an rms $\\sim$ 17 $\\mu$Jy beam$^{-1}$ at $\\sim$ 5.5 GHz. Additionally, the Protostellar Outflow at the EarliesT Stage (POETS) survey is undertaking a search of radio-jets  using the VLA with an angular resolution of $\\sim$0\\rlap.$^{\\prime \\prime}$1 and an image rms of $\\sim$10 $\\mu$Jy beam$^{-1}$ \\citep{2018A&A...619A.107S, 2019A&A...623L...3S}. In our radio continuum Jansky VLA survey we observed 58 high-mass star forming regions and detected 70 radio sources with a \n5$\\sigma$ detection limit and an rms $\\sim$ 5 $\\mu$Jy beam$^{-1}$ at $\\sim$ 6 GHz \\citep{2016ApJS..227...25R}. \\citealt*{2018A&ARv..26....3A} is a recent comprehensive review of ionized jets in star forming regions. \\\\\n\nThe main goal of this  paper is to investigate the nature of the 70 detected radio sources \nreported in \\citetalias{2016ApJS..227...25R}. The observations along with the complete list of targets, coordinates, radio  detections and derived observational parameters are presented in \\citetalias{2016ApJS..227...25R}. In Section \\ref{sec:models} we examine several scenarios to explain the origin of the ionized gas emission and we study the physical properties of the detected sources.  Section \\ref{discussion_paperII}  contains a discussion of the viability of the different scenarios. In Section \\ref{conclusions_paperII}\nwe summarize our findings. Additionally, Appendix \\ref{app:lum} shows the bolometric luminosity estimates for these high-mass star forming regions using {\\it Herschel}\/Hi--GAL data and Appendix \\ref{mom_rate_appe} shows a study of the momentum rate of ionized jets.\n\n\n\n\n\n\n\n\n\n\n\\section[Models  for the Radio Emission]{ Models Considered for the Radio Emission} \\label{sec:models}\n\n\\subsection{Low-mass Young Stellar Objects}\\label{yso}\n\nThe main goal of our high sensitivity continuum survey presented in  \\citetalias{2016ApJS..227...25R} \nwas to detect radio emission from high-mass protostars. However, there exists a variety of sources that could also appear as radio detections in our images.\nIn \\citetalias{2016ApJS..227...25R}  we considered  contamination by extragalactic radio\nsources, and found that only a small number of extragalactic sources are expected to be observed within\nthe typical dust clump size of $\\sim$30$^{\\prime \\prime}$ (8 and 2 sources in the 6 and $1.3\\,$cm bands, respectively for the entire sample). \nA more likely source of contamination would be the presence of low-mass YSOs which are expected\nto be present in regions of high-mass star formation  \\citep[e.g.,][]{2013A&A...554A..48R}. We are thus interested in identifying possible low-mass class 0 -- class III YSOs  that could have \nbeen detected in our survey toward high-mass star forming regions. \n\nA large sample of low-mass YSOs has been  observed with the VLA at 4.5 and 7.5 GHz as part of  the Gould Belt survey (i.e.,  Ophiuchus at a distance of 120 pc:  \n\\citealp{2013ApJ...775...63D}; Orion at 414 pc: \\citealp{2014ApJ...790...49K}; Serpens at 415 pc: \\citealp{2015ApJ...805....9O}; Taurus-Auriga at 140 pc: \\citealp{2015ApJ...801...91D}, and\nPerseus at 235 pc: \\citealp{2016ApJ...818..116P}). \nThe brightest low-mass YSO in the entire Gould Belt survey (excluding the Orion region) was found in the Ophiuchus region (source J162749.85--242540.5,\na class III YSO, i.e., weak-lined T-Tauri star, with S$_{7.5\\,GHz} = 8.51\\,$mJy, \\citealt{2013ApJ...775...63D}).  To determine whether such an object would have been detected in our survey,\nwe scaled its flux density to the assumed distance\\footnote{Distances were taken from the literature, and are listed in Table \\ref{SED_Parameters}. \nMost distances are kinematic; only a few regions have trigonometric parallax measurements.}\nof each of our targets, and compared its scaled flux density to our adopted detection limit of  $\\geq$ 5 times the image rms\nat 7.4 GHz for each of our regions. We found that such a YSO would not be detected in any of our targets located at distances beyond $2\\,$kpc.\nSince the majority\nof our targets exceed this distance (see Figure \\ref{T_Tauri_hist}), we conclude that for most of our observed regions the detected radio sources are not low-mass YSOs.\\\\\n\n\nThere are 10 regions in our survey that are located at distances $\\leq$ 2 kpc. However, given the 7.4 GHz image rms for these regions, only in 7 of them would we have detected the brightest low-mass YSO of Ophiuchus.\nThese regions are  five  HMCs: 18517$+$0437, 20126$+$4104, 20293$+$3952, 20343$+$4129, G34.43$+$00.24mm1, and  two CMC--IRs: LDN1657A$-$3  and\nUYSO1. In these 7 regions we detected a total of 13 radio sources within the FWHM of the mm clumps: 10 towards HMCs and 3 towards CMC--IRs. That some of these \nsources are  possibly low-mass YSOs can be seen in the case of  IRAS 20126$+$4104: Besides the well-studied high-mass protostars associated\nwith radio sources 20126$+$4104 A and 20126$+$4104 B, the radio source G$78.121+3.632$ in this region (see  \\citetalias{2016ApJS..227...25R} , Table 4) corresponds to the source I20var, which was\n discussed by \\citet{2007A&A...465..197H}. This is a highly \nvariable radio source and  has observational properties consistent with a flaring T-Tauri star. In the same region, we have also detected a new object of similar characteristics. Radio source 20126$+$4104 C \nwas detected for the first time in our survey although several high sensitivity observations of this region have been made in the past \\citep[][]{2007A&A...465..197H}.\n\nHence, 20126$+$4104 C is clearly \nvariable in the radio regime, and is a candidate for a low-mass pre-main sequence star.\nAdditionally, the radio source LDN1657A$-$3 A, which has a negative spectral index ($\\alpha=-1.2$), is also a candidate for a variable radio source, where the emission is probably\ncaused by non-thermal processes on the surface of a T-Tauri star.  While the observational properties of these sources are consistent with low mass YSOs, we note that alternative explanations are possible \\citep[e.g.,][]{2018A&A...612A.103C}.\n\nIn summary, while some degree of contamination by low-mass YSOs probably exists in our survey for the nearest sources,\nfor the majority of our targets the detected radio sources are \nvery likely not contaminated by emission from low-mass YSOs. \n\n\n\\begin{figure}[!h]%\n    \\centering\n\n    \\includegraphics[width=0.7\\linewidth, clip=True]{Regions_T_Tauri}%\n     \\caption[The distance distribution of our targets is shown as a black line]{\\small{The distance distribution of our targets is shown as a black line. The color histogram shows the number of targets where we expect the detection of T-Tauri \n     stars which are as bright as J162749.85--242540.5 in Ophiuchus. This is the case for 5 HMCs and 2 CMC-IRs.}\n}%\n    \\label{T_Tauri_hist}%\n\\end{figure}\n\n\\subsection{H{\\small II} Regions}\nIn \\citetalias{2016ApJS..227...25R}  we reported the detection of 70 radio continuum sources associated with three different types of mm clumps and we calculated their 5--25 GHz spectral index ($\\alpha$) using power-law fits of the form $S_{\\nu}$ $\\propto$ $\\nu^{\\alpha}$.  The spectral index values and the fits to the data for all the radio detections are reported in \\citetalias{2016ApJS..227...25R} in Table 4 (electronic version) and in Figure 4, respectively. The range of spectral indices found was $-$1.2 to 1.8 (see Figure 5 in \\citetalias{2016ApJS..227...25R}). Based on their radio spectra, we classify  these sources as flat spectral index ($-$0.25$<\\alpha<$0.2),  positive spectral index ($\\alpha \\geq$ 0.2), and negative spectral index ($\\alpha <$-0.25). Thus, we have 10 sources with flat, 44 sources with positive, and 9 sources with negative spectral index. For the remaining 7, there is not a clear estimate of the spectral index.\n\nThe radio sources have a variety of morphologies. Excluding the sources without spectral index information, there are 6 extended sources, 8 sources with elongated structures,\nand the majority of sources (49) are compact with respect to our synthesized beam. In this section we consider whether a family of H{\\small II} region models could  explain sources with flat and positive spectral index.\n\n\n\\subsubsection{Extended Sources}\\label{ext_sources}\n\nAmong the sources detected in our survey associated with $1.2\\,$mm dust emission, there are six sources that are clearly extended at cm wavelengths with\nrespect to the $\\sim$0\\rlap.$^{\\prime \\prime}$4 resolution of the maps, hence they are candidates for  H{\\small II} regions,\ni.e. photoionized gas.\n These sources are relatively bright (S$_{25.5\\,GHz} \\approx$ 1\\,mJy), and are found mostly toward HMCs.\nMoreover, they generally show a flat spectral index, indicative of optically thin free-free emission. \nFor five of these sources, we calculate the physical properties from the $25.5\\,$GHz continuum flux using the formulae from \\citet{1994ApJS...91..659K},\nwhich assume spherical symmetry, and optically thin emission from a uniform density plasma with T$_{e}= 10^{4}$\\,K. The results are listed in Table \\ref{HII_Parameters}, where column \n1 is the region name, column 2 is the \nspecific radio source, and columns 3 and 4 are the frequency ($\\nu$) and radio flux (S$_{\\nu}$), respectively.  Column 5 is the observed linear size (diameter)\nof the radio source ($\\Delta s$) at 3$\\sigma$ rms level in the image, column 6 is the emission measure (EM), column 7 is the electron density (n$_e$),\ncolumn 8 is the \nexcitation parameter (U) and column 9 is the logarithm of the Lyman continuum  flux (N$^{\\prime}_{Ly}$)  required for ionization.   We use log\\,N$^{\\prime}_{Ly}$ to \nestimate \nthe spectral type of the ionizing star (listed in column 10) using the tabulation in \\citet{1973AJ.....78..929P}, further assuming that a single ZAMS star is photoionizing the nebula and \nproducing the \nLyman continuum flux. The distances used for these calculations are listed in Table \\ref{SED_Parameters}, and the near kinematic distance is adopted  when the region has a distance ambiguity.\n\n\n\n\n\\begin{deluxetable}{l c c c c c c c c c}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Extended Sources:  Parameters  from Radio Continuum  \\label{HII_Parameters}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Region}                  & \n\\colhead{Radio }  &\n\\colhead{$\\nu$}   &\n\\colhead{S$_{\\nu}$ }               & \n\\colhead{$\\Delta s$}  & \n\\colhead{EM\/$10^{5}$ }             & \n\\colhead{n$_{e}\/10^{3}$ } & \n\\colhead{U } & \n\\colhead{log\\,N$^{\\prime}_{Ly}$ } &\n\\colhead{Spectral } \n \\\\[2pt]\n\\colhead{}                            & \n\\colhead{Source}                      & \n\\colhead{(GHz)}                      & \n\\colhead{($\\mu$Jy)}                      & \n\\colhead{(pc)}                      & \n\\colhead{(pc\\,cm$^{-6}$)}                      &  \n\\colhead{(cm$^{-3}$)}                      & \n\\colhead{(pc\\,cm$^{-2}$)}                     & \n\\colhead{(s$^{-2}$)}  &\n\\colhead{Type\\tablenotemark{a}}                         \\\\[-20pt]\\\\}\n\\startdata\n\\input{latex_classic_HII_table_mod.txt}\n\\enddata\n\\tablenotetext{\\text{a}}{Using the tabulation in  \\citet{1973AJ.....78..929P}.}\n\\tablenotetext{\\text{b}}{Includes radio source 20293$+$3952 B (see Figure 2 in \\citetalias{2016ApJS..227...25R}).}\n\\end{deluxetable}\n\nThe measured sizes of the five sources listed in Table \\ref{HII_Parameters} are all below $0.1\\,$pc, which according to  \\citet{2005IAUS..227..111K} would suggest a classification as ultra (UC)- or hypercompact (HC)\nH{\\small II} regions. However, the calculated emission measures and electron densities are an order of magnitude or more  smaller than  typical values for such H{\\small II} regions. The typical values of the emission measure and electron density for UCH{\\small II} regions are EM$\\gtrsim 10^{7}$ pc cm$^{-6}$ and n$_{e}\\gtrsim 10^{4}$ cm$^{-3}$ and for HCH{\\small II} regions are EM$\\gtrsim 10^{10}$ pc cm$^{-6}$ and n$_{e}\\gtrsim 10^{6}$ cm$^{-3}$ \\citep{2005IAUS..227..111K}.\nWe note that four of these resolved extended radio sources (18470--0044 A, 18521$+$0134 B, 19035$+$0641 B  and 20293$+$3952 C) are offset $\\sim$ 2\/3 of the radius from the center of the mm clumps,\nand are thus located at their outskirts. A plausible explanation for the lower electron densities and emission measures is that early B-type stars have\nformed near the edge of the dust clumps where the density of the surrounding medium is much lower than in the center.\n\nWe will comment on two additional resolved and extended radio sources detected in this survey. The first one is  18470--0044 A, which was previously observed by \\citet{2011ApJ...739L..17H} at 25.5 GHz using the VLA in the C-configuration.  This radio source has an offset of 7\\rlap.$^{\\prime \\prime}$2 with respect to the peak of the mm clump associated with IRAS 18470--0044  reported by \\citet{2002ApJ...566..945B}. Our image towards this region is affected by sidelobes due to the large flux and extended emission of a \nnearby radio source, and we were not able to accurately measure the radio flux at 1.3 cm. This source is among the detections without a clear estimate of its spectral index. However, based on the flux \nreported by \\citet{2011ApJ...739L..17H},  and our measured flux at 6 cm (see \\citetalias{2016ApJS..227...25R}), this radio source has a flat spectrum, and it is likely an H{\\small II} region ionized by a \nB2 ZAMS star. \n\nThe second bright, and resolved, radio source is G34.43$+$00.24mm2 A, which is the only extended source detected towards an IRDC clump in our survey. Interestingly, this radio source \nhas a spectral index of $\\alpha= -0.5$ (see \\citetalias{2016ApJS..227...25R}) and is associated with a 24 $\\mu$m point source, and  at least two molecular outflows \\citep{2007ApJ...669..464S}. This radio \nsource was originally detected at 6 cm by \\citet[labeled by them as Mol 74]{1998A&A...336..339M}  and \\citet{2004ApJ...602..850S} using the VLA (FWHM $\\sim$6\\rlap.$^{\\prime \\prime}$$\\times\n$3\\rlap.$^{\\prime \\prime}$). The  flux densities at 6 cm reported in those studies are consistent with our data.  This radio source has an offset of 7\\rlap.$^{\\prime \\prime}$8 with respect to the \ncenter of the $1.3\\,$mm clump G34.43$+$00.24mm2  detected by \\citet{2006ApJ...641..389R}.\\\\\n\nWe found that at least 36$\\%$ of the HMCs have an extended radio source within the 1\\rlap.$^{\\prime}$8 FWHM primary beam at 25.5 GHz. However, most of them are located slightly outside the FWHM of the mm clump, and are not discussed in this study. These extended radio sources are: 18089$-$1732   G12.890$+$0.495, 18182$-$1433   G16.584$-$0.053, 18470$-$0044  G32.113$+$0.097, 18521$+$0134  G34.749$+$0.021, 19012$+$0536  \nG39.389$-$0.143 and  19266$+$1745 G53.037$+$0.115.    All these sources and their radio continuum parameters are reported in \\citetalias{2016ApJS..227...25R}. \n\n\n\\subsubsection{Compact Sources}\\label{compact_sources_sect}\n\nIn \\citetalias{2016ApJS..227...25R} we characterized the change in flux density with frequency using a power law. \nWe reported the detection of  36 compact radio sources  (51$\\%$ of total detections) with a rising spectrum ($\\alpha >$0.2), of which \n7 were not detected at 6 cm, thus a lower limit for their spectral index was estimated. We also detected 5 compact radio sources with a flat cm spectrum.\nIn this section we investigate whether a uniform density UC\/HC H{\\small II} region can explain the observed fluxes and spectral indices for compact sources with rising spectra.\n\nWhile optically thin free-free emission results in a flat spectral index ($\\alpha = -0.1$), a rising spectrum implies appreciable optical depth in the emitting gas.\nTo fit our spectra, we thus will require a turnover, (i.e. $\\tau_{\\nu} \\sim 1$) near the intermediate frequency of our observing bands, around $\\nu_{t}=$14.7 GHz,\nwhich in turn requires an emission measure near 9$\\times$10$^{8}$ pc\\,cm$^{-6}$. For fitting the data we use a uniform density, spherical H{\\small II} region model\nwith electron temperature of $T_{e}=$ 10$^{4}$\\,K\nas shown in equation (11) from \\citet{1975A&A....39..217O}. We find that \nfor all compact, rising spectrum sources detected in our survey, within the given uncertainties, a uniform density H{\\small II} region spectrum can be reasonably fit to the radio continuum data.\nExamples of the fits are shown by the continuous blue line in Figure \\ref{HII_fit_examples}. \n The fits for all the 36 compact radio sources with rising spectral index is shown in Appendix \\ref{all_HII_fit} Figure \\ref{HII_fit_app}.\n\n\n\n\\begin{figure}[!h]%\n    \\centering\n  \\hspace*{-0.5cm} \n    \\includegraphics[width=0.35\\linewidth, clip]{g23_01_alpha_HII_A}%\n        \\includegraphics[width=0.35\\linewidth, clip]{18440_alpha_HII_A}%\n        \\includegraphics[width=0.35\\linewidth, clip]{20343_alpha_HII_B}%\n     \\caption[Spectra of the compact radio sources G23.01--0.41 A  (left), 18440--0148 A (center) and 20343$+$4129 B  (right)]{\\small{Spectra of the compact radio sources G23.01--0.41 A  (left), 18440--0148 A (center) and 20343$+$4129 B  (right). Error bars are an assumed uncertainty of 10$\\%$ from the flux densities added in quadrature with an assumed 10$\\%$ error in calibration. The  continuous blue line is the H{\\small II} region fit using a spherical, constant density model. The red arrow indicates the frequency where $\\tau_{\\nu}=$1 in each model fit. The dashed line is the best fit to the data from a power-law of the form $S_{\\nu}$ $\\propto$ $\\nu^{\\alpha}$.}}\\label{HII_fit_examples}%\n\\end{figure}\n\n\nOur fitting results show that the generally quite weak emission from these very compact, rising spectrum sources implies a very small size\nfor the emitting regions. The sizes are much smaller than our angular resolution, and are on the order of the initial Str\\\"{o}mgren sphere radius.\nThe initial Str\\\"{o}mgren sphere radius ($R_{s}$) depends on the Lyman continuum (N$_{Ly}$) flux and the ambient molecular density (n$_{H_{2}}$) as\nstated in equation 1 from \\citet{1996ApJ...473L.131X}:\n\n\\begin{equation}\nR_{s}= 4104.7 \\left( \\frac{N_{Ly}}{10^{49}s^{-1}}\\right)^{1\/3}\\left(\\frac{n_{H_{2}}}{10^{5}cm^{-3}}\\right)^{-2\/3}  \\text{[au]}  .\n\\end{equation}\n\nWe show the relation R$_{s}$ versus N$_{Ly}$, represented by the solid lines, for n$_{H_{2}}=$ 10$^{5}$, 10$^{6}$,  10$^{7}$ and 10$^{8}$ cm$^{-3}$ in Figure \n\\ref{Stromgren_sphere}. To place our sources in this diagram, we estimated the Lyman continuum flux (N$_{Ly}$) from our $25.5\\,$GHz flux density, with\nthe formulae of \\citet{1994ApJS...91..659K}, and use the radii ($\\Delta\\,s\/2$) derived from the spectral fitting.  These data are listed in Table~\\ref{tab:fig3}.\nOur sources, represented as solid purple dots\nin Figure \\ref{Stromgren_sphere}, cluster around the expected Str\\\"{o}mgren radius for an initial density of $10^{6}$\\,cm$^{-3}$. \n\n\\begin{figure}[!h]%\n    \\vspace{1.2cm}\n    \\centering\n    \\includegraphics[width=0.5\\linewidth, clip]{R_strongrem}%\n     \\caption{\\small{Initial Str\\\"{o}mgren sphere  radius as a function of the Lyman continuum for compact sources with rising spectra. The solid lines represent the  ambient molecular density for n$_{H_{2}}=$ 10$^{5}$ --  10$^{8}$ cm$^{-3}$. The solid purple dots represent the radius of the\n     H{\\small II} regions ($\\Delta\\,s\/2$) as implied by the spherical, constant density H{\\small II} region model fits.  The dashed green line represents the lower limit of  $R_{Turb}$ (for $\\xi =$1) and the  dashed blue line presents $R_{Th}$ if the initial ambient molecular density for both cases is n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$. Shaded areas  represent the path from their initial  radius up to their final Str\\\"{o}mgren sphere radius if the sources were born at a density of n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$ (green and blue shaded areas for turbulence and thermal pressure confinement, respectively).\nThe error bars in the bottom right corner correspond to a 20$\\%$ calibration uncertainty. \n}}%\n    \\label{Stromgren_sphere}%\n\\end{figure}\n\n\n\nWhen nuclear burning begins and a high-mass star produces enough UV photons to photo-ionize the surrounding material, the initial Str\\\"{o}mgren sphere radius is reached (to within a few percent) in a recombination timescale, $t_{r}= $(n$_{H_{2}}$$\\beta_{2}$)$^{-1}$[s], where $\\beta_{2}$= 2.6$\\times$10$^{-13}$\\,cm$^{3}$s$^{-1}$ is the recombination coefficient \\citep{1980pim..book.....D}. For any reasonable initial density this time scale is extremely short ($< 1\\,$yr), and the initial\nStr\\\"{o}mgren sphere radius is reached almost instantaneously. The highly over-pressured ionized region will then begin to expand, and hence the initial\nStr\\\"{o}mgren sphere is a very short-lived configuration, and therefore it is unlikely that the large number of sources detected represent this evolutionary stage.\n\n\n\n\nAfter  formation of the initial Str\\\"{o}mgren sphere around a star, the UC H{\\small II} region is highly overpressured,  and as a result, it expands approximately at the sound speed until approaching pressure equilibrium with the ambient medium. \n\\citet{1995RMxAA..31...39D} and \\citet{1996ApJ...473L.131X} have studied  the confinement of UC H{\\small II} regions in a molecular core by thermal, and thermal plus turbulent pressure, respectively. \nAssuming pressure equilibrium between ionized and surrounding molecular gas, \\citet{1996ApJ...473L.131X}  gives the final radius of the ionized region (R) as :\n\n\n\\begin{equation}\nR = R_{s} \\left(   \\frac{2k\\xi T_{H^{+}}}{m_{H_{2}}\\sigma_{v}^{2} + kT_{k}}\\right)^{2\/3}  ,\n\\end{equation}\n\n\\noindent\nwhere $T_{H^{+}}$ is the temperature of the ionized region, $\\xi$ is a turbulence factor ($>$ 1) that takes into account the pressure due to stellar winds and turbulence in the ionized gas, $\\sigma_{v}$ is the velocity dispersion produced by turbulence and $T_{k}$ is the kinetic temperature of the surrounding molecular gas. \nUsing typical values for the physical conditions in regions where high-mass stars form, we can  test whether the sources discussed in this section could be ionized regions in pressure equilibrium with the surrounding molecular gas.\nWhile molecular line observations with single dish instruments indicate average densities of n$_{H_{2}}=$ 10$^{5}$\\,cm$^{-3}$ over the $1\\,$pc clump sizes \\citep[e.g.,][]{2000ApJ...536..393H}, interferometric measurements of high-mass star forming cores\nhave revealed central densities of n$_{H_{2}}= 10^7 - 10^{10} $\\,cm$^{-3}$ on scales $< 0.1\\,$pc \\citep[e.g.,][]{1990ApJ...362..191G, 2015A&A...573A.108G}. Following \\citet{1996ApJ...473L.131X}, we adopt values of $T_{H^{+}}=$ 10$^{4}$\\,K, $\\sigma_{v}=$ 2 km\\,s\n$^{-1}$ (FWHM$\\,\\sim$\\,5 km\\,s$^{-1}$) and $T_{k}=$ 100 K. Assuming $\\xi = 1$, and evaluating the above equation with these numbers we get $R_{turb} = 11.2 R_{s} $ for the case of thermal plus turbulent pressure, and\n$R_{th} = 54.3 R_{s}$ for thermal pressure only. These relations are shown in Figure \\ref{Stromgren_sphere} for n$_{H_{2}}= 10^7$\\,cm$^{-3}$ as green and blue dashed lines, respectively. \nConsidering  the location of our data points in Figure \\ref{Stromgren_sphere}, we can exclude the extremely high densities of $10^{10} $\\,cm$^{-3}$ as found by \\citet{2015A&A...573A.108G}, which would predict\nmuch smaller source sizes. On the other hand, our data points are located within the  shaded areas that represent the path from their initial Str\\\"omgren radius up to their final radius in pressure equilibrium,\nif the sources were born at a density of n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$ (green and blue shaded areas for turbulence and thermal pressure confinement, respectively).\n\n\nAn estimate of the expansion time $\\tau_{expansion}$ for an ionized region can be obtained assuming that it  expands at its sound speed ($C_{s} \\sim$ 10 km\\,s$^{-1}$). To expand to $\\sim$\\,200 au,\nthen $\\tau_{expansion} \\sim R\/C_{s} \\sim $ 100 yr. Thus, an initial Str\\\"omgren sphere will expand fairly quickly and, as suggested by \\citet{1995RMxAA..31...39D} and \\citet{1996ApJ...473L.131X}, \nthe ionized regions can remain compact as long as the molecular core provides the outside pressure. Observations of  UCH{\\small II} regions and HMCs suggest that this time is on order $10^5\\,$yr\n\\citep{1989ApJ...340..265W,2001ApJ...550L..81W}, and it hence appears that  our sources could be ionized regions around newly formed stars in pressure equilibrium in their\nmolecular cores. While our calculations have not been fitted to a particular source, the observed scatter in Figure \\ref{Stromgren_sphere} can be accounted for with a varying amount of turbulence in the molecular gas,\ni.e. if the molecular line FWHM varies between $\\sim$ 7 -- 20 km\\,s$^{-1}$ for the case of an  ambient molecular density of n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$.\n\n\nIt is  interesting to note that we found that the radius of the extended sources discussed above are within the pressure equilibrium zone for an initial density of n$_{H_{2}}=$ 10$^{5}$\\,cm$^{-3}$. \nThese sources are located on the outskirts of the mm core, and one might ask whether they have migrated out of the molecular core center, or if they were born in their current location. Assuming stellar velocities between 2 and 12 km\\,s$^{-1}$ \n\\citep{2007ApJ...660.1296F} in order for them to travel to the half power point of the cores (FWHM median angular size for HMC $=$18\\rlap.$^{\\prime \\prime}$ at a distance of 4 kpc), times between\naround 10$^{5}$ yr and 10$^{4}$ yr, respectively, are needed. While migration toward lower density regions is thus possible, we note that we do not find any strong evidence for cometary regions which would be predicted due to\nbow shocks between molecular and ionized gas \\citep{1990ApJ...353..570V}.\n\n\\subsubsection{Lyman Continuum}\n\nAn additional point to consider to understand the nature of our detections is the Lyman continuum photon rate  as a function of the bolometric luminosity. We analyze this relation for all the sources \nwith a flat or a rising spectrum (including extended, elongated structure, as well as compact morphology) as shown in Figure \\ref{Lyman_cont_plot}. The Lyman continuum photon rate is estimated from the radio \ncontinuum flux at 25.5 GHz and the bolometric luminosities for our regions are estimated  from  {\\it Herschel}\/Hi--GAL fluxes, and from ancillary data (see \\S \\ref{app:lum}).  We list these data in Table~\\ref{tab:fig4}.\nFor data taken from the literature, care was taken that the  Lyman continuum flux and bolometric luminosities refer to the same distance. For sources with distance ambiguity, we use the near kinematic distance.\nIn Figure \\ref{Lyman_cont_plot}, compact and elongated sources are represented by filled circles if the bolometric luminosities are estimated in this work  (see \\S \\ref{app:lum}), or open circles if the luminosity\nis taken from the literature. The extended sources from \\S \\ref{ext_sources} are represented by the  $\\color{blue} \\times $  symbol. \nThe continuous black line is the expected Lyman continuum photon rate from a single zero-age main-sequence (ZAMS) star at a given luminosity,  and the shaded area\nbounded by the solid black line shows the expected Lyman continuum from a stellar cluster of the same N$_{Ly}$. For more details on these curves see \\citet{2013A&A...550A..21S}.\nThus, H{\\small II} regions ionized by stellar UV photons from a single early-type star are expected to lie on the black line. If,  on the other hand, the Lyman continuum comes from a cluster of stars (a likely scenario for high-mass \nstars) rather than from a single ZAMS star, the expected  N$_{Ly}$ is lower, and should be located within the shaded area \\citep{2015A&A...579A..71C}.\n\n\n\n As seen in Figure \\ref{Lyman_cont_plot}, only a small fraction of our sources fall in the shaded area of the plot  indicating direct stellar photoionization. Most of the HMC sources (red open\/filled circles) lie\n below the curve of the expected Lyman continuum flux, and hence are underluminous at radio wavelengths.\n On the other hand, the majority of the sources detected towards CMCs and CMC--IRs are located in the so-called ``forbidden area'' above the Lyman continuum line, showing  an excess of Lyman continuum compared \n to the expected value based \non their luminosities. This is true even if the sources are corrected by the distance, i.e., when there is ambiguity in the kinematic distance of the source or the value of the distance is incorrect. For reference, the arrow in the plot indicates the amount \nthat a point will move if the distance increases by a factor of 2. If the distance changes by any other factor the point will move parallel to the arrow. Additionally, there is  a possibility that some  bolometric luminosities  are underestimated (see \\S \\ref{app:lum}), however if this is the case we believe that the luminosities will shift to the right by less than 0.5 dex.\n\n\\begin{figure}[!h]\n    \n    \\centering\n    \\includegraphics[width=0.5\\linewidth, clip]{Lyman_plot_v3}\n     \\caption{\\small{Lyman continuum measured at 25.5 GHz as a \n     function of the bolometric luminosity for all detected sources with flat or rising spectra in our sample. The bolometric luminosity is mainly estimated from  {\\it Herschel}\/Hi--GAL data (except for the open circles for\n     whose the bolometric luminosity  is from the literature). The circles  represent the compact sources with flat or rising spectra, while the blue  $\\color{blue} \\times $ symbol represents the flat spectrum extended \n     sources from \\S \\ref{ext_sources}. UC H{\\small II} regions from \\citet{1994ApJS...91..659K} are represented by the gray $\\color{gray} \\times $ symbol. The continuous black line is the expected Lyman continuum photon rate of a \n     single  ZAMS star at a given luminosity, and the shaded area gives these quantities for the case of a cluster \\citep{2013A&A...550A..21S}. The arrow indicates how much a point would move if the distance were increased by \n     a factor of 2. \nThe error bars in the bottom right corner correspond to a 20$\\%$ calibration uncertainty. \n}}\n    \\label{Lyman_cont_plot}\n\n\\end{figure}    \n    \n\n\n\n\\citet{2013A&A...550A..21S} have reported Lyman continuum excess for several sources in an 18 and 22.8 GHz survey of high-mass star forming regions with the Australian Telescope Compact Array (ATCA).\nInterestingly, $\\sim$70$\\%$ of their H{\\small II} regions with Lyman excess are associated with molecular clumps belonging to two types of sources that are in the earliest evolutionary stages of high-mass stars based on their classification (equivalent to our CMCs and CMC-IRs clumps).  \nAdditionally, \\citet{2015A&A...579A..71C}  found  Lyman  continuum excess for about 1\/3 of their  sample of 200 compact and UC H{\\small II} regions selected from the CORNISH survey \\citep{2013ApJS..205....1P}.  Their sources with Lyman continuum excess are also in an earlier evolutionary phase within their sample. Both studies argued that the Lyman excess is not easily justified, leaving room for two possible scenarios, invoking additional\nsources of  UV photons from an  ionized jet, or from an accretion shock in the protostar\/disk system. \\citet{2016A&A...588L...5C} suggested that the Lyman excess is  produced by  accretion shocks, based on outflow (SiO) and infall (HCO$^{+}$) tracer observations towards the 200 H{\\small II} regions studied in \\citet{2015A&A...579A..71C}. \n\nIt is important to mention that due to our selection criteria (see \\citetalias{2016ApJS..227...25R}) the sources studied by \\citet{2013A&A...550A..21S} and \\citet{2015A&A...579A..71C} are much brighter at radio wavelengths than the ones from our work, with radio luminosities at 5 GHz of $\\sim$ 10$^{2}$--10$^{6}$ mJy kpc$^{2}$ vs 10$^{-2}$--10 mJy kpc$^{2}$ in our sample. In Figure \\ref{Lyman_cont_plot} we also show several UC H{\\small II} regions from  \\citet{1994ApJS...91..659K}, denoted by the $\\color{gray} \\times $ symbol. These sources seem to be produced by higher free-free emission compared with our sample, suggesting that our sources represent a different population of radio sources. \\citet{1999RMxAA..35...97C} based on selection criteria similar to ours,  detected sources with low radio luminosities like the ones in this work. Furthermore, these low radio luminosities are typical of thermal jets, with UV photons that are produced by shocks from collimated winds from the protostar  with the surrounding material \\citep[e.g.,][]{1996ASPC...93....3A}. Thus, while the above analysis of the cm SEDs suggests a model of pressure confined H{\\small II} regions for our compact sources, the Lyman continuum photon rate  as a function of the bolometric luminosity\nshown in Figure \\ref{Lyman_cont_plot} does not lend strong support to this model. A further possible explanation for the compact sources with rising spectra, as well as for several elongated sources\ndetected in our survey is that they arise from thermal jets. We explore this scenario in the following section.   \n\n\\subsection{Ionized Jets}\\label{ionized_jet_sect}\n\nBased on the low radio luminosities (S$_{5\\,GHz}\\,$d$^{2}$ $\\sim$ 10$^{-2}$ -- 10 mJy kpc$^{2}$) of our detected sources, we need to consider the possibility that the source of ionization is not a ZAMS star, but rather \nthat their nature is that of a thermal, ionized jet produced by shock ionization as described in \\S \\ref{sec:intro}.\nSupport for this hypothesis comes from a subset of resolved sources from  our survey. We have characterized 12 jet candidates based on their elongated, or string-like morphology in conjunction with an\nassociation with a molecular outflow. These sources are listed in Table \\ref{jet_cand_list}, where column 1 is the name of the region, column 2 are the radio sources that are thought to be part of the ionized jet, and \ncolumn 3 lists the approximate direction of the ionized jet. Column 4  shows the approximate direction of the molecular outflows associated with the centimeter continuum emission as found in the literature. Column 5 \nindicates if the centimeter continuum emission is a new detection or if it has been  detected in previous studies.  Column 6 lists the references for the molecular outflow detections and previous centimeter continuum \ndetections if any.  Examples for these sources are shown in Figures \\ref{fig:spitzer_examples}a, \\ref{fig:spitzer_examples}b, and \\ref{fig:UKIDSS_examples}a, \\ref{fig:UKIDSS_examples}b. To our knowledge, 6 of these ionized jet candidates are new detections. In the cases of previous detections of centimeter continuum emission towards the listed regions, our high-sensitivity observations described in Paper I\ngenerally show the elongation or string-like morphology of a jet for the first time \\citep[e.g.,][]{2017ApJ...843...99H}. Furthermore, this subset of resolved jet candidates have the expected spectral index ( $0.2 \\leq \\alpha \\leq 1.2$) for ionized jets,  several of them are associated with  6.7 GHz CH$_{3}$OH masers and H$_{2}$O \nmasers, and they have excess emission at 4.5 $\\mu$m,  which may trace shocked gas via H$_{2}$ emission in outflows or  scattered continuum from an outflow cavity \\citep[e.g.,][]{2011ApJ...729..124C, \n2013ApJS..208...23L}. In some cases, like towards the ionized jet in 18182$-$1433, some of the radio sources  have  negative spectral indices, consistent with non-thermal lobes, since it is thought that when very strong shock waves from a fast jet move through a magnetized medium, some of the electrons are accelerated to relativistic velocities producing synchrotron emission \\citep{2003ApJ...587..739G, 2010Sci...330.1209C}. \\citet{2016MNRAS.460.1039P} and \\citep{2018A&A...619A.107S, 2019A&A...623L...3S} also reported the detection of  ionized jets with non-thermal lobes (see also review by \\citealt{2018A&ARv..26....3A}). \n\n\n\n\n\\begin{deluxetable}{l c c c c c c}\n\\tabletypesize{\\scriptsize}\n \\renewcommand*{\\arraystretch}{1.5}\n\\tablecaption{Ionized Jets  \\label{jet_cand_list}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Region}                  & \n\\colhead{Radio Source}        &\n\\colhead{Jet Direction}          &\n\\colhead{Outflow Direction}   &\n\\colhead{ H$_{2}-$Jet Direction}   &\n\\colhead{New Detection}             &\n \\colhead{Reference}      \\\\[-20pt]\\\\}\n\\startdata\n\\setcounter{iso}{0}\t\nG11.11$-$0.12P1     &    A, C, D  &   NE$-$SW   &   E$-$W, NE$-$SW\\tablenotemark{a}   &  E$-$W   &    y   & \\rxn \\label{rxn:Wang2014}  \\rxn \\label{rxn:Rosero2014} \\rxn \\label{rxn:Lee2013}  \\\\\n18089$-$1732         &    A           &   N$-$S        &    N$-$S                 &   no\/very weak\\tablenotemark{b}     & n          & \\rxn \\label{rxn:Beuther2004} \\rxn \\label{rxn:Beuther2010} \\rxn \\label{rxn:Zapata2006}      \\\\\n18151$-$1208         &    B          &   NE$-$SW   &   NW$-$SE\\tablenotemark{c}    &  NW-SE  &    n     &  \\rxn \\label{rxn:Fallscheer2011} \\rxn \\label{rxn:Hofner2011}  \\rxn \\label{rxn:Davis2004}   \\rxn \\label{rxn:Varricat2010} \\\\\n18182$-$1433         &    A$-$C\\tablenotemark{d}   &   E$-$W        &   NE$-$SW, NW$-$SE     &   E$-$W   &    n       &  \\rxn \\label{rxn:Beuther2006}   \\rxn \\label{rxn:Moscadelli2013}  \\rxn \\label{rxn:Lee2012}        \\\\\nIRDC18223$-$3      &    A$-$B\\tablenotemark{e}       &    NE$-$SW  &  NW$-$SE\\tablenotemark{f}      &  SE-NW  &  y     &  \\rxn \\label{rxn:Fallscheer2009}  \\rxn \\label{rxn:Beuther2005}   \\\\   \n G23.01$-$0.41      &    A    &    NE$-$SW         &  NE$-$SW  &  non-detection   & n & \\rxn \\label{rxn:Sanna2016} \\rxn \\label{rxn:Araya2008} \\rxn \\label{rxn:Sanna2018} \\osref{rxn:Lee2013}\\\\\n18440$-$0148         &    A          &    NW-SE       &  \\nodata\\tablenotemark{g}        &  non-detection  &    y    &     \\rxn \\label{rxn:Navarete2015}      \\\\\n18566$+$0408        &    A$-$D\\tablenotemark{h}  &    E$-$W        &   NW$-$SE   &  non-detection  &    n    &    \\rxn \\label{rxn:Zhang2007}\\rxn \\label{rxn:Araya2007}  \\rxn \\label{rxn:Hofner2017} \\osref{rxn:Lee2013}                                   \\\\\n19035$+$0641        &    A          &    NE$-$SW   &   NW$-$SE  &  no\/very weak\\tablenotemark{b}    &  y   &    \\rxn \\label{rxn:Lopez_Sep2010}       \\\\\n19411$+$2306        &    A          &    NE$-$SW   &   NE$-$SW  &   detection\\tablenotemark{b}    &  y   &    \\rxn \\label{rxn:Beuther2002bb}            \\\\\n20126$+$4104        &   A$-$B   &    NW$-$SE   &   NW$-$SE, S$-$N   &   NW$-$SE  &   n  &  \\rxn \\label{rxn:Su2007}  \\rxn \\label{rxn:Shepherd2000} \\rxn \\label{rxn:Hofner2007} \\rxn \\label{rxn:Cesaroni1999}  \\rxn \\label{rxn:Cesaroni2013}  \\\\\n20216$+$4107        &    A          &    NE$-$SW   &  NE$-$SW   &   NE$-$SW   &    y   &   \\osref{rxn:Lopez_Sep2010}  \\osref{rxn:Navarete2015}     \\\\\n\\enddata\n\\tablenotetext{\\text{a}}{ALMA unpublished data (Rosero et al. in prep).}\n\\tablenotetext{\\text{b}}{ T. Stanke and H. Beuther (private communication).}\n\\tablenotetext{\\text{c}}{A blue-shifted component of a molecular outflow going in the direction of 18151$-$1208 B is seen in Figure 4 of \\citet{2011ApJ...729...66F} but it is not discussed by the authors.}\n\\tablenotetext{\\text{d}}{Radio source B has a negative spectral index and radio source A has an upper limit value in its spectral index. Their fluxes are not included in \n Figures \\ref{fig:rad_bol_lum} and \\ref{fig:Tanaka_tracks} (right panel). \n}\n\\tablenotetext{\\text{e}}{Radio source B has an upper limit value for the flux at 4.9 GHz and its value is not included in \n Figures \\ref{fig:rad_bol_lum} and \\ref{fig:Tanaka_tracks} (right panel). \n}\n\\tablenotetext{\\text{f}}{A blue-shifted component of a molecular outflow going in the direction of IRDC18223$-$3  is seen in Figure 5 of \\citet{2011ApJ...729...66F} but it is not discussed by the authors.}\n\n\\tablenotetext{\\text{g}}{\\citet{2002ApJ...566..931S} report the presence of CO (2--1) wings towards this region, but contour maps of the molecular outflow are not available.}\n\\tablenotetext{\\text{h}}{Radio sources C and D have  upper limit spectral indices that are consistent with being negative and their fluxes were not included in \n Figures \\ref{fig:rad_bol_lum} and \\ref{fig:Tanaka_tracks} (right panel). \n}\n\\tablecomments{Generally there are multiple molecular outflows in each of these high-mass star forming region. We reference the ones that are located closest to the centimeter continuum emission. The  `y' and `n'  indicates if the centimeter radio continuum detection  is new or if it has been detected in a previous study, respectively.\\\\\n \\osref{rxn:Wang2014}  \\citet{2014MNRAS.439.3275W};  \\osref{rxn:Rosero2014}  \\citet{2014ApJ...796..130R}; \\osref{rxn:Lee2013} \\citet{2013ApJS..208...23L}; \\osref{rxn:Beuther2004} \\citet{2004ApJ...616L..23B}; \\osref{rxn:Beuther2010} \\citet{2010ApJ...724L.113B};   \\osref{rxn:Zapata2006} \\citet{2006AJ....131..939Z}; \\osref{rxn:Fallscheer2011}  \\citet{2011ApJ...729...66F}; \\osref{rxn:Hofner2011}  \\citet{2011ApJ...739L..17H}; \\osref{rxn:Davis2004}  \\citet{2004A\\string&A...425..981D}; \\osref{rxn:Varricat2010}  \\citet{2010MNRAS.404..661V}; \\osref{rxn:Beuther2006}  \\citet{2006A\\string&A...454..221B};  \\osref{rxn:Moscadelli2013}  \\citet{2013A\\string&A...558A.145M};  \\osref{rxn:Lee2012}  \\citet{2012ApJS..200....2L}; \\osref{rxn:Fallscheer2009} \\citet{2009A\\string&A...504..127F}; \\osref{rxn:Beuther2005} \\citet{2005ApJ...634L.185B}; \\osref{rxn:Sanna2016} \\citet{2016A\\string&A...596L...2S}; \\osref{rxn:Araya2008} \\citet{2008ApJS..178..330A}; ; \\osref{rxn:Sanna2018} \\citet{2019A\\string&A...623A..77S}; \\osref{rxn:Navarete2015}  \\citet{2015MNRAS.450.4364N}; \\osref{rxn:Zhang2007}  \\citet{2007A\\string&A...470..269Z}; \\osref{rxn:Araya2007}  \\citet{2007ApJ...669.1050A}; \\osref{rxn:Hofner2017} \\citet{2017ApJ...843...99H}; \\osref{rxn:Lopez_Sep2010}  \\citet{2010A\\string&A...517A..66L}; \\osref{rxn:Beuther2002bb}  \\citet{2002A\\string&A...383..892B}; \\osref{rxn:Su2007} \\citet{2007ApJ...671..571S}; \\osref{rxn:Shepherd2000}  \\citet{2000ApJ...535..833S}; \\osref{rxn:Hofner2007}  \\citet{2007A\\string&A...465..197H}; \\osref{rxn:Cesaroni1999} \\citet{1999A\\string&A...345..949C};  \\osref{rxn:Cesaroni2013} \\citet{2013A\\string&A...549A.146C}\n }\n\\end{deluxetable}\n\n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{tabular}{c}\n\n\\includegraphics[width=0.7\\textwidth,clip=true, trim = 10 240 10 80, clip, angle = 0]{18089}  \\\\\n\\includegraphics[width=0.68\\textwidth,clip=true, trim = 10 240 10 60, clip, angle = 0]{19411}   \n\\end{tabular}\n\\caption{\\small{\\emph{Spitzer} IRAC GLIMPSE three-color (3.6$\\mu m$-blue, 4.5$\\mu m$-green and 8.0$\\mu m$-red )\nimages of two ionized jet candidates, overlayed with VLA 6 cm continuum emission contours. \nNote that both regions show 4.5 $\\mu$m excess emission. In the right panel we show\nan enlarged version of the radio continuum from \\citetalias{2016ApJS..227...25R}.\n{\\bf Top: 18089-1732 A:}  The arrows represent the direction of a the north-south bipolar SiO outflow detected by  \\citet{2004ApJ...616L..23B, 2010ApJ...724L.113B}. The black circle and the square are the 6.7 GHz CH$_{3}$OH and H$_{2}$O masers reported in \\citep{2002A&A...390..289B}, respectively.\nVLA 6 cm contour levels are ($-$2.0, 3.0, 10.0, 25.0, 40.0) $\\times $6 $\\mu$Jy beam$^{-1}$, and 1.3 cm contour levels ($-$1.5, 3.0, 5.0, 7.5, 15.0, 25.0, 45.0, 95.0) $\\times$  10 $\\mu$Jy beam$^{-1}$.\n{\\bf Bottom: 19411+2306 A:} The arrows represent the direction of the detected CO  outflow by \\citet{2002A&A...383..892B}. \\citet{2002ApJ...566..931S} reported that 6.7 GHz CH$_{3}$OH and H$_{2}$O\n masers were not detected for this source in their survey.  VLA 6 cm contour levels are ($-$2.0, 2.0, 3.0, 6.0, 10.0, 13.0, 15.0) $\\times$ 5.5 $\\mu$Jy beam$^{-1}$, and 1.3 cm contour levels ($-$2.0, 2.0, 3.0, 6.0, 8.0, 10.0, 12.0) $\\times$  8 $\\mu$Jy beam$^{-1}$.  }}\n \\label{fig:spitzer_examples}\n\\end{figure}\n\n\nAs listed in Table~\\ref{jet_cand_list}, at least 5 of the ionized jet candidates are aligned in the same direction as a large scale molecular outflow (see Figures \\ref{fig:spitzer_examples} for examples). The other \nionized jet candidates appear to be associated with molecular outflows where the directions are approximately perpendicular. In Figure \\ref{fig:UKIDSS_examples} we present the examples of 18151$-$1208 B and 19035+0641 \nA where we show VLA~6$\\,$cm continuum emission contours overlayed on a UKIDSS\\footnote{United Kingdom Infrared Telescope (UKIRT) Infrared Deep Sky Survey (UKIDSS) Galactic Plane Survey \\citep{2007MNRAS.379.1599L}.} {\\emph K}-band (2.2 $\\mu m$) image. It is interesting to note that the putative ionized jets and the \nUKIDSS {\\emph K}-band emission in both cases are elongated in the same direction. This together with the fact that the ionized jets are located nearly at the peak of the UKIDSS {\\emph K}-band emission could indicate \nthat the latter is tracing scattered light from the central protostar that is escaping from an outflow cavity \\citep{2013ApJS..208...23L}. The observed misalignment between cm continuum emission and  the dominating molecular flow in the region could be explained by the existence of two flows, where the molecular outflow associated\nwith the jet is weaker, and hence undetected. This could in fact be the case for 18151$-\n$1208 B, where a blue-shifted component of a CO molecular outflow observed with the Submillimeter Array appears to be aligned in the direction of the ionized jet\n(see Figure 4 of \\citealt{2011ApJ...729...66F}), although this outflow component is not discussed by the authors. Another possible explanation for the misalignment in the directions of the ionized jet and the molecular outflow is that they are subjected to precession, where \nthe flow axis changes from the small to the large scale as suggested by e.g., \\citet{2000ApJ...535..833S} and \\citet{2005A&A...434.1039C} for the case of 20126$+$4104,  \\citet{2013A&A...558A.145M} to explain the \ncase of  18182$-$1433 and \\citet{2007ApJ...669.1050A} for 18566$+$0408. \n\nFor a further test of their jet nature, we have also attempted to estimate the deconvolved sizes of the central jet components using the CASA task  {\\tt imfit}. This was possible for 3 sources within the subsample of jet candidates listed in\nTable~\\ref{jet_cand_list}. Figure \\ref{fig:size_freq} shows the deconvolved major axis as a function of frequency for 18151$-$1208 B and 18440$-$0148 A (the case of 18566$+$0408 B is reported in \\citealt{2017ApJ...843...99H}). Within the uncertainties these radio sources follow the relation $\\theta_{maj} \\propto \\nu^{\\gamma}$, where a major axis index of $\\gamma = -0.7$ is expected for a biconical ionized wind or jet \\citep{1986ApJ...304..713R}. Therefore, at least in these 3 cases, we have further evidence for the jet nature of these specific radio sources. \n\nIn summary, for the subsample of elongated continuum sources listed in Table~\\ref{jet_cand_list} it is very likely that the nature of these sources are ionized jets at the base of a molecular outflow.\n\n\n\n\n\n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{tabular}{c}\n\\includegraphics[width=0.7\\textwidth,clip=true, trim = 10 200 10 80, clip, angle = 0]{18151} \\\\\n\\includegraphics[width=0.68\\textwidth, clip=true, trim = 10 150 10 80, clip, angle = 0]{19035}  \n\\end{tabular}\n\\caption{\\small{UKIDSS {\\emph K}-band images of two ionized jet candidates, overlayed with VLA 6 cm continuum emission contours. In the right panel we show\nan enlarged version of the radio continuum from \\citetalias{2016ApJS..227...25R}. \n{\\bf Top: 18151$-$1208 B:} The arrows represent the direction of the two nearly perpendicular  CO  outflows detected by \\citet{2011ApJ...729...66F}. A blue-shifted component of a molecular outflow going in the direction\nof 18151$-$1208 B is seen in Figure 4 of \\citet{2011ApJ...729...66F} but it is not discussed by the authors. The black circle  is the 6.7 GHz CH$_{3}$OH  maser from \\citet{2002A&A...390..289B}. The x symbol represents the position of an \nadditional radio source detected at 1.3 cm reported in \\citet{2016ApJS..227...25R}. VLA 6 cm contour levels are ($-$2.0, 3.0, 9.0, 15.0) $\\times$ 6 $\\mu$Jy beam$^{-1}$, and 1.3 cm contour levels ($-$2.0, 3.0, 6.0, 8.5, 20.0, 40.0, 60.0)   $\\times$  8 $\\mu$Jy beam$^{-1}$.\n{\\bf Bottom: 19035+0641 A:} The arrows represent the direction of the detected CO and HCO$^{+}$ outflows \\citep{2002A&A...383..892B, 2010A&A...517A..66L}. The black circle  and the square are the 6.7 GHz CH$_{3}$OH   and  \nH$_{2}$O masers from \\citet{2002A&A...390..289B}, respectively. VLA 6 cm contour levels are ($-$2.5, 3.0, 10.0, 25.0, 120.0, 280.0, 380.0) $\\times$  4 $\\mu$Jy beam$^{-1}$, and 1.3 cm contour levels ($-$2.0, 5.0, 10.0, 20.0, 30.0, 50.0, 90.0, 170.0) $\\times$  8 $\\mu$Jy beam$^{-1}$.}}\n \\label{fig:UKIDSS_examples}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[!h]\n\\centering\n\\begin{tabular}{cc}\n\n\\includegraphics[width=0.34\\textwidth,  clip, angle = 0]{18151alpha_size_freq}  &\n\\includegraphics[width=0.34\\textwidth,  clip, angle = 0]{18440alpha_size_freq}  \\\\\n \\vspace{-1.cm} \n\\end{tabular}\n\\caption{\\small{Deconvolved major angular axis as a function of frequency for the ionized jet candidates 18151$-$1208 B, 18440$-$0148 A and 18566$+$0408 B. The arrows represent the size limit value from the synthesized beam of the map at the given frequency. The dashed line is the power law fit of the form $\\theta_{maj} \\propto \\nu^{\\gamma}$.}}\n \\label{fig:size_freq}\n\\end{figure}\n\n\n\\defcitealias{2011MNRAS.415..893A}{AMI Consortium: Scaife et al. (2011}\n\\defcitealias{2012MNRAS.420.1019A}{2012)}\n\nAs mentioned above, most of our detected radio sources with a rising spectrum are compact, i.e., spatially unresolved, or marginally resolved. Several of these sources are associated with molecular \noutflows, and 6.7 GHz CH$_{3}$OH  and 22 GHz H$_{2}$O masers as found in the literature. More precisely, only   6 of the 25 regions where we detected a radio source with a rising spectrum, are not \nassociated with molecular outflows, or adequate data that would trace such outflows do not seem to exist.\nFurthermore, after taking into account the shape of the synthesized beam of our VLA observations  some of \nthese radio sources appear slightly elongated in a certain direction. Examples are 18264$-$1152 F and G53.25$+$00.04mm2 A (see \\citetalias{2016ApJS..227...25R}, Figure~2). Therefore, we now investigate the \npossibility that  the compact radio continuum sources with a rising spectrum represent ionized jets. \n\n\\begin{figure}[h]\n\\centering\n\\begin{tabular}{ccc}\n\\hspace*{\\fill}%\n\\includegraphics[width=0.5\\textwidth, clip=true, angle = 0]{Rad_LumVSBol_Lum_paper}\n\\end{tabular}\n\\caption{\\small{Radio luminosity at 4.9 GHz  as a function of the bolometric luminosity. The red stars and octagons are our ionized jet and jet candidates listed in Table \\ref{jet_cand_list} and \\ref{candidates}  towards HMCs and CMC-IRs, respectively.  The bolometric luminosity for the red symbols is mainly estimated from our {\\it Herschel}\/Hi--GAL data (except for the open symbols whose bolometric luminosity information is from the literature).  The green circles represent ionized jets associated with low-mass protostars  ($1\\,L_{\\odot} \\leq L_{bol} \\leq 1000\\,L_{\\odot}$)   from \\citet{2018A&ARv..26....3A} and the yellow circles are the very low luminosity objects (VeLLOs) and  low-mass protostars from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}.\nThe purple triangles represent ionized jets from high-mass stars as found in the literature,\nfrom \\citet{2008AJ....135.2370R} and \\citet{2016A&A...585A..71M}. The  $\\times$ symbols are UC and HC H{\\small II} regions from \\citet{1994ApJS...91..659K}. \nThe dashed line relation shows the positive correlation found by \\citet{2015aska.confE.121A} derived for jets from low-mass stars. The red dotted line is our best fit to the data including ionized jets from low, intermediate and high-mass YSOs, but excluding the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}.\nThe error bars in the bottom right corner correspond to a 20$\\%$ calibration uncertainty.\n}}\n \\label{fig:rad_bol_lum}\n\\end{figure}\n\n\nA  statistical way of investigating the nature of our compact sources is to study the energy contained in the ionized gas. In Figure \\ref{fig:rad_bol_lum} we show the \nradio luminosity S$_\\nu\\,$d$^2$ of all the components of the ionized jet (or the jet candidate) as a function of the bolometric luminosity of the region. As in Figure \\ref{Lyman_cont_plot} above, the black line is the radio luminosity expected from the Lyman continuum\nflux at a given bolometric luminosity if it arises from photoionization of a single ZAMS star. In addition to the compact, rising spectra sources from our survey we also show in Figure \\ref{fig:rad_bol_lum} as green circles\nthe radio luminosity from low-mass stars ($1\\,L_{\\odot} \\leq L_{bol} \\leq 1000\\,L_{\\odot}$) associated with ionized jets from \\citet{2018A&ARv..26....3A} and as yellow circles the radio luminosity of very low luminosity objects (VeLLOs) and  low-mass protostars detected at 1.8 cm, and reported in \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}. In order to compare the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A} and \\citet{2018A&ARv..26....3A} with our $4.9\\,$GHz data, we scaled their\nfluxes using a factor of 0.48 and 0.76, respectively, assuming that those sources have a spectral index $\\alpha=0.6$, which is the canonical value of ionized jets. \n The scaling factors are calculated using \n$\\frac{S_{\\lambda_{1}}}{S_{\\lambda_{2}}}= \\left(\\frac{\\lambda_{2}}{\\lambda_{1}} \\right)^{\\alpha}$. \n\nIt is well known that for low mass YSOs the radio luminosities are correlated with the bolometric luminosity, and we show the correlation\n$\\frac{S_{\\nu}d^{2}}{\\text{mJy kpc}^{2}}= 8.7 \\times 10^{-3} \\left( \\frac{L_{\\text{bol}}}{L_{\\odot}} \\right)^{0.54}$ first found by \\citet{1995RMxAC...1...67A} and recently updated by \\citet{2018A&ARv..26....3A}. The black dashed line is the best fit to the green circles in Figure \\ref{fig:rad_bol_lum}, which are  the low-mass ionized jets presented by \\citet{2018A&ARv..26....3A}.\nIt is clear from Figure \\ref{fig:rad_bol_lum} that the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A} also follow this relation, although their data were observed at low resolution ($\\sim$ 30\\rlap.$^{\\prime \\prime}$) and there is not enough information that proves that they correspond to ionized jets. \n\\citet{1995RMxAC...1...67A} used this observed correlation to explain the apparent excess ionization levels from low mass YSOs by shock induced ionization from jets,\nas modeled by \\citet*{1987RMxAA..14..595C} and \\citet{1989ApL&C..27..299C}. \nA handful of detections of ionized jets towards high-mass stars  in recent years suggested that this correlation appears to also hold for stars with luminosities up to $\\sim$ 10$^{5}$ L$_{\\odot}$ (see \n\\citealt{2016MNRAS.460.1039P}). We have added these objects from \\citet{2008AJ....135.2370R} and  \\citet{2016A&A...585A..71M} as purple triangles in Figure \\ref{fig:rad_bol_lum} and the data has been properly scaled to our frequency of 4.9~GHz assuming $\\alpha=0.6$. The data from our survey (\\citetalias{2016ApJS..227...25R}) in conjunction with improved estimates of the \nluminosities based on Herschel data (see \\S \\ref{app:lum}) allow us to further populate this plot and test if a correlation exists. In Figure \\ref{fig:rad_bol_lum} the red stars and octagons are our radio sources with rising spectrum detected\ntowards HMCs and CMC-IRs, respectively, and we see that most sources are located very close to the relation found by \\citet{1995RMxAC...1...67A} up to luminosities of $\\sim$ 10$^{5}$ L$_{\\odot}$. In fact, a fit of the data \nincluding low, intermediate and high-mass YSOs is shown as a red dotted line and the result is similar to what was found by  \\citet{1995RMxAC...1...67A}. We excluded the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A} from our fit since it is unclear if those source are indeed  ionized jets.\n We take this result as a strong indication that the weak, and compact radio sources which we found in our survey are caused by the same mechanism which causes the radio emission the low mass YSOs, namely it is caused by ionized jets.  We also note that of the 6 compact radio sources where currently no observational association with molecular flows is known,\n 5 match our fit (red dotted line in Fig 8) of the $S_\\nu\\,d^2$ vs $L_{bol}$ relationship.\n \n \n\\defcitealias{2011MNRAS.415..893A}{AMI Consortium: Scaife et al. (2011}\n\\defcitealias{2012MNRAS.420.1019A}{2012)} \n \nFurthermore, in Figure \\ref{fig:Curiel_plot} we show the momentum rate ($\\dot{P}$) of the molecular outflow as a function of the radio luminosity (S$_\\nu\\,$d$^2$) of the ionized jet estimated from our flux values at 4.9 GHz (symbols and colors are the same as  in Figure  \\ref{fig:rad_bol_lum}). The momentum rate of the molecular outflows comes from information from the literature for our ionized jets (and jet candidates), if available, and the values are in most cases from single dish observations.  For consistency, we have scaled the physical values, so that they are based on the same distance. However,\nmany uncertainties remain due to the inhomogeneity of the data set. In particular, the values for the momentum rate come from observations taken\nby different authors, using different spectral lines, as well as different telescopes. Hence, the large scatter in Figure \\ref{fig:Curiel_plot} is not unexpected,\nand the creation of a homogenous data set for the $\\dot{P}$ versus $S_\\nu\\,d^2$ relation will be an important future task.\n\nIn spite of the large scatter, the correlation seen in  Figure \\ref{fig:Curiel_plot} indicates that the more radio luminous the protostar are the more powerful they are in pushing outflowing material. This correlation, which has been studied by several authors (e.g., \\citealt{1995RMxAC...1...67A}, \\citealt{2008AJ....135.2370R}, \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}, \\citealt{2018A&ARv..26....3A}), follows the shocked-induced ionization model introduced by \\citet{1987RMxAA..14..595C, 1989ApL&C..27..299C}, suggesting that the ionization of thermal jets is due to shocks. The  shocked-induced ionization model implies $\\left(\\frac{S_{\\nu}d^{2}}{mJy\\,kpc^{2}}\\right) = 10^{3.5} \\eta \\left(\\frac{\\dot{P}}{M_{\\odot}\\,yr^{-1}\\,km\\,s^{-1}}\\right)$ at $\\nu=5$ GHz where $\\eta$ is the shock efficiency fraction or the fraction of material that gets ionized by the shocks, which for low-mass protostars has been observationally found to be around 10$\\%$ (or $\\eta=0.1$).  \\citet{2018A&ARv..26....3A} suggested that the ionization fraction of jets in general is low ($\\sim 1 - 10\\%$). With the current data, and due to the large scatter seen in the correlation of  $\\dot{P}$ vs S$_\\nu\\,$d$^2$, we cannot yet properly quantify how the efficiency fraction changes with the luminosity of the protostar (e.g., if the ionization in thermal jets associated with high-mass protostars is higher than for low-mass protostars). Therefore, a uniform survey to measure the momentum rate of the molecular outflows associated with  ionized jets (ideally with  comparable resolutions) will be fundamental to further constrain this model. \\citet{2018A&ARv..26....3A} discussed both  correlations shown in Figure~\\ref{fig:rad_bol_lum} and Figure~\\ref{fig:Curiel_plot}  in great detail  and they interpreted them as an indication that the mechanism of ionization, accretion and ejection of outflows associated with protostars do not depend on their luminosities.\n\n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.5\\textwidth, clip=true, angle = 0]{Momentum_Rate_Survey}\n\\caption{\\small{Momentum rate of the molecular outflow  as a function of the radio luminosity at 4.9 GHz. The red stars and octagons are our ionized jets and jet candidates listed in Table \\ref{jet_cand_list} and \\ref{candidates}  towards HMCs and CMC-IRs, respectively, symbols for which there is information of the momentum rate.  The momentum rate values of the molecular outflow for all the sources including our data are collected from the literature.  The green circles represent ionized jets associated with low-mass protostars  ($1\\,L_{\\odot} \\leq L_{bol} \\leq 1000\\,L_{\\odot}$)   from \\citet{2018A&ARv..26....3A}  and the yellow circles are the very low luminosity objects (VeLLOs) and  low-mass protostars from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}.\nThe purple triangles represent ionized jets from high-mass stars as found in the literature,\nfrom \\citet{2008AJ....135.2370R} and \\citet{2016A&A...585A..71M}. The  $\\times$ symbols are UC and HC H{\\small II} regions from \\citet{1994ApJS...91..659K}. \nThe dashed line relation shows the positive correlation found by \\citet{1995RMxAC...1...67A} derived for jets from low-mass stars. The red dotted line is our best fit to the data including ionized jets from low, intermediate and high-mass YSOs, but excluding the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}. The gray shaded area corresponds to the momentum rate as predicted by the shock ionization model from \\citet*{1987RMxAA..14..595C} for values of the shock efficiency fraction of $\\eta = 0.1$ and $\\eta = 1.0$. \n The error bar in the middle right  corresponds to a 20$\\%$ calibration uncertainty. The error in $\\dot{P}$  is not represented in the figure because it \nvaries widely, and depends strongly on how different authors have gathered the data.\n}}\n \\label{fig:Curiel_plot}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Discussion}\\label{discussion_paperII}\nResults reported in \\citetalias{2016ApJS..227...25R} of detection rates  of CMC (6$\\%$), CMC-IR (53$\\%$) and HMCs (100$\\%$) provide further evidence for an evolutionary sequence in the formation of high-mass stars, from a very early stage type of cores (i.e., CMCs) to relatively more evolved ones (i.e., HMCs). The fraction of centimeter wavelength sources detected towards HMCs is  higher  than previously expected towards this type of cores and suggests that radio continuum may be detectable at weak levels in all HMCs. The lack of radio detections for some objects in the sample (including most CMCs) provides interesting constraints and are ideal follow up candidates for studies of the earliest stages of high-mass stars.\nIt is important to note that it is likely that the ionized material from  jets or  HC H{\\small II} regions associated with these type of cores remains undetected at our sensitivity, thus in order to rule out these regions as pre-stellar cores deeper observations are required or alternative tracers for ongoing star formation in these cores need to be identified.\n\nHere we consider some constraints on the nature of the  centimeter continuum emission detected in these cores and clumps towards high-mass star forming regions. As described in \\S \\ref{yso}, most of our radio detections arise from high-mass YSOs and at least for 7 regions some of the radio detections could potentially arise from (solar-like mass) T-Tauri stars. Also, we detected at least 10 radio sources associated with the mm cores\/clumps with a flat spectral index, most of them resolved sources, which are most likely UC H{\\small II} regions.  Understanding the nature of the rising spectral index sources has proven to be more challenging. These compact radio sources appear to be well fitted (within the uncertainties) when using either a homogeneous H{\\small II} region or a power-law fit  (as shown in Figure \\ref{HII_fit_examples}).  \nTherefore, we will discuss two plausible scenarios, that the radio sources are either  UC\/HC H{\\small II} regions or that the emission arises from shock ionized jets. \n\n\n\\subsection{H{\\small II} regions}\n\nFor the first scenario, when fitting the sources in terms of a homogeneous H{\\small II} region, the solutions required a significantly smaller size (several of them an order of magnitude smaller) for the H{\\small II} region than the upper limit given by the FWHM synthesized beam. However, since these calculations assumed a pure hydrogen nebula, we must consider whether internal dust absorption can make the regions as small as the H{\\small II} region model is predicting.   \\citet{1989ApJS...69..831W} suggested that, even if the dust absorbs 90$\\%$ of the UV photons, the radius of the H{\\small II} region is reduced by only a factor of 0.46. Thus, dust absorption alone appears insufficient to explain the small region sizes predicted by the H{\\small II} region model used to fit our data. As shown in Figure \\ref{Stromgren_sphere}, these sources could be explained as turbulence-pressured confined  H{\\small II} regions if they are born in a clump with density of n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$ and assuming  velocity dispersions of $\\sigma \\sim$ 3--8 km~s$^{-1}$ (FWHM$\\sim$7--20 km~s$^{-1}$). However, it is not clear if high velocity dispersions are common towards  the dense clumps harboring high-mass stars, since   $\\sigma \\sim$ 2 km s$^{-1}$ seems to be more typical. Measuring the line width of an optically thin tracer on $\\sim$100 au scale would provide a decisive constraint on the velocity dispersion. We also found that the sources could be consistent with having been born in a denser environment of $n_{H_{2}}\\approx$ 10$^{8}$\\,cm$^{-3}$. Arguably, \\citetalias{2016ApJ...818...52T}  predicts  that such a density for the ionized region is already too high for a protostar of  8 M$_{\\odot}$ to 24 M$_{\\odot}$.  Recently,  \\citet{2016A&A...585A..71M} detected  compact radio sources towards high-mass YSOs with similar physical characteristics to the ones found in this survey, where the Lyman continuum derived from the bolometric luminosities always exceeds the one obtained from the radio luminosities (as seen in Figure \\ref{fig:rad_bol_lum}). From their analysis they conclude that those sources cannot be HC or UC H{\\small II} regions, unless the ionized gas has a density gradient (e.g., model IV of \\citealt{1975A&A....39..217O}). \n\n\nAdditionally, for the extended UC H{\\small II} regions detected at the outskirts of the mm cores there are two scenarios: either they were born in a low density clump of $n_{H_{2}}\\approx$ 10$^{5}$\\,cm$^{-3}$ or  they were born at a higher density and have migrated out of the center potential.  The latter scenario requires large stellar dispersion velocities ($\\gtrsim$10 km~s$^{-1}$), which are not typical unless the source is a runaway OB star that has been dynamically ejected.   Observed stellar dispersion velocities, for instance for Orion's brightest  population is  only $\\lesssim$3 km~s$^{-1}$  \\citep[e.g.,][]{2005AJ....129..363S,2009ApJ...697.1103T}, which makes the former scenario more plausible. However, \\citet{2007ApJ...660.1296F} predicted that stellar velocities up to $\\lesssim$13 km~s$^{-1}$ are likely for core densities of 10$^{7}$\\,cm$^{-3}$, and  that these high stellar velocities carry the star to lower density regions of the core\/clump, where the H{\\small II} region is free to expand. We are leaning to favor the scenario  where the sources have migrated since it allows \nto explain the occurrence of both compact and extended emission in the same protocluster (e.g., \\citealt{2007prpl.conf..181H}). \n\\newpage         \n\\subsection{Radio Jets}\n         \nNow we discuss the second scenario where the radio emission of the radio compact sources with rising spectrum is due to shock ionization. \nThe observable properties of several of our radio detections indicate that they likely have a jet nature: one can argue that the low centimeter emission from the majority of the sources detected in this survey, their free-free\nspectral index being in the range $0.2 \\leq \\alpha \\leq 1.8$  and their association with molecular outflows indicate that even those sources without an elongated radio morphology are also ionized jets or stellar winds that are conical, accelerating and\/or recombining.  From the analysis in \\S\\ref{compact_sources_sect} and \\S \\ref{ionized_jet_sect} we inferred that from the 44 sources with rising spectral index, approximately 12 of them are ionized jets (see Table \\ref{jet_cand_list}) and 13 are jet\/wind candidates (see Table \\ref{candidates}). \nIn fact, half of the jet candidates in Table \\ref{candidates} have a spectral index \n$\\alpha \\approx 0.6$ and all but two of them (UYSO1 A and 18521$+$0134 A) have a spectral index $\\alpha \\leq 1.0$, which is consistent with the expected value of a spherical, isothermal and constant velocity ionized wind \\citep[e.g.,][]{1975A&A....39....1P}. As stated before, the deviation from the value $\\alpha = 0.6$ could be due to acceleration or recombination within the flow. \n\n\n\n\\begin{deluxetable}{l c c c c}\n\\tabletypesize{\\scriptsize}\n \\renewcommand*{\\arraystretch}{1.5}\n\\tablecaption{Ionized Jet\/Wind Candidates  \\label{candidates}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Region}                  & \n\\colhead{Radio Source}        &\n\\colhead{Outflow Direction}   &\n\\colhead{ H$_{2}-$Jet Direction}   &\n \\colhead{Reference}      \\\\}\n\\startdata\n\\setcounter{iso}{0}\t\nUYSO1                    &    A   &   NW$-$SE    &   \\nodata  & \\rxn \\label{rxn:Forbrich04}   \\\\\n18264$-$1152         &    F  &   NW$-$SE     &  E$-$W  & \\rxn \\label{rxn:Sanchez-Monge2013} \\rxn \\label{rxn:Navarete2015}    \\\\\n18345$-$0641        &    A   &    NW$-$SE    &   very weak &\\rxn \\label{rxn:Beuther2002} \\rxn \\label{rxn:Varricat2013}  \\rxn \\label{rxn:Varricat2010}  \\\\\n18470$-$0044        &    B   &   E$-$W          &   no\/very weak\\tablenotemark{a}      &  \\osref{rxn:Beuther2002}   \\\\\n18517$+$0437       &    A    &   N$-$S          &   very weak   & \\rxn \\label{rxn:Lopez_Sep10} \\osref{rxn:Varricat2010} \\\\\n18521$+$0134       &    A   &    \\nodata\\tablenotemark{b}     &     non-detection       &   \\rxn \\label{rxn:Cooper13}   \\\\\n G35.39$-$00.33mm2        &    A   &  \\nodata            &  \\nodata   &    \\nodata   \\\\\n18553$+$0414       &    A   &    \\nodata\\tablenotemark{c}     &   non-detection         &    \\osref{rxn:Navarete2015}   \\\\\n19012$+$0536       &    A   &    NE$-$SW      &   non-detection   &   \\osref{rxn:Beuther2002} \\osref{rxn:Navarete2015}   \\\\\nG53.25$+$00.04mm2        &    A   & \\nodata    &     \\nodata       &    \\nodata   \\\\\n 19413$+$2332      &    A   &      \\nodata\\tablenotemark{d}     &     \\nodata      &   \\osref{rxn:Beuther2002}  \\\\\n20293$+$3952      &    E\\tablenotemark{e}    &     NE$-$SW    &     detection    &  \\rxn \\label{rxn:Beuther04} \\rxn \\label{rxn:Palau07_a} \\osref{rxn:Varricat2010} \\\\        \n 20343$+$4129      &    B   &     E$-$W      &   non-detection     &  \\rxn \\label{rxn:Palau07}  \\osref{rxn:Cooper13}\\\\        \n\\enddata\n\\tablenotetext{\\text{a}}{ T. Stanke and H. Beuther (private communication).}\n\\tablenotetext{\\text{b}}{\\citet{2002ApJ...566..931S} reports non-detection of CO (2--1) wings towards this region, although an outflow  could be present at an inclination angle of $< 10^{\\circ}$ to the plane of the sky.}\n\\tablenotetext{\\text{c}}{\\citet{2002ApJ...566..931S} reports the presence of CO (2--1) wings towards this region, but contour maps of the molecular outflow are not available.}\n\\tablenotetext{\\text{d}}{CO outflow is detected in the region, but the data does not show a clear bipolar structure.}\n\n\\tablenotetext{\\text{e}}{Radio source E has an upper limit value for the flux at 4.9 GHz and its value is not included in Figures \\ref{fig:rad_bol_lum} and \\ref{fig:Tanaka_tracks} (right panel).}\n\\tablecomments{The dots indicates that there is not enough information available about observations of the molecular outflow in the literature.\\\\\n\\osref{rxn:Forbrich04} \\citet{2004ApJ...602..843F}; \\osref{rxn:Sanchez-Monge2013}  \\citet{2013A\\string&A...557A..94S};  \\osref{rxn:Navarete2015}  \\citet{2015MNRAS.450.4364N}; \\osref{rxn:Beuther2002}  \\citet{2002A\\string&A...383..892B};  \\osref{rxn:Varricat2013}  \\citet{2013A\\string&A...554A...9V}; \\osref{rxn:Varricat2010}  \\citet{2010MNRAS.404..661V}; \\osref{rxn:Lopez_Sep10} \\citet{2010A\\string&A...517A..66L}; \\osref{rxn:Cooper13} \\citet{2013MNRAS.430.1125C}, \\osref{rxn:Beuther04} \\citet{2004ApJ...608..330B}; \\osref{rxn:Palau07_a} \\citet{2007A\\string&A...465..219P}; \\osref{rxn:Palau07} \\citet{2007A\\string&A...474..911P}.}\n\n\\end{deluxetable}\n\n\n\\subsection{H{\\small II} regions vs Radio Jets}\n\nIn Figure \\ref{fig:rad_bol_lum} we compared the radio luminosity with the bolometric luminosity using the radio flux at 4.9 GHz.  When fitting the ionized jets (and jet candidates) from low, intermediate and high-mass protostars using a power-law (represented by the dotted red line) we find an index of 0.63$\\pm$0.04 with a correlation coefficient  of $r=$0.89 which yields the relation $S_{\\nu}d^{2}$ [mJy kpc$^{-2}$]=$ 6.5 \\times 10^{-3}$ (L$_{bol}$\/L$_{\\odot}$)$^{0.63}$.  This result is comparable with the index found by \\citet{2016MNRAS.460.1039P} of 0.64$\\pm$0.04 for jets spanning luminosities from $\\sim 10^{-1}$ to $10^{5}$ L$_{\\odot}$, although their fit has a lower correlation coefficient ($r=$0.73). Their estimates for bolometric luminosities, which include {\\it Herschel}\/Hi--GAL data for most of their sources, are similar to ours. Therefore, the scatter in their data may come from the radio fluxes. \\citet{2016MNRAS.460.1039P} have stated that some of their jets have high flux densities probably because those objects represent a transition  between a jet and H{\\small II} region stages. \n Further, it is important to note that a similar relation between the bolometric luminosity and the luminosity of shocked H$_2$ emission from molecular jets\nhas been reported by \\citet{2015A&A...573A..82C} for sources with a wide range of bolometric luminosities. These studies together with our\nrefined relation point to a common flow mechanism from YSOs of any luminosity.\n\nUntil very recently, the stellar evolutionary models that have been used to analyze this type of sources  correspond to more evolved objects (i.e., ZAMS star). However, the recently introduced \\citetalias{2016ApJ...818...52T} model predicts the ionizing luminosity of a protostar which will allow us to compare our data with a more appropriate part of the evolutionary track.  These evolutionary stellar models mainly depend on the accretion history, this is the mass of the core (M$_{c}$) and the mass surface density of the ambient clump ($\\Sigma_{cl}$). Figure \\ref{fig:Tanaka_tracks} shows the same relations as those in Figs. \\ref{Lyman_cont_plot} and \\ref{fig:rad_bol_lum}, but now we also consider an evolutionary track for a YSO which is represented by the cyan continuous line for an initial core mass of M$_{c}=$ 60 M$_{\\odot}$ and a mass surface density of ambient clump of $\\Sigma_{cl}=$~1~g~cm$^{-2}$ (\\citetalias{2016ApJ...818...52T} fiducial case). This cyan track shows the evolutionary sequence of the ionizing photon luminosity as a function of the protostellar luminosity. Its shape shows each of the physical stages in the evolution of the protostar: accretion stage, swelling stage (as seen with the decrease in the ionizing luminosity as the temperature decreases), contraction stage (increase of the ionizing luminosity as the temperature also increases) and nuclear burning stage when the protostar reaches the ZAMS (represented by the black continuous line; for more discussion on this evolutionary track see \\citealt{2014ApJ...788..166Z}; \\citetalias{2016ApJ...818...52T}).  The left panel of Figure \\ref{fig:Tanaka_tracks} shows that\n for the majority of the radio sources detected  towards CMCs and CMC--IRs, the  Lyman continuum excess (for the fiducial case L$_{bol} \\sim$10$^{2}$--10$^{3}$) is still evident and it is not likely due to photoionization. Additionally, the evolutionary  track for a YSO shows how the ionizing luminosity decreases as the protostar swells while accreting its mass and before it enters the Kelvin-Helmholz contraction (for the fiducial case L$_{bol} \\sim$10$^{3}$--10$^{4}$). This further indicates that the measured radio flux for most of our radio sources detected towards HMCs are also very unlikely to be photoionized by the central object.\n \nModel calculations presented by \\citet{2002ApJ...568..754K,2003ApJ...599.1196K,2007ApJ...666..976K}  predict that high accretion rates on the order of  $10^{-4} - 10^{-3}$ M$_{\\odot}$yr$^{-1}$ can choke off the H{\\small II} region to very small sizes producing very low radio continuum; see also Section 5 of \\citet{1995RMxAC...1..137W}. This might be a possible scenario for some of our more compact sources, but additional evidence is necessary such as high-resolution mm observations of infall tracers to determine mass infall rates for these sources. Based on the analysis and discussion presented above, we are inclined to favor the scenario that most of our compact sources (see Table \\ref{candidates}) with rising spectrum are ionized jets.  However, the confirmation of these radio sources as shocked ionized gas requires further observational and theoretical work. Additional  observations and tests are necessary in  order to have conclusive information of the nature of these detections. Higher resolution data ($\\lesssim$ 0\\rlap.$^{\\prime \\prime}$1) of the radio continuum is required to resolve the ionized jets and estimate their degree of  collimation. Additionally, high resolution millimeter data will help us to disentangle multiple outflows, possibly being driven by protostellar clusters as expected toward high-mass star forming regions and to study the  kinematics of the outflow material. Masers arise from the hot core regions and their  association with ionized material is very important. They indicate the evolutionary stage \\citep[e.g.,][]{2018A&A...619A.107S} of the exciting object and allow detailed studies of the kinematics at a smaller scale, very close to the powering high-mass YSO and the disk\/jet interface.   For resolved sources, long-term monitoring of the ionized jet is necessary in order to estimate proper motions,\nvelocities of the radio jets and evolution in the morphology of the jet. \n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{tabular}{cc}\n\\vspace{-0.7cm}\n\\includegraphics[width=0.46\\textwidth, clip=true, angle = 0]{Lyman_plot_Tanaka} &\\hspace{-1.6em} \n\\includegraphics[width=0.46\\textwidth,  clip=true, angle = 0]{Tanaka_M60}  \n\\end{tabular}\n\n\\caption{\\small{Lyman continuum (left) and radio luminosity (right) as a function of the bolometric luminosity. Symbols and colors are the same as used in Figures \\ref{Lyman_cont_plot} and \\ref{fig:rad_bol_lum}, except that now we also show the estimated Lyman continuum from the \\citetalias{2016ApJ...818...52T} model for an optically thin H{\\small II} region based on the ionization of a protostar (cyan continuous line). The stellar model evolution  starts with a  core mass of M$_{c}=$ 60 M$_{\\odot}$ and a mass surface density of ambient clump of $\\Sigma_{cl}=$~1~g~cm$^{-2}$. The black continuous line is the Lyman continuum from a ZAMS star.\n The error bars in the bottom right corner correspond to a 20$\\%$ calibration uncertainty. \n}}\n \\label{fig:Tanaka_tracks}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Summary and Conclusions}\\label{conclusions_paperII}\nIn this work we investigate the nature of the 70  radio sources\nreported in \\citetalias{2016ApJS..227...25R}.  These radio sources were observed using the VLA at 6 and 1.3 cm towards a sample of high-mass star forming region candidates having either no previous radio continuum detection or a relatively weak detection at the 1 mJy level. We have explored several scenarios such as pressure confined H{\\small II} regions and ionized jets to explain the origin of the ionized gas emission and we have studied the physical properties of the detected sources. Based on our results we favor the scenario that $\\sim 30 - 50 \\%$ of our radio detections are ionized jets and\/or jet knots. These sources, listed in Tables \\ref{jet_cand_list} and \\ref{candidates}, have observational properties that are not expected towards regular H{\\small II} regions such as the correlation of their radio luminosity and bolometric luminosity and the correlation of the momentum rate of the molecular outflow with the radio luminosity of the ionized jet. Such correlations have been found observationally towards ionized jets associated with high-mass protostars of different luminosities and are also predicted in recent theoretical models such as the \\citetalias{2016ApJ...818...52T} model. However, for the most compact radio continuum detections we cannot rule out the scenario that they correspond to pressure confined H{\\small II} regions.  Our main results from this survey are summarized below: \n\n\\begin{itemize}\n\\item We detected centimeter wavelength sources in 100$\\%$ of our HMCs, which is a higher fraction than previously expected and suggests that radio continuum may be detectable at weak levels in all HMCs.  The lack of radio detections for some objects in the sample (including most CMCs) contributes  evidence that these clumps are in an earlier evolutionary stage than HMCs, providing interesting constraints and ideal follow up candidates for studies of the earliest stages of high-mass stars.\n\n\\item At least 10$\\%$ of our detected radio sources are consistent with non-thermal emission and likely due to either  active magnetospheres in T-Tauri stars (possibly for the few regions located at a distance $<$ 2 kpc) or synchrotron emission from fast shocks in disks or jets.\n\n\\item For the most compact radio detections, the sources are consistent with being small pressure confined H{\\small II} regions. Also, we cannot completely exclude the possibility that these sources are gravitationally trapped H{\\small II} regions. \n\n\\item The majority of our detected radio continuum sources ($\\sim$80$\\%$) have spectral indices ($-$0.1$<\\alpha<$2) that are consistent with thermal (free-free) emission from ionized gas. \n\n\\item Most of the radio sources with a rising spectrum detected towards clumps at an earlier evolutionary stage (i.e., CMCs and CMC--IRs) show Lyman continuum excess, consistent with previous results.  This can be explained either by  UV photons from shocks producing an ionized jet or shocks in an accretion flow onto the disk.\n\n\\item For most of the radio sources with a rising spectrum detected towards HMCs, the estimated Lyman continuum is lower than  expected if the radio flux comes from a single ZAMS star. This could indicate that the origin of the measured radio flux is not from HC\/UC H{\\small II} regions but  shock ionized jets.\n\n\\item We detected at least 12 ionized jets (6 of them are new detections to the\nknowledge of the authors) based on their spectral index, morphology and molecular outflow associations. For several of the previously detected jets, we detected additional knots or lobes that are part of the collimated structure. Additionally, we detected at least 13 jet\/wind candidates. \n\n\\item We found that ionized jets from low and high-mass stars are very well correlated. This is consistent with previous studies and is further evidence of a common origin for jets of any luminosity.\n\n\n\n\\end{itemize}\n\n\n\n\n\\acknowledgments\n We thank the anonymous referee, whose comments improved this manuscript.\nSupport for this work was provided by the NSF through the Grote Reber Fellowship Program administered by Associated Universities, Inc.\/National Radio Astronomy Observatory.  P. H. acknowledges support from NSF grant AST--1814011. C. C-G. acknowledges support from UNAM DGAPA--PAPIIT grant number IA102816, IN10818. E. D. A. is partially supported by NSF grant AST--1814063. We thank K. E. I. Tanaka for providing the stellar model evolutionary tracks predicted by  the \\citetalias{2016ApJ...818...52T} model. We thank K. E. I. Tanaka, K. Johnston and J. Marvil for  useful discussions. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. This work is based in part on observations made with the Spitzer Space telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. P.H. acknowledges support from NSF grant AST$-$0908901 for this project.  Some of the data reported here were obtained as part of the UKIRT Service Program. The United Kingdom Infrared Telescope is operated by the Joint Astronomy Centre on behalf of the UK Particle Physics and Astronomy Research Council.\nThis research made use of APLpy, an open-source plotting package for Python hosted at http:\/\/aplpy.github.com.\n\n\\vspace{5mm}\n\n\\software{CASA \\citep{2007ASPC..376..127M}, APLpy \\citep{2012ascl.soft08017R}.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction }\nThe Batalin--Vilkovisky formalism (BV--formalism) is the most\ngeneral method of quantization\nof gauge field theories~\\cite{bat1},~\\cite{bat2}.\nIn recent years the interest\nin studying  its geometrical nature has increased.\nIt was stimulated by  Witten's paper \\cite{witten},\nwhere the necessity of such an investigation is pointed out and\nparticularly for the\nformulation of the background independent open-string\nfield theory on the base of the BV-formalism.\nThe realization of this program  began in \\cite{witten1}, \\cite{witten2}.\n\nIt is known that the BV-formalism uses unusual structures --  odd\nPoisson brackets (antibrackets) and the operator $\\Delta$.\nAs one of the main obstacles to the construction of\na background independent open\nstring field theory,  indicated by Witten \\cite{witten},\n was the nonexistence  of an invariant\ndefinition of the operator $\\Delta$, and of a\nnaturally defined integral measure.\nHowever, such a definition of the operator $\\Delta$\nwas shown by one of us (O. K.) in\n\\cite{khud} before the cited Witten's paper  appeared (see also \\cite{km}).\nIts realization on K\\\"ahlerian supermanifolds  \\cite{ners} and its\nsimplest properties\n\\cite{knjinr} were considered.\n\nIn the present paper we study the operator $\\Delta$ in more detail.\nWe propose an invariant definition of this operator and show that\nthe condition of its nilpotency defines (in some sense) the choice of the\nintegration measure.\n\nIn Section 2 we propose an invariant definition of the operator $\\Delta$\n on supermanifolds, given by the odd symplectic structure\nand by the volume form as the divergence of the Hamiltonian vector field.\n We show that  all the relations  between the antibrackets and the operator\n$\\Delta$, which  are satisfied for canonical ones in the BV-formalism, are\nsatisfied for\nany generalized operator $\\Delta$  and the corresponding odd brackets.\nHowever\nthe nilpotency condition for such an operator holds only for\na certain class of the integral density.\n\nIn Section 3 we consider the realization of the operator $\\Delta $ on\nsupermanifolds, given by the odd (and even) K\\\"ahlerian structure,\nand show that it is\nnilpotent, if the integral density is the\ncharacteristic class of the basic K\\\"ahlerian manifold (the function from\nChern classes). Then it corresponds to the\ndivergence operator  $\\delta =* d *$\nof the basic manifold.\n\nIn Section 3 we discuss the geometrical nature of\nthe Bat\\-al\\-in-Vil\\-kov\\-isky formalism.\\\\\n\nWhen this paper was in preparation, we received two very important papers of\nA.S. Schwarz \\cite {schwarz}, \\cite{schwarz2}  where the geometry of\nthe BV-formalism is analyzed in detail and in particular the same definition\nof the operator $\\Delta$, as in \\cite{khud}, \\cite{ners}, \\cite{knjinr} and\nin the present paper is given.\n       \\setcounter{equation}0\n\\section{Odd Poisson Brackets and Operator $\\Delta$}\nThe  odd Poisson bracket (odd bracket, antibracket, Buttin bracket)\n  of the functions $f$ and $g$ on the supermanifold ${\\cal M}$ is\ndefined by the following  conditions \\cite{ber}, \\cite{leit} :\n\\begin{equation}\n\\begin{array}{l}\n\\{ f, ga +hb \\}_{1} = \\{ f, g\\}_{1}a + \\{ f, h \\}_{1}b,\n         \\quad {\\rm where}\\quad a,b =  const   \\\\[3mm]\np(\\{ f, g \\}_{1} )= p(f)+ p(g) + 1\n         \\quad {\\rm (grading \\ condition)}  \\label{eq:bgrad} \\\\[3mm]\n\\{ f, g \\}_1 = -(-1)^{(p(f)+1)(p(g)+1)}\\{ g, f \\}_1\n         \\quad {\\rm ( \"antisymmetricity\"\\ condition )} \\label{eq:anti} \\\\[3mm]\n\\{ f, gh \\}_1 =\\{ f, g \\}_{1}h +(-1)^{(p(f)+1)p(g)} g\\{ f, h \\}_1\n         \\quad {\\rm ( Leibnitz \\ rule )} \\label{eq:bLieb} \\\\[3mm]\n ( -1)^{(p(f)+1)(p(h)+1)}\\{ f,\\{ g, h \\}_{1}\\}_1\n                  +{\\rm {cycl. perm.}}( f,g,h) = 0\n         \\quad{\\rm {(Jacobi\\ id.)}}\n\\end{array}\n\\label{eq:bjac}\n\\end{equation}\n  Locally, the  odd bracket can be written as:\n\\begin{equation}\n\\{ f, g \\}_1 = \\frac{\\partial ^R f}{\\partial x^A} \\Omega^{AB}\n            \\frac{\\partial  ^L g}{\\partial x^B}\n\\label{eq:bloc}\n\\end{equation}\nwhere $\\Omega^{AB}$ satisfies to conditions\n\\begin{eqnarray}\n& & p(\\Omega^{AB} )= p(A)+ p(B) + 1\n\\quad  {\\rm (grading\\ condition)} \\nonumber  \\\\[3mm]\n& & \\Omega^{AB}  = -(-1)^{(p(A)+1)(p(B)+1)} \\Omega^{BA}\n\\quad {\\rm ( \"antisymmetricity\" \\ condition )} \\nonumber \\\\[3mm]\n& & ( -1)^{(p(A)+1)(p(C)+1)}\\frac{\\partial^{R}\n \\Omega^{AB}}{\\partial x^D }\\Omega^{DC}\n   +{\\rm {cycl.\\  perm.}}(A,B,C ) = 0\n  \\quad{\\rm {(Jacobi\\ id.)}}\n\\nonumber\n\\end{eqnarray}\nwhere\n$x^A$ are the local coordinates of ${\\cal M}$, $p_A \\equiv p( x^A ) $.\n$ \\frac{\\partial ^R }{\\partial x^A}$ and\n$\\frac{\\partial ^L}{\\partial x^A} $ denote\ncorresponding right and left derivatives.\nThey are connected with each other by:\n$$\\frac{\\partial ^R f}{\\partial x^A}=\n  (-1)^{(p(f)+1)p_A}\\frac{\\partial ^l f}{\\partial x^A}. $$\nIf ${\\cal M}$ has an equal number of\neven and odd coordinates, the odd bracket\ncan be nondegenerate. Then one can associate with it the\nodd symplectic structure\n\\begin{equation}\n\\Omega =dx^A \\Omega^{AB}dx^B   \\label{eq:symp}\n\\end{equation}\n where $\\Omega_{AB}\\Omega^{BC} =\\delta_{A}^{C}$.\nThis form is closed because of the Jacobi identities (\\ref{eq:bjac}).\nLocally, one can reduce  (\\ref{eq:symp}) to the canonical form \\cite{leit}:\n\\begin{equation}\n\\Omega^{\\rm can} =\\sum_{i=1}^{N} dx^i \\wedge d \\theta_i\n   \\label{eq:sympcan}\n\\end{equation}\nwhere $( x^i,\\theta _i) $ are some local coordinates (Darboux coordinates)\n($p( x^i )= 0, p(\\theta_i)= 1 $).\nThe corresponding odd bracket takes the form\n\\begin{equation}\n\t\t  \\{f,g\\}_1 =\n\t\t\t\\sum_{i=1}^{N}\\left(\n\t\t\\frac{\\partial^{R} f}{\\partial  x^i}\n\t\t\\frac{\\partial^{L} g}{\\partial\\theta_i}\n\t\t\t +\n              \\frac{\\partial^{R} f}{\\partial  \\theta_i}\n\t\t\\frac{\\partial^{L} g}{\\partial  x^i}\n\t\t   \\right)       \\label{eq:bcan}\n\\end{equation}\n Transformations preserving odd symplectic structures (or odd brackets)\n have (locally) the hamiltonian form:\n  \\begin{equation}\n{\\cal L}_{{\\bf V}}\\Omega = O\n\\quad {\\rm iff}\\quad {\\bf V}=\\{.,H\\}_1 \\equiv {\\bf D}_{H}     \\label{ham}\n\\end{equation}\n where $ H$ is an arbitrary function (Hamiltonian) on ${\\cal M}$,\n${\\cal L}_{{\\bf V}}$ denotes the Lie derivative\nalong the vector field {\\bf V}.\n\nIt is well known that any supermanifold  can be associated with some vector\n bundle \\cite{ber}. The odd symplectic structure\ncan be globally defined on the supermanifolds\n which are associated with the cotangent bundles of manifolds.\n\n Let $T^* M$ be the cotangent bundle of the manifold $M$.\n  $x^{i}$ are local coordinates on $M$\n  and $( x^{i}, v_{i} )$ are the corresponding\n  local coordinates on $T^* M$.\n From map to map\n  \\begin{equation}\nx^{i} \\rightarrow {\\tilde x^{i}} = {\\tilde x^{i}} ( x),\\quad\n      v_{i} \\rightarrow {\\tilde v_{i}} = \\sum_{i=1}^N\n  \\frac{\\partial x^{j}}{\\partial {\\tilde x^{i}}} v_{j}.\n     \\label{eq:trans}\n\\end{equation}\n Considering for every map the superalgebra generated by\n  $(x^{i}, \\theta _{i})$, where the $x^{i}$ are even and\nthe  $\\theta_{i}$ are odd coordinates,  transforming from map\n to map like $(x^{i}, v_{i})$ in ref{eq:trans})\n  ($v\\leftrightarrow \\theta$), we come to the\n supermanifold ${\\cal M}$ which is associated with $T^*M$ in the\n  coordinates $(x^{i}, \\theta_{i})$.\n\n Obviously, on this supermanifold, in the coordinates $(x^i, \\theta_{i})$,\none can globally define the\n canonical odd symplectic structure (\\ref{eq:sympcan}) \\cite{leit}.\n\n\nFor the coordinates $(x^i, \\theta_{i})$ on ${\\cal M}$,\none can admit a more general class  of transformations:\n                           $$\nx^i\\rightarrow {\\tilde x}^i(x,\\theta) \\quad\n\t\\theta_{i}\\rightarrow {\\tilde \\theta}_{i}(x,\\theta )    $$\n   which do not correspond to (\\ref{eq:trans}).\n In particular,\nif $\\theta_i \\to {\\tilde \\theta}^i = \\omega^{ij}\\theta _j $,\n  where $\\omega _{ij}$ is the matrix of some nondegenerate Poisson bracket\n on $M$, then the supermanifold\n  ${\\cal M}$ in the coordinates $(x^i,{\\tilde \\theta}^i )$ is\n  associated with the tangent bundle $TM$ of $M$,\ne.g. ${\\tilde \\theta^i}$ transform\n  under (\\ref{eq:trans}) as $dx^i$.\n\n On the supermanifolds which can be associated in some\n  coordinates with the tangent or cotangent bundle the\n  superstructures are evidently reduced to  standard geometrical objects.\n\n   Concerning the integration, the\nproperties of the odd brackets strongly\ndiffer from the properties of odd brackets \\cite {khud}, \\cite {km},\nsuch as:\\\\\n\n -- the odd bracket hasn't an invariant volume form and invariant\n   integral densities;\\\\\n\n -- it has semidensities, which depend on higher order derivatives.\\\\\n\nThe first of this properties plays an essential role in\nthe Batalin--Vilkovisky quantization\nformalism.\nUsing this property, we can construct on ${\\cal M}$\nan invariant generalization of the\nimportant object of the BV-formalism  -- the operator $\\Delta$.\n\nLet the supermanifold ${\\cal M}$ be\n   provided with the odd symplectic structure (\\ref{eq:symp})  and\n the volume form\n\\begin{equation}\ndv=\\rho(x, \\theta)d^{N}x d^{N}\\theta.  \\label{eq:vol}\n\\end{equation}\nHere $\\rho (x,\\theta)$\nis some integral density. Under coordinate transformation\n${\\tilde x}^{A} = {\\tilde x}^{A}(x)$,\nit  transforms as:\n\\begin{equation}\n   {\\tilde \\rho}({\\tilde x}) = \\rho (x ({\\tilde x})) {\\rm Ber}\n \\frac{\\partial^{R} x^A}{\\partial {\\tilde x}^B}  \\label{eq:denstrans}\n\\end{equation}\nOn this supermanifold  one can invariantly define a second order odd\ndifferential operator, which we call the \"generalized operator $ \\Delta $\",\nand which is invariant\n under the transformations preserving the symplectic structure\n  and the volume form \\cite {khud}. Its action on a function $f(x,\\theta)$\n  is the divergence  of the Hamiltonian\n  vector field ${\\bf D}_{f}$\n  with the volume form $dv$:\n\\begin{equation}\n\\Delta _{\\rho} f =\\frac{1}{2}div_{\\rho} {\\bf D}_{f}\n\\equiv \\frac{1}{2}\\frac{{\\cal L}_{{\\bf D}_f} dv}{dv}, \\label{eq:delta}\n\\end{equation}\nwhere  ${\\cal L}_{{\\bf D}_f}$ is the Lie derivative\nalong ${{\\bf D}_f}$ \\cite {ber}, \\cite {voronov}.\nIn coordinate form:\n\\begin{equation}\n           \\Delta f=\\frac{1}{2\\rho}\n     \\frac{\\partial^R}{\\partial x^A}\n\\left(\\rho\\{x^A,f\\}_1\\right)   \\label{eq:deltaloc}\n\\end{equation}\n   It has no analog within even symplectic structures.\nThe oddness of the Poisson bracket (\\ref{eq:bloc})  forces\n the nontrivial grading of $\\Delta $,\nand the \"antisymmetricity\" condition (\\ref{eq:anti}) forces\nits dependence on second derivatives.\n\nIf the Poisson bracket  in (\\ref{eq:delta}) is canonical,\nand $\\rho =constant$,\nthe generalized operator $\\Delta$ takes the canonical  form\n\\begin{equation}\n     \\Delta^{\\rm can} = \\frac{\\partial^R}{\\partial x^i}\n\\frac{\\partial ^L}{\\partial \\theta_i} \\label{eq:deltacan}\n\\end{equation}\nused in the BV-formalism.\n\n From the Leibnitz rule (\\ref{eq:bLieb}) and the\ndefinition (\\ref{eq:delta}) follows\n\\begin{equation}\n(-1)^{p(g)}\\{f,g \\}_1 = \\Delta(fg) - f\\Delta g\n -(-1)^{p(g)}(\\Delta f)g  \\label{eq:Liebdelta}\n\\end{equation}\n\n From the Jacobi identity (\\ref{eq:bjac}) and the\ndefinition (\\ref{eq:delta}) follows\n\\begin{equation}\n\\Delta \\{f,g \\}_1 = \\{f,\\Delta g \\}_1\n+(-1)^{p(g)+1}\\{\\Delta f ,g \\}_1     \\label{eq:jacdelta}\n\\end{equation}\nThe density transformation rule (\\ref{eq:denstrans}) implies for\nthe generalized operator $\\Delta$\nthe following transformation rule\nunder canonical transformations:\n\\begin{equation}\n \\Delta'f = \\Delta f +\\frac{1}{2}\n\\{\\log {\\cal J} ,f \\}_1 , \\label{eq:transdelta}\n\\end{equation}\nwhere ${\\cal J}$ is the Jacobian of the canonical transformation of\nthe odd bracket, $\\Delta'$ is the\ngeneralized operator $\\Delta$ in the new coordinates.\nFor example, let us demonstrate the derivation of (\\ref{eq:jacdelta}):\n\\begin{eqnarray}\n&&\\Delta \\{f,g \\}_{1}dv = {\\cal L}_{{\\bf D}_{\\{f,g \\}_1}}dv =\\nonumber\\\\[3mm]\n&&=\\left ( {\\cal L}_{{\\bf D}_f}{\\cal L}_{{\\bf D}_g} -\n(-1)^{(p(f)+1)(p(g)+1)}{\\cal L}_{{\\bf D}_g}{\\cal L}_{{\\bf D}_f} \\right )dv\n= \\nonumber\\\\[3mm]\n&&={\\cal L}_{{\\bf D}_f} {\\Delta g}dv\n-(-1)^{(p(f)+1)(p(g)+1)}{\\cal L}_{{\\bf D}_g} {\\Delta f}dv = \\nonumber\\\\[3mm]\n&& = \\left ( \\{f,\\Delta g \\}_{1}\n+(-1)^{p(g)+1}\\{\\Delta f ,g \\}_1 \\right ) dv \\nonumber\n\\end{eqnarray}\nLet us write the following useful expressions, too:\n\\begin{equation}\n \\Delta f(g)=f'(g) \\Delta g + \\frac{1}{2}f''(g)\n\\{g,g \\}_1,  \\label{eq:difdelta}\n\\end{equation}\n where $f(g)$ is an even complete function, and $g$ is\nan even function on ${\\cal M}$.\n\nThe properties (\\ref{eq:Liebdelta}) -\n(\\ref{eq:difdelta}) are satisfied for any\n$\\rho$ and in the same manner as\nthe relations between canonical Poisson brackets\n(\\ref{eq:bcan}) and (\\ref{eq:deltacan}) in the\nBV-formalism \\cite{bat2}, \\cite{witten}.\nThis can br derived in the same way as (\\ref{eq:jacdelta}).\n\nHowever (\\ref{eq:deltacan}) satisfies the\n nilpotency  condition\n\\begin{equation}\n\t       \\Delta^2=0     \\label{eq:nilp}\n\\end{equation}\nwhich is very important in the BV-formalism.\n\nThe latter condition is violated for arbitrary $\\rho (x,\\theta)$.\n Indeed, if we have two densities $\\rho$ and ${\\tilde \\rho}$,\nand ${\\tilde \\rho} = \\lambda\\rho,\\ p(\\lambda )=0$,\nthen the corresponding operators $\\Delta$ are related by\n\\begin{equation}\n    \\Delta_{\\tilde \\rho} f =\\Delta_{\\rho} f +\n\\frac{1}{2} \\{\\log \\lambda, f\\}_{1} \\label{eq:deltacon}\n\\end{equation}\nIt is easy to see that\n \\begin{equation}\n \\Delta^{2}_{\\tilde \\rho} f =\\Delta^{2}_{\\rho} f\n+ \\{ \\Gamma_{\\lambda}, f \\}_{1} ,  \\label{eq:deltasqcon}\n\\end{equation}\nwhere\n\\begin{equation}\n  \\Gamma_{\\lambda}\n=\\lambda^{-\\frac{1}{2}}\\Delta_{\\rho} \\lambda^{\\frac{1}{2}},\n\\quad p(\\Gamma_{\\lambda}) = 1\n\\end{equation}\nIf for some $\\Delta_{\\rho}$  the nilpotency condition\n(\\ref{eq:nilp}) is satisfied,\nthen it is also satisfied for  $\\Delta_{\\tilde \\rho}$ (\\ref{eq:deltacon})\nif  $ \\Gamma_{\\lambda} =$odd constant$=0$.\n\nFor example, if the  symplectic structure is canonical,\n(\\ref{eq:nilp}) holds if\n$\\rho (x,\\theta)$ satisfies the equation\n\\begin{equation}\n\\Delta^{{\\rm can}} \\sqrt \\rho =0.\\label{eq:sqrt}\n\\end{equation}\nBut this is the master equation of the\nBV-formalism for the action $ S= -i\\frac{1}{2}\\log\\rho $.\n\n  The geometrical meaning of the\nnilpotency condition (and correspondingly of the master equation)\nwill be illustrated on a simple example\nin the next Section.\n\\setcounter{equation}0\n\\section{Example: The operator $\\Delta$ on K\\\"ahlerian Supermanifolds}\n\nAs we saw in the previous section, in contrary to the case of an even\nsymplectic structure, on supermanifolds with an odd symplectic\nstricture there arises\na nontrivial differential geometry.\n\n It is sufficient to show\nthe correspondence between the generalized operator\n$ \\Delta$ and geometrical objects on basic manifolds in the case of a\nK\\\"ahlerian basic manifold,\nbecause\non K\\\"ahlerian manifolds the symplectic structure corresponds to a\nRiemannian one,\nand a Riemmannian structure has a rich differential geometry.\nMoreover, in this case there also  exists\n on ${\\cal M}$ an even K\\\"ahlerian\nstructure, and, using it, we can construct a natural\nintegral density \\cite {ners}, \\cite{knjinr}.\n\nLet ${\\cal M}$ be a\ncomplex supermanifold, and $z^A$ local complex coordinates\non ${\\cal M}$.\nA symplectic structure $\\Omega^{\\kappa}$ -- here and further\n$\\kappa =0(1)$ if\nthe symplectic structure is even (odd) --  on ${\\cal M}$ is\ncalled K\\\"ahlerian,\nif in local coordinates $z^{A}$ it takes the following form:\n\\begin{equation}\n\\Omega^{\\kappa}=i(-1)^{p(A)(p(B)+\\kappa+1)}g^\\kappa_{A {\\bar B}}\n           dz^A \\wedge d{\\bar z}^B,\n\\end{equation}\n      where\n$$  g^\\kappa_{A {\\bar B}}  =\n     (-1)^{(p(A)+\\kappa+1)(p(B)+\\kappa+1)+\\kappa +1}\n         \\overline {g^\\kappa _{B {\\bar A}}},\n           \\quad p(g^\\kappa _{A\\bar B})=p_A +p_B+\\kappa$$\n\nThen there exists a local real even (odd) function\n$K^\\kappa(z,{\\bar z})$\n    (K\\\"ahlerian potential), such that\n\\begin{equation}\n\t        g^\\kappa_{A {\\bar B}}  =\n              \\frac{\\partial ^L}{\\partial z^A}\n\t\t\\frac{\\partial ^R}{\\partial {\\bar z}^B}\n\t\t\t       K^\\kappa  (z,{\\bar z})\n\\end{equation}\n         To  $\\Omega^{\\kappa}$ there corresponds\n    the  Poisson bracket\n   \\begin{equation}\n\t\t     \\{ f,g\\}_\\kappa\n                        =\n\t\t     i\\left(\n\t     \\frac{\\partial ^R f}{\\partial \\bar z^A}\n\t\t\t   g^{{\\bar A}B}_\\kappa\n\t        \\frac{\\partial  ^L g}{\\partial z^B}\n\t\t           -\n            (-1)^{(p(A)+\\kappa)(p(B)+\\kappa)}\n      \t\t \\frac{\\partial  ^R f}{\\partial z^A}\n\t\t     g^{{\\bar A}B}_\\kappa\n\t\t \\frac{\\partial ^L g }{\\partial \\bar z^B}\n\t\t       \\right),\n\\end{equation}\n where\n$$g^{{\\bar A}B}_\\kappa g_{B{\\bar C}}^\\kappa=\n  \\delta^{\\bar A}_{\\bar C} \\;\\;,\\;\\;\\;\\; \\overline{g^{{\\bar A}B}_\\kappa}\n = (-1)^{(p(A)+\\kappa)(p(B)+\\kappa)}g^{{\\bar B}A}_\\kappa.$$\n Its  satisfies the conditions of reality and \"antisymmetricity\"\n       \\begin{equation}\n \\overline{\\{ f, g\\}_\\kappa }\n     =\\{\\bar f,\\bar g \\}_\\kappa,\\;\\;\\;\\{ f, g \\}_\\kappa = -\n(-1)^{(p(f)+\\kappa)(p(g)+\\kappa)}\\{ g, f \\}_\\kappa,\n\\end{equation}\nand the Jacobi identities :\n\\begin{equation}\n( -1)^{(p(f)+\\kappa)(p(h)+\\kappa)}\n\\{ f,\\{ g, h \\}_{\\kappa}\\}_\\kappa +{\\rm {cycl. perm.}}(f,g,h) = 0\n\\end{equation}\nLet ${\\cal M}$ be associated with the tangent bundle $TM$ of the\nK\\\"ahlerian manifold $M$,\nand $z^A = (w^{a}, \\theta^a )$  local coordinates on it,\n$\\theta^a$ transforming from map to map\nlike $dw^a$.\nLet\n\\begin{equation}\ng_{a{\\bar b}}(w,{\\bar w})\n =\\frac{\\partial^{2} K(w,{\\bar w})}\n{{\\partial \\omega^{a}}{\\partial {\\bar\\omega^b}}}\n\\end{equation}\nbe a K\\\"ahlerian metric on $M$, with $K$ its K\\\"ahlerian\npotential \\cite{kobnom}.\nThen  the local functions\n\\begin{eqnarray}\nK_0(w,{\\bar w},\\sigma,{\\bar \\sigma})&=& K(w,{\\bar w})+\nig_{a{\\bar b}}(w,{\\bar w})\\sigma^a{\\bar \\sigma}^b,\n\\quad p( K_0 )=0 \\label{eq:evenpot}\\\\[3mm]\n  K_{1}(w,{\\bar w},\\sigma,{\\bar \\sigma})&=&\n    \\epsilon\\frac{\\partial K(w,{\\bar w})}{\\partial w^a}\\sigma^a+\n  {\\bar \\epsilon}\\frac{\\partial K(w,{\\bar w})}\n   {\\partial {\\bar w^a}}{\\bar \\sigma^a} \\quad p( K_1 )=1\n \\label{eq:oddpot}\n   \\end{eqnarray}\n(where  $\\epsilon$ is\nan arbitrary complex constant) correctly define an even and an\nodd symplectic structures on ${\\cal M}$\n(this is not the most general form of K\\\"ahlerian potentials\non such supermanifolds \\cite {ners})\n\nThe odd K\\\"ahlerian potential (\\ref{eq:oddpot})\ndefines on ${\\cal M}$ the following odd bracket:\n\\begin{equation}\n  \\{ f,g\\}_1 = \\frac{i}{\\epsilon}\\left (\n\t     \\frac{\\partial ^R f}{\\partial \\theta^a} \\nabla^{a}g -\n   \\nabla^{a}f \\frac{\\partial^L g }{\\partial\n\\theta^a } \\right ) +{\\rm c.c}  \\label {eq:oddbk}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\overline {\\nabla^a}}=g^{{\\bar a}b}\\nabla_b \\quad \\nabla_a=\n\\frac{\\partial}{\\partial w^a} -\n        \\Gamma^c_{ab}\\theta^b\\frac{\\partial^L}{\\partial\\theta^c},\n\\end{equation}\nand $\\Gamma^c_{ab}=g^{\\bar d c}g_{a \\bar d,b} $ are\nthe Christoffel symbols of the K\\\"ahlerian metric  on $M$.\n\nIt is easy to see, that in the coordinates\n$(w^a, \\theta_a =i g_{a\\bar b}{\\bar \\sigma}^b )$,\nin which ${\\cal M}$ is associated with $ T^* M$, the\nodd Poisson bracket takes the canonical form.\n\nThe generalized operator $\\Delta$ corresponding to (\\ref{eq:oddbk}) takes the\nform\n    \\begin{equation}\n \\Delta f = \\left(\\frac{1}{\\epsilon}\\nabla^a\n\\frac{\\partial^L}{\\partial\\theta^a} +\n\t\t\\frac{1}{\\bar \\epsilon}\n{\\overline {\\nabla^a}}\\frac{\\partial^L}{\\partial{\\bar \\theta}^a}\\right)f +\n       \\frac{1}{2}\\{\\log \\rho,f \\}_{1}\n\\end{equation}\nIf $\\nabla_{a} \\rho =0$ (or, in fact, if $\\rho$ is a characteristic class\nof $M$) then\n   \\begin{equation}\n\t    \\Delta f = \\frac{1}{\\sqrt \\rho}\n  \\left(\\frac{1}{\\epsilon}\\nabla^a \\frac{\\partial^L}{\\partial\\theta^a} +\n   \\frac{1}{\\bar \\epsilon} {\\overline {\\nabla^a}}\\frac{\\partial^L}\n    {\\partial{\\bar \\theta}^a}\\right)(\\sqrt\\rho f),  \\label{eq:deltan}\n\\end{equation}\nis obviously nilpotent.\n\nThe invariant density which corresponds to (\\ref{eq:evenpot}) is\n    \\begin{equation}\n\t     \\rho=  det(\\delta^a_b+i\n\t      R^a_{bc{\\bar d}}\\theta^c \\bar\\theta^d)\n\\end{equation}\nwhere $ R^a_{bc\\bar d}=(\\Gamma^a_{bc})_{,\\bar d}$ is\nthe curvature tensor on $M$.\nIt is associated with the\ngenerating functions of the Chern classes of the underlying\n K\\\"ahlerian manifold \\cite{kobnom}.\n\nObviously (\\ref {eq:deltan}) corresponds  to the operator\nof covariant divergence $\\delta = \\ast d \\ast$ on $M$ with some effective\nweight.\n\n \\setcounter{equation}0\n\\section { Discussion}\n\nAs we have seen in the previous Sections, the operator $\\Delta$ has\na simple geometrical\nnature on the  supermanifolds associated with the\ncotangent bundles of manifolds.\nObviously, the same construction holds, if we replace the basic\nmanifold $M$ in the previous Sections by some\nsupermanifold ${\\cal M}_{0}$. Then if $x^i$ are local coordinates on\n${\\cal M}_{0}$ ($p( x^i)\\neq 0$), then, on the supermanifold ${\\cal M}$ which\nis associated with the cotangent bundle $T^*{\\cal M}_{0}$,\none can naturally define\nthe odd Poisson bracket (\\ref{eq:bcan})\n where $\\theta_i$ corresponds to coordinates of the bundle, $p(\\theta_i) =\n p( x^i ) +1$.\nThese coordinates are the  analogs of the antifields of the BV-formalism.\n\nBut what is the reason for the introduction of antifields\n(and, correspondingly,\nfor the structure of supermanifolds with an odd symplectic structure)\n in the BV-formalism ?\n\nIn our opinion, this is connected to the\npeculiarity of the integration on supermanifolds.\nIndeed, if we have some differential form $\\omega ( x^i, dx^i )$\non the supermanifold ${\\cal M}_{0}$, its integral over ${\\cal M}_{0}$\ndefined in the following way \\cite{bernleit}, \\cite{voronov}:\n\\begin{equation}\n\\int_{{\\cal M}_{0}}\\omega \\equiv \\int_{\\hat{\\cal M}_{0}}\n\\omega (x^i, \\theta^i ) D(x,\\theta)[dxd\\theta]   \\label{eq:blint}\n\\end{equation}\nwhere $ p(\\theta^{i})=p(x^{i}) + 1$, and $\\hat{\\cal M}_{0}$\ndenotes the supermanifold\nassociated with the tangent bundle of (supermanifold) ${\\cal M}_{0}$\n(i.e. $\\theta^{i}$ transforms\nlike $dx^{i}$), then $D(x,\\theta)$ is\nthe natural density on $\\hat{\\cal M}$ \\cite{bernleit2}.\n\nTransiting from the description on $\\hat{\\cal M}_{0}$ to that on\n${\\cal M} ={\\cal T}_{*}{\\cal M}_{0}$ --\nthe supermanifold associated with the cotangent bundle of\nthe supermanifold ${\\cal M}_{0}$ -- we saw that\nthe integral (4.2) takes the form of the partition function of the\nBV-formalism.\n\nCorrespondingly, the master-equation of the BV-formalism\ncorresponds to the closeness of the\ninitial differential form. This is clearly seen in the case where\nthe  basic manifold,\nis K\\\"ahlerian (in the general case this proposition was\nstrongly proved in \\cite{schwarz}.\nThen the gauge invariance of the partition function in the BV-formalism\nfollows from  Stokes' theorem \\cite{bernleit2}, \\cite{voronov}.\n\n\\section {Acknowledgments}\nWe are very indebted to I. A. Batalin for valuable discussions.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{intro}\n\nRecent discovery of charge density waves (CDW) in hole-doped cuprates has raised a new wave of interest to the physics of these materials. The CDWs have been detected in underdoped samples of YBCO \\cit\n{y123REXS-3,y123REXS-4,y123REXS-5,y123REXS-6,y123XRD-3,y123XRD-4\n, Bi-2201 \\cite{bi2201STM-1,bi2201STM-2}, Bi-2212 \\cit\n{Bi2212REXS-2,Bi2212STM-2,Bi2212STM-3}\nand Hg-1201 \\cite{HgREXS,HgXRD} by direct methods such as resonant X-ray\nscattering, hard X-ray diffraction and scanning tunneling microscopy(STM).\nA wide variety of techniques, that can be sensitive to a CDW indirectly, also confirm its presence, among them transport measurements \\cite{y123Transp,HgTransp}, nuclear magnetic resonance \\cite{y123NMR-2}, ultrasound propagation \\cite{y123US} and pump-probe experiments \\cite{y123Refl}.\n\n\nSeveral common properties of this CDW state in hole-doped cuprates have been identified. The transition temperature $T_{CDW}$ is higher than $T_c$ but lower or equal to the pseudogap temperature $T^*$. The temperature and magnetic filed dependence of the CDW amplitude({\\it e.g.} \\cite{y123XRD-3}) is consistent with the CDW state competing with the superconductivity.\n\nThe CDW wave vectors seen in the experiments \\cite{y123REXS-5,y123XRD-3,HgXRD,Bi2212STM-0} are directed along the $Cu-O-Cu$ bonds of the $CuO_2$ plane (axes of the Brillouin zone, axial CDW). The CDW period is approximately equal along both the axes and increases with doping \\cite{y123XRD-3,bi2201STM-1,Bi2212REXS-2}.\n\n\nRecent studies have also revealed important information about the distribution of the modulated charge inside the unit cell, {\\it i.e.} the CDW form factor. It has been found for Bi-2212 \\cite{Bi2212STM-2} and YBCO \\cite{y123REXS-5} that\nthe charge is modulated approximately in antiphase at two oxygen sites of the unit cell with the charge at $Cu$ site being constant. In other words, the CDW form factor is characterized by a dominant d- component. The properties mentioned up to now are quite different from the stripe state of the La-based compounds \\cite{StripeRev,y123REXS-6}.\n\n\nConsiderable attention has been also drawn to the nanoscale structure of the CDW. Quantum resistance oscillation experiments \\cite{y123QO-1,y123QO-2,HgQO} have been interpreted \\cite{seb1} as being due to a checkerboard modulation, where CDWs with two orientations uniformly coexist throughout the sample. Results of studies \\cite{Bi2212STM-0,y123REXS-4} suggest, however, show that the charge ordered state consists of domains where CDW is unidirectional.\n\n\nThere have been a number of attempts to obtain the CDW state with the properties discussed above from microscopic calculations. In a model of fermions interacting with antiferromagnetic critical spin fluctuations \\cite{SFREV} (spin fermion (SF) model) a charge order appears in perturbation theory as a subleading instability \\cite{MetSach} hindered by the curvature of the Fermi surface. This order is a checkerboard CDW with d-form factor and wave vectors directed along the diagonals of the BZ \\cite{Efetov2013,Sach2013}. The nearest-neighbor Coulomb interaction can, in principle, make this state leading as has been shown in Ref. \\cite{SachSau}. Moreover, thermal fluctuations between this charge order and SC have been shown to be able to destroy both the orders while pertaining a single-particle gap \\cite{Efetov2013}, which can explain the pseudogap phase. Qualitative aspects of CDW-SC competition are also well-captured in the SF model: moderate magnetic fields suppressing the superconductivity have been shown to favor CDW \\cite{MEPE} resembling the experiment \\cite{y123US}. The vortex cores in the SC state have been shown to contain CDW \\cite{EMPE} which is seen in STM \\cite{hoffman,hamidian}. The diagonal direction of modulation wavevectors contrasting the experiments, however, has proved to be quite robust.\n\n\nSome proposals have been put forward to overcome this contradiction. A CDW with the correct wavevector direction has been\nobtained in Refs. \\cite{cascade,WangChub2014}, however the form factor has been found to lack the dominant d-symmetry with a large s- component. In Ref. \\cite{pepin} a mixture of the states proposed in Ref. \\cite{Efetov2013} and Ref. \\cite{WangChub2014} has been considered, which should contain either the diagonal modulation or an axial CDW with a non d-form factor. CDW considerations using other models \\cite{Punk2015,DavisDHLee2013,Kampf2013,yamakawa2015,Kampf2014,ChowdSach2014,ThomSach}also do not seem to explain the robustness of the axial d-form factor CDW in the cuprates.\n\nIn this contribution we review the treatment of the SF model allowing the neighboring hot spots to overlap, such that eight hot spots merge into two hot regions entirely covering the antinodal portions of the Fermi\nsurface. This corresponds to sufficiently small values $|\\varepsilon\n(\\pi ,0)-E_{F}|\\lesssim \\Gamma $, where $E_{F}$ is the Fermi energy, $\\varepsilon \\left( \\pi ,0\\right) $ is the energy in the middle of the Brillouin zone edge, and $\\Gamma $ is a characteristic energy of the\nfermion-fermion interaction due to the antiferromagnetic fluctuations.\nConsideration of this limit is motivated by ARPES data \\cite{Bi2201ARPES-1,Bi2201ARPES-2,Bi2212ARPES-1,Bi2212ARPES-2} showing that the energy separation between the hot spots and $(\\pi ,0);(0,\\pi )$ is actually\nquite small. In addition to the electron-electron interaction via\nparamagnons, we consider also the effects of low-energy (low-momentum) part of\nthe Coulomb interaction, which should not contradict the philosophy of the\nlow energy SF model. A detailed derivation and discussion of the results can\nbe found in our paper \\cite{preprint}.\n\n\\section{Model and main equations}\n\\label{sec1}\n\nWe consider a single band of fermions interacting through critical antiferromagnetic (AF) fluctuations (paramagnons) represented by a spinful bosonic field as well as the Coulomb force. As the AF fluctuations peak at momentum transfer $(\\pi,\\pi)$ we restrict our model to two regions of the Fermi surface connected with this vector represented in Fig. \\ref{fig1}. Inside these regions we do not specify individual hot spots, {\\it i.e.} points on the FS connected by $(\\pi,\\pi)$ as we assume the interaction to be important in all the whole region. This assumption is supported by ARPES experiments \\cite{Bi2201ARPES-1,Bi2201ARPES-2,Bi2212ARPES-1,Bi2212ARPES-2} showing that $|\\varepsilon (\\pi ,0)-E_{F}|$ is actually smaller than the pseudogap energy, which can be taken as the interaction scale.\n\\begin{figure}[tbp]\n\\includegraphics[width=0.5\\linewidth]{1.eps}\n\\centering\n\\caption{A typical cuprate Fermi Surface with two regions connected\nby the antiferromagnetic wavevector $(\\pi,\\pi)$ }\n\\label{fig1}\n\\end{figure}\n\nThe fermion-paramagnon part of the Lagrangian takes the form:\n\\begin{equation}\n\\begin{split}\nL_{\\mathrm{SF}} =\\sum_{\\mathbf{p,}\\nu =1,2} \\chi\n_{\\mathbf{p}}^{\\nu \\dagger } \\left[\\partial _{\\tau }+\\varepsilon _{\\nu}\\left(\\mathbf{p}\\right) -\\mu_0 \\right] \\chi _{\\mathbf{p}}^{\\nu }+\n\\\\\n+\\sum_{q}\\vec{\\varphi}_{-\\mathbf{q}}(-v_{s}^{-2}\\partial _{\\tau }^{2}+\n\\mathbf{q}^{2}+\\xi^{-2})\\vec{\\varphi}_{\\mathbf{q}} \\\\ +\\lambda\n^{2}\\sum_{\\mathbf{p,q}}\\left[ \\chi _{\\mathbf{p+q}}^{1\\dagger }\n\\vec{\\varphi}_{\\mathbf{q}}\\vec{\\sigma}\\chi _{\\mathbf{p}}^{2}+\\chi _{\\mathbf{\np+q}}^{2\\dagger }\\vec{\\varphi}_{\\mathbf{q}}\\vec{\\sigma}\\chi\n_{\\mathbf{p}}^{1} \\right].\n\\end{split}\n\\label{sf_h}\n\\end{equation}\nwhere $\\varepsilon _{\\nu}(\\mathbf{p})$ is the electron dispersion in region $\\nu=1,\\;2$ (including the chemical potential), $v_s$ is the velocity of spin waves and $\\xi$ is the magnetic correlation length. We shall not write explicitly the terms corresponding the Coulomb interaction as we will take their effect into account qualitatively.\n\nAssuming that the regions 1 and 2 occupy a small portion of the BZ we expand $\\varepsilon _{1(2)}(\\mathbf{p})$ around $[\\pi,0]([0,\\pi])$ resulting in $\\varepsilon _{p}^{1}=\\alpha p_{x}^{2}-\\beta p_{y}^{2}-\\mu_0,\\;\\varepsilon_{p}^{2}=\\alpha p_{y}^{2}-\\beta p_{x}^{2}-\\mu_0$, where $\\mu _{0}$ is the chemical potential counted from $\\varepsilon (\\pi ,0)=\\varepsilon (0,\\pi )$. Moreover, we will average the curvature term (the one with $\\beta$) inside each region leading to the final form:\n\\begin{equation}\n\\varepsilon _{p}^{1}=\\alpha p_{x}^{2}-\\mu ,\\quad \\varepsilon _{p}^{2}=\\alpha\np_{y}^{2}-\\mu ,  \\label{mod_disp}\n\\end{equation}\nwhere $\\mu =\\mu _{0}+\\langle \\beta p_{\\parallel }^{2}\\rangle $. To study particle-hole instabilities we define the order parameter:\n\\begin{equation}\nW_{\\mathbf{Q}}(\\tau-\\tau',\\mathbf{k})=\\langle \\chi_{\\mathbf{k}-\\mathbf{Q}\/2,\\sigma}^{\\dagger}(\\tau')\n\\chi_{\\mathbf{k}+\\mathbf{Q}\/2,\\sigma }(\\tau)\\rangle  \\label{mod_op}\n\\end{equation}\nAs has been shown in \\cite{preprint} this order parameter is related to density modulations at the three atoms of the unit cell in the following way:\n\\begin{equation}\n\\begin{split}\n\\delta n_{Cu}(\\mathbf{r}) =2e^{i\\mathbf{Q}\\mathbf{r}}\\sum_{\\mathbf{k}}W_{\\mathbf{Q}}(0,\\mathbf{k})+c.c,  \\\\\n\\delta n_{O_{x}}(\\mathbf{r}) =\\frac{p}{4}e^{i\\mathbf{Q}\\mathbf{r}}\\sum_{\\mathbf{k}}\\cos(k_{x}a_{0})W_{\\mathbf{Q}}(0,\\mathbf{k})+c.c.,  \\\\\n\\delta n_{O_{y}}(\\mathbf{r}) =\\frac{p}{4}e^{i\\mathbf{Q}\\mathbf{r}}\\sum_\n\\mathbf{k}}\\cos (k_{y}a_{0})W_{\\mathbf{Q}}(0,\\mathbf{k})+c.c.\n\\end{split}\n\\label{mod_dens}\n\\end{equation}\nAs both the regions we consider yield approximately $\\cos (k_{x}a_{0})+\\cos (k_{y}a_{0})\\approx 0$ we have $\\delta n_{O_{x}}(\\mathbf{r}) + \\delta n_{O_{y}}(\\mathbf{r})\\approx 0$ in or model, {i.e.} charge is modulated in antiphase at the two oxygen sites of the unit cell.\n\n\nNow we can discuss the qualitative effects of the Coulomb interaction in the $CuO_2$ plane. The strong on-site repulsion prohibits any real charge modulations on the $Cu$ sites leading to the constraint: $\\delta n_{Cu}=0$ for the order parameter. Together with $\\delta n_{O_{x}}(\\mathbf{r}) + \\delta n_{O_{y}}(\\mathbf{r})\\approx 0$ discussed above this leads to the conclusion that the charge modulations obtained in our model will have the d-form factor in accord with the experiments \\cite{Bi2212STM-2,y123REXS-5}.\n\nThe nearest-neighbor Coulomb interaction has been shown in \\cite{SachSau} to suppress superconductivity and support charge ordering, explaining $T_{CDW}>T_c$. This allows one to consider the particle-hole channel of the model separately from the particle-particle one.\n\n\n\\section{Pomeranchuk instability and intra-cell charge modulation.}\nOur main finding is that for sufficiently small $\\mu$ the leading particle-hole instability is the one with ${\\bf Q}=0$. The ordered state is then characterized not by a CDW, but rather a deformation of the FS (this type of transition is known as Pomeranchuk instability \\cite{pomeranchuk}, \\cite{yamase2005}). Moreover, it follows from (\\ref{mod_dens}) that such a deformation leads to a redistribution of charge between the oxygen sites of the unit cell (see Fig.\\ref{figpom}).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{6.eps}\n\\caption{Pictorial representation of two possible shapes of the Fermi\nsurface below the Pomeranchuk transition and the corresponding\nintra-unit-cell charge redistributions. Grey arrows mark the emergent nesting vectors for regions $1$ and $2$.}\n\\label{figpom}\n\\end{figure}\n\nOne can obtain this result analytically for a simplified BCS-like model where the paramagnon propagator is replaced by a constant. For that case a mean-field analysis yields that if $\\mu\/T_{Pom}\\leq1.1$ then it is the leading instability. $T_{Pom}$ is given in this case by $\\frac{1}{2\\alpha }\\left( \\frac{\\lambda _{0}\\Lambda }{4\\pi ^{2}}\\right) ^{2}$ where $\\lambda_0$ is the dimensionless coupling constant and $\\Lambda$ is the size of a single region in the momentum space. This expression contrasts the usual exponential dependence obtained in BCS-like theories. A detailed account on this simplified case is presented in \\cite{preprint}.\n\n\nNow let us turn to the model presented here. As a starting approximation we will use the self-consistent equations represented by diagrams in Fig. \\ref{figFeyn}\n\\begin{figure}[h]\n\\includegraphics[trim = 100 150 0 0,width=0.8\\linewidth]{7.eps}\n\\caption{Feynman diagrams for fermionic and bosonic propagators illustrating\nthe approximations used.}\n\\label{figFeyn}\n\\end{figure}\nThe integral over momentum in the fermionic self-energy can be greatly simplified provided $\\mu v_{s}^{2}\/\\alpha \\ll (v_{s}\/\\xi )^{2}$, i.e. that the correlation length is not too large. Then the self-energy and polarization operator do not depend on the momentum. To analyze the FS deformation we distinguish the 'even' $(\\Sigma_1+\\Sigma_2)\/2\\equiv i\\varepsilon _{n}-if(\\varepsilon _{n})$ and 'odd' $(\\Sigma_1-\\Sigma_2)\/2\\equiv P$ contributions to self-energy, with the latter being zero in the normal state. After the momentum integration one can introduce an energy scale $\\Gamma =\\left( \\frac{\\lambda ^{2}v_{s}}{\\sqrt{\\alpha }\\hbar ^{2}}\\right)^{2\/3}$ and write the self-consistency equations in the dimensionless form (see Eq. \\ref{sf_Pom}), where $\\bar{a}$ denotes $\\overline{(v_s\/\\xi)}$.\n\\begin{figure*}\n\\begin{centering}\n\\begin{equation}\n\\begin{gathered}\n\\bar{f}(\\bar{\\varepsilon}_{n})-\\bar{\\varepsilon}_{n}=\n0.75\\bar{T}\\sum_{\\bar{\\varepsilon}_{n}^{\\prime }}\n\\frac{1}{\\sqrt{\\bar{\\Omega}(\\bar{\\varepsilon}_{n}-\\bar{\\varepsilon}_{n}^{\\prime})+\\bar{a}}}\n\\frac{\\mathrm{sgn}(\\mathrm{Re}[f(\\bar{\\varepsilon}_{n}^{\\prime })])}{2}\n\\left[\n\\frac{1}{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}+\\bar{P}(\\bar{\\varepsilon}_{n}^{\\prime })}}\n\\frac{1}{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}-\\bar{P}(\\bar\n\\varepsilon}_{n}^{\\prime })}}\n\\right]\n\\\\\n\\bar{P}(\\bar{\\varepsilon}_{n})=i\\cdot 0.75\\bar{T}\\sum_{\\bar{\\varepsilon\n_{n}^{\\prime }}\\frac{1}{\\sqrt{\\bar{\\Omega}(\\bar{\\varepsilon}_{n}-\\bar\n\\varepsilon}_{n}^{\\prime })+\\bar{a}}}\\frac{\\mathrm{sgn}(\\mathrm{Re}[f(\\bar\n\\varepsilon}_{n}^{\\prime })])}{2}\\left[ \\frac{1}{\\sqrt{i\\bar{f}(\\bar\n\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}-\\bar{P}(\\bar{\\varepsilon}_{n}^{\\prime })}}\n\\frac{1}{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}+\\bar{P}(\\bar\n\\varepsilon}_{n}^{\\prime })}}\\right]\n\\\\\n\\bar{\\Omega}(\\bar{\\omega}_{n})-\\bar{\\omega}_{n}^{2}=-\\frac{\\bar{T}}{2}\\sqrt{\\frac{v_{s}^{2}\/\\alpha }{\\Gamma }}\n\\sum_{\\bar{\\varepsilon}_{n}}\n\\left[\n\\frac\n{\\mathrm{sgn}(\\mathrm{Re}[f(\\varepsilon_{n})])}\n{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_n)+\\bar{\\mu}+\\bar{P}(\\bar{\\varepsilon}_n)}}\n\\frac\n{\\mathrm{sgn}(\\mathrm{Re}[f(\\varepsilon _{n}+\\omega _{n})])}\n{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}+\\omega _{n})+\\bar{\\mu}-\\bar{P}(\\bar{\\varepsilon}_{n}+\\omega _{n})}}\n+\n\\right.\n\\\\\n\\left.\n\\frac\n{\\mathrm{sgn}(\\mathrm{Re}[f(\\varepsilon_{n})])}\n{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n})+\\bar{\\mu}-\\bar{P}(\\bar{\\varepsilon}_n)}}\n\\frac\n{\\mathrm{sgn}(\\mathrm{Re}[f(\\varepsilon _{n}+\\omega _{n})])}\n{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}+\\omega _{n})+\\bar{\\mu}+\\bar{P}(\\bar{\\varepsilon}_{n}+\\omega _{n})}}\n\\right]\n.\n\\end{gathered}\n\\label{sf_Pom}\n\\end{equation}\n\\end{centering}\n\\end{figure*}\nNote that the polarization operator $\\bar{\\Omega}(\\bar{\\omega}_{n})-\\bar{\\omega}_{n}^{2}$ contains a factor $\\sqrt{v_{s}^{2}\/\\alpha\\Gamma}$ absent in the fermionic self-energy part. This factor will also arise if one calculates the vertex correction, as there one has to integrate a product of fermionic Green's functions like in the polarization operator. This allows us to use $\\sqrt{v_{s}^{2}\/\\alpha\\Gamma}$ as a small parameter to justify the Eliashberg-like approximation given by Fig. \\ref{figFeyn}. We shall not neglect, however, the polarization operator, as it behaves linearly at low frequencies and might outpower the initial quadratic dispersion.\n\n\n\nThe equations (\\ref{sf_Pom}) have been numerically solved by an iteration scheme, yielding the transition temperature $T_{Pom}$ where the 'odd' self-energy $P$ becomes non-zero. To show that this transition can be indeed leading we have also computed the transition temperature for a CDW with wavevector along the BZ diagonal. This instability has been found to be universally leading in previous studies. The transition temperature can be found from the linearized equation for the CDW order parameter $W_{diag}(\\varepsilon_{n})$:\n\\begin{equation}\n\\begin{gathered}\n\\bar{W}_{diag}(\\bar{\\varepsilon}_{n})\n=\\frac{0.75\\bar{T}}{2}\\sum_{\\varepsilon _{n}^{\\prime\n}}\\frac{\\bar{W}_{diag}(\\bar{\\varepsilon}_{n}^{\\prime\n})}{\\sqrt{\\bar{\\Omega}(\\bar{\\varepsilon}_{n}-\\bar{\\varepsilon _{n}^{\\prime\n}})+\\bar{a}}}\n\\\\\n\\times \\frac{\\mathrm{sgn}\\left(\n\\mathrm{Re}[f(\\bar{\\varepsilon}_{n}^{\\prime })]\\right)\n}{\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime\n})\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}}}.\n\\end{gathered}\n\\label{sf_tdiag}\n\\end{equation}\nThe results of the numerical solutions are presented in Fig.\\ref{figpomdiag}.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=\\linewidth]{9.eps}\n\\caption{$\\bar{T}_{Pom}(\\bar{\\protect\\mu})$ (dashed line) and $\\bar{T}_{diag}(\\bar{\\protec\n\\mu})$ (dotted line) for $\\overline{(v_s\/\\xi)}=0.1$, $\\sqrt{\\frac{v_{s}^{2}\/\\alpha }\n\\Gamma }}=0.5(a),\\;0.1(b)$.}\n\\label{figpomdiag}\n\\end{figure}\nOne can clearly see that for $\\bar{\\mu}$ less than a certain value Pomeranchuk instability is the leading one. Note that the ratio $\\mu\/T_{Pom}$ can be as high as $12$ for $\\sqrt{v_{s}^{2}\/\\alpha\\Gamma}=0.5$ and $9$ for $\\sqrt{v_{s}^{2}\/\\alpha\\Gamma}=0.1$.\n\nAs the Fermi Surface seen in ARPES experiments is universally found to be $C_4$-symmetric and in the light of the domained CDW structure \\cite{Bi2212STM-3,y123REXS-4}, we assume that Pomeranchuk order should also be organized in domains with different sign of the order parameter. This constitutes a way of 'masking' a $C_4$ breaking alternative to the one proposed in \\cite{yamase2009}.\n\\section{Incommensurate charge modulation.}\nThe deformed Fermi surface of Fig.\\ref{figpom} can be unstable to CDW formation at lower temperatures. The direction and the magnitude of the wavevector are directly related to the sign and the magnitude of $P$. We assume that the CDW wavevector should yield nesting in the region where the FS 'expands' due to the FS deformation. Then one has:\n\\begin{equation}\nQ^{SF}(T)=2\\sqrt{(\\mu +0.5\\left\\vert P(-\\pi T)+P(\\pi T)\\right\\vert )\/\\alpha }.  \\label{sf_q}\n\\end{equation}\nIn our model the FS in the second region 'closes' moving out of the considered region for $P>\\mu$. However, as is seen from Fig.\\ref{figpom}, in reality such a deformation can lead to emergent nesting in this region with the same vector direction as in the first one. The best-case scenario is that the nesting vectors in both regions coincide also in magnitude, {\\it i.e.} $Q_1=Q_2$ (see Fig.\\ref{figpom}). We shall assume that this is indeed the case, thus providing an upper limit on the $T_{CDW}$. In this case the equation for the CDW transition is:\n\\begin{eqnarray}\n&&\\bar{W}(\\bar{\\varepsilon}_{n})=0.75\\;i\\frac{\\bar{T}_{CDW}}{2\n\\sum_{\\varepsilon _{n}^{\\prime }}\\frac{\\bar{W}(\\bar{\\varepsilon}_{n}^{\\prime\n})}{\\sqrt{\\bar{\\Omega}(\\bar{\\varepsilon}_{n}-\\bar{\\varepsilon _{n}^{\\prime }\n)+\\bar{a}}}  \\notag \\\\\n&&\\times \\frac{\\mathrm{sgn}(\\mathrm{Re}[f(\\bar{\\varepsilon}_{n}^{\\prime })]\n}{(\\left[ i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{P}(\\bar{\\varepsilon\n_{n}^{\\prime })-P(0)\\right] g\\left( \\bar{\\varepsilon}^{\\prime }\\right) }.\n\\label{c11}\n\\end{eqnarray\nThe results of numerical calculations are presented in Fig.\\ref{figCDW}. It turns out that the CDW transition can closely follow the onset of the FS deformation.\n\\begin{figure}[tbp]\n\\includegraphics[width=\\linewidth]{12.eps}\n\\caption{$T_{Pom}(\\bar{\\protect\\mu})$ (dashed line) and $\\bar{T}_{CDW}(\\bar\n\\protect\\mu})$ (dotted line) determined from Eq. (\\protect\\ref{c11}) for $\\overline{(v_s\/\\xi)}=0.1$, $\\protect\\sqrt{\\frac{v_{s}^{2}\/\\protect\\alpha }{\\Gamma }\n=0.5(a),\\;0.1(b)$.}\n\\label{figCDW}\n\\end{figure}\n\n\\section{Comparison with experiments and conclusions.}\n\n\\label{sec4}\n\nMotivated by the existing ARPES data \\cit\n{Bi2201ARPES-1,Bi2201ARPES-2,Bi2212ARPES-1} we have considered the SF model\nwith overlapping hotspots and demonstrated that the d-wave Fermi\nsurface distortion can be the leading instability. The\ntransition is further followed at a lower temperature by a transition into\na state with a d-form factor CDW directed along one of the BZ axes. The corresponding transition temperatures $T_{Pom}$ and $T_{CDW}$ can be not far away from each other.\n\nThe results obtained allow us to draw the following qualitative picture of\nthe charge order formation:\n\n$\\bullet $ At $T_{Pom}\\geq T^{\\ast }$ $C_{4}$ symmetry is broken by a Pomeranchuk\ntransition. The Fermi surface is deformed(see Fig. \\ref{figpom}) and doped holes\nare redistributed between the oxygen orbitals of the unit cell. The sample consists\nof domains with different signs of the order parameter corresponding to two alternatives presented in Fig.\\ref{figpom}.\n\n$\\bullet $ At $T_{CDW}<T_{Pom}$ a uniaxial d-form factor CDW forms in each domain.  CDW wave vector\nis along one of the BZ axes dependinig on the sign of the Pomeranchuk order parameter inside each domain (see Fig.\\ref{figpom}). The CDW period generally exceeds the one corresponding to antinodal nesting and is determined\nself-consistently by the interaction and parameters of the Fermi surface. Qualitatively, the CDW wavevector tracks the FS and should decrease with hole doping (thus the CDW period should increase).\n\n\nOur findings help us in understanding the results of recent experiments. The\nPomeranchuk deformation explains well why the C$_{4}$-symmetry at\ncommensurate peaks in Fourier transformed STM data \\cite{Bi2212STM-0}. Formation of domains with different directions of the C$_{4}$-symmetry breaking is seen in STM experiments \\cite{Bi2212STM-3} and can\nalso help explaining results of the transport measurements in YBCO \\cit\n{y123Transp-2}. It is important to note, though, that the effects of the\ndeformation of the Fermi surface on transport can be masked by existence of\nthe domains. This may also resolve the apparent contradiction to the ARPES\ndata \\cite{Bi2201ARPES-1,Bi2212ARPES-1} always showing a C$_{4}$-symmetric\nFermi surface.\n\nThe most important aspect of the Pomeranchuk order is that it explains the\nrobustness of the axial d-form factor CDW in the cuprates. We also note that\nthe organization of the CDW phase in the unidirectional domains is indeed\nseen in STM \\cite{Bi2212STM-3} and XRD \\cite{y123REXS-4,HgXRD} measurements.\nThe coexistence of the unidirectional CDW and Pomeranchuk order also allows one to resolve\na seeming contradiction to results obtained in experiments on quantum\noscillations \\cite{seb1}. Although the unidirectional CDW leads\nto an open Fermi surface that does not support quantum oscillations, the\nsimultaneous presence of a C$_{4}$-symmetry breaking can indeed \\cit\n{kivelson2011} close the Fermi surface leading to quantum oscillations in\nhigh magnetic fields.\n\n\\begin{acknowledgements}\nThe authors gratefully acknowledge the financial support of the Ministry of\nEducation and Science of the Russian Federation in the framework of Increase\nCompetitiveness Program of NUST~\\textquotedblleft MISiS\\textquotedblright\\\n(Nr.~K2-2014-015).\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\nThis paper is devoted to the question of building an approximate sparse solution of a given convex optimization problem.\nA typical setting is to find an approximate minimum of a real-valued convex function $E$ defined on the Banach space $X$, i.e.\n\\begin{equation}\\label{eq:opt}\n\t\\text{find}\\ \\ x^* = \\mathop{\\operatorname{argmin}}_{x \\in X} E(x).\n\\end{equation}\nWhen optimization is performed over the whole space $X$, it is called an {\\it unconstrained optimization problem}.\nUsually in practice it is desirable to obtain a minimizer that possesses a certain structure or belongs to a given domain $S \\subset X$, in which case problem~\\eqref{eq:opt} becomes a {\\it constrained optimization problem}.\n\nIn particular, it is often preferable that the constructed solution $x^*$ is sparse with respect to a given set of elements $\\mathcal{D} \\subset X$.\nA conventional approach to such a task is to impose an additional $\\ell_1$-regularization on the original problem (see e.g.~\\cite{FNW}) and, instead of~\\eqref{eq:opt}, solve the problem\n\\begin{equation}\\label{eq:opt_reg}\n\t\\text{find}\\ \\ x^* = \\mathop{\\operatorname{argmin}}_{x \\in X} \\big( E(x) + \\lambda \\|x\\|_\\mathcal{D} \\big),\n\\end{equation}\nwhere $\\lambda > 0$ is an appropriate regularization parameter and $\\|\\cdot\\|_\\mathcal{D}$ is the atomic norm with respect to the set $\\mathcal{D}$ (see e.g.~\\cite{chandrasekaran2012convex}), i.e.\n\\begin{equation}\\label{eq:D_norm}\n\t\\|x\\|_\\mathcal{D} := \\inf \\left\\{ \\sum_{g \\in \\mathcal{D}} |c_g| : x = \\sum_{g \\in \\mathcal{D}} c_g \\, g \\right\\}.\n\\end{equation}\nWhile such an approach is quite popular and has its uses, it might not always result in the most appropriate solution since it essentially changes the target function in order to promote sparsity of the solution.\n\nAnother way of obtaining a sparse minimizer (without changing the optimization problem) is to procedurally construct a sequence of minimizers with an increasing support, or, more generally, to design an algorithm that after $m$ iterations provides a point $x_m$ such that $E(x_m)$ is close to the $\\inf_{x \\in S} E(x)$ and that $x_m$ is $m$-sparse with respect to $\\mathcal{D}$, i.e.\n\\[\n\tx_m = \\sum_{j=1}^m c_j \\, g_j\n\t\\ \\ \\text{with}\\ \\ \n\tg_1, \\ldots, g_m \\in \\mathcal{D}\n\t\\ \\ \\text{and}\\ \\ \n\tc_1, \\ldots, c_m \\in \\mathbb{R}.\n\\]\nA wide class of algorithms that fit such requirements is the greedy algorithms in approximation theory, see e.g. \\cite{D}, \\cite{VTbook}.\nA typical problem of greedy approximation is the following.\nLet $X$ be a Banach space with the norm $\\|\\cdot\\|$ and let $\\mathcal{D}$ be a dictionary, i.e. a dense set of semi-normalized elements of $X$.\nThe goal of a greedy algorithm is to obtain a sparse (with respect to the dictionary $\\mathcal{D}$) approximation of a given element $f \\in X$.\nGreedy algorithms are iterative by design and generally after $m$ iterations a greedy algorithm constructs an $m$-term linear combination with respect to $\\mathcal{D}$ that approximates the element $f$.\n\nIt is easy to reframe a greedy approximation problem as a convex optimization problem.\nIndeed, for a given dictionary $\\mathcal{D}$ consider the set of all $m$-term linear combinations with respect to $\\mathcal{D}$ ($m$-sparse with respect to $\\mathcal{D}$ elements):\n\\[\n\t\\Sigma_m(\\mathcal{D}) := \\left\\{ x \\in X: x = \\sum_{j=1}^m c_j \\, g_j, \\ \\ g_1, \\ldots, g_m \\in \\mathcal{D} \\right\\}.\n\\]\nGreedy algorithms in approximation theory are designed to provide a simple way to build good approximants of $f$ from $\\Sigma_m(\\mathcal{D})$, hence the problem of greedy approximation is the following:\n\\begin{equation}\\label{eq:opt_ga}\n\t\\text{find}\\ \\ x_m = \\mathop{\\operatorname{argmin}}_{x \\in \\Sigma_m} \\|f - x\\|.\n\\end{equation}\nClearly, problem~\\eqref{eq:opt_ga} is a constrained optimization problem of the real-valued convex function $E(x) := \\|f - x\\|$ over the manifold $\\Sigma_m(\\mathcal{D}) \\subset X$.\n\nAt first glance the settings of approximation and optimization problems appear to be very different since in approximation theory our task is to find a sparse approximation of a given element $f \\in X$, while in optimization theory we want to find an approximate sparse minimizer of a given target function $E : X \\to \\mathbb{R}$ (for instance, energy function or loss function).\nHowever it is now well understood that similar techniques can be used for solving both problems.\nNamely, it was shown in~\\cite{VT140} and in follow up papers (see, for instance, \\cite{DT}, \\cite{GP}, \\cite{NP}, \\cite{VT141}, and~\\cite{VT148}) how methods developed in nonlinear approximation theory (greedy approximation techniques in particular) can be adjusted to find an approximate sparse (with respect to a given dictionary $\\mathcal{D}$) solution to the optimization problem~\\eqref{eq:opt}.\nMoreover, there is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms, for example, \\cite{BD1}, \\cite{BD2}, \\cite{chandrasekaran2012convex}, \\cite{Cl}, \\cite{Ja2}, \\cite{JS}, \\cite{SSZ}, \\cite{TRD}, and~\\cite{Z}.\n\nWith an established framework it is straightforward to adjust a greedy strategy to a context of convex optimization; however, each of these modified techniques requires an individual analysis to guarantee a desirable performance.\nOn the other hand, it is known that the behavior of a greedy method is largely determined by the underlying geometry of the problem setting.\nIn particular, in~\\cite{VT165} we present a unified way of analyzing different greedy-type algorithms in Banach spaces.\nSpecifically, we define the class of Weak Biorthogonal Greedy Algorithms ($\\mathcal{WBGA}$) and prove convergence and rate of convergence results for algorithms from this class.\nSuch an approach allows for a simultaneous analysis of a wide range of seemingly different greedy algorithms based on the smoothness characteristic of the problem.\n\nIn this paper we adopt the approach of unified analysis for the setting of convex minimization.\nIn Section~\\ref{sec:wbga} we adjust the class $\\mathcal{WBGA}$ of algorithms designed for greedy approximation in Banach spaces and derive the class of Weak Biorthogonal Greedy Algorithms for convex optimization ($\\mathcal{WBGA}$(co)), which consists of greedy algorithms designed for convex optimization.\nWe prove convergence and rate of convergence results for algorithms from the class $\\mathcal{WBGA}$(co) in Theorems~\\ref{thm:wbga_conv} and~\\ref{thm:wbga_rate} respectively.\nThus, results in Section~\\ref{sec:wbga} address two important characteristics of an algorithm~--- convergence and rate of convergence.\n\nThe rate of convergence is an essential characteristic of an algorithm, though in certain practical applications resistance to various perturbations might be of equal importance.\nA systematic study of the stability of greedy algorithms in Banach spaces was started in~\\cite{T7} and further advanced in~\\cite{De}, where necessary and sufficient conditions for the convergence of a certain algorithm were obtained.\nA transition to the optimization setting was performed in~\\cite{DT} and~\\cite{VT148}, where stability results for greedy-type algorithms for convex optimization were obtained.\nIn Section~\\ref{sec:awbga} we discuss the stability of the algorithms from the $\\mathcal{WBGA}$(co) by analyzing convergence properties of the algorithms from $\\mathcal{WBGA}$(co) under the assumption of imprecise calculations in the steps of the algorithms.\nWe call such algorithms {\\it approximate greedy algorithms} or {\\it algorithms with errors}.\nWe prove convergence and rate of convergence results for the Weak Biorthogonal Greedy Algorithms with errors, which describes the stability of the algorithms from the class $\\mathcal{WBGA}$(co)~--- an important characteristic that is crucial for practical implementation.\n\nSince theoretical analysis cannot always predict the practical behavior of an algorithm, it is of interest to observe its actual implementation for particular problems.\nIn Section~\\ref{sec:numerics} we demonstrate the performance of some algorithms from the class $\\mathcal{WBGA}$(co) by employing them to solve various minimization problems.\nAdditionally, we compare these algorithms with a conventional method of obtaining sparse minimizers~--- optimization with $\\ell_1$-regularization~\\eqref{eq:opt_reg}.\nLastly, in Sections~\\ref{sec:proofs_wbga} and~\\ref{sec:proofs_awbga} we prove the results stated in Sections~\\ref{sec:wbga} and~\\ref{sec:awbga} respectively.\n\n\n\n\\section{Weak Biorthogonal Greedy Algorithms for Convex Optimization}\\label{sec:wbga}\nIn this section we introduce and discuss the class of Weak Biorthogonal Greedy Algorithms for convex optimization, denoted as $\\mathcal{WBGA}$(co).\nWe begin by recalling the relevant terminology.\n\n\\subsection{Preliminaries}\nLet $X$ be a real Banach space with the norm $\\|\\cdot\\|$.\nWe say that a set of elements $\\mathcal{D}$ from $X$ is a dictionary if each $g \\in \\mathcal{D}$ has the norm bounded by one and $\\mathcal{D}$ is dense in $X$, that is\n\\[\n\t\\|g\\| \\le 1\n\t\\ \\ \\text{for any}\\ \\ \n\tg \\in \\mathcal{D},\n\t\\ \\ \\text{and}\\ \\\n\t\\overline{\\operatorname{span}\\mathcal{D}} = X.\n\\]\nFor notational convenience in this paper we consider {\\it symmetric dictionaries}, i.e. such that\n\\[\n\tg\\in \\mathcal{D} \\ \\ \\text{implies} \\ \\ -g \\in \\mathcal{D}.\n\\]\nWe denote the closure (in $X$) of the convex hull of $\\mathcal{D}$ by $\\mathcal{A}_1(\\mathcal{D})$:\n\\begin{equation}\\label{eq:A_1(D)}\n\t\\mathcal{A}_1(\\mathcal{D}) := \\overline{\\mathrm{conv} \\mathcal{D}},\n\\end{equation}\nwhich is the standard notation in relevant greedy approximation literature. \n\nThe modulus of smoothness $\\rho(E,S,u)$ of a function $E : X \\to \\mathbb{R}$ on a set $S \\subset X$ is defined as\n\\begin{equation}\\label{eq:mod_smt}\n\t\\rho(E,S,u) := \\frac{1}{2} \\sup_{x\\in S, \\|y\\|=1} \\Big| E(x + uy) + E(x - uy) - 2E(x) \\Big|.\n\\end{equation}\nWe note that, in comparison to the modulus of smoothness of a norm (see, for instance,~\\cite[Part~3]{beauzamy2011introduction}), the modulus of smoothness of a function additionally depends on the chosen set $S \\subset X$.\nThat is because a norm is a positive homogeneous function, thus its smoothness on the whole space is determined by its smoothness on the unit sphere, which is not the case for a general function on a Banach space.\n\nThe function $E$ is uniformly smooth on $S \\subset X$ if $\\rho(E,S,u) = o(u)$ as $u \\to 0$.\nWe say that the modulus of smoothness $\\rho(E,S,u)$ is of power type $1 \\leq q \\leq 2$ if $\\rho(E,S,u) \\leq \\gamma u^q$ for some $\\gamma > 0$.\nNote that the class of functions with the modulus of smoothness of a nontrivial power type is completely different from the class of uniformly smooth Banach spaces with the norms of a nontrivial power type since any uniformly smooth norm is not uniformly smooth as a function on any set containing $0$.\nHowever, it is shown in~\\cite{borwein2009uniformly} that if a norm $\\|\\cdot\\|$ has the modulus of smoothness of power type $q \\in [1,2]$, then the function $E(\\cdot) := \\|\\cdot\\|^q$ has the modulus of smoothness $\\rho(E,S,u)$ of power type $q$ for any set $S \\subset X$.\nIn particular, it implies (see e.g.~\\cite[Lemma B.1]{donahue1997rates}) that for any $1 \\le p < \\infty$ the function $E : L_p \\to \\mathbb{R}$ defined as\n\\[\n\tE_p(x) = \\|x\\|_{L_p}^p\n\\]\nhas the modulus of smoothness that satisfies\n\\[\n\\rho_p(E,X,u) \\le\n\t\\left\\{\\begin{array}{ll}\n\t\t\\frac{1}{p} u^p & 1 \\le p \\le 2,\n\t\t\\\\\n\t\t\\frac{p-1}{2} u^2 & 2 \\le p < \\infty,\n\t\\end{array}\\right.\n\\]\ni.e. $\\rho_p(E,X,u)$ is of power type $\\min\\{p,2\\}$.\n\\\\\nA typical smoothness assumption in convex optimization is of the form\n\\[\n\t|E(x + uy) - E(x) - \\<E'(x),uy\\>| \\le Cu^2\n\\]\nwith some constant $C > 0$ and any $u \\in \\mathbb{R}$, $x \\in X$, $\\|y\\| = 1$.\nIn terms of the modulus of smoothness~\\eqref{eq:mod_smt} such an assumption corresponds to the case $\\rho(E,X,u) \\le Cu^2 \/ 2$, i.e. that the modulus of smoothness of $E$ is of power type $2$.\n\nThroughout the paper we assume that the target function $E$ is Fr{\\'e}chet-differentiable, i.e. that at any $x \\in X$ there is a bounded linear functional $E'(x) : X \\to \\mathbb{R}$ such that\n\\[\n\t\\sup_{\\|y\\|=1} \\Big( \\lim_{u \\to 0} \\frac{E(x + uy) - E(x)}{u} - \\< E'(x), y \\> \\Big) = 0.\n\\]\nThen the convexity of $E$ implies that for any $x,y \\in D$\n\\begin{equation}\\label{eq:E'_conv1}\n\tE(y) \\ge E(x) + \\<E'(x),y-x\\>,\n\\end{equation}\nor, equivalently,\n\\begin{equation}\\label{eq:E'_conv2}\n\tE(x) - E(y) \\le \\<E'(x),x - y\\> = \\<-E'(x),y-x\\>.\n\\end{equation}\n\n\\begin{Remark}\nThe condition of Fr{\\'e}chet-differentiability is not necessary and can be relaxed by considering support functionals in place of the derivative of $E$, as is done in~\\cite[Chapter~5]{dereventsov2017convergence}.\nAlthough the existence of support functionals is guaranteed by the convexity of the target function, we additionally impose the assumption of differentiability for the convenience of presentation.\n\\end{Remark}\n\n\n\\subsection{Weak Biorthogonal Greedy Algorithms}\nTypically in greedy approximation one has to perform a greedy selection from a given dictionary $\\mathcal{D}$, which might not always be possible.\nIn order to guarantee the feasibility of algorithms, it is conventional to perform a {\\it weak} greedy step where the greedy search is relaxed.\nSuch relaxations are represented by a given sequence $\\tau := \\{t_m\\}_{m=1}^\\infty$, referred to as a {\\it weakness sequence}.\n\nFor a convex Fr{\\'e}chet-differentiable target function $E : X \\to \\mathbb{R}$ we define the following class of greedy algorithms.\n\\\\[.5em]\\noindent\n{\\bf Weak Biorthogonal Greedy Algorithms ($\\boldsymbol{\\mathcal{WBGA}}$(co)).\\\\}\nWe say that an algorithm belongs to the class $\\mathcal{WBGA}$(co) with a weakness sequence $\\tau = \\{t_m\\}_{m=1}^\\infty$, $t_m\\in[0,1]$, if sequences of approximators $\\{G_m\\}_{m=0}^\\infty$ and selected elements $\\{\\varphi_m\\}_{m=1}^\\infty$ of the dictionary $\\mathcal{D}$ satisfy the following conditions at every iteration $m \\ge 1$:\n\\begin{enumerate}[label=\\bf(\\arabic*), leftmargin=.5in]\n\t\\item\\label{wbga_gs}\n\t\tGreedy selection: ${\\displaystyle \\<-E'(G_{m-1}), \\varphi_m\\> \\ge t_m \\sup_{\\varphi\\in\\mathcal{D}} \\< -E'(G_{m-1}), \\varphi \\>}$;\n\t\\item\\label{wbga_er}\n\t\tError reduction: ${\\displaystyle E(G_m) \\le \\inf_{\\lambda\\ge0} E(G_{m-1} + \\lambda\\varphi_m)}$;\n\t\\item\\label{wbga_bo}\n\t\tBiorthogonality: ${\\displaystyle \\<E'(G_{m}), G_m\\> = 0}$.\n\\end{enumerate}\n\n\\smallskip\\noindent\nWe assume that for a given target function $E : X \\to \\mathbb{R}$ the set\n\\[\n\tD = D(E) := \\big\\{ x \\in X : E(x) \\le E(0) \\big\\} \\subset X\n\\]\nis bounded.\nCoupled with the assumption that $E$ is convex and Fr{\\'e}chet-differentiable, boundedness of $D$ guarantees that\n\\[\n\t\\inf_{x \\in X} E(x) = \\inf_{x \\in D} E(x) > -\\infty\n\t\\ \\ \\text{and}\\ \\\n\t\\mathop{\\operatorname{argmin}}_{x \\in X} E(x) = \\mathop{\\operatorname{argmin}}_{x \\in D} E(x) \\in D,\n\\]\ni.e. there is a nontrivial and attainable minimum of $E$.\nThen by condition~\\ref{wbga_er} the sequence of $m$-sparse approximants $\\{G_m\\}_{m=0}^\\infty$ constructed by an algorithm from the $\\mathcal{WBGA}$(co) satisfies the relation\n\\[\n\tE(0) = E(G_0) \\ge E(G_1) \\ge E(G_2) \\ge \\dots,\n\\]\nwhich guarantees that $G_m \\in D$ for all $m \\ge 0$.\n\n\\begin{Remark}\nIn the case $E(x) := \\|f - x\\|^q$ with any $f\\in X$ and $q \\ge 1$, the class $\\mathcal{WBGA}$(co) coincides with the class $\\mathcal{WBGA}$ from the approximation theory, which is introduced and analyzed in~\\cite{VT165}.\n\\end{Remark}\n\n\n\\subsection{Examples of algorithms from the $\\mathcal{WBGA}$(co)}\\label{sec:wbga_ga}\nIn this section we briefly overview a few particular algorithms from the class $\\mathcal{WBGA}$(co) that will be utilized in the numerical experiments presented in Section~\\ref{sec:numerics}.\nBy $\\tau := \\{t_m\\}_{m=1}^\\infty$ we denote a weakness sequence, i.e. a given sequence of non-negative numbers $t_m \\le 1$, $m = 1,2,3,\\dots$.\n\nWe first define the Weak Chebyshev Greedy Algorithm for convex optimization that is introduced and studied in~\\cite{VT140}.\n\\\\[.5em]\\noindent\n{\\bf Weak Chebyshev Greedy Algorithm (WCGA(co)).\\\\}\nSet $G^c_0 = 0$ and for each $m \\ge 1$ perform the following steps:\n\\begin{enumerate}\n\t\\item Take any $\\varphi^{c}_m \\in \\mathcal{D}$ satisfying\n\t\t\\[\n\t\t\t\\< -E'(G^c_{m-1}), \\varphi^c_m \\> \\ge t_m \\sup_{\\varphi\\in\\mathcal{D}} \\< -E'(G^c_{m-1}), \\varphi \\>;\n\t\t\\]\n\t\\item Denote $\\Phi_m^c = \\operatorname{span} \\{\\varphi^c_k\\}_{k=1}^m$ and find $G_m^c \\in \\Phi_m^c$ such that\n\t\t\\[\n\t\t\tE(G_m^c) = \\inf_{G \\in \\Phi_m^c} E(G).\n\t\t\\]\n\\end{enumerate}\n\n\\smallskip\\noindent\nAnother algorithm, which utilizes a simpler approach to updating the approximant is the Weak Greedy Algorithm with Free Relaxation for convex optimization (see~\\cite{VT140}).\n\\\\[.5em]\\noindent\n{\\bf Weak Greedy Algorithm with Free Relaxation (WGAFR(co)).\\\\}\nSet $G^f_0 = 0$ and for each $m \\ge 1$ perform the following steps:\n\\begin{enumerate}[label=\\bf(\\arabic*), leftmargin=.5in]\n\t\\item Take any $\\varphi^f_m \\in \\mathcal{D}$ satisfying\n\t\t\\[\n\t\t\t\\< -E'(G^f_{m-1}), \\varphi^f_m \\> \\ge t_m \\sup_{\\varphi\\in\\mathcal{D}} \\< -E'(G^f_{m-1}), \\varphi \\>;\n\t\t\\]\n\t\\item Find $\\omega_m \\in \\mathbb{R}$ and $ \\lambda_m \\in \\mathbb{R}$ such that\n\t\t\\[\n\t\t\tE\\big( (1-\\omega_m) G^f_{m-1} + \\lambda_m \\varphi^f_m \\big)\n\t\t\t= \\inf_{ \\lambda, \\omega \\in \\mathbb{R}} E\\big( (1 - \\omega) G^f_{m-1} + \\lambda \\varphi^f_m \\big)\n\t\t\\]\n\t\tand define $G^f_m = (1 - \\omega_m) G^f_{m-1} + \\lambda_m \\varphi^f_m$.\n\\end{enumerate}\n\n\\smallskip\\noindent\nThe next algorithm~--- the Rescaled Weak Relaxed Greedy Algorithm for convex optimization~--- is an adaptation of its counterpart from the approximation theory (see~\\cite{VT165}) that can be viewed as a generalization of the Rescaled Pure Greedy Algorithm, introduced in~\\cite{Pet} and adapted for convex optimization in~\\cite{GP}.\n\\\\[.5em]\\noindent\n{\\bf Rescaled Weak Relaxed Greedy Algorithm (RWRGA(co)).\\\\}\nSet $G^r_0 = 0$ and for each $m \\ge 1$ perform the following steps:\n\\begin{enumerate}\n\t\\item Take any $\\varphi^r_m \\in \\mathcal{D}$ satisfying\n\t\t\\[\n\t\t\t\\< -E'(G^r_{m-1}), \\varphi^r_m \\> \\ge t_m \\sup_{\\varphi\\in\\mathcal{D}} \\< -E'(G^r_{m-1}), \\varphi \\>;\n\t\t\\]\n\t\\item Find $\\lambda_m \\ge 0$ such that\n\t\t\\[\n\t\t\tE(G^r_{m-1} + \\lambda_m \\varphi^r_m) = \\inf_{\\lambda \\ge 0} E(G^r_{m-1} + \\lambda \\varphi^r_m);\n\t\t\\]\n\t\\item Find $\\mu_m \\in \\mathbb{R}$ such that\n\t\t\\[\n\t\t\tE\\big( \\mu_m (G^r_{m-1} + \\lambda_m \\varphi^r_m) \\big)\n\t\t\t= \\inf_{\\mu \\in \\mathbb{R}} E\\big( \\mu (G^r_{m-1} + \\lambda_m \\varphi^r_m) \\big)\n\t\t\\]\n\t\tand define $G^r_m = \\mu_m (G^r_{m-1} + \\lambda_m \\varphi^r_m)$.\n\\end{enumerate}\n\n\\begin{Proposition}\\label{prp:ga_wbga}\nThe WCGA(co), the WGAFR(co), and the RWRGA(co) belong to the class $\\mathcal{WBGA}$(co).\n\\end{Proposition}\n\n\n\\subsection{Convergence results for the $\\mathcal{WBGA}(co)$}\nIn this section we state the results related to convergence and the rate of convergence for algorithms from the class $\\mathcal{WBGA}$(co).\n\nOur setting of an infinite dimensional Banach space makes the formulation of convergence results nontrivial, and thus we require a special sequence which is defined for a given modulus of smoothness $\\rho(u) := \\rho(E,D,u)$ and a given weakness sequence $\\tau = \\{t_m\\}_{m=1}^\\infty$.\n\nLet $E : X \\to \\mathbb{R}$ be a convex uniformly smooth function, then $\\rho(u) := \\rho(E,D,u) : \\mathbb{R} \\to \\mathbb{R_+}$ is an even convex function.\nAssume that $\\rho(u)$ has the property $\\rho(1\/\\theta_0) \\ge 1$ for some $\\theta_0 \\in (0,1]$ and\n\\[\n\t\\lim_{u\\to 0} \\rho(u)\/u = 0.\n\\]\nNote that assumptions on uniform smoothness of $E$ and boundedness of domain $D \\subset X$ guarantee the above properties.\nThen for a given $0 < \\theta \\le \\theta_0$ define $\\xi_m := \\xi_m(\\rho,\\tau,\\theta)$ as the solution of the equation\n\\begin{equation}\\label{eq:theta}\n\t\\rho(u) = \\theta t_m u.\n\\end{equation}\nNote that conditions on $\\rho(u)$ imply that the function\n\\[\n\ts(u) := \\left\\{\\begin{array}{ll}\n\t\t\\rho(u)\/u, & u \\neq 0\n\t\t\\\\\n\t\t0, & u = 0\n\t\\end{array}\\right.\n\\]\nis continuous and increasing on $[0,\\infty)$ with $s(1\/\\theta_0) \\ge \\theta_0$.\nThus equation~\\eqref{eq:theta} has the unique solution $\\xi_m = s^{-1}(\\theta t_m)$ such that $0 < \\xi_m \\le 1\/\\theta_0$.\n\n\\noindent\nWe now formulate our main convergence result for the $\\mathcal{WBGA}$(co).\n\\begin{Theorem}\\label{thm:wbga_conv}\nLet $E$ be a uniformly smooth on $D \\subset X$ convex function with the modulus of smoothness $\\rho(E,D,u)$.\nAssume that a sequence $\\tau := \\{t_m\\}_{m=1}^\\infty$ satisfies the condition that for any $\\theta \\in(0,\\theta_0]$ we have\n\\[\n\t\\sum_{m=1}^\\infty t_m \\xi_m(\\rho,\\tau,\\theta) = \\infty.\n\\]\nThen for any algorithm from the class $\\mathcal{WBGA}$(co) we have\n\\[\n\t\\lim_{m\\to\\infty} E(G_m) = \\inf_{x\\in D} E(x).\n\\]\n\\end{Theorem}\n\n\\noindent\nHere are two simple corollaries of Theorem~\\ref{thm:wbga_conv}.\n\\begin{Corollary}\nLet $E$ be a uniformly smooth on $D \\subset X$ convex function.\nThen any algorithm from the class $\\mathcal{WBGA}$(co) with a constant weakness sequence $\\tau = t \\in (0,1]$ converges, i.e.\n\\[\n\t\\lim_{m\\to\\infty} E(G_m) = \\inf_{x\\in D} E(x).\n\\]\n\\end{Corollary}\n\n\\begin{Corollary}\nLet $E$ be a convex function with the modulus of smoothness of power type $1 < q \\le 2$, that is, $\\rho(E,D,u) \\le \\gamma u^q$.\nLet a sequence $\\tau := \\{t_m\\}_{m=1}^\\infty$, $t_m \\in (0,1]$ for $m = 1,2,3,\\ldots$ be such that \n\\[\n\t\\sum_{m=1}^\\infty t_m^p = \\infty, \\ \\ p = \\frac{q}{q-1}.\n\\]\nThen any algorithm from the class $\\mathcal{WBGA}$(co) with the weakness sequence $\\tau$ converges, i.e.\n\\[\n\t\\lim_{m \\to \\infty} E(G_m) = \\inf_{x\\in D} E(x).\n\\]\n\\end{Corollary}\n\n\\noindent\nWe now proceed to the rate of convergence estimates, which are of interest in both finite dimensional and infinite dimensional settings.\nA typical assumption in this regard is formulated in terms of the convex hull $\\mathcal{A}_1(\\mathcal{D})$ of the dictionary $\\mathcal{D}$, defined by~\\eqref{eq:A_1(D)}.\n\\begin{Theorem}\\label{thm:wbga_rate}\nLet $E$ be a convex function with the modulus of smoothness of power type $1 < q \\le 2$, that is, $\\rho(E,D,u) \\le \\gamma u^q$.\nTake an element $f^\\epsilon \\in D$ and a number $\\epsilon \\ge 0$ such that\n\\[\n\tE(f^\\epsilon) \\le \\inf_{x\\in D} E(x) + \\epsilon, \\ \\ f^\\epsilon\/A(\\epsilon) \\in \\mathcal{A}_1(\\mathcal{D})\n\\]\nwith some number $A(\\epsilon) \\ge 1$.\nThen for any algorithm from the class $\\mathcal{WBGA}$(co) we have\n\\[\n\tE(G_m) - \\inf_{x\\in D} E(x) \\le \\max\\left\\{ 2\\epsilon, C(q,\\gamma) A(\\epsilon)^q \\left(C(E,q,\\gamma) + \\sum_{k=1}^m t_k^p\\right)^{1-q} \\right\\},\n\\]\nwhere $p = q\/(q-1)$.\n\\end{Theorem}\n\n\\begin{Corollary}\nLet $E$ be a convex function with the modulus of smoothness of power type $1 < q \\le 2$, that is, $\\rho(E,D,u) \\le \\gamma u^q$.\nIf $\\operatorname{argmin}_{x \\in D} E(x) \\in \\mathcal{A}_1(\\mathcal{D})$ then for any algorithm from the class $\\mathcal{WBGA}$(co) we have\n\\[\n\tE(G_m) - \\inf_{x \\in D} E(x) \\le C(q,\\gamma) \\left(C(E,q,\\gamma) + \\sum_{k=1}^m t_k^p\\right)^{1-q},\n\\]\nwhere $p = q\/(q-1)$.\n\\end{Corollary}\n\n\\begin{Remark}\nWhile the results stated in this section are known for the WCGA(co) and the WGAFR(co) (see~\\cite{VT140}), they are novel for the RWRGA(co).\n\\end{Remark}\n\n\n\n\\section{Weak Biorthogonal Greedy Algorithms with errors for Convex Optimization}\\label{sec:awbga}\nIn this section we address the question of the stability of algorithms from the class $\\mathcal{WBGA}$(co) by introducing the wider class $\\mathcal{WBGA}(\\Delta,\\text{co})$, which allows for imprecise calculations in the realization of algorithms.\nSuch an approach is of a practical interest since computational inaccuracies often occur naturally in applications.\nTo account for imprecise computations we introduce a sequence $\\Delta := \\{\\delta_m, \\epsilon_m\\}_{m=1}^\\infty$, where $\\delta_m \\in [0,1]$ and $\\epsilon_m \\ge 0$ for $m = 1,2,3,\\dots$, that represents the allowed inaccuracies in the steps of the algorithms.\nIn accordance with the conventional notation (see e.g.~\\cite{gribonval2001approximate}, \\cite{galatenko2003convergence}), we refer to a given sequence $\\Delta := \\{\\delta_m, \\epsilon_m\\}_{m=1}^\\infty$ as an {\\it error sequence}.\n\nFor a convex Fr{\\'e}chet-differentiable target function $E : X \\to \\mathbb{R}$ we define the following class of greedy algorithms with errors.\n\\\\[.5em]\\noindent\n{\\bf Weak Biorthogonal Greedy Algorithms with errors ($\\boldsymbol{\\mathcal{WBGA}(\\Delta,\\text{co})}$).\\\\} \nWe say that an algorithm belongs to the class $\\mathcal{WBGA}(\\Delta,\\text{co})$ with a weakness sequence $\\tau = \\{t_m\\}_{m=1}^\\infty$, $t_m\\in[0,1]$ and an error sequence $\\Delta = \\{\\delta_m,\\epsilon_m\\}_{m=1}^\\infty$, $\\delta_m\\in[0,1], \\epsilon_m\\ge0$, if sequences of approximators $\\{G_m\\}_{m=0}^\\infty$ and selected elements $\\{\\varphi_m\\}_{m=1}^\\infty$ of the dictionary $\\mathcal{D}$ satisfy the following conditions at every iteration $m \\ge 1$:\n\\begin{enumerate}[label=\\bf(\\arabic*), leftmargin=.5in]\n\t\\item\\label{awbga_gs}\n\t\tGreedy selection: ${\\displaystyle \\<-E'(G_{m-1}), \\varphi_m\\> \\ge t_m \\sup_{\\varphi\\in\\mathcal{D}} \\<-E'(G_{m-1}), \\varphi\\>}$;\n\t\\item\\label{awbga_er}\n\t\tError reduction: ${\\displaystyle E(G_m) \\le \\inf_{\\lambda\\ge0} E(G_{m-1} + \\lambda\\varphi_m) + \\delta_m}$;\n\t\\item\\label{awbga_bo}\n\t\tBiorthogonality: ${\\displaystyle |\\<E'(G_{m}), G_m\\>| \\le \\epsilon_m}$;${\\displaystyle\\phantom{\\inf_\\lambda}}$\n\t\\item\\label{awbga_bd}\n\t\tBoundedness: ${\\displaystyle E(G_{m}) \\le E(0) + C_0}$.\n\\end{enumerate}\n\n\\smallskip\\noindent\nNote that in addition to conditions~\\ref{wbga_gs}--\\ref{wbga_bo} from the definition of the class $\\mathcal{WBGA}$(co), for the $\\mathcal{WBGA}(\\Delta,\\text{co})$ we require the boundedness condition~\\ref{awbga_bd} to account for the magnitude of allowed errors $\\Delta$.\nIn particular, if the error sequence $\\Delta$ is summable, i.e. $\\sum_{m=1}^\\infty \\delta_m < \\infty$, then condition~\\ref{awbga_bd} follows directly from~\\ref{awbga_er} with $C_0 = \\sum_{m=1}^\\infty \\delta_m$.\n\\\\\nMoreover, we assume that the set\n\\[\n\tD \\subset D_1 := \\{x \\in X : E(x) \\le E(0) + C_0\\} \\subset X,\n\\]\nwhere $C_0$ is the constant from condition~\\ref{awbga_bd}, is bounded.\nThen condition~\\ref{awbga_bd} guarantees that $G_m \\in D_1$ for all $m \\ge 0$ for any algorithm from the $\\mathcal{WBGA}(\\Delta,\\text{co})$.\n\n\\begin{Remark}\nIn the error reduction condition~\\ref{awbga_er} from the definition of the class $\\mathcal{WBGA}(\\Delta,co$) the infimum is taken over all $\\lambda \\ge 0$.\nIn order to simplify this problem, one can consider a wider than the $\\mathcal{WBGA}(\\Delta,co)$ class~--- the class $\\mathcal{WBGA}(\\Delta,[0,1],co)$ of algorithms satisfying conditions~\\ref{awbga_gs}, \\ref{awbga_bo}, \\ref{awbga_bd}, and the following condition instead of~\\ref{awbga_er}:\n\\[\n\t\\text{{\\bf (2')} {\\rm Restricted error reduction: }}\n\tE(G_m) \\le \\inf_{\\lambda\\in[0,1]} E(G_{m-1} + \\lambda\\varphi_m) + \\delta_m.\n\\]\nThen finding such $\\lambda\\in[0,1]$ is a line search problem, which is known to be a simple one-dimensional convex optimization problem (see e.g.~\\cite{boyd2004convex}, \\cite{N}).\n\\end{Remark}\n\n\n\\subsection{Examples of algorithms from the $\\mathcal{WBGA}(\\Delta,co)$}\\label{sec:awbga_ga}\nIn this section we briefly overview particular algorithms from the class $\\mathcal{WBGA}(\\Delta,\\text{co})$ that correspond to the approximate versions of the algorithms considered in Section~\\ref{sec:wbga_ga}.\nDenote by $\\tau := \\{t_m\\}_{m=1}^\\infty$ and $\\Delta := \\{\\delta_m,\\epsilon_m\\}_{m=1}^\\infty$ a weakness sequence and an error sequence respectively, i.e. given sequences of numbers $t_m \\in [0,1]$, $\\delta_m \\in [0,1]$, and $\\epsilon_m \\ge 0$ for $m = 1,2,3,\\ldots$.\n\nWe begin with the Weak Chebyshev Greedy Algorithm with errors for convex optimization. \n\\\\[.5em]\\noindent\n{\\bf Weak Chebyshev Greedy Algorithm with errors (WCGA($\\Delta,\\text{co}$)).\\\\}\nSet $G^c_0 = 0$ and for each $m \\ge 1$ perform the following steps:\n\\begin{enumerate}\n\t\\item Take any $\\varphi^{c}_m \\in \\mathcal{D}$ satisfying\n\t\t\\[\n\t\t\t\\< -E'(G^c_{m-1}), \\varphi^c_m \\> \\ge t_m \\sup_{\\varphi\\in\\mathcal{D}} \\< -E'(G^c_{m-1}), \\varphi \\>;\n\t\t\\]\n\t\\item Denote $\\Phi_m^c = \\operatorname{span} \\{\\varphi^c_k\\}_{k=1}^m$ and find $G_m^c \\in \\Phi_m^c$ such that\n\t\t\\[\n\t\t\tE(G_m^c) \\le \\inf_{G \\in \\Phi_m^c} E(G) + \\delta_m.\n\t\t\\]\n\\end{enumerate}\n\n\\smallskip\\noindent\nNext, we state the Weak Greedy Algorithm with Free Relaxation and errors for convex optimization, introduced and studied in~\\cite{DT}.\n\\\\[.5em]\\noindent\n{\\bf Weak Greedy Algorithm with Free Relaxation and errors (WGAFR($\\Delta,\\text{co}$)).\\\\}\nSet $G^f_0 = 0$ and for each $m \\ge 1$ perform the following steps:\n\\begin{enumerate}\n\t\\item Take any $\\varphi^f_m \\in \\mathcal{D}$ satisfying\n\t\t\\[\n\t\t\t\\< -E'(G^f_{m-1}), \\varphi^f_m \\> \\ge t_m \\sup_{\\varphi\\in\\mathcal{D}} \\< -E'(G^f_{m-1}), \\varphi \\>;\n\t\t\\]\n\t\\item Find $\\omega_m \\in \\mathbb{R}$ and $ \\lambda_m \\in \\mathbb{R}$ such that\n\t\t\\[\n\t\t\tE\\big( (1-\\omega_m) G^f_{m-1} + \\lambda_m \\varphi^f_m \\big)\n\t\t\t\\le \\inf_{\\lambda, \\omega \\in \\mathbb{R}} E\\big( (1 - \\omega) G^f_{m-1} + \\lambda \\varphi^f_m \\big) + \\delta_m\n\t\t\\]\n\t\tand define $G^f_m = (1 - \\omega_m) G^f_{m-1} + \\lambda_m \\varphi^f_m$.\n\\end{enumerate}\n\n\\smallskip\\noindent\nLastly, we introduce a new algorithm~--- the Rescaled Weak Relaxed Greedy Algorithm with errors for convex optimization.\n\\\\[.5em]\\noindent\n{\\bf Rescaled Weak Relaxed Greedy Algorithm with errors (RWRGA($\\Delta,\\text{co}$)).\\\\}\nSet $G^r_0 = 0$ and for each $m \\ge 1$ perform the following steps:\n\\begin{enumerate}\n\t\\item Take any $\\varphi^r_m \\in \\mathcal{D}$ satisfying\n\t\t\\[\n\t\t\t\\< -E'(G^r_{m-1}), \\varphi^r_m \\> \\ge t_m \\sup_{\\varphi\\in\\mathcal{D}} \\< -E'(G^r_{m-1}), \\varphi \\>;\n\t\t\\]\n\t\\item Find $\\lambda_m \\ge 0$ such that\n\t\t\\[\n\t\t\tE(G^r_{m-1} + \\lambda_m \\varphi^r_m) \\le \\inf_{\\lambda \\ge 0} E(G^r_{m-1} + \\lambda \\varphi^r_m) + \\delta_m\/2;\n\t\t\\]\n\t\\item Find $\\mu_m \\in \\mathbb{R}$ such that\n\t\t\\[\n\t\t\tE\\big( \\mu_m (G^r_{m-1} + \\lambda_m \\varphi^r_m) \\big)\n\t\t\t\\le \\inf_{\\mu \\in \\mathbb{R}} E\\big( \\mu (G^r_{m-1} + \\lambda_m \\varphi^r_m) \\big) + \\delta_m\/2\n\t\t\\]\n\t\tand define $G^r_m = \\mu_m (G^r_{m-1} + \\lambda_m \\varphi^r_m)$.\n\\end{enumerate}\n\n\\begin{Proposition}\\label{prp:ga_awbga}\nThe WCGA($\\Delta$,co), the WGAFR($\\Delta$,co), and the RWRGA($\\Delta$,co) belong to the class $\\mathcal{WBGA}(\\Delta,co)$ with\n\\[\n\t\\epsilon_m = \\inf_{u > 0} \\frac{\\delta_m + 2\\rho(E,D_1,u \\|G_m\\|)}{u}.\n\\]\n\\end{Proposition}\n\n\n\\subsection{Convergence results for the $\\mathcal{WBGA}(\\Delta,co)$}\nIn this section we discuss the convergence and rate of convergence results for algorithms from the class $\\mathcal{WBGA}(\\Delta,\\text{co})$.\n\n\\noindent\nFirst, we state the convergence result.\n\\begin{Theorem}\\label{thm:awbga_conv}\nLet $E$ be a uniformly smooth on $D_1 \\subset X$ convex function.\nAssume that an error sequence $\\Delta := \\{\\delta_m,\\epsilon_m\\}_{m=1}^\\infty$ is such that $\\delta_m \\to 0$ and $\\epsilon_m \\to 0$ as $m \\to \\infty$.\nThen any algorithm from the class $\\mathcal{WBGA}(\\Delta,\\text{co})$ with a constant weakness sequence $\\tau = t \\in (0,1]$ converges, i.e.\n\\[\n\t\\lim_{m\\to\\infty} E(G_m) = \\inf_{x\\in D_1} E(x).\n\\]\n\\end{Theorem}\n\n\\noindent\nSecond, we provide the rate of convergence estimate.\n\\begin{Theorem}\\label{thm:awbga_rate}\nLet $E$ be a convex function with the modulus of smoothness of power type $1 < q \\le 2$, that is, $\\rho(E,D_{1},u) \\le \\gamma u^q$.\nTake an element $f^\\epsilon \\in D_1$ and a number $\\epsilon \\ge 0$ such that\n\\[\n\tE(f^\\epsilon) \\le \\inf_{x \\in D_1} E(x) + \\epsilon, \\ \\ \n\tf^\\epsilon\/A \\in \\mathcal{A}_1(\\mathcal{D}),\n\\]\nwith some number $A := A(\\epsilon) \\ge 1$.\nThen for any algorithm from the class $\\mathcal{WBGA}(\\Delta,\\text{co})$ with a constant weakness sequence $\\tau = t \\in (0,1]$ and an error sequence $\\Delta = \\{\\delta_m,\\epsilon_m\\}_{m=1}^\\infty$ with $\\delta_m + \\epsilon_m \\le cm^{-q}$, $m = 1,2,3,\\dots$ we have\n\\[\n\tE(G_m) - \\inf_{x\\in D_1} E(x) \\le \\epsilon + C(E,q,\\gamma,t,c) A(\\epsilon)^q \\, m^{1-q}.\n\\]\n\\end{Theorem}\n\n\\begin{Corollary}\nUnder the conditions of Theorem~\\ref{thm:awbga_rate}, specifying \n\\[\n\tA(\\epsilon) := \\inf \\Big\\{ A > 0 : \\exists f \\in D_1 : f\/A \\in \\mathcal{A}_1(\\mathcal{D}),\\ \\ E(f) \\le \\inf_{x\\in D_1}E(x) + \\epsilon \\Big\\}\n\\]\nand denoting\n\\[\n\t\\eta_m := \\inf \\big\\{ \\epsilon > 0: A(\\epsilon)^q \\, m^{1-q} \\le \\epsilon \\big\\},\n\\]\nwe obtain for any algorithm from the class $\\mathcal{WBGA}(\\Delta,co)$\n\\[\n\tE(G_m) - \\inf_{x\\in D_1} E(x) \\le C(E,q,\\gamma,t) \\, \\eta_m.\n\\]\n\\end{Corollary}\n\n\\begin{Remark}\nIt follows from the proofs of Theorems~\\ref{thm:awbga_conv} and~\\ref{thm:awbga_rate}, given in Section~\\ref{sec:proofs_awbga}, that the results stated in this section also hold for the class $\\mathcal{WBGA}(\\Delta,[0,1],\\text{co})$.\n\\end{Remark}\n\n\n\n\\section{Numerical experiments}\\label{sec:numerics}\nIn this section we demonstrate the performance of the algorithms from the class $\\mathcal{WBGA}$(co) that are discussed in Section~\\ref{sec:wbga_ga}: the Weak Chebyshev Greedy Algorithm (WCGA(co)), the Weak Greedy Algorithm with Free Relaxation (WGAFR(co)), and the Rescaled Weak Relaxed Greedy Algorithm (RWRGA(co)).\n\nFor each of the numerical experiments presented below we consider the Banach space $X = \\ell_1^{(\\mathrm{dim})}$ of dimensionality $\\mathrm{dim}$, a target function $E : X \\to \\mathbb{R}$, and a dictionary $\\mathcal{D} \\in X$.\nWe then employ the aforementioned algorithms to solve the optimization problem~\\eqref{eq:opt}, i.e. to find a sparse (with respect to the dictionary $\\mathcal{D}$) minimizer\n\\[\n\tx^* = \\mathop{\\operatorname{argmin}}_{x \\in X} E(x).\n\\]\nSince greedy algorithms are iterative by design, in Examples~1--2 we obtain and present the trade-off between the sparsity of the solution $x^*$ and the value of $E(x^*)$.\nIn Examples~3--4, we additionally compare the greedy algorithms for convex optimization with a conventional method of finding sparse solutions~--- the optimization with $\\ell_1$-regularization, see~\\eqref{eq:opt_reg}.\nSpecifically, we solve the problem\n\\[\n\t\\text{find}\\ \\ x^* = \\mathop{\\operatorname{argmin}}_{x \\in X} \\Big( E(x) + \\lambda \\|x\\|_\\mathcal{D} \\Big),\n\\]\nwhere $\\|\\cdot\\|_\\mathcal{D}$ is the atomic norm with respect to the dictionary $\\mathcal{D}$, defined by~\\eqref{eq:D_norm}.\nTo obtain minimizers of different sparsities, the values of the regularization parameter $\\lambda$ are taken from the sequence $\\{0.1 \\times (0.9)^k\\}_{k=0}^{49}$, i.e. $50$ regularized optimization problems are solved in every setting.\n\nTo avoid an unintentional bias in the selection of dictionary $\\mathcal{D}$ and target function $E$, we generate those randomly, based on certain parameters that are described in the setting of each example.\nIn order to provide a reliable demonstration that is independent of a particular random generation, we compute $100$ simulations for each presented example and provide the distribution of the optimization results (shown in Figures~1--4).\nIn the presented pictures the solid lines represent the mean minimization values for each algorithm and the filled areas represent the minimization distribution across all $100$ simulations.\nFinally, to make the results consistent across simulations, we rescale the optimization results to be in the interval $[0,1]$, i.e. instead of reporting the value of $E(x^*)$ we report\n\\[\n\t\\frac{E(x^*) - \\inf_{x \\in X} E(x)}{E(0) - \\inf_{x \\in X} E(x)} \\in [0,1].\n\\]\n\nNumerical experiments presented in this section are performed in Python~3.6 with the use of NumPy and SciPy libraries.\nThe source code is available at~\\url{https:\/\/github.com\/sukiboo\/wbga_co_2020}.\n\n\n\\subsection{Example 1}\n\\begin{figure}[t]\n\t\\includegraphics[width=\\linewidth]{.\/images\/ex1.pdf}\n\t\\caption{Distribution of optimization results for Example~1.}\n\\end{figure}\nIn this example we consider the space $X = \\ell_1^{(500)}$, and construct a dictionary $\\mathcal{D}$ of size $1000$ as linear combinations of the canonical basis $\\{e_i\\}_{i=1}^{500}$ of $X$ with uniformly distributed coefficients, i.e.\n\\[\n\t\\mathcal{D} = \\{\\varphi_j\\}_{j=1}^{1000},\n\t\\ \\ \\text{where}\\ \\ \n\t\\varphi_j = \\sum_{i=1}^{500} c^i_j \\, e_i\n\t\\ \\ \\text{with}\\ \\ \n\tc^i_j \\sim \\mathcal{U}(0,1).\n\\]\nThe target function $E : X \\to \\mathbb{R}$ is chosen as\n\\[\n\tE(x) = \\|x - f\\|_p^p,\n\\]\nwhere $p = 1.2$ and $f \\in X$ is randomly generated as a linear combination of $60$ randomly selected elements of $\\mathcal{D}$ with normally distributed coefficients, i.e.\n\\[\n\tf = \\sum_{k=1}^{60} a_k \\, \\varphi_{\\sigma(k)},\n\t\\ \\ \\text{where}\\ \\ \n\ta_k \\sim \\mathcal{N}(0,1)\n\t\\ \\ \\text{and}\\ \\\n\t\\sigma\\ \\ \\text{is a permutation of}\\ \\ \\{1,\\ldots,1000\\}.\n\\]\nPerformance of the greedy algorithms in this setting is presented in Figure~1.\nThe average number of iterations required to obtain a minimizer of sparsity $50$ is $185$ for the RWRGA(co), $93$ for the WGAFR(co), and $50$ for the WCGA(co).\n\n\n\\subsection{Example 2}\n\\begin{figure}[t]\n\t\\includegraphics[width=\\linewidth]{.\/images\/ex2.pdf}\n\t\\caption{Distribution of optimization results for Example~2.}\n\\end{figure}\nIn this example we once again consider the space $X = \\ell_1^{(500)}$ and a dictionary $\\mathcal{D}$ of size $1000$, constructed as linear combinations of the canonical basis $\\{e_i\\}_{i=1}^{500}$ of $X$ with uniformly distributed coefficients, i.e.\n\\[\n\t\\mathcal{D} = \\{\\varphi_j\\}_{j=1}^{1000},\n\t\\ \\ \\text{where}\\ \\ \n\t\\varphi_j = \\sum_{i=1}^{500} c^i_j \\, e_i\n\t\\ \\ \\text{with}\\ \\ \n\tc^i_j \\sim \\mathcal{U}(0,1).\n\\]\nThe target function $E : X \\to \\mathbb{R}$ is chosen as\n\\[\n\tE(x) = \\|x - f\\|_p^p \\, \\|g\\|_q^q + \\|x - g\\|_q^q \\, \\|f\\|_p^p,\n\\]\nwhere $p = 3$, $q = 1.2$, and the elements $f,g \\in X$ are each randomly generated as linear combinations of $30$ randomly selected elements of $\\mathcal{D}$ with normally distributed coefficients, i.e.\n\\begin{gather*}\n\tf = \\sum_{k=1}^{30} a^1_k \\, \\varphi_{\\sigma_1(k)}\n\t\\ \\ \\text{and}\\ \\ \n\tg = \\sum_{k=1}^{30} a^2_k \\, \\varphi_{\\sigma_2(k)},\n\t\\\\\n\t\\text{where}\\ \\ \n\ta^1_k, a^2_k \\sim \\mathcal{N}(0,1)\n\t\\ \\ \\text{and}\\ \\\n\t\\sigma_1, \\sigma_2\\ \\ \\text{are permutations of}\\ \\ \\{1,\\ldots,1000\\}.\n\\end{gather*}\nPerformance of the greedy algorithms in this setting is presented in Figure~2.\nThe average number of iterations required to obtain a minimizer of sparsity $50$ is $125$ for the RWRGA(co), $78$ for the WGAFR(co), and $50$ for the WCGA(co).\n\n\n\\subsection{Example 3}\n\\begin{figure}[t]\n\t\\includegraphics[width=\\linewidth]{.\/images\/ex3.pdf}\n\t\\caption{Distribution of optimization results for Example~3.}\n\\end{figure}\nIn this example we additionally compare the greedy algorithms with conventional optimization with $\\ell_1$-regularization, see~\\eqref{eq:opt_reg}.\nSince obtaining the minimization-sparsity trade-off with $\\ell_1$-regularization is more expensive computationally than it is for the greedy algorithms, we restrict ourselves to work in a space of smaller dimensionality.\nNamely, we consider the space $X = \\ell_1^{(100)}$, and construct a dictionary $\\mathcal{D}$ of size $200$ as linear combinations of the canonical basis $\\{e_i\\}_{i=1}^{100}$ of $X$ with uniformly distributed coefficients, i.e.\n\\[\n\t\\mathcal{D} = \\{\\varphi_j\\}_{j=1}^{200},\n\t\\ \\ \\text{where}\\ \\ \n\t\\varphi_j = \\sum_{i=1}^{100} c^i_j \\, e_i\n\t\\ \\ \\text{with}\\ \\ \n\tc^i_j \\sim \\mathcal{U}(0,1).\n\\]\nThe target function $E : X \\to \\mathbb{R}$ is chosen as\n\\[\n\tE(x) = \\|x - f\\|_p^p \\, \\|g\\|_q^q + \\|x - g\\|_q^q \\, \\|f\\|_p^p,\n\\]\nwhere $p = 4$, $q = 1.5$, and the elements $f,g \\in X$ are each randomly generated as linear combinations of $30$ randomly selected elements of $\\mathcal{D}$ with normally distributed coefficients, i.e.\n\\begin{gather*}\n\tf = \\sum_{k=1}^{30} a^1_k \\, \\varphi_{\\sigma_1(k)}\n\t\\ \\ \\text{and}\\ \\ \n\tg = \\sum_{k=1}^{30} a^2_k \\, \\varphi_{\\sigma_2(k)},\n\t\\\\\n\t\\text{where}\\ \\ \n\ta^1_k, a^2_k \\sim \\mathcal{N}(0,1)\n\t\\ \\ \\text{and}\\ \\\n\t\\sigma_1, \\sigma_2\\ \\ \\text{are permutations of}\\ \\ \\{1,\\ldots,200\\}.\n\\end{gather*}\nPerformance of the greedy algorithms and optimization with $\\ell_1$-regularization in this setting is presented in Figure~3.\nThe average number of iterations required to obtain a minimizer of sparsity $50$ is $105$ for the RWRGA(co), $74$ for the WGAFR(co), and $50$ for the WCGA(co).\n\n\\subsection{Example 4}\n\\begin{figure}[t]\n\t\\includegraphics[width=\\linewidth]{.\/images\/ex4.pdf}\n\t\\caption{Distribution of optimization results for Example~4.}\n\\end{figure}\nIn this example we compare the greedy algorithms with conventional optimization with $\\ell_1$-regularization in a classical setting of canonical basis instead of a randomly-generated dictionary.\nNamely, we consider the space $X = \\ell_1^{(200)}$, and set a dictionary $\\mathcal{D}$ to be the canonical basis $\\{e_i\\}_{i=1}^{200}$ of $X$, i.e.\n\\[\n\t\\mathcal{D} = \\{e_j\\}_{j=1}^{200},\n\t\\ \\ \\text{where}\\ \\ \n\te_j = (\\underbrace{0,\\ldots,0}_{j-1},1,\\underbrace{0,\\ldots,0}_{200-j}).\n\\]\nThe target function $E : X \\to \\mathbb{R}$ is chosen as\n\\[\n\tE(x) = \\|x - f\\|_p^p \\, \\|g\\|_q^q + \\|x - g\\|_q^q \\, \\|f\\|_p^p,\n\\]\nwhere $p = 7$, $q = 3$, and $f,g$ are randomly generated as elements of $X$ with normally distributed coefficients, i.e.\n\\[\n\tf = \\sum_{k=1}^{200} a^1_k \\, e_k\n\t\\ \\ \\text{and}\\ \\ \n\tg = \\sum_{k=1}^{200} a^2_k \\, e_k,\n\t\\ \\text{where}\\ \\ \n\ta^1_k, a^2_k \\sim \\mathcal{N}(0,1).\n\\]\nPerformance of the greedy algorithms and optimization with $\\ell_1$-regularization in this setting is presented in Figure~4.\nNote that in this case all greedy algorithms~--- the RWRGA(co), the WGAFR(co), and the WCGA(co)~--- coincide due to the fact that elements of the dictionary $\\mathcal{D}$ are mutually disjoint.\nHence the number of iterations required to obtain a minimizer of sparsity $100$ is exactly $100$ for all three greedy algorithms.\n\n\n\n\\section{Proofs for Section~\\ref{sec:wbga}}\\label{sec:proofs_wbga}\nIn this section we provide the proofs of the results from Section~\\ref{sec:wbga}.\nWe begin with a known lemma.\n\\begin{Lemma}[{\\cite[Lemma~6.1]{VT140}}]\\label{lem:E'_L=0}\nLet $E$ be a uniformly smooth Fr{\\'e}chet-differentiable convex function on a Banach space $X$ and $L$ be a finite-dimensional subspace of $X$.\nLet $x_L$ denote the point from $L$ at which $E$ attains the minimum, i.e.\n\\[\n\tx_L = \\mathop{\\operatorname{argmin}}_{x \\in L} E(x) \\in L.\n\\]\nThen for any $\\phi \\in L$ we have\n\\[\n\t\\<E'(x_L), \\phi\\> = 0.\n\\]\n\\end{Lemma}\n\n\\noindent\nWe now prove that the algorithms stated in Section~\\ref{sec:wbga_ga} belong to the class $\\mathcal{WBGA}$(co).\n\\begin{proof}[Proof of Proposition~\\ref{prp:ga_wbga}]\nIt is easy to see that conditions~\\ref{wbga_gs} and~\\ref{wbga_er} from the definition of the class $\\mathcal{WBGA}$(co) are satisfied for all three algorithms.\nCondition~\\ref{wbga_bo} for any $m \\ge 1$ follows directly from Lemma~\\ref{lem:E'_L=0} with $x_L = \\phi = G_m$ and\n\\[\n\tL = \\Phi^c_m = \\operatorname{span}\\{\\varphi_1^c, \\ldots, \\varphi_m^c\\}\n\\]\nfor the WCGA(co), and\n\\[\n\tL = \\operatorname{span}\\{G_{m-1}^f, \\varphi_m^f\\}\n\t\\ \\ \\text{or}\\ \\ \n\tL = \\operatorname{span}\\{G_{m-1}^r, \\varphi_m^r\\}\n\\]\nfor the WGAFR(co) \/ RWRGA(co) respectively.\n\\end{proof}\n\n\n\\noindent\nWe proceed by listing the lemmas that will be utilized later in the proofs of the main results.\nThe following simple lemma is well-known (see, for instance, \\cite{VT140}).\nFor the reader's convenience we present its proof here.\n\\begin{Lemma}[{\\cite[Lemma~6.3]{VT140}}]\\label{lem:E'_rho}\nLet $E$ be a Fr{\\'e}chet-differentiable convex function.\nThen the following inequality holds for any $x \\in S \\subset X$, $y \\in X$, and $u \\in \\mathbb{R}$\n\\[\n\t0 \\le E(x + uy) - E(x) - u\\<E'(x),y\\> \\le 2\\rho(E,S,u\\|y\\|).\n\\]\n\\end{Lemma}\n\\begin{proof}\nThe left inequality follows directly from~\\eqref{eq:E'_conv1}.\nNext, from the definition of modulus of smoothness~\\eqref{eq:mod_smt} it follows that\n\\[\n\tE(x + uy) + E(x - uy) \\le 2\\big( E(x) + \\rho(E,S,u\\|y\\|) \\big).\n\\]\nFrom inequality \\eqref{eq:E'_conv1} we get\n\\[\n\tE(x - uy) \\ge E(x) - u\\<E'(x),y\\>. \n\\]\nCombining the above two estimates, we obtain\n\\[\n\tE(x + uy) \\le E(x) + u\\<E'(x),y\\> + 2\\rho(E,S,u\\|y\\|),\n\\]\nwhich proves the second inequality. \n\\end{proof}\n\n\n\\begin{Lemma}[{\\cite[Lemma~6.10]{VTbook}}]\\label{lem:F_A1(D)}\nFor any bounded linear functional $F$ and any dictionary $\\mathcal{D}$, we have\n\\[\n\t\\sup_{g\\in \\mathcal{D}} \\<F,g\\> = \\sup_{f\\in\\mathcal{A}_1(\\mathcal{D})} \\<F,f\\>.\n\\]\n\\end{Lemma}\n\n\n\\noindent\nThe following lemma is similar to the result from~\\cite{T13}.\nFor the reader's convenience we present a brief proof of this lemma here.\n\\begin{Lemma}\\label{lem:y_k}\nSuppose that a sequence $y_1 \\ge y_2 \\ge y_3 \\ge \\ldots > 0$ satisfies inequalities\n\\[\n\ty_k \\le y_{k-1} (1 - w_k y_{k-1}), \\ \\ w_k \\ge 0\n\\]\nfor any $k > n$.\nThen for any $m > n$ we have\n\\[\n\t\\frac{1}{y_m} \\ge \\frac{1}{y_n} + \\sum_{k=n+1}^m w_k.\n\\]\n\\end{Lemma}\n\\begin{proof}\nThe proof follows directly from the chain of inequalities\n\\[\n\t\\frac{1}{y_k} \\ge \\frac{1}{y_{k-1}} (1 - w_k y_{k-1})^{-1}\n\t\\ge \\frac{1}{y_{k-1}} (1 + w_k y_{k-1})\n\t= \\frac{1}{y_{k-1}} + w_k.\n\\]\n\\end{proof}\n\n\n\\noindent\nThe following lemma is our key tool for establishing convergence and rate of convergence of algorithms from the class $\\mathcal{WBGA}$(co).\n\\begin{Lemma}[{{\\bf Error Reduction Lemma}}]\\label{lem:erl}\nLet $E$ be a uniformly smooth on $D \\subset X$ convex function with the modulus of smoothness $\\rho(E,D,u)$.\nTake a number $\\epsilon\\ge 0$ and an element $f^\\epsilon \\in D$ such that\n\\[\n\tE(f^\\epsilon) \\le \\inf_{x \\in X} E(x) + \\epsilon, \\ \\ \n\tf^\\epsilon \/ A \\in \\mathcal{A}_1(\\mathcal{D}),\n\\]\nwith some number $A := A(\\epsilon) \\ge 1$.\nThen for any algorithm from the class $\\mathcal{WBGA}$(co) we have for any $m \\ge 1$\n\\begin{multline*}\n\tE(G_m) - E(f^\\epsilon) \\le E(G_{m-1}) - E(f^\\epsilon)\n\t\\\\\n\t+ \\inf_{\\lambda\\ge0} \\Big(-\\lambda t_m A^{-1} (E(G_{m-1})-E(f^\\epsilon)) + 2\\rho(E,D,\\lambda) \\Big).\n\\end{multline*}\n\\end{Lemma}\n\\begin{proof}\nThe main idea of the proof is the same as in the proof of the corresponding one-step improvement inequality for the WCGA (see, for instance, \\cite[Lemma~6.11]{VTbook}).\nIt follows from~\\ref{wbga_er} of the definition of the class $\\mathcal{WBGA}$(co) that\n\\[\n\tE(0) \\ge E(G_1) \\ge E(G_2) \\ldots.\n\\]\nThus if $E(G_{m-1}) - E(f^\\epsilon) \\le 0$ then the claim of Lemma~\\ref{lem:erl} is trivial.\nAssuming $E(G_{m-1}) - E(f^\\epsilon) > 0$, Lemma~\\ref{lem:E'_rho} provides for any $\\lambda \\ge 0$\n\\[\n\tE(G_{m-1} + \\lambda \\varphi_m) \\le E(G_{m-1}) - \\lambda \\<-E'(G_{m-1}),\\varphi_m\\> + 2 \\rho(E,D,\\lambda)\n\\]\nand by~\\ref{wbga_gs} from the definition of the class $\\mathcal{WBGA}$(co) and Lemma~\\ref{lem:F_A1(D)} we get\n\\begin{align*}\n\t\\<-E'(G_{m-1}),\\varphi_m\\> \n\t&\\ge t_m \\sup_{g\\in \\mathcal{D}} \\<-E'(G_{m-1}),g\\> \n\t\\\\\n\t&= t_m\\sup_{\\phi \\in \\mathcal{A}_1(\\mathcal{D})} \\<-E'(G_{m-1}),\\phi\\>\n\t\\ge t_m A^{-1} \\<-E'(G_{m-1}),f^\\epsilon\\>.\n\\end{align*}\nBy~\\ref{wbga_bo} from the definition of the class $\\mathcal{WBGA}$(co) and by convexity~\\eqref{eq:E'_conv2} we obtain\n\\[\n\t\\<-E'(G_{m-1}),f^\\epsilon\\> = \\<-E'(G_{m-1}),f^\\epsilon-G_{m-1}\\> \\ge E(G_{m-1})-E(f^\\epsilon).\n\\]\nThus, by~\\ref{wbga_er} from the definition of the $\\mathcal{WBGA}$(co) we deduce\n\\begin{align*}\n\tE(G_m) &\\le \\inf_{\\lambda\\ge0} E(G_{m-1} + \\lambda\\varphi_m)\n\t\\\\\n\t&\\le E(G_{m-1}) + \\inf_{\\lambda\\ge0} \\Big( -\\lambda t_m A^{-1} (E(G_{m-1}) - E(f^\\epsilon)) + 2\\rho(E,D,\\lambda) \\Big),\n\\end{align*}\nwhich proves the lemma.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:wbga_conv}]\nThe error reduction property~\\ref{wbga_er} of the class $\\mathcal{WBGA}$(co) implies that the sequence of minimizers $\\{G_m\\}_{m=0}^\\infty$ is in $D$ and the sequence $\\{E(G_m)\\}_{m=0}^\\infty$ is non-increasing.\nTherefore, we have\n\\[\n\t\\lim_{m\\to \\infty} E(G_m) = a \\ge \\inf_{x\\in D}E(x).\n\\]\nDenote\n\\[\n\tb := \\inf_{x\\in D} E(x)\n\t\\ \\ \\text{and}\\ \\ \n\t\\alpha := a - b.\n\\]\nWe prove that $\\alpha = 0$ by contradiction.\nIndeed, assume that $\\alpha > 0$.\nThen for any $m \\ge 0$ we have\n\\[\n\tE(G_m) - b \\ge \\alpha.\n\\]\nWe set $\\epsilon = \\alpha\/2$ and find $f^\\epsilon \\in D$ such that\n\\[\n\tE(f^\\epsilon) \\le b + \\epsilon \\ \\ \\text{and}\\ \\ f^\\epsilon\/A \\in \\mathcal{A}_1(\\mathcal{D})\n\\]\nwith some $A := A(\\epsilon) \\ge 1$.\nThen by Lemma~\\ref{lem:erl} we get\n\\[\n\tE(G_m) - E(f^\\epsilon) \\le E(G_{m-1}) - E(f^\\epsilon) + \\inf_{\\lambda\\ge0} (-\\lambda t_m A^{-1}\\alpha\/2 + 2\\rho(E,D,\\lambda)).\n\\]\nSpecify $\\theta := \\min\\left\\{ \\theta_0,\\frac{\\alpha}{8A} \\right\\}$ and take $\\lambda = \\xi_m(\\rho,\\tau,\\theta)$ given by~\\eqref{eq:theta}.\nThen we obtain\n\\[\n\tE(G_m) \\le E(G_{m-1}) - 2\\theta t_m\\xi_m.\n\\]\nThe assumption\n\\[\n\t\\sum_{m=1}^\\infty t_m\\xi_m =\\infty\n\\]\nimplies a contradiction, which proves the theorem.\n\\end{proof}\n\n \n\\begin{proof}[Proof of Theorem~\\ref{thm:wbga_rate}]\nDenote\n\\[\n\ta_n := E(G_n) - E(f^\\epsilon),\n\\]\nthen the sequence $\\{a_n\\}_{n=0}^\\infty$ is non-increasing.\nIf for some $n \\le m$ we have $a_n \\le 0$ then $E(G_m) - E(f^\\epsilon) \\le 0$, which implies\n\\[\n\tE(G_m) - \\inf_{x \\in D} E(x) \\le \\epsilon,\n\\]\nand hence the statement of the theorem holds.\nThus we assume that $a_n > 0$ for $n \\le m$.\nBy Lemma~\\ref{lem:erl} we have\n\\begin{equation}\\label{B3}\n\ta_m \\le a_{m-1} + \\inf_{\\lambda\\ge0} \\left(-\\frac{\\lambda t_m a_{m-1}}{B} + 2\\gamma \\lambda^q\\right).\n\\end{equation}\nChoose $\\lambda$ from the equation\n\\[\n\t\\frac{\\lambda t_m a_{m-1}}{A} = 4\\gamma \\lambda^q,\n\\]\nwhich implies that\n\\[\n\t\\lambda = \\left(\\frac{ t_m a_{m-1}}{4\\gamma A}\\right)^{\\frac{1}{q-1}} .\n\\]\nLet\n\\[\n\tA_q := 2(4\\gamma)^{\\frac{1}{q-1}}.\n\\]\nUsing the notation $p := q\/(q-1)$ we get from~\\eqref{B3}\n\\[\n\ta_m \\le a_{m-1}\\left(1-\\frac{\\lambda t_m}{2A} \\right)\n\t= a_{m-1}\\left(1 - \\frac{t_m^p}{A_q A^{p}} a_{m-1}^{\\frac{1}{q-1}}\\right).\n\\]\nRaising both sides of this inequality to the power $1\/(q-1)$ and taking into account the inequality $x^r\\le x$ for $r\\ge 1$, $0\\le x\\le 1$, we obtain\n\\[\n\ta_m^{\\frac{1}{q-1}} \\le a_{m-1}^{\\frac{1}{q-1}} \\left(1 - \\frac{t^p_m}{A_q A^{p}} a_{m-1}^{\\frac{1}{q-1}}\\right).\n\\]\nThen Lemma~\\ref{lem:y_k} with $y_k := a_k^{\\frac{1}{q-1}}$, $n=0$, $w_k=t^p_m\/(A_qA^{p})$, which provides\n\\[\n\ta_m^{\\frac{1}{q-1}} \\le C(q,\\gamma) A^{p}\\left(C(E,q,\\gamma) + \\sum_{k=1}^m t_k^p\\right)^{-1},\n\\]\nthat implies\n\\[\n\ta_m \\le C(q,\\gamma) A^q\\left(C(E,q,\\gamma) + \\sum_{k=1}^m t_k^p\\right)^{1-q},\n\\]\nwhich proves the theorem.\n\\end{proof}\n\n\n\n\\section{Proofs for Section~\\ref{sec:awbga}}\\label{sec:proofs_awbga}\nIn this section we state the proofs for the results from Section~\\ref{sec:awbga}.\nWe begin with the proof that the algorithms stated in Section~\\ref{sec:awbga_ga} belong to the class $\\mathcal{WBGA}(\\Delta,\\text{co})$.\n\\begin{proof}[Proof of Proposition~\\ref{prp:ga_awbga}]\nIt is easy to see that conditions~\\ref{awbga_gs} and~\\ref{awbga_er} from the definition of the class $\\mathcal{WBGA}(\\Delta,\\text{co})$ are satisfied for all three algorithms.\nCondition~\\ref{awbga_bd} holds with $C_0 = 1$ since for all three algorithms we have for any $m \\ge 1$\n\\[\n\tE(G_m) \\le E(0) + \\delta_m \\le E(0) + 1.\n\\]\nTo guarantee condition~\\ref{awbga_bo}, first note that for any $m \\ge 1$ and any $u > 0$ the definition of modulus of smoothness~\\eqref{eq:mod_smt} provides\n\\[\n\tE((1+u) G_m) + E((1-u) G_m) \\le 2E(G_m) + 2\\rho(E,D_1,u\\|G_m\\|).\n\\]\nAssume that $\\<E'(G_m), G_m\\> \\ge 0$ (the case $\\<E'(G_m), G_m\\> < 0$ is handled similarly).\nThen from convexity~\\eqref{eq:E'_conv1} we get\n\\[\n\tE((1+u) G_m) \\ge E(G_m) + u \\<E'(G_m), G_m\\>\n\\]\nand from the definitions of the corresponding algorithms we obtain\n\\[\n\tE((1-u) G_m) \\ge E(G_m) - \\delta_m.\n\\]\nCombining the above estimates we deduce\n\\[\n\t\\<E'(G_m), G_m\\> \\le \\frac{\\delta_m + 2\\rho(E,D_1, u \\|G_m\\|)}{u}.\n\\]\nTaking infimum over $u > 0$ completes the proof.\n\\end{proof}\n\n\n\\noindent\nNext, we state necessary technical lemmas that will be utilized in the proof of main results.\n\\begin{Lemma}[{\\cite[Lemma~3.2]{VT148}}]\\label{lem:a_delta_0}\nLet $\\rho(u)$ be a non-negative convex on $[0,1]$ function with the property $\\rho(u)\/u\\to0$ as $u\\to 0$.\nAssume that a nonnegative sequence $\\{\\alpha_k\\}_{k=1}^\\infty$ is such that $\\alpha_k\\to0$ as $k\\to\\infty$.\nSuppose that a nonnegative sequence $\\{a_k\\}_{k=0}^\\infty$ satisfies the inequalities\n\\[\n\ta_m \\le a_{m-1} + \\inf_{0\\le\\lambda\\le1}(-\\lambda va_{m-1} + B\\rho(\\lambda)) + \\alpha_m, \\ \\ m = 1,2,3,\\dots\n\\]\nwith positive numbers $v$ and $B$.\nThen\n\\[\n\t\\lim_{m\\to\\infty} a_m = 0.\n\\]\n\\end{Lemma}\n\n\n\\begin{Lemma}[{\\cite[Lemma~3.3]{VT148}}]\\label{lem:a_delta_q}\nSuppose a nonnegative sequence $a_0,a_1,\\dots$ satisfies the inequalities for $m = 1,2,3,\\dots$\n\\[\n\ta_m\\le a_{m-1} + \\inf_{0\\le \\lambda\\le 1}(-\\lambda va_{m-1}+B\\lambda^q) + \\alpha_m, \\ \\ \\alpha_m \\le cm^{-q},\n\\]\nwhere $q\\in (1,2]$, $v\\in(0,1]$, and $B > 0$.\nThen\n\\[\n\ta_m \\le C(q,v,B,a_0,c) \\, m^{1-q} \\le C'(q,B,a_0,c) \\, v^{-q} \\, m^{1-q}.\n\\]\n\\end{Lemma}\n\n\n\\noindent\nLastly, we establish a generalized version of Lemma~\\ref{lem:erl}.\n\\begin{Lemma}[{{\\bf General Error Reduction Lemma}}]\\label{lem:gerl}\nLet $E$ be a uniformly smooth on $S \\subset X$ convex function with the modulus of smoothness $\\rho(E,S,u)$.\nTake a number $\\epsilon \\ge 0$ and an element $f^\\epsilon \\in S$ such that\n\\[\n\tE(f^\\epsilon) \\le \\inf_{x\\in X} E(x) + \\epsilon, \\ \\ \n\tf^\\epsilon\/B \\in \\mathcal{A}_1(\\mathcal{D}),\n\\]\nwith some number $B \\ge 1$.\nSuppose that $G \\in S$ and $\\varphi \\in \\mathcal{D}$ satisfy the following conditions\n\\begin{gather}\n\t\\label{C1}\n\t\\<-E'(G),\\varphi\\> \\ge \\theta \\sup_{g\\in \\mathcal{D}} \\<-E'(G),g\\>, \\ \\ \\theta \\in (0,1];\n\t\\\\\n\t\\label{C2}\n\t|\\<E'(G),G\\>| \\le \\delta, \\ \\ \\delta \\in [0,1].\n\\end{gather}\nThen we have\n\\begin{align*}\n\t\\inf_{0\\le\\lambda\\le1} E(G+\\lambda\\varphi)\n\t&\\le E(G)\n\t\\\\\n\t&+ \\inf_{0\\le\\lambda\\le1} (-\\lambda \\theta B^{-1} (E(G) - E(f^\\epsilon)) + 2\\rho(E,S,\\lambda)) + \\delta.\n\\end{align*}\n\\end{Lemma}\n\\begin{proof}\nIf $E(G_{m-1}) - E(f^\\epsilon) \\le 0$ then the claim of Lemma~\\ref{lem:gerl} is trivial.\nAssuming $E(G_{m-1}) - E(f^\\epsilon) > 0$, Lemma~\\ref{lem:E'_rho} provides for any $\\lambda \\ge 0$\n\\[\n\tE(G + \\lambda \\varphi) \\le E(G) - \\lambda \\<-E'(G),\\varphi\\> + 2 \\rho(E,S,\\lambda).\n\\]\nBy~\\eqref{C1} and Lemma~\\ref{lem:F_A1(D)} we get\n\\begin{align*}\n\t\\<-E'(G),\\varphi\\>\n\t&\\ge \\theta \\sup_{g \\in \\mathcal{D}} \\<-E'(G),g\\>\n\t\\\\\n\t&= \\theta\\sup_{\\phi \\in \\mathcal{A}_1(\\mathcal{D})} \\<-E'(G),\\phi\\> \\ge \\theta B^{-1} \\<-E'(G),f^\\epsilon\\>.\n\\end{align*}\nBy~\\eqref{C2} and by convexity~\\eqref{eq:E'_conv2} we obtain\n\\[\n\t\\<-E'(G),f^\\epsilon\\> = \\<-E'(G),f^\\epsilon-G\\> + \\<-E'(G),G\\> \\ge E(G)-E(f^\\epsilon)-\\delta.\n\\]\nThus\n\\[\n\tE(G + \\lambda\\varphi) \\le E(G) - \\lambda \\theta B^{-1} (E(G) - E(f^\\epsilon)) + 2\\rho(E,S,\\lambda)) + \\delta,\n\\]\nwhich proves the lemma.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:awbga_conv}]\nAssumption~\\ref{awbga_bd} from the definition of the class $\\mathcal{WBGA}(\\Delta,\\text{co})$ implies that for any $m \\ge 0$\n\\[\n\tE(G_m) \\le E(0) + C_0\n\t\\ \\ \\text{and}\\ \\\n\tG_m \\in D_1.\n\\]\nThen from Lemma~\\ref{lem:gerl} with $S = D_1$, $G = G_{m-1}$, $\\varphi = \\varphi_m$, $\\delta = \\epsilon_m$, $\\theta = t$, $B = A(\\epsilon)$ and property~\\ref{awbga_er} from the definition of the class $\\mathcal{WBGA}(\\Delta,\\text{co})$ we obtain\n\\begin{align}\\nonumber\n\tE(G_m)\n\t&\\le \\inf_{0\\le\\lambda\\le1} E(G_{m-1} + \\lambda\\varphi_m) + \\delta_m\n\t\\\\\\nonumber\n\t&\\le E(G_{m-1}) + \\inf_{0\\le\\lambda\\le 1} \\big(-\\lambda t A^{-1} (E(G_{m-1}) - E(f^\\epsilon))\n\t\\\\\\label{eq:E(G_m)}\n\t&\\phantom{E(G_{m-1}) + \\inf_{0\\le\\lambda\\le 1} \\big(-\\lambda}\n\t+ 2\\rho(E,D_1,\\lambda) \\big) + \\delta_m + \\epsilon_m.\n\\end{align}\nDenote\n\\[\n\ta_n := \\max\\big\\{ E(G_n) - E(f^\\epsilon), 0 \\big\\}.\n\\]\nNote that under our assumptions $t \\in (0,1]$ and $A := A(\\epsilon) \\ge 1$ we always have\n\\[\n\ta_{m-1} + \\inf_{0\\le\\lambda\\le 1}(-\\lambda t A^{-1} a_{m-1} + 2\\rho(E,D_1,\\lambda)) \\ge 0.\n\\]\nTherefore estimate~\\eqref{eq:E(G_m)} implies\n\\begin{equation}\\label{eq:a_m}\n\ta_m \\le a_{m-1} + \\inf_{0\\le\\lambda\\le 1} (-\\lambda t A^{-1} a_{m-1} + 2\\rho(E,D_1,\\lambda)) + \\delta_m + \\epsilon_m.\n\\end{equation}\nWe apply Lemma~\\ref{lem:a_delta_0} with $v = tA^{-1}$, $B = 2$, $\\rho(u) = \\rho(E,D_1,\\lambda)$, and $\\alpha_m = \\delta_m + \\epsilon_m$ to obtain\n\\[\n\t\\lim_{m\\to\\infty} a_m = 0,\n\\]\nwhich implies\n\\[\n\t\\limsup_{m\\to\\infty} E(G_m) \\le \\epsilon + \\inf_{x\\in D_1} E(x)\n\\]\nand, due to the arbitrary nature of choice of $\\epsilon > 0$,\n\\[\n\t\\lim_{m\\to\\infty} E(G_m) = \\inf_{x\\in D_1} E(x),\n\\]\nwhich completes the proof.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:awbga_rate}]\nFrom estimate~\\eqref{eq:a_m} we get\n\\begin{align*}\n\ta_m\n\t&\\le a_{m-1} + \\inf_{0\\le\\lambda\\le 1} (-\\lambda t A^{-1} a_{m-1} + 2\\rho(E,D_1,\\lambda)) + \\delta_m + \\epsilon_m\n\t\\\\\n\t&\\le a_{m-1} + \\inf_{0\\le\\lambda\\le 1}(-\\lambda t A^{-1} a_{m-1} + 2\\gamma\\lambda^q) + \\delta_m + \\epsilon_m.\n\\end{align*}\nApplying Lemma~\\ref{lem:a_delta_q} with $v = t A^{-1}$, $B = 2\\gamma$, and $\\alpha_m = \\delta_m + \\epsilon_m$ completes the proof.\n\\end{proof}\n\n\n\n\n\n\\section*{Acknowledgments}\nThe first author acknowledges support given by the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC., for the U.S. Department of Energy under Contract DE-AC05-00OR22725.\n\nThe work was supported by the Russian Federation Government Grant N{\\textsuperscript{\\underline{o}}}14.W03.31.0031. The paper contains results obtained in frames of the program \"Center for the storage and analysis of big data\", supported by the Ministry of Science and High Education of Russian Federation (contract 11.12.2018 N{\\textsuperscript{\\underline{o}}}13\/1251\/2018 between the Lomonosov Moscow State University and the Fond of support of the National technological initiative projects).\n\n\\section*{References}\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis article is a complementary part to the work done in \\cite{CMFixed} with \nits own independent interest. We discuss geometric conditions under which \nthere are no invariant Beltrami differentials supported on the dissipative set \nof a rational map $R$. \n\nIn this paper we will always assume that the conservative set of the action of \n$R$ belongs to the Julia set.\n\nNow, let us introduce the geometric objects to be treated in this \npaper. \n\nWe denote by $P(R)$ the closure of the postcritical set of $R$ and consider \nthe surface $S_R:=\\bar{\\C}\\setminus P(R)$.  The surface $S_R$ is not always \nconnected, however, on each connected component of $S_R$ we fix a \nPoincar\\'e hyperbolic metric  and denote by $\\lambda$ the family of all these \nmetrics.\n\nLet $Q(S_R)$ be the subspace of $ L_1(S_R)$ of holomorphic \nintegrable functions on $S_R.$\n\nA rational map $R$ defines a complex Push-Forward map on $L_1(\\C)$, with \nrespect to the Lebesgue measure $m$, which is a contracting endomorphism and \nis called the complex Ruelle-Perron-Frobenius, for shortness  \nRuelle operator. The Ruelle operator has the following formula:\n\n\\[R^*(\\phi)(z)=\\sum_{y \\in R^{-1}(z)} \\frac{\\phi(y)}{R'(y)^2}\nR(\\zeta)\\]\n\\[=\\sum_i \\phi(\\zeta_i)(z)\\zeta'_i(z)\\] where $\\zeta_i$ is any local complete \nsystem of branches of $R^{-1}.$ The space $Q(S_R)$ is invariant under the action \nof the Ruelle operator. \nThe Beltrami operator $Bel:L_\\infty(\\C)\\rightarrow L_\\infty(\\C)$ given by \n\\[Bel(\\mu)=\\mu(R)\\frac{\\overline{R'}}{R'}\\] is dual to the Ruelle operator \nacting on $L_1(\\C)$. \n\nThe fixed point space $Fix(B)$ of the Beltrami operator is called the \n\\textit{space of invariant Beltrami differentials}. An element $\\alpha \\in \nL_\\infty(\\C)$ is called non trivial if and only if the functional given by \n\\[v_\\alpha(\\phi) = \\int \\phi \\alpha\\]  is non zero on $Q(S_R).$ The norm of \n$v_\\alpha$ in $Q^*(S_R)$, for a non trivial element $\\alpha$,  is called the \n\\textit{Teichm\\\"uller norm} of $\\alpha$ and it is denoted by $\\| \\alpha \n\\|_{T}.$ \n\nA non trivial element $\\alpha$ is called \\textit{extremal} if and only if the \n$\\|\\alpha\\|_\\infty=\\|\\alpha\\|_T.$ \n\nA sequence of unit vectors $\\{\\phi_i\\}$ is called a \n\\textit{Hamilton-Krushkal} sequence, for short HK-sequence, for an extremal  \nelement $\\alpha$ if and only if \\[ \\lim_{i\\rightarrow \n\\infty}|v_\\alpha(\\phi_i)|=\\|\\alpha\\|_\\infty.\\]\n\nA HK sequence $\\{\\phi_i\\}$ is called \\textit{degenerated} if converge to $0$ \nuniformly on compact sets.\n \nLet $T:\\mathcal{B}\\rightarrow \\mathcal{B}$ be a linear contraction of a Banach \nspace $\\mathcal{B}$. An element \n$b\\in \\mathcal{B}$ is called \\textit{mean ergodic} with respect to $T$ if and \nonly if the sequence of Ces\\`aro averages with respect to $T$, given by\n$C_n(b)=\\frac{1}{n}\\sum_{i=0}^{n-1} T^i(b)$, forms a weakly precompact family. \nIndeed (see Krengel \\cite{Krengel}), when $\\mathcal{B}$ is weakly complete then, for \na mean ergodic element $b$,  the sequence $C_n(b)$ converges in norm to its limit, \nthis limit always is  a fixed element of $T$. If every element $b\\in \\mathcal{B}$ \nis mean ergodic with respect to $T$ then the operator $T$ is called \nmean-ergodic.\n\nBy the Bers Representation Theorem, the space $Q^*(S_R)$ is linearly \nquasi-isome\\-trically isomorphic to the \\textit{Bergman} space $B(S_R)$ which \nis \nthe space of holomorphic functions $\\phi$ on $S_R$ with the \nnorm $\\|\\lambda^{-2}\\phi\\|_{L_\\infty(S_R)}.$\n\nIn the case where $S_R$ has finitely many components, a classical theorem, \nsee for example \\cite{Matsuzaki} and references within,  states that \n$Q(S_R)\\subset B(S_R)$ if and only if the infimum of the length of simple \nclosed geodesics is bounded away from $0$.\n\n\\section{Main Theorem}\nLet $X$ be an $R$ invariant measurable set, then the set $W:=\\bigcup \nR^{-n}(X)$ is completely invariant. In the following theorem we will only \nconsider Ces\\`aro averages with respect to the Ruelle operator $R^*$ in \n$L_1(W).$\n\n\n\\begin{theorem}\\label{MainTechnicalThm}\nLet $X$ be an $R$ invariant measurable subset such that the restriction map \n$r(\\phi)= \\phi|_X$ from $Q(S_R)$  to $L_1(X) $ is weakly precompact.  Then \nevery $\\phi \\in Q(S_R)$ is mean ergodic with respect to $R^*$ in  $L_1(W)$. \n\\end{theorem}\n\n\\begin{proof}\nIf $X$ is $R$ invariant then the Ruelle operator $R^*$ defines an endomorphism \nof $L_1(X)$. Given $\\phi \\in Q(S_R)$, the family of Ces\\`aro averages \n$C_n(\\phi)$ restricted on $X$ forms a weakly precompact subset of $L_1(X).$ \nWe claim that $C_n(\\phi)$ converges in norm  on $L_1(X).$  Indeed, first \nwe show that every weak accumulation point  of $C_n(\\phi)$ is a fixed point for \nthe Ruelle operator. Let $f$ be  the weak limit of $C_{n_i}(\\phi)$ for some \nsubsequence $\\{n_i\\},$ then $R^*(f)$ is the weak limit of $R^*(C_{n_i}(\\phi))$.\nBy the Fatou Lemma \n$$\\int_X |f-R^*(f)| \\leq \\liminf \\int_X |C_{n_i}(\\phi)-R^*(C_{n_i}(\\phi))|$$\n$$\\leq \\liminf \\| C_{n_i}(\\phi)-R^*(C_{n_i}(\\phi))\\|_{L_1(S_R)}$$ \n$$\\leq \\limsup \\|C_{n_i}(I-R^*)(\\phi)\\|_{L_1(S_R)}.$$\n\nBut \n$$ \\|C_{n_i}(I-R^*)(\\phi)\\|_{L_1(S_R)}\\leq \\frac{2}{n_i}\\|\\phi\\|_{L_1(S_R)}.$$\nThen $f$ is a non zero fixed point of Ruelle operator. As in \\cite{MakRuelle} \nwe have that $|f|$ defines a finite absolutely continuous invariant measure. \nHence, the support of $f$ is a non trivial subset of the conservative set of \n$R.$ By Lyubich's Ergodicity theorem (see \\cite{Mc1} and \\cite{LyuTypical}) and \nthe fact \nthat $X$ does not intersect the postcritical set we have $X=W=S_R$. But, \nMcMullen's Theorem (Theorem 3.9 of \\cite{Mc1}) implies that in this case $R$ is \na, so called, \\textit{flexible Latt\\`es} map. Furthermore, \nthe space $Q(S_R)$ is finitely dimensional and hence $R^*$ is a compact \nendomorphism of $Q(S_R)$, it follows that $R^*$ is mean ergodic on $Q(S_R)$. \n\nTherefore, if $R$ is not a flexible Latt\\`es map then any weak limit of \n$C_n(\\phi)$ is $0$. Since the weak closure of convex bounded sets is equal to \nthe closure in norm of convex bounded sets, we conclude our claim.\n\nNow let $W_n=R^{-n}(X)$, one can inductively prove that  \n$\\phi|_{W_n}$ is mean ergodic on $L_1(W_n)$. Indeed, let $\\psi_n=\\phi|_{W_n}$, \nsince $R^*:L_1(W_n)\\rightarrow L_1(W_{n-1}) \\subset L(W_n)$ \nand $R^*(\\psi_n)=R^*(\\phi)|_{W_{n-1}}$, then by arguments above we are done. \n\nNow consider $\\phi|_{W}-\\phi|_{W_n}$, the $L_1$ norm of this difference \nconverges to $0$ in $L_1(W)$, since the Ces\\`aro averages does not expand the \n$L_1$ norm we have $$\\|C_k(\\phi|_{W}-\\phi|_{W_n})\\| \\leq \n\\|\\phi|_{W}-\\phi|_{W_n}\\|.$$\n\nHence $C_k(\\phi|_{W})$ converges to $0$ and $\\phi$ is mean ergodic on $L_1(W)$. \n\n\\end{proof}\n\nNow we state our Main Theorem.\n\n\\begin{theorem}\\label{MainTheorem}\nLet $R$ be a rational map and let $X\\subset S_R$ be an invariant \nmeasurable set of positive Lebesgue measure.\nAssume that the restriction map $r(\\phi)=\\phi|_X$ from $Q(S_R)$ into \n$L_1(X)$ is weakly precompact. If $\\mu$ is a  non trivial invariant Beltrami \ndifferential, then $m(supp (\\mu)\\cap X)>0$ if and only if $R$ is a flexible \nLatt\\`es map.\n\\end{theorem}\n\n\\begin{proof}\nAssume that $R$ is a flexible Latt\\`es map. Then $R$ is ergodic on the Riemann \nsphere and therefore the support of any invariant Beltrami differential $\\mu$ \nis the whole Riemann sphere. Hence, if $X$ is invariant of positive Lebesgue \nmeasure then $m(supp(\\mu)\\cap X)=m(X)>0.$ \n\nAgain let $W=\\bigcup R^{-n}(X)$. Now let $\\mu$ be a non \ntrivial invariant Beltrami differential supported on $W$. If $R$ is not \nLatt\\`es, then for any $\\phi \\in Q(S_R)$  we have $$\\int_{S_R} \\phi \\mu \n=\\int_{S_R} \\mu C_k(\\phi)=\\int_{W} \\mu C_k(\\phi).$$ \n\nBy Theorem \\ref{MainTechnicalThm}, the right hand side converges to $0$ as $k$ \nconverges to $\\infty$. Hence  $\\int \\phi \\mu=0$ for every quadratic \ndifferential $\\phi$ and the functional $\\phi\\mapsto \\int \\phi \\mu$ is $0$ on \n$Q(S_R).$ Which contradicts the assumption that $\\mu$ is non trivial. \n\n\\end{proof}\n\n\nIn the proofs of the previous theorems, the only ingredient was the\nprecompactness of the Ces\\`aro averages $C_n(\\phi)$. Hence, \nit is enough to assume the weak precompactness only of \nCes\\`aro averages on elements of $Q(S_R)$. \nBy results of the second author in \\cite{MakRuelle}, see also a related work \non \\cite{CMFixed}, it  is enough to consider the Ces\\`aro averages  of \nrational functions in $Q(S_R)$ having poles only on the set of critical values. \n\n\\section{Compactness}\nWe want to discuss conditions under which the restriction map $\\phi\\mapsto \n\\phi|_A$ is weakly precompact. Unfortunately, so far we have not found \nconditions where the restriction is weakly precompact but not compact.\nLet us start with the following observations and definitions. \n\n\\begin{definition}\nA rational map $R$ satisfies the $B$-condition if and only if for any $\\phi\\in \nQ(S_R)$ we have $$\\|\\lambda^{-2}(z) \n\\phi(z)\\|_{L_\\infty(S_R)}\\leq C \\|\\phi(z)\\|_{L_1(S_R)},$$ where $C$ \nis a constant independent of $\\phi.$  \n\\end{definition}\nIn other words, if $R$ satisfies the $B$-condition, then\n$Q(S_R)\\subset B(S_R)$  and the inclusion\nmap $Q(S_R)\\rightarrow B(S_R)$ is continuous. As it was noted on the \nintroduction, this happens when $S_R$ has finitely many components and \nthe infimum of the length of the simple closed geodesics is bounded away from \n$0.$\n\n\\begin{proposition}\\label{prop.Bcond.comp}\nIf $R$ satisfies the $B$-condition and $\\lambda(X)<\\infty$ then the \nrestriction map is compact.\n\\end{proposition}\n\n\\begin{proof}\nIf $R$ satisfies the $B$-condition then $$\\lambda^{-2}|\\phi(z)|\\leq sup_{z\\in \nS_R} |\\lambda^{-2}(z) \\phi(z) | \\leq C \\|\\phi \\|_1,$$ hence $|\\phi(z)|\\leq \nC\\| \n\\phi\\|_1 \\lambda^{2}(z),$  by Lebesgue Theorem \nthe restriction map is compact.\n\\end{proof}\n\nUsing Theorem \\ref{MainTheorem} and Proposition \\ref{prop.Bcond.comp} \nwe have the following.\n\\begin{corollary}\\label{cor.area}\nIf $R$ satisfies the $B$-condition and $X$ is an invariant set of \npositive Lebesgue measure with $Area_\\lambda(X)<\\infty$. If $\\mu$ is a non \nzero invariant Beltrami differential, then $m(supp(\\mu)\\cap X)>0$ if and only \nif $R$ is a flexible Latt\\`es map. \n\\end{corollary}\nIn general,  the finiteness of the hyperbolic area of $X$ does not \nimply the finiteness of hyperbolic area of $W$. Generically, it \ncould be that the hyperbolic area of $W$ is infinite regardless of the area of \n$X$.  \nOn the other hand, by Corollary \\ref{cor.area}, if $R$ satisfies the \n$B$-condition and the hyperbolic area \n$Area_\\lambda(J(R))$ is bounded then $R$ satisfies Sullivan's conjecture. \nHowever, in this situation, we believe that the following stronger statement \nholds true:\n\nThe $Area_\\lambda(J(R))<\\infty$ if \nand only if either $m(J(R))=0$ or $R$ is postcritically finite. \n\nIn fact, we do not know if the $B$-condition is sufficient on this statement.\n\nNow we consider the more general condition when the restriction map $r_X$ is \ncompact. This condition, in some sense,  reflects the geometry of the \npostcritical set.\n \nOn the product $S_R\\times S_R\\subset \\C^2$  there exist a unique \nfunction $K(z,\\zeta)$ which is characterized by the following conditions.\n\n\\begin{enumerate}\n \\item $K(\\zeta,z)=-\\overline{K(z,\\zeta)}$\n \\item For any $\\zeta_0\\in S_R$, the function $\\phi_{\\zeta_0}(z)=K(z,\\zeta_0)$\nbelongs to the intersection $Q(S_R)\\cap B(S_R).$ \n \\item If $z_0,\\zeta_0$  belong to different components of $S_R$, then \n$K(z_0,\\zeta_0)=0.$\n\\item The operator $P(f)(z)=\\int \\lambda^{-2}(\\zeta) K(z,\\zeta)f(\\zeta) d\\zeta \nd\\bar{\\zeta}$ from $L_1(S_R)$ to $Q(S_R)$ is a continuous surjective \nprojection.\n\\end{enumerate}\n\nIn fact, the function $K(z,\\zeta)$ is defined on any planar hyperbolic Riemann \nsurface $S$. In particular, when the surface $S$ is the unit disk $\\mathbb{D}$ \nthe function  $K(z,\\zeta)$ has the formula $$K(z,\\zeta)=\\frac{3}{2}\\pi i \nK_\\mathbb{D}(z,\\zeta)^2$$ where \n$K_\\mathbb{D}(z,\\zeta)=[\\pi(1-z\\bar{\\zeta})^2]^{-1}$ is the classical Bergman \nKernel function on the unit disk.  For further details on these facts see for \nexample Chapter 3, \\S 7 of the book of I. Kra \n\\cite{KraBook} .\n\n\nNow we consider the following function $$\\omega(\\zeta,z)=\\lambda^{-2}(\\zeta) \nK(z,\\zeta)$$ and $$w(z)=\\omega(z,z).$$  \n\n\nThe following proposition is a consequence of H\\\"older inequality and appear as \nLemma 2 on Ohtake's paper \\cite{Ohtakedeform}.\n\n\\begin{proposition}\\label{prop.bound.comp}\n If $X$ has positive measure and $$\\int_X |w|<\\infty$$ then the restriction \n$r_X:\\phi\\mapsto \\phi|_X$ from $Q(S_R)$ to $L_1(X)$ is compact.\n\\end{proposition}\n\\begin{proof}\n\n\nWe follow arguments of Lemma 2 in \\cite{Ohtakedeform}. If $D$ is a \ncomponent of $S_R$, then by H\\\"older's \ninequality as in Lemma 2 of \\cite{Ohtakedeform},   we have that $$|(\\phi|_D)(z)| \n\\leq \nC|(w|_D)(z)|\\int_D |\\phi| $$ where the constant $C$ does not depend on $D$.  \nSince $S_R$ is a countable union of components, then \n\n$$|\\phi(z)|\\leq C |w| \\|\\phi(z)\\|.$$ As $w$ is integrable on $X$ then by \napplying once again the Lebesgue Theorem we complete the proof.\n\\end{proof}\n\nAs a consequence we have:\n\\begin{corollary}\\label{cor.finite}\n If $\\int_{J(R)} |w|<\\infty$ then $R$ satisfies Sullivan's conjecture.\n\\end{corollary}\n\n\\begin{proof}\n Follows from Theorem \\ref{MainTheorem} and Proposition \\ref{prop.bound.comp}.\n\\end{proof}\n\n\nRemarks: \n\\begin{enumerate}\n\\item If $R$ satisfies the $B$ condition then by Classical results, see the \ncomments before Proposition 1 in \\cite{Ohtake}, we have that $w(z)\\leq C \n\\lambda^2(z)$ where \n$C$ does not depend on $z.$ Partially, if $X$ has bounded hyperbolic area then \n$w(z)$ is integrable on $X$, hence the conditions of Proposition \n\\ref{prop.bound.comp} implies Proposition \\ref{prop.Bcond.comp}.\nAs it is mentioned in \\cite{Ohtake}, the conditions in Proposition \n\\ref{prop.Bcond.comp} are strictly weaker than conditions of Proposition \n\\ref{prop.bound.comp}.\n\\item Moreover, by other result of Ohtake (Proposition 3 in \n\\cite{Ohtakedeform}) we note \nthat in general, the boundedness of the hyperbolic area is not a quasiconformal \ninvariant.\n\n\n\\end{enumerate}\n\n\nIn other words, Proposition \\ref{prop.Bcond.comp} and Proposition \n\\ref{prop.bound.comp} states that if $X$ is completely invariant positive \nmeasure set and satisfying an integrability condition then $X$ can not support \nextremal \ndifferentials with  Hamilton-Krushkal degenerated sequences.\n\nHence, Corollary \\ref{cor.area} and Corollary \\ref{cor.finite}, in the case \nwhen $X$ is a completely invariant, derive from results in \n\\cite{CMFixed}. Together, the corollaries mean that \nif a map $R$ has an invariant line field which does not allow a  \nHamilton-Krushkal degenerated sequences on $Q(S_R)$, then $R$ is a Latt\\`es map \nif and only if the postcritical set has Lebesgue measure zero.\n\nLet $Y_n$ be an exhaustion of $S_R$ by compact subsets such that the Lebesgue \nmeasure of \n$Y_{n+1}\\setminus Y_n$ converge to zero. Let $P_n$ be the sequence of \nrestrictions \n$P_n:L_1(S_R)\\rightarrow L_1(S_R)$ given by $P_n(f)=\\chi_{n} P(f)$ where \n$\\chi_{n}$ is the characteristic function on $Y_n$. \nImmediately from the definition we have the following facts:\n\n\\begin{enumerate}\n \\item For each $n$, the map  $P_n$ is a compact operator.\n \\item The limit \\[\\lim_{n\\rightarrow \\infty} \n\\|P_n(f)-P(f)\\|_{L_1(S_R)}\\rightarrow 0\\] for all $f$ on $L_1(S_R)$.\n\\end{enumerate}\n\nWe have the following Theorem:\n\n\\begin{theorem}\\label{th.exhaustion}\n Let $\\mu\\neq 0$ be an extremal invariant Beltrami differential, then the \nfollowing conditions are equivalent:\n\n\\begin{itemize}\n \\item The map $R$ is a flexible Latt\\`es map. \n \\item  There exist an exhaustion of compact sets $Y_n$ as defined above \nsuch that the following inequality is true:  \\[\\inf_{n} \\|P_n -P\\|_{L_1(supp \n(\\mu))}<1.\\]\n\\end{itemize}\n  \n\\end{theorem}\n\n\\begin{proof}\nAssume that $R$ is a flexible Latt\\`es map, then $Q(S_R)$ is finitely \ndimensional then the operators $P_n$ converge to $P$ by norm. Hence,\nthe infimum $\\inf_{n} \\|P_n -P\\|_{L_1(supp(\\mu))}=0.$ \n\nNow, let us assume that $\\inf_{n} \\|P_n -P\\|_{L_1(supp (\\mu))}<1$. We show \nthat this condition implies that $\\mu$ does not accept degenerated \nHamilton-Krushkal sequences. Indeed, assume that  $\\{\\phi_n\\}$ is a degenerated \nHamilton-Krushkal sequence for $\\mu$. By assumption, there exist $n_0$ such \nthat \n\\[\\sup_{f\\in L_1(supp(\\mu)),\\|f\\|=1} \\int |P_{n_0}(f)-P(f)|=r<1.\\] Since \n$\\phi_n$ is degenerated and by the compactness of $P_{n_0}$ we have that \n\\[\\lim_{j\\rightarrow \\infty} \\|P_{n_0}(\\phi_j)\\|_{L_1(S_R)}\\rightarrow 0.\\] \nHence\n\\[\\|\\mu\\|_\\infty=\\lim_{j} \\bigg |\\int \\mu \\phi_j\\bigg |=\\lim_j \\bigg \n|\\int_{supp(\\mu) }\\mu \n\\phi_j\\bigg |\\]\n\\[=\\lim_{j} \\bigg |\\int_{supp(\\mu)} \\mu(P_{n_0}(\\phi_j)-P(\\phi_j))\\bigg |\\]\n\\[\\leq \\|\\mu\\|_\\infty \\sup_{f\\in L_1(supp(\\mu)),\\|f\\|=1} \\int \n|P_{n_0}(f)-P(f)|=r \\|\\mu\\|_\\infty < \\|\\mu\\|_\\infty.\\] \nWhich is a contradiction. \n\nApplying the Corollary 1.5 in \\cite{EarleLi}, the extremal differential  $\\mu$ \ndoes \nnot accept Hamilton{-}Krushkal degenerated sequences if \nand only if there exist $\\phi$ in $Q(S_R)$ and a suitable constant $K$ such \nthat \n\\[v_\\mu(\\gamma)=K \\int \\frac{|\\phi|}{\\phi} \\gamma.\\]\n\n\n\nHence, for any $\\gamma$ in $Q(S_R)$ we have\n\n$$\\int \\frac{|\\phi|}{\\phi}R^*(\\gamma)=\\int\\frac{|\\phi|}{\\phi} \\gamma$$ and\n$$1=\\int\\frac{|\\phi|}{\\phi}R^*(\\phi).$$ This implies that \n$$\\frac{|R^*(\\phi)|}{R^*(\\phi)}=\\frac{|\\phi|}{\\phi}$$ but since $\\phi$ is \nholomorphic then $\\phi$ is a non zero fixed point on $Q(S_R)$. \nUsing arguments of the proof of Theorem \\ref{MainTechnicalThm} we are \ndone. \n\\end{proof}\n\nThe following Proposition is an illustration of when the conditions of Theorem \n\\ref{th.exhaustion} are fulfilled. \n\n\\begin{proposition} If $R$ is a rational map satisfying the $B$ condition.\nIf $A$ is a measurable subset of $S_R$ so that \n\\[\\int_A \\int_{S_R} |K(z,\\zeta)| dz\\wedge d\\bar{z} \\wedge d\\zeta \\wedge \nd\\bar{\\zeta}<\\infty\\]\nthen for any exhaustion of $S_R$ by compact sets $Y_n$ and operators \n$P_n$ defined as above we have $\\lim \\|P_n-P\\|_{L_1(A)}=0.$ \n\\end{proposition}\n\n\\begin{proof} Let $Y_n$ be an exhaustion of compact sets as above.  Since \n$K(z,\\zeta)$ is absolutely integrable on $A\\times S_R$ then \n$$|\\chi_{n} K(z,\\zeta)|\\leq |K(z,\\zeta)|$$ and\n$\\chi_{n}K(z,\\zeta)\\rightarrow K(z,\\zeta)$ pointwise on $A\\times S_R.$ By the \nLebesgue theorem $$\\inf \\int_{A}\\int_{S_R} |K(z,\\zeta)-\\chi_{n} \nK(z,\\zeta)|=0.$$\nFor all $\\phi \\in Q(S_R)$, we have $$\\|P_n(\\phi)-P(\\phi)\\|_{L_1(A)}$$  \n\n\\[\\leq \\int_A |P_n(\\phi)-P(\\phi)|\\leq \\int_{A} \\int_{S_R} \n|\\lambda^{-2}(\\zeta)\\phi(\\zeta)(K(z,\\zeta)-\\chi_n K(z,\\zeta))|d\\zeta dz\\]\n\n\n\\[\\leq \\|\\lambda^{-2} \\phi\\|_\\infty \\int_{A} \\int_{S_R} |K(z,\\zeta)-\\chi_n \nK(z,\\zeta)| d\\zeta dz \n\\]\nwhich by the $B$-condition we have that the latter is \n\\[\\leq C \\|\\phi\\|_{L_1(S_R)}\\int_{A} \\int_{S_R} |K(z,\\zeta)-\\chi_n \nK(z,\\zeta)| d\\zeta dz .\\] For some constant $C$ which does not depend on \n$\\phi$.\n\nNow let $f\\in L_1(A)$, since $P$ is a projection then $f=\\phi+\\omega$ where \n$\\phi\\in \nQ(S_R),$ $P(\\omega)=P_n(\\omega)=0$ and $$\\|\\phi\\|_{Q(S_R)}\\leq \\|P\\| \n\\|f\\|_{L_1(A)}.$$\n Hence $\\lim \\|P_n-P\\|_{L_1(A)}= 0.$ \n\n\n\\end{proof}\nFinally we characterize a Latt\\`es map in terms of the geometry of $Q^*(S_R).$ \nWe start with the following definitions.\n\\begin{definition} \n\\begin{enumerate} \n \\item A set $L$ in $Q^*(S_R)$ is called a geodesic ray if $L$ \nis an isometric image of the non negative real numbers $\\mathbb{R}_+$. \n \\item Let   $L_1$ and $L_2$ be geodesic rays with parameterizations \n$\\psi_1:\\mathbb{R}_+\\longrightarrow L_1$ and \n$\\psi_2:\\mathbb{R}_+\\longrightarrow L_2$ respectively. The pair of rays $L_1$ \nand $L_2$ are  called equivalent if \n$$\\limsup_{t\\rightarrow \\infty} \\| \\psi_1(t)-\\psi_2(t) \\|_T\\leq d <\\infty$$ for \nsome $d.$\n\\item An element $v$ in $Q^*(S_R)$ is called asymptotically finite if the \nnumber of equivalence classes of geodesic rays in $Q^*(S_R)$ containing $0$ and \n$v$ is finite.\n\\end{enumerate}\n \n\n \n\\end{definition}\n\n\nNow we characterize rational maps which have asymptotically finite non trivial \ninvariant Beltrami differentials.\n\n\\begin{theorem}\\label{thm. uniq.equiv}\nAssume that $S_R$ is connected and let $\\mu$ be non trivial an invariant \nBeltrami differential for $R$ supported on $S_R$. Then the functional \n$v_\\mu(\\phi)=\\int \\phi\\mu$ is asymptotically finite if and only if $R$ is \nLatt\\`es.\n\\end{theorem}\n \n\n\\begin{proof}\n If $R$ is a Latt\\`es map then $Q^*(S_R)$ is finitely dimensional and then  \nthere is only a unique geodesic ray passing through any \npair of points in $Q^*(S_R)$ see \\cite{EarleLi} and \\cite{GardLakic}. \n\nReciprocally, suppose that the functional $v_\\mu$ is asymptotically \nfinite. Let us first assume \nthat $\\|v_\\mu \\|_{Q^*(S_R)}=\\|\\mu\\|_{L_\\infty}.$ Then by Corollary 6.4 in \n\\cite{EarleLi}, if $\\mu$ accept degenerated \nHamilton-Krushkal sequences there exist $\\C$-linear isometry \n$I:\\ell_\\infty\\rightarrow Q^*(S_R)$ such that if $m$ is the constant \nsequence \nwith value $\\|\\mu\\|_\\infty$ then $I(m)=v_\\mu.$\n \n\n\nNow let $\\{e_i\\}$ be the canonical basis of $\\ell_\\infty$. Then as in \n\\cite{EarleLi}, we define geodesic rays in $\\ell_\\infty$ as follows:\n\nFor any $r\\geq \\|\\mu\\|_\\infty$ and \n\n$$\\psi_{r,i}(t)=\\bigg \\{ \\begin{array}{l} \nt\\cdot m \\textnormal{ for } t\\leq r. \\\\\nr\\cdot m + (t-r)\\|\\mu\\|_\\infty e_i \\textnormal{ for } t>r.\n\\end{array}$$\n\nBut for all $i_0, r_1,r_2$,  $$\\limsup_{t\\rightarrow \\infty}\n\\|\\psi_{i_0,r_1}(t)-\\psi_{i_0,r_2}(t)\\|_{\\ell_\\infty}\\leq \n|r_1-r_2|\\|\\mu\\|_\\infty.$$\n\n\nAlso for all $i\\neq j$ and all $r$ we have \n\\[\\limsup_{t\\rightarrow \\infty} \\|\\psi_{j,r}(t)-\\psi_{i,r}(t)\\|_\\infty\\]\n\\[=\\limsup_{t\\rightarrow \\infty}\\|\\mu\\|_\\infty t \\|e_i-e_j\\|=\\infty.\\]\n\n\nBut the existence of the isometry $I$ gives a contradiction. \nHence $\\mu$ does not accept Hamilton-Krushkal degenerated sequences.\n\nNow using similar arguments as in the proof of Theorem \\ref{th.exhaustion}, we \ncomplete the proof in the case where $\\mu$ is extremal.\n\n\nFinally we show that if $\\mu$ is a non trivial invariant Beltrami differential, \nthen there exist an extremal invariant differential $\\nu$ such that \n$v_\\nu(\\gamma)=v_\\mu(\\gamma)$ for all $\\gamma$ in $Q(S_R).$ \n\n\nIndeed, if $\\mu$ is not extremal then by the Banach Extension Theorem  and \nRiesz Representation Theorem there exist is another Beltrami differential \n$\\alpha$  which is extremal satisfying  \n$\\|\\alpha\\|_\\infty=\\|\\mu\\|_T<\\|\\mu\\|_\\infty$ and \nsuch that defines the same functional as $\\mu$ in $Q(S_R)$. Let $\\beta$ be a \n$*$-weak limit of the Ces\\`aro averages $C_n(\\alpha)=\\frac{1}{n} \n\\sum_{i=0}^{n-1} \n\\alpha(R^i)\\frac{\\overline{(R^i)'}}{(R^i)'}$, then \n$\\beta(R)\\frac{\\overline{R'}}{R'}=\\beta$ and $\\|\\beta\\|_\\infty\\leq \n\\|\\alpha\\|_\\infty.$  Then we claim that $v_\\beta=v_\\mu$.\nLet $\\{C_{n_i}(\\alpha)\\}$ be a sequence of averages $*$-weakly converging to \n$\\beta$. For any $\\gamma \\in Q(S_R)$ we have $$\\int \\gamma \\beta=\\lim \\int \nC_{n_i}(\\alpha)\\gamma$$ by duality the previous limit is \nequal to $$\\lim \\int \\alpha \\frac{1}{n_i}\\sum_{k=0}^{n_i-1} R^{*k}(\\gamma)=\\lim \n\\int \\mu \\frac{1}{n_i}\\sum_{k=0}^{n_i-1} R^{*k}(\\gamma)$$ but $\\mu$ is an \ninvariant differential and again using duality the previous limit becomes \n$$\\lim \\int \\mu \\gamma=\\int \\mu \\gamma.$$\n\nHence for any $\\gamma$ in $Q(S_R)$ we have $$\\int \\beta \\gamma= \\lim \n\\int C_{n_i}(\\alpha)\\gamma=\\int \\mu \\gamma.$$\nSince $\\alpha$ is extremal we have \n$\\| \n\\beta \\|_\\infty=\\| \\alpha \\|_\\infty=\\|\\mu\\|_T$. Thus \n$\\beta$ is the desired  extremal invariant differential.  \n\n\\end{proof}\n\n\nTo conclude, let us note that the arguments of the theorems in this paper \nwork for entire and meromorphic functions in the class\nof Eremenko-Lyubich. This is the class of all entire or meromorphic functions \nwith finitely many critical and singular values. It is not completely clear \nwhether this arguments can be carried on entire or meromorphic functions whose \nasymptotic value set contains a compact set of positive Lebesgue measure. \n\n\n\n \\bibliographystyle{amsplain} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\n\\IEEEPARstart{D}{iabetic} \nretinopathy (DR) has become a worldwide major medical concern for the large population of diabetic patients and has been the leading cause of blindness in the working-age population today\\cite{thomas2019idf,ciulla2003diabetic,raman2016diabetic}. DR lesions often present as microaneurysms (MAs), hemorrhages (HEs), soft exudates (SEs), and hard exudates (EXs) which can be observed in colorful fundus images and are the basis of diagnosis for ophthalmologists. \n\nHowever, until now there has been no valid treatment to cure this disease completely. The most recognized treatment is the early diagnosis and intervention to controll the progression of the disease and to avoid eventual loss of vision\\cite{wong2018guidelines}. Thus, many national health institutions are promoting DR screening, which has been proven effective in reducing the rate of blindness caused by DR\\cite{ciulla2003diabetic,ting2016diabetic}. However, screening is a heavy burden for primary care systems during the promotion, since the ophthalmologists are in very short supply and have already engaged in post-DR treatment. For this reason, automatic segmentation technology for DR lesions has become a trend towards assisting ophthalmologists in diagnosis.\n\n\n\n\\begin{figure}[!t]\n\\centerline{\\includegraphics[width=\\columnwidth]{fig1.png}}\n\\caption{Illustration of fundus image with characteristics of DR lesion. \nSE: soft exudate; HE: hemorrhage; EX: hard exudate; MA: microaneurysm.\n(a) Original image with different clusters of lesions denoted by red and blue bounding boxes;\n(b) magnified lesion regions where green, purple, red and blue area represent SE, HE, EX and MA, respectively;\n(c) location statistics of certain lesions on IDRiD dataset in pixels. Specifically, from left to right are the distances from SE center to nearest HE center, from EX cluster center to nearest MA center, from SE center to the nearest vascular tree midline, and from MA center to the nearest vascular tree midline.}\n\\label{fig1}\n\\end{figure}\n\n\nRecent research efforts have been directed towards automatic DR segmentation based on deep learning (DL). \nGuo et al. \\cite{guo2019seg} adopted a pretrained Vgg16\\cite{simonyan2014very} as backbone with multi-scale feature fusion block for DR lesion segmentation. Specifically, they extracted side features from each convolution layer in Vgg16 and fused them in a weighted way. Further, a multi-channel bin loss was proposed to alleviate class-imbalance and loss-imbalance problems.\nZhou et al. \\cite{zhou2019collaborative} designed a collaborative learning network to jointly improve the performance of DR grading and DR lesion segmentation with attention mechanism. The attention mechanism allowed features with image-level annotations to be refined by class-specific information, and generated pixel-level pseudo-masks for the training of segmentation model.\nHowever, previously published studies paid too much attention to the designs of networks and many of the researches up to now have only achieved the segmentation of just one or two lesions \\cite{tavakoli2020automated,mamilla2017extraction,wu2017automatic,khojasteh2019novel,mo2018exudate}. It should be noted that as a complex medical lesion segmentation task, the pathological connections have not received enough attention.\n\nAfter comprehensive investigation in possible causes of DR lesions, we found two interesting presentation phenomena shown in Fig.\\ref{fig1}: 1) lesions are usually closed to specific veins and arteries. For example, SEs are generally distributed at the margins of the main trunk of the upper and lower arteries, and MAs are generally distributed at the margins of the capillaries; 2) most lesions have certain spatial interactions with each other. Specifically, SEs commonly appear at the edge of HEs, while EXs are usually arranged in a circular pattern around one or several MAs, which is consistent with the occurrence of pathology.\n\nMotivated by the above observations, we propose a relation transformer block (RTB), comprised of a cross-attention and a self-attention head, to explore dependencies among lesions and other fundus tissues. For specific, \nthe cross-attention head is designed to capture implicit relations between lesions and vessels. We design a dual-branch network to employ both lesion and vascular information, and cross-attention head is integrated between the two branches to make effective use of vascular information in lesion segmentation branch. \nAs we know, the fundus tissues, \\textit{e.g.,} vessels, optic disc, nerves and other lesions, are complex and easily confused with DR lesions of interest, but considering that vascular information describes certain distribution patterns of the above tissues, the cross-attention head is able to locate more lesions through the layout provided and eliminate the false positives far from certain vessels.\nThe self-attention head is employed to investigate the relationships of multi-lesion themselves. Since some of the lesions look similar, \\textit{e.g.,} HE and MA (both red-lesion), SE and EX (both exduate-lesion), misclassifications between lesions occur frequently. Through impactful information emphasized by self-attention head, the distinction and connection of lesions play roles in reducing confusion in segmentation.\nNote that before DL was utilized in medical imaging, vessels were easily mistaken for red lesions, and vessel detection was regarded as a routine step in the segmentation with detected vessels being removed straight away. However, as no fundus dataset is annotated with both vessels and DR lesions, the DL-based lesion segmentation task no longer extracts vessel features separately. To our best knowledge, this is the first trial that utilizes vascular information for deep based fundus lesion segmentation.\n\n\n\nIn addition, some special lesion patterns, such as MA with small size and SE with blurred border, are hard to be situated accurately due to the lack of fine-grained details in high-level features.\nTo alleviate this, we propose a Global Transformer Block (GTB) inspired by GCNet\\cite{GCNet} to further extract detail information, which can preserve the detailed lesion information and suppress the less useful information of channels in each position. In our network, GTB is also adopted to generate dual branches. Specifically, after the backbone, the shared fundus features are obtain as input of the two GTBs, and then GTBs generate more specific features of vessels and lesions respectively, which would be further investigated at the pathological connection level by RTB.\n\nWe have evaluated our network on two publicly available datasets - IDRiD and DDR. Experimental results show that our network outperforms the state-of-the-art DR lesion segmentation reports and achieves the best performance in EX, MA and SE. Furthermore, we also implement ablative experiments on IDRiD dataset and validate the effectiveness of RTB and GTB in improving DR lesion segmentation outcomes.\n\nIn summary, our contributions are as follows:\n\\begin{itemize}\n    \\item We propose a dual-branch architecture to obtain vascular information, which contributes to locate the position of DR lesions. For effective use of vascular information in multi-lesion segmentation, we design a relation transformer block (RTB) based on transformer mechanism. To our best knowledge, this is the first work to employ multi-heads transformer structure in lesion segmentation in fundus medical images.\n    \\item We present global transformer block (GTB) and relation transformer block (RTB) to detect the special medical patterns with small size or blurred border. The design explores the internal relationship between DR lesions which improves the performance in capture the details of interest.\n    \\item Experiments on the IDRiD dataset show that our method achieves a front row finish on DR multi-lesion segmentation. Specifically, our method achieves the best performance in exduates segmentation and ranks second in HE lesion segmentation.\n    Experiments on the DDR dataset show that our method outperforms other methods on EX, MA and SE segmentation task and ranks second on HE segmentation task.\n\\end{itemize}\n\n\n\\section{Related Work} \n\\subsection{Pathological Analysis of the DR Lesions}\nDR lesions segmentation is a complex topic due to large intra-class variance.\nFurthermore, DR lesions vary with different stages of disease as well, which also brings challenges to segmentation. \nHowever, instead of discovering lesions directly, we notice that there are pathological associations between these lesions, which can be depicted in the spatial distribution. \n\nWe first investigate the possible pathological causes of DR lesions. Briefly, MA is the earliest lesion of DR observed as spherical lateralized swelling which is produced by vascular atresia; EX looks like yellowish-white well-defined waxy patch, generally thought to be lipid produced by the rupture of the retinal nerve tissues, incidentally, which is also resulted from the vascular atresia. Additionally, when the rupture of vessels happens after the vascular atresia, bloods leak out from the vessels, which leads to the lipoproteins in vessels leaking into the retina as well. The leaking bloods form the HE patterns and the leaking lipoproteins form the SE patterns  with poorly defined borders. In summary, as shown in Fig.\\ref{fig1}(b), most of the EXs are observed arranged in a circular pattern around one or several MAs (the bottom line) and most of the SEs appear at the edge of HEs (the upper line).\n\nIn addition to the intra-class dependencies among lesions, the inter-class relations between DR lesions and vessels also make great sense. As mentioned above, vascular abnormalities are the direct or indirect causes of DR lesions, specifically, we found the fact that SEs are often distributed near the trunk of upper and lower arteries, and MAs are generally distributed among the capillaries. Furthermore, intricate fundus tissues often confuse the identification of lesions, but we notice that there are certain pattern rules of fundus, especially in the distribution of various veins and arteries, which would provide valuable prior information.  \n\nTo confirm the above pathological analysis, we count the distances in pixel between different fundus tissues on the IDRiD dataset. Fig.\\ref{fig1} (c) illustrates the distance between cluster center of EX and the nearest neighbor MA, SE and the nearest neighbor HE, MA and the closet capillaries, SE and closest upper and lower arteries, respectively. The statistical results verify that there is an exploitable pattern in the distribution of DR lesions. \n\n\\subsection{Deep Neural Networks in DR Lesion Segmentations}\n\nDR lesion segmentation task based on traditional image processing techniques\\cite{walter2007automatic,Automatic2005,alipour2012analysis} is facing two main challenges: the great morphological differences of the same lesions in different disease stages and the confusions of DR lesions and similar structures in the fundus. These two problems have not been effectively solved until \ndeep neural networks (DNNs) exploded in the field of computer vision (CV) \\cite{krizhevsky2012imagenet} and have also been widely applied to DR lesion segmentations\\cite{CABNet,CANet,CENet}.\n\nHowever, DNNs also raise new difficulties. For instance, the detailed information is easily lost by deep networks but most of the DR lesions are very small and even just one or two pixels. Besides, considering the balance of different characteristics of red and exudate lesions, the accuracy of multi-task model is limited. \n\nTo deal with the above issues, researchers have proposed many improvements, which can be summarised as two directions:\n\nFirstly, some researchers focus on the designs of attention models fusing low-level and high-level features together to avoid details lost in the deep network.\nZhang et al. \\cite{zhang2019detection} fused multiple features with distinct target features in each layer based on attention mechanism and achieved preliminary MA detection.\nWang et al. \\cite{wang2017zoom} designed a dual-branch attention network, with one producing a 5-graded score map and the other producing an attention gate map combined to the score map to highlight suspicious regions.\nZhou et al. \\cite{zhou2019collaborative}  applied low-level and high-level guidances to different lesion features and obtained the  refined multi-lesion attention maps, which were further employed as pseudo-masks to train the segmentation model.\n\nSecondly, the task of segmenting DR lesions is divided into segmenting red lesions and exudate lesions separately, which evades the balanced cost of inter-class disparities and enables fully learning of the same type of lesions.\nMo et al. \\cite{mo2018exudate} designed a fully convolutional residual network incorporating multi-level hierarchical information to segment the exudates without taking the red lesions into account.\nXie et al. \\cite{xie2020sesv} built a general framework to predict the errors generated by existing models and then correct them, which performed well in MA segmentation.\n\nHowever, the first direction pays much attention to the network designs, with seldom considering the pathological connections of DR lesions, while the second direction leads to time consuming and large memory requirements. In order to take advantages of the pathological connections and improve the efficiency of the multi-task model, we propose a RTB consisting of self-attention and cross-attention head to segment DR lesions simultaneously.\n\n\\subsection{Transformer in Medical Images}\n\nTransformer network has been one of the fundamental architecture for natural language processing (NLP) since 2017 due to the efficient and effective self-attention mechanism\\cite{vaswani2017attention}. It improves the performance on many NLP tasks, such as text classification, general language understanding and question answering. Compared with recurrent networks, transformer network achieves parallel computation and reduces the computational complexity. \nIn a basic transformer attention block, Query ($Q$), Key ($K$), Value ($V$) are the three typical inputs to a attention operation. At first, $Q$ and $K$ are  computed in the form of pairwise function to obtain the corresponding attention of all points on $K$ for each point on $Q$. The pairwise function can optionally be Gaussian, Embedding Guassian, Dot-Product, Concatenation and \\textit{etc.}. Then, the product is multiplied by $V$ and passes through a column-wise softmax operator to ensure every column sum to 1. Every position on the output contains the recoded global information by attention mechanism. In self-attention operator, $Q = K = V$, so that the output has the same shape with input.\n\nInspired by the success in the domain of NLP, a standard transformer was applying to CV in 2020\\cite{dosovitskiy2020image} with the fewest modifications, called as Vision Transformer (ViT). The input to a ViT is a sequence of cropping images which are linearly encoded by aliquoting the original image. The image patches are treated the same way as tokens (words) in an NLP application.\n\n\nRecently, the transformer architecture has also been applied to the field of medical image processing. \n Liu et al. \\cite{GPT} proposed a global pixel transformer (GPT) to predict several target fluorescent labels in microscopy images. The GPT is similar to a three-headed transformer with different sizes of query inputs, which allows it to adequately capture features at different scales.\nGuo et al. \\cite{guo2021transformer} applied the ViT to anisotropic 3D medical image segmentation, with the self-attention model arranged at the bottom of the Unet architecture.\nSong et al.\\cite{DRT} built a Deep Relation Transformer (DRT) to combine OCT and VF information for glaucoma diagnosis. They modified the standard transformer to an interactive transformer that utilizes a relationship map of VF features interacting with OCT features. \n\n\\section{Methodology}\nIn this section, we first give a brief overview of our proposed network, and then elaborate on the key network components, \\textit{i.e.,} global transformer block (GTB) and relation transformer block (RTB). Finally, designed loss function is further provided. \n\n\n\\subsection{Overview}\nGiven an input fundus image, the proposed network is designed to output one vascular mask and four lesion masks in parallel.\nFig.\\ref{fig2} depicts its overall architecture, which is comprised of four key components: backbone, global transformer block (GTB), relation transformer block (RTB), and segmentation head.\nA dual-branch architecture is employed upon the backbone to explore vascular and pathological features separately, where the transformers based on GTB and RTB\nare incorporated to reason about interactions among both features.\n\n\n\nTo be specific, the fundus image first passes through a backbone to obtain an abstracted feature map $\\mathbf{F}$, with a spatial resolution of $W\\times H$ and $C$ number of channels.\nThen, two parallel branches comprised of global transformer block (GTB) are incorporated to exploit long-range dependencies among pixels in $\\mathbf{F}$, resulting in specific vessel features $\\mathbf{F}_v$ and primary lesion features $\\mathbf{F}_l$ fueled with global contextual information, respectively.\nUpon the branch providing lesion features, we further integrate a relation transformer block (RTB) to model spatial relations between vessels and lesions due to their inherent pathological connections using a self-attention and a cross-attention head:\nthe self-attention head inputs only the lesion features $\\mathbf{F}_l$, and exploits long-range contextual information to generate self-attentive features $\\mathbf{F}_{s}$ through a self-attention mechanism; the cross-attention head inputs both the lesion and vessel features $\\mathbf{F}_{l}$, $\\mathbf{F}_v$,\nand incorporates beneficial fine-grained vessel structural information into $\\mathbf{F}_v$, producing cross-attentive features $\\mathbf{F}_{c}$. \nThe resulting $\\mathbf{F}_{s}$ and $\\mathbf{F}_{c}$ are concatenated together to form the output of the RTB. \nFinally, two sibling heads, each of which contains a Norm layer and a $1\\times 1$ convolution, are used to predict vascular and pathology masks based on the vessel features and concatenated lesion features, respectively.\n\nGTB contains one head while RTB contains two heads. Although the basic heads of GTB and RTB are based on the transformer structure that generates query, key and value for relation reasoning,\nthey are structurally different in our work. \nThe query of head in GTB is similar to a channel-wise weights and the one in RTB has the same size in spatial dimension with the input.\nIn a training process, GTB is employed to generate specific multi-lesion and vessel features independently which maintain more details of interest, and RTB further exploits the inherent pathogenic relationships between multi-lesion and vessels, which eliminate noise and imply the location information. \n\n\\begin{figure*}[htbp]\n\\centerline{\\includegraphics[width=\\textwidth]{fig2.png}}\n\\caption{Pipeline of the proposed method. The input image passes through a backbone to obtain the shared feature $\\mathbf{F}$. Then the shared feature $\\mathbf{F}$ takes two branches to achieve vessels and multi-lesion segmentation respectively. Two Global Transformer Blocks (GTB) are applied to both branches to generate specific features, and a Relation Transformer Block (RTB) is incorporated after GTB to explore the inherent pathological connections among multi-lesion and between multi-lesion and vessels.}\n\\label{fig2}\n\\end{figure*}\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=\\columnwidth]{fig3.png}}\n\\caption{The overall structure of Global Transformer Block (GTB).}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=\\columnwidth]{fig4.png}}\n\\caption{The details of Relation Transformer Block (RTB).}\n\\label{fig4}\n\\end{figure}\n\n\n\\subsection{Global Transformer Block}\nThe Global Transformer Block (GTB) contains two parallel branches of the same architecture to extract features for lesions and vessels separately. Such a dual-branch design owns to the fact that lesions and vessels generally have dramatically different visual patterns. To be concrete, lesions are discrete patterns, with nearly random spatial distribution. While vessels are topological connected structures, and the layout of vascular trunks, containing central retinal artery, ciliary artery and etc, generally follows some common rules. \nIt is hence necessary to use specialized branches to learn specific characteristics of different objects of interest.\n\n \nFig.\\ref{fig3} presents the detailed structure of each GTB branch.\nIt takes an input $\\mathbf{F}\\in \\mathbb{R}^{\\: C\\times W\\times H}$ generated from the backbone, and outputs attentively-refined feature maps $\\mathbf{F}_{i}\\in \\mathbb{R}^{\\: C\\times W\\times H},i \\in\\left\\{l,v\\right\\}$ of lesions and vessels respectively.\nSpecifically, GTB follows the typical framework of transformer networks. \nThree generators, denoted as $\\mathcal{Q}$, $\\mathcal{K}$ and $\\mathcal{V}$ are first employed to transform the input $\\mathbf{F}$ into query, key and value, respectively. In GTB, the generator  $\\mathcal{Q}$ is implemented with a $3 \\times 3$ convolution followed by a global average pooling, and outputs a query vector $\\mathcal{Q}(\\mathbf{F}) \\in \\mathbb{R}^{\\: C^{'}\\times 1}$ with the channel number designed as $C^{\\prime}=C\/8$; the generator $\\mathcal{K}$ and $\\mathcal{V}$ have the same architecture as $\\mathcal{Q}$ expect for replacing the global averaging pooling with a reshape operation, leading to the key and value $\\mathcal{K}(\\mathbf{F}),\\mathcal{V}(\\mathbf{F})\\in \\mathbb{R}^{\\:C^{'}\\times HW}$.\n\nWe define the pairwise function of query and key as a matrix multiplication:\n\\begin{equation}\n    \\mathcal{F}(\\mathbf{F})=\\mathcal{K}(\\mathbf{F})^T\\mathcal{Q}(\\mathbf{F}) ,\n\\end{equation}\nwhere the superscript $T$ denotes a transpose operator for matrix. \nNote that the query of GTB acts as a channel-wise query instead of the position query as NLNet\\cite{wang2018NLnet}. To be specific, the query vector is considered as a feature selector for channels of key matrix.  \nSubsequently, the product $\\mathcal{F}(\\mathbf{F})\\in\\mathbb{R}^{\\:HW\\times 1}$ also acts as a feature selector for spatial positions of value matrix. \nIn summary, the GTB can be roughly described as a attention mechanism which fuses channel-wise first and then spatial-wise weighted features together with input information.\n\nNext, we consider the global transform operation defined as:\n\\begin{equation}\n    \\mathcal{G}(\\mathbf{F}) = \\mathcal{V}(\\mathbf{F})softmax(\\mathcal{F}(\\mathbf{F}))\\in \\mathbb{R}^{\\:C^{'}\\times 1},\n\\end{equation}\nwhere $softmax$ is a softmax function to normalize the $\\mathcal{F}(\\mathbf{F})$.\n\nThen, We take the obtained attentive features $\\mathcal{G}(\\mathbf{F})$ with a linear embedding as a residual term to the input $\\mathbf{F}$, and get the final output through a residual connection:\n\\begin{equation}\n    \\mathbf{F}_{i} = W\\mathcal{G}(\\mathbf{F})+\\mathbf{F},\\ i\\in\\left \\{ l,v\\right \\},\n\\end{equation}\nwhere the $+$ operation denotes broadcasting element-wise sum operation; the $W$ is a linear embedding, implemented as $1 \\times 1$ convolution to convert the channel number of intermediate feature map from $C^{'}$ back to $C$. As a result, the output features are received with the same format as the input, but have been enriched with specialized vessel and lesion features, respectively.\n\n\n\n\n\n\nThe GTB structure is inspired by the GCNet\\cite{GCNet}. They both follow the idea of transformer mechanism but generate the per-channel weights. The weight vectors in GCNet and GTB both obtained by a matrix multiplication, but different from GCNet, GTB attains the two multipliers with three generator to further highlight the useful channels in each position, realizing both channel-wise and spatial-wise attention. Due to the fundus lesion features, especially the small discrete ones, are easily confused with artifacts or idiosyncratic tissues, \nuseful information tends to exist in only a few pixels of certain channels. It is an improved method aimed at the feasibility of small discrete pattern segmentation in fundus images.\n\n\\subsection{Relation Transformer Block}\nRelation Transformer Block (RTB) consists of a self-attention and a cross-attention head, used to capture intra-class dependencies among lesions and inter-class relations between lesions and vessels, respectively, as shown in Fig.\\ref{fig4}. In each head, three trainable linear embeddings, implemented with a $3 \\times 3$ convolution followed by a reshape operation, are employed as the query, key and value generator $\\mathcal{G}_i, \\mathcal{K}_i, \\mathcal{V}_i,i \\in\\left\\{s,c\\right\\}$, respectively.\nThe pairwise computations of query and key in self-attention head and cross-attention head are described as:\n\\begin{equation}\n    \\begin{split}\n        \\mathcal{F}_{s}(\\mathbf{F}_l)&=\\mathcal{K}_{s}(\\mathbf{F}_l)^T\\mathcal{Q}_{s}(\\mathbf{F}_l)  \\\\\n        \\mathcal{F}_{c}(\\mathbf{F}_l,\\mathbf{F}_v)&=\\mathcal{K}_{c}(\\mathbf{F}_v)^T\\mathcal{Q}_{c}(\\mathbf{F}_l),\n    \\end{split}\n\\end{equation}\nwhere the subscripts $s$ and $c$ denote the self-attention and the cross-attention head, respectively.\nIt is important to emphasize that different from the self-attention head that derives the query, key all from input lesion features $\\mathbf{F}_l$, the cross-attention head generates the key from the vessel features $\\mathbf{F}_v$ instead to integrate vascular information. \n\n\n\nNext, the individual attentive features of the two heads are computed respectively as:\n\\begin{equation}\n\\begin{split}\n    \\mathcal{G}_{s}(\\mathbf{F}_{l})&=\\mathcal{V}_{s}(\\mathbf{F}_{l})softmax(\\mathcal{F}_{s}(\\mathbf{F}_{l}))\\\\\n    \\mathcal{G}_{c}(\\mathbf{F}_{l},\\mathbf{F}_v)&=\\mathcal{V}_{c}(\\mathbf{F}_{v})softmax(\\mathcal{F}_{c}(\\mathbf{F}_{l},\\mathbf{F}_v)).\n\\end{split}\n\\end{equation}\n\nWe adopt residual learning to each head as well and get the outputs:\n\n\n\\begin{equation}\n\\begin{split}\n    \\mathbf{F}_{i} = W_{i}\\mathcal{G}_{i}(\\mathbf{F}_{l},\\mathbf{F}_v)\\oplus\\mathbf{F}_l\\\\i\\in \\left\\{s, c \\right\\},\n\\end{split}\n\\end{equation}\nwhere the $W_i$ is a linear embedding implemented as $1 \\times 1$ convolution, and the $\\oplus$ operation is performed by a residual connection of element-wise addition. \n\nAs such, the self-attention head computes the response in a position as a weighted sum of the features in all positions, and thus well captures long-range dependencies. \nGiven the fact that DR lesions are usually dispersed over a broad range, the self-attention can exchange message among multiple lesions, regardless of their positional distance, and thus allows the modeling of intra-class pairwise relations of lesions. The head is supposed to distinguish the mixtures of more than two lesions and further refine the edges of large patterns in lesion segmentation.\n\n\nThe cross-attention head queries global vascular structures from the vessel features, thus incorporating interactions between lesions and vessels.\nConsidering that lesions and vessels have strong inherent pathogenic connections, the cross attention help to better locate MA and SE, and meanwhile eliminate false positives of EX caused by vessel reflection and MA caused by capillary confusion.\n\nWe concatenate the resulting features $\\mathbf{F}_{s}$ from the self-attention head and that $\\mathbf{F}_{c}$ from the cross-attention head, leading to the final RTB output:\n\n\\begin{equation}\n    \\mathbf{F}_{out} = [\\mathbf{F}_s;\\mathbf{F}_c],\n\\end{equation}\nwhere the $[\\cdot\\ ;\\ \\cdot]$ denotes the concatenation at channel dimension.\n\n\n\n\\subsection{Loss Function}\nWe employ two loss functions, \\textit{i.e.}, $\\mathcal{L}_{lesion}$ and $\\mathcal{L}_{vessel}$ for the multi-lesion and vessel segmentation branches respectively, and the total loss of our network is defined as:\n\\begin{equation}\n    \\mathcal{L} =\\mathcal{L}_{lesion}+\\lambda \\mathcal{L}_{vessel},\n\\end{equation}\nwhere the $\\mathcal{L}_{lesion}$ denotes the 5-class weighted cross-entropy loss for multi-lesion segmentation and the $\\mathcal{L}_{vessel}$ is a binary weighted cross-entropy loss to learn vascular features; the $\\lambda$ is set as the weight in the loss function. When $\\lambda = 0.0$, the network is optimized by the multi-lesion features only, and as $\\lambda$ grows, vascular information plays an increasing role in optimization.\n\n\n\\section{Experiments And Results}\n\\subsection{Datasets}\n\\textbf{IDRiD Dataset} is available for the segmentation and grading of retinal image challenge 2018\\cite{porwal2018indian, porwal2020idrid}. The segmentation part of the dataset contains 81 $4288 \\times 2848$ sized fundus images, accompanied by four pixel-level annotations, \\textit{i.e.,} EX, HE, MA and SE if the image has this type of lesion. In total, there are 81 EX annotations, 81 MA annotations, 80 HE annotations, and 40 SE annotations. The partition of training set and testing set is provided on IDRiD already, with 54 images for training and the rest 27 images for testing. \n\n\\textbf{DDR Dataset} is provided by Ocular Disease Intelligent Recognition (ODIR-2019) for lesion segmentation and lesion detection\\cite{LI2019}. This dataset consists 13,673 fundus images from 147 hospitals, covering 23 provinces in China. For segmentation task, 757 fundus images are provided with pixel-level annotation for EX, HE, MA and SE if the image has this type of lesion. In total, there are 486 EX annotations, 570 MA annotations, 601 HE annotations, and 239 SE annotations. The partition of training set, validation set and testing set is provided on DDR already, with 383 images for training, 149 images for validation and the rest 225 images for testing.\n\n\\begin{table*}\n    \\centering\n    \\caption{Performance Comparison with the state-of-the-art works reported on the IDRiD dataset, where the \\textbf{separate} and \\textbf{same} indicates the way in which the method segments the lesions separately by different models or at the same time by one model}\n    \\label{tab1}\n    \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n    \\hline\n        \\multirowcell{2}{Method} & \\multirowcell{2}{sepa-\\\\rate} & \\multirowcell{2}{same}  & \\multicolumn{2}{c|}{Hard   Exudates} &\\multicolumn{2}{c|}{Haemorrhages} \n        & \\multicolumn{2}{c|}{Microaneurysms} \n        & \\multicolumn{2}{c|}{Soft   Exudates}  \\\\ \\cline{4-11}\n         & & & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC \\\\ \\hline\n        VRT(1st) & \\checkmark & & 0.7127  & - & 0.6804  & - & 0.4951  & - & 0.6995  & - \\\\ \n        PATech(2nd) & \\checkmark & & 0.8850  & - & 0.6490  & - & 0.4740  & - & - & - \\\\ \n        iFLYTEK-MIG(3rd) & \\checkmark & & 0.8741  & - & 0.5588  & - & 0.5017  & - & 0.6588  & - \\\\ \n         \n        DRUNet\\cite{kou2019microaneurysms} & \\checkmark & & - & - & - & -  & - & 0.9820  & - & - \\\\ \n        SESV\\cite{xie2020sesv} & \\checkmark & & - & - & - & - & \\textbf{0.5099}  & - & - & - \\\\\n        L-Seg\\cite{guo2019seg} & & \\checkmark & 0.7945  & - & 0.6374  & - & 0.4627  & - & 0.7113  & - \\\\\n        SSCL\\cite{zhou2019collaborative} & & \\checkmark & 0.8872  & 0.9935  & \\textbf{0.6936}  & \\textbf{0.9779}  & 0.4960  & 0.9828  & 0.7407  & 0.9936  \\\\ \n        RTN(Ours) & & \\checkmark & \\textbf{0.9024}  & \\textbf{0.9980}  & 0.6880  & 0.9731  & 0.4897  & \\textbf{0.9952}  & \\textbf{0.7502}  & \\textbf{0.9938}  \\\\ \\hline\n    \\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Performance Comparison with state-of-the-art segmentation methods on the DDR dataset, where * denotes the results are reproduced by ourselves}\n\\label{tab2}\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n    \\hline\n        \\multirowcell{2}{Method} & \\multicolumn{2}{c|}{Hard   Exudates} &\\multicolumn{2}{c|}{Haemorrhages} \n        & \\multicolumn{2}{c|}{Microaneurysms} \n        & \\multicolumn{2}{c|}{Soft   Exudates} \\\\ \\cline{2-9}\n         & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC \\\\ \\hline\n        HED\\cite{xie2015holistically} & 0.4252 & 0.9612 & 0.2014 & 0.8878 & 0.0652 &0.9299 & 0.1301 & 0.8215 \\\\ \n        DeepLab v3+\\cite{chen2018deeplabv3} & 0.5405 & 0.9641 & 0.3789 & 0.9308 & 0.0316 & 0.9245 & 0.2185 & 0.8642 \\\\\n        UNet\\cite{guan2019fully,Yakubovskiy2019} & 0.5505 & 0.9741 & \\textbf{0.3899} & \\textbf{0.9387} & 0.0334 & 0.9366 & 0.2455 & 0.8778 \\\\\n        L-seg*\\cite{guo2019seg} & 0.5645 & 0.9726 & 0.3588 & 0.9298 & 0.1174 & 0.9423 & 0.2654 & 0.8795\\\\ \n        RTN(Ours) & \\textbf{0.5671} & \\textbf{0.9751} & 0.3656 & 0.9321 & \\textbf{0.1176} & \\textbf{0.9452} & \\textbf{0.2943} & \\textbf{0.8845} \\\\ \\hline\n    \\end{tabular}\n\\end{table*}\n\n\\subsection{Implementation Details}\n\n\\subsubsection{Data Preparation}\nTo prepare more trainable data, some operations are performed on the original images. First, the images are input into the a segmentation model pretrained on DRIVE\\cite{staal2004ridgeb91} and STARE\\cite{hoover2000locatingb92} dataset with vessel annotations and the pseudo vascular masks are obtained. Next, \nlimited by the memory received, the large images are random resized and cropped into small pieces of $512 \\times 512$ size,  additionally we apply random horizontal flips, vertical flips, and random rotation as forms of data augmentation to reduce overfitting. Then, in order to enhance image contrast while preserving local details, we process Contrast Limited Adaptive Histogram Equalization (CLAHE) on all input images with ClipLimit=2 and GridSize=8 by setting. The CLAHE is proven to be effective due to the anomalousness distinguished from the background in diabetic fundus, and the quantitative results are presented in Table \\ref{tab5}.\n\n\\subsubsection{Model Settings}\nThe typical UNet architecture\\cite{Yakubovskiy2019} is  \na popular method for medical images segmentation, constructed by an encoder and a decoder with skip connections in channel-wise concatenation manner. In this paper, we apply DenseNet-161\\cite{huang2017densely} pretrained on ImageNet dataset as the backbone of UNet encoder\\cite{guan2019fully} to achieve better performance. The channel number $C$ of output of UNet is set to 32.\n\n\\subsubsection{Experiment settings}\nOur framework is implemented using pytorch backend and performed on NVIDIA GeForce RTX 3090 GPU with 24GB of memory. During the training, the batch-size is set to 16. The initial learning rate is set to 0.001 and is decay in a step-wise manner to 0.1 times of the previous every 120 epochs. All models are trained for 250 epochs with the SGD optimizer with momentum 0.9 and weight decay 0.0005.\n\nThe loss function settings are as follows: a) the return loss ratio $\\lambda$ is set to 0.1; b) the weights of $\\mathcal{L}_{lesion}$ are set as 0.001, 0.1, 0.1, 1.0, 0.1 for background, EX, HE, MA and SE respectively; c) The coefficients of background and vessels in $\\mathcal{L}_{vessel}$ are set as 0.01 and 1.0.\n\n\\subsection{Evaluation Metrics}\nTo evaluate the performance of the proposed method, we employ the area-under-the-curve (AUC) of both the precision and recall (PR) curve and receiving operating characteristic (ROC) curve \\cite{scikit-learn}, which are also recognized as metrics of fundus image segmentation in previous competitions and researches. The former is more concerned with the accuracy of the true data in prediction, while the latter reflects the performance of the data predicted positively. There is more emphasis on recall metric in medical images, which indicates the performance of true samples being successfully predicted, \\textit{i.e.,} the AUC\\_PR drawn by recall metric shows more practical value, and the AUC\\_ROC also characterizes the effectiveness of the model.\n\n\\subsection{Comparisons on Other State-of-the-art Methods}\nWe compare our method with previous works reported on the IDRiD dataset (Table \\ref{tab1}). Our method ranks first in AUC\\_ROC of EX, MA and SE and AUC\\_PR of EX and SE, ranks second in AUC\\_ROC and AUC\\_PR of HE. \nNote that the first five methods of Table \\ref{tab1} all employed individual models segmenting the four lesions separately: according to the conference reports of the top 3 IDRiD competition teams, four models were developed for segmentation of four lesions respectively; DRUNet\\cite{kou2019microaneurysms} and SESV\\cite{xie2015holistically} proposed a specific network to segment MA only. Although SESV achieved the best AUC\\_PR of MA, such individual designs require modifying a large number of hyper-parameters in training stage and lead to time consuming in inference.\nThe rest methods of Table \\ref{tab1}: L-Seg\\cite{zhou2019collaborative}, SSCL\\cite{zhou2019collaborative} and our network, all propose one model to segment the four lesions at the same time. SSCL performs better than ours in AUC\\_PR of MA segmentation but worse in AUC\\_ROC, that might result from the fact that although RTB reduces false positives of MA away from the vessels, it also introduces the false negatives of MA.\n\nWe also verify the effectiveness of the proposed method on DDR dataset. Compared with the IDRiD dataset, the DDR dataset is an updated dataset with few results reported on it, so we apply some state-of-the-art segmentation methods on the DDR dataset to make comparisons. As shown in Table \\ref{tab2}, in the comparison with other state-of-the-arts, our method achieves the best performance in EX, MA and SE and ranks second in HE. It is obvious that our method is not doing the best job in HE segmentation both in IDRiD and DDR dataset, which may be explained by the fact that the large HE patterns are formed by blood irregularly haloing on the retina, and the specific bleeding points have been blurred. In this case, RTB, an approach that is more concerned with theoretical correlations, does not contribute useful information in the segmentation of large HEs. As mentioned in the dataset section, considering there are many low quality images with uneven illumination, underexposure, overexposure, image blurring, retinal artifacts and other disturbing lesion tissues in DDR dataset, DDR dataset is more challenging than IDRiD dataset, which results in the performance of the former lagging behind that of the latter. \n\n\n\n\n\\begin{table}[]\n  \\centering\n  \\caption{Performance Comparison of the different backbones with our network}\n  \\scalebox{0.9}{\n    \\begin{tabular}{|c|c|c|c|c|c|}\n    \\hline\n    \\multirowcell{2}{Framework}  &\n    \\multirowcell{2}{Encoder}&\n    \\multicolumn{4}{c|}{AUC\\_PR}\\\\\\cline{3-6}\n      & & EX & HE & MA & SE \\\\\n    \\hline\n     \\multirowcell{5}{UNet\\cite{guan2019fully}} & ResNet-34\\cite{he2016deep} & 0.8778    & 0.6764  & 0.4659   & 0.7407   \\\\\n     & ResNet-50\\cite{he2016deep} & 0.8858    & 0.6874   & 0.4692   & 0.7475   \\\\\n     & Xception\\cite{chollet2017xception} & 0.8924     & 0.6806 & 0.4851 & 0.7424 \\\\\n     & Vgg19-bn\\cite{simonyan2014very} & 0.8821    & 0.6832  & 0.4789    & 0.7423    \\\\\n     & DenseNet-161\\cite{huang2017densely} & \\textbf{0.9024}    & \\textbf{0.6880}   & \\textbf{0.4897}    & \\textbf{0.7502}   \\\\\n    \\hline\n    \\end{tabular}}\n  \\label{tab4}\n\\end{table}\n\n\\subsection{Ablation Studies on IDRiD Dataset}\nWe conduct ablation studies to better understand the impact of each component of our network. First, results of several encoding architectures are available to select the proper one as backbone. Then, considering the topology of vessels, regularization terms of loss function are discussed. Next, we analyze the effect of GTB based on the baseline, which is defined as the complete workflow without GTB and RTB. In order to identify whether vascular information contributes to lesion segmentation in advanced, we apply concatenation operation to the two outputs of GTBs directly. Finally, the roles of both self-attention head and cross-attention head in RTB are discussed thoroughly.\nSince AUC\\_PR is considered as the most important clinical evaluation metric for lesion segmentation in fundus images, we simplify the metrics to AUC\\_PR only for ablative comparisons. The loss and PR curves obtained from different experiments are set out in Fig.\\ref{fig5} and detailed AUC\\_PR values can be compared in Table \\ref{tab3}.\n\n\n\\subsubsection{Analysis on the Backbone}\nTo compare the effectiveness of backbone models, we perform experiments to select proper encoding architecture.    \nResNet-34\\cite{he2016deep}, ResNet-50\\cite{he2016deep}, Xception\\cite{chollet2017xception}, Vgg19-bn\\cite{simonyan2014very} and DenseNet-161\\cite{huang2017densely} integrated with UNet are implemented with the segmentation heads of our network. As can be seen from the Table \\ref{tab4}, the DenseNet-161 integrated with UNet achieves the best performance in all lesions and is utilized as the following backbone.\n\n\\begin{figure*}[htbp]\n\\centerline{\\includegraphics[width=\\textwidth]{fig5.png}}\n\\caption{Loss and PR curves for segmentation over four DR lesions. Ablation studies are compared to explore the effectiveness of the baseline itself and stacked by GTB, self-attention head (sah), cross-attention head (rah) and RTB one by one.}\n\\label{fig5}\n\\end{figure*}\n\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=\\columnwidth]{fig6.png}}\n\\caption{Visualization of (a) Original images with annotations; (b) spatial attention features with Convolutional Block Attention Module (CBAM) and (c) query attention features $\\mathcal{F}$ with Global Transformer Block (GTB) for three images from IDRiD dataset. The GTB picks more discrete and small lesions up and fine-grains the patterns of interest.}\n\\label{fig6}\n\\end{figure}\n\n\n\n\\subsubsection{Analysis on the Regularization Term}\nConsidering the topology of vessels, an extension of loss function regularization terms has been conducted. In the result of vessel segmentation, there are two common cases, the neglected ends and the truncated trunks. \nBased on the above cases, we propose two regularization terms $R_{thin}$ and $R_{cl}$. The former inspired by \\cite{yang2021hybrid} applies focal-loss function specifically on peripheral vessels, and the latter utilizes the center-line idea of \\cite{shit2021cldice} to ensure the connectivity. Table \\ref{tab_loss} indicates that the $R_{thin}$ improves the performance of MA segmentation and the $R_{cl}$ achieves the best grades in AUC\\_PR of HE and MA. However, the alterations are unremarkable, probably due to the fact that the groundtruths of vessels are pseudo-masks generated by semi-supervision, rather than manually annotated masks. The upper bound on the performance of the vessel segmentation restricts the improvement of the regularization terms on the final results.\n\n\\begin{table}[]\n  \\centering\n  \\caption{Performance Comparison of Different Regularization Terms on IDRiD Dataset}\n    \\begin{tabular}{|c|c|c|c|c|c|}\n    \\hline\n    \\multirowcell{2}{$R_{thin}$}  &\\multirowcell{2}{$R_{cl}$}  & \\multicolumn{4}{c|}{AUC\\_PR}\\\\\\cline{3-6}\n &    & EX & HE & MA & SE \\\\\n    \\hline\n&&\\textbf{0.9024}    & 0.6880   & 0.4897   & \\textbf{0.7502}\\\\\n\\checkmark&&0.8975&0.6845&0.4899&0.7432\\\\\n&\\checkmark&0.8912&\\textbf{0.6891}&\\textbf{0.4901}&0.7435\\\\\n\\checkmark&\\checkmark&0.8923&0.6808&0.4895&0.7426\\\\\n    \\hline\n    \\end{tabular}\n  \\label{tab_loss}\n\\end{table}\n\n\n\\begin{figure*}[htbp]\n\\centerline{\\includegraphics[width=\\textwidth]{fig7.png}}\n\\caption{Visualization of the query position (red points) of different lesions and their two query-specific attention maps with Relation Transformer Block (RTB). The red borders denote the self-attention maps, and the blue denote the cross-attention maps. The attention of different query positions in EX, HE, MA and SE varies.}\n\\label{fig7}\n\\end{figure*}\n\n\\begin{table*}\n    \\centering\n    \\caption{Performance Comparison of different attention blocks on the IDRiD dataset}\n    \\label{tab5}\n    \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n    \\hline\n        lesion & \\multicolumn{2}{|c|}{Hard   Exudates} &\\multicolumn{2}{|c|}{Haemorrhages} \n        & \\multicolumn{2}{|c|}{Microaneurysms} \n        & \\multicolumn{2}{|c|}{Soft   Exudates}  \\\\ \\hline\n        method & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC & AUC\\_PR & AUC\\_ROC \\\\ \\hline\n        baseline(without CLAHE) & 0.8025 & 0.9912 & 0.6031 & 0.9498 & 0.3912 & 0.9803 & 0.5478 & 0.9120\\\\ \\hline\n        baseline & 0.8593  & 0.9919  & 0.6284  & 0.9553  & \\textbf{0.4279}  & 0.9830  & 0.5766  & 0.9171  \\\\ \n        baseline+SENet\\cite{hu2018squeeze} & 0.8653  & 0.9931  & 0.6408  & 0.9497  & 0.3861  & 0.9869  & 0.5683  & 0.9299  \\\\ \n        baseline+CBAM\\cite{woo2018cbam} & 0.8606  & 0.9918  & 0.6470  & 0.9480  & 0.3979  & 0.9871  & 0.5525  & 0.9373  \\\\ \n        baseline+GC\\cite{GCNet} & 0.8633  & 0.9929  & 0.6406  & 0.9488  & 0.4031  & 0.9809  & 0.5540  & 0.9251  \\\\ \n        baseline+GTB(Ours) & \\textbf{0.8659}  & \\textbf{0.9933}  & \\textbf{0.6570}  & \\textbf{0.9534}  & 0.4071  & \\textbf{0.9879}  & \\textbf{0.5968}  & \\textbf{0.9458}  \\\\ \\hline\n    \\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}[]\n  \\centering\n  \\caption{Performance Comparison of different components of our network on the IDRiD dataset, where \\textbf{cat} denotes a simple concatenate of multi-lesion and vessel features in channel-wise, \\textbf{cah} and \\textbf{sah} is abbreviations for cross-attention head and self-attention head respectively}\n    \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n    \\hline\n    \\multirowcell{2}{Framework} & \\multirowcell{2}{GTB} &\\multirowcell{2}{cat} & \\multicolumn{2}{c|}{RTB} & \\multicolumn{4}{c|}{AUC\\_PR}\\\\\\cline{4-9}\n     & & & cah & sah & EX & HE & MA & SE \\\\\n    \\hline\n    \\multirowcell{6}{baseline}     &       &       &       &       & 0.8593    & 0.6284  & 0.4071   & 0.5766   \\\\\n          & \\checkmark     &       &       &       & 0.8659   & 0.6570   & 0.4279   & 0.5968   \\\\\n          & \\checkmark     & \\checkmark     &       &       & 0.8672    & 0.6756  & 0.4294    & 0.6663    \\\\\n          & \\checkmark     &       & \\checkmark     &       & 0.8682    & 0.6818   & 0.4847    & 0.7463   \\\\\n          & \\checkmark     &       &       & \\checkmark     & 0.8862    & 0.6846    & 0.4663   & 0.7422   \\\\\n          & \\checkmark     &       & \\checkmark     & \\checkmark     & \\textbf{0.9024}  & \\textbf{0.6880} \n        & \\textbf{0.4897}  & \\textbf{0.7502}  \\\\\n    \\hline\n    \\end{tabular}\n  \\label{tab3}\n\\end{table*}\n\n\n\n\\subsubsection{Analyze the Effect of GTB}\n\nTable \\ref{tab3} shows that GTB improves the performance on the basis of baseline, which indicates the weights assigned to channels in each position by GTB benefit the segmentation of all four lesions. To highlight the performance of GTB further, we compare it with other popular attention blocks under the same model parameters. \nTable \\ref{tab5} illustrates that the performance of GTB on IDRiD dataset is better than others. Different from the pooling operation as other attention blocks do, channel-wise weights of GTB are specific in each pixel, so that different channels are enlightened in different pixels.\n\nAs shown in Fig.\\ref{fig6}, the snapshots of attentive features after softmax function are taken in color. In the comparison of the spatial attention features of CBAM\\cite{woo2018cbam} and the attention features $\\mathcal{F}$ of GTB, many discrete and small patterns overlooked by the former are noticed by the latter.\n\n\\subsubsection{Analyze the Effect of RTB}\nTable \\ref{tab3} lists the ablation results. We first reaffirm the idea that vascular information contributes to multi-lesion segmentation by concatenating the vascular and multi-lesion features in channel-wise. In the comparison with and without concatenations, the greatest performance gains are obtained for HE and SE, which is consistent with the previous analysis that the HE and SE have strong relations with vessels. In order to make full use of vascular information, RTB is applied instead of simple concatenation.\nCompared with the simple concatenation, cross-attention head incorporating multi-lesion and vascular features in a transformer way enhances the scores of all four lesions further. Likewise, with the integration of self-attention head, the scores get a huge improvement as well. \n\nAs visualized in Fig.\\ref{fig7}, query-specific attention maps focus on specialized tissues. Take the query pixel on MA as an example, the self-attention tends to smaller patterns, and the cross-attention specializes the vascular tributaries, which are supposed to assist in reducing false negatives mistaken for tributaries and eliminating false alarms far from tributaries.\nBoth self-attention head and cross-attention head play a role in improving the network performance, evincing the fact that exploring the internal relationships of multi-lesion and vessels makes sense.\n\nFinally both GTB and RTB are incorporated and the network achieves the highest results. As shown in the bottom half of the Fig.\\ref{fig5}, the curve corresponding to the complete network wraps almost entirely around the others. \n\n\n\n\\begin{figure*}[htbp]\n\\centerline{\\includegraphics[width=\\textwidth]{fig8.png}}\n\\caption{Visualization of segmentation results for multi-lesion segmentation on IDRiD dataset. The different columns represent the original images, the segmentation results generated by baseline, baseline+GTB, baseline+GTB+RTB and groundtruths respectively. The yellow boxes denote the improvements over baseline brought about by GTB, in the form of pickups of missing detections, while the green boxes denote the improvements over baseline+GTB brought about by RTB, mainly in the form of fewer false alarms.\n}\n\\label{fig8}\n\\end{figure*}\n\n\\subsection{Generalization Studies on DDR and IDRiD Dataset}\nFor medical images, it is challenging but meaningful to realize the generalization over different domains under different imaging conditions. In purpose to validate the generalization capability, models are trained with the images from the train set of DDR dataset and tested on test set of IDRiD dataset which is captured from another source. Table \\ref{tab6} compares the results obtained from the preliminary analysis of generalization. From the chart, it can be seen that our method achieves the best performance by narrowing down the gap between images under different conditions.\n\n\n\n\n\n\n\n\\subsection{Qualitative Results}\nTo better illustrate the effect of GTB and RTB, we visualize the results of certain images. \nFig.\\ref{fig8} compares the segmentation results with corresponding original images and groundtruths. We take the segmentation maps of baseline, baseline with GTB and baseline with GTB and RTB to present the improvements of different components of our network.\nThe yellow boxes present the improvements of GTB, where the missing detections are discovered. The green boxes are steps up from the yellow boxes by RTB, which picks up missing detections further and reduces false alarms. Additionally, the edges of the large lesion patterns are more precisely fine-tuned by RTB, especially for the SE with blurred edges.\n\n\n\\begin{table}[]\n  \\centering\n  \\caption{Performance Comparison of different methods on the generalization from DDR dataset to IDRiD dataset}\n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n    \\multirowcell{2}{Framework}  & \\multicolumn{4}{c|}{AUC\\_PR}\\\\\\cline{2-5}\n     & EX & HE & MA & SE \\\\\n    \\hline\n     HED\\cite{xie2015holistically} & 0.5420    & 0.2104  & 0.1245   & 0.1278   \\\\\n     DeepLab v3+\\cite{chen2018deeplabv3} & 0.6480    & 0.4472   & 0.1823   & 0.2926   \\\\\n     UNet\\cite{guan2019fully,Yakubovskiy2019} & 0.6472 & 0.4452 & 0.1965 & 0.2845 \\\\\n     Lseg\\cite{guo2019seg} & 0.6501    & 0.4405  & 0.1986    & 0.3059    \\\\\n     RTN(Ours) & \\textbf{0.6799}    & \\textbf{0.4504}   & \\textbf{0.2114}    & \\textbf{0.3401}   \\\\\n    \\hline\n    \\end{tabular}\n  \\label{tab6}\n\\end{table}\n\n\n\n\\section{Conclusion And Discussion}\nIn this paper, we present a novel network that employs a dual-branch architecture with GTB and RTB to segment the four DR lesions simultaneously. \nOutstanding experiment results of our network can be attributed to GTB and RTB, which investigate the intra-class dependencies among multi-lesion and inter-class relations of multi-lesion and vessels.\n\n\nHowever, limited to the considerable cost of expertise pixel-level annotations, the vessel pseudo masks provided by semi-supervised learning are inevitably coarse-grained and lead to the inadequacy of our network. Therefore,\nin our future work, we will further modify the vascular semi-supervised learning  strategy and keep improving the transformer structures to achieve better performance in DR multi-lesion segmentation with less memory requirement. \n\n\n\n\n\n\\input{ref.bbl}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction and statement of results}\n\nThis paper gives a systematic treatment of two topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space.\n\nIn particular, we establish the continuity of certain mappings between spaces of smooth mappings, e.g.\\ the continuity of the joint composition map.\nAs a first application we prove that the bisection group of an arbitrary Lie groupoid (with finite-dimensional base) is a topological group.\nFor the most part, these results are generalizations of well known constructions to spaces of smooth functions with infinite-dimensional range.\nWe refer to \\cite{illman,michor,hirsch} for topologies on spaces of smooth functions between finite-dimensional manifolds.\n\\medskip\n\nTo understand these results of the present article, recall first the situation for spaces of smooth functions between finite-dimensional manifolds. \nFor $0 \\leq r \\leq \\infty$, let $C^r (M,N)$ denote the set of $r$-times continuously differentiable functions between manifolds $M$ and $N$\nIn the case where $r$ is finite, the standard choice for a topology on $C^r (M,N)$ is the well known Whitney $C^r$-topology (cf.\\ \\cite{illman,hirsch}). \nFor $r=\\infty$ and $M$ non-compact there are several choices for a suitable topology. \nOne can for example choose the topology generated by the union of all Whitney $C^r$-topologies. \nWe call this topology the strong $C^\\infty$-topology and write $C^\\infty_S (M,N)$ for the smooth functions with this topology.\\footnote{The strong topology is in the literature often also called the ``Whitney $C^\\infty$-topology''. Following Illman in \\cite{illman}, we will not use this naming convention as it can be argued that the strong $C^\\infty$-topology is not a genuine $C^\\infty$-topology. See ibid. for more information.}\nNote that each basic neighborhood of the strong $C^\\infty$-topology allows one to control derivatives of functions only up to a fixed upper bound.\nHowever, in applications one wants to control the derivatives of up to arbitrary high order (this is made precise in Section \\ref{sect: vsTop}).\nTo achieve this one has to refine the strong topology, obtaining the \\emph{very strong} topology\\footnote{In \\cite{michor} this topology is called the $\\mathcal{D}$-topology.} in the process (cf.\\ \\cite{illman} for an exposition).\nWe denote by $C_{\\textup{vS}}^\\infty (M,N)$ the smooth functions with the very strong topology and note that this topology is fine enough for many questions arising from differential topology.\n\nUnfortunately, as is argued in \\cite{michor} this topology is still not fine enough, if one wants to obtain manifold structures on $C^\\infty (M,N)$ (and subsequently on the group of diffeomorphisms $\\Diff (M)$).\nHence Michor constructed a further refinement of the very strong topology, called the $\\mathcal{FD}$-topology. \nIn the present paper, we call this topology the \\emph{fine very strong} topology and denote the space of smooth functions with this topology by $C_{\\textup{fS}}^\\infty (M,N)$.\n\nNote that the topologies discussed so far coincide if the source manifold is compact. \nIn fact in this case, all of these topologies coincide with the compact open $C^\\infty$-topology (see e.g.\\ \\cite[Definition 5.1]{neeb}). \nThe compact open $C^\\infty$-topology for infinite-dimensional target manifolds is already well understood and has been used in many investigations, for example in infinite-dimensional Lie theory, e.g.\\ \\cite{glock1}.\nHence our investigation will only turn up new results for non-compact source manifolds and infinite-dimensional target manifolds.\n\\smallskip\n\nWe will now go into some more detail and explain the main results of the present paper.\nOur aim is now to generalize the construction of the very strong and fine very strong topology to the set of smooth functions $C^\\infty (M,X)$, where $X$ is a locally convex manifold.\nHere smooth maps are understood in the sense of Bastiani's calculus \\cite{bastiani} (often also called Keller's $C^r$-theory~\\cite{keller}).\nWe refer to \\cite{milnor1983,glock1,neeb} for streamlined expositions, but have included a brief recollection in Appendix \\ref{calculus}.\n\nWorking in this framework we construct the very strong and the fine very strong topology for $C^\\infty (M,X)$, where $M$ is finite-dimensional and $X$ is a locally convex manifold.\nOur exposition mostly follows Illman's article \\cite{illman} and we adapt his arguments to our setting. \nIn particular, we describe the topology in terms of local charts as in \\cite{illman} (cf.\\ also \\cite{hirsch}).\nFor finite-dimensional manifolds one can alternatively introduce the topology using jet bundles and it is well known that both approaches yield the same topology.\nThis fact seems to be a folklore theorem, but we were not able to locate a proof in the literature. \nAs this fact is needed later on, a proof is given in Appendix \\ref{folklore}.  \nThe advantage of the approach using local charts can be summarized as follows: Arguments and proofs often split into two distinct steps.\nFirst one establishes a property of the function space topology only for the (easier) special case of vector space valued smooth mappings. \nThen a localization argument involving manifold charts allows one to establish the result for smooth maps between manifolds.\n\nTo our knowledge the topologies discussed in the present paper have so far only been studied for finite-dimensional manifolds. A topology somewhat similar to the very strong topology but for infinite-dimensional manifolds can be found in \\cite[Section 41]{KM97}. Albeit the similar look, be aware that the jet bundles used in the construction are only manifolds in the inequivalent convenient setting of calculus. In particular, the topology in loc.cit.\\ does not coincide with the one constructed here if $M$ is non-compact (cf.\\ \\cite[42.2 Remarks]{KM97}). We refer to Remark \\ref{rem: topgeneral} for related topologies on function spaces between Banach manifolds.\nFor finite-dimensional manifolds, our construction recovers exactly the ones in the literature.  \nWe exploit this and recall that the set $\\Prop (N,N) \\subseteq C^\\infty (M,N)$ of all proper maps is open in the very strong and the fine very strong topology.\nThen one can establish continuity of certain composition mappings, in particular our results subsume the following theorem.\\smallskip\n\n\\textbf{Theorem A} \\emph{Let $M$, $N$ be finite-dimensional manifolds, $X$ and $Y$ be (possibly infinite-dimensional) manifolds. In the following, endow all function spaces either with the very strong or the fine very strong topology. Then \n the joint composition \n  \\begin{displaymath}\n   \\Gamma \\colon \\Prop (M,N) \\times C^\\infty (N,X) \\rightarrow C^\\infty (M,X) ,\\quad  (f,g) \\mapsto g\\circ f\n  \\end{displaymath}\n  is continuous.}\n  \n \\emph{Further, for any smooth map $h \\colon X \\rightarrow Y$, the pushforward $$h_* \\colon C^\\infty (M,X) \\rightarrow C^\\infty (M,Y) , \\quad f \\mapsto h\\circ f$$ \n  is continuous.}\n\nHaving this theorem at our disposal, we construct an interesting class of topological groups: \nSuppose $\\mathcal{G} = (G \\ensuremath{\\nobreak\\rightrightarrows\\nobreak} M)$ is a Lie groupoid. This means that $G,M$ are smooth manifolds, equipped with submersions $\\alpha,\\beta \\colon G\\rightarrow M$ and an associative and smooth multiplication $G\\times _{\\alpha,\\beta}G \\rightarrow G$ that\n admits a smooth identity map $1 \\colon M\\rightarrow G$ and a smooth inversion $\\iota\\colon G\\rightarrow G$. \n Then the bisections $\\Bis(\\mathcal{G})$ of $\\mathcal{G}$ are the sections\n $\\sigma\\colon M\\rightarrow G$ of $\\alpha$ such that $\\beta \\circ \\sigma$ is a\n diffeomorphism of $M$. This becomes a group with respect to \n \\begin{equation*}\n  (\\sigma \\star \\tau ) (x) := \\sigma ((\\beta \\circ \\tau)(x))\\tau(x)\\text{ for }  x \\in M.\n \\end{equation*}\nMany interesting groups from differential geometry such as diffeomorphism groups, automorphism groups and gauge transformations of principle bundles can be realised as bisection groups of suitable Lie groupoids.  \nBy construction $\\Bis (\\mathcal{G}) \\subseteq C^\\infty (M,G)$ and with respect to the topologies on the space of smooth functions we obtain the following.\n\n\\textbf{Theorem B} \n\\emph{Let $\\mathcal{G} = (G\\ensuremath{\\nobreak\\rightrightarrows\\nobreak} M)$ be a Lie groupoid with finite-dimensional base $M$. Then $(\\Bis (\\mathcal{G}),\\star)$ is a topological group with respect to the subspace topology induced by either the very strong or the fine very strong topology on $C^\\infty (M,G)$.}\n\nThis result is a first step needed to turn the bisection group into an infinite-dimensional Lie group. \nIn fact, it turns out that one can establish this result quite easily (see below) once Theorem B is available.\nThe key step to establish the applications mentioned below, is to work out the continuity of certain composition mappings (which has been done in Theorem A).\nThen Proposition C and Theorem D below can be established using standard techniques from the literature.\nIn the present paper we will be only concerned with properties of the topology on function spaces. \nHence the next results are stated without a proof. We provide only some references to the literature and hope to provide details in future work.\n\n\n\\textbf{Proposition C} \\emph{\nLet $M$ be a finite-dimensional manifold and $X$ be a possibly infinite-dimensional manifold which admits a local addition.\\footnote{This is for example  satisfied if $X$ is a Lie group, see also \\cite[Section 42.4]{KM97} for a definition of local additions and more examples.} \nThen $C_{\\textup{fS}}^\\infty (M,X)$ can be turned into a manifold modeled on spaces of compactly supported sections of certain bundles.}\n\nIt turns out that once the space of smooth functions is endowed with the correct topology it is not hard to prove Proposition C.\nMore details and references to literature containing the necessary auxiliary facts can be found at the end of Section \\ref{sect: vsTop}.\nProposition C generalizes \\cite[Theorem 10.4]{michor} in so far as it admits arbitrary infinite-dimensional manifolds as target manifolds (whereas loc.cit.\\ was confined to finite-dimensional targets). \nWe remark that in \\cite[42.4 Theorem]{KM97} the smooth functions $C^\\infty (M,X)$ for $M$ and $X$ as in Proposition C have been endowed with a manifold structure in the inequivalent convenient setting of calculus.\nHowever, following \\cite[42.2 Remarks]{KM97} the topology on $C^\\infty (M,X)$ used in the construction does not coincide with the fine very strong topology if $M$ is non-compact. \nHence both constructions are inequivalent even if both $M$ and $X$ are finite-dimensional (and \\( M \\) is non-compact).\n\nThe manifold structure provided by Proposition C allows one to establish the Lie group structure for a general class of bisection groups. Adapting arguments from \\cite{michor} and \\cite{Schmeding2015} one can prove that \n\n\\textbf{Theorem D} \\emph{\nThe group of bisections of a Lie groupoid $\\mathcal{G} = (G\\ensuremath{\\nobreak\\rightrightarrows\\nobreak} M)$ with $M$ finite-dimensional and $G$ a Banach manifold\\footnote{Assuming certain mild conditions on $G$ (i.e.\\ an adapted local addition, cf. \\cite{Schmeding2015}), it is not necessary to assume that $G$ is a Banach manifold.} is an infinite-dimensional Lie group. }\n\nThis generalizes the construction from \\cite{Schmeding2015}, where the group of bisections of a Lie groupoid with \\emph{compact} base was turned into an infinite-dimensional Lie group. \nThus one obtains a conceptual approach to the Lie group structures of many groups which are of interest in differential geometry (e.g.\\ automorphism groups and gauge transformation groups of principle bundles over a \\emph{non compact} base).\nMoreover, Theorem D is a crucial ingredient if one wants to extend the strong connection between Lie groupoids and infinite-dimensional Lie groups which was developed in \\cite{SchmedingWockel15}.\n\n\\newpage\n\\section{The very strong topology}\\label{sect: vsTop}\nIn this section, we introduce the \\emph{very strong topology} on the space $ C^\\infty (M,X) $ of smooth maps from a finite-dimensional smooth manifold $ M $ to a possibly infinite-dimensional smooth manifold $ X $. \nThe very strong topology allows us to control derivatives of smooth maps up to arbitrarily high order on certain families of compact sets.\nThis is a straightforward generalization of the very strong topology on the space of smooth maps between finite-dimensional manifolds as described in \\cite{illman}.\n\n\\textbf{Notation and conventions.} We write $ \\mathbb{N} := \\lbrace 1,2,\\dots \\rbrace $ and $ \\mathbb{N}_0 := \\lbrace 0,1,\\dots \\rbrace $, and will only work with vector spaces over the field of real numbers $ \\RR $. Finite-dimensional manifolds are always assumed to be $\\sigma$-compact, i.e. a countable union of compact subspaces (which for finite-dimensional manifolds is equivalent to being second countable). We always endow \\( \\RR^n \\) with the supremum norm $ \\Vert \\cdot \\Vert_\\infty $ unless otherwise stated. We define $ B_\\epsilon^n (x) := \\lbrace y \\in \\mathbb{R}^n : \\Vert y - x \\Vert_\\infty < \\epsilon \\rbrace $. Notation and conventions regarding locally convex vector spaces, smooth maps, and infinite-dimensional manifolds is covered in Appendix \\ref{calculus}. Typically, $ M $ and $ N $ will be finite-dimensional smooth manifolds, $ X $ a smooth manifold modeled on a locally convex vector space, and $ E $ a locally convex vector space.\n\n\\begin{definition}\n\t\\label{norm}\n\tLet $E$ be a locally convex vector space, $p$ a continuous seminorm on $E$, $f \\colon \\mathbb{R}^m \\to E $ smooth, $A \\subseteq \\mathbb{R}^m $ compact, $r \\in \\mathbb{N}_0 $, and $e_1, \\dots, e_m $ the standard basis vectors in $\\mathbb{R}^m$. Then define\n\t\\begin{displaymath}\n\t\\Vert f \\Vert (r,A,p) = \\sup \\lbrace p(\\dd^{(k)} f(a;\\alpha)) : a \\in A, \\alpha \\in \\lbrace e_1, \\dots , e_m \\rbrace^k, 0 \\leq k \\leq r \\rbrace .\n\t\\end{displaymath}\n\\end{definition}\n\\begin{remark}\n\t\\label{normremark}\n\tThe symbol $ \\dd^{(k)} f $ is defined in Definition \\ref{Crmap}. Elsewhere in the literature, $ \\dd^{(k)} f (x;y) = \\dd^{(k)} f (x;y_1,\\dots,y_k) $ is often denoted\n\t\\begin{align*}\n\t\\frac{\\partial^k}{\\partial y_k \\cdots \\partial y_1}f(x) && \\mbox{or} && \\frac{\\partial}{\\partial y} f(x),\n\t\\end{align*}\n\twhere $ y = (y_1,\\dots,y_k) $.\n\t\n\tIn the definition above we require $ \\alpha \\in \\lbrace e_1,\\dots,e_m \\rbrace^k $. But for any $ \\alpha \\in B_1^n (0) $ and $ a \\in A $ and $ k \\leq r $ we have $ p (\\dd^{(k)} f (a;\\alpha)) \\leq K \\Vert f \\Vert (r,A,p) $ for some constant $ K $ depending only on $ r $ and $ m $, by \\eqref{schwarzrule} in Proposition \\ref{chainruleprop}.\n\t\n\tIf $ E = \\RR^n $, any norm generates the topology on $ E $ and norms are in particular seminorms. By Proposition \\ref{generatingfamilies}, the very strong topology is not affected if we always assume that the seminorm $p$ on $ \\RR^n $ is the supremum norm $ \\Vert\\cdot \\Vert_\\infty $. In this case we simply write $ \\Vert f \\Vert (r,A) $ for $ \\Vert f \\Vert (r,A,\\Vert\\cdot \\Vert_\\infty ) $.\n\\end{remark}\n\n\\begin{lemma}[Triangle inequality]\n\t\\label{triangleinequality}\n\tLet $ E,p,A,r $ be as in Definition \\ref{norm}. Then the map\n\t\\begin{displaymath}\n\t\t\\Vert\\cdot \\Vert (r,A,p) : C^\\infty (\\mathbb{R}^m,E) \\to \\mathbb{R}\n\t\\end{displaymath}\n\tsatisfies the triangle inequality. In fact it is a seminorm on $ C^\\infty (\\mathbb{R}^m,E) $.\n\\end{lemma}\n\\begin{proof}\n\t\tUse linearity of $ d(-)(a,\\alpha) $ for fixed $ (a, \\alpha) $, and the fact that $ p $ satisfies the triangle inequality.\n\\end{proof}\n\n\\begin{definition}[Elementary neighborhood]\n\t\\label{elementarynbh}\n\tLet $E$, $p$, and $r$ be as in Definition \\ref{norm}, $M$ an $m$-dimensional smooth manifold, $ X $ a smooth manifold modeled on $ E $. Consider \\( f \\colon M \\to X \\) smooth, $(U,\\phi)$ a chart on $M$, $ (V,\\psi) $ a chart on $ X $, $ A \\subseteq U $ compact such that $ f(A) \\subseteq V $, and $ \\epsilon > 0 $. Define\n\t\\begin{align*}\n\t\\mathcal{N}^r (f; A,(U,\\phi),(V,\\psi),p,\\epsilon ) = \\lbrace h &\\in C^\\infty (M,E) : \\mbox{$ h(A) \\subseteq V $ and } \\\\\n\t&\\Vert \\psi \\circ h \\circ \\phi^{-1} - \\psi \\circ f \\circ \\phi^{-1} \\Vert (r, \\phi (A), p) < \\epsilon \\rbrace .\n\t\\end{align*}\n\tWe call this set an \\emph{elementary ~$C^r$-neighborhood of }~$ f $ in ~$ C^\\infty (M,E) $.\n\\end{definition}\n\n\\textbf{Conventions for elementary neighborhoods}\nIf $ X = \\RR^n $, we will assume that $ p $ is the supremum norm and omit the $ p $ when writing down the elementary neighborhoods.\n\nWhen there is a canonical choice of charts for our manifolds, e.g.\\ if $ X = E $ is a locally convex vector space, we omit the obvious charts when writing down elementary $ C^r $-neighborhoods.\nThus for $ f \\colon M \\to E $ we write e.g.\\ $ \\mathcal{N}^r (f;A,(U,\\phi),p,\\epsilon) := \\mathcal{N}^r (f;A,(U,\\phi),(X,\\id ),p,\\epsilon ) .$ \n\n\n\\begin{remark}\n\t\\begin{enumerate}\n\t \\item The conditions $ f(A) \\subseteq V $ and $ h(A) \\subseteq V $ ensure that the map $ \\psi \\circ h \\circ \\phi^{-1} - \\psi \\circ f \\circ \\phi^{-1} $ makes sense. \n\tFurther, the conditions enable us to control the open sets into which a (given) compact set is mapped, i.e.\\ the kind of control provided by the well known compact open topology (cf.\\ \\cite[Definition I.5.1]{neeb}).\n        Indeed, by restricting to elementary $C^0$-neighborhoods, one would recover a subbase of the compact open topology on $C^\\infty (M,X)$.\n\t \\item We define elementary neighborhoods only for finite-dimensional source manifolds as the seminorms in Definition \\ref{norm} make only sense for these manifolds. \n\t Compare Remark \\ref{rem: topgeneral} for more information on alternative approaches to the topology which avoid this problem. \n\t\\end{enumerate}\n\\end{remark}\n\nWe now define what will become the basis sets in the very strong topology on $ C^\\infty (M,X) $. \n\\begin{definition}[Basic neighborhood]\n\t\\label{basicnbh}\n\tLet $ f \\colon M \\to X $ be a smooth map from a finite-dimensional smooth manifold $M$ to a smooth manifold $ X $ modeled on a locally convex vector space $E$. A \\emph{basic neighborhood of $f$ in ~$ C^\\infty (M,X) $} is a set of the form\n\t\\begin{displaymath}\n\t\\bigcap_{i \\in \\Lambda } \\mathcal{N}^{r_i} (f; A_i,(U_i, \\phi_i),(V_i,\\psi_i),p_i, \\epsilon_i),\n\t\\end{displaymath}\n\twhere $ \\Lambda $ is a possibly infinite indexing set, for all \\( i \\) the other parameters are as in Definition \\ref{elementarynbh}, and $ \\lbrace A_i \\rbrace_{i \\in \\Lambda} $ is locally finite. We call $ \\lbrace A_i \\rbrace_{i \\in \\Lambda} $ the \\emph{underlying compact family} of the neighborhood.\n\\end{definition}\nWithout loss of generalization, \\( \\Lambda = \\mathbb{N} \\), since every locally finite family over a \\( \\sigma \\)-compact space is countable.\n\nAs Proposition \\ref{basicnbhsisbasis} show, the basic neighborhoods in ~$ C^\\infty (M,X) $ form a basis for a topology on ~$ C^\\infty (M,X) $. In order to prove the proposition we need the following lemma.\n\\begin{lemma}\n\t\\label{trianglelemma}\n\tLet $ f : M \\to X $ be smooth, and ~$ g \\in \\mathcal{N} := \\mathcal{N}^r (f; A, (U,\\phi),(V,\\psi),p,\\epsilon) $. \n\tThen there exists $ \\epsilon' > 0 $ such that ~$ \\mathcal{N}' := \\mathcal{N}^r (g; A, (U, \\phi ),(V,\\psi), p, \\epsilon') \\subseteq \\mathcal{N} $.\n\\end{lemma}\n\\begin{proof}\n\tFor $ h,\\tilde{h} \\in C^\\infty (M,X) $ with $ h(A),\\tilde{h}(A) \\subseteq V $, let\n\t\\begin{displaymath}\n\t\td(h,\\tilde{h}) = \\Vert \\psi \\circ \\tilde{h} \\circ \\phi^{-1} - \\psi \\circ h \\circ \\phi^{-1} \\Vert (r,\\phi(A),p).\n\t\\end{displaymath}\n\tNote that $ d $ satisfies the triangle inequality by Lemma \\ref{triangleinequality}, and that $ h \\in \\mathcal{N} $ is equivalent to $ d(f,h)<\\epsilon $.\n\t\n\tSet $ \\epsilon' = \\epsilon - d(f,g) $, and let $ \\mathcal{N}' $ be as in the statement of the lemma. If $ h \\in \\mathcal{N}' $, then \n\t\\begin{displaymath}\n\td(f,h) \\leq d(f,g) + d(g,h) < d(f,g) + (\\epsilon - d(f,g)) = \\epsilon.\n\t\\end{displaymath}\n\tHence $ h \\in \\mathcal{N} $, and $ \\mathcal{N}' \\subseteq \\mathcal{N} $.\n\\end{proof}\n\\begin{proposition}\n\t\\label{basicnbhsisbasis}\n\tLet $ \\mathcal{U} $ and $ \\mathcal{U}' $ be basic neighborhoods of $ f $ and $ f' $ in $ C^\\infty (M,X) $, respectively. If $ g \\in \\mathcal{U} \\cap \\mathcal{U}' $, then there exists a basic neighborhood $ \\mathcal{V} $ of $ g $ such that $ \\mathcal{V} \\subseteq \\mathcal{U} \\cap \\mathcal{U}' $.\n\t\n\tHence the basic neighborhoods form a basis for a topology on $ C^\\infty (M,E) $, called \\emph{the very strong topology on $ C^\\infty (M,E) $}.\n\\end{proposition}\n\\begin{proof}\n\tWe may write\n\t\\begin{align*}\n\t\\mathcal{U} = \\bigcap_{i \\in \\Lambda} \\mathcal{N}_i && \\mbox{and} && \\mathcal{U}' = \\bigcap_{j \\in \\Lambda'} \\mathcal{N}'_j \n\t\\end{align*}\n\tfor some sets $ \\Lambda $ and $ \\Lambda' $, where $ \\mathcal{N}_i $ and $ \\mathcal{N}'_i $ are elementary neighborhoods of $ f $ and $ f' $, respectively.\n\tFor all $ i \\in \\Lambda $ and $ j \\in \\Lambda $ choose as in Lemma \\ref{trianglelemma} elementary neighborhoods $ \\mathcal{M}_i $ and $ \\mathcal{M}'_j $ of $ g $ such that $ \\mathcal{M}_i \\subset \\mathcal{N}_i $ and $ \\mathcal{M}'_j \\subset \\mathcal{N}'_j $. Then \n\t\\begin{displaymath}\n\t\\mathcal{V} := \\left( \\bigcap_{i \\in \\Lambda} \\mathcal{M}_i \\right) \\cap \\left( \\bigcap_{j \\in \\Lambda'} \\mathcal{M}'_i \\right) \\subseteq \\mathcal{U} \\cap \\mathcal{U}'. \n\t\\end{displaymath} \n\tIt remains to check that $ \\mathcal{V} $ is in fact a basic neighborhood of $ g $. The set $ \\mathcal{V} $ is a basic neighborhood of $ g $ provided that the underlying compact family of $ \\mathcal{V} $ is locally finite. This is indeed the case since the underlying compact families of $ \\mathcal{U} $ and $ \\mathcal{U}' $ are locally finite and finite unions of locally finite families are locally finite.\n\\end{proof}\nThe preceding proposition justifies the following definition.\n\\begin{definition}[Very strong topology]\n\tThe \\emph{very strong topology on $ C^\\infty (M,X) $} is the topology on $ C^\\infty (M,X) $ with basis the basic neighborhoods in $ C^\\infty (M,X) $.\\\\\n\tThe set $ C^\\infty (M,X) $ equipped with the very strong topology will be denoted by $ C_{\\textup{vS}}^\\infty (M,X) $.\n\\end{definition}\n\n\\begin{remark}\n We will work later on with $C_{\\textup{vS}}^\\infty (M,E)$, where $E$ is a locally convex space. To this end, we considered $E$ as a manifold with the canonical atlas given by the identity.\n This may seem artificial at first glance as one in principle needs to take all ``manifold charts'' of $E$ into account.\n Note however that by Lemma \\ref{lem: atlaschoice} the very strong topology on $C^\\infty (M,E)$ is generated by all basic neighborhoods of the form \n \\begin{equation}\\label{loc: charts}\n  \\bigcap_{i \\in \\Lambda } \\mathcal{N}^{r_i} (f; A_i,(U_i, \\phi_i),(E,\\id_E),p_i, \\epsilon_i),\n \\end{equation}\n i.e.\\ it suffices to consider elementary neighborhoods with respect to the identity chart.\n Hence the topology on $C^\\infty (M,E)$ is quite natural.\n \n Similarly for $C^\\infty (\\mathbb{R}^n,E)$ the charts $(U_i,\\phi_i)$ in \\eqref{loc: charts} can be replaced by $(\\mathbb{R}^n,\\id_{\\mathbb{R}^n})$ by Lemma \\ref{lem: atlaschoice2}. \n In the following, we will always assume that our elementary and basic neighborhoods are constructed with respect to the identity if one (or both) of the manifolds are a locally convex space.  \n\\end{remark}\n\n\n\\begin{remark}\n\tThere are other well-known topologies on $ C^\\infty (M,X) $. The \\emph{strong} topology (or \\emph{Whitney} $C^\\infty$-topology) and the \\emph{compact open} $C^\\infty$-topology (or \\emph{weak} topology) have as bases neighborhoods of the form described in Definition \\ref{basicnbh}, with some additional restrictions. For the strong topology the collection $\\lbrace  r_i \\rbrace_{i \\in \\Lambda} $ of indices giving differentiation order is bounded, and for the compact open $C^\\infty$-topology we require that the indexing set $ \\Lambda $ is finite. \n\t\n\tThe very strong topology is finer than the strong topology which is finer than the compact open $C^\\infty$-topology, and in the case that $ M $ is compact all of these topologies coincide (since every locally finite family meets a compact set only finitely many times). We refer the reader to section 2.1 in \\cite{hirsch} for information about the strong and compact open $C^\\infty$ topologies in the case that $ X $ is finite-dimensional. A comparison of the strong topology and the very strong topology can be found in the introduction of \\cite{illman}.\n\t\n\tSince the very strong topology is finer than the strong topology, subsets of $ C^\\infty (M,X) $ that are open in the strong topology are also open in the very strong topology. \\cite[Section 2.1]{hirsch} has several results stating that certain subsets of $ C^\\infty (M,N) $ are open in the strong topology, consequently also in the very strong topology. In particular, the set $ \\Prop (M,N) $ of proper smooth maps is open in $ C_{\\textup{vS}}^\\infty (M,N) $. We write $ \\Prop_{\\textup{vS}} (M,N) $ for the subspace $ \\Prop (M,N) $ of $ C_{\\textup{vS}}^\\infty (M,N) $ equipped with the subspace topology.\n\\end{remark}\n\n\\begin{remark}\\label{rem: topgeneral}\nOne can also define the very strong topology on the space $ C^\\infty (X,Y) $ where $ X $ and $ Y $ are Banach manifolds (i.e. modeled on Banach spaces). \nTo this end one needs to redefine the seminorms generating the topology, which in the vector space case will take the following form:\n\nIf \\( X,Y \\) are Banach spaces, \\( f \\colon X \\to Y \\) smooth, \\( r \\in \\mathbb{N}_0 \\), and \\( A \\subseteq X \\) compact define\n\\begin{equation}\\label{semi:Frechet}\n\t\\Vert f \\Vert (r,A) = \\sup \\left\\lbrace \\Vert D^k f (x) \\Vert_Y : \\mbox{ \\( 0 \\leq k \\leq r \\) and \\( x \\in A \\)} \\right\\rbrace,\n\\end{equation}\nwhere \\( D^k f \\) denotes the \\( k \\)-th \\emph{\\Frechet  derivative} of \\( f \\).\n\nHere we use that every smooth Bastiani map is also smooth in the sense of \\Frechet differentiability by \\cite[Lemma 2.10]{milnor1983}. It is easy to see that all statements made on elementary neighborhoods in the present section remain valid. Hence we obtain a very strong topology on smooth functions between Banach manifolds. \n\n\tNote that one can prove as in Appendix \\ref{folklore} that the ``very strong topology'' constructed with respect to the seminorms \\eqref{semi:Frechet} induces again the (original) very strong topology on $ C^\\infty (X,Y) $ if $ X $ is finite-dimensional. \n\tUnfortunately, for an infinite-dimensional Banach manifold $ X $ this topology does not allow us to control the behavior of functions ``at infinity'' (or anywhere for that matter since compact subsets of infinite-dimensional  Banach spaces have empty interior).\nTo see this recall that manifolds modeled on infinite-dimensional Banach space don't have a locally finite compact exhaustion by the Baire category theorem. \n\n\tRecall however, that one can define a Whitney $ C^\\infty $-topology for $X,Y$ Banach manifolds via jet bundles (see e.g.\\ \\cite{michor,KM97} or Appendix \\ref{folklore} for a short exposition). As shown in \\cite[Chapter 9]{marg}, this topology then allows one to control the behavior of a function on all of $ X $. \n\tThe key difference is that the Whitney topology defined in this way controls the behavior of jets on locally finite families of \\textbf{closed} sets. \n\tObviously, one can not hope to describe it via the seminorms as the existence of the suprema in the seminorms is tied to the compactness of the sets. Even worse, for an infinite-dimensional manifold $ X $ and $Y=E$ a locally convex space, the largest topological vector space contained in $C^\\infty(X,E)$ with respect to this topology is trivial (cf. \\cite[437]{KM97}). For these reasons we work exclusively with the very strong topology for finite-dimensional source manifolds.\n\\end{remark}\n\n\\textbf{Additional facts about the very strong topology.} Sometimes it is convenient to assume that the continuous seminorms $ p $ used in constructing very strong neighborhoods are of a certain form, as we have already remarked. There is no loss of generality in making such assumptions if the family of seminorms that we restrict to is ``big enough''.\n\n\\begin{proposition}\n\t\\label{generatingfamilies}\n\tLet $ M $ be a finite-dimensional smooth manifold and $ X $ a smooth manifold modeled on a locally convex vector space $ E $. Suppose $ \\mathcal{P} $ is a generating family of seminorms for $ E $ (see Definition \\ref{seminormfamilydef}). \n\t\n\tIf we replace every instance of ``$p$ is a continuous seminorm on $ E $'' in the definitions and results earlier in this section with ``$ p \\in \\mathcal{P} $'', then the resulting very strong topology on $ C^\\infty (M,X) $ is unaffected.\n\\end{proposition}\n\\begin{proof}\n\tLet $ \\mathcal{T} $ be the very strong topology on $ C^\\infty (M,X) $ constructed with respect to all continuous seminorms on $ E $, and let $ \\mathcal{T}' $ be the very strong topology on $ C^\\infty (M,X) $ obtained by restricting to seminorms in $ \\mathcal{P} $. Then $ \\mathcal{T}' $ is obviously coarser than $ \\mathcal{T} $ since every $ p \\in \\mathcal{P} $ is continuous, so it suffices to show that $ \\mathcal{T} $ is coarser than $ \\mathcal{T}' $. This will be the case if for every basic $ \\mathcal{T} $-very strong neighborhood $ \\mathcal{U} = \\bigcap_{i\\in\\Lambda} \\mathcal{N}_i $ of $ f \\in C^\\infty (M,X) $, where each $ \\mathcal{N}_i $ is an elementary $ \\mathcal{T} $-very strong neighborhood $$ \\mathcal{N}_i = \\mathcal{N}^{r_i} (f;A_i,(U_i,\\phi_i),(V_i,\\psi_i),p_i,\\epsilon_i ), $$ there exists a basic $ \\mathcal{T}'$-very strong neighborhood $ \\mathcal{U}' $ of $ f $ such that $ \\mathcal{U}' \\subseteq \\mathcal{U} $.\n\t\n\tFix $ i \\in \\Lambda $. By \\eqref{generating family criterion} in Proposition \\ref{locally convex prop} there exist $ n_i \\in \\mathbb{N} $ and $ p_{i,1},\\dots,p_{i,n_i} \\in \\mathcal{P} $ and $ c_i > 0 $ such that $ p_i \\leq c_i \\sup_{1\\leq j \\leq n_i} p_{i,j} $. And then\n\t\\begin{align*}  \n\t\\mathcal{V}_i := \\bigcap_{j=1}^{n_i} \\mathcal{N}^{r_i} \\left( f;A_i,(U_i,\\phi_i),(V_i,\\psi_i),p_{i,j}, \\frac{ \\epsilon_i }{2c_i} \\right) \\subseteq \\mathcal{N}_i.\n\t\\end{align*}\n\tIndeed, if $ g \\in \\mathcal{V}_i $, then for $ a \\in A_i $, $ \\alpha \\in \\lbrace e_1,\\dots,e_m \\rbrace^k $, $ 0 \\leq k \\leq r_i $, and $ 0 \\leq j \\leq n_i $, we have $$ c_i p_{i,j} ( \\dd (\\psi_i \\circ g \\circ \\phi_i^{-1} - \\psi_i \\circ f \\circ \\phi_i^{-1})^{(k)}(a,\\alpha) ) < \\frac{ \\epsilon_i}{2}, $$ which together with $ p_i \\leq c_i \\sup p_{i,j} $ clearly implies that $ g \\in \\mathcal{N}_i $.\n\t\n\tNow set $ \\mathcal{U}' := \\bigcap_{i \\in \\Lambda} \\mathcal{V}_i $. This is a basic $ \\mathcal{T}' $-very strong neighborhood of $ f $ such that $ \\mathcal{U}' \\subseteq \\mathcal{U} $.\n\\end{proof}\n\nThe following lemma is useful when constructing certain basic neighborhoods. The proof given here is fairly detailed, but throughout the remainder of this text the details of similar arguments will be omitted.\n\\begin{lemma}\n\t\\label{hackinglemma}\n\tLet \\( M \\) be a finite-dimensional smooth manifold, \\( X \\) a locally convex manifold, and \\( f \\colon M \\to X \\) a smooth map. Suppose \\( \\lbrace K_n \\rbrace_{n \\in \\mathbb{N}} \\) is a locally finite family of compact subsets of \\( M \\). Then there exist families of charts \\( \\lbrace (V_i,\\psi_i) \\rbrace_{i\\in\\mathbb{N}} \\) for \\( X \\) and \\( \\lbrace (U_i,\\phi_i) \\rbrace_{i\\in\\mathbb{N}} \\) for \\( M \\), and a locally finite family \\( \\lbrace A_i \\rbrace_{i\\in\\mathbb{N}} \\) of compact subsets of \\( M \\) such that \n\t\\begin{enumerate}\n\t\t\\item \\( \\bigcup_{i\\in\\mathbb{N}} A_i = \\bigcup_{n \\in \\mathbb{N}} K_n \\),\n\t\t\\item \\( A_i \\subseteq U_i \\) for all \\( i \\in \\mathbb{N} \\),\n\t\t\\item \\( f(U_i) \\subseteq V_i \\) for all \\( i \\in \\mathbb{N} \\).\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\tFix \\( n \\in \\mathbb{N} \\). For every \\( x \\in K_n \\) choose a chart \\( (V_{n,x},\\psi_{n,x}) \\) around \\( f(x) \\) and a chart \\( (U_{n,x},\\phi_{n,x}) \\) around \\( x \\). By shrinking \\( U_{n,x} \\) we may assume that \\( f(U_{n,x}) \\subseteq V_{n,x} \\). Since \\( M \\) is locally compact there exists a compact neighborhood \\( A_{n,x}' \\) around \\( x \\) such that \\( A_{n,x}' \\subseteq U_{n,x} \\). Now set \\( A_{n,x} = K_n \\cap A_{n,x}' \\). By compactness of \\( K_n \\) there exist finitely many \\( x_{n,1},\\dots,x_{n,k_n} \\in K_n \\) such that \\( \\lbrace A_{n,x_{n,j}} \\rbrace_{i=1}^{k_n}  \\) covers \\( K_n \\). \n\t\n\tThe families \\( \\lbrace (V_{n,x_{n,i}},\\psi_{n,x_{n,i}}) \\rbrace_{n,i} \\), \\( \\lbrace (U_{n,x_{n,i}},\\phi_{n,x_{n,i}}) \\rbrace_{n,i} \\) and \\( \\lbrace A_{n,x_{n,i}} \\rbrace_{n,i} \\) have the desired properties. By relabeling the indices we can take the indexing set to be \\( \\mathbb{N} \\).\n\\end{proof}\n\n\\begin{lemma}\n\t\\label{biglemma}\n\tLet $ M $ be a finite-dimensional smooth manifold, $ X $ a smooth manifold modeled on a locally convex vector space $ E $, and let \\( U \\subseteq M \\) and \\( V \\subseteq X \\) be open subsets. Consider the subspace \\( C^\\infty_{\\text{vS,sub}} (U,V) := \\left\\lbrace f \\in C^\\infty (M,X) : f(U) \\subseteq V \\right\\rbrace \\subseteq C_{\\textup{vS}}^\\infty (M,X) \\).\n\t\\begin{enumerate}\n\t\t\\item \\( C^\\infty_{\\text{vS,sub}} (U,V) \\) is an open subset of \\( C_{\\textup{vS}}^\\infty (M,X) \\).\n\t\t\\item The restriction \\( \\res_{\\text{vS}} \\colon C^\\infty_{ \\text{vS,sub}} (U,V) \\to C_{\\textup{vS}}^\\infty (U,V) \\) is continuous.\n\t\t\\item If $f \\in C^\\infty (U,V)$ and $\\mathcal{N}^r (f; A,(U_\\phi,\\phi),(V_\\psi,\\psi),p,\\epsilon )$ is an elementary neighborhood of $f$ such that $\\psi (V_\\psi)$ is a convex set, then there exists $g \\in C^\\infty (M,X)$ with\n\t\t\t  \\begin{equation}\\label{eq: elres}\n\t\t   \\res_{\\text{vS}}^{-1} (\\mathcal{N}^r (f; A,(U_\\phi,\\phi),(V_\\psi,\\psi),p,\\epsilon )) = \\mathcal{N}^r (g; A,(U_\\phi,\\phi),(V_\\psi,\\psi),p,\\epsilon )\n\t\t  \\end{equation}\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\t\\begin{enumerate}\n\t\t\\item Suppose \\( f \\in C^\\infty(U,V) \\).\n\t\t\tSince $ U $ is an open subset of $ M $, the subspace \\( U \\) is metrizable and locally compact, hence $ \\sigma $-compact.\n\t\t\tCombining Lemma \\ref{dugundjifact} and Lemma \\ref{hackinglemma}, we find a locally finite exhaustion \\( \\lbrace A_n \\rbrace_{n \\in \\mathbb{N}} \\) of \\( U \\) by compact sets, charts \\( \\lbrace (U_n,\\phi_n) \\rbrace_{n \\in \\mathbb{N}} \\) for \\( M \\), and charts \\( \\lbrace (V_n,\\psi_n) \\rbrace_{n \\in \\mathbb{N}} \\) for \\( X \\) such that \\( A_n \\subseteq U_n \\) and \\( f(A_n) \\subseteq V_n \\) for all \\( n \\in \\mathbb{N} \\).\n\t\t\tSince \\( f(A_n) \\subseteq V \\), shrink the \\( V_n \\) if necessary such that \\( V_n \\subseteq V \\) for all \\( n \\in \\mathbb{N} \\) (while still \\( f( A_n ) \\subseteq V_n \\)).\n\t\t\tTake any continuous seminorm $ p $ on $ E $ and define \n\t\t\t\\[ \\mathcal{U} = \\bigcap_{n \\in \\mathbb{N}} \\mathcal{N}^0 \\left( f; A_n,(U_n,\\phi_n),(V_n,\\psi_n),p,1 \\right). \\] \n\t\t\tIf $ g \\in \\mathcal{U} $, then $ g(A_n) \\subseteq V_n \\subseteq V $ for all $ n \\in \\mathbb{N} $, from which it follows that $ g(U) = g \\left( \\bigcup A_n \\right) \\subseteq V $. So $ \\mathcal{U} $ is a neighborhood of $ f $ in $ C_{\\textup{vS}}^\\infty (M,X) $ such that $ \\mathcal{U} \\subseteq C^\\infty (U,V) $.\n\t\t\\item Take an arbitrary basic neighborhood\n\t\t\t\\[\n\t\t\t\t\\mathcal{U} = \\bigcap_{i\\in\\Lambda} \\mathcal{N}^{r_i} \\left( f;A_i,(U_i,\\phi_i),(V_i,\\psi_i),p_i,\\epsilon_i \\right)\n\t\t\t\\]\n\t\t\tin \\( C_{\\textup{vS}}^\\infty (U,V) \\). We will show that given \\( g \\in \\res_{\\text{vS}}^{-1} (\\mathcal{U}) \\), there exists a basic neighborhood \\( \\mathcal{V} \\) of \\( g \\) in \\( C_{\\textup{vS}}^\\infty (M,X) \\) such that \\( \\mathcal{V} \\subseteq \\res_{\\text{vS}}^{-1} (\\mathcal{U}) \\), i.e. \\( \\res_{\\text{vS}}^{-1} (\\mathcal{U}) \\) is open. By Lemma \\ref{basicnbhsisbasis} there are elementary neighborhoods of $\\res_{\\text{vS}} (g)$ such that \n\t\t\t\\[\n\t\t\t\t\\mathcal{V} = \\bigcap_{i\\in\\Lambda} \\mathcal{N}^{r_i} \\left( \\res_{\\text{vS}} (g) ;A_i,(U_i,\\phi_i),(V_i,\\psi_i),p_i,\\delta_i \\right).\n\t\t\t\\]\n\t\t\tis contained in $\\mathcal{U}$. \n\t\t\tNow clearly $\\bigcap_{i\\in\\Lambda} \\mathcal{N}^{r_i} \\left( g ;A_i,(U_i,\\phi_i),(V_i,\\psi_i),p_i,\\delta_i \\right)$  is contained in $\\res_{\\text{vS}}^{-1} (\\mathcal{V}) \\subseteq \\res_{\\text{vS}}^{-1} (\\mathcal{U})$.\n \t\t\\item Since $M$ is finite-dimensional, whence paracompact, we can choose a neighborhood $W$ of $A$ and a smooth cutoff function $\\rho \\colon M \\rightarrow \\RR$ with $\\rho|_{W} \\equiv 1$ and $\\rho_{M \\setminus f^{-1} (V_\\psi) \\cap U_\\phi}\\equiv 0$.\n \t\tComposing with a suitable translation, we may assume without loss of generality that $\\psi(V_\\psi)$ is a convex $0$-neighborhood.\n \t\tSuppressing the translation we can thus define $$g \\colon M \\rightarrow X , \\quad x \\mapsto \\begin{cases}\n \t\t                                                             \\psi^{-1} ( \\rho(x) \\cdot \\psi \\circ f (x)) & \\text{if } x \\in U_\\phi, \\\\\n \t\t                                                             \\psi^{-1} (0) & \\text{else}.\n \t\t                                                            \\end{cases}\n\t\t      $$\n\t\tNow as $g|_W = f|_W$ the identity \\eqref{eq: elres} is satisfied.    \\qedhere\n\t\\end{enumerate}\n\\end{proof}\n\n\n\\section{Composition of maps in the very strong topology}\\label{sect: compo}\nThroughout this section, $ M $ and $ N $ are finite-dimensional smooth manifolds and $ X $ denotes a smooth manifold modeled on a locally convex vector space $ E $. \n\nIt is a desirable property of the very strong topology on \\( C^\\infty (M,X) \\) that composition\n\\begin{align*}\n\t\\Gamma : C_{\\textup{vS}}^\\infty (M,N) \\times C_{\\textup{vS}}^\\infty (N,X) &\\to C_{\\textup{vS}}^\\infty (M,X) \\\\\n\t(f,h) &\\mapsto h \\circ f\n\\end{align*}\nis continuous. But this is not the case in general, a counterexample can be found in Example \\ref{counterexample}. However, the restriction of the composition map\n\\begin{displaymath}\n\\Gamma \\colon \\Prop_{\\textup{vS}} (M,N) \\times C_{\\textup{vS}}^\\infty (N,X) \\to C_{\\textup{vS}}^\\infty (M,X)\n\\end{displaymath}\n\\textit{is} continuous, where $ \\Prop_{\\textup{vS}} (M,N) $ denotes the subspace of $ C_{\\textup{vS}}^\\infty (M,N) $ consisting of all the proper maps. This is precisely what Theorem \\ref{compiscont} says, and proving it is the main goal of this section.\n\nAs we will see, the crucial property of proper maps needed is the fact that if $ \\lbrace A_i \\rbrace $ is a locally finite family of subsets of $ M $ and $ f : M \\to N $ is proper, then $ \\lbrace f(A_i) \\rbrace $ is locally finite. This will enable us to choose for a basic neighborhood $ \\mathcal{V} $ of some composition $ h \\circ f $ in $ C_{\\textup{vS}}^\\infty (M,X) $ basic neighborhoods $ \\mathcal{U} $ and $ \\mathcal{U}' $ of $ f $ and $ h $, respectively, such that $ \\Gamma (\\mathcal{U} \\times \\mathcal{U}' ) \\subseteq \\mathcal{V} $. The challenge is to choose $ \\mathcal{U}' $ in such a way that the underlying compact family of the neighborhood is locally finite.\n\nWe now give the promised counterexample to the statement that composition of maps in the very strong topology is continuous in general. This example is inspired by the proof of \\cite[Proposition 2.2(b)]{counterglock}.\n\\begin{example}\n\t\\label{counterexample}\n\tThe composition map\n\t\\begin{align*}\n\t\t\\Gamma \\colon C_{\\textup{vS}}^\\infty (\\mathbb{R},\\mathbb{R}) \\times C_{\\textup{vS}}^\\infty (\\mathbb{R},\\mathbb{R}) \\to C_{\\textup{vS}}^\\infty (\\mathbb{R},\\mathbb{R}), \\quad (f,h) \\mapsto h \\circ f\n\t\\end{align*}\n\tis not continuous.\n\\end{example}\n\\begin{proof}\n\tNote that for every basic neighborhood $ \\mathcal{U} $ of $ f \\in C_{\\textup{vS}}^\\infty (\\mathbb{R},\\mathbb{R}) $ there exists a basic neighborhood $ \\mathcal{U}' $ of $ f $ with underlying compact family $ \\lbrace [2n-1,2n+1] \\rbrace_{n\\in\\mathbb{Z}} $ such that $ f \\in \\mathcal{U}' \\subseteq \\mathcal{U} $, since each compact interval $ [2n-1,2n+1] $ intersects only finitely many sets belonging to the locally finite underlying compact family of $ \\mathcal{U} $.\n\t\n\tTo show discontinuity of $ \\Gamma $ it suffices to show discontinuity at $ (0,0) $. Let $ \\mathcal{V} $ be the basic neighborhood of 0 given by $$ \\mathcal{V} := \\bigcap_{n\\in\\mathbb{N}} \\mathcal{N}^{n} (0;[2n-1,2n+1],1) . $$ We will show that for any pair of basic neighborhoods\n\t\\begin{align*}\n\t\\mathcal{U} = \\bigcap_{n\\in\\mathbb{Z}} \\mathcal{N}^{r_n} (0;[2n-1,2n+1],\\epsilon_n), \\\\\n\t\\mathcal{U}' = \\bigcap_{n\\in\\mathbb{Z}} \\mathcal{N}^{r_n'} (0;[2n-1,2n+1],\\epsilon_n'),\n\t\\end{align*}\n\tthere exists a pair of functions $ (f,h) \\in \\mathcal{U}' \\times \\mathcal{U} $ such that $ h \\circ f \\notin \\mathcal{V} $. \n\t\n\tConstruct $ h \\in C^\\infty (\\mathbb{R},\\mathbb{R}) $ such that in a neighborhood of $ 0 $, $ h $ is given by the equation $ h(x) = x^{r_0 + 1} $, and such that $ \\operatorname{supp} h \\subseteq ]-1,1[ $. For some sufficiently small $ k > 0 $ we will have $ kh \\in \\mathcal{U} $. For every $ m \\in \\mathbb{N} $ define $$ h_m (x) := \\frac{k}{m^{r_0}} h(mx) $$ and note that $ h_m \\in \\mathcal{U} $, since $$ | h_m^{(j)} (x) | = \\frac{k m^j}{m^{r_0}} | h^{(j)} (mx) | \\leq k | h^{(j)}(mx) | < \\epsilon_0 $$ for $ j \\leq r_0 $, where we use the notation \\( g^{(j)} (y) = \\dd^{(j)} g(y;1,\\dots,1) \\) for smooth maps \\( g \\colon \\RR \\to \\RR \\).\n\t\n\tLet $ 2n \\geq r_0 + 1 $ and construct $ \\tilde{f} \\in C^\\infty (\\mathbb{R},\\mathbb{R}) $ such that $ \\tilde{f}(x) = x - 2n $ in a neighborhood of $ 2n $ and $ \\operatorname{supp} \\tilde{f} \\subseteq ]2n-1,2n+1[ $. Then for some sufficiently small $ s > 0 $ we have $ f := s \\tilde{f} \\in \\mathcal{U}' $.\n\t\n\tSo far we have a sequence $ \\lbrace h_m \\rbrace_{m\\in\\mathbb{N}} \\subset \\mathcal{U} $ and $ f \\in \\mathcal{U}' $. By construction, $ h_m \\circ f (x) = kms^{r_0+1} (x-2n)^{r_0+1} $ in a neighborhood of $ 2n $. Hence $$ |(h_m \\circ f)^{(r_0+1)}(2n)| = kms^{r_0+1} (r_0+1)! \\geq 1 $$ for large enough $ m $, in which case $ h_m \\circ f \\notin \\mathcal{V} $.\n\\end{proof}\nHaving given the example above, we return our focus to the main task of the section, which is proving Theorem \\ref{compiscont}. Leading up to the theorem is a sequence of lemmata. \n\nAlthough we are actually interested in mapping spaces between manifolds, we first give a lemma that only applies to vector spaces. In a sense, this lemma resolves the main difficulty, and generalizing to manifolds is only a matter of dealing with charts.\n\\begin{lemma}\n\t\\label{realdomains}\n\tConsider the composition map\n\t\\begin{displaymath}\n\t\t\\Gamma : C_{\\textup{vS}}^\\infty (\\mathbb{R}^m,\\mathbb{R}^n) \\times C_{\\textup{vS}}^\\infty (\\mathbb{R}^n,E) \\to C_{\\textup{vS}}^\\infty (\\mathbb{R}^m,E),\\quad (f,h) \\mapsto h \\circ f.\n\t\\end{displaymath}\n\tLet $ (f,h) \\in C_{\\textup{vS}}^\\infty (\\mathbb{R}^m,\\mathbb{R}^n) \\times C_{\\textup{vS}}^\\infty (\\mathbb{R}^n,E) $, and consider an arbitrary elementary neighborhood\n\n\t$\t\\mathcal{N} = \\mathcal{N}^r (h\\circ f;A,p,\\epsilon) \\subseteq C_{\\textup{vS}}^\\infty (\\mathbb{R}^m,E)$\n\n\tof $ h \\circ f $. For all compact neighborhoods \\( A' \\) of \\( f(A) \\) there exist \\( \\delta, \\delta' > 0 \\) such that the elementary neighborhoods\n\t\\begin{align*}\n\t\t\\mathcal{M} = \\mathcal{N}^r (f;A,\\delta) && \\mbox{and} && \\mathcal{M}' = \\mathcal{N}^r (h;A',p,\\delta')\n\t\\end{align*}\n\tsatisfy $ \\Gamma (\\mathcal{M} \\times \\mathcal{M}') \\subseteq \\mathcal{N} $.\n\\end{lemma}\n\\begin{proof}\n\tLet $ A' $ be any compact neighborhood of $ f(A) $. We proceed in several steps.\n\t\n\t\\textbf{Step 1.} Our first goal is to define $ \\mathcal{M} $. We want a $ \\delta > 0 $ such that for all $ \\hat{f} \\in C^\\infty (\\mathbb{R}^m, \\mathbb{R}^n) $, the inequality $ \\Vert \\hat{f} - f \\Vert (r,A) < \\delta $ implies\n\t\\begin{align*}\n\t\t\\Vert h \\circ (\\hat{f} - f) \\Vert (r,A,p) < \\frac{\\epsilon}{2} && \\mbox{and} && \\hat{f} (A) \\subseteq A'.\n\t\\end{align*}\n\tBy Lemma \\ref{greenlemma} it is possible to choose $ \\delta $ such that the first property holds. We may choose $ \\delta $ such that the second property also holds, because $ \\hat{f}(A) = (\\hat{f}-f)(A) + f(A) \\subseteq B_\\delta^n (0) + f(A) $. Pick such a $ \\delta $ and define\n\t\\begin{displaymath}\n\t\t\\mathcal{M} := \\mathcal{N}^r (f;A,\\delta).\n\t\\end{displaymath}\n\t\n\tObserve that by the triangle inequality (Lemma \\ref{triangleinequality}) there exists an $ R > 0 $ such that every $ \\hat{f} \\in \\mathcal{M} $ satisfies $ \\Vert \\hat{f} \\Vert (r,A) \\leq R $.\n\t\n\t\\textbf{Step 2.} Our second goal is to define $ \\mathcal{M}' $. We want a $ \\delta' > 0 $ such that for all $ \\hat{h} \\in C^\\infty (\\mathbb{R}^n,E) $ and all $ \\hat{f} \\in \\mathcal{M} $,\n\t\\begin{align*}\n\t\t\\Vert \\hat{h} - h \\Vert (r,A',p) < \\delta' && \\implies && \\Vert (\\hat{h} - h) \\circ \\hat{f} \\Vert (r,A,p) < \\frac{\\epsilon}{2}.\n\t\\end{align*}\n\tA $ \\delta' $ having this property exists by Lemma \\ref{greenlemma} and the observation at the end of step 1. Now define\n\t\\begin{displaymath}\n\t\t\\mathcal{M}' := \\mathcal{N}^r (h;A',p,\\delta').\n\t\\end{displaymath}\n\t\n\t\\textbf{Step 3.} Now we must show that $ \\mathcal{M} $ and $ \\mathcal{M}' $ have the desired property. Let $ \\hat{f} \\in \\mathcal{M} $ and $ \\hat{h} \\in \\mathcal{M}' $. By the triangle inequality,\n\t\\begin{displaymath}\n\t\t\\Vert \\hat{h} \\circ \\hat{f} - h \\circ f \\Vert (r,A,p) \\leq \\Vert (\\hat{h} - h) \\circ \\hat{f} \\Vert (r,A,p) + \\Vert h \\circ (\\hat{f} - f) \\Vert (r,A,p) < \\epsilon.\n\t\\end{displaymath} \n\tSo $ \\hat{h}\\circ \\hat{f} \\in \\mathcal{N} $. Thus $ \\Gamma (\\mathcal{M} \\times \\mathcal{M}') \\subseteq \\mathcal{N} $.\n\\end{proof}\n\nA version of the preceding lemma still holds if we replace $ \\mathbb{R}^m $ and $ \\mathbb{R}^n $ with finite-dimensional smooth manifolds and $ E $ with an infinite-dimensional smooth manifold. This is our next result.\n\n\\begin{lemma}\n\t\\label{illmannslemma}\n\tGiven $ f \\in C^\\infty (M,N) $ and $ h \\in C^\\infty (N,X) $ and an arbitrary elementary neighborhood\n\t\\begin{displaymath}\n\t\t\\mathcal{N} := \\mathcal{N}^r (h\\circ f;A,(U,\\phi),(W,\\eta),p,\\epsilon) \\subseteq C_{\\textup{vS}}^\\infty (M,X)\n\t\\end{displaymath}\n\tof $ h\\circ f $ there exist finitely many\n\t\\begin{align*}\n\t\t\\mathcal{M}_j := \\mathcal{N}^r (f;A_j,(U,\\phi),(V_j,\\psi_j),\\delta_j) && \\mbox{and} && \\mathcal{M}_j' := \\mathcal{N}^r (h;A_j',(V_j,\\psi),(W,\\eta),p,\\delta_j')\n\t\\end{align*}\n\tsuch that\n\t\\begin{enumerate} \n\t\t\\item $\\displaystyle\\Gamma \\left( \\bigcap_j \\left( \\mathcal{M}_j \\times \\mathcal{M}_j' \\right) \\right) \\subseteq \\mathcal{N} $,\n\t\t\\item $\\displaystyle \\bigcup_j A_j = A $,\n\t\t\\item $ f(A_j) \\subseteq \\interior A_j' \\subseteq A_j' \\subseteq V_j $ for all $ j $.\n\t\\end{enumerate}\n\t\n\tMoreover, given any neighborhood $ Q $ of $ f(A) $ we may choose the $ V_j $ such that all $ V_j \\subseteq Q $.\n\\end{lemma}\n\\begin{proof}\n\tSince $ f(A) $ is compact we may choose finitely many sets\n\t\\begin{align*}\n\t\tD_j \\subseteq \\interior A_j' \\subseteq A_j' \\subseteq V_j \\subseteq N && \\mbox{such that} && f(A) \\subseteq \\bigcup_j D_j,\n\t\\end{align*}\n\twhere $ D_j $ and $ A_j' $ are compact, and $ V_j $ is a chart domain for a chart $ (V_j, \\psi_j) $ on $ N $. Shrinking the $ V_j $ we may assume that every $ V_j \\subseteq Q $. Set\n\t\\begin{displaymath} \n\t\tA_j := A \\cap f^{-1} (D_j),\n\t\\end{displaymath} \n\tto obtain compact sets that satisfy\n\t\\begin{align*}\n\t\t\\bigcup_j A_j = A && \\mbox{and} && f(A_j) \\subseteq D_j \\mbox{, for all $ j $.}\n\t\\end{align*}\n\t\n\tLet\n\t\\( \\mathcal{N}_j := \\bigcap_j \\mathcal{N}^r (h\\circ f ; A_j, (U,\\phi),(W,\\eta),p,\\epsilon) \\),\n\tand note that $ \\mathcal{N} = \\bigcap_j \\mathcal{N}_j $. For each $ j $ apply Lemma \\ref{realdomains} to the maps $ \\psi_j \\circ f \\circ \\phi^{-1} $ and $ \\eta \\circ h \\circ \\psi_j^{-1} $ and the elementary neighborhood\n\t\\begin{displaymath}\n\t\t\\tilde{\\mathcal{N}}_j := \\mathcal{N}^r (\\eta \\circ h \\circ f \\circ \\phi^{-1};\\phi(A_j),p,\\epsilon)\n\t\\end{displaymath}\n\tto obtain elementary neighborhoods\n\t\\begin{align*}\n\t\t\\tilde{\\mathcal{M}}_j = \\mathcal{N}^r (\\psi_j \\circ f \\circ \\phi^{-1};\\phi(A_j),\\delta_j) && \\mbox{and} && \\tilde{\\mathcal{M}}_j' = \\mathcal{N}^r (\\eta \\circ h \\circ \\psi_j^{-1};\\psi_j (A_j'),p,\\delta_j')\n\t\\end{align*}\n\tsuch that $ \\Gamma (\\tilde{\\mathcal{M}}_j \\times \\tilde{\\mathcal{M}}_j') \\subseteq \\tilde{\\mathcal{N}}_j $. \n\t\n\tThe elementary neighborhoods $ \\tilde{\\mathcal{M}}_j $ and $ \\tilde{\\mathcal{M}}_j' $ of $ \\psi_j \\circ f \\circ \\phi^{-1} $ and $ \\eta \\circ h \\circ \\psi_j^{-1} $, respectively, induce elementary neighborhoods\n\t\\begin{align*}\n\t\t\\mathcal{M}_j = \\mathcal{N}^r (f;A_j,(U,\\phi),(V_j,\\psi_j),\\delta_j) && \\mbox{and} && \\mathcal{M}_j' = \\mathcal{N}^r (h;A_j',(V_j,\\psi_j),(W,\\eta),p,\\delta_j')\n\t\\end{align*}\n\tof $ f $ and $ h $, respectively. These neighborhoods correspond to each other in the sense that $ \\hat{f} \\in \\mathcal{M}_j $ if and only if $ \\psi_j \\circ \\hat{f} \\circ \\phi^{-1} \\in \\tilde{\\mathcal{M}}_j $, and $ \\hat{h} \\in \\mathcal{M}_j' $ if and only if $ \\eta \\circ \\hat{h} \\circ \\psi_j^{-1} \\in \\tilde{\\mathcal{M}}_j' $. Similarly for $ \\tilde{\\mathcal{N}}_j $ and $ \\mathcal{N}_j $. Since $ \\Gamma (\\tilde{\\mathcal{M}}_j \\times \\tilde{\\mathcal{M}}_j') \\subseteq \\tilde{\\mathcal{N}}_j $, one has by the correspondence described here that $ \\Gamma (\\mathcal{M}_j \\times \\mathcal{M}_j') \\subseteq \\mathcal{N}_j $. \n\t\n\tNow just observe that\n\t\\begin{displaymath}\n\t\t\\Gamma \\left( \\bigcap_j \\left( \\mathcal{M}_j \\times \\mathcal{M}_j' \\right) \\right) \\subseteq \\bigcap_j \\Gamma \\left( \\mathcal{M}_j \\times \\mathcal{M}_j' \\right) \\subseteq \\bigcap_j \\mathcal{N}_j = \\mathcal{N}.\n\t\\end{displaymath}\n\\end{proof}\n\\begin{lemma}\n\t\\label{compislemma}\n\tConsider smooth maps $ f \\in C_{\\textup{vS}}^\\infty (M,N) $ and $ h \\in C_{\\textup{vS}}^\\infty (N,X) $ and a basic neighborhood $ \\mathcal{U} = \\bigcap_{i \\in \\Lambda} \\mathcal{N}_i $, where each\n\t\\begin{displaymath} \n\t\\mathcal{N}_i = \\mathcal{N}^{r_i} (h\\circ f;A_i,(U_i,\\phi_i),(W_i,\\eta_i),p_i,\\epsilon_i)\n\t\\end{displaymath}\n\tis an elementary neighborhood of $ h\\circ f $. \n\t\n\tIf $ \\lbrace f(A_i) \\rbrace_{i \\in \\Lambda} $ is locally finite, then there exist basic neighborhoods $ \\mathcal{V} $ and $ \\mathcal{V}' $ of $ f $ and $ h $, respectively, such that $ \\Gamma (\\mathcal{V} \\times \\mathcal{V}') \\subseteq \\mathcal{U} $.\n\\end{lemma}\n\\begin{proof}\n\tSince $ \\lbrace f(A_i) \\rbrace_{i\\in\\Lambda} $ is locally finite, there exist compact neighborhoods $ Q_i $ of $ f(A_i) $ such that $ \\lbrace Q_i \\rbrace_{i \\in \\Lambda} $ is locally finite, by \\cite[30.C.10]{cech}. Here we use our assumption that finite-dimensional manifolds are $ \\sigma $-compact.\n\t\n\tFor each $ i \\in \\Lambda $, Lemma \\ref{illmannslemma} implies that there exist\n\t\\begin{align*}\n\t\t\\mathcal{W}_i :=& \\bigcap_{j=1}^{n_i} \\mathcal{N}^{r_i} (f;A_{i,j},(U_i,\\phi_i),(V_{i,j},\\psi_{i,j}),\\delta_{i,j}), \\\\\n\t\t\\mathcal{W}_i' :=& \\bigcap_{j=1}^{n_i} \\mathcal{N}^{r_i} (h;A_{i,j}',(V_{i,j},\\psi_{i,j}),(W_i,\\eta_i),p_i,\\delta_{i,j}')\n\t\\end{align*}\n\tsuch that\n\t\\begin{enumerate}\n\t\t\\item $\\displaystyle \\Gamma (\\mathcal{W}_i \\times \\mathcal{W}_i') \\subseteq \\mathcal{N}_i $,\n\t\t\\item $\\displaystyle \\bigcup_{j=1}^{n_i} A_{i,j} = A_i $,\n\t\t\\item $ A_{i,j}' \\subseteq V_{i,j} \\subseteq Q_i $ for all \\( j \\).\n\t\\end{enumerate}\n\tThen $ \\lbrace A_{i,j} \\rbrace_{i,j} $ is locally finite by (2) and since $ \\lbrace A_i \\rbrace_i $ is locally finite, and $ \\lbrace A_{i,j}' \\rbrace_{i,j} $ is locally finite by (3) and since $ \\lbrace Q_i \\rbrace_i $ is locally finite.\tHence\n\t\\begin{align*}\n\t\t\\mathcal{V} := \\bigcap_{i\\in\\Lambda} \\mathcal{W}_i && \\mbox{and} && \\mathcal{V}' := \\bigcap_{i\\in\\Lambda} \\mathcal{W}_i'\n\t\\end{align*}\n\tare basic neighborhoods of $ f $ and $ h $, respectively, such that\n\t\\begin{displaymath}\n\t\t\\Gamma \\left( \\mathcal{V} \\times \\mathcal{V}' \\right) = \\Gamma \\left( \\bigcap_{i\\in\\Lambda} \\left( \\mathcal{W}_i \\times \\mathcal{W}_i' \\right) \\right) \\subseteq \\bigcap_{i\\in\\Lambda} \\Gamma (\\mathcal{W}_i \\times \\mathcal{W}_i') \\subseteq \\bigcap_{i\\in\\Lambda} \\mathcal{N}_i = \\mathcal{U}.\n\t\\end{displaymath}\n\\end{proof}\n\\begin{theorem}\n\t\\label{compiscont}\n\tLet $M$ and $N$ be finite-dimensional smooth manifolds and let $ X $ be a smooth manifold modeled on a locally convex vector space $E$. Then the composition map\n\t\\begin{displaymath}\n\t\\Gamma \\colon \\Prop_{\\textup{vS}} (M,N) \\times C_{\\textup{vS}}^\\infty (N,X) \\to C_{\\textup{vS}}^\\infty (M,X)\n\t\\end{displaymath}\n\tsending $ (f,h) $ to $ h \\circ f $ is continuous.\n\\end{theorem}\n\\begin{proof}\n\tIt suffices to show that given maps $ f \\in \\Prop_{\\textup{vS}} (M,N) $ and $ h \\in C_{\\textup{vS}}^\\infty (N,E) $ and a basic neighborhood\n\t\\begin{displaymath}\n\t\t\\mathcal{U} = \\bigcap_{i\\in\\Lambda} \\mathcal{N}^{r_i} (h\\circ f;A_i,(U_i,\\phi_i),(V_i,\\psi_i),p_i,\\epsilon_i)\n\t\\end{displaymath}\n\tof $ h\\circ f $ in $ C_{\\textup{vS}}^\\infty (M,X) $, there exist basic neighborhoods $ \\mathcal{V} $ and $ \\mathcal{V}' $ of $ f $ and $ h $, respectively, such that $ \\Gamma (\\mathcal{V} \\times \\mathcal{V}') \\subseteq \\mathcal{U} $.\n\t\n\tSo suppose that we are given $ f , h $ and $ \\mathcal{U} $ as above. Then $ \\lbrace f(A_i) \\rbrace_{i \\in \\Lambda} $ is locally finite since $ f $ is proper, by \\cite[Lemma 3.10.11]{engelking}. Thus we may apply Lemma \\ref{compislemma} to obtain the desired neighborhoods $ \\mathcal{V} $ and $ \\mathcal{V}' $.\n\\end{proof}\nUnfortunately, precomposition is not continuous in general as an examination of Example \\ref{counterexample} reveals. However, precomposition by a proper map is continuous. \n\\begin{proposition}\n\t\\label{precomposition}\n\tLet $ f \\in \\Prop_{\\textup{vS}} (M,N)$. Then the following map is continuous \n\t\\begin{align*}\n\t\tf^* \\colon C_{\\textup{vS}}^\\infty (N,X) &\\to C_{\\textup{vS}}^\\infty (M,X), \\quad\t\th \\mapsto h \\circ f \n\t\\end{align*}\n\\end{proposition}\n\\begin{proof}\n\tThe map $ \\iota_f \\colon C_{\\textup{vS}}^\\infty (N,X) \\to \\Prop (M,N) \\times C_{\\textup{vS}}^\\infty (N,X) $ given by $ \\iota_f (h) = (f,h) $ is continuous. Hence $ f^* $, which is the composition \n\t\\begin{displaymath}\n\t\tC_{\\textup{vS}}^\\infty (N,X) \\xrightarrow{\\iota_f} \\Prop_{\\textup{vS}} (M,N) \\times C_{\\textup{vS}}^\\infty (N,X) \\xrightarrow{\\Gamma} C_{\\textup{vS}}^\\infty (M,X),\n\t\\end{displaymath}\n\tis also continuous. \n\\end{proof}\n\nWe will now prove that postcomposition is always continuous. \nThis result is needed even for postcomposition by a map $f\\colon X \\rightarrow Y$ between infinite-dimensional manifolds.\nThus the next proposition can not readily be deduced from Theorem \\ref{compiscont} (or the other results in this section)\\footnote{For the finite-dimensional case, a proof along these lines can be found in \\cite{illman}.}. \nInstead we have to take a detour using results on the (coarser) compact open $C^\\infty$-topology (cf.\\ to the proof of Lemma \\ref{lem: atlaschoice}). \n\n\\begin{proposition}\\label{postcomposition}\n Let $f \\colon X \\rightarrow Y$ be smooth, where $Y$ is a (possibly infinite-dimensional) manifold. \n Then for any finite-dimensional manifold $M$ the following map is continuous \n      \\begin{displaymath}\n       f_* \\colon C_{\\textup{vS}}^\\infty (M,X) \\rightarrow C_{\\textup{vS}}^\\infty (M,Y) , \\quad h \\mapsto f\\circ h\n      \\end{displaymath}\n\\end{proposition}\n\n\\begin{proof}\n To see that  $f_*$ is continuous, we proceed in several steps.\n \n \\textbf{Step 1} \\emph{Special elementary neighborhoods.} \n Consider first an arbitrary elementary neighborhood $\\mathcal{N} = \\mathcal{N}^{r} (f\\circ h; A,(U, \\phi),(V,\\psi),q, \\epsilon)$ in $C_{\\textup{vS}}^\\infty (M,Y)$.\n Since $h(A)$ is compact, there are finitely many manifold charts $(W_i,\\kappa_i)$ of $X$ with $h(A) \\subseteq \\bigcup_{i} W_i$.\n Now the open sets $h^{-1} (W_i)$ cover $A$ and thus there are finitely many compact sets $K_j$ such that $A=\\bigcup_{j} K_j$ and $h(K_j) \\subseteq W_{i_j}$.\n Thus we replace $A$ by the finitely many compact sets. Note that this will ensure that the families of compact sets considered later remain locally finite.\n To shorten the notation, assume without loss of generality that there is a manifold chart $(W,\\kappa)$ of $X$ such that $h(A) \\subseteq W$ and $f(W) \\subseteq V$.\n In particular, we can thus consider the mapping $f_{\\kappa}^\\psi := \\psi \\circ f \\circ \\kappa^{-1} \\colon \\kappa (W) \\rightarrow \\psi (V)$\n \n \\textbf{Step 2} \\emph{The preimage of a special elementary neighborhood of $f\\circ h$ is a neighborhood of $h$.} We work locally in charts.\n Let $Y$ be modeled on the locally convex space $F$ and $X$ be modeled on the locally convex space $E$.\n Recall that the compact open $C^\\infty$-topology (see \\cite[Definition I.5.1]{neeb}) controls the derivatives of functions on compact sets.\n Moreover, the elementary neighborhoods of the very strong topology form a subbase of the compact open $C^\\infty$-topology.\n We denote by $C^\\infty (M,F)_{\\text{co}}$ the vector space of smooth functions with the compact open $C^\\infty$-topology. \n \n Choose a compact neighborhood $C \\subseteq U$ of $A$ such that $h(C) \\subseteq W$ (this entails $f\\circ h(C)\\subseteq V$). \n We endow the subset $\\lfloor C,\\kappa(W)\\rfloor := \\{g\\in C^\\infty (M,E) \\mid g(C) \\subseteq \\kappa(W)\\}$ with the subspace topology induced by the compact open $C^\\infty$-topology.\n As $\\kappa (W) \\subseteq E$ is open, we note that $\\lfloor C,\\kappa(W)\\rfloor$ is open in $C^\\infty (M,E)_{\\text{co}}$.\n Now \\cite[Proposition 4.23 (a)]{glockomega} shows that \n \\begin{displaymath}\n  (f_{\\kappa}^\\psi)_* \\colon \\lfloor C,\\kappa(W) \\rfloor \\rightarrow C^\\infty (\\interior C,F)_{\\text{co}},\\quad  h \\mapsto (\\psi\\circ f\\circ \\kappa^{-1}) \\circ h\n \\end{displaymath}\n is continuous. \n Moreover, $\\mathcal{N}_{loc} := \\mathcal{N}^{r} (\\psi \\circ f\\circ h|_{\\interior C}; A,(U\\cap \\interior C, \\phi),(F,\\id_F),q, \\epsilon)$ is open in $C^\\infty (\\interior C,F)_{\\text{co}}$.\n Further, $f_{\\kappa}^\\psi \\circ \\kappa \\circ h|_{\\interior C} = \\psi \\circ f \\circ h|_{\\interior C}$. \n Observe that thus $\\kappa \\circ h \\in  ((f_{\\kappa}^\\psi)_*)^{-1} (\\mathcal{N}_{loc})$. \n As the elementary neighborhoods form a subbase of the compact open $C^\\infty$-topology,  Lemma \\ref{lem: atlaschoice} together with continuity of $(f_{\\kappa}^\\psi)_*$ yields\n \\begin{equation}\\label{eq: locpush}\n   \\lfloor C, \\kappa (W)\\rfloor \\cap \\bigcap_{k=1}^N \\mathcal{N}^{r} (\\kappa \\circ h; A_k,(U_k, \\phi_k),(E,\\id_E),p_k, \\epsilon_k)  \\subseteq  ((f_{\\kappa}^\\psi)_*)^{-1} (\\mathcal{N}_{loc}).\n \\end{equation}\n Recall from the proof of \\cite[Proposition 4.23 (a)]{glockomega} that the compact sets $A_k$ are contained by construction in $\\interior C$. \n Thus one easily deduces from \\eqref{eq: locpush} that \n \\begin{displaymath}\n  \\mathcal{N}^{0} (h; C,(U, \\phi),(W,\\kappa),p_1, 1) \\cap \\bigcap_{k=1}^N \\mathcal{N}^{r} (h; A_k,(U_k, \\phi_k),(W,\\kappa),p_k, \\epsilon_k) \\subseteq (f_*)^{-1} (\\mathcal{N})\n \\end{displaymath}\n Summing up, we see that $(f_*)^{-1} (\\mathcal{N})$ is a neighborhood of $h$. \n\n\n Further, this \\emph{finite} family of neighborhoods controls the behavior of mappings only on a pre chosen compact set $C$ (which depends of course on $h$). \n \n \\textbf{Step 3} \\emph{Preimages of basic neighborhoods are open}  \n Let $\\mathcal{M} = \\bigcap_{i \\in \\mathbb{N}} \\mathcal{N}_i$ be a basic neighborhood of $f\\circ h \\in C^\\infty (M,Y)$ with $\\{A_k\\}_{k \\in \\mathbb{N}}$ its the underlying compact family.\n We will prove that for arbitrary $g \\in (f_*)^{-1} (\\mathcal{M})$ the preimage is a neighborhood of $g$.\n Choose with Proposition \\ref{basicnbhsisbasis} a basic neighborhood of $f\\circ g$ which is contained in $\\mathcal{M}$. \n Replacing $\\mathcal{M}$ with this basic neighborhood, it suffices thus to consider the case $g=h$. \n Splitting each $A_k$ as in Step 1 we may assume without loss of generality that each $\\mathcal{N}_i$ is of the form considered in Step 2.\n Use \\cite[30.C.10]{cech} to construct for every $A_k$ a compact neighborhood $C_k$ such that $\\{C_k\\}_{k \\in \\mathbb{N}}$ is locally finite.\n Now we proceed for every elementary neighborhood $\\mathcal{N}_k$ as above (replace $C$ in Step 2 by $C_k$ and shrink $C_k$ if necessary!). \n Since the family $\\{C_k\\}_{k \\in \\mathbb{N}}$ is locally finite, we thus end up with a basic neighborhood $\\mathcal{M}_h$ around $h$ which mapped by $f_*$ to $\\mathcal{M}$.\n We conclude that $(f_*)^{-1} (M)$ is a neighborhood of $h$, whence of every of its elements.\n Hence preimages of basic neighborhoods under $f_*$ are open in $C_{\\textup{vS}}^\\infty (M,X)$, whence $f_*$ is continuous.\n\\end{proof}\n\n\n\nAs an application, we can now identify (as topological spaces) spaces of maps into a product with products of spaces of mappings to the factors.\n\n\\begin{theorem}\n\t\\label{product theorem}\n\tLet $ M $ be a finite-dimensional manifold, and let $ X_1 $ and $ X_2 $ be smooth manifolds modeled on locally convex vector spaces $ E_1 $ and $E_2$, respectively. Then \n\t\\begin{align*}\n\t\t\\iota : C_{\\textup{vS}}^\\infty (M, X_1 \\times X_2) \\to C_{\\textup{vS}}^\\infty (M, X_1) \\times C_{\\textup{vS}}^\\infty (M, X_2), \\quad f \\mapsto (\\pr_1 \\circ f, \\pr_2 \\circ f)\n\t\\end{align*}\n\tis a homeomorphism, where for \\( i \\in \\lbrace 1,2 \\rbrace \\) \\( \\pr_i \\colon X_1 \\times X_2 \\to X_i \\) is the canonical projection.\n\\end{theorem}\n\\begin{proof}\n\tClearly $ \\iota $ is a bijection, and it is continuous by Proposition \\ref{postcomposition}. We will prove that $ \\iota^{-1} $ is continuous, i.e. that $ \\iota $ is open.\n\tBy \\eqref{locally convex product} in Proposition \\ref{locally convex prop}, the set \n\t\t\\begin{displaymath}\n\t\t\t\\mathcal{P} := \\lbrace p \\circ \\pr_i : p \\mbox{ is a continuous seminorm on } E_i \\rbrace\n\t\t\\end{displaymath} \n\tis a generating family of seminorms on $ E_1 \\times E_2 $.\n\tConsider a basic neighborhood $ \\mathcal{U} = \\bigcap_{i\\in\\Lambda} \\mathcal{N}_i $ of $ f \\in C_{\\textup{vS}}^\\infty (M,X_1 \\times X_2) $, where each\n\t$\n\t\t\\mathcal{N}_i = \\mathcal{N}^{r_i} (f;A_i,(U_i,\\phi_i),(V_i,\\psi_i),p_i,\\epsilon_i).\n\t$\n\tBy Proposition \\ref{generatingfamilies} we may assume that each $ p_i \\in \\mathcal{P} $. Take an arbitrary $ i \\in \\Lambda $. If $ p_i = p\\circ \\pr_1 $ for some continuous seminorm $ p $ on $ E_1 $, let\n\t\\begin{align*}\n\t\t\\mathcal{M}_i = \\mathcal{N}^{r_i} (\\pr_1 \\circ f; A_i,(U_i,\\phi_i),(V_i,\\psi_i),p,\\epsilon_i) && \\mbox{and} && \\mathcal{M}_i' = C_{\\textup{vS}}^\\infty (M,X_2).\n\t\\end{align*}\n\tIf $ p_i = q\\circ \\pr_2 $ for a continuous seminorm $ q $ on $ E_2 $, reverse the roles of $ \\mathcal{M}_i $ and $ \\mathcal{M}_i' $.\n\t\n\tNow suppose without loss of generalization that $ p_i = p \\circ \\pr_1 $ for some continuous seminorm $ p $ on $ E_1 $. For $ g \\colon \\mathbb{R}^m \\supseteq \\phi_i (A_i) \\to \\psi_i (V_i) \\subseteq E_1 \\times E_2 $, one has\n\t\\begin{displaymath}\n\t\\dd^{(k)} g = \\dd^{(k)} (\\pr_1 \\circ g , \\pr_2 \\circ g) = ( \\dd^{(k)} \\pr_1 \\circ g, \\dd^{(k)} \\pr_2 \\circ g ),\n\t\\end{displaymath}\n\tso the condition\n\t\\( p_i \\left( \\dd^{(k)}g (a;\\alpha) \\right) < \\epsilon_i \\)\n\tis equivalent to\n\t\\( p \\left( \\dd^{(k)} (\\pr_1 \\circ g) (a;\\alpha) \\right) < \\epsilon_i \\).\n\tHence $ \\mathcal{M}_i \\times \\mathcal{M}_i' = \\iota (\\mathcal{N}_i) $. Since $ \\iota $ is bijective one has\n\t\\begin{displaymath}\n\t\t\\iota (\\mathcal{U}) = \\bigcap_{i\\in\\Lambda} \\iota (\\mathcal{N}_i) = \\bigcap_{i\\in\\Lambda} \\mathcal{M}_i \\times \\bigcap_{i\\in\\Lambda} \\mathcal{M}_i'.\n\t\\end{displaymath}\n\tSo $ \\iota $ is open, and a homeomorphism.\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{product corollary}\n\tIf $ Q $ is a compact smooth manifold, then following map is continuous\n\t\\begin{align*}\n\t\t\\chi \\colon C_{\\textup{vS}}^\\infty (M,X) \\to C_{\\textup{vS}}^\\infty (Q \\times M, Q \\times X), \\quad f \\mapsto \\id \\times f.\n\t\\end{align*}\n\\end{corollary}\n\\begin{proof}\n\tBy Theorem \\ref{product theorem} it suffices to show that the maps\n\t\\begin{align*}\n\t\t\\chi_1 \\colon C_{\\textup{vS}}^\\infty (M,X) &\\to C_{\\textup{vS}}^\\infty (Q\\times M, Q) && &\\mbox{and} && \\chi_2 \\colon C_{\\textup{vS}}^\\infty(M,X) &\\to C_{\\textup{vS}}^\\infty (Q \\times M, X) \\\\\n\t\tf &\\mapsto \\pr_1 \\circ (\\id \\times f) && & && f &\\mapsto \\pr_2 \\circ (\\id \\times f)\n\t\\end{align*}\n\tare continuous. For $ (q,m) \\in Q \\times M $, one has\n\t\\begin{align*}\n\t\t\\chi_1(f)(q,m) &= \\pr_1 \\circ (\\id \\times f) (q,m) = \\pr_1 (q,f(m)) = q = \\pr_1 (q,m), \\\\ \n\t\t\\chi_2(f)(q,m) &= \\pr_2 \\circ (\\id \\times f) (q,m) = \\pr_2 (q,f(m)) = f(m) \\\\ &= f \\circ \\pr_2 (q,m) = \\pr_2^* (f) (q,m).\n\t\\end{align*}\n\tThe map \\( \\chi_1 \\) is constant in $ f $, hence continuous, and the map \\( \\chi_2 = \\pr_2^* \\). Since $ Q $ is compact, $ \\pr_2 $ is proper, so Proposition \\ref{precomposition} implies that $ \\chi_2 = \\pr_2^* $ is also continuous.\n\\end{proof}\n\n\\section{The fine very strong topology}\n\nIn the end, we would like a structure on \\( C^\\infty (M,X) \\) as a locally convex manifold, where \\( M \\) is a finite-dimensional smooth manifolds and \\( X \\) is a manifold modeled on a locally convex vector space \\( E \\), but for this purpose the very strong topology is not fine enough. A first step in the direction of making \\( C^\\infty (M,X) \\) into a locally convex manifold would be having a similar structure on \\( C^\\infty (M,E) \\). One might hope that \\( C_{\\textup{vS}}^\\infty (M,E) \\) itself with the vector space structure induced by pointwise operations would be a locally convex vector space. But as Corollary \\ref{supportcorollary} points out, this is not the case when $ E $ is a (non-trivial) locally convex vector space and $ M $ is a non-compact manifold. However, we will see in the next section that the subspace of \\( C_{\\textup{vS}}^\\infty (M,E) \\) consisting of maps with compact support, denoted \\( C_{\\text{vS,c}}^\\infty (M,E) \\), is a locally convex vector space. Following \\cite{michor}, we refine the topology on \\( C_{\\textup{vS}}^\\infty (M,E) \\) to obtain a structure on \\( C^\\infty (M,E) \\) as a smooth manifold modeled on \\( C_{\\text{vS,c}}^\\infty (M,E) \\). The resulting topology on \\( C^\\infty (M,E) \\), or more generally \\( C^\\infty (M,X) \\), is called the \\emph{fine very strong} topology on $ C^\\infty (M,X) $. The space \\( C^\\infty (M,X) \\) equipped with the fine very strong topology is denoted $ C_{\\textup{fS}}^\\infty (M,X) $. \n\nFortunately, the results of the previous sections are easily extended to hold in the fine very strong topology. This is done in Proposition \\ref{fsmoothprop}.\n\nIt is a folklore fact (Proposition \\ref{prop: topologiescoincide}) that in the finite-dimensional case, the very strong topology is equivalent to the $ \\mathcal{D} $-topology as described in \\cite[36]{michor}.\\footnote{The $\\mathcal{D}$-topology was defined using jet bundles (also reviewed in Appendix \\ref{folklore}). Our treatment of the topology has the advantage that only elementary arguments are needed. Further, only our approach generalizes to arbitrary locally convex target manifolds.}  Consequently, the fine very strong topology is equivalent to the $ \\mathcal{FD} $-topology defined in \\cite[40]{michor}.\n\n\\begin{proposition}\n\t\\label{supportprop}\n\tLet $ M $ be a finite-dimensional smooth manifold and $ E $ be a locally convex vector space. Consider a sequence $ \\lbrace f_n \\rbrace_{n\\in \\mathbb{N}} \\subseteq C_{\\textup{vS}}^\\infty (M,E) $ which converges in the very strong topology towards \\( f \\in C^\\infty (M,E) \\). \n\tThen there exist a compact $ K \\subseteq M $ and an $ N \\in \\mathbb{N} $ such that for all $ n \\geq N $ we have $$ \\osupp{f}{f_n} :=  \\lbrace y \\in M : f_n (y) \\neq f(y) \\rbrace \\subseteq K. $$\n\\end{proposition}\n\\begin{proof}\n\tFor $ f \\in C_{\\textup{vS}}^\\infty (M,X) $, we will show that $ f $ cannot be a limit of $ \\lbrace f_n \\rbrace $ if for all compact $ K \\subseteq M $ and all $ N \\in \\mathbb{N} $ there exists $ n \\geq N $ such that $ \\osupp{f}{f_n} \\nsubseteq K $.\n\n\tLet \\( \\lbrace A_n \\rbrace_{n \\in \\mathbb{N}} \\) be a locally finite exhaustion of \\( M \\) by compact sets (exists by Lemma \\ref{dugundjifact} since \\( M \\) is \\( \\sigma \\)-compact), and for \\( n \\in \\mathbb{N} \\) set \\( K_n = \\bigcup_{i=1}^n A_i \\).\n\n\tConstruct a basic neighborhood of \\( f \\) recursively, using the following procedure. Let \\( n_0 = 1 \\), \\( m_0 = 1 \\). For \\( i \\in \\mathbb{N} \\), choose \\( n_i > n_{i-1} \\) such that \\( \\osupp{f}{f_{n_i}} \\nsubseteq K_{m_{i-1}} \\). By construction there exists \\( m_i > m_{i-1} \\) such that \\( \\osupp{f}{f_{n_i}} \\cap \\left( M \\setminus K_{m_{i-1}} \\right) \\cap A_{m_i} \\neq \\emptyset \\). Take any \\( x \\) in this nonempty set. Since \\( f (x) \\neq f_{n_i} (x) \\), there exists a continuous seminorm \\( p_i \\) on \\( E \\) such that \\( 2 \\epsilon_i := p_i (f_{n_i} (x) - f(x)) > 0 \\), and then\n\t\\[\n\t\tf_{n_i} \\notin \\mathcal{N}_i := \\mathcal{N}^0 \\left( f; A_{m_i}, p_i, \\epsilon_i \\right).\n\t\\]\n\tNow \\( \\mathcal{U} := \\bigcap_{i \\in \\mathbb{N}} \\mathcal{N}_i \\) is a basic neighborhood of \\( f \\) such that for all \\( N \\in \\mathbb{N} \\) there exists \\( n \\geq N \\) such that \\( f_n \\notin \\mathcal{U} \\). So the sequence \\( \\lbrace f_n \\rbrace_{n \\in \\mathbb{N}} \\) does not converge to \\( f \\).\n\\end{proof}\n\\begin{remark}\n\tOne can easily prove the proposition above for \\( E \\) a locally convex manifold rather than a locally convex vector space, by ``hacking'' the compact sets \\( A_i \\) in the proof into smaller compact sets that are contained in charts.\n\\end{remark}\n\\begin{corollary}\n\t\\label{supportcorollary}\n\tLet $ M $ be a finite-dimensional non-compact manifold and $ E \\neq \\lbrace 0 \\rbrace $ a locally convex vector space. Then $ C_{\\textup{vS}}^\\infty (M,E) $ with the vector space structure induced by pointwise operations is not a topological vector space.\n\\end{corollary}\n\\begin{proof}\n\tLet $ f \\in C_{\\textup{vS}}^\\infty (M,E) $ be a non-zero constant map. Then Proposition \\ref{supportprop} shows that $ \\lim_{\\lambda \\to 0} (\\lambda f) \\neq 0 = (\\lim_{\\lambda \\to 0} \\lambda) f $, hence scalar multiplication is not continuous.\n\\end{proof}\n\\begin{remark}\n\t\\label{supportcorollaryremark}\n\tAlthough $ C_{\\textup{vS}}^\\infty (M,E) $ is not a topological vector space, it is a topological group under pointwise addition by Lemma \\ref{continuous addition}. And $ C_{\\textup{vS}}^\\infty (M,\\mathbb{R}) $ is a topological ring under the pointwise operations induced by addition and multiplication in $ \\mathbb{R} $.\n\\end{remark}\n\\begin{definition}[The fine very strong topology]\n\tDefine an equivalence relation $ \\sim $ on $ C^\\infty (M,X) $ by declaring that $ f \\sim g $ whenever $$ \\csupp{f}{g} := \\overline{\\lbrace y \\in M : f(y) \\neq g(y) \\rbrace} $$ is compact. Now refine the very strong topology on $ C^\\infty (M,X) $ by demanding that the equivalence classes are open in $ C^\\infty (M,X) $. In other words, equip $ C^\\infty (M,X) $ with the topology generated by the very strong topology and the equivalence classes. This is the \\emph{fine very strong topology} on $ C^\\infty (M,X) $. We write $ C_{\\textup{fS}}^\\infty (M,X) $ for $ C^\\infty (M,X) $ equipped with the fine very strong topology.\n\\end{definition}\n\\begin{remark}\n\t\\label{fvs remark}\n\tHere is another way to look at the fine very strong topology. Start with $ C_{\\textup{vS}}^\\infty (M,X) $ and equip the equivalence classes $ [f] $ with the subspace topology. Then $$ C_{\\textup{fS}}^\\infty (M,X) = \\bigsqcup_{[f] \\in C^\\infty (M,X)\/\\sim} [f] $$ as topological spaces.\n\tTaking the family of all sets of the form $ \\mathcal{U} \\cap [f] $, where $ \\mathcal{U} $ runs through the basic neighborhoods in $ C^\\infty (M,X) $ and $ [f] $ runs through the equivalence classes, yields a basis for the fine very strong topology on $ C^\\infty (M,X) $.\n\\end{remark}\n\\begin{remark}\n\tIf $ f \\in C^\\infty (M,X) $ is a proper map and $ f \\sim \\hat{f} $, then $ \\hat{f} $ is also proper. Indeed, if $ K \\subseteq X $ is compact, then $ \\hat{f}^{-1} (K) \\subseteq f^{-1}(K) \\cup \\csupp{f}{\\hat{f}} $. Since closed subspaces of compact spaces are compact, $ \\hat{f}^{-1} (K) $ is compact.\n\\end{remark}\nWe would obviously like the results of the previous sections to remain true in the fine very strong topology. Fortunately, it is easy to extend the results to this case using the following lemma.\n\\begin{lemma}\n\t\\label{corollarylemma}\n\tLet $ T $ be a topological space, and $ \\zeta \\colon T \\to C^\\infty (M,X) $ a function. If $ \\zeta $ is continuous as a map to $ C_{\\textup{vS}}^\\infty (M,X) $ and $ \\zeta^{-1} ([f]) \\subseteq T $ is open for all equivalence classes $ [f] \\subseteq C^\\infty (M,X) $, then $ \\zeta $ is continuous as a map to $ C_{\\textup{fS}}^\\infty (M,X) $.\n\\end{lemma}\n\\begin{proof}\n\tThe map $ \\zeta $ is continuous if preimages of basis elements are open. Basis elements for $ C_{\\textup{fS}}^\\infty (M,X) $ are of the form $ \\mathcal{U} \\cap [f] $ for some basic neighborhood $ \\mathcal{U} $ and some equivalence class $ [f] $, and $ \\zeta^{-1}(\\mathcal{U}\\cap [f]) = \\zeta^{-1}(\\mathcal{U}) \\cap \\zeta^{-1} ([f]) $.\n\\end{proof}\n\\begin{proposition}\n\t\\label{fsmoothprop}\n\tTheorem \\ref{compiscont}, Proposition \\ref{precomposition}, Proposition \\ref{postcomposition}, Theorem \\ref{product theorem}, and Corollary \\ref{product corollary} still hold if we in every case replace the very strong topology with the fine very strong topology.\n\t\n\tIn the cases that we consider $ \\Prop_{\\textup{vS}} (M,N) $, replace this with $ \\Prop_{\\mbox{fS}} (M,N) $, by which is meant the subset $ \\Prop (M,N) \\subseteq C_{\\textup{fS}}^\\infty (M,N) $ equipped with the subspace topology.\n\\end{proposition}\n\\begin{proof}\n\tThe proof is case by case. In all cases except for the generalization of Theorem \\ref{product theorem} and its corollary, it suffices by \\ref{corollarylemma} to check that preimages of equivalence classes are open. Unless otherwise stated, letters (such as $ f $ or $ N $) are always assumed to have the same role here as in the statement of the corresponding result.\n\n\t\\textit{Theorem \\ref{compiscont} (the full composition map is continuous).} Suppose that $ f \\sim \\hat{f} $ and $ h \\sim \\hat{h} $. We have $ \\csupp{h \\circ f}{ \\hat{h} \\circ \\hat{f} } \\subseteq \\csupp{f}{\\hat{f}} \\cup f^{-1}\\left( \\csupp{h}{\\hat{h}} \\right) $. The right hand side  is compact since $ f $ is proper, so $ \\csupp{h \\circ f}{ \\hat{h} \\circ \\hat{f} } $ is a closed subset of a compact space, hence compact. By definition this means that $ h \\circ f \\sim \\hat{h} \\circ \\hat{f} $. \n\t\t\n\tConsider an equivalence class $ [g] \\subseteq C_{\\textup{fS}}^\\infty (M,X) $. By what we just observed, if $ h \\circ f \\sim g $ and $ \\hat{f} \\sim f $ and $ \\hat{h} \\sim h $, then $ \\hat{h} \\circ \\hat{f} \\sim g $. Hence $$ \\Gamma^{-1} ([g]) = \\bigcup_{h \\circ f \\sim g} [f] \\times [h], $$ which is open.\n\t\n\t\\textit{Proposition \\ref{precomposition} (precomposition is continuous).} If $ h \\sim \\hat{h} $, then $ h \\circ f \\sim \\hat{h} \\circ f $ by the same argument as before. So for any equivalence class $ [g] \\subseteq C^\\infty (M,X) $, we have $$ (f^*)^{-1} ([g]) = \\bigcup_{h \\circ f \\sim g} [h]. $$\n\t\n\t\\textit{Proposition \\ref{postcomposition} (postcomposition is continuous).} If $ h, \\hat{h} \\in C^\\infty (M,X) $, then it is easy to see that $ \\csupp{f \\circ h}{f \\circ \\hat{h}} \\subseteq \\csupp{h}{\\hat{h}} $. So if $ h \\sim \\hat{h} $, then $ f \\circ h \\sim f \\circ \\hat{h} $, since closed subsets of compact spaces are compact. It follows that for any equivalence class $ [g] \\subseteq C^\\infty (M,X) $, we have $ (f_*)^{-1}([g]) = \\bigcup_{f \\circ h \\sim g} [h]. $ \n\t\n\t\\textit{Theorem \\ref{product theorem} (the product theorem).} For the same reasons as in the proof of the very strong version of the theorem, $ \\iota $ is clearly a bijective continuous map. So by Lemma \\ref{corollarylemma} it suffices to show that images of equivalence classes are open. \n\t\n\tObserve that for $ f, \\hat{f} \\in C^\\infty (M, X_1 \\times X_2) $, we have $ \\csupp{f}{\\hat{f}} = \\csupp{\\pr_1 \\circ f}{\\pr_1 \\circ \\hat{f}} \\cup \\csupp{\\pr_2 \\circ f}{\\pr_2 \\circ \\hat{f}} $. Hence $ f \\sim \\hat{f} $ if and only if $ \\pr_1 \\circ f \\sim \\pr_1 \\circ \\hat{f} $ and $ \\pr_2 \\circ f \\sim \\pr_2 \\circ \\hat{f} $. Another way of stating this fact is $ \\iota ([f]) = [\\pr_1 \\circ f] \\times [\\pr_2 \\circ f] $ for all $ f \\in C^\\infty (M,X_1 \\times X_2) $.\n\t\n\t\\textit{Corollary \\ref{product corollary}.} Same proof as in the very strong case.\n\\end{proof}\n\n\n\\section{The manifold structure on smooth vector valued functions}\\label{sect: smmfd}\nThroughout this section, $ M $ is a finite-dimensional manifold, $ E $ is a locally convex vector space, and \\( X \\) is a locally convex manifold.\n\nRecall from Corollary \\ref{supportcorollary} that \\( C_{\\textup{vS}}^\\infty (M,E) \\) with pointwise operations is not a locally convex vector space, in fact it is not even a topological vector space. Neither is \\( C_{\\textup{fS}}^\\infty (M,E) \\). However, we will in this section make \\( C_{\\textup{fS}}^\\infty (M,E) \\) into a locally convex manifold. This is a first step towards making \\( C_{\\textup{fS}}^\\infty (M,X) \\) into a locally convex manifold (but we will not do this). The modeling space for \\( C_{\\textup{fS}}^\\infty (M,E) \\) as a locally convex manifold is \\( C_{\\text{vS,c}}^\\infty (M,E) \\), defined below.\n\n\\begin{definition}\n\tWe define $ C_{\\text{vS,c}}^\\infty (M,E) $ to be the subspace of $ C_{\\textup{vS}}^\\infty (M,E) $ consisting of the functions with compact support, i.e. \n\t$$ C_{\\text{vS,c}}^\\infty (M,E) = \\lbrace f \\in C_{\\textup{vS}}^\\infty (M,E) : \\mbox{$\\csupp{f}{0}$ is compact} \\rbrace $$ equipped with the subspace topology from $ C_{\\textup{vS}}^\\infty (M,E) $.\n\tNote that $ C_{\\text{vS,c}}^\\infty (M,E) = [0] $ in $ C_{\\textup{fS}}^\\infty (M,E) $.\n\\end{definition}\n\nAs a first step towards proving that \\( C_{\\text{vS,c}}^\\infty (M,E) \\) with pointwise operations is a locally convex vector space, we show that \\( C^\\infty (M,E) \\) with pointwise addition is a topological group in the very strong and fine very strong topologies.\n\n\\begin{lemma}\n\t\\label{continuous addition}\n\tAddition\n\t\\begin{align*}\n\t\t\\Sigma \\colon C^\\infty (M,E) \\times C^\\infty (M,E) &\\to C^\\infty (M,E), \\quad (f,g) &\\mapsto f+h = \\left[ m \\mapsto f(m) + h(m) \\right]\n\t\\end{align*}\n\tis continuous when $ C^\\infty (M,E) $ is equipped with the very strong topology or fine very strong topology.\n\\end{lemma}\n\\begin{proof}\n\tWe prove the assertion only for the very strong topology as the proof carries over verbatim to the fine very strong topology.\n\tBy Theorem \\ref{product theorem} there is a canonical homeomorphism $ \\iota \\colon C_{\\textup{vS}}^\\infty (M,E) \\times C_{\\textup{vS}}^\\infty (M,E) \\cong C_{\\textup{vS}}^\\infty (M,E \\times E ) $. Since addition $ S \\colon E \\times E \\to E $ in $ E $ is smooth, induced postcomposition $ S_* \\colon C_{\\textup{vS}}^\\infty (M,E\\times E) \\to C_{\\textup{vS}}^\\infty (M,E) $ is continuous.\n\tHence $ \\Sigma = S_* \\circ \\iota $ is continuous.\n\\end{proof}\n\nOnce we have established the following proposition, it will be easy to make \\( C_{\\textup{fS}}^\\infty (M,E) \\) into a locally convex manifold modeled on \\( C_{\\text{vS,c}}^\\infty (M,E) \\). The hard work lies here.\n\\begin{proposition}\n\t\\label{locally convex vs prop}\n\tThe topological space $ C_{\\text{vS,c}}^\\infty (M,E) $ with vector space structure induced by pointwise operations in $ E $ is a locally convex vector space.\n\\end{proposition}\n\\begin{proof}\n\tIn Lemma \\ref{continuous addition} we showed that addition is continuous, and the topological space $ C_{\\text{vS,c}}^\\infty (M,E) $ is Hausdorff since the compact open \\( C^\\infty \\)-topology on \\( C^\\infty(M,E) \\) is Hausdorff and the very strong topology is finer than the compact open \\( C^\\infty \\)-topology. It is therefore only necessary to check that scalar multiplication is continuous in order to conclude that $ C_{\\text{vS,c}}^\\infty (M,E) $ is a topological vector space. Finally, we must verify that this topological vector space is locally convex.\n\t\n\t\\textit{Scalar multiplication $\\mu \\colon \\mathbb{R} \\times C_{\\text{vS,c}}^\\infty (M,E) \\to C_{\\text{vS,c}}^\\infty (M,E),\\  \n\t\t(\\lambda, f) \\mapsto \\lambda f $ is continuous.} \n\tLet $ (\\lambda,f) \\in \\mathbb{R} \\times C_{\\text{vS,c}}^\\infty (M,E) $, and consider a basic neighborhood $ \\mathcal{V} = \\bigcap_{i \\in \\Lambda} \\mathcal{N}_i $ of $ \\lambda f $, where each $\\mathcal{N}_i = \\mathcal{N}^{r_i} (\\lambda f; A_i, (U_i,\\phi_i),p_i,\\epsilon_i) $ is an elementary neighborhood of $ \\lambda f $. We will show that there exists open sets $ I \\subseteq \\RR $ and $ \\mathcal{U} \\subseteq C_{\\textup{vS}}^\\infty (M,E) $ such that $ \\mu (I \\times \\mathcal{U}) \\subseteq \\mathcal{V} $. \n\t\n\tSince $ \\csupp{f}{0} $ is compact, only finitely many $ A_i $ intersect $ \\csupp{f}{0} $, say only for $ i = i_1, \\dots , i_n $. Define $ \\epsilon := \\min (\\epsilon_{i_1},\\dots,\\epsilon_{i_n}) $. \n\t\n\tSet $m_1 := \\max\\left\\{\\sup_{1 \\leq j \\leq n} \\Vert f \\circ \\phi_{i_j}^{-1} \\Vert (r_{i_j}, \\phi_{i_j}(A_{i_j}),p_{i_j}), 1\\right\\}$ and \n\t\\[ I := B_{\\frac{\\epsilon}{2 m_1 }}^1 (\\lambda) = \\left] \\lambda - \\frac{\\epsilon}{2 m_1}, \\lambda + \\frac{\\epsilon}{2 m_1} \\right[ . \\] \n\t\n\tDefine $ m_2 := \\sup \\lbrace | t | : t \\in I \\rbrace $, and set $ \\mathcal{U} := \\bigcap_{i \\in \\Lambda} \\mathcal{N}^{r_i} \\left( f;A_i,(U_i,\\phi_i),p_i,\\frac{\\epsilon_i}{2 m_2} \\right). $\n\tSuppose $ (\\lambda',f') \\in I \\times \\mathcal{U} $. \n\tFor all $ i \\in \\Lambda $, $ x \\in A_i $, $ 1 \\leq k \\leq r_i $, and $ \\alpha \\in \\lbrace e_1, \\dots , e_{\\dim M} \\rbrace^k $, we have\n\t\\begin{align*}\n\t\t& p_i \\left( \\dd^{(k)} (\\lambda' f' \\circ \\phi_i^{-1} - \\lambda f \\circ \\phi_i^{-1})(\\phi_i(x);\\alpha) \\right) \\\\\n\t\t\\leq& | \\lambda' | p_i \\left( \\dd^{(k)} (f' \\circ \\phi_i^{-1} - f \\circ \\phi_i^{-1})(\\phi_i(x);\\alpha) \\right) + | \\lambda' - \\lambda | p_i \\left( \\dd^{(k)} (f \\circ \\phi_i^{-1})(\\phi_i(x);\\alpha) \\right) \\\\\n\t\t<& \\frac{\\epsilon_i}{2} + \\frac{\\epsilon}{2 m_1} p_i \\left( \\dd^{(k)} (f \\circ \\phi_i^{-1})(\\phi_i(x);\\alpha) \\right) =: C.\n\t\\end{align*}\n\tIf $ i \\notin \\lbrace i_1,\\dots,i_n \\rbrace $, then $ p_i \\left( \\dd^{(k)} (f \\circ \\phi_i^{-1})(\\phi_i(x);\\alpha) \\right) = 0 $, in which case $ C \\leq \\epsilon_i $. And if $ i \\in \\lbrace i_1, \\dots,i_n \\rbrace $, then $ \\epsilon \\leq \\epsilon_i $ and $ p_i \\left( \\dd^{(k)} (f \\circ \\phi_i^{-1})(\\phi_i(x);\\alpha) \\right) \\leq m_1 $, in which case we still have $ C \\leq \\epsilon_i $. Hence $ \\lambda' f' \\in \\mathcal{V} $, and $ \\mu (I \\times \\mathcal{U}) \\subseteq \\mathcal{V} $. Consequently, $\\mu$ is continuous.\n\t\n\t\\textit{The space is locally convex.} We have now established that $ C_{\\mbox{vS,c}}^\\infty (M,E) $ is a topological vector space. It remains to see that this topological vector space is locally convex. For $ r \\in \\mathbb{N}_0 $, $ (U,\\phi) $ a chart on $ M $, $ A \\subseteq U $ compact, and $ p $ a continuous seminorm on $ E $, define\n\t\\begin{align*}\n\t\t\\Vert\\cdot \\Vert(r,A,(U,\\phi),p) \\colon C_{\\mbox{vS,c}}^\\infty (M,E) \\to [0,\\infty), \\quad f \\mapsto \\Vert f \\circ \\phi^{-1} \\Vert (r,\\phi_i (A_i),p) \n\t\\end{align*}\n\tThis is a seminorm on $ C_{\\mbox{vS,c}}^\\infty (M,E) $. Consider a family $ \\lbrace \\Vert\\cdot \\Vert (r_i,A_i,(U_i,\\phi_i),p_i) \\rbrace_{i\\in \\Lambda} $ of such seminorms, where $ \\lbrace A_i \\rbrace_{i \\in \\Lambda} $ is locally finite. For some family $ \\lbrace \\epsilon_i \\rbrace_{i \\in \\Lambda} $ define $ q \\colon C_{\\mbox{vS,c}}^\\infty (M,E) \\to [0,\\infty) $ by \\[ q(f) = \\sup_{i\\in\\Lambda} \\epsilon_i \\Vert f \\Vert (r_i,A_i,(U_i,\\phi_i),p_i). \\] \n\tEvery $ f \\in C_{\\mbox{vS,c}}^\\infty (M,E) $ has compact support, so $ \\operatorname{supp}(f,0) $ intersects only finitely many of the $ A_i $, from which it follows that $ \\Vert f \\Vert (r_i,A_i,(U_i,\\phi_i),p_i) \\neq 0 $ for only finitely many $ i \\in \\Lambda $. Hence $ q(f) < \\infty $, so $ q $ is well-defined. Clearly $ q $ is a seminorm. Also $ q $ is continuous as for all $ \\lambda > 0 $, the preimage $ q^{-1} [0,\\lambda) $ is a basic neighborhood of $ 0 $, e.g.\\ \n\t\\[ q^{-1}[0,1) = \\left( \\bigcap_{i \\in \\Lambda } \\mathcal{N}^{r_i} (0;A_i,(U_i,\\phi_i),p_i,\\epsilon_i) \\right) \\cap C_{\\text{vS,c}}^\\infty (M,E). \\] So every basic neighborhood of 0 arises as a preimage of a continuous seminorm. Consequently, $ C_{\\text{vS,c}}^\\infty (M,E) $ is locally convex (see \\cite[\\S 18]{koethe}).\n\\end{proof}\n\nWe will now provide an alternative description of the topology on $ C_{\\text{vS,c}}^\\infty (M,E) $ as an inductive limit of certain locally convex spaces.\nThis characterization also implies that $ C_{\\text{vS,c}}^\\infty (M,E) $ is a locally convex space (thus providing an elegant proof of Proposition \\ref{locally convex vs prop}).\nNote however: Though the proof of Proposition \\ref{locally convex vs prop} is a bit cumbersome, it is also completely elementary and does not use auxiliary results on inductive limits.\n\n\n\\begin{definition}\nLet $K \\subseteq M$ be a compact subset and $E$ be a locally convex space.\nThen we define \n  \\begin{displaymath}\n  C^\\infty_K (M,E) := \\{f\\in C^\\infty (M,E) \\mid \\operatorname{supp} (f,0) \\subseteq K\\}                                                          \n  \\end{displaymath}\n and topologize this space with the compact open $C^\\infty$-topology, i.e.\\ the topology generated by the subbase $\\mathcal{N} \\cap C^\\infty_K (M,E)$ where $\\mathcal{N}$ runs through all elementary neighborhoods of $C^\\infty_{\\text{vS}} (M,E)$.\n Recall from \\cite[Proposition 4.19]{glockomega} that $C^\\infty_K (M,E)$ is a locally convex vector space.\n\\end{definition}\n\n\\begin{remark}\n Since all functions $C^\\infty_K (M,E)$ have compact support contained in $K$ one can prove that the compact open $C^\\infty$-topology coincides with the subspace topologies induced by $C_{\\textup{vS}}^\\infty (M,E)$ and $C_{\\textup{fS}}^\\infty (M,E)$.\n However, we will not need this. \n\\end{remark}\n\nDenote by $\\mathcal{K} (M)$ the set of compact subsets of $M$. \nObserve that as sets $C^\\infty_{\\text{vS,c}} (M,E) = \\bigcup_{K \\in \\mathcal{K} (M)} C^\\infty_K (M,E)$.\nWe claim that the topology on the compactly supported functions is determined by the smaller locally convex spaces:\nTo see this, recall that with respect to inclusion, $\\mathcal{K} (M)$ is a directed set. \nFurther, for $K,L \\in \\mathcal{K} (M)$ with $K \\subseteq L$ the canonical inclusion $\\iota_K^L\\colon C^\\infty_K (M,E) \\rightarrow C^\\infty_L (M,E)$ is continuous linear by definition of the topology.\nHence we can form the locally convex inductive limit $\\displaystyle \\lim_{\\rightarrow} C^\\infty_K (M,E)$ (cf.\\ \\cite[\\S 19 3.]{koethe}) of the family $\\{C^\\infty_K (M,E)\\}_{\\mathcal{K} (M)}$ (with respect to the canonical inclusions).\n\n\\begin{lemma}\\label{lem: indlim}\n Let $E$ be a locally convex space, then as locally convex spaces\n  \\begin{displaymath}\n   C^\\infty_{\\text{vS,c}} (M,E) = \\lim_{\\rightarrow} C^\\infty_K (M,E).\n  \\end{displaymath}\n\\end{lemma}\n\n\\begin{proof}\n Since as sets $C^\\infty_{\\text{vS,c}} (M,E) = \\displaystyle\\lim_{\\rightarrow} C^\\infty_K (M,E)$, we only have to prove that the topologies coincide.\n However, since $M$ is $\\sigma$-compact, \\cite[Proposition 8.13 (d)]{glockomega} implies that a basis for the inductive limit topology on $C^\\infty_{\\text{vS,c}} (M,E) = \\displaystyle\\lim_{\\rightarrow} C^\\infty_K (M,E)$ is given by the basic neighborhoods of the very strong topology.\n\\end{proof}\n\n\n\\begin{proposition}\n\tFor each class $ [f] $ in $ C_{\\textup{fS}}^\\infty (M,E) $ define $ \\phi_{[f]} \\colon [f] \\to C_{\\text{vS,c}}^\\infty (M,E) $ by $ \\phi_{[f]}(g) = g-f $. \n\tThen $ \\mathcal{A} = \\lbrace (\\phi_{[f]},[f]) \\rbrace_{f \\in C^\\infty (M,E)} $ is a smooth atlas for $ C_{\\textup{fS}}^\\infty (M,E) $. Hence $ C_{\\textup{fS}}^\\infty(M,E) $ is a smooth manifold modeled on $ C_{\\text{vS,c}}^\\infty (M,E) $.\n\\end{proposition}\n\\begin{proof}\n\tWe will first show that every chart $ \\phi_{[f]} $ is a homeomorphism. First of all, note that $ \\phi_{[f]} $ is well-defined since $ g - f $ is smooth and compactly supported for $ g \\in [f] $. It is bijective with inverse $ \\phi_{[f]}^{-1} (h) = h + f $. Both $ \\phi_{[f]} $ and $ \\phi_{[f]}^{-1} $ are continuous by Lemma \\ref{continuous addition}.\n\t\n\tThe chart domains of $ \\mathcal{A} $ cover $ C_{\\textup{fS}}^\\infty $, whence we have to check that chart transformations are smooth. Let $ (\\phi_{[f]},[f]) $ and $(\\phi_{[g]},[g]) $ be charts with $ [f] \\cap [g] \\neq \\emptyset $.\n\tThen $ [f] = [g] $ and $ \\phi_{[g]} \\circ \\phi_{[f]}^{-1}(h) = h + f - g $, whence it is smooth in $ h $ as addition in $C_{\\text{vS,c}}^\\infty (M,E)$ is so.\n\\end{proof}\nStructurally, the manifold $ C_{\\textup{fS}}^\\infty (M,E) $ is just a collection of (affine) copies of $ C_{\\text{vS,c}}^\\infty (M,E)$. For this reason, it is also called in \\cite{michor} a \\emph{local topological affine space}.\n\nTo construct a manifold structure on $ C_{\\textup{fS}}^\\infty (M,X) $ for an arbitrary locally convex manifold $X$ one needs a so called \\emph{local addition} on $X$ (cf.\\ \\cite{michor,KM97}).\nA local addition replaces the vector space addition. \nIt allows to ``smoothly choose'' charts on $X$ (see \\cite{stacey} for more information). \nThe details are similar to \\cite[Section 10]{michor} but require certain analytical tools (e.g.\\ a suitable version of the $\\Omega$-Lemma, \\cite[Appendix F]{glockomega})\\footnote{To apply the $\\Omega$-Lemma as stated in \\cite{glockomega}, one needs a topology on spaces of compactly supported sections in vector bundles. In ibid.\\ the compact open $C^\\infty$-topology is used, however by arguments similar to Lemma \\ref{lem: indlim} one proves that this topology coincides with the very strong topology.}.\n\n\\section{Application to bisection groups}\nIn this section we use our results on the very strong and the fine very strong topology to turn certain groups into topological groups.\nThe groups envisaged here are the bisection groups associated to certain Lie groupoids.\nA reference on (finite-dimensional) Lie groupoids is \\cite{mackenzie}, see \\cite{Schmeding2015,SchmedingWockel15} for infinite-dimensional Lie groupoids.\n\\begin{definition}[Lie groupoid]\n\tLet $ M $ be a finite-dimensional smooth manifold and $ G $ a smooth manifold modeled on a locally convex vector space. Then a groupoid $ \\mathcal{G} = (G \\rightrightarrows M) $ with source projection $ \\alpha \\colon G \\to M $ and target projection $ \\beta \\colon G \\to M $ is a \\emph{(locally convex) Lie groupoid} if $ \\alpha $ and $ \\beta $ are smooth submersions (i.e. locally projections), partial multiplication $ m \\colon G \\times_{\\alpha,\\beta} G \\to G $ is smooth, object inclusion $ 1 \\colon M \\to G $ is smooth, and inversion $ i \\colon G \\to G $ is smooth.\n\\end{definition}\n\\begin{definition}[Bisection group]\n\tThe \\emph{group of bisections} $ \\Bis (\\mathcal{G}) $ of a Lie groupoid $ \\mathcal{G} = (G \\rightrightarrows M) $ is the set of sections $ \\sigma \\colon M \\to G $ of $ \\alpha $ such that $ \\beta \\circ \\sigma $ is a diffeomorphism of $ M $. The group operation $ \\star $ is given by $$ (\\sigma \\star \\tau)(x) := \\sigma ((\\beta \\circ \\tau)(x)) \\tau (x). $$ With this operation, the object inclusion $ 1 \\colon M \\to G $ becomes the neutral element and the inverse of a section $ \\sigma $ is $ \\sigma^{-1}(x) = i(\\sigma((\\beta \\circ \\sigma)^{-1}(x))). $\n\\end{definition}\n\\begin{example}\n\t\\begin{enumerate}\n\t\t\\item For a finite-dimensional manifold $ M $, the \\emph{unit Lie groupoid} is the groupoid $ (M \\rightrightarrows M) $ with both source and target projection $ \\id_M $. The bisection group of this groupoid is trivial.\n\t\t\\item Let $ M $ be a finite-dimensional smooth manifold. Then $ \\mathcal{P}(M) := (M \\times M \\rightrightarrows M) $ with source projection $ \\alpha = \\pr_2 $ and target projection $ \\beta = \\pr_1 $ is a Lie groupoid. Multiplication in the groupoid is given by $ (x,y)(y,z) = (x,z) $. Postcomposition $ \\beta_* $ induces an isomorphism $ \\Bis (\\mathcal{P}(M)) \\cong \\Diff(M) $ of groups, where \\( \\Diff (M) \\) is the group of smooth diffeomorphisms of \\( M \\).\n\t\t\\item Suppose $ G $ is a locally convex Lie group, and $ * $ is the one-point space. Then $ (G \\rightrightarrows *) $ is a Lie groupoid with bisection group $ G $.\n\t\\end{enumerate}\n\\end{example}\n\nTo prepare the construction of a topological group structure on bisection groups, recall the following facts on diffeomorphism groups of finite-dimensional manifolds.\n\\begin{remark}\\label{rem: topgroup}\n Let $M$ be a finite-dimensional manifold and $\\Diff (M)$ be the group of smooth diffeomorphisms of $M$.\n As $\\Diff (M) \\subseteq C^\\infty (M,M)$, we can endow $\\Diff (M)$ either with the subspace topology induced by the very strong topology (write $\\Diff_{\\text{vS}} (M)$) or with respect to the fine very strong topology (we write $\\Diff_{\\text{fS}} (M)$).\n Now as a consequence of \\cite[Corollary 7.7]{michor} and Proposition \\ref{prop: topologiescoincide}, both $\\Diff_{\\text{vS}} (M)$ and $\\Diff_{\\text{fS}} (M)$ are topological groups.\n Note that $\\Diff_{\\text{fS}} (M)$ is even a locally convex Lie group by \\cite[Theorem 11.11]{michor}.\n In particular, we remark that the (subspace topology induced by the) fine very strong topology is the Lie group topology of $\\Diff (M)$.\n\\end{remark}\n\n\\begin{proposition}\\label{prop: topgp}\n\tIf $ \\Bis (\\mathcal{G}) $ is equipped with the subspace topology with respect to $ C_{\\textup{vS}}^\\infty (M,G) $ or $ C_{\\textup{fS}}^\\infty (M,G) $, then $ \\Bis (\\mathcal{G}) $ becomes a topological group.\n\\end{proposition}\n\\begin{proof}\n\tWe will prove that $ \\Bis (\\mathcal{G}) $ becomes a topological group when equipped with the subspace topology with respect to $ C_{\\textup{vS}}^\\infty (M,G) $. The case where we consider the subspace topology with respect to $ C_{\\textup{fS}}^\\infty (M,G) $ can be proven identically, since we only use results that hold in both topologies.\n\tLet $ \\Omega \\colon \\Bis (\\mathcal{G}) \\times \\Bis (\\mathcal{G}) \\to \\Bis (\\mathcal{G}) $ be the multiplication map defined by $ \\Omega (\\sigma,\\tau) = \\sigma \\star \\tau $, and let $ \\iota $ be the inclusion $ \\Bis (\\mathcal{G})  \\to C_{\\textup{vS}}^\\infty (M,G) $. Observe that we can write $$ \\Omega(\\sigma,\\tau)(x) = \\sigma((\\beta \\circ \\tau)(x))\\tau(x) = m(\\Gamma(\\beta \\circ \\tau, \\sigma)(x),\\tau(x)). $$ So $ \\iota \\circ \\Omega $ can be written as a composition of continuous maps; the diagram\n\t\\begin{align*}\n\t\t\\xymatrix{\n\t\t\t\\Bis (\\mathcal{G}) \\times \\Bis (\\mathcal{G}) \\ar[d]^{(\\beta_* \\circ \\pr_2, \\iota \\times \\iota )} \\ar@{.>}[rr]^{\\iota \\circ \\Omega} &&C_{\\textup{vS}}^\\infty (M,G)\t\\\\\n\t\t\t\\Prop_{\\textup{vS}} (M,M) \\times C_{\\textup{vS}}^\\infty(M,G) \\times C_{\\textup{vS}}^\\infty (M,G) \\ar[d]^{\\Gamma \\times \\id}  &&  \\\\\n\t\t\tC_{\\textup{vS}}^\\infty (M,G) \\times C_{\\textup{vS}}^\\infty (M,G) \\ar[rr]^-{\\cong} & &  C_{\\textup{vS}}^\\infty (M,G \\times G) \\ar[uu]^{m_*}\n\t\t\t}\n\t\\end{align*}\n\tcommutes. Here we have used that $ \\beta_* (\\Bis (\\mathcal{G})) \\subseteq \\Diff (M) \\subseteq \\Prop (M,M) $ by definition of bisections. All of the maps represented by normal arrows in the diagram are continuous by results in the previous sections. Since $ \\iota \\circ \\Omega $ is continuous, so is $ \\Omega $.\n\t\n\tLet $ \\Phi \\colon \\Bis (\\mathcal{G}) \\to \\Bis (\\mathcal{G}) $ be the inversion map. \n\tInversion $ \\Inv \\colon \\Diff_{\\text{vS}} (M) \\to \\Diff_{\\text{vS}}(M) $ is continuous by \\cite[Theorem 7.6]{michor} and Proposition \\ref{prop: topologiescoincide}. The diagram\n\t\\begin{align*}\n\t\t\\xymatrix{\n\t\t\t\\Bis(\\mathcal{G}) \\ar[d]^{\\beta_*} \\ar@{.>}[rr]^{\\iota \\circ \\Phi} && C_{\\textup{vS}}^\\infty(M,G) \\\\\n\t\t\t\\Diff_{\\text{vS}}(M) \\ar[r]^{\\Inv} & \\Diff_{\\text{vS}}(M) \\ar[r]^{\\sigma_*} & C_{\\textup{vS}}^\\infty (M,G) \\ar[u]^{i_*}\n\t\t\t}\n\t\\end{align*}\n\tcommutes as $ \\Phi(\\sigma)(x) = i(\\sigma((\\beta \\circ \\sigma)^{-1}(x))) = (i_*(\\sigma_*(\\Inv(\\beta_*(\\sigma))))(x)$. \n\tAll maps represented by normal arrows are continuous. Thus $ \\iota \\circ \\Phi $ and also $ \\Phi $ are continuous.\n\\end{proof}\n\nAs noted in Remark \\ref{rem: topgroup}, $\\Diff (M)$ is a topological group with respect to the subspace topologies induced by the (fine) very strong topology on $C^\\infty (M,M)$.\nThus we obtain the following morphisms of topological groups.\n\\begin{corollary}\n\tThe target projection \\( \\beta \\colon G \\to M \\) of a locally convex Lie groupoid \\( \\mathcal{G} = (G \\rightrightarrows M) \\) induces a map \\( \\beta_* \\colon \\Bis (\\mathcal{G}) \\to \\Diff(M) \\) given by postcomposition. This is a homomorphism of topological groups with respect to the very strong and fine very strong topologies on both groups.\n\\end{corollary}\n\\begin{proof}\n\tSince $\\beta_* \\colon C_{\\textup{vS}}^\\infty (M,G) \\rightarrow C_{\\textup{vS}}^\\infty (M,M)$ is continuous, so is the (co)restriction of \\( \\beta_* \\) to $\\Bis (\\mathcal{G})$ and \\( \\Diff (M) \\). The same argument holds in the fine very strong topology.\n\tThe map \\( \\beta_* \\) is also a group homomorphism, since \n\t\\[ \\left( \\beta_* ( \\sigma \\star \\tau ) \\right) (x) = \\beta \\left( \\sigma ((\\beta \\circ \\tau)(x)) \\tau (x) \\right) = \\beta ( \\sigma ( \\beta \\circ \\tau ) (x)) = \\left( \\beta_*(\\sigma) \\circ \\beta_* (\\tau)\\right) (x).\\qedhere \\]\n\\end{proof}\n\nThe results of this section enable the construction of a Lie group structure on $\\Bis (\\mathcal{G})$. \nIt is worth noting that the key step in constructing the Lie group structure is sorting out the topology of the function spaces.\nUsing the manifold structure on $C_{\\textup{fS}}^\\infty (M,G)$ (see comments in Section \\ref{sect: smmfd}) one establishes the smoothness of joint composition and postcomposition with respect to these structures.\nSince Theorem A and the $\\Omega$-Lemma \\cite[Appendix F]{glockomega} are at our disposal, one can copy exactly the arguments from the finite-dimensional case outlined in \\cite[\\S 10 and \\S 11]{michor}.\nAfter that one can proceed as in \\cite{Schmeding2015} and establish smoothness of the group operations following the proof of Proposition \\ref{prop: topgp}.\nAgain, results of this type are beyond the scope of the present paper. \n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHigh-fidelity numerical simulations play a critical role in modern-day engineering and scientific investigations. The computational cost of high-fidelity or full-order models (FOMs) is, however, often prohibitively expensive. This limitation has led to the emergence of reduced-order modeling techniques. Reduced-order models (ROMs) are formulated to \\textit{approximate} solutions to a FOM on a low-dimensional manifold. Common reduced-order modeling techniques include balanced truncation~\\cite{balanced_truncation_moore,balanced_truncation_roberts}, Krylov subspace techniques~\\cite{krylov_rom}, reduced-basis methods~\\cite{Hesthaven2016}, and the proper orthogonal decomposition approach~\\cite{chatterjee_pod_intro}.\nReduced-order models based on such techniques have been implemented in a wide variety of disciplines and have been effective in reducing the computational cost associated with high-fidelity numerical simulations~\\cite{kerschen_mech_pod,padhi_neural_net_pod,cao_meteorology_pod}.\n\nProjection-based reduced-order models constructed from proper orthogonal decomposition (POD) have proved to be an effective tool for model order reduction of complex systems. In the POD-ROM approach, snapshots from a high-fidelity simulation (or experiment) are used to construct an orthonormal basis spanning the solution space. A small, truncated set of these basis vectors forms the \\emph{trial} basis.\nThe POD-ROM then seeks a solution within the range of the trial basis via projection. Galerkin projection, in which the FOM equations are projected onto the same trial subspace, is the simplest type of projection. The Galerkin ROM (G ROM) has been used successfully in a variety of problems. When applied to general non-self-adjoint and non-linear problems, however, theoretical analysis and numerical experiments have shown that Galerkin ROM lacks \\textit{a priori} guarantees of stability, accuracy, and convergence~\\cite{rowley_pod_energyproj}. This last issue is particularly challenging as it demonstrates that enriching a ROM basis does not necessarily improve the solution~\\cite{huang_combustion_roms}. The development of stable and accurate reduced-order modeling techniques for complex non-linear systems is the motivation for the current work.\n\\begin{comment}\nResearch examining the stability and accuracy of ROMs is typically approached from either a stabilization viewpoint or from a closure modeling viewpoint. \n\\end{comment}\n\nA significant body of research aimed at producing accurate and stable ROMs for complex non-linear problems exists in the literature. These efforts include, but are not limited to, ``energy-based\" inner products~\\cite{rowley_pod_energyproj,Kalashnikova_sand2014}, symmetry transformations~\\cite{sirovich_symmetry_trans}, basis adaptation~\\cite{carlberg_hadaptation,adeim_peherstorfer}, $L^1$-norm minimization~\\cite{l1}, projection subspace rotations~\\cite{basis_rotation}, and least-squares residual minimization approaches~\\cite{bui_resmin_steady,bui_unsteady,rovas_thesis,carlberg_thesis,bui_thesis,carlberg_lspg,carlberg_lspg_v_galerkin,carlberg_gnat}. The Least-Squares Petrov--Galerkin (LSPG)~\\cite{carlberg_lspg} method comprises a particularly popular least-squares residual minimization approach and has been proven to be an effective tool for non-linear model reduction. Defined at the fully-discrete level (i.e., after spatial and temporal discretization), LSPG relies on least-squares minimization of the FOM residual at each time-step. While the method lacks \\textit{a priori} stability guarantees for general non-linear systems, it has been shown to be effective for complex problems of interest~\\cite{carlberg_gnat, carlberg_lspg_v_galerkin, huang_scitech19}. Additionally, as it is formulated as a minimization problem, physical constraints such as conservation can be naturally incorporated into the ROM formulation~\\cite{carlberg_conservative_rom}. At the fully-discrete level, LSPG is sensitive to both the time integration scheme as well as the time-step. For example, in Ref.~\\cite{carlberg_lspg_v_galerkin} it was shown that LSPG produces optimal results at an intermediate time-step. Another example of this sensitivity is that, when applied to explicit time integration schemes, the LSPG approach reverts to a Galerkin approach. This limits the scope of LSPG to implicit time integration schemes, which can in turn increase the cost of the ROM~\\cite{carlberg_lspg_v_galerkin}\\footnote{It is possible to use LSPG with an explicit time integrator by formulating the ROM for an implicit time integration scheme, and then time integrating the resulting system with an explicit integrator.}. This is particularly relevant in the case where the optimal time-step of LSPG is small, thus requiring many time-steps of an implicit solver. Despite these challenges, the LSPG approach is arguably the most robust  technique that is used for ROMs of non-linear dynamical systems.\n\n\\begin{comment}\nOne method of producing stable and accurate ROMs is through stabilization, or the addition of dissipative effects to otherwise-unstable dynamics. Research on ROM stabilization seeks to address the issue that, in general, Galerkin ROMs lack any \\textit{a priori} guarantees of stability~\\cite{Kalashnikova_sand2014}. A variety of model reduction techniques have been developed to address this issue, including ``energy-based\" inner products~\\cite{Kalashnikova_sand2014,rowley_pod_energyproj}, symmetry transformations~\\cite{sirovich_symmetry_trans}, projection subspace rotations~\\cite{basis_rotation}, and the Least-Squares Petrov--Galerkin (LSPG) approach~\\cite{carlberg_lspg}.  Typically defined at the fully-discrete level (i.e., after spatial and temporal discretization), the LSPG approach relies on least-squares minimization of the FOM residual at each time-step. While the method lacks \\textit{a priori} stability guarantees for general non-linear systems, it has been shown to be effective for complex problems of interest~\\cite{carlberg_gnat, carlberg_lspg_v_galerkin, huang_scitech19}. Additionally, as it is formulated as a minimization problem, physical constraints such as conservation can be naturally incorporated into the ROM formulation~\\cite{carlberg_conservative_rom}. At the fully-discrete level, LSPG is sensitive to both the time integration scheme as well as the time-step. For example, in Ref.~\\cite{carlberg_lspg_v_galerkin} it was shown that LSPG produces optimal results at an intermediate time-step. Another example of this sensitivity is that, when applied to explicit time integration schemes, the LSPG approach reverts to a Galerkin approach. This limits the scope of LSPG to implicit time integration schemes, which can in turn increase the cost of the ROM~\\cite{carlberg_lspg_v_galerkin}\\footnote{It is possible to use LSPG with an explicit time integrator by formulating the ROM for an implicit time integration scheme, and then time integrating the resulting system with an explicit integrator.}. This is particularly relevant in the case where the optimal time-step of LSPG is small, thus requiring many time-steps of an implicit solver. Despite these drawbacks, the LSPG approach is arguably the most robust  technique that is used for ROMs of non-linear dynamical systems.\n\\end{comment}\n\nA second school of thought addresses stability and accuracy of ROMs from a closure modeling viewpoint. This follows from the idea that instabilities and inaccuracies in ROMs can, for the most part, be attributed to the truncated  modes. While these truncated  modes may not contain a significant portion of the system energy, they can play a significant role in the dynamics of the ROM~\\cite{Wang_ROM_thesis}.  This is analogous to the closure problem encountered in large eddy simulation. Research has examined the construction of mixing length~\\cite{aubry_mixlength_pod}, Smagorinsky-type~\\cite{Wang_ROM_thesis,Ullmann_smag,wang_smag,smag_ROM}, and variational multiscale (VMS) closures~\\cite{Wang_ROM_thesis,san_iliescu_geostrophic,Bergmann_pod_vms,Stabile2019} for POD-ROMs. The VMS approach is of particular relevance to this work. Originally developed in the context of finite element methods, VMS is a formalism to derive stabilization\/closure schemes for numerical simulations of multiscale problems. The VMS procedure is centered around a sum decomposition of the solution $u$ in terms of resolved\/coarse-scales $\\tilde{u}$ and unresolved\/fine-scales ${u}^{\\prime}$. The impact of the fine-scales on the evolution of the coarse-scales is then accounted for by devising an approximation to the fine-scales. This approximation is often referred to as a ``subgrid-scale'' or ``closure'' model. \n\nResearch has examined the application of both phenomenological and residual-based subgrid-scale models to POD-ROMs. In Refs.~\\cite{san_iliescu_geostrophic,iliescu_pod_eddyviscosity,iliescu_vms_pod_ns}, Iliescu and co-workers examine the construction of eddy-viscosity-based ROM closures via the VMS method. These eddy-viscosity methods are directly analogous to the eddy-viscocity philosophy used in turbulence modeling. While they do not guarantee stability \\textit{a priori}, these ROMs have been shown to enhance accuracy on a variety of problems in fluid dynamics. However, as eddy-viscosity methods are based on phenomenological assumptions specific to three-dimensional turbulent flows, their scope may be limited to specific types of problems. Residual-based methods, which can also be derived from VMS, constitute a more general modeling strategy. The subgrid-scale model emerging from a residual-based method typically appears as a term that is proportional to the residual of the full-order model; if the governing equations are exactly satisfied by the ROM, then the model is inactive. While residual-based methods in ROMs are not as well-developed as they are in finite element methods, they have been explored in several contexts. In Ref.~\\cite{Bergmann_pod_vms}, ROMs of the Navier-Stokes equations are stabilized using residual-based methods. This stabilization is performed by solving a ROM stabilized with a method such as streamline upwind Petrov--Galerkin (SUPG) and augmenting the POD basis with additional modes computed from the residual of the Navier-Stokes equations. In Ref.~\\cite{iliescu_ciazzo_residual_rom}, residual-based stabilization is developed for velocity-pressure ROMs of the incompressible Navier-Stokes equations. Both eddy-viscosity and residual-based methods have been shown to improve ROM stability and performance. The majority of existing work on residual-based stabilization (and eddy-viscosity methods) is focused on ROMs formulated from continuous projection (i.e., projecting a continuous PDE using a continuous basis). In this instance, the ROM residual is defined at the continuous level and is directly linked to the governing partial differential equation. In many applications (arguably the majority~\\cite{Kalashnikova_sand2014}), however, the ROM is constructed through discrete projection (i.e., projecting the spatially discretized PDE using a discrete basis). In this instance, the ROM residual is defined at the semi-discrete level and is tied to the \\textit{spatially discretized} governing equations. Residual-based methods for ROMs developed through discrete projections have, to the best of the authors' knowledge, not been investigated.\n \nAnother approach that displays similarities to the variational multiscale method is the Mori-Zwanzig (MZ) formalism. Originally developed by Mori~\\cite{MoriTransport} and Zwanzig~\\cite{ZwanzigLangevin} and reformulated by Chorin and co-workers~\\cite{ChorinOptimalPrediction,ChorinOptimalPredictionMemory,Chorin_book,ProblemReduction}, the MZ formalism is a type of model order reduction framework. The framework consists of decomposing the state variables in a dynamical system into a resolved (coarse-scale) set and an unresolved (fine-scale) set. An exact reduced-order model for the resolved scales is then derived in which the impact of the unresolved scales on the resolved scales appears as a memory term. This memory term depends on the temporal history of the resolved variables. In practice, the evaluation of this memory term is not tractable. It does, however, serve as a starting point to develop closure models. As MZ is formulated systematically in a dynamical system setting, it promises to be an effective technique for developing stable and accurate ROMs of non-linear dynamical systems. A range of research examining the MZ formalism as a multiscale modeling tool exists in the community. Most notably, Stinis and co-workers~\\cite{stinisEuler,stinisHighOrderEuler,Stinis-rMZ,stinis_finitememory,PriceMZ,PriceMZ2} have developed several models for approximating the memory, including finite memory and renormalized models, and examined their application to the semi-discrete systems emerging from Fourier-Galerkin and Polynomial Chaos Expansions of Burgers' equation and the Euler equations. Application of MZ-based techniques to the classic POD-ROM approach has not been undertaken.\n\nThis manuscript leverages work that the authors have performed on the use of the MZ formalism to develop closure models of partial differential equations~\\cite{parishAIAA2016,parishMZ1,parish_dtau,GouasmiMZ1,parishVMS}. In addition to focusing on the development and analysis of MZ models, the authors have examined the formulation of the MZ formalism within the context of the VMS method~\\cite{parishVMS}. By expressing MZ models within a VMS framework, similarities were discovered between MZ and VMS models. In particular, it was discovered that several existing MZ models are residual-based methods. \n\n The contributions of this work include:\n\\begin{enumerate}\n\\item The development of a novel projection-based reduced-order modeling technique, termed the Adjoint Petrov--Galerkin (APG) method. The method leads to a ROM equation that is driven by the residual of the discretized governing equations. The approach is equivalent to a Petrov--Galerkin ROM and displays similarities to the LSPG approach.  The method can be evolved in time with explicit integrators (in contrast to LSPG). This potentially lowers the cost of the ROM.\n\n\\item Theoretical error analysis examining conditions under which the \\textit{a priori} error bounds in APG may be smaller than in the Galerkin method.\n\n\\item Computational cost analysis (in FLOPS) of the proposed APG method as compared to the Galerkin and LSPG methods. This analysis shows that the APG ROM is twice as expensive as the G ROM for a given time step, for both explicit and implicit time integrators. In the implicit case, the ability of the APG ROM to make use of Jacobian-Free Newton-Krylov methods suggests that it may be more efficient than the LSPG ROM.\n\n\\item Numerical evidence on ROMs of compressible flow problems demonstrating that the proposed method is more accurate and stable than the G ROM on problems of interest. Improvements over the LSPG ROM are observed in most cases. An analysis of the computational cost shows that the APG method can lead to lower errors than the LSPG and G ROMs for the same computational cost.\n\n\\item Theoretical results and numerical evidence that provides a relationship between the time-scale in the APG ROM and the spectral radius of the right-hand side Jacobian. Numerical evidence suggests that this relationship also applies to the selection of the optimal time-step in LSPG.\n\n\\end{enumerate}\n\n\nThe structure of this paper is as follows: Section~\\ref{sec:FOM}  outlines the full-order model of interest and its formulation in generalized coordinates. Section~\\ref{sec:ROM}  outlines the reduced-order modeling approach applied at the semi-discrete level. Galerkin, Petrov--Galerkin, and VMS ROMs will be discussed. Section~\\ref{sec:MZ} details the Mori-Zwanzig formalism and the construction of the Adjoint Petrov--Galerkin ROM. Section~\\ref{sec:analysis} provides theoretical error analysis. Section~\\ref{sec:cost} discusses the implementation and computational cost of the Adjoint Petrov--Galerkin method. Numerical results and comparisons with Galerkin and LSPG ROMs are presented in Section~\\ref{sec:numerical}. Conclusions are provided in Section~\\ref{sec:conclude}.\n\nMathematical notation in this manuscript is as follows: matrices are written as bold uppercase letters (e.g. ${\\mathbf{V}}$), vectors as lowercase bold letters (e.g. ${\\mathbf{u}}$), and scalars as italicized lowercase letters (e.g. $a_i$). Calligraphic script may denote vector spaces or special operators (e.g. $\\mathcal{V}$, $\\mathcal{L}$). Bold letters followed by parentheses indicate a matrix or vector function (e.g. $\\mathbf{R}(\\cdot)$, ${\\mathbf{u}} (\\cdot)$), and those followed by brackets indicate a linearization about the bracketed argument (e.g. $\\mathbf{J}[\\cdot]$).\n\n\\section{Full-Order Model and Generalized Coordinates}\\label{sec:FOM}\nConsider a full-order model that is described by the dynamical system,\n\\begin{equation}\\label{eq:FOM}\n\\frac{d }{dt}{\\mathbf{u}}(t) = \\mathbf{R}({\\mathbf{u}}(t)), \\qquad {\\mathbf{u}}(0) = {\\mathbf{u}}_0, \\qquad t \\in [0,T], \n\\end{equation}\nwhere $T \\in \\mathbb{R}^+$ denotes the final time, ${\\mathbf{u}} : [0,T] \\rightarrow \\RR{N}$ denotes the state, and ${\\mathbf{u}}_0 \\in \\mathbb{R}^N$ the initial conditions. The function $\\mathbf{R}: \\mathbb{R}^N  \\rightarrow \\mathbb{R}^N$ with $\\mathbf{y} \\mapsto \\mathbf{R}(\\mathbf{y})$ is a (possibly non-linear) function and will be referred to as the ``right-hand side\" operator. Equation~\\ref{eq:FOM} arises in many disciplines, including the numerical discretization of partial differential equations. In this context, $\\mathbf{R}(\\cdot)$ may represent a spatial discretization scheme with source terms and applicable boundary conditions.  \n\nIn many practical applications, the computational cost associated with solving Eq.~\\ref{eq:FOM} is prohibitively expensive due to the high dimension of the state. The goal of a ROM is to transform the $N$-dimensional dynamical system presented in Eq.~\\ref{eq:FOM} into a $K$ dimensional dynamical system, with $K \\ll N$. To achieve this goal, we pursue the following agenda:\n\\begin{enumerate}\n\\item Develop a weak form of the FOM in generalized coordinates.\n\\item Decompose the generalized coordinates into a $K$-dimensional resolved coarse-scale set and an $N-K$ dimensional unresolved fine-scale set.\n\\item Develop a $K$-dimensional ROM for the coarse-scales by making approximations to the fine-scale coordinates.\n\\end{enumerate}\nThe remainder of this section will address task 1 in the above agenda.\n\n\nTo develop the weak form of Eq.~\\ref{eq:FOM}, we start by defining a trial basis matrix comprising $N$ orthonormal basis vectors,\n\\begin{equation*}\n\\mathbf{V} \\equiv \\begin{bmatrix}\n\\mathbf{v}_1 & \\mathbf{v}_2 & \\cdots & \\mathbf{v}_N\n\\end{bmatrix},\n\\end{equation*}\nwhere $\\mathbf{v}_i \\in \\mathbb{R}^N$, $\\mathbf{v}_i^T \\mathbf{v}_j = \\delta_{ij}$.\nThe basis vectors may be generated, for example, by the POD approach. \nWe next define the \\textit{trial space} as the range of the trial basis matrix,\n$${\\MC{V}} \\trieq \\text{Range}({\\mathbf{V}}).$$\nAs $\\mathbf{V}$ is a full-rank $N \\times N$ matrix, $\\mathcal{V} \\equiv \\RR{N}$ and the state variable can be exactly described by a linear combination of these basis vectors,\n\\begin{equation}\\label{eq:genCord}\n{\\mathbf{u}}(t) = \\sum_{i=1}^N \\mathbf{v}_i a_i(t).  \n\\end{equation}\nFollowing~\\cite{carlberg_lspg}, we collect the basis coefficients $a_i(t)$ into $\\mathbf{a} : [0,T] \\rightarrow \\RR{N}$ and refer to $\\mathbf{a}$ as the \\emph{generalized coordinates}. We similarly define the test basis matrix, ${\\mathbf{W}}$, whose columns comprise linearly independent basis vectors that span the \\textit{test space}, $\\mathcal{W}$,\n\\begin{equation*}\n{\\mathbf{W}} \\equiv \\begin{bmatrix}\n\\mathbf{w}_1 & \\mathbf{w}_2 & \\cdots& \\mathbf{w}_N\n\\end{bmatrix}, \\qquad \n\\mathcal{W} \\trieq \\text{Range}({\\mathbf{W}}) ,\n\\end{equation*}\nwith $\\mathbf{w}_i \\in \\mathbb{R}^N$. \n\nEquation~\\ref{eq:FOM} can be expressed in terms of the generalized coordinates by inserting Eq.~\\ref{eq:genCord} into Eq.~\\ref{eq:FOM}, \n\\begin{equation}\\label{eq:FOM2}\n{\\mathbf{V}} \\frac{d }{dt}\\mathbf{a}(t)= \\mathbf{R}({\\mathbf{V}} \\mathbf{a}(t)).\n\\end{equation}\nThe weak form of Eq.~\\ref{eq:FOM2} is obtained by taking the $L^2$ inner product with ${\\mathbf{W}}$,\\footnote{The authors recognize that many types of inner products are possible in formulating a ROM. To avoid unnecessary abstraction, we focus here on the simplest case.}\n\\begin{equation}\\label{eq:FOM3}\n{\\mathbf{W}}^T  {\\mathbf{V}} \\frac{d }{dt}\\mathbf{a}(t)  = {\\mathbf{W}}^T  \\mathbf{R}({\\mathbf{V}} \\mathbf{a}(t)).\n\\end{equation}\nManipulation of Eq.~\\ref{eq:FOM3} yields the following dynamical system,\n\\begin{equation}\\label{eq:FOM_generalized}\n\\frac{d }{dt}\\mathbf{a}(t) =  [{\\mathbf{W}}^T {\\mathbf{V}} ]^{-1} {\\mathbf{W}}^T \\mathbf{R}({\\mathbf{V}} \\mathbf{a}(t)), \\qquad \\mathbf{a}(t=0) = \\mathbf{a}_0, \\qquad t \\in [0,T],\n\\end{equation}\nwhere $\\mathbf{a}_0 \\in \\RR{N}$ with $\\mathbf{a}_0 =  [{\\mathbf{W}}^T {\\mathbf{V}} ]^{-1} {\\mathbf{W}}^T {\\mathbf{u}}_0.$ \nNote that Eq.~\\ref{eq:FOM_generalized} is an $N$-dimensional ODE system and is simply Eq.~\\ref{eq:FOM} expressed in a different coordinate system. It is further worth noting that, since ${\\mathbf{W}}$ and ${\\mathbf{V}}$ are invertible (both are square matrices with linearly independent columns), one has $[{\\mathbf{W}}^T {\\mathbf{V}} ]^{-1} {\\mathbf{W}}^T = {\\mathbf{V}}^{-1}.$ This will not be the case for ROMs.\n\n\\section{Reduced-Order Models}\\label{sec:ROM}\n\\subsection{Multiscale Formulation}\nThis subsection addresses task 2 in the aforementioned agenda. Reduced-order models seek a low-dimensional representation of the original high-fidelity model. To achieve this, we examine a multiscale formulation of Eq.~\\ref{eq:FOM_generalized}. Consider sum decompositions of the trial and test space,\n\\begin{equation}\n{\\MC{V}} = \\tilde{\\MC{V}} \\oplus {\\MC{V}^{\\prime}}, \\qquad \\mathcal{W} = \\mathcal{\\tilde{W}} \\oplus \\mathcal{W}'.\n\\end{equation}\nThe space $\\tilde{\\MC{V}}$ is referred to as the coarse-scale trial space, while ${\\MC{V}^{\\prime}}$ is referred to as the fine-scale trial space. We refer to $\\tilde{\\MC{W}}$ and ${\\MC{W}^{\\prime}}$ in a similar fashion.\nFor simplicity, define $\\tilde{\\MC{V}}$ to be the column space of the first $K$ basis vectors in ${\\mathbf{V}}$ and ${\\MC{V}^{\\prime}}$ to be the column space of the last $N-K$ basis vectors in ${\\mathbf{V}}$. This approach is appropriate when the basis vectors are ordered in a hierarchical manner, as is the case with POD. Note the following properties of the decomposition:\n\\begin{enumerate}\n\\item The coarse-scale space is a subspace of $\\mathcal{V}$, i.e., $\\tilde{\\MC{V}} \\subset {\\MC{V}}$.\n\\item The fine-scale space is a subspace of $\\mathcal{V}$, i.e., ${\\MC{V}^{\\prime}} \\subset {\\MC{V}}$.\n\\item The fine and coarse-scale subspaces do not overlap, i.e.,  $\\tilde{\\MC{V}} \\cap {\\MC{V}^{\\prime}} = \\{ 0 \\}.$\n\\item The fine-scale and coarse-scale subspaces are orthogonal, i.e., $\\tilde{\\MC{V}} \\perp {\\MC{V}^{\\prime}}.$ This is due to the fact that the basis vectors that comprise $\\mathbf{V}$ are orthonormal.\n\\end{enumerate}\nFor notational purposes, we make the following definitions for the trial and test spaces:\n\\begin{equation*}\n{\\mathbf{V}} \\equiv \\begin{bmatrix} \\tilde{\\mathbf{V}} & ; & {\\mathbf{V}^{\\prime}} \\end{bmatrix}, \\quad {\\mathbf{W}} \\equiv \\begin{bmatrix} \\tilde{\\mathbf{W}} & ; & {\\mathbf{W}^{\\prime}} \\end{bmatrix}, \n\\end{equation*}\nwhere $[ \\cdot \\hspace{0.05 in}; \\hspace{0.05 in} \\cdot ]$ denotes the concatenation of two matrices and,\n\\begin{alignat*}{2}\n&\\tilde{\\mathbf{V}} \\equiv \\begin{bmatrix}\n\\mathbf{v}_1 & \\mathbf{v}_2 & \\cdots & \\mathbf{v}_K\n\\end{bmatrix},  && \\quad \\tilde{\\MC{V}} \\trieq \\text{Range}(\\tilde{\\mathbf{V}}) , \\\\\n&{\\mathbf{V}^{\\prime}}  \\equiv \\begin{bmatrix}\n\\mathbf{v}_{K+1} & \\mathbf{v}_{K+2} & \\cdots &  \\mathbf{v}_{N}\n\\end{bmatrix}, && \\quad  {\\MC{V}^{\\prime}} \\trieq \\text{Range}({\\mathbf{V}^{\\prime}}). \\\\\n&\\tilde{\\mathbf{W}} \\equiv \\begin{bmatrix}\n\\mathbf{w}_1 & \\mathbf{w}_2 & \\cdots & \\mathbf{w}_K\n\\end{bmatrix},  && \\quad \\tilde{\\MC{W}} \\trieq \\text{Range}(\\tilde{\\mathbf{W}}) , \\\\\n&{\\mathbf{W}^{\\prime}}  \\equiv \\begin{bmatrix}\n\\mathbf{w}_{K+1} & \\mathbf{w}_{K+2} & \\cdots & \\mathbf{w}_{N}\n\\end{bmatrix}, && \\quad {\\MC{W}^{\\prime}} \\trieq \\text{Range}({\\mathbf{W}^{\\prime}}) .\n\\end{alignat*}\nThe coarse and fine-scale states are defined as,\n\\begin{equation*}\n\\tilde{\\mathbf{u}}(t) \\trieq \\sum_{i=1}^K \\mathbf{v}_i a_i(t) \\equiv \\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}(t), \\qquad {\\mathbf{u}^{\\prime}}(t) \\trieq \\sum_{i = K+1}^N \\mathbf{v}_i a_i(t) \\equiv {\\mathbf{V}^{\\prime}} {\\mathbf{a}^{\\prime}}(t),\n\\end{equation*}\nwith $\\tilde{\\mathbf{u}} : [0,T] \\rightarrow \\tilde{\\mathcal{V}}$, ${\\mathbf{u}^{\\prime}} : [0,T] \\rightarrow \\mathcal{V}', \\tilde{\\mathbf{a}}: [0,T] \\rightarrow \\RR{K},$ and ${\\mathbf{a}^{\\prime}} : [0,T] \\rightarrow \\RR{N-K}.$\n\\begin{comment}\nWe make similar definitions for the test space,\n\\begin{equation*}\n{\\mathbf{W}} \\overset{\\Delta}{=} \\begin{bmatrix} \\tilde{\\mathbf{W}} & ; & {\\mathbf{W}^{\\prime}} \\end{bmatrix},   \n\\end{equation*}\nwhere,\n\\begin{alignat*}{2}\n&\\tilde{\\mathbf{W}} \\overset{\\Delta}{=} \\begin{bmatrix}\n\\mathbf{w}_1, \\mathbf{w}_2, \\hdots, \\mathbf{w}_K\n\\end{bmatrix},  && \\quad \\text{Range}(\\tilde{\\mathbf{W}}) \\overset{\\Delta}{=} \\tilde{\\MC{W}}, \\\\\n&{\\mathbf{W}^{\\prime}}  \\overset{\\Delta}{=} \\begin{bmatrix}\n\\mathbf{w}_{K+1}, \\mathbf{w}_{K+2}, \\hdots, \\mathbf{w}_{N}\n\\end{bmatrix}, && \\quad \\text{Range}({\\mathbf{W}^{\\prime}})\\overset{\\Delta}{=} {\\MC{W}^{\\prime}}.\n\\end{alignat*}\n\\end{comment}\nThese decompositions allow Eq.~\\ref{eq:FOM3} to be expressed as two linearly independent systems,\n\\begin{equation}\\label{eq:FOM_VMS_coarse}\n\\tilde{\\mathbf{W}}^T \\tilde{\\mathbf{V}} \\frac{d }{dt}\\tilde{\\mathbf{a}}(t) + \\tilde{\\mathbf{W}}^T {\\mathbf{V}^{\\prime}} \\frac{d }{dt}{\\mathbf{a}^{\\prime}}(t) =\\tilde{\\mathbf{W}}^T  \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}(t) + {\\mathbf{V}^{\\prime}} {\\mathbf{a}^{\\prime}}(t)),\n\\end{equation}\n\\begin{equation}\\label{eq:FOM_VMS_fine}\n{\\mathbf{W}^{\\prime}}^T \\tilde{\\mathbf{V}} \\frac{d }{dt} \\tilde{\\mathbf{a}}(t) + {\\mathbf{W}^{\\prime}}^T {\\mathbf{V}^{\\prime}} \\frac{d }{dt}{\\mathbf{a}^{\\prime}}(t) ={\\mathbf{W}^{\\prime}}^T  \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}(t) + {\\mathbf{V}^{\\prime}} {\\mathbf{a}^{\\prime}}(t)).\n\\end{equation}\nEquation~\\ref{eq:FOM_VMS_coarse} is referred to as the coarse-scale equation, while Eq.~\\ref{eq:FOM_VMS_fine} is referred to as the fine-scale equation. It is important to emphasize that the system formed by Eqs.~\\ref{eq:FOM_VMS_coarse} and~\\ref{eq:FOM_VMS_fine} is still an exact representation of the original FOM.\n\nThe objective of ROMs is to solve the coarse-scale equation. The challenge encountered in this objective is that the evolution of the coarse-scales depends on the fine-scales. This is a type of ``closure problem\" and must be addressed to develop a closed ROM.\n\n\\subsection{Reduced-Order Models}\nAs noted above, the objective of a ROM is to solve the (unclosed) coarse-scale equation. We now develop ROMs of Eq.~\\ref{eq:FOM} by leveraging the multiscale decomposition presented above. This section addresses task 3 in the mathematical agenda.\n\n The most straightforward technique to develop a ROM is to make the approximation,\n\\begin{equation*}\\label{eq:ufine_ansatz}\n{\\mathbf{u}^{\\prime}} \\approx \\mathbf{0}.\n\\end{equation*}\nThis allows for the coarse-scale equation to be expressed as,\n\\begin{equation}\\label{eq:FOM_VMS_coarse_2}\n\\tilde{\\mathbf{W}}^T \\tilde{\\mathbf{V}} \\frac{d }{dt}\\tilde{\\mathbf{a}}(t) =\\tilde{\\mathbf{W}}^T  \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}(t) ).\n\\end{equation}\nEquation~\\ref{eq:FOM_VMS_coarse_2} forms a $K$-dimensional reduced-order system (with $K \\ll N$) and provides the starting point for formulating several standard ROM techniques. The Galerkin and Least-Squares Petrov--Galerkin ROMs are outlined in the subsequent subsections.\n\n\\subsubsection{The Galerkin Reduced-Order Model}\nGalerkin projection is a common choice for producing a reduced set of ODEs. In Galerkin projection, the test basis is taken to be equivalent to the trial basis, i.e.  $\\tilde{\\mathbf{W}} = \\tilde{\\mathbf{V}}$. The Galerkin ROM is then,\n\\begin{equation}\\label{eq:GROM}\n\\tilde{\\mathbf{V}}^T \\frac{d }{dt}\\tilde{\\mathbf{u}}(t) = \\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{u}}(t)), \\qquad \\tilde{\\mathbf{u}}(0) = \\tilde{\\mathbf{u}}_0, \\qquad t \\in [0,T].\n\\end{equation}\nGalerkin projection can be shown to be optimal in the sense that it minimizes the $L^2$-norm of the FOM ODE residual over $\\text{Range}(\\tilde{\\mathbf{V}})$~\\cite{carlberg_lspg}. As the columns of $\\tilde{\\mathbf{V}}$ no longer spans ${\\MC{V}}$, it is possible that the initial state of the full system, ${\\mathbf{u}}_0$, may differ from the initial state of the reduced system, $\\tilde{\\mathbf{u}}_0$. For simplicity, however, it is assumed here that the initial conditions lie fully in the coarse-scale trial space, i.e. ${\\mathbf{u}}_0 \\in \\tilde{\\MC{V}}$ such that,\n\\begin{equation}\\label{eq:ROM_IC}\n\\tilde{\\mathbf{u}}_0 = {\\mathbf{u}}_0.\n\\end{equation}\n Note that this issue can be formally addressed by using an affine trial space to ensure that $\\tilde{\\mathbf{u}}_0 = {\\mathbf{u}}_0$.\n\nEquation~\\ref{eq:GROM} can be equivalently written for the generalized coarse-scale coordinates, $\\tilde{\\mathbf{a}}$,\n\\begin{equation}\\label{eq:GROM_modal}\n\\frac{d }{dt}\\tilde{\\mathbf{a}}(t) = \\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}(t)), \\qquad \\tilde{\\mathbf{a}}(0) = \\tilde{\\mathbf{a}}_0, \\qquad t \\in [0,T],\n\\end{equation}\nwhere $\\tilde{\\mathbf{a}}_0 \\in \\RR{K}$ with  $\\tilde{\\mathbf{a}}_0 =\\tilde{\\mathbf{V}}^T {\\mathbf{u}}_0.$\nEquation~\\ref{eq:GROM_modal} is a $K$-dimensional ODE system (with $K\\ll N$) and is hence of lower dimension than the FOM.  Note that, similar to Eq.~\\ref{eq:FOM_generalized}, the projection via $\\tilde{\\mathbf{V}}^T$ would normally be $\\big[ \\tilde{\\mathbf{V}}^T \\tilde{\\mathbf{V}} \\big]^{-1} \\tilde{\\mathbf{V}}$. When $\\tilde{\\mathbf{V}}$ is constructed via POD, its columns are orthonormal and $\\tilde{\\mathbf{V}}^T \\tilde{\\mathbf{V}} = \\mathbf{I}$; the projector has been simplified to reflect this. Non-orthonormal basis vectors will require the full computation of $\\big[ \\tilde{\\mathbf{V}}^T \\tilde{\\mathbf{V}} \\big]^{-1} \\tilde{\\mathbf{V}}$.\n\nIt is important to note that, in order to develop a computationally efficient ROM, some ``hyper-reduction'' method must be devised to reduce the cost associated with evaluating the matrix-vector product, $\\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{u}}(t))$. Gappy POD~\\cite{everson_sirovich_gappy} and the (discrete) empirical interpolation method~\\cite{eim,deim} are two such techniques. More details on hyper-reduction are provided in Appendix~\\ref{appendix:hyper}\n\nWhen applied to unsteady non-linear problems, the Galerkin ROM is often inaccurate and, at times, unstable. Examples of this are seen in Ref.~\\cite{carlberg_lspg_v_galerkin}. These issues motivate the development of more sophisticated reduced-order modeling techniques. \n\n\\subsubsection{Petrov--Galerkin and Least-Squares Petrov--Galerkin Reduced-Order Models}\nIn the Petrov--Galerkin approach, the test space is  different from the trial space. Petrov--Galerkin approaches have a rich history in the finite element community~\\cite{brooks_supg,hughes_petrovgalerkin} and can enhance the stability and robustness of a numerical method. In the context of reduced-order modeling for dynamical systems, the Least-Squares Petrov--Galerkin method (LSPG)~\\cite{carlberg_lspg} is a popular approach. The LSPG approach is a ROM technique that seeks to minimize the fully discrete residual (i.e., the residual after spatial and temporal discretization) at each time-step. The LSPG method can be shown to be optimal in the sense that it minimizes the $L^2$-norm of the \\textit{fully discrete} residual at each time-step over $\\text{Range}(\\tilde{\\MC{V}})$. To illustrate the LSPG method, consider the algebraic system of equations for the FOM obtained after an implicit Euler temporal discretization,\n\\begin{equation}\\label{eq:coarse_implicit_euler_0}\n\\frac{{\\mathbf{u}}^{n} - {\\mathbf{u}}^{n-1}  }{\\Delta t} - \\mathbf{R}({\\mathbf{u}}^{n}) = \\mathbf{0},\n\\end{equation}\nwhere ${\\mathbf{u}}^n \\in \\RR{N}$ denotes the solution at the $n^{th}$ time-step.\nThe FOM will exactly satisfy Eq.~\\ref{eq:coarse_implicit_euler_0}. The ROM, however, will not. The LSPG method minimizes the residual of Eq.~\\ref{eq:coarse_implicit_euler_0} over each time-step. For notational purposes, we define the residual vector for the implicit Euler method,\n\\begin{equation*}\n\\mathbf{r}_{\\text{IE}}: (\\mathbf{y};{\\mathbf{u}}^{n-1}) \\mapsto  \\frac{\\mathbf{y} - {\\mathbf{u}}^{n-1}  }{\\Delta t} -  \\mathbf{R}(\\mathbf{y}).\n\\end{equation*}\nThe LSPG method is defined as follows,\n\\begin{equation*}\n{\\mathbf{u}}^n = \\underset{\\mathbf{y} \\in \\text{Range}(\\tilde{\\mathbf{V}})  }{\\text{arg min}}|| \\mathbf{A}(\\mathbf{y}) \\mathbf{r}_{\\text{IE}}(\\mathbf{y};{\\mathbf{u}}^{n-1}) ||_2^2,\n\\end{equation*}\nwhere $\\mathbf{A}(\\cdot) \\in \\mathbb{R}^{z \\times n}$ with $z \\le N$ is a weighting matrix. The standard LSPG method takes $\\mathbf{A} = \\mathbf{I}$. For the implicit Euler time integration scheme (as well as various other implicit schemes) the LSPG approach can be shown to have an equivalent continuous representation using a Petrov--Galerkin projection~\\cite{carlberg_lspg_v_galerkin}. For example, the LSPG method for any backward differentiation formula (BDF) time integration scheme can be written as a Petrov--Galerkin ROM with the test basis,\n\\begin{equation*}\n\\mathbf{\\tilde{\\mathbf{W}}} =\\big( \\mathbf{I} - \\alpha \\Delta t \\mathbf{J}[\\tilde{\\mathbf{u}}(t)] \\big) \\tilde{\\mathbf{V}},\n\\end{equation*}\nwhere $\\mathbf{J}[\\tilde{\\mathbf{u}}(t)] = \\frac{\\partial \\mathbf{R}}{\\partial \\mathbf{y}}(\\tilde{\\mathbf{u}}(t))$ is the Jacobian of the right-hand side function evaluated about the coarse-scale state and $\\alpha$ is a constant, specific to a given scheme (e.g. $\\alpha = 1$ for implicit Euler, $\\alpha = \\frac{2}{3}$ for BDF2, $\\alpha = \\frac{6}{11}$ for BDF3, etc.).\nWith this test basis, the LSPG ROM can be written as,\n\\begin{equation}\\label{eq:LSPGROM}\n \\tilde{\\mathbf{V}}^T \\bigg( \\frac{d }{dt}\\tilde{\\mathbf{u}}(t) -   \\mathbf{R}(\\tilde{\\mathbf{u}}(t)) \\bigg) = \\tilde{\\mathbf{V}}^T \\mathbf{J}^T [\\tilde{\\mathbf{u}}(t)]  \\alpha  \\Delta t\\bigg(  \\frac{d }{dt}\\tilde{\\mathbf{u}}(t) - \\mathbf{R}(\\tilde{\\mathbf{u}}(t)) \\bigg)  , \\qquad \\tilde{\\mathbf{u}}(0) = \\tilde{\\mathbf{u}}_0, \\qquad t \\in [0,T].\n\\end{equation}\nIn writing Eq.~\\ref{eq:LSPGROM}, we have coupled all of the terms from the standard Galerkin ROM on the left-hand side, and have similarly coupled the terms introduced by the Petrov--Galerkin projection on the right-hand side. One immediately observes that the LSPG approach is a residual-based method, meaning that the stabilization added by LSPG is proportional to the residual. The LSPG method is similar to the Galerkin\/Least-Squares (GLS) approach commonly employed in the finite element community~\\cite{hughes_GLS,hughes0}. This can be made apparent by writing Eq.~\\ref{eq:LSPGROM} as,\n\\begin{equation*}\n \\bigg(  \\mathbf{v}_i ,  \\frac{d }{dt}\\tilde{\\mathbf{u}}(t) -   \\mathbf{R}(\\tilde{\\mathbf{u}}(t))   \\bigg)= \\bigg(\\mathbf{J}  [\\tilde{\\mathbf{u}}(t)] \\mathbf{v}_i   , \\tau \\big[  \\frac{d }{dt}\\tilde{\\mathbf{u}}(t) - \\mathbf{R}(\\tilde{\\mathbf{u}}(t)) \\big] \\bigg) , \\qquad i = 1,2,\\hdots,K,  \n\\end{equation*}\nwhere $(\\mathbf{a},\\mathbf{b}) = \\mathbf{a}^T \\mathbf{b}$ and $\\tau = \\alpha \\Delta t$ is the stabilization parameter. Compare the above to, say, Eq. 70 and 71 in Ref.~\\cite{hughes0}. A rich body of literature exists on residual-based methods, and viewing the LSPG approach in this light helps establish connections with other methods.\nWe highlight several important aspects of LSPG. Remarks 1 through 3 are derived by Carlberg et al. in Ref.~\\cite{carlberg_lspg_v_galerkin}:\n\\begin{enumerate}\n\\item The LSPG approach is inherently tied to the temporal discretization. For different time integration schemes, the ``stabilization\" added by the LSPG method will vary. For optimal accuracy, the LSPG method requires an intermediary time-step size.\n\\item In the limit of $\\Delta t \\rightarrow 0$, the LSPG approach recovers a Galerkin approach.\n\\item For explicit time integration schemes, the LSPG and Galerkin approach are equivalent.\n\\item For backwards differentiation schemes, the LSPG approach is a type of GLS stabilization for non-linear problems. \n\\item While commonalities exist between LSPG and multiscale approaches, the authors believe that the LSPG method should \\textit{not} be viewed as a subgrid-scale model. The reason for this is that it is unclear how Eq.~\\ref{eq:LSPGROM} can be derived from Eq.~\\ref{eq:FOM_VMS_coarse}. This is similar to the fact that, in Ref.~\\cite{hughes0}, \\textit{adjoint} stabilization is viewed as a subgrid-scale model while GLS stabilization is not.  The challenge in deriving Eq.~\\ref{eq:LSPGROM} from Eq.~\\ref{eq:FOM_VMS_coarse} lies primarily in the fact that the Jacobian in Eq.~\\ref{eq:LSPGROM} contains a transpose operator. We thus view LSPG as mathematical stabilization rather than a subgrid-scale model.\n\\end{enumerate}\nWhile the LSPG approach has enjoyed much success for constructing ROMs of non-linear problems, remarks 1, 2, 3, and 5 suggest that improvements over the LSPG method are possible. Remark 1 suggests improvements in computational speed and accuracy are possible by removing sensitivity to the time-step size. Remark 3 suggests that improvements in computational speed and flexibility are possible by formulating a method that can be used with explicit time-stepping schemes. Lastly, remark 5 suggests that improvements in accuracy are possible by formulating a method that accounts for subgrid effects.\n\n\n\\subsection{Mori-Zwanzig Reduced-Order Models}\\label{sec:MZ}\nThe optimal prediction framework formulated by Chorin et al.~\\cite{ChorinOptimalPrediction,ChorinOptimalPredictionMemory,Chorin_book}, which is a (significant) reformulation of the Mori-Zwanzig (MZ) formalism of statistical mechanics, is a model order reduction tool that can be used to develop representations of the impact of the fine-scales on the coarse-scale dynamics. In this section, the optimal prediction framework is used to derive a compact approximation to the impact of the fine-scale POD modes on the evolution of the coarse-scale POD modes. For completeness, the optimal prediction framework is first derived in the context of the Galerkin POD ROM. It is emphasized that the content presented in Sections~\\ref{sec:liouville} and \\ref{sec:projOpsLangevin} is simply a formulation of Chorin's framework, with a specific projection operator, in the context of the Galerkin POD ROM. \n\n\nWe pursue the MZ approach on a Galerkin formulation of Eq.~\\ref{eq:FOM_generalized}. Before describing the formalism, it is beneficial to re-write the original FOM in terms of the generalized coordinates with the solution being defined implicitly as a function of the initial conditions,\n\\begin{equation}\\label{eq:FOM_generalized_b}\n\\frac{d }{dt} \\mathbf{a} (\\mathbf{a}_0,t) =  {\\mathbf{V}}^T \\mathbf{R}({\\mathbf{V}} \\mathbf{a} (\\mathbf{a}_0,t)), \\qquad \\mathbf{a}(0) = \\mathbf{a}_0, \\qquad t \\in [0,T],\n\\end{equation}\nwith $\\mathbf{a}: \\mathbb{R}^N \\times [0,T] \\rightarrow \\mathbb{R}^N$, $\\mathbf{a} \\in \\RR{N} \\otimes \\RRC{N} \\otimes \\MC{T}$ the time-dependent generalized coordinates, $\\RRC{N}$ the space of (sufficiently smooth) functions acting on $\\RR{N}$, $\\MC{T}$ the space of (sufficiently smooth) functions acting on $[0,T]$, and $\\mathbf{a}_0 \\in \\mathbb{R}^N$ the initial conditions. Here, $\\mathbf{a}(\\mathbf{a}_0,t)$ is viewed as a function that maps from the coordinates $\\mathbf{a}_0$ (i.e., the initial conditions) and time to a vector in $\\mathbb{R}^N$. It is assumed that the right-hand side operator $\\mathbf{R}$ is continuously differentiable on $\\mathbb{R}^N$.\n\\subsubsection{The Liouville Equation}\\label{sec:liouville}\nThe starting point of the MZ approach is to transform the non-linear FOM (Eq.~\\ref{eq:FOM_generalized_b}) into a linear partial differential equation. \nEquation~\\ref{eq:FOM_generalized_b} can be written equivalently as the following partial differential equation in $\\mathbb{R}^N \\times [0,T]$~\\cite{ChorinOptimalPredictionMemory},\n\\begin{equation}\\label{eq:Liouville}\n\\frac{\\partial }{\\partial t}v(\\mathbf{a}_0,t) = \\mathcal{L} v(\\mathbf{a}_0,t); \\qquad\nv(\\mathbf{a_0},0) = g(\\mathbf{a}_0),\n\\end{equation}\nwhere $v:  \\mathbb{R}^N  \\times [0,T] \\rightarrow \\mathbb{R}^{N_v}$ with $v \\in \\RR{N_v} \\otimes \\RRC{N} \\otimes \\MC{T}$ is a set of $N_v$ observables and $g: \\mathbb{R}^N  \\rightarrow \\mathbb{R}^{N_v}$ is a state-to-observable map. The operator $\\mathcal{L}$ is the Liouville operator, also known as the Lie derivative, and is defined by,\n\\begin{align*}\n\\mathcal{L} &: \\mathbf{q} \\mapsto \\bigg[ \\frac{\\partial  }{\\partial \\mathbf{a_0}} \\mathbf{q} \\bigg] {\\mathbf{V}}^T \\mathbf{R}( {\\mathbf{V}} \\mathbf{a_0} ),\\\\\n       &: \\RR{q} \\otimes \\RRC{N} \\rightarrow \\RR{q} \\otimes \\RRC{N},\n\\end{align*}\nfor arbitrary q.\nEquation~\\ref{eq:Liouville} is referred to as the Liouville equation and is an exact statement of the original dynamics. The Liouville equation describes the solution to Eq.~\\ref{eq:FOM_generalized_b} for \\textit{all} possible initial conditions. The advantage of reformulating the system in this way is that the Liouville equation is linear, allowing for the use of superposition and aiding in the removal of the fine-scales.\n\nThe solution to Eq.~\\ref{eq:Liouville} can be written as,\n\\begin{equation*}\nv(\\mathbf{a_0},t) = e^{t \\mathcal{L}}  g(\\mathbf{a_0}) .\n\\end{equation*}\nThe operator $e^{t \\mathcal{L}}$, which has been referred to as a ``propagator\", evolves the solution along its trajectory in phase-space~\\cite{ZwanzigBook}. The operator $e^{t \\mathcal{L}}$ has several interesting properties. Most notably, the operator can be ``pulled\" inside of a non-linear functional~\\cite{ZwanzigBook},\n\\begin{equation*}\ne^{t \\mathcal{L}}  g(\\mathbf{a}_0) =   g( e^{t \\mathcal{L}} \\mathbf{a}_0).\n\\end{equation*}\nThis is similar to the composition property inherent to Koopman operators~\\cite{Koopman}. With this property, the solution to Eq.~\\ref{eq:Liouville} may be written as,\n\\begin{equation*}\nv(\\mathbf{a_0},t) =  g( e^{t \\mathcal{L}}  \\mathbf{a}_0).\n\\end{equation*\nThe implications of $e^{t \\mathcal{L}}$ are significant. It demonstrates that, given trajectories $\\mathbf{a}(\\mathbf{a_0},t)$, the solution $v$ is known for any observable $g$. \n\nNoting that $\\mathcal{L}$ and $e^{t \\mathcal{L}}$ commute, Eq.~\\ref{eq:Liouville} may be written as,\n\\begin{equation*}\n\\frac{\\partial }{\\partial t} v(\\mathbf{a_0},t) = e^{t \\mathcal{L}}  \\mathcal{L} v(\\mathbf{a_0},0).\n\\end{equation*}\nA set of partial differential equations for the resolved generalized coordinates can be obtained by taking $g(\\mathbf{a_0}) = \\tilde{\\mathbf{a}}_0$,\n\\begin{equation}\\label{eq:Liouville_sg_res}\n\\frac{\\partial }{\\partial t}  e^{t \\mathcal{L}} \\tilde{\\mathbf{a}}_0 = e^{t \\mathcal{L}}  \\mathcal{L}  \\tilde{\\mathbf{a}}_0.\n\\end{equation}\nThe remainder of the derivation is performed for $g(\\mathbf{a_0}) = \\tilde{\\mathbf{a}}_0$, thus $N_v = K$. \n\\subsubsection{Projection Operators and the Generalized Langevin Equation}\\label{sec:projOpsLangevin}\nThe objective now is to remove the dependence of Eq.~\\ref{eq:Liouville_sg_res} on the fine-scale variables.\nSimilar to the VMS decomposition, $\\RRC{N}$ can be decomposed into resolved and unresolved subspaces,\n\\begin{equation*}\n\\RRC{N} = \\tilde{\\MC{H}} \\oplus \\MC{H}',\n\\end{equation*} \nwith $\\tilde{\\MC{H}}$ being the space of all functions of the resolved coordinates, $\\tilde{\\mathbf{a}}_0$, and $\\MC{H}'$ the complementary space.\nThe associated projection operators are defined as $\\mathcal{P}: \\RRC{N} \\rightarrow \\tilde{\\MC{H}}$ and $\\mathcal{Q} = I - \\mathcal{P}$. Various types of projections are possible, and here we consider,\n\\begin{equation*}\n\\mathcal{P}f( \\mathbf{a}_0 ) = \\int_{\\RR{N}} f( \\mathbf{a}_0 ) \\delta({\\mathbf{a}^{\\prime}}_0) d {\\mathbf{a}^{\\prime}}_0,\n\\end{equation*}\nwhich leads to\n\\begin{equation*}\n\\mathcal{P}f(\\mathbf{a}_0 )= f([\\tilde{\\mathbf{a}}_0;\\mathbf{0}]).\n\\end{equation*}\n\nThe projection operators can be used to split the Liouville equation,\n\\begin{equation}\\label{eq:Liouville_sg_split}\n\\frac{\\partial }{\\partial t}  e^{t \\mathcal{L}} \\tilde{\\mathbf{a}}_0 = e^{t \\mathcal{L}}  \\mathcal{PL} \\tilde{\\mathbf{a}}_0 +  e^{t \\mathcal{L}}\\mathcal{QL}\\tilde{\\mathbf{a}}_0.\n\\end{equation}\nThe objective now is to remove the dependence of the right-hand side of Eq.~\\ref{eq:Liouville_sg_split} on the fine-scales, ${\\mathbf{a}_0^{\\prime}}$ (i.e. $\\mathcal{QL}\\tilde{\\mathbf{a}}_0$). This may be achieved by Duhamel's principle,\n\\begin{equation}\\label{eq:duhamel}\ne^{t \\mathcal{L}} = e^{t \\mathcal{Q} \\mathcal{L}} + \\int_0^t e^{(t - s)\\mathcal{L}} \\mathcal{P}\\mathcal{L} e^{s \\mathcal{Q} \\mathcal{L}} ds.\n\\end{equation}\nInserting Eq.~\\ref{eq:duhamel} into Eq.~\\ref{eq:Liouville_sg_split}, the generalized Langevin equation is obtained,\n\\begin{equation}\\label{eq:MZ_Identity}\n\\frac{\\partial }{\\partial t}  e^{t \\mathcal{L}} \\tilde{\\mathbf{a}}_0 =   \\underbrace{e^{t\\mathcal{L}}\\mathcal{PL} \\tilde{\\mathbf{a}}_0}_{\\text{Markovian}} +   \\underbrace{e^{t\\mathcal{QL}}\\mathcal{QL}  \\tilde{\\mathbf{a}}_0}_{\\text{Noise}} + \n \\underbrace{ \\int_0^t e^{{(t - s)}\\mathcal{L}} \\mathcal{P}\\mathcal{L} e^{s \\mathcal{Q} \\mathcal{L}} \\mathcal{QL}\\tilde{\\mathbf{a}}_0 ds}_{\\text{Memory}}.\n\\end{equation}\nBy the definition of the initial conditions (Eq.~\\ref{eq:ROM_IC}), the noise-term is zero and we obtain,\n\\begin{equation}\\label{eq:MZ_Identity3}\n\\frac{\\partial}{\\partial t}    e^{t \\mathcal{L}} \\tilde{\\mathbf{a}}_0 = e^{t\\mathcal{L}}\\mathcal{PL}  \\tilde{\\mathbf{a}}_0+   \\int_0^t e^{{(t - s)}\\mathcal{L}} \\mathcal{P}\\mathcal{L} e^{s \\mathcal{Q} \\mathcal{L}} \\mathcal{QL}  \\tilde{\\mathbf{a}}_0 ds.\n\\end{equation}\nThe system described in Eq.~\\ref{eq:MZ_Identity} is precise and not an approximation to the original ODE system. For notational purposes, define,\n\\begin{equation}\\label{eq:kerndef}\n\\mathbf{K}(\\tilde{\\mathbf{a}}_0,t) \\equiv \\mathcal{PL}e^{t\\mathcal{QL}}\\mathcal{QL}\\tilde{\\mathbf{a}}_0.\n\\end{equation}\nThe term $\\mathbf{K}: \\mathbb{R}^K \\times [0,T] \\rightarrow \\mathbb{R}^K$ with $\\mathbf{K} \\in \\RR{K} \\otimes  \\tilde{\\MC{H}} \\otimes \\MC{T}$ is referred to as the memory kernel. \n\nUsing the identity $e^{t \\mathcal{L}} \\mathcal{PL}\\tilde{\\mathbf{a}}_0 = \\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{u}}(t))$ and Definition~\\eqref{eq:kerndef}, Equation~\\ref{eq:MZ_Identity3} can be written in a more transparent form,\n\\begin{equation}\\label{eq:MZ_Identity_VMS}\n\\tilde{\\mathbf{V}}^T \\bigg( \\frac{\\partial }{ \\partial t}\\tilde{\\mathbf{u}}(t) - \\mathbf{R}(\\tilde{\\mathbf{u}}(t)) \\bigg) = \\int_0^t \\mathbf{K}(\\tilde{\\mathbf{a}}(t-s),s) ds,\n\\end{equation}\nNote that the time derivative is represented as a partial derivative due to the Liouville operators embedded in the memory.\n\nThe derivation up to this point has cast the original full-order model in generalized coordinates (Eq.~\\ref{eq:FOM_generalized_b}) as a linear PDE. Through the use of projection operators and Duhamel's principle, an \\textit{exact} equation (Eq.~\\ref{eq:MZ_Identity_VMS}) for the coarse-scale dynamics \\textit{only in terms of the coarse-scale variables} was then derived. The effect of the fine-scales on the coarse-scales appeared as a memory integral. This memory integral may be thought of as the closure term that is required to exactly account for the unresolved dynamics.\n \n\\subsubsection{The $\\tau$-model and the Adjoint Petrov--Galerkin Method}\\label{sec:tau-model}\nThe direct evaluation of the memory term in Eq.~\\ref{eq:MZ_Identity_VMS} is, in general, computationally intractable. To gain a reduction in computational cost, an approximation to the memory must be devised. A variety of such approximations exist, and here we outline the $\\tau$-model~\\cite{parish_dtau,BarberThesis}. The $\\tau$-model can be interpreted as the result of assuming that the memory is driven to zero in finite time and approximating the integral with a quadrature rule. This can be written as a two-step approximation,\n\\begin{equation*}\n\\int^t_0 \\mathbf{K}(\\tilde{\\mathbf{a}}(t-s),s) ds \\approx \\int^t_{t-\\tau} \\mathbf{K}(\\tilde{\\mathbf{a}}(t-s),s) ds \\approx \\tau \\mathbf{K}(\\tilde{\\mathbf{a}}(t),0).\n\\end{equation*}\nHere, $\\tau \\in \\RR{}$ is a stabilization parameter that is sometimes referred to as the ``memory length.\" It is typically static and user-defined, though methods of dynamically calculating it have been developed in \\cite{parish_dtau}. The \\textit{a priori} selection of $\\tau$ and sensitivity of the model output to this selection are discussed later in this manuscript. \n\nThe term $\\mathbf{K}(\\tilde{\\mathbf{a}}(t),0)$ can be shown to be~\\cite{parishVMS},\n\\begin{equation*}\n\\mathbf{K}(\\tilde{\\mathbf{a}}(t),0) =  \\tilde{\\mathbf{V}}^T \\mathbf{J}[\\tilde{\\mathbf{u}}(t)]  {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}(t)),\n\\end{equation*}\nwhere ${\\Pi^{\\prime}}$ is the ``orthogonal projection operator,\" defined as ${\\Pi^{\\prime}} \\equiv \\big(\\mathbf{I}-\\tilde{\\mathbf{V}} \\tilde{\\mathbf{V}}^T\\big)$. We define the corresponding coarse-scale projection operator as $\\tilde{\\Pi} \\equiv \\tilde{\\mathbf{V}} \\tilde{\\mathbf{V}}^T$. The coarse-scale equation with the $\\tau$-model reads,\n\\begin{equation}\\label{eq:MZ_coarse_tau_NL}\n\\tilde{\\mathbf{V}}^T \\bigg( \\frac{d }{dt}\\tilde{\\mathbf{u}}(t) - \\mathbf{R}(\\tilde{\\mathbf{u}}(t)) \\bigg) = \\tau \\tilde{\\mathbf{V}}^T \\mathbf{J}[\\tilde{\\mathbf{u}}(t)]  {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}(t)).\n\\end{equation}\nEquation~\\ref{eq:MZ_coarse_tau_NL} provides a closed equation for the evolution of the coarse-scales. The left-hand side of Eq.~\\ref{eq:MZ_coarse_tau_NL} is the standard Galerkin ROM, and the right-hand side can be viewed as a subgrid-scale model. \n\nWhen compared to existing methods, the inclusion of the $\\tau$-model leads to a method that is analogous to a non-linear formulation of the \\textit{adjoint} stabilization technique developed in the finite element community. The ``adjoint\" terminology arises from writing Eq.~\\ref{eq:MZ_coarse_tau_NL} in a Petrov--Galerkin form,\n\\begin{equation}\\label{eq:adjoint_Galerkin}\n \\bigg[ \\bigg( \\mathbf{I} + \\tau {\\Pi^{\\prime}}^T \\mathbf{J}^T[\\tilde{\\mathbf{u}}(t)]\\bigg)  \\tilde{\\mathbf{V}} \\bigg]^T  \\bigg( \\frac{d }{dt}\\tilde{\\mathbf{u}}(t) - \\mathbf{R}(\\tilde{\\mathbf{u}}(t)) \\bigg) = \\mathbf{0}.\n\\end{equation}\nIt is seen that Eq.~\\ref{eq:adjoint_Galerkin} involves taking the inner product of the coarse-scale ODE with a test-basis that contains the adjoint of the coarse-scale Jacobian. Unlike GLS stabilization, adjoint stabilization can be derived from the multiscale equations~\\cite{hughes0}. Due to the similarity of the proposed method with adjoint stabilization techniques, as well as the LSPG terminology, the complete ROM formulation will be referred to as the Adjoint Petrov--Galerkin (APG) method. \n\n\\subsubsection{Comparison of APG and LSPG}\nThe APG method displays similarities to LSPG. From Eq.~\\ref{eq:adjoint_Galerkin}, it is seen that the test basis for the APG ROM is given by,\n\\begin{equation}\\label{eq:MZ_testbasis}\n\\tilde{\\mathbf{W}}_{A}  = \\bigg( \\mathbf{I} + \\tau {\\Pi^{\\prime}}^T \\mathbf{J}^T[\\tilde{\\mathbf{u}}]\\bigg)  \\tilde{\\mathbf{V}} .\n\\end{equation}\nRecall the LSPG test basis for backward differentiation schemes,\n\\begin{equation}\\label{eq:LSPG_testbasis}\n\\tilde{\\mathbf{W}}_{LSPG} = \\big( \\mathbf{I} - \\alpha \\Delta t \\mathbf{J}[\\tilde{\\mathbf{u}}] \\big) \\tilde{\\mathbf{V}}.\n\\end{equation}\nComparing Eq.~\\ref{eq:LSPG_testbasis} to Eq.~\\ref{eq:MZ_testbasis}, we can draw several interesting comparisons between the LSPG and APG method. Both contain a time-scale: $\\tau$ for APG and $\\alpha \\Delta t$ for LSPG. Both include Jacobians of the non-linear function $\\mathbf{R}(\\tilde{\\mathbf{u}})$. The two methods differ in the presence of the orthogonal projection operator in APG, a transpose on the Jacobian, and a sign discrepancy on the Jacobian. These last two differences are consistent with the discrepancies between GLS and adjoint stabilization methods used in the finite element community. See, for instance, Eqs. 71 and 73 in Ref~\\cite{hughes0}.\n\n\n\n\n\\section{Analysis}\\label{sec:analysis}\nThis section presents theoretical analyses of the Adjoint Petrov--Galerkin method. Specifically, error and eigenvalue analyses are undertaken for linear time-invariant (LTI) systems.\nSection~\\ref{sec:error_bound} derives \\textit{a priori} error bounds for the Galerkin and Adjoint Petrov--Galerkin ROMs. Conditions under which the APG ROM may be more accurate than the Galerkin ROM are discussed. Section~\\ref{sec:selecttau} outlines the selection of the parameter $\\tau$ that appears in APG. \n\\subsection{A Priori Error Bounds}\\label{sec:error_bound}\nWe now derive \\textit{a priori} error bounds for the Galerkin and Adjoint Petrov--Galerkin method for LTI systems. Define ${\\mathbf{u}}_F$ to be the solution to the FOM, $\\tilde{\\mathbf{u}}_G$ to be the solution to the Galerkin ROM, and $\\tilde{\\mathbf{u}}_A$ the solution to the Adjoint Petrov--Galerkin ROM. The full-order solution, Galerkin ROM, and Adjoint Petrov--Galerkin ROMs obey the following dynamical systems,\n\\begin{equation}\\label{eq:fom_error}\n\\frac{d }{dt}{\\mathbf{u}}_F(t) =  \\mathbf{R}({\\mathbf{u}}_F(t)), \\qquad {\\mathbf{u}}_F(0) = {\\mathbf{u}}_0,\n\\end{equation}\n\\begin{equation}\\label{eq:grom_error}\n\\frac{d }{dt}\\tilde{\\mathbf{u}}_G(t) = \\mathbb{P}_G \\mathbf{R}(\\tilde{\\mathbf{u}}_G(t)), \\qquad {\\mathbf{u}}_G(0) = {\\mathbf{u}}_0,\n\\end{equation}\n\\begin{equation}\\label{eq:ag_error}\n\\frac{d }{dt}\\tilde{\\mathbf{u}}_A(t) = \\mathbb{P}_{A} \\mathbf{R}(\\tilde{\\mathbf{u}}_A(t)), \\qquad {\\mathbf{u}}_A(0) = {\\mathbf{u}}_0,\n\\end{equation}\nwhere the Galerkin and Adjoint Petrov--Galerkin projections are, respectively,\n\\begin{equation*}\n\\mathbb{P}_G = \\tilde{\\Pi}, \\qquad \\mathbb{P}_A =  \\tilde{\\Pi} \\big[ \\mathbf{I} + \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}_A] {\\Pi^{\\prime}} \\big].    \n\\end{equation*}\nThe residual of the full-order model is defined as,\n\\begin{equation*}\n\\mathbf{r}_F : {\\mathbf{u}} \\mapsto \\frac{d {\\mathbf{u}}}{dt} - \\mathbf{R}({\\mathbf{u}}).\n\\end{equation*}\nWe define the error in the Galerkin and Adjoint Petrov--Galerkin method as,\n\\begin{equation*}\n\\mathbf{e}_G \\overset{\\Delta}{=} {\\mathbf{u}}_F - \\tilde{\\mathbf{u}}_G, \\qquad \\mathbf{e}_A \\trieq {\\mathbf{u}}_F - \\tilde{\\mathbf{u}}_A.   \n\\end{equation*}\nSimilarly, the coarse-scale error is defined as,\n\\begin{equation*}\n\\mathbf{\\tilde{e}}_G \\overset{\\Delta}{=} \\tilde{\\Pi} {\\mathbf{u}}_F - \\tilde{\\mathbf{u}}_G, \\qquad \\tilde{\\mathbf{e}}_A \\trieq \\tilde{\\Pi}  {\\mathbf{u}}_F - \\tilde{\\mathbf{u}}_A.   \n\\end{equation*}\nIn what follows, we assume Lipschitz continuity of the right-hand side function: there exists a constant $\\kappa > 0$ such that $\\forall \\mathbf{x},\\mathbf{y} \\in \\mathbb{R}^N$,\n\\begin{equation*}\n\\norm{ \\mathbf{R}(\\mathbf{x}) - \\mathbf{R}(\\mathbf{y})} \\le \\kappa \\norm{ \\mathbf{x} - \\mathbf{y}}. \n\\end{equation*}\nTo simplify the analysis, the Adjoint Petrov--Galerkin projection is approximated to be stationary in time. Note that the Galerkin projection is stationary in time.  For clarity, we suppress the temporal argument on the states when possible in the proofs.\n\\begin{theorem}\n\\textit{A priori} error bounds for the Galerkin and Adjoint Petrov--Galerkin ROMs are, respectively,\n\\begin{equation}\\label{eq:g_nlbound} \\norm{\\mathbf{e}_G(t)} \\le \\int_0^t e^{ \\norm{\\mathbf{P}_G} \\kappa s }\\norm{\\big[\\mathbf{I} - \\mathbb{P}_G \\big]  \\mathbf{R}( {\\mathbf{u}}_F(t-s) ) } ds .\\end{equation}\n\\begin{equation}\\label{eq:ag_nlbound} \\norm{\\mathbf{e}_A(t)} \\le \\int_0^t e^{\\norm{\\mathbf{P}_A} \\kappa s }\\norm{\\big[\\mathbf{I} - \\mathbb{P}_A \\big]  \\mathbf{R}({\\mathbf{u}}_F(t-s))} ds .\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nWe prove only Eq.~\\ref{eq:ag_nlbound} as Eq.~\\ref{eq:g_nlbound} is obtained through the same arguments. Following~\\cite{carlberg_lspg_v_galerkin}, start by subtracting Eq.~\\ref{eq:ag_error} from Eq.~\\ref{eq:fom_error}, and adding and subtracting $\\mathbb{P}_A\\mathbf{R} ( {\\mathbf{u}}_F$),\n\\begin{equation*}\\label{eq:ea_lti_1}\n\\frac{d \\mathbf{e}_A}{dt} = \\mathbf{R} ({\\mathbf{u}}_F) + \\mathbb{P}_A \\mathbf{R} ({\\mathbf{u}}_F) - \\mathbb{P}_A \\mathbf{R} ({\\mathbf{u}}_F)  - \\mathbb{P}_A \\mathbf{R}(\\tilde{\\mathbf{u}}_G), \\qquad \\mathbf{e}_A(0) = \\mathbf{0}.\n\\end{equation*}\nTaking the $L^2$-norm,\n\\begin{equation*}\\label{eq:ea_lti_2}\n\\norm{ \\frac{d \\mathbf{e}_A}{dt} } = \\norm{ \\mathbf{R} ({\\mathbf{u}}_F) + \\mathbb{P}_A \\mathbf{R} ({\\mathbf{u}}_F) - \\mathbb{P}_A \\mathbf{R} ({\\mathbf{u}}_F)  - \\mathbb{P}_A \\mathbf{R}(\\tilde{\\mathbf{u}}_G)}.\n\\end{equation*}\nApplying the triangle inequality,\n\\begin{equation*}\\label{eq:ea_lti_3}\n\\norm{ \\frac{d \\mathbf{e}_A}{dt} }\\le \\norm{\\big[\\mathbf{I} - \\mathbb{P}_A \\big]  \\mathbf{R}( {\\mathbf{u}}_F) }  + \\norm{  \\mathbb{P}_A \\big( \\mathbf{R} {\\mathbf{u}}_F)    -  \\mathbf{R}(\\tilde{\\mathbf{u}}_G) \\big) } .\n\\end{equation*}\nInvoking the assumption of Lipschitz continuity,\n\\begin{equation*}\\label{eq:ea1}\n\\norm{ \\frac{d \\mathbf{e}_A}{dt} }\\le \\norm{\\big[\\mathbf{I} - \\mathbb{P}_A \\big]  \\mathbf{R}( {\\mathbf{u}}_F)}  +\\norm{ \\mathbb{P}_A} \\kappa  \\norm{\\mathbf{e}_A}.\n\\end{equation*}\nNoting that $\\frac{d \\norm{ \\mathbf{e}_A } }{dt} \\le \\norm{ \\frac{d \\mathbf{e}_A}{dt}}$ \nwe have\n\\footnote{ \n$$\\frac{d \\norm{ \\mathbf{e}_A } }{dt} = \\frac{1}{\\norm{\\mathbf{e}_A}} \\mathbf{e}_A^T \\frac{d \\mathbf{e}_A}{dt} \\le  \\norm{ \\frac{1}{\\norm{\\mathbf{e}_A}} \\mathbf{e}_A} \\norm{ \\frac{d \\mathbf{e}_A}{dt}} \\le \\norm{ \\frac{d \\mathbf{e}_A}{dt}}\n$$},\n\\begin{equation}\\label{eq:ea2}\n\\frac{d  \\norm{\\mathbf{e}_A }}{dt} \\le \\norm{\\big[\\mathbf{I} - \\mathbb{P}_A \\big]  \\mathbf{R}( {\\mathbf{u}}_F)}  +\\norm{ \\mathbb{P}_A} \\kappa  \\norm{\\mathbf{e}_A}.\n\\end{equation}\n\\begin{comment}\nNext, we show  $\\int_0^t \\norm{ \\frac{d \\mathbf{e}_A}{ds} }ds$ is bounded from below by $\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds }$. Expanding $\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds }^2$,\n\\begin{equation*}\n\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds }^2 = \\bigg[ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds  \\bigg]^T\\bigg[ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds  \\bigg] .\n\\end{equation*}\nPulling the first integral on the right-hand side into the second integral as it is independent of $s$,\n\\begin{equation*}\n\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds }^2 =  \\int_0^t  \\bigg[ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds  \\bigg]^T \\bigg[\\frac{d \\mathbf{e}_A}{ds} \\bigg] ds   \n\\end{equation*}\nUsing $\\mathbf{a}^T \\mathbf{b} \\le \\norm{\\mathbf{a}}\\norm{\\mathbf{b}}$,\n\\begin{equation*}\n\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds }^2\\le  \\int_0^t  \\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds  } \\norm{ \\frac{d \\mathbf{e}_A}{ds} } ds  .\n\\end{equation*}\nPulling the first term on the right-hand side outside of the integral,\n\\begin{equation*}\n\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds }^2 \\le  \\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds  }  \\int_0^t  \\norm{ \\frac{d \\mathbf{e}_A}{ds} } ds   \n\\end{equation*}\nDividing through by $\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds  }$ gives,\n\\begin{equation*}\n\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds }  \\le \\int_0^t \\norm{ \\frac{d \\mathbf{e}_A}{ds} } ds.\n\\end{equation*}\nNext, it is noted that,\n\\begin{align*}\n\\norm{ \\int_0^t \\frac{d \\mathbf{e}_A}{ds}  ds } &=  \\norm{\\mathbf{e}_A(t) - \\mathbf{e}_A(0)} \\\\\n&=   \\norm{\\mathbf{e}_A(t)} , \\\\\n&= \\int_0^t  \\frac{d \\norm{\\mathbf{e}_A}} {ds}  ds .\n\\end{align*}\nTherefore,\n\\begin{equation}\\label{eq:norm_inequality}\n\\int_0^t  \\frac{d \\norm{\\mathbf{e}_A}} {ds}  ds  \\le \\int_0^t \\norm{ \\frac{d \\mathbf{e}_A}{ds} } ds.\n\\end{equation}\n\\end{comment}\nAn upper bound on the error for the Adjoint Petrov--Galerkin method is then obtained by solving Eq.~\\ref{eq:ea2} for $\\norm{\\mathbf{e}_A}$, which yields,\n\\begin{equation*}\n\\norm{ \\mathbf{e}_A(t) } \\le \\int_0^t e^{\\norm{ \\mathbb{P}_A } \\kappa s }\\norm{\\big[\\mathbf{I} - \\mathbb{P}_A \\big]  \\mathbf{R} ({\\mathbf{u}}_F(t-s))} ds .\n\\end{equation*}\n\n\n \\end{proof}\nNote that, for the Adjoint Petrov--Galerkin method, the error bound provided in Eq.~\\ref{eq:ag_nlbound} is not truly an \\textit{a priori} bound as  $\\mathbb{P}_A$ is a function of $\\tilde{\\mathbf{u}}_A$.\nEquation~\\ref{eq:g_nlbound} (and~\\ref{eq:ag_nlbound}) indicates an exponentially growing error and contains two distinct terms. The term $e^{\\norm{\\mathbb{P}_A} \\kappa s }$ indicates the exponential growth of the error in time.  The second term of interest is $\\norm{\\big[\\mathbf{I} - \\mathbb{P}_A \\big]  \\mathbf{R} ({\\mathbf{u}}_F(t))}$. This term corresponds to the error introduced at time $t$ due to projection.  It is important to note that the first term controls how the error will grow in time, while the second term controls how much error is added at a given time. \n\nUnfortunately, for general non-linear systems, \\textit{a priori} error analysis provides minimal insight beyond what was just mentioned. To obtain a more intuitive understanding of the APG method, error analysis in the case that $\\mathbf{R}(\\cdot)$ is a linear time-invariant operator is now considered. \n\n\\input{proofs}\n\n\\begin{comment}\n\\begin{corollary}\\label{cor:1}\nIf $\\norm{ \\mathbb{P}_A} \\le \\norm{\\mathbb{P}_G }$, then the upper bound on the error will accumulate at a slower rate in the Adjoint Petrov--Galerkin method than in the Galerkin method.\n\\end{corollary}\n\\begin{proof}\nIf $\\norm{ \\mathbb{P}_A} \\le \\norm{\\mathbb{P}_G }$ then $e^{\\norm{\\mathbb{P}_A} s} < e^{ \\norm{\\mathbb{P}_G} s}$ $\\forall s > 0$, and the desired result is obtained.\n\\end{proof}\nCorollary~\\ref{cor:1} shows that it is possible for the APG ROM to be more accurate than the G ROM. Theorem 2 demonstrates one possible situation in which this will occur.\n\n\\begin{theorem}\\label{theorem:symmetric}\nIf $ \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} $ is a diagonalizable symmetric matrix with negative eigenvalues, then \n$$ \\norm{ \\mathbb{P}_A} \\le \\norm{\\mathbb{P}_G }; \\qquad \\forall  \\tau \\in \\bigg[0, \\frac{2}{ \\rho( \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} )} \\bigg],$$\nand errors in the Adjoint Petrov--Galerkin method grow slower than in the Galerkin method. \\footnote{Note that the function $\\rho(\\cdot)$ indicates the spectral radius, not to be confused with the physical density $\\rho$ which appears later in this manuscript.}\n\\end{theorem}\n\n\\begin{proof}\nStart by defining an upper bound on $\\mathbb{P}_A$. It is helpful to note that by the orthonormality of $\\tilde{\\mathbf{V}}$,\n\\begin{equation}\\label{eq:picoarse_norm}\n\\norm{\\mathbb{P}_G} = \\norm{\\tilde{\\Pi}} = 1.\n\\end{equation}\nBy the sub-multiplicative property of the $L^2$-norm,\n$$\\norm{ \\mathbb{P}_A } \\le \\norm{\\tilde{\\Pi}} \\norm{\\big[ \\mathbf{I} + \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} \\big]}.$$\nBy Eq.~\\ref{eq:picoarse_norm},\n$$\\norm{ \\mathbb{P}_A }\\le  \\norm{\\big[ \\mathbf{I} + \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} \\big]}.$$\nInvoking the assumption that $ \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} $ is diagonalizable, the eigendecomposition can be written as,\n$$ \\mathbf{I} + \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} = \\mathbf{S}  (\\tau \\Lambda + \\mathbf{I})  \\mathbf{S}^{-1},$$\nwhere $\\Lambda$ is a diagonal matrix containing the $N-K$ non-zero eigenvalues of $\\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}}$. \nInvoking the assumption that $ \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} $ is symmetric, one has $\\norm{\\mathbf{S}}=\\norm{\\mathbf{S}^{-1}} = 1$, obtaining the upper bound,\n$$\\norm{ \\mathbb{P}_A }\\le   \\norm{ (\\tau \\Lambda + \\mathbf{I}) }.$$\nThe norm of a diagonal matrix is the maximum absolute value of its diagonal elements.\nKnowing that the eigenvalues of $\\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}}$ are all negative, this can be written as,\n$$\\norm{ \\mathbb{P}_A }\\le  |\\tau \\cdot \\underset{i}{min} (\\lambda_i) + 1|.$$\nNoting that $\\norm{ \\mathbb{P}_G } = 1$, we can enforce that $\\norm{ \\mathbb{P}_A } \\le \\norm{ \\mathbb{P}_G}$ via the inequality,\n$$ |\\tau \\cdot \\underset{i}{min} (\\lambda_i) + 1| \\leq 1 $$\nRecognizing that $\\underset{i}{min} (\\lambda_i) = - \\rho( \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} )$, we can solve for bounds on $\\tau$ to arrive at,\n$$\\norm{ \\mathbb{P}_A } \\le \\norm{ \\mathbb{P}_G};  \\qquad \\forall  \\tau \\in \\bigg[0, \\frac{2}{ \\rho( \\mathbf{J}[\\tilde{\\mathbf{u}}_A]{\\Pi^{\\prime}} ) } \\bigg] .$$\n\\end{proof}\n\n\n\n\nTheorem~\\ref{theorem:symmetric} is interesting. First, it provides theoretical results showing the conditions under which the Adjoint Petrov--Galerkin method will have a lower rate of error growth than the Galerkin method. Further, it provides valuable insight into the selection of the stabilization parameter $\\tau$. Specifically, it is an upper bound on $\\tau$ and shows that this bound varies inversely with the spectral radius of the Jacobian.\n\\end{comment}\n\n\\subsection{Selection of Memory Length $\\tau$}\\label{sec:selecttau}\nThe APG method requires the specification of the parameter $\\tau$. Theorem~\\ref{theorem:errorbound_symmetric} showed that, for a self-adjoint linear system, bounds on the value of $\\tau$ are related to the eigenvalues of the Jacobian of the full-dimensional right-hand side operator. While such bounds provide intuition into the behavior of $\\tau$, they are not particularly useful in the selection of an optimal value of $\\tau$ as they 1.) are conservative due to repeated use of inequalities and 2.) require the eigenvalues of the full right-hand side operator, which one does not have access to in a ROM. Further, the bounds were derived for a self-adjoint linear system, and the extension to non-linear systems is unclear. \n\nIn practice, it is desirable to obtain an expression for $\\tau$ using only the coarse-scale Jacobian, $\\tilde{\\mathbf{V}}^T \\mathbf{J}[\\tilde{\\mathbf{u}}]\\tilde{\\mathbf{V}}$.\nIn Ref.~\\cite{parishMZ1}, numerical evidence showed a strong correlation between the optimal value of $\\tau$ and this coarse-scale Jacobian. Based on this numerical evidence and the analysis in the previous section, the following heuristic for selecting $\\tau$ is used:\n\\begin{equation}\\label{eq:taueq}\n\\tau = \\frac{C}{\\rho (\\tilde{\\mathbf{V}}^T \\mathbf{J}[\\tilde{\\mathbf{u}}] \\tilde{\\mathbf{V}})},\n\\end{equation}\nwhere $C$ is a model parameter and $\\rho(\\cdot)$ indicates the spectral radius. In Ref.~\\cite{parishMZ1}, $C$ was reported to be $0.2.$  In the numerical experiments presented later in this manuscript, the sensitivity of APG to the value of $\\tau$ and the validity of Eq.~\\ref{eq:taueq} are examined. \n\nSimilar to the selection of $\\tau$ in the APG method, the LSPG method requires the selection of an appropriate time-step~\\cite{carlberg_lspg_v_galerkin}. In practice, this fact can be problematic as finding an optimal time-step for LSPG which minimizes error may result in a small time-step and, hence, an expensive simulation. The selection of the parameter $\\tau$, on the other hand, does not impact the computational cost of the APG ROM.\n\n\\section{Implementation and Computational Cost of the Adjoint Petrov--Galerkin Method}\\label{sec:cost}\n\nThis section details the implementation of the Adjoint Petrov--Galerkin ROM for simple time integration schemes. Algorithms for explicit and implicit time integration schemes are provided, and the approximate cost of each method in floating-point operations (FLOPs) is analyzed. Here, a FLOP refers to any floating-point addition or multiplication; no distinction is made between the computational cost of either operation. The notation used here is as follows: $N$ is the full-order number of degrees of freedom, $K$ is the number of modes retained in the POD basis, and $\\omega N$ is the number of FLOPs required for one evaluation of the right-hand side, $\\mathbf{R}(\\tilde{\\mathbf{u}}(t))$. For sufficiently complex problems, $\\omega$ is usually on the order of $\\mathcal{O}(10) < \\omega < \\mathcal{O}(1000)$. The analysis presented in this section does not consider hyper-reduction. The analysis can be approximately extended to hyper-reduction by replacing the full-order degrees of freedom with the dimension of the hyper-reduced right-hand side.\\footnote{An accurate cost-analysis for hyper-reduced ROMs should consider over sampling of the right-hand side and the FOM stencil.}\n\n\\subsection{Explicit Time Integration Schemes}\nThis section explores the cost of the APG method within the scope of explicit time integration schemes. For simplicity, the analysis is carried out only for the explicit Euler scheme. The computational cost of more sophisticated time integration methods, such as Runge-Kutta and multistep schemes, is generally a  proportional scaling of the cost of the explicit Euler scheme. Algorithm~\\ref{alg:alg_apg_exp} provides the step-by-step procedure for performing an explicit Euler update to the Adjoint Petrov--Galerkin ROM. Table~\\ref{tab:alg_apg_exp} provides the approximate floating-point operations for the steps reported in Algorithm~\\ref{alg:alg_apg_exp}. The algorithm for an explicit update to the Galerkin ROM, along with the associated FLOP counts, is provided in Algorithm~\\ref{alg:alg_g_exp} and Table~\\ref{tab:alg_g_exp} in Appendix~\\ref{appendix:algorithms}. As noted previously, LSPG reverts to the Galerkin method for explicit schemes, and so is not detailed in this section. Table~\\ref{tab:alg_apg_exp} shows that, in the case that $K \\ll N$ (standard for a ROM) and $\\omega \\gg 1$ (sufficiently complex right-hand side), the Adjoint Petrov--Galerkin ROM is approximately twice as expensive as the Galerkin ROM.\n\n\n\n\n\n\\begin{algorithm}\n\\caption{Algorithm for an explicit Euler update for the APG ROM}\n\\label{alg:alg_apg_exp}\nInput: $\\tilde{\\mathbf{a}}^n$\\;\n\\newline\nOutput: $\\tilde{\\mathbf{a}}^{n+1}$\\;\n\\newline\nSteps:\n\\begin{enumerate}\n\\item Compute the state from the generalized coordinates, $\\tilde{\\mathbf{u}}^n =\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^{n+1}$\n\\item Compute the right-hand side from the state, $\\mathbf{R}(\\tilde{\\mathbf{u}}^n)$\n\\item Compute the projection of the right-hand side, $\\tilde{\\Pi} \\mathbf{R}(\\tilde{\\mathbf{u}}^n) = \\tilde{\\mathbf{V}} \\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{u}}^n)$\n\\item Compute the orthogonal projection of the right-hand side, ${\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}^n) =  \\mathbf{R}(\\tilde{\\mathbf{u}}^n) -  \\tilde{\\Pi} \\mathbf{R}(\\tilde{\\mathbf{u}}^n)$\n\\item Compute the action of the Jacobian on ${\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}^n)$ using either of the two following strategies:\n    \\begin{enumerate}\n    \\item Finite difference approximation:\n    \\begin{equation*}\n    \\mathbf{J}[\\tilde{\\mathbf{u}}^n] {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}^n) \\approx \\frac{1}{\\epsilon} \\Big[ \\mathbf{R}\\big(\\tilde{\\mathbf{u}}^n + \\epsilon {\\Pi^{\\prime}}  \\mathbf{R}(\\tilde{\\mathbf{u}}^n) \\big) - \\mathbf{R}(\\tilde{\\mathbf{u}}^n  ) \\Big], \n    \\end{equation*}\n    where $\\epsilon$ is a small constant value, usually  $\\sim \\mathcal{O}(10^{-5})$.\n    \\item Exact linearization:\n    \\begin{equation*}\n    \\mathbf{J}[\\tilde{\\mathbf{u}}^n] {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}^n) = \\mathbf{R'}[\\tilde{\\mathbf{u}}^n]({\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}^n)),\n    \\end{equation*}\n    where $\\mathbf{R}'[\\tilde{\\mathbf{u}}^n]$ is right-hand side operator linearized about $\\tilde{\\mathbf{u}}^n$.\n    \\end{enumerate}\n\\item Compute the full right-hand side:  $\\mathbf{R}(\\tilde{\\mathbf{u}}^n) +  \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}^n] {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}^n)$\n\n\\item Project: $\\tilde{\\mathbf{V}}^T \\bigg[  \\mathbf{R}(\\tilde{\\mathbf{u}}^n) +  \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}^n] {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}^n) \\bigg]$\n\\item Update the state $\\tilde{\\mathbf{a}}^{n+1} = \\tilde{\\mathbf{a}}^n + \\Delta t \\tilde{\\mathbf{V}}^T \\bigg[  \\mathbf{R}(\\tilde{\\mathbf{u}}^n) +  \\tau\\mathbf{J}[\\tilde{\\mathbf{u}}^n] {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}^n) \\bigg]$\n\\end{enumerate}\n\\end{algorithm}\n\n\\begin{table}\n\\begin{tabular}{p{7cm} p{8cm}}\n\\hline\nStep in Algorithm~\\ref{alg:alg_apg_exp}& Approximate FLOPs \\\\\n\\hline\n1    & $2 N K - N$  \\\\\n2    & $\\omega N$  \\\\\n3    & $4 N K - N - K$  \\\\\n4    & $N $  \\\\\n5    & $(\\omega + 4)N  $  \\\\\n6    & $2N $  \\\\\n7    & $2NK - K $  \\\\\n8    & $2K $  \\\\\n\\hline\nTotal    & $8 N K + (2\\omega + 5) N$ \\\\\nTotal for Galerkin Method & $4 N K + (\\omega-1) N + K$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Approximate floating-point operations for an explicit Euler update to the Adjoint Petrov--Galerkin method reported in Algorithm~\\ref{alg:alg_apg_exp}. The total FLOP count for the Galerkin ROM with an explicit Euler update is additionally reported for comparison. A full description of the Galerkin update is provided in Appendix~\\ref{appendix:algorithms}.}\n\\label{tab:alg_apg_exp}\n\\end{table}\n\n\n\n\\subsection{Implicit Time Integration Schemes}\nThis section evaluates the computational cost of the Galerkin, Adjoint Petrov--Galerkin, and Least-Squares Petrov--Galerkin methods for implicit time integration schemes. For non-linear systems, implicit time integration schemes require the solution of a non-linear algebraic system at each time-step. Newton's method, along with a preferred linear solver, is typically employed to solve the system. For simplicity, the analysis provided in this section is carried out for the implicit Euler time integration scheme along with Newton's method to solve the non-linear system. Before proceeding, the full-order residual, Galerkin residual, and APG residual at time-step $(n + 1)$ are denoted as,\n\\begin{align*}\n &\\mathbf{r}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^{n+1}) = \\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^{n+1} - \\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^n - \\Delta t \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^{n+1}),\\\\\n&\\mathbf{r}_G(\\tilde{\\mathbf{a}}^{n+1}) = \\tilde{\\mathbf{a}}^{n+1} - \\tilde{\\mathbf{a}}^n - \\Delta t \\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^{n+1}),\\\\\n&\\mathbf{r}_{A}(\\tilde{\\mathbf{a}}^{n+1}) = \\tilde{\\mathbf{a}}^{n+1} - \\tilde{\\mathbf{a}}^n - \\Delta t \\tilde{\\mathbf{V}}^T\\bigg[\\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^{n+1}) + \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}]{\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^{n+1} ) \\bigg].\n\\end{align*}\nAs the future state, $\\tilde{\\mathbf{a}}^{n+1}$, is unknown, we denote an intermediate state, $\\tilde{\\mathbf{a}}_k$, that is updated after every Newton iteration until some convergence criterion is met.\nNewton's method is defined by the iteration,\n\\begin{equation}\\label{eq:newton_linear}\n    \\frac{\\partial \\mathbf{r}(\\tilde{\\mathbf{a}}_k)}{\\partial \\tilde{\\mathbf{a}}_k} \\big[ \\tilde{\\mathbf{a}}_{k+1} - \\tilde{\\mathbf{a}}_k\\big] = - \\mathbf{r}(\\tilde{\\mathbf{a}}_k).\n\\end{equation}\nNewton's method solves Eq.~\\ref{eq:newton_linear} for the change in the state, $\\tilde{\\mathbf{a}}_{k+1} - \\tilde{\\mathbf{a}}_k$, for $k = 1,2,\\hdots$, until the residual converges to a sufficiently-small number. For a ROM, both the assembly and solution of this linear system is the dominant cost of an implicit method.\n\nTwo methods are considered for the solution to the non-linear algebraic system arising from implicit time discretizations of the G and APG ROMs: Newton's method with direct Gaussian elimination and Jacobian-Free Newton-Krylov GMRES. The Gauss-Newton method with Gaussian elimination is considered for the solution to the least-squares problem arising in LSPG.\n\nAlgorithm~\\ref{alg:alg_apg_imp} provides the step-by-step procedures for performing an implicit Euler update to the Adjoint Petrov--Galerkin ROM with the use of Newton's method and Gaussian elimination. Table~\\ref{tab:alg_apg_imp} provides the approximate floating-point operations for the steps reported in these algorithms. Analogous results for the Galerkin and LSPG ROMs are reported in Algorithms~\\ref{alg:alg_g_imp} and~\\ref{alg:alg_LSPG}, and Tables~\\ref{tab:alg_g_imp} and~\\ref{tab:alg_LSPG} in Appendix~\\ref{appendix:algorithms}.\nIn the limit that $K \\ll N$ and $\\omega \\gg 1$, the total FLOP counts reported show that APG is twice as expensive as both the LSPG and Galerkin ROMs. It is observed that the dominant cost for all three methods lies in the computation of the low-dimensional residual Jacobian. Computation of the low-dimensional Jacobian requires $K$ evaluations of the unsteady residual. Depending on values of $\\omega$, $N$, and $K$, this step can consist of over $50\\%$ $(K \\ll N)$ of the CPU time.\\footnote{It is noted that the low-dimensional Jacobian can be computed in parallel.}\n\nTo avoid the cost of computing the low-dimensional Jacobian required in the linear solve at each Netwon step, the Galerkin and APG ROMs can make use of Jacobian-Free Netwon-Krylov (JFNK) methods to solve the linear system, opposed to direct methods such as Gaussian elimination. JFNK methods are iterative methods that allow one to circumvent the expense associated with computing the full low-dimensional Jacobian. Instead, JFNK methods only compute the \\textit{action} of the Jacobian on a vector at each iteration of the linear solve. This can drastically decrease the cost of the implicit solve. JFNK utilizing the Generalized Minimal Residual (GMRES) method~\\cite{gmres}, for example, is guaranteed to converge to the solution $\\tilde{\\mathbf{a}}_k$ in at most $K$ iterations. It takes $K$ residual evaluations just to form the Jacobian required for direct methods.\n\nThe LSPG method is formulated as a non-linear least-squares problem. The use of Jacobian-free methods to solve non-linear least-squares problems is significantly more challenging. The principle issue encountered in attempting to use Jacobian-free methods for such applications it that one requires the action of the \\textit{transpose} of the residual Jacobian on a vector. This quantity cannot be computed via a standard finite difference approximation or linearization. It is only recently that true Jacobian-free methods have been utilized for solving non-linear least-squares problems. In Ref~\\cite{nlls_JacobianFree}, for example, automatic differentiation is utilized to compute the action of the transposed Jacobian on a vector. Due to the challenges associated with Jacobian-free methods for non-linear least-squares problems, this method is not considered here as a solution technique for LSPG.\n\nAlgorithm~\\ref{alg:alg_apg_jfnk} and Table~\\ref{tab:alg_apg_jfnk} report the algorithm and FLOPs required for an implicit Euler update to APG using JFNK GMRES. The term $\\eta \\le K$ is the number of iterations needed for convergence of the GMRES solver at each Newton iteration. For a concise presentation, the same update for the Galerkin ROM is not presented. Figure~\\ref{fig:implicitcost} shows the ratio of the cost of the various implicit ROMs as compared to the Galerkin ROM solved with Gaussian elimination. The standard LSPG method is seen to be approximately the same cost of Galerkin, while APG is seen to be approximately $2$x the cost of Galerkin. The success of the JFNK methods depends on the number of GMRES iterations required for convergence. If $\\eta = K$, which is the maximum number of iterations required for GMRES, the cost of JFNK methods is seen to be the same as their direct-solve counterparts. For cases where JFNK converges at a rate of $\\eta < K,$ the iterative methods out-perform their direct-solve counterparts.\n\nThe analysis presented here shows that, for a given basis dimension, the Adjoint Petrov--Galerkin ROM is approximately twice the cost of the Galerkin ROM for both implicit and explicit solvers. In the implicit case, the APG ROM utilizing a direct linear solver is approximately 2x the cost of LSPG. It was highlighted, however, that APG can be solved via JFNK methods. For cases where one either doesn't have access to the full Jacobian, or the full Jacobian can't be stored, JFNK methods can significantly decrease the ROM cost. The use of JFNK methods within the LSPG approach is more challenging due to the presence of the transpose of the residual Jacobian. Lastly it is noted that, although hyper-reduction can decrease the cost of a residual evaluation, it does not entirely alleviate the cost of forming the Jacobian.\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.65\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{implicit_cost.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Estimates of the G, APG, and LSPG reduced-order models for an implicit Euler update. This plot is generated for values of $N=1000$ and $\\omega = 50$, and $\\eta = \\{K,K\/2,K\/5\\},$ where $\\eta$ is the total number of iterations required for the GMRES solver at each Newton step.}\n\\label{fig:implicitcost}\n\\end{figure}\n\n\n\n\n\n\\begin{comment} \n\nLSPG, on the other hand, cannot leverage these methods as the full residual Jacobian must be formed in computing the test basis. Unless hyper-reduction methods are implemented, this fact severely limits the practical application LSPG to problems with many degrees of freedom.\n\nThis linear system solve is generally the dominant cost of the implicit method, and many methods are available for computing the solution are available. In calculating the computational cost for this step, we use Gaussian elimination, which is the simplest but most expensive method (and is also potentially unstable).\n\nAlgorithms~\\ref{alg:alg_g_imp},~\\ref{alg:alg_apg_imp}, and~\\ref{alg:alg_LSPG} provide step-by-step procedures for performing an implicit Euler update to the Galerkin, Adjoint Petrov--Galerkin, and Least-Squares Petrov--Galerkin ROMs, respectively. Tables~\\ref{tab:alg_g_imp},~\\ref{tab:alg_apg_imp}, and~\\ref{tab:alg_LSPG} provide the approximate floating-point operations for the steps reported these algorithms. \n\nThe analysis reveals a couple key points in comparing the three ROM methods. Again, we see that in the case of $N \\gg K$ and $\\omega \\gg 1$, the cost of the implicit APG ROM is roughly double the cost of implicit Galerkin ROM. Additionally, we see that the cost of LSPG suffers from the fact that the original problem is posed as a minimization of the full-order residual. This results in the evaluation of an $N\\times K$ residual Jacobian. In the case of $N \\gg K$, as is generally the case for most ROMs, this cost far outpaces the extra cost incurred by computing the modified right-hand side of the APG method. In general, note that the computational cost of all methods is dominated by computing the residual Jacobian. In fact, for $N \\gg K$, it is likely that computing this Jacobian is even more expensive than computing $\\Delta \\tilde{\\mathbf{a}}_k$ via Gaussian elimination. This leads to another critical point, as we scrutinize what linear system is solved to compute the Newton update $\\Delta \\tilde{\\mathbf{a}}$ in each method. For a residual Jacobian $\\mathbf{J}_k$, the linear system for the Galerkin and APG methods takes the form,\n\\begin{equation}\n    \\mathbf{J}_k \\Delta \\tilde{\\mathbf{a}} = - \\mathbf{r}(\\tilde{\\mathbf{a}}_k),\n\\end{equation}\nwhile the LSPG linear system takes the form,\n\\begin{equation}\n    \\big[ \\tilde{\\mathbf{V}}^T \\mathbf{J}_k^T \\mathbf{J}_k \\tilde{\\mathbf{V}}] \\Delta \\tilde{\\mathbf{a}}_k = - \\tilde{\\mathbf{V}}^T \\mathbf{J}_k^T \\mathbf{r}(\\tilde{\\mathbf{u}}_k)\n\\end{equation}\nThat is, the APG\/Galerkin methods result in a strict Newton's method form. Instead of first computing the residual Jacobian and then solving a generic linear system, indirect Jacobian-free Newton-Krylov methods may be used to solve the system without the explicit computation of the residual Jacobian. The popular Generalized Minimal Residual (GMRES) method, for example, is guaranteed to converge to the solution $\\tilde{\\mathbf{a}}_k$ in $K$ iterations, although the residual Jacobian is never actually generated. LSPG, on the other hand, cannot leverage these methods as the full residual Jacobian must be formed in computing the test basis. Unless hyper-reduction methods are implemented, this fact severely limits the practical application LSPG to problems with many degrees of freedom.\n\\end{comment} \n\n\n\n\n\n\n\\begin{algorithm}\n\\caption{Algorithm for an implicit Euler update for the APG ROM using Newton's Method with Gaussian Elimination}\n\\label{alg:alg_apg_imp}\nInput: $\\tilde{\\mathbf{a}}^n$, residual tolerance $\\xi$ \\;\n\\newline\nOutput: $\\tilde{\\mathbf{a}}^{n+1}$\\;\n\\newline\nSteps:\n\\begin{enumerate}\n\\item Set initial guess, $\\tilde{\\mathbf{a}}_k$\n\\item  Loop while $\\mathbf{r}^k > \\xi$\n\\begin{enumerate}\n    \\item Compute the state from the generalized coordinates, $\\tilde{\\mathbf{u}}_k = \\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k$\n    \\item Compute the right-hand side from the full state, $\\mathbf{R}(\\tilde{\\mathbf{u}}_k)$\n    \\item Compute the projection of the right-hand side, $\\tilde{\\Pi} \\mathbf{R}(\\tilde{\\mathbf{u}}^n) = \\tilde{\\mathbf{V}} \\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{u}}^n)$\n    \\item Compute the orthogonal projection of the right-hand side, ${\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}_k) =  \\mathbf{R}(\\tilde{\\mathbf{u}}_k) -  \\tilde{\\Pi} \\mathbf{R}(\\tilde{\\mathbf{u}}_k)$\n    \\item Compute the action of the right-hand side Jacobian on ${\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}_k)$, as in Alg.~\\ref{alg:alg_apg_exp}.\n    \\item Compute the modified right-hand side, $ \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k) + \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}]{\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k )$\n    \\item Project the modified right-hand side,  $\\tilde{\\mathbf{V}}^T\\Big[\\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k) + \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}]{\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k ) \\Big]$\n    \\item Compute the APG residual, $\\mathbf{r}_A(\\tilde{\\mathbf{a}}_k) = \\tilde{\\mathbf{a}}_k - \\tilde{\\mathbf{a}}^n - \\Delta t \\tilde{\\mathbf{V}}^T\\Big[\\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k) + \\tau \\mathbf{J}[\\tilde{\\mathbf{u}}]{\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k ) \\Big]$\n    \\item Compute the residual Jacobian, $\\frac{\\partial \\mathbf{r}(\\tilde{\\mathbf{a}}_k)}{\\partial \\tilde{\\mathbf{a}}_k}$\n    \\item Solve the linear system via Gaussian Elimination: $\\frac{\\partial \\mathbf{r}(\\tilde{\\mathbf{a}}_k)}{\\partial \\tilde{\\mathbf{a}}_k} \\Delta  \\tilde{\\mathbf{a}} = - \\mathbf{r}(\\tilde{\\mathbf{a}}_k)$\n    \\item Update the state: $\\tilde{\\mathbf{a}}_{k+1} = \\tilde{\\mathbf{a}}_k + \\Delta \\tilde{\\mathbf{a}}$\n    \\item $k = k + 1$\n\\end{enumerate}\n\\item Set final state, $\\tilde{\\mathbf{a}}^{n+1} = \\tilde{\\mathbf{a}}_k$\n\\end{enumerate}\n\\end{algorithm}\n\n\n\\begin{table}[]\n\\centering\n\\begin{tabular}{p{7cm} p{8cm}}\n\\hline\nStep in Algorithm~\\ref{alg:alg_apg_imp}& Approximate FLOPs \\\\\n\\hline\n2a    & $2 N K - N $ \\\\\n2b    & $ \\omega N $ \\\\\n2c    & $4 N K - N - K$  \\\\\n2d    & $ N $  \\\\\n2e    & $ (\\omega + 4) N $ \\\\\n2f    & $ 2N $ \\\\\n2g    & $ 2NK - K $ \\\\\n2h    & $ 3K $ \\\\\n2i    & $ (2\\omega + 5) NK + K^2 + 8NK^2 $ \\\\\n2j    & $ K^3 $ \\\\\n2k    & $ K $ \\\\\n\\hline\nTotal & $ (2\\omega + 5)N + 2K + (2\\omega + 13) NK + K^2 + 8NK^2 + K^3 $ \\\\\nGalerkin ROM FLOP count & $ (\\omega - 1)N + 3K + (\\omega + 3)NK + 2K^2 + 4NK^2 + K^3 $ \\\\\nLSPG ROM FLOP count& $ (\\omega + 2)N + (\\omega + 6) NK - K^2 + 4NK^2 + K^3 $ \\\\\n\n\\end{tabular}\n\\caption{Approximate floating-point operations for one Newton iteration for the implicit Euler update to the Adjoint Petrov--Galerkin method reported in Algorithm~\\ref{alg:alg_apg_imp}. FLOP counts for the Galerkin ROM and LSPG ROM with an implicit Euler update are additionally reported for comparison. A full description of the Galerkin and LSPG ROM updates are provided in Appendix~\\ref{appendix:algorithms}}\n\\label{tab:alg_apg_imp}\n\\end{table}\n\n\n\n\n\n\n\n\n\\begin{algorithm}\n\\caption{Algorithm for an implicit Euler update for the APG ROM using JFNK GMRES}\n\\label{alg:alg_apg_jfnk}\nInput: $\\tilde{\\mathbf{a}}^n$, residual tolerance $\\xi$ \\;\n\\newline\nOutput: $\\tilde{\\mathbf{a}}^{n+1}$\\;\n\\newline\nSteps:\n\\begin{enumerate}\n\\item Set initial guess, $\\tilde{\\mathbf{a}}_k$\n\\item  Loop while $\\mathbf{r}^k > \\xi$\n\\begin{enumerate}\n\n    \\refstepcounter{enumii}\\item[$(a\\text{--}h)$] Compute steps 2a through 2h in Algorithm~\\ref{alg:alg_apg_imp}\n    \\setcounter{enumii}{8}\n    \\item Solve the linear system,  $\\frac{\\partial \\mathbf{r}(\\tilde{\\mathbf{a}}_k)}{\\partial \\tilde{\\mathbf{a}}_k} \\Delta \\tilde{\\mathbf{a}}_k = \\mathbf{r}_A(\\tilde{\\mathbf{a}}_k)$ using Jacobian-Free GMRES\n    \\item Update the state: $\\tilde{\\mathbf{a}}_{k+1} = \\tilde{\\mathbf{a}}_k + \\Delta \\tilde{\\mathbf{a}}$\n    \\item $k = k + 1$\n\\end{enumerate}\n\\item Set final state, $\\tilde{\\mathbf{a}}^{n+1} = \\tilde{\\mathbf{a}}_k$\n\\end{enumerate}\n\\end{algorithm}\n\n\n\\begin{table}[]\n\\centering\n\\begin{tabular}{p{6cm} p{9cm}}\n\\hline\nStep in Algorithm~\\ref{alg:alg_apg_jfnk}& Approximate FLOPs \\\\\n\\hline\n2a    & $2 N K - N $ \\\\\n2b    & $ \\omega N $ \\\\\n2c    & $4 N K - N - K$  \\\\\n2d    & $ N $  \\\\\n2e    & $ (\\omega + 4) N $ \\\\\n2f    & $ 2N $ \\\\\n2g    & $ 2NK - K $ \\\\\n2h    & $ 3K $ \\\\\n2i    & $ (2\\omega + 5) N \\eta + K \\eta  + 8N K \\eta  + \\eta^2 K $\\\\\n2k    & $ K $ \\\\\n\\hline\nTotal & $ \\big((2\\eta + 2) \\omega + 5\\eta + 5) \\big)N + (\\eta^2 + \\eta + 2)K + (8\\eta + 8) NK $ \\\\\n\\end{tabular}\n\\caption{Approximate floating-point operations for one Newton iteration for the implicit Euler update to the Adjoint Petrov--Galerkin method using Jacobian-Free GMRES reported in Algorithm~\\ref{alg:alg_apg_jfnk}.}\n\\label{tab:alg_apg_jfnk}\n\\end{table}\n\n\n\n\\begin{comment}\nThe Adjoint Petrov--Galerkin method is straightforward to implement in a Jacobian-free fashion and requires minimal modifications to a standard Galerkin ROM. The Jacobian-free implementation of the additional terms in the Adjoint Petrov--Galerkin method is as follows:\n\\begin{enumerate}\n    \\item Compute the coarse-scale right-hand side, $\\mathbf{R}(\\tilde{\\mathbf{u}})$.\n    \\item Compute the orthogonal projection of the right-hand side, ${\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}) =  \\mathbf{R}(\\tilde{\\mathbf{u}}) - \\tilde{\\Pi} \\mathbf{R}(\\tilde{\\mathbf{u}})$.\n    \\item Compute the action of the Jacobian on ${\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}})$. This can be done without explicitly forming the Jacobian using either of the two following strategies:\n    \\begin{enumerate}\n    \\item Finite difference approximation:\n    \\begin{equation*}\n    \\mathbf{J}[\\tilde{\\mathbf{u}}] {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}) = \\frac{1}{\\epsilon} \\Big[ \\mathbf{R}\\big(\\tilde{\\mathbf{u}} + \\epsilon {\\Pi^{\\prime}}  \\mathbf{R}(\\tilde{\\mathbf{u}}) \\big) - \\mathbf{R}(\\tilde{\\mathbf{u}}  ) \\Big] + \\mathcal{O}(\\epsilon^2),\n    \\end{equation*}\n    where $\\epsilon$ is a small constant value, usually  $\\sim \\mathcal{O}(10^{-5})$.\n    \\item Exact linearization:\n    \\begin{equation*}\n    \\mathbf{J}[\\tilde{\\mathbf{u}}] {\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}}) = \\mathbf{R'}[\\tilde{\\mathbf{u}}]({\\Pi^{\\prime}} \\mathbf{R}(\\tilde{\\mathbf{u}})),\n    \\end{equation*}\n    where $\\mathbf{R}'[\\tilde{\\mathbf{u}}]$ is right-hand side operator linearized about $\\tilde{\\mathbf{u}}$.\n    \\end{enumerate}\n    \\item Multiply by $\\tau \\tilde{\\mathbf{V}}^T$ to compute the subgrid-scale term.\n\\end{enumerate}\nThe Adjoint Petrov--Galerkin method is observed to require an extra right-hand side (or linearized right-hand side) evaluation. Assuming the right-hand side evaluation is the dominant cost of the ROM, the Adjoint Petrov--Galerkin method is roughly twice as expensive as the standard Galerkin ROM.\n\\end{comment}\n\n\n\\section{Numerical Examples}\\label{sec:numerical}\n\nApplications of the APG method are presented  for ROMs of compressible flows: the 1D Sod shock tube problem and 2D viscous flow over a cylinder. In both problems, the test bases are chosen via POD. The shock tube problem highlights the improved stability and accuracy of the APG method over the standard Galerkin ROM, as well as improved performance over the LSPG method. The impact of the choice of $\\tau$ (APG) and $\\Delta t$ (LSPG) time-scales are also explored. The cylinder flow experiment examines a more complex problem and assesses the predictive capability of APG in comparison with Galerkin and LSPG ROMs. The effect of the choice of $\\tau$ on simulation accuracy is further explored.  \n\n\\subsection{Example 1: Sod Shock Tube with reflection}\nThe first case considered is the Sod shock tube, described in more detail in \\cite{sod}. The experiment simulates the instantaneous bursting of a diaphragm separating a closed chamber of high-density, high-pressure gas from a closed chamber of low-density, low pressure gas. This generates a strong shock, a contact discontinuity, and an expansion wave, which reflect off the shock tube walls at either end and interact with each other in complex ways. The system is described by the one-dimensional compressible Euler equations with the initial conditions,\n\\begin{comment}\n\\begin{equation}\\label{eq:euler_1D}\n    \\frac{\\partial {\\mathbf{u}}}{\\partial t} + \\frac{\\partial \\mathbf{f}}{\\partial x} = 0, \\quad\n    {\\mathbf{u}} = \n    \\begin{Bmatrix} \\rho \\\\ \\rho u \\\\ \\rho E \\end{Bmatrix}, \\quad \n    \\mathbf{f} = \\begin{Bmatrix} \\rho u \\\\ \\rho u^2 + p \\\\  u(\\rho E + p) \\end{Bmatrix}.\n\\end{equation}\nThe problem setup is given by the initial conditions,\n\\end{comment} \n\n\\begin{equation*}\n\\rho = \n\\begin{cases} \n      1 & x\\leq 0.5 \\\\\n      0.125 & x > 0.5 \n   \\end{cases},\n\\qquad\np = \n\\begin{cases} \n      1 & x\\leq 0.5 \\\\\n      0.1 & x > 0.5 \n   \\end{cases},\n\\qquad\nu = \n\\begin{cases} \n      0 & x\\leq 0.5 \\\\\n      0 & x > 0.5 \n   \\end{cases},\n\\end{equation*}  \nwith $x \\in [0,1]$. Impermeable wall boundary conditions are enforced at x = 0 and x = 1.\n\n\\subsubsection{Full-Order Model}\nThe 1D compressible Euler equations are solved using a finite volume method and explicit time integration. The domain is partitioned into 1,000 cells of uniform width. The finite volume method uses the first-order Roe flux~\\cite{roescheme} at the cell interfaces. A strong stability-preserving RK3 scheme~\\cite{SSP_RK3} is used for time integration. The solution is evolved for $t \\in [0.0,1.0]$ with a time-step of $\\Delta t = 0.0005$, ensuring CFL$\\leq 0.75$ for the duration of the simulation. The solution is saved every other time-step, resulting in 1,000 solutions snapshots for each conserved variable.\n\n\\subsubsection{Solution of the Reduced-Order Model}\nUsing the FOM data snapshots, trial bases for the ROMs are constructed via the proper orthogonal decomposition (POD) approach. A separate basis is constructed for each conserved variable. The complete basis construction procedure is detailed in Appendix~\\ref{appendix:basisconstruction}. Once a coarse-scale trial basis $\\tilde{\\mathbf{V}}$ is built, a variety of ROMs are evaluated according to the following formulations:\n\n\\begin{enumerate}\n\\item Galerkin ROM:\n\\begin{equation*}\\label{eq:galerkin_ROM}\n\\tilde{\\mathbf{V}}^T \\bigg( \\frac{d \\tilde{\\mathbf{u}}}{dt} - \\mathbf{R}(\\tilde{\\mathbf{u}}) \\bigg) = 0, \\qquad t \\in [0,1].\n\\end{equation*}\n\n\\item Adjoint Petrov--Galerkin ROM:\n\\begin{equation*}\\label{eq:MZPG_ROM}\n\\tilde{\\mathbf{V}}^T\\bigg(\\mathbf{I} + \\tau  \\mathbf{J}[\\tilde{\\mathbf{u}}] {\\Pi^{\\prime}} \\bigg)  \\bigg( \\frac{d \\tilde{\\mathbf{u}}}{dt} - \\mathbf{R}(\\tilde{\\mathbf{u}}) \\bigg) = 0 , \\qquad t \\in [0,1].\n\\end{equation*}\n\\textit{Remark: The Adjoint Petrov--Galerkin ROM requires specification of $\\tau$.}\n\n\\item Least-Squares Petrov--Galerkin ROM (Implicit Euler Time Integration):\n\\begin{equation*}\n{\\mathbf{u}}^n = \\underset{\\mathbf{y} \\in \\text{Range}(\\tilde{\\mathbf{V}})  }{\\text{arg min}}\\norm{ \\frac{\\mathbf{y} - \\tilde{\\mathbf{u}}^{n-1}}{\\Delta t} - \\mathbf{R}(\\mathbf{y}) }^2, \\qquad \\text{for } n = 1,2,\\hdots , \\text{ceil}\\big(\\frac{1}{\\Delta t}\\big).\n\\end{equation*}\n\\textit{Remark: The LSPG approach is strictly coupled to the time integration scheme and time-step.}\n\n\n\\end{enumerate}\n\n\\subsubsection{Numerical Results}\nThe first case considered uses $50$ basis vectors each for the conserved variables $\\rho, \\rho u,$ and $\\rho E$. The total dimension of the reduced model is thus $K = 150$. Roughly 99.9--99.99\\% of the POD energy is captured by this 150-mode basis. In fact, 99\\% of the energy is contained in the first 5-12 modes of each conserved variable.\n\nThe Adjoint Petrov--Galerkin ROM requires specification of the memory length $\\tau$. Similarly, LSPG requires the selection of an appropriate time-step. The sensitivity of both methods to this selection will be discussed later in this section.\nThe simulation parameters are provided in Table~\\ref{tab:sod_tab1}.\n\nDensity profiles at $t = 0.25$ and $t = 1.0$ for explicit Galerkin and APG ROMs, along with an implicit LSPG ROM, are displayed in Fig.~\\ref{fig:sod_density}. All three ROMs are capable of reproducing the shock tube density profile in Fig.~\\ref{fig:sod_density_0p25}; a normal shock propagates to the right and is followed closely behind by a contact discontinuity, while an expansion wave propagates to the left. All three methods exhibit oscillations at $x = 0.5$, the location of the imaginary burst diaphragm, and near the shock at $x = 0.95$. At $t = 1.0$, when the shock has reflected from the right wall and interacted with the contact discontinuity, much stronger oscillations are present, particularly near the reflected shock at $x = 0.45$. These oscillations are reminiscent of Gibbs phenomenon, and are an indicator of the inability to accurately reconstruct sharp gradients. The Galerkin ROM exhibits the largest oscillations of the ROMs considered, while LSPG exhibits the smallest.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ l  l  l  l  l}\\hline\n  ROM Type & Time Scheme & $\\Delta t$ & $\\tau$ & $\\int ||e||_2 dt$\\\\ \\hline\n  Galerkin  & SSP-RK3    & 0.0005 & N\/A     & 1.5752 \\\\\n  Galerkin  & Imp. Euler & 0.0005 & N\/A     & 1.0344 \\\\\n  APG       & SSP-RK3    & 0.0005 & 0.00043 & 1.0637 \\\\\n  APG       & Imp. Euler & 0.0005 & 0.00043 & 0.8057 \\\\\n  APG       & Imp. Euler & 0.001  & 0.00043 & 0.7983 \\\\\n  LSPG      & Imp. Euler & 0.0005 & N\/A     & 1.1668 \\\\\n  LSPG      & Imp. Euler & 0.001  & N\/A     & 1.4917 \\\\ \\hline\n\\end{tabular}\n\\caption{Computational details for Sod shock tube ROM cases, $K = 150$}\n\\label{tab:sod_tab1}\n\\end{table}\n\n\\begin{comment}\n\\begin{figure}\n    \\centering\n    \\begin{minipage}{0.49\\linewidth}\n    \\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/PODspectrumZoomed.png}\n    \\caption{Sod shock tube POD energy spectrum}\n    \\label{fig:pod_spectrum}\n    \\end{minipage}\\hfill\n    \\begin{minipage}{0.49\\linewidth}\n    \\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/err_consVars_k50_apgExplicit.png}\n    \\caption{Conserved variable error profiles, APG w\/ SSP-RK3 time integration, $K = 150$, $\\Delta t = 0.0005$}\n    \\label{fig:sod_error_consVars}\n    \\end{minipage}\\hfill\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/rho_k50_dt0p0005_t0p25__explicit.png}\n\\caption{$t=0.25$.}\n\\label{fig:sod_density_0p25}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/rho_k50_dt0p0005_t1p00_explicit.png}\n\\caption{$t=1$.}\n\\label{fig:sod_density_1p0}\n\\end{subfigure}\n\\end{center}\n\\caption{Density profiles for the Sod shock tube with $K = 150$, $\\Delta t = 0.0005$.}\n\\label{fig:sod_density}\n\\end{figure}\n\n\nFigure~\\ref{fig:sod_error} shows the evolution of the error for all of the ROMs listed in Table~\\ref{tab:sod_tab1}. The $L^2$-norm of the error is computed as,\n\\begin{equation*}\n||e||_2 = \\sqrt{ \\sum_{i=1}^{1000} \\Big[(\\tilde{\\rho}_{i,ROM} - \\tilde{\\rho}_{i,FOM} )^2 + (\\widetilde{\\rho u}_{i,ROM} - \\widetilde{\\rho u}_{i,FOM} )^2  + (\\widetilde{\\rho E}_{i,ROM} - \\widetilde{\\rho E}_{i,FOM})^2 \\Big] }.\n\\end{equation*}\n Here, the subscript $i$ denotes each finite volume cell. The FOM values used for error calculations are projections of the FOM data onto $\\tilde{\\MC{V}}$, e.g. $\\tilde{\\rho}_{FOM} = \\tilde{\\Pi} \\rho_{FOM}$. This error measure provides a fair upper bound on the accuracy of the ROMs, as the quality of the ROM is generally dictated by the richness of the trial basis and the projection of the FOM data is the maximum accuracy that can be reasonably hoped for.\n\nIn Figure~\\ref{fig:sod_error}, it is seen that the APG ROM exhibits improved accuracy over the Galerkin ROM. The LSPG ROM for $\\Delta t = 0.0005$ performs slightly better than the explicit Galerkin ROM, and worse than the implicit Galerkin ROM. Increasing the time-step to $\\Delta t = 0.001$ results in a significant increase in error for the LSPG ROM. This is due to the fact that the performance of LSPG is influenced by the time-step. For a trial basis containing much of the residual POD energy, LSPG will generally require a very small time-step to improve accuracy; this sensitivity will be explored later. Lastly, it is observed that the APG ROM is \\textit{not} significantly affected by the time-step. The APG ROM with $\\Delta t = 0.001$ shows moderately increased error prior to $t = 0.3$ and similar error afterwards when compared against the $\\Delta t = 0.0005$ APG ROM case. \n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/err_k50_explicit.png}\n\\caption{Explicit Galerkin\/APG, implicit LSPG, $\\Delta t = 0.0005$}\n\\label{fig:sod_err_explicit}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/err_k50_implicit.png}\n\\caption{All implicit, various $\\Delta t$}\n\\label{fig:sod_err_implicit}\n\\end{subfigure}\n\\end{center}\n\\caption{$L^2$-norm error profiles for the Sod shock tube with $150$ basis vectors.}\n\\label{fig:sod_error}\n\\end{figure}\n\nFigure~\\ref{fig:sod_mode_study} studies the effect of the number of modes retained in the trial basis on the stability and accuracy, over the range $K = 60,75,90,\\ldots,180$. Missing data points indicate an unstable solution. Values of $\\tau$ for the APG ROMs are again selected by user choice. The most striking feature of these plots is the fact that even though the explicit Galerkin ROM is unstable for $K \\leq 135$ and the implicit Galerkin ROM is unstable for $K \\leq 75$, the APG and LSPG ROMs are stable for all cases. Furthermore, the APG and LSPG ROMs are capable of achieving stability with a time-step twice as large as that of the Galerkin ROM. The cost of the APG and LSPG ROMs are effectively halved, but they are still able to stabilize the simulation. Interestingly, the Galerkin and APG ROMs both exhibit abrupt peaks in error at $K = 120$, while the LSPG ROMs do not. The exact cause of this is unknown, but displays that a monotonic decrease in error with enrichment of the trial space is not guaranteed. \n\nSeveral interesting comparisons between APG and LSPG arise from Figure~\\ref{fig:sod_mode_study}. First, with the exception of the $K = 120$ case, Fig.~\\ref{fig:sod_err_explicit} shows that the APG ROM with explicit time integration exhibits accuracy comparable to that of the LSPG ROM with implicit time integration. As can be seen in comparing Tables~\\ref{tab:alg_apg_exp} and~\\ref{tab:alg_LSPG}, the cost of APG with explicit time integration is significantly lower than the cost of LSPG. This is an attractive feature of APG, as it is able to use inexpensive explicit time integration while LSPG is restricted to implicit methods. Additionally, we draw attention to the poor performance of LSPG at high $K$ for a moderate time-step in Fig.~\\ref{fig:sod_modeSens_implicit}. Increasing the time-step to $\\Delta t = 0.001$ to decrease simulation cost only exacerbates this issue; as the trial space is enriched, LSPG requires a smaller time-step to yield accurate results. If we wish to improve the LSPG solution for $K = 150$, we must decrease the time-step below that of the FOM. The accuracy of the APG ROM does not change when the time-step is doubled from $\\Delta t = 0.0005$ to $\\Delta t = 0.001$. This halves the cost of the APG ROM with no significant drawbacks.  \n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/modeSensExplicit.png}\n\\caption{Explicit Galerkin\/APG, implicit LSPG, fixed $\\Delta t$}\n\\label{fig:sod_modeSens_explicit}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/modeSensImplicit.png}\n\\caption{All implicit, various $\\Delta t$}\n\\label{fig:sod_modeSens_implicit}\n\\end{subfigure}\n\\end{center}\n\\caption{Integrated error mode sensitivity study for the Sod shock tube.}\n\\label{fig:sod_mode_study}\n\\end{figure}\n\n\\subsubsection{Optimal Memory Length Investigations}\nAs mentioned previously, the success of LSPG is tied to the physical time-step and the time integration scheme, while the parameter $\\tau$ in the APG method may be chosen independently from these factors. In minimizing ROM error, finding an optimal value of $\\tau$ for the APG ROM may permit the choice of a much larger time-step than the optimal LSPG time-step. Further, the APG method may be applied with explicit time integration schemes, which are generally much less expensive than the implicit methods which LSPG is restricted to. To demonstrate this, the APG ROM and LSPG ROM with $K=150$ are simulated for a variety of time scales ($\\tau$ for APG and $\\Delta t$ for LSPG). \n\nFigure~\\ref{fig:sod_memlength_sensitivity} shows the integrated error of the ROMs versus the relevant time scale. For this case, the optimal value of $\\Delta t$ for LSPG is less than $0.0001$ and is not shown. The optimal value of $\\tau$ is not greatly affected by the choice of time integration scheme (implicit or explicit) or time-step. Furthermore, because $\\tau$ can be chosen independently from $\\Delta t$ for APG, the APG ROM can produce low error at a much larger time-step ($\\Delta t = 0.001$) than the optimal time-step for the LSPG ROM. This highlights the fact that the ``optimal\" LSPG ROM may be computationally expensive due to a small time-step, whereas the ``optimal\" APG ROM requires only the specification of $\\tau$ and can use, potentially, much larger time-steps than LSPG. It has to be mentioned, however, that the choice of $\\tau$  has an impact on performance --- selections of $\\tau$ larger than those plotted in Fig.~\\ref{fig:sod_memlength_sensitivity} caused the ROM to lose stability. \n\nAlso discussed previously, computing an optimal value of $\\tau$ \\textit{a priori} may be linked to the spectral radius of the coarse-scale Jacobian, (i.e. $\\rho \\big( \\tilde{\\mathbf{V}}^T \\mathbf{J}[\\tilde{\\mathbf{u}}_0]\\tilde{\\mathbf{V}} \\big)$, not to be confused with the physical density $\\rho$). We consider the APG ROM for basis sizes of  $K=30,60,90,150,180,240$. For each case, an ``optimal\" $\\tau$ is found by minimizing the misfit between the ROM solution and the projected FOM solution. The misfit is defined as follows,\n\\begin{equation}\\label{eq:misfit}\n\\mathcal{J}(\\tau) = \\sum_{i=1}^{200} ||e(\\tau,t = i10\\Delta t)||_2.\n\\end{equation}\nEquation~\\ref{eq:misfit} corresponds to summing the $L^2$-norm of the error at every $10^{th}$ time-step. Figure~\\ref{fig:sod_tau_specRad} shows the resulting optimal $\\tau$ for each case plotted against the inverse of the spectral radius of the coarse-scale Jacobian evaluated at $t=0$. \nThe strong linear correlation suggests that a near-optimal value of $\\tau$ may be chosen by evaluating the spectral radius, $\\rho \\big( \\tilde{\\mathbf{V}}^T \\mathbf{J}[\\tilde{\\mathbf{u}}_0]\\tilde{\\mathbf{V}} \\big)$ and using the above linear relationship to $\\tau$. \n\nTwo points are emphasized here:\n\\begin{enumerate}\n\\item The spectral radius plays an important role in both implicit and explicit time integrators and is often the determining factor in the choice of the time-step. Theoretical analysis on the stability of explicit methods (and convergence of implicit methods) shows a similar dependence to the spectral radius. Choosing the memory length to be $\\tau = \\Delta t$ is one simple heuristic that may be used.\n\\item\nWhile a linear relationship between $\\tau$ and the spectral radius of the coarse-scale Jacobian has been observed in every problem the authors have examined, the slope of the fit is somewhat problem dependent. For the purpose of reduced-order modeling, however, this is only a minor inconvenience as an appropriate \nvalue of $\\tau$ can be selected by assessing the performance of the ROM on the training set, i.e., on the simulation used to construct the POD basis. \n\\item Finally, more complex methods may be used to compute $\\tau$. A method to dynamically compute $\\tau$ based on Germano's identity, for instance, was proposed in~\\cite{parish_dtau} in the context of the simulation of turbulent flows with Fourier-Galerkin methods. Extension of this technique to projection-based ROMs and the development of additional techniques to select $\\tau$ will be the subject of future work.\n\\end{enumerate}\n\n\\begin{figure}\n    \\centering\n    \\begin{minipage}{0.49\\linewidth}\n    \\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/dt_tau_sensitivity_semilog.png}\n    \\caption{Error as a function of time scale, $K = 150$.}\n    \\label{fig:sod_memlength_sensitivity}\n    \\end{minipage}\\hfill\n    \\begin{minipage}{0.49\\linewidth}\n    \\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{sodFigs\/tauSpecRadCorr_revis.png}\n    \\caption{Optimal $\\tau$ as a function of the spectral radius evaluated at $t=0$.}\n    \\label{fig:sod_tau_specRad}\n    \\end{minipage}\\hfill\n\\end{figure}\n\n\\input{cyl_new}\n\n\n\n\n\\section{Conclusion}\\label{sec:conclude}\nThis work introduced the Adjoint Petrov--Galerkin method for non-linear model reduction. Derived from the variational multiscale method and Mori-Zwanzig formalism, the Adjoint Petrov--Galerkin method is a Petrov--Galerkin projection technique with a non-linear time-varying test basis. The method is designed to be applied at the semi-discrete level, i.e., after spatial discretization of a partial differential equation, and is compatible with both implicit and explicit time integration schemes. The method displays commonalities with the adjoint-stabilization method used in the finite element community as well as the Least-Squares Petrov--Galerkin approach used in non-linear model-order reduction. Theoretical error analysis was presented that showed conditions under which the Adjoint Petrov--Galerkin ROM may have lower \\textit{a priori} error bounds than the Galerkin ROM. The theoretical cost of the Adjoint Petrov--Galerkin method was considered for both explicit and implicit schemes, where it was shown to be approximately twice that of the Galerkin method. In the case of implicit time integration schemes, the Adjoint Petrov--Galerkin ROM was shown to be capable of being more efficient than Least-Squares Petrov--Galerkin when the non-linear system is solved via Jacobian-Free Newton-Krylov methods. \n\nNumerical experiments with the Adjoint Petrov--Galerkin, Galerkin, and Least-Squares Petrov--Galerkin method were presented for the Sod shock tube problem and viscous compressible flow over a cylinder parameterized by the Reynolds number. In all examples, the Adjoint Petrov--Galerkin method provided more accurate predictions than the Galerkin method for a fixed basis dimension. Improvements over the Least-Squares Petrov--Galerkin method were observed in most cases. In particular, the Adjoint Petrov--Galerkin method was shown to provide relatively accurate predictions for the cylinder flow at Reynolds numbers outside of the training set used to construct the POD basis. The Galerkin method, with both an equivalent and an enriched trial space, failed to produce accurate results in these cases. Additionally, numerical evidence showed a correlation between the spectral radius of the reduced Jacobian and the optimal value of the stabilization parameter appearing in the Adjoint Petrov--Galerkin method.\n\nWhen augmented with hyper-reduction, the Adjoint Petrov--Galerkin ROM was shown to be capable of producing accurate predictions within the POD training set with computational speedups up to 5000 times compared to the full-order models. This speed-up is a result of hyper-reduction of the right-hand side, as well as the ability to use explicit time integration schemes at large time-steps. A study of the Pareto front for simulation error versus relative wall time showed that, for the compressible cylinder problem, the Adjoint Petrov--Galerkin ROM is competitive with the Galerkin ROM, and more efficient than the LSPG ROM for the problems considered. \n\n\n\n\n\n\\section{Acknowledgements}\\label{sec:acknowledge}\nThe authors acknowledge support from the US Air Force Office of Scientific Research through the Center of Excellence Grant FA9550-17-1-0195 (Tech. Monitors: Mitat Birkan \\& Fariba Fahroo) and the project LES Modeling of Non-local effects using Statistical Coarse-graining (Tech. Monitors: Jean-Luc Cambier \\& Fariba Fahroo). E. Parish acknowledges an appointment to the Sandia National Laboratories John von Neumann fellowship. This paper describes objective technical results and analysis. Any subjective\nviews or opinions that might be expressed in the paper do not necessarily\nrepresent the views of the U.S. Department of Energy or the United States\nGovernment Sandia National Laboratories is a multimission laboratory managed\nand operated by National Technology and Engineering Solutions of Sandia, LLC.,\na wholly owned subsidiary of Honeywell International, Inc., for the U.S.\nDepartment of Energy's National Nuclear Security Administration under contract\nDE-NA-0003525.\n\n\\begin{appendices}\n\n\\section{Hyper-reduction for the Adjoint Petrov--Galerkin Reduced-Order Model}\\label{appendix:hyper}\n\nIn the numerical solution of non-linear dynamical systems, the evaluation of the non-linear right-hand side term usually accounts for a large portion (if not the majority) of the computational cost. Equation~\\ref{eq:GROM_modal} shows that standard projection-based ROMs are incapable of reducing this cost, as the evaluation of $\\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}})$ still scales with the number of degrees of freedom $N$. If this issue is not addressed, and there is no reduction in temporal dimensionality, the ROM will typically be \\emph{more} expensive than the FOM due to additional matrix-vector products from projection onto the reduced-order space. Techniques for overcoming this bottleneck are typically referred to as hyper-reduction methods.  The thesis of hyper-reduction methods is that, instead of computing the entire right-hand side vector, only a few entries are calculated. The missing entries can either be ignored (as done in collocation methods) or reconstructed (as done in the discrete interpolation and gappy POD methods). This section outlines the Gappy POD method, selection of the sampling points through QR factorization, and the algorithm for the hyper-reduced Adjoint Petrov--Galerkin ROM.\n\n\\subsection{Gappy POD}\nThe gappy POD method~\\cite{everson_sirovich_gappy} seeks to find an approximation for $\\mathbf{R}(\\cdot)$ that evaluates the right-hand side term at a reduced number of spatial points $r \\ll N$. This is achieved through the construction of a trial space for the right-hand side and least-squares reconstruction of a sampled signal. The offline steps required in the Gappy POD method are given in Algorithm~\\ref{alg:gappy_hyper_offline}.\n\n\\begin{algorithm}\n\\caption{Algorithm for the offline steps required for hyper-reduction via gappy POD.}\n\\label{alg:gappy_hyper_offline}\nOffline Steps:\n\\begin{enumerate}\n    \\item Compute the full-order solution, storing $n_t$ time snapshots in the following matrices,\n    \\begin{alignat*}{2}\n       \n        &\\text{Right-hand side snapshots:} \\; &&\\mathbf{F} = [\\mathbf{R}({\\mathbf{u}}_1) \\quad \\mathbf{R}({\\mathbf{u}}_2) \\quad ... \\quad \\mathbf{R}({\\mathbf{u}}_{n_t})] \\in \\mathbb{R}^{N \\times n_t}\n    \\end{alignat*}    \n   \n    \\item Compute the right-hand side POD basis $\\mathbf{U} \\in \\mathbb{R}^{N \\times r}$, from $\\mathbf{F}$.\n    \\item Compute the sampling point matrix $\\mathbf{P} = [\\mathbf{e}_{p_1} \\quad \\mathbf{e}_{p2} \\quad ... \\mathbf{e}_{p_r}] \\in \\mathbb{R}^{N_{p} \\times N}$, where $\\mathbf{e}_i$ is the $i$th cannonical unit vector and $N_p$ is the number of sampling points.\n    \\item Compute the stencil matrix $\\mathbf{P}_s = [\\mathbf{e}_{s_1} \\quad \\mathbf{e}_{s2} \\quad ... \\mathbf{e}_{s_r}] \\in \\mathbb{R}^{N_{s} \\times N}$, where $\\mathbf{e}_i$ is the $i$th cannonical unit vector and $N_s$ is the number of stencil points required to reconstruct the residual at the sample points. Note $N_s \\ge N_p$.\n\n    \\item Compute the least-squares reconstruction matrix: $\\big[ \\mathbf{P}^T \\mathbf{U} \\big]^{+}$, where the superscript $+$ denotes the pseudo-inverse. \n Note that this matrix corresponds to the solution of the least-squares problem for a gappy signal $\\mathbf{f} \\in \\mathbb{R}^N$:\n$$ \\mathbf{a}_{\\mathbf{f}} = \\underset{\\mathbf{b} \\in \\mathbb{R}^r}{\\text{argmin}} || \\mathbf{P}^T \\mathbf{U} \\mathbf{b}   - \\mathbf{P}^T \\mathbf{f}||,$$\nwhich has the solution,\n$$\\mathbf{a}_{\\mathbf{f}} = \\bigg[ \\mathbf{P}^T \\mathbf{U} \\bigg]^{+} \\mathbf{f}.$$\n\\end{enumerate}\n\\end{algorithm}\n\n\nNote that, because the gappy POD approximation of the right-hand side only samples the right-hand side term at $N_p$ spatial points, the cost of the ROM no longer scales with the full-order degrees of freedom $N$, but instead with the number of POD basis modes $K$ and the number of sample points $N_p$. Thus, the cost of evaluating the full right-hand side may be drastically reduced at the price of storing another snapshot matrix $\\mathbf{F}$ and computing another POD basis $\\mathbf{U}$ in the offline stage of computation. Furthermore, the product $\\tilde{\\mathbf{V}}^T \\mathbf{U} \\big[ \\mathbf{P}^T \\mathbf{U} \\big]^{+}$ may be precomputed during the offline stage if both $\\mathbf{P}$ and $\\mathbf{U}$ remain static throughout the simulation. This results in an relatively small $K\\times N_p$ matrix. As such, an increase in offline computational cost may produce a significant decrease in online computational cost.\n\n\\subsection{Selection of Sampling Points}\nStep 3 in Algorithm~\\ref{alg:gappy_hyper_offline} requires the construction of the sampling point matrix, for which several methods exist. In the discrete interpolation method proposed by Chaturantabut and Sorensen~\\cite{deim}, the sample points are selected inductively from the basis $\\mathbf{U}$, based on an error measure between the basis vectors and approximations of the basis vectors via interpolation. The method proposed by Drmac and Gugercin~\\cite{qdeim_drmac} leverages the rank-revealing QR factorization to compute $\\mathbf{P}$. Dynamic updates of the $\\mathbf{U}$ and $\\mathbf{P}$ via periodic sampling of the full-order right-hand side term is even possible via the methods developed by Peherstorfer and Willcox~\\cite{adeim_peherstorfer}. \n\nThe 2D compressible cylinder simulations presented in this manuscript uses a modified version of the rank-revealing QR factorization proposed in~\\cite{qdeim_drmac} to obtain the sampling points. The modifications are added to enhance the stability and accuracy of the hyper-reduced ROM within the discontinuous Galerkin method. Algorithm~\\ref{alg:qdeim} outlines the steps used in this manuscript to compute the sampling points.\n\n\\begin{algorithm}\n\\caption{Algorithm for QR-factorization-based selection of sampling matrix.}\n\\label{alg:qdeim}\nInput: Right-hand side POD basis $\\mathbf{U} \\in \\mathbb{R}^{N \\times r}$ \\;\n\\newline\nOutput: Sampling matrix $\\mathbf{P}$ \\;\n\\newline\nSteps:\n\\begin{enumerate}\n   \n    \\item Transpose the right-hand side POD basis, $\\mathbf{U}' = \\mathbf{U}^T$\n    \\item Compute the rank-revealing QR factorization of $\\mathbf{U}'$, generating a permutation matrix $\\Gamma \\in \\mathbb{R}^{N \\times N}$, unitary $\\mathbf{Q} \\in \\mathbb{R}^{r \\times r}$, and orthonormal $\\mathbf{R} \\in \\mathbb{R}^{r \\times N}$ such that\n    \\begin{equation*}\n        \\mathbf{U}' \\Gamma = \\mathbf{QR}\n    \\end{equation*}\n    Details on computing rank-revealing QR decompositions can be found in~\\cite{qr_decomp_gu}, though many math libraries include optimized routines for this operation.\n    \\item From the permutation matrix $\\Gamma$, select the first $r$ columns to form the interpolation point matrix $\\mathbf{P}$. \n    \\item When applied to systems of equations, sampling approaches that select only specific indices of the residual can be inaccurate due to the fact that, at a given cell, the residual of all unknowns at that cell may not be calculated. This issue is further exacerbated in the discontinuous Galerkin method, where each cell has a number of quadrature points. As such, we perform the additional step: \n    \\begin{enumerate}\n    \\item Augment the sampling point matrix, $\\mathbf{P}$, with additional columns such that all unknowns are computed at the mesh cells selected by Step 3. In the present context, these additional unknowns correspond to each conserved variable and quadrature point in the selected cells. This step leads to $N_p > r$.\n    \\end{enumerate}\n\\end{enumerate}\n\n\n\\end{algorithm}\n\n\n\\subsection{Hyper-Reduction of the Adjoint Petrov--Galerkin ROM}\nLastly, the online steps required for an explicit Euler update to the Adjoint Petrov--Galerkin ROM with Gappy POD hyper-reduction is provided in Algorithm~\\ref{alg:alg_ag_hyper}. It is worth noting that Step 3 in Algorithm~\\ref{alg:alg_ag_hyper} requires one to reconstruct the right-hand side at the stencil points. Hyper-reduction via a standard collocation method, which provides no means to reconstruct the right-hand side, is thus not compatable with the Adjoint Petrov--Galerkin ROM. \n\\begin{algorithm}\n\\caption{Algorithm for an explicit Euler update for the Adjoint Petrov--Galerkin ROM with gappy POD hyper-reduction.}\n\\label{alg:alg_ag_hyper}\nInput: $\\tilde{\\mathbf{a}}^n$ \\;\n\\newline\nOutput: $\\tilde{\\mathbf{a}}^{n+1}$\\;\n\\newline\nOnline steps at time-step $n$:\n\\begin{enumerate}\n\\item Compute the state at the stencil points: $\\tilde{\\mathbf{u}}_s^n = \\mathbf{P}_s^T \\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^n,$ with $\\tilde{\\mathbf{u}}_s^n \\in \\mathbb{R}^{N_s}$ \n\\item Compute the generalized coordinates to the right-hand side evaluation via,\n\\begin{equation*}\\label{eq:RHSapprox}\n   \\mathbf{a}_{\\mathbf{R}}^n = \\big[ \\mathbf{P}^T \\mathbf{U} \\big]^{+} \\mathbf{P}^T \\mathbf{R}(\\tilde{\\mathbf{u}}_s^n),\n\\end{equation*}\nwith $\\mathbf{a}_{\\mathbf{R}}^n \\in \\mathbb{R}^r$.\nNote that the product $\\mathbf{P}^T \\mathbf{R}(\\tilde{\\mathbf{u}}_s^n)$ requires computing $\\mathbf{R}(\\tilde{\\mathbf{u}}_s^n)$ \\emph{only at the sample points}, as given by the unit vectors stored in $\\mathbf{P}$.\n\\item Reconstruct the right-hand side at the stencil points: $\\overline{\\mathbf{R}_s(\\tilde{\\mathbf{u}}_s^n)} = \\mathbf{P}_s^T \\mathbf{U} \\mathbf{a}_{\\mathbf{R}}^n$\n\\item Compute the orthogonal projection of the approximated right-hand side at the stencil points:\n$${\\Pi^{\\prime}} \\overline{\\mathbf{R}_s(\\tilde{\\mathbf{u}}^n)} =  \\overline{\\mathbf{R}_s(\\tilde{\\mathbf{u}}^n)} -  \\tilde{\\mathbf{V}} \\tilde{\\mathbf{V}}^T \\overline{ \\mathbf{R}_s(\\tilde{\\mathbf{u}}^n)}$$\n\\item Compute the generalized coordintes for the action of the Jacobian on ${\\Pi^{\\prime}} \\overline{\\mathbf{R}_s(\\tilde{\\mathbf{u}}^n)}$ at the sample points using either finite difference or exact linearization. For finite difference:\n    \\begin{equation*}\n    \\mathbf{a}_{\\mathbf{J}}^n \\approx \\frac{1}{\\epsilon} \\big[ \\mathbf{P}^T \\mathbf{U} \\big]^{+} \\mathbf{P}^T  \\Big[ \\mathbf{R}_s\\big(\\tilde{\\mathbf{u}}^n_s + \\epsilon {\\Pi^{\\prime}}  \\overline{\\mathbf{R}_s}(\\tilde{\\mathbf{u}}^n_s) \\big) - {\\mathbf{R}_s(\\tilde{\\mathbf{u}}^n_s  )} \\Big], \n    \\end{equation*}\n    with $\\mathbf{a}_{\\mathbf{J}}^n \\in \\mathbb{R}^r$. Note $\\epsilon$ is a small constant value, usually  $\\sim \\mathcal{O}(10^{-5})$.\n\\item Compute the combined sampled right-hand side:  $\\tilde{\\mathbf{V}}^T \\mathbf{U} \\bigg[ \\mathbf{a}_{\\mathbf{R}}^n + \\tau \\mathbf{a}_{\\mathbf{J}}^n \\bigg]$\n\\item Update the state: $\\tilde{\\mathbf{a}}^{n+1} = \\tilde{\\mathbf{a}}^n + \\Delta t \\tilde{\\mathbf{V}}^T \\mathbf{U} \\bigg[ \\mathbf{a}_{\\mathbf{R}}^n + \\tau \\mathbf{a}_{\\mathbf{J}}^n \\bigg] $\n\\end{enumerate}\n\n\\end{algorithm}\n\n\n\n\\section{POD Basis Construction}\\label{appendix:basisconstruction}\n\nFor the 1D Euler case detailed in this manuscript, the procedure for constructing separate POD bases for each conserved variables is as follows:\n\n\\begin{enumerate}\n\\item Run the full-order model for $t \\in (0,1)$ at a time-step of $\\Delta t = 0.0005$. The state vector is saved at every other time step to create 1000 state snapshots.\n\\item Collect the snapshots for each state into three state snapshot matrices:\n$$\n\\mathbf{S}_{\\rho}  = \\begin{bmatrix}\n\\mathbf{\\rho}_1 & \\mathbf{\\rho}_2 & \\hdots & \\rho_{1000}\n\\end{bmatrix},\n\\mathbf{S}_{\\rho u}  = \\begin{bmatrix}\n\\mathbf{\\rho u}_1 & \\mathbf{\\rho u}_2 & \\hdots & \\mathbf{\\rho u}_{1000}\n\\end{bmatrix},\n\\mathbf{S}_{\\rho E}  = \\begin{bmatrix}\n\\mathbf{ \\rho E}_1 & \\mathbf{ \\rho E}_2 & \\hdots & \\mathbf{\\rho E}_{1000}\n\\end{bmatrix},\n$$\nwhere $\\mathbf{\\rho}_i, \\mathbf{\\rho u}_i, \\mathbf{\\rho E}_i \\in \\mathbb{R}^{1000}$.\n\\item Compute the singular-value decomposition (SVD) of each snapshot matrix, e.g. for $\\mathbf{S}_{\\rho}$,\n$$\\mathbf{S}_{\\rho} \\mathrel{\\overset{\\makebox[0pt]{\\mbox{\\normalfont\\tiny\\sffamily SVD}}}{=}} \\mathbf{V}_{\\rho} {\\Sigma}_{\\rho} \\mathbf{U}^T_{\\rho}.$$\nThe columns of $\\mathbf{V}_{\\rho}$ and $\\mathbf{U}_{\\rho}$ are the left and right singular vectors of $\\mathbf{S}_{\\rho}$, respectively. ${\\Sigma}_{\\rho}$ is a diagonal matrix of the singular values of $\\mathbf{S}_{\\rho}$. The columns of $\\mathbf{V}_{\\rho}$ form a basis for the solution space of $\\rho_i$. \n\n\\textit{Remarks}\n\\begin{enumerate}\n\\item In this example, a separate basis is computed for each conserved quantity. It is also possible to construct a global basis by stacking $\\mathbf{S}_{\\rho}$, $\\mathbf{S}_{\\rho \\mathbf{u}}$, and $\\mathbf{S}_{\\rho \\mathbf{E}}$ into one snapshot matrix and computing one ``global\" SVD.\n\\end{enumerate}\n\\item Decompose each basis into bases for the resolved and unresolved scales by selecting the first $K$ columns and last $1000-K$ columns, respectively, e.g.\n$$\\mathbf{V}_\\rho = \\begin{bmatrix} \\tilde{\\mathbf{V}}_{\\rho} & ; & {\\mathbf{V}^{\\prime}}_{\\rho} \\end{bmatrix},$$\nwhere $\\tilde{\\mathbf{V}}_{\\rho} \\in \\mathbb{R}^{1000 \\times K}$ and ${\\mathbf{V}^{\\prime}} \\in \\mathbb{R}^{1000 \\times 1000 - K}.$  \\\\\n\\textit{Remarks}\n\\begin{enumerate}\n\\item Each basis vector is orthogonal to the others, hence the coarse and fine-scales are orthogonal.\n\\item In this example, we have selected 1000 snapshots such that the column space of ${\\mathbf{V}}$ spans ${\\MC{V}}$. In general, this is not the case. As the APG method requires no processing of the fine-scale basis functions, however, this is not an issue.\n\\end{enumerate}\n\\item Construct a global coarse-scale basis,\n$$\\tilde{\\mathbf{V}} = \\begin{bmatrix}\n\\tilde{\\mathbf{V}}_{\\rho} & \\mathbf{0} & \\mathbf{0} \\\\\n\\mathbf{0} & \\tilde{\\mathbf{V}}_{\\rho u} & \\mathbf{0} \\\\\n\\mathbf{0} & \\mathbf{0} & \\tilde{\\mathbf{V}}_{\\mathbf{\\rho E}} \\\\\n\\end{bmatrix}.$$\n\\end{enumerate}\n\n\n\\section{Algorithms for the Galerkin and LSPG ROMs}\\label{appendix:algorithms}\nSection~\\ref{sec:cost} presented an analysis on the computational cost of the Adjoint Petrov--Galerkin ROM. This appendix presents similar algorithms and FLOP counts for the Galerkin and LSPG ROMs. The following algorithms and FLOP counts are reported:\n\\begin{enumerate}\n    \\item An explicit Euler update to the Galerkin ROM (Algorithm~\\ref{alg:alg_g_exp}, Table~\\ref{tab:alg_g_exp}).\n    \\item An implicit Euler update to the Galerkin ROM using Newton's method with Gaussian elimination (Algorithm~\\ref{alg:alg_g_imp}, Table~\\ref{tab:alg_g_imp}).\n    \\item An implicit Euler update to the Least-Squares Petrov--Galerkin ROM using the Gauss-Newton method with Gaussian elimination (Algorithm~\\ref{alg:alg_LSPG}, Table~\\ref{tab:alg_LSPG}).\n\\end{enumerate}\n\n\\begin{algorithm}[h!]\n\\caption{Algorithm for an explicit Euler update for the Galerkin ROM.}\n\\label{alg:alg_g_exp}\nInput: $\\tilde{\\mathbf{a}}^n$ \\;\n\\newline\nOutput: $\\tilde{\\mathbf{a}}^{n+1}$\\;\n\\newline\nSteps:\n\\begin{enumerate}\n\\item Compute the state from the generalized coordinates, $\\tilde{\\mathbf{u}}^n =\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}^{n}$\n\\item Compute the right-hand side from the state, $\\mathbf{R}(\\tilde{\\mathbf{u}}^n)$\n\\item Project the right-hand side, $\\tilde{\\mathbf{V}}^T\\mathbf{R}(\\tilde{\\mathbf{u}}^n)$\n\\item Update the state $\\tilde{\\mathbf{a}}^{n+1} = \\tilde{\\mathbf{a}}^n + \\Delta t \\tilde{\\mathbf{V}}^T\\mathbf{R}(\\tilde{\\mathbf{u}}^n)$\n\\end{enumerate}\n\\end{algorithm}\n\n\\begin{table}[h!]\n\\begin{tabular}{p{7cm} p{8cm}}\n\\hline\nStep in Algorithm~\\ref{alg:alg_g_exp} & Approximate FLOPs \\\\\n\\hline\n1    & $2NK - N$  \\\\\n2    & $\\omega N$  \\\\\n3    & $2NK - K$  \\\\\n4    & $2K $  \\\\\n\\hline\nTotal    & $4 N K + (\\omega-1) N + K$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Approximate floating-point operations for an explicit Euler update to the Galerkin method reported in Algorithm~\\ref{alg:alg_g_exp}.}\n\\label{tab:alg_g_exp}\n\\end{table}\n\n\n\\begin{algorithm}[h!]\n\\caption{Algorithm for an implicit Euler update for the Galerkin ROM using Newton's Method with Gaussian Elimination}\n\\label{alg:alg_g_imp}\nInput: $\\tilde{\\mathbf{a}}^n$, residual tolerance $\\xi$ \\;\n\\newline\nOutput: $\\tilde{\\mathbf{a}}^{n+1}$\\;\n\\newline\nSteps:\n\\begin{enumerate}\n\\item Set initial guess, $\\tilde{\\mathbf{a}}_k$\n\\item  Loop while $\\mathbf{r}^k > \\xi$\n\\begin{enumerate}\n    \\item Compute the state from the generalized coordinates, $\\tilde{\\mathbf{u}}_k = \\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k$\n    \\item Compute the right-hand side from the full state, $\\mathbf{R}(\\tilde{\\mathbf{u}}_k)$\n    \\item Project the right-hand side, $\\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{u}}_k)$\n    \\item Compute the Galerkin residual, $\\mathbf{r}_G(\\tilde{\\mathbf{a}}_k) = \\tilde{\\mathbf{a}}_k - \\tilde{\\mathbf{a}}^n - \\Delta t \\tilde{\\mathbf{V}}^T \\mathbf{R}(\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_k)$\n    \\item Compute the residual Jacobian, $\\frac{\\partial \\mathbf{r}(\\tilde{\\mathbf{a}}_k)}{\\partial \\tilde{\\mathbf{a}}_k}$\n    \\item Solve the linear system via Gaussian Elimination: $\\frac{\\partial \\mathbf{r}(\\tilde{\\mathbf{a}}_k)}{\\partial \\tilde{\\mathbf{a}}_k} \\Delta  \\tilde{\\mathbf{a}} = - \\mathbf{r}(\\tilde{\\mathbf{a}}_k)$\n    \\item Update the state: $\\tilde{\\mathbf{a}}_{k+1} = \\tilde{\\mathbf{a}}_k + \\Delta \\tilde{\\mathbf{a}}$\n    \\item $k = k + 1$\n\\end{enumerate}\n\\item Set final state, $\\tilde{\\mathbf{a}}^{n+1} = \\tilde{\\mathbf{a}}_k$\n\\end{enumerate}\n\\end{algorithm}\n\n\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{p{7cm} p{8cm}}\n\\hline\nStep in Algorithm~\\ref{alg:alg_apg_imp}& Approximate FLOPs \\\\\n\\hline\n2a    & $2 N K - N $ \\\\\n2b    & $ \\omega N $ \\\\\n2c    & $2 N K - K$  \\\\\n2d    & $ 3K $  \\\\\n2e    & $ 4 N K^2 +  (\\omega - 1) N K + 2K^2 $ \\\\\n2f    & $ K^3 $ \\\\\n2g    & $ K $ \\\\\n\\hline\nTotal & $ (\\omega - 1)N + 3K + (\\omega + 3)NK + 2K^2 + 4NK^2 + K^3 $\n\\end{tabular}\n\\caption{Approximate floating-point operations for one Newton iteration for the implicit Euler update to the Galerkin method reported in Algorithm~\\ref{alg:alg_g_imp}.}\n\\label{tab:alg_g_imp}\n\\end{table}\n\n\\clearpage\n\n\\begin{algorithm}[H]\n\\caption{Algorithm for an implicit Euler update for the LSPG ROM using a Gauss-Newton method with Gaussian Elimination}\n\\label{alg:alg_LSPG}\nInput: $\\tilde{\\mathbf{a}}^n$, residual tolerance $\\xi$ \\;\n\\newline\nOutput: $\\tilde{\\mathbf{a}}^{n+1}$\\;\n\\newline\nSteps:\n\\begin{enumerate}\n\\item Set initial guess, $\\tilde{\\mathbf{a}}_k$\n\\item  Loop while $\\mathbf{r}_k > \\xi$\n\\begin{enumerate}\n    \\item Compute the state from the generalized coordinates, $\\tilde{\\mathbf{u}}_k =\\tilde{\\mathbf{V}} \\tilde{\\mathbf{a}}_{k}$\n    \\item Compute the right-hand side from the full state, $\\mathbf{R}(\\tilde{\\mathbf{u}}_k)$\n    \\item Compute the residual, $\\mathbf{r} (\\tilde{\\mathbf{u}}_k) = \\tilde{\\mathbf{u}}_k - \\tilde{\\mathbf{u}}^n - \\Delta t \\mathbf{R}(\\tilde{\\mathbf{u}}_k)$\n    \\item Compute the test basis, $\\mathbf{W}_k = \\frac{\\partial \\mathbf{r}(\\tilde{\\mathbf{u}}_k)}{\\partial \\tilde{\\mathbf{u}}_k} \\tilde{\\mathbf{V}} = \\frac{\\partial \\mathbf{r}(\\tilde{\\mathbf{u}}_k)}{\\partial \\tilde{\\mathbf{a}}_k} $\n    \\item Compute the product, $\\tilde{\\mathbf{W}}_k^T \\tilde{\\mathbf{W}}_k$\n    \\item Project the residual onto the test space, $\\tilde{\\mathbf{W}}^T \\mathbf{r}(\\tilde{\\mathbf{a}}_k)$\n    \\item Solve $\\tilde{\\mathbf{W}}^T \\tilde{\\mathbf{W}} \\Delta \\tilde{\\mathbf{a}} = - \\tilde{\\mathbf{W}}^T \\mathbf{r}(\\tilde{\\mathbf{a}}_k) $ for $\\Delta \\tilde{\\mathbf{a}}$ via Gaussian elimination\n    \\item Update solution, $\\tilde{\\mathbf{a}}_{k+1} = \\tilde{\\mathbf{a}}_k + \\Delta \\tilde{\\mathbf{a}}$\n    \\item k = k + 1\n\\end{enumerate}\n\\item Set final state, $\\tilde{\\mathbf{a}}^{n+1} = \\tilde{\\mathbf{a}}_k$ \n\\end{enumerate}\n\\end{algorithm}\n\n\n\\begin{table}[H]\n\\begin{tabular}{p{7cm} p{8cm}}\n\\hline\nStep in Algorithm~\\ref{alg:alg_LSPG}& Approximate FLOPs \\\\\n\\hline\n2a    & $ 2NK - N $ \\\\\n2b    & $ \\omega N $  \\\\\n2c    & $ 3N $  \\\\\n2d    & $ (\\omega + 2)NK + 2NK^2  $  \\\\\n2e    & $ 2NK^2 - K^2 $ \\\\\n2f    & $ 2NK - K $ \\\\\n2g    & $ K^3 $ \\\\\n2h    & $ K $ \\\\\n\\hline\nNewton Iteration Total & $ (\\omega + 2)N + (\\omega + 6) NK - K^2 + 4NK^2 + K^3 $ \\\\\n\n\\end{tabular}\n\\caption{Approximate floating-point operations for one Newton iteration for the implicit Euler update to the LSPG method reported in Algorithm~\\ref{alg:alg_LSPG}.}\n\\label{tab:alg_LSPG}\n\\end{table}\n\n\n\n\\end{appendices}\n\n\\clearpage\n\n\\bibliographystyle{aiaa}\n\n\n\n\\subsection{Example 2: Flow Over Cylinder}\\label{sec:cylinder}\nThe second case considered is viscous compressible flow over a circular cylinder. The flow is described by the two-dimensional compressible Navier-Stokes equations. A Newtonian fluid and a calorically perfect gas are assumed.\n\\begin{comment} \nThe flow is described by the two-dimensional compressible Navier-Stokes equations,\n\\begin{equation}\\label{eq:compressible_ns}\n\\frac{\\partial \\mathbf{u}}{\\partial t} + \\nabla \\cdot \\big( \\mathbf{F}(\\mathbf{u} ) - \\mathbf{F}_v (\\mathbf{u},\\nabla \\mathbf{u} )      \\big) =0,\n\\end{equation}\nwhere $\\mathbf{F}$ and $ \\mathbf{F}_v$ are the inviscid and viscous fluxes, respectively. For a two-dimensional flow the state vector and inviscid fluxes are,\n$$\n\\mathbf{u} = \\begin{Bmatrix}\n\\rho \\\\ \\rho u_1 \\\\ \\rho u_2 \\\\ \\rho E \\end{Bmatrix}, \\qquad \\mathbf{F}_{1} = \\begin{Bmatrix} \\rho u_1 \\\\ \\rho u_1^2 +      p \\\\ \\rho u_1 u_2 \\\\ u_1(E + p) \\end{Bmatrix}, \n\\qquad \\mathbf{F}_{2} = \\begin{Bmatrix} \\rho u_2 \\\\ \\rho u_1 u_2  \\\\ \\rho u_2^2 + p \\\\ u_2(E + p) \\end{Bmatrix}.\n$$\nThe viscous fluxes are given by,\n$$\n\\qquad \\mathbf{F}_{v_1} = \\begin{Bmatrix} 0 \\\\ \\tau_{11} \\\\ \\tau_{12}  \\\\ u_j \\tau_{j1} + c_p \\frac{\\mu}{\\text{Pr}} \\frac{\\partial T}{\\partial x_1}  \\end{Bmatrix}, \n\\qquad \\mathbf{F}_{v_2} = \\begin{Bmatrix} 0 \\\\ \\tau_{21} \\\\ \\tau_{22}  \\\\ u_j \\tau_{j2} + c_p \\frac{\\mu}{\\text{Pr}} \\frac{\\partial T}{\\partial x_2}  \\end{Bmatrix}.\n$$\nWe assume a Newtonian fluid, which leads to a viscous stress tensor of the form,\n\\begin{equation*}\n\\tau_{ij} = 2\\mu S_{ij},\n\\end{equation*}\nwhere,\n\\begin{equation*}\n S_{ij} = \\frac{1}{2} \\big( \\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i} \\big) - \\frac{1}{3} \\frac{\\partial      u_k}{\\partial x_i} \\delta_{ij}.\n\\end{equation*}\nThe Navier-Stokes equations are closed with a constitutive relationship for a calorically perfect gas,\n$$p = (\\gamma - 1)( \\rho E - \\frac{1}{2} \\rho u_1^2 - \\frac{1}{2} \\rho u_2^2 \\big),$$\nwhere $\\gamma = 1.4$ is the heat-capacity ratio. \n\\end{comment}\n\\subsubsection{Full-Order Model}\\label{sec:cylinder_fom}\nThe compressible Navier-Stokes equations are solved using a discontinuous Galerkin (DG) method and explicit time integration. Spatial discretization with the discontinuous Galerkin method leads to a semi-discrete system of the form,\n\\begin{equation*}\n \\frac{d {\\mathbf{u}}}{dt} = \\mathbf{M}^{-1}\\mathbf{f}({\\mathbf{u}}), \\qquad {\\mathbf{u}}(t=0) = {\\mathbf{u}}_0,\n\\end{equation*}\nwhere $\\mathbf{M}\\in \\mathbb{R}^{N \\times N}$ is a block diagonal mass matrix and $\\mathbf{f}({\\mathbf{u}}) \\in \\mathbb{R}^N$ is a vector containing surface and volume integrals. Thus, using the notation defined in Eq.~\\ref{eq:FOM}, the right-hand side operator for the DG discretization is defined as,\n$$\\mathbf{R}({\\mathbf{u}}) = \\mathbf{M}^{-1}\\mathbf{f}(\\tilde{\\mathbf{u}}).$$\n\nFor the flow over cylinder problem considered in this section, a single block domain is constructed in polar coordinates by uniformly discretizing in $\\theta$ and by discretizing in the radial direction by,\n$$r_{i+1} = r_i + r_i (R_g - 1),$$\nwhere $R_g$ is a stretching factor and is defined by,\n$$R_g = r_{max}^{1\/N_r}.$$\n\nThe DG method utilizes the Roe flux at the cell interfaces and uses the first form of Bassi and Rebay~\\cite{BR1} for the viscous fluxes. Temporal integration is again performed using a strong stability preserving RK3 method. Far-field boundary conditions and linear elements are used. Details of the FOM are presented in Table~\\ref{tab:cylinder}. \n\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|c | c | c | c | c | c | c | c | c | c |}\\hline\n $r_{max}$ & $N_r$ & $N_{\\theta}$ & $p_r$ & $p_{\\theta}$ & $\\Delta t$  & Mach & $a_{\\infty}$ & $p_{\\infty}$ & $T_{\\infty}$ \\\\ \\hline\n 60 & 80 & 80 &  3 & 3  & $5e-3$ & 0.2 & $1.0$ & $1.0$ & $\\gamma^{-1}$ \\\\\\hline\n\\end{tabular}\n\\caption{Details used for flow over cylinder problem. In the above table, $N_r$ and $N_{\\theta}$ are the number of cells in the radial and $\\theta$ direction, respectively. Similarly, $p_r$ and $p_{\\theta}$ are the polynomial orders in the radial and $\\theta$ direction. Lastly, $a_{\\infty}$, $p_{\\infty}$, and $T_{\\infty}$ are the free-stream speed of sound, pressure, and temperature.}\n\\label{tab:cylinder}\n\\end{table}\n\n\n\\subsubsection{Solution of the Full-Order Model and Construction of the ROM Trial Space}\\label{sec:cyl_romsteps}\nFlow over a cylinder at Re=$100,200,$ and $300$, where Re=$ \\rho_{\\infty} U_{\\infty} D \/ \\mu$ is the Reynolds number, are considered. These Reynolds numbers give rise to the well studied von K\\'arm\\'an vortex street. Figure~\\ref{fig:vonkarman} shows the FOM solution at Re=100 for several time instances to illustrate the vortex street.\n\nThe FOM is used to construct the trial spaces used in the ROM simulations. The process used to construct these trial spaces is as follows:\n\\begin{enumerate}\n    \\item Initialize FOM simulations at Reynold's numbers of Re=$100,200,$ and $300.$ The Reynold's number is controlled by raising or lowering the viscosity.\n    \\item Time-integrate the FOM at each Reynolds number until a statistically steady-state is reached.\n    \\item Once the flow has statistically converged to a steady state, reset the time coordinate to be $t=0$, and solve the FOM for $t \\in [0,100]$.\n    \\item Take snapshots of the FOM solution obtained from Step 3 at every $t=0.5$ time units over a time-window of $t \\in [0,100]$ time units, for a total of $200$ snapshots at each Reynolds number. This time window corresponds to roughly two cycles of the vortex street, with $100$ snapshots per cycle.\n    \\item  Assemble the snapshots from each case into one global snapshot matrix of dimension $N \\times 600$. This snapshot matrix is used to construct the trial subspace through POD. Note that only one set of basis functions for all conserved variables is constructed. \n    \\item Construct trial spaces of dimension $N\\times 11$, $N\\times 43$, and $N \\times 87$. These subspace dimensions correspond to an energy criterion of $99, 99.9,$ and $99.99\\%$. The different trial spaces are summarized in Table~\\ref{tab:rom_basis_1}.\n\\end{enumerate}\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.05\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/colorbar.png}\n\\label{fig:vonkarman0}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.3\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/cyl0000.png}\n\\caption{$x$-velocity at $t=0.0$}\n\\label{fig:vonkarman1}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.3\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/cyl0005.png}\n\\caption{$x$-velocity at $t=25.0$}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.3\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/cyl0010.png}\n\\caption{$x$-velocity at $t=50.0$}\n\\label{fig:vonkarman3}\n\\end{subfigure}\n\\end{center}\n\\caption{Evolution of the von K\\'arm\\'an vortex street at Re=100.}\n\\label{fig:vonkarman}\n\\end{figure}\n\n\\subsubsection{Solution of the Reduced-Order Models}\nThe G ROM, APG ROM, and LSPG ROMs are considered. Details on their implementation are as follows:\n\\begin{enumerate}\n\\item Galerkin ROM: The Galerkin ROM is evolved in time using both explicit and implicit time integrators. In the explicit case, a strong stability RK3 method is used. In the implicit case, Crank-Nicolson time integration is used. The non-linear algebraic system is solved using SciPy's Jacobian-Free Netwon-Krylov solver. LGMRES is employed as the linear solver. The convergence tolerance for the max-norm of the residual is set at the default ftol=6e-6. \n\n\\item Adjoint Petrov-Galerkin ROM: The APG ROM is evolved in time using the same time integrators as the Galerkin ROM. The extra term appearing in the APG model is computed via finite difference\\footnote{It is noted that, when used in conjunction with a JFNK solver that utilizes finite difference to approximate the action of the Jacobian on a vector, approximating the extra RHS term in APG via finite difference leads to computing the finite difference approximation of a finite difference approximation. While not observed in the examples presented here, this can have a detrimental effect on accuracy and\/or convergence.} with a step size of $\\epsilon = $1e-5. Unless noted otherwise, the memory length $\\tau$ is selected to be $\\tau = \\frac{0.2}{\\rho(\\tilde{\\mathbf{V}}^T \\mathbf{J}[\\tilde{\\mathbf{u}}(0)] \\tilde{\\mathbf{V}} ) } .$ The impact of $\\tau$ on the numerical results is considered in the subsequent sections. \n\n\\item LSPG ROM: The LSPG ROM is formulated from an implicit Crank-Nicolson temporal discretization. The resulting non-linear least-squares problem is solved using SciPy's least-squares solver with the `dogbox' method~\\cite{scipy_leastsquares_dogbox}. The tolerance on the change to the cost function is set at ftol=1e-8. The tolerance on the change to the generalized coordinates is set at xtol=1e-8. The SciPy least-squares solver is comparable in speed to our own least-squares solver that utilizes the Gauss-Newton method with a thin QR factorization to solve the least-squares problem. The SciPy solver, however, was observed to be more robust in driving down the residual than the basic Gauss-Newton method with QR factorization, presumably due to SciPy's inclusion of trust-regions, and hence results are reported with the SciPy solver.\n\\end{enumerate}\nAll ROMs are initialized with the solution of the Re=$100$ FOM at time $t=0$, the $x$-velocity of which is shown in Figure~\\ref{fig:vonkarman1}.\n\n\n\n\\begin{table}[]\n\\begin{center}\n\\begin{tabular}{c c c c}\n\\hline\nBasis \\# & Trial Basis Dimension ($K$) &  Energy Criteria  & $\\tau$ (Adjoint Petrov-Galerkin) \\\\\n\\hline\n1    & $ 11$ & $99\\%$  & $1.0$ \\\\\n2    & $ 42$ & $99.9\\%$  & $0.3$   \\\\\n3    & $ 86$ & $99.99\\%$  & $0.1$  \\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of the various basis dimensions used for Example 2}\n\\label{tab:rom_basis_1}\n\\end{center}\n\\end{table}\n\n\n\n\\subsubsection{Reconstruction of Re=100 Case}\\label{sec:re100_rec}\nReduced-order models of the Re=100 case are first considered. This case was explicitly used in the construction of the POD basis and tests the ability of the ROM to reconstruct previously ``seen\" dynamics. Unless otherwise noted, the default time-step for all ROMs is taken to be $\\Delta t = 0.5$. The values of $\\tau$ used in the APG ROMs are selected from the spectral radius heuristic and are given in Table~\\ref{tab:rom_basis_1}. Figures~\\ref{subfig:re100a} and~\\ref{subfig:re100b} show the lift coefficient as well as the mean squared error (MSE) of the full-field ROM solutions for the G ROM, APG ROM, and LSPG ROMs for Basis \\#2, while Figure~\\ref{subfig:re100c} shows the integrated MSE for $t \\in [0,200]$ for Basis \\#1, 2, and 3. Figure~\\ref{subfig:re100d} shows the integrated error as a function of relative CPU time for the various ROMs. The relative CPU time is defined with respect to the FOM, which is integrated with an explicit time-step 100 times lower than the ROMs. The lift coefficients predicted by all three ROMs are seen to overlay the FOM results. The mean squared error shows that, for a given trial basis dimension, the APG ROM is more accurate than both the Galerkin and LSPG ROMs. This is the case for both explicit and implicit time integrators. As shown in Figure~\\ref{subfig:re100c}, the APG ROM converges at a similar rate to the G ROM as the dimension of the trial space grows. For Basis \\#2 and \\#3, the implicit time-marching schemes are slightly less accurate than the explicit time-marching schemes. Finally, Figure~\\ref{subfig:re100d} shows that, for a given CPU time, the G ROM with explicit time-marching produces the least error. The APG ROM with explicit time-marching is the second-best performing method. In the implicit case, both the G and APG ROMs lead to lower error at a given CPU time than LSPG. This decrease in cost is due to the fact that the G and APG ROMs utilize Jacobian-Free Netwon-Krylov solvers. As discussed in Section~\\ref{sec:cost}, it is much more challenging for LSPG to utilize Jacobian-Free methods. Due to the increased cost associated with implicit solvers, only explicit time integration is used for the G ROM and APG ROM beyond this point. \n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/cl_999case.pdf}\n\\caption{Lift Coefficient as a function of time for Basis \\# 2.}\n\\label{subfig:re100a}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/mse_vs_t_999case.pdf}\n\\caption{Normalized error as a function of time for Basis \\# 2.}\n\\label{subfig:re100b}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/converge.pdf}\n\\caption{Integrated normalized error vs trial subspace dimension.}\n\\label{subfig:re100c}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/rom_pareto.pdf}\n\\caption{Integrated normalized error vs relative CPU time.}\n\\label{subfig:re100d}\n\\end{subfigure}\n\\end{center}\n\\caption{ROM results for flow over cylinder at Re=100. The lift coefficient is defined as $C_L = \\frac{2L}{\\rho U_{\\infty}^2 D}$, with $L$ being the integrated force on the cylinder perpendicular to the free-stream velocity vector.}\n\\label{fig:re100}\n\\end{figure}\n\nNext, we investigate the sensitivity of the different ROMs to the time-step size. Reduced-order models of the Re=$100$ case using Basis \\#2 are solved using time-steps of $\\Delta t = \\big[0.1,0.2,0.5,1]$. Note that the largest time-step considered is $200$ times larger than the FOM time-step, thus reducing the temporal dimensionality of the problem by 200 times. The mean-squared error of each ROM solution is shown in Figure~\\ref{fig:re100_dtvary}. The G and APG ROMs are stable for all time-steps considered. Further, it is seen that varying the time-step has a minimal effect on the accuracy of the G and APG ROMs. In contrast, the accuracy of LSPG deteriorates if the time-step grows too large. This is due to the fact that, as shown in Ref.~\\cite{carlberg_lspg_v_galerkin}, the stabilization added by LSPG depends on the time-step size. Optimal accuracy of the LSPG method requires an intermediate time-step. The ability of the APG and G ROMs to take large time-steps without a significant degradation in accuracy allows for significant computational savings. This advantage is further amplified when large time-steps can be taken with an explicit solver, as is the case here. This will be discussed in more detail in Section~\\ref{sec:cylhyper}.\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/mse_vs_t_dtvary}\n\\caption{Normalized error as a function of time.}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/converge_dtvary.pdf}\n\\caption{Integrated normalized error as a function of the time-step.}\n\\end{subfigure}\n\\end{center}\n\\caption{Results for time-step study of flow over cylinder at Re$=100$.}\n\\label{fig:re100_dtvary}\n\\end{figure}\n\nLastly, we numerically investigate the sensitivity of APG to the parameter $\\tau$ by running simulations for $\\tau=[0.001,0.01,0.1,0.3,0.5,1.]$. All simulations are run at $\\Delta t = 0.5$. The results of the simulations are shown in Figure~\\ref{fig:re100_tau}. It is seen that, for all values of $\\tau$, the APG ROM produces a better solution than the G ROM. The lowest error is observed for an intermediate value of $\\tau$, in which case the APG ROM leads to over a $50\\%$ reduction in error from the G ROM. As $\\tau$ approaches zero, the APG ROM solution approaches the Galerkin ROM solution. Convergence plots for LSPG as a function of $\\Delta t$ are additionally shown in Figure~\\ref{subfig:re100b_tau}. It is seen that the optimal time-step in LSPG is similar to the optimal value of $\\tau$ in APG.\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.45\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/mse_vs_t_tauvary.pdf}\n\\caption{Normalized error as a function of time}\n\\label{subfig:re100a_tau}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.45\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/converge_tauvary.pdf}\n\\caption{Integrated normalized error as a function of $\\tau$. Results for how the time-step $\\Delta t$ impacts LSPG are included for reference.}\n\\label{subfig:re100b_tau}\n\\end{subfigure}\n\\end{center}\n\\caption{Summary of numerical results investigating the impact of the parameter $\\tau$ on the performance of the Adjoint Petrov-Galerkin method.}\n\\label{fig:re100_tau}\n\\end{figure}\n\n\n\\begin{comment}\n\n\\subsubsection{Prediction at Re$=150$}\nSimulations at a Reynolds number of Re$=150$ are now considered. The Re$=150$ case was not considered in the construction of the POD basis, and thus this case tests the predictive ability of the ROM. The case is initialized using the flow field from the Re$=100$ simulation. The Reynolds number is modified by lowering the viscosity. Figure~\\ref{fig:re150_unsteady} shows the temporal evolution of the lift coefficient as predicted by the FOM, G ROM, APG ROM, and LSPG ROM. The G ROM and LSPG ROMs are seen to behave similarly for this case, both under-predicting the growth in amplitude of the lift coefficient and shedding frequency. The solution generated by the APG ROM provides a better prediction for both the lift coefficient, as well as the full-field mean-squared error. By $t=200$, the instantaneous MSE of the APG ROM is approximately an order of magnitude better than both the G ROM and LSPG ROM. It is interesting to again observe that, at early time, the APG ROM is less accurate than both the G ROM and LSPG ROM. This further supports the theoretical analysis in Section~\\ref{sec:analysis}. \n \\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/re150_cl999case.pdf}\n\\caption{Lift coefficient prediction}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/re150_mse_vs_t_999case.pdf}\n\\caption{Extended time integration of ROM cases}\n\\label{fig:re150_longtime}\n\\end{subfigure}\n\\end{center}\n\\caption{Temporal evolution of cylinder case for Re$=150$.}\n\\label{fig:re150_unsteady}\n\\end{figure}\n\n\\end{comment}\n\n\\subsubsection{Parametric Study of Reynolds Number Dependence}\nNext, the ability of the different ROMs to interpolate between between different Reynolds numbers is studied. Simulations at Reynolds numbers of Re=100,150,200,250, and 300 with Basis $\\#2$ and $\\#3$ are performed.  All cases are initialized from the Re=100 simulation. Note that the trial spaces in the ROMs were constructed from statistically steady-state FOM simulations of Re=100,200,300. The Reynolds number is modified by changing the viscosity.\n\nFigure~\\ref{fig:ROM_summary} summarizes the amplitude of the lift coefficient signal as well as the shedding frequency for the various methods. The values reported in Figure~\\ref{fig:ROM_summary} are computed from the last 150s of the simulations\\footnote{Not all G ROMs reached a statistically steady state over the time window considered}. The Galerkin ROM is seen to do poorly in predicting the lift coefficient amplitude for both Basis $\\#2$ and Basis $\\#3$. Unlike in the Re=100 case, enhancing the basis dimension does not improve the performance of the ROMs. Both the LSPG and APG ROMs are seen to offer much improved predictions over the Galerkin and ROM. This result is promising, as the ultimate goal of reduced-order modeling is to provide predictions in new regimes.\n\nThe results presented in this example highlight the shortcomings of the Galerkin ROM. To obtain results that are even qualitatively correct for the Re=\\{150,200,250,300\\} cases, either APG or LSPG must be used. As reported in Figure~\\ref{subfig:re100d}, explicit APG is over an order of magnitude faster than LSPG, and implicit APG with a JFNK solver is anywhere from 2x to 5x faster than LSPG. Therefore, APG is the best-performing method for this example. \n\n\n\n \\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/re_vs_amp.pdf}\n\\caption{Prediction for lift coefficient amplitudes}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/re_vs_st.pdf}\n\\caption{Prediction for shedding frequency}\n\\end{subfigure}\n\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/cl_re150.pdf}\n\\caption{Re=$150$}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/cl_re200.pdf}\n\\caption{Re=$200$}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/cl_re250.pdf}\n\\caption{Re=$250$}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/cl_re300.pdf}\n\\caption{Re=$300$}\n\\end{subfigure}\n\n\n\\end{center}\n\\caption{Summary of ROM simulations}\n\\label{fig:ROM_summary}\n\\end{figure}\n\n\n\n\\subsubsection{Flow Over Cylinder at Re=$100$ with Hyper-Reduction}\\label{sec:cylhyper}\nThe last example considered is again flow over a cylinder, but this time the reduced-order models are augmented with hyper-reduction. The purpose of this example is to examine the performance of the different reduced-order models when fully equipped with state-of-the-art reduction techniques. For hyper-reduction, an additional snapshot matrix of the right-hand side is generated. This additional snapshot matrix is generated by following steps one through five provided in Section~\\ref{sec:cyl_romsteps}. Hyper-reduction for the G ROM and the APG ROM is achieved through the Gappy POD method~\\cite{everson_sirovich_gappy}. Hyper-reduction for LSPG is achieved through collocation using the same sampling points.\\footnote{It is noted that collocated LSPG out-performed the GNAT method for this example, and thus GNAT is not considered.} When augmented with hyper-reduction, the trial basis dimension ($K$), right-hand side basis dimension ($r$), and number of sample points $(N_s)$ can impact the performance of the ROMs. Table~\\ref{tab:rom_basis_details} summarizes the various permutations of $K$, $r$, and $N_s$ considered in this example. The sample points are selected through a QR factorization of the right-hand side snapshot matrix~\\cite{qdeim_drmac}. These sample points are then augmented such that they contain every conserved variable and quadrature point at the selected cells. The sample mesh corresponding to Basis numbers 4,5, and 6 in Table~\\ref{tab:rom_basis_details} is shown in Figure~\\ref{fig:re100_sample}. Details on hyper-reduction and its implementation in our discontinuous Galerkin code are provided in Appendix~\\ref{appendix:hyper}.  \n\n\\begin{table}[]\n\\begin{tabular}{c c c c c}\n\\hline\nBasis \\# & Trial Basis Dimension ($K$) &  RHS Basis Dimension ($r$)  & Sample Points ($N_S$) & Maximum Stable $\\Delta t$\\\\\n\\hline\n1    & $ 11$ & $103$ & $4230$ & 4.0\\\\\n2    & $ 42$ & $103$ & $4230$ & 2.0\\\\\n3    & $ 86$ & $103$ & $4230$ & 1.0\\\\\n4    & $ 11$ & $268$ & $8460$ & 4.0\\\\\n5    & $ 42$ & $268$ & $8460$ & 2.0\\\\\n6    & $ 86$ & $268$ & $8460$ & 1.0\\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of the various reduced-order models evaluated on the flow over cylinder problem. The maximum stable $\\Delta t$ for each basis is reported for the SSP-RK3 explicit time-marching scheme and was empirically determined.}\n\\label{tab:rom_basis_details}\n\\end{table}\n\n\n\nFlow at Re=$100$ is considered. All simulations are performed at the maximum stable time-step for a given basis dimension, as summarized in Table~\\ref{tab:rom_basis_details}. Figure~\\ref{fig:re100_qdeim_pareto} shows the integrated error as a function of relative wall time for the various ROM techniques and basis numbers. All methods  show significant computational speedup, with the G and APG ROMs producing wall-times up to 5000 times faster than the FOM while retaining a reasonable MSE. It is noted that the majority of this speed up is attributed to the increase in time-step size.\nFor a given level of accuracy, LSPG is significantly more expensive than the G and APG ROMs. The reason for this expense is three-fold. First, LSPG is inherently implicit. For a given time-step size, an implicit step is more expensive than an explicit step. Second, LSPG requires an intermediate time-step for optimal accuracy. In this example, this intermediate time-step is small enough that the computational gains that could be obtained with an implicit method are negated. The third reason is that, for each Gauss-Newton iteration, LSPG requires the computation of the action of the Jacobian on the trial space basis, $\\tilde{\\mathbf{V}}$. This expense can become significant for large basis dimensions as it requires the computation of a dense Jacobian. It is noted that it may be possible to achieve computational speed-ups in our implementation of LSPG through the development of a least-squares solver more tailored to LSPG, sparse Jacobian updates, or Jacobian approximation techniques.\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\includegraphics[trim={4cm 0cm 4cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/samplemesh1.png}\n\\caption{Full sample mesh}\n\\label{subfig:re100_sampling_full}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\includegraphics[trim={0cm 0cm 4cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/samplemesh_wake2.png}\n\\caption{Close up of wake}\n\\label{subfig:re100_sampling_zoom}\n\\end{subfigure}\n\\end{center}\n\\caption{Mesh used for hyper-reduction. Cells colored in red are the sampled cells. Note that all conserved variables and quadrature points are computed within a cell.}\n\\label{fig:re100_sample}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/qdeim_pareto.pdf}\n\\caption{Wall time vs normalized error for hyper-reduced ROMs.}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/qdeim_mse_999case.pdf}\n\\caption{Normalized error as a function of time for Basis \\#5}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/qdeim_cl_999case.pdf}\n\\caption{Lift coefficient as a function of time for Basis \\# 5.}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\n\\includegraphics[trim={0cm 0cm 0cm 0cm},clip,width=1.\\linewidth]{figs_cylnew\/qdeim_cl_9999case.pdf}\n\\caption{Lift coefficient as a function of time for Basis \\# 6.}\n\\end{subfigure}\n\n\\end{center}\n\\caption{Results for hyper-reduced ROMs at Re$=100$.}\n\\label{fig:re100_qdeim_pareto}\n\\end{figure}\n\n\n\n\\subsubsection{Discussion}\\label{sec:theorem_discussion}\nTheorems~\\ref{theorem:apg_error} and~\\ref{theorem:errorbound_symmetric} contain three interesting results that are worth discussing. First, as discussed in Corollary~\\ref{corollary:resid_bound}, Theorem~\\ref{theorem:apg_error} shows that, in the limit $\\tau \\rightarrow 0^+$, the upper-bound provided in Theorem~\\ref{theorem:residualbounds} on the error introduced at time $t$ in the APG ROM due to the fine-scales is less than that introduced in the Galerkin ROM (per unit $\\tau$). This is an appealing result as APG is derived as a subgrid-scale model. Two remarks are worth making regarding this result. First, while the result was demonstrated in the limit $\\tau \\rightarrow 0^+$, it will hold so long as the norm of the truncation error in the quadrature approximation is less than the norm of the integral it is approximating; i.e. the approximation is doing a better job than neglecting the integral entirely. Second, the result derived in Theorem~\\ref{theorem:apg_error} does not \\textit{directly} translate to showing that,\n\\begin{equation}\\label{eq:resid_discussion}\n\\norm{\\tilde{\\mathbf{V}}^T \\mathbb{P}_A \\mathbf{r}_F(\\tilde{\\mathbf{u}}_F(t)) } \\le \\norm{\\tilde{\\mathbf{V}}^T \\mathbb{P}_G \\mathbf{r}_F(\\tilde{\\mathbf{u}}_F(t)) }.\n\\end{equation}\nThis is a consequence of Theorem~\\ref{theorem:residualbounds}, in which the integral that defines the fine-scale solution was split into two intervals. The APG ROM attempts to approximate the first integral, while the Galerkin ROM ignores both terms. Theorem~\\ref{theorem:apg_error} showed that, in the limit $\\tau \\rightarrow 0^+$, the APG approximation to the first integral is better than in the case of Galerkin (i.e., the APG approximation is better than no approximation). The only time that the result provided in Theorem~\\ref{theorem:apg_error} will \\textit{not} translate to APG providing a better approximation to the \\textit{entire} integral (and thus proving Eq.~\\ref{eq:resid_discussion}), is when integration over the second interval ``cancels out\" the integration over the first interval. To make this idea concrete, consider the integral,\n$$\\int_0^{2 \\pi} \\sin(x) dx = \\int_0^{\\pi} \\sin(x) dx  + \\int_{\\pi}^{2 \\pi} \\sin(x) dx .$$\nClearly, $\\int_0^{2 \\pi} \\sin(x) dx = 0$. If the entire integral was approximated using just the interval $0$ to $\\pi$ (which is analogous to APG), then one would end up with the approximation  $\\int_0^{2 \\pi} \\sin(x) dx \\approx \\int_0^{\\pi} \\sin(x) dx = 2$. Alternatively, if one were to ignore the integral entirely (which is analogous to Galerkin) and make the approximation $\\int_0^{2 \\pi} \\sin(x) dx \\approx 0$  (which in this example is exact), a better approximation would be obtained.\n\n The next interesting result is presented in Theorem~\\ref{theorem:errorbound_symmetric}, where it is shown that for a self-adjoint system with negative eigenvalues, the eigenvalues associated with the APG ROM error equation are \\textit{greater} than the Galerkin ROM. This implies that APG is \\textit{less} dissipative than Galerkin, and means that errors may be slower to decay in time. Thus, while Theorem~\\ref{theorem:apg_error} shows that the \\textit{a priori} contributions to the error due to the closure problem may be smaller in the APG ROM than in the Galerkin ROM, the errors that \\textit{are} incurred may be slower to decay. Finally, Corollary~\\ref{corollary:errorbound_symmetric} shows that, for self-adjoint systems with negative eigenvalues, the bounds on the parameter $\\tau$ such that all eigenvalues associated with the evolution of the error in APG remain negative depends on the spectral content of the Jacobian of $\\mathbf{A}$. This observation has been made heuristically in Ref~\\cite{parishMZ1}. Although the upper bound on $\\tau$ in Eq.~\\ref{eq:tau_bound} is very conservative due to the repeated use of inequalities, it provides insight into the selection and behaviour of $\\tau$.\n\n\n\n\n\\section{Significance}\n\n\nThis work develops a new reduced-order modeling technique for discretely projected ROMs. We show that the method outperforms the standard Galerkin-ROM and, in some cases, the popular least-squares Petrov-Galerkin approach. This work is novel to both the reduced-order modeling community and the Mori-Zwanzig community. For the reduced-order modeling community, the novel aspects and contributions of this work are:\n\\begin{enumerate}\n\\item A new reduced-order model derived from the Mori-Zwanzig formalism and variational multiscale method that is developed specifically for discretely projected ROMs. The method leads to a coarse-scale ROM equation that is driven by the coarse-scale residual. The method can be evolved in time with explicit integrators (in contrast to the popular least-squares Petrov Galerkin approach), potentially lowering the cost of the ROM. \n\n\\item An analysis of the present approach as a Petrov-Galerkin Method. This setting exposes similarities between the present approach and the least-squares Petrov-Galerkin (LSPG) approach. We find that the proposed method displays similarities to the well known adjoint stabilization technique, while LSPG displays similarities with the Galerkin least squares technique. \n\n\\item A comprehensive analysis of the algorithmic implementation of the proposed method as well as the estimation of the computational cost (in FLOPS) for the proposed method, Galerkin, and least-squares Petrov-Galerkin approaches. Implicit and explicit time marching schemes are considered.\n\n\\item Detailed $\\textit{a priori}$ error analysis, where we show circumstances in which the presented technique is expected to be more accurate than the Galerkin method.\n\n\\item Numerical evidence on ROMs of compressible flow problems demonstrating that the proposed method is more accurate and stable than a Galerkin-ROM. Improvements over the LSPG ROM are observed in most cases. \n\n\\item A thorough examination of errors vs CPU time for the presented method, Galerkin method, and LSPG method. We show that the presented method leads to lower errors for a given CPU time than the LSPG method. Improvements over Galerkin are observed in some cases. These results are presented for cases both with and without hyper-reduction.\n\n\\end{enumerate}\nThis work is also novel to the Mori-Zwanzig research community. The contributions include:\n\\begin{enumerate}\n\\item This is the first application of an analytical MZ method for a POD ROM. All previous work with analytic MZ models have focused on analytical basis.\n\\item By formulating MZ in the context of the variational multiscale method, we express the proposed model in a general \"residual-based\" context that allows us to apply it to complex systems. The applications presented in this work (POD-ROMs of the Sod shocktube and flow over a cylinder) mark a significant step up in complexity as compared to existing numerical problems that have been explored in the MZ community. In the authors' viewpoint, this is one of the first MZ methods that can, out of the box, be directly applied to complex systems of engineering interest.\n\\item We provide numerical evidence of a relationship between the time-scale in the proposed model and the spectral radius of the right-hand side Jacobian. This relationship applies to the selection of the optimal time-step in LSPG as well.\n\\end{enumerate}\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intr}\n\nThe Kontsevich integral was invented by M.~Kontsevich \\cite{Kon}\nas a tool to prove the fundamental theorem of the theory of\nfinite type (Vassiliev) invariants (see \\cite{BN1,BNe}).\nIt provides an invariant exactly as strong as the totality\nof all Vassiliev knot invariants.\n\nThe Kontsevich integral is defined for oriented tangles (either framed or\nunframed) in $\\R^3$, therefore it is also defined in the particular cases\nof knots, links and braids.\n\n$$\n  \\includegraphics[width=12cm]{tang_br_link_knot.eps}\n$$\n\nAs a starter, we give two examples where simple versions of the Kontsevich\nintegral have a straightforward geometrical meaning. In these examples, as\nwell as in the general construction of the Kontsevich integral, we represent\n3-space $\\R^3$ as the product of a real line $\\R$ with coordinate $t$ and a\ncomplex plane $\\C$ with complex coordinate $z$.\n\\smallskip\n\n\\noindent\n\\begin{minipage}{4.5in}\n\\textbf{Example 1.} The number of twists in a braid with two strings $z_1(t)$ \nand $z_2(t)$ placed in the slice $0\\le t\\le 1$ is equal to\n$$\\frac{1}{2\\pi i}\\int_0^1\\frac{dz_1-dz_2}{z_1-z_2}.$$\n\\end{minipage}\n\\quad\n\\raisebox{-12mm}{\\includegraphics[height=25mm]{2braid.eps}}\n\n\\medskip\n\n\\noindent\n\\begin{minipage}{4.5in}\n\\textbf{Example 2.} The linking number of two spatial curves\n$K$ and $K'$ can be computed as\n$$  \n  lk(K,K')=\\frac{1}{2\\pi i}\n          \\int_{m<t<M}\\sum_j \\varepsilon_j\n          \\frac{d(z_j(t)-z'_j(t))}{z_j(t)-z'_j(t)},\n$$\nwhere $m$ and $M$ are the minimum and the maximum values of $t$ on the\nlink $K\\cup K'$, $j$ is the index that enumerates all possible choices\nof a pair of strands of the link as functions $z_j(t)$, $z'_j(t)$ corresponding\nto $K$ and $K'$, respectively, and $\\varepsilon_j=\\pm1$ according to the parity\nof the number of chosen strands that are oriented downwards.\n\\end{minipage}\n\\quad\n\\raisebox{-17mm}{\\includegraphics[height=35mm]{lk_num.eps}}\n\\smallskip\n\nThe Kontsevich integral can be regarded as a far-going generalization of\nthese formulas. It aims at encoding all information about how the horizontal\nchords on the knot (or tangle) rotate when moved in the vertical direction.\nFrom a more general viewpoint, the Kontsevich integral represents the\nmonodromy of the Knizhnik--Zamolodchikov connection in the complement\nto the union of diagonals in $\\C^n$ (see \\cite{BN1,Oht}).\n\n\\section{Chord diagrams and weight systems}\n\\label{ch_diag}\n\n\\subsection{Algebras $\\A(p)$}\nThe Kontsevich integral of a tangle $T$ takes values in the space of chord\ndiagrams supported on $T$.\n\nLet $X$ be an oriented one-dimensional manifold, that is, a collection\nof $p$ numbered oriented lines and $q$ numbered oriented circles. \nA chord diagram of order $n$\nsupported on $X$ is a collection of $n$ pairs of unordered points in $X$,\nconsidered up to an orientation- and component-preserving diffeomorphism.\nIn the vector space formally generated by all chord diagrams of order $n$\nwe distinguish the subspace spanned by all \\textit{four-term relations}\n\\begin{center}\n\\includegraphics[width=8cm]{4Trel.eps}\n\\end{center}\nwhere thin lines designate chords, while thick lines are pieces\nof the manifold $X$. Apart from the shown fragments, all the four diagrams\nare identical. The quotient space over all such combinations\nwill be denoted by $\\A_n(X)=\\A_n(p,q)$. \nLet $\\A(p,q)=\\oplus_{n=0}^\\infty\\A_n(p,q)$\nand let $\\hat\\A(p,q)$ be the graded completion of $\\A(p,q)$ (i.e. the space of \nformal infinite series $\\sum_{i=0}^\\infty a_i$ with $a_i\\in\\A_i(p,q)$.\nIf, moreover, we divide $\\A(p,q)$ by all \\textit{framing independence}\nrelations (any diagram with an isolated chord, i.e. a chord joining\ntwo adjacent points of the same connected component of $X$, is set to 0),\nthen the resulting space is denoted by $\\A'(p,q)$, and its graded completion \nby $\\hat\\A'(p,q)$.\n\nThe spaces $\\A(p,0)=\\A(p)$ have the structure of an algebra (the product of\nchord diagrams is defined by concatenation of underlying manifolds in\nagreement with the orientation). Closing a line component into a\ncircle, we get a linear map $\\A(p,q)\\to\\A(p-1,q+1)$ which is an isomorphism\nwhen $p=1$. In particular, $\\A(S^1)\\cong\\A(\\R^1)$ has the structure of an\nalgebra; this algebra is denoted simply by $\\A$; the Kontsevich integral \nof knots takes its values in its graded completion $\\hat\\A$. \nAnother algebra of special importance is $\\hat\\A(3)=\\hat\\A(3,0)$, because it\nis where the Drinfeld associators live.\n\n\\subsection{Hopf algebra structure}\nThe algebra $\\A(p)$ has a natural structure of a Hopf algebra with\nthe coproduct $\\delta$ defined by all ways to split the set of chords\ninto two disjoint parts. To give a \nconvenient description of its primitive space, one can use\ngeneralized chord diagrams. We now allow \ntrivalent vertices not belonging to\nthe supporting manifold and use STU relations\n\\begin{center}\n\\includegraphics[width=5cm]{STU.eps}\n\\end{center}\nto express the generalized diagrams as linear combinations of conventional\nchord diagrams, e.g.\n\\begin{center}\n\\includegraphics[width=6cm]{STUex.eps}\n\\end{center}\n\nThen the primitive space coincides with the subspace of $\\A(p)$ spanned by\nall connected generalized chord diagrams (\\textit{connected} means that they\nremain connected when the supporting manifold $X$ is disregarded).\n\n\\subsection{Weight systems}\nA \\textit{weight system} of degree $n$ is a linear function on the\nspace $\\A_n$. Every Vassiliev invariant $v$ of degree $n$ defines a weight\nsystem  $\\mbox{symb}(v)$ of the same degree called its \\textit{symbol}.\n\n\\subsection{Algebras $\\B(p)$}\nApart from the spaces of chord diagrams modulo four-term relations,\nthere are closely related spaces of Jacobi diagrams.\nA \\textit{Jacobi diagram} is defined as a\nuni-trivalent graph, possibly disconnected, having at least one vertex of\nvalency 1 in each connected component and supplied with two additional\nstructures: a cyclic order of edges\nin each trivalent vertex and a labelling of univalent vertices taking values\nin the set $\\{1,2,\\dots,p\\}$.\nThe space $\\B(p)$ is defined\nas the quotient of the vector space formally generated by all \n$p$-coloured Jacobi diagrams modulo the two types of relations:\n\n\\begin{center}\nAntisymmetry: \\raisebox{-5mm}{\\includegraphics[height=11mm]{asf.eps}}\n\\qquad\nIHX: \\raisebox{-5mm}{\\includegraphics[height=11mm]{ihx.eps}}\n\\end{center}\n\nThe disjoint union of Jacobi diagrams makes the space $\\B(p)$ into an\nalgebra.\n\nThe symmetrization map $\\chi_p:\\B(p)\\to\\A(p)$, defined as the average\nover all ways to attach the legs of colour $i$ to $i$-th connected component\nof the underlying manifold:\n\n\\begin{center}\n\\raisebox{-7mm}{\\includegraphics[height=15mm]{isom_chi.eps}}\n\\end{center}\nis an isomorphism of vector spaces (the formal PBW isomorphism \\cite{BN1,LM2}) \nwhich is not\ncompatible with the multiplication. The relation between $A(p)$ and $\\B(p)$\nvery much resembles the relation between the universal enveloping algebra\nand the symmetric algebra of a Lie algebra.\nThe algebra $\\B=\\B(1)$ is used to write out the explicit formula for the\nKontsevich integral of the unknot (see \\cite{BLT} and below).\n\n\\section{The construction}\n\\label{defki}\n\n\\subsection{Kontsevich's formula}\nWe will explain the construction of the Kontsevich integral in the classical\ncase of (closed) oriented knots; for an arbitrary tangle $T$ the formula is \nthe same, only the result is interpreted as an element of $\\hat\\A(T)$.\nAs above, represent three-dimensional space $\\R^3$ as a direct product of a\ncomplex line $\\C$ with coordinate $z$ and a real line $\\R$ with coordinate\n$t$.\nThe integral is defined for Morse knots,\ni.~e. knots $K$ embedded in $\\R^3=\\C_z\\times\\R_t$\nin such a way that the coordinate $t$ restricted to $K$\nhas only nondegenerate (quadratic) critical points. (In fact, this condition\ncan be weakened, but the class of Morse knots is broad enough and \nconvenient to work with.)\n\nThe  Kontsevich integral $Z(K)$ of the knot $K$ is the following element\nof the completed algebra $\\hat\\A'$:\n$$\n  Z(K) = \\sum_{m=0}^\\infty \\frac{1}{(2\\pi i)^m}\n         \\int\\limits_{\\substack{\n    t_{\\mbox{\\tiny min}}<t_m<\\dots<t_1<t_{\\mbox{\\tiny max}}\\\\\n    t_j\\mbox{\\tiny\\ are noncritical}}}\\quad\n         \\sum_{P=\\{(z_j,z'_j)\\}} (-1)^{\\downarrow_P} D_P\n         \\bigwedge_{j=1}^m \\frac{dz_j-dz'_j}{z_j-z'_j} \\ .\n$$\n\n\\subsection{Explanation of the constituents}\n\nThe real numbers $t_{\\mbox{\\scriptsize min}}$ and \n$t_{\\mbox{\\scriptsize max}}$ are the minimum and the maximum of \nthe function $t$ on $K$.\n\nThe integration domain is the $m$-dimensional simplex\n$t_{\\mbox{\\scriptsize min}}<t_m<\\dots<t_1<t_{\\mbox{\\scriptsize max}}$ \ndivided by the critical values into a certain number of \n{\\it connected components}. For example,\nthe following picture shows an embedding of the unknot where, for $m=2$,\nthe integration domain has six connected components:\n \\begin{center}\n     \\begin{picture}(280,95)(0,0)\n      \\put(0,0){\\includegraphics[width=280pt]{idomf.eps}}\n     \\end{picture}\n  \\end{center}\n\nThe number of summands in the integrand is constant in each connected \ncomponent of the integration domain, but can be different for different \ncomponents.\nIn each plane $\\{t=t_j\\}\\subset\\R^3$ choose an unordered pair of distinct\npoints $(z_j,t_j)$ and $(z'_j,t_j)$ on $K$, so that $z_j(t_j)$\nand $z'_j(t_j)$ are continuous branches of the knot.\nWe denote by $P=\\{(z_j,z'_j)\\}$ the collection of such pairs for $j=1,\\dots,m$.\nThe integrand is the sum over all choices of the pairing $P$.\nIn the example above for the component\n$\\{t_{\\mbox{\\scriptsize min}}<t_1<c_1$, \n$c_2<t_2<t_{\\mbox{\\scriptsize max}}\\}$\nwe have only one possible pair of points on the levels $\\{t=t_1\\}$ and\n$\\{t=t_2\\}$. Therefore, the sum over $P$ for this component consists of\nonly one summand. Unlike this, in the component\n$\\{t_{\\mbox{\\scriptsize min}}<t_1<c_1$, $c_1<t_2<c_2\\}$\nwe still have only one possibility for the level $\\{t=t_1\\}$, but the plane\n$\\{t=t_2\\}$ intersects our knot $K$ in four points. So we have\n$\\binom{4}{2}=6$ possible pairs $(z_2,z'_2)$ and the total number\nof summands is six (see the picture below).\n\nFor a pairing $P$ the symbol `$\\downarrow_P$' denotes the number of points\n$(z_j,t_j)$ or $(z'_j,t_j)$ in $P$ where the coordinate $t$ decreases\nalong the orientation of $K$.\n\nFix a pairing $P$. Consider the knot $K$ as an oriented circle and connect\nthe points $(z_j,t_j)$ and $(z'_j,t_j)$ by a chord. Up to a diffeomorphism, \nthis chord does not depend on the value of $t_j$ within a connected\ncomponent. We obtain a chord diagram with $m$ chords.\nThe corresponding element of the algebra $\\A'$ is denoted by $D_P$.\nIn the picture below, for each connected component in our example, we show\none of the possible pairings, the corresponding chord diagram with the sign\n$(-1)^\\downarrow_P$ and the number of summands of the integrand (some of which\nare equal to zero in $\\A'$ due to the framing independence relation).\n\n\\begin{center}\n  \\includegraphics[width=12cm]{intdom.eps}\n\\end{center}\n\nOver each connected component, $z_j$ and $z'_j$ are smooth functions\nof $t_j$.\\\\ By\n${\\displaystyle \\bigwedge_{j=1}^m \\frac{dz_j-dz'_j}{z_j-z'_j}}$ we mean the\n{pullback} of this form to the integration domain of variables\n$t_1,\\dots,t_m$. The integration domain is considered with the \n{orientation} of the space $\\R^m$ defined by the natural order of the \ncoordinates $t_1$, ..., $t_m$.\n\nBy convention, the term in the Kontsevich integral corresponding to $m=0$\nis the (only) chord diagram of order 0 with coefficient one. It represents\nthe unit of the algebra $\\A'$.\n\n\\subsection{Framed version of the Kontsevich integral}\n\nLet $K$ be a framed oriented Morse knot with writhe number $w(K)$.\nDenote the corresponding knot without framing by $\\bar{K}$.\nThe framed version of the Kontsevich integral can be defined\nby the formula\n$$\n Z^{\\mbox{\\scriptsize fr}}(K) = \n e^{\\frac{w(K)}{2}\\Theta}\\cdot Z(\\bar{K})\\in\\hat\\A,\n$$\nwhere $\\Theta$ is the chord diagram with one chord and \nthe integral  $Z(\\bar{K})\\in\\hat\\A'$ is\nunderstood as an element of the completed algebra\n$\\hat\\A$\n (without 1-term relations) by virtue of a natural\ninclusion $\\A'\\to\\A$ defined as identity on the primitive subspace\nof $\\A'$.  \n\nSee \\cite{Gor,LM2} for other approaches.\n\n\\section{Basic properties}\n\\label{prop}\n\n\\subsection{Constructing the universal Vassiliev invariant}\nThe Kontsevich integral $Z(K)$\n\\begin{enumerate}\n\\item\n    converges for any Morse knot $K$,\n\\item\n   is invariant under deformations of the knot in the class \nof Morse knots,\n\\item\n   behaves in a predictable way under the deformation that adds a pair\nof new critical points to a Morse knot:\n$$\n  Z\\biggl(\\,\\raisebox{-12pt}{\\includegraphics[width=30pt]{humpt.eps}}\\,\\biggr)\n  = Z(H) \\cdot \n  Z\\biggl(\\,\\raisebox{-12pt}{\\includegraphics[width=30pt]{humpl.eps}}\\,\\biggr).\n$$\n\\end{enumerate}\n\nHere the first and the third pictures depict two embeddings of an \narbitrary knot, differing only in the shown fragment, \n$H = \\raisebox{-3mm}{\\includegraphics[width=7mm]{humpk.eps}}$\nis the {\\it hump} (unknot embedded in $\\R^3$ in the specified way),\nand the product is the product in the completed algebra $\\hat\\A'$ \nof chord diagrams. The last equality allows one to define a genuine knot\ninvariant by the formula \n$$\\label{I}\n  I(K) = Z(K)\/Z(H)^{c\/2},\n$$\nwhere $c$ denotes the number of critical points of $K$ and the ratio\nmeans the division in the algebra $\\hat\\A'$ according to the rule\n$(1+a)^{-1} = 1-a+a^2-a^3+\\dots$\n\nThe expression $I(K)$ is sometimes referred to as the `final' Kontsevich\nintegral as opposed to the `preliminary' Kontsevich integral\n$Z(K)$. It represents a universal Vassiliev invariant in the following\nsense:\n\\textit{Let $w$ be a weight system, i.e. a linear functional\non the algebra $\\hat\\A'$. Then the composition $w(I(K))$\nis a numerical Vassiliev invariant, and any Vassiliev invariant can \nbe obtained in this way.}\n\nThe final Kontsevich integral for framed knots is defined in the same\nway, using the hump $H$ with zero writhe number.\n\n\\subsection{Is universal Vassiliev invariant universal?}\nAt present, it is not known whether the Kontsevich integral separates knots,\nor even if it can tell the orientation of a knot. However, the\ncorresponding problem is solved, in the affirmative, in the case of\nbraids and string links (theorem of Kohno--Bar-Natan (\\cite{BN2,Koh}).\n\n\\subsection{Omitting long chords}\nWe will state a technical lemma which is highly important\nin the study of the Kontsevich integral. It is used in the proof of\nthe multiplicativity, in the combinatorial construction etc.\n\nSuppose we have a Morse knot $K$ with a distinguished tangle $T$.\n\n\\begin{center}\n\\includegraphics[width=250pt]{untaf.eps}\n\\end{center}\n\n\\noindent\nLet $m$ and $M$ be the maximal and\nminimal values of $t$ on the tangle $T$. \nIn the horizontal planes between the levels $m$ and $M$ we can \ndistinguish two kinds of chord: `short' chords that lie either inside\n$T$ or inside $K\\setminus T$, and `long' chords that connect\na point in $T$ with a point in $K\\setminus T$.\nDenote by $Z_T(K)$ the expression \ndefined by the same formula as the Kontsevich integral $Z(K)$ \nwhere only short chords are taken into consideration.\nMore exactly, if $C$ is a\nconnected component of the integration domain (see Section \\ref{defki})\nwhose projection on the coordinate axis $t_j$ is\nentirely contained in the segment $[m,M]$, then in the sum\nover the pairings $P$ we\ninclude only those pairings that include short chords.\n\n\\textbf{Lemma.}\n`Long' chords can be omitted when computing the Kontsevich integral:\n$Z_T(K)=Z(K)$.\n\n\\subsection{Kontsevich's integral and operations on knots}\n\nThe Kontsevich integral behaves in a nice way with respect to the natural\noperations on knots, such as mirror reflection,\nchanging the orientation of the knot, mutation of knots (see \\cite{CD}),\ncabling (see \\cite{Wil1}). We give some details regarding the first two\nitems.\n\\smallskip\n\n\\textbf{Fact 1.}\nLet $R$ be the operation that sends a knot to its mirror image. Define\nthe corresponding operation $R$ on chord diagrams as\nmultiplication by $(-1)^n$ where $n$ is the order of the diagram.\nThen the Kontsevich integral commutes with\nthe operation $R$: $Z(R(K))=R(Z(K))$,\nwhere by $R(Z(K))$ we mean simultaneous application of $R$ to all the\nchord diagrams participating in $Z(K)$.\n\n\\textsl{Corollary.}\nThe Kontsevich integral $Z(K)$ and the universal Vassiliev invariant \n$I(K)$ of an amphicheiral knot $K$ consist only of even order terms.\n(A knot $K$ is called {\\it amphicheiral},\nif it is equivalent to its mirror image: $K=R(K)$.)\n\\smallskip\n\n\\textbf{Fact 2.}\nLet $S$ be the operation on knots which inverts their orientation.\nThe same letter will also denote the analogous operation on chord\ndiagrams (inverting the orientation of the outer circle or, which is the \nsame thing, axial symmetry of the diagram).\nThen the Kontsevich integral commutes with \nthe operation $S$ of inverting the orientation:\n$Z(S(K))=S(Z(K))$.\n\n\\textsl{Corollary.} The following two assertions are equivalent:\n\n--- Vassiliev invariants do not distinguish the orientation of knots,\n\n--- all chord diagrams are symmetric: $D=S(D)$ modulo four-term relations.\n\nThe calculations of \\cite{Kn} show that up to order 12 all chord \ndiagrams are symmetric. For bigger orders the problem is still open.\n\n\\subsection{Multiplicative properties}\n\nThe Kontsevich integral for tangles is multiplicative:\n$$\n  Z(T_1)\\cdot Z(T_2) = Z(T_1\\cdot T_2)\n$$\nwhenever the product $T_1\\cdot T_2$, defined by vertical concatenation of\ntangles, exists. Here the product in the left-hand side is understood\nas the image of the element $Z(T_1)\\otimes Z(T_2)$ under the natural map\n$\\A(T_1)\\otimes\\A(T_2)\\to\\A(T_1\\cdot T_2)$.\n\nThis simple fact has two important corollaries:\n\n\\begin{enumerate}\n\\item\nFor any knot $K$ the Kontsevich integral $Z(K)$\nis a group-like element of the Hopf algebra $\\hat\\A'$, i.e.\n$$\n\\delta(Z(K)) = Z(K)\\otimes Z(K)\\,,\n$$\nwhere $\\delta$ is the comultiplication in $\\A$ defined above.\n\\item\nThe final Kontsevich integral, taken in a different normalization\n$$\\label{Iprime}\n  I'(K)=Z(H)I(K)=\\frac{Z(K)}{Z(H)^{c\/2-1}}.\n$$\nis multiplicative with respect to the connected sum of knots:\n$$\n  I'(K_1\\#K_2)=I'(K_1)I'(K_2),\n$$\n\\end{enumerate}\n\n\n\\subsection{Arithmetical properties}\n\nFor any knot $K$ the coefficients in the expansion of $Z(K)$\nover an arbitrary basis consisting of chord diagrams are rational \n(see \\cite{Kon,LM2} and below).\n\n\n\\section{Combinatorial construction of the Kontsevich integral}\n\\label{comb}\n\n\\subsection{Sliced presentation of knots} \nThe idea is to cut the knot into a number of standard simple\ntangles, compute the Kontsevich integral for each of them and then\nrecover the integral of the whole knot from these simple pieces. \n\n\nMore exactly, we represent the knot by a family of plane diagrams\ncontinuously depending on a parameter $\\varepsilon\\in(0,\\varepsilon_0)$ and cut\nby horizontal planes into a number of slices with the following\nproperties.\n\n\\begin{enumerate}\n\\item\nAt every boundary level of a slice (dashed lines in the pictures below) \nthe distances between various strings are asymptotically\nproportional to different whole powers \nof the parameter $\\varepsilon$.\n\\item\nEvery slice contains exactly one special event and\nseveral strictly vertical strings which\nare farther away (at lower powers of $\\varepsilon$) from any string\nparticipating in the event than its width. \n\\item\nThere are three types of special events:\n\n\\begin{center}\n\\begin{tabular}{lrcrc}\nmin\/max: & $m=$ & \n\\raisebox{-3mm}{\\includegraphics[width=25mm]{evmin.eps}} & $M=$ \n\\raisebox{-3mm}{\\includegraphics[width=25mm]{evmax.eps}} \\\\[2em]\nbraiding:  & $B_+=$ &\n\\raisebox{-7mm}{\\includegraphics[height=15mm]{ecbradp.eps}} &  $B_-=$\n\\raisebox{-7mm}{\\includegraphics[height=15mm]{ecbradm.eps}}\\\\[2em]\nassociativity: & $A_+=$ &\n\\raisebox{-7mm}{\\includegraphics[height=15mm]{evass1.eps}} &  $A_-=$\n\\raisebox{-7mm}{\\includegraphics[height=15mm]{evass2.eps}}\n\\end{tabular}\n\\end{center}\nwhere, in the two last cases, the strings may be replaced by bunches of\nparallel strings which are closer to each other than the width of this\nevent.\n\\end{enumerate}\n\n\n\\subsection{Recipe of computation of the Kontsevich integral}\n\nGiven such a sliced representation of a knot,\nthe combinatorial algorithm to compute its Kontsevich integral\nconsists in the following:\n\\begin{enumerate}\n\\item\n   Replace each special event by a series of chord diagrams\nsupported on the corresponding tangle according to the rule\n\\begin{eqnarray*}\n   m,M &\\mapsto& 1,\\\\\n   B_+ &\\mapsto& R,\\quad B_+ \\mapsto R^{-1},\\\\\n   A_+ &\\mapsto& \\Phi,\\quad A_- \\mapsto \\Phi^{-1},\n\\end{eqnarray*}\nwhere\n\\begin{align*}\nR\n  &= \\raisebox{-4mm}{\\includegraphics[height=11mm]{R0.eps}}\\cdot\n\\exp\\left(\\frac{\\raisebox{-1.3mm}{\\includegraphics[height=4mm]{doc.eps}}}{2}\\right)\n  =  \\raisebox{-4mm}{\\includegraphics[height=11mm]{R0.eps}}\n  + \\frac{1}{2}\\,\\raisebox{-4mm}{\\includegraphics[height=11mm]{R1.eps}}\n  + \\frac{1}{2\\cdot 2^2}\\, \\raisebox{-4mm}{\\includegraphics[height=11mm]{R2.eps}}\n  + \\frac{1}{3!\\cdot 2^3}\\, \\raisebox{-4mm}{\\includegraphics[height=11mm]{R3.eps}} +\\dots\\,\\\\\n\\Phi &= 1-\\frac{\\zeta(2)}{(2\\pi i)^2} [a,b]\n    -\\frac{\\zeta(3)}{(2\\pi i)^3} ([a,[a,b]]+[b,[a,b])+\\dots\n\\end{align*}\n($\\Phi\\in\\hat\\A(3)$ is the Knizhnik-Zamolodchikov Drinfeld associator \ndefined below;\nit is an infinite series in two variables\n$a=\\raisebox{-2mm}{\\includegraphics[height=6mm]{vstod.eps}}$, \n$b=\\raisebox{-2mm}{\\includegraphics[height=6mm]{vstdt.eps}}$).\n\\item\n  Compute the product of all these series from top to bottom taking \ninto account the connection of the strands of different tangles, thus \nobtaining an element of the algebra $\\hat\\A'$.\n\\end{enumerate}\n\nTo accomplish the algorithm, we need two auxiliary operations\non chord diagrams:\n\\begin{enumerate}\n\\item\n$S_i:\\A(p)\\to\\A(p)$ defined as multiplication by $(-1)^k$\non a chord diagram containing $k$ endpoints of chords on the string\nnumber $i$. This is the correction term in the computation\nof $R$ and $\\Phi$ in the case when the tangle contains some strings\noriented downwards (the upwards orientation is considered as positive).\n\\item\n$\\Delta_i:\\A(p)\\to\\A(p+1)$\nacts on a chord diagram $D$ by doubling the $i$-th string of $D$\nand taking the sum over all possible lifts of the endpoints of chords\nof $D$ from the $i$-th string to one of the two new strings.\nThe strings are counted by their bottom points from left to right.\nThis operation can be used to express the combinatorial Kontsevich\nintegral of a generalized associativity tangle (with strings replaced\nby bunches of strings) in terms of the\ncombinatorial Kontsevich integral of a simple associativity tangle.\n\\end{enumerate}\n\n\\subsection{Example.}\nUsing the combinatorial algorithm, we compute the Kontsevich integral \nof the trefoil knot $3_1$ to the terms of degree 2.\nA sliced presentation for this knot \nshown in the picture implies that\n$Z(3_1)= S_3(\\Phi)R^{-3}S_3(\\Phi^{-1})$\n(here the product from left to right corresponds\nto the multiplication of tangles from top to bottom).\n\n\\noindent\n\\begin{minipage}{4in}\nUp to degree 2, we have:\n$\\Phi=1+\\frac{1}{24}[a,b]+\\dots$,\n$R=X(1+\\frac{1}{2}a+\\frac{1}{8}a^2+\\dots)$,\nwhere $X$ means that the two strands in each term of the series\nmust be crossed over at the top.\nThe operation $S_3$ changes the orientation of the third strand,\nwhich means that $S_3(a)=a$ and $S_3(b)=-b$.\nTherefore,\n$S_3(\\Phi)=1-\\frac{1}{24}[a,b]+\\dots$,\n$S_3(\\Phi^{-1})=1+\\frac{1}{24}[a,b]+\\dots$,\n$R^{-3}=X(1-\\frac{3}{2}a+\\frac{9}{8}a^2+\\dots)$\nand\n$Z(3_1) = (1-\\frac{1}{24}[a,b]+\\dots)\nX(1-\\frac{3}{2}a+\\frac{9}{8}a^2+\\dots)(1+\\frac{1}{24}[a,b]+\\dots)\n=1-\\frac{3}{2}Xa-\\frac{1}{24}abX\n+\\frac{1}{24}baX+\\frac{1}{24}Xab-\\frac{1}{24}Xba+\\frac{9}{8}Xa^2+\\dots$\nClosing these diagrams into the circle, we see that in the algebra $\\A$ \nwe have $Xa=0$ (by the framing independence relation),\nthen $baX=Xab=0$ (by the same relation, because these diagrams consist\nof two parallel chords) and $abX=Xba=Xa^2=\\raisebox{-2mm}{\\includegraphics[height=6mm]{cd22.eps}}$.\nThe result is:\n$Z(3_1)=1+\\frac{25}{24}\\raisebox{-2mm}{\\includegraphics[height=6mm]{cd22.eps}}+\\dots$.\nThe final Kontsevich integral \nof the trefoil (in the multiplicative normalization, see page\n\\pageref{Iprime}) is thus equal to\n$I'(3_1)=Z(3_1)\/Z(H)=(1+\\frac{25}{24}\\raisebox{-2mm}{\\includegraphics[height=6mm]{cd22.eps}}+\\dots)\/\n(1+\\frac{1}{24}\\raisebox{-2mm}{\\includegraphics[height=6mm]{cd22.eps}}+\\dots)=1+\\raisebox{-2mm}{\\includegraphics[height=6mm]{cd22.eps}}+\\dots$ \n\\end{minipage}\n\\qquad\n\\raisebox{-24mm}{\\includegraphics[height=55mm]{nat3-1f.eps}}\n\n\\subsection{Drinfeld associator and rationality}\n\nThe Drinfeld associator used as a building block in the combinatorial\nconstruction of the Kontsevich integral, can be defined as the limit\n$$\n\\Phi_{\\mbox{\\scriptsize KZ}}=\\lim_{\\varepsilon\\to0}\n\\varepsilon^{-b} Z(AT_\\varepsilon) \\varepsilon^a,\n$$\nwhere \n$a=\\raisebox{-2mm}{\\includegraphics[height=6mm]{vstod.eps}}$,\n$b=\\raisebox{-2mm}{\\includegraphics[height=6mm]{vstdt.eps}}$),\nand \n$AT_\\varepsilon$ is the positive associativity tangle (special event $A_+$\nshown above)\nwith the distance between the vertical strands constant 1\nand the distance between the close endpoints equal to $\\varepsilon$.\nAn explicit formula for $\\Phi_{\\mbox{\\scriptsize KZ}}$ was found by Le and Murakami \n\\cite{LM2};\nit is written as a nested summation over four variable multiindices and\ntherefore does not provide an immediate insight into the structure of the\nwhole series;\nwe confine ourselves by quoting\nthe beginning of the series (note that $\\Phi_{\\mbox{\\scriptsize KZ}}$ is a group-like \nelement\nin the free associative algebra with 2 generators, hence its logarithm\nbelongs to the corresponding free Lie algebra):\n\\begin{align*}\n  \\log(\\Phi_{\\mbox{\\scriptsize KZ}})\n &= -\\zeta(2) [x,y]\n    -\\zeta(3) ([x,[x,y]]+[y,[x,y])\\\\\n & -\\frac{\\zeta(2)^2}{10}(4[x,[x,[x,y]]]+[y,[x,[x,y]]]\n      +4[y,[y,[x,y]]])\\\\\n &-\\zeta(5) ([x,[x,[x,[x,y]]]]+[y,[y,[y,[x,y]]]])\\\\\n&+(\\zeta(2)\\zeta(3)-2\\zeta(5))([y,[x,[x,[x,y]]]]+ [y,[y,[x,[x,y]]]])\\\\\n&+(\\frac{1}{2}\\zeta(2)\\zeta(3)-\\frac{1}{2}\\zeta(5)) [[x,y],[x,[x,y]]]\\\\\n&+(\\frac{1}{2}\\zeta(2)\\zeta(3)-\\frac{3}{2}\\zeta(5)) [[x,y],[y,[x,y]]]+\\dots\n\\end{align*}\nwhere $x=\\frac{1}{2\\pi i}a$ and $y=\\frac{1}{2\\pi i}b$.\nIn general, $\\Phi_{\\mbox{\\scriptsize KZ}}$ is an infinite series whose coefficients\nare \\textit{multiple zeta values} (\\cite{LM2,Zag}) \n$$\n  \\zeta(a_1,\\dots,a_n)=\n\\sum_{0<k_1<k_2<\\dots<k_n} k_1^{-a_1}\\dots k_n^{-a_n}.\n$$\n\nThere are other equivalent definitions of $\\Phi_{\\mbox{\\scriptsize KZ}}$, in particular \none in terms of the asymptotical behaviour of solutions of the simplest\nKnizhnik--Zamolodchikov equation\n$$\n\\frac{dG}{dz} =\\left(\\frac{a}{z} + \\frac{b}{z-1}\\right)G,\n$$\nwhere $G$ is a function of a complex variable taking values in the\nalgebra of series in two non-commuting variables $a$ and $b$ \n(see \\cite{Dri}).\n\nIt turns out (theorem of Le and Murakami \\cite{LM2}) that the combinatorial\nKontsevich integral does not change if $\\Phi_{\\mbox{\\scriptsize KZ}}$ is replaced by another\nseries in $\\hat\\A(3)$ provided it satisfies certain axioms (among which the\npentagon and hexagon relations are the most important, see \\cite{Dri,LM2}).\n\nDrinfeld \\cite{Dri} proved the existence of an associator\n$\\Phi_{\\Q}$ with rational coefficients. Using it instead of $\\Phi_{\\mbox{\\scriptsize KZ}}$\nin the combinatorial construction, we obtain the following\n\n\\textbf{Theorem.} (\\cite{LM2}) The coefficients of the Kontsevich integral\nof any knot (tangle) are rational when $Z(K)$ is expanded over an arbitrary\nbasis consisting of chord diagrams.\n\n\\section{Explicit formulas for the Kontsevich integral}\n\\label{explic}\n\n\\subsection{The wheels formula}\nLet $O$ be the unknot; the expression $I(O)=Z(H)^{-1}$ is referred to\nas the \\textit{Kontsevich integral of the unknot}. A closed form\nformula for $I(O)$ was proved in \\cite{BLT}:\n\n\\textbf{Theorem.}\n$$  I(O) = \\exp \\sum_{n=1}^\\infty b_{2n} w_{2n}\n       = 1 + (\\sum_{n=1}^\\infty b_{2n}w_{2n})\n           + \\frac{1}{2}(\\sum_{n=1}^\\infty b_{2n}w_{2n})^2 + \\dots\n$$\nHere $b_{2n}$ are modified Bernoulli numbers, i.e.\nthe coefficients of the Taylor series\n$$\n  \\sum_{n=1}^\\infty b_{2n}x^{2n} = \\frac{1}{2}\n           \\ln\\frac{e^{x\/2}-e^{-x\/2}}{x},\n$$\n($b_2=1\/48$, $b_4=-1\/5760$, $b_6=1\/362880$,\\dots),\nand $w_{2n}$ are the {\\it wheels}, i.~e. Jacobi diagrams\n of the form\n$$\n  w_2=\\raisebox{-3.5mm}{\\includegraphics[height=8mm]{wh2.eps}},\\quad\n  w_4=\\raisebox{-3.5mm}{\\includegraphics[height=8mm]{wh4.eps}},\\quad\n  w_6=\\raisebox{-5mm}{\\includegraphics[height=11mm]{wh6.eps}},\\quad\\dots\n$$\nThe sums and products are understood as operations in the \nalgebra of Jacobi diagrams $\\B$, and the result is then carried over\nto the algebra of chord diagrams $\\A$ along the isomorphism $\\chi$ \n(see Section \\ref{ch_diag}).\n\n\n\\subsection{Generalizations}\n\nThere are several generalizations of the wheels formula:\n\n1. Rozansky's \\textit{rationality conjecture} \\cite{Roz}\nproved by A.Kricker \\cite{Kri} affirms that the Kontsevich integral\nof any (framed) knot can be written in a form resembling the wheels formula.\nLet us call the \\textit{skeleton} of a Jacobi diagram the regular 3-valent\ngraph obtained by `shaving off' all univalent vertices.\nThen the wheels formula says that all diagrams in the expansion of\n$I(O)$ have one and the same skeleton (circle), and the generating function\nfor the coefficients of diagrams with $n$ legs is a certain analytic\nfunction, more or less rational in $e^x$. In the same way, the theorem of\nRozansky and Kricker states that the terms in $I(K)\\in\\hat\\B$, when arranged\nby their skeleta, have the generating functions of the form\n$p(e^x)\/A_K(e^x)$, where $A_K$ is the Alexander polynomial of $K$ and $p$ is\nsome polynomial function.  Although this theorem does not give an explicit\nformula for $I(K)$, it provides a lot of information about the structure of\nthis series.\n\n2. J.March\\'e \\cite{Mar} gives a closed form formula for the\nKontsevich integral of torus knots $T(p,q)$.\n\nThe formula of March\\'e, although explicit, is rather intricate, and \nhere, by way of example, we only write out the first several terms of the \nfinal Kontsevich integral $I'$ \nfor the trefoil (torus knot of type (2,3)), following \\cite{Wil2}:\n$$\n  I'(\\raisebox{-3mm}{\\includegraphics[height=9mm]{k3-1m.eps}})=\n \\raisebox{-2.5mm}{\\includegraphics[height=7mm]{cd0.eps}}\n-\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd2.eps}}\n+\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd3.eps}}\n-\\frac{31}{24}\\,\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd41.eps}}\n+\\frac{5}{24}\\,\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd42.eps}}\n+\\frac{1}{2}\\,\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd43.eps}}\n+\\dots\n$$\n\n\\subsection{First terms of the Kontsevich integral}\n\nA Vassiliev invariant $v$ of degree $n$ is called \n\\textit{canonical} if it can be recovered from the Kontsevich integral\nby applying a homogeneous weight system, i.e. if\n$v = \\mbox{symb}(v)\\circ I$.\nCanonical invariants define a grading in the filtered space of Vassiliev\ninvariants which is consistent with the filtration.\nIf the Kontsevich integral is expanded over a fixed basis\nin the space of chord diagrams $\\hat\\A'$, then the coefficient\nof every diagram is a canonical invariant.\nAccording to \\cite{Sta,Wil2}, the expansion of the final Kontsevich integral\nup to degree 4 can be written as follows:\n\\begin{align*}\n  I'(K) &= \n \\raisebox{-2.5mm}{\\includegraphics[height=7mm]{cd0.eps}}\n-c_2(K)\\,\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd2.eps}}\n-\\frac{1}{6}j_3(K)\\,\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd3.eps}}\\\\\n&+\\frac{1}{48}\\bigl(4j_4(K)+36c_4(K)-36c_2^2(K)+3c_2(K)\\bigr)\n\\,\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd41.eps}}\\\\\n&+\\frac{1}{24}\\bigl(-12c_4(K)+6c_2^2(K)-c_2(K)\\bigr)\n\\,\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd42.eps}}\\\\\n&+\\frac{1}{2}c_2^2(K)\\,\n\\,\\raisebox{-2.5mm}{\\includegraphics[height=7mm]{fd43.eps}}\n+\\dots\n\\end{align*}\nwhere $c_n$ are coefficients of the Conway polynomial \n$\\nabla_K(t)=\\sum c_n(K)t^n$\nand $j_n$ are modified coefficients of the \nJones polynomial  $J_K(e^t)=\\sum j_n(K)t^n$.\nTherefore, up to degree 4, the basic canonical Vassiliev invariants\nof unframed knots are $c_2$, $j_3$, $j_4$, $c_4+\\frac{1}{12}c_2$\nand $c_2^2$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nWeyl semimetals (WSMs) are three dimensional topological systems that host an even number of band-touching points (termed as \\textit{Weyl nodes}) in the bulk spectrum, near which the low-energy excitations follow the relativistic Weyl equation~\\cite{WSMreview}. Such Weyl quasiparticles have definite chirality and the chirality of these quasiparticles are given by the nature of the Weyl nodes, which can act as either sources or sinks of Berry curvature in the Brillouin zone. In a finite geometry, WSMs also host unique surface states, named as \\textit{Fermi arc} states, whose projected Fermi surfaces are open arcs on each of the surfaces. Numerous materials have been predicted to be suitable candidates for WSMs , and a variety of experiments demonstrate their novel character.\n\nCorrelations functions impact many properties of these systems, and are of special interest because of\nthe unique helical nature of low-energy excitations in WSMs. Within this class of properties,\nthe effective interaction between two localized impurity spins introduced in such a system, mediated by the WSM conduction electrons, is described by the Ruderman-Kittel-Kasuya-Yoshida (RKKY) theory~\\cite{RKKY}, and is  directly related to the spin-spin correlation function of  electrons within the system. Among solid-state materials, spin-orbit coupled systems ~\\cite{RKKYSO1,RKKYSO2}, particularly topological ones, are particularly interesting in the way they mediate long-range -- and sometimes controllable -- coupling\\cite{RKKYQSH1,RKKYQSH2,RKKYElectrical1,RKKY2DTI,RKKY2DTI2,RKKYTI,Reja_2017,Reja_2019,SonuRKKY} among spins. Work on RKKY interactions through bulk Weyl fermions~\\cite{RKKYWSM1,RKKYWSM2,RKKYWSM3} show the interactions can be anisotropic and are in some circumstances weak, but generally carry signatures of the chiral nodes.\n\nSignificant attention has also been given to RKKY interactions on the surface of WSMs~\\cite{RKKYWSMSur1,RKKYWSMSur2,RKKYWSMSur3}.\nSurface states, at the Fermi energy of a WSM in a slab geometry, typically reside\nat wavevectors which form arcs in the surface Brillouin zone.  These arcs join one Weyl node to the other, and typically disperse energetically perpendicular to a given arc, with different signs of the dispersion for each of the two physical surfaces. The essentially one dimensional character of the surface states results in a strong, highly anisotropic spin-spin correlation function, with similarly anisotropic RKKY interactions among spin\nimpurities adsorbed on the surface of a WSM.  Previous studies\nhave been largely confined to semi-infinite geometries, for which coupling {\\it between}\nsurfaces cannot be modeled.  Such coupling is potentially significant,\nas the Fermi arc states can be relatively weakly localized at their surfaces; moreover,\nthe penetration length of a Fermi arc state diverges as the surface wavevector\napproaches the projection of a Weyl node. This can induce interesting physics due to\nnon-negligible coupling between spins on opposite surfaces of the WSM.\n\n\nIn this paper, we investigate the effects of such intersurface coupling by analyzing a slab geometry of {finite thickness}. Specifically, we examine effective spin-spin interactions due to the RKKY mechanism for two different situations. Firstly, when two spins are on the {\\it same} surface of the WSM, surface electrons on the opposite surface can participate in their coupling. The resulting RKKY interaction reflects the anisotropy of the Fermi surface, and in the thick slab limit recovers previous results in which only a single surface was modeled~\\cite{RKKYWSMSur1}. Secondly, when the spin impurities are on {\\it opposite} surfaces of the slab, the resulting coupling depends strongly on the overlap of the Fermi arc states. We find that the resulting coupling is a non-monotonic function of the slab thickness, and a thin-film limit can be defined by observing when the coupling between the spins is strongest. In order to compute the coupling, we have developed a recursive Green's function scheme in which the elements of a Green's function on the surfaces can be computed essentially exactly with relatively high numerical efficiency. We show that analytical solutions for the WSM wavefunctions in a slab geometry verify the numerical results, and offer some insight into their qualitative behavior.\n\n\nThis manuscript is organized as follows. In Section II, we introduce the simple WSM model  used for our work and find analytical solutions for wavefunctions in a slab geometry with appropriate boundary conditions. In Section III, we briefly discuss the formal expression for RKKY interactions and our numerical scheme for computing them in a slab geometry of a tight-binding model.  Our numerical results are presented in Section IV, along with a comparison with analytical results. Finally we conclude with a summary and discussion\nin Section V.\n\n\n\n\\section{Weyl Semimetal - Thin Film}\\label{sec:wsmtf}\n\\subsection{Model Hamiltonian}\nA minimal model of a WSM has two Weyl nodes at the Fermi energy and breaks time-reversal (TR) symmetry. For such a model, the low-energy Hamiltonian can be written using a two-band model. If the two bands represent spin states, then for a slab geometry, with the Weyl nodes separated along the momentum of one of the translational invariant (in-plane) directions,\nthe surface states (Fermi arcs) are  spin-polarized, resulting in completely spin-polarized surfaces of the slab. As the indirect spin-exchange interaction is only interesting when the ground-state is spin-{\\it unpolarized}, the minimal model we consider must have at least two Fermi arcs on each surface, with the spin-polarizations of\neach oriented such that the net spin density on either surface vanishes. If the two Fermi arcs on a surface have distinct locations in the surface Brillouin zone, then one has a total of four Weyl nodes in the bulk, each with a distinct location in momentum space. If the Fermi arcs join two {\\it Dirac} nodes, then the Fermi\narcs will overlap in the surface Brillouin zone.  This latter situation was considered, for example, in Ref.~\\cite{RKKYWSMSur1}.  In our work we confine our studies to the former case (i.e., Weyl semimetals).\n\nOur starting point is a model Hamiltonian defined on a cubic lattice~\\cite{WSMModel}.\nThe Hamiltonian preserves time-reversal symmetry (defined by the time reversal operator $ T= \\textit{i} \\sigma_y K $, with $ K $ the complex conjugation operator and $\\sigma_y$  a Pauli matrix acting in the spin-space), but breaks inversion symmetry, and so has four degenerate Weyl nodes.  Specifically, we take\n\\begin{equation}\\label{eq:H1}\nH(k)=\\lambda \\sum_{\\alpha =x,y,z} \\sigma^\\alpha \\sin{k_\\alpha} -\\mu + \\tau^y \\sigma^y M_k.\n\\end{equation}\nHere $M_k=m+2-\\cos{k_x}-\\cos{k_z} $, and $ \\tau^\\alpha $ are Pauli matrices acting in an orbital space. For $ | m | \\leqslant \\lambda $ the four Weyl nodes are located at $ \\textbf{\\textit{k}}=(0,\\pm \\pi\/2 \\pm k_0 ,0) $, where $k_0 = \\pi\/2-\\sin^{-1}(m\/\\lambda)$.\nOn a given surface, the two Fermi arcs join the four Weyl nodes in a pairwise fashion, as illustrated in Fig.~(\\ref{fig:bulk}). States of the two Fermi arcs are spin-polarized along $\\sigma_x$ in opposite directions (i.e, they are eigenvectors of $\\sigma_x$ with opposite eigenvalues for the two Fermi arcs). Furthermore, the two Fermi arcs, at low-energy, are dispersionless along the $k_y$ direction and have opposite velocities along the $x$ direction.\n\nThe Hamiltonian can be brought into a block diagonal form.  Writing\n$H' = U H U^{\\dagger}$, with the unitary matrix $U$ defined by\n\\begin{align}\nU = \\frac12\\left(\\begin{array}{cccc}\n-1 & -i & -i & 1\\\\\n1 & -i & i & 1\\\\\n1 & i & -i & 1\\\\\n-1 & i & i & 1\n\\end{array}\\right),\\label{eq:U}\n\\end{align}\none finds $H^{\\prime}$ has two $2 \\times 2$ blocks, where for each block (labeled by $\\eta =\\pm1$), the two-band Hamiltonian is\n\\begin{align}\\label{eq:blokH}\nH'_{\\eta} &= \\lambda(\\sigma_y \\sin k_x - \\sigma_x \\sin k_z) \\nonumber \\\\\n&+\\eta \\sigma_z (2 + m - \\cos k_x - \\cos k_z) - \\lambda\\sigma_z \\sin k_y.\n\\end{align}\nThis is a particularly useful form, in which each block individually breaks time-reversal (TR) symmetry, while $T$ maps $H'_+$ to $H'_-$ (and vice-versa), so that the total Hamiltonian is TR symmetric. Each of the blocks has two Weyl nodes separated in momentum space, and on\na given surface they are joined by one Fermi arc. In principle, a system hosting many Fermi arcs on a surface should be structured in such a way that each joins two Weyl nodes; an\neffective model of such a multi-Weyl node system could be written effectively as $H = H_1 \\otimes H_2 \\otimes \\cdots$, where each of the blocks contains two Weyl nodes. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{sx.pdf}\\\\\n\t\\caption{Top: For the WSM, Eq.~(\\ref{eq:H1}), in a slab geometry with finite thickness in $z$ direction and for $k_x=0$, the band-structure (in the unit of $\\lambda$) as a function of $k_y$ shows the four Weyl nodes in the bulk and the two Fermi arc joining them (the lattice spacing $a$ is taken to be unity). Bottom: The spin densities of the Fermi arc states are shown along various $z$ for two values of $k_y$ as pointed in the top figure. Parameters: $m=0.5~\\lambda$, thickness $N_z=45$ lattice spacings.}\\label{fig:bulk}\n\\end{figure}\n\nTo focus on the physics of the Fermi arcs, we expand the Hamiltonian to lowest non-trivial\norder in $k_x$ and $k_z$, writing\n$k_x \\rightarrow q_x$ and $k_z \\rightarrow q_z$.  Then from Eq.~(\\ref{eq:blokH}) we obtain\n\\begin{align}\\label{eq:newH}\nH'_{\\eta} &\\approx \\lambda(\\sigma_y q_x - \\sigma_x q_z) + \\sigma_z M_{\\eta}(k_y),\n\\end{align}\nwith $M_{\\eta}(k_y) = m\\eta - \\lambda\\sin k_y$. The four Weyl nodes are at\n${\\bf K}_{\\eta,\\xi} = (0,\\eta\\frac{\\pi}{2}+\\xi k_0,0)$ with $\\eta, \\xi =\\pm 1$ and $k_0 = \\cos^{-1}(m\/\\lambda)$.\nFor the $\\eta=+1$ block, $M_+ < 0 $ between $k_y\\in (\\pi\/2-k_0,\\pi\/2+k_0)$.\nFor a surface perpendicular to the $z$ direction, along the $k_y$ axis\nthese two points are connected by a Fermi arc.  For the $\\eta=-1$ block, $M_- > 0 $ between $k_y \\in (-\\pi\/2-k_0,-\\pi\/2+k_0)$, and again there is a Fermi arc connecting these points on the $k_y$ axis for the same surface. This situation is illustrated in Fig. \\ref{fig:bulk}.\nNote that for $H'$ (i.e., after the unitary transformation), states on the Fermi arcs are eigenvectors of $\\sigma_y$ rather than $\\sigma_x$.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{E_n0.pdf}\\\\\n\t\\caption{(Top) Minimum solution of $\\chi'$ for values of $M^{\\prime}_{\\eta}$. When $M'_{\\eta} \\rightarrow -\\infty$, $\\chi' \\rightarrow 0$, when $M'_{\\eta} =0 $, $\\chi' = (\\pi\/2)^2$ and for large $M'_{\\eta}$, $\\chi \\approx M^{\\prime 2}_{\\eta} + \\pi^2$. (Bottom) The lowest energy solution for $q_x=0$ for various values of $L_z$ from 5 to 30 are shown for half of the Brillouin zone, containing two Weyl nodes. The energy values decreases exponentially with increasing $L_z$ depicting surface states for $k_y$ between the Weyl nodes.}\\label{fig:solchi}\n\\end{figure}\n\n\nNear the Weyl nodes, if we can write the low-energy Hamiltonian in the form of $H=k_{\\mu}A_{\\mu\\nu}\\sigma_{\\nu}$, then the chirality of the node is given by sgn(Det[$\\mathbf{A}$]). Writing, $\\textbf{\\textit{k}} = (q_x,\\eta\\frac{\\pi}{2}+\\xi k_0 +q_y,q_z) $ and expanding to first order in $q_i$, we arrive at the low-energy Hamiltonian\n\\begin{align}\nH^{{\\rm low}}_{\\eta\\xi} &\\approx \\lambda(\\sigma_y q_x - \\sigma_x q_z) + \\eta\\xi  \\alpha \\sigma_z q_y,\n\\end{align}\nwith $\\alpha =\\sqrt{1-(m\/\\lambda)^2}$.  The chiralities of the four nodes may\nthen be written as ${\\rm sgn}({\\rm Det}[\\mathbf{A}_{\\eta,\\sigma}])= -\\eta\\xi$.\n\n\\subsection{Infinite mass boundary condition}\nTo make progress analytically, we need to construct appropriate boundary conditions of the Dirac Hamiltonian Eq.~(\\ref{eq:blokH}) for a slab geometry, such that the properties of the Fermi arc can be recovered. In general boundary conditions for Dirac equation can be cumbersome [REF], but our goal is to recover the properties of the surface modes (i.e, Fermi arc states). We construct boundary conditions by taking the Hamiltonian of the vacuum\n(outside the slab, which extends from $z=0$ to $z=L_z$), similar to Eq.~(\\ref{eq:newH}), except for the mass term, whose form is taken as $M^{\\rm vac}_{\\eta} = \\eta m_0$, with $m_0 \\rightarrow \\infty$. This construction is required to ensure that for momentum between the Weyl nodes the effective mass term ($M_{\\eta}(k_y)$) for the Weyl semimetal and the vacuum ($M_{\\eta}^{\\rm vac}$) are oppositely signed.  \n\nThe eigenfunctions for the Hamiltonian $H_{\\rm vac} = \\lambda(q_x\\sigma_y - q_z\\sigma_x) + M_{\\eta}^{{\\rm vac}}\\sigma_z$ are\n\\begin{align}\n\\psi_{\\rm vac} \\propto \\left(\\begin{array}{c}\n\\lambda(q_z+i q_x) \\\\\nM_{\\eta}^{{\\rm vac}}- E\n\\end{array} \\right)e^{i(q_z z + q_x x)},\n\\end{align}\nwith eigenvalue $E =\\pm \\sqrt{m_0^2 +\\lambda^2(q_z^2 + q_x^2)}$. For $m_0 \\gg E$, the eigenfunctions are normalizeable if\n\\begin{align}\n& q_z = i\\kappa, \\quad {\\rm for} ~~ z\\ge L_z, \\nonumber\\\\\n&q_z = -i\\kappa, \\quad {\\rm for} ~~ z\\le 0,\\nonumber\n\\end{align}\nwith $\\kappa = \\sqrt{m_0^2 +q_x^2 -E^2}$. Thus, in the limit $m_0 \\rightarrow \\infty$, we have $\\kappa \\rightarrow m_0$. For $z>L_z$,\n\\begin{align}\\label{eq:vacone}\n\\psi_{>} \\propto \\left(\\begin{array}{c}\nim_0+iq_x \\\\\n\\eta m_0 -E\n\\end{array} \\right)e^{-m_0z} \\approx \\left(\\begin{array}{c}\ni\\\\\n\\eta\n\\end{array} \\right)e^{-m_0z}.\n\\end{align}\nFor $z<0$,\n\\begin{align}\\label{eq:vactwo}\n\\psi_{<} \\propto \\left(\\begin{array}{c}\n-im_0+iq_x \\\\\n\\eta m_0 -E\n\\end{array} \\right)e^{m_0 z} \\approx \\left(\\begin{array}{c}\ni\\\\\n-\\eta\n\\end{array} \\right)e^{m_0z}.\n\\end{align}\nAt $z=0, L_z$, these spinors become the Fermi arc wavefunctions, and are recognizable as eigenvectors of $\\sigma_y$.\n\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{Top.pdf}\n\t\\caption{ The RKKY coupling between two spins (connected to the same orbital) put on the same surface (along $x$-direction \\textit{i.e.} $\\mathbf{R}=(R,0,0)$) of the WSM slab with (a) the analytical wave-functions and keeping only the $n=0$ bands, (b), (c) numerically evaluated Green's function in the real-space. With increasing thickness, the all components except $J_{xx}$ becomes essentially thickness independent after certain thickness, as shown in (d). Inset of (d) shows the RKKY coupling vs slab thickness calculated using analytical wave-functions and $n=0$ bands. Results shown are for $\\mu=0$ (i.e., Fermi wavevector $k_F=0$)\nand $m=0.5\\lambda$.  $R\/a=40$ in panel (d).\n}\\label{fig:top}\n\\end{figure*}\n\n\n\nMatching the wavefunction $\\psi(z)$ within the slab to these boundary forms yields the conditions\n\\begin{align}\\label{bc}\n\\psi(z=0) \\propto \\psi_{{\\rm}<}(z=0)~~{\\rm and}~~\\psi(z=L) \\propto \\psi_{{\\rm}>}(z=L_z),\n\\end{align}\nwhere\n\\begin{align}\\label{eq:states}\n\\psi(z) =& a\\left(\\begin{array}{c}\n\\lambda(q_z+iq_x) \\\\\nM_{\\eta}(k_y) - E\n\\end{array} \\right)e^{iq_z z}\\nonumber\\\\\n& + b\\left(\\begin{array}{c}\n\\lambda(-q_z+iq_x) \\\\\nM_{\\eta}(k_y) - E\n\\end{array} \\right)e^{-iq_z z},\n\\end{align}\nwith $q_z =(1\/\\lambda)\\sqrt{E^2 - M_{\\eta}^2 -\\lambda^2q_x^2}$. Non-trivial solutions of Eq.~\\ref{bc} exists if\n\n\\begin{widetext}\n\\begin{align}\n{\\rm Det}\\left(\\begin{array}{cccc}\ni & \\lambda(q_z+iq_x) & \\lambda(-q_z+iq_x)& 0\\\\\n-\\eta & M_{\\eta}(k_y) - E & M_{\\eta}(k_y) - E & 0\\\\\n0 & \\lambda(q_z+iq_x)e^{iq_z L_z} & \\lambda(-q_z+iq_x)e^{-iq_z L_z}& i\\\\\n0 & (M_{\\eta}(k_y) - E)e^{iq_zL_z} & (M_{\\eta}(k_y) - E)e^{-iq_z L_z} & \\eta\n\\end{array} \\right) = 0.\n\\end{align}\n\\end{widetext}\nSimplifying this condition, we obtain a transcendental equation,\n\\begin{align}\n&\\frac{{\\rm tanh}\\left(L_z\\sqrt{(M_{\\eta}\/\\lambda)^2 - \\chi}\\right)}{L_z\\sqrt{(M_{\\eta}\/\\lambda)^2 -\\chi}} = -\\frac{\\lambda}{L_z\\eta M_{\\eta}},\n\\label{eq:chi}\n\\end{align}\nwhere $\\chi = (E\/\\lambda)^2 - q_x^2$. For all real solutions $\\chi$ of this equation, the energy has values $E = \\pm\\lambda\\sqrt{\\chi+q_x^2}$. No solutions of Eq. \\ref{eq:chi} exist with $\\chi<0$.\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=0.99\\textwidth]{Two.pdf}\n\t\\caption{The RKKY coupling between two spins (connected to same orbital) on opposite surfaces of the WSM slab, with the positions of the two spins at $(x,y,z) = (0,0,0)$ and $ (R,0,L_z=N_za)$. (a), (c) and (e) are results for the analytical wavefunctions keeping only the $n=0$ bands. (b), (d) and (f) show numerically evaluated results from the Green's function approach. With increasing thickness, all components decrease rapidly (shown in more detail in Fig.~\\ref{fig:withN} and tabulated in Table II). These results are for $\\mu=0$\n($k_F=0$). For all panels, $m=0.5~\\lambda$.  \n}\\label{fig:two}\n\\end{figure*}\n\n\nFor bound-state solutions, i.e, when $q_z$ is imaginary, $\\chi < (M_{\\eta}\/\\lambda)^2$.  The left hand side of Eq. \\ref{eq:chi} is a positive function with values between 0 and 1. Thus, such bound-state solutions are only possible when $\\eta M_{\\eta}<0$ as well as when $|L_zM_{\\eta}\/\\lambda|>1$, i.e., when $|M_{\\eta}|>\\lambda\/L_z$.\nDefining $\\chi' = L_z^2\\chi$ and $M'_{\\eta} = \\eta L_zM_{\\eta}\/\\lambda$, we rewrite\nEq. \\ref{eq:chi} as\n\\begin{align}\\label{eq:chim}\n&\\frac{{\\rm tanh}(\\sqrt{M_{\\eta}^{\\prime 2} - \\chi'})}{\\sqrt{M^{\\prime 2}_{\\eta} -\\chi'}} = -\\frac{1}{M^{\\prime }_{\\eta}}.\n\\end{align}\nThe various solutions of $\\chi'$ from Eq. \\ref{eq:chim} can be labeled by an index $n=0,1,..$ (with increasing values of $n$ corresponding to larger values of $\\chi'$) and the corresponding energy solutions $E_{n,\\pm}(q_x,q_y) = \\pm \\lambda \\sqrt{\\chi_n + q_x^2}$ gives rise to particle-hole symmetric bands. The minimum solution of $\\chi'$ is shown in Fig.~\\ref{fig:solchi}. The bands with $n=0$ contains all the Fermi arc states (when $k_y$ is between the Weyl nodes, in the Fermi arc interval) and low-energy bulk states (when $k_y$ is outside the interval).\n\nThe coefficient $(a\/b)$ for the states, Eq.~(\\ref{eq:states}), can be found from the boundary conditions at $z=0$ to be\n\\begin{align}\n\\frac{a}{b} = -\\frac{M_{\\eta} -E + \\eta \\lambda(q_x +i q_z)}{M_{\\eta} -E + \\eta \\lambda(q_x - i q_z)}.\n\\end{align}\nWe can then write down the wavefunctions. Defining\n$K = M_{\\eta}(k_y) -E, f=\\lambda(q_x-iq_z), g=\\lambda(q_x+iq_z)$, one finds\n\\begin{widetext}\n\\begin{align}\\label{eq:wf1}\n|\\psi \\rangle &= \\frac{1}{\\sqrt{N}} \\left\\{ (K+\\eta g)\\left(\\begin{array}{c}\ni f \\\\\nK\n\\end{array} \\right)e^{iq_z z} +(K+\\eta f)\\left(\\begin{array}{c}\n-i g \\\\\n-K\n\\end{array} \\right)e^{-iq_z z} \\right\\}.\n\\end{align}\n\\end{widetext}\nFor real $q_z = \\sqrt{\\chi - M_{\\eta}(k_y)^2}$ (when $\\chi>m^2$, $f=g^*$) the normalization factor has the form\n\\begin{align}\nN =& 2|K+\\eta f|^2(K^2+|f|^2)L\\nonumber \\\\\n&+{\\rm Im}\\left[(K+\\eta f)^2(K^2+g^2)\\left(\\frac{e^{-2iLq_z}-1}{q_z}\\right) \\right].\n\\end{align}\nFor purely imaginary $q_z=i\\kappa$ (when $\\chi<m^2$),$f=q_x+\\kappa$, $g=q_x-\\kappa$,\n\\begin{align}\nN =&-2(K+\\eta f)(K+\\eta g)(K^2+g f) L\\nonumber \\\\\n&+[(K+\\eta g)^2(f^2+K^2)e^{-\\kappa L}\\nonumber\\\\\n&+(K+\\eta f)^2(g^2+K^2) e^{\\kappa L}]\\frac{\\sinh(\\kappa L)}{\\kappa}.\n\\end{align}\nThese are the full solutions of the low-energy states of the WSM slab in the rotated basis (Eq.~(\\ref{eq:U})). Once written in the original basis, these solutions correctly reproduce the spin configuration of the Fermi arc states.\n\n\n\n\n\n\n\\section{RKKY interaction and Recursive Green's function}\\label{sec:recgf}\nRuderman-Kittel-Kasuya-Yoshida (RKKY) theory~\\cite{RKKY} describes the effective coupling between\ntwo impurity spins ${\\bf S}_1$ and ${\\bf S}_2$ in a metal mediated by the conduction electrons.\nThe spins, located respectively at ${\\bf r}_1$ and ${\\bf r}_2$,\nare typically treated as classical magnetic moments, and\nare assumed to be coupled to the electrons by $sd$ Hamiltonians,\n$H_{sd}=J{\\bf S}_i \\cdot {\\bf s}({\\bf r}_i)$ ($i=1,2$), where ${\\bf s}({\\bf r}_i)$ is\nthe conduction electron spin density at the location of impurity spin $i$.\nFor small $J$ the resulting impurity spin interaction becomes\n\\begin{align}\nH_{\\text{RKKY}} &= -\\frac{J^2}{\\pi} \\int_{-\\infty}^{E_F} d\\omega \\text{Tr}[(\\mathbf{S}_1.\\mathbf{\\sigma})G(\\mathbf{r}_{12};\\omega + i0+)(\\mathbf{S}_2.\\mathbf{\\sigma})\\nonumber\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~\\times G(-\\mathbf{r}_{12};\\omega + i0+)]\\label{eq:rkky}\\\\\n& \\equiv \\sum_{i,j = x,y,z} J_{ij} S_{1i}S_{2j}\\label{eq:corr},\n\\end{align}\nwhere $\\mathbf{r}_{12}$ is the separation of the two spins and\n$G(\\mathbf{r}_{12};\\omega + i0+)$ is the real space Green's function for the unperturbed electron system. The resultant $J_{ij}$ is essentially the electronic spin-spin correlation matrix. In all of our results we show $J_{ij}$ in the unit of $J^2$.\n\nDetails of the particular electron system in which the impurity spins are embedded enter\nthe calculation through $G(\\mathbf{r}_{12};\\omega + i0+)$.\nFor our WSM system, we will proceed in two ways.\nFirst, we will directly compute $G$ in momentum space from the low-energy Hamiltonian wavefunctions Eq.~(\\ref{eq:wf1}), and then Fourier transform the expression\nto obtain the needed real-space Green's function. Our second approach is more\nnumerical, and involves inverting the tight-binding model, Eq.~(\\ref{eq:H1}).\nIn this approach the discrete translational invariance in the $x-y$ plane of the\nslab geometry allows, for each two dimensional wavevector, independent computation of the Green's function.\nA computation of the real space Green's function then follows from a Fourier transform.\nFor the results we present in the next section, we also restrict ourselves to considering impurities which are exchange-coupled to the same orbital of the two-orbital model, Eq.~\\ref{eq:H1}, which captures the essential physics of interest. For the case of the semi-analytical model of the last section, for which the atomic-scale structure is not\nincluded, we assume\nthe impurities to be exchange-coupled to the conduction electrons within a small region (of thickness of one lattice spacing) on each surface.\n\nBefore proceeding to our results, we use the remainder of this section to outline the\nrecursive Green's function method we use for our fully numerical studies.\nWe are interested in the coupling between impurities placed on the surfaces,\nso that in the computation of $G(\\mathbf{r}_{12};\\omega + i0+)$ one only actually\nneeds the Green's function for sites ${\\bf r}_1$ and ${\\bf r}_2$ on the slab surfaces.\nFollowing Ref.~\\onlinecite{Keshav}, we can compute the two dimensional Fourier transform of this,\n$G_{ij}(\\omega,k_x,k_y)$, where $i$ and $j$ label the surfaces of the slab\non which ${\\bf r}_1$ and ${\\bf r}_2$ reside, respectively.\n\n\n\n\nFor a slab geometry of $N_z$ number of sites in the $z$ direction, we re-write the\ntight-binding Hamiltonian (Eq. \\ref{eq:H1}) in the form\n\\begin{align}\\label{eq:slab}\nH(\\vec{k}_{||})=\\sum_j &\\left(\\psi_{j}^{\\dagger}(\\vec{k}_{||})A(\\vec{k}_{||})\\psi_{j+1}(\\vec{k}_{||}) + {\\rm h.c.}\\right.\\nonumber \\\\\n&\\left.+ \\psi_{j}^{\\dagger}(\\vec{k}_{||})h_{j,j}(\\vec{k}_{||})\\psi_{j}(\\vec{k}_{||}) \\right),\n\\end{align}\nwhere $\\vec{k}_{||}=(k_x,k_y)$, which are good quantum numbers. This allows us to write the Hamiltonian in the form of\n\\begin{align}\nH &= \\left(\\begin{array}{ccccc}\nh & A & 0 & .  & 0\\\\\nA^{\\dagger} & h & A &  . & 0\\\\\n0 & A^{\\dagger} & h &  . & 0\\\\\n. & . & . & .  & .\\\\\n0 & 0 & 0 &  . & h\\\\\n\\end{array}\\right)\n\\end{align}\nand the Green's function is evaluated from the equation\n\\begin{align}\\label{eq:gdef}\n\\left(\\omega\\mathbb{I}-H(\\vec{k}_{||})\\right)G(\\vec{k}_{||},\\omega)=\\mathbb{I}.\n\\end{align}\nWhen $N_z = 1+2^k$, the above set of equations can be recast in the form\n\\begin{align}\n\\left( \\omega\\mathbb{I} -h' \\right)G' = \\mathbb{I},\n\\end{align}\nwith\n\\begin{align}\nh' = \\left(\\begin{array}{cc}\nh^{(k)} & A^{(k)} \\\\\nA^{\\dagger(k)} & h^{(k)} \\\\\n\\end{array}\\right), \\quad G' = \\left(\\begin{array}{cc}\nG_{11} & G_{1N_z} \\\\\nG_{N_z1} & G_{1N_z} \\\\\n\\end{array}\\right),\n\\end{align}\nwhere the $h_t^{(k)},h_b^{(k)}$ and $A^{(k)}$ are found by recursively solving\n\\begin{align}\nA^{(i+1)} =A^{(i)}(\\omega-h^{(0)})^{-1}A^{(i)}, \\\\\nh^{(i+1)} =h^{(i)}+A^{\\dagger(i)}(\\omega-h^{(i)})A^{(i)}, \\\\\nh_{t}^{(i+1)} = h_{t}^{(i)}+A^{(i)}(\\omega-h^{(i)})^{-1}A^{\\dagger(i)},\\\\\nh_{b}^{(i+1)} = h_{b}^{(i)}+ A^{\\dagger(i)}(\\omega-h^{(i)})^{-1}A^{(i)},\n\\end{align}\nwith $h_t^{(0)}=h_{b}^{(0)}=h^{(0)}=h$ and $A^{(0)}=A$. This yields the two surface Green's functions $G(\\vec{k}_{||})_{1,1}$ and $G(\\vec{k}_{||})_{N_z,N_z}$ as well as their connections $G(\\vec{k}_{||})_{1,N_z}$ and $G(\\vec{k}_{||})_{N_z,1}$ without requiring\na solution for the full Green's function.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{withL.pdf}\n\t\\caption{Top main panel: The RKKY coupling between two spins (connected to same orbital) on  opposite surfaces of the WSM slab of thickness $L_z=N_za$ and placed at (0,0,0) and $(R = 40a,0,L_z)$  as a function of $N_z$, evaluated using numerical Green's function method. Inset: Results from analytical wavefunctions, keeping only the $n=0$ bands. With increasing thickness, all components decrease rapidly (see discussion in the main text). Bottom panel: The RKKY coupling between two spins (connected to same orbital) on opposite surfaces of the WSM slab of thickness $N_za$ placed at three lateral distances $(R = 20a,0,L_z)$, $(R = 40a,0,L_z)$, $(R = 60a,0,L_z)$  as a function of $N_z$.  These results are computed using the analytical wavefunctions keeping only the $n=0$ bands. Note that the spin-spin couplings peak at slightly different slab thickness for different lateral separations between them. Other parameters are same as in Fig.~\\ref{fig:two}.}\\label{fig:withN}\n\\end{figure}\n\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{xzsurface.pdf}\n\t\\caption{RKKY coupling for slab with surfaces perpendicular to the inter-Weyl node separation ($y$ direction in our model) for which there are no Fermi arcs.  $m=0.5\\lambda$, $N_z=33$.\nTop panel: Spins on the same surface.  Bottom panel: Spins on opposite surfaces. In comparison with Figs.~\\ref{fig:top} and ~\\ref{fig:two} the couplings are very small. For larger $R$, as the $J_{ii}$ become very small, the oscillations are possibly due to the numerical inaccuracy.\n}\\label{fig:xzsurface}\n\\end{figure}\n\n\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=0.95\\textwidth]{Finitemu.pdf}\n\t\\caption{The diagonal elements of coupling matrix at a finite chemical potential (given by $\\mu$ in units of $\\lambda$), showing $2k_F$ oscillation. The left column shows the results for the spins on the same surface of the slab and the right column shows the results when the spins are on opposite surfaces. The slab thickness $N_z=33$ and all other parameters are same as in Figs.~\\ref{fig:top} and ~\\ref{fig:two}.}\\label{fig:finitemu}\n\\end{figure*}\n\n\n\\section{Numerical results}\\label{sec:num}\nIn this section, we present our results for the specific cases when ({\\it i}) the magnetic impurities are on the same surface of the WSM, and ({\\it ii}) when the magnetic impurities are on opposite surfaces of the WSM. In both cases we compute the RKKY interaction using the recursive Green's function method outlined in the last section, as well as using the analytical wavefunctions of the $n=0$ bands of Sec.~\\ref{sec:wsmtf}, which contain the Fermi arc states, and compare the results.\n\n\\subsection{Impurities on a single surface}\nAs discussed above, the Fermi arc states disperse in energy along $\\vec{k}$\nperpendicular to the arc itself.\nAs a result these states have a highly asymmetric velocity, with $v_y\\ll v_x$.\nThe effective interaction between spin impurities placed on the same surface reflects this\nstrong asymmetry. For a semi-infinite system, it can be shown for large $r$ that the elements of the Green's function $G(\\vec{r})$ asymptotically vanish as $\\sim 1\/r^2$ when $\\vec{r}$ is on the surface and\n$\\vec{r} \\parallel \\hat{y}$. By contrast, they fall off as $~\\sim 1\/r$ when $\\vec{r}$\npoints along the $\\hat{x}$ direction~\\cite{RKKYWSMSur1}; the difference is a consequence of the (nearly)\nunidirectional dispersion of the Fermi arc energies.\nThis results in the strongest RKKY coupling for impurities\nseparated along\nthe $\\hat{x}$ direction, and in what follows we focus on separations along this direction.\nMoreover,\nfor a thick enough sample (when the presence of the other surface may be neglected), states in each of the Fermi arcs are spin-polarized (along the direction of $\\sigma_x$ in our model) and are chiral in their dispersion (i.e, the energy is proportional to $\\pm k_x$ for $\\eta=\\pm1$). One expects in this case that the RKKY interaction $J_{xx}$ will be vanishingly small~\\cite{RKKYQSH1,SonuRKKY,RKKYWSMSur1}.\n\nThese expectations may be understood as follows.\nThe Fermi arcs, for a thick sample, are exponentially confined to a surface at $z=0$ with an approximate wave function (see Eqs ~(\\ref{eq:vacone}) and (\\ref{eq:vactwo})):\n\\begin{align}\n\\psi_{{\\rm FA}}(\\textbf{r}) \\approx e^{i k_x x} e^{i k_y y} e^{-M_{\\eta}(k_y) \\frac{z}{\\lambda}}\\left(\\begin{array}{c}\ni \\\\\n\\eta \\\\\n\\end{array}\\right),\n\\end{align}\nwhere $M_{\\eta}(k_y) = m\\eta -\\lambda\\sin(k_y)$.  These wavefunctions allow us to write an effective Green's function from the Fermi arc on this two-dimensional surface in the form\n\\begin{align}\nG_{\\eta}^{{\\rm FA}}(\\omega + i\\delta;\\textbf{r}) &= (\\sigma_0-\\eta\\sigma_y)\\int \\frac{d^2 k}{(2\\pi)^2}\\frac{e^{i k_x x} e^{i k_y y}}{\\omega -\\eta v_F k_x +i\\delta}\\nonumber\\\\\n&\\times\\frac{M_{\\eta}(k_y)}{\\lambda}\\theta(k_y-k_{\\eta,1})\\theta(k_{\\eta,2}-k_y),\n\\end{align}\nwhere the Fermi arcs exists between $k_{\\eta,1}= \\eta\\frac{\\pi}{2} - k_0$ and $k_{\\eta,2} = \\eta\\frac{\\pi}{2} + k_0$. The $k_y$ momentum can be integrated between $k_{\\eta,1}$ and $k_{\\eta,2}$ and performing the $k_x$ integral one obtains,\n\\begin{align}\\label{eq:GFFA}\nG^{{\\rm FA}}_{\\eta} (\\omega + i\\delta; \\textbf{\\textit{r}}) = i\\eta\\frac{\\eta \\sigma_y - \\sigma_0}{v_F^2} e^{i(\\omega+i\\delta) \\frac{|x|}{v_F}}I(y)\\theta(\\eta x),\n\\end{align}\nwhere, the Fermi velocities of the Weyl nodes $v_F = \\lambda$. In the limit $\\delta \\rightarrow 0+$,\n\\begin{align}\n&I(y) = e^{i\\pi y\/2}\\Gamma(y),\\\\\n&\\text{with}~~ \\Gamma(y)=\\frac{yv_F\\sin k_0 \\cos (yk_0) - m\\sin (yk_0)}{y(y^2-1)},\n\\end{align}\nand \\textbf{\\textit{r}} = $(x,y)$. This approximate form for the Green's function is useful in determining the asymptotic behavior of the RKKY interaction, as we show briefly in the next section (see also Ref.~\\onlinecite{RKKYWSMSur1}). Note that the step function in Eq.~(\\ref{eq:GFFA}) implements the chiralities of the Fermi arcs. As $\\eta=\\pm$, in the full 4$\\times$4 orbital\/spin space, the Green's function is then\n\\begin{align}\nG^{{\\rm FA}}( \\omega; \\mathbf{r}) = \\left(\\begin{array}{cc}\nG^{{\\rm FA}}( \\omega; \\mathbf{r})_+ & 0\\\\\n0 & G^{{\\rm FA}}( \\omega; \\mathbf{r})_-\n\\end{array}\\right).\n\\end{align}\nWith this expression\nit is straightforward to work out the RKKY integral (Eq. \\ref{eq:rkky})\nand show that, in the original basis of the Hamiltonian, Eq.~(\\ref{eq:H1}), the elements of the correlation matrix, Eq.~(\\ref{eq:corr}), are given by\n\\begin{align}\n& J_{xx}=0; ~~ J_{yy}=J_{zz} = \\frac{J^2\\Gamma (y)^2}{2\\pi^3xv_F^4}\\cos(2k_F x +\\pi y),\\\\\n& J_{zy}=-J_{yz} =  \\frac{J^2\\Gamma (y)^2}{2\\pi^3xv_F^4}\\sin(2k_F x +\\pi y).\n\\label{eq:FAJs}\n\\end{align}\nOther off-diagonal components vanish. In this limit, the separation of the \\textit{pairs} of the Weyl nodes (taken as $\\pi\/a$ in our model) does not alter the result.\n\nThe symmetries apparent in Eqs. \\ref{eq:FAJs} can be understood from the original tight-binding system as we discuss in Appendix A, and are captured by our results, as\nillustrated in Fig.~\\ref{fig:top}.\nIn particular\nthe value of $J_{xx}$ decreases with increasing thickness of the sample whereas the\nother diagonal components become constant with increasing thickness. In contrast to the analytical result, $J_{zz}$ and $J_{yy}$ fall off asymptotically roughly as $1\/x^2$, as listed in Table 1.  This difference from Eqs. \\ref{eq:FAJs} is expected due to the curvature of the Fermi arcs~\\cite{RKKYWSMSur1}.  Off-diagonal components other than $J_{zy}$ and $J_{yz}$ (not shown) are several orders of magnitude smaller than these quantities, as expected from the above\nanalysis. Note also the qualitative agreement between results from our semi-analytical model and the tight-binding computation.\n\n\n\n\\begin{center}\n\t\\begin{table}[h]\n\t\t\\begin{tabular}{ | c | c | c |}\n\t\t\t\\hline\n\t\t\t& $N_z$ & $\\alpha$  \\\\ \\hline\n\t\t\t$J_{xx}$  & 9 & $\\approx$ 2.53 \\\\\n\t\t\t\\hline\n\t\t\t$J_{xx}$  & 17 & $\\approx$ 2.29 \\\\\n\t\t\t\\hline\n\t\t\t$J_{xx}$  & 33 & $\\approx$ 2.26 \\\\\n\t\t\t\\hline\t\t\t\n\t\t\\end{tabular}~~~~~~~\n\t\t\t\\begin{tabular}{ | c | c | c |}\n\t\t\\hline\n\t\t& $N_z$ & $\\alpha$  \\\\\n\t\t \\hline\n\t\t$J_{zz}$  & 9 & $\\approx$ 2.48 \\\\\n\t\t\\hline\n\t\t$J_{zz}$  & 17 & $\\approx$ 2.10 \\\\\n\t\t\\hline\n\t\t$J_{zz}$  & 33 & $\\approx$ 2.04 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\n\t\t\\caption{Assuming the couplings $J_{ii}$ between the impurity spins on the same surface go as $\\sim R^{-\\alpha}$, the best fitted value of $\\alpha$ is quoted for the result of Fig.~\\ref{fig:top}(b) for $R\/a$ between 50 and 80.}\n\t\\end{table}\n\\end{center}\n\nThe results for $J_{xx}$ obtained from the tight-binding computation are particularly interesting.  In contrast to the result for straight Fermi arcs in a single surface\nsystem derived above, in the slab geometry $J_{xx}$\nremains non-zero and falls off rather slowly (see Table 1) with the distance between the impurity spins.  (Similar behavior is found in our analytical model at small thickness.)  The discrepancy can be attributed to two possible effects: (i) interactions mediated by the bulk states which were not included in the simple Fermi arc analysis, and (ii) the presence of the second surface. Interestingly, Fig.~\\ref{fig:top}(b) shows that $J_{xx}$ vanishes rapidly with increasing thickness, which clearly favors mechanism (ii). Fig.~\\ref{fig:top}(d) also manifests a critical thickness $L_c$, defined by the width $L_z$ at which $J_{xx}$ attains its maximum value before decaying sharply with further increase. This critical thickness can be used to define a ``thin-film limit'' of the system, for which the effects of having two surfaces are maximal.  Noting that $k_0$ is the only relevant momentum scale, we expect the thin film limit to scale as $L_c \\sim 1\/k_0$. A numerical verification of this hypothesis is presented in Appendix B.\n\n\n\n\n\\subsection{Impurities on opposite surfaces}\nWhen the impurity spins are put on opposite surfaces of the slab, they may communicate via electron states that are present in the bulk of the Weyl semimetal.  To examine this effect\nnumerically, we place the two spins on different surfaces of a WSM slab with various thicknesses and vary their separation along the $x$ axis (i.e., the direction in which the Fermi arcs states disperse). Results from these are illustrated in Figs.~\\ref{fig:two} and ~\\ref{fig:withN}.\n\n\n\\begin{center}\n\t\\begin{table}[h]\n\t\t\\begin{tabular}{ | c | c | c |}\n\t\t\t\\hline\n\t\t $N_z$ & $\\alpha$ & $\\beta$  \\\\ \\hline\n\t\t9 & $\\approx$ 3.28 & $\\approx$ 2.36\\\\\n\t\t\t\\hline\n\t\t17 & $\\approx$ 3.33 & $\\approx$ 2.39 \\\\\n\t\t\t\\hline\n\t\t33 & $\\approx$ 3.24 & $\\approx$ 2.32 \\\\\n\t\t\t\\hline\t\t\t\n\t\t\\end{tabular}~~~~~~\n\t\t\t\\begin{tabular}{ | c | c |}\n\t\t\\hline\n\t\t$R$ & $\\gamma$  \\\\ \\hline\n\t\t20 &  $\\approx$ 4.96\\\\\n\t\t\\hline\n\t\t40 &  $\\approx$ 5.02 \\\\\n\t\t\\hline\n\t\t60 & $\\approx$ 5.06\\\\\n\t\t\\hline\t\t\t\n\t\\end{tabular}\n\t\t\\caption{Assuming the couplings $J_{ii}$ between the impurity spins on two opposite surfaces go as $\\sim (R^2+L_Z^2)^{-\\alpha\/2}$, the best fitted value of $\\alpha$ is quoted in the left-most column for the result of Fig.~\\ref{fig:two}(b) for $R\/a$ between 50 and 80. For fixed $L_z$, the same can be fitted with $R^{-\\beta}$, which is shown in the middle column. For fixed $R$, the results from Fig.~\\ref{fig:withN}(b) can be fitted (for $N_z$ between 13 and 21) with $L_z^{-\\gamma}$, which is shown in the right-most column.}\n\t\\end{table}\n\\end{center}\n\nFor the range of parameters we examined, the symmetry properties of the spin coupling matrix turn out to be the same as when the spins are situated in the bulk and are\nseparated along the $z$ direction (see Appendix A). Numerically, when the two spins are located at sites $(x,y,z) = (0,0,0)$ and $(R,0,L_z=N_za)$,  the coupling between them is surprisingly strong despite the fact that they reside on different surfaces, decaying as $\\sim (R^2+L_z^2)^{-\\alpha\/2}$, where $\\alpha$ is between 3 and 4, for fixed $L_z$ and increasing $R$, showing a rather slow decay of the RKKY coupling and possibility of mean-field magnetic ordering. Fitting the decay with $\\sim R^{-\\beta}$, where the thickness $L_z$ is fixed, the coupling decay in even slower manner, with $\\beta $ between 2 and 3. Results from the tight-binding simulation and from the low-energy wavefunctions both support these results.\n\nAs a function of the thickness $L_z$, for fixed $R$ the couplings initially increase and after attaining maximum values decrease rapidly. \nResults for both the analytical and tight-binding approaches for varying $N_z$ \nare illustrated in Fig.~\\ref{fig:withN}. For fixed $R$, as a function of the thickness the coupling decays as $\\sim L_z^{-\\gamma}$ with $\\gamma\\approx 5$. This is the same falloff as for RKKY coupling in the bulk of the WSM~\\cite{RKKYWSMSur1}, suggesting\nthat for large enough $L_z$ the coupling between spins on opposite surfaces is eventually\ndominated by the bulk states.\nThe clear differences among the parameters $\\alpha,\\beta,\\gamma$ capture the essential physics of the WSM system with Fermi arcs, and have been listed in the Table II.\n\nThe maximum coupling as a function of the thickness can be qualitatively understood as follows. As the thickness increases, the Fermi arcs localize increasingly firmly on the surfaces, increasing the surface density of states near the Fermi energy. This leads to an increase in the coupling between the impurity spins on the surfaces and the conduction electrons,\nwhich can mediate intersurface interactions effectively when $L_z$ is not too large.\nOn the other hand, as $L_z$ increases, the \nnumber of conduction electron states which are sensitive to both surfaces decreases,\nresulting in weaker coupling between spins on opposite surfaces. With increasing thickness, the competition between these two mechanisms gives rise to a critical thickness for which the coupling between spins placed on opposite surfaces maximizes.\nAs in our earlier argument for impurities on the same surface, with $k_0$ the only relevant momentum scale we expect this thickness to scale as $\\sim 1\/k_0$. We explore this in Appendix B.  Again this critical thickness also defines a thin-film limit; the values of this critical thickness obtained from inter-surface coupling are of the same scale as those obtained from the intra-surface coupling.\n\n\n\n\\section{SUMMARY AND DISCUSSION}\\label{sec:disc}\nIn this work we have examined RKKY interactions among impurity spins on the surfaces of Weyl semimetal (WSM) slabs, using both an approach in which the wavefunctions of the WSM electrons are found in an analytical form, and a more fully numerical recursive Green's function technique.  We find that Fermi arc surface states play an important role in the RKKY coupling, creating couplings that are stronger and more long-range than\nis found for impurities well-inside the bulk of the system.  Surprisingly, even the\ncoupling between spins on opposite surfaces can be relatively strong.  As a function of\nfilm thickness, we find that the RKKY couplings are non-monotonic, with maxima that\ncan define a ``thin-film'' limit, in which the effects of {\\it both} surfaces are\nin some sense maximal.  The relative strengths and signs of different components of the RKKY couplings $J_{ij}$ can be understood using a simple model in which only surface\nstates associated with the Fermi arcs are retained, and in which the Fermi arcs\nare perfectly straight.\n\nThe importance of electron states with strong support on the surfaces can be examined by comparing results\nfor geometries with Fermi arcs to ones without them.  Fig. ~\\ref{fig:xzsurface}\nillustrates RKKY coupling for spins on the same and opposite surfaces which are perpendicular\nto the direction of separation between the Weyl nodes in the bulk, for which surface states\nare not present.  The generally smaller scale of the resulting couplings supports the\nidea that the Fermi arc states play a large quantitative role in setting the coupling scale.\n\nThe results presented to this point have been for vanishing chemical potential $\\mu$,\nwhere the only extended Fermi surfaces are due to the Fermi arcs, and the Fermi energy\npasses directly through the Weyl nodes in the bulk.  In general, when $\\mu \\ne 0$ and\nthe Fermi wavevector $k_F \\ne 0$ in the bulk, one expects $2k_F$ oscillations in the RKKY\ncoupling.  Results for $\\mu \\ne 0$ are presented in Fig. ~\\ref{fig:finitemu}, for which\nthe oscillations are apparent.  The envelopes within which these oscillations occur\nbehave rather similarly to the results for $\\mu=0$.\n\nWhen a system is of order or thinner than a critical thickness $\\sim 1\/k_0$, our results show that\na proper treatment of RKKY interactions requires one to retain states from the Fermi arcs\nof {\\it both} surfaces, even if the two spins reside on the same surface.  For real systems,\nsuch as TaAs~\\cite{taas}, the typical separation of Weyl nodes is rather small (of the order of $k_0\\approx 0.1\\pi\/a$) and thus we expect the critical thickness to be of the order of several tens to a hundred lattice spacings.  Such thicknesses are quite reasonable for thin-film semiconductor systems.\n\nWe conclude with some speculations about the kind of magnetic order these RKKY interactions might induce in the low temperature state of spin impurities on the surfaces of a WSM thin film. At large distances, the strongest couplings we find are for $J_{yy} = J_{zz}<0$ within\na single surface, suggesting the system will form a planar ferromagnet in its ground state.\nThe non-vanishing $J_{yz}$ and $J_{zy}$ couplings if large enough could induce spiral order;\nwhile at short distances these can be larger than the diagonal elements, at long distances\nthe latter are significantly larger.  Given the relatively slow spatial decay of the RKKY interaction, it seems likely that the system will favor ferromagnetism.  Furthermore, the\nsign of coupling for impurities on different surfaces suggests that the magnetization\nof the two surfaces will be parallel to one another in the groundstate.  In principle at\nlow temperature such magnetic order should be detectable.  Moreover, with this type of order\none expects a magnetic disordering transition at finite temperature in the Kosterlitz-Thouless universality class,\nwhich might be detected in thermal measurements or via spin transport in the system.  Finally,\nthe importance of the Fermi arc states in supporting such magnetic order could be tested\nby comparing the behavior of slabs in which the surfaces support them to ones in which they do not.\nWe leave the investigation of these questions to future work.\n\n{\\it Acknowledgements} -- HAF thanks the NSF for support through Grant Nos. DMR-1914451, DMR-1506263 and DMR-1506460, by the US-Israel Binational Science Foundation. AK thanks support from SERB (Gov. of India) through grant ECR\/2018\/001443 and BRNS (Gov. of India) support through grant 58\/20\/15\/2019-BRNS.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s:intro}\n\nElectron microscope images done on annealed amorphous materials generally reveal a tessellation of crystalline grains with a distribution of sizes and shapes. The inhomogeneous end product of the crystallization process has its origin in the formation of nuclei and their growth into grains as schematically shown in Fig.~\\ref{fig:tessellation}.\n\\begin{figure}[h] \n\\begin{center}\n\\includegraphics[scale=0.55]{Lokovic_Fig1.eps}\n\\end{center}\n\\caption[Grain distribution at intermediate stages of crystallization]{Schematic representation of the grain distribution at intermediate stages of crystallization. The smallest dots are nuclei created randomly in space and time. The dark grains are completely surrounded by other grains and have thus stopped growing because of impingement, absence of coalescence and secondary growth. In this figure, unimpinged grains are disk like, corresponding to an isotropic growth rate. Impingement changes the shape of grains. In Sec.~\\ref{s:anisotropic} we generalize the theory to anisotropic growth rates.}\n\\label{fig:tessellation}\n\\end{figure}\n\nThese can occur in a variety of ways, one of which is the random formation of stable clusters with respect to thermodynamic fluctuations within the sample, and the growth of these nuclei into grains. An important example of such random nucleation and growth (RNG) process is observed in the crystallization of silicon thin films used for solar cells (see, {\\it e.g.}, Refs.~\\onlinecite{wagnerPSSA10,*HandbookPhotovoltaicv11} and references therein). Characterizing quantitatively the grain size distribution (GSD) is important since many physical properties directly depend on this distribution. For example, electrical, magnetic, optical or even superconducting properties are affected by the granularity of a sample.\n\nWe have developed recently a theory for the non-equilibrium GSD during the crystallization of a solid in $d$ dimensions\\cite{bergmannJCG08,teranPRB10} and applied the theory successfully to the solid phase crystallization of Silicon.\\cite{billMRS09} Theoretical studies discussing the grain size distribution in various contexts use analytical tools mainly in one dimension,\\cite{gelbardJCIS79,shiJMR91,shiJMR91b,shiMCP94,sekimotoPA84,sekimotoIJMPB91,axePRB86,bennaimPRE96,junPRE05} or numerical techniques\\cite{brownJAP95,fayadSM99,riosSM99,wangISIJI03,axePRB86,crespoPRB96,pinedaPRB99,pinedaPRE04,brunaJAP06,soderlundPRL98,farjasPRB07,farjasPRB08} to describe the formation of grains during crystallization. The time-dependent GSD derived in Refs.~\\onlinecite{bergmannJCG08,teranPRB10} and extended here is obtained analytically for $d$ dimensions as a solution of a differential equation that accounts for the random creation of nuclei over the volume of the sample and their growth into grains. Besides the ability to describe the distribution during the crystallization process, one other important outcome of the theory is that the lognormal distribution of grain sizes generally postulated to fit experimental data at full crystallization results naturally from our theory when the effective nucleation {\\it and} growth rates are time-decaying quantities.\\cite{bergmannJCG08,teranPRB10} As discussed in Ref.~\\onlinecite{teranPRB10}, the proposed differential equation for the distribution is rather general and the theory may thus apply to a variety of phenomena not related to crystallization.\n\nThe main purpose of this paper is to extend the theory in two major ways and demonstrate that the non-equilibrium GSD derived previously is robust under various generalizations. We also provide an expression for the GSD (Eq.~\\eqref{solanistropic} below) that can be easily used to analyze experimental data.  The first generalization  allows for {\\it anisotropic} grain growth. In Refs.~\\onlinecite{bergmannJCG08,teranPRB10} we had assumed that unimpinged grains remain spherical (or disk-like in 2D thin films). Yet, it is known experimentally that unimpinged grains can display many different shapes such a ellipsoidal or filamentary (see {\\it e.g.} Ref.~\\onlinecite{kakinumaJVSTA95,*hartman73,*gentryPRB09}). Such non-spherical shapes result from the anisotropic growth of grains. In the second extension of the theory we introduce a nucleation rate that accounts for the fact that local thermodynamic and structural fluctuations allow clusters of various sizes to become stable nuclei. Previously we had assumed that only nuclei of a well defined size are formed. We show below that the theory is robust against relaxing this condition to allow for the formation of nuclei with sizes varying within an experimentally reasonable range of a given average.\n\nThe theory presented in Refs.~\\onlinecite{bergmannJCG08,teranPRB10} applies when nucleation occurs randomly over the sample at a constant microscopic rate $I_0$ and nuclei grow into grains at constant microscopic rate $v_0$. We assume that grains do not coalesce to form larger grains. We also assume the absence of secondary grain growth.  Our assumptions imply that the total number of grains per unit volume of the sample, $N(t)$ is a monotonously increasing function of time; at fixed number of nuclei the crystallization process consists in the growth of a constant number of grains. The distribution of grain sizes then obeys a continuity equation. Since we assume simultaneous nucleation and growth of grains the differential equation must include a source term for random nucleation. The resulting equation for the non-equilibrium GSD $N(\\mathbf{r},t)$ is\\cite{bergmannJCG08,teranPRB10}\n\\begin{eqnarray}\n\\label{PDE}\n\\frac{\\partial N(\\mathbf{r},t)}{\\partial t} + \\boldsymbol{\\nabla}_{\\mathbf{r}}\\cdot\\left[N(\\mathbf{r},t)\\,\\mathbf{v}(\\mathbf{r},t)\\right] = \\mathcal{D}(\\mathbf{r},t)\\,,\n\\end{eqnarray}\nwhere the right hand side of the equation is the source term for nucleation and $\\mathbf{v}$ is the general growth rate of grains. The source term is written as a product $\\mathcal{D}(\\mathbf{r},t) = J(t) D(\\mathbf{r})$, where $J(t)$ is the time dependent effective nucleation rate and $D(\\mathbf{r})$ is the source term of nuclei. Both quantities are defined below. $\\mathbf{r} = (r_1,\\dots,r_d)$ is a $d-$dimensional vector characterizing the size of a grain. As discussed in the next section we describe grains as $d-$dimensional ellipsoids and the components $r_i$ ($i=1,\\dots,d$) of the vector are the semi-axes of the ellipsoid. For isotropic growth rates\\cite{teranPRB10} we have $r_1=\\dots =r_d=\\rho$, the radius of the $d-$dimensional sphere. Other possible definitions of $\\mathbf{r}$ are suggested in Ref.~\\onlinecite{teranPRB10}. In section \\ref{s:anisotropic} we assume $r_1 > r_2 \\geq r_3$ and in Sec.~\\ref{s:gaussian} we take $r_i=\\rho$ ($i=1,\\dots,d$).\n\nSolving Eq.~\\eqref{PDE} requires the knowledge of the effective nucleation and growth rates, $J(t)$ and $\\mathbf{v}(\\mathbf{r},t)$, respectively. It is important to realize that though the microscopic rates $I_0$ and $v_0$ are constant, the {\\it effective} rates are time-dependent. Assuming that no nuclei can form in the volume of an existing grain, the probability of formation of a new nucleus must decay in time since the volume available for its formation decreases in time. Kolmogorov, Mehl, Johnson and Avrami (KMJA) have derived an exact expression for the effective nucleation rate $J(t)$ in the case of RNG processes in $d$ dimensions\\cite{kolmogorov37,mehljohnson,avrami39,*avrami40,*avrami41}\n\\begin{eqnarray}\\label{Jt}\nJ(t) = I_0 e^{-\\left(\\frac{t-t_0}{t_{cI}}\\right)^{d+1}}\\Theta\\left(\\frac{t-t_o}{t_{cI}}\\right) \\equiv I_0 f(t).\n\\end{eqnarray}\nThe expression for $f(t)$ is well-established as discussed in numerous publications (see references in Ref.~\\onlinecite{teranPRB10}) and applies to RNG processes. $t_0$ is the incubation time, $t_{cI} = \\left[(d+1)\/I_0v_0\\omega_d\\right]^{1\/d+1}$ is the critical time for nucleation and $\\Theta$ is the Heaviside function. $\\omega_d$ is the constant appearing in the volume $\\Omega_d = \\omega_d r_1\\cdots r_d$ of the hypersphere.\\cite{teranPRB10}  For example for $d=1,2,3$ we have $\\omega_1 = 2$, $\\omega_2 = \\pi$, $\\omega_3 = 4\\pi\/3$, respectively.\n\nThe effective growth rate similarly decreases in time since impingement stops the growth of a grain in the direction perpendicular to the contact line between grains. Once completely surrounded by other grains a given grain cannot grow anymore (see Fig.~\\ref{fig:tessellation}). The time-dependence of the effective growth rate is postulated to have a form similar to the one for $J(t)$. The  discussion of Refs.~\\onlinecite{bergmannJCG08,teranPRB10,billMRS09} lead to consider an exponential time decay of the effective growth rate\n\\begin{eqnarray}\\label{vt}\nv(t) = v_0 e^{-\\left(\\frac{t-t_0}{t_{cv}}\\right)} \\Theta\\left(\\frac{t-t_0}{t_{cv}}\\right) \\equiv v_0 g(t)\\,.\n\\end{eqnarray}\nBoth the microscopic growth rate $v_0$ and the critical time $t_{cv}$ can be determined experimentally.\\cite{bergmannJCG08}\n\nFinally, to solve Eq.~\\eqref{PDE} we need an expression for the source term of nuclei, $D(\\mathbf{r})$. We consider two cases. In one case, we assume that only nuclei of a specific critical volume $\\Omega_c$ can form. In this case, $D(\\mathbf{r}) = \\delta\\left(\\Omega - \\Omega_c\\right)$ is given by a Dirac distribution. This assumption allows for an analytic treatment of the equation\\cite{bergmannJCG08,teranPRB10} and will be considered in the next section, Sec.~\\ref{s:anisotropic}, dealing with anisotropic growth rates. In Sec.~\\ref{s:gaussian} we generalize the theory by relaxing this condition. We consider the more physical standpoint according to which a cluster is thermodynamically stable when within a range of volumes around a mean value. We consider the case of a Gaussian distribution\n\\begin{eqnarray}\\label{DrgaussianOm}\nD(\\mathbf{r}) = \\frac{1}{\\epsilon\\,\\sqrt{2\\,\\pi}}\\,e^{-\\frac{1}{2\\epsilon^2}(\\Omega - \\Omega_c)^2},\n\\end{eqnarray}\nwhere $\\epsilon$ is a small real number and $\\Omega_c$ is now the mean value of a nucleus' volume. The study of the nucleation barrier that has to be overcome by a cluster of atoms or molecules and stabilizes the nucleus has been studied in detail in Ref.~\\onlinecite{shiJMR91,*shiJMR91b,*shiMCP94}. Their study gives an expression for $\\epsilon$ that can be used to estimate its value for specific systems.\n\n\n\\section{Anisotropic growth rate}\\label{s:anisotropic}\n\n\\begin{figure*} \n\\begin{center}\n\\includegraphics[scale=0.5]{Lokovic_Fig2a.eps}\n\\includegraphics[scale=0.5]{Lokovic_Fig2b.eps}\n\\end{center}\n\\caption[Anisotropic grain and space of grain sizes]{(a) Ellipsoidal grains resulting from an anisotropic growth rate (see text). (b) Space of grain sizes spanned by the semi-axes of the ellipsoid. The non-equilibrium GSD $N(\\mathbf{r},t)$ is calculated in this space. The growth rate $\\mathbf{v}$ points along a radial line in that space since we assume that the anisotropic growth changes the volume of the grain but keeps its shape invariant (see text).}\n\\label{fig:ellipsoid}\n\\end{figure*}\n\nIn the theory developed in Refs.~\\onlinecite{bergmannJCG08,teranPRB10,billMRS09} we assumed that the growth of grains is isotropic, leading to spherical grains when they do not impinge on each other. This assumption is not always valid as even isolated grains often appear non-spherical in shape. The shape is determined by either extrinsic or intrinsic factors. A patterned substrate containing steps may lead to filamentary grains and is an example of an extrinsic factor. On the other hand, when the interaction between atoms or molecules is directional, the grain may also become non-spherical. For example, in Silicon the growth rate is different along different principal axes of the crystal.\\cite{kakinumaJVSTA95,*hartman73} Another example is the case of planar molecules that often experience a strong binding when stacked, but only weakly interact with each other through the edges of the molecules.\\cite{*gentryPRB09} Although thermodynamic fluctuations will favor the stacking, the binding through the edges may not be completely neglected, leading to an ellipsoidal shape of grains (see Fig.~\\ref{fig:ellipsoid}).\\\\\nIn compounds made of atoms rather than molecules ({\\it e.g.} polycrystalline Si) the formation of  periodic bond chains and the mobility of atoms determine the crystal axis along which the growth preferably occurs.\\cite{hartman73} Hence, interactions between the entities constitute an intrinsic factor leading to anisotropic growth of grains. It is important to note that while extrinsic factors generally impose the growth of grains along specific directions of space, the intrinsic factors allow for a random orientation of the ellipsoids in the volume of the sample. Furthermore, since microscopic interactions between molecules are responsible for the shape of the grain, the growth of a grain will mainly consist of an increase in volume and keep the shape invariant. That is, the ratio of semi-axes of the ellipse (for $d=2$) or the ellipsoid ($d=3$) is constant: $r_j\/r_1 = r_{cj}\/r_{c1}$ for $j=2,3$, $r_1$ ($r_{c1}$) being the major semi-axis of the grain (nucleus). In spherical coordinates it means that $\\varphi$ and $\\theta$ are constant in $\\mathbf{r}$-space (see Fig.~\\ref{fig:ellipsoid}). The calculations performed below rely on these physical assumptions.  When considering anisotropic growth we will assume it is due to intrinsic factors. We also assume the absence of nucleation centers in the amorphous sample, and thus a homogeneous and isotropic nucleation rate.\n\nThe effective growth rate can be written in the form\n\\begin{eqnarray}\\label{vrt}\n\\mathbf{v}(\\mathbf{r}, t)&= v_1(t)\\hat{r}_1 + v_2(t)\\hat{r}_2 + v_3(t)\\hat{r}_3 = v(t)\\,\\hat{r},\n\\end{eqnarray}\nwhere the last equality is obtained by introducing spherical coordinates in $\\mathbf{r}$-space and using the fact mentioned above that the shape of the ellipsoid does not change as the grain grows. $v(t)$ is given by Eq.~\\eqref{vt}. We emphasize that the vector is written in the space of semi-axes of the ellipsoid $(r_1,r_2,r_3)$ and not in real space $(x,y,z)$. This expression for $\\mathbf{v}(\\mathbf{r},t)$ is valid in absence of diffusion of atoms during the crystallization process as is for example the case in solid phase crystallization.\\cite{bergmannJCG08} We also write the nucleation source term in polar (spherical) coordinates\n\\begin{eqnarray}\\label{Drdelta}\nD(\\mathbf{r}) = \\delta(\\Omega_d - \\Omega_{c,d}) = \\frac{1}{A_{c,d}}\\delta(r-r_c)\n\\end{eqnarray}\nwhere $A_{c,d}$ is the area of the nucleus in $d$ dimensions\n\\begin{subequations}\\label{Acd}\n\\begin{eqnarray}\nA_{c,2} &=& 2\\,\\pi\\,r_c\\,\\cos\\varphi\\,\\sin\\varphi,\\\\\nA_{c,3} &=& 4\\,\\pi\\,r_c^2\\,\\sin^2\\theta \\cos\\theta \\sin\\varphi \\cos\\varphi.\n\\end{eqnarray}\n\\end{subequations}\n\nWith Eqs.~(\\ref{Jt},\\ref{vrt}-\\ref{Acd}) the partial differential equation, Eq.~\\eqref{PDE}, can be solved analytically for the anisotropic case. The calculation is more involved than for the isotropic growth rate but follows the procedure presented in Ref.~\\onlinecite{teranPRB10}. Hence, we do not repeat the derivation here. The general result for the non-equilibrium GSD with anisotropic growth rate is\n\\begin{widetext}\n\\begin{eqnarray}\\label{solanistropic}\nN(\\gamma,\\tau) &=& \\left(\\frac{I_o}{v_0}\\right) \\frac{1}{A_{\\infty,d}}\\,\\gamma^{d-1}\\,\\frac{f[\\sigma(\\gamma,\\tau)]}{g[\\sigma(\\gamma,\\tau)]\n\\left\\{\\Theta\\left(\\frac{\\gamma-\\gamma_c}{1-\\gamma_c}\\right) - \\Theta\\left[\\frac{\\gamma-\\gamma_{max}(\\tau)}{1-\\gamma_c}\\right]\\right\\},\n\\end{eqnarray}\n\\end{widetext}\nwhere $f$ and $g$ are given by Eqs.~(\\ref{Jt},\\ref{vt}) with time replaced by $\\sigma(\\gamma,\\tau)$ below.  This expression is written in terms of the dimensionless quantities\n\\begin{eqnarray} \\label{dim}\n\\gamma=\\frac{r}{r_\\infty},\\quad \\tau=\\frac{t}{\\sqrt{t_{cv}t_{cI}}},\n\\end{eqnarray}\nwhere $r_\\infty$ is the magnitude of $\\mathbf{r}$ for the largest grain found at full crystallization, $\\gamma_c = r_c\/r_\\infty$, and $\\sigma(t)$ is given by\\cite{teranPRB10}\n\\begin{eqnarray}\\label{sigmart}\n\\sigma(\\gamma,\\tau) = \\tau_0 + t_r\\,\\ln\\left( \\frac{\\gamma-\\gamma_c}{\\mathcal{V}_0} + e^{-(\\tau-\\tau_0)\/t_r} \\right)^{-1},\n\\end{eqnarray}\nwith $t_r^2 = t_{cv}\/t_{cI}$ and ${\\mathcal V}_0 = t_{cv}v_0\/r_\\infty$.\n\nThe central result, Eq.~\\eqref{solanistropic}, is a generalization of Eq.~(21) in Ref.~\\onlinecite{teranPRB10} to the case of anisotropic growth rates.  It can be implemented in a variety of ways to describe experimental data or extract specific parameters of the model by fitting to experimental findings.\\cite{billMRS09}  The result has several interesting features we now describe.\n\nThe term in curly parenthesis is easily understood. It states that the non-equilibrium GSD $N(\\gamma,\\tau)$ is only non-zero in the interval $\\gamma_c\\leq \\gamma \\leq \\gamma_{\\rm max}$, that is, for grain sizes between those of a nucleus ($\\gamma_c$) and of the largest grain found at time $t$ in the sample [$\\lim_{\\tau\\to\\infty} \\gamma_{\\rm max}(\\tau) = 1$ since $r_{\\rm max}(t\\to\\infty) = r_\\infty$)].\nFurther, the non-equilibrium GSD is proportional to the ratio of microscopic nucleation and growth rates $I_0\/v_0$. It is also proportional to the rate $f(\\sigma)\/g(\\sigma)$. Note, however, that since $\\sigma(\\gamma,\\tau)$ is a non-linear function of time and grain size, the ratio is {\\it not} simply the ratio of Eqs.~\\eqref{Jt} and \\eqref{vt}. Finally, the non-equilibrium GSD is inversely proportional to the surface area $A_{\\infty,d}$ of the largest grain found at full crystallization. The expression for $A_{\\infty,d}$ is the same as in Eq.~\\eqref{Acd} with $r_c$ replaced by $r_\\infty$.\n\nEq.~\\eqref{solanistropic} is a remarkable result. It states that the only difference between isotropic and anisotropic growth rates is the presence of $A_{\\infty,d}$ in the prefactor. Thus, the results derived in Refs.~\\onlinecite{bergmannJCG08,teranPRB10} are robust against the generalization to anisotropic growth rates!\n\n\\begin{figure}[htb] \n\\begin{center}\n\\includegraphics[scale=0.55]{Lokovic_Fig3.eps}\n\\end{center}\n\\caption[Normalized GSD, anisotropic growth, $\\tau$ and $d=3$]{Normalized grain size distribution in the early stage ($\t\\tau = 1$), intermediate stage ($\\tau=2.4$) and late stage ($\\tau=5$) of crystallization for $d=3$. In the early stage only the \tnucleation peaks is observed. At intermediate stage the nucleation peak appears concomitantly with the broader growth \tpeak. At late stages of crystallization, nucleation has disappeared and only the growth peak remains. The curve at full \tcrystallization $\\tau \\geq 5$ is lognormal like.\\cite{teranPRB10} Parameters are $\\gamma_c = 0.0001, \\tau_0 = 0, t_r = \t0.75, \\mathcal{V}_0 = 1$ (see Refs.~\\onlinecite{bergmannJCG08,billMRS09}).}\n\\label{fig:GSDtau}\n\\end{figure}\n\nFig.~\\ref{fig:GSDtau} displays the normalized non-equilibrium GSD $\\bar{N}(\\gamma,\\tau) = N(\\gamma,\\tau)\/N(\\tau)$, where\n\\begin{eqnarray}\\label{Nt}\nN(\\tau) = \\int_{\\gamma_c}^{\\gamma_{\\rm max}(\\tau)} N(\\gamma,\\tau')\\,d\\tau',\n\\end{eqnarray}\nfor early, intermediate and late stages of crystallization\\cite{teranPRB10} in three dimensions ($d=3$).\nIn the early stage, only a nucleation peak appears as the initial process is dominated by the formation of nuclei. The intermediate stage is characterized by the simultaneous appearance of a nucleation peak (near $\\gamma_c$) and a growth peak. That stage occurs over a very short period; in the present calculation, full crystallization is reached for $\\tau \\gtrsim 5$ and the time interval of the intermediate stage is $2\\lesssim \\tau \\lesssim 3$. The late stage of crystallization is obtained for $\\tau \\gtrsim 3$ where nucleation is almost completely suppressed and grains continue growing in areas of untransformed material until complete tessellation of the sample is achieved. Thus, in the late stage of crystallization the shape of the distribution remains unchanged but shifts towards larger grain sizes. As discussed in Refs.~\\onlinecite{bergmannJCG08,teranPRB10} the GSD for $\\tau \\geq 5$ in Fig.~\\ref{fig:GSDtau} displays a lognormal like distribution.\nThe results of the GSD for $d=2$, that is for a thin film where the average grain size at full crystallization is larger than the thickness of the film, are qualitatively similar to the $d=3$ case depicted in Fig.~\\ref{fig:GSDtau} and is therefore not shown here (see Ref.~\\onlinecite{teranPRB10}). \n\nThe remarkable conclusion of this section is that the non-equilibrium GSD for isotropic and anisotropic growth rates are qualitatively similar as they only differ by the constant prefactor $A_{\\infty,d}^{-1}$.\n\n\n\\section{Gaussian nucleation rate}\\label{s:gaussian}\n\nThe theory developed in Refs.~\\onlinecite{bergmannJCG08,teranPRB10,billMRS09}  and used in the previous section considered the  simplified expression, Eq.~\\eqref{Drdelta}, for the source term of nuclei $D(\\mathbf{r})$. We assumed that only nuclei of a well-defined specific volume $\\Omega_c$ are stable in the system. Such assumption is physically not  entirely realistic as grains with a few more or a few less molecules are likely to be thermodynamically stable as well. In the present section we analyze how the non-equilibrium GSD is affected if one relaxes the condition imposed by Eq.~\\eqref{Drdelta}.  Contrary to the previous section but in accordance with Refs.~\\onlinecite{bergmannJCG08,teranPRB10,billMRS09} we assume isotropic growth of grains, leading to spherically shaped unimpinged grains.\n\nTo generalize the theory we assume that the Dirac distribution appearing in Eq.~\\eqref{Drdelta} is replaced by the Gaussian distribution, Eq.~\\eqref{DrgaussianOm}. In dimensionless quantities and using spherical coordinates $D(\\mathbf{r})$ reads\n\\begin{eqnarray}\nD(\\gamma) = \\frac{1}{r_\\infty\\, \\epsilon'\\,\\sqrt{2\\,\\pi}}\\,e^{-\\frac{1}{2\\,\\epsilon'^2}(\\gamma - \\gamma_c)^2}\\,\n\\end{eqnarray}\nwhere $\\epsilon' \\equiv \\epsilon\/r_\\infty$. This term has to be inserted into the right hand side of Eq.~\\eqref{PDE}. Contrary to all previous calculations the present form of the differential equation allows solving for the non-equilibrium GSD analytically except for one integral. Defining\n\\begin{eqnarray}\\label{Ntilde}\n\\tilde{N}(\\gamma,\\tau) = \\gamma^{d-1}\\,A_{\\infty, d}\\,r_\\infty^2\\,N(\\gamma,\\tau),\n\\end{eqnarray}\nthe normalized GSD for a Gaussian nucleation rate becomes\n\\begin{widetext}\n\\begin{eqnarray}\\label{solgaussian}\n\\tilde{N}(\\gamma,\\tau) &=& \\frac{2\\,\\epsilon'\\,t_r^2}{\\gamma_c\\sqrt{2\\,\\pi}}e^{-\\frac{\\gamma_c^2}{2\\,\\epsilon'^2}}\\,\\frac{I(\\sigma)}{v(\\sigma)}\\,\\delta_{d,2\n\\mp \\frac{t_r}{2\\,\\gamma_c^{d-1}}\\int_{\\tau_0}^\\tau \\frac{I(\\tau')}{\\epsilon' \\sqrt{2\\,\\pi}}\\left[\\gamma - u(\\tau',\\tau)\\right]^{d-1\n\\exp\\left\\{-\\frac{\\left[\\gamma \\mp \\gamma_c - u(\\tau',\\tau)\\right]^2}{2\\,\\epsilon'^2}\\right\\}d\\tau',\n\\end{eqnarray}\n\\end{widetext}\nwhere the minus (plus) sign is for $d=2$ ($d=3$), Eq.~\\eqref{sigmart} defines $\\sigma$, and\n\\begin{eqnarray}\nu(\\tau',\\tau) = \\mathcal{V}_0 \\left[\\exp\\left(-\\frac{\\tau-\\tau_0}{t_r}\\right) - \\exp\\left(-\\frac{\\tau'-\\tau_0}{t_r}\\right)\\right].\n\\end{eqnarray}\nBecause of the Kronecker symbol $\\delta_{d,2}$ the first term is absent for $d=3$. That term vanishes in the limit $\\epsilon\\to 0$, in which case the Gaussian in the second term reduces to the Dirac distribution and the integral can be solved, leading to the result previously established in Refs.~\\onlinecite{bergmannJCG08,teranPRB10}.\n\nSince for $\\epsilon\\neq 0$ the above integral cannot be solved analytically we performed numerical calculations to study the time dependent form of the GSD. To comply with typical experimental results (see, {\\it e.g.}, Ref.~\\onlinecite{shiJMR91,*shiJMR91b,*shiMCP94}) we assumed that the width of the Gaussian is of the order of a tenth of the average nuclei radius, $\\epsilon\\sim 0.1 r_c$. The results for the normalized distribution differ from those of the previous section represented in Fig.~\\ref{fig:GSDtau} only for early stages of crystallization, and only marginally. The nucleation peak is slightly wider in the early and intermediate stages. Hence, Eq.~\\eqref{solgaussian} should only be used to discuss experimental studies of early stages of nucleation. For all other cases, a nucleation source term with a Dirac distribution is sufficient.\n\n\\section{Conclusion}\\label{s:conclusion}\nWe studied two major generalizations of the theory established in Refs.~\\onlinecite{bergmannJCG08,teranPRB10} to determine the non-equilibrium grain size distribution during the crystallization of a solid. In the first, we assumed that the growth of grains can be anisotropic, as observed in some experiments. We showed that the final non-equilibrium grain size distribution remains essentially unaffected by this generalization [except for a constant prefactor in $N(\\mathbf{r},t)$]. In the second extension of the theory we considered the case where nuclei with variable size within a physically realistic range can be thermodynamically stable and develop into grains. Using a Gaussian distribution to model the source of nuclei, the partial differential equation could not be completely solved analytically. The numerical results showed that only for early stages of crystallization, when nucleation dominates the process, did the generalization affect the grain size distribution quantitatively. The main conclusion of this work is that the theory established in Refs.~\\onlinecite{bergmannJCG08,teranPRB10} is very robust against these important generalizations. This is particularly surprising for the generalization to anisotropic grain growth. This is explained in part by the fact that we assume the shape of unimpinged grains to remain unaltered during the growth.  This  assumption of our model is physically justified by the fact that the formation of a nucleus and its growth into a grain are determined by the microscopic interactions between atoms and molecules, the intrinsic factors described in the introduction.\n\nIn both generalizations we found the previously obtained behavior of the non-equilibrium grain size distribution, which goes through three stages during the crystallization: a nucleation dominated early stage, an intermediate stage where nucleation and growth have similar strength and a late stage of crystallization where growth dominates.\\cite{teranPRB10} We also note that in all cases studied the non-equilibrium grain size distribution is found to depend on the ratio of the effective nucleation and growth rates, $N(r,t) \\propto I[\\sigma(r,t)]\/v[\\sigma(r,t)]$ with, however, a non-trivial function $\\sigma(r,t)$ of grain size and time.  The assumptions leading to that result are the random nucleation and growth of grains at fixed thermodynamic conditions and in the absence of coalescence or secondary grain formation.  When using our expression to describe the crystallization of specific materials it is important to carefully consider the latter assumption since coalescence may affect the grain size distribution.\n In conclusion, Eq.~\\eqref{solanistropic} provides an excellent description of the non-equilibrium grain size distribution in systems where random nucleation and growth occurs and can be used to analyze experimental data.\n\n\\begin{acknowledgements}\nWe gratefully acknowledge the support of the Research Corporation, the Army Research Laboratory and the CNSM block grant at CSU Long Beach.\n\\end{acknowledgements}\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Combining Gaussian variables}\n\\label{sec:combining_gauss}\n\\input{gauss_appendix.tex}\n\n\\section{Variational Bayes}\n\\label{sec:variational_bayes}\nVariational Bayes is an approximation technique commonly used when evaluation of the posterior distribution or evaluation of expectations with respect to the posterior distribution is computationally intractable. This can be because of high dimensionality of the posterior distribution or the analytic intractability of its expression.\n\nTo resolve the computational expense the posterior distribution $p(z|x)$ \nis approximated with an analytically tractable distribution $q(z|x)$. \nIn this setting the log-likelihood of the data is given by,\n\\begin{equation}\n  \\log p(x) = \\text{KL}(q||p) + \\mathcal{L}(q)\n\\end{equation}\nwhere the first term is the KL-divergence,\n\\begin{equation*}\n   \\text{KL}(q||p) = -\\int q(z) \\log \\frac{p(x|z)}{q(z)} dz\n\\end{equation*}\nand the second term is the lower bound,\n \\begin{equation*}\n   \\mathcal{L}(q) = \\int q(z) \\log \\frac{p(x,z)}{q(z)} dz\n \\end{equation*}\n\n\\section{Derivations for transition dynamics (Sec \\ref{sec:op_estim})}\n\\input{derivation_op_estim}\n\n\\section{Derivations for PPCA (Sec \\ref{sec:im_op_estim})}\n\\input{derivation_img_op_estim}\n\n\\section{Derivations for PNPCA (Sec \\ref{sec:nonlin_im_op_estim})}\n\\input{derivation_nonlinear_img_op_estim}\n\n\\section{Introduction}\\label{sec:introduction}}\n\n\n\n\n\n\\IEEEraisesectionheading{\\section{Introduction}\n\\label{sec:learing_lie_generators_intro}}\n\\input{intro.tex}\n\n\\section{Estimating transition dynamics}\n\\label{sec:op_estim}\n\\label{sec:learning_lie_generators_operator_estim}\n\\input{operator_estim.tex}\n\n\\section{Joint estimation of PPCA image representations and transition dynamics}\n\\label{sec:im_op_estim}\n\\label{sec:learning_lie_generators_image_op_estim}\n\\input{image_op_estim.tex}\n\n\\section{Joint estimation of PNPCA image representations and transition dynamics}\n\\label{sec:nonlin_im_op_estim}\n\\label{sec:learning_lie_generators_nonlin_image_op_estim}\n\\input{nonlin_image_op.tex}\n\n\\section{Related Work}\n\\label{sec:learning_lie_generators_related_work}\n\\input{rw.tex}\n\n\\section{Conclusion}\n\\input{conclusion.tex}\n\\appendices\n\\label{sec:appendix}\n\\input{appendix.tex}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcompsoc\n \n  \\section*{Acknowledgments}\n\\else\n \n  \\section*{Acknowledgment}\n\\fi\nThe authors would like to thank Andrew Jaegle and Camillo J. Taylor for their insights and invaluable conversation. Support was provided by the following grants: NSF IIS 1703319, NSF\nTRIPODS 1934960, NSF CPS 2038873, ARL RCTA W911NF-10-2-0016, ARL DCIST\nCRA W911NF-17-2-0181, and ONR N00014-17-1-2093. \n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n\n\\bibliographystyle{unsrt}\n\n\n\n\\subsection{Posterior distribution of the image representation}\n\\label{sec:posterior_image}\nThe posterior distribution of the latent image representation given in equation (\\ref{eq:im_op_im_rep_posterior}) is derived using the equations in Section \\ref{sec:combining_gauss}. The conditional distribution of the image given\nthe latent image representation is a linear Gaussian distribution and the prior distribution on the latent\nimage representation is Gaussian. Using the equations in Section \\ref{sec:combining_gauss}, the posterior distribution is given by\n\\begin{equation*}\n  p(z^i|x^i, \\theta_R) = \\mathcal{N}(z^i|u, \\sigma^{-2}\\Sigma)\n\\end{equation*}\nwhere $\\theta_R = (W, \\mu, \\sigma)$ are the representation embedding parameters and\n\\begin{equation*}\n  \\Sigma = (I + W^TW)^{-1}, \\quad u = \\Sigma^{-1} W^T x^i_\\mu\n\\end{equation*}\nwhere $x^i_\\mu=x^i - \\mu$ is $x_i$ mean-centered.\n\n\\subsection{Complete-data likelihood}\nThe complete-data likelihood is the product of the data and latent variable distributions,\n\\begin{align*}\n  \\prod_i& p(\\lambda^i, z^{i+1}, x^{i+1}, z^i| x^i, \\theta_T, \\Lambda, \\theta_R) = \\\\&\\prod_i p(x^{i+1} | z^{i+1}, \\theta_R)\\, p(z^{i+1} | z^i, \\lambda^i , \\theta_T, \\Lambda)\\, p(\\lambda^i| \\Lambda) \\,p(z^i | x^i, \\theta_R).\n\\end{align*}\nwhere $\\theta_T=(G,\\Omega)$ are the transformation parameters.\nUsing the result from equation (\\ref{eq:deriv_op_estim_complete_data})\nthe complete-data likelihood can be expressed\n\\begin{align*}\n  \\prod_i p(&\\lambda^i, z^{i+1}, x^{i+1}, z^i | x^i, \\theta_T, \\Lambda, \\theta_R) \n  =\\\\& \\prod_i p(x^{i+1} | z^{i+1}, \\theta_R) \\,p(\\lambda^i, z^{i+1} | z^i, \\theta_T, \\Lambda)\\, p(z^i | x^i, \\theta_R)\n\\end{align*}\nThe distribution\n\\begin{align*}\n  p(\\lambda^i, z^{i+1} | z^i, \\theta_T, \\Lambda) &= p(z^{i+1} | z^i, \\lambda^i, \\theta_T, \\Lambda)p(\\lambda^i| \\Lambda) \\\\&= \\mathcal{N}(\\lambda^i, z^{i+1} | m, R^{-1})\n\\end{align*}\nwith\n\\begin{equation*}\n  R^{-1} =\n  \\begin{pmatrix}\n    \\Lambda & \\Lambda A^T\\\\\n    A\\Lambda & \\Omega + A\\Lambda A^T\n  \\end{pmatrix}, \\quad\n  m = \\begin{pmatrix}\n    0\\\\\n    z^i\n    \\end{pmatrix}\n\\end{equation*}\nand\n\\begin{equation*}\nA_{\\cdot,j} = G^j z^i,\n\\end{equation*}\nis given in equation (\\ref{eq:deriv_op_estim_complete_data}).\n\n\\subsection{Posterior distribution of the latent variables}\n\\label{sec:im_op_estim_latent}\nThe latent posterior distribution of the combination coefficients $\\lambda^i$ and the latent image distributions $z^i$ and $z^{i+1}$ given in equation \n(\\ref{eq:latent_posterior})\nis derived using the equations in Section \\ref{sec:combining_gauss}.\n\nThe conditional distribution of the transformed image $x^{i+1}$ given the latent representation of the transformed image $z^{i+1}$ is a linear Gaussian distribution and the conditional distribution of the latent representation of the transformed image $z^{i+1}$ given the latent representation of the initial image $z^i$ is Gaussian. Using the equations in Section \\ref{sec:combining_gauss}, these distributions are combined to give the posterior distribution of the latent representation of the transformed image $z^{i+1}$ as\n\\begin{align*}\n  p(& z^{i+1} | x^{i+1}, z^i, \\lambda^i, \\theta_T, \\theta_R) =\\\\ &\\frac{ p(x^{i+1} | z^{i+1}, \\theta_R)p(z^{i+1} | z^i, \\lambda^i, \\theta_T) }{p(x^{i+1} | \\theta_R)} = \\mathcal{N}(z^{i+1}| \\gamma, \\Gamma)\n\\end{align*}\nwhere\n\\begin{align*}\n  \\gamma = \\Gamma \\{ \\sigma^{-2}W^T x^{i+1}_\\mu + \\Omega^{-1}(z^i + A\\lambda^i)\\}, \\\\\n  \\Gamma = (\\Omega^{-1} + \\sigma^{-2} W^T W)^{-1}, \\quad A_{\\cdot,m} = G^j z^i.\n\\end{align*}\nThe latent posterior distribution can then be expressed,\n\\begin{align*}\n   p(&\\lambda^i, z^{i+1}, z^i | x^{i+1}, x^i, \\theta_T, \\Lambda, \\theta_R) = \\\\& p( z^{i+1} | x^{i+1}, z^i, \\lambda^i, \\theta_T, \\theta_R )\\, p(\\lambda^i| \\Lambda)\\, p(z^i | x^i, \\theta_R).\n\\end{align*}\nThis latent posterior distribution is non-Gaussian, however, \nit is still in the exponential family\nand can be expressed in the canonical form,\n\\begin{equation*}\np(x|\\eta) = h(x)\\exp\\{\\eta(\\theta)^T T(x)-A(\\theta)\\},\n\\end{equation*}\nwhere,\n\\begin{align*}\n  h(x) =& 3\\log \\frac{1}{(2\\pi)^{d\/2}}\\\\\n  A(\\theta) =& \\Gamma\\sigma^{-2}W^T  x^{i+1}_\\mu(x^{i+1}_\\mu)^T W\\sigma^{-2} \\\\&+ \\sigma^2\\Sigma^{-2}\\Sigma^{-1}W^T x^{i+1}_\\mu (x^{i+1}_\\mu)^T W\\Sigma^{-1}\\\\& + \\log\\frac{1}{|\\Gamma|^{1\/2}} + \\log\\frac{1}{|\\Lambda|^{1\/2}} + \\log\\frac{1}{|\\sigma^{-2}\\Sigma|^{1\/2}}\\\\\n  \\eta(\\theta)= & (\\Gamma^{-1}, -2\\sigma^{-2}W^T x^{i+1}_\\mu, -2\\Omega^{-1}, -2\\Omega^{-1}G^j,\\\\& \\Omega^{-1}\\Gamma\\Omega^{-1} + \\sigma^2\\Sigma^{-1}, (G^k)^T\\Omega^{-1}\\Gamma\\Omega^{-1}G^j,\\\\& 2\\Omega^{-1}\\Gamma\\sigma^{-2}W^T x^{i+1}_\\mu-2\\sigma^2\\Sigma^{-1}\\Sigma^{-1}W^T x^i_\\mu,\\\\& 2(G^j)^T\\Omega^{-1}\\Gamma\\sigma^{-2}W^T x^{i+1}_\\mu, 2(G^j)^T\\Omega^{-1}\\Gamma\\Omega^{-1}, \\Lambda)\\\\\n  T(x) = & (z^{i+1}(z^{i+1})^T,  z^{i+1},  z^i(z^{i+1})^T,  \\lambda^i_j z^i(z^{i+1})^T,  z^i(z^i)^T,\\\\&  \\lambda^i_j\\lambda^i_k z^i(z^i)^T,  z^i,  \\lambda^i_j(z^i)^T,  \\lambda^i_j z^i(z^i)^T,  \\lambda^i(\\lambda^i)^T)\n\\end{align*}\n\nThe sufficient statistics, $T(x)$, are computed by taking partial derivatives of $A(\\theta)$ with respect to $\\eta(\\theta)$.\n\n\\subsection{Update equations for the model parameters}\nThe update equation for each distribution parameter is found by computing the partial derivative of the complete-data log-likelihood function with respect to the parameter and solving for the parameter value at the critical point.\nThe complete-data log-likelihood is given by\n\\begin{align*}\n  \\log p(&\\lambda^i, z^{i+1}, x^{i+1}, z^i | \\theta_T, \\Lambda, \\theta_R) =\\\\& \\log \\mathcal{N}(\\lambda^i, z^{i+1} | m, R^{-1}) + \\log \\mathcal{N}(z^{i+1}|u^{i+1}, \\Sigma) \\\\&+ \\log \\mathcal{N}(z^i|u^i, \\Sigma)\n\\end{align*}\nSetting the partial derivative of the complete-data log-likelihood function with respect to parameter $G$ equal to zero gives, \n\\begin{equation}\n  \\label{eq:deriv_im_op_estim_G}\nG = \\left(\\sum_i \\Delta z^i(z^i\\otimes\\lambda^i)^T\\right)\\left(\\sum_i z^i(z^i)^T\\otimes \\lambda^i(\\lambda^i)^T \\right)^{-1}\n\\end{equation}\nwhere $\\Delta z^i = z^{i+1} - z^i$ is the difference between sequential image representations.\n\nSetting the partial derivative of the complete-data log-likelihood function with respect to parameter $\\Omega$ equal to zero gives, \n\\begin{align}\n\\label{eq:deriv_im_op_estim_Omega}\n  \\Omega = \\frac{1}{M}\\Big( \\sum_i& \\Delta z^i(\\Delta z^i)^T - 2A\\lambda^i(\\Delta z^i)^T + A\\lambda^i(\\lambda^i)^TA^T\\Big).\n\\end{align}\nUpdate equations \\ref{eq:deriv_im_op_estim_G} and \\ref{eq:deriv_im_op_estim_Omega} are derived similarly to the update equations in Section \\ref{sec:op_estim}.\nSetting the partial derivative of the complete-data log-likelihood function with respect to parameter $W$ equal to zero gives, \n\\begin{align*}\n  0 = \\sum_i& \\frac{\\partial}{\\partial W}\\log \\mathcal{N}(x^{i+1}|W z^{i+1} + \\mu, \\sigma^{2}I) \\\\&+ \\frac{\\partial}{\\partial W}\\log \\mathcal{N}(x^i|W z^i + \\mu, \\sigma^2I).\n\\end{align*}\nReordering terms gives\n\\begin{align*}\n  W&\\left(\\sum_i z^{i+1}(z^{i+1})^T + z^i(z^i)^T\\right) =\\\\ &\\left(\\sum_i x^{i+1}_\\mu (z^{i+1})^T + x^i_\\mu (z^i)^T\\right),\n\\end{align*}\nand solving for $W$ gives\n\\begin{align*}\n  W = &\\left(\\sum_i x^{i+1}_\\mu (z^{i+1})^T + x^i_\\mu (z^i)^T\\right) \\\\&\\left(\\sum_i z^{i+1}(z^{i+1})^T + z^i(z^i)^T\\right)^{-1}.\n\\end{align*}\nSetting the partial derivative of the complete-data log-likelihood function with respect to parameter $\\sigma$ equal to zero gives, \n\\begin{align*}\n  0 =& \\sum_i \\frac{\\partial}{\\partial \\sigma^{-2}}\\log \\mathcal{N}(x^{i+1}|W z^{i+1} + \\mu, \\sigma^{2}I) \\\\&\\qquad+ \\frac{\\partial}{\\partial \\sigma^{-2}}\\log \\mathcal{N}(x^i|W z^i + \\mu, \\sigma^2I)\\\\\n  =& \\sum_i \\frac{\\partial}{\\partial \\sigma^{-2}} \\log|\\sigma^{-2}I| + \\frac{\\partial}{\\partial \\sigma^{-2}}\\sigma^{-2}I(x^i_\\mu - W z^i)(x^i_\\mu - W z^i)^T \\\\ &\\qquad+ \\frac{\\partial}{\\partial \\sigma^{-2}}\\sigma^{-2}I (x^{i+1}_\\mu - W z^{i+1})(x^{i+1}_\\mu - W z^{i+1})^T.\n\\end{align*}\nReordering terms gives,\n\\begin{align*}\n  N\\sigma^2I = \\sum_i&(x^i_\\mu - W z^i)(x^i_\\mu - W z^i u)^T \\\\&+ (x^{i+1}_\\mu - W z^{i+1})(x^{i+1}_\\mu - W z^{i+1})^T\n\\end{align*}\nand application of the trace operator gives the update,\n\\begin{align*}\n    \\sigma^2 =& \\frac{1}{ND} \\text{tr}\\Big(\\sum_i x^i_\\mu (x^i_\\mu)^T + x^{i+1}_\\mu (x^{i+1}_\\mu)^T \\\\&\\qquad\\qquad\\qquad+ W(z^i(z^i)^T + z^{i+1}(z^{i+1})^T)W^T\\Big) \\\\&-\\frac{2}{ND} \\text{tr}\\left(\\sum_i W(z^i(x^i_\\mu)^T + z^{i+1}(x^{+1}i_\\mu)^T) - \\mu\\mu^T\\right)\n\\end{align*}\n\n\\subsection{Posterior distribution of the image representation}\nComputation of the posterior distribution of the latent image representation given in Section \\ref{sec:nonlin_im_op_estim} is analytically intractable. A common approach in this setting is to approximate the distribution by an analytically tractable variational distribution. In Section \\ref{sec:nonlin_im_op_estim} the posterior distribution of the latent image representation is approximated by the variational posterior\n\\begin{equation}\n  \\label{eq:deriv_nonlin_var_posterior}\n  q(z^k|x^k, w_\\phi) = \\mathcal{N}(z^k | \\phi(x^k)_\\mu, \\phi(x^k)_\\sigma)\n\\end{equation}\nwhere the mean and standard deviation of the distribution are functions of the encoding network parameters and the image $x^k$.\n\\subsection{Complete-data likelihood}\nThe complete-data likelihood is the product of the data and latent variable distributions,\n\\begin{align*}\n  \\prod_i &p(\\lambda^i, z^{i+1}, x^{i+1}, z^i | x^i, \\theta_T, \\Lambda, w_\\psi, w_\\phi) = \\\\&\\prod_i p(x^{i+1} | z^{i+1}, w_\\psi)\\, p(z^{i+1} | z^i, \\lambda^i ,\\theta_T)\\, p(\\lambda^i| \\Lambda)\\, p(z^i | x^i, w_\\psi\n\\end{align*}\nSince the posterior distribution is intractable, the\nvariational posterior in equation (\\ref{eq:deriv_nonlin_var_posterior}) is used instead. To ensure the variational\nposterior is a good approximation of the true posterior distribution\nrequires that the KL-divergence between the variational posterior and the true posterior (see Section \\ref{sec:variational_bayes}) is added to the complete-data log-likelihood function. The resulting form of the complete-data log-likelihood is\n\\begin{align*}\n  \\sum_i &\\log p(\\lambda^i, z^{i+1}, x^{i+1}, z^i | x^i, \\theta_T, \\Lambda, w_\\psi, w_\\phi) = \\\\\n  &\\sum_i \\log p(x^{i+1} | z^{i+1}, w_\\psi) + \\log p(z^{i+1} | z^i, \\lambda^i , \\theta_T) \\\\ &\\qquad + \\log p(\\lambda^i| \\Lambda) + \\log p(z^i | x^i, w_\\psi)\\\\\n  \\geq & \\sum_i \\log p(x^{i+1} | z^{i+1}, w_\\psi) + \\log p(z^{i+1} | z^i, \\lambda^i , \\theta_T) \\\\ &\\qquad +\\log p(\\lambda^i| \\Lambda) + \\mathcal{L}(q)\\\\\n  = & \\sum_i \\log p(x^{i+1} | z^{i+1}, w_\\psi) + \\log p(z^{i+1} | z^i, \\lambda^i , \\theta_T) \\\\ &\\qquad +\\log p(\\lambda^i| \\Lambda) - \\text{KL}(q(z|x, w_\\phi)\\,||\\,p(z | w_\\psi)) \\\\&\\qquad + \\mathbb{E}_{q(z|x, w_\\phi)}[p(x|z, w_\\psi)],\n\\end{align*}\nwhere, for convenience, $p(z | w_\\psi)$ is chosen to be the standard-Gaussian.\n\n\\subsection{Posterior distribution of the latent variables}\n\\label{sec:nonlin_op_estim_latent}\nThe latent posterior distribution of the combination coefficients $\\lambda^i$ and the latent image distributions $z^i$ and $z^{i+1}$ given in equation \n(\\ref{eq:nonlin_op_estim_latent_posterior}) is given by\n\\begin{align*}\n   p(&\\lambda^i, z^{i+1}, z^i | x^i, x^{i+1}, \\theta_T, \\Lambda, w_\\psi, w_\\phi) = \\\\&  \\frac{ p(x^{i+1} | z^{i+1}, w_\\psi)}{p(x^{i+1} | w_\\psi)} p(z^{i+1} | z^i, \\lambda^i, \\theta_T) p(\\lambda^i| \\Lambda) p(z^i | x^i, w_\\phi)=\\\\\n  &  \\frac{p(z^{i+1} | x^{i+1},w_\\psi) }{p(z^{i+1} | w_\\psi)} p(z^{i+1} | z^i, \\lambda^i, \\theta_T)p(\\lambda^i| \\Lambda) p(z^i | x^i, w_\\psi)\n\\end{align*}\nwhere the substitution\n\\begin{equation*}\n  p(x^{i+1} | z^{i+1}, w_\\psi) = \\frac{p(z^{i+1} | x^{i+1}, w_\\psi) p(x^{i+1} | w_\\psi)}{p(z^{i+1} | w_\\psi)}\n\\end{equation*}\nis due to Bayes rule.\nSince the posterior distribution of the latent image representation is analytically intractable, the posterior distribution of the latent variables is approximated by\n\\begin{align*}\n   p(&\\lambda^i, z^{i+1}, z^i | x^i, x^{i+1}, \\theta_T, \\Lambda, w_\\psi, w_\\psi) \\approx \\\\&  \\frac{q(z^{i+1} | x^{i+1}, w_\\phi)}{p(z^{i+1} | w_\\psi)} p(z^{i+1} | z^i, \\lambda^i, \\theta_T) p(\\lambda^i| \\Lambda) q(z^i | x^i, w_\\phi).\n\\end{align*}\n  This latent posterior distribution is non-Gaussian, however, \nit is still in the exponential family\nand can be expressed in the canonical form,\n\\begin{equation*}\np(x|\\eta) = h(x)\\exp\\{\\eta(\\theta)^T T(x)-A(\\theta)\\},\n\\end{equation*}\nwhere,\n\\begin{align*}\n  h(x) =& 3\\log \\frac{1}{(2\\pi)^{d\/2}}\\\\\n  A(\\theta) =& \\text{tr}(\\phi_\\sigma(x^{i+1})^{-1}\\phi_\\mu(x^{i+1})\\phi_\\mu(x^{i+1})^T \\\\&+ \\phi_\\sigma(x^i)^{-1}\\phi_\\mu(x^i)\\phi_\\mu(x^i)^T) \\\\&+ \\log\\frac{1}{|\\phi_\\sigma(x^{i+1})|^{1\/2}}+ \\log\\frac{1}{|\\phi_\\sigma(x^i)|^{1\/2}} \\\\&+ \\log\\frac{1}{|\\Lambda|^{1\/2}} - \\log\\frac{1}{|I|^{1\/2}}\\\\\n  \\eta(\\theta) =& (\\phi_\\sigma(x^{i+1})^{-1} - I + \\Omega,\\, -2\\phi_\\mu(x^{i+1})\\phi_\\sigma(x^{i+1})^{-1},\\,\\\\& -2\\Omega^{-1},\\, -2\\Omega^{-1}G^j,\\, \\phi_\\sigma(x^i)^{-1} + \\Omega^{-1},\\, \\\\&-2\\Omega^{-1}G^j,\\, (G^j)^T\\Omega^{-1}G^k,\\, \\Lambda,\\, -2\\phi_\\mu(x^i)\\phi_\\sigma(x^i)^{-1})\\\\\n  T(x) =& (z^{i+1}(z^{i+1})^T,\\, z^{i+1},\\, z^i(z^{i+1})^T,\\, \\lambda^i_j z^i(z^{i+1})^T,\\, z^i(z^i)^T,\\,\\\\& \\lambda^i_j z^i(z^i)^T,\\, \\lambda^i_j\\lambda^i_k z^i(z^i)^T,\\, z^i,\\,\n           \\lambda^i(\\lambda^i)^T,\\, z^i)\n\\end{align*}\nThe sufficient statistics, $T(x)$, are computed by taking partial derivatives of $A(\\theta)$ with respect to $\\eta(\\theta)$.\n\n\\subsection{Update equations for the model parameters}\nThe update equation for each distribution parameter is found by computing the partial derivative of the complete-data log-likelihood function with respect to the parameter and solving for the parameter value at the critical point.\nThe complete-data log-likelihood is given by\n\\begin{align*}\n  \\sum_i &\\log p(\\lambda^i, z^{i+1}, x^{i+1}, z^i | x^i, \\theta_T, \\Lambda, w_\\psi, w_\\phi) \\approx \\\\\n  &\\sum_i \\log p(x^{i+1} | z^{i+1}, w_\\psi) + \\log p(z^{i+1} | z^i, \\lambda^i , \\theta_T) \\\\ &\\qquad + \\log p(\\lambda^i| \\Lambda) + \\log q(z^i | x^i, w_\\phi) + \\mathcal{L}(q).\n\\end{align*}\nSetting the partial derivative of the complete-data log-likelihood function with respect to parameters $G$ and $\\Omega$ equal to zero give the same results as in \nSection \\ref{sec:complete_data_op} where\n\\begin{equation*}\nG = \\left(\\sum_i \\Delta z^i(z^i\\otimes \\lambda^i)^T\\right)\\left(\\sum_i z^i(z^i)^T \\otimes \\lambda^i(\\lambda^i)^T \\right)^{-1}\n\\end{equation*}\nand\n\\begin{align*}\n  \\Omega = \\frac{1}{M}\\Big( \\sum_i& \\Delta z^i(\\Delta z^i)^T - 2A\\lambda^i(\\Delta z^i)^T + A\\lambda^i(\\lambda^i)^TA^T\\Big).\n\\end{align*}\nThe parameters $w_\\phi$ and $w_\\psi$ are updated by backpropagation using the reparameterization trick (\\cite{kingma2013auto}).\n\n\\subsection{Complete-data likelihood}\n\\label{sec:complete_data_op}\nThe complete-data likelihood is the product of the data and latent variable distributions,\n\\begin{align*}\n  \\prod_i p(\\lambda^i, z^{i+1} | z^i, \\theta_T, \\Lambda) &= \\prod_i p(z^{i+1} | z^i, \\lambda^i, \\theta_T)\\, p(\\lambda^i | \\Lambda)\n \\\\ &= \\prod_i\\mathcal{N}(z^{i+1}| z^i + A\\lambda^i, \\Omega)\\, \\mathcal{N}(\\lambda^i| 0, \\Lambda)\n\\end{align*}\nwhere $\\theta_T=(G, \\Omega)$ are the transformation parameters.\nThe form given in\nequation (\\ref{eq:op_estim_complete-data_td}) is derived using the equations in Section \n\\ref{sec:combining_gauss}. \nThe transformed image distribution is a linear Gaussian distribution and the prior distribution on the combination coefficients is Gaussian. Using the equations in Section \\ref{sec:combining_gauss}, the complete-data likelihood can be expressed\n\\begin{equation}\n  \\label{eq:deriv_op_estim_complete_data}\n  p(\\lambda^i, z^{i+1} | z^i, \\theta_T, \\Lambda) = \\mathcal{N}(\\lambda^i, z^{i+1} | m, R^{-1})\n\\end{equation}\nwhere the parameters $R^{-1}$ and $m$ are given by\n\\begin{equation*}\n  R^{-1} =\n  \\begin{pmatrix}\n    \\Lambda & \\Lambda A^T\\\\\n    A\\Lambda & \\Omega + A\\Lambda A^T\n  \\end{pmatrix}, \\quad\n  m = \\begin{pmatrix}\n    0\\\\\n    z^i\n    \\end{pmatrix}\n\\end{equation*}\nand\n\\begin{equation*}\n  A_{\\cdot,j} = G^j z^i.\n\\end{equation*}\n\n\\subsection{Posterior distribution of the latent variables}\n\\label{sec:latent_posterior_op}\nThe latent posterior distribution of the combination coefficients $\\lambda^i$ given in equation (\\ref{eq:op_estim_latent_posterior_td}) is derived using the equations in Section \\ref{sec:combining_gauss}. \nThe transformed image distribution is a linear Gaussian distribution and\nthe prior distribution on the combination coefficients is Gaussian.\nUsing the equations in Section \\ref{sec:combining_gauss}, the latent posterior distribution on the combination coefficients is given by\n\\begin{equation*}\n  p(\\lambda^i | z^{i+1}, z^i, \\theta_T,\\Lambda) = \\mathcal{N}(\\lambda | q, K )\n\\end{equation*}\nwhere $\\theta_T=(G, \\Omega)$ are the transformation parameters, and where\n\\begin{equation*}\n  K = (\\Lambda^{-1} + A^T\\Omega^{-1} A)^{-1}, \\quad q = K A^T\\Omega^{-1}\\Delta z^i\n\\end{equation*}\nwhere $\\Delta z_i=z^{i+1} - z^i$ is the difference between sequential image representations, and\n\\begin{equation*}\n  A_{\\cdot,j} = G^j z^i.\n\\end{equation*}\n\n\\subsection{Update equations for the model parameters}\n\\label{sec:optim_estim_m_step}\nThe update equation for each distribution parameter is found by computing the partial derivative of the complete-data log-likelihood function with respect to the parameter and solving for the parameter value at the critical point. The complete-data log-likelihood is given by\n\\begin{equation*}\n\\sum_i \\log  p(\\lambda^i, z^{i+1} | z^i, \\theta_T, \\Lambda) = \\sum_i \\log \\mathcal{N}(\\lambda^i, z^{i+1} | m, R^{-1})\n\\end{equation*}\nwith\n\\begin{equation*}\n  R^{-1} =\n  \\begin{pmatrix}\n    \\Lambda & \\Lambda A^T\\\\\n    A\\Lambda & \\Omega + A\\Lambda A^T\n  \\end{pmatrix}, \\quad\n  m = \\begin{pmatrix}\n    0\\\\\n    z^i\n    \\end{pmatrix}\n\\end{equation*}\nand\n\\begin{equation}\n  \\label{eq:deriv_op_estim_A_def}\n  A_{\\cdot,j} = G^j z^i.\n\\end{equation}\nSetting the partial derivative of the complete-data log-likelihood function with respect to parameter $G$ equal to zero gives, \n\\begin{equation*}\n  0 =\\sum_i  \\frac{\\partial}{\\partial R} \\log \\mathcal{N}(\\lambda^i, z^{i+1} | m, R^{-1})\\frac{\\partial R}{\\partial A}\\frac{\\partial A}{\\partial G}\n\\end{equation*}\nyielding\n\\begin{align}\n  \\label{eq:partialA}\n  0 &= \\sum_i \\frac{\\partial}{\\partial R} \\log \\mathcal{N}(\\lambda^i, z^{i+1} | m, R^{-1}) \\\\&\\propto -\\sum_i\\frac{\\partial}{\\partial R}\\log|R| + \\frac{\\partial}{\\partial R}\\text{tr}(\\sum_i R y y^T) \\\\&= -\\sum_i R^{-1} + \\sum_i y y^T\n\\end{align}\nwith\n\\begin{equation*}\n  y = \\begin{pmatrix}\n    \\lambda^i\\\\\n    \\Delta z^i\n  \\end{pmatrix}\n\\end{equation*}\nwhere $\\Delta z^i=z^{i+1}-z^i$ is the difference between sequential image representations.\nThe right most terms give a system of equations,\n\\begin{align}\n  \\label{eq:system}\n  &\\begin{pmatrix}\n    \\Lambda & \\Lambda A^T\\\\\n    A\\Lambda & \\Omega + A\\Lambda A^T\n  \\end{pmatrix}=\n                    \\begin{pmatrix}\n                      \\lambda^i(\\lambda^i)^T & \\lambda^i(\\Delta z^i)^T\\\\\n                      \\Delta z^i(\\lambda^i)^T & \\Delta z^i(\\Delta z^i)^T\n                      \\end{pmatrix}\n\\end{align}\nand by substitution,\n\\begin{equation*}\n  \\sum_i A \\lambda^i(\\lambda^i)^T = \\sum_i \\Delta z^i(\\lambda^i)^T.\n\\end{equation*}\nSubstituting equation (\\ref{eq:deriv_op_estim_A_def}) gives,\n\\begin{equation*}\n \n  \\sum_i G (z^i \\otimes \\lambda^i) (\\lambda^i)^T = \\sum_i \\Delta z^i(\\lambda^i)^T.\n\\end{equation*}\nBy a property of the Kronecker product, multiplying both sides by $(z^i \\otimes \\lambda^i)$ gives, \n\\begin{equation*}\nG = \\left(\\sum_i \\Delta z^i(z^i\\otimes \\lambda^i)^T\\right)\\left(\\sum_i z^i(z^i)^T\\otimes \\lambda^i (\\lambda^i)^T\\right)^{-1}\n\\end{equation*}\n\nThe update equation for $\\Omega$ is derived similarly.\nTaking the relevant terms in the system of equations in (\\ref{eq:system}) gives,\n\\begin{align*}\n  \\sum_i A\\lambda^i(\\Delta z^i)^T =& \\sum_i A\\lambda^i(\\lambda^i)^TA^T\\\\\n  M \\Omega + \\sum_i A\\lambda^i(\\Delta z^i)^T =& \\sum_i \\Delta z^i(\\Delta z^i)^T\n\\end{align*}\ncombining the above gives,\n\\begin{align*}\n  \\Omega = \\frac{1}{M}\\Big( \\sum_i& \\Delta z^i(\\Delta z^i)^T - 2A\\lambda^i(\\Delta z^i)^T + A\\lambda^i(\\lambda^i)^TA^T\\Big).\n\\end{align*}\n\n\n\\subsection{Linear Gaussian variable}\nIn this section a form for the marginal, conditional and joint distributions\nare given for the case where the conditional distribution $p(y|x)$\nis a linear Gaussian model, that is, when $p(y|x)$\nhas a mean that is a linear function of $x$ and a covariance that\nis independent of $x$. The material in this section is adapted\nprimarily from \\cite{bishop2006pattern}, details of the derivations are given in Section 2.3 of the same text.\n\nBeginning with the linear Gaussian $p(y|x)$ and $p(x)$,\n\\begin{align*}\n  p(x) &= \\mathcal{N}(x|\\mu, \\Lambda^{-1})\\\\\n  p(y|x) &= \\mathcal{N}(y| A x + b, L^{-1}),\n\\end{align*}\nthe marginal distribution $p(y)$ and conditional distribution $p(x|y)$\nare given by\n\\begin{align*}\n  p(y) &= \\mathcal{N}(y| A\\mu + b, L^{-1} + A\\Lambda^{-1}A^T)\\\\\n  p(x|y) &= \\mathcal{N}(x|\\Sigma\\{A^TL(y-b)+\\Lambda\\mu\\}, \\Sigma)\n\\end{align*}\nwhere\n\\begin{align*}\n  \\Sigma =& (\\Lambda + A^TLA)^{-1}.\n\\end{align*}\nThe joint distribution $p(x,y)$ is given by\n\\begin{align*}\n  p(x,y) &=\\mathcal{N}(x, y | m, R^{-1})\n\\end{align*}\nwhere\n\\begin{align*}\n  R^{-1} =\n  \\begin{pmatrix}\n    \\Lambda^{-1} & \\Lambda^{-1}A^T\\\\\n    A\\Lambda^{-1} & L^{-1} + A\\Lambda^{-1}A^T\n  \\end{pmatrix}, \\quad\n  m = \\begin{pmatrix}\n    \\mu\\\\\n    A\\mu + b\n    \\end{pmatrix}.\n\\end{align*}\n\n\\subsubsection{Gaussian joint distribution}\nIn this section the conditional and marginal distributions of two sets of variables are given for the case when their joint distribution is Gaussian. The material\nin this section is adapted primarily from \\cite{bishop2006pattern}, details of the derivations are given in\nSection 2.3 of that text.\n\nFor the variable $x\\sim\\mathcal{N}(\\mu, \\Lambda^{-1})$ partitioned into two disjoint subsets $x_a$ and\n$x_b$ so that\n\\begin{equation*}\n  x = \\begin{pmatrix}\n    x_a\\\\\n    x_b\n  \\end{pmatrix}.\n\\end{equation*}\nThe corresponding partitions of the mean and precision matrix are given by\n\\begin{equation*}\n  \\mu =\n  \\begin{pmatrix}\n    \\mu_a\\\\\n    \\mu_b\n  \\end{pmatrix}, \\quad\n  \\Lambda =\n  \\begin{pmatrix}\n    \\Lambda_{aa} & \\Lambda_{ab}\\\\\n     \\Lambda_{ab} & \\Lambda_{bb}\n  \\end{pmatrix}\n\\end{equation*}\nand the conditional distribution $p(x_a|x_b)$ is Gaussian with\nsufficient statistics\n\\begin{equation*}\n  \\mu_{a|b} = \\mu_a - \\Lambda_{aa}^{-1}\\Lambda_{ab}(x_b - \\mu_b), \\quad\n  \\Sigma_{a|b}=\\Lambda_{aa}^{-1}.\n\\end{equation*}\n\n\n\n\n\n\n\n\\subsubsection{Estimating Lie generators}\n\\cite{freeman1991design, perona1995deformable, teo1998design}, and \\cite{bansal2014steerability}\npropose techniques for estimating a steering basis on linear Lie groups. \nSteering can be described as transforming a function defined on a group \nby the group action using a linear combination of basis functions and combination coefficients soley dependent on the steering direction.\n The steering basis is determined from a known Lie transformation group and not from\n a sequence of images.\n\n\\subsubsection{Probabilistic modeling with Lie dynamics}\n\\cite{miao2007learning} estimate Lie transition generators and combination coefficients from an image sequence in an EM framework.\nThe image representations from which\nthe dynamics are estimated are assumed to be given and sequential images are assumed to be close in the transformation space.\n\n\\cite{sohl-dickstein2010unsupervised} use a similar framework to estimate Lie transition generators and their combination coefficients but do not require sequential images to be close in the transformation space. The resulting nonconvexity in inference is addressed using a coarse-to-fine estimation of the transformation generators. The generators themselves are constrained to be diagonalizable and consequently do not capture transformations such as constrast, scaling and translation without periodic boundary. By relaxing the Lie group assumption, however, \\cite{sohl-dickstein2010unsupervised} demonstrate how their technique can be used to capture a fuller set of transformations.\n\n\\cite{cohen2014learning} introduce an approach\nfor probabilistic estimation of special orthogonal transition dynamics\nfrom an image sequence. The authors model the transformation coefficients\nusing the von-Mises distribution and show that the posterior distribution of the transformation coefficients is also von-Mises.\n\n\\cite{falorsi2018explorations} introduce an approach for probabilistic estimation of low-dimensional image representations that are compatible with the action of special orthogonal transformations in 3D. Images are mapped to the Fourier domain where they are transformed by a group action. The representation of the group action is given. \\cite{falorsi2019reparameterizing} extends \\cite{falorsi2018explorations} to accomodate other Lie transformations groups but also requires the group representation to be provided.\n\n\\subsubsection{Probabilistic modeling of transition dynamics}\nThere are many other approaches for estimating\ndisentangled image representations from video sequences\nusing deep neural networks. These approaches typically do not\ndirectly model the trasition dynamics in matrix form\nand when they do, the dynamics are not constrained to have the Lie group structure.\n\nFor example,\n\\cite{watter2015embed} learn to estimate state dependent\nlocally linear discrete-time transition dynamics. \nA learned transformation of\nan image gives the (low-dimensional) state vector, and\na learned embedding of the state vector gives the\nlocally linear transition dynamics. During training,\nthe authors must regularize their loss function \nto ensure estimates in the state space correspond\nto embeddings of the observation space.\n\nAnother example comes from\n\\cite{whitney2019disentangling} where latent representations\nof varying factors are learned from a video sequence by defining a set of\nfactors which which evolve in time\nand from which reconstruction of an image is possible.\nThe time evolution of each factor is\ndetermined by an MLP and is conditionally\ndependent on its representation at the prevous time step.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\\label{sec:app}\n\n\\subsection{Propositional Logic}\nWe briefly present propositional logic, which underpins the dataset generation. Let $S$ be a (finite) set $S$ of propositional variables. A \\emph{literal} is defined as $v$, or $\\bar{v}$ (resp., $\\neg v$), where $v \\in S$. A disjunction of literals is a \\emph{clause}. The \\emph{width} of a clause is defined as the number of literals it contains. \nA formula $\\phi$ is in \\emph{conjunctive normal form (\\CNF)} if it is a conjunction of clauses. A  \\CNF has width $k$ if it contains clauses of width at most $k$, and is referred to as a $k-$\\CNF. To illustrate, the formula ${\\phi=(x_1 \\lor \\neg x_3) \\land (x_4 \\lor x_1)}$ is a \\CNF with clauses of width 2.\n\n\nAn assignment $\\nu: S \\mapsto \\{0,1\\}$ maps variables to False (0), or True (1), and \\emph{satisfies} $\\phi$, which we denote by $\\nu\\models \\phi$, in the usual sense, where $\\models$ is propositional entailment. \nGiven a propositional formula $\\phi$, the \\emph{satisfiability problem}, commonly known as \\SAT, consists of determining whether $\\phi$ admits a satisfying assignment, and is NP-complete \\cite{cook1971complexity}. \n\n\\subsection{Proof of \\Cref{theo:uni}}\n\n\nWe first prove a Boolean version of the theorem.\n\n\\begin{lemma}\\label{lem:boolean}\n  Let $n\\ge 1$, and let $f:\\CG_n\\to\\{0,1\\}$ be an\n  invariant\n  Boolean\n  function. Then, for all $\\epsilon,\\delta>0$ there is a MPNN with RNI\n  that $(\\epsilon,\\delta)$-approximates $f$.\n\\end{lemma}\n\nTo prove this lemma, we use a logical characterization of the\nexpressiveness of MPNNs, which we always assume to admit global\nreadouts. Let $\\LC$ be the extension of first-order predicate logic using\ncounting quantifiers of the form $\\exists^{\\ge k}x$ for $k\\ge 0$,\nwhere $\\exists^{\\ge k}x\\phi(x)$ means that there are at least $k$\nelements $x$ satisfying $\\phi$.\n\nFor example, consider the formula\n\\begin{equation}\n  \\label{eq:a1}\n  \\phi(x):=\\neg\\exists^{\\ge 3}y\\big(\n  E(x,y)\\wedge \\exists^{\\ge 5} z E(y,z)\\big).\n\\end{equation}\nThis is a formula in the language of graphs; $E(x,y)$ means that there\nis an edge between the nodes interpreting $x$ and $y$.\nFor a graph $G$ and a vertex $v\\in V(G)$, we have\n$G\\models\\phi(v)$ (``$G$ satisfies $\\phi$ if the variable $x$ is\ninterpreted by the vertex $v$\") if and only if $v$ has at most $2$\nneighbors in $G$ that have degree at least $5$.\n\nWe will not only\nconsider formulas in the language of graphs, but also formulas in\nthe \\emph{language of colored graphs}, where in addition to the binary\nedge relation we also have unary relations, that is, sets of\nnodes, which we may view as colors of the nodes. For example, the\nformula\n\\[\n  \\psi(x):=\\exists^{\\ge 4}y\\big(E(x,y)\\wedge \\textit{RED}(y)\\big)\n\\]\nsays that node $x$ has at least $4$ red neighbors (more precisely,\nneighbors in the unary relation $\\textit{RED}$). Formally, we assume\nwe have fixed infinite list $R_1,R_2,\\ldots$ of color symbols that we\nmay use in our formulas. Then a \\emph{colored graph} is a graph together\nwith a mapping that assigns a finite set $\\rho(v)$ of colors $R_i$ to\neach vertex (so we allow one vertex to have more than one, but only\nfinitely many, colors).\n\nA \\emph{sentence} (of the logic $\\LC$ or any other logic) is a formula\nwithout free variable. Thus a sentence expresses a property of a\ngraph, which we can also view as a Boolean function. For a sentence\n$\\phi$ we denote this function by $\\llbracket\\phi\\rrbracket$. If\n$\\phi$ is a sentence in the language of (colored) graphs, then for\nevery (colored) graph\n$G$ we have  $\\llbracket\\phi\\rrbracket(G)=1$ if $G\\models\\phi$ and\n$\\llbracket\\phi\\rrbracket(G)=0$ otherwise.\n\nIt is easy to see that $\\LC$ is only a syntactic extension of first\norder logic $\\FO$---for every $\\LC$-formula there is a logically\nequivalent $\\FO$-formula. To see this, note that we can simulate   \n$\\exists^{\\ge k}x$ by $k$ ordinary existential quantifiers:\n$\\exists^{\\ge k}x$ is equivalent to $\\exists x_1\\ldots\\exists\nx_k\\Big(\\bigwedge_{1\\le i<j\\le k}x_i\\neq x_j\\wedge\\bigwedge_{1\\le i\\le\n  k}\\phi(x_i)\\Big)$. However, counting quantifiers add expressiveness if we restrict the number of\nvariables. The $\\LC$-formula $\\exists^{\\ge k}x\\;(x=x)$ (saying that there\nare at least $k$ vertices) with just one variable is not equivalent to any $\\FO$-formula\nusing less than $k$ variables. By $\\LC^k$ we denote the fragment of\n$\\LC$ consisting of all formulas with at most $k$ variables.\n\nFor example, the formula $\\phi(x)$ in (\\ref{eq:a1}) is in $\\LC^3$, but\nnot in $\\LC^2$. But $\\phi(x)$ is equivalent to the following formula\n$\\phi'(x)$ in $\\LC^2$:\n\\begin{equation}\n  \\label{eq:a2}\n  \\phi'(x):=\\neg\\exists^{\\ge 3}y\\big(\n  E(x,y)\\wedge \\exists^{\\ge 5} x E(y,x)\\big).\n\\end{equation}\nThe fragments $\\LC^k$ are interesting for us, because their\nexpressiveness corresponds to that of $(k-1)$-WL and hence to that of\n$k$-GNNs. More precisely, for all $k\\ge 2$, two graphs $G$ and $H$\nsatisfy the same $\\LC^{k}$-sentences if and only if $(k-1)$-WL does\nnot distinguish them \\cite{CaiFI92}. By the results of \\cite{MorrisAAAI19,Keyulu18} this implies,\nin particular, that two graphs are indistinguishable by all MPNNs if\nand only if they satisfy the same $\\LC^2$-sentences.\n\\cite{BarceloKM0RS20} strengthened this result and showed that every\n$\\LC^2$-sentence can be simulated by an MPNN. \n\n\\begin{lemma}[\\cite{BarceloKM0RS20}]\\label{lem:barcelo}\n  For every $\\LC^2$-sentence $\\phi$ and every $\\epsilon>0$ there is an\n  MPNN that $\\epsilon$-approximates $\\llbracket\\phi\\rrbracket$.\n\\end{lemma}\n\nSince here we are talking about deterministic MPNNs, there is no\nrandomness involved, and we just say \\emph{``$\\epsilon$-approximates''}\ninstead of ``$(\\epsilon,1)$-approximates''.\n\nLemma~\\ref{lem:barcelo} not only holds for sentences in the language\nof graphs, but also for sentences in the language of colored\ngraphs. Let us briefly discuss the way MPNNs access such colors. We\nencode the colors using one-hot vectors that are part of the\ninitial states of the nodes. For example, if we have a formula that\nuses color symbols among $R_1,\\ldots,R_k$, then we reserve $k$\nplaces in the initial state $\\vec x_v=(x_{v1},\\ldots,x_{v\\ell})$ of each\nvertex $v$ (say, for convenience, $x_{v1},\\ldots,x_{vk}$) and we\ninitialize $\\vec x_v$ by letting $x_{vi}=1$ if $v$ is in $R_i$ and\n$x_{vi}=0$ otherwise.\n\nLet us call a colored graph $G$ \\emph{individualized} if for any two\ndistinct vertices $v,w\\in V(G)$ the sets $\\rho(v),\\rho(w)$ of colors\nthey have are distinct.\nLet us say that a sentence $\\chi$ \\emph{identifies} a (colored) graph $G$ if for\nall (colored) graphs $H$ we have $H\\models\\chi$ if and only if $H$ is\nisomorphic to $G$.\n\n\\begin{lemma}\n  For every individualized colored graph $G$ there is a\n  $\\LC^2$-sentence $\\chi_G$ that identifies $G$. \n\\end{lemma}\n\n\\begin{proof}\n  Let $G$ be an individualized graph. For every vertex $v\\in V(G)$,\n  let\n  \\[\n    \\alpha_v(x):=\\bigwedge_{R\\in\\rho(v)}R(x)\\wedge\\bigwedge_{R\\in\\{R_1,\\ldots,R_k\\}\\setminus\\rho(x)}\\neg\n    R(x).\n  \\]\n  Then $v$ is the unique vertex of $G$ such that\n  $G\\models\\alpha_v(v)$. For every pair $v,w\\in V(G)$ of\n  vertices, we let\n  \\[\n    \\beta_{vw}(x,y):=\n    \\begin{cases}\n      \\alpha_v(x)\\wedge\\alpha_w(y)\\wedge E(x,y)&\\text{if }(v,w)\\in\n      E(G),\\\\\n      \\alpha_v(x)\\wedge\\alpha_w(y)\\wedge \\neg E(x,y)&\\text{if }(v,w)\\not\\in\n      E(G).\n    \\end{cases}\n  \\]\n  We let\n  \\begin{align*}\n    \\chi_G:=&\\bigwedge_{v\\in V(G)}\\big(\\exists\n    x\\alpha_v(x)\\wedge\\neg\\exists^{\\ge\n      2}x\\alpha_v(x)\\big)~\\wedge \\\\\n      &\\bigwedge_{v,w\\in V(G)}\\exists x\\exists y\\beta_{vw}(x,y).\n  \\end{align*}\n  It is easy to see that $\\chi_G$ identifies $G$.\n\\end{proof}\n\nFor $n,k\\in\\Nat$, we let $\\CG_{n,k}$ be the class of\nall individualized colored graphs that only use colors among $R_1,\\ldots,R_k$.\n\n\\begin{lemma}\n  Let $h:\\CG_{n,k}\\to\\{0,1\\}$ be an invariant Boolean function. Then\n  there exists a $\\LC^2$-sentence $\\psi_h$ such that for all\n  $G\\in\\CG_{n,k}$ it holds that $\\llbracket\\psi_h\\rrbracket(G)=h(G)$.\n\\end{lemma}\n\n\\begin{proof}\n  Let $\\mathcal H\\subseteq\\CG_{n,k}$ be the subset consisting of all graphs\n  $H$ with $h(H)=1$. We let\n  \\[\n    \\psi_h:=\\bigvee_{H\\in\\mathcal H}\\chi_H.\n  \\]\n  We eliminate duplicates in the disjunction. Since up to isomorphism,\n  the class $\\CG_{n,k}$ is finite, this makes the disjunction finite and\n  hence $\\psi_h$ well-defined.\n\\end{proof}\n\nThe \\emph{restriction} of a colored graph $G$ is the underlying plain\ngraph, that is, the graph $G^\\vee$ obtained from the colored graph $G$ by\nforgetting all the colors. Conversely, a colored graph $G^\\wedge$ is an\n\\emph{expansion} of a plain graph $G$ if $G=(G^\\wedge)^\\vee$. %\n\n\\begin{corollary}\\label{cor:inv}\n  Let $f:\\CG_{n}\\to\\{0,1\\}$ be an invariant Boolean function. Then\n  there exists a $\\LC^2$-sentence $\\phi^\\wedge_f$ (in the language\n  of colored graphs) such that for all $G\\in\\CG_{n,k}$ it holds that\n  $\\llbracket\\psi^\\wedge_f\\rrbracket(G)=f(G^\\vee)$.\n\\end{corollary}\n\nTowards proving Lemma~\\ref{lem:boolean}, we fix an $n\\ge 1$ and a\n$\\epsilon,\\delta>0$. We let \n\\[\n  c:=\\left\\lceil\\frac{2}{\\delta}\\right\\rceil\n  \\quad\n  \\text{and}\n  \\quad\n  k:=c^2\\cdot n^3\n\\]\nThe technical\ndetails of the proof of Lemma~\\ref{lem:boolean} and\nTheorem~\\ref{theo:uni} depend on the exact choice of the random\ninitialization and the activation functions used in the neural\nnetworks, but the idea is always the same. For simplicity, we assume\nthat we initialize the states $\\vec x_v=(x_{v1},\\ldots,x_{v\\ell})$ of all\nvertices to $(r_v,0,\\ldots,0)$, where $r_v$ for $v\\in V(G)$ are chosen\nindependently uniformly at random from $[0,1]$. As our activation\nfunction $\\sigma$, we choose the linearized sigmoid function defined\nby $\\sigma(x)=0$ for $x<0$, $\\sigma(x)=x$ for $0\\le x<1$, and\n$\\sigma(x)=1$ for $x\\ge 1$.\n\n\\begin{lemma}\\label{lem:prob}\n   Let $r_1,\\ldots,r_n$ be chosen\n   independently uniformly at random from the interval $[0,1]$.\n   For $1\\le i\\le n$  and $1\\le j\\le c\\cdot n^2$, let\n   \\[\n     s_{ij}:=k\\cdot r_i-(j-1)\\cdot\\frac{k}{c\\cdot n^2}.\n   \\]\n   Then with probability greater than $1-\\delta$, the following\n   conditions are satisfied.\n   \\begin{enumerate}\n   \\item[(i)] For all $i\\in\\{1,\\ldots,n\\},j\\in\\{1,\\ldots,c\\cdot n^2\\}$ it\n     holds that $\\sigma(s_{ij})\\in\\{0,1\\}$.\n   \\item[(ii)] For all distinct $i,i'\\in\\{1,\\ldots,n\\}$ there exists a\n     $j\\in\\{1,\\ldots,c\\cdot n^2\\}$ such that\n     $\\sigma(s_{ij})\\neq\\sigma(s_{i'j})$.\n   \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n  For every $i$, let $p_i:=\\lfloor r_i\\cdot k\\rfloor$. Since $k\\cdot\n  r_i$ is uniformly random from the interval $[0,k]$, the integer\n  $p_i$ is uniformly random from $\\{0,\\ldots,k-1\\}$. Observe that\n  $0<\\sigma(s_{ij})<1$ only if $p_i- (j-1)\\cdot\\frac{k}{c\\cdot n^2}=0$ (here\n  we use the fact that $k$ is divisible by $c\\cdot n^2$). The probability that\n  this happens is $\\frac{1}{k}$. Thus, by the Union Bound,\n  \\begin{equation}\n    \\label{eq:a3}\n    \\Pr\\big(\\exists i,j:\\;0<\\sigma(s_{ij})<1\\big)\\le\\frac{c\\cdot n^3}{k}.\n  \\end{equation}\n  Now let $i,i'$ be distinct and suppose that\n  $\\sigma(s_{ij})=\\sigma(s_{i'j})$ for all $j$. Then for all $j$ we\n  have $s_{ij}\\le 0\\iff s_{i'j}\\le 0$ and therefore $\\lfloor\n  s_{ij}\\rfloor\\le 0\\iff \\lfloor\n  s_{i'j}\\rfloor\\le 0$. This implies that\n  \\begin{align}\n  \\begin{split}\n    &\\forall j\\in\\{1,\\ldots,c\\cdot n^2\\}:\\\\\n    &\\quad p_i\\le\n    (j-1)\\cdot\\frac{k}{c\\cdot n^2}\\iff  p_{i'}\\le\n    (j-1)\\cdot\\frac{k}{c\\cdot n^2}. \\label{eq:a4}\n    \\end{split}\n  \\end{align}\n  Let $j^*\\in\\{1,\\ldots, c\\cdot n^2\\}$ such that\n  \\[p_i\\in\\Big\\{(j^*-1)\\cdot\\frac{k}{c\\cdot\n    n^2},\\ldots,j^*\\cdot\\frac{k}{c\\cdot n^2}-1\\Big\\}.\\] Then\n  by (\\ref{eq:a4}) we have: \\[p_i'\\in\n  \\Big\\{(j^*-1)\\cdot\\frac{k}{c\\cdot n^2},\\ldots,j^*\\cdot\\frac{k}{c\\cdot n^2}-1\\Big\\}.\\] As\n  $p_{i'}$ is independent of $p_i$ and hence of $j^*$, the probability\n  that this happens is at most $\\frac{1}{k}\\cdot\n  \\frac{k}{c\\cdot n^2}=\\frac{1}{c\\cdot n^2}$.\n  This proves that for all distinct $i,i'$ the probability that \n  $\\sigma(s_{ij})=\\sigma(s_{i'j})$ is at most $\\frac{1}{c\\cdot\n    n^2}$. Hence, again by the Union Bound,\n  \\begin{equation}\n    \\label{eq:a5}\n    \\Pr\\big(\\exists i\\neq i'\\forall\n    j:\\;\\sigma(s_{ij})=\\sigma(s_{i'j})\\big)\\le \\frac{1}{c}.\n  \\end{equation}\n  (\\ref{eq:a3}) and (\\ref{eq:a5}) imply that the probability that either (i) or (ii) is\n  violated is at most\n  \\[\n    \\frac{c\\cdot n^3}{k}+\\frac{1}{c}\\le \\frac{2}{c}\\le\\delta.\n    \\qedhere\n  \\]\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:boolean}]\n   For given function $f:\\CG_n\\to\\{0,1\\}$, we choose the sentence\n   $\\psi^\\wedge_f$ according to Corollary~\\ref{cor:inv}. Applying\n   Lemma~\\ref{lem:barcelo} to this sentence and $\\epsilon$, we obtain\n   an MPNN $\\CN_f$ that on a colored graph $G\\in \\CG_{n,k}$ computes\n   an $\\epsilon$-approximation of $f(G^\\vee)$.\n  \n   Without loss of generality, we assume that the vertex set of the\n   input graph to our MPNN is $\\{1,\\ldots,n\\}$. We choose $\\ell$ (the\n   dimension of the state vectors) in such a way that\n   $\\ell\\ge c\\cdot n^2$ and $\\ell$ is at least as large as the\n   dimension of the state vectors of $\\CN_f$. Recall that\n   the state vectors are initialized as $\\vec\n   x^{(0)}_i=(r_i,0,\\ldots,0)$ for values $r_i$ chosen independently\n   uniformly at random from the interval $[0,1]$.\n\n   In the first step, our MPNN computes the purely local\n   transformation (no messages need to be passed) that maps\n   $\\vec x^{(0)}_i$ to\n   $\\vec x^{(1)}_i=(x^{(1)}_{i1},\\ldots,x^{(1)}_{i\\ell})$ with\n   \\[\n     x^{(1)}_{ij}=\n     \\begin{cases}\n       \\sigma\\Big(k\\cdot r_i-(j-1)\\cdot\\frac{k}{c\\cdot n^2}\\Big)&\\text{for }1\\le j\\le\n       c\\cdot n^2,\\\\\n       0&\\text{for }c\\cdot n^2+1\\le j\\le\\ell.\n     \\end{cases}\n   \\]\n   Since we treat $k,c,n$ as constants, the mapping \\[r_i\\mapsto k\\cdot\n   r_i-(j-1)\\cdot\\frac{k}{c\\cdot n^2}\\] is just a linear mapping\n   applied to $r_i=x^{(0)}_{i1}$.\n\n   By Lemma~\\ref{lem:prob}, with probability at least $1-\\delta$, the\n   vectors $\\vec x_i^{(1)}$ are mutually distinct $\\{0,1\\}$-vectors,\n   which we view as encoding a coloring of the input graph with colors\n   from $R_1,\\ldots,R_k$. Let $G^\\wedge$ be the resulting colored\n   graph. Since the vectors $\\vec x^{(0)}_i$ are\n   mutually distinct, $G^\\wedge$ is individualized and thus in the\n   class $\\CG_{n,k}$. We now apply the MPNN $\\CN_f$, and it computes a\n   value $\\epsilon$-close to \\[\\llbracket\\psi_f^\\wedge\\rrbracket(G^\\wedge)=f((G^\\wedge)^\\vee)=f(G).\\]\n \\end{proof}\n\n \\begin{proof}[Proof of Theorem~\\ref{theo:uni}]\n   Let $f:\\CG_n\\to \\Real$ be invariant. Since $\\CG_n$ is finite, the range\n   $Y:=f(\\CG_n)$ is finite. To be precise, we have\n   $N:=|Y|\\le|\\CG_n|=2^{\\binom{n}{2}}$. \n\n   Say, $Y=\\{y_1,\\ldots,y_N\\}$. For\n   $i=1,\\ldots,N$, let $g_i:\\CG_n\\to\\{0,1\\}$ be the Boolean function\n   defined by\n   \\[\n     g_i(G)=\n     \\begin{cases}\n       1&\\text{if }f(G)=y_i,\\\\\n       0&\\text{otherwise}.\n     \\end{cases}\n   \\]\n   Note that $g_i$ is invariant. Let $\\epsilon,\\delta>0$ and $\\epsilon':=\\frac{\\epsilon}{\\max Y}$\n   and $\\delta':=\\frac{\\delta}{N}$. By Lemma~\\ref{lem:boolean}, for\n   every $i\\in\\{1,\\ldots,N\\}$ there is an MPNN with RNI $\\CN_i$ that\n   $(\\epsilon',\\delta)$-approximates $g_i$. Putting all the $\\CN_i$\n   together, we obtain an invariant MPNN $\\CN$ that computes a\n   function $g:\\CG_n\\to\\{0,1\\}^N$. We only need to apply the linear\n   transformation\n   \\[\n     \\vec x\\mapsto\\sum_{i=1}^N x_i\\cdot y_i\n   \\]\n   to the output of $\\CN$ to obtain an approximation of $f$.\n \\end{proof}\n\n\\begin{remark}\n  Obviously, our construction yields MPNNs with a prohibitively large state\n  space. In particular, this is true for the brute force step from\n  Boolean to general functions. We doubt that there are much more\n  efficient approximators, after all we make no assumption whatsoever\n  on the function $f$.\n\n  The approximation of Boolean functions\n  is more interesting. It may still happen that\n  the GNNs get exponentially large in $n$; this seems\n  unavoidable. However, the nice thing here is that our construction\n  is very adaptive and tightly linked to the descriptive complexity of\n  the function we want to approximate. This deserves a more thorough\n  investigation, which we leave for future work.\n\n  As opposed to other universality results for GNNs, our construction\n  needs no higher-order tensors defined on tuples of nodes, with\n  practically infeasible space requirements on all but very small\n  graphs. Instead, the complexity of our construction goes entirely\n  into the dimension of the state space. The advantage of this is that\n  we can treat this dimension as a hyperparameter that we can easily\n  adapt and that gives us more fine-grained control over the space\n  requirements. Our experiments show that usually in practice a small\n  dimension already yields very powerful networks.\n\\end{remark}\n\n\\begin{remark}\n  In our experiments, we found that partial RNI, which assigns random values to a fraction of all node embedding vectors, often yields very good results, sometimes better than a full RNI. There is a theoretical plausibility\n  to this. For most graphs, we do not\n  lose much by only initializing a small fraction of vertex embeddings, because in a few message-passing rounds GNNs can propagate the randomness and individualize the full input\n  graph with our construction. On the other hand, we reduce the amount of noise our models\n  have to handle when we only randomize partially.\n\\end{remark}\n\n\n\n\\subsection{Details of Dataset Construction}\n\\label{app:corePair}\nThere is an interesting universality result for functions defined on planar graphs. It is known that 3-WL can distinguish between planar graphs \\cite{KieferPS19}. Since 4-GCNs can simulate 3-WL, this implies that functions over planar graphs can be approximated by 4-GCNs. This result can be extended to much wider graph classes, including all graph classes excluding a fixed graph as a minor \\cite{GroheWL}. \n\nInspired by this, we generate planar instances, and ensure that they can be distinguished by 2-WL, by carefully constraining these instances further. Hence, any GNN with 2-WL expressive power can approximate solutions to these planar instances. This, however, does not imply that these GNNs will solve \\EXP in practice, but only that an appropriate approximation function exists and can theoretically be learned.\n\n\n\n\\subsubsection{Construction of \\EXP}\n\n\n\\EXP consists of two main components, (i) a pair of \\emph{cores}, which are non-isomorphic, planar, 1-WL indistinguishable,  2-WL distinguishable, and decide the satisfiability of every instance, and (ii) an additional randomly generated and satisfiable \\emph{planar component}, identically added to the core pair, to add variability to \\EXP and make learning more challenging. We first present both components, and then provide further details about graph encoding and planar embeddings. \n\n\\paragraph{Core pair.}\n\n\\begin{figure*}[t]  %\n\t\\centering\n\t\\begin{subfigure}[t]{\\linewidth} \n\t\\centering\n\t\t\\begin{tikzpicture}[node distance = 1cm,line width=0.8pt,shorten >=2pt, shorten <=2pt,-, scale=0.3]\n\t\t\\tikzstyle{var} = [text width=1.2em, text centered, text=black, circle, inner sep=2pt,  draw=black, fill=white, thick]\n\t\t\\tikzstyle{disj} = [text width=1.2em ,fill=gray!120,text centered, text=white, circle, inner sep=2pt,  draw=black,  thick]\n\t\t\\node[var] (x0) {$x_0$};\n\t\t\\node[var, right=0.5cm of x0] (x0n) {$\\bar{x_0}$};\n\t\t\\node[var, right = 0.5cm of x0n] (x1) {$x_1$};\n\t\t\\node[var, right=0.5cm of x1] (x1n) {$\\bar{x_1}$};\n\t\t\\node[var, right = 0.5cm of x1n] (x2) {$x_2$};\n\t\t\\node[var, right=0.5cm of x2] (x2n) {$\\bar{x_2}$};\n\t\t\\node[var, right = 0.5cm of x2n] (x3) {$x_3$};\n\t\t\\node[var, right= 0.5cm of x3] (x3n) {$\\bar{x_3}$};\n\t\t\\node[disj, above= 0.5cm of  x0n] (d0) {$d_0$};\n\t\t\\node[disj, above= 0.5cm of  x1n] (d1) {$d_1$};\n\t\t\\node[disj, above= 0.5cm of  x2n] (d2) {$d_2$};\n\t\t\\node[disj, above right = 1.5cm and 0.2 of x1] (d6) {$d_6$};\n\t\t\\node[disj, above right = 2cm and 0.2cm of x0] (d4) {$d_4$};\n\t\t\\node[disj, above = 2.5cm of x0] (d3) {$d_3$};\n\t\t\\node[disj, below = 0.5cm  of x1n] (d7) {$d_7$};\n\t\t\\node[disj, below = 1cm of x0n] (d5) {$d_5$};\n\t\t\n\t\t\\draw[color=black,dashed] (x0.east) -- (x0n.west);\n\t\t\\draw[color=black,dashed] (x1.east) -- (x1n.west);\n\t\t\\draw[color=black,dashed] (x2.east) -- (x2n.west);\n\t\t\\draw[color=black,dashed] (x3.east) -- (x3n.west);\n\t\t\\draw[color=black] (x0n.north) |- (d0.south);\n\t\t\\draw[color=black] (x1.north) |- (d0.east);\n\t\t\\draw[color=black] (x1n.north) -- (d1.south);\n\t\t\\draw[color=black] (x2.north) |- (d1.east);\n\t\t\\draw[color=black] (x2n.north) -- (d2.south);\n\t\t\\draw[color=black] (x3.north) |- (d2.east);\n\t\t\\draw[color=black] (x0.north) -- (d3.south);\n\t\t\\draw[color=black] (x3n.north) |- (d3.east);\n\t\t\n\t\t\\draw[color=black] (x0.60) |- (d4.west);\n\t\t\\draw[color=black] (x3.60) |- (d4.east);\n\t\t\\draw[color=black] (x0n.south) -- (d5.north);\n\t\t\\draw[color=black] (x3n.south) |- (d5.east);\n\t\t\n\t\t\\draw[color=black] (x1.60) |- (d6.west);\n\t\t\\draw[color=black] (x2.60) |- (d6.east);\n\t\t\\draw[color=black] (x1n.south) --  (d7.north);\n\t\t\\draw[color=black] (x2n.south) |- (d7.east);\n\t\t\\end{tikzpicture}\n\t\t\\caption{The encoding of the formula $\\phi_1$.}\\label{fig:phi1}\t\t\n\t\\end{subfigure}\n\t\\\\ \\vspace{0.25cm}\n\t\\begin{subfigure}[t]{\\linewidth}  \n\t\\centering\n\t\t\\begin{tikzpicture}[node distance = 1cm,line width=0.8pt,shorten >=2pt, shorten <=2pt,-, scale=0.3]\n\t\t\\tikzstyle{var} = [text width=1.2em, text centered, text=black, circle, inner sep=2pt,  draw=black, fill=white,  thick]\n\t\t\\tikzstyle{disj} = [text width=1.2em ,fill=gray!120,text centered, text=white, circle, inner sep=2pt,  draw=black,  thick]\n\t\t\\node[var] (x0) {$x_0$};\n\t\t\\node[var, right=0.5cm of x0] (x0n) {$\\bar{x_0}$};\n\t\t\\node[var, right =0.5cm of x0n] (x1) {$x_1$};\n\t\t\\node[var, right=0.5cm of x1] (x1n) {$\\bar{x_1}$};\n\t\t\\node[var, right =0.5cm of x1n] (x2n) {$\\bar{x_2}$};\n\t\t\\node[var, right=0.5cm of x2] (x2) {$x_2$};\n\t\t\\node[var, right = 0.5cm of x2] (x3n) {$\\bar{x_3}$};\n\t\t\\node[var, right= 0.5cm of x3] (x3) {$x_3$} ;\n\t\t\\node[disj, below = 0.5cm of  x0n] (d0) {$d_0$};\n\t\t\\node[disj, below = 1cm of  x0] (d1) {$d_1$};\n\t\t\\node[disj, below = 0.5cm of  x2] (d3) {$d_3$};\n\t\t\\node[disj, below = 1cm of  x2n] (d2) {$d_2$};\n\t\t\\node[disj, above = 0.5cm of x1n] (d7) {$d_7$};\n\t\t\\node[disj, above = 1cm of x1] (d6) {$d_6$};\n\t\t\\node[disj, above= 1.5cm of x0n] (d5) {$d_5$};\n\t\t\\node[disj, above= 2cm of x0] (d4) {$d_4$};\n\t\t\n\t\t\\draw[color=black,dashed] (x0.east) -- (x0n.west);\n\t\t\\draw[color=black,dashed] (x1.east) -- (x1n.west);\n\t\t\\draw[color=black,dashed] (x2.west) -- (x2n.east);\n\t\t\\draw[color=black,dashed] (x3.west) -- (x3n.east);\n\t\t\\draw[color=black] (x0.north) -- (d4.south);\n\t\t\\draw[color=black] (x0.south) -- (d1.north);\n\t\t\\draw[color=black] (x0n.north) -- (d5.south);\t\t\n\t    \\draw[color=black] (x0n.south) -- (d0.north);\n\t    \\draw[color=black] (x1.north) -- (d6.south);\t\t\n\t\t\\draw[color=black] (x1.south) |- (d0.east);\n        \\draw[color=black] (x1n.north) -- (d7.south);\t\t\n\t\t\\draw[color=black] (x1n.south) |- (d1.east);\t\n\t    \\draw[color=black] (x2.north) |- (d6.east);\t\t\n\t\t\\draw[color=black] (x2.south) -- (d3.north);\n        \\draw[color=black] (x2n.north) |- (d7.east);\t\t\n\t\t\\draw[color=black] (x2n.south) -- (d2.north);\n\t    \\draw[color=black] (x3.north) |- (d4.east);\t\t\n\t\t\\draw[color=black] (x3.south) |- (d2.east);\n        \\draw[color=black] (x3n.north) |- (d5.east);\t\t\n\t\t\\draw[color=black] (x3n.south) |- (d3.east);\t\t\n\t\t\\end{tikzpicture}\n\t\t\\caption{The encoding of the formula $\\phi_2$.}\\label{fig:phi2}\t\t\n\t\\end{subfigure}\n\t\\caption{Illustration of planar embeddings for the formulas $\\phi_1$ and $\\phi_2$ for $n=2$.}\n\\label{fig:planEmb}\n\\end{figure*}\nIn \\EXP, a core pair consists of two \\CNF formulas $\\phi_1, \\phi_2$, both defined using $2n$ variables, $n \\in \\Nbb^{+}$, such that $\\phi_1$ is unsatisfiable and $\\phi_2$ is satisfiable, and such that their graph encodings are 1-WL indistinguishable and planar. $\\phi_1$ and $\\phi_2$ are constructed using two structures which we refer to as \\emph{variable chains} and \\emph{variable bridges} respectively. \n\nA \\emph{variable chain} $\\phi_{chain}$ is defined over a set of $n \\geq 2$ Boolean variables%\n, and imposes that all variables be equally set. The variable chain can be defined in increasing or decreasing order over these variables. More specifically, given variables $x_i, ..., x_j$, \n\\begin{align}\n\\text{Chain}_\\text{Inc}(i, j) &= \\bigwedge_{k=i}^{j-1} (\\bar{x_k} \\vee x_{i+(k+1)\\%(j-i+1)}),~\\text{and}\\\\\n\\text{Chain}_\\text{Dec}(i, j) &= \\bigwedge_{k=i}^{j-1} ({x_k} \\vee \\bar{x}_{i+(k+1)\\%(j-i+1)}).\n\\end{align}\nAdditionally, a \\emph{variable bridge} is defined over an even number of variables $x_0,..., x_{2n-1}$, as \n\\begin{align}\n{\\phi_{bridge} = \\bigwedge_{i=0}^{n-1} \\big((x_i \\vee x_{2n-1-i}) \\land (\\bar{x_i} \\vee \\bar{x}_{2n-1-i})\\big)}.\n\\end{align}\nA variable bridge makes the variables it connects forcibly have opposite values, e.g., $x_0 = \\bar{x_1}$ for $n=1$. We denote a variable bridge over $x_0,..., x_{2n-1}$ as $\\text{Bridge}(2n)$. \n\nTo get $\\phi_1$ and $\\phi_2$, we define $\\phi_1$ as a variable chain and bridge on all variables, yielding contrasting and unsatisfiable constraints. To define $\\phi_2$, we ``cut'' the chain in half, such that the first $n$ variables can differ from the latter $n$, satisfying the bridge. The second half of the ``cut'' chain is then flipped to a decrementing order, which preserves the satisfiability of $\\phi_2$, but maintains the planarity of the resulting graph. More specifically, this yields:\n\\begin{align}\n\\phi_1 &= \\text{Chain}_\\text{Inc}(0, 2n) \\land \\text{Bridge}(2n)\\text{, and} \\\\\n\\phi_2 &= \\text{Chain}_\\text{Inc}(0, n) \\land \\text{Chain}_\\text{Dec}(n, 2n) \\land \\text{Bridge}(2n).\n\\end{align}\n\\paragraph{Planar component.}\nFollowing the generation of $\\phi_1$ and $\\phi_2$, a disjoint satisfiable planar graph component $\\phi_\\text{planar}$ is added. $\\phi_\\text{planar}$ shares no variables or disjunctions with the cores, so is primarily introduced to create noise and make learning more challenging. $\\phi_\\text{planar}$ is generated starting from random 2-connected (i.e., at least 2 edges must be removed to disconnect a component within the graph) bipartite planar graphs from the Plantri tool \\cite{brinkmann2007fast}, such that (i)\nthe larger set of nodes in the graph is the variable set\\footnote{Ties are broken arbitrarily if the two sets are equally sized.}, (ii)\nhighly-connected disjunctions are split in a planarity-preserving\nfashion to maintain disjunction widths not exceeding 5, (iii) literal signs for variables are uniformly randomly assigned, and (iv) redundant disjunctions,\nif any, are removed. If this $\\phi_\\text{planar}$ is satisfiable, then it is accepted and used. Otherwise, the formula is discarded and a new $\\phi_\\text{planar}$ is analogously generated until a satisfiable formula is produced.\n\nSince the core pair and $\\phi_{\\text{planar}}$ are disjoint, it clearly follows that the graph encodings of $\\phi_\\text{planar} \\land \\phi_1$ and $\\phi_\\text{planar} \\land \\phi_2$ are planar and 1-WL indistinguishable. Furthermore, $\\phi_\\text{planar} \\land \\phi_1$ is satisfiable, and $\\phi_\\text{planar} \\land \\phi_2$ is not. Hence, the introduction of $\\phi_\\text{planar}$ maintains all the desirable core properties, all while making any generated \\EXP dataset more challenging. \n\nThe structural properties of the cores, combined with the combinatorial difficulty of \\SAT, make \\EXP a challenging dataset. For example, even minor formula changes, such as flipping a literal, can lead to a change in the \\SAT outcome, which enables the creation of near-identical, yet semantically different instances. Moreover, \\SAT is NP-complete \\cite{cook1971complexity}, and remains so on planar instances \\cite{hunt1998complexity}. Hence, \\EXP is cast to be challenging, both from an expressiveness and computational perspective. \n\n\\paragraph{Remark 3.} Intuitively, $\\phi_1$ and $\\phi_2$, generated as described, can be distinguished by 2-WL, as 2-WL can detect the break in cycles resulting from the ``cut''. In other words, 2-WL can identify that the chain has been broken in between these two formulas, and thus will return distinct colorings. Hence, $\\phi_1$ and $\\phi_2$ can be distinguished by 3-GCNs.\n\\paragraph{Graph encoding.}\nWe use the following graph encoding, denoted by $Enc$: (i) Every variable is encoded by two nodes, representing its positive and negative literals, and connected by an edge, (ii) Every disjunction is represented by a node, and an edge connects a literal node to a disjunction node if the literal appears in the disjunction, and (iii) Variable and disjunction nodes are encoded with their respective types. We opt for this encoding, as it is commonly used in the literature \\cite{Selsam-ICLR2019}, and, for the sake of our empirical evaluation, yields planar encodings for \\EXP graph pairs. \n\n\\paragraph{Planar embeddings for core pair.}\nWe show planar embeddings for $Enc(\\phi_1)$ and $Enc(\\phi_2)$ for $n=2$ in \\Cref{fig:planEmb}, and these embeddings can naturally be extended to any $n$. %\n $Enc(\\phi_1)$ and $Enc(\\phi_2)$ can also be shown to be 1-WL indistinguishable. This can be observed intuitively, as node neighborhoods in both graphs are identical and very regular: all variable nodes are connected to exactly one other variable node and two disjunction nodes, and all disjunction nodes are connected to exactly two variables. %\n\n\n\\subsubsection{Construction of \\CEXP}\n\\label{app:CExpConstruction}\nGiven an \\EXP dataset with $N$ pairs of graphs, we create \\CEXP by selecting $N \/ 2$ graph pairs and modifying them to yield \\Corrupt. The unmodified graph pairs are therefore exactly identical in type to \\EXP instances, and we refer to these instances within \\CEXP as \\EXPTwo.  Then, for every graph pair, we discard the satisfiable graph and construct a new graph from a copy of the unsatisfiable graph as follows:\n\\begin{enumerate}\n    \\item Randomly introduce new literals to the existing disjunctions of the copy of the unsatisfiable graph, such that no redundancies are created (e.g., adding $x$ to a disjunction when $x$ or $\\bar{x}$ is already present), until 3 literals are added \\emph{and} the formula becomes satisfiable. Literal addition is done by creating new edges in the graph between disjunction and literal nodes. To do this, disjunctions with less than 5 literals are uniformly randomly selected, and the literal to add is uniformly randomly sampled from the set of all non-redundant literals for the disjunction.\n    \\item Once a satisfiable formula is reached, iterate sequentially over all added edges, and eliminate any edge whose removal does not restore unsatisfiability. This ensures that a minimal number of new edges, relative to the original unsatisfiable graph, are added. \n\\end{enumerate}\n\nObserve that these modifications have several interesting effects on the dataset. First, they preserve the existing UNSAT core nodes and edges, while flipping the satisfiability of their overall formulas, which makes the learning task go beyond structure identification. Second, they introduce significant new variability to the dataset, in that the planar component and cores can share edges. Finally, they make the graph pairs 1-WL distinguishable, which gives standard GNNs a chance to perform well on \\Corrupt. \n\n\\subsubsection{Dataset generation for experiments}\nTo create the \\EXP dataset, we randomly generate 600 core pairs, where $n$ is uniformly randomly set between 2 and 4 inclusive. Then, we generate the additional planar component using Plantri, such that 500 $\\phi_\\text{planar}$ formulas are generated from 12-node planar bipartite planar graphs, and the remaining 100 from planar bipartite graphs with 15 nodes. \n\nThis generation process implies that every formula has a number of variables ranging between 10 (4 core variables when $n=2$ plus a minimum 6 variables from the larger bipartite set during $\\phi_\\text{planar}$ generation from 12-node graphs) and 22 variables (8 core variables for $n=4$ plus a maximally-sized variable subset of 14 nodes for $\\phi_\\text{planar}$ generation from 15-node graphs). \n\nFurthermore, the number of disjunctions also ranges from 10 (8 core disjunctions for $n=2$ plus the minimum 2 disjunctions for the case where $\\phi_\\text{planar}$, generated from 12-node graphs, has 10 variables and 2 disjunctions) to 30 disjunctions (16 core disjunctions for $n=4$ plus at most 14 disjunctions for the case where $\\phi_\\text{planar}$, generated from 15-node graphs, initially has 8 variables and 7 disjunctions, which can at most lead to 14 final disjunctions following step (ii)).\n\n\n\n\\subsection{Standard Deviation of GCN-50\\%RNI on \\EXP over training}\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=0.7]\n\n\\definecolor{color0}{rgb}{0.12156862745098,0.466666666666667,0.705882352941177}\n\\definecolor{color4}{rgb}{1,0.498039215686275,0.0549019607843137}\n\\definecolor{color2}{rgb}{0.172549019607843,0.627450980392157,0.172549019607843}\n\\definecolor{color3}{rgb}{0.83921568627451,0.152941176470588,0.156862745098039}\n\\definecolor{color1}{rgb}{0.4,0.4,0.4}\n\n\\begin{axis}[\nlabel style={font=\\large},\ntick label style={font=\\normalsize},\ntick align=outside,\ntick pos=left,\nx grid style={white!69.01960784313725!black},\nxlabel={Training Epoch},\nxmajorgrids,\nxmin=0, xmax=500, \ny grid style={white!69.01960784313725!black},\nymajorgrids,\nymin=0, ymax=0.10\n]\n\n\\addplot [line width=0.4mm, color4]\ntable [row sep=\\\\]{%\n5 0.020480342879074177\\\\\n10 0.027588242262078077\\\\\n15 0.05413460589637239\\\\\n20 0.05555277770832988\\\\\n25 0.059773879468097674\\\\\n30 0.07255744850346749\\\\\n35 0.043652669512362643\\\\\n40 0.05161287522400674\\\\\n45 0.04330127018922194\\\\\n50 0.024209731743889924\\\\\n55 0.050826502273256365\\\\\n60 0.04133198922545748\\\\\n65 0.03128054063187242\\\\\n70 0.04916666666666667\\\\\n75 0.04119735698102758\\\\\n80 0.04444097208657795\\\\\n85 0.06182412330330469\\\\\n90 0.03993049516903648\\\\\n95 0.03227486121839516\\\\\n100 0.047316898555261305\\\\\n105 0.03497022543059546\\\\\n110 0.038151743807531995\\\\\n115 0.041965594373380564\\\\\n120 0.037277115410575073\\\\\n125 0.037043517951488115\\\\\n130 0.040858835573770895\\\\\n135 0.02688710967483614\\\\\n140 0.02929732638541157\\\\\n145 0.025603819159562044\\\\\n150 0.021730674684008827\\\\\n155 0.022684429314693673\\\\\n160 0.028625940062196122\\\\\n165 0.02334820954353649\\\\\n170 0.02370243775554648\\\\\n175 0.013017082793177762\\\\\n180 0.01615893285805442\\\\\n185 0.01643590243609669\\\\\n190 0.010172129679778086\\\\\n195 0.013565683830083088\\\\\n200 0.010833333333333339\\\\\n205 0.01699673171197596\\\\\n210 0.00870025542409214\\\\\n215 0.012801909579781013\\\\\n220 0.009464847243000464\\\\\n225 0.01118033988749895\\\\\n230 0.007637626158259735\\\\\n235 0.002499999999999991\\\\\n240 0.010034662148993576\\\\\n245 0.005527707983925673\\\\\n250 0.0055901699437494795\\\\\n255 0.0033333333333333214\\\\\n260 0.0076829537144107425\\\\\n265 0.005335936864527384\\\\\n270 0.005335936864527384\\\\\n275 0.005527707983925671\\\\\n280 0.002499999999999991\\\\\n285 0.0033333333333333214\\\\\n290 0.005527707983925673\\\\\n295 0.002499999999999991\\\\\n300 0.005527707983925673\\\\\n305 0.002499999999999991\\\\\n310 0.0038188130791298527\\\\\n315 0.007637626158259738\\\\\n320 0.0076829537144107425\\\\\n325 0.005335936864527385\\\\\n330 0.002499999999999991\\\\\n335 0.002499999999999991\\\\\n340 0.0033333333333333214\\\\\n345 0.002499999999999991\\\\\n350 0.005335936864527385\\\\\n355 0.0\\\\\n360 0.010034662148993576\\\\\n365 0.005335936864527385\\\\\n370 0.005527707983925673\\\\\n375 0.0055901699437494795\\\\\n380 0.0\\\\\n385 0.010034662148993576\\\\\n390 0.005527707983925671\\\\\n395 0.002499999999999991\\\\\n400 0.002499999999999991\\\\\n405 0.005000000000000016\\\\\n410 0.002499999999999991\\\\\n415 0.002499999999999991\\\\\n420 0.002499999999999991\\\\\n425 0.005000000000000016\\\\\n430 0.002499999999999991\\\\\n435 0.002499999999999991\\\\\n440 0.0\\\\\n445 0.0\\\\\n450 0.005335936864527384\\\\\n455 0.0\\\\\n460 0.0033333333333333214\\\\\n465 0.005335936864527384\\\\\n470 0.007637626158259738\\\\\n475 0.005000000000000016\\\\\n480 0.0\\\\\n485 0.0033333333333333214\\\\\n490 0.0\\\\\n495 0.007637626158259738\\\\\n500 0.002499999999999991\\\\\n};\n\n\\end{axis}\n\\end{tikzpicture}\n\\caption{Standard deviation of test accuracy over all 10 validation splits for GCN-50\\%RNI on \\EXP.}\\label{fig:stdev}\n\\end{figure}\n\\begin{figure*}[t!]\n\t\\centering\n\t\\begin{subfigure}{.48\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_core}\n\t\t\\caption{Learning curves on \\EXP.}\n\t\t\\label{app:fig:modelConvergence}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.48\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_coreSparse}\n\t\t\\caption{Learning curves on \\SparseEXP.}\n\t\t\\label{app:fig:modelConvergenceSparse}\n\t\\end{subfigure}%\n\t\\caption{Model convergence results for Experiment 1 on the datasets \\EXP and \\SparseEXP.}\n\t\\label{app:fig:resEXP}\n\\end{figure*}\n In this subsection, we investigate the variability of \\GCNRNI~learning across validation folds, and do so with a representative model and dataset, namely the semi-randomized GCN-50\\%RNI model and the standard \\EXP dataset. The standard deviation of the test accuracy of GCN-50\\%RNI over \\EXP, across all 10 cross-validation folds relative to the number of epochs, is shown in \\Cref{fig:stdev}. From this figure, we see that standard deviation spikes sharply at the start of training, and only begins dropping after 100 epochs. This suggests that the learning behavior of GCN-50\\% RNI is quite variable, sometimes requiring few epochs to converge, and in other cases requiring a very high number of epochs. Furthermore, standard deviation converges to almost zero following 200 epochs, corresponding to the phase where all validation folds have achieved near-perfect test performance. From these findings, we further confirm that RNI introduces volatility to GCN training, this time manifesting in variable convergence times across validation folds, but that this volatility does not ultimately hinder convergence and performance, as all folds eventually reach satisfactory performance within a reasonable amount of epochs, and subsequently stabilize. \n\n\\subsection{Additional Experiments}\nIn addition to the experiments in the main body of the paper, we additionally evaluate RNI on sparser analog datasets to \\EXP and \\CEXP, namely \\SparseEXP and \\SparseCEXP. These datasets only contain 25\\% of the number of instances of their original counterparts, and are used to study the behavior and impact of RNI when data is sparse.\n\n\\subsubsection{Experiment 1: \\SparseEXP}\nIn this experiment, we generate \\SparseEXP analogously to \\EXP, except that this dataset only consists of 150 graph pairs, i.e., 300 graphs in total. We then train \\ThreeGNNFull for 200 epochs, and all other systems for 1000 epochs on \\SparseEXP, as opposed to 100 and 500 respectively for \\EXP, to give all evaluated models a better opportunity to compensate for the smaller dataset size. We show the learning curves for all models on \\SparseEXP, and reproduce the original figure for \\EXP, in \\Cref{app:fig:resEXP} for easier comparison.\n\nFirst, we observe that all models converge slower on \\SparseEXP compared to \\EXP. This is not surprising, as a lower data availability makes learning a well-performing function slower and more challenging. More specifically, sparsity implies that (i) fewer weight updates are made per epoch, and (ii) these updates are of lower quality, as they are computed from a less complete dataset. Nonetheless, the same convergence patterns for \\GCNRNI~models and \\ThreeGNNFull are also visible in this setting, further highighting the increased convergence time required by \\GCNRNI~models. \n\nWe also observe that all \\GCNRNI~models, though also eventually converging, do so in a more volatile fashion. Indeed, \\GCNRNI~models suffer from the sparseness of the dataset, as this makes them more sensitive to RNI. As a result, these models require more training to effectively learn robustness against RNI values, and learn this from a smaller sample set, increasing their variability further. Moreover, the nature of \\SparseEXP makes learning more difficult, as it fully relies on RNI for MPNNs to have a chance of achieving above-random performance, and thus encourages MPNNs to fit specific RNI values. Hence, RNI introduces significant volatility and variability to training, particularly with sparser data, and requires substantial training and epochs for \\GCNRNI~models to effectively develop a robustness to RNI instantiations. \n\n\\subsubsection{Experiment 2: \\SparseCEXP}\n\\begin{figure*}[t!]\n\t\\centering\n\t\\begin{subfigure}{.48\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_coreplus}\n\t\t\\caption{Learning curves on \\CEXP.}\n\t\t\\label{app:fig:modelConvergencePlus}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.48\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_LRNEXP}\n\t\t\\caption{Learning curves on \\EXPTwo (\/E) and \\Corrupt (\/C).}\n\t\t\\label{app:fig:modelConvergenceSplit}\n\t\\end{subfigure}%\n\t\\break\n\t\\begin{subfigure}{.48\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_coreplusSparse}\n\t\t\\caption{Learning curves on \\SparseCEXP.}\n\t\t\\label{app:fig:modelConvergencePlusSparse}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.48\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_LRNEXPSparse}\n\t\t\\caption{Learning curves on\\EXPTwo (\/E) and \\Corrupt (\/C).}\n\t\t\\label{app:fig:modelConvergenceSplitSparse}\n\t\\end{subfigure}%\n\t\\caption{Model convergence results for Experiment 2 on \\CEXP and \\SparseCEXP.}\n\t\t\\label{app:fig:res}\n\\end{figure*}\n\nAnalogously to Experiment 1, we generate a \\SparseCEXP dataset similarly to \\CEXP, but only generate 150 graph pairs. Then, we select 75 graph pairs and modify them, as described in \\Cref{app:CExpConstruction}. We report the learning curves for all models on \\SparseCEXP, as well as the original curves for \\CEXP from the main body, in \\Cref{app:fig:res}.\n\nAs in the previous subsection, similar behavior is observed on \\SparseCEXP compared with \\CEXP, only differing by slower convergence in the former case. However, we note that the ``struggle'' phase described in the main paper, which only occurs during the first 100 epochs over \\CEXP, lasts for around 500 epochs on \\SparseEXP. Intuitively, this ``struggle'' phenomenon is due to conflicting learning requirements, stemming from \\Corrupt and \\EXPTwo, which effectively require models to ``isolate''  deterministic dimensions  for \\Corrupt, and other randomized dimensions for \\EXPTwo. This in itself is already challenging on \\CEXP, but is made even more difficult on \\SparseCEXP due to its sparsity. Indeed, sparsity makes that further samples are needed in expectation to find a reasonable solution, leading to a lengthy ``struggle'' phase, in which both \\Corrupt and \\EXPTwo data points conflict with one another during optimization. \n\n\n\\subsection{Hyper-parameter details}\nAll GCN models with (partially or completely) deterministic initial node embeddings map a 2-dimensional one-hot encoding of node type (literal or disjunction) to a $k$-dimensional embedding space, where $k$ corresponds to the dimensionality of the deterministic embeddings. Furthermore, the final prediction for every graph is computed by aggregating all node embeddings using the $\\max$ function, and then passing the result through a multi-layer perceptron (MLP) of 3 layers with dimensionality $x$, 32 and 2 respectively, where $x$ is the embedding dimensionality used in the given model. The activation function for the first two MLP layers is the ELU function \\cite{ClevertUH15}, and the softmax function is used to make a final prediction at the final MLP layer.\n\nAll neural networks in this work are optimized using the Adam optimizer \\cite{Kingma-ICLR2014}. All training is conducted with a fixed learning rate $\\lambda$, for fairer comparison between all models. Initially, decaying learning rates were used, but these were discarded, as they yielded sub-optimal convergence for all \\GCNRNI~models. Finally, all experiments were run on a V100 GPU. Detailed hyper-parameters, namely learning rate $\\lambda$ and RNI distribution $p$, per model on every evaluation dataset are shown in \\Cref{tab:hyperParams}.\n\n\\begin{table}[t!]\n\t\\centering\n\t\\caption{Hyper-parameter configurations for all experiments.} \n\t\\begin{tabular}{lcccc}\n\t\t\\toprule \n\t\tDataset & \\multicolumn{2}{c}{\\EXP} & \\multicolumn{2}{c}{\\CEXP}\\\\\n\t\t  & $\\lambda$ & $p$ & $\\lambda$ & $p$\\\\\n\t\t \\cmidrule(r){2-3}\n\t\t \\cmidrule(r){4-5}\n\t\t GCN & $1\\times10^{-4}$ & N\/A & $1\\times10^{-4}$ & N\/A \\\\\n\t\t GCN-12.5\\%RNI & $2\\times10^{-4}$ & N & $2\\times10^{-4}$ & N \\\\\n\t\t GCN-50\\%RNI & $2\\times10^{-4}$ & N & $2\\times10^{-4}$ & N \\\\\n\t\t GCN-87.5\\%RNI & $2\\times10^{-4}$ & N & $5\\times10^{-4}$ & N \\\\\n\t\t \\GCNRNI & $5\\times10^{-4}$ & N & $5\\times10^{-4}$ & N \\\\\n\t\t \\ThreeGNNFull & $5\\times10^{-4}$ & N\/A & $2\\times10^{-4}$ & N\/A \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:hyperParams}\n\\end{table}\n\n\\subsubsection{Results for \\GCNRNI~with hyperbolic tangent activation}\n\n\\begin{table}[t!]\n\\centering\n\\caption{Performance of \\GCNRNI on \\EXP dataset with tanh.}\n\\begin{tabular}{lHc} \n\\toprule\nModel & Training Accuracy (\\%) & Testing  Accuracy (\\%)\\\\\n\\midrule\n \\GCNRNI(U) & 93.0 $\\pm$ 6.32 & 92.7 $\\pm$ 5.61 \\\\ \n\\textbf{\\GCNRNI(N)} & \\textbf{95.7 $\\pm$ 2.64} & \\textbf{96.0 $\\pm$ 2.11} \\\\ \n\\GCNRNI(XU) & 65.1 $\\pm$ 20.6 & 64.6 $\\pm$ 19.9 \\\\ \n\\GCNRNI(XN) & 63.2 $\\pm$ 21.2 & 63.0 $\\pm$ 20.9 \\\\ \n\\bottomrule\n\\end{tabular}\n\\label{tab:exp1Tanh}\n\\end{table}\n\nIn addition to experimenting with the RNI probability distribution, we also experimented with different activation functions for the GCN message passing iterations. Results are shown in \\Cref{tab:exp1Tanh}. Performance with $tanh$ is significantly more variable across distributions than ELU, which shows that RNI is highly sensitive to  choices of hyper-parameters.\n\n\n\n\n\n  \n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nGraph neural networks (GNNs) \\cite{Scarselli09,Gori2005} are neural architectures designed for learning functions over graph domains, and naturally encode desirable properties such as permutation invariance (resp., equivariance) relative to graph nodes, and node-level computation based on message passing. These properties provide GNNs with a strong inductive bias, enabling them to effectively learn and combine both local and global graph features \\cite{BattagliaGraphNetworks}. GNNs have been applied to a multitude of tasks, ranging from protein classification \\cite{GilmerSRVD17} and synthesis \\cite{YouLYPL18}, protein-protein interaction \\cite{FoutBSB17}, and social network analysis \\cite{HamiltonYL17}, to recommender systems \\cite{YingHCEHL18} and combinatorial optimization \\cite{BengioTourDHorizon}. %\n\nWhile being widely applied, popular GNN architectures, such as message passing neural networks (MPNNs), are limited in their expressive power. Specifically, MPNNs are at most as powerful as the Weisfeiler-Leman (1-WL) graph isomorphism heuristic \\cite{MorrisAAAI19,Keyulu18}, and thus cannot discern between several families of non-isomorphic graphs, e.g., sets of regular graphs \\cite{CaiFI92}. \nTo address this limitation, alternative GNN architectures with provably higher expressive power, such as $k$-GNNs~\\cite{MorrisAAAI19} and invariant (resp., equivariant) graph networks~\\cite{MaronInvEqui19}, have been proposed. These models, which we refer to as \\emph{higher-order GNNs}, are inspired by the generalization of 1-WL to $k-$tuples of nodes, known as $k$-WL  \\cite{CaiFI92}. \nWhile these models are very expressive, they are computationally very demanding. As a result, MPNNs, despite their limited expressiveness, remain the standard for graph representation learning. \n\nIn a rather recent development, MPNNs have achieved empirical improvements using \\emph{random node initialization} (RNI), in which initial node embeddings are randomly set. Indeed, RNI enables MPNNs to detect \\emph{fixed} substructures,  so extends their power beyond 1-WL, and also allows for a better approximation of a class of combinatorial problems \\cite{SatoRandom2020}. While very important, these findings do not explain the overall theoretical impact of RNI on GNN learning and generalization for \\emph{arbitrary} functions. \n\nIn this paper, we thoroughly study the impact of RNI on MPNNs. Our main result states that MPNNs enhanced with RNI are \\emph{universal}, and thus can approximate every function defined on graphs of any fixed order. This follows from a logical characterization of the expressiveness of MPNNs \\cite{BarceloKM0RS20} combined with an argument on order-invariant definability. \nImportantly, MPNNs enhanced with RNI preserve the \\emph{permutation-invariance} of MPNNs in expectation, and possess a strong inductive bias. \nOur result strongly contrasts with 1-WL limitations of deterministic MPNNs, and provides a foundation for developing expressive and memory-efficient MPNNs with strong inductive bias.\n\nTo verify our theoretical findings, we carry out a careful empirical study. We design \\EXP, a synthetic dataset requiring 2-WL expressive power for models to achieve above-random performance, and run MPNNs with RNI on it, to observe \\emph{how well} and \\emph{how easily} this model can learn and generalize. Then, we propose \\CEXP, a modification of \\EXP with partially 1-WL distinguishable data, and evaluate the same questions in this more variable setting. Overall, the contributions of this paper are as follows:\n\\begin{enumerate}[-,leftmargin=*]\n\n\t\\item We prove that MPNNs with RNI are universal, while being permutation-invariant in expectation. This is a significant improvement over the 1-WL limit of standard MPNNs and, to our knowledge, a first universality result for  memory-efficient GNNs.\n\t\n\t\\item We introduce two carefully designed datasets, \\EXP and \\CEXP, based on graph pairs only distinguishable by 2-WL or higher, to rigorously evaluate the impact of RNI.\n\t\n\t\\item  We analyze the effects of RNI on MPNNs on these datasets, and observe that (i)~MPNNs with RNI closely match the performance of higher-order GNNs, (ii)~the improved performance of MPNNs with RNI comes at the cost of slower convergence, and (iii)~partially randomizing initial node features improves model convergence and accuracy.\n\t\\item We additionally perform the same experiments with analog sparser datasets, with longer training, and observe a similar behavior, but more volatility.\n\\end{enumerate}\nThe proof of the main theorem, as well as further details on datasets and experiments, can be found in the appendix of this paper.\n\n\\section{Graph Neural Networks}\n\n\n\nGraph neural networks (GNNs) \\cite{Gori2005,Scarselli09} are neural models for learning functions over graph-structured data.\nIn a GNN, graph nodes are assigned vector representations, which are updated iteratively through series of \\emph{invariant} or \\emph{equivariant} computational layers. \nFormally, a function $f$ is \\emph{invariant} over graphs if, for isomorphic graphs $G,H\\,{\\in}\\,\\CG$ it holds that $f(G)\\,{=}\\,f(H)$. Furthermore, a function $f$ mapping a graph $G$ with vertices $V(G)$ to vectors ${\\boldsymbol x\\in\\Rbb^{|V(G)|}}$ is \\emph{equivariant} if,  for every permutation $\\pi$ of $V(G)$, it holds that ${f(G^\\pi)=f(G)^\\pi}$.\n\n\n\\subsection{Message Passing Neural Networks} \nIn MPNNs \\cite{GilmerSRVD17}, node representations aggregate \\emph{messages} from their neighboring nodes, and use this information to iteratively update their representations. Formally, given a node $x$, its vector representation $v_{x,t}$ at time $t$, and its\nneighborhood $N(x)$, an update can be written as:\n\\[\nv_{x,t+1} = combine \\Big(v_{x,t},aggregate\\big(\\{v_{y,t}|~y \\in N(x)\\}\\big)\\Big),\n\\]\nwhere \\emph{combine} and \\emph{aggregate} are functions, and \\emph{aggregate} is typically permutation-invariant. \nOnce message passing is complete, the final node representations are then used to compute target outputs. Prominent MPNNs include graph convolutional networks~(GCNs)~\\cite{Kipf16} and graph attention networks~(GATs)~\\cite{VelickovicCCRLB18}.\n\nIt is known that standard MPNNs have the same power as the 1-dimensional Weisfeiler-Leman algorithm (1-WL) \\cite{Keyulu18,MorrisAAAI19}. This entails that graphs (or nodes) cannot be distinguished by MPNNs if $1$-WL does not distinguish them. For instance, 1-WL cannot distinguish between the graphs $G$ and $H$, shown in Figure \\ref{fig:indistinguishable}, despite them being clearly non-isomorphic. Therefore, MPNNs cannot learn functions with different outputs for $G$ and $H$. \n\n\\begin{figure}[t!]\n\t\\centering\n\t\\begin{tikzpicture}[\n\tvertex\/.style = {draw,fill=black!70,circle,inner sep=0pt, minimum height= 2mm},\n\t]\n\t\\begin{scope}\n\t\\node[vertex] (v0) at (90.00:1cm) {};\n\t\\node[vertex] (v1) at (141.43:1cm) {};\n\t\\node[vertex] (v2) at (192.86:1cm) {};\n\t\\node[vertex] (v3) at (244.29:1cm) {};\n\t\\node[vertex] (v4) at (295.71:1cm) {};\n\t\\node[vertex] (v5) at (347.14:1cm) {};\n\t\\node[vertex] (v6) at (398.57:1cm) {};\n\t\\draw[thick] (v0) edge (v1) edge (v6) (v1) edge (v6) (v2) edge\n\t(v3) edge (v5) (v4) edge (v5) edge (v3);\n\t\\node at (-1,1) {$G$};\n\t\\end{scope}\n\t\n\t\\begin{scope}[xshift=3cm]\n\t\\node[vertex] (v0) at (90.00:1cm) {};\n\t\\node[vertex] (v1) at (141.43:1cm) {};\n\t\\node[vertex] (v2) at (192.86:1cm) {};\n\t\\node[vertex] (v3) at (244.29:1cm) {};\n\t\\node[vertex] (v4) at (295.71:1cm) {};\n\t\\node[vertex] (v5) at (347.14:1cm) {};\n\t\\node[vertex] (v6) at (398.57:1cm) {};\n\t\\draw[thick] (v0) edge (v1) edge (v6) (v2) edge (v1) edge (v3)\n\t(v4) edge (v3) edge (v5) (v5) edge (v6);\n\t\\node at (-1,1) {$H$};\n\t\\end{scope}\n\t\\end{tikzpicture}\n\t\\caption{$G$ and $H$ are indistinguishable by $1$-WL} %\n\t\\label{fig:indistinguishable}\n\\end{figure}\nAnother somewhat trivial limitation in the expressiveness of MPNNs is that information is only propagated along edges, and hence can never be shared between distinct connected components of a graph \\cite{BarceloKM0RS20,Keyulu18}. \nAn easy way to overcome this limitation is by adding \\emph{global readouts}, that is, permutation-invariant functions that aggregate the current states of all nodes. Throughout the paper, we therefore focus on MPNNs with global readouts, referred to as \\emph{ACR-GNNs} \\cite{BarceloKM0RS20}. \n\n\\subsection{Higher-order Graph Neural Networks}\nWe now present the main classes of higher-order GNNs.\n\n\\paragraph{Higher-order MPNNs.} The $k-$WL hierarchy has been directly emulated in GNNs, such that these models learn embeddings for \\emph{tuples} of nodes, and perform message passing between them, as opposed to individual nodes. This higher-order message passing approach resulted in models such as $k$-GNNs \\cite{MorrisAAAI19}, which have $(k-1)$-WL expressive power.\\footnote{In the literature, different versions of the Weisfeiler-Leman algorithm have inconsistent dimension counts, but are equally expressive. For example, $(k+1)$-WL and $(k+1)$-GNNs in \\cite{MorrisAAAI19} are equivalent to $k$-WL of \\cite{CaiFI92,GroheWL}. We follow the latter, as it is the standard in the  literature on graph isomorphism testing.} These models need $O(|V|^k)$ memory to run, leading to \\emph{excessive memory requirements}. \n\n\\paragraph{Invariant (resp., equivariant) graph networks.} Another class of higher-order GNNs is invariant (resp., equivariant) graph networks \\cite{MaronInvEqui19}, which represent graphs as a tensor, and implicitly pass information between nodes through invariant (resp., equivariant) computational blocks.  %\nFollowing intermediate blocks, \\emph{higher-order} tensors are typically returned, and the order of these tensors correlates directly with the expressive power of the overall model. Indeed, invariant networks \\cite{MaronFSL19}, and later equivariant networks \\cite{KerivenP19}, are shown to be universal, but with tensor orders of $O(|V|^2)$, where $|V|$ denotes the number of graph nodes. Furthermore, invariant (resp., equivariant) networks with intermediate tensor order $k$ are shown to be equivalent in power to $(k-1)$-WL \\cite{MaronBSL19}, which is strictly more expressive as $k$ increases \\cite{CaiFI92}. Therefore, universal higher-order models require \\emph{intractably-sized intermediate tensors} in practice. \n\n\\paragraph{Provably powerful graph networks.} A special class of invariant GNNs is provably powerful graph networks (PPGNs)\\cite{MaronBSL19}. PPGNs are based  on ``blocks'' of multilayer perceptrons (MLPs) and matrix multiplication, which theoretically have 2-WL expressive power, and only require memory $O(|V|^2)$ (compared to $O(|V|^3)$ for 3-GNNs). However, PPGNs theoretically require \\emph{exponentially many samples} in the number of graph nodes to learn necessary functions for 2-WL expressiveness \\cite{PunyLRGA}.\n\n\\section{MPNNs with Random Node Initialization}\n\\label{sec:RNIUniversal}\nWe present the main result of the paper, showing that RNI makes MPNNs universal, in a natural sense. \nOur work is a first positive result for the universality of MPNNs. This result is not based on a new model, but rather on random initialization of node features, which is widely used in practice, and in this respect, it also serves as a theoretical justification for models that are empirically successful.\n\n\\subsection{Universality and Invariance}\nIt may appear somewhat surprising, and even counter-intuitive, that randomly initializing node features on its own would deliver such a gain in expressiveness.\nIn fact, on the surface, random initialization no longer preserves the invariance of MPNNs, since the result of the computation of an MPNN with RNI not only depends on the structure (i.e., the isomorphism type) of the input graph, but also on the random initialization.\nThe broader picture is, however, rather subtle, as we can view such a model as computing a random variable (or as generating an output distribution), and this random variable would still be invariant. \nThis means that the outcome of the computation of an MPNN with RNI does still \\emph{not} depend on the specific representation of the input graph, which fundamentally maintains invariance. Indeed, the mean of random features, in expectation, will inform GNN predictions, and is identical across all nodes, as randomization is i.i.d. However, the variability between different samples and the variability of a random sample enable graph discrimination and improve expressiveness. Hence, in expectation, all samples fluctuate around a unique value, preserving invariance, whereas sample variance improves expressiveness. \n\n\n\nFormally, let $\\mathcal G_n$ be the class of all $n$-vertex graphs, i.e., graphs that consist of at most $n$ vertices, and let ${f:\\Gmc_n\\to\\Rbb}$. \nWe say that a randomized function $\\Xmc$ that associates with every graph $G\\in \\Gmc_n$ a random variable $\\Xmc (G)$ is an \\emph{$(\\epsilon,\\delta)$-approximation} of $f$ if for all $G\\in\\Gmc_n$ it holds that ${\\Pr\\big(|f(G)-\\Xmc(G)|\\le\\epsilon\\big)\\ge 1-\\delta}$. \nNote that an MPNN $\\Nmc$ with RNI computes such functions $\\Xmc$. If $\\Xmc$ is computed by $\\Nmc$, we say that \\emph{$\\Nmc$ $(\\epsilon,\\delta)$-approximates $f$}.\n\\begin{theorem}[Universal approximation]\n\\label{theo:uni}\n\tLet $n\\ge 1$, and let $f:\\mathcal G_n\\to\\mathbb R$ be invariant. Then, for all\n\t$\\epsilon,\\delta>0$, there is an MPNN with RNI that $(\\epsilon,\\delta)$-approximates $f$.\n\\end{theorem}\nFor ease of presentation, we state the theorem only for real-valued functions, but note that it can be extended to equivariant functions. The result can also be extended to weighted graphs, but then the function $f$ needs to be continuous.\n\n\\subsection{Result Overview}\nTo prove \\Cref{theo:uni}, we first show that MPNNs with RNI can capture arbitrary Boolean functions, by building on the result of \\cite{BarceloKM0RS20}, which states that any logical sentence in $\\LC^2$ can be captured by an MPNN (or, by an ACR-GNN in their terminology). The logic $\\LC$ is the extension of first-order predicate logic using counting quantifiers of the form $\\exists^{\\ge k}x$ for $k\\ge 0$, where $\\exists^{\\ge k}x\\phi(x)$ means that there are at least $k$ elements $x$ satisfying $\\phi$, and $\\LC^2$  is the two-variable fragment of $\\LC$.\n\nWe establish that any graph with identifying node features, which we call \\emph{individualized graphs}, can be represented by a sentence in $\\LC^2$. Then, we extend this result to sets of individualized graphs, and thus to Boolean functions mapping these sets to True, by showing that these functions are represented by a $\\LC^2$ sentence, namely, the disjunction of all constituent graph sentences. \nFollowing this, we provide a construction with node embeddings based on RNI, and show that RNI individualizes input graphs w.h.p. \nThus, RNI makes that MPNNs learn a Boolean function over individualized graphs w.h.p. Since all such functions can be captured by a sentence in $\\LC^2$, and an MPNN can capture any Boolean function, MPNNs with RNI can capture arbitrary Boolean functions. \nFinally, the result is extended to real-valued functions via a natural mapping, yielding universality.\n\n\n\n\nThe concrete implications of \\Cref{theo:uni} can be summarized as follows. First, MPNNs with RNI can distinguish individual graphs with an embedding dimensionality polynomial in the inverse of desired confidence $\\delta$ (namely, $O(n^2 \\delta^{-1})$, where $n$ is the number of graph nodes). \nSecond, universality also holds with partial RNI, and even with only one randomized dimension. \nThird, the theorem is adaptive and tightly linked to the descriptive complexity of the approximated function. That is, for a more restricted class of functions, there may be more efficient constructions than the disjunction of individualized graph sentences, and our proof does not rely on a particular construction. \nFinally, our construction provides a \\emph{logical characterization}for MPNNs with RNI, and substantiates how randomization improves expressiveness. This construction therefore also enables a more logically grounded theoretical study of randomized MPNN models, based on particular architectural or parametric choices.\n\nSimilarly to other universality results, Theorem~\\ref{theo:uni} can potentially result in very large constructions. This is a simple consequence of the generality of such results: \\Cref{theo:uni} applies to families of functions, describing problems of \\emph{arbitrary} computational complexity, including problems that are computationally hard, even to approximate.\nThus, it is more relevant to empirically verify the formal statement, and test the capacity of MPNNs with RNI relative to higher-order GNNs. \nHigher-order GNNs typically suffer from prohibitive space requirements, but this not the case for MPNNs with RNI, and this already makes them more practically viable. \nIn fact, our experiments demonstrate that MPNNs with RNI indeed combine expressiveness with efficiency in practice.\n\n\n\\section{Datasets for Expressiveness Evaluation} \n\\label{sec:dataset} \n\nGNNs are typically evaluated on real-world datasets \\cite{KKMMN2016}, which are not tailored for evaluating expressive power, as they do not contain instances indistinguishable by $1$-WL. In fact, higher-order models only marginally outperform MPNNs on these datasets \\cite{BenchmarkingGNNs}, which further highlights their unsuitability.\nThus, we developed the synthetic datasets \\EXP and \\CEXP. \\EXP explicitly evaluates GNN expressiveness, and consists of graph instances $\\{G_1, \\ldots ,G_n$, $H_1, \\ldots ,H_n\\}$, where each instance encodes  a propositional formula. The classification task is to determine whether the formula is satisfiable (\\SAT). %\nEach pair $(G_i$, $H_i)$ respects the following properties: \n(i)~$G_i$ and $H_i$ are non-isomorphic, \n(ii)~$G_i$ and $H_i$ have different \\SAT outcomes, that is, $G_i$ encodes a satisfiable formula, while $H_i$ encodes an unsatisfiable formula, \n(iii)~$G_i$ and $H_i$ are 1-WL indistinguishable, so are \\emph{guaranteed} to be classified in the same way by standard MPNNs, and \n(iv)~$G_i$ and $H_i$ are 2-WL distinguishable, so \\emph{can} be classified differently by higher-order GNNs. \n\nFundamentally, every $(G_i, H_i)$ is carefully constructed on top of a basic building block, the \\emph{core pair}. \nIn this pair, both cores are based on propositional clauses, such that one core is satisfiable and the other is not, both \\emph{exclusively} determine the satisfiability of $G_i$ (resp., $H_i$), and have graph encodings enabling all aforementioned properties. \nCore pairs and their resulting graph instances in \\EXP are \\emph{planar} and are also carefully constrained to ensure that they are 2-WL distinguishable. Thus, core pairs are key substructures within \\EXP, and distinguishing these cores is essential for a good performance. \n\n\nBuilding on \\EXP, \\CEXP includes instances with varying expressiveness requirements.  Specifically, \\CEXP is a standard \\EXP dataset where 50\\% of all satisfiable graph pairs are made 1-WL distinguishable from their unsatisfiable counterparts, only differing from these by a small number of added edges.\nHence, \\CEXP consists of 50\\% ``corrupted'' data, distinguishable by MPNNs and labelled \\Corrupt,  and 50\\% unmodified data, generated analogously to \\EXP, and requiring expressive power beyond 1-WL, referred to as \\EXPTwo. \nThus, \\CEXP contains the same core structures as \\EXP, but these lead to different \\SAT values in \\EXPTwo and \\Corrupt, which makes the learning task more challenging than learning \\EXPTwo or \\Corrupt in isolation.\n\n\\section{Experimental Evaluation} \n\nIn this section, we first evaluate the effect of RNI on MPNN expressiveness based on \\EXP, and compare against established higher-order GNNs. \nWe then extend our analysis to \\CEXP. \nOur experiments use the following models: \n\\paragraph{1-WL GCN (\\OneGCN).} A GCN with 8 distinct message passing iterations, ELU non-linearities \\cite{ClevertUH15}, 64-dimensional embeddings, and deterministic learnable initial node embeddings indicating node type. \n\tThis model is guaranteed to achieve 50\\% accuracy on \\EXP. \n\t\n\\paragraph{GCN - Random node initialization (\\GCNRNI).} A 1-GCN enhanced with RNI. We evaluate this\n\tmodel with four initialization distributions, namely, \n\tthe standard normal distribution $\\mathcal N(0,1)$ (N),\n\tthe uniform distribution over $[-1,1]$ (U), Xavier normal (XN), and the Xavier uniform distribution (XU)   \\cite{GlorotB10}. %\n\tWe denote the respective models \\GCNRNI($D$), where $D \\in \\{\\text{N,\\,U,\\,XN,\\,XU}\\}$.  \n\t\n\\paragraph{GCN - Partial RNI (GCN-$x$\\%RNI).}\n\tA \\GCNRNI~model, where $\\floor{\\frac{64x}{100}}$ dimensions are initially randomized, and all remaining dimensions are set deterministically from one-hot representation of the two input node types (literal and disjunction). We set $x$ to the extreme values 0 and 100\\%, 50\\%, as well as  near-edge cases of 87.5\\% and 12.5\\%, respectively.\n\t\n\\paragraph{PPGN.} A higher-order GNN with 2-WL expressive power \\cite{MaronBSL19}. We set up PPGN using its original implementation, and use its default configuration of eight 400-dimensional computational blocks.\n\t\n\\paragraph{\\OneTwoThreeLocal.} A higher-order GNN~\\cite{MorrisAAAI19} emulating $2$-WL on 3-node tuples. \\OneTwoThreeLocal operates at increasingly coarse granularity, starting with single nodes and rising to 3-tuples. This model uses a \\emph{connected} relaxation of 2-WL, which slightly reduces space requirements, but comes at the cost of some theoretical guarantees.  We set up \\OneTwoThreeLocal with 64-dimensional embeddings, 3 message passing iterations at level 1, 2 at level 2, and 8 at level 3.\n\t\n\\paragraph{\\ThreeGNNFull.} A GCN analog of the \n\t\\emph{full} 2-WL procedure over 3-node tuples, thus preserving all theoretical guarantees.\n\n\n\\subsection{How Does RNI Improve Expressiveness?}\n\nIn this experiment, we evaluate GCNs using different RNI settings on \\EXP, and compare with standard GNNs and higher-order models.\nSpecifically, we generate an \\EXP dataset consisting of 600 graph pairs. Then, we evaluate all models on \\EXP using 10-fold cross-validation. We train \\ThreeGNNFull for 100 epochs per fold, and all other systems for 500 epochs, and report \\emph{mean test accuracy} across all folds.\n\n\\begin{table}[t!]\n\t\\centering\n\t\\begin{tabular}{HlHc} \n\t\t\\toprule\n\t\t\\multicolumn{2}{c}{Model} & Training Accuracy (\\%) & Test Accuracy (\\%)\\\\\n\t\t\\midrule\n\t\t\\multirow{4}{*}{\\textit{\\GCNRNI}} & \\GCNRNI(U) & 97.9 $\\pm$ 0.64 & 97.3 $\\pm$ 2.55 \\\\ \n\t\t& \\textbf{\\GCNRNI(N)} & \\textbf{98.0 $\\pm$ 0.53} & \\textbf{98.0 $\\pm$ 1.85} \\\\ \n\t\t& \\GCNRNI(XU) & 97.0 $\\pm$ 0.83 & 97.0 $\\pm$ 1.43 \\\\\n\t\t& \\GCNRNI(XN) & 97.0 $\\pm$ 1.20 & 96.6 $\\pm$ 2.20 \\\\ \n\t\t\\cmidrule{1-4}\n\t\t\\multirow{3}{*}{\\textit{HO-GNNs}} & PPGN & 50.0 & 50.0\\\\ \n\t\t& \\OneTwoThreeLocal & 50.0 & 50.0\\\\ \n\t\t& \\textbf{\\ThreeGNNFull} & \\textbf{99.9 $\\pm$ 0.002} & \\textbf{99.7 $\\pm$ 0.004} \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\n\t\t\\caption{Accuracy results on \\EXP.}\n\t\\label{tab:exp1Results}\n\\end{table}%\n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\begin{subfigure}{.32\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_core}\n\t\t\\caption{\\EXP.}\n\t\t\\label{fig:modelConvergence}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.32\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_coreplus}\n\t\t\\caption{\\CEXP.}\n\t\t\\label{fig:modelConvergencePlus}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.32\\textwidth}\n\t\t\\centering\n\t\t\\input{fig\/fig_LRNEXP}\n\t\t\\caption{\\EXPTwo (\/E) and \\Corrupt (\/C).}\n\t\t\\label{fig:modelConvergenceSplit}\n\t\\end{subfigure}%\n\t\\label{fig:res}\n\t\\caption{Learning curves across all experiments for all models.}\n\\end{figure*}\nFull test accuracy results for all models are reported in Table \\ref{tab:exp1Results}, and model convergence for \\ThreeGNNFull and all \\GCNRNI~models are shown in \\Cref{fig:modelConvergence}.\nIn line with \\Cref{theo:uni}, \\GCNRNI~achieves a near-perfect performance on \\EXP, substantially surpassing 50\\%. Indeed, \\GCNRNI~models achieve above 95\\% accuracy with all four RNI distributions. %\nThis finding further supports observations made with rGNNs \\cite{SatoRandom2020}, and shows that RNI is also beneficial in settings beyond structure detection. \nEmpirically, we observed that \\GCNRNI~is highly sensitive to changes in learning rate, activation function, and\/or randomization distribution, and required delicate tuning to achieve its best performance.\n\n\nSurprisingly, PPGN does not achieve a performance above 50\\%, despite\nbeing theoretically 2-WL expressive.  \nEssentially, PPGN learns an approximation of 2-WL, based on power-sum multi-symmetric polynomials (PMP), but fails to distinguish \\EXP graph pairs, despite extensive training. This suggests that PPGN struggles to learn the required PMPs, and we could not improve these results, both for training and testing, with hyperparameter tuning. \nFurthermore, PPGN requires exponentially many data\nsamples in the size of the input graph \\cite{PunyLRGA} for learning. Hence, PPGN is likely struggling to discern between \\EXP graph pairs due to the smaller sample size and variability of the dataset. \n\\OneTwoThreeLocal also only achieves 50\\% accuracy, which can be attributed to theoretical model limitations. Indeed, this algorithm only considers 3-tuples of nodes that form a connected subgraph, thus discarding disconnected 3-tuples, where the difference between \\EXP cores lies. This further highlights the difficulty of \\EXP, as even relaxing 2-WL reduces the model to random performance.\nNote that \\ThreeGNNFull achieves near-perfect performance, as it explicitly has the necessary theoretical power, irrespective of learning constraints, and must only learn appropriate injective aggregation functions for neighbor aggregation \\cite{Keyulu18}.\n\nIn terms of convergence, we observe that \\ThreeGNNFull converges\nsignificantly faster than \\GCNRNI~models, for all randomization\npercentages. Indeed, \\ThreeGNNFull only requires about 10 epochs to\nachieve optimal performance, whereas \\GCNRNI~models all require over 100 epochs. \nIntuitively, this slower convergence of \\GCNRNI~can be attributed to a harder learning task compared to \\ThreeGNNFull: Whereas\n\\ThreeGNNFull learns from deterministic embeddings, and can naturally discern between dataset cores, \\GCNRNI~relies on RNI to discern between \\EXP data points, via an artificial node ordering. This implies that \\GCNRNI~must leverage RNI to detect structure, then subsequently learn robustness against RNI variability, which makes its learning task especially challenging.\n\n\n\nOur findings suggest that RNI practically improves  MPNN expressiveness, and makes them competitive with higher-order models, despite being less demanding computationally. Indeed, for a 50-node graph, GCN-RNI only requires 3200 parameters (using 64-dimensional embeddings), whereas \\ThreeGNNFull requires 1,254,400 parameters. Nonetheless, GCN-RNI performs comparably to \\ThreeGNNFull, and, unlike the latter, can easily scale to larger instances. This increase in expressive power, however, comes at the cost of slower convergence. Even so, RNI proves to be a promising direction for building scalable yet powerful MPNNs. \n\\subsection{How Does RNI Behave on Variable Data?} \n\nIn the earlier experiment, RNI practically improves the expressive power of GCNs over \\EXP. However, \\EXP solely evaluates expressiveness, and this leaves multiple questions open: How does RNI impact learning when data contains instances with varying expressiveness requirements, and how does RNI affect generalization on more variable datasets? We experiment with \\CEXP to explicitly address these questions. \n\n\nWe generated \\CEXP by generating another 600 graph pairs, then selecting 300 of these and modifying their satisfiable graph, yielding \\Corrupt.\n\\CEXP is well-suited for holistically evaluating the efficacy of RNI, as it evaluates the contribution of RNI on \\EXPTwo conjointly with a second learning task on \\Corrupt involving very similar core structures, and assesses the effect of different randomization degrees on overall and subset-specific model performance. \n\nIn this experiment, we train \\GCNRNI~(with varying randomization degrees) and \\ThreeGNNFull on \\CEXP, and compare their accuracy. For \\GCNRNI, we observe the effect of RNI on learning \\EXPTwo and \\Corrupt, and the interplay between these tasks. %\nIn all experiments, we use the normal distribution for RNI, given its strong performance in the earlier experiment.\n\n\n\nThe learning curves of all \\GCNRNI~ and \\ThreeGNNFull on \\CEXP are shown in \\Cref{fig:modelConvergencePlus}, and the same curves for the \\EXPTwo and \\Corrupt subsets are shown in \\Cref{fig:modelConvergenceSplit}. As on \\EXP, \\ThreeGNNFull converges very quickly, exceeding 90\\% test accuracy within 25 epochs on \\CEXP. By contrast, \\GCNRNI, for all randomization levels, converges much slower, around after 200 epochs, despite the small size of input graphs ($\\sim$70 nodes at most). Furthermore, fully randomized \\GCNRNI~ performs worse than partly randomized \\GCNRNI, particularly on \\CEXP, due to its weak performance on \\Corrupt.\n\nFirst, we observe that partial randomization significantly improves performance. This can clearly be seen on \\CEXP, where GCN-12.5\\%RNI and GCN-87.5\\%RNI achieve the best performance, by far outperforming GCN-RNI, which struggles on \\Corrupt.\nThis can be attributed to having a better inductive bias than a fully randomized model.  Indeed, GCN-12.5\\%RNI has mostly deterministic node\nembeddings, which simplifies learning over \\Corrupt. This also applies to GCN-87.5\\%RNI, where the number of deterministic dimensions, though small, remains sufficient.  Both models also benefit from randomization for \\EXPTwo, similarly to a fully randomized GCN.\nGCN-12.5\\%RNI and GCN-87.5\\%RNI effectively achieve the best of both worlds on \\CEXP, leveraging inductive bias from deterministic node embeddings, while harnessing the power of RNI to perform strongly on \\EXPTwo. This is best shown in \\Cref{fig:modelConvergenceSplit}, where standard GCN fails to learn \\EXPTwo, fully randomized GCN-RNI struggles to learn \\Corrupt, and the semi-randomized GCN-50\\%RNI achieves perfect performance on both subsets. We also note that partial RNI, when applied to several real datasets, where 1-WL power is sufficient, did not harm performance \\cite{SatoRandom2020}, and thus at least preserves the original learning ability of MPNNs in such settings. Overall, these are surprising findings, which suggest that MPNNs can viably improve across all possible data with partial and even small amounts of randomization. \n\nSecond, we  observe that the fully randomized \\GCNRNI~ performs substantially worse than its partially randomized counterparts. Whereas fully randomized \\GCNRNI~ only performs marginally worse on \\EXP (cf. \\Cref{fig:modelConvergence}) than partially randomized models, this gap is very large on \\CEXP, primarily due to \\Corrupt. \nThis observation concurs with the earlier idea of inductive bias: Fully randomized \\GCNRNI~ loses all node type information, which is key for \\Corrupt, and therefore struggles. Indeed, the model fails to achieve even 60\\% accuracy on \\Corrupt, where other models are near perfect, and also relatively struggles on \\EXPTwo, only reaching 91\\% accuracy and converging slower. \n\nThird, all \\GCNRNI~models, at all randomization levels, converge significantly slower than \\ThreeGNNFull on both \\CEXP and \\EXP. However, an interesting phenomenon can be seen on \\CEXP: All \\GCNRNI ~models fluctuate around 55\\% accuracy within the first 100 epochs, suggesting a struggle jointly fitting both \\Corrupt and \\EXPTwo, before they ultimately improve. This, however, is not observed with \\ThreeGNNFull. Unlike on \\EXP, randomness is not necessarily beneficial on \\CEXP, as it can hurt performance on \\Corrupt.\nHence, RNI-enhanced models must additionally learn to isolate  deterministic dimensions for \\Corrupt, and randomized dimensions for \\EXPTwo. \nThese findings consolidate the earlier observations made on \\EXP, and highlight that the variability and slower learning for RNI also hinges on the complexity of the input dataset.  \n\nFinally, we observe that both fully randomized \\GCNRNI, and, surprisingly, \\OneGCN, struggle to learn \\Corrupt relative to partially randomized \\GCNRNI. We also observe that \\OneGCN does not ``struggle'', and begins improving consistently from the start of training. \nThese observations can be attributed to key conceptual , but very distinct hindrances impeding both models. For \\OneGCN, the model is jointly trying to learn both \\EXPTwo and \\Corrupt, when it provably cannot fit the former. This joint optimization severely hinders \\Corrupt learning, as data pairs from both subsets are highly similar, and share identically generated UNSAT graphs. \nHence, \\OneGCN, in attempting to fit SAT graphs from both subsets, knowing it cannot distinguish \\EXPTwo pairs, struggles to learn the simpler difference in \\Corrupt pairs. For \\GCNRNI, the model discards key type information, so must only rely on structural differences to learn \\Corrupt, which impedes its convergence. All in all, this further consolidates the promise of partial RNI as a means to combine the strengths of both deterministic and random features.\n\n\n\n\\section{Related Work}\n\\label{sec:rw}\n\nMPNNs have been enhanced with RNI \\cite{SatoRandom2020}, such that the model trains and runs with partially randomized initial node features. These models, denoted rGNNs, are shown to near-optimally approximate solutions to specific combinatorial optimization problems, and can distinguish between 1-WL indistinguishable graph pairs based on fixed local substructures. Nonetheless, the precise impact of RNI on GNNs for learning arbitrary functions over graphs remained open. Indeed, rGNNs are only shown to admit parameters that can detect a \\emph{unique, fixed} substructure, and thus tasks requiring \\emph{simultaneous} detection of multiple combinations of structures, as well as problems having no locality or structural biases, are not captured by the existing theory. \n\nOur work improves on Theorem 1 of \\cite{SatoRandom2020}, and shows \\emph{universality} of MPNNs with RNI. Thus, it shows that arbitrary real-valued functions over graphs can be learned by MPNNs with RNI. Our result is distinctively based on a logical characterization of MPNNs, which allows us to link the size of the MPNN with the descriptive complexity of the target function to be learned.\nEmpirically, we highlight that the power of RNI in a significantly more challenging setting, using a target function (\\SAT) which does not rely on local structures, is \\emph{hard} to approximate. %\n\nSimilarly to RNI, random pre-set color features have been used to disambiguate between nodes \\cite{DasoulasSSV20}. This approach, known as CLIP, introduces randomness to node representations, but explicitly makes graphs distinguishable by construction. By contrast, we study random features produced by RNI, which (i)~are not designed a priori to distinguish nodes, (ii)~do not explicitly introduce a fixed underlying structure, and (iii)~yield potentially infinitely many representations for a single graph. In this more general setting, we nonetheless show that RNI adds expressive power to distinguish nodes with high probability, leads to a universality result, and performs strongly in challenging problem settings. \n\n\\section{Summary and Outlook}\nWe studied the expressive power of MPNNs with RNI, and showed that these models are universal and  preserve MPNN invariance in expectation. \nWe also empirically evaluated these models on carefully designed datasets, and observed that RNI improves their learning ability, but slows their convergence. \nOur work delivers a theoretical result, supported by practical insights, to quantify the effect of RNI on GNNs. An interesting topic for future work is to study whether polynomial functions can be captured via efficient constructions; see, e.g., \\cite{GroheLogicGNN} for related open problems.\n\n\\section*{Acknowledgments}\nThis work was supported by the Alan Turing Institute under the UK EPSRC grant EP\/N510129\/1, by the AXA Research Fund, and by the EPSRC grants EP\/R013667\/1 and EP\/M025268\/1. Ralph Abboud is funded by the Oxford-DeepMind Graduate Scholarship and the Alun Hughes Graduate Scholarship. Experiments  were conducted on the Advanced Research Computing (ARC) cluster administered by the University of Oxford.\n\n\n\\bibliographystyle{named}\n{\\small\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nFor a graph $G=(V,E)$ with vertex set $V$ and edge set $E$, a \\textit{binary edge-labeling} is a surjection $f : E \\to \\{ 0,1 \\}$. Let $i \\in \\{ 0,1 \\}$. An edge labeled $i$ is called an \\textit{$i$-edge} and let $e(i)$ denote the total number of $i$-edges in $G$ with respect to a binary edge-labeling $f$. In the case where $|e(1)-e(0)| \\leq 1$, a binary edge-labeling is called \\textit{edge-friendly}. Call the number of $i$-edges incident with a vertex $v$ the \\textit{$i$-degree} of $v$, denoted $\\deg_i(v)$, so that the degree of $v$ is $\\deg(v) = \\deg_1(v) + \\deg_0(v)$. An edge-friendly labeling of $G$ will induce a (possibly partial) \\textit{vertex-labeling} where a vertex $v$ is labeled $1$ when $\\deg_1(v) > \\deg_0(v)$, is labeled $0$ when $\\deg_0(v) > \\deg_1(v)$, and is unlabeled when $\\deg_1(v) = \\deg_0(v)$. Call a vertex labeled $i$ an \\textit{$i$-vertex} and let $v(i)$ denote the total number of $i$-vertices in $G$ with respect to an edge-friendly labeling $f$. The \\textit{edge-balanced index set} of $G$ is defined as\n\\begin{align*}\nEBI(G) = \\big\\{ |v(1)-v(0)|: \\text{over all edge-friendly labelings of $G$} \\big\\}.\n\\end{align*} \nMore information about graph labelings can be found in Gallian's dynamic survey~\\cite{GallianYYYY}.\n\nThe idea of a balanced labeling was introduced in 1992 by Lee, Liu, and Tan~\\cite{LLT1992}. In 1995, Kong and Lee provided results concerning edge-balanced graphs~\\cite{KL1995}. In~\\cite{KWL2009}, Kong, Wang, and Lee introduced the problem of finding the $EBI$ of complete bipartite graphs by solving the cases where the smaller part has cardinality 1, 2, 3, 4, or 5, and the special case where both parts have the same cardinality, but left all other cases open. In~\\cite{KMPR2014}, Krop, Minion, Patel, and Raridan concluded the edge-balanced index set problem for complete bipartite graphs with both parts of odd cardinality. \n\nA natural next step in the problem is to find the $EBI$ of the complete bipartite graphs where at least one part has even cardinality. In this paper, we conclude the problem for complete bipartite graphs where the larger part is of odd cardinality and the smaller is of even cardinality.\n\nFor positive integers $a,b$, where $a \\leq b$, let $[a]$ denote the set of integers $\\{ 1, \\dots, a \\}$ and let $[a,b]$ denote the set of integers $\\geq a$ but $\\leq b$; in the case where $a=1$, $[a]=\\{1\\}$, and in the case where $a=b$, $[a,a]=\\{a\\}$.  \n\nThroughout the rest of this paper, let $K_{m,n}$ be a complete bipartite graph with part $A$ of cardinality $m$ and part $B$ of cardinality $n$, where $m$ is odd, $n$ is even, and $m>n \\geq 2$. Let $q$ be the quotient when $m$ is divided by $\\frac{n}{2}+1$ and let $r$ be the remainder. If $r=0$, then we further partition $A$ into sets $A_i$ and denote the vertices of $A_i$ by $v_j^i$, where $i \\in [q]$ and $j \\in \\left[ \\frac{n}{2}+1 \\right]$. If $r \\geq 1$, then we partition $A$ as in the $r=0$ case with the addition of another partition $A_*$ having vertices denoted $v_i^*$, where $i \\in [r]$. Denote the vertices of $B$ by $u_i$, where $i \\in [n]$. \n\nIn the case where $n=2$, we have $q = \\frac{m-1}{2} \\geq 1$ and $r=1$. If $n \\geq 4$ and $q=1$, then $m = \\frac{n}{2}+1+r$, where $r \\geq 2$; for, if $r=0,1$, then $m \\leq \\frac{n}{2}+2 \\leq n$, a contradiction. In fact, if $n \\geq 4$, $q=1$, and $r \\geq 2$ is even, say $r=2j$ for some $j \\geq 1$, then $\\frac{n}{2}$ is even since $m$ is odd. That is, $n = 4k$ for some $k \\geq 1$. We have $m>n$ if and only if $j \\geq k$ and $r \\leq \\frac{n}{2}$ if and only if $j \\leq k$; hence, $j=k$ and $ r = \\frac{n}{2}$. Similarly, if $n \\geq 4$, $q=1$, and $r \\geq 3$ is odd, we have $r = \\frac{n}{2}$. Thus, for $n \\geq 4$, if $q=1$, then $m=n+1$ and $r = \\frac{n}{2}$. \n\nAny edge-friendly labeling of $K_{m,n}$ has $e(1) = e(0) = \\frac{mn}{2}$. Every vertex in $A$ has even degree so each vertex in this part is a $1$-vertex, a $0$-vertex, or is unlabeled, and every vertex in $B$ has odd degree so each is labeled either $1$ or $0$. Let $v_A(i)$ and $v_B(i)$ represent the number of $i$-vertices in $A$ and $B$, respectively, so that $v(i) = v_A(i)+v_B(i)$, where $i=0,1$. Without loss of generality, we assume that $v_A(1) \\geq v_A(0)$ and $v_B(1) \\geq v_B(0)$, which implies that $v(1) \\geq v(0)$; with this assumption, every element in $EBI(K_{m,n})$ can be computed as $v(1)-v(0)$. Note that not every vertex in $B$ can be a $1$-vertex. If every vertex in $B$ were a $1$-vertex, then the number of $1$-edges incident with these vertices would be at least $n\\left(\\frac{m+1}{2}\\right) > \\frac{mn}{2}$, a contradiction. However, it is possible to have only one $0$-vertex in $B$ since $(n-1)\\left(\\frac{m+1}{2}\\right) < \\frac{mn}{2}$. We have similar results for the vertices in $A$.\n\n\\section{Two Particular Edge-Friendly Labelings}\n\\label{sec:2-e-f}\n\nIn this section, we describe two edge-friendly labelings $f,f'$ of $K_{m,n}$. The labeling $f$ will show that $0 \\in EBI(K_{m,n})$ and $f'$ will show that $n-2 \\in EBI(K_{m,n})$. When $n=2$, the labelings $f$ and $f'$ give the same index $0$, so we do not construct $f'$ when $n=2$. For larger values of $n$, we obtain two distinct edge-friendly labelings that give different indices. \n\n\\subsection{Initializing the Labelings $f,f'$}\n\\label{subsec:initialize-f-f'}\n\nSet $f(v_1^i u_j)=f'(v_1^i u_j) = 1$, where $i \\in [q]$ and $j \\in \\left[ \\frac{n}{2} \\right]$, and label the remaining edges incident with each vertex $v_1^i$ by $0$. Then $v_1^i$ is unlabeled. If $r \\geq 1$, set $f(v_1^* u_j)=f'(v_1^* u_j) = 1$, where $j \\in \\left[ \\frac{n}{2} \\right]$, and label the remaining edges incident with vertex $v_1^*$ by $0$, so that $v_1^*$ is unlabeled.\n\n\\subsection{The Labeling $f$}\n\\label{subsec:f}\n\nAfter initializing $f$ as described above, we continue to label the edges of $K_{m,n}$ as follows to create an edge-friendly labeling $f$: For $i \\in [q]$, $j \\in \\left[ 2, \\frac{n}{2}+1 \\right]$, and $k \\in \\left[ \\frac{n}{2} \\right]$, set $f(v_j^i u_k) = 0$ and label the remaining edges incident with each vertex $v_j^i$ by $1$, so that $v_j^i$ is unlabeled. If $r \\geq 2$, for $j \\in [2, r]$ and $k \\in \\left[ \\frac{n}{2} \\right]$, set $f(v_j^* u_k) = 0$ and label the remaining edges incident with vertex $v_j^*$ by $1$, so that $v_j^*$ is unlabeled. Note that under $f$, all edges in the graph have been labeled either $0$ or $1$ and $f$ is edge-friendly by construction. \n\nAll vertices in $A$ are unlabeled. If $n=2$, then $\\deg_1(u_1) = \\deg_0(u_2) = q+1$ and $\\deg_0(u_1) = \\deg_1(u_2) = q$, so vertex $u_1$ is a $1$-vertex and $u_2$ is a $0$-vertex. For $n \\geq 4$ and $i \\in \\left[ \\frac{n}{2} \\right]$, we have $\\deg_0(u_i) = \\deg_1(u_{i+\\frac{n}{2}}) > \\deg_1(u_i) = \\deg_0(u_{i+\\frac{n}{2}})$, so vertex $u_i$ is a $0$-vertex and $u_{i+\\frac{n}{2}}$ is a $1$-vertex. Thus, $v(1)=v(0)=\\frac{n}{2}$, which gives $0 \\in EBI(K_{m,n})$. \n\n\\subsection{The Labeling $f'$}\n\\label{subsec:f'}\n\nBy previous remarks, we know that it is possible to have an edge-friendly labeling of $K_{m,n}$ where one vertex of $B$, say $u_n$, is a $0$-vertex and the remaining vertices $u_1, \\dots, u_{n-1}$ are $1$-vertices. \n\nAfter initializing $f'$ as described above, we continue to label the edges of $K_{m,n}$, where $n \\geq 4$, as follows to create an edge-friendly labeling $f'$. We begin by labeling the edges (which are not already labeled) incident with vertex $u_n$ by $0$ so that $\\deg_0(u_n)=m$ and $u_n$ is a $0$-vertex. The remaining edge labels will be determined based on the parity of $i \\in [q]$. For $i \\in [q]$ with $i$ odd, $j \\in \\left[2,\\frac{n}{2}+1\\right]$, $k \\in \\left[\\frac{n}{2}-1\\right]$, and $\\ell=j+k-2$, set $f'(v_j^i u_\\ell)=0$ and label the remaining unlabeled edges incident with each vertex $v_j^i$ by $1$, so that $v_j^i$ is unlabeled. For $i \\in [q]$ with $i$ even, $j \\in \\left[2,\\frac{n}{2}\\right]$, $k \\in \\left[\\frac{n}{2}\\right]$, and $\\ell = j+k-1$, set $f'(v_j^i u_\\ell)=1$ and label the remaining unlabeled edges incident with each vertex $v_j^i$ by $0$, so that $v_j^i$ is unlabeled. For $i \\in [q]$ with $i$ even, $j = \\frac{n}{2}+1$, $k \\in \\left[\\frac{n}{2}-1\\right]$, and $\\ell=j+k-1$, set $f'(v_j^i u_\\ell)=1$ and $f'(v_j^i u_1)=1$, and label the remaining unlabeled edges incident with each vertex $v_j^i$ by $0$, so that $v_j^i$ is unlabeled. If $r \\geq 2$, then similar to the even $i$ case, we set $f'(v_j^* u_\\ell)=1$, where $j \\in \\left[2,r\\right]$, $k \\in \\left[\\frac{n}{2}\\right]$, and $\\ell = j+k-1$, and label the remaining unlabeled edges incident with each vertex $v_j^*$ by $0$, so that $v_j^*$ is unlabeled. Under $f'$, all edges in the graph have been labeled either $0$ or $1$, and since $e(0)=e(1)$, the constructed labeling $f'$ is edge-friendly. \n\nAll vertices in $A$ are unlabeled, vertex $u_n$ is a $0$-vertex, and $\\deg_1(u_i) > \\deg_0(u_i)$ for $i \\in [n-1]$, so vertex $u_i$ is a $1$-vertex. Thus, $v(1)=n-1$ and $v(0)=1$, which gives $n-2 \\in EBI(K_{m,n})$. \n\n\\section{Main Result}\n\\label{sec:main-result}\n\nWe are now ready to prove the following:\n\\begin{thm}\n\\label{thm:max-EBI-odd-even}\nLet $K_{m,n}$ be a complete bipartite graph with parts of cardinality $m$ and $n$, where $m$ is odd, $n$ is even, and $m>n \\geq 2$. Then $EBI(K_{m,2}) = \\{0\\}$. For $n\\geq 4$, let $q$ be the quotient when $m$ is divided by $\\frac{n}{2}+1$ and let $r$ be the remainder. Then\n\\begin{align}\nEBI(K_{m,n}) =\n\\begin{cases}\n\\left\\{ 0,1, \\dots, m+n-2q-2 \\right\\}, &\\text{if~$r = 0$}, \\\\\n\\left\\{ 0,1, \\dots, m+n-2q-3 \\right\\}, &\\text{if~$r = 1$}, \\\\\n\\left\\{ 0,1, \\dots, m+n-2q-4 \\right\\}, &\\text{if~$r \\geq 2$}.\n\\end{cases}\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nLet $n=2$. Then the labeling $f$ given in Section~\\ref{subsec:f} shows that $0 \\in EBI(K_{m,2})$. To see that $0$ is the only index in the edge-balanced index set of $K_{m,2}$, consider switches on pairs of edges incident with a vertex $u \\in B$, say $e=uv$ and $e'=uv'$, where $f(e)=1$ and $f(e')=0$. Such switches will not alter the edge-friendliness of the labeling, nor alter the label on vertex $u$, but each switch will change the unlabeled vertex $v$ to a $0$-vertex and the unlabeled vertex $v'$ to a $1$-vertex. No matter how many switches are performed (up to labeling all but one vertex in part $A$), we will always have $v_A(1)=v_A(0)$ and $v_B(1)=v_B(0)=1$. It is impossible to label all the vertices in part $A$, so $EBI(K_{m,2}) = \\{ 0 \\}$.\n\n\nFor the remainder of the proof, let $n \\geq 4$, let $q$ be the quotient when $m$ is divided by $\\frac{n}{2}+1$, and let $r$ be the remainder.\n\n\nConsider the labeling $f'$ given in Section~\\ref{subsec:f'}, which provides an edge-friendly labeling of $K_{m,n}$ and shows that $n-2 \\in EBI(K_{m,n})$. We perform edge label switches on pairs of $0$-edges and $1$-edges incident with the same vertex in part $B$, noting that such a switch will not alter the edge-friendliness of the labeling. For $i \\in [q]$, switch the label on edge $v_1^i u_1$ with the label on edge $u_1 v_2^i$. These edge label switches will not change the label on vertex $u_1$, but will cause $v_1^i$ to change from an unlabeled vertex to a $0$-vertex and will cause $v_2^i$ to change from unlabeled vertex to a $1$-vertex. After performing these edge label switches, we note that the number of $1$-vertices increased by $q$ and the number of $0$-vertices increased by $q$, so we still have $n-2 \\in EBI(K_{m,n})$. Continuing our $\\{0,1\\}$-edge-pair switches, for $i \\in [q]$ and $j \\in \\left[2,\\frac{n}{2}\\right]$, switch the label on edge $v_1^i u_j$ with the label on edge $u_j v_{j+1}^i$. Each such switch increases the edge-balanced index by one. Moreover, after all of these $\\{0,1\\}$-edge-pair switches, we have that $\\deg_0(v_1^i)=n$, implying that each $v_1^i$ is a $0$-vertex with no incident $1$-edges, and that $\\deg_1(v_j^i) = \\frac{n}{2}+1$, where $j \\in \\left[2,\\frac{n}{2}+1\\right]$, implying that each $v_j^i$ is a $1$-vertex (but just barely). Thus, all vertices in $A_i$, where $i \\in [q]$, are labeled either $0$ or $1$, and we have attained each index from $n-2$ to $n-2+q\\left(\\frac{n}{2}-1\\right)=m+n-2q-2-r$ in the edge-balanced index set. If $r=0$, then we are done as we cannot increase the edge-balanced index further. That is, if $r=0$, then the maximal index in the edge-balanced index set is $m+n-2q-2$. Similarly, if $r=1$, then we do not have any extra $1$-edges incident with vertices $v_j^i$, where $i \\in [q]$ and $j \\in \\left[2,\\frac{n}{2}+1\\right]$, that could be used to change vertex $v_1^*$ into a $1$-vertex, so we cannot increase the edge-balanced index further. That is, for $r=1$, the maximal index is $m+n-2q-3$. For values of $r \\geq 2$, we may perform additional $\\{0,1\\}$-edge-pair switches to force all vertices in part $A$ to be labeled, increasing the number of $0$-vertices in $A$ by one and the number of $1$-vertices by $r-1$. In particular, for $j \\in [r-1]$, switch the label on edge $v_1^* u_j$ with the label on edge $u_j v_{j+1}^*$. Then $v_1^*$ is a $0$-vertex and $v_{j+1}^*$ is a $1$-vertex. In this case, we have that $v_A(0)=q+1$ and $v_A(1)=m-q-1$, which means that the maximal index in the edge-balanced index set is $v(1)-v(0)=v_A(1)+v_B(1)-v_A(0)-v_B(0)=m-q-1+n-1-(q+1)-1=m+n-2q-4$. \n\n\nNow, consider the labeling $f$ given in Section~\\ref{subsec:f}. Performing the same $\\{0,1\\}$-edge-pair switches described above, we find that we are able to achieve subsets of the edge-balanced index set based on the value of $r$. If $r=0$, then we achieve the indices $\\{ 0,1, \\dots, m-2q \\}$. If $r=1$, then we achieve the indices $\\{ 0,1, \\dots, m-2q-1 \\}$. If $r \\geq 2$, then we achieve the indices $\\{ 0,1, \\dots, m-2q-2 \\}$.  \n\n\nFor the last part of the proof, note that if $r=0$, then we have that $\\{ 0,1, \\dots, m-2q \\} \\cup \\{ n-2, \\dots, m+n-2q-2 \\} = \\{ 0,1, \\dots, m+n-2q-2 \\}$, since $q = \\frac{2m}{n+2}$ and $m > n+1$ implies $m-2q = \\frac{m(n-2)}{n+2}> n-3 + \\frac{4}{n+2} > n-3$. That is, if $r=0$, then $EBI(K_{m,n}) = \\{ 0,1, \\dots, m+n-2q-2 \\}$. Now, if $r=1$, then $q = \\frac{2m-2}{n+2}$ and $m \\geq n+3$ implies $m-2q-1 = \\frac{(m-1)(n-2)}{n+2} \\geq n-2$, and $EBI(K_{m,n}) = \\{ 0,1, \\dots, m+n-2q-3 \\}$. Finally, for values of $r \\geq 2$, we consider three cases: (i)~$m=n+1$, (ii)~$m=n+3$, and (iii)~$m \\geq n+5$. For case~(i), if $m=n+1$, then $q=1$ and $m-2q-2=n-3$. For case~(ii), if $m=n+3$, then $q=2$ and $m-2q-2=n-3$. For case~(iii), if $m \\geq n+5$, then $m-2q-2 = \\frac{(m-2)(n-2)}{n+2} + \\frac{4r-8}{n+2} \\geq \\frac{(n+3)(n-2)}{n+2} > n-2$. Thus, if $r \\geq 2$, then $m-2q-2 \\geq n-3$ and $EBI(K_{m,n}) = \\{ 0,1, \\dots, m+n-2q-4 \\}$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nEnd of 1998 the concept of ``Grid computing\" was introduced in the monograph ``The Grid: Blueprint for a New\nComputing Infrastructure\" by I. Foster and C. Kesselman \\cite{Foster:1998:GBN}. Two years earlier, in 1997, the\ndevelopment of the UNICORE - Uniform Interface to Computing Resources - system was initiated to enable German\nsupercomputer centers to provide their users with a {\\em seamless, secure, and intuitive access} to their\nheterogeneous computing resources. Like in the case of the Globus Toolkit\\textsuperscript{\\textregistered}\n\\cite{Foster:1997:GMI} UNICORE was started before ``Grid Computing\" became the accepted new paradigm for\ndistributed computing.\n\nThe UNICORE vision was proposed to the German Ministry for Education and Research (BMBF) and received funding. A\nfirst prototype was developed in the UNICORE\\footnote{funded in part by BMBF grant 01 IR 703, duration: August\n1997 - December 1999} project \\cite{Erwin:2000:UNI}. The foundations for the current production version were laid\nin the follow-up project UNICORE Plus\\footnote{funded in part by BMBF grant 01 IR 001 A-D, duration: January 2000\n- December 2002} \\cite{Erwin:2003:UNI}, which was successfully completed in 2002. Since then UNICORE was used in\noperation at German supercomputing centers and became a solid basis for numerous European projects. In this paper\nwe will describe the evolution of UNICORE from a prototype software developed in research projects to a Grid\nmiddleware used today in the daily operation of production Grids.\n\nAlthough already set out in the initial UNICORE project proposal in 1997, the goals and objectives of the UNICORE\ntechnology are still valid:\n\\bi\n\\item Foremost, the aim of UNICORE is to hide the rough edges resulting from different hardware architectures, vendor\nspecific operating systems, incompatible batch systems, different application environments, historically grown\ncomputer center practices, naming conventions, file system structures, and security policies -- just to name the\nmost obvious.\n\\item Equally, security is a constituent part of UNICORE's design relying on X.509 certificates for the\nauthentication of users, servers, and software, and the encryption of the communication over the internet.\n\\item Finally, UNICORE is usable by scientists and engineers without having to study vendor or site-specific\ndocumentation. A Graphical User Interface (GUI) is available to assist the user in creating and managing jobs.\n\\ei\n\nAdditionally, several basic conditions are met by UNICORE: the Grid middleware supports operating systems and\nbatch systems of all vendors present at the partner sites. In 1997 these were for instance large Cray T3E systems,\nNEC and Hitachi vector machines, IBM SP2s, and smaller Linux clusters. Nowadays the spectrum is even broader, of\ncourse with modern hardware, such as IBM p690 systems. The deployed software has to be non-intrusive, so that it\ndoes not require changes in the computing centers hard- and\/or software infrastructure. Maintaining site autonomy\nis still a major issue in Grid computing, when aspects of acceptability and usability in particular from the\nsystem administrator's point of view are addressed. In addition to UNICORE's own security model, site-specific\nsecurity requirements (\\eg firewalls) are supported.\n\nNear the end of the initial funding period of the UNICORE Plus project, a working prototype was available, which\nshowed that the initial concept works. By combining innovative ideas and proven components over the years, this\nfirst prototype evolved to a {\\em vertically integrated} Grid middleware solution.\n\nThe remainder of this paper is structured as follows. In section 2 the architecture of UNICORE and its core\nfeatures are described. European funded projects, which use UNICORE as a basis for their work are described in\nSection 3, and in Section 4 the usage of UNICORE in production is described. Section 5 gives an outlook on the\nfuture development of UNICORE. The paper closes with conclusions and acknowledgements.\n\n\n\n\\section{The Architecture of UNICORE}\n\\label{sec:arch}\n\\figref{fig:UNICORE-arch} shows the layered Grid architecture of UNICORE consisting of user, server and target\nsystem tier \\cite{Romber:2002:UGI}. The implementation of all components shown is realized in Java. UNICORE meets\nthe Open Grid Services Architecture (OGSA) \\cite{Foster:2003:TPG} concept following the paradigm of 'Everything\nbeing a Service'. Indeed, an analysis has shown that the basic ideas behind UNICORE already realizes this paradigm\n\\cite{Snelling:2002:UGI,Snelling:2003:UOG}.\n\n\\subsection{User Tier}\n\\label{sec:arch-userT}\nThe UNICORE Client provides a graphical user interface to exploit the entire set of services offered by the\nunderlying servers. The client communicates with the server tier by sending and receiving Abstract Job Objects\n(AJO) and file data via the UNICORE Protocol Layer (UPL) which is placed on top of the SSL protocol. The AJO is\nthe realization of UNICORE's job model and central to UNICORE's philosophy of abstraction and seamlessness. It\ncontains platform and site independent descriptions of computational and data related tasks, resource information\nand workflow specifications along with user and security information. AJOs are sent to the UNICORE Gateway in form\nof serialized and signed Java objects, followed by an optional stream of bytes if file data is to be transferred.\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth]{UNICORE-architecture.eps}\n\\caption{The UNICORE architecture.} \\label{fig:UNICORE-arch}\n\\end{figure}\n\nThe UNICORE client assists the user in creating complex, interdependent jobs that can be executed on any UNICORE\nsite (Usite) without requiring any modifications. A UNICORE job, more precisely a job group, may recursively\ncontain other job groups and\/or tasks and may also contain dependencies between job groups to generate job\nworkflows. Besides the description of a job as a set of one or more directed a-cyclic graphs, conditional and\nrepetitive execution of job groups or tasks are also included. For the monitoring of jobs, their status is\navailable at each level of recursion down to the individual task. Detailed log information is available to analyze\npotential error conditions. At the end of the execution of the job it is possible to retrieve the {\\sf stdout} and\n{\\sf stderr} output of the job. Data management functions like import, export, and transfer are available through\nthe GUI as explicit tasks. This allows the user to specify data transfer from one target system to another (\\eg\nfor workflows), from or to the local workstation before or after the execution of a job, or to store data\npermanently in archives.\n\nThe previously described features already provide an effective tool to use resources of different computing\ncenters both for capacity or capability computing, but many scientists and engineers use application packages. For\napplications without a graphical user interface, a tool kit simplifies the development of a custom built UNICORE\nplug-in. Over the years many plug-ins were developed, so that plug-ins already exist for many standard scientific\napplications, as \\eg for CPMD (Car-Parrinello Molecular Dynamics) \\cite{Huber:2001:SCP}, Fluent or MSC Nastran.\n\n\\subsection{Server Tier}\n\\label{sec:arch-serverT}\nThe server tier contains the Gateway and the Network Job Supervisor (NJS). The Gateway controls the access to a\nUsite and acts as the secure entry point accepting and authenticating UPL requests. A Usite identifies the\nparticipating organization (\\eg a supercomputing center) to the Grid with a symbolic name that resolves into the\nURL of the Gateway. An organization may be part of multiple Grids offering the same or different resources to\ndifferent communities. The Gateway forwards incoming requests to the underlying Network Job Supervisor (NJS) of a\nvirtual site (Vsite) for further processing. The NJS represents resources with a uniform user mapping scheme and\nno boundaries like firewalls between them.\n\nA Vsite identifies a particular set of resources at a Usite and is controlled by a NJS. A Vsite may consist of a\nsingle supercomputer, \\eg a IBM p690 System with LoadLeveler, or a Linux cluster with PBS as resource management\nsystem. The flexibility of this concept supports different system architectures and gives the organization full\ncontrol over its resources. Note that, there can be more than one Vsite inside each USite as depicted in\n\\figref{fig:UNICORE-arch}.\n\nThe NJS is responsible for the virtualization of the underlying resources by mapping the abstract job on a\nspecific target system. This process is called ``incarnation\" and makes use of the Incarnation Database (IDB).\nSystem-specific data are stored in the IDB describing the software and hardware infrastructure of the system.\nAmong others, the available resources like software, incarnation of abstract commands (standard UNIX command like\nrm, cp, ...) and site-specific administrative information are stored. In addition to the incarnation the NJS\nprocesses workflow descriptions included in an AJO, performs pre- and post-staging of files and authorizes the\nuser via the UNICORE User Database (UUDB). Typically the Gateway and NJS are running on dedicated secure systems\nbehind a firewall, although the Gateway could be placed outside a firewall or in a demilitarized zone.\n\n\\subsection{Target System Tier}\n\\label{sec:arch-targetsysT}\nThe Target System Interface (TSI) implements the interface to the underlying supercomputer with its resource\nmanagement system. It is a stateless daemon running on the target system and interfacing with the local resource\nmanager realized either by a batch system like PBS \\cite{openPBS:url} or CCS \\cite{Hovestadt:2003:SHR}, a batch\nsystem emulation on top of \\eg Linux, or a Grid resource manager like Globus' GRAM\n\\cite{GRAM:url,Menday:2003:GEU}.\n\n\\subsection{Single Sign-On}\n\\label{sec:arch-signon}\nThe UNICORE security model relies on the usage of permanent X.509 certificates issued by a trusted Certification\nAuthority (CA) and SSL based communication across `insecure' networks. Certificates are used to provide a single\nsign-on in the client. The client unlocks the user's keystore when it is first started, so that no further\npassword requests are handed to the user. All authentication and authorization is done on the basis of the user\ncertificate. At each UNICORE site user certificates are mapped to local accounts (standard UNIX uid\/gid), which\nmay be different at each site, due to existing naming conventions. The sites retain full control over the\nacceptance of users based on the identity of the individual -- the distinguished name -- or other information that\nmight be contained in the certificate. UNICORE can handle multiple user certificates, \\ie it permits a client to\nbe part of multiple, disjoint Grids. It is also possible to specify project accounts in the client allowing users\nto select different accounts for different projects on one execution system or to assume different roles with\ndifferent privileges.\n\nThe private key in the certificate is used to sign each job and all included sub-jobs during the transit from the\nclient to sites and between sites. This protects against tampering while the job is transmitted over insecure\ninternet connections and it allows to verify the identity of the owner at the receiving end, without having to\ntrust the intermediate sites which forwarded the job.\n\n\n\n\\section{UNICORE Based Projects}\n\\label{sec:projects}\nDuring the evolutionary development of the UNICORE technology, many European and international projects have\ndecided to base their Grid software implementations on UNICORE or to extend the growing set of core UNICORE\nfunctions with new features specific to their project focus. The goals and objectives of projects using UNICORE\nare not limited to the computer science community alone. Several other scientific domains such as bio-molecular\nengineering or computational chemistry are using the UNICORE technology as the basis of their work. In the\nfollowing we present short overviews of goals and objectives of UNICORE-based projects and describe additional\nfunctions and services contributed to the UNICORE development.\n\n\\subsection{EUROGRID -- Application Testbed for European Grid Computing}\n\\label{sec:projects-eurogrid}\nIn the EUROGRID\\footnote{funded in part by EC grant IST-1999-20247, duration: November 2000 - January 2004}\nproject \\cite{eurogrid:url} a Grid network of leading European High Performance Supercomputing centers was\nestablished. Based on the UNICORE technology application-specific Grids were integrated, operated and\ndemonstrated: \\bi\n\\item Bio-Grid for biomolecular science\n\\item Meteo-Grid for localized weather prediction\n\\item CAE-Grid for coupling applications\n\\item HPC-Grid for general HPC end-users\n\\ei\n\nAs part of the project, the UNICORE software was extended by an efficient data transfer mechanism, resource\nbrokerage mechanisms, tools and services for Application Service Providers (ASP), application coupling methods,\nand an interactive access feature \\cite{Mallmann:2001:EAT}. Efficient data transfer is a important issue, as Grids\ntypically rely on public parts of Internet connections. The available limited bandwidth has to be used efficiently\nto reduce the transfer time and the integrity of the transferred data has to be maintained, even if the transfer\nis interrupted. Depending on the application domain, additional security and confidentiality concerns need to be\nconsidered. This UNICORE high performance data transfer also uses X.509 certificates for authentication and\nencryption. To achieve not only a fast and secure transfer of data, but also high-performance capabilities,\nnetwork Quality of Service (QoS) aspects, overlapping of streamed data transfers, and packet assembling and\ncompression techniques are included.\n\nIn order to optimize the selection of resources -- either done by the users manually or by a metascheduler\nautomatically -- resource brokerage mechanisms and detailed resource description abilities are important. Within\nthe EUROGRID project, mechanisms were added to UNICORE, which allow users to specify their jobs in an abstract way\nimproving the overall resource selection and accounting. In particular for the benefit of the industrial user\naspects of security, convenience, and cost efficiency were addressed. To this end, the already existing security\nconcepts of UNICORE were thoroughly evaluated and assessed as being adequate, hence no additional development had\nto be done. The task of the developed resource broker is to match the abstract specification of the users jobs and\ntheir requirements with the available resources in the Grid. The resource broker reports the best match back to\nthe user including an estimate of the costs, which than allows the user to assign the appropriate resources to the\njob. For the suppliers of Grid resources (\\eg supercomputing centers) the resource broker allows to specify\ninformation about computational resources, architectures, processing power, storage and archiving facilities,\npost-processing facilities like visualization equipment, available software packages, and security guarantees. All\nthis data is enhanced by billing information.\n\nSupercomputing centers converge from pure providers of raw supercomputing power to Application Service Providers\n(ASP) running relevant scientific applications. For accounting and billing purposes the ASP needs to know the\nexact resources consumed by each customer in each run. For measuring the usage of supercomputers standard\nmechanisms provided by the resource management and operating system can be used, but measuring the usage of\nlicenses requires a sophisticated approach. For some applications, \\eg from the Computer Aided Engineering (CAE)\ndomain, this includes a connection to the applications licence manager. Establishing a link to the above mentioned\nresource broker is required to influence their decisions.\n\nFor solving complex problems applications from different domains, \\eg fluid-structure or\nelectromagnetism-structure, need to be coupled. This is established by using the EUROGRID resource broker\nfunctionality and combining it with the available Metacomputing functionality developed in the UNICORE Plus\nproject (\\cf \\secref{sec:intro}), which allows different schedulers of compute and application resources to\ncooperate. Finally, an interactive access to control and steer running application is needed for many scientific\napplications. The interactive use includes an interactive shell to actually login to computing resources using the\nUNICORE technology and security infrastructure.\n\nEUROGRID used the UNICORE technology to provide the above described services and functionalities by developing new\ncomponents. After the project ended, the developed components were revised and useful additions to the core\nUNICORE functions are now part of the available UNICORE software.\n\n\n\\subsection{GRIP -- Grid Interoperability Project}\n\\label{sec:projects-grip}\nGrid computing empowers users and organizations to work effectively in an information-rich environment. Different\ncommunities and application domains have developed distinct Grid implementations some based on published open\nstandards or on domain and community specific features. GRIP\\footnote{funded in part by EC grant IST-2001-32257,\nduration: January 2002 - February 2004} \\cite{GRIP:url} had the objective to demonstrate that the different\napproaches of two distinct grids can successfully complement each other and that different implementations can\ninteroperate. Two prominent Grid systems were selected for this purpose: UNICORE and Globus\\texttrademark\n\\cite{globus:url}, a toolkit developed in the United States. In contrast to UNICORE, Globus provides a set of APIs\nand services which requires more in-depth knowledge from the user. Globus is widely used in numerous international\nprojects and many centers have Globus installed as Grid middleware.\n\nThe objectives of GRIP were: \\bi\n\\item Develop software to enable the interoperation of independently developed Grid solutions\n\\item Build and demonstrate prototype inter-Grid applications\n\\item Contribute to and influence international Grid standards\n\\ei\n\nDuring the runtime of the GRIP project the Open Grid Service Architecture was proposed by the Global Grid Forum\n(GGF) \\cite{GGF:url}. The arrival of OGSA also was an opportunity to influence the standards directly which were\nto be created and to start developments that allow UNICORE to interoperate not only with Globus but with services\non the Grid in general, once the definition of the services and their interfaces became mature. OGSA did not\nchange the overall objectives of GRIP, however, it influenced directly some of the technical results.\n\nA basic requirement of GRIP was that the Grid interoperability layer should not change the well-known UNICORE user\nenvironment. As developers from both communities cooperated in the GRIP project, this goal was reached with only\nlittle changes of the UNICORE server components and no changes of the Globus Toolkit. This was achieved by the\ndevelopment of the so called Globus Target System Interface (Globus TSI), which provides UNICORE-access to\ncomputational resources managed by Globus. The Globus TSI was integrated into a heterogeneous UNICORE and Globus\ntestbed.\n\nTo achieve the main objective of GRIP, the interoperability between UNICORE and Globus and initial OGSA services,\nthe following elements had to be implemented: \\bi\n\\item The interoperability layer between UNICORE and Globus Version 2\n\\item The interoperability layer between UNICORE and Globus Version 3\n\\item The Access from UNICORE to simple Web services as a first step towards full integration of Web services\n\\item The Interoperability of the certificate infrastructures of UNICORE and Globus\n\\item A resource broker capable of brokering between UNICORE and Globus resources\n\\item The Ontology of the resource description on an abstract level\n\\ei\n\nIn GRIP, two important application areas were selected to prove that the interoperability layers work as\nspecified: \\bi\n\\item Bio-molecular applications were instrumented in such a way that they are Grid-aware in any\nGrid environment and capable to seamlessly use UNICORE and Globus managed resources. The techniques developed in\nGRIP were designed and implemented in a generalized way to ensure that they can be used in other application\ndomains as well.\n\\item A meteorological application, the Relocatable Local Model (RLM), was decomposed in such\na way that the components could execute on the most suitable resources in a Grid, independent of the middleware.\n\\ei\n\nThe results of the GRIP project are important for understanding general interoperability processes between Grid\nmiddleware systems. The experience and knowledge of the GRIP partners allowed to work in many relevant areas\nwithin GGF, like security, architecture, protocols, workflow, production management, and applications, and to\ninfluence the work in GGF.\n\n\n\\subsection{OpenMolGRID -- Open Computing Grid for Molecular Science and Engineering}\n\\label{sec:projects-openmolgrid}\nThe OpenMolGRID\\footnote{funded in part by EC grant IST-2001-37238, duration: September 2002 - February 2005}\nproject \\cite{OpenMolGrid:url} was focused on the development of Grid enabled molecular design and engineering\napplications. {\\em In silico} testing \\cite{Sild:2005:OAW} has become a crucial part in the molecular design\nprocess of new drugs, pesticides, biopolymers, and biomaterials. In a typical design process $O(10^5)$ to\n$O(10^6)$ candidate molecules are generated and their feasibility has to be tested. It is not economical to carry\nout experimental testing on all possible candidates. Therefore, computational screening methods provide a cheap\nand cost effective alternative to reduce the number of candidates. Over the years Quantitative Structure\nActivity\/Property Relationship (QSAR\/QSPR) methods have been shown to be reliable for the prediction of various\nphysical, chemical, and biological activities \\cite{Karelson:2000:MDQ}.\n\nQSPR\/QSAR relies on the observation that molecular compounds with similar structure have similar properties. For\neach specific application a set of molecules is needed for which the target property is known. This requires\nsearching globally distributed information resources for appropriate data. For the purpose of exploring molecular\nsimilarity, descriptors are calculated from the molecular structure. Thousands of molecular descriptors have been\nproposed and are used to characterize molecular structures with respect to different properties. Their calculation\nputs high demands on computer resources and requires high-performance computing.\n\nBased on this complex application the objectives of the OpenMolGRID project were defined as:\n\\bi\n\\item Development of tools for secure and seamless access to distributed information and computational methods\nrelevant to molecular engineering within the UNICORE frame\n\\item Provision of a realistic testbed and reference application in life science\n\\item Development of a toxicity prediction model validated with a large experimental set\n\\item Provision of design principles for next generation molecular engineering systems.\n\\ei\nIn particular this included to use UNICORE to automatize, integrate, and speed-up the drug discovery pipeline.\n\nThe OpenMolGRID project addressed the objectives above by defining abstraction layers for data sources (databases)\nand methods (application software), and integrating all necessary data sources (\\eg ECOTOX \\cite{ecotox:url}) and\nmethods (\\eg 2D\/3D Molecular Structure Conversion and Optimization, Descriptor Calculation, Structure Enumeration)\ninto UNICORE. The project developed application specific user interfaces (plug-ins) and a mechanism to generate a\ncomplete UNICORE Job from an XML workflow specification. This so called Meta-Plug-in takes care of including all\nauxiliary steps like data format transformation and data transfers into the job, distributing data parallel tasks\nover available computational resources, and allocating resources to the tasks. Thereby the molecular design\nprocess was significantly improved as the time to build QSAR\/QSPR models, the probability for mistakes, and the\nvariability of results was reduced. In addition a command line client (CLC) for UNICORE was developed to enable\nthe data warehouse to use Grid resources for its data transformation processes. The CLC offers the generation of\nUNICORE jobs from XML workflow description as well as the job submission, output retrieval, status query, and job\nabortion. The CLC consists of commands, an API, and a queuing component.\n\nBesides the technical achievements of OpenMolGRID and the added value for pharmaceutical companies its results\nwill contribute to the standardization of QSAR models.\n\n\n\\subsection{VIOLA -- Vertically Integrated Optical Testbed for Large Applications}\n\\label{sec:projects-viola}\nThe aim of the VIOLA\\footnote{funded in part by BMBF grant 01AK605F, duration: May 2004 - April 2007} project\n\\cite{VIOLA:url} is to build up a testbed with the latest optical network technology (multi 10 Gigabit Ethernet\nlinks). The goals and objectives of VIOLA are: \\bi\n\\item Testing of new network components and network architectures\n\\item Development and testing of software for dynamic bandwidth management\n\\item Interworking of network technology from different manufacturers\n\\item Development and testing of new applications from the Grid and Virtual Reality (VR) domain\n\\ei\n\nThe performance of the new network technology is evaluated with different scientific applications that need a very\nhigh network performance and network flexibility. UNICORE is used to build up the Grid on top of the hardware\nwithout taking fundamental software modifications. Only an interface to the meta-computer software library\nMetaMPICH \\cite{METAMPICH:url} needs to be integrated into UNICORE. Grid applications from the High Performance\nSupercomputing and Virtual Reality domain are enhanced for an optimized usage of the available bandwidth and the\nprovided Quality of Service classes. In this context a Meta-Scheduler framework is developed, which is able to\nhandle complex workflows and multi-site jobs by coordinating supercomputers and the network connecting them.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth]{VIOLA-superscheduler.eps}\n\\caption{The VIOLA Meta-Scheduler architecture.} \\label{fig:VIOLA-supersched}\n\\end{figure}\n\nVIOLA's first generation Meta-Scheduler architecture focuses on the scheduling functionality requiring only\nminimal changes to the UNICORE system. As depicted in \\figref{fig:VIOLA-supersched}, the system comprises the\nAgreement Manager, the Meta-Scheduler itself \\cite{Quecke:2000:MAR}, and a Meta-Scheduling plug-in (which is part\nof the client and not pictured separately). Before submitting a job to a Usite (\\cf \\secref{sec:arch-serverT}),\nthe Meta-Scheduling plug-in and the Meta-Scheduler exchange the data necessary to schedule the resources needed.\nThe Meta-Scheduler is then (acting as an Agreement Consumer in WS--Agreement terms \\cite{GRAAP:url}) contacting\nthe Agreement Manager to request a certain level of service, a request which is translated by the Manager into the\nappropriate resource management system commands. In case of VIOLA's computing resources the targeted resource\nmanagement system is the EASY scheduler. Once all resources are reserved at the requested time the Meta-Scheduler\nnotifies the UNICORE Client via the Meta-Scheduling plug-in to submit the job. This framework will also be used to\nschedule the interconnecting network, but potentially any resource can be scheduled if a respective Agreement\nManager is implemented and the Meta-Scheduling plug-in generates the necessary scheduling information. The\nfollow-on generation of the Meta-Scheduling framework will then be tightly integrated within UNICORE\/GS (\\cf\n\\secref{sec:future-unigrids-GS}).\n\n\n\\subsection{NaReGI -- National Research Grid Initiative}\n\\label{sec:projects-naregi}\nThe Japanese NaReGI project \\cite{NAREGI:url} includes the UNICORE technology as the basic middleware for research\nand development. NaReGI is a collaboration project between industry, academia, and government. The goals and\nobjectives are: \\bi\n\\item Establishment of a national Japanese research Grid infrastructure\n\\item Revitalization of the IT industry through  commercialization of Grid middleware and strengthened\ninternational competitiveness\n\\item Dissemination of Grid environments throughout industry\n\\item Trailblazing the standardization of Grid technology\n\\item Cultivation of human resources specializing in IT technology for Grids\n\\ei\n\nSimilar to the GRIP project (\\cf \\secref{sec:projects}) where an interoperability layer between UNICORE and Globus\nToolkit 2 and 3 was developed, the NaReGI project plans to implement such a layer between UNICORE and Condor\n\\cite{CONDOR:url}, called UNICONDORE. This interoperability layer will allow to submit jobs from the UNICORE\nclient to Condor pools and to use Condor commands to submit jobs to UNICORE managed resources.\n\nIn the first phase of the NaReGI testbed UNICORE provides access to about 3000 CPUs in total with approximately 17\nTFlops of peak performance. It is expected to increase the integrated peak performance to 100+ TFlops by the end\nof the project in 2007.\n\n\n\n\\section{UNICORE in Production}\n\\label{sec:production}\nFrom its birth in two German BMBF-funded projects to its extensive use and further development in a variety of EU\nand BMBF research projects, the UNICORE technology ran through an evolutionary process transforming from an\ninitial prototype software to a powerful production Grid middleware.\n\n\\subsection{UNICORE@SourceForge}\n\\label{sec:production-SF}\nSince May 2004, the UNICORE technology with all its components is available as open source software under the BSD\nlicense. It can be downloaded from the SourceForge repository. Besides the core developers of UNICORE (namely\nFujitsu Laboratories of Europe, Intel Germany and the Research Center J\\\"{u}lich), there are numerous contributors\nfrom all over the world, \\eg Norway, Poland, China and Russia. The Web site \\cite{UNICORE-sourceforge:url} offers\na convenient entry point for interested users and developers. In the download section the UNICORE software is\nbundled in different packages, \\eg the client package and individual packages for the different server components\nGateway, NJS, TSI\/IDB, UUDB (\\cf \\secref{sec:arch}), and plug-ins. Until January 2005 more than 2800 downloads of\nUNICORE are counted.\n\nA tracker section linked on the Web site establishes a communication link to the core developer community. The\ncorresponding mailing lists allow users to report bugs, to request new features, and to get informed about bug\nfixes or patches. For the announcement of new software releases a separate mailing list was created. The Grid team\nat the Research Center J\\\"{u}lich is responsible for UNICORE@SourceForge. Its work includes coordinating and driving\nthe development effort, and producing consolidated, stable, and tested releases of the UNICORE software.\n\n\\subsection{Production System on Jump}\n\\label{sec:production-Jump}\nSince July 2004 UNICORE is established as production software to access the supercomputer resources of the John\nvon Neumann-Institute for Computing (NIC) at the Research Center J\\\"{u}lich. These are the 1312-processor IBM p690\ncluster (Jump) \\cite{Jump:url}, the Cray SV1 vector machine, and a new Cray XD1 cluster system. As an alternative\nto the standard SSH login, UNICORE provides an intuitive and easy way for submitting batch jobs to the systems.\nThe academic and industrial users come from all over Germany and from parts of Europe. The applications come from\na broad field of domains, \\eg astrophysics, quantumphysics, medicine, biology, chemistry, and climate research,\njust to name the largest user communities. A dedicated, pre-configured UNICORE client with all required\ncertificates and accessible Vsites is available for download. This alleviates the installation and configuration\nprocess significantly. Furthermore, an online installation guide including a certificate assistant, an user\nmanual, and example jobs help users getting started.\n\nTo provide the NIC-users with adequate certificates and to ease the process of requesting and receiving a\ncertificate, a certificate authority (CA) was established. User certificate requests are generated in the client\nand have to be send to the CA. Since introduction of UNICORE at NIC, more than 120 active users requested a\nUNICORE user certificate.\n\nA mailing list serves as a direct link of the users to UNICORE developers in the Research Center J\\\"{u}lich. The list\nallows to post problems, bug reports, and feature requests. This input is helpful in enhancing UNICORE with new\nfeatures and services, in solving problems, identifying and correcting bugs, and influences new releases\nof UNICORE available at SourceForge.\n\n\\subsection{DEISA -- Distributed European Infrastructure for Scientific Applications}\n\\label{sec:production-DEISA}\nTraditionally, the provision of high performance computing resources to researchers has traditionally been the\nobjective and mission of national HPC centers.On the one hand, there is an increasing global competition between\nEurope, USA, and Japan with growing demands for compute resources at the highest performance level, and on the\nother hand stagnant or even shrinking budgets. To stay competitive major investments are needed every two years --\nan innovation cycle that even the most prosperous countries have difficulties to fund.\n\nTo advance science in Europe, eight leading European HPC centers devised an innovative strategy to build a\nDistributed European Infrastructure for Scientific Applications (DEISA) \\cite{DEISA:url}. The centers join in\nbuilding and operating a tera-scale supercomputing facility. This becomes possible through deep integration of\nexisting national high-end platforms, tightly coupled by a dedicated network and supported by innovative system\nand grid software. The resulting virtual distributed supercomputer has the capability for natural growth in all\ndimensions without singular procurements at the European level. Advances in network technology and the resulting\nincrease in bandwidth and lower latency virtually shrink the distance between the nodes in the distributed\nsuper-cluster. Furthermore, DEISA can expand horizontally by adding new systems, new architectures, and new\npartners thus increasing the capabilities and attractiveness of the infrastructure in a non-disruptive way.\n\nBy using the UNICORE technology, the four core partners of the projects have coupled their systems using virtually\ndedicated 1 Gbit\/s connections. The DEISA super-cluster currently consists of over 4000 IBM Power 4 processors and\n416 SGI processors with an aggregated peak performance of about 22 teraflops. UNICORE provides the seamless,\nsecure and intuitive access to the super-cluster.\n\nThe Research Center J\\\"{u}lich is one of the DEISA core partners and is responsible for introducing UNICORE as Grid\nmiddleware at all partner sites and for providing support to local UNICORE administrators.\n\nAll DEISA partners have installed the UNICORE server components Gateway, NJS, TSI, and UUDB to access the local\nsupercomputer resources of each site via UNICORE. \\figref{fig:DEISA-architecture} shows the DEISA UNICORE\nconfiguration. For clarity only four sites are shown. At each site, a Gateway exists as an access to the DEISA\ninfrastructure. The NJSs are not only registered to their local Gateway, but to all other Gateways at the partner\nsites as well. Local security measures like firewall configurations need to consider this, by permitting access to\nall DEISA users and NJSs. This fully connected architecture has several advantages. If one Gateway has a high\nload, access to the high performance supercomputers through DEISA is not limited. Due to the fully connected\narchitecture, no single point of failure exists and the flexibility is increased.\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{DEISA-architecture.eps}\n\\caption{The DEISA architecture.} \\label{fig:DEISA-architecture}\n\\end{figure}\n\nThe DEISA partners operate different supercomputer architectures, which are all accessible through UNICORE.\nInitially all partners with IBM p690 clusters are connected to one large virtual supercomputer. In a second step\nother supercomputers of different variety are connected to DEISA, making the virtual supercomputer heterogeneous.\nUNICORE can handle this, as it is designed to serve such heterogeneous architectures in a seamless, secure, and\nintuitive way.\n\nIn December 2004 a first successful UNICORE demonstration between the four DEISA core sites FZJ (Research Center\nJ\\\"{u}lich, Germany), RZG (Computing Center Garching, Germany), CINECA (Italian Interuniversity Consortium, Italy) and\nIDRIS (Institute for Development and Resources in Intensive Scientific Computing, France) was given. Different\nparts of a distributed astrophysical application were generated and submitted with UNICORE to all four sites.\n\nThe experience and knowledge of the researchers, developers, users, and administrators in working with UNICORE in\nthe DEISA project on a large production platform will be used as useful input for future developments of the\nUNICORE technology. A close synchronization with the UniGrids project (\\cf \\secref{sec:future-unigrids}) is\nforeseen.\n\n\n\n\\section{Future of UNICORE}\n\\label{sec:future}\nThe current UNICORE software implements a vertically integrated Grid architecture providing seamless access to\nvarious resources. Every resource is statically integrated into the UNICORE Grid by providing an interface to the\nappropriate resource manager.\n\nOne of the benefits Web services will bring to Grid computing is the concept of loosely coupled distributed\nservices. Merging the idea of ``everything being a service'' with the achievements of the Grid community led to\nGrid services, enabling a new approach to the design of Grid architectures. The adoption of XML and the drive for\nstandardization of the Open Grid Service Architecture provide the tools to move closer to the promise of\ninteroperable Grids. A demonstrator validated the correspondence of UNICORE's architectural model with the\nOGSA\/OGSI (Open Grid Service Infrastructure \\cite{Tuecke:2003:OGSI}) approach, which encouraged the development of\nan OGSA\/OGSI compliant UNICORE Grid architecture in the GRIP project (\\cf \\secref{sec:projects-grip}).\n\nIn \\cite{Menday:2003:GEU} UNICORE is examined for the evolution of a Grid system towards a service oriented Grid,\nprimarily focussing on architectural concepts and models. Based on the current architecture and the enhancements\nprovided by GRIP, first steps already integrate Web services into UNICORE. This included the provision of OGSI\ncompliant port types parallel to the proprietary ones as well as the design of XML based protocols. This work was\ncontinued in the UniGrids project.\n\nAs mentioned above the development of a Grid middleware is an continuous process of integrating new features,\nservices, and adapting to emerging standards, and UNICORE is no exception. In the following we present new\ndevelopments, some technical details, and report on projects, which enhance the UNICORE technology to serve the\ndemands of the Grid in the future \\cite{Jeffrey:2004:NGG}.\n\n\n\\subsection{UniGrids -- Uniform Interface to Grid Services}\n\\label{sec:future-unigrids}\nThe strength of the UNICORE architecture is well-proven as described above. The rapid definition and adoption of\nOGSA allow the UNICORE development community to re-cast and extend the concepts of UNICORE through the use of Web\nservices technologies. The goal of the UniGrids\\footnote{funded in part by EC grant IST-2002-004279, duration:\nJuly 2004 - June 2006} project \\cite{UniGrids:url} is to lift UNICORE on an architecture of loosely-coupled\ncomponents while keeping its 'end-to-end' nature.\n\nThus, the integration of Web services techniques and UNICORE, which already started in the GRIP project (\\cf\n\\secref{sec:projects-grip}), will continue in the \\mbox{UniGrids} project. Interoperability, through adopting and\ninfluencing standards, form the philosophical foundation for UniGrids. The project aims to transform UNICORE into\na system with interfaces that are compliant with the Web Services Resource Framework (WS-RF) \\cite{WSRF:url} and\nthat interoperate with other WS-RF compliant software components.\n\nSuch an approach offers great advantages both for the ease of development of new components by aggregation of\nservices and through the integration of non-UNICORE components into the standards-based infrastructure.\n\nIn this sense, work is continuing in the following areas: \\bi\n\\item Development of a compliant WS-RF hosting environment used for publishing UNICORE job and file services as\nWeb services.\n\\item Support of dynamic virtual organizations by enhancing the UNICORE security infrastructure to allow different\nusage models such as delegation and collective authorization.\n\\item Development of translation mechanisms, such as resource ontologies, to interoperate with other OGSA compliant\nsystems. Support for Grid economics by developing a Service Level Agreement (SLA) framework and cross-Grid\nbrokering services.\n\\item Development and integration of generic software components for visualization and steering of simulations (VISIT\n\\cite{visit:url}), device monitoring and control, and tools for accessing distributed data and databases.\n\\ei\n\nApplications from the scientific and industrial domain, like biomolecular and computational biology, geophysical\ndepth imaging by oil companies, automotive, risk-management, energy, and aerospace are used to prove the\ndevelopments in UniGrids.\n\nThe development in the UniGrids project will lead to UNICORE\/GS, which follows the architecture of OGSA through\nthe standardization of WS-RF and related work like \\eg the Web Services Notification technology \\cite{WSN:url}.\nThe results will be made available under an open source BSD license.\n\n\\subsubsection{UNICORE\/GS}\n\\label{sec:future-unigrids-GS}\nWeb service technology, and in particular the WS-RF, forms the basis for the UNICORE\/GS software. WS-RF is the\nfollow-on to OGSI, but more in line with mainstream Web services architecture \\cite{WSARCH:url}. Based on this new\ntechnology, UNICORE\/GS will retain its key characteristics of seamlessness, security, and intuitiveness from both\nthe user and administrative perspective, but will be built on a service oriented framework. This means that there\nis a loosening of the coupling between the components of the system. UNICORE\/GS keeps the classical UNICORE\ntopology of Usites, each containing a number of Vsites, but provides a new framework for integrating other\nservices and providing common infrastructure functionality as services. This has the implication that new services\nwill be easily integrated into the UNICORE\/GS environment. Conversely, UNICORE\/GS will be well-prepared to make\nuse of external services.\n\nThe WS-RF technology is used to model core functionalities such as job submissions and file transfers as\nWS--Resources. These services are accessible via web service interfaces and thus establishing the UniGrids atomic\nservices layer. This layer will be realized making extensive use of existing UNICORE server components.\n\nAll services in a Usite are accessible through the UniGrids Gateway that provides a secure entrance into the\nUNICORE\/GS infrastructure. The principal is exactly the same as for classic UNICORE, however, the Gateway now\nroutes messages according to Web Services Addressing (WS--Addressing) \\cite{WSA:url}. Authentication is based on\ntransport level HTTPS security, although the intention is to move to Web Services Security (WS--Security)\n\\cite{WSS:url}. Regarding authorized access to resources, the UNICORE User Database (UUDB) will be available as a\nservice to other services in the Usite, and will form the basis for future work concerning virtual organizations\nand fine-grained authorization schemes.\n\nThe underlying UniGrids atomic services layer will provide an excellent framework to deploy higher-level services\nsuch as co-allocation schedulers, workflow engines, and services for provision and easy access to data-intensive,\nremotely-steerable simulations.\n\n\\subsection{NextGrid -- Architecture for Next Generation Grids}\n\\label{sec:future-nextgrids}\nIn comparison to the UniGrids project which evolves the existing UNICORE Grid system to a service-oriented one,\nthe NextGRID\\footnote{funded in part by EC grant IST-2002-511563, duration: September 2004 - August 2007}\n\\cite{nextgrid:url} project aims for the future: The goal is to provide the foundations for the next generation of\nGrids. NextGRID is not a project based on the UNICORE architecture or Grid system as-is, but institutions and\npeople involved in the UNICORE development from the beginning on contribute expertise and experience to NextGRID.\n\nSince it is obvious that there is no such thing as the one and only next generation Grid, and experts envisage the\nco-existence of multiple Grids with well-defined boundaries and access points, NextGRID is going to define a Grid\narchitecture which can be seen as building blocks for Grids. It does not only provide interoperability by-design\nbetween entities which exist within one instantiation of such an architecture, but it also facilitates the\ninteroperability between different Grids developed according to the NextGRID architecture.\n\nAlthough developing a Grid one generation ahead, NextGRID is not starting from scratch. Properties to incarnate\nand functions to realize future Grids are expertly described in \\cite{Priol:2003:NGG} and \\cite{Jeffrey:2004:NGG}.\nThese reports frame NextGRID's architectural development while the Open Grid Services Architecture is going to\ndefine Grid services and their interactions and does therefore make up a staring point for the conceptualization\nand design of NextGRID. In addition, regarding the underlying technology and architectural model, NextGRID\npropagates the usage of Web Services and the adoption of Service-Oriented Achitecture (SOA) \\cite{Erl:2004:SOA}\nconcepts and models.\n\nNextGRID focuses on security, economic sustainability, privacy\/legacy, scalability and usability. The following\nproperties have the highest priorities when carrying out the following work: \\bi\n\\item Developing an architecture for next generation Grids\n\\item Implementing and testing prototypes aligned with the concepts and design of the NextGRID architecture\n\\item Creating reference applications which make use of the NextGRID prototypes\n\\item Facilitating the transition from scientific- to business-oriented Grids by integrating the means to negotiate\na certain Quality of Service (QoS) level\n\\item Specifying the methods, processes, and services necessary to dynamically operate Grids across multiple\norganizations which comprise heterogeneous resources\n\\ei\n\nSince the ongoing UNICORE development in projects like UniGrids shares resources as well as the technological\nfoundation with NextGRID there is a high chance that the outcome of NextGRID will also represent the next step of\nUNICORE's evolution.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper we presented the evolution of the UNICORE technology from a Grid software with prototype character\ndeveloped in two German projects to a full-grown, well-tested, widely used and accepted Grid middleware. UNICORE\n-- Uniform Interface to Computing Resources -- provides a {\\em seamless, secure and intuitive} access to\ndistributed Grid resources. Although the UNICORE vision was already coined in 1997, the then stated goals and\nobjectives of hiding the seams of resource usage, incorporating a strong security model, and providing an easy to\nuse graphical user interface for scientists and engineers are still valid today: to achieve these goals and\nobjectives, UNICORE is designed as a vertically integrated Grid middleware providing components at all layers of a\nGrid infrastructure, from a graphical user interface down to the interfaces to target machines.\n\nInitially developed in the German projects UNICORE and UNICORE Plus, UNICORE was soon established as a promising\nGrid middleware in several European projects. In the GRIP project an interoperability layer between UNICORE and\nthe Globus Toolkit 2 and 3 was developed to demonstrate the interoperability of independently developed Grid\nsolutions, allowing to build and to demonstrate inter-Grid applications from the bio-molecular and meteorological\ndomain. In the EUROGRID project, European high performance supercomputing centers joined to extend UNICORE with an\nefficient data transfer, resource brokerage mechanisms, ASP services, application coupling methods, and an\ninteractive access. In addition, a Bio-Grid, Meteo-Grid, CAE-Grid, and HPC-Grid were established to integrate a\nvariety of application domains. The main objective of the OpenMolGRID project is to provide a unified and\nextensible information-rich environment based on UNICORE for solving problems from molecular science and\nengineering. In the VIOLA project a vertically integrated testbed with the latest optical network technology is\nbuilt up. UNICORE is used as the Grid middleware for enabling the development and testing of new applications in\nthe optical networked testbed, which provides advanced bandwidth management and QoS features.\n\nWith these developments UNICORE grew to a software system usable in production Grids. In this context UNICORE is\ndeployed in the large German supercomputing centers to provide access to their resources. At the John von\nNeumann-Institute for Computing, Research Center J\\\"{u}lich, many users submit their batch jobs through UNICORE to the\n1312-processor 8.9 TFlop\/s IBM p690 cluster and the Cray SV1 vector machine. Leading European HPC centers joined\nin the project DEISA to build a distributed European infrastructure for scientific applications based on UNICORE\nto build and operate a distributed multi tera-scale supercomputing facility.\n\nThe future of UNICORE is promising and follows the trend of ``Everything being a Service\" by adapting to Open Grid\nService Architecture (OGSA) standards. In this context, the UniGrids project continues the effort of the GRIP\nproject in integrating the Web Services and UNICORE technology to enhance UNICORE to an architecture of\nloosely-coupled components while keeping its ``end-to-end\" nature. To this end UNICORE\/GS will be developed, which\nmakes UNICORE compliant with the Web Services Resource Framework (WS-RF).\n\nToday the UNICORE software is available as open source under a BSD licence from SourceForge for download. This\nenables the community of core UNICORE developers to grow and makes future development efforts open to the public.\n\n\n\n\\section{Acknowledgments}\n\\label{sec:ack}\nThe work summarized in this paper was done by many people. We gratefully thank them for their past, present, and\nfuture contributions in developing the UNICORE technology. Most of the work described here was supported and\nfunded by BMBF and different programmes of the European Commission under the respective contract numbers mentioned\nabove.\n\n\n\\newcommand{\\noopsort}[1]{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $X$ be a scheme over an algebraically closed field $k$ of\ncharacteristic $p$, with $p\\, >\\, 0$. Fix a $k$-point $x$ of $X$. Nori\nintroduced the notion of fundamental group scheme $\\pi(X,x)$ in \\cite{N1}\nand further developed it in \\cite{N2}. Since then the\nfundamental group scheme is being\nstudied and in the process has turned into\nan important tool in algebraic geometry of positive\ncharacteristic. In \\cite{N2} Nori proves that\n$\\pi(X,x)$ is trivial for proper rational normal varieties. More generally,\n$\\pi(X,x)\\,=\\, 0$ if $X$ is separably rationally connected \\cite{Bi}.\nZhu proves that a general Fano (proper, smooth, connected with\nample anticanonical bundle) hypersurfaces in projective spaces are\nseparably rationally connected \\cite{Zh}. Therefore, the fundamental\ngroup schemes of general Fano hypersurfaces in a projective space\nare trivial.\n\nIn \\cite{CL} Chambert-Loir proves that every proper rationally\nchain connected normal variety has finite \\'etale\nfundamental group, and its order is coprime to $p$\n(the characteristic of $k$) \\cite{CL2}. This result can also be obtained as a\nconsequence of \\cite[Theorem 1.5]{Ko} and \\cite[Theorem 1.6]{Ko}. Shioda gave an example of a rationally\nconnected variety over a field of characteristic $p\\,\\neq\\, 5$ whose \\'etale\nfundamental group is $\\mathbb{Z}\/5\\mathbb{Z}$ \\cite{Sh}. Examples of rationally chain connected varieties whose local \nfundamental group scheme is not trivial are also known.\nFor example, a supersingular Enriques surface $E$ over an algebraically \nclosed field of characteristic $2$ is unirational (see\n\\cite[Corollary 1.3.1]{CD}), hence it is rationally \nchain connected. It is known that there exists a nontrivial\n$\\alpha_2$--torsor over $E$ (see \\cite[Chapter I, \\S 3]{CD}). \n\nWe prove the following (see Theorem \\ref{teoMAIN} and Remark \\ref{remFANO}):\n\n\\begin{theorem}\nLet $k$ be an algebraically closed field and $X$\na proper normal, rationally chain connected $k$--scheme. Let $x\\,\\in\\, X(k)$ be a point.\nThen the fundamental group scheme $\\pi(X,x)$ is finite.\n\\end{theorem}\n\nThe strategy of the proof is similar to that in \\cite{CL}, adapted to the \nnew setting.\n\n\\section{Preliminaries}\\label{sez:PREM}\n\nWe will write $\\pi(X)$ instead of $\\pi(X,x)$ to simplify the notation. \nHowever all the schemes for which we will compute the fundamental group \nscheme are meant to be pointed and all the morphisms between them take the \nmarked point in the domain space to the marked point in the target space. \nThe same convention will be applied to torsors: we assume \nthey are pointed and morphisms between them take the marked point in the \ndomain space to the marked point in the target space.\n\nLet $k$ be an algebraically closed field of any characteristic.\nA proper variety $X$ over $k$ is said to be rationally chain connected if for every algebraically\nclosed field $\\Omega$ containing $k$, for any two points in $X(\\Omega)$ there is a\nproper and connected curve passing through them such that its normalization is a\ndisjoint union of projective lines. If this union consists of only one projective line\nwe say that $X$ is rationally connected.\n\nLet $X$ be a rationally chain connected variety over $k$.\nWe recall \\cite[Lemma 1]{CL} and we sketch its proof:\n\n\\begin{lemma}\\label{lemCL1}\nLet $k\\,\\subseteq\\, \\Omega$ be a field extension\nwhere $\\Omega$ is algebraically closed. Let $$F_{\\Omega}\\,:\\, \\mathbb{P}_{\\Omega}^1\n\\,\\longrightarrow\\, X_{\\Omega}$$ be a rational curve of $X_{\\Omega}$. Let\n$x_0\\,:=\\,F_{\\Omega}(0)$ and $x_{\\infty}\\,:=\\,F_{\\Omega}(\\infty)$ be points of\n$X_{\\Omega}$ then let $V_0$ and $V_{\\infty}$ be their Zariski closure in $X$.\nThen there exist a normal integral $k$-scheme $T$, a morphism\n$$F\\,:\\, \\mathbb{P}_{T}^1\\,\\longrightarrow\\, X$$ such that the morphisms defined as\n$$F_0(t)\\,:=\\,F(0,t)\\,:\\,T\\,\\longrightarrow\\, X\\ ~ \\text{ and }~\\ F_{\\infty}(t)\\,:=\\,\nF(\\infty,t)\\,:\\,T\\,\\longrightarrow\\, X$$\nare dominant over $V_0$ and $V_{\\infty}$ respectively.\n\\end{lemma}\n\n\\begin{proof}\nThere exists a finitely generated $k$-algebra $k\\,\\subseteq\\, A$ contained\nin $\\Omega$, and there is a morphism $$F_{A}\\,:\\,\n\\mathbb{P}_{A}^1\\,\\longrightarrow\\, X_{A}$$ such that $F_{\\Omega}\\,=\\,F_A\n\\otimes_A\n\\Omega$. We set $T\\,:=\\,{\\rm Spec}(A)$ that we assume to be normal (otherwise we replace it with a finite extension). We now consider the morphism\n$F\\,:\\, \\mathbb{P}_{T}^1\\,\\longrightarrow\\, X$ obtained from $F_{A}$ after composing with the projection $X_A\\,\\longrightarrow\\, X$. In \\cite[Lemma 1]{CL} it has been proved that $F_0$ and $F_{\\infty}$ are\ndominant over $V_0$ and $V_{\\infty}$; we briefly recall this last part for the\nconvenience of the reader: We study $F_{0}$ (it will\nbe the same for $F_{\\infty}$). The\nimage by $F_{0}$ of the generic point of ${\\rm Spec}(A)$ is the generic\npoint of $V_0$. Since $V_0$ is closed in $X$, the inverse image\n$F_{0}^{-1}(V_0)$ is closed in ${\\rm Spec}(A)$ and dense. Thus\n$F_{0}^{-1}(V_0)$ coincides with\n${\\rm Spec}(A)$, and hence the image of $F_{0}$ is contained in $V_0$ and contains\nits generic point. Therefore it contains an open dense subset of $V_0$. \n\\end{proof}\n\n\\section{The main theorem}\n\nThe following lemma is well-known. We include\na short proof of it for the convenience of the reader.\n\n\\begin{lemma}\\label{lemGROUPS}\nLet $G$ be a finite $k$-group scheme and let $G^{\\text{\\'et}}$ and $G^{\\rm loc}$ be\nrespectively the maximal \\'etale quotient and the maximal connected quotient. Then the\nnatural morphism $$\\alpha\\,:\\,G\\,\\longrightarrow\\, G^{\\text{\\'et}}\\times G^{\\rm loc}$$\nis faithfully flat. \n\\end{lemma}\n\n\\begin{proof}\nThe field being perfect the reduced subscheme $G_{\\text{red}}$ is a subgroup\nscheme of $$N^{\\text{loc}}\\,:=\\,\\text{kernel} (G\\to G^{\\text{loc}})\\, ,$$ while the\nconnected component $G^0$ of $G$ is $\\text{kernel}(G\\to \nG^{\\text{\\'et}})$. If $\\alpha$ is not faithfully flat we can factor it as in\nthe following diagram:\n$$\\xymatrix{ & G\\ar@{->>}[ldd]\\ar@{->>}[d]^{q}\\ar@{->>}[rdd] & \\\\ &\nG'\\ar@{->>}[ld]\\ar@{^{(}->}[d]^{j}\\ar@{->>}[rd] & \\\\G^{\\text{\\'et}} &\nG^{\\text{\\'et}}\\times G^{\\text{loc}}\\ar@{->>}[l]\\ar@{->>}[r] & G^{\\text{loc}} }\n$$\nwhere $q\\,:\\,G\\,\\twoheadrightarrow\\, G'$ is faithfully flat and $j\\,:\\,G'\\,\n\\hookrightarrow\\, G^{\\text{\\'et}}\\times G^{\\text{loc}}$ is a closed immersion. Clearly\n$G^{\\text{\\'et}}$ and $G^{\\text{loc}}$ are still the maximal \\'etale\nquotient and the maximal connected quotient of $G'$ respectively. So we can\nassume $\\alpha$ is a closed immersion. Therefore, the lemma is equivalent to the\nassertion that $\\alpha$ is an isomorphism.\n\n{}From \\cite[\\S~6, Ex. 9]{WW} it follows that $G^{\\text{\\'et}}$\nis isomorphic to a subgroup-scheme of $G$ which we identify with\n$G^{\\text{\\'et}}$, so in particular $G^{\\text{\\'et}}\\,\\leq\\, \nG_{\\text{red}}\\,\\leq\\, N^{\\text{loc}}$. Therefore, we have \n$$\\vert G^{\\text{\\'et}}\\vert\\vert G^{\\text{loc}}\\vert\\,=\\,\n\\vert G^{\\text{\\'et}}\\vert\\frac{\\vert G\\vert }{\\vert N^{\\text{loc}}\\vert\n}\\,\\leq\\,\\vert G^{\\text{\\'et}}\\vert\\frac{\\vert G\\vert }{\\vert G^{\\text{\\'et}}\n\\vert } \\,=\\, \\vert G\\vert $$\nwhich implies that $\\alpha$ is an isomorphism.\n\\end{proof}\n\nWe recall that when $X$ is a reduced and connected scheme over a field $k$\nthen the fundamental group scheme can be defined. In this case a finite $G$-torsor $Y\\,\\longrightarrow\\, X$ is called Nori-reduced if the canonical morphism $\\pi(X)\n\\,\\longrightarrow\\, G$ is faithfully flat. \n\n\\begin{lemma}\\label{lemTORS}\nLet $X$ be a connected and reduced scheme over $k$. Let $G$ (respectively,\n$H$) be a finite local (respectively, finite \\'etale) $k$--group\nscheme. Let $$Y\\,\\longrightarrow\\, X$$ and $T\\,\\longrightarrow\\, X$ be a $G$--torsor and\nan $H$--torsor respectively. We assume that both $Y$ and $T$ are Nori-reduced. Then\nthe $H\\times G$--torsor $$T\\times_X Y\\,\\longrightarrow\\, X$$ is also Nori-reduced. \n\\end{lemma}\n\n\\begin{proof}\nIf $\\pi(X)\\,\\longrightarrow\\, H\\times G$ is not faithfully flat then there\nexists a triple $(M\\, , Z\\, ,\\iota)$, where\n\\begin{itemize}\n\\item $M\\,\\hookrightarrow\\, H\\times G$ is a subgroup-scheme,\n\n\\item $Z\\,\\longrightarrow\\, X$ is a $M$--torsor, and\n\n\\item $\\iota\\, :\\, Z\\,\\hookrightarrow\\, T\\times_X Y$ is a reduction of structure\ngroup-scheme, to $M$, of the $H\\times G$--torsor $T\\times_X Y$.\n\\end{itemize}\nLet\n$$T'\\,\\longrightarrow\\, X~ \\ \\text{ and }~\\ Y'\\,\\longrightarrow\\, X$$ be the\n$M^{\\text{\\'et}}$ and $M^{\\text{loc}}$--torsors respectively,\nobtained from the $M$--torsor $Z\\,\\longrightarrow\\, X$ using the projections\nof $M$ to $M^{\\text{\\'et}}$ and $M^{\\text{loc}}$ respectively (the notation\nis as in Lemma \\ref{lemGROUPS}).\nWe have a closed immersion $$Z\\,\\hookrightarrow\\, T'\\times_X Y'$$ induced by\nthe closed immersion $M\\,\\hookrightarrow \\,M^{\\text{\\'et}}\\times M^{\\text{loc}}$.\nThe latter is an isomorphism by Lemma \\ref{lemGROUPS}, so the same is true for\n$Z\\,\\hookrightarrow\\, T'\\times_X Y'$.\n\nThe projection $M\\,\\twoheadrightarrow\\, H$ (respectively, $M\\,\\twoheadrightarrow\n\\, G$) clearly factors through $M^{\\text{\\'et}}$ (respectively, $M^{\\text{loc}}$).\nNote that the projections $$M\\,\\longrightarrow\\, H \\ \\ \n\\text{ and } \\ \\ M\\,\\longrightarrow \\,G$$\nare faithfully flat morphisms because the two torsors $Y$ and $T$ are\nNori-reduced. Consequently, the two homomorphisms $M^{\\text{\\'et}}\\,\\longrightarrow\\,\nH$ and $M^{\\rm loc}\\,\\longrightarrow\\, G$ are isomorphisms. Now using\nLemma \\ref{lemGROUPS} it follows that the inclusion $M\\,\\hookrightarrow\\, H\\times G$\nis an isomorphism. Consequently, the $H\\times G$--torsor $T\\times_X Y\\,\\longrightarrow\\,\nX$ is Nori-reduced.\n\\end{proof}\n\nThe following result was proved in \\cite[Proposition 3.6]{EPS} under the assumption\nthat $X$ is proper.\n\n\\begin{corollary}\\label{corTORS}\nLet $X$ be a connected and reduced scheme over $k$. Let $G$ (respectively, $H$)\nbe a finite local (respectively, finite \\'etale) $k$-group\nscheme. Let $Y\\,\\longrightarrow\\, X$ be a $G$--torsor and\n$T\\,\\longrightarrow\\, X$ an $H$--torsor. We assume that both the torsors are\nNori-reduced. Then the $G$--torsor\n$$T\\times_X Y \\,\\longrightarrow\\,T$$ is also Nori-reduced. In particular, the morphism\n$\\pi^{\\rm loc}(T)\\,\\longrightarrow\\, \\pi^{\\rm loc}(X)$ is faithfully flat.\n\\end{corollary}\n\n\\begin{proof}\nLet us assume that there is a finite local $k$-group\nscheme $G_1\\,\\subset\\,G$, and $G_1$--torsor $U\\,\\longrightarrow\\, T$ and a\nreduction\n$$i\\,:\\,U\\,\\hookrightarrow\\, T\\times_X Y$$ of structure group to $G_1$.\nLet $S$ be any $k$-scheme. For any $x\\,\\in\\, X(S)$ we choose $u_x\\,\\in\\, U(S)$ whose\nimage in $X(S)$ is $x$. We set $(t_x\\, ,y_x)\\,:=\\,i(u_x)$, then $$T(S)\\times_{X(S)}Y(S)\n\\,=\\,\\{(ht_x,gy_x)\\, ,~\\ \\forall ~x\\,\\in\\, X(S)\\, ,~\\ \\forall ~ g\\,\\in\\, G(S)\\, ,\n~\\ \\forall ~ h\\,\\in\\, H(S) \\}$$ so the image\nof $U(S)$ by $i_S$ can be identified with the subset \n$$\\{(ht_x\\, ,gy_x)\\, ,~\\ \\forall ~x\\,\\in\\, X\\, ,~\\ \\forall ~g\\,\\in\\,\nG_1\\, ,~\\ \\forall ~h\\,\\in\\, H \\}\\, ;$$ this gives $U$ the structure of an $H\\times\nG_1$--torsor over $X$, contained in the $H\\times G$--torsor $T\\times_X Y$. This\nimplies that $G_1\\,=\\,G$ by Lemma \\ref{lemTORS}.\n\\end{proof}\n\n\\begin{corollary}\\label{corTORS2}\nLet $X$ be a connected reduced scheme over $k$ and $G$ a finite local $k$-group\nscheme. Let $T\\,\\longrightarrow\\, X$ be a finite \\'etale cover, and let $Y\n\\,\\longrightarrow\\, X$ be a $G$--torsor. If\n$Y\\,\\longrightarrow\\, X$ is Nori-reduced and $T$ is connected, then the $G$--torsor\n$$T\\times_X Y\\,\\longrightarrow\\, T$$ is also Nori-reduced. In particular the homomorphism\n$\\pi^{\\rm loc}(T)\\,\\longrightarrow\\, \\pi^{\\rm loc}(X)$ is faithfully flat.\n\\end{corollary}\n\n\\begin{proof} This follows from Corollary \\ref{corTORS} and the fact that there exist a\nfinite \\'etale $k$--group scheme $H'$ and an $H'$--torsor $T'\\,\\longrightarrow\\, X$\nthat dominates $T\\,\\longrightarrow\\, X$.\n\\end{proof}\n\n\\begin{remark}\\label{remNORI}\nIn \\cite{N2}, Nori proved that if\n$i\\,:\\,U\\,\\longrightarrow\\, Y$ is an open immersion between connected and\nreduced schemes with $Y$ normal, then the morphism $\\pi(U)\\,\\longrightarrow\\,\n\\pi(Y)$ induced by $i$ is faithfully flat (see \\S~II, Proposition 6 and\nits corollaries). Consequently, the homomorphism\n$\\pi^{\\rm loc}(U)\\,\\longrightarrow\\, \\pi^{\\rm loc}(Y)$ induced by $i$ is\nalso faithfully flat. \n\\end{remark}\n\n\\begin{notation}\\label{notFINDEX}\nLet $k$ be a field and $u\\,:\\,M\\,\\longrightarrow\\, G$ a $k$-group scheme\nhomomorphism. We say that $u$ is of finite index if the following\nproperty is satisfied: for any $k$-group scheme $Q$ and any faithfully\nflat morphism of $k$-group schemes $G\\,\\longrightarrow\\, Q$, if the group\nscheme image of $M\\to Q$ is finite then $Q$ is also finite.\n\\end{notation}\n\n\\begin{lemma}\\label{lemCLinsep}\nLet $f\\,:\\,X\\,\\longrightarrow\\, Y$ be a finite purely inseparable morphism between normal\nintegral schemes. Then the homomorphism $$\\pi(X)\\,\\longrightarrow\\,\n\\pi(Y)$$ is of finite index, while $\\pi^{\\rm \\acute{e}t}(X)\\,\\longrightarrow\\,\n\\pi^{\\rm \\acute{e}t}(Y)$ is faithfully flat. So in particular $\\pi^{\\rm loc}(X)\\,\\longrightarrow\\,\n\\pi^{\\rm loc}(Y)$ is of finite index. \n\\end{lemma}\n\n\\begin{proof}\nWe assume that $char(k)\\,=\\,p\\,>\\,0$. We observe that under the above assumptions the morphism\n$f$ is surjective (see \\cite[Ex. 5.3.9]{Li}). Let us first consider the case where $Y\\,=\\,X$ with $f\\,:=\\,F_Y$ being \nthe absolute Frobenius morphism of $Y$. Let $T \\,\\longrightarrow\\, Y$ be the universal \n$\\pi(Y)$--torsor of $Y$ (it is a scheme, as all the transition morphisms\nare affine), where $\\pi(Y)$ is the fundamental group scheme of Nori. We set\n$$T^{(p)}\\,:= \\, T\\times_Y Y$$ via the Frobenius $F_Y$ of $Y$. As\nusual, $F_{T\/Y}\\,:\\, T\\,\\longrightarrow\\, T^{(p)}$ is the \nrelative Frobenius. The relative Frobenius commutes with base change, so if we \npull back over $x\\,:\\,{\\rm Spec}(k) \\,\\longrightarrow\\, Y$ (a fixed closed point) what we obtain is the relative\nFrobenius\n$$ F_{\\pi(Y)\/{\\rm Spec}(k)}\\,:\\, \\pi(Y) \\,\\longrightarrow\\, \\pi(Y)^{(p)}\n\\,\\simeq\\, \\pi(Y)\\, ,$$\nwhere the last isomorphism clearly follows from the fact that $k$ is algebraically closed, whence perfect; thus,\nin particular, $$F_{T\/Y}\\,:\\, T \\,\\longrightarrow\\, T^{(p)}$$\nis the natural morphism from the universal torsor to\nthe pro-finite torsor obtained after pulling back. The same holds for any\ntorsor: so let $P$ be a $Q$-torsor, where $\\pi(Y)\\,\\longrightarrow\\, Q$ is a\nfaithfully flat $k$-group scheme homomorphism; then we have the\nrelative Frobenius\n$$ F_{Q\/{\\rm Spec}(k)}\\,:\\, Q \\,\\longrightarrow\\, Q^{(p)}\\,\\simeq\\, Q\\, ,$$ which\nfactors as $Q\\,\\longrightarrow\\, F\\,\\longrightarrow\\, Q$ (where $Q\\,\n\\longrightarrow\\, F$ is faithfully flat and $F\\,\n\\longrightarrow\\, Q$ is a closed immersion). Since the kernel is finite, if\nwe assume $F$ to be finite then $Q$ is also finite thus\n$F_{\\pi(Y)\/{\\rm Spec}(k)}$ is of finite index.\n\nNow $F_{\\pi(Y)\/{\\rm Spec}(k)}$ is a finite endomorphism and this is sufficient to conclude that it is of finite index. As \\'etale\ntorsors are not \nmodified by the Frobenius, it follows that $ F_{\\pi^{\\rm \\acute{e}t}(Y)\/{\\rm\nSpec}(k)}$ is an isomorphism. What has been \nproved for $f\\,=\\,F_Y$ still holds, of course,\nfor $f\\,=\\,F_Y^m$, the Frobenius iterated $m$ times. So now we consider the general case where $f\n\\,:\\,X\\,\\longrightarrow\\,Y$ is the given purely \ninseparable morphism. Then there exist a positive integer $m$ and a morphism $h\\,:\\, Y\\,\\longrightarrow\\, X$ such that \n$$f\\circ h \\,=\\,F_Y^m \\,:\\, Y \\,\\longrightarrow\\, Y$$\n(the absolute Frobenius morphism iterated $m$ times). We consider the pullback \n$$T_X \\,:=\\, T\\times_Y X$$ and the universal $\\pi(X)$--torsor $P\\,\n\\longrightarrow\\, X$ on $X$. There are natural morphisms\n$P\\,\\longrightarrow\\, T_X$ and $$u\\,:\\,\\pi(X)\\,\\longrightarrow\\, \\pi(Y)\\, .$$\nPulling back further to $h\\,:\\, Y\n\\,\\longrightarrow\\,X$, the following factorization is obtained:\n$$\\xymatrix{\\pi(Y) \\ar@\/_1pc\/[rr]_{F^m_{\\pi(Y)\/{\\rm Spec}(k)}} \\ar[r] & \\pi(X) \\ar[r]^u & \\pi(Y).}$$\n\nThe previous discussion yield the following:\n\n\\begin{itemize}\n \\item if we assume that $\\pi(Y)$ and $\\pi(X)$ are both \\'etale, this implies\nthat $$u\n\\,:\\, \\pi(X)\\,\\longrightarrow\\, \\pi(Y)$$ is faithfully flat;\n \\item otherwise we can only conclude that $u$ is of finite index, which is all\nthat we can expect.\n\\end{itemize}\nThis is enough to conclude the proof.\n\\end{proof}\n\n\\begin{lemma}\\label{lemCL2}\nLet $f\\,:\\,X\\,\\longrightarrow\\, Y$ be a dominant morphism between normal\nintegral schemes. Then the homomorphisms\n$$\\varphi^{\\rm loc}:\\pi^{\\rm loc}(X)\\,\\longrightarrow\\,\n\\pi^{\\rm loc}(Y)\\ \\ \\text{ and } \\ \\ \\varphi^{\\rm \\acute{e}t}:\\pi^{\\rm \\acute{e}t}(X)\\,\n\\longrightarrow\\,\n\\pi^{\\rm \\acute{e}t}(Y)$$ induced by $f$ are of finite index. \n\\end{lemma}\n\n\\begin{proof}\nThis is inspired by \\cite[Lemme 2]{CL} (see also \\cite[Lemme 4.4.17]{De}\nfor the zero characteristic case). \n\nLet $t$ be a closed point of the generic fiber of $f$, and let $T$ denote its Zariski\nclosure in $X$. The morphism $f$ induces a generically finite morphism $f_{\\vert T}\\,\n:\\,T \\longrightarrow\\, Y$: indeed its generic fiber has relative dimension\nzero and it this thus a finite number of points; therefore, there exists an\nopen dense subscheme $U\\,\\subseteq\\, Y$ such that\n$$f'\\,:\\,V\\,\\longrightarrow\\, U\\, ,$$\nwhere $V\\,:=\\,T\\times_Y U$, is a finite morphism. Hence there exist a scheme $W$ and two\nfinite morphisms\n$$e\\,:\\,V\\,\\longrightarrow\\, W\\ \\ \\text{ and }\\ \\ i\\,:\\,W\\,\\longrightarrow\\, V$$ such that $i\\circ e\\,=\\,f'$, where $i$ is purely inseparable\nand $e$ is generically \\'etale. This implies\nthat there exists an open dense subscheme $W'\\,\\subseteq\\, W$ such that\n$$e'\\,:\\,V'\\,\\longrightarrow\\, W'\\, ,$$ where $V':=V\\times_W W'$, is a finite \\'etale cover. In what follows \nwe study the morphism\n$$\\varphi^{\\rm loc}\\,:\\,\\pi^{\\rm loc}(X)\\,\\longrightarrow\\,\n\\pi^{\\rm loc}(Y)\\, ,$$ the (similar) details for \n$\\varphi^{\\rm \\acute{e}t}$ are left to the reader. By Corollary\n\\ref{corTORS2} the morphism $\\pi^{\\rm loc}(V')\\,\\longrightarrow\\,\n\\pi^{\\rm loc}(W')$ induced by $e'$ is faithfully flat while the morphism \n$$\\pi^{\\rm loc}(W)\\,\\longrightarrow\\, \\pi^{\\rm loc}(U)$$\ninduced by $i$ is of finite index by Lemma \\ref{lemCLinsep} and clearly \n$\\pi^{\\rm loc}(W')\\,\\longrightarrow\\, \\pi^{\\rm loc}(W)$ is faithfully flat (see Remark \\ref{remNORI})\nso the composition $$u:\\pi^{\\rm loc}(V')\\,\\longrightarrow\\, \\pi^{\\rm loc}(U)$$ is of finite index. Now\nwe have the diagram of homomorphisms\nof local fundamental group schemes \n$$\\xymatrix{\\pi^{\\rm{loc}}(V')\\ar[r]\\ar[d]_{u} & \\pi^{\\rm{loc}}(X)\\ar[d]^{\\varphi^{\\rm loc}}\n\\\\ \\pi^{\\rm{loc}}(U)\\ar[r]^{v} & \\pi^{\\rm{loc}}(Y). }$$\nNow $u$ is of finite index and the homomorphism $v$ is\nfaithfully flat (see, again, Remark \\ref{remNORI}). Hence $\\varphi^{\\rm loc}$ is finite index.\n\\end{proof}\n\nIn \\cite{MS} Mehta and\nSubramanian proved that $$\\pi(X\\times Y)\\,=\\, \\pi(X)\\times \\pi(Y)$$ for two connected, proper and reduced schemes \n$X$ and $Y$. If one of the two schemes ($X$ or $Y$) is not proper anymore then the previous formula may not hold. \nHowever a weaker result will be sufficient for our purposes; the following proposition can be found in \n\\cite{N2}, Chapter II, Proposition 9, here we suggest a different approach :\n\n\\begin{proposition}\\label{propNEWPROD}\nLet $Y$ be an integral scheme over $k$, then the homomorphism $$\\varphi\\,:\\,\n\\pi(\\mathbb{P}^1\\times Y)\\,\\longrightarrow\\, \\pi(Y)$$ induced by \nthe projection $p_2\\,:\\,\\mathbb{P}^1\\times Y\\,\\longrightarrow\\, Y$ is an\nisomorphism.\n\\end{proposition}\n\n\\begin{proof}\nIt is clear that \n$\\varphi$ is faithfully flat as $p_2$ has a section, so we only need to prove\nthat given a finite $k$--group scheme \n$G$ and a $G$--torsor $T\\,\\longrightarrow\\, \\mathbb{P}^1\\times Y$, there exists a\n$G$--torsor $$T'\\,\\longrightarrow\\,\nU$$ whose pullback to $\\mathbb{P}^1\\times Y$ is the given one. First we briefly recall that the fundamental group scheme of $Y$ at a $k$-point $y$ is the automorphism group scheme of the fiber functor $y^{\\ast}$ on the category of essentially finite vector\nbundles, as described in \\cite{N1}. Let $Rep_k(G)$ \ndenote the category of $k$--linear finite dimensional representations of $G$. Then \nassociated to our $G$--torsor $T\\,\\longrightarrow\\, \\mathbb{P}^1\\times Y$ there is a\nfiber functor \n$$F_T\\,:\\,Rep_k(G)\\,\\longrightarrow\\, \\mathcal{Q}coh(\\mathbb{P}^1\\times Y)$$ by a\nfundamental result in \nTannakian theory (recalled for instance in \\cite[Proposition (2.9)]{N1}). From this \nwe will construct a functor $$F\\,:\\,Rep_k(G)\\,\\longrightarrow\\,\n\\mathcal{Q}coh(Y)\\, .$$ For \nany $G$--module $V$, set $$F(V)\\,=\\,(p_2)_*(F_T(V))\\, .$$ We first observe that $F(V)$ is a \nvector bundle: when restricted to $\\mathbb{P}^1$, clearly $F_T(V)$ is an essentially finite \nvector bundle over the projective line, thus trivial, whence $H^1(\\mathbb{P}^1, \nF_T(V))\\,=\\,0$, and the evaluation homomorphism $$(p_2)^*(p_2)_* (F_T(V))\n\\,\\longrightarrow\\, (F_T(V))$$ is an isomorphism.\nMoreover $F$ is compatible with the operations of taking tensor \nproducts, direct sums and duals. Hence $F$ is a fiber functor and we can \nassociate to it a $G$--torsor $T'\\,\\longrightarrow\\, U$ which is the desired one\nsince its pullback to $\\mathbb{P}^1\\times Y$ is isomorphic to\n$T\\,\\longrightarrow\\, \\mathbb{P}^1\\times Y$.\n\\end{proof}\n\nWe now recall that for any $0\\, \\leq\\, \\,\\nu\\, \\leq\\, \\, \\dim(X)$, \nthere is a point in $X(\\Omega)$, where $\\Omega$ in the\nalgebraic closure of the function field of $X$, whose Zariski closure in $X$\nis of dimension $\\nu$.\n\n\\begin{theorem}\\label{teoMAIN}Let $k$ be an algebraically closed field and $X$\na normal, rationally chain connected $k$--scheme. Then $\\pi^{\\rm loc}(X)$ is finite.\n\\end{theorem}\n\n\\begin{proof}\nSince $X$ is rationally chain connected, there exists a chain of rational curves\nconnecting a rational point $x_0\\,\\in\\, X(k)$ to a generic point $x_m\\,\\in\\, X(\\Omega)$,\nwhere $\\Omega$ is the algebraic closure of the function field of $X$. According to Lemma\n\\ref{lemCL1} there exists a sequence of integral subvarieties $V_0\\, , \\cdots\\, , V_m$\nof $X$ where $V_0\\,=\\,x_0$ and $V_m\\,=\\,X$ and for every integer $i\\,\\in\\,\n\\{0, \\cdots , m-1\\}$ a family of rational curves\n$$\nF^i\\,:\\,\\mathbb{P}^1_k\\times T_i \\,\\longrightarrow\\, X\n$$\nwith $T_i$ normal and projective, such that the morphisms\n$$F^i_0\\,:\\,T_i\\,\\longrightarrow\n\\, X \\ \\ \\text{ and }\\ \\ F^i_{\\infty}\\,:\\, T_i\\,\\longrightarrow\\, X\\, ,$$\ndefined by $F^i_0(t)\\,:=\\,\nF^i(0,t)$ and $F^i_{\\infty}(t)\\,:=\\,F^i(\\infty,t)$, are dominant on $V_i$ and $V_{i+1}$\nrespectively. If $V_i$ is not normal then we can consider an open normal\nsubscheme $V_i'\\,\\subset\\, V_i$ and the pullback \n$$\\xymatrix{T_i'\\ar[r]\\ar[d] & V_i'\\ar[d] \\\\ T_i\\ar[r] & V_i. }$$\nIn a similar way, if $V_{i+1}$ is not normal then we can consider an open normal\nsubscheme $V_{i+1}'\\,\\subset\\, V_{i+1}$ and its pullback, as before, that we will call\n$T_i^{''}$. This will not affect $V_0$ and $V_m$ of course. So this induces the\nfollowing commutative diagram on local group schemes:\n$$\\xymatrix{ & & \\pi^{\\text{loc}}(T_i')\\ar[ld]_{\\alpha}\\ar[rd]^{u}\n & & \\\\ & \\pi^{\\text{loc}}(T_i)\\ar[ld]_{\\beta} & &\n\\pi^{\\text{loc}}(V_i')\\ar[dr]^{v} & \\\\ \\pi^{\\text{loc}}(\\mathbb{P}^1_k\\times\nT_i) \\ar[rrrr]^{\\pi(F^i)} & & & & \\pi^{\\text{loc}}(X) \\\\ &\n\\pi^{\\text{loc}}(T_i)\\ar[lu]^{\\gamma} & & \\pi^{\\text{loc}}(V_{i+1}')\\ar[ru]_{w}\n& \\\\ & & \\pi^{\\text{loc}}(T_i^{''})\\ar[lu]^{\\delta}\\ar[ru]_{z} & & }$$\nWe avoid to put the index $i$ on the morphisms not to make notation too\nheavy. We know that $\\pi^{\\text{loc}}(V_0)\\,=\\,0$, both $u$ and $z$ are of finite index by Lemma \\ref{lemCL2}, \nboth $\\alpha$ and $\\delta$ are faithfully flat by \nRemark \\ref{remNORI} and both $\\beta$ and $\\gamma$ are isomorphisms by Proposition \n\\ref{propNEWPROD}. So at each step we prove that the image of\n$\\pi^{\\text{loc}}(V_{i+1}')$ in $\\pi^{\\text{loc}}(X)$ is finite. The last step\nwill finally prove that $\\pi^{\\text{loc}}(X)$ is finite.\n\\end{proof}\n\n\\begin{lemma}\\label{lemLAST}Let $X$ be a rationally chain connected variety and let $f : Y \\longrightarrow X$ be an \\'etale Galois cover. Then $Y$ is rationally chain connected.\n\\end{lemma}\n\n\\begin{proof}\nFix a point $y$ of $Y$. Let $U_y$ be the subset of $Y$ that is rationally chain connected to the point $y$. Let $\\Omega$ be any algebraically closed field containing $k$, then any morphism  $g : \\mathbb{P}^1_{\\Omega}\\longrightarrow X_{\\Omega}$ lifts to a morphism\n$g' : \\mathbb{P}^1_{\\Omega}\\longrightarrow Y_{\\Omega}$ using the homotopy lifting property  because $\\mathbb{P}^1_{\\Omega}$ is simply connected. Since $X$ is also rationally chain connected, these two together imply that\n$U_y$ is both open and closed. Hence $U_y = Y$, and $Y$ is rationally chain connected.\n\\end{proof}\n\n\\begin{remark}\\label{remFANO}\nLet notations be as in Theorem \\ref{teoMAIN}, then from \\cite[Th\\'eor\\`eme]{CL} and Lemma \\ref{lemLAST} we also obtain that $\\pi(X)$ is finite. If $char(k)\\,=\\,p\\,>\\,0$, and $X$ is moreover smooth and proper, then $\\vert\\pi(X)^{\\rm \\acute{e}t}\\vert$ is coprime to $p$, as proved in \\cite{CL2}. In particular all this holds when $X$ is a Fano variety since in this case it is rationally chain connected (cf. \\cite{Ca} and \\cite{KMM}). Furthermore when $X$ is a general hypersurface of a projective space then then it is separably\nrationally connected, and by \\cite{Bi} this implies that $\\pi(X)\\,=\\,0$.\n\\end{remark}\n\n\\section*{Acknowledgments}\n\nWe thank Antoine Chambert-Loir for a useful communication. We thank the two\nreferees for comments that helped us in improving the paper. The first-named author would like to thank T.I.F.R. for \nits hospitality and Cinzia Casagrande for useful discussion. The second-named author \nthanks Universit\\'e Lille 1 and Niels Borne for hospitality. He also acknowledges the \nsupport of a J. C. Bose Fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLife processes such as cellular signaling, control and regulation\narise from complex interactions and reactions between biomolecules.\nA fundamental challenge of understanding and controlling life processes\nis that they are inherently multiscale \u2013 cellular signaling alone\ninvolves $6$ orders of magnitude in lengthscales ($0.1$ nanometers\nto $100$ micrometers) and $18$ orders of magnitude in timescales\n(femtoseconds to hours). Unfortunately, these scales are tightly coupled\n\u2013 a single-point mutation in a protein can disturb the biochemical\ninteractions such that this results in disease or death of the organism.\nNo single experimental or simulation technique can probe all time-\nand lengthscales at a resolution required to understand such a process\ncomprehensively. \n\nIn computer simulations, this dilemma can be mitigated by multiscale\ntechniques \u2013 different parts of the system are described by a high-resolution\nand a low-resolution model, and these parts are coupled to give rise\nto a hybrid simula\\textcolor{black}{tion. A famous example of such\na multiscale model in biophysical chemistry is the coupling of quantum\nmechanics and molecular mechanics (QM\/MM) \\citep{WarshelLevitt_JMB76_QMMM}.\nHere we lay the foundations for a hybrid simulation technique that\ncouples two scales that are particularly useful to model intracellular\ndynamics: a Markov state model (MSM) of the molecular dynamics (MD)\nscale that describes structural changes of biomolecules and their\ncomplexes, and the reaction-diffusion scale that describes diffusion,\nassociation and dissociation on the lengthscale of a cell. We call\nthis approach MSM\/RD, due to the combination of the simulation models\nchosen at these scales:}\n\\begin{enumerate}\n\\item \\textcolor{black}{MSMs of the molecular scale: MD simulation allows\nus to probe molecular processes at atomic detail, but its usefulness\nhas long been limited by the sampling problem. Recently, the combination\nof hard- and software for high-throughput MD simulations \\citep{ShirtsPande_Science2000_FoldingAtHome,BuchEtAl_JCIM10_GPUgrid,Shaw_Science10_Anton,DoerrEtAl_JCTC16_HTMD}\nwith MSMs \\citep{PrinzEtAl_JCP10_MSM1,BowmanPandeNoe_MSMBook,SarichSchuette_MSMBook13}\nhas enabled the extensive statistical description of p}rotein folding\nand conformation changes \\citep{NoeSchuetteReichWeikl_PNAS09_TPT,Bowman_JCP09_Villin,LindorffLarsenEtAl_Science11_AntonFolding,KohlhoffEtAl_NatChem14_GPCR-MSM},\nas well as the association of proteins with ligands \\citep{BuchFabritiis_PNAS11_Binding,SilvaHuang_PlosCB_LaoBinding,PlattnerNoe_NatComm15_TrypsinPlasticity,deSancho2015identification,kubas2016mechanism}\nand even other proteins \\citep{PlattnerEtAl_NatChem17_BarBar}. Using\nmulti-ensemble Markov models (MEMMs) \\citep{WuMeyRostaNoe_JCP14_dTRAM,RostaHummer_DHAM,WuEtAL_PNAS16_TRAM,MeyWuNoe_xTRAM},\nMSMs can be derived that even capture the kinetics of ultra-rare events\nbeyond the seconds timescale at atomistic resolution \\citep{PaulEtAl_PNAS17_Mdm2PMI,CasasnovasEtAl_JACS17_UnbindingKinetics}.\nMSM approaches can thus model the long-lived states and transition\nrates of molecular detail interactions, but the cost of atomistic\nMD sampling limits them to relatively small biomolecules and complexes. \n\\item Reaction-diffusion (RD) scale: While atomic detail is relevant for\nsome processes that affect the cellular scale, it is neither efficient\nnor insightful to maintain atomic resolution at all times for cellular\nprocesses. We choose particle-based reaction-diffusion (PBRD) dynamics\nkinetics as a reference model for the cellular scale. PBRD simulates\nparticles, representing individual copies of proteins, ligands or\nother metabolites. Particles move in space via diffusion and reactive\nspecies will react with a probability according to their reaction\nrate when being clo\\textcolor{black}{se by. Here, a reaction may represent\nmolecular processes such as binding, dissociation, conformational\nchange, or actual enzymatic reactions. PBRD acknowledges that chemical\nreactions are inherently discrete and stochastic in nature \\citep{qian2010cellular},\nand that diffusion in cells is often not fast enough to justify well-stirred\nreaction kinetics \\citep{erban2009stochastic,fange2010stochastic,takahashi2010spatio}.\nA large number of recent software packages and codes implement some\nform of PBRD \\citep{andrews2004stochastic,BiedermannEtAl_BJ15_ReaddyMM,donev2010first,donev2017efficient,hattne2005stochastic,SchoenebergNoe_PlosOne13_ReaDDy,van2005green,ZonTenWolde_PRL05_GFRD},\nsee also the reviews \\citep{mereghetti2011diffusion,SchoenebergUllrichNoe_BMC14_RDReview}.\nHydrodynamic interactions at this scale could be incorporated by particle-based\ncoupling terms \\citep{ermak1978brownian,geyer2009n}. The effect of\ncrowders and complicated boundaries such as membranes on the particle\ndiffusion can be represented by including interaction forces on the\nRD scale \\citep{SchoenebergNoe_PlosOne13_ReaDDy}.}\n\\end{enumerate}\n\\textcolor{black}{In the limit that the conformational transitions\nof all molecules are fast, the MSM dynamics of each molecule effectively\naverages, and the interaction between the molecules (e.g. association)\noccurs with suitably averaged rates, reducing the problem to PBRD.\nHowever, when the lifetimes of some conformations are long compared\nto the typical time between two molecular interactions, or even the\ntime between successive rebinding events of two molecules, the conformation\ndynamics of molecules described by the MSM part couples with the RD\ndynamics. MSM\/RD opens up the possibility to simulate and analyze\nsuch effects quantitatively. For example, bimolecular binding rates\nfrom MD-derived MSMs can be inaccurate due to periodic boundary effects\nand a short-lived dissociated state in comparison to the MSM lag-time\n\\citep{PlattnerEtAl_NatChem17_BarBar}. MSM\/RD can overcome these\nissues by extending the diffusion domain available lessening the periodic\nboundary effects and increasing the lifetime of the dissociated state. }\n\n\\textcolor{black}{The ultimate aim of MSM\/RD is to produce an efficient\nmultiscale simulation that reproduces the essential statistical behavior\nof a practically unaffordable large-scale MD simulation by employing\nonly statistics obtained from simulations of the constituent biomolecules\nin small solvent boxes. As developing a full theory involving rotational\ndiffusion, three- or more-body interactions, hydrodynamics will be\nhighly complex, we here aim to make a first step towards this goal\nby coupling MSM and RD scales for bimolecular systems without large-scale\nhydrodynamic interactions.}\n\n\\textcolor{black}{We first derive a theory of MSM\/RD for bimolecular\nsystems, as depicted in Fig. \\ref{fig:schemeMSMRD-myoglobin}. When\nthe two molecules are far from each other, they both undergo a diffusion\nprocess. When they come close to each other, molecular interactions,\nmodeled with MD-derived MSMs, need to be taken into account. We further\ndevelop an algorithm to couple the MSM and RD scales for the special\ncase of a protein interacting with a ligand, which is one of the main\nadvances in this paper. This is not a trivial undertaking since one\nneeds to solve two problems: to couple the MSM and RD part in such\na way that the correct macroscopic rates and equilibrium probabilities\nare recovered, and to develop a suitable MSM discretization such that\nthis coupling can be made. We demonstrate the validity of our theory\nand algorithms on a toy model of p}rotein-ligand interaction and on\nbinding of carbon monoxide to myoglobin.\n\n\\begin{figure}\n\\centering \n\n\\includegraphics[width=1\\columnwidth]{figs\/multiscale_scheme_simple}\n\n\\caption{Sketch of the MSM\/RD scheme. When molecules $A$ and\\textcolor{black}{{}\n$B$ are not in close proximity, they diffuse freely. When $A$ and\n$B$ are close, they merge into a complex particle $C$ which itself\ndiffuses and whose internal dynamics are encoded by coupled MSM state\ntransitions. When the molecules transition into a dissociated state,\nthey are again separated into two separately diffusing particles $A$\nand $B$ with initial positions depending on the last MSM state. Note\nthat in the dissociated state, molecules $A$ and $B$ could also\npotentially undergo conformational changes encoded in independent\nMSM state transitions.}}\n\n\\label{fig:schemeMSMRD-myoglobin} \n\\end{figure}\n\nIn related work, \\citep{sbailo2017efficient,vijaykumar2015combining}\nhave coupled MD with a diffusion scheme. The work \\citep{VotapkaAmaro_SEEKR_SPCB17}\nfurther incorporates milestoning theory \\citep{faradjian2004computing}\nto compute the local kinetic information in terms of transitions between\nmilestones via short MD runs. In contrast with their work, we do not\nemploy direct MD simulations at the ``small'' scale, but represent\nthe small scale by an MSM as this allows us to operate on roughly\nthe same timesteps for the small and the large scales. Other works\nhave proposed alternative schemes to couple random walks (MSMs) with\nBrownian diffusion schemes, some examples can be seen in \\citep{del2016discrete,flegg2012two,flegg2015convergence}.\nHowever, these works focus on specific contexts that are not directly\napplicable for coupling MD-derived MSMs with reaction-diffusion schemes.\n\n\\section{MSM\/RD: coupling Markov state models and reaction-diffusion}\n\n\\textcolor{black}{We develop a theoretical description for MSM\/RD.\nThe relevant scenarios for MSM\/RD can be classified by the number\nof interacting particles, or the related reaction order:}\n\\begin{enumerate}\n\\item \\textcolor{black}{First-order reactions: isolated diffusing particles\ncan be modeled by an MSM obtained from MD simulations in a solvent\nbox. The MSM directly translates into a set of unimolecular reactions\nthat can be implemented in standard PBRD software. As long as the\nparticles don't interact, the only effect of different states on the\ndynamics are changes between different diffusion constants\/tensors.}\n\\item \\textcolor{black}{Second-order reactions: interactions between two\nmolecules that can be modeled as bimolecular reactions including protein-ligand\nor protein-protein association ($A+B\\rightarrow C$). As soon as the\ncomplex $C$ has been formed, its dynamics may be described by state\ntransitions of an MSM of the complex.}\n\\item \\textcolor{black}{Higher-order reactions: simultaneous interactions\nbetween more than two molecules. }\n\\end{enumerate}\n\\textcolor{black}{In this work, we will focus on second-order reactions.\nFirst-order reactions are trivial state changes of a particle that\nare occurring as part of the MSM dynamics. Consistent with current\nconventions in PBRD frameworks, we follow the convention of breaking\ndown higher-order reactions to second-order reactions, although in\nSec. \\ref{sec:Conclusion} we suggest possible extensions to treat\nthese explicitly.}\n\n\\textcolor{black}{In order to derive the theory for second-order reactions,\nwe concentrate on the dynamics of two molecules, $A$ and $B$. For\nthe sake of simplicity, we assume the two molecules do not have conformational\nchanges of their own, so they can only diffuse and interact with each\nother. However, it is straightforward to extend MSM\/RD to include\nconformational changes (first-order reactions) coupled with second-order\nreactions.}\n\n\\subsection{The \\emph{ground truth} model with full dynamics}\n\n\\textcolor{black}{\\emph{Ground truth}}\\textcolor{black}{{} is a term\noften used in machine learning that refers to a reference model with\nrespect to which modeling errors are measured. In the present context,\nthe }\\textcolor{black}{\\emph{ground truth}}\\textcolor{black}{{} model\ncontains the two (or more) solute molecules whose interactions will\nbe later approximated by an MSM in a }\\textcolor{black}{\\emph{large-scale}}\\textcolor{black}{{}\nsimulation, i.e. a simulation box that is not truncated after a small\nsolvent boundary as customary for MD simulation. Importantly, there\nis no universally correct ground truth, but this model employs the\nMD simulation setup and dynamical model chosen by the user for the\nmodeling task at hand. This choice includes the MD force field, solvation\nconditions and ion concentration, the protonation state at the pH\nof interest or even constant-pH simulations \\citep{DonniniEtAl_JCTC2011_ConstantPH},\nthe treatment of electrostatics, the thermostat, the integrator and\ntime step, etc.}\n\n\\textcolor{black}{If such a large-scale model were simulated for a\nlong time or with many trajectories, it would give rise to statistical\nproperties of the solute molecules that we want to reproduce, such\nas their equilibrium constants and association rates. However, such\na simulation is in general inefficient or infeasible, and our aim\nis that to reproduce its statistical properties using an MSM\/RD model\nthat is parametrized only using small MD simulations of the constituent\nsolute molecules and complexes.}\n\n\\textcolor{black}{For simplicity, we derive the MSM\/RD theory using\nall-atom explicit solvent MD simulations with a Langevin thermostat\nas the ground truth, as this setup is frequently used for MD simulations.\nHowever, the MSM\/RD results apply more generally, e.g. to different\nchoices of thermostats or integrators, as the MSM limit for long-time\ndescription of the dynamics and the overdamped limit for long-time\nand large-scale description of the solute transport are achieved from\na large family of ground truth models.}\n\n\\textcolor{black}{Langevin dynamics evolve as:}\n\\begin{equation}\nm_{k}\\frac{d^{2}}{dt^{2}}x_{k}(t)=-\\nabla_{k}U(\\mathbf{x}_{t})-\\gamma_{k}\\frac{d}{dt}x_{k}+\\sqrt{2k_{B}T\\gamma_{k}}\\boldsymbol{\\xi}_{k}(t),\\label{eq:Langevin}\n\\end{equation}\nwhere $x_{k}$ represents the three-dimensional position of the $k^{th}$\natom in the system (including the solvent), $\\mathbf{x}_{t}=[x_{1}(t),\\dots,x_{k}(t),\\dots,x_{N}(t)]$,\n$N$ the total number of atoms, $U$ is the potential energy and $-\\nabla_{k}U$\nis the force acting on the $k^{th}$ particle, $m_{k}$ is the $k^{th}$\nparticle mass, $\\gamma_{k}$ is the $k^{th}$ damping coefficient,\nand $\\boldsymbol{\\xi}_{k}(t)$ is a Gaussian random force such that\nthe expectations of its components satisfy $E[\\xi_{k,i}(t)]=0$ (zero\nmean) and $E[\\xi_{k,i}(t)\\xi_{k,j}(s)]=\\delta_{ij}\\delta(t-s)$ (white\nnoise) with $k_{B}T$ being the thermal energy. In simulations, we\nuse finite-time-step approximations of (\\ref{eq:Langevin}) and use\nit to generate stochastic trajectories. For the theoretical analysis,\nit is more useful to look at the ensemble dynamics, i.e., the propagation\nof probability densities in time. For this, we can ask: If we start\nthe dynamical system in phase space point $\\mathbf{y}$ and let it\nrun, with which probability will we find it in a point $\\mathbf{x}$\na time $\\tau$ later? We call this probability the transfer probability\n$p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)$, and we will use it to\ndescribe the action of the ground truth dynamics \\citep{SchuetteFischerHuisingaDeuflhard_JCompPhys151_146}.\nThe transfer probability $p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)$\nsubsumes the full complexity of the MD model, in\\textcolor{black}{cluding\ninteraction energies of the molecules with each other and external\nfields, and it can be constructed regardless of which thermostat or\nintegrator is used. The propagation of prob}ability densities $\\rho(\\mathbf{x};\\,t)$\nin time is formally described by the propagator $\\mathcal{P}_{\\tau}$:\n\\begin{align}\n\\rho(\\mathbf{x};\\,t+\\tau) & =\\mathcal{P}_{\\tau}\\rho(\\mathbf{x};\\,t)\\nonumber \\\\\n & =\\int p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)\\rho(\\mathbf{y};\\,t)\\,\\mathrm{d}\\mathbf{y}\\label{eq:propagation_propagator}\n\\end{align}\nWe want to find an efficient algorithm to approximate these dynamics.\nMore specifically we want to approximate certain aspects of these\ndynamics, such as the long-time behavior.\n\nIt is often useful to consider densities relative to the stationary\ndensity $\\pi(\\mathbf{x})$ given by\n\\[\nu(\\mathbf{x};\\,t)=\\frac{\\rho(\\mathbf{x};\\,t)}{\\pi(\\mathbf{x})},\n\\]\nwhich defines the propagator relative to the stationary density, or\ntransfer operator \\citep{SchuetteFischerHuisingaDeuflhard_JCompPhys151_146}:\n\\begin{align}\nu(\\mathbf{x};\\,t+\\tau) & =\\mathcal{T}_{\\tau}u(\\mathbf{x};\\,t)\\nonumber \\\\\n & =\\int\\frac{\\pi(\\mathbf{y})}{\\pi(\\mathbf{x})}p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)u(\\mathbf{y};\\,t)\\,\\mathrm{d}\\mathbf{y}\\nonumber \\\\\n & =\\int p(\\mathbf{x}\\rightarrow\\mathbf{y};\\,\\tau)u(\\mathbf{y};\\,t)\\,\\mathrm{d}\\mathbf{y}\\label{eq:propagation_transfer_operator}\n\\end{align}\nThe third row follows from detailed balance. For reversible systems,\nwhere detailed balance is fulfilled, $\\mathcal{T}_{\\tau}$ is often\ncalled backward propagator, as it appears to evolve densities backward\nin time.\n\nWe will now introduce a scale separation by treating molecules $A$\nand $B$ different when they are close (interacting) and far apart\n(non-interacting). More specifically these scales are defined by the\ndistance between the centers of mass of $A$ and $B$, $r_{AB}$:\n\\begin{enumerate}\n\\item MSM domain: molecules are in the \\emph{interaction} region $I=\\{\\mathbf{x}\\mid r_{AB}(\\mathbf{x})<R\\}$. \n\\item RD domain: molecules are in the \\emph{outside} region $O=\\{\\mathbf{x}\\mid r_{AB}(\\mathbf{x})\\ge R\\}$.\n\\end{enumerate}\nThe definition of $R$ will be investigated later. Next, we take a\ncloser look at the dynamics valid in these respective domains.\n\n\\subsection{Markov state models for interacting molecules\\label{subsec:innerMSM}}\n\nWe consider molecules that are closer than $R$ to be interacting,\nhence we call the corresponding subset of state space $I$. The kinetics\nin $I$ are fully described by Eq. (\\ref{eq:propagation_transfer_operator}),\nwhich can be approximat\\textcolor{black}{ed by an MSM derived from\na MD simulation that fully includes $I$ (usually plus some extra\nspace, because MD simulations typically employ periodic rather than\nspherical boundary conditions). We implicitly assume that the interaction\nforces between  proteins or protein-ligand pairs have decayed to zero\nat distances $R$ or greater. Note that this assumption requires that\nthe MD simulation conducted to parametrize an MSM\/RD model has a sufficiently\nlarge simulation box, suitable electrostatics treatment and solvation\nconditions (ions etc.) such that in the dissociated state the solutes\ncan be in any orientation without significantly interacting with each\nother or with their periodic images.}\n\n\\textcolor{black}{The interaction region $I$ will here be approximated\nby an MSM. We perform a spectral decomposition of (\\ref{eq:propagation_transfer_operator}),\nassuming that there exists a unique stationary density $\\pi$ and\nthe dynamics obey detailed balance. Furthermore, we truncate the spectral\ndecomposition after a finite number of $k$ terms:\n\\begin{equation}\n\\rho(\\mathbf{x};\\,t+\\tau)\\approx\\pi(\\mathbf{x})\\sum_{i=1}^{k}\\lambda_{i}(\\tau)\\langle\\psi_{i}(\\mathbf{x}),\\rho(\\mathbf{x};\\,t)\\rangle\\psi_{i}(\\mathbf{x})\\label{eq:transfSpectral}\n\\end{equation}\nHere, $\\langle,\\rangle$ denotes the scalar product with respect to\n$\\pi(\\mathbf{x})$, $\\psi_{i}$ are the eigenfunctions of $\\mathcal{T}_{\\tau}$\nand its leading eigenvalues have the form\n\\[\n\\lambda_{i}(\\tau)=\\mathrm{e}^{-\\tau\/t_{i}},\n\\]\nwhere $t_{i}$ is a characteristic relaxation timescale. The truncation\nafter $k$ terms in Eq. (\\ref{eq:transfSpectral}) assumes that $\\tau$\nis long compared to $t_{k+1}$, where $3t_{k}<\\tau$ is sufficient\nfor practical purposes. Most of the solvent dynamics correspond to\nthe fast coordinates that are averaged out (large-scale hydrodynamics\nare not part of the MSM term, while solvent molecules with long-lived\ninteractions with the solute molecule can be considered to be part\nof that solute molecule). Now we can perform a Galerkin projection\nof the transfer operator by discretizing the phase space using basis\nfunctions $\\chi_{i}(\\mathbf{x}),\\,i=1,...,n$. In MSMs, these are\ncharacteristic functions \n\\[\n\\chi_{i}(\\mathbf{x})=\\begin{cases}\n1 & \\mathbf{x}\\in S_{i}\\\\\n0 & \\mathbf{x}\\notin S_{i}\n\\end{cases}\n\\]\nwhere the $S_{i}$ form a complete partition of phase space, i.e.\n$\\Omega=\\{S_{1}\\cup S_{2}\\cup\\cdots\\cup S_{n}\\}$. The phase space\nhas now been discretized into a finite state space. The local densities\nbecome vectors simply given by\n\\[\n\\pi_{j}=\\int_{x\\in S_{j}}\\pi(\\mathbf{x})d\\mathbf{x},\\qquad\\rho_{j}(t)=\\int_{x\\in S_{j}}\\rho(\\mathbf{x};\\,t)d\\mathbf{x}.\n\\]\nFurthermore, we want the transfer operator to be approximated by a\nmatrix. We can obtain this matrix by noting that the eigenfunctions\nof the transfer operator also become vectors in state space}\n\n\\textcolor{black}{\n\\[\n\\psi_{i}^{j}=\\frac{1}{\\int_{\\Omega}\\chi_{j}(\\mathbf{x})d\\mathbf{x}}\\int_{x\\in S_{j}}\\psi_{i}(\\mathbf{x})d\\mathbf{x}.\n\\]\nInserting these equations into (\\ref{eq:transfSpectral}), and rewriting\nit in matrix form, we obtain the Chapman-Kolmogorow equation}\n\n\\textcolor{black}{\n\\begin{equation}\n\\boldsymbol{\\rho}(t+\\tau)=\\mathbf{T}^{\\top}(\\tau)\\,\\boldsymbol{\\rho}(t),\\label{eq:MSM}\n\\end{equation}\nwith $\\lambda_{i}(\\tau)$ and $\\boldsymbol{\\psi_{i}}=[\\psi_{i}^{1},\\dots,\\psi_{i}^{n}]$\nthe $i^{th}$ eigenvalue and eigenvector of the transition probability\nmatrix $\\mathbf{T}(\\tau),$ respectively, and with $\\boldsymbol{\\rho}(t)=[\\rho_{1}(t),\\dots,\\rho_{n}(t)]$\nthe probability mass function \\citep{PrinzEtAl_JCP10_MSM1}.}\n\n\\textcolor{black}{Estimating a high-quality MSM from MD simulation\ndata can be quite complex. It typically involves (i) mapping the MD\ncoordinates to a set of features, such as residue distances, contact\nmaps or torsion angles, (ii) reducing the dimension to slow collecti}ve\nvariables (CVs), often based on the variational approach or conformation\ndynamics \\citep{NoeNueske_MMS13_VariationalApproach,NueskeEtAl_JCTC14_Variational}\nor its special case time-lagged independent component analysis (TICA)\n\\citep{PerezEtAl_JCP13_TICA,SchwantesPande_JCTC13_TICA} \u2013 see \\citep{NoeClementi_COSB17_SlowCVs,KlusEtAl_JNS17_DataDriven}\nfor an overview, (iii) optionally, embedding the resulting coordinates\nin a metric space whose distances correspond to some form of dynamical\ndistance \\citep{NoeClementi_JCTC15_KineticMap,NoeClementi_JCTC16_KineticMap2},\n(iv) discretizing the result space using data-based clustering \\citep{BowmanPandeNoe_MSMBook,SchererEtAl_JCTC15_EMMA2,HarriganEtAl_BJ17_MSMbuilder},\ntypically resulting in 100-1000 discrete states, and (v) estimating\nthe transition matrix $\\mathbf{T(\\tau)}$ or a transition rate matrix\n$\\mathbf{K}$ with $\\mathbf{T}(\\tau)=\\exp\\left(\\tau\\mathbf{K}\\right)$\nat some lag time $\\tau$, and validating it \\citep{Bowman_JCP09_Villin,PrinzEtAl_JCP10_MSM1,BucheteHummer_JPCB08,TrendelkampSchroerEtAl_InPrep_revMSM}.\nFinally, the MSM may be coarse-grained to few metastable states \\citep{KubeWeber_JCP07_CoarseGraining,HummerSzabo_JPCB15_CoarseGraining,OrioliFaccioli_JCP16_CoarseMSM,NoeEtAl_PMMHMM_JCP13}.\nThe MSM software packages PyEMMA \\citep{SchererEtAl_JCTC15_EMMA2}\nand MSMbuilder \\citep{HarriganEtAl_BJ17_MSMbuilder} can greatly help\nto simplify this process and make it reproducible.\n\nIn the case where there are well-defined meta-stable regions in phase\\textcolor{black}{{}\nspace, we can greatly reduce the number of states in the MSM. One\nway to simplify the MSM construction process above and to directly\nend up with a few-state MSM is to employ VAMPnets, where the complex\nMSM construction pipeline is replaced by a neural network that is\ntrained using the variational approach for Markov processes \\citep{MardtEtAl_VAMPnets}.\nAlternatively, one can replace the discretization step (iv) above\nby employing a core set approach that was derived in \\citep{BucheteHummer_JPCB08}\nand further analyzed in \\citep{schutte2011markov}. The essential\nidea is to define the states as cores around the metastable regions.\nDue to the metastability, the probability of finding the system outside\nof the metastable regions is very small, so to a good approximation\nthe kinetics can be described as a core-to-core jump process \\citep{schutte2011markov}.\nThis approach will be employed throughout this paper and and explained\nin more detail in Sec. \\ref{sec:MSM\/RD-implementation}.}\n\n\\subsection{Reaction-diffusion dynamics for noninteracting molecules}\n\nWhen molecules are far apart, and thus in the RD domain defined by\n$r_{AB}(\\mathbf{x})\\ge R$, they are not directly interacting. As\nthe dynamics of the two molecules are independent, it is convenient\nto only track the net diffusion of the centers of mass, $\\boldsymbol{r}_{A}$\nand $\\boldsymbol{r}_{B}$. Furthermore, we assume that the dynamics\nin the RD domain can be tracked by coarse timesteps of at least $\\Delta t$\nwhich exceeds the typical velocity autocorrelation time (picoseconds).\nAt such timescales, the fast dynamics corresponding to the solvent\nare averaged out. It is possible that even longer timesteps are made\nusing an event-based integration scheme such as first-passage kinetic\nMonte Carlo (FPKMC) algorithm, Green's function reaction dynamics\n(GFRD) or MD-GFRD \\citep{donev2010first,takahashi2010spatio,vijaykumar2015combining,van2005green,ZonTenWolde_PRL05_GFRD}.\nAt such timesteps, the Langevin equation (\\ref{eq:Langevin}) becomes\nan overdamped Langevin equation for the centers of mass of the two\nmolecules, i.e. the motion is governed by pure diffusion: \n\n\\begin{equation}\n\\frac{d\\boldsymbol{r}_{A}(t)}{dt}=\\sqrt{2D_{A}}\\boldsymbol{\\xi}_{A}(t),\\quad\\frac{d\\boldsymbol{r}_{B}(t)}{dt}=\\sqrt{2D_{B}}\\boldsymbol{\\xi}_{B}(t),\\label{eq:overdampMassCenters}\n\\end{equation}\nwhere $\\boldsymbol{\\xi}_{A}(t)$ and $\\boldsymbol{\\xi}_{B}(t)$ are\nindependent white noise vectors with each of their components satisfying\n$E[\\xi_{K,i}(t)]=0$ and $E[\\xi_{K,i}(t)\\xi_{K,j}(s)]=\\delta_{ij}\\delta(t-s)$.\n$D_{A}$ and $D_{B}$ are the net diffusion coefficients for the centers\nof mass, which can be obtained from MD simulations. In general, as\nwe are tracking the center of mass, we also need to track the rotational\ndiffusion of the molecules. However, as rotational diffusion is not\nrelevant for the examples discussed in this manuscript, we refer to\n\\citep{SchluttigEtAl_JCP08_PatchyParticleAssociation,VijaykumarEtAl_Arxiv16_AnisotropicMultiscaleGFRD}. \n\nIn the present case, we can simply fix the frame of reference in $\\boldsymbol{r}_{A}(t)$,\nassume the rotation of $A$ is slower than the diffusion of $B$,\nwhich is true for protein-ligand systems, and fix the orientation\nof the axis to that of molecule $A$. We further assume that $B$\nis a small molecule such that its orientation is not very relevant,\nas it will be the case in our implementation of the scheme. This simplifies\nEqs. \\ref{eq:overdampMassCenters} into a simple diffusion in $\\boldsymbol{r}_{B}$\nonly \n\n\\begin{equation}\n\\frac{d\\boldsymbol{r}_{B}(t)}{dt}=\\sqrt{2(D_{B}+D_{A})}\\boldsymbol{\\xi}(t),\\label{eq:COM_diffusion}\n\\end{equation}\nwith the components of $\\boldsymbol{\\xi}(t)$ satisfying $E[\\xi_{i}(t)]=0$\nand $E[\\xi_{i}(t)\\xi_{,j}(s)]=\\delta_{ij}\\delta(t-s)$. \n\n\\subsection{MSM\/RD coupled dynamics\\label{subsec:MSM\/RD-coupled}}\n\nThe present coupled model only considers interactions between up to\ntwo molecules. This is a frequent assumption in PBRD \\citep{ZonTenWolde_PRL05_GFRD,donev2010first,SchoenebergNoe_PlosOne13_ReaDDy,BiedermannEtAl_BJ15_ReaddyMM}\nbut may be restrictive from a molecular standpoint. We assume that\nsimultaneous reactions between three or more molecules such as $A+B+C\\rightarrow D$\ncan always be broken down into $A+B\\rightarrow AB;\\,AB+C\\rightarrow D$\nor other bimolecular pathways, and therefore focus on MSM\/RD involving\ntwo molecules. In order to do the coupling, as the dynamics in the\n$I$ and $O$ region are given in terms of states and coordinates\nrespectively, we need to recognize that $\\mathbf{x}$ and $\\mathbf{y}$\nin the transfer density $p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)$\ncan be either coordinates $\\mathbf{c}$ (center-of-mass position and\nperhaps orientation of the molecule) or states $s$ (metastable regions\nin the coordinate space). In order to implement the coupling, we suggest\ndefining two quantities: \n\\begin{itemize}\n\\item \\texttt{$p_{\\mathrm{entry}}\\left[\\mathbf{c}_{t}\\rightarrow\\mathbf{x}_{t+\\Delta};\\,\\Delta\\right]$},\ntransfer probability of starting in coordinates $\\mathbf{c}_{t}$\njust inside the MSM domain ($r_{AB}(\\mathbf{c}_{t})<R$) conditioned\non hitting only one state $\\mathbf{x}_{t+\\Delta}=s_{t+\\Delta}$ in\nthe MSM domain (transition event) OR on exiting once the MSM domain\n$\\mathbf{x}_{t+\\Delta}=\\mathbf{c}_{t+\\Delta}$ (return event)\n\\item $p_{\\mathrm{exit}}\\left[s_{t}\\rightarrow\\mathbf{x}_{\\mathrm{t}+\\Delta};\\,\\Delta\\right]$,\ntransfer probability of starting in state $s_{t}$ conditioned on\nexiting once the MSM domain $\\mathbf{x}_{t+\\Delta}=\\mathbf{c}_{t+\\Delta}$\n(exit event) OR hitting once any other state $(\\mathbf{x}_{t+\\Delta}=s_{t+\\Delta})$. \n\\end{itemize}\nOnce we know these transfer probabilities, we can introduce the basic\nalgorithm where $\\tau_{\\mathrm{RD}}$ and $\\tau_{\\mathrm{MSM}}$ correspond\nto the diffusion and MSM time-step, respectively: \n\n\\texttt{\\noindent}\\texttt{\\textbf{Input}}\\texttt{: Initial mode (RD\nor MSM), initial condition (coordinates $\\mathbf{c}_{0}$ or state\n$s_{0}$, respectively) and $t=0$:}\n\n\\texttt{While $t\\leq t_{\\mathrm{final}}:$}\n\\begin{enumerate}\n\\item \\texttt{If in RD mode:}\n\\begin{enumerate}\n\\item \\texttt{Propagate $\\mathbf{c}_{t}\\rightarrow\\mathbf{c}_{t+\\tau_{\\mathrm{RD}}}$\nby diffusion}\n\\item \\texttt{Update time $t\\mathrel{{+}{=}}\\tau_{\\mathrm{RD}}$}\n\\item \\texttt{If $r_{AB}(\\mathbf{c}_{t})<R$ (enter MSM domain):}\n\\begin{itemize}\n\\item \\texttt{Sample next event ($\\mathbf{x}_{t+\\Delta}$,$\\Delta$)from}~\\\\\n\\texttt{ $p_{\\mathrm{entry}}[\\mathbf{c}_{t}\\rightarrow\\mathbf{x}_{t+\\Delta};\\,\\Delta]$.}\n\\item \\texttt{If transition event:}~\\\\\n\\texttt{Map to state }$s_{t+\\Delta}=\\mathbf{x}_{t+\\Delta}$\\texttt{}~\\\\\n\\texttt{Update time $t\\mathrel{{+}{=}}\\Delta$ }~\\\\\n\\texttt{Switch to MSM mode}\n\\item \\texttt{Else (return event):}~\\\\\n\\texttt{Map to coordinates $\\mathbf{c}_{t+\\Delta}=\\mathbf{x}_{t+\\Delta}$}~\\\\\n\\texttt{Update time $t\\mathrel{{+}{=}}\\Delta$}\n\\end{itemize}\n\\end{enumerate}\n\\item \\texttt{Else (MSM mode):}\n\\begin{enumerate}\n\\item \\texttt{If $s_{t}\\neq s_{t-\\tau_{MSM}}$or previous mode $\\neq$ MSM\nmode:}\n\\begin{itemize}\n\\item \\texttt{Sample next event ($\\mathbf{x}_{t+\\Delta}$,$\\Delta$) from\n}~\\\\\n$p_{\\mathrm{exit}}\\left[s_{t}\\rightarrow\\mathbf{x}_{\\mathrm{t}+\\Delta};\\,\\Delta\\right]$\\texttt{. }\n\\item \\texttt{If exit event:}~\\\\\n\\texttt{Map to coordinates $\\mathbf{c}_{t+\\Delta}=\\mathbf{x}_{t+\\Delta}$}~\\\\\n\\texttt{Update time $t\\mathrel{{+}{=}}\\Delta$}~\\\\\n\\texttt{Switch to RD mode}\\texttt{\\textcolor{blue}{{} }}\\texttt{\\textcolor{black}{and\nbreak current loop iteration}}\n\\end{itemize}\n\\item \\texttt{Propagate $s_{t}\\rightarrow s_{t+\\tau_{MSM}}$ using the MSM}\n\\item \\texttt{Update time} $t\\mathrel{{+}{=}}\\tau_{MSM}$\n\\end{enumerate}\n\\end{enumerate}\nThere are additional issues in specific scheme implementations, such\nas estimating the unknown conditional transfer probabilities, and\nchoosing the MSM discretization and $R$ such that the overall discretization\nerror is small, among others. These issues are non-trivial and could\npotentially be tackled with different approaches. In order to quantify\nthe accuracy of a given approach, we quantify how well is our scheme\napproximating the ground truth by comparing relevant macroscopic observables.\nWe present one possible implementation of the scheme in Sec. \\ref{sec:MSM\/RD-implementation}.\n\n\\section{An MSM\/RD implementation for protein-ligand systems\\label{sec:MSM\/RD-implementation}}\n\nNow we develop an implementation of the MSM\/RD scheme for a special\nclass of systems: th\\textcolor{black}{e binding of a small ligand\nto a protein \u2013 a case that is relevant in the study of protein-drug\nbinding kinetics \\citep{schuetz2017kinetics}. While the theory described\nbefore is more general, implementations to more challenging systems\nsuch as protein-protein interaction will be treated in future contributions.\nWe begin by considering the macromolecule $A$ fixed at the origin\nwith fixed orientation and the ligand $B$ freely diffusing around\nit with an overall diffusion constant $D=D_{A}+D_{B}$. The macromolecule\nhas several possible binding sites given by some interaction po}tential.\nIn order to present the MSM\/RD scheme in detail, we distinguish three\ndifferent simulations: \n\\begin{enumerate}\n\\item \\textbf{Reference simulation} (ground truth, if available): MD simulation\nof $B$ and its interaction with $A$ in a large spherical domain\nwith radius $R_{s}$. Unfortunately, reference simulations of realistic\nsystems are in general not computationally feasible due to the time\nand lengthscales of the simulation. Nonetheless, reference simulations\nof simple systems are used to verify the MSM\/RD scheme and validate\nits use in more complex systems. \n\\item \\textbf{Small-scale simulation} (MD simulation): analogous to the\nreference simulation with the difference that $B$ is constrained\nto a small box with periodic boundary conditions, see Fig. \\ref{fig:trajectoriesIllustration}a.\nAs the potential is negligible outside this box, the main interaction\ndynamics are extracted from this simulation's data into an MSM. This\nsimulation is used to parametrize the MSM\/RD model.\n\\item \\textbf{MSM\/RD simulation} (hybrid model): couples the MSM for short-range\ninteractions derived from the MD simulation 2. with a diffusion scheme\nfor the long-range, see Fig. \\ref{fig:trajectoriesIllustration}c.\nThe goal of the scheme is to approximate the ground truth dynamics\ngiven by the reference simulation 1. \n\\end{enumerate}\n\\begin{figure*}\n\\centering\n\n(a)\\includegraphics[width=0.3\\textwidth]{figs\/cartoons\/trajectory_types_black}\n(b)\\includegraphics[width=0.3\\textwidth]{figs\/cartoons\/trajectory_types_color}\n(c) \\includegraphics[width=0.3\\textwidth]{figs\/cartoons\/MSMRD_cartoonV3}\n\n\\caption{Illustrations of a MD trajectory, its classification to extract the\nrelevant dynamics and the MSM\/RD scheme.\\textbf{ a)} Illustration\nof a trajectory of a ligand in a MD simulation within a box with periodic\nboundaries (small-scale simulation). Note that there are two metastable\nregions, e.g. binding sites on a protein, where the ligand stays for\na longer time.\\textbf{ b)} Illustration of truncation and classification\nof the trajectory. The MSM domain is chosen so the interaction potential\nis effectively zero outside this region (bath state); the cores $X$\nand $Y$ are chosen to represent the metastable regions in phase space.\nThe truncated trajectories are classified into entry trajectories\n(green), transition trajectories (red) and exit trajectories (blue),\nwhich are used for the coupling in the MSM\/RD scheme. In order to\nobtain the MSM for the MSM\/RD scheme, the system is also classified\ninto three states, the bath state and the two cores $X$ and $Y$;\nit is also shown when the transition between these states occur along\na trajectory. \\textbf{c)} Representation of the MSM\/RD scheme. The\nfull trajectories from the MD-simulation are used to derive an MSM\nto model the dynamics in the MSM domain. The entry and exit trajectories\nfrom the MD-simulation are used to couple the Brownian dynamics in\nthe diffusion domain with the dynamics in the MSM domain. }\n\n\\label{fig:trajectoriesIllustration} \n\\end{figure*}\n\n\\subsection{Estimation}\n\nIn order to parametrize the MSM\/RD scheme, we need to estimate quantities\nfrom the small-scale simulation 2. that characterize the state-to-state\ndynamics and the coupling between the MSM and RD domain. The state\nto state dynamics are estimated using an MSM, and the coupling is\ngiven in terms of entry and exit events from the MSM domain. These\nmight happen on different timescales, so we would like to be free\nfrom the fixed time-step the MSM requires to be well equilibrated.\nTherefore, we use trajectory statistics for entry and exit events.\n\n\\subsubsection{MSM}\n\nAs a first step for the construction of the Markov model and MSM\/RD\nparametrization, we need to find a discrete representation of the\nunderlying data. In this work we use the core MSM approach \\citep{schutte2011markov},\nwhich requires the definition of cores as metastable regions of phase\nspace. Cores are given by spherical domains around the metastable\nregions in the MD simulation and can be found using a clustering algorithm.\nIn the core MSM approach a discrete trajectory is constructed by assigning\nthe last visited state-index to each point in the trajectory. Note\nthe trajectory may leave the core of a given state and re-enter multiple\ntimes without transitioning to other states. Using this discretization\ntechnique we truncate the discrete trajectory into three types of\ntrajectories as shown in Fig. \\ref{fig:trajectoriesIllustration}b:\n$i)$ entry trajectories that start just inside the MSM domain and\neither leave the domain next or hit a core inside the domain. $ii)$\ntransition trajectories that start in a state and hit another state\nas next event and $iii)$ exit trajectories that start in a state\nand leave the MSM domain as next event. These trajectories are used\nto estimate the transfer densities and to parametrize the MSM\/RD simulation.\n\nThe MSM for the short-range interactions is built using the full discrete\ntrajectories and the exit trajectories (Fig. \\ref{fig:trajectoriesIllustration}a).\nWe follow the methods from \\citep{PrinzEtAl_JCP10_MSM1} to estimate\na transition matrix $\\mathbf{T}(\\tau)$, where the entries are the\ntransition probabilities $T_{i,j}$ from state $i$ to $j$. Using\nthe discrete trajectories, we create count matrices $C_{i,j}^{\\mathrm{full}}(\\tau)$\nfrom the complete data set and $C_{i,j}^{\\mathrm{exit}}(\\tau)$ from\nthe exit trajectories, which count all the transitions from state\n$i$ to $j$ at a lag time $\\tau$ observed in the respective datasets.\nAs the coupling between the MSM and RD domain is handled separately,\nthe MSM dynamics only accounts for transitions amongst the cores and\ntherefore the counts arising from exit trajectories have to be subtracted\n\\begin{align}\nC_{i,j}(\\tau)=C_{i,j}^{\\mathrm{full}}(\\tau)-\\delta_{i,j}C_{i,j}^{\\mathrm{exit}}(\\tau),\\label{eqn:countMatrix}\n\\end{align}\nwhere $\\delta_{i,j}$ denotes the Kronecker delta. We then use a maximum\nlikelihood estimator to obtain a transition matrix from the given\ncounts $C_{ij}$. Note that here we have chosen an irreversible estimator,\nas we can no longer assume that detailed balance holds for this count\nmatrix.\n\n\\subsubsection{Entering the MSM domain\\label{subsec:EnterMSM}}\n\n The protocol to enter the MSM domain from the RD domain is constructed\nwith the entry trajectories as defined above. It consists of generating\na list $L_{\\mathrm{entry}}=\\{\\mathbf{c}_{\\mathrm{entry}},\\mathbf{x}_{\\mathrm{end}},\\Delta\\}$\nof all start coordinates $\\mathbf{c}_{\\mathrm{entry}}$(just inside\nthe MSM domain) and endpoints $\\mathbf{x}_{\\mathrm{end}}$ of entry\ntrajectories and their corresponding times $\\Delta$. The endpoints\nmay be either MSM states or coordinates in the RD domain, see Fig.\n\\ref{fig:trajectoriesIllustration}b. The ensemble of trajectories\nin this list estimates the conditional transfer probability \\texttt{$p_{\\mathrm{entry}}\\left[\\mathbf{c}_{t}\\rightarrow s_{t+\\Delta};\\,\\Delta\\right]$}\n(Sec. \\ref{subsec:MSM\/RD-coupled}) for several times $\\Delta$. In\nthe MSM\/RD simulation samples are drawn from this list of entry points.\n\n\\subsubsection{Exiting the MSM domain\\label{subsec:ExitMSM}}\n\nFor each state $\\mathbf{s}$ of the MSM, we collect all exit and transition\ntrajectories and save their end coordinate or state along with their\nrespective exit time in the lists $L_{\\mathrm{exit},s}=\\{\\mathbf{c}_{exit},\\Delta\\}$\nand $L_{\\mathrm{trans},s}=\\{s_{\\mathrm{trans}},\\Delta\\}$. The ensemble\nof trajectories in these list estimates the conditional transfer probability\n\\texttt{$p_{\\mathrm{exit}}\\left[s_{t}\\rightarrow\\mathbf{c}_{\\mathrm{t}+\\Delta};\\,\\Delta\\right]$\n}(Sec. \\ref{subsec:MSM\/RD-coupled}) for several times $\\Delta$.\nThe probability of an exit event $P_{\\mathrm{exit},s}$ is simply\nestimated as the ratio of exiting trajectories over the total numbers\nof trajectories,\n\n\\begin{align}\nP_{\\mathrm{exit},s}=\\frac{\\#\\text{\\text{ of trajectories in} }L_{\\mathrm{exit},s}}{\\#\\text{\\text{ of trajectories in} }L_{\\mathrm{exit},s}\\text{ and }L_{\\mathrm{trans},s}}.\\label{eqn:exitProbability-1}\n\\end{align}\n\nThe MSM\/RD scheme for this implementation is based on the scheme from\nSec. \\ref{subsec:MSM\/RD-coupled} and is shown in the Appendix.\n\n\\subsection{Verification of the MSM\/RD scheme}\n\nIn order to verify the MSM\/RD scheme, we use systems where a reference\nsimulation is available. We verify the internal dynamics by comparing\nthe first passage times (FPTs) distributions and mean first passage\ntimes (MFPTs) for each pair of metastable states within region $I$\nbetween the MSM\/RD and reference simulations. We estimate the ground\ntruth MFPTs by computing the FPTs $t_{i,j}^{\\textrm{ref}}$, where\nthe initial conditions are chosen as the minima $\\mu_{i}$ and the\nsystem is propagated following the reference simulation until hitting\nstate $j$ (conditioned on not leaving the MSM domain). For the MSM\/RD\nscheme, we compute the FPTs $t_{i,j}^{\\textrm{MSM}}$ by placing the\nparticle in state $i$ and propagating the system following the MSM\/RD\nscheme until state $j$ is hit. If the particle exits the MSM domain\nbefore reaching state $j$, the trajectory is not taken into account.\nWhen a sufficiently large sample is generated, we can estimate the\ndistributions of FPTs by histograms. The MFPTs are estimated as $\\tau_{ij}^{\\textrm{ref}}$=$\\overline{t_{ij}^{\\textrm{ref}}}$\nand $\\tau_{ij}^{\\textrm{MSM}}$=$\\overline{t_{ij}^{\\textrm{MSM}}}$,\nrespectively. The MFPT relative error between the MSM\/RD and the reference\nsimulations is estimated as \n\\begin{align}\n(E_{\\text{rel}})_{\\mathrm{ij}}=\\frac{\\tau_{ij}^{\\textrm{ref}}-\\tau_{ij}^{\\mathrm{MSM}\\mathscr{}}}{\\tau_{ij}^{\\textrm{ref}}}.\\label{eqn:relativeError}\n\\end{align}\nIn order to verify the coupling between the RD and MSM domain, we\nalso estimate and compare the unbinding rate, binding rate and equilibrium\nconstant. The two latter are calculated for different particle concentrations\n$c$ by fixing the radius $R_{s}$ of the simulation domain such that\n$c=1\/V_{\\text{RD}},$with $V_{\\mathrm{RD}}$ the volume of the RD\ndomain.\n\n\\section{Results\\label{sec:results}}\n\nIn this section, we implement the MSM\/RD scheme from Sec. \\ref{sec:MSM\/RD-implementation}\nin two systems. The first is a simple model of a ligand diffusing\nin a potential landscape, which is used to verify that the MSM\/RD\nscheme reproduces the correct dynamics. The second corresponds to\na more realistic MD system, where we study the binding of carbon monoxide\nto myoglobin.\n\n\\subsection{Ligand diffusion in potential landscape\\label{subsec:3D-diff-pot}}\n\nWe implement the MSM\/RD scheme in a simple model, where the reference\nsimulation is available. The model consists of a ligand $B$ under\nover-damped Langevin dynamics in a three-dimensional potential landscape\n\\begin{equation}\n\\frac{d\\mathbf{x}(t)}{dt}=-\\frac{1}{\\gamma}\\nabla U(\\mathbf{x})+\\sqrt{2D}\\boldsymbol{\\xi}(t),\\label{eq:overdLangevin}\n\\end{equation}\nwith $U$ the interaction potential with some macromolecule $A$ fixed\nat the origin, $\\gamma$ the damping, and each component of the noise\nsatisfies $E[\\xi_{i}(t)]=0$ and $E[\\xi_{i}(t)\\xi_{j}(s)]=\\delta_{ij}\\delta(t-s)$\nwith $D=k_{B}T\/\\gamma$ the diffusion coefficient. A trajectory density\nplot of the potential landscape chosen is shown in Fig. \\ref{fig:diff3Dpot}a,\nand it consists of nine Gaussians with different depths and widths\n\\begin{align}\nU(\\mathbf{r})=-\\sum_{i=1}^{9}s_{i}\\mathcal{N}(\\mathbf{\\boldsymbol{\\mu}}_{i},\\mathbf{\\boldsymbol{\\Sigma}}_{i}),\\label{eqn:potentialLandscape}\n\\end{align}\nwhere $\\mathcal{N}(\\boldsymbol{\\mu}_{i},\\boldsymbol{\\Sigma}_{i})$\ndenotes a Gaussian centered at minimum $\\boldsymbol{\\mu}_{i}$ with\ncovariance matrix $\\boldsymbol{\\Sigma}_{i}$, $s_{i}$ denotes a scale\nfactor. The small-scale simulation consists of Euler-Maruyama numerical\nrealizations of Eq. (\\ref{eq:overdLangevin}) under this potential\nconstrained to a box with an edge length of $6$ units with periodic\nboundary conditions. The reference simulation is analogous to the\nsmall-scale simulation with the difference that it uses a larger spherical\ndomain with reflective boundary conditions at a range of radii corresponding\nto simulations at different ligand concentrations. \n\n\\subsubsection{Parametrization of the MSM\/RD scheme}\n\nWe use a radiu\\textcolor{black}{s of $R=2.5\\:\\mathrm{nm}$ for the\nMSM domain ($I$ region) since outside this domain the potential (Eq.\n\\ref{eqn:potentialLandscape}) is essentially zero. We generate $120$\nsmall-scale simulation trajectories, each with a length of $10^{7}$\nsteps, a time-step of $\\Delta t=10^{-4}\\:\\mathrm{ns}$, and sampled\nevery tenth step. This results in a total simulation time of $t=1.2\\cdot10^{5}\\:\\mathrm{ns}$. }\n\n\\textcolor{black}{The cores are defined as spheres with radius $0.2\\:\\mathrm{nm}$\naround the minima $\\boldsymbol{\\mu}_{i}$, and the count matrix of\ntransition between cores is generated from the trajectories following\nEq. (\\ref{eqn:countMatrix}). A maximum likelihood estimator (implemented\nin PyEMMA \\citep{SchererEtAl_JCTC15_EMMA2}) is then applied to the\ncount matrix to yield the MSM. From the trajectories, we also generate\nthe lists $L_{\\mathrm{entry}}$, $L_{\\mathrm{exit},s}$, $L_{\\mathrm{trans},s}$\nand $P_{\\mathrm{exit},s}$, introduced in Secs. \\ref{subsec:EnterMSM}\nand \\ref{subsec:ExitMSM}. We then estimate the timescales of the\neigenmodes for different MSM lag times to test how well the underlying\nprocess is estimated by the MSM. The timescales have small variations\nfor different lag times (Fig. \\ref{fig:diff3Dpot}d), which means\nthe system can be considered Markovian for all lag times. However,\nwe have to be careful not to choose the lag time too large, such that\nrelevant fast timescales are neglected resulting in significant errors.\nFor all further analyses, we consider a lag time of $\\tau_{MSM}=500\\Delta t=0.05\\:\\mathrm{ns}$\nto be a}n optimal compromise.\n\n\\begin{figure*}\n\\centering\n\n(a) \\includegraphics[width=0.3\\textwidth]{figs\/cartoons\/meatballs_potential}\n(b)\\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/internalDynamicsWhiteCenter_notitle}\n\\textcolor{blue}{(c)}\\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/internalDynamicsHistograms_units}\n\n\\textcolor{blue}{(d)} \\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/impliedTimescales_resamplingError_units}\n\\textcolor{blue}{(e)}\\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/Kon_units}\n\\textcolor{blue}{(f)}\\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/Keq_units}\n\n\\caption{Visualization and verification results for the simple model of ligand\ndiffusing in a potential landscape.\\textbf{ (a)} Density plot of the\nposition of the ligand in the three dimensional potential. Red indicates\nregions of higher density while blue indicates regions of lower density.\n\\textbf{(b)} Relative error of MFPTs conditioned on not leaving the\nMSM domain between the MSM\/RD and the reference simulation. \\textbf{(c)\n}Comparison of first passage times distribution histograms for the\ntransitions with the highest error in (b). The left pane corresponds\nto transitions with negative relative error, and the right pane to\ntransitions with positive relative error. \\textbf{(d)} Implied timescales\nof the MSM. The shaded area represents the standard deviation of the\nbootstrapping sample. We observe well converged timescales for all\nconsidered lag times.\\textbf{ (e)} The rate $k_{on}$ as function\nof the concentration of the system for the MSM\/RD and reference simulations.\n\\textbf{(f)} Same plot as e) but for the logarithm of the equilibrium\nconstant $\\log(K_{eq})$. The error bars in (e) and (f) represent\nthe 95\\% confidence interval using a bootstrapping approach.}\n\n\\label{fig:diff3Dpot}\n\\end{figure*}\n\n\\subsubsection{Comparison of dynamic properties}\n\nIn order to compute the binding rate, we calculate the first passage\ntime from a uniforml\\textcolor{black}{y sampled location close to\nthe boundary $r=R_{S}-\\delta$ to any MSM state. We choose $\\delta=0.05\\:\\mathrm{nm}$\nand use $10^{4}$ simulations to average and estimate the $\\text{MFPT}_{\\mathrm{on}}$,\nfrom which we calculate the binding rate as $k_{\\mathrm{on}}^{*}=1\/\\text{MFPT}_{\\mathrm{on}}$.\nThis procedure is performed for both the MSM\/RD and the reference\nsimulation, and we observe excellent agreement between the two (Fig.\n\\ref{fig:diff3Dpot}e).}\n\n\\textcolor{black}{For the unbinding rate, we consider the inverse\nprocess by starting in an MSM state and propagating the dynamics until\ncrossing a boundary defined by a sphere with radius $2.7\\:\\mathrm{nm}>R$.\nWe obtain a reference value of $0.402_{0.400}^{0.404}\\:\\mathrm{ns^{-1}}$\n(Sub- and superscript indicate lower and upper bound of the $95\\%$\nprecentile) and an MSM\/RD simulation value of $k_{\\mathrm{off}}=0.400_{0.398}^{0.402}\\:\\mathrm{ns^{-1}}$.\nWe further compute the logarithm of the equilibrium constant $\\log(K_{\\mathrm{eq}})=\\log(k_{\\mathrm{off}}\/k_{\\mathrm{on}}^{*})$\nfor both models and for the chosen values of concentrations, resulting\nin accurate reproduction of the reference values by the MSM\/RD scheme(Fig.\n\\ref{fig:diff3Dpot}f). Thus we verify that the coupling between the\nMSM domain and the RD domain works consistently in the MSM\/RD simulation\nscheme.}\n\n\\textcolor{black}{Next, we want to ensure that also the dynamics between\nthe states inside the MSM are reproduced to a high accuracy. We compare\nMFPTs between all pairs of states conditioned on not leaving the MSM\ndomain. In the reference simulation this is done by placing the particle\nat position $\\boldsymbol{\\mu}_{i}$ and propagating the system until\nstate $j$ is reached. If the particle leaves the MSM domain before\nreaching state $j$, this trajectory is discarded. For the MSM\/RD\nsimulation, we simply start in state $i$ and propagate until state\n$j$ is hit, while discarding trajectories that leave the MSM domain.\nThis procedure is repeated until $10^{4}$ successful trajectories\nare found for both simulations, w}hich are averaged to obtain the\nMFPTs. The relative errors are calculated with Eq. (\\ref{eqn:relativeError});\nall relative errors are below $9\\%$ (Fig. \\ref{fig:diff3Dpot}b).\nWe further observe that negative errors arise for state pairs that\nare close together and thus have short passage times. For these transitions,\nwe tend to overestimate the MFPT in the MSM\/RD simulation as short\nprocesses are truncated in the MSM estimation. Moreover, we observe\nthat the highest positive errors arise for transitions which are far\napart. These are the hardest to sample since for these transitions\nthere are a very high number of possible long and non-direct transition\ntrajectories, which are less likely to be observed . We chose the\nfour transitions with the highest relative error and compared their\nFPTs distribution histograms (Fig. \\ref{fig:diff3Dpot}c). Even though\nthese transitions have the highest errors, we observe the distributions\nmatch well. Therefore, we verify MSM\/RD scheme also describes the\ninternal dynamics accurately.\n\n\\subsection{Binding of CO to myoglobin\\label{subsec:BindingCO}}\n\nAs an application of the MSM\/RD scheme, we study the binding of carbon\nmonoxide (CO) to myoglobin. Myoglobin is a globular protein which\nis responsible for the transport of oxygen in muscle tissue. The binding\nprocess of CO to myoglobin has recently be\\textcolor{black}{en studied\nby de Sancho et al. \\citep{deSancho2015identification}, whose data\nwe use to parametrize the MSM\/RD scheme. The dataset consist of MD\ntrajectories of $20$ CO molecules and one myoglobin protein for a\ntotal simulation time of $500\\:\\mathrm{ns}$. The MD simulation is\nconfined to a periodic box with edge length of $5\\:\\mathrm{nm}$.\nDespite the fact that only one CO molecule can reside in the binding\npocket, the error of treating $20$ CO molecules as being statistically\nindependent is small within statistical uncertainty (see  \\citep{deSancho2015identification}\nfor details). We therefore extract$20$ independent CO trajectories,\neffectively  increasing the total simulation time to $10\\:\\mu\\mathrm{s}$.}\n\n\\subsubsection{Parametrization of MSM\/RD scheme}\n\nIn order to parametrize the scheme, all frames are first \\textcolor{black}{aligned\nusing the $C_{\\alpha}$ atoms of the myoglobin as reference. On the\naligned data, we run the density-based spatial clustering of applications\nwith noise algorithm (DBSCAN) \\citep{ester1996density}, which finds\na total of 16 metastable regions\/cores. The positions and size of\nthe cores are shown in Fig. \\ref{fig:Myoglobin}a, where it can be\nobserved that the algorithm correctly identifies regions of high ligand\ndensity, including the myoglobin bound state indicated in red. The\nradius of the spherical cores is the radius at which $80\\:\\%$ of\nthe datapoints that were assigned to the respective state are inside\nthe core. Four states are discarded as they are not part of the largest\nconnected set. As the simulation box had been set up to just contain\nthe protein and a $1\\:\\mathrm{nm}$ solvent layer, we choose the largest\nMSM domain that still fits inside the box $(R=2.5\\:\\mathrm{nm)}$.\nAnalogous to the previous example, we follow Sec. \\ref{sec:MSM\/RD-implementation}\nto estimate an MSM for the close-range dynamics and generate $L_{\\mathrm{entry}}$,\n$L_{\\mathrm{exit},s}$, $L_{\\mathrm{trans},s}$ and $P_{\\mathrm{exit},s}$\nto couple the dynamics in the two domains. }\n\n\\textcolor{black}{We compute the implied timescales for the MSM and\nchoose a lag time of $150\\:\\mathrm{ps}$ where timescales are sufficiently\nconverged (Fig. \\ref{fig:Myoglobin}b). The diffusion constant is\ncomputed using the mean squared displacement (MSD) of the parts of\nthe CO trajectories that are far from the protein, with $D=\\Delta\\text{MSD}(t)\/6\\Delta t$.\nWe find a diffusion constant of $\\ensuremath{D_{\\text{CO}}=2.5\\:\\mathrm{nm}^{2}\\mathrm{ns}^{-1}}$,\nwhich is comparable to the experimental value which is in the range\nof $D_{\\text{CO}}=2.03\\:\\mathrm{nm}^{2}\\mathrm{ns}^{-1}$ (at $20\\:C\\text{\\textdegree}$)\nto $D_{\\text{CO}}=2.43\\:\\mathrm{nm}^{2}\\mathrm{ns}^{-1}$ (at $30\\:C\\text{\\textdegree}$)\n\\citep{wise1968diffusion}.}\n\n\\begin{figure}\n\\centering\n\n(a) \\includegraphics[width=0.8\\columnwidth]{figs\/myoglobin\/myoglobin_centers_pbc_high_contrast}\n\n(b)\\includegraphics[width=0.8\\columnwidth]{figs\/myoglobin\/impliedTimescales_myoglobin_pbc}\n\n(c)\\includegraphics[width=0.8\\columnwidth]{figs\/myoglobin\/reactionRateFit_pbc}\n\n\\caption{\\textcolor{black}{Discretization and results of the CO-myoglobin system.}\\textbf{\\textcolor{black}{{}\n(a)}}\\textcolor{black}{{} Definition of the cores (wire frame spheres)\nwithin the myoglobin. The red sphere indicates the bound state. The\ngray spheres correspond to the states that were not in the connected\nset and therefore discarded. The blue dots are positions of the CO\nmolecules for every 50th frame in the vicinity of the protein. }\\textbf{\\textcolor{black}{b)}}\\textcolor{black}{{}\nImplied timescales of the dynamics of the CO myoglobin system. The\ndatapoints and shaded area denote the sample mean and standard deviation\nof the bootstrapping sample over the trajectories: from the 20 given\ntrajectories we resample 20 with replacement. Over this sample we\nrun our discretization process which returns a sample of timescales.\nThe trajectory-samples which are not ergodic or do not lead to a connected\ncount matrix are considered invalid and discarded. Solid lines are\nfound using the full dataset. }\\textbf{\\textcolor{black}{c)}}\\textcolor{black}{{}\nReaction rate as estimated from multiple simulations at different\nconcentrations.}}\n\n\\label{fig:Myoglobin}\n\\end{figure}\n\n\\subsubsection{Comparison of dynamic properties}\n\n\\textcolor{black}{As in the previous example, we compute the binding\nrate by sampling positions sampled uniformly in the RD domain and\nsimulating the MSM\/RD model until it reaches the bound state. For\neach concentration, 200 trajectories are run to estimate the binding\nrate $k_{\\mathrm{on}}^{*}$. These rates are plotted against the concentration\nand shown in Fig. \\ref{fig:Myoglobin}c. The reaction rate $k_{\\mathrm{on}}=57_{52}^{62}\\:\\mu\\mathrm{M}^{-1}\\mathrm{s}^{-1}$\nis obtained as the slope of the linear fit. For the unbinding rate,\nwe start simulations in the bound state and collect MFPTs for leaving\nthe MSM domain; we find a rate of $k_{\\mathrm{off}}=19.0_{18.8}^{19.2}\\:\\mu\\mathrm{s}^{-1}$.\nThe resulting equilibrium constant $K_{\\mathrm{eq}}=k_{\\mathrm{on}}\/k_{\\mathrm{off}}=3.0_{2.7}^{3.3}\\:\\mathrm{M}^{-1}$\nis similar to $3.6\\,\\mathrm{M}^{-1}$ found by de Sancho et al. \\citep{deSancho2015identification},\nboth of which are close to the experimental value of $2.2\\:\\mathrm{M}^{-1}$\\citep{carver1990analysis}\n(see Tab. \\ref{tab:Myoglobin_kinetics} for comparison). The binding\nrate and unbinding rate found by de Sancho et al. \\citep{deSancho2015identification},\nalthough yielding a similar equilibrium constant, are both nearly\nan order of magnitude faster than the ones obtained with MSM\/RD (Tab.\n\\ref{tab:Myoglobin_kinetics}). The first indication that the present\nrates are an improved estimate is the fact that the kinetics (both\nthe MSM relaxation timescales and $k_{\\mathrm{on}}$) are independent\nof the lag time (Fig. \\ref{fig:Myoglobin}b, c). }\n\n\\textcolor{black}{To validate that the MSM\/RD estimates of $k_{\\mathrm{off}}$\nand $k_{\\mathrm{on}}$ have been estimated without significant bias,\nit must be shown that they are statistically consistent with the ground\ntruth (in this case a sufficiently large and sufficiently long MD\nsimulation). Here, $k_{\\mathrm{off}}$ can be estimated directly by\ncounting the frequency of ligand dissociation events from the binding\npocket in the underlying MD simulations. Since there are not sufficient\nfull dissociation pathways from the bound to the dissociated states\nin the MD data in order to make a statistically relevant comparison,\nwe obtain a more precise estimate by computing the MFPT using an MSM\ndirectly constructed from the original MD data with the same discretization\nas used in the MSM\/RD model. This resulted in a reference estimate\nof $23.4_{11.6}^{46.6}\\:\\mu\\mathrm{s}^{-1}$ (95\\% percentile computed\nwith 1000 bootstrap samples), which is consistent with the MSM\/RD\nestimate (Tab. \\ref{tab:Myoglobin_kinetics}).}\n\n\\textcolor{black}{Unfortunately, this method is not as accurate for\nthe binding rate $k_{\\mathrm{on}}$, which is notoriously difficult\nto estimate from small MD simulation boxes, where the length of trajectory\nsegments in which the ligand stays in the dissociated state without\ntouching the protein or crossing the periodic boundary are short compared\nto lagtimes $\\tau$ used in an MSM approach, resulting in biased estimates\n\\citep{PlattnerEtAl_NatChem17_BarBar}. Therefore, we performed another\nMyoglobin MD simulation in an eightfold larger periodic box (edge\nlength $10\\,\\mathrm{nm}$) with the same CO concentration as in the\nsmall MD simulation (resulting in $160$ CO molecules) for a total\nsimulation time of $405\\:\\mathrm{ns}$. For this data, a direct MSM\nestimate of the binding rate yields $74.7_{29.9}^{130.9}\\:\\mu\\mathrm{M}^{-1}\\mathrm{s}^{-1}$\n(95\\% percentile computed with 1000 bootstrap samples). As a result,\nthe MSM\/RD binding and dissociation rates are consistent with standard\nestimates computed directly from MD simulation, and the MSM\/RD modeling\nerror can be concluded to be statistically insignificant.}\n\n\\textcolor{black}{Given the consistency of the model,  we also compare\nthe results to experimental measurements, which is essentially a test\nof the MD model (e.g. force field, thermostat, integrator). These\nare yet a factor 4-5 slower than our estimates ($k_{\\mathrm{on}}=12\\:\\mu\\mathrm{M}^{-1}\\mathrm{s}^{-1}$\nand $k_{\\mathrm{off}}=5.3\\:\\mu\\mathrm{s}^{-1}$ found in \\citep{carver1990analysis}),\nconfirming that the major part of the difference between the estimates\nin \\citep{deSancho2015identification} and theexperimental values\ncould be removed by the fact that MSM\/RD is a significantly more accurate\nmodel of the binding kinetics. }\n\n\\begin{table*}\n\\begin{tabular}{|c|>{\\centering}p{2cm}|>{\\centering}p{3cm}|c|>{\\centering}p{2cm}|>{\\centering}p{2cm}|}\n\\hline \n & MSM\/RD  & Reference (approx. ground truth) & MSM in \\textcolor{black}{\\citep{deSancho2015identification}} & Experiment \\textcolor{black}{\\citep{carver1990analysis}} & Unit\\tabularnewline\n\\hline \n\\hline \n$k_{\\mathrm{on}}$ & $57.0_{52.0}^{62.0}$ & $74.7_{27.9}^{130.9}$ & 647 & $12$ & $\\mathrm{M}^{-1}\\mu\\mathrm{s}^{-1}$\\tabularnewline\n\\hline \n$k_{\\mathrm{off}}$ & $19.0_{18.8}^{19.2}$ & $23.4_{11.6}^{46.6}$ & 179 & $5.3$ & $\\mu s^{-1}$\\tabularnewline\n\\hline \n$K_{\\mathrm{eq}}$ & $3.0_{2.7}^{3.3}$ & $3.19_{2.6}^{3.8}$ & $3.6$ & $2.2$ & $\\mathrm{M}^{-1}$\\tabularnewline\n\\hline \n\\end{tabular}\n\n\\caption{\\textcolor{black}{\\label{tab:Myoglobin_kinetics}Rates and equilibrium\nconstants for Myglobin-CO estimated from different methods. The reference\nvalues approximate the ground truth by conducting a standard MSM-based\nMFTP estimate from the MD simulation (for $k_{\\mathrm{on}}$ a larger\nsimulation box was used to allow for a generous definition of the\ndissociated state).}}\n\\end{table*}\n\n\\section{Conclusion\\label{sec:Conclusion}}\n\nWe introduced and developed the MSM\/RD scheme, which couples MD-derived\nMSMs with RD simulations. We showed an implementation for protein-ligand\nsystems and applied it to two simple systems. The main advantage of\nthe algorithm is that it can simulate large time- and lengthscales\nwhile conserving molecular resolution and computational efficiency.\nThis is achieved by extracting the characteristic features of the\ndynamics fro\\textcolor{black}{m several short MD simulations into\nan MSM, which can produce new data with great accuracy and at a much\nfaster rate than the original MD simulations. This is a clear advantage\nin comparison to previous works, like \\citep{vijaykumar2015combining,VijaykumarEtAl_Arxiv16_AnisotropicMultiscaleGFRD},\nsince it does not require running MD simulations every time two particles\nare close to each other. It can further yield more accurate binding\nrates than traditional MSM methods by extending the diffusion domain\navailable, lessening the periodic boundary effects and increasing\nthe lifetime of the dissociated state. The scheme can be, in principle,\ncoupled to any RD scheme, like over-damped Langevin dynamics, Langevin\ndynamics, GFRD \\citep{van2005green,ZonTenWolde_PRL05_GFRD} and FPKMC\nalgorithm \\citep{donev2010first}, which could yield additional efficiency\nand accuracy or even incorporate long-range hydrodynamic interactions.}\n\n\\textcolor{black}{We first implemented the MSM\/RD scheme for a simple\nligand diffusion model (Sec. \\ref{subsec:3D-diff-pot}), which served\nto verify the scheme. It reproduced the expected dynamics and binding\/unbinding\nrates of the reference simulation. It was also able to generate an\naccurate MSM for the internal dynamics with a relatively small amount\nof data, which hints that it is feasible to extract the characteristic\ndynamics of a computationally feasible amount of MD simulations. Moreover,\nwe implemented the MSM\/RD scheme for the binding of CO to myoglobin\nsystem. After successfully extracting a self-consistent MSM and a\ncoupling scheme, we found that the equilibrium constant is consistent\nwith previous experimental and computational results \\citep{carver1990analysis,deSancho2015identification}.\nWe also showed that the MSM\/RD estimates are consistent with the underlying\nMD simulations \u2013 in particular our estimated association rate is consistent\nwith the association rate estimated from a reference MD simulation\nconducted in a large simulation box that was not used to parametrize\nthe MSM\/RD model. This is a significant improvement over Ref. \\citep{deSancho2015identification},\nwhere tenfold higher rates were estimated.}\n\n\\textcolor{black}{The MSM\/RD theory we introduced provides the framework\nupon which schemes for more complex systems can be constructed. In\nparticular, the next steps are to include association of two macromolecules,\nwhich may require to account for rototranslational diffusion, and\nthe coupling between protein-ligand association and conformational\nchanges. With the addition of these features, biologically relevant\nscenarios can be simulated. For example, if conformational changes\nof the protein are rare events and have different ligand association\n\/ dissociation rates, then the conformational dynamics and the ligand\nbinding dynamics are nontrivially coupled at high ligand concentrations\n\u2013 see \\citep{PlattnerNoe_NatComm15_TrypsinPlasticity} for the example\nof Trypsin and Benzamidine. A biological relevant example is the activation\nof the Calcium sensor Synaptotagmin in neuronal synapses \\citep{Suedhof_Neuron13_Neurotransmission}.\nHere, a locally very high Calcium concentration is created by the\nopening of voltage-gated Calcium channels as a response to an electric\nsignal. Synaptotagmin then binds up to five Calcium ions while going\nthrough different conformations, while the local Calcium concentration\nis reduced by diffusion. If Synaptotagmin successfully binds enough\nCalcium ions and transitions into an active conformation, it can catalyze\nthe fission of neuronal vesicles, which transduces the signal to the\npostsynaptic side. Such scenarios can be simulated with MSM\/RD simulations,\nin which the channels, the Synaptotagmin proteins and the ions are\nresolved as individual particles, and the binding\/dissociation kinetics\nand conformational changes of Synaptotagmin is encoded in an MSM.}\n\n\\textcolor{black}{MSM\/RD could be extended to deal with higher-order\nreactions. The most direct approach is to treat interactions of order\n2, 3, etc., by different MSMs which are then coupled in a regular\nMSM\/RD framework. The question then is how the higher-order MSMs are\nobtained. The brute-force approach would be to simulate the dynamics\nbetween three or more molecules with MD \u2013 e.g. with the help of enhanced\nsampling methods \u2013 and to extract corresponding higher-order MSMs.\nA cheaper, but approximate approach would be to ignore coupling between\ndifferent states and assume that multiple ligands can bind and transition\nbetween binding sites independently, perhaps except for multiple occupation\nof the same binding site. Based on such an assumption, higher-order\nMSMs could be constructed by tensor products of MSMs with one protein\nand one ligand. In practice, conducting }\\textcolor{black}{\\emph{some}}\\textcolor{black}{{}\nbut not all higher-order simulations and combining them to a generative\nmodel via machine learning methods may present a feasible pathway.}\n\n\\textcolor{black}{Finally, when considering protein interactions at\nhigh concentrations, the diffusion dynamics and long-range interactions\nof proteins are expected to be more complicated and involve hydrodynamic\neffects and anomalous diffusion. To include such effects, appropriate\ndynamical schemes should be included in the RD part.}\n\n\\textcolor{black}{In future developments, we will extend the MSM\/RD\nscheme to address these issues; however, it should be acknowledged\nthat some of these extensions come with their own set of challenges\nthat are not trivial to address.}\n\n\\section*{Acknowledgments}\n\nWe gratefully acknowledge support by the Deutsche Forschungsgemeinschaft\n(grants SFB1114, projects C03 and A04), the Einstein Foundation Berlin\n(ECMath grant CH17) and the European research council (ERC starting\ngrant 307494 \\textquotedbl{}pcCell\\textquotedbl{}). David De Sancho\nwas supported by grants CTQ2015-65320- R and RYC-2016- 19590 from\nthe Spanish Ministry of Economy, Industry and Competitiveness (MINECO).\nWe also thank Tim Hempel and Nuria Plattner for helpful discussions\nand software tutorials.\n\n\\section*{Appendix: MSM\/RD scheme for Sec. \\ref{sec:MSM\/RD-implementation}\\label{sec:Appendix:MSM\/RDscheme}}\n\n\\textcolor{black}{Based on the estimated quantities defined in the\nSec. \\ref{sec:MSM\/RD-implementation}, we introduce an implementation\nof the MSM\/RD algorithm from Sec. \\ref{subsec:MSM\/RD-coupled}.}\n\n\\texttt{\\textcolor{blue}{\\noindent}}\\texttt{\\textcolor{black}{Input: Initial\nmode (RD or MSM), initial condition (coordinates $\\mathbf{c}_{0}$\nor state $s_{0}$, respectively) and $t=0$:}}\n\n\\texttt{\\textcolor{black}{While $t\\leq t_{\\mathrm{final}}:$}}\n\\begin{enumerate}\n\\item \\texttt{\\textcolor{black}{If in RD mode:}}\n\\begin{enumerate}\n\\item \\texttt{\\textcolor{black}{Propagate $\\mathbf{c}_{t}\\rightarrow\\mathbf{c}_{t+\\tau_{\\mathrm{RD}}}$\nby diffusion }}\n\\item \\texttt{\\textcolor{black}{Update time $t\\mathrel{{+}{=}}\\tau_{\\mathrm{RD}}$}}\n\\item \\texttt{\\textcolor{black}{If $r_{AB}(\\mathbf{c}_{t})<R$ (enter MSM\ndomain):}}\n\\begin{itemize}\n\\item \\texttt{\\textcolor{black}{Select trajectory from }}\\textcolor{black}{$L_{\\mathrm{entry}}=\\{\\mathbf{c}_{\\mathrm{entry}},\\mathbf{x}_{\\mathrm{end}},\\Delta\\}$}\\texttt{\\textcolor{black}{{}\nwith $\\mathbf{c}_{\\mathrm{entry}}$ closest to $\\mathbf{c}_{t}$}}\n\\item \\texttt{\\textcolor{black}{If $\\mathbf{x}_{\\mathrm{end}}$ is a state:}}~\\\\\n\\texttt{\\textcolor{black}{Map to state $s_{t+\\Delta}=\\mathbf{x}_{\\mathrm{end}}$}}~\\\\\n\\texttt{\\textcolor{black}{Update time $t\\mathrel{{+}{=}}\\Delta$}}~\\\\\n\\texttt{\\textcolor{black}{Switch to MSM mode}}\n\\item \\texttt{\\textcolor{black}{If $\\mathbf{x}_{\\mathrm{end}}$ are coordinates:}}~\\\\\n\\texttt{\\textcolor{black}{Map to coordinates $\\mathbf{c}_{t+\\Delta}=\\mathbf{x}_{\\mathrm{end}}$}}~\\\\\n\\texttt{\\textcolor{black}{Update time $t\\mathrel{{+}{=}}\\Delta$}}\n\\end{itemize}\n\\end{enumerate}\n\\item \\texttt{\\textcolor{black}{Else (MSM mode):}}\n\\begin{enumerate}\n\\item \\texttt{\\textcolor{black}{If $s_{t}\\neq s_{t-\\tau_{MSM}}$or previous\nmode $\\neq$ MSM mode:}}\n\\begin{itemize}\n\\item \\texttt{\\textcolor{black}{Sample exit event with }}\\textcolor{black}{$P_{\\mathrm{exit},s}$}\\texttt{\\textcolor{black}{{} }}\n\\item \\texttt{\\textcolor{black}{If exit event:}}~\\\\\n\\texttt{\\textcolor{black}{Uniformly select trajectory from }}\\textcolor{black}{$L_{\\mathrm{exit},s}=\\{\\mathbf{c}_{\\mathrm{exit}},\\Delta\\}$}\\texttt{\\textcolor{black}{}}~\\\\\n\\texttt{\\textcolor{black}{Map to coordinates $\\mathbf{\\mathbf{c}_{\\mathrm{t+\\Delta}}=c}_{\\mathrm{exit}}$}}~\\\\\n\\texttt{\\textcolor{black}{Update time $t\\mathrel{{+}{=}}\\Delta$}}~\\\\\n\\texttt{\\textcolor{black}{Switch to RD mode and break current loop\niteration}}\n\\end{itemize}\n\\item \\texttt{\\textcolor{black}{Propagate $s_{t}\\rightarrow s_{t+\\tau_{\\mathrm{MSM}}}$\nusing}}\\textcolor{black}{{} $\\mathbf{T}(\\tau_{\\mathrm{MSM}})$}\n\\item \\texttt{\\textcolor{black}{Update time}}\\textcolor{black}{{} $t\\mathrel{{+}{=}}\\tau_{\\mathrm{MSM}}$.}\n\\end{enumerate}\n\\end{enumerate}\n\\textcolor{black}{The diffusion in the RD domain is done using a Euler-Maruyama\ndiscretization of Eq. (\\ref{eq:COM_diffusion}) \\citep{higham2001algorithmic}.\nNote the diffusion step can be simulated more efficiently with event-based\nalgorithms, like FPKMC or eGFRD \\citep{donev2010first,van2005green,takahashi2010spatio,vijaykumar2015combining}\nfor systems with low particle concentrations. In order to optimize\nthe efficiency of the algorithm, the entry points of entry trajectories\nare classified into equal area bins on the sphere. This allows the\nalgorithm to find the closest trajectory to a given entry point more\nefficiently. The partition of the sphere was done following \\citep{leopardi2006partition}.}\n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{S:1}\n\nAs early as the 2008 Beijing Olympic Games, the concept of Smart Sports Hall was put forward, which is the choice of sports modernization development and the choice of deepening scientific and technological innovation. Increasing the construction of intelligent stadiums can better meet the needs of sports competitions on the site. In 2014, China's State Council issued a No. 46 documents to encourage national fitness and vigorously build public sports infrastructure. With the continuous improvement of people's requirements for the quality of life, the traditional stadiums and gymnasiums only used for competition and sports can no longer meet people's needs. The emergence of intelligent stadiums and gymnasiums is an inevitable trend. An intelligent stadium system has a variety of functions, such as live events, sports data collection, and analysis, spectator navigation, and spectator statistics \\cite{wang2016study}. Therefore, the intelligent stadium system based on position tracking can collect athletes' data faster, which significantly promotes sports information. It is of great practical significance to construct an intelligent gymnasium that can improve the scientific nature of physical education. First of all, the use of intelligent facilities can improve the scientific nature of sports. Secondly, sports venues are important places for all kinds of athletic activities. Compared with traditional sports venues, intelligent sports venues have inherent advantages in order management, risk protection, and other aspects, which can improve the order and safety of sports. Finally, intelligent technology can improve the use efficiency of sports venues, the utilization rate of physical education resources, reduce the waste of resources, and provide students with personalized physical exercise guidance and guide students to carry out sports activities with higher efficiency.\n\n\nLocation tracking is an essential problem in intelligent stadiums. At present, there are a lot of indoor positioning technologies based on different algorithms and technologies that have been proposed, such as ultra-wide bandwidth (UWB) positioning, Bluetooth positioning, WiFi positioning, etc. \\cite{zhu2020indoor}. Ubisense proposed a UWB indoor positioning technology that is not affected by line-of-sight (LoS) and has high positioning accuracy \\cite{s16050707}, but the hardware cost is very high. IBeacon is a protocol that can be used for indoor positioning systems proposed by Apple. It has low power consumption and comprehensive coverage, but it is easy to receive interference, leading to RSSI instability \\cite{jianyong2014RSSI}. Microsoft proposed location fingerprint positioning technology based on WiFi. With the help of wireless local area network (WLAN) infrastructure, mobile intelligent terminal positioning can be realized without additional special equipment and only needs positioning software. \n\n\n\nAfter years of construction, most large stadiums have completed the construction of a WiFi wireless network. WiFi technology provides an environmental basis for the application and development of intelligent sports construction. WiFi signal is undoubtedly a cheap and effective solution to complete positioning compared with other indoor positioning methods, such as sensors or Bluetooth, which often require additional equipment \\cite{yang2012locating}. However, there are still many problems with intelligent stadium WiFi location technology. When serving athletes with fast movement speed and uncertain direction, receiving signal strength indicator (RSSI) of the access point (AP) with long-distance and unstable signal tends to fluctuate greatly \\cite{jun2013social}. At the same time, in order to ensure positioning accuracy, the equipment often needs a faster sampling speed \\cite{shu2015gradient} to ensure that there will be no noticeable calculation delay in the process of rapid data collection. \n\n\n\n\n\\subsection{Related work}\n\\label{S:2}\nAt present, indoor positioning technology based on WiFi is most suitable for large-scale use in intelligent stadiums. Wireless infrastructure has been widely deployed in all indoor places. The two key technologies of existing WiFi fingerprint locations are fingerprint database construction and fingerprint matching. The location design and collection of the initial fingerprint database usually adopt two modes. One is to locate reference points, and the corresponding fingerprint information through intensive collection of fingerprints in the location scene \\cite{serodio2012lightweight}. Second, the location and fingerprint information of representative and relatively sparse reference points are collected. Then the fingerprint database is reconstructed by interpolating or predicting fingerprint information of other locations in the positioning scene through a particular model \\cite{guo2019robust,nikitaki2013localization}. A method using user feedback and rating mechanism to update the fingerprint database is proposed, but it can not effectively identify valid information and error feedback \\cite{luo2017smartphone}. Moreover, a crowd-sourcing fingerprint database updating method selectively extracts reliable information through a clustering algorithm to update fingerprint data, which is an effective method to improve the quality of the fingerprint database \\cite{niu2015wicloc}.\n\n\nFor WiFi indoor positioning, the mainstream positioning method is the classical k-nearest neighbor (KNN), and its variants, such as weighted k-nearest neighbor (WKNN) \\cite{miao2014novel,lin2014group}. The location effect of these algorithms depends on data acquisition. In addition, even for a very accurate database, these methods tend to rudely discard all data except the predicted location and its neighbors, leading to inaccurate positioning. Although these methods ensure low computation and real-time, the utilization efficiency of the database is very low, and the accuracy is also limited. \n\\par\nSince the nature of the localization problem still belongs to the classification problem, the neural network will become potential direct competitors \\cite{abdullah2016k,chen2017indoor}. The intuitive idea is that a three-dimensional image can be constructed from the X-axis, Y-axis, and WiFi signal intensity within the physical plane. By processing the database in the way of image processing \\cite{zhang2017discriminative} and taking the positioning position as the classification result, each classification output can make use of other data of the database, thus ensuring high precision \\cite{zhang2017elastic-net}. The self-organizing map SOM neural network further eliminates the randomness of the mapping, provides the stability of the algorithm, and at the same time ensures the low delay of the algorithm \\cite{zhang2018deterministic}. Although there is no application of SOM in indoor positioning, it will be a very potential solution if a sufficient amount of special database information in specific scenarios such as smart stadium is provided to enhance SOM's generalization ability. But for the trained neural network, the dimension of input and output is fixed, which determines the upper limit of the algorithm to a certain extent. Input imprecise RSSI will reduce accuracy, also difficult to reduce the amount of computation. The positioning accuracy entirely depends on the output dimension. For a large stadium, the system must ensure a very high output dimension and a very high delay or accept a very low accuracy.\n\\par\nUnlike the previous two algorithms, I-WKNN combines the specific signal distribution model and proposes asymmetric Gaussian filtering to improve the generalization ability. As an algorithm that can meet the requirements of real-time and high precision, it obtains lower delay by discarding some useless information and improving the database's utilization rate to ensure higher accuracy.\n\n\n\n\\subsection{Contributions}\nThe main contributions of the paper can be summarized as follows:\n\\begin{itemize}\n    \\item This paper puts forward the application scheme of WiFi location in the sports field. As far as we know, this was the first time that indoor real-time positioning technology was applied to real-time positioning in intelligent stadiums.\n    \\item The scheme of AP signal selection is established. The scheme filters out the AP with an unstable signal value and corrects the relatively stable AP with a Gaussian filter.\n    \\item This paper proposes a new algorithm I-WKNN to improve the positioning accuracy and speed. This algorithm improves the speed and reliability of positioning, mainly by optimizing fingerprint collection methods and reducing the search space for fingerprint matching. \n    \\item The algorithm's performance is analyzed and compared from the perspective of delay and positioning accuracy. The comparison algorithm is the triangulation algorithm, the conventional WKNN algorithm, which is based on a fingerprint database.\n\\end{itemize}\n\n\\section{System Model}\n\\subsection{WiFi fingerprint positioning principle}\nThe principle of WiFi fingerprint positioning technology is mainly to match the test point's WiFi signal strength information with the geographic coordinates \\cite{8255985}. The strength of the WiFi signal, described by RSSI, monotonically decreases with distance. There are three stages in WiFi indoor positioning, the preparation phase, the offline training stage, and the real-time positioning stage. Note that the AP signal selection mechanism is introduced in the latter two stages.\n\nPreparation stage: Divide the stadium into blocks, a total of $M$ blocks, one block as a reference point and corresponding to a geographic coordinate. Moreover, $N$ WiFi devices are fixed indoors, and each WiFi device is an AP.\n\nOffline training stage: Developers select reference points and collect multiple RSSI values from different APs at each reference point. Then record the geographic coordinates of test points, the MAC addresses of APs, and the RSSI values as fingerprints \\cite{hu2018toward,hu2018experimental}. The RSSI value should be processed before real-time positioning \\cite{bisio2014trainingless}. More specifically, $\\overrightarrow{\\text{RSSI}}_{m}\\left( n \\right)$  represents the RSSI value of the ${{m}^{{th}}}$ reference point corresponding to the ${{n}^{{th}}}$ AP and $\\overrightarrow{r}_{m,n}$  represents the RSSI value at $S$ the time slot obtained by sampling the ${{n}^{{th}}}$ AP at the ${{m}^{{th}}}$ reference point. In the $S$ time slot, the WiFi signal strength RSSI values $\\overrightarrow{r}_{m,n}$ of $N$ AP signals are received, and then AP signals are selected in this stage at each reference point. In the $S$ time slot, the WiFi signal strength RSSI values $\\overrightarrow{r}_{m,n}$ of $N$ AP signals are received, and then AP signals are selected in this stage at each reference point. In the offline training stage, the AP signals with a high signal loss rate and strong signal jitter are eliminated first, and then the AP signals are corrected by asymmetric Gaussian filtering. After the selection of AP signals, the RSSI value $\\overrightarrow{\\text{RSSI}}_{m}\\left( n \\right)$  of $N$ APs corresponding to $M$ reference points in the stadium is obtained, and the value range of $N$ is $\\left[ 1,N \\right]$, and the value range of $M$ is $\\left[ 1,M\\right]$. The WiFi fingerprint of the ${{m}^{{th}}}$ reference point is stored in the fingerprint database together with $N$  RSSI value $\\overrightarrow{\\text{RSSI}}_{m}\\left( n \\right)$  \n and corresponding APs' MAC addresses. \n\nReal-time positioning stage: Users measure the RSSI value of $N$ AP signals in real-time at the current position in the stadium and then select the AP signals in the real-time positioning stage. In this stage, the AP signals with a high signal loss rate and strong signal jitter are eliminated first, and then the AP signals are corrected by asymmetric Gaussian filtering. After that, the RSSI value of the AP signal selected online is matched with the WiFi fingerprint recorded in the database. Then, the location estimation algorithm is run according to the matching results to estimate the user's geographic coordinates. The specific process is shown in the following subsection.\n\n\\subsection{A typical fingerprint location algorithm: WKNN}\n\nThe principle of WiFi fingerprint positioning technology matches the WiFi signal strength received by the test point with the geographic coordinates. The received WiFi signal strength is described by RSSI, which decreases monotonically with increasing distance. The WiFi indoor positioning is divided into two stages, and the specific process is shown in Figure~\\ref{f1}.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f1.png}\n    \\caption{The Principle of WiFi Fingerprint Positioning.}\n    \\label{f1}\n\\end{figure}\n\nOffline training stage: The developers divide the indoor area into blocks, and there are $M$ blocks in total. One block is a reference point and corresponding to a geographic coordinate $\\left( {{x}_{i}},{{y}_{i}} \\right),\\text{ }i\\text{ }=1,2,\\cdots ,M$. Fixed $N$ WiFi devices indoors, and each WiFi device is an AP.  Multiple RSSI values for different APs are collected at each reference point. Then, the geographic coordinates of the reference point, the MAC address of APs, and the RSSI value are taken as the WiFi fingerprint of the reference point $\\left( {{x}_{i}},{{y}_{i}} \\right)$ is $\\{{{\\text{MAC}}_{1}},{{\\text{RSSI}}_{1}};\\ldots ;\\text{ }{{\\text{MAC}}_{N}},{{\\text{RSSI}}_{N}}\\}$, which is stored in the fingerprint database. \n\nReal-time positioning stage: The user (the object to be positioned) measure the RSSI  \\\\$\\{{{\\text{MAC}}_{1}}^{(1)},{{\\text{RSSI}}_{1}}^{(1)};\\ldots ;\\text{ }{{\\text{MAC}}_{N}}^{(1)},{{\\text{RSSI}}_{N}}^{(1)}\\}$, $\\{{{\\text{MAC}}_{1}}^{(2)},{{\\text{RSSI}}_{1}}^{(2)};\\ldots ; \\text{ }{{\\text{MAC}}_{N}}^{(2)},{{\\text{RSSI}}_{N}}^{(2)}\\}$, $\\cdots $, $\\{{{\\text{MAC}}_{1}}^{(t)},{{\\text{RSSI}}_{1}}^{(t)};\\ldots ;\\text{ }{{\\text{MAC}}_{N}}^{(t)},{{\\text{RSSI}}_{N}}^{(t)}\\}$ at time slot $t$ in real-time at the current position. After filtering, the RSSI will carry out Matching Algorithm with the WiFi fingerprint recorded in the database. The location estimation algorithm is run to estimate the user's geographic coordinates based on the Matching results.\n\nThen the WKNN algorithm is described in detail. At the time slot  $t$, analyze the geographical location of the stadium according to the positioning results of the last few moments. From the ${{i}^{{th}}}$ reference point in the fingerprint database, all of the physically adjacent reference points are selected, and the Euclidean geometric distance from them is calculated. The Euclidean geometric distance between the t moment and the ${{i}^{{th}}}$ reference point is denoted as $D(i,t)$,\n\\begin{equation}\nD(i,t)=||{{\\overrightarrow{\\text{RSSI}}}_{i}}-\\overrightarrow{\\text{RSSI}}_{u}^{^{(t)}}|{{|}_{2}},\n\\end{equation}\nwhere ${{\\left\\| \\centerdot  \\right\\|}_{2}}$ represents the 2-norm of the vector, ${{\\overrightarrow{\\text{RSSI}}}_{i}}$ is the signal strength value of the ${{i}^{{th}}}$ reference point, which is an n-dimensional vector with an immutable order. $\\overrightarrow{\\text{RSSI}}_{u}^{^{(t)}}$ is the signal intensity value collected by the user at time slot $t$, and the vector is also n-dimensional, and the order should correspond to the order of APs in the database. If some APs' RSSI values can not be measured, a small value $\\text{RSSI}_{\\min}$ is substituted (e.g. $-100$~dB). \n\nAfter obtaining the Euclidean geometric distance between the user and the ${{i}^{{th}}}$ reference point, we order the Euclidean distance values, leaving $K$ minimum values. Generally, the idea of weighted Euclidean algorithm in the Landmarc algorithm is adopted. The reciprocal of Euclidean distance square is used as the weighted factor, and the weight $w(k,t)$ of the reference point $i$ is denoted as,\n\n\\begin{equation}\nw(k,t)=\\left\\{ \\begin{matrix}\n   \\frac{1}{{{D}^{2}}(k,t)}\/\\sum\\limits_{i}{\\frac{1}{{{D}^{2}}(i,t)}}, & i\\text{ is selected},  \\\\\n   0, & \\text{otherwise},  \\\\\n\\end{matrix} \\right.\n\\end{equation}\nthe final positioning result is,\n\\begin{equation}\n({{x}^{(t)}},{{y}^{(t)}})=\\sum\\limits_{i}{\\text{w}(i,t) \\cdot ({{x}_{i}},{{y}_{i}})},\n\\end{equation}\nwhere $({{x}^{(t)}},{{y}^{(t)}})$ and $({{x}_{i}},{{y}_{i}})$ are all binary groups, representing the horizontal and vertical coordinate values in physical locations. However, in large venues, intelligent functions such as real-time positioning based on WKNN algorithm still face the problems of poor positioning accuracy and large positioning delay.\n\n\\subsection{Asymmetric Gaussian filtering}\n\nFor WiFi signal, if the distance between AP and user remains constant, the user receives a stable RSSI signal. However, when positioning in the stadium, the WiFi signal will be subject to any interference such as noise and blocking by athletes, and the intensity of the AP signal will also fluctuate. Assuming that the noise at each moment is independent of each other, and has a similar distribution, the stability of the RSSI time average can be approximately described by the Law of Large Numbers,\n\\begin{equation}\n    \\underset{n\\to \\infty }{\\mathop{\\lim }}\\,\\mathbb{P} \\left[ \\left| \\frac{1}{n}\\sum\\limits_{i=1}^{n}{{\\overrightarrow{\\text{RSSI}}}_{\\text{i}}}-\\mu  \\right|<\\sigma  \\right]=1,\n\\end{equation}\nwhere $E({\\overrightarrow{\\text{RSSI}}})=\\mu ,\\text{ }D({\\overrightarrow{\\text{RSSI}}})={{\\sigma }^{2}}\\text{ }(i=1,2,\\cdots )$.  Under the finite sample, it can be approximately considered that if the sample value is less than $\\mu -3\\sigma $ or greater than $\\mu +3\\sigma $, it can not be accepted. At the same time, the occurrence probability of this event is only around 0.3 percent.\n\nRSSI is enhanced only because of the protocol in the measurement of the stadium signal, and it is not significant. However, because of complex scenes inside the stadium, depth decline often occurs, so it can be considered that the signal does not meet the Symmetry Gaussian distribution. The signal which is less than $\\mu -{{g}_{\\inf }}\\sigma $ or greater than $\\mu +{{g}_{\\text{sup}}}\\sigma $ should be eliminated, with g as the parameters to be determined. Therefore, the problem of Gaussian Filtering can be translated into,\n\\begin{equation}\n    \\left\\{ \\begin{matrix}\n   \\arg \\min  & {{g}_{\\text{inf}}}+{{g}_{\\text{sup}}}  \\\\\n   \\text{s}.t. & \\int_{\\mu -{{g}_{\\text{inf}}}\\sigma }^{\\mu +{{g}_{\\text{sup}}}\\sigma }{f(x)dx>1-\\text{ }\\!\\!\\epsilon\\!\\!\\text{ }}  \\\\\n\\end{matrix} \\right.,\n\\end{equation}\nwhere $\\epsilon$ is the allowable deviation of the algorithm, $f(.)$ is the probability density function of the sample.\n\nIn other words, the goal of asymmetric Gaussian filtering is to filter out as few signals as possible on the premise of tolerating errors. Since the problem is non-convex and requires extensive computation, this optimization problem is only used in the offline phase. In the online phase, we directly utilize the offline phase's solution results to reduce the solution time. \n\n\\section{I-WKNN Algorithm}\n\\subsection{Offline stage of the Algorithm}\n\nAssuming that there are S values of RSSI, they are obtained from sampling the ${{n}^{{th}}}$ AP at the $m$ reference point. They are denoted as ${{\\overrightarrow{r}}_{m,n}}$. In the offline stage, if the signal is lost seriously or fluctuates violently, the data of the source will be removed.\nWhen the signal cannot be measured, it will be replaced by a smaller value $\\text{RSSI}_{\\min}$. If the signal has not received data, the value will be zero after subtracting this value. The proportion of unreceived data can describe the degree of signal loss, that is to say, the ratio of 0-norm value to vector length in step (B). The elimination of the signal is replaced by $\\text{RSSI}_{\\min}$, which is equivalent to the data value when the signal is not received. The normalized variance in step (C) is used to describe the fluctuation. Asymmetric Gaussian filtering in step (D) eliminates partial signals that deviate entirely from the expected value. Then the AP selection algorithm in the offline stage is:\n{\n\\begin{algorithm}[h]\n    \\caption{AP selection in offline phase}\n   \n    \\begin{algorithmic}\n    \\STATE \\textbf{(A) Initialize:} $m \\leftarrow 1,n \\leftarrow 1$, initialize ${{\\Theta}_{1}}$ and ${{\\Theta }_{2}}$.\n    \n    \n    \\FOR{m = 1 \\textbf{to} M}\n    \\FOR{n = 1 \\textbf{to} N}\n    \n    \\STATE \\textbf{(B) Eliminated part of AP by the signal loss rate:} \n    \\IF{$\\left \\|{{\\overrightarrow{\\text{r}}}_{m,n}}+\\text{RSSI}_{\\min} \\right \\|_{0} \\geq S{{\\Theta }_{1}}$}\n    \\STATE ${{\\overrightarrow{\\text{RSSI}}}_{m}}(n) \\leftarrow \\text{RSSI}_{\\min}$.\n    \\ENDIF\n    \n    \\STATE \\textbf{(C) Eliminated part of AP by fluctuation:}\n    \\IF{$\\left \\|\\text{S}\\cdot {{\\overrightarrow{\\text{r}}}_{m,n}} - \\left \\| {\\overrightarrow{\\text{r}}_{m,n}} \\right \\|_{1}\\right \\|_{2} \\geq {{\\Theta }_{2}} \\left \\|{{\\overrightarrow{\\text{r}}}_{m,n}}\\right \\|_{1}$}\n    \\STATE ${{\\overrightarrow{\\text{RSSI}}}_{m}}(n) \\leftarrow \\text{RSSI}_{\\min}$.\n    \\ENDIF\n    \n    \\STATE \\textbf{(D) Perform asymmetric Gaussian filtering:}\n    \\IF{${{\\overrightarrow{\\text{RSSI}}}_{m}}(n)\\ne \\text{RSSI}_{\\min}$}\n    \\STATE ccording to criterion of asymmetric Gaussian filtering, eliminate the RSSI value which is less than $\\mu -{{g}_{\\inf }}\\sigma $ or larger than $\\mu +{{g}_{\\text{sup}}}\\sigma $ in ${{\\overrightarrow{\\text{r}}}_{m,n}}$, then take the mean of the subsectors as ${{\\overrightarrow{\\text{RSSI}}}_{m}}(n)$.\n    \\ENDIF\n    \n    \\ENDFOR\n    \\ENDFOR\n    \n\\end{algorithmic}\n\\label{a1}\n\\end{algorithm}\n\\par\nIn step (A), ${{\\Theta}_{1}}$ is the miss rate threshold,  and ${{\\Theta }_{2}}$ is the jitter peak average ratio threshold. $\\text{RSSI}_{\\min}$ in step (B) is a minimum preset value. ${{\\left\\| \\centerdot  \\right\\|}_{p}}$ is the p-norm of a vector, where $p=0,1,2$.\n}\n\n\\subsection{I-WKNN Algorithm with Its Online stage}\n\nIn the offline stage, the parameter selection of the Gaussian filter is related to the data distribution in the database. To some extent, it requires human intervention. If the filtering effect is not effective, the parameters need to be changed. In the offline stage, each reference point will have a set of parameters of asymmetric Gaussian filtering for each AP. For the sake of saving time, this set of parameters will be directly used in the online stage. \n\n{\n\\begin{algorithm}[h]\n    \\caption{AP selection in offline phase}\n   \n    \\begin{algorithmic}\n    \\STATE \\textbf{(A) Initialize:} $n \\leftarrow 1$, initialize the threshold ${{\\Theta }_{1}}$ and ${{\\Theta }_{2}}$.\n    \n    \\FOR{n = 1 \\textbf{to} N}\n    \\STATE \\textbf{(B) Eliminated part of AP by the signal loss rate:} \n    \n    \\IF{$\\left \\|   \\sum\\limits_{\\tau =t-T+1}^{t} {\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( \\tau  \\right)}}\\left( n \\right)}-\\text{RSSI}_{\\min}   \\right \\|_{0}\\leq T{{\\Theta }_{1}}$,$\\tau \\in [t-T+1,t]$}\n    \n        \\STATE $\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}}\\left( n \\right)=\\frac{1}{T}\\sum\\limits_{\\tau =t-T+1}^{t}{\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( \\tau  \\right)}}\\left( n \\right)}$,\n        \n        \\ELSE\n        \\STATE Abandoned the $n^{th}$ AP information in time $\\tau$.\n        \n    \\ENDIF\n    \\ENDFOR\n    \n    \\STATE \\textbf{(D) obtain the RSSI of $N$ APs:} \\\\ $\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}} \\leftarrow \\left( \\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}}\\left( 1 \\right),\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}}\\left( 2 \\right),\\cdots ,\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}}\\left( N \\right) \\right)$.\n\\end{algorithmic}\n\\label{a2}\n\\end{algorithm}\nThe algorithm assumes that the measurements at $T$ moments before the current moment are still valid. At time $T$, the AP selection algorithm in the online stage is as follows: In step (B), $T$ is the impact time on the measurement results at the time slot $t$, $\\overrightarrow{\\text{RSSI}}_{\\text{u}}^{(\\tau )}\\left( n \\right)$ is the signal strength received by the user at the time moment $\\tau$, and the value range of $\\tau$ is $[t-T+1,t]$. Step (B) filters out part of the signal values according to the signal strength values, filter the remaining signal according to the criterion of asymmetric Gaussian filtering. The ${{n}^{{th}}}$ AP information received by the user at time slot $t$ after asymmetric Gaussian filtering is the mean value of filtered results after $T$ time slots before $t$ time slot.\n\n\n\\par\nIn addition to standard WKNN, the algorithm introduces the AP selection mechanism and asymmetric Gaussian filtering algorithm. The overall flow chart of the I-WKNN algorithm is shown in Figure~\\ref{f2}, containing the offline stage and online stage. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f2.png}\n    \\caption{Flow Chart for Positioning.}\n    \\label{f2}\n\\end{figure}\n}\n\n\n\\section{The Application in Intelligent Stadiums}\n\n\\subsection{Experimental Scene}\n\nIn this article, the intelligent stadium is based on the client\/server (C\/S) architecture, uses JAVA as the server development language, and MySQL database as the fingerprint database. Our server implements three main functions: it can efficiently complete the database storage and socket communication. \n\n\nThe typical attenuation of WiFi signal is caused by the superposition of signal propagation in space. RSSI can present the effect of dynamic distribution, and the dynamic change and the nature of the most common Gaussian distribution are exceptionally similar under the free space. In addition, when pedestrians are excluded, the collection of RSSI presents a Gaussian distribution, as shown in Figure~\\ref{f3}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{f3.png}\n    \\caption{The Distribution of RSSI in An Ideal Situation.}\n    \\label{f3}\n\\end{figure}\n\nIn an ideal channel, the data obtained by RSSI sampling at the same place at different times is approximately Gaussian. Under multipath channels, the distribution with maximum information entropy is Gaussian distribution. Under the condition of known mean value and variance, the Gaussian distribution model is first introduced. However, in the actual scene, the RSSI of the WiFi signal will fluctuate, superposition, and disappear due to the influence of shielding, personnel movement, and multipath in the sampling process. Thus, instead of presenting as a Gaussian distribution, it is the double-peak situation in the actual sampling.\n\nDatabase and real-time measured RSSI information and geographical location are uploaded and processed by MATLAB software on the computer. We set up 250 points in the test area as clustering points, then numbered the position of each training point from right to left and from top to bottom. Run the WiFi location client program at each training point, measured each address and signal strength 1000 times. The data was processed, and after Gaussian filtering, the remaining records were written to the MySQL database.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f5.jpg}\n    \\caption{The Result of Gaussian Filtering for One AP.}\n    \\label{f5}\n\\end{figure}\n\nThe result of Gaussian filtering for one AP is shown in Figure~\\ref{f5}. The blue data should be abandoned, and the red data will remain. From the signal distribution of the AP, the fading ratio is significantly higher than the enhancement ratio, and occasionally the signal cannot be detected, which is represented by $-70$~dB in this example.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f6.jpg}\n    \\caption{The Proportion of RSSI for One AP.}\n    \\label{f6}\n\\end{figure}\n\nMoreover, we show the corresponding histogram of the proportion of RSSI in Figure~\\ref{f6}. It can be seen that through the Gaussian filter, the signal becomes noticeably more concentrated. Moreover, the adaptive Gaussian filter can better describe the current RSSI distribution of WiFi signals.\n\n\\subsection{The Performance of Accuracy}\n\nIn the online stage, we set $K=5$ and placed 10 APs. The device stores information for the last 20 slots. The following two comparison algorithms are given.\n\\begin{itemize}\n    \\item WKNN algorithm: No extra processing for fingerprint database, just find five largest RSSI values, and get the gravity center of the corresponding five points in every moment.\n    \\item KNN algorithm: A sample belongs to a category if most of the K most similar (that is, closest to each other in the feature space) samples in the feature space belong to that category.\n\\end{itemize}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f7.jpg}\n    \\caption{The CDF Graph of Accuracy for Three Algorithms.}\n    \\label{f7}\n\\end{figure}\n\nThe CDF graph is used to observe the accuracy of each positioning since the result of each positioning is equally important and the positioning error of each positioning needs to be known. It can be seen from the Figure~\\ref{f7}, that the algorithm proposed in this paper is superior to the other two. Although the elimination of some APs seems to discard some information, the accuracy is greatly improved. In addition, it can be seen from the figure that the proportion of deviation of the four positioning algorithms below 2m is 95\\% (I-WKNN), 79\\% (WKNN), 39\\% (WKNN), and 46\\% (KNN), respectively. Similarly, the mean positioning deviations were 1.14~m (I-WKNN), 1.42~m (WKNN), and 2.32~m (KNN), respectively.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f8.jpg}\n    \\caption{The PDF Graph of Accuracy for Three Algorithms.}\n    \\label{f8}\n\\end{figure}\n\nThe corresponding PDF of location error is shown in Figure~\\ref{f8}. The proposed method has the highest probability of location error of about 1~m, and the maximum error will not exceed 3~m. The error of WKNN is slightly larger than that of our method. The performance of KNN is the worst, with a maximum error of almost 6~m.\n\n\\subsection{The Performance of Time Delay}\n\nTo reduce the influence of various factors such as device heterogeneity and time uncertainty on the positioning accuracy, we used a single device to run the two algorithms and completed 40 positioning. The overall time-consuming and average time-consuming of the experiment are shown in Figure~\\ref{f9}. Because positioning speed is related to hardware and software performance, this chart is for reference only.\n\nUnlike the positioning accuracy, the time delay only needs to be lower than an acceptable value, and the samples with a significant delay should be paid special attention. So the bar chart shows the average delay, the best 20\\% experiment, and the worst 20\\% experiment, rather than every sample.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f9.png}\n    \\caption{The Comparison of Average Time Delay for Three Algorithms.}\n    \\label{f9}\n\\end{figure}\n\nAs can be seen from the figure, the KNN method, which has a significant delay, is entirely unsuitable for the rapidly changing scene such as an intelligent stadium. For the best cases, the delay of the I-WKNN algorithm has the smallest time delay. In terms of average delay, I-WKNN has obvious improvement over traditional WKNN. For the worst 20 percent, I-WKNN is still a small advantage over WKNN.\n\n\\section{Conclusion}\n\nAn improved WKNN algorithm is proposed in this paper, called I-WKNN. The improved AP selection algorithm and asymmetric Gaussian filter algorithm optimize the offline and online stage of fingerprint location. In the experiment, the triangulation algorithm, the traditional WKNN algorithm, is used to compare the proposed one. The accuracy of I-WKNN has obvious advantages compared with the other three algorithms. Its average deviation is 1.14~m, and the proportion of the deviation lower than 2~m is 95\\%. I-WKNN algorithm is only worse than triangulation in the time delay, thus ranks second, with an average delay of 326~ms and a maximum delay of 432~ms, which can meet the requirements of rapid positioning of the stadium. It can be seen that in complex environments, it has high precision and fast positioning speed, which will make it suitable for the scene of the stadium.\n\n\n\n\n\\input{ccs.bbl}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nSensors are employed by unmanned autonomous vehicles to navigate through their surroundings,  with a substantial dependence on vision-based sensors like RGB cameras. As these sensors are impacted by bad weather conditions, perception pipelines require considerable training on diverse data to increase the robustness on downstream tasks. One specific scenario which causes distortion of images is adverse weather conditions, like heavy snowfall, haze and dust tornados. In these critical situations, weather corruptions can hinder the object detectability and pose a serious threat to navigation and reliability. Thus, there is a need for efficient denoising, deraining and restoration techniques.\n\nHowever, denoising techniques are often evaluated using Image similarity metrics like PSNR, SSIM \\cite{a37} and not by their effectiveness in achieving results for the targeted application. It is possible that the output image of these methods has high image quality but is contextually irrelevant for the object detection task. In this work we evaluate the effectiveness of restoration techniques for denoising images with the intention of better object detection. By introducing a contrastive approach towards restoration evaluation,  a method for guiding the training of GANs and restoration progress is proposed. Additionally,  attention maps are leveraged for understanding  why these techniques assist object detection and why certain classes are easily recognized or ignored.\n\nPrimary contributions of this work are :\n\\begin{enumerate}\n    \\item Exploring two new data-based generative adversarial network techniques for denoising weather corrupted images. \n    \\item Proposing a novel contrastive approach using a weighted loss for evaluating the training progress and post-training performance of the highlighted restoration techniques. \n    \\item Explaining the training progress of the proposed generative denoising methods using attention maps and validating the results using object detection evaluation metrics.\n    \\item Evaluating the optimal noise level of the trained Restormer denoising methods (color and grayscale) using attention maps and validating the results using object detection evaluation metrics.\n\\end{enumerate}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth,height=5in]{images\/image1.png}\n\\end{center}\n  \\caption{RestoreX-AI: Proposed contrastive approach for evaluating restoration models.}\n\\label{fig1}\n\\end{figure*}\n\n\n\n\n\n\\section{Related Work}\n\nThe rapid development and deployment of autonomous vehicles have exposed several critical challenges in computer vision, one of which being object detection robust to weather corruptions. At the foundation of this challenge is the classic vision task of object detection, the progress of which we summarize in the following sub-section. Ultimately, when these modules are integrated in autonomous driving systems, their suggestively \"black-box\" nature, triggers the concerns of industry professionals, car-makers and users alike. Without human-tailored explanations for the vehicle's behavior, even the most advanced systems fail to benefit from widespread adoption due to concerns about safety and reliability. Especially in adverse weather, when even human drivers exercise extra caution, these robust models need to provide irrefutable explanations for the detections made at every state. We discuss the progress of previous works in these 3 directions which set the background of our work in restoration, detection and explainability. \n\n\n\n\n\\subsection{Multi-weather corruption and restoration}\n\nThe thermal variations accompanying weather change can adversely impact the optical, electronic, and mechanical components used in capturing visual data, thus harming the performance of visual recognition systems \\cite{a4}. Frigid temperatures, snowfall or dense fog, for example, can cause condensation on the lens, further blurring the view and obscuring the object boundaries; rain streaks on car windows can generate glares or act as a double lens \\cite{a36}. For an autonomous car, it is critical and essential to overcome the effects of weather conditions to ensure reliability. Number of approaches for this have been reported. For example, one study \\cite{a5} looked at the performance of gated cameras, while another \\cite{a6} expanded the research to include stereo, gated, and thermal cameras, as well as Radar and LiDAR scanners, and found considerable increases in car recognition in varying levels of fog, and other adverse weather conditions. The use of unique methodologies like domain adaptation to transform the weather conditions while keeping objects of interest intact. For example, \\cite{a7} investigates the consequences of synthetic weather images on road segmentation and traffic object detection, whereas \\cite{a8} shows that using synthetic time-of-day (night imagery) improves localization, and \\cite{a9} proposes a de-raining model to improve semantic segmentation. For their efficiency in image restoration and effectiveness in removing weather corruptions, image restoration employing restomers \\cite{a10} and image denoising algorithms \\cite{a11} are becoming increasingly popular.\n\n\n\\subsection{Object Detection}\n\nThe introduction of fast detectors like SSD, Faster RCNN, and YOLO transformed the face of object detection \\cite{a22,a23,a24}. Taking 2D detectors forward, 3D detection expanded with Stereo RCNN \\cite{a25}, AVOD \\cite{a26}, MVLidarNet \\cite{a27} and MVF algorithm \\cite{a28} bringing new perspectives to the task of object detection. Specific tasks like multiscale object detection \\cite{a29}, pedestrian detection in crowds \\cite{a30} and detection under adverse weather \\cite{a31} have also been solved using ensemble methods and data augmentation.\n\n\n\n\\subsection{Explainability in Object Detection}\n\nThe early rise of deployable machine learning technologies was accompanied by criticism of the \"black-box\"-like nature of ML models. Particularly in sensitive applications like medical diagnosis, self-driving cars and algorithmic test checking, there is a need for thorough explainability in models to ensure the trust and safety of users. To meet these requirements, many techniques were proposed for explaining models \\cite{a12,a13,a14,a15,a16}. In \\cite{a17}, explainability is demonstrated by studying what each of its neurons has learned to detect. \\cite{a18} focused on individual predictions, using the technique of heatmaps to highlight important pixels. Some works also interpret classifiers by identifying representative training examples \\cite{a19,a20}. \\cite{a21} introduced a new perspective to this challenge by making CNN-based models more transparent by producing visual explanations. As newer machine learning systems are rapidly adopted, the demand for explainable models which incorporate diverse approaches is gaining attention of the community. \n\n\n\\section{Methodology}\n\nThis work proposes a contrastive approach (loss) for monitoring the training progress of restoration models in the context of object detection. To put forward a diverse range of training samples, new restoration techniques using GANs have also been proposed. First, the restoration models were trained on different tasks and tested on the DAWN dataset \\cite{a35}. As the models trained, their progress was monitored using the contrastive approach and simultaneously attention maps were generations to support the detection task. Finally, the OD performance of all the methods is compared using standard evaluation metrics (class AP and mAP). The 4 restoration techniques experimented with in this study are:\n\n\\begin{enumerate}\n    \\item Weather-NightGAN:Conditional GANs trained on night-to-day task for multi-weather corruption restoration.\n    \\item Weather-RainGAN:Conditional GANs trained on rain-to-clear task for multi-weather corruption restoration.\n    \\item Restormer (Gaussian color denoising)  for multi-weather corruption restoration.\n    \\item Restormer (Gaussian grayscale denoising)  for multi-weather corruption restoration.\n\\end{enumerate}\n\n\n\n\\subsection{Conditional Generative Adversarial Networks}\n\nGANs or Generative Adversarial networks are generative models that learn mapping between noisy z and output image y, G : z \u2192 y. Conditional GANs learn a mapping from observed image x and random noise vector z, to y, G : {x, z} \u2192 y \\cite{a1}. The generator G is trained to produce images similar to the \"real\" images, as compared by an adversarially trained discriminator, D, which is used for detecting the \"fakes\". The final objective of the conditional GAN can be expressed as: \n\n\\begin{equation*}\n   G^{*} = arg  min_{G} max_{D} \\mathcal{L}_{cGAN} (G, D) +  \\lambda \\mathcal{L}_{L1} (G) \n\\end{equation*}\n\nwhere G tries to minimize this objective against an adversarial D that tries to maximize it, i.e. $G^{*} = arg  min_{G} max_{D} L_{cGAN} (G, D).$\n\n\nIn this study, we propose 2 new use-cases of the conditional GAN for restoration purposes. In the first case, the GAN is trained on night-to-day images and tested for multi-weather corruption tasks. The intuition behind this is the similarity in corruptions of night and bad weather images, like poor lighting and condensation on lens. In the second use-case, the GAN is trained on only rain-to-clear images (synthetically generated) and tested for multi-weather corruption tasks. The intuition behind this is the similarity in corruptions of rain and bad weather images, like snow and rain streaks. There is an additional challenge which notes if the single-weather trained conditional GAN can adapt to multi-weather corruptions. \n\n\n\n\\subsection{Restormer}\n\nThe restormer is a highly efficient transformer that was proposed for denoising tasks in image restoration \\cite{a2}. It consists of a multi-Dconv head transposed attention (MDTA) and a gated-Dconv feed-forward network (GDFN). These proposed architectural changes gave it the ability to capture long-range pixel interactions, while still remaining applicable to large images. It is both computationally efficient, and has the capacity to handle high-resolution images, a feature critical for a task like adverse weather object detection. In this work, we study the effects of trained noise levels (15, 25 and 50) on the denoising performance of color and grayscale Restormers on the DAWN dataset. The goal is to study how the noise level affects the model performance in OD task and which level is optimal for generating explainable detections.\n\n\\subsection{Grad-CAM}\n\nGradient-weighted Class Activation Mapping (Grad-CAM) is a technique that produces visual explanations for the purpose of making CNN-based models more transparent \\cite{a3}. For getting the class discriminative localization map Grad-CAM   ${L_{Grad-CAM}^c}$   $\\epsilon  R ^ {u * v}$ of width u and height v for any class c , first comes the computation of gradient of the score for class c, $y^c$ (before the softmax), according to feature maps $A^k$ of a convolutional layer, i.e. $\\frac{\\partial{\\mathbf{y^{c}}}}{\\partial A_{}^k}$ . The neuron importance weights $a_c^k$ are attained by global-average-pooling these gradients flowing back. The 'importance' of feature map k for a target class c is captured by this weight $a_c^k$.  A weighted combination of forward activation maps is performed, and followed by a ReLU to obtain,\nThis results in a coarse heat-map of the same size as the convolutional feature maps. Grad-CAM is used for the purpose of explaining object detection in the restored images of different techniques compared in this study. We additionally use the Grad-CAM model's detection probability in calculating the contrastive metric for monitoring training progress.\n\n\n\\subsection{Proposed Approach: RestoreX-AI}\n\nDue to the instability of training of GANs, over-or-under training does not always lead to the perfect solution images for object detection. Parallelly, tuning on high noise levels does not always provide the best images from the Restormer model. Even after producing good images by standard metrics (PSNR, SSIM), their applicability for the OD task remains uncertain, which brings the need for a new evaluation standard. We propose using a weighted sum of the explainability results (detection probability of class provided by the Grad-CAM model) and the similarity of the predicted and actual label to define this new standard. This weighted sum is calculated for every stage of training (one stage can be a user-defined set of epochs), and then used to monitor the progress of the model. We introduce this new parameter for assessing the quality of restoration which is calculated using equation 1. \n\n\\begin{equation}\n    \\Delta \\phi  = \\Delta ( \\Sigma (S(p,a)*d)\/N )\n \\end{equation}\n \n Here S is the similarity of labels that be measured either by grouping the objects (cars, race cars and taxis have similarity 1, person, groom have 1 and so on), or by strict parameters (cars and race cars have similarity 0). p and a are the predicted and actual labels of the object under detection. N is the number of training samples generated in that stage, which are used to evaluate the current progress of that model. Refer to Appendix \\ref{appendix:a} Table \\ref{tab2} to view the measure of similarity of objects grouped together for this study. The explanation probability is d or the value returned by the Grad-CAM model, which is its prediction of what is present in the image. The quality of restoration between stages can be denoted as $ \\Delta  \\phi  $, or the difference between qualities at consecutive stages.\n\n\n\\subsection{Datasets}\n\nFor training the Weather-RainGAN and Weather-NightGAN, corresponding images of the same scene in rain-clear and night-day conditions were required. For the Weather-RainGAN, we used Rain 100L \\cite{a32}, which is a synthesized data of rain streaks with corresponding rain-free images. For the Weather-NightGAN we used Transient Attributes dataset \\cite{a34}, which used a high-level image editing method which allows a user to adjust the attributes of a scene, e.g. change a scene to be \"night\" or \"day\". The final testing of all restoration methods required a multi-weather dataset with high-resolution images, for which the DAWN dataset was selected. The DAWN dataset is a large vehicle detection dataset which has captured images of driving scenes in adverse weather conditions \\cite{a35}. It consists of 1000 images from real-traffic scenes as seen in multiple adverse weather conditions including fog, snow, rain, and sandstorms. The images have been annotated with 2D annotations(boxes) with 6 object classes namely  car, bus, truck, motorcycle, person and bicycle.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth,height=8.5in]{images\/image2.png}\n\\end{center}\n  \\caption{Foggy weather condition: Grad-CAM Attention Maps for (a) Restormer Grayscale Denoising (b) Restormer Colour Denoising (c) Weather-RainGAN and (d) Weather-NightGAN.}\n\\label{fig2}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth,height=8.5in]{images\/image3.png}\n\\end{center}\n  \\caption{Snowfall weather condition: Grad-CAM Attention Maps for (a) Restormer Grayscale Denoising (b) Restormer Colour Denoising (c) Weather-RainGAN and (d) Weather-NightGAN.}\n\\label{fig3}\n\\end{figure*}\n\n\n\\begin{table*}[h]\n \\centering\n\\begin{tabular}{llllll} \\hline \n\\textbf{Image Restoration Technique} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Class 1 AP\\\\ {[}car{]}\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Class 2 AP\\\\ {[}bus{]}\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Class 3 AP\\\\ {[}person{]}\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Class 4 AP\\\\ {[}motorcycle{]}\\end{tabular}} & \\textbf{mAP} \\\\ \\hline \n\\multicolumn{6}{l}{\\textbf{Gaussian Gray Denoising Restormer}}                                                                                                                                                                                                                                                                                                         \\\\\\hline \nNoise 15                             & 1                                                                       & 5                                                                       & 22                                                                         & 0                                                                              & 6            \\\\\nNoise 25                             & 1                                                                       & 9                                                                       & 23                                                                         & 0                                                                              & 6            \\\\\nNoise 50                             & 1                                                                       & 0                                                                       & 29                                                                         & 0                                                                              & 6            \\\\ \\hline \n\\multicolumn{6}{l}{\\textbf{Gaussian Color Denoising Restormer}}                                                                                                                                                                                                                                                                                                       \\\\ \\hline  \nNoise 15                             & 1                                                                       & 3                                                                       & 21                                                                         & 0                                                                              & 5            \\\\\nNoise 25                             & 1                                                                       & 5                                                                       & 22                                                                         & 0                                                                              & 6            \\\\\nNoise 50                             & 1                                                                       & 11                                                                      & 24                                                                         & 0                                                                              & 7            \\\\ \\hline \n\\multicolumn{6}{l}{\\textbf{Weather-RainGAN}}                                                                                                                                                                                                                                                                                                                        \\\\\\hline   \nStage 1                              & 1                                                                       & 0                                                                       & 0                                                                          & 0                                                                              & 0            \\\\\nStage 2                              & 1                                                                       & 17                                                                      & 0                                                                          & 0                                                                              & 4            \\\\\nStage 3                              & 1                                                                       & 0                                                                       & 0                                                                          & 0                                                                              & 0            \\\\\nStage 4                              & 1                                                                       & 0                                                                       & 0                                                                          & 76                                                                             & 15           \\\\\nStage 5                              & 1                                                                       & 0                                                                       & 0                                                                          & 0                                                                              & 0            \\\\ \\hline \n\\multicolumn{6}{l}{\\textbf{Weather-NightGAN}}                                                                                                                                                                                                                                                                                                                          \\\\\\hline\nStage 1                              & 11                                                                      & 0                                                                       & 0                                                                          & 0                                                                              & 2            \\\\\nStage 2                              & 1                                                                       & 0                                                                       & 0                                                                          & 0                                                                              & 0            \\\\\nStage 3                              & 1                                                                       & 0                                                                       & 0                                                                          & 0                                                                              & 0            \\\\\nStage 4                              & 17                                                                      & 0                                                                       & 0                                                                          & 0                                                                              & 3            \\\\\nStage 5                              & 48                                                                      & 0                                                                       & 0                                                                          & 0                                                                              & 10      \\\\ \\hline    \n\\end{tabular}\n\\caption{Object detection performance of all 4 restoration methods as measured at the designed stages.}\n\\label{tab1}\n\\end{table*}\n\n\n\n\n\\section{Experiments and Results}\n\n\\subsection{Restormer-based Grayscale Denoising}\n\nSection 1 of Table \\ref{tab1} shows the object detection scores of Gaussian grayscale image denoising using the Restormer model. To compare the clarity and detection scores,  models trained on different noise levels 15, 25 and 50 are included in testing. The purpose of this experimentation is to identify which noise level is optimal for the object detection task, additionally verified by the explainability setup shown in Figures \\ref{fig2} and \\ref{fig3}. The overall mAP of object detection remains constant on all noise levels, however individual class scores are observed to fluctuate. The AP for bus and person class increases from noise 15 to 25 by 4 AP and 1AP respectively. However the bus AP drops to 0 and person AP boosts to 29 when noise is set to 50. The increase in noise level would correspond to smoother images produced by restormer, however,  it is observed that the bus AP drops when noise level is set to 50. This is very interesting to note and also observable in Figure \\ref{fig2}, where the attention maps shift from the bus to the car as the noise level increases.\n\n\\subsection{Restormer-based Colour Denoising}\n\n\nSection 2 of Table \\ref{tab1} shows the object detection scores of Gaussian color image denoising using the Restormer model. To compare the clarity and detection scores, models trained on different noise levels 15, 25 and 50 are included in testing. The purpose of this experimentation is to identify which noise level is optimal for the object detection task, additionally verified by the explainability setup. The overall mAP of object detection increases steadily by 1 with all increasing noise levels, and individual class scores are observed to increase as well. The AP for bus and person class increases from noise 15 to 25 by 2 AP and 1AP respectively. The bus AP further increases to 11 and person AP boosts to 24 when noise is set to 50. The increase in noise level is improving the image quality and object detection. As visible in Figure \\ref{fig3}, the attention maps on the people are focusing in the region of interest as the noise level increases.\n\n\n\\subsection{Weather-RainGAN}\n\nThe utilization of Pix2Pix GAN \\cite{a1} for deraining purposes in mapped rain-clear images is proposed, and validated using our proposed technique. The dataset used for this purpose was Rain 100L \\cite{a32}, which is a synthesized data of rain streaks with corresponding rain-free images. The images in Rain 100 L are originally from BSD 200 dataset \\cite{a33}. The GAN model is trained with the rainy images as source and clear images as target, expecting this technique to produce denoised images on test images of the DAWN dataset. The intuition behind this idea was the inherent similarity between weather corruptions and synthetic rain streaks, with the goal of the GAN learning to work similarly on the 2 tasks. The training epochs are divided into stages(1,2,3,4,5) to monitor the training progress of the GANs and produce results of OD and explainability at each stage. As expected, the GANs are producing highly unstable behavior with increasing and 0 AP at most of the stages. However, the sudden boost in AP at specific epochs inspired the formulation of our proposed weighted explainability measure. It can be seen in Section 3 of Table \\ref{tab1}, that the bus AP is 17 at Stage 4, but 0 at all other stages and the motorcycle AP is 76 at Stage 8 and 0 at other Stages. The car AP however, remains constant at 1 and overall mAP increases over time due to the individual class performance boosts. As these results are vague, we take a closer look at Figures \\ref{fig2} and \\ref{fig3}, to determine possible causes for this object detection performance. It can be observed that  the GAN is actually performing quite well in denoising weather conditions like fog and snow, and the attention maps are converging towards the relevant objects as the training progresses. But while the object detection results are very disjoint, the attention maps progress continuously and show improvement. The effectiveness of this GAN can be observed as a deraining solution as the image not only gets clearer, but the object detection and attention maps get more precise over training as well. \n\n\n\n\n\n\n\n\\subsection{Weather-NightGAN}\n\nA new use-case of the PixtoPix GAN \\cite{a1} is proposed, trained on a different use-case for deraining purposes in mapped rain-clear images, and validated our proposed technique using the experimental procedure. The dataset used for this purpose was  Transient Attributes dataset \\cite{a34}, which used a high-level image editing method which allows a user to adjust the attributes of a scene, e.g. change a scene to be \"night\" or \"day\". The GAN model is trained with the night images as source and day images as target, expecting this technique to produce denoised images on test images of the DAWN dataset. The intuition behind this idea was the inherent similarity between weather corruptions and dark night images. As expected, the GANs are producing highly unstable behavior with increasing and 0 AP at most of the stages as shown in Table \\ref{tab1}. This technique actually worked out only for 1 class (car) and boosted its AP from 11 to 48 over the training period. The remaining classes had a constant 0 AP. By taking a closer look at Figures \\ref{fig2} and \\ref{fig3}, possible causes for this object detection performance can be determined. The GAN is actually performing quite randomly in denoising weather conditions like fog and snow using night training images, but the attention maps are still converging towards the relevant objects as the training progresses. This is exactly aligned with the initial analysis and intuition for the discovery i.e. cases in which object detection is clear to a CV detector, although imperceptible to the human eye. While the object detection results are very disjoint, the attention maps progress continuously and show improvement. The effectiveness of this GAN can be observed in denoising weather conditions for cars, but not for other classes. The purpose of our methodology is not just to observe the restoration progress in a positive light, but to also stop training in case the GAN goes too far. As can be seen in this particular case, the GAN distorts the images after Stage 2, which may confuse detectors when tested against it. In Stage 4, the GAN produces images which are heavily distorted but oddly easier for car detection than even the original image. \n\nThe attention maps and results for all four restoration techniques and four weather conditions: fog, rain, snow and dust tornado have been displayed in Appendix \\ref{appendix:a}.\n\n\\section{Conclusion}\n\n\nThe goal of our work was to study the relationship between object detection performance, image clarity, explainability and training time in the context of novel versus established restoration techniques. The scope of this study covered 4 different restoration techniques, aiming to denoise images corrupted by over 6 different weather conditions presented in the DAWN dataset, namely fog, snow storm, haze, dust tornadoes, rainfall and mist. All 4 techniques worked differently, with the Restormer-based methods providing stable all-rounded results, while the GANs provided class-specific boosts in performance. The overall rise in mAP before and after applying the techniques was 0\\%, 40\\%, 275\\%, 400\\% respectively. Contrary to popular beliefs, greater denoising does not always guarantee better object detection results as observed in this paper. And conversely, poor denoising does not always guarantee worse object detection results. Particularly for specific tasks like bus and car detection, it can be seen that newer approaches like our proposed method (Weather-RainGAN and Weather-NightGAN) can boost the detector's performance with its resultant images. We present conditional GANs cases that perform superbly on diverse weather conditions ranging from dust tornadoes to snowfall, in spite of having trained on limited single weather conditions.\n\nWe present 2 very interesting observations obtained through this study:\n\\begin{enumerate}\n    \\item GANs can generate images complex to the human eye, but comparatively interpretable for vision models post processing. This opens the possibility of exploring the capabilities of vision models beyond the scope of human vision and also warding off potential attacks which can cripple modern detectors. \n    \\item Restoration and denoising methods which produce clearer images (as measured using standard image quality metrics like PSNR), may in fact present a greater challenge for machine perception in object detection.\n    \n\\end{enumerate}\nUnderstanding how differently humans and detectors perceive information in the same image demands greater exploration of explainability. We would like to open the discussion for countless possibilities arising from these disparate perspectives, E.g. if a model sees a pedestrian on a rainy road which a human cannot see, or conversely  a pedestrian visible to a passenger's eye which a model cannot capture. While using denoising techniques are a popular choice for tackling corruptions, it must also be acknowledged that not all techniques are suitable for all use-cases as seen for car detection. Certain classes may respond better to denoising depending on their physical characteristics as perceived by the detectors. Going ahead with building robust, explainable models, this problem must be studied from multiple perspectives in the future. We hope to inspire a new line of research in this direction which deals with the complexity of the task of weather corruptions and can solve it using diverse generalizable solutions. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{APPENDIX}\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n\n\\section{Introduction}\n\nA very popular research discipline in control theory is the extension\nof control methodologies originally developed for systems that are\ndescribed by ordinary differential equations (ODEs) to systems governed\nby partial differential equations (PDEs); however, with regard to\nstability investigations this extension is accompanied by a significant\nrise of complexity, see e.g. \\cite{Luo1998} for a comprehensive framework\nfor the stability analysis of infinite-dimensional systems. Therefore,\na lot of research effort has been invested in this topic, where for\nexample the stability analysis of mechanical systems with certain\nboundary conditions has been addressed. For instance, in \\cite{Miletic2015}\nthe stability of an Euler-Bernoulli beam subjected to nonlinear damping\nand a nonlinear spring at the tip is analysed, whereas \\cite{Stuerzer2016}\nis concerned with the stability behaviour of a gantry crane with heavy\nchain and payload. Furthermore, the proof of stability of a Lyapunov-based\ncontrol law as well as a Lyapunov-based observer design for an in-domain\nactuated Euler-Bernoulli beam has been presented in \\cite{Henikl2012}.\n\nA well-known methodology, that has also been extended to the infinite-dimensional\nscenario, is the combination of the port-Hamiltonian (pH) system representation\nwith energy-based control. In this regard, in particular a pH-system\nrepresentation based on an underlying jet-bundle structure, see e.g.\n\\cite{Ennsbrunner2005,Schoeberl2012,Schoeberl2014a}, as well as a\nformulation exploiting Stokes-Dirac structures, see e.g. \\cite{Schaft2002,Gorrec2005},\nhave turned out to be especially suitable. For a detailed comparison\nof these approaches, where the main difference is the choice of the\nvariables, the interested reader is referred to \\cite{Schoeberl2013b}\nor \\cite{Malzer2020}. In fact, with respect to boundary-control systems,\na lot of literature is available, see e.g. \\cite{Schoeberl2011,Rams2017a}\nand \\cite{Macchelli2004,Macchelli2017}, where boundary controllers\nbased on the well-known energy-Casimir method are designed within\nthe jet-bundle and the Stokes-Dirac approach, respectively. Moreover,\nrecently the pH-system description has also been exploited with regard\nto the observer design, see e.g. \\cite{Toledo2019}, where a pH-based\nobserver-design procedure for boundary-control systems has been developed\nwithin the Stokes-Dirac scenario. In light of the observer design,\nof course stability investigations play an important role, since it\nmust be ensured that the observer error tends to zero.\n\nIn \\cite{Malzer2020}, a control-design procedure based on the energy-Casimir\nmethod together with an observer design exploiting the pH-system representation\nhas been presented within the jet-bundle framework as well as within\nthe Stokes-Dirac scenario for infinite-dimensional systems with in-domain\nactuation. Furthermore, the design procedures have been demonstrated\nand compared by means of an in-domain actuated vibrating string; however,\nthe investigation regarding the asymptotic stability of the observer\nerror -- which is of course essential -- has only been sketched.\nTherefore, the aim of this paper is to carry out the stability investigation\nof the observer error of this system in detail. To this end, first\nof all, in Section \\ref{sec:Observer_Design} we summarise the observer\ndesign that exploits the pH-system representation based on a jet-bundle\nstructure, while in Section \\ref{sec:Vibrating_String} the observer\ndesign is explicitly demonstrated for an in-domain actuated vibrating\nstring. Thus, the main contribution of this paper is to verify the\nasymptotic stability of the observer error, where i) it is necessary\nto investigate the well-posedness, see Subsection \\ref{subsec:Wellposedness},\nand ii) to apply LaSalle's invariance principle for infinite-dimensional\nsystems, see Subsection \\ref{subsec:LaSalle}.\n\n\\section{Observer Design based on a Port-Hamiltonian Framework\\label{sec:Observer_Design}}\n\nWith respect to the observer design, see \\cite[Sec. V]{Malzer2020},\nwe intend to exploit a pH-system description for infinite-dimensional\nsystems with $1$-dimensional spatial domain, which is equipped with\nthe spatial coordinate $z\\in[0,L]$. The system representation is\nbased on an underlying jet-bundle structure, and therefore, first\nof all we introduce the bundle $\\pi:\\mathcal{E}\\rightarrow\\mathcal{B}$,\nwhere the total manifold $\\mathcal{E}$ is equipped with the coordinates\n$(z,x^{\\alpha})$, with $x^{\\alpha}$, $\\alpha=1,\\ldots,n$, denoting\nthe dependent variables, while the base manifold $\\mathcal{B}$ possesses\nthe independent (spatial) coordinate $(z)$ solely. Next, we consider\nthe so-called vertical tangent bundle $\\nu_{\\mathcal{E}}:\\mathcal{V}(\\mathcal{E})\\rightarrow\\mathcal{E}$,\nequipped with the coordinates $(z,x^{\\alpha},\\dot{x}^{\\alpha})$,\nwhich is a subbundle of the tangent bundle $\\tau_{\\mathcal{E}}:\\mathcal{T}(\\mathcal{E})\\rightarrow\\mathcal{E}$,\npossessing the coordinates $(z,x^{\\alpha},\\dot{z},\\dot{x}^{\\alpha})$\ntogether with the fibre bases $\\partial_{z}=\\partial\/\\partial z$\nand $\\partial_{\\alpha}=\\partial\/\\partial x^{\\alpha}$. Thus, a vertical\nvector field $v=\\mathcal{E}\\rightarrow\\mathcal{V}(\\mathcal{E})$,\nin local coordinates given as $v=v^{\\alpha}\\partial_{\\alpha}$ with\n$v^{\\alpha}\\in C^{\\infty}(\\mathcal{E})$, i.e. $v^{\\alpha}$ is a\nsmooth function on $\\mathcal{E}$, is defined as a section. A further\nimportant differential-geometric object is the so-called co-tangent\nbundle $\\tau_{\\mathcal{E}}^{*}:\\mathcal{T}^{*}(\\mathcal{E})\\rightarrow\\mathcal{E}$,\npossessing the coordinates $(z,x^{\\alpha},\\dot{z},\\dot{x}_{\\alpha})$\ntogether with the fibre bases $\\mathrm{d}z$ and $\\mathrm{d}x^{\\alpha}$,\nwhich allows to introduce a one-form $w:\\mathcal{E}\\rightarrow\\mathcal{T}^{*}(\\mathcal{E})$\nas a section that can locally be given as $w=\\breve{w}\\mathrm{d}z+w_{\\alpha}\\mathrm{d}x^{\\alpha}$\nwith $\\breve{w},w_{\\alpha}\\in C^{\\infty}(\\mathcal{E})$. With respect\nto the pH-system representation, we are interested in densities $\\mathfrak{H}=\\mathcal{H}\\mathrm{d}z$\nwith $\\mathcal{H}\\in C^{\\infty}(\\mathcal{J}^{1}(\\mathcal{E}))$, where\nthese densities can be formed by sections of certain pullback bundles,\nwhose use is omitted here for ease of presentation. That is, $\\mathcal{H}$\nis a smooth function on the first jet manifold $\\mathcal{J}^{1}(\\mathcal{E})$,\nwhich is equipped with the coordinates $(z,x^{\\alpha},x_{z}^{\\alpha})$,\nwhere the $1$st-order jet variable $x_{z}^{\\alpha}$ corresponds\nto the derivative of $x^{\\alpha}$ with respect to $z$. Moreover,\nthe first prolongation of a vertical vector field reads as $j^{1}(v)=v^{\\alpha}\\partial_{\\alpha}+d_{z}(v^{\\alpha})\\partial_{\\alpha}^{z}$,\nwith $\\partial_{\\alpha}^{z}=\\partial\/\\partial x_{z}^{\\alpha}$, where\nwe exploit the total derivative $d_{z}=\\partial_{z}+x_{z}^{\\alpha}\\partial_{\\alpha}+x_{zz}^{\\alpha}\\partial_{\\alpha}^{z}+\\ldots$.\n\nHaving discussed this essential preliminaries, we are able to introduce\nthe pH-system representation including inputs and outputs on the spatial\ndomain as\\begin{subequations}\\label{eq:pH_sys_jetbundle}\n\\begin{align}\n\\dot{x} & =(\\mathcal{J}-\\mathcal{R})(\\delta\\mathfrak{H})+u\\rfloor\\mathcal{G}\\,,\\label{eq:pH_sys_dynamics}\\\\\ny & =\\mathcal{G}^{*}\\rfloor\\delta\\mathfrak{H}\\,,\n\\end{align}\n\\end{subequations}see e.g. \\cite{Ennsbrunner2005,Schoeberl2008a,Schoeberl2014},\nwhere $\\rfloor$ denotes the so-called Hook operator allowing for\nthe natural contraction between tensor fields. In (\\ref{eq:pH_sys_jetbundle}),\nthe variational derivative $\\delta\\mathfrak{H}=\\delta_{\\alpha}\\mathcal{H}\\mathrm{d}x^{\\alpha}\\wedge\\mathrm{d}z$,\nwith $\\wedge$ denoting the exterior (wedge) product, locally reads\nas $\\delta_{\\alpha}\\mathcal{H}=\\partial_{\\alpha}\\mathcal{H}-d_{z}(\\partial_{\\alpha}^{z}\\mathcal{H})$.\nFurthermore, the linear operators $\\mathcal{J},\\mathcal{R}:\\mathcal{T}^{*}(\\mathcal{E})\\wedge\\mathcal{T}^{*}(\\mathcal{B})\\rightarrow\\mathcal{V}(\\mathcal{E})$\ndescribe the internal power flow and the dissipation effects of the\nsystem, respectively. The coefficients $\\mathcal{J}^{\\alpha\\beta}$\nof the interconnection tensor $\\mathcal{J}$ meet $\\mathcal{J}^{\\alpha\\beta}=-\\mathcal{J}^{\\beta\\alpha}\\in C^{\\infty}(\\mathcal{J}^{2}(\\mathcal{E}))$,\nwhile we have $\\mathcal{R}^{\\alpha\\beta}=\\mathcal{R}^{\\beta\\alpha}\\in C^{\\infty}(\\mathcal{J}^{2}(\\mathcal{E}))$\nand $[\\mathcal{R}^{\\alpha\\beta}]\\geq0$ for the coefficient matrix\nof the symmetric and positive semi-definite dissipation mapping $\\mathcal{R}$.\nWith respect to the dual input and output bundles $\\rho:\\mathcal{U}\\rightarrow\\mathcal{J}^{2}(\\mathcal{E})$\nand $\\varrho:\\mathcal{Y}\\rightarrow\\mathcal{J}^{2}(\\mathcal{E})$,\nwe have the input map and its adjoint output map $\\mathcal{G}:\\mathcal{U}\\rightarrow\\mathcal{V}(\\mathcal{E})$\nand $\\mathcal{G}^{*}:\\mathcal{T}^{*}(\\mathcal{E})\\wedge\\mathcal{T}^{*}(\\mathcal{B})\\rightarrow\\mathcal{Y}$,\nrespectively, and thus, the relation $(u\\rfloor\\mathcal{G})\\rfloor\\delta\\mathfrak{H}=u\\rfloor(\\mathcal{G}^{*}\\rfloor\\delta\\mathfrak{H})=u\\rfloor y$\nholds, see \\cite[Sec. 4]{Ennsbrunner2005} or \\cite[Sec. 3]{Schoeberl2008a}.\nTo be able to determine the formal change of the Hamiltonian functional\n$\\mathscr{H}=\\int_{0}^{L}\\mathcal{H}\\mathrm{d}z$ along solutions\nof (\\ref{eq:pH_sys_dynamics}), we make use of the Lie-derivative\n$\\mathrm{L}_{j^{1}(v)}$, where we set $v=\\dot{x}$ with (\\ref{eq:pH_sys_dynamics}),\nsee \\cite[Sec. IV-A]{Schoeberl2011}, and thus, we obtain\n\\begin{equation}\n\\dot{\\mathscr{H}}=-\\int_{0}^{L}\\mathcal{R}(\\delta\\mathfrak{H})\\rfloor\\delta\\mathfrak{H}+\\int_{0}^{L}u\\rfloor y+\\left.(\\dot{x}\\rfloor\\delta^{\\partial}\\mathfrak{H})\\right|_{0}^{L}\\,\\label{eq:H_p}\n\\end{equation}\nby means of integration by parts and Stoke's theorem. If $\\mathscr{H}$\ncorresponds to the total energy of the system, then (\\ref{eq:H_p})\nstates a power-balance relation, where the first expression describes\nthe energy that is dissipated for example due to damping effects.\nMoreover, the expression $\\int_{0}^{L}u\\rfloor y$ denotes a collocation\nterm distributed over (a part of) the spatial domain. The last term\ncorresponds to collocation restricted to the boundary, which is indicated\nby $(\\cdot)|_{0}^{L}$, where the boundary operator locally reads\nas $\\delta_{\\alpha}^{\\partial}\\mathcal{H}=\\partial_{\\alpha}^{z}\\mathcal{H}$.\nNote that here we consider systems with trivial boundary conditions,\nimplying that the boundary ports $(\\dot{x}^{\\alpha}\\delta_{\\alpha}^{\\partial}\\mathcal{H})|_{0}^{L}$\nvanish.\n\nNext, the intention is to exploit the pH-formulation with respect\nto the observer design. In particular, the copy of the plant (\\ref{eq:pH_sys_dynamics})\nis extended by an error-injection term, and thus, by means of the\nobserver-energy density $\\hat{\\mathcal{H}}$, the observer system\nis locally given by\\begin{subequations}\\label{eq:observer_JB}\n\\begin{align}\n\\dot{\\hat{x}}^{\\hat{\\alpha}} & =(\\mathcal{J}^{\\hat{\\alpha}\\hat{\\beta}}-\\mathcal{R}^{\\hat{\\alpha}\\hat{\\beta}})\\delta_{\\hat{\\beta}}\\hat{\\mathcal{H}}+\\mathcal{G}_{\\xi}^{\\hat{\\alpha}}u^{\\xi}+\\mathcal{K}_{\\eta}^{\\hat{\\alpha}}u_{o}^{\\eta}\\,,\\label{eq:observer_JB_dynamics}\\\\\n\\hat{y}_{\\xi} & =\\mathcal{G}_{\\xi}^{\\hat{\\alpha}}\\delta_{\\hat{\\alpha}}\\hat{\\mathcal{H}}\\,,\\label{eq:observer_JB_output_densities}\n\\end{align}\n\\end{subequations}with $\\hat{\\alpha},\\hat{\\beta}=1,\\ldots,n$ and\n$\\xi,\\eta=1,\\ldots,m$, where we use Einstein's convention on sums.\nIn (\\ref{eq:observer_JB_dynamics}), we have the additional input\n$u_{o}^{\\eta}=\\delta^{\\eta\\xi}(\\bar{y}_{\\xi}-\\hat{\\bar{y}}_{\\xi})$\n-- with the Kronecker-Delta symbol meeting $\\delta^{\\xi\\eta}=1$\nfor $\\xi=\\eta$ and $\\delta^{\\xi\\eta}=0$ for $\\xi\\neq\\eta$ --,\nwhere $\\bar{y}_{\\xi}$ corresponds to the integrated output density\nof the plant according to $\\bar{y}_{\\xi}=\\int_{0}^{L}y_{\\xi}\\mathrm{d}z$,\nwhich is assumed to be available as measurement quantity, while $\\hat{\\bar{y}}_{\\xi}$\nrepresents the copy of the integrated plant-output according to $\\hat{\\bar{y}}_{\\xi}=\\int_{0}^{L}\\hat{y}_{\\xi}\\mathrm{d}z$\nwith (\\ref{eq:observer_JB_output_densities}). The aim is to design\nthe observer gain $\\mathcal{K}_{\\eta}^{\\hat{\\alpha}}$ such that the\nobserver error $\\tilde{x}=x-\\hat{x}$ tends to $0$, where it is beneficial\nto reformulate the observer-error dynamics $\\dot{\\tilde{x}}=\\dot{x}-\\dot{\\hat{x}}$\nas pH-system according to\\begin{subequations}\\label{eq:observer_error_system_JB}\n\\begin{align}\n\\dot{\\tilde{x}}^{\\tilde{\\alpha}} & =(\\mathcal{J}^{\\tilde{\\alpha}\\tilde{\\beta}}-\\mathcal{R}^{\\tilde{\\alpha}\\tilde{\\beta}})\\delta_{\\tilde{\\beta}}\\tilde{\\mathcal{H}}-\\mathcal{K}_{\\xi}^{\\tilde{\\alpha}}u_{o}^{\\xi}\\,,\\\\\n\\tilde{y}_{\\xi} & =-\\mathcal{K}_{\\xi}^{\\tilde{\\alpha}}\\delta_{\\tilde{\\alpha}}\\tilde{\\mathcal{H}}\\,.\\label{eq:observer_error_output_JB}\n\\end{align}\n\\end{subequations}with (\\ref{eq:observer_error_output_JB}) denoting\nthe collocated output density. If we investigate the formal change\nof the error-Hamiltonian $\\tilde{\\mathscr{H}}=\\int_{0}^{L}\\tilde{\\mathcal{H}}\\mathrm{d}z$,\nwhich follows to\n\\begin{multline*}\n\\dot{\\tilde{\\mathscr{H}}}=-\\int_{0}^{L}\\delta_{\\tilde{\\alpha}}(\\tilde{\\mathcal{H}})\\mathcal{R}^{\\tilde{\\alpha}\\tilde{\\beta}}\\delta_{\\tilde{\\beta}}(\\tilde{\\mathcal{H}})\\mathrm{d}z+\\ldots\\\\\n-\\int_{0}^{L}\\delta_{\\tilde{\\alpha}}(\\tilde{\\mathcal{H}})\\mathcal{K}_{\\xi}^{\\tilde{\\alpha}}\\delta^{\\xi\\eta}(\\bar{y}_{\\eta}-\\hat{\\bar{y}}_{\\eta})\\mathrm{d}z\\,,\n\\end{multline*}\nwe find that by means of a proper choice for the components $\\mathcal{K}_{\\xi}^{\\hat{\\alpha}}$\nwe are able to render $\\dot{\\tilde{\\mathscr{H}}}\\leq0$ . Hence, the\ntotal energy of the observer error $\\tilde{\\mathscr{H}}$ is an appropriate\ncandidate for a Lyapunov functional and therefore serves as basis\nwith respect to the stability analysis. Next, the observer-design\nprocedure is demonstrated by an example.\n\n\\section{Observer Design for an In-Domain Actuated Vibrating String\\label{sec:Vibrating_String}}\n\nIn this chapter, we design an infinite-dimensional observer for an\nin-domain actuated vibrating string by exploiting energy considerations.\nThe governing equation of motion of the system under consideration\nreads as\\begin{subequations}\n\\begin{equation}\n\\rho\\frac{\\partial^{2}w}{\\partial t^{2}}=T\\frac{\\partial^{2}w}{\\partial z^{2}}+f(z,t)\\,,\\label{eq:vib_String_eom}\n\\end{equation}\nwhere $w$ describes the vertical deflection of the string, $\\rho$\nthe mass density and $T$ Young's modulus. Regarding the boundary\nconditions, we have that the string is clamped at $z=0$ and free\nat $z=L$, i.e.\n\\begin{align}\nw(0,t) & =0\\,,\\quad T\\frac{\\partial w}{\\partial z}(L,t)=0\\,.\\label{eq:BC_VS}\n\\end{align}\n\\end{subequations}In (\\ref{eq:vib_String_eom}), the distributed\nforce $f(z,t)=g(z)u(t)$ is generated by an actuator behaving like\na piezoelectric patch, where the applied voltage $u(t)$ serves as\nmanipulated variable. The spatially dependent function $g(z)=h(z-L_{p_{1}})-h(z-L_{p_{2}})$,\nwhere $h(\\cdot)$ denotes the Heaviside function, describes the placement\nof the actuator between $z=L_{p_{1}}$ and $z=L_{p_{2}}$. In fact,\nthe force-distribution on the domain $L_{p_{1}}\\leq z\\leq L_{p_{2}}$\nis supposed to be constant and is scaled by $u(t)$.\n\nFirst, the intention is to find a pH-system representation that can\nbe exploited for the observer design. To this end, we introduce the\nunderlying bundle structure based on $\\pi:(z,w,p)\\rightarrow(z)$\ntogether with the generalised momenta $p=\\rho\\dot{w}$, and thus,\n(\\ref{eq:vib_String_eom}) can be rewritten as\n\\begin{equation}\n\\dot{p}=Tw_{zz}+g(z)u\\,.\\label{eq:VS_JB}\n\\end{equation}\nIf we use the Hamiltonian density $\\mathcal{H}=\\frac{1}{2\\rho}p^{2}+\\frac{1}{2}T(w_{z})^{2}\\in\\mathcal{J}^{1}(\\mathcal{E})$,\nwe obtain the appropriate pH-system formulation\\begin{subequations}\\label{eq:pH_formulation_VS}\n\\begin{align}\n\\left[\\begin{array}{c}\n\\dot{w}\\\\\n\\dot{p}\n\\end{array}\\right] & =\\left[\\begin{array}{cc}\n0 & 1\\\\\n-1 & 0\n\\end{array}\\right]\\left[\\begin{array}{c}\n\\delta_{w}\\mathcal{H}\\\\\n\\delta_{p}\\mathcal{H}\n\\end{array}\\right]+\\left[\\begin{array}{c}\n0\\\\\ng(z)\n\\end{array}\\right]u\\,,\\\\\ny & =\\left[\\begin{array}{cc}\n0 & g(z)\\end{array}\\right]\\left[\\begin{array}{c}\n\\delta_{w}\\mathcal{H}\\\\\n\\delta_{p}\\mathcal{H}\n\\end{array}\\right]=g(z)\\frac{p}{\\rho}\\,.\\label{eq:pH_form_VS_output_density}\n\\end{align}\n\\end{subequations}By taking the boundary conditions (\\ref{eq:BC_VS})\ninto account, one finds that the formal change of the Hamiltonian\nfunctional $\\mathscr{H}$ follows to $\\dot{\\mathscr{H}}=\\int_{0}^{L}g(z)\\frac{p}{\\rho}u\\mathrm{d}z$,\ni.e. we have a distributed port that can be used for control purposes.\nIn fact, for the system under consideration, in \\cite{Malzer2020}\na dynamic controller based on the energy-Casimir method has been designed.\nHowever, with regard to this control methodology, it should be mentioned\nthat it yields unsatisfactory results for uncertain initial conditions,\nsee e.g. \\cite{Rams2017b} where this problem is briefly discussed\nfor a boundary-control system. Therefore, in the following we intend\nto design an infinite-dimensional observer in order to overcome this\nobstacle.\n\nConcerning the observer design, it is assumed that the spatial integration\nof the distributed output density (\\ref{eq:pH_form_VS_output_density})\naccording to $\\bar{y}=\\int_{0}^{L}g(z)\\frac{p}{\\rho}\\mathrm{d}z$,\nwhich can be interpreted as the current through the actuator, is available\nas measurement quantity. Thus, if we use the observer density $\\hat{\\mathcal{H}}=\\frac{1}{2\\rho}\\hat{p}^{2}+\\frac{1}{2}T(\\hat{w}_{z})^{2}$\nand the copy of the plant output according to $\\hat{\\bar{y}}=\\int_{0}^{L}g(z)\\frac{\\hat{p}}{\\rho}\\mathrm{d}z$,\nwe are able to introduce an observer for the in-domain actuated vibrating\nstring in the form\n\\begin{align*}\n\\left[\\begin{array}{c}\n\\dot{\\hat{w}}\\\\\n\\dot{\\hat{p}}\n\\end{array}\\right]\\! & =\\!\\left[\\!\\begin{array}{cc}\n0\\! & \\!1\\\\\n-1\\! & \\!0\n\\end{array}\\!\\right]\\!\\left[\\begin{array}{c}\n\\delta_{\\hat{w}}\\hat{\\mathcal{H}}\\\\\n\\delta_{\\hat{p}}\\hat{\\mathcal{H}}\n\\end{array}\\right]\\!+\\!\\left[\\begin{array}{c}\n0\\\\\ng(z)\n\\end{array}\\right]\\!u\\!+\\!\\left[\\begin{array}{c}\nk_{1}\\\\\nk_{2}\n\\end{array}\\right]\\!(\\bar{y}-\\hat{\\bar{y}}),\n\\end{align*}\nwhere the governing equations are restricted to the boundary conditions\n$\\hat{w}(0)=0$ and $T\\hat{w}_{z}(L)=0$. Next, by means of the error\ncoordinates $\\tilde{w}=w-\\hat{w}$, $\\tilde{p}=p-\\hat{p}$, the observer-error\ndynamics can be deduced to\\begin{subequations}\\label{eq:observer_error_dynamics_VS}\n\\begin{align}\n\\dot{\\tilde{w}} & =\\dot{w}-\\dot{\\hat{w}}=\\frac{1}{\\rho}\\tilde{p}-k_{1}(\\bar{y}-\\hat{\\bar{y}})\\,,\\label{eq:observer_error_dyn_w}\\\\\n\\dot{\\tilde{p}} & =\\dot{p}-\\dot{\\hat{p}}=T\\tilde{w}_{zz}-k_{2}(\\bar{y}-\\hat{\\bar{y}})\\,,\\label{eq:observer_error_dyn_p}\n\\end{align}\nwhere the boundary conditions\n\\begin{align}\n\\tilde{w}(0) & =0\\,,\\quad T\\tilde{w}_{z}(L)=0\\label{eq:BC_observer_error_left}\n\\end{align}\n\\end{subequations}hold. With respect to the determination of $k_{1}$\nand $k_{2}$, it is beneficial to reformulate (\\ref{eq:observer_error_dyn_w})\nand (\\ref{eq:observer_error_dyn_p}) as the pH-system\\begin{subequations}\\label{eq:observer_error_VS_JB-1}\n\\begin{align}\n\\left[\\begin{array}{c}\n\\dot{\\tilde{w}}\\\\\n\\dot{\\tilde{p}}\n\\end{array}\\right]\\! & =\\!\\left[\\begin{array}{cc}\n0 & 1\\\\\n-1 & 0\n\\end{array}\\right]\\!\\left[\\begin{array}{c}\n\\delta_{\\tilde{w}}\\tilde{\\mathcal{H}}\\\\\n\\delta_{\\tilde{p}}\\tilde{\\mathcal{H}}\n\\end{array}\\right]\\!-\\!\\left[\\begin{array}{c}\nk_{1}\\\\\nk_{2}\n\\end{array}\\right]\\!(\\bar{y}-\\hat{\\bar{y}}),\\\\\n\\tilde{y} & \\!=\\!-\\!\\left[\\begin{array}{cc}\nk_{1} & k_{2}\\end{array}\\right]\\!\\left[\\begin{array}{c}\n\\delta_{\\tilde{w}}\\tilde{\\mathcal{H}}\\\\\n\\delta_{\\tilde{p}}\\tilde{\\mathcal{H}}\n\\end{array}\\right]\\!=\\!k_{1}T\\tilde{w}_{zz}\\!-\\!k_{2}\\frac{\\tilde{p}}{\\rho},\\label{eq:pH_observer_error_output}\n\\end{align}\n\\end{subequations}where the energy density of the observer error\nreads as $\\tilde{\\mathcal{H}}=\\frac{1}{2\\rho}\\tilde{p}^{2}+\\frac{1}{2}T(\\tilde{w}_{z})^{2}$\nand (\\ref{eq:pH_observer_error_output}) states the corresponding\noutput density. If we investigate the formal change of the error-Hamiltonian\nfunctional $\\tilde{\\mathscr{H}}$, which can be deduced to\n\\begin{align}\n\\dot{\\tilde{\\mathscr{H}}} & =\\int_{0}^{L}(T\\tilde{w}_{zz}k_{1}(\\bar{y}-\\hat{\\bar{y}})-\\frac{\\tilde{p}}{\\rho}k_{2}(\\bar{y}-\\hat{\\bar{y}}))\\mathrm{d}z\\,,\\label{eq:H_tilde_p}\n\\end{align}\nand take into account that $(\\bar{y}-\\hat{\\bar{y}})=\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z$,\nwe find that the choice $k_{1}=0$ and $k_{2}=kg(z)$ with $k>0$,\nyields\n\\begin{equation}\n\\dot{\\tilde{\\mathscr{H}}}(\\tilde{w},\\tilde{p})=-k(\\bar{y}-\\hat{\\bar{y}})^{2}\\leq0\\,.\\label{eq:H_tilde_p_fin}\n\\end{equation}\nHowever, the fact that $\\tilde{\\mathscr{H}}>0$ and $\\dot{\\tilde{\\mathscr{H}}}\\leq0$\nhold is not sufficient for the convergence of the observer, and therefore,\nin the following, detailed stability investigations are carried out\nto verify that the observer error is asymptotically stable.\n\n\\section{Observer Convergence\\label{sec:Stability_Analysis}}\n\nIn this section, based on functional analysis the convergence of the\nobserver error is proven in two steps. First, we address the well-posedness\nof the observer-error system making heavy use of the well-known Lumer-Phillips\ntheorem, see e.g. \\cite{Liu1999}. Afterwards, LaSalle's invariance\nprinciple for infinite-dimensional systems is applied to show the\nasymptotic stability of the observer error, where beforehand it is\nnecessary to verify the precompactness of the solution trajectories.\n\n\\subsection{Well-posedness of the Observer-Error System\\label{subsec:Wellposedness}}\n\nNow, a careful investigation of the well-posedness of the observer-error\nsystem is carried out. To this end, we reformulate (\\ref{eq:observer_error_dynamics_VS})\nas an abstract Cauchy problem and show that the operator under consideration\ngenerates a $C_{0}$-semigroup of contractions.\n\nFirst, we define the state vector $\\chi=\\left[\\chi^{1},\\chi^{2}\\right]^{T}=\\left[\\tilde{w},\\tilde{p}\\right]^{T}$\ntogether with the state space $\\mathcal{X}=H_{C}^{1}(0,L)\\times L^{2}(0,L)$,\nwhere $H_{C}^{1}(0,L)=\\{\\chi^{1}\\in H^{1}(0,L)|\\chi^{1}(0)=0\\}$,\nwith $H^{l}(0,L)$ denoting a Sobolev space of functions whose derivatives\nup to order $l$ are square integrable, see \\cite{Adams2003} for\na detailed introduction of Sobolev spaces. Thus, the state space $\\mathcal{X}$\nis equipped with the standard norm\n\\begin{equation}\n\\left\\Vert \\chi\\right\\Vert _{n}^{2}=\\left\\langle \\tilde{w},\\tilde{w}\\right\\rangle _{L^{2}}+\\left\\langle \\tilde{w}_{z},\\tilde{w}_{z}\\right\\rangle _{L^{2}}+\\left\\langle \\tilde{p},\\tilde{p}\\right\\rangle _{L^{2}}\\,.\\label{eq:standard_norm}\n\\end{equation}\nNext, to be able to rewrite the observer-error dynamics as an abstract\nCauchy problem of the form $\\dot{\\chi}(t)=\\mathcal{A}\\chi(t)$ with\n$\\chi(0)=\\chi_{0}$, we introduce the linear operator $\\mathcal{A}:\\mathcal{D}(\\mathcal{A})\\subset\\mathcal{X}\\rightarrow\\mathcal{X}$\naccording to\n\\[\n\\mathcal{A}:\\left[\\begin{array}{c}\n\\tilde{w}\\\\\n\\tilde{p}\n\\end{array}\\right]\\rightarrow\\left[\\begin{array}{c}\n\\frac{1}{\\rho}\\tilde{p}\\\\\nT\\tilde{w}_{zz}-kg(z)\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z\n\\end{array}\\right]\\,,\n\\]\nwhere the (dense) domain of $\\mathcal{A}$ is defined as\n\\begin{multline}\n\\mathcal{D}(\\mathcal{A}):=\\{\\chi\\in\\mathcal{X}|\\tilde{w}\\in(H^{2}(0,L)\\cap H_{C}^{1}(0,L)),\\\\\n\\tilde{p}\\in H_{C}^{1}(0,L),T\\tilde{w}_{z}(L)=0\\}\\,.\\label{eq:domain_A}\n\\end{multline}\nThus, the intention is to investigate the operator $\\mathcal{A}$\nregarding some properties such that a variant of the well-known Lumer-Phillips\ntheorem \\cite[Thm. 1.2.4]{Liu1999} can be applied. With respect to\nthis forthcoming investigations, it is beneficial to introduce\n\\begin{equation}\n\\left\\Vert \\chi\\right\\Vert _{\\mathcal{\\mathcal{X}}}^{2}=T\\left\\langle \\tilde{w}_{z},\\tilde{w}_{z}\\right\\rangle _{L^{2}}+\\frac{1}{\\rho}\\left\\langle \\tilde{p},\\tilde{p}\\right\\rangle _{L^{2}}\\,,\\label{eq:energy_norm}\n\\end{equation}\nwhich is called energy norm due to the equivalence $\\tilde{\\mathscr{H}}=\\frac{1}{2}\\left\\Vert \\chi\\right\\Vert _{\\mathcal{\\mathcal{X}}}^{2}$.\nBecause $\\tilde{w}(0)=0$ and further $\\tilde{w}(z)=\\int_{0}^{z}\\tilde{w}_{z}\\mathrm{d}y_{1}$\nholds, we find constants $c_{1},c_{2}$, which have to meet $0<c_{1}\\leq\\mathrm{min}\\left(\\frac{T}{L+1},\\frac{1}{\\rho}\\right)$\nand $c_{2}\\geq\\mathrm{max}\\left(T,\\frac{1}{\\rho}\\right)>0$, such\nthat $c_{1}\\left\\Vert \\chi\\right\\Vert _{n}^{2}\\leq\\left\\Vert \\chi\\right\\Vert _{\\mathcal{X}}^{2}\\leq c_{2}\\left\\Vert \\chi\\right\\Vert _{n}^{2}$\nis fulfilled, and hence, the energy norm (\\ref{eq:energy_norm}) is\nequivalent to the standard norm (\\ref{eq:standard_norm}). Similar\nto the proof of Lemma 2.2 in \\cite{Stuerzer2016}, where they exploit\nthe dense inclusion $H^{2}(0,L)\\subset H^{1}(0,L)$ and modify the\nboundary values of $\\tilde{w}$ and its derivatives in a proper manner,\nit can be shown that the domain $\\mathcal{D}(\\mathcal{A})$ given\nin (\\ref{eq:domain_A}) is dense in $\\mathcal{X}$. Thus, according\nto \\cite[Def. 1.1.1]{Liu1999}, -- since we have the equivalence\n$\\tilde{\\mathscr{H}}=\\frac{1}{2}\\left\\Vert \\chi\\right\\Vert _{\\mathcal{\\mathcal{X}}}^{2}$\n-- the relation (\\ref{eq:H_tilde_p_fin}) implies that $\\mathcal{A}$\nis dissipative.\n\nIn the following, we show that the inverse operator $\\mathcal{A}^{-1}$\nexists and is bounded, i.e. for every $\\bar{\\chi}=\\left[f,h\\right]^{T}\\in\\mathcal{X}$\nand $\\chi=\\left[\\tilde{w},\\tilde{p}\\right]^{T}\\in\\mathcal{D}(\\mathcal{A})$,\nwe can uniquely solve\n\\begin{equation}\n\\mathcal{A}\\!\\left[\\!\\begin{array}{c}\n\\tilde{w}\\\\\n\\tilde{p}\n\\end{array}\\!\\right]\\!=\\!\\left[\\begin{array}{c}\n\\frac{1}{\\rho}\\tilde{p}\\\\\nT\\tilde{w}_{zz}-kg(z)\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z\n\\end{array}\\right]\\!=\\!\\left[\\begin{array}{c}\nf\\\\\nh\n\\end{array}\\right],\\label{eq:Calc_A-1}\n\\end{equation}\nand prove that $\\mathcal{A}^{-1}$ maps bounded sets in $\\mathcal{X}$\ninto bounded sets in $\\mathcal{K}:=(H^{2}(0,L)\\cap H_{C}^{1}(0,L))\\times H_{C}^{1}(0,L)$.\nFrom the $1$st line of (\\ref{eq:Calc_A-1}) it follows that $\\tilde{p}=\\rho f\\in H_{C}^{1}(0,L)$.\nMoreover, an integration of the $2$nd line of (\\ref{eq:Calc_A-1})\nyields\n\\begin{multline}\n\\tilde{w}_{z}(z)=-\\frac{1}{T}(\\int_{z}^{L}h(y_{2})\\mathrm{d}y_{2}+\\ldots\\\\\n+\\int_{z}^{L}kg(y_{2})\\int_{0}^{L}g(y_{1})f(y_{1})\\mathrm{d}y_{1}\\mathrm{d}y_{2})\\label{eq:w_z_calc}\n\\end{multline}\nas $\\tilde{w}_{z}(L)=0$ holds. If we further integrate (\\ref{eq:w_z_calc}),\nwe obtain\n\\begin{multline}\n\\tilde{w}(z)=-\\frac{1}{T}(\\int_{0}^{z}\\int_{y_{3}}^{L}h(y_{2})\\mathrm{d}y_{2}\\mathrm{d}y_{3}+\\ldots\\\\\n+\\int_{0}^{z}\\int_{y_{3}}^{L}kg(y_{2})\\int_{0}^{L}g(y_{1})f(y_{1})\\mathrm{d}y_{1}\\mathrm{d}y_{2}\\mathrm{d}y_{3})\\label{eq:w_calc}\n\\end{multline}\nas $\\tilde{w}(0)=0$, and thus, $\\tilde{w}(z)$ is uniquely defined\nby $\\bar{\\chi}$. Since we have shown that the inverse operator $\\mathcal{A}^{-1}$\nexists, it remains to investigate the boundedness. To this end, it\nis verified that the norm of $\\chi=\\mathcal{A}^{-1}\\bar{\\chi}$ in\n$\\mathcal{K}$ is bounded by $\\left\\Vert \\bar{\\chi}\\right\\Vert _{\\mathcal{X}}$.\nFirst, we state an inequality that is often used in the sequel; in\nfact, for a -- basically arbitrary -- function $f$, by means of\nthe Cauchy-Schwarz inequality we find the important relation\n\\begin{equation}\n(\\int_{0}^{L}f\\mathrm{d}z)^{2}\\leq C\\int_{0}^{L}\\left|f\\right|^{2}\\mathrm{d}z\\,,\\label{eq:inequalitiy_square}\n\\end{equation}\nwhere it should be mentioned that here and in the following $C$\ndenotes positive, not necessarily equal constants. Next, we investigate\nthe norm $\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}$. Therefore,\nwe substitute (\\ref{eq:w_z_calc}) in $\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}=(\\int_{0}^{L}\\left|\\tilde{w}_{z}\\right|^{2}\\mathrm{d}z)^{1\/2}$\nand apply the Triangle inequality, which yields\n\\begin{multline*}\n\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}\\leq(\\int_{0}^{L}\\frac{1}{T^{2}}(\\int_{z}^{L}h(y_{2})\\mathrm{d}y_{2})^{2}\\mathrm{d}z)^{\\frac{1}{2}}+\\\\\n(\\int_{0}^{L}\\frac{1}{T^{2}}(\\int_{z}^{L}kg(y_{2})\\int_{0}^{L}g(y_{1})f(y_{1})\\mathrm{d}y_{1}\\mathrm{d}y_{2})^{2}\\mathrm{d}z)^{\\frac{1}{2}}.\n\\end{multline*}\n Thus, by means of (\\ref{eq:inequalitiy_square}) and due to the fact\nthat $\\int_{z}^{L}h^{2}\\mathrm{d}y_{2}\\leq\\int_{0}^{L}h^{2}\\mathrm{d}z=\\left\\Vert h\\right\\Vert _{L_{2}}^{2}$\nholds, we obtain\n\\begin{multline}\n\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}\\leq C\\left\\Vert h\\right\\Vert _{L_{2}}\\frac{1}{T}L^{\\frac{1}{2}}+\\\\\nC(\\int_{0}^{L}\\frac{1}{T^{2}}\\int_{z}^{L}k^{2}g^{2}(y_{2})(\\int_{0}^{L}g(y_{1})f(y_{1})\\mathrm{d}y_{1})^{2}\\mathrm{d}y_{2}\\mathrm{d}z)^{\\frac{1}{2}}.\\label{eq:ineq_w_z_L_2}\n\\end{multline}\nNext, we apply the Cauchy-Schwarz inequality to the second term of\nthe right-hand side in (\\ref{eq:ineq_w_z_L_2}), which enables us\nto find the estimate $\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})$.\nSimilarly, by means of the $2$nd line of (\\ref{eq:Calc_A-1}) we\nare able to deduce $\\left\\Vert \\tilde{w}_{zz}\\right\\Vert _{L^{2}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})$.\nMoreover, if we substitute (\\ref{eq:w_calc}) in $\\left\\Vert \\tilde{w}\\right\\Vert _{L^{2}}=(\\int_{0}^{L}\\left|\\tilde{w}\\right|^{2}\\mathrm{d}z)^{1\/2}$,\nwe find $\\left\\Vert \\tilde{w}\\right\\Vert _{L^{2}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})$,\nand hence, we have $\\left\\Vert \\tilde{w}\\right\\Vert _{H^{2}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})$.\nSince from the first line in (\\ref{eq:Calc_A-1}) we immediately get\n$\\left\\Vert \\tilde{p}\\right\\Vert _{H^{1}}=\\rho\\left\\Vert f\\right\\Vert _{H^{1}}$,\nwe can state the important estimate\n\\[\n\\left\\Vert \\tilde{w}\\right\\Vert _{H^{2}}+\\left\\Vert \\tilde{p}\\right\\Vert _{H^{1}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})\\,,\n\\]\nwhich shows that $\\mathcal{A}^{-1}$ maps bounded sets in $\\mathcal{X}$\ninto bounded sets in $\\mathcal{K}$. \n\nThe boundedness of $\\mathcal{A}^{-1}$ implies that $\\lambda=0$ cannot\nbe an eigenvalue of $\\mathcal{A}$, and hence, it follows that $0\\in\\rho(\\mathcal{A})$,\nthe resolvent set of $\\mathcal{A}$. Furthermore, since $\\mathcal{D}(\\mathcal{A})$\nis dense in $\\mathcal{X}$ and $\\mathcal{A}$ is dissipative, all\nrequirements for the variant of the Lumer-Phillips theorem according\nto \\cite[Thm. 1.2.4]{Liu1999} are met, and therefore, we are able\nto show that $\\mathcal{A}$ is the infinitesimal generator of a $C_{0}$-semigroup\nof contractions on $\\mathcal{X}$. That is, the norm $\\left\\Vert \\chi(t)\\right\\Vert _{\\mathcal{X}}$\nremains bounded for $t\\rightarrow\\infty$; however, with respect to\nthe observer error it is necessary that it tends to $0$, which is\nshown in the following subsection.\n\n\\subsection{Asymptotic Stability of the Observer-Error System\\label{subsec:LaSalle}}\n\nNow, the objective is to apply LaSalle's invariance principle in order\nto prove the asymptotic stability of the observer error, where the\nproof follows the intention of \\cite[Sec. 3]{Guo2011}. However, the\napplicability of LaSalle's invariance principle according to \\cite[Thm. 3.64]{Luo1998}\nrequires the precompactness of the solution trajectories, which is\nnot ensured in the infinite-dimensional scenario. Since in the previous\nsection we have shown that $\\mathcal{A}^{-1}$ is bounded, by means\nof the Sobolev embedding theorem, it follows that $\\mathcal{A}^{-1}$\nis compact (see proof of Lemma 2.4 in \\cite{Stuerzer2016} or \\cite[p. 201]{Luo1998}),\nwhich further implies the precompactness of the trajectories, see\n\\cite[Rem. 4.2]{Miletic2015}. \n\nIn light of LaSalle's invariance principle, we investigate the set\n$\\mathcal{S}=\\{\\chi\\in\\mathcal{X}|\\dot{\\tilde{\\mathscr{H}}}=0\\}$,\nwhere $\\dot{\\tilde{\\mathscr{H}}}(\\tilde{w},\\tilde{p})=-k(\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z)^{2}=0$\nimplies $\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z=0$. In\nthe set $\\mathcal{S}$ we have\\begin{subequations}\\label{eq:eom_S}\n\\begin{align}\n\\rho\\tilde{w}_{tt} & =T\\tilde{w}_{zz}\\,,\\label{eq:eom_S_pde}\\\\\n\\tilde{w}(0,t) & =0\\,,\\label{eq:eom_S_BC_left}\\\\\nT\\tilde{w}_{z}(L,t) & =0\\,,\\label{eq:eom_S_BC_right}\n\\end{align}\n\\end{subequations}which is similar to the problem considered in \\cite[Sec. 3]{Guo2011};\nhowever, the restriction describing the set $\\mathcal{S}$, which\nis constrained to the boundary there, is completely different. To\nbe able to show that the only possible solution in $\\mathcal{S}$\nis the trivial one, we need to investigate the general solution of\n(\\ref{eq:eom_S}). To this end, like in \\cite[Sec. 3]{Guo2011}, we\nfirst focus on determining the eigenvalues and eigenfunctions of (\\ref{eq:eom_S}),\ni.e. we consider\n\\begin{equation}\n\\bar{\\mathcal{A}}\\left[\\begin{array}{cc}\n\\phi & \\kappa\\end{array}\\right]^{T}=\\left[\\begin{array}{cc}\n\\frac{\\kappa}{\\rho} & T\\phi_{zz}\\end{array}\\right]=\\lambda\\left[\\begin{array}{cc}\n\\phi & \\kappa\\end{array}\\right]^{T}\\,.\\label{eq:eigen_equation}\n\\end{equation}\nFrom (\\ref{eq:eigen_equation}) we obtain $\\kappa=\\rho\\lambda\\phi$\nand furthermore\\begin{subequations}\\label{eq:eigen_problem}\n\\begin{align}\n\\phi_{zz} & =\\frac{\\lambda^{2}}{\\vartheta^{2}}\\phi\\,,\\\\\n\\phi(0) & =0\\,,\\label{eq:eigen_problem_BC_left}\\\\\n\\phi_{z}(L) & =0\\,,\\label{eq:eigen_problem_BC_right}\n\\end{align}\n\\end{subequations}where $\\vartheta^{2}=\\frac{T}{\\rho}$. To find\nthe solution of (\\ref{eq:eigen_problem}), we have to investigate\nthe three cases $\\lambda^{2}>0$, $\\lambda^{2}=0$ and $\\lambda^{2}<0$\nin the following. For $\\lambda^{2}>0$ and $\\lambda^{2}=0$, we have\nthe ansatz $\\phi(z)=Ae^{\\frac{\\lambda}{\\vartheta}z}+Be^{-\\frac{\\lambda}{\\vartheta}z}$\nand $\\phi(z)=Az+B$, respectively, where by means of the boundary\nconditions (\\ref{eq:eigen_problem_BC_left}) and (\\ref{eq:eigen_problem_BC_right}),\none can easily deduce that for both cases only the trivial solution\n$\\phi(z)=0$ exists. Thus, we focus on the case $\\lambda^{2}<0$,\nand consequently, due to the fact that $\\lambda$ has an imaginary\ncharacter then, as ansatz for the eigenfunctions we have $\\phi(z)=A\\sin(\\frac{\\left|\\lambda\\right|}{\\vartheta}z)+B\\cos(\\frac{\\left|\\lambda\\right|}{\\vartheta}z)$.\nTo fulfil the boundary condition (\\ref{eq:eigen_problem_BC_left}),\n$B=0$ must be valid, and hence, the ansatz simplifies to $\\phi(z)=A\\sin(\\frac{\\left|\\lambda\\right|}{\\vartheta}z)$.\nFurthermore, by means of the boundary condition (\\ref{eq:eigen_problem_BC_right}),\nwe find $\\partial_{z}\\phi(L)=\\frac{\\left|\\lambda\\right|}{\\vartheta}A\\cos(\\frac{\\left|\\lambda\\right|}{\\vartheta}L)=0$,\nwhich exhibits infinitely many non-trivial solutions for\n\\begin{equation}\n\\left|\\lambda_{k}\\right|=(k-\\frac{1}{2})\\frac{\\pi}{L}\\vartheta\\,,\\label{eq:eigenvalues}\n\\end{equation}\nwith $k=1,2,\\ldots$. With regard to the investigation of the set\n$\\mathcal{S}$, the velocity of the vibrating string is of particular\ninterest. Consequently, since we deduced the (imaginary) eigenvalues\n$\\lambda_{k}=\\pm i\\omega_{k}\\vartheta$ with $\\omega_{k}=(k-\\frac{1}{2})\\frac{\\pi}{L}$,\nthe ansatz for the general solution of the velocity can be given according\nto\\begin{subequations}\\label{eq:solution_a_b}\n\\begin{align}\n\\tilde{w}_{t}(z,t) & \\!=\\!\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}\\!(a_{k}\\cos(\\omega_{k}\\vartheta t)\\!+\\!b_{k}\\sin(\\omega_{k}\\vartheta t))\\varphi_{k}(z),\\label{eq:solution_w_t}\n\\end{align}\nwhere the coefficients $A_{k}$ are hidden in $a_{k}$ and $b_{k}$,\nand therefore, for the eigenfunctions we use $\\varphi_{k}(z)=\\sin(\\omega_{k}z)$\nhere and in the sequel. Hence, an integration of (\\ref{eq:solution_w_t})\nyields\n\\begin{equation}\n\\tilde{w}(z,t)\\!=\\!\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}\\!(a_{k}\\sin(\\omega_{k}\\vartheta t)\\!-\\!b_{k}\\cos(\\omega_{k}\\vartheta t))\\frac{\\varphi_{k}(z)}{\\omega_{k}\\vartheta}.\\label{eq:solution_w}\n\\end{equation}\n\\end{subequations}By means of $\\sin(x)=\\frac{1}{2i}(e^{ix}-e^{-ix})$\nand $\\cos(x)=\\frac{1}{2}(e^{ix}+e^{-ix})$, after a straightforward\ncomputation we can beneficially rewrite (\\ref{eq:solution_a_b}) according\nto\n\\begin{multline*}\n\\left[\\begin{array}{c}\n\\tilde{w}(z,t)\\\\\n\\tilde{w}_{t}(z,t)\n\\end{array}\\right]=\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}c_{k}e^{i\\omega_{k}\\vartheta t}\\left[\\begin{array}{c}\n-i\\frac{\\varphi_{k}}{\\omega_{k}\\vartheta}\\\\\n\\phi_{k}\n\\end{array}\\right]+\\ldots\\\\\n\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}c_{-k}e^{-i\\omega_{k}\\vartheta t}\\left[\\begin{array}{c}\ni\\frac{\\varphi_{k}}{\\omega_{k}\\vartheta}\\\\\n\\phi_{k}\n\\end{array}\\right]\\,,\n\\end{multline*}\nwhere the coefficients $c_{k}=\\frac{1}{2}\\left(a_{k}-ib_{k}\\right)$\nand $c_{-k}=\\frac{1}{2}\\left(a_{k}+ib_{k}\\right)$ fulfil (see \\cite[Eq. (3.19)]{Guo2011})\n\\begin{equation}\n\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}\\left|c_{\\pm k}\\right|^{2}=\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}\\left(a_{k}^{2}+b_{k}^{2}\\right)<\\infty\\,,\\label{eq:coefficients_bounded}\n\\end{equation}\nwhich will play an important role later. Thus, we are able to write\n$\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z=\\int_{L_{p_{1}}}^{L_{p_{2}}}\\tilde{w}_{t}\\mathrm{d}z=0$\nas\n\\begin{equation}\n\\int_{L_{p_{1}}}^{L_{p_{2}}}\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}(c_{k}e^{i\\omega_{k}\\vartheta t}+c_{-k}e^{-i\\omega_{k}\\vartheta t})\\sin(\\omega_{k}z)\\mathrm{d}z=0\\,.\\label{eq:condition_S}\n\\end{equation}\n\nNow, we show that the only solution in $\\mathcal{S}$ is the trivial\none, i.e. $c_{\\pm k}=0\\forall k\\geq1$ is valid. Otherwise, if there\nexists a $k_{0}$ with $\\left|c_{k_{0}}\\right|\\neq0$, due to (\\ref{eq:coefficients_bounded})\nwe can find a $K>k_{0}$ such that\\begin{subequations}\\label{eq:bound_inf_coeffs}\n\\begin{align}\n\\left|\\int_{L_{p_{1}}}^{L_{p_{2}}}\\overset{\\infty}{\\underset{\\mathop{k=K}}{\\mathop{\\sum}}}c_{k}\\varphi_{k}\\mathrm{d}z\\right| & <\\left|\\frac{c_{k_{0}}}{4}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\varphi_{k_{0}}\\mathrm{d}z\\right|\\\\\n\\left|\\int_{L_{p_{1}}}^{L_{p_{2}}}\\overset{\\infty}{\\underset{\\mathop{k=K}}{\\mathop{\\sum}}}c_{-k}\\varphi_{k}\\mathrm{d}z\\right| & <\\left|\\frac{c_{k_{0}}}{4}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\varphi_{k_{0}}\\mathrm{d}z\\right|\n\\end{align}\n\\end{subequations}holds, i.e. the sum of the coefficients from $K$\nto $\\infty$ multiplied with their corresponding eigenfunctions can\nbe bounded by $c_{k_{0}}$ and $\\varphi_{k_{0}}$. Here, it is assumed\nthat $\\int_{L_{p_{1}}}^{L_{p_{2}}}\\sin(\\omega_{k_{0}})\\mathrm{d}z\\neq0$,\ni.e. $\\omega_{k_{0}}\\neq\\frac{2\\pi}{L_{p_{2}}-L_{p_{1}}}j$ with $j\\in\\mathbb{N}_{+}$.\nHowever, if we consider the absolute value of the eigenvalues (\\ref{eq:eigenvalues}),\nwe find that this is ensured for a proper choice of the length of\nthe in-domain actuator according to $L_{p_{2}}-L_{p_{1}}\\neq\\frac{4L}{2k-1}$.\nConsequently, because $\\omega_{k}\\neq\\omega_{l}\\forall k\\neq l$,\nfor $t>0$ we can reformulate (\\ref{eq:condition_S}) as\n\\begin{multline}\n-c_{k_{0}}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\varphi_{k_{0}}\\mathrm{d}z=\\int_{L_{p_{1}}}^{L_{p_{2}}}\\{\\overset{K}{\\underset{\\mathop{k=1,k\\neq k_{0}}}{\\mathop{\\sum}}}c_{k}e^{i(\\omega_{k}-\\omega_{k_{0}})\\vartheta t}\\varphi_{k}+\\\\\n+\\overset{\\infty}{\\underset{\\mathop{k=K+1}}{\\mathop{\\sum}}}c_{k}e^{i(\\omega_{k}-\\omega_{k_{0}})\\vartheta t}\\varphi_{k}+\\overset{K}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}c_{-k}e^{-i(\\omega_{k}+\\omega_{k_{0}})\\vartheta t}\\varphi_{k}+\\\\\n+\\overset{\\infty}{\\underset{\\mathop{k=K+1}}{\\mathop{\\sum}}}c_{-k}e^{-i(\\omega_{k}+\\omega_{k_{0}})\\vartheta t}\\varphi_{k}\\}\\mathrm{d}z\\,.\\label{eq:condition_S_ref}\n\\end{multline}\nNext, the idea is to integrate (\\ref{eq:condition_S_ref}) with respect\nto the time $t$ and to investigate the absolute value. Hence, we\nfind that the right-hand side of\n\\begin{multline}\n\\left|c_{k_{0}}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\varphi_{k_{0}}\\mathrm{d}z\\right|t\\leq\\\\\n2\\left|\\int_{0}^{t}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\{\\overset{K}{\\underset{\\mathop{k=1,k\\neq k_{0}}}{\\mathop{\\sum}}}c_{k}e^{i(\\omega_{k}-\\omega_{k_{0}})\\vartheta\\tau}\\varphi_{k}\\}\\mathrm{d}z\\mathrm{d}\\tau\\right|+\\\\\n+2\\left|\\int_{0}^{t}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\{\\overset{K}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}c_{-k}e^{-i(\\omega_{k}+\\omega_{k_{0}})\\vartheta\\tau}\\varphi_{k}\\}\\mathrm{d}z\\mathrm{d}\\tau\\right|\\,,\\label{eq:inequaltiy_coefficients}\n\\end{multline}\nwhere we used (\\ref{eq:bound_inf_coeffs}) to obtain an estimation\nfor the sums from $k=K+1$ to $k=\\infty$, is bounded for all $t\\geq0$.\nSince for an appropriate choice of the actuator-length it is ensured\nthat the integral on the left-hand side cannot vanish, the only possibility\nthat inequality (\\ref{eq:inequaltiy_coefficients}) holds for $t\\rightarrow\\infty$\nis that $c_{k_{0}}=0$ is valid. Thus, it is shown that the only possible\nsolution in $\\mathcal{S}$ is the trivial one, which finally proves\nthe asymptotic stability of the observer error and therefore justifies\nthe application of the observer developed in \\cite{Malzer2020}. Furthermore,\nin Figure (\\ref{fig:Comp_w_state_obs}), the comparison of the string\ndeflection $w(L,t)$ and the observer state $\\hat{w}(L,t)$ is depicted,\nwhere the tip of the string is moved from $w(L,0)=0$ to $w(L,t_{end})=0.1$\nand the observer state is initialised as $\\hat{w}(L,0)=0.1$.\n\\begin{figure}\n\\input{w_L_Obs.tex}\\caption{\\label{fig:Comp_w_state_obs}Comparison of the string deflection $w(L,t)$\nand the observer state $\\hat{w}(L,t)$.}\n\\end{figure}\n\n\n\\section{Conclusion and Outlook}\n\nIn this paper, the asymptotic stability of an observer error of an\nin-domain actuated vibrating string, where the observer has been developed\nin \\cite{Malzer2020}, was investigated. First, we showed that the\nlinear operator, which describes the observer error as an abstract\nCauchy problem, is the infinitesimal generator of a contraction semigroup.\nSecond, by means of LaSalle's invariance principle the asymptotic\nstability of the observer error was proven. In fact, by choosing the\nlength of the actuator properly, it was shown that the only possible\nsolution for $\\dot{\\tilde{\\mathscr{H}}}=0$ is the trivial one, which\nimplies that the observer error tends to zero. Future-research tasks\nmight deal with the stability analysis of the closed loop obtained\nby the controller design presented in \\cite{Malzer2020}, or even\nwith the stability investigation of the combination of controller\nand observer.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe observation of the sky with space-born instruments, equipped with detectors working at those electro-magnetic frequencies that cannot be accessed from the ground, revealed the existence of several classes of high energy radiation sources. With their location in distant galaxies, Active Galactic Nuclei (AGNs, in brief) turned out to be the most powerful non transient sources of such radiation. Although AGNs appear with many different properties, they share an extreme intrinsic luminosity, ranging between $10^{41}\\, {\\rm erg\\, s^{-1}}$ and $10^{46}\\, {\\rm erg\\, s^{-1}}$, comparable to or greater than the energy output of large galaxies, but released from a region that is smaller than 1~pc in radius. To explain this property we assume that large amounts of matter, in the order of some solar masses per year, are conveyed to the nuclear regions of active galaxies, where they are accreted by a Super Massive Black Hole (SMBH). It is now well established that SMBHs with masses between $10^6\\, {\\rm M_\\odot}$ and some $10^9\\, {\\rm M_\\odot}$ reside in the nuclei of every massive galaxy (Ferrarese \\& Merrit, 2000; Shankar, 2009). In addition, we know that accretion of fuel into their gravitational field leads to the conversion of gravitational binding energy into radiation with very high efficiency (Blandford \\& Znajek, 1977; Shields, 1978).\n\nDue to the presence of relativistic plasmas and strong magnetic fields in the vicinity of the black hole's accretion flow, the spectrum of the emitted radiation results from a combination of thermal and non-thermal components that cover several orders of magnitude in frequency, sometimes extending from radio wavelengths all the way up to $\\gamma$-ray energies. The Fermi Large Area Telescope (Fermi-LAT; see Atwood et al., 2009) gave new life to the study of the $\\gamma$-ray sky, producing, after 4 years of scientific observations, a map of $\\gamma$-ray detections with unprecedented resolution and sensitivity in the energy range between 100~MeV and 300~GeV (Acero et al., 2015). Thanks to this result, a large number of $\\gamma$-ray sources can now be associated with lower energy counterparts. In the extra-Galactic environment, it turned out that the most commonly detected objects are AGNs belonging to the blazar class. Blazars are extremely variable, highly polarized, radio-loud sources, dominated by power-law continuum spectra of type $F_\\nu \\propto \\nu^{-\\alpha}$, with a typical radio spectral slope $\\alpha \\leq 0.5$. Their properties are the consequence of the relativistic beaming of the synchrotron radiation produced by a jet that is collimated and accelerated close to our line of sight (Blandford \\& K\\\"onigl, 1979). They are classically separated into BL Lac objects (BLL), whose optical spectra show a nearly featureless power-law continuum, and Flat Spectrum Radio Quasars (FSRQs), which, instead, are characterized by strong emission lines.\n\nIn addition to the blazars that dominate the extra-Galactic $\\gamma$-ray population, other types of AGNs, together with a large number of sources without a firm classification, are detected as well. In this contribution we describe our investigation on the nature of $\\gamma$-ray AGNs of undetermined type, through the observation of their optical spectra. We focus our attention on targets whose spectral energy distributions (SED) are consistent with those of blazars, although they still lack firm spectroscopic classification, and they are therefore called Blazar Candidates of Undetermined type (BCU in 3FGL terminology, Acero et al., 2015). In this report, we discuss the spectral classification of some BCUs, with respect to the associated SEDs, rather than giving a full list of observations. In the following sections, we describe the techniques used in the attempt to identify the low energy counterparts of the $\\gamma$-ray sources, the details of how we collected the optical spectra and their classification criteria. Finally, we draw a sketch of the $\\gamma$-ray emission in the different classes of objects that we observe, compared with the multiple wavelength properties of their SEDs.\n\n\\section{Association of $\\gamma$-ray sources}\nAt the energies of $\\gamma$-ray photons it is not possible to focus radiation through reflecting or refracting optical devices. The Fermi-LAT, instead, measures the production of $e^\\pm$ pairs, through the conversion of photons in the detector (Atwood et al., 2009). The pair properties are used to reconstruct the energy and the direction of the incoming photon. The precision that can be achieved in the measurement depends mainly on the incoming photon incidence angle and on its energy. It can be roughly estimated that, for a photon of energy between 10 GeV and 100 GeV, hitting the detector at normal incidence, the 95\\% containment angle is approximately 0.5$^\\circ$ (Ackermann et al., 2012). Although the detection of multiple photons from a source can improve the performances, the contribution of nearby sources and background noise, combined with less than optimal detection conditions, results in a still significant uncertainty in the localization of the sources. Thus, in general, the task to associate a low energy counterpart to $\\gamma$-ray sources is not trivial.\n\nIn the case of AGNs, the expected broad band emission provides a reliable way to better constrain the source position. Since $\\gamma$-rays from AGNs arise from Inverse Compton (IC) scattering of low energy seed photons by relativistic plasma particles confined in strong magnetic fields, the $\\gamma$-ray production occurs together with synchrotron radiation. If the energy distribution of the plasma particles is extended enough to support significant $\\gamma$-ray emission, we expect that powerful synchrotron radiation is produced at radio and x-ray energies, as well. Taking advantage from the angular resolution of instruments working at these frequencies, which can measure the position of radiation sources down to a few arcseconds, we are able to identify candidate counterparts to $\\gamma$-ray emission by looking for coincident x-ray and radio sources within the $\\gamma$-ray detection uncertainty radius (Ackermann et al., 2011; Gasparrini et al., 2012). Furthermore, the existence of a connection between $\\gamma$-ray emission and lower frequency radiation implies that large flux variations, that characterize AGNs particularly in the high energies, can be observed in different frequencies. Thanks to the monitoring strategy of Fermi-LAT observations, covering the entire sky approximately every 3~hr, the detection of important flaring activity can be matched with follow-up observations that are able to identify the possible correlated variations of the source in other frequencies. Once the source position is determined by either of the techniques down to a few arcseconds, it is possible to search for the optical counterpart and to obtain its spectrum.\n\n\\section{Optical observations}\nThe latest catalog of Fermi-LAT detected AGNs (3LAC, see Ackermann et al., 2015) includes blazars, radio galaxies, steep spectrum radio quasars (SSRQs), Seyfert galaxies and Narrow Line Seyfert 1 galaxies. In addition to the classified sources, however, a number of undetermined type objects, generally called AGNs or BCUs, still exists. Focusing our attention on these unclassified objects, we carried out a spectroscopic study that combines publicly available spectra, like those extracted from the latest data release of the Sloan Digital Sky Survey (SDSS-DR 12; Alam et al., 2015) and of the 6dF Galaxy Redshift Survey (6dFGRS-DR 3; Jones et al., 2004, 2009), together with new observations performed at the 1.22m and the 1.82m telescopes of the Asiago Astrophysical Observatory (Ciroi et al., 2014).\n\nFig. 1 illustrates examples of spectra obtained in this study. The main purpose of this optical spectroscopic analysis is to identify the AGN class for the source associated to the $\\gamma$-ray emission and to determine its redshift, through the identification of known emission or absorption lines. In general, the detection of emission lines ensures the most reliable measurement of the source redshift, because they are easier to detect and they are originated very close to central SMBH. Absorption lines, on the contrary, are produced either in the host galaxy or along the light path towards us. They are much more difficult to detect and can only be seen if the AGN is not overwhelmingly dominant. It follows that the presence of emission lines characterizes AGNs with radiatively efficient accretion activity, therefore surrounded by nuclear and circum-nuclear regions of ionized gas, like quasars and Seyfert galaxies. BLLs, on the contrary, just give raise to a featureless power-law continuum, where some absorption features can show up, if the jet power is relatively low with respect to the host galaxy luminosity, or when some intervening medium happens to lie along the line of sight to the source.\n\n\\begin{table}[t]\n\\caption{List of sources included in this report. The table provides the $\\gamma$-ray source identifier, the associated counterpart name, the counterpart position (J2000.0), the object redshift and its spectroscopic classification.}\n\\begin{footnotesize}\n\\begin{center}\n\\begin{tabular}{llcccr}\n\\hline\n\\hline\nId. & Counterpart name & R. A. & Dec. & {\\it z} & Class \\\\\n\\hline\n3FGL J$0134.5+2638$ & 1RXS J$013427.2+263846$ & $01:34:28.3$ & $+26:38:45.0$ & $\\geq$0.108 & BL Lac \\\\\n3FGL J$0339.2-1738$ & PKS $0336-177$ & $03:39:13.7$ & $-17:36:00.6$ & 0.065 & Elliptical \\\\\n3FGL J$0904.3+4240$ & S4 $0900+42$ & $09:04:17.1$ & $+42:37:59.0$ & 1.342 & FSRQ \\\\\n3FGL J$1031.0+7440$ & S5 $1027+74$ & $10:31:22.0$ & $+74:41:58.3$ & 0.122 & FSRQ \\\\\n3FGL J$1315.4+1130$ & 1RXS J$131531.9+113327$ & $13:15:32.0$ & $+11:33:27.0$ & $\\geq$0.730 & BL Lac \\\\\n3FGL J$1412.0+5249$ & SBS $1410+530$ & $14:11:49.4$ & $+52:49:00.2$ & 0.076 & Elliptical \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{footnotesize}\n\\end{table}\n\\section{Results and models}\nA summary of our spectroscopic observations is presented in Table 1. According to the spectra collected in Fig. 1, our targets belong to different families of optical sources. Together with examples of classic high luminosity blazar objects, characterized by prominent emission lines (FSRQs) or dominant power-law featureless spectra (BLLs), we notice associations with some weaker sources, where the spectrum of the host galaxy, with its characteristic stellar continuum and absorption features, plays an increasing role. Looking at the spectra presented here, we note that in the high redshift regime (approximately from $z \\geq 0.15$) we are able to detect $\\gamma$-ray emission only from objects with powerful jets. At smaller distances, however, $\\gamma$-ray sources likely arising from jets of weaker power or misaligned orientation, which do not significantly contribute to the optical spectra, can be identified as well.\n\nIn order to estimate the power of the jets, which are likely producing the observed $\\gamma$-ray signal, and to compare it with the characteristics of the corresponding optical spectra, we reconstructed the SEDs associated to our targets. We used the ASI Space Data Center SED Builder tool to retrieve multiple frequency data points and to interpolate them with jet radiation models.\\footnote{The SED Builder tool is available at \\texttt{http:\/\/tools.asdc.asi.it\/SED\/}} We took into account archival data available from the literature, including in particular observations performed within extensive sky survey programs, to cover as an extended frequency range as possible. The results of this selection are plotted in Fig. 2. The broad band electro-magnetic emission is generally consistent with the classic two-hump blazar SED, very likely connected with jet activity. There are, however, some objects where an IR to optical radiation excess clearly occurs. The presence of such an excess is made particularly evident by comparing the observations with simple Synchrotron Self-Compton (SSC) models, which we used to interpolate the data. Due to the possibility that external radiation fields may give a relevant contribution to the IC effect (a scenario that is referred to as External Compton, or EC, and is very likely more appropriate, especially in the case of FSRQs), the SSC approach might be an oversimplified interpretation of the SEDs. However, the non simultaneous nature of the different data points collected in this work did not allow to consider more advanced models.\n\nTable 2 reports details of the SED models applied to our targets. The residuals were calculated without taking into account the thermal components. The SSC models are able to reproduce most of the sources with acceptable residuals ($\\chi^2_{red} \\leq 1.5$), while the most notable exceptions are probably due to the difficulties that such models have in explaining the high energy IC tail. Intrinsic source variability at the different observation epochs is also expected to affect the scatter. In general, we observe that jets of large scale and power are required to interpolate classic BLL and FSRQ blazars, while weaker targets are also explained by less powerful jets.\n\n\\subsection{Notes on single sources}\n{\\bf 3FGL J0134.5+2638}, observed with the Asiago 1.22m telescope, reveals a power law continuum spectrum with faint absorption lines. Absence of emission lines having equivalent width $EW \\geq 5\\,$\\AA leads to a BLL classification. Its SED shows the characteristic two-hump behavior of blazars and it is appreciably well interpolated by a SSC model. \\\\\n{\\bf 3FGL J0339.2--1738} is associated with an elliptical galaxy, with spectrum available from 6dFGRS. Flux calibration of the spectrum was obtained deriving an average sensitivity curve for the 6dF instrument, based on the flux calibrated spectra of IC 5135 and UGC 842. The resulting spectrum shows the characteristic continuum and absorption lines of an old stellar population, typical of ellipticals. The associated SED is a two-hump distribution with a prominent radiation excess in the optical window. \\\\\n{\\bf 3FGL J0904.3+4240}, detected by the SDSS, shows the highest redshift in this sample, which brings the strong UV emission lines of C~IV~$\\lambda$1549, C~III]~$\\lambda$1909 and Mg~II~$\\lambda$2798 of quasar spectra into the optical domain. The SED is characteristic of blazars, but with a dominant IC component over the Synchrotron part. \\\\\n{\\bf 3FGL J1031.0+7440} was observed in Asiago, with the 1.82m telescope. It shows the prominent emission lines of a Seyfert 1 galaxy, with a full width at half the maximum ${\\rm FWHM}({\\rm H}\\beta) = 2286 \\pm 350\\, {\\rm km\\, s^{-1}}$. In a standard $\\Lambda$CDM cosmology with $H_0 = 70\\, {\\rm km\\, s^{-1}\\, Mpc^{-1}}$, $\\Omega_\\Lambda = 0.7$ and $\\Omega_M = 0.3$, its redshift corresponds to a distance of 569.8~Mpc. From an apparent magnitude $V = 17.2$, we infer an absolute magnitude $M_V = -21.6$, placing this object on the border between faint quasar and bright Seyfert 1 activity. The blazar SED is accompanied by a small radiation excess in the optical domain, suggesting that the jet power and the thermal contribution from the central engine are comparable in this object. \\\\\n{\\bf 3FGL J1315.4+1130}, also detected by the SDSS, is characterized by the BLL power law continuum and by faint absorption lines. Identification of the strongest features as a Ca~II doublet is consistent with the presence of an absorption feature at the predicted wavelength of Mg~I. Fewer data points are available to reconstruct the SED of this source and we do not appreciate any deviation from a two component  blazar SED. \\\\\n{\\bf 3FGL J1412.0+5249} is detected by the SDSS and it shows the characteristics of an elliptical galaxy. Its counterpart is actually a giant elliptical located in a galaxy cluster. The associated SED is the most complex of this sample, featuring a strong optical excess, emitted by the bright host galaxy, and a high energy IC component that is hardly reproduced by SSC models. \\\\\n\n\\begin{table}[t]\n\\caption{SSC model parameters. The table columns report, respectively, the 3FGL source name, the electron energy distribution power-law index before break $\\alpha_{el}^{(1)}$, the electron distribution index after break $\\alpha_{el}^{(2)}$, the logarithm of the break energy (in units of $m_e c^2$), the magnetic field $B$ (expressed in Gauss), the Doppler factor $\\delta$, the jet radius expressed in parsec, and the reduced residuals.}\n\\begin{footnotesize}\n\\begin{center}\n\\begin{tabular}{cccccccc}\n\\hline\n\\hline\nId. & $\\alpha_{el}^{(1)}$ & $\\alpha_{el}^{(2)}$ & $\\log E_{break}$ & $B$ & $\\delta$ & $R_{jet}$ & ${\\chi_{red}^2}^{\\rm a}$ \\\\\n\\hline\n3FGL J$0134.5+2638$ & 1.5 & 4.7 & 4.0 & 1.00 & 10 & 0.001 & 1.180 \\\\\n3FGL J$0339.2-1738$ & 1.5 & 5.0 & 4.0 & 0.75 & 15 & 0.001 & 1.097 \\\\\n3FGL J$0904.3+4240$ & 2.3 & 3.6 & 3.5 & 0.05 & 30 & 0.003 & 1.455 \\\\\n3FGL J$1031.0+7440$ & 1.8 & 5.0 & 4.1 & 1.00 & 15 & 0.001 & 1.339 \\\\\n3FGL J$1315.4+1130$ & 1.6 & 4.8 & 4.8 & 0.60 & 20 & 0.002 & 1.732 \\\\\n3FGL J$1412.0+5249$ & 1.5 & 4.7 & 4.0 & 1.00 & 10 & 0.001 & 1.889 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n$^{\\rm a}$ Residuals of SSC models are computed without taking into account the thermal excess data points.\n\\end{footnotesize}\n\\end{table}\n\\section{Conclusions}\nIn this study we presented the optical spectra of six targets that have been associated to $\\gamma$-ray sources of still undetermined type in 3LAC. In some cases, the spectra of this type of objects can still be obtained from large spectroscopic surveys, such as the SDSS, above all, but, in others, we need specifically planned observations. The increase of sensitivity and resolution achieved by the Fermi-LAT at $\\gamma$-ray energies is now providing better opportunities to identify candidate counterparts to extra-Galactic $\\gamma$-ray emission and, therefore, to improve the selection of targets.\n\nWith an improved ability to detect $\\gamma$-rays from faint sources we can now investigate the occurrence of nuclear activity on different power scales. The detection of faint blazar-like activity in low luminosity AGNs or even apparently normal galaxies opens a new window on the demographics of $\\gamma$-ray sources, as well as on the mechanisms that contribute to black hole growth and jet formation. The possibility that such objects may represent an important contribution to the $\\gamma$-ray radiation of undetermined origin deserves further investigation. Searching for radiation from hidden AGNs is a fundamental science case for instruments designed to observe high energy photons and other hints of jet activity, such as light polarization. Therefore, we plan to further investigate the spectroscopic properties of candidate counterparts to $\\gamma$-ray emission, taking possibly into account light polarization studies as well, through an extensive observational campaign designed for middle class telescopes.\n\n\\section*{Acknowledgements}\nThe \\textit{Fermi}-LAT Collaboration acknowledges support for LAT development, operation and data analysis from NASA and DOE (United States), CEA\/Irfu and IN2P3\/CNRS (France), ASI and INFN (Italy), MEXT, KEK, and JAXA (Japan), and the K.A.~Wallenberg Foundation, the Swedish Research Council and the National Space Board (Sweden). Science analysis support in the operations phase from INAF (Italy) and CNES (France) is also gratefully acknowledged.\n\nThis work is based on observations collected at Copernico (or\/and Schmidt) telescope(s) (Asiago, Italy) of the INAF - Osservatorio Astronomico di Padova. Part of this work is based on archival data, software or online services provided by the ASI SCIENCE DATA CENTER (ASDC).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPositive feedback is ubiquitous in biochemical networks and can lead to a bifurcation from a monostable to a bistable cellular state \\cite{mitrophanov2008positive, tkavcik2012optimizing, das2009digital, vogel2016dichotomy}. Near the bifurcation point, the bistable state often reflects a choice between two accessible but opposing cell fates. For example, in T cells, the distribution of doubly phosphorylated ERK (ppERK) can be bimodal \\cite{vogel2016dichotomy}. ppERK is a protein that initiates cell proliferation and is implicated in the self\/non-self decision between mounting an immune response or not \\cite{vogel2016dichotomy, altan2005modeling}.\n\nThe bifurcation point is similar to an Ising-type critical point in physical systems such as fluids, magnets, and superconductors, where a disordered state transitions to one of two ordered states at a critical temperature \\cite{goldenfeld1992lectures}. In fact, universality tells us that the two should not just be similar, they should be the same: because they are both bifurcating systems, both types of systems should exhibit the same critical scaling exponents and therefore belong to the same universality class \\cite{goldenfeld1992lectures}. Although this powerful idea has allowed diverse physical phenomena to be united into specific behavioral classes, the application of universality to biological systems is still developing \\cite{mora2011biological, munoz2018colloquium, salman2012universal, brenner2015universal, pal2014non, ridden2015entropy, qian2016framework, hidalgo2014information}.\n\nBiological tools such as flow cytometry, fluorescence microscopy, and RNA sequencing allow reliable experimental estimates of abundance distributions, inspiring researchers to seek to apply insights from statistical physics to biological data. In particular, recent studies have demonstrated that biological systems on many scales, from molecules \\cite{mora2010maximum}, to cells \\cite{kastner2015critical, krotov2014morphogenesis, de2017critical, chen2012scale, aguilar2018critical, wan2018time}, to populations \\cite{bialek2014social, attanasi2014finite, cavagna2017dynamic}, exhibit signatures consistent with physical systems near a critical point. However, some of these studies have come under scrutiny because some of the signatures, particularly scaling laws, can arise far from or independent of a critical point \\cite{schwab2014zipf, touboul2017power, newman2005power}. Part of the problem is that the identification of appropriate scaling variables from data can be ambiguous, and one is often left looking for scaling relationships in an unguided way. \n\nTypical approaches to the interpretation of abundance distributions include fitting to either detailed mechanistic models of the underlying reaction scheme, or to an effective description of the data such as a Gaussian or lognormal mixture model. The former approach is usually difficult to parameterize and difficult to generalize to other systems. The latter approach often suffers from numerical issues (the likelihood is unbounded and the expectation-maximization algorithm can lead to spurious solutions \\cite{biernacki2003choosing}). Moreover, the vicinity of a bifurcation point is precisely where a mixture analysis is most likely to fail. In contrast, mapping to a statistical physics framework is expected to be universal, in the sense that the precise microscopic details of a broad range of biochemical models are unimportant near the bifurcation point, as they are coarse-grained rather than particular reaction parameters.\n\nHere we provide a framework for mapping well-mixed stochastic models of biochemical feedback to the mean-field Ising model and apply it to published data on T cells. This allows us to extract effective thermodynamic quantities from experimental data without needing to fit to a parametric model of the system. This makes the theory applicable to a broad class of biological datasets without worrying about model selection or goodness-of-fit criteria. The theory provides insights on how T cells respond to drugs and reveals distinctions between one type of drug response and another. Furthermore, we find that one of the thermodynamic quantities (the heat capacity) provides a novel way to estimate absolute molecule number from fluorescence level in bifurcating systems. We demonstrate that our results can be extended to cases where feedback is indirect and discuss further extensions, including to spatiotemporal dynamics.\n\n\n\\section{Results}\n\nWe consider a reaction network in a cell where $X$ is the molecular species of interest, and the other species $A$, $B$, $C$, etc.\\ form a chemical bath for $X$ [Fig.\\ \\ref{fig:setup}(a)]. The reactions of interest produce or degrade an $X$ molecule, can involve the bath species, and in principle are reversible. We allow for nonlinear feedback on $X$, meaning that the production of an $X$ molecule in a particular reaction might require a certain number of $X$ molecules as reactants. This leads to an arbitrary number of reactions of the form\n\\begin{equation}\n\\label{eq:rxns}\nj_rX + Y_r^+ \\xrightleftharpoons[k_r^-]{k_r^+} (j_r+1)X + Y_r^-,\n\\end{equation}\nwhere in the $r$th reaction, $j_r$ are stoichiometric integers describing the nonlinearity, $k_r^\\pm$ are the forward ($+$) and backward ($-$) reaction rates, and $Y_r^\\pm$ represent bath species involved as reactants ($+$) or products ($-$). A simple and well-studied special case of Eq.\\ \\ref{eq:rxns} is Schl\\\"ogl's second model \\cite{schlogl1972chemical, dewel1977renormalization, nicolis1980systematic, brachet1981critical, grassberger1982phase, prakash1997dynamics, liu2007quadratic, vellela2009stochastic}, in which $X$ is either produced spontaneously from bath species $A$, or in a trimolecular reaction from two existing $X$ molecules and bath species $B$ (i.e., $R = 2$, $j_1 = 0$, $j_2 = 2$, $Y_1^+ = A$, $Y_2^+ = B$, and $Y_1^- = Y_2^- = \\emptyset$).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig1}\n\\caption{Setup and behavior of the model. (a) We consider well-mixed stochastic biochemical networks described by an effective feedback function $f_n$. (b) Feedback produces either one or two stable steady states. (c) The molecule number distribution is peaked around these states or flat at the bifurcation point. (d) Mapping to the Ising model reveals that the effective reduced temperature drives the distribution to the unimodal ($\\theta > 0$) or bimodal ($\\theta < 0$) state (see c), while the effective field $h$ biases the distribution toward high ($h > 0$) or low ($h < 0$) molecule number. Parameters: $H=3$ and $n_c = 100$ in b, c, and d; $h = 0$ in b and c; and $\\theta = 0$ in d (see also Appendix \\ref{app:param}).}\n\\label{fig:setup}\n\\end{figure}\n\nWe assume that molecules are well-mixed and that the numbers of bath molecules are constant. The latter assumption is equivalent to integrating out all species but $X$, such that the feedback on $X$ arises directly from $X$ itself (Eq.\\ \\ref{eq:rxns}). However, in general the feedback will be indirect, with $X$ regulating dynamic species in the bath that in turn regulate $X$ (this is almost certainly the case in the T cells we study here). Therefore, we consider this more general case later in Section \\ref{sec:indirect} and show that the results discussed below remain unchanged.\n\nThe master equation for the probability of observing $n$ molecules of species $X$ according to Eq.\\ \\ref{eq:rxns} is\n\\begin{equation}\n\\label{eq:me}\n\\dot{p}_n = b_{n-1}p_{n-1} + d_{n+1}p_{n+1} - (b_n + d_n)p_n,\n\\end{equation}\nwhere $b_n = \\sum_{r=1}^R J_{rn}^+$ and $d_n = \\sum_{r=1}^R J_{rn}^-$ are the total birth and death propensities, and $J_{rn}^+ = k_r^+ n_r^+ n!\/(n-j_r)!$ and $J_{rn}^- = k_r^- n_r^- n!\/(n-j_r-1)!$ are the forward and backward propensities of each reaction pair. Here $n_r^\\pm$ are the numbers of molecules of the bath species involved in reaction $r$, and the factorials account for the number of ways that $X$ molecules can meet in a reaction. The steady state of Eq.\\ \\ref{eq:me} is \\cite{van1992stochastic, gardiner1985handbook}\n\\begin{equation}\n\\label{eq:pn}\np_n = p_0 \\prod_{j = 1}^n \\frac{b_{j-1}}{d_j} = \\frac{p_0}{n!} \\prod_{j=1}^n f_j,\n\\end{equation}\nwhere $p_0^{-1} = \\sum_{n=0}^\\infty(1\/n!)\\prod_{j=1}^n f_j$ is set by normalization. In the second step of Eq.\\ \\ref{eq:pn} we define an effective birth propensity $f_n \\equiv nb_{n-1}\/d_n$ corresponding to spontaneous death with propensity $n$ [Fig.\\ \\ref{fig:setup}(a)]. In general, $f_n$ is an arbitrary, nonlinear feedback function governed by the reaction network. For the Schl\\\"ogl model, it is $f_n = [aK^2 + s(n-1)(n-2)]\/[(n-1)(n-2)+K^2]$, where we have introduced the dimensionless quantities $a \\equiv k_1^+n_A\/k_1^-$, $s \\equiv k_2^+ n_B\/k_2^-$, and $K^2 \\equiv k_1^-\/k_2^-$. As a ubiquitous example we also consider the Hill function $f_n = a + sn^H\/(n^H+K^H)$ with coefficient $H$. Importantly, the inverse of Eq.\\ \\ref{eq:pn},\n\\begin{equation}\n\\label{eq:fn}\nf_n = \\frac{np_n}{p_{n-1}},\n\\end{equation}\nallows calculation of the feedback function from the distribution \\cite{walczak2009stochastic}, as utilized when analyzing the experimental data later in Section \\ref{sec:immune}.\n\nThe quantity $f_n-n$ determines the dynamic stability: there can be either one or two stable states $n_*$ [Fig.\\ \\ref{fig:setup}(b)], and the transition from a monostable to a bistable regime occurs at a bifurcation point [Fig.\\ \\ref{fig:setup}(c) inset]. These deterministic regimes correspond stochastically to unimodal and bimodal distributions $p_n$, respectively, with maxima at $n_*$, while the bifurcation point corresponds to a distribution that is flat on top [Fig.\\ \\ref{fig:setup}(c)].\n\n\n\\subsection{Ising mapping and scaling exponents}\n\nTo understand the scaling behavior near the bifurcation point, we expand the stability condition $f_{n_*}-n_*=0$ to third order around a point $n_c$ satisfying $f''_{n_c} = 0$. This choice of $n_c$ eliminates the quadratic term in the dynamic forcing $f_n-n$, equivalent to eliminating the cubic term in an effective potential as in Ginzburg--Landau theory \\cite{kopietz2010introduction}. Defining the parameters\n\\begin{equation}\n\\label{eq:cparam}\nm \\equiv \\frac{n_*-n_c}{n_c}, \\quad\nh \\equiv \\frac{2(f_{n_c} - n_c)}{-f'''_{n_c}n_c^3}, \\quad\n\\theta \\equiv \\frac{2(1-f'_{n_c})}{-f'''_{n_c}n_c^2},\n\\end{equation}\nthe expansion $f_{n_c} + f'_{n_c}(n_*-n_c) + f'''_{n_c}(n_*-n_c)^3\/3! - n_* = 0$ becomes $h - \\theta m - m^3\/3 = 0$. This expression is equivalent to the expansion of the Ising mean field equation $m = \\tanh[(m+h)\/(1+\\theta)]$ for small magnetization $m$, where $\\theta = (T-T_c)\/T_c$ is the reduced temperature, and $h$ is the dimensionless magnetic field \\cite{kopietz2010introduction}. Therefore, in our system we interpret $m$ as the order parameter, $\\theta$ as an effective reduced temperature, and $h$ as an effective field. Explicit expressions for $n_c$, $\\theta$, and $h$ in terms of the  biochemical parameters and vice versa are given for the Schl\\\"ogl and Hill models in Appendix \\ref{app:param}.\n\nWe see in Fig.\\ \\ref{fig:setup}(c) and (d) that $n_c$ determines where the distribution is centered, that $\\theta$ drives the system to the unimodal ($\\theta > 0$) or bimodal ($\\theta < 0$) state, and that $h$ biases the system to high ($h > 0$) or low ($h < 0$) molecule numbers. Note that unlike in the Ising model, even when $h=0$ an asymmetry persists between the high and low states [see the purple distribution in Fig.\\ \\ref{fig:setup}(c)]. The reason is that in the master equation (Eq.\\ \\ref{eq:me}), unlike in Ginzburg--Landau theory, fluctuations scale with molecule number, such that the high state is wider than the low state.\n\nThe equivalence between our system and the Ising mean-field equation near the critical point (Eq.\\ \\ref{eq:cparam}) implies that our system has the same scaling exponents $\\beta=1\/2$, $\\gamma=1$, and $\\delta=3$ as the Ising universality class in its mean-field limit \\cite{kopietz2010introduction}. For completeness, we verify in Appendix \\ref{app:scalings} that these scalings are indeed obeyed by the Schl\\\"ogl and Hill models.\n\nHowever, Eq.\\ \\ref{eq:cparam} does not explicitly determine the value of the exponent $\\alpha$. The reason is that, unlike $\\beta$, $\\gamma$, and $\\delta$, the exponent $\\alpha$ depends on the entire distribution $p_n$, not just the maxima.  Specifically, $\\alpha$ concerns the heat capacity, $C|_{h=0}\\sim |\\theta|^{-\\alpha}$, which depends on the entropy $S$ and thus $p_n$. The equilibrium definition $C = T\\partial_T S$ generalizes to a nonequilibrium system like ours when one uses the Shannon entropy $S = -k_{\\rm B}\\sum_n p_n \\log p_n$ \\cite{mandal2013nonequilibrium}. Since $T = (1+\\theta)T_c$, we have $C = (1+\\theta)\\partial_\\theta S$, or\n\\begin{equation}\n\\label{eq:C}\n\\frac{C}{k_{\\rm B}} = -(1+\\theta)\\sum_{n=0}^\\infty p_n (1+\\log p_n) \\Bigg( \\psi_n - \\sum_{j=0}^\\infty p_j \\psi_j \\Bigg),\n\\end{equation}\nwhere $\\psi_n \\equiv (1\/2)f'''_{n_c}n_c^2\\sum_{j=1}^n(j-n_c)\/f_j$. Eq.\\ \\ref{eq:C} follows from performing the $\\theta$ derivative using the expression in Eq.\\ \\ref{eq:pn}, the expansion below Eq.\\ \\ref{eq:cparam}, and the definition of $\\theta$ (Eq.\\ \\ref{eq:cparam}). We see in Fig.\\ \\ref{fig:heat1}(a) that when $h=0$, $C$ exhibits a minimum at $\\theta^*$. We see in Fig.\\ \\ref{fig:heat1}(b) that $\\theta^*$ vanishes as the system size increases, $n_c\\to\\infty$. This implies that $C|_{h=0}\\sim |\\theta|^0$ to sub-quadratic order in $\\theta$, or $\\alpha = 0$, again consistent with the Ising universality class in its mean-field limit. Interestingly, whereas $C$ is discontinuous in the mean-field Ising model \\cite{kopietz2010introduction} and constant in the van der Waals model of a fluid \\cite{goldenfeld1992lectures}, it is minimized here; nevertheless, in all cases $\\alpha = 0$. Note from Fig.\\ \\ref{fig:heat1}(a) that $C$ is negative near $\\theta = 0$; negative heat capacity is a well-known feature of nonequilibrium steady states \\cite{zia2002getting, boksenbojm2011heat, bisquert2005master}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig2}\n\\caption{(a) Heat capacity (Eq.\\ \\ref{eq:C}) is minimized at the bifurcation point, corresponding to exponent $\\alpha = 0$. (b) The location of the minimum approaches $\\theta^*\\to0$ as $n_c\\to\\infty$, as expected. Parameters: $H=3$, $n_c = 500$, and $h=0$.}\n\\label{fig:heat1}\n\\end{figure}\n\n\n\n\\subsection{Application to immune cell data}\n\\label{sec:immune}\n\nTo demonstrate the utility of our theory, we apply it to published data from T cells \\cite{vogel2016dichotomy}. In these experiments, chemotherapy drugs inhibit the enzymes MEK and SRC in the biochemical networks of the cells. The inhibition results in bimodal (low dose) or unimodal (high dose) distributions of ppERK abundance, which is measured as fluorescence intensity $I$ by flow cytometry. The distributions are shown for a range of drug doses in Fig.\\ \\ref{fig:experiment}(a) and (b) (the insets show distributions of log intensity for clarity). Experimental details are given in the original publication \\cite{vogel2016dichotomy} and are summarized in Appendix \\ref{app:expt}, along with the drugs and dose amounts.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig3}\n\\caption{Application of the theory to immune cell data. Upon administration of either (a) MEK or (b) SRC inhibitor, experimental distributions of T cell ppERK fluorescence intensity are unimodal (bimodal) for high (low) doses. Insets show distributions of log intensity for clarity. (c, d) Feedback functions calculated from the experimental distributions correspondingly exhibit either one or two stable states. (e-g) Effective thermodynamic quantities calculated from the data vary with drug dose in distinct ways for each drug. The results in (c-g) corroborate those in \\cite{vogel2016dichotomy}, but with a much simpler framework that has three parameters instead of five and requires no fitting or prior biological knowledge of the system. Error bars: standard error from filter windows $25 \\le W \\le 35$ (see Appendix \\ref{app:analysis}).}\n\\label{fig:experiment}\n\\end{figure}\n\nFirst, we compute the feedback function $f$ from each distribution using Eq.\\ \\ref{eq:fn} (see Appendix \\ref{app:analysis}). Fig.\\ \\ref{fig:experiment}(c) and (d) show the corresponding forcing functions [compare to Fig.\\ \\ref{fig:setup}(b)]. As expected, in each case we see that the forcing function transitions from two stable states to one stable state as the drug is applied.\n\nThen, we compute $I_c$ (the analog of $n_c$ in units of fluorescence intensity), $\\theta$, and $h$ from the feedback function using Eq.\\ \\ref{eq:cparam} (see Appendix \\ref{app:analysis}). These quantities are shown as a function of drug dose in Fig.\\ \\ref{fig:experiment}(e)-(g). We see that the behavior is different depending on whether MEK inhibitor (MEKi) or SRC inhibitor (SRCi) is applied. Specifically, MEKi decreases $I_c$, increases $\\theta$, and decreases $h$; whereas SRCi only decreases $h$, leaving the other quantities unchanged. Thus, the effective thermodynamic quantities can differentiate cellular responses to different perturbations, such as the application of different drugs.\n\nFurthermore, the mapping provides an intuitive interpretation of the drug responses. MEKi causes a transition from a bimodal to a unimodal state in the expected way: by increasing the reduced temperature $\\theta$ from a negative to a positive value [Fig.\\ \\ref{fig:experiment}(f)]. In the process, $I_c$ decreases [Fig.\\ \\ref{fig:experiment}(e)], meaning that the unimodal state is shifted to lower molecule number, near the lower mode of the bimodal state [Fig.\\ \\ref{fig:experiment}(a) inset]. In contrast, SRCi causes a transition from a bimodal to a unimodal state in a different way: by decreasing the field while leaving $\\theta$ and $I_c$ unchanged [Fig.\\ \\ref{fig:experiment}(e)-(g)]. In essence, the distribution remains bimodal and unshifted, except that the field causes the high mode to diminish in weight [Fig.\\ \\ref{fig:experiment}(b) inset]. Interestingly, the mean dose-response curves are similar for the two drugs \\cite{vogel2016dichotomy}, but our mapping elucidates precisely how the transitions are different at the distribution level. Related conclusions were drawn in \\cite{vogel2016dichotomy}, but those conclusions relied on fitting the distributions to a five-parameter Gaussian mixture model, which is expected to fail near the bifurcation point. Here we use only three parameters and no fitting, and we emerge with an intuitive interpretation in terms of thermodynamic quantities.\n\nFinally, we note that for both drugs the effective field is negative at all doses [Fig.\\ \\ref{fig:experiment}(g)]. The reason is that the fluorescence distributions have long tails (which is why they are often easier to visualize in log space); see Fig.\\ \\ref{fig:experiment}(a) and (b). In the theory, a long tail is indistinguishable from a low-molecule-number bias in the peak, which corresponds to $h < 0$. We address the possible origins and implications of the long tails in the Discussion (Section \\ref{sec:discussion}).\n\n\n\\subsection{Estimation of molecule number}\n\nWe now apply the theory to compute the heat capacity from the T cell data.\nSpecifically, we compute $C$ using Eq.\\ \\ref{eq:C} (see Appendix \\ref{app:analysis}) for all drugs and doses used in the experiments \\cite{vogel2016dichotomy} (Appendix \\ref{app:expt}). Unlike the other thermodynamic quantities, $C$ requires a conversion from fluorescence intensity to molecule number because it depends explicitly on the distribution $p_n$ (Eq.\\ \\ref{eq:C}). Therefore we compute $C$ for various values of the conversion factor $I_1$, where $n = I\/I_1$. The results are shown in Fig.\\ \\ref{fig:heat2}. We see that irrespective of $I_1$ over four orders of magnitude, the data closest to $h=0$ (yellow) exhibit a global minimum in $C$ at $\\theta = 0$, as expected from Fig.\\ \\ref{fig:heat1}(a). However, we also see that the depth of the minimum agrees with that of the theory only for the particular choice $I_1 \\approx 0.1$ [Fig.\\ \\ref{fig:heat2}(c)].\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig4}\n\\caption{Estimation of molecule number by comparing heat capacity between theory and experiments. (a-d) Rough estimate of fluorescence-to-molecule-number conversion factor $I_1$ (see titles) obtained by comparing depths of theory and experimental minima. ``Hill'' refers to the theoretical curve produced by Hill-function feedback as in Eq.~\\ref{eq:Hill}. Different symbols correspond to different drugs. See Appendix \\ref{app:expt} for drugs (shape) and doses (size). (e) More precise estimate obtained from plotting sum of squared errors (SSE) for data within $-\\Delta\\theta \\le \\theta \\le \\Delta\\theta$ and fitting to parabola (see Appendix D for details). Here $\\Delta\\theta = 0.05$. (f) Estimate is insensitive to value of $\\Delta\\theta$. Theory parameters: $H = 4$, $h = 0$, and $n_c = \\bar{I}_c\/I_1$, where $\\bar{I}_c = 730$ is the average value across all experiments.}\n\\label{fig:heat2}\n\\end{figure}\n\nTo obtain a more precise estimate of $I_1$, we plot the sum of squared errors between the data and the theory as a function of $I_1$ in Fig.\\ \\ref{fig:heat2}(e). We focus on the bifurcation region by considering only values of $\\theta$ within $-\\Delta\\theta \\le \\theta \\le \\Delta\\theta$, and we find that our results are not sensitive to the choice of $\\Delta\\theta$ [Fig.\\ \\ref{fig:heat2}(f)]. This procedure (see the details in Appendix \\ref{app:analysis}) results in an estimate of $I_1 = 0.5$ $\\pm$ $0.2$, as seen in Fig.\\ \\ref{fig:heat2}(f). This value of $I_1$ corresponds to $\\bar{n}_*=$ 170,000 $\\pm$ 70,000 ppERK molecules in the high mode averaged across all cases with no inhibitor. It is possible to compare this value with previous measurements on these cells. In two separate experiments, it was estimated that there are approximately 100,000  \\cite{altan2005modeling} and 214,000 \\cite{hukelmann2016cytotoxic} ERK molecules per cell, and that only about 50\\% of these molecules are doubly phosphorylated during T cell receptor activation \\cite{altan2005modeling} (see Appendix \\ref{app:analysis}). These considerations give a range of roughly 50,000$-$107,000 ppERK molecules, which is consistent with our estimate of 170,000 $\\pm$ 70,000. The agreement is especially notable given that T cell protein abundances generally span six orders of magnitude, from tens to tens of millions of molecules per cell \\cite{hukelmann2016cytotoxic}.\n\nWhy does the heat capacity extract the conversion between fluorescence intensity and molecule number? As mentioned above, $\\alpha$ is the only exponent that is a function of $p_n$ instead of just its maxima. This means that the plot of $C$ vs.\\ $\\theta$ contains information not only about means or modes, but also about fluctuations. The notion that fluctuation information is essential for converting from intensity to molecule number can be seen with a simpler example: a Poisson distribution. Here we would have $\\sigma_I^2\/\\bar{I}^2 = \\sigma_n^2\/\\bar{n}^2 = 1\/\\bar{n} = I_1\/\\bar{I}$. From this relation it is clear that information about not only the mean ($\\bar{I}$) but also the fluctuations ($\\sigma_I^2$) in intensity is necessary and sufficient to infer the conversion factor $I_1$. In our case, the heat capacity is extracting similar information, but for a bifurcating system.\n\n\n\\subsection{Generalization to indirect feedback}\n\\label{sec:indirect}\n\nIn the T cells, it is well known that ppERK does not apply feedback to its own activation directly, but rather indirectly via upstream components \\cite{vogel2016dichotomy, shin2009positive, altan2005modeling}. Therefore, we seek to determine the extent to which the above results are sensitive to our assumption in the theory that the feedback is direct. To this end, we construct a minimal extension of the model in Eq.\\ \\ref{eq:rxns} in which the feedback is indirect:\n\\begin{align}\n& \\emptyset \\xrightleftharpoons[k_2]{k_1} X, \\qquad\n2X \\xrightleftharpoons[k_4]{k_3} D, \\nonumber \\\\\n& D \\xrightarrow{k_5} D + A, \\qquad\nA \\xrightarrow{k_6} A + X, \\qquad\nA \\xrightarrow{k_7} \\emptyset, \\nonumber \\\\\n\\label{eq:indirect}\n& D \\xrightarrow{k_8} D + B, \\qquad\nB + X \\xrightarrow{k_9} B, \\qquad\nB \\xrightarrow{k_{10}} \\emptyset.\n\\end{align}\nHere $X$ is produced, is degraded, and reversibly dimerizes (first line); the dimer $D$ produces a species $A$ that produces $X$ and is degraded (second line); and the dimer also produces a species $B$ that degrades $X$ and is degraded (third line). Eq.\\ \\ref{eq:indirect} is an extension of Eq.\\ \\ref{eq:rxns} because there are multiple stochastic variables ($X$, $D$, $A$, and $B$), there are irreversible reactions, and $X$ feeds back on itself indirectly through $D$, $A$, and $B$ instead of directly.\n\nThe deterministic steady state of Eq.\\ \\ref{eq:indirect} is\n\\begin{equation}\n\\label{eq:multidet}\n0 = \\dot{n}\/k_2 = c_0 - n_* + c_2 n_*^2 - c_3 n_*^3,\n\\end{equation}\nwhere $c_0 \\equiv k_1\/k_2$, $c_2 \\equiv k_3k_5k_6\/(k_2k_4k_7)$, $c_3 \\equiv k_3k_8k_9\/(k_2k_4k_{10})$, and the molecule numbers of $D$, $A$, and $B$ have been eliminated in favor of $n_*$ by setting their own time derivatives to zero. Because Eq.\\ \\ref{eq:multidet} is cubic in $n_*$, we see immediately that it has the same form as the expanded Ising mean field equation $h - \\theta m - m^3\/3 = 0$ (see Eq.\\ \\ref{eq:cparam}). Specifically, defining $m = (n_*-n_c)\/n_c$ as in Eq.\\ \\ref{eq:cparam}, the choice $n_c = c_2\/(3c_3)$ eliminates the term quadratic in $m$ and implies $\\theta = 3c_3\/c_2^2 - 1$ and $h = 9c_0c_3^2\/c_2^3 - 3c_3\/c_2^2 +2\/3$. It immediately follows that this model has the same exponents  $\\beta=1\/2$, $\\gamma=1$, and $\\delta=3$ as the mean-field Ising universality class.\n\nTo test whether the heat capacity for this model exhibits the same features as that for the direct feedback model in Fig.\\ \\ref{fig:heat1}(a), we compute the steady state marginal distribution $p_n$ using stochastic simulations \\cite{gillespie1977exact} of Eq.\\ \\ref{eq:indirect}. Specifically, we set $k_3\/k_4 = 1\/n_c$ and $k_5\/k_7 = k_8\/k_{10} = 1$ to ensure that the numbers of $D$, $A$, and $B$ molecules, respectively, are on the order of $n_c$. We then set $k_4\/k_2 = k_7\/k_2 = k_{10}\/k_2 = \\rho$, where $\\rho$ is a free parameter that determines whether the degradation timescales of $D$, $A$, and $B$, respectively, are faster ($\\rho > 1$) or slower ($\\rho < 1$) than that of $X$. These conditions, along with the definitions of $n_c$, $\\theta$, and $h$ above, constitute nine equations for nine reaction rates, plus $k_2$ which sets the units of time. Solving these equations yields expressions for the rates in terms of $n_c$, $\\theta$, $h$, and $\\rho$ that we use in the simulations.\n\nFig.\\ \\ref{fig:indirect}(a) shows the heat capacity $C$ as a function of $\\theta$ for $h = 0$, $n_c = 100$, and $\\rho = \\{0.1, 1, 10\\}$, where $C = (1+\\theta)\\partial_\\theta S$ is computed from the entropy $S = -k_{\\rm B}\\sum_n p_n \\log p_n$ by numerical derivative. We see that for all $\\rho$ values, the curves exhibit a minimum at $\\theta = 0$, implying $\\alpha = 0$, and they rise more steeply for negative than for positive $\\theta$ as in Fig.\\ \\ref{fig:heat1}(a).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig5}\n\\caption{Verification that indirect feedback does not qualitatively change modeling assumptions or results. (a) $C$ and $\\theta$ calculated from extended model with indirect feedback. (b) $C$ and $\\theta$ inferred assuming the feedback is direct (Eq.\\ \\ref{eq:fn}). Compare with Fig.\\ \\ref{fig:heat1}(a). Parameters: $n_c=100$ and $h=0$.}\n\\label{fig:indirect}\n\\end{figure}\n\nWe then investigate whether Eq.\\ \\ref{eq:rxns} remains valid as a coarse-grained description of the extended model in Eq.\\ \\ref{eq:indirect}. To answer this question, we infer values of $n_c$, $\\theta$, $h$, and $C$ directly from the simulation data $p_n$ using the same protocol as for the experimental data. That is, we compute $f_n$ via Eq.\\ \\ref{eq:fn}, and then compute $\\theta$, $h$, and $C$ from its derivatives at $n_c$ according to Eqs.\\ \\ref{eq:cparam} and \\ref{eq:C}, where $n_c$ satisfies $f''_{n_c} = 0$. As with the experimental data (see Appendix \\ref{app:analysis}), derivatives are calculated using a Savitsky-Golay filter \\cite{savitzky1964smoothing}, although here we apply the filter directly to $f_n$ and perform the analysis directly in $n$ space, not log space.\n\nFig.\\ \\ref{fig:indirect}(b) shows the result of this procedure for the inferred heat capacity $C$ as a function of the inferred $\\theta$. We see that, as with the exact $C$ and $\\theta$ [Fig.\\ \\ref{fig:indirect}(a)], the data exhibit a minimum at $\\theta = 0$ and rise more steeply for negative than for positive $\\theta$. Note that the values of $C$ and $\\theta$ are different in (a) and (b), which is expected because the shape of $p_n$ is not quantitatively the same in the two models of Eqs.\\ \\ref{eq:rxns} and \\ref{eq:indirect}; nonetheless, the shape of the $C$ vs.\\ $\\theta$ curves remains the same. We have checked that the inferred values of $n_c$ and $h$ are distributed around their known values of $100$ and $0$, respectively, and that the shape persists across a range of filter window sizes.\n\nThese results suggest that the main findings above are not sensitive to our assumption that feedback is direct, and therefore that we are justified in using Eq.\\ \\ref{eq:rxns} as a coarse-grained model to analyze the T cell data.\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe have employed the fact that a feedback-induced bifurcation exhibits the scaling properties of the mean-field Ising universality class to provide a simple prescription for modeling and analyzing biological data. Contrary to existing mixture-model approaches, our method is most valuable near the bifurcation point, which is where biologically significant cell-fate decisions are expected to take place. Our approach provides the effective order parameter, reduced temperature, magnetic field, and heat capacity from experimental distributions without fitting or needing to know the molecular details. By applying the approach to T cell flow cytometry data, we discovered that these quantities discriminate between cellular responses in an intuitive, interpretable way, and that the heat capacity allows estimation of the molecule number from fluorescence intensity for a bifurcating system. By generalizing the theory to include indirect feedback, we demonstrated the capacity to model realistic signaling cascades where indirect feedback is common. Our approach should be applicable to other systems observed to undergo a pitchfork-like bifurcation and the associated unimodal-to-bimodal transition in abundance distributions, but not to systems which have an absorbing or extinction state, as they are expected to fall under a different universality class \\cite{ohtsuki1987nonequilibrium, grassberger1978reggeon}.\n\nThe theory assumes only birth-death reactions and neglects more complex mechanisms such as bursting \\cite{friedman2006linking, mugler2009spectral} or parameter fluctuations \\cite{shahrezaei2008colored, horsthemke1984noise}. These mechanisms are known to produce long tails and may be responsible for the long tails observed in the experimental data [Fig.\\ \\ref{fig:experiment}(a) and (b)]. Cell-to-cell variability (CCV) may also contribute to the long tails, as it is known to be present in T cell populations \\cite{cotari2013cell}. Our theory neglects CCV and instead assumes that the distribution of molecule numbers across the population is the same as that traced out by a single cell over time. Although CCV may play an important role, one generically expects the role of intrinsic fluctuations to be amplified near a critical point, and models that ignore CCV have been shown to be sufficient to explain both the bimodality \\cite{das2009digital} and variance properties \\cite{prill2015noise} of ppERK in T cells. Moreover, the fact that our theory provides an estimate of the molecule number that is consistent with other estimates suggests that intrinsic fluctuations play a large role. Distinguishing between intrinsic fluctuations and long-lived CCV is an important topic for future work.\n\nOur work provides key tools that can be used for a broader exploration of biological systems. The approach is applicable to any experimental dataset that exhibits unimodal and bimodal abundance distributions, and could lead to a unified picture of diverse cell types and environmental perturbations in terms of effective thermodynamic quantities. At the same time, several extensions of our work are natural. For example, the dynamics of the theory could be probed to investigate the consequences of critical slowing down for driven or dynamically perturbed systems with feedback. Alternatively, the theory could be generalized to systems that are not well-mixed, such as intracellular compartments or communicating populations, to investigate space-dependent universal behavior and its biological implications.\n\n\n\n\\section{Data availability}\nData and code for all figures and the MIFlowCyt record are available at\\\\\n\\url{https:\/\/github.com\/AmirErez\/UniversalImmune}.\n\n\n\n\\section*{Acknowledgments}\nThis work was supported by Human Frontier Science Program grant LT000123\/2014 (Amir Erez), National Institutes of Health (NIH) grant R01 GM082938 (A.E.), Simons Foundation grant 376198 (T.A.B.\\ and A.M.), and the Intramural Research Program of the NIH, Center for Cancer Research, National Cancer Institute.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\nThe use of Low Temperature Detectors (LTDs) for sensing  \n X-ray and $\\gamma$-ray signals is quite widespread and well \nestablished~\\cite{Ullom-2015}.\nLTDs are also widely used in the field of fundamental physics, especially for \nDouble Beta Decay (DBD), and Dark Matter (DM) searches~\\cite{Pirro-2017}.\nIn these surveys the need for a hybrid detector, in which an energy \nrelease can be measured through different mechanisms,  is of primary importance in order \nto distinguish the nature of interacting particles.  For instance hybrid detectors can help identify \nand reject events caused by the natural background. With thermal detectors this can be \nachieved  using scintillating or luminescent crystals. The simultaneous and \n\\it independent \\rm readout of the heat  and the (escaping) light produced by the \ninteraction  reveals the nature of the interacting particles thanks to \nthe different  scintillation yields of $n$, $\\alpha$ and $\\gamma\/\\beta$ events.\nThis discrimination technique is presently used for DM \nsearches~\\cite{CRESST-2016,Angloher:2017sft,Angloher:2016hbv}, \nDBD searches~\\cite{CUPID-0-2018,Cupid-Mo-2017,Amore-2017}, and it can be  \nalso implemented for rare nuclear decays~\\cite{Alfa-1,Alfa-2,Pattavina:2018nhk}.\n\nAt milli-Kelvin temperatures, the light detectors are usually bolometers \nthemselves: a \\it dark \\rm thin crystal absorbs the photons, producing heat (phonons) that is\nmeasured by a suitable thermometer.   \nThe main difference among the various Bolometric Light Detector (BLD) instruments currently in use  \nis the choice of the thermometer element, e.g. Transition Edge Sensors \n(TES)~\\cite{TES_LD_CRESST}, Neutron  Transmutation Doped  (NTD) \nthermistors~\\cite{NTD_LD_Lucifer-2013} or Micro Magnetic Calorimeters (MMC)~\\cite{MMC_LD-2015}.  \n\nThe work presented here was performed within the CUPID \nframework~\\cite{CUORE-IHE-2014,CUPID-2015}, the future follow up of \nCUORE~\\cite{CUORE-2018} that represents the largest world-wide bolometric  experiment to date. \nThe aim was to  develop NTD-based BLDs with improved performance in terms of \nsensitivity, time response and simplified packaging for large arrays. \nUsing the tiny Cherenkov light emission of TeO$_2$~\\cite{Tabarelli-2010,Enriched-TeO2-Cherenkov-2017}  to \ndecrease by two order of magnitude the  $\\alpha$-induced background, requires a BLD with a S\/N ratio \nof the order of  $\\sim$5~\\cite{CUPID-2015}:  this corresponds to a RMS baseline resolution of the BLD of the \norder of $\\sim$20~eV  being the Cherenkov  light signal of the order of 100 eV.\nActually one can work towards the optimization  of the light collection~\\cite{Casali-2017} and\/or \ntowards the energy resolution of the BLD or -as we made in this work- both.\nAdditionally, in case of $^{100}$Mo-based compounds, beside the  same need to suppress the surface \n$\\alpha$-induced background,  a fast time response of the BLD ($\\leq$~1~ms)  is mandatory to suppress\nthe background  induced the pile-up of the 2$\\nu$ DBD~\\cite{2_nu-Pile_up-2012}: also in \nthis case the S\/N ratio will play an important role~\\cite{2_nu-Pile_up-2016}.\n\n\n\nOur work has therefore focused on two aspects of BLD performance: (1) improving the response of \nthe NTD thermometer and (2) increasing the light collection.\nWhile the first aspect is strictly related to a specific technique, the second aspect \nis worthy of additional remarks. The working principle of a BLD is \nirrespective  of the sensor: a  thin  crystal wafer (usually  Si or Ge)  absorbs \nthe emitted photons and  converts them into heat. \nUnlike a conventional bolometric approach, we have to avoid the optical coupling \nbetween crystal and BLD made with optical grease or similar substance since the unavoidable \nheat flow through the optical coupling and the increase of the heat capacity of the system would  reduce the  \nindependence of the two  detectors, eliminating the possibility of particle \ndiscrimination afforded by the different scintillation yields.  Therefore the thermal contact  between \nthe luminescent crystal and BLD has to be avoided,\nespecially in the case of extremely low scintillation yields.  This is true for  most \nof the Mo-based compounds~\\cite{Cupid-Mo-2017} and, even more importantly in  case of \nCherenkov signals. A 2615~keV $\\gamma$-ray energy release in a CUORE-like TeO$_2$ absorber \nproduces a light signal in the BLD on the order of  $\\sim$100~eV~\\cite{Casali-2017}. \nFor this reason the BLD is always facing the scintillating crystal without directly contacting \nit via a coupling medium.\n \nIn the following section it is shown that  if the BLD is simply resting on the crystal \nsurface, held in  position only by gravity, the thermal coupling between the BLD and \nthe crystal is almost negligible and the leakage of the BLD thermal signal through the scintillating crystal vanishes. \nThis fact can be explained considering the acoustic mismatch described in \nthe diffused mismatch model whereby the  heat carriers (phonons) in insulating materials are \nscattered at the interfaces~\\cite{Matsumoto-1977,Swartz-1989}. This approach shows that the \nthermal resistance between two dielectric crystals is strongly dependent on the surface \nstate, on the different phonon characteristics in the two materials (density and Debye \ntemperature), and on the applied force. This latter parameter has a significant effect.  \nWhen two solids are placed in contact with each other, the actual contact area can be much smaller \nthan the cross sections involved due to surface irregularities.  By rising the applied force \nbetween the materials, a plastic or permanent deformation occurs and the \"real\" contact surface \narea increases. The result of this action is that the thermal conductance of the contact is directly \nproportional to the applied force~\\cite{Barucci-2001,Ventura-2008}.\n\nAlthough such  simple stand will  clearly not produce  a so-called \"optical matching,\" the \nlight collection  will be  definitively larger due to geometrical factors~\\footnote{For instance if \nthe BLD is held  in its own structure, depending on the mounting scheme, there are generally a \nfew mm of distance from the BLD to the scintillating crystal. \nThis increases the chance for photon escape or absorption by the holding structure rather than the BLD.}.\nIn addition, removing the BLD mounting structure decreases the presence of materials and surfaces close\nto the detector which reduces possible radioactive contamination, a fundamental aspect of dealing with rare event searches.\n\n\n\\section{Bolometric Light Detectors}\n\nOur BLDs are usually constituted by electronic grade undoped Ge wafers, coupled with  Ge NTD  thermistors. \nWe started to develop these detectors coupled with several scintillating DBD \ncrystals~\\cite{PIRRO-2005} and we deeply characterized  their operation and \nperformances~\\cite{light-detectors-2013} to finally realize  the  LUCIFER~\\cite{LUCIFER-2013} experiment, \nwhich has been renamed CUPID-0~\\cite{CUPID-0-detector_2018}.\n\nEach   BLD  of CUPID-0 (totalling 26 detectors) was made by a double side polished electronic \ngrade undoped Ge wafer (44.5~mm diameter, 0.17~mm thick). The NTD thermistor, with dimension of \n(2.85~$\\times$~2~$\\times$~0.5)~mm$^3$, \nis glued through  six small glue dots ($\\sim$~0.5~mm diameter, 0.05~mm height) made with \nAraldit\\textsuperscript{\\textregistered} Rapid glue.\nThe performance of six of these detectors was evaluated in a dedicated test run~\\cite{LUCIFER-2016} and \nthe results are summarized in Tab.~\\ref{tab-cupid-0-LD}.\n\nTo further optimize our BLDs, we  produced a set of devices based on the pioneering work of \nCoron et.al.~\\cite{Coron-2004}.  For this study we (1) decreased the heat capacity (size) of the \nthermistor, (2) increased the thermal conductance between the thermistor and the Ge wafer, and (3) \ndecreased the  thermal conductance to  the thermal bath. \nWith respect to the  thermistor size, we used thermistors with a dimension of\n(2.85~$\\times$~1~$\\times$ 0.4)~mm$^3$, roughly 2.5 times smaller than the CUPID-0 devices. \nWe also decided to replace the six glue dots with an uniform glue layer, thus increasing the thermal \nconductance between the thermistors and light-absorbing Ge wafer. \n\nIt should be noted that in our experience the use of glue dots \ninstead of a \\it more effective \\rm  thin gluing layer is  preferred when coupling inherently different \nmaterials (e.g. TeO$_2$ crystals and Ge thermistors).  The dot approach reduces the mechanical stresses \ninduced by  differential thermal contraction of the materials when cooled. \nIn such cases, and especially when working with larger-sized thermistors, we sometimes observed cracks on \nthe crystal surface after a cooling cycle. This phenomenon is greatly reduced in our case since we glue \nGermanium thermistors to Germanium light absorbers and use smaller thermistors.  Even in this \ncase, however, there are some small unavoidable stresses due to misorientation between the thermistor\nand absorber crystallographic planes, but we have found that these effects never led to visible cracks.\n\nWith respect to the mounting (i.e. the conductance to the thermal bath), there are many ways  \nto hold the BLD in place.  In earlier work we  adopted  two~\\cite{light-detectors-2013} or \nthree~\\cite{CUPID-0-detector_2018} small PTFE  clamps that squeeze the edge of the Ge, keeping \nit fixed in a Cu standalone holder. \nPTFE is a  common material also used by other  groups working with NTD \nsensors~\\cite{Lumineu-2017} and  with MMC detectors~\\cite{MMC_LD-2015}. \nOther clamping schemes and material choices have been demonstrated by the CRESST group.  \nThese include bronze clamps and Silicon  or  CaWO$_4$-based sticks~\\cite{Strauss-2018}. \nThe design used in~\\cite{Coron-2004}, however, is probably the most complex from a construction point \nof view, using several ultra thin superconductive wires to suspend the Ge wafer from a copper frame to \nproduce a negligible thermal link that maximizes the heat flow from the wafer to the NTD.  \n\\begin{table}[t]\n\\centering\n\\caption{Mean performance of six  CUPID-0-like light detectors~\\cite{LUCIFER-2016}.\n\\textbf{R$_{work}$} refers to the resistance of the NTD Ge thermistor in working conditions, \n\\textbf{Response} refers to the absolute voltage \ndrop (in $\\mu$V) produced by an energy release of 1\\,keV,  \\textbf{Baseline RMS} is the resolution\nafter signal filtering~\\cite{Gatti-1986:1,Alduino-2016:045503}. {\\bf$\\tau_{r}$} and {\\bf$\\tau_{d}$} are the \nrise and  decay  times, computed as the time difference  between the 90$\\%$ and 10$\\%$ of the leading\nedge and as the time difference between the 30$\\%$ and 90$\\%$ of the trailing edge,\nrespectively. The Bessel cut-off frequency is 200 Hz (see last remarks of \nSec.~\\ref{sec:results}).}\n\\label{tab-cupid-0-LD}\n\\begin{tabular}{ccccc}\n\\hline\\noalign{\\smallskip}\nR$_{work}$      &Response     &Baseline RMS     &$\\tau_{r}$       &$\\tau_{d}$  \\\\\n\n[M$\\Omega$]      &[$\\mu$V\/keV]     &[eV]             &[ms]              &[ms]\\\\\n\\hline\n   0.87          & 1.36            & 43             & 1.77              & 5.06 \\\\\n\\hline\n\\end{tabular}\n\\end{table} \n\nWe decided to avoid any kind of holding structure whatsoever so we laid the BLD \ndirectly on the crystal, kept in position only by its weight ($\\sim$1.1 g). \nIn this configuration the main thermal link between the BLD and the cryostat is represented by \nthe thin gold NTD thermistor wires ( 2 $\\times$ 15 mm length, 25 $\\mu$m diameter).  As mentioned \nabove, the expected thermal conductance to the scintillating crystal is negligible.  The  crystal  \nchosen for this test was a  (50.5~$\\times$~50.5~$\\times$~50.5)~mm$^3$ \nTeO$_2$ crystal.  The aim was to test  the new setup with a light signal on the order \nof few tens of eV. The Ge light-absorbing wafer \nbelongs to the batch used for CUPID-0, which include a  70~nm SiO anti-reflecting coating~\\cite{Mancuso-2014} \nthat was deposited on the side that rests on the TeO$_2$ crystal.\n\n\\section{Experimental details}\n\\label{sec:experimental_details}\nThe TeO$_2$ crystal  was mounted in a similar way as described \nin~\\cite{Enriched-TeO2-Cherenkov-2017,Casali-2014} with the only exception that the TeO$_2$ crystal was standing on \nthe reflecting foil and both TeO$_2$ and BLD were not equipped with Si heaters.\nThese heaters were normally glued \non the bolometer  to inject pulsed thermal signals for gain stabilization.\nThe TeO$_2$ face supporting the BLD and the opposite one were polished at (nearly)  optical level. \nThe remaining four lateral faces were matted in order to increase light collection~\\cite{Casali-2017}. \n\nThe  TeO$_2$  crystal is held by four S-shaped PTFE supports that are fixed to Cu columns. \nThe PTFE contracts upon cooling, creating a tensioned support that maintains the crystal \nposition. \n\nIn order to maximize light collection, the crystal is completely surrounded by a plastic \nreflecting sheet  (3M Vikuiti\\textsuperscript{TM}), in the same way as \nin~\\cite{Enriched-TeO2-Cherenkov-2017,Casali-2014}. A photograph of the detectors is presented \nin Fig.~\\ref{fig_0-setup}.\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_0-setup.pdf}\n\\caption{Photograph of the detectors. The BLD is simply  resting on the TeO$_2$ and the four \nPTFE supports (as well as the thermistor glued on the  TeO$_2$) do not hold the BLD in any way: they   \nsimply avoid the BLD to lean out from the top surface, as a mere translation constraints. \nThe gold wires of both NTDs are then crimped within micro Cu tubes to ensure the electrical contact \nas well as the thermal conductance to the heat sink. The $^{55}$Fe X-ray source is attached \nto the top reflecting cover sheet that encloses the detectors (with a clearance of $\\sim$4~mm from the BLD) \nand can be observed -reflected by the Ge wafer surface- between the two NTDs.}\n\\label{fig_0-setup}\n\\end{figure}\nThe  entire  setup was  enclosed  in a Cu box and thermally  coupled  to the  mixing  chamber  \nof the  CUPID R\\&D cryostat, a $^3$He\/$^4$He  dilution  refrigerator installed  \ndeep underground  within  Hall  C  of the  Laboratori Nazionali  del Gran Sasso, Italy. \nTo  avoid vibrations reaching the detectors, the box is mechanically decoupled from the \ncryostat by utilizing a two-stage pendulum system~\\cite{Pirro-2006}.  \n\n\nThe thermistors of the detectors are biased  with a quasi-constant current produced by applying a \nfixed voltage through large (27+27 or 2+2 G$\\Omega$) load resistors~\\cite{Arnaboldi-2002:1808}.   \nWhen light is absorbed in the Ge wafer, a thermal pulse is produced which is subsequently \ntransferred to the NTD sensor, changing the resistance of the thermistor.  This, in turn, creates a\nvoltage change across the current-biased NTD which is amplified using  \nfront end electronics located just outside the cryostat~\\cite{Arnaboldi-2004}. The signals are \nthen filtered by an anti-aliasing 6-pole Bessel filter (with a cutoff  frequency of 16~Hz \nfor the TeO$_2$ crystal and 550~Hz  for the BLD) and finally fed into a NI PXI-6284 18-bit ADC.\n\nThe sampling rate of the ADC was 1~kHz  for the TeO$_2$ crystal and 8 kHz for the BLD. \nThe two independent  triggers are software generated such that when a trigger fires, the \ncorresponding waveform is recorded. Moreover, when the trigger of the \nTeO$_2$ crystal fires, the corresponding waveform  of the  BLD is always \nrecorded, irrespective of its trigger. A detailed description of the DAQ system can \nbe found in~\\cite{DiDomizio:2018ldc}.\nThe amplitude and the shape of the voltage pulses are then determined via off-line analysis. \nThe pulse amplitude of the thermal signals is estimated by  the Optimum Filtering (OF)  \ntechnique~\\cite{Gatti-1986:1,Alduino-2016:045503}, that maximizes the signal-to-noise ratio \nin a way that improves the energy resolution and lowers the threshold of the detector. \nThe amplitude of the  light signal, however, is evaluated from the filtered waveform \nat a fixed time delay  with respect to the TeO$_2$ bolometer, as described in detail \nin~\\cite{Piperno-2001:10005}.\\newline\nThe amplitude of the acquired TeO$_2$ heat signals is energy-calibrated using several \n$\\gamma$-ray peaks from a $^{228}$Th source. \nThe BLD, on the contrary, is calibrated thanks to the  5.9~keV  and 6.5~keV X-ray \nquanta  produced by a $^{55}$Fe X-ray source permanently faced  to the detector.\n\n\\section{Data analysis and results}\n\\subsection{BLD performance}\n\\label{sec:results}\nThe crystals were  tested at a  cryostat base temperature of $\\sim$11~mK. \nIn order to obtain a fast response, we operated the BLD in the so-called \"over-biased\" \nconfiguration whereby \nthe biasing current of the circuit is set much larger than the current that would ensure the highest \nabsolute thermal response~\\cite{NTD_LD_Lucifer-2013}. This choice ensures a small working \nresistance, thus minimizing the effect of  the low pass filtering  induced by the overall \ncapacity  ($\\sim$200 pF) of the front end readout wires. \n\nIn Fig.~\\ref{fig_1-55Fe} we show the $^{55}$Fe calibration spectrum obtained with the BLD.\nThe baseline energy resolution (ie, the absolute sensitivity) of the BLD is given by \nthe width of randomly acquired baselines (noise) after the  application of OF. \nAs is typical for this style of detectors, the energy resolution of monochromatic energy \nabsorption events is much worse than the baseline resolution, irrespective \nof the type of sensor~\\cite{NTD_LD_Lucifer-2013,TES_LD_CRESST}.  \n\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_1-55Fe.pdf}\n\\caption{Energy distribution of the random sampled noise. The  width of the distribution  \n($\\sigma\\approx$20 eV) represents the baseline energy resolution of our BLD. The right inset shows\nthe $^{55}$Fe calibration spectrum of the BLD. The x-axis units represent the absolute voltage drop \nacross the thermistor. \nThe RMS resolution on the 5.9 keV and 6.5 keV X-ray peaks is 59 eV (see text).}\n\\label{fig_1-55Fe}\n\\end{figure}\nThe noise and signal power spectra of the BLD are presented in Fig.~\\ref{fig_2-NPS}.\n\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_2-NPS.pdf}\n\\caption{Noise power spectrum (black line) and signal power spectrum (blue line) of the BLD. \nThe y-axis scale is in absolute values  for the noise. The signal spectrum is scaled  in \narbitrary units, being the roll-off induced by the Bessel filter  the same between noise and signal.\nThe working resistance of the thermistor is 1.47 M$\\Omega$, biased with a current of 3.7 nA \nthorough (2+2) G$\\Omega$ metallic load resistors. The peaks are due to the microphonic \nnoise induced by the vibration of the readout wires.}\n\\label{fig_2-NPS}\n\\end{figure}\nThe bump that can be observed in Fig.~\\ref{fig_2-NPS} at  $\\sim$400 Hz arises from a \nresonance that enhances the thermal noise generated within the thermistor.  This occurs  \nwhen the impedance of the parasitic capacitance of the link becomes smaller than that of the \nthermistor, which is a fed-backed device~\\cite{Arnaboldi-2005}. \nThe bump is found at the border of  the bandwidth of the signal and is rejected from the \noptimum filter algorithm.\n\n\nFig.~\\ref{fig_3-rise-decay} shows the corresponding rise and decay times of $^{55}$Fe X-rays absorption events.\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_3-rise-decay.pdf}\n\\caption{Rise and decay times distributions corresponding to the $^{55}$Fe X-rays. The \nBessel cut-off frequency of the Front-End is 550~Hz.}\n\\label{fig_3-rise-decay}\n\\end{figure}\nThe measured rise time shown in Fig.~\\ref{fig_3-rise-decay} is most likely slower than the intrinsic \nrise time of the detector since it contains contributions from the Bessel filter (independent from the \nthermistor impedance) and from the capacitance of the readout wires. This last \ncontribution is difficult to measure since it involves the dynamic resistance of the \nthermistor. The contribution of the 550~Hz Bessel filter to the rise time was evaluated \nin~\\cite{NTD_LD_Lucifer-2013} and reported as 0.65~ms. Thus, after applying a quadratic deconvolution, the \n\\it intrinsic \\rm rise time of our BLD should be of the order of 0.5~ms, compatible with the \nexpectation of ~\\cite{Coron-2004}.\nThe overall performance of the BLD is summarized in Tab.~\\ref{tab-new-BLD}.\n\n\\begin{table}\n\\centering\n\\caption{Performances of the BLD of this work, to be compared with the ones of \nTab.~\\ref{tab-cupid-0-LD}.}\n\\label{tab-new-BLD}\n\\begin{tabular}{ccccc}\n\\hline\nR$_{work}$      &Response     &Baseline RMS     &$\\tau_{r}$       &$\\tau_{d}$ \\\\\n\n[M$\\Omega$]     &[$\\mu$V\/keV]     &[eV]             &[ms]              &[ms]\\\\\n\\hline\n1.47             &3.86          &20              & 0.83             &1.63 \\\\\n\\hline\n\\end{tabular}\n\\end{table} \n\\subsection{Heat and Light measurement}\n\\label{scatter-section}\nIn order to evaluate the long-term discriminatory performance of our BLD, we performed a 70 h run that \nincluded two event-generating calibration sources embedded into the setup.  A $^{228}$Th source was placed\na few cm away from the TeO$_2$ crystal and \na \\it smeared \\rm $^{238}$U $\\alpha$ source was applied to the inside of the light reflector facing the TeO$_2$. \nThe aim of the $\\alpha$ source was to directly measure the discrimination capability between \n$\\alpha$ and $\\beta\/\\gamma$ in the DBD region of interest of $^{130}$Te. \nThe source was made using 2 $\\mu l$ of a standard calibrated solution (0.1 \\%) of $^{238}$U, and the dried \nsource deposition was covered with a 6 $\\mu m$ aluminized Mylar foil to smear the \n$\\alpha$ energy.\n\nThe light vs heat scatter plot is presented in Fig.~\\ref{fig_4-scatter-plot} and shows an \nunexpected feature.\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_4-scatter-plot.pdf}\n\\caption{Light vs heat scatter plot obtained in a 70 h measurement with the TeO$_2$ exposed \nto a $^{228}$Th source and a smeared $^{238}$U $\\alpha$ source. Unfortunately $\\alpha$ \nenergy loss in the Mylar -constituting the smearing  medium- results in a tiny, but \nmeasurable, light emission that increases towards lower energies, i.e. at larger energy \nloss in the Mylar. The events above 4~MeV, on the contrary, are due to internal and\/or \nsurface contaminations and their light emission is compatible with zero \n(see text).}\n\\label{fig_4-scatter-plot}\n\\end{figure}\nThe $^{238}$U $\\alpha$-events arising from the smeared source clearly show a tiny light \nemission that increases towards lower energies. This feature can only be ascribed to an \nenergy loss in the  Mylar which emits few scintillation photons. To avoid this effect \nwe usually face the aluminized surface of the Mylar towards the crystal so as to reflect the \n(very few) photons that could be produced in this plastic. This time however, \nwe mistakenly mounted the Mylar with the uncoated side towards the detector.  This was confirmed after\nsubsequently opening the cryostat and checking.\n\nThe  result is shown in Fig.~\\ref{fig_4-scatter-plot}: the amount of Cherenkov \nlight, produced by a 2615~keV $\\gamma$, that is collected with this new set-up is \n(151~$\\pm$~4)~eV, 50 \\% larger with respect to all our previous measurements with \nmassive crystals~\\cite{Casali-2017}, as well as roughly 50 \\% larger  with respect to a \nmeasurement recently performed with a NTD-based light detector~\\cite{Lumineu-2017} of the \nsame type (considering the 40 \\% reduced transmission area between BLD and crystal, as \ndeclared in the article).\nThe light distribution of  the 74 events belonging to the internal\n$^{210}$Po $\\alpha$ at 5407~keV (5304~keV  $\\alpha$ + 103~keV nucleus recoil) shows a \nmean value of (5.8~$\\pm$~3.3)~eV, still compatible with zero (see Sec.~\\ref{sec:thermal_interference}) \nas it should be if the light only arises from the Cherenkov effect. More importantly, the width of \nthe light distribution of $\\alpha$'s is $\\sigma_{\\alpha}$=(22.7~$\\pm$~ 2.7)~eV, fully compatible \nwith the RMS noise of the BLD of Tab.~\\ref{tab-new-BLD}.\nThe light signal induced by the 2615~keV $\\gamma$ -on the contrary-  shows a width of \n$\\sigma_{\\gamma\/\\beta}$=(31.5~$\\pm$~4.3)~eV   which is \na result of the photostatistics and the light collection. \n\nIn order to evaluate the Discrimination Power (DP) that can be obtained between the \n$\\alpha$ and $\\beta\/\\gamma$ distributions at 2528~keV (the Q$_{\\beta\\beta}$-value of \nthe DBD  of $^{130}$Te) we use the same formula and arguments used \nin~\\cite{Enriched-TeO2-Cherenkov-2017,Lumineu-2017}: the  DP can be quantified as the \ndifference between the average values of the two distributions normalized to the square \nroot of the quadratic sum of their  widths: \n\\begin{equation} \nDP = \\frac{|\\mu_{\\gamma\/\\beta}-\\mu_{\\alpha}|}{\\sqrt{\\sigma^{2}_{\\gamma\/\\beta}+\n\\sigma^{2}_{\\alpha}}}.\n\\label{eq:DP}\n\\end{equation} \nRe-scaling the light signal from 2615 to 2528~keV, we obtain DP=3.6, using one highly likely \nassumption that an $\\alpha$ particle at 2528~keV will show a light signal equal than the same \nparticle at~5304 keV ($^{210}$Po).\nThis DP is the best ever achieved with large mass TeO$_2$ crystals (M $>$ 7 g) and without \nthe need for additional Neganov-Luke \namplification~\\cite{Lumineu-2017,Casali:2015gya,Gironi:2016nae}, or\nmore sophisticated TES sensors~\\cite{Karo-2014} or both~\\cite{Willers-2014}.\n\n\\section{Thermal conductance}\\label{sec:thermal_interference}\nAs stated in Sec.~\\ref{sec:Introduction}, the actual goal of this work was to experimentally \ndemonstrate that the BLD can rest on the scintillating or luminescent crystal without heat sinking to it.  \nUsing the results in the previous section we can now calculate a limit on the heat flow through \nthe Ge wafer and the TeO$_2$. If one assumes that a 5407 keV energy release in the TeO$_2$ produces\na mean value BLD signal that only depends on the heat flow (assuming no light emission), then we have \nan upper limit for the ratio of the heat flow through TeO$_2$ and Ge: 5.8~eV\/5407~keV$\\sim$10$^{-6}$.\n\nIn our case, an extremely low heat conductance was determined experimentally using static conditions. We measured \nthe base resistance of the BLD  as 223.5 M$\\Omega$ (corresponding to 11.8~mK), keeping the \nTeO$_2$ thermistor unbiased (i.e.  no power dissipation in it). We then gave the maximum  \n(allowed by our biasing set-up) bias to the TeO$_2$ thermistor, corresponding to \n4.8 nA, and the TeO$_2$ thermistor changed its resistance from 626 M$\\Omega$ \n(bias~$\\rightarrow$~0) to 1.71 M$\\Omega$.  The power dissipated on the TeO$_2$ was therefore 40 pW. \nThe base resistance of the BLD decreased to 222.8 M$\\Omega$, which corresponds to a temperature increase of \nonly $\\approx$~4.3 $\\pm$ 0.2 ~$\\mu$K. \nThe same operation was performed with the BLD in working condition, i.e. bias current \nof 3.7 nA and a resistance of 1.47~M$\\Omega$ (corresponding to $\\sim$23~mK), and no variation\nof the baseline of the BLD was registered.\nA further investigation of the thermal conductance between a Ge-BLD and a TeO$_2$ crystal was performed\nby exploiting a small TeO$_2$ crystal ($20~\\times~20~\\times~14$~mm$^{3}$, 34~g mass).\nWe used a standard BLD, i.e., the same thickness and height as in the previous discussion, but with the Ge wafer\nheld with PTFE clamps in a  stand-alone Cu mounting~\\cite{NTD_LD_Lucifer-2013}.\nFor this experiment we rested the $20\\times20$~mm$^{2}$ surface of the 34~g crystal \n on the Ge wafer. The NTD thermistor-equipped TeO$_2$ crystal was surrounded  with the same reflecting \nfoil and we performed the same measurement described in Sec.~\\ref{scatter-section} with the same \noverall setup.\nThis time a 5304~keV $^{210}$Po decay occurring in the TeO$_2$ created a mean signal in the BLD \nof (317~$\\pm$~29)~eV, definitively not compatible with the result of Sec.~\\ref{scatter-section}.\nThe mean (light) signal registered in coincidence with the 2615~keV $\\gamma$-line of $^{208}$Tl \nwas (336~$\\pm$~5)~eV. \nThe $\\alpha$-induced signal in the BLD, therefore, has to be ascribed to an effective thermal \ntransfer from the TeO$_2$ to the BLD.\nWe can make a very rough estimation of the size of this transfer using the \nresults of  the measurement of Sec.~\\ref{scatter-section}.  If we assume the heat conductance \nto be linearly proportional to the pressure force  between the two mediums, then we may\nsimply compare the weight differences: 1.1 g in the case of the wafer resting onto the TeO$_2$ \ncrystal versus 34~g in this last configuration.\nTheir ratio, i.e. 31, should be, in first approximation, the ratio between the thermal \nconductance in the two setups.  Ascribing the $\\alpha$ signal of Sec.~\\ref{scatter-section} \nexclusively to thermal transfer we would expect a thermal transfer signal of \n(180~$\\pm$~90)~eV, which is compatible with the 317~eV observed during this measurement. On the other\nhand, under the same assumption, we can evaluate the 2615-keV induced Cherenkov light signal\nof this crystal as the difference between the observed signal and the re-scaled thermal transfer \nevaluated from  the $\\alpha$. In this way we observe that the energy of the Cherenkov light \nemission in this 34~g  crystal is (185~$\\pm$~15)~eV.\n\n\\section{Conclusions}\nWe have demonstrated the possibility of mounting BLDs by simply resting them on the surface of the \ncorresponding scintillating crystal. With this new mounting method the light collection can increase up\nto 50\\% with respect to standard setups. We do not observe appreciable heat flow between the\nscintillating crystal and BLD. \nWe also improved the time response of our thermistor-based light detectors, reaching a rise\ntime of 0.8 ms and demonstrating that 0.5 ms is  achievable. This time response is necessary \nto remove the background induced by the pile-up of the 2$\\nu$-DBD mode in the case \nof $^{100}$Mo-based crystals.  We reached a baseline resolution\nof 20~eV RMS, more than 2 times better than the average value  our previous CUPID-0-like detectors. \nThanks to these developments, we definitively demonstrated that standard thermistor-based\nBLDs can be used for CUPID, both to read out the tiny Cherenkov light of TeO$_2$ as well as to\nread out the Mo-based scintillating crystals.  \n\nWe do believe that this simplified technique could be applied to any kind of BLD, irrespective\nof the sensor type. The first approximation thermal conductance between crystal and BLD \ndoes not depend upon the energy of the phonons, so we would expect that thermal transfer \nwould be as negligible in TES or MMC devices as it is in our NTDs.  \nMore generally this new technique could be also  applied  in  the case of stacked, standard small\nbolometers, provided that the weight does not exceed a\nfew grams. However, since the measured thermal transfer is rather small, the weight of the\nbolometer will not be a significant limiting factor in low energy threshold applications.  \n\n\\section{Acknowledgments}\nThis work was performed within the CUPID experiment founded by INFN and supported by\nthe National Science Foundation under Grant NSF-PHY-1614611.\n\nWe thank the CUPID-0 and the CUORE collaborations for the overall support and for sharing their \nDAQ and software.\nWe  express our gratitude to LNGS for the generous hospitality  and, in particular, to\nthe mechanical workshop personnel including E. Tatananni, A. Rotilio, A. Corsi, and B.\nRomualdi for their continuous and  constructive help. We are  also grateful to  M. Guetti \nfor his invaluable support and expertise  in the cryostat facility maintenance. \nWe acknowledge Dr. C. Arnaboldi for his precious \nsupport, even though he has left this field of research many years ago.  We are especially \ngrateful to E. Ferri for her kind support in the thermistor wire-bonding.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Additional Definitions}\n\n\\begin{definition}[Pairwise Independence Hash Functions]\\label{def:pairwise}\nLet $\\mathcal{H}$ be a family of functions from $\\{1,\\ldots, N\\}$ to $\\{1,\\ldots, M\\}$. The family $\\mathcal{H}$ is \\emph{pairwise independent} if for every $x,y \\in \\{1,\\ldots, N\\}$ such that $x \\neq y$ and for every $a,b \\in \\{1,\\ldots, M\\}$ it holds that \n$$\\Pr_{h \\in \\mathcal{H}}[h(x)=a \\wedge h(y)=b]=1\/M^2~.$$\nThat is, if $h$ is chosen uniformly at random from $\\mathcal{H}$, then the random variable $h(x)$ and $h(y)$ are uniformly distributed and pairwise independent. \n\\end{definition}\n\n\\begin{fact}\\label{fc:pairwise}\\cite{TCS-010}\nThere is an explicit family $\\mathcal{H}$ of pairwise independent has functions from $\\{0,1\\}^n \\to \\{0,1\\}^m$ constructed using $O(\\max\\{\nm,n\\})$ bits and computable in $\\operatorname{\\text{{\\rm poly}}}(n,m)$ time.\n\\end{fact}\n\\section{Overview of the Cycle Space Sampling Technique} \\label{sec:cycle_space_overview}\n\nThe cycle space sampling technique allows to detect cuts in a graph using a connection between cuts and cycles in a graph.\nThis beautiful technique was introduced by Pritchard and Thurimella \\cite{pritchard2011fast}, that showed its applicability for distributed algorithms identifying small cuts in a graph. We next give a short overview of the technique, for full details see \\cite{pritchard2011fast}.\n\nThe \\emph{cycle space} of a graph is the family of all subsets of edges $F$ that have even degree at each vertex, any such subset $\\phi \\subseteq E$ is called a \\emph{binary circulation}. The \\emph{cut space} is the family of all induced edge cuts. It is easy to see that if we take a cycle $C$ in a graph and an induced edge cut, then the number of edges of the cycle that cross the cut is even. The cycle space technique extends this observation and shows that the cycle space and cut space are orthogonal vector spaces. Using this, they show the following (see Propositions 2.2 and 2.5 in \\cite{pritchard2011fast}).\n\n\\begin{claim} \\label{claim_cycle} \nLet $\\phi$ be a uniformly random binary circulation and $F \\subseteq E$. Then\n$$Pr[|F \\cap \\phi| \\ is \\ even] = \\left\\{\n                \\begin{array}{ll}\n                  1,\\ if\\ F\\ is\\ an\\ induced\\ edge\\ cut\\\\\n                  1\/2,\\ otherwise\n                \\end{array}\n              \\right. $$ \n\\end{claim}   \n\nHence, sampling a random binary circulation allows to detect if a subset of edges is an induced edge cut with probability $1\/2$. To reduce the failure probability to $1\/2^b$ we can choose $b$ random binary circulations. To use this technique, the authors provide an efficient way to sample a random binary circulation, we describe next. Let $T$ be a spanning tree of the graph. For any non-tree edge $e$, adding $e$ to the graph creates a cycle. These cycles are the \\emph{fundamental cycles}, and it is shown that the \\emph{fundamental cycles} are a basis for the cycle space. Based on this, they show that sampling a random binary circulation can be done by choosing each fundamental cycle with probability $1\/2$, or equivalently choosing each non-tree edge with probability $1\/2$. The binary circulation $\\phi$ sampled has all the non-tree edges sampled, and each tree edge that appears in odd number of sampled cycles. Given the sampled non-tree edges in $\\phi$, the tree edges in $\\phi$ can be identified using a simple scan of the tree, as shown in \\cite{pritchard2011fast}. Choosing $b$ random binary circulations, is equivalent to choosing a $b$-bit random string $\\phi(e)$ for each non-tree edge. For a tree edge $t$, we define $\\phi(t) = \\oplus_{e \\in C_t} \\phi(e)$, where $C_t$ are all non-tree edges $e$ such that $t$ is in the fundamental cycle of $e$. This again can be computed by a simple scan of the tree, and takes $O((n+m)b)$ time if the labels have size $b$. This gives the following.\n\n\\cycle*\n\nTo see this, let $\\phi_1,...,\\phi_b$ be the sampled binary circulations. If $F$ is an induced edge cut, then from Claim \\ref{claim_cycle}, for every sampled circulation $\\phi_i$, we have that $|F \\cap \\phi_i|$ is even, and hence for all $i$, the $i$'th bit of $\\Moplus_{e \\in F} \\phi(e)$ is equal to 0 as needed. Otherwise, for all $i$, the $i$'th bit $\\Moplus_{e \\in F} \\phi(e)$ equals $0$ with probability $1\/2$, hence the probability that the whole vector equals $0$ is $1\/2^b$, as needed.\n\\section{Fault-Tolerant Approximate Distance Labels}\\label{sec:ft-distance}\nGiven integer parameters $f,k \\geq 1$, an $(f,k)$ \\emph{FT approximate distance labeling scheme} assigns labels $\\mathsf{DistLabel}\\xspace: V \\cup E \\to \\{0,1\\}^{q}$ such that given the labels of $s,t$ and a subset $F \\subseteq E$, $|F|\\leq f$, there exists a decoding algorithm that outputs a distance estimate $\\delta_{G \\setminus F}(s,t)$ satisfying:\n$$\\mbox{\\rm dist}_{G\\setminus F}(s,t) \\leq \\delta_{G \\setminus F}(s,t) \\leq k\\cdot\\mbox{\\rm dist}_{G\\setminus F}(s,t)~.$$\n\nWe next show that there is an efficient transformation from any FT connectivity labeling scheme into an FT approximate distance labeling scheme. This transformation increases the label size by a multiplicative factor of $\\widetilde{O}(n^{1\/k})$. This technique was first introduced by \\cite{chechik2012f} in the context of distance sensitivity oracles, and it is based on the notion of tree covers.\n\n\\begin{definition}[Tree Covers]\\label{def:tree-cover}\nLet $G=(V,E)$ be an undirected graph with edge weights $\\omega$, and let $\\rho,k$ be two integers. Define $B_{\\rho}(v)=\\{ u \\in V ~\\mid~ \\mbox{\\rm dist}_G(u,v)\\leq \\rho\\}$. A tree cover $\\mathsf{TC}\\xspace(G, \\omega, \\rho,k)$ is a collection of rooted trees $\\mathcal{T}=\\{T_1,\\ldots, T_\\ell\\}$ with root $r(T)$ for every $T \\in \\mathcal{T}$ such that:\n\\begin{enumerate}[noitemsep]\n\\item For every vertex $v$ there exists a tree $T \\in \\mathcal{T}$ such that $B_{\\rho}(v) \\subseteq T$.\n\\item The radius of each tree $T$ is at most $(2k-1)\\cdot \\rho$.\n\\item Each vertex participates in $(k \\cdot n^{1\/k})$ trees.\n\\end{enumerate}\nLet $|\\mathsf{TC}\\xspace(G, \\omega, \\rho,k)|$ denote the number of trees in the tree cover $\\mathsf{TC}\\xspace(G, \\omega, \\rho,k)$.\n\\end{definition}\n\n\\begin{proposition}\\cite{Peleg:2000}\nFor any $n$-vertex graph $G=(V,E, \\omega)$, and any parameters $\\rho,k$, one can compute tree covers $\\mathsf{TC}\\xspace(G, \\omega, \\rho,k)$ in time $\\widetilde{O}(|E(G)| \\cdot n^{1\/k})$.\n\\end{proposition}\n\n\\begin{lemma}[From Connectivity Labels to Approximate Distance Labels]\\label{lem:reduction}\nLet $G=(V,E, \\omega)$ be a weighted undirected $n$-vertex graph where $\\omega(e)\\in [1,W]$, and let \n$\\mathsf{ConnLabel}\\xspace: V \\cup E \\to \\{0,1\\}^s$ be an $f$-FT connectivity labeling scheme for $G$ with decoding time $t$. Then for every integer $k\\geq 1$, there is an $(f,(8k-2)(|F|+1))$ FT approximate distance labeling scheme $\\mathsf{DistLabel}\\xspace: V \\cup E \\to \\{0,1\\}^{q}$ for $G$, where $q=O(s \\cdot k \\cdot n^{1\/k}\\cdot \\log (nW))$, and with decoding time $\\widetilde{O}(t \\log{(nW)})$.\n\\end{lemma}\n\n\\paragraph{The labeling algorithm.}\nFor every vertex $u$, the label $\\mathsf{DistLabel}\\xspace(u)$ consists of $K=\\log(nW)$ sub-labels of FT connectivity labels in distinct subgraphs of $G$ defined as follows. The $i^{th}$ sub-label addresses all distances that are at most $2^i$ in $G$. Let $H_i$ be set of heavy edges in $G$ of weight at least $2^i$, and define the $i^{th}$ tree-cover by \n\\begin{equation}\\label{eq:TC-i}\n\\mathsf{TC}\\xspace_i=\\mathsf{TC}\\xspace(G\\setminus H_i,\\omega, 2^i,k)~.\n\\end{equation}\nFor each tree $T_{i,j} \\in \\mathsf{TC}\\xspace_i$, the algorithm applies the FT connectivity scheme on the graph $G_{i,j}=G[V(T_{i,j})]$. For every vertex $u$ and $i \\in \\{1,\\ldots, K\\}$, let $i^*(u)$ be an index of a tree in $\\mathsf{TC}\\xspace_i$ that covers the $2^i$-ball of $u$. I.e., $B_{2^i}(v) \\subseteq T_{i,i^*(u)}$. \nThe label of every $u \\in V$ is then given by:\n$$\\mathsf{DistLabel}\\xspace(u)=\\{\\langle \\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(u), i,j \\rangle ~\\mid~ i \\in [1,K], j \\in \\{1,\\ldots, |\\mathsf{TC}\\xspace_i|\\}, u \\in G_{i,j}\\} \\bigcup \\{i^*(u) ~\\mid~ i \\in [1,K]\\}~.$$\n\nSimilarly, the label of each edge $e \\in G$ contains the FT connectivity label of $e$ in each of the instances $(G_{i,j}, T_{i,j})$:\n$$\\mathsf{DistLabel}\\xspace(e)=\\{\\langle \\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e), i,j \\rangle ~\\mid~ i \\in [1,K], j \\in \\{1,\\ldots, |\\mathsf{TC}\\xspace_i|\\}, e \\in G_{i,j}\\}.$$ \n\nThe time for assigning the labels is the time for constructing the tree cover and computing the indexes $i^{*}(v)$, and the time for assigning the connectivity labels on each one of the trees. The first part requires polynomial time. The second depends on the connectivity labels. For example, using our scheme from Section \\ref{sec:ftconn-sketch} the time complexity of the second part is $\\widetilde{O}(mn^{1\/k})$, as it is linear in the total number of vertices and edges in the trees.\n\n\n\n\\paragraph{The decoding algorithm.}\nConsider the query $\\langle s,t,F\\rangle$. \nThe algorithm has $K$ phases, in each phase $i \\in [1,K]$ the decoding algorithm of the FT connectivity labels is applied on the instance $G_{i,i^*(s)}, T_{i,i^*(s)}$ where $G_{i,i^*(s)}$ contains the $2^i$ ball of $s$ in $G$. \nIf $t \\notin G_{i,i^*(s)}$, the phase $i$ ends and we continue to phase $i+1$. \nOtherwise, the algorithm decides if $s$ and $t$ are connected in $G_{i,i^*(s)} \\setminus F$ in the following manner. \nLet $F_i=F \\cap G_{i,i^*(s)}$, this subset of edges can be obtained from the labels of the $F$ edges. \nSince the labels of $s,t$ and $F_i$ contain the FT connectivity labels in the subgraph $G_{i,i^*(s)}$ and the tree $T_{i,i^*(s)}$, the algorithm can apply the decoding algorithm of the FT connectivity scheme.\nIf $s$ and $t$ are indeed connected in $G_{i,i^*(s)}\\setminus F_i$, the algorithm returns the estimate $\\delta_{G \\setminus F}(s,t)= (4k-1) \\cdot (|F|+1) \\cdot 2^{i}$. Otherwise, it proceeds to the next phase.\n\nOverall, let $i$ be the minimum index in $\\{1,\\ldots, K\\}$ for which $s$ and $t$ are connected in the subgraph $G_{i,i^*(s)} \\setminus F$. Then the decoding algorithm returns the distance estimate $\\delta_{G \\setminus F}(s,t)=(4k-1) \\cdot (|F|+1) \\cdot 2^{i}$. If no such $i$ exists, the decoding algorithm returns $\\delta_{G \\setminus F}(s,t)=\\infty$, which implies that $s$ and $t$ are not connected in $G \\setminus F$. \n\nThe decoding time is $\\widetilde{O}(t \\log{(nW)})$, where $t$ is the decoding time of the connectivity labels, as we use the decoding algorithm of the connectivity labels $K$ times on the graphs $G_{i,i^*(s)}$. To obtain this, we need to make sure that given the labels of $s,t,F$ we can easily find their connectivity label in the graph $G_{i,i^*(s)}$ if exist. This can be easily done if we store the connectivity labels in a sorted order.\n\n\\paragraph{Analysis.}\nWe now analyze the construction, and start by bounding the size of the labels. By the properties of the tree-cover in Def. \\ref{def:tree-cover}, each vertex appears in $O(K \\cdot k \\cdot n^{1\/k})$ subgraphs. Thus, $\\mathsf{DistLabel}\\xspace(u)$consists of $O(K \\cdot k n^{1\/k})$ FT connectivity labels and the label size is bounded by $O(K \\cdot k n^{1\/k} \\cdot s)$ bits, as desired. Next, we show correctness. By the correctness of the FT connectivity labeling scheme, it is sufficient to show the following. Let $P_{s,t,F}$ be an $s$-$t$ shortest path in $G \\setminus F$ of length $(2^{i-1}, 2^{i}]$. By the properties of the tree cover, there is a tree $T_{i,i^*(s)} \\in \\mathsf{TC}\\xspace_i$ that contains all the vertices of the path $P_{s,t,F}$. Therefore, we have that $s$ and $t$ are connected in $G_{i,i^*(s)}\\setminus F$. Since the labels of $s, t$ and $F_i=F \\cap G_{i,i^*(s)}$ contain the FT connectivity labels in $G_{i,i^*(s)}$, we get that the distance estimate returned by the algorithm satisfies that\n$$\\mbox{\\rm dist}_{G \\setminus F}(s,t)\\leq \\delta_{G \\setminus F}(s,t) \\leq (4k-1)(|F|+1)  \\cdot 2^i \\leq (8k-2)(|F|+1) \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)~.$$\nTo see this, let $j \\leq i$ be the first index such that $s$ and $t$ are connected in $G_{j,j^*(s)}\\setminus F$. The algorithm returns the estimate $(4k-1)(|F|+1) \\cdot 2^j \\leq (4k-1)(|F|+1) \\cdot 2^i = (8k-2)(|F|+1) \\cdot 2^{i-1} \\leq (8k-2)(|F|+1) \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. To prove the left inequality, we show that if $s$ and $t$ are connected in $G_{j,j^*(s)}\\setminus F$, there is indeed a path between them in $G \\setminus F$ of length at most $\\delta_{G \\setminus F}(s,t) = (4k-1)(|F|+1) \\cdot 2^j$. First, from the tree cover properties, the radius of $T_{j,j^*(s)}$ is at most $(2k-1)2^j$, implying that any two vertices in $T_{j,j^*(s)}$ are at distance at most $(4k -2) \\cdot 2^j$ from each other. Now the graph $T_{j,j^*(s)} \\setminus F$ has at most $|F|+1$ connected components. Since $G_{j,j^*(s)}\\setminus F$ is connected, it implies that there is a path between $s$ and $t$ in $G_{j,j^*(s)}\\setminus F$. This path traverses at most $|F|+1$ different components in $T_{j,j^*(s)} \\setminus F$, and at most $|F|$ edges connecting them, each one of weight at most $2^j$. As the diameter of each component is bounded by $(4k - 2) \\cdot 2^j$, the length of the path is at most $(4k - 2) \\cdot 2^j \\cdot (|F|+1) + 2^j \\cdot |F| \\leq (4k - 1) \\cdot 2^j \\cdot (|F| + 1)$, as needed.\n\n\\remove{\nLet $\\mathcal{T}=\\bigcup_{i=1}^{K}\\mathsf{TC}\\xspace_i$ be the collection of tree covers with $K=O(\\log (nW))$ scales of distances. We call an edge $(u,v)$ a \\emph{tree edge} if it appears on at least one of the trees in $\\mathcal{T}$. \nUsing Lemma \\ref{lem:useful-recovery-edges}, we have the following decoding algorithm, which becomes useful in the context of routing schemes, as described in the next section. \\mtodo{there could be a problem with treating edges globally as tree edges or non-tree edges, I think for each edge $e \\in G_{i,j}$ we need to know its label in this graph (either tree or non-tree label), extended identifiers are also different for different trees because of the ancestry labels.}\n\\begin{lemma}\\label{lem:approx-dist-recovery}\nConsider the $(f,(8k-2)(f+1))$ approximate distance labels $\\mathsf{DistLabel}\\xspace$ obtained by using Lemma \\ref{lem:reduction} with the FT connectivity labeling scheme of Sec. \\ref{sec:ftconn-sketch}. For any triplet $s,t,F \\subseteq E$ and $|F|\\leq f$, let $F_T$ be the tree edges of $F$. Then, given the labels $\\{\\mathsf{DistLabel}\\xspace(w), w \\in \\{s,t\\} \\cup F_T\\}$ and the extended edge identifiers of $F \\setminus F_T$, the decoding algorithm can be modified to return a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of an $s$-$t$ path in $G \\setminus F$, along with indices $i,j$. Each $G$-edge $e$ of $\\widehat{P}$ is augmented with port information and the extended identifier of $e$; and each non-$G$ edge $e'=(u,v)$ corresponds to a $u$-$v$ path in $T_{i,j} \\setminus F$. In addition, the length of the $s$-$t$ path encoded by $\\widehat{P}$ is bounded by $(8k-2)(|F|+1)\\cdot \\mbox{\\rm dist}_{G\\setminus F}(s,t)$. \n\\end{lemma}\n\\begin{proof}\n\n\n\n\\end{proof}\n}\n\n\\section{Introduction}\nDistributed graph representation is concerned with augmenting each vertex (and possibly also edges) with useful and low-space information in order to efficiently address various graph queries in a distributed manner. As the\nvertices and edges of the network may occasionally fail or malfunction, it is desirable\nto make these representations robust against failures. In this paper, we provide new constructions of succinct \\emph{labeled-based distributed data structures} that can handle connectivity, distance queries and routing in the presence of edge failures. \n\nConnectivity labels are short names attached to each vertex in the $n$-vertex input graph $G$, such that given the labels of a pair of vertices $s$ and $t$ (and no any other information), it is possible to deduce if $s$ and $t$ are connected in $G$. The primary complexity measure of the  labeling scheme is the label length (maximum length of a label). In general, labels can be viewed as the \\emph{logical} names of the vertices \\cite{kannan1992implicat,peleg2005informative}, as they are considerably more informative than the physical names that usually correspond to arbitrary $O(\\log n)$-bit identifiers. For example, in routing applications the label of the vertex is treated as its ``address\". It is quite immediate to provide connectivity labeling schemes of logarithmic length. Over the years, these labels have served the basis for devising also approximate distance labels, and compact routing schemes, which are arguably the \\emph{grand finale} of the distributed representation schemes. \n\n\nOur goal in this paper is to provide \\emph{fault-tolerant} analogs for the above mentioned schemes, while paying a small overhead in terms of space and other complexity aspects. Several notions of fault-tolerant labeling and routing schemes have been addressed in the literature; starting with the earlier introduction of FT routing schemes by Dolev \\cite{dolev1984new}, to the more recent formulations of forbidden-set labeling and routing schemes by Courcelle et al. \\cite{courcelle2007forbidden,CourcelleT07}. Despite much activity revolving these topics, FT labeling and routing schemes with \\emph{sub-linear} space are currently known only for a limited collection of graph families. We next elaborate more on the state-of-the-art affairs, and our main objectives.\n\n\n\\paragraph{Fault-Tolerant Connectivity and Distance Labeling.} \nFT connectivity labeling schemes, also known in the literature as \\emph{forbidden-set} labeling \\cite{CourcelleT07}, assign labels to the vertices and the edges of the graph such that given the labels of a vertex pair $s,t$, and the labels of the faulty edges $F$, one can determine if $s$ and $t$ are connected in $G \\setminus F$. \n\n\n\n %\n\nSince their introduction, efficient FT labeling schemes have been devised only for a restricted collection of graph families such as graphs with bounded tree-width and planar graphs \\cite{CourcelleT07,AbrahamCGP16}. In the lack of any FT connectivity labeling schemes for general graphs with sub-linear label length (for any $f\\geq 2$ faults\\footnote{While there is no \\emph{explicit} construction of FT labeling for general graphs, for $f=1$, the centralized distance sensitivity oracle of \\cite{khanna2010approximate} might be modified to provide approximate distance labels against a single fault.}), we ask:\n\n\\begin{question}\\label{q:label}\nIs it possible to design FT connectivity labeling scheme resilient to at most $f$ edge faults, for general graphs with label length of $\\operatorname{\\text{{\\rm poly}}}(\\log n)$ bits, or even $\\operatorname{\\text{{\\rm poly}}}(\\log n,f)$ bits? \n\\end{question}\n\nFT connectivity labels are also closely related to \\emph{connectivity sensitivity oracles} \\cite{patrascu2007planning}, which are low-space centralized data-structures that handle efficiently $\\langle s, t, F \\rangle$ connectivity queries using $S(n)$ space. Our main goal is in providing a \\emph{distributed} variant of such constructions, e.g., where each vertex or edge in the graph ``holds\" only $S(n)\/n$ bits of information, such that an $\\langle s,t, F \\rangle$ query can be addressed using only the information stored by $s,t$ and $F$. \n\nAn important step towards designing FT compact routing schemes involves the computation of \\emph{FT approximate distance labels}. In this setting, given the labels of $s,t$ and the faulty edges $F$, it is required to report an approximation for the $s$-$t$ shortest path distance in $G \\setminus F$.\nFT approximate distance labels can be viewed as the distributed analog of $f$-FT \\emph{distance sensitivity oracles} \\cite{khanna2010approximate,WeimannY10}. \nThese are global succinct data-structures that given an $\\langle s,t,F \\rangle$ query report fast an estimate for the approximate $s$-$t$ distance in $G \\setminus F$. Our goal is to provide FT approximate labeling schemes that match the state-of-the-art space vs. stretch tradeoff of the centralized data structures.\n\n\n\n\\paragraph{Fault-Tolerant Routing.} A desirable requirement in most communication networks is to provide efficient routing protocols in the presence of faults. Specifically, an $f$-FT routing protocol is a distributed algorithm that, for any set of at most $f$ faulty edges $F$, allows a vertex $s$ to route a message to a destination vertex $t$ along an approximate $s$-$t$ shortest path in $G \\setminus F$ (without knowing $F$ in advance). The routing scheme consists of two algorithms: (i) a preprocessing algorithm which computes (succinct) routing tables and labels for each vertex in the graph; and (ii) a routing algorithm that given the received message and the routing table of vertex $v$ determines the next-hop (specified as a port number) on the $v$-$t$ (approximate) shortest path in $G \\setminus F$. The efficiency of the scheme is determined by the tradeoff between the \\emph{stretch} (i.e., the ratio between the weighted length of the $s$-$t$ route in $G\\setminus F$ to the corresponding shortest path distance) and the \\emph{space} of the routing tables, labels and messages. \nWhile the stretch vs. space tradeoff of routing schemes is fully understood in the non-faulty setting, the corresponding bounds in the FT setting are still far from optimal.\nSo far, in all the prior schemes, the space of the individual routing tables could be linear in the worst case, even when allowing a large stretch bound. This is in strike contrast to the standard (non-faulty) compact routing schemes, e.g., by Thorup and Zwick \\cite{thorup2001compact}, which provide each vertex a table of $\\widetilde{O}(n^{1\/k})$ bits, while guaranteeing a route stretch of $2k-1$. The current large gap in the quality of FT routing schemes compared to their non-faulty counterparts leads to the following question.\n\n\\begin{question}\\label{q:route}\nIs it possible to design $f$-fault-tolerant compact routing scheme for general graphs with \\emph{sub-linear} table size and with a sub-logarithmic stretch?\n\\end{question}\n\n\n\n\n\\input{result.tex}\n\n\n\n\\section{Introduction}\nDistributed graph representation is concerned with augmenting each vertex (and possibly also edges) with useful and low-space information in order to efficiently address various graph queries in a distributed manner. As the\nvertices and edges of the network may occasionally fail or malfunction, it is desirable\nto make these representations robust against failures. In this paper, we provide new constructions of succinct \\emph{labeled-based distributed data structures} that can handle connectivity, distance queries and routing in the presence of edge failures. \n\nConnectivity labels are short names attached to each vertex in the $n$-vertex input graph $G$, such that given the labels of a pair of vertices $s$ and $t$ (and no any other information), it is possible to deduce if $s$ and $t$ are connected in $G$. The primary complexity measure of the  labeling scheme is the label length (maximum length of a label). In general, labels can be viewed as the \\emph{logical} names of the vertices \\cite{kannan1992implicat,peleg2005informative}, as they are considerably more informative than the physical names that usually correspond to arbitrary $O(\\log n)$-bit identifiers. For example, in routing applications the label of the vertex is treated as its ``address\". It is quite immediate to provide connectivity labeling schemes of logarithmic length. Over the years, these labels have served the basis for devising also approximate distance labels, and compact routing schemes, which are arguably the \\emph{grand finale} of the distributed representation schemes. \n\nOur goal in this paper is to provide \\emph{fault-tolerant} analogs for the above mentioned schemes, while paying a small overhead in terms of space and other complexity aspects. Fault-tolerant (FT) connectivity labeling scheme, also known in the literature as \\emph{forbidden-set} labeling \\cite{CourcelleT07}, assigns labels to the vertices and the edges of the graph such that given the labels of a vertex pair $s,t$, and the labels of the faulty edges $F$, one can determine if $s$ and $t$ are connected in $G \\setminus F$. Several notions of fault-tolerant labeling and routing schemes have been addressed in the literature; starting with the earlier introduction of FT routing schemes by Dolev \\cite{dolev1984new}, to the more recent formulations of forbidden-set labeling and routing schemes by Courcelle et al. \\cite{courcelle2007forbidden,CourcelleT07}. Despite much activity revolving these topics, FT labeling and routing schemes with \\emph{sub-linear} space are currently known only for a limited collection of graph families. We next elaborate more on the state-of-the-art affairs, and our main objectives. \\mtodo{the discussion of related work in the next few pages is quite long. We can consider to have a shorter intro, mainly defining the main problems addressed, and have later a detailed related work section.}\n\n\n\n %\n\n\\paragraph{Fault-Tolerant Connectivity Labeling.} FT labels for connectivity were introduced by \\cite{courcelle2007forbidden} \\mtext{under the term \\emph{forbidden-set labeling}. Forbidden set refers to a subset $F$ of at most $f$ edges, such that given the labels of $s,t$ and $F$ one should determine if $s$ and $t$ are connected in $G \\setminus F$. The forbidden edge set can be treated in this context as faulty edges\\footnote{For routing, the forbidden-set scheme is slightly weaker than FT scheme as explained later.}.} \\mtodo{not sure that this is the best place for discussing the forbidden-set name, maybe it can fit better in a related work section.} Since their introduction, efficient FT labeling schemes have been devised only for a restricted collection of graph families. \\textbf{MP: For example, Courcelle et al. \\cite{CourcelleT07} presented a labeling scheme with logarithmic label length for the families of $n$-vertex graphs with bounded clique-width, tree-width and planar graphs. For $n$-vertex graphs with doubling dimension at most $\\alpha$, Abraham et al. \\cite{AbrahamCGP16} designed FT labeling schemes with label length $O((1 + 1\/\\epsilon)^{2\\alpha}\\log n)$ that output $(1+\\epsilon)$ approximation of the shortest path distances under faults.} In the lack of any FT connectivity labeling schemes for general graphs with sub-linear label length (for any $f\\geq 2$ faults\\footnote{While there is no \\emph{explicit} construction of FT labeling for general graphs, for $f=1$, the centralized distance sensitivity oracle of \\cite{khanna2010approximate} might be modified to provide approximate distance labels against a single fault.}), we ask:\n\n\\begin{question}\\label{q:label}\nIs it possible to design FT connectivity labeling scheme resilient to at most $f$ edge faults, for general graphs with label length of $\\operatorname{\\text{{\\rm poly}}}(\\log n)$ bits, or even $\\operatorname{\\text{{\\rm poly}}}(\\log n,f)$ bits? \n\\end{question}\n\n\\mtodo{The next pargarph is a bit long, we can consider shortning it, and move the more detailed discussion to a related work section. Maybe just focus on the centralized data structures here and not on the certificates?} \\mertodo{Yes, I agree, modifying accordingly.} \nFT connectivity labels are also closely related to \\emph{sensitivity connectivity oracles}, which are low-space centralized data-structure that handle efficiently $\\langle s, t, F \\rangle$ connectivity queries. \\textbf{MP: The first construction of these oracles was given by Patrascu and Thorup \\cite{patrascu2007planning} providing an $S(n)=\\widetilde{O}(fn)$ space oracle that answers $\\langle s,t, F \\rangle$ connectivity queries in $\\widetilde{O}(f)$ time. The state-of-the-art bounds of these oracles are given by Duan and Pettie \\cite{DuanConnectivitySODA17}.} \nOur main goal is in providing a \\emph{distributed} variant of such constructions, e.g., where each vertex or edge in the graph ``holds\" only $S(n)\/n$ bits of information, such that an $\\langle s,t, F \\rangle$ query can be addressed using only the information stored by $s,t$ and $F$. \n\n\\paragraph{Fault-Tolerant Approximate Distance Labeling.} An important step towards designing FT compact routing schemes involves the computation of \\emph{FT approximate distance labels}. In this setting, given the labels of $s,t$ and the faulty edges $F$, it is required to report an approximation for the $s$-$t$ shortest path distance in $G \\setminus F$. \\mtodo{Also the part that starts here may fit better in a related work section (in this case, we will probably need to remove also the question from here - seems that one of the reviwers supported removing the questions in any case).} FT approximate distance labels can be viewed as the distributed analog of $f$-FT \\emph{distance sensitivity oracles} \\cite{khanna2010approximate,WeimannY10}. \nThese are global succinct data-structures that given an $\\langle s,t,F \\rangle$ query report fast an estimate for the approximate $s$-$t$ distance in $G \\setminus F$. \\textbf{MP: Chechik et al. \\cite{chechik2012f} presented the first randomized construction resilient to $f$ edge faults. \nSpecifically, for any $n$-vertex weighted graph, stretch parameter $k$, and a fault bound $f$, they provide a data-structure with $O(f k n^{1+1\/k}\\log(nW))$ space, query time of $\\widetilde{O}(|F|)$, and $O(f k)$ stretch, where $W$ is the weight of the heaviest edge in the graph. Their solution is based on an elegant transformation that converts the FT connectivity oracle of \\cite{patrascu2007planning} into an FT approximate distance oracle.} Our goal is to provide FT approximate labeling schemes that match the state-of-the-art space vs. stretch tradeoff provided by the oracles of \\cite{chechik2012f}, we ask:\n\n\\begin{question}\\label{q:dist-label}\nIs it possible to design FT approximate distance labels with space vs. stretch tradeoff that match the state-of-the-art bounds of the \\emph{centralized} sensitive oracles, e.g., of \\cite{chechik2012f}?\n\\end{question}\n\n\\textbf{MP: While the main focus of this paper is in approximate distances, sensitivity oracles that report (possibly near) exact distances under faults have been studied also thoroughly in e.g., \\cite{demetrescu2002oracles,bernstein2008improved,duan2009dual,WeimannY10,GrandoniW12,ChechikCFK17,van2019sensitive}. Since reporting exact distances requires linear label length already in the fault-free setting \\cite{gavoille2004distance}, we focus on the approximate relaxation, where there is still hope to obtain labels of polylogarithmic length.}\n\\mtodo{Same comment for this paragraph. Also, there is a recent paper about FT exact distance labels in planar graphs \\cite{DBLP:journals\/corr\/abs-2102-07154} that we should probably mention, they show that any directed weighted planar\ngraph admits fault-tolerant distance labels of size $O(n^{2\/3})$. There are also some references cited in their paper that maybe we should discuss as well, for example see the paragraph ``Forbidden-set distance labeling schemes'' in page 2 here: https:\/\/arxiv.org\/pdf\/2102.07154.pdf} \\mertodo{I actually preferred not open up the discussion on special graph families, e.g., planar graphs, graphs with bounded dimension, etc. We do mention it for the direct setting of routing or labeling, but I do not think it should be mentioned for oracles for the following reason. For labels, there was no prior work for general graphs, but for oracles we do have such works so no need to add the extra overhead of special graph families.} \n\n\\paragraph{Fault-Tolerant Routing.} A desirable requirement in most communication networks is to provide efficient routing protocols in the presence of faults. Specifically, an $f$-FT routing protocol is a distributed algorithm that, for any set of at most $f$ faulty edges $F$, allows a vertex $s$ to route a message to a destination vertex $t$ along an approximate $s$-$t$ shortest path in $G \\setminus F$ (without knowing $F$ in advance). The routing scheme consists of two algorithms: (i) a preprocessing algorithm which computes (succinct) routing tables and labels for each vertex in the graph; and (ii) a routing algorithm that given the received message and the routing table of vertex $v$ determines the next-hop (specified as a port number) on the $v$-$t$ (approximate) shortest path in $G \\setminus F$. The efficiency of the scheme is determined by the tradeoff between: \n\\begin{enumerate}[noitemsep]\n\\item the \\emph{stretch} of the route, i.e., the ratio between the length of the route to the $s$-$t$ distance in $G \\setminus F$. \n\\item the \\emph{space} of the routing tables, routing labels and messages. \n\\end{enumerate}\nWhile the stretch vs. space tradeoff of routing schemes is fully understood in the non-faulty setting, the corresponding bounds in the FT setting are still far from optimal. \\mtodo{Also the part that starts here can fit better in a related work section. Also, a reviwer suggested to discuss also \\cite{rajan2012space}. It seems that this work appeard after Chechik et al., and focus on the case of a single edge failure. They show (Theorem 1) a routing scheme with routing tables of size $\\widetilde{O}(k \\deg(v)+ n^{1\/k})$ size per vertex, $O(k^2)$ stretch and $O(k+\\log{n})$ size header that handle a failure of one edge.} \\textbf{MP: The first formalization of FT routing schemes was given by the influential works of Dolev \\cite{dolev1984new} and Peleg \\cite{peleg1987fault}. These earlier works presented the first non-trivial solutions for general graphs supporting at most $\\lambda$ faulty edges, where $\\lambda$ is the edge-connectivity of the graph. Their routing labels had linear size, providing $s$-$t$ routes of possibly linear length (even in cases where the surviving $s$-$t$ path is of $O(1)$ length). In competitive FT routing schemes, it is required to provide $s$-$t$ routes of length that competes with the shortest $s$-$t$ path in $G \\setminus F$, even in cases where $G \\setminus F$ is not connected. Competitive FT routing schemes \\cite{peleg2009good} for general graphs were given by Chechik et al. \\cite{ChechikLPR10,chechik2012f} for the special case of $f\\leq 2$ faults. \nSpecifically, for a given stretch parameter $k$, they gave a routing scheme with a total space bound of $\\widetilde{O}(n^{1+1\/k})$ bits, polylogarithmic-size labels and messages, and a routing \\emph{stretch} of $O(k)$. \nThis scheme was extended later on for any $f$ by Chechik \\cite{chechik2011fault}, at the cost of increasing the routing stretch to $O(f^2(f+\\log^2 n)k)$.}\nSo far, in all these prior schemes, the space of the individual routing tables could be linear in the worst case, even when allowing a large stretch bound. This is in strike contrast to the standard (non-faulty) compact routing schemes, e.g., by Thorup and Zwick \\cite{thorup2001compact}, which provide each vertex a table of $\\widetilde{O}(n^{1\/k})$ bits, while guaranteeing a route stretch of $2k-1$. The current large gap in the quality of FT routing schemes compared to their non-faulty counterparts leads to the following question.\n\n\\begin{question}\\label{q:route}\nIs it possible to design $f$-fault-tolerant compact routing scheme for general graphs with \\emph{sub-linear} table size and with a sub-logarithmic stretch?\n\\end{question}\n\n\\textbf{MP: A more relaxed setting of FT routing scheme which has been studied in the literature is given by the \\emph{forbidden set routing schemes}, introduced by Courcelle and Twigg \\cite{CourcelleT07}. In that setting, it is assumed that the routing protocol knows in advance the set of faulty edges $F$. In contrast, in the FT routing setting, the failing edges are a-priori unknown to the routing algorithm,  and can only be detected upon arriving one of their endpoints. Forbidden set routing schemes have been devised to the same class of restricted graph families as obtained for the forbidden set labeling setting \\cite{CourcelleT07,AbrahamCGP16,abraham2012fully}.}\n\n\n\n\\input{result.tex}\n\n\n\n\\subsection{Connectivity Labels Based on Graph Sketches}\\label{sec:ftconn-sketch}\nIn this section, we show the following:\n\\begin{theorem}\nFor every undirected $n$-vertex graph $G=(V,E)$, a positive integer $f$, there is a randomized $f$-FT connectivity labels $\\mathsf{ConnLabel}\\xspace_{G}: V \\cup E \\to \\{0,1\\}^{\\ell}$ of length $\\ell=O(\\log^3 n)$ bits. The decoding time of the scheme is $\\widetilde{O}(f)$, and the computation time for assigning the labels is $\\widetilde{O}(m+n)$.\n\\end{theorem}\nIn Section \\ref{sec:label-alg}, we present the labeling algorithm which assigns labels based on the notion of graph sketches. In Section \\ref{sec:dec-alg} we present the decoding algorithm that given the label information determines if $s$ and $t$ are connected in $G \\setminus F$. When the graph $G$ is clear from the context, we may omit it and simply write $\\mathsf{ConnLabel}\\xspace$. \n\n\n\n\\subsubsection{The Labeling Algorithm}\\label{sec:label-alg}\nGiven a connected graph $G$, let $T$ be an arbitrary rooted spanning tree in $G$ that is used throughout this section. In our future applications of this labeling scheme (e.g., routing), both the graph $G$ and the tree $T \\subseteq G$ will be given as input to the labeling algorithm. In the latter case, we denote the output labels by $\\mathsf{ConnLabel}\\xspace_{G,T}$. Throughout, all vertices have unique ids $\\operatorname{ID}(v)$ between $\\{1,\\ldots,n \\}$.  \n\n\\paragraph{Extended Edge Identifiers.} In our algorithm it is important to distinguish between an identifier of a single edge to the bitwise XOR of several edges. For this purpose, we define for each edge $e$ an extended edge identifier $\\operatorname{EID}_T(e)$ that allows distinguishing between these cases, and serves as the identifier of the edge.\nThe extended edge identifier $\\operatorname{EID}_T(e)$ consists of a (randomized) unique distinguishing identifier $\\operatorname{UID}(e)$, as well as additional tree related information that facilitates the decoding procedure. The computation of $\\operatorname{UID}(e)$ is based on the notion of $\\epsilon$-\\emph{bias} sets \\cite{naor1993small}. The construction is randomized and guarantees that, w.h.p., the XOR of the $\\operatorname{UID}$ part of each given subset of edges $S \\subseteq E$, for $|S|\\geq 2$, is not a legal $\\operatorname{UID}$ identifier of any edge.\nLet $\\mathsf{XOR}\\xspace(S)$ be the bitwise XOR of the extended identifiers of edges in $S$, i.e., $\\mathsf{XOR}\\xspace(S)=\\oplus_{e \\in S} \\operatorname{EID}_T(e)$. In addition, let $\\mathsf{XOR}\\xspace_U(S)=\\oplus_{e \\in S} \\operatorname{UID}(e)$. Missing proofs are deferred to Appendix \\ref{sec:miss-proof}.\n\n\\begin{lemma}[Modification of Lemma 2.4 in \\cite{GhaffariP16}]\n\\label{cl:epsbias}\nThere is an algorithm that creates a collection $\\mathcal{I}=\\{\\operatorname{UID}(e_1), \\ldots, \\operatorname{UID}(e_{M})\\}$ of $M=\\binom{n}{2}$ random identifiers for all possible edges $(u,v)$, each of $O(\\log n)$-bits using a seed $\\mathcal{S}_{ID}$ of $O(\\log^2 n)$ bits. These identifiers are such that for each subset $E' \\subseteq E$, where $|E'|\\neq 1$, we have $\\Pr[\\mathsf{XOR}\\xspace_U(E') \\in \\mathcal{I}] \\leq 1\/n^{10}$. In addition, given the identifiers $\\operatorname{ID}(u), \\operatorname{ID}(v)$ of the edge $e=(u,v)$ endpoints, and the seed $\\mathcal{S}_{ID}$, one can determine $\\operatorname{UID}(e)$ in $\\widetilde{O}(1)$ time.\n\\end{lemma}\n\\def\\APPENDUNIQUEID{\n\\begin{proof}[Proof of Lemma \\ref{cl:epsbias}]\nThe lemma is proved in \\cite{GhaffariP16}, the only part that is not discussed there is the time to determine $\\operatorname{UID}(e)$ that follows from \\cite{naor1993small}. \nBy Theorem 3.1 of \\cite{naor1993small}, given the seed $\\mathcal{S}_{ID}$ and the edge identifier $e_j=(\\operatorname{ID}(u), \\operatorname{ID}(v))$, determining the $i^{th}$ bit of $\\operatorname{UID}(e_{j})$ can be done in $O(\\log n)$ time. Thus, determining all $O(\\log n)$ bits, takes $O(\\log^2 n)$ time. \n\\end{proof}\n\nFor every vertex $v \\in G$, let $\\mathsf{ANC}\\xspace_T(v)$ be the ancestor label of $v$ computed for the given tree $T$ using Lemma \\ref{anc_labels}. The extended identifier $\\operatorname{EID}_T(e)$ is given by\n\\begin{equation}\\label{eq:extend-ID}\n\\operatorname{EID}_T(e)=[\\operatorname{UID}(e), \\operatorname{ID}(u), \\operatorname{ID}(v), \\mathsf{ANC}\\xspace_T(u), \\mathsf{ANC}\\xspace_T(v)]~.\n\\end{equation}\nThe identifiers of $\\operatorname{ID}(u), \\operatorname{ID}(v)$ are used in order to verify the validity of the unique identifier $\\operatorname{UID}(e)$.  \nWhen the tree $T$ is clear from the context, we might omit it and simply write $\\operatorname{EID}(e)$. As we will see, the labeling scheme will store the seed $\\mathcal{S}_{ID}$ as part of the labels of the tree edges. \n\n\n\\paragraph{Fault-Tolerant Labels via Graph Sketches.} \nGraph sketches are a tool to identify outgoing edges. We start by providing an intuition for them. Say that $S$ is a connected component, and that there are $2^j$ edges outgoing from $S$. If we sample all edges in the graph with probability $1\/2^j$, there is a constant probability that exactly one outgoing edge from $S$ is sampled, and our goal is to find it using local information stored at the vertices of $S$. This information is the \\emph{sketch}. \nThe sketch of each vertex stores the bitwise XOR of sampled edges adjacent to it. Now looking at the XOR of all the sketches of vertices of $S$ allows to detect an outgoing edge. This holds as any sampled edge that has both endpoints in $S$ gets cancelled out, and we are left with the XOR of sampled edges outgoing from $S$. If there is exactly one outgoing edge, we find it. To increase the success probability we can repeat the process $O(\\log{n})$ times. We define sets of vertices $E_{i,j}$, where for $i \\in \\{1, \\ldots, c \\log n\\}$, the set $E_{i,j}$ is obtained by sampling each edge with probability $2^{-j}$. Since we repeat the process $O(\\log{n})$ times for each $j$, then w.h.p we can use the sketches to identify outgoing edge from any component. To use this approach in our context, it is crucial to be able to simulate the sampling process using a small random seed. To do this, we follow \\cite{DuanConnectivityArxiv16,DuanConnectivitySODA17} and use pairwise independent hash functions to decide whether to include edges in sampled sets.\nWe choose $L=c\\log n$ \npairwise independent hash functions $h_1, \\ldots, h_{L}:\\{0,1\\}^{\\Theta(\\log n)} \\to \\{0, \\ldots, 2^{\\log m}-1\\}$,\nand for each $i \\in \\{1, \\ldots, L\\}$ and $j \\in [0,\\log m]$, define the edge set \n$$E_{i,j} =\\{ e \\in E ~\\mid~ h_i(e) \\in [0,2^{\\log m-j})\\}~.$$ \nEach of these hash functions can be defined using a random seed of logarithmic length \\cite{TCS-010}. Thus, a \nrandom seed $\\mathcal{S}_h$ of length $O(L \\log n)$ can be used to determine the collection of all these $L$ functions. As observed in \\cite{DuanConnectivityArxiv16,GibbKKT15}, pairwise independence is sufficient to guarantee that for any set $E' \\subset E$ and any $i$, there exists an index $j$, such that with constant probability $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ is the name (extended identifier) of one edge in $E'$, for a proof see Lemma 5.2 in  \\cite{GibbKKT15}.\n\\begin{lemma}\\label{lem:hitting-pairwise}\nFor any edge set $E'$ and any $i$, with constant probability there exists a $j$ satisfying that $|E' \\cap E_{i,j}|=1$.\n\\end{lemma}\n\n\nWe also need to be able to tell that a bit string of $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ is a legal edge ID or not. Here we exploit the extended ids. See Appendix \\ref{sec:miss-proof} for a proof.\n\\begin{lemma} \\label{lemma_unique}\nGiven the seed $\\mathcal{S}_{ID}$, one can determine in $\\widetilde{O}(1)$ time if $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ corresponds to a single edge ID in $G$ or not, w.h.p.\n\\end{lemma}\n\\def\\APPENDLEMMUNIQUE{\n\\begin{proof}[Proof of Lemma \\ref{lemma_unique}]\nLet $X=\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$. Letting $E''=E' \\cap E_{i,j}$, then $X$ can be written as the concatenation of $\\mathsf{XOR}\\xspace_1(E'')$ and $\\mathsf{XOR}\\xspace_2(E'')$, where $\\mathsf{XOR}\\xspace_1(E'')=\\mathsf{XOR}\\xspace_U(E'')$ is the bit-wise XOR of the unique identifiers $\\operatorname{UID}(e)$ for $e \\in E''$ and $\\mathsf{XOR}\\xspace_2(E'')$ is the bit-wise XOR of the remaining information in the extended identifiers of $E''$.  We now show how using the seed and $\\mathsf{XOR}\\xspace_2(E'')$, one can test the validity of $\\mathsf{XOR}\\xspace_1(E'')$.\nThe algorithm detects the case that $|E''| \\geq 2$ as follows. First, in the case that $E''$ is a single edge, $\\mathsf{XOR}\\xspace_2(E'')$ should contain legal ids $\\operatorname{ID}(u),\\operatorname{ID}(v)$. If this is not the case, it follows that $|E''| \\neq 1$. If $\\mathsf{XOR}\\xspace_2(E'')$ contains legal ids $\\operatorname{ID}(u),\\operatorname{ID}(v)$, we use them and the seed $\\mathcal{S}_{ID}$ to determine $\\operatorname{UID}(e)$ for $e = (u,v)$, and we check if $\\mathsf{XOR}\\xspace_1(E'')=\\operatorname{ID}_1(e)$. We have two options, either $E'' = \\{e\\}$ is the single edge $e$, in which case $\\mathsf{XOR}\\xspace_U(E'')=\\operatorname{UID}(e) \\in \\mathcal{I}$, and the verification succeeds. Otherwise $|E''| \\geq 2$, in which case, from Lemma \\ref{cl:epsbias}, $\\Pr[\\mathsf{XOR}\\xspace_U(E'') \\in \\mathcal{I}] \\leq 1\/n^{10}$, hence w.h.p $\\mathsf{XOR}\\xspace_U(E'') \\neq \\operatorname{UID}(e) \\in \\mathcal{I}$ and we identify that $|E''| \\geq 2$.\n\\end{proof}\n\n\nFor each vertex $v$ and indices $i,j$, let $E_{i,j}(v)$ be the edges incident to $v$ in $E_{i,j}$. \nThe $i^{th}$ \\emph{basic sketch unit} of each vertex $v$ is then given by:\n\\begin{equation}\n\\label{eq:vsketch}\n\\mathsf{Sketch}\\xspace_{G,i}(v)=[\\mathsf{XOR}\\xspace(E_{i,0}(v)),\\ldots,\\mathsf{XOR}\\xspace(E_{i,\\log m}(v))].\n\\end{equation}\nThe sketch of each vertex $v$ is defined by a concatenation of $L=\\Theta(\\log n)$ basic sketch units: \n$$\\mathsf{Sketch}\\xspace_G(v)=[\\mathsf{Sketch}\\xspace_{G,1}(v),\\mathsf{Sketch}\\xspace_{G,2}(v), \\ldots\\mathsf{Sketch}\\xspace_{G,L}(v)]~.$$ \nFor every subset of vertices $S$, let \n$\\mathsf{Sketch}\\xspace_G(S)=\\oplus_{v \\in S}\\mathsf{Sketch}\\xspace_G(v).$ When the graph $G$ is clear from the context, we may omit it and write $\\mathsf{Sketch}\\xspace_{i}(v)$ and $\\mathsf{Sketch}\\xspace(v)$. \n\nWe are now ready to define the fault-tolerant connectivity labels of vertices and edges. \nThe label of each vertex $u$ is given by:\n\\begin{equation}\\label{eq:conn-vertex}\n\\mathsf{ConnLabel}\\xspace_{G,T}(u)=\\langle \\mathsf{ANC}\\xspace_T(u), \\operatorname{ID}(u) \\rangle~,\n\\end{equation}\nwhere $\\mathsf{ANC}\\xspace_T(u)$ is the ancestry label of $u$ with respect to the tree $T$. \nFor every $u \\in V(T)$, let $T_u$ be the subtree rooted at $u$.  The label $\\mathsf{ConnLabel}\\xspace_{G,T}(e)$ of each \\emph{edge} $e=(u,v)$ is given by:\n\\begin{equation*}\n    \\mathsf{ConnLabel}\\xspace_{G,T}(e)=\n    \\begin{cases}\n      \\langle \\operatorname{EID}_T(e), \\mathsf{Sketch}\\xspace(V(T_u)), \\mathsf{Sketch}\\xspace(V(T_v)), \\mathsf{Sketch}\\xspace(V), \\mathcal{S}_{ID}, \\mathcal{S}_h\\rangle ,& \\mbox{~for~} e \\in T \\\\\n     \\langle \\operatorname{EID}_T(e) \\rangle,& \\mbox{~Otherwise}.\n    \\end{cases}\n\\end{equation*}\n\nWe complete this subsection by bounding the label size and computation time of the labeling algorithm. For proofs see Appendix \\ref{sec:miss-proof}. \n\\begin{claim}\\label{cl:label-length}\nThe label length is $O(\\log^3 n)$ bits.\n\\end{claim}\n\\def\\APPENDLABELCONSISE{\n\\begin{proof}[Proof of Claim \\ref{cl:label-length}]\nThe label size is dominated by the sketching information $\\mathsf{Sketch}\\xspace(V(T_u))$, which is made of a concatenation of the bitwise XOR of $O(\\log n)$ basic sketch units $\\mathsf{Sketch}\\xspace_i(u)$. By Eq. (\\ref{eq:vsketch}), each unit has $O(\\log^2 n)$ bits, and thus overall, the label has $O(\\log^3 n)$ bits.\n\\end{proof}\n\n\nWe show that assigning the labels takes  $\\widetilde{O}(m+n)$ time.\n\\begin{claim}\\label{cl:time-conn-labelsketch}\nThe time complexity of the labeling algorithm is $\\widetilde{O}(m+n).$\n\\end{claim}\n\\def\\APPENDCONNLABELSKETCH{\n\\begin{proof}[Proof of Claim \\ref{cl:time-conn-labelsketch}]\nTo compute the labels of vertices we assign ids to vertices in $O(n)$ time, and compute ancestry labels in $O(n)$ time using Lemma \\ref{anc_labels}. To compute the extended identifiers $\\operatorname{EID}_T(e)$, we also choose the random seed $\\mathcal{S}_{ID}$ and compute $\\operatorname{UID}(e)$ using Lemma \\ref{cl:epsbias}, this takes $\\widetilde{O}(1)$ time per edge, and $\\widetilde{O}(m)$ time for all edges. Lastly, we should compute the sketch values $\\mathsf{Sketch}\\xspace(V(T_u))$. For this, first, we choose the random seed $\\mathcal{S}_h$, and compute the values $\\mathsf{Sketch}\\xspace_G(v)$. For this, we should identify for each vertex the adjacent edges in $E_{i,j}$. For each edge we can identify the sets it belongs to in $\\widetilde{O}(1)$ time using Fact \\ref{fc:pairwise}. This allows us computing the sketch values of all vertices in $\\widetilde{O}(m+n)$ time. We can then compute the values $\\mathsf{Sketch}\\xspace(V(T_u))$ by scanning the tree in $\\widetilde{O}(n)$ time.   \n\\end{proof}\n\nFinally, the subsequent decoding algorithm will be based on the following useful property of the graph sketches, stored by our labels. \n\\begin{lemma}\\label{lem:sketch-property}\nFor any subset $S$, given one basic sketch unit $\\mathsf{Sketch}\\xspace_i(S)$ and the seed $\\mathcal{S}_{ID}$ one can compute, with constant probability, an outgoing edge $E(S, V \\setminus S)$ if such exists. The complexity is $\\widetilde{O}(1)$ time.\n\\end{lemma}\n\\def\\APPENDSKETCHPROP{\n\\begin{proof}[Proof of Lemma \\ref{lem:sketch-property}]\nThe proof follows from Lemma \\ref{lem:hitting-pairwise}. Note that by definition of the sketch values $\\mathsf{Sketch}\\xspace_i(S)=\\oplus_{v \\in S}\\mathsf{Sketch}\\xspace_i(v)=[\\mathsf{XOR}\\xspace(E_{i,0}(S)),\\ldots,\\mathsf{XOR}\\xspace(E_{i,\\log m}(S))],$ where $E_{i,j}(S)$ are the outgoing edges from $S$ in $E_{i,j}$ (edges that have both endpoints in $S$ are cancelled out by the XOR operation). Let $E'$ be all the outgoing edges from $S$. From Lemma \\ref{lem:hitting-pairwise}, with constant probability there exists a $j$ such that $|E' \\cap E_{i,j}|=1$. In this case, $\\mathsf{XOR}\\xspace(E_{i,j}(S))$ corresponds to an extended id of a single outgoing edge from $S$. We can check if this happens in $\\widetilde{O}(1)$ time using Lemma \\ref{lemma_unique}.   \n\\end{proof}\n\n\n\n\n\\subsubsection{The Decoding Algorithm} \\label{sec:dec-alg}\nWe next describe the decoding algorithm where given a triplet $s,t, F \\in V \\times V \\times E^f$ along with their labels, it determines whether $s$ and $t$ are connected in $G\\setminus F$, w.h.p. \nThe decoding algorithm has four key steps: The first step identifies the at most $f+1$ components $\\mathcal{C}_0=\\{C_1,\\ldots, C_\\ell\\}$ of $T \\setminus F$, as well as the components of $s$ and $t$ in $\\mathcal{C}_0$. The second step uses the label information to compute the sketch value $\\mathsf{Sketch}\\xspace(C_i)$ of each component $C_i \\in \\mathcal{C}_0$. The third step modifies this sketch information into $\\mathsf{Sketch}\\xspace_{G \\setminus F}(C_i)$, by subtracting the information related to the faulty edges. The forth and final step uses the sketch information in order to simulate $L=O(\\log n)$ steps of the Boruvka algorithm. At the end of these steps, the decoding algorithm identifies the connected components of both $s$ and $t$ in $G \\setminus F$. In the case where $s$ and $t$ are indeed connected in $G \\setminus F$, the algorithm also outputs a succinct representation of an $s$-$t$ path in $G \\setminus F$. This extra information would be used later on by our compact routing scheme. We next describe these steps in details. \n\n\\paragraph{Step 1: Identification of the connected components $\\mathcal{C}_0$ in $T \\setminus F$.} \nLet $F_T=F \\cap T$ be the faulty tree edges and let $F_{NT}=F \\setminus F_T$ be the faulty non-tree edges. Let $Q=\\{s,t\\} \\cup V(F_T)$.  Each component $C_i$ of $T \\setminus F$ will be identified by the maximum vertex ID in $C_i \\cap V(F_T)$. Note that in the case where $F_T=\\emptyset$, $T \\setminus F=T$ and thus $s$ and $t$ are connected iff $s,t \\in V(T)$.  From now on, we therefore assume that $F_T \\neq \\emptyset$. \n\nWe next show that although we do not have full information about the tree $T$ and the vertices of each connected component, the ancestry labels of $V(F_T)$ give us enough information to identify the connected components of $T \\setminus F$. Additionally, given an ancestry label of a vertex $u$, we can identify the connected component of $u$. To obtain this, it is helpful to look at the \\emph{component tree} that is obtained by contracting each connected component of $T \\setminus F$ to one vertex, as follows. Let $\\ell = |F_T|+1.$ The component tree $T_C = (\\mathcal{C}_0, E_C)$ is a tree of $\\ell$ vertices representing the connected components in $T \\setminus F$, and $|F_T|=\\ell-1$ edges corresponding to the edges of $F_T$. There is an edge $(C_i,C_j) \\in E_C$ iff there is an edge $(u,v) \\in F_T$ where $u \\in C_i, v \\in C_j$. See Figure \\ref{componentTreePic} for an illustration.  \n\n\n\\setlength{\\intextsep}{0pt}\n\\begin{figure}[h]\n\\centering\n\\setlength{\\abovecaptionskip}{-2pt}\n\\setlength{\\belowcaptionskip}{6pt}\n\\includegraphics[scale=0.55]{componentTree.pdf}\n \\caption{Illustration of the component tree where $F=\\{e_1,e_2,e_3,e_4\\}$. Each connected component of $T \\setminus F$ is contracted to one vertex on the right.}\n\\label{componentTreePic}\n\\end{figure}\n\nWe can construct the tree $T_C$ using the ancestry labels of the edges $F_T$.  For this, for each edge $e \\in F_T$ we just need to identify the set of edges from $F_T$ above $e$ in $T$. Moreover, for a given vertex $v$, its connected component is exactly determined by the set of edges in $F_T$ above it in $T$, which can again be identified using the ancestry labels of $v \\cup V(F_T)$. In particular, we can identify the connected components of $s$ and $t$. The component tree can be constructed in $O(f^2)$ time by checking for any pair of edges $e,e' \\in F_T$, if $e$ is above $e'$ in the tree. We next show a faster algorithm taking only $\\widetilde{O}(f)$ time by exploiting properties of the ancestry labels. Moreover, we show that the component of each vertex can be identified in $O(\\log{f})$ time.\n\n\\begin{claim} \\label{claim_component_tree}\nThe component tree can be constructed in $O(f \\log{f})$ time. Additionally, given $\\mathsf{ANC}\\xspace_T(v)$, we can identify the connected component of $v$ in $T \\setminus F$ in $O(\\log{f})$ time.\n\\end{claim}\n\n\\begin{proof}\nOur algorithm uses ancestry labels based on DFS from \\cite{kannan1992implicat}. In this scheme, the label of each vertex $v$ is composed of two numbers $(DFS_1(v),DFS_2(v))$ that represent the first and last times a DFS scan of the tree visits $v$. A vertex $u$ is an ancestor of a vertex $v$ iff the interval $(DFS_1(u),DFS_2(u))$ contains the interval $(DFS_1(v),DFS_2(v))$. To build the component tree, we sort the labels of $V(F_T)$, as described next. First, for each component $C \\in T \\setminus F$, we use the highest vertex in the component to represent the component. For the highest component, this is the root $r$. For any other component, we have that the highest vertex of the component, $v$, is in $V(F_T)$. This holds as the edge connecting $v$ to its parent $p(v)$ is necessarily in $F_T$ (otherwise, $v$ is not the highest vertex in its component), see Figure \\ref{componentTreePic} for illustration. Hence, for any edge $(v,p(v)) \\in F_T$, we have that the vertex $v$ represents one component (we can identify which of the vertices is the parent using the ancestry labels). Hence, we have $|F_T|+1$ vertices $v_i$ representing the components $C_i$ of the component tree, and we also know the ancestry labels $(DFS_1(v_i),DFS_2(v_i))$ of all vertices $v_i$, except $r$. For $r$ we can use the label $(1,M)$ where $M$ is a number greater than all values $DFS_2(v_i)$ of other vertices. We next use these labels to determine the structure of the component tree.\nFor this, we create for each vertex $v_i$ two tuples: $(DFS_1(v_i),v_i,1),(DFS_2(v_i),v_i,2)$, and we sort the $2(|F_T|+1)$ tuples according to their first coordinate. This takes $O(f \\log{f})$ time. We next scan the sorted list, and when we reach the tuple $(DFS_1(v_i),v_i,1)$, we identify the parent of $v_i$ in the component tree, as follows. The first tuple is $(1,r,1)$ and $r$ is set to be the root of the component tree. For a vertex $v_i \\neq r$, we identify its parent when we reach $(DFS_1(v_i),v_i,1)$. Let $(DFS_b(u),u,b)$ be the last tuple before $(DFS_1(v_i),v_i,1)$ in the sorted order. If $b=1$, then $u$ is the parent of $v_i$ in the component tree. If $b=2$, let $w$ be the parent of $u$ in the component tree, then $w$ is also the parent of $v$ in the component tree. Additionally, $w$ was already computed as $(DFS_1(u),u,1)$ appears before $(DFS_1(v_i),v_i,1)$. Hence, we can find the parent of $v$ in $O(1)$ time using the tuple before it. Scanning the list takes $O(f)$ time, and after it we know for each component its parent in the component tree, which gives the complete structure of the tree. We next prove the correctness of the algorithm.  \n\nWe first discuss the case that $b=1$. Here $(DFS_1(u),u,1)$ is the last tuple before $(DFS_1(v_i),v_i,1)$. This means that $u$ is necessarily an ancestor of $v$, because the entry $(DFS_1(v_i),v_i,1)$ is between the entries $(DFS_1(u),u,1)$ and $(DFS_2(u),u,2)$, and the DFS scan traverses exactly the subtree of $u$ in the time interval $(DFS_1(u),DFS_2(u))$, implying that $v_i$ is a child of $u$. Moreover, this is the closest ancestor to $v_i$ among the vertices $\\{v_1,v_2,...,v_{\\ell}\\} \\setminus \\{v_i\\}$, as the DFS scan traverses the ancestors of $v_i$ from the highest to the lowest. It follows that $u$ represents the closest component $C$ above $v_i$ in the component tree, as needed.  \n\nWe next discuss the case that $b=2$. Here $(DFS_2(u),u,2)$ is the last tuple before $(DFS_1(v_i),v_i,1)$. Note that now $u$ is not an ancestor of $v_i$, as the DFS scan finished scanning the subtree of $u$ before reaching $v_i$, but we claim that $u$ and $v_i$ have the same parent in the component tree. For this, we show they have exactly the same ancestors in the set $\\{v_1,v_2,...,v_{\\ell}\\} \\setminus \\{u,v_i\\}.$ For any ancestor $w\\neq u$ of $u$, we have that $DFS_1(w) < DFS_1(u) < DFS_2(u) < DFS_2(w)$. As $(DFS_1(v_i),v_i,1)$ is the first tuple after $(DFS_2(u),u,2)$, it must hold that $DFS_1(w) < DFS_1(v_i) <  DFS_2(w)$, implying that $v_i$ is a child of $w$ as needed. Similarly, any ancestor $w \\neq v_i$ of $v_i$ is also an ancestor of $u$, as we have $DFS_1(w) < DFS_2(u) < DFS_1(v_i) < DFS_2(v_i) < DFS_2(w)$.\nHence, the parent of $u$ in the component tree is also the parent of $v_i$ in the component tree, as needed.  \n \nLastly, we show that using similar ideas we can also identify the component of a vertex $v$ in $T \\setminus F$. We create for $v$ the tuple, $(DFS_1(v),v,1)$, and use binary search to find the last tuple smaller or equal to it in the sorted list we computed before, denote it by $(DFS_b(u),u,b)$. \nIf $b=1$ then $v$ is in the component of $u$, and else it is in the component of the parent of $u$ (that was computed before). The complexity of the binary search is $O(\\log{f})$, we next prove correctness. \nOne special case is that $v$ is a root of one of the components in the component tree. In this case, the entry $(DFS_b(u),u,b)$ we find is equal to $(DFS_1(v),v,1)$, and $u=v$ is indeed the component of $v$. Otherwise, $v$ is an internal vertex in its component, and the root of the component is the closest ancestor to $v$ in $\\{v_1,...,v_{\\ell}\\}$.\nIf $b=1$, then as shown before, $u$ is the closest ancestor to $v$ in the component tree, as needed. If $b=2$, then as shown before, $u$ is not an ancestor of $v$, but has exactly the same ancestors in the component tree. Hence, the root $w$ of the component above $u$ is the root of the component of $v$, as needed.    \n\\end{proof} \n\n\n\\paragraph{Step 2: Computing the sketch values of each component $\\mathcal{C}_0$ in $G$.} \nFor each component $C_j \\in \\mathcal{C}_0$ the algorithm computes $\\mathsf{Sketch}\\xspace_G(C_j)$ using the sketch information of the vertices in $V(F_T)$.  The basic observation here is the following. Given $S' \\subset S$ and $\\mathsf{Sketch}\\xspace(S), \\mathsf{Sketch}\\xspace(S')$, it holds that $\\mathsf{Sketch}\\xspace(S \\setminus S')=\\mathsf{Sketch}\\xspace(S) ~\\oplus~ \\mathsf{Sketch}\\xspace(S')$. To compute the sketch values, first, we define for each component a temporary value $\\mathsf{Sketch}\\xspace'_G(C_j)$ as follows. Let $v_j$ be the highest vertex (closest to the root in $T$) in the component $C_j$. For the component of the root $r$, this is $r$. For any other component $C_j$, let $(C_j,p(C_j))$ be the edge connecting $C_j$ to its parent in the component tree. This edge corresponds to an edge $(v_j,p(v_j)) \\in F_T$, where $v$ is the highest vertex in $C_j$. We define $\\mathsf{Sketch}\\xspace'_G(C_j) = \\mathsf{Sketch}\\xspace_G(V(T_{v_j}))$.\nSince $(v_j,p(v_j)) \\in F_T$, the sketch information $\\mathsf{Sketch}\\xspace'_G(C_j)$ can be obtained from the label of the tree edge $(v_j,p(v_j))$. We also know the temporary sketch value of the component of $r$, as $\\mathsf{Sketch}\\xspace_G(V_{r})=\\mathsf{Sketch}\\xspace_G(V)$ is part of the labels of all tree edges (and we assume that $F_T \\neq \\emptyset$). We next use the temporary sketch values to compute the sketch values of components using the following claim.\n\n\\begin{claim}\nLet $C_j$ be a component in $T \\setminus F$. If $C_j$ is a leaf in the component tree, we have $\\mathsf{Sketch}\\xspace_G(C_j) = \\mathsf{Sketch}\\xspace'_G(C_j).$ Otherwise, let $D=\\{D_1,...,D_t\\}$ be the children of $C_j$ in the component tree and let $\\mathsf{Sketch}\\xspace'(D)=\\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace'_G(D_i)$, then $\\mathsf{Sketch}\\xspace_G(C_j) = \\mathsf{Sketch}\\xspace'_G(C_j) \\oplus \\mathsf{Sketch}\\xspace'(D).$ \n\\end{claim}\n\n\\begin{proof}\nIt holds that $\\mathsf{Sketch}\\xspace_G(C_j) = \\oplus_{v \\in C_j} \\mathsf{Sketch}\\xspace_G(v)$. By definition, $\\mathsf{Sketch}\\xspace'_G(C_j)=\\mathsf{Sketch}\\xspace_G(V(T_{v_j})) = \\oplus_{v \\in V(T_{v_j})} \\mathsf{Sketch}\\xspace_G(v)$ is the XOR of sketches of all vertices in the subtree of $v_j$. As $v_j$ is the highest vertex in $C_j$, if $C_j$ is a leaf component in the component tree, then the vertices in $C_j$ are exactly the vertices in $T_{v_j}$, and the claim follows. Otherwise, the vertices in $C_j$ are all vertices in $T_{v_j}$ that are not contained in any component below $C_j$. Hence, to compute the value $\\mathsf{Sketch}\\xspace_G(C_j)$, we should subtract from $\\mathsf{Sketch}\\xspace_G(V(T_{v_j}))$ the sketch values of vertices in components below $C_j$. Let $D_1,\\ldots,D_t$ be the children of $C_j$ in the component tree, and let $u_1,\\ldots,u_t$ be the highest vertices in the components $D_1,\\ldots,D_t$, respectively. Any vertex that is in some component below $C_j$ is in exactly one of the subtrees $T_{u_1},\\ldots,T_{u_t}$. Hence the sketch value of vertices in components below $C_j$ equals $\\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace_G(V(T_{u_i}))= \\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace'_G(D_i)=\\mathsf{Sketch}\\xspace'(D)$. To conclude, we get $\\mathsf{Sketch}\\xspace_G(C_j)=\\mathsf{Sketch}\\xspace_G(V(T_{v_j})) \\oplus \\mathsf{Sketch}\\xspace'(D)=\\mathsf{Sketch}\\xspace'_G(C_j) \\oplus \\mathsf{Sketch}\\xspace'_G(D)$, as needed.\n\\end{proof}\n\nTo conclude, from the values $\\mathsf{Sketch}\\xspace'_G(C_j)$, we can easily compute the values $\\mathsf{Sketch}\\xspace_G(C_j)$. The complexity is $\\widetilde{O}(f)$, as for each component, the sketch $\\mathsf{Sketch}\\xspace'(C_j)$ participates in two computations, and we have at most $O(f)$ components and the sketches have poly-logarithmic size.\n\n\n\\paragraph{Step 3: Computing the sketch values of each component $\\mathcal{C}_0$ in $G \\setminus F$.} \nFor each faulty edge $e \\in F$ (both tree and non-tree edges), our goal is to subtract the sketch information of $e$ from the corresponding components of the endpoint of $e$. The step does not require the label information of the edges, and it would be sufficient to know only the seed $\\mathcal{S}_h$ that determines the sampling of edges into the sketches, and the extended identifier of the failing edges. Since $F_T \\neq \\emptyset$, the algorithm holds the seed $\\mathcal{S}_h$ (from the label of an edge $e \\in F_T$), and it has the extended identifiers of all edges in $F$ as part of their labels. \n\nUsing the extended identifier of the faulty edge $e=(u,v)$, one can determine in $O(\\log{f})$ time the components in $\\mathcal{C}_0$ to which its endpoints belong, from Claim \\ref{claim_component_tree}. Using the identifier $\\operatorname{EID}(e)$ and the seed $\\mathcal{S}_h$, one can determine all the indices of the sketch to which the edge $e$ was sampled in $\\widetilde{O}(1)$ time using Fact \\ref{fc:pairwise}.\nLetting $C_u, C_v$ be the components of $u$ and $v$ in $T \\setminus F$, respectively. If $C_u \\neq C_v$, then the values $\\mathsf{Sketch}\\xspace_G(C_u),\\mathsf{Sketch}\\xspace_G(C_v)$ are updated by XORing them with the matrix that contains the extended identifier $\\operatorname{EID}(e)$ in the relevant positions. The complexity is poly-logarithmic, as the matrix has poly-logarithmic size. In the case that $C_u=C_v$, as $e$ is an internal edge in the component, it is not part of $\\mathsf{Sketch}\\xspace_G(C_u)$, and there is no need to update the value. Overall, doing the computation for all edges in $F$ takes $\\widetilde{O}(f)$ time.\nFrom that point on, all sketches of the components $\\mathcal{C}_0$ can be treated as sketches that have been computed in $G \\setminus F$.  \n\n\n\\paragraph{Step 4: Simulating the Boruvka algorithm.} Finally, our goal is to determine the identifiers of the maximal connected components of $s$ and $t$ of $G \\setminus F$. The input to this step is the identifiers of the components $\\mathcal{C}_0=\\{C_{1}, \\ldots, C_k\\}$ in $T \\setminus F$, along with their sketch information in $G \\setminus F$. While the algorithm does not have information on the vertices of each component, it knows the component identifier of each vertex in $Q$. \n\nThe algorithm consists of $L=O(\\log n)$ phases of the Boruvka algorithm. Each phase $i \\in \\{1,\\ldots, L\\}$ will be given as input a partitioning $\\mathcal{C}_i=\\{C_{i,1}, \\ldots, C_{i,k_i}\\}$ of (not necessarily maximal) connected components in $G \\setminus F$.\nThese components are identified by an $O(\\log n)$ bit identifier, where for each vertex in $Q$, the algorithm receives its unique component identifier in  $\\mathcal{C}_i$. In addition, the algorithm receives the sketch information of the components $\\mathcal{C}_i$ in $G \\setminus F$. The output of the phase is a partitioning $\\mathcal{C}_{i+1}$, along with their sketch information in $G \\setminus F$ and the identifiers of the components for each vertex in $U$. A component $C_{i,j} \\in \\mathcal{C}_i$ is \\emph{growable} if it has at least one non-faulty outgoing edge to a vertex in $V \\setminus C_{i,j}$. That is, the component is growable if it is strictly contained in some maximal connected component in $G \\setminus F$. Letting $N_i$ denote the number of growable components in $\\mathcal{C}_i$, the output partitioning $\\mathcal{C}_{i+1}$ of the $i^{th}$ step guarantees that $N_{i+1}\\leq N_i \/2$ w.h.p. To obtain outgoings edges from the growable components in $\\mathcal{C}_i$, the algorithm uses the $i^{th}$ basic-unit sketch $\\mathsf{Sketch}\\xspace_i(C_{i,j})$ of each $C_{i,j} \\in \\mathcal{C}_i$. By Lemma \\ref{lem:sketch-property}, from every growable component in $\\mathcal{C}_i$, we get one outgoing edge $e'=(x,y)$ with constant probability. Using the extended edge identifier of $e'$ the algorithm can also detect the component $C_{i,j'}$ to which the second endpoint, say $y$, of $e'$ belongs using Claim \\ref{claim_component_tree}.\nThat label allows us to compute the component of $y$ in the initial partitioning $T \\setminus F$, i.e., the component $C_{0,q}$ of $y$ in $\\mathcal{C}_0$. Thus $y$ belongs to the unique component $C_{i,j'} \\in \\mathcal{C}_i$ that contains \n$C_{0,q}$. \n\n\n\nAs noted in prior works \\cite{ahn2012analyzing,kapron2013dynamic,DuanConnectivityArxiv16}, it is important to use fresh randomness (i.e., independent sketch information) in each of the Boruvka phases. The reason is that the cut query, namely, asking for a cut edge between $S$ and $V \\setminus S$, should not be correlated with the randomness of the sketches. Note that indeed the components of $\\mathcal{C}_i$ are correlated with the randomness of the first $(i-1)$ basic sketch units of the vertices. Thus, in phase $i$ the algorithm uses the $i^{th}$ basic sketch units of the vertices (which are independent of the other sketch units) to determine the outgoing edges of the components in $\\mathcal{C}_i$.\n\n\nThe algorithm then computes the updated sketches of the merged components. This is done by XORing over the sketches of the components in $\\mathcal{C}_i$ that got merged into a single component in \n$\\mathcal{C}_{i+1}$. In expectation, the number of growable components is reduced by factor $2$ in each phase. Thus after $O(\\log n)$ phases, the expected number of growable components is at most $1\/n^5$, and using Markov inequality, we conclude that w.h.p there are no growable components. The final partitioning $\\mathcal{C}_L$ corresponds w.h.p to the maximal connected components in $G \\setminus F$. The pair $s$ and $t$ are connected in $G \\setminus F$ only if the components $C_s,C_t$ of $s,t$ respectively in $T \\setminus F$ are connected in the final component decomposition.\nWe next show that the complexity of the algorithm is $\\widetilde{O}(f)$. This is also the decoding time of the whole algorithm, as all steps take $\\widetilde{O}(f)$ time, as discussed above. \n\n\\begin{claim}\\label{cl:complexity-step-four}\nThe complexity of step 4 is $\\widetilde{O}(f)$.\n\\end{claim}\n\n\\begin{proof\nThe algorithm has $O(\\log{n})$ phases, where in each phase the following is computed. First, given the sketch values of the current components we identify outgoing edges from the components. This takes $\\widetilde{O}(1)$ time per component from Lemma \\ref{lem:sketch-property}, and $\\widetilde{O}(f)$ time for all components, as we have at most $f+1$ components. Next, for each outgoing edge we identify the components it connects using its ancestry labels, this takes $\\widetilde{O}(1)$ time per edge using Claim \\ref{claim_component_tree}. Then, we merge components accordingly and compute the sketch values of the new components by XORing the sketch values of merged components. Overall this takes $\\widetilde{O}(f)$ time, as we have at most $O(f)$ merges. In more detail, we can use a union-find data structure to implement the merges, where every time we merge components we compute the sketch value of the new component. We also maintain for each original component in $T \\setminus F$ its current component in phase $i$, this allows us to learn the current components connected by an outgoing edge $e$. This information can be maintained as follows. Let $C$ be a component in $T \\setminus F$, and assume we know the component $C_{i,j}$ it belongs to at the beginning of phase $i$. After the merges of phase $i$, $C_{i,j}$ joins some component $C_{i+1,j'}$ of phase $i+1$. We can use the find operation to identify the id of the new component. Overall, we have $O(f)$ merges and $O(f)$ find operations to identify for each component $C \\in T \\setminus F$, the corresponding component $C_{i+1,j'}$ it belongs to, hence the complexity is bounded by $\\widetilde{O}(f)$. \n\\end{proof}\n\nFinally, we show that the decoding algorithm can be slightly modified to output a compressed encoding of an $s$-$t$ path in $G \\setminus F$, using $O(f\\log n)$ bits. This encoding is represented by an $s$-$t$ path $\\widehat{P}$ that has two type of edges, appearing in an alternate manner on  $\\widehat{P}$: $G$-edges and edges $e'=(u,v)$ such that the $u$-$v$ tree path is intact in $T \\setminus F$. See Figure \\ref{fig:succ-paths}. \n\\begin{lemma}\\label{lem:useful-recovery-edges}\nConsider a triplet $s,t,F$ such that $s$ and $t$ are connected in $G \\setminus F$. \nThe decoding algorithm can also output a set of at most $f$ recovery edges $R$ such $(T \\setminus F) \\cup R$ is a spanning tree. In addition, it outputs a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of the $s$-$t$ path. The edges of $\\widehat{P}$ are labeled by $0$ and $1$, where $0$-labeled edges correspond to $G$-edges and $1$-labeled edges $e=(x,y)$ correspond to $x$-$y$ paths in $T \\setminus F$. \n\\end{lemma}\n\\begin{proof}\nLet $C_s, C_t$ be the components of $s$ and $t$ in the initial partitioning $\\mathcal{C}_0$. In Step $4$ of the decoding algorithm, the Boruvka algorithm is simulated up to the point that $C_s$ and $C_t$ are connected. Therefore, the algorithm has computed a path $P$ that connects the components $C_s$ and $C_t$. Each vertex on that path corresponds to a component in $\\mathcal{C}_0$, and each edge corresponds to an outgoing edge (discovered using the sketch information). Since $\\mathcal{C}_0$ has at most $f+1$ components, $|P|\\leq f+1$. \nEach such edge $e' \\in P$ corresponds to an edge in $G$. Let $e_1=(x_1,y_1),\\ldots, e_k=(x_k,y_k)$ be the $G$-edges corresponding to the edges of $P$ ordered from $C_s$ to $C_t$. Letting $y_0=s$ and $x_{k+1}=t$, we get that \n$y_i$ and $x_{i+1}$ belong to the same component in $\\mathcal{C}_0$, for every $i \\in \\{0,\\ldots, k\\}$. \nThe labeled path is given by $\\widehat{P}=[s, x_1,y_1, x_2, y_2, \\ldots y_k,t]$ where the edges $(y_i,x_{i+1})$ are labeled $1$ and the edges $(x_{i}, y_i)$ are labeled $0$. Each $0$-labeled edge is a real edge in $G$, and each $1$-labeled edge $(x_{i}, y_i)$ corresponds to a tree path $\\pi(x_i, y_i)$ in $T \\setminus F$. \n\\end{proof}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.40]{suc-path.pdf}\n\\caption{\\sf Shown is a tree $T$ with faulty edges $e_1,\\ldots, e_4$. The $s$-$t$ path in $G \\setminus F$ is represented by the path $\\widehat{P}=[s,v_1]\\circ (v_1,v_2) \\circ [v_2,v_3] \\circ (v_3,r) \\circ [r,v_4] \\circ (v_4, v_5) \\circ [v_5,t]$. The recovery edges $(v_1,v_2), (v_3,r)$ and $(v_4, v_5)$ are shown in dashed lines.  \\label{fig:succ-paths}\n}\n\\end{center}\n\\end{figure}\n\\subsection{Connectivity Labels Based on Graph Sketches}\\label{sec:ftconn-sketch}\n\n\\mtodo{I'm not sure if we want to have the statement of the theorem with respect to a tree. Maybe only have a variant with a tree where it's needed (for the routing)}\n\nIn this section, we show the following:\n\\begin{theorem}\nFor every undirected $n$-vertex graph $G=(V,E)$ and a spanning tree $T \\subseteq G$, a positive integer $f$, there is a randomized $f$-FS connectivity labels $\\mathsf{ConnLabel}\\xspace_{G,T}: V \\cup E \\to \\{0,1\\}^{Q}$ of length $Q=O(\\log^3 n)$ bits. \n\\textbf{MP: add later the computation time of the scheme and the decoding time.}\n\\end{theorem}\nIn Section \\ref{sec:label-alg}, we present the labeling algorithm which assigns labels based on the notion of graph sketches. In Section \\ref{sec:dec-alg} we present the decoding algorithm that given the label information determines if $s$ and $t$ are connected in $G \\setminus F$. When the graph $G$ and the spanning tree $T$ are clear from the context, we may omit it and simply write $\\mathsf{ConnLabel}\\xspace$. \n\n\n\n\\subsubsection{The Labeling Algorithm}\\label{sec:label-alg}\nGiven a graph $G$, let $T$ be an arbitrary rooted tree in $G$ that is used throughout this section.\nFor any node $u \\in V(T)$, let $V_u$ be the subset of nodes in the subtree of $T$ rooted at $u$. The algorithm starts by computing ancestry labels $\\mathsf{ANC}\\xspace(u)$ for all nodes $u$ given the tree $T$ using Lemma \\ref{anc_labels}. Additionally, we assign to all vertices unique ids $\\operatorname{ID}(v)$ between $\\{1,...,n \\}.$\n\n\\paragraph{Extended Edge Identifiers.} Each edge $e$ in $G$ is assigned to a unique identifier $ID_T(e)$ that is made of two parts: a \\emph{distinguishing} part $ID_{1,T}(e)$ (that does not depend on $T$) and a \\emph{logical part} $ID_{2,T}(e)$ that depends on $T$; each of $O(\\log n)$ bits. The distinguishing part $ID_{1,T}(e)$ is defined in a way that guarantees that the XOR of several identifiers does not correspond to a legal identifier of any edge $e \\in G$ w.h.p. The logical part $ID_{2,T}(e)$ contains auxiliary information that aids the decoding algorithm. \nThe computation of the distinguishing parts $ID_{1,T}(e)$ is based on the notion of $\\epsilon$-\\emph{bias} sets \\cite{naor1993small}. The construction is randomized and guarantees that, w.h.p., the XOR of the $ID_{1,T}$ part of each given subset of edges $S \\subseteq E$, for $|S|\\geq 2$, is not a legal $ID_{1,T}$ identifier of any edge.\nLet $\\mathsf{XOR}\\xspace(S)$ be the bitwise XOR of the extended identifiers of edges in $S$, i.e., $\\mathsf{XOR}\\xspace(S)=\\oplus_{e \\in S} \\operatorname{ID}_T(e)$. In addition, let $\\mathsf{XOR}\\xspace_1(S)=\\oplus_{e \\in S} \\operatorname{ID}_{1,T}(e)$.\n\n\\begin{lemma}[Modification of Lemma 2.4 in \\cite{GhaffariP16}]\n\\label{cl:epsbias}\nThere is an algorithm that creates a collection $\\mathcal{I}=\\{\\operatorname{ID}_1(e_1), \\ldots, \\operatorname{ID}_1(e_{M})\\}$ of $M=\\binom{n}{2}$ random identifiers for all possible edges $(u,v)$, each of $O(\\log n)$-bits using a seed $\\mathcal{S}_{ID}$ of $O(\\log^2 n)$ bits. These identifiers are such that for each subset $E' \\subseteq E$, where $|E'|\\neq 1$, we have $\\Pr[\\mathsf{XOR}\\xspace_1(E') \\in \\mathcal{I}] \\leq 1\/n^{10}$. In addition, given the identifiers $\\operatorname{ID}(u), \\operatorname{ID}(v)$ of the edge $e=(u,v)$ endpoints, and the seed $\\mathcal{S}_{ID}$, one can determine $\\operatorname{ID}_1(e)$ in $\\widetilde{O}(1)$ time.\n\\end{lemma}\n\\begin{proof}\nThe lemma is proved in \\cite{GhaffariP16}, the only part that is not discussed there is the time to determine $\\operatorname{ID}_1(e)$ that follows from \\cite{naor1993small}. \nBy Theorem 3.1 of \\cite{naor1993small}, given the seed $\\mathcal{S}_{ID}$ and the edge identifier $e_j=(\\operatorname{ID}(u), \\operatorname{ID}(v))$, determining the $i^{th}$ bit of $\\operatorname{ID}_1(e_{j})$ can be done in $O(\\log n)$ time. Thus, determining all $O(\\log n)$ bits, takes $O(\\log^2 n)$ time. \n\\end{proof}\nThe second part of the identifier $\\operatorname{ID}_{2,T}(e)$ is given by $\\operatorname{ID}_{2,T}(e)=[\\operatorname{ID}(u), \\operatorname{ID}(v), \\mathsf{ANC}\\xspace_T(u), \\mathsf{ANC}\\xspace_T(v)]$. \nThe identifiers of $\\operatorname{ID}_T(u), \\operatorname{ID}_T(v)$ are used in order to verify the validity of the first part of the label $\\operatorname{ID}_1(e)$. When the tree $T$ is clear from the context, we might omit it and simply write $\\operatorname{ID}(e)$. \n\n\\paragraph{Graph Sketches.}\nGraph sketches are a powerful tool to identify outgoing edges \\cite{kapron2013dynamic,ahn2012analyzing}. \\mtodo{I can add some references later, maybe in the related work section} We start by illustrating the intuition behind them. For a vertex $v$, let $\\mathsf{BaseSketch}\\xspace(v)$ be the bitwise xor of all IDs of edges adjacent to $v$. If we take a subset of vertices $S$, and define  $\\mathsf{BaseSketch}\\xspace(S) = \\oplus_{v \\in S} \\mathsf{BaseSketch}\\xspace(v)$, we can see that all edges that have both endpoints in $S$ are cancelled, and we are left with the xor of outgoing edges from $S$. If there is only one such edge, we get its id. In the case there are more outgoing edges, we can use sampling to identify one outgoing edge. We next formalize this idea and show how to use it in our labels. \n\n\\paragraph{Forbidden-Set Labels via Graph Sketches.} \nThe graph sketches are based on random sub-sampling of the graph edges with logarithmic number of scales, i.e., with probability of $2^{-i}$ for every $i \\in \\{1,\\ldots, m\\}$. For our purposes and similarly to \\cite{DuanConnectivityArxiv16,DuanConnectivitySODA17}, we use pairwise independent hash functions to decide whether to include edges in sampled sets. Choose $L=c\\log n$ \npairwise independent hash functions $h_1, \\ldots, h_{L}:\\{0,1\\}^{2\\log n} \\to \\{0, \\ldots, 2^{\\log m}-1\\}$, and for each $i \\in \\{1, \\ldots, L\\}$ and $j \\in [0,\\log m]$, define the edge set \n$$E_{i,j} =\\{ e \\in E ~\\mid~ h_i(e) \\in [0,2^{\\log m-j})\\}~.$$ \nEach of these hash functions can be defined using a random seed of logarithmic length \\cite{TCS-010}. Thus, a \nrandom seed $\\mathcal{S}_h$ of length $O(L \\log n)$ can be used to determine the collection of all these $L$ functions. As observed in \\cite{DuanConnectivityArxiv16,GibbKKT15}, pairwise independence is sufficient to guarantee that for any set $E' \\subset E$ and any $i$, there exists a $j$ such that with constant probability $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ is the name (extended identifier) of one edge in $E'$, for a proof see Lemma 5.2 in  \\cite{GibbKKT15}.\n\\begin{lemma}\\label{lem:hitting-pairwise}\nFor any edge set $E'$ and any $i$, with constant probability there exists a $j$ satisfying that $|E' \\cap E_{i,j}|=1$.\n\\end{lemma}\n\n\nWe also need to be able to tell that a bit string of $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ is a legal edge ID or not.\n\\textbf{MP: all these lemmas and claims can (should) be moved to the subsequent analysis section.}\n\n\\begin{lemma}\nGiven the seed $\\mathcal{S}_{ID}$, one can determine in $\\widetilde{O}(1)$ time if $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ corresponds to a single edge ID in $G$ or not, w.h.p.\n\\end{lemma}\n\\begin{proof}\nLet $X=\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$. Letting $E''=E' \\cap E_{i,j}$, then $X$ can be written as the concatenation of $\\mathsf{XOR}\\xspace_1(E'')$ and $\\mathsf{XOR}\\xspace_2(E'')$. \nUsing the seed and $\\mathsf{XOR}\\xspace_2(E'')$, one can test the validity of $\\mathsf{XOR}\\xspace_1(E'')$.\nThe algorithm detects the case that $|E''| \\geq 2$ as follows. First, in the case that $E''$ is a single edge, $\\mathsf{XOR}\\xspace_2(E'')$ should contain legal ids $\\operatorname{ID}(u),\\operatorname{ID}(v)$. If this is not the case, it follows that $|E''| \\neq 1$. If $\\mathsf{XOR}\\xspace_2(E'')$ contains legal ids $\\operatorname{ID}(u),\\operatorname{ID}(v)$, we use them and the seed $\\mathcal{S}_{ID}$ to determine $\\operatorname{ID}_1(e)$ for $e = (u,v)$, and we check if $\\mathsf{XOR}\\xspace_1(E'')=\\operatorname{ID}_1(e)$. We have two options, either $E'' = \\{e\\}$ is the single edge $e$, in which case $\\mathsf{XOR}\\xspace_1(E'')=\\operatorname{ID}_1(e) \\in \\mathcal{I}$, and the verification succeeds. Otherwise $|E''| \\geq 2$, in which case, from Lemma \\ref{cl:epsbias}, $\\Pr[\\mathsf{XOR}\\xspace_1(E'') \\in \\mathcal{I}] \\leq 1\/n^{10}$, hence w.h.p $\\mathsf{XOR}\\xspace_1(E'') \\neq \\operatorname{ID}_1(e) \\in \\mathcal{I}$ and we identify that $|E''| \\geq 2$.\n\\end{proof}\nFor each vertex $v$ and indices $i,j$, let $E_{i,j}(v)$ be the edges incident to $v$ in $E_{i,j}$. \nThe $i^{th}$ \\emph{basic sketch unit} of each node $v$ is then given by:\n\\begin{equation}\n\\label{eq:vsketch}\n\\mathsf{Sketch}\\xspace_i(v)=[\\mathsf{XOR}\\xspace(E_{i,0}(v)),\\ldots,\\mathsf{XOR}\\xspace(E_{i,\\log m}(v))].\n\\end{equation}\nThe sketch of each node $v$ is defined by a concatenation of $L=\\Theta(\\log n)$ basic sketch units: \n$$\\mathsf{Sketch}\\xspace(v)=[\\mathsf{Sketch}\\xspace_{1}(v),\\mathsf{Sketch}\\xspace_{2}(v), \\ldots\\mathsf{Sketch}\\xspace_{L}(v)]~.$$ \nFor every subset of vertices $S$, let \n$\\mathsf{Sketch}\\xspace(S)=\\oplus_{v \\in S}\\mathsf{Sketch}\\xspace(v).$ \n\nWe are now ready to define the forbidden set \\mtodo{fault-tolerant?} connectivity labels of vertices and edges. \nThe label of each vertex $u$ is given by:\n$$\\mathsf{ConnLabel}\\xspace_{G,T}(u)=\\langle \\mathsf{Sketch}\\xspace(V(T(u))), \\mathsf{Sketch}\\xspace(V), \\mathsf{ANC}\\xspace_T(u), \\mathcal{S}_{ID}, \\mathcal{S}_h\\rangle,$$ \nwhere $\\mathsf{ANC}\\xspace_T(u)$ is the ancestry label of $u$ with respect to the tree $T$.\nThe label $\\mathsf{ConnLabel}\\xspace_{G,T}(e)$ of each \\emph{edge} $e=(u,v)$ is as given by:\n\\begin{equation*}\n    \\mathsf{ConnLabel}\\xspace(e)=\n    \\begin{cases}\n      \\langle \\mathsf{ConnLabel}\\xspace_{G,T}(u), \\mathsf{ConnLabel}\\xspace_{G,T}(v)\\rangle ,& \\mbox{~for~} e \\in T \\\\\n     \\langle ID_T(e) \\rangle,& \\mbox{~Otherwise}.\n    \\end{cases}\n\\end{equation*}\n\n\n\\begin{claim}\nThe label length is $O(\\log^3 n)$ bits.\n\\end{claim}\n\\begin{proof}\nThe label size is dominated by the sketching information $\\mathsf{Sketch}\\xspace(V_u)$, which is made of a concatenation of the bitwise XOR of $O(\\log n)$ basic sketch units $\\mathsf{Sketch}\\xspace_i(u)$. By Eq. (\\ref{eq:vsketch}), each unit has $O(\\log^2 n)$ bits, and thus overall, the label has $O(\\log^3 n)$ bits.\n\\end{proof}\n\n\\mtodo{I think maybe now we want something slightly different, that given a basic sketch unit, we can find with constant probability an outgoing edge? Also, should add a proof.}\n\n\\begin{lemma}\\label{lem:sketch-property}\nFor any subset $S$, given $\\mathsf{Sketch}\\xspace(S)$ one can compute, w.h.p., an outgoing edge $E(S, V \\setminus S)$ if such exists. \n\\end{lemma}\n\n\n\\subsubsection{The Decoding Algorithm} \\label{sec:dec-alg}\nWe next describe the decoding algorithm where given every triplet $s,t, F \\in V \\times V \\times E^f$ along with their labels, it determines whether $s$ and $t$ are connected in $G\\setminus F$, w.h.p. For our decoding algorithm it would actually be sufficient to get as input:\n\\begin{enumerate}[noitemsep]\n\\item the connectivity labels of $s,t$, the labels of the faulty tree-edge $F \\cap E(T)$, and in addition,\n\\item the extended identifiers of the faulty non-tree edges $F \\setminus T$ (which are part of the labels)\\footnote{This property will be important later on for obtaining the compact routing schemes.}. \n\\end{enumerate}\nThe decoding algorithm has four key steps: The first step identifies the at most $f+1$ components $\\mathcal{C}_0=\\{C_1,\\ldots, C_\\ell\\}$ of $T \\setminus F$, as well as the components of $s$ and $t$ in $\\mathcal{C}_0$. The second step uses the label information to compute the sketch value $\\mathsf{Sketch}\\xspace(C_i)$ of each component $C_i \\in \\mathcal{C}_0$. The third step modifies this sketch information into $\\mathsf{Sketch}\\xspace_{G \\setminus F}(C_i)$, by subtracting the information related to the faulty edges. The forth and final step uses the sketch information in order to simulate $L=O(\\log n)$ steps of the Boruvka algorithm. At the end of these steps, the decoding algorithm identifies the connected components of both $s$ and $t$ in $G \\setminus F$. In the case where $s$ and $t$ are indeed connected in $G \\setminus F$, the algorithm also outputs a succinct representation of an $s$-$t$ path in $G \\setminus F$. This extra information would be used later on by our compact routing scheme. We next describe these steps in details. \n\n\\paragraph{Step 1: Identification of the connected components $\\mathcal{C}_0$ in $T \\setminus F$.} \nLet $F_T=F \\cap T$ be the faulty tree edges and let $F_{NT}=F \\setminus F_T$ be the faulty non-tree edges. Let $U=\\{s,t\\} \\cup V(F_T)$.  Each component $C_i$ of $T \\setminus F$ will be identified by the maximum vertex ID in $C_i \\cap U$. Note that the non-tree faulty edges $F_{NT}$ have no impact on the components of $T \\setminus F$ (thus their labels are indeed not needed for that step).\nWe next show that although we do not have full information about the tree $T$ and the vertices of each connected component, the ancestry labels of $V(F_T)$ give us enough information to identify the connected components of $T \\setminus F$. Additionally, given an ancestry label of a vertex $u$, we can identify the connected component of $u$. To obtain this, it is helpful to look at the \\emph{component tree} that is obtained by contracting each connected component of $T \\setminus F$ to one vertex, as follows. Let $\\ell = |F_T|+1.$ The component tree $T_C = (\\mathcal{C}_0, E_C)$ is a tree of $\\ell$ vertices representing the connected components in $T \\setminus F$, and $|F_T|=\\ell-1$ edges corresponding to the edges of $F_T$. There is an edge $\\{C_i,C_j\\} \\in E_C$ iff there is an edge $\\{u,v\\} \\in F_T$ where $u \\in C_i, v \\in C_j$. See Figure \\ref{componentTreePic} for an illustration.  We can construct the tree $T_C$ using the ancestry labels of the edges $F_T$. For this we just need to identify for any edge in $F_T$ the set of edges from $F_T$ above it in $T$. Moreover, for a given vertex $v$, its connected component is exactly determined by the set of edges in $F_T$ above it in $T$, which can again be identified using the ancestry labels of $v \\cup V(F_T)$. In particular, we can identify the connected components of $s$ and $t$. \\mtodo{add the time complexity, we can probably get a faster algorithm if needed.}\n\n\\setlength{\\intextsep}{0pt}\n\\begin{figure}[h]\n\\centering\n\\setlength{\\abovecaptionskip}{-2pt}\n\\setlength{\\belowcaptionskip}{6pt}\n\\includegraphics[scale=0.55]{componentTree.pdf}\n \\caption{Illustration of the component tree where $F=\\{e_1,e_2,e_3,e_4\\}$. Each connected component of $T \\setminus F$ is contracted to one vertex on the right.}\n\\label{componentTreePic}\n\\end{figure}\n \n\n\\paragraph{Step 2: Computing the sketch values of each component $\\mathcal{C}_0$ in $G$.} \nFor each component $C_j \\in \\mathcal{C}_0$ the algorithm computes $\\mathsf{Sketch}\\xspace_G(C_j)$ using the label information of the nodes in $U$.  The basic observation here is the following. Given $S' \\subset S$ and $\\mathsf{Sketch}\\xspace(S), \\mathsf{Sketch}\\xspace(S')$, it holds that $\\mathsf{Sketch}\\xspace(S \\setminus S')=\\mathsf{Sketch}\\xspace(S) ~\\mathsf{XOR}\\xspace~ \\mathsf{Sketch}\\xspace(S')$. To compute the sketch values, first, we define for each component a temporary value $\\mathsf{Sketch}\\xspace'_G(C_j)$ as follows. Let $v_j$ be the highest vertex in the component $C_j$. For the component of the root $r$, this is $r$. For any other component $C_j$, let $\\{C_j,p(C_j)\\}$ be the edge connecting $C_j$ to its parent in the component tree. This edge corresponds to an edge $\\{v,p(v)\\} \\in F_T$, where $v$ is the highest vertex in $C_j$. We define $\\mathsf{Sketch}\\xspace'_G(C_j) = \\mathsf{Sketch}\\xspace_G(V_{v_j})$. Note that this value is part of the label of the vertex $v_j$. For any $v_j \\neq r$, we have $v_j \\in V(F_T)$, and we also learn the identity of $v_j$ when constructing the component tree in Step 1, hence we know $\\mathsf{Sketch}\\xspace'_G(C_j)$. We also know the temporary sketch value of the component of $r$, as $\\mathsf{Sketch}\\xspace_G(V_{r})=\\mathsf{Sketch}\\xspace_G(V)$ is part of the labels of all vertices. We next use the temporary sketch values to compute the sketch values of components using the following claim.\n\n\\begin{claim}\nLet $C_j$ be a component in $T \\setminus F$. If $C_j$ is a leaf in the component tree, we have $\\mathsf{Sketch}\\xspace_G(C_j) = \\mathsf{Sketch}\\xspace'_G(C_j).$ Otherwise, let $D=\\{D_1,...,D_t\\}$ be the children of $C_j$ in the component tree and let $\\mathsf{Sketch}\\xspace'(D)=\\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace'_G(D_i)$, then $\\mathsf{Sketch}\\xspace_G(C_j) = \\mathsf{Sketch}\\xspace'_G(C_j) \\oplus \\mathsf{Sketch}\\xspace'(D).$ \n\\end{claim}\n\n\\begin{proof}\nIt holds that $\\mathsf{Sketch}\\xspace_G(C_j) = \\oplus_{v \\in C_j} \\mathsf{Sketch}\\xspace_G(v)$. By definition, $\\mathsf{Sketch}\\xspace'_G(C_j)=\\mathsf{Sketch}\\xspace_G(V_{v_j}) = \\oplus_{v \\in V_{v_j}} \\mathsf{Sketch}\\xspace_G(v)$ is the xor of sketches of all vertices in the subtree of $v_j$. As $v_j$ is the highest vertex in $C_j$, if $C_j$ is a leaf component in the component tree, then the vertices in $C_j$ are exactly the vertices in $V_{v_j}$, and the claim follows. Otherwise, the vertices in $C_j$ are all vertices in $V_{v_j}$ that are not contained in any component below $C_j$. Hence, to compute the value $\\mathsf{Sketch}\\xspace_G(C_j)$, we should subtract from $\\mathsf{Sketch}\\xspace_G(V_{v_j})$ the sketch values of vertices in components below $C_j$. Let $D_1,...,D_t$ be the children of $C_j$ in the component tree, and let $u_1,...,u_t$ be the highest vertices in the components $D_1,...,D_t$. Any vertex that is in some component below $C_j$ is in exactly one of the subtrees $V_{u_1},...,V_{u_t}$. Hence the sketch value of vertices in components below $C_j$ equals $\\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace_G(V_{u_i})= \\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace'_G(D_i)=\\mathsf{Sketch}\\xspace'(D)$. To conclude, we get $\\mathsf{Sketch}\\xspace_G(C_j)=\\mathsf{Sketch}\\xspace_G(V_{v_j}) \\oplus \\mathsf{Sketch}\\xspace'(D)=\\mathsf{Sketch}\\xspace'_G(C_j) \\oplus \\mathsf{Sketch}\\xspace'(D)$, as needed.\n\\end{proof}\n\nTo conclude, from the values $\\mathsf{Sketch}\\xspace'_G(C_j)$, we can easily compute the values $\\mathsf{Sketch}\\xspace_G(C_j)$. The complexity is $\\tilde{O}(f)$, as for each component, the sketch $\\mathsf{Sketch}\\xspace'(C_j)$ participates in two computations, and we have at most $O(f)$ components and the sketches have poly-logarithmic size.\n\n\n\\paragraph{Step 3: Computing the sketch values of each component $\\mathcal{C}_0$ in $G \\setminus F$.} \nFor each faulty edge $e \\in F$ (both tree and non-tree edges), our goal is to subtract the sketch information of $e$ from the corresponding components of the endpoint of $e$. The step does not require the label information of the edges, and it would be sufficient to know only the seed $\\mathcal{S}_h$ that determines the sampling of edges into the sketches, and the extended identifier of the failing edges. \n\nUsing the extended identifier of the faulty edge $e=(u,v)$, one can determine the components in $\\mathcal{C}_0$ to which its endpoints belong from the ancestry labels of $u$ and $v$, as explained in Step 1. Using the identifier $\\operatorname{ID}(e)$ and the seed $\\mathcal{S}_h$, one can determine all the indices of the sketch to which the edge $e$ was sampled. \\mtodo{what is the complexity of this? MP: It should be $\\widetilde{O}(1)$ using Fact \\ref{fc:pairwise}.}\nLetting $C_u, C_v$ be the components of $u$ and $v$ in $T \\setminus F$, respectively. The values $\\mathsf{Sketch}\\xspace_G(C_u),\\mathsf{Sketch}\\xspace_G(C_v)$ are updated by XORing them with the matrix that contains the identifier $\\operatorname{ID}(e)$ in the relevant positions. \\mtext{The complexity is poly-logarithmic, as the matrix has poly-logarithmic size.}\nFrom that point on, all sketches of the components $\\mathcal{C}_0$ can be treated as sketches that have been computed in $G \\setminus F$.  \n\n\n\\paragraph{Step 4: Simulating the Boruvka algorithm.} Finally, our goal is to determine the identifiers of the maximal connected components of $s$ and $t$ of $G \\setminus F$. The input to this step is the identifiers of the components $\\mathcal{C}_0=\\{C_{1}, \\ldots, C_k\\}$ in $T \\setminus F$, along with their sketch information in $G \\setminus F$. While the algorithm does not have information on the nodes of each component, it knows the component identifier of each node in $U$. \n\nThe algorithm consists of $L=O(\\log n)$ phases of the Boruvka algorithm. Each phase $i \\in \\{1,\\ldots, L\\}$ will be given as input a partitioning $\\mathcal{C}_i=\\{C_{i,1}, \\ldots, C_{i,k_i}\\}$ of (not necessarily maximal) connected components in $G \\setminus F$.\nThese components are identified by an $O(\\log n)$ bit identifier, where for each vertex in $U$, the algorithm receives its unique component identifier in  $\\mathcal{C}_i$. In addition, the algorithm receives the sketch information of the components $\\mathcal{C}_i$ in $G \\setminus F$. The output of the phase is a partitioning $\\mathcal{C}_{i+1}$, along with their sketch information in $G \\setminus F$ and the identifiers of the components for each node in $U$. A component $C_{i,j} \\in \\mathcal{C}_i$ is \\emph{growable} if it has at least one non-faulty outgoing edge to a node in $V \\setminus C_{i,j}$. That is, the component is growable if it is strictly contained in some maximal connected component in $G \\setminus F$. Letting $N_i$ denote the number of growable components in $\\mathcal{C}_i$, the output partitioning $\\mathcal{C}_{i+1}$ of the $i^{th}$ step guarantees that $N_{i+1}\\leq N_i \/2$ w.h.p. To obtain outgoings edges from the growable components in $\\mathcal{C}_i$, the algorithm uses the $i^{th}$ basic-unit sketch $\\mathsf{Sketch}\\xspace_i(C_{i,j})$ of each $C_{i,j} \\in \\mathcal{C}_i$. By Lemma \\ref{lem:sketch-property}, from every growable component in $\\mathcal{C}_i$, we get one outgoing edge $e'=(x,y)$ with constant probability. Using the extended edge identifier of $e'$ the algorithm can also detect the component $C_{i,j'}$ to which the second endpoint, say $y$, of $e'$ belongs. Specifically, this is done using the ancestry label of the detected edge $e'$. That label allows us to compute the component of $y$ in the initial partitioning $T \\setminus F$, i.e., the component $C_{0,q}$ of $y$ in $\\mathcal{C}_0$. Thus $y$ belongs to the unique component $C_{i,j'} \\in \\mathcal{C}_i$ that contains \n$C_{0,q}$. \n\n\n\nAs noted in prior works \\cite{ahn2012analyzing,kapron2013dynamic,DuanConnectivityArxiv16}, it is important to use fresh randomness (i.e., independent sketch information) in each of the Boruvka phases. The reason is that the cut query, namely, asking for a cut edge between $S$ and $V \\setminus S$, should not be correlated with the randomness of the sketches. Note that indeed the components of $\\mathcal{C}_i$ are correlated with the randomness of the first $(i-1)$ basic sketch units of the vertices. Thus, in phase $i$ the algorithm uses the $i^{th}$ basic sketch units of the vertices (which are independent of the other sketch units) to determine the outgoing edges of the components in $\\mathcal{C}_i$.\n\n\nThe algorithm then computes the updated sketches of the merged components. This is done by xoring over the sketches of the components in $\\mathcal{C}_i$ that got merged into a single component in \n$\\mathcal{C}_{i+1}$. In expectation, the number of growable components is reduced by factor $2$ in each phase. Thus after $O(\\log n)$ phases, the expected number of growable components is at most $1\/n^5$, and using Markov inequality, we conclude w.h.p there are no growable components. The final partitioning $\\mathcal{C}_L$ corresponds w.h.p to the maximal connected components in $G \\setminus F$. The pair $s$ and $t$ are connected in $G \\setminus F$ only if the components $C_s,C_t$ of $s,t$ respectively in $T \\setminus F$ are connected in the final component decomposition.\n\\mtodo{add complexity.}\n\nFinally, we show that the decoding algorithm can be slightly modified to output a compressed encoding of an $s$-$t$ path in $G \\setminus F$, using $O(f\\log n)$ bits. This encoding is represented by an $s$-$t$ path $\\widehat{P}$ that has two type of edges, appearing in an alternate manner on  $\\widehat{P}$: $G$-edges and edges $e'=(u,v)$ such that the $u$-$v$ tree path is intact in $T \\setminus F$. See Figure \\ref{fig:succ-paths}. \n\\begin{lemma}\\label{lem:useful-recovery-edges}\nConsider a triplet $s,t,F$ such that $s$ and $t$ are connected in $G \\setminus F$. \nThe decoding algorithm can also output a set of at most $f$ recovery edges $Q$ such $(T \\setminus F) \\cup Q$ is a spanning tree. In addition, it outputs a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of the $s$-$t$ path. The edges of $\\widehat{P}$ are labeled by $0$ and $1$, where $0$-labeled edges correspond to $G$-edges and $1$-labeled edges $e=(x,y)$ correspond to $x$-$y$ paths in $T \\setminus F$. \n\\end{lemma}\n\\begin{proof}\nLet $C_s, C_t$ be the components of $s$ and $t$ in the initial partitioning $\\mathcal{C}_0$. In Step $4$ of the decoding algorithm, the Boruvka algorithm is simulated up to the point that $C_s$ and $C_t$ are connected. Therefore, the algorithm has computed a path $P$ that connects the components $C_s$ and $C_t$. Each node on that path corresponds to a component in $\\mathcal{C}_0$, and each edge corresponds to an outgoing edge (discovered using the sketch information). Since $\\mathcal{C}_0$ has at most $f+1$ components, $|P|\\leq f+1$. \nEach such edge $e' \\in P$ corresponds to an edge in $G$. Let $e_1=(x_1,y_1),\\ldots, e_k=(x_k,y_k)$ be the $G$-edges corresponding to the edges of $P$ ordered from $C_s$ to $C_t$. Letting $y_0=s$ and $x_{k+1}=t$, we get that \n$y_i$ and $x_{i+1}$ belong to the same component in $\\mathcal{C}_0$, for every $i \\in \\{0,\\ldots, k\\}$. \nThe labeled path is given by $\\widehat{P}=[s, x_1,y_1, x_2, y_2, \\ldots y_k,t]$ where the edges $(y_i,x_{i+1})$ are labeled $1$ and the edges $(x_{i}, y_i)$ are labeled $0$. Each $0$-labeled edge is a real edge in $G$, and each $1$-labeled edge $(x_{i}, y_i)$ corresponds to a tree path $\\pi(x_i, y_i)$ in $T \\setminus F$. \n\\end{proof}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.40]{suc-path.pdf}\n\\caption{\\sf Shown is a tree $T$ with faulty edges $e_1,\\ldots, e_4$. The $s$-$t$ path in $G \\setminus F$ is represented by the path $\\widehat{P}=[s,v_1]\\circ (v_1,v_2) \\circ [v_2,v_3] \\circ (v_3,r) \\circ [r,v_4] \\circ (v_4, v_5) \\circ [v_5,t]$. The recovery edges $(v_1,v_2), (v_3,r)$ and $(v_4, v_5)$ are shown in dashed lines.  \\label{fig:succ-paths}\n}\n\\end{center}\n\\end{figure}\n\\section{Fault-Tolerant (FT) Connectivity Labels}\n\n\nWe next discuss two labeling schemes for connectivity that are based on two different approaches. The first one uses the \\emph{cycle space sampling} technique to try to find cuts that disconnect $s$ and $t$. The second one uses \\emph{graph sketches} to try to find a path that connects $s$ and $t$. Since the second approach allows to find a path between $s$ and $t$ if exists, it is also useful later for routing. In terms of label size, the first approach gives labels of size $O(f + \\log{n})$, which is near-optimal if the number of failures is $f=O(\\log{n})$. On the other hand, the second scheme gives labels of size $O(\\log^3{n})$, which is better when the number of failures is large.\nWe next discuss the labeling schemes. During this section, we assume that the input graph $G$ is connected. If not, we can add to the label of each vertex and edge the id of their connected component in $G$, and apply the labeling scheme to each one of the connected components separately. \n\n\\subsection{Connectivity Labels Based on Cycle Space Sampling}\n\n\\subsubsection{The Labeling Algorithm}\n\nOur labels are composed of two ingredients, that we review next.\n\n\\paragraph{Cycle Space Labels.}\nThe cycle space sampling technique, introduced in \\cite{pritchard2011fast}, allows to give the edges of a graph short labels that allow to detect cuts in the graph. For a set of vertices $S$, $\\delta(S)$ is the set of edges with exactly one endpoint in $S$. A subset of edges $F$ is called an \\emph{induced edge cut} if $F = \\delta(S)$ for some $S$.\nThe following is shown in \\cite{pritchard2011fast} (see Corollary 2.9). \n\n \\cycle*\n\n\\remove{\n\\begin{restatable}{lemma}{cycle} \\label{cycle_space_lemma}\nThere is an algorithm that assigns the edges of a graph $G=(V,E)$ $b$-bit labels $\\phi(e)$ such that given a subset of edges $F \\subseteq E$, we have:\n$$Pr[\\Moplus_{e \\in F} \\phi(e) = 0] = \\left\\{\n                \\begin{array}{ll}\n                  1,\\ if\\ F\\ is\\ an\\ induced\\ edge\\ cut\\\\\n                  2^{-b},\\ otherwise\n                \\end{array}\n              \\right. $$ \nWhere $0$ is the all-zero vector. The time complexity for assigning the labels is $O((m+n)b)$.\n\\end{restatable}\n}\n\nFor an overview of the technique, see Appendix \\ref{sec:cycle_space_overview}. \nIn our algorithm, given a subset of edges $F$ of size at most $f$, we want to be able to check for any subset $F' \\subseteq F$ if $F'$ is an induced edge cut. To support all these $2^f$ queries w.h.p we choose $b=f+ c \\log{n}$ for a constant $c$. This guarantees that the probability of error is at most $\\frac{2^f}{2^{f+c\\log{n}}}=\\frac{1}{n^c}$. This will guarantee that given a query $\\langle s,t,F \\rangle$, our algorithm answers correctly w.h.p. We remark that if we increase the size of labels to $O(f \\log{n})$ we can get an algorithm that is correct for \\emph{all} queries w.h.p. \nThe reason is that we can then check for any subset of edges $F$ of size at most $f$ if $F$ is an induced edge cut. As the number of subsets of size at most $f$ is bounded by $O(n^f)$, we get that the labels are correct for all such subsets w.h.p.\n\n\n\\paragraph{Ancestry Labels.} Our second ingredient are ancestry labels for trees.\nTo use them, we first fix a spanning tree $T$ of the graph rooted at $r$. The goal is to assign vertices short labels, such that given the labels of $u$ and $v$, we can infer if $u$ is an ancestor of $v$ in $T$. A simple labeling scheme based on a DFS scan solves the problem with labels of size $2 \\lceil \\log{n} \\rceil$ per vertex \\cite{kannan1992implicat}, the time for assigning the labels is $O(n)$ for the DFS scan of the tree. Labeling schemes with improved label size appear in \\cite{abiteboul2006compact,alstrup2002improved,fraigniaud2010compact,fraigniaud2010optimal}.\n\n\\begin{lemma} \\label{anc_labels}\nFor every tree $T$, there is an algorithm that assigns the vertices $u$ of the tree labels $\\mathsf{ANC}\\xspace_T(u)$ of $O(\\log{n})$ bits, such that given the labels of $u$ and $v$ we can infer if $u$ is an ancestor of $v$ in $T$ in $O(1)$ time. The time for assigning the labels is $O(n)$. \n\\end{lemma}\n\n\\paragraph{The Final Labels.}\nOur final labels contain the following ingredients:\n\\begin{enumerate}\n\\item The label of the edge $e=(u,v)$ is composed of $(\\phi(e),\\mathsf{ANC}\\xspace_T(u),\\mathsf{ANC}\\xspace_T(v),j)$, where $j$ is a bit indicating if $e$ is a tree edge in $T$. In total, the label size is $O(f + \\log{n})$.\n\\item The label of a vertex $v$ is its ancestry label $\\mathsf{ANC}\\xspace_T(v)$ of size $O(\\log{n})$ bits.\n\\end{enumerate}\n\nAs discussed, the time for assigning the labels is $O((m+n)b)=\\widetilde{O}((m+n)f)$, as $b=f+c\\log{n}$.\nWe next explain how we use these labels to check FT connectivity.\n\n\n\\subsubsection{The Decoding Algorithm}\n\nWe next discuss several observations that allow us to check if $s$ and $t$ are disconnected by $F$.\n\n\\begin{claim} \\label{obs_induced}\nThe vertices $s$ and $t$ are disconnected by $F$ if an only if there is an induced edge cut $F' \\subseteq F$ that disconnects $s$ and $t$.\n\\end{claim}\n\n\\begin{proof}\nFirst, if $F' \\subseteq F$ disconnects $s$ and $t$, then clearly $F$ disconnects $s$ and $t$.\nOn the other hand, if $s$ and $t$ are disconnected by $F$, let $F' \\subseteq F$ be a minimal set of edges whose removal disconnects $s$ and $t$. We show that $F'$ is an induced edge cut. Let $V_s$ be the vertices in the connected component of $s$ in $G \\setminus F'$. We show that all edges in $F'$ are between $V_s$ and $V \\setminus V_s$, implying that $F'$ is an induced edge cut. Assume to the contrary that there is an edge $e \\in F'$ with both endpoints in one of the sides, say $V_s$, then $F' \\setminus \\{e\\}$ is still a cut that disconnects $s$ and $t$ (as $V_s$ is still disconnected from the rest of the graph if we add $e$), contradicting the minimality of $F'$. A symmetric argument shows that $e$ cannot have both its endpoints in $V \\setminus V_s$.\n\\end{proof}\n\nWe next show that given an induced edge cut $F'$, there is a simple way to determine the two sides of the cut induced by $F'$ (see Figure \\ref{cutSidesPic} for illustration). For a vertex $v$ and an induced edge cut $F'$, we denote by $n_v(F')$ the number of edges from $F'$ in the path from the root $r$ to $v$ in the spanning tree $T$. We show the following.\n\n\\remove{\n\\begin{claim}\nLet $F'$ be an induced edge cut, and let $T$ be a spanning tree with root $r$. Let $V_0$ be all the vertices $v$ where in the path from $r$ to $v$ there is an even number of edges from $F'$, and let $V_1 = V \\setminus V_0$. Then $(V_0,V_1)$ is the induced edge cut defined by $F'$.\n\\end{claim}\n}\n\n\\begin{claim} \\label{obs_cut_sides}\nLet $F'$ be an induced edge cut. Let $$V_0=\\{v \\in V|\\ n_v(F')\\ is \\ even \\},$$ $$V_1=\\{v \\in V|\\ n_v(F')\\ is \\ odd \\}.$$ Then $(V_0,V_1)$ is the induced edge cut defined by $F'$.\n\\end{claim}\n\n\\begin{proof}\nSince $F'$ is an induced edge cut, the endpoints of every edge in $F'$ are on different sides of the cut. Hence, if we scan the tree $T$ from the root to the leaves, every time we reach an edge from $F'$ we change the side of the cut. It follows that one side of the cut contains all vertices $v$ such that $n_v(F')$ is even, and the other side has all vertices $v$ such that $n_v(F')$ is odd. Hence $V_0,V_1$ are the two sides of the cut.\n\\end{proof}\n\n\\setlength{\\intextsep}{0pt}\n\\begin{figure}[h]\n\\centering\n\\setlength{\\abovecaptionskip}{-2pt}\n\\setlength{\\belowcaptionskip}{6pt}\n\\includegraphics[scale=0.55]{cutSides2.pdf}\n \\caption{Here $F'=\\{e_1,e_2,e_3,e_4\\}$ is an induced edge cut. On the right, you can see the partition into sides in the tree. Every time we reach an edge from $F'$, we change the side of the cut.}\n\\label{cutSidesPic}\n\\end{figure}\n\nFrom Claims \\ref{obs_induced} and \\ref{obs_cut_sides}, we get the following.\n\n\\begin{corollary} \\label{cor_ft_connectivity}\nThe vertices $s$ and $t$ are disconnected by $F$ if an only if there is an induced edge cut $F' \\subseteq F$, such that one of the values $n_s(F'),n_t(F')$ is even and the other is odd. \n\\end{corollary}\n\nThis gives a simple approach to detect if $s$ and $t$ are disconnected by $F$. We go over all subsets $F' \\subseteq F$, for each one of them we first check if $F'$ is an induced edge cut using the cycle space labels. Second, if $F'$ is an induced edge cut, we compute the values $n_s(F'),n_t(F')$, if the number is even for one of them and odd for the second, we deduce that $F'$ disconnects $s$ and $t$. Note that we can use the ancestry labels to compute the values $n_s(F'),n_t(F')$. For example, for computing $n_s(F')$ we should check how many edges in $F'$ are in the tree path between $r$ to $s$. For this, for each tree edge $e=(u,v)$ in $F'$, we check if it is above $s$ in the tree, which happens if and only if both $u$ and $v$ are ancestors of $s$.\nThis simple approach requires time exponential in $|F|$ for going over all subsets of $F$, we next show a faster way to check the same condition.\n\n\\subsubsection{Faster Decoding Algorithm}\n\nWe next show that checking the condition from Corollary \\ref{cor_ft_connectivity} boils down to solving a system of linear equations.\nFirst, note that from Lemma \\ref{cycle_space_lemma}, w.h.p, a set of edges $F' \\subseteq F$ is an induced edge cut iff $\\Moplus_{e \\in F'} \\phi(e) = 0$. Hence, if we want to check if there is a non-empty subset $F' \\subseteq F$ that is an induced edge cut it is equivalent to checking if there exists a binary vector $x=(x_1,...,x_f) \\neq 0$ such that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi(e_i) = 0$, where $\\{e_1,...e_f\\}$ are the edges of $F$. Or equivalently checking if the vectors $\\{\\phi(e)\\}_{e \\in F}$ are linearly dependant. To check the condition from Corollary \\ref{cor_ft_connectivity}, we generalize this idea. \n\nLet $b = O(f + \\log{n})$ be the size of the cycle space labels.\nGiven a triplet $(s,t,F)$, we assign for each edge $e \\in F$, a binary vector $\\phi'(e)$ of length $b+2$, as follows.\n\\begin{enumerate}\n\\item If $e$ is a tree edge which is in the tree path $r-s$ but not in the path $r-t$, then $\\phi'(e)=10\\phi(e)$.\n\\item If $e$ is a tree edge which is in the tree path $r-t$ but not in the path $r-s$, then $\\phi'(e)=01\\phi(e)$.\n\\item In all other cases, $\\phi'(e)=00\\phi(e).$ \n\\end{enumerate}\n\nWe denote by $w_1,w_2$ binary vectors of length $b+2$ such that $w_1=100..0,w_2=010...0$ (all right entries are equal to 0).\nWe show that the condition from Corollary \\ref{cor_ft_connectivity} holds iff there is a binary vector $x=(x_1,...,x_f)$ and $j \\in \\{1,2\\}$ such that $$\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i) = w_j.$$ This holds iff there is a solution to at least one of the linear systems $Ax=w_1,Ax=w_2$, where $A$ is a $(b+2) \\times f$ matrix that has the vectors $\\{\\phi'(e)\\}_{e \\in F}$ as its column vectors, and $x,w_1,w_2$ are column vectors. All operations are modulo 2.\n\n\\begin{lemma}\nWith high probability, the vertices $s$ and $t$ are disconnected by $F$ if an only if there is a binary vector $x=(x_1,...,x_f)$ and $j \\in \\{1,2\\}$ such that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)  = w_j.$\n\\end{lemma}\n\n\\begin{proof}\nWe assume for the proof that the cycle space labels are correct, i.e., a set of edges $F' \\subseteq F$ is an induced edge cut iff $\\Moplus_{e \\in F'} \\phi(e) = 0$. This happens w.h.p from Lemma \\ref{cycle_space_lemma} and the choice of $b=O(f + \\log{n})$.\n\nFirst we show that if $s$ and $t$ are disconnected by $F$, the condition of the lemma holds.\nFrom Corollary \\ref{cor_ft_connectivity}, $s$ and $t$ are disconnected by $F$ iff there is an induced edge cut $F' \\subseteq F$, such that one of the values $n_s(F'),n_t(F')$ is even and the other is odd. Denote by $n'_s(F')$ the number of edges from $F'$ in the $r-s$ tree path that are not in the $r-t$ path, and denote by $n'_t(F')$ the number of edges from $F'$ in the $r-t$ tree path that are not in the $r-s$ path. Note that if one of the values $n_s(F'),n_t(F')$ is even and the other is odd, then also one of $n'_s(F'),n'_t(F')$ is even and the other is odd, as if we denote by $y$ the number of edges from $F'$ that are in both $r-s$ and $r-t$, we get that $n'_s(F') = n_s(F') - y, n'_t(F') = n_t(F') - y$. Assume first that $n'_s(F')$ is even and $n'_t(F')$ is odd. Let $x$ be the characteristic vector of $F'$. We show that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)  = w_2$. First, as $F'$ is an induced edge cut, we have that $\\Moplus_{e \\in F'} \\phi(e) = 0$. Hence, the $b$ last bits of $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)$ are equal to 0 as needed. $F'$ has even number of edges that are in the path $r-s$ and not $r-t$, as the labels $\\phi'(e)$ of all these edges start in $10$, the XOR of the first 2 bits of these edges sums to $00$.  $F'$ has odd number of edges that are in the path $r-t$ but not $r-s$. The labels of all these edges start in $01$, as there is an odd number of them, the XOR of the first 2 bits of these edges sums to $01$. All other edges have labels that start in $00$, hence the XOR of their first 2 bits sums to $00$. Overall we get that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)=\\Moplus_{e \\in F'} \\phi'(e)=010...0=w_2$. The case that $n'_s(F')$ is odd and $n'_t(F')$ is even is symmetric and results in the equation $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)=100...0=w_1.$\n\nOn the other hand, if we have that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)=w_j$ for a binary vector $x=(x_1,...,x_f)$ and $j \\in \\{1,2\\}$, we can build from it $F'$ that satisfies the condition in Corollary \\ref{cor_ft_connectivity}, as follows. We define $F'$ to be all edges $e_i \\in F$ such that $x_i =1$. Since $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)=\\Moplus_{e \\in F'} \\phi'(e)=w_j$, we have that $\\Moplus_{e \\in F'} \\phi(e) = 0$, hence $F'$ is an induced edge cut. Additionally if $w_j=w_2$, it implies that the XOR of the first 2 bits of labels $\\{\\phi'(e)\\}_{e \\in F'}$ are equal to $01$. By the definition of the labels, this can only happen if $n'_s(F')$ is even and $n'_t(F')$ is odd. Similarly, if $w_j=w_1$, then $n'_s(F')$ is odd and $n'_t(F')$ is even. In both cases we get that one of the values $n_s(F'),n_t(F')$ is even and the other is odd, hence $s$ and $t$ are disconnected by $F$ from Corollary \\ref{cor_ft_connectivity}. \n\\end{proof}\n\nTo conclude, the question if $s$ and $t$ are disconnected by $F$ boils down to checking if there is a solution to at least one of the linear systems $Ax=w_1,Ax=w_2$, where $A$ is a $(b+2) \\times f$ matrix, and $b=O(f + \\log{n})$. Note that we can construct the labels $\\phi'(e)$ and hence the matrix $A$ given the labels of $s,t,F$. For this, we need the labels $\\phi(e)$ of edges in $F$, and also to distinguish for each edge in $F$ if it is in the $r-s,r-t$ paths in the tree. The latter can be deduced from the ancestry labels of $s,t,F$ and from the bits indicating which edges in $F$ are tree edges. A tree edge $e=(u,v) \\in F$ is in the $r-s$ path iff both $u$ and $v$ are ancestors of $s$, this can be checked in $O(1)$ time using the ancestry labels of $u,v,s$. Hence we can build the matrix $A$ in $O(fb)$ time. To check if the linear systems have a solution we can use Gaussian elimination, that takes $O(MN^2)$ time for $M \\times N$ matrix, in our case this is $O((f+\\log{n})f^2)$. Alternatively, we can use $O(N^{\\omega})$ algorithms for $N \\times N$ matrices, where $\\omega$ is the exponent of matrix multiplication.\nFor this, we add zero columns to our matrix $A$ to make it a $(b+2) \\times (b+2)$ matrix $A'$ and increase the length of $x$ to $b+2$, the new system $A'x=w_i$ has a solution iff the original system $Ax=w_i$ has a solution. The complexity here is $O((b+2)^{\\omega})=O((f+\\log{n})^{\\omega})$.\nThis gives the following. \n \n\\begin{theorem}\nThere is a randomized $f$-FT connectivity labeling scheme that assigns the edges and vertices of the graph labels of size $O(\\log{n})$ bits per vertex and $O(f + \\log{n})$ bits per edge. The decoding time of the scheme is $\\min\\{O((f+\\log{n})f^2),O((f+\\log{n})^{\\omega})\\}$. The time complexity for assigning the labels is $\\widetilde{O}((m+n)f).$\n\\end{theorem}  \n\n\\remove{\n\\begin{theorem}\nWe can assign the edges and vertices of the graph labels of size $O(\\log{n})$ bits per vertex and $O(f + \\log{n})$ bits per edge, such that given the labels of $(s,t,F)$ we can check if $s$ and $t$ are disconnected by $F$ in $O(f^3 \\log{n})$ \\mtodo{check the complexity} time, w.h.p.\n\\end{theorem} \n} \n\n\\section{Missing Proofs}\\label{sec:miss-proof}\n\n\n\\APPENDUNIQUEID\n\n\\APPENDLEMMUNIQUE\n\n\\APPENDLABELCONSISE\n\n\\APPENDSKETCHPROP\n\n\\APPENDCONNLABELSKETCH\n\\section{Preliminaries}\nGiven a graph $G=(V,E)$, and vertex $u \\in V$, let $\\deg(u,G)$ be the degree of $u$ in $G$. \nGiven a tree $T$ and $u, v \\in T$, denote the $u$-$v$ path in $T$ by $\\pi(u,v,T)$. When the tree $T$ is clear from the context, we may omit it and write $\\pi(u,v)$. For a (possibly weighted) subgraph $G' \\subseteq G$ and a vertex pair $s,t \\in V$, let $\\mbox{\\rm dist}_{G'}(s,t)$ denote the length of the $s$-$t$ shortest path in $G'$. \n\n\\paragraph{Fault-Tolerant Labeling Schemes.}\nFor a given graph $G$, let $\\Pi: V\\times V \\times \\mathcal{G} \\to \\mathbb{R}_{\\geq 0}$\nbe a function defined on pairs of vertices and a subgraph $G' \\subset G$, where $\\mathcal{G}$ is the family of all subgraphs of $G$. For an integer parameter $f\\geq 1$, an $f$-\\emph{fault-tolerant labeling scheme} for a function $\\Pi$ and a graph family $\\mathcal{F}$ is a pair of functions $(L_{\\Pi},D_{\\Pi})$. The function $L_{\\Pi}$ is called the \\emph{labeling function}, and $D_{\\Pi}$ is called the \\emph{decoding function}. For every graph $G$ in the family $\\mathcal{F}$, the labeling function $L_{\\Pi}$ associates with each vertex $u \\in V(G)$ and every edge $e \\in E(G)$, a label $L_{\\Pi}(u,G)$ (resp., $L_{\\Pi}(e,G)$). It is then required that given the labels of any triplets $s,t, F \\in V \\times V \\times E^f$, the decoding function $D_{\\Pi}$ computes $\\Pi(s,t, G \\setminus F)$.  The primary complexity measure of a labeling scheme is the \\emph{label length}, measured by the length (in bits) of the largest label it assigns to some vertices (or edges) in $G$ over all graphs $G \\in \\mathcal{F}$. An $f$-FT connectivity labeling scheme is required to output YES iff $s$ and $t$ are connected in $G \\setminus F$.  In $f$-FT \\emph{approximate distance labeling scheme} it is required to output an estimate for the $s$-$t$ distance in the graph $G \\setminus F$. Formally, an $f$-FT labeling scheme is $q$\\emph{-approximate} if the value $\\delta(s,t,F)$ returned by the decoder algorithm satisfies that $\\mbox{\\rm dist}_{G \\setminus F}(s,t)\\leq \\delta(s,t,F) \\leq q \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$.  Throughout the paper we provide randomized labeling schemes which provide a high probability guarantee of correctness for any fixed triplet $\\langle s,t, F \\rangle$. \n\n\n\\paragraph{Fault-Tolerant Routing Schemes.} In the setting of FT routing scheme, one is given a pair of source $s$ and destination $t$ as well as $F$ edge faults, which are initially unknown to $s$. The routing scheme consists of \\emph{preprocessing} and \\emph{routing} algorithms. The preprocessing algorithm defines labels $L(u)$ to each of the vertices $u$, and a header $H(M)$ to the designated message $M$. In addition, it defines for every vertex $u$ a routing table $R(u)$. The labels and headers are usually required to be short, i.e., of poly-logarithmic bits. \nThe routing procedure determines at each vertex $u$ the port-number on which $u$ should send the messages it receives. The computation of the next-hop is done by considering the header of the message $H(M)$, the label of the source and destination $L(s)$ and $L(t)$ and the routing table $R(u)$. The routing procedure at vertex $u$ might also edit the header of the message $H(M)$. The failing edges are not known in advance and can only be revealed by reaching (throughout the message routing) one of their endpoints. The \\emph{space} of the scheme is determined based on maximal length of message headers, labels and the individual routing tables. The stretch of the scheme is measured by the ratio between the length of the path traversed until the message arrived its destination and the length of the shortest $s$-$t$ path in $G \\setminus F$. In the more relaxed setting of \\emph{forbidden-set routing schemes} the failing edges are given as input to the routing algorithm.\n\n\n\\subsection{Additional Related Work}\n\n\\paragraph{Fault-Tolerant Labeling Schemes.} FT labels for connectivity were introduced by \\cite{courcelle2007forbidden} under the term \\emph{forbidden-set labeling}. Forbidden set refers to a subset $F$ of at most $f$ edges, such that given the labels of $s,t$ and $F$ one should determine if $s$ and $t$ are connected in $G \\setminus F$. The forbidden edge set can be treated in this context as faulty edges\\footnote{For routing, the forbidden-set scheme is slightly weaker than FT scheme as explained later.}.\nPrevious works study FT connectivity labels only in restricted graph families. For example, Courcelle et al. \\cite{CourcelleT07} presented a labeling scheme with logarithmic label length for the families of $n$-vertex graphs with bounded clique-width, tree-width and planar graphs. For $n$-vertex graphs with doubling dimension at most $\\alpha$, Abraham et al. \\cite{AbrahamCGP16} designed FT labeling schemes with label length $O((1 + 1\/\\epsilon)^{2\\alpha}\\log n)$ that output $(1+\\epsilon)$ approximation of the shortest path distances under faults. Recently, \\cite{DBLP:journals\/corr\/abs-2102-07154} studied FT exact distance labels in planar graphs, and show that any directed weighted planar graph admits fault-tolerant distance labels of size $O(n^{2\/3})$.\n\n\\paragraph{Connectivity and Distance Sensitivity Oracles.} \nConnectivity and distance sensitivity oracles are centralized data structures that support connectivity or distance queries in the presence of failures. \nThe first construction of connectivity sensitivity oracles was given by Patrascu and Thorup \\cite{patrascu2007planning} providing an $S(n)=\\widetilde{O}(fn)$ space oracle that answers $\\langle s,t, F \\rangle$ connectivity queries in $\\widetilde{O}(f)$ time. The state-of-the-art bounds of these oracles are given by Duan and Pettie \\cite{DuanConnectivitySODA17}.\nChechik et al. \\cite{chechik2012f} presented the first randomized construction of distance sensitivity oracle resilient to $f$ edge faults. \nSpecifically, for any $n$-vertex weighted graph, stretch parameter $k$, and a fault bound $f$, they provide a data-structure with $O(f k n^{1+1\/k}\\log(nW))$ space, query time of $\\widetilde{O}(|F|)$, and $O(f k)$ stretch, where $W$ is the weight of the heaviest edge in the graph. Their solution is based on an elegant transformation that converts the FT connectivity oracle of \\cite{patrascu2007planning} into an FT approximate distance oracle.\n\nWhile the main focus of this paper is in approximate distances, sensitivity oracles that report (possibly near) exact distances under faults have been studied also thoroughly in e.g., \\cite{demetrescu2002oracles,bernstein2008improved,duan2009dual,WeimannY10,GrandoniW12,ChechikCFK17,van2019sensitive}. Since reporting exact distances requires linear label length already in the fault-free setting \\cite{gavoille2004distance}, we focus on the approximate relaxation, where there is still hope to obtain labels of polylogarithmic length.\n\n\\paragraph{Fault-Tolerant Routing Schemes.}\nThe first formalization of FT routing schemes was given by the influential works of Dolev \\cite{dolev1984new} and Peleg \\cite{peleg1987fault}. These earlier works presented the first non-trivial solutions for general graphs supporting at most $\\lambda$ faulty edges, where $\\lambda$ is the edge-connectivity of the graph. Their routing labels had linear size, providing $s$-$t$ routes of possibly linear length (even in cases where the surviving $s$-$t$ path is of $O(1)$ length). In competitive FT routing schemes, it is required to provide $s$-$t$ routes of length that competes with the shortest $s$-$t$ path in $G \\setminus F$, even in cases where $G \\setminus F$ is not connected. Competitive FT routing schemes \\cite{peleg2009good} for general graphs were given by Chechik et al. \\cite{ChechikLPR10,chechik2012f} for the special case of $f\\leq 2$ faults. \nSpecifically, for a given stretch parameter $k$, they gave a routing scheme with a total space bound of $\\widetilde{O}(n^{1+1\/k})$ bits, polylogarithmic-size labels and messages, and a routing \\emph{stretch} of $O(k)$. \nThis scheme was extended later on for any $f$ by Chechik \\cite{chechik2011fault}, at the cost of increasing the routing stretch to $O(f^2(f+\\log^2 n)k)$. For a single edge failure, \\cite{rajan2012space} showed a routing scheme with routing tables of size $\\widetilde{O}(k \\deg(v)+ n^{1\/k})$ size per vertex, $O(k^2)$ stretch and $O(k+\\log{n})$ size header.\n\n\\paragraph{Forbidden Set Routing.}\nA more relaxed setting of FT routing scheme which has been studied in the literature is given by the \\emph{forbidden set routing schemes}, introduced by Courcelle and Twigg \\cite{CourcelleT07}. In that setting, it is assumed that the routing protocol knows in advance the set of faulty edges $F$. In contrast, in the FT routing setting, the failing edges are a-priori unknown to the routing algorithm,  and can only be detected upon arriving one of their endpoints. Forbidden set routing schemes have been devised to the same class of restricted graph families as obtained for the forbidden set labeling setting \\cite{CourcelleT07,AbrahamCGP16,abraham2012fully}.\n\\subsection{Our Results}\nWe provide space-efficient labeling and routing schemes for any $n$-vertex graph. Our schemes are \\emph{randomized} and provide a high probability guarantee\\footnote{As standard, we use the term high-probability to indicate success guarantee of $1-1\/n^c$ for any given constant $c>1$.} for any given triplet $\\langle s, t, F\\rangle$. In other words, the schemes can faithfully support polynomially many queries\\footnote{The same type of guarantee is provided in the centralized sensitivity oracles, e.g., of \\cite{DuanConnectivitySODA17}. Providing a high probability guarantee over all possible triplets is possible upon increasing the space bound by a factor of $f$ (largest number of faults supported).}. \n\nOur first key result presents two independent schemes for FT connectivity labels. These are the first FT connectivity labels for \\emph{general graphs}.\nThese two constructions yield the following theorem, addressing Question \\ref{q:label}: \n\n\\begin{theorem}\\label{thm:conn-labels}[FT Connectivity Labeling Schemes, Informal]\nFor any $n$-vertex graph and a bound $f$ on the number of edge faults, there is a \\emph{randomized} $f$-FT connectivity labeling scheme with label length of $O(\\min\\{f+\\log n, \\log^3 n\\})$ bits. The labels are computed in $\\widetilde{O}(m)$ time, and the decoding algorithm takes $\\operatorname{\\text{{\\rm poly}}}(f,\\log n)$ time. \n\\end{theorem}\n\nBy the tightness of the label length of fault-free connectivity labels,\nour scheme is optimal for $f=O(\\log n)$. Moreover, the label length is nearly-optimal for any $f$. Our actual scheme provides more information then merely a single bit (connected or not connected). Specifically, we augment the connectivity labels with additional information so that the decoding algorithm, given the labels of $s,t$ and $F$, can also output a succinct description of an $s$-$t$ path in $G\\setminus F$ (if such a path exists). This succinct path representation finds applications in the context of our FT routing schemes. \n\nWe next consider the task of reporting also approximate $s$-$t$ distances in $G\\setminus F$ using the labels of $s,t$ and $F$. We employ the reduction of Chechik et al. \\cite{chechik2012f} to convert the FT connectivity labels into FT approximate distance labels, providing nearly the same space vs. stretch tradeoff as in the centralized data-structures of \\cite{chechik2012f}. Specifically, we show:\n\n\\begin{theorem}\\label{thm:dist-labels}[FT Approximate Distance Labeling Schemes]\nFor any $n$-vertex (possibly weighted) graph, a bound $f$ on the number of edge faults, and a stretch parameter $k$, there is a randomized $f$-FT approximate distance labeling scheme with label length of $O(k \\cdot n^{1\/k}\\cdot \\log (nW) \\cdot \\log^3 n)$. Given the labels of $s,t$ and $F$ the scheme returns a distance estimate \n$$\\mbox{\\rm dist}_{G\\setminus F}(s,t)\\leq \\delta(s,t,F) \\leq (8k-2)(|F|+1)\\mbox{\\rm dist}_{G\\setminus F}(s,t)~.$$\n\\end{theorem}\n\nFor the purpose of routing, we exploit the extra information provided by our connectivity labels, in order to output, in addition to the distance estimate $\\delta(s,t,F)$, also a succinct description of the approximate $s$-$t$ \nshortest path in $G \\setminus F$. Our second key result provides FT compact routing schemes, with an almost optimal tradeoff between the space and stretch, for constant number of faults $f$. We answer Question \\ref{q:route} by showing:\n\n\\begin{theorem}\\label{thm:routing}[FT Compact Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$ and the routing label of the destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The routing labels have $\\widetilde{O}(f)$ bits, the table size of each vertex is $\\widetilde{O}(f^3 \\cdot n^{1\/k} \\log(nW))$, the header size (also known as message size) is bounded by $\\widetilde{O}(f^3)$ bits. \n\\end{theorem}\nThis improves over the state-of-the-art construction of Chechik \\cite{chechik2011fault} that \nobtained routing schemes with stretch of $O(f^2(f+\\log^2 n)k)$ and tables of size $O(\\deg(v)n^{1\/k}\\log{(nW)})$ for every vertex $v$. We note that the construction of Chechik \\cite{chechik2011fault} has a bounded global space of $\\widetilde{O}(n^{1+1\/k} \\log{(nW)})$, but the individual tables might have even super-linear space (e.g., when $k=O(1)$ and $\\deg(v)=O(n)$).  For the special case of $f=2$, Chechik et al. \\cite{ChechikLPR10,chechik2012f} provide a stretch bound of $O(k)$, and total space of $\\widetilde{O}(n^{1+1\/k} \\log{(nW)})$, where the space of each table is bounded by $O(\\deg(v)n^{1\/k})$, thus super-linear in the worst case. Our scheme provides an improved bound on the individual tables, nearly matching the fault-free constructions for $f=O(1)$. We also show an improved scheme if one only aims to optimize for the global space, rather than optimizing for the largest table size for a vertex. For comparison of our results to prior work see Table \\ref{table_routing}.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{ |p{4.2cm}|p{4cm}|p{6cm}|p{0.5cm}|}\n \\hline\t\n \\multicolumn{4}{|c|}{Constructions of Fault-Tolerant Routing Schemes}\\\\\n \\hline\n  Reference & Stretch & Table Size & $|F|$ \\\\\n \\hline\n  Rajan \\cite{rajan2012space} & $O(k^2)$ & $\\widetilde{O}(k \\deg(v)+ n^{1\/k})$ per vertex & 1\\\\\n  Chechik et al. \\cite{chechik2012f} & $O(k)$ & $\\widetilde{O}(n^{1+1\/k} \\log(nW))$ total size & 2\\\\\n  Chechik \\cite{chechik2011fault}  & $O(|F|^2(|F|+\\log^2 n)k)$ & $\\widetilde{O}(n^{1+1\/k} \\log(nW))$ total size & $f$\\\\\n  Chechik \\cite{chechik2011fault}  & $O(|F|^2(|F|+\\log^2 n)k)$ & $\\widetilde{O}(\\deg(v)n^{1\/k} \\log(nW))$ per vertex & $f$\\\\\n  \\textbf{Here} & $O(|F|^2 k)$ & $\\widetilde{O}(f \\cdot n^{1+1\/k} \\log(nW))$ total size & $f$\\\\\n  \\textbf{Here} & $O(|F|^2 k)$ & $\\widetilde{O}(f^3 \\cdot n^{1\/k} \\log(nW))$ per vertex & $f$\\\\\n \\hline\n\n\n \\end{tabular}\n \\caption{Comparison between FT routing schemes with a set of failures $F$}\n\\label{table_routing}\n\\end{table}\n\nFinally, we provide a lower bound result on the minimal stretch regardless for the \\emph{space} of the routing scheme, e.g., even if all vertices store all the graph edges. \n\n\\begin{theorem}[Stretch Lower-Bound for FT Routing]\\label{thm:lb-routing}\nAny FT routing randomized scheme resilient to $f$ faults induces an expected stretch of $\\Omega(f)$ regardless of the size of the routing tables and labels. In particular, this holds even if each routing table contains a complete information on the graph. \n\\end{theorem}\n\n\n\\paragraph{Open Problems.} Our work leaves several interesting open ends. One natural direction is to provide labeling and routing schemes resilient to \\emph{vertex} faults. The major challenge in handling vertex failure is that even a single faulty vertex might disconnect the graph into $\\Omega(n)$ disconnected components. Another interesting direction is to derandomize our constructions. Currently there are no deterministic constructions of FT labeling schemes for general graphs. Finally, it will be also important to provide FT distance approximate labeling schemes whose stretch bound is independent in the number of faults $f$. This problem is also open in the corresponding setting of approximate distance sensitive oracles. \n\n\\section{Compact Routing Schemes}\nIn this section, we explain how to use our FT distance labels to provide compact and low stretch routing schemes. This is the first scheme to provide an almost tight tradeoff between the space and the multiplicative stretch, for a constant number of faults $f=O(1)$.  Throughout this section, tree routing operations are performed by using the tree routing scheme of Thorup and Zwick \\cite{thorup2001compact}.\n\\begin{fact}\\label{fc:route-trees}[Routing on Trees]\\cite{thorup2001compact}\nFor every $n$-vertex tree $T$, there exists a routing scheme that assigns each vertex $v \\in V(T)$ a label $L_T(v)$ of $(1+o(1))\\log n$ bits. Given the label of a source vertex\nand the label of a destination, it is possible to compute, in constant time, the port number of the edge from the source that heads in the direction of the destination.\n\\end{fact}\n\n\n\nWe slightly modify the connectivity label of the edges and vertices by augmenting them with routing information. \nFirst, we augment the extended identifier of an edge (see Eq. (\\ref{eq:extend-ID})) with port information and tree routing information, by having:\n\\begin{equation}\\label{eq:edge-extended-routing}\n\\operatorname{EID}_T(e)=[\\operatorname{UID}(e), \\operatorname{ID}(u), \\operatorname{ID}(v), \\mathsf{ANC}\\xspace_T(u), \\mathsf{ANC}\\xspace_T(v), \\mbox{\\tt port}(u,v), \\mbox{\\tt port}(v,u), L_T(u), L_T(v)]~,\n\\end{equation}\nwhere $\\mbox{\\tt port}(u,v)$ is the port number of the edge $(u,v)$ for $u$, and the labels $L_T(u), L_T(v)$ are the tree routing labels taken from Fact \\ref{fc:route-trees}. \nWe then slightly modify the connectivity label of Eq. (\\ref{eq:conn-vertex}) to include also the tree label $L_T(u)$from Fact \\ref{fc:route-trees}, by defining \n\\begin{equation}\\label{eq:conn-vertex-label-routing}\n\\mathsf{ConnLabel}\\xspace_{G,T}(u)=\\langle \\mathsf{ANC}\\xspace_T(u), \\operatorname{ID}(u), L_T(u)\\rangle~.\n\\end{equation}\n\nThroughout this section, when applying the connectivity labels from Section \\ref{sec:ftconn-sketch} on a graph $G$ with a spanning tree $T$, we use these modified extended identifiers and labels. This will also be the basis for the application of the distance labels of Section \\ref{sec:ft-distance}. \nSimilarly to the distance labels of Section \\ref{sec:ft-distance}, we will apply the connectivity labels with respect to the different trees of the tree cover as discussed in Section \\ref{sec:ft-distance}. \nLet $T_{i,j} \\in \\mathsf{TC}\\xspace_i$, recall that $G_{i,j}=G[V(T_{i,j})]$ and that $\\mathcal{T}=\\bigcup_{i=1}^K \\mathsf{TC}\\xspace_i$ for $K=O(\\log (nW))$. \n\n\\begin{lemma}\\label{lem:succint_path_routing}\nConsider a triplet $s,t,F$ such that $s,t,F \\in G_{i,j}$. \\\\\nGiven the connectivity labels $\\{\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(w)\\}_{w \\in F \\cup \\{s,t\\}}$, we can determine w.h.p if $s$ and $t$ are connected in $G_{i,j} \\setminus F$. If they are connected, we can output a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of the $s$-$t$ path in $G_{i,j} \\setminus F$. The edges of $\\widehat{P}$ are labeled by $0$ and $1$, where $0$-labeled edges correspond to $G_{i,j}$-edges and $1$-labeled edges $e=(x,y)$ correspond to $x$-$y$ paths in $T_{i,j} \\setminus F$. For each $G_{i,j}$-edge, the succinct path description has the port information of the edge, and for each $x-y$ path, the description has the tree routing labels $L_{T_{i,j}}(x),L_{T_{i,j}}(y)$.\nThe length of the $s$-$t$ path encoded by $\\widehat{P}$ is bounded by $(4k-1)(|F|+1)\\cdot 2^i$. \n\\end{lemma}\n\n\\begin{proof}\nThe proof follows the proof of Lemma \\ref{lem:useful-recovery-edges}.\nUsing $\\{\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(w)\\}_{w \\in F \\cup \\{s,t\\}}$, the decoding algorithm of Section \\ref{sec:ftconn-sketch} determines if $s$ and $t$ are connected in $G_{i,j} \\setminus F$. If they are connected, then from Lemma \\ref{lem:useful-recovery-edges}, we get a succinct description of the $s$-$t$ path in $G_{i,j} \\setminus F$. We next show that the algorithm indeed has the relevant port and tree routing information. For this note that all the vertices in the path $\\widehat{P}$ obtained by Lemma \\ref{lem:useful-recovery-edges} are either $s$ and $t$ or endpoints of the $|F|$ recovery edges found in the algorithm. The labels of $s$ and $t$ contain the tree routing information $L_{T_{i,j}}(s)$ and $L_{T_{i,j}}(t)$, and when the algorithm finds a recovery edge, it learns about its extended id $\\operatorname{EID}_{T_{i,j}}(e)$ that has the port information and tree routing information of its endpoints. Any $G_{i,j}$-edge in $\\widehat{P}$ is a recovery edge, hence the algorithm has its port information, and for any $x$-$y$ path in $T_{i,j} \\setminus F$, the algorithm has the tree routing labels $L_{T_{i,j}}(x),L_{T_{i,j}}(y)$, as needed.\nThe stretch analysis follows the stretch analysis in Section \\ref{sec:ft-distance}. It is based on the fact that $\\widehat{P}$ has as most $|F|+1$ subpaths in $T_{i,j} \\setminus F$, each of length at most $(4k-2)2^i$, and at most $|F|$ recovery edges of weight at most $2^i$.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Forbidden Set Routing (Faulty Edges are Known)}\\label{sec:routing-known}\nWe start by describing the routing scheme in the forbidden set setting, where the faulty edges $F$ are known to the source vertex $s$. We show the following.\n\n\\begin{theorem}\\label{thm:routing-known}[Forbidden-Set Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$, a label of the destination $t$, and labels of at most $f$ forbidden edges $F$ (known to $s$), routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $(8k-2)(|F|+1)\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The table size of each vertex is bounded by $\\widetilde{O}(n^{1\/k} \\log{(nW)})$. The header size of the messages is bounded by $\\widetilde{O}(f)$ bits. The labels of vertices and edges have size $\\widetilde{O}(n^{1\/k} \\log(nW))$.\n\\end{theorem}\n\n\\begin{proof}\nThe algorithm is based on the distance labels from Section \\ref{sec:ft-distance} using the slightly modified connectivity labels (augmented with port and tree roting information). Recall that the distance labels are based on applying fault-tolerant connectivity labels on different graphs $G_{i,j}$, we use the slightly modified connectivity labels and the corresponding distance labels. \nThe routing table of each vertex $u$ consists of its distance label $\\mathsf{DistLabel}\\xspace(u)$. The label of an edge $e$ is $\\mathsf{DistLabel}\\xspace(e)$. Each distance label has $\\widetilde{O}(n^{1\/k} \\log(n W))$ bits. \n\nIn the routing algorithm, the vertex $s$ is given the label $\\mathsf{DistLabel}\\xspace(t)$, and the labels $\\{\\mathsf{DistLabel}\\xspace(e)\\}_{e \\in F}$, and it needs to route a message to $t$ in the graph $G \\setminus F$. \nRecall that the algorithm from Section \\ref{sec:ft-distance} works in $K$ phases, where in phase $i$ it checks if $s$ and $t$ are connected in the graph $G_{i,i^*(s)} \\setminus F$ that contains the $2^i$-ball around $s$. Let $i$ be the first iteration where $s$ and $t$ are connected in $G_{i,i^*(s)} \\setminus F$ according to the algorithm, and denote $G_i = G_{i,i^*(s)},T_i = T_{i,i^*(s)}$, and let $F_i = F \\cap G_i$. The algorithm can also give a succinct description of an $s$-$t$ path in $G_i \\setminus F_i$ following Lemma \\ref{lem:succint_path_routing}. For this, note that we indeed have all the required information. The distance labels of edges in $F$ in particular contain the labels $\\{\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)\\}_{e \\in F_i}$, and we can also tell which edges of $F$ are in $G_i$ from the labels. Also, the labels of $s,t$ contain the information $\\operatorname{ID}_{T_i}(s),\\operatorname{ID}_{T_i}(t)$ if they are both in $T_i$ (otherwise, they are not connected in level $i$).\n\nThe path $\\widehat{P}$ as described in Lemma \\ref{lem:succint_path_routing} is composed of $O(|F|)$ parts, where segment $(x,y)$ in the path corresponds either to an edge in $G_i$ or to a tree path in $T_i \\setminus F$, it also has the relevant port and tree routing information. Our goal is to route a message according to this path. For this we add to the header of the message the description of $\\widehat{P}$, the indexes $(i,i^*(s))$ of the tree we explore and an index $1 \\leq q \\leq 2|F|+1$ that represents the segment of $\\widehat{P}$ we currently explore, initially $q=1$. Overall, the header size is $\\widetilde{O}(f)$. To route a message according to the path, we work as follows. The header specifies the current segment in $\\widehat{P}$. If the current segment corresponds to an edge $(x,y) \\in G$, then $x$ uses the port information to route the message to $y$ and increases the index $q$. Otherwise, the current segment represents a tree path $(x,y) \\in T_i$ and a vertex $u$ in this path uses its routing label in $T_i$ and the routing label of $y$ in $T_i$ (that is part of the header) to route the message towards $y$. When the message reaches $y$, it increases the index $q$. This completes the description of the routing process. The length of the path described is at most $(8k-2)(|F|+1)\\cdot \\mbox{\\rm dist}_{G\\setminus F}(s,t)$, as shown in Section  \\ref{sec:ft-distance}. \n\\end{proof}\n\n\n\n\\subsection{Fault-Tolerant Routing (Faulty Edges are Unknown)}\\label{sec:route-unknown}\nWe now consider the more involved setting where the set of failed edges $F$ are unknown to $s$. In this case, an edge $(u,v) \\in F$ is detected only when the message arrives, during the routing procedure, to one of the endpoints of $e$. Note that the routing scheme should, by definition, be prepared to any set of faulty edges $F$. However, the space bound of our scheme is required to be bounded by $\\widetilde{O}(f n^{1+1\/k})$, which is possibly much smaller than the number of graph edges $m$. This in particular implies that we cannot store the FT distance labels of all the graph edges. Nevertheless, we show that it is sufficient to explicitly store the labeling information for the tree edges in $\\mathcal{T}=\\bigcup_{i=1}^K \\mathsf{TC}\\xspace_i$. The required information for the failed non-tree edges would be revealed throughout the process, by applying the decoding algorithm of Lemma \\ref{lem:succint_path_routing}.\nOur routing scheme eventually routes the message along the $s$-$t$ path encoded by the FT distance labels of $s,t$ and $F$. However, since the labels of $F$ are unknown in advance, the routing scheme will detect these edges in a trail and error fashion which induces an extra factor of $f$ in the final multiplicative stretch. This extra $f$ factor is also shown to be essential, in the end of the section.\nWe proceed by describing the routing tables.  \n\n\n\n\\paragraph{The routing labels and tables.} For ease of presentation, we first describe a solution with a multiplicative stretch of $O(kf^2)$, and \\emph{global} space of $\\widetilde{O}(f K \\cdot n^{1+1\/k})$, but the individual tables of some of the vertices might be large. We later on improve the space of each table to $\\widetilde{O}(f^3 K \\cdot n^{1\/k})$ bits.\n\nRecall that $\\mathcal{T}=\\bigcup_i^{K} \\mathsf{TC}\\xspace_i$, for $K=O(\\log (nW))$ is a collection of tree covers in all $K=\\lceil \\log (nW) \\rceil$ distance scales, see Eq. (\\ref{eq:TC-i}). For every vertex $v$, let $\\deg_{\\mathcal{T}}(v)=\\sum_{T_{i,j} \\in \\mathcal{T}}\\deg(u,T_{i,j})$ be the sum of degrees of $u$ in the collection of trees  $\\mathcal{T}$. Recall that $G_{i,j}=G[V(T_{i,j})]$.\nFor the routing we apply the FT connectivity labels on the graphs $G_{i,j}$, similarly to Section \\ref{sec:ft-distance}. \n\\\\ \\\\\n\\noindent \\textbf{Routing labels.} The routing process uses at most $f'=f+1$ independent applications of randomized FT connectivity labels from Section \n\\ref{sec:ftconn-sketch}, applied on each one of the graphs $G_{i,j}$. \nIn more details, when we apply the labeling scheme on the graph $G_{i,j}$ with spanning tree $T_{i,j}$, we use $f'$ independent random seeds $\\mathcal{S}_h$ to determine the randomness of the sketches. \nHowever, the seed $\\mathcal{S}_{ID}$ used to determine the extended ids of edges in $G_{i,j}$ is fixed in the $f'$ applications, hence the extended identifiers of the edges (see Eq. (\\ref{eq:extend-ID})) are fixed in all the $f'$ applications, and we only use fresh randomness to compute the sketch information using $f'$ independent seeds $\\mathcal{S}^1_h,\\ldots, \\mathcal{S}^{f'}_h$. \nThis process is done independently on each one of the graphs $G_{i,j}$. \n\nDenote the output connectivity labels obtained by the ${\\ell}^{th}$ application of the scheme (using $\\mathcal{S}^\\ell_h$) on the graph $G_{i,j}$ by $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_{i,j},T_{i,j}}(w)$ for every $w \\in E(G_{i,j})\\cup V(G_{i,j})$. For every edge $e \\in G_{i,j}$, define its $T_{i,j}$ routing label by\n\\begin{equation}\\label{eq:route-edge-label}\n    L_{route,i,j}(e)=\n    \\begin{cases}\n      (\\mathsf{ConnLabel}\\xspace^1_{G_{i,j},T_{i,j}}(e),\\ldots,\\mathsf{ConnLabel}\\xspace^{f'}_{G_{i,j},T_{i,j}}(e)),& \\mbox{~for~} e \\in T_{i,j} \\\\\n     \\operatorname{EID}_{T_{i,j}}(e),& e \\in G_{i,j}\\setminus E(T_{i,j})~.\n    \\end{cases}\n\\end{equation}\nEvery $L_{route,i,j}(e)$ label has $O(f \\log^3 n)$ bits. \nIn our routing algorithms, the $T_{i,j}$ routing labels of the discovered faulty edges will be added to the header for the message in order to guide the routing process. \nWe now turn to define the routing labels of vertices. Recall that for a vertex $v$ and index $1 \\leq i \\leq K$, we denote by $i^*(v)$ an index such that the $2^i$-ball around $v$ is contained in $G_{i,i^*(v)}$.\nThe routing label $L_{route}(v)$ of $v$ \nFor every \\emph{vertex} $v$, the routing label of $v$ is given by\n\\begin{equation}\\label{eq:Label-route-vertex}\nL_{route}(v) = \\{(i^*(v), \\mathsf{ConnLabel}\\xspace^1_{G_{i,i^*(v)},T_{i,i^*(v)}}(v) | i \\in [1,K]\\}~.\n\\end{equation}\n\n\nNote that by definition, the connectivity labels of the \\emph{vertices} are the same in all $f'$ applications of the labeling algorithm, and therefore it is sufficient to include only one of these copies in the label. The size of the label is $O(K\\log{n})=O(\\log{n} \\log{nW})$.\n\\\\ \\\\\n\\noindent \\textbf{Routing tables.} The routing table $R_{route}(v)$ of a vertex $v$ has the following information for every tree \n$T_{i,j}$ such that $v \\in T_{i,j}$:\n\\begin{equation}\\label{eq:route-table-ij}\nR_{route,i,j}(v)=\\{L_{route,i,j}(e), e \\in E(v,T_{i,j})\\} \\cup \\{\\mathsf{ConnLabel}\\xspace^1_{G_{i,j},T_{i,j}}(v)\\}~,\n\\end{equation}\nwhere $E(v,T_{i,j})$ is the set of edges incident to $v$ in the tree $T_{i,j}$. \nThe final routing table is given by $R_{route}(v)=\\{R_{route,i,j}(v), (i,j) ~\\mid~ T_{i,j} \\in \\mathcal{T}, v\\in T_{i,j}\\}$.\n\nSince the connectivity labels are of size $\\widetilde{O}(f)$, and as each $v$ appears in $\\deg_{\\mathcal{T}}(v)$ trees, the size of the table is $\\widetilde{O}(f \\deg_{\\mathcal{T}}(v)).$ Since the total number of tree edges in $\\mathcal{T}$ is bounded by $\\widetilde{O}(K \\cdot n^{1+1\/k})$, this provides a global space bound of $\\widetilde{O}(fK \\cdot n^{1+1\/k})$ bits.\n\n\n\\paragraph{The routing algorithm.} In the routing algorithm, the source vertex $s$ initially gets the routing label $L_{route}(t)$ (Eq. (\\ref{eq:Label-route-vertex})) of the destination $t$ and its own routing table, $R_{route}(s)$, and its goal is to find the smallest radius graph $G_{i,j}$ such that $s$ and $t$ are connected in $G_{i,j} \\setminus F$, and use it for routing. \nAs the set $F$ is \\emph{not} known in advance, the algorithm works in $K= O(\\log{nW})$ phases, where in phase $i$ it tries to route a message in the graph $G_{i,i^*(t)}$ (which contains the entire $2^i$-radius ball of $t$). If $s$ and $t$ are connected in $G_{i,i^*(t)} \\setminus F$ the algorithm succeeds, and otherwise we proceed to the next phase, corresponding to the distance scale of $2^{i+1}$. \nWe next describe the algorithm for a single phase $i$, we denote $G_i = G_{i,i^*(t)}, T_i = T_{i,i^*(t)}$. Note that $s$ can deduce the index $i^*(t)$ from the routing label of $t$, and it can check if $s \\in T_i$ using its routing table. If $s \\not \\in T_i$, we proceed to the next phase.\n\nIf $s \\in T_i$, the routing procedure for phase $i$ has at most $|F|+1$ iterations. We maintain the following invariant in the beginning of each iteration $\\ell \\in \\{1,\\ldots, |F|+1\\}$: (i) the iteration starts at vertex $s$, (ii) the algorithm has already detected a subset of $\\ell-1$ faulty edges $F_\\ell \\subseteq F$, and (iii) the header contains the labels $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$ of all the edges $e \\in F_\\ell$. Each iteration $\\ell \\leq |F|+1$ terminates either at the destination vertex $t$, or at the source vertex $s$. In addition, w.h.p., if $s$ and $t$ are connected in $G_i \\setminus F$, iteration $|F|+1$ terminates at $t$. The invariant holds vacuously for iteration $1$.\n\nWe now describe the $\\ell^{th}$ iteration (of the $i^{th}$ phase) of the routing procedure given the invariant. The source vertex $s$ considers the $\\ell^{th}$ copy of the FT connectivity labels, $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_i,T_i}(e)$ for every $e\\in F_\\ell$. \nUsing the routing labels of the edges, that are part of the header, the routing label $L_{route}(t)$ (of Eq. (\\ref{eq:Label-route-vertex})) and the routing table $R_{route}(s)$, $s$ can apply the decoding algorithm of \nLemma \\ref{lem:succint_path_routing} to determine if $s$ and $t$ are connected in $G_i \\setminus F_\\ell$. \nIf the answer is no, the algorithm proceeds to the next phase $i+1$.\nOtherwise, by applying the decoding algorithm of Lemma \\ref{lem:succint_path_routing}, it computes the succinct path $\\widehat{P}_\\ell$. The path $\\widehat{P}_\\ell$ encodes an $s$-$t$ path in $G_i \\setminus F_\\ell$, that includes the relevant port and tree routing information of its vertices. The header of the message $H_\\ell$ then contains \n$$H_\\ell=\\langle \\widehat{P}_\\ell, i, i^*(t), \\{L_{route, i, i^*(t)}(e)\\}_{e \\in F_{\\ell}}, q \\rangle~,$$ where $q = O(f)$ is an index indicating the current segment of $\\widehat{P}_\\ell$ we explore. Note that the header $H_\\ell$ contains the $f$ copies of connectivity labels of the $F_{\\ell}$ edges, and not only the $\\ell^{th}$ copy.\nThe size of the header is $\\widetilde{O}(f^2)$, as the description of the path has size $\\widetilde{O}(f)$, and additionally we have at most $f$ faulty edges with labels of size $\\widetilde{O}(f)$.\nLet $P_\\ell$ be the $G$-path encoded by the path $\\mathcal{P}_\\ell$. The algorithm then routes the message along $P_\\ell$ in the same manner as in Sec. \\ref{sec:routing-known}. In the case where $P_{\\ell}\\cap F=\\emptyset$, the iteration successfully terminates at the destination vertex $t$. From now on, we consider the case that $P_{\\ell}$ contains at least one faulty edge. \n\nLet $e=(u,v)$ be the first edge (closest to $s$) on the path $P_\\ell$ that belongs to $F$. Since $P_\\ell \\cap F_\\ell=\\emptyset$, it holds that $e \\in F \\setminus F_\\ell$. Without loss of generality, assume that $u$ is closer to $s$ on $P_\\ell$. Thus the faulty edge $e$ is detected upon arriving to the vertex $u$. \nIn the case where $e$ is a \\emph{non-tree edge}, then it must be a $G$-edge on $\\widehat{P}_\\ell$. Since this path has the extended ids $\\operatorname{EID}_{T_i}(e)$ of its $G$-edges, and since the connectivity label of a non-tree edge $e$ is its extended identifier $\\operatorname{EID}_{T_i}(e)$ in all the $f'$ applications of the scheme on $G_i$\\footnote{This is because we use the same random seed $\\mathcal{S}_{ID}$ in all these applications.}, $u$ can add $L_{route,i,i^*(t)}(e)=\\operatorname{EID}_{T_i}(e)$ to the header of the message. Assume now that $e$ is a tree edge in $T_i$. The vertex $u$ then adds the routing label $L_{route,i,i^*(t)}(e)$ to the header of the message, as $e$ is a tree edge adjacent to $u$ it has this information in its routing table. Finally, it marks the header with the sign $R$, indicating that the message should now be routed in the reverse direction, until arriving $s$ again. This completes the description of iteration $\\ell$. It is easy to see that the invariant is maintained. If $s$ and $t$ are connected in $G_i \\setminus F$, after at most $f$ iterations all faulty edges are detected. In the last iteration, the path computed based on the labeling information is free from faulty edges, and the routing is completed (in the same manner as in Sec. \\ref{sec:routing-known}) at the destination $t$. We next bound the multiplicative stretch of the routing.\n\n\n\n\n\\begin{claim}\\label{cl:route-length}\nFix a set of faulty edges $F$, and let $s,t$ be vertices that are connected in $G \\setminus F$. Then, the message is routed from $s$ to $t$ within $32k (|F|+1)^2 \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ steps, w.h.p.\n\\end{claim}\n\\begin{proof}\nFirst note that since each iteration and each graph $G_{i}$ uses an independent set of FT connectivity labels, then in each phase and each iteration the decoding algorithm succeeds w.h.p. and outputs an $s$-$t$ path $\\widehat{P}_\\ell$ if exists. \n\nAssume that $\\mbox{\\rm dist}_{G \\setminus F}(s,t) \\in (2^{i-1},2^i]$. Then, $s$ and $t$ are connected in $G_i \\setminus F$, as $T_i = T_{i,i^*(t)}$ contains the $2^i$-ball around $t$. We show that the algorithm terminates at $t$ in phase $i$ or before it, and that in any phase $j \\leq i$, the routing algorithm traverses a path of length at most $2(4k-1)(|F|+1)^2 \\cdot 2^j$.\n\nLet $j \\leq i$. In the $\\ell$'th iteration of phase $j$, the algorithm first checks if $s$ and $t$ are connected in $G_j \\setminus F_{\\ell}$, where $F_{\\ell}$ is the set of currently detected faults. If the answer is no, the algorithm proceeds to the next phase. Otherwise, it tries to route a message from $s$ to $t$ on the path encoded by $\\widehat{P}_{\\ell}$. The length of the path is bounded by $(4k-1)(|F|+1)\\cdot 2^j$ from Lemma \\ref{lem:succint_path_routing}. The algorithm either succeeds, or finds a faulty edge on the way in which case it returns to $s$ by traversing the same path on the reverse direction. Overall, the algorithm traverses a path of length at most $2(4k-1)(|F|+1)\\cdot 2^j$, in this iteration. In all $|F|+1$ iterations of phase $j$, the length of the path explored is at most $2(4k-1)(|F|+1)^2 \\cdot 2^j$. Summing over all iterations $j \\leq i$, the stretch is bounded by $$\\sum_{j=1}^{i} 2(4k-1)(|F|+1)^2 \\cdot 2^j = 2(4k-1)(|F|+1)^2 \\sum_{j=1}^i 2^j \\leq 2^{i+2} (4k-1)(|F|+1)^2 \\leq 32k (|F|+1)^2 \\mbox{\\rm dist}_{G \\setminus F}(s,t).$$ The last inequality uses the fact that $2^{i-1} \\leq \\mbox{\\rm dist}_{G \\setminus F}(s,t).$ \n\nIn the $i$'th phase, since $s$ and $t$ are connected in $G_i \\setminus F$, then for any $F_{\\ell} \\subseteq F$, $s$ and $t$ are connected in $G_i \\setminus F_{\\ell}$, hence the algorithm always finds a path $\\widehat{P}_{\\ell}$. Hence, it either succeeds in routing the message to $t$ in one of the iterations (or one of the previous phases), or learns about all the failures $F$. In the latter case, in iteration $|F|+1$ it learns about a failure-free path $\\widehat{P}_{|F|+1}$, and the routing terminates at $t$. This completes the proof.\n\\end{proof}\nTo conclude, we have the following.\n\n\\begin{theorem}\nFor every integers $k,f$, there exists an $f$-FT compact routing scheme that given a message $M$ at the source vertex $s$ and a label $L_{route}(t)$ of the destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. \nThe global table size is $\\widetilde{O}(f \\cdot n^{1+1\/k} \\log{(nW)})$.\nThe header size of the messages is bounded by $\\widetilde{O}(f^2)$ bits, and the label size of vertices is $O(\\log{(nW)} \\log{n})$. \n\\end{theorem}\n\n\n\n\n\n\\paragraph{Improving the size of the routing tables.} \nSo far, we have described a routing scheme that consumes a total space of $\\widetilde{O}(f\\cdot n^{1+1\/k}\\log (nW))$ bits, and multiplicative stretch\n$32(|F|+1)^2 k$. We now explain the required modifications needed to providing routing tables with $\\widetilde{O}(f^3\\cdot n^{1\/k})$ bits per vertex. The most space consuming information for a vertex $u$ is the connectivity labeling\ninformation of the edges incident to $u$ in each of the trees $T_{i,j} \\in \\mathcal{T}$. As the degree of $u$ in some of the trees might be $\\Theta(n)$, it leads to tables of possible super-linear size. To reduce the space of the individual tables, we apply a load balancing idea which distributes the labeling information incident to \\emph{high}-degree vertices among their neighbors. \n\nInstead of storing the labeling information of $e=(u,v)$ at the routing tables of $u$ and $v$, we define \nfor every tree $T \\in \\mathcal{T}$ and an edge $e=(u,v) \\in T$, a subset $\\Gamma_T(e)$ of vertices that store the connectivity labeling information of $e$ in $T$. We will make sure that the information on some vertex in $\\Gamma_T(e)$ can be easily extracted in the routing procedure upon arriving one of its endpoints. In addition, we will make sure that each vertex stores the information only for a small number of edges in each of its trees. Consider an edge $e=(u,v)$ in a tree $T$, and assume, without loss of generality, that $u$ is the parent of $v$ in the tree $T$. In the case where $\\deg(u,T)\\leq f+1$, we simply let $\\Gamma_T(e)=\\{u,v\\}$. That is, the label of $e$ is stored by both endpoints of $e$ (as before). The interesting case is where $\\deg(u,T)\\geq f+2$, in which case, $u$ might not be able to store the label of $e$, and will be assisted by its other children as follows.  Let $Child(u,T)=[v_1,\\ldots, v_\\ell]$ be the lexicographically ordered list of the children of $u$ in $T$.  The algorithm partitions $Child(u,T)$ into consecutive blocks of size $f+1$ (the last block might have $2f+1$ vertices). Letting $[v_{q,1}, \\ldots, v_{q,f+1}] \\subseteq Child(u,T)$ be the block containing $v$, define\n$$\\Gamma_T(e)=\\{v_{q,1}, \\ldots, v_{q,f+1}\\}~.$$\nNote that in particular, $v \\in \\Gamma_T(e)$. Thus, the label of $e$ is stored by $v$ and $\\ell \\in [f,2f-1]$ additional children of $u$ in $T$. \n\nWe then modify the tree labels from Fact \\ref{fc:route-trees} to contain the port information of $\\Gamma_T(e)$. \nIn order to do that, we will be using the more relaxed variant of Fact \\ref{fc:route-trees}, we have:\n\\begin{claim}\\label{cl:route-trees-port}\nFor every $n$-vertex tree $T$, there exists a (deterministic) routing scheme that assigns each vertex $v \\in V(T)$ a label $L_T(v)$ of $O(f\\log^2 n)$ bits and table $R_T(v)$ of $O(f\\log n)$ bits. Given the label $L_T(t)$ of the target $t$  \nand the routing table $R_T(u)$, the vertex $u$ can compute in $\\widetilde{O}(f)$ time: (i) the port number of the edge $e=(u,v)$ on its tree path to $t$, and (ii) the port numbers of the neighbors of $u$ in the set $\\Gamma_T(e=(u,v))$. \n\\end{claim}\n\\begin{proof}\nThe proof follows by slightly modifying the simpler scheme of Fact \\ref{fc:route-trees} by \\cite{thorup2001compact}. Specifically, we will be using the routing scheme based on heavy-light tree decomposition. This scheme assigns each vertex $v$ labels of $O(\\log^2 n)$ bits that contain the port information of the at most $O(\\log n)$ light edges on the root to $v$ path in $T$.  The vertices are enumerated in DFS ordering, and the label of each vertex  contains its DFS range, and the specification of all light edges on its path in $T$ from the root, along with a port information of these edges.  The routing table of $v$ stores its DFS range, the port number of the (unique) heavy child of $v$ and also the port to its parent. In our modification, we augment the label of each vertex $u$ with the port information of $\\Gamma_T(e')$ for every light edge $e'$ appearing on the root to $u$ path in $T$. Since there are $O(\\log n)$ such light edges, the total label information is encoded in $O(f\\log^2 n)$ bits. The routing table $R_T(u)$ is augmented with the port information for the set $\\Gamma_T(e'')$, where $e''$ is the (unique) heavy child of $u$.  The routing scheme is then exactly as described at \\cite{thorup2001compact}, only that in addition to the port of the next-hop $e=(u,v)$, we also obtain the port information of $\\Gamma_T(e)$. This increases the labels and tables in the scheme of \\cite{thorup2001compact} by a factor of $O(f)$, the claim follows.  \n\\end{proof}\n\nSince the modified claim of tree routing defines now both tree routing labels and tables, we employ the following modifications. The extended identifier $\\operatorname{EID}_T(e)$ of an edge $e=(u,v)$ from Eq. (\\ref{eq:edge-extended-routing})\ncontains the modified tree labels and thus has $O(f\\log^2 n)$ bits.\n\nThe \\emph{routing labels} of Eq. (\\ref{eq:route-edge-label}) are defined in the same manner only using the modified extended edge identifiers. The routing label of each edge has $\\widetilde{O}(f^2)$ bits, and routing label of every vertex has $\\widetilde{O}(f)$ bits. \nWe are now ready to describe the more succinct \\emph{routing tables} of each vertex $v$. We modify the definition of Eq. (\\ref{eq:route-table-ij}) by letting:\n\\begin{equation*}\\label{eq:route-table-ij-mod}\nR_{route,i,j}(v)=\\{L_{route,i,j}(e), e \\in \\Gamma_{T_{i,j}}(e)\\} \\cup \\mathsf{ConnLabel}\\xspace^1_{G_{i,j},T_{i,j}}(v) \\cup R_{T_{i,j}}(v)~,\n\\end{equation*}\nthus the routing table $R_{route,i,j}(v)$ is augmented the tree routing tables $R_{T_{i,j}}(v)$ of Claim \\ref{cl:route-trees-port}. In addition, $R_{route}(v)=\\{R_{route,i,j}(v), (i,j) ~\\mid~ T_{i,j} \\in \\mathcal{T}, v\\in T_{i,j}\\}$ as before.\nWe therefore have:\n\\begin{claim}\\label{cl:route-balance-table}\nThe size of each routing table $R_{route}(v)$ is bounded by $\\widetilde{O}(f^3 K n^{1\/k})$ bits.\n\\end{claim}\n\\begin{proof}\nFor every tree $T_{i,j}$ containing $v$, $v$ stores the routing labels for the tree $T_{i,j}$ of all edges in the set $E'(v,T_{i,j})=\\{e \\in T_{i,j} ~\\mid~ v \\in  \\Gamma_{T_{i,j}}(e)\\}$. Since each connectivity label of an edge contains the modified tree labels from Fact \\ref{cl:route-trees-port}, it has $\\widetilde{O}(f)$ bits, and as the routing label for $T_{i,j}$ contains $O(f)$ copies of this label, overall each routing label of an edge has $\\widetilde{O}(f^2)$ bits. Observe that $|E'(v,T_{i,j})|=O(f)$ as each vertex stores the label of its parent in the tree, $O(f)$ child edges, and $O(f)$ child edges of its parent in the tree. Since each $v$ participates in $\\widetilde{O}(K n^{1\/k})$ trees, overall its routing table has $\\widetilde{O}(f^3 K n^{1\/k})$ bits, as required. \n\\end{proof}\n\nIt remains to explain the required modifications for the routing procedure over a tree $T_i=T_{i,i^*(t)}$. Upon arriving to a vertex $u$ incident to a faulty \\emph{tree} edge $e=(u,v)$ the procedure is as follows. If $e$ is a non-tree edge or if $u$ stores the connectivity label $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$\\footnote{This covers the cases where $v$ is either a parent of $u$ or else, it is one of the at most $f+1$ children of $u$ in $T_i$.}, then $u$ adds the routing label of the edge to the header, as before. In the remaining case it must hold that $e$ is the edge incident to $u$ on its tree path to some vertex $y$. By using the tree routing scheme of Claim \\ref{cl:route-trees-port} we have that given the tree routing labels  $L_{T_{i}}(u)$ and $L_{T_{i}}(y)$, the vertex $u$ can also obtain the port numbers of its $\\ell \\in [f,2f-1]$ children in $\\Gamma_{T_{i}}(e)$ that store the label $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$. Since there are at most $f$ edge faults in the network, and $\\Gamma_{T_{i,j}}(e)$ contains information on at least $f+1$ ports of $u$'s neighbors that contain the label of $e$, the vertex $u$ can access a non-faulty neighbor, say $w$, that has the label information of $e$. That vertex can then add the labeling information of $e$ to the header of the message, and the routing algorithm proceeds as before. Since we use the modified tree labels of Claim \\ref{cl:route-trees-port}, each connectivity label has $\\widetilde{O}(f)$ bits, and each routing label of an edge for a tree $T_{i,j}$ has $\\widetilde{O}(f^2)$ bits. Since the header stores the routing labels of $O(f)$ edges, it consists of $\\widetilde{O}(f^3)$ bits. \n\n\nThe stretch is still bounded by $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$, as we next explain. Recall that in the proof of Claim \\ref{cl:route-length}, we bounded the length of the path we explore in one iteration of the algorithm of phase $j$ by $2(4k-1)(|F|+1)2^j.$ In the new scheme, when we discover a faulty edge, the vertex $u$ may send messages to $|F|+1$ neighbors until it finds the label of the edge. This adds at most $2(|F|+1)2^j$ to the stretch, as the weight of edges in the tree of phase $j$ is at most $2^j$, and we may send messages in both directions. This gives that the length of the path we explore in one iteration is now at most $2(4k-1)(|F|+1)2^j+2(|F|+1)2^j=8k(|F|+1)2^j.$ The rest of the analysis proceeds as in the proof of Claim \\ref{cl:route-length}, and gives that the stretch is bounded by $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ (we get the same bound as in the original proof we bounded $2(4k-1)$ with $8k$ during the analysis).\nWe therefore have:\n\\begin{theorem}\\label{thm:routing-unknown}[Fault-Tolerant Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$ and a label $L_{route}(t)$ of the destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The routing labels have $\\widetilde{O}(f)$ bits, the table size of each vertex is $\\widetilde{O}(f^3 \\cdot n^{1\/k} \\log(nW))$. The header size of the messages is bounded by $\\widetilde{O}(f^3)$ bits. \n\\end{theorem}\n\n\n\\paragraph{Lower Bound.} Finally, we show that the price of not knowing the set of faulty edges $F$ in advance might indeed incur a multiplicative stretch of $\\Omega(f)$. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:lb-routing}]\nConsider a graph that consists of $f+1$ vertex disjoint $s$-$t$ paths, each of length $L=\\Theta(n\/f)$. The last edge of each of the paths, except for one, is faulty. Assume that the non-faulty path is chosen uniformly at random. Since the routing scheme is oblivious to the faulty edges, it can discover a faulty edge only upon sending the message to one of the edge endpoints. The expected length of the routing is given by:\n$$\\frac{L}{f+1} +2L \\cdot \\left(1-\\frac{1}{f+1} \\right)\\cdot \\frac{1}{f} + \\ldots+ \\left(f+1\\right)L\\cdot \\prod_{i=0}^{f-1} \\left(1-\\frac{1}{f+1-i}\\right)=\\Omega(f L)~.$$ \nSince the $s$-$t$ shortest path under these faults is $L$, the proof follows. See Fig. \\ref{fig:LB-stretch} for an illustration.\n\\end{proof}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.40]{lb.pdf}\n\\caption{\\sf Illustration for a stretch lower bound for any FT routing schemes. The $s$-$t$ pair are connected by $f+1$ vertex disjoint paths of length $L$. Since the faulty-edge is the last edge of the path, the routing requires $\\Omega(L)$ steps to discover a single faulty edge. As the non-faulty path is chosen uniformly at random, in expectation, the routing requires $\\Omega(fL)$ steps.  \\label{fig:LB-stretch}\n}\n\\end{center}\n\\end{figure}\n\n\n\\section{Compact Routing Schemes}\nIn this section, we explain how to use our FT-distance labels to provide compact and low stretch routing schemes. This is the first scheme to provide an almost tight tradeoff between the space and the multiplicative stretch, for a constant number of faults $f=O(1)$.  Throughout this section, tree routing operations are performed by using the tree routing scheme of Thorup and Zwick \\cite{thorup2001compact}.\n\\begin{fact}\\label{fc:route-trees}[Routing on Trees]\\cite{thorup2001compact}\nFor every $n$-vertex tree $T$, there exists a routing scheme that assigns each vertex $v \\in V(T)$ a label $L_T(v)$ of $(1+o(1))\\log n$ bits. Given the label of a source vertex\nand the label of a destination, it is possible to compute, in constant time, the port number of the edge from the source that heads in the direction of the destination.\n\\end{fact}\n\n\\textbf{MP: check if needed.} \\mtodo{I've added a discussion of the port information later.}\nIn our routing scheme, each vertex $u$ is required to compute the port number of the next-hop towards the target destination. Towards that goal, we modify the FT-distance labels of Sec. \\ref{sec:ft-distance}, by adding the port information to the extended identifier of the edges. That is, the extended identifier of each edge $e=(u,v)$ is augmented with a third field containing the port number of $v$ w.r.t $u$ and vice-versa. To avoid cumbersome notation, we still refer to this port-augmented FT-distance label of $u$ by $\\mathsf{DistLabel}\\xspace(u)$. \n\n\n\\mtodo{Added the part from here. This adds port and tree routing information to the ids, and adapts Lemma \\ref{lem:useful-recovery-edges} (finding succinct path description) to the context of routing.}\n\nWe slightly modify the connectivity label of the edges and vertices by augmenting them with routing information. \nSpecifically, we define the vertex identifier $\\operatorname{ID}_T(u)=[\\operatorname{ID}(u), L_T(u)]$ where $L_T(u)$ is taken from Fact \\ref{fc:route-trees}. We then slightly modify the \nthe connectivity label of Eq. (\\ref{{eq:conn-node}}) by defining \n$$\\mathsf{ConnLabel}\\xspace_{G,T}(u)=\\langle \\mathsf{ANC}\\xspace_T(u), \\operatorname{ID}_T(u) \\rangle~.$$\nIn addition, we also augment the extended identifier of an edge Eq. (\\ref{eq:extend-ID}) with port information and tree routing information, by having:\n$$\\operatorname{EID}(e)=[\\operatorname{UID}(e), \\operatorname{ID}(u), \\operatorname{ID}(v), \\mathsf{ANC}\\xspace_T(u), \\mathsf{ANC}\\xspace_T(v), \\mbox{\\tt port}(u,v), \\mbox{\\tt port}(v,u), L_T(u), L_T(v)]~,$$\nwhere $\\mbox{\\tt port}(u,v)$ is the port number of the edge $(u,v)$ for $u$. \nThroughout this section, when applying the connectivity labels from Section \\ref{sec:ftconn-sketch} on a graph $G$ with spanning tree $T$, we use these labels and extended identifiers to support routing.\nWe later apply the connectivity labels with respect to the different trees of the tree cover as discussed in Section \\ref{sec:ft-distance}. \nLet $T_{i,j} \\in \\mathsf{TC}\\xspace_i$, recall that $G_{i,j}=G[V(T_{i,j})]$ and that $\\mathcal{T}=\\bigcup_{i=1}^K \\mathsf{TC}\\xspace_i$ for $K=O(\\log (nW))$. \n\n\\begin{lemma}\\label{lem:succint_path_routing}\nConsider a triplet $s,t,F$ such that $s,t,F \\in G_{i,j}$. \nGiven $ID_{T_{i,j}}(s),ID_{T_{i,j}}(t)$ and the connectivity labels $\\{\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e)\\}_{e \\in F}$, we can determine w.h.p if $s$ and $t$ are connected in $G_{i,j} \\setminus F$. If they are connected, we can output a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of the $s$-$t$ path in $G_{i,j} \\setminus F$. The edges of $\\widehat{P}$ are labeled by $0$ and $1$, where $0$-labeled edges correspond to $G_{i,j}$-edges and $1$-labeled edges $e=(x,y)$ correspond to $x$-$y$ paths in $T_{i,j} \\setminus F$. For each $G_{i,j}$-edge, the succinct path description has the port information of the edge, and for each $x-y$ path, the description has the tree routing labels $L_{T_{i,j}}(x),L_{T_{i,j}}(y)$.\nThe length of the $s$-$t$ path encoded by $\\widehat{P}$ is bounded by $(4k-1)(f+1)\\cdot 2^i$. \n\\end{lemma}\n\n\\mtodo{For the proof I use the fact that the decoding algorithm only uses the ancestry labels of $s,t$, we should probably add some remark on the topic also on the labels section.}\n\n\\begin{proof}\nThe proof follows the proof of Lemma \\ref{lem:useful-recovery-edges}. To see this, first note that the decoding algorithm from Section \\ref{sec:ftconn-sketch} mostly uses the connectivity labels of the faulty edges $F$, and it needs only the labels $\\mathsf{ANC}\\xspace(s),\\mathsf{ANC}\\xspace(t)$ of $s$ and $t$, to detect the connected components of $s$ and $t$ in $T_{i,j} \\setminus F$. \\mtodo{do we want to edit this there? at least have some remark} \nHence, using $ID_{T_{i,j}}(s),ID_{T_{i,j}}(t)$ and the connectivity labels $\\{\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e)\\}_{e \\in F}$, we can first check if $s$ and $t$ are connected in $G_{i,j} \\setminus F$ using the decoding algorithm of Section \\ref{sec:ftconn-sketch}. If they are connected, then from Lemma \\ref{lem:useful-recovery-edges}, we get a succinct description of the $s$-$t$ path in $G_{i,j} \\setminus F$. We next show that the algorithm indeed has the relevant port and tree routing information. For this note that all the vertices in the path $\\widehat{P}$ obtained by Lemma \\ref{lem:useful-recovery-edges} are either $s$ and $t$ or endpoints of the $f$ recovery edges found in the algorithm. The ids $ID_{T_{i,j}}(s),ID_{T_{i,j}}(t)$ of $s$ and $t$ have their tree routing information, and when the algorithm finds a recovery edge, it learns about its extended id $ID_{T_{i,j}}(e)$ that has the port information and tree routing information of its endpoints. Any $G_{i,j}$-edge in $\\widehat{P}$ is a recovery edge, hence the algorithm has its port information, and for any $x$-$y$ path in $T_{i,j} \\setminus F$, the algorithm has the tree routing labels $L_{T_{i,j}}(x),L_{T_{i,j}}(y)$, as needed.\nThe stretch analysis follows the stretch analysis in Section \\ref{sec:ft-distance}. It is based on the fact that $\\widehat{P}$ has as most $f+1$ subpaths in $T_{i,j} \\setminus F$, each of length at most $(4k-2)2^i$, and at most $f$ recovery edges of weight at most $2^i$.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Forbidden Set Routing (Faulty Edges are Known)}\\label{sec:routing-known}\nWe start by describing the routing scheme in the forbidden set setting, where the faulty edges $F$ are known to the source vertex $s$. We show the following.\n\n\\begin{theorem}\\label{thm:routing-known}[Forbidden-Set Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$, a label of the destination $t$, and labels of at most $f$ forbidden edges $F$ (known to $s$), routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $(8k-2)(f+1)\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The table size of each vertex is bounded by $\\widetilde{O}(n^{1\/k} \\log{(nW)})$. The header size of the messages is bounded by $\\widetilde{O}(f)$ bits. The labels of vertices and edges have size $\\widetilde{O}(n^{1\/k} \\log{(nW)})$.\n\\end{theorem}\n\n\\mtodo{I edited a bit the proof so it's more similar to the next section and uses Lemma \\ref{lem:succint_path_routing}. The routing tables here still use the distance labels. Also, added a bit more details on the routing process.}\n\n\\begin{proof}\nThe algorithm is based on the distance labels from Section \\ref{sec:ft-distance}. Recall that the distance labels are based on applying fault tolerant connectivity labels on different graphs $G_{i,j}$, we use the connectivity labels from Section \\ref{sec:ftconn-sketch}.\nThe routing table of each vertex $u$ consists of its distance label and the tree routing labels from Fact \\ref{fc:route-trees} for every tree $T_{i,j} \\in \\mathcal{T}$ containing $u$. That is,\n$$R_{route}(u)=\\mathsf{DistLabel}\\xspace(u) \\cup \\{ \\langle L_{T_{i,j}}(u), (i,j) \\rangle ~\\mid~ i \\in [1,K], j \\in \\{1,\\ldots, |\\mathsf{TC}\\xspace_{i}|\\}, u \\in T_{i,j}\\}~.$$\nSince each vertex $u$ participates in $O(k n^{1\/k} \\log(n W))$ trees of $\\mathcal{T}$, and since the distance labels are of size $O(k n^{1\/k} \\log(n W) \\cdot \\log^3{n})$, the table $R_{route}(u)$ has $O(k n^{1\/k} \\log(n W) \\cdot \\log^3{n})$ bits.\nThe label of a vertex $u$ is just $R_{route}(u)$, and the label of an edge $e$ is $\\mathsf{DistLabel}\\xspace(e)$. The labels have size $O(k n^{1\/k} \\log(n W) \\cdot \\log^3{n})$. \\mtodo{we can have shorter labels for vertices as in the next section (but then we need to \"open the box\" of the distance labels), as the labels of edges are anyway large I'm not sure if we want to do it in this section.}\n\n\\mtext{\nIn the routing algorithm, the vertex $s$ is given the label $R_{route}(t)$ of $t$, and the labels $\\{\\mathsf{DistLabel}\\xspace(e)\\}_{e \\in F}$, and it needs to route a message to $t$ in the graph $G \\setminus F$. First, from the distance labels of $s,t$ and $F$, $s$ can check using the algorithm from Section \\ref{sec:ft-distance} if $s$ and $t$ are connected in $G \\setminus F$, and if so get a succinct description of a path between them. In more detail, recall that the algorithm from Section \\ref{sec:ft-distance} works in $K$ phases, where in phase $i$ it checks if $s$ and $t$ are connected in the graph $G_{i,i^*(s)} \\setminus F$ that contains the $2^i$-ball around $s$. Let $i$ be the first iteration where $s$ and $t$ are connected in $G_{i,i^*(s)} \\setminus F$ according to the algorithm, and denote $G_i = G_{i,i^*(s)},T_i = T_{i,i^*(s)}$, and let $F_i = F \\cap G_i$. The algorithm can also give a succinct description of an $s$-$t$ path in $G_i \\setminus F_i$ following Lemma \\ref{lem:succint_path_routing}. For this, note that we indeed have all the required information. The distance labels of edges in $F$ in particular contain the labels $\\{\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)\\}_{e \\in F_i}$, and we can also tell which edges of $F$ are in $G_i$ from the labels. Also, the labels of $s,t$ contain the information $ID_{T_i}(s),ID_{T_i}(t)$ if they are both in $T_i$ (otherwise, they are not connected in level $i$).\n\nThe path $\\widehat{P}$ as described in Lemma \\ref{lem:succint_path_routing} is composed of $O(f)$ parts, where segment $(x,y)$ in the path corresponds either to an edge in $G_i$ or to a tree path in $T_i \\setminus F$, it also has the relevant port and tree routing information. Our goal is to route a message according to this path. For this we add to the header of the message the description of $\\widehat{P}$, the indexes $(i,i^*(s))$ of the tree we explore and an index $1 \\leq q \\leq 2f+1$ that represents the segment of $\\widehat{P}$ we currently explore, initially $q=1$. Overall, the header size is $\\widetilde{O}(f)$. To route a message according to the path, we work as follows. The header specifies the current segment in $\\widehat{P}$. If the current segment corresponds to an edge $(x,y) \\in G$, then $x$ uses the port information to route the message to $y$ and increases the index $q$. Otherwise, the current segment represents a tree path $(x,y) \\in T_i$ and a vertex $u$ in this path uses its routing label in $T_i$ and the routing label of $y$ in $T_i$ (that is part of the header) to route the message towards $y$. When the message reaches $y$, it increases the index $q$. This completes the description of the routing process. The length of the path described is at most $(8k-2)(f+1)\\cdot \\mbox{\\rm dist}_{G\\setminus F}(s,t)$, as shown in Section  \\ref{sec:ft-distance}. } \n\\end{proof}\n\n\n\n\\subsection{Fault Tolerant Routing (Faulty Edges are Unknown)}\\label{sec:route-unknown}\nWe now consider the more involved setting where the set of failed edges $F$ are unknown to $s$. In this case, an edge $(u,v) \\in F$ is detected only when the message arrives, during the routing procedure, to one of the endpoints of $e$. Note that the routing scheme should, by definition, be prepared to any set of faulty edges $F$. However, the space bound of our scheme is required to be bounded by $\\widetilde{O}(f n^{1+1\/k})$, which is possibly much smaller than the number of graph edges $m$. This in particular implies that we cannot store the FT-distance labels of all the graph edges. Nevertheless, we show that it is sufficient to explicitly store the labeling information for the tree edges in $\\mathcal{T}$. The required information for the failed non-tree edges would be revealed throughout the process, by applying the decoding algorithm of Lemma \\ref{lem:succint_path_routing}.\nOur routing scheme eventually routes the message along the $s$-$t$ path encoded by the FT-distance labels of $s,t$ and $F$. However, since the labels of $F$ are unknown in advance, the routing scheme will detect these edges in a trail and error fashion which induces to an extra factor of $f$ in the final multiplicative stretch. This extra $f$ factor is also shown to be essential, in the end of the section.\nWe proceed by describing the routing tables.  \n\n\\remove{\n\\paragraph{The routing tables.} For ease of presentation, we first describe a solution with a multiplicative stretch of $O(kf^2)$, and \\emph{global} space of $\\widetilde{O}(f K \\cdot n^{1+1\/k})$, \\mtodo{should it be $n^{1+2\/k}$?} but the individual tables of some of the vertices might be of linear space. \nRecall that $\\mathcal{T}=\\bigcup_i^{K} \\mathsf{TC}\\xspace_i$, for $K=O(\\log (nW))$ is a collection of tree covers in all $K=\\lceil \\log (nW) \\rceil$ distance scales, see Eq. (\\ref{eq:TC-i}). For every vertex $u$, let $\\deg_T(u)=\\sum_{T_{i,j} \\in \\mathcal{T}}\\deg(u,T_{i,j})$ be the sum of degrees of $u$ in the collection of trees  $\\mathcal{T}$.\nThe algorithm computes each vertex $u$ a table of $\\widetilde{O}(f \\cdot \\deg_T(u) \\cdot n^{1\/k})$ bits.\nSince the total number of tree edges in $\\mathcal{T}$ is bounded by $\\widetilde{O}(K \\cdot n^{1+1\/k})$, this provides a global space bound of $\\widetilde{O}(fK \\cdot n^{1+2\/k})$ bits. We later on improve the space of each table to $\\widetilde{O}(fK \\cdot n^{2\/k})$ bits. \\mtodo{maybe extra $k,K$ are needed in some places.}\n\nThe routing process has at most $f+1$ phases, and each phase uses an independent set of randomized FT-distance labels. Therefore, the preprocessing algorithm employs $f$ \\mtodo{$f$ or $f+1$?} independent applications of the $(f,(8k-2)(f+1))$ FT-distance labels scheme. Denote the output labels obtained by the $i^{th}$ application by $\\mathsf{DistLabel}\\xspace_i(w)$ for every $w \\in V(G)\\cup E(G)$. Let $E_T=\\bigcup_{T_{i,j} \\in \\mathcal{T}} E(T_{i,j})$ be the collection of tree edges. The \\emph{routing label} \\mtodo{maybe we want to have shorter labels, and have this information only in the routing tables?} of each $w \\in V\\cup E_T$ is given by \n$$L_{route}(w)=\\{\\mathsf{DistLabel}\\xspace_1(w),\\ldots, \\mathsf{DistLabel}\\xspace_{f+1}(w)\\}~.$$ \nThe routing label $L_{route}(e)$ of a non-tree edge $e \\in E \\setminus E_T$ is simply the extended identifier of $e$. For each vertex $u$, the routing table $R_{route}(u)$ consists of the following:\n\\begin{enumerate\n\\item Tree labels $L_{T_{i,j}}(u)$ for every tree $T_{i,j}$ in $\\mathcal{T}$ that contains $u$ (from Fact \\ref{fc:route-trees}). These labels are augmented with their corresponding tree index $(i,j)$.\n\\item The routing labels $L_{route}(u)$ and $\\{L_{route}(e=(u,v)) ~\\mid~ e \\in T_{i,j}, T_{i,j} \\in \\mathcal{T}\\}$.\n\\end{enumerate}\nThe header information initially contains the succinct path information\\footnote{\\textbf{MP: This is not essential, we can always make the header contain the labels of the currently detected faults.}} obtained by the decoding algorithm of the FT-distance labeling when given the labels $L_{route}$ of $s$ and $t$. \\mtodo{do we want to have $f$ paths? Or only one of them that is relevant for the current phase}  As we will see, throughout the routing procedure, the header information will be augmented with the labels of the currently detected faulty edges. \n}\n\n\n\\paragraph{The routing tables.} For ease of presentation, we first describe a solution with a multiplicative stretch of $O(kf^2)$, and \\emph{global} space of $\\widetilde{O}(f K \\cdot n^{1+1\/k})$, but the individual tables of some of the vertices might be large. We later on improve the space of each table to $\\widetilde{O}(fK \\cdot n^{1\/k})$ bits.\n\nRecall that $\\mathcal{T}=\\bigcup_i^{K} \\mathsf{TC}\\xspace_i$, for $K=O(\\log (nW))$ is a collection of tree covers in all $K=\\lceil \\log (nW) \\rceil$ distance scales, see Eq. (\\ref{eq:TC-i}). For every vertex $v$, let $\\deg_T(v)=\\sum_{T_{i,j} \\in \\mathcal{T}}\\deg(u,T_{i,j})$ be the sum of degrees of $u$ in the collection of trees  $\\mathcal{T}$. Recall that $G_{i,j}=G[V(T_{i,j})]$.\n\\mtext{For the routing we apply the FT-connectivity labels on the graphs $G_{i,j}$, similarly to Section \\ref{sec:ft-distance}.\nThe routing process uses $f$ independent applications of randomized FT-connectivity labels from Section \\ref{sec:ftconn-sketch}, applied on each one of the graphs $G_{i,j}$. \\mtodo{There is some delicate issue here. In my understanding we do want to have the extended ids fixed in different applications, and only change the randomness of sketches, as I try to explain here.} In more detail, when we apply the labeling scheme on the graph $G_{i,j}$ with spanning tree $T_{i,j}$, we use $f$ independent random seeds $\\mathcal{S}_h$ to determine the randomness of the sketches. However, the seed $\\mathcal{S}_{ID}$ used to determine the extended ids of edges in $G_{i,j}$ is fixed in the $f$ applications, hence the ids are fixed and we only use fresh randomness to compute the sketch information. This process is done independently on each one of the graphs $G_{i,j}$. }  \nDenote the output labels obtained by the ${\\ell}^{th}$ application of the scheme on the graph $G_{i,j}$ by $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_{i,j},T_{i,j}}(w)$ for every $w \\in V(G_{i,j})\\cup E(G_{i,j})$, and we denote by $$\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(w)=(\\mathsf{ConnLabel}\\xspace^1_{G_{i,j},T_{i,j}}(w),...,\\mathsf{ConnLabel}\\xspace^f_{G_{i,j},T_{i,j}}(w)).$$\nSince in each application the labels are of size $O(\\log^3{n})$, the total size of each label is $O(f \\log^3{n})$. \\mtext{Recall that for a non-tree edge $e \\in G_{i,j}$, we have that  $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_{i,j},T_{i,j}}(e)=\\operatorname{ID}_{T_{i,j}}(e)$. This label does not depend on the sketch information, and hence is the same in all $f$ applications, we will exploit this later in the algorithm.}\n\n\\mtodo{I changed a bit the definition of routing tables and labels. The labels currently are of polylog size (no extra $f$ or $n^{1\/k}$).}\n\\mtext{\nThe routing table $R_{route}(v)$ of a vertex $v$ has the following information for any tree $T_{i,j}$ such that $v \\in T_{i,j}$:\n\\begin{enumerate}\n\\item The values $(ID_{T_{i,j}}(v),i,j)$.\n\\item The connectivity labels $\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e)$ for any tree edge $e \\in T_{i,j}$ adjacent to $v$. \n\\end{enumerate}\n\nSince the connectivity labels are of size $\\widetilde{O}(f)$, and since $v$ appears in $\\deg_T(v)$ trees, the size of the table is $\\widetilde{O}(f \\deg_T(v)).$ Since the total number of tree edges in $\\mathcal{T}$ is bounded by $\\widetilde{O}(K \\cdot n^{1+1\/k})$, this provides a global space bound of $\\widetilde{O}(fK \\cdot n^{1+1\/k})$ bits.\n\nRecall that for a vertex $v$ and index $1 \\leq i \\leq K$, we denote by $i^*(v)$ an index such that the $2^i$-ball around $v$ is contained in $T_{i,i^*(v)}$.\nThe routing label $L_{route}(v)$ of $v$ has the id of $v$ in the trees $T_{i,i^*(v)}$.\n$$L_{route}(v) = \\{(i^*(v),ID_{T_{i,i^*(v)}}(v)) | i \\in [1,K]\\}$$\n\nThe size of the label is $O(K\\log{n})=O(\\log{n} \\log{nW})$.}\n\n\\mtodo{In the routing algorithm the main difference is that we have log iterations for the different trees, in each tree the algorithm is similar to before.}\n\n\\paragraph{The routing algorithm.} \\mtext{In the routing algorithm, the source vertex $s$ gets the routing label $L_{route}(t)$ of the destination $t$ and its goal is to find a graph $G_{i,j}$ such that $s$ and $t$ are connected in $G_{i,j} \\setminus F$, and use it for routing. As the set $F$ is not known in advance, the algorithm works in $K= O(\\log{nW})$ iterations, where in iteration $i$ it tries to route a message in $G_{i,i^*(t)}$. If $s$ and $t$ are connected in $G_{i,i^*(t)} \\setminus F$ the algorithm succeeds, and otherwise we proceed to the next iteration. \nWe next describe the algorithm for one iteration $i$, we denote $G_i = G_{i,i^*(t)}, T_i = T_{i,i^*(t)}$. Note that $s$ learns the index $i^*(t)$ from the label of $t$, and it can check if $s \\in T_i$ using its routing table. If $s \\not \\in T_i$, we proceed to the next iteration.}\n\nIf $s \\in T_i$, the routing procedure for iteration $i$ has at most $f+1$ phases. Each phase will start at the source vertex $s$. We will maintain the following invariant in the beginning of each phase $\\ell \\in \\{1,\\ldots, f+1\\}$: (i) the algorithm has already detected a subset of $\\ell-1$ faulty edges $F_\\ell \\subseteq F$, and (ii) the header contains the labels $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$ of all the edges $e \\in F_\\ell$. Each phase $\\ell \\leq f+1$ will terminate either at the destination vertex $t$, or at the source vertex $s$. In addition, w.h.p., if $s$ and $t$ are connected in $G_i \\setminus F$, phase $f+1$ will terminate at $t$. The invariant holds vacuously for phase $1$.\n\nWe now describe the $\\ell^{th}$ phase of the routing procedure given the invariant. The source vertex $s$ considers the $\\ell^{th}$ copy of the FT-connectivity labels, $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_i,T_i}(e)$ for every $e \\in F_\\ell$. \nUsing the connectivity labels, that are part of the header, and the ids $ID_{T_i}(s),ID_{T_i}(t)$ that are part of the routing table of $s$ and routing label of $t$, the algorithm first checks if $s$ and $t$ are connected in $G_i \\setminus F_\\ell$ using Lemma \\ref{lem:succint_path_routing}. If the answer is no, the algorithm moves to the next iteration $i+1$.\nOtherwise, by applying the decoding algorithm of Lemma \\ref{lem:succint_path_routing}, it computes the succinct path $\\widehat{P}_\\ell$. The path $\\widehat{P}_\\ell$ encodes an $s$-$t$ path in $G_i \\setminus F_\\ell$, that includes the relevant port and tree routing information of its vertices. The header of the message $H_\\ell$ then contains \n$$H_\\ell=\\langle \\widehat{P}_\\ell, i, i^*(t), \\{\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)\\}_{e \\in F_{\\ell}}, q \\rangle~,$$ where $q = O(f)$ is an index indicating the current segment of $\\widehat{P}_\\ell$ we explore. \\mertodo{$H_\\ell$ should contain $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_i,T_i}(e)$, right?}\n\n\n\\mtext{The size of the header is $\\widetilde{O}(f^2)$, as the description of the path has size $\\widetilde{O}(f)$, and additionally we have at most $f$ faulty edges with labels of size $\\widetilde{O}(f)$.} \\mtodo{in some sense we don't really need to have all this $O(f^2)$ information on the header at once because it's enough that $s$ learns in each phase on one of the labels and then stores it locally, but this I guess doesn't work with the formal definition of routing.}\nLet $P_\\ell$ be the $G$-path encoded by the path $\\mathcal{P}_\\ell$. The algorithm then routes the message along $P_\\ell$ in the same manner as in Sec. \\ref{sec:routing-known}. In the case where $P_{\\ell}\\cap F=\\emptyset$, the phase successfully terminates at the destination vertex $t$. From now on, we consider the case that $P_{\\ell}$ contains at least one faulty edge. \n\nLet $e=(u,v)$ be the first edge (closest to $s$) on the path $P_\\ell$ that belongs to $F$. Since $P_\\ell \\cap F_\\ell=\\emptyset$, it holds that $e \\in F \\setminus F_\\ell$. Without loss of generality, assume that $u$ is closer to $s$ on $P_\\ell$. Thus the faulty edge $e$ is detected upon arriving to the vertex $u$. \nIn the case where $e$ is a \\emph{non-tree edge}, then it must be a $G$-edge on $\\widehat{P}_\\ell$. Since this path has the ids $ID_{T_i}(e)$ of its $G$-edges, and since the connectivity label of a non-tree edge $e$ is $ID_{T_i}(e)$ in all the $f$ applications of the scheme on $G_i$, $u$ can add $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$ to the header of the message. \\mtodo{here we use the fact that the extended ids are the same in all $f$ phases (of the same tree)} Assume now that $e$ is a tree edge in $T_i$. The vertex $u$ then adds the label $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$ to the header of the message, as $e$ is a tree edge adjacent to $u$ it has this information in its routing table. Finally, it marks the header with the sign $R$, indicating that the message should now be routed in the reverse direction, until arriving $s$ again. This completes the description of phase $\\ell$. It is easy to see that the invariant is maintained. If $s$ and $t$ are connected in $G_i \\setminus F$, after at most $f$ phases all faulty edges are detected. In the last phase, the path computed based on the labeling information is free from faulty edges, and the routing is completed (in the same manner as in Sec. \\ref{sec:routing-known}) at the destination $t$. We next bound the multiplicative stretch of the routing.\n\n\n\n\n\\mtodo{updated the stretch analysis.}\n\n\\mtext{\n\\begin{claim}\\label{cl:route-length}\nFix a set of faulty edges $F$, and let $s,t$ be vertices that are connected in $G \\setminus F$. Then, the message is routed from $s$ to $t$ within $32k (f+1)^2 \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ steps, w.h.p.\n\\end{claim}\n\\begin{proof}\nFirst note that since each phase and each graph $G_{i}$ uses an independent set of FT-connectivity labels, then in each iteration and each phase the decoding algorithm succeeds w.h.p. and outputs an $s$-$t$ path $\\widehat{P}_\\ell$ if exists. \n\nAssume that $\\mbox{\\rm dist}_{G \\setminus F}(s,t) \\in (2^{i-1},2^i]$. Then, $s$ and $t$ are connected in $G_i \\setminus F$, as $T_i = T_{i,i^*(t)}$ contains the $2^i$-ball around $t$. We show that the algorithm terminates at $t$ in iteration $i$ or before it, and that in any iteration $j \\leq i$, the routing algorithm traverses a path of length at most $2(4k-1)(f+1)^2 \\cdot 2^j$.\n\nLet $j \\leq i$. In the $\\ell$'th phase of iteration $j$, the algorithm first checks if $s$ and $t$ are connected in $G_j \\setminus F_{\\ell}$, where $F_{\\ell}$ is the set of currently detected faults. If the answer is no, the algorithm proceeds to the next iteration. Otherwise, it tries to route a message from $s$ to $t$ on the path encoded by $\\widehat{P}_{\\ell}$. The length of the path is bounded by $(4k-1)(f+1)\\cdot 2^j$ from Lemma \\ref{lem:succint_path_routing}. The algorithm either succeeds, or finds a faulty edge on the way in which case it returns to $s$ by traversing the same path on the reverse direction. Overall, the algorithm traverses a path of length at most $2(4k-1)(f+1)\\cdot 2^j$, in this phase. In all $f+1$ phases of iteration $j$, the length of the path explored is at most $2(4k-1)(f+1)^2 \\cdot 2^j$. Summing over all iterations $j \\leq i$, the stretch is bounded by $$\\sum_{j=1}^{i} 2(4k-1)(f+1)^2 \\cdot 2^j = 2(4k-1)(f+1)^2 \\sum_{j=1}^i 2^j \\leq 2^{i+2} (4k-1)(f+1)^2 \\leq 32k (f+1)^2 \\mbox{\\rm dist}_{G \\setminus F}(s,t).$$ The last inequality uses the fact that $2^{i-1} \\leq \\mbox{\\rm dist}_{G \\setminus F}(s,t).$ \n\nIn the $i$'th iteration, since $s$ and $t$ are connected in $G_i \\setminus F$, then for any $F_{\\ell} \\subseteq F$, $s$ and $t$ are connected in $G_i \\setminus F_{\\ell}$, hence the algorithm always finds a path $\\widehat{P}_{\\ell}$. Hence, it either succeeds in routing the message to $t$ in one of the phases (or one of the previous iterations), or learns about all the failures $F$. In the latter case, in phase $f+1$ it learns about a failure-free path $\\widehat{P}_{f+1}$, and the routing terminates at $t$. This completes the proof.\n\\end{proof}\n}\n\n\\remove{\n\\begin{claim}\\label{cl:route-length}\nFix a set of faulty edges $F$. Then, the message is routed from $s$ to $t$ within $(16k-4)(f+1) f\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ steps, w.h.p.\n\\end{claim}\n\\begin{proof}\nFirst note that since each phase and each graph $G_{i}$ uses an independent set of FT-connectivity labels, w.h.p., the decoding algorithm of phase $\\ell$ applied on the label set $\\{\\mathsf{DistLabel}\\xspace(w), w \\in \\{s,t\\} \\cup F_\\ell\\}$ succeeds w.h.p. and outputs an $s$-$t$ path $\\widehat{P}_\\ell$. \n\nLet $P_{\\ell}$ be the $s$-$t$ path in $G \\setminus F_\\ell$ encoded by $\\widehat{P}_\\ell$. By Lemma \\ref{lem:approx-dist-recovery}, the length of $P_\\ell$ is bounded by $(8k-2)(f+1)\\cdot \\mbox{\\rm dist}_{G \\setminus F_\\ell}(s,t)$. Since in phase $\\ell$ the message is routed along a (possibly) partial path $P_\\ell$ and back to $s$, we have $2|P_\\ell|$ routing hops \\mtodo{change to capture weighted graphs?} in that phase. Overall, we have at most $2\\sum_{i=1} |P_\\ell|$ routing hops. Since \n$F_\\ell \\subseteq F$ for every $\\ell$, it holds that $\\mbox{\\rm dist}_{G \\setminus F_\\ell}(s,t)\\leq \\mbox{\\rm dist}_{G \\setminus F}(s,t)$, and consequently $(16k-4)(f+1) f \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ routing steps over all. The claim follows. \n\\end{proof}\n}\n\nTo conclude, we have the following.\n\n\\begin{theorem}\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ the source vertex $s$ and a label $L_{route}(t)$ of the destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k (f+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. \nThe global table size is $\\widetilde{O}(f \\cdot n^{1+1\/k} \\log{(nW)})$.\nThe header size of the messages is bounded by $\\widetilde{O}(f^2)$ bits, and the label size of vertices is $O(\\log{(nW)} \\log{n})$. \n\\end{theorem}\n\n\n\n\n\n\\mtodo{I didn't change the next section, as some of the definitions of routing tables\/labels changed, there are probably also changes here.}\n\n\\paragraph{Improving the size of the routing tables.} \nSo far, we have described a routing scheme that consumes a total space of $\\widetilde{O}(f\\cdot n^{1+1\/k}\\log (nW))$ bits, and multiplicative stretch\n$32(f+1)^2 k$. We now explain the required modifications needed to providing routing tables with $\\widetilde{O}(f^2\\cdot n^{1\/k})$ bits per vertex. The most space consuming information for a vertex $u$ is the connectivity labeling\ninformation of the edges incident to $u$ in each of the trees $T_{i,j} \\in \\mathcal{T}$. As the degree of $u$ in some of the trees might be $\\Theta(n)$, it leads to tables of linear size. To reduce the space of the individual tables, we apply a load balancing idea which distributes the labeling information incident to \\emph{high}-degree vertices among their neighbors. \n\nInstead of storing the labeling information of $e=(u,v)$ at the routing tables of $u$ and $v$, we define \nfor every tree $T \\in \\mathcal{T}$ and an edge $e=(u,v) \\in T$, a subset $\\Gamma_T(e)$ of vertices that store the connectivity labeling information of $e$ in $T$. We will make sure that the information on some vertex in $\\Gamma_T(e)$ can be easily extracted in the routing procedure, and that each vertex stores the information for a small number of edges in each tree. Consider an edge $e=(u,v)$ in a tree $T$, and assume, without loss of generality, that $u$ is the parent of $v$ in the tree $T$. In the case where $\\deg(u,T)\\leq f+1$, we simply let $\\Gamma_T(e)=\\{u,v\\}$. That is, the label of $e$ is stored by both endpoints of $e$ (as before). The interesting case is where $\\deg(u,T)\\geq f+2$, in which case, $u$ might not be able to store the label of $e$, and will assist its other children as follows.  Let $Child(u,T)=[v_1,\\ldots, v_\\ell]$ be the lexicographically ordered list of the children of $u$ in $T$.  The algorithm partitions $Child(u,T)$ into consecutive blocks of size $f+1$ (the last block might have $2f+1$ vertices). Letting $[v_{q,1}, \\ldots, v_{q,f+1}] \\subseteq Child(u,T)$ be the block containing $v$, define\n$$\\Gamma_T(e)=\\{v_{q,1}, \\ldots, v_{q,f+1}\\}~.$$\nNote that in particular, $v \\in \\Gamma_T(e)$. Thus, the label of $e$ is stored by $v$ and $\\ell \\in [f,2f-1]$ additional children on $u$ in $T$. \n\nWe then modify the tree labels from Fact \\ref{fc:route-trees} to contain the port information of $\\Gamma_T(e)$. \nTo do that, we re-define the identifier of an edge $e$ to include the port information of $\\Gamma_T(e)$. Thus, the identifier of an edge has $O(f\\log n)$ bits. Using Fact \\ref{fc:route-trees}, we then have tree labels of $O(f\\log n)$ such that given the labels of $u$ and $t$, the algorithm can compute the port information $\\Gamma_T(e)$ where $e$ is the edge incident to $u$ on the $u$-$t$ tree path in $T$. We have:\n\\begin{claim}\\label{cl:tree-label-port}\n\n\n\\end{claim}\n\nIn addition, we modify the routing tables of each vertex $v$, so that its includes the connectivity labels $\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e)$ for any tree edge $e \\in T_{i,j}$ such that $v \\in \\Gamma_{T_{i,j}}(e)$ for every tree $T_{i,j} \\in \\mathcal{T}$. This should be compared with the solution of the previous subsection, where $v$'s table contained the information for all its incident edges in $T_{i,j}$. We therefore have:\n\\begin{claim}\\label{cl:route-balance-table}\nThe size of each routing table $R_{route}(v)$ is bounded by $\\widetilde{O}(f K n^{1\/k})$ bits.\n\\end{claim}\n\\begin{proof}\nFor every tree $T_{i,j}$ containing $v$, $v$ stores that connectivity label of $\\widetilde{O}(f)$ bits of all edges in the set $E(v,T_{i,j})=\\{e \\in T_{i,j} ~\\mid~ v \\Gamma_{T_{i,j}}(e)\\}$. Note that $|E(v,T_{i,j})|\\leq f$ as each vertex stores the label of its parent in the tree, $O(f)$ child edges, and $O(f)$ child edges of its parent in the tree. Since each $v$ participates in $\\widetilde{O}(K n^{1\/k})$ trees, overall its routing table has $\\widetilde{O}(f K n^{1\/k})$ bits, as required. \n\\end{proof}\n\nIt remains to explain the required modifications for the routing procedure. Upon arriving to a vertex $u$ incident to a faulty \\emph{tree} edge $e=(u,v)$ the procedure is as follows. The interesting case is where $e$ is a tree edge in some path $\\pi(x,y,T_{i,j})$ where $(x,y)$ is an edge in the succinct path $\\widehat{P}$ indicated on the header of the message. \nIn this case, the tree routing algorithm applied at $u$ is given the tree labels $L_{T_{i,j}}(u)$ and $L_{T_{i,j}}(y)$ and output the port information of $\\Gamma_{T_{i,j}}(e)$ for the edge $e=(u,v)$ that lies on the tree path $\\pi(u,y, T_{i,j})$. Since there are at most $f$ edge faults, and $\\Gamma_{T_{i,j}}(e)$ contains information on at least $f+1$ ports of $u$'s neighbors that contain the label of $e$, the vertex $u$ can access a non-faulty neighbor, say $w$, that has the label information of $e$. That vertex can then add the labeling information of $e$ to the header of the message, and the routing algorithm proceeds as before. \n\n\n\nWe now bound the size of each routing table. \n\\begin{claim}\nThe routing table of each vertex has $\\widetilde{O}(f^2 n^{2\/k} \\log (nW))$ bits.\n\\end{claim}\n\\begin{proof}\nEach vertex appears in $O(k n^{1\/k} \\log (nW))$ trees in $\\mathcal{T}$. For each such tree, it might store the routing labels of at $O(f)$ edges. Since each label $L_{route}(e)$ has $\\widetilde{O}(f n^{1\/k} \\log (nW))$ bits, over all it stores $\\widetilde{O}(f^2 \\cdot n^{2\/k}\\log (nW))$ bits. \n\\end{proof}\n\n\\mtodo{how is the stretch affected from the process?}\n\nWe have:\n\\begin{theorem}\\label{thm:routing-unknown}[Fault-Tolerant Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$ and a destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k f^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The table size of each vertex is $\\widetilde{O}(f^2 \\cdot n^{1\/k} \\log(nW))$. The header size of the messages is bounded by $\\widetilde{O}(f^2)$ bits. \n\\end{theorem}\n\n\\paragraph{Lower Bound.} Finally, we show that the price of not knowing the set of faulty edges $F$ in advance might indeed incur a multiplicative stretch of $\\Omega(f)$. \n\n\\begin{theorem}[Stretch Lower-Bound for FT-Routing]\\label{thm:lb-routing}\nAny FT-routing randomized scheme resilient to $f$ faults induces an expected stretch of $\\Omega(f)$ regardless of the size of the routing tables and labels. In particular, this holds even if each routing table contains a complete information on the graph. \n\\end{theorem}\n\\begin{proof}\nConsider a graph that consists of $f+1$ vertex disjoint $s$-$t$ paths, each of length $L=\\Theta(n\/f)$. The last edge of each of the paths, except for one, is faulty. Assume that the non-faulty path is chosen uniformly at random. Since the routing scheme is oblivious to the faulty edges, it can discover a faulty edge only upon sending the message to one of the edge endpoints. The expected length of the routing is given by:\n$$\\frac{L}{f+1} +2L \\cdot \\left(1-\\frac{1}{f+1} \\right)\\cdot \\frac{1}{f} + \\ldots+ \\left(f+1\\right)L\\cdot \\prod_{i=0}^{f-1} \\left(1-\\frac{1}{f+1-i}\\right)=\\Omega(f L)~.$$ \nSince the $s$-$t$ shortest path under these faults is $L$, the proof follows. See Fig. \\ref{fig:LB-stretch} for an illustration.\n\\end{proof}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.40]{lb.pdf}\n\\caption{\\sf Illustration for a stretch lower bound for any FT-routing schemes. The $s$-$t$ pair are connected by $f+1$ vertex disjoint paths. Since the faulty-edge is the last edge of the path, the routing requires $\\Omega(L)$ steps to discover a single faulty edge. As the non-faulty path is chosen uniformly at random, in expectation, the routing requires $\\Omega(fL)$ steps.  \\label{fig:LB-stretch}\n}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Our Techniques} \\label{sec:techniques}\n\nFor our FT labeling schemes, we present two constructions based on different techniques. \nThe first construction uses the \\emph{cycle-space sampling} technique of Pritchard and Thurimella \\cite{pritchard2011fast} to determine if $s$ and $t$ are disconnected by a set of failures $F$. This technique has been applied in the past mainly in the context of computing small cuts in the distributed setting. \nThe second construction uses the tool of \\emph{linear sketches} by Ahn et al. \\cite{ahn2012analyzing} to try to find a path that connects $s$ and $t$ in $G \\setminus F$. This scheme is also useful for routing.\nWe next give an overview of the two approaches, and the applications for routing. Throughout, we assume that the graph $G$ is originally connected, otherwise the scheme can be applied to each connected component of $G$, which can be indicated in the label of the vertex.\n\n\\paragraph{Connectivity Labels Based on Cycle Space Sampling.} The cycle space sampling technique, introduced by Pritchard and Thurimella \\cite{pritchard2011fast}, allows one to detect cuts in a graph by exploiting the interesting connection between cuts and cycles in a graph. This technique was used in \\cite{pritchard2011fast} to design distributed algorithms for identifying small cuts in a graph. In more details, the technique is based on the relation between \\emph{induced edge cuts} and \\emph{binary circulations}, defined as follows. For a subset of vertices $S$, we denote by $\\delta(S)$ the set of edges with exactly one endpoint in $S$. An \\emph{induced edge cut} is a set of edges of the form $\\delta(S)$ for some $S$. A \\emph{binary circulation} is a set of edges in which every vertex has an even degree. For example, a cycle is a binary circulation. Note that if $F$ is an induced edge cut, and $\\phi$ is a cycle, the number of edges in the intersection $|F \\cap \\phi|$ is even, as the cycle crosses the cut even number of times. This is also true for any binary circulation $\\phi$. The cycle space technique extends this observation and shows that if $\\phi$ is a random binary circulation and $F \\subseteq E$, then\n$$Pr[|F \\cap \\phi| \\ is \\ even] = \\left\\{\n                \\begin{array}{ll}\n                  1,\\ if\\ F\\ is\\ an\\ induced\\ edge\\ cut\\\\\n                  1\/2,\\ otherwise\n                \\end{array}\n              \\right. $$ \nHence, by choosing a \\emph{random} binary circulation, one can detect if a set of edges $F$ is an induced edge cut with probability $1\/2$.  To increase the success probability, we can choose $b$ random binary circulations. \nBased on these ideas, \\cite{pritchard2011fast} showed how to assign the edges of the graph $b$-bit labels with the following property.  See Appendix \\ref{sec:cycle_space_overview} for an overview.\n\n\\begin{restatable}{lemma}{cycle} \\label{cycle_space_lemma}\nThere is an algorithm that assigns the edges of a graph $G=(V,E)$, $b$-bit labels $\\phi(e)$ such that given a subset of edges $F \\subseteq E$, we have:\n$$Pr[\\Moplus_{e \\in F} \\phi(e) = 0] = \\left\\{\n                \\begin{array}{ll}\n                  1,\\ if\\ F\\ is\\ an\\ induced\\ edge\\ cut\\\\\n                  2^{-b},\\ otherwise\n                \\end{array}\n              \\right. $$ \nWhere $0$ is the all-zero vector. The time complexity for assigning the labels is $O((m+n)b)$.\n\\end{restatable}\n\n\\noindent\\textbf{The connectivity labels.} We next explain how to use this technique to build FT connectivity labels. Our goal is to assign labels to the vertices and edges of the graph, such that given the labels of two vertices $s,t$ and a set of failures $F$, we can check if $s$ and $t$ are disconnected by $F$. It is easy to show that $s$ and $t$ are disconnected by $F$ iff there is an \\emph{induced edge cut} $F' \\subseteq F$ that disconnects $s$ and $t$. While we can use the cycle space labels to check if a subset of edges $F' \\subseteq F$ is an induced edge cut, this is still not enough to solve FT connectivity. To do so, we should check if an induced edge cut $F'$ \\emph{disconnects} the vertices $s$ and $t$. To check this, we bring to our construction \\emph{ancestry labels} in trees, and show that we can determine if $s$ and $t$ are in the same side of cut (induced by $F'$) based on the ancestry labels of $s,t$ and $F'$. The key observation is that a spanning tree $T$ of the graph is disconnected to at most $|F'|+1$ connected components, upon removing $F'$, where for any $e \\in F'$ both its endpoints reside on two different sides of the induced edge cut defined by $F'$.\nWe can use this to identify which components of $T \\setminus F'$ are on the same side of the induced edge cut. Moreover, we show that the ancestry labels allow us to determine the connected components of $s$ and $t$ in $T \\setminus F'$. A brute-force implementation of this approach leads to a decoding time that is \\emph{exponential} in $|F|$. I.e., the algorithm should check for any subset $F' \\subseteq F$ if $F'$ is an induced edge cut. To overcome it, we show an efficient way to find $F' \\subseteq F$ that disconnects $s$ and $t$ if exists, by translating our problem to a system of linear equations. This results in a decoding time polynomial in $|F|$ and $\\log{n}$. The size of the labels is $O(f+\\log{n})$, to guarantee that the cycle space labels are correct for any $F' \\subseteq F$ w.h.p.      \n\n\\paragraph{Connectivity Labels Based on Graph Sketches.} We next provide some flavor of our labels based graph sketches. The length of the labels obtained in this technique is $O(\\log^3{n})$ bits, which is dominated by the sketching information. A \\emph{graph sketch} of a vertex $v$ is a randomized string of $\\widetilde{O}(1)$ bits that compresses $v$'s edges. The linearity of these sketches allows one to infer, given the sketches of subset of vertices $S$, an outgoing cut edge $(S, V \\setminus S)$. Graph sketches have numerous applications in the context of connectivity computation under various computational settings, e.g., \\cite{kapron2013dynamic,kapralov2014spanners,GibbKKT15,DBLP:conf\/podc\/KingKT15,DBLP:conf\/wdag\/MashreghiK18,GhaffariP16,DuanConnectivitySODA17}. More concretely, our sketch-based labels are inspired by the centralized connectivity sensitivity oracles of Duan and Pettie \\cite{DuanConnectivitySODA17}.   A common approach for deducing the graph connectivity merely from the sketches of the individual vertices is based on the well-known Boruvka algorithm \\cite{Boruvka}. This algorithm works in $O(\\log n)$ phases, where in each phase, from each growable component an outgoing edge is selected. All these outgoing\nedges are added to the forest, while ignoring cycles. Each such phase reduces the number of\ngrowable components by a $2$ factor, thus within $O(\\log n)$ phases, a maximal forest is computed. Since this algorithm only requires the computation of outgoing edges it can simulated using $O(\\log n)$ independent sketches for each of the vertices. \n\nOur high level approach for determining the $s$-$t$ connectivity in $G \\setminus F$ mimics this above mentioned procedure. For simplicity assume that $G$ is connected and let $T$ be some spanning tree in $G$. Using ancestry labels, one can infer the components of $T \\setminus F$. Moreover, by augmenting the labels with graph sketching information, one can also deduce the sketch of each component in $T \\setminus F$. \nNote however that these sketches are in $G$ and therefore might encode outgoing edges that belong to $F$. To overcome this technicality, our sketching scheme allows us to cancel out the effect of the faulty edges $F$ from the sketching information. Consequently, we obtain the sketches of each $T \\setminus F$ component in the surviving graph $G \\setminus F$. We can then apply the Boruvka's algorithm on the components of $T \\setminus F$, and infer the $s$-$t$ connectivity in $G \\setminus F$. The actual implementation of this labeling scheme is somewhat more delicate. We note that some of these technicalities are for the sake of our later extension of these labels into compact routing schemes.\n\n\n\n\\remove{\n\\mertodo{This text can be omitted now, see if want to move elsewhere. We start by illustrating the underlying intuition for sketch. For a vertex $v$ and a subset of edges $E' \\subseteq E$, let $\\mathsf{Sketch}\\xspace_{E'}(v)$ be the bitwise XOR of all the IDs of $E'$ edges adjacent to $v$.  \nFor a subset of vertices $S$, define $\\mathsf{Sketch}\\xspace_{E'}(S) = \\oplus_{v \\in S} \\mathsf{Sketch}\\xspace_{E'}(v)$. The useful property of sketches is that all edges of $E'$ that have both endpoints in $S$ are cancelled, and thus $\\mathsf{Sketch}\\xspace_{E'}(S)$ corresponds to the XOR of the identifiers of the $E'$ edges outgoing from $S$. If there is only one such edge, then\nits ID corresponds to the value of $\\mathsf{Sketch}\\xspace_{E'}(S)$. By combining this idea with a basic sampling trick one can \nidentify one outgoing edge from any subset $S$. But here we should also require special edge IDs in order to distinguish between an illegal ID, obtained by XORing IDs of several edges, and a true ID of a single edge.\nIntuitively, we first define $O(\\log{m})$ sets of edges $E_j$, where $E_j$ is obtained from $E$ by sampling each edge with probability $1\/2^j$. Next, we define $\\mathsf{Sketch}\\xspace(v) = (\\mathsf{Sketch}\\xspace_{E_0}(v),...,\\mathsf{Sketch}\\xspace_{E_{\\log{m}}}(v))$, and $\\mathsf{Sketch}\\xspace(S) = (\\mathsf{Sketch}\\xspace_{E_0}(S),...,\\mathsf{Sketch}\\xspace_{E_{\\log{m}}}(S))$. These $O(\\log^2{n})$-bit sketch units have the property that given $\\mathsf{Sketch}\\xspace(S)$, with constant probability there is a sketch unit that holds the identifier of exactly one outgoing edge of $S$. In our algorithm, we use $\\Theta(\\log{n})$ sketch units (each time with different sampled sets $E_j$) to be able to eventually find outgoing edges w.h.p. \nA crucial point in this regard is to be able to distinguish between sketch units that hold the XOR of at least two edge identifiers vs. units that store the identifier of \\emph{exactly} one edge (i.e., an outing edge). For that purpose we employ the computation of edge identifiers by \\cite{GhaffariP16}, that have the property that the XOR of any two edge identifiers is not a \\emph{legal} edge identifier of a single edge. We also show that identification of the legal edge can be done efficiently, with no global information.\n\n\\noindent\\textbf{FT Connectivity from graph sketches.} We use the graph sketches together with the well-known Boruvka algorithm \\cite{Boruvka} to identify the connected components of the graph $G \\setminus F$. This eventually allows us to check if $s$ and $t$ are connected in $G \\setminus F$ based on their components. We start by describing our general approach, and then explain how to simulate it based only on labels of $s,t$ and $F$. Let $T$ be a spanning tree of the graph $G$. If the edges $F$ are removed from $G$, it breaks the tree $T$ to at most $|F|+1$ connected components. To figure out the connected components in $G \\setminus F$, we apply Boruvka algorithm on the connected components of $T \\setminus F$. In this algorithm, at each iteration we have a set of components, and our goal is to find an outgoing edge from each connected component, and then merge components connected by an edge. To find outgoing edges, we use the sketches of the components. After repeating the process for $O(\\log{n})$ iterations, we find the connected components of $G \\setminus F$ w.h.p. The vertices $s$ and $t$ are connected in $G \\setminus F$ iff they are in the same component. \n\\noindent\\textbf{The connectivity labels.} At a high-level, to simulate the algorithm using only the information provided by the connectivity labels, we include in the labels of vertices and edges ancestry labels in a spanning tree $T$. In addition, for any tree edge $e=\\{u,v\\} \\in T$ we include in its label the sketch information of $T_v$ and $T_u$, as well as the sketch information of $T$, where $T_v,T_u$ are the subtrees of $T$ rooted at $v$ and $u$, respectively. \nWe show that this information allows us to identify the connected components in $T \\setminus F$, and compute the sketch information of each one of the components. Moreover, the ancestry labels give us an efficient way to identify the connected component of any vertex $v$. This is useful both for identifying the connected components of $s$ and $t$, and to find the connected components of any outgoing edge that the algorithm finds. When we merge components, the sketch information of the new component can be obtained by XORing the sketches of the components we merge. One delicate point in the algorithm is that we originally compute sketches in the original graph $G$, where our algorithm works in the graph $G \\setminus F$, and so we need the sketch information in $G \\setminus F$. To obtain this, we cancel the information about edges from $F$ in the sketches by XORing their IDs in the relevant places. This can be done efficiently by using pairwise independence hash functions to generate the sketches. \n}\n}\n\n\n\\paragraph{Applications for Routing Schemes.} The starting point to our routing scheme is given by our (sketch-based) labeling scheme. These labels allows one to deduce also a succinct description of an $s-t$ path in $G \\setminus F$ if exists, by following the component merging procedure of the Boruvka algorithm. This description is composed of $O(f)$ path segments, where each segment $\\{u,v\\}$ either corresponds to an outgoing (non-tree) edge found in the algorithm using the sketch information, or to a tree path between two vertices $u$ and $v$ in the same connected component in $T \\setminus F$. Given the connectivity labels of $s,t$ and $F$, we can find this description, and use it for routing. Routing across an edge $\\{u,v\\}$ just requires sending a message over the edge, while routing on a tree path between $u$ and $v$ can be done using a routing scheme for trees. \nWhile this approach allows to send a message from $s$ to $t$, there is no bound on the length of the path traversed. Additionally, this approach assumes that the set of failures $F$ is known in advance. We next explain how to overcome these issues.\n\\\\\n\\noindent\\textbf{Bounding the stretch.} To route messages on low-stretch paths we use the notion of \\emph{tree covers}, following the approach in \\cite{chechik2012f}. This approach also allows us to translate our connectivity labels to approximate distance labels as we discuss in Section \\ref{sec:ft-distance}. Here, instead of applying our connectivity scheme on just one graph $G$, we apply it on many subgraphs $G_{i,j}$ of $G$ with the following properties. \n\\begin{enumerate}\n\\item Each vertex $v$ is contained in $\\widetilde{O}(n^{1\/k})$ subgraphs.\n\\item For any $1 \\leq i \\leq \\log(nW)$, and any vertex $v$, there is a subgraph $G_{i,i^*(v)}$ that contains all the vertices in the $2^i$-neighborhood of $v$.\n\\item If $v$ and $u$ are connected in the graph $G_{i,i^*(v)} \\setminus F$, then there is a path between them of length at most $O(k|F| 2^i)$ in the graph $G_{i,i^*(v)} \\setminus F$.\\label{prop_path}\n\\end{enumerate}\nBy applying our connectivity scheme on each one of the subgraphs $G_{i,j}$, we can route a message from $s$ to $t$ on a path of stretch $O(k|F|)$. The size of the labels and routing tables of vertices is $\\widetilde{O}(n^{1\/k})$ as each vertex and edge participate in $\\widetilde{O}(n^{1\/k})$ subgraphs.\n\n\\\\\n\\noindent\\textbf{Faulty edges are unknown.} The scheme we described assumes that the routing algorithm knows the labels of $s,t$ and $F$ in advance, we next explain how to avoid this assumption. Our general approach is to work in phases, where in each phase we try to route a message from $s$ to $t$ according to the currently set of known faults. We either succeed, or learn about the label of a new faulty edge $e \\in F$ and try again. The stretch of the scheme increases to $O(k|F|^2)$ because of the $|F|+1$ phases. Direct application of this approach may require large routing tables, as each vertex may need to know the labels of all edges adjacent to it, to be able to learn the labels of faulty edges found in the algorithm. To overcome it we use the following ingredients. \n\nFirst, recall that in our connectivity labeling scheme we use a spanning tree $T$. In the routing scheme, these are the trees of the tree cover. We show that it is enough for each vertex to store labels only of its adjacent \\emph{tree} edges. \nConsequently, the total size of all routing tables can be bounded by $\\widetilde{O}(fn^{1+1\/k})$.\\footnote{The $f$ term in the size comes from the fact we apply the connectivity labels $f+1$ times to support the $|F|+1$ phases.}\nHowever, this alone is not enough to bound the size of individual routing tables of vertices, as the degree of a vertex in a tree may be linear. To overcome this, we show a clever way to load balance the labels' information between $v$ and its children in the tree. This results in tables of size $\\widetilde{O}(f^3 n^{1\/k})$ per vertex, while keeping the same stretch of the scheme. The increase in the total size of tables comes from the fact we now duplicate labels $f+1$ times, to be able to recover them in the presence of $f$ failures. ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s_intro}\n\\addtocounter{footnote}{9}\n\n\nPolycyclic aromatic hydrocarbons (PAHs) are large carbon molecules that can be thought of as the transition between the gaseous and solid phases of the interstellar medium (ISM).  They are commonly identified as the source of emission for multiple broad emission features in the mid-infrared, including those at 3.3, 6.2, 7.7, 11.3, 12.7~$\\mu$m \\citep{2008tielens}.  While these complex molecules can be excited by optical and infrared photons \\citep{2002li}, excitation by ultraviolet photons is more efficient \\citep{2008tielens}.  \n\n\nMany authors have sought to calibrate PAH emission as an extragalactic star formation tracer, mainly because it is relatively unaffected by dust extinction and because PAHs emit at shorter wavelengths than hot dust and can therefore be imaged with better angular resolutions.  While early analyses with the Infrared Space Observatory \\citep{1996kessler} found some evidence for a relation between PAH emission and other star formation tracers \\citep{2001roussel,2004forster}, later work, including work with the {\\it Spitzer} Space Telescope \\citep{2004werner}, demonstrated that the PAH emission was actually poorly correlated with other star formation tracers.  PAH emission appeared suppressed relative to other star formation tracers in star forming regions or overluminous in diffuse regions \\citep{2004boselli,2004helou,2005calzetti,2007prescott,2008bendo}.  The ratio of PAH to hot dust continuum emission was also found to decrease as metallicity decreased \\citep{2005engelbracht,2006madden,2007calzetti,2008engelbracht,2008galliano,2008gordon,2013galametz}. The common explanations are either that low metallicity regions contain fewer PAHs or that the PAHs are exposed to harder ultraviolet radiation in low-metallicity environments with less dust attenuation.  None the less, globally-integrated PAH emission has been shown to be correlated with other globally-integrated star formation tracers, and methods have been developed for calculating extinction-corrected star formation rates using a combination of H$\\alpha$ and PAH emission in the 8~$\\mu$m {\\it Spitzer} band \\citep{2008zhu,2009kennicutt}, although \\citet{2007calzetti} warns that even global measurements could be affected by metallicity effects.\n\n\nIn contrast, a few authors have found that PAH emission was associated with emission at $>100~\\mu$m that will primarily trace $\\ltsim30$~K large dust grains.  One of the first groups to draw attention to this was \\citet{2002haas}, who demonstrated that PAH emission was better correlated with 850~$\\mu$m from cold dust than 15~$\\mu$m emission from hot dust, although this analysis was limited to regions with high infrared surface brightnesses.  The ISO-based analysis by \\citet{2004boselli} also implied that PAHs were associated with diffuse dust rather than star forming regions.  Later work by \\citet{2006bendo} and \\cite{2008bendo} demonstrated that the PAH emission showed a strong correlation with 160~$\\mu$m emission and that the 8\/160~$\\mu$m surface brightness ratio was dependent upon the 160~$\\mu$m surface brightness.  These results combined with the breakdown in the relation between the PAH and hot dust emission implied that the PAHs were primarily associated with dust in the diffuse ISM or in cold molecular clouds near star forming regions and that both the PAHs and the large dust grains were heated by the same radiation field.\n\n\nData from the {\\it Herschel} Space Observatory \\citep{2010pilbratt} can be used to further study the relation between PAHs and large dust grains.  The telescope is able to resolve emission at 160 and 250~$\\mu$m on $<18$~arcsec scales, which is a marked improvement in comparison to the 38~arcsec scales that could be resolved by {\\it Spitzer} at 160~$\\mu$m.  Moreover, \\citet{2010bendo}, \\citet{2011boquien}, and \\citet{2012bendo} have demonstrated that $\\leq160$ and $\\geq250$~$\\mu$m emission from nearby galaxies may originate from dust heated by different sources.  The 70\/160 and 160\/250~$\\mu$m surface brightness ratios were typically correlated with star formation tracers such as ultraviolet, H$\\alpha$, and 24~$\\mu$m emission and peaked in locations with strong star formation, suggesting that the dust seen at $\\leq160$~$\\mu$m is primarily heated locally by star forming regions.  Meanwhile, the 250\/350 and 350\/500~$\\mu$m ratios were more strongly correlated with near-infrared emission and generally varied radially in the same way as the emission from the total stellar population (including both young, intermediate-aged, and evolved stars), demosntrating that the dust was primarily heated by the diffuse interstellar radiation field (ISRF) from these stars.  PAH emission can be compared to dust emission observed by {\\it Herschel} to determine which of these two dust components are more closely associated with PAH emission, which would ultimately lead to a better understanding of how the PAHs are excited and how they survive in certain environments in the ISM.\n\n\nSo far, the relation between PAH and dust emission has been investigated using {\\it Herschel} data for only two galaxies.  \\cite{2014calapa} have shown that 8~$\\mu$m emission from PAHs in M33 is well correlated with 250~$\\mu$m emission.  They go on to further demonstrate the 8\/250~$\\mu$m ratio is correlated with the 3.6~$\\mu$m band tracing the total stellar population, implying that the PAHs are excited by the diffuse ISRF.  \\citet{2014lu} present an alternative analysis with M81 in which they divide the PAH emission into components heated by two sources: a component heated by star forming regions traced by H$\\alpha$ emission and a component heated by the diffuse ISRF traced by the cold dust emission at 500~$\\mu$m emission.  The results show that most ($\\sim85$\\%) of the 8~$\\mu$m emission from diffuse regions is associated with the cold dust emission, while in star forming regions, most ($\\sim60$\\%) of the 8~$\\mu$m emission is excited by young stars.\n\n\nThe goal of this paper, which is a continuation of the work by \\citet{2013jones}, is to further study the relationship between PAH emission at 8~$\\mu$m and far-infrared emission from large dust grains using {\\it Herschel} Space Observatory \\citep{2010pilbratt} far-infrared images of NGC 2403 and M83.  These are two of the fourteen nearby galaxies within the Very Nearby Galaxies Survey (VNGS; PI: C. Wilson), a {\\it Herschel}-SPIRE Local Galaxies Guaranteed Time Program.  The VNGS was meant to sample galaxies with multiple morphological and active galactic nucleus types, and includes several well-studied galaxies including the Antennae Galaxies, Arp 220, Centaurus A, M51, and NGC 1068.  These two specific galaxies were selected because they are non-interacting nearby ($<10$~Mpc) spiral galaxies with an inclination from face on $\\leq~60^{o}$ and major axes $>10$~arcmin\\footnote{M81 is also in the VNGS, but the analysis of PAH emission from that galaxy is covered by \\citet{2014lu}.}  The basic properties of these galaxies are given in Table \\ref{t_galaxies}.\n\n\nNGC 2403 is an SAB(s)cd galaxy \\citep{1991devaucouleurs} with no clear bulge and flocculent spiral structure \\citep{1987elmegreen}.  Since the brightest star forming regions are found well outside the centre of the galaxy, it is easy to differentiate between effects related to star forming regions and either effects related to the evolved stellar population (which peaks in the centre of the galaxy) or effects tied to galactocentric radius.  This has been exploited previously to illustrate how PAH emission is inhibited relative to hot dust emission in star forming regions \\citep{2008bendo} and to differentiate between different heating sources for the dust seen at 70-500~$\\mu$m \\citep{2012bendo}.  M83 (NGC 5236) is an SAB(s)c galaxy \\citep{1991devaucouleurs} with a bright starburst nucleus \\citep{1983bohlin} and two strongly defined grand design spiral arms \\citep{1998elmegreen}.  Since we can resolve the spiral structure with {\\it Herschel}, we can compare the properties of arm and interarm regions quite effectively.  Both galaxies are at similar distances; we can resolve structures of $<400$~pc in the {\\it Herschel} data.  Although both of these galaxies are late-type spiral galaxies, they have the potential to yield different information on how PAHs relate to the far-infrared emission from large dust grains.\n\n\nWe focus our analysis on the {\\it Spitzer} 24~$\\mu$m data, which trace emission from very small grains and hot dust heated locally by star forming regions, and {\\it Herschel} 160 and 250~$\\mu$m data, which trace emission from large dust grains.  The prior analysis by \\citet{2008bendo} had shown an association between the 8~$\\mu$m and 160~$\\mu$m emission, but as stated above, the 160~$\\mu$m band may contain significant emission from large dust grains heated by star forming regions, while the 250~$\\mu$m band, at least for NGC~2403 and M83, originates more from dust heated by the diffuse ISRF \\citep{2012bendo} and could be better associated with PAH emission if PAHs are destroyed in star forming regions.  The next shortest waveband for which we have data for these two galaxies is at 70~$\\mu$m, but the available 70~$\\mu$m data have a lower signal to noise ratio, and the {\\it Spitzer} data are strongly affected by latent image artefacts.  Moreover, the 70~$\\mu$m emission may include emission from the same sources as the 24~$\\mu$m band.  The available 350 and 500~$\\mu$m data trace the same thermal component of dust seen at 250~$\\mu$m, but because the resolution of those data are coarser compared to the 250~$\\mu$m waveband, using the data would provide no additional benefit. \n\n\nFor this analysis, we use the techniques developed by \\citet{2008bendo} and \\citet{2012bendo} based upon qualitative analyses of surface brightness ratio maps based on images with matching point spread functions (PSFs) and quantitative analyses of the surface brightnesses and surface brightness rations measured in rebinned versions of these images.  Section \\ref{s_data} introduces the data and the data preparation steps.  We then present the analysis in Section~\\ref{s_analysis_ratios} and then use these results to identify the PAH excitation sources in Section~\\ref{s_pahexcitation}.  Following this, we discuss the implications of these results in Section~\\ref{s_discussion} and provide a summary in Section \\ref{s_conclusions}. \n\n\n\\begin{table*}\n\\centering\n\\begin{minipage}{94mm}\n\\caption{Properties of the sample galaxies$^a$.}\n\\label{t_galaxies}\n\\begin{tabular}{@{}lccccc@{}}\n\\hline\nName \t&\n    RA & \n    Dec &\n    Hubble &\n    Distance &\n    Size of Optical \\\\ \n&\n    (J2000) & \n    (J2000) &\n    Type &\n    (Mpc)$^b$ &\n    Disc (arcmin) \\\\ \n\\hline\nNGC 2403 & \n    07 36 54.5 &\n   +65 35 58 &\n   SAB(s)cd &\n   3.2 $\\pm$0.3 &\n   $22.0 \\times 12.3$ \\\\\nM83 &\n    13 37 00.3 &\n    -29 52 04 &\n    SAB(s)c &\n    4.5$\\pm$ 0.2 &\n    $12.9 \\times 11.3$ \\\\\n\\hline\n\\end{tabular}\n$^{a}$ Data are taken from \\cite{1991devaucouleurs} unless otherwise specified.\\\\\n$^{b}$ Distances are taken from \\cite{2001freedman}.\\\\ \n\\end{minipage}\n\\end{table*}\n\n\n\\section{Data}\n\\label{s_data}       \n\n\nThe 3.6, 4.5, 5.8 and 8.0~$\\mu$m data for NGC~2403 were observed with the Infrared Array Camera \\citep[IRAC; ][]{2004fazio} on {\\it Spitzer} as part of the {\\it Spitzer} Infrared Nearby Galaxies Survey \\citep[SINGS; ][]{2003kennicutt}, and the 3.6-8.0~$\\mu$m images for M83 were observed with IRAC by the Local Volume Legacy (LVL) Survey \\citep{2009dale}.  Both groups used similar drizzle techniques to mosaic basic calibrated data to produce final images with 0.75~arcsec pixels. The full-width at half maxima (FWHMs) of the PSFs are listed in Table~\\ref{t_IRAC}.  We also applied correction factors that optimise the data for photometry of extended source emission as suggested by the IRAC Instrument Handbook; these correction factors are listed in Table~\\ref{t_IRAC}.  The calibration uncertainty of the data is 3\\% \\citep{2013irac}.\n\n\n\\begin{table}\n\\caption{Properties of the IRAC instrument$^a$.}\n\\label{t_IRAC}\n\\begin{center}\n\\begin{tabular}{lcc}\n\\hline\nChannel &\n  FWHM  &\n  Correction Factors$^b$ \\\\\n\\hline\n3.6~$\\mu$m &\n  1.7 &\n  0.91 \\\\\n4.5~$\\mu$m &\n  1.7 &\n  0.94 \\\\\n5.8~$\\mu$m &\n  1.9 &\n  0.66 \\\\\n8.0~$\\mu$m &\n  2.0  &\n  0.74 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n$^a$ These data are given by the IRAC Instrument Handbook \\citep{2013irac}\\footnote{http:\/\/irsa.ipac.caltech.edu\/data\/SPITZER\/docs\/irac\\\\ \/iracinstrumenthandbook\/IRAC\\_Instrument\\_Handbook.pdf}.\\\\\n$^b$ These correction factors are for extended, diffuse emission. \n\\end{table}\n\n\nThe 24~$\\mu$m data were acquired using the Multiband Imaging Photometer for Spitzer \\citep[MIPS; ][]{2004rieke} on {\\it Spitzer} and were reprocessed by \\citet{2012bendomips} using the MIPS Data Analysis Tools \\citep{2005gordon} along with multiple modifications.  The final images have pixel scales of 1.5~arcsec, PSF with FWHM of 6~arcsec \\citep{2007engelbracht}, and calibration uncertainties of 4\\% \\citep{2007engelbracht}.\n\n\nThe 160~$\\mu$m data are updated versions of the 160~$\\mu$m data published by \\citet{2012bendo} and \\citet{2012foyle}.  The galaxies were observed at 160~$\\mu$m with the Photodetector Array Camera and Spectrometer \\citep[PACS; ][]{2010poglitsch} on {\\it Herschel} in four pairs of orthogonal scans performed at the 20 arcsec s$^{-1}$ rate.  The observations of NGC~2403 covered a $40\\times40$ arcmin region, while the observations of M83 covered a $25\\times25$ arcmin region.  The data were processed using the {\\it Herschel} Interactive Processing Environment \\citep[\\small{HIPE}; ][]{2010ott} version 11.1.  We used the standard data processing pipeline, which includes cosmic ray removal and cross-talk corrections, for the individual data frames.  We then remapped the data using {\\small SCANAMORPHOS} version 23 \\citep{2013roussel}, which also removes additional noise in the data and drift in the background signal.  We applied a colour correction of $1.01 \\pm\n0.07$, which has a mean value and uncertainty appropriate for emission from a modified blackbody with a temperature between 15 and 40~K and an emissivity function that scales as $\\lambda^{-\\beta}$ where $\\beta$ is between 1 and 2 \\citep{2011muller}\\footnote{http:\/\/herschel.esac.esa.int\/twiki\/pub\/Public\/PacsCalibrationWeb\\\\ \/cc\\_report\\_v1.pdf}.  The FWHM of the PSF is $\\sim12$~arcsec \\citep{2012lutz}\\footnote{\n  https:\/\/herschel.esac.esa.int\/twiki\/pub\/Public\/PacsCalibrationWeb\\\\ \/bolopsf\\_20.pdf}, and the flux calibration uncertainty is 5\\%\n\\citep{2013altieri}\\footnote{http:\/\/herschel.esac.esa.int\/Docs\/PACS\/pdf\/pacs\\_om.pdf}.\n\n\nThe 250~$\\mu$m images, produced using data from the Spectral and Photometric Imaging REceiver \\citep[SPIRE; ][]{2010griffin} on {\\it Herschel}, are also updated versions of the 250~$\\mu$m images originally published by \\citet{2012bendo} and \\citet{2012foyle}.  The observations consisted of one pair of orthogonal scans using the 30 arcsec s$^{-1}$ scan rate and nominal bias voltage settings.  The maps cover a $30\\times30$ arcmin region around NGC~2403 and $40\\times40$~arcmin region around M83.  The data were reprocessed using HIPE version 12.1 through a pipeline that includes the standard signal jump correction, cosmic ray removal, low pass filter correction, and bolometer time response corrections, but we used the BRIght Galaxy ADaptive Element method \\citep[][Smith et al., in preparation]{2012smith, 2013auld} to remove drift in the background signal and to destripe the data.  The final maps were produced using the naive mapmaker in HIPE and have pixel scales of 6~arcsec.  The FWHM of the PSF is specified by the SPIRE Handbook \\citep{2014valtchanov}\\footnote{herschel.esac.esa.int\/Docs\/SPIRE\/spire\\_handbook.pdf} as 18.1~arcsec, and the calibration uncertainty is 4\\% \\citep{2013bendo}.  To optimise the data for extended source photometry, we multiplied the data by the point source to extended source conversion factor of 91.289 MJy sr$^{-1}$ (Jy beam$^{-1}$)$^{-1}$ \\citep{2014valtchanov} and then applied a colour correction of $0.997 \\pm 0.029$, which should be appropriate for a modified blackbody with a temperature between 10 and 40~K and a $\\beta$ between 1.5 and 2 \\citep{2014valtchanov}.\n\n\nFor a discussion on the spiral density waves in M83 in Section~\\ref{s_m83pah}, we also included 0.23~$\\mu$m data from the Galaxy Evolution Explorer \\citep[GALEX; ][]{2005martin} produced by \\citet[][ see also \\citealt{2011lee}]{2009dale}.  The images have pixel scales of 1.5~arcsec, PSF FWHM of $\\sim6$~arcsec \\citep{2005martin}, and calibration uncertainties of $<1$\\% \\citep{2007morrissey}.  We applied a foreground extinction correction based on $A_{0.23\\mu\\text{m}}=0.56$ given by \\citet{2011lee} based on the \\citet{1989cardelli} extinction law with $R_V=3.1$.\n\n\nFor measuring quantitative star formation rates (so that we could identify locations that are strongly influenced by star formation using quantitative criteria), we included H$\\alpha$ data for these two galaxies in our analysis.  The H$\\alpha$ image for NGC~2403 was originally produced by \\citet{2002boselli} using data from the 1.20~m Newton Telescope at the Observatoire de Haute Provence.  The H$\\alpha$ image for M83 was produced by \\citet{2006meurer} using observations from the Cerro Tololo 1.5 Meter Telescope taken as part of the Survey for Ionization in Neutral Gas Galaxies.  We applied extinction corrections for dust attenuation within the Milky Way using calculations performed by the NASA\/IPAC Extragalactic Database\\footnote{http:\/\/ned.ipac.caltech.edu\/} based on data from \\citet{1998schlegel}, and we also use data from the literature to correct for [N{\\small II}] emission falling within the wavebands covered by the H$\\alpha$ filters.  Details on the data are given in Table~\\ref{t_hadata}.\n\n\\begin{table}\n\\caption{Properties of and corrections for the H$\\alpha$ images}\n\\label{t_hadata}\n\\begin{center}\n\\begin{tabular}{lp{1.8cm}p{1.8cm}}\n\\hline\nGalaxy &                              NGC~2403 &                        M83 \\\\ \\hline\nSource &                              \\citet{2002boselli} &             \\citet{2006meurer} \\\\\nPixel Scale  (arcsec ~pixel$^{-1}$) & 0.69 &                            0.43 \\\\\nPSF FWHM (arcsec) &                   3 &                               1.6 \\\\\nCalibration uncertainty &             5\\% &                             4\\% \\\\\nForground extinction ($A_R$) &        0.87 &                            0.144 \\\\ \\relax\n[N{\\small II}] \/ H$\\alpha$ ratio &      $0.28 \\pm 0.05^a$ &               $0.40 \\pm 0.13^b$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n$^a$ Both the 6548 and 6583 {\\AA} [N{\\small II}] lines fall within the band covered by the H$\\alpha$ filter used in the NGC~2403 observations.  This number represents the ratio of emission from both lines to H$\\alpha$ emission measured in the radial strip data from \\citet{2010moustakas}.\\\\\n$^b$ The H$\\alpha$ filter used in the M83 observations includes emission from only the [N{\\small II}] 6583 {\\AA} line.  This ratio is based on the ratio of ony that line to H$\\alpha$ emission and is calculated using data from \\citet{2005boissier}.\n\\end{table}\n\n\n\n\n\n\\subsection{Data preparation}\n\\label{s_data_prep}\n\n\nTo study the relation of PAH emission to far-infrared emission from large dust grains, we perform analyses using maps in which the PSFs have been matched to the PSF of the 250~$\\mu$m data (with a FWHM of 18~arcsec), and we plot data from images with matching PSFs that have been resampled into 18~arcsec bins that represent individual resolution elements within the maps.  The data from these bins should be statistically independent.  See \\citet{2008bendo} and \\citet{2012bendo} for additional discussion on this topic.\n\n\nIn the first step of preparing the data, foreground stars were identified by eye and removed from the H$\\alpha$, 3.6, 4.5, 5.8, 8 and 24~$\\mu$m data; these were typically sources that appeared unresolved and that had 3.6\/24~$\\mu$m flux density ratios $~\\rlap{$>$}{\\lower 1.0ex\\hbox{$\\sim$}}$10.  Next, the data were convolved with kernels from \\citet{2011aniano}\\footnote{Available from http:\/\/www.astro.princeton.edu\/$\\sim$ganiano\/Kernels.html .  Note that the kernels are created using circularised versions of instrumental PSFs.  In the case of the IRAC data, the circularised PSFs have FWHM ranging from 1.9 to 2.8~arcsec, which is larger than the original PSFs.} to match the PSFs of the H$\\alpha$, 3.6, 4.5, 5.8, 8, 24 and 160~$\\mu$m data to the 18~arcsec PSF of the 250~$\\mu$m data. This was done to preserve the colour variations across the data when it was rebinned, and it eliminated the need to perform additional aperture corrections.  The median background was then measured outside of the optical disc of each galaxy in each waveband and subtracted from the data.  For the qualitative map-based analyses, the 3.6 and 8 $\\mu$m images were shifted to match the world coordinate systems of the 24, 160 and 250~$\\mu$m maps so that we could create 8\/24, 8\/160, and 8\/250~$\\mu$m surface brightness ratio maps; the pixel size of each ratio map is set to the pixel size of the image for the longer-wavelength data used in the ratios.  For the analyses on binned data and for producing the profiles in Section~\\ref{s_m83pah}, the images were all shifted to match the world coordinate system of the 250~$\\mu$m data and then rebinned into 18~arcsec pixels to match the size of the PSF of the 250~$\\mu$m data.  The rebinning was done so that the centre of each galaxy was located at the centre of an 18~arcsec bin.  \n\n\nThe emission observed in the 4.5-24~$\\mu$m bands contains stellar continuum emission.  We remove this stellar emission by subtracting a rescaled version of the IRAC 3.6~$\\mu$m image \\citep[e.g. ]{2004helou, 2010marble, 2014ciesla}.  The IRAC 3.6~$\\mu$m band is suitable for this step because it generally contains unobscured stellar emission \\citep{2003lu}.  While hot dust emission may produce 3.6~$\\mu$m emission \\citep{2009mentuch, 2010mentuch} and while emission from PAHs at 3.3~$\\mu$m also falls within the IRAC 3.6~$\\mu$m band, the comparison of 3.6~$\\mu$m emission to H-band emission by Bendo et al. (2014, submitted) suggests that, on the spatial scales of our data, local enhancements in hot dust and 3.3~$\\mu$m PAH emission have a very minor effect on the total 3.6~$\\mu$m emission.  The continuum subtraction equations derived by \\citet{2004helou} were based on using an earlier version of Starburst99 \\citep{1999leitherer} to simulate the infrared stellar SED of a stellar population with a Salpeter initial mass function \\citep[IMF; ][]{1955salpeter} and two different metallicities.  From this analysis, Helou et al. derived mean 3.6\/8 and 3.6\/24~$\\mu$m stellar surface brightness ratios that could be used to rescale the 3.6~$\\mu$m emission and subtract it from the 8, 4.5, 5.8 and 24~$\\mu$m data.  We re-derived these values using a newer version of Starburst99 (version 6.0.3) to simulate a solar metallicity stellar population with a Kroupa IMF \\citep{2001kroupa}, which is now becoming more popular to use than the Salpeter IMF.  We also examined the differences resulting from using both the Geneva and Padova stellar evolutionary tracks and found that the selection of one set of tracks over the other did not significantly affect the results.  From these tests, we derive the following equations to subtract the stellar continuum from the 4.5-24~$\\mu$m data:\n\\begin{equation}\n\\begin{multlined}\nI_{\\nu}(4.5 \\mu \\mbox{m}~\\mbox{(SCS)} ) \\\\\n  = I_\\nu(4.5 \\mu \\mbox{m}) - (0.60 \\pm 0.02) I_\\nu(3.6 \\mu \\mbox{m}) \n\\end{multlined}\n\\end{equation}\n\n\n\\begin{equation}\n\\begin{multlined}\nI_{\\nu}(5.8 \\mu \\mbox{m}~\\mbox{(SCS)} ) \\\\\n  = I_\\nu(5.8 \\mu \\mbox{m}) - (0.40 \\pm 0.03) I_\\nu(3.6 \\mu \\mbox{m}) \n\\end{multlined}\n\\end{equation}\n\n\n\\begin{equation}\n\\begin{multlined}\nI_{\\nu}(8 \\mu \\mbox{m}~\\mbox{(SCS)} ) \\\\\n  = I_\\nu(8 \\mu \\mbox{m}) - (0.246 \\pm 0.015) I_\\nu(3.6 \\mu \\mbox{m}) \n\\end{multlined}\n\\end{equation}\n\n\n\\begin{equation}\n\\begin{multlined}\nI_\\nu(24 \\mu \\mbox{m} ~\\mbox{(SCS)} ) \\\\\n  = I_\\nu(24 \\mu \\mbox{m}) - (0.033 \\pm 0.003) I_\\nu(3.6 \\mu \\mbox{m}). \n\\end{multlined}\n\\end{equation}\nIn these equations, \"SCS\" stands for stellar continuum subtracted.  The scaling terms are based on calculations performed at time intervals equally spaced in logarithm space between $10^{7}$ and $10^{10}$ yr.  The values of the scaling terms are based on the mean of the results from using the Geneva and Padova tracks.  The uncertainties are the greater of either the difference in the mean values measured between the results for the two tracks or the larger of the standard deviations measured in the scaling terms derived for the separate tracks. Changing the metallicities to $Z=0.008$ changed the factors by $\\ltsim1\\sigma$.  The uncertainties in the coefficients translate to a $\\ltsim1$\\% uncertainty in the corrected 8 and 24~$\\mu$m maps, which is negligible compared to the calibration uncertainties.  The 4.5-8.0~$\\mu$m coefficients derived here are typically within $1\\sigma$ of equivalent coefficients derived in other studies \\citep[e.g.][]{2004helou, 2010marble, 2014ciesla}.  The coefficients for the 24~$\\mu$m data may disagree with coefficients from other papers by up to 0.012 or $4\\sigma$, although the values derived in these other papers differ among each other by 0.018.  However, this correction is so small for the 24~$\\mu$m data (typically $\\sim1$\\% in NGC~2403 and M83) that the relatively high disagreement among the values should not have a major impact on our analysis or on other analyses relying upon this type of stellar continuum subtraction.\n\n\nThe 8~$\\mu$m band still contains continuum emission from very hot grains.  In most solar-metallicity galaxies, this continuum emission may constitute $\\sim20$\\% of the total stellar-continuum-subtracted 8~$\\mu$m emission \\citep[e.g.][]{2007smith}, although in locations with very weak PAH emission, such as star-forming regions or metal-poor dwarf galaxies, a much higher percentage of the 8~$\\mu$m emission may be thermal continuum emission \\citep[e.g.][]{2005engelbracht, 2006cannon, 2008engelbracht, 2008gordon}.   To remove the excess dust continuum emission, we use the following equation derived in an empirical analysis of photometric and spectroscopic data by \\citet{2010marble}\\footnote{The equation given by \\citet{2010marble} also includes a term that integrates the emission in frequency and converts the data into units of erg s$^{-1}$ cm$^{-2}$.  Since we are comparing the PAH emission in the 8~$\\mu$m band to continuum emission in other bands that is measured in Jy arcsec$^{-2}$, it is easier to keep the 8~$\\mu$m data in units of Jy arcsec$^{-2}$, so we do not include the unit conversion term in this equation.}:\n\\begin{equation}\n\\label{e_pahdustsub}\n\\begin{multlined}\nI_{\\nu}(8 \\mu \\mbox{m}~{\\mbox{(PAH)})} =  (I_{\\nu}(8 \\mu \\mbox{m}~{\\mbox{(SCS)})}\\\\\n  - (0.091 + .314 I_{\\nu}(8 \\mu \\mbox{m})\/I_{\\nu}(24 \\mu \\mbox{m}))  \\\\\n  \\times (I_{\\nu}(4.5 \\mu \\mbox{m}~{\\mbox{(SCS)})} + I_{\\nu}(5.8 \\mu \\mbox{m}~{\\mbox{(SCS)})})^{0.718} \\\\\n  \\times I_{\\nu}(24 \\mu \\mbox{m}~{\\mbox{(SCS)})}^{0.282}) \n\\end{multlined}\n\\end{equation}\nWhen this equation is applied to our data, the 8~$\\mu$m surface brightnesses typically decrease by $15-20$\\%.  Based on the analysis from \\citet{2010marble}, the percentage difference between the 8~$\\mu$m PAH fluxes calculated using this equation and the fluxes of the spectral features measured spectroscopically is 6\\%. Throughout the rest of this paper, when we refer to 8~$\\mu$m emission, we are referring to the 8~$\\mu$m emission calculated using Equation~\\ref{e_pahdustsub}.\n\n\nFor the binned analysis, we wanted to illustrate which bins were more strongly influenced by emission from star forming regions and which regions tend to trace emission from dust predominantly heated by evolved stars. To do this, we created specific star formation rate (SSFR) maps.  We first applied an intrinsic extinction correction to the H$\\alpha$ intensities (measured in erg cm$^{-2}$ s$^{-1}$ arcsec$^{-2}$) using \n\\begin{equation}\n\\label{e_hacorr}\n\\begin{multlined}\nI(\\mbox{H$\\alpha$ (corrected)}) = I(\\mbox{H$\\alpha$ (observed)})\\\\\n\t+ 2.0 \\times 10^{-25} (12.5~\\mbox{THz})I_{\\nu}(24 \\mu \\mbox{m}) \\left( \\frac{\\mbox{erg cm$^{-2}$ s$^{-1}$}}{\\mbox{Jy}} \\right),\n\\end{multlined}\n\\end{equation}\nwhich is a variant of the correction equation given by \\citet{2009kennicutt}.  Since 24~$\\mu$m emission has been shown to be associated with H$\\alpha$ emission and other star formation tracers \\citep[e.g.][]{2005calzetti, 2007calzetti, 2007prescott, 2014bendo}, it is the best band to use when correcting H$\\alpha$ emission for intrinsic dust extinction.  The 24~$\\mu$m band may also contain emission from diffuse dust heated by the radiation field from evolved stars \\citep{2009kennicutt}, which we would expect to affect low surface brightness regions in these galaxies, so low star formation rates derived using Equation~\\ref{e_hacorr} should be treated cautiously.\n\n\nAfter converting the corrected H$\\alpha$ intensities to units of erg s$^{-1}$ pc$^{-2}$ (written as $L(\\mbox{H}\\alpha)\/A$, with $A$ representing the area per pixel in pc$^2$), we used \n\\begin{equation}\n\\Sigma(\\mbox{SFR}) = 7.9 \\times 10^{-42}\\left(\\frac{L(\\mbox{H}\\alpha)}{A}\\right)\\left(\\frac{\\mbox{erg s$^{-1}$}}{\\mbox{M$_\\odot$ yr$^{-1}$}}\\right)\n\\end{equation}\nfrom \\citet{1998kennicutt} to calculate star formation rate surface densities $\\Sigma(\\mbox{SFR})$.  To produce maps of the total stellar surface mass density $\\Sigma(\\mbox{M$_{\\star}$})$, we used\n\\begin{equation}\n\\label{e_totalstellar}\n\\begin{multlined}\n\\Sigma(\\mbox{M$_{\\star}$})\n   =10^{5.65}\n   \\left(\\frac{I_{\\nu}(3.6 \\mu \\mbox{m})^{2.85}I_{\\nu}(4.5 \\mu \\mbox{m})^{-1.85}\\Omega}\n   {A}\\right)\\\\  \n   \\left(\\frac{D}{0.05}\\right)^{2}\n   \\left(\\frac{\\mbox{M$_{\\odot}$ arcsec$^{2}$}}\n   {\\mbox{Jy Mpc$^2$ pc$^2$}}\\right)\n\\end{multlined}\n\\end{equation}\nbased on the equation from \\citet{2012eskew}.  In this equation, $\\Omega$ is the angular area of the bin in the map, and $D$ distance to the source.  We then divided $\\Sigma(\\mbox{SFR})$ by $\\Sigma(\\mbox{M$_{\\star}$})$ to calculate the SSFR.\n\n\n\n\n\\section{Analysis of 8\/24, 8\/160, and 8\/250~$\\mu$\\lowercase{m} ratios}\n\\label{s_analysis_ratios}\n\n\n\\subsection{Map-based analysis}\n\\label{s_analysis_maps}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_01.ps}\n\\caption{The 21 $\\times$ 18~arcmin images of NGC 2403 used in the analysis. North is up and east is to the left in each image. The 3.6~$\\mu$m maps trace the intermediate-age and older stars. The 8~$\\mu$m image mainly shows the PAH 7.7~$\\mu$m emission feature but may also contain small amounts of emission from hot dust and stellar sources.  The 24~$\\mu$m band traces emission from hot (100~K) dust, and the 160 and 250~$\\mu$m trace emission from colder (15-30~K) dust. The FWHM for each image is shown as a green circle in the lower left corner of each panel, and the light blue ellipse in the 3.6~$\\mu$m image outlines the optical disk of the galaxy.}\n\\label{f_ngc2403_maps}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_02.ps}\n\\caption{The 20 $\\times$ 20~arcmin images of M83 used in the analysis.  See Figure \\ref{f_ngc2403_maps} for additional information on the image format.}\n\\label{f_m83_maps}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure}\n\\epsfig{file=jonesa_fig_03.ps}\n\\caption{The 8\/24, 8\/160 and 8\/250~$\\mu$m surface brightness ratio maps for NGC 2403.  These images are based on data where the PSFs are matched to the PSF of the 250~$\\mu$m data. The FWHM for the 250~$\\mu$m PSF is shown by the green circle in the lower left corner of each panel.  The colour scales in the images have been adjusted to show the structure in the surface brightness ratios; some of the red or purple pixels may be outside the range of values shown in the colour bars.  Data not detected at the $5\\sigma$ level in either band is left blank.  The 8\/24~$\\mu$m ratio is low in star forming regions where the 24~$\\mu$m emission is brightest.  The 8\/160 and 8\/250~$\\mu$m images are similar in that the ratios generally decrease with radius.  However, the 8\/160~$\\mu$m ratio shows more structure, while the 8\/250~$\\mu$m map is generally smoother.}\n\\label{f_ngc2403_ratio}\n\\end{figure}\n\n\n\\begin{figure}\n\\epsfig{file=jonesa_fig_04.ps}\n\\caption{The 8\/24, 8\/160 and 8\/250~$\\mu$m surface brightness ratio maps for M83.  These images are based on data where the PSFs are matched to the PSF of the 250~$\\mu$m data.   The maps are formatted in the same way as the maps in Figure~\\ref{f_ngc2403_ratio}.  We see spiral arm structure in all images.  The arm structure in the 8\/24~$\\mu$m image is traced by a series of red point-like sources where the ratio decreases in star forming regions.  However, the filamentary spiral structures in the 8\/160 and 8\/250~$\\mu$m maps are locations offset from the 160 and 250~$\\mu$m emission in Figure~\\ref{f_m83_maps} where PAH emission is enhanced relative to cold dust emission.}\n\\label{f_m83_ratio}\n\\end{figure}\n\n\\begin{figure}\n\\epsfig{file=jonesa_fig_05a.ps}\n\\epsfig{file=jonesa_fig_05b.ps}\n\\caption{ Maps showing the specific star formation rate (SSFR) for NGC~2403 and M83. The data are formatted the same as the corresponding ratio maps for each galaxy in Figures \\ref{f_ngc2403_ratio} and \\ref{f_m83_ratio}}\n\\label{f_ssfr}\n\\end{figure}\n\nFigures \\ref{f_ngc2403_maps} and \\ref{f_m83_maps} show the 3.6, 8, 24, 160 and 250~$\\mu$m images used in the analysis (before the application of the convolution or rebinning steps described in Section~\\ref{s_data_prep}).  Figures \\ref{f_ngc2403_ratio} and \\ref{f_m83_ratio} show the 8\/24, 8\/160 and 8\/250~$\\mu$m surface brightness ratios of the data after the convolution step but before rebinning.  The 8, 24, 160, and 250~$\\mu$m images all look very similar, demonstrating that the PAHs, hot dust, and cold dust are found in the same large-scale structures.  However, the ratio maps demonstrate how the PAH emission varies with respect to the dust traced by the other infrared bands. For comparison to these figures, we also show maps of the SSFR in Figure \\ref{f_ssfr}.\n\n\nIn both galaxies, we see a decrease in the 8\/24~$\\mu$m ratio in locations where the 24~$\\mu$m emission peaks. If the PAH emission was tracing star formation in the same way as the hot dust emission, we would see little variation across the 8\/24~$\\mu$m ratio maps. The disparity indicates either that the 24~$\\mu$m emission is enhanced in regions with high SSFR, that the PAH emission is inhibited where the hot dust emission peaks in the centres of regions with high SSFR, or that both effects occur within the star forming regions. In NGC 2403, we see the 8\/24~$\\mu$m ratio is higher in the diffuse regions outside the regions with high SSFR, particularly in the southern half of the galaxy.  In M83, we see the enhancement of PAHs relative to the 24~$\\mu$m emission in not only the interarm regions but also between high SSFR regions in the spiral arms.  The 8\/24~$\\mu$m ratio is also very low in the starburst nucleus of M83, as is also seen by Wu et al. (2014, submitted). \n\n\nThe 8\/160 and 8\/250~$\\mu$m ratio maps for NGC~2403 and M83 present different results for each galaxy.  In NGC~2403, the 8\/160 and 8\/250~$\\mu$m ratios peak near the centre and decrease with radius, although the 8\/160~$\\mu$m map looks more noisy than the 8\/250~$\\mu$m map.  Instead of seeing the PAH emission decrease relative to the cold dust emission in individual regions with high SSFR, as was the case in the 8\/24~$\\mu$m ratio maps, we see the PAH emission enhanced relative to the 160 and 250~$\\mu$m emission at the location of the infrared-brightest star forming region in the northeast side of the disc. The 8\/160 and 8\/250~$\\mu$m ratios generally do not change significantly near most other star forming regions.  With the exception of the infrared-brightest star forming region, the radial gradients in the 8\/160 and 8\/250~$\\mu$m ratios look similar to the radial gradients in the 3.6~$\\mu$m image seen in Figure~\\ref{f_ngc2403_maps}.  In NGC~2403, \\citet{2012bendo} found that the 160\/250~$\\mu$m surface brightness ratios were correlated with H$\\alpha$ emission and peaked in locations with strong star formation, while the 250\/350~$\\mu$m ratios were more strongly correlated with near-infrared emission and generally varied radially in the same way as the older stellar populations.  These results demonstrated that the 160~$\\mu$m emission is dominated by dust heated locally in star forming regions but the dust seen at 250~$\\mu$m is heated by the diffuse ISRF.  The similarity between the 8\/160, 8\/250, 250\/350, and 3.6~$\\mu$m radial gradients suggests that the PAHs are intermixed with the cold large dust grains and that the enhancement of PAH emission relative to the large dust grains depends on the surface brightness of the evolved stellar population.  If this is the case, the 8\/160~$\\mu$m map may looks noisy compared to the 8\/250~$\\mu$m map because emission in the 8 and 160~$\\mu$m bands is affected by different stellar populations while emission in the 8 and 250~$\\mu$m bands is affected by mainly the evolved stellar population.\n\n\nThe dust emission for the different wavebands peak in slightly different places in these profiles of the M83's spiral arms.  The 250~$\\mu$m emission, which trace most of the dust mass in the spiral arms, peak on the downstream (or inner) side of the spiral arm.  The profile of the 24~$\\mu$m emission tends to appear narrower and peaks $0-7$~arcsec further towards the upstream (or outer) side of the spiral arms (although this is small relative to the 18~arcsec resolution of the data used to create these plots).  This is consistent with the classical description of star formation within spiral arms \\citep[e.g.]{1969roberts, 1979elmegreen}.  First the gas flows into the spiral arms, then the gas is shocked by the spiral density waves and collapses into stars, and finally young stars emerge on the upstream side of the spiral arms.  \n\n\nThe 8~$\\mu$m emission peaks slightly further downstream from the 250~$\\mu$m emission, and the profiles of the 8~$\\mu$m emission on the downstream side of the spiral arms is broader than the profiles on the upstream side.  This is particularly pronounced for profiles A, C, and F.  While the 8\/24~$\\mu$m ratio drops sharply near the star forming regions as expected, the 8\/250~$\\mu$m ratio peak 10-30~arcsec (or $\\sim$200-650 pc) downstream from the dust lane, as is also seen in the countour overlays in Figure~\\ref{f_m83_overlay}. This demonstrates that the PAHs emission is enhanced relative to the cold dust on the downstream side of the spiral arms well outside the dust lanes.  The ultraviolet emission also peaks downstream from the 250~$\\mu$m emission in many of these profiles and that the profiles of the ultraviolet emission in B and C look broader on the downstream side.  The possible connection of these profiles to the ultraviolet emission and to the 160\/250~$\\mu$m ratios is discussed further in Section~\\ref{s_m83pah}.\n\n\nThe offset enhancements in PAH emission in the arms of M83 could appear because of astrometry problems, but we have checked the astrometry among the images using foreground and background sources outside the optical disc of the galaxy and found no significant offsets greater than $\\sim1$~arcsec in the sources between images.  It is also possible that the broader PAH emission could result from issues related to the PSF matching step, but usually these types of artefacts will appear symmetric around bright sources, whereas the enhanced PAH emisson appears asymmetric.  It is more likely that the phenomenon is real and has been difficult to detect before because of limitations in the angular resolution of far-infrared data.\n\n\n\\begin{figure}\n\\epsfig{file=jonesa_fig_06.ps}\n\\caption{The 8~$\\mu$m image of M83 (after the PSF has been matched to the PSF of the 250~$\\mu$m data) showing locations where we produced additional plots to illustrate the offset between the 8\/250~$\\mu$m ratio and the dust emission from the spiral arms.  The blue boxes show the locations in Figure~\\ref{f_m83_overlay} where we overlay contours of the 8\/250~$\\mu$m and 160\/250~$\\mu$m ratios on the 250~$\\mu$m data.  The cyan lines show the locations of the surface brightness profiles plotted in Figure~\\ref{f_m83_line}.  The image is formatted in the same way as Figure~\\ref{f_m83_maps}.}\n\\label{f_m83_line_map}\n\\end{figure}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_07a.ps,height=7.1cm}\n\\epsfig{file=jonesa_fig_07b.ps,height=7.1cm}\n\\epsfig{file=jonesa_fig_07c.ps,height=7.1cm}\n\\epsfig{file=jonesa_fig_07d.ps,height=7.1cm}\n\\caption{The 250~$\\mu$m images of the two spiral arms in M83 with the 8\/250~$\\mu$m ratio and 160\/250~$\\mu$m ratio overlaid as contours.  The contours for the 8\/250~$\\mu$m ratio start at 0.07 and increase upwards in increments of 0.01.  The contours for the 160\/250~$\\mu$m ratio start at 2.4 and increase upwards in increments of 0.2.  The images are formatted in the same way as Figure~\\ref{f_m83_maps}.  The 8\/250~$\\mu$m ratios themselves are discussed in Section~\\ref{s_analysis_maps}, while both the 8\/250 and 160\/250~$\\mu$m ratios are discussed in Section~\\ref{s_m83pah}.}\n\\label{f_m83_overlay}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_08.ps}\n\\caption{Profiles of the H$\\alpha$, 0.23, 8, 24, 160, and 250~$\\mu$m surface brightnesses and 8\/24, {8\/250, and 160\/250}~$\\mu$m surface brightness ratios measured in 18~arcsec wide regions along the locations shown in Figure~\\ref{f_m83_line_map}.  The x-axis shows the distance from the peak of the 250~$\\mu$m emission; negative numbers are for the upstream side of the arms, while positive materials are for the downstream side.  The profiles were measured in 18~arcsec wide regions in images where the PSF had been matched to the PSF of the 250~$\\mu$m data, which has a FWHM of 18~arcsec (or $\\sim400$~pc), but the data are supersampled at 1~arcsec resolutions to produce smooth curves.  All surface brightnesses are normalised so that the peak values are 1, and all ratios are normalised so that they range between 0 and 1.  The uncertainties in the normalised surface brightnesses are $~\\rlap{$<$}{\\lower 1.0ex\\hbox{$\\sim$}} 1$\\%, and the uncertainties in the normalised ratios are $~\\rlap{$<$}{\\lower 1.0ex\\hbox{$\\sim$}} 5$\\%.  The H$\\alpha$ data shown here are corrected for foreground dust attenuation but not corrected for dust extinction within M83.}\n\\label{f_m83_line}\n\\end{center}\n\\end{figure*}\n\n\n\n\n\\subsection{Analysis of binned data}\n\\label{s_binned_analysis}\n\nIn Figures \\ref{f_8-24}-\\ref{f_8-250}, we plot the relations between the PAH emission at 8~$\\mu$m and either the hot dust emission at 24~$\\mu$m or the cold dust emission at 160 and 250~$\\mu$m.  Pearson correlation coefficients for these relations are given in Table \\ref{t_correlation}.  To first order, the 8~$\\mu$m surface brightness is well correlated with the 24, 160 and 250~$\\mu$m surface brightnesses.  However, the 8\/24, 8\/160 and 8\/250~$\\mu$m ratios reveal the presence of both scatter in the relations between the 8~$\\mu$m emission and emission in other bands as well as systematic variations in these relations. \n\n\nThe variations in the 8\/24~$\\mu$m ratio in Figure~\\ref{f_8-24} show that the 8\/24~$\\mu$m ratio decreases in areas where the 24~$\\mu$m emission is strongest in both galaxies.  However, the relations between the 8\/24~$\\mu$m ratio and the 24~$\\mu$m emission differ somewhat between the two galaxies.  In NGC~2403, we found that we could see different trends in the data when we separated the 18~arcsec binned data into two subsets where the SSFR was either $\\geq1\\times10^{-10}$~yr$^{-1}$ or $<1\\times10^{-10}$~yr$^{-1}$.  The data with low SSFR follow a relation in which $\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(24\\mu\\mbox{m}))$ increases slightly from $\\sim 0.0$ to $\\sim 0.2$ as $\\log (I_\\nu(24\\mu\\mbox{m}))$ increases from -6 to -4.  The emission from these regions may orginate mainly from locations in the diffuse ISM with relatively soft radiation fields where the PAH emission is well-correlated with emission from hot, diffuse dust heated by the diffuse ISRF.  As both the PAHs and the hot, diffuse dust are stochastically heated, the ratio of PAH to hot dust emission is expected to be roughly constant.  The slight decrease in the 8\/24~$\\mu$m ratio as the diffuse 24~$\\mu$m surface brightness decreases is potentially a result of an increase in the hardness of the radiation field as the 24~$\\mu$m surface brightness decreases (possibly as a result of changes in metallicity with radius as found by multiple authors \\citep[e.g. ][]{1994zaritsky,2010moustakas}), which could lead to either PAH emission being suppressed or 24~$\\mu$m emission being enhanced.  Data points tracing locations with high SSFR fall below the relation between the 8\/24~$\\mu$m ratio and 24~$\\mu$m surface brightness.  These locations would be expected to have harder radiation fields that may enhance the 24~$\\mu$m emission or suppress the PAH emission.  While we are able to empirically separate data into regions with high and low 8\/24~$\\mu$m ratios using a SSFR cutoff value of $1\\times10^{-10}$~yr$^{-1}$, additional work in modelling the stellar populations, the PAH excitation, and the dust heating is needed to understand the details of how the SSFR of the stellar populations affects the variations in the 8\/24~$\\mu$m ratio within NGC~2403.\n\n\nIn M83, the relationship between the 8\/24~$\\mu$m ratio and 24~$\\mu$m emission is close to linear at $\\log (I_\\nu(24\\mu\\mbox{m}))>-4$, but it flattens at $\\log (I_\\nu(24\\mu\\mbox{m})) < -4$. Some of the lowest 8\/24~$\\mu$m ratios corresponds to regions with high SSFR in the nucleus and spiral arms where, again, the 24~$\\mu$m emission may be strongly enhanced or the PAH emission is suppressed.  The relation between 24~$\\mu$m emission and the 8\/24~$\\mu$m ratio at $\\log (I_\\nu(24\\mu\\mbox{m}))>-4$ in M83 is similar to the relation seen in NGC~2403.  Unlike NGC~2403, however, we found that we could not readily separate data in the plots of $\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(24\\mu\\mbox{m}))$ versus $\\log (I_\\nu(24\\mu\\mbox{m}))$ for M83 simply by selecting data by SSFR, as some regions with low SSFR have low 8\/24~$\\mu$m ratios.  These regions are mostly locations within radii of 1.5~kpc.  The exact reason why we see this is unclear, although it is possible that hard ultraviolet photons from the starburst nucleus leak into the diffuse ISM in this region and destroy the PAHs in the diffuse ISM.\n\n\nFigure \\ref{f_8-160} shows good correlations between the 8~$\\mu$m and 160~$\\mu$m surface brightnesses.  The plot of the 8\/160~$\\mu$m vs 160~$\\mu$m for M83 shows that the 8\/160~$\\mu$m ratio is close to constant over a range of infrared surface brightnesses that vary by a factor of 100, indicating that the relation between 8 and 160~$\\mu$m emission is very close to a one-to-one relationship.  Some scatter is seen in the 8\/160~$\\mu$m ratio at high 160~$\\mu$m surface brightnesses.  Some of these data points are for locations around the infrared-bright centre of M83 where the outer regions of the PSF were not matched perfectly in the convolution step, while other data points sample regions along the spiral arms where the enhancement in the 8\/160~$\\mu$m ratio is offset from the 160~$\\mu$m surface brightness as discussed in Section~\\ref{s_analysis_maps}.  In NGC 2403, the 8\/160~$\\mu$m ratio increases with 160~$\\mu$m surface brightness, and the relation exhibits more scatter, indicating that the relation of 8~$\\mu$m emission to 160~$\\mu$m emission in NGC~2403 is different from the relation for M83. \n\n\nThe relations between the 8 and 250~$\\mu$m emission in Figure \\ref{f_8-250} are similar to the relations between the 8 to 160~$\\mu$m emission.  For both galaxies, the 8\/250~$\\mu$m ratio increases with the 250~$\\mu$m surface brightness, and the correlation coefficients are relatively strong.  In NGC~2403, the correlation coefficient between the 8\/250~$\\mu$m ratio and the 250~$\\mu$m surface brightness is 0.83, which is much higher than the correlation coefficient of 0.66 for the relation between the 8\/160~$\\mu$m ratio and the 160~$\\mu$m surface brightness.  Given that the square of the Pearson correlation coefficient indicates the fraction of variance in one quantity that depends upon the other quantity, the difference in the correlation coefficients is equivalent to a $\\sim25$\\% difference in being able to describe the variance in the relations.  This suggests that the PAHs are more strongly associated with the colder dust seen at 250~$\\mu$m than the warmer dust seen at 160~$\\mu$m.  In M83, the relation between the 8\/250~$\\mu$m ratio and 250~$\\mu$m surface brightness is sloped and also shows significant scatter at high surfaces brightnesses in the same way as the relationship between the 8\/160~$\\mu$m ratio and the 8~$\\mu$m emission.\n\n\nBecause M83 is at a distance $\\sim$1.5 further than NGC 2403, the 18~arcsec bins used in this analysis will cover regions with different spatial scales.  In Appendix~\\ref{a_binsizecheck}, we examined how the results for the analysis on NGC~2403 would change if we used 27~arcsec bins, which cover approiximately the same spatial scales as the 18~arcsec bins used for the M83 data. We see no noteable difference in the results using the 27~arcsec bins compared to the 18~arcsec bins; most correlation coefficients change by $\\leq0.05$.  Hence, adjusting the bin sizes for the two galaxies to similar spatial scales is unimportant.  We will therefore use data measured in the smaller bins as it takes full advantage of the capabilities of the {\\it Herschel} data that we are using and as it allows us to illustrate how the relations are still found in smaller structure in NGC 2403.\n\n\n\\begin{table}\n\\caption{Pearson correlation coefficients for the binned data.}\n\\label{t_correlation}\n\\begin{tabular}{p{5.6cm}cc}\n\\hline\n       & \n   \tNGC &\n\tM83 \n\t\\\\ \n       & \n   \t2403 &\n\t\\\\ \\hline\n$\\log (I_\\nu(8\\mu\\mbox{m}))$ vs $\\log(I_\\nu(24\\mu\\mbox{m}))$ &\n   \t0.96\t&\n\t0.97\t\\\\\n$\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(24\\mu\\mbox{m}))$ vs $\\log(I_\\nu(24\\mu\\mbox{m}))$ &\n\t-0.13\t&\n\t-0.74\t\\\\\n$\\log (I_\\nu(8\\mu\\mbox{m}))$ vs $\\log(I_\\nu(160\\mu\\mbox{m}))$ &\n\t0.98\t&\n\t0.98\t\\\\ \n$\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(160\\mu\\mbox{m}))$ vs $\\log(I_\\nu(160\\mu\\mbox{m}))$ &\t\n\t0.66\t&\n\t0.17\t\\\\\n$\\log (I_\\nu(8\\mu\\mbox{m}))$ vs $\\log(I_\\nu(250\\mu\\mbox{m}))$ &\n\t0.98\t&\t\n\t0.97\t\\\\ \n$\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(250\\mu\\mbox{m}))$ vs $\\log(I_\\nu(250\\mu\\mbox{m}))$ &\n\t0.83\t&\n\t0.44\t\\\\\t\t\n\t\\hline\n\\end{tabular}\t\n\\end{table}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_09.ps}\n\\caption{The 8~$\\mu$m surface brightness and the 8\/24~$\\mu$m ratios as a function of 24~$\\mu$m emission for the 18~arcsec binned data for both galaxies.  Only data detected at the $5\\sigma$ level are displayed.  The best fitting linear functions between the surface brightnesses (weighted by the errors in both quantities) are shown as black lines in the top panels. The blue points are locations with high SSFR, and the red points and error bars are locations which are predominantly heated by the diffuse ISRF; see Section~\\ref{s_data_prep} for more details. In M83 we highlight locations within a 1.5~kpc radius of the centre in black. }\n\\label{f_8-24}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_10.ps}\n\\caption{The 8~$\\mu$m surface brightness and the 8\/160~$\\mu$m ratios as a function of 160~$\\mu$m emission for the 18~arcsec binned data for both galaxies.  The data are formatted in the same way as in Figure~\\ref{f_8-24}.}\n\\label{f_8-160}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_11.ps}\n\\caption{The 8~$\\mu$m surface brightness and the 8\/250~$\\mu$m ratios as a function of 250~$\\mu$m emission for the 18~arcsec binned data for both galaxies.  The data are formatted in the same way as in Figure~\\ref{f_8-24}.}\n\\label{f_8-250}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Identification of PAH excitation sources}\n\\label{s_pahexcitation}\n\n\n\\subsection{PAH excitation in NGC 2403}\n\\label{s_ngc2403pah}\n\nWe can conclude from our analysis that the 8~$\\mu$m emission that we observe from NGC~2403 does not originate from PAHs excited locally within the centres of star forming regions, as the relationship between 8 and 24~$\\mu$m emission shows that PAH 8~$\\mu$m emission decreases relative to 24~$\\mu$m emission within regions with high SSFR.  This is partly because the 24~$\\mu$m emission is very sensitive to dust heating and increases significantly within star forming regions \\citep[e.g.][]{2001dale, 2002dale}.  However, the strong ultraviolet radiation from these massive young stars probably photodissociates the PAHs in the centres of these regions including species that produce features other than the 7.7~$\\mu$m feature, as has been seen spectroscopically in other galactic and extragalactic star forming regions \\citep{2007berne, 2007lebouteiller, 2007povich, 2008gordon}.  PAH emission has been observed in the outer regions of photodissociation regions \\citep{2007berne, 2007lebouteiller, 2007povich}; the photons that excite the PAHs in these locations can also heat the very small grains that produce the 24~$\\mu$m emission.  In our data, the emission from the inner and outer regions of these regions will be blended.  Integrating over the centres of these regions, the 8\/24~$\\mu$m ratio will still appear low comapred to diffuse regions outside these regions because the PAH emission is suppressed in parts of the regions while the 24~$\\mu$m emission is not, a result also obtained by \\citet{2005calzetti}.  The dust emitting at 160~$\\mu$m is also heated by light from star forming regions \\citep{2012bendo}.  While the 8~$\\mu$m emission is better correlated with the 160~$\\mu$m band than with the 24~$\\mu$m band, the 8\/160~$\\mu$m ratio still shows significant scatter as a function of 160~$\\mu$m surface brightness, possibly because the PAH emission is still inhibited in the regions from which the 160~$\\mu$m emission is originating.\n\n\nHowever, we see a strong correlation between the 8 and 250~$\\mu$m surface brightnesses, and the relation between the 8\/250~$\\mu$m ratio and the 250~$\\mu$m surface brightness shows that the residuals in the relation between the 8 and 250~$\\mu$m are very small, especially compared to the equivalent residuals for the relations between the 8~$\\mu$m emission and emission in either the 24~$\\mu$m or 160~$\\mu$m bands. This is particularly evident when comparing the correlation coefficients for the 8\/24~$\\mu$m ratio versus 24~$\\mu$m emission, the 8\/160~$\\mu$m versus 160~$\\mu$m emission, and the 8\/250~$\\mu$m ratio versus 250~$\\mu$m emission in Table~\\ref{t_correlation}.  This indicates that the PAH emission is much more strongly tied to the dust emitting in the 250~$\\mu$m band.  \\citet{2012bendo} demonstrated that the dust emitting at $\\geq 250$~$\\mu$m in NGC~2403 was heated mainly by the diffuse ISRF from the total stellar population.  This implies that the PAHs in NGC~2403 are also mainly excited by the diffuse ISRF.  Moreover, the map of the 8\/250~$\\mu$m ratio in Figure~\\ref{f_ngc2403_ratio} looks very similar to both the 3.6~$\\mu$m map in Figure~\\ref{f_ngc2403_maps} that traces the light from the total stellar population and the 250\/350~$\\mu$m ratio map from \\citet{2012bendo} that shows the variations in the colour temperatures of the large dust grains heated by the ISRF from these stars.\n\n\nTo examine this relation further, we plot the 8\/250~$\\mu$m ratio versus the 3.6~$\\mu$m surface brightness for NGC 2403 in Figure \\ref{f_2403-8-250}. We find a strong correlation between 8\/250~$\\mu$m ratio and the 3.6~$\\mu$m surface brightness; the Pearson correlation coefficient for the relation between these data in logarithmic space is 0.89.  This shows that the enhancement of PAH emission relative to cold dust emission scales with the stellar surface brightness, which implies that the PAHs are primarily mixed in with the large dust grains in the diffuse ISM and that the PAHs are predominantly heated by the diffuse ISRF from the total stellar population.  \n\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=jonesa_fig_12a.ps}\n\\epsfig{file=jonesa_fig_12b.ps}\n\\caption{The 8\/250$~\\mu$m surface brightness ratio plotted as a function of the 3.6~$\\mu$m surface brightness and galactocentric radius for the 18~arcsec binned data for NGC~2403. The data are formatted in the same way as in Figure~\\ref{f_8-24}.  The radii are based on using an inclination of $62.9\\deg$ from \\citet{2008deblok}.}\n\\label{f_2403-8-250}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=jonesa_fig_13a.ps}\n\\epsfig{file=jonesa_fig_13b.ps}\n\\caption{To examine whether the relation between the 3.6~$\\mu$m emission and the 8\/250~$\\mu$m ratio is related to hot dust or PAH emission in the 3.6~$\\mu$m band, we first show the relation between 3.6 and 8~$\\mu$m emission.  The relation in the top panel could be the result of either the 3.6 and 8~$\\mu$m bands tracing emission from similar sources or the 3.6~$\\mu$m band tracing starlight exciting the PAHs seen in the 8~$\\mu$m band.  Next, we plot the 4.5\/3.6$~\\mu$m ratio as a function of the 24$~\\mu$m emission in the bottom panel to examine whether the slope of the SED at 3.6-4.5~$\\mu$m is influenced by hot dust emission.  The absence of such a relation as well as the relative invariance of the 4.5\/3.6~$\\mu$m ratio implies that the 3.6 and 4.5~$\\mu$m bands are relatively unaffected by non-stellar emission.  The data are formatted in the same way as in Figure~\\ref{f_8-24}.}\n\\label{f_2403_stars}\n\\end{center}\n\\end{figure}\n\n\nIt is also possible that, because the correlation coefficient for the relation between the logarithm of the 3.6~$\\mu$m surface brightness and radius is -0.95, the 8\/250~$\\mu$m ratio actually depends on radius rather than 3.6~$\\mu$m surface brightness.  Any radial dependence would be expected to be related to metallicity, which decreases with radius, and this could influence the PAHs.  However, the results from \\citet{2008engelbracht} and \\citet{2008gordon} indicated that the apparent dependence of PAH emission on metallicity is really the result of changes in the radiation field illuminating the ISM in lower metallicity environments.  The radiation field would be harder in low metallicity systems first because of increases in the stellar temperatures of O and B stars \\citep{2004massey, 2005massey, 2007trundle} and second because of decreases of extinction related to a decrease in the gas to dust ratio. This in turn leads to harder interstellar radiation fields in low-metallicity environments that potentially destroy PAHs.  To examine this, we plot the 8\/250~$\\mu$m ratio as a function of radius in the bottom panel of Figure~\\ref{f_2403-8-250}.  As expected, the 8\/250~$\\mu$m ratio decreases as radius increases.  The relation between the logarithm of the 8\/250~$\\mu$m ratio and radius has a Pearson correlation coefficient of -0.82, which has an absolute value that is similar to the value of 0.89 for the relation between the lorarithms of the 8\/250~$\\mu$m ratio and the 3.6~$\\mu$m emission.  However, at the resolution of these data, we do see non-axisymmetric substructures in the 8\/250~$\\mu$m image in Figure~\\ref{f_ngc2403_ratio} that correspond to similar substructure in the 3.6~$\\mu$m images, but these structures are mainly visible at radii of $<10$ kpc.  When we look at data within this inner 10~kpc region, we get a correlation coefficient of -0.77 for the relation with radius and 0.91 for the relation with 3.6~$\\mu$m surface brightness.  This implies that the total stellar surface brightness is more influentual on PAH excitation than any effects related to radius.\n\n\nAlthough it is unlikely, the correlation between the 8\/250~$\\mu$m ratio and 3.6~$\\mu$m emission could be the result of a correlation between emission in the 3.6 and 8~$\\mu$m bands themselves.  Both bands may contain thermal continuum dust emission (although the thermal continuum emission should have been removed from the 8~$\\mu$m data when we applied Equation~\\ref{e_pahdustsub}), and the 3.6~$\\mu$m band may also contain emission from the 3.3~$\\mu$m PAH emission feature, although previous work by \\citet{2003lu} indicated that emission at $<5$~$\\mu$m from nearby galaxies is dominated by stellar emission.  To investigate this further, we plot the relation between the 8~$\\mu$m PAH emission and 3.6~$\\mu$m stellar surface brightness in Figure~\\ref{f_2403_stars}.  The data are well correlated; the correlation coefficient for the relation in logarithm space is 0.94.  While this could indicate that the same emission sources are seen at 3.6 and 8~$\\mu$m, it is also possible that the 8~$\\mu$m emission is correlated with 3.6~$\\mu$m emission because the stars seen at 3.6~$\\mu$m excite the PAHs seen at 8~$\\mu$m.  Hence, the correlation between 3.6 and 8~$\\mu$m does not necessarily prove anything about the relation between the emission in these bands.  \\citet{2010mentuch} illustrated that it was possible to identify the influence of non-stellar emission at near-infrared wavenegths by examining the 4.5\/3.6~$\\mu$m surface brightness ratio.  This ratio would be relatively invariant for stellar emission because it traces emission from the Rayleigh-Jeans side of the stellar SED, but if hot dust emission influences the bands, the ratio should increase.  We plot the 4.5\/3.6~$\\mu$m ratio versus 24~$\\mu$m emission in the lower panel of Figure \\ref{f_2403_stars}. This relationship is almost flat.  Most of the data points have log($I_\\nu$(4.5~$\\mu$m)\/$I_\\nu$(3.6~$\\mu$m)) values that lie within a range of -0.14 to -0.23, which would be consistent with what was observed for evolved stellar populations by \\citet{2010mentuch}.  The absence of significant variations in the 4.5\/3.6~$\\mu$m ratio with 24~$\\mu$m implies that the 3.6 and 4.5~$\\mu$m bands are largely uninfluenced by hot dust emission.  We do see a few data points with values of log($I_\\nu$(4.5~$\\mu$m)\/$I_\\nu$(3.6~$\\mu$m))$>-0.14$ where the 3.6 and 4.5~$\\mu$m may be more strongly influenced by non-stellar emission, but these data only weakly influence our results.  If we exclude these data, the correlation coefficient for the relation between the 3.6~$\\mu$m data and the 8\/250~$\\mu$m data changes by $<0.01$.  This shows that the 3.6~$\\mu$m emission in NGC~2403 is largely dominated by the stellar population and is relatively unaffected by hot dust or PAH emission. Therefore, the most likely explanation for the correlation between the 3.6 and 8~$\\mu$m emission as well as the correlation between the 3.6~$\\mu$m emission and the 8\/250~$\\mu$m ratio is that the PAHs are excited by the stellar population seen at 3.6~$\\mu$m.\n\n\n\\subsection{PAH excitation in M83}\n\\label{s_m83pah}\n\n\nThe results from the 8~$\\mu$m to 24~$\\mu$m relationship in M83 are similar to NGC 2403.  We see the 8~$\\mu$m PAH emission is low in regions where the 24~$\\mu$m emission peaks.  Again, PAHs are probably being destroyed locally in regions with high SSFR.  However, we find that 8~$\\mu$m emission is more strongly related to the 160 and 250~$\\mu$m emission.  We also see offsets between the 8\/250~$\\mu$m ratios and the dust mass (as traced by the 250~$\\mu$m band).  \\citet{2012bendo} and \\citet{2014bendo} also found that the 160\/250~$\\mu$m colours appeared offset relative to the star forming regions in the spiral arms, and \\citet{2012foyle} found a related offset in the dust colour temperatures.  This implies that the enhancement in PAH emission relative to cold dust emission is related to the enhancement of the temperature of the dust seen at 160~$\\mu$m.  To examine this relationship further, we map the 250~$\\mu$m emission from the spiral arms overlaid with contours showing the 160\/250~$\\mu$m ratio in Figure~\\ref{f_m83_overlay}, and we show profiles of the 160\/250~$\\mu$m ratio across the spiral arms in Figure~\\ref{f_m83_line}.  These plots show that the 8\/250 and 160\/250~$\\mu$m ratios trace similar structures offset from the dust mass as well as the H$\\alpha$ and 24~$\\mu$m emission associated with star formation.  To check how well the 8\/250 and 160\/250~$\\mu$m ratios are correlated, we plot the two ratios in Figure~\\ref{f_8-250-160-250}.  The relation has a Pearson correlation coefficient of 0.65, implying that the excitation of PAH emission and the heating of the dust seen at 160~$\\mu$m are, to some degree, linked.\n\n\nIn spiral density waves, as mentioned before, large quantities of gas and dust are expected where the ISM is shocked on the upsteam sides of the spiral arms, star forming regions would be found immediately downstream of the shocks, and older stars would be expected further downstream \\citep[e.g.][]{1969roberts, 1979elmegreen, 2008tamburro, 2009martinezgarcia, 2011sanchezgil}.  This could cause offset enhancement of PAH emission relative to spiral arm dust lanes, as seen in Figure~\\ref{f_m83_line} and as also implied by the relation in Figure~\\ref{f_8-250-160-250}, in two possible ways.  \n\n\nOne possible explanation is that the dense dust lanes on the upstream edge of the spiral arms severely attenuate the starlight escaping from the photoionising stars within star forming regions, but light easily escapes across the downstream side of the spiral arms where the dust density is lower.  Such a geometrical arrangement of the star forming regions relative to the dust would produce the slight offsets between the 24~$\\mu$m emission (tracing obscured star formation) and H$\\alpha$ emission (tracing unobscured star formation) seen in most of the profiles in Figure~\\ref{f_m83_line} and may also explain the downstream areas with enhanced H$\\alpha$ emission in profiles C and F.  If photoionising light is primarily travelling asymmetrically from the star forming regions and if the 160~$\\mu$m band traces dust heated by the light escaping from the star forming regions into the diffuse ISM, the 160~$\\mu$m emission would appear enhanced relative to 250~$\\mu$m emission along the downstream side of the spiral arms.  Similarly, PAHs mixed in with the dust emitting at 160~$\\mu$m would be excited by the ultraviolet light escaping from star forming regions and appear enhanced relative to the dust emission in the same locations, although the total PAH emission itself will peak along the spiral arms where the total mass of the PAHs is greater (which is also true for dust emission observed in any single band).\n\n\nThe other possible explanation is that the PAHs and the dust seen at 160~$\\mu$m are locally heated by a young, non-ionising population of stars (stars with ages older than 4 Myr) that have left the dusty star forming regions in the spiral arms.  As shown by \\citet{1999leitherer}, such a population would still produce a substantial amount of ultraviolet and blue light that could strongly enhance the PAH emission and the temperature of the large dust grains on the downstream side of the arms, but the radiation from these stars may not include higher energy photons that destroy PAHs.  In the profiles in Figure~\\ref{f_m83_line}, the ultraviolet emission either peaks downstream of the dust mass or has a profile on the downstream side that is broader than the dust emission profile.  Additionally, the ultraviolet emission appears to stronger relative to the H$\\alpha$ emission in most downstream locations.  This provides additional support for the possibility that the 8~$\\mu$m emission observed in M83 originates from PAHs excited locally by young, non-ionising stars, although additional analysis would be needed to confirm this.\n\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=jonesa_fig_14.ps}\n\\caption{The 8\/250$~\\mu$m surface brightness ratio plotted as a function of the 160\/250~$\\mu$m surface brightness ratio for the 18~arcsec binned data. The data are formatted in the same way as in Figure~\\ref{f_8-24}.}\n\\label{f_8-250-160-250}\n\\end{center}\n\\end{figure}\n\n\nIt is also worth briefly noting that we do not see variations in the 8\/160 or 8\/250~$\\mu$m ratios in M83 that imply a dependence upon radius and hence a dependence upon the metallicity, which decreases with radius \\citep{1994zaritsky}.  The variations in the PAH emission with respect to the cold dust emission are driven mainly by local excitation of the PAHs.\n\n\n\n \n\\section{Discussion}\n\\label{s_discussion}\n\n\nThe results show that the 8\/24~$\\mu$m ratio decreases in many regions with high SSFR, in agreement with previous findings from \\citet{2004helou}, \\citet{2005calzetti}, \\citet{2006bendo}, \\citet{2006madden}, \\citet{2007berne}, \\citet{2007lebouteiller}, \\citet{2007povich}, \\citet{2008bendo}, \\citet{2008gordon}, and \\citet{2014calapa}.  Even though our analysis is mainly focused on the 7.7~$\\mu$m PAH feature that falls within IRAC channel 4, the results from \\citet{2007lebouteiller}, \\citet{2007povich}, and \\citet{2008gordon} suggest that other PAH emission features may also decrease relative to hot dust emission within star forming regions.\n\n\nOur results showing the correlation of the 8~$\\mu$m emission from PAHs with the far-infrared emission from large dust grains is largely in agreement with the results from \\citet{2002haas}, \\cite{2008bendo}, and \\citet{2014calapa}.  However, the general conclusion from these papers had been that the PAHs are mixed with large grains heated by the diffuse ISRF.  While NGC~2403 certainly fits that scenario, M83 does not.  Instead, PAH emission in M83 is more strongly associated with large dust grains heated either by light escaping from star forming regions and travelling hundreds of pc away from the spiral arms or locally by young stars that produce substantial non-ionising ultraviolet radiation, which was unexpected.  Further study would be needed to determine whether such variations in PAH excitation are seen among other nearby galaxies as well.\n\n\nPAH may be excited by different radiation fields in NGC 2403 and M83 because of the differences in the spiral structure in the two galaxies.  Because NGC~2403 is a flocculent spiral galaxy, star formation is expected to be triggered by clouds collapsing in local gravitational instabilities \\citep[e.g.][]{2003tosaki,2010dobbs}.  In such a scenario, inflowing dust on scales of tens or hundreds of parsecs may be roughly symmetrically distributed around star forming regions, although more modelling work on cloud collapse in flocculent galaxies is needed to confirm this.  If the dust is distributed this way, dust near the centres of these shells would absorb the ultraviolet and blue light from the star forming regions inside, and any PAHs within these central regions would be destroyed.  Meanwhile, dust and PAHs in the outer shells would be shielded from the light from the star forming regions and would instead be heated by the diffuse ISRF.  Also note that the stars in spiral arm filaments in flocculent spiral galaxies are not expected to exhibit any age gradients like grand-design spiral galaxies \\citep{2010dobbs}.  If young, non-photoionising stars contribute significantly to PAH excitation, then the relatively homogeneous distribution of these stars may result in the PAHs appearing enhanced over broad areas rather than appearing enhanced near the spiral filaments.  In contrast to NGC~2403, M83 is a grand design spiral galaxy in which cold, dusty gas flows into star forming sites mainly from one side of the spiral arms and star forming regions emerge from the other side \\citep[e.g. ][]{1979elmegreen, 1993garciaburillo, 2008tamburro, 2009egusa, 2013vlahakis}.  Hence, dust will preferentially be located upstream of individual star forming regions within M83.  As described in Section~\\ref{s_m83pah}, PAHs are destroyed within the centres of these star forming regions, but PAHs in the diffuse ISM could be excited by starlight diffusing out of the optically-thin side of the star forming regions.  Additionally, multiple studies \\citep[e.g. ][]{2009martinezgarcia, 2011sanchezgil} have found gradients in the ages of the stellar populations downstream of spiral arms in grand design spiral galaxies.  PAH emission could appear enhanced downstream of star forming regions if the PAHs are destroyed in photoionising regions but strongly excited locally in regions with soft ultraviolet emission from young, non-photoionising stars.  If these geometrical descriptions for the relation between PAHs and excitation sources is accurate, then we should find that PAHs are excited by the diffuse ISRF in other flocculent late-type spiral galaxies while PAHs are excited in regions offset from star forming regions in other grand design spiral galaxies.\n\n\nSome dust emission models \\citep[e.g. ][]{2007draine} and radiative transfer models \\citep[e.g. ][]{2011popescu} show PAHs as excited by the radiation fields from all stellar populations regardless of the hardness or intensity of the fields\\footnote{The version of the \\citet{2007draine} model typically applied to infrared SEDs is usually based on dust heated by a radiation field with the same spectral shape as the local ISRF as specified by \\cite{1983mathis}.  When applying the dust model to data, only the amplitude of the radiation field is treated as a free parameter.  However, \\citet{2014draine} includes an example of SED fitting with the \\citet{2007draine} model in which the spectral shape of the illuminating radiation field is also allowed to vary.}.  Our results show that this approach is an oversimplification of PAH excitation.  New refinements in dust emission and radiative transfer models are needed to replicate how PAHs are excited by radiation fields from different stellar populations within different galaxies and how the PAH\/dust mass ratio may change with variations in the hardness of the illuminating radiation field.  For example, \\citet{2013crocker} used stellar population synthesis and simplified models of dust and PAH absorption to predict the contributions of different stellar populations to PAH excitation in NGC 628 and found that $\\sim$40\\% of the PAHs are excited by stars $<10$~Myr in age, $\\sim20$\\% are excited by stars with ages of 10-100~Myr, and the remainder are excited by stars $>$100~Myr in age.  It is also apparent that PAH excitation changes across spiral density waves (either because of details in the geometry of the star forming regions or because of variations in the stellar populations on either side of the waves), and it would be appropriate to make improvements to radiative transfer models so that they can replicate these effects.\n\n\nThese results have multiple implications for using PAH emission as a proxy for other quantities.  While PAH emission cannot be used on sub-kpc scales to measure accurate star formation rates, groups such as \\citet{2008zhu} and \\citet{2009kennicutt} have suggested using globally-integrated PAH emission to estimate extinction corrections for optical star formation tracers such as H$\\alpha$ emission, thus producing extinction corrected global star formation metrics.  When PAHs are excited by star forming regions, globally-integrated PAH emission should more accurately represent the light attenuated by dust in star forming regions and should provide fairly accurate star formation rates.  When PAHs are excited by the diffuse ISRF, however, the connection between star formation and PAH emission is less clear, and star formation rates calculated using PAH emission could be less reliable.\n\n\nPrevious results showing a relation between PAH emission and far-infrared emission from large dust grains had implied that PAHs could be used as a proxy of dust mass \\citep[e.g.][]{2008bendo}.  In cases where the PAHs are associated with dust heated by the diffuse ISRF, this should still be appropriate, although metallicity-related effects would still need to be taken into account.  In cases where the PAHs are heated by diffuse light from star forming regions or from young, non-photoionising stars that have emerged from star forming regions, the PAH emission will still scale approximately with dust mass but will also vary depending upon the radiation field from the young stars.  In this situation, using PAH emission to trace dust mass may be less reliable.  \n\n\nMultiple authors have identified an empirical relation between either radially-averaged or globally-integrated PAH and CO emission \\citep{2006regan,2010bendoco, 2013tan, 2013vlahakis}, implying that the PAHs are, to some degree, correlated with molecular gas.  This would be expected if the PAHs also trace the cold dust that is found associated with the molecular gas.  However, the relation between PAH and CO emission breaks down on small spatial scales, including in NGC~2403 (\\citealt{2010bendoco}, but also see \\citealt{2013tan}).  In M51, \\citet{2013vlahakis} found an offset between PAH and CO emission in the spiral arms, with the molecular gas associated with the cold dust in the locations where material is entering the spiral arms and the PAH emission appearing enhanced further downstream where it is excited by light from young stars.  Our results imply that, in future work, we may be able to measure a similar offset between PAH and CO emission in M83 as well as other grand design spiral galaxies.  While PAH emission was already shown to be a poor tracer of molecular gas on sub-kpc scales, the phenomenology of PAHs excitation in spiral density waves causes even more problems with using it as a proxy for molecular gas.\n\n\n\\section{Conclusions}\n\\label{s_conclusions}\n\n\nWe identified different relations between PAH emission and far-infrared emission from large dust grains in the two galaxies we examined.  For NGC 2403, we find the 8~$\\mu$m emission is most strongly associated with emission from cold dust at 250~$\\mu$m.  In particular, we find that the 8\/250~$\\mu$m ratio shows a very strong dependence upon the 3.6~$\\mu$m emission from the total stellar population, indicating that the PAHs are mixed in with the diffuse dust and heated by the diffuse ISRF from the total stellar population.  Star forming regions play a much less significant role in the excitation of the PAHs observed in the 8~$\\mu$m band. In contrast, we see in M83 that the PAH emission is more strongly associated with the 160~$\\mu$m emission from large grains heated by star forming regions as implied by the strong correlation between the 160\/250~$\\mu$m and 8\/250~$\\mu$m ratios. This illustrates that PAHs in M83 are excited either by starlight escaping asymmetrically from star forming region so that locations towards the downstream edges of the spiral arms show enhancement in 8~$\\mu$m emission compared to the dust mass or that the PAHs are excited locally by young, non-photoionising stars that have migrated downstream from the spiral arms.  \n\n\nMany dust emission and radiative transfer models currently treat PAHs as though they are excited by all radiation fields of all intensities from all stellar populations within the galaxies to which they are applied, much in the same way that emission from silicate and large carbonaceous dust grains is modelled.  The results from just these two galaxies show that this assumption is not universally applicable.  These dust models need to be adjusted to account for the observational results showing that PAHs are sometimes excited by the diffuse ISRF from the total stellar population and sometimes excited either by young, non-photoionising stars that have emerged from star forming regions or light escaping from these regions and travelling hundreds of pc away (although the PAH emission may be inhibited within the star forming regions themselves).  Additionally, some models rely upon using a single SED shape (such as the SED of the local ISRF) for the radiation field illuminating PAHs and dust.  Such models cannot account for the possibility that PAH emission could be inhibited if the radiation fields are excessively hard.  To properly characterise the PAH excitation, it is necessary to model the PAHs as being illuminated by radiation fields with different spectral shapes.\n\n\nOur data here show differences between the PAH excitation within two galaxies with similar Hubble types.  We should expand the analysis to include galaxies with a wider range of Hubble types, including E, S0, and Sa galaxies where the evolved stellar populations may play a larger role in dust heating and therefore may be more responsible for PAH excitation.  We should also examine other grand-design spiral galaxies to determine whether PAH emission from the spiral arms in these galaxies is offset from the star forming regions in the same way that it is in M83. This research will lead to a better understanding of PAH excitation mechanisms as well as the relation of PAHs to star formation and large dust grains. \n\n\n\\section*{Acknowledgments}\n\n\nThis work has ben produced as part of a MSc Thesis for the University of Manchester. SPIRE has been developed by a consortium of institutes led by Cardiff Univ. (UK) and including: Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).  IDL is a postdoctoral researcher of the FWO-Vlaanderen (Belgium).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{1. Introduction}\nIn recent years the effect of the spin of electrons on the\ntransport properties of nanostructures have been studied\nintensively, both theoretically and experimentally. In the context\nof spin-based electronics (spintronics) the possibility to control\nelectrical currents by a weak external magnetic field using the\nZeeman and\/or the spin-orbit interaction is one of the main goals.\n\nMagnetic materials and especially half-metals are natural sources of\nspin-polarized electrons for spintronics. Transport of spin-polarized\nelectrons in nanostructures (quantum dots, suspended nanowires,\netc.) in external magnetic field results in new phenomena where spin,\ncharge and mechanical degrees of freedom are strongly inter-related.\nIn this new field of investigations (spintromechanics, see\nRef.~\\onlinecite{pulkin}) the presence of a mechanically ``soft\"\nsubsystem results both in a strong enhancement of spintronic effects\nand in magnetic control of the mechanical subsystem in the\nclassical as well as in the quantum transport regimes.\n\nVibrational effects are known to be important for the transport\nproperties of molecular transistors (see, e.g., the reviews in\nRefs.~\\onlinecite{galperin} and \\onlinecite{krive}). In single-molecule\ntransistors a strong electron-vibron coupling was observed\nin a $C_{60}$-based transistor with nonmagnetic (gold)\nleads \\cite{park}. The measured current-voltage characteristics in\nthis experiment revealed low-energy periodic step-like features.\nThey were interpreted as a signature of vibron-assisted electron\ntunneling via the fullerene molecule. Experimental $I-V$ curves were\ntheoretically explained \\cite{shekhter,flensberg} in the frames of a\nsimple model of a single-level quantum dot strongly coupled to a\nsingle vibrational mode and weakly coupled to the source and drain\nelectrodes.\n\nLater on $C_{60}$-based molecular transistors with magnetic (Ni)\nleads were fabricated \\cite{pasupathy}. In samples where the\ntunneling coupling to the ferromagnetic electrodes were\nrelatively strong ($\\sim$ tens of meV), Kondo-assisted tunneling via the\n$C_{60}$ molecule was observed. These measurements also proved the\npresence of a strong inhomogeneous magnetic field produced by the\nferromagnetic electrodes in the nano-gap between them. In samples with weak\ntunneling couplings the usual Coulomb blockade picture for a single-electron \ntransistor was observed.\n\nIn the present paper we formulate the conditions for the\nappearance of a vibrational instability of a fullerene molecule\nsuspended in the gap between two magnetic leads with opposite\nmagnetization. Electron shuttling of spin-polarized electrons\nproduced by magnetic (exchange) forces was\npredicted in Ref.~\\onlinecite{kul} for the case of 100\\%\npolarization of the leads. In this limit (realized for\nhalf-metals) the electric current is blocked (spin blockade) in\nthe absence of spin-flips induced by, e.g., an external magnetic\nfield. It was shown that in the absence of dissipation in the\nmechanical subsystem such a magnetic field triggers a shuttle\ninstability even for vanishingly small fields \\cite{kul}. In the\npresence of dissipation a threshold magnetic field is determined\nby the rate of dissipation and it is small for a weak dissipation.\n\nOne of\nour aims here is to develop a theory of\nmagnetic shuttling for conditions corresponding to the experimental\nset-up of Ref.~\\onlinecite{pasupathy}, where the electrons in the\nferromagnetic leads were partially polarized ($\\sim 30\\%$). The\nabsence of a spin blockade in this case qualitatively changes the criterion for\nelectron shuttling. We will show that even in the absence of mechanical\ndissipation, a shuttling regime of electron transport occurs in a\nfinite interval of external magnetic field strengths,\n$H_{\\text{min}}<H<H_{\\text{max}}$, where $H_{\\text{min}}$ is\ndetermined by the degree of spin polarization $\\eta$.\n\nIn particular, for a high degree of spin polarization\n$(\\eta\\rightarrow 1)$ and if $\\Gamma\\gg\\hbar\\omega$ (where $\\Gamma\/\\hbar$\nis the tunneling\nrate of majority spin electrons and $\\omega\/2\\pi$ is the\nmechanical vibration frequency of the fullerene) the threshold magnetic field for\nreaching the shuttling regime of electron transport reads:\n$H_{\\text{min}}\\sim\\sqrt{1-\\eta}\\,\\Gamma$. In the limit of ``hard\"\nvibrons, $\\Gamma\\ll\\hbar\\omega$, the threshold field is determined\nby the vibron energy, $H_{\\text{min}}\\sim\\sqrt{1-\\eta}\\,\\hbar\\omega$.\n\nThe calculations outlined below aim at determining the rate of change\n$r(H)$ (to be defined later) of the\ncenter-of-mass coordinate of the single-molecule shuttle, the sign of\nwhich then allows us to formulate the\nconditions required to observe shuttling of spin-polarized\nelectrons in $C_{60}$-based molecular transistors.\n\n\\section{2. Hamiltonian and Equations of Motion}\nThe Hamiltonian of a magnetically driven single electron shuttle (see\nRefs.~\\onlinecite{kul} -- \\onlinecite{parafilo}) consists of four terms,\n\\begin{equation}\\label{4}\n\\hat H=\\sum_{j=S,D}\\left(\\hat H_j+\\hat H_{t,j}\\right)+\\hat H_d+\\hat\nH_v,\n\\end{equation}\nwhere $\\hat H_j$ is the standard Hamiltonian of noninteracting\nelectrons in the source ($j=S$) and drain ($j=D$) electrodes, $\\hat\nH_{t,j}$ is a tunneling Hamiltonian with coordinate dependent\ntunneling amplitudes $t_j(\\hat x)$ and $\\hat H_d$ is the Hamiltonian\nof a single level ($\\varepsilon_0$) quantum dot (QD) magnetically coupled to\nleads of spin-polarized electrons by coordinate dependent exchange\ninteractions $J_j(\\hat x)$. It has been shown \\cite{krive} that the shuttling\nregime of single-electron transport can be realized in the presence\nof an external magnetic field $H_{\\text{ext}}$ for oppositely\nmagnetized source and drain electrodes. If\nthe external\nmagnetic field is directed perpendicular to the antiparallel polarization vectors\nof the leads, the QD Hamiltonian reads\n\\begin{eqnarray}\\label{5}\n&&\\hat H_d= \\left[\\varepsilon_0-\\frac{J(\\hat x)}{2} \\right]\na^\\dagger_\\uparrow a_\\uparrow+\\left[\\varepsilon_0+\\frac{J(\\hat\nx)}{2} \\right] a^\\dagger_\\downarrow\na_\\downarrow\\nonumber\\\\&&\\hspace{0.65cm}-\\frac{g\\mu\nH}{2}\\left(a^\\dagger_\\uparrow a_\\downarrow+a^\\dagger_\\downarrow\na_\\uparrow\\right)+Ua^\\dagger_\\uparrow a_\\uparrow\na^\\dagger_\\downarrow a_\\downarrow.\n\\end{eqnarray}\nHere $a^\\dag_\\sigma$ $(a_\\sigma)$ is the creation (annihilation)\noperator for an electron with spin projection $\\sigma=\\uparrow,\\,\n\\downarrow$ on the dot, $J(\\hat x)=J_S(\\hat x)-J_D(\\hat x)$, $\\mu$\nis the Bohr magneton ($g$ is the gyromagnetic ratio), and $U$ is the\nCoulomb repulsion energy. QD vibrations are described by the\nharmonic oscillator Hamiltonian\n\\begin{equation}\\label{6}\n\\hat H_v=\\frac{\\hat p^2}{2m}+\\frac{m\\omega^2\\hat x^2}{2},\n\\end{equation}\nwhere $\\hat x$ is the displacement operator, $\\hat p$ is the\ncanonical conjugated momentum ($\\left[\\hat x, \\hat p\n\\right]={i}\\hbar$), $m$ is the mass and $\\omega$ is the\n(angular) vibration frequency of the QD.\n\nThe aim of the present paper is to find the conditions under\nwhich magnetically driven shuttling in experiments with\nfullerene-based single-molecule transistors \\cite{park,pasupathy}\ncan be realized.\nIn this case the ferromagnetic leads are characterized by a certain\ndegree $0<\\eta<1$ of spin polarization ($\\sim 30\\%$ in the\nexperiment of Ref.~\\onlinecite{pasupathy}) and the characteristic\nvibron energy $\\hbar\\omega$ ($\\sim$ several meV,\nRef.~\\onlinecite{park}) is larger or of the order of the energy\nscale $\\Gamma$ that characterizes the tunneling coupling to the leads,\n$\\Gamma\\leq\\hbar\\omega$.\n\nIt follows from the above considerations that we are not in the adiabatic regime\nof mechanical motion  (which is said to be antiadiabatic when\n$\\Gamma\\ll \\hbar\\omega$) and the physical picture of ``magnetic\nshuttling\" developed in Ref.~\\onlinecite{kul} (see also\nRef.~\\onlinecite{parafilo}) for the adiabatic regime does not hold\nhere. As in Ref.~\\onlinecite{ilin} we will solve the problem using\nequations of motion for the reduced density operator. In the limit\n$eV\\gg\\Gamma,\\, \\hbar\\omega,\\, \\mu H,\\, k_BT$ ($T$ is the\ntemperature, $V$ is the bias voltage) the density operator can be\nfactorized into the product of a QD density operator and an\nequilibrium density matrix for the leads. In the Coulomb blockade\nregime, $eV,\\,T\\ll U$, the matrix elements of the QD density\noperator ($\\rho_0=\\langle 0\\vert\\hat \\rho_d\\vert0 \\rangle$,\n$\\rho_\\sigma=\\langle \\sigma\\vert \\hat \\rho_d\\vert\\sigma\\rangle$,\n$\\rho_{\\uparrow\\downarrow}=\n\\langle\\uparrow\\vert\\hat\\rho_d\\vert\\downarrow\\rangle$) in the\nHilbert space of a singly occupied dot level (they are still\noperators in the Hilbert space of a harmonic oscillator) are\ndetermined by the following set of equations \\cite{mis},\n\\begin{eqnarray}\\label{baseq}\n&&\\frac{\\partial \\rho_0}{\\partial t}=-i\\left[ H_v,\n\\rho_0\\right]-\\frac{1}{2}\\left\\{\\Gamma_S^\\uparrow(\\hat\nx)+\\Gamma_S^\\downarrow(\\hat x),\n\\rho_0\\right\\}\\nonumber\\\\\n&&\\hspace{0.7cm}\\label{baseq1} +\\sqrt{\\Gamma_D^\\uparrow(\\hat\nx)}\\rho_\\uparrow\\sqrt{\\Gamma_D^\\uparrow(\\hat x)}\n+\\sqrt{\\Gamma_D^\\downarrow(\\hat x)}\n\\rho_\\downarrow\\sqrt{\\Gamma_D^\\downarrow(\\hat x)},\\\\\n&&\\frac{\\partial \\rho_\\uparrow}{\\partial t}=-i\\left[\nH_v,\\rho_\\uparrow\\right]+\\frac{{i}}{2}\\left[J(\\hat x),\n\\rho_\\uparrow\\right]+ \\frac{{i} h}{2}\n\\left(\\rho_{\\uparrow\\downarrow}\n-\\rho^{\\dag}_{\\uparrow\\downarrow}\\right)\n\\nonumber\\\\&&\\hspace{0.7cm}+\\sqrt{\\Gamma_S^\\uparrow(\\hat x)}\\rho_0\n\\sqrt{\\Gamma_S^\\uparrow(\\hat x)}-\n\\frac{1}{2}\\left\\{\\Gamma_D^\\uparrow\n(\\hat x), \\rho_\\uparrow\\right\\},\\\\\n&& \\frac{\\partial \\rho_\\downarrow}{\\partial t}=-i\\left[\nH_v,\\rho_\\downarrow\\right]-\n\\frac{{i}}{2}\\left[J(x),\\rho_\\downarrow\\right]-\\frac{{i}\nh}{2}\\left(\\rho_{\\uparrow\\downarrow}-\\rho^\\dag_{\\uparrow\\downarrow}\n\\right)\\nonumber\\\\\n&&\\hspace{0.7cm}+\\sqrt{\\Gamma_S^\\downarrow(\\hat\nx)}\\rho_0\\sqrt{\\Gamma_S^\\downarrow(\\hat x)} -\\frac{1}{2}\n\\left\\{\\Gamma_D^\\downarrow(\\hat x),\\rho_\\downarrow\\right\\},\\\\\n&& \\frac{\\partial \\rho_{\\uparrow\\downarrow}}{\\partial t}= -i\\left[\nH_v, \\rho_{\\uparrow\\downarrow}\\right]+\\frac{{i}}{2}\\left\\{J(\\hat\nx),\\rho_{\\uparrow \\downarrow}\\right\\}+\\frac{{i} h}{2}\n\\left(\\rho_\\uparrow-\\rho_\\downarrow\\right)\\nonumber\\\\&&\\hspace{0.7cm}\n-\\frac{\\rho_{\\uparrow\\downarrow} }{2}\\Gamma_D^\\downarrow(\\hat x)\n-\\Gamma_D^\\uparrow(\\hat x)\\frac{\\rho_{\\uparrow\\downarrow}}{2}.\n\\label{baseq2}\n\\end{eqnarray}\nIn order to solve Eqs.~(\\ref{baseq1}) -- (\\ref{baseq2})\nit is convenient to introduce dimensionless variables\nfor time ($t\\omega\\rightarrow t$), dot displacement ($\\hat\nx\/x_0\\rightarrow \\hat x$, where $x_0=\\sqrt{\\hbar\/m\\omega}$ is the\nzero-point oscillation amplitude), momentum ($\\hat p\nx_0\/\\hbar\\rightarrow \\hat p$) and for various characteristic\nenergies ($\\hbar\\omega\\rightarrow 1$, $g\\mu\nH\/\\hbar\\omega\\rightarrow h$, $\\Gamma_j^\\sigma(\\hat\nx)\/\\hbar\\omega\\rightarrow\\Gamma_j^\\sigma (\\hat x)$, where\n$\\Gamma_j^\\sigma(\\hat x)=2\\pi\\nu\\vert t_{j,\\sigma}(\\hat x)\\vert^2$\nis the level width, $\\nu$ is the density of states).\n\n\\section{3. Influence of Spin Polarization on the Magnetic Shuttle\nInstability} We are interested in the conditions under which a\nmagnetic shuttle instability occurs. Therefore it is sufficient to\nexpand tunneling amplitudes and exchange energies up to linear\nterms in the coordinate operator $\\hat x$,\n\\begin{equation}\\label{7}\nJ(\\hat x)=J_0-\\alpha \\hat x,\\,\\Gamma_{S\/D}^\\sigma(\\hat\nx)=\\Gamma_{S\/D}^\\sigma\\left(1\\mp\\frac{\\hat x}{l}\\right),\n\\end{equation}\nwhere $l$ is the characteristic electron tunneling length (of the\norder of 1~{\\AA}).\n\nNote that the zero-point fluctuation amplitude of a fullerene molecule \nin the vicinity of the Lennard-Jones potential minimum is\nmuch smaller than the electron tunneling length, $l\\gg 1$. Since the\nspatial scale of the exchange interaction is also determined by the\ntunneling length $l$, we can treat the mechanical coordinate $\\hat x$ as\na classical variable. The equation of motion for the classical\ncoordinate $x_c$ takes the form\n\\begin{equation}\\label{8}\n\\frac{d^2 x_c}{d\nt^2}+x_c=-\\frac{\\alpha}{2}\\text{Tr}\\left(\\rho_\\uparrow-\n\\rho_\\downarrow\\right)\\,.\n\\end{equation}\nThe r.h.s. of Eq. (\\ref{8}) can readily be found from the set of\nlinear equations (\\ref{baseq1}) -- (\\ref{baseq2}) when the operator $\\hat\nx$ is replaced by $x_c$.\n\nTo simplify analytical calculations we consider a symmetric junction for which\n$J_0=0$, $\\Gamma_S^\\uparrow=\\Gamma_D^\\downarrow=\\Gamma$, and\n$\\Gamma_S^\\downarrow=\\Gamma_D^\\uparrow=\\gamma$. To first order in\nthe small dimensionless parameter\n$\\alpha\/l\\sim\\left(J\/\\hbar\\omega\\right) \\left(x_0\/l\\right)^2\\ll 1$\nEq.~(\\ref{8}) is reduced to a linear equation. For the time\ndependent coordinate displacement $x_1=x_1(t)$ this equation reads\n\\begin{equation}\\label{9}\n\\frac{d^2\nx_1}{dt^2}+x_1=-\\frac{\\alpha}{2l}\\int_{-\\infty}^tdt'\\langle e_0\\vert\ne^{\\hat A(t-t')}\\vert e\\rangle x_1(t').\n\\end{equation}\nHere the matrix $\\hat A$ takes the form\n\\begin{equation}\\label{90}\n\\hat A=\\frac{1}{2}\\left(\n\\begin{array}{ccc}\n-\\Gamma_1&-\\Gamma_2& -2h \\\\\n\\Gamma_2 &-3\\Gamma_1 & 0 \\\\\n2h & 0& -\\Gamma_1\n\\end{array}\\right),\n\\end{equation}\n$\\Gamma_{1,2}=\\Gamma\\pm\\gamma$, $\\vert e_0\\rangle=(1,0,0)^T$, and the\nvector $\\vert e\\rangle=(e_1,e_2,e_3)^T$ has the components\n\\begin{eqnarray}\n&&e_1= -\\frac{2\\Gamma_2}{\\Delta}(\\Gamma_1^2-\\Gamma_2^2),\\,\ne_2=-\\frac{2\\Gamma_1}\n{\\Delta}\\left(\\Gamma_1^2-\\Gamma_2^2+4h^2\\right) \\nonumber\\\\\n&&e_3=-\\frac{4h\\Gamma_1\\Gamma_2}{\\Delta},\\,\n\\Delta=3\\Gamma_1^2+\\Gamma_2^2+12h^2.\n\\end{eqnarray}\n\nA shuttle instability occurs when the oscillatory solution of\nEq.~(\\ref{9}), $x_1(t)\\sim\\text{exp} ({i}\\Omega t)$ becomes\nunstable, that is when $\\text{Im}\\,\\Omega<0$. The smallness of the\nr.h.s. of Eq.~(\\ref{9}) allows one to obtain the imaginary part of\n$\\Omega$ by perturbation theory. The values of the parameters\n$\\Gamma_1, \\Gamma_2$ and $h$ for which $\\text{Im}\\,\\Omega<0$,\ncorresponding to the\nshuttle regime of electron transport, satisfy the inequality\n\\begin{equation}\\label{10}\nh^4+A_1h^2+A_2<0,\n\\end{equation}\nwhere\n\\begin{eqnarray}\nA_1&=&-3\\left(\\Gamma_1^2+1\\right)-\\frac{\\Gamma_1^2\n-\\Gamma_2^2}{8\\Gamma_1^2}\\left(5\\Gamma_1^2+4\\right),\\\\\nA_2&=&\\frac{\\left(\\Gamma_1^2-\\Gamma_2^2\\right)\n\\left(\\Gamma_1^2+4\\right)}{8\\Gamma_1^2}\\left(\\Gamma_1^2+1+\n\\frac{\\Gamma_1^2-\\Gamma_2^2}{4}\\right).\n\\end{eqnarray}\nNote that in the absence of an external magnetic field the\nnanoelectromechanical coupling results in additional damping of the\nmechanical subsystem (since the coefficient $A_2>0$).\n\nIt is evident from Eq.~(\\ref{10}), which is biquadratic in the magnetic field ($h$),\nthat a shuttle instability occurs in a finite interval\nof magnetic fields, $h_{\\text{min}}<h<h_{\\text{max}}$. Now we\nintroduce the degree of spin polarization as\n\\begin{equation}\\label{100}\n\\eta=\\frac{\\vert N_\\uparrow-N_\\downarrow\\vert}{N_\\uparrow+\nN_\\downarrow}\\simeq\\frac{\\Gamma-\\gamma}{\\Gamma+\\gamma}\n\\end{equation}\nwhere $N_\\sigma$ is the number of particles with spin projection\n$\\sigma$. The last relation in Eq.~(\\ref{100}) comes from the\ndefinition of tunneling rates and our assumption that tunneling\namplitudes do not depend on spin projections. We analyze\nEq.~(\\ref{10}) in two limiting cases: (i) $\\Gamma\\gg 1$, $1-\\eta\\ll 1$,\nand (ii) $\\Gamma\\ll 1$, $1-\\eta\\ll 1$.\n\nNote that in the limit of weak polarization, $\\eta\\rightarrow 0$,\nmagnetic forces are small, $J\\rightarrow 0$, and electron shuttling\nis supported by Coulomb forces. This case was considered in\nRef.~\\onlinecite{ilin}, where  it was shown that in the limit of\nweak polarization a magnetic field ceases to influence single electron\nshuttling.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{Fig1_Ilinskaya.eps}\n\\caption {Upper critical magnetic field (upper set of curves) and\nlower critical field (lower set of curves) for the shuttle\ntransport regime plotted as a function of the normalized tunneling\nrate $\\Gamma\/\\hbar\\omega$ of majority spin electrons for different\nvalues of spin polarization $\\eta$. The solid curves were plotted\nfor $\\eta =1$ [100\\% spin polarization, here the lower curve\ncoincides with the $x$-axis], the dashed curves were plotted for\n$\\eta = 0.8$, and the short-dashed curves for $\\eta = 0.3$. The\nshuttle regime corresponds to the area between the lower and upper\ncritical fields (dark region for the case of $\\eta = 0.8$), while\noutside this area one is in the vibronic regime.}\n\\end{figure}\n\nIn the adiabatic limit, (i), one finds (reverting to using\nparameters with dimensions) that the\nshuttle instability region is defined by the double inequality\n\\begin{equation}\\label{11}\n\\frac{\\Gamma}{2}\\sqrt{\\frac{1-\\eta}{3}}<g\\mu H<\\sqrt{3}\\,\\Gamma.\n\\end{equation}\nIn the antiadiabatic limit, (ii), the critical magnetic fields are\ndetermined by the vibron energy $\\hbar\\omega$ rather than the\ntunneling rate $\\Gamma$, so that\n\\begin{equation}\\label{12}\n\\hbar\\omega\\sqrt{\\frac{1-\\eta}{3}}<g\\mu H<\\sqrt{3}\\,\\hbar\\omega.\n\\end{equation}\n\nFor the general case that part of the $\\Gamma, H$ parameter space\nwhich corresponds to a magnetic shuttle instability is shown in\nFig.~1 for different degrees of spin polarization.\nFor a given (normalized) tunneling coupling $\\Gamma\/\\hbar\\omega$ \nthe range of magnetic fields for which an instability occurs is shifted \ntowards higher fields as the spin polarization decreases.\nThe upper critical\nfield for high spin polarizations, $\\eta\\rightarrow 1$, depends\nlinearly on $\\eta$,\n\\begin{equation}\\label{13}\ng\\mu\nH_{\\text{max}}\\sim\\sqrt{3\\left(\\Gamma^2+(\\hbar\\omega)^2\\right)}\n\\left[1+K(1-\\eta)\\right]\\,,\n\\end{equation}\nwhere $K\\sim 1$ is a positive constant.\n\nThe dependence of the lower critical magnetic field on $\\eta$ is\nweaker. In the antiadiabatic regime one finds that\n\\begin{equation}\\label{14}\ng\\mu H_{\\text{min}}\\sim\n\\hbar\\omega\\sqrt{1-\\eta},\\quad \\Gamma\\ll\\hbar\\omega,\n\\end{equation}\nand\nhence $H_{\\text{min}}$ rapidly saturates to a constant value of\norder $\\hbar\\omega\/g\\mu$ with decreasing spin polarization (see\nFig.~2, short-dashed curve).\nIn the adiabatic regime,\n$\\Gamma\\gg\\hbar\\omega$, the\nlower critical field\ndecreases linearly with increasing spin polarization, except in\nthe close vicinity of complete spin polarization, where\n$H_{\\text{min}} \\propto \\sqrt{1-\\eta}$ (compare the solid curve in\nFig.~2).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{Fig2_Ilinskaya.eps}\n\\caption {Plots of the lower critical magnetic field\n$H_{\\text{min}}$, which defines the border between the vibronic\nregime ($H<H_{\\text{min}}$) and the shuttle transport regime\n($H>H_{\\text{min}}$; compare Fig.~1) vs. spin polarization $\\eta$\nfor different values of the normalized tunneling rate\n$\\Gamma\/\\hbar\\omega$ of majority spin electrons (solid curve:\n$\\Gamma\/\\hbar\\omega=10$;  dashed curve: $\\Gamma\/\\hbar\\omega=3$;\nshort-dashed curve: $\\Gamma\/\\hbar\\omega=0.1$).}\n\\end{figure}\n\nThe appearance of an upper and a lower critical magnetic field has a\nsimple physical explanation. When $\\mu H$ is the largest energy\nscale in our problem, $\\mu H\\gg \\Gamma,\\,\\hbar\\omega$, the fast\nprecession of the electron spin of the dot in a perpendicular external magnetic field\nnullifies the average spin and the magnetic shuttle instability\ndisappears. To estimate the upper field one may compare the\ncharacteristic spin precession frequency, $\\mu\nH_{\\text{max}}\/\\hbar$, with the electron tunneling rate,\n$\\Gamma\/\\hbar$, or the frequency of vibrations $\\omega$. That is\n$\\mu H_{\\text{max}}\\sim\\text{max}\\left(\\Gamma, \\hbar\\omega\\right)$.\nThe lower critical field can be readily estimated for\na high degree of spin polarization, $1-\\eta\\ll 1$.\nIn this case we have to compare the average time between spin\nflips, $\\tau_f$, induced by a constant magnetic field $H$ in the\npresence of an electron tunneling coupling $\\Gamma$ with the\ncharacteristic life-time of minority spin electrons on the dot,\n$\\sim \\hbar\/\\gamma$. The spin-flip rate $\\nu_f$ in weak magnetic\nfields $H$ can be estimated by perturbation theory\nwith the result that $\\hbar\\nu_f\\sim(\\mu H)^2\/\\text{max}(\\Gamma,\n\\hbar\\omega)$. Therefore the lower magnetic field is strongly\nsensitive to spin polarization,\n\\begin{equation}\\label{15}\n\\mu H_{\\text{min}}\\sim\n\\sqrt{\\Gamma\\gamma}\\,\\text{max}(\\Gamma,\\hbar\\omega)\\sim\n\\sqrt{1-\\eta}\\,\\text{max}(\\Gamma,\\hbar\\omega),\n\\end{equation}\nand disappears for 100\\% spin-polarized electrons ($\\eta=1)$.\n\nNext we estimate the maximum rate of (exponential) increase,\n$r_m=-\\text{Im}\\{\\Omega (H_{\\text{opt}})\\}$, of the QD oscillation\namplitude in the shuttle regime.\nIn the adiabatic limit, $\\Gamma\\gg \\hbar\\omega$,\none finds that $g\\mu H_{\\text{opt}}\\simeq 0.4\\, \\Gamma$ and that\n\\begin{equation}\\label{16}\nr_m\\simeq C\\frac{\\omega J}{\\Gamma}\\left(\\frac{x_0}{l}\\right)^2,\n\\end{equation}\nwhere $C\\sim 0.1$ is a small numerical factor. In the case\n$\\Gamma\\ll \\hbar\\omega$, which we are interested in here, the\nmaximum rate is realized when $g\\mu H_{\\text{opt}}\\simeq\n\\hbar\\omega$, corresponding to\n\\begin{equation}\\label{17}\nr_m\\simeq\n\\frac{\\Gamma}{\\hbar}\\frac{J}{\\hbar\\omega}\\left(\\frac{x_0}{l}\\right)^2\\,,\n\\end{equation}\nwhere we omit a numerical factor of the order of one.\n\nIn the presence of dissipation in the mechanical subsystem, which\ncan be described by adding a phenomenological friction term\n$\\gamma_d \\dot{x}_c(t)$ to the equation of motion (\\ref{8})\n($\\gamma_d=\\omega\/Q$, where $Q$ is the quality factor), the\nshuttling regime appears when $r_m>\\omega\/Q$. Therefore electron\nshuttling in a $C_{60}$-based molecular transistor with magnetic\nelectrodes could be realized if the quality factor $Q$ of the\nmechanical resonator obeys the inequality\n\\begin{equation}\n\\label{18}\nQ>Q_{\\text{opt}}=\\frac{\\left(\\hbar\\omega\\right)^2}{J\\Gamma}\n\\left(\\frac{l}{x_0}\\right)^2.\n\\end{equation}\n\nFor the experimental setup in Ref.~\\onlinecite{park}, where\nfullerene vibrations were observed, the factor\n$\\left(l\/x_0\\right)^2\\simeq 10^3$ and\n$\\Gamma\\ll\\hbar\\omega\\sim$~5~meV (one can estimate\n$\\Gamma\\sim$~0.1 -- 0.5~meV from the maximal current measured in\nRef.~\\onlinecite{park}). In the $C_{60}$-based transistor with\nmagnetic (Ni) leads $J\\sim\\Gamma\\sim$~10~meV (see\nRef.~\\onlinecite{pasupathy}). From Eq.~(\\ref{18}) one can estimate\nthat the required quality factor is $Q\\geq 10^3-10^4$. However the\noptimal external magnetic field in this case,\n$H_{\\text{opt}}\\simeq$~50~T, is too high. Instead, we therefore\nestimate $Q$ for magnetic fields in the vicinity of the lower\ncritical magnetic field $H\\geq H_{\\text{min}}$ where magnetic\nfields for a very high degree of electron spin polarization ($\\sim\n99\\%$) could be of the order of a few tesla. In this case\n($\\hbar\\omega\\gg \\Gamma$, $1-\\eta\\ll 1$)\n\\begin{equation}\\label{19}\nr(\\eta)\\simeq \\omega\\frac{J \\Gamma\n(1-\\eta)}{\\Gamma^2+4(1-\\eta)(\\hbar\\omega)^2}\\left(\\frac{l}{x_0}\\right)^2.\n\\end{equation}\nAssuming that $\\Gamma\\simeq\\sqrt{1-\\eta}\\,\\hbar\\omega$ we find\n$Q\\sim Q_{\\text{opt}}\/(1-\\eta)$.\n\n\\section{4. Conclusions}\nIn summary we have considered the feasibility of observing magnetically\ndriven single-electron shuttling under\nrealistic conditions\ncorresponding to an already experimentally realized $C_{60}$-based\nsingle-molecule transistor with magnetic leads. The main\nrequirement for magnetic shuttling is the presence of an external\nmagnetic field that induces electron\nspin flips. We have shown that the optimal magnetic field, defined\nas the field that maximizes the rate of increase of the shuttling\namplitude, is determined by the vibration frequency $\\omega$. For\nfullerene-based single-electron transistors this frequency could\nbe in the THz region \\cite{park}\nwith\ncorresponding\nmagnetic fields\nin the region of several tenths of teslas. For magnetic electrodes\nwith a very high degree of spin polarization one needs less strong\n(by an order of magnitude) magnetic fields. However, the quality\nfactor of the corresponding mechanical resonator has to be\nexceptionally high, $Q\\geq 10^5$.\n\n{\\bf Acknowledgements:} Financial support from the Leading Foreign\nResearch Institutes Recruitment Program (2009-00514) of NRF,\nKorea, and the Swedish Research Council (VR) is gratefully\nacknowledged. OI, IK and SK acknowledge financial support from\nNational Academy of Science of Ukraine, Grant No 4\/15--N. IK and\nSK thank the Department of Physics at the University of Gothenburg\nand the Department of Physics and Astronomy at Seoul National\nUniversity for their hospitality.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nComplex systems of a dispersed phase in a solid matrix can behave very differently from one of their components taken alone. Their broad range of properties explains that examples of dispersions such as composites~\\cite{matthews1999composite} or porous media~\\cite{coussy2011mechanics} are widespread in nature and in the industry. In all dispersions, interfacial forces can appear at the boundary between the dispersed phase and the continuous matrix. A coupling of surface tension forces to the bulk elasticity of a solid has been evidenced in soft systems like biological tissues~\\cite{clements1961pulmonary}, or through the deformation of soft substrates like polymers at the contact line with a drop resting on the solid~\\cite{pericet2008effect}. Capillary forces also affect the overall mechanical properties of nanoporous media~\\cite{duan2005size}. For larger pores, because of the hardness of the matrix in usual porous media, the influence of interfacial effects on the overall properties of the saturated material is negligible~\\cite{dormieux2006microporomechanics}. Dispersions in softer materials could allow for observable coupling of interfacial forces to the bulk elasticity of the solid at larger scales than the nanometer. Many dense suspensions~\\cite{coussot2005rheometry} of geological interest, like muds, or with industrial applications, like fresh concrete or emulsions, behave as soft elastic solids below a critical level of stress~\\cite{coussot2012rheophysique}. To study the role of surface tension forces in soft elastic materials, we investigate the elastic behaviour of dispersions of bubbles in concentrated emulsions. Those aerated emulsions, which have applications in the food~\\cite{vanAken2001333} and cosmetic~\\cite{balzer1991alkylpolyglucosides} industry, have been the subject of stability and rheology studies~\\cite{C1SM06537H, PhysRevLett.104.128301, kogan2013mixtures}. However, their overall elastic properties have not yet been studied in detail.\n\nIn dispersions of bubbles in a soft material, coupling between the elasticity of the matrix and capillary effects is expected to occur through bubble deformation. The elastic deformation of the matrix tends to deform the bubbles and surface tension forces will thus act to minimize the area of the bubble by maintaining a spherical shape. The limit case of negligible surface tension forces is a soft porous medium. Theoretical work shows that adding holes in a solid softens it~\\cite{dormieux2006microporomechanics}. In the limit case of predominant surface tension forces compared to the matrix elasticity, a bubble should no longer be deformable and should behave as a rigid inclusion with no shear stiffness. Experimental and theoretical work have shown that rigid beads in a soft solid strengthen the solid~\\cite{mahaut2008yield}. The case of rigid bubbles is similar except for the boundary condition, changed from no-slip for beads to full-slip for bubbles. Theoretical models in the dilute limit predict a strengthening of the dispersion when adding rigid bubbles~\\cite{dormieux2006microporomechanics}. Between those two limit cases, more work is needed to investigate the elastic response of the soft aerated solid. In this work, we restrain to the range of gas volume fraction $\\phi<50\\%$, so that we do not consider foams of those materials, in which the bubbles are deformed by geometrical constraints. We design model systems and appropriate experimental methods that allow us to measure the shear modulus of dispersions of monodisperse bubbles embedded in a medium of chosen elasticity. We compare our experimental results to estimates of the elastic modulus through a micro-mechanical approach. \n\\section*{Experimental aspects}\nThe dispersion matrices we choose are concentrated oil in water emulsions of shear moduli ranging from 100 to 1000Pa. Concentrated emulsions behave as soft elastic solids for stresses well below their yield stress~\\cite{mason1995elasticity}. In the experimental systems, unless otherwise indicated, the radius of the droplets is around 1 to 2$\\mathrm{\\mu m}$ (the poydispersity is around 20\\%), which, at the considered gas volume fractions, should ensure that there is scale separation between the drops and the bubbles, and consequently validate the use of the emulsion as an elastic continuous medium embedding the bubbles~\\cite{goyon2008spatial}. In all the systems, the yield stress of the emulsion is high enough to ensure that no bubble rise occurs at rest or during measurements~\\cite{Dubash2007123}. Most dispersions are prepared by gently mixing the emulsion with a separately produced monodisperse foam. The foams are obtained by blowing nitrogen plus a small amount of perfluorohexane ($\\mathrm{C_{6}F_{14}}$) through a porous glass frit or through needles: we are able to produce nearly monodisperse foams with average bubble radii $R_{b}$ ranging from 40$\\mathrm{\\mu m}$ to 800$\\mathrm{\\mu m}$. Coarsening is strongly reduced by the presence of $\\mathrm{C_{6}F_{14}}$~\\cite{gandolfo1997interbubble}, meaning that the bubble size is stable during measurements. The continuous phase of the foam is the same as the one in the emulsion, ensuring that mixing is easy and does not induce any chemical effect in the dispersions. The mixing with the foam adds a small amount of continuous phase to the emulsion. To ensure that for a series of experiments at different $\\phi$ in a given emulsion, the elastic modulus of the matrix in the dispersions remains the same, we add controlled amounts of pure continuous phase in order to reach the same the oil volume fraction in the emulsion~\\cite{PhysRevLett.104.128301, kogan2013mixtures}. An example of a dispersion of bubbles in an emulsion is shown on figure~\\ref{fig:photo}. The composition of all the tested emulsions is indicated in table~\\ref{tab:recap_systs}, and illustrates the variety of chemical compositions, surface tensions and elastic properties of the matrix that were used to perform the study.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{graph1_leg1_2.png}\n\\caption{Microphotograph of a dispersion of monodisperse bubbles (R=200$\\mu$m) in emulsion (2). The emulsion is transparent, allowing for the visualisation of in-depth bubbles that thus do not have the same apparent radius. Inset: close-up of droplets of emulsion (2) at the interface with a bubble.\\label{fig:photo}}\n\\end{figure}\n\nThe shear modulus of the dispersions is measured on a control stress rheometer by imposing small amplitude oscillations at a frequency of typically 1Hz. The oscillatory stress is chosen to be well below the yield stress of the systems, so that the oscillations are performed in the linear elastic regime of each material. At this frequency, the loss modulus of the systems is negligible. The geometry used to perform the rheometrical measurements is chosen according to the bubble size : for $R_{b}\\le 50\\mathrm{\\mu m}$, the material is sheared between parallel  plates (radius $R$=25mm, gap $h$=2.5mm). The planes are serrated to prevent slippage of the dispersion~\\cite{coussot2005rheometry}. Dispersions containing bigger bubbles require a larger thickness of sheared material and are studied in Couette-like devices : for $50\\mathrm{\\mu m}< R_{b}< 800\\mathrm{\\mu m}$, we use a vane in cup (exceptionally a serrated bob in cup) geometry (inner radius $R_i$=12.5mm, outer radius $R_o$=18mm), and for $R_{b}\\ge 800\\mathrm{\\mu m}$, we use vane in cup geometries (either $R_i$=12.5mm and $R_o$=25mm or $R_i$=22.5mm and $R_o$=45mm). \n\\begin{center}\n\\begin{table*}\n{\\renewcommand{\\arraystretch}{1.5}\n\\renewcommand{\\tabcolsep}{0.2cm}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n& \\textbf{oil - vol. fraction} & \\textbf{continuous phase} & \\textbf{$\\mathbf{G'(0)}$ (Pa)} & \\textbf{$\\mathbf{\\gamma}$ (mN.$\\mathbf{m^{-1}}$)}\\\\\n\\hline\nemulsion (1a) & silicon (V20) - 75\\% & Forafac$\\textregistered$ ($\\mathrm{{DuPont}^{TM}}$) 4\\% w. in water & 230 &15.5 $\\pm$ 0.1 \\\\\n\\hline\nemulsion (1b) & silicon (V20) - 73\\% & Forafac$\\textregistered$ ($\\mathrm{{DuPont}^{TM}}$) 4\\% w. in water & 163 &15.5 $\\pm$ 0.1 \\\\\n\\hline\nemulsion (2) & silicon (V350) - 79\\% & TTAB 3\\% w. in water\/glycerol 50\/50 w\/w & 650 & 35.5 $\\pm$ 0.1 \\\\\n\\hline\nemulsion (3) & dodecane - 73\\% & SDS 2.7\\% w. in water & 285 & 36 $\\pm$ 1 \\\\\n\\hline\nemulsion (4) & silicon (V350) - 70\\% & TTAB 3\\% w. in water\/glycerol 36\/64 w\/w & 799 & 35 $\\pm$ 1 \\\\\n\\hline\n\\end{tabular}}\n\\caption{Synthetic description of all the emulsions used as matrices in the bubble dispersions: nature and volume fraction of the oil dispersed phase, composition of the aqueous continuous phase (including the surfactant) and relevant physical constants for the determination of the capillary number: elastic modulus of the matrix, and surface tension between air and the continuous phase. The composition given is the one of the matrix actually embedding the bubbles.\\label{tab:recap_systs}}\n\\end{table*}\n\\end{center}\n\\section*{Results}\nWe start by studying the influence of the bubble radius $R_b$. In this aim, we prepare dispersions of bubbles in emulsion (3) (see table~\\ref{tab:recap_systs} for details). In a first series of experiments, we add bubbles of $R_b=(50\\pm10)\\mathrm{\\mu m}$ ($10\\mathrm{\\mu m}$ being the width of the volume-weighed bubble radius distribution) at various gas volume fractions $\\phi$ in the emulsion. Those bubbles are slightly more polydisperse than is generally used for this study, because of the foam production technique. The shear modulus $G'(\\phi)$ of the dispersions is measured to be slightly decreasing with $\\phi$. This result is reported in dimensionless quantities $\\hat{G}(\\phi)=G'(\\phi)\/G'(0)$ as a function of $\\phi$ on figure~\\ref{fig:1}. We then prepare dispersions of larger bubbles in emulsion (3): a series with $R_b=(143\\pm17)\\mathrm{\\mu m}$ and another one with $R_b=(800\\pm40)\\mathrm{\\mu m}$. The results for $\\hat{G}(\\phi)$ are also reported on figure~\\ref{fig:1}. The measurements show that the larger the bubbles, the softer the dispersion. This result can be understood as a manifestation of a simple physical effect, already evidenced in~\\cite{kogan2013mixtures} (see also~\\cite{rust2002effects} and~\\cite{llewellin2002rheology} for the effect of bubble deformation on the viscosity of bubbly Newtonian fluids), that the interfacial energy to volume ratio is lower in larger bubbles, resulting in least bubble resistance to deformation. \n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.29]{graph1.pdf}\n\\caption{Dimensionless elastic modulus $\\hat{G}$ as a function of the gas volume fraction $\\phi$ for dispersions with three different bubble radii $R_b$ in emulsion (3) [see legend]. The full lines are the computed $\\hat{G}_{homog}(\\phi)$ for $Ca=0.23$ (dark blue), $Ca=0.57$ (green) and $Ca=3.2$ (pink); experimentally measured $Ca$: $0.23\\pm0.05$, $0.57\\pm0.08$, $3.2\\pm0.4$. Inset: $\\hat{G}$ as a function of $\\phi$ for dispersions of $R_b\\approx 150\\mathrm{\\mu m}$ in \\textcolor{spinach}{$\\bullet$} emulsion (3) and \\textcolor{orange}{$\\blacklozenge$} emulsion (4). The full lines are the computed $\\hat{G}_{homog}(\\phi)$ for $Ca=0.57$ (green) and $Ca=1.65$ (orange); experimentally measured $Ca$: $0.57\\pm0.08$, $1.65\\pm0.15$.\\label{fig:1} }\n\\end{figure}\nWe now keep the bubble size constant, and vary the elastic modulus of the matrix: we prepare dispersions of $R_b=143\\mathrm{\\mu m}$ bubbles in emulsion (3) and of $R_b=(150\\pm10)\\mathrm{\\mu m}$ bubbles in emulsion (4) (see table~\\ref{tab:recap_systs}). In the two series of experiments, the bubble sizes are close and the surface tension is similar, but $G'(0)$ is almost three times higher in emulsion (4). $\\hat{G}(\\phi)$ is plotted for both systems on the inset in figure~\\ref{fig:1}. As observed on the previous suspensions, $\\hat{G}(\\phi)$ is a decreasing function of $\\phi$, and this decrease is all the stronger as $G'(0)$ is high.\nTo quantify the competition between the matrix elasticity and the bubble's resistance to deformation, we introduce a capillary number \n\\begin{equation}\nCa=\\frac{G'(0)}{2\\gamma\/R_b}\n\\label{eq:Ca}\n\\end{equation} \nwhich compares the shear modulus of the dispersion medium to the interfacial stress scale, the capillary pressure in the bubbles. This capillary number is equally affected by an increase in $R_b$ or a decrease in $G'(0)$. To quantify the relevance of $Ca$ on the overall elastic response of the dispersion at a given $\\phi$, we perform two series of experiments with close $R_b$, but very different capillary pressure because of very different surface tension, and we adjust the elastic modulus in one of the emulsions so that $Ca$ is similar in both systems. The two experimental systems are as follow: the first one is dispersions of $R_b=143\\mathrm{\\mu m}$ of radius bubbles in emulsion (3), which leads to $Ca=0.57\\pm0.08$, and the second one dispersions of (129$\\pm$10)$\\mathrm{\\mu m}$ of radius bubbles in emulsion (1b) for which $Ca=0.70\\pm0.08$. We observe that the measured values of $\\hat{G}(\\phi, Ca)$ are very close, as can be seen on figure~\\ref{fig:2}. The value of $Ca$ unequivocally determines the elastic behaviour of the dispersion at a given $\\phi$.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.30]{graph2.pdf}\n\\caption{Effect of a change in the surface tension: dimensionless elastic modulus $\\hat{G}$ as a function of $\\phi$ for dispersions of \\textcolor{spinach}{$\\bullet$} $R_b=143\\mathrm{\\mu m}$ bubbles in emulsion (3) and \\textcolor{red}{$\\bigstar$} $R_b=129\\mathrm{\\mu m}$ bubbles in emulsion (1). The surface tension is much lower in emulsion (1b), but $G'(0)$ has been chosen to get close values for $Ca$ in both systems. This experimentally measured $Ca$ are $0.57\\pm0.08$ and $0.70\\pm0.08$. The full line is the computed $\\hat{G}_{homog}(\\phi, Ca)$ at $Ca=0.63$, which is compatible with both systems, given the uncertainty on the value of $Ca$.\\label{fig:2}}\n\\end{figure}\n\nWe now investigate the limit value of $Ca\\to \\infty$, for which surface tension forces are negligible and the bubbles can be assimilated to holes in the matrix. This is the case in usual porous materials. $\\hat{G}(\\phi, Ca\\to\\infty)$ can then be computed in the dilute limit~\\cite{dormieux2006microporomechanics}: $\\hat{G}(\\phi, Ca\\to\\infty)=1-\\frac{5}{3}\\phi$. To compare this prediction to experimental data, we design a system in which surface tension effects are bound to be poor: we include the biggest bubbles of this study, of radius (1 $\\pm$ 0.1) mm, in emulsion (2), which has a high elastic modulus (see table~\\ref{tab:recap_systs}). Note that for this system the bubbles are injected directly in the emulsion in a tee-junction in a milli-fluidic device. As before, we measure the elastic modulus of the dispersion at various $\\phi$. The experimental data points for the dimensionless modulus $\\hat{G}(\\phi)$ are compared to the dilute limit for dispersions of holes in an elastic medium on figure~\\ref{fig:3}. We observe that $\\hat{G}$ is a decreasing function of $\\phi$. The exact value of $Ca$ in this system is $9.0\\pm1.2$. The theoretical dilute limit for spherical holes in an elastic medium is already a good estimate of $\\hat{G}(\\phi\\to 0, Ca)$at $Ca\\sim10$.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.30]{graph3.pdf}\n\\caption{Remarkable values of $Ca$: $\\mathbf{Ca\\to \\infty}$: $\\hat{G}(\\phi)$ for dispersions of \\textcolor{lgtblue}{$\\blacktriangledown$} $R_b$=1mm bubbles in emulsion (2). The full line is the computed $\\hat{G}_{homog}(\\phi, Ca)$ at $Ca=9.0$ (light blue); experimentally measured $Ca$: $9.0\\pm1.2$. $\\mathbf{Ca\\approx 0.25}$: dimensionless elastic modulus $\\hat{G}$ for dispersions of \\textcolor{darkblue}{$\\bullet$} $R_b$=50$\\mu$m bubbles in emulsion (3) and \\textcolor{gray}{$\\bigstar$} $R_b$=41$\\mu$m in emulsion (1a). The full lines are the computed $\\hat{G}_{homog}(\\phi, Ca)$ at $Ca=0.23$ (dark blue) and $Ca=0.30$ (grey); experimentally measured $Ca$: $0.23\\pm0.05$, $0.30\\pm0.05$. The dashed lines are the dilute limits for rigid (top) and fully deformable (bottom) spheres, with a full slip boundary condition.\\label{fig:3} }\n\\end{figure}\n\nThe limit case of $Ca\\to 0$ also leads to a simplification: the bubbles are stiff compared to the matrix and the dispersion is made of rigid spheres with a full slip boundary condition in an elastic medium. The theoretical dilute limit can be computed as $\\hat{G}(\\phi, Ca=0)=1+\\phi$~\\cite{dormieux2006microporomechanics} and is plotted on figure~\\ref{fig:3}. An experimental validation of this limit with our systems may be biased, because increasing the capillary pressure would mean reducing $R_b$, and we might no longer assume scale separation between the bubbles and the oil droplets. We thus choose not to investigate this limit.\nFrom $Ca\\to 0$ to $Ca\\to \\infty$, $\\hat{G}(\\phi, Ca)$ turns from an increasing to a decreasing function of $\\phi$. Between these two extreme values, we have observed on the dispersion of the smallest bubbles in emulsion (3), already discussed above on figure~\\ref{fig:1}, that $G'(\\phi)$ has little variation with $\\phi$ and is comparable to $G'(0)$. The capillary number in this system is $Ca=0.23\\pm0.05$. To further check the peculiarity of this value of $Ca$, we prepare another dispersion of small bubbles $R_b$=(41$\\pm$5)$\\mathrm{\\mu m}$ in emulsion (1a) (see table~\\ref{tab:recap_systs}), with a close capillary number: $Ca=0.30\\pm0.05$. $\\hat{G}(\\phi, Ca)$ for both dispersions of small bubbles is plotted on figure~\\ref{fig:3}. We observe that in both systems, $\\hat{G}(\\phi, Ca)$ exhibit little dependence on the gas volume fraction, and is of order 1. The non-perturbative effect of bubble addition in the matrix can be seen as an experimental validation of previous micro-mechanical calculations~\\cite{palierne1991rheologica, doi:10.1061\/9780784412992.224} which have shown that a spherical bubble of radius $R_b$ and surface tension $\\gamma$ in an elastic medium can be described as an equivalent elastic sphere of radius $R_b$ and no surface tension. Indeed, the deformation of the bubble under a strain $\\epsilon$ leads to an increase in the bubble area that is proportional to $\\epsilon^2$. The stored interfacial energy scales as $\\gamma \\epsilon^2$, which is analogous to an elastic energy. If the equivalent elasticity of the sphere is equal to that of the matrix, the bubbles are non-perturbative and $\\hat{G}(\\phi)= 1$. The equivalent elasticity of a bubble in a matrix $G'(0)$ can be written as a function of $G'(0)$ and $Ca$~\\cite{palierne1991rheologica, doi:10.1061\/9780784412992.224}:\n\\begin{equation}\nG^{eq}=G'(0)\\frac{8}{3+20Ca}\n\\end{equation}\nwith $Ca$ defined in equation~\\ref{eq:Ca}. The expression of $G^{eq}$ shows that $Ca$ introduced above does not actually compare the equivalent elasticity of the bubble to that of the matrix. This explains why the overall elasticity of the dispersion is unperturbed by the presence of the bubbles for a somewhat unnatural value of $Ca$ around 0.2 to 0.3, which can be understood thanks to the computation of $G^{eq}$: $G^{eq}=G'(0)$ for $Ca=1\/4$. Relying on the equivalent elastic sphere model for a bubble, a micro-mechanical approach allows to compute the overall elastic properties of the dispersions at finite $Ca$. The overall elasticity of a composite material made of elastic spheres in a matrix of another elastic material in the semi-dilute limit can be computed as a function of $Ca$ and $\\phi$, in the framework of the Mori-Tanaka scheme~\\cite{doi:10.1061\/9780784412992.224, palierne1991rheologica}: \n\\begin{equation}\n\\hat{G}_{homog}(\\phi, Ca)=1-\\frac{\\phi(4Ca-1)}{1+\\frac{12}{5}Ca-\\frac{2}{5}\\phi(1-4Ca)}\n\\end{equation}\nNote that this expression is compatible with the previously discussed limits of $Ca \\to \\infty$, $Ca \\to 0$ and $Ca=1\/4$. Predictions of the model for $\\hat{G}(\\phi, Ca)$ at the experimentally measured $Ca$ are plotted in full coloured lines on figures~\\ref{fig:1} to~\\ref{fig:3}. A comparison of $\\hat{G}_{homog}(\\phi, Ca)$ to $\\hat{G}(\\phi, Ca)$ for all the systems we used, at all tested gas volume fractions is presented on figure~\\ref{fig:4}. Experimental measurements and computations are generally in good agreement all over the range of systems we investigated. \n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{graph4.pdf}\n\\caption{Consistency between $G'$ and ${G'}_{homog}$ for all tested systems. $\\bigstar$: emulsion (1), $\\blacktriangledown$: emulsion (2), $\\bullet$: emulsion (3), $\\blacklozenge$: emulsion (4).\\label{fig:4} }\n\\end{figure}\n\nAs we have seen that the two dimensionless parameters $Ca$ and $\\phi$ are enough to understand and predict the elasticity of the dispersions, we now plot $\\hat{G}(\\phi, Ca)$ as a function of $Ca$, for 4 values of $\\phi$, on figure~\\ref{fig:5}. As can be noticed on the graphs~\\ref{fig:1} to~\\ref{fig:3}, the achieved values of $\\phi$ are different for all tested systems. To be able to plot $\\hat{G}(\\phi,Ca)$ at a given $\\phi$, we interpolate the experimental data at the exact values of $\\phi$ used for plotting on figure~\\ref{fig:5}. The full lines are computations of $\\hat{G}_{homog}(\\phi, Ca)$. As expected, $\\hat{G}(\\phi, Ca)$ is a decreasing function of $Ca$: higher values of $Ca$ correspond to more deformable bubbles that lower the overall elastic modulus of the dispersions. The non-perturbative effect of the bubbles for $Ca=1\/4$ is evidenced by the crossing of $\\hat{G}_{homog}(\\phi, Ca)$ at 1 for $Ca=0.25$, whatever the gas volume fraction. Below this value, the increase of $\\hat{G}_{homog}(\\phi, Ca)$ is consistent with previously discussed theoretical limits, but could not be investigated with our experimental systems. The series of data points at $Ca=0.23\\pm0.05$ does not fit in the increasing $\\hat{G}_{homog}(\\phi, Ca)$ regime, perhaps because of broader polydispersity: the uncertainty on the value of $Ca$ mainly arises from the width of the bubble radius distribution and the value of $Ca$ for the largest bubbles in the foam is for instance higher than 0.25. A model computing $\\hat{G}_{homog}(\\phi, Ca)$ as a function of the whole measured distribution of radii may better represent the experimental data.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.32]{graph5.pdf}\n\\caption{Dimensionless elastic modulus $\\hat{G}$ as a function of $Ca$ for four different values of the gas volume fraction. The dots are interpolated experimental data points, the full lines are $\\hat{G}_{homog}(\\phi, Ca)$. \\label{fig:5} }\n\\end{figure}\n\n\\section*{Conclusions}\nAs a conclusion, we have designed model systems in which precise control of the bubble stiffness and the matrix elasticity allows to experimentally determine the elastic modulus of dispersions of bubbles in a soft matrix. The results show that the addition of bubbles leads to a softening of the dispersion that is finely tuned by the capillary number. Those model systems enable us to compare our experimental results to estimates of the shear modulus through a micro-mechanical approach. Precise control of the capillary number provides experimental data validating the theoretical description of the bubble as an equivalent elastic sphere. The good agreement between theoretical and measured elastic moduli confirms the generality of the study, which demonstrates that $\\phi$ and $Ca$ entirely govern the overall response of the dispersions. More work remains to do in the limit of small capillary numbers ($Ca<1\/4$), for which the predicted regime of increasing $\\hat{G}(\\phi, Ca)$ could not be experimentally investigated. Emulsions may not be the most relevant matrices for that study and dedicated experiments on even softer media could be more appropriate. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nUltraluminous X-ray sources (ULXs) are off-nuclear X-ray sources in galaxies with an X-ray luminosity above the Eddington luminosity of a 10 \\msun~black hole, or $\\sim 10^{39}$ erg s$^{-1}$. Several scenarios have been proposed to explain their high luminosities. Geometrical \\citep{king01} or relativistic \\citep{kording02} beaming may allow for the observation of super-Eddington luminosities. In some sources there is evidence for a new state with truly super-Eddington accretion rates \\citep{gladstone09}. Recent investigations into the X-ray luminosity function (XLF) of ULXs suggest that the majority of ULXs are formed by the high-luminosity tail of X-ray binaries (XRBs) and contain stellar mass black holes (BHs) \\citep{swartz11, mineo12}. The best-fitting XLF exhibits a cut-off around $10^{40}$ erg s$^{-1}$, suggesting that this may be the effective upper limit for the luminosity of the most massive objects in the sample.\n\nHowever, \\citet{swartz11} argue that if they extrapolate their best-fitting XLF, based on a complete sample of ULXs within 14.5 Mpc, to larger distances, they can not explain the relatively large number of ULXs with luminosities above $10^{41}$ erg s$^{-1}$~that have been observed. Hence these ULXs may belong to a different class of objects. These sources would have to exceed the Eddington limit by more than a factor 100 if they contained stellar-mass black holes. They may be good candidates to host the predicted but thus far elusive intermediate-mass black holes (IMBHs). These IMBHs may form in the collapse of a dense stellar cluster \\citep{portegieszwart02}, the collapse of population III stars in the early universe \\citep{madau01} or the direct collapse of massive gas clouds \\citep{begelman06}. They may reside in globular clusters \\citep{maccarone07}, but conclusive evidence for their existence there has not yet been found. For a review on IMBHs and their formation mechanisms see \\citet{vandermarel04}. The best candidate for an IMBH to date is the extremely bright source HLX-1 in ESO 243-49, which reaches maximum X-ray luminosities of $\\sim 10^{42}$ erg s$^{-1}$ \\citep{farrell09}. The recent work by \\citet{sutton12} provides more evidence for extreme ULXs as IMBHs.\n\nMost ULX candidates are discovered by searching for off-nuclear X-ray point sources in galaxies (e.g. from the {\\it Chandra}~or XMM-{\\it Newton}~serendipitous source catalogs; see for example \\citealt{walton11b, liu11}). The ULX catalogs compiled in this way are contaminated with objects that also show up as off-nuclear, bright X-ray sources but are not accreting IMBHs or stellar-mass BHs. Background active galactic nuclei (AGN) and quasars are obvious examples, but some X-ray bright supernovae (most likely type IIn, \\citealt{immler03}) and active foreground stars may also contaminate the catalogs. One way to identify these contaminants if the ULX candidate has a bright optical counterpart is to take an optical spectrum. If emission or absorption lines are present the redshift to the source can be measured. In this way we can determine whether the source is associated with the galaxy or is a background or foreground object (compare e.g. \\citealt{gutierrez13}).\n\nIf the X-ray source is associated with the galaxy, optical spectra can give us additional information to classify the object. Some ULXs \\citep{pakull02, kaaret09} are surrounded by bubbles of ionized gas, which can act as calorimeters and as such tell us if the emission is strongly beamed or not. The intensity ratios of the emission lines from these regions provide information on the source of the ionizing radiation, e.g. whether they are shock ionized or X-ray photo-ionized (e.g. \\citealt{abolmasov07}).\n\nWe selected four high-luminosity (L$_X \\geq 10^{40}$ erg s$^{-1}$) ULX candidates from the catalog of \\citet{walton11b} with accurate positions that we measured using archival {\\it Chandra}~observations and optical counterparts that are sufficiently bright for optical spectroscopy. Two of the ULX candidates are situated in elliptical galaxies (NGC~533~and NGC~741). ULX candidates in elliptical galaxies have a higher chance to be background AGN (39\\%, compared to 24\\% for all sources in the catalog of \\citet{walton11b}). On the other hand, IMBHs may form in dense (globular) clusters \\citep{portegieszwart02, miller02} and the optical counterparts to these ULX candidates could well be just that, making them interesting targets for further investigation. AM~0644-741~is a ring galaxy with a ULX candidate situated in between the nucleus and the ring. ESO~306-003~is a spiral galaxy with a ULX in the outer edge of the disk, apparently associated with an extended optical source. We obtained optical spectra of these four sources with the FOcal Reducer and low dispersion Spectrograph (FORS2) mounted on the Very Large Telescope (VLT) \\citep{appenzeller98}. The observations and data reduction steps are described in Section 2; Section 3 contains the results. In Section 4 we discuss our findings.\n\n\\section{Observations and data reduction}\n\\subsection{X-ray observations}\nWe use archival {\\it Chandra}~observations to get exact source positions for the ULX candidates in NGC~533, NGC~741, AM~0644-741~and ESO~306-003. Table \\ref{chantab} lists the details of all observations. \n\n\\begin{table*}\n\\begin{minipage}{135mm}\n\\caption{The \\textit{Chandra} observations of the four ULX candidates.}\\label{chantab}\n \\begin{tabular}{lccccc}\n \\hline\n Galaxy & Observation ID & Exposure time & Source on CCD & Off-axis angle & Observation date \\\\\n & & (kiloseconds)& & (arcmin) & (UT)\\\\\n \\hline\nNGC~533~& 2880 & 38.1 & ACIS S3 & 0.85 & 2002-07-28\\\\\n NGC~741~& 2223 & 30.74 & ACIS S3 & 2.74 & 2001-01-28\\\\\n AM~0644-741~& 3969 & 39.97 & ACIS S3 & 0.57 & 2003-11-17\\\\\n ESO~306-003~& 4994 & 22.75 & ACIS I3 & 6.60 & 2004-03-10\\\\\n \\hline\n \\end{tabular}\n \n\\medskip\nNotes: {\\it Chandra}~observation ID number, exposure time in kiloseconds, CCD on which the source was detected, the off-axis angle of the source in arcminutes and the observation date.\n\\end{minipage}\n\\end{table*}\n\nWe use \\textsc{Ciao} version 4.4 to process the {\\it Chandra}~observations, with the calibration files from \\textsc{caldb} version 4.5.0. We treat the {\\it Chandra}~observations as follows: first we update the event files with ACIS\\_{}PROCESS\\_{}EVENTS, then we use WAVDETECT to find the position of the ULX candidate. Sources within 3 arcmins of one of the ACIS aimpoints have a 90\\% confidence error circle around the absolute position with a radius of 0.6''; this is valid for the ULX candidates in NGC~533, NGC~741~and AM~0644-741. The candidate in ESO~306-003~has 25 counts and was observed at 6.6' off-axis, which means it has a 95\\% confidence error circle with a radius of $\\sim$1.5'' \\citep{hong05}. For the sources in NGC~533, NGC~741~and AM~0644-741~we extract the source counts in a circle with 6 pixel radius (90\\% encircled energy fraction) around the source positions using SPECEXTRACT. For the ULX candidate in ESO~306-003~we use a circle with a radius of 10 pixels to get the same encircled energy fraction, since it was observed at 6.6' off-axis. As background regions we use circles with 80 pixel radius on the same CCD but not containing any sources. We use \\textsc{XSpec} version 12.6.0 to fit an absorbed powerlaw (pegpwrlw) to the data in the 0.3-8 keV range. We then extrapolate to get the 0.2-12 keV flux to compare this with the values reported by \\citet{walton11b}. For consistency we adopt the same model parameters: a photon index of 1.7 and $N_H = 3 \\times 10^{20}$ cm$^{-2}$, and allow only the flux to vary. We find that all \\textit{Chandra} fluxes are consistent with those from XMM-{\\it Newton}~as reported by \\citet{walton11b}. The positions of the X-ray sources and their fluxes are summarized in Table \\ref{chansrc}.\n\n\\begin{table*}\n\\begin{minipage}{90mm}\n\\caption{The positions and unabsorbed 0.2-12 keV X-ray fluxes of the ULX candidates.}\\label{chansrc}\n \\begin{tabular}{ccccc}\n \\hline\n Host galaxy & Right Ascension & Declination & Source flux \\\\\n &&& (erg cm$^{-2}$ s$^{-1}$)\\\\\n \\hline\nNGC~533~& 01:25:33.63 & +01:46:42.6& $2.9 \\pm 0.2 \\times 10^{-14}$ \\\\\n NGC~741~& 01:56:16.14 & +05:38:13.2 & $2.5 \\pm 0.3 \\times 10^{-14}$ \\\\\n AM~0644-741~& 06:43:02.24 & -74:14:11.1 & $3.5 \\pm 0.2 \\times 10^{-14}$ \\\\\n ESO~306-003~& 05:29:07.21 & -39:24:58.4 & $2.4 \\pm 0.4 \\times 10^{-14}$ \\\\\n \\hline\n \\end{tabular}\n \n\\medskip\nNotes: the positions of the ULX candidates in NGC~533, NGC~741~and AM~0644-741~are accurate to within 0.6''(90\\% confidence level), for the source in ESO~306-003~this value is 1.3''. We fit the fluxes assuming an absorbed powerlaw with photon index 1.7 and $N_H = 3 \\times 10^{20}$ cm$^{-2}$ for consistency with the method used by \\citet{walton11b}. \n\\end{minipage}\n\\end{table*}\n\n\\subsection{Optical images and photometry}\nTo find the optical counterparts of the ULX candidates we use archival optical observations of their host galaxies. NGC~533~and NGC~741~were observed as part of the Sloan Digital Sky Survey (SDSS), and we use the SDSS $r'$-band images to identify the optical counterparts to the ULX candidates in these galaxies (Figure \\ref{fig:agn}). There is no photometric information for the source in NGC~533, so we use the aperture photometry tool in \\textsc{GAIA} to estimate the $r'$-band magnitude. SDSS does provide \\textit{u'}, \\textit{g'}, \\textit{r'}, \\textit{i'} and \\textit{z'} magnitudes for the object in NGC~741, but these are incorrect because the source is too close to the edge of the frame. Therefore we also use \\textsc{GAIA} to estimate the $r'$-band magnitude for this source. For both optical counterparts we find that $r' = 21 \\pm 1$.\n\nThe Hubble Space Telescope (HST) archive contains several observations of AM~0644-741~made with the Advanced Camera for Surveys (ACS). We use the V-band (F555W) image with exposure identifier j8my05o2q, observed on 2004-01-16 with an exposure time of 2200 seconds (see Figure \\ref{fig:agn}). We visually compare the position of point sources from the USNO CCD Astrograph Catalog (UCAC) 3 \\citep{zacharias09} with their counterparts in the HST image and find that the astrometric calibration of the image does not need to be improved. The ULX candidate has a counterpart that is in the DAOPHOT source list of this HST image in the Hubble Legacy Archive (HLA\\footnote{http:\/\/hla.stsci.edu}). It has a V-band magnitude of $21.79 \\pm 0.05$.\n\nWe identify the optical counterpart to the ULX candidate in ESO~306-003~in a 480 seconds R-band observation made on 2004-01-25 UT with VLT\/VIMOS that we retrieved from the ESO archive. Its R-band magnitude is approximately 21, with the caveat that this is an extended source in a region with a very high background level due to the galaxy, which means that this measurement is not very accurate. We also obtained a \\textit{g'}-band, 120 seconds exposure of this galaxy in our VLT\/FORS2 run (see Figure \\ref{fig:eso306}), of which we visually inspected the astrometric solution by comparing the positions of bright stars with those in the UCAC 3.\n\n\\subsection{Optical spectroscopy}\nWe obtained VLT\/FORS2 observations of NGC~741~($3 \\times 1800$ s), AM~0644-741~($3 \\times 1800$ s) and ESO~306-003~($2 \\times 2700$ s) on 2011-12-03 UT under programme 088.B-0076A using the GRIS\\_600V grism and a 1\\arcsec~slit width. This configuration covers the wavelength range 4430-7370 \\AA{}~with a dispersion of 0.74 \\AA{}\/pixel,  yielding a resolution of 4.25 \\AA{}~for the 1'' slit (measured at 6300 \\AA). This allows us to observe the H$\\alpha$, [N{\\sc II}] complex and the H$\\beta$ and [O{\\sc III}] lines if the sources are located at the same distance as their apparent host galaxies, with high enough resolution to separate them. The night was photometric so we also observed several spectrophotometric standard stars to perform a flux calibration. The seeing varied between 0.7 and 1.1\\arcsec.\nThe spectra of NGC~533~($3 \\times 1500$ s) were made in service mode on 2012-01-16 UT with the GRIS\\_300V+10 grism and a 0.5\\arcsec~slit width, giving a wavelength coverage from 4450-8700 \\AA{}~with a dispersion of 1.68 \\AA{}\/pixel and a spectral resolution of 6.4 \\AA{}~for the 0.5'' slit (measured at 6300 \\AA{}). The seeing varied during the night and we have no observations of spectrophotometric standards.\n\nTo reduce the spectra we use the \\textsc{starlink} software package \\textsc{Figaro} and the \\textsc{Pamela} package developed by Tom Marsh\\footnote{http:\/\/deneb.astro.warwick.ac.uk\/phsaap\/software}. We follow the steps outlined in the \\textsc{Pamela} manual to extract the spectra, using Keith Horne's optimal extraction algorithm \\citep{horne86}. We then use the software package \\textsc{Molly}, also by Tom Marsh\\footnotemark[\\value{footnote}], to perform the wavelength calibration and, for the data taken on 2011-12-03, the flux calibration. We do not correct for telluric absorption. Because we have multiple spectra of each source we average them to get a better signal-to-noise ratio. The two observations of ESO~306-003~were taken under varying seeing conditions. Because of this the continuum level is different in the two spectra, so we normalize these spectra before averaging them.\nWe use \\textsc{Molly}'s MGFIT task to fit Gaussian profiles to the emission lines in the spectra to determine the full width at half maximum (FWHM) of the lines and the redshift to the sources. \n\n\n\n\\section{Results}\n\n\\begin{table*}\n\\begin{minipage}{150mm}\n\\caption{Source properties of the background AGN}\\label{optinfo}\n \\begin{tabular}{lccccc}\n \\hline\n Source name & In galaxy & z & Line & FWHM & Log(F$_X$\/F$_{\\textrm{opt}}$) \\\\\n &&&& km\/s &\\\\\n \\hline\nCXOU J012533.3+014642 & NGC~533~& $1.8549 \\pm 0.0003$ & C\\textsc{IV} & $2300 \\pm 70 $ & $0.0 \\pm 0.5$\\\\\n& & & C\\textsc{III}] & $7800 \\pm 200$ & \\\\\n& & & Mg{\\sc II} & $5700 \\pm 200$ & \\\\\nCXOU J015616.1+053813 & NGC~741~& $0.8786 \\pm 0.0006$ or & Mg{\\sc II} or & $8400 \\pm 200$ & $0.0 \\pm 0.5$\\\\\n& & $1.7535 \\pm 0.0009$ & C{\\sc III}] & & \\\\\nCXO J064302.2-741411 & AM~0644-741~& $1.3993 \\pm 0.0001$ & C{\\sc III}] & $5100 \\pm 70$ &  $0.7 \\pm 0.1$\\\\\n & & & Mg{\\sc II} & $4220 \\pm 40$ &\\\\\n \\hline\n \\end{tabular}\n \n\\medskip\nNotes: Lines used for the redshift determination to the quasars, their FWHM in km\/s and the X-ray to optical flux ratio of these sources. The X-ray to optical flux ratios are calculated using the XMM-{\\it Newton}~0.2-12 keV fluxes from \\citet{walton11b} and the \\textit{r'}-band (for NGC~533~and NGC~741) or V-band (for AM~0644-741) optical fluxes.\n\\end{minipage}\n\\end{table*}\n\n\\begin{figure*}\n\\hbox{\n\\includegraphics[width=0.33\\textwidth]{ngc533_sdss}\n\\includegraphics[width=0.33\\textwidth]{ngc741_sdss}\n\\includegraphics[width=0.33\\textwidth]{a0644_hst}\n}\n\\hbox{\n\\includegraphics[width=0.33\\textwidth]{ngc533_avspec}\n\\includegraphics[width=0.33\\textwidth]{ngc741_avspec}\n\\includegraphics[width=0.33\\textwidth]{a0644_avspec}\n}\n\\caption{The finders and FORS2 spectra of the three ULX candidates that are background AGN. The 90\\% confidence error circles around the X-ray positions have a radius of 0.6'', for NGC~533~and NGC~741~we plot a larger circle for visual clarity. \\emph{Left:} The SDSS \\textit{r'}-band image of NGC~533~with a 1.2'' radius circle around the {\\it Chandra}~position of the ULX candidate and the spectrum in which the C\\textsc{IV}, C{\\sc III}] and Mg{\\sc II} emission lines, redshifted by $z=1.85$, are marked. The absorption features at 6200 \\AA{}~are caused by interstellar absorption, and those at 6900 \\AA{}~and 7600 \\AA{}~are telluric in origin. \\emph{Middle:} The SDSS \\textit{r'}-band image of NGC~741~with a 1.2'' radius circle around the {\\it Chandra}~position of the ULX candidate and the spectrum of the optical counterpart. The marked emission line can be either Mg{\\sc II} $\\lambda2798$ line redshifted by $z=0.88$ or C{\\sc III}] at $z = 1.75$. \\emph{Right:} An HST ACS V-band image of AM~0644-741~with the 0.6'' radius error circle around the {\\it Chandra}~position of the ULX candidate, and the spectrum with the Mg{\\sc II} and C{\\sc III}] lines, redshifted by $z=1.40$, marked.}\\label{fig:agn}\n\\end{figure*}\n\n\\subsection{NGC~533}\nNGC~533~is the dominant elliptical galaxy in a group with the same name at $z = 0.0185$ \\citep{smith00}. The ULX candidate is located at 78'' from the center of the galaxy that has a semi-minor axis of 90'' (based on the D25 isophote, \\citealt{nilson73}). The X-ray source has an unresolved optical counterpart that is visible in the image of the SDSS, with \\textit{r'}-band magnitude $\\approx 21$. Figure \\ref{fig:agn} shows the galaxy with the position of the ULX candidate and the FORS2 spectrum of the source. \n\nThree broad emission lines are visible. We identify these as C\\textsc{IV}, C{\\sc III}] and Mg{\\sc II} at $z = 1.8549 \\pm 0.0003$. This proves the ULX candidate to be a background AGN, not associated with NGC~533. The 0.2-12 keV X-ray luminosity calculated for this source by \\citet{walton11b} was $(2 \\pm 1) \\times 10^{40}$ erg s$^{-1}$, assuming a distance to the ULX of 73.8 Mpc. The true distance to this source is 4730 Mpc (using $H_0 = 75$ km\/s\/Mpc for consistency with \\citealt{walton11b}), which gives this AGN an X-ray luminosity of $(7 \\pm 4) \\times 10^{43}$ erg s$^{-1}$~using the flux as measured with XMM-{\\it Newton}.\n\n\n\\subsection{NGC~741}\nNGC~741~is an elliptical galaxy located at $z = 0.0185$ with a (D25) semi-major axis of 92.7'' \\citep{devaucouleurs91}. The ULX candidate is located 78'' West of the center of NGC~741~and has a counterpart that is unresolved in the SDSS image. Its \\textit{r'}-band magnitude is $\\sim 21$. Figure \\ref{fig:agn} shows the SDSS \\textit{r'}-band image of NGC~741~with the position of the ULX candidate and the FORS2 spectrum of the counterpart.\n\nThe spectrum shows one broad emission line, with a FWHM of 147 \\AA{}. We cannot say with certainty which line this is. The most likely options are that it is either the Mg{\\sc II} $\\lambda2798$ line or the C{\\sc III}] $\\lambda1909$ line. In the first case this ULX candidate would be a background AGN at a redshift of $z = 0.8786 \\pm 0.0006$ with an X-ray luminosity of $(1.6 \\pm 0.6) \\times 10^{43}$ erg s$^{-1}$. In the second case it would be at $z = 1.7535 \\pm 0.0009$, with $L_X = (4.2 \\pm 1.5) \\times 10^{43}$ erg s$^{-1}$. In both cases the source is not a ULX but a background AGN, unconnected to NGC~741.\n\n\\subsection{AM~0644-741}\nAM~0644-741~is a ring galaxy at $z = 0.022$ that shows signs of recent interaction with a smaller galaxy \\citep{few82, lauberts89}. The ULX candidate in this galaxy is located in between the core of the galaxy and the ring. A point-like object with a V-band magnitude of 21.8 is visible at the position of the X-ray source in archival HST images (see Figure \\ref{fig:agn}).\n\nThe FORS2 spectrum of the counterpart shows two emission lines that we identify as Mg{\\sc II} and C{\\sc III}] at $z=1.3993 \\pm 0.0001$. This ULX candidate is another background AGN with an 0.2-12 keV luminosity of $(8.1 \\pm 0.8) \\times 10^{43}$ erg s$^{-1}$.\n\n\n\\subsection{ESO~306-003}\nThe spiral galaxy ESO~306-003, at $z \\approx 0.016$ \\citep{dasilva06}, contains a ULX candidate that is located on the edge of the spiral structure (see Figure \\ref{fig:eso306}). An optical source is visible on the edge of the error circle. Visual inspection shows the profile of the counterpart to be  more extended than that of point sources in the same image, but because of the high background level and steep gradient it is not possible to perform an acceptable fit to the profile. The full width at half maximum (FWHM) of point sources in this image (provided by the seeing) is 0.8''. At the distance of ESO~306-003~this yields a lower limit to the size of the source of 240 pc. The two spectra that we obtained of this source show slightly different line ratios and continuum levels (for example, the H$\\beta$\/H$\\alpha$ ratio changes by 10\\%). This can be explained by seeing variations if the optical counterpart to this source is extended: then slit losses can cause the small changes in the line ratios if there are intrinsic spatial variations in the line ratios in the extended source.\n\nThe spectrum is similar to that of an H{\\sc II} region, with narrow Hydrogen emission lines and strong forbidden lines. The redshift of the lines equals that of the center of the galaxy, indicating that if the X-rays are associated with this optical source, this is a bona fide ULX with a luminosity of $1.4 \\pm 0.3 \\times 10^{40}$ erg s$^{-1}$~based on the XMM-{\\it Newton}~flux measured by \\citet{walton11b}. The X-ray flux is constant between the XMM-{\\it Newton}~and {\\it Chandra}~observations. The X-ray to optical flux ratio of the source is log(F$_X$\/F$_{\\textrm{opt}}$)$ = 0.3 \\pm 0.5$, based on the XMM-{\\it Newton}~0.2-12 keV flux from \\citet{walton11b} and the \\textit{r'}-band flux. The line ratios, especially the [O{\\sc I}] $\\lambda$6300\/H$\\alpha$ ratio, place the source among the transition objects in the diagnostic diagrams of \\citet{ho08} (see Figure \\ref{fig:dd}). The He{\\sc II} $\\lambda4686$ emission line has been detected in several ULX nebulae \\citep{pakull02, kaaret09}, but we do not detect it here, possibly because the sensitivity of the detector drops off steeply towards the blue end. The 2-$\\sigma$ upper limit for the equivalent width of this line is 1.0 \\AA{}. This corresponds to a flux of $\\sim 10^{-17}$ erg cm$^{-2}$ s$^{-1}$ or a luminosity of $\\sim 5 \\times 10^{36}$ erg s$^{-1}$.\n\n\\begin{figure*}\n\\hbox{\n\\includegraphics[width=0.4\\textwidth]{eso306_vlt_im}\n\\includegraphics[width=0.6\\textwidth]{eso306_avspec}\n}\n\\caption{\\emph{Left:} The FORS2 \\textit{g'}-band acquisition image of ESO~306-003~with the 1.3'' radius (90\\% confidence) error circle around the position of the ULX candidate. \\emph{Right:} The FORS2 spectrum of the candidate optical counterpart to the X-ray source. Several emission lines, redshifted by $z=0.016$, are marked.}\\label{fig:eso306}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{dds_OI}\n\\caption{[O{\\sc I}] $\\lambda$6300\/H$\\alpha$ versus [O{\\sc III}] $\\lambda$5007\/H$\\beta$ line ratios for H{\\sc II} regions, AGN (LINERs\nand Seyferts) and transition objects (figure adapted from \\citealt{ho08}). The black dot represents the line ratios for the ULX in ESO~306-003.}\\label{fig:dd}\n\\end{figure}\n\n\\section{Discussion}\nWe obtained VLT\/FORS2 spectra of the optical counterparts of four bright ULX candidates with accurate positions obtained by us from archival {\\it Chandra}~observations. Two of these are located in elliptical galaxies NGC~533~and NGC~741. Another candidate is situated in AM~0644-741, a ring galaxy that recently interacted with a small elliptical galaxy, and in the spiral galaxy ESO~306-003. Three of our four targets turn out to be background AGN with X-ray luminosities ranging from 1 to 8 $\\times 10^{43}$ erg s$^{-1}$; one (in ESO~306-003) seems to be a bona fide ULX. \n\nThe fraction of background AGN in our sample is higher than the fraction estimated by \\citet{walton11b} for their catalog. Although this can be due to small number statistics since we only investigate four sources, it is in line with results from other spectroscopic studies of ULX candidates. Optical spectroscopy of a sample of 23 ULX candidates in total yielded 20 background AGN and three foreground stars (\\citealt{gutierrez05, gutierrez06, gutierrez07, gutierrez13}). Another study that targeted 17 ULX candidates from the catalog of \\citet{colbert02} found that 15 were background AGN and the other two objects were foreground stars \\citet{wong08}.\n\nAll these studies mainly target ULX candidates that are relatively isolated and have a bright optical counterpart, a selection effect induced by the relative ease with which spectroscopic observations can be carried out for these sources. Sources located in crowded areas, like the spiral arms of late type galaxies, are more difficult targets for ground based optical spectroscopic observations. This means that spectroscopic studies are aimed at ULX candidates that have a low X-ray to optical flux ratio and that are situated relatively far away from their suspected host galaxies. As the authors of these previous papers also note, these selection criteria introduce a bias towards background AGN.\n\nA possible method to select ULX candidates that are most likely to be real ULXs is to calculate the expected contribution of background sources based on the known density of AGN in X-ray and optical observations (\\citealt{lopez06, sutton12}). Alternatively it may be possible to use the X-ray to optical flux ratios of ULX candidates to select targets for future spectroscopic studies. All our sources have X-ray to optical flux ratios log(F$_X$\/F$_{\\textrm{opt}}$) in the range between -1 and 1, typical for AGN (e.g. \\citealt{barger03}). Most ULXs show values for log(F$_X$\/F$_{\\textrm{opt}}$) ranging from 2 - 3 (\\citealt{tao11, tao12, sutton12}). The low value that we find for the ULX in ESO~306-003~can be explained if we assume that we do not resolve the ULX counterpart but instead observe the optical flux of the entire HII region that it resides in, thus lowering log(F$_X$\/F$_{\\textrm{opt}}$).\n\nHowever, if we were to select candidates for spectroscopy on the basis of their X-ray to optical flux ratios only we run the risk of missing interesting sources. For instance, ULXs may display different values for log(F$_X$\/F$_{\\textrm{opt}}$) when observed in the high and low states, as was shown for M101 ULX-1 and M81 ULS1 (\\citealt{tao11}). For both sources log(F$_X$\/F$_{\\textrm{opt}}$) is between 2 and 3 during the high state, but around 0 during the low state, well inside the range of optical to X-ray flux ratios found for AGN. Therefore other source properties should be taken into account as well, such as galaxy morphology, the distance of the ULX to its apparent host galaxy and the absolute magnitude of its optical counterpart. \nThe source in AM~0644-741{} is a good example of a candidate with such favorable properties: situated in a ring galaxy, which is a strong sign of a recent interaction phase that triggered star formation, often linked to ULXs (e.g. \\citealt{swartz04}), and close to the center of its apparent host galaxy, decreasing the chance that it is a background AGN (\\citealt{wong08}). It has an optical counterpart of such magnitude that it is consistent with being a bright globular cluster if it is at the distance of AM~0644-741{}. Nevertheless our optical spectrum showed it to be a background object.\n\n\\subsection*{The ULX in ESO~306-003}\nThe X-ray source in ESO~306-003~is the only one of the four candidates in our sample that appears to be a bona fide ULX.  The extended nature of the source is confirmed by the fact that the emission line spectrum is consistent with that of an HII region. However, the [OI]\/H$\\alpha$ ratio indicates that some of the ionizing flux could come from an X-ray source. Potentially, we have found a ULX embedded in an HII region. \nAnother possibility is that this ULX candidate is a background AGN shining through an HII region in ESO~306-003. The X-ray to optical flux ratio is similar to that of the other AGN in our sample, so we would expect to see a contribution of redshifted emission lines from the AGN in the optical spectrum. The fact that we do not detect this makes this scenario implausible. \n\nWe find a 2-$\\sigma$ upper limit for the flux of a  HeII $\\lambda4686$ line of $10^{-17}$ erg cm$^{-2}$ s$^{-1}$. This corresponds to an upper limit to the luminosity in the line of $\\sim 5 \\times 10^{36}$ erg s$^{-1}$. The presence of this line would be a strong indication of ionization by an X-ray source. We can compare this upper limit with the strength of the HeII $\\lambda4686$ line in other ULX nebulae. For Holmberg II X-1, \\citet{pakull02} report a luminosity of $2.5 \\times 10^{36}$ erg s$^{-1}$. \\citet{kaaret09} report a flux for this line from the ULX in NGC 5408 of $3.3 \\times 10^{-16}$ erg cm$^{-2}$ s$^{-1}$, which translates to a luminosity of $9 \\times 10^{35}$ erg s$^{-1}$~at the distance to NGC 5408 (4.8 Mpc, \\citealt{karachentsev02}). Both these ULXs have an X-ray luminosity of $\\sim 10^{40}$ erg s$^{-1}$ -- similar to ESO~306-003~-- and a HeII $\\lambda4686$ to X-ray luminosity ratio of $\\sim 10^{-4}$. If the same is true for ESO~306-003~then we would expect a HeII $\\lambda4686$ flux of a few times $10^{-18}$ erg cm$^{-2}$ s$^{-1}$, which is just below our 2-$\\sigma$ upper limit. New observations of this source with greater sensitivity at the wavelength of the HeII $\\lambda4686$ line are needed to determine if the nebula is X-ray photo-ionized or not.\n\n\\section*{Acknowledgements}\nPGJ and MAPT acknowledge support from the Netherlands Organisation for Scientific Research. GM acknowledges support from the Spanish Plan Nacional de Astronom\\'{\\i}a y Astrof\\'{\\i}sica under grant AYA2010-21490-C02-02. This research is based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 088B-0076A. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application package CIAO and of the software packages Pamela and Molly provided by Tom Marsh.\n\n \\bibliographystyle{mn_new}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:int}\n\n\nRecently, since the developments of both hardware and software\nin computer science enable us to simulate complex\nphysical processes numerically, such computer simulations\nbecome more important from industrial viewpoints.\nEspecially \nthe computation of the incompressible multi-phase fluid dynamics\nhas crucial roles in order to evaluate the behavior of several\ndevices and materials in a micro-region,\n{\\it {e.g.}}, ink-jet printers, solved toners and so on.\nIn the evaluation, it is strongly required that\nthe fluid interfaces with multiple junctions are \nstably and naturally computed from these practical reasons.\n\nIn this article, \nin order to handle the fluid interfaces with multiple junctions \nin a three dimensional micro-region, \nwe investigate a surface tension of an incompressible multi-phase \nflow with multiple junctions as a numerical computational method\nunder the assumption that the Reynolds number is not so large.\nIn the investigation, we encounter many interesting mathematical\nobjects and results,\nwhich are associated with\nlow dimensional interface geometry having singularities, and\nwith the infinite dimensional geometry of incompressible fluid dynamics.\nFurther since even in a macroscopic theory,\nwe introduce artificial intermediate regions in the material\ninterfaces among different fluids or among a solid and fluids,\nthe regions give a resolution of the singularities in the interfaces\nto provide extended Euler equations naturally.\nThus even though we consider the multi-phase fluid model as\na computational model,\nwe believe that it must be connected with mathematical nature of \nreal fluid phenomena as their description.\nWe will mention the background, the motivation and the strategy of\nthis study more precisely as follows.\n\nFor a couple of decades, \nin order to represent the physical process with\nthe interfaces of the multi-phase fluids, the computational\nschemes have been studied well.\nThese schemes are mainly classified into two types.\nThe first type is based on the level-set method \\cite{S}\ndiscovered by H-K. Zhao, T. Chan, B. Merriman, S. Osher\nand L. Wang \\cite{ZCMO,ZMOW}.\nThe second one\nis based on the phase-field theory, which was found by\n J. U. Brakbill, D. B. Kothe and C. Zemach \\cite{BKZ},\nand B. Lafaurie, C. Nardone, R. Scardovelli, S. Zaleski, and G. Zanetti\n\\cite{LZZ}.\nThe authors in Reference \\cite{LZZ} called the scheme SURFER.\nFollowing them, there are many studies on the SURFER scheme,\n{\\it{e.g.}}, \\cite[references therein]{AMS,Cab,Jac}.\n\nThe level-set method is a computational method in which we describe a\n(hyper-)surface in terms of zeros of the level-set function, {\\it{i.e.}}, a\nreal function whose value is a signed distance from the surface,\nsuch as $q(x)$ in Section \\ref{sec:two-one}.\nUsing the scheme based upon the level-set method\nin the three dimensional Euclidean space,\nwe can deal well with  topology changes,\ngeometrical objects with singularities, {\\it{e.g.}}, cusps,\nthe multiple junctions of materials, and so on.\nHowever in the computation, we need to deal with the\nconstraint conditions even for two-phase fluids\n\\cite{ZCMO,ZMOW}.\nA dynamical problem with constraint conditions\nis basically complicate and sometimes gives difficulties\nto find its solution since the constraint conditions sometimes\ngenerate an ill-posed problem in the optimization. \nIn the numerical computation for incompressible fluid,\nwe must check the consistency between \nthe incompressible condition and the constraint condition.\nThe check generally requires a complicate implementation\nof the algorithm, and increases computational cost.\nIts failure sometimes makes the computation unstable, especially\nwhen we add some other physical conditions.\nSince instability disturbs the evaluation of a complex\nsystem as a model of a real device, it must be avoided.\n\nOn the other hand, \nusing the SURFER scheme \\cite{LZZ}, \nwe can easily compute effects of the surface tension of a two-phase fluid\nin the Navier-Stokes equation.\nThe phase field model is the model that we represent materials in terms\nof supports of smooth functions which roughly correspond to\nthe partition of unity in pure mathematics \\cite[I p.272]{KN}\nas will be mentioned in Sections \\ref{sec:four} and \\ref{sec:five}.\nWe call these functions\n{\\lq\\lq}color functions{\\rq\\rq} or {\\lq\\lq}phase fields{\\rq\\rq}.\nThe phase fields have artificial intermediate regions which\nrepresent their interfaces  approximately.\nIn the SURFER scheme \\cite{LZZ},\nthe surface tension is given as a kind of stress force, or volume force\ndue to the intermediate region.\nHence the scheme makes the numerical computations of the surface tension \nstable.\nHowever\nit is not known how to consider a multi-phase ($N$-phase, $N\\ge2$) flow\n in their scheme.\nIn Reference \\cite{BKZ}, the authors propose a method as an\nextension of the SURFER scheme \\cite{LZZ}\n to the contact angle problem\nby imposing a constraint to fix its angle.\nIn this article, we will generalize \n the SURFER scheme to multi-phase flow\nwithout any constraints.\n\nNature must not impose any constraints even at such a triple junction,\nwhich is governed by a physical principle. If it is a \nHamiltonian system, its determination must obey the minimal\nprinciple or the variational principle.\nWe wish to find a theoretical framework\nin which we can consistently\nhandle the incompressible flows with interfaces including \nthe surface tensions and the multiple junctions without any constraints.\nAs the multiple junctions should be treated as singularities in \na mathematical framework which are very difficult to be handled\nin general, \nit is hard to extend\nmathematical approaches for fluid interface problems without\na multiple junction\n\\cite{BG,SZ} to a theory for the problem with multiple junctions.\nOur purpose of this article is to find such a theoretical\nframework which enables us to solve the fluid interface problems with \nmultiple junctions numerically\nas an extension of the SURFER scheme.\n\n\nFor the purpose, we employ the phase field model.\nThe thickness of the actual intermediate region \nin the interface between a solid and a fluid or between\ntwo fluids is of atomic order and \nis basically negligible in the macroscopic theory. \nHowever the difference\nbetween zero and {\\lq\\lq}the limit to zero{\\rq\\rq} \nsometimes brings a crucial difference in physics and\nmathematics; for example, in the Sato hyperfunction theory,\nthe delta function is regarded as a function in the boundary\nof the holomorphic functions \\cite{KKK,II}, {\\it{i.e.}},\n$\\displaystyle{\\delta(x) =\n \\lim_{\\epsilon \\to 0}\\frac{1}{2\\pi \\ii}\n\\left(\\frac{1}{x - \\ii \\epsilon}\n-\\frac{1}{x + \\ii \\epsilon}\\right)}$\n$\\displaystyle{\n\\equiv \\lim_{\\epsilon \\to 0}\\frac{1}{\\pi }\n\\frac{\\epsilon}{x^2 +  \\epsilon^2}}$.\nAs mentioned above,\nthe phase field model has the artificial intermediate region which is\ncontrolled by a small parameter $\\epsilon$ \nand appears explicitly even as a macroscopic\ntheory. We regard that it represents the effects coming from\nthe actual intermediate region of materials.\nNamely, we regard that the stress force expression in \nthe SURFER scheme\nis caused by the artificial intermediate region of the phase-fields\nand it represents well the surface effect\ncoming from that of real materials.\n\nIn order to extend the stress force expression of the two-phase\nflow to that of the multi-phase ($N$-phase, $N\\ge2$) flow,\nwe will first reformulate \nthe SURFER scheme in the framework of the variational theory.\nIn Reference \\cite{Jac}, a similar attempt was reported\nbut unfortunately there were not precise derivations. Our\ninvestigations in Section \\ref{sec:four}  \nshow that the surface tension expression of\nthe SURFER scheme\nis derived as a momentum conservation in \nNoether's theorem \\cite{BGG,IZ} and its derivation\n requires a generalization of the Laplace equation \\cite{LL} as\nthe Euler-Lagrange equation \\cite{AM,BGG}, which is not trivial\neven for a static case.\n\nIn order to deal with this problem in a dynamics case consistently,\nwe should also consider the Euler equation in the framework\nof the variational principle.\nIt is well-known that the incompressible fluid dynamics is\ngeometrically interpreted as \na variational problem of an infinite\ndimensional Lie group, related to diffeomorphism, due to\nV. I. Arnold \\cite{Ar,AK}, D. Ebin and J. Marsden \\cite{EM}, H. Omori \\cite{O}\nand so on. Following them, there are so many related works\n\\cite{AF,B,K,NHK,Schm,Shk,Shn,V}.\n\nOn the reformulation of the SURFER scheme \\cite{LZZ} for the dynamical case,\nwe introduce an action integral including \nthe kinematic energy of the incompressible fluid and the surface energy.\nThe variational method reproduces the governing equation in\nthe SURFER scheme.\n\nAfter then, we extend the surface energy to that of multi-phase fields \nand add the energy term to the action integral.\nThe variational principle of the action integral\nleads us to a novel expression of the surface tension\nand the extended Euler equation which we require.\nUsing the extended Euler equation,\nwe can deal with the surface tensions of the \nmulti-phase flows, the multiple junctions of the \nof phase fields including singularities, the topology changes and so on.\nWe can also compute a wall effect naturally and a contact\nangle problem.\nThe computation of the governing equation is freed from any\nconstraints, except the incompressible condition.\n\nIn other words, in this article, we completely unify\nthe theory of the multi-phase ($N$-phase, $N\\ge2$) field\nand the theory of the incompressible fluid dynamics of Euler equation\nas an infinite dimensional geometrical problem.\n\n\nContents are as follows:\nSection \\ref{sec:two} is devoted to the preliminaries\n of the theory of surfaces\nin our Euclidean space from a low-dimensional\ndifferential geometrical viewpoint\n\\cite{M,GMO,FW} and Noether's theorem in the  \nclassical field theory \\cite{AM,BGG,IZ}.\nSection \\ref{sec:three} reviews the derivation of the Euler equation\nto the incompressible fluid dynamics following the variational method\nfor an infinite-dimensional Lie algebra based upon Reference \\cite{EM}. \nIn Section \\ref{sec:four}, we reformulate the SURFER scheme \\cite{LZZ}.\nThere the Laplace equation for the surface tension \nand the Euler equation in Reference \\cite{LZZ}\nare naturally obtained by the variational method\nin Propositions \\ref{prop:4-4} and \\ref{prop:4-6}.\nSection \\ref{sec:five} is our main section in which we extend\nthe theory in Reference \\cite{LZZ}  to that for a multi-phase flow\nand obtain the Euler equation with the surface tension\nof the multi-phase field in Theorem \\ref{th:5-2}.\nThe extended Euler equation for the multi-phase flow\nis derived from the variational\nprinciple of the action integral in Theorem  \\ref{th:5-1}.\nAs a special case, we also derive the \nEuler equation to a two-phase field with wall effects\nin Theorem \\ref{th:5-3}.\nIn Section 6, \nusing these methods in the computational fluid dynamics \\cite{Ch,H,HN},\nwe consider numerical computations \nof the contact angle problem of a two-phase field\nbecause the contact angle problem \nfor the two-phase field circumscribed in a wall\nis the simplest non-trivial triple junction problem.\nBy means of our scheme, for given surface tension coefficients, \nwe show two examples of the numerical computations in which\nthe contact angles automatically appeared\nwithout any geometrical constraints\n and any difficulties for the singularities at triple junctions.\nThe computations were very stable.\nPrecisely speaking, as far as we computed, \nthe computations did not collapse for any boundary conditions\nand for any initial conditions.\n\n\n\n\n\\section{Mathematical Preliminaries} \\label{sec:two}\n\n\\subsection{Preliminary of surface theory} \\label{sec:two-one}\n\nIn this subsection, we review the theory of surfaces \nfrom the viewpoint of low-dimensional differential geometry.\nThe interface problems have been also studied for last three decades\nin pure mathematics,\nwhich are considered as a revision of the \nclassical differential geometry \\cite{E} from a modern point of view\n\\cite{ES,FW,GMO,M,T},\n{\\it{ e.g.}}, generalizations of the Weierstrass-Ennpper\ntheory of the minimal surfaces, isothermal surfaces,\nconstant curvature surfaces, constant mean curvature surfaces,\nWillmore surfaces and so on.  \nThey are also closely connected with\nthe harmonic map theory \nand the theory of the variational principle \\cite{FW,GMO}.\n\n\nWe consider a smooth surface $S$ embedded in \nthree dimensional Euclidean space $\\EE^3$.\nLet $x = (x^1,x^2,x^3)$ be of the Cartesian coordinate system\nand represent a point in $\\EE^3$, and \nlet the surface $S$ be locally expressed by a local parameter $(s^1,s^2)$.\nWe assume that the surface $S$ is expressed by\nzeros of a real valued smooth function $q$ over $\\EE^3$, {\\it{i.e.}},\n$$\n\tq(x)=0,\n$$\nsuch that in the region whose $|q|$ is sufficiently small\n($|q| < \\epsT$ for a positive number $\\epsT>0$),\n$|dq|$ agrees with the infinitesimal length in the Euclidean space.\nThen $dq$ means the normal co-vector field (one-form), {\\it{ i.e.}},\nfor the tangent vector field\n$e_\\alpha:=\\partial_\\alpha:=\\partial\/\\partial s^\\alpha$\n($\\alpha = 1, 2$) of $S$,\n\\begin{equation}\n\t\\langle\\partial_\\alpha, dq\\rangle=0 \\quad \\mbox{ over }\\quad\n               S = \\{ x \\in \\EE^3 \\ | q(x) = 0\\}.\n\\label{eq:ddq}\n\\end{equation} \nHere $\\langle, \\rangle$ means the pointwise pairing \nbetween the cotangent bundle and the tangent bundle of $\\EE^3$.\nThe function $q$\n can be locally regarded as so-called the level-set function \n\\cite{S,ZMOW}.\nWe could redefine the domain of $q$ such that it is restricted to\na tubular neighborhood $T_S$ of $S$,\n$$\n      T_S:=\\{x \\in \\EE^3 \\ | \\ |q(x)|<\\epsT \\}.\n$$ \nOver $T_S$, $q$\nagrees with the level-set function of $S$.\nThere we can naturally define a projection map\n$\\pi: T_S \\to S$ and then we can regard $T_S$ as a fiber bundle over\n$S$, which is homeomorphic to the normal bundle $N_{S}\\to S$.\nHowever the level-set function is defined as a signed \ndistance function which is a global function over $\\EE^3$\nas a continuous function \\cite{S} and thus it has no natural\nprojective structure in general;\nfor example,\nthe level-set function $L$ of a sphere with radius $a$ is given by\n$$\nL(x^1,x^2,x^3) = \\sqrt{(x^1)^2 +(x^2)^2 + (x^3)^2} - a,\n$$ \nwhich induces the natural projective (fiber) structure but \nthe origin $(0, 0, 0)$ in the sphere case.  \nThe level-set function has no projective structure \nat $(0, 0, 0)$ in this case, and\nwe can not define its differential there.\nIn other words,\nthe level-set function is not a global function over $\\EE^3$ \nas a smooth function in general.\n\n  When we use the strategy of the \nfiber bundle and its connection, we restrict ourselves to\nconsider the function $q$ in  $T_S$. \nThen the relation (\\ref{eq:ddq}) and the parameter\n$(s_1, s_2)$ are naturally lifted \nto $T_S$ as an inverse image of $\\pi$.\n\nFurther for $e_q:=\\partial_q:=\\partial\/\\partial q$, we have\n$$\n\t\\partial_\\alpha (e_q) = \\sum_\\beta \\Gamma^\\beta_{\\ \\alpha q} e_\\beta\n     \\mbox{ over } S.\n$$\nHere \n$(\\Gamma^\\beta_{\\ \\alpha q})$ is the Weingarten map, which is\na kind of a point-wise $2\\times2$-matrix \n$((\\Gamma^\\beta_{\\alpha q})_{\\alpha\\beta})$ \\cite[Chapter VII]{KN}.\nThe eigenvalue of \n$(\\Gamma^\\beta_{\\alpha q})$ is the principal\ncurvature, whereas a half of its trace\n$\\mbox{tr}(\\Gamma^\\beta_{\\alpha q})\/2$ is known as the \nmean curvature and its determinant \n$\\mbox{det}(\\Gamma^\\beta_{\\alpha q})$\nmeans the Gauss curvature \\cite[Chapter VII]{KN}.\n\n\t\n\nNoting  the relation,\n$\\langle e_\\beta, d s^\\alpha\\rangle = \\delta_\\beta^\\alpha$ \nfor $\\alpha, \\beta = 1, 2$, \n the twice of the mean curvature, $\\kappa$, is given by,\n$$\n\t\\sum_{\\alpha}  \\partial_\\alpha (e_q) d s^\\alpha = \\kappa\n        \\quad \\mbox{over}\\quad S.\n$$\nFurther noting the relation $\\partial_q e_q d q =0$, we obtain\n$$\n\t \\sum_{\\alpha}  \\partial_\\alpha (e_q) d s^\\alpha \n\t + \\partial_q(e_q) d q = \\kappa\n        \\quad \\mbox{over}\\quad S.\n$$\nDue to  the flatness of the Euclidean space, we identify $e_q$ with \n$\\nabla q \/|\\nabla q|$ and then we have  the following proposition.\n\n\\begin{proposition} \\label{prop:2-1}\nThe following relation holds at a point over $S$,\n$$\n\t\\mathrm{div}\\left( \\frac{\\nabla q }{|\\nabla q|}\\right) = \\kappa.\n$$\n\\end{proposition}\nFor the case $|\\nabla q| = 1$,\nusing the Hodge star operator \\cite{AM,N} and the exterior\nderivative $d$, we also have \nan alternative expression $*d * dq =\\kappa$ over the surface $S$.\nHere the Hodge star operator is $*: \\Lambda^p(T_S) \\to \\Lambda^{3-p}(T_S)$\nand the exterior derivative $d: \\Lambda^p(T_S) \\to \\Lambda^{p+1}(T_S)$\n$(d \\omega = \\sum_{i=1}^3\\partial_i \\omega d x^i)$,\nwhere $\\Lambda^p(T_S)$ is the set of smooth $p$-forms over\n$T_S$ \\cite{N}.\n\nNoting that as the left hand side of formula in Proposition \\ref{prop:2-1}\ncan be lifted to $T_S$,\nthe formula plays an important role in References \\cite{BKZ,LZZ,ZCMO}\nand in this article.\n\n\\subsection{Preliminary of Noether's theorem} \\label{sec:two-two}\n\nIn this subsection,\n we review Noether's theorem in the variational method which\nappears in a computation of the energy-momentum tensor-field\nin the classical field theory \\cite{AM,BGG,IZ}.\n\nLet  the set of $\\ell$ smooth real-valued functions\n over $n$-dimensional Euclidean space $\\EE^n$\n be denoted by $\\cC^{\\infty}(\\EE^n)^{\\otimes\\ell}$,\nwhere $n$ is mainly three.\nLet $x = (x^1,x^2,\\ldots,x^n)$ be of the Cartesian coordinate system\nof $\\EE^n$.\nWe consider the functional \n$I: \\cC^{\\infty}(\\EE^n)^{\\otimes\\ell} \\to \\RR$,\n\\begin{equation}\n\tI = \\int_{\\EE^n} d^n x{ \\cF}(\\phi_a(x),\\partial_i \\phi_a(x)),\n\\label{eq:IinEL}\n\\end{equation}\nwhere\n$\\cF$ is a local functional,\n$\\cF: \\cC^{\\infty}(\\EE^n)^{\\otimes\\ell}|_x \\to \\Lambda^n(\\EE^n)|_x$,\n\\begin{equation*}\n\\begin{split}\n\\cF: (\\phi_a)_{a=1,\\ldots,\\ell}|_x \\mapsto &\n{\\cF}(\\phi_a(x),\\partial_i \\phi_a(x))d^n x\n\\equiv{ \\cF}(\\phi_a(x),\\partial_1 \\phi_a(x),\n\\ldots, \\partial_n \\phi_a(x))d^n x \\\\\n&\n\\equiv{ \\cF}(\\phi_1(x), \\ldots, \\phi_\\ell(x), \n\\partial_1 \\phi_1(x), \\ldots, \\partial_n \\phi_\\ell(x))d^n x \\\\\n\\end{split}\n\\end{equation*}\nand $\\partial_i := \\partial\/ \\partial x^i$, $(i=1, \\cdots, n)$.\nThen we obviously have the the following\nproposition.\n\n\\begin{proposition}\\label{prop:A-1} \nFor the functional $I$ in (\\ref{eq:IinEL}) over\n$\\cC^{\\infty}(\\EE^n)^{\\otimes\\ell}$,\nthe Euler-Lagrange equation coming from the variation\nwith respect to $\\phi_a$ of $(\\phi_b)_{b=1, \\ldots, \\ell}\n \\in \\cC^{\\infty}(\\EE^n)^{\\otimes\\ell}$,\n{\\it{i.e.}}, \n\n\t\\frac{\\delta I}{\\delta \\phi_a(x)} = 0\n$,\n is given by\n\\begin{equation}\n\t\\frac{\\delta \\cF}{\\delta \\phi_a(x)}\n\t-\\sum_{i=1}^n\n\\partial_i \\frac{\\delta \\cF}{\\delta \\partial_i \\phi_a(x)} = 0.\n\\label{eq:AppeA1}\n\\end{equation}\n\\end{proposition}\n\nUsing the equation (\\ref{eq:AppeA1}), \nwe consider an effect of a small translation\n  $x$ to $x'= x + \\delta x$ on the functional $I$.\nThe following proposition is known as Noether's theorem\nwhich plays crucial roles in this article.\n\n\\begin{proposition}\\label{prop:A-2} \nThe functional derivative $I$ with respect to $\\delta x_i$ is\ngiven by\n\\begin{equation}\n\\frac{\\delta I}{\\delta x^i}\n   = \\sum_{j=1}^n\\partial_j\\left[ \\sum_{a=1}^\\ell\n   \\frac{\\delta \\cF}{\\delta\\partial_j \\phi_a}\\partial_i \\phi_a \\right]\n         - \\partial_i  \\left[  { \\cF}\\right].\n\\label{eq:AppeA2}\n\\end{equation}\nIf $I$ is invariant for the translation, \n(\\ref{eq:AppeA2}) gives the conservation of the momentum.\n\\end{proposition}\n\n\n\\begin{proof}\nFor the variation $x'= x + \\delta x$, the scalar function becomes\n\\begin{equation*}\n\t\\phi_a(x') = \\phi_a(x) +\\sum_{i=1}^n  \\partial_i \\phi_a(x)\n\t\\delta x^i + O(\\delta x^2).\n\\end{equation*}\nFrom the relations on the Jacobian and each component,\n\\begin{equation*}\n\t\\frac{\\partial x'}{\\partial x} \n          = 1 +\\sum_{i=1}^n \\partial_i \\delta x^i + O(\\delta x^2),\n\\quad\n\t\\frac{\\partial x^k}{\\partial x'{}^i}=\\delta^k_i\n\t -\\partial_i \\delta x^k + O(\\delta x^2),\n\\end{equation*}\nwe have\n\\begin{equation*}\n\\begin{split}\n\t\\frac{\\partial \\phi_a(x')}{\\partial x'{}^i}\n      &=\\frac{\\partial \\phi_a(x) +\n         \\sum_{j=1}^n \\partial_j \\phi_a(x) \\delta x^j}{\\partial x^k}\n       \\frac{\\partial x^k}{\\partial x'{}^i} + O(\\delta x^2)\\\\\n      &=\\partial_i \\phi_a+\n       \\sum_{j=1}^n (\\partial_i\\partial_j \\phi_a) \\delta x^j  + O(\\delta x^2).\n\\end{split}\n\\end{equation*}\nThen up to $\\delta x^2$, we obtain \n\\begin{equation*}\n\\begin{split}\n&  \\int_{\\EE^n} d^n x' {\\cF}(\\phi_a(x'),\\partial_i' \\phi_a(x'))\n-  \\int_{\\EE^n} d^n x {\\cF}(\\phi_a(x),\\partial_i \\phi_a(x))\\\\\n&= \\int_{\\EE^n}\\Bigr[\n\\sum_{i=1}^n\\sum_{a=1}^\\ell \n\\frac{\\delta {\\cF}}{\\delta \\phi_a} \\partial_i \\phi_a(x) \n\\delta x^i +\\sum_{i,j = 1}^n \\sum_{a=1}^\\ell\n\\frac{\\delta \\cF}{\\delta\\partial_j \\phi_a} \n\\partial_i \\partial_j\\phi_a(x) \\delta x^i\n+\\sum_{j=1}^n{\\cF} \\partial_i \\delta x^i\\Bigr] d^n x\\\\\n &=\\int_{\\EE^n}\\left(\n\\sum_{i=1}^n \\partial_i\\left[ \\sum_{j=1}^n \\sum_{a=1}^\\ell\n\\frac{\\delta \\cF}{\\delta\\partial_i \\phi_a}\\partial_i \\phi_a \n            -{ \\cF} \\right] \\delta x^i \\right) d^n x.\\\\\n            \\end{split}\n\\end{equation*}\nHere we use the Euler-Lagrange equation (\\ref{eq:AppeA1}) and then\nwe have  (\\ref{eq:AppeA2}).\nIf we assume that $I$ is invariant for the variation, it vanishes.\n\\qed\n\\end{proof}\n\n\n\n\n\n\n\\section{Variational principle for incompressible fluid dynamics}\n\\label{sec:three}\n\nAs we will derive the governing equation as the variational\nequation of an incompressible multi-phase flow with interfaces\nusing the variational method,\nlet us review the variational theory of the\nincompressible fluid to obtain the Euler equation\nfollowing References \\cite{Ar,AK,EM,K,Ko,MW,NHK}.\n\nLet $\\Omega$ be a smooth domain in $\\EE^3$.\nThe incompressible fluid dynamics can be interpreted\nas a geometrical problem associated with an infinite dimensional Lie group\n\\cite{AK,EM,O}.\nIt is related to the volume-preserving diffeomorphism  \ngroup $\\SDiff(\\Omega)$\nas a subgroup of  the diffeomorphism group\n$\\Diff(\\Omega)$. The diffeomorphism group $\\Diff(\\Omega)$\nis generated by a smooth coordinate transformation of $\\Omega$.\nThe Lie algebras $\\sdiff(\\Omega) \\equiv T_e\\SDiff(\\Omega)$ of $\\SDiff(\\Omega)$\nand $\\diff(\\Omega) \\equiv T_e\\Diff(\\Omega)$ of $\\Diff(\\Omega)$ are\n the infinite dimensional real vector spaces.\n The $\\sdiff(\\Omega)$  is a linear subspace of $\\diff(\\Omega)$.\n\nFollowing Ebin and Marsden \\cite{EM}, we consider \nthe geometrical meaning of the action integral of\nan incompressible fluid,\n\\begin{equation}\n\t\\int_T dt \\int_{\\Omega} d^3 x \n\\left(\\frac{1}{2}\\rho |u|^2\\right).\n\\label{eq:Eue}\n\\end{equation}\nHere \n$T:=(0, T_0)$ is a subset of the set of real numbers $\\RR$,\n$(x, t)$ is the Cartesian coordinate of the space-time \n$\\Omega\\times T$, $\\rho$ is the density of the fluid\nwhich is constant in this section, and $u=(u^1, u^2, u^3)$ is\nthe velocity field of the fluid.\n\nGeometrically speaking,\na flow obeying the incompressible fluid dynamics is\nconsidered  as a section of\na principal bundle\n$\\IFluid(\\Omega\\times T)$ over the absolute time axis $T \\subset \\RR$ \nas its base space,\n\\begin{equation}\n\\begin{CD}\n\t\\SDiff(\\Omega) @>>> \\IFluid(\\Omega\\times T) \\\\\n         @.                   @V{\\varpi}VV\\\\\n        @.                     T.\\\\\n\\end{CD}\n\\label{eq:PBIFluid}\n\\end{equation}\nThe projection $\\varpi$ is induced from the\ntrivial fiber structure $\\varpi_\\Omega: \\Omega \\times T \n\\to T$, $((x, t) \\to t)$. In the classical (non-relativistic) mechanics,\nevery point of space-time has a unique absolute time $t\\in \\RR$,\nwhich is contrast to one in the relativistic theory.\n\nDue to the Weierstrass polynomial approximation theorem \\cite{Y},\n we can locally approximate\na smooth function by a regular function.\nLet  the set of smooth functions\nover $\\Omega$ be denoted by $\\cC^\\infty(\\Omega)$\nand the set of the regular real functions \nby $\\cC^\\omega(\\Omega)$\nwhose element can be expressed by the Taylor expansion\nin terms of local coordinates.\n\nThe action of $\\Diff(\\Omega)$ on \n$\\cC^\\omega(\\Omega) \\subset\\cC^\\infty(\\Omega)$\nis given by\n$$\n\t\\ee^{s u^i \\partial_i} f(x) = f(x + s u),\n$$\nfor an element $f \\in \\cC^\\omega(\\Omega)$, and small $s>0$,\nwhere $\\partial_i := \\partial \/ \\partial x^i$ and \nwe use the Einstein convention; when an index $i$ appears twice,\nwe sum over the index $i$.\nThus the action $\\ee^{s u^i \\partial_i}$\nis regarded as an element of $\\Diff(\\Omega)$.\n\nAs a frame bundle of the principal bundle $\\IFluid(\\Omega\\times T)$, we\nconsider a vector bundle $\\Coor(\\Omega\\times T)$ with infinite rank,\n$$\n\\begin{CD}\n\t\\cC^\\infty(\\Omega) @>>> \\Coor(\\Omega\\times T) \\\\\n         @.                   @V{\\varpi'}VV\\\\\n        @.                     T.\\\\\n\\end{CD}\n$$\nSince $\\cC^\\infty(\\Omega)$ is regarded as a non-countably infinite\ndimensional linear space over $\\RR$,\nwe should regard  \n$\\Diff(\\Omega)$ and $\\SDiff(\\Omega)$  as subgroups of\nan infinite dimensional general linear group if defined.\n\nMore rigorously, we should\nconsider the ILH space (inverse limit of Hilbert space) \n(or ILB space (inverse limit of Banach space)) introduced in \nReference \\cite{O} by adding a certain\ntopology to (a subspace of) $\\cC^\\infty(\\Omega\\times T)$,\nand then we also should regard $\\Diff$ and $\\SDiff$  \nas an ILH Lie group.\nHowever our purpose is to obtain an extended Euler equation from a more\npractical viewpoint.\nThus we formulate the theory primitively even though\nwe give up to consider a general solution for a general initial\ncondition.\n\nWe consider smooth sections of $\\Coor(\\Omega\\times T)$ and\n$\\IFluid(\\Omega\\times T)$.\nSmooth sections of $\\Coor(\\Omega\\times T)$ can be\nrealized as $\\cC^\\infty(\\Omega \\times T)$.\nIn the meaning of the Weierstrass polynomial approximation theorem \\cite{Y},\nan appropriate topology in $\\cC^\\infty(\\Omega\\times T)$ makes\n$\\cC^\\omega(\\Omega\\times T)$ dense in  $\\cC^\\infty(\\Omega\\times T)$\nby restricting the region $\\Omega \\times T$ appropriately.\nUnder the assumption, we also deal with a smooth section of\n$\\IFluid(\\Omega\\times T)$.\n\n\t\nLet us consider a coordinate function \n$(\\gamma^i(x, t))_{i=1,2,3} \\in \\cC^\\omega(\\Omega \\times T)$\nsuch that\n$$\n\t\\frac{d}{d t} \\gamma^i(x, t) = u^i(x, t), \\quad\n\t\\gamma^i(x, t) = x^i \\ \\ \\mbox{at } \\ t \\in T,\n$$\nwhich means \n$$\n\\gamma^i(x, t + \\delta t) = x^i + u^{i}(x,t) \\delta t + O(\\delta t^2),\n$$\nfor a small $\\delta t$.\nHere the addition is given as a Euclidean move in $\\EE^3$.\nAs an inverse function of $\\gamma=\\gamma(u,t)$, \nwe could regard $u$ as a function of $\\gamma$ and $t$,\n$$\nu(x,t) =u(\\gamma(x,t),t).\n$$\nFurther we introduce a small quantity modeled on $\\delta t\\cdot u^i$,\n\\begin{equation}\n\t\\tgamma^i(x,t) := \\gamma^i(x,t) - x^i.\n\\label{eq:tgamma}\n\\end{equation}\nThen a section $g$\nof $\\IFluid(\\Omega\\times T)$ at $t \\in T$ can written by,\n\\begin{equation}\n\tg(t) = \\ee^{\\tgamma^i \\partial_i} \\in \\IFluid(\\Omega\\times T)\\Bigr|_{t}\n        \\approx \\SDiff(\\Omega) \\subset \\Diff(\\Omega).\n\\label{eq:etgamma}\n\\end{equation}\nHere we consider $g$ as an element of $\\SDiff(\\Omega)$ and thus\nit satisfies the condition of\nthe volume preserving, which appears as the constraint that the Jacobian,\n$$\n\\frac{\\partial \\gamma}{\\partial x} \n:=\n\\det\\left(\\frac{\\partial \\gamma^i}{\\partial x^j} \\right)\n= (1 + \n\\tr(\\partial_j u^i) \\delta t) + O(\\delta t^2),\n$$\nmust preserve $1$, {\\it{i.e.}}, the well-known condition\nthat $\\tr (\\partial_j u^i) = \\mathrm{div}(u)$ must vanish,\nor $\\displaystyle{\\frac{d}{dt}\\frac{\\partial \\gamma}{\\partial x} = 0}$.\n\nFollowing Reference \\cite{EM}, we reformulate the\naction integral (\\ref{eq:Eue})\nas  {\\lq\\lq}the energy functional{\\rq\\rq}\nin the frame work of the harmonic\nmap theory.\nIn the harmonic map theory \\cite{GMO}\nby considering a smooth map $h : M \\to G$\nfor a $n$-smooth base manifold $M$ and its target group manifold $G$,\n{\\lq\\lq}the energy functional{\\rq\\rq} is given by\n\\begin{equation}\n    E = \\frac{1}{2}\\int_M \\tr \\left((h^{-1} d h )*(h^{-1} d h )\\right).\n    \\label{eq:B-1}\n\\end{equation}\nHere $*$ means the Hodge star operator, which is\nfor $*:TG\\otimes\\Lambda^p(M) \\to TG\\otimes\\Lambda^{n-p}(M)$ where\n$\\Lambda^p(M)$ is the set of the smooth $p$-forms over $M$ \\cite{N},\nand \n$TG\\otimes\\Lambda^p(M)$ is the set of the tangent bundle $TG$ valued\nsmooth $p$-forms over $M$ \\cite{N}.\nThe term {\\lq\\lq}energy functional{\\rq\\rq} in the harmonic map theory\nmeans that it is an invariance of the system and thus it sometimes\ndiffers from an actual energy in physics.\n\n\nSince in (\\ref{eq:PBIFluid}), the base space $T$ is one-dimensional and\nthe target space $\\IFluid(\\Omega\\times T)|_{t}$ at $t \\in T$ is \nthe infinite dimensional space,\n{\\lq\\lq}the energy functional{\\rq\\rq} (\\ref{eq:B-1})\nin the harmonic map theory corresponds to\nthe action integral $\\cS_{\\free} [\\gamma]$ which is defined by\n$$ \n\\cS_{\\free} [\\gamma]=\n\\frac{1}{2} \\int _T \\int_\\Omega \n\\frac{\\partial \\gamma}{\\partial x} \n\\rho d^3 x \\cdot dx^i \\otimes dx^i \\left(\n\t\\left( \\ee^{-\\tgamma^k \\partial_k}\n       dt \\frac{d}{dt}\\ee^{\\tgamma^\\ell \\partial_\\ell} \\right)\n\t\\left( \\ee^{-\\tgamma^j \\partial_j}\n        \\frac{d}{dt}\\ee^{\\tgamma^n \\partial_n} \\right)\n         \\right).\n$$\nHere $d x^i (\\partial_j ) := \n\\langle\\partial_j ,  d x^i \\rangle\n = \\delta^i_{\\ j}$ is the natural pairing  between\n $T \\Omega$ and $T^* \\Omega$.\nThe trace in  (\\ref{eq:B-1}) corresponds to the \nintegral over $\\Omega$ with \n$ \\frac{\\partial \\gamma}{\\partial x} \n\\rho d^3 x \\cdot dx^i \\otimes dx^i$.\nThe Hodge $*$ operator acts on the element such as\n$*\\left( \\ee^{-\\tgamma^k \\partial_k}\n dt \\frac{d}{dt}\\ee^{\\tgamma^\\ell \\partial_\\ell} \\right)\n= \\left( \\ee^{-\\tgamma^k \\partial_k}\n        \\frac{d}{dt}\\ee^{\\tgamma^\\ell \\partial_\\ell} \\right)$\nas the natural map from\n$\\diff(\\Omega)$ valued 1-form to 0-form.\nFurther we assume that $\\rho$ is a constant function in this section.\nThen the  action integral $\\cS_{\\free} [\\gamma]$\nobviously agrees with (\\ref{eq:Eue}).\n\nWe investigate the functional derivative and the\nvariational principle of this $\\cS_\\free[\\gamma]$.\nLet us consider the variation,\n$$\n\t\\gamma^j(x,t') = \\gamma^j(x, t') + \\delta \\gamma^j(x,t'), \\quad\n\\mbox{and} \\quad\n\t\\tgamma^j(x,t') = \\tgamma^j(x, t') + \\delta \\gamma^j(x,t'), \\quad\n$$\nwhere we implicitly assume that $\\delta \\gamma^j$ is proportional\nto the Dirac $\\delta$ function,\n$\\delta(t' - t)$, for some $t$ and $\\delta \\gamma^j$ vanishes at\n$\\partial \\Omega$.\nAs we have concerns only for local effects or differential\nequations, we implicitly assume that\nwe can neglect the boundary effect arising from  $\\partial \\Omega$ on\nthe variational equation.\nIf one needs the boundary effect, he would follow the study of\nShkoller \\cite{Shk}.\nFurther one could use the language of the sheaf theory to\ndescribe the local effects \\cite{KKK}.\nAs we are concerned only with differential equation and thus\nour theory is completely local except Section \\ref{sec:VOF}, we could\ndeal with germs of related bundles \\cite{AGLV} as in Reference \\cite{M},\nwhich is also naturally connected with\n a computational method of fluid dynamics \\cite{M2}.\n\nLet us consider the extremal point of the action integral (\\ref{eq:Eue})\nfollowing the variational principle.\nNoting that $\\partial \\gamma \/\\partial x=1$, \nthe above Jacobian becomes\n$$\n\\frac{\\partial (\\gamma + \\delta \\gamma)}{\\partial x} \n=\n\\frac{\\partial \\gamma}{\\partial x} \n(1 + \\partial_k \\delta \\gamma^k) + O((\\delta \\gamma)^2).\n$$\nSince we  employ the projection method, we firstly\nconsider a variation\nin $\\diff(\\Omega)$ rather than $\\sdiff(\\Omega)$.\nFor the variation,\nthe  action integral $\\cS_{\\free} [\\gamma]$ with (\\ref{eq:etgamma})\nbecomes \n\\begin{equation}\n\\begin{split}\n\\cS_\\free [\\gamma+\\delta  \\gamma] -&\n\\cS_\\free [\\gamma] =\\\\\n    &-\\int_T dt \\int_\\Omega \n\\frac{\\partial \\gamma}{\\partial x} \nd^3 x \\cdot dx^i \\otimes dx^i \n \\left(\n\t\\delta \\gamma^k\n        \\frac{d}{dt}\\left( \\rho\n         g^{-1} \\frac{d}{dt} g \\right)\n+ \\delta \\gamma^k \\partial_k \\frac{1}{2}\\rho |u|^2 \n \\right).\\\\\n\\end{split}\n\\nonumber\n\\end{equation}\n\nNow we have the following proposition.\n\n\\begin{proposition} \\label{prop:3-1}\nUsing the above definitions, the variational principle\nin $\\SDiff(\\Omega)$,\n$$\n\\frac{\\delta \\cS_\\free[\\gamma]}{\\delta \\gamma(x,t)}\n   \\Bigr|_{\\SDiff(\\Omega)|_t} = 0,\n$$\nis reduced to the Euler equation,\n\\begin{equation}\n \\frac{\\partial}{\\partial t} \\rho u^i + u^j \\partial_j \\rho u^i\n          + \\partial_i p = 0,\n\\label{eq:EulerEq}\n\\end{equation}\nwhere $p$ comes from the projection from $T\\Diff(\\Omega)|_{\\SDiff(\\Omega)}\n\\to T\\SDiff(\\Omega)$.\n\\end{proposition}\n\n\\begin{proof}\nBasically we leave the rigorous proof\nand especially the derivation of $p$ to \\cite{AK,EM}.\nThe existence of $p$ was investigated well in \nAppendix of Reference \\cite{EM}\nas the Hodge decomposition \\cite{AM,N}. (See also\n the following Remark \\ref{rmk:3-2}.)\nExcept the derivation of $p$,\nwe use the above relations and the following relations,\n\\begin{equation*}\n\\begin{split}\n        \\frac{d}{dt}\n\t\\left(\\rho  \\ee^{-\\tgamma^j \\partial_j}\n        \\frac{d}{dt}\\ee^{\\tgamma^n \\partial_n} \\right) \n    &=\n\t\\frac{d}{dt} \\left(\\rho u^i(\\gamma(t),t) \\partial_i \\right) \\\\\n   &= \\left( \\frac{\\partial}{\\partial t}\\rho u^i|_{x=\\gamma}\n       + (\\frac{d}{dt}\\tgamma^j) \\partial_j \\rho u^i \\right) \\partial_i \\\\\n   &= \\left( \\frac{\\partial}{\\partial t}\\rho u^i\n       + u^j \\partial_j\\rho u^i \\right) \\partial_i \\\\\n   &=:\\left(\\frac{D}{Dt} \\rho u^i \\right) \\partial_i. \\\\\n\\end{split}\n\\end{equation*}\nThen we obtain the Euler equation.\n\\qed\n\\end{proof}\n\n\\begin{myremark}\\label{rmk:3-2}\n {\\rm{\nThe Euler equation was obtained by the simple variational principle.\nPhysically speaking, the conservation of \nthe momentum in the sense of Noether's theorem \\cite{BGG,IZ} led to\nthe Euler equation.\nHowever, we could introduce the pressure $p_L$ term as the\nLagrange multiplier of the constraint of the volume preserving.\nIn the case, instead of $\\cS_\\free$, we deal with \n$$\n\\cS_{\\free,p} = \\cS_\\free\n + \\int_T dt \\int_\\Omega p_L(x,t) \\frac{\\partial \\gamma}{\\partial x} \n                    d^3 x.\n$$\nThen  noting the term coming from the Jacobian, the relation,\n$$\n\t\\frac{\\delta \\cS_{\\free,p}\n             [\\gamma]}{\\delta\\gamma(x,t)}\\Bigr|_{\\SDiff(\\Omega)|_t} = 0,\n$$\nis reduced to the Euler equation,\n$$\n \\frac{\\partial}{\\partial t} \\rho u^i + u^j \\partial_j \\rho u^i\n          + \\partial_i (p_L + \\frac{1}{2}\\rho |u|^2) = 0.\n$$\nAs the pressure is determined by the (divergence free) condition of $u$,\nwe  renormalize \\cite[(25)]{Ko},\n$$\n\tp := p_L + \\frac{1}{2}\\rho |u|^2.\n$$\nMore rigorous arguments are left to  References \\cite{EM,O} \nand physically interpretations are, {\\it {e.g.}}, in References\n\\cite{AF,B,K,NHK,Schm,V}.\n\n\nWe give a comment on \nthe projection from $T\\Diff(\\Omega)|_{\\SDiff(\\Omega)}\n\\to T\\SDiff(\\Omega)$ in (\\ref{eq:EulerEq}),\n which is known as the projection method.\nFirst we note that\nthe divergence free condition $\\mathrm{div} (u) =0$ \nsimplifies the Euler equation (\\ref{eq:EulerEq}),\n$$\n\\rho \\frac{D u}{D t} + \\nabla p = 0, \\quad \n\\frac{\\partial u^i}{\\partial t} + u^j \\partial_j u^i\n+\\frac{1}{\\rho}\\partial_i p=0.\n$$\nAs mention in Section \\ref{sec:VOF}, in the difference equation \nwe have a natural interpretation of the projection method \\cite{Cho}.\nWe, thus, regard $D u \/ Dt$ in $T\\Diff(\\Omega)|_{\\SDiff(\\Omega)}$ as \n$\\lim_{\\delta t \\to 0}\\frac{u(t + \\delta t)-u(t)}{\\delta t}$\nfor $u(t+\\delta t):=u(t+\\delta t,\\gamma(t+\\delta t)) \\in \\diff(\\Omega)$ \nand $u(t):=u(t,\\gamma(t)) \\in \\sdiff(\\Omega)$, {\\it{i.e.}},\n$\\mathrm{div} \\left(u(t)\\right) = 0$ by considering\n$T\\Diff(\\Omega)$ at the unit $e$ of $\\Diff(\\Omega)$\n up to $\\delta x^2$, as we did in (\\ref{eq:tgamma}) and (\\ref{eq:etgamma}).\nIn order to find the deformation \n$u^\\parallel(t + \\delta t)$ in $\\sdiff(\\Omega)$ by a natural projection\nfrom $\\diff(\\Omega)$ to $\\sdiff(\\Omega)$ \\cite[,p.36]{CM},\nwe decompose $u(t + \\delta t)$ into $u^\\parallel(t + \\delta t)$ and\n$u^\\perp(t + \\delta t)$ such that \n$\\partial_i u^{\\perp i}(t + \\delta t) := \\partial_i u^i(t + \\delta t)$.\nThen $u^\\parallel(t + \\delta t):= u(t + \\delta t)-$\n$u^\\perp(t + \\delta t)$ belongs to $\\sdiff(\\Omega)$.\nThus the pressure $p$ is determined by \\cite{CM}\n\\begin{equation}\n\\partial_i u^i(t + \\delta t)\n+ \\delta t\\partial_i \\frac{1}{\\rho}\\partial_i p =0.\\quad\n\\label{eq:ProjM}\n\\end{equation}\nIn other words, since $u^\\parallel(t + \\delta t)\\equiv$\n$u^i(t + \\delta t) + \\delta t \\frac{1}{\\rho}\\partial_i p$ belongs\nto $\\sdiff(\\Omega)$,\nthe deformation of\n$u^{\\parallel i}(t + \\delta t)-  u^i(t)$ which gives\n$D u^\\parallel\/D t$ and the Euler equation (\\ref{eq:EulerEq}) \nis the deformation in $\\IFluid(\\Omega \\times T)$.\nAfter taking the continuous limit $\\delta t \\to 0$,\nthe equation for the pressure\n(\\ref{eq:ProjM}) can be written as \\cite{Cho},\n$$\n(\\partial_i u^j) (\\partial_j u^i)\n+\\partial_i\\frac{1}{\\rho}\\partial_i p=0,\n$$\nby noting the relations $[\\partial_t, \\partial_i]=0$\nand $\\mathrm{div}(u(t))=0$, {\\it{i.e.}},\n$\\partial_i u^i(t + \\delta t) = \\partial_i[ u^i(t) $\n$+ \\frac{\\partial}{\\partial t} u^i(t) \\delta t$\n$+ u^j(t)\\partial_j u^i(t)\\delta t] + O(\\delta t^2)$.\nThe Poisson equation with (\\ref{eq:EulerEq})\nguarantees the divergence free condition.\nHence the pressure $p$ in the incompressible fluid\nis determined geometrically.\n}}\n\\end{myremark}\n\n\n\n\n\\section{Reformulation of Surface tension as a minimal surface energy}\n\\label{sec:four}\n\nIn this section we  reformulate  \nthe SURFER scheme \\cite{LZZ} following the \nvariational principle and the arguments of previous sections.\n\n\n\\subsection{Analytic expression of surface area}\\label{subsec:fourOne}\n\nWe first should note that in general, the higher dimensional \ngeneralized function like the Dirac delta function has some\ndifficulties in its  definition \\cite{Y}. \nFor the difficulties,\nin the Sato hyperfunctions theory \\cite{KKK},\nthe sheaf theory and the cohomology theory are\nnecessary to the descriptions of the higher dimensional generalized functions,\nwhich are too abstract to be applied to a problem with an arbitrary\ngeometrical setting.\nEven for the generalized function in the framework of\nSchwartz distribution theory, we should pay attentions on its treatment.\nHowever since the surface $S$ in this article is \na hypersurface and its codimension is one, \nthe situation makes the problems much easier.\n\nWe assume that the smooth surface $S$ is orientable and compact such\n that we could\ndefine its inner side and outer side. In other words, there\nis a three dimensional subspace (a manifold with boundary) $B$ such that\nits boundary $\\partial B$ agrees with $S$ and $B$ \nis equal to the inner side of $S$ with $S$ itself.\nThen we consider a generalized function $\\theta$ over \n$\\Omega \\subset \\EE^3$ such that\nit vanishes over the complement $B^c = \\Omega \\setminus B$\n and is unity for the interior $B^\\circ:=\nB \\setminus \\partial B$;\n$\\theta$ is known as a characteristic function of $B$.\n\nWe consider the global function \n$\\theta(x)$ and its derivative $d\\theta(x)$ in the sense of\nthe generalized function, which is given by\n\\begin{equation*}\nd \\theta(x) = \\sum_{i}\\partial_i \\theta(x) d x^i =\\partial_q \\theta(x) d q.\n\\end{equation*}\nHere we use the notations in Section \\ref{sec:two-one}.\nUsing the nabla symbol $\\nabla \\theta=(\\partial_i \\theta(x) )_{i=1,2,3}$,\n$|\\nabla \\theta| d^3 x$ is interpreted as\n\\begin{equation*}\n\t|\\nabla \\theta|d^3 x=|(* d \\theta)\\wedge dq|. \\quad\n\\end{equation*}\nHere due to the Hodge star operation \n$*: \\Lambda^p(\\Omega) \\to \\Lambda^{3-p}(\\Omega)$,\n$* d \\theta = \\tilde e \\partial_q \\theta d s^1\\wedge d s^2$\nwhere $\\tilde e$ is the Jacobian between the coordinate systems\n $(d s^1, d s^2, d q)$ and $(d x^1, d x^2, d x^3)$.\nThen we have the following proposition;\n\\begin{proposition}\nIf the integral,\n\\begin{equation*}\n\\cA:=\t\\int_{\\Omega} |\\nabla \\theta| d^3 x \\equiv \\int_{\\Omega} \n       |(* d \\theta)\\wedge dq|,\n\\end{equation*}\nis finite, $\\cA$ agrees with the area of the surface $S$.\n\\end{proposition}\n\nIt should be noted that due to the codimension of \n$S \\subset \\Omega$,\nwe have used the fact that the Dirac $\\delta$ function \nalong $q \\in T_S$ is the integrable\n function whose integral is the Heaviside function.\nThis fact is a key of this approach.\n\n\\subsection{Quasi-characteristic function for surface area}\n\n\nFor the later convenience, we introduce a support of \na function over $\\Omega$, which \nis denoted by {\\lq\\lq}supp{\\rq\\rq}, {\\it{i.e.}},\nfor a function $g$ over $\\Omega$, its support is defined by\n$$\n\t\\supp(g) =\\overline{\\{x \\in \\Omega \\ | \\ g(x) \\neq 0\\}},\n$$\nwhere {\\lq\\lq}$\\ \\bar{\\  } \\ ${\\rq\\rq}\n means the closure as the topological space $\\Omega$.\n\n\nOne of our purposes is to express the surface $S$ by means of \nnumerical methods, approximately.\nSince it is difficult  to deal with the generalized function $\\theta$\nin a discrete system\nlike the structure lattice \\cite{Ch},\nwe introduce a smooth function $\\xi$ over $\\Omega$\nas a quasi-characteristic function\nwhich approximates the function\n$\\theta$\n\\cite{BKZ,LZZ},\n\\begin{equation}\n     \\xi(x) =\\left\\{ \n\\begin{matrix} \n              0 & \\mbox{for } x \\in B^c \\bigcap \\{x \\in \\Omega \\ | \\ \n                                 |q(x)| < \\epsX\/2\\}^c, \\\\\n              1 & \\mbox{for } x \\in B \\bigcap \\{x  \\in \\Omega\\ | \\\n                                 |q(x)| < \\epsX\/2\\}^c, \\\\\n              \\mbox{monotonically increasing in } q(x)& \\ \\mbox{otherwise}. \n                          \\end{matrix}\n    \\right.\n\\label{eq:epsX}\n\\end{equation}\nWe note that along the line of $d q$ for $q \\in (-\\epsX\/2, \\epsX\/2)$,\n$\\xi$ is a monotonically increasing function which interpolates \nbetween $0$ and $1$.\nWe now implicitly assume that $\\epsX$ is much smaller than\n$\\epsT$ defined in Section \\ref{sec:two-one} so that\nsupport of $|\\nabla \\xi|$ is in the tubular neighborhood $T_S$.\nHowever after formulating\nthe theory, we extend the geometrical setting in Section \\ref{sec:two-one}\nto more general ones which include singularities; there\n$\\epsT$ might lose its mathematical meaning but $\\epsX$\nsurvives as a control parameter which governs the system.\nFor example, as in Reference \\cite{LZZ}, \nwe can also deal with a topology change well.\n\nBy letting $\\xi^c (x) := 1 - \\xi(x)$, $\\xi^c$ and $\\xi$ are\nregarded as the partition of unity \\cite[I p.272]{KN}, or\n$$\n\t\\xi(x) + \\xi^c(x) \\equiv 1.\n$$\nWe call these $\\xi$ and $\\xi^c$  {\\lq\\lq}color functions{\\rq\\rq} or \n{\\lq\\lq}phase fields{\\rq\\rq} in the following sections.\nWe have an approximation\nof the area of the surface $S$ by the following proposition.\n\n\\begin{proposition}\nDepending upon $\\epsX$, we define the integral,\n\\begin{equation*}\n     \\cA_{\\xi}:=\\int_{\\Omega} |\\nabla \\xi| d^3 x, \n\\end{equation*}\nand then the following inequality holds,\n\\begin{equation*}\n\\quad |\\cA_\\xi -\\cA | < \\epsX \\cdot \\cA.\n\\end{equation*}\n\\end{proposition}\n\nHere we note that\n$\\cA_\\xi$ is regarded as the approximation of the area $\\cA$ of $S$\ncontrolled by $\\epsX$. In other words,\nwe use $\\epsX$ as the parameter which controls the\ndifference between the characteristic function $\\theta$ and \nthe quasi-characteristic function $\\xi$\nin the phase field model \\cite{BKZ,LZZ}.\n\nLet us consider its extremal point following the \nvariational principle in a purely geometrical sense.\n\n\\begin{proposition} \\label{prop:4-3}\nFor sufficiently small $\\epsX$, we have\n\\begin{equation*}\n\\begin{split}\n\t\\frac{\\delta}{\\delta \\xi(x)} \\cA_\\xi \n         &=-\\partial_i\\frac{\\partial_i \\xi }{|\\nabla \\xi| }(x)\n                        \\\\\n       &=\\kappa(x),\n\\end{split}\n\\end{equation*}\nwhere $x \\in S$ or $q=0$. \n\\end{proposition}\n\n\\begin{proof}\nNoting the facts that $\\partial \\xi\/\\partial q <0$ at $q=0$ and\n\\begin{equation*}\n\t|\\nabla \\xi| = \\sqrt{\\nabla \\xi\\cdot \\nabla \\xi},\n\\end{equation*}\nProposition \\ref{prop:A-1} and\nthe equality in Proposition \\ref{prop:2-1} show the relation.\n\\qed\n\\end{proof}\n\n\nIn the vicinity of $S$, $q$ in Section \\ref{sec:two-one}\ncould\nbe identified with the level-set function and \nthe authors in References \\cite{ZCMO,ZMOW} also used this relation.\nSince all of geometrical quantities on $S$ are\nlifted to $T_S$ as the inverse image of $\\pi$,\nthe relation in Proposition \\ref{prop:4-3} is also\ndefined over $(\\supp(|\\nabla \\xi|))^\\circ \\subset  T_S$\nand we redefine the $\\kappa$ by the relation from here.\n\n\\bigskip\n\\subsection{Statics}\nLet us consider physical problems as we finish the geometrical\nsetting.\nBefore we consider dynamics of the phase field flow,\nwe consider a statical surface problem.\nLet $\\sigma$ be the surface tension coefficient between two fluids\ncorresponding to $\\xi$ and $\\xi^c$.\nNow let us call $\\xi$ and $\\xi^c$ {\\lq\\lq}color functions{\\rq\\rq} or \n{\\lq\\lq}phase fields{\\rq\\rq}. \nMore precisely, we say that a color function with individual\nphysical parameters is a phase field.\nThe surface energy $\\cE:=\\sigma \\cA$ is, then, approximately given by\n\\begin{equation}\n\t\\cE_\\two := \\sigma \\cA_\\xi = \\sigma\\int_{\\Omega} |\\nabla \\xi| d^3 x.\n\\end{equation}\nAs a statical mechanical problem, we consider\nthe variational method of this system following Section \\ref{sec:two-two}.\n\nSince a statical surface phenomenon is caused by the difference\nof the pressure of each material, we now consider a free energy functional\n\\cite{MM},\n\\begin{equation}\n\t\\cF_\\two :=\\int_{\\Omega} \n     \\left(\\sigma|\\nabla \\xi| - (p_1 \\xi + p_2 \\xi^c)\\right) d^3 x,\n  \\label{eq:ELs0}\n\\end{equation}\nwhere $p_a$ ($a = 1, 2$) is the proper pressure of each material.\n\n\\begin{proposition}\\label{prop:4-4}\nThe variational problem with respect to $\\xi$, $\\delta \\cF_\\two\/\\delta \\xi =0$,\nreproduces the Laplace equation {\\rm{\\cite[Chap.7]{LL}}},\n\\begin{equation}\n  (p_1 - p_2)  - \\sigma \\kappa(x) = 0, \\quad x\\in \n        (\\supp(|\\nabla \\xi|))^\\circ.\n  \\label{eq:ELs}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nAs in Proposition \\ref{prop:A-1}, direct computations give the relation.\n\\qed\n\\end{proof}\n\nThis proposition implies that the functional $\\cF_\\two$ is natural. The\nsolutions of (\\ref{eq:ELs}) are given by the constant mean curvature\nsurfaces studied in References \\cite{FW,GMO,T}. \n\nFurthermore we also have another static equation,\nwhose relation to the Laplace equation (\\ref{eq:ELs})\nis written in Remark \n\\ref{rmk:4-2}.\n\\begin{proposition}\\label{prop:4-5}\nFor every point $x \\in \\Omega$,\nthe variation principle,\n$\\delta \\cF_\\two\/\\delta x^i=0$, gives\n\\begin{equation}\n  \\sigma\\left(  \\sum_j \\partial_i\\frac{\\partial_j \\xi \n              \\partial_j \\xi }{|\\nabla \\xi| }\n     -  \\sum_j\\partial_j\\frac{\\partial_j \\xi \\partial_i \\xi }{|\\nabla \\xi| }\n \\right)\n  - (p_1 - p_2) \\partial_i \\xi=0,\n\\label{eq:Amini0}\n\\end{equation}\nor\n\\begin{equation}\n\\partial_j \\tau_{ij}(x) - (p_1 - p_2) \\partial_i \\xi(x)=0,\n\\label{eq:Amini}\n\\end{equation}\nwhere\n$$\n\\tau(x) := \\sigma\\left(I - \\frac{\\nabla \\xi}{|\\nabla \\xi|}\n\\otimes\n\\frac{\\nabla \\xi}{|\\nabla \\xi|}\\right)\n|\\nabla \\xi|(x).\n$$\n\\end{proposition}\n\n\\begin{proof}\nWe are, now, concerned with  the variation \n$x \\to x + \\delta x$ for every point $x \\in \\Omega$.\nWe apply Proposition \\ref{prop:A-2} to this case, {\\it{i.e.}},\n\\begin{equation*}\n\\begin{split}\n\\frac{\\delta \\cF_\\two}{\\delta x^i}\n&= -\\sigma \\left[\\partial_i |\\nabla \\xi|- \\partial_j\\left(\n         \\partial_i \\xi(x) \\cdot\\frac{\\delta}{\\delta\\partial_j \\xi(x)} \n           |\\nabla \\xi|  \\right) \\right](x)\n + (p_1 - p_2) \\partial_i \\xi(x),\n\\end{split}\n\\end{equation*}\nby using (\\ref{eq:ELs}) as its Euler-Lagrange equation\n(\\ref{eq:AppeA1}). Further for $x\\not\\in (\\supp(|\\nabla \\xi|))^\\circ$,\nits Euler-Lagrange equation (\\ref{eq:AppeA1}) gives a trivial\nrelation, {\\it{i.e.}}, {\\lq\\lq}$0=0${\\rq\\rq}.\nThen we have (\\ref{eq:Amini}).\n\\qed\n\\end{proof}\n\n\n\\begin{myremark}\\label{rmk:4-0}\n{\\rm{\nIt is worthwhile noting that (\\ref{eq:Amini0}) and (\\ref{eq:Amini}) \nare defined over $\\Omega$\nrather than $(\\supp(|\\nabla \\xi|))^\\circ$ because due to the relation,\n$$\n  | \\partial_i \\xi | \\le |\\nabla \\xi|,\n$$\neven at the point at which denominators in the first term in (\\ref{eq:Amini0}) \nvanish,\nthe first term is well-defined and vanishes.\n\nHence (\\ref{eq:Amini0}) and (\\ref{eq:Amini}) \ncould be regarded as an extension of the defined region\nof (\\ref{eq:ELs}) to $\\Omega$ and thus  \n (\\ref{eq:Amini0}) and (\\ref{eq:Amini}) \nhave the advantage over (\\ref{eq:ELs}).\nThe extension makes the handling of the surface tension much easier.\n}}\n\\end{myremark}\n\n\\begin{myremark}\\label{rmk:4-1}\n{\\rm{\nIn the statical mechanics, there appears a force $\\partial_i \\tau_{ij}$,\nwhich agrees with one in  \n(33) and (34) in Reference \\cite{LZZ}\nand (2.11) in Reference \\cite{Jac}. \nWe should note that in Reference \\cite{Jac},\nit was also stated that \n this term is derived from the momentum conservation \nhowever there was not its derivation in detail. \nThe derivation of the above $\\tau$ needs the \nEuler-Lagrange equation (\\ref{eq:AppeA1}), which corresponds to\n the Laplace equation (\\ref{eq:ELs}) in this case,\nwhen we apply Proposition \\ref{prop:A-2} to\nthis system, though these objects did not appear in Reference \\cite{Jac}. \n}}\n\\end{myremark}\n\n\\begin{myremark}\\label{rmk:4-2}\n{\\rm{\nIn this remark, we comment on the identity between \n  (\\ref{eq:ELs}) and (\\ref{eq:Amini}).\nComparing these, we have the identity,\n$$\n\t\\partial_i \\tau_{ij} = \\sigma \\partial_j \\xi \\cdot \\kappa,\n$$\nwhich is, of course, obtained from the primitive computations.\nIt implies that (\\ref{eq:Amini}) can be derived from the \nLaplace equation (\\ref{eq:ELs}) with this relation. \nHowever it is worthwhile noting that both come from the \nvariational principle in this article.\n In fact, when we handle multiple junctions,\nwe need a generalization of the Laplace equations over there \nlike (\\ref{eq:ELM3}), which is not easily obtained by taking the primitive\napproach. \nFurther the derivations from the variational principle\nshow their geometrical meaning in the sense of References \\cite{Ar,AM,BGG}.\n}}\n\\end{myremark}\n\\bigskip\n\\subsection{Dynamics}\n\nNow we investigate the dynamics of the two-phase field.\nThere are two different liquids which are expressed by\nphase fields $\\xi$ and $\\xi^c$ respectively.\nWe assume that they obey the incompressible fluid\ndynamics. \nAs in the previous section,\nwe consider the action of the volume-preserving diffeomorphism  \ngroup $\\SDiff(\\Omega)$\non the color functions $\\xi$ and $\\xi^c$.\nWe extend the domain of $\\xi$ and $\\xi^c$  to $\\Omega \\times T$\nand they are smooth sections of $\\Coor(\\Omega\\times T)$.\nFor the given $t$, we will regard\n$\\xi$ and $\\xi^c$ as functions of $\\gamma^i$ in the previous section,\n{\\it{i.e.},} $\\xi=\\xi(\\gamma(x,t))$.\nFor example, the density of the fluid is expressed by the relation,\n $$\n\\rho = \\rho_1 \\xi^c + \\rho_2 \\xi\n$$ \nfor constant proper densities $\\rho_1$ and $\\rho_2$ of the individual\nliquids.\nThe density $\\rho$, now, differs from a constant function\nover $\\Omega\\times T$ in general.\n\nWe consider the action integral $\\cS_{\\two}$ \nincluding the surface energy,\n\\begin{equation}\n\t\\cS_{\\two}[\\gamma]\n     = \\int_T d t\\int_{\\Omega}\\left( \\frac{1}{2}\\rho |u|^2 -\n               \\sigma|\\nabla \\xi| + (p_1\\xi + p_2 \\xi^c) \\right) d^3 x.\n\\label{eq:AI2d}\n\\end{equation}\nThe ratio between $\\rho$ and $\\sigma$\ndetermines the ratio between the contributions\nof the kinematic part and the potential (or surface energy) part\nin the dynamics of the fluid.\nSince the integrand in \n(\\ref{eq:AI2d}) contains no $\\partial \\xi\/\\partial t$ term,\nwe obtain the same terms in the variational calculations\nfrom the second and the third term in (\\ref{eq:AI2d})\nas those in (\\ref{eq:ELs}) and (\\ref{eq:Amini}) in \n the static case even if we regard $n$ as $4$\nand $x^4$ as $t$ in Section \\ref{sec:two-two}.\nBy applying Proposition \\ref{prop:A-1} to this system, we have\nthe following proposition as the Euler-Lagrange equation for $\\xi$.\n\n\\begin{lemma}\nThe function derivative of $\\cS_{\\two}$ with respect to $\\xi$ gives\n\\begin{equation}\n\t\\frac{1}{2}(\\rho_1 - \\rho_2) |u(x,t)|^2\n       + (p_1 - p_2) - \\sigma \\kappa (x,t)= 0, \n   \\quad x\\in \n        (\\supp(|\\nabla \\xi|))^\\circ,\n\\label{eq:EL2d}\n\\end{equation}\nup to the volume preserving condition.\n\\end{lemma}\nThis could be interpreted as a generalization of the \nLaplace equation (\\ref{eq:ELs}) as in the following remark.\n\n\\begin{myremark}{\\rm{\nHere we give some comments on the\ngeneralized Laplace equation (\\ref{eq:EL2d})\nup to the volume preserving condition.\nThis relation (\\ref{eq:EL2d})\ndoes not look invariant for \nGalileo's transformation, $u \\to u + u_0$ for a constant velocity\n$u_0$.\nHowever for the simplest problem of  Galileo's boost,\n{\\it{i.e.}}, static state on a system with a constant velocity $u_0$,\nthe equation (\\ref{eq:EL2d}) gives\n\\begin{equation}\n\t\\frac{1}{2}(\\rho_1 - \\rho_2) |u_0|^2\n       + (p_1 - p_2) - \\sigma \\kappa (x,t)= 0, \n   \\quad x\\in \n        (\\supp(|\\nabla \\xi|))^\\circ,\n\\end{equation}\nwhich might differ from the  Laplace equation (\\ref{eq:ELs}).\nHowever for the boost, we should transform $p_a$ into\n\\begin{equation}\n\t\\tilde p_a := p_a + \\frac{1}{2} \\rho_a |u_0|^2.\n\\label{eq:GalileoB}\n\\end{equation}\nThen the above equation of $\\tilde p_a$ agrees with the static one\n (\\ref{eq:ELs}). In other words (\\ref{eq:GalileoB}) makes\n our theory invariant for the Gaililio's transformation.\n\nFor a more general case, we should regard $p_a$ as a function over\n$\\Omega \\times T$ rather than a constant number due to\nthe volume preserving condition.\nThese values are contained in the pressure as mentioned in \n(\\ref{eq:p_PLTwo}). \nThe statement {\\lq\\lq}up to the volume preserving condition{\\rq\\rq}\n has the meaning in this sense.\nIn fact, in the numerical computation,\nthese individual pressures $p_a$'s are  not so important\nas we see in Remark \\ref{rmk:4-11}.\nDue to the constraint of the incompressible\n(volume-preserving) condition, the pressure $p$ is determined\nas mentioned in Remark \\ref{rmk:3-2}.\nThere are no contradictions with the \nGalileo's transformation and $\\SDiff(\\Omega)$-action.\n}}\n\\end{myremark}\n\n\n\n\n\nWe consider the infinitesimal action of\n$\\SDiff(\\Omega)$ around its identity.\nAs did in Section \\ref{sec:three},\nwe apply the variational method to this\nsystem in order to obtain the Euler equation with the surface tension.\n\n\n\\begin{proposition}\\label{prop:4-6}\nFor every $(x,t) \\in \\Omega \\times T$,\nthe variational principle,\n$\\delta \\cS_\\two\/\\delta \\gamma^i(x,t) = 0$, gives the equation of motion,\nor the Euler equation with the surface tension,\n\\begin{equation}\n\\begin{split}\n        \\frac{D \\rho u^i}{D t} +\n            \\sigma \\left( \\sum_j \\partial_i\\frac{\\partial_j \\xi \n              \\partial_j \\xi }{|\\nabla \\xi| }\n     -  \\sum_j  \\partial_j\\frac{\\partial_j \\xi \\partial_i \\xi }{|\\nabla \\xi| }\n     \\right)\n          + \\partial_i p = 0.\n\\label{eq:Eulxi}\n\\end{split}\n\\end{equation}\nHere $p$ is also the pressure coming from the effect of the \nvolume-preserving.\n\\end{proposition}\n\n\\begin{proof}\nThe measure $d^3 x$ is regarded as \n$\\displaystyle{\\frac{\\partial \\gamma}{\\partial x} d^3 x}$ with\n$\\displaystyle{\\frac{\\partial \\gamma}{\\partial x} = 1}$.\nNoting $\\displaystyle{\\frac{d}{dt}\\frac{\\partial \\gamma}{\\partial x} = 0}$,\nthe proof in Proposition \\ref{prop:3-1} and Remark \\ref{rmk:3-2}\nprovide  the kinematic part with pressure term and \nProposition \\ref{prop:4-5} gives the remainder.\nIn this proof, the total pressure $p$ is defined in Remark \\ref{rmk:4-11}.\n\\qed\n\\end{proof}\n\n\\begin{myremark} \\label{rmk:4-11}\n{\\rm{\nMore rigorous speaking, as we did in Remark \\ref{rmk:3-2},\nwe also renormalize the pressure,\n\\begin{equation}\n\\begin{split}\n\tp &= p_L + \\frac{1}{2}\\rho |u|^2 + p_1 \\xi + p_2 \\xi^c \\\\\n\t&= p_L + \\frac{1}{2}(\\rho_1 - \\rho_2) \\xi |u|^2 \n+ (p_1 - p_2) \\xi + \\frac{1}{2}\\rho_2 |u|^2 + p_2. \n\\end{split}\n\\label{eq:p_PLTwo}\n\\end{equation}\nAs in Section \\ref{sec:two-two}, the third term  in \n(\\ref{eq:Eulxi}) includes the effects from $p_a$'s via the\ngeneralized Laplace equation (\\ref{eq:EL2d}) as the Euler-Lagrange\nequation  (\\ref{eq:AppeA1}).\n}}\n\\end{myremark}\n\n\\begin{myremark} \\label{rmk:7} {\\rm{\n\\begin{enumerate}\n\\item\nThe equation of motion (\\ref{eq:Eulxi}) is \nthe same as (24) in Reference \\cite{LZZ} basically.\nWe emphasize that  it is \nreproduced by   the variational principle.\n\\item\nAs in Reference \\cite{LZZ}, in our framework, we can deal with the \ntopology\nchanges and the singularities which are controlled by the parameter\n$\\epsX$. The above dynamics is well-defined as a field\nequation provided that $\\epsX$ is finite. If needs,\none can evaluate\nits extrapolation for vanishing of $\\epsX$.\n\n\\item\nIn general, $\\epsX$ is not constant for the time\ndevelopment. Due to the equation of motion, it changes.\nAt least, in numerical computation, the numerical\ndiffusion makes the intermediate region wider in general. \nHowever even when the time passes \nbut we regard it as a small parameter,\nthe approximation is justified.\n\n\\item\nSince from Remark \\ref{rmk:4-0}, the surface tension is defined over \n$\\Omega$, the Euler equation is defined over $\\Omega$ without any \nassumptions.\n\n\\item\nIt should be noted that the surface force is\nnot difficult to be computed as in Reference \\cite{LZZ} but\nthere sometimes appear so-called parasite current problems in the \ncomputations even though we will not touch the problem \nin this article.\n\n\\end{enumerate}\n}}\n\\end{myremark}\n\n\\section{Multi-phase flow with multiple junctions}\n\\label{sec:five}\n\nIn this section,\nwe extend the SURFER scheme \\cite{LZZ}\nof two-phase flow  to multi-phase ($N$-phase, $N\\ge2$) flow.\n\n\\subsection{Geometry of color functions}\n\nIn order to extend the geometry of the color functions in the previous\nsection, we introduce several geometrical tools.\nFirst let us define a geometrical object similar to smooth $d$-manifold \nwith boundary.\nHere we note that $d$-manifold means  $d$-dimensional manifold,\nand $d$-manifold with boundary means that \nits interior is a $d$-manifold and its boundary is\na $(d-1)$-dimensional manifold. We distinguish\na smooth (differential) manifold from a topological manifold here.\n\nWhen we consider multi-junctions in $\\EE^3$,\nwe encounter\na geometrical object with smooth {\\lq\\lq}boundaries{\\rq\\rq} whose dimensions\nare two, one and zero \neven though it is regarded as a topological $3$-manifold with boundary.\n\n\\begin{definition} \\label{def:5-1}\nWe say that a path-connected topological $d$-manifold with boundary $V$ is \n{\\it{a path-connected interior smooth $d$-manifold}}\nif $V$ satisfies the followings:\n\\begin{enumerate}\n\n\\item \nThe interior $V^\\circ$ is a path-connected smooth $d$-manifold, and\n\n\\item\n $V$ has finite path-connected subspaces \n$V_\\alpha$, $(\\alpha = 1, \\cdots, \\ell)$\nsuch that\n\\begin{enumerate}\n\\item $V\\setminus V^\\circ$ is decomposed by $V_\\alpha$, \n{\\it{i.e.}},\n$$\nV\\setminus V^\\circ = \\coprod_{\\alpha = 1}^\\ell V_\\alpha,\n$$\n\n\\item Each $V_\\alpha$ is a path-connected \nsmooth $k$-manifold in $\\Omega$ $(k<d)$.\n\\end{enumerate}\n\\end{enumerate}\nWe say that $V_\\alpha$ is a {\\it{singular-boundary}} of $V$ and\nlet their union $V\\setminus V^\\circ$ denoted by \n$\\partial_\\sing V := V\\setminus V^\\circ$.\n\\end{definition}\n\nHere the disjoint union is denoted by $\\coprod$, {\\it{i.e.}},\nfor subsets $A$ and $B$  of $\\Omega$, \n$A \\coprod B := A \\bigcup B$\n if $A \\bigcap B = \\emptyset$.\n\nBy letting $V^{(n)}:=V$ and\n$V^{[k]}:=\\{V_\\alpha  \\subset V\\ | \\ \\dim V_\\alpha \\le k\\}$,\nand by picking up an appropriate path-connected part $V^{(k)}\\subset V^{[k]}$\neach $k$,\nwe can find a natural stratified structure,\n$$\nV^{(n)} \\supset V^{(n-1)} \\supset \\cdots \\supset \nV^{(2)} \\supset V^{(1)} \\supset V^{(0)},\n$$\nwhich is known as a stratified submanifold in the \nsingularity theory \\cite{AGLV}. \n\nIn terms of \npath-connected interior smooth $d$-manifolds, we\nexpress subregions corresponding to materials in a regions\n$\\Omega \\subset \\EE^3$ as extensions of \n$B$ and $B^c$ in Section \\ref{subsec:fourOne}.\n\n\n\\begin{definition} \\label{def:5-2}\nFor a smooth domain $\\Omega \\subset \\EE^3$,\nwe say that $N$ path-connected interior smooth $3$-manifolds \n$\\{ B_a\\}_{a=0, \\cdots, N-1}$\nare {\\it{colored decomposition of $\\Omega$}} \nif $\\{ B_a\\}_{a=0, \\cdots, N-1}$ satisfy the followings:\n\\begin{enumerate}\n\\item every $B_a$ is a closed subset in $\\Omega$,\n\n\\item $\\Omega = \\bigcup_{a=0, \\cdots, N-1} B_a$, and\n\n\\item  $\\Omega \\setminus (\n\\bigcup_{a <b } B_a \\cap B_b) =\n \\coprod_{a=0, \\cdots, N-1} B_a^\\circ$.\n\\end{enumerate}\n\\end{definition}\n\n\nRoughly speaking, each $B_a$ corresponds to a material in $\\Omega$;\nDefinition \\ref{def:5-2}\n1.  means that $B_a$ is surrounded by singular boundary or the boundary\nof $\\Omega$,  2. implies that there is no {\\lq\\lq}vacuum{\\rq\\rq} \nin $\\Omega$ and 3. guarantees that the interiors of these materials \ndon't overlap.\n\n\\bigskip\nIn general,\nfor $ a \\neq b$, $ B_a \\cap B_b$ is a singular geometrical\nobject if it is not the empty set.\nSingularity basically makes its treatment difficult in mathematics.\nIn order to avoid such difficulties, we introduce \n  color functions $\\xi_a(x)$ $(a=0, 1, 2, \\cdots, N-1)$ \nover a region $\\Omega$, which are modeled on $\\xi$ and $\\xi^c$\nas in Section \\ref{subsec:fourOne},\nare controlled by\na small parameter $\\epsX > 0$ and approximate \nthe characteristic functions over $B_a$.\n\n\\bigskip\nTo define color functions $\\xi_a(x)$ $(a=0, 1, 2, \\cdots, N-1)$, \n we introduce another geometrical object, \n{\\it{$\\epsilon$-tubular neighborhood}} in $\\EE^3$:\n\\begin{definition} \\label{def:5-3}\nFor a closed subspace $U \\subset \\Omega$ and a positive number $\\epsilon$, \n{\\it{$\\epsilon$-tubular neighborhood $T_{U, \\epsilon}$ of $U$}} is \ndefined by\n$$\n T_{U, \\epsilon} := \\{ x \\in \\Omega \\ |\\ \\dist(x, U) < \\frac{\\epsilon}{2}\\},\n$$\nwhere $\\dist(x, U)$ is the distance between $x$ and $U$ in $\\EE^3$.\n\\end{definition}\n\nWe assume that each $T_{\\partial_\\sing B_a, \\epsilon}$ has a\nfiber structure over $\\partial_\\sing B_a$ as topological manifolds\nas mentioned in Section \\ref{sec:two-one}.\nUsing the $\\epsilon$-tubular neighborhood, we define \n$\\epsX$-controlled color functions.\n\n\\begin{definition} \\label{def:5-4}\nWe say that \n$N$ smooth non-negative functions  $\\{ \\xi_a\\}_{a=0, \\cdots, N-1}$ \nover $\\Omega \\subset \\EE^3$ are\n{\\it{$\\epsX$-controlled color functions associated with\na colored decomposition $\\{ B_a\\}_{a=0, \\cdots, N-1}$ of $\\Omega$}},\nif they satisfy the followings:\n\\begin{enumerate}\n\\item $\\xi_a$ belongs to $\\cC^\\infty(\\Omega)$ and for $x \\in \\Omega$,\n$$\n   \\sum_{a=0, 1 \\cdots, N-1} \\xi_a(x) \\equiv 1.\n$$\n\\item For every $ M_a := \\supp(\\xi_a)$ and  \n          $L_a := \\supp(1-\\xi_a)$, $(a=0, 1, \\cdots, N-1)$,\n\\begin{enumerate}\n \\item $B_a \\varsubsetneqq M_a$,\n\n \\item $L_a^c \\varsubsetneqq B_a$,\n\n \\item $(M_a \\setminus L_a^c)^\\circ \\subset T_{\\partial_\\sing B_a, \\epsX}$,\n\n \\item $(M_a \\setminus L_a^c)^\\circ \\supset  \\partial_\\sing B_a$.\n\\end{enumerate}\n\n\\item For $x \\in (M_a \\setminus L_a^c)$,\n we define the smooth function $q_a$ by\n$$\n\tq_a(x) = \\left\\{ \\begin{matrix} \n            \\dist(x, \\partial_\\sing B_a), & x \\in (M_a \\setminus B_a), \\\\\n            -\\dist(x, \\partial_\\sing B_a), & \n       \\mbox{ otherwise. }\n     \\end{matrix} \\right. \n$$\nThen for the flow $\\exp( - t \\frac{\\partial}{\\partial q_a})$\non $\\cC^\\infty(\\Omega) |_{ (M_a \\setminus L_a^c)}$,\n$\\xi_a$ monotonically increases along $t \\in U \\subset \\RR$ at\n$x \\in  (M_a \\setminus L_a^c)$.\n\\end{enumerate}\nWhen $(M_a \\setminus L_a^c)^\\circ = T_{ \\partial_\\sing B_a, \\epsX}$\nfor every $a$,\n $\\{ \\xi_a\\}_{a=0, \\cdots, N-1}$ are called \n{\\it{proper $\\epsX$-controlled color functions associated with\nthe colored decomposition of $\\Omega \\subset \\EE^3$,\n$\\{ B_a\\}_{a=0, \\cdots, N-1}$}} or merely {\\it{proper}}.\n\\end{definition}\n\n\\bigskip\nThe functions $\\xi_a$'s are, geometrically, the partition of unity \n\\cite[I p.272]{KN} \nand a quasi-characteristic function, roughly speaking, which\nis equal to $1$ in the far inner side of $B_a$, vanishes at the \nfar outer side of\n$B_a$ and monotonically behaves in the artificial intermediate region.\nNoting that the flow $\\exp( - t \\frac{\\partial}{\\partial q})$ corresponds\nto the flow from the outer side to the inner side,  \n$\\xi_a$ decreases from the inner side to the outer side.\n\\bigskip\n\nFrom here, let us go on to use the notations $B_a$, $M_a$, $L_a$, and\n$\\xi_a$ in Definition \\ref{def:5-4}.\nFurther for later convenience, we employ the following assumptions\nwhich are not essential in our theory but make the arguments simpler. \n\\begin{assumption}\\label{assump:one} {\\rm{\nWe assume the following:\n\\begin{enumerate}\n\\item {\\it{The colored decomposition $\\{ B_a\\}_{a=0, \\cdots, N-1}$ of $\\Omega$\nand $\\epsX$ satisfy the condition\nthat every $L_a^c$ is not the empty set.}}\n\nThis assumption means that the singularities that we consider can be resolved \nby the above procedure. Since $\\epsX$ can be small enough,\nthis assumption does not have crucial effects on our theory.\n\n\\item {\\it{The colored decomposition $\\{ B_a\\}_{a=0, \\cdots, N-1}$ of $\\Omega$\nand $\\epsX$ satisfy the relation,\n$$\n\t\\partial \\Omega \\bigcap\n        \\left(\\bigcup_{a \\neq b; a, b \\neq 0} M_a\\bigcap M_b\\right)\n        = \\emptyset,\n$$\nand every intersection $B_a \\bigcap B_0$ perpendicularly intersects\nwith $\\partial \\Omega$.}}\n\nThis describes the asymptotic behavior of the materials.\nFor example $M_0$ will be assigned to a wall in Section \\ref{sec:VOF}.\nThis assumption is neither so essential in this\nmodel but makes the arguments easy of the boundary effect. \nAs mentioned in Section \\ref{sec:three},\nwe neglect the boundary effect because we are concerned only with\na local theory or differential equations. \nIf one wishes to remove this assumption, he could consider smaller\nregion $\\Omega'\\subset \\Omega$ after formulates the problems in \n$\\Omega$.\n\n\\item\n{\\it{ The volume of every $B_a$, the \narea of every $\\partial_\\sing B_a$,\nand the length defined over every one-dimensional object\n in $\\partial_\\sing B_a$ \nare finite.}}\n\nAs our theory is basically local, this assumption is not essential, either.\n\\end{enumerate}\n}}\n\\end{assumption}\n\n\nUnder the assumptions,\nwe fix  colored decomposition $\\{ B_a\\}_{a=0, \\cdots, N-1}$ and \n $\\epsX$-controlled color functions $\\{ \\xi_a\\}_{a=0, \\cdots, N-1}$.\n\nAs mentioned in the previous section, we have an approximate\ndescription of the area of $\\partial_\\sing B_a$.\n\n\\begin{proposition} \\label{prop:5-1}\nBy letting\nthe area of $\\partial_\\sing B_a$ be $\\cA_a$, the integral\n$$\n     \\cA_{\\xi_a}:=\\int_{\\Omega} |\\nabla \\xi_a| d^3 x,\n$$\napproximates $\\cA_a$ by\n$$\n\t|\\cA_{\\xi_a} - \\cA_a | < \\epsX \\cA_a.\n$$\n\\end{proposition}\n\n\n\nHere we notice that \n$M_{ab}:=M_a \\bigcap M_b$ $(a, b = 0, 1, 2, \\cdots, N-1, a\\neq b)$ means\nthe intermediate region whose interior\nis  a $3$-manifold.\nSimilarly we define $M_{abc}:=M_a \\bigcap M_b \\bigcap M_c$\n$(a, b, c = 0, 1, 2, \\cdots, N-1; a\\neq b, c; b\\neq c)$ and so on.\nSince the relation, $\\bigcup M_a = \\Omega$, holds,\nwe look on the intersections of $M_a$ as an approximation of\nthe intersections of $B_a$ which is parameterized by $\\epsX$.\nEven though there exist some singular geometrical\nobjects in $\\{B_a\\}$ \\cite{AGLV},\nwe can avoid its difficulties due to finiteness of $\\epsX$.\n(We expect that the\ncomputational result of a physical process \nmight have weak dependence on $\\epsX$\nwhich is small enough.\nMore precisely the actual value is obtained by the extrapolation of \n$\\epsX = 0$ for series results of different $\\epsX$'s\napproaching to $\\epsX = 0$.)\n\n\\subsection{Surface energy}\n\nLet us define the surface energy $\\cE_\\exact^{(N)}$\nby \n$$\n\\cE_\\exact^{(N)} = \\sum_{a > b} \\sigma_{ab} \\Area(B_a \\bigcap B_b),\n$$  \nwhere $\\sigma_{ab}$ is the surface tension coefficient\n$(\\sigma_{ab}>0$, $\\sigma_{ab} = \\sigma_{b a})$ between\nthe materials corresponding to $B_a$ and $B_b$,\nand $\\Area(U)$  is the area of an interior smooth $2$-manifold $U$.\n\nWe have an approximation of the surface energy $\\cE_\\exact^{(N)}$\nby the following proposition.\n\\begin{proposition} \\label{prop:5-2}\nThe free energy,\n\\begin{equation}\n\t\\cE^{(N)} = \n  \\sum_{a > b}\n\\sigma_{ab}\\int_\\Omega d^3x\\ \n\\sqrt{|\\nabla \\xi_a(x)| |\\nabla \\xi_b(x)|}\n(\\xi_a(x) + \\xi_b(x)),\n\\label{eq:cEN}\n\\end{equation}\nhas a positive number $M$ such that\n$$\n |\\cE^{(N)} - \\cE_\\exact^{(N)}| < \\epsX M.\n$$\n\\end{proposition}\n\n\n\\begin{proof}\nFor $a\\neq b$, \n$B_a \\bigcap B_b$ consists of the union of some interior smooth\n$2$-manifolds. Their singular-boundary parts\n$\\partial_\\sing (B_a \\bigcap B_b)\n\\equiv \\{V_\\alpha\\}_{\\alpha \\in I_{ab}}$ are\nunion of some smooth $1$-manifolds and\nsmooth $0$-manifolds.\nThus  $\\{V_\\alpha\\}_{\\alpha \\in I_{ab}}$ has no \neffect on the surface energy $\\cE_\\exact^{(N)}$\nbecause they are measureless.\n\nOver the subspace,\n\\begin{equation}\nM_{ab}^\\prop := \\{ x \\in M_{ab} \\ | \\ \n\\xi_a(x) + \\xi_b(x) = 1 \\}^\\circ,\n\\label{eq:Mabprop}\n\\end{equation}\nand for a positive number $\\ell$, we have identities, \n\\begin{equation}\n\\begin{split}\n          |\\nabla \\xi_a(x)| (\\xi_a(x) + \\xi_b(x))^{\\ell}\n         & = |\\nabla \\xi_b(x)| (\\xi_a(x) + \\xi_b(x))^{\\ell}\\\\\n         & = \\sqrt{|\\nabla \\xi_a(x)| |\\nabla \\xi_b(x)|}\n(\\xi_a(x) + \\xi_b(x))^{\\ell}.\\\\\n\\end{split}\n\\label{eq:sym}\n\\end{equation}\nThe sum of the integrals over $M_{ab}^\\prop$ dominates $\\cE^{(N)}$\nif $\\epsX$ is sufficiently small.\n\nWe evaluate the remainder.\nFor example, for different $a$, $b$ and $c$, \nthe part in $\\cE^{(N)}$ coming from\n\\begin{equation}\nM_{abc}^\\prop := \\{ x \\in M_{abc} \\ | \\ \n               \\xi_a(x) + \\xi_b(x)  + \\xi_c(x) = 1 \\}^\\circ\n\\label{eq:Mabcprop}\n\\end{equation}\nis order of $\\epsX^2$.  Namely we have\n\\begin{equation*}\n\\begin{split}\n&\\left|\\int_{M_{abc}} d^3x\\ \n\\sqrt{|\\nabla \\xi_a(x)| |\\nabla \\xi_b(x)|}\n(\\xi_a(x) + \\xi_b(x)) \n-\\Length(B_a\\cap B_b \\cap B_c) \\right|\\\\\n&\\qquad\\qquad\\qquad <  \\epsX^2 \\Length(B_a\\cap B_b \\cap B_c),\\\\\n\\end{split}\n\\end{equation*}\nwhere $\\Length(C)$ is the length of a curve $C$.\nThus we find a number $M$ satisfying the inequality.\n\\qed\n\\end{proof}\n\n\n\\begin{myremark}\\label{rmk:5-1} {\\rm{\n\\begin{enumerate}\n\\item\n$M$ is bound by\n$$\n\tM \\le \\max(\\sigma_{ab})\\left(\n \\sum_{a<b}\\left(\\Area(B_a\\cap B_b) \n+ \\epsX \n \\Length\\left(\\partial_\\sing(B_a\\cap B_b)\\right)\\right) + K \\epsX^2\\right),\n$$\nwhere $K$ is the number of isolated points in all\nof singular-boundary parts of $\\{B_a\\}$.\n\n\n\\item\nIt should be noted that $\\cE^{(N)}$ becomes the surface\nenergy of the system exactly when $\\epsX$ vanishes.\n\n\\item\nUsing the identities (\\ref{eq:sym}), we can also approximate \n$\\cE^{(N)}$ by \n\\begin{equation*}\n  \\sum_{a> b}\n\\sigma_{ab}\\int_\\Omega d^3x\\ \n|\\nabla \\xi_a(x)| (\\xi_a(x) + \\xi_b(x))^\\ell,\n\\end{equation*}\nusing a positive number $\\ell$. In such a way, there are so many \nvariants which, approximately, represent the surface energy \nin terms of $\\xi_a$'s.\n\\end{enumerate}\n}}\n\\end{myremark}\n\n\n\\subsection{Statics}\nLet us consider the statics of the multi-phase fields.\nIn the above arguments in this section,\nwe have given the geometrical objects, first, and\ndefined the functions $\\xi_a$, functional energy\n$\\cE^{(N)}$ and so on.  \nIn this subsection on\nthe static mechanics of the multi-fields, \nwe consider the deformation of these geometrical objects\nand determine a configuration whose corresponds to an extremal point\nof the functional, {\\it{i.e.}}, $\\cF_\\mul$ in the following\nproposition. In other words, we derive the Euler-Lagrange equation\nwhich governs the extremal point of the functional\nand characterizes the configuration of $M_a$, $L_a$ and\napproximately $B_a$ for every $a = 0, \\cdots, N-1$.\n\nLet us introduce the proper pressure\n\\begin{equation}\n\tp_P(x) := \\sum p_a \\xi_a(x),\n\\label{eq:proppress}\n\\end{equation}\nwhere $p_a$ is a certain pressure of each material.\n\n\\begin{proposition}\n\\label{prop:5-8}\nThe Euler-Lagrange equation\nof the static free energy integral,\n\\begin{equation*}\n\t\\cF_\\mul= \\int_{\\Omega}\\left( \n\\sum_{a> b}\n\\sigma_{ab} \n\\sqrt{|\\nabla \\xi_a(x)| |\\nabla \\xi_b(x)| }\n(\\xi_a(x) + \\xi_b(x)) - p_P\n                \\right) d^3 x,\n\\end{equation*}\n with respect to $\\xi_a$, {\\it{i.e.}},\n $\\delta \\cF_\\mul\/\\delta \\xi_a = 0$, is given as follows:\n\\begin{enumerate}\n\\item\nFor a point $ x \\in {M_{ab}}^\\prop$ of (\\ref{eq:Mabprop}),\n\\begin{equation}\n        (p_a - p_b) - \\sigma_{ab}\\kappa_a (x)= 0, \n\\label{eq:ELM2}\n\\end{equation}\nwhere \n$$\n\t\\kappa_a := - \\partial_i \\frac{\\partial_i \\xi_a}{|\\nabla\\xi_a|}.\n$$\n\n\\item\nFor a point $x \\in M_{abc}^\\prop$ of (\\ref{eq:Mabcprop}),\n\\begin{equation}\n        (p_a - p_b  - p_c) - \\tilde\\kappa_{abc} (x)= 0, \n\\label{eq:ELM3}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{split}\n \\tilde\\kappa_{abc} &:= \n\\sigma_{bc}\\sqrt{|\\nabla \\xi_b(x)| |\\nabla \\xi_c(x)| }\n-\\sigma_{ab}\\sqrt{|\\nabla \\xi_a(x)| |\\nabla \\xi_b(x)| }\n-\\sigma_{ac}\\sqrt{|\\nabla \\xi_a(x)| |\\nabla \\xi_c(x)| }\\\\\n& + \\partial_i \n\\left[\\frac{\\partial_i \\xi_a}{\\sqrt{|\\nabla\\xi_a|}^3}.\n\\left(\\sigma_{ab} \\sqrt{|\\nabla\\xi_b|}( \\xi_a + \\xi_b)+\n\\sigma_{ac} \\sqrt{|\\nabla\\xi_c|}( \\xi_a + \\xi_c) \\right) \\right].\n\\end{split}\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\n\n\n\\begin{proof}\nFor a point $ x \\in {M_{ab}}^\\prop$ of (\\ref{eq:Mabprop}),\nwe have  $ \\xi_a(x) + \\xi_b(x) = 1 $, and thus  the\nEuler-Lagrange equation\n(\\ref{eq:AppeA1}) leads (\\ref{eq:ELM2}).\n\nSimilarly for a point $ x \\in {M_{abc}}^\\prop$ of (\\ref{eq:Mabcprop}),\nwe have  $ \\xi_a(x) + \\xi_b(x) + \\xi_c(x)= 1 $, and thus  the\nconcerned terms of the integrand in the energy functional are given by\n\\begin{equation}\n\\begin{split}\n\\cdots &\n+\\sqrt{|\\nabla \\xi_a(x)| |\\nabla \\xi_b(x)| }(\\xi_a + \\xi_b)\n+\\sqrt{|\\nabla \\xi_a(x)| |\\nabla \\xi_c(x)| }(\\xi_a + \\xi_b) \\\\\n&+\\sqrt{|\\nabla \\xi_b(x)| |\\nabla \\xi_c(x)| }(1 - \\xi_a) +\\cdots. \\\\\n\\end{split}\n\\end{equation}\nThe Euler-Lagrange equation \n(\\ref{eq:AppeA1})\ngives (\\ref{eq:ELM3}).\n\\qed\n\\end{proof}\n\n\\begin{myremark} \n {\\rm{\n\\begin{enumerate} \n\n\\item  It is noticed that \n(\\ref{eq:ELM2}) agrees with the Laplace equation (\\ref{eq:ELs}) and thus\nwe also reproduce the Laplace equation locally.\n\n\\item (\\ref{eq:ELM3})\ncould be regarded as another generalization of the Laplace equation\nthough $M_{abc}^\\prop$ does not contribute to the surface energy\nwhen $\\epsX$ vanishes and has a negligible effect even for\na finite $\\epsX$ if $\\epsX$ is sufficiently small.\nIndeed, (\\ref{eq:ELM3}) does not appear in the theory of surface tension\n\\cite{LL}. However \n(\\ref{eq:ELM3}) is necessary and plays a role\n to guarantee the stability\nin the numerical computations\nand to preserve the consistency in numerical approach with\nfinite intermediate regions for $\\epsX \\neq 0$.\n\n\\item Similarly we have similar equations\nfor a higher intersection regions.\n\n\\end{enumerate} \n}}\n\\end{myremark} \n\n\nAs a generalization of (\\ref{eq:Amini0})\nwe immediately have the following.\n\n\\begin{proposition} \\label{prop:5-9}\nFor every point $x \\in \\Omega$,\nthe variational principle, $\\delta \\cF_\\mul\/ \\delta x^i = 0$,\ngives \n\\begin{equation}\n\\begin{split}\n           & \\partial_i p_P \n   -\\sum_{a > b} \\sigma_{a,b}\\Bigr[\n           \\partial_i \\left(\n\\sqrt{|\\nabla \\xi_a| |\\nabla \\xi_b|}\n (\\xi_a+\\xi_b) \\right) \\\\\n    &\n       - \\partial_j\\left(\n         \\frac{\\partial_i \\xi_a \\partial_j \\xi_a }\n              {\\sqrt{|\\nabla \\xi_a|}^{3}}\n              \\sqrt{|\\nabla \\xi_b|}\n(\\xi_a+\\xi_b) \\right) \\Bigr] = 0.\n\\end{split}\n\\label{eq:MSFe}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nIt is the same as Proposition \\ref{prop:4-5},\nwhich essentially comes from \nProposition \\ref{prop:A-2}.\n\\qed\n\\end{proof}\n\n\\begin{myremark} \n {\\rm{\nIn Proposition \\ref{prop:5-9}, we can apply the equation without \nany classification of geometry like (\\ref{eq:Mabprop}) and (\\ref{eq:Mabcprop}).\nIt is also noted that (\\ref{eq:MSFe}) is globally defined  over \n$\\Omega$ as mentioned in Remark \\ref{rmk:4-0}.\n}}\n\\end{myremark} \n\n\n\n\n\n\n\n\n\n\\subsection{Dynamics}\nUsing these equations, let us consider\nthe dynamics of the multi-phase flow.\nWe extend the colored-decomposition of $\\Omega$ and\nthe $\\epsX$-controlled color functions of $\\{\\xi_a\\}_{a=0,\\cdots,N-1}$\nto those of $\\Omega \\times T$ and $\\cC^\\infty(\\Omega \\times T)$\nusing another fiber structure of $\\Coor(\\Omega\\times T)$.\nMathematically speaking, since our space-time is a\ntrivial bundle $\\Omega\\times T$ and has the fiber structure\n$\\Omega\\times (t_a, t_b) \\to \\Omega$ for a small interval $(t_a, t_b)$\ndue to the integrability,\nwe can consider the pull-back of the map $\\xi_a : \\Omega \\to \\RR$.\nIf we consider a global behavior of $\\xi_a$ with respect to time $t$,\nwe should pay more attentions on the Lagrange picture $\\gamma(x,t)$\nand the integrability.\nHowever as our theory is local, we can regard\n $(t_a, t_b)$ as $T$ with an infinitesimal interval.\n\nThus $\\xi_a$ is redefined as $\\xi_a:=\\xi_a(\\gamma(x,t))$\nfor $(x,t) \\in \\Omega \\times T$ and it is denoted by \n$\\xi_a(x,t)$.\nIn the time development of $\\xi_a$, the control parameter $\\epsX$\nis not necessary to be constant. However in this article,\nwe assume that $\\epsX$ is sufficiently small for every $t \\in T$.\n\nLet the density of each $\\xi_a$ be denoted by $\\rho_a$.\nWe have the global density function $\\rho(x,t)$ and pressure\n$p_P(x,t)$ given by\n$$\n\t\\rho(x,t) = \\sum \\rho_a \\xi_a(x,t), \\quad\n\tp_P(x,t) = \\sum p_a \\xi_a(x,t).\n$$\n\nIn contrast to the previous subsection, in this subsection,\nwe investigate an initial problem. In other words,\nevery configuration of the geometrical objects,  \n$M_a$, $L_a$ and approximately $B_a$ ($a = 0, \\cdots, N-1$),\nwith divergence free velocity $u$, ($\\mathrm{div}(u)=0$)\ncan be an initial condition to the dynamics of the multi-phase fields.\nThe following equations which we will derive in this subsection\ngovern the deformations of these\ngeometrical objects as their time-development.\nFurther it is noticed that\nin this subsection,\nthe proper pressure $p_P(x,t)$ has no mathematical nor\nphysical meaning because it becomes a part of the total pressure $p$,\nwhich is determined by the divergence free condition $\\mathrm{div}(u) =0$\nas mentioned in Remark \\ref{rmk:3-2}.\n\n\\bigskip\n\nWe have the first theorem;\n\n\\begin{theorem} \\label{th:5-1}\nThe action integral of the multi-phase fields, or\nthe $\\epsX$-controlled color functions $\\xi_a$\n with physical parameters $\\rho_a$, $\\sigma_{ab}$, $p_a$\n$(a, b = 0, 1, \\cdots, N-1)$ defined above, is given by\n\\begin{equation}\n\t\\cS_\\mul= \\int_T d t\\int_{\\Omega}\\left( \\frac{1}{2}\\rho |u|^2 -\n  \\sum_{a> b}\n\\sigma_{ab} \n\\sqrt{|\\nabla \\xi_a| |\\nabla \\xi_b| }\n (\\xi_a + \\xi_b)\n               + p_P \\right) d^3 x,\n\\label{eq:actionM}\n\\end{equation}\nunder the volume-preserving deformation.\n\\end{theorem}\n\n\\begin{proof}\nThe action integral is additive.\nThe first term exhibits the kinematic energy of the fluids.\nThe second term represents the surface energy up to $\\epsX$ as in\nProposition \\ref{prop:5-2}.\nThe proper pressure $p_P$ in (\\ref{eq:proppress}) leads the Laplace equations.\nWe can regard it as the action integral of \nthe multi-phase fields with these parameters.\n\\qed\n\\end{proof}\n\nThen we have further generalization of (\\ref{eq:EL2d}) as follows:\n\\begin{lemma} \\label{lemma:5-11}\nAssume that every $M_a(t)$, $M_{ab}^\\prop(t)$ and $M_{abc}^\\prop(t)$ deform\nfor the time-development following a certain equation.\nThe Euler-Lagrange equation of the action integral\n with respect to $\\xi_a$, $\\delta \\cS_\\mul\/ \\delta \\xi_a =0$,\nis given, up to the volume preserving condition, as follows:\n\\begin{enumerate}\n\\item\nFor a point $x \\in M_{ab}^\\prop$, we have\n\\begin{equation}\n\t\\frac{1}{2}(\\rho_a - \\rho_b) |u(x,t)|^2\n       + (p_a - p_b) - \\sigma_{ab}\\kappa_a (x,t)= 0. \n\\label{eq:ELM2d}\n\\end{equation}\n\\item\nFor a point $x \\in M_{abc}^\\prop$, we have\n\\begin{equation}\n\t\\frac{1}{2}(\\rho_a - \\rho_b - \\rho_c) |u(x,t)|^2\n       + (p_a - p_b - p_c) - \\tilde\\kappa_{abc} (x,t)= 0. \n\\label{eq:ELM3d}\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\nSimilarly we have the similar equations for  higher intersection regions.\n\\begin{proof}\nIt is the same as proof of Proposition\n\\ref{prop:5-8}.\n\\qed\n\\end{proof}\n\nUsing these equations, we have the second theorem,\nwhich is our main theorem:\n\\begin{theorem} \\label{th:5-2}\nFor every $(x, t) \\in \\Omega \\times T$,\nthe variational principle, $\\delta \\cS_\\mul\/ \\delta \\gamma(x,t) = 0$,\nprovides the equation of motion,\n\\begin{equation}\n\\begin{split}\n       \\frac{D \\rho u^i}{D t} +\n           & \\partial_i p \n   +\\sum_{a> b} \\sigma_{a,b}\\Bigr[\n           \\partial_i \\left(\n\\sqrt{|\\nabla \\xi_a| |\\nabla \\xi_b|}\n (\\xi_a+\\xi_b) \\right) \\\\\n    &\n       - \\partial_j\\left(\n         \\frac{\\partial_i \\xi_a(x) \\partial_j \\xi_a(x) }\n              {\\sqrt{|\\nabla \\xi_a|}^{3}}\n              \\sqrt{|\\nabla \\xi_b|}\n(\\xi_a+\\xi_b) \\right) \\Bigr] = 0.\n\\end{split}\n\\label{eq:EeqM}\n\\end{equation}\nHere $p$ is the pressure coming from the effect of the \nvolume-preserving or incompressible condition, which\nincludes the proper pressure $p_P$  (\\ref{eq:proppress}).\n\\end{theorem}\n\n\n\\begin{proof}\nWe naturally obtain it by using 1) \nProposition \\ref{prop:3-1} and its proof,\n2) Remark \\ref{rmk:3-2},\n3) Lemma \\ref{lemma:5-11}\nand 4) Proposition \\ref{prop:A-2}.\n\\qed\n\\end{proof}\n\nHere we note that\nby expressing the low-dimensional geometry in terms\nof the global smooth functions $\\xi$'s with finite $\\epsX$,\nwe have unified the infinite dimensional geometry\nor the incompressible fluid dynamics\ngoverned by $\\IFluid(\\Omega \\times T)$, and the $\\epsX$-parameterized\nlow dimensional geometry with singularities to obtain  \nthe extended Euler equation (\\ref{eq:EeqM}).\nWhen $\\epsX$ approaches to zero, we must consider the \nhyperfunctions \\cite{KKK,II} instead of $\\cC^\\infty(\\Omega \\times T)$,\nbut we conjecture that our results would be justified even under the limit;\nthe unification would have more rigorous meanings.\n\nIt should be noted that\non the unification,\nit is very crucial that we express the low-dimensional geometry in terms\nof the global smooth functions $\\xi$'s as \nthe infinite-dimensional vector spaces.\nThe $\\SDiff(\\Omega)$ naturally acts on $\\xi$'s and thus \nwe could treat the low-dimensional geometry and the incompressible\nfluid dynamics in the framework of the infinite dimensional\nLie group \\cite{AK,EM,O}.\nIt is contrast to the level-set method.\nAs mentioned in Section \\ref{sec:two-one},\nthe level-set function does not belong to $\\cC^\\infty(\\Omega)$\nand thus we can not consider $\\SDiff(\\Omega)$ action and\ntreat it in the framework.\n\n\\begin{myremark}\\label{rmk:11}{\\rm{\n\\begin{enumerate}\n\\item\n(\\ref{eq:EeqM}) is the Euler equation with the surface tension\nto multi-phase fields\nwhich gives the equation of motion of the multi-phase flow even with\nthe multiple junctions.\nAs we will illustrate examples in Section \\ref{sec:VOF}, \nthe dynamics with the triple junction \ncan be solved without any geometrical constraints.\nIt should also noted that for a point in $M_{ab}^\\prop$, \n(\\ref{eq:EeqM}) is reduced to the original Euler equation in \nReference \\cite{LZZ}\nor (\\ref{eq:Eulxi}).\n\n\\item\nThe Euler equation (\\ref{eq:EeqM}) appears as the momentum conservation in\nthe sense of Noether's theorem (Section \\ref{sec:two-two}).\nIt implies that (\\ref{eq:EeqM}) is natural from the geometrical viewpoint\n\\cite{Ar,AK,EM,K,Ko,MW,NHK}.\n\n\\item\nFurther even though we set $\\{\\xi_a(\\cdot, t)\\}$ as \nproper $\\epsX$-controlled\ncolored functions as an initial state,\n their time-development is not guaranteed \nthat $\\{\\xi_a(\\cdot, t)\\}$,  $(t>0)$,\nis  proper $\\epsX$-controlled.\nIn general $\\epsX$ may become large for the time development,\nat least, numerically due to the numerical diffusion. (See examples in\nSection \\ref{sec:VOF}).\nHowever even for $t>0$,\nwe can find $\\epsX(t)$ such that\n $\\{\\xi_a(\\cdot, t)\\}$ are $\\epsX(t)$-controlled colored functions\nand if $\\epsX(t)$ is sufficiently small, our approximation is\nguaranteed by $\\epsX(t)$.\n\n\\item\nThe surface tension is also defined over $\\Omega \\times T$ and thus\n the Euler equation is defined over $\\Omega \\times T$ without any \nassumptions due to Remark \\ref{rmk:4-0}.\n\n\\item We may set $\\epsX$ depending upon the individual\nintermediate region between these fields by \nletting $\\epsilon_{ab}$ mean that for\n$\\xi_a$ and $\\xi_b$, $a\\neq b$. Then if we recognize $\\epsX$ as \n$\\displaystyle{\\max_{a, b =0}^{N-1}\\epsilon_{ab}}$, \nabove arguments are applicable for the\ncase.\n\n\\item We defined the $\\epsX$-controlled colored functions\nusing the $\\epsT$-tubular neighborhood $T_{U, \\epsT}$ and\nthe colored decomposition of $\\Omega$\nin Definition \\ref{def:5-4} by letting $\\epsT = \\epsX$.\nOn the other hand,\nas in Reference \\cite{LZZ},\nour formulation can describe a topology change well following\nthe Euler equation (\\ref{eq:EeqM}) such as a split of\na bubble into two bubbles in a liquid. \nThe $\\epsX$-controlled colored functions \ncan represents the geometry for such a topology change without any\ndifficulties. However on the topology change,\nthe path-connected region and \nthe $\\epsX$-tubular neighborhood \nlose their mathematical meaning and thus, more rigorously,\nwe should redefine the $\\epsX$-controlled colored functions. \nSince\nthe  $\\epsX$-controlled colored functions represent the geometry\nas an analytic geometry, it is not difficult to modify the definitions\nthough it is too abstract.\nIn other words, we should first define the $\\epsX$-controlled colored \nfunctions $\\xi$'s without the base geometry, and \ncharacterize geometrical objects using\nthe functions $\\xi$'s.\nHowever since such a way is too abstract to find these geometrical meanings,\nwe avoided a needless confusion in these definitions and employed\nDefinition \\ref{def:5-4}.\n\n\\end{enumerate}\n}}\n\\end{myremark}\n\n\n\\subsection{Equation of motion of triple-phase flow}\n\nLet us concentrate ourselves on a triple-phase flow problem,\nnoting (\\ref{eq:sym}).\nFrom the symmetry of the triple phase,\nwe introduce {\\lq\\lq}proper{\\rq\\rq} surface tension coefficients,\n$$\n\t\\sigma_0 = \\frac{\\sigma_{01} + \\sigma_{02} - \\sigma_{12}}{2}, \\quad\n\t\\sigma_1 = \\frac{\\sigma_{01} + \\sigma_{12} - \\sigma_{02}}{2}, \\quad\n\t\\sigma_2 = \\frac{\\sigma_{02} + \\sigma_{12} - \\sigma_{01}}{2}, \\quad\n$$\nor $\\sigma_{ab} = \\sigma_a + \\sigma_b$.\nHere it should be noted that\nthe {\\lq\\lq}proper{\\rq\\rq} surface tension coefficient \nis based upon the speciality of the triple-phase and\ndoes not have more physical meaning than above definition.\n\n\\begin{lemma} \\label{lemma:5-2}\nFor different $a, b,$ and $c$, we have the following\napproximation,\n\\begin{equation}\n\\left|\\int_\\Omega\\left( \\sqrt{|\\nabla \\xi_a| |\\nabla \\xi_b|}\n             (\\xi_a + \\xi_b)\n\t+ \\sqrt{|\\nabla \\xi_a| |\\nabla \\xi_c|}\n             (\\xi_a + \\xi_c)\n\t- |\\nabla \\xi_a|\\right)d^3 x\\right|  < \\epsX \\cA_a.\n\\label{eq:Appr3}\n\\end{equation}\n\\end{lemma}\n\nUsing the relation,\nthe free energy (\\ref{eq:cEN}) has a simpler expression up to\n$\\epsX$.\n\\begin{proposition} \\label{prop:5-3}\n By letting\n\\begin{equation*}\n\t\\cE^{(3)}_{\\mathrm{sym}} := \\sigma_{0}\\int_\\Omega d^3x\\ |\\nabla \\xi_0(x)| \n          + \\sigma_{1}\\int_\\Omega d^3x\\ |\\nabla \\xi_1(x)|  \n          + \\sigma_{2}\\int_\\Omega d^3x\\ |\\nabla \\xi_2(x)| ,\n\\end{equation*}\nwe have a certain number $M$ related to area of the surfaces $\\{B_a\\}$\nsuch that\n\\begin{equation*}\n   |\\cE^{(3)} - \\cE^{(3)}_{\\mathrm{sym}}| < \\epsX M.\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\nDue to Lemma \\ref{lemma:5-2}, it is obvious.\n\\qed\n\\end{proof}\n\nThe action integral (\\ref{eq:actionM}) also becomes\n\\begin{equation*}\n\t\\cS_\\tri= \\int_T d t\\int_{\\Omega}\\left( \\frac{1}{2}\\rho |u|^2 -\n\t\\sum_a( \\sigma_a |\\nabla\\xi_a| - p_a \\xi_a )\\right) d^3 x.\n\\end{equation*}\nFor a practical reason, we consider a  simpler expression by\nspecifying the problem.\n\n\n\\subsection{Two-phase flow and wall with triple-junction}\n\\label{subsec:5-6}\n\nMore specially we consider the case \nthat $\\xi_o$ corresponds to the wall which does not move.\nFor the case, we can neglect the wall part of the equation, because\nit causes a mere energy-shift of $\\cE^{(3)}_{\\mathrm{sym}}$.\nThen the action integral and the Euler equation become simpler.\nWe have the following theorem as a corollary.\n\n\\begin{theorem} \\label{th:5-3}\nThe action integral of two-phase flow with  wall is given by\n\\begin{equation*}\n\t\\cS_\\wall= \\int_T d t\\int_{\\Omega}\n     \\left( \\frac{1}{2}\\rho |u|^2 -\n\t\\sum_{a=1}^2( \\sigma_a |\\nabla\\xi_a| - p_a \\xi_a )\\right) d^3 x,\n\\end{equation*}\nand the equation of motion is given by\n\\begin{equation}\n \\frac{D \\rho u^i}{D t} \n          + \\partial_i p \n          -\\partial_j (\\overline \\tau_{ij}) = 0,\n\\label{eq:Eeq3}\n\\end{equation}\nwhere\n\\begin{equation}\n            \\overline \\tau = \\sum_{a=1}^2\\sigma_a \n\\left(I - \n\\frac{\\nabla \\xi_a}{|\\nabla \\xi_a|}\n\\otimes\n\\frac{\\nabla \\xi_a}{|\\nabla \\xi_a|}\\right)\n|\\nabla \\xi_a|.\n\\label{eq:overtau}\n\\end{equation}\n\\end{theorem}\n\nPractically this Euler equation (\\ref{eq:Eeq3})\nis more convenient due to the proper surface tension coefficients.\nHowever this quite differs from the original (\\ref{eq:Amini0}) \nand (\\ref{eq:Amini}) in Reference \\cite{LZZ}\nand governs the motion of two-phase flow with a wall completely. \n\n\\begin{myremark}\\label{rmk:5-2}\n{\\rm{\nEquation (\\ref{eq:Eeq3})\nis the Euler equation with the surface tension\nfor two-phase fields with a wall or triple junctions\nin our theoretical framework.\nWe should note that \nunder the approximation (\\ref{eq:Appr3}),\n(\\ref{eq:Eeq3}) is equivalent to (\\ref{eq:EeqM}),\neven though \n(\\ref{eq:Eeq3}) is far simpler than (\\ref{eq:EeqM}).\n\n From Remark \\ref{rmk:4-0},\n it should be noted that\n$\\overline \\tau$ \nand the Euler equation (\\ref{eq:Eeq3})\nare defined over $\\Omega \\times T$. This property\nas a governing equation is very important for the \ncomputations to be stable, which is\nmentioned in Introduction.\nSince the non-trivial part of $\\overline{\\tau}$\nis localized in $\\Omega$ of each $t\\in T$,\n$\\overline \\tau$ vanishes and has no effect on the equation \n in the other area.\n}}\n\\end{myremark}\n\n\n\nWe will show some numerical computational results\nof this case in the following section.\nThere we could also consider\nthe viscous stress forces and the wall shear stress.\n\n\\section{Numerical computations}\n\\label{sec:VOF}\n\nIn this section, we show some numerical\ncomputations of two-phase flow surrounded by a wall obeying the extended Euler\nequation in Theorem \\ref{th:5-3}.\nAs in Theorem \\ref{th:5-3},\n the wall is expressed by the color function $\\xi_0$ and\nhas the intermediate region $(M_0 \\setminus L_0^c)^\\circ$ \nwhere $\\xi_0$ has its value $(0,1)$.\nAs dynamics of the incompressible two-phase flow with a static wall, \nwe numerically\nsolve the equations,\n\\begin{equation}\n\\begin{split}\n    &\\mathrm{div}(u) = 0,\n \\\\\n    &\\frac{D \\rho u^i}{D t} + \n    (\\partial_i p - K_i) = 0 ,\n \\\\\n   &\\frac{D \\rho}{D t} = 0.\n\\end{split}\n\\label{eq:NumEq}\n\\end{equation}\nHere for the numerical computations,\nwe assume that the force $K$ consists of\nthe surface tension, the viscous stress forces,\nand the wall shear stress,\n\\begin{equation}\nK_j = \\partial_i  \\bar \\tau_{ij} + \\partial_i  \\tau_{ij} + \\hat \\tau_j.\n\\label{eq:Ki}\n\\end{equation}\nHere \n$\\bar \\tau$ is given by (\\ref{eq:overtau}),\n$\\tau$ is the viscous tensor,\n$$\n \\tau_{ij} := 2 \\eta \\left(E_{ij} - \\frac{1}{3} \n           \\mathrm{div} (u)\\right), \\quad\n E_{ij} := \n \\frac{1}{2}\\left(\\frac{\\partial u^i}{\\partial x_j}\n +\\frac{\\partial u^j}{\\partial x_i}\\right)\n$$\nwith the viscous constant\n$$\n\\eta(x)  := \\eta_1 \\xi_1 + \\eta_2 \\xi_2,\n$$\nand $\\hat \\tau_j$ is the wall shear stress which is localized at\nthe intermediate region $(M_0 \\setminus L_0^c)^\\circ$ \nwhere $\\xi_0$ has its value $(0, 1)$.\n\nThe boundary condition of the interface between the fluid $\\xi_a$ \n$(a = 1, 2)$ and the wall $\\xi_0$ is generated dynamically in this\ncase. In other words, in order that\nthe wall shear stress term suppress the slip over the intermediate region\n$(M_0 \\setminus L_0^c)^\\circ$ asymptotically $t \\to \\infty$ due to\ndamping, we let $\\hat \\tau_j$ be proportional to\n$j$-component of $\\partial u^\\parallel \/\\partial q_0$\nfor the parallel velocity $u^\\parallel$ to the wall and relevant to\n$(1 - \\xi_0(x))$, and make $u$ vanish over $L_0$.\nHere $q_0$, $M_0$, and $L_0$ are of Definition \\ref{def:5-4}.\n\nThe viscous force can not be dealt with in the framework of the \nHamiltonian system because it has dissipation.\nHowever from the conventional consideration of the balance of \nthe momentum \\cite[Sec.13]{EM},\nit is not difficult to evaluate it.\nThe viscosity basically makes the numerical computations stable.\n\n\n\n\nIn the numerical computations,\nwe consider the problem in the structure lattice $\\cL$ \nmarked by $a \\ZZ^3$,\nwhere $\\ZZ$ is the set of the integers and $a$ is a positive number.\nThe lattice consists of cells and faces of each cell.\nLet every cell be a cube with sides of the length $a$.\nWe deal with a subspace $\\Omega_\\cL$\nof the lattice as $\\Omega_\\cL :=\\Omega \\cap \\cL \\subset \\EE^3$.\nThe fields $\\xi$'s are defined over the cells as cellwise constant\nfunctions and \nthe velocity field $u$ is defined over faces as\nfacewise constant functions \\cite{Ch};\n$\\xi$ is a constant function in each cell and depends on the\nposition of the cell,\nand similarly the components of the velocity field,\n$u^1$, $u^2$, and $u^3$ are facewise constant functions\ndefined over $x^2x^3$-faces, \n$x^3x^1$-faces, and $x^1x^2$-faces of each cell respectively.\n\nAs we gave a comment in Remark \\ref{rmk:11} 5, we make the parameter\n$\\epsX$ depend on the intermediate region in this section.\nLet $\\epsilon_{12}$ be the parameter\n for the two-phase field or the liquids, and \n$\\epsilon_0:= \\epsilon_{01}\\equiv\\epsilon_{02}$ be\none for the intermediate region \n$(M_0 \\setminus L_0^c)^\\circ$ between liquids and the wall.\n\nAs mentioned in Introduction,\nwe assume that $\\epsilon_{12}$ for the two-phase field\nin our method is given as $\\epsilon_{12} \\ge a$\nso that we could estimate the intermediate effect in our model\n following References \\cite{AMS,BKZ,Cab,Jac,LZZ},\neven though the thickness of the intermediate region among real liquids\nis of atomic order and is basically negligible in the macroscopic theory. \n\nIn the computational fluid dynamics, the VOF (volume of fluid) method\ndiscovered by Hirt and his coauthors \\cite{Ch,HN,H} \nis  well-established when we deal with fluid with a wall.\nSince we handled  triple-junction problems as in Section\n \\ref{subsec:5-6}, we reformulate our model in the VOF-method.\nIt implies that\nwe identify $1-\\xi_0$ with the so-called $V$-function $V := 1 - \\xi_0$\nin the VOF method \nbecause $V$ in the VOF method\nmeans the volume fraction of the fluid and corresponds to $1-\\xi_0$\nin our formulation.\n\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[scale=0.50,angle=270]{Fig1.ps}\n\\caption{VOF with porous matter expression: \nFor the consistency between the color function method and\nVOF-method,  \nwe consider each cell as a fictitious \nporous material whose volume ratio and \nopen fraction are a value in $[0,1]$ without\nimposing any wall shear stress on fictitious\nsurface of the porous parts in each cell.\nThis expression represents purely  geometrical effects.\n}\n\\label{fig:VOF}\n\\end{center}\n\\end{figure}\n\nAs the convention in Reference \\cite{H}, $V$ is also defined as\na cellwise constant function.\nIn the following examples, we will set \n$\\epsilon_0$ to be $a$ or the unit cell basically.\nHowever we can also make it $\\epsilon_0 > a$ as for two-phase field.\nIt means that for the case $\\epsilon_0 > a$,\nwe consider each cell as a fictitious \nporous material whose volume ratio $V \\in [0,1]$ \nwithout imposing any wall shear stress on the fictitious\nsurface of the porous parts itself in each cell as in Figure 1.\n(As mentioned above, we set the wall shear stress $\\hat \\tau_j$ from the\nphysical wall $\\xi_0$. The porous parts are purely fictitious.)\nThe region where $V$ is equal to 1 means the region where fluid freely exists\nwhereas\nthe region where $V$ vanishes means the region where existence of fluid \nis prohibited. The region with $V \\in (0,1)$ is the intermediate\nregion $(M_0 \\setminus L_0^c)^\\circ$. \nHere we emphasize that \nthe fictitious porous in each cell brings\npurely geometrical effects to this model.\n\nThen we could go on to\nconsider the problem in consistency between\nVOF-method and $\\xi_0$ function in the phase-field model.\nLet functions $f_1\\equiv f$ and $f_2$\nover $\\supp(V)$ be defined by the relations,\n\\begin{equation*}\n\t\\xi_1 = V f_1, \\quad \\xi_2 = V f_2, \\quad f_1 + f_2 = 1.\n\\end{equation*}\n\nFurther we also modify the open fraction $A$ in the VOF-method,\nwhich is defined over each face. We interpret $A$ as the open area of \nthe fictitious porous material of each face of each cell,\nwhich also has a value in $[0,1]$ as in Figure 1.\nWe also use the open area fraction $A$ of each face of each cell\n\\cite{H,HN}.\nFor a face belonging to the cell whose $V=1$, $A$ is also\nequal to 1.\nFollowing the convention in discretization by Hirt \\cite{H}, \n$A$ is regarded as an operator acting\non the face-valued functions \nlike\n\\begin{equation*}\n\tA \\circ u \\equiv  Au = (A_{1} u^1, \n                                A_{2} u^2,A_{3} u^3),\n\\end{equation*}\n\\begin{equation}\n       (Au)^{1} = A_{1} u^1, \\quad\n       (Au)^{2} = A_{2} u^2, \\quad\n       (Au)^{3} = A_{3} u^3. \\quad\n\\label{eq:Au}\n\\end{equation}\nHere we note that $A_i a^2$ implicitly appearing in\n(\\ref{eq:Au}) can be interpreted as a \ntwo-chain of homological base associated with a face of a cell.\nFor example, for a velocity field $\\mu := u^i(x) d x^i$ defined\nover a cell in the continuous theory \nand a piece of the boundary element of the\ncell $A_1 a^2$, \nthe discretized $u^1$ defined over the face is given by  \n$$\n   (A u)^1 := \\frac{1}{a^2} \\int_{A_1 a^2} * \\mu  = A_1 u^1,\n$$\nwhere $*$ is the Hodge star operator, {\\it{i.e.}},\n$*\\mu := u^1(x) d x^2 d x^3 +\nu^2(x) d x^3 d x^1 +\nu^3(x) d x^1 d x^2$.\nThus the discretization (\\ref{eq:Au}) is very natural even\nfrom the point of view of the modern differential geometry. \n\nHence $\\mathrm{div} (u) \\equiv \\nabla u$ reads\n$\\nabla A u$ as the difference equation in VOF-method \\cite{H}\nand we employ this discretization method.\n\n\nWe give our algorithm to compute (\\ref{eq:NumEq})\nprecisely as follows.\nAs a convention, we specify the quantities with {\\lq\\lq}old{\\rq\\rq}\nand  {\\lq\\lq}new{\\rq\\rq} corresponding to the previous states\nand the next states at each time step respectively\nin the computation.\nIn other words, we give the algorithm that we construct the\nnext states using the previous data by regarding the current\nstate as an intermediate state in the time step.\nWe use the project-method \\cite{Cho,Ch}; \n\\begin{equation*}\n\\begin{split}\n   \\mbox{I}  &: \\frac{\\rho \\tilde\\uu - \\rho \\uu^\\old }{\\Delta t }\n                 = -(\\uu^\\old\\cdot \\nabla ) \\rho \\uu^\\old ,\n   \\nonumber \\\\\n    \\mbox{II}  &:\\frac{\\uu^\\new-\\tilde\\uu  }{\\Delta t }\n                 = -\\frac{1}{\\rho}( \\nabla  p -\\KK),\n   \\nonumber \\\\\n   \\mbox{III}  &: \\nabla \\uu^\\new =0. \\\\\n\\end{split}\n\\end{equation*}\nThe step I is the part of the advection of the \nvelocity $\\uu^\\old$.\nIn the step I, we define an intermediate velocity\n$\\tilde\\uu$ and after then,\nwe compute $\\uu^\\new$ and $p$ in the steps II and III.\n\nThe time-development of $\\rho$ is given by the equation,\n\\begin{equation*}\n f^\\new = f^\\old + \\Delta t \\nabla ((A \\uu^\\old) f^\\old),\n\\end{equation*}\nand \n\\begin{equation*}\n\\rho = V(\\rho_{1} f + \\rho_{2} (1-f))\n\\end{equation*}\nfor the proper densities $\\rho_{a}$ of $\\xi_a$ $(a = 1, 2)$.\n\nEven for the case that we can deal with multi-phase flow with \nlarge density difference, we evaluate its time-development.\nPrecisely speaking,  when we evaluate $\\tilde \\uu$,\nfollowing the idea of Rudman \\cite{R} we \nemploy the momentum advection $\\tilde\\uu$ of $\\uu$,\n$$\n\\tilde\\uu:=\n   \\frac{1}{\\rho^\\new}[\\rho^\\old\\uu^\\old\n -\\Delta t(\\uu^\\old\\cdot \\nabla ) \\rho^\\old \\uu^\\old].\n$$\nOur derivation of the Euler equation shows that the Rudman's\nmethod is quite natural.\n\nFollowing the conventional notation, the \nguessed-value of the velocity is denoted by $\\uu^*$, which\nis the initial value for the steps in II and III.\nLet us define\n\\begin{equation*}\n\\uu^* := \\tilde\\uu + \n \\Delta t\\frac{1}{\\rho^\\new} \\KK(\\rho^\\old,f^\\old, \\uu^\\old).\n\\end{equation*}\nIn order to evaluate the guessed velocity,\nwe compute the force $\\KK$ from\n(\\ref{eq:Ki}) noting that \n$\\mathrm{div} \\tau$ and $\\mathrm{div} \\overline\\tau$ \nread\n$\\nabla A \\tau$ and   $\\nabla A \\overline\\tau$ respectively.\n\nFollowing the SMAC (Simplified-Marker-and-Cell) method \\cite{AH,Cho,Ch}, \nwe numerically determine the new velocity $\\uu^\\new$\nand the pressure $p$ in a certain boundary condition\nusing the preconditioned conjugate gradient method (PCGM):\n\\begin{equation*}\n\\begin{split}\n\\mbox{(IIa) Evaluate } p \\mbox{ using the PCGM}\n&:\\frac{1}{\\Delta t}\\nabla( A\\circ\\uu^*) \n = \\nabla A \\circ \\frac{1}{\\rho^\\new} \\nabla p, \\\\\n\\mbox{(IIb) By using }p \\mbox{ determine } \\uu^\\new\n&: \\uu^\\new = \\uu^* - \\Delta t \\frac{1}{\\rho^\\new} \\nabla p.\\\\\n\\end{split}\n\\end{equation*}\nMore precisely speaking, (III) $\\nabla(A\\circ \\uu^\\new)=0$ means\nthat we numerically solve the Poisson equation,\n$$\n\\nabla\\left(A \\Delta t \\frac{1}{\\rho^\\new} \\nabla p\\right)\n=\\nabla(A\\circ \\uu^*).\n$$\nThen we obtain $\\uu^\\new$, which obviously\nsatisfies (III) $\\nabla(A\\circ \\uu^\\new)=0$,\nwhich is known as the Hodge decomposition method \\cite{AH,Cho,CM} \nas mentioned in Remark \\ref{rmk:3-2}.\n\n\nFollowing the algorithm, we computed the two-phase flow with\na wall and triple junctions.\nWe illustrate two examples of the numerical\nsolutions of the triple junction problems as follows.\n\n\n\n\n\\bigskip \n\n\\subsection{Example 1}\n\\bigskip \n\nHere we show a computation of\na capillary problem, or the meniscus oscillation,\nin Figure 2.\nWe set two liquids in a parallel wall with the physical parameters;\n$\\eta_1 = \\eta_2 = 0.1$[cp],\n$\\rho_1 = \\rho_2 = 1.0$[pg\/$\\um^3$],\n$\\sigma_1 = 3.349$[pg\/$\\mu$sec${}^2$], \n$\\sigma_2 = 46.651$[pg\/$\\mu$sec${}^2$]. \n\n\nWe used $\\cL:=12[\\um] \\times 0.5[\\um]\\times 16 [\\um]$ lattice whose\nunit length $a$ is $0.125[\\um]$. \nThe first liquid exists in the down side and the second liquid does \nin the upper side in the region  $10[\\um] \\times 0.5[\\um]\\times 15 [\\um]$\nsurrounded by the wall and the boundaries with the boundary conditions.\nAs the boundary conditions,\nat the upper side from the bottom of the wall by $15[\\um]$,\nwe fix the constant pressure as $100$[KPa] and,\nalong $x^2$-direction, we set the periodic boundary condition.\n\nWe set $\\epsilon_{12} = \\epsilon_0 = 1$ mesh for the intermediate regions,\nat least, as its initial condition.\nEach time interval is 0.001 [$\\mu$sec]. \n\n\nAs the initial state, we start the state that\nthe fluid surface is flat as in Figure 2 (a)\nand the first liquid exists in the box region\n$10[\\um] \\times 0.5[\\um] \\times 7.0 [\\um]$, which\nis not stable.\nDue to the surface tension, it moves and starts\nto oscillate but due to viscosity, the oscillation decays.\nThough we did not impose the contact angle as a geometrical constraint,\nthe dynamics of the contact angle was calculated due to a balance between\nthe kinematic energy and the potential energy or the surface energy.\nThe oscillation converged to the stable shape with the proper contact\nangle, which is given by\n\\begin{equation}\n\\cos \\varphi = \\frac{\\sigma_2 - \\sigma_1} {\\sigma_2 + \\sigma_1}\n  \\equiv \\frac{\\sigma_{02} - \\sigma_{01}} {\\sigma_{12}}.\n\\label{eq:sigma_phi}\n\\end{equation}\nThe angle given by $\\sigma$'s are designed as $30$ [degree] whereas it\nin the numerical experiment in Figure 2 is a little bit\nlarger than $30$ [degree],\nthough it is very difficult to determine it precisely.\nHowever since we could tune the parameters $\\sigma$'s so that we obtain\nthe required state, our formulation is very practical.\n\nDue to the numerical diffusions and others, \nthe thickness of the intermediate regions changes\nin the time development and also depends on the positions of\nthe interfaces, even though it is fixed the same at the \ninitial state. However we consider that it is thin enough\nto evaluate the physical system since the contact angle\nis reasonably estimated.\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[scale=0.50,angle=270]{Fig2.ps}\n\\caption{ The meniscus oscillation:\nEach figure shows the time development.}\n\\label{fig:OS}\n\\end{center}\n\\end{figure}\n\n\\subsection{Example 2}\nThis example is on the computations of the contact angles for different \nsurface tension coefficients displayed in Figure 3.\n\nEven in this case, in order to see the difference between\n the designed contact angle and computed one,\nwe go on to handle two-dimensional symmetrical\nproblems though we used three-dimensional computational software. \nIn other words, we set that\n$x^2$-direction is periodic.\n\nSince the contact angle $\\varphi$ in our convention is given by \nthe formula (\\ref{eq:sigma_phi}).\nBy setting $\\sigma$'s\n$$\n\t\\frac{\\sigma_1}{\\sigma_2} =\n \\frac{1- \\cos\\varphi}{1+ \\cos\\varphi},\n$$\nfor given the contact angle $\\varphi$, we computed\nfive triple junction problems without any geometrical constraints;\neach $\\sigma$ is given in the caption in Figure 3.\nThe other physical parameters are\ngiven by $\\eta_1 = \\eta_2 = 0.1$[cp] \nand $\\rho_1 = \\rho_2 = 1.0$[pg\/$\\um^3$].\n\nIn this computation we used a $240\\times 4\\times 112$ lattice whose\nunit length $a$ is $0.125[\\um]$; $\\Omega= 30[\\um] \\times 0.5[\\um] \n\\times 14[\\um]$.\nWe set the flat layer as a wall\nby thickness $3[\\um]$ from the bottom of $\\Omega$ along the $z$-axis. \nAs the boundary conditions,\nat the upper side from the bottom of the wall by $9[\\um]$,\nwe fix the constant pressure as $100$[KPa].\n\nAs the initial state for each computation.\nwe set a semicylinder with radius $5[\\mu m]$\n in the flat wall like Figure 3 (d).\nWe also set $\\epsilon_{12} = \\epsilon_0 = 1$ mesh for the intermediate regions.\nEach time step also corresponds to 0.001 [sec]. \n\nDue to the viscosity, after time passes sufficiently $50[\\mu$sec], the\nstatic solutions were obtained as illustrated in Figure 3, which \nrecover the contact angles under our approximation within\ngood agreements.\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[scale=0.50,angle=270]{Fig3.ps}\n\\caption{ The different contact angles are illustrated due to \nthe different surface energy: By fixing $\\sigma_1 =1.0000$[pg\/$\\mu$sec${}^3$],\n(a): $\\varphi = 30$ [degree], $\\sigma_2 = 13.9282 $[pg\/$\\mu$sec${}^3$],\n(b): $\\varphi = 45$ [degree], $\\sigma_2 =  5.8284 $[pg\/$\\mu$sec${}^3$],\n(c): $\\varphi = 60$ [degree], $\\sigma_2 =  3.0000$[pg\/$\\mu$sec${}^3$], \n(d): $\\varphi = 90$ [degree], $\\sigma_2 =  1.0000$[pg\/$\\mu$sec${}^3$], \nand\n(e): $\\varphi = 120$ [degree], $\\sigma_2 = 0.3333$[pg\/$\\mu$sec${}^3$].} \n\\label{fig:CA}\n\\end{center}\n\\end{figure}\n\n\\bigskip \n\\bigskip \n\\section{Summary}\n\nBy exploring an incompressible fluid with a phase-field\ngeometrically \\cite{Ar,AK,EM,K,Ko,MW,NHK},\nwe reformulated the expression of the surface tension\nfor the two-phase flow found by \nLafaurie, Nardone, Scardovelli, Zaleski and Zanetti\n\\cite{LZZ} as a variational problem.\nWe reproduced the Euler equation of two-phase flow (\\ref{eq:Eulxi})\nfollowing the variational principle of the action integral\n(\\ref{eq:AI2d}) in Proposition \\ref{prop:4-6}.\n \nThe new formulation along the line of the variational principle\nenabled us to extend (\\ref{eq:Eulxi}) to that for the \n multi-phase ($N$-phase, $N\\ge2$) flow.\nBy extending (\\ref{eq:Eulxi}), we obtained the novel Euler equation \n(\\ref{eq:EeqM}) with the surface tension of the multi-phase fields\nin Theorem \\ref{th:5-2} from the action integral of\nTheorem \\ref{th:5-1}\nas the conservation of momentum in the sense of Noether's theorem.\nThe variational principle for the infinite dimensional system\nin the sense of References \\cite{Ar,AK,EM} gives the equation of motion\nof multi-phase flow controlled by the small parameter $\\epsX$\nwithout any geometrical\nconstraints and any difficulties for the singularities at\nmultiple junctions.\n\nFor the static case, we gave governing equations \n(\\ref{eq:ELM2}), (\\ref{eq:ELM3}) and (\\ref{eq:MSFe}) \nwhich generate  the\nlocally constant mean curvature surfaces with triple junctions\nby controlling a parameter $\\epsX$ to avoid these singularities.\nAs the solutions of (\\ref{eq:ELs}) has been studied well\nas the constant mean curvature surfaces  \nfor last two decades \\cite{ES,FW,GMO,T}, our extended equations \n(\\ref{eq:ELM2}), (\\ref{eq:ELM3}) and (\\ref{eq:MSFe}) \nmight shed new light on treatment of singularities of their extended\nsurfaces, or a set of locally constant mean \ncurvature surfaces.\n (Even though we need an interpretation of our scheme,\nfor example, \nit can be applied to a soap film problem with triple junction.) \nIt implies that our method might give a method of resolutions\nof singularities in the framework of analytic geometry.\n\n\nBy specifying the problem of the multi-phase flow\n to the contact angle  problems\nat triple junctions with a static wall,\nwe obtained the simpler Euler equation (\\ref{eq:Eeq3})\nin Theorem \\ref{th:5-3}.\nUsing the VOF method \\cite{H,HN},\nwe showed two examples of the numerical computations\nin Section \\ref{sec:VOF}.\nIn our computational method,\nfor given surface tension coefficients, \nthe contact angle is automatically generated\nby the surface tension\nwithout any geometrical constraints\n and any difficulties for the singularities at\ntriple junctions.  The computations were very stable.\nIt means that the computations did not\ncollapse nor behave wildly for every initial and the boundary\nconditions.\n\nIn our theoretical framework,\nwe have unified the infinite dimensional geometry\nor an incompressible fluid dynamics\ngoverned by $\\IFluid(\\Omega \\times T)$, and the $\\epsX$-parameterized\nlow dimensional geometry with singularities given by the multi-phase fields.\nWe obtained all of equations following the same \nvariational principle.\nWe naturally reproduced the Laplace equations, (\\ref{eq:ELs}) and\n(\\ref{eq:ELM2}), and obtained their generalizations\n(\\ref{eq:EL2d}), \n(\\ref{eq:ELM2}), (\\ref{eq:ELM3}),\n(\\ref{eq:ELM3d}) and (\\ref{eq:MSFe}),\nand the Euler equations,\n(\\ref{eq:Eulxi}),\n(\\ref{eq:EeqM}), and (\\ref{eq:Eeq3})\nin Proposition \\ref{prop:4-6} and Theorems \\ref{th:5-2} and  \\ref{th:5-3}.\nThese equations are derived\nfrom the same action integrals by choosing the physical parameters. \nIn the sense of References \\cite{Ar,AM,BGG}, it implies that\nwe gave geometrical interpretations of the multi-phase flow.\nEven though the phase-field model has\nthe artificial intermediate regions with unphysical thickness $\\epsX$, \nour theory supplies a model which shows how to evaluate their effects on\nthe  surface tension forces, from geometrical viewpoints.\nThe key fact of the model\nis that we express the low-dimensional geometry in terms\nof the infinite-dimensional vector spaces, or \n{\\it{global functions $\\xi$'s}}\nwhich have natural $\\Diff$ and $\\SDiff$ actions.\nThus\nwe can treat them in the framework of infinite dimensional\nLie group \\cite{AK,EM,O} to consider its Euler equation.\nIt is contrast to the level-set method;\nin analytic geometry and algebraic geometry, zeros of a function\nexpresses a geometrical object and thus the level-set method is \nso natural from the point of view. However as mentioned in Section\n\\ref{sec:two-one}, the level-set function cannot be a global functions \nas $\\cC^\\infty(\\Omega)$ and thus it is difficult to handle the method in the\nframework of the infinite dimensional Lie group $\\SDiff(\\Omega)$.\n\nAs our approach gives a resolution of the singularities\nby a parameter $\\epsX$,\nin future we will explore topology changes, geometrical objects\nwith singularities and so on, more concretely in our theoretical framework.\nWhen $\\epsX$ approaches to zero, we need more rigorous arguments in \nterms of hyperfunctions \\cite{KKK}\nbut we conjecture that our results would be correct for the \nvanishing limit of $\\epsX$ because the Heaviside function  is expressed by\n$\\displaystyle{\\theta(q) = \\lim_{\\epsX \\to 0} \\frac{1}{\\pi}\\tan^{-1}\n\\left(\\frac{q}{\\epsX}\\right)}$ in the Sato hyperfunction theory,\nwhich could be basically identified with $\\xi(q)$ of the finite $\\epsX$.\nSince an application of the Sato hyperfunction theory to fluid\ndynamics was reported by Imai on vortex layer and so on \\cite{II},\nwe believe that this approach might give another collaboration\nbetween pure mathematics and fluid mechanics.\n\n\\bigskip \n\\bigskip \n\nAcknowledgements:\n\nThis article is written by the authors in memory\nof their colleague, collaborator and leader Dr. Akira Asai\nwho led to develop this project.\nThe authors are also grateful to Mr. Katsuhiro Watanabe\nfor critical discussions and to the anonymous referee for\nhelpful and crucial comments.\n\n\\bigskip \n\\bigskip \n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\n\nFinancial support by the DFG is gratefully acknowledged. AUJL acknowledges financial support by the Swiss SNF and the\nNCCR Quantum Science and Technology.\nComputation time on the Cray XE6 system Hermit and\nthe NEC Nehalem cluster Laki at the HLRS, and the bwGRiD cluster\nare gratefully acknowledged.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMost massive stars are found in binary systems \\citep{Chini2012,Sana2012}.\nWhen the companion is an early type star, the collision of their stellar winds (wind-wind collision: WWC) \nproduces strong shocks and thermalizes gas to tens of millions of degrees Kelvin.\nThis hot gas emits X-rays, which are a good probe of the wind nature and interaction mechanism.\nThe shocks can also accelerate electrons to GeV energies, which produce radio synchrotron emission \\citep[e.g.,][]{Pittard2006}.\nThese non-thermal electrons are also suspected to up-scatter UV emission from the stars through the inverse-Compton process to X-ray (and higher) energies.\n\nEta Carinae \\citep[$d \\sim$2.3~kpc,][]{Smith2006b} is a nearby example of an extremely massive binary system with energetic WWC activity \\citep{Corcoran1997,Damineli1997,Ishibashi1999}.\nThe primary star is suspected to have had an initial mass of $\\gtrsim$100~\\UNITSOLARMASS\\ \\citep[see][]{Davidson1997,Hillier2001}\nand is currently in the poorly understood Luminous Blue Variable (LBV) stage.\nSince a series of eruptions between 1838$-$1890,\nthe two stars have been enshrouded by bipolar ejecta called the Homunculus Nebula (HN),\nbut their highly eccentric orbit ($e\\sim$0.9) with a period of 5.54 years can be measured from periodic variations at various wavelengths \\citep{Corcoran2005,Damineli2008}.\nThe companion star has not been detected directly, but it is believed to be an O supergiant or WN star \\citep{VernerE2005a}.\nThe primary star has a thick slow wind with $v_{wind} \\sim$420~\\UNITVEL\\ and \\Mdot\\ $\\sim$8.5$\\times$10$^{-4}$~\\UNITSOLARMASSYEAR\\ \\citep{Groh2012},\nwhile the secondary star has a thin fast wind with $v_{wind} \\sim$3000~\\UNITVEL\\ and \\Mdot\\ $\\sim$ 10$^{-5}$~\\UNITSOLARMASSYEAR\\ \\citep{Pittard2002}.\n\nThe WWC of $\\eta$~Car\\ produces luminous X-ray emission from hot plasma up to \\KT\\ $\\sim$4~keV, which has been observed mostly in the 2$-$10~keV band.\nThe emission increases inversely-proportional to the stellar separation, as suggested by WWC theory \\citep{Stevens1992}.\nHowever, the X-ray flux suddenly drops to a minimum level \\citep{Corcoran2010} after reaching a maximum brightness.\nDetailed studies \\citep{Hamaguchi2007b,Hamaguchi2014a} revealed two distinct phases during the X-ray minimum ---\nthe deep X-ray minimum, which has the lowest observed flux level and lasts approximately three weeks,\nand the shallow X-ray minimum, where the emission abruptly increases three-fold from the deep minimum level.\nThe deep minimum is probably produced by an eclipse of the WWC apex by the primary stellar body or wind, \nwhile the shallow minimum probably indicates the intrinsic decline of the WWC activity \\citep{Hamaguchi2014a}.\n\n\\input{tab1}\n\nThere have been several observations of $\\eta$~Car\\ in the hard X-ray band above 10~keV, up to $\\sim$100~keV.\n\\citet{Viotti2002,Viotti2004} claimed a detection of extremely hard X-ray emission from $\\eta$~Car\\ with the PDS instrument on \\SAX, \nbut the measured flux was significantly higher than those of later measurements,\nso source confusion in the wide PDS field of view ($\\sim$1.3$^{\\circ}$ FWHM) was suspected.\n\\citet{Leyder2008,Leyder2010} detected a flat power-law ($\\Gamma \\sim$1$-$2) source between $\\sim$20$-$100~keV with \\INTEGRAL\/ISGRI.\nThey constrained the source position to within 1.6\\ARCMIN\\ of $\\eta$~Car.\nSince they found no X-ray source in a \\CHANDRA\\ image consistent with the observed spectrum above 20~keV,\nthey identified the source as $\\eta$~Car.\n\\citet{Sekiguchi2009} analyzed the first two \\SUZAKU\\ \\citep{Mitsuda2007} observations of $\\eta$~Car\\ around apastron in 2005\nand detected X-ray emission between 15$-$40~keV with the HXD\/PIN instrument.\nThey showed that the spectrum below $\\sim$20~keV can be reproduced by \\KT\\ $\\sim$4~keV plasma emission observed below $\\sim$10~keV,\nwhile the spectrum above 10~keV requires a flat power-law of $\\Gamma \\sim$1.4.\nThese papers suggested that the power-law component may originate from the inverse-Compton up-scattering of stellar UV photons \nby non-thermal GeV electrons accelerated at the WWC region.\nOn the other hand, the \\AGILE\\ and \\FERMI\\ $\\gamma$-ray observatories discovered \na relatively stable $\\gamma-$ray source between 0.1$-$100~GeV \\citep{Tavani2009,Abdo2010},\nwhose spectrum may be connected to this extremely hard X-ray source \\citep{Farnier2011,Reitberger2012}.\n\nThe \\SUZAKU\\ observatory monitored $\\eta$~Car\\ 10 times between 2005$-$2011 and throughout one orbital cycle of $\\eta$~Car.\n\\SUZAKU\\ has the lowest background in the 15$-$40 keV band of any X-ray observatory launched before 2012, so that\nit gives the most reliable results on the orbital modulation of extremely hard X-ray emission from $\\eta$~Car.\nIt also has good sensitivity and spectral resolution between 5$-$9~keV, providing detailed profiles of\nthe $K\\alpha$ and $K\\beta$ line complexes of highly ionized Fe and Ni atoms.\nIn this paper, we present the flux and spectral variation of $\\eta$~Car\\ between 5$-$40 keV with orbital phase,\nfit all the spectra with a consistent model, and discuss the nature of the observed emission components.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{fig1.eps}\n\\caption{{\\it Left}: Composite XIS3 image of $\\eta$~Car\\ between 5$-$10~keV.\nThe solid blue circle at the center and the solid red regions are the source and background regions for the XIS analysis, respectively.\nThe dotted squares show the XIS {\\FOV}s of individual observations.\n{\\it Right}: Approximate HXD\/PIN {\\FOV}s overlaid on a mosaic \\XMM\/MOS image of the Carina nebula between 2$-$7~keV.\nThe inner and outer boxes are boundaries of half and zero photons to the PIN detector from an on-axis source, respectively.\n\\label{fig:obsimg}\n}\n\\end{center}\n\\end{figure*}\n\n\\section{Observations and Analysis}\n\n\\subsection{Observations}\n\n\\begin{figure}[h]\n\\plotone{fig2.eps}\n\\caption{\nXIS0+3 spectrum of $\\eta$~Car\\ during the deep minimum (SUZ$_{090125}$).\nThe best-fit models of the \\CHANDRA\\ deep minimum spectra of the HN ({\\it dash}, Hamaguchi et al. in prep.) and the CCE \\citep[{\\it dash-dot},][]{Hamaguchi2014a}, \nthe best-fit model of the residual by a thermal ({\\tt apec}) model ({\\it dot}),\nand the sum of these models ({\\it solid grey}) are shown for comparison.\n\\label{fig:deepmin_spec}\n}\n\\end{figure}\n\nSince its launch in 2005, \\SUZAKU\\  has observed $\\eta$~Car\\ fourteen times.\nTable~\\ref{tbl:obslogs} summarizes the former ten observations, which were performed before 2011.\nThe first two observations were performed during the performance verification (PV) phase and their earlier result is summarized in \\citet{Sekiguchi2009}.\nThe subsequent 8 observations were obtained through the guest observer program (AO-2, 3, 4, 6, PI: Kenji Hamaguchi).\nIndividual \\SUZAKU\\  observations are designated SUZ, subscripted with the year, month and day of the observation.\n\nDuring these observations, \\SUZAKU\\ ran two types of instruments:\nthe X-ray Imaging Spectrometer \\citep[XIS,][]{Koyama2006} in the focal plane of the thin-foil X-Ray Telescope \\citep[XRT,][]{Serlemitsos2007}\nand the Hard X-ray Detector \\citep[HXD,][]{Takahashi2007,Kokubun2007}.\nThe XIS consists of four X-ray CCD cameras, XIS0$-$3, three of which (XIS0, 2 and 3)\nuse front-illuminated (FI) CCD chips, while one (XIS1) uses a back-illuminated (BI) chip.\nThe FI chips have good hard X-ray sensitivity, covering $\\sim$0.5$-$10~keV,\nwhile the BI chip has good soft X-ray sensitivity down to $\\sim$0.3~keV.\nThe XIS2 was fatally damaged on 2006 Nov 9 by a mirco-meteorite, so this camera was unavailable after the 3rd observation (SUZ$_{070623}$).\nAnother micro-meteorite damaged one eighth of the XIS0 imaging area in 2009,\nwhile multiple micro-meteorites probably produced small holes on optical blocking filters of all the XISs,\nbut these did not significantly degrade the data quality.\nThe XISs initially had good spectral resolution (FWHM $\\sim$150~eV at 5.9~keV)\\footnote{http:\/\/heasarc.gsfc.nasa.gov\/docs\/astroe\/prop\\_tools\/suzaku\\_td\/node10.html}, \nbut the resolution has gradually degraded with age due to radiation damage,\nwith a substantial recovery in 2006 October after initiating the Spaced Charge Injection (SCI) operation\nwith a sacrifice of the effective imaging area.\nThe XRT has a butterfly-shaped point spread function (PSF) with half power diameter (HPD) of $\\sim$2$'$.\nThe effective area decreases as the off-axis angle increases, due to mirror vignetting.\nThe HXD consists of two types of detectors, the PIN with sensitivity between 15$-$70~keV and the GSO between 40$-$600~keV.\nThe GSO did not detect any significant signal above the non-X-ray background (NXB) level, so\nwe only used the PIN detector. The PIN detector has a collimator with a 34\\ARCMIN$\\times$34\\ARCMIN\\ \\FOV,\non the bottom of which are PIN Si diodes.\nThe depletion voltage to the diodes has been reduced gradually to mitigate the increase of detector noise,\nso that the detection efficiency has gradually decreased since launch.\n\nThe \\SUZAKU\\ point source observations have two default pointing positions --- the XIS nominal position,\nwhich puts the main target at the XRT+XIS focus, and the HXD nominal position, which maximizes the\nHXD collimator opening to the target.\nThe HXD nominal position is at 3.5\\ARCMIN\\ off-centered from the XIS nominal position.\nIn the PV observations, $\\eta$~Car\\ was placed at the XIS nominal position partly for instrument calibration.\nIn AO-2, 3 and 4, we placed $\\eta$~Car\\ at the HXD nominal position to maximize the HXD\/PIN sensitivity to the star.\nIn AO-6, we put again $\\eta$~Car\\ at the XIS nominal position \nbecause a failure of a spacecraft gyro began to affect the XIS flux measurement at the HXD nominal position.\nThe satellite roll angles during the AO observations were optimized within the operational constraints\nsuch that contamination from the nearby high energy sources AXP 1E~1048.1$-$5937 and IGR~J10447-6027 in the HXD\/PIN \\FOV\\ are minimized ($\\lesssim$1~\\%, see the right panel of Figure~\\ref{fig:obsimg}).\nOnly the HXD\/PIN observation in SUZ$_{050829}$ included 5\\% of the emission from 1E~1048.1$-$5937.\n\nAll the XIS observations were operated with the normal mode (no window option) \nbecause the count rates of $\\eta$~Car\\ for each XIS are $\\lesssim$7~\\UNITCPS, a factor of 2 below the threshold of significant photon pile-up.\nHowever, the XIS pileup estimator \\citep{Yamada2012} derived small pile-up for relatively high count rate observations\nsuch as SUZ$_{081210}$ ($\\sim$3\\% pileup at the \\PSF\\ core).\nBecause of this artificial effect,\nthe XIS spectra in SUZ$_{080610}$ and SUZ$_{081210}$ significantly flatten above $\\sim$9~keV.\nWe therefore excluded XIS spectra of these observations above 9~keV.\nThe XIS FI data had an anomaly at the first $\\sim$9~ksec of SUZ$_{070623}$, whose interval we did not analyze.\n\n\\begin{figure*}[hdtp]\n\\epsscale{1.5}\n\\plotone{fig3.eps}\n\\caption{{\\it Top}: \nXIS light curve of the WWC X-rays between 5$-$9~keV (triangle), compared to the \\RXTE\\ light curves between 2$-$10 keV \\citep[solid line,][]{Corcoran2010}.\nThe XIS0 and XIS3 count rates are normalized by the detector efficiency at SUZ$_{050829}$ and their count rates are averaged to derive the XIS count rate.\nThe fore- and back-ground emissions are estimated from the deep X-ray minimum data and subtracted, i.e.\\ the count rate during the deep minimum is zero.\n{\\it Bottom}: \nHXD\/PIN light curves between 15$-$25~keV (diamond) and 25$-$40 keV (open circle).\nThe 25-40 keV data points are slightly shifted to the right to show the error bars clearly.\nThe vertical bars show 1$\\sigma$ errors, including the PIN systematic uncertainty of 1.3\\%.\nThe dashed line shows the 25$-$40~keV flux of the \\INTEGRAL\\ point source in \\citet{Leyder2008}.\nIn both panels, black and grey colors show intervals between 2005$-$2010 July and after 2010 August, respectively.\n\\label{fig:obstiming}\n}\n\\end{figure*}\n\n\\input{tab2}\n\n\n\\subsection{Extraction of the WWC Emission Data}\n\nIn this paper, we analyze hard X-ray data above 5~keV to study the highest energy phenomena of $\\eta$~Car.\nFor consistent analysis, we use data only from the XIS0 and XIS3 among the XISs,\nboth of which are FI sensors running through the $\\eta$~Car\\ observations.\nWe used the HEASoft version 6.14 and the CALDB version hxd20110913, xis20130724, and xrt20110630 for the data calibration.\nThe left panel of Figure~\\ref{fig:obsimg} displays a 5$-$10~keV image from all the XIS3 data.\nThe brightest source at the center is $\\eta$~Car, the second brightest source to the west of $\\eta$~Car\\ is the Wolf-Rayet (WR) binary system WR~25,\nand the third to the north is the O4 star HD~93250.\nThe field includes more unresolved faint point sources, but no serendipitously bright X-ray source appeared during the observations in the XIS \\FOV.\n\nIn the XIS analysis,\nwe defined a source region with a 2.5\\ARCMIN\\ radius circle centered at $\\eta$~Car\\ to minimize contamination from WR~25 and HD~93250.\nThe source region includes $\\sim$90\\% of X-ray photons from the star.\nWe extracted the background from an annulus with a 5\\ARCMIN\\ inner radius and a 8\\ARCMIN\\ outer radius centered at $\\eta$~Car,\nexcluding areas within 3\\ARCMIN\\ from WR~25, HD~93250 and the centers of the X-ray clusters in \\citet{Feigelson2011}.\nThe source region includes hard X-ray sources other than the central point source,\nsuch as X-ray reflection at the HN \\citep{Corcoran2004} and multiple young stars \\citep[e.g.,][]{Wolk2011}.\n\n\\citet{Hamaguchi2014a} indicated that the WWC X-ray emission completely disappeared below 10~keV between 2009 Jan 12 and 28.\nThe SUZ$_{090125}$ observation was performed during this interval, so that the XIS spectrum should originate from the surrounding X-ray components.\nThe known hard X-ray components other than the WWC are the stable hot X-ray plasma, possibly in the foreground wind cavity \\citep[the CCE component,][]{Hamaguchi2007b,Hamaguchi2014a},\nwhich accounts for $\\sim$55\\% of the 5$-$10 keV emission, and X-ray reflection at the HN, which accounts for $\\sim$33\\% of the 5$-$10 keV emission (Figure~\\ref{fig:deepmin_spec}).\nThe remaining $\\sim$12\\% probably originates from hard X-ray point sources within the source region.\nThe CCE component, which can be measured only during the X-ray minimum, did not vary more than $\\sim$10\\%\nbetween 2003 and 2009 (Hamaguchi et al. in prep.), suggesting its stability over an orbital cycle.\nThe HN reflection emission is expected to decline by a factor of $\\sim$4 from periastron (around SUZ$_{090125}$) to apastron (Hamaguchi et al. in prep.).\nIn this paper, we assume that the XIS data at SUZ$_{090125}$ represent the contaminating emission in all the XIS spectra.\nHowever, this assumption significantly overestimates contribution of the fluorescent iron $K$ line from the HN to the apastron spectra\nsince the line flux around apastron is comparable to that at SUZ$_{090125}$.\nWe therefore defer the discussion of the fluorescent iron $K$ line emission to a later paper.\n\n\nThe HXD\/PIN data include significant contamination from NXB, point sources,\nGalactic Ridge X-ray Emission (GRXE) and cosmic ray background (CXB).\nThe NXB is estimated from the tuned background model with 1.3\\% systematic uncertainty (1$\\sigma$)\n(JX-ISAS-SUZAKU-MEMO-2007-09\\footnote{ftp:\/\/legacy.gsfc.nasa.gov\/suzaku\/data\/background\/pinnxb\\_ver2.0\\_tuned\/}).\nThe only high-energy point source that could contaminate the HXD\/PIN data is the AXP 1E~1048.1$-$5937.\nHowever, the HXD\/PIN count rate of 1E~1048.1$-$5937 on 2008 Nov.~30 (Obs ID: 403005010)\nthat excludes the NXB and the typical CXB spectrum was only 3.4$\\times$10$^{-3}$~\\UNITCPS (15$-$40~keV),\nwhich is $\\lesssim$1\/5 of the $\\eta$~Car\\ count rate.\nAn extrapolation of the XIS spectrum to the HXD band accounts for only one-fifth of the detected HXD count rate:\nthe rest probably originates from GRXE.\nIn addition, the HXD band flux should not increase by more than a factor of two at any \\SUZAKU\\ observation of $\\eta$~Car, \nconsidering a factor of $\\lesssim$4 variation of this AXP in the soft band since 1996 \\citep{Dib2014} and no strong color variation observed from the AXPs \\citep{Enoto2010}.\nFurthermore, $<$5\\% of this AXP emission contributes to the HXD\/PIN spectra of $\\eta$~Car.\nConsidering all these results, contamination of this AXP, 1E~1048.1$-$5937, of the HXD spectra of $\\eta$~Car\\ should be negligible.\nThe GRXE emission around $\\eta$~Car\\ is estimated at 1.4$\\times$10$^{-11}$~\\UNITFLUX deg$^{-2}$ between 3$-$20~keV \\cite[see section~5.2 in][]{Hamaguchi2007a}.\nThis is consistent with the remaining HXD\/PIN spectrum of 1E~1048.1$-$5937, assuming an absorbed thermal plasma model ({\\tt apec} $\\times$ {\\tt TBabs})\nwith \\KT = 10 keV and \\NH = 5.0$\\times$10$^{22}$~\\UNITNH.\nSince the GRXE emission is not expected to vary strongly in 30\\ARCMIN,\nwe use this spectrum as GRXE contamination in the HXD\/PIN spectra of $\\eta$~Car.\nThe CXB is estimated from the typical CXB emission \\citep{Boldt1987}, which may \nfluctuate by $\\lesssim$30\\% from region to region \\citep{Miyaji1998}.\nThe CXB\/GRXE contributions are estimated at 1.4$\\times$10$^{-2}$\/4.1$\\times$10$^{-3}$~\\UNITCPS\\ [15$-$25~keV] and \n3.7$\\times$10$^{-3}$\/4.8$\\times$10$^{-4}$~\\UNITCPS\\ [25$-$40~keV], while \nthe CCE and HN contribution should be negligible ($\\lesssim$10$^{-3}$~\\UNITCPS\\ [15$-$25~keV], $\\lesssim$10$^{-4}$~\\UNITCPS\\ [25$-$40~keV]).\nThe CXB and GRXE contribution is excluded from the PIN count rates and spectra as background.\nIn the light curve analysis, \nwe assume that the statistical noise errors are Gaussian and assume a systematic uncertainty of 1.3\\% in the HXD\/PIN NXB model.\n\nThe detector effective areas to $\\eta$~Car\\ varied by up to $\\sim$30\\% for the XIS and $\\sim$12\\% for the HXD\/PIN between the observations\nbecause of changes in observing conditions ---\nthe nominal pointing position, the XIS SCI operation and the sensitivity degradation of the PIN sensors.\nThe efficiency variation in the spectral analysis is automatically considered with spectral responses generated with {\\tt xisrmfgen} and {\\tt xisarfgen} in the HEAsoft tools for \nthe XIS and provided by the calibration team through the calibration database\\footnote{http:\/\/heasarc.gsfc.nasa.gov\/docs\/heasarc\/caldb\/suzaku\/} for the HXD.\nIn the light curve analysis, the average efficiency in a given energy band is calculated from the generated spectral response,\nand the count rates of each observation are normalized at the detector efficiency at SUZ$_{050829}$.\n\n\\begin{figure*}[t]\n\\epsscale{1.3}\n\\plotone{fig4a.eps}\n\\caption{XIS and HXD spectra of $\\eta$~Car\\ and the best-fit model in Tables~\\ref{tbl:specfit_variable}, \\ref{tbl:specfit_constant}.\nThe HXD spectra exclude expected contributions from CXB and GRXE, i.e.\\ both the XIS and HXD spectra should originate within 2.5\\ARCMIN\\ of $\\eta$~Car.\nThe red, blue, green and grey lines in each panel represent the WWC thermal component, the power-law component,\nthe stable foreground thermal component (i.e.\\ CCE, HN and surrounding point sources), and their total, respectively.\nEach bottom panel shows the residuals of the $\\chi^{2}$ fit.\n\\label{fig:obsspec}\n}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\figurenum{4}\n\\plotone{fig4b.eps}\n\\caption{Continued.\n\\label{fig:obsspec1}\n}\n\\end{figure*}\n\n\n\\section{Result}\n\n\\subsection{Light curve}\n\nFor each observation,\nwe produced an XIS light curve between 5$-$9~keV and an HXD\/PIN light curve between 15$-$25~keV with 500~sec time bins.\nThe XIS light curves at SUZ$_{050829}$, SUZ$_{081210}$, SUZ$_{090610}$, SUZ$_{091121}$, and SUZ$_{110724}$\nwith good photon statistics reject a constant model at a confidence limit above 95\\%.\nThese variations are apparently caused by small flux fluctuations on timescales of $\\lesssim$2~ksec and not by a systematic variation on long timescales,\nas seen in the \\XMM\\ observations in 2003 \\citep{Hamaguchi2007b}.\nThe HXD\/PIN light curves at SUZ$_{050829}$, SUZ$_{060203}$, and SUZ$_{070623}$ reject a constant model at above 95\\% confidence, \nbut they did not show any apparent long-term variation, either.\n\nWe measured from each observation the net XIS count rate between 5$-$9~keV \nand the net HXD\/PIN count rates between 15$-$25~keV and 25$-$40~keV (Table~\\ref{tbl:cnt_rate}).\nThe top panel of Figure~\\ref{fig:obstiming} shows the XIS light curve between 5$-$9~keV.\nIn this plot, we subtracted the XIS count rate in SUZ$_{090125}$ (0.05~\\UNITCPS) from \nXIS count rates of all the XIS observations as contamination from surrounding X-ray sources.\nWe also compare this to the \\RXTE\\ light curve after 2005 between 2$-$10~keV \\citep{Corcoran2010}.\nThe amount of contamination of X-ray sources in the \\RXTE\\ \\FOV\\ is also estimated from the \ndeep X-ray minimum observations \\citep[see details in][]{Hamaguchi2014a} and subtracted from the \\RXTE\\ light curve.\nThis means that both light curves should be of the WWC X-ray emission and directly comparable.\nTheir vertical axes are scaled such that the same height gives the same energy flux\nin the typical $\\eta$~Car\\ spectrum (\\KT\\ $\\sim$4.5~keV, \\NH\\ $\\sim$5$\\times$10$^{22}$~\\UNITNH\\ and $Z \\sim$0.8~solar).\nThese two light curves match very well.\n\nThe bottom panel of Figure~\\ref{fig:obstiming} shows the HXD\/PIN light curves between 15$-$25~keV and 25$-$40~keV.\nTheir vertical axes are scaled such that the data points at SUZ$_{050829}$ overlap.\nThe 15$-$25~keV light curve varied similarly to the 5$-$9~keV light curve outside the minimum: it increased gradually toward periastron.\nThis result indicates that the 15$-$25~keV emission has the same origin as the 5$-$10~keV emission, i.e.\\ the WWC X-rays.\nIt, however, varied differently during the minimum.\nIt declined only by a factor of 3 from SUZ$_{081210}$ at SUZ$_{090125}$ when the 5$-$9~keV flux dropped to zero.\nThe minimum observed flux occurred during the next observation (SUZ$_{090215}$) during the shallow minimum phase.\n\nOn the other hand, the 25$-$40~keV light curve did not show any significant variation near the X-ray maximum,\naccepting a constant flux model (reduced $\\chi^{2}=$ 1.11 for d.o.f. = 9).\nWe converted the 22$-$100~keV flux of the \\INTEGRAL\\ source \\citep[][0.15~\\UNITCPS]{Leyder2008} to the HXD\/PIN count rate \nbetween 25$-$40~keV, assuming a $\\Gamma =1.1$ power-law spectrum, and plotted it with a dotted line.\nThis flux level matches quite well all the data points except SUZ$_{050829}$ and SUZ$_{080610}$.\nThe result suggests that the 25$-$40~keV emission does not originate from the WWC thermal plasma, but from \nthe same component as the \\INTEGRAL\\ source.\n\nIn summary, the light curve analysis suggests two major X-ray emission components between 5$-$40~keV:\nstrongly variable emission below $\\sim$25~keV and stable emission above $\\sim$25~keV.\nThe former component probably corresponds to the WWC thermal plasma emission and the latter to the power-law component \n\\citep{Leyder2008,Sekiguchi2009,Leyder2010}.\n\n\\begin{figure*}[t]\n\\epsscale{1.5}\n\\plotone{fig5.eps}\n\\caption{\nXIS spectrum in SUZ$_{081210}$ fit by an absorbed CE plasma model ({\\tt apec}, {\\it top left}), \nan absorbed NEI plasma model ({\\tt nei}, {\\it top right})\nand an absorbed CE + NEI plasma model ({\\tt apec}+{\\tt nei}, {\\it bottom left}),\nand the \\CHANDRA\\ HETG spectrum obtained between 2008 December 8$-$13 overlaying the best-fit {\\tt apec}+{\\tt nei} model for the SUZ$_{081210}$ spectrum ({\\it bottom right}).\nThe \\CHANDRA\\ spectrum above $\\sim$7.75~keV is contaminated by the METG grating data and therefore is not extracted.\nThe model for the \\CHANDRA\\ spectrum includes the CCE component, but not the HN nor the surrounding point source components,\nwhich are outside of the \\CHANDRA\\ event extracting region.\n\\label{fig:xis_spec_issue}\n}\n\\end{figure*}\n\n\\subsection{Spectrum}\n\nFigures~\\ref{fig:obsspec} shows the XIS0+3 and HXD\/PIN spectra of all observations.\nThe XIS spectra varied as in the 2003 orbital cycle \\citep{Hamaguchi2007b};\nthe hard band slope between 7$-$10~keV, which reflects the hottest temperature of the WWC plasma,\ndid not vary significantly through the cycle.\nThe HXD spectra below 25~keV seem to connect smoothly to the XIS hard band.\nThe spectra above 25~keV do not show prominent features within the limited photon statistics.\n\nThe Helium-like iron line complex at $\\sim$6.7~keV distorted strongly toward the low energy side\naround periastron, as seen in the previous cycle \\citep{Hamaguchi2007b}.\nTo show this distortion clearly,\nwe first fit the XIS spectrum in SUZ$_{081210}$ above 5~keV by an absorbed 1$T$ collision equilibrium (CE) plasma model \n({\\tt apec}, {\\it top left} panel of Figure~\\ref{fig:xis_spec_issue}).\nThe spectrum also shows a fluorescent line from cold iron at 6.4 keV, for which we assume a narrow Gaussian line,\nbased on a \\CHANDRA\\ grating observation of $\\eta$~Car\\ around apastron in 2000 \\citep{Corcoran2001a}.\nWe also add another narrow Gaussian line for Fe K$\\beta$ fluorescence at 7.06~keV with the intensity tied to 12.2\\% of the Fe K$\\alpha$ line \\citep{Yamaguchi2014a}.\nThe spectrum also shows an iron $K$ absorption edge at 7.1~keV, for which the column density of cold iron (\\NFE) is varied independently\nfrom the hydrogen column density (\\NH).\nThe best-fit model has a strong excess at $\\sim$6.5~keV, for which the 6.4~keV line is overestimated to compensate --- \nthis result suggests more emission from lowly ionized iron.\nA marginal enhancement between Ni I $K\\alpha$ and Ni XXVII $K\\alpha$ lines also support the presence of lowly ionized nickel.\n\nHowever, the {\\tt nei} model, a non-equilibrium ionization (NEI) plasma model in {\\tt xspec} (NEIVERS version 3.0),\ndoes not reproduce the Hydrogen-like iron line at 6.9 keV in the spectrum ({\\it top right} panel of Figure~\\ref{fig:xis_spec_issue}).\nThis model still does not reproduce the excess at $\\sim$6.5~keV.\nThe {\\tt pshock} model, which considers plasma distribution in different ionization timescales in the plane parallel shock, gives a similar result.\nThus, there has to be a significant amount of CE plasma emission to reproduce the Hydrogen-like lines.\n\nWe, therefore, fit this spectrum by an absorbed {\\tt apec} plus {\\tt nei} model as a testbed.\nThe plasma temperatures of the {\\tt apec} and {\\tt nei} components cannot be independently determined and therefore are tied.\nTheir elemental abundances are also tied together.\nThe best-fit model reproduces the XIS spectrum well ({\\it bottom left} panel of Figure~\\ref{fig:xis_spec_issue}).\nThe excess at $\\sim$6.5~keV totally disappears, while the Hydrogen-lines are reproduced well.\nThis best-fit model also reproduces well the \\CHANDRA\\ HETG grating spectrum obtained quasi simultaneously\nto SUZ$_{081210}$ between 2009 Dec 8$-$13\n({\\it bottom right} panel of Figure~\\ref{fig:xis_spec_issue}, Observation ID: 10831, 8930, 10827, Total exposure: 74.6~ksec, PI: Corcoran, M.~F.).\nWith a factor of $\\sim$5 better spectral resolution than the XISs, the NEI component is clearly seen as a red wing of the Helium-like Fe $K\\alpha$ line.\nThe residual at the blue side of the Helium-like Fe $K\\alpha$ line, which cannot be resolved with the CCD resolution,\ncan be reproduced by a Doppler broadening of $\\Delta v\\sim$800~\\UNITVEL.\n\nWe therefore use this model for all the spectra but SUZ$_{090125}$, which does not show WWC emission below 10~keV.\nThe spectra outside of the X-ray minimum and maximum (SUZ$_{050829}$, SUZ$_{060203}$, SUZ$_{070623}$, SUZ$_{091121}$ and SUZ$_{110724}$) \ndo not show clear distortions in the Fe $K\\alpha$ line but small excesses at $\\sim$6.5~keV, so that we tie their ionization parameters.\nThe elemental abundances of all the emission components are tied together and the abundance ratios between elements\nabove Helium are fixed at the values of the {\\tt aspl} solar abundance model \\citep{Asplund2009} in the XSPEC modeling.\nThis is a reasonable approach because the hot plasma is heated by the secondary stellar winds and so is expected to \nreflect the elemental abundance of the secondary star, which has an unknown evolutionary status.\nWe also assume the elemental abundance of the absorber to be solar except for iron.\nThe absorbing material should originate from the primary star, which is depleted in hydrogen, carbon and oxygen but rich in nitrogen.\nHowever, these elements do not affect the spectral structure above 5~keV.\nThe other high-$Z$ elements are considered to be solar \\citep{Hillier2001}.\nThe absolute \\NH\\ depends on the hydrogen abundance, but it can be easily adjusted for any abundance models later.\n\nTo this variable WWC model, we add an absorbed power-law model for the extremely hard X-ray component above $\\sim$25~keV.\nThe HXD\/PIN spectra between 25$-$40 keV do not have enough statistics to determine the power-law index and normalization, individually.\nWe therefore fixed the power-law index at 1.4.\nThe result does not change substantially for power law indices in the range 1.0$-$1.8.\nSince the 25$-$40~keV light curve did not show significant flux variations,\nwe tied the normalization of the power-law component between observations.\n\\citet{Leyder2010} constrained the position of this power-law source to within 1.6\\ARCMIN\\ from $\\eta$~Car\\ from the \\INTEGRAL\\ observations.\nSince the XIS source region is within 2.5\\ARCMIN\\ from $\\eta$~Car, the XIS spectra should also include this power-law source.\nA simple extrapolation of this $\\Gamma\\sim$1.0$-$1.8 power-law spectrum should show significant emission below $\\sim$10~keV during the deep X-ray minimum,\nbut neither the XIS spectrum at SUZ$_{090125}$, nor any \\CHANDRA, nor \\XMM\\ spectra during the minimum suggest the presence of this power-law source \\citep{Hamaguchi2014a}.\n\\citet{Leyder2010} did not find any promising candidate of this counterpart other than $\\eta$~Car\\ in a \\CHANDRA\\ image, either.\nThis means that this power-law component is heavily absorbed, at least during the X-ray minimum, and cannot be seen below 10~keV.\nThe absorption to this power-law component cannot be constrained outside the X-ray minimum.\nWe, therefore, assume a constant absorption to this power-law component through the orbital cycle.\n\n\\input{tab3}\n\\input{tab4}\n\nThe 15$-$25~keV flux at SUZ$_{090125}$ is too high for either the thermal component seen in the XIS band \nor the power-law component above 25~keV.\nWe therefore assume the excess as the deeply embedded WWC emission and reproduce it with the model for the WWC thermal component.\nSince the statistics are limited, we fixed the plasma temperature at 4~keV, the typical temperature of the WWC plasma outside of the X-ray minima.\nWe also tied \\NH\\ and \\NFE\\ of SUZ$_{090125}$ because the Fe $K$ absorption edge cannot be measured.\nTo all the spectral models, we add the best-fit model of the XIS spectrum in SUZ$_{090125}$ \nto account for emission from the CCE, the HN and the surrounding point sources.\n\nIn this model fit, the model normalizations for the HXD\/PIN spectra were multiplied by 1.15 for the XIS nominal pointing observations\nand 1.19 for the HXD nominal pointing observations,\naccording to the Suzaku Data Reduction Guide\\footnote{http:\/\/heasarc.gsfc.nasa.gov\/docs\/suzaku\/analysis\/abc\/}.\n\nThe best-fit model reproduced all the spectra very well (reduced $\\chi^2$ =1.08, d.o.f =3437, Tables~\\ref{tbl:specfit_variable}, \\ref{tbl:specfit_constant} and Figure~\\ref{fig:obsspec}).\nThe hottest plasma temperatures of the thermal WWC component are stable at \\KT\\ $\\sim$4~keV outside the minimum\nand the HXD\/PIN spectra showed no signature of hotter plasma.\nThe fit to the SUZ$_{090215}$ spectrum resulted in a low plasma temperature of $\\sim$2~keV, similarly to spectral fits\nto the shallow minimum spectra in 2003 \\citep{Hamaguchi2007b}.\nHowever, \\KT\\ is degenerate with \\NH\\ in fits to strongly absorbed spectra,\nso that this variation may not suggest an actual decline in temperature of the hottest plasma.\nThe elemental abundance of the plasma is close to solar.\nThe \\NH\\ goes down to zero for SUZ$_{060203}$ and SUZ$_{110724}$.\nHowever, the extrapolations of these models significantly overestimate the spectra below 5~keV;\nthe soft band spectra suggest higher \\NH~$\\sim$3$-$5$\\times$10$^{22}$~\\UNITNH.\nThe \\NH\\ probably does not correctly represent the absorption column to the hot plasma \nbecause of lower temperature plasma emission important around 5~keV.\nOn the other hand, \\NFE\\ is determined from the iron edge feature and is therefore a more reliable estimator of the absorption to the hot plasma.\nThe \\NFE\\ increased toward the deep minimum and reached the maximum of $\\approx$8$\\times$10$^{24}$~\\UNITNH\\ at SUZ$_{090125}$.\nThe power-law component also required a very high value of \\NH~$\\sim$2$\\times$10$^{24}$~\\UNITNH.\n\n\\section{Discussion}\n\n\\subsection{Orbital Modulation of the Physical Parameters}\n\n\\SUZAKU\\ sampled a whole orbital cycle of $\\eta$~Car\\ between 2005 and 2011.\nThough the X-ray minimum in 2009 was significantly shorter than those in previous cycles \\citep{Corcoran2010},\nthe latest \\SUZAKU\\ spectrum was very similar to that in the previous cycle, again suggesting a cyclic variation.\n\nThe combined fit to the XIS and HXD\/PIN spectra confirmed that the hottest temperatures of the WWC plasma\nare stable at \\KT~$\\sim$4$-$5~keV through the orbit outside the X-ray minimum (Figure~\\ref{fig:spec_param} {\\it top}).\nThe same conclusion was deduced by \\citet{Ishibashi1999} using the \\RXTE\\ data obtained between 1996 April and 1998 October\nand \\citet{Hamaguchi2007b} using the \\XMM\\ and \\CHANDRA\\ data obtained between 2000 July and 2003 September.\nThe new \\SUZAKU\\ result is important in two ways.\nThe set of observations has a long baseline between 2005 August and 2011 July and samples the whole orbital cycle.\nThe HXD\/PIN provided the best quality measure of the extremely high energy spectra above 15~keV with a smaller \\FOV\\ and\nlower background than the \\RXTE\\ Proportional Counter Array (PCA).\nThe wide band coverage realized with the HXD provided a better measurement\nof the continuum slope in the very high energy range, and therefore increased the sensitivity to the hotter temperature plasma.\nThe \\SUZAKU\\ results show that the plasma temperature does not change prominently outside the minimum,\nin line with predictions by WWC theories.\n\nThe distortion of the Helium-like Fe $K\\alpha$ line was first recognized during the X-ray minimum in 2003\nand discussed as caused by the NEI effect \\citep{Hamaguchi2007b}.\nThe new \\SUZAKU\\ result demonstrates that an NEI plasma with $\\tau~\\sim$0.6$-$1.3$\\times$10$^{11}$~cm$^{-3}$~s$^{-1}$\ncan reproduce this distortion in spectral fits and \nthis NEI plasma component can be present through the orbital cycle, with an increased ratio around periastron.\nThe {\\tt pshock} model, which considers the ionization timescale distribution in a plane parallel shock,\nfails to reproduce the whole He-like Fe $K\\alpha$ line profile.\nThis result suggests that the plasma really has two peaks in the ionization timescale distribution at\n$\\sim$10$^{11}$~cm$^{-3}$~s$^{-1}$ (the NEI component) and above $\\sim$10$^{12}$~cm$^{-3}$~s$^{-1}$ (the CE component).\nThis result probably means that the NEI plasma heats up in $\\sim$1000 (10$^{8}$~\\UNITPPCC$\/n$) sec, where $n$ is the plasma density,\nand then quickly cools down without reaching the thermal equilibrium.\nThe \\EM\\ ratio of the NEI plasma increases from $\\sim$25\\% around apastron to $\\sim$75\\% around periastron (Figure~\\ref{fig:spec_param} {\\it middle}).\nThe ratio is high with a large uncertainty during the shallow minimum (SUZ$_{090215}$), \nbut it is also high after the recovery (SUZ$_{090610}$) as well.\nThis result does not suggest that the ratio is correlated with the X-ray luminosity.\nThe SUZ$_{090610}$ observation is when the WWC apex is wrapped inside the primary wind, and therefore in a high density environment.\nThe NEI plasma may quickly contact the thick cool primary wind and cool down, while the CE plasma may rapidly leave the system in the\npart of the shocked secondary wind which does not come into direct contact with the higher density primary wind.\n\nThe absorption column density of cold iron (\\NFE) increases by a factor of 2$-$4 toward periastron (Figure~\\ref{fig:spec_param} {\\it bottom}).\nThis variation is very similar to the \\NH\\ variation measured from hard band spectra above 5 keV in 2003 \\citep{Hamaguchi2007b}; \nthe absorption to the WWC apex varies periodically as well.\nInterestingly, the lowest \\NFE\\ observed around apastron is still a factor of 2$-$3 higher than absorption to soft X-rays ($\\sim$4$-$5$\\times$10$^{22}$~\\UNITNH),\nthough the WWC apex should be seen through the thin secondary wind.\nSince there is no evidence of iron overabundance in $\\eta$~Car\\ \\citep[e.g.,][]{Hamaguchi2007b,Hillier2001},\nthis perhaps could be explained if the secondary wind piles up over the WWC contact surface.\n\n\\begin{figure*}[t]\n\\plotone{fig6.eps}\n\\caption{Variations of the selected spectral parameters in the best-fit model \n({\\it Top}: plasma temperature, {\\it Middle}: emission measure ratio of the NEI ({\\tt nei}) plasma,\n{\\it Bottom}: absorption column density of cold iron in units of an equivalent hydrogen column density\nin the case of solar abundance matter.\nThe open circle at the bottom panel is of SUZ$_{090125}$, measured from the soft band cut-off.\nThe vertical bars depict 90\\% error ranges.\n\\label{fig:spec_param}\n}\n\\end{figure*}\n\n\\subsection{High 15$-$25~keV Flux during the Deep X-ray Minimum}\n\nThe relatively strong 15$-$25~keV emission at SUZ$_{090125}$ can be reproduced by the WWC emission\nviewed through extremely high photoelectric absorption (\\NH\\ $\\approx$8$\\times$10$^{24}$~\\UNITNH).\nThe \\CHANDRA\\ spectrum obtained at the end of the X-ray eclipse on 2009 Feb 3 also suggested \na very high \\NH\\ of $\\sim$10$^{24}$~\\UNITNH\\ \\citep{Hamaguchi2014a}, \nand the WWC apex should be more embedded at the middle of the X-ray eclipse.\nA peak \\NH\\ of $\\sim$several$\\times$10$^{24}$~\\UNITNH\\ during periastron is also suggested by simulations of WWC X-ray emission from $\\eta$~Car\\ \n\\citep[][Russell et al.\\ in preparation]{Parkin2011}.\nThe large \\NH\\ at SUZ$_{090125}$ is consistent with the picture that \nthe WWC X-ray emission peered through very thick intervening material that totally blocked X-ray emission below 10 keV.\nIf this interpretation is correct,\nthe intervening material would be the inner primary wind, and not the primary stellar body.\n\nIn this interpretation, the WWC activity during the deep X-ray minimum is still strong behind the absorber.\nThe \\EM\\ at SUZ$_{090125}$ is as large as that at the maximum in SUZ$_{081210}$.\nThis is consistent with the WWC theory, in which the luminosity is inversely proportional to the distance between the two stars.\nHowever, the Compton scattering process becomes important at this absorption column.\nSince emission scattered off the line of sight may end up reaching us after another scattering,\nthe amount of attenuation by the Compton scattering also depends on the shape of the surrounding \nintervening material.\nA broad-band spectrum above 10~keV at this phase with good photon statistics is required to correctly \nmeasure the amount of the Compton scattering and hence the intrinsic luminosity.\n\n\n\\subsection{Origin of the Power-law Component}\nThe PIN count rates between 25$-$40~keV did not vary strongly.\nThe $\\gamma$-ray source, 1FGL~J1045.2-5942, detected by the \\FERMI\\ $\\gamma$-ray observatory also only varied by a factor of $\\lesssim$2\nincluding the X-ray minimum, with a possible weak decline after the recovery \\citep{Abdo2010,Farnier2011,Reitberger2012}.\nThis result strengthens the hypothesis \nthat the 25$-$40~keV power-law source is connected to the \\FERMI\\ $\\gamma$-ray source \\citep{Leyder2008,Leyder2010,Abdo2010}.\n\nIn this interpretation, the power-law component originates from emission up-scattered by GeV particles accelerated at the WWC region.\nHowever, our results show that this power-law component does not change significantly around the maximum when the WWC head-on\ncollision is the strongest and around shallow minimum when emission near the WWC apex apparently shuts off \\citep[][and additional references therein]{Hamaguchi2014a}.\nOur result does not suggest that the power-law component does not originate near the WWC stagnation point.\nIt remains to be seen whether the power-law component can be reproduced by models of particle acceleration in the WWC.\n\nTo satisfy low X-ray flux below 10~keV of $\\eta$~Car\\ during the deep X-ray minimum,\nthe power-law source should suffer extremely strong absorption of \\NH~$\\approx$2$\\times$10$^{24}$~\\UNITNH.\nThis high \\NH\\ does not favor the foreground shock region such as the CCE plasma cavity nor the HN lobe,\nwhose extinctions are less than \\NH~$\\sim$5$\\times$10$^{22}$~\\UNITNH.\nOne obvious but less interesting hypothesis is an unrelated neutron star or an active galactic nuclei behind $\\eta$~Car,\nthough the chance of this coincidence is not high.\nA provocative but more interesting hypothesis may be the presence of an active compact object associated with the binary system,\nwhich was once bound to the binary system but ejected by events such as the 1840 eruption. \nThe flat power-law spectrum is similar to those of high-mass X-ray binaries, \nand the luminosity is within the range of systems with wind-fed accretion.\nThe presence of a compact object in the system would require a progenitor with an initial mass greater than the initial mass of \n$\\eta$~Car, i.e. $>$150~\\UNITSOLARMASS.  \nThe evolution of such a massive progenitor would probably result in the creation of a black hole.\n\n\\section{Conclusion}\n\nWe analyzed datasets of the 10 \\SUZAKU\\ observations of the super massive star $\\eta$~Car\\ and studied\nthe variation of the extremely hard X-ray emission above 15~keV through the orbital cycle for the first time.\nOur study suggests that the 15$-$25~keV emission originates in the tail of the thermal emission seen below 10~keV,\nwhile the emission above 25~keV is the power-law component observed with \\INTEGRAL.\nThe origin of the power-law component is mysterious.\nThe $K\\alpha$ and $K\\beta$ lines of Fe and Ni ions need emission from both CE and NEI plasmas.\nThe NEI plasma ratio increases toward periastron; this result may suggest an increase of gas density around the WWC apex around periastron.\nIn the summer of 2014, another X-ray observing campaign for the latest periastron passage is being performed with\nmultiple X-ray observatories including \\NUS, which provides focussed imaging up to 80 keV.\nThese observations should provide the best measure of the presence of the deeply embedded X-ray component and the power-law component\nand their spectral properties (e.g., \\NH, \\KT, $\\Gamma$).\nThey should help understand the nature of the WWC emission around periastron and the mysterious power-law source.\n\n\\acknowledgments\n\nThis research has made use of data obtained from the High Energy Astrophysics Science Archive\nResearch Center (HEASARC), provided by NASA's Goddard Space Flight Center.\nThis research has made use of NASA's Astrophysics Data System Bibliographic Services.\nWe appreciate the \\SUZAKU\\ operations team for optimizing the observations, \nand Masahiro Tsujimoto, Keith Arnaud and Adam Foster for suggestions on the XIS data analysis and the appropriate NEI model.\n\nFacilities: \\facility{Suzaku (XIS,HXD)}, \\facility{RXTE (PCA)}, \\facility{CHANDRA (HETG)}\n\n\\input{ms_apjstyle.bbl}\n\n\\end{document}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Linear $k_\\perp$ factorization is broken}\n\nHeavy nuclei are strongly absorbing targets and bring a new scale into\nthe perturbative QCD (pQCD) description of hard processes \\cite{Mueller}.\nThis has severe consequences for the relations between various \nhard scattering observables. In a regime of small absorption,\nsmall--$x$ processes are adequatly described by the \nlinear $k_\\perp$--factorization, and the pertinent observables\nare linear functionals of a universal unintegrated gluon distribution.\nNot so for the strongly absorbing target, where the linear \n$k_\\perp$--factorization is broken \\cite{DIS_Dijets}, \nand has to be replaced by a new, nonlinear \n$k_\\perp$--factorization \\cite{DIS_Dijets,Nonuniversality}, \nwhere observables are in general\nnonlinear functionals of a properly defined unintegrated glue.\nThe relevant formalism has been worked out for all interesting \nprocesses \\cite{Nonuniversality,Nonlinear} (see \\cite{CGC} \nfor references on related work), but in this \nvery short contribution we concentrate on deep\ninelastic scattering (DIS). Here, in the typical inelastic DIS event \nthe nuclear debris will be left in a state with \nmultiple color excited nucleons after the $q\\bar q$ dipole exchanged\nmany gluons with the target. The partial cross sections for final states \nwith a fixed number of color excited nucleons are the\ntopological cross sections. It is customary to describe \nthem in a language of unitarity cuts through\nmultipomeron exchange diagrams \\cite{AGK}.  \nIn our approach \\cite{Cutting_Rules}, color excited \nnucleons in the final state give a clear--cut definition of a cut pomeron.\nTopological cross sections carry useful information on the correlation between \nforward or midrapidity\njet\/dijet production and multiproduction in the nuclear fragmentation region\nas well as on the centrality of a collision.\n\n\\section{Nuclear collective glue and its unitarity cut interpretation}\n\nThe basic ingredient of the nonlinear $k_\\perp$--factorization is\nthe collective nuclear unintegrated glue, which made its first appearance\nin our work on the diffractive breakup of pions into jets $\\pi A \\to \n\\mathrm{jet}_1 \\mathrm{jet}_2 A$ \\cite{NSS}.\nIndeed, in the high energy limit, the nearly back--to--back\njets acquire their large transverse momenta directly from gluons.\nIt is then natural use the diffractive $S$--matrix of a $q \\bar q$--dipole\n$S_A({\\bf{b}},x,{\\bf{r}})$ for defining the nuclear unintegrated glue:\n\\begin{eqnarray}\n\\int {d^2 {\\bf{r}} \\over (2 \\pi)^2} \\, \nS_A({\\bf{b}},x,{\\bf{r}}) \\exp(-i{\\bf{p}} {\\bf{r}}) =\nS_A({\\bf{b}},x,{\\bf{r}} \\to \\infty) \\delta^{(2)}({\\bf{p}}) + \\phi({\\bf{b}},x, {\\bf{p}})\n\\equiv \\Phi({\\bf{b}},x,{\\bf{p}}) \\, .\n\\end{eqnarray}\nNotice that it resums multiple scatterings of a dipole, so that there \nis no straightforward relation to the conventional parton distribution\nwhich corresponds to just two partons in the $t$--channel.\nIt is still meaningful to call it an unintegrated glue \n-- one reason was given above -- another one, besides its role in \nfactorization formulas is its small-$x$ evolution property:\nThe so--defined $\\phi({\\bf{b}},x,{\\bf{p}})$ can be shown \\cite{NS_LPM} \nto obey \\footnote{Strictly speaking\nonly a few iterations of this equation make good sense.} \nthe Balitskii--Kovchegov \\cite{BK} evolution equation. \n\nClose to $x_A \\sim (m_N R_A)^{-1}$, for heavy nuclei, the dipole $S$--matrix is the familiar Glauber--Gribov exponential\n$S_A({\\bf{b}},x_A,{\\bf{r}}) = \\exp[-\\sigma(x_A,{\\bf{r}})T({\\bf{b}})\/2]$; for large dipole sizes it \ncan be expressed as $S_A({\\bf{b}},x_A,{\\bf{r}} \\to \\infty) = \\exp[-\\nu_A(x_A,{\\bf{b}})]$.\nHere the nuclear opacity \n$\\nu_A(x_A,{\\bf{b}}) = {1 \\over 2 } \\sigma_0(x_A) T({\\bf{b}})$, is\ngiven in terms of the dipole cross section for large dipoles \n$\\sigma_0(x) = \\sigma(x_A,{\\bf{r}} \\to \\infty)$.\nIn momentum space, a useful expansion is in terms of multiple convolutions\nof the free--nucleon unintegrated glue\n(we use a notation \n$f(x,{\\bf{p}}) \\propto {\\bf{p}}^{-4} \\partial G(x,{\\bf{p}}^2)\/ \\partial \\log({\\bf{p}}^2)$):\n\\begin{eqnarray}\n\\phi({\\bf{b}},x_A,{\\bf{p}}) = \\sum_{j \\geq 1}  w_j \\big(\\nu_A(x_A,{\\bf{b}})\\big)\n f^{(j)}(x_A,{\\bf{p}}) \\, . \n\\label{Nuc_glue}\n\\end{eqnarray}\nHere \n\\begin{equation}\nw_j(x_A,\\nu_A) = {\\nu_A^j(x_A,{\\bf{b}}) \\over j! } \\exp[-\\nu_A(x_A,{\\bf{b}})]\n, \\,\nf^{(j)} (x_A,{\\bf{p}}) = \\displaystyle \\int \\big[ \\prod^j d^2 {\\bf{\\kappa}}_i \nf(x_A,{\\bf{\\kappa}}_i) \\big] \\delta^{(2)}( {\\bf{p}} - \\sum \\kappa_i)   \\, . \n\\end{equation}\n\nCuriously, the very same collective nuclear glue is proportional to the\nspectrum of quasielastically scattered quarks:\n\\begin{equation}\n{d\\sigma(qA\\to qX) \\over d^2 {\\bf{b}} d^2 {\\bf{p}}} \n\\propto \\phi({\\bf{b}},x_A,{\\bf{p}}) \\, .\n\\end{equation}\n\nNow, we can state the {\\bf{first unitarity cutting rule}} in momentum space:\nthe $k$--th order term in the expansion (\\ref{Nuc_glue}) \ncorresponds to the topological cross section for the quark--nucleus \nscattering with $k$ color excited nucleons in the final state:\n\\begin{equation}\n{d \\sigma^{(k)} (qA \\to qX) \\over d^2 {\\bf{b}} d^2 {\\bf{p}}} \\propto \nw_k \\big(\\nu_A({\\bf{b}}) \\big) f^{(k)} ({\\bf{p}})  \\, .\n\\end{equation}\nThis simple substitution rule forms at the heart of the cutting\nrules applied to the nonlinear quadratures of \\cite{Nonlinear}.\n\n\\section{Standard AGK vs. QCD}\n\nGiven the close relation between the nuclear unintegrated glue and\nthe Glauber--Gribov scattering theory from color dipoles, one may\nbe tempted to play around with various expansions of the exponential. \nTaking\ninspiration from 1970's hadronic models one may then 'derive' \nexpressions for topological cross sections.\nFor example, the inelastic cross section of the \n$q \\bar{q}$-dipole-nucleus interaction is certainly obtained from:\n\\begin{eqnarray}\n\\Gamma^{inel} ({\\bf{b}}, {\\bf{r}}) &=& 1 - \\exp[- \\sigma({\\bf{r}}) T({\\bf{b}})] \n= \\sum_k \\Gamma^{(k)}({\\bf{b}},{\\bf{r}}) \\, ,\n\\end{eqnarray}\nand $\\Gamma^{(k)}({\\bf{b}},{\\bf{r}}) = \\exp[-\\sigma({\\bf{r}}) T({\\bf{b}})]\n(\\sigma({\\bf{r}}) T({\\bf{b}}))^k\/k!$ is then interpreted as the $k$--cut Pomeron\ntopological cross section. This is entirely incorrect, \nthe reason is that this result neglects the color--coupled channel\nstructure of the intranuclear evolution of the color dipole.\nInterestingly, a simple closed expression can be obtained\nwith full account for color \\cite{Cutting_Rules}:\n\\begin{eqnarray}\n\\Gamma^{(k)}({\\bf{b}},{\\bf{r}}) = \\sigma({\\bf{r}}) T({\\bf{b}}) \\, w_{k-1}(2 \\nu_A({\\bf{b}}) )\n {e^{-2 \\nu_A({\\bf{b}})} \\over \\lambda^k} \\gamma(k,\\lambda) \n\\, ,\n\\nonumber\n\\end{eqnarray}\nwhere $ \\lambda = 2 \\nu_A({\\bf{b}}) - \\sigma({\\bf{r}}) T({\\bf{b}})$, \nand $\\gamma(k,x)$ is an incomplete Gamma--function.\nFor a more quantitative comparison, consult fig 1. We see that\nthe standard Glauber--AGK predicts a strong hierarchy: $k$ cuts \nare suppressed by the $k$--th power of the dipole cross section.\nIn the QCD--cutting rules there is an additional dimensionful parameter,\nthe opacity of a nucleus for large dipoles $\\nu_A$, \nand the distribution over $k$ is substantially broader.\nThis difference will be more dramatic the smaller the dipole and\nreflects itself in the predicted $Q^2$--dependence of \nDIS structure functions with fixed multiplicity of cut Pomerons. More\nfigures, as well as another example for the failure of standard AGK,\ncan be found on the conference website.\n\n\\section{Conclusions}\nTopological cross sections can be obtained from nonlinear $k_\\perp$ \nfactorization formulas by straightforward substitution (cutting) rules.\nFor a correct isolation of topological cross sections a careful\ntreatment of the color coupled channel properties of the color(!) dipole\nintranuclear evolution is mandatory. Don't be misguided by \nsimple formulas derived in a single channel context, or by a too\nliteral analogy between color transparency and the Chudakov--Perkins\nsuppression of multiple ionisation by small size $e^+ e^-$ pairs in QED.\n \n\n\\begin{figure}\n  \\includegraphics[height=.4\\textheight, angle=270]{Gl_larger2_proc}\n  \\includegraphics[height=.4\\textheight, angle=270]{QCD_larger2_proc}\n  \\caption{{\\bf{Left}}: the profile function for $k$ cut Pomerons according to standard Glauber--AGK for a fairly large dipole $r = 0.6$ fm at $x=0.01$ for $A=208$. {\\bf{Right}}: the same for the QCD cutting rules.}\n\\end{figure}\n\n\\begin{theacknowledgments}\n It is a pleasure to thank the organizers for the kind invitation.\nThis work was partially supported by the Polish Ministry for Science\nand Higher Education (MNiSW) under contract 1916\/B\/H03\/2008\/34.\n\\end{theacknowledgments}\n\n\n\n\\bibliographystyle{aipproc}  \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\\input{introduction.tex}\n\n\\section{Related Work}\n\\label{sec:relatedwork}\n\\input{relatedwork.tex}\n\n\\section{Methodology}\n\\label{sec:methodology}\n\\input{methodology.tex}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\input{experiment.tex}\n\n\\section{Conclusion}\\label{conclusion}\n\nWe have presented an easy-to-implement yet effective channel pruning algorithm, \\ie, PEEL, to acquire a desired compact model via reallocating resources from less informative layers to more crucial layers. The intuition of PEEL~is that layers do not contribute equally to the model performance and those having an indispensable influence on the performance should be assigned more resources to magnify their positive effect. We conduct extensive experiments to verify the efficacy of PEEL~on ImageNet dataset with modern CNN architectures, \\ie, ResNet-18, ResNet-50, MobileNetV2, MobileNetV3-small and EfficientNet-B0. Experimental results suggest the effectiveness of PEEL~in uncovering compact yet accurate architectures consistently under various pruning levels, compared with state-of-the-art channel pruning methods. Besides, PEEL~yields stable searching results with small performance variance, and the searching cost is evidently smaller than contemporary channel pruning algorithms (\\eg, DMCP~\\cite{guo2020dmcp}).\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\subsection{Results}\n\n\\begin{comment}\n\\begin{table\n\\caption{Performance of PEEL~and various channel pruning approaches on ImageNet validation set. Models \\textit{are all} trained with knowledge distillation (KD) for all algorithms, using either the official codes or our reproduced implementation. If the reproduced result with KD is worse than their reported value, we will just use the results reported in their original paper.}\n\\label{result_table}\n\\centering\n\\footnotesize{\n\\begin{tabular}{c|c|c|c}\n\\hline\nBackbone & Method & Top-1 Acc (\\%) & FLOPs \\\\\n\\hline \n\\multirow{10}*{ResNet-18} & Uniform 1.0 $\\times$ & 70.3 & 1.8G \\\\\n\\cline{2-4}\n~ & Uniform 0.85 $\\times$ & 69.0 & 1.33G \\\\\n~ & NR~\\cite{qiao2019neural} & 68.6 & 1.33G \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 70.0 & 1.33G \\\\\n~ & \\textbf{PEEL} & \\textbf{70.6} & 1.33G \\\\\n\\cline{2-4}\n~ & Uniform 0.75 $\\times$ & 68.2 & 1.05G \\\\\n~ & NR~\\cite{qiao2019neural} & 68.1 & 1.04G \\\\\n~ & FPGM~\\cite{he2019filter} & 68.4 & 1.04G  \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 69.4 & 1.04G \\\\\n~ & \\textbf{PEEL} & \\textbf{69.9} & 1.04G \\\\\n\\hline\n\\multirow{18}*{ResNet-50} & Uniform 1.0 $\\times$ & 76.4 & 4.1G \\\\\n\\cline{2-4}\n~ & Uniform 0.85 $\\times$ & 75.8 & 3.0G \\\\\n~ & NR~\\cite{qiao2019neural} & 75.2 & 3.0G \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 76.2 & 3.0G \\\\\n~ & AutoSlim~\\cite{yu2019autoslim} & 76.0 & 3.0G \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 76.5 & 3.0G \\\\\n~ & \\textbf{PEEL} & \\textbf{76.9} & 3.0G \\\\\n\\cline{2-4}\n~ & Uniform 0.75 $\\times$ & 74.6 & 2.3G \\\\\n~ & NR~\\cite{qiao2019neural} & 74.3 & 2.4G \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 75.4 & 2.3G \\\\\n~ & FPGM~\\cite{he2019filter} & 75.6 & 2.4G \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 76.3 & 2.2G \\\\\n~ & \\textbf{PEEL} & \\textbf{76.8} & 2.2G \\\\\n\\cline{2-4}\n~ & Uniform 0.5 $\\times$ & 72.9 & 1.1G \\\\\n~ & NR~\\cite{qiao2019neural} & 73.1 & 1.1G \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 73.4 & 1.1G \\\\\n~ & AutoSlim~\\cite{yu2019autoslim} & 74.0 & 1.1G \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 74.6 & 1.1G \\\\\n~ & \\textbf{PEEL} & \\textbf{75.1} & 1.1G \\\\\n\\hline\n\\multirow{21}*{MobileNetV2} & Uniform 1.0$\\times$ & 72.3 & 300M \\\\\n~ & NR~\\cite{qiao2019neural} & 73.0 & 300M \\\\\n~ & AutoSlim~\\cite{yu2019autoslim} & 74.2 & 300M \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 74.6 & 300M \\\\\n~ & \\textbf{PEEL} & \\textbf{74.8} & 300M \\\\\n\\cline{2-4}\n~ & Uniform 0.75 $\\times$ & 70.1 & 210M \\\\\n~ & NR~\\cite{qiao2019neural} & 70.2 & 211M \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 71.2 & 217M \\\\\n~ & AMC~\\cite{he2018amc} & 70.8 & 211M \\\\\n~ & AutoSlim~\\cite{yu2019autoslim} & 73.0 & 211M \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 73.5 & 211M \\\\\n~ & \\textbf{PEEL} & \\textbf{73.9} & 211M \\\\\n\\cline{2-4}\n~ & Uniform 0.5 $\\times$ & 64.8 & 97M \\\\\n~ & NR~\\cite{qiao2019neural} & 64.2 & 87M \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 63.8 & 87M \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 66.1 & 87M \\\\\n~ & \\textbf{PEEL} & \\textbf{66.6} & 87M \\\\\n\\cline{2-4}\n~ & Uniform 0.35$\\times$ & 60.1 & 59M \\\\\n~ & NR~\\cite{qiao2019neural} & 59.7 & 59M \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 62.7 & 59M \\\\\n~ & \\textbf{PEEL} & \\textbf{62.9} & 59M \\\\\n\\cline{2-4}\n\n\\hline\n\\end{tabular}\n}\n\\end{table}\n\\end{comment}\n\n\n\\begin{table\n\\caption{Performance of PEEL~and various channel pruning approaches with and without knowledge distillation (KD) on ImageNet validation set.}\n\\label{result_table}\n\\centering\n\\footnotesize{\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n\\multirow{2}*{Backbone} & \\multirow{2}*{Method} & \\multicolumn{2}{|c|}{Top-1 Acc (\\%)} & \\multirow{2}*{FLOPs} \\\\\n\\cline{3-4}\n~ & ~ & w\/ KD & w\/o KD & ~ \\\\\n\\hline \n\\multirow{10}*{ResNet-18} & Uniform 1.0 $\\times$ & \\multicolumn{2}{|c|}{70.3} & 1.8G \\\\\n\\cline{2-5}\n~ & Uniform 0.85 $\\times$ & 69.0 & 68.5 & 1.33G \\\\\n~ & NR~\\cite{qiao2019neural} & 68.6 & 68.2 & 1.33G \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 70.0 & 69.7 & 1.33G \\\\\n~ & \\textbf{PEEL} & \\textbf{70.6} & 70.1 & 1.33G \\\\\n\\cline{2-5}\n~ & Uniform 0.75 $\\times$ & 68.2 & 67.8 & 1.05G \\\\\n~ & NR~\\cite{qiao2019neural} & 68.1 & 67.9 & 1.04G \\\\\n~ & FPGM~\\cite{he2019filter} & 68.4 & 68.1 & 1.04G  \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 69.4 & 69.0 & 1.04G \\\\\n~ & \\textbf{PEEL} & \\textbf{69.9} & 69.3 & 1.04G \\\\\n\\hline\n\\multirow{18}*{ResNet-50} & Uniform 1.0 $\\times$ & \\multicolumn{2}{|c|}{76.4} & 4.1G \\\\\n\\cline{2-5}\n~ & Uniform 0.85 $\\times$ & 75.8 & 75.4 & 3.0G \\\\\n~ & NR~\\cite{qiao2019neural} & 75.2 & 74.8 & 3.0G \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 76.2 & 76.0 & 3.0G \\\\\n~ & AutoSlim~\\cite{yu2019autoslim} & 76.0 & 75.8 & 3.0G \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 76.5 & 76.4 & 3.0G \\\\\n~ & \\textbf{PEEL} & \\textbf{76.9} & 76.6 & 3.0G \\\\\n\\cline{2-5}\n~ & Uniform 0.75 $\\times$ & 74.6 & 74.1 & 2.3G \\\\\n~ & NR~\\cite{qiao2019neural} & 74.3 & 74.0 & 2.4G \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 75.4 & 75.2 & 2.3G \\\\\n~ & FPGM~\\cite{he2019filter} & 75.6 & 75.5 & 2.4G \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 76.3 & 76.2 & 2.2G \\\\\n~ & \\textbf{PEEL} & \\textbf{76.8} & 76.5 & 2.2G \\\\\n\\cline{2-5}\n~ & Uniform 0.5 $\\times$ & 72.9 & 72.7 & 1.1G \\\\\n~ & NR~\\cite{qiao2019neural} & 73.1 & 72.6 & 1.1G \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 73.4 & 73.2 & 1.1G \\\\\n~ & AutoSlim~\\cite{yu2019autoslim} & 74.0 & 73.6 & 1.1G \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 74.6 & 74.4 & 1.1G \\\\\n~ & \\textbf{PEEL} & \\textbf{75.1} & 74.7 & 1.1G \\\\\n\\hline\n\\multirow{21}*{MobileNetV2} & Uniform 1.0$\\times$ & \\multicolumn{2}{|c|}{72.3} & 300M \\\\\n\\cline{2-5}\n~ & NR~\\cite{qiao2019neural} & 73.0 & 72.2 & 300M \\\\\n~ & AutoSlim~\\cite{yu2019autoslim} & 74.2 & 73.1 & 300M \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 74.6 & 73.9 & 300M \\\\\n~ & \\textbf{PEEL} & \\textbf{74.8} & 74.2 & 300M \\\\\n\\cline{2-5}\n~ & Uniform 0.75 $\\times$ & 70.1 & 69.2 & 210M \\\\\n~ & NR~\\cite{qiao2019neural} & 70.2 & 69.4 & 211M \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 71.2 & 70.1 & 217M \\\\\n~ & AMC~\\cite{he2018amc} & 70.8 & 70.0 & 211M \\\\\n~ & AutoSlim~\\cite{yu2019autoslim} & 73.0 & 72.2 & 211M \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 73.5 & 72.4 & 211M \\\\\n~ & \\textbf{PEEL} & \\textbf{73.9} & 73.0 & 211M \\\\\n\\cline{2-5}\n~ & Uniform 0.5 $\\times$ & 64.8 & 64.3 & 97M \\\\\n~ & NR~\\cite{qiao2019neural} & 64.2 & 63.8 & 87M \\\\\n~ & MetaPruning~\\cite{liu2019metapruning} & 63.8 & 63.4 & 87M \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 66.1 & 65.6 & 87M \\\\\n~ & \\textbf{PEEL} & \\textbf{66.6} & 66.0 & 87M \\\\\n\\cline{2-5}\n~ & Uniform 0.35$\\times$ & 60.1 & 59.7 & 59M \\\\\n~ & NR~\\cite{qiao2019neural} & 59.7 & 59.2 & 59M \\\\\n~ & DMCP~\\cite{guo2020dmcp} & 62.7 & 62.4 & 59M \\\\\n~ & \\textbf{PEEL} & \\textbf{62.9} & 62.6 & 59M \\\\\n\\cline{2-5}\n\n\\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\\noindent\\textbf{Comparisons with state of the arts:}\nThe detailed performance comparison of PEEL~and contemporary channel pruning algorithms is shown in Table~\\ref{result_table}. We also record the floating-point operations (FLOPs) to approximate the forward time of different architectures. From Table.~\\ref{result_table}, we observe that PEEL~achieves 25$\\%$ and 45$\\%$ FLOPs reduction for ResNet-18 and ResNet-50, respectively, with almost no loss of accuracy. \nAs to MobileNetV2, the result is more encouraging as our algorithm can find a strong architecture with \\textbf{2.5\\%} higher in top-1 accuracy than the original model while possessing the same FLOPs. In addition, PEEL~consistently finds better architectures whose performance outperforms those searched by other competitive channel pruning techniques, \\eg, DMCP~\\cite{guo2020dmcp}, NR~\\cite{qiao2019neural} and MetaPruning~\\cite{liu2019metapruning}. For instance, when the backbone model is ResNet-50 and the target FLOPs is 2.2G, the architecture produced by PEEL~is \\textbf{0.5\\%} higher in top-1 accuracy than that found by DMCP. Moreover, the network structure obtained via PEEL~consistently outperforms the uniformly pruned model in all experiments, suggesting the effectiveness of the proposed resource reallocation approach.\n\\begin{comment}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{.\/figure\/FLOPs_acc_res18_50.pdf}\n \\vskip -0.1cm\n \\caption{FLOPs-accuracy spectrum of PEEL, DMCP and uniform pruning on (a) ResNet-18 and (b) ResNet-50.}\n \\centering\n \\vskip -0.2cm\n \\label{fig:flops_acc_res18}\n\\end{figure}\n\n\\noindent\\textbf{A detailed comparison with DMCP:} \nWe compare PEEL~against the recent and representative DMCP~\\cite{guo2020dmcp}.\nFigure~\\ref{fig:flops_acc_res18} shows the accuracy of architectures pruned by PEEL and DMCP under diverse FLOPs constraints on ResNet-18 and ResNet-50. Here, we take the uniform pruning as reference. It is observed that PEEL~can always find better architectures than DMCP in terms of top-1 accuracy. On all FLOPs levels, PEEL~outperforms DMCP by least 0.4\\% in terms of top-1 accuracy. Thanks to the resource reallocation, the top-1 accuracy gap between PEEL~and uniform pruning lies between 1.0\\% and 1.7\\%. \n\\end{comment}\n\n\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{.\/arch_comparison_res18.pdf}\n \\vskip -0.1cm\n \\caption{Comparison between structures searched by PEEL~and DMCP on ResNet-18 when $\\#$FLOPs=1.04G. The vertical bar and vertical line denote the mean and variance of the channel numbers in each layer, respectively. Results are averaged over five runs.}\n \\centering\n\n \\label{fig:arch_compare}\n\\end{figure}\n\n\\begin{table}[!t]\n\\caption{Pruning MobileNetV3-small and EfficientNet-B0.}\n\\label{prune_harder_model_table}\n\\centering\n\\small{\n\\begin{tabular}{c|c|c|c}\n\\hline\nBackbone & Method & Top-1 Acc (\\%) & FLOPs \\\\\n\\hline\n\\multirow{2}*{MobileNetV3} & Uniform 1.0 $\\times$ & 67.4 & 56M \\\\\n\\cline{2-4}\n~ & PEEL & 67.2 & 49M \\\\\n\\hline\n\\multirow{2}*{EfficientNet-B0} & Uniform 1.0 $\\times$ & 77.1 & 390M \\\\\n\\cline{2-4}\n~ & PEEL & 77.0 & 346M \\\\\n\\hline\n\\end{tabular}\n}\n\\vspace{-5ex}\n\\end{table}\n\n\\noindent \\textbf{Pruning MobileNetV3-small and EfficientNet-B0:} As shown in Table~\\ref{prune_harder_model_table}, PEEL~saves \\textbf{12.5\\%} FLOPs for MobileNetV3-small and \\textbf{11.3\\%} FLOPs for EfficientNet-B0 without incurring severe performance drops. Note that MobileNetV3-small and EfficientNet-B0 are two extremely lightweight classification models and removing redundancy in these models is nontrivial. The experimental results explicitly showcase the effectiveness and generality of PEEL~on network pruning.\n\nWe also visualize the structures uncovered by PEEL~and DMCP. As depicted in Fig.~\\ref{fig:arch_compare}, both architectures have more channels as the layer goes deeper. This is expected since more channels will be leveraged to compensate for the loss of spatial information incurred by the downsampling operations. Nevertheless, PEEL~tends to place more channels in the early stages while DMCP chooses to put more channels in the later stages. Putting more resources at shallow layers brings more gains as there is more redundancy in the deep layers. Besides, the variance of the number of channels in each layer of PEEL~is much smaller than that of DMCP. For example, the averaged channel variance of DMCP on the last four layers of ResNet-18 is \\textbf{38.8} while the channel variance of PEEL~is \\textbf{21.8}. The gap between the structural variance of DMCP and our algorithm is more evident in ResNet-50 and MobileNetV2. In ResNet-50, the averaged channel variance of PEEL~is almost half of the variance of DMCP for all layers. We provide more details in the supplementary material.\nThe searching instability of DMCP also leads to high performance variance in the searched architectures. The performance of PEEL is  much stabler relatively. We summarize the performance of architectures searched by PEEL~and DMCP in Table~\\ref{performance_var_table}. \n\n\n\\begin{table}[t]\n\\caption{Performance variance of searched architectures of PEEL and DMCP on ResNet-50. Results are averaged over five runs.}\n\\vskip 0.1cm\n\\label{performance_var_table}\n\\centering\n\\small{\n\\begin{tabular}{c|c|c|c}\n\\hline\n\\multirow{2}*{Method} & \\multicolumn{3}{c}{$\\#$FLOPs} \\\\\n\\cline{2-4}\n~ & 2.2G & 1.8G & 1.1G \\\\\n\\hline\n\\hline\n\\textbf{PEEL~} & \\textbf{76.7\\% $\\pm$ 0.2\\%} & \\textbf{75.3\\% $\\pm$ 0.3\\%} & \\textbf{75.0\\% $\\pm$ 0.2\\%} \\\\\n\\hline\nDMCP & 75.7\\% $\\pm$ 0.7\\% & 74.2\\% $\\pm$ 0.8\\% & 73.8\\% $\\pm$ 0.8\\% \\\\\n\\hline\n\\end{tabular}\n}\n\\vspace{-3ex}\n\\end{table}\n\n\n\\begin{table}[!t]\n\\caption{Comparison between the training time and memory usage of PEEL~and DMCP on ResNet-50 ($\\#$FLOPs=1.1G). Here, the memory usage is measured on one GPU and the batch size is set as 64 for each GPU.}\n\\vskip 0.1cm\n\\label{train_cost_table}\n\\centering\n\\small{\n\\begin{tabular}{c|c|c}\n\\hline\nAlgorithm & Train Time (H) & Memory usage (G) \\\\\n\\hline\n\\hline\n\\textbf{PEEL~} & \\textbf{37} & \\textbf{4.3} \\\\\n\\hline\nDMCP~\\cite{guo2020dmcp} & 98 & 9.5 \\\\\n\\hline\n\\end{tabular}\n}\n\\vspace{-5ex}\n\\end{table}\n\n\nThe proposed PEEL~not only takes shorter training durations to find and train the desired model, but also consumes less GPU memory during the searching phase. \nHere, we examine the training cost of PEEL and DMCP, \\ie, the total training hours and the GPU memory usage during the searching phase. The original unpruned model is ResNet-50 and the target FLOPs is 1.1 G. We keep all other hyperparameters the same, including batch size (512), GPU types (NVIDIA TITAN X) and number of used GPUs (8). From Table~\\ref{train_cost_table}, the total number of training hours of PEEL~is roughly a third of DMCP. The result is not surprising as DMCP requires training on the original cumbersome model whilst PEEL~merely entails training of a much slimmer model. As to the GPU memory usage, since DMCP needs to collect gradients of several sampled architectures while PEEL~only computes the gradient of one compact architecture, the memory consumption of DMCP is almost twice as that of PEEL~. We also provide a comparison to another representative USNet~\\cite{yu2019universally} in the supplementary material. \n\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{.\/FLOPs_assign_res18.pdf}\n \\vskip -0.1cm\n \\caption{Where the resources go. Percentage of FLOPs assigned by PEEL~to different groups of (a) ResNet-18 and (b) ResNet-50 under different target FLOPs constraints. X-axis denotes group indices and y-axis represents the target FLOPs. A darker color indicates a larger quantity of assigned resources. Since ResNet-18 and ResNet-50 have only one layer in group 1 and resources assigned to one layer are very limited, we put this layer into group 2 and obtain the above figure.}\n \\centering\n \\vskip -0.4cm\n \\label{fig:flops_assign_res18}\n\\end{figure}\n\n   \n\n\\subsection{Ablation study}\n\n\n\\noindent \\textbf{Where the resources go:}\nTo have a deeper understanding on the effect of the resource reallocation module, we visualize the percentage of resources distributed by PEEL~to different groups of layers on ResNet-18 and ResNet-50. As shown in Fig.~\\ref{fig:flops_assign_res18}, layers in the first three groups are given much more resources than the last two groups when the network is slightly trimmed. It is natural as there are more redundant channels in the deep layers in the mild pruning level~\\cite{liu2019metapruning}. As the pruning becomes more aggressive, the percentages of FLOPs assigned to different groups become more even since all layers do not have sufficient channels and call for more resources from the resource pool. And we can observe the same patterns in the resource reallocation of ResNet-18 and ResNet-50 when the FLOPs budget becomes tighter.\n\n\\noindent \\textbf{Effect of $\\lambda$:} The resources available in the resource pool is $(1-\\lambda)M$, thus the pool size is controlled by the hyperparameter $\\lambda$. We select the value of $\\lambda$ from $\\{0.5, 0.6, 0.7, 0.8, 0.9, 1\\}$ and compare the model performance under these settings. As illustrated in Fig.~\\ref{fig:single_multi_round} (a), the performance of the searched model is relatively stable when $\\lambda$ ranges from 0.7 to 0.9. When $\\lambda$ decreases from 0.7, the performance of the final pruned architecture gradually drops. The trend is expected since PEEL~only performs the evaluation of layer importance once on the over-pruned model. The importance evaluation would become inaccurate given a small $\\lambda$, as more resources are assigned to the pool while the over-pruned backbone is too slim. Another phenomenon we observe is that as the pruning becomes more aggressive, the performance of the pruned model is more susceptible to the value of $\\lambda$ (see the green line in Fig.~\\ref{fig:single_multi_round} (a)). We conjecture that the performance drop is caused by the challenging pruning condition. Given very limited FLOPs resources, one has to allocate resources in a very careful manner so that the pruned model can exhibit satisfactory performance.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{.\/single_multi_round.pdf}\n \\vskip -0.1cm\n \\caption{One-round v.s. multi-round reallocation. We show the $\\lambda$-accuracy spectrum of (a) one-round and (b) three-round resource reallocation of PEEL~on ResNet-50. Different color denotes different target FLOPs.}\n \\centering\n \\vskip -0.2cm\n \\label{fig:single_multi_round}\n\\end{figure}\n\n\\noindent \\textbf{One-round reallocation v.s. multi-round reallocation:} Recall that we adopt the one-round resource reallocation in our algorithm, \\ie, allocating the resources by just performing a single pass of layer importance estimation. An alternative strategy is to evenly divide the resources into several parts and reallocate these resources successively in multiple rounds. Multi-round reallocation is more expensive as one needs to repeat Step-2 and Step-4 iteratively (see Sec.~\\ref{sec:methodology}), and each iteration involves the fine-tuning of the backbone. As depicted in Fig.~\\ref{fig:single_multi_round}, the performance of the multi-round resource reallocation is more stable than the one-round version when the $\\lambda$ is set 0.6 and 0.5. For instance, when the target FLOPs is 2.2G and $\\lambda$ is set as 0.5, the performance of the one-round reallocation decreases from 76.8\\% to 75.3\\% while the multi-round reallocation can still achieve 76.0\\% in top-1 accuracy. The more stable performance may come from the fact that multi-round resource reallocation reevaluates the importance of each layer in each round, thus allowing a more appropriate resource reallocation.\n\n\\noindent \\textbf{Robustness to the layer importance indicator:} The original PEEL~adopts BN statistics to reflect the importance of each layer. Here, we explore different importance indicators and check whether the proposed PEEL~is sensitive to the chosen indicator. We choose the widely-used filter norm~\\cite{li2017pruning} and reconstruction errors~\\cite{luo2017thinet} as the criteria. Layers with large filter norm or large reconstruction errors are considered as important. However, directly taking filter norm as the evaluating metric is biased since the filter norm of different layers have diverse scales in magnitude~\\cite{chin2020towards}. To correct such bias, we follow~\\cite{chin2020towards} and learn the layer-wise affine transformations for the filter norm. Under this circumstance, filter norm can better reflect the value of each layer. From Table~\\ref{diff_indicator_table}, when pruning ResNet-50 under different FLOPs requirements, architectures searched by PEEL~using BN statistics, filter norm and reconstruction errors achieve similar performance. The results suggest the robustness of PEEL, irrespective of the methods used in evaluating layer importance.  \n\n\\begin{table}[t]\n\\caption{Performance of PEEL~with different importance criteria on ResNet-50.}\n\\vskip 0.1cm\n\\label{diff_indicator_table}\n\\centering\n\\small{\n\\begin{tabular}{c|c|c|c}\n\\hline\n\\multirow{2}*{Criterion} & \\multicolumn{3}{c}{$\\#$FLOPs} \\\\\n\\cline{2-4}\n~ & 2.2 G & 1.8 G & 1.1 G \\\\\n\\hline\n\\hline\nBN statistics (Ours) & 76.8\\% & 75.5\\% & 75.1\\% \\\\\n\\hline\nFilter norm~\\cite{li2017pruning} & 76.7\\% & 75.2\\% & 75.2\\% \\\\\n\\hline\nReconstruction errors~\\cite{luo2017thinet} & 76.6\\% & 75.4\\% & 75.0\\% \\\\\n\\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{.\/different_backbone.pdf}\n \\vskip -0.1cm\n \\caption{Visualization of the architecture found by PEEL~on a different backbone model that has the same number of channels for all layers. The base model is ResNet-18, and target FLOPs is 1.04G.}\n \\centering\n \\vskip -0.4cm\n \\label{fig:diff_backbone}\n\\end{figure}\n\n\\noindent \\textbf{Effect of knowledge distillation:} By comparing the two columns of the Top-1 Acc in Table~\\ref{result_table}, we find that PEEL~consistently yields better performance than baseline pruning approaches with or without knowledge distillation (KD). For instance, when performing pruning on ResNet-18 and target FLOPs is 1.33G, PEEL~with KD is 0.6\\% higher than DMCP with KD. One interesting phenomenon we observe is that KD brings more gains when the backbone model has fewer parameters. For example, KD can bring approximately 0.3\\%, 0.5\\% and 1.0\\% to PEEL~on ResNet-50, ResNet-18 and MobileNetV2, respectively when the pruning is not aggressive. We conjecture that the increased gains are attributed to the deficiency of small models in grasping knowledge in labels by themselves and hence they more eagerly call for the guidance of the original model.\n\n\\noindent \\textbf{Efficacy of PEEL~on a different backbone model:} Recall that we treat the uniformly trimmed model as the backbone and conduct resource reallocation on it. It is natural to wonder whether the resource reallocation would still work if a totally distinct architecture is used. Here, we use an architecture that has the same number of channels across all layers as the new starting point. \nAll configurations are the same as the previous experiments. From Fig.~\\ref{fig:diff_backbone}, we can observe that in this new backbone, groups 1, 2 and 3 are assigned with few parameters while the majority of the resources are distributed to groups 4 and 5. Overall, PEEL~puts more parameters in the deeper layers, and the resulting architecture is \\textbf{8.7\\%} higher in terms of top-1 accuracy in comparison to the original model (62.4\\% v.s. 53.7\\%). The result demonstrates the effectiveness of resource reallocation as well as the insensitivity of our algorithm to the backbone model.     \n\n\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{.\/different_policy.pdf}\n \\vskip -0.1cm\n \\caption{Policy on resource reallocation. Performance of PEEL~with importance-guided, winner-take-all, uniform and random policies on (a) ResNet-18 and (b) ResNet-50, respectively.}\n \\centering\n \\vskip -0.2cm\n \\label{fig:diff_policy}\n\\end{figure}\n\n\n\\noindent \\textbf{Policy on resource reallocation:} Here, we compare our `importance-guided' resource reallocation strategy with other parameter reallocation policies, \\ie, winner-take-all, uniform and random reassignment. The winner-take-all policy determines the most important group according to the computed group significance and puts all resources in that group. \nThe uniform policy adds the same number of channels to all layers while the random policy stochastically selects several layers and increases their channels. Here, we name the original reallocation strategy as importance-guided policy as it distributes resources based upon the estimated layer importance. From Table~\\ref{fig:diff_policy}, our importance-guided reallocation policy evidently outperforms the other three policies on ResNet-18 and ResNet-50. For instance, when the target FLOPs is 1.8G on ResNet-50, the top-1 accuracy of the importance-guided policy is 75.5\\% while the top-1 accuracy of the winner-take-all, uniform and random policy is 74.9\\%, 74.1\\% and 74.0\\%, respectively. The superior performance demonstrates the effectiveness of the importance-guided policy. \n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Platform and Evaluation}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.96\\linewidth]{Images\/tasks_simple_to_complex.png} \\vspace{-3mm}\n\t\\caption[FishGym Tasks]{The control results for benchmark tasks using our trained policies and two-way coupled fluid-structure interaction solver. Top row: cruising in a shallow fluid; second row: cruising in a deep fluid (with buoyancy control); third row: pose control with a U-turn; fourth row: two-fish schooling; bottom row: path following with an arbitrarily specified path.}\n\t\\label{fig:tasksequence} \n\t\\vspace{-15pt}\n\\end{figure}\n\nIn the following, we first describe our setup on simulation and evaluate our fluid simulation platform with our learning algorithms for robot fish swimming in multiple aspects.\n\n\\subsection{Platform setup}\n\\paragraph{Simulation}\nIn most our training tasks, all the fish robots have a density of 1080$kg\/m^3$, and we used local non-inertial fluid-structure interaction solver for simulating robot fish dynamics, with a grid resolution of $100\\times100\\times100$ and a physical time step of 0.004s.\nThe solver costs around 3.5 seconds for simulating one physical second on an NVidia TitanXp GPU with 12G memory.\nFor two-fish schooling, we extended the local domain horizontally to simultaneously include two fishes, with a grid resolution of $150\\times50\\times100$ and the same physical time step, which costs around 4 seconds for simulating one physical second on the same GPU.\n\n\\paragraph{Training}\nThe policy network consists of two layers, each containing 256 units, with an ReLU activation function. \nFor each task, the policy is trained using SAC \\cite{Haarnoja2018}.\nWe train each policy for a total of 2000 simulation rollouts, each of which contains 50 time steps, where a candidate policy executes a new action at each time step. \nWe train the policy network with a batch size of 256.\nThe model parameters are updated for each step, and one gradient step is performed after each rollout.\nWe train all policies on a machine with an NVidia TitanXP GPU, and the training process usually starts to converge after 6 hours (1000 episodes), and have a smooth convergence within 10 hours. \nThe training could be several times faster if we use the most state-of-the-art GPUs, such as NVidia GeForce RTX3090.\n\n\\subsection{Comparison for different types of robot fishes}\nOur platform can support robot fishes designed with different skeleton connectivity and skin shapes.\nFig.~\\ref{fig:different_fish_type_result} shows the training results for three types of robot fishes swimming along an arbitrarily given path.\nThe average distances from the path (around 7 meters long) are as close as 0.03m, 0.05m, 0.08m, respectively, indicating the capability of our platform in supporting a variety of robot fishes.\n\\subsection{Comparison for different simulation models}\nIn the literature, a simple empirical model was proposed as the simulator for robot fish swimming~\\cite{Terzopoulos1994}, which has been used until now~\\cite{grzeszczuk1998neuroanimator,si2014realistic,min2019softcon}.\nIt models the instantaneous force on the surface of the robot due to viscous fluid as:\n\\vspace{-5pt}\n\\begin{equation}\n\tF = -k\\int_{S}(\\mathbf{n} \\cdot \\mathbf{v}) \\mathbf{n} d s ,\n\t\\label{empirical_model_eq}\n\t\\vspace{-5pt}\n\\end{equation}\nwhere $\\mathbf{n}$ is the unit outward normal; $\\mathbf{v}$ is the relative velocity between the surface and the fluid (since there is no fluid simulation, the fluid velocity is assumed to be zero), and $k$ is a constant manually tuned for different robot fishes and the surrounding fluids.\nNote that for a specific system, $k$ can only be determined either by real measurement data or from other physically more accurate simulators.\n\nTo examine the similarity and difference between the empirical model and our physical simulator for robot fish swimming, we conduct two test cases with analyses below.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.96\\linewidth]{\"Images\/different_robot_fishes\"} \\vspace{-3mm}\n\t\\caption[Fish Gym Tasks]{Different types of robot fishes trained using local policy learning for arbitrary path following. Top: koi robot fish; middle: flatfish robot fish with a different skeleton topology (with branching during modeling); bottom: eel robot fish with a long concatenated skeleton.}\\vspace{-3mm}\n\t\\label{fig:different_fish_type_result}\n\\end{figure}\n\\paragraph{Path following}\nPath following is a primitive task for robot control.\nHere, we compare the similarity and difference of path following using an empirical model and our physical simulator.\nSince there is no clue on how to tune the parameter $k$ in an empirical model, we arbitrarily choose one and learn a policy. \n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.96\\linewidth]{\"Images\/Parameter_need_tune\"} \\vspace{-3mm}\n\t\\caption{Comparison for path following using (left) empirical model with an arbitrarily chosen parameter and (right) our physical simulator. Note that due to improper boundary force, serious drifting happens for empirical model when turning. The numbers indicate the order during path following.} \\vspace{-3mm}\n\t\\label{fig:empirical_failed_path}\n\\end{figure}\n\nFig.~\\ref{fig:empirical_failed_path} (left) shows the control result, indicating serious drifting when turning, while our physical simulator does not require any parameter turning (we directly specify the physical parameter for the fluid as $\\rho$=1000$kg\/m^3$ and $\\nu$=0.00089$m^2\/s$ with a zero-velocity initialization to match that used in the empirical model), leading to a reasonable path following result as shown in Fig.~\\ref{fig:empirical_failed_path} (right).\nNote that the mean path deviation for the empirical model with an arbitrary parameter is as large as 0.5m, while our physical simulator only has a mean path deviation of 0.03m.\nThe empirical model can be much improved if we collect data from our simulator and fit the parameter $k$, leading to a very similar result in a static fluid environment but runs much faster.\nHowever, in some cases where we cannot assume static fluid background, empirical model can completely fail no matter how we fit the parameter $k$, and we demonstrate this case in the following two-fish schooling task.\n\\paragraph{Two-fish schooling}\nFish schooling describes a common phenomenon where fishes tend to swim in a group and one fish follows the other.\nIt has been revealed by scientists that due to the vortex ring generated behind the leader fish, the follower fish tries to utilize the vortex ring to reduce drag and pass through it in order to catch up with the leader fish more efficiently \\cite{Novati2017,Verma2018}.\nWe demonstrate this behavior in Fig.~\\ref{fig:teaser} (bottom row).\nDue to more accurate modeling to capture complex fluid flows, the training can successfully obtain a policy that utilizes the vortex ring, see Fig.~\\ref{fig:teaser} (bottom), while empirical model, on the other hand, fails to learn such a policy due to lack of a real fluid-structure interaction.\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{Images\/density_compare} \\vspace{-3mm}\n\t\\caption{Control behaviors of a koi-like robot fish for different fluid densities. Top: $\\rho=1000$; bottom: $\\rho=10$.}\n\t\\label{fig:densitycompare}\n\t\\vspace{-18pt}\n\\end{figure} \n\n\\subsection{Comparison for different fluid settings}\nIn contrast to the empirical model, our simulator can freely set physical parameters, leading to different control behaviors for a trained robot fish.\nHere, we change the fluid density (with a higher density $\\rho=1000$ and a lower density $\\rho=10$, see Fig.~\\ref{fig:densitycompare}) for simulation and train with the same learning parameters to achieve a cruising task.\nIt is observed that the robot fish in a higher density fluid seems to move with less swing amplitude, while the one in a lower density fluid has a larger swing amplitude to generate greater propulsion force, which is consistent with the physical expectations.\n\n\\subsection{Effects of reward weighting}\n\nIn this experiment we investigate effects of different reward weighting. Table~\\ref{energy_table} summarizes the total energy cost by varying weights $w_v$ and  $w_e$ in the cruising task, where $w_v$ encourages fish to reach the target position while $w_e$ encourages fish to save its energy.\nA good balance between energy preservation and the time to accomplish the task can be made, e.g., $w_v=1.0$ and $w_e=0.5$.\n\n\\begin{table}[h]\n\\vspace{-3mm}\n\t\\caption{Impact of of weights in the reward} \\vspace{-3mm}\n\t\\label{energy_table}\n\t\\begin{center}\n\t\\begin{tabular}{c|c|c|c}\n\t\\hline\n\t$w_v$ & $w_e$ & Total Energy Cost & Total Time (sec.)\\\\\n\t\\hline\n \t0.00 & 1.00 & 0.0413 & 10.0 \\\\\n \t0.20 & 1.00 & 12.354 & 5.2 \\\\\n \t1.00 & 1.00 & 15.046 & 4.6 \\\\\n \t1.00 & 0.50 & 17.705 & 4.2 \\\\\n \t1.00 & 0.00 & 40.969 & 4.2 \\\\\n\t\\hline\n\t\\end{tabular}\n\t\\end{center}\n\\vspace{-15pt}\n\\end{table}\n\n\n\n\\section{Conclusion}\nIn this paper, we propose a new open-to-use simulation platform for training underwater fish-like robots.\nThe whole platform consists of a new modeling for fish-like underwater robot, a GPU-based non-inertial high-performance fluid-structure interaction solver (as a training environment), and reinforcement learning algorithms with both global and local policy learning.\nFour different benchmark tasks were proposed and trained with our platform, with expected results.\nWe compared and analyzed the new training platform in terms of different results in multiple aspects to evaluate the advantages.\n\nThere are also some limitations.\nFirst, the fish model is a reduced model that may deviate from the real robot design, and is now hence difficult to directly transfer to a real robot once learned.\nSecond, since we use local simulation, the platform is unable to training fish robot with a more complex external environment, e.g., a large vortex.\nFinally, the grid resolution around the fish is not fine enough (otherwise, it will become very slow for training), and the accuracy is not sufficiently high.\nSupporting simulation in more complex fluid environment and developing more efficient training method with higher accuracy deserve our future work.\n\\section{Introduction}\n\\label{sec:intro}\n\nBio-inspired underwater robots often demonstrate strong maneuverability, propulsion efficiency, and deceptive visual appearance.\nThese advantages have motivated a set of academic studies on bio-inspired soft robots and biomimetric fish-like robots in the past years~\\cite{Du2015,Paley2021,Duraisamy2019}.\nIt also opens up some important applications, such as marine education, navigation and rescue, seabed exploration, scientific surveying, etc.~\\cite{Kopman2012,Picardi2020,Berlinger2021,Li2021,Katzschmann2018}.\nHowever, due to lack of sufficient datasets and high physical cost, training their intelligent behaviors in real environments that at least mimic the bionic creatures or even exceed their capabilities in accomplishing complex tasks is still quite challenging.\n\nAlternatively, simulation has been considered as a viable and important tool for acquiring a large number of datasets in different scenarios~\\cite{AvilaBelbutePeres2018,Lee2019,James2019,Bergamin2019}.\nMost of the currently available simulators for robot training mainly target rigid and soft body systems~\\cite{Coumans2015,Lee2018,Todorov2012}.\nExisting simulators for fluid environment are either highly inaccurate (e.g., based on an empirical model~\\cite{Terzopoulos1994}), too restrictive to support different agents or  environments~\\cite{Song2017,Song2020,Verma2018}, or expensive to generate a large amount of training data \\cite{Verma2018,Novati2017}.\nThere is currently a dearth of simulation platform which is able to provide a versatile, efficient yet accurate results that could be used for training control policies of underwater robots. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{\"Images\/school_comparision_teaser_2fig.png\"} \\vspace{-3mm}\n\t\\caption{\\textbf{Two-fish schooling behaviors.}  With the empirical model (1st row), the follower fish always stays in the same line as the leader fish. However, with our physical simulator (2nd row), the follower fish can utilize the wake vortex ring (blue) and gradually pass through the vortex ring to preserve energy. This motion behavior cannot be acquired through a simple empirical model, highlighting the advantages of the proposed FishGym platform. }  \\vspace{-3mm}\n\t\\label{fig:teaser}\n\t\\vspace{-10pt}\n\\end{figure}\nWe propose \\textbf{FishGym}\\footnote{https:\/\/github.com\/fish-gym\/gym-fish}, a high-performance simulation platform targeting the two-way interaction dynamics between fish-like underwater robots and the surrounding fluid environment. The robots are modeled by the skeletons of arbitrary topology with surface skinning, whose motion is driven by the articulated rigid body dynamics~\\cite{Weinstein2006}, while the fluid-structure interaction is achieved using a recently proposed GPU-optimized lattice Boltzmann solver~\\cite{Li2020,Chen2021gpu}, where immersed boundary method~\\cite{li2016immersed,wu2010improved} is employed for efficient two-way coupling.\nTo support simulations in a local fluid domain around the robot to enable training in an infinitely large physical domain, we propose a modification of the original lattice Boltzmann solver that enables simulations in a local frame of reference with acceleration, with higher flexibility in acquiring various training environments. \nThe whole simulation module is then coupled with a reinforcement learning module implemented using PyTorch~\\cite{Raffin2019,Paszke2019}.   \n\nTo demonstrate the capability of the proposed simulator, we evaluated existing reinforcement learning algorithm with reward functions and training procedures tailored for underwater robots.\nWe compare the learned control policies with that from the empirical model on several underwater planning and control tasks to assess the feasibility and advantages of our framework. \nAnalyses on the emerged behaviors also indicate consistency with previous studies on fish motion in nature.   \nIn summary, we have made the following contributions to training bionic underwater robots: \n\\begin{itemize}\n\\item A GPU-accelerated lattice Boltzmann solver that enables high-performance fluid-structure interaction in a local moving frame of reference to allow robot swimming in an infinitely large domain;\n\\item A high-performance simulation platform to help explore training bionic underwater robots;\n\\item A learning algorithm tailored for bionic under-water robots that is able to acquire natural and efficient control policies for swimming;\n\\item A collection of benchmark tasks for underwater robot to evaluate and compare different learning methods and control policies. \n   \n\n\n\n\n\\end{itemize}\n\t \n\n\n\n\n\n\n\n\n\n\\section{FishGym Framework}\nOur physics-based robot learning framework  consists of three components:\n1) the robot model for which we focus on fish-like robots, 2) \nthe simulation method for predicting fluid-robot interaction, \nand 3) the robot learning method that leverages our simulation method.\nWe now present their details.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{\"Images\/two_fish_skeletons\"}\n\t\\caption[Fish Skeletons]{Illustration of modeled fishes with different skeletons and skins, where the number of joints, the length of each skeleton edge as well as the topology of the skeleton can vary for different types of fishes.}\n\t\\label{fig:fish_model} \n\t\\vspace{-20pt}\n\\end{figure}\n\n\\subsection{Robot model}\nMotivated by the anatomy of the fish structure,\nwe model a fish robot's locomotion by its skeletal structure, that is, bones connected by joints.\nCovering the skeleton are flesh and skin. Provided a skeleton configuration (e.g., with a certain set of joint angles), we use linear blend skinning~\\cite{magnenat1988joint} to determine \nthe fish's surface shape. In our fluid-robot simulation, the fish surface is assumed to be inelastic, and the flesh is treated as a rigid body under the current skeleton configuration. We make this assumption for the sake of computational efficiency.\nBy changing its joint angles, a fish robot can adjust its skeleton pose,\nwhich in turn determines its surface shape.\nThree examples of fish robots with different skeletal structures are shown in Fig.~\\ref{fig:fish_model}.\n\n\n\n\n\nThe bone skeleton is driven by the articulated rigid body dynamics \\cite{Weinstein2006}:\n\\vspace{-8pt}\n\\begin{equation}\n\\mathbf{M}(\\mathbf{q})\\ddot{\\mathbf{q}}+\\mathbf{C}(\\mathbf{q}, \\dot{\\mathbf{q}})=\\boldsymbol{\\tau}_{int}+\\boldsymbol{\\tau}_{ext} ,\n\\vspace{-8pt}\n\\end{equation}\nwhere $\\mathbf{q}, \\dot{\\mathbf{q}}$ and $\\ddot{\\mathbf{q}}$ are respectively the vectors of generalized positions, velocities and accelerations of all joints. $\\mathbf{M}(\\mathbf{q})$ is the mass matrix, and $\\mathbf{C}(\\mathbf{q}, \\dot{\\mathbf{q}})$ accounts for the Coriolis and centrifugal forces;\ndetails of these forces will be described shortly.\n$\\boldsymbol{\\tau}_{ {int }}$ and $\\boldsymbol{\\tau}_{ {ext }}$ are the vectors representing the generalized internal forces (including the spring forces on joints to enable elasticity, damping forces due to velocities, friction forces, as well as the actuation given by the controller) and the generalized external forces (caused by gravity, possible collisions and the surrounding fluids) exerted on the multi-body system.\nWe employ DART \\cite{Lee2018} to solve the above multi-body system, and for each time step, we control the bone shape by applying generalized actuation forces on joints.\nThe skin surface is achieved by employing linear blending method proposed by \\cite{magnenat1988joint}.\n\n\\subsection{GPU-accelerated localized fluid-structure interaction}\nThere exist many simulation methods that may \npredict fluid-robot interaction~\\cite{klingner2006fluid,lv2010novel,dai2005adaptive}. \nThese methods, however, require a fixed simulation domain, inside which the \nunderwater robot moves. \nWhen the simulation domain is large,\nsimulation is costly.\nTo reduce the cost,\nwe assume that the fluid \nfurther away from the robot by a certain distance will not influence the robot motion. \nThereby, we can fix the size of the simulation domain centered around the fish robot and allow the domain to move along with the fish. \nThis setup allows the fish robot to move in an \ninfinite spatial domain while keeping the simulation domain limited. \nBut then, to capture fluid dynamics correctly, we need to simulate fluid-structure interaction in a moving frame of reference.\nBeing able to swim in an infinitely large domain is very important for training fish-like underwater robots; Also crucial is the simulation performance, as\ntraining the learning algorithm will often run fluid simulations\nmany times (see Section~\\ref{sec:fish_learning}).\nWe tackle both problems next.\n\n\\subsubsection{Formulation}\nFluid in a fixed frame of reference is often governed by the following NS equation:\n\\vspace{-6pt}\n\\begin{equation}\n\\frac{\\partial \\mathbf{u}}{\\partial t}+(\\mathbf{u} \\cdot \\nabla) \\mathbf{u}=-\\frac{1}{\\rho} \\nabla p+\\nu \\nabla^{2} \\mathbf{u} + \\mathbf{F},\n\\vspace{-6pt}\n\\end{equation}\nwhere $\\rho$, $\\mathbf{u}$, $p$ and $\\mathbf{F}$ represent the density, velocity, pressure and external force fields, and $\\nu$ is the kinematic viscosity.\nThis equation cannot be used to solve flows in a moving frame of reference around the fish, which should be reformulated in a frame of reference with acceleration.\nAccording to \\cite{Asmuth2016}, by transforming with time-dependent relative translation $\\mathbf{p}$ and rotation $\\mathbf{r}$ (represented as Euler angles) between consecutive frames, the NS equation in an accelerating frame of reference results in an additional virtual force added to the system:\n\\vspace{-6pt}\n\\begin{equation}\n\t\\vspace{-6pt}\n\\mathbf{F}_{ni}=-\\ddot{\\mathbf{p}}-\\ddot{\\mathbf{r}} \\times \\mathbf{x}^{\\prime}-\\dot{\\mathbf{r}} \\times\\left(\\dot{\\mathbf{r}} \\times \\mathbf{x}^{\\prime}\\right)-2 \\dot{\\mathbf{r}} \\times \\mathbf{u}^{\\prime} ,\n\\end{equation}\nwhere all the physical quantities are measured in a moving frame of reference.\nIn case of any immersed solid, e.g., the swimming robot, we apply Neumann boundary condition (i.e., slipping) as an approximation.\n\n\\subsubsection{Simulation}\nTo simulate the above dynamics in an efficient manner, we discard the traditional NS solver; instead, we employ a GPU-optimized LBM solver with immersed boundary (IB) method \\cite{Li-2020} for fluid-structure interaction, whose high efficiency has been demonstrated.\nThe fish-like robot surface is uniformly sampled before simulation, and the external virtual force due to acceleration can be added into the system very easily.\nThe difficulty we need to address is the domain boundary, which in theory should be set according the large-domain simulation.\nHowever, in practice, we do not know the exact values of the domain boundary, and when fish moves with different velocities and accelerations, the flow can go into and outside from any portion of the domain boundary.\nThus, we need a domain boundary treatment which can adapt to this situation, and through a set of experiments, we found that the method described in \\cite{ZhaoLi2002} satisfy this requirement and has been employed in our IB-LB simulation.\nFig.~\\ref{fig:complocalglobal} compares the accuracy between a full global fluid domain simulation (left) where we directly use IB-LB method in \\cite{Li-2020} and the proposed localized fluid-structure interaction (right). \nBoth methods produce nearly identical robot motion trajectories.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{Images\/comp_local_global} \\vspace{-3mm}\n\t\\caption{Comparison between fluid-structure interaction in a global static fluid domain (left) and the non-inertial counterpart in a moving local fluid domain (right), indicating the closeness of the paths with a known policy.} \\vspace{-5mm}\n\t\\label{fig:complocalglobal}\n\t\\vspace{-3pt}\n\\end{figure}\n\n\\subsection{Reinforcement learning for fish-like robot control}\n\\label{sec:fish_learning}\nDue to complex fish dynamic model, the control of fish-like robot swimming cannot be simply achieved by a model-based controller.\nIn addition, since the control input is usually high-dimensional and cannot be fully decoupled, the PID controller cannot be used either.\nThus, reinforcement learning (RL) is a common choice to train complex control policies for swimming.\nIn this paper, we propose four benchmark tasks that will be trained using RL in order to demonstrate the power of the new simulator together with the learning algorithms.\nThese benchmark tasks are:\n\\begin{itemize}\n\t\\item \\textbf{Cruising.}\n\tThe robot fish tries to swim to reach a given target location that is a distant away from the robot.\n\t\\item \\textbf{Pose control.}\n\tA robot fish tries to control its pose in order to make a U-turn.\n\t\\item \\textbf{Two-fish schooling.}\n\tA robot fish follows a leader fish as closely as possible, where the leader fish is controlled to swim in a straight path.\n\t\\item \\textbf{Path following.}\n\tA robot fish follows a given arbitrary path as closely and efficiently as possible.\n\\end{itemize}\n\nIn RL, an agent learns a policy for a specific task through repeated interaction with the environment.\nGiven a state $\\mathbf{s}_i$, the RL tries to learn a parametric policy $\\pi_{\\boldsymbol{\\theta}}$, which is usually represented by a fully-connected neural network, to produce an action $\\mathbf{a}_i$;\nThe action can be taken by the agent to transit to the next state $\\mathbf{s}_{i+1}$, where a reward $r_i$ is evaluated.\nThe agent iterates transitions until it satisfies one of the exit conditions, e.g., finite time horizon, or success\/failure of a given task.\nA parametric policy $\\pi_{\\boldsymbol{\\theta}}$ is learned by finding the optimal parameters $\\boldsymbol{\\theta}^{*}$ that maximize the expected return:\n\\vspace{-8pt}\n\\begin{equation}\n\t\\vspace{-6pt}\nJ(\\theta)=\\mathbb{E}_{\\tau \\sim p_{\\theta}(\\tau)}\\left[\\sum_{t=0}^{T} \\gamma^{t} r_{t}\\right] ,\n\\end{equation}\nwhere $T$ is the maximum number of control time steps, $\\gamma$ is the discounting factor, and $\\tau$ is the sampled trajectory containing a sequence of states and actions, i.e., $\\tau = (s_0, a_0 , s_1, a_1, ..., s_i, a_i, ..., s_t, a_t)$.\nThe policy learning can be achieved in two different ways.\nDepending on the available resources, we can sample the trajectories from one single task, or from multiple tasks to learn a \\textit{global policy}.\nHowever, if the variety of tasks is large, it is time consuming especially when the simulator is not fast enough.\nOn the other hand, if the tasks can be subdivided into small and simple sub-tasks, we can gather all these sub-tasks together and sample from them to learn a \\textit{local policy}, which is expected to be much more generalizable given a relatively small training set, and could be affordable for limited resources.\nWe adopt both approaches to train different tasks.\n\n\nIn the following, we specify in detail the specific designs on how we train these benchmark tasks.\n\\paragraph{State} \nFor all benchmark tasks, we consider the following state variable:\n\\vspace{-8pt}\n$$\n\\mathbf{s}=(\\mathbf{s}_d, \\mathbf{s}_p, \\mathbf{s}_r, \\mathbf{s}_{task}) ,\n\\vspace{-6pt}\n$$\nwhere $\\mathbf{s}_d = (\\mathbf{q},\\dot{\\mathbf{q}})$ contains the dynamic states, with $\\mathbf{q}$ the vector of generalized joint positions, and $\\dot{\\mathbf{q}}$ is the vector of generalized joint velocities;\n$\\mathbf{s}_p=(\\mathbf{p},\\dot{\\mathbf{p}})$ contains translation, where $\\mathbf{p}$ is the relative translation vector between simulation time steps, and $\\dot{\\mathbf{p}}$ is the relative velocity vector;\n$\\mathbf{s}_r = \\mathbf{r} $ contains rotation (Euler angles);\nand $\\mathbf{s}_{task}$ contains task-specific states. \nIn practice, to reduce input dimension, all state variables are expressed in a local coordinate system. \n\n\\paragraph{Action}\nFor all benchmark tasks, the action can be generally defined as:\n\\vspace{-10pt}\n$$\n\\mathbf{a} = (\\boldsymbol{\\sigma},\\Delta v) ,\n\\vspace{-6pt}\n$$\nwhere $\\boldsymbol{\\sigma}$ is the vector containing the actuation forces applied to the joints, and\n$\\Delta v$ is the change of the bladder's volume inside the robot fish, which controls buoyancy to enable going up and down in a fluid.\n\\paragraph{Reward}\nThe reward for training all benchmark tasks can also be written in a general mathematical form as:\n\\vspace{-4pt}\n$$\nr = w_pr_p + w_vr_v + w_rr_r + w_er_e + w_{task}r_{task} ,\n\\vspace{-4pt}\n$$\nwhere $r_p= \\exp( -\\|\\mathbf{p}\\|_2)$ and $r_v = \\|\\dot{\\mathbf{p}}\\|_2$ drive the robot towards its target position and velocity as fast as possible;\n$r_r= 1-\\|\\mathbf{r}\\|_2$ drives the robot towards its target rotation (pose);\n$r_e =\\|\\mathbf{\\tau}\\|^2$ measures the effort exhausted during the swimming;\n$r_{task}$ is a task-specific reward, which will be specified later;\nand $w_p,w_v,w_r,w_e,w_{task}$ are the corresponding weights for different components in the reward.\nThe weights for each benchmark task are listed in Table.~\\ref{weight_table}.\n\\begin{table}[t]\t\n\t\\caption{Weights used in the reward of each task} \n\t\\vspace{-10pt}\n\t\\label{weight_table}\n\t\\begin{center}\n\t\\begin{tabular}{c|c|c|c|c|c}\n\t\t& $w_v$ & $w_p$ & $w_r$ & $w_e$ & $w_{task}$ \\\\ \\hline\n\t\tCruising           & 1  &0   & 0.2   & 0.5   & 0          \\\\\n\t\tPose Control       & 0  &0   & 1     & 0     & 0          \\\\\n\t\tTwo Fish Schooling & 0  &1   & 0     & 0.1   & 0          \\\\\n\t\tPath Following     & 1  &0   & 0     & 0.5   & 1       \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{center}\n\\vspace{-25pt}\n\\end{table}\nOur proposed four benchmark tasks are trained using either global or local policy learning approaches we described.\n\\paragraph{Global policy learning}\nFor cruising, pose control and two-fish schooling tasks, we use global policy learning with a single input task, meaning that we train robot fish separately for each task, where $r_{task}=0$. Fig.~\\ref{fig:tasksequence} shows the snapshots of the swimming results, where the first two rows show cruising inside a shallow and a deep fluid; the third row shows the pose control for U-turn, and the fourth row shows the two-fish schooling result, which has not been demonstrated in previous works.\n\n\\paragraph{Local policy learning}\n\\begin{figure}[htb]\n\t\\vspace{-6pt}\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{Images\/local_policy.png}\n\t\\vspace{-12pt}\n\t\\caption{Illustration of local policy training for arbitrary path following. Left: a local target is sampled given a random $d$ and $\\theta$; right: when applying learned policy for path following, we always select a local target ahead on the desired path, which changes for each time step.}\n\t\\label{fig:local_policy}\n\t\\vspace{-10pt}\n\\end{figure}\nTraining robot fish following an arbitrarily long path is more difficult, and global policy learning could be resource demanding and time consuming.\nTo make the training easier while also retaining generalizability, we use local policy learning instead.\nIn fact, following an arbitrarily long path can be viewed as following a sequence of short and straight local paths along nearly the tangent direction of the global path given a robot location, greatly simplifying the training process. \nDuring training, we randomly sample local paths (parameterized by $(d,\\theta)$, see Fig.~\\ref{fig:local_policy} (left), where $d$ is the distance to the local path and $\\theta$ is the angle to the target; note that we restrict the robot fish to swim in a horizontal 3D plane) and form a set of trajectories that are representative of the local conditions of a global path; then we can train the local policy once and apply it to any specified path at any time step, similar in idea to \\cite{Peng2017}. \nIn our local policy learning, the task specific state is defined as $s_{task} = \\mathbf{d}$, and the task specific reward $r_{task}$ is:\n\\vspace{-4pt}\n\\begin{equation}\nr_{task} = \\|\\dot{\\mathbf{d}}\\|_2+\\exp( -\\|\\mathbf{d}\\|_2) ,\n\\vspace{-4pt}\n\\end{equation}\nwhich encourages fast and stable convergence to the local path.\nHere, $\\mathbf{d}$ is a vector containing relative distance to the path.\nThe training is initialized randomly by the technique proposed in \\cite{Peng2018}, and on each trial, random initial velocity is enforced on the robot fish and random angles and velocities are set on the joints.\nOnce learned, we apply the policy every some time steps based on the input state and a local target location that is 0.5m ahead on the local path, see Fig.~\\ref{fig:local_policy} (right), and Fig.~\\ref{fig:tasksequence} (bottom) shows a path following result.\n\\section{Related Work}\nWe herein review the relevant work in the literature for both simulation and learning algorithms before we dig into the details of our whole framework.\n\n\\subsection{Simulation environments in robotics}\n\nRobot training can be achieved following OpenAI Gym~\\cite{Brockman2016}, which is an open-source robot learning framework with general definitions, and can be implemented for training a variety of robots with different environments.\nHowever, both the simulator and learning framework should be provided separately.\nAt present, the most commonly used physics simulators in robot are based on rigid-body, soft-body and cloth dynamics~\\cite{Coumans2015,Todorov2012}. \nIn particular, for articulated rigid body systems, DART~\\cite{Lee2018} can be a good choice.\n\nFluid environment was traditionally provided by solving N\\\"avier-Stoke (NS) equation coupled with a rigid body simulator~\\cite{Tan2011}; but efficiency limited their application especially for vortical flows. Recently, a new simulation environment for underwater soft-body creatures appeared relying on the finite-element method and projection dynamics~\\cite{min2019softcon}; however, its choice of empirical formula~\\cite{Terzopoulos1994} on hydrodynamics makes it impossible to create complex flow environment involving vortices and turbulence. The same issue also applies to some marine vehicle simulators~\\cite{cieslak2019stonefish,manhaes2016uuv} based on Fossen model\\cite{fossen2011handbook}.\nVery recently, Gan et al.~\\cite{Gan2020} proposed a fluid environment, but only for limited tasks and accuracy.\n\nThere is lack of versatile and efficient yet accurate fluid simulation environment upon which more general underwater robot training can be performed, and our proposed ``FishGym'' tries to fill the gap by providing highly efficient GPU-based simulator for two-way coupled fluid-structure interaction.\nThere are also some learning frameworks that can be used based on OpenAI Gym, e.g., rllib~\\cite{Liang2018}, Coach~\\cite{Caspi2017} and stable-baselines3~\\cite{Raffin2019}, and we adopted ``stable-baselines3'' for training our fish-like underwater robots. \n\n\n\n\n\\iffalse\n\\subsection{Fish dynamics and actuation}\\label{sec:fish_model_work}\nIn this paper, we focus on the fish-like underwater robots, and we briefly discuss the dynamics and actuation of fishes in order to motivate our later design.\n\n\\paragraph{Fish modeling}\nFish modeling targets the description of fish structure in a proper way for later simulation, and various models have been proposed with different complexity and accuracy.\nOne of the simplest models is to regard fish as a surface mesh with a mass-spring system to approximate the functionality of muscles \\cite{Terzopoulos1994}.\nAnother type of simple fish model is to employ non-deformable rigid\/flexible foil \\cite{Lu2013,Kaya2007,Hover2004,Wang2018}.\nHowever, these simple models cannot produce accurate fish dynamics. \nTo improve accuracy for simulation, articulated rigid body systems involving connected links with joint constraints were considered for underwater creatures \\cite{Tan2011, Liu2015}, but it is inaccurate for fishes.\nA better deformable fish model which is commonly used employs triangle meshes with center-line curvature driven deformation~\\cite{Ming2018,Song2017,Song2020,Novati2017,Verma2018}, but it is less flexible to allow different types of fishes and control behaviors.\nA more accurate physically-based approach is to model fish as a general volumetric, elastically deformable body \\cite{Pan2018}, but it is very costly during fish dynamics simulation.\n\nIn contrast, our fish is modeled by the skeletons of arbitrary topology with surface skinning, whose motion is driven by the articulated rigid body dynamics. This intuitive approach is easy to extend to other skeleton-based bionic robot shape. Secondly, compared to deformable approaches, the utilization of a mature articulated solver can be expected to step into real robotics. Third, our later coupling method which treat the skin as point samples which is physically more accurate and adaptive, compared to surface voxelization\\cite{Tan2011}.\n\nFish dynamics aims to describe the dynamic interaction between fishes and the surrounding fluid.\nLighthill proposed a large-amplitude elongated-body theory \\cite{Lighthill1975}, which first explains mathematically how fish moves in a fluid. \nStarting from that, researchers either used empirical force model \\cite{Terzopoulos1994} or more accurate fluid-structure interaction simulation ~\\cite{Song2017,Song2020,Verma2018,Novati2017} for fish dynamics description. \nWhile empirical model is fast, it is usually inaccurate; in addition, hyper-parameters inside the model are also difficult to tune in order to match the real physical system.\nDirect numerical simulation ~\\cite{dai2012dynamic,song2014three,Novati2017}, on the other hand, is more accurate and reliable, but it is usually very slow, leading to a very long training period. \nWhen fish is immersed inside fluid, the fish surface is often marked as the solid boundary condition for fluid simulation \\cite{Ming2018,Song2017,Song2020,Tan2011, Liu2015}, the fluid forces calculated at the surface are applied to drive the fish motion.\n\nThere are several models could be used for defining a fish's actuation. \nFor example, the torso of the fish is deformed through controlling the contraction ratio of springs in \\cite{Terzopoulos1994}.\nAn alternative way is to manipulate the centerline curvatures using mathematical functions ranging from interpolation \\cite{Song2017}, sinusoidal \\cite{Song2020}, exponential and polynomial \\cite{Ming2018} to weighted sum of cubic splines \\cite{Novati2017,Verma2018}. \nIn \\cite{Tan2011}, a robot is actuated by specifying expected joint poses for a PD controller to output torque signals. \n\\fi \n\n\n\n\\subsection{Fluid-structure interaction}\n\nFluid simulation has been studied for decades.\nTwo different fields have intensively progressed its development.\nIn computational fluid dynamics (CFD), fluid simulation mostly targets accuracy, and a set of fluid solvers are available, from finite difference \\cite{godunov1959finite,Rai-1991,smolarkiewicz1998mpdata}, to finite volume \\cite{eymard2000finite,versteeg2007introduction,pinelli2010immersed}, as well as to finite elements \\cite{wilson1983finite,girault2012finite,elman2014finite}.\nThese algorithms are typically very expensive, which are difficult for training underwater robots.\nIn computer graphics (CG), fluid simulation concerns efficiency more than accuracy, and a large number of more efficient yet less accurate solvers were proposed~\\cite{Stam2001,Kim2005FlowFixer,Becker2007Weakly,Ihmsen2014,Jiang2015Affine,Zehnder2018,Qu2019}.\nHowever, even though GPU acceleration has been used in some of these solvers, efficiency is still not high enough.\nWhen rigid body dynamics is coupled for fluid-structure interaction \\cite{klingner2006fluid,lv2010novel,dai2005adaptive}, the efficiency can be even lower.\n\nIn recent years, lattice Boltzmann method (LBM) has been considered as a very promising alternative to traditional fluid solvers~\\cite{Liu-2012,Daniel-2014,Rosis-2017,Li-2018,Li-2020}, exhibiting excellent efficiency and accuracy (usually an order of magnitude faster than the NS counterpart with comparable accuracy on GPU).\nIts pure local dynamics without solving global equations greatly benefits the highly parallel implementation~\\cite{Li-2018,Li-2020,Chen2021gpu}.\nWhen LBM is coupled with immersed boundary (IB) method~\\cite{Li-2020}, it can be easily used to simulate two-way coupled fluid-structure interaction.\nIn particular, Chen et al.~\\cite{Chen2021gpu} proposed a GPU-optimized implementation of IB-LBM, which provides a super-efficient solver for fluid-structure interaction, making the originally expensive fluid simulation now affordable for robot training purposes.\nOur proposed platform is based on such a solver, with modifications to allow it for simulating fish dynamics in a local moving domain for higher flexibility, which is not supported in any previous works. \n\n\n\n\n\\subsection{Reinforcement learning for robot control}\nReinfocement learning (RL) is a branch of machine learning which aims to train agents using data collected through interaction with the surrounding environment. \nFor real world problems in robotics, model-free RL algorithms are often used~\\cite{Nian2020}. \nThere are two main approaches of model-free RL: policy optimization and Q-learning. \nPolicy optimization algorithms, like PPO~\\cite{Schulman2017a} and A2C\/A3C~\\cite{Mnih2016}, are stable but sample-inefficient. \nQ-learning methods, like DQN~\\cite{Mnih2013} and C51~\\cite{Bellemare2017}, are more sample-efficient but less stable.\nBoth of them have wide applications. \nFor example, PPO was used in multi-robot collision avoidance task~\\cite{long2018towards}, bipedal robot locomotion~\\cite{li2021reinforcement} etc. \nDQN also proves to work well on a certain type of tasks in real robots~\\cite{kato2017autonomous,xin2017application,chen2021non}.\nRecently, SAC~\\cite{Haarnoja2018} emerges to combine the strengths of the above two main approaches and has proved its capability in real robot problems like Dexterous manipulation~\\cite{haarnoja2018soft}, mobile robot navigation~\\cite{de2021soft}, robot arm control~\\cite{wong2021motion}, multi-legged robot~\\cite{haarnoja2018learning}, etc.\nDue to its sample efficiency and wide applications, we adopt SAC in this paper.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAlthough the minimal Standard Model (SM) agrees well with \ncurrent electroweak experiments~\\cite{lep_slc_97}, \nit is important to examine consequences of new physics \nmodels beyond the SM at current or future collider experiments. \nOne of the simplest extensions of the SM is to introduce an \nadditional U(1) gauge symmetry, ${\\rm U(1)'}$, \nwhose breaking scale is close to the electroweak scale. \nThe ${\\rm U(1)'}$ symmetry is predicted in a certain class \nof grand unified theories (GUTs) with gauge group whose rank \nis higher than that of the SM. \nIn general, the additional ${\\rm U(1)'}$ gauge boson \n$Z'$ can mix with the hypercharge ${\\rm U(1)}_Y$ gauge boson \nthrough the kinetic term at above the electroweak scale, \nand also it can mix with the SM $Z$ boson after the \nelectroweak symmetry is spontaneously broken.  \nThrough those mixings, the $Z'$ boson can affect the electroweak \nobservables at the $Z$-pole and the $W$ boson mass $m_W$. \nBoth the $Z$-$Z'$ mixing and the direct $Z'$ contribution \ncan affect neutral current experiments off the $Z$-pole. \nThe presence of an additional $Z'$ boson can be explored \ndirectly at $p \\bar{p}$ collider experiments. \n\nThe supersymmetric (SUSY) $E_6$ models are the promising candidates \nwhich predict an additional $Z'$ boson at the weak scale (for \na review, see~\\cite{hewett_rizzo}). \nThe gauge group $E_6$ can arise from the perturbative heterotic \nstring theory as a consequence of its compactification.  \nIn the $E_6$ models, the SM matter fields in each generation \nare embedded into its fundamental representation ${\\bf 27}$ \nthat also contains several exotic matter fields -- two SM singlets, \na pair of weak doublets and color triplets. \nBecause $E_6$ is a rank-six group, it can have two extra \n${\\rm U(1)}$ factors besides the SM gauge group. \nA superposition of the two extra ${\\rm U(1)}$ groups may \nsurvive as the ${\\rm U(1)'}$ gauge symmetry at the GUT scale. \nThe ${\\rm U(1)'}$ symmetry may break spontaneously at the weak \nscale through the radiative corrections to the mass term of the \nSM singlet scalar field~\\cite{radiative_u1prime}. \n\nIn this paper, we study constraints on the $Z'$ bosons predicted \nin the SUSY $E_6$ models. \nAlthough there are several previous works~\\cite{altarelli,\nlangacker_luo,chm,gherghetta,zprime_lep}, we would like to \nupdate their studies by using the recent results of \nelectroweak experiments, and by allowing for an arbitrary \nkinetic mixing~\\cite{holdom, eta_model,general_zzmixing} \nbetween the $Z'$ boson and the hypercharge $B$ boson. \nIn our study, we use the results of $Z$-pole experiments \nat LEP1 and SLC, and the $m_W$ measurements at Tevatron and LEP2 \nwhich were reported at the summer conferences in \n1997~\\cite{lep_slc_97}. \nWe also study the constraints from low-energy neutral current \n(LENC) experiments: lepton-quark, lepton-lepton scattering \nexperiments and atomic parity violation measurements. \n\nWe find that the lower mass limit of the heavier mass \neigenstate $Z_2$ is obtained as a function of the effective $Z$-$Z'$ \nmixing term $\\zeta$, which is a combination of the mass and kinetic \nmixings. \nIn principle, $\\zeta$ is calculable, together with the gauge \ncoupling $g_E$, once the particle spectrum of the $E_6$ model \nis specified. \nWe show the theoretical prediction for $\\zeta$ and $g_E$ \nin the SUSY $E_6$ models by assuming the minimal particle \ncontent which satisfies the anomaly free condition and the \ngauge coupling unification.  \nFor those models, the electroweak data give stringent \nlower mass bound on the $Z_2$ boson. \n\nThis paper is organized as follows. \nIn the next section, we briefly review the additional $Z'$ \nboson in the SUSY $E_6$ models and the generic feature of \n$Z$-$Z'$ mixing in order to fix our notation. \nWe show that the effects of $Z$-$Z'$ mixing and direct \n$Z'$ boson contribution are parametrized \nby the following three terms: \n(i) a tree-level contribution to the $T$ parameter~\\cite{peskin_takeuchi}, \n$T_{\\rm new}$, \n(ii) the effective $Z$-$Z'$ mass mixing angle $\\bar{\\xi}$ \nand \n(iii) a contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$ which appears in the low-energy \nprocesses. \nIn Sec.~3, we collect the latest results of electroweak experiments. \nThere, the theoretical predictions for the electroweak \nobservables are shown together with the SM radiative corrections. \nIn Sec.~4, we show constraints on the $Z'$ bosons from the \nelectroweak data. \nThe presence of non-zero kinetic mixing between the \n${\\rm U(1)}_Y$ and ${\\rm U(1)'}$ gauge bosons \nmodifies the couplings between the $Z'$ boson and the SM \nfermions. \nWe discuss impacts of the kinetic mixing term \non the $\\chi^2$-analysis. \nThe 95\\% CL lower mass limit of the heavier mass eigenstate $Z_2$ \nis given as a function of the effective $Z$-$Z'$ mixing parameter \n$\\zeta$. \nThe $\\zeta$-independent constraints from the low-energy \nexperiments and those from the direct search experiments \nat Tevatron are also discussed. \nIn Sec.~5, \nwe find the theoretical prediction for $\\zeta$ in some \nSUSY $E_6$ models ($\\chi, \\psi, \\eta, \\nu$) \nby assuming the minimal particle content.  \nStringent $Z_2$ boson mass bounds are found for most models.\nSec.~6 summarizes our findings. \n\\section{$Z$-$Z'$ mixing in supersymmetric $E_6$ model}\n\\subsection{$Z'$ boson in supersymmetric $E_6$ model}\n\\setcounter{equation}{0} \nSince the rank of $E_6$ is six, \nit has two ${\\rm U(1)}$ factors besides the SM gauge \ngroup which arise from the following decompositions: \n\\begin{eqnarray}\n\t\\begin{array}{rl}\n\tE_6 &\\supset {\\rm SO(10)} \\times {\\rm U(1)}_\\psi  \n\\\\\n\t&\\supset {\\rm SU(5)} \\times {\\rm U(1)}_\\chi \\times {\\rm U(1)}_\\psi. \n\t\\end{array}\n\\end{eqnarray}\n\\label{eq:e6_breaking}\n\\hsp{-0.3}\nAn additional $Z'$ boson in the electroweak scale \ncan be parametrized as \na linear combination of the ${\\rm U(1)}_\\psi$ gauge boson \n$Z_\\psi$ and the ${\\rm U(1)}_\\chi$ gauge boson \n$Z_\\chi$ as~\\cite{PDG} \n\\begin{equation}\nZ' = Z_\\chi \\cos \\beta_E + Z_\\psi \\sin \\beta_E. \n\\end{equation}\nIn this paper, we study the following four $Z'$ models \nin some detail: \n\\begin{eqnarray}\n\\begin{array}{|c||c|c|c|c|} \\hline \n~~~\\beta_E~~~ & ~~~0~~~ &~~~\\pi\/2~~~ & \\tan^{-1}(-\\sqrt{5\/3}) \n& \\tan^{-1}(\\sqrt{15}) \\\\ \\hline\n{\\rm model} & \\chi      & \\psi         & \\eta      & \\nu      \n\\\\  \\hline \n\\end{array}\n\\end{eqnarray}\nIn the SUSY-$E_6$ models, each generation of the SM quarks \nand leptons is embedded into a {\\bf 27} representation. \nIn Table~1, we show all the matter fields \ncontained in a {\\bf 27} and their classification in SO(10) \nand SU(5). \nThe ${\\rm U(1)'}$ charge assignment on the matter fields \nfor each model is also given in the same table. \nThe normalization of the ${\\rm U(1)'}$ charge follows \nthat of the hypercharge. \n\\esix_contents\nBesides the SM quarks and leptons, there are two SM singlets \n$\\nu^c$ and $S$, a pair of weak doublets $H_u$ and $H_d$, \na pair of color triplets $D$ and $\\overline{D}$ in each generation. \nThe $\\eta$-model arises when $E_6$ breaks into a rank-5 group \ndirectly in a specific compactification of the heterotic \nstring theory~\\cite{eta_ellis}. \nIn the $\\nu$-model, the right-handed neutrinos $\\nu^c$ are \ngauge singlet~\\cite{nu_model} and can have large Majorana \nmasses to realize the see-saw mechanism~\\cite{see-saw}. \n\nThe ${\\rm U(1)'}$ symmetry breaking occurs if the scalar \ncomponent of the SM singlet field develops the vacuum \nexpectation value (VEV). \nIt can be achieved at near the weak scale via radiative \ncorrections to the mass term of the SM singlet scalar \nfield. \nFor example, the terms $SD\\overline{D}$ and $S H_u H_d$ appear \nin the ${\\rm SU(3)}_C \\times {\\rm SU(2)}_L \\times \n{\\rm U(1)}_Y \\times {\\rm U(1)'}$ invariant superpotential. \nIf the Yukawa couplings of the $SD\\overline{D}$ term and\/or \n$S H_u H_d$ term are $O(1)$, the squared mass of the \nscalar component of $S$ can become negative at the weak \nscale through the renormalization group equations (RGEs) \nwith an appropriate boundary condition at the GUT scale. \nRecent studies of the radiative ${\\rm U(1)'}$ symmetry \nbreaking can be found, {\\it e.g.}, in ref.~\\cite{radiative_u1prime}. \n\nSeveral problems may arise in the $E_6$ models from view \nof low-energy phenomenology~\\cite{hewett_rizzo}.  \nFor example, the scalar components of extra colored triplets \n$D, \\overline{D}$ in ${\\bf 27}$ could mediate an instant proton \ndecay. \nIt should be forbidden by imposing a certain discrete symmetry \non the general \n${\\rm SU(3)}_C \\times {\\rm SU(2)}_L \\times {\\rm U(1)}_Y \\times\n{\\rm U(1)'}$ invariant superpotential. \nExcept for the $\\nu$-model~\\cite{nu_model}, the large Majorana \nmass of $\\nu^c$ is forbidden by the ${\\rm U(1)'}$ gauge \nsymmetry, and the fine-tuning is needed to make the Dirac neutrino \nmass consistent with the observation. \nFurther discussions can be found in ref.~\\cite{hewett_rizzo}.  \nIn the following, we \nassume that these requirements are satisfied by an unknown \nmechanism. \nMoreover we assume that all the super-partners of the SM \nparticles and the exotic matters do not affect the radiative \ncorrections to the electroweak observables significantly,  \n{\\it i.e.}, they are assumed to be heavy enough to decouple from \nthe weak boson mass scale. \n\n\\subsection{Phenomenological consequences of $Z$-$Z'$ mixing}\nIf the SM Higgs field carries a non-trivial ${\\rm U(1)'}$ \ncharge, its VEV induces the $Z$-$Z'$ mass mixing. \nOn the other hand, the kinetic mixing between the hypercharge \ngauge boson $B$ and the ${\\rm U(1)'}$ gauge boson $Z'$ \ncan occur through the quantum effects below the GUT scale. \nAfter the electroweak symmetry is broken, the effective \nLagrangian for the neutral gauge bosons in the \n${\\rm SU(2)}_L \\times {\\rm U(1)}_Y \\times {\\rm U(1)'}$ \ntheory is given by~\\cite{eta_model} \n\\begin{eqnarray}\n{\\cal L}_{gauge} \n\t&=&  -\\frac{1}{4}Z^{\\mu\\nu}Z_{\\mu\\nu}\n            -\\frac{1}{4}Z'^{\\mu\\nu}Z'_{\\mu\\nu} \n\t    -\\frac{\\sin \\chi}{2}B^{\\mu\\nu}Z'_{\\mu\\nu}\n\t    -\\frac{1}{4}A^{0\\mu\\nu}A^{0}_{\\mu\\nu} \n\\nonumber \\\\ \n\t& & + m^2_{ZZ'} Z^{\\mu}Z'_{\\mu}\n\t    +\\frac{1}{2} m^2_Z Z^{\\mu}Z_{\\mu}\n\t    +\\frac{1}{2} m^2_{Z'} Z'^\\mu Z'_{\\mu}, \n\\label{eq:l_gauge}\n\\end{eqnarray} \nwhere $F^{\\mu\\nu} (F=Z,Z',A^0,B)$ represents the gauge field strength. \nThe $Z$-$Z'$ mass mixing and the kinetic mixing are characterized \nby $m^2_{ZZ'}$ and $\\sin \\chi$, respectively. \nIn this basis, the interaction Lagrangian for the neutral current \nprocess is given as \n\\begin{eqnarray}\n{\\cal L}_{NC} &=& -\\sum_{f,\\, \\alpha}  \\left\\{ \n\t\\; e \\, Q_{f^{}_{\\alpha}} \\overline{f^{}_{\\alpha}}\n\t\\gamma^{\\mu}f^{}_{\\alpha} A^0_{\\mu} +\n\tg^{}_Z \\overline{f^{}_{\\alpha}} \\gamma^{\\mu}\n\t\\left( I^3_{f_L} - Q_{f^{}_{\\alpha}} \\sin^2\\theta_W \\right)\n\tf^{}_{\\alpha} Z^{}_{\\mu} \\right. \\nonumber \\\\ \n\t& & \\left. + g^{}_E Q^{f^{}_{\\alpha}}_E \n\t\\overline{f^{}_{\\alpha}}\\gamma^{\\mu}f^{}_{\\alpha} \n\tZ'_{\\mu} \\right\\}, \n\\label{eq:neutraC}\n\\end{eqnarray}\nwhere $g_Z = g\/\\cos\\theta_W = g_Y\/\\sin\\theta_W$. \nThe ${\\rm U(1)'}$ gauge coupling constant is denoted by $g_E$ \nin the hypercharge normalization. \nThe symbol $f_\\alpha$ denotes the quarks or leptons with \nthe chirality $\\alpha$ ($\\alpha = L$ or $R$). \nThe third component of the weak isospin, the electric charge \nand the ${\\rm U(1)'}$ charge of $f_\\alpha$ are given by \n$I^3_{f_\\alpha}$, $Q_{f_\\alpha}$ and $Q_E^{f_\\alpha}$, respectively. \nThe ${\\rm U(1)'}$ charge of the quarks and leptons listed in Table~1 \nshould be read as \n\\begin{eqnarray}\n\t\\left.\n\t\\begin{array}{l}\nQ_E^Q = Q_E^{u_L} = Q_E^{d_L},~~~Q_E^L = Q_E^{\\nu_L} = Q_E^{e_L}, \n\\\\ \nQ_E^{f^c} = -Q_E^{f_R} ~~~(f=e,u,d),\n\t\\end{array}\n\t\\right \\}. \n\\label{eq:charge_rule}\n\\end{eqnarray}\nThe mass eigenstates $(Z_1,Z_2,A)$ is obtained by \nthe following transformation; \n\\begin{equation}\n\\left( \\begin{array}{c}  Z \\\\  Z' \\\\  A^0  \n\\end{array}\\right) \n= \n\\left(\n\t\\begin{array}{ccc}\n\\cos \\xi + \\sin \\xi \\sin \\theta_W \\tan \\chi &\n-\\sin \\xi + \\cos \\xi \\sin\\theta_W \\tan \\chi & 0 \\\\\n\\sin \\xi \/ \\cos \\chi & \\cos \\xi \/ \\cos \\chi & 0 \\\\\n-\\sin\\xi \\cos \\theta_W \\tan \\chi & \n- \\cos \\xi \\cos \\theta_W \\tan \\chi & 1   \n\t\\end{array}\n\\right) \n\\left( \\begin{array}{c} {Z_1} \\\\ \n{Z_2} \\\\ {A} \\end{array}\\right). \n\\end{equation}\nHere the mixing angle $\\xi$ is given by \n\\begin{equation}\n\\tan 2\\xi = \\frac{-2c^{}_{\\chi}(m^2_{ZZ'}+s^{}_W s^{}_{\\chi}\n            m^2_Z)}{m^2_{Z'} - (c^2_{\\chi}-s^2_W s^2_{\\chi})m^2_Z+\n            2s^{}_W s^{}_{\\chi} m^2_{ZZ'}}~, \n\\label{eq:angle_xi}\n\\end{equation}\nwith the short-hand notation, \n$c_\\chi =  \\cos\\chi$, $s_\\chi = \\sin\\chi$ and $s_W = \\sin\\theta_W$. \nThe physical masses $m_{Z_1}$ and $m_{Z_2}$ ($m_{Z_1} < m_{Z_2}$) \nare given as follows; \n\\begin{subequations}\n\\begin{eqnarray}\nm_{Z_{1}}^2 \n\t &=& m_Z^2 (c_\\xi + s_\\xi s_W t_\\chi)^2 \n\t+ m_{Z'}^2 \\biggl( \\frac{s_\\xi}{c_\\chi} \\biggr)^2\n\t+ 2 m^2_{ZZ'} \\frac{s_\\xi}{c_\\chi} (c_\\xi + s_\\xi s_W t_\\chi),  \n\\label{eq:light_Z1}\n\\\\\nm_{Z_{2}}^2 \n\t &=& m_Z^2 (c_\\xi s_W t_\\chi - s_\\xi)^2 \n\t+ m_{Z'}^2 \\biggl( \\frac{c_\\xi}{c_\\chi} \\biggr)^2\n\t+ 2 m^2_{ZZ'} \\frac{c_\\xi}{c_\\chi} (c_\\xi s_W t_\\chi -s_\\xi), \n\\end{eqnarray}\n\\end{subequations}\nwhere $c_\\xi = \\cos\\xi$, $s_\\xi = \\sin\\xi$ and $t_\\chi = \\tan\\chi$. \nThe lighter mass eigenstate $Z_1$ should be identified with \nthe observed $Z$ boson at LEP1 or SLC. \nThe excellent agreement between the current experimental results \nand the SM predictions at the quantum level implies that the \nmixing angle $\\xi$ have to be small. \nIn the limit of small $\\xi$, the interaction Lagrangians \nfor the processes \n$Z_{1,2} \\rightarrow f_\\alpha \\overline{f_\\alpha}$ are expressed as \n\\begin{subequations}\n\\begin{eqnarray}\n{\\cal L}_{Z_1} &=& -\\sum_{f,\\, \\alpha} g^{}_Z \n\t\\overline{f_{\\alpha}} \\gamma^{\\mu} \\left[\n\t\\left( I^{3}_{f_{L}}-Q_{f_{\\alpha}}\\sin^2\\theta_W \n\t\\right) + \\tilde{Q}^{f_{\\alpha}}_E \\bar{\\xi} \\right]\n\tf_{\\alpha}  Z_{1 \\mu}, \n\\label{eq:neutral1}\\\\ \n{\\cal L}_{Z_2} &=& -\\sum_{f,\\,\\alpha}\\frac{g^{}_E}{c^{}_{\\chi}}\n\t\\overline{f_{\\alpha}}\\gamma^{\\mu} \\left[ \\tilde{Q}^{f_{\\alpha}}_E\n\t-\\left( I^3_{f_{\\alpha}} - Q_{f_{\\alpha}}\n\t\\sin^2\\theta_W \\right) \\frac{g^{}_Z c^{}_{\\chi}}{g^{}_E}\n\t\\xi \\right]f_{\\alpha} Z_{2 \\mu}, \n\\label{eq:neutral2}\n\\end{eqnarray}\n\\label{eq:neutralboth}\n\\end{subequations}\n\\hsp{-0.3} \nwhere the effective mixing angle $\\bar{\\xi}$ \nin eq.~(\\ref{eq:neutral1}) is given as \n\\begin{equation}\n\\bar{\\xi} = \\frac{g_E}{g_Z\\cos \\chi } \\xi. \n\\end{equation}\nIn eq.~(\\ref{eq:neutralboth}), the effective ${\\rm U(1)'}$ \ncharge $\\tilde{Q}_E^{f_\\alpha}$ is introduced as \na combination of $Q_E^{f_\\alpha}$ and the hypercharge $Y_{f_\\alpha}$: \n\\begin{subequations}\n\\begin{eqnarray}\n\\tilde{Q}^{f_\\alpha}_E &\\equiv& Q^{f_\\alpha}_E + Y_{f_\\alpha} \\delta, \n\\label{eq:effective_charge}\\\\\n\\delta &\\equiv&  -\\frac{g^{}_Z}{g^{}_E}s^{}_W s^{}_{\\chi}, \n\\end{eqnarray}\n\\label{eq:u1_charge}\n\\end{subequations}\n\\hsp{-0.3}\nwhere the hypercharge $Y_{f_\\alpha}$ should be read from Table~1 \nin the same manner with $Q_E^{f_\\alpha}$ \n(see, eq.~(\\ref{eq:charge_rule})).  \nAs a notable example, one can see from Table~1 \nthat the effective charge $\\tilde{Q}_E^{f_\\alpha}$ of \nthe leptons ($L$ and $e^c$) disappears in the \n$\\eta$-model if $\\delta$ is taken to be $1\/3$~\\cite{eta_model}. \n\nNow, due to the $Z$-$Z'$ mixing, the observed \n$Z$ boson mass $m_{Z_1}$ at LEP1 or SLC is shifted from \nthe SM $Z$ boson mass $m_Z$: \n\\begin{equation}\n\\Delta m^2 \\equiv m_{Z_1}^2 - m_Z^2 \\le 0. \n\\label{eq:mass_shift}\n\\end{equation}\nThe presence of the mass shift affects the \n$T$-parameter~\\cite{peskin_takeuchi} at tree level. \nFollowing the notation of ref.~\\cite{hhkm}, the $T$-parameter \nis expressed in terms of the effective form factors $\\bar{g}_Z^2(0), \n\\bar{g}_W^2(0)$ and the fine structure constant $\\alpha$:\n\\begin{subequations}\n\\begin{eqnarray}\n\\alpha T &\\equiv& 1 - \\frac{\\bar{g}^2_{W}(0)}{m^2_W}\n\t\\frac{m^2_{Z_1}}{\\bar{g}^2_Z(0)}  \\\\ \n\t&=&\n\t\\alpha \\left(T_{\\rm SM}^{}+  T_{\\rm new}^{}\\right), \n\\end{eqnarray}\n\\end{subequations}\nwhere \n$T_{\\rm SM}^{}$ and the new physics contribution \n$T_{\\rm new}$ are given by: \n\\begin{subequations}\n\\begin{eqnarray}\n\\alpha T_{\\rm SM} \n\t&=& 1 - \\frac{\\bar{g}^2_{W}(0)}{m^2_W}\n\t\t\\frac{m^2_{Z}}{\\bar{g}^2_Z(0)}, \\\\\n\\alpha T_{\\rm new}\n\t& = & -\\frac{ \\Delta m^2}{m^2_{Z_1}} \\geq 0. \n\\end{eqnarray}\n\\end{subequations}\nIt is worth noting that the sign of $T_{\\rm new}$ is \nalways positive. \nThe effects of the $Z$-$Z'$ mixing in the $Z$-pole experiments \nhave hence been parametrized by the effective mixing angle \n$\\bar{\\xi}$ and the positive parameter $T_{\\rm new}$. \n\nWe note here that we retain \nthe kinetic mixing term $\\delta$ as a part of the effective $Z_1$ \ncoupling $\\tilde{Q}_E^{f_\\alpha}$ in eq.~(\\ref{eq:effective_charge}). \nAs shown in refs.~\\cite{eta_model,general_zzmixing,holdom2}, \nthe kinetic mixing term $\\delta$ can be absorbed into a further \nredefinition of $S$ and $T$. \nSuch re-parametrization may be useful \nif the term $Y_{f_\\alpha} \\delta$ in eq.~(\\ref{eq:effective_charge}) \nis much larger than the $Z'$ charge $Q_E^{f_\\alpha}$. \nIn the $E_6$ models studied in this paper, we find no merit in \nabsorbing the $Y_f \\delta$ term because, the remaining \n$Q_E^{f_\\alpha}$ term is always significant. \nWe therefore adopt $\\tilde{Q}_E^{f_\\alpha}$ as the effective \n$Z_1$ couplings and $T_{\\rm new}$ accounts only for the \nmass shift (\\ref{eq:mass_shift}). \nAll physical consequences such as the bounds on $\\bar{\\xi}$ and \n$m_{Z_2}$ are of course independent of our choice of the \nparametrization. \n\nThe two parameters $T_{\\rm new}$ and $\\bar{\\xi}$ \nare complicated functions of the parameters of the effective \nLagrangian (\\ref{eq:l_gauge}). \nIn the small mixing limit, \nwe find the following useful expressions \n\\begin{subequations}\n\\begin{eqnarray}\n\\bar{\\xi} &=& -\\biggl( \\frac{g_E}{g_Z}\\frac{m_Z}{m_{Z'}} \n\t\\biggr)^2 \\zeta \\biggl[ 1+ O(\\frac{m_Z^2}{m_{Z'}^2})\\biggr], \n\\\\\n\\alpha T_{\\rm new} &=& \\hphantom{-} \\biggl( \\frac{g_E}{g_Z}\n\t\t\\frac{m_Z}{m_{Z'}} \\biggr)^2 \\zeta^2\n\t\\biggl[ 1+ O(\\frac{m_Z^2}{m_{Z'}^2})\\biggr], \n\\end{eqnarray}\n\\label{eq:tnew_xibar}\n\\end{subequations}\n\\hsp{-0.3}\nwhere we introduced an effective mixing parameter $\\zeta$ \n\\begin{equation}\n\\zeta = \\frac{g_Z}{g_E}\\frac{m_{ZZ'}^2}{m_Z^2} - \\delta.  \n\\label{eq:zeta}\n\\end{equation}\nThe $Z$-$Z'$ mixing effect disappears at $\\zeta = 0$. \nStringent limits on $m_{Z'}$ and hence on $m_{Z_2}$ \ncan be obtained through the mixing effect if $\\zeta$ is \n$O(1)$. \nWe will show in Sec.~5 that $\\zeta$ is calculable \nonce the particle spectrum of the model is specified. \nThe parameter $\\zeta$ plays an essential role in \nthe analysis of $Z'$ models. \n\nIn the low-energy neutral current processes, effects of \nthe exchange of the heavier mass eigenstate $Z_2$ can be \ndetected. \nIn the small $\\bar{\\xi}$ limit, they constrain the contact \nterm $g_E^2\/c_\\chi^2 m_{Z_2}^2$. \n\\section{Electroweak observables in the $Z'$ model}\n\\setcounter{equation}{0}\nIn this section, we give the theoretical predictions for the \nelectroweak observables which are used in our analysis. \nThe experimental data of the $Z$-pole experiments and the $W$ \nboson mass measurement~\\cite{lep_slc_97} \nare summarized in Table~2. \nThose for the low-energy experiments~\\cite{chm} are listed in \nTable~\\ref{table:low_energy}. \n\\lep_table \nL.E.N.C.\\ _table \n\\subsection{Observables in $Z$-pole experiments}\nThe decay amplitude for the process \n$Z^{}_1 \\rightarrow f_\\alpha \\overline{f_\\alpha}$ is written as\n\\begin{equation}\nT(Z_1 \\rightarrow f_{\\alpha} \\overline{f_{\\alpha}}) \n\t= M^f_{\\alpha}~\\epsilon_{Z_1} \\cdot J_{f_{\\alpha}}, \n\\label{eq:decay_amp}\n\\end{equation}\nwhere $\\epsilon_{Z_1}^\\mu$ is the polarization vector of the \n$Z^{}_1$ boson and \n$J^{\\mu}_{f_{\\alpha}} = \\overline{f_\\alpha}\\gamma^\\mu f_\\alpha$ is \nthe fermion current without the coupling factors. \nThe pseudo-observables of the $Z$-pole experiments are \nexpressed in terms of the real scalar amplitudes $M_\\alpha^f$ \nwith the following normalization~\\cite{lep_slc_97} \n\\begin{equation}\ng^f_{\\alpha} = \\frac{M^f_{\\alpha}}{\\sqrt{4\\sqrt{2}G_Fm^2_{Z_1}}} \n\t\\approx \\frac{M^f_{\\alpha}}{0.74070}. \n\\end{equation}\n\nFollowing our parametrization of the $Z$-$Z'$ mixing in \neq.~(\\ref{eq:neutral1}), the effective coupling $g^f_{\\alpha}$ \nin the $Z'$ models can be expressed as \n\\begin{eqnarray}\ng_\\alpha^f = (g_\\alpha^f)_{\\rm SM} + \\tilde{Q}_E^{f_\\alpha} \\bar{\\xi}. \n\\end{eqnarray}\nThe SM predictions~\\cite{hhkm, hhm} for the effective \ncouplings $(g_\\alpha^f)_{\\rm SM}$ can be parametrized as \n\\begin{subequations}\n\\begin{eqnarray}\n(g^{\\nu}_L)_{\\rm SM} &=& \\makebox[3.3mm]{ } 0.50214 \n\t+ 0.453\\,{\\Delta \\bar{g}^2_Z},\n\\label{eq:amp_nu} \\\\\n(g^e_L)_{\\rm SM} &=& - 0.26941 - 0.244\\,{\\Delta \\bar{g}^2_Z} \n\t+ 1.001\\,{\\Delta \\bar{s}^2},  \n\\label{eq:amp_el}\\\\\n(g^e_R)_{\\rm SM} &=& \\makebox[3.3mm]{ } 0.23201 + 0.208\\,\n\t{\\Delta \\bar{g}^2_Z} + 1.001\\,{\\Delta \\bar{s}^2}, \n\\label{eq:amp_er}\\\\\n(g^u_L)_{\\rm SM} &=&  \\makebox[3.3mm]{ } 0.34694 + 0.314\\,\n\t{\\Delta \\bar{g}^2_Z} - 0.668\\,{\\Delta \\bar{s}^2}, \n\\label{eq:amp_ul}\\\\\n(g^u_R)_{\\rm SM} &=& - 0.15466 - 0.139\\,{ \\Delta \\bar{g}^2_Z}  \n\t- 0.668 \\,{\\Delta \\bar{s}^2},  \n\\label{eq:amp_ur}\\\\\n(g^d_L)_{\\rm SM} &=& - 0.42451 - 0.383\\,{\\Delta \\bar{g}^2_Z} \n\t+ 0.334\\,{\\Delta \\bar{s}^2}, \n\\label{eq:amp_dl}\\\\\n(g^d_R)_{\\rm SM} &=& \\makebox[3.3mm]{ }  0.07732\n\t+ 0.069\\,{ \\Delta \\bar{g}^2_Z} + 0.334\\,{\\Delta \\bar{s}^2},  \n\\label{eq:amp_dr}\\\\\n(g^b_L)_{\\rm SM} &=& - 0.42109 - 0.383\\,{ \\Delta \\bar{g}^2_Z} \n\t+ 0.334\\,{\\Delta \\bar{s}^2} \n\t+ 0.00043 x_t, \n\\label{eq:amp_bl}\n\\end{eqnarray}\n\\label{eq:amp_sm}\n\\end{subequations}\n\\hsp{-0.3}\nwhere the SM radiative corrections are expressed in terms of \nthe effective couplings $\\Delta \\gzbar$ and $\\Delta \\sbar$~\\cite{hhkm, hhm} \nand the top-quark mass dependence of the $Zb_L^{} b_L^{}$ vertex \ncorrection in $(g^b_L)_{\\rm SM}$ is parametrized by \nthe parameter $x_t^{}$ \n\\begin{equation}\nx_t^{} \\equiv \\frac{m_t - 175~{\\rm GeV}}{10~{\\rm GeV}}. \n\\end{equation}\nThe gauge boson propagator corrections, $\\Delta \\gzbar$ and $\\Delta \\sbar$, \nare defined as the shift in the effective couplings \n$\\bar{g}_Z^2(m_{Z_1}^2)$ and $\\bar{s}^2(m_{Z_1}^2)$~\\cite{hhkm} \nfrom their SM reference values at $m_t^{} = 175~{\\rm GeV}$ and \n$m_H^{} = 100~{\\rm GeV}$. \nThey can be expressed in terms of the $S$ and $T$ parameters as \n\\begin{subequations}\n\\begin{eqnarray}\n\\!\\!\\!\\!\\!\\!\\!\\! \\Delta \\bar{g}^2_Z &=&  \\bar{g}_Z^2(m_{Z_1}^2) - 0.55635 \n\t= 0.00412 \\Delta T + 0.00005[1-(100~{\\rm GeV}\/m^{}_H)^2], \\\\  \n\\!\\!\\!\\!\\!\\!\\!\\! \\Delta \\bar{s}^2 &=& \\bar{s}^2(m_{Z_1}^2) - 0.23035 \n\t= 0.00360 \\Delta S - 0.00241 \\Delta T - 0.00023 x_\\alpha^{}, \n\\end{eqnarray}\n\\end{subequations}\nwhere the expansion parameter $x_\\alpha^{}$ is introduced to estimate \nthe uncertainty of the hadronic contribution to the QED coupling \n$1\/\\overline{\\alpha}(m_{Z_1}^2) = 128.75 \\pm 0.09$~\\cite{eidelman}:\n\\begin{equation}\nx_\\alpha^{} \\equiv \\frac{1\/\\overline{\\alpha}(m_{Z_1}^2) - 128.75}{0.09}. \n\\label{eq:xa_qed}\n\\end{equation}\nHere, $\\Delta S,\\Delta T,\\Delta U$ parameters are also measured from their \nSM reference values and they are given as the sum of the SM \nand the new physics contributions \n\\begin{equation}\n\\Delta S = \\Delta S^{}_{\\rm SM}+S^{}_{\\rm new}, \\;\\;\n\\Delta T = \\Delta T^{}_{\\rm SM}+T^{}_{\\rm new}, \\;\\; \n\\Delta U = \\Delta U^{}_{\\rm SM}+U^{}_{\\rm new}. \n\\label{eq:stu_delta}\n\\end{equation}\nThe SM contributions can be parametrized as~\\cite{hhm}\n\\begin{subequations}\n\\begin{eqnarray}\n\\Delta S^{}_{\\rm SM} &=& - 0.007 x^{}_t +0.091 x^{}_H -0.010 x^2_H , \\\\\n\\Delta T^{}_{\\rm SM} &=& (0.130 - 0.003 x^{}_H) x^{}_t +0.003 x^{}_t \n- 0.079 x^{}_H -0.028 x^2_H \\nonumber \\\\ & & +0.0026 x^3_H, \\\\\n\\Delta U^{}_{\\rm SM} &=& 0.022 x^{}_t -0.002x^{}_H, \n\\end{eqnarray}\n\\end{subequations}\nwhere $x_H^{}$ is defined by\n\\begin{eqnarray}\nx_H^{} &\\equiv& \\log (m_H^{}\/100~{\\rm GeV}). \n\\label{eq:higgs_xh}\n\\end{eqnarray}\n\nThe pseudo-observables of the $Z$-pole experiments \nare given by using the above eight effective couplings $g_\\alpha^f$ \nas follows. \nThe partial width of $Z_1$ boson is given by \n\\begin{eqnarray}\n\\Gamma_f &=& \\frac{G_Fm_{Z_1}^{3}}{3\\sqrt{2} \\pi} \\left\\{ \n\t\\left| g^f_L + g^f_R \\right|^2\\frac{C_{fV}}{2} \n\t+ \\left| g^f_L - g^f_R \\right|^2 \\frac{C_{fA}}{2}  \\right\\}\n\t\\left( 1+\\frac{3}{4}Q^2_f\\frac{\\bar{\\alpha}(m^2_{Z_1})}{\\pi}\\right)\n        \\makebox[10mm][l]{,}\n\\label{eq:partial_width}\n\\end{eqnarray}\nwhere the factors $C_{fV}$ and $C_{fA}$ account for the \nfinite mass corrections and the final state QCD corrections \nfor quarks. \nTheir numerical values are listed in Table~\\ref{tab:cvca}. \nThe $\\alpha_s$-dependence in $C_{qV}, C_{qA}$ \nis parametrized in terms of the parameter $x_s^{}$\n\\begin{equation}\nx_s^{} \\equiv \\frac{\\alpha_s(m_{Z_1})-0.118}{0.003}. \n\\end{equation}\nThe last term proportional to $\\bar{\\alpha}(m^2_{Z_1})\/\\pi$ \nin eq.~(\\ref{eq:partial_width}) accounts for the \nfinal state QED correction. \n\\cvca_tab\nThe total decay width $\\Gamma_{Z_1}$ and the hadronic decay \nwidth $\\Gamma_h$ are given in terms of $\\Gamma_f$: \n\\begin{subequations}\n\\begin{eqnarray}\n\\Gamma_{Z_1} &=& 3\\Gamma_{\\nu} + \\Gamma_e \n\t+ \\Gamma_{\\mu} + \\Gamma_{\\tau} + \\Gamma_h, \n\\label{eq:total_width}\\\\\n\\Gamma_h &=& \\Gamma_u + \\Gamma_c + \\Gamma_d + \\Gamma_s + \\Gamma_b. \n\\label{eq:hadron_width}\n\\end{eqnarray}\n\\end{subequations}\nThe ratios $R_\\ell^{}, R_c^{}, R_b^{}$ and the hadronic peak \ncross section $\\sigma_h^0$ are given by: \n\\begin{equation}\nR_{\\ell} = \\frac{\\Gamma_h}{\\Gamma_e},\\;\nR_c      = \\frac{\\Gamma_c}{\\Gamma_h},\\;\nR_b      = \\frac{\\Gamma_b}{\\Gamma_h},\\;\n\\sigma^0_h = \\frac{12\\pi}{m^2_{Z_1}}\n\t\\frac{\\Gamma_e\\Gamma_h}{\\Gamma_{Z_1}^2}. \n\\end{equation}\n\nThe left-right asymmetry parameter $A^{f}$ is also \ngiven in terms of the effective couplings $g_\\alpha^f$ as  \n\\begin{equation}\nA^f \t= \\frac{(g^{f}_L)^2-(g^{f}_R)^2}{(g^{f}_L)^2+(g^{f}_R)^2}. \n\\end{equation}\nThe forward-backward (FB) asymmetry $A^{0,f}_{FB}$ \nand the left-right (LR) asymmetry $A^{0,f}_{LR}$ \nare then given as follows: \n\\begin{subequations}\n\\begin{eqnarray}\nA^{0,f}_{FB} &=& \\frac{3}{4}A^{e}A^{f}, \\\\\nA^{0,f}_{LR} &=&  A^{f}. \n\\end{eqnarray}\n\\end{subequations}\n\n\\subsection{$W$ boson mass}\nThe theoretical prediction of $m^{}_W$\ncan be parametrized as~\\cite{hhkm,hhm}\n\\begin{equation}\nm_W^{}{\\rm (GeV)} =80.402-0.288\\,{\\it \\Delta S}+0.418\\,{\\it \\Delta T} \n+0.337\\,{\\it \\Delta U}+0.012\\,{\\it x_{\\alpha}}, \n\\end{equation}\nby using the same parameters, $\\Delta S, \\Delta T, \\Delta U$ (\\ref{eq:stu_delta}) \nand $x_\\alpha^{}$ (\\ref{eq:xa_qed}). \n\n\\subsection{Observables in low-energy experiments} \n\\setcounter{equation}{0}\nIn this subsection, we show the theoretical predictions for the \nelectroweak observables in the low-energy neutral current \nexperiments (LENC) --- \n(i) polarization asymmetry of the charged lepton scattering off \nnucleus target (\\ref{section_slac}--\\ref{section_mainz}), \n(ii) parity violation in cesium atom (\\ref{section_apv}),  \n(iii) inelastic $\\nu_\\mu$-scattering off nucleus target \n(\\ref{section_nq}) and \n(iv) neutrino-electron scattering (\\ref{section_ne}). \nThe experimental data are summarized in Table~\\ref{table:low_energy}. \nTheoretical expressions for the observables of \n(i) and (ii) are conveniently given in terms of \nthe model-independent parameters $C_{1q}, C_{2q}$~\\cite{jekim} \nand $C_{3q}$~\\cite{chm}.  \nThe $\\nu_\\mu$-scattering data (iii) and (iv) are expressed \nin terms of the parameters $g_{L\\alpha}^{\\nu_\\mu f}$. \nAll the model-independent parameters can be expressed \ncompactly in terms of the reduced helicity amplitudes \n$M_{\\alpha\\beta}^{f f'}$~\\cite{chm,hhkm} of the \nprocess $f_\\alpha f'_\\beta \\rightarrow f_\\alpha f'_\\beta$: \n\\begin{subequations}\n\\begin{eqnarray}\nC_{1q} &=& \\frac{1}{2 \\sqrt{2} G_F} ( \\hphantom{-} M_{LL}^{\\ell q}\n\t+ M_{LR}^{\\ell q} - M_{RL}^{\\ell q} - M_{RR}^{\\ell q} ), \n\\\\\nC_{2q} &=& \\frac{1}{2 \\sqrt{2} G_F} ( \\hphantom{-} M_{LL}^{\\ell q}\n\t- M_{LR}^{\\ell q} + M_{RL}^{\\ell q} - M_{RR}^{\\ell q} ), \n\\\\\nC_{3q} &=& \\frac{1}{2 \\sqrt{2} G_F} ( \t-M_{LL}^{\\ell q}\n\t+ M_{LR}^{\\ell q} + M_{RL}^{\\ell q} - M_{RR}^{\\ell q} ), \n\\\\\ng_{L\\alpha}^{\\nu_\\mu f} &=& \\frac{1}{2 \\sqrt{2} G_F} \n\t(-M_{L\\alpha}^{\\nu_\\mu f} ). \n\\label{eq:nutrino_amplitude}\n\\end{eqnarray}\n\\end{subequations}\nBelow, we divide these model-independent parameters into two pieces as \n\\begin{subequations}\n\\begin{eqnarray}\nC_{iq} &=& (C_{iq})_{\\rm SM} + \\Delta C_{iq}, \\\\\ng_{L\\alpha}^{\\nu_\\mu f} &=& (g_{L\\alpha}^{\\nu_\\mu f})_{\\rm SM} + \n\t\\Delta g_{L\\alpha}^{\\nu_\\mu f}, \n\\end{eqnarray}\n\\end{subequations}\nwhere the first term in each equation is the SM contribution\nwhich is parametrized conveniently by $\\Delta S$ and $\\Delta T$ in \nref.~\\cite{chm}. \nThe terms $\\Delta C_{iq}$ and $\\Delta g_{L\\alpha}^{\\nu_\\mu f}$ \nrepresent the additional contributions from the $Z$-$Z'$ mixing \nand the $Z_2$ exchange:\n\\begin{subequations}\n\\begin{eqnarray}\n\\Delta C^{}_{1u} &=& \n\t(-0.19s^{}_\\beta-0.15c^{}_\\beta+0.65\\delta )\\bar{\\xi} \n\t-\\frac{g^2_E}{c^2_\\chi}\n\t\\frac{(\\tilde{Q}^L_E-\\tilde{Q}^E_E)\n\t(\\tilde{Q}^Q_E+\\tilde{Q}^U_E)}\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\t\n\\Delta C^{}_{1d} &=& \n\t(0.36s^{}_\\beta-0.54c^{}_\\beta+0.17\\delta )\\bar{\\xi}\n\t-\\frac{g^2_E}{c^2_\\chi}\n\t\\frac{(\\tilde{Q}^L_E-\\tilde{Q}^E_E)(\\tilde{Q}^Q_E+\\tilde{Q}^D_E)}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta C^{}_{2u} &=& \n\t(0.02s^{}_\\beta-0.84c^{}_\\beta+1.48\\delta )\\bar{\\xi} \n\t-\\frac{g^2_E}{c^2_\\chi}\\frac{(\\tilde{Q}^L_E+\\tilde{Q}^E_E)\n\t(\\tilde{Q}^Q_E-\\tilde{Q}^U_E)}{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta C^{}_{2d} &=&\n\t(0.02s^{}_\\beta+0.84c^{}_\\beta-1.48\\delta )\\bar{\\xi}\n\t-\\frac{g^2_E}{c^2_\\chi}\\frac{(\\tilde{Q}^L_E+\\tilde{Q}^E_E)\n\t(\\tilde{Q}^Q_E-\\tilde{Q}^D_E)}{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta C^{}_{3u} &=&\n\t(-0.82c^{}_\\beta+1.00\\delta )\\bar{\\xi}\n\t-\\frac{g^2_E}{c^2_\\chi}\\frac{(\\tilde{Q}^L_E-\\tilde{Q}^E_E)\n\t(\\tilde{Q}^U_E-\\tilde{Q}^Q_E)}{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta C^{}_{3d} &=&\n\t(1.06s^{}_\\beta-0.82c^{}_\\beta-1.00\\delta )\\bar{\\xi}\n\t-\\frac{g^2_E}{c^2_\\chi}\\frac{(\\tilde{Q}^L_E-\\tilde{Q}^E_E)\n\t(\\tilde{Q}^D_E-\\tilde{Q}^Q_E)}{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta g_{LL}^{\\nu u} &=& \n\t(0.44s^{}_\\beta+0.22c^{}_\\beta-0.18\\delta )\\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^Q_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta g_{LR}^{\\nu u} &=& \n\t(-0.35s^{}_\\beta+0.01c^{}_\\beta+0.82\\delta )\\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^U_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta g_{LL}^{\\nu d} &=& \n\t(0.04s^{}_\\beta-0.72c^{}_\\beta+0.59\\delta )\\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^Q_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta g_{LR}^{\\nu d} &=& \n\t(-0.22s^{}_\\beta-0.52c^{}_\\beta-0.41\\delta )\\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^D_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}},\\\\\n\\Delta g_{LL}^{\\nu e} &=& (0.12 s^{}_\\beta + 0.28 c^{}_\\beta \n\t- 0.23 \\delta) \\bar{\\xi} \n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^L_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}},\n\\\\\n\\Delta g_{LR}^{\\nu e} &=& (-0.14 s^{}_\\beta + 0.49 c^{}_\\beta \n\t- 1.23 \\delta) \\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^e_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}. \n\\end{eqnarray}\n\\label{eq:lenc_extra}\n\\end{subequations}\nwhere $c^{}_\\beta = \\cos\\beta_E$ and $s^{}_\\beta = \\sin\\beta_E$. \n\n\\subsubsection{SLAC $e$D experiment}\n\\label{section_slac}\nThe parity asymmetry in the inelastic scattering of polarized \nelectrons from the deuterium target was measured at SLAC~\\cite{slac}. \nThe experiment constrains the parameters \n$2C_{1u}-C_{1d}$ and $2C_{2u}-C_{2d}$. \nThe most stringent constraint shown in Table~\\ref{table:low_energy} \nis found for the following combination\n\\begin{subequations}\n\\begin{eqnarray}\nA_{\\rm SLAC} &=& 2C_{1u}-C_{1d} +0.206(2C_{2u}-C_{2d}) \\\\\n\t&=& 0.745 - 0.016\\,{\\it \\Delta S} + 0.016\\,{\\it \\Delta T} \n\t\\nonumber \\\\\n\t&&~~~~\n\t+ 2\\Delta C_{1u}- \\Delta C_{1d} \n\t+ 0.206(2\\Delta C_{2u}- \\Delta C_{2d}), \n\\end{eqnarray}\n\\label{eq:slac_sm1}\n\\end{subequations}\nwhere the theoretical prediction~\\cite{chm} is evaluated \nat the mean momentum transfer $\\langle Q^2 \\rangle = $ 1.5 GeV$^2$. \n\n\\subsubsection{CERN $\\mu^\\pm$C experiment}\nThe CERN $\\mu^\\pm$C experiment~\\cite{cern} measured \nthe charge and polarization asymmetry of deep-inelastic \nmuon scattering off the ${}^{12}$C target.\nThe mean momentum transfer of the experiment may be estimated \nat $\\langle Q^2 \\rangle = $ 50 GeV$^2$~\\cite{souder}.\nThe experiment constrains the parameters \n$2C_{2u}-C_{2d}$ and $2C_{3u}-C_{3d}$. \nThe most stringent constraint is found for the following \ncombination~\\cite{chm}\n\\begin{subequations}\n\\begin{eqnarray}\nA_{\\rm CERN} &=& 2C_{3u}-C_{3d}+0.777(2C_{2u}-C_{2d}) \\\\\n\t &=& -1.42-0.016\\,{\\it \\Delta S}+0.0006\\,{\\it \\Delta T}  \n\t\\nonumber \\\\\n\t&&~~~+ 2\\Delta C_{3u}- \\Delta C_{3d}\n\t+ 0.777(2\\Delta C_{2u}- \\Delta C_{2d}). \n\\end{eqnarray}\n\\label{eq:cern_sm1}\n\\end{subequations}\n\\subsubsection{Bates $e$C experiment} \nThe polarization asymmetry of the electron elastic scattering\noff the ${}^{12}$C target was measured at Bates \\cite{bates}.\nThe experiment constrains the combination\n\\begin{subequations}\n\\begin{eqnarray}\nA_{\\rm Bates} &=& C_{1u}+C_{1d} \\\\\n\t&=& - 0.1520 - 0.0023\\,{\\it \\Delta S} \n\t+ 0.0004\\,{\\it \\Delta T} \n\t+ \\Delta C_{1u} + \\Delta C_{1d}, \n\\end{eqnarray}\n\\end{subequations}\nwhere the theoretical prediction~\\cite{chm} is evaluated \nat $\\langle Q^2 \\rangle = $ 0.0225 GeV$^2$.\n\\subsubsection{Mainz $e$Be experiment }\n\\label{section_mainz}\nThe polarization asymmetry of electron quasi-elastic scattering\noff the ${}^9$Be target was measured at Mainz \\cite{mainz}.\nThe data shown in Table~\\ref{table:low_energy} is for \nthe combination\n\\begin{subequations}\n\\begin{eqnarray}\nA_{\\rm Mainz} &=& -2.73 C_{1u} + 0.65 C_{1d} - 2.19 C_{2u} \n\t+ 2.03 C_{2d} \n\\\\\n\t&=&  -0.876 + 0.043\\Delta S - 0.035\\Delta T\n\t\\nonumber \\\\\n\t&&~~~\n\t-2.73 \\Delta C_{1u} + 0.65 \\Delta C_{1d} \n\t- 2.19 \\Delta C_{2u} + 2.03 \\Delta C_{2d}, \n\\end{eqnarray}\n\\end{subequations}\nwhere the theoretical prediction~\\cite{chm} is evaluated \nat $\\langle Q^2 \\rangle = $ 0.2025 GeV$^2$. \n\\subsubsection{Atomic Parity Violation}\n\\label{section_apv}\nThe experimental results of parity violation in the atom \nare often given in terms of the weak charge $Q^{}_W(A,Z)$\nof nuclei. By using the model-independent parameter $C_{1q}^{}$,\nthe weak charge of a nuclei can be expressed as \n\\begin{equation}\nQ^{}_W(A,Z)=2ZC_{1p}^{}+2(A-Z)C_{1n}^{}. \n\\end{equation}\nBy taking account of the long-distance photonic \ncorrection~\\cite{apv_photonic}, \nwe find $C_{1p}$ and $C_{1n}$ as \n\\begin{subequations}\n\\begin{eqnarray}\nC_{1p} &=& \\hphantom{-} 0.03601  -0.00681 \\Delta S + 0.00477\\,\\Delta T \n\t+ 2\\Delta C_{1u} + \\Delta C_{1d}, \\\\\nC_{1n} &=& -0.49376 - 0.00366\\,{\\it \\Delta T} \n\t+ \\Delta C_{1u} + 2\\Delta C_{1d}. \n\\end{eqnarray}\n\\end{subequations}\nThe data for cesium atom $^{133}_{55}Cs$~\\cite{noecker,wood} \nis given in Table~\\ref{table:low_energy} and \nthe theoretical prediction of the weak charge \nis found to be~\\cite{chm} \n\\begin{equation}\nQ_{W}(^{133}_{55}Cs) =  -73.07 -0.749\\,{\\it \\Delta S}\n\t- 0.046\\,{\\it \\Delta T} \n\t+ 376 \\Delta C_{1u} + 422 \\Delta C_{1d}. \n\\end{equation}\n\\subsubsection{Neutrino-quark scattering} \n\\label{section_nq}\nFor the $\\nu_\\mu$-quark scattering, the experimental results \nup to the year 1988 were summarized in ref.~\\cite{fh} in terms \nof the model-independent parameters $g_L^2, g_R^2, \\delta_L^2, \\delta_R^2$. \nThe most stringent constraint on the result in ref.~\\cite{fh} \nis found for the \nfollowing combination: \n\\begin{eqnarray}\nK_{\\rm FH} &=& g_L^2 + 0.879 g_R^2 -0.010 \\delta_L^2 -0.043 \\delta_R^2.\n\\end{eqnarray}\nMore recent CCFR experiment at Tevatron measured the following \ncombination~\\cite{ccfr}\n\\begin{eqnarray}\nK_{\\rm CCFR} &=&  1.7897 g_L^2 + 1.1479 g_R^2 - 0.0916 \\delta_L^2 \n\t- 0.0782 \\delta_R^2. \n\\end{eqnarray}\nThe data are shown in Table~\\ref{table:low_energy} and the \nSM predictions are calculated from our reduced \namplitudes (\\ref{eq:nutrino_amplitude}) as \nfollows~\\cite{chm,hhkm}\n\\begin{eqnarray}\ng_\\alpha^2 = (g_{L\\alpha}^{\\nu_\\mu u})^2 \n\t+ (g_{L\\alpha}^{\\nu_\\mu d})^2, ~~~ \n\\delta_\\alpha^2 = (g_{L\\alpha}^{\\nu_\\mu u})^2 \n\t- (g_{L\\alpha}^{\\nu_\\mu d})^2,  \n\\end{eqnarray}\nfor $\\alpha = L$ and $R$, respectively, where \n\\begin{subequations}\n\\begin{eqnarray}\ng_{LL}^{\\nu_\\mu u} &=& \n\t\\hphantom{-} 0.3468 - 0.0023 \\Delta S + 0.0041 \\Delta T, \n\\\\\ng_{LR}^{\\nu_\\mu u} &=& \n\t-0.1549 - 0.0023 \\Delta S + 0.0004 \\Delta T, \n\\\\\ng_{LL}^{\\nu_\\mu d} &=& \n\t-0.4299 + 0.0012 \\Delta S - 0.0039 \\Delta T, \n\\\\\ng_{LR}^{\\nu_\\mu d} &=& \n\t\\hphantom{-} 0.0775 + 0.0012 \\Delta S - 0.0002 \\Delta T.\n\\end{eqnarray}\n\\end{subequations}\nThe above predictions are obtained at the momentum transfer \n$\\langle Q^2 \\rangle = 35~{\\rm GeV}^2$ relevant for the \nCCFR experiment~\\cite{ccfr}. \nThe estimations are found to be valid~\\cite{chm} also \nfor the data of ref.~\\cite{fh}, whose typical scale is \n$\\langle Q^2 \\rangle = 20~{\\rm GeV}^2$. \n\n\\subsubsection{Neutrino-electron scattering} \n\\label{section_ne}\nThe $\\nu_\\mu$-$e$ scattering experiments measure the neutral \ncurrents in a purely leptonic channel. \nThe combined results~\\cite{chm,charm-II} are given in \nTable~\\ref{table:low_energy}. \nThe theoretical predictions \n\\begin{subequations}\n\\begin{eqnarray}\ng_{LL}^{\\nu_\\mu e} &=& -0.273 + 0.0033 \\Delta S - 0.0042 \\Delta T \n\t+ \\Delta g_{LL}^{\\nu_\\mu e}, \n\\\\\ng_{LR}^{\\nu_\\mu e} &=& \\hphantom{-} 0.233 + 0.0033 \\Delta S - 0.0006 \\Delta T \n\t+ \\Delta g_{LR}^{\\nu_\\mu e}, \n\\end{eqnarray}\n\\end{subequations}\nare evaluated at $\\langle Q^2 \\rangle = 2m_e E_\\nu$ \nwith $E_\\nu = 25.7~{\\rm GeV}$ for the CHARM-II \nexperiment~\\cite{charm-II}. \n\\section{Constraints on $Z'$ bosons from electroweak experiments}\n\\setcounter{equation}{0} \nFollowing the parametrization presented in Sec.~3, \nwe can immediately obtain the constraints on $T_{\\rm new}, \\bar{\\xi}$ \nand $g_E^2\/c^2_\\chi m_{Z_2}^2$ from the data listed in Table~2 \nand Table~\\ref{table:low_energy}. \nSetting $S_{\\rm new} = U_{\\rm new} = 0$, we find that \nthe $Z$-pole measurements constrains $T_{\\rm new}$ and $\\bar{\\xi}$ \nwhile $m_W$ data constrains $T_{\\rm new}$. \nThe contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$ is constrained from the LENC data. \nThe number of the free parameters is, therefore, six: \nthe above three parameters and the SM parameters, \n$m_t^{}, \\alpha_s(m_{Z_1})$ and $\\bar{\\alpha}(m_{Z_1}^2)$. \nThroughout our analysis, we use \n\\begin{subequations}\n\\begin{eqnarray}\nm_t^{} &=& 175.6 \\pm 5.5~{\\rm GeV}~\\cite{mt96}, \\\\\n\\alpha_s (m_{Z_1}^{}) &=& 0.118 \\pm 0.003~\\cite{PDG}, \\\\\n1\/\\bar{\\alpha}(m_{Z_1}^2) &=& 128.75 \\pm 0.09~\\cite{eidelman}, \n\\end{eqnarray}\n\\end{subequations}\nas constraints on the SM parameters. \nThe Higgs mass dependence of the results are parametrized \nby $x_H^{}$ (\\ref{eq:higgs_xh}) \nin the range $77~{\\rm GeV} <  m_H^{} ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, 150~{\\rm GeV}$. \nThe lower bound is obtained at the LEP experiment~\\cite{higgs_direct}. \nThe upper bound is the theoretical limit on the lightest Higgs \nboson mass in any supersymmetric models that accommodate \nperturbative unification of the gauge couplings~\\cite{kane}. \nWe first obtain the constraints from the $Z$-pole experiments \nand $W$ boson mass measurement only, and then obtain \nthose by including the LENC experiments.  \n\n\\subsection{Constraints from $Z$-pole and $m_W$ data}\nLet us examine first the constraints from the $Z$-pole and \n$m_W$ data by performing the five-parameter fit for $T_{\\rm new}, \n\\bar{\\xi}, m_t^{}, \\alpha_s(m_{Z_1})$ and $\\bar{\\alpha}(m_{Z_1}^2)$. \nThe results for the $\\chi, \\psi, \\eta$ and $\\nu$ models at \n$\\delta = 0$ are summarized as follows: \n\\def(\\roman{enumi}){(\\roman{enumi})}\n\\def\\roman{enumi}{\\roman{enumi}}\n\\begin{enumerate}\n\\item $\\chi$-model ($\\delta = 0$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.040 + 0.15x_H^{} \\pm 0.12 \\\\\n\t\\bar{\\xi} &=& \\hphantom{-} 0.00017 - 0.00005x_H^{} \\pm 0.00046\n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.28,  \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (16.5 + 0.7 x_H^{})\/(12), \n\\end{array}\n\\label{eq:const_chi}\n\\end{eqnarray}\n\\item $\\psi$-model ($\\delta = 0$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.043 + 0.16x_H^{} \\pm 0.11 \\\\\n\t\\bar{\\xi} &=& \\hphantom{-} 0.00019 + 0.00012x_H^{} \\pm 0.00050\n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.20, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (16.5 + 0.4 x_H^{})\/(12), \n\\end{array}\n\\label{eq:const_psi}\n\\end{eqnarray}\n\\item $\\eta$-model ($\\delta = 0$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.053 + 0.14x_H^{} \\pm 0.11 \\\\\n\t\\bar{\\xi} &=& -0.00014 - 0.00062x_H^{} \\pm 0.00108 \n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.09, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (16.6 + 0.4 x_H^{})\/(12), \n\\end{array}\n\\label{eq:const_eta}\n\\end{eqnarray}\n\\item $\\nu$-model ($\\delta = 0$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.042 + 0.15x_H^{} \\pm 0.11 \\\\\n\t\\bar{\\xi} &=& \\hphantom{-} 0.00016 +  0.00007x_H^{} \\pm 0.00042 \n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.23, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (16.5 + 0.5 x_H^{})\/(12). \n\\end{array}\n\\label{eq:const_nu}\n\\end{eqnarray}\n\\end{enumerate}\nIn the above four $Z'$ models, the results for $T_{\\rm new}$ \nand $\\bar{\\xi}$ are consistent with zero for $x_H^{} = 0$. \nMoreover, the best fits of $T_{\\rm new}$ in all the $Z'$ models \nare in the unphysical \nregion, $T_{\\rm new} < 0$. \nThe parameter $T_{\\rm new}$ could be positive for the large $x_H^{}$: \nFor example, $x_H^{} = 0.41$ ($m_H^{} = 150~{\\rm GeV}$) makes $T_{\\rm new}$ \nin all the four $Z'$ models positive. \nThe allowed range of the effective mixing angle $\\bar{\\xi}$ is \norder of $10^{-3}$ for the $\\eta$-model and $10^{-4}$ for \nthe other three models in 1-$\\sigma$ level. \nThe $x_H^{}$-dependence of $\\bar{\\xi}$ in the $\\eta$-model is \nlarger than the other three models. \nFor comparison, we show the result for the leptophobic \n$\\eta$-model ($\\delta=1\/3$) \n\\begin{enumerate}\n\\addtocounter{enumi}{4}\n\\item leptophobic $\\eta$-model ($\\delta = 1\/3$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.049 + 0.15x_H^{} \\pm 0.11 \\\\\n\t\\bar{\\xi} &=& \\hphantom{-} 0.00269 +  0.00026x_H^{} \\pm 0.00309 \n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.03, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (15.9 + 0.5 x_H^{})\/(12).\n\\end{array}\n\\label{eq:zpole_leptophobic}\n\\end{eqnarray}\n\\end{enumerate}\nBy comparing the $\\eta$-model with no kinetic mixing ($\\delta = 0$) \nin eq.~(\\ref{eq:const_eta}), we find significantly \nweaker constraint on $\\bar{\\xi}$. \n\\tnew_xi \nIn Fig.~\\ref{allowed_eta}, \nwe show the 1-$\\sigma$ and 90\\% CL allowed region on \nthe $(\\bar{\\xi}, T_{\\rm new})$ plane \nin the $\\eta$-model with $\\delta = 0$ and $1\/3$ for \n$m_H^{} = 100~{\\rm GeV}$. \n\nThe best fit results at $m_H^{} = 100~{\\rm GeV}$ under the \nconstraint $T_{\\rm new} \\geq 0$ are shown in \nTable~2. \nWe can see from Table~2 that \nthere is no noticeable improvement of the fit \nfor the $\\chi,\\psi,\\eta$ and $\\nu$ models at $\\delta = 0$. \nThe $\\chi^2_{\\rm min}$ remains almost the same as that of the SM, \neven though each model has two new free parameters, \n$T_{\\rm new}$ and $\\bar{\\xi}$. \nThe fit slightly improves for the leptophobic $\\eta$-model \n($\\delta = 1\/3$) because of the smaller pull factor \nfor the $R_b$ data. \nThe probability of the fit, 18.7\\% CL, is still less \nthan that of the SM, 26.2\\% CL, because the $\\chi^2_{\\rm min}$ \nreduces only 0.8 despite two additional free parameters. \n\nWe explore the whole range of the parameters, $\\beta_E$ and $\\delta$. \nIn Fig.~\\ref{chisq_distribution}, we show the improvement in \n$\\chi^2_{\\rm min}$ over the SM value, $\\chi^2_{\\rm min}({\\rm SM})\n= 16.9$ (see Table~2): \n\\begin{eqnarray}\n\\Delta \\chi^2 \\equiv \\chi_{\\rm min}^2(\\beta_E, \\delta) \n\t- \\chi^2_{\\rm min}({\\rm SM}), \n\\end{eqnarray}\nwhere $\\chi^2_{\\rm min}(\\beta_E, \\delta)$ is evaluated \nat the specific value of $\\beta_E$ and $\\delta$ for \n$m_H^{} = 100~{\\rm GeV}$.\n\\fig_beta_delta \nAs we seen from Fig.~\\ref{chisq_distribution}, the $\\chi^2_{\\rm min}$ \ndepends very mildly in the whole range of the $\\beta_E$ and $\\delta$ \nplane, except near the leptophobic $\\eta$-model \n($\\beta_E = \\tan^{-1}(\\sqrt{5\/3})$ and $\\delta = 1\/3$)~\\cite{eta_model}. \nEven for the best choice of $\\beta_E$ and $\\delta$, the improvement in \n$\\chi^2_{\\rm min}$ is only 1.5 over the SM. \nBecause each model has two additional parameters $T_{\\rm new}$ and \n$\\bar{\\xi}$, we can conclude that no $Z'$ model in this framework \nimproves the fit over the SM. \nThe ``$\\times$'' marks plotted in Fig.~\\ref{chisq_distribution} show \nthe specific models which we will discuss in the next section. \n\n\\subsection{Constraints from $Z$-pole + $m_W$ + LENC data}\nNext we find constraints on the contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$ \nby including the low-energy data in addition to the \n$Z$-pole and $m_W$ data.\nBecause $T_{\\rm new}$ and $\\bar{\\xi}$ are already constrained \nseverely by the $Z$-pole and $m_W$ data, only the contact \nterms proportional to $g_E^2\/c_\\chi^2 m_{Z_2}^2$ \ncontribute to \nthe low-energy observables, except for the special case \nof the leptophobic $\\eta$-model ($\\delta = 1\/3$). \n\nWe summarize the results of the six-parameter \nfit for the $\\psi, \\chi, \\eta$ and $\\nu$ models:\n\\begin{flushleft}\n(i) $\\chi$-model\n\\begin{subequations}\n\\begin{eqnarray}\n& & \\!\\!\\!\\!\\left. \\!\\!\n\\begin{array}{r c l c l c l }\nT_{\\rm new} &\\!\\!=\\!\\!&\\!\\! -0.063\\!\\!&\\!\\!+\\!\\!&\\!\\! 0.14  x^{}_H\\!\\!\n&\\!\\!\\pm\\!\\!&\\!\\!0.11\\!\\!\\!\\!  \\\\\n\\bar{\\xi}   &\\!\\!=\\!\\!&\\!\\! -0.00005\\!\\!&\\!\\!-\\!\\!&\\!\\!0.00006x^{}_H\\!\\!\n&\\!\\!\\pm\\!\\!&\\!\\! 0.00044\\!\\!\\! \\\\\ng^2_E\/c^2_{\\chi}m^2_{Z_2} &\\!\\!=\\!\\!& \n\\makebox[3.3mm]{}\\!\\!0.26\\!\\!&\\!\\!+\\!\\!&\n\\!\\!0.01 x^{}_H\\!\\! &\\!\\!\\pm\\!\\!&\\!\\!0.21\\!\\!\n\\end{array} \\right\\} \n\\rho_{\\rm corr} \\!=\\! \\left(\n\\begin{array}{rrr}\n\\!\\!1.00\\!&\\!0.25\\!& \\!0.09\\!\\! \\\\\n        &\\!1.00\\!& \\!0.15\\! \\\\\n        &        & \\!1.00\\!\n\\end{array} \\right)\\!\\!,\\makebox[6mm]{ }  \n \\\\ \\!\\!\\!\\!\\!&&\\chi^2_{\\rm min}\/ ({\\rm d.o.f.}) \n= (19.9 + 0.9 x^{}_H)\/(20), \n\\end{eqnarray}\n\\end{subequations}\n\\end{flushleft}\n\\begin{flushleft}\n(ii) $\\psi$-model\n\\begin{subequations}\n\\begin{eqnarray}\n& & \\!\\!\\!\\!\\left. \\!\\!\n\\begin{array}{r c l c l c l }\nT_{\\rm new} &\\!\\!=\\!\\!&\\!\\! -0.065 \\!\\!  &\n\\!\\!+\\!\\!&\\!\\! 0.15  x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.11\\!\\! \\\\\n\\bar{\\xi}   &\\!\\!=\\!\\!&\\!\\! -0.00014\\!\\! &\n\\!\\!+\\!\\!&\\!\\! 0.00012x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.00050\\!\\!\\! \\\\\ng^2_E\/c^2_{\\chi}m^2_{Z_2} &\\!\\!=\\!\\!&\\!\\! \\makebox[3.3mm]{}1.66\\!\\!&\n\\!\\!+\\!\\!&\\!\\!0.19 x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\! 2.90\\!\\!\n\\end{array} \\right\\} \n\\rho^{}_{\\rm corr} \\!=\\! \\left( \n\\begin{array}{rrr}\n\\!\\!1.00\\! & \\!0.19\\! & \\!0.07\\!\\! \\\\\n         & \\!1.00\\! & \\!0.03\\! \\\\\n         &          & \\!1.00\\! \n\\end{array} \\right)\\!\\!,\\makebox[6mm]{ }  \n \\\\ \\!\\!\\!\\!\\!&&\\chi^2_{\\rm min}\/ ({\\rm d.o.f.}) \n\t= (21.1+0.8 x^{}_H)\/(20), \n\\end{eqnarray}\n\\end{subequations}\n\\end{flushleft}\n\\begin{flushleft}\n(iii) $\\eta$-model\n\\begin{subequations}\n\\begin{eqnarray}\n\\hsp{-2.5}& & \\!\\!\\!\\!\\left. \\!\\!\n\\begin{array}{r c l c l c l }\nT_{\\rm new}\\! &\\!\\!\\!=\\!\\!&\\!\\! -0.074\\!\\! &\\!\\!+\\!\\!&\n\\!\\! 0.14  x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.11\\!\\!\\! \\\\\n\\bar{\\xi}\\!   &\\!\\!\\!=\\!\\!&\\!\\! -0.00038\\!\\! &\\!\\!-\\!\\!& \n\\!\\!0.00063x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.00106\\!\\!\\!\\! \\\\\ng^2_E\/c^2_{\\chi}m^2_{Z_2}\\! &\\!\\!=\\!\\!&\\!\\!-0.62\\!\\!&\\!\\!+\\!\\!&\n\\!\\!0.08 x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.87\\!\\!\\!\n\\end{array} \\right\\}\\! \n\\rho^{}_{\\rm corr} \\!=\\! \\left( \n\\begin{array}{rrr}\n\\!\\!1.00\\! & \\!0.06\\! & \\!-0.05\\!\\! \\\\\n         & \\!\\!1.00\\! & \\!-0.22\\!\\! \\\\\n         &          & \\!1.00\\!  \n\\end{array} \\right)\\!\\!,\\makebox[6mm]{ }  \n \\\\ \\!\\!\\!\\!\\!&&\\chi^2_{\\rm min}\/ ({\\rm d.o.f.}) \n\t= (20.8+0.5 x^{}_H)\/(20), \n\\end{eqnarray}\n\\end{subequations}\n\\end{flushleft}\n\\begin{flushleft}\n(iv) $\\nu$-model\n\\begin{subequations}\n\\begin{eqnarray}\n\\hsp{-1.5}& & \\!\\!\\!\\!\\left. \\!\\!\n\\begin{array}{r c l c l c l }\nT_{\\rm new} &\\!\\!=\\!\\!&\\!\\! -0.061\\!\\! &\\!\\!+\\!\\!&\n\\!\\! 0.15  x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.11\\!\\! \\\\\n\\bar{\\xi}   &\\!\\!=\\!\\!&\\!\\! \\makebox[3.3mm]{} 0.00010\\!\\! &\\!\\!+\\!\\!&\n\\!\\! 0.00006x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.00041\\!\\!\\!\\! \\\\\ng^2_E\/c^2_{\\chi}m^2_{Z_2}\\! &\\!\\!=\\!\\!&\\!\\!-0.65\\!\\!&\\!\\!+\\!\\!&\n\\!\\!0.04 x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\! 0.54\\!\\!\n\\end{array} \\right\\} \n\\rho^{}_{\\rm corr} \\!=\\! \\left( \n\\begin{array}{rrr}\n\\!\\!1.00\\! & \\!0.21\\! & \\!0.07\\! \\\\\n         & \\!1.00\\! & \\!0.03\\! \\\\\n         &          & \\!1.00\\!\n\\end{array} \\right)\\!\\!,\\makebox[6mm]{ }  \n \\\\ \\!\\!\\!\\!\\!&&\\chi^2_{\\rm min}\/ ({\\rm d.o.f.}) \n\t= (20.1+0.8 x^{}_H)\/(20).\n\\end{eqnarray}\n\\end{subequations}\n\\end{flushleft}\nThe contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$ in the $\\psi$ and $\\eta$ models \nis consistent with zero in 1-$\\sigma$ level. \nBoth the best fit and the 1-$\\sigma$ error of \nthe parameters $T_{\\rm new}$ and $\\bar{\\xi}$ in all the \n$Z'$ models are slightly affected \nby including the LENC data: The best fit value of $T_{\\rm new}$ \nin all the $Z'$ models cannot be positive \neven for the $m_H^{} = 150~{\\rm GeV}$ $(x_H^{} = 0.41)$. \nSince the leptophobic $\\eta$-model does not have the contact term, \nthe low-energy data constrain the same parameters $T_{\\rm new}$ \nand $\\bar{\\xi}$. \nAfter taking into account both the high-energy and low-energy \ndata, we find \n\\begin{enumerate}\n\\addtocounter{enumi}{4}\n\\item leptophobic $\\eta$ model ($\\delta = 1\/3$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\\left.\n\\begin{array}{lrl}\nT_{\\rm new} &=&  -0.074 + 0.148 x_H^{} \\pm 0.110\\\\ \n\\bar{\\xi}      &=&  \\hphantom{-} 0.00157 + 0.00019 x_H^{} \\pm 0.00279\n\\end{array}\n\t\\right \\}\\rho_{\\rm corr} = 0.02, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) \n\t= (21.2 + 1.0 x_H^{})\/(21). \n\\end{array}\n\\end{eqnarray}\n\\end{enumerate}\nThe allowed range of $\\bar{\\xi}$ is slightly severe as \ncompared to eq.~(\\ref{eq:zpole_leptophobic}). \n\nThe best fit results for $m_H^{} = 100~{\\rm GeV}$ under the \ncondition $T_{\\rm new} \\geq 0$ are shown in Table~\\ref{table:low_energy}. \nIt is noticed that the best fit values for the weak charge of \ncesium atom $^{133}_{55}Cs$ in the $\\chi, \\eta$ and $\\nu$ models \nare quite close to the experimental data. \nThese models lead to $\\Delta \\chi^2 = -1.8$ ($\\chi$), \n$-0.8$ $(\\eta)$ and $-1.6$ $(\\nu)$. \nNo other noticeable point is found in the table. \n\\mass_lenc \n\nThe above constraints on $g_E^2\/c^2_\\chi m_{Z_2}^2$ from the LENC data give \nthe lower mass bound of the heavier mass eigenstate $Z_2$ \nin the $Z'$ models except for the leptophobic $\\eta$-model. \nIn Fig.~\\ref{mass_distribution}, the contour plot of \nthe 95\\% CL lower mass limit of $Z_2$ boson from the \nLENC experiments \nare shown on the $(\\beta_E, \\delta)$ plane by setting \n$g_E = g_Y$ and $m_H^{} = 100~{\\rm GeV}$ \nunder the condition $m_{Z_2} \\geq 0$. \nIn practice, we obtain the 95\\% CL lower limit of \nthe $Z_2$ boson mass $m_{95}$ in the following way: \n\\begin{eqnarray}\n0.05 =\n \\frac{\n\\int^\\infty_{m_{95}} dm_{Z_2} P(m_{Z_2})}\n{\\int^\\infty_{0} dm_{Z_2} P(m_{Z_2})}, \n\\end{eqnarray}\nwhere we assume that the probability density function \n$P(m_{Z_2})$ is proportional to ${\\rm exp}(-\\chi^2(m_{Z_2})\/2)$. \n\nWe can read off from Fig.~\\ref{mass_distribution} that \nthe lower mass bound of the $Z_2$ boson in the $\\psi$ model \nat $\\delta = 0$ is much weaker than those of the other $Z'$ \nmodels. \nIt has been pointed out that the most stringent \nconstraint on the contact term is the APV measurement \nof cesium atom~\\cite{chm}. \nSince  all the SM matter fields in the $\\psi$ model \nhave the same ${\\rm U(1)'}$ charge (see Table~1), \nthe couplings of contact interactions are Parity conserving, \nwhich makes constraint from the APV measurement useless. \nWe also find in Fig.~\\ref{mass_distribution} that the lower mass \nbound of the $Z_2$ boson disappears near the leptophobic $\\eta$-model \n($\\beta_E = \\tan^{-1}(\\sqrt{5\/3})$ and $\\delta = 1\/3$)~\\cite{eta_model}. \n\nWe summarize the 95\\% CL lower bound on $m_{Z_2}$ \nfor the $\\chi,\\psi,\\eta$ and $\\nu$ models ($\\delta = 0$) \nin Table~\\ref{mzelenc}. \nFor comparison, we also show the lower bound of $m_{Z_2}$  \nin the previous study~\\cite{cvetic_langacker_review} \nin the same table. \nThe bounds on the $Z_\\chi$ and $Z_\\nu$ masses \nare more severely constrained \nas compared to ref.~\\cite{cvetic_langacker_review}.  \nAlthough we used the latest electroweak data, our result \nfor the $Z_\\psi$ boson mass is somewhat weaker than that of \nref.~\\cite{cvetic_langacker_review}.  \nIn the analysis of ref.~\\cite{cvetic_langacker_review}, \nthe $e^+e^- \\rightarrow \\mu^+\\mu^-, \n\\tau^+\\tau^-$ data below the $Z$ pole \nare also used besides the $Z$-pole, $m_W$ and the LENC data. \nAs we mentioned before, the lower mass bound of the $Z'$ boson \nis obtained from the LENC data, not from the $Z$-pole data.  \nBecause the APV measurement which is most stringent constraint \nin the LENC data does not well constrain the $\\psi$ model, \nit is expected that the $e^+ e^-$ annihilation data below \nthe $Z$-pole play an important role to obtain the bound of \n$Z_\\psi$ boson mass. \n\\bound_lenc \n\nOur results in Table~\\ref{mzelenc} are also slightly weaker than \nthose in ref.~\\cite{chm}. \nThe results in ref.~\\cite{chm} have been obtained \nwithout including the $Z$-$Z'$ mixing effects and by setting \n$m_t^{} = 175~{\\rm GeV}, m_H^{} = 100~{\\rm GeV}, x_\\alpha^{} = 0$ \nand $T_{\\rm new} = 0$. \n\n\\subsection{Lower mass bound of $Z_2$ boson}\nWe have found that the $Z$-pole, $m_W$ and the LENC data \nconstrain ($T_{\\rm new}, \\bar{\\xi}$), $T_{\\rm new}$ and $g_E^2\/c^2_\\chi m_{Z_2}^2$, \nrespectively. \nWe can see from eq.~(\\ref{eq:zeta}) that, for a given $\\zeta$, \nconstraints on $T_{\\rm new}, \\bar{\\xi}$ and $g_E^2\/c^2_\\chi m_{Z_2}^2$ can be \ninterpreted as the bound on $m_{Z_2}$. \nWe show the 95\\% CL lower mass bound of the $Z_2$ boson \nfor $m_H^{} = 100~{\\rm GeV}$ \nin four $Z'$ models as a function of $\\zeta$. \nThe bound is again found under the condition $m_{Z_2} \\geq 0$. \nResults are shown in Fig.~\\ref{mass_95cl}.(a) $\\sim$  \n\\ref{mass_95cl}.(d) for the $\\chi,\\psi,\\eta,\\nu$ models, respectively. \nThe lower bound from the $Z$-pole and $m_W$ data, and \nthat from the LENC data are separately plotted in the same figure. \nIn order to see the $g_E$-dependence of the $m_{Z_2}$ bound \nexplicitly, we show the lower mass bound for the combination \n$m_{Z_2}g_Y\/g_E$. \nWe can read off from Fig.~\\ref{mass_95cl} that the bound on \n$m_{Z_2}g_Y\/g_E$ is approximately independent of $g_E$ for \n$g_E\/g_Y = 0.5 \\sim 2.0$ in each model. \n\\zprime_mass \nAs we expected from the formulae for \n$T_{\\rm new}$ and $\\bar{\\xi}$ in the small \nmixing limit (eq.~(\\ref{eq:tnew_xibar})), \nthe $Z_2$ mass is unbounded from the $Z$-pole data at \n$\\zeta = 0$. \nFor models with very small $\\zeta$, the lower bound on \n$m_{Z_2}$, therefore, comes from the LENC experiments \nand the direct search experiment at Tevatron. \nFor comparison, we plot the 95\\% CL lower bound on \n$m_{Z_2}$ obtained from the direct search experiment~\\cite{direct_search} \nin Fig.~\\ref{mass_95cl}. \nIn the direct search experiment, the $Z'$ decays into the exotic \nparticles, {\\it e.g.}, the decays into the light right-handed neutrinos \nwhich are expected for some models, are not taken into account. \nWe summarize the 95\\% CL lower bound on $m_{Z_2}$ \nfor the $\\chi,\\psi,\\eta$ and $\\nu$ models ($\\delta = 0$) \nobtained from the low-energy data and \nthe direct search experiment~\\cite{direct_search} \nin Table~\\ref{mzelenc}. \n\nThe lower bound of $m_{Z_2}$ is affected by the Higgs boson mass \nthrough the $T$ parameter. \nAs we seen from eqs.~(\\ref{eq:const_chi}) $\\sim$ (\\ref{eq:const_nu}),  \n$T_{\\rm new}$ tends to be in the physical region ($T_{\\rm new} \\geq 0$) \nfor large $m_H^{}$ $(x_H^{})$. \nThen, we find that the large Higgs boson mass decreases the lower \nbound of $m_{Z_2}$. \nFor $\\zeta = 1$, \nthe lower $m_{Z_2}$ bound in the $\\chi,\\psi,\\nu$ ($\\eta$) models \nfor $m_H^{} = 150~{\\rm GeV}$ is weaker than that for $m_H^{} = 100~{\\rm GeV}$ \nabout 7\\% (11\\%). \nOn the other hand, the Higgs boson with $m_H^{} = 80~{\\rm GeV}$ \nmakes the lower $m_{Z_2}$ bound in all the $Z'$ models \nsevere about 5\\% as compared to the case for $m_H^{} = 100~{\\rm GeV}$.  \nBecause $T_{\\rm new}$ and $\\bar{\\xi}$ are proportional to \n$\\zeta^2$ and $\\zeta$, respectively (see eq.~(\\ref{eq:tnew_xibar})), \nand it is unbounded at $|\\zeta| \\simeq 0$, \nthe lower bound of $m_{Z_2}$ may be independent of $m_H^{}$ \nin the small $|\\zeta|$ region. \nThe $m_H^{}$-dependence of the lower mass bound obtained \nfrom the LENC data is safely negligible. \n\n\nIt should be noted that,  at $\\zeta = 0$, \nonly the leptophobic $\\eta$-model ($\\delta = 1\/3$) is \nnot constrained from both the $Z$-pole and the low-energy \ndata. \nThe precise analysis and discussion for the lower mass bound \nof the $Z_2$ boson in the leptophobic $\\eta$-model can be \nfound in ref.~\\cite{uch}. \nIt is shown in ref.~\\cite{uch} that \nthe $\\zeta$-dependence of the lower mass bound is slightly \nmilder than that of the $\\eta$-model with $\\delta = 0$ \nin Fig.~\\ref{mass_95cl}.(c). \n\nIt has been discussed that the presence of $Z_2$ boson \nwhose mass is much heavier than the SM $Z$ boson mass,  \nsay 1 TeV, may lead to a find-tuning problem to stabilize \nthe electroweak scale against the ${\\rm U(1)'}$ scale~\\cite{drees}. \nThe $Z_2$ boson with $m_{Z_2} \\leq 1~{\\rm TeV}$ for $g_E = g_Y$ \nis allowed by the electroweak data only if $\\zeta$ satisfies \nthe following condition: \n\\begin{eqnarray} \n\\begin{array}{ll}\n-0.6 ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, \\zeta ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, +0.3 & ~~{\\rm for~the}~\\chi,\\psi,\\nu~{\\rm models}, \n\\\\\n-0.7 ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, \\zeta ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, +0.6 & ~~{\\rm for~the}~\\eta~{\\rm model}. \n\\end{array}\n\\label{eq:zeta_condition}\n\\end{eqnarray}\nIn principle, the parameter $\\zeta$ is calculable, together \nwith the gauge coupling $g_E$, once the particle spectrum \nof the $E_6$ model is specified. \nIn the next section, we calculate the $\\zeta$ parameter in \nseveral $E_6$ $Z'$ models. \n\n\n\\section{Light $Z'$ boson in minimal SUSY $E_6$-models}\n\\setcounter{equation}{0}\nIt is known that the gauge couplings are not unified \nin the $E_6$ models with three generations of {\\bf 27}. \nIn order to guarantee the gauge coupling unification, \na pair of weak-doublets, $H'$ and $\\overline{H'}$, \nshould be added into  \nthe particle spectrum at the electroweak scale~\\cite{dienes}. \nThey could be taken from ${\\bf 27} + {\\bf \\overline{27}}$ \nor the adjoint representation {\\bf 78}. \nThe ${\\rm U(1)'}$ charges of the additional weak doublets \nshould have the same magnitude and opposite sign, $a$ and $-a$,  \nto cancel the ${\\rm U(1)'}$ anomaly. \nIn addition, a pair of the complete SU(5) multiplet such as \n${\\bf 5 + \\overline{5}}$ can be added without spoiling the unification \nof the gauge couplings~\\cite{eta_model, dienes}. \n\nThe minimal $E_6$ model which have three generations of {\\bf 27} \nand a pair ${\\bf 2} + {\\bf \\overline{2}}$ depends in principle on the \nthree cases; $H'$ has the same quantum number as $L$ or $H_d$ of \n{\\bf 27}, or $\\overline{H_u}$ of ${\\bf\\overline{27}}$. \nAll three cases will be studied below. \n\\e6_extra \nThe hypercharge and ${\\rm U(1)'}$ charge of \nthe extra weak doublets for the $\\chi,\\psi,\\eta,\\nu$ models \nare listed in Table~\\ref{table:extra_higgs}.  \nFor comparison, we also show those in the model of Babu \n{\\it et al.~}~\\cite{eta_model}, where two pairs of \n${\\bf 2 + \\overline{2}}$ from {\\bf 78} and a pair of ${\\bf 3 + \\overline{3}}$ \nfrom ${\\bf 27 + \\overline{27}}$ are introduced \nto achieve the quasi-leptophobity at the weak scale. \n\nLet us recall the definition of $\\zeta$; \n\\begin{equation}\n\\zeta = \\frac{g_Z}{g_E}\\frac{m_{ZZ'}^2}{m_Z^2} - \\delta.  \n\\label{eq:zetadef}\n\\end{equation}\nIn the minimal model, \nthe following eight scalar-doublets can develop VEV to \ngive the mass terms $m_Z^2$ and $m_{ZZ'}^2$ in eq.~(\\ref{eq:l_gauge}): \nthree generations of $H_u,  H_d$, \nand an extra pair, $H'$ and $\\overline{H'}$. \nThen, $m_Z^2$ and $m_{ZZ'}^2$ are written in terms of \ntheir VEVs as \n\\begin{subequations}\n\\begin{eqnarray}\nm_Z^2 &=& \\frac{1}{2} g_Z^2\\biggl[ \n\t \\sum_{i=1}^3 \\biggl\\{ \n\t\\langle H_u^i \\rangle^2 + \\langle H_d^i \\rangle^2 \n\t\\biggr\\} + \\langle H' \\rangle^2 \n\t+ \\langle \\overline{H'} \\rangle^2 \\biggr], \n\\\\\nm_{ZZ'}^2 &=& \\! g_Z g_E \\biggl[ \n\t\\sum_{i=1}^{3} \\biggl\\{ \n\t-Q_E^{H_u} \\langle H_u^i \\rangle^2 \n\t+ Q_E^{H_d} \\langle H_d^i \\rangle^2 \n\t\\biggr \\}\n\t+ Q_E^{H'} \\langle H' \\rangle^2 \n\t- Q_E^{\\overline{H'}} \\langle \\overline{H'} \\rangle^2 \n\t\\biggr]\\!, \n\\end{eqnarray}\n\\end{subequations}\nwhere $i$ is the generation index. \nThe third component of the weak isospin $I_3$ for the Higgs \nfields are\n\\begin{equation}\nI_3(H_d) = I_3(H') = -I_3(H_u) = -I_3(\\overline{H'}) = 1\/2. \n\\end{equation}\nTaking account of the ${\\rm U(1)'}$ charges of \nthe extra Higgs doublets, $Q_E^{H'} = - Q_E^{\\overline{H'}}$, \nwe find from eq.~(\\ref{eq:zetadef}) \n\\begin{eqnarray}\n\\zeta &=& 2 \\frac{\\displaystyle{ \\sum_{i=1}^3 }\\biggl\\{  \n\t-Q_E^{H_u} \\langle H_u^i \\rangle^2 \n\t+Q_E^{H_d} \\langle H_d^i \\rangle^2 \n\t\\biggr \\}\n\t+ Q_E^{H'} \\biggl( \\langle H' \\rangle^2 \n\t+  \\langle \\overline{H'}\\rangle^2  \\biggr) }\n\t{\\displaystyle{ \\sum_{i=1}^3 }\\biggl\\{ \n\t\\langle H_u^i \\rangle^2 + \\langle H_d^i \\rangle^2 \n\t\\biggr\\}\n\t+ \\langle H'\\rangle^2 + \\langle \\overline{H'} \\rangle^2 } \n\t- \\delta.  \n\\label{eq:zeta_final}\n\\end{eqnarray}\nWe note here that the observed $\\mu$-decay constant \nleads to the following sum rule \n\\begin{eqnarray}\nv_u^2 + v_d^2 + v_{H'}^2 + v_{\\overline{H'}}^2 \\equiv \nv^2 = \\frac{1}{\\sqrt{2}G_F} \\approx (246~{\\rm GeV})^2,  \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\left.\n\\begin{array}{rr}\n\t\\displaystyle{\\sum_{i=1}^3\\langle H_u^i \\rangle^2 = \\frac{v_u^2}{2},}\n\t&\n\t\\displaystyle{\\sum_{i=1}^3\\langle H_d^i \\rangle^2 = \\frac{v_d^2}{2}, }\n\t\\\\\n\t\\displaystyle{\\langle H' \\rangle^2 = \\frac{v_{H'}^2}{2}, }\n\t&\n\t\\displaystyle{\\langle \\overline{H'} \\rangle^2 = \\frac{v_{\\overline{H'}}^2}{2}}. \n\\end{array}\n\\right.\n\\end{eqnarray}\nBy further introducing the notation\n\\begin{subequations}\n\\begin{eqnarray}\n\\tan \\beta &=& \\frac{v_u}{v_d}, \n\\\\\nx^2 &=& \\frac{v_{H'}^2 + v_{\\overline{H'}}^2}{v^2}, \n\\end{eqnarray}\n\\end{subequations}\nwe can express eq.~(\\ref{eq:zeta_final}) as \n\\begin{eqnarray}\n\\zeta &=& 2 \\biggl\\{\n\t-Q_E^{H_u}(1-x^2)\\sin^2\\beta +Q_E^{H_d}(1-x^2)\\cos^2\\beta \n\t+Q_E^{H'}x^2\n\t\\biggr\\} \n\t- \\delta. \n\\end{eqnarray}\nBecause $H'$ and $\\overline{H'}$ are taken from {\\bf 27} + \n{$\\bf\\overline{27}$}, the ${\\rm U(1)'}$ charge of \n$H'$, $Q_E^{H'}$, is identified with that of $L$, \n$H_d$ or $\\overline{H_u}$. \n\nAmong all the models, only in the $\\chi$-model one can have \nsmaller number of matter particles. \nIn the $\\chi$-model, three generations of the matter fields \n{\\bf 16} and  a pair of Higgs doublets make the model \nanomaly free. \nIn this case, $\\zeta$ is found to be independent of $\\tan\\beta$: \n\\begin{subequations}\n\\begin{eqnarray}\n\\zeta &=& 2 \\frac{ Q_E^{H_d} - Q_E^{H_u} \\tan^2 \\beta  }\n\t{1 + \\tan^2 \\beta}\n\t- \\delta \n\\\\\n\t&=& 2 Q_E^{H_d} - \\delta. \n\\end{eqnarray}\n\\end{subequations}\n\n\\coeff_rge\nLet us now examine the kinetic mixing parameter $\\delta$ in each model. \nThe boundary condition of $\\delta$ at the GUT scale is $\\delta = 0$. \nThe non-zero kinetic mixing term can arise at low-energy \nscale through the following RGEs: \n\\begin{subequations}\n\\begin{eqnarray}\n\\frac{d}{dt} \\alpha_i &=& \\frac{1}{2\\pi}b_i \\alpha_i^2,  \n\\label{eq:rgea}\n\\\\\n\\frac{d}{dt} \\alpha_4 &=& \\frac{1}{2\\pi}\n( b_E + 2 b_{1E} \\delta + b_1 \\delta^2 ) \\alpha_4^2 ,\n\\label{eq:rgeb}\\\\\n\\frac{d}{dt} \\delta &=& \\frac{1}{2\\pi}\n( b_{1E} + b_1 \\delta ) \\alpha_1, \n\\label{eq:rgec}\n\\end{eqnarray}\n\\label{eq:rge}\n\\end{subequations}\nwhere $i=1,2,3$ and $t=\\ln \\mu$. \nWe define $\\alpha_1$ and $\\alpha_4$ as \n\\begin{eqnarray}\n\\alpha_1 \\equiv \\frac{5}{3}\\frac{g_Y^2}{4\\pi}, \n~~~~~~~\n\\alpha_4 \\equiv \\frac{5}{3}\\frac{g_E^2}{4\\pi}. \n\\end{eqnarray}\nThe coefficients of the $\\beta$-functions for $\\alpha_1, \n\\alpha_4$ and $\\delta$ are: \n\\begin{eqnarray}\nb_1 = \\frac{3}{5} {\\rm Tr} (Y^2), \n~~~~\nb_E = \\frac{3}{5} {\\rm Tr} (Q_E^2), \n~~~~\nb_{1E} = \\frac{3}{5} {\\rm Tr} (Y Q_E). \n\\end{eqnarray}\nFrom eq.~(\\ref{eq:rgec}), we can clearly see that the non-zero $\\delta$ \nis generated at the weak scale if $b_{1E} \\neq 0$ holds. \nIn Table~\\ref{table:coeff_rge}, we list $b_1, b_E$ and $b_{1E}$ \nin the minimal $\\chi,\\psi,\\eta,\\nu$ models and the $\\eta_{\\rm BKM}$ \nmodel~\\cite{eta_model}. \nAs explained above, the $\\chi(16)$ model has three generations \nof {\\bf 16}, and the $\\chi(27)$ model has three generations of \n{\\bf 27}. \nWe can see from Table~\\ref{table:coeff_rge} that the \nmagnitudes of the differences $b_1 - b_2$ and $b_2 - b_3$ are \ncommon among all the models including the minimal supersymmetric \nSM. \nThis guarantees the gauge coupling unification \nat $\\mu = m_{GUT} \\simeq 10^{16}~{\\rm GeV}$. \n\n\\ge_value \nIt is straightforward to obtain $g_E(m_{Z_1})$ and \n$\\delta(m_{Z_1})$ for each model. \nThe analytical solutions of eqs.~(\\ref{eq:rgea})$\\sim$\n(\\ref{eq:rgec}) are as follows:\n\\begin{subequations}\n\\begin{eqnarray}\n\\frac{1}{\\alpha_i(m_{Z_1})} &=& \\frac{1}{\\alpha_{GUT}} \n\t+ \\frac{1}{2\\pi}b_i \\ln \\frac{m_{GUT}}{m_{Z_1}}, \n\\\\\n\\delta(m_{Z_1}) &=& -\\frac{b_{1E}}{b_1}\\biggl( \n\t1 - \\frac{\\alpha_1(m_{Z_1})}{\\alpha_{GUT}} \\biggr), \n\\\\\n\\frac{1}{\\alpha_4(m_{Z_1})} &=& \\frac{1}{\\alpha_{GUT}} \n\t+ \\biggl\\{ \\frac{b_E}{b_1} - \n\t\\biggl(\\frac{b_{1E}}{b_1} \\biggr)^2 \n\t\\biggr \\} \n\t  \\biggl\\{ \\frac{1}{\\alpha_1(m_{Z_1})} - \\frac{1}{\\alpha_{GUT}}\n\t\\biggr \\} \n\\nonumber \\\\\n\t&& \n\t- \\biggl( \\frac{b_{1E}}{b_1} \\biggr)^2 \n\t\\frac{ \\alpha_1(m_{Z_1}) - \\alpha_{GUT}}{\\alpha_{GUT}^2}, \n\\end{eqnarray}\n\\end{subequations}\nwhere $\\alpha_{GUT}$ denotes the unified gauge coupling at \n$\\mu = m_{GUT}$. \nIn our calculation, \n$\\alpha_3(m_{Z_1}) = 0.118$ and \n$\\alpha(m_{Z_1}) = e^2(m_{Z_1})\/4\\pi = 1\/128$ \nare used as example. \nThese numbers give $g_Y(m_{Z_1}) = 0.357$. \nWe summarize the predictions for $g_E$ \nand $\\delta$ at $\\mu = m_{Z_1}$ in the \nminimal $E_6$ models and the $\\eta_{\\rm BKM}$ model \nin Table~\\ref{table:ge_delta}. \nIn all the minimal models, the ratio $g_E\/g_Y$ is approximately \nunity and $|\\delta|$ is smaller than about 0.07. \nOn the other hand, the $\\eta_{\\rm BKM}$ model \npredicts \n\\tnb_zeta\n$g_E\/g_Y \\sim 0.86$ and $\\delta \\sim 0.29$, which is close \nto the leptophobic-$\\eta$ model at $\\delta = 1\/3$. \nIn Figs.~\\ref{chisq_distribution} and \\ref{mass_distribution}, \nwe show the predictions of all the models by ``$\\times$'' symbol.\n\nNext, we estimate the parameter $\\zeta$ for several \nsets of $\\tan\\beta$ and $x$. \nIn Table~\\ref{table:zetasummary}, we show the predictions for \n$\\zeta$ in the minimal $\\chi,\\psi,\\eta,\\nu$ models and the \n$\\eta_{\\rm BKM}$ model. \nThe results are shown for $\\tan\\beta = 2$ and $30$, and \n$x^2 = 0$ and $0.5$. \nWe find from the table that the parameter $\\zeta$ is \nin the range $|\\zeta| ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, 1.35$ for all the models \nexcept for the $\\eta_{\\rm BKM}$ model, where the \npredicted $\\zeta$ lies between $-2.0$ and $-1.2$.  \nIt is shown in Fig.~\\ref{mass_95cl}\nthat $m_{Z_2}g_Y\/g_E$ is approximately \nindependent of $g_E\/g_Y$.  \nActually, we find in Table~\\ref{table:ge_delta} and \nTable~\\ref{table:zetasummary} that \nthe predicted $|\\delta|$ is smaller than about 0.1 \nand $g_E\/g_Y$ is quite close to unity in all the minimal models.  \nWe can, therefore, read off from Fig.~\\ref{mass_95cl} \nthe lower bound of $m_{Z_2}$ in the minimal models at $g_E = g_Y$. \nIn Table~\\ref{table:mass95_zeta}, we summarize the 95\\% CL lower \n$m_{Z_2}$ bound for the minimal $\\chi,\\psi,\\eta,\\nu$ models and \nthe $\\eta_{\\rm BKM}$ model which correspond to the predicted $\\zeta$ \nin Table~\\ref{table:zetasummary}. \n\\masszeta\nMost of the lower mass bounds in Table~\\ref{table:mass95_zeta} \nexceed 1 TeV. \nThe $Z_2$ boson with $m_{Z_2} \\sim O(1~{\\rm TeV})$ should be \nexplored at the future collider such as LHC. \nThe discovery limit of the $Z'$ boson in the $E_6$ models at LHC \nis expected as~\\cite{cvetic_bound}\n\\begin{eqnarray}\n\\begin{array}{cccc}\\hline \n\\chi & \\psi & \\eta & \\nu \\\\ \\hline \n3040 & 2910 & 2980 & *** \\\\ \\hline \n\\end{array}\n\\end{eqnarray}\nAll the lower bounds of $m_{Z_2}$ listed in \nTable~\\ref{table:mass95_zeta} are smaller than 2 TeV \nand they are, therefore, in the detectable range of LHC. \nBut, it should be noticed that most of them \n($1~{\\rm TeV} ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, m_{Z_2}$) may require the fine-tuning to stabilize \nthe electroweak scale against the ${\\rm U(1)'}$ scale~\\cite{drees}. \n\nThe lower bound of the $Z_2$ boson mass in the $\\eta_{\\rm BKM}$ \nmodel for the predicted $\\zeta$ can be read off from Fig.~2 in \nref.~\\cite{uch}. \nBecause somewhat large $\\zeta$ is predicted in the $\\eta_{\\rm BKM}$ \nmodel, $1 ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, |\\zeta|$, the lower mass bound is also large \nas compared to the minimal models. \n\n\\section{Summary}\nWe have studied constraints on $Z'$ bosons in the SUSY \n$E_6$ models. \nFour $Z'$ models --- the $\\chi,\\psi,\\eta$ and $\\nu$ models \nare studied in detail. \nThe presence of the $Z'$ boson affects the electroweak processes \nthrough the effective $Z$-$Z'$ mass mixing angle $\\bar{\\xi}$, \na tree level contribution $T_{\\rm new}$ which is a positive \ndefinite quantity, and the contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$. \nThe $Z$-pole, $m_W$ and LENC data constrain ($T_{\\rm new}, \n\\bar{\\xi}$), $T_{\\rm new}$ and $g_E^2\/c^2_\\chi m_{Z_2}^2$, respectively. \nThe convenient parametrization of the electroweak \nobservables in the SM and the $Z'$ models are presented. \nFrom the updated electroweak data, we find that the $Z'$ models \nnever give the significant improvement of the $\\chi^2$-fit \neven if the kinetic mixing is taken into accounted. \nThe 95\\% CL lower mass bound of the heavier mass eigenstate $Z_2$ \nis given as a function of the effective $Z$-$Z'$ mixing \nparameter $\\zeta$. \nThe approximate scaling low is found for the $g_E\/g_Y$-dependence \nof the lower limit of $m_{Z_2}$. \nBy assuming the minimal particle content of the $E_6$ model,  \nwe have found the theoretical predictions for $\\zeta$. \nWe have shown that the $E_6$ models \nwith minimal particle content which is consistent with \nthe gauge coupling unification predict the non-zero kinetic mixing \nterm $\\delta$ and the effective mixing parameter $\\zeta$ of \norder one. \nThe present electroweak experiments lead to the lower \nmass bound of order 1 TeV or larger for those models. \n\n\\section*{Acknowledgment}\nThis work is supported in part by Grant-in-Aid for Scientific \nResearch from the Ministry of Education, Science and Culture of Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s1}\n\nIn this paper, we will study the loop quantum cosmology (LQC)\n\\cite{aa-rev,mbrev} of the Bianchi type II model. These models are\nof special interest to the issue of singularity resolution\nbecause of the intuition derived from the body of results related to\nthe Belinksii, Khalatnikov, Lifshitz (BKL) conjecture\n\\cite{bkl1,bkl2} on the nature of generic, spacelike singularities\nin general relativity (see, e.g., \\cite{bb}). Specifically, as the\nsystem enters the Planck regime, dynamics at any fixed spatial point\nis expected to be well described by the Bianchi I evolution.\nHowever, there are transitions in which the parameters\ncharacterizing the specific Bianchi I space-time change and the\ndynamics of these transitions mimics the Bianchi II time evolution.\nIn a recent paper \\cite{awe2}, we studied the Bianchi I model in the\ncontext of LQC. In this paper we will extend that analysis to the\nBianchi II model. We will follow the same general approach and use\nthe same notation, emphasizing only those points at which the\npresent analysis differs from that of \\cite{awe2}.\n\nBianchi I and II models are special cases of type A Bianchi models\nwhich were analyzed already in the early days of LQC (see in\nparticular \\cite{mb-hom, bdv}). However, as is often the case with\npioneering early works, these papers overlooked some important\nconceptual and technical issues. At the classical level,\ndifficulties faced by the Hamiltonian (and Lagrangian) frameworks in\nnon-compact, homogeneous space-times went unnoticed. In these cases,\nto avoid infinities, it is necessary to introduce an elementary cell\nand restrict all integrals to it \\cite{as,abl}. The Hamiltonian\nframeworks in the early works did not carry out this step. Rather,\nthey were constructed simply by dropping an infinite volume integral\n(a procedure that introduces subtle inconsistencies). In the quantum\ntheory, the kinematical quantum states were assumed to be periodic\n---rather than almost-periodic--- in the connection, and the quantum\nHamiltonian constraint was constructed using a ``pre-$\\mu_o$''\nscheme. Developments over the intervening years have shown that\nthese strategies have severe limitations (see, e.g.,\n\\cite{aps3,acs,cs1,aa-badhonef,cs2}). In this paper, they will be\novercome using ideas and techniques that have been introduced in the\nisotropic and Bianchi I models in these intervening years. Thus, as\nin \\cite{awe2} the classical Hamiltonian framework will be based on\na fiducial cell, quantum kinematics will be constructed using almost\nperiodic functions of connections and quantum dynamics will use the\n``$\\bar\\mu$ scheme.'' Nonetheless, the space-time description of\nBianchi II models in \\cite{mb-hom, bdv}, tailored to LQC, will\nprovide the point of departure of our analysis.\n\nNew elements required in this extension from the Bianchi I model\ncan be summarized as follows. Recall first that the spatially\nhomogeneous slices $M$ in Bianchi models are isomorphic to\n3-dimensional group manifolds. The Bianchi I group is the\n3-dimensional group of translations. Hence the the three Killing\nvectors ${}^o\\xi^a_i$ on $M$ ---the left invariant vector fields on\nthe group manifold--- commute and coincide with the right\ninvariant vector fields ${}^o\\!e^a_i$ which constitute the fiducial\northonormal triads on $M$. In LQC one mimics the strategy used in\nLQG and spin foams and defines the curvature operator in terms of\nholonomies around plaquettes whose edges are tangential to these\nvector fields. The Bianchi II group, on the other hand, is\ngenerated by the two translations and the rotation on a null\n2-plane. Now the Killing vectors ${}^o\\xi^a_i$ no longer commute and\nneither do the fiducial triads ${}^o\\!e^a_i$. Therefore we have to\nfollow another strategy to build the elementary plaquettes.\nHowever, this situation was already encountered in the\nk=$1$, isotropic models \\cite{warsaw,apsv}. There, the desired\nplaquettes can be obtained by alternating between the integral\ncurves of right and left invariant vector fields which do commute.\nHowever, in the isotropic case, the gravitational connection is\ngiven by $A_a^i = c\\,\\, {}^o\\!\\omega_a^i$, where ${}^o\\!\\omega_a^i$ are the covectors\ndual to ${}^o\\!e^a_i$ and the holonomies around these plaquettes turned\nout to be almost periodic functions of the connection component\n$c$ \\cite{warsaw,apsv}. By contrast, in the Bianchi II model we\nhave three connection components $c^i$ because of the presence of\nanisotropies, and, unfortunately, the holonomies around our\nplaquettes are no longer almost periodic functions of $c^i$. (This\nis also the case in more complicated Bianchi models.) Since the\nstandard kinematical Hilbert space of LQC consists of almost\nperiodic functions of $c^i$, these holonomy operators are not\nwell-defined on this Hilbert space. Thus, the strategy \\cite{abl}\nused so far in LQC to define the curvature operator is no longer\nviable.\n\nOne could simply enlarge the kinematical Hilbert space to\naccommodate the new holonomy functions of connections. But then the\nproblem quickly becomes as complicated as full LQG. To solve the\nproblem within the standard, symmetry reduced kinematical framework\nof LQC, one needs to generalize the strategy to define the curvature\noperator. Of course, the generalization must be such that, when\napplied to all previous models, it is compatible with the procedure\nof computing holonomies around suitable plaquettes used there. We\nwill carry out this task by suitably modifying ideas that have\nalready appeared in the literature. This generalization will enable\none to incorporate \\emph{all} class A Bianchi models in the LQC\nframework.\n\nOnce this step is taken, one can readily construct the quantum\nHamiltonian constraint and the physical Hilbert space, following\nsteps that were introduced in the analysis \\cite{awe2} of the\nBianchi I model. However, because Bianchi II space-times have\nspatial curvature, the spin connection compatible with the\northonormal triad is now non-trivial. It leads to two new terms in\nthe Hamiltonian constraint that did not appear in the Bianchi I\nHamiltonian. We will analyze these new terms in some detail. In\nspite of these differences, the big bang singularity is resolved in\nthe same precise sense as in the Bianchi I model \\cite{awe2}: If a\nquantum state is initially supported only on classically\nnon-singular configurations, it continues to be supported on\nnon-singular configurations throughout its evolution.\n\nThe paper is organized as follows. Section \\ref{s2} summarizes the\nclassical Hamiltonian theory describing Bianchi II models. Section\n\\ref{s3} discusses the quantum theory. We first define a non-local\nconnection operator $\\hat{A}_a^i$ and use it to obtain the\nHamiltonian constraint. We then show that the singularity is\nresolved and the Bianchi I quantum dynamics is recovered in the\nappropriate limit. In Section \\ref{s4}, we introduce effective\nequations for the model (with the same caveats as in the Bianchi I\ncase \\cite{awe2})\nFinally, in section V we summarize our results and discuss the new\nelements that appear in the Bianchi II model. In Appendix A we\nimprove on the discussion of discrete symmetries presented in\n\\cite{awe2}. The results on the Bianchi I model obtained in\n\\cite{awe2} carry over without any change. But the change of\nviewpoint is important to the LQC treatment of the Bianchi II model\nand more general situations.\n\n\n\\section{Classical Theory}\n\\label{s2}\n\nThis section is divided into two parts. In the first we recall the\nstructure of Bianchi II space-times and in the second we summarize\nthe phase space formulation, adapted to LQC.\n\n\\subsection{Diagonal Bianchi II Space-times}\n\\label{s2.1}\n\nBecause the issue of discrete symmetries is subtle in background\nindependent contexts, and because it plays a conceptually important\nrole in the quantum theory of Bianchi II models, we will begin with\na brief summary of how various fields are defined\n\\cite{alrev,aa-dis}. This stream-lined discussion brings out the\nassumptions which are often only implicit, making the discussion of\ndiscrete symmetries clearer.\n\nIn the Hamiltonian framework underlying loop quantum gravity (LQG),\none fixes an \\emph{oriented} 3-manifold $M$ and a 3-dimensional\n`internal' vector space $I$ equipped with a positive definite metric\n$q_{ij}$. The internal indices $i,j,k,\\ldots$ are then freely\nlowered and raised by $q_{ij}$ and its inverse. A spatial triad\n$e^a_i$ is an isomorphism from $I$ to tangent space at each point of\n$M$ which associates a vector field $v^a:= e^a_i v^i$ on $M$ to each\nvector $v^i$ in $I$.%\n\\footnote{Thus, in LQG one begins with non-degenerate triads and\nmetrics, passes to the Hamiltonian framework and then, at the end,\nextends the framework to allow degenerate geometries.}\nThe dual co-triads are denoted by $\\omega_a^i$. Given a triad, we\nacquire a positive definite metric $q_{ab}:= q_{ij} \\omega_a^i\n\\omega_b^j$ on $M$. The metric $q_{ab}$ in turn singles out a 3-form\n$\\epsilon_{abc}$ on $M$ which has \\emph{positive orientation} and\nsatisfies $ \\epsilon_{abc}\\epsilon_{def}\\, q^{ad} q^{be}q^{cf}= 3!$.\nOne can then define a 3-form $\\epsilon_{ijk}$ on $I$ via\n$\\epsilon_{ijk} = \\epsilon_{abc} e^a_i e^b_j e^c_k$. Note that\n$\\epsilon_{ijk}$ is automatically compatible with $q_{ij}$, i.e.,\n$\\epsilon_{ijk}\\epsilon_{lmn}\\, q^{il} q^{jm} q^{kn}= 3!$. If a\ntriad $\\bar{e}^a_i$ is obtained by flipping an odd number of the\nvectors in the triad $e^a_i$, then $\\bar{e}^a_i$ and $e^a_i$ have\nopposite orientations and the fields they define satisfy\n$\\bar{q}_{ab} = {q}_{ab},\\, \\bar\\epsilon_{abc} = \\epsilon_{abc}$ but\n$\\bar\\epsilon_{ijk} = - \\epsilon_{ijk}$. Had we fixed\n$\\epsilon_{ijk}$ once and for all on $I$, then $\\epsilon_{abc}$\nwould have flipped sign under this operation and volume integrals on\n$M$ computed with the unbarred and barred triads would have had\nopposite signs. With our conventions, these volume integrals will\nnot change and the parity flips will be symmetries of the symplectic\nstructure and the Hamiltonian constraint.\n\nThe triad also determines an unique spin connection $\\Gamma_a^i$ via\n\\begin{equation} \\label{sc} D_{[a} \\omega_{b]}^i\\, \\equiv \\,\n\\partial_{[a}\\omega_{b]}^i + \\epsilon^{i}{}_{jk} \\Gamma_{[a}^j\n\\omega_{b]}^k \\, =\\, 0\\, .\\end{equation}\nThe gravitational configuration variable $A_a^i$ is then given by\n$A_a^i = \\Gamma_a^i + \\gamma K_a^i$ where $K_{ab} := K_a^i\n\\omega_{bi}$ is the extrinsic curvature of $M$ and $\\gamma$ is the\nBarbero-Immirzi parameter, representing a quantization ambiguity.\n(The numerical value of $\\gamma$ is fixed by the black hole entropy\ncalculation.) The momenta $E^a_i$ carry, as usual, density weight 1\nand are given by: $E^a_i = \\sqrt{q} e^a_i$. The fundamental Poisson\nbracket is:\n\\begin{equation} \\{A_a^i(x), \\, E^b_j(y)\\} = 8\\pi G\\gamma\\,\\, \\delta_a^b\\,\n\\delta^i_j\\, \\delta^3(x,y)\\, .\\end{equation}\n\nIn Bianchi models \\cite{taub,bianchi,atu}, one restricts oneself to\nthose phase space variables admitting a 3-dimensional group of\nsymmetries which act simply and transitively on $M$. Thus, the\n3-metrics $q_{ab}$ under consideration admit a 3-parameter group of\nisometries and $M$ is diffeomorphic to a 3-dimensional Lie group\n$G$. (However, there is no canonical diffeomorphism, so that there\nis no preferred point on $M$ corresponding to the identity element\nof $G$.) To avoid a proliferation of spaces and types of indices, it\nis convenient to identify the internal space $I$ and the Lie-algebra\n$\\mathcal{L} G$ of $G$ via a fixed isomorphism. Then, there is a natural\nisomorphism ${}^o\\xi^a_i$ between $\\mathcal{L} G \\equiv I$ and Killing vector\nfields on $M$: for each internal vector $v^i$, ${}^o\\xi^a_i v^i$ is a\nKilling field on $M$. For brevity we will refer to ${}^o\\xi^a_i$ as\n(left invariant) vector fields on $M$. There is a canonical triad\n${}^o\\!e^a_i$\n---the right invariant vector fields--- which is Lie dragged by the\n${}^o\\xi^a_i$. This triad and the dual co-triad ${}^o\\!\\omega_a^i$ satisfy:\n\\begin{eqnarray} [ {}^o\\xi_i,\\, {}^o\\!e_j ] &=&0, \\quad\\quad  [{}^o\\!e_i,\\, {}^o\\!e_j] =\n- {}^o C_{ij}^k\\, {}^o\\!e_k,\\nonumber\\\\\n\\mathcal{L}_{{}^o\\xi_i}\\,( {}^o\\!\\omega^j) &=&0, \\quad\\quad {\\rm d}\\,{}^o\\!\\omega^k = \\frac{1}{2}\\,\n{}^o C_{ij}^k {}^o\\!\\omega^i\\wedge{}^o\\!\\omega^j,\\end{eqnarray}\nwhere ${}^o C_{ij}^k$ denotes the structure constants of $\\mathcal{L} G$. It is\nconvenient to use the fixed fields ${}^o\\!e^a_i$ and ${}^o\\!\\omega_a^i$ as\n\\emph{fiducial} triads and co-triads.\n\nIn the case when $G$ is the Bianchi II group, we have ${}^o C_{ik}^k =0$\nas in all class A Bianchi models and, furthermore, the symmetric\ntensor $k^{kl}:={}^o C_{ij}^k \\, \\epsilon^{ijl}$ has signature +,0,0.\nTherefore, we can fix, once and for all an orthonormal basis\n${}^o b_1^i, {}^o b_2^i, {}^o b_3^i$ in $I$ such that the only non-zero\ncomponents of ${}^o C_{ij}^k$ are\n\\begin{equation} {}^o C_{23}^1 = - {}^o C_{32}^1 = \\tilde{\\alpha}\\, ,\\end{equation}\nwhere $\\tilde{\\alpha}$ is a non-zero real number.%\n\\footnote{Without loss of generality $\\tilde{\\alpha}$ can be chosen to be 1. We\nkeep it general because we will rescale it later (see Eq.\n(\\ref{tilde})) and because we want to be able to pass to the Bianchi\nI case by taking the limit $\\tilde{\\alpha}\\to0$.}\nWe will assume that this basis is so oriented that\n\\begin{equation} \\label{ve1} \\epsilon_{123}\\, :=\\, \\epsilon_{ijk} \\, {}^o b^i_1\\,\n{}^o b^j_2\\, {}^o b^k_3\\, \\, =\\, \\varepsilon\\end{equation}\nwhere $\\varepsilon = \\pm 1$ depending on whether the frame $e^a_i$ (which\ndetermines the sign of $\\epsilon_{ijk}$) is right or left handed.\nThroughout this paper we will set ${}^o\\xi^a_1 = {}^o\\xi^a_i {}^o b^i_1,\\,\n{}^o\\!e^a_1 = {}^o\\!e^a_i{}^o b^i_1,\\, {}^o\\!\\omega_a^1 = {}^o\\!\\omega_a^i{}^o b^1_i$, etc.\n\nThe form of the components of ${}^o C^k_{ij}$ in this basis implies that\n$M$ admits global coordinates $x,y,z$ such that the Bianchi II\nKilling vectors have the fixed form\n\\begin{equation} {}^o\\xi_1^a = \\left(\\frac{\\partial}{\\partial x}\\right)^a, \\qquad\n {}^o\\xi^a_2 = \\left(\\frac{\\partial}{\\partial y}\\right)^a, \\qquad\n {}^o\\xi^a_3 = \\tilde{\\alpha} y\\left(\\frac{\\partial} {\\partial x}\\right)^a+\n \\left(\\frac{\\partial}{\\partial z}\\right)^a. \\end{equation}\nThese expressions bring out the fact that, if we were to attempt to\ncompactify the spatial slices to pass to a $\\mathbb{T}^3$ topology\n---as one can in the Bianchi I model--- we will no longer have\nglobally well-defined Killing fields. Thus, in the Bianchi II model,\nwe are forced to deal with the subtleties associated with\nnon-compactness of the spatially homogeneous slices.\n\nIn the $x,y,z$ chart, the right invariant triad is given by\n\\begin{equation} {}^o\\!e^a_1 = \\left(\\frac{\\partial}{\\partial x}\\right)^a, \\qquad {}^o\\!e^a_2\n= \\tilde{\\alpha} z \\left(\\frac{\\partial}{\\partial\nx}\\right)^a+\\left(\\frac{\\partial}{\\partial y} \\right)^a, \\qquad {}^o\\!e^a_3\n= \\left(\\frac{\\partial}{\\partial z}\\right)^a, \\end{equation}\nand the dual co-triad by\n\\begin{equation} {}^o\\!\\omega_a^1=({\\rm d} x)_a-\\tilde{\\alpha} z({\\rm d} y)_a, \\qquad{}^o\\!\\omega_a^2=({\\rm d} y)_a,\n\\qquad{}^o\\!\\omega_a^3=(dz)_a. \\end{equation}\nThey determine a fiducial 3-metric ${}^o\\!q_{ab}:= q_{ij}{}^o\\!\\omega_a^i{}^o\\!\\omega_b^j$\nwith Bianchi II symmetries:\n\\begin{equation} {}^o\\!q_{ab} {\\rm d} x^a {\\rm d} x^b =  ({\\rm d} x-\\tilde{\\alpha} z\\:{\\rm d} y)^2\\,+\\,{\\rm d}\ny^2\\,+\\, {\\rm d} z^2. \\end{equation}\n\nIn the diagonal models, the physical triads $e^a_i$ are related to\nthe fiducial ones by%\n\\footnote{There is no sum if repeated indices are both covariant or\ncontravariant.  As usual, the Einstein summation convention holds if a\ncovariant index is contracted with a contravariant index.}\n\\begin{equation} \\label{edef} \\omega_a^i = a^i(\\tau){}^o\\!\\omega_a^i, \\qquad \\mathrm{and}\n\\qquad a_i(\\tau) e^a_i = {}^o\\!e^a_i\n \\end{equation}\nwhere the $a_i$ are the three directional scale factors. Since the\nphysical spatial metric is given by  $q_{ab} =\n\\omega_a^i\\omega^{}_{bi}$, the space-time metric can be expressed as\n\\begin{equation} \\label{metric} {\\rm d} s^2= -N {\\rm d}\\tau^2 + a_1(\\tau)^2\\:({\\rm d} x-\\tilde{\\alpha}\nz\\:{\\rm d} y)^2+a_2(\\tau)^2\\:{\\rm d} y^2+a_3(\\tau)^2\\:{\\rm d} z^2 \\end{equation}\nwhere $N$ is the lapse function adapted to the time coordinate\n$\\tau$.\n\nFor later use, let us calculate the spin connection (\\ref{sc})\ndetermined by triads $e^a_i$. From the definition of $\\Gamma_a^i$ it\nfollows that\n\\begin{equation} \\Gamma_a^i =\n-\\epsilon^{ijk}\\,e^b_j\\,\\left(\\partial_{[a}\\omega_{b]k}+\\frac{1}{2}e^c_k\n\\omega^l_a\\partial_{[c}\\omega_{b]l}\\right)\\, . \\end{equation}\nUsing (\\ref{ve1}), the components of $\\Gamma_a^i$ in the internal\nbasis ${}^o b^i_1, {}^o b^i_2, {}^o b^i_3$ can be expressed as\n\\begin{equation} \\Gamma_a^1 = \\frac{\\tilde{\\alpha}\\varepsilon a_1^2}{2a_2a_3}\\:{}^o\\!\\omega_a^1; \\qquad \\Gamma_a^2\n=-\\frac{\\tilde{\\alpha}\\varepsilon a_1}{2a_3}\\:{}^o\\!\\omega_a^2; \\qquad \\Gamma_a^3 = -\\frac{\\tilde{\\alpha}\\varepsilon\na_1}{2a_2}\\:{}^o\\!\\omega_a^3. \\end{equation}\n\nBefore studying the dynamics of the model, let us examine the action\nof internal parity transformation $\\Pi_k$ which flips the $k$th\ntriad vector and leaves the orthogonal vectors alone. (For details\nsee Appendix and \\cite{aa-dis}). Under the parity transformation\n$\\Pi_1$, for example, we have: $e^a_1\\,\\to \\, -e^a_1,\\, e^a_2\\, \\to\ne^a_2,\\, e^a_ 3\\, \\to \\, e^a_3$ and $a_1\\to -a_1,\\, a_2\\to a_2, a_3\n\\to a_3$ whence $\\Gamma_a^1 \\to -\\Gamma^a_1,\\, \\Gamma_a^2 \\to\n\\Gamma^a_2,\\, \\Gamma_a^3 \\to \\Gamma^a_3$. Thus, both $e^a_i$ and\n$\\Gamma_a^i$ are \\emph{proper} internal vectors. $\\varepsilon$ on the other\nhand is a pseudo internal scalar, $\\varepsilon \\to -\\varepsilon$ under every\n$\\Pi_k$. Note that the fiducial quantities carrying a label $o$ do\nnot change under this transformation; it affects only the physical\nquantities.\n\n\\subsection{The Bianchi II Phase space}\n\\label{s2.2}\n\nAs is usual in LQC, we will now use the fiducial triads and\nco-triads to introduce a convenient parametrization of the phase\nspace variables, $E^a_i, A_a^i$. Because we have restricted\nourselves to the diagonal model and these fields are symmetric under\nthe Bianchi II group, from each equivalence class of gauge related\nphase space variables we can choose a pair of the form\n\\begin{equation} \\label{var} E^a_i = \\tilde{p}_i\\sqrt{|{}^o\\!q|}\\,{}^o\\!e^a_i \\qquad \\mathrm{and}\n\\qquad A_a^i = \\tilde{c}^i \\,{}^o\\!\\omega_a^i, \\end{equation}\nwhere, as spelled out in footnote 3, there is no sum over $i$. Thus,\na point in the phase space is now coordinatized by six real numbers\n$\\tilde{p}_i,\\tilde{c}^i$. One would now like to use the symplectic structure in\nfull general relativity to induce a symplectic structure on our\nsix-dimensional phase space. However, because of spatial homogeneity\nand the ${\\mathbb{R}}^3$ spatial topology, the integrals defining the\nsymplectic structure, the Hamiltonian (and the action) all diverge.\nTherefore we have to introduce a fiducial cell $\\mathcal{V}$ and\nrestrict integrals to it \\cite{as,abl}. We will take the fiducial\ncell to be rectangular with edges along the coordinate axes and\nlengths of $L_1, L_2$ and $L_3$ with respect to the \\emph{fiducial}\nmetric ${}^o\\!q_{ab}$. It then follows that the volume of the fiducial\ncell with respect to ${}^o\\!q_{ab}$ is $V_o=L_1L_2L_3$. Then the non-zero\nPoisson brackets are given by:\n\\begin{equation} \\label{pb1} \\{\\tilde{c}^i,\\, \\tilde{p}_j\\} \\, = \\, \\frac{8\\pi G \\gamma}{V_o}\\,\n\\delta^i_j \\end{equation}\nwhere $\\gamma$ is the Barbero-Immirzi parameter. As in the Bianchi I\ncase, we have a 1-parameter ambiguity in the symplectic structure\nbecause of the explicit dependence on $V_o$ and we have to make sure\nthat the final physical results are either independent of $V_o$ or\nremain well-defined as we remove the `regulator' and take the limit\n$V_o \\to \\infty$.\n\nIt is convenient to rescale variables to absorb this dependence in\nthe phase space coordinates (as was done in the treatment of Bianchi\nI model in \\cite{awe2}). Let us set\n\\begin{equation} p_1= L_2L_3\\tilde{p}_1, \\qquad p_2=L_3L_1\\tilde{p}_2, \\qquad p_3= L_1L_2\\tilde{p}_3,\n\\end{equation}\n\\begin{equation} \\label{tilde} c_1=L_1\\tilde{c}_1, \\qquad c_2=L_2\\tilde{c}_2, \\qquad\nc_3=L_3\\tilde{c}_3 \\qquad \\mathrm{and} \\qquad \\alpha =\n\\frac{L_2L_3}{L_1}\\tilde{\\alpha}\\, , \\end{equation}\nwhere the last rescaling has been introduced to absorb factors of\n$L_i$ which would otherwise unnecessarily obscure the expression of\nthe Hamiltonian constraint. The Poisson brackets between these new\nphase space coordinates is given by%\n\\begin{equation} \\label{pb2}\\{c^i,\\, p_j\\} \\, = \\, 8\\pi G \\gamma \\,\\delta^i_j \\,\n. \\end{equation}\nThese variables have direct physical interpretation. For example,\n$p_1$ is the (oriented) area of the 2-3 face of the elementary cell\nwith respect to the \\emph{physical} metric $q_{ab}$ and $h^{(1)} =\n\\exp c_1\\tau_1$ is the holonomy of the physical connection $A_a^i$\nalong the first edge of the elementary cell.\n\n\nOur choice (\\ref{var}) of physical triads and connections has fixed\nthe internal gauge as well as the diffeomorphism freedom.\nFurthermore, it is easy to explicitly verify that, thanks to\n(\\ref{var}), the Gauss and the diffeomorphism constraints are\nautomatically satisfied. Thus, as in \\cite{awe2}, we are left just\nwith the Hamiltonian constraint\n\\begin{equation} \\label{Hgen} \\mathcal{C}_H = \\int_\\mathcal{V}\n\\Big[\\frac{NE^a_iE^b_j}{16\\pi G\\sqrt{|q|}}\n\\big(\\epsilon^{ij}{}_kF_{ab}{}^k-2(1+\\gamma^2)K_{[a}^iK_{b]}^j \\Big)\n+ N \\mathcal{H}_{{\\rm matt}}\\big]\\, {\\rm d}^3x\\, , \\end{equation}\nwhere\n\\begin{equation} F_{ab}{}^k=2\\partial_{[a}A_{b]}^k+\\epsilon_{ij}{}^kA_a^iA_b^j \\end{equation}\nis the curvature of $A_a^i$ and $\\mathcal{H}_{\\rm matt}$ is the matter\nHamiltonian density. As in \\cite{awe2}, our matter field will\nconsist only of a massless scalar field $T$ which will later serve\nas a relational time variable a la Liebniz. (Additional matter\nfields can be incorporated in a straightforward manner, modulo\npossible intricacies of essential self-adjointness.) Thus,\n\\begin{equation} \\mathcal{H}_{{\\rm matt}} = \\frac{1}{2}\\frac{p_T^2}{\\sqrt{|q|}}. \\end{equation}\n\nSince we want to use the massless scalar field as relational time,\nit is convenient to use a harmonic-time gauge, i.e., assume that the\ntime coordinate $\\tau$ in (\\ref{metric}) satisfies $\\Box \\tau=0$.\nThe corresponding lapse function is $N=\\sqrt{|p_1p_2p_3|}$. With\nthis choice, the Hamiltonian constraint simplifies considerably.\nNote first that the basic canonical variables can be expanded as\n\\begin{equation} E^a_i = \\frac{p_i}{V_o}L_i\\sqrt{|{}^o\\!q|}{}^o\\!e^a_i \\qquad {\\rm and} \\qquad\nA_a^i = \\frac{c^i}{L^i}{}^o\\!\\omega_a^i, \\end{equation}\nand the extrinsic curvature is given by\n $$\\qquad  K_a^i = \\gamma^{-1} (A_a^i-\\Gamma_a^i).$$\nNext, using $p_1 = ({\\rm sgn}a_1)\\, |a_2a_3|\\,L_2L_3$ etc, the\ncomponents of the spin connection become:\n\\begin{equation} \\Gamma_a^1 = \\frac{\\alpha\\varepsilon p_2p_3}{2p_1^2}\\frac{{}^o\\!\\omega_a^1}{L_1}, \\qquad\n\\Gamma_a^2= -\\frac{\\alpha\\varepsilon p_3}{2p_1}\\frac{{}^o\\!\\omega_a^2}{L_2}, \\qquad\n\\Gamma_a^3=-\\frac{\\alpha\\varepsilon p_2}{2p_1} \\frac{{}^o\\!\\omega_a^3}{L_3} \\, .\\end{equation}\nCollecting terms, the Hamiltonian constraint (\\ref{Hgen}) becomes\n\\begin{align} \\label{Hcl} \\mathcal{C}_H&=-\\frac{1}{8\\pi G\\gamma^2}\n\\Big[p_1p_2c_1c_2+p_2p_3 c_2c_3+p_3p_1c_3c_1+\\alpha\\varepsilon p_2p_3c_1\n\\nonumber \\\\\n&\\qquad\\qquad-(1+ \\gamma^2)\\,\\big(\\frac{\\alpha p_2p_3}{2p_1}\\big)^2\\Big]\n + \\frac{1}{2}p_T^2 \\\\\n& = \\mathcal{C}_H^{\\rm (BI)} - \\frac{1}{8\\pi G\\gamma^2}\\Big[\\alpha\\varepsilon\np_2p_3c_1-(1+ \\gamma^2)\\,\\big(\\frac{\\alpha p_2p_3}{2p_1}\\big)^2\\Big],\n\\end{align}\nwhere $\\mathcal{C}_H^{\\rm (BI)}$ is the Hamiltonian constraint\n(including the matter term) for Bianchi I space-times which has\nalready been studied in \\cite{awe2}.  Note that this constraint is\nrecovered in the limit $\\alpha\\to 0$, as it must be.\n\n\nKnowing the form of the Hamiltonian constraint, it is now possible to derive\nthe time evolution of any classical observable $\\mathcal{O}$ by taking its\nPoisson bracket with $\\mathcal{C}_H$:\n\\begin{equation} \\dot{\\mathcal{O}} = \\{\\mathcal{O},\\mathcal{C}_H\\}\\, , \\end{equation}\nwhere the `dot' stands for derivative with respect to harmonic time\n$\\tau$. This gives\n\\begin{equation} \\label{ceom1} \\dot{p_1}=\\gamma^{-1}(p_1p_2c_2+p_1p_3c_3+\\alpha\\varepsilon p_2p_3), \\end{equation}\n\\begin{equation} \\dot{p_2}=\\gamma^{-1}(p_2p_1c_1+p_2p_3c_3), \\end{equation}\n\\begin{equation} \\dot{p_3}=\\gamma^{-1}(p_3p_1c_1+p_3p_2c_2), \\end{equation}\n\\begin{equation}\n\\dot{c_1}=-\\frac{1}{\\gamma}\\Big(p_2c_1c_2+p_3c_1c_3+\\frac{1}{2p_1}(1+\\gamma^2)\n\\big(\\frac{\\alpha p_2p_3}{p_1}\\big)^2\\Big), \\end{equation}\n\\begin{equation} \\dot{c_2}=-\\frac{1}{\\gamma}\\Big(p_1c_2c_1+p_3c_2c_3+\\alpha\\varepsilon\np_3c_1-\\frac{1}{2p_2} (1+\\gamma^2)\\big(\\frac{\\alpha\np_2p_3}{p_1}\\big)^2\\Big), \\end{equation}\n\\begin{equation} \\label{ceom2}\n\\dot{c_3}=-\\frac{1}{\\gamma}\\Big(p_1c_3c_1+p_2c_3c_2+\\alpha\\varepsilon p_2\nc_1-\\frac{1}{2p_3}(1+\\gamma^2)\\big(\\frac{\\alpha p_2p_3}{p_1}\\big)^2\\Big).\n\\end{equation}\nAny initial data satisfying the Hamiltonian constraint can be\nevolved by using the six equations above. It is straightforward to\nextend these results if there are additional matter fields.\n\nFinally, let us consider the parity transformation $\\Pi_k$ which\nflips the $k$th \\emph{physical} triad vector $e^a_k$. (As noted\nbefore, this transformation does not act on any of the fiducial\nquantities which carry a label $o$.) Under this map, we have:\n$q_{ab} \\to q_{ab}, \\, \\epsilon_{abc} \\to \\epsilon_{abc}\\,$ but\n$\\epsilon_{ijk} \\to -\\epsilon_{ijk}, \\, \\varepsilon \\to -\\varepsilon$. The canonical\nvariables $c^i, p_i$ transform as proper internal vectors and\nco-vectors: For example\n\\begin{equation} \\Pi_1(c_1,c_2,c_3) \\rightarrow (-c_1, c_2, c_3) \\qquad {\\rm and}\n\\qquad \\Pi_1(p_1,p_2,p_3) \\rightarrow (-p_1, p_2, p_3)\\, . \\end{equation}\nConsequently, both the symplectic structure and the Hamiltonian\nconstraint are left invariant under any of the parity maps $\\Pi_k$.\n\nThis Hamiltonian description will serve as the point of departure\nfor loop quantization in the next section.\n\n\n\\section{Quantum Theory}\n\\label{s3}\n\nThis section is divided into three parts.  In the first, we discuss\nthe kinematics of the model, in the second we define an operator\ncorresponding to the connection $A_a^i$ using holonomies and in the\nthird we introduce the Hamiltonian constraint operator and describe\nits action on states.\n\n\n\\subsection{LQC Kinematics}\n\nThe kinematics for the LQC of Bianchi II models is almost identical\nto that for Bianchi I models. Therefore, in the sub-section we\nclosely follow \\cite{awe2}.\n\nLet us begin by specifying the elementary functions on the\nclassical phase space which will have unambiguous analogs in the\nquantum theory. As in the Bianchi I model, the elementary\nvariables are the momenta $p_i$ and holonomies of the\ngravitational connection $A_a^i$ along the integral curves of the\nright invariant vector fields ${}^o\\!e^a_i$. Let $\\tau_i$ be a basis of\nthe Lie algebra of SU(2), satisfying $\\tau_i \\tau_j =\n\\frac{1}{2}\\epsilon_{ij}{}^k \\tau_k- \\frac{1}{4} \\delta_{ij}\\mathbb{I}$\nwhere $\\mathbb{I}$ is the unit $2\\times2$ matrix. Consider an edge\nof length $\\ell L_k$ with respect to the fiducial metric\n${}^o\\!q_{ab}$, parallel to ${}^o\\!e^a_k$. The holonomy $h_k^{(\\ell)}$ along\nit is given by\n\\begin{equation} \\label{hol} h_k^{(\\ell)}(c_1,c_2,c_3) = \\exp\\left(\\ell\nc_k\\tau_k\\right) = \\cos\\frac{\\ell c_k}{2} \\mathbb{I} + 2\\sin\\frac{\\ell\nc_k}{2}\\tau_k. \\end{equation}\n(Note that $\\ell$ depends of the fiducial cell but not on the\nfiducial metric.) This family of holonomies is completely\ndetermined by the almost periodic functions $\\exp(i\\ell c_k)$ of\nthe connection. These almost periodic functions will be our\nelementary configuration variables which will be promoted\nunambiguously to operators in the quantum theory.\n\nIt is simplest to use the $p$-representation to  specify the\ngravitational sector $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ of the kinematic Hilbert space. The\northonormal basis states $|p_1,p_2,p_3\\rangle$ are eigenstates of\nquantum geometry. For example, in the state $|p_1,p_2,p_3\\rangle$\nthe face $S_{23}$  of the fiducial cell $\\mathcal{V}$ (given by $x$\n={\\rm const}) has area $|p_1|$.\nThe basis is orthonormal in the sense\n\\begin{equation} \\langle p_1,p_2,p_3|p_1',p_2',p_3'\\rangle = \\delta_{p_1^{}p_1'}\n\\delta_{p_2^{}p_2'}\\delta_{p_3^{}p_3'}\\, , \\end{equation}\nwhere the right side features Kronecker symbols rather than the\nDirac delta distributions. Hence kinematical states can consist only\nof \\emph{countable} linear combinations\n\\begin{equation} |\\Psi\\rangle \\,=\\,\n\\sum_{p_1,p_2,p_3}\\Psi(p_1,p_2,p_3)|p_1,p_2,p_3\\rangle\\ \\end{equation}\nof these basis states for which the norm\n\\begin{equation} \\label{norm} ||\\Psi ||^2\\, =\\, \\sum_{p_1,p_2,p_3}\\,\n|\\Psi(p_1,p_2,p_3)|^2 \\end{equation}\nis finite. Because the right side features a sum over a countable\nnumber of points on ${\\mathbb{R}}^3$, rather than a 3-dimensional integral,\nLQC kinematics are inequivalent to those of the Schr\\\"odinger\napproach used in Wheeler-DeWitt quantum cosmology.\n\nNext, recall that on the classical phase space the three reflections\n$\\Pi_i:\\,\\,e^a_i\\,\\to\\, -e^a_i$ are large gauge transformations\nunder which physics does not change (since both the metric and the\nextrinsic curvature are left invariant). These large gauge\ntransformations have a natural induced action, denoted by\n$\\hat\\Pi_i$, on the space of wave functions $\\Psi(p_1,p_2,p_3)$. For\nexample,\n\\begin{equation} \\hat\\Pi_1\\Psi(p_1,p_2,p_3)=\\Psi(-p_1,p_2,p_3). \\end{equation}\nSince $\\hat\\Pi_i^2$ is the identity, for each $i$, the group of\nthese large gauge transformations is simply $\\mathbb{Z}_2$. As in Yang-Mills\ntheory, physical states belong to its irreducible representation.\nFor definiteness, as in the isotropic and Bianchi I models, we will\nwork with the symmetric representation. It then follows that\n$\\mathcal{H}_{\\mathrm{kin}}^{\\mathrm{grav}}$ is spanned by wave\nfunctions $\\Psi(p_1,p_2,p_3)$ which satisfy\n\\begin{equation} \\label{parity} \\Psi(p_1,p_2,p_3)=\\Psi(|p_1|,|p_2|,|p_3|) \\end{equation}\nand have a finite norm (\\ref{norm}).\n\nThe action of the elementary operators on\n$\\mathcal{H}_{\\mathrm{kin}}^{\\mathrm{grav}}$ is as follows: the\nmomenta act by multiplication whereas the almost periodic\nfunctions in $c_i$ shift the $i$th argument. For example,\n\\begin{equation} [\\hat p_1 \\Psi](p_1,p_2,p_3) = p_1\\, \\Psi(p_1,p_2,p_3) \\,\\quad\n\\mathrm{and} \\,\\quad \\Big[\\widehat{\\exp(i\\ell c_1)}\\Psi\\Big](p_1,\np_2, p_3) = \\Psi(p_1-8\\pi\\gamma G\\hbar \\ell, p_2, p_3)\\, . \\end{equation}\nThe expressions for $\\hat p_2, \\widehat{\\exp(i\\ell c_2)}, \\hat\np_3$ and $\\widehat{\\exp(i\\ell c_3)}$ are analogous.  Finally, we\nneed to define the operator $\\hat{\\varepsilon}$ since $\\varepsilon$ features in\nthe expression of the Hamiltonian constraint. In the classical\ntheory, $\\varepsilon$ is unambiguously defined on non-degenerate triads,\ni.e., when $p_1p_2p_3 \\not= 0$. In quantum theory, wave functions\ncan have support also on degenerate configurations. We will extend\nthe definition to degenerate triads using the basis\n$|p_1,p_2,p_3\\rangle$:\n\\begin{equation} \\label{ve2} \\hat{\\varepsilon}\\,|p_1,p_2,p_3\\rangle := \\left\\{\n\\rlap{\\raise2ex\\hbox{\\,\\,$\\quad|p_1,p_2,p_3 \\rangle$ if $p_1p_2p_3\n\\ge 0$,}}{\\lower2ex\\hbox{\\,\\,$ -\\,|p_1,p_2,p_3 \\rangle$ if\n$p_1p_2p_3<0$.}} \\right. \\end{equation}\nFinally, the full kinematical Hilbert space\n$\\mathcal{H}_{\\mathrm{kin}}$ will be the tensor product\n$\\mathcal{H}_{\\mathrm{kin}}=\\mathcal{H}_{\\mathrm{kin}}^\n{\\mathrm{grav}}\\otimes\\mathcal{H}_{\\mathrm{kin}}^{\\mathrm{matt}}$,\nwhere $\\mathcal{H}_{\\mathrm{kin}}^{\\mathrm{matt}}=L^2({\\mathbb{R}},dT)$ is\nthe matter kinematical Hilbert space for the homogeneous scalar\nfield.  On $\\mathcal{H}_ {\\mathrm{kin}}^{\\mathrm{matt}}$, $\\hat T$\nwill act by multiplication and $\\hat p_T:=-i\\hbar \\mathrm{d}_T$\nwill act by differentiation. As in isotropic and Bianchi I models,\nour final results would remain unaffected if we use a ``polymer\nrepresentation'' also for the scalar field.\n\n\n\\subsection{The connection operator $\\hat{A}_a^i$}\n\\label{s3.2}\n\nTo define the quantum Hamiltonian constraint, we cannot directly use\nthe symmetry reduced classical constraint (\\ref{Hcl}) because it\ncontains connection components $c_k$ themselves and in LQC only\nalmost periodic functions of $c_k$ have unambiguous operator\nanalogs. Indeed, in all LQC models considered so far\n\\cite{abl,aps3,warsaw,apsv,kv1,ls,bp,awe2}, we were led to return to\nthe expression (\\ref{Hgen}) in the full theory and mimic the\nprocedure used in LQG \\cite{tt}. More precisely, the key strategy\nwas to follow full LQG (and spin foams) and define a ``field\nstrength operator'' using holonomies around suitable closed loops.\nIn the Bianchi I model, these closed loops were formed by following\nintegral curves of right invariant vector fields (which are also\nleft invariant). As mentioned in section \\ref{s2}, in the Bianchi II\nmodel the right invariant vector fields define the fiducial triads\n${}^o\\!e^a_i$, the left invariant vector fields, the Killing fields\n${}^o\\xi^i$. Neither constitutes a commuting set, whence their integral\ncurves cannot be used to form closed loops. However, as in the k=1\ncase \\cite{warsaw,apsv}, one can hope to exploit the fact that the\nright invariant vector fields do commute with the left invariant\nones and construct the closed loops by alternately following right\nand left invariant vector fields. But, as mentioned in section\n\\ref{s1}, a new problem arises: unlike in the k=1 (or Bianchi I)\nmodel the resulting holonomies are no longer almost periodic\nfunctions of $c_k$, whence the Hilbert space $\\mathcal{H}_{\\rm kin}^{\\rm\ngrav}$ does not support these holonomy operators. For completeness\nwe will first show this fact explicitly and then introduce a new\navenue to bypass this difficulty.\n\nThe problematic curvature component turns out to be $F_{yz}{}^1$. To\nconstruct the corresponding operator, following the strategy used in\nthe k=1 case \\cite{warsaw,apsv}, we will construct a closed loop\n$\\Box_{yz}$ as follows. In the coordinates $(x,y,z)$,\\,\\, i) Move\nfrom $(0,0,0)$ to $(0,\\bar\\mu_2L_2,0)$ following $\\xi^a_2$;\\,\\, ii)\nthen move from $(0,\\bar\\mu_2L_2,0)$ to\n$(0,\\bar\\mu_2L_2,\\bar\\mu_3L_3)$ following ${}^o\\!e^a_3$;\\,\\, iii) then\nmove from $(0,\\bar\\mu_2L_2,\\bar\\mu_3L_3)$ to $(0,0,\\bar\\mu_3L_3)$\nfollowing $-\\xi^a_2$;\\,\\, and, finally, iv) move from\n$(0,0,\\bar\\mu_3L_3)$ to $(0,0,0)$ following $-{}^o\\!e^a_3$. The\nparameters $\\bar\\mu_i$ which determine the `lengths' of these edges\ncan be fixed by the semi-heuristic correspondence between LQC and\nLQG exactly as in the Bianchi I model \\cite{awe2} because the\ngeometric considerations used in that analysis continue to hold\nwithout any modification in all Bianchi models with $\\mathbb{R}^3$ spatial\ntopology:\n\\begin{equation} \\label{mubar} \\bar\\mu_1 =\n\\sqrt\\frac{|p_1|\\Delta\\,\\ell_{\\mathrm{Pl}}^2}{|p_2p_3|}, \\qquad  \\bar\\mu_2 =\n\\sqrt\\frac{|p_2|\\Delta\\,\\ell_{\\mathrm{Pl}}^2}{|p_1p_3|}, \\qquad \\bar\\mu_3 =\n\\sqrt\\frac{|p_3| \\Delta\\,\\ell_{\\mathrm{Pl}}^2}{|p_1p_2|} \\end{equation}\nwhere $\\Delta\\,\\ell_{\\mathrm{Pl}}^2 = 4\\sqrt{3}\\pi\\gamma\\,\\ell_{\\mathrm{Pl}}^2$ is the `area gap'.\nThe holonomy around this closed loop $\\Box_{yz}$ is given by\n\\begin{equation} {h}_{\\Box_{yz}} = \\frac{2}{c\\,\\,\\bar\\mu_2\\bar\\mu_3L_2L_3}\\cos\\left(\n\\frac{\\bar\\mu_2c_2}{2}\\right)\\sin\\left(\\frac{\\bar\\mu_2 c}{2}\\right)\n\\Big(c_2\\sin(\\bar\\mu_3c_3)+\n\\alpha\\bar\\mu_3c_1\\cos(\\bar\\mu_3c_3)\\Big) \\end{equation}\nwhere\n\\begin{equation} \\label{c12} c = \\sqrt{\\alpha^2\\bar\\mu_3^2c_1^2+c_2^2}. \\end{equation}\nIf we were to shrink the loop so that the area it encloses goes to\nzero, we do indeed recover the classical expression of $F_{yz}{}^1$.\nHowever, because of presence of the term $c$, if $\\alpha\\not=0$ the\nright side fails to be almost periodic in $c_1$ and $c_2$. Hence\nthis holonomy operator fails to exist on $\\mathcal{H}_{\\rm kin}$. It is clear\nfrom the expression (\\ref{c12}) of $c$ that the problem is\nindependent of the specific way $\\bar\\mu_i$ are fixed.\n\nWe will bypass this difficulty by mimicking another strategy used in\nfull LQG \\cite{tt}: We will use holonomies along segments parallel\nto ${}^o\\!e^a_i$ to define an operator corresponding to the connection\nitself. This is a natural strategy because holonomies along these\nsegments suffice to separate the Bianchi II connections (\\ref{var}).\nLet us set $A_a := A_a^k\\tau_k$. Then we have the identity:\n\\begin{equation} \\label{classA} A_a =  \\lim_{\\ell_k \\to 0}\\, \\sum_k\n\\,\\frac{1}{2\\ell_kL_k}\\,\\, \\Big(h_k^{(\\ell_k)} -\n(h_k^{(\\ell_k)})^{-1}\\Big) \\end{equation}\nwhere $h_k^{(\\ell_k)}$ is given by (\\ref{hol}). In the expressions\nof physically interesting operators such as the Hamiltonian\nconstraint of full LQG, one often replaces $A_a$ with the (analog of\nthe) right side of (\\ref{classA}). But because of the specific forms\nof these operators, the limit trivializes on diffeomorphism\ninvariant states of LQG. In LQC, we have gauge fixed the system and\ntherefore cannot appeal to diffeomorphism invariance. Indeed, while\nthe holonomies are well-defined for each non-zero $\\ell_k$, the\nlimit fails to exist on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$. A natural\nstrategy is to shrink $\\ell_k$ to a judiciously chosen non-zero\nvalue. But what would this value be? In the case of plaquettes, we\ncould use the interplay between LQG and LQC directly because the\nargument $p_i$ of LQC quantum states refers to \\emph{quantum} areas\nof faces of the elementary cell $\\mathcal{V}$ \\cite{awe2}. For edges\nwe do not have such direct guidance. There is, nonetheless a natural\nprinciple one can adopt: Normalize $\\ell_k$ such that the numerical\ncoefficient in front of the curvature operator constructed from the\nresulting connection agrees with that in the expression of the\ncurvature operator constructed from holonomies around closed loops,\nin all cases where the second construction is available. We will use\nthis strategy. Let us apply it to the Bianchi I model where\n$F_{ab}{}^k = \\epsilon_{ij}{}^k\\, A_a^i A_b^j$. Using holonomies\naround closed loops one obtains the field strength operator\n\\begin{equation} \\hat{F}_{ab}{}^k = \\epsilon_{ij}{}^k\\,\n\\big(\\frac{\\sin\\bar{\\mu}c}{\\bar{\\mu}L}\\, {}^o\\!\\omega_a\\big)^i\\,\n\\big(\\frac{\\sin\\bar{\\mu}c}{\\bar{\\mu}L}\\, {}^o\\!\\omega_b\\big)^j \\end{equation}\nwhere\n\\begin{equation} \\big(\\frac{\\sin\\bar{\\mu}c}{\\bar{\\mu}L}\\, {}^o\\!\\omega_a\\big)^i =\n\\big(\\frac{\\sin\\bar{\\mu_i}c_i}{\\bar{\\mu_i}L_i}\\, {}^o\\!\\omega_a^i\\big) \\quad\\quad\n\\hbox{\\rm (no sum over i)} \\nonumber \\end{equation}\n(see Eqs (3.12) and (3.13) in \\cite{awe2}). Therefore, our strategy\nyields $\\ell_k = 2\\bar\\mu_k$, that is,\n\\begin{equation} \\label{Aop} \\hat{A}_a^k =\n\\frac{\\widehat{\\sin(\\bar\\mu^kc^k)}}{\\bar\\mu^kL_k}\\,\\,{}^o\\!\\omega_a^k, \\end{equation}\nwhere there is no sum over $k$. Note that the principle stated above\nleads us unambiguously to the factor $2$ in $\\ell_k = 2\\bar\\mu_k$;\nwithout recourse to a systematic strategy, one may have naively set\n$\\ell_k =\\bar\\mu_k$.\n\nIf we compare the expression (\\ref{Aop}) of the connection operator\nwith the expression (\\ref{var}) of the classical connection, we have\neffectively defined an operator $\\hat{c}$ via\n\\begin{equation} \\hat{c}_k = \\frac{\\widehat{\\sin(\\bar\\mu^kc^k)}}{\\bar\\mu^k} \\end{equation}\nwhere there is again no sum over $k$. In the literature such a\nquantization of $c$ is often called ``polymerization.'' Our approach\nis an improvement over such strategies in two respects. First, we\ndid not just make the substitution $c \\rightarrow \\sin \\ell c\/\\ell$\nby hand; a priori one could have used another substitution such as\n$c \\rightarrow \\tan \\ell c\/\\ell$. Rather, as in full LQG, we used\nthe strategy of expressing the connection in term of holonomies,\n`the elementary variables'. But this still leaves open the question\nof what $\\ell$ one should use. We determined this by requiring that\nthe overall normalization of $\\hat{F}_{ab}{}^k$ constructed from\n$\\hat{A}_a^i = c^i (L^i)^{-1}\\,{}^o\\!\\omega_a^i$ should agree with that of\n$\\hat{F}_{ab}{}^k$ constructed from holonomies around appropriate\nclosed loops, when the second construction is possible. Therefore,\nour construction is a bona-fide generalization of the previous\nconstructions used successfully in LQC.\n\nThis strategy has some applications beyond the Bianchi II model\nstudied in this paper. First, the k=$-1$ isotropic case has been\nstudied in detail in \\cite{kv1,ls}. The analysis uses the $\\bar\\mu$\nscheme, carries out a numerical simulation using exact LQC equations\nand shows that the effective equations of the ``embedding approach\"\n\\cite{jw,vt} (discussed in section \\ref{s4}) provide an excellent\napproximation to the quantum evolution. While this is an essentially\nexhaustive treatment, as \\cite{kv1,ls} itself points out, the\ntreatment has a conceptual limitation in that it builds holonomies\naround the closed loops using the extrinsic curvature $K_a^i$\n---rather than $A_a^i$--- as a ``connection''. This limitation can\nbe overcome in a straightforward fashion using our current strategy.\nMore importantly, this strategy is applicable to all class A Bianchi\nmodels, including type IX. Thus, it opens the door to the LQC\ntreatment of all these models in one go.\n\n\n\\subsection{The quantum Hamiltonian constraint}\n\\label{s3.3}\n\nWith the connection operator at hand, one can construct the\nHamiltonian constraint operator starting either from the general LQG\nexpression (\\ref{Hgen}) or the symmetry reduced expression\n(\\ref{Hcl}). We will begin by a small change in the representation\nof kinematical states which will facilitate this task.\n\n\\subsubsection{A more convenient representation}\n\\label{s3.3.1}\n\nIgnoring factor ordering ambiguities for the moment, the\nconstraint operator $\\hat{\\mathcal{C}}_H$ is given by\n\\begin{align} \\label{qHam1} \\hat{\\mathcal{C}}_H = -\\frac{1}{8\\pi\nG\\gamma^2\\Delta\\ell_{\\mathrm{Pl}}^2}&\\Big[p_1 p_2|p_3|\\sin\\bar\\mu_1c_1\n\\sin\\bar\\mu_2c_2+|p_1|p_2p_3\\sin\\bar\\mu_2c_2\\sin\\bar\\mu_3c_3 \\nonumber \\\\\n&+p_1|p_2|p_3\\sin\\bar\\mu_3c_3\\sin\\bar\\mu_1c_1\\Big]-\n\\frac{1}{8\\pi G\\gamma^2}\\Big[\\alpha\\hat{\\varepsilon}p_2p_3\\sqrt\\frac{|p_2p_3|}{|p_1|\\Delta\n\\ell_{\\mathrm{Pl}}^2}\\sin\\bar\\mu_1c_1\\nonumber \\\\ & -(1+\\gamma^2)\\left(\\frac{\\alpha\np_2p_3}{2p_1}\\right)^2\\Big]+\\frac{1}{2}\\hat{p}_T^2 \\end{align}\nwhere for simplicity of notation here and in what follows we have\ndropped the hats on the $p_i$ and $\\sin\\bar\\mu_ic_i$ operators.\nRecall that, classically, the Bianchi II symmetry group reduces to\nthe Bianchi I symmetry group if we set $\\alpha=0$. If one sets\n$\\alpha=0$ in (\\ref{qHam1}), the last two terms disappear and the\noperator $\\hat{\\mathcal{C}}_H$ reduces to that of the Bianchi I\nmodel \\cite{awe2} showing explicitly that our construction is a\nnatural generalization of the strategy used there.\n\nTo obtain the action of operators corresponding to terms of the form\n$\\sin\\bar\\mu_ic_i$ we use the same strategy as in \\cite{awe2}. As\nshown there, it is simplest to introduce dimensionless variables\n\\begin{equation}\n\\lambda_i=\\frac{\\mathrm{sgn}(p_i)\\sqrt{|p_i|}}{(4\\pi\\gamma\\sqrt\\Delta\\ell_{\\mathrm{Pl}}^3)^{1\/3}}\\,\n. \\end{equation}\nThen the kets $|\\lambda_1,\\lambda_2,\\lambda_3\\rangle$ constitute an orthonormal\nbasis in which the operators $p_k$ are diagonal\n\\begin{equation} p_k|\\lambda_1,\\lambda_2,\\lambda_3\\rangle\\, =\\,\n[\\mathrm{sgn}(\\lambda_k)(4\\pi\\gamma\\sqrt\\Delta\\ell_{\\mathrm{Pl}}^3)^{2\/3}\n\\lambda_k^2]\\,\\,|\\lambda_1,\\lambda_2,\\lambda_3\\rangle\\, . \\end{equation}\nQuantum states will now be represented by functions\n$\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)$. The operator $e^{i\\bar\\mu_1c_1}$ acts on\nthem as follows\n\\begin{align} \\big[e^{i\\bar\\mu_1c_1}\\,\\Psi\\big] (\\lambda_1,\\lambda_2,\\lambda_3)\n&= \\Psi(\\lambda_1- \\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3) \\nonumber \\\\\n&= \\Psi(\\frac{v-2\\mathrm{sgn}(\\lambda_2\\lambda_3)}{v}\\cdot\\, \\lambda_1,\\lambda_2,\\lambda_3),\n\\end{align}\nwhere we have introduced the variable $v=2\\lambda_1\\lambda_2\\lambda_3$ which is\nproportional to the volume of the fiducial cell:\n\\begin{equation} \\hat{V}\\,\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\\, =\\,\n[2\\pi\\gamma\\sqrt\\Delta\\,|v|\\,\\ell_{\\mathrm{Pl}}^3]\\, \\Psi(\\lambda_1, \\lambda_2,\\lambda_3). \\end{equation}\n(The $e^{i\\bar\\mu_1c_1}$ operator is well-defined in spite of the\nappearance of $|\\lambda_2\\lambda_3|$ in the denominator; see \\cite{awe2}.)\nThe operators $e^{i\\bar\\mu_2 c_2}$ and $e^{i\\bar\\mu_3c_3}$ have\nanalogous action.\n\nWe are now ready to write the Hamiltonian constraint explicitly in\nthe $\\lambda_i$-representation. As noted above, the three terms in the\nfirst square bracket on the right hand side of Eq. (\\ref{qHam1})\nconstitute the gravitational part of $\\hat{\\mathcal{C}}_H$ for the\nLQC of Bianchi I model%\n\\footnote{There are some minor changes in the action of these three\nterms since $\\gamma$ is no longer treated as a pseudoscalar (see\nAppendix \\ref{a1}), but these do not affect the discussion.}\nand have been discussed in \\cite{awe2}. In the next two\nsub-sections we will now discuss the last two terms, which are\nspecific to the Bianchi II model.\n\n\\subsubsection{The Fourth term in $\\hat{\\mathcal{C}}_H$}\n\\label{s3.3.2}\n\nUsing a symmetric factor ordering, the fourth term becomes\n\\begin{equation} \\label{hc4} \\hat{\\mathcal{C}}_H^{(4)} = -\\frac{\\alpha\np_2p_3\\sqrt{|p_2p_3|}} {16\\pi\nG\\gamma^2\\sqrt\\Delta\\ell_{\\mathrm{Pl}}}\\,\\,\\widehat{|p_1|^{-1\/4}}\\,(\\hat{\\varepsilon}\\,\n\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_1c_1\\,\\hat{\\varepsilon})\\,\\widehat{|p_1|^{-1\/4}}\n\\, . \\end{equation}\n(Note that $p_2$ and $p_3$ commute with the other terms in\n$\\hat{\\mathcal{C}}_H^{(4)}$). The operator $p_1$ is self-adjoint on\n$\\mathcal{H}_{\\rm kin}^{\\rm grav}$ whence any measurable function of $p_1$ is\nalso a well-defined self-adjoint operator. However, since kets\n$|\\lambda_1=0, \\lambda_2,\\lambda_3\\rangle$ are normalizable in $\\mathcal{H}_{\\rm kin}^{\\rm\ngrav}$, the naive inverse powers of $\\hat{p}_1$ fail to be densely\ndefined and cannot be self-adjoint. To define inverse powers, as is\nusual in LQG, we will use a variation on the Thiemann inverse triad\nidentities \\cite{tt}. Classically, we have the identity\n\\begin{equation} \\label{class} |p_1|^{-1\/4} = -\\frac{i\\,\\mathrm{sgn}(p_1)}{2\\pi G\\gamma}\n\\sqrt\\frac{|p_2p_3|}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\,\\,\ne^{-i\\bar\\mu_1c_1}\\,\\{e^{i\\bar\\mu_1c_1},|p_1|^{1\/4}\\}\\, . \\end{equation}\nwhich holds for any choice of $\\bar\\mu_1$. Since it is most natural\nto use the same $\\bar\\mu_1$ that featured in the definition of the\nconnection operator, we will make this choice. Eq (\\ref{class})\nsuggests a natural quantization strategy for $|p_1|^{-1\/4}$. Using\nit and the parity considerations, we are led to the following factor\nordering:%\n\\footnote{In the classical theory, $(L_2L_3)^{1\/4}\\,|p_1|^{-1\/4}$ is\nindependent of the choice of the elementary cell. As pointed out in\n\\cite{kv1} the inverse triad operators, by contrast, depend on the\nchoice of the cell. However, one can verify that as we remove the\nregulator, i.e., take the limit $\\mathcal{V} \\to \\mathbb{R}^3$, as in the\nclassical theory, the limiting\n$(L_2L_3)^{1\/4}\\,\\widehat{|p_1|^{-1\/4}}$ has a well defined limit.}\n\\begin{equation} \\widehat{|p_1|^{-1\/4}} = - \\frac{i\\,\\mathrm{sgn}(p_1)}{2\\pi\nG\\gamma}\\sqrt\\frac{|p_2p_3|}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\,\\,e^{-i\\bar\\mu_1c_1\/2}\\,\\,\n\\frac{1}{i\\hbar}[e^{i\\bar\\mu_1c_1},|p_1|^{1\/4}]\\,\\,\ne^{-i\\bar\\mu_1c_1\/2}\\, , \\end{equation}\nwhere, as is common in LQC, $\\mathrm{sgn}(p_1)$ is defined as\n\\begin{equation} \\mathrm{sgn}(p_1) = \\left\\{\\rlap{\\rlap{\\raise4ex\\hbox{\\,\\,$+1$ if $p_1>0$,}}\n{\\raise0ex\\hbox{\\,\\,$0$ if $p_1=0$,}}}\n{\\lower4ex\\hbox{\\,\\,$-1$ if $p_1<0$.}} \\right. \\end{equation}\n\nAt first it may seem surprising that the expression of\n$\\widehat{|p_1|^{-1\/4}}$ involves operators other than ${p_1}$. It\nis therefore important to verify that it has the standard desirable\nproperties. First, as one would hope, it is indeed diagonal in the\neigenbasis of the operators $\\hat{p}_k$:\n\\begin{equation} \\label{inv} \\widehat{|p_1|^{-1\/4}}\\, |\\lambda_1,\\lambda_2,\\lambda_3\\rangle =\n\\frac{\\sqrt2 \\mathrm{sgn}(\\lambda_1)\\,\\sqrt{|\\lambda_2\\lambda_3|}}\n{(4\\pi\\gamma\\sqrt\\Delta\\ell_{\\mathrm{Pl}}^3)^{1\/6}}\n\\left(\\sqrt{|v+\\mathrm{sgn}(\\lambda_2\\lambda_3)|}-\\sqrt{|v-\\mathrm{sgn}(\\lambda_2\\lambda_3)|}\\right)\\,\n|\\lambda_1,\\lambda_2,\\lambda_3\\rangle. \\end{equation}\nSecond, on eigenkets with large volume, the eigenvalue is indeed\nwell-approximated by $p_1^{-1\/4}$, whence on semi-classical states\nit behaves as the inverse of $\\hat{p}^{1\/4}$, just as one would\nhope. Thus, (\\ref{inv}) is a viable candidate for\n$\\widehat{|p_1|^{-1\/4}}$. But there are interesting\nnon-trivialities in the Planck regime. In particular, although\ncounter-intuitive, as is common in LQC the operator annihilates\nstates $|\\lambda_1,\\lambda_2,\\lambda_3\\rangle$ with $v = 2\\lambda_1\\lambda_2\\lambda_3 =0$\n\nFinally, note that the operator $\\hat{\\varepsilon}$ appearing in the\nexpression (\\ref{hc4}) of $\\hat{\\mathcal{C}}_H^{(4)}$ either\noperates immediately before or after $\\widehat{|p_1|^{-1\/4}}$. Since\n$\\widehat{|p_1|^{-1\/4}}$ annihilates all zero volume states and\n$\\hat{\\varepsilon}$ acts on such states as the identity operator, we only\nneed to consider the action of $\\hat{\\varepsilon}$ on states with nonzero\nvolume. In this case, $\\hat{\\varepsilon}$ acts as $\\mathrm{sgn}(v)$. Therefore the\naction of $\\hat{\\mathcal{C}}_H^{(4)}$ can be written as:\n\\begin{align}\n\\Big[\\hat{\\mathcal{C}}_H^{(4)}\\,\\Psi\\Big](\\lambda_1,\\lambda_2,\\lambda_3) =&\n-\\frac{i\\alpha\\pi\\sqrt\\Delta\\hbar\\ell_{\\mathrm{Pl}}^2}{(4\\pi\\gamma\\sqrt\\Delta)^{1\/3}}\\,\\,\n\\mathrm{sgn}(v)\\,\\, (\\lambda_2\\lambda_3)^4\\nonumber\\\\\n\\left(\\sqrt{|v+\\mathrm{sgn}(\\lambda_2\\lambda_3)|}-\\sqrt{|v-\\mathrm{sgn}(\\lambda_2\\lambda_3)|} \\right)\n& \\quad \\Big[\\Phi^+(\\lambda_1,\\lambda_2,\\lambda_3) -\\Phi^-(\\lambda_1,\\lambda_2,\\lambda_3)\\Big]\n\\label{c4}\n\\end{align}\nwhere\n\\begin{align} \\Phi^\\pm(\\lambda_1,\\lambda_2,\\lambda_3) =& \\Big(\\sqrt{\\left|v\\pm2\\mathrm{sgn}(\\lambda_2\n\\lambda_3)+\\mathrm{sgn}(\\lambda_2\\lambda_3))\\right|}\n-\\sqrt{\\left|v\\pm2\\mathrm{sgn}(\\lambda_2\\lambda_3)-\\mathrm{sgn}(\\lambda_2\\lambda_3)\\right|}\\, \\Big)\n\\nonumber \\\\  & \\quad\\: \\times\n\\big(\\mathrm{sgn}(v)+\\mathrm{sgn}(v\\pm2 \\mathrm{sgn}(\\lambda_2\\lambda_3))\\big)\\,\\,\n\\Psi(\\frac{v\\pm2\\mathrm{sgn}(\\lambda_2\\lambda_3)}{v}\\lambda_1,\\lambda_2,\\lambda_3).\\label{phi} \\end{align}\n\nRecall that in the classical theory the singularity corresponds\nprecisely to the phase space points at which the volume vanishes.\nTherefore, as in the Bianchi I model, states with support only on\npoints with $v=0$ will be called `singular' and those which vanish\nat points with $v=0$ will be called regular. The total Hilbert space\n$\\mathcal{H}_{\\rm kin}^{\\rm grav}$ is naturally decomposed as a direct sum $\\mathcal{H}_{\\rm kin}^{\\rm grav} = \\mathcal{H}^{\\rm\ngrav}_{\\rm sing}\\oplus \\mathcal{H}^{\\rm grav}_{\\rm reg}$ of singular and\nregular sub-spaces. We will conclude this discussion by examining\nthe action of $\\hat{\\mathcal{C}}_H^{(4)}$ on these sub-spaces. Note\nfirst that in the action (\\ref{hc4}) of $\\hat{\\mathcal{C}}_H^{(4)}$,\nthe state is first acted upon by the operator\n$\\widehat{|p_1|^{-1\/4}}$. Since this operator annihilates states\n$|\\lambda_1 \\lambda_2,\\lambda_3\\rangle$ with $v = 2\\lambda_1\\lambda_2\\lambda_3 =0$, singular\nstates are simply annihilated by $\\hat{\\mathcal{C}}_H^{(4)}$. In\nparticular this implies that the singular sub-space is mapped to\nitself under this action. It is clear from (\\ref{phi}) that if\n$\\Psi$ is regular, i.e. vanishes on all points with $v =0$,\n$\\Phi^\\pm$ also vanish at these points. Thus the regular sub-space\nis also preserved by this action. This fact will be used in the\ndiscussion of singularity resolution in section \\ref{s3.3.4}.\n\n\n\\emph{Remark:}\\, Our definition of the operator\n$\\widehat{|p|^{-1\/4}}$ is not unique; as is common with non-trivial\nfunctions of elementary variables, there are factor ordering\nambiguities. For example, for $0<n<1\/2$, we have the classical\nidentity\n\\begin{equation}\n|p_1|^{n-1\/2}=\\frac{-i\\mathrm{sgn}(p_1)\\sqrt{|p_2p_3|}}{8\\pi\\gamma\\sqrt\\Delta\nG\\ell_{\\mathrm{Pl}} n} e^{-i\\bar\\mu_1c_1}\\left\\{e^{i\\bar\\mu_1c_1},|p_1|^n\\right\\}\\,\n. \\nonumber \\end{equation}\nHence, it is possible to instead define $\\widehat{p_1^{-1\/4}}$ as\n\\begin{equation} \\widehat{p_1^{-1\/4}} =\n\\left(\\widehat{|p_1|^{n-1\/2}}\\right)^{-1\/(4n-2)} \\nonumber \\end{equation}\nwhere\n\\begin{equation} \\widehat{|p_1|^{n-1\/2}} =\n-\\frac{(4\\pi\\gamma\\sqrt\\Delta\\ell_{\\mathrm{Pl}}^3)^{(2+2n)\/3}} {4^n(8\\pi\\gamma\nG\\sqrt\\Delta\\ell_{\\mathrm{Pl}})^3n}\\mathrm{sgn}(\\lambda_1)|\\lambda_2\\lambda_3|^{1-2n}\\Big[|v+\n\\mathrm{sgn}(\\lambda_2\\lambda_3)|^{2n}-|v-\\mathrm{sgn}(\\lambda_2\\lambda_3)|^{2n}\\Big]. \\nonumber \\end{equation}\nFor $n\\ne1\/4$, this choice for the operator $\\widehat{p_1^{-1\/4}}$\nis not equivalent to the one we chose.  These two choices are both\nwell-defined and admit the same classical limit but they differ in\nthe Planck regime. It is also possible to construct other such\ninequivalent $\\widehat{p_1^{-1\/4}}$ candidate operators. For\ndefiniteness we have made the `simplest' choice.\n\n\n\n\\subsubsection{The fifth term in $\\hat{\\mathcal{C}}_H$}\n\\label{s3.3.3}\n\nLet us now consider the last term in the expression of the\ngravitational part of the Hamiltonian constraint\n\\begin{equation} \\hat{\\mathcal{C}}_H^{(5)} = \\frac{\\alpha^2}{32\\pi\nG\\gamma^2}(1+\\gamma^2)\\,(p_2 p_3)^2\\,\\,\\widehat{p_1^{-2}}. \\end{equation}\nThis term is simpler since it only involves powers of $p_k$ and we\nare working in a representation where $p_k$ are diagonal. From our\ndiscussion of the last section, it is natural to set\n\\begin{equation} \\widehat{p_1^{-2}}:=\\left(\\widehat{p_1^{-1\/4}}\\right)^8\\, , \\end{equation}\nthen we have\n\\begin{align} \\label{c5} \\hat{\\mathcal{C}}_H^{(5)}\\,\\Psi(\\lambda_1,\\lambda_2,\n\\lambda_3)\\, =\\, & \\frac{8\\pi\\alpha^2\\Delta(1+\\gamma^2)\\hbar\\ell_{\\mathrm{Pl}}^2}{(4\\pi\\gamma\\sqrt\n\\Delta)^{2\/3}}\\,\\mathrm{sgn}(\\lambda_1)^8\\lambda_2^8\\lambda_3^8 \\nonumber \\\\ & \\times \\left(\n\\sqrt{|v+\\mathrm{sgn}(\\lambda_2\\lambda_3)|}-\\sqrt{|v-\\mathrm{sgn}(\\lambda_2\\lambda_3)|}\\right)^8 \\Psi(\\lambda_1,\\lambda_2,\n\\lambda_3). \\end{align}\nAgain, it is clear that if $v=0$, the wave function is annihilated\nby this part of the constraint.  Also, it follows by inspection that\nthe singular and regular subspaces are both mapped to themselves by\nthe action of $\\hat{\\mathcal{C}}_H^{(5)}$.\n\n\\subsubsection{Singularity resolution}\n\\label{s3.3.4}\n\nWe can now determine the gravitational part $\\hat{\\mathcal{C}}_{\\rm grav}$\nof the Hamiltonian constraint by combining the results of\n\\cite{awe2} and Eqs. (\\ref{c4}) and (\\ref{c5}). We have:\n\\begin{equation} \\label{qHam2} \\hat{\\mathcal{C}}_{\\rm grav} = \\hat{\\mathcal{C}}_{\\rm grav}^{\\rm\n(BI)} + \\hat{\\mathcal{C}}_H^{(4)} + \\hat{\\mathcal{C}}_H^{(5)} \\end{equation}\nwhere $\\hat{\\mathcal{C}}_{\\rm grav}^{\\rm (BI)}$ is the gravitational part of\nthe Hamiltonian constraint in the Bianchi I model \\cite{awe2}. There\nis however, a conceptual subtlety. In the classical theory the\nHamiltonian density $\\mathcal{C}_{\\rm grav}\/(L_1L_2L_3)^2$ is independent of\nthe choice of the elementary cell (where we have to divide by\n$(L_1L_2L_3)^2$ because the lapse corresponding to harmonic time\nscales as $(L_1L_2L_3)$ and the Hamiltonian constraint is obtained\nby integration over the elementary cell $\\mathcal{V}$). As shown in\nthe section V of \\cite{awe2}, $\\hat{\\mathcal{C}}_{\\rm grav}^{\\rm\n(BI)}\/(L_1L_2L_3)^2$ is again independent of the choice of the\nelementary cell $\\mathcal{V}$. However, the two additional terms\nthat are special to the Bianchi II model are not independent of\n$\\mathcal{V}$ because they involve the inverse-triad operators\n\\cite{kv1}. Nonetheless, in the limit as we take the regulator away,\ni.e., $\\mathcal{V} \\to \\mathbb{R}^3$, the operator\n$\\hat{\\mathcal{C}}_{\\rm grav}\/(L_1L_2L_3)^2$ has a well-defined limit (see\nfootnote 5). Strictly speaking, in the discussion of Bianchi II\nquantum dynamics, we have to work with this limit, rather than with\noperators defined using a fixed cell.\n\nAs in the Bianchi I model, the action simplifies if we replace one\nof the $\\lambda_i$ by $v$.  In the Bianchi I model, it does not matter\nwhich of the $\\lambda_i$ is replaced because of the additional symmetry\nof that model. In the Bianchi II case, while it remains possible to\nreplace any of the $\\lambda_i$, it is simplest to replace $\\lambda_1$ by $v$\nand represent quantum states as $\\Psi=\\Psi(\\lambda_2,\\lambda_3,v;T)$. This\nchange of variables would be nontrivial if, as in the Wheeler-DeWitt\ntheory, we had used the Lesbegue measure in the gravitational\nsector. However, it is quite tame here because the norms are defined\nusing a discrete measure. The inner product on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ is now given\nby\n\\begin{equation} \\langle\\Psi_1|\\Psi_2\\rangle_{\\rm kin} = \\sum_{\\lambda_2,\\lambda_3,v}\n\\,\\,\\bar{\\Psi}_1(\\lambda_2,\\lambda_3,v)\\,\\Psi_2(\\lambda_2,\\lambda_3,v) \\end{equation}\nand states are symmetric under the action of $\\hat\\Pi_k$. In\nAppendix \\ref{a1}, we show that, under the action of reflections\n$\\hat\\Pi_i$, the operators $\\sin\\bar\\mu_ic_i$ have the same\ntransformation properties that $c_i$ have under reflections $\\Pi_i$\nin the classical theory. As a consequence, $\\hat{\\mathcal{C}}_{\\rm grav}$ is\nalso reflection symmetric. Therefore, its action is well defined on\n$\\mathcal{H}_{\\rm kin}^{\\rm grav}$: $\\hat{\\mathcal{C}}_{\\rm grav}$ is a densely defined, symmetric\noperator on this Hilbert space. In the isotropic case, its analog\nhas been shown to be essentially self-adjoint \\cite{warsaw2}. In\nwhat follows we will assume that (\\ref{qHam2}) is essentially\nself-adjoint on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ and work with its self-adjoint extension.\n\nIt is now straightforward to write down the full Hamiltonian\nconstraint on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$:\n\\begin{equation} \\label{qHam3} -\\hbar^2\\, \\partial^2_T \\, \\Psi(\\lambda_2,\\lambda_3,v; T) =\n\\Theta\\, \\Psi(\\lambda_2,\\lambda_3,v; T)\\quad {\\rm where}\\quad \\Theta = -2\n\\hat{\\mathcal{C}}_{\\rm grav} \\end{equation}\nAs in the isotropic case \\cite{aps2}, one can obtain the physical\nHilbert space $\\mathcal{H}_{\\rm phy}$ by a group averaging procedure and the\nfinal result is completely analogous. Elements of $\\mathcal{H}_{\\rm phy}$\nconsist of `positive frequency' solutions to (\\ref{qHam3}), i.e.,\nsolutions to\n\\begin{equation} \\label{qHam4} -i\\hbar \\partial_T \\Psi(\\lambda_2,\\lambda_3,v; T)\\, = \\,\n\\sqrt{|\\Theta|}\\, \\Psi(\\lambda_2,\\lambda_3,v; T)\\, ,\\end{equation}\nwhich are symmetric under the three reflection maps $\\hat\\Pi_i$,\ni.e. satisfy\n\\begin{equation} \\label{sym} \\Psi(\\lambda_2,\\lambda_3,v;\\, T) = \\Psi(|\\lambda_2|,|\\lambda_3|,|v|;\\,\nT)\\, . \\end{equation}\nThe scalar product is given simply by:\n\\begin{eqnarray} \\label{ip1} \\langle \\Psi_1|\\Psi_2\\rangle_{\\rm phys} &=& \\langle\n\\Psi_1(\\lambda_2,\\lambda_3, v; T_o)|\\Psi_2(\\lambda_2,\\lambda_3,v; T_o) \\rangle_{\\rm\nkin} \\nonumber\\\\\n&=& \\sum_{\\lambda_1,\\lambda_2,\\lambda_3} \\bar\\Psi_1(\\vec{\\lambda}, T_o)\\,\n\\Psi_2(\\vec{\\lambda}, T_o) \\end{eqnarray}\nwhere $T_o$ is any ``instant'' of internal time $T$.\n\nWe can now address the issue of singularity resolution using general\nproperties of various operators. Recall that the gravitational part\nof the Hamiltonian constraint operator in the Bianchi I model shares\ntwo properties with the fourth and the fifth terms studied above\nwhich are specific to the Bianchi II model. First, it annihilates\nsingular states and, second, singular states decouple from the\nregular states under its action. Therefore the full Bianchi II\nHamiltonian constraint also has these two properties. Since the\nsingular states decouple from regular states%\n\\footnote{Singular states are in the kernel of $\\Theta$ and regular\nstates are orthogonal to the singular ones. From spectral\ndecomposition one expects $\\sqrt{\\Theta}$ to have the same property.\nHowever, to complete this argument, one would have to establish that\n$\\hat{\\mathcal{C}}_{\\rm grav}$ is essentially self-adjoint and its self\nadjoint extension also shares this property.},\nan initial state in the regular sub-space cannot become singular\nduring evolution. It is in this precise sense that the classical\nsingularity is resolved. Sometimes one considers weaker forms of\nsingularity resolution. For example, it could happen that the\nevolution of the wave function is always well defined but a regular\nstate can evolve to the singular sub-space. For the Bianchi I and II\nmodels, the singularity is resolved in a stronger sense: \\emph{Not\nonly is the evolution well defined at all times, but the singular\nstates (are stationary and) decouple entirely from the regular\nones.}\n\n\\subsubsection{The explicit form of the Hamiltonian constraint}\n\\label{s.3.5}\n\nWe will conclude by providing an explicit form of the full quantum\nconstraint equation that will be needed in numerical simulations.\n\nRecall that in the Bianchi I model \\cite{awe2} symmetries enabled us\nto restrict our attention to the positive octant of the\n3-dimensional space spanned by $(\\lambda_1,\\lambda_2,\\lambda_3)$. This is again the\ncase for the Bianchi II model. More precisely, elements of $\\mathcal{H}_{\\rm kin}^{\\rm grav}$\nare invariant under the three parity maps $\\hat{\\Pi}_k$ and, as\nshown in the Appendix \\ref{a1}, the Hamiltonian constraint satisfies\n$\\hat{\\Pi}_k\\, \\hat{\\mathcal{C}}_{\\rm grav} \\hat{\\Pi}_k =\n\\hat{\\mathcal{C}}_{\\rm grav}$. Therefore, knowledge of the restriction of\nthe image $\\hat{ \\mathcal{C}}_{\\rm grav}\\Psi$ of $\\Psi$ to the positive\noctant suffices to determine $\\hat{\\mathcal{C}}_{\\rm grav}\\Psi$ completely.\nIn the positive octant, $\\mathrm{sgn}(\\lambda_k)$ can only be 0 or 1 which\nsimplifies the action of operators. Therefore, in the remainder of\nthis section we will restrict the argument of $\\hat{\n\\mathcal{C}}_H\\Psi$ to the positive octant. The full action is given\nsimply by\n\\begin{equation} \\big(\\hat{\\mathcal{C}}_{\\rm grav}\\Psi\\big)(\\lambda_2,\\lambda_3, v)=\\big(\\hat{\n\\mathcal{C}}_{\\rm grav}\\Psi\\big)(|\\lambda_2|,|\\lambda_3|,|v|). \\end{equation}\n\nSince the singular states are annihilated by\n$\\hat{\\mathcal{C}}_{\\rm grav}$, their evolution is trivial:\n\\begin{equation} \\partial_T^2\\, \\Psi(\\lambda_2,\\lambda_3,v=0;T) = 0\\, . \\end{equation}\nNon-singular states are physically more relevant. On them, the\nexplicit form of the full constraint is given by:\n\\begin{align} \\partial_T^2\\, \\Psi(\\lambda_2,\\lambda_3,v;T) =& \\frac{\\pi G}{2}\\Bigg[\\sqrt{v}\n\\bigg((v+2)\\sqrt{v+4}\\,\\Psi^+_4(\\lambda_2,\\lambda_3,v;T) - (v+2)\\sqrt v\\,\n\\Psi^+_0( \\lambda_2, \\lambda_3,v;T)\\nonumber \\\\& -(v-2)\\sqrt\nv\\,\\Psi^-_0(\\lambda_2,\\lambda_3,v;T)+(v-2)\\sqrt{|v-4|}\n\\,\\Psi^-_4(\\lambda_2,\\lambda_3,v;T)\\bigg)\\nonumber \\\\ &\n+\\frac{2i\\alpha\\sqrt\\Delta}{(4\\pi\n\\gamma\\sqrt\\Delta)^{1\/3}}(\\lambda_2\\lambda_3)^4\\left(\\sqrt{v+1}-\\sqrt{|v-1|}\\right)\\bigg(\n\\Phi^--\\Phi^+\\bigg)(\\lambda_2,\\lambda_3,v;T) \\nonumber\\\\& \\label{qHamfin} +\n\\frac{16\n\\alpha^2\\Delta(1+\\gamma^2)}{(4\\pi\\gamma\\sqrt\\Delta)^{2\/3}}(\\lambda_2\\lambda_3)^8\\:\n(\\sqrt{v+1}-\\sqrt{|v-1|})^8\\:\\Psi(\\lambda_2,\\lambda_3,v;T)\\Bigg] \\end{align}\nwhere $\\Psi^\\pm_{0,4}$ are defined as follows:\n\\begin{align} \\Psi^\\pm_4(\\lambda_2,\\lambda_3,v;T)=& \\:\\Psi\\left(\\frac{v\\pm4}{v\\pm2}\\cdot\n\\lambda_2,\\frac{v\\pm2}{v}\\cdot\\lambda_3,v\\pm4;T\\right)+\\Psi\\left(\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_2,\n\\lambda_3,v\\pm4;T\\right)\\nonumber\\\\& +\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_2,\\frac{v\\pm4}{v\n\\pm2}\\cdot\\lambda_3,v\\pm4;T\\right)+\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_2, \\lambda_3,v\\pm4;T\n\\right) \\nonumber \\\\ & +\\Psi\\left(\\lambda_2,\\frac{v\\pm2}{v}\\cdot\\lambda_3,v\\pm4;T\\right)+\n\\Psi\\left(\\lambda_2,\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_3,v\\pm4;T\\right), \\end{align}\nand\n\\begin{align} \\Psi^\\pm_0(\\lambda_2,\\lambda_3,v;T)= & \\:\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_2,\n\\frac{v}{v\\pm2}\\cdot\\lambda_3,v;T\\right)+\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_2,\\lambda_3,v;T\n\\right) \\nonumber \\\\ & +\\Psi\\left(\\frac{v}{v\\pm2}\\cdot\\lambda_2,\\frac{v\\pm2}{v}\\cdot\\lambda_3,\nv;T\\right)+\\Psi\\left(\\frac{v}{v\\pm2}\\cdot\\lambda_2,\\lambda_3,v;T\\right) \\nonumber \\\\ & +\n\\Psi\\left(\\lambda_2,\\frac{v}{v\\pm2}\\cdot\\lambda_3,v;T\\right)+\\Psi\\left(\\lambda_2,\\frac{v\\pm2}{v}\n\\cdot\\lambda_3,v;T\\right)\\, ,\\end{align}\nwhile $(\\Phi^--\\Phi^+)$ is given by\n\\begin{align} \\big(\\Phi^--\\Phi^+\\big)(\\lambda_2,\\lambda_3,v;T)\\, =& \\,(\\sqrt{|v-2+\n\\mathrm{sgn}(v-2)|}-\\sqrt{|v-2-\\mathrm{sgn}(v-2)|}) \\nonumber \\\\ & \\qquad \\times (1+\\mathrm{sgn}(v-2))\n\\Psi(\\lambda_2,\\lambda_3,v-2;T) \\nonumber \\\\ & \\: -2(\\sqrt{v+3}-\\sqrt{v+1})\n\\Psi(\\lambda_2,\\lambda_3,v+2;T). \\end{align}\n(The imaginary coefficients in (\\ref{qHamfin}) come from the\naction of single $\\sin\\bar\\mu_ic_i$ terms.)\n\nEq. (\\ref{qHamfin}) immediately implies that, as in the Bianchi I\nmodel, the steps in $v$ are uniform: the argument of the wave\nfunction only involves $v-4, v-2, v, v+2$ and $v+4$. Thus, there is\na superselection in $v$. For each $\\epsilon\\in[0,2),$ let us\nintroduce a lattice $\\mathcal{L}_\\epsilon$ of points $v=2n+\\epsilon$\nif $\\epsilon$ is 0 or 1 or $v=n+\\epsilon$ otherwise.\n\\footnote{The lattice for $\\epsilon \\not= 0,1$ is twice as large as\nthat for $\\epsilon=0$ or $\\epsilon=1$ due to the symmetry properties\nof the wave function.}\nThen the quantum evolution ---and the action of the Dirac\nobservables $\\hat{p}_T$ and $\\hat{V}|_{T}$ commonly used in\nLQC--- preserves the subspaces $\\mathcal{H}^\\epsilon_{\\mathrm{phy}}$\nconsisting of states with support in $v$ on $\\mathcal{L}_\\epsilon$.\nThe most interesting lattice is the one corresponding to\n$\\epsilon=0$\nsince it includes the classically singular points $v=0$.\n\nFinally, it is obvious from (\\ref{qHamfin}) that in the limit\n$\\alpha\\to0$ quantum dynamics of the Bianchi II model reduces to\nthat of the Bianchi I model discussed in \\cite{awe2}. In particular,\nit is possible to obtain the LQC dynamics for the $k$=0 FRW\ncosmology from this model by first taking $\\alpha\\to0$ and then\nfollowing the projection map defined in section IVA in \\cite{awe2}.\n\n\n\n\\section{Effective Equations}\n\\label{s4}\n\nIn the isotropic models, effective equations have been introduced\nvia two different approaches ---the embedding and the truncation\nmethods. Both start by regarding the space of quantum states as an\ninfinite dimensional symplectic manifold ---the quantum phase\nspace--- which is also equipped with a K\\\"ahler structure that\ndescends from the Hermitian inner product on the Hilbert space. In\nthe first method, one finds a judicious embedding of the classical\nphase space into the quantum phase space which is approximately\npreserved by the quantum evolution vector field \\cite{jw,vt}. By\nprojecting this vector field into the image of the embedding one\nobtains quantum corrected effective equations. In the isotropic case\nthese effective equations provide an excellent approximation to the\nfull quantum evolution of states which are Gaussians at late times,\neven in the $\\Lambda\\not=0$ as well as k=$\\pm 1$ cases where the\nmodels are not exactly soluble. In the second method one uses\nexpectation values, uncertainties, and higher moments to define a\nconvenient system of coordinates on the infinite dimensional phase\nspace. The exact quantum evolution equations are then a set of\ncoupled non-linear ordinary differential equations for these\ncoordinates. By a judicious truncation of this system one obtains\neffective equations containing quantum corrections \\cite{bs}. In its\nspirit the first method is analogous to the `variational principle\ntechnique' used in perturbation theory, in that it requires a\njudicious combination of art (of selecting the embedding) and\nscience. It is often simpler to use and can be surprisingly\naccurate. The second method is more systematic, similar in our\nanalogy to the standard, order by order perturbation theory. It is\nalso more general in the sense that it is applicable to a wide\nvariety of states. In this section we will use the first method to\ngain qualitative insights into leading order quantum effects.\n\nTo obtain the effective equations, without loss of generality we can\nrestrict our attention to the positive octant of the classical phase\nspace (where $\\varepsilon=1$). Then the quantum corrected Hamiltonian\nconstraint is given by the classical analogue of (\\ref{qHam1}):\n\\begin{equation} \\label{Heff}\n\\frac{p_T^2}{2}+\\mathcal{C}^{\\mathrm{eff}}_{\\mathrm{grav}} = 0, \\end{equation}\nwhere\n\\begin{align} \\mathcal{C}^{\\mathrm{eff}}_{\\mathrm{grav}} =&\n-\\frac{p_1p_2p_3}{8\\pi G\\gamma^2\\Delta\\ell_{\\mathrm{Pl}}^2}\n\\Bigg[\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_2c_2\n\\sin\\bar\\mu_3c_3+\\sin\\bar\\mu_3c_3\\sin\\bar\\mu_1c_1\\Bigg] \\nonumber\n\\\\ & \\quad - \\frac{1}{8\\pi\nG\\gamma^2}\\Bigg[\\frac{\\alpha(p_2p_3)^{3\/2}}{\\sqrt\\Delta\\ell_{\\mathrm{Pl}}\n\\sqrt{p_1}}\\sin\\bar\\mu_1c_1 -(1+\\gamma^2)\\left(\\frac{\\alpha\np_2p_3}{2p_1}\\right)^2 \\Bigg]. \\end{align}\nUsing the expressions (\\ref{mubar}) of $\\bar\\mu_k$, it is easy to\nverify that far away from the classical singularity ---more\nprecisely in the regime in which the (gauge fixed) spin connection\nand the extrinsic curvature are sufficiently small so that\n$c_k\\bar\\mu_k \\ll 1$---  the effective Hamiltonian constraint\n(\\ref{Heff}) is well-approximated by the classical one (\\ref{Hcl}).\n\nSince $\\sin\\theta$ is bounded by 1 for all $\\theta$, these equations\nimply that the matter density $\\rho_{\\mathrm{matt}}=\np_T^2\/2V^2=p_T^2\/2p_1p_2p_3$ satisfies\n\\begin{equation} \\rho_{\\mathrm{matt}} \\le \\frac{3}{8\\pi\\gamma^2\\Delta G\\ell_{\\mathrm{Pl}}^2}+\n\\frac{1}{8\\pi \\gamma^2 G} \\left[\\frac{x}{\\sqrt\\Delta\\ell_{\\mathrm{Pl}}} -\n\\frac{(1+\\gamma^2)x^2}{4} \\right] \\end{equation}\nwhere we have introduced $x:=\\alpha\\sqrt{p_2p_3\/p_1^3}$. The maximum\nof the expression in square brackets is attained at\n$x=2\/(1+\\gamma^2)\\sqrt\\Delta\\ell_{\\mathrm{Pl}}$, whence\n\\begin{equation} \\rho_{\\mathrm{matt}} \\le \\frac{3+(1+\\gamma^2)^{-1}}\n{8\\pi\\gamma^2\\Delta G \\ell_{\\mathrm{Pl}}^2} \\approx 0.54 \\rho_{\\mathrm{Pl}}. \\end{equation}\nThus, on the constraint surface in the phase space defined by\n(\\ref{Heff}), the matter energy density is bounded by $0.54\n\\rho_{\\mathrm{Pl}}$. But this bound may be far from being optimal.\nIn all isotropic models, the optimal bound on matter density was\nfound to be $0.41 \\rho_{\\rm Pl}$. In the Bianchi I model, available\nsimulations by Vandersloot (private communication) show that the\n`volume bounce' occurs when matter density is \\emph{lower} than\n$0.41 \\rho_{\\rm Pl}$ because there is also energy density in\ngravitational waves. It would be interesting to use numerical\nsimulations to find out what happens for generic solutions to the\nBianchi II effective equations.\n\nFinally, to obtain the effective equations for each variable, one\nsimply takes its Poisson bracket with the effective Hamiltonian\nconstraint.  This gives the effective equations\n\\begin{equation} \\dot{p_1} = \\gamma^{-1}\\left(\\frac{p_1^2}{\\bar\\mu_1}(\\sin\\bar\\mu_2c_2+\n\\sin\\bar\\mu_3c_3)+\\alpha p_2p_3\\right)\\cos\\bar\\mu_1c_1, \\end{equation}\n\\begin{equation} \\dot{p_2} = \\frac{p_2^2}{\\gamma\\bar\\mu_2}(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_3c_3)\n\\cos\\bar\\mu_2c_2, \\end{equation}\n\\begin{equation} \\dot{p_3} = \\frac{p_3^2}{\\gamma\\bar\\mu_3}(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_2c_2)\n\\cos\\bar\\mu_3c_3, \\end{equation}\n\\begin{align} \\dot{c_1} &= -\\frac{1}{\\gamma}\\Big[\\frac{p_2p_3}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\big(\n\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_3c_3+\\sin\\bar\\mu_2\nc_2\\sin\\bar\\mu_3c_3 \\nonumber \\\\ & \\qquad +\\frac{\\bar\\mu_1c_1}{2}\\cos\\bar\\mu_1c_1(\n\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_3c_3)-\\frac{\\bar\\mu_2c_2}{2}\\cos\\bar\\mu_2c_2(\\sin\\bar\n\\mu_1c_1+\\sin\\bar\\mu_3c_3) \\nonumber \\\\ & \\qquad -\\frac{\\bar\\mu_3c_3}{2}\\cos\\bar\n\\mu_3c_3(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_2c_2)\\big)+(1+\\gamma^2)\\alpha^2\\frac{(p_2\np_3)^2}{2p_1^3} \\nonumber \\\\ & \\qquad +\\frac{\\alpha}{2\\sqrt\\Delta\\ell_{\\mathrm{Pl}}}\\left(\n\\frac{p_2p_3}{p_1}\\right)^{3\/2}\\!\\!(\\bar\\mu_1c_1\\cos\\bar\\mu_1c_1-\\sin\\bar\\mu_1c_1)\n\\Big], \\end{align}\n\\begin{align} \\dot{c_2} &= -\\frac{1}{\\gamma}\\Big[\\frac{p_1p_3}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\big(\n\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_3c_3+\\sin\\bar\\mu_2\nc_2\\sin\\bar\\mu_3c_3 \\nonumber \\\\ & \\qquad -\\frac{\\bar\\mu_1c_1}{2}\\cos\\bar\\mu_1c_1(\n\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_3c_3)+\\frac{\\bar\\mu_2c_2}{2}\\cos\\bar\\mu_2c_2(\\sin\\bar\n\\mu_1c_1+\\sin\\bar\\mu_3c_3) \\nonumber \\\\ & \\qquad -\\frac{\\bar\\mu_3c_3}{2}\\cos\\bar\n\\mu_3c_3(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_2c_2)\\big)-(1+\\gamma^2)\\alpha^2\\frac{p_2\np_3^2}{2p_1^2} \\nonumber \\\\ & \\qquad +\\frac{\\alpha p_3}{2\\bar\\mu_1}(3\\sin\\bar\n\\mu_1c_1-\\bar\\mu_1c_1\\cos\\bar\\mu_1c_1) \\Big], \\end{align}\n\\begin{align} \\dot{c_3} &= -\\frac{1}{\\gamma}\\Big[\\frac{p_1p_2}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\big(\n\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_3c_3+\\sin\\bar\\mu_2\nc_2\\sin\\bar\\mu_3c_3 \\nonumber \\\\ & \\qquad -\\frac{\\bar\\mu_1c_1}{2}\\cos\\bar\\mu_1c_1(\n\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_3c_3)-\\frac{\\bar\\mu_2c_2}{2}\\cos\\bar\\mu_2c_2(\\sin\\bar\n\\mu_1c_1+\\sin\\bar\\mu_3c_3) \\nonumber \\\\ & \\qquad +\\frac{\\bar\\mu_3c_3}{2}\\cos\\bar\n\\mu_3c_3(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_2c_2)\\big)-(1+\\gamma^2)\\alpha^2\\frac{p_2^2\np_3}{2p_1^2} \\nonumber \\\\ & \\qquad +\\frac{\\alpha p_2}{2\\bar\\mu_1}(3\\sin\\bar\n\\mu_1c_1-\\bar\\mu_1c_1\\cos\\bar\\mu_1c_1) \\Big]. \\end{align}\nIn the ``embedding approach'' these effective equations provide the\nleading-order quantum corrections to the classical equations of\nmotion Eqs.~(\\ref{ceom1})~--~(\\ref{ceom2}). It would be very\ninteresting to numerically test if the accuracy they display in the\nisotropic case for states which are Gaussians at late times carries\nover to the Bianchi II case.\n\n\n\\section{Discussion}\n\\label{s5}\n\nIn this paper, we analyzed the ``improved'' LQC dynamics of the\nBianchi II model.  As in the isotropic and Bianchi I cases, we chose\nthe matter source to be a massless scalar field since it continues\nto serve as a viable relational time parameter in the classical as\nwell as the quantum theory. It is again rather straightforward to\naccommodate additional matter fields in this framework.\n\nOur broad strategy is the same as that used in the Bianchi I model\n\\cite{awe2}. However, because Bianchi II models have anisotropies\n\\emph{as well as} spatial curvature, holonomies around closed curves\nare no longer guaranteed to be almost periodic functions of the\nconnection. Hence, one cannot use them to construct the field\nstrength operator on the LQC Hilbert space; a new conceptual and\ntechnical input is necessary to define the quantum Hamiltonian\nconstraint operator. We overcame this difficulty by generalizing the\nstrategy used so far \\cite{abl,aps3,warsaw,apsv,kv1,ls,bp,awe2}.\nSpecifically, we used holonomies around open segments parallel to\nthe fiducial triads ${}^o\\!e^a_i$ to define a connection operator. This\nstrategy is also inspired by methods introduced by Thiemann in the\nfull theory \\cite{tt}. However, because of gauge fixing LQC does not\nenjoy the manifest diffeomorphism invariance of full LQG. As a\nconsequence, in LQC one needs a principle to fix the `length' of the\nopen segment along which holonomy is evaluated. We required that the\n`length' be so chosen that the field strength operator constructed\nfrom the resulting connection should agree with that constructed\nfrom holonomies around closed loops whenever the second construction\nis available. This guarantees that (apart from `tame' factor\nordering ambiguities) the new procedure reduces to the one used in\nthe LQC literature before. Moreover, the strategy of defining the\nHamiltonian constraint through this connection operator can be used\nalso in more general contexts. In particular, it enables one to\novercome a conceptual limitation of the otherwise complete treatment\nof the isotropic, k=$-1$ model given in \\cite{kv1,ls}. More\nimportantly, it extends to more general class A Bianchi models. A\nsystematic treatment of the Bianchi IX model along the lines of this\npaper would be especially interesting.\n\nThere is a second ---but primarily technical--- difference from the\nBianchi I case: The Hamiltonian operator now contains inverse powers\nof $p_1$. This was handled following a general method introduced by\nThiemann to define inverse triad operators in LQG \\cite{tt}. As\nusual, there is a factor ordering ambiguity. In the main discussion\nwe used the simplest operator which has the same symmetries with\nrespect to parity as its classical counterpart.\n\nAfter addressing these two issues, we obtained a well defined\nquantum Hamiltonian constraint and showed that the singularity in\nBianchi II models is resolved in the same precise sense as in the\nFRW and Bianchi I models. The Kinematical Hilbert space $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ can\nbe decomposed as $\\mathcal{H}_{\\rm kin}^{\\rm grav} = \\mathcal{H}_{\\rm sing}^{\\rm grav} \\oplus \\mathcal{H}^{\\rm\ngrav}_{\\rm reg}$ where states in the singular subspace have support\nonly on configurations with zero volume, while those in the regular\nsub-space have no support on these singular configurations.\n\\emph{The Hamiltonian constraint annihilates states in $\\mathcal{H}_{\\rm\nsing}^{\\rm grav}$ and maps $\\mathcal{H}^{\\rm grav}_{\\rm reg}$ to itself.} We\nalso provided an explicit form of the Hamiltonian constraint which\nshould be helpful in performing numerical simulations.\n\nFinally, we obtained effective equations using the ``embedding\nmethod'' introduced by Willis \\cite{jw} and further developed by\nTaveras \\cite{vt} in the isotropic case. There, although the\nassumptions made in the derivation fail in the deep Planck regime,\nthe final equations provide a surprisingly accurate approximation to\nthe full quantum evolution of states which are Gaussians at late\ntimes. This holds not only for the exactly soluble k=0, $\\Lambda=0$\nmodel but also for the much more complicated $\\Lambda\\not=0$ and\nk=$\\pm 1$ models. It would be interesting to see if this phenomenon\nextends also to Bianchi II models. Furthermore, numerical solutions\nof these effective equations themselves may be of considerable\ninterest because the simplest upper bound on matter density they\nlead to is higher than that in all other models studied so far,\nincluding Bianchi I. Numerical simulations of effective equations\nwill answer several questions within this approximation. Is the\nupper bound optimal, i.e., do generic solutions to effective\nequations come close to saturating it? In the Bianchi I case,\nnumerical simulations by Vandersloot (private communication)\nrevealed that, unlike in the isotropic model, there are several\ndistinct kinds of `bounces.' Roughly, anytime a shear ---or a Weyl\ncurvature--- scalar enters the Planck regime, quantum geometry\nrepulsion comes into pay in a dominant manner and `dilutes' that\nscalar, preventing a blow up. How do additional terms in the Bianchi\nII effective equations affect this scenario? Qualitative lessons\nfrom numerical simulations would be valuable in developing further\nintuition for various quantum geometry effects.\n\n\n\\section*{Acknowledgements:}\n\nWe would like to thank Gianluca Calcagni, Alex Corichi, Jerzy\nLewandowski, Simone Mercuri, Tomasz Pawlowski and Hanno Sahlmann for\nhelpful discussions. This research was supported in part by NSF\ngrant PHY0854743, the George A. and Margaret M. Downsbrough\nEndowment, the Eberly research funds of Penn State, Le Fonds\nqu\\'eb\\'ecois de la recherche sur la nature et les technologies and\nthe Edward A. and Rosemary A. Mebus Fellowship.\n\n\n\\begin{appendix}\n\n\n\\section{Parity Symmetries}\n\\label{a1}\n\nIn non-gravitational physics, parity transformations are normally\ntaken to be discrete diffeomorphisms $x_i \\rightarrow -x_i$ in the\nphysical space which are isometries of the flat 3-metric thereon. In\nthe phase space formulation of general relativity,  we do not have a\nflat metric ---or indeed, any fixed metric. Therefore these discrete\nsymmetries are no longer meaningful (except in the weak field\nlimit). However, if the dynamical variables have internal indices\n---such as the triads and connections used in LQG--- we can use the\nfact that the internal space $I$ is a vector space equipped with a\nflat metric $q_{ij}$ to define parity operations on the internal\nindices. Associated with any unit internal vector $v^i$, there is a\nparity operator $\\Pi_v$ which reflects the internal vectors across\nthe 2-plane orthogonal to $v^i$. This operation induces a natural\naction on triads $e^a_i$, the connections $A_a^i$ and the conjugate\nmomenta $P^a_i =: (1\/8\\pi G\\gamma)\\, E^a_i$ (since they are internal\nvectors or co-vectors).\n\nThe triads $e^a_i$ are proper internal co-vectors. In previous\nreferences \\cite{bd,awe2}, conventions were such that the spin\nconnection $\\Gamma^a_i$ turned out to be an internal \\emph{pseudo}\nvector. It was then natural to regard the Barbero-Immirzi parameter\n$\\gamma$ to be a pseudo quantity so that the connection $A_a^i$ has\ndefinite parity namely, it transforms as an internal pseudo-vector.\nThis in turn led to the conclusion that $P^a_i$ is also an internal\npseudo-vector (as one would expect because it is canonically\nconjugate to $A_a^i$) \\cite{awe2}. While this is all\nself-consistent, these conventions lead to two undesirable\nconsequences. First, in the classical theory, it is not possible to\nreconstruct the triads $e^a_i$ unambiguously starting from the\nmomenta $P^a_i$. Therefore, one cannot recover the space-time\ngeometry unambiguously starting from the Hamiltonian theory. Second,\nthe momenta $P^a_i$ are subject to a non-holonomic constraint which\nobstructs the passage to quantum theory a la LQG. However, if one\nsets conventions as in section \\ref{s2.1}, then $\\Gamma_a^i, \\gamma,\nA_a^i$ and $P_a^i$ are all \\emph{proper} quantities and the two\ndifficulties disappear \\cite{aa-dis}. In the main text we have used\nthis strategy. We now summarize the differences from the Appendix of\n\\cite{awe2} that it leads to.\n\nIn diagonal Bianchi models, we can restrict ourselves just to three\nparity operations $\\Pi_i$. Under their action, the canonical\nvariables $c_i,p_i$ transform as follows:\n\\begin{equation} \\label{P1} \\Pi_1 (c_1,c_2, c_3) = (-c_1, c_2, c_3), \\quad \\quad\n\\Pi_1 (p_1, p_2, p_3) = (-p_1, p_2, p_3)\\, , \\end{equation}\nand the action of $\\Pi_2, \\Pi_3$ is given by cyclic permutations.\nThus, $c^i$ and $p_i$ are \\emph{proper} internal vectors and\nco-vectors. Under any of these maps $\\Pi_i$, the symplectic\nstructure (\\ref{pb2}), the Hamiltonian (\\ref{Hcl}), and hence also\nthe Hamiltonian vector field, are left invariant. This is just as\none would expect because $\\Pi_i$ are simply large gauge\ntransformations of the theory under which the physical metric\n$q_{ab}$ and the extrinsic curvature $K_{ab}$ do not change. Also,\nit is clear from the action of (\\ref{P1}) that if one knows the\ndynamical trajectories on the octant $p_i\\ge 0$ of the phase space,\nthen dynamical trajectories on any other octant can be obtained just\nby applying a suitable (combination of) $\\Pi_i$. Therefore, in the\nclassical theory one can restrict one's attention just to the\npositive octant.\n\nLet us now turn to the quantum theory. We now have three operators\n$\\hat\\Pi_i$. Their action on states is given by\n\\begin{equation} \\hat\\Pi_1 \\Psi(\\lambda_1,\\lambda_2,\\lambda_3) = \\Psi(-\\lambda_1,\\lambda_2\\lambda_3)\\, , \\end{equation}\netc. What is the induced action on operators? Since\n\\begin{align} \\label{P2}\n\\hat{\\Pi}_1\\hat{\\lambda}_1\\hat{\\Pi}_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\n&= \\hat{\\Pi}_1 \\Big({\\lambda}_1\\,\\Psi(-\\lambda_1,\\lambda_2,\\lambda_3)\\Big) \\nonumber \\\\\n&= -\\lambda_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3), \\end{align}\nwe have\n\\begin{equation} \\label{P3} \\hat{\\Pi}_1\\hat{\\lambda}_1\\hat{\\Pi}_1  = -\\hat{\\lambda}_1. \\end{equation}\nThe Hamiltonian constraint operator, modulo factor ordering which is\nnot important here, is given by Eq. (\\ref{qHam1}). To calculate its\ntransformation property under parity maps, in addition to\n(\\ref{P3}), we also need the transformation property of the\noperators $\\sin \\bar\\mu_ic_i$ and $\\hat{\\varepsilon}$ and operators\ncorresponding to inverse powers of $p_1$. Due to the symmetries of\ntype A Bianchi models, to know the properties of $\\sin \\bar\\mu_ic_i$\nunder parity transformations, it is sufficient to calculate\n$\\hat{\\Pi}_i \\sin \\bar\\mu_1c_1\\hat{\\Pi}_i$.  We have:\n\\begin{align} \\hat{\\Pi}_1\\sin\\bar\\mu_1c_1\\hat{\\Pi}_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\n&= \\frac{1}{2i}\\, \\hat\\Pi_1\\,\\Big[\\Psi(-\\lambda_1+\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)-\n\\Psi(-\\lambda_1-\\frac{1} {|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)\\Big] \\nonumber \\\\\n&= \\frac{1}{2i}\\Big[\\Psi(\\lambda_1+\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)-\\Psi(\\lambda_1-\n\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)\\Big] \\nonumber \\\\ &=\n-\\sin\\bar\\mu_1c_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3), \\end{align}\nwhence\n\\begin{equation} \\hat{\\Pi}_1 \\sin\\bar\\mu_1c_1\\hat\\Pi_1 = -\\sin\\bar\\mu_1c_1. \\end{equation}\nAn identical calculation shows that\n\\begin{align} \\hat{\\Pi}_2\\sin\\bar\\mu_1c_1\\hat{\\Pi}_2\\,\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\n&= \\frac{1}{2i}\\, \\hat\\Pi_2\\, \\Big[ \\Psi(\\lambda_1-\\frac{1}{|\\lambda_2\\lambda_3|},\n-\\lambda_2,\\lambda_3)-\\Psi(\\lambda_1+\\frac{1}{|\\lambda_2\\lambda_3|},-\\lambda_2,\\lambda_3)\\Big] \\nonumber\\\\\n&= \\frac{1}{2i}\\Big[\\Psi(\\lambda_1-\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)-\n\\Psi(\\lambda_1+\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)\\Big] \\nonumber \\\\\n&= \\sin\\bar\\mu_1c_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\\, , \\end{align}\nand similarly for $\\hat\\Pi_3$. Therefore, we have:\n\\begin{equation} \\hat{\\Pi}_2 \\sin\\bar\\mu_1c_1\\hat\\Pi_2 = \\sin\\bar\\mu_1c_1, \\quad\n{\\rm and} \\quad \\hat{\\Pi}_3 \\sin\\bar\\mu_1c_1\\hat{\\Pi}_3 =\n\\sin\\bar\\mu_1c_1.\\end{equation}\nAs expected, these transformation properties of $\\sin\\bar\\mu_1c_1$\nunder $\\hat\\Pi_i$ mirror those of $c_1$ under the three parity\noperations $\\Pi_i$ in the classical theory. (Note that, because of\nthe absolute value signs in the expressions (\\ref{mubar}),\n$\\bar\\mu_i$ do not change under any of the parity maps.)  Finally,\nit is clear from Eq. (\\ref{ve2}) that\n\\begin{equation}  \\hat\\Pi_i\\,\\hat{\\varepsilon}\\,\\hat\\Pi_i = \\left\\{ \\rlap{\\raise2ex\\hbox{\n$\\hat{\\varepsilon}$ if $v=0$,}}{\\lower2ex\\hbox{$-\\hat{\\varepsilon}$ otherwise,}}\n\\right. \\end{equation}\nand from Eq. (\\ref{inv}) that\n\\begin{equation} \\hat\\Pi_i\\,\\widehat{|p_1|^{-1\/4}}\\,\\hat\\Pi_i =\n\\widehat{|p_1|^{-1\/4}}. \\end{equation}\n(Note incidentally that this need not be the case for different\nfactor ordering choices in Eq. (\\ref{inv}).)\n\nWe can now collect these results to study the transformation\nproperty of the Hamiltonian constraint. Consider first the regular\nsubspace $\\mathcal{H}_{\\rm reg}^{\\rm grav}$ of $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ spanned by states which\nhave no support on points with $v=0$. From Eq. (\\ref{qHam1}) it\nfollows that the restriction to $\\mathcal{H}_{\\rm reg}^{\\rm grav}$ of the\ngravitational part of the Hamiltonian constraint is left invariant\nunder $\\hat\\Pi_i$. Since $\\hat{p}_T^2$ is manifestly invariant, on\nthe regular sub-space we have\n\\begin{equation} \\label{Pham} \\hat{\\Pi}_i\\,\\, \\hat{\\mathcal{C}}_H \\,\\,\\hat{\\Pi}_i\n= \\hat{\\mathcal{C}}_H \\end{equation}\nNext, since the gravitational part of the Hamiltonian constraint\nannihilates the states in the singular sub-space (i.e. those with\nsupport only on those points at which $v=0$), we have\n\\begin{equation} \\hat{\\mathcal{C}}_H\\Psi=-\\hbar^2\\partial_T^2\\Psi=\\hat{\\Pi}_i\\,\\,\n\\hat{\\mathcal{C}}_H \\,\\, \\hat{\\Pi}_i\\Psi. \\end{equation}\nThus, the Hamiltonian constraint operator is left invariant by all\nthe parity operators, mirroring the behavior of its classical\ncounterpart.\n\nThis invariance implies that, given any state $\\Psi \\in\n\\mathcal{H}_{\\rm kin}^{\\rm grav}$, the restriction to the positive\noctant of its image under $\\hat{\\mathcal{C}}_{\\rm grav}$ determines\nits image everywhere on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$. This\nproperty simplifies calculations and was used to arrive at the form\nof the Hamiltonian constraint given in (\\ref{qHamfin}).\n\n\n\\end{appendix}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Statement of Results}\n\nA level $p$ modular function $f(\\tau)$ is a holomorphic function\non the complex upper half-plane which satisfies \\[\nf\\left(\\frac{a\\tau+b}{c\\tau+d}\\right)=f(\\tau)\\text{ for all }\\left(\\begin{smallmatrix}a & b\\\\\nc & d\\end{smallmatrix}\\right)\\in\\Gamma_{0}(p)\\]\n and is meromorphic at the cusps of $\\Gamma_{0}(p)$. Equivalently,\n$f(\\tau)$ is a weakly holomorphic modular form of weight $0$ on\n$\\Gamma_{0}(p)$. Such a function will necessarily have a $q$-expansion\nof the form $f(\\tau)=\\sum_{n=n_{0}}^{\\infty}a(n)q^{n}$, where $q=e^{2\\pi i\\tau}$.\n\nOf particular interest in the study of modular forms is the\nclassical $j$-invariant, $j(\\tau)=q^{-1}+744 + \\sum_{n=1}^{\\infty}\nc(n)q^{n}$, which is a modular function of level $1$. The\ncoefficients $c(n)$ of the $j$-function, like the Fourier\ncoefficients of many other modular forms, are of independent\narithmetic interest; for instance, they appear as dimensions of a\nspecial graded representation of the Monster group.\n\nIn 1949 Lehner showed \\cite{Lehner:1,Lehner:2} that \\[\nc(2^{a}3^{b}5^{c}7^{d}n)\\equiv0\\pmod{2^{3a+8}3^{2b+3}5^{c+1}7^{d}},\\]\n proving that the coefficients $c(n)$ are often highly divisible\nby small primes. Similar results have recently been proven for other\nmodular functions in~\\cite{Griffin}, and for modular forms of level\n1 and small weight in~\\cite{Doud:padic}, \\cite{DoudJenkinsLopez}. It\nis natural to ask whether such congruences hold for the Fourier\ncoefficients of modular functions of higher level, such as those\nstudied by Ahlgren~\\cite{Ahlgren:theta} in his work on Ramanujan's\n$\\theta$-operator.\n\nLehner's results for $j(\\tau)$ are in fact more general; in~\\cite{Lehner:2}\nhe pointed out that for $p=2,3,5,7$, similar congruences hold for\nthe coefficients of level $p$ modular functions which have integral\ncoefficients at both cusps and have poles of order less than $p$\nat the cusp at infinity.\n\nIn this paper, for $p\\in\\{2,3,5,7\\}$, we examine canonical bases for\nspaces of level $p$ modular functions which are holomorphic at the\ncusp $0$. To construct these bases, we introduce the level $p$\nmodular function $\\psi^{(p)}(\\tau)$, defined as \\[\n\\psi^{(p)}(\\tau)=\\left(\\frac{\\eta(\\tau)}{\\eta(p\\tau)}\\right)^{\\frac{24}{p-1}}\n\\text{ where }\\eta(\\tau) =\nq^{\\frac{1}{24}}\\prod_{n=1}^{\\infty}(1-q^{n}).\\] The integer\n$\\frac{24}{p-1}$ for $p=2,3,5,7$ will appear frequently, so we will\ndenote it $\\lambda^{(p)}$, or simply $\\lambda$ where no confusion\narises. The function $\\psi^{(p)}(\\tau)$ is a modular function of\nlevel $p$ with a simple pole at $\\infty$ and a simple zero at 0.  We\nwill also use the modular function \\[\\phi^{(p)}(\\tau) =\n(\\psi^{(p)}(\\tau))^{-1}.\\]\n\nFollowing Ahlgren \\cite{Ahlgren:theta}, and using the notation of\nDuke and Jenkins \\cite{Duke:zeros}, for $p=2,3,5,7$ we construct a\nbasis $\\{f_{0,m}^{(p)}(\\tau)\\}_{m=0}^{\\infty}$ for the space of\nlevel $p$ modular functions which are holomorphic at 0 as follows:\n\\[ f_{0,0}^{(p)}(\\tau)=1,\\]\n\\[f_{0,m}^{(p)}(\\tau)=q^{-m}+O(1) =\n\\psi^{(p)}(\\tau)^{m}-Q(\\psi^{(p)}(\\tau)),\\] where $Q(x)$ is a\npolynomial of degree $m-1$ with no constant term, chosen to\neliminate all negative powers of $q$ in $\\psi^{(p)}(\\tau)^{m}$\nexcept for $q^{-m}$. Since $\\psi^{(p)}(\\tau)$ vanishes at $0$ and\nthe polynomial $Q$ has no constant term, we see that the functions\n$f_{0,m}^{(p)}$ also vanish at $0$ when $m>0$. We write\\[\nf_{0,m}^{(p)}=q^{-m}+\\sum_{n=0}^{\\infty}a_{0}^{(p)}(m,n)q^{n}\\]\n so that for $n\\geq0$, the symbol $a_{0}^{(p)}(m,n)$ denotes the coefficient\nof $q^{n}$ in the $m^{th}$ basis element of level $p$.  Note that\nthe function $f_{0, m}^{(p)}$ corresponds to Ahlgren's $j_m^{(p)}$.\n\nFor an example of some of these functions, consider the case $p=2$:\n\\begin{align*}\nf_{0,1}^{(2)}(\\tau) & =\\psi^{(2)}(\\tau)\\\\\n & =q^{-1}-24+276q-2048q^{2}+11202q^{3}-49152q^{4}+\\ldots\\\\\nf_{0,2}^{(2)}(\\tau) & =\\psi^{(2)}(\\tau)^{2}+48\\psi^{(2)}(\\tau)\\\\\n & =q^{-2}-24-4096q+98580q^{2}-1228800q^{3}+10745856q^{4}+\\ldots\\\\\nf_{0,3}^{(2)}(\\tau) & =\\psi^{(2)}(\\tau)^{3}+72\\psi^{(2)}(\\tau)^{2}+900\\psi^{(2)}(\\tau)\\\\\n & =q^{-3}-96+33606q-1843200q^{2}+43434816q^{3}-648216576q^{4}+\\ldots\\end{align*}\nThe function $f_{0,m}^{(p)}$ is a level $p$ modular function that\nvanishes at $0$ (if $m\\neq0$) and has a pole of order $m$ at\n$\\infty$. The conditions at the cusps determine this function\nuniquely; if two such functions exist, their difference is a\nholomorphic modular function, which must be a constant. Since both\nfunctions vanish at 0, this constant must be $0$.\n\nThe functions comprising these bases for $p=2,3,5,7$ have\ndivisibility properties which bear a striking resemblance to the\ndivisibility properties of $j(\\tau)$; in many cases they are\nidentical. As an example of some of the divisibility properties we\nencounter with this basis, we experimentally examine the $2$-adic\nvaluation of the even indexed coefficients of $f_{0,m}^{(2)}(\\tau)$\nfor $m=1,3,5,7$ in Table \\ref{tab:2-Adic-Table}.  As the data in the\ntable suggest, the $2$-divisibility which $j(\\tau)$ exhibits gives\nus a lower bound on the $2$-divisibility of the odd-indexed $p=2$\nbasis elements.\n\n\n\\begin{table}[h]\n\\noindent \\begin{centering}\n\\label{Flo:2-Adic-Float}\n\\par\\end{centering}\n\\noindent \\begin{centering}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{1}{|c}{} &  & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,2)$} & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,4)$} & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,6)$} & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,8)$} & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,10)$} & $a_{0}^{(2)}(m,12)$\\tabularnewline\n\\hline\n & $1$ & $11$ & $14$ & $13$ & $17$ & $12$ & $16$\\tabularnewline\n\\cline{2-8}\n & $3$ & $13$ & $16$ & $15$ & $19$ & $14$ & $18$\\tabularnewline\n\\cline{2-8}\n$m$ & $5$ & $12$ & $15$ & $14$ & $18$ & $13$ & $17$\\tabularnewline\n\\cline{2-8}\n & $7$ & $14$ & $17$ & $16$ & $20$ & $15$ & $19$\\tabularnewline\n\\cline{2-8}\n & min & $11$ & $14$ & $13$ & $17$ & $12$ & $16$\\tabularnewline\n\\hline\n\\multicolumn{8}{|c|}{}\\tabularnewline\n\\hline\n$j(\\tau)$ &  & $11$ & $14$ & $13$ & $17$ & $12$ & $16$\\tabularnewline\n\\hline\n\\end{tabular}\n\\par\\end{centering}\n\\noindent \\centering{}\\caption{\\label{tab:2-Adic-Table}$2$-adic\nvaluation of $a_{0}^{(2)}(m,n)$ compared to corresponding\ncoefficients in $j(\\tau)$}\n\\end{table}\n\nNote that these functions form a basis for $M_0^\\infty(p)$, the\nspace of modular forms of weight 0 and level $p$ with poles allowed\nonly at the cusp at $\\infty$.  A full basis for the space $M_0^!(p)$\nof weakly holomorphic modular forms of weight $0$ and level $p$ is\ngenerated by the $f_{0, m}^{(p)}(\\tau)$ and the functions\n$(\\phi^{(p)}(\\tau))^n$ for integers $n\\geq 1$.\n\nRecall that the concluding remarks of Lehner's second paper\n\\cite{Lehner:2} state that the coefficients of certain level $p$\nmodular functions having a pole of order less than $p$ at $\\infty$\nhave the same $p$-divisibility properties as the coefficients $c(n)$\nof $j(\\tau)$. More precisely, we have the following theorem.\n\\begin{thm}[Lehner]\n\\label{thm:Lehner-Main} Let $p\\in\\{2,3,5,7\\}$ and let $f(\\tau)$ be a\nmodular function on $\\Gamma_{0}(p)$ having a pole at $\\infty$ of\norder $<p$ and $q$-series of the form\n\\[f(\\tau)=\\sum_{n=n_{0}}^{\\infty}a(n)q^{n},\\]\n\\[ f(-1\/p\\tau) = \\sum_{m=m_0}^\\infty b(n) q^n,\\]\nwhere $a(n), b(n) \\in\\mathbb{Z}$. Then the coefficients $a(n)$\nsatisfy the following congruence properties:\n\\[\n\\begin{array}{rll}\na(2^{a}n)\\equiv0 & \\pmod{2^{3a+8}} & \\text{if }p=2\\\\\na(3^{a}n)\\equiv0 & \\pmod{3^{2a+3}} & \\text{if }p=3\\\\\na(5^{a}n)\\equiv0 & \\pmod{5^{a+1}} & \\text{if }p=5\\\\\na(7^{a}n)\\equiv0 & \\pmod{7^{a}} & \\text{if }p=7.\\end{array}\\]\n\\end{thm}\nNote that Lehner's original statement of this theorem mistakenly\nstates that a function on $\\Gamma_{0}(p)$ inherits the\n$p$-divisibility property for \\emph{every }prime in $\\{2,3,5,7\\}$,\nnot just the prime matching the level.\n\nA necessary condition in the statement of Lehner's theorem is that\nthe function must have an integral $q$-expansion at $0$. This\ncondition is quite strong; in fact, neither the function\n$\\phi^{(p)}(\\tau)$ nor any of its powers satisfy it, although the\nfunctions $f_{0, m}^{(p)}(\\tau)$ do.\n\nFurther, Lehner's theorem assumes that the order of the pole at\n$\\infty$ must be less than $p$. In this paper, we remove this\nrestriction on the order of the pole to show that every function in\nthe $f_{0,m}^{(p)}$ basis has divisibility properties similar to\nthose in Theorem \\ref{thm:Lehner-Main}.  Specifically, we prove the\nfollowing theorem.\n\\begin{thm}\n\\label{thm:Andersen-Main}Let $p\\in\\{2,3,5,7\\}$, and let \\[\nf_{0,m}^{(p)}(\\tau)=q^{-m}+\\sum\\limits _{n=0}^{\\infty}a_{0}^{(p)}(m,n)q^{n}\\]\n be an element of the basis described above, with $m=p^{\\alpha}m'$\nand $(m',p)=1$. Then, for $\\beta > \\alpha$,\n\\[\n\\begin{array}{rll}\na_{0}^{(2)}(2^{\\alpha}m',2^{\\beta}n)\\equiv0 & \\pmod{2^{3(\\beta-\\alpha)+8}} & \\text{if }p=2\\\\\na_{0}^{(3)}(3^{\\alpha}m',3^{\\beta}n)\\equiv0 & \\pmod{3^{2(\\beta-\\alpha)+3}} & \\text{if }p=3\\\\\na_{0}^{(5)}(5^{\\alpha}m',5^{\\beta}n)\\equiv0 & \\pmod{5^{(\\beta-\\alpha)+1}} & \\text{if }p=5\\\\\na_{0}^{(7)}(7^{\\alpha}m',7^{\\beta}n)\\equiv0 & \\pmod{7^{(\\beta-\\alpha)}} & \\text{if }p=7.\\end{array}\\]\n\\end{thm}\n\nNote that for basis elements $f_{0,m}^{(p)}$ with $(m,p)=1$, these\ndivisibility properties match those in Theorem\n\\ref{thm:Lehner-Main}; in fact, Lehner's proof is easily extended to\nprove the congruences in these cases. For basis elements with\n$m=p^{\\alpha}m'$ and $\\alpha\\ge1$, the divisibility is ``shifted.''\nThis shifting occurs in the $(\\beta-\\alpha)$ factor in the exponent\nof the modulus.\n\nFor the coefficients $a_0^{(p)}(p^\\alpha m', p^\\beta n)$ with\n$\\alpha > \\beta$, computations suggest that similar congruences\nshould hold.  Additionally, it appears that powers of the function\n$\\phi^{(p)}(\\tau)$ have Fourier coefficients with slightly weaker\ndivisibility properties, despite the fact that their Fourier\ncoefficients at $0$ are not integral. It would be interesting to\nmore fully understand these congruences.\n\n\\section{Preliminary Lemmas and Definitions}\n\nIn this section, we provide the necessary definitions and background\nfor the proof of the main theorem.\n\nFor a prime $p$ we define the level $p$ Hecke operator $U_{p}$ by\\[\nU_{p}f(\\tau)=\\frac{1}{p}\\sum_{\\ell=0}^{p-1}f\\left(\\frac{\\tau+\\ell}{p}\\right),\\]\nusing the notation $U_{p}^{n}f=U_{p}U_{p}\\cdots U_{p}f$ for repeated\napplication of $U_{p}$. When $f$ has the Fourier expansion\n$f(\\tau)=\\sum_{n=n_{0}}^{\\infty}a(n)q^{n}$, this operator takes the\nform\\[ U_{p}f(\\tau)=\\sum_{n=n_{0}}^{\\infty}a(pn)q^{n},\\] essentially\n{}``pulling out'' all of the coefficients of $f$ whose index is\ndivisible by $p$. This operator preserves modularity:\nif $f$ is a level $p$ modular function, then $U_{p}f$ is also a level\n$p$ modular function.\n\nFor the primes $p=2,3,5,7$ the topological genus of\n$\\Gamma_{0}(p)\\backslash\\mathcal{H}$ is zero, so the field of level\n$p$ modular functions is generated by a single modular function\ncalled a Hauptmodul.  For the primes in consideration, one such\nfunction is $\\psi^{(p)}(\\tau)$. Note that the modular function\n$\\phi^{(p)}(\\tau)=\\psi^{(p)}(\\tau)^{-1}=q+O(q^{2})$ is also a\nHauptmodul.\n\nFurther, for these primes, the fundamental domain for\n$\\Gamma_{0}(p)$ has precisely two cusps, which may be taken to be at\n$\\infty$ and at $0$. Hence, we are most concerned with the behavior\nof these functions at $\\infty$ and at $0$. To switch between cusps,\nwe make the substitution $\\tau\\mapsto-1\/p\\tau$. The following lemma\ngives relations for $\\psi^{(p)}(\\tau)$ and $\\phi^{(p)}(\\tau)$ at\n$0$, and makes clear that powers of $\\phi^{(p)}$ do not satisfy\nLehner's integrality condition.\n\\begin{lem}\n\\label{lem:Psi-At-0}The functions $\\psi^{(p)}(\\tau)$ and $\\phi^{(p)}(\\tau)$\nsatisfy the relations \\begin{align}\n\\psi^{(p)}(-1\/p\\tau) & =p^{\\lambda\/2}\\phi^{(p)}(\\tau)\\label{eq:Psi-At-0}\\\\\n\\phi^{(p)}(-1\/p\\tau) & =p^{-\\lambda\/2}\\psi^{(p)}(\\tau)\\label{eq:Phi-At-0}\\end{align}\n\\end{lem}\n\\begin{proof}\nThe functional equation for $\\eta(\\tau)$ is\n$\\eta(-1\/\\tau)=\\sqrt{-i\\tau}\\eta(\\tau)$. Using this, we find that \\[\n\\psi^{(p)}\\left(\\frac{-1}{p\\tau}\\right) =\n\\left(\\frac{\\eta(-1\/p\\tau)} {\\eta(-1\/\\tau)}\\right)^{\\lambda} =\n\\left(\\frac{\\sqrt{-ip\\tau}\n \\eta(p\\tau)}{\\sqrt{-i\\tau} \\eta(\\tau)}\\right)^{\\lambda} = (\\sqrt{p})^{\\lambda}\n\\left(\\frac{\\eta(p\\tau)}{\\eta(\\tau)}\\right)^{\\lambda} =\np^{\\lambda\/2}\\phi^{(p)}(\\tau).\\] The second statement follows after\nreplacing $\\tau$ by $-1\/p\\tau$ in the first statement.\n\\end{proof}\nWe next state a well-known lemma which gives a formula for\ndetermining the behavior of a modular function at $0$ after $U_{p}$\nhas been applied. A proof can be found in \\cite{Apostol:modular}.\n\\begin{lem}\n\\label{lem:Main-Up-Formula}Let $p$ be prime and let $f(\\tau)$ be\na level $p$ modular function. Then\\begin{equation}\np(U_{p}f)(-1\/p\\tau)=p(U_{p}f)(p\\tau)+f(-1\/p^{2}\\tau)-f(\\tau).\\label{eq:Main-Up-Formula}\\end{equation}\n\n\\end{lem}\nLehner's original papers included the following lemma and its proof, which gives\ntwo important equations. The first gives a formula for $U_{p}\\phi^{(p)}$\nas a polynomial with integral coefficients in $\\phi^{(p)}$; the second\ngives an algebraic relation which is satisfied by $\\phi^{(p)}(\\tau\/p)$.\n\\begin{lem}\n\\label{lem:Modular-Eq-Phi}Let $p\\in\\{2,3,5,7\\}$. Then there exist\nintegers $b_{j}^{(p)}$ such that\n\n\\[\n\\begin{array}{cc}\n\\text{(a)} & U_{p}\\phi^{(p)}(\\tau)=p\\sum\\limits _{j=1}^{p}b_{j}^{(p)}\\phi^{(p)}(\\tau)^{j}.\\end{array}\\]\n\n\nFurther, let $h^{(p)}(\\tau)=p^{\\lambda\/2}\\phi^{(p)}(\\tau\/p).$ Then\\[\n\\begin{array}{cc}\n\\text{(b)} & \\big(h^{(p)}(\\tau)\\big)^{p}+\\sum\\limits _{j=1}^{p}(-1)^{j}g_{j}(\\tau)\\big(h^{(p)}(\\tau)\\big)^{p-j}=0\\end{array}\\]\n\n\nwhere $g_{j}(\\tau)=(-1)^{j+1}p^{\\lambda\/2+2}\\sum\\limits _{\\ell=j}^{p}b_{\\ell}\\phi^{(p)}(\\tau)^{\\ell-j+1}$.\\end{lem}\n\\begin{proof}\n\n(a) Since $\\phi$ vanishes at $\\infty$, $U_{p}\\phi$ also vanishes\nat $\\infty$; we will now consider its behavior at $0$. Using (\\ref{eq:Main-Up-Formula})\nand replacing $\\tau$ by $p\\tau$ in (\\ref{eq:Psi-At-0}) we obtain\\begin{align*}\nU_{p}\\phi(-1\/p\\tau) & =U_{p}\\phi(p\\tau)+p^{-1}\\phi(-1\/p^{2}\\tau)-p^{-1}\\phi(\\tau)\\\\\n & =U_{p}\\phi(p\\tau)+p^{-1}\\psi(p\\tau)-p^{-1}\\phi(\\tau)\\\\\n & =O(q^{p})+p^{-\\lambda\/2-1}q^{-p}+O(1)-p^{-1}q+O(q^{2})\\\\\np^{\\lambda\/2+1}U_{p}\\phi(-1\/p\\tau) & =q^{-p}+O(1)\\end{align*}\n\n\nThe right side of this equation is a level $p$ modular function with\ninteger coefficients, so we may write it as a polynomial in\n$\\psi(\\tau)$ with integer coefficients. The polynomial will not have\na constant term since the left side vanishes at $0$. Therefore,\\[\np^{\\lambda\/2+1}U_{p}\\phi(-1\/p\\tau)=\\sum_{j=1}^{p}c_{j}\\psi(\\tau){}^{j}.\\]\n\n\nNow, replacing $\\tau$ by $-1\/p\\tau$, we find\\[\np^{\\lambda\/2+1}U_{p}\\phi(\\tau)=\\sum_{j=1}^{p}c_{j}p^{\\lambda j\/2}\\phi(\\tau){}^{j}.\\]\n\n\nAfter cancelling the $p^{\\lambda\/2+1}$, we find that\n$U_{p}\\phi(\\tau)=\\sum\\limits _{j=1}^{p}c_{j}'\\phi(\\tau)^{j}$ and we\ncompute the coefficients $c_{j}'$ (the authors used\n\\noun{mathematica}). The computation is finite, and we find that\neach coefficient $c_{j}'$ has a factor of $p$, so the coefficients\n$b_{j}^{(p)}$ are integral. A complete table of values of the\n$b_{j}^{(p)}$ is found in Table \\ref{tab:b_j-Table}.\n\n\\begin{table}[h]\n\\noindent \\begin{centering}\n\\label{Flo:b_j-Float}\n\\par\\end{centering}\n\n\\noindent \\begin{centering}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\cline{3-6}\n\\multicolumn{1}{c}{} &  & \\multicolumn{4}{c|}{$p$}\\tabularnewline\n\\cline{3-6}\n\\multicolumn{1}{c}{} &  & 2 & 3 & 5 & 7\\tabularnewline\n\\hline\n & 1 & $3\\cdot2^{2}$ & $10\\cdot3^{1}$ & $63\\cdot5^{0}$ & $82\\cdot7^{0}$\\tabularnewline\n\\cline{2-6}\n & 2 & $2^{10}$ & $4\\cdot3^{6}$ & $52\\cdot5^{3}$ & $176\\cdot7^{2}$\\tabularnewline\n\\cline{2-6}\n & 3 &  & $3^{10}$ & $63\\cdot5^{5}$ & $845\\cdot7^{3}$\\tabularnewline\n\\cline{2-6}\n$j$ & 4 &  &  & $6\\cdot5^{8}$ & $272\\cdot7^{5}$\\tabularnewline\n\\cline{2-6}\n & 5 &  &  & $5^{10}$ & $46\\cdot7^{7}$\\tabularnewline\n\\cline{2-6}\n & 6 &  &  &  & $4\\cdot7^{9}$\\tabularnewline\n\\cline{2-6}\n & 7 &  &  &  & $7^{10}$\\tabularnewline\n\\hline\n\\end{tabular}\n\\par\\end{centering}\n\n\\noindent \\centering{}\\caption{\\label{tab:b_j-Table}Values of $b_{j}^{(p)}$}\n\n\\end{table}\n\n\n(b) We again apply (\\ref{eq:Main-Up-Formula}) to $\\phi(\\tau)$, this\ntime using what we know from $(a)$.\\[\npU_{p}\\phi(-1\/p\\tau)=pU_{p}\\phi(p\\tau)+\\phi(-1\/p^{2}\\tau)-\\phi(\\tau)\\]\n\\[\np^{2}\\sum_{j=1}^{p}b_{j}\\phi(-1\/p\\tau)^{j}=p^{2}\\sum_{j=1}^{p}b_{j}\\phi(p\\tau)^{j}+\\phi(-1\/p^{2}\\tau)-\\phi(\\tau).\\]\nWe now use Lemma \\ref{lem:Psi-At-0} with the knowledge that $\\psi(\\tau)=\\phi(\\tau)^{-1}$\nto obtain\n\\[p^{2}\\sum_{j=1}^{p}b_{j}p^{-\\lambda j\/2}\\phi(\\tau)^{-j}-p^{2}\\sum_{j=1}^{p}b_{j}\\phi(p\\tau)^{j}+\\phi(\\tau)-p^{-\\lambda\/2}\\phi(p\\tau)^{-1}=0.\\]\nAfter replacing $\\tau$ by $\\tau\/p$ and multiplying by $p^{\\lambda\/2}$,\nthis becomes\\begin{equation}\np^{\\lambda\/2+2}\\sum_{j=1}^{p}b_{j}\\big(h(\\tau)^{-j}-\\phi(\\tau)^{j}\\big)+h(\\tau)-\\phi(\\tau)^{-1}=0.\\label{eq:Intermediate-Modular-Eq-Phi}\\end{equation}\n\n\nWe now divide by $h^{-1}-\\phi$. Note two facts: \\[\nh^{-j}-\\phi^{j}=(h^{-1}-\\phi)\\sum_{\\ell=0}^{j-1}h^{-\\ell}\\phi^{j-\\ell-1}\\]\n\\[\n\\frac{h-\\phi^{-1}}{h^{-1}-\\phi}=\\frac{h(h\\phi-1)}{\\phi(1-h\\phi)}=-\\frac{h}{\\phi}.\\]\nSo (\\ref{eq:Intermediate-Modular-Eq-Phi}) becomes\\[\np^{\\lambda\/2+2}\\sum_{j=1}^{p}b_{j}\\sum_{\\ell=0}^{j-1}h^{-\\ell}\\phi^{j-\\ell-1}-\\phi^{-1}h=0\\]\nwhich, after multiplying by $\\phi h^{p-1}$, becomes\\[\np^{\\lambda\/2+2}\\sum_{j=1}^{p}b_{j}\\sum_{\\ell=0}^{j-1}h^{p-\\ell-1}\\phi^{j-\\ell}-h^{p}=0.\\]\nWe now change the order of summation and rearrange terms to obtain\nthe desired formula:\\[\nh(\\tau)^{p}=\\sum_{j=1}^{p}\\big(p^{\\lambda\/2+2}\\sum_{\\ell=j}^{p}b_{\\ell}\\phi(\\tau)^{\\ell-j+1}\\big)h(\\tau){}^{p-j}.\\]\n\n\\end{proof}\nThe next lemma states that when you apply $U_{p}$ to a certain type\nof polynomial in $\\phi_{p}$, you get a similar polynomial back which\nhas picked up a power of $p$. The details of this lemma are found in\nboth \\cite{Lehner:1} and \\cite{Lehner:2}, scattered throughout the\nproofs of the main theorems. For our purposes, it will be more\nuseful in the following form.\n\\begin{lem}\n\\label{lem:Phi-Polynomials}Let $p\\in\\{2,3,5,7\\}$ and let $R^{(p)}$\ndenote the set of polynomials in $\\phi^{(p)}$ of the form\\[\nd_{1}\\phi^{(p)}(\\tau)+\\sum_{n=2}^{N}d_{n}p^{\\gamma}\\phi^{(p)}(\\tau)^{n}\\]\n\\[\n\\begin{array}{ll}\n\\text{where }\\gamma= & \\begin{cases}\n8(n-1) & \\text{if }p=2\\\\\n4(n-1) & \\text{if }p=3\\\\\nn & \\text{if }p=5\\\\\nn & \\text{if }p=7.\\end{cases}\\end{array}\\] Then $U_{p}$ maps\n$R^{(p)}$ to $p^{\\delta}R^{(p)}$ where $\\delta=3,2,1,1$ for\n$p=2,3,5,7$, respectively. That is, applying $U_{p}$ to a polynomial\nof the above form yields a polynomial of the same form with an extra\nfactor of $p^{\\delta}$.\\end{lem}\n\\begin{proof}\nConsider the function\\[\nd_{1}U_{p}\\phi(\\tau)+\\sum_{n=2}^{r}d_{n}p^{\\gamma}U_{p}\\phi(\\tau)^{n}.\\]\nFor the first term, Lemma \\ref{lem:Modular-Eq-Phi}(a) shows that\n$U_{p}\\phi(\\tau)\\in p^{\\delta}R_{p}$ since, by inspection, the $b_{j}^{(p)}$\nintegers are divisible by sufficiently high powers of $p$.\n\nFor the remaining terms, we will prove\\begin{equation}\np^{\\gamma}U_{p}\\phi^{n}=p^{\\delta}r\\label{eq:Up-Phi^t}\\end{equation}\nwhere $r\\in R_{p}$. The result will immediately follow.\n\nBy the definition of $U_{p}$ we have\\begin{equation}\nU_{p}\\phi^{n}=p^{-1}\\sum_{\\ell=0}^{p-1}\\phi\\left(\\frac{\\tau+\\ell}{p}\\right)^{n}=p^{-1-\\lambda t\/2}\\sum_{\\ell=0}^{p-1}h_{\\ell}(\\tau)^{n}\\label{eq:Up-Def-W-Sum}\\end{equation}\nwhere $h_{\\ell}(\\tau)=p^{\\lambda\/2}\\phi\\left(\\frac{\\tau+\\ell}{p}\\right)$\nis related to $h$ from Lemma \\ref{lem:Modular-Eq-Phi}(b). Let $S_{n}$\nbe the sum of the $n^{th}$ powers of the $h_{\\ell}$ so that \\[\nS_{n}=\\sum_{\\ell=0}^{p-1}h_{\\ell}^{n}.\\]\n\n\nDefine the polynomial\n$F(x)=\\sum_{j=0}^{p}(-1)^{j}g_{j}(\\tau)x^{p-j}$ where the\n$g_{j}(\\tau)$ are as in Lemma \\ref{lem:Modular-Eq-Phi}. In the same\nlemma, if we replace $\\tau$ with $\\tau+\\ell$, the $g_{j}(\\tau)$ are\nunaffected since $\\phi(\\tau+1)=\\phi(\\tau)$. Therefore, that lemma\ntells us that the $p$ roots of the polynomial $F(x)$ are precisely\nthe $h_{\\ell}$. Using Newton's formula for the $n^{th}$ power sum of\nthe roots of a polynomial, we obtain\\begin{equation}\nS_{n}=\\sum_{\\ell=0}^{p-1}h_{\\ell}^{n}=\\sum_{j=1}^{n}(-1)^{j+1}g_{j}S_{n-j}\\label{eq:Newtons-Formula}\\end{equation}\nwhere $g_{j}=0$ for $j>p$ and $S_{0}=n$.\n\nWe now proceed case-by-case. The $p=2$ case illustrates the method,\nso we will only include the intermediate steps in the $p=3,5,7$ cases.\n\\begin{caseenv}\n\\item $p=2$. Then, using (\\ref{eq:Up-Def-W-Sum}), equation (\\ref{eq:Up-Phi^t}) is\nequivalent to\\[\n2^{8(n-1)}\\big(2^{-1-12n}S_{n}\\big)=2^{3}r,\\text{ or}\\]\n\\begin{equation}\nS_{n}=2^{4n+12}r.\\label{eq:St-Powers-of-2}\\end{equation}\nWe now use (\\ref{eq:Newtons-Formula}) to calculate $S_{1}$ and $S_{2}$:\\[\nS_{1}=g(1)\\]\n\\[\nS_{2}=g_{1}S_{1}-2g_{2}=g_{1}^{2}-2g_{2}.\\]\nFrom Lemma \\ref{lem:Modular-Eq-Phi} we can compute the values of\nthe $g_{j}$. Using the $b_{j}$ values from the table in that lemma,\nwe have\\[\ng_{1}=2^{14}(b_{1}\\phi_{2}+b_{2}\\phi_{2}^{2})=2^{16}(3\\phi_{2}+2^{8}\\phi_{2}^{2})\\]\n\\[\ng_{2}=-2^{14}b_{2}\\phi_{2}=-2^{24}\\phi_{2}.\\]\nWe can now see that \\[\nS_{1}=g_{1}=2^{16}(3\\phi_{2}+2^{8}\\phi_{2}^{2})\\]\n\\[\nS_{2}=2^{32}(3\\phi_{2}+2^{8}\\phi_{2}^{2})^{2}+2^{25}\\phi_{2}=2^{20}(2^{5}\\phi_{2}+2^{12}3^{2}\\phi_{2}^{2}+2^{21}3\\phi_{2}^{3}+2^{28}\\phi_{2}^{4}).\\]\nThus (\\ref{eq:St-Powers-of-2}) is satisfied for $n=1,2$. We proceed\nby induction. Assume (\\ref{eq:St-Powers-of-2}) is satisfied for all\nintegers $<n$. We show that it is satisfied for $n$. For ease of\ncomputation, we introduce the set \\[\nR^{*}=2^{8}R^{(2)}=\\left\\{ \\sum_{i=1}^{m}d_{i}2^{8i}\\phi_{2}^{i}\\big|d_{i}\\in\\mathbb{Z},m\\in\\mathbb{Z}^{+}\\right\\} \\]\nwhich, the reader will notice, is a ring. From (\\ref{eq:Newtons-Formula})\nwe obtain\\[\nS_{n}=g_{1}S_{n-1}-g_{2}S_{n-2}=2^{8}r_{1}^{*}\\cdot2^{4n}r_{2}^{*}+2^{16}r_{3}^{*}\\cdot2^{4(n-1)}r_{4}^{*}=2^{4n+8}r_{5}^{*}=2^{4n+16}r\\]\nwhere $r_{i}^{*}\\in R^{*}$ and $r\\in R^{(2)}$.\n\\item $p=3$. We want to show \\begin{equation}\nS_{n}=3^{2n+7}r\\label{eq:St-Powers-of-3}\\end{equation}\n where $r\\in R^{(3)}$. We compute the $g_{j}$ and $S_{n}$ as follows,\nusing the $b_{j}$ from the table:\\[\n\\begin{array}{ccc}\ng_{1}=3^{9}(3^{9}\\phi_{3}^{3}+3^{5}4\\phi_{3}^{2}+10\\phi_{3}), &\ng_{2}=3^{14}(-3^{4}\\phi_{3}^{2}-4\\phi_{3}), &\ng_{3}=3^{18}\\phi_{3},\\end{array}\\]\n\\[\n\\begin{array}{ccc}\nS_{1}=g_{1}, & S_{2}=g_{1}^{2}-2g_{2}, &\nS_{3}=g_{1}{}^{3}-3g_{1}g_{2}+3g_{3}.\\end{array}\\] From this, we\nobtain\\[\nS_{1}=3^{9}(3^{9}\\phi_{3}^{3}+3^{5}4\\phi_{3}^{2}+10\\phi_{3})\\]\n\\[\nS_{2}=3^{14}(8\\phi_{3}+3^{5}34\\phi_{3}^{2}+3^{9}80\\phi_{3}^{3}+3^{13}68\\phi_{3}^{4}+3^{18}8\\phi_{3}^{5}+3^{25}\\phi_{3}^{6})\\]\n\\[\nS_{3}=3^{19}(\\phi_{3}+3^{5}40\\phi_{3}^{2}+3^{8}1174\\phi_{3}^{3}+3^{15}136\\phi_{3}^{4}+3^{18}581\\phi_{3}^{5}+3^{25}16\\phi_{3}^{6}+3^{27}58\\phi_{3}^{7}+3^{32}4\\phi_{3}^{8}+3^{35}\\phi_{3}^{9})\\]\nwhich proves (\\ref{eq:St-Powers-of-3}) for $n=1,2,3$. For the inductive\nstep, let $R^{*}$ be the ring $3^{4}R^{(3)}$ so that\\begin{align*}\nS_{n} & =g_{1}S_{n-1}-g_{2}S_{n-2}+g_{3}S_{n-3}\\\\\n & =3^{5}r_{1}^{*}3^{2n+1}r_{2}^{*}+3^{10}r_{3}^{*}3^{2n-1}r_{4}^{*}+3^{14}r_{5}^{*}3^{2n-3}r_{6}^{*}\\\\\n & =3^{2n+6}r_{7}^{*}\\\\\n & =3^{2n+10}r\\end{align*}\nwhere $r_{i}^{*}\\in R^{*}$ and $r\\in R^{(3)}$.\n\\item $p=5$. We want\\begin{equation}\nS_{n}=5^{2n+2}r\\label{eq:St-Powers-of-5}\\end{equation}\nwhere $r\\in R^{(5)}$. Computing the $S_{n}$ we find\\[\n\\begin{array}{ccc}\nS_{1}=5^{5}r_{1} & S_{2}=5^{8}r_{2} & S_{3}=5^{10}r_{3}\\end{array}\\]\n\\[\n\\begin{array}{cc}\nS_{4}=5^{13}r_{4} & S_{5}=5^{16}r_{5}\\end{array}\\]\nfor some $r_{1},\\ldots,r_{5}\\in R^{(5)}$. This proves (\\ref{eq:St-Powers-of-5})\nfor $n=1,\\ldots,5$. For the inductive step, let $R^{*}$ be the ring\n$5R^{(5)}$ so that\\begin{align*}\nS_{n} & =g_{1}S_{n-1}-g_{2}S_{n-2}+g_{3}S_{n-3}-g_{4}S_{n-4}+g_{5}S_{n-5}\\\\\n & =5^{4}r_{1}^{*}5^{2n-1}r_{2}^{*}-\\ldots+5^{14}r_{9}^{*}5^{2n-9}r_{10}^{*}\\\\\n & =5^{2n+3}r_{11}^{*}\\\\\n & =5^{2n+4}r\\end{align*}\nwhere $r_{i}^{*}\\in R^{*}$ and $r\\in R^{(5)}$ .\n\\item $p=7$. We want\\begin{equation}\nS_{n}=7^{n+2}r\\label{eq:St-Powers-of-7}\\end{equation}\nwhere $r\\in R^{(7)}$. Computing the $S_{n}$ we find\\[\n\\begin{array}{cccc}\nS_{1}=7^{4}r_{1} & S_{2}=7^{6}r_{2} & S_{3}=7^{7}r_{3} & S_{4}=7^{9}r_{4}\\end{array}\\]\n\\[\n\\begin{array}{ccc}\nS_{5}=7^{11}r_{5} & S_{6}=7^{13}r_{6} & S_{7}=7^{15}r_{7}\\end{array}\\]\nfor some $r_{1},\\ldots,r_{7}\\in R^{(7)}$. This proves (\\ref{eq:St-Powers-of-7})\nfor $n=1,\\ldots,7$. For the inductive step, let $R^{*}$ be the ring\n$7R^{(7)}$ so that\\begin{align*}\nS_{n} & =\\sum\\limits _{i=1}^{7}(-1)^{i+1}g_{i}S_{n-i}\\\\\n & =7^{3}r_{1}^{*}7^{n}r_{2}^{*}-\\ldots+7^{13}r_{13}^{*}7^{n-6}r_{14}^{*}\\\\\n & =7^{n+3}r_{15}^{*}\\\\\n & =7^{n+4}r\\end{align*}\nwhere $r_{i}^{*}\\in R^{*}$ and $r\\in R^{(7)}$ .\n\\end{caseenv}\n\\end{proof}\n\n\\section{Proof of the Theorem}\n\nTo remind the reader of the main result of the paper, we include it\nhere.\n\\begin{thm*}\nLet $p\\in\\{2,3,5,7\\}$, and let $f_{0,m}^{(p)}(\\tau)=q^{-m}+\\sum\na_{0}^{(p)}(m,n)q^{n}$ be an element of the basis described above,\nwith $m=p^{\\alpha}m'$ and $(m',p)=1$. Then, for $\\beta > \\alpha$,\n\\[\n\\begin{array}{rll}\na_{0}^{(2)}(2^{\\alpha}m',2^{\\beta}n)\\equiv0 & \\pmod{2^{3(\\beta-\\alpha)+8}} & \\text{if }p=2\\\\\na_{0}^{(3)}(3^{\\alpha}m',3^{\\beta}n)\\equiv0 & \\pmod{3^{2(\\beta-\\alpha)+3}} & \\text{if }p=3\\\\\na_{0}^{(5)}(5^{\\alpha}m',5^{\\beta}n)\\equiv0 & \\pmod{5^{(\\beta-\\alpha)+1}} & \\text{if }p=5\\\\\na_{0}^{(7)}(7^{\\alpha}m',7^{\\beta}n)\\equiv0 & \\pmod{7^{(\\beta-\\alpha)}} & \\text{if }p=7.\\end{array}\\]\n\\end{thm*}\nThe proof is in three cases. The first illustrates the method for\nthe simplest basis elements, namely those with $(m,p)=1$. The second\ndemonstrates the ``shifting'' property at its first occurence,\n$f_{0,p}^{(p)}$. The third is the general case; it builds\ninductively upon the methods of the first two cases.\n\n\n\\subsection{Case 1: $(m,p)=1$}\n\\begin{proof}\nThis proof is almost identical to Lehner's proof of Theorem 3 in\n\\cite{Lehner:2}, however it applies not only to functions which\nhave poles of order bounded by $p$, but to all basis elements with\n$(m,p)=1$. For ease of notation, let $f(\\tau)=f_{0,m}^{(p)}(\\tau)$.\n\nWe will demonstrate the method with $m=1$, then generalize it to\nall $m$ relatively prime to $p$. First, we will write $U_{p}f(\\tau)$\nas a polynomial in $\\phi(\\tau)$ with integral coefficients, all of\nwhich are divisible by the desired power of $p$. Since $U_{p}$ isolates\nthe coefficients whose index is divisible by $p$, we will have proven\nthe theorem for $\\beta=1$. We will then apply $U_{p}$ repeatedly\nto the polynomial, showing that the result is always another polynomial\nin $\\phi$ with integral coefficients, all of which are divisible\nby the desired higher power of $p$.\n\nConsider the level $p$ modular function $g(\\tau)=pU_{p}f(\\tau)+p^{\\lambda\/2}\\phi(\\tau)$.\nNotice that $g(\\tau)$ is holomorphic at $\\infty$ since both $U_{p}f(\\tau)$\nand $\\phi(\\tau)$ are holomorphic there. The $q$-expansion at $0$\nfor $g(\\tau)$ is given by\\[\ng(-1\/p\\tau)=p(U_{p}f)(-1\/p\\tau)+p^{\\lambda\/2}\\phi(-1\/p\\tau)\\]\nwhich, by Lemmas \\ref{lem:Psi-At-0} and \\ref{lem:Main-Up-Formula} becomes\\[\ng(-1\/p\\tau)=p(U_{p}f)(p\\tau)+f(-1\/p^{2}\\tau)-f(\\tau)+\\psi(\\tau).\\]\nWhen we notice that $f(\\tau)=\\psi(\\tau)$ in this $m=1$ case, we\nobtain\\begin{align*}\ng(-1\/p\\tau) & =p(U_{p}f)(p\\tau)+\\psi(-1\/p^{2}\\tau)-\\psi(\\tau)+\\psi(\\tau)\\\\\n & =p(U_{p}f)(p\\tau)+p^{\\lambda\/2}\\phi(p\\tau),\\end{align*}\nwhich is holomorphic at $\\infty$. Hence, $g(\\tau)$ is a holomorphic\nmodular function on $\\Gamma_{0}(p)$, so it must be constant. Therefore,\n\\begin{equation}\nU_{p}f(\\tau)=c_{0}-p^{\\lambda\/2-1}\\phi(\\tau)\\label{eq:Up-f1=00003DPhi}\\end{equation}\nfor some constant $c_{0}$. The proof is complete for $\\beta=1$.\n\nNote: the prime $13$, having genus zero, would work in this construction;\nhowever, in that case $\\lambda=\\frac{24}{13-1}=2$, so $13^{\\lambda\/2-1}=1$,\nand we gain no new information.\n\nWe now iterate the above process to prove the theorem for $\\beta>1$.\nNotice that \\[\nU_{p}(U_{p}f(\\tau))=c^{(p)}-p^{\\lambda\/2-1}U_{p}\\phi(\\tau).\\] We\nknow from Lemma \\ref{lem:Modular-Eq-Phi} that $U_{p}\\phi$ is a\npolynomial in $\\phi$; in fact, by inspection of the $b_{j}^{(p)}$\nvalues we see that we may write\\begin{align*}\nU_{2}\\phi^{(2)}(\\tau) & =2^{3}\\big(d_{1}^{(2)}\\phi^{(2)}(\\tau)+\\sum_{n=2}^{2}d_{n}^{(2)}2^{8(n-1)}\\phi^{(2)}(\\tau)^{n}\\big)\\\\\nU_{3}\\phi^{(3)}(\\tau) & =3^{2}\\big(d_{1}^{(3)}\\phi^{(3)}(\\tau)+\\sum_{n=2}^{3}d_{n}^{(3)}3^{4(n-1)}\\phi^{(3)}(\\tau)^{n}\\big)\\\\\nU_{5}\\phi^{(5)}(\\tau) & =5\\big(d_{1}^{(5)}\\phi^{(5)}(\\tau)+\\sum_{n=2}^{5}d_{n}^{(5)}5^{n}\\phi^{(5)}(\\tau)^{n}\\big)\\\\\nU_{7}\\phi^{(7)}(\\tau) &\n=7\\big(d_{1}^{(7)}\\phi^{(7)}(\\tau)+\\sum_{n=2}^{7}d_{n}^{(7)}7^{n}\\phi^{(7)}(\\tau)^{n}\\big)\\end{align*}\nfor some integers $d_{n}^{(p)}$. This shows that the second $U_{p}$\niteration is divisible by the correct power of $p$. Further, it\ngives us a polynomial of a suitable form to iterate the process\nusing Lemma \\ref{lem:Phi-Polynomials}. In each of the polynomials\nabove, notice that $U_{p}\\phi(\\tau)=p^{\\delta}r$ for some $r\\in\nR^{(p)}$. Using Lemma \\ref{lem:Phi-Polynomials}, we find that\\[\nU_{p}(U_{p}\\phi)(\\tau)=p^{2\\delta}r'\\] for some $r'\\in R^{(p)}$, and\nfurther\\[ U_{p}^{\\beta}\\phi(\\tau)=p^{\\beta\\delta}r^{(\\beta)}\\] for\nsome $r_{\\beta}\\in R^{(p)}$. This completes the proof for $m=1$.\n\nNow, if $(m,p)=1$, then $U_{p}f(\\tau)$ is holomorphic at $\\infty$,\njust as it was with $m=1$. Moving to the cusp at $0$ we find\nthat $(U_{p}f)(-1\/p\\tau)$ can be written as a polynomial in $\\psi(\\tau)$\nwhich appears as a polynomial in $\\phi(\\tau)$ when we return to $\\infty$.\nSimilar to (\\ref{eq:Up-f1=00003DPhi}), we obain the equality\\[\nU_{p}f(\\tau)=c_{0}+\\sum_{i=1}^{M}p^{\\lambda i\/2-1}c_{i}\\phi(\\tau)^{i}\\]\nfor some $c_{i}\\in\\mathbb{Z}$ and $M\\in\\mathbb{Z}^{+}$. The only\ndifference between this equation and (\\ref{eq:Up-f1=00003DPhi}) is\nthat in this more general case, we find that $U_{p}f$ is a higher\ndegree polynomial in $\\phi$. This formula can easily be iterated\nas before to obtain the desired result.\n\\end{proof}\n\n\\subsection{Case 2: $m=p$}\n\\begin{proof}\nAgain, for ease of notation, denote $f_{0,p}^{(p)}(\\tau)$ by\n$f(\\tau)$. For the $m=p$ case, we will proceed as before; however,\nwe will find that $U_{p}f(\\tau)$ has poles at both $\\infty$ and $0$,\nand that $U_{p}f(\\tau)$ does not possess any interesting\ndivisibility properties, but $U_{p}^{2}f(\\tau)$ does. This property\nwill manifest itself as the {}``shifting'' previously mentioned.\n\nNotice first that $U_{p}f(\\tau)=q^{-1}+O(1)$ has a simple pole at\n$\\infty$. Therefore, we shall deal primarily with the function $U_{p}f(\\tau)-\\psi(\\tau)$,\nwhich is holomorphic at $\\infty$. We can use Lemmas \\ref{lem:Psi-At-0}\nand \\ref{lem:Main-Up-Formula} to view this function at $0$:\n\n\\begin{align*}\np(U_{p}f)(-1\/p\\tau)-p\\psi(-1\/p\\tau) & = p(U_{p}f)(p\\tau)+f(-1\/p^{2}\\tau)-f(\\tau)-p^{\\lambda\/2+1}\\phi(\\tau) \\\\\n & =pq^{-p}+O(1)+O(1)-q^{-p}+O(1)+O(q)\\\\\n & =c_{0}+\\sum_{i=1}^{p}c_{i}\\psi(\\tau)^{i}\n\\end{align*}\nfor some integers $c_{i}$. Replacing $\\tau$ by $-1\/p\\tau$, we obtain\n\\begin{equation}\n(U_{p}f)(\\tau)=\\frac{c_{0}}{p}+\\psi(\\tau)+\\sum_{i=1}^{p}c_{i}p^{\\lambda\ni\/2-1}\\phi(\\tau)^{i}.\\label{eq:Case-2-Upf}\n\\end{equation}\n\n\nThe $\\psi(\\tau)$ term in the equation makes any attempt at\n$p$-divisibility fail; for example, computation shows that the\n$7^{th}$ coefficient of $\\psi^{(2)}(\\tau)$ is odd. However,\n$\\psi(\\tau)$ satisfies Lehner's divisibility properties, so $U_{p}f$\ninherits its $p$-divisibility from $\\psi(\\tau)$. So the function\\[\nU_{p}^{2}f(\\tau)=c_{0}+U_{p}\\psi(\\tau)+\\sum_{i=1}^{p}c_{i}p^{\\lambda\ni\/2-1}U_{p}\\phi(\\tau)^{i}\\] has the same $p$-divisibility as\n$f_{0,1}^{(p)}$; hence, the shift.\n\\end{proof}\n\n\\subsection{Case 3: $m=p^{\\alpha}m'$}\n\\begin{proof}\nWe prove this case using induction on $\\alpha$. Case 1 showed that the theorem is true for\nall $m'$ relatively prime to $p$, so the $\\alpha=0$ base case is complete. Assume Theorem\n\\ref{thm:Andersen-Main}\nholds for all $m$ of the form $m=p^{\\ell}m'$ with $\\ell<\\alpha$.\nWe will show it holds for $m=p^{\\alpha}m'$.\nTo simplify notation,\nlet $f_{\\alpha}(\\tau)=f_{0,p^{\\alpha}m'}^{(p)}(\\tau)$.\n\nSince $f_{\\alpha}(\\tau)=q^{-p^{\\alpha}m'}+O(1)$, we find that\n$U_{p}f_{\\alpha}(\\tau)=q^{-p^{\\alpha-1}m'}+O(1)$ has a pole of order\n$p^{\\alpha-1}m'$ at $\\infty$. So we focus our attention on\n$U_{p}f_{\\alpha}(\\tau)-f_{\\alpha-1}(\\tau)$, which is holomorphic at\n$\\infty$. Using (\\ref{eq:Main-Up-Formula}) we examine this function\nat $0$:\\begin{align*}\np(U_{p}f_{\\alpha})\\left(\\frac{-1}{p\\tau}\\right) -\npf_{\\alpha-1}\\left(\\frac{-1}{p\\tau}\\right) &\n=p(U_{p}f_{\\alpha})(p\\tau)+f_{\\alpha}\\left(\\frac{-1}{p^{2}\\tau}\\right)\n-\nf_{\\alpha}(\\tau)-pf_{\\alpha-1}\\left(\\frac{-1}{p\\tau}\\right)\\\\\n & =pq^{-p^{\\alpha}m'}+O(1)+O(1)-q^{-p^{\\alpha}m'}-O(1)-O(1)\\\\\n & =(p-1)q^{-p^{\\alpha}m'}+O(1).\\end{align*}\n\n\nAs before, we write this function as a polynomial in $\\psi(\\tau)$\nwith integral coefficients $c_{i}$:\\[\np(U_{p}f_{\\alpha})(-1\/p\\tau)-pf_{\\alpha-1}(-1\/p\\tau)=c_{0}+\\sum_{i=1}^{p^{\\alpha}m'}c_{i}\\psi(\\tau)^{i},\\]\nwhich, after switching back to the $q$-expansion at $\\infty$, becomes\\begin{equation}\nU_{p}f_{\\alpha}(\\tau)=\\frac{c_{0}}{p}+f_{\\alpha-1}(\\tau)+\\frac{1}{p}\\sum_{i=1}^{p^{\\alpha}m'}c_{i}p^{\\lambda i\/2}\\phi(\\tau)^{i}.\\label{eq:Case-3-Upf}\\end{equation}\n\n\nNotice that (\\ref{eq:Case-3-Upf}) looks very similar to\n(\\ref{eq:Case-2-Upf}), where $\\psi(\\tau)$ is replaced by\n$f_{\\alpha-1}(\\tau)$, so $U_{p}f_{\\alpha}(\\tau)$ inherits whatever\ndivisibility properties $f_{\\alpha-1}(\\tau)$ has. Our inductive\nhypothesis states that $f_{\\alpha-1}(\\tau)$ exhibits Lehner's\ndivisibility properties only after $U_{p}$ is applied $\\alpha-1$\ntimes. Therefore, applying $U_{p}$ to (\\ref{eq:Case-3-Upf})\n$\\alpha-1$ times, we obtain\\[\nU_{p}^{\\alpha}f_{\\alpha}(\\tau)=\\frac{c_{0}}{p}+U_{p}^{\\alpha-1}f_{\\alpha-1}(\\tau)\n+\\frac{1}{p}\\sum_{i=1}^{p^{\\alpha}m'}c_{i}p^{\\lambda\ni\/2}U_{p}^{\\alpha-1}\\phi(\\tau)^{i}\\] showing that\n$U_{p}^{\\alpha}f_{\\alpha}(\\tau)$ exhibits Lehner's divisibility\nproperties.\n\\end{proof}\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{Introduction}}\n\nThe radar head echo is a signal that reflects from the plasma surrounding the\nfast-descending meteoroid and is doppler-shifted by approximately the\nmeteoroid velocity. Only a small volume of the dense plasma sufficiently close\nto the meteoroid contributes to the corresponding radar wave reflection.\nQuantitative knowledge of the spatial structure of the near-meteoroid plasma\nis crucial for accurate modeling the head echo radar reflections\n\\citep{Bronshten:Physics83,Ceplecha:Meteor1998,Close:NewMethod2005,Campbell-Brown:Meteoroid2007}.\n\nIn the companion paper \\citep{Dimant:Formation1_17}, hereinafter referred to as\nPaper~1, we developed a first-principle kinetic theory of the plasma formed\naround a small meteoroid as it moves through the atmosphere at hypersonic\nspeeds. Using a number of easily justified assumptions, we obtained\napproximate analytic expressions describing velocity distributions of meteoric\nions and neutrals. In this paper, we calculate the spatial structure of the\nplasma density that follows from the kinetic theory developed in Paper~1. This\ncalculation demonstrates that this spatial structure differs dramatically from\na simple Gaussian or exponential distribution currently employed for modeling\nradar wave scattering from the meteor plasma \\citep{Close:NewMethod2005,Marshall:FDTD15}. \nThis research does not describe the distribution of\nplasma or neutrals in the meteoroid tail where particles lag well behind the\nmeteoroid after having collided more than once.\n\nSimple analysis of individual collisions between particles indicates that\nheavy meteoric particles in the near-meteor sheath consist predominantly of\nthe `primary' and `secondary' particles. By a primary particle we mean an\nablated meteoroid particle that moves freely with a ballistic trajectory until\nit collides with an atmospheric molecule. These primary particles are\npredominantly neutral. A secondary particle is a former primary particle that\nexperienced exactly one collision, either scattering or ionizing. Most of the\nnear-meteoroid ions responsible for head echoes belong to the group of\nsecondary particles. The vast majority of ions that experienced multiple\ncollisions since the original ablation lag behind the fast-moving meteoroid\nand form a long-lived extended column of plasma visible to radars through\nspecular or non-specular echoes.\n\nGiven the velocity distributions developed in Paper 1 as a function of spatial\ncoordinates, one can integrate over velocity variables to find the\ncorresponding particle density. However, the complexity of these analytic\nexpressions makes this non-trivial. This paper makes an additional simplifying\nassumption about the collision model (the isotropic differential cross-section\nof ionization) and then integrates over the velocities to obtain the meteor\ngas and plasma density as a function of distance from the meteoroid.\n\nThe paper is organized as follows.\nSection~\\ref{Summary of the ion distribution function} summarizes the results\nof Paper~1 on the ion distribution function.\nSection~\\ref{Plasma density calculations} performs the calculations of the\nnear-meteoroid plasma density. Section~\\ref{Discussion} discusses implications\nof our theory and some caveats. Section~\\ref{Conclusions} lists the major\nunderlying assumptions and discusses the paper results.\n\n\\section{Summary of the ion distribution\nfunction\\label{Summary of the ion distribution function}}\n\nPaper~1 does all our calculations in the rest frame of a meteoroid moving\nthrough the atmosphere with the local velocity $-\\vec{U}$, so that in this\nframe the impinging atmospheric particles move with the opposite velocity,\n$\\vec{U}$. We define the coordinate system with the major axis passing through\nthe meteoroid center and parallel to $\\vec{U}$. Due to the axial symmetry\nabout $\\vec{U}$, we characterize the real space by two spherical coordinates:\nthe radial distance from the meteoroid center, $R$, and the polar angle,\n$\\theta$, measured from the major axis ($\\theta=0$ corresponds to the major\nsemi-axis behind the meteoroid, while $\\theta=\\pi$ corresponds to the opposite\nsemi-axis in front of it). Figure~\\ref{Fig:Cartoon_reproduced}, reproduced\nfrom Paper~1, explains all relevant notations.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=30pc]{Cartoon_5_20_2015.pdf}\n\\caption{Nomenclature of spatial coordinates and velocity variables. The\nspatial variables $R=|\\vec{R}|$, $\\theta$, $\\varphi$ denote the radius and two\nangles of the spherical coordinate system with the origin at the meteoroid\ncenter and the major axis anti-parallel to the meteoroid velocity (shown on\nthe left). All other variables pertain to the particle velocity space:\n$V=|\\vec{V}|$ is the particle speed, $\\vartheta$ is the polar angle of\n$\\vec{V}$ with respect to the local axis parallel to $\\vec{U}$, $\\Phi$ is the\naxial angle measured from the common $\\vec{U}$-$\\vec{R}$ plane; $\\Theta$ is\nthe polar angle of $\\vec{V}$ with respect to the local radial distance\n$\\vec{R}$.}\n\\label{Fig:Cartoon_reproduced}\n\\end{figure}\n\n\nThe velocity distribution of secondary ions, $f^{(2)}$, is expressed as a\nfunction of three velocity variables that are invariants of the ion\ncollisionless motion. These variables include the ion speed, $V$, the cosine\nof the angle between the ion velocity vector $\\vec{V}$ and $\\vec{U}$,\n$\\mu=\\cos\\vartheta$, and a normalized angular momentum variable, $R_{0}$,\nwhich to the minimum distance between the ion trajectory and the meteoroid\ncenter, $R_{0}=R\\sin\\Theta$, where $\\Theta$ is the polar angle of $\\vec{V}$\nwith respect to the local radius vector $\\vec{R}$. The entire set of\nvelocity-space variables also includes a discrete variable $\\sigma_{R}$ which\ntakes two values, $\\pm1$, depending on the sign of the particle radial\nvelocity,%\n\\begin{equation}\nV_{R}=\\frac{dR}{dt}\\equiv V\\cos\\Theta=\\sigma_{R}\\sqrt{1-\\frac{R_{0}^{2}}%\n{R^{2}}}\\ V. \\label{V_r_snova}%\n\\end{equation}\nThe value of $\\sigma_{R}=+1$ corresponds to the outgoing particles, $V_{R}>0$,\nwhile $\\sigma_{R}=-1$ corresponds to the incoming particles, $V_{R}<0$. At any\nlocation, the entire distribution function is given by a sum of the two\ncorresponding functions,%\n\\begin{equation}\nf^{(2)}(V,\\mu,R_{0};R,\\theta)=\\sum_{\\sigma_{R}=\\pm1}f_{\\sigma_{R}}^{(2)}%\n(V,\\mu,R_{0};R,\\theta). \\label{sum_f}%\n\\end{equation}\nThe functions $f_{\\sigma_{R}}^{(2)}$ are non-zero provided $\\mu=\\cos\n\\vartheta>0$; otherwise $f_{\\sigma_{R}}^{(2)}=0$,%\n\\begin{align}\n&  \\left.  f_{\\sigma_{R}}^{(2)}(V,R,\\theta)\\right\\vert _{\\mu>0}=L_{\\sigma_{R}%\n}\\ \\delta\\left(  V-\\frac{2m_{\\beta}\\mu U}{m+m_{\\beta}}\\right)  ,\\nonumber\\\\\nL_{\\sigma_{R}}  &  =\\frac{G_{\\mathrm{ion}}(U,1-2\\mu^{2})n_{0}n_{\\mathrm{A}}%\n}{\\sqrt{3}\\ \\mu U^{2}}\\left(  1+\\frac{m}{m_{\\beta}}\\right)  ^{3}%\nI(R,R_{0}),\\label{f^(2)_via_I}\\\\\n&  \\left.  f_{\\sigma_{R}}^{\\left(  2\\right)  }(V,R,\\theta)\\right\\vert _{\\mu\n<0}=0.\\nonumber\n\\end{align}\nThe quantities $n_{0}$ and $n_{\\mathrm{A}}$ are the densities of the ablated\nparticles at the meteoroid surface and of the atmospheric particles at a given\naltitude, respectively. The quantity $G_{\\mathrm{ion}}(U,1-2\\mu^{2})$\noriginates from the differential cross-section of ionizing collisions,\n$G_{\\mathrm{ion}}$, expressed as a function of the relative speed between the\ntwo colliding particles, $u=\\left\\vert \\vec{u}\\right\\vert $, and the cosine of\nthe scattering angle, $\\Theta_{\\mathrm{sc}}$. In this paper, we simplify our\ntreatment by assuming $G_{\\mathrm{ion}}$ to be a function of only $u\\approx\nU$. The corresponding angular dependence in the relevant energy range is\ngenerally unknown, but the assumption of isotropic $G_{\\mathrm{ion}}(U)$ is reasonable.\n\nThe condition $\\mu>0$ is fulfilled if either\n\\begin{subequations}\n\\label{sgn}%\n\\begin{align}\n&  \\sigma_{R}=\\mathrm{sgn}(\\cos\\theta)\\qquad\\text{and}\\qquad0<R_{0}%\n<R_{c}(\\theta,\\Phi)\\label{sigma=sgn}\\\\\n\\text{or}  & \\nonumber\\\\\n&  \\sigma_{R}=-\\mathrm{sgn}(\\cos\\theta)\\qquad\\text{and}\\qquad R_{c}%\n(\\theta,\\Phi)<R_{0}<R, \\label{sigma=-sgn}%\n\\end{align}\nwhere $\\mathrm{sgn}(x)$ means the sign of $x$ and%\n\\end{subequations}\n\\begin{equation}\nR_{c}(\\theta,\\Phi)=\\frac{R\\left\\vert \\cos\\theta\\right\\vert }{\\sqrt{1-\\sin\n^{2}\\theta\\sin^{2}\\Phi}}. \\label{R_c(Phi)}%\n\\end{equation}\nHere $\\Phi$ is the axial angle of the particle velocity $\\vec{V}$ around the\ndirection of the local radius-vector $\\vec{R}$ (see\nFigure~\\ref{Fig:Cartoon_reproduced}) and we set the origin $\\Phi=0$ where\n$\\vec{V}$ lies in the common $\\vec{R}$-$\\vec{U}$ plane.\n\nIn this paper, we consider the meteor plasma located not too close to the\nmeteoroid, $R\\gg r_{\\mathrm{M}}$. The corresponding multiplier $I(R,R_{0})$ in\nequation~(\\ref{f^(2)_via_I}) has the following piecewise definition:%\n\\begin{equation}\nI(R,R_{0})=\\left\\{\n\\begin{array}\n[c]{ccc}%\nJ_{R}^{\\infty} & \\text{for} & \\sigma_{R}=-1,\\\\\n&  & \\\\\nJ_{R_{0}}^{\\infty}+J_{R_{0}}^{R}=2J_{R_{0}}^{\\infty}-J_{R}^{\\infty} &\n\\text{for} & \\sigma_{R}=+1,\n\\end{array}\n\\right.  \\label{III}%\n\\end{equation}\nwhere, under constraints of $R,R_{0}>3r_{\\mathrm{M}}$, the well-convergent\nintegral $J_{a}^{b}$, taken as a function of its integration limits, $b>a\\geq\nR_{0}$, is given by%\n\\begin{equation}\nJ_{a}^{b}\\approx r_{\\mathrm{M}}^{2}\\int_{a}^{b}\\left[  1+\\left(\n\\frac{R^{\\prime}}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}\\right]  \\exp\\left[\n-\\ \\frac{3}{2}\\left(  \\frac{R^{\\prime}}{\\lambda_{T}^{(1)}}\\right)\n^{2\/3}\\right]  \\frac{dR^{\\prime}}{R^{\\prime}\\sqrt{\\left(  R^{\\prime}\\right)\n^{2}-R_{0}^{2}}}. \\label{J_a^b_R>3epsilon}%\n\\end{equation}\nHere $\\lambda_{T}^{(1)}$ is the mean free path of the primary (ablated)\nparticles,%\n\\begin{equation}\n\\lambda_{T}^{(1)}=\\frac{V_{T}}{\\nu_{T}^{(1)}},\\qquad V_{T}=\\left(  \\frac\n{T_{M}}{m_{\\mathrm{M}}}\\right)  ^{1\/2},\\qquad\\nu_{T}^{(1)}\\approx2\\pi\nn_{\\mathrm{A}}U\\int_{-1}^{1}G^{\\left(  1\\right)  }(U,\\Lambda)d\\Lambda,\n\\label{lambda_T^(1)}%\n\\end{equation}\nwhere $T_{M}$ and $m_{\\mathrm{M}}$ are the temperature and mass of the primary\nmeteor particles. The quantity $G^{\\left(  1\\right)  }(U,\\Lambda)$ includes\nall collisions that result in scattering of the primary neutral particles. The\nexpression for $G^{\\left(  1\\right)  }$, as that for $G_{\\mathrm{ion}}$, takes\ninto account that $V_{T}\\ll U$, so that the collision frequency $\\nu_{T}%\n^{(1)}$ depends only on the meteoroid speed, $U$, and hence is the same for\nall particles. This reduces $\\lambda_{T}^{(1)}$ to a constant value which\nbecomes the characteristic length-scale of the near-meteoroid plasma.\n\nThe general integral $J_{a}^{b}$ cannot be calculated exactly, but the\nparticular integral $J_{R_{0}}^{\\infty}$ has an almost perfect analytic\napproximation,%\n\\begin{equation}\nJ_{R_{0}}^{\\infty}\\approx\\frac{\\pi r_{\\mathrm{M}}^{2}}{2R_{0}}\\sqrt{1+\\frac\n{2}{\\pi}\\left(  \\frac{R_{0}}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}}\\ \\exp\\left[\n-\\ \\frac{3}{2}\\left(  \\frac{R_{0}}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}\\right]  ,\n\\label{J_R_0_interpo}%\n\\end{equation}\naccurate within $1\\%$ for all $R_{0}$. As we demonstrate below, depending on\nthe specific calculation, it may become beneficial to use either the exact\noriginal integral expression for $J_{a}^{b}$ given by\nequation~(\\ref{J_a^b_R>3epsilon}) or (only for $J_{R_{0}}^{\\infty}$) its\napproximation given by equation~(\\ref{J_R_0_interpo}).\n\nFor local calculations of the ion density it is more convenient to pass from\nthe invariant velocity variables $V,R_{0},\\mu$ to local variables\n$V,R_{0},\\Phi$, where $\\Theta$ and $\\Phi$ are the polar and axial angles of\nthe ion velocity about the direction of the local radius-vector $\\vec{R}$, as\ndepicted by Figure~\\ref{Fig:Cartoon_reproduced}.\n\n\\section{Plasma density calculations\\label{Plasma density calculations}}\n\nRadar head echo is determined by the spatial distribution of the electron\ndensity around the meteoroid. The near-meteoroid plasma is quasi-neutral, so\nthat the electron density almost equals that of ions, $n_{e}\\approx n_{i}=n$.\nWe calculate the spatial distribution of the ion density based on the\ndistribution function explained in\nsection~\\ref{Summary of the ion distribution function}.\n\nThe ion density can be easily calculated in the far region of $R\\gg\\lambda\n_{T}^{(1)}$. Albeit less simple, but $n^{(2)}(R,\\theta)$ can also be\nexplicitly calculated in the opposite limit of $R\\ll\\lambda_{T}^{(1)}$. In the\nentire space of arbitrary $R$, we were unable to find a unified purely\nalgebraic expression for $n^{(2)}(R,\\theta)$. However, we have reduced the\ngeneral 3D velocity-space integral to a much simpler expression for $n^{(2)}$\nin terms of normalized variables and parameters, as explained below. This\nuniversal expression involves only treatable analytical functions and two 1D\nintegral functions suitable for simple numerical integration and tabulation.\nThe resultant universal expression for $n^{(2)}$ makes the future analysis and\ncomputer modeling of the radar signal much easier.\n\n\\subsection{Preliminary remarks}\n\nAt a given location determined by $R$ and $\\theta$, the ion density is given\nby $n^{\\left(  2\\right)  }=\\sum_{\\sigma_{R}=\\pm1}\\int f_{\\sigma_{R}}%\n^{(2)}V^{2}dVd\\Omega$, where $d\\Omega=d\\left(  \\cos\\Theta\\right)  d\\Phi$\ndenotes the elementary volume of the local solid angle. Choosing instead of\n$\\Theta$ the new variable $R_{0}=R\\sin\\Theta$, we obtain%\n\\begin{equation}\nn^{\\left(  2\\right)  }=\\frac{1}{R}\\sum_{\\sigma_{R}=\\pm1}\\int_{0}^{R}%\n\\frac{R_{0}dR_{0}}{\\sqrt{R^{2}-R_{0}^{2}}}\\int_{0}^{2\\pi}d\\Phi\\int_{0}%\n^{\\infty}f_{\\sigma_{R}}^{(2)}V^{2}dV. \\label{n^(2)_snova}%\n\\end{equation}\n\n\nFirst, we integrate over $V$ to eliminate the $\\delta$-function in equation\n(\\ref{f^(2)_via_I}),%\n\\begin{equation}\n\\int_{0}^{\\infty}f_{\\sigma_{R}}^{(2)}V^{2}dV=\\frac{4\\mu G_{\\mathrm{ion}}%\nn_{0}n_{\\mathrm{A}}}{\\sqrt{3}}\\left(  1+\\frac{m}{m_{\\beta}}\\right)\nI(R,R_{0}). \\label{yields}%\n\\end{equation}\nAs a result of this simple integration, the previously singular factor $\\mu$\nin the expression for $L_{\\sigma_{R}}$ has moved from its denominator to the\nnumerator, reducing dramatically the relative contribution of the\n`small-angle' ($\\Theta_{\\mathrm{sc}}=1-2\\mu^{2}\\approx1$, $\\mu\\ll1$)\nionization where some assumptions of our general theory are invalid\n\\citep{Dimant:Formation1_17}. Since we assumed above isotropic $G_{\\mathrm{ion}%\n}=G_{\\mathrm{ion}}(U)$, the only $\\Phi$-dependent quantity in the right-hand\nside (RHS) of equation (\\ref{yields}) is $\\mu$ in the numerator. This variable\nis expressed in terms of $R_{0}$ and $\\Phi$ as%\n\\begin{equation}\n\\mu=\\sigma_{R}\\sqrt{1-\\frac{R_{0}^{2}}{R^{2}}}\\cos\\theta+\\frac{R_{0}\\sin\n\\theta}{R}\\ \\cos\\Phi. \\label{mumumu}%\n\\end{equation}\nThe function $I(R,R_{0})$, along with the corresponding integral expressions\nfor $J_{a}^{b}$, are described by equations~(\\ref{III}) to\n(\\ref{J_R_0_interpo}).\n\n\\subsection{Long-distance asymptotics, $R\\gg\\lambda_{T}^{(1)}$, behind the\nmeteoroid\\label{Long-distance asymptotics}}\n\nWe start by calculating the ion density at the simplest limit of long radial\ndistances, $R\\gg\\lambda_{T}^{(1)}$, behind the meteoroid. Ignoring\nexponentially small densities (as explained below), we will consider only the\nspace behind the meteoroid, $\\cos\\theta>0$. Outgoing particles within the\ndominant beam-like (along $\\vec{R}$) velocity distribution make the major\ncontribution to $n^{\\left(  2\\right)  }$. In equations (\\ref{n^(2)_snova}) and\n(\\ref{yields}), setting $R_{0}\\ll R$, $\\vartheta\\approx\\theta$, $\\mu\n\\approx\\cos\\theta$, while neglecting the exponentially small quantity\n$J_{R}^{\\infty}$ in equation~(\\ref{III}), we obtain $I(R,R_{0})\\approx\n2J_{R_{0}}^{\\infty}$. This allows us to easily integrate the RHS of\nequation~(\\ref{n^(2)_snova}) over $\\Phi$. In the exact integral expression\ngiven by (\\ref{J_a^b_R>3epsilon}), the primary contribution to $J_{R_{0}%\n}^{\\infty}$ arises from components of $f_{+}^{(2)}$ near the meteoroid,\n$R^{\\prime}\\lesssim\\lambda_{T}^{(1)}\\ll R$. This allows us to extend the upper\nintegration limit to infinity. This yields%\n\\begin{align}\n&  \\left.  n^{\\left(  2\\right)  }\\right\\vert _{R\\gg\\lambda_{T}^{(1)}}%\n\\approx\\frac{16\\pi r_{\\mathrm{M}}^{2}G_{\\mathrm{ion}}n_{0}n_{\\mathrm{A}}%\n\\cos\\theta}{R^{2}\\sqrt{3}}\\left(  1+\\frac{m}{m_{\\beta}}\\right) \\nonumber\\\\\n&  \\times\\int_{0}^{\\infty}R_{0}dR_{0}\\int_{R_{0}}^{\\infty}\\left[  1+\\left(\n\\frac{R^{\\prime}}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}\\right]  \\exp\\left[\n-\\ \\frac{3}{2}\\left(  \\frac{R^{\\prime}}{\\lambda_{T}^{(1)}}\\right)\n^{2\/3}\\right]  \\frac{dR^{\\prime}}{R^{\\prime}\\sqrt{\\left(  R^{\\prime}\\right)\n^{2}-R_{0}^{2}}}. \\label{n^(2)_long_start}%\n\\end{align}\nChanging here the order of integration with the corresponding adjustment of\nthe integration limits, $\\int_{0}^{\\infty}dR_{0}\\int_{R_{0}}^{\\infty}\\left(\n\\cdots\\right)  dR^{\\prime}=\\int_{0}^{\\infty}dR^{\\prime}\\int_{0}^{R^{\\prime}%\n}\\left(  \\cdots\\right)  dR_{0}$, and using the simple identities\n\\begin{align*}\n\\int_{0}^{R^{\\prime}}\\frac{R_{0}dR_{0}}{\\sqrt{\\left(  R^{\\prime}\\right)\n^{2}-R_{0}^{2}}}  &  =R^{\\prime},\\\\\n\\int_{0}^{\\infty}\\left[  1+\\left(  \\frac{R^{\\prime}}{\\lambda_{T}^{(1)}%\n}\\right)  ^{2\/3}\\right]  \\exp\\left[  -\\ \\frac{3}{2}\\left(  \\frac{R^{\\prime}%\n}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}\\right]  dR^{\\prime}  &  =\\sqrt{\\frac{2\\pi\n}{3}}\\ \\lambda_{T}^{(1)},\n\\end{align*}\nwe obtain for $\\cos\\theta>0$:\n\\begin{equation}\n\\left.  n^{\\left(  2\\right)  }\\right\\vert _{R\\gg\\lambda_{T}^{(1)}}\\approx\n\\frac{kr_{\\mathrm{M}}^{2}\\lambda_{T}^{(1)}G_{\\mathrm{ion}}n_{0}n_{\\mathrm{A}}%\n}{R^{2}}\\left(  1+\\frac{m}{m_{\\beta}}\\right)  \\cos\\theta, \\label{n_R>>lambda}%\n\\end{equation}\nwhere $k=16\\sqrt{2}\\ \\pi^{\\frac{3}{2}}\/3$. Equation~(\\ref{n_R>>lambda}) shows\nthat at $R\\gg\\lambda_{T}^{(1)}$ the density of the major ion population fall\noff as $(\\cos\\theta)\/R^{2}$.\n\nIf, however, instead of using the exact integral expression for $J_{R_{0}%\n}^{\\infty}$ given by equation (\\ref{J_a^b_R>3epsilon}), we employed its\napproximation given by equation~(\\ref{J_R_0_interpo}) then, applying the\nidentity\n\\begin{equation}\n\\int_{0}^{\\infty}\\sqrt{\\left(  1+\\frac{4x}{3\\pi}\\right)  x}\\ \\exp\\left(\n-\\ x\\right)  dx=\\frac{\\sqrt{3\\pi}}{4}\\exp\\left(  \\frac{3\\pi}{8}\\right)\nK_{1}\\left(  \\frac{3\\pi}{8}\\right)  , \\label{K_1}%\n\\end{equation}\ndeduced from \\citep[][equation~3.372]{Gradshteyn:Table94}, with the modified\nBessel function of the second kind $K_{\\nu}(x)$, we would obtain\nequation~(\\ref{n_R>>lambda}) with $k$ replaced by a formally different\ncoefficient, $k_{1}=(2^{\\frac{3}{2}}\\pi^{\\frac{5}{2}}\/\\sqrt{3})\\exp\\left(\n3\\pi\/8\\right)  K_{1}\\left(  3\\pi\/8\\right)  $. However, the numerical values of\n$k\\approx42$ and $k_{1}\\approx41.74$ are so close to each other that can be\nconsidered as essentially the same. This confirms that the approximate\nexpression $J_{R_{0}}^{\\infty}$ given by equation~(\\ref{J_R_0_interpo}) is\nreasonably accurate and can be successfully used in other occasions, as done below.\n\n\\subsection{General distances \\label{Moderate distances}}\n\nFor all but largest meteors the radar head echo is formed within moderate\nradial distances of $R\\sim\\lambda_{T}^{(1)}$, the most difficult domain to\ntreat analytically. Below, we reduce the general expression for $n^{\\left(\n2\\right)  }$ to a simpler form, more suitable for a further analytic or\nnumerical treatment, and then obtain the explicit spatial distribution of\n$n^{\\left(  2\\right)  }$ for $R\\ll\\lambda_{T}^{(1)}$, the limit opposite to\nthat considered in section~\\ref{Long-distance asymptotics}. After that, we\nwill discuss the general case, using numerical integrations.\n\n\\subsubsection{Reduction of the general ion density \\label{Reduction}}\n\nUnder assumption of the isotropic differential cross-section, $G_{\\mathrm{ion}%\n}(U)$, equation~(\\ref{yields}) involves $\\mu$ only as a linear multiplier. For\nthe further analysis, equation~(\\ref{n^(2)_snova}) with the integration over\n$\\Phi$ is no longer convenient. More advantageous is integrating over $\\mu$,\nwhere $\\mu$ is given by equation~(\\ref{mumumu}). Introducing a dimensionless\nvariable%\n\\begin{equation}\n\\xi_{0}=\\frac{R_{0}}{R}=\\sin\\Theta\\leq1 \\label{xi_0}%\n\\end{equation}\nand changing variables $R_{0},\\Phi$ to $\\xi_{0},\\mu$, we arrive at\n\\begin{subequations}\n\\label{n^(2),I_muha}%\n\\begin{align}\n&  n^{\\left(  2\\right)  }=2\\times\\frac{4n_{0}n_{\\mathrm{A}}}{\\sqrt{3}}\\left(\n1+\\frac{m}{m_{\\beta}}\\right)  G_{\\mathrm{ion}}M\\label{n^(2)_prome_2}\\\\\n&  M\\equiv\\sum_{\\sigma_{R}=\\pm1}\\int_{0}^{1}\\frac{I(R,\\xi_{0}R)\\xi_{0}d\\xi\n_{0}}{\\sqrt{1-\\xi_{0}^{2}}}\\ I_{\\mu}(\\xi_{0}),\\label{M}\\\\\nI_{\\mu}(\\xi_{0})  &  =\\int\\frac{\\mathrm{H}\\left(  \\mu\\right)  \\mu d\\mu}%\n{\\sqrt{\\xi_{0}^{2}\\sin^{2}\\theta-\\left(  \\mu-\\sigma_{R}\\sqrt{1-\\xi_{0}^{2}%\n}\\cos\\theta\\right)  ^{2}}}, \\label{I_muha}%\n\\end{align}\nwhere the function $I(R,R_{0})$ is given by equation~(\\ref{III}) and\n$\\mathrm{H}\\left(  x\\right)  $ is the Heaviside step-function ($\\mathrm{H}%\n\\left(  x\\right)  =1$ for $x\\geq0$ and $\\mathrm{H}\\left(  x\\right)  =0$ for\n$x<0$). The latter takes into account the fact that the distribution function\nof secondary particles is non-zero only for positive $\\mu$, as discussed in\nPaper~1. The integration in $I_{\\mu}(\\xi_{0})$ is performed over the entire\n$\\mu$-range where the expression under the square root is non-negative. The\nfactor `$2$' in front of the RHS of equation~(\\ref{n^(2)_prome_2}) takes into\naccount the fact that each value of $\\mu$ corresponds to two symmetric values\nof $\\Phi$ with the same $\\cos\\Phi$ but opposite $\\sin\\Phi$. Introducing%\n\\end{subequations}\n\\begin{equation}\n\\mu_{1}=\\sigma_{R}\\sqrt{1-\\xi_{0}^{2}}\\cos\\theta-\\xi_{0}\\sin\\theta,\\qquad\n\\mu_{2}=\\sigma_{R}\\sqrt{1-\\xi_{0}^{2}}\\cos\\theta+\\xi_{0}\\sin\\theta,\n\\label{mu_1,2}%\n\\end{equation}\nand eliminating the step-function, we can rewrite $I_{\\mu}(\\xi_{0})$ in\nequation~(\\ref{I_muha}) as%\n\\begin{equation}\nI_{\\mu}(\\xi_{0})=\\int_{\\max(\\mu_{1},0)}^{\\max(\\mu_{2},0)}\\frac{\\mu d\\mu}%\n{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)  \\left(  \\mu-\\mu_{1}\\right)  }},\n\\label{I_mu}%\n\\end{equation}\nwhere the integration limits take into account that in general case $\\mu\n_{1,2}$ can be negative. For all signs of $\\cos\\theta$, the relations between\n$\\mu_{1,2}$ and $0$ are listed in this table:%\n\\begin{equation}%\n\\begin{tabular}\n[c]{|l|l|l|}\\hline\n& $\\xi_{0}<\\left\\vert \\cos\\theta\\right\\vert $ & $\\xi_{0}>\\left\\vert \\cos\n\\theta\\right\\vert $\\\\\\hline\n$\\sigma_{R}\\cos\\theta<0$ & $\\mu_{1}<\\mu_{2}<0$ & $\\mu_{1}<0<\\mu_{2}$\\\\\\hline\n$\\sigma_{R}\\cos\\theta>0$ & $\\mu_{2}>\\mu_{1}>0$ & $\\mu_{1}<0<\\mu_{2}$\\\\\\hline\n\\end{tabular}\n\\ \\ \\ \\ \\ . \\label{Table:mu>>}%\n\\end{equation}\nAll this yields for $I_{\\mu}$, defined by equation~(\\ref{I_muha}), a\npiece-wise expression:%\n\\begin{equation}\nI_{\\mu}(\\xi_{0})=\\left\\{\n\\begin{array}\n[c]{ccc}%\n\\int_{\\mu_{1}}^{\\mu_{2}}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)\n\\left(  \\mu-\\mu_{1}\\right)  }} & \\text{if} & \\xi_{0}<\\left\\vert \\cos\n\\theta\\right\\vert \\text{ and }\\sigma_{R}\\cos\\theta>0,\\\\\n\\int_{0}^{\\mu_{2}}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)  \\left(\n\\mu-\\mu_{1}\\right)  }} & \\text{if} & \\xi_{0}>\\left\\vert \\cos\\theta\\right\\vert\n\\text{ for all }\\sigma_{R}\\cos\\theta,\\\\\n0 & \\text{if} & \\xi_{0}<\\left\\vert \\cos\\theta\\right\\vert \\text{ and }%\n\\sigma_{R}\\cos\\theta<0.\n\\end{array}\n\\right.  . \\label{I_mu_proma}%\n\\end{equation}\nUsing the definitions given by equation~(\\ref{mu_1,2}), we obtain:%\n\\begin{equation}\n\\int_{\\mu_{1}}^{\\mu_{2}}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)\n\\left(  \\mu-\\mu_{1}\\right)  }}=\\pi\\sigma_{R}\\sqrt{1-\\xi_{0}^{2}}\\cos\\theta,\n\\label{Int_mu_full_again}%\n\\end{equation}\nand (for $\\xi_{0}>\\left\\vert \\cos\\theta\\right\\vert $):%\n\\begin{align}\n&  \\int_{0}^{\\mu_{2}}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)  \\left(\n\\mu-\\mu_{1}\\right)  }}\\nonumber\\\\\n&  =\\sigma_{R}\\left(  \\frac{\\pi}{2}+\\arcsin\\frac{\\sigma_{R}\\sqrt{1-\\xi_{0}%\n^{2}}\\cos\\theta}{\\xi_{0}\\sin\\theta}\\right)  \\sqrt{1-\\xi_{0}^{2}}\\cos\n\\theta+\\sqrt{\\xi_{0}^{2}-\\cos^{2}\\theta}. \\label{Int_mu_partial_again}%\n\\end{align}\nRecalling equation~(\\ref{III}), for the quantity $M$, defined by\nequation~(\\ref{M}), we obtain%\n\\begin{align}\nM  &  =\\int_{0}^{1}\\frac{\\left(  2J_{R_{0}}^{\\infty}-J_{R}^{\\infty}\\right)\n_{\\sigma_{R>0}}\\xi_{0}d\\xi_{0}}{\\sqrt{1-\\xi_{0}^{2}}}\\int_{\\max(\\mu_{1}%\n,0)}^{\\max(\\mu_{2},0)}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)\n\\left(  \\mu-\\mu_{1}\\right)  _{\\sigma_{R=+1}}}}\\label{K}\\\\\n&  +\\int_{0}^{1}\\frac{\\left(  J_{R}^{\\infty}\\right)  _{\\sigma_{R<0}}\\xi\n_{0}d\\xi_{0}}{\\sqrt{1-\\xi_{0}^{2}}}\\int_{\\max(\\mu_{1},0)}^{\\max(\\mu_{2}%\n,0)}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)  \\left(  \\mu-\\mu\n_{1}\\right)  _{\\sigma_{R=-1}}}}\\nonumber\n\\end{align}\nRegrouping the terms, and using equations~(\\ref{I_mu_proma}%\n)--(\\ref{Int_mu_partial_again}), for all $\\theta$ we obtain%\n\\begin{align}\n&  M=\\pi\\left\\vert \\cos\\theta\\right\\vert \\int_{0}^{\\left\\vert \\cos\n\\theta\\right\\vert }J_{R_{0}}^{\\infty}\\xi_{0}d\\xi_{0}+\\left(  \\pi\\cos\n\\theta\\right)  \\int_{0}^{1}J_{R_{0}}^{R}\\xi_{0}d\\xi_{0}\\nonumber\\\\\n&  +2\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{1}J_{R_{0}}^{\\infty}\\xi\n_{0}\\sqrt{\\frac{\\xi_{0}^{2}-\\cos^{2}\\theta}{1-\\xi_{0}^{2}}}\\ d\\xi\n_{0}\\label{K_snova}\\\\\n&  +2\\left\\vert \\cos\\theta\\right\\vert \\int_{\\left\\vert \\cos\\theta\\right\\vert\n}^{1}J_{R_{0}}^{\\infty}\\xi_{0}\\arcsin\\frac{\\sqrt{1-\\xi_{0}^{2}}\\left\\vert\n\\cos\\theta\\right\\vert }{\\xi_{0}\\sin\\theta}\\ d\\xi_{0}\\ ,\\nonumber\n\\end{align}\nwhere $J_{R_{0}}^{R}=J_{R_{0}}^{\\infty}-J_{R}^{\\infty}$. All terms in the RHS\nof equation~(\\ref{K_snova}) are symmetric with respect to the sign of\n$(\\cos\\theta)$, except the second term which is antisymmetric. This term is\nresponsible for the entire asymmetry between the locations in front of the\nmeteoroid ($\\cos\\theta<0$) and behind it ($\\cos\\theta>0$).\n\nTo simplify further, we introduce other variables and parameters,%\n\\begin{equation}\n\\eta=\\frac{R^{\\prime}}{R},\\qquad\\beta=\\left(  \\frac{R}{\\lambda_{T}^{(1)}%\n}\\right)  ^{\\frac{2}{3}},\\qquad q=\\frac{r_{\\mathrm{M}}^{2}}{R},\n\\label{beta_snova}%\n\\end{equation}\nwhere $R^{\\prime}$ is the integration variable in $J_{a}^{b}$, defined by\nequation~(\\ref{J_a^b_R>3epsilon}). With use of these dimensionless quantities,\nequation~(\\ref{J_a^b_R>3epsilon}) yields\n\\begin{subequations}\n\\label{J_R_0^infty,1}%\n\\begin{align}\nJ_{R_{0}}^{\\infty}  &  =q\\int_{\\xi_{0}}^{\\infty}\\left(  1+\\beta\\eta^{\\frac\n{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)\n\\frac{d\\eta}{\\eta\\sqrt{\\eta^{2}-\\xi_{0}^{2}}},\\label{J_R_0^infty}\\\\\nJ_{R_{0}}^{R}  &  =q\\int_{\\xi_{0}}^{1}\\left(  1+\\beta\\eta^{\\frac{2}{3}%\n}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)\n\\frac{d\\eta}{\\eta\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}. \\label{J_R_0^1}%\n\\end{align}\nFor some calculations, we will also need approximate\nequation~(\\ref{J_R_0_interpo}) for $J_{R_{0}}^{\\infty}$,%\n\\end{subequations}\n\\begin{equation}\nJ_{R_{0}}^{\\infty}\\approx\\frac{\\pi q}{2\\xi_{0}}\\sqrt{1+\\frac{2}{\\pi}\\ \\beta\n\\xi_{0}^{\\frac{2}{3}}}\\exp\\left(  -\\ \\frac{3}{2}\\ \\beta\\xi_{0}^{\\frac{2}{3}%\n}\\right)  . \\label{J_R_)_approx_dimensionless}%\n\\end{equation}\nFor $J_{R_{0}}^{R}$ we need no approximations, as will become clear soon.\n\nBefore proceeding, we check that the long-distance limit of $R\\gg\\lambda\n_{T}^{(1)}$ (i.e., $\\beta\\gg1$) for the above equations provides a smooth\ntransition to the range of long distances considered in\nsection~\\ref{Long-distance asymptotics}. Beyond a narrow vicinity around\n$\\theta=\\pi\/2$, given by $\\left\\vert \\cos\\theta\\right\\vert \\lesssim\n\\beta^{-3\/2}$, we can easily see that in the RHS of equation~(\\ref{K_snova})\nthe third and fourth terms are exponentially small. Neglecting them and\nextending the same accuracy to the upper integration limit in the first and\nsecond terms, $J_{R_{0}}^{R}\\approx J_{R_{0}}^{\\infty}$, we obtain\n\\begin{equation}\n\\left.  M\\right\\vert _{\\beta\\gg1}\\approx\\left\\{\n\\begin{array}\n[c]{ccc}%\n2\\pi\\left(  \\cos\\theta\\right)  \\int_{0}^{\\infty}J_{R_{0}}^{\\infty}\\xi_{0}%\nd\\xi_{0} & \\text{if} & \\cos\\theta>0,\\\\\n&  & \\\\\n0 & \\text{if} & \\cos\\theta<0.\n\\end{array}\n\\right.  \\label{K_beta>>1}%\n\\end{equation}\nUsing for $J_{R_{0}}^{\\infty}$ the exact equation~(\\ref{J_R_0^infty}) and\napplying for the double integration the same approach as in\nsection~\\ref{Long-distance asymptotics}, we obtain\n\\[\n\\left.  M\\right\\vert _{\\beta\\gg1}\\approx\\frac{2\\pi q\\cos\\theta}{\\beta\n^{\\frac{3}{2}}}\\sqrt{\\frac{2\\pi}{3}}.\n\\]\nReturning from the temporary dimensionless parameters $\\beta,q$ to the\noriginal coordinate $R$ and inserting the corresponding $M$ to\nequation~(\\ref{n^(2)_prome_2}), for $\\cos\\theta>0$ we recreate\nequation~(\\ref{n_R>>lambda}).\n\n\\subsubsection{Short distances, $R\\ll\\lambda_{T}^{(1)}$%\n\\label{Very short distances}}\n\nNow we consider the short-distance limit of $R\\ll\\lambda_{T}^{(1)}$ ($\\beta\n\\ll1$) which is opposite to that discussed in\nsection~\\ref{Long-distance asymptotics}. For not too large integration\nvariables $R^{\\prime}$, $\\eta=R^{\\prime}\/R\\ll\\beta^{-3\/2}$, all factors with\n$\\beta\\eta^{\\frac{2}{3}}$ in equation~(\\ref{J_R_0^infty,1}) can be neglected.\nSince this range of $R^{\\prime}$ makes the dominant contribution to all\nintegrals, we extend this approximation to the entire range of $\\eta$, so\nthat\n\\begin{subequations}\n\\label{J_R,R_0_reduced}%\n\\begin{align}\nJ_{R_{0}}^{\\infty} &  \\approx q\\int_{\\xi_{0}}^{\\infty}\\frac{d\\eta}{\\eta\n\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}=\\frac{\\pi q}{2\\xi_{0}},\\label{J_R_0_reduced}\\\\\nJ_{R_{0}}^{R} &  \\approx q\\int_{\\xi_{0}}^{1}\\frac{d\\eta}{\\eta\\sqrt{\\eta\n^{2}-\\xi_{0}^{2}}}=\\frac{q}{\\xi_{0}}\\arccos\\xi_{0}.\\label{J_R_reduced}%\n\\end{align}\nIn this limit, the first two terms in the RHS\\ of equation~(\\ref{K_snova}) can\nbe easily integrated, yielding $\\pi q[\\pi(\\cos^{2}\\theta)\/2+\\cos\\theta]$. The\ntwo remaining terms can be expressed in terms of the complete elliptic\nintegrals of the 1st and 2nd kind,%\n\\end{subequations}\n\\begin{equation}\n\\mathrm{K}\\left(  k\\right)  =\\int_{0}^{1}\\frac{dt}{\\sqrt{\\left(\n1-t^{2}\\right)  \\left(  1-k^{2}t^{2}\\right)  }},\\qquad\\mathrm{E}\\left(\nk\\right)  =\\int_{0}^{1}\\sqrt{\\frac{1-k^{2}t^{2}}{1-t^{2}}}%\n\\ dt,\\label{F,E_complete}%\n\\end{equation}\nrespectively, where $0\\leq k<1$. Indeed, for constant $\\xi_{0}J_{R_{0}%\n}^{\\infty}$, as in equation~(\\ref{J_R_0_reduced}), the third term in\nequation~(\\ref{K_snova}) becomes proportional to%\n\\[\nI_{1}=\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{1}\\sqrt{\\frac{\\xi_{0}^{2}%\n-\\cos^{2}\\theta}{1-\\xi_{0}^{2}}}\\ d\\xi_{0}.\n\\]\nThis integral already resembles an elliptic integral, but reducing $I_{1}$ to\nthose with the real arguments requires additional efforts. Substituting\n$\\xi_{0}=\\sqrt{1-z^{2}\\sin^{2}\\theta}$, we reduce $I_{1}$ to $\\mathrm{E}%\n\\left(  \\sin\\theta\\right)  -\\left(  \\cos^{2}\\theta\\right)  \\mathrm{K}\\left(\n\\sin\\theta\\right)  $. In a similar way, we can also calculate the fourth term\nin equation~(\\ref{K_snova}). With constant $J_{R_{0}}^{\\infty}\\xi_{0}$, this\nterm becomes proportional to%\n\\[\nI_{2}=\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{1}\\arcsin\\frac{\\sqrt{1-\\xi\n_{0}^{2}}\\left\\vert \\cos\\theta\\right\\vert }{\\xi_{0}\\sin\\theta}\\ d\\xi_{0}.\n\\]\nIntegration of the corresponding indefinite integral by parts gives%\n\\begin{align*}\n&  \\int\\arcsin\\frac{\\sqrt{1-\\xi_{0}^{2}}\\left\\vert \\cos\\theta\\right\\vert }%\n{\\xi_{0}\\sin\\theta}\\ d\\xi_{0}\\\\\n&  =\\xi_{0}\\arcsin\\frac{\\sqrt{1-\\xi_{0}^{2}}\\left\\vert \\cos\\theta\\right\\vert\n}{\\xi_{0}\\sin\\theta}+\\left\\vert \\cos\\theta\\right\\vert \\int\\frac{d\\xi_{0}%\n}{\\sqrt{\\left(  1-\\xi_{0}^{2}\\right)  \\left(  \\xi_{0}^{2}-\\cos^{2}%\n\\theta\\right)  }}.\n\\end{align*}\nMaking\\ the same substitution for the remaining integral in the RHS as done\nfor $I_{1}$ and evaluating everything over the proper integration limits, we\nobtain $I_{2}=(\\mathrm{K}\\left(  \\sin\\theta\\right)  -\\pi\/2)|\\cos\\theta|$. When\nadding all terms in the RHS of equation~(\\ref{K_snova}), the $\\mathrm{K}%\n\\left(  \\sin\\theta\\right)  $-terms in $I_{1,2}$ cancel and equation~(\\ref{M})\nreduces to a simple expression: $M\\approx\\pi q[\\cos\\theta+\\mathrm{E}\\left(\n\\sin\\theta\\right)  ]$. As a result, the ion density in the short-distance\nlimit reduces to%\n\\begin{equation}\n\\left.  n^{\\left(  2\\right)  }\\right\\vert _{R\\ll\\lambda_{T}^{(1)}}\\approx\n\\frac{8\\pi r_{\\mathrm{M}}^{2}G_{\\mathrm{ion}}n_{0}n_{\\mathrm{A}}}{\\sqrt{3}%\n\\,R}\\left(  1+\\frac{m}{m_{\\beta}}\\right)  \\left[  \\cos\\theta+\\mathrm{E}\\left(\n\\sin\\theta\\right)  \\right]  .\\label{n_R<<lambda}%\n\\end{equation}\n\n\nComparing equation~(\\ref{n_R<<lambda}) with the opposite limit given by\nequation~(\\ref{n_R>>lambda}) shows that the $1\/R^{2}$-dependency of\n$n^{\\left(  2\\right)  }|_{R\\gg\\lambda_{T}^{(1)}}$ transforms to the\n$1\/R$-dependency for $n^{\\left(  2\\right)  }|_{R\\ll\\lambda_{T}^{(1)}}$. The\nangular $\\theta$-dependency also changes significantly. While in the\nlong-distance limit of $R\\gg\\lambda_{T}^{(1)}$ ions occupy almost entirely the\nhalf-space behind the meteoroid ($0\\leq\\theta<\\pi\/2$), in the short-distance\nlimit of $R\\ll\\lambda_{T}^{(1)}$ ions show a noticeable presence in front of\nthe meteoroid ($\\pi\/2\\leq\\theta\\leq\\pi$) as well. Red dashed curves in\nFigure~\\ref{Fig:M_versus_theta} show the corresponding angular dependencies\nnormalized to their maximum values at $\\theta=0$.\n\n\\subsubsection{Arbitrary distances \\label{General case}}\n\nThe case of moderate distances $R\\sim\\lambda_{T}^{(1)}$ is covered by general\nequations (\\ref{n^(2),I_muha}) and (\\ref{K_snova}) with $J_{R_{0}}^{\\infty}$\nand $J_{R_{0}}^{R}$ expressed in the original integral form by\nequation~(\\ref{J_R_0^infty,1}), or in an approximate, but explicit, form for\n$J_{R_{0}}^{\\infty}$ by equation~(\\ref{J_R_)_approx_dimensionless}). Unlike\n$J_{R_{0}}^{\\infty}$, the integral $J_{R_{0}}^{R}$ is involved only in the\nsecond term in the RHS of equation~(\\ref{K_snova}). As we show below, this\nterm can be calculated exactly by using the integral form of\nequation~(\\ref{J_R_0^1}). Below we obtain the explicit analytic expressions\nfor the first and second terms in the RHS of equation~(\\ref{K_snova}). Being\nunable to obtain a general analytic approximation for the two last integral\nterms, we will integrate them numerically.\n\n\\paragraph{First term in\\ equation~(\\ref{K_snova}).}\n\nUsing~equation~(\\ref{J_R_0^infty}), for the integral in the first term of the\nexpression for $M$ in~(\\ref{K_snova}), we have%\n\\begin{align}\n&  Q_{1}\\equiv\\frac{1}{q}\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert }J_{R_{0}%\n}^{\\infty}\\xi_{0}d\\xi_{0}\\nonumber\\\\\n&  =\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert }\\left[  \\int_{\\xi_{0}}%\n^{\\infty}\\left(  1+\\beta\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac\n{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)  \\frac{d\\eta}{\\eta}\\right]  \\frac{\\xi\n_{0}d\\xi_{0}}{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}, \\label{Q_1}%\n\\end{align}\nwhere the dimensionless variables $\\eta$, $\\beta$, and $q$ are defined by\nequation~(\\ref{beta_snova}). Changing the order of integration, we obtain%\n\\begin{align*}\n&  Q_{1}=\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert }\\left(  1+\\beta\n\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}%\n{2}\\right)  \\frac{d\\eta}{\\eta}\\ \\int_{0}^{\\eta}\\frac{\\xi_{0}d\\xi_{0}}%\n{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}\\\\\n&  +\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{\\infty}\\left(  1+\\beta\n\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}%\n{2}\\right)  \\frac{d\\eta}{\\eta}\\ \\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert\n}\\frac{\\xi_{0}d\\xi_{0}}{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}.\n\\end{align*}\nThese integrations yield%\n\\begin{align}\n&  Q_{1}=\\sqrt{\\frac{2\\pi}{3\\beta^{3}}}\\operatorname{erf}\\left(  \\sqrt\n{\\frac{3\\beta}{2}}\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{1}{3}}\\right)\n+J_{1}\\nonumber\\\\\n&  -\\left(  \\left\\vert \\cos\\theta\\right\\vert ^{\\frac{2}{3}}+\\frac{2}{\\beta\n}\\right)  \\left\\vert \\cos\\theta\\right\\vert ^{\\frac{1}{3}}\\exp\\left(\n-\\ \\frac{3\\beta\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{2}{3}}}{2}\\right)  ,\n\\label{First_term}%\n\\end{align}\nwhere $\\operatorname{erf}(x)=(2\/\\sqrt{\\pi})\\int_{0}^{x}e^{-x^{2}}dx$ is the\nstandard error-function and%\n\\begin{equation}\nJ_{1}=\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{\\infty}\\left(  1+\\beta\n\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}%\n{2}\\right)  \\left(  1-\\sqrt{1-\\frac{\\left\\vert \\cos\\theta\\right\\vert ^{2}%\n}{\\eta^{2}}}\\right)  d\\eta. \\label{J_1_snova}%\n\\end{equation}\nThe integral $J_{1}$ cannot be taken exactly, but directly below we obtain its\napproximate value. However, even without doing this, one can easily verify\nthat equations~(\\ref{First_term}) and (\\ref{J_1_snova}) provide both the\ncorrect limit of short distances, $\\lim_{\\beta\\rightarrow0}Q_{1}=\\pi\\left\\vert\n\\cos\\theta\\right\\vert \/2$, and the large-distance asymptotics of $\\beta\\gg1$,\n$Q_{1}\\approx(2\\pi\/3)^{1\/2}\\beta^{-3\/2}$.\n\nNow we find an approximate expression for $J_{1}$ by constructing a proper\nanalytic interpolation between two limiting cases. For small $\\beta$, we have%\n\\begin{equation}\n\\left.  J_{1}\\right\\vert _{\\beta\\rightarrow0}=\\left(  \\frac{\\pi}{2}-1\\right)\n\\left\\vert \\cos\\theta\\right\\vert . \\label{J_1_beta->0}%\n\\end{equation}\nIn the opposite limit of large $\\beta$, the major contribution to the integral\n$J_{1}$ is made in the small vicinity of the lower integration limit. This\nyields the following asymptotics,%\n\\begin{equation}\nJ_{1}\\approx\\left(  1-\\sqrt{\\frac{\\pi}{2\\beta\\left\\vert \\cos\\theta\\right\\vert\n^{\\frac{2}{3}}}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\left\\vert \\cos\n\\theta\\right\\vert ^{\\frac{2}{3}}}{2}\\right)  \\left\\vert \\cos\\theta\\right\\vert\n. \\label{J_1_beta>>1}%\n\\end{equation}\nInterpolating between equations ~(\\ref{J_1_beta->0}) and (\\ref{J_1_beta>>1})\nas%\n\\begin{equation}\nJ_{1}\\approx\\left(  1-\\frac{\\left(  4-\\pi\\right)  \\sqrt{2\\pi}}{2\\sqrt\n{2\\pi+\\left(  4-\\pi\\right)  ^{2}\\beta\\left\\vert \\cos\\theta\\right\\vert\n^{\\frac{2}{3}}}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\left\\vert \\cos\n\\theta\\right\\vert ^{\\frac{2}{3}}}{2}\\right)  \\left\\vert \\cos\\theta\\right\\vert\n, \\label{J_1_approxy}%\n\\end{equation}\nwe obtain a reasonably good approximation for $J_{1}$, valid in the entire\nrange of $\\beta$. Even in the worst case of $\\beta\\left\\vert \\cos\n\\theta\\right\\vert ^{\\frac{2}{3}}\\sim6$, the mismatch between the actual\nintegral value and this approximation is only $\\simeq6\\%$.\n\nCombining equations~(\\ref{First_term}) with (\\ref{J_1_approxy}), for the\ndouble integral $Q_{1}$ defined by equation~(\\ref{Q_1}), to a good accuracy we\nobtain%\n\\begin{align}\n&  Q_{1}\\approx\\sqrt{\\frac{2\\pi}{3\\beta^{3}}}\\ \\operatorname{erf}\\left(\n\\sqrt{\\frac{3\\beta}{2}}\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{1}{3}}\\right)\n\\nonumber\\\\\n&  -\\left[  \\frac{\\left(  4-\\pi\\right)  \\sqrt{2\\pi}\\left\\vert \\cos\n\\theta\\right\\vert }{2\\sqrt{2\\pi+\\beta\\left(  4-\\pi\\right)  ^{2}\\left\\vert\n\\cos\\theta\\right\\vert ^{\\frac{2}{3}}}}+\\frac{2\\left\\vert \\cos\\theta\\right\\vert\n^{\\frac{1}{3}}}{\\beta}\\right]  \\exp\\left(  -\\ \\frac{3\\beta\\left\\vert\n\\cos\\theta\\right\\vert ^{\\frac{2}{3}}}{2}\\right)  . \\label{First_term_final}%\n\\end{align}\n\n\n\\paragraph{Second term in\\ equation~(\\ref{K_snova}).}\n\nNow we calculate the integral%\n\\begin{align}\n&  Q_{2}\\equiv\\frac{1}{q}\\int_{0}^{1}J_{R_{0}}^{R}\\xi_{0}d\\xi_{0}\\nonumber\\\\\n&  =\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert }\\left[  \\int_{\\xi_{0}}%\n^{1}\\left(  1+\\beta\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\n\\eta^{\\frac{2}{3}}}{2}\\right)  \\frac{d\\eta}{\\eta}\\right]  \\frac{\\xi_{0}%\nd\\xi_{0}}{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}. \\label{Q_2}%\n\\end{align}\nUnlike $Q_{1}$, this integral can be calculated exactly. Indeed, changing the\norder of integration, we obtain%\n\\begin{align}\n&  Q_{2}=\\int_{0}^{1}\\left(  1+\\beta\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(\n-\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)  \\frac{d\\eta}{\\eta}\\ \\int%\n_{0}^{\\eta}\\frac{\\xi_{0}d\\xi_{0}}{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}\\nonumber\\\\\n&  =\\int_{0}^{1}\\left(  1+\\beta\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(\n-\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)  d\\eta\\nonumber\\\\\n&  =\\sqrt{\\frac{2\\pi}{3\\beta^{3}}}\\ \\operatorname{erf}\\left(  \\sqrt\n{\\frac{3\\beta}{2}}\\right)  -\\left(  1+\\frac{2}{\\beta}\\right)  \\exp\\left(\n-\\ \\frac{3\\beta}{2}\\right)  . \\label{Second_term_final}%\n\\end{align}\n\n\n\\paragraph{Density along the major axis.}\n\nNow we consider two particular positions along the major axis: strictly behind\nthe meteoroid ($\\theta=0$) and strictly in front of it ($\\theta=\\pi$). In both\nthese positions, we have $\\left\\vert \\cos0\\right\\vert =1$, so that the third\nand fourth terms in the RHS\\ of equation~(\\ref{K_snova}) become zero. The\ncombination of the two first terms there is given by%\n\\begin{equation}\n\\pi\\left\\vert \\cos\\theta\\right\\vert \\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert\n}J_{R_{0}}^{\\infty}\\xi_{0}d\\xi_{0}+\\pi\\left(  \\cos\\theta\\right)  \\int_{0}%\n^{1}J_{R_{0}}^{R}\\xi_{0}d\\xi_{0}=\\pi q\\left(  \\left\\vert \\cos\\theta\\right\\vert\nQ_{1}+\\left(  \\cos\\theta\\right)  Q_{2}\\right)  \\label{combination}%\n\\end{equation}\nAs a result, at the major axis behind the meteoroid we obtain\n\\begin{align}\n&  \\left.  n^{\\left(  2\\right)  }\\right\\vert _{\\theta=0}=\\frac{8\\pi\nr_{\\mathrm{M}}^{2}n_{0}n_{\\mathrm{A}}}{R\\sqrt{3}}\\left(  1+\\frac{m}{m_{\\beta}%\n}\\right)  G_{\\mathrm{ion}}\\left\\{  2\\sqrt{\\frac{2\\pi}{3}}\\frac{\\lambda\n_{T}^{(1)}}{R}\\operatorname{erf}\\left[  \\sqrt{\\frac{3}{2}}\\left(  \\frac\n{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}\\right]  \\right. \\nonumber\\\\\n&  \\left.  -\\left[  \\frac{\\left(  4-\\pi\\right)  \\sqrt{2\\pi}}{2\\sqrt\n{2\\pi+\\left(  4-\\pi\\right)  ^{2}(R\/\\lambda_{T}^{(1)})^{\\frac{2}{3}}}%\n}+1+4\\left(  \\frac{\\lambda_{T}^{(1)}}{R}\\right)  ^{\\frac{2}{3}}\\right]\n\\exp\\left[  -\\ \\frac{3}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)\n^{\\frac{2}{3}}\\right]  \\right\\}  . \\label{n^(2)_theta=0}%\n\\end{align}\nSimilarly, at the major axis in front of of the meteoroid we obtain\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=30pc]{M_versus_theta.pdf}\n\\caption{Density versus the polar angle $\\theta$ for different distances\n(black solid curves from the top to the bottom: $R\/\\lambda_{T}^{(1)}%\n=0.1;0.3;1;3;10$). Red dashed curves show asymptotic solutions given by\nequation~(\\ref{K_beta>>1}) (the top curve) and by equation~(\\ref{n_R<<lambda}%\n). All density distributions are normalized to the maximum values strictly\nbehind the meteoroid, $\\theta=0$ (see\nFig.~\\ref{Fig:Density_vs_Radius_logarithm}).}\n\\label{Fig:M_versus_theta}%\n\\end{figure}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=30pc]{Density_vs_Radius_logarithm.pdf}\n\\caption{Normalized density versus the relative radial distance,\n$R\/\\lambda_{T}^{(1)}$, for several equally separated angles $\\theta$. The\nnormalization corresponds to the factor $M$ with $q\\rightarrow\\lambda\n_{T}^{(1)}\/R$. Red dashed curves for $\\theta=0$ (the top curve) and for\n$\\theta=\\pi$ (the bottom curve) are described analytically by\nequations~(\\ref{n^(2)_theta=0}) and (\\ref{n^(2)_theta=pi}), respectively.\nBlack solid curves show $M(R,\\theta)$ obtained by combining analytic\nequations~(\\ref{Q_1}) and (\\ref{Q_2}) for the two first terms of\nequation~(\\ref{K_snova}) with numerically calculated two remaining integral\nterms. The black solid curves, from the top to the bottom, correspond to\n$\\theta=\\pi\/8$; $\\pi\/4$; $3\\pi\/8$; $\\pi\/2$; $5\\pi\/8$; $3\\pi\/4;$ and $7\\pi\/8$,\nrespectively. The black solid curves with $\\theta=\\pi\/8$ and $\\theta=7\\pi\/8$\nare almost indistinguishable from the top and bottom red dashed curves,\n$\\theta=0,\\pi$.}%\n\\label{Fig:Density_vs_Radius_logarithm}\n\\end{figure}\n\\begin{align}\n&  \\left.  n^{\\left(  2\\right)  }\\right\\vert _{\\theta=\\pi}=\\frac{8\\pi\nr_{\\mathrm{M}}^{2}n_{0}n_{\\mathrm{A}}}{R\\sqrt{3}}\\left(  1+\\frac{m}{m_{\\beta}%\n}\\right)  G_{\\mathrm{ion}}\\nonumber\\\\\n&  \\times\\left[  1-\\frac{\\left(  4-\\pi\\right)  \\sqrt{2\\pi}}{2\\sqrt\n{2\\pi+\\left(  4-\\pi\\right)  ^{2}(R\/\\lambda_{T}^{(1)})^{\\frac{2}{3}}}}\\right]\n\\exp\\left[  -\\ \\frac{3}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)\n^{\\frac{2}{3}}\\right]  . \\label{n^(2)_theta=pi}%\n\\end{align}\n\n\nFigure~\\ref{Fig:Density_vs_Radius_logarithm} shows these two radial\ndependencies with $\\cos\\theta=1$ ($\\theta=0$) and $\\cos\\theta=-1$ ($\\theta\n=\\pi$) are actually representative for slightly more general $\\cos\\theta$. The\nradial-distance dependencies for positive and negative $\\cos\\theta$ look\nqualitatively different, as we discuss below.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=30pc]\n{Density_Contours_3D.pdf}\n\\caption{Spatial distribution of plasma density around a small meteoroid. The\nfigure shows a cross-section of an axially symmetric distribution by a plane\nthat includes the symmetry axis (blue dashed line). The meteoroid is shown by\na small white circle at the symmetry axis. The color coding signifies the\nplasma density in relative units; $r$ is the distance to the symmetry axis.}\n\\label{Fig:3Ddensity}\n\\end{figure}\n\nBehind the meteoroid, $\\cos\\theta>0$, the radial dependence of the density\ngradually changes from $n^{\\left(  2\\right)  }\\varpropto1\/R$ for $R\\ll\n\\lambda_{T}^{(1)}$, as described by equation~(\\ref{n_R<<lambda}), to\n$n^{\\left(  2\\right)  }\\varpropto1\/R^{2}$ for $R\\gg\\lambda_{T}^{(1)}$, as\ndescribed by equation~(\\ref{n_R>>lambda}). This change in the power-law radial\ndependence of $n^{\\left(  2\\right)  }$ occurs for the following reason. The\nsource for the secondary ions are the primary neutral particles, whose density\nfalls off near the meteoroid roughly as $1\/R^{2}$. At a given location $R$,\nthe number of ions moving in a certain direction is determined by the total\ncollisional ionization over the preceding segment of the straight-line\ntrajectory aligned with that direction. For $R\\ll\\lambda_{T}^{(1)}$, the total\nintegration over the entire ionization path acquires an additional factor\n$\\varpropto R$ which gradually transforms $1\/R^{2}$ to $1\/R$. On the other\nhand, for $R\\gg\\lambda_{T}^{(1)}$ only localized ionization within $R^{\\prime\n}\\lesssim\\lambda_{T}^{(1)}$ plays a role, resulting in `saturation' of the\nprevious additional factor $R$ at a constant value $\\sim\\lambda_{T}^{(1)}$.\nThis leaves the $\\varpropto1\/R^{2}$ dependence of $n^{\\left(  2\\right)  }$\nessentially untouched. This transition works only for locations behind the\nmeteoroid, $\\cos\\theta>0$, because almost all freshly born ions have\nvelocities $\\vec{V}$ with positive $\\mu=\\cos\\vartheta$, where $\\vartheta$ is\nthe angle between $\\vec{V}$ and $-\\vec{U}$, as shown in\nFigure~\\ref{Fig:Cartoon_reproduced}. Regardless of how far away from the\nmeteoroid this $R,\\theta$-point is located, all preceding straight-line\ntrajectory segments with $\\mu>0$ always cross the near-meteoroid volume\n$R^{\\prime}\\lesssim\\lambda_{T}^{(1)}$,\n\nA significantly different situation takes place in front of the meteoroid,\n$\\cos\\theta<0$. At $R\\ll\\lambda_{T}^{(1)}$, straight-line trajectory segments\nwith $\\mu>0$ also cross a part of the near-meteoroid volume $R^{\\prime\n}\\lesssim\\lambda_{T}^{(1)}$. That is why here $\\left.  n^{\\left(  2\\right)\n}\\right\\vert _{\\theta=\\pi}$ is quite noticeable, although a few times smaller\nthan $\\left.  n^{\\left(  2\\right)  }\\right\\vert _{\\theta=0}$. On the other\nhand, at larger distances, $R\\gtrsim\\lambda_{T}^{(1)}$, there are almost no\npreceding trajectory segments with positive $\\mu$ that would cross the near\nregion of $R^{\\prime}\\lesssim\\lambda_{T}^{(1)}$. These trajectories cross the\nregions where the number of the primary particles is itself exponentially\nsmall, so that $\\left.  n^{\\left(  2\\right)  }\\right\\vert _{\\theta=\\pi\n}\\varpropto\\exp[-\\ (3\/2)(R\/\\lambda_{T}^{(1)})^{2\/3}]$. This exponentially\ndecreasing density remains much less than that behind the meteoroid where\n$n^{\\left(  2\\right)  }$ decreases largely by a power law.\n\n\\paragraph{General case.\\label{General case_again}}\n\nFor the general case of $R\\sim\\lambda_{T}^{(1)}$ with $\\cos\\theta\\neq\\pm1$, we\nwere unable to find acceptable analytic approximations for the two last\n(integral) terms in the RHS\\ of equation~(\\ref{K_snova}). Therefore, we will\nintegrate those 1D integrals numerically.\n\nNow we summarize the entire expression for $n^{\\left(  2\\right)  }$ by\ncombining equation~(\\ref{n^(2)_prome_2}) with (\\ref{K_snova}), where in the\nfirst two terms the integrals $\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert\n}J_{R_{0}}^{\\infty}\\xi_{0}d\\xi_{0}=qQ_{1}$ and $\\int_{0}^{1}J_{R_{0}}^{R}%\n\\xi_{0}d\\xi_{0}=qQ_{2}$ were calculated with $Q_{1,2}$ explicitly given by\nequations~(\\ref{First_term_final}) and (\\ref{Second_term_final}) (as it was\ndone above when calculating the ion density at the major axis). For the two\nremaining integral terms in equation~(\\ref{K_snova}) we use the approximation\nfor $J_{R_{0}}^{\\infty}$ given by equation~(\\ref{J_R_)_approx_dimensionless}).\nThis gives%\n\\begin{align}\nn^{\\left(  2\\right)  }  &  =\\frac{8\\pi r_{\\mathrm{M}}^{2}n_{0}n_{\\mathrm{A}}%\n}{\\sqrt{3}R}\\left(  1+\\frac{m}{m_{\\beta}}\\right)  G_{\\mathrm{ion}%\n}(U)\\nonumber\\\\\n&  \\times\\left[  f_{1}\\left(  R,\\cos\\theta\\right)  \\left\\vert \\cos\n\\theta\\right\\vert +f_{2}\\left(  R\\right)  \\cos\\theta+f_{3}\\left(  R,\\cos\n\\theta\\right)  \\right]  , \\label{n^(2)_general}%\n\\end{align}\nwhere%\n\\begin{align}\n&  f_{1}\\left(  R,\\cos\\theta\\right)  =\\frac{\\lambda_{T}^{(1)}}{R}\\sqrt\n{\\frac{2\\pi}{3}}\\operatorname{erf}\\left[  \\sqrt{\\frac{3}{2}}\\left(  \\frac\n{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{1}{3}}\\left\\vert \\cos\\theta\\right\\vert\n^{\\frac{1}{3}}\\right] \\nonumber\\\\\n&  -\\left[  \\frac{\\left(  4-\\pi\\right)  \\left\\vert \\cos\\theta\\right\\vert\n}{2\\sqrt{1+\\left.  \\left(  4-\\pi\\right)  ^{2}(R\/\\lambda_{T}^{(1)})^{\\frac\n{2}{3}}\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{2}{3}}\\right\/  (2\\pi)}%\n}+2\\left(  \\frac{\\lambda_{T}^{(1)}}{R}\\right)  ^{\\frac{2}{3}}\\left\\vert\n\\cos\\theta\\right\\vert ^{\\frac{1}{3}}\\right] \\nonumber\\\\\n&  \\times\\exp\\left[  -\\ \\frac{3\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{2}{3}%\n}}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}\\right]  ,\n\\label{f_1}%\n\\end{align}\n\n\\begin{align}\nf_{2}\\left(  R\\right)   &  =\\frac{\\lambda_{T}^{(1)}}{R}\\sqrt{\\frac{2\\pi}{3}%\n}\\operatorname{erf}\\left[  \\sqrt{\\frac{3}{2}}\\left(  \\frac{R}{\\lambda\n_{T}^{(1)}}\\right)  ^{\\frac{1}{3}}\\right] \\nonumber\\\\\n&  -\\left[  1+2\\left(  \\frac{\\lambda_{T}^{(1)}}{R}\\right)  ^{\\frac{2}{3}%\n}\\right]  \\exp\\left[  -\\ \\frac{3}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}%\n}\\right)  ^{\\frac{2}{3}}\\right]  , \\label{f_2}%\n\\end{align}\n\n\\begin{align}\nf_{3}(R,\\cos\\theta)  &  =\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{1}%\n\\sqrt{1+\\frac{2\\xi_{0}^{\\frac{2}{3}}}{\\pi}\\left(  \\frac{R}{\\lambda_{T}^{(1)}%\n}\\right)  ^{\\frac{2}{3}}}\\exp\\left[  -\\ \\frac{3\\xi_{0}^{\\frac{2}{3}}}%\n{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}\\right]\n\\nonumber\\\\\n&  \\times\\sqrt{\\frac{\\xi_{0}^{2}-\\cos^{2}\\theta}{1-\\xi_{0}^{2}}}\\ d\\xi\n_{0}\\nonumber\\\\\n&  +\\left\\vert \\cos\\theta\\right\\vert \\int_{\\left\\vert \\cos\\theta\\right\\vert\n}^{1}\\sqrt{1+\\frac{2\\xi_{0}^{\\frac{2}{3}}}{\\pi}\\left(  \\frac{R}{\\lambda\n_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}}\\exp\\left[  -\\ \\frac{3\\xi_{0}^{\\frac{2}{3}%\n}}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}\\right]\n\\nonumber\\\\\n&  \\times\\arcsin\\frac{\\sqrt{1-\\xi_{0}^{2}}\\left\\vert \\cos\\theta\\right\\vert\n}{\\xi_{0}\\sqrt{1-\\cos^{2}\\theta}}\\ d\\xi_{0}. \\label{f_3}%\n\\end{align}\nFor the isotropic differential cross-section the mean free path defined\nequation by equation~(\\ref{lambda_T^(1)}) reduces to\n\\begin{equation}\n\\lambda_{T}^{(1)}=\\frac{1}{4\\pi Un_{\\mathrm{A}}G(U)}\\left(  \\frac\n{T_{\\mathrm{M}}}{m_{\\mathrm{M}}}\\right)  ^{1\/2}. \\label{lambda_new}%\n\\end{equation}\n\n\nFigures~\\ref{Fig:M_versus_theta}, \\ref{Fig:Density_vs_Radius_logarithm}, and\n\\ref{Fig:3Ddensity}\\ illustrate the general $\\theta,R$-dependences of\n$n^{\\left(  2\\right)  }$.\n\nAs might be expected, in Figure~\\ref{Fig:M_versus_theta} the normalized curves\nwith intermediate values of $R\/\\lambda_{T}^{(1)}$ smoothly and uniformly fill\nthe gap between the two asymptotic solutions corresponding to the long,\n$R\\gg\\lambda_{T}^{(1)}$, and short, $R\\ll\\lambda_{T}^{(1)}$, distances, as\ndescribed by equations~(\\ref{n_R>>lambda}) and (\\ref{n_R<<lambda}). Similarly,\nin Figure~\\ref{Fig:Density_vs_Radius_logarithm} the curves with intermediate\nvalues of $\\cos\\theta$ fill the gap between the solutions strictly in front of\nthe meteoroid ($\\cos\\theta=-1$) and behind it ($\\cos\\theta=1$), as described\nby equations~(\\ref{n^(2)_theta=pi}) and (\\ref{n^(2)_theta=0}).\n\nFigure~\\ref{Fig:3Ddensity} shows the entire 3D structure of the ion density in\ncolor coding. Since the spatial distribution of the plasma density is axially\nsymmetric, this figure shows a meridional cross-section that includes the\nmajor axis. Behind the meteoroid, at $z\\equiv R\\cos\\theta\\gg\\lambda_{T}^{(1)}$\nthe presented distribution may be inaccurate because it does not properly\ndescribe the extended trail formed lagging behind multiply scattered ions.\nHowever, around the meteoroid this distribution should be reasonably accurate.\nFigure~\\ref{Fig:3Ddensity} shows a significant span in the density values\n(seven orders of magnitude). Notice visible bulges in constant density\ncontours in front of the meteoroid. This feature has not been predicted in\nprevious PIC simulations \\citep{Dyrud:Plasma2008EMaP,Dyrud:Plasma2008AdSpR}.\n\n\\section{Discussion\\label{Discussion}}\n\nThe spatial structure of the near-meteoroid plasma shown in\nFigure~\\ref{Fig:3Ddensity} scales with the collisional mean free path of the\nprimary (ablated) particles, $\\lambda_{T}^{(1)}$. The latter, in turn, scales\nwith the altitude($h$)-dependent atmospheric density as $\\lambda_{T}%\n^{(1)}\\propto1\/n_{\\mathrm{A}}(h)$. The frontal edge of the meteoroid plasma\nhas a bulge located around the major axis at the distance $\\sim\\lambda\n_{T}^{(1)}$ from the meteoroid.\n\nThe most striking feature of the meteor-plasma spatial structure is not the\nangular dependence of $n^{(2)}$ but rather the fact that near the meteoroid,\n$R\\lesssim\\lambda_{T}^{(1)}$, the plasma density behaves as $n^{\\left(\n2\\right)  }\\simeq F(\\theta)\/R$. The singular $1\/R$ dependence holds almost to\nthe meteoroid surface, $R\\sim r_{\\mathrm{M}}$. This fact has serious\nimplications for head-echo observations and for the corresponding modeling of\nthe radar wave propagation.\n\nTo analyze radio wave propagation through the dense meteoroid plasma with\n$n_{e}\\approx n_{i}\\approx n^{(2)}\\gg n_{\\mathrm{A}}$, we apply the simplest\ncriterion of geometrical optics applicable (the WKB approximation):\n\\begin{equation}\n\\frac{\\lambda_{0}}{2\\pi}\\ \\frac{\\left\\vert \\nabla\\tilde{n}\\right\\vert\n}{|\\tilde{n}|^{2}}\\ll1 \\label{WKB}%\n\\end{equation}\n\\citep{Ginzburg:Propagation71}. Here $\\lambda_{0}=c\/f$ is the vacuum wavelength\nwith the frequency $f$, and index of refraction (neglecting the geomagnetic\nfield) given by $\\tilde{n}=\\left(  1-n_{e}\/n_{\\mathrm{cr}}\\right)  ^{1\/2}$,\nwhere%\n\\begin{equation}\nn_{\\mathrm{cr}}=\\frac{4\\pi^{2}\\epsilon_{0}m_{e}f^{2}}{e^{2}}\\approx\n1.24\\times10^{4}%\n\\operatorname{cm}%\n^{-3}\\left(  \\frac{f}{1%\n\\operatorname{MHz}%\n}\\right)  ^{2} \\label{n_cr}%\n\\end{equation}\nis the critical plasma density corresponding to $\\tilde{n}=0$.\n\nMost of small meteoroids have a radius within the hundreds-of-microns range,\ni.e., many orders of magnitude less than the mean free path, $\\lambda\n_{T}^{(1)}$. This means that the radar beam with the wave frequency in the\ntens-of-MHz range (e.g., Jicamarca with $f=50~$MHz) will almost certainly\ncross the boundary $R=R_{\\mathrm{cr}}(\\theta)$ where $n_{\\mathrm{cr}}%\n=n_{e}\\approx n^{\\left(  2\\right)  }(R_{\\mathrm{cr}},\\theta)$. Under the\ngeometric-optics approximation, the boundary $R\\approx R_{\\mathrm{cr}}%\n(\\theta)$ represents the wave reflection level.\n\nAccording to equation~(\\ref{WKB}), geometric optics applies beyond the wave\nreflection level of $\\tilde{n}=0$ and, for $n_{e}\\propto1\/R$, where\n\\begin{equation}\nR\\gg\\sqrt{\\frac{\\lambda_{0}R_{\\mathrm{cr}}}{4\\pi|\\tilde{n}|^{3}}}.\n\\label{proximity}%\n\\end{equation}\nFor small radii, $R\\ll R_{\\mathrm{cr}}$, for which $|\\tilde{n}|\\approx\n(R_{\\mathrm{cr}}\/R)^{1\/2}\\gg1$, this requires $R\\gg\\lambda_{0}^{2}\/(16\\pi\n^{2}R_{\\mathrm{cr}})$, meaning that geometric optics applicability will\nnecessarily fail in the close proximity to the meteoroid.\n\nIf $R_{\\mathrm{cr}}\\gg\\lambda_{0}\/(4\\pi)$ then the radio wave propagating from\nlarger to smaller $R$ will reach the reflection level of the plasma density,\n$R=R_{\\mathrm{cr}}$, well before reaching the volume where the WKB becomes\ninvalid. This corresponds to normal overdense reflection of the radar wave, so\nthat the $1\/R$ dependence of $n^{\\left(  2\\right)  }$ will not prevent using\nthe regular geometric-optics technique. If, however, $R_{\\mathrm{cr}}%\n\\lesssim\\lambda_{0}$ then the applicability of the geometric-optics\npropagation breaks down before the radio wave reaches its reflection level.\nThis requires modeling the radar wave propagation using full Maxwell's\nequations \\citep{Marshall:FDTD15}. The characteristics of the radar signal\nobtained in this way may differ from those obtained in the framework of the\nregular ray tracing.\n\nTo conclude this discussion, we note that in the underlying kinetic theory we\nhave neglected the effect of fields on the ion collisionless motion. Directly\nnear the meteoroid, where the calculated density sharply increases with\ndecreasing $R$ this assumption may become invalid. This is especially true\nwithin the Debye length from the meteoroid surface, where one may expect a\nsignificant net positive charge formed by slow ions after fast electrons moved\naway from that region. This charge separation can create a significant\npotential barrier for electrons that, at the same time, will accelerate ions\naway from the meteoroid. This may modify the $1\/R$ rate near the meteoroid\nsurface. However, the relevant distances are typically much smaller than the\nradar wavelength, so that this localized modification should not affect the\noverall radio wave propagation and formation of the radar head echo.\n\n\\section{Conclusions\\label{Conclusions}}\n\nBased on the kinetic theory developed in the companion paper\n\\citep{Dimant:Formation1_17}, we have calculated the spatial distribution of\nthe meteor plasma responsible for the radar head echo.\n\nThe underlying theory assumes that (1) plasma electrons obey the Boltzmann\ndistribution; (2) most plasma ions originate from the ablated meteoroid\nmaterial after exactly one ionizing collision, while further collisions can be\nneglected; (3) the ion motion between collisions is largely unaffected by\nfields. In order to make plasma density calculations easier, in this paper we\nhave additionally assumed the isotropic differential cross-section of the\nmeteor-particle ionization.\n\nEquations~(\\ref{n^(2)_general}) to (\\ref{f_3}) describe the entire spatial\ndistribution of the near-meteoroid plasma density.\nFigures~\\ref{Fig:M_versus_theta}, \\ref{Fig:Density_vs_Radius_logarithm}, and\n\\ref{Fig:3Ddensity}\\ illustrate the solution. The major feature of the\nnear-meteoroid plasma density spatial distribution is its dominant $1\/R$\ndependence in most of the plasma. This quasi-singular behavior makes the radar\npenetrated near-meteoroid plasma to be overdense at locations sufficiently\nclose to the meteoroid. This and other features are important for modeling the\nradar head echo. In near future, we are planning to employ the obtained plasma\ndensity distribution for such modeling.\n\n\\section*{Acknowledgments}\n\nThis work was supported by NSF Grant AGS-1244842.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nWhen the Lord gives you a revelation of the enemy's toxic plots, it empowers you to overcome every attack. As I always say, a devil exposed is a devil defeated. In _Hell's Toxic Trio_ Ryan exposes a threefold demonic cord that many believers have not yet discerned. He also shows you how it can be easily broken with heaven's strategy. You can't see what you can't see, but this book will open your eyes to the root of toxic attacks in your life. Get this book!\n\n\u2014JENNIFER LECLAIRE\n\nSENIOR LEADER, AWAKENING HOUSE OF PRAYER\n\nRyan LeStrange has an unusual and unique mantle of apostolic\/prophetic revelation. He captures key issues hindering breakthrough, spiritual freedom, and inheritance. In _Hell's Toxic Trio_ he both explains the hindrances as well as shows how to move into freedom. A much-needed book!\n\n\u2014BARBARA J. 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Ryan LeStrange identifies three of these high-level pollutants in his book _Hell's Toxic Trio_ and shows ways to cleanse the soul so you can emanate a clean and righteous spirit and even permeate the atmosphere around you. The only thing missing from this book is a warning label: \"If you partake of these contents, you might react and spit out some old stuff!\" This book can change you from the inside out!\n\n\u2014JAMES W. GOLL\n\nFOUNDER, GOD ENCOUNTERS MINISTRIES\n\nAUTHOR, EQUIPPER, PERFORMING ARTIST,\n\nCERTIFIED LIFE LANGUAGE TRAINER\n\nThroughout church history God has been faithful to raise up apostolic and prophetic leaders in hours of confusion. For such a time as this Ryan LeStrange has been raised up. _Hell's Toxic Trio_ is the _rhema_ word for now! Not only does he expose the antichrist spirit, he points the reader to the feet of Jesus. This book will be considered a classic.\n\n\u2014BRIAN WILLIAMS\n\nFOUNDER AND SENIOR LEADER, HOPE CITY\n\nAre you fearful, anxious, or rejected? There is lasting healing for the wounded soul. In this powerful book apostle Ryan LeStrange delivers a revelatory remedy with typical refreshing candor. Through each page God will expose the toxic trio that has arrested the destinies of millions, and you will be catapulted into your divine fulfillment. As you read, release your faith and expect personal revival.\n\n\u2014KYNAN BRIDGES\n\nPASTOR, APOSTOLIC LEADER, BEST-SELLING AUTHOR\n\nGRACE & PEACE GLOBAL FELLOWSHIP\n\nMost CHARISMA HOUSE BOOK GROUP products are available at special quantity discounts for bulk purchase for sales promotions, premiums, fund-raising, and educational needs. For details, write Charisma House Book Group, 600 Rinehart Road, Lake Mary, Florida 32746, or telephone (407) 333-0600.\n\nHELL'S TOXIC TRIO by Ryan LeStrange\n\nPublished by Charisma House\n\nCharisma Media\/Charisma House Book Group\n\n600 Rinehart Road\n\nLake Mary, Florida 32746\n\nwww.charismahouse.com\n\nThis book or parts thereof may not be reproduced in any form, stored in a retrieval system, or transmitted in any form by any means\u2014electronic, mechanical, photocopy, recording, or otherwise\u2014without prior written permission of the publisher, except as provided by United States of America copyright law.\n\nUnless otherwise noted, all Scripture quotations are taken from the Modern English Version. Copyright \u00a9 2014 by Military Bible Association. Used by permission. All rights reserved.\n\nScripture quotations marked KJV are from the King James Version of the Bible.\n\nScripture quotations marked NASB are from the New American Standard Bible, copyright \u00a9 1960, 1962, 1963, 1968, 1971, 1972, 1973, 1975, 1977, 1995 by The Lockman Foundation. Used by permission. (www.Lockman.org)\n\nScripture quotations marked NIV are taken from the Holy Bible, New International Version\u00ae, NIV\u00ae. Copyright \u00a9 1973, 1978, 1984, 2011 by Biblica, Inc.\u2122 Used by permission of Zondervan. All rights reserved worldwide. www.zondervan.com. The \"NIV\" and \"New International Version\" are trademarks registered in the United States Patent and Trademark Office by Biblica, Inc.\u2122\n\nScripture quotations marked NKJV are taken from the New King James Version\u00ae. Copyright \u00a9 1982 by Thomas Nelson. Used by permission. All rights reserved.\n\nScripture quotations marked THE MESSAGE are from _The Message: The Bible in Contemporary English_ , copyright \u00a9 1993, 1994, 1995, 1996, 2000, 2001, 2002. Used by permission of NavPress Publishing Group.\n\nCopyright \u00a9 2018 by Ryan LeStrange\n\nAll rights reserved\n\nCover design by Vincent Pirozzi\n\nDesign Director: Justin Evans\n\nVisit the author's website at RyanLeStrange.com.\n\nLibrary of Congress Cataloging-in-Publication Data:\n\nAn application to register this book for cataloging has been submitted to the Library of Congress.\n\nInternational Standard Book Number: 978-1-62999-488-8\n\nE-book ISBN: 978-1-62999-489-5\n\nWhile the author has made every effort to provide accurate Internet addresses at the time of publication, neither the publisher nor the author assumes any responsibility for errors or for changes that occur after publication. Further, the publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.\n\nI dedicate this book to the radical warriors\n\nwho have fought through so many attacks\n\nof the enemy! I dedicate it to the fearless\u2014\n\nthe pioneers and planters who forge\n\nahead in spite of the opposition. In every\n\nkingdom assignment, whether it be family,\n\nbusiness, or ministry, there will typically\n\nbe spiritual resistance. Victory belongs to\n\nthe brave! In order to do exploits, we must\n\nkeep pressing. I commend all of those who\n\nhave daringly faced the opposition and\n\nkept going. Your greatest days are ahead!\n\nCONTENTS\n\nAcknowledgments\n\nForeword by John Eckhardt\n\nIntroduction\n\nChapter 1 The Reality of the Unseen Realm\n\nChapter 2 The Power of the Anointing\n\nChapter 3 Jezebel, the Spiritual Assassin\n\nChapter 4 Jezebel's Seduction, Schemes, and Strategies\n\nChapter 5 Casting Down the Demon Queen Jezebel\n\nChapter 6 Python, the Heat-Seeking Deceiver\n\nChapter 7 Breaking Python's Grip\n\nChapter 8 The DNA of the Religious Spirit\n\nChapter 9 First-Love Lifestyle: Freedom From the Religious Spirit\n\nChapter 10 Enforcing the Kingdom: Living Free\n\nNotes\nACKNOWLEDGMENTS\n\nTHANKS TO MY forever love, the amazing Joy LeStrange, who fearlessly pioneers with me, and to my talented son, Joshua. Thanks to my mom, Eileen, and all of my extended family for all the love and support. Thank you to my staff and team, who work so hard to help me circle the globe with fire. Thanks to my editing team and those who help bring these writing projects together. Thank you to my supporters, friends, and partners for standing, believing, sowing, and building with me\u2014it takes a team!\n\nThanks to all of those who have labored to build, enlarge, and expand the kingdom. Thanks to the generals who have gone ahead\u2014we honor you. Thanks to those in my generation who have dared to carry the mandate of radical prophetic preaching, supernatural miracles, and nation-shaking faith\u2014I am honored to serve alongside you. Thanks to the next generation for being brave and daring\u2014I want to stand with, walk alongside, and partner with you to see a great awakening.\nFOREWORD\n\nRYAN LESTRANGE HAS a passion to see the church walk in power and liberty. He is committed to seeing the church experience revival and glory. There are evil spirits that attempt to block the church from experiencing God's best. In this book Ryan LeStrange exposes some of the strategies of darkness that prevent the church from experiencing the liberty of the Holy Spirit. Many leaders are not aware of the tremendous battle in the spirit realm between good and evil. We should not be ignorant of Satan's devices.\n\nRyan LeStrange is a fresh voice for revival, prayer, the apostolic ministry, and the prophetic ministry. He is ministering around the world, and he challenges believers to move in power and authority. He has a burden to see the church break free from controlling powers that stop and block revival. This book is a product of his determination to see the fire of revival burn continuously.\n\nEvery generation has authors who will help us break through for kingdom advancement. Ryan LeStrange has experienced the truths written in this book. He is not echoing what others have said but is giving fresh revelation for this new generation. There are thousands of believers worldwide who desire to see the church grow and experience God's choice blessings. If you desire these blessings, then take to heart the insight given in this book. I have dealt with the spirits of Jezebel and python. These are deadly enemies to leaders and churches worldwide.\n\nThere are many leaders who are not equipped to deal with these adversaries. I recommend this book to leaders and intercessors. A greater knowledge in this area is essential if we desire victory. Some of the truths in this book may be new to many, but consider what is said, and the Lord will give you understanding. Prayer is one of the most effective ways to overcome these opponents. There are powerful prayers in this book that will help any believer or church wage the good warfare.\n\nReligious spirits can be very deceptive. These spirits are often difficult to discern because they are religious. We need discernment to separate the true from the false. Ryan exposes spirits that mask themselves as being from God but are from the enemy.\n\nI have been a proponent of the prophetic ministry for many years. I have seen the great blessings prophets and prophetic people bring to the church. Ministries that embrace the prophetic ministry are a threat to the demonic realm. These ministries are a key to seeing the church move forward and advance.\n\nSatan hates the prophetic ministry and will attempt to block it. Prophets and prophetic people always bring fresh moves and attract revival. This is why it is so important to keep the church free from demons that hate the prophetic. Those who promote the prophetic ministry must have insight into what stops and blocks the prophetic flow.\n\nDo you have a passion for seeing and experiencing revival and the glory of God? Do you desire to see churches filled with light and power? Do you want to see revival spread across the land? Do you desire to have the keys that unlock doors and open the way for salvation, healing, and deliverance? If your answer is yes, then I believe you have the right book in your hand. This book will challenge you to rise up and be the overcomer you are called to be.\n\n\u2014JOHN ECKHARDT\n\nBEST-SELLING AUTHOR, _P RAYERS THAT ROUT DEMONS_\nINTRODUCTION\n\nWHAT IF I told you that there was a secretive conspiracy in operation, lurking just beneath the surface of your daily life and affairs? I know, you would most likely roll your eyes and say something like, \"Oh no, not another crazy conspiracy theory!\" I understand that the word _conspiracy_ brings all kinds of emotions and thoughts, but go with me on a journey for just a moment.\n\nWhen you were born again, you were re-created in the image and likeness of God. You were suddenly and powerfully brought into the world of the supernatural. There are spiritual forces at work in every element of your life. You have been given access to the realm of seeing, hearing, and knowing. Jesus said that He would send us a comforter to help and assist in our walk with God. Holy Spirit came to reside on the inside of us, providing the force and the power of the anointing of God in every area of our lives so that we become conquerors!\n\nThe anointing creates unstoppable power and momentum that bring explosive results in the kingdom of God. The devil and his minions absolutely hate the anointing. They want to shut down the flow of God's power and ability in your life.\n\nHell has launched a dark, malicious conspiracy against you. The word _conspire_ means \"to join in a secret agreement to do an unlawful or wrongful act.\" In this book I will unmask three of hell's unlawful conspirators. I will show you how these particular spirits have similarities in their operations to bind and quench the life-giving breath and anointing of God. This evil toxic trio twists, turns, seduces, and restricts in order to hinder the spiritual progress of God's people.\n\nHowever, exposure creates opportunity for deliverance! As you embark on this journey into revealing truths, your spiritual eyes will be opened to see and recognize the hidden operations of these evil spirits and the pathways of deliverance. It is time for you personally to come under the refreshing winds of heaven and be released into your new season. It is time for the heaviness to go. It is time for your destiny to manifest. It is time to see your prophetic promises realized.\n\nI will rip the veil off the Jezebel spirit that has been at work throughout history to seduce God's people and kill the prophets. I will lay bare the evil operations of this foul anti-anointing spirit. You will see how it has robbed mankind throughout the ages and is currently at work in various facets of society. I will share personal experiences with you to arm you with strategies to overcome its power and walk free.\n\nI will expose the python spirit with its multifaceted operations. Many people fall prey to the attack of this spirit by being ignorant of what they are actually facing. This spirit hits on various fronts to choke out the wind of God in the life of a believer. A python attack can create spiritual, emotional, and physical suffocation. Revealing this vile serpent brings tremendous liberty to the people of God. You will be fully armed to break the grip of this demon.\n\nI will reveal to you the personal and corporate operations of the religious spirit. This spirit comes in subtly to sow legalism and bondage into the hearts of God's people. It dilutes identity and purpose to stifle the fresh prophetic revelation in the lives of believers. You will see it clearly and receive strategy to overcome it on every level.\n\nExposing the conspiracy tactics of this toxic trio is not just about studying these spirits\u2014it is about setting you on course for a greater walk in the Spirit. It is about entering God's prophetic times and seasons for your life. It is about enjoying the invigorating winds of heaven. Together we will go on a journey of revelation and impartation. We will uncover, expose, bind, break, decree, and release. We will stir up the fresh winds of personal revival and forge a brave path ahead into destiny. The toxic trio will fall, and the will of heaven will be manifested in your life!\n\nTherefore do not fear them, for there is nothing concealed that will not be revealed, or hidden that will not be known.\n\n\u2014MATTHEW 10:26, NASB\nChapter 1\n\nTHE REALITY OF THE UNSEEN REALM\n\nGOD TOOK ME into a vision of the future as I was ministering with a dear friend of mine who carries a strong prophetic mantle for the nation of America. We were ministering to people prophetically, laying hands on them, and releasing the word of the Lord. I walked onto the platform next to my friend, and I was immediately caught up into another dimension. It came in an instant! It was as though a whirlwind were carrying me higher and higher up into the spirit realm.\n\nSuddenly I was swept up into this powerful encounter in the realm of the spirit. I was caught up, and God was about to show me something of great significance. This is how prophetic communication and experiences often occur in my life; they come in a moment, carrying great purpose. In a single instant God can give a message that changes the course of a person, ministry, family, or nation. We must learn to recognize, emphasize, and value these types of experiences. They carry communication from another realm.\n\nAs I was swept into the realm of the spirit, I saw the stadiums in America erupting with uncontrollable fire. The fire of deliverance was hitting the masses. It was not calm but violent in nature as the purging fires of heaven burned out demon powers. There was an intermingling of fire and glory, bringing mass deliverance. I asked the Lord about what I saw; He told me that there is a dimension of glory so heavy that it displaces demons. People can be set free in this glory.\n\nIn this vision I saw the fire of Pentecost knocking multitudes off their feet with Upper Room power. There was an extreme outpouring of fire like what fell in chapter 2 of the book of Acts. What I saw was unlike anything I ever witnessed in large gatherings. It was much more intense and radical. The Lord told me, \"It is coming!\"\n\nThere is a fiery move coming forth. The Lord told me this fire and glory will be carried by the pure. He is going to use those who have refused to change their message to please religious spirits. He is mantling those who have chosen not to bow to political systems in the church but have stayed tuned in to what heaven is declaring. They will operate in this next level of fire and glory.\n\nDIMENSIONS AND VIEWPOINTS\n\nAlmost a year later I was hosting a conference where one of the great prophets in our nation was speaking. He began to weep as he shared dreams the Lord had given him for America. He shared about the move of mass deliverance he saw coming to the nation. As he poured out from his spirit, it was like lightning hit me. I heard him speak of the very same things I had seen that day when I was shown the coming move of fire.\n\nThese encounters may sound strange or unusual, but they are typical for born-again people filled with the Spirit of God. Prophetic experiences and encounters should be a daily occurrence for Christians. It should not seem strange or unusual that people are having dreams and visions of that which is and that which is to come. It was these types of supernatural encounters that fueled the move of God in the early church. As heaven spoke, the people moved.\n\nWe are living and breathing here on earth, yet we are born of another dimension, called to occupy another space. We are spiritual beings temporarily parked in this natural realm.\n\nBrothers, become fellow imitators with me and observe those who walk according to our example. For many are walking in such a way that they are the enemies of the cross of Christ. I have told you of them often and tell you again, even weeping. Their destination is destruction, their god is their appetite, their glory is in their shame, their minds are set on earthly things. But our citizenship is in heaven, from where also we await for our Savior, the Lord Jesus Christ, who will transform our body of humiliation, so that it may be conformed to His glorious body, according to the working of His power even to subdue all things to Himself.\n\n\u2014PHILIPPIANS 3:17\u201321\n\nPaul paints a vivid picture of people living with two drastically different perspectives. In one group the people have their minds set on earthly things, therefore missing the heavenly prize. They are so focused on this present world that their lives are void of spiritual power and significance. Paul boldly declares that this would not be his pathway, nor would it be the journey of the Philippian church. They were navigating two spaces, this present world and that which is to come. They were active and effective in living, working, and building during their time on earth, but their walk was empowered by the realization that they were effectively representing another dimension. They were heavenly citizens, ambassadors, equipped to subdue the powers of darkness, shipwreck the strategies of hell, and release the kingdom.\n\nThis is a picture of how we are to walk. We are to be representatives of the heavenly sphere with supernatural power and authority functioning in the _now_. We are not waiting for a one-day breakthrough or blessing. We are to be moving, living, working, and functioning in it right now!\n\nThere is an often unseen realm of the spirit. Many pastors and ministry leaders are facing conflict and challenge in their churches. They think they are just having difficulty with a rebellious person, but there is something deeper at work in these situations. Many families are facing sudden struggle that is so overwhelming that family members are battling the urge to leave and break up the family. Their minds are under siege, and their commitment is being tested. The devil wants to hide behind natural reasoning and human wisdom.\n\nLet me be clear: I believe in counseling, renewing the mind, and growing in wisdom. Not every challenge can be resolved by casting out a demon. Sometimes people must work to grow in wisdom, to advance in understanding, and to change some things about themselves. We are three-part beings.\n\nMay the very God of peace sanctify you completely. And I pray to God that your whole spirit, soul, and body be preserved blameless unto the coming of our Lord Jesus Christ.\n\n\u20141 THESSALONIANS 5:23\n\nTo be sanctified wholly, we must deal with our threefold nature. We are a spirit. The spirit part of our being contains the DNA of heaven. My spirit is not ever depressed. My spirit does not desire the nature of the enemy or lust after the works of hell. When I was born again, the kingdom came to rest in my spirit with all power and authority. If believers learned how to live and dwell in the spirit realm, they would live their lives in victorious exploits. The enemy works to bombard and overwhelm them, to trap them in the beggarly elements of earth, but God has called them to soar high above the works of darkness.\n\nThe second part of man is the soul. I believe this complex part of our being is the mind, will, and emotions. It is the part that feels, understands, and chooses. This part can become toxic without the washing of the water by the Word of God.\n\nThat He might sanctify and cleanse it with the washing of water by the word.\n\n\u2014EPHESIANS 5:26\n\nThe Word of God is like a Holy Spirit washing machine that removes the stains and pains of past experiences, turmoil, and torment. The reality of the Word of God can and will annihilate bondage in a human soul.\n\nHEALING WOUNDED AND DAMAGED SOULS\n\nA wounded and damaged soul becomes a breeding ground for wrong spiritual influences. In fact, wounded emotions can act as gateways to demonic powers. The devil understands that your beliefs determine your direction. One of his warfare strategies is to infiltrate the way you think. He wants to empower false belief systems, toxic thinking, and reckless emotional upheaval.\n\nThat you put off the former way of life in the old nature, which is corrupt according to the deceitful lusts, and be renewed in the spirit of your mind; and that you put on the new nature, which was created according to God in righteousness and true holiness.\n\n\u2014EPHESIANS 4:22\u201324\n\nHealing wounds in the soul is a critical step in personal freedom. It firmly closes an open door to the powers of the enemy. Here are some simple steps that serve as keys to help you dismantle emotional bondages that the enemy has constructed in your life.\n\nRecognize the problem.\n\nThe first step to recovery is recognition. You can identify patterns, emotional outbursts (when the wound is triggered), and unhealthy cycles. Seeing that there is a problem empowers confrontation and healing. Denial empowers the bondage to remain. Many people with soul wounds live in a continual state of denial and false blame. This allows the hurt to remain and the problem to be camouflaged.\n\nGet to the root.\n\nYou need Holy Spirit's guidance, deliverance, healing, and cleansing. Once you have recognized a problem, allow Holy Spirit to shine a bright light on the root system. How did this begin? What was the point of trauma? What is the root system that I need to deal with? For example, a wound of rejection can manifest as anger, but that is just the fruit, not the root. The healing must go down all the way to the root. The entry wound needs to be dealt with. Often this process will require assistance from other skilled ministers, depending on the depth of the issue.\n\nConfront inner thoughts and lies.\n\nLies come to block the working of truth. The Word of God is truth! It is your foundation and must be the basis from which you draw your identity. Every deep emotional wound will form a series of lies that empower the existence of the wound. You must boldly confront the lies and make a decision to uproot them.\n\nReplace lies with truth.\n\nThere is healing in the Word of God. If the enemy has planted a lie in you that says you are not valuable, then you must get Scripture that contradicts that lie. The Word acts as an antidote to the toxic infection of lies and bondage.\n\nMeditate on the Word.\n\nRelease transformation through meditation. Meditation is the system of spiritual digestion. You must get in the truth and pray over those verses. Speak them, ponder them, and allow them to penetrate to the deepest recesses of your mind and emotions. This process draws out the healing properties of the Word of God. Sometimes you will spend days and months in the same verses because you are chewing them up and digesting them. You may need to write them and post them on the walls as reminders of your promise.\n\nBreak behavior patterns and cycles tied to lies.\n\nEach and every wound will create bad behavior patterns and cycles. Once you have dealt with the root, you have displaced the power of the wound, but there may be residual patterns that need to be broken. If you have acted a certain way for years, it is now routine and you have to make a conscious decision not to allow that behavior anymore. When you catch yourself falling back into an old pattern, be quick to do a course correct. Sometimes, if the wound was severe, you may actually have to learn how to act normally. This is particularly true in the area of relationships.\n\nResist, worship, and walk it out.\n\nComing out of these patterns can be a process. There must be resistance on a regular basis. Worship and inviting the presence of God into your life bring oil into what can feel like a dry process. Then you must walk it out\u2014day by day, bit by bit. Don't get discouraged if you have a thought that is not healthy\u2014this is a temptation. Cast it down and keep pressing on. Give yourself permission to walk out the process.\n\nRest in the grace of God.\n\nYou must come back to the perfect work of the Cross. One of the things Jesus did was take all emotional upheaval and wounds in your place. His unfailing love and perfect work empower your freedom. You must not and cannot get trapped in condemnation and self-righteousness. Having a strong understanding of God's grace and goodness will empower your recovery.\n\nPart of maturing in the new nature is accessing new ways of thinking. The Word of God becomes the road map to proper thinking and therefore opens the gateways to kingdom living. Once you think differently, you live differently. Renewing the mind is not a single-step process but a journey of continual change, growth, and renewal.\n\nHere is a list of ten types of damaged souls, their symptoms, and Scripture passages to renew your mind and put you on the path toward proper thinking.\n\n1. Conflicted soul\u2014unable to make rational decisions, tossed to and fro, highly unstable and overly emotional\n\nA sound mind makes for a robust body, but runaway emotions corrode the bones.\n\n\u2014PROVERBS 14:30, THE MESSAGE\n\n2. Tormented soul\u2014high levels of accusation, fear of punishment, dread over past mistakes, a heavy lack of peace\n\nThere is therefore now no condemnation for those who are in Christ Jesus, who walk not according to the flesh, but according to the Spirit.\n\n\u2014ROMANS 8:1\n\nAnd the peace of God, which surpasses all understanding, will protect your hearts and minds through Christ Jesus.\n\n\u2014PHILIPPIANS 4:7\n\n3. Fearful soul\u2014fearful and anxious on a regular basis, unable to do certain activities because of abnormal levels of fear\n\nFor God has not given us the spirit of fear, but of power, and love, and self-control.\n\n\u20142 TIMOTHY 1:7\n\nThere is no fear in love, but perfect love casts out fear, because fear has to do with punishment. Whoever fears is not perfect in love.\n\n\u20141 JOHN 4:18\n\n4. Anxious soul\u2014nervous and easily panicked, void of peace and rest, anxiety attacks and bouts with a racing heart and\/or heavy breathing\n\nBe anxious for nothing, but in everything, by prayer and supplication with gratitude, make your requests known to God.\n\n\u2014PHILIPPIANS 4:6\n\n5. Rejected soul\u2014easily offended and continually needing unnatural levels of affirmation\n\nIf my father and my mother forsake me, then the LORD will take me in.\n\n\u2014PSALM 27:10\n\nAll whom the Father gives Me will come to Me, and he who comes to Me I will never cast out.\n\n\u2014JOHN 6:37\n\n6. Weary soul\u2014tired and frustrated, a lack of flow in the Spirit, feeling frustrated and assuming that kingdom works will not produce promised harvest\n\nFor I satiate the weary souls and I replenish every languishing soul.\n\n\u2014JEREMIAH 31:25\n\nCome to Me, all you who labor and are heavily burdened, and I will give you rest. Take My yoke upon you, and learn from Me. For I am meek and lowly in heart, and you will find rest for your souls.\n\n\u2014MATTHEW 11:28\u201329\n\n7. Offended soul\u2014holding grudges and refusing to let go of past wrongs; this type of emotional wound is yoked to a spirit of bitterness and infirmity that can manifest in various crippling illnesses\n\nBear with one another and forgive one another. If anyone has a quarrel against anyone, even as Christ forgave you, so you must do.\n\n\u2014COLOSSIANS 3:13\n\nPursue peace with all men, and the holiness without which no one will see the Lord, watching diligently so that no one falls short of the grace of God, lest any root of bitterness spring up to cause trouble, and many become defiled by it.\n\n\u2014HEBREWS 12:14\u201315\n\n8. Lost soul\u2014vagabond mentality, refusing to be established in the Word of God and the will of God\n\nBlessed is the man who walks not in the counsel of the ungodly, nor stands in the path of sinners, nor sits in the seat of scoffers; but his delight is in the law of the LORD, and in His law he meditates day and night. He will be like a tree planted by the rivers of water, that brings forth its fruit in its season; its leaf will not wither, and whatever he does will prosper.\n\n\u2014PSALM 1:1\u20133\n\n9. Wounded soul\u2014enduring hurts and long-term wounds; this creates a toxic mental environment that sees everything in a negative light\n\nThe LORD is my shepherd; I shall not want. He makes me lie down in green pastures; He leads me beside still waters. He restores my soul; He leads me in paths of righteousness for His name's sake.\n\n\u2014PSALM 23:1\u20133\n\n[The LORD] gathers together the outcasts of Israel. He heals the broken in heart, and binds up their wounds.\n\n\u2014PSALM 147:2\u20133\n\n10. Confused soul\u2014vacillating between ideas, unsure of identity, lacking clarity, not resolute and firm in thought, decision, or direction\n\nThe steps of a man are made firm by the LORD; He delights in his way.\n\n\u2014PSALM 37:23\n\nYOUR EARTHLY HOUSE\n\nThe third part of man is the body. Your body permits you to be here on planet Earth, the house for your spirit man. Hebrews 9:27 says, \"As it is appointed for men to die once, but after this comes the judgment.\" This verse is not referring to spiritual death but to physical death. It teaches a basic Bible doctrine that every human being will eventually experience death in the flesh and then enter eternity.\n\nInstead, I say that we are confident and willing to be absent from the body and to be present with the Lord.\n\n\u20142 CORINTHIANS 5:8\n\nThe moment our natural bodies take their final breaths, we are instantly swept into another dimension. We do not die in terms of the spiritual dimensions; we simply shift arenas. We are no longer confined by the limitations of earthly bodies but are swallowed up into the realm of the spirit to receive our eternal rewards. This is one of the reasons why we must boldly teach and preach that the spirits of the dead are not hanging around on earth. Grandma is not coming back to talk with you\u2014it is a demon spirit that is mimicking her. She has entered the realm of eternity. No loved one who passed away is in your present; he is in your future.\n\nThe physical body has appetites that must be bridled in order to do the will of God. To have a rich and vibrant spirit life, you must conquer fleshly living.\n\nI say then, walk in the Spirit, and you shall not fulfill the lust of the flesh. For the flesh lusts against the Spirit, and the Spirit against the flesh. These are in opposition to one another, so that you may not do the things that you please. But if you are led by the Spirit, you are not under the law. Now the works of the flesh are revealed, which are these: adultery, sexual immorality, impurity, lewdness, idolatry, sorcery, hatred, strife, jealousy, rage, selfishness, dissensions, heresies, envy, murders, drunkenness, carousing, and the like. I warn you, as I previously warned you, that those who do such things shall not inherit the kingdom of God.\n\n\u2014GALATIANS 5:16\u201321\n\nYou must learn to live from the realm of the spirit and subdue the nature of your flesh. Discipline is a key ingredient to victory. Our bodies were created to serve us; we were not created to serve our bodies. There is often a struggle between the desires of the flesh and the realm of the spirit. It is conquered by being intentional and consistent in developing the spiritual life. As you learn to live from the place of the spirit, you will override the natural appetites of the flesh.\n\nSee then that you walk circumspectly, not as fools but as wise, redeeming the time, because the days are evil. Therefore do not be unwise, but understand what the will of the Lord is. And do not be drunk with wine, in which is dissipation; but be filled with the Spirit, speaking to one another in psalms and hymns and spiritual songs, singing and making melody in your heart to the Lord, giving thanks always for all things to God the Father in the name of our Lord Jesus Christ, submitting to one another in the fear of God.\n\n\u2014EPHESIANS 5:15\u201321, NKJV\n\nMAXIMIZING SPIRITUAL MOMENTS\n\nPaul encourages us to live rich spiritual lives, redeeming the time. He gives us a picture of maximizing moments and reaching the full spiritual intention of each season. How is this accomplished? It is reached through a daily infilling of Holy Spirit! He instructs us not to be drunk on wine but rather to be filled with the Spirit. The indication here is that there is a dwelling place in the glory of God that produces a similar effect to drunkenness.\n\nHave you ever been in a restaurant enjoying a quiet evening, only to be interrupted by people who have had too much to drink? They forget about everyone else. They are louder, bolder, and brutally honest because they are drunk. It is the same picture that we have in the Upper Room. Newly filled disciples came staggering out of a prayer meeting and were accused by the religious of being drunk. (See Acts 2:13.) They acted and spoke differently. Their level of boldness increased. There is a realm in the spirit that produces the intoxication of heaven and fuels personal, regional, and global revival!\n\nThis is the type of radical spirit living that comes from a fresh flow of Holy Spirit power in our lives. We are no longer timid but free to be as bold as lions (Prov. 28:1). Intentional communion with Holy Spirit brings greater awareness of the realm of the spirit and supernatural living. The picture in Paul's writings is not of a single encounter but a daily walk. We are called to receive a fresh flow of heaven daily. We are called to receive heavenly wisdom, heavenly insight, and heavenly empowerment. We were _never_ called to live spiritually stagnant.\n\nThe word _dwell_ means \"to remain for a time; to live as a resident; to keep the attention directed; to speak or write insistently.\" This is not a picture of passive pursuit but of an intentional lifestyle. We are called to live in the dimension of the spirit. One of the schemes of hell is to cause our minds, attention, and eyes to be fixed upon the natural realm, which dilutes our spiritual sensitivity and shipwrecks our faith.\n\nPRAYER IS A CORNERSTONE\n\nOne of the cornerstones of spirit living is prayer. At its heart, prayer is communication with God. There are many types of prayer that are valid and important. For example, the prayer of agreement is when two believers simply agree on a promise. A prayer of binding is when a child of God exercises legal authority over the works of darkness, forbidding their operation and advancements. There is a prayer of deep groaning and travail in the spirit, a gut-wrenching birthing in which the spirit dimension takes over the faculties of a believer to cry out in unknown languages and sounds until victory is realized in the spirit. A seasoned believer learns and utilizes the different facets of prayer. To engage the realm of the spirit, a person must be seasoned in prayer and active in daily fellowship with God.\n\nRejoice always. Pray without ceasing. In everything give thanks, for this is the will of God in Christ Jesus concerning you. Do not quench the Spirit. Do not despise prophecies. Examine all things. Firmly hold onto what is good.\n\n\u20141 THESSALONIANS 5:16\u201321\n\nThese verses illustrate the lifestyle that unfolds the spirit dimension. We are instructed to \"pray without ceasing.\" What does this look like practically? Is this stating we should neglect our family responsibilities and commitments and remain in a prayer closet 24\/7? I don't believe these verses communicate that at all. I believe they are telling us that in all things we are to maintain an attitude of continual prayer. I say it like this: \"Live tuned in!\" We live with an ear on the heartbeat of Jesus, continually leaning in and looking into the realm of the spirit, listening, and being quick to heed His instructions.\n\nIn John Eckhardt's best-selling book _Prayers That Rout Demons_ he writes, \"Our source of power is the Holy Spirit and the Word of God. We build ourselves up in faith when we confess the Word of God. We experience greater confidence when we understand the Word and walk in revelation. Prayer plugs us into the power source. Prayer connects us to God and allows His power to flow to us in any situation.\"\n\nWe were made to live in communion with God. We are called to dwell in His presence. We are marked and mantled as carriers of the glory. You can never carry the glory if you have not first heard His voice! The accuracy of hearing Him is contingent upon the strength of listening. Prayer is a two-way conversation. It is a dialogue in which you offer your heart up to God through intimate conversation and come into His presence, allowing Him to reveal His heart to you.\n\nEvery world changer has a red-hot prayer life! Prayer is the lifeblood of spiritual progress, breakthrough, and exploits. Without clear direction, intimate fellowship, and radical pursuit, there will be no supernatural displays of power.\n\nConfess your faults to one another and pray for one another, that you may be healed. The effective, fervent prayer of a righteous man accomplishes much. Elijah was a man subject to natural passions as we are, and he prayed earnestly that it might not rain, and it did not rain on the earth for three years and six months. And he prayed again, and the sky gave rain, and the earth brought forth its fruit.\n\n\u2014JAMES 5:16\u201318\n\nThe word _fervent_ in James 5:16 comes from a Greek word meaning to be active, operative, or put forth power. Another definition is \"energize, working in a situation which brings it from one stage (point) to the next, like an electrical current energizing a wire, bringing it to a shining light bulb.\" This type of fervent prayer releases spiritual power that opens the heavens and releases the strength of the kingdom. It is vital in spiritual warfare because things are never moved in the realm of the spirit without prayer. Prayer is the power that turns on the light of revelation. Without prayer there is no revelation, and without revelation there is no effective warfare.\n\nSome time ago I was preaching and felt prompted to follow in the footsteps of the apostle Paul. Paul prayed a powerful and effective prayer on behalf of the church at Ephesus, as chronicled in Ephesians 1. He asked that the eyes of their inner man be opened. I prayed this prayer over a lady, and she told me that almost immediately she became aware of the spiritual realm in a much more profound way. (This is one of the things that apostolic people and leaders must do\u2014release life-giving impartation and activate the spiritual sight of the church.) Her spiritual vision surged as she became acutely aware of atmospheres, bondages, needs, and potential breakthroughs. At first she was almost overwhelmed by all that her spiritual senses were picking up, which is often the case when people have their spiritual senses kicked into high gear. She shared with me the struggle that she was having sitting in a church service when she began to see and sense things. She lost track of what was happening in the meeting, distracted by the tangibility of the spirit realm and all that was going on around her.\n\nMature prophetic people come to understand that seeing or knowing something is not an automatic release to act upon it. Once your eyes are opened, you find yourself seeing frequently because the veil between the spiritual and natural realms is torn down. Prophetic seeing and knowing must be weighed by the Word of God. They must be weighed by the spiritual authority in your life and ministry. All things in the kingdom are subject to authority. This means that if I am under another person's authority, I must be subject to his or her chain of command and protocol. Many believe that once their spiritual senses become active, it is a license to release anything at any time, but that is an extremely immature prophetic paradigm.\n\nLet every person be subject to the governing authorities, for there is no authority except from God, and those that exist are appointed by God. Therefore whoever resists the authority resists what God has appointed, and those who resist will incur judgment.\n\n\u2014ROMANS 13:1\u20132\n\nIn the early church the apostles were on the front lines exposing demon rulers, confronting them, and mobilizing the church to tear them down. Paul fervently sought an impartation of insight and heavenly vision for his spiritual children. He wanted them to be keenly aware of the spirit realm. Without this awareness there is no action plan for victory. The enemy will reduce you to a realm of carnality and defeat. This is part of his strategy\u2014to bombard your natural senses, physical desires, and emotional well-being until you become oblivious to the realm of the spirit.\n\nDemon powers find satisfaction in the destruction of humanity. They war against us, the children of God, because our very existence reminds them of the downtrodden and cursed place in which they have been assigned to live. They strive to kill the God nature in us and degrade our lives to a series of monotonous, carnal activities with no spiritual impact. They infiltrate our thinking with humanistic arguments and fleshly wisdom, denying the realm of the supernatural. This is a master strategy that keeps demonic entities under the cover of darkness without exposure and empowers their wicked schemes.\n\nPROPHETIC PEOPLE\n\nProphets and prophetic people are like spotlights. They violently break through the murky abyss and uncloak the weapons of the enemy. James W. Goll masterfully describes these vital roles in his book _The Seer_ :\n\nAs an anointed people, we as the Body of Christ have been called to carry a prophetic-type burden that will cause us to live on the cutting edge of God's eternal purpose. The prophetic ministry is but one aspect of the fivefold calling of apostles, prophets, evangelists, pastors, and teachers. As such, it is resident to a degree in every ministry and more evident and active in certain ones. God's ultimate weapon is a man or a woman who has encountered the prophetic anointing. God does not anoint _projects_ ; He anoints _people_!\n\nPeople anointed with a prophetic ministry speak the word of the Lord in the name of the Lord. They carry weight in the Church by virtue of the ethical, moral, and spiritual urgency of their message. Their credentials, credibility, and status as prophetic vessels stem not from birth or official designation, but by the power of their inner call and by the response of those who hear them.\n\nI often say that all prophets and prophetic people live with giant targets on their backs because the forces of hell realize they can see them. The enemy cannot afford to be exposed because exposure is one of the first steps of deliverance. Without proper revelation a person, family, ministry, or region will remain firmly within the grip of demonic forces. The hidden demons must be uncovered, the plans made manifest, the attacks revealed, and the schemes divulged. This is one of the life-giving aspects of healthy prophetic ministry. Prophets and prophetic anointing shine light in the dark places and work in tandem with apostolic leaders to bring revelation that empowers building.\n\nLearning to live with a keen sensitivity to the realm of the spirit is imperative to living a life of victory. Heaviness manifesting is often a sign of a hidden demonic power. God has provided a series of prophetic tools and spiritual gifts by which He speaks to His people. As children of God, we must educate ourselves on His methodology of speaking and be quick to recognize the message. Once we have received communication, we must effectively steward it. Learning to live with spirit eyes and ears open is key!\n\nProphetic power tools are given to reveal and release the will of God in the lives of people. Where the will of God is known, strength is provided. Prophecy releases power.\n\nBut he who prophesies speaks to men for their edification and exhortation and comfort.\n\n\u20141 CORINTHIANS 14:3\n\nFar too often the gifts are abused to unnecessarily wound God's people. Young and immature believers mistake what they hear and see as permission to bring accusation and manifest a spirit of rebellion.\n\nRecently I had an encounter with a group like that. I was conducting a multiday training seminar and activating people in prophetic gifts. As time went by, I sensed an extreme heaviness hitting the room. I began to pray, and the Lord immediately highlighted 1 Corinthians 13, the love chapter. He told me to read those scriptures the following day and ask the participants to repent for any areas where they were not walking in love. He told me to have them minister to one another and let go of anything hindering them. As I obeyed the instruction of the Lord, it became obvious that there were some heavy offenses brewing in the group.\n\nI confronted a few people and found out they were faultfinding and criticizing people in the prophetic classes. As I gave them instruction and wisdom, they had a defense for every correction. They were blinded to their own wrongdoing and walking in open rebellion. They were deceived by experiencing things they deemed spiritual; they believed they were right and everyone else was wrong, including those in authority over the seminar.\n\nThis stance violates several simple prophetic protocols, but deeper than that it opens the door for the enemy by partnering with a spirit of rebellion. If indeed they were picking up on issues with others, the first step should have been to develop the heart of an intercessor for those who were hurting. This is key to changing atmospheres and lives. God releases and reveals to bring us into a place of prayer and petition. Intercession is vital.\n\nFor this reason we also, since the day we heard it, do not cease to pray for you and to ask that you may be filled with the knowledge of His will in all wisdom and spiritual understanding; that you may walk in a manner worthy of the Lord, pleasing to all, being fruitful in every good work, and increasing in the knowledge of God, strengthened with all might according to His glorious power, enduring everything with perseverance and patience joyfully, giving thanks to the Father, who has enabled us to be partakers in the inheritance of the saints in light. He has delivered us from the power of darkness and has transferred us into the kingdom of His dear Son, in whom we have redemption through His blood, the forgiveness of sins.\n\n\u2014COLOSSIANS 1:9\u201314\n\nPaul was a mighty apostle of God moving in strength and demonstration, yet he was a passionate intercessor who gave much of his time and energy to praying for others. Intercession is at the forefront of transformation. God will often reveal something as an invitation to pray. Prayer releases angels, strengthens people, allows us to partner with the heavenly realm, moves things in the heavens, and lifts people out of bondage. Prayer releases power.\n\nIn understanding the dimensions of the spirit, we must comprehend atmospheres. The atmosphere in a place is the emotional climate, the vibe, the feeling. Have you ever walked into a room where people were having a heated disagreement and stopped as soon as you walked in? Even though the verbal communication ended, the emotional and spiritual temperature was at a boiling point. We would say, \"You could cut the tension with a knife.\" That is a reference to the atmosphere. Atmospheres are manifestations of spiritual forces.\n\nATMOSPHERIC SHIFTS\n\nI remember the first time I visited Disneyland. They call it a place where dreams come true. As a small child, I had never experienced anyplace like it. My senses were on overload. There were costumed characters everywhere. The smell of carnival foods filled the park. Every section had a theme, and even the shrubs were cut in the shapes of Disney characters. None of that happened by accident. A huge team of people worked around the clock to cultivate an atmosphere that made guests feel happy.\n\nJust as Disneyland creates an atmosphere, churches and ministries create atmospheres. I have spoken in places where the atmosphere was so filled with hunger that you could just say the name of Jesus, and people would begin to shout! It is easy to move in the supernatural and release strong prophetic preaching in places like that because they are like spiritual wombs, alive with power and presence.\n\nRecently this was illustrated to me in a profound way. God had given me a strong prophetic message to preach. I released it in one place with great power and authority. As I preached, it was as though layers of revelation were being released. Words, statements, and thoughts blazed across my consciousness with ease. The place erupted with shouts as I spoke. I traveled to another place and preached along the same lines, releasing the same revelation, but it felt as though every word fell flat with no impact. When I finished, I couldn't wait to leave. I felt heavy and extremely frustrated. I asked the Lord why. He told me it was the atmosphere. He said that some places have the right atmosphere to release strong prophetic insight, while others do not, as the people have not prepared themselves and the word will not be received.\n\nWhen a home falls into sudden strife, it can be the manifestation of a demon spirit. When this happens, people are swept into the atmosphere of the thing without praying and discerning the source behind the heaviness. If we would be more sensitive, we would avoid a lot of unnecessary trouble. One of the most prophetic tools we have is our use of words. All of creation came into existence as a result of what God spoke in Genesis 1. We must guard and appropriate our words to create the right atmosphere and to shut the door on the wrong ones. When we announce things such as \"I am in a bad mood,\" we are inviting in spirits of heaviness and anxiety. When we say, \"I am on edge,\" we open the door for torment and fear. When we proclaim, \"The stress is killing me,\" we release a two-edged attack. We empower spirits of stress, trauma, and confusion, and we also release a death decree over our bodies. Many times we say things without any thought of the consequences. These are all proclamations that shift atmospheres.\n\nMany people just accept the current vibes in their homes as the reality without attempting to discern what is happening behind the scenes. Climates and atmospheres are typically the result of spiritual laws and principles. Atmospheres must be governed. We have authority to cleanse our homes. We can declare the promises of God, bind the forces of hell, and release the glory of God in our homes. The same principles can be applied in the workplace. I am convinced that many kingdom people are being sent into the marketplace to release and demonstrate the kingdom of God. Entire companies could shift as a result of prayer and decree!\n\nI have seen churches and ministries suddenly come under a cloud of heaviness. This is often the result of anti-anointing spirits lurking undercover, creating false accusations, releasing division, and plundering the camp. It takes rich prophetic insight, powerful intercession, and strong apostolic authority to turn back the tide and break the attack. Powerful churches are closed today because Jezebel walked in the back door and was not discerned or dealt with. Properly discerning the spiritual climate and identifying and effectively breaking the driving force of an attack can save the life of a ministry.\n\nPOINTS TO CONSIDER\n\n\u00bb Are you sensitive to the realm of the spirit?\n\n\u00bb Are you aware of the atmosphere around you?\n\n\u00bb When you sense something negative about another, do you pray or criticize?\n\n\u00bb Have you developed regular prayer habits? If not, then when, where, and how will you start?\n\nWHERE TO BEGIN\n\n\u00bb Become aware of atmospheres in various places.\n\n\u00bb Be intentional about praying over the atmospheres where you have authority.\n\n\u00bb Pay attention to dreams, promptings, and leadings.\n\n\u00bb Don't brush off heaviness. Ask for discernment about what is behind it, and take authority over it.\n\nPRAYER\n\n_Father, I thank You that the eyes of my inner man are open. I thank You that I hear, see, and know in the realm of the spirit. I thank You that I do not overlook Your subtle promptings and clear directions. I ask for divine insight and spirit leading in my life. I commit my steps to You, and I will be aware of Your instructions. In Jesus's name, amen._\nChapter 2\n\nTHE POWER OF THE ANOINTING\n\nONE OF THE most life-giving forces in the world is the anointing of God. It is like heavenly dew and rocket fuel mixed into one. There is nothing better than teaching that is anointed, preaching that is anointed, prayers that are anointed, and atmospheres that are filled with the presence of God. When you step into a church, ministry, or meeting that is void of the anointing, it is like walking into a spiritual desert. The atmosphere is sorely lacking, and there is no sign of victory.\n\nGOD'S POWER\n\nIn _Operating in the Anointing_ , Pastor Benny Hinn writes, \"Acts 1:8 promises that we will receive power after the Holy Ghost comes upon us. This promise is essential to understanding the anointing. When the Holy Spirit comes upon us, we are in God's presence. But God's presence is not the same as His anointing. God's presence is His glory and His person; His anointing is His power. In Hebrew, the word 'anoint' is _mashach_ , which means 'to rub in.' The Greek word is _chrism_ and means 'to smear.' When we receive the anointing, we are, in a sense, 'smeared' with His power, which means that He is not only upon us but within us, as oil would be rubbed into and absorbed by the skin. Simply put, the anointing is God's power. It is the manifestation and the result of His presence.\"\n\nThe anointing is the power of God upon and within people to fulfill spiritual assignments, break oppression, and catapult people into destiny. God never calls someone to do something without providing an anointing! We are completely unable to do the works of Jesus, move in the supernatural, or achieve exploits without Holy Ghost power.\n\nWhen the anointing hits a person or a gathering, there is a divine charge! Explosive spiritual shock waves are released, and things far beyond human achievement are accomplished. It is imperative that believers learn to value and recognize the anointing. The reality is that we are crippled in the realm of the supernatural if we do not move in the anointing.\n\nThe anointing is always connected to purpose. Prophets are anointed to prophesy and reveal. Apostles are anointed to break through, build, guide, and govern. Evangelists are anointed to work miracles and impact nations. Intercessors are anointed to pray fiery prayers that shake the heavens. Pastors are anointed to love, lead, and feed sheep. For every mandate, there is divine ability.\n\nYou shall take the garments and clothe Aaron with the tunic and the robe of the ephod and the ephod and the breastplate, and gird him with the skillfully woven band of the ephod. And you shall put the turban on his head and put the holy crown on the turban. Then shall you take the anointing oil and pour it on his head and anoint him. You shall bring his sons and put tunics on them. You shall gird them with sashes, Aaron and his sons, and put the headbands on them, and the priest's office shall be theirs for a perpetual statute. Thus you shall consecrate Aaron and his sons.\n\n\u2014EXODUS 29:5\u20139\n\nThe Lord gave very strong instructions concerning the priesthood. They had to be set apart, clothed, and then anointed to fulfill their priestly duties. This, of course, was an Old Testament example. Under the old covenant everything manifested through type and shadow. We see things in a picture, because God was not indwelling people as He is in our day.\n\nYou also, as living stones, are being built up into a spiritual house as a holy priesthood to offer up spiritual sacrifices that are acceptable to God through Jesus Christ.\n\n\u20141 PETER 2:5\n\nWe are called priests to be set apart for the Lord. When we become followers of Jesus, there is a full surrender. It is one of the key elements of living a life full of spiritual power. Holy Spirit leads us into a deeper and more intense relationship with the Father.\n\nWhat kind of clothing do we wear in the spirit? What are our priestly garments? Are they outward or spiritual? Many in the church debate what you can and cannot wear as children of God. Fashion has radically changed and evolved through the generations. While I believe in decency, the need for a person to dictate what others should look like often comes from strong roots in a religious mentality of bondage. I believe our priestly garments are not natural but spiritual. We are given powerful weapons of war and armor that thwart the attacks of the wicked one.\n\nFinally, my brothers, be strong in the Lord and in the power of His might. Put on the whole armor of God that you may be able to stand against the schemes of the devil.\n\n\u2014EPHESIANS 6:10\u201311\n\nIn the spirit realm we have mantles, or supernatural garments, that bring life, power, and authority according to our unique assignments. We have garments of righteousness through our acceptance of the full work of Jesus at Calvary. We have spiritual armor to protect and equip us for battle.\n\nOur anointing is not natural. It is not just oil from a bottle but divine oil from heaven filled with life, strength, and breakthrough. We all have a unique and powerful anointing available to us according to our assignments.\n\nI will never forget one of my early experiences with the anointing. I was in Bible college and so hungry for God! I spent hours in private worship and prayer. I so desperately wanted His presence. It was as if I were in a Holy Ghost washing machine. He was powerfully cleansing and delivering me.\n\nI had decided to join a student choir, and we were scheduled to sing at an area church. I was asked to do something special during the program, and I felt completely unqualified. I told my leader I did not feel ready. I told him I didn't have the skill to do it, and that was the truth! I did not realize it was a God setup.\n\nAs my time approached, my stomach was tied in knots. My mind was swirling; I just knew I would mess up. I got up to do my part, and sure enough, I began to mess up. Then something shifted. I felt something like liquid fire bursting forth from deep inside of me and surging from my head to my toes. It was an indescribable sensation. It was as if in a moment every high point I had ever experienced in prayer suddenly merged to become one overwhelming moment in the spirit.\n\nSuddenly words started flowing from another realm. What was happening? What was I saying? Would I get in trouble? I was off the plan and off course, but I was in another realm. My eyes were closed, and I was saying what was coming to my spirit supernaturally, with no fear. All nervousness and fear were swallowed up in this heavenly surge.\n\nWhen I finished speaking, I opened my eyes, and the altar was filled with people. Some were weeping; others, kneeling; but all were encountering God and His transforming power. The whole event switched gears and became a spontaneous revival meeting as God poured out on His people. That was the night I was totally transformed as I got a taste of both my mantle and the anointing that would catapult me into nations. The breaker showed up that night to tear down walls, remove bondages, and liberate hearts. My fears, shortcomings, and own thoughts were effortlessly swallowed up in the anointing. That was the first of many such experiences in the school of the spirit that God used to train me.\n\nDIFFERENT FLOWS OF ANOINTING AVAILABLE TO BELIEVERS\n\nWherever there is anointing, there is divine flow! To accomplish the will of God, there is a need for various types of anointings. Let's explore a few anointings that need to be active in the church and in the lives of God's people.\n\nMiracle anointing\n\nMiracles are instant turnarounds! They are manifestations of the supernatural in everyday life. Miracles can happen in every arena of life. We can experience financial, family, debt-cancellation, and healing miracles. God is a miracle-working God! Miracles are commonplace in the kingdom of God. The miracle anointing is the divine flow of God to perform miracles. It must be recognized and maximized. Typically there are instructions from God that open up this realm of anointing.\n\nGod worked powerful miracles by the hands of Paul. So handkerchiefs or aprons he had touched were brought to the sick, and the diseases left them, and the evil spirits went out of them.\n\n\u2014ACTS 19:11\u201312\n\nHealing anointing\n\nHealing is one of the works Jesus accomplished on the cross. The Bible tells us He sent His word and healed us (Ps. 107:20). The healing anointing is the divine presence of God to accomplish supernatural healing. Healing can be a process. It is not unusual for someone to have a sense of strength come into his or her physical body and begin driving out the weakness.\n\nNow when the sun was setting, all those who had anyone sick with various diseases brought them to Him. And He laid His hands on every one of them and healed them.\n\n\u2014LUKE 4:40\n\nProphetic anointing\n\nThe prophetic anointing is the power of God to prophesy, see in the spirit, receive revelation, and function prophetically. There are multiple ways that God speaks prophetically. When there is a prophetic anointing, the level of revelation and divine insight becomes explosive. When Saul came into the company of prophets, it caused the prophetic anointing to hit him, and he began to speak prophetically. That is what happens when a prophetic anointing is at work. It becomes easy to see, know, hear, and declare. Oil of revelation begins to flow from heaven.\n\nWhen they came to the hill, a group of prophets met him. And the Spirit of God came upon him, and he prophesied among them. So when all who previously knew him saw that he prophesied among the prophets, the people said one to another, \"What is this that has come upon the son of Kish? Is Saul also among the prophets?\"\n\n\u20141 SAMUEL 10:10\u201311\n\nFinancial anointing\n\nThere is a realm in the anointing that attracts increase; it helps create wealth and tap into the abundant blessings of God. When it is an anointing, there is an uncommon ease. Things happen suddenly and supernaturally. Many believers miss this realm because they do not understand that there is power (anointing) to get wealth!\n\nBut you must remember the LORD your God, for it is He who gives you the ability to get wealth, so that He may establish His covenant which He swore to your fathers, as it is today.\n\n\u2014DEUTERONOMY 8:18\n\nBuilding anointing\n\nThroughout Scripture God places value upon building. David told his son that he would inherit the task of building God a magnificent house.\n\nThen David called Solomon his son and commanded him to build a house for the LORD, the God of Israel.\n\n\u20141 CHRONICLES 22:6\n\nThe Lord revealed Himself in this passage as the founder and builder of the church:\n\nAnd I tell you that you are Peter, and on this rock I will build My church, and the gates of Hades shall not prevail against it.\n\n\u2014MATTHEW 16:18\n\nHe also sent Holy Spirit to live in us and pray through us. Praying in the Spirit builds up our spirit man and the realm of faith in us. The gift of prophecy builds people up. When the word of the Lord is released, the people of God are strengthened, encouraged, and established.\n\nGod is into building, and He has provided a strong grace to build. Apostles are foundational gifts to the church. They are sent ones\u2014sent to build! The church is called to be an advancing, prevailing, and building people. When He tells people to do particular exploits, there is divine ability to build. Leaders need to tap into the building anointing. Businesspeople need to tap the building anointing. Apostles and prophets are called to help build\u2014they need to flow in that grace, claim it, and stir it up.\n\nPrayer anointing\n\nI was preaching a revival meeting when suddenly strong intercession broke out all over the building. People began to run to the altar, praying under a supernatural leading from heaven. I didn't urge it or create it; heaven just showed up and people began to move into deep realms of prayer. Whether it is in our personal lives or as a body of believers, there is an anointing for prayer that we can tap into. We should look for that river and dive deep into it!\n\nEven them I will bring to My holy mountain and make them joyful in My house of prayer. . . . For My house shall be called a house of prayer for all people.\n\n\u2014ISAIAH 56:7\n\nLeadership anointing\n\nLeadership\u2014people placed in roles of authority\u2014is a heavenly concept. Many people struggle with this because of ignorance and rebellion, but it is still the truth. God never appoints a leader without the anointing to serve. Leaders need to learn the value of the anointing in this area and make a conscious effort to ask God for it and maximize it. The people who are being served by a leader need to pray and ask the Lord to anoint the leader.\n\nLet every person be subject to the governing authorities, for there is no authority except from God, and those that exist are appointed by God. Therefore whoever resists the authority resists what God has appointed, and those who resist will incur judgment.\n\n\u2014ROMANS 13:1\u20132\n\nHelps and service anointing\n\nHelps is listed among the supernatural ministry gifts in 1 Corinthians 12:28. There is an anointing to help. Every purpose of God, work of God, leader, and ministry needs help. There is a blessing in helping. Many churches and ministries need to pray and teach on the anointing to help. People's lives can be supernaturally changed when they begin to serve the works of the Lord.\n\nAs everyone has received a gift, even so serve one another with it, as good stewards of the manifold grace of God.\n\n\u20141 PETER 4:10\n\nSonship anointing\n\nThere is an anointing for us to function as sons and daughters of God. We were once orphans, but the power of the blood of Jesus brought us into total freedom and established us in the family of God. The enemy will attempt to bring accusation and heaviness, but we do not have to receive it. We can rest in what heaven has already said and enjoy the freedom of sonship.\n\nYet to all who received Him, He gave the power to become sons of God, to those who believed in His name, who were born not of blood, nor of the will of the flesh, nor of the will of man, but of God.\n\n\u2014JOHN 1:12\u201313\n\nBlessed be the God and Father of our Lord Jesus Christ, who has blessed us with every spiritual blessing in the heavenly places in Christ, just as He chose us in Him before the foundation of the world, to be holy and blameless before Him in love; He predestined us to adoption as sons to Himself through Jesus Christ according to the good pleasure of His will.\n\n\u2014EPHESIANS 1:3\u20135\n\nFamily anointing\n\nThere is divine power and assistance to raise families. There is supernatural ability for marriage and relationships. Too many people only see certain things as anointed, but every covenant carries an anointing. We can tap into the realm of God's power for our families. God can teach us how to be better spouses and parents through the anointing. This is one facet of the anointing\u2014to release revelation and insight.\n\nBut from the beginning of the creation, God \"made them male and female.\" \"For this cause shall a man leave his father and mother, and cleave to his wife, and the two shall be one flesh.\" So then they are no longer two, but one flesh. What therefore God has joined together, let not man put asunder.\n\n\u2014MARK 10:6\u20139\n\nMusical anointing\n\nMusic is deeply spiritual. It can open realms and create atmospheres. A simple lyric becomes lodged in our brains for days. A certain rhythm causes our bodies to move without thinking about it. Music carries something that affects us profoundly. There is an anointing for the right kind of music. There is an anointing on songs, lyrics, and sounds. There are realms of worship that push back the powers of hell and open the heavens. We need anointed musicians, minstrels, and psalmists!\n\nIt happened that when the evil spirit from God came on Saul, David would take the lyre in his hand and play. So Saul was refreshed and was well, and the evil spirit departed from him.\n\n\u20141 SAMUEL 16:23\n\nKing David raised up an army of full-time worshippers that included four thousand musicians. He said:\n\nFour thousand shall offer praises to the LORD with the instruments that I have made for praise.\n\n\u20141 CHRONICLES 23:5\n\nWorship was a weapon of choice in Israel. It was radical worship that propelled David into the palace. He was a prophetic worshipper who knew how to tap into the anointing to push back darkness. He created an army of prophetic worshippers who staffed a throne room type of prayer-and-worship model in his kingdom. This is a model that is still relevant today. David valued the call to radical worship, prayer, and intercession. He established his own company of worship leaders who kept the incense of prayer and longing rising up to the throne of God.\n\nThese are the singers, heads of the fathers' houses of the Levites, who lodged in the chambers, and were free from other duties; for they were employed in that work day and night.\n\n\u20141 CHRONICLES 9:33, NKJV\n\nDeliverance anointing\n\nOne of the most elementary aspects of the anointing is the bondage-breaking power. Demon spirits entangle and ensnare, but the anointing can destroy those ties! When the anointing of God begins to move, it breaks the cords that have held God's people. There is power to set the captives free.\n\nThe Spirit of the Lord GOD is upon me because the LORD has anointed me to preach good news to the poor; He has sent me to heal the broken-hearted, to proclaim liberty to the captives, and the opening of the prison to those who are bound.\n\n\u2014ISAIAH 61:1\n\nHe said to them, \"I saw Satan as lightning fall from heaven. Look, I give you authority to trample on serpents and scorpions, and over all the power of the enemy. And nothing shall by any means hurt you.\"\n\n\u2014LUKE 10:18\u201319\n\nRegional anointing\n\nThere is an anointing to impact regions. This is one of the cornerstones of an apostolic ministry. There is an established sphere of influence. There are regional plans, regional purposes, and supernatural power to establish the kingdom of God within regions.\n\nWe are not boasting of things beyond our measure in other men's labors. But we have hope that when your faith is increased, our region shall be greatly enlarged by you, to preach the gospel in the regions beyond you and not to boast in another man's accomplishments.\n\n\u20142 CORINTHIANS 10:15\u201316\n\nGlobal anointing\n\nSome have an anointing for global exploits. Their platform is the world, and they are called to go forth, shaking nations. There are global prophets who are given mandates for nations. They speak in a higher dimension of authority and insight. They are releasing prophetic insight not just for people but for nations. There are evangelists who have been supernaturally propelled forth into the nations of the world. They have faith, a word, and favor for nations. There are apostles who have been given a global platform to transform nations. They are given strong words for the nations accompanied by an earth-shaking anointing. God sends them with supernatural power and authority.\n\nBefore I formed you in the womb I knew you; and before you were born I sanctified you, and I ordained you a prophet to the nations.\n\n\u2014JEREMIAH 1:5\n\nCorporate anointing\n\nThere is an anointing that is unique and different when God's people gather. It is the fruit of various gifts coming together under one united banner. As believers worship, pray, and pursue heaven together, there is precious dew that flows down, bringing life, strength, and breakthrough. In many ways it is a multiplication of the individual anointing upon the lives of people.\n\nBehold, how good and how pleasant it is for brothers to dwell together in unity! It is like precious oil upon the head, that runs down on the beard\u2014even Aaron's beard\u2014and going down to the collar of his garments; as the dew of Hermon, that descends upon the mountains of Zion, for there the LORD has commanded the blessing, even life forever.\n\n\u2014PSALM 133:1\u20133\n\nBeliever's anointing\n\nThere is a well that dwells on the inside of every believer. It is part of our new nature and the complete work of Jesus at Calvary. We are not only appointed for dynamic kingdom works but anointed for them also. The realm of healing, miracles, casting out demons, spirit leading, and divine insight is available in the life of every believer. This has nothing to do with an office or the grace available for that specific work. This is the fruit of the kingdom in our lives. We can tap into the believer's anointing to be a powerful witness to the world and help change the lives of those around us.\n\nHe said to them, \"Go into all the world, and preach the gospel to every creature. He who believes and is baptized will be saved. But he who does not believe will be condemned. These signs will accompany those who believe: In My name they will cast out demons; they will speak with new tongues; they will take up serpents; if they drink any deadly thing, it will not hurt them; they will lay hands on the sick, and they will recover.\"\n\n\u2014MARK 16:15\u201318\n\nTHE SECRET IS IN THE SURRENDER\n\nGod anointed Jesus of Nazareth with the Holy Spirit and with power, who went about doing good and healing all who were oppressed by the devil, for God was with Him.\n\n\u2014ACTS 10:38\n\nJesus walked in a tangible anointing that had maximum impact everywhere He went. How did He maintain such a high level of spiritual operation? The secret is in the surrender! He was completely surrendered to the will of the Father\u2014clinging to His voice for instruction, continually listening, looking, and leaning into the realm of the spirit.\n\nFor I came down from heaven, not to do My own will, but the will of Him who sent Me.\n\n\u2014JOHN 6:38\n\nJesus was not doing His own thing. The devil did his own thing and was swiftly kicked out of heaven. Many people follow in the footsteps of Lucifer, seeking to live their own lives, fulfill their own dreams, build their own ministries, and compete for their own voices to be heard. This is the epitome of demonic behavior. There are many in the ministry who are camouflaged rebels. They have the right language, yet they lack the surrender.\n\nAnd He said to them, \"I was watching Satan fall from heaven like lightning.\"\n\n\u2014LUKE 10:18, NASB\n\nDon't get me wrong\u2014it is fantastic to have big, bold dreams and wild adventures of faith. I do not believe God created any human to live a boring and mundane life. Anointed people go big! The difference is the surrender. Are we allowing God to dream enormous dreams through us? Are we allowing the Father to place His desires in our hearts? Are we willing to change course at a moment's notice if the Lord asks us to? These are all questions we must ask if we dare to do exploits.\n\nGod is the ultimate dreamer and master planner. He left absolutely nothing to chance and authored every step of your destiny. He meticulously planned your entrance, your life, and your graduation from the earth realm into glory. He has ordained exploits beyond your wildest imagination.\n\nThere are too many people crouched in the corner, shackled with the bonds of disappointment. They have checked out of dreaming and are living on autopilot, merely existing. God wants to heal the dream nature inside of them. He wants to touch the broken places and command the dead bones to live again. He wants to speak to the broken dreams and declare purpose in the midst of pain.\n\nSome had a false start. They launched into a thing without the proper timing. The Bible declares that every purpose has an appointed time (Eccles. 3:1). They heard a word, dreamed a dream, saw a vision, and stepped out with little or no thought about the timing and the plan. There is always divine timing. It is an illegal spiritual operation to advance into a thing without obtaining the wisdom of God for the timing.\n\nAnointed people are sensitive to times and seasons. They have a strong Holy Ghost GPS. They know the dangers of a false start. Don't misunderstand me\u2014everyone has missed it in the area of timing. Peter is perhaps one of the greatest biblical examples of a passionate disciple who kept making false starts. His passion would often get ahead of his character, yet God kept restoring him and reviving him.\n\nI hear the Lord saying, \"For every false start there is an authentic launch. Some moved before the time and received flattering words, only to be greatly disappointed, but I am binding up their wounds and reviving them. I am reigniting the fire of destiny within them. I am relaunching them for the next phase. The time shall be redeemed. The enemy has been screaming, 'It's over,' but I am saying it is just beginning!\"\n\nI see some relaunches on the horizon. It is time to launch again with fresh purpose. There is a new beginning for you. There are things to be set in motion in your life and destiny. The devil keeps intimidating you and telling you that it is over, but God is going to launch you once again with renewed passion, purpose, and momentum.\n\nSurrender is the secret weapon of anointed people. The anointing flowing in abundance is the confirmation of surrender. Jesus was not striving to be anointed. He was doing the will of the One who sent Him. The lower He went in His own ambitions, the higher He rose in the spirit. This is a great mystery, but it is a power secret of anointed people\u2014first is last, and last is first (Matt. 20:16).\n\nWhy is surrender so significant? Because all covenants center on sacrifice and surrender. Every covenant demands a shedding of blood. It is one of the signs of the surrender. There must be a sacrifice in order to have a legitimate covenant.\n\nAnd not through the blood of goats and calves, but through His own blood, He entered the holy place once for all, having obtained eternal redemption.\n\n\u2014HEBREWS 9:12, NASB\n\nCOVENANT VIRTUE\n\nEach covenant offers certain unique rights and privileges. It also creates boundaries and alliances. One of the strongest pictures of covenant thinking is in the life of Abraham. After walking through the blood, the Lord declares to Abraham that He will place a generational blessing upon his lineage (his seed). (See Genesis 15.) The obedience of Abraham and his friendship with God would have generational implications.\n\nAll too often ministers spend hours focusing on every possible generational curse, the need for repentance, and the possibility of bondage without ever emphasizing the power of a covenant with God and the generational blessings available. I believe that God is establishing strong emphasis on engaging the realm of the spirit with a covenant mind-set\u2014in other words, stepping into a battle from a place of victory and absolutely gutting the powers of hell with the name, the Word, and the blood. If God's people awaken to their true identity and the strength of their covenant, they will be undefeatable, unstoppable, and immovable!\n\nThe Lord used strong covenant language with Abraham that makes many in the modern Christian world uncomfortable.\n\nI will bless them who bless you and curse him who curses you, and in you all families of the earth will be blessed.\n\n\u2014GENESIS 12:3\n\nThis is a lost reality; nonetheless it is a spiritual law. Two people in covenant fight the same battles and share the same enemies. This is the bedrock of effective spiritual warfare. Many people have passive spirits. They want to remain neutral in every matter and avoid conflict at all costs. They create covenant alignments to get the perks yet are unwilling to defend those they are supposed to be aligned with or fight the battles with them. This is a transaction mentality and not a covenant mentality. Transactional people enter relationships to get a benefit, but they lack commitment and endurance. They are fair-weather friends and leaders who run at the first sign of trouble.\n\nMany churches, ministries, and organizations are weak and spiritually diluted because of this false thinking. People come and go with no commitment and no covenant. These people abandon their posts at the first sign of battle. All covenants are proved in the thick of battle. When war erupts, the covenant is tested.\n\nLet's expand our grasp on these concepts. In today's church culture we are adamant about creating converts. We send out teams to pray with people on the streets in hopes of converting them. We plan big events to draw in the lost; we design the whole thing to talk them out of going to hell and accepting Jesus. I applaud all these efforts and cheer them on\u2014yet there is more!\n\nWas the church instructed to simply make converts? Is this the highest expression of _ekklesia_ (the church)? Should this be the ultimate aim of the church?\n\nGo therefore and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all things I have commanded you. And remember, I am with you always, even to the end of the age.\n\n\u2014MATTHEW 28:19\u201320\n\nImagine a groom going from place to place, having numerous wedding ceremonies with different women, leaving suddenly from each, and moving to the next bride. This seems crazy, but in many ways it is what we are doing in modern Christianity. We ask people to wear the ring without fully engaging in the covenant. Conversion is only the beginning of the matter. The highest expression of the kingdom is disciples who are reproducing, representing, and establishing kingdom culture.\n\nIt was at this time that He went off to the mountain to pray, and He spent the whole night in prayer to God. And when day came, He called His disciples to Him and chose twelve of them, whom He also named as apostles.\n\n\u2014LUKE 6:12\u201313, NASB\n\nSomething was so deep in the heart of Jesus that He was up all night praying, pondering, and seeking wisdom. What could be that important? What could be that serious? He was about to choose the twelve men whom He would pour into day and night, the twelve who would change the world. He was about to select a ragtag group of men to cut a covenant with and share the deepest part of Himself. These were not casual relationships\u2014these were the very foundations of the ministry of the Lord of glory! He was going to choose His disciples, and there was no matter deeper in the heart of Jesus than this one.\n\nThese men would eat, travel, dream, and weep together; they would fight for one another, fight against one another, and do mind-blowing exploits. Jesus began a discipleship ministry, and relationship with these men was one of His most demanding tasks. Yet His limited time with these men built the foundations of Christianity as we know it.\n\nJesus preached to the masses, healed the multitudes, and shook cities and regions. He spent countless hours teaching the deep things of the kingdom. He taught the unlearned using pictures, parables, and stories. But when the crowds left, He was making disciples. He was building covenant relationships and investing His very person into those men destined to change the world.\n\nOften in this world we strive for bigger and better. We measure ourselves by our achievements, and we feel absolutely defeated if someone next to us seems to be outdoing us. Sometimes we live our whole lives doing outrageous things but never experience true covenant relationships.\n\nCovenant places boundaries upon our lives. When we become disciples, we become surrendered ones. I am often asked what I think about particular social issues and evolving morality in our society. In the past I have been far too quick to state an opinion. I have come to understand that as I am a bondservant of Jesus, my opinion does not matter in such affairs. When I became a disciple, I became tied to the value system of heaven. I am not allowed the freedom to decide what I think is right and wrong. I have divine guidance. At times the integrity of Jesus challenges my human heart and understanding, but it is my covenant responsibility to uphold the values of Jesus. There are things that I don't do because heaven has not authorized them. A life of purpose is a life of focus and surrender.\n\nThe anointing was one of the primary things that made a difference in Jesus's ministry. As He walked the face of the earth, the tangible touch of God was upon Him. Those with issues were drawn to Him in a supernatural way. The sick, the tormented, the afflicted, and the bound came. Not only did they come, but they left forever changed.\n\nJesus was anointed. He was supernaturally charged to break the back of demon powers and bring miraculous results. This is one of the pillars of purpose in the anointing. The anointing provides supernatural power to break the works of hell.\n\nAnd it shall come to pass in that day, that his burden shall be taken away from off thy shoulder, and his yoke from off thy neck, and the yoke shall be destroyed because of the anointing.\n\n\u2014ISAIAH 10:27, KJV\n\nA yoke is a piece of equipment used to connect oxen to a plow. The spiritual picture here is an instrument that attaches a person to a heavy burden. When the anointing comes, it breaks and destroys not only the evil weight but also every device that connects a person to it. Wherever there is great anointing, there is great freedom!\n\nNow the Lord is the Spirit. And where the Spirit of the Lord is, there is liberty.\n\n\u20142 CORINTHIANS 3:17\n\nPhilip the evangelist was leading a tremendous revival. As the anointing was moving, demons were fleeing.\n\nWhen the crowds heard Philip and saw the miracles which he did, they listened in unity to what he said. For unclean spirits, crying with a loud voice, came out of many who were possessed. And many who were paralyzed or lame were healed. So there was much joy in that city.\n\n\u2014ACTS 8:6\u20138\n\nPhilip was leading a tremendous move of God with mass deliverance. There is nothing like a heaven-sent revival. Historically revival has been used to change the destiny of a generation and call people back into deep intimacy with God. There have been various types of revivals, but all have featured spectacular displays of God's power. There is something about miracles, signs, and wonders that pierces the hearts of people.\n\nAN EMAIL LEADS TO REVIVAL\n\nA strong wave of revival can overturn a church, a community, a state, or even a nation. The wave comes in like a flood, subduing the powers of hell and releasing the reality of the glorious kingdom of God. I remember a very special revival in which I was privileged to have a role. The Lord sent the pastor of a small church into my life through an email. At the time, I was leading a regional ministry, and we had regular gatherings to stoke the fires of revival. This pastor felt tugged by the Lord to reach out to me. He sent me an email stating that he knew he was supposed to connect with me, and the Lord told him that his city would be changed when I came there. As I read the words, I felt a bit overwhelmed and questioned their authenticity. After all, maybe he was just exaggerating.\n\nThis pastor began attending some of my meetings, and the Lord connected us through a series of events. We built a relationship, ministered alongside one another, and strategized about the coming days. After a long period of time, the Lord reminded me of his initial email\u2014it was about me coming into his region. We had preached, traveled, and prayed together, but I had not yet been to his region. I felt a quickening to go. I went for what was supposed to be a weekend of meetings, but heaven swept into the building, and the miraculous of God began to break forth. There were many miracles. I vividly recall one that would become a catalyst for revival. There was an older man who said he had extreme bone problems. He had evident mobility issues and was unable to walk without major assistance. He had two canes and could barely move. As I stood before him praying, the gift of faith hit me, and I began to command him to rise up and walk. God's power overcame him, and he immediately jumped up and took off walking, completely without issue. It was a phenomenal sight. He lifted up the two canes and held them high as he walked with full mobility. Revival had begun!\n\nI stayed in that region for months, praying, preaching, prophesying, and holding revival meetings alongside the host pastor. There were many prophetic confirmations about what God was doing. The Lord had sent many other trailblazers ahead to seed the ground with the Word of the Lord. People began to come from various states and regions, and we outgrew the small church where the meetings began and had to rent a larger facility. Everything was supernatural! People were healed, saved, and filled with Holy Spirit. Lives were forever changed, and kingdom plans were birthed.\n\nIn a \"suddenly\" moment God fulfilled words given years before over that pastor, his church, and the region. He partnered with prophetic words given years before about the name of and strategy for this move. The Lord grew his ministry far beyond what it was when the revival began.\n\nThis is one of the reasons why demon spirits hate revival. Revival can bring a sudden fulfillment of dormant promises. Revival releases tremendous velocity in the spirit that breaks the clutch of demons. Revival can refresh the driest people and penetrate the hardest hearts. Revival releases a torrent of anointing and glory that plunders the structures of hell. The devil hates revival and strives to attack any person or ministry pressing in for it.\n\nThe anointing obliterates demonic powers. This is why there are demon spirits that have an assignment against the anointing. They want to quench the fresh flow of heaven in ministries, believers, and worship times. They want to stifle and bind up men and women of God. They want to create dry, dead, religious atmospheres that pose no threat to their kingdom. They launch strategic attacks against the people of God and against anointed places. They do not strike randomly but with intent.\n\nThese demons are called anti-anointing spirits. They fight and hinder the flow of life and power in the realm of the spirit. In 1 John 2:18 there is a warning about antichrists: \"Little children, it is the last hour. As you have heard that the antichrist will come, even now there are many antichrists. By this we know that it is the last hour.\" What is the meaning of _antichrists_? _Christ_ represents the Anointed One and His anointing. So the picture here is of a family of demonic entities that have a target to fight the anointing. They are on assignment to subdue, restrict, and overwhelm.\n\nWhen a ministry begins bringing in a fresh anointing of life and power, these antichrist spirits take notice and conceive heinous plans to stop the flow. They will send strife, division, accusation, hindrance, and financial attacks. They launch an all-out assault to stop the flow.\n\nAs we expound on the toxic trio of demons in this book, we will identify their operations and expose their agenda. While each of the three spirits is unique, they are similar in their type.\n\nThey are all spirits that work to destroy the anointing of God.\n\nPOINTS TO CONSIDER\n\n\u00bb Have you experienced resistance when you have pressed in to particular spiritual assignments?\n\n\u00bb Your anointing is connected to your purpose.\n\n\u00bb Resistance is confirmation that what you are doing is dangerous to the kingdom of darkness.\n\n\u00bb What areas of your life need to be surrendered?\n\n\u00bb What is the Lord dealing with you about concerning devotion, obedience, and surrender?\n\nWHERE TO BEGIN\n\n\u00bb Learn to recognize spiritual resistance in your personal life, ministry assignments, and corporate body of believers.\n\n\u00bb Bind the enemy when you sense the resistance.\n\n\u00bb Tap into the various types of anointings that are available for your life. Pray over them\u2014confess that they are yours, and ponder the scriptures associated with them.\n\nPRAYER\n\n_Father, I thank You that I am created for a purpose and destined for kingdom assignments. I ask You to help me recognize the resistance of the enemy and be strong in You to break the powers of darkness. I confess that I am fully committed to kingdom assignments, mandates, and relationships. I am spiritually sensitive, activated, and aware, in Jesus's name._\nChapter 3\n\nJEZEBEL, THE SPIRITUAL ASSASSIN\n\nI WAS A YOUNG ministry pioneer embarking on a brand-new adventure. God had given me direction, and I was ready to go. I could not wait to fulfill the dreams that were in my heart. My sense of expectation was surging as I stepped forward to answer the call to birth something new. It was a faith adventure in every way. I felt the Lord leading me into unknown territory with few resources. I had a word, I had a dream, and I had faith\u2014with that I jumped into the deep and launched a brand-new ministry.\n\nSHE WANTS THE HEADS OF FORERUNNERS\n\nIn the early stages of the journey the Lord brought someone before me whom I had only casually met a few times through some ministry friends. I felt as though I was in over my head, trying to launch something with no help. I reached out to that casual acquaintance to see if there was any interest in joining our ragtag team and launching into the deep. To my surprise, the answer was yes! We added a new team member.\n\nAs our team grew, there were exciting new relationships. People were being added to assist in the various areas, and God was doing all types of miracles. We experienced financial miracles, healing miracles, family miracles, and strong waves of deliverance. Our meetings were always unique. God showed up in power each and every time.\n\nTwo of the team members were a gracious couple who seemed to have a strong hospitality gifting and to be ready to serve in any way. As time progressed, I had some subtle checks but nothing too drastic. There were some questions, little concerns, things that I would find a way to brush off because I felt as though I needed every helper I could get. What would we do if we lost a key person?\n\nOne night while alone with this couple, I witnessed something that startled me and shook my confidence. The wife began to have a conversation with a member of my family that seemed to take a \"prophetic\" turn, probing into past wounds and pains. It happened suddenly and without warning. There were some accurate details and pieces of information, but as it unfolded, I knew in my spirit something was terribly off. What was it?\n\nThe event had some of the markings of a prophetic encounter, but there was something underlying. It was forceful, was obtrusive, and ended in confusion with no peace. Even deeper, when it was unfolding, there was not an awareness of God's love and redemptive nature. There was instead a sense of unlawful operation. There was a spiritual dynamic taking place, but it wasn't at all good or comforting. I knew that something was terribly amiss, and I knew that I would never allow anything like that again on my team. Now I had to discern what was taking place.\n\nI maintained a heart of pursuit with an attitude of seeking wisdom to resolve what I was sensing. Soon it would all play out in vivid detail. There was an underlying agenda with a power play that would unfold. What I sensed during that encounter was far greater than what I knew at that time. I was about to have a battle with a spirit that I had encountered before. This would be one of the first of many rounds with this spirit in that phase of my life and ministry. The same demonic entity would show up again and again with a determination to shut down our ministry and back me into a corner. It tried, but it did not prevail.\n\nAs time passed, there were further confirmations of the suspicions we had about the couple on our team. One of the most vivid skirmishes was when this spirit confronted me and my wife late one night in the couple's home. It was an actual spirit that showed up and began to unload a massive attack. We both sensed the evil presence, and our minds started to swirl as the demon unloaded garbage and accusation. I knew in that moment that there was an unclean nature to the spirit in that home. It had an assignment of seduction. I would come to find out that seduction is one of the primary tools of the devil that we were about to confront.\n\nThe entire thing came to a head after a series of small rebellions. The couple would choose to ignore an instruction, to do something they were not permitted to do, or to take charge of a corporate gathering without my permission. The once seemingly kind and loving couple morphed into something very different. There were major emotional and spiritual instabilities manifesting in the wife. The husband seemed firm at times but uncommonly passive at other times. There was something about the wife that appeared lustful but remained cloaked under a stream of \"God said\" statements, kind acts, and prophetic declarations.\n\nFinally there was an ultimatum delivered. I was told that the word of the Lord had been given that the wife was supposed to be my co-leader. The Lord had spoken to her that she was not to be under my authority but on an equal playing field. There was much pressure given to make the case, and her husband dutifully went along with it. Eventually we were in a state of strong disagreement, and a power play was under way. It was only after I discerned what was glaringly apparent behind the scenes that the full extent of the spiritual force was revealed\u2014this was a wrestling for control, illegal access, and illegitimate authority. This was a power play by the Jezebel spirit.\n\nI could now clearly see the roles of Ahab and Jezebel in this scenario. I want to be clear that this spirit can and will operate through any gender. It is simply looking for a host to seduce, control, and manifest through. I drew a firm line in the sand and stood up to the spirit. There was only one choice: repent and be free, or exit the team. After much posturing, attempted manipulation, and exaggerated emotions, the couple exited. This was the first of many battles we had in that ministry with that particular spirit.\n\nWARRIORS WIN!\n\nThrough the years, I have often been criticized for my preaching on demons and the Jezebel spirit, in particular. Many times I have wished that I wouldn't have to see, hear, or deal with these wicked enemies. I wish that I could live as others do, like ostriches with their heads in the sand, but I cannot. I am a warrior! I am a justice person who sees things as absolute. I hate the devil and the powers of hell. I cannot afford to be passive in spiritual warfare, or the enemy will take me out. _You_ cannot afford to be passive either.\n\nMy education on the Jezebel spirit was born in the belly of the fight. I never intended to learn about or preach on this spirit, but I fought it many times, as will any prophetic person. It is one of the major enemies of all prophets, prophetic people, and prophetic churches, hubs, and ministries. You cannot and must not leave Jezebel unchecked! There are wise strategies and sure methods for dealing with this destructive demon, but it must be addressed.\n\nThe Jezebel spirit is a spiritual assassin that takes particular aim at all things prophetic. To more fully understand this, let's dive deeper into one of the key components of prophetic ministry and one of the Bible's most legendary assassinations orchestrated by the Jezebel spirit.\n\nAs they departed, Jesus began to say to the crowds concerning John, \"What did you go out into the wilderness to see? A reed shaken by the wind? If not, what did you go out to see? A man dressed in soft clothing? Indeed, those who wear soft clothing are in kings' houses. Then what did you go out to see? A prophet? Yes, I say to you, and more than a prophet. For this is he of whom it is written:\n\n'Look, I am sending My messenger before Your face, who will prepare Your way before You.'\n\nTruly I say to you, among those who are born of women, there has risen no one greater than John the Baptist. But he who is least in the kingdom of heaven is greater than he.\"\n\n\u2014MATTHEW 11:7\u201311\n\nJohn the Baptist was known for his preaching of repentance and his declaration of Jesus as the Messiah. It is interesting that Jesus asks the people in this passage, \"What did you go . . . to see? A reed shaken by the wind?\" This could mean a variety of things, but I want to take a prophetic spin on this. I believe that John was shaken by the winds of the Spirit, shaken by prophetic utterance, shaken by a realm of _nabi_ (bubble up) preaching. Words and messages burst forth from his belly with authority and power.\n\nJohn was untamed. He occupied the wilderness places. He was not a polished prophet. He did not give a variety of prophetic predictions as other Old Testament prophets did. He released a relatively simple message, yet it catapulted him to the front of the line in Jesus's assessment of Old Testament prophets. He topped the list!\n\nJohn was a messenger. All prophets are some type of messenger. The means of communication and the style of release differ greatly based on the particular mantle, but they are messengers. There are various realms of the prophetic, different functions and diverse flows, but every prophetic revelation contains a message that at the end of the day points back to Jesus.\n\nProphets were and are those who prepare the ground for the seed. They speak what we have not yet seen, revealing that which can be seen in the realm of the spirit. They are by nature forerunners. This is critical in our understanding of why Jezebel hates the prophetic anointing. This evil spirit despises the work of true prophetic ministry because it prepares generations, regions, peoples, and nations for the move of God. It prepares the way. John prepared the way for Jesus, and this is what I believe Jesus was commending. Jesus referenced John as the greatest because he was chosen in the gap between covenants to prepare for the greatest spiritual transition since the fall of Adam. He was chosen to be the messenger to make the decree of all decrees. He was the forerunner to usher in the Christ.\n\nWhat is a forerunner? It is someone who goes out ahead. Anything and everything prophetic has a forerunner nature. Prophets, prophecy, and prophetic flow always reveal what is in the heart and mind of God. As God's heart and mind come into the space that we occupy, they burst forth carrying divine power. The release of a forerunner word is the momentum in the spirit to carry a person into the manifestation of what has been spoken. The revelation carries the power of manifestation. A prophetic word is not an abstract promise but a spiritual reality. When God speaks prophetically, you are carried beyond the human and natural dimension into the limitless realm of the spirit, where all things are possible. Prophetic forerunners move out ahead in order to make a way for those that will come. They blaze spiritual trails. They open up revelations.\n\nSEDUCTION IS HER GAME\n\nFor Herod had laid hold of John, bound him, and put him in prison for the sake of Herodias, his brother Philip's wife. For John said to him, \"It is not lawful for you to have her.\" When Herod would have put him to death, he feared the crowd, because they counted him as a prophet.\n\nBut when Herod's birthday was celebrated, the daughter of Herodias danced before them and pleased Herod. Therefore he promised with an oath to give her whatever she would ask. Being previously instructed by her mother, she said, \"Give me John the Baptist's head on a platter.\" The king was sorry. Nevertheless, for the oath's sake and those who sat with him at supper, he commanded it to be given to her. He sent and beheaded John in the prison. His head was brought on a platter and given to the girl, and she brought it to her mother. His disciples came and took up the body and buried it. And they went and told Jesus.\n\n\u2014MATTHEW 14:3\u201312\n\nJohn the Baptist was murdered at the request of a demon-possessed woman. This was not natural anger but spiritual rage. John refused to endorse the unlawful union of the king and his new love. This is one of the attributes of rugged prophets\u2014they will not bow to spiritual pressure when they perceive that something is wrong. They cannot be bought, intimidated, or manipulated. When they have heard the clear voice of the Lord, they become resolute. John was unwavering in his condemnation of this relationship.\n\nThe Jezebel spirit is an immoral demon. It uses various means to seduce its victims. One of its manifestations is sexual seduction. In this story we see a twofold perversion. First, Herodias captured the attention of the king so he would come into immoral relationship with her. Second, she used the allure of her daughter's dance to capture the attention of Herod and several men to ask for the head of the leading prophet.\n\nSeduction was used to order the assassination of a mighty forerunner. These same powers are at work today, attempting to annihilate the destinies, ministries, and spiritual exploits of prophetic leaders. Jezebel has strategic plans of distraction and deception. The Jezebel spirit had John's head cut off! There is so much revelation in this assassination. This spirit understands that if it successfully attacks the head, then it can kill the body. Many times men and women, chosen by God, begin to lead a charge toward change and reformation, but the Jezebel spirit bombards them with a multipronged attack in order to cut off the head of destiny!\n\nJezebel loves to intimidate prophets and prophetic people. This spirit wants to send prophets into hiding. It accomplishes this in a variety of ways. As Jezebel was empowered by a weak and wicked leader named Ahab, the true prophets suffered greatly.\n\nWhen Jezebel killed the prophets of the LORD, Obadiah took a hundred prophets and hid them in groups of fifty in a cave and fed them with bread and water.\n\n\u20141 KINGS 18:4\n\nThe prophets were forced into the cave. Jezebel was on a murderous rampage, determined to slaughter the voice of the Lord in the land. God's prophets had to go into hiding. They could not minister. They could not flow. They could not give revelation to the people or the land because they were hiding in the cave.\n\nThe Jezebel spirit comes to push the prophetic anointing into the cave. It comes to tear down prophetic ministries and overwhelm prophetic leaders. It comes to steal the joy, rob the life, zap the strength, and quench the power. The prophets were struggling and had little provision. They survived but did not thrive. Jezebel will try to cut off the finances of prophetic ministry and prophetic people. Jezebel does not want you sowing into the prophetic voice of the Lord in the earth. It wants the prophets dead or hidden and struggling.\n\nThere is another dimension of the cave that we must think about with extreme honesty. Prophets can be peculiar people depending on their call and makeup. The way they function, the way they process, or their bold personalities can cause people to misunderstand them. As with any forerunner call, there is also a level of frontline warfare that emerges.\n\nJezebel may not be able to put you in a literal cave, but it will try to get you to live in a cave of rejection. Many prophets wear rejection as a badge. Instead of becoming intimate glory carriers, they manifest wounds and bitterness. Jesus came to set you and me free from the cave. We must determine not to put ourselves in a cave of rejection, a cave of isolation, or a cave of fear.\n\nWhen the Jezebel spirit runs rampant, it pushes prophets into the cave. This spirit sends waves of depression over prophets so that they end up alone, contemplating their demise. It releases words of witchcraft and spirits of fear to send the prophets and prophetic people running for their lives.\n\nBut I have a few things against you: You permit that woman Jezebel, who calls herself a prophetess, to teach and seduce My servants to commit sexual immorality and eat food sacrificed to idols.\n\n\u2014REVELATION 2:20\n\nHundreds of years after the actual human being named Jezebel died, Jesus gave this strong indictment of the church in Thyatira because of their allowance of Jezebel's illegal prophetic activities. He was railing against the acceptance of witchcraft in place of authentic prophetic ministry. This realm of the Jezebel spirit is particularly toxic.\n\nA Jezebel spirit does its best to establish its own renegade prophetic ministry through seduction. It seduces leaders with gifts, flattery, finances, and false loyalty. Jezebel proves its spirituality with grandiose proclamations of fasting, prayer, and revelation. The Jezebel spirit is a deeply rebellious spirit that will not submit to ordained leadership in any way, shape, or form.\n\nA person operating in a Jezebel spirit can \"fake it\" for a time in order to be handed the reins of authority. It is a dangerous game of deception and seduction in an attempt to wrestle power away from God-ordained leaders, only to fulfill its own wicked agenda. This is the overthrow nature of Jezebel\u2014to take the reins of power from those whom God has appointed. This spirit will move through people pledging loyalty and alliance, but it is deceptive.\n\nSTRANGE FIRE BURNS\n\nOne of the tactics that the Jezebel spirit employs is false prophetic witchcraft. Jezebel is a bewitching spirit. To more fully understand this, we must explore the nature of witchcraft.\n\nFor rebellion is as the sin of witchcraft, and stubbornness is as iniquity and idolatry. Because you have rejected the word of the LORD, He has also rejected you from being king.\n\n\u20141 SAMUEL 15:23\n\nThe first component of witchcraft is rebellion. This was the Lord's harsh indictment against Saul as he openly disobeyed the instructions of heaven. He refused to do what God said. The Lord exposes a root desire of witchcraft in rebellion. The dictionary offers a simple definition of the word _rebellion_ : \"opposition to one in authority or dominance.\"\n\nIt is no mistake that God's government is established as a theocracy. It is not a democracy with voting, individual rights, opinions, and preferences. In fact, when we become disciples of Jesus, we are invited into a lifestyle of surrender. The apostle Paul, who we know was a powerful early church apostle, referred to himself as a bondservant (Rom. 1:1, NKJV). A bondservant was one who had been set free legally but chose to stay in the house. To commemorate this decision, the bondservant's ear would be pierced straight through at the doorpost of the house. (See Exodus 21:5\u20136 and Deuteronomy 15:16\u201317.) This is a powerful prophetic picture. One who was freed decided to remain. I believe Paul more deeply demonstrated this surrendered lifestyle when he wrote, \"'All things are lawful for me,' but not all things are helpful\" (1 Cor. 10:23). He was teaching the church. He was stating that he did not live a life of navigating the narrow road because of law and legalism but because of grace. Through Jesus he was set free from the law, but he chose to remain attached to the ways of the Lord. He said no to some things because they were not wise for him as an apostle.\n\nThe bondservant has his ear pressed to the doorpost of the house. He is listening to the heartbeat of the house. This is another prophetic picture. God wants to bring us so in tune with His voice and His desires that we hear the secret whispers of His heart and know His ways. This type of intimacy fuels a life of power and exploits.\n\nA bondservant surrenders. Witchcraft refuses surrender. A bondservant submits to authority. Witchcraft lusts for quick power and dominance. A bondservant is disciplined. A witch is unbridled. A bondservant is humble. A witch is arrogant and boastful. A bondservant lives to please the master. A witch lives to serve and please self.\n\nWitchcraft is self-focused, self-promoting, self-serving, and self-pleasing. Jezebel is a witch.\n\nThe book of Leviticus tells of Nadab and Abihu, who were sons of Aaron and therefore priests.\n\nNow Nadab and Abihu, the sons of Aaron, each took his censer and put fire in it, and put incense on it, and offered strange fire before the LORD, which He did not command them to do. Then a fire came out from the LORD and devoured them, and they died before the LORD. Then Moses said to Aaron, \"This is what the LORD spoke, saying: 'I will be sanctified by those who come near Me; and before all the people I will be glorified.'\" And Aaron held his peace.\n\n\u2014LEVITICUS 10:1\u20133\n\nThe priests were struck dead for playing with strange fire. This is a picture of the power of witchcraft being judged by God. Witchcraft operates in the realm of strange fire. Witchcraft taps into spiritual dimensions that are not holy. Priests are called to be holy\u2014separated and set apart.\n\nI have had several alarming examples of the Jezebel spirit and strange fire. Not long ago I traveled to another nation to preach. While standing on the platform, I was suddenly swept up into the realm of heavenly vision. I saw witchcraft intermingling with the prophetic. I saw prophets being seduced by the spirit of witchcraft. I began to say what I saw and to intercede for that nation.\n\nAfter the meeting, I was eating in the back room with a prophet who shared startling confirmation of what I had seen. He told me about a group of prophets who had sold themselves to the powers of the enemy for fame, money, and power. They were preaching in large meetings, with masses of people coming to see them. They had detailed information from the realm of the spirit, but they had obtained it through illegal access.\n\nThis is an imperative prophetic truth! The motive behind a word is as critical in discerning it as the content of the word itself. You must listen with wisdom when prophetic things are declared. Jezebel spirits will use flattering words to gain access to people. If a person gives himself over strongly enough to false fire, he can tap into information. He receives communication from the spirit realm and knows things, but it is the communication of devils and cannot produce life or reveal the heart of God. If you listen to the word and discern the motive, you can identify that it is strange fire.\n\nI had a very interesting encounter with a witch some time ago. The woman started coming to our church, telling people she was very prophetic. She tried her best to gain access to me. She was very nice, but I had a subtle check. It was not a big check, just something down deep on the inside that did not feel right. She began to pursue a member of my family. She did nice things, volunteered, helped, and gave gifts. She shared little words of encouragement that were accurate and got the attention of my family member.\n\nI was riding in the car with my wife one day, and we were having a conversation about our future and possible ministry assignments. We were talking details, places, times, etc. Within moments this person messaged someone in my family and attempted to prophesy about our conversation. She knew some of the things we had discussed. That could easily have been a prophetic operation if the mind and the character of the Lord were in it, but in this case they weren't. I believe demons relayed the information.\n\nAs time unfolded, the Lord made it very clear to me that we were dealing with a witch who was trying to infiltrate our prophetic circle. One night I lay in bed deeply grieved. The Lord began to speak to me about the person and her past. He told me what to look up and even gave me instructions to research the person's background online. In twenty-four hours, based on the word of the Lord, massive corruption was uncovered. The Lord divinely enlightened me and protected me from lying powers of witchcraft.\n\nThis is why discerning the spirits is so important. Strange fire has spiritual power, manifestation, and even information, but it is not holy. The heart of God is not in it. The character of God is not in it. Anytime someone appears very powerful in the realm of the spirit but is dishonest, is greedy, has poor family relationships, or has a bad track record, you must be cautious. I fear that often strange fire manifests, deceiving God's people to unloose evil plans against them.\n\nI do not write this to put fear in you but to arm you for battle and effectively equip you. There is no time to be distracted or delayed. Strange fire works through Jezebel, releasing false prophetic utterances and unholy encounters. We must rightly identify and avoid these spirits and their agendas.\n\nYears ago I was asked to be part of a business opportunity. It sounded like a dream come true. One of the people involved was highly skilled in this arena. There was just one problem: something deep inside me was unsettled. After wrestling with this but not having any confirmation on my hesitancy, I decided to override my concerns and go forward. I determined that my reservations were just my mind wavering. I announced my decision, and we were going forward.\n\nThat same night I was lying in bed and could not sleep. At about 12:30 a.m. my phone went off. A dear friend messaged me with an urgent request to speak to me. I knew that I needed to hear what my friend had to say. I felt compelled to get on the phone right then. When God speaks, you must obey, no matter how foolish or extreme it seems.\n\nI called my friend, who began to warn me about doing business with this person. Not only was there a horrible track record of deception, but there was also a history of manipulation and Jezebelian activity. Now all my concerns were validated, and I knew that I had heard from God. That night I reversed course and canceled the agreement. A warning saved me years of possible trouble.\n\nHow can you properly discern the Jezebel spirit, strange fire, witchcraft, or false prophetic ministry? There are a few pieces of advice I would give. Number one: examine the motive. Proverbs 4:23 warns us to guard our hearts. We are to purify our hearts, examine our motives, and allow Holy Spirit to search us and reveal any improper agenda.\n\nThe Jezebel spirit always has an agenda. There is a wrong motive behind every activity of Jezebel. One of the ways you can properly discern the Jezebel spirit is by examining motive. Another way is by examining fruit. Most of the encounters I have had with this spirit have been with people who have had a track record of sowing discord, dishonor, and bad character. We should look at the fruit in the life of a person who wants to speak into our lives.\n\nAlso examine the methodology. We will explore this deeper in the book. A Jezebel spirit does things in a sneaky way to avoid proper oversight because it is a rebellious spirit. A Jezebel spirit is a forceful rank breaker that regards no one but itself. It will dishonor leaders and breach protocol every time. Whenever these behaviors are rampant, there is a wrong spirit.\n\nPOINTS TO CONSIDER\n\n\u00bb Have you encountered someone with bad character or a bad track record trying to dominate you with prophetic words, leadings, etc., and had a concern? Don't override the check! It may save your life.\n\n\u00bb Jezebel goes after the prophetic. It tries to cut off the head. Pray over your own life. Pray over the lives of those you know who carry a prophetic anointing.\n\n\u00bb Are you keeping yourself in a surrendered lifestyle as a bondservant? This is a freedom key.\n\nWHERE TO BEGIN\n\n\u00bb Learn to examine motives. Recognize that the _why_ is as important as the _what_.\n\n\u00bb Be a person of integrity. Value character, not just gifting.\n\n\u00bb Beware of people who have no stability but always want to give a word to others.\n\n\u00bb Ministry flows from covenant and relationship. Ask the Lord for a covenant heart.\n\nPRAYER\n\n_I thank You, Lord, that I am not deceived. I hear Your voice and obey Your leading. I ask for clear discernment of any and all wrong relationships and motives coming into my life. I want to love and value Your bride. I want to be a person of mercy and grace. I do not want to accuse, attack, and criticize everyone, but I do want to walk in wisdom. Help me walk in wisdom and clear discernment. In Jesus's name, amen._\nChapter 4\n\nJEZEBEL'S SEDUCTION, SCHEMES, AND STRATEGIES\n\nEVERYTHING THAT SATAN does is a counterfeit. He cannot create anything and is powerless over the Most High God. In order to buffet the life of a believer who is hidden in Christ, he looks for an entry point. He and his minions continually study the lives of human beings, paying close attention to their words, actions, pain, and temptation. Demonic powers utilize human weakness to create entry points for domination and strongholds. It is the ambition of the kingdom of darkness to overthrow the plans of the kingdom of God. Nothing brings greater pleasure to the hounds of hell than tormenting God's people and interrupting heavenly plans.\n\nCOUNTERFEIT REALITIES\n\nOne of the most evident examples of an attempt by evil powers to interrupt kingdom plans is the spiritual battle the apostle Paul faced. Paul was converted through a dramatic encounter with Jesus on the Damascus Road. He was launched into a series of supernatural experiences with the Lord that sent him forth into the calling on his life. This was not a simple conversion and release of a student into the ministry. This was a man supernaturally changed and carrying a mantle to help father the early church. His ministry was turning the known world upside down as the spirit of revival spread from place to place with bold preaching, in-depth teaching, radical miracles, signs, wonders, salvations, and Holy Ghost baptisms.\n\nThe mantle that Paul was carrying quickly became a target for the artillery of hell. Paul was spending countless hours praying in the spirit and receiving heavy revelation from God.\n\nPaul decreed in 1 Corinthians 14:18, \"I thank my God that I speak in tongues more than you all.\" As he spent time in prayer unraveling the mysteries of God, he was given foundational truths to be used as building blocks for the early church.\n\nBut I reveal to you, brothers, that the gospel which was preached by me is not according to man. For I neither received it from man, neither was I taught it, except by a revelation of Jesus Christ.\n\n\u2014GALATIANS 1:11\u201312\n\nPaul was carrying uncommon revelation and authority. His teaching was fueling the move of God because the truths were being captured directly from the heart and mind of God. This type of powerful ministry posed a significant threat to the plans of darkness. The enemy could not allow Paul's ministry to go unchallenged, so hell mobilized a counterattack.\n\nAnd lest I should be exalted above measure by the abundance of revelations, a thorn was given me in the flesh, a messenger of Satan, to torment me, lest I be exalted above measure.\n\n\u20142 CORINTHIANS 12:7\n\nThere was a demonic entity assigned to the life of Paul to harass him and shut him down. There was a demon stalking him, creating havoc, struggle, and turmoil. This provides stunning clarity about spiritual warfare strategies. Let's examine a few foundational truths in order to more clearly comprehend satanic tactics.\n\nThings in the realm of the spirit are governed by laws. There is divine order in the kingdom of God. Nothing is done randomly or without authorization. God always has a clear-cut chain of command. He reveals this to us over and over in Scripture.\n\nThen God said, \"Let us make man in our image, after our likeness, and let them have dominion over the fish of the sea, and over the birds of the air, and over the livestock, and over all the earth, and over every creeping thing that creeps on the earth.\" So God created man in His own image; in the image of God He created him; male and female He created them.\n\n\u2014GENESIS 1:26\u201327\n\nWhen God created mankind, He placed the inhabitants of the earth under the authority of His creation. Man had all power and dominion. The enemy had absolutely no authority whatsoever; that is why he had to persuade man to cooperate with his plans to breach authority and gain influence over mankind. Satan could do nothing without the willing cooperation of Adam and Eve. When they yielded to sin and disobeyed God, it changed the order in the earth.\n\nAnd to Adam He said, \"Because you have listened to the voice of your wife and have eaten from the tree about which I commanded you, saying, 'You shall not eat of it,' cursed is the ground on account of you; in hard labor you will eat of it all the days of your life. Thorns and thistles it will bring forth for you, and you will eat the plants of the field. By the sweat of your face you will eat bread until you return to the ground, because out of it you were taken; for you are dust, and to dust you will return.\"\n\n\u2014GENESIS 3:17\u201319\n\nMan was put under a curse because of the unlawful activities of Adam and Eve. It would take a redemptive plan to restore the authority.\n\nSo it is written, \"The first man Adam was made a living soul.\" The last Adam was made a life-giving spirit.\n\n\u20141 CORINTHIANS 15:45\n\nKINGDOM GOVERNMENT\n\nJesus is called the second Adam. He was birthed to redeem, or to buy back, mankind and undo the curse of sin. He was of divine origin, as was the first Adam, yet a man living in a fleshly body and enduring temptation. He was the sacrifice for sin and the Lamb that was slain. He came to break the power of hell and reverse the curse.\n\nChrist has redeemed us from the curse of the law by being made a curse for us\u2014as it is written, \"Cursed is everyone who hangs on a tree\"\u2014so that the blessing of Abraham might come on the Gentiles through Jesus Christ, that we might receive the promise of the Spirit through faith.\n\n\u2014GALATIANS 3:13\u201314\n\nIn every area there is an authority structure. In our homes there is a godly order and chain of command.\n\nWives, be submissive to your own husbands as unto the Lord. For the husband is the head of the wife, just as Christ is the head and Savior of the church, which is His body. But as the church submits to Christ, so also let the wives be to their own husbands in everything. Husbands, love your wives, just as Christ also loved the church and gave Himself for it.\n\n\u2014EPHESIANS 5:22\u201325\n\nHusbands are commanded to love and cherish their wives. They are instructed to arise as spiritual leaders, providing prayer, protection, and godly direction in union with their wives. Children are admonished to obey the instructions of their parents.\n\nChildren, obey your parents in the Lord, for this is right. \"Honor your father and mother,\" which is the first commandment with a promise, \"so that it may be well with you and you may live long on the earth.\"\n\n\u2014EPHESIANS 6:1\u20133\n\nWe are encouraged to pray for those in authority over us.\n\nTherefore I exhort first of all that you make supplications, prayers, intercessions, and thanksgivings for everyone, for kings and for all who are in authority, that we may lead a quiet and peaceful life in all godliness and honesty.\n\n\u20141 TIMOTHY 2:1\u20132\n\nIn the government of the kingdom of God there are established roles, responsibilities, and ranks. There is not a random placement of people but a strategic development of gifts, callings, and offices to lead and guide God's people. This point is further illustrated with the language used in this scripture:\n\nAnd God hath set some in the church, first apostles, secondarily prophets, thirdly teachers, after that miracles, then gifts of healings, helps, governments, diversities of tongues.\n\n\u20141 CORINTHIANS 12:28, KJV\n\nThe word _set_ here means fixed, placed, or established. The Lord Himself places men and women in roles of divine leadership. When He establishes them in those roles, there is a measure of authority granted to fulfill the responsibility. There is also an anointing or divine enablement released upon their lives. Prophets are anointed to prophesy. Teachers are anointed to teach; it flows out of them with ease because they have been given a spiritual portion. With these ranks and responsibilities there comes a demand for character. All too often we witness the tragic fall of highly gifted men and women who had tremendous power but very poor fruit.\n\nAt the forefront of the government of God are two offices: the apostle and the prophet. This does not in any way mean that they are more valuable or important than any other office or gift. It simply means that they bear a greater responsibility and have a higher level of governing authority. To illustrate this point, the Bible says God set the apostle first (1 Cor. 12:28). The word _first_ here is the Greek word _pr\u014dton_ , meaning \"firstly (in time, place, order, or importance):\u2014before, at the beginning, chiefly (at, at the) first (of all).\" It is quite clear that the Bible is establishing rank. Apostles are like generals tasked with receiving and releasing revelation that pushes the body forward. They are given plans for regions, peoples, and nations. They are called to teach and clear up confusion in key areas. They are God's generals. Beside them are the prophets, great and dynamic spiritual forerunners who go out ahead in revelation, clearing the path for the people of God.\n\nGod works on the earth through His ekklesia, His ruling and governing people. He has a rank, authority, and structure. He releases divine revelation, powerful anointing, and angelic assistance to get the job done\u2014on earth as it is in heaven! This is the goal: to release the kingdom of God on earth, to release the authority of the risen Christ on earth, subduing and overwhelming the powers of hell. The Lord is mantling people who will fight against and overcome the powers of hell.\n\nSatan has a counterfeit government. He delegates various powers and strategies to his dark army.\n\nFor we wrestle not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places.\n\n\u2014EPHESIANS 6:12, KJV\n\n\u2022 **Principalities** are the highest-ranking ruling spirits assigned to territories, regions, nations, and people groups. The Greek word indicates the primary or highest authority. These demonic forces are given charge to bind and hold territories. We see a picture of this in the Old Testament in Daniel 10 concerning the prince of Persia, who is a ruling spirit. Principalities will come in direct conflict with apostles and apostolic ministry because they are also assigned to regions and peoples.\n\n\u2022 **Powers** are the second-highest-ranking spirits that carry a manifestation of supernatural power to enforce the plans of the principalities. They influence the thoughts and feelings of human beings, seeding the thought realm with demonic lies. They work hard to destroy kingdom order, rule, and reign. The meaning of the word _powers_ indicates supernatural force and manifestation. These spirits operate in heavy divination, false fire, counterfeit spiritual manifestation, and evil supernatural operations. Powers attempt to overthrow prophets and prophetic ministry.\n\n\u2022 **Rulers of darkness** release deception, false religion, and the occult. They work to bind the souls of men and women in foul deception. These demons sow the seeds of false belief systems, directly combatting the office of the teacher.\n\n\u2022 **Spiritual wickedness** releases lust for sin and depravity. Spiritual wickedness dwells in the high places, the airspace above the earth. Both angels and demons travel through a great highway known as the first and second heavens. If our spiritual eyes were opened, we would see the spiritual activity taking place in the heavens and all around us. These demonic beings mirror the evangelist, serving as the recruiting arm of the kingdom of darkness. These spirits use supernatural enticement to draw people in to dark plans and schemes. They draw people to darkness much as the evangelist draws people to the kingdom of God.\n\n\u2022 **Demons** are the foot soldiers in the kingdom of darkness, carrying out the plans of the ruling spirits. They are sent on assignment and vary in category depending on their particular purpose. Demons are continually on the move, implementing the plans of darkness and attempting to create bondage in the lives of human beings. Just as a pastor tends to the needs of individual people, so these demons tend to evil assignments against the lives of people.\n\nDEMONIC PARTNERSHIPS\n\nThe Jezebel spirit is a ruling spirit that partners with evil powers to manifest manipulation and witchcraft. One of the powers the Jezebel spirit employs is seduction. Wherever the Jezebel spirit is at work, there is seduction at hand. The word _seduce_ means \"to persuade to disobedience or disloyalty; to lead astray usually by persuasion or false promises; to carry out the physical seduction of.\" The Jezebel spirit comes with enticement to lure people away from ordained plans, purposes, and order. There are layers in the seduction strategy of Jezebel.\n\nIn her book _Jezebel's Puppets_ , Jennifer LeClaire wrote a plot summary for Jezebel: \"A wicked spirit that has roamed the earth for thousands of years seeking someone to entice into sin, Jezebel is more than the spirit of control and manipulation that many make it out to be. Jezebel is a spirit of seduction that works to woo people into immorality and idolatry\u2014but it doesn't work alone. With the help of Ahab, false prophets, and faithful servants, Jezebel has been sinking its seductive teeth into society for ages\u2014and has infiltrated the pulpit and the pews of the church. Since few see Jezebel for what it is, this spirit continues to secure the hearts and minds of those who claim Jesus as their first and true love. Will Jezebel ultimately lead the church into a great falling away, or will the church wake up to this perversion before it's too late?\"\n\nOne of the ways that Jezebel seduces people is by speaking to their ambitions. When a Jezebel spirit comes into a ministry, it will often work to entangle key leaders and people by speaking to their ambitions. It will prophesy to a key leader that he should really be the senior leader. It will speak to the wounds in the heart of a person and validate pain in order to gain access. It can also reveal certain hidden desires that seem to have been forgotten. It will speak accurately to those desires but from an unholy place to draw people away from their current assignments and entangle them in an evil agenda.\n\nThe spirit of Jezebel is a sexually impure spirit. The spirit of whoredom is one of Jezebel's defiled cohorts. The whoredom of Jezebel includes prostitution, fornication, adultery, and all manner of evil sexual sins. Much of the rampant sexual sin that we see in the world and in the church today is rooted in the seduction of Jezebel.\n\nAnd it came to pass, when Joram saw Jehu, that he said, Is it peace, Jehu? And he answered, What peace, so long as the whoredoms of thy mother Jezebel and her witchcrafts are so many?\n\n\u20142 KINGS 9:22, KJV\n\nJezebel partnered with the prince of the power of the air to release constant bombardment of filthy entertainment devised to lower inhibitions and seduce people. At the heart of witchcraft and idol worship are impure sexual practices. The very act of sex itself creates a deep and lasting bond in the participants' hearts, and even an emotional or mental affair can cause irreparable damage to a marriage.\n\nYou have heard that it was said by the ancients, \"You shall not commit adultery.\" But I say to you that whoever looks on a woman to lust after her has committed adultery with her already in his heart.\n\n\u2014MATTHEW 5:27\u201328\n\nThere is an emotional, soulish seduction that precedes a physical act. The seduction of Jezebel will always begin in the mind. In the arts and entertainment industry Jezebel has systematically lowered the resistance to sex and perversion for generations until almost anything goes now. As the web of seduction has been cast over minds, confusion and bondage have unfolded with devastating effects. I have personally prayed with countless young people who have been gravely addicted to pornography with easy access on their smartphones. It has been a progressive and evil plan intended to chip away biblical values and covenant relationships.\n\nThis whoredom spirit is not only a sexual issue but a covenant issue. Where you find the spirit of whoredom there is a breaking of vows, a lack of faithfulness, and self-serving behavior. Whoredom sells itself, its gift, its talent, for gain. It will quickly break a relationship for another opportunity. This spirit linked with Jezebel has defiled ministry relationships and aborted the concept of family in the kingdom.\n\nTHE PIT\n\nJezebel creates a deep and dark pit of sorrow as it seduces its victims and brings them into a place of defilement. In ministry this spirit lacks any kind of faithfulness and will cause people to jump from place to place with no root system. As they yield to this spirit, they will eventually wind up trapped in its pit.\n\nFor a whore is a deep ditch; and a strange woman is a narrow pit. She also lieth in wait as for a prey, and increaseth the transgressors among men.\n\n\u2014PROVERBS 23:27\u201328, KJV\n\nOver the years, I have often seen a spirit of Jezebel come into a ministry that I was leading or the ministry of one of my friends with an agenda to disrupt and cause problems. I have prayed through and worked through deep and troubling challenges brought on by people with this spirit. When Jezebel comes into a church or ministry, it is sneaky and subtle, yet intensely wicked. Jezebel thrives where there is an Ahab type of leader who will overlook its cunning manipulation. People with Jezebel spirits are often highly gifted individuals who offer goods, services, finances, or assistance, but there is always a price to pay.\n\nSome years ago I took a small team with me to a powerful church where I was going to be preaching. I will never forget the encounter we had with a boisterous individual who attended that church. This person seemed unstable, had been married several times, and kept stalking the ministers. At the time, this person was focused on an determined to marry a powerful minister. That type of behavior in and of itself freaks me out!\n\nI was at this church for a series of meetings, and the person of whom I am speaking kept finding a way into the back room, into conversations, into events. This individual's presence became increasingly uncomfortable. Finally one of my team members pulled me aside to ask me if I had observed these behaviors. Of course I said yes, and we both agreed that it was a spirit of Jezebel. This individual kept trying to gain access to me, offering gifts, money, etc., but I discerned the motive and wanted no part of it.\n\nFinally my team and I asked the leaders of the church if they had discerned the spirit. To our amazement, they said yes but indicated they were afraid to deal with it. You could have knocked me over with a feather, I was so shocked! They knew that there was a Jezebel spirit slithering and maneuvering in their midst, yet they had determined to allow it? Why? The answer is simple. They recognized the severity of the manifestation if they confronted it.\n\nI have seen entire ministries destroyed by retaliation from a Jezebel spirit. It is imperative that you not allow that to bring fear but simply arm yourself with the wisdom of God. If you pick a fight with a demon, you need to be ready. It is not something that you just casually do. It takes authority, strategy, and wisdom to win the war.\n\nTHREE PHASES OF THE JEZEBEL SPIRIT\n\nI believe that every spirit of Jezebel that enters a ministry will go through three phases. I have developed this teaching through the years to help equip church and ministry leaders when discerning and confronting this manipulative spirit.\n\n1. Entry\n\nJezebel looks for vulnerable people to influence and disciple. In a church it will search out the weak, the hurting, those who can be manipulated through prophetic words. In a family or relational setting it will arrive looking to gain avenues of influence.\n\nIn ministries Jezebel will begin a network of gossip with secret \"prayer\" calls and conversations, laying the groundwork for a future rebellion. It keeps these contacts hidden beneath the cover of darkness to avoid exposure.\n\nIt attempts to infiltrate any and all prayer and prophetic ministries. It will attempt to join the intercession team and become a part of the prophetic team.\n\nIt uses flattery. In a ministry it will flatter the leader with big compliments and grandiose spiritual pronouncements. In family and relationships it will offer self-serving commitments to gain favor with the intended target(s).\n\nIt will babysit the leader's kids, wash the cars, run errands, and give gifts. A Jezebel spirit will release funds, offer money, and lavish gifts on a person, people, or leaders, all in an attempt to gain influence. This spirit lusts for power!\n\nIt boasts of revelation. It will reveal how spiritual it is\u2014how much the person fasts, prays, and seeks God. It will exalt itself in the realm of spirituality far above others. This is a conquer strategy.\n\nIt searches for a position and always has a hidden agenda.\n\n2. Exposure\n\nThis is the messy phase of the Jezebel ordeal. In a church or ministry, this phase is critical.\n\nAfter many hidden schemes, the behaviors of a Jezebel spirit begin to unravel, and those involved start to see the problem at hand. Once this spirit recognizes it is at risk of being exposed, it will work overtime to rally allies. In a church or ministry it will try to build a coalition of people to stand with it. In a family struggle it will pit one member against the other.\n\nIt will boldly state that the ministry or church leaders are not spiritual enough and do not accurately hear from God. It will say that God is not pleased with the ministry and make wicked pronouncements over a place.\n\nThis spirit then exerts strong control and emotional manipulation over followers, demanding loyalty. In a family it will create a split, and in a ministry it will quickly build a coalition of offended, weak people to stand with it. It will prophesy to their wounds and gain their loyalty.\n\nWhen the demand for submission comes from a leader, Jezebel will boldly and unashamedly refuse to submit. This spirit is a foul, rank-breaking demon that recognizes no authority and exists to tear down godly structure and rank.\n\n3. Confrontation\n\nWhen the confrontation happens, the person with the Jezebel spirit will voice opposition and stand firm in a position of independence. The Jezebel spirit will not bow to any type of leader.\n\nPeople with the Jezebel spirit will boldly declare that God is on their side. This is all part of the deception. Even though people with the Jezebel spirit are acting in direct contradiction to the Word of God, Jezebel will not bend. It does not recognize the authority of the Word of God and will twist Scripture to justify its case.\n\nThis spirit will display huge emotional outbursts and tantrums to disrupt and distract. Every time I have ever encountered a Jezebel spirit in operation, I have witnessed a strong measure of emotional manipulation. This will manifest through tears, anger, sorrow, and blame shifting. The Jezebel spirit will attempt to place guilt on the one who is righteous and get the attention off itself.\n\nThe end result is an intense power struggle. In a church or ministry there must be a choice: either the people will follow the senior leader or they will not. Jezebel and the leader cannot peacefully coexist, as it is the mission of this spirit to destroy the leader. This is the harsh reality. There will be a showdown, and the people must make a choice. This is where strong prophetic ministry is vital. The prophets can expose what is happening and empower the people to see it from a spiritual rather than an emotional standpoint.\n\nIn a family or relationship the process is much the same. Choices have to be made. Will people allow and tolerate the maneuvering of Jezebel? Lines must be drawn. Humble repentance needs to take place to bring deliverance and freedom.\n\nWhen a person with a Jezebel spirit is forced out of a church or ministry, he or she will let loose the powers of witchcraft on the way out. The Jezebel spirit will release a multitude of curses, creating confusion and a demonic swirl that must be broken with strong prayer and authority. This spirit will often attempt to reach back into a church and snatch people out with manipulation and accusation.\n\nThose who are not strong in the spirit can easily get sucked in to Jezebel's witchcraft with misplaced sympathy. They can easily watch the tantrums and feel that there has been a lack of compassion. Compassion is one of the most powerful forces in the world, but it must be properly placed. Demons will latch on to compassion and use it as a means to stay. Jezebel wants to take over! It does not just want a piece; it wants the entire cake. This is the harsh reality, but it must be recognized.\n\nPOINTS TO CONSIDER\n\n\u00bb Have you experienced emotional manipulation from the Jezebel spirit?\n\n\u00bb Do you know someone who exhibits these characteristics?\n\n\u00bb Launching out in a confrontation without prayer and wisdom can lead to utter chaos!\n\n\u00bb The place to begin is prayer. Ask the Lord for insight, strength, and wisdom.\n\n\u00bb Demons only respond to firm faith and authority.\n\n\u00bb As a leader, have you been confronted by the operation of a Jezebel spirit on some level? Have you experienced its seduction drawing you away from kingdom purposes?\n\n\u00bb You must never allow this spirit to continue to maneuver in your midst.\n\nWHERE TO BEGIN\n\n\u00bb First review some of the symptoms listed in this chapter.\n\n\u00bb Ask Holy Spirit to shine His light on your own heart. Proper self-examination is key.\n\n\u00bb Pray for greater discerning of spirits and wisdom in relationships.\n\n\u00bb Recognize the reality of the spirit realm and its impact on your life.\n\nPRAYER\n\n_Father, I come before You seeking purity. I do not want my life mixed with any impure spiritual influence. Search my heart and reveal anything that is impure. I ask that You guide me in my family, in my private life, and in my kingdom calling. Make me aware of any seducing powers of Jezebel. I come against those powers in Jesus's name and forbid their operation in my life. I claim purity over my life, and I claim wisdom and discernment. In Jesus's name, amen._\nChapter 5\n\nCASTING DOWN THE DEMON QUEEN JEZEBEL\n\nJEZEBEL IS FOUND as a character in the Old Testament. She was used to defile God's people, tear down the altars of God, release demonic Baal worship, engage in seduction, destroy the prophetic ministry in the land, and pollute the kingdom. She was a demon-possessed witch masquerading as a prophet. She was no true prophet but one who carried strange fire and evil utterances.\n\nCONQUEST BRINGS CONTAMINATION\n\nAhab the son of Omri did more evil in the sight of the LORD than all who were before him. The sins of Jeroboam the son of Nebat were seen as minor for him to walk in, for he took Jezebel the daughter of Ethbaal, king of the Sidonians, as his wife and went and served Baal and worshipped him. He raised an altar for Baal in the house of Baal, which he had built in Samaria. Ahab made an Asherah and did more to provoke the LORD God of Israel to anger than all the kings of Israel who preceded him.\n\n\u20141 KINGS 16:30\u201333\n\nIt is impossible to fully grasp the depth of evil that took place during the reign of Ahab without proper inspection of the history of Jezebel. She was a false prophetess of the demon god Baal. She operated in powerful seduction to accomplish her will and the will of Baal. Her alignment with Ahab sent the nation into a downward spiral of wickedness and idolatry.\n\nWhen Ahab came into covenant with Jezebel, he bowed to Baal. Not only did he surrender his life to false gods, but he also connected Israel to demonic idolatry. Jezebel was an instrument of the enemy used to defile God's people and corrupt the kingdom. This picture gives us insight into the plots this spirit launches today. It is a perverse spirit of domination that works to open up gateways for demonic activity to overthrow godly plans, families, ministries, and people. Jezebel is an overthrow spirit that works to overcome and subdue people for destructive purposes. An alliance with this spirit is the entry point of disaster!\n\n_Jezebel_ means \"unmarried; un-husbanded or un-exalted; without dwelling or cohabitation.\" This is a picture of a defiled woman who carries strong evil spirits and lacks any capacity for covenant relationship. What type of man would enter into a relationship with this woman? Jezebel was literally Jeze-baal\u2014an ambassador and servant of the god Baal, who stood in direct opposition to Yahweh. Ahab had no business marrying this woman and forming a covenant with a covenant breaker.\n\nThe Jezebel spirit neither honors nor recognizes covenant. When this spirit operates in families, it manipulates and forms alliances to fulfill its own evil agenda. When this spirit operates in ministries, it totally disregards authority, honor, and relationship to pursue its own desires. It disguises its ambition in terminology using the name of the Lord. It is totally deceptive and demon-inspired. It is a rank-breaking devil that abhors authority and refuses any type of biblical submission. Jezebel has no regard for loyalty and typically has a long track record of bad relationships.\n\nThe Jezebel spirit cannot and will not operate without the permission of an Ahab. Jezebel will always test the strength of a leader, pushing the boundaries. In the church world those with a Jezebel spirit will hop from place to place, leaving a long wake of destruction and deception behind. It is crucial that kingdom leaders inspect fruit and know the track record of a new leader before positioning him or her in power. When you place a person with a Jezebel spirit in a position of authority, you will live to regret it; this spirit will create confusion, calamity, and demonic disruption in pursuit of its own will. Those with a Jezebel spirit may act as if they are listening, but they will not accept your instruction or correction.\n\nHEED THE WARNINGS\n\nNot only does the spirit of Jezebel employ seduction and sexual deviancy, but it also opens the door for spirits of infirmity and catastrophic illness.\n\nLook! I will throw her onto a sickbed, and those who commit adultery with her into great tribulation, unless they repent of their deeds.\n\n\u2014REVELATION 2:22\n\nAt the end of a person's journey with the Jezebel spirit there is extensive sickness. I have tragically seen this happen twice in my life. The Lord will gracefully woo people over and over again to repentance. He will repeatedly send prophetic warnings, but they must heed the warnings, humble themselves, cut all affiliation with Jezebel, and truly repent, which means to completely change their thinking. Controlling personalities must dramatically change their methods of operation and thought patterns. They must undo years of learned behavior and get emotional healing along with spiritual deliverance. Cast out, reform, rethink, renew, and undo the yoke of evil spirits!\n\nSome time ago I encountered a powerful minister who had a personality that could capture the attention of anyone. This leader could persuade you to do anything. When this person's skills were put to good use, it was like an iron in the hand of the Lord. People were forever changed. This minister took joy in pouring into people hungry for training and development in their call, and in the hands of the Lord this was a powerful gift! As I got up close, though, I also saw something else, a dark underside.\n\nWhen a person did not do what this minister wanted, the minister would lash out with accusations and attacks. I had heard this person call other pastors to defame some traveling ministers because they did not do what the leader desired. The strength that I saw used in a good way was also used in a toxic capacity. As time passed, it seemed as though no one would stand up to this bullying behavior, so it went unchecked, and increased. The Jezebel spirit was running rampant with no accountability or repentance.\n\nThis minister was caught in lies on many occasions. These deceptions were tools used to manipulate the outcomes of situations to get the minister's own way. At first the lies were in private, but eventually they became bolder. Public messages were filled with deception. Stories that never happened in the person's life were made up. Struggles that were current were claimed as victories that had already been won. As Jezebel gained a stronghold, the behavior spiraled out of control. Lives were destroyed, people were broken, and hearts were wounded. This pattern of bondage would eventually create a choke hold, stifling the anointing and destroying a once-vibrant ministry. It would eventually end in ruin. Watching it from a distance was like watching a tragic play unfold, but there was little anyone could do because the Jezebel spirit had its claws deeply embedded.\n\nThis person would turn and twist facts to achieve the desired outcome. As time went by, deep paranoia and loneliness set in. Jezebel isolated the minister and created a deep well of distrust of people. Instead of love, there was bitterness. What had been a vibrant and powerful personality became weak, mean, and bitter. Occasionally you would catch a glimpse of the old flame, but it was increasingly buried deep beneath a hard exterior shell. Close personal relationships were in shambles as the Jezebel spirit grew in strength in the life of this person. God sent countless people with the mercy and compassion of the Lord to call for repentance, yet these calls went unheeded, viewed as a breach of authority. There is a lesson here: no matter who we are or what position we hold, we are all accountable on some level and need the input, oversight, and correction of godly leaders. Removing this counsel and creating a team of yes-men creates a breeding ground for the Jezebel spirit.\n\nThe story turned tragic as the minister fell prey to a series of illnesses. Eventually a final blow claimed the life of this once-powerful servant of God. This individual died filled with wounds, paranoia, and bitterness. This is the intended outcome of the evil powers of Jezebel. It wants to strip the anointing, the life, the zeal, and the purpose out of God's people. It wants to overwhelm and conquer. If allowed, Jezebel will lead a person to a bed of affliction. I am certainly not saying that all illness comes from this\u2014there are also physical issues and attacks from the enemy. But it is a reality that Jezebel will lead a person to a sickbed.\n\nDON'T BECOME AN AHAB\n\nI was leading a young ministry when a man came to me to share his story of deep marital problems. This man carried such a sweet presence of the Lord. He was compassionate, sincere, and kind. He began to tell me the story of recent troubles with his wife.\n\nThis man told me that his wife had become enamored with a woman claiming to be a prophet and would not make any decision without her counsel. His wife was spending most of their money on lengthy phone conversations and then giving what was left to the woman. (This was a number of years ago, before the technology we have now, when cell phones charged for long distance and we paid by the minute.) His wife ignored his repeated warnings, refusing to submit to his instructions.\n\nAs he shared this story, my stomach began to turn, and I could feel the powers of evil attacking his family. I strongly and firmly warned him that this was not a prophet but a Jezebel spirit. I pointed out scriptures about marriage and how this spirit was ignoring the Word of God. Anything prophetic that disregards the Word is a demon spirit! I told him that he needed to stand firm, cut off the money for the phone calls, and put an end to this. I told him that I would meet with him and his wife to minister deliverance to her and break the powers of Jezebel.\n\nHe agreed with what I was saying, but there was an issue\u2014he didn't have the strength. As I talked with him, I could see that he was indeed a man like Ahab, lacking the backbone to stand up to this demonic power. Authority is always required to break the grip of demon spirits and get them out. Jezebel is the same! You cannot be nice and passive with a Jezebel spirit. It is a vile demonic power sent to abort destinies and destroy lives. You must see it, confront it, and remove it. A leader who allows Jezebel to stay in position is guilty of its sins. This man could not and would not operate in his authority. Instead, he allowed the situation to continue.\n\nHis wife talked him into cutting ties with our ministry because she recognized that I would not tolerate the Jezebel spirit. Years later I received devastating news. I was told the wife ended up in a sickbed and eventually died. She had drained all of their money, and their lives ended in disaster, all because of the sinister plot of Jezebel.\n\nAhab becomes a gateway for the operation of the Jezebel spirit. Jezebel, who married King Ahab, was the daughter of King Ethbaal of Sidon. His very name demonstrates deep-rooted idol worship. His name means \"a man of Baal.\" He was a high priest of the Canaanite fertility goddess Astarte. Many scholars agree that Astarte and the Babylonian goddess Ishtar are the same idol. Both Astarte and Ishtar were revered as fertility goddesses. They were associated with fertility, life, and sensuality.\n\nJezebel was raised from childhood to be a servant of demon gods. She was a loyal ally of Baal, determined to represent and worship him, along with the goddess of fertility, wherever she went, including the land of Yahweh's people. She could only do this with the submission of a wicked king who yielded to her seductive powers.\n\nIn his landmark book _Unmasking the Jezebel Spirit_ , John Paul Jackson writes:\n\nA spirit of Ahab symbolizes the abdication of authority, or at the very least, passive authority. It bespeaks of a mind-set that avoids confrontation and denies fault. The spirit of Ahab loves the position it has and fears confrontation. Someone with an Ahab spirit would rather make peace at any cost, even if it leads to making an unholy alliance.\n\nAn individual under the influence of an Ahab spirit often makes truces instead of covenants, thus prostituting rather than sanctifying relationships. But how can you have a truce with someone whose goal is to destroy you? It is impossible! Nonetheless, an Ahab spirit will always sacrifice the future good for the sake of peace today.\n\nWorking in tandem, the spirits of Ahab and Jezebel will quietly form a codependent relationship. Both will need and feed off the other in order to accomplish each one's goals. A pastor who is influenced by an Ahab spirit will need the help of someone influenced by a Jezebel spirit to maintain position and enlarge or entrench a powerbase.\n\nUNHOLY ALLIANCES\n\nThe kingdom of Israel was sent into a time of great trial by the unholy alliance of Ahab and Jezebel. They defiled the ways of Yahweh, exalted the altar of Baal, and permitted the unholy to take dominance. Jezebel instituted the building of stone idols atop all the high places and even in the holy Temple. She was a harlot and a witch, according to 2 Kings 9:22.\n\nStrong, life-giving churches are built upon two dominant anointings: the apostolic and the prophetic.\n\nNow, therefore, you are no longer strangers and foreigners, but are fellow citizens with the saints and members of the household of God, having been built upon the foundation of the apostles and prophets, Jesus Christ Himself being the chief cornerstone.\n\n\u2014EPHESIANS 2:19\u201320\n\nAs the early church was advancing with rapid success, it was the apostles who were at the forefront. The apostolic gift is a strong, bold gift! The word _apostle_ means \"sent one.\" Apostles are sent to places and to people with a divine commission. They are governors who rule with supernatural wisdom, strength, and authority. They teach and train people to redeem and rebuild the mighty works of God. They activate gifts, callings, and destinies. They do not build comfortable places of rest but active centers of kingdom activity and training.\n\nThe apostolic spirit is one of strength and power. When apostles preach, they challenge the demonic powers and push back the darkness. They are not content to just leave things as they are. They are mantled to shake things up! They make demonic powers nervous. Now, there are some who falsely teach that there are no modern-day apostles. This is in error, as the book of Ephesians makes no distinction when it identifies the five ministry offices:\n\nHe gave some to be apostles, prophets, evangelists, pastors, and teachers, for the equipping of the saints, for the work of service, and for the building up of the body of Christ.\n\n\u2014EPHESIANS 4:11\u201312\n\nThe work of equipping is ongoing. It is not finished. People are being called, trained, appointed, and released today. The kingdom is growing and expanding even now. Apostles were listed first in 1 Corinthians 12:28. They were sent in as generals to lead the body into heavenly battle plans and prevail over the powers of hell. There were twelve original apostles of the Lamb who hold a unique place in the history of the church and the heavens. They are irreplaceable. Yet the office and calling of apostles continued after the original twelve and is still alive and active today, leading the mighty army of God.\n\nThe church also desperately needs prophets and prophetic ministry. It is absolutely impossible to obtain the victory without prophetic ministry. These two anointings work together to unlock kingdom plans and purposes. Prophets see in high definition. They are lifted high into heavenly places to see coming attacks, to warn, to unravel mysteries, and to be exposed to huge dreams in the heart of God for His kids. They shine bright light on the pathway for the people of God. Prophets confirm and affirm. They speak God's plans and His heart over people's lives, connecting them to eternal purposes. When prophetic ministry is functioning in a strong and healthy capacity, it creates such a life-giving flow of heaven on earth.\n\nSurely the Lord GOD does nothing without revealing His purpose to His servants the prophets.\n\n\u2014AMOS 3:7\n\nProphets have one of the most special invitations of all\u2014to share in the secrets of the Lord. I love to ponder Amos 3:7 because it takes me back to childhood. I can remember the game where a friend would whisper in my ear, and then I would share that same secret to another, and we'd laugh and laugh. You could not hear the secret unless you were close! Prophets and prophetic ministry require intimacy. As we live close to the Father, we are invited to explore all the vast chambers of His wonderful imagination and its immeasurable creativity. We are also invited to explore the depths of His vibrant, rich, and compassionate heart of love. Prophets ultimately reveal the heart, the mind, and the dreams of God to His people. Prophets are revealers!\n\nThese two gifts, along with the other wonderful gifts of God, powerfully equip the church to inherit the position that Jesus paid for: rulership.\n\nFor if by one man's trespass death reigned through him, then how much more will those who receive abundance of grace and the gift of righteousness reign in life through the One, Jesus Christ.\n\n\u2014ROMANS 5:17\n\nStrong churches, strong ministries, and strong believers enforce the reign of Christ. They use the authority that Jesus granted and stand boldly upon the promises of God. They do not shrink back when the enemy rears his head. It takes people who know who they are in the spirit to conquer the operations of the enemy.\n\nApostles and apostolic ministries guard the gates. They do not allow demonic powers to keep their hold on people, places, or regions. They march forth with a strong word from God and an authority in the Spirit. The demons in the early church era knew who the apostles were because they were constantly dismantling their structures and kicking them out.\n\nThe evil spirit answered, \"I know Jesus, and I know Paul, but who are you?\"\n\n\u2014ACTS 19:15\n\nThis demon knew Paul because he was terrorizing the courts of hell. Paul was training believers to cast out demons. Paul was teaching people the truth. Paul was leading an advancing army that was arising and standing against the plans of darkness. Modern-day apostles pose a great threat to the kingdom of darkness. Regions cannot and will not be changed without spiritual warfare. There is a struggle for which kingdom will prevail, and many ministries give up before the victory comes. They need a strong apostolic anointing to push them over the top.\n\nDON'T RUN AND DON'T HIDE\n\nProphets are an enemy to the Jezebel spirit. One of the primary functions of the Jezebel spirit is a false prophetic ministry. During the reign of Ahab, Jezebel murdered prophets of the Lord, released tremendous intimidation, and had the remaining prophets hiding in a cave.\n\nTrue prophets know how to see in the dark. They have night vision. They can discern what is going on beneath cover. They see Jezebel lurking beneath a polished exterior. When the Jezebel spirit encounters a true prophet or prophetic voice, it will unload warfare.\n\nOne of the most vivid examples of the attack of Jezebel against the prophetic is the plight of Elijah when she released a demonic word curse over him.\n\nAnd Ahab told Jezebel all that Elijah had done and how he had executed all the prophets with the sword. Then Jezebel sent a messenger to Elijah, saying, \"So let the gods do to me and more also, if I do not make your life as the life of one of them by tomorrow about this time.\"\n\nWhen he saw that she was serious, he arose and ran for his life to Beersheba, which belongs to Judah, and left his servant there. But he went a day's journey into the wilderness and came and sat down under a juniper tree and asked that he might die, saying, \"It is enough! Now, O LORD, take my life, for I am not better than my fathers.\"\n\nAs he lay and slept under the juniper tree, an angel touched him and said to him, \"Arise and eat.\" He looked, and there was a cake baked on coals and a jar of water at his head. And he ate and drank and then lay down again.\n\nThe angel of the LORD came again a second time and touched him and said, \"Arise and eat, because the journey is too great for you.\" He arose and ate and drank and went in the strength of that food forty days and forty nights to Horeb, the mountain of God.\n\n\u20141 KINGS 19:1\u20138\n\nElijah stood atop the mountain, challenging all the false prophets of Baal, and God answered with supernatural fire. In a single act of uncommon bravery, he openly exposed the corruption of Jezebel. Her prophets were killed as a result of his victory that day. But the witch was not about to go away quietly. The prophet had prevailed over her kingdom, and she was going to counterpunch him. This is symptomatic of the Jezebel spirit; it fights the prophetic anointing tooth and nail. If exposure comes forth, it will hurl a flurry of demonic word curses and intimidating accusations. It wants to send the prophets fleeing for their very lives.\n\nJezebel sent a vile and corrupt message to the servant of God. She wasn't even there, but someone else carried her contaminated words upon his lips. Jezebel always works through messengers. This spirit will build demonic networks inside of ministries, teams, families, and organizations. It will seduce, lie, and intimidate to create its coalition, and then conspire with its team to overthrow true prophetic anointing.\n\nWhen the utterance was released, Elijah was sent into an abnormal tailspin. He sank deep into a pit of depression. This was an absolutely shocking transformation. He went from a fearless warrior of Yahweh to a broken and intimidated man. What could have changed so suddenly? There were a couple of factors at work.\n\nFirst, he had just come from one of the most intense moments of ministry in his entire life. People often do not realize the toll that extreme times of outpouring can take on the body. When the raw and powerful anointing of God pours through people, they can feel zapped in their minds and bodies afterward. This is why protection in prayer and the right people surrounding a man or woman of God are vital. I believe Elijah was unusually susceptible to a spiritual attack because every part of his being was absolutely drained as he had warred with his mind, body, and spirit being lifted into the heavenlies and crushing the throne of darkness. Jezebel sent her message at the most opportune time for her. This spirit will come when it finds people at their most vulnerable.\n\nSecond, I believe Elijah allowed the power of Jezebel's words to overtake him by giving them consideration. The Bible says that \"he saw\" that she was serious. Her witchcraft opened up evil imaginations, and he did not cast them down. He could see himself being slaughtered. He could see himself being captured. He then ran for his life.\n\nIn a moment he was sent packing! This is what Jezebel wants\u2014to plant evil seeds of corruption and defeat in the minds of prophets. This spirit works through curses in an attempt to shut down the prophetic. It is key that we stand in opposition by the authority of the name of Jesus and the power of almighty God. There is no place for fear in the heart of a warrior who has been authorized by the mighty King Jesus! Jezebel is no match for the blood of Jesus.\n\nThe Lord sent angelic assistance to minister to Elijah and bring him out of the deep slumber of attack. Jezebel wanted to mute his senses and zap his strength. Prophets are fierce warriors who are wired with a high regard for justice. They tend to see things as right or wrong and refuse to compromise. They are one of the greatest threats to this foul spirit. That is why it will fight them tooth and nail.\n\nFEAR NOT\n\nIt is critical that we realize there is no room for fear or defeat in the kingdom of God. We have been granted all power and authority through the name of Jesus! We are never to shrink back from the enemies of the Lord. We are to get wisdom and strategy to dismantle both the kingdom and powers of hell. Fear not, for the Lord God Almighty fights for you and with you!\n\nDo not fear, for I am with you; do not be dismayed, for I am your God. I will strengthen you, I will help you, yes, I will uphold you with My righteous right hand.\n\n\u2014ISAIAH 41:10\n\nLook, I give you authority to trample on serpents and scorpions, and over all the power of the enemy. And nothing shall by any means hurt you.\n\n\u2014LUKE 10:19\n\nWhile Jezebel is one of the nastiest spirits you will ever encounter, it is vital that you maintain a kingdom perspective on the authority that God has given you. We are to study and discern demons, not to be afraid of them but to be empowered to conquer them. \"For we are not ignorant of [Satan's] devices\" (2 Cor. 2:11).\n\nWe are always to fight these spiritual battles from a position of faith, wisdom, and authority. Faith is empowered by focusing more on the answer than on the current problem. Yes, we know there is an attack, and yes, we know where it is coming from. But we also know where we stand with God. We have the answers. We boldly stand upon the Word of God, unafraid. It is imperative that you keep your mind planted in the Word of God. Don't be moved by Jezebel's evil report. Don't give in to the cloud of confusion that you may be feeling. It all has to bow!\n\nWISDOM IS DERIVED BY TWO METHODS\n\nThere are two methods for gaining wisdom. First, wisdom comes from study. Second Timothy 2:15 says, \"Study to show yourself approved by God, a workman who need not be ashamed, rightly dividing the word of truth.\" Becoming a student of God's Word is vital for successful kingdom living. You have to know what God says about you, your current situation, and the intended outcome.\n\nSecond, wisdom comes from intimate relationship and encounter with God.\n\nThe Spirit of the LORD shall rest upon him, the Spirit of wisdom and understanding, the Spirit of counsel and might, the Spirit of knowledge and of the fear of the LORD.\n\n\u2014ISAIAH 11:2\n\nThe Holy Spirit provides wisdom, counsel, understanding, and ability. You cannot spend time with Holy Spirit and not get insight! He is the Spirit of wisdom. Wisdom unlocks strategy. Strategy is the how-to. You know where you are to go, but you must have a strategy for how to get there. Spiritual warfare demands wise strategy.\n\nOr what king, going to wage war against another king, does not sit down first and take counsel whether he is able with ten thousand to meet him who comes against him with twenty thousand?\n\n\u2014LUKE 14:31\n\nPeople are often ignorant concerning spiritual warfare. They pick fights without a solid battle plan and end up sidelined by defeat and discouragement. Apostolic people are strategic people. They pray strategic prayers, they implement strategic plans, and they embark on strategic journeys.\n\nYou fight by enforcing your authority. Jesus won it all. You are boxing in a match that is fixed. You have been granted the ultimate legal jurisdiction over the enemy. The key is to war from the heavenlies and the place of your authority, not on an equal plane. That is the lie of the enemy, trying to call you down from heavenly places and get you stuck and vulnerable.\n\nThen Elisha the prophet called one of the sons of the prophets, \"Prepare yourself. Take this flask of oil in your hand, and go to Ramoth Gilead. When you get there, look for Jehu the son of Jehoshaphat, the son of Nimshi. Go in and make him rise from among his brothers, and bring him into an inner chamber. Then take the flask of oil, pour it on his head, and say, 'Thus says the LORD: I have anointed you king over Israel.' Then open the door and flee. Do not wait.\"\n\nSo the young man, the prophet, went to Ramoth Gilead. When he arrived, the commanders of the army were sitting, and he said, \"I have a word for you, Commander.\"\n\nJehu said, \"Which one of us?\"\n\nAnd he said, \"For you, Commander.\"\n\nSo he arose, went into the house, poured the oil on his head, and said to him, \"Thus says the LORD, God of Israel: I am anointing you king over the people of the LORD, over Israel. You will strike the house of Ahab your master, and I will avenge the blood of my servants the prophets and the blood of all the servants of the LORD from the hand of Jezebel. The whole house of Ahab will perish, and I will cut off from Ahab all the males in Israel, both imprisoned and free. I will make the house of Ahab like the house of Jeroboam son of Nebat and like the house of Baasha the son of Ahijah. Dogs will eat Jezebel in the territory of Jezreel, and no one will bury her.\" Then he opened the door and fled.\n\n\u20142 KINGS 9:1\u201310\n\nThe word of the Lord came forth powerfully that Jezebel's rule was over. She had resisted God's plan, defiled God's people, and withstood all prophetic correction. She would be killed and consumed by the dogs. Her end would be as violent as her evil reign.\n\nBORN FOR WAR\n\nProphets and prophetic people are born for war. They are bold, radical, spiritual warriors who see the attacks of the enemy but also see the power that is available for God's people. They do not back down from a fight. Jezebel reigned through terror, witchcraft, seduction, and intimidation, but God's prophet spoke her death sentence. The time was at hand for a deliverance in the kingdom, and her evil witchcraft could not stop it.\n\nJehu heard the word of the Lord and arose. He got in his chariot and went out for war. He was empowered by the directive of the Almighty. He did not sit in fear, wondering what would happen. What we need in this hour is a Jehu anointing! We need people to arise with the spirit that Jehu had upon him to break demon powers. It takes the spirit of Elijah to expose Jezebel but a Jehu anointing to break the spell. I pray that God raises up Jehus in ministries, in families, in regions, and in nations. May the holy boldness of the Lord be released, and may a supernatural anointing for victorious warfare be poured out upon God's people in this time!\n\nJehu armed himself for battle. He had the spirit of might upon him as he tapped into the justice of the Lord. We need a mantle of justice on our lives that becomes offended by the trespassing and meddling of Jezebel. We need a boldness to say enough is enough! We must get firm with the powers of hell and say NO to witchcraft over our families, NO to demonic forces in our ministries, and NO to devilish confusion over our minds. A Jehu anointing causes you to rise up in holy boldness and deal with the devil.\n\nThe spirit of Elijah recognized, exposed, and confronted Jezebel, but it was Jehu who defeated the witch. The Elijah anointing empowers us to see and expose Jezebel, but we must not stop there. We need the strength, boldness, and power of Jehu to rid our lives of this foul demon.\n\nWhen Jehu came to Jezreel, Jezebel heard about it. She put black paint on her eyes, adorned her head, and looked down through the window. As Jehu entered in at the gate, she said, \"Is everything all right, Zimri, murderer of his master?\"\n\nAnd he lifted up his face toward the window and said, \"Who is on my side? Who?\" And two or three eunuchs looked down to him. He said, \"Drop her down.\" So they dropped her down and some of her blood splattered on the wall and on the horses. Then he trampled her.\n\nThen he entered, ate and drank, and said, \"Attend to that cursed woman and bury her, for she is a king's daughter.\" So they went to bury her, but they found nothing of her except a skull, the feet, and the palms of her hands. They returned and told Jehu, and he said, \"This is the word of the LORD, which He spoke by His servant Elijah the Tishbite, saying, 'On the property of Jezreel dogs will eat the flesh of Jezebel. The corpse of Jezebel will be like dung in the field on the property of Jezreel, so that they cannot say, This is Jezebel.'\"\n\n\u20142 KINGS 9:30\u201337\n\nJezebel attempted one last seduction. She looked out through the window, having painted her eyes and adorned her head. She was hoping to use her seductive powers to escape her execution. Jehu was not entangled in Jezebel's erotic enchantment; instead, he was armed with the word of the Lord and a righteous cause. The Jehu anointing is unafraid of the powers of Jezebel. It is angered by the presence of unholy seduction and offended by bold manipulation.\n\nThe anointing upon Jehu brought the fear of the Lord. Those who served Jezebel were suddenly unafraid of her. Those around her who had been in her web of deceit were the very ones who would obey the command of Jehu to throw her down. People often put up with witchcraft and bondage despite repeated warnings from the Lord. They need the bold mantle of Jehu to come and set them free. Jehu did not ask or plead; he demanded justice. \"Throw the witch down!\" That was his unwavering demand, and in a single moment the demon-possessed queen met her death. She was thrown down, trampled upon, and eaten by the dogs. She was defeated, and the power of her spell, broken.\n\nSTRATEGIES TO BREAK THE HOLD OF A JEZEBEL SPIRIT\n\n\u2022 **Discern.** Properly identifying this spirit empowers you to break free from it.\n\n\u2022 **Pray.** Warfare requires wisdom. Do not just immediately address it; you will come under counterattack and warfare. Allow the Lord to give you divine wisdom and insight.\n\n\u2022 **Establish godly boundaries of authority.** Don't be an Ahab! Jezebel pushes the boundaries, disrespecting all manner of authority. Everything in the kingdom has rank and order. It is a breach of biblical understanding to claim that Holy Spirit is against order and authority. Establish Spirit-led boundaries in your personal life, in your family, and in ministry. Do not be pushed outside of what the Lord has shown you.\n\n\u2022 **Use strong spiritual authority.** Demons will laugh at weakness. Fight from a position of strength. Jesus already won the victory! Be like Jehu and make a bold demand. Do not ask demons anything. Cast them out and break their power.\n\n\u2022 **Confront, offer repentance, then deliver.** After you have prayed and sought the Lord for clarity, if He has released you to deal with it, then confront the person and the situation. Offer opportunity for growth, repentance, and deliverance. Most people bound with Jezebel refuse to even examine their hearts because they are deeply deceived. If repentance is shunned, then their influence, position, or relationship in your life must be severed.\n\nTypically there will be a power struggle. Be prepared, and know that if you are walking in righteousness and integrity, then the Lord will defend you. Don't be manipulated or moved by the emotional manifestations or spiritual pressure. Stand strong! Stand on the truth! Stand on the Word!\n\nPOINTS TO CONSIDER\n\n\u00bb Is there any area in your life where you have taken on an Ahab mentality?\n\n\u00bb Have you allowed a Jezebel spirit to operate?\n\n\u00bb If you discern that Ahab spirit, kick it out. Call forth the fire of God to cleanse and purge.\n\n\u00bb Jezebel and Ahab are codependent and rely on each other.\n\n\u00bb Demons work in packs. Find the leader and kick it out.\n\n\u00bb Do not fear; use your authority.\n\nWHERE TO BEGIN\n\n\u00bb Determine to close any and all doors of manipulation.\n\n\u00bb Refuse to take on an Ahab personality and create illegal agreements in the spirit.\n\n\u00bb Study covenant thinking and living. Be strong and bold.\n\n\u00bb Realize that God has boundaries. It is important to honor His ways.\n\nPRAYER\n\n_Lord, I thank You that I am a covenant person. Teach me how to live with Your heart and perspective. I refuse to take on an Ahab personality. I thank You that I walk in truth and kingdom identity. I thank You that I take a stand for what is right. You have granted me authority over the powers of hell. I am unafraid of the powers of Jezebel. I thank You that all witchcraft, confusion, and curses of Jezebel are broken over my life in Jesus's name. I boldly declare that I am free through the shed blood of Jesus. I release freedom over my entire family in Jesus's name. Amen._\nChapter 6\n\nPYTHON, THE HEAT-SEEKING DECEIVER\n\nTHERE IS NOTHING like the fresh wind of God blowing upon your life. I can think of so many times I have been in seasons when the refreshing winds of God were blowing. Other times the healing wind of God was blowing. At the first church I planted some years ago, the Lord gave me very specific instructions about conducting a monthly healing meeting. Those meetings became the lifeblood of our young ministry.\n\nThe Lord sent people from all over the region and surrounding areas to experience the miracle-working power of God. There was such a wind of God's Spirit on the concept and the gatherings. You could feel God's sense of approval on the direction that we were taking. Of all the things that we had done at the time, those healing meetings were my absolute favorite!\n\nI can vividly remember one evening when a mother brought her teenage son who had major issues with one of his legs. She said he was born with a severe condition that demanded he wear a brace to walk. The mother explained to me that if he attempted to walk without the brace, his leg would just buckle and snap because it lacked the necessary parts to stand. As she shared that with me, I clearly heard the voice of the Lord say to pray and take the brace off. God was going to heal him! I related to the mother what the Lord was prompting me to do. I wanted her approval and cooperation before taking such a daring step of faith. She agreed. We prayed and took the brace off, and the young man took off running! It was a great day of celebration. God touched his body and caused the impossible to be suddenly possible.\n\nDuring that season outstanding miracles were commonplace in the meetings. We would get so caught up in the healing presence of God that the miracle realm would just open, and it became easy to get people healed. The ripples from those meetings went out throughout the territory; we became known as a group who believed in miracles. We would frequently receive phone calls asking us to come to the hospital and pray for someone in critical condition. We went and saw wonderful healings and miracles. It was a rich time for our ministry.\n\nThen suddenly the flow in the healing meetings changed. It was as if the life and vitality of the meetings diminished. At first I wondered what we had done to cause this. Was something wrong? Did we grieve Holy Spirit? What was happening? It was clear that the wind was not blowing on these gatherings as strongly as it had been. As I began to pray, the Lord told me that there was a seasonal shift taking place and He wanted to do some new things. He began to pour out strategy. It became clear that the winds of redirection were upon us. As we obeyed the Lord's instructions, we experienced wonderful new assignments, and the hand of God brought even more miracles. The key is that my team and I determined to ride the wind! Wherever the wind of God was blowing, that was where I was going to be. I wasn't going to be out on my own just doing something because it was a good idea.\n\nAt this point in my ministry I have experienced so many different seasons. I have had wonderful seasons of pioneering, daring seasons of growth, stunning prophetic seasons of major God encounters, difficult wilderness seasons of refining, and glorious seasons of revival and renewing. I have also had seasons of rapid advance when God suddenly launched me forward. I have learned to be thankful for each season and have determined to maximize the moment. People who ride the wind refuse to stay stuck! God is not stuck, and He never planned for you to be stuck. When the wind blows upon an area of your life, follow that wind!\n\nThen the LORD will appear over them, and His arrow will go forth like lightning; and the Lord GOD will blow the trumpet, and will march in the storm winds of the south.\n\n\u2014ZECHARIAH 9:14, NASB\n\nThe Lord spoke to Israel and told them that He would defend them. He said that He would march in the wind and their help was in the wind. Some time ago I was praying and reading about great men and women of God. I found that many of them learned how to ride the winds. They were not so stuck in their ways that they could not change course when heaven mandated it. This is a spirit life key! You must be willing to go where the wind of God is sending you. You cannot just go by your own ideas or thinking. I read about one great revivalist who said that she always kept a bag packed. She would get caught up in the Spirit, and God would tell her to go to a nation, so she would immediately gather her things and head out. God used her to shake nations. She carried His glory in a significant manner and ministered under the glorious winds of heaven.\n\nGLORY IS IN THE WIND\n\nGlory people are Spirit people! Spirit people are wind people; they move with the blowing of the wind of God. They discern the winds of the Spirit. There are various winds of God available to blow on people for various reasons: healing winds, winds of change, winds of transformation, holy winds, and winds of transition.\n\nThe Hebrew word for \"spirit\" is _ruach_ , which means \"breath\" or \"wind.\" Genesis 2:7 says, \"Then the LORD God formed man from the dust of the ground and breathed into his nostrils the breath of life, and man became a living being.\" Notice the order of events: God gathered the dust in order to create mankind, but it was not until the _ruach_ , the breath of God and the wind of the Spirit, was released that man lived! The breath of God in us and upon us is what fuels our being. We are a spirit who lives in a body and possesses a soul. We are not a body with a spirit. The essence of who and what we are is the very breath of God Himself.\n\nIt was this breath that left Adam and Eve in the garden so that they suddenly died. Their bodies lived, but they were spiritually dead, as they were cursed to be people of the dust and not people of the spirit. The spirit is the life! We are carriers of the winds of heaven. When we pray, wind is released! When we decree, wind is released! When we prophesy, wind is released! When we worship, wind is released! It comes from our inner man, swirls all around us, and then catches us up into heavenly wind.\n\nHow did God birth the church? Did He assemble everyone together to be taught theology lessons? Did He begin with an organizational meeting? Did He have a heavenly board meeting to vote on what to do and how to do it? The answer to all these questions is a resounding no!\n\nGod urged a hungry people to gather in a room and to wait there. They were to assemble and patiently remain on standby until something miraculous happened. They were asked to believe in something they had never seen or experienced. They were promised something revolutionary and remarkable. They were given the task of prayer. Prayer fuels the arrival and release of the wind in our lives.\n\nAs this band of faithful ones was gathered in the Upper Room, there was a suddenly! In a moment, God blew through that assembly with a wild display of His power. The early church was launched in the wind!\n\nWhen the day of Pentecost had come, they were all together in one place. Suddenly a sound like a mighty rushing wind came from heaven, and it filled the whole house where they were sitting. There appeared to them tongues as of fire, being distributed and resting on each of them, and they were all filled with the Holy Spirit and began to speak in other tongues, as the Spirit enabled them to speak.\n\n\u2014ACTS 2:1\u20134\n\nThe sound of the wind hit, and everything changed. Holy Spirit showed up in the wind. His breath began to flow through that place, and things were turned upside down. No one was the same. It wasn't a sound effect. They were hearing the entrance of the Spirit of God; the wind had arrived. Tongues broke forth, and fire fell. New wine was poured out as men and women were intoxicated in the glory of God. They left that room forever changed.\n\nPeter, who had denied Christ and was wracked with fear, became bold as a lion. He had been baptized in the wind and fire. He was chosen by God in a redemptive act to preach the first sermon of the church and to win three thousand people to Jesus in a single moment. The wind that blew that day shifted the course of human history. That same wind is still blowing and rewriting history. That wind is still lifting men and women out of obscurity and mantling them for nations. That same wind is still releasing flames of consuming fire. That same wind is still sweeping gatherings of the hungry and reaching the very deepest places in the hearts of men.\n\nThere is a spirit from the depths of hell that hates the wind. It comes to quench the breath and choke its victim to death. It is in the trio of anti-anointing demons that we are revealing in this book. It hides, waiting for the perfect moment to strike and implement its diabolical plans. It is a spirit that is determined to find people and places full of the wind of God and suffocate them.\n\nSOUNDS RIGHT BUT WRONG SPIRIT\n\nThe apostle Paul was on a preaching assignment when he encountered this evil demon:\n\nOn one occasion, as we went to the place of prayer, a servant girl possessed with a spirit of divination met us, who brought her masters much profit by fortune-telling. She followed Paul and us, shouting, \"These men are servants of the Most High God, who proclaim to us the way of salvation.\"\n\n\u2014ACTS 16:16\u201317\n\nPaul was on his way to prayer. He was on assignment, determined to see the gospel released in a powerful capacity. As he was focused on his mission, a great distraction arose. This young woman, who was totally possessed with an evil spirit, began to follow him. She was constantly crying out as she followed him. The content of her message was not wrong, but there was a wicked entity involved in her pursuit.\n\nPaul had arrived in the region with a clear mandate from the Lord to spread the gospel. He was sent as an apostle to bring liberty, truth, and salvation. He was a general deployed into a crucial mission field. Who met him? I believe it was the ruling spirit of that region. I will elaborate as we continue. This young lady followed him and would not relent. She was possessed by a spirit of divination. The literal translation of _divination_ is \"python.\" The girl was being controlled by the python spirit. It was binding the region with fortune-telling, incantations, and evil spiritual practices over the land. Paul was there to break its grip, but it was determined to stop him. Paul was being hunted by a serpent!\n\nHe wasn't sitting around doing nothing. He was in hot pursuit of God's presence and plan. He was on his way to prayer when this snake showed up. This spirit comes on the scene when a person is full of the life-giving breath of God. To more fully understand the tactics of this spirit, we must recognize the type of serpent a python is.\n\nA python is a fierce, suffocating warrior that quietly and methodically stalks its prey. It does not kill a victim with venom. The danger of the python is not its bite but its strength. When it sinks its teeth into its victim, the python is able to wrap its muscular body around the victim, constricting slowly. As the victim struggles for breath, the python tightens its hold. Every time the victim exhales, the pressure is increased, breaking bones and totally overwhelming the respiratory system. Finally the victim concedes to the massive grip of the serpent, dying of suffocation. A python kills by removing the breath\u2014this is the picture of this vile spirit.\n\nA python is a heat seeker. It identifies life by locating heat. Dead things are cold, but living things are full of warmth. The python spirit looks for the move of God. It looks for spiritual breakthrough and those pursuing the more. It finds those on their way to higher places and slithers through their lives and ministries, waiting to impose its deadly grasp.\n\nShe did this for many days. But becoming greatly troubled, Paul turned to the spirit and said, \"I command you in the name of Jesus Christ to come out of her.\" And it came out at that moment.\n\nWhen her masters saw that the hope of their profits was gone, they seized Paul and Silas, and dragged them into the marketplace to the rulers. And they brought them to the magistrates, saying, \"These men, being Jews, greatly trouble our city and teach customs which are not lawful for us, being Romans, to receive or observe.\"\n\nThe crowd rose up together against them. And the magistrates tore the garments off them and gave orders to beat them. After they had laid many stripes on them, they threw them into prison, commanding the jailer to guard them securely. Having received such an order, he threw them into the inner prison and fastened their feet in the stocks.\n\nAt midnight Paul and Silas were praying and singing hymns to God, and the prisoners were listening to them. Suddenly there was a great earthquake, so that the foundations of the prison were shaken. And immediately all the doors were opened and everyone's shackles were loosened.\n\n\u2014ACTS 16:18\u201326\n\nThis young woman persistently followed Paul and his team around. She was loosing evil witchcraft against them. Though her words were not necessarily incorrect, Paul discerned the spirit behind what she was saying. This is how the devil works\u2014when you are pressing in to the fire of God, the passion of prayer, and the longing for revival, the enemy sends a demonic attack to distract and disrupt your pursuit.\n\nPaul was grieved in his spirit. He could sense the presence of evil in this young lady. On the outside it may have looked good, but there was a foul serpent lurking just beneath the surface. Thank God for the discerning of spirits and revelation knowledge. If only all Christians realized what a powerful position they occupy in the spirit through the completed work of Jesus. Not only did Jesus win the spiritual battles for us and position us in Him, but He also paved the way for the wonderful Holy Spirit to come and indwell us. We can hear from heaven and avoid devilish traps through the leading of the Spirit. Galatians 5:25 says, \"If we live in the Spirit, let us also walk in the Spirit.\" We can walk with Holy Spirit into truth, illumination, and victory! We do not have to be blinded by the powers of hell.\n\nThe python spirit is connected to fortune-telling and the working of familiar spirits. This young lady brought great financial rewards to her masters through her fortune-telling. She was empowered by a false spirit. The Greeks believed in the connection between Python and Apollo. They also embraced an \"oracle,\" a demon-possessed false prophet whose body would be taken over by familiar spirits to release the utterances provided from evil spirits. The Greeks believed wholeheartedly in these evil spiritual practices.\n\nThis culture of a python and the oracle predicting events had invaded Philippi. This evil python spirit and its practices influenced everyday life in Macedonia. According to Greek mythology, the serpent, who guarded the temple at Delphi, was killed by Apollo (and then the oracle at Delphi took on the name Pythia, and Apollo became known as Pythian Apollo). The word Pythian was later applied to diviners or soothsayers inspired by Apollo. The python spirit empowers evil sorceries and false spirit communication. As Paul was ministering throughout the region of Macedonia, he undoubtedly saw and discerned the reach and grip of this evil spirit.\n\nThere are historical accounts of violent physical reactions as false prophetic voices yielded to this demon in Greek culture and released utterances. Their bodies were overtaken by this serpent. In much the same way as the power of God affected people's physical bodies in the Upper Room, this demonic entity overwhelms a person who opens the door and creates evil manifestations as it releases its venomous prophecies.\n\nThe python spirit is alive and well in the world today, working through psychics, fortune-tellers, mediums, and all manner of false spiritualists. These underworld practices are clearly and strongly forbidden in the Bible. Leviticus 20:6 states, \"The person who turns to spirits through mediums and necromancers in order to whore after them, I will even set My face against that person and will cut him off from among his people.\" Christians are not to seek out spiritual assistance from any source other than God's Word, His gifts, His Spirit, and those being led by His Spirit. Seeking spiritual assistance elsewhere is a sure way to open doors to the enemy in your life.\n\nA number of years ago I encountered a woman who was totally overrun with demonic powers. She would sit in church meetings and begin to manifest demons. Her body would contort in a way that looked as if a serpent was operating through her. As the process unfolded, she would twitch with strange movements and then belch repeatedly.\n\nThe manifestations became a tremendous distraction. The leaders at the church were ministering to her, but she refused spiritual help. She would have to be removed from meetings because of the commotion she would cause. I will never forget one episode in church. There was a guest speaker preaching quite powerfully. In the blink of an eye, the demon-possessed woman turned and slapped her husband across the face. It made a huge noise, and everyone looked around, but she had moved under supernatural power, and very few people saw it because it was so fast. I happened to catch it from where I was sitting, but she moved uncommonly fast.\n\nThis woman had gotten wrapped up in false spiritual practices and became overwhelmed by the power of darkness. She opened a door for demons to come inhabit her. The devil doesn't care how small the crack in the door is\u2014he will exploit it to come in. This is why governing our spiritual lives by the Word of God and obeying the promptings of Holy Spirit are critical.\n\nThe python spirit seeks to ensnare people in a false prophetic capacity, to choke the very life of God out of them. The python spirit will release false fire over your life. The python spirit will release false words over your future. The python spirit will loose strange fire upon you. The python spirit will entice you and then cunningly take you into its deadly lair to defeat you.\n\nVISION INVADER\n\nOne of the words for \"prophet\" in the Old Testament is translated \"seer\" (e.g., 2 Sam. 24:11). The Scriptures often refer to Old Testament prophets as seers. This is a particular type of prophetic function that indicates a high level of visionary encounters in the realm of the spirit. Seers receive detailed prophetic encounters in visions, dreams, and night visions. They process the realm of the spirit in high definition. They are often totally swept up into the spirit realm, far beyond the confines of this natural world. They feel and sense things in a way that is deeply impacting. This is true of all prophetic people. They are extremely engaged in the realm of the spirit.\n\nMany young prophets and seers struggle to find anyone who can assist them in developing their gift, to understand the very real experiences they are having. Oftentimes churches shun the prophetic ministry due to lack of understanding, controlling demons, or religious traditions. It's sad to say, but many churches today have made a conscious decision to program out the move of God. They view prophets and prophetic people as troublemakers.\n\nThese abandoned prophets in the making are having spiritual encounters and experiences that they don't understand. If they cannot find help in the kingdom of God, they can be drawn into false spiritual practices because they are embraced by people who have spiritual language and an understanding of what they are experiencing. Sadly I believe that many God-called prophets have been deceived by the python spirit and are functioning outside of God's call and blessing upon their lives. This is one reason that I am passionate about prophetic teaching, training, and activation. I don't want to see python tangle up and deceive future world changers.\n\nI pray that God will raise up an army of spiritual mothers and fathers who will accept, love, and groom prophets, seers, and those with uncommon prophetic abilities. It is also key that those being trained are humble and submissive. One of the quickest ways for a deeply prophetic person to get off track is pride. Seeing and knowing in the spirit realm is a God-given gift. Pride comes in when people begin to use their gifts as a means of lifting themselves up and demanding people recognize their greatness. It becomes very demonic behavior that draws people away from Jesus and does damage to the people of God instead of strengthening them and serving them.\n\nPride goes before destruction, and a haughty spirit before a fall.\n\n\u2014PROVERBS 16:18\n\nThe python spirit comes to choke the life, the wind, and the breath out of you. When you are under attack from a python spirit, there is an uncommon sense of heaviness and oppression. As much as it is a false prophetic spirit, it is also a wicked spirit of bondage. It was no mistake that it was attempting to distract and derail the apostle Paul. It wanted to stall out the mission that God had sent him on. It wanted to abort the apostolic assignment. Python comes to rob the life-giving force of anointing and power.\n\nIn _The Spiritual Warfare Battle Plan_ , Jennifer LeClaire sums it up perfectly:\n\nThe python spirit had a stronghold in Philippi. When the man of prayer started heading for the house of prayer, this spirit launched its first attack against him\u2014a distraction followed by a full-blown trial that aimed to take him out of his purpose. Python knows it has no authority in a city that prays in the presence of God, so it works to distract people from praying so they can't fulfill their purpose.\n\nPython would rather watch you lick your wounds than pray to a healing God. Python would rather hear you complain or gossip than take your problems to a miracle-working God. Python would rather distract you with attacks, trials, and persecutions than see you press into a gracious God for deliverance. Again, python's ultimate goal is to put you in bondage and thwart your purpose. You may be going through the motions but feel dead on the inside because python has squeezed the life out of you.\n\nThe python spirit is an enemy of revival! It shows up to squeeze those who are pressing in to receive fresh flames of fire, strong anointing, and life-giving outpouring. Python has an assignment in regions to clearly identify those who pose a spiritual threat. When new ministries are established in a lukewarm territory and begin to engage the heavenlies, releasing the life of God over dry people, python senses the heat and comes lurking in the darkness, ready to strike. Understanding the operation of this spirit is essential for those who are going to shake a territory. It is certain that python will show up. Recognizing its evil plots empowers you to overcome.\n\nThe python spirit comes into families to put them under such a heavy squeeze that they quit moving forward. There is an anointing for families. In the same way that God empowers a person for exploits or a church for great things, He can release fresh oil over a family to bond together, to love one another, to experience the kingdom together, and to defeat the operation of hell. Python will come to bind up that power and create a cold atmosphere void of the life of God.\n\nPython comes against prayer. This spirit wants to kill prayer lives. It knows that prayer is one of the most potent weapons in the arsenal of a believer. It doesn't mind if people go through the motions of religion with no spiritual power, but when they begin to hunger for intimate communication with God, this demon gets nervous. Paul was raising up a kingdom movement that centered on prayer and the presence of God, and python was determined to shut it down. If Paul had not properly discerned this demon, it is hard to say where the church would be today. Thank God the apostle of God refused to give in. Thank God he didn't move away from the promise. Thank God he rebuked that demon. Thank God he was a warrior.\n\nPOINTS TO CONSIDER\n\n\u00bb Have you been distracted, particularly in prayer?\n\n\u00bb Have you opened any doors to familiar spirits?\n\n\u00bb Have you battled heaviness?\n\n\u00bb Do you feel as if you just cannot go forward in what God has called you to do?\n\n\u00bb Are there any enduring spiritual battles that could be the operation of a python spirit?\n\nWHERE TO BEGIN\n\n\u00bb Understand that the python spirit comes after things that are alive.\n\n\u00bb When you press in to God, this demon will attempt to take you out.\n\n\u00bb Concerning prophetic ministry, it is extremely important to guard yourself, making sure that you are drinking from a pure stream. How can you know that? By a person's fruit, alignments, and integrity.\n\n\u00bb Don't just accept every prophetic word; examine it by the written Word, and make sure the vessel is pure. This does not mean to be critical, but it means to honestly evaluate. Deception begins with a small seed.\n\nPRAYER\n\n_I thank You, Lord, that I am not deceived. I hear Your voice and obey Your leading. I ask for clear direction. Lord, I come before You, thanking You for Your purity in my life. I receive the washing of Your blood, and I thank You for freedom from all deception. I embrace Your wind in my life! I am hungry for Your anointing, for Your power, for Your fire! I want Your life moving in me and ministering to others through me. I break the grip of every operation of the python spirit in Jesus's name. I break its power right now and command it to go! I thank You, Lord, that I am free indeed. In Jesus's name, amen._\nChapter 7\n\nBREAKING PYTHON'S GRIP\n\nWE'VE ESTABLISHED THAT the python spirit attacks and chokes the breath out of its victims. When Paul exposed this spirit, it was hiding in a young lady. She was saying and doing things that seemed right to others, yet there was something that disturbed Paul. He was looking beyond the natural realm, glimpsing the unseen realm of spirits. One of the keys to defeating this spirit is recognition. You must be able to see it to defeat it.\n\nKNOW YOUR ENEMY\n\nThe gift of discerning of spirits empowers a believer to see the spirit motivating a person or situation. During the operation of this gift, spiritual eyes are opened so that truth is illuminated. This is imperative in dealing with the enemy, who is a master at hiding. Prophetic exposure brings violent confrontation.\n\nWhen Paul cast the python spirit out, there was a massive counterattack, but his faith never wavered. After being attacked, mistreated, and falsely accused, he held a spontaneous worship service in a prison cell. His praise brought about liberty.\n\nPaul's reaction reveals keys to freedom from this spirit: worship, praise, and faith. We will explore each of these keys to understand how they bring deliverance from the grip of an evil serpent. We will also examine the appearance of demonic spirits throughout the Bible and the promises of God's power over their attacks. We will draw lessons from the life of Paul by discussing his demonic encounters and victory strategies.\n\nTo grasp the concept of the python spirit, it is important to identify the hunting techniques of an actual python. These serpents are not known to kill their prey with deadly venom. They hide, lurking in the darkness and patiently waiting to capture their victim. They are proficient in identifying life by the presence of heat. Where there is heat, there is vitality and breath. As we established earlier, breath represents life in the spirit and the move of God. A python spirit, then, comes after that which is alive and moving. It has no interest in stalking lukewarm believers\u2014they are already in its grip. It comes after people who are manifesting the heat of God and releasing the breath of heaven. It lies in wait for the perfect moment to strike.\n\nWhen does python strike? As a person is pressing in to the things of God. This demon stalked the apostle Paul on his way to prayer. Its purpose was to extinguish the move of God. A python spirit suddenly strikes its prey, coiling its massive body around the victim, putting it in a deadly squeeze, and slowly killing it. This demon functions much like a natural python in that it grips its victim, choking the spiritual wind out of it.\n\nThe python spirit hates the life-giving anointing of God and comes to contaminate, diminish, and quench the anointing. It is hunting every person who is moving in the Spirit. It is on the prowl, looking for fiery places of revival, with the intent of shutting down the move of God. It is actively searching the earth for ministries that are moving in the Spirit. It will release plans to stall out the move of God.\n\nWhen the python spirit strikes, vitality is lost, and the life of God is suffocated. It loves to choke out the move of God. It loves to remove the life-giving flow of heaven. Under attack by a python spirit, you feel lifeless. It squeezes you with fierce pressure to get you to give up on hopes, dreams, plans, and purposes. It wants you to settle for a mediocre spiritual life void of the rich power of heaven.\n\nThe python spirit will get you second-guessing the things that heaven clearly and accurately revealed to you in the secret place. This is one of the cornerstones of a python attack. It grips the mind and does all that it can to stall out your pursuit of heaven's agenda. It is an anti-anointing demon. The anointing is connected to purpose. Python is on assignment to dilute your purpose and thereby hinder the anointing in your life.\n\nHeavy mind traffic and swirling thoughts of confusion are elements of the python attack plan. The enemy knows that the mind is the point of logic, thought, and reason. If he can sow seeds of destruction and confusion in the human mind, then he can create questioning of the call of God, stealing clarity.\n\nI will set no wicked thing before my eyes. I hate the work of those who turn aside; it shall not have part of me.\n\n\u2014PSALM 101:3\n\nThis is a strong word! Guarding our eyes ultimately helps protect our minds. The enemy recognizes the power of the human mind. It has often been said that if you can imagine it, you can become it. The Bible declares in Proverbs 23:7 that as a man thinks in his heart, so is he. This means that our inner thought life dictates the direction of our outward living. When a spiritual attack strikes the mind, the ultimate goal is to pollute thinking and cause the person to sink down into a pit of despair. This is especially true when the python spirit is at work. Its twisting and deceitful lies put purpose under siege in an attempt to derail destiny.\n\nTherefore guard your minds, be sober, and hope to the end for the grace that is to be brought to you at the revelation of Jesus Christ.\n\n\u20141 PETER 1:13\n\nWe are to be intentional about guarding our thought lives. We cannot and must not allow every negative thought to take root in our imaginations. Our imaginations are the creative engines of our beings. When our imaginations are not protected, we begin to see, dream, and declare destruction. The aim of a spiritual attack is to get people so bound in confusion and fear that they prophesy their own defeat.\n\nPYTHON POLLUTES\n\nA python spirit overwhelms its prey with a barrage of negative thoughts. It attempts to pollute the creative flow with damaging negativity. We must be quick to recognize this scheme, casting down contrary thoughts with the name of Jesus and the Word of God as two of our weapons. In 2 Corinthians 10:5 we are instructed to cast down imaginations. This is not passive on our parts. This is aggressive warfare. This demands action from us. In the spirit realm words carry authority and power. We must combat the lies of the enemy with a strong confession. We need to immediately command those thoughts to leave in the name of Jesus. We must exercise bold authority over lying thoughts with the words of our mouths.\n\nWhoever guards his mouth and his tongue keeps his soul from trouble.\n\n\u2014PROVERBS 21:23\n\nKeeping our mouths full of the Word of God, the promises of God, and the will of God is the key to victory. We are to combat the lies of the enemy with the words of our mouths. We are also to use the Word of God as a weapon. For every lie, we need to answer with a truth. Answer the attack with a promise. Jesus revealed a tremendously effective warfare strategy when the devil came to tempt Him. Each time the enemy released warfare and lies against Him, He answered with the Word of God.\n\nThen Jesus said to him, \"Get away from here, Satan! For it is written, 'You shall worship the Lord your God, and Him only shall you serve.'\"\n\n\u2014MATTHEW 4:10\n\nThe devil hurled lies and temptations at Jesus, but at every point he was met with truth. Jesus stood on the Word of God. Jesus spoke the Word of God. Jesus fought the devil with the Word of God. This is our example when under attack\u2014we need to meet the enemy with the Word of God. God's Word acts as a defensive weapon, repelling the aggressive advances of the enemy. We can stand firmly upon the promises in the Word of God. As we dive deep into God's Word, our faith is built, and we are empowered to come out of the crippling effect of spiritual attack.\n\nWhen the apostle Paul discerned, confronted, and cast out the python spirit, there was an uproar. Paul and Silas were brutally attacked, stripped, beaten, and imprisoned as a result of a major demonic counterattack. Demons and demonic powers understand the art of war. One of the things that a successful army must do is stop the advance of the enemy. The army cannot allow an advance to go forth without opposing it. Many times when you engage the realm of the spirit and catch the attention of the enemy, you will receive a counterattack. Knowing this prepares you to face the onslaught and come out on the other side with victory. You are not to have a spirit of fear; you are to become a skillful warrior.\n\nHe trains my hands for war, so that my arms bend a bow of bronze. You have given me the shield of Your salvation, and Your right hand has held me up, and Your gentleness has made me great.\n\n\u2014PSALM 18:34\u201335\n\nDavid penned this powerful declaration of strength. He was known as a fierce warrior who understood the art of battle and had tasted the sweet cup of victory. Time and again God delivered and defended this radical lover of His presence. David was a worshipping warrior. He knew how to hide deeply in the arms of the Father. His unwavering affection for the Lord brought the presence of God into his life like a tidal wave. He dwelled in the courts of the Lord, enjoying immeasurable favor and supernatural protection. He faced massive spiritual attacks but kept coming out on the other side with the victory.\n\nUSE YOUR SECRET WEAPONS\n\nThe journey into destiny is not completed without a fight. The python spirit comes to undo the sense of purpose in our lives. It comes to squeeze out the fresh breath of heaven. It comes to hinder our prayer lives and dilute the anointing we carry. Python launched a severe retaliation against Paul and his apostolic team, but they had a secret weapon.\n\nAt midnight Paul and Silas were praying and singing hymns to God, and the prisoners were listening to them. Suddenly there was a great earthquake, so that the foundations of the prison were shaken. And immediately all the doors were opened and everyone's shackles were loosened.\n\n\u2014ACTS 16:25\u201326\n\nDespite the unjust beatings, the public humiliation they faced by being stripped naked in front of everyone, and their imprisonment, Paul and Silas decided to lift their voices in praise at the midnight hour. It is revealing that they began their spontaneous worship session at the darkest time of night. There is a message here. When the enemy blasts or retaliates against you and everything is falling apart, praise will bring you supernatural strength. Praise is a weapon!\n\nWhen we enter the realm of praise, we are fervently expressing our gratitude to the Lord and remembering His greatness. Praise establishes Jesus as Lord over every trial, every storm, every attack, and every present circumstance in our lives. As we fill our mouths with thanksgiving and fervent declarations of the goodness of God, we break through the fog of battle and encounter the warmth of His love and the force of His power. In 2 Chronicles 20 the men of Judah supernaturally won the battle against the enemy by praising and worshipping God. God defended them and set ambushes against their enemies because of their affection for Him. Praise was the choice weapon!\n\nIt was Paul's and Silas's praise that brought the presence of God into the prison and shook them loose. Praise can break the chains of bondage. Praise can shatter the lies of the enemy. Praise can chip away at the heaviness and release the glory. Once the glory comes in, you cannot stay bound. It is impossible. When God's presence fills a place, bondage has no choice but to go. Praise brings liberty.\n\nI remember a time when the power of God was healing people as I was ministering in a small church. A man came forward who had been battling severe back pain and had very limited movement. I felt the anointing and the power of God as I began to pray for him. After I prayed, I asked him to check his movement and the pain. Nothing had changed as far as he could tell. I began to praise the Lord. As I thanked God and worshipped Him, I got totally swept up in His presence. I was getting so blessed by the presence of God that I really forgot about the man and our prayer request. My hands were still on the man because I had been praying for him, but now I was lost in the presence of God, having my own praise party. After several minutes a surge of God's anointing came. The man began to move, and he said he was without pain and had full restoration of movement. Praise had overtaken affliction!\n\nDISCERNING A PYTHON ATTACK\n\nThere are two ways that you can properly detect the presence of a python spirit. First, you can identify it by becoming aware of its tactics. Secondly, it can be supernaturally revealed to you through the gifts of the Spirit. We will discuss both methods, but we will begin by listing the symptoms of a python attack and exploring each one to be equipped to detect, confront, and defeat this slithering demon.\n\nIt comes on the scene during a spiritual advance.\n\nThe python spirit shows up in the life of a person during his or her pursuit of God. Python comes on the scene in the life of a church or ministry as it is pressing in to the deeper things of God. Python has an assignment to bind and hinder the anointing. It is drawn to Holy Spirit heat. This attack will come at times of spiritual movement and progress.\n\nIt says things that sound right but grieve you on the inside.\n\nWe do not have a clear record of exactly how this demonic stalker was acting around Paul, but we do know what she was saying. This account provides us with some keen insight. In judging spiritual things, it is vital to examine not only the content but also the spiritual motivation behind it. For example, a minister can preach a sermon on faithfulness, but if his motive is to control people, then it is in error. A person with a python spirit manifesting may say all types of things that are right biblically but does not have a right spirit behind him. If you have a check in your spirit, pay attention. Stop and pray over it. Do not just shun the warning that God is giving you. Paul saw what others could not see.\n\nIt makes you feel restricted, overwhelmed, and bound.\n\nAll spiritual attacks will release heaviness and bondage upon your life. One of the telltale signs of an attack is when you experience extreme heaviness. The python spirit coils around you and puts you in a tight squeeze. You feel absolutely restricted and bound.\n\nIt can come with flattery.\n\nThe woman following Paul was saying all types of nice things with an undertone of flattery. Demons will often attempt to gain access by telling us what we want to hear. Again, this is why examination and prayer are key. Flattery overemphasizes things with an ulterior motive. It produces exaggerated kindness and grandiose spiritual announcements. It is designed to entangle your soul in order to get a foothold.\n\nIt is a stalker of the anointing.\n\nThe python pursues the anointing. It will follow people who are anointed. It will show up in places where there is a strong flow of heaven.\n\nIt can partner with religious spirits, the Jezebel spirit, witchcraft, and other demons to twist and squeeze the life out.\n\nDemons like to tag-team and partner for dark conquests. Typically when a major attack is launched against a leading ministry or person, there is more than one hidden devil. A python spirit can work in tandem with a religious spirit to squeeze the power out of someone's life. It can also act with a spirit of witchcraft and Jezebel to manifest a false and contaminated spiritual flow.\n\nIt persecutes the function and flow of anointing and comes against the move of God.\n\nWhen python comes to squeeze out the flow of heaven, it will persecute and mock the anointing of God. If it comes into a ministry, it will come against strong moves of God, finding fault and complaining. It will do the same in someone's personal life. It will bring railing accusation about spiritual encounters. We should and must judge all things by the Word of God, including heavenly encounters. Python raises unfounded concerns to totally shut down the sweet presence of God.\n\nIt is drawn to people with the fire of God.\n\nPython looks for those with the flames of God on their lives. It comes to put out the flame. It does not want the people of God burning with cleansing fire.\n\nIt is one of the spirits that fuel false prophetic flow, psychics, and so on.\n\nPython empowers occult activity, false predictions, false prophetic flow, and psychic powers. It is an evil demon that manifests false fire. It can accurately relay information that it unlawfully obtains from the spirit realm, but it is totally demonic. It cannot reveal the heart of God. False prophetic ministry always reveals things, details, and pieces of information, but it does not reveal Jesus. Real prophets reveal Jesus! They may share pieces of information or identify something that has happened in your life. Prophets see and know. They hear and encounter. They have supernatural access and can be highly accurate in their prophetic words over our lives, but each and every prophetic encounter will always end up drawing a person closer to Jesus. That is what real prophetic ministry does; anything less than that is false.\n\nIt can attempt to access position through impure prophetic ministry.\n\nThis is a common attribute of demonic spirits manifesting false fire. They will often attempt to gain an audience with a leader or access into the life of a person with impure prophetic ministry. The python spirit manifests in this capacity looking for influence. It will release information and illegitimate words to bait a person into allowing it into his circle of influence. Authentic prophetic ministry and prophets do not struggle to be noticed or to gain position. Many prophets would rather be in their prayer closets seeking God than on a stage or occupying a position of influence.\n\nIt manifests a false anointing and power.\n\nThe python spirit can be a carrier of strange fire. It can manifest realms of supernatural power and flow that come from the demonic realm.\n\nIt can attack your physical body.\n\nAs we know, man is a three-part being. We are a spirit, we live in a body, and we have a soul (mind, will, and emotions). Spiritual attacks typically show up on multiple fronts in these various dimensions of our lives. For example, it is not uncommon for someone under an attack of heaviness to experience physical symptoms of exhaustion. The python spirit is a constrictive spirit that comes to choke off the flow of wind and air. In the spiritual sense it will choke fresh life, prophetic anointing, and Holy Spirit wind. In the physical arena it can manifest as choking, severe respiratory problems, or even coiling of the physical body when being cast out. Many times in deliverance a python spirit being confronted will manifest with serpentlike moves in the physical body. I have seen many prophets and prophetic people get hit with the python spirit and have extraordinary bouts of respiratory, breathing, and throat attacks. These are uncommon attacks with a twofold purpose to manifest in both the spiritual and natural realm. To be clear\u2014not every physical issue is a spirit, but some are! We know that demonic powers can attack the physical realm, and this one does. It attacks the breath, wind, lungs, and voice.\n\nLearning the schemes and operations of the python spirit empowers believers for victory. We can quickly detect its operation and develop a prayer strategy to counteract its unlawful operations. Being armed with proper knowledge brings advantage in the arena of spiritual warfare. As we grow in our understanding of demonic spirits, their wicked desires, their battle tactics, and how to defeat them, we are skillfully equipped for victory.\n\nUSE THE GIFTS\n\nGaining knowledge regarding the python spirit is only one way that we can be prepared. There is another vital tool in uncovering hidden demonic powers. The realm of Spirit leading and the gifts of the Spirit are power weapons to properly expose demon spirits.\n\nEvery Christian has the right to tap into the realm of revelation and the leading of Holy Spirit. One of the wonderful gifts that has been provided in this set of gifts is the discerning of spirits.\n\nBut the manifestation of the Spirit is given to everyone for the common good. To one is given by the Spirit the word of wisdom, to another the word of knowledge by the same Spirit, to another faith by the same Spirit, to another gifts of healings by the same Spirit, to another the working of miracles, to another prophecy, to another discerning of spirits, to another various kinds of tongues, and to another the interpretation of tongues. But that one and very same Spirit works all these, dividing to each one individually as He will.\n\n\u20141 CORINTHIANS 12:7\u201311\n\nBelievers have been granted unlimited access to the realm of revelation and the all-knowing mind of Christ. Included in the ministry of Holy Spirit are these nine gifts that access spiritual breakthroughs and supernatural power. One of the greatest weapons against hidden devils is the discerning of spirits.\n\nWhat precisely is the gift of discerning of spirits? It is the ability to see, discern, and recognize the spirit motivating a person or situation. This gift can manifest in a variety of ways. All spiritual gifts operate as a means of transmitting divine information and power. Each one accesses different parts of the nature of God. The discerning of spirits allows a person to see and recognize a spirit being. This gift provides much-needed spiritual intelligence. It can function in a variety of ways. For example, people can discern a spirit by having an inner vision in which they see something with the eyes of their spirit and know that it is a spiritual being. People can experience a scent\u2014they actually smell something that reveals the presence of a demon. In the realm of the spirit, demons are foul! I have had experiences dealing with a person with an unclean spirit when I supernaturally smelled an awful stench that let me know there was a demonic power in operation. People can hear the voice of the Lord speaking to them about a spirit in operation. There are many methodologies of communicating the information, but the end result is the same\u2014a spirit being is revealed.\n\nEvery Christian needs to thank God for the nine gifts of the Spirit. When I was first born again, one of my mentors taught me that the Holy Ghost lives in my belly, and therefore all nine gifts of the Spirit live in me. My mentor also told me that if I forgot to claim them, they would not be in operation. Faith is in the believing, speaking, and doing. All exploits of faith have those three components. You believe, and therefore you speak; as you speak, you do! Faith creates movement and momentum.\n\nSpiritual gifts need to be activated. Begin to claim the discerning of spirits in your life. Take time to thank God and tell Him that you believe this gift is in you and active. Speak it out by faith. Then be quick to pay attention when God speaks. One encounter with the discerning of spirits can literally save you years of trouble.\n\nRecently I went to speak for a pastor whom I did not know very well. Right before I was scheduled to go, a dear friend of mine gave me a warning about the pastor and the ministry, but it was too late for me to cancel. When I arrived, I immediately sensed something was not right. The pastor was saying all the right words, but there was a void in the realm of the spirit. I prayed throughout my time there, and my eyes were opened to see a stronghold of rejection that caused the pastor to do immature, even offensive, things. Rejection, control, and even lying manifested while I was there. The discerning of spirits, along with wisdom from understanding spiritual warfare and people, helped me avoid a long-term relationship with this pastor that would have only brought great distraction and damage. I felt bad for the pastor, the church, and the family because they were all hurting as a result of these spirits buffeting this leader. The pastor needed deliverance and emotional healing. Although I was thrilled to minister to the people there, the alignment was bad, and I was glad to get out of that atmosphere. When you are in a spiritually toxic atmosphere, it depletes you. If you are a prophetic person, you really feel the swirling of evil powers, and your heart yearns for deliverance and healing for those involved.\n\nA strategy is an action plan with specific steps to achieve an objective. Let's talk about a strategy to conquer the python spirit in your life, your ministry, your family, or any other facet of daily living. How can we conquer this intruding force?\n\nFirst, it begins with recognition. All spiritual warfare is rooted in the ability to see what is trying to remain hidden. When you are dealing with any demon, you must see it for what it is. You cannot downplay it or ignore it. Ignoring a demon power in your life can result in horrific consequences. Recognition and identification are critical. Look at the symptoms and the manifestations, and then tap into any leading that you have, but if you are not forced into a timeline, then wait on the Lord for instruction.\n\nPython struck Paul on the way to prayer. Prayer is the force that often unravels the mind of God in a situation. We are given a powerful war strategy in the book of Ephesians, chapter 6, as Paul teaches on the armor of God. No spiritual warrior is complete without armor. Many of us have memorized and know the symbolism behind each piece. We pray and put on our armor by faith. But there is a vital instruction right after the portion of Scripture about our armor. Paul makes a statement that must not be overlooked:\n\nPray in the Spirit always with all kinds of prayer and supplication. To that end be alert with all perseverance and supplication for all the saints.\n\n\u2014EPHESIANS 6:18\n\nSpend time praying and listening. It is amazing what God can show you when you seek Him. Prayer is the foundation of productive warfare. Then use your authority! Command the spirit of python to break its hold in the name of Jesus. Paul directly confronted the spirit and cast it out. Command python to go. Stand boldly in the place of authority through the sacrifice of Jesus, and tell the python it has to go. Break its power with the name of Jesus. Break its heaviness with the name of Jesus. Break its divination with the name of Jesus. If python has made any false decrees over you, break them. Take authority over them, and call them broken.\n\nUse the weapon of praise. What did Paul do? He praised God. He boldly lifted his voice to the Lord. Praise will keep the momentum going. If you feel that serpent slithering, twisting, and attacking, start praising God. As you are praising Him, remember the times in your life when His goodness prevailed, and then thank the Lord for them. Get swept up in the goodness of God, and the anointing will flow! When the anointing flows, it will begin smashing the burdens and removing the bondage.\n\nFinally stand upon the Word of God. Follow in the footsteps of Jesus and use the written Word as your promise book. Find key verses that minister to your situation, meditate on them, and then speak them out of your mouth. Answer the enemy with the Word of God! He cannot defeat you when you align yourself with the Word, the blood, and the name of Jesus. Those are victory master keys. Cut off the head of the python and go free.\n\nPOINTS TO CONSIDER\n\n\u00bb Have you been discouraged, overwhelmed, or stagnant?\n\n\u00bb Recognize that the python spirit wants to choke out the life in you.\n\n\u00bb Increasing your prayer life will keep you built up in the Spirit.\n\nWHERE TO BEGIN\n\n\u00bb Praise and worship are keys to overcoming any demonic attack.\n\n\u00bb You have authority over demons, so fight back from a place of victory.\n\n\u00bb Jesus taught us to use the Word of God: \"It is written.\"\n\nPRAYER\n\n_Thank You, Lord, that I am free from all the power of the enemy. Show me any areas of my life or ministry where I have allowed any demonic influence. Show me if I am compromising in any area, and help me walk in purity and holiness. I decree that I will fulfill the purpose for which I was created and will not be stopped or hindered by the spirit of python or any other spirit. In Jesus's name, amen._\nChapter 8\n\nTHE DNA OF THE RELIGIOUS SPIRIT\n\nONE OF THE ultimate aims of anti-anointing spirits is to trap a person in dead works with no spiritual power or thrust. The anointing is like wind that empowers and accelerates the purposes of God. It is jet fuel in the realm of the spirit. Building in the kingdom without the presence of the anointing will lead to discouragement and bondage. The enemy wants to create a dry spiritual climate with no breakthrough or freedom. One of his favorite weapons is the religious spirit.\n\nWhat exactly is a religious spirit? Simply defined, it is a demon power that uses religious structures to entangle people in bondage, prevent transformation, and enforce the spiritual status quo. It fights revelation, defies prophetic insight, hates apostolic authority, and uses tradition to build structures of limitation.\n\nDON'T LET THE OIL RUN DRY!\n\nLiving in the kingdom demands the precious oil of Holy Spirit on a regular basis. This was one of the most distinguishing aspects of Jesus's ministry. Everywhere He went, there were supernatural results. You could not encounter the ministry of Jesus without encountering the power of God. The two went hand in hand. Hebrews 13:8 boldly declares, \"Jesus Christ is the same yesterday, and today, and forever.\" That means we should expect the same realm of power to show up now when we preach and declare Jesus and His glorious kingdom. The power of God is a witness to the person of Jesus. The power of God is part of the authenticity of the kingdom. The power of God is evidence of our sonship and position of redemption through the blood of the Lamb. We have been granted power over the devil. We have been given supernatural power over disease. We have been given power over temporary circumstances and challenges. Our inheritance brings access to the full majesty of heaven.\n\nWe are called to live in the anointing. We are called to move in the supernatural. We should not bow to the lies of the enemy or just give up in defeat. There is oil to help in every situation. There is Holy Spirit power and refreshing. The religious spirit comes to void those realities and crush life in the spirit. It comes to stifle and hinder. It comes to mock and challenge. It comes to bind and restrict.\n\nMany people begin with vibrant spiritual lives but end up dry and empty because of the religious spirit. Some years ago the Lord sent me to a specific region to establish ministries and labor for awakening. We began to experience explosive results. God did spectacular and supernatural things in our midst. After a period of time I began to meet some of the pastors who had sizeable ministries in the region. I noticed a common thread: they were all dry. I heard story after story of how their ministries began with the raw power of God, but over time they were worn down by criticism, complaints, and attacks, only to retreat and build ministries that were lukewarm. As I stepped into a higher perspective, I realized that I was seeing the stronghold in that region\u2014the religious spirit. It was active in every fiber of people's thinking and ruled the churches. It manifested through the sound of the region. It manifested through the government of the region. It manifested through the ekklesia of the region. Any ministry or voice that came in to challenge it was swiftly opposed. Over a period of years I tangled with this spirit many times. I had several prophetic dreams about the religious spirit and its plans in the territory. The bottom line is that it had established a successful stronghold and was choking the life out of the people of God. That is the aim of this horrendous demon.\n\nThen the Pharisees went and took counsel to entangle Him in His words. They sent their disciples to Him with the Herodians, saying, \"Teacher, we know that You are truthful and teach the way of God truthfully, and are swayed by no one. For You do not regard the person of men. Tell us then, what do You think? Is it lawful to pay taxes to Caesar, or not?\"\n\nBut Jesus perceived their wickedness and said, \"Why test Me, you hypocrites? Show Me the tax money.\" They brought Him a denarius. He said to them, \"Whose is this image and inscription?\"\n\nThey said to Him, \"Caesar's.\"\n\nThen He said to them, \"Render therefore to Caesar the things that are Caesar's, and to God the things that are God's.\"\n\nWhen they heard these words, they were amazed, and left Him and went on their way.\n\n\u2014MATTHEW 22:15\u201322\n\nThe religious elite of the day were trying to ensnare Jesus in a demonic trap. They were furious because He was preaching a message of power, love, and the kingdom. They were preaching a message of defeat, condemnation, and bondage. Jesus was loosing the captives and delivering the bound. They were exalting themselves while trampling upon the destiny of sons and daughters. Jesus was moving in great power and authority while they were dry and void of breakthrough. This is a picture of the stark contrast between the gospel and religion. The gospel will always point humanity to the grace of the Cross and the complete sacrifice of Jesus. By moving through that reality, we can ultimately find our positions as sons and daughters, demonstrating the kingdom, relating to God as Abba (our Father), and enjoying the carefree life of one who has been gloriously ransomed. It is easy to move in the power of God when we know our roles in His kingdom. Religion preaches a message of death, destruction, and entanglement. Religion strives without results. It points to human weakness instead of divine solutions. Religion works aggressively to deny and nullify the absolute price that was paid at Calvary. Religion continues to demand more without acknowledging that the price has been paid. Religion builds walls instead of doors and gates. Freedom will always offend religion.\n\nIn the passage in Matthew 22 there are some telltale signs of the religious spirit. These markers help us properly understand the makeup of this evil power.\n\nThey conspired against Jesus.\n\nThe religious spirit will rally allies in offense and anger. The Pharisees were outraged by Jesus's preaching and ministry, so they began to strategize in secret. Secrecy is one of the devil's great strategies. The enemy slithers around under the cover of darkness. His minions work best in hiding. The prophetic spirit boldly shines the light on hidden devils. That is why demons hate prophets, prophetic people, and prophetic ministry.\n\nThe god of this world has blinded the minds of those who do not believe, lest the light of the glorious gospel of Christ, who is the image of God, should shine on them. For we do not preach ourselves, but Christ Jesus the Lord, and ourselves your servants for Jesus' sake.\n\n\u20142 CORINTHIANS 4:4\u20135\n\nThe gospel brings the light of God on the scene! Light exposes the power of hell. The religious spirit operates with secret councils, secret committees, and backroom deal makers. They met in secret to attack Jesus. This is the game plan of the religious spirit.\n\nThey were offended by truth yet honored dead traditions.\n\nThe religious spirit does not honor the Word of God. It twists the Word of God to strip away the power of God. Religion establishes traditions without power. People go through the motions without any results or breakthrough. Where the anointing is, there is breakthrough and freedom!\n\nFor laying aside the commandment of God, you hold the tradition of men\u2014the washing of pitchers and cups, and many other such things you do.\n\n\u2014MARK 7:8\n\nThey held secret meetings joined together by a common offense.\n\nWhen you remove the oil of Holy Spirit, there is a dry atmosphere. Dry people are easily offended. I have never seen a religious person, religious church, or religious ministry where there is not an abundance of offense. God unites people by spiritual purpose, assignment, and legacy, but demons rally people around anger, bitterness, and offense.\n\nThey used flattery to His face, yet their ultimate goal was to take Him out.\n\nDevils will use flattery to speak to the egos of men and drive in a nail of bondage. They spoke to Jesus with the intent of flattering Him while secretly plotting against Him. Religious devils operate with a smooth tongue and a wicked plan.\n\nThe actions of the Pharisees displayed many attributes of the religious spirit. Let's explore some of the religious spirit attributes and how they manifest in the lives of believers.\n\nNEGATIVE WORDS\n\nThe religious spirit comes with negative words. It loves to debate, argue, and pick apart every revelation and legitimate kingdom concept. It will twist Scripture in order to back its point. It will rip apart any person or ministry with a powerful anointing. It will defy all those with a prophetic spirit because it hates the flow. It wants to build religious camps of slavery and bondage. It will engage in endless disputes without ever moving in the power of God or anointing. Religious demons tell you why you cannot pray for healing, yet they have never prayed for a sick person in spite of the clear call of all believers to do so. Religious spirits will tell you why you cannot prophesy or how you did it incorrectly, yet they never have a prophetic word. Religious spirits will invalidate current cutting-edge ministries and wrap their offense in biblical language to hide their true intent.\n\nThe religious spirit condemns. Jesus came to set each of us gloriously free from condemnation and the works of the law. As we awaken to all that He is and all that He has called us to be, we step into a bold, uncommon realm of liberty. The religious spirit continually harasses and condemns people, placing value on judgment above mercy and blinding the eyes of its victims to the real power of the gospel.\n\nThere is therefore now no condemnation for those who are in Christ Jesus, who walk not according to the flesh, but according to the Spirit.\n\n\u2014ROMANS 8:1\n\nThe religious spirit also loves to complain and criticize. Wherever there is a religious spirit in operation, there will be critical attitudes in operation with much murmuring, complaining, and wrong speaking. Complaining is the language of hell. It empowers curses, bondage, fear, and contamination. Our mouths were created to be powerful weapons in the hands of the Lord. The religious spirit has no joy or victory; therefore it complains on a continual basis. Religious churches are always murmuring. They are never satisfied and manage to find something to be discouraged about. They create a toxic environment that kills dreams.\n\nDo not grumble against one another, brothers, lest you be condemned. Look, the Judge is standing at the door.\n\n\u2014JAMES 5:9\n\nPERVERSION AND COMPROMISE\n\nThe religious spirit is a spirit of perversion! One definition of the word _pervert_ is \"to twist the meaning or sense of.\" Religious devils love to twist scriptures, bend truth, and alter the true meaning behind many kingdom concepts. The religious demon is a twisted devil. It binds people and turns the truth in order to deceive them. It leads to compromise by focusing on works without power. It creates a heart that is trapped in lifeless activity, easily opening the door to compromise. It abandons the plans of God in pursuit of building its own kingdom. Religion compromises the plans of heaven, the pathway of destiny, and the dream of God. It becomes so stuck in dead traditions that the life of the spirit is stomped out!\n\nBeware lest anyone captivate you through philosophy and vain deceit, in the tradition of men and the elementary principles of the world, and not after Christ.\n\n\u2014COLOSSIANS 2:8\n\nWhere there is a religious demon, there is often an abundance of sexual perversion. People are so yoked with heavy bondage in every area of their lives that they begin to manifest deviant behaviors. The religious spirit robs people of the oil of spiritual intimacy and passion. It dominates their lives with ritual and no relationship. It overwhelms their souls with hardship and oppression. In the absence of relationship and an abundance of rules, it's easy to give way to impure desires and appetites.\n\nThere are some groups that will not allow you to preach without a uniform, yet it is no secret that some of their leaders engage in improper sexual relationships. In fact, sexual sin runs rampant in many of these groups because they are bound by the oppressive powers of the religious spirit. It is not what they wear, their tradition, or their practices that increase the level of carnality; it is the yoke of the religious spirit. Having a particular dress code is not a sin, but when a person or group places a demand on others to conform to their preferences and preaches hell on them if they don't, then you have the working of a demon of bondage. When the oil of Holy Spirit is removed, something moves in to fill the vacuum.\n\nPOLITICS AND CONTROL\n\nThe religious spirit is a political spirit. I am not talking about governing or world affairs but about political methodologies in kingdom ministry. The religious spirit values soulish ideas and control above strong Spirit leading. It plays politics in the church instead of hearing and releasing the fresh word of the Lord. When a place or a people becomes gripped by the religious spirit, it will become overly political, consumed and driven by the opinions of men. This type of atmosphere discourages daring spiritual leadership and radical obedience. It shuns strong prophetic preaching and bold ministry. It will stifle the flow and ministry of Holy Spirit to placate people and keep them in a false state of peace. The religious spirit is a master at control, binding and hindering regions.\n\nThe religious spirit places far too great an emphasis on keeping everyone happy. The reality is that when you boldly preach the gospel, prophesy, and cast out demons, there will be waves in the ministry. An apostolic or prophetic ministry will have warfare. It is impossible to advance in the spirit personally or corporately and not have some type of resistance. The religious spirit will try to contain and confine. Spirit life is about freedom and identity. The religious spirit places unholy limitations on people's lives, trying to mask their identity in Christ. This demon boxes in people, ministries, and groups. It places a lid on spiritual life. Religious spirits hate freedom! Their goal is to capture and contain people. The anointing brings freedom. The power of God breaks prison doors open!\n\nNow the Lord is the Spirit. And where the Spirit of the Lord is, there is liberty.\n\n\u20142 CORINTHIANS 3:17\n\nNO JOY\n\nReligious people are angry people. They are offended by happiness, joy, and the preaching of the goodness of God.\n\nClap your hands, all you people! Shout to God with a joyful voice.\n\n\u2014PSALM 47:1\n\nThe Scripture tells us to worship God with exuberance and energy. Have you ever noticed how religious people value quiet and are offended by noise? Real praise is filled with emotion! There are emotions of thankfulness, gratitude, and excitement. As praise goes up, God's glory comes down, and people are shifted into the realm of the spirit. We were not created to offer God somber, unemotional praise and adoration. It is a shame that many people are OK with screaming, holding up signs, and dressing in uniform at a sporting event but totally irritated by bold and aggressive praise at a worship gathering. Oh, how shocked they will be when they get to heaven.\n\nCry out and shout for joy, O inhabitant of Zion. For great is the Holy One of Israel in your midst.\n\n\u2014ISAIAH 12:6\n\nAnd those the LORD has rescued will return. They will enter Zion with singing; everlasting joy will crown their heads. Gladness and joy will overtake them, and sorrow and sighing will flee away.\n\n\u2014ISAIAH 35:10, NIV\n\nThe Bible says that joy will crown our heads. We are to walk in the joy of the Lord. In fact, Nehemiah tells us that the joy of the Lord is our strength (Neh. 8:10). We can get through the battle with joy. Joy is a fruit of the Spirit, according to Galatians 5:22. As we spend time with Holy Spirit and live in the realm of the spirit, an overwhelming joy will come forth.\n\nJoy is not connected to the ups and downs of life. There is a joy in the realm of the spirit that can be continual. The religious spirit stifles the joy of the Lord. It creates an atmosphere void of joy. It despises joy. Where there is joy, there is laughter. There is celebration. There is happiness and rejoicing\u2014acting out the presence of joy. Religious demons create oppression, fear, and anger. In the camp of religion there is no room for joy or celebration.\n\nUNGENEROUS\n\nOne time the Lord told me that if I wanted to locate the depth of the religious spirit in a region, just preach on the goodness of God and generosity. Angry people are not generous. Religious people get upset with stories of God's goodness and His provision. Religious demons hate preaching on provision because it is part of the revelation of God as our Father. A father provides, protects, and guides. Religious demons paint a false picture of identity.\n\nFor the LORD your God has blessed you in all the works of your hands. He knows your wanderings through this great wilderness. These forty years the LORD your God has been with you. You have lacked nothing.\n\n\u2014DEUTERONOMY 2:7\n\nOver and over in Scripture God establishes Himself as our provider. One of the names of God, _Jehovah Jireh_ , means \"the Lord will provide.\" When we understand the role of our heavenly Father as provider, our faith is inspired, and we can dream big, believe big, sow big, and harvest big! Financial exploits are empowered by moving out of limitation and into the realm of generosity.\n\nBut my God shall supply your every need according to His riches in glory by Christ Jesus.\n\n\u2014PHILIPPIANS 4:19\n\nWe are called to live generously and give generously. This type of kingdom mentality breaks the back of poverty and fear in our lives. We no longer trust in ourselves or in this world's economy but rather in the abundant provision of heaven. We begin to move from fear and withholding the seed into faith and planting the seed with an expectation of supernatural harvest!\n\nThere is one who scatters, yet increases; and there is one who withholds more than is right, but it leads to poverty. The generous soul will be made rich, and he who waters will be watered also himself. The people will curse him who withholds grain, but blessing will be upon the head of him who sells it.\n\n\u2014PROVERBS 11:24\u201326\n\nWhat leads a man or woman to transformative change?\n\nDo you despise the riches of His goodness, tolerance, and patience, not knowing that the goodness of God leads you to repentance?\n\n\u2014ROMANS 2:4\n\nIt is the preaching of the goodness of God that positions a heart at the gateway of change. It is the revelation of the goodness of God that unfolds transformation. Understanding the abundance of God's goodness is a major key to moving into kingdom plans.\n\nThink about it\u2014how can you enter into exploits believing that God is mad at you and views you as a worm? The answer is simple: you cannot! A mentality of religion, fear, and false identity disempowers faith and massively fuels a fear mentality. You live your life as a series of works to earn a redemption that has already been paid for by Jesus in all of His glory.\n\nReligion shackles you to the whipping post of failure. It provides an erroneous narrative of the greatest love story ever written. It convinces people who were eternally pursued, bought back, freed, and forgiven that God is angry and the plan unfulfilled. Many people are totally paralyzed by this toxic mind-set.\n\nWhen we begin to hear, know, and understand the true nature of our Father, we become gloriously free. What do free men do? They dream immense dreams. They daringly step out to go for the seemingly impossible. This is why the religious spirit hates the message of the goodness of God\u2014it leads people to freedom. When you discover what has already been done and who you are called to be, you will be well situated for an exhaustive metamorphosis.\n\n. . . making the word of God of no effect through your tradition, which you have delivered. And you do many similar things.\n\n\u2014MARK 7:13\n\nOPERATIONS OF THE RELIGIOUS SPIRIT\n\nThe religious spirit can operate on many different levels. There are two common operations. One is the level of personal attack, in which this demon goes after the hearts and minds of people who are on the verge of personal breakthrough. The other level is corporate attack, when the religious spirit targets those who hold power to transform the corporate body of Christ, the ekklesia.\n\nPersonal attack\n\nHunger is always that catalyst of change in the life of an individual. A person gets so hungry for the more of God that he or she becomes desperate. This is the gateway to progress. Our individual spiritual lives should look like a journey of personal advancement. We should never be content to just camp out where God met us last month or last week. Our prayer lives should be in motion. Our devotional lives should be in motion. Our prophetic lives (seeing, hearing, knowing) should be in motion. Our revelation should be increasing and evolving as the face of Jesus becomes clearer to us.\n\nThe Bible teaches in Matthew 5:6 that the hungry shall be filled. What about the satisfied? There is no account of provision for those who are complacent. Hunger and thirst are the picture of pursuit. Hungry people will travel. Hungry people will get up early in the morning. Hungry people will invest their lives in pursuit of heaven. I was recently reading the story of one of the great healing ministers of the previous generation. This man would preach with mind-blowing miracles erupting like a volcano. I wanted to know more. What was his secret? What was it that caused him to touch God in a way that touched a generation? What did he know about God that I did not yet know? What was the key that unlocked the door to another realm? So I started reading about his life. I quickly discovered the secret. He decided to pursue God until he found Him in all His miracle-working glory. He locked himself away for days and hours, praying until the Miracle Worker showed up. He got hungry and began to pursue God\n\nMake no mistake about it, that kind of pursuit infuriates the religious spirit. It will hurl condemning thoughts against a person who begins to chase God. It will release vile accusation and mind-tampering heaviness. It confines people on a personal basis to block their access to their spiritual inheritance. This is one level of operation. This is boots-on-the-ground combat.\n\nCorporate attack\n\nThere is another level of warfare that the religious spirit engages in. It attacks people, leaders, revelation, offices, and wineskins that hold power to transform the corporate body of Christ. This is a higher level of strategic operation of the religious spirit. There are reforms that heaven releases on earth to suddenly shift the function, insight, and strength of the ekklesia. There are religious spirit high-level operations that create mind-sets, cultures, and concepts to keep the corporate body of Christ from moving forward.\n\nUltimately you cannot change the flow of the ekklesia until you change the function. Many times people cry, beg, and scream for an Upper Room outpouring. They desperately want Holy Spirit to pour forth fresh rivers and new wine, but they do not want to change the systems or structure of how they do things. Revival typically leads to a reform\u2014a new order and new methods. Jesus taught about this very thing. He said that there could not be new wine without a new wineskin.\n\nAnd no one pours new wine into old wineskins, or else the new wine bursts the wineskins, and the wine is spilled, and the wineskins will be marred. But new wine must be poured into new wineskins.\n\n\u2014MARK 2:22\n\nNew wine cannot be contained in an old wineskin. The old wineskin represents a fixed way of doing things. The reality is that God is continually opening our eyes to discover where He is moving and how we must move with Him. He shifts the wineskin before sending forth the burst of new wine. If He just sent the new wine, it would actually bring destruction to what has been built. There must be revelation of how to function. God raises up forerunners to tune in to His instruction and help His people embrace revelation to empower them to receive new wine.\n\nWe can see this pattern in Scripture when John the Baptist came on the scene preaching the message of the kingdom. He was preparing the people for a seismic shift, as Jesus would soon come announcing that the kingdom of heaven was at hand. The kingdom was there, standing right in front of them. The wineskin was shifting, but many would not shift. Then in the Upper Room the outpouring came and birthed the age of the ekklesia. There was a new wineskin. It would take apostolic fathers and mothers to articulate and teach the systems and structures that would appropriately steward the new wine.\n\nThis pattern continues today. Reform is birthed as God touches men and women to see in the Spirit, revealing His methods. There is reform happening with the purpose of abandoning the pathway of religion and embracing the plans of awakening. The unfolding of a new wineskin empowers God's people to receive a sustained release of new wine.\n\nThe religious demonic structures work overtime against the ekklesia to attack and persecute agents of change. They cite traditions and practices of man to overwhelm scriptural truth and prophetic leading. They fight like a religious mafia, debating, belittling, and tearing down to maintain business as usual. Many who are giving voice to it are deceived, just as Paul was before his encounter with Jesus on the road to Damascus. They believe that they are actually protecting the kingdom of God when, in fact, they are manifesting a religious spirit.\n\nThe late C. Peter Wagner masterfully articulates these truths in his book _Freedom From the Religious Spirit_ :\n\nDaniel 2:21 tells us, \"[God] changes the times and the seasons.\" Part of the character of God, then, is to continually produce new wine and to provide new wineskins for it. But Satan, of course, does not like that a bit, and later in the same book of Daniel, Satan reveals his purposes through the \"fourth beast\" who, according to the text, \"shall persecute the saints of the Most High, and shall intend to change times and the law\" (Dan. 7:25). What God designs for His glory and the advance of His kingdom, Satan constantly attacks in order to turn it back around.\n\nWhat device does Satan use to attempt to roll back God's new times and seasons? Look at that phrase \"persecute the saints.\" The Aramaic word for \"persecute\" is _belah,_ which means to \"wear out\" the mind. In other words, the corporate spirit of religion, as Satan's agent in this case, does not play so much on the heart or on the emotions or on personal holiness or on the fruit of the Spirit, but rather on the _mind_.\n\nTEACHING BRINGS LIBERTY\n\nThe work of the religious spirit on this level must be counteracted by teaching and training. Many moves of God are aborted too quickly by the attacks of a religious spirit. In the void of proper teaching, people succumb to the mental attacks of this demonic power.\n\nMy brothers, not many of you should become teachers, knowing that we shall receive the greater judgment. We all err in many ways. But if any man does not err in word, he is a perfect man and able also to control the whole body. See how we put bits in the mouths of horses that they may obey us, and we control their whole bodies. And observe ships. Though they are so great and are driven by fierce winds, yet they are directed with a very small rudder wherever the captain pleases. Even so, the tongue is a little part of the body and boasts great things. See how great a forest a little fire kindles. The tongue is a fire, a world of evil. The tongue is among the parts of the body, defiling the whole body, and setting the course of nature on fire, and it is set on fire by hell.\n\n\u2014JAMES 3:1\u20136\n\nThose in the office of the teacher act as the rudder of a ship, turning the entire body. We will live and manifest what we believe. The way we change what we believe is through study\u2014learning and renewing our minds. The teacher plays a key role in renewing the corporate mind of the body of Christ. High-level operations of the religious spirit will attack and attempt to discredit any teaching that holds the power to set people free. This demon knows that if people begin to think and believe differently, then they will move into transformation.\n\nTeaching is essential to the formation, maintenance, and healthy establishment of new wineskins and new wine. Apostles receive heavenly communication and instruction to implement and build new-wineskin ministries for revelation in the body of Christ. Prophets release the revelation and decree of critical components of the new wineskins, but teachers fortify the wineskins with teaching, training, and building the walls, brick by brick, nugget by nugget.\n\nIn today's media-driven society, one of the religious spirit's favorite outlets is social media and online platforms. It will release lies against emerging voices and powerful kingdom leaders. It will attack their character, criticize their words, and attempt to tear down the revelation. It sees the threat and acts swiftly to stop it. Religious spirits attack any messenger who challenges their systems of bondage. People with religious demons will attack when they feel threatened by the revelation or ministry of another. Insecurity is always a breeding ground of demonic influence.\n\nWhile most of these subjects relate to the attacks of the religious spirits on Christians individually and corporately, there is yet another dimension of the religious spirit. It binds the minds of the unsaved and keeps their hearts trapped in deception by partnering with a power called \"doctrines of devils,\" actual demonic spirits that carry the message of false teaching to humanity to trap them outside of salvation. All people who dedicate their lives to the rules and regulations of any spiritual practice not built upon the cornerstone of Jesus and His salvation are being influenced by the religious spirit. The religious spirit works outside of the Christian church to build strong fortresses of false religion, creating men and women who are deceived and bound for hell but think they are on the road to heaven. It infiltrates politics and media and creates statements like, \"All roads lead to heaven.\" These words are evil pronouncements of deception fostered by the powers of hell to convince mankind that they can be saved without the Savior and redeemed without the Redeemer.\n\nFalse religions are empowered and established by demonic rulers. They create belief systems to keep people outside of heaven. They penetrate the thinking and believing of people to bind them to false ideas. Only the power of almighty God can break this type of religious deception. That is why it took a bright light and a Damascus road experience to reach Paul. He had to be convinced by the supernatural power of a living God. A living God speaks and moves!\n\nPOINTS TO CONSIDER\n\n\u00bb Do I get jealous of others whom God is promoting?\n\n\u00bb Am I spiritually dry? Does there seem to be an invisible lid on my spiritual life?\n\n\u00bb Am I overly critical and too quickly distracted by meaningless debates?\n\n\u00bb Am I stuck in dead traditions that do not lead me to the life of Jesus?\n\n\u00bb Do I allow fear of man to hinder me and hold me back from moving in my calling?\n\n\u00bb Am I easily offended?\n\n\u00bb Do I really have the joy of the Lord in my life, or do I just do what is required of me to please others?\n\nWHERE TO BEGIN\n\n\u00bb Do a heart check every day.\n\n\u00bb Learn to do everything for God alone, not to be noticed or praised by man.\n\n\u00bb Ask God to reveal areas of bondage in your life and to show you how to escape.\n\n\u00bb Encounter Holy Spirit on a continual basis to keep your oil flowing and your purpose pure.\n\n\u00bb Be purposeful about recognizing patterns of religious activity with no spirit life.\n\n\u00bb Release the oil of Holy Spirit in prayer and worship.\n\n\u00bb Meditate on the goodness, generosity, and grace of God. Allowing His new nature to permeate your thinking will eradicate the power of the religious spirit in your life.\n\nPRAYER\n\n_Father, keep me pure in all my motives. Show me if I ever begin to allow religion or bondage to enter my life. I want to serve You and You alone. I want only to please You, to do all I do for You. I am not a man pleaser but a God pleaser! You are the Author and Finisher of my faith. I break free from religious tradition and bondage in Jesus's name. I rip every demonic lid off my spiritual life, and I release Your abundant grace over me. I know that You are good, and I rest in Your goodness. I receive Your goodness. I confess Your goodness. I say that I am filled to overflowing with Your joy. Thank You, Lord, that I walk close to You all the days of my life, I hear Your voice, and You alone do I follow._\nChapter 9\n\nFIRST-LOVE LIFESTYLE: FREEDOM FROM THE RELIGIOUS SPIRIT\n\nIT IS IMPORTANT when examining properly the many-faceted dimensions of the religious spirit to begin with the understanding that there is a company of demons that make up this family of religious spirits. There are the religious demons that sow seeds of the world's false religions, as we established in the previous chapter. These devils work to abort the magnificent revelation of the man named Jesus Christ. In some regions of the world these demonic powers are so strong that the only way Jesus can break through is by visiting people in person through a vision, a dream, or even an encounter. The stronghold of false religion chokes out the preaching of the gospel, even threatening believers with acts of violence. This is one class of demon operating at a high level to restrict and rule a territory.\n\nDON'T BE DECEIVED\n\nThen there are the types of demonic beings that are most often associated with the term _religious spirits._ These spirits sow seeds of bondage into the lives of Christian believers. There are levels of assignment in that class of demon as well. There are the foot soldiers, or little demons, that influence the minds, thoughts, emotions, and belief systems of people. I remember a number of years ago dealing with one such devil. I was sharing with someone about the powerful ministry of Holy Spirit and speaking in other tongues. Without hesitation the person with whom I was sharing said that speaking in tongues was of the devil.\n\nI asked why this person thought like that, and he rattled off what his church had taught and gave me a few opinions. I let him finish, and then I got out my Bible. I began to expound on the outpouring in the book of Acts. Then I went to commonly misunderstood scriptures in 1 Corinthians 14 and explained the diversities of tongues and the laws that govern each operation. I shared my personal testimony about all the miracles and breakthroughs I have had from praying in the Spirit. When I finished, the person was unwavering. \"What about the Bible?\" I asked. He had no answer. It was then I realized that his mind was twisted in knots and blocked to the truth by a religious spirit. This was the work of a demon that blocked the transforming knowledge of the Word of God.\n\nEvery one of us has had personal attacks from the religious spirit in our lives. We have been spoon-fed lies to keep us from kingdom living and victorious believing. I remember a time when I was in a revival meeting and Holy Ghost laughter broke out. The joy of the Lord hit the building as God began to minister to His people. I had never been in a meeting like that. It scared me, and I thought it must be of the devil. As I was trying to figure it out, the Lord rebuked me and told me it was Him moving! I quickly repented and jumped in the flow. That night my life was forever changed. As a matter of fact, nearly every breakthrough I have had in my journey with God has required me to face religious thinking and believing in order to go to the next level. I have had to determine that my heart is after God and I will be in the midst of His times, seasons, and activities in the earth. If I must change my mind-set, then I will! I will not get stuck.\n\nThere is an even greater measure of the religious spirit that acts to enslave groups of people in false belief systems and lifeless works. This type of spirit is a high-level demon most likely on the level of a principality, much like the queen of heaven (Jer. 7:18). This is a far greater plot than afflicting an individual with lust, rage, or despair. This is a territorial demon that infiltrates churches, ministries, groups, organizations, cities, and regions to establish firm strongholds of religious bondage. This demon has a collective assignment to ensnare and entangle a large group of people, thereby preventing the light of the gospel from spreading.\n\nThis ruling demon does not manifest boldly but hides beneath religious language and subtle attitudes. It seeks to establish a lid on a territory and people. It places a lid on the realm of power. Jesus preached the kingdom with power! The disciples preached the kingdom with power! The early church preached the kingdom with power! This high-ranking demon does all it can to dismantle kingdom thinking and structures as well as the power that they generate.\n\nThis demon deceives people into thinking they are doing God's will. It creates guardians that will fight with great intensity to maintain the religious status quo. One of the answers for this type of ruler is a true apostle who comes into a territory with a breaker anointing. (See Micah 2.) Apostles then break open the tough ground, preach with power, build a strong kingdom-based ministry, and establish spiritual sons and daughters. This demon, of course, will fight that type of ministry tooth and nail. Many regions with this type of religious spirit in operation have no strong apostolic or prophetic ministry. The religious churches and groups have run them all out of town.\n\nWHERE THE SPIRIT OF THE LORD IS, THERE IS LIBERTY\n\nThere is no bigger aim of the religious spirit than to create enduring bondage. The aim of this demonic spirit is to undo the radical freedom that was obtained through Jesus's sacrifice at the cross.\n\nNow the Lord is the Spirit. And where the Spirit of the Lord is, there is liberty.\n\n\u20142 CORINTHIANS 3:17\n\nThe presence of the Lord breaks chains, tears down walls, and removes shackles. Religion hates spiritual freedom. It works overtime to reinforce evil and restrictive barriers.\n\nThe religious spirit creates a lack of freedom in the following:\n\n\u2022 **Receiving God's love** \u2014When you are bound, it is so difficult to know and believe the undying love of God.\n\n\u2022 **Receiving God's goodness** \u2014The nature of the Father toward His sons and daughters is that of goodness. Religious bondage empowers a theology that severely limits receptivity of the goodness of God. Believing, accepting, and receiving the goodness of God bring tremendous liberty.\n\nFor freedom Christ freed us. Stand fast therefore and do not be entangled again with the yoke of bondage.\n\n\u2014GALATIANS 5:1\n\n\u2022 **Believing God's promises and who He says we are** \u2014Bondage attempts to hold you away from the promises of God. We access breakthrough by accepting and acting on His promises. If you can find a promise from God, you can find a breakthrough.\n\nReligious bondage keeps people stuck in slavery when Jesus provided adoption into the family of God. A bound heart is a wounded heart! Breaking the grip of bondage releases tremendous healing and freedom of identity.\n\n\u2022 **Prayer** \u2014The religious spirit creates powerless patterns of prayer that resemble routine more than relationship. Prayer is intended to be an intimate method of communication. Freedom from religious bondage releases a heart to enter a deeper and more glorious communication with God. Breaking the bondage will release new dimensions of intimacy. One of the highest points of prayer is the release of glory. When we come into the presence of God and His glory shows up, He is there in person. When God steps in, anything is possible.\n\n\u2022 **Relationship with others** \u2014The religious spirit issues a constant stream of criticism and judgment, eradicating the grace of God. Bound people are continually judging others without any type of grace. True grace empowers love. When we walk in love, we are not easily offended. We are quick to forgive and full of mercy.\n\n\u2022 **Walking in the power of God** \u2014Religious bondage annihilates the flow of the power of God. It is impossible to believe God for miracles and breakthroughs when you are stuck in religious bondage. This demon is an anti-anointing demon. Reading and studying the miracles of Jesus build faith and create a paradigm of the power of God.\n\nHow can I personally overcome the prison of the religious spirit and the patterns of bondage? As with any deliverance, it begins with recognizing the patterns we have listed. These are the symptoms of a prison that the religious spirit has placed you in illegally. You must realize that this spirit infiltrates your thinking. In order to break free, you must be willing to think differently and look at the Word of God through a fresh lens.\n\nDemon spirits travel through breaches. The dictionary defines _breach_ as \"a broken, ruptured, or torn condition or area.\" It is essentially an opening. Deliverance requires expelling a spirit, breaking its grip by using the authority of the Word, the name, and the blood of Jesus to command it to go, and slamming every door shut to close the breach.\n\nI remember a lady who struggled with condemnation and fear. She was invited to attend a meeting with a preacher who operated in a heavy religious spirit and released a very condemning message. One of my ministry team members advised her not to go and sit under that teaching. Why? Because it would only reinforce a mental and emotional opening that allowed demon spirits to travel back and forth. These openings must be closed by prayer and the Word of God. Every lie has to be addressed by the truth!\n\nOne of the great examples of this is the confrontation between Jesus and the devil in the wilderness. As Satan raged on with temptation after temptation, Jesus kept meeting his lies with the Word of God. Over and over again He stated, \"It is written.\" He did not offer up His opinion or suggestion. He stood firmly upon the written Word of God. This is a victory key! Meet the lies with the Word. As you are breaking free from a religious spirit, you will have to refresh and renew your mind with the Word of God. You will have to speak God's promises of sonship and adoption over and over again.\n\nBreaking free from the religious spirit will require purposeful pursuit of sonship. You must dive deep into the Scriptures to discover your identity as a son or daughter of God. You must build your faith by listening to teaching that reinforces your freedom. This will lead you to a place of unfettered worship that only sons and daughters enjoy. As you come into His presence from a place of freedom, you will soar above the bondage of the past.\n\nSTAY THE COURSE\n\nOne way you can judge the strength of a revelation or a prophetic concept is the measure of resistance that it provokes. It is vital that you view the resistance as a confirmation and press on. The religious spirit blocks kingdom breakthrough by pressuring apostolic and prophetic ministry gifts to bow to dead traditions and humanistic opinions. The reality is that those very traditions act as tools to restrict and contain the move of God. We need daring apostolic leaders who refuse to bow and keep blazing the trail!\n\nWhen they had traveled through Amphipolis and Apollonia, they came to Thessalonica, where there was a synagogue of the Jews. According to his custom, Paul went in, and on three Sabbaths he lectured to them from the Scriptures, explaining and proving that the Christ had to suffer and to rise from the dead, and saying, \"This Jesus, whom I preach to you, is the Christ.\" Some of them were persuaded and joined with Paul and Silas, including a great crowd of devout Greeks and many leading women.\n\nBut the Jews who did not believe became jealous and, taking some evil men from the marketplace, gathered a crowd, stirred up the city, and attacked the house of Jason, trying to bring them out to the mob. But when they did not find them, they dragged Jason and some brothers to the city officials, crying out, \"These men who have turned the world upside down have come here also.\"\n\n\u2014ACTS 17:1\u20136\n\nPaul and his team were directly challenging the ruling religious spirit, its false mind-set, its teaching, and the demonic grip it had on the region. What was the weapon of choice? Teaching! Paul expounded the truths of the Word of God in order to set the captives free. Breaking a ruling religious power will take teaching. There must be a release of heavenly foundations, a ripping apart of false foundations, and a building of new foundations. Apostles and prophets can work together to accomplish this goal in short order.\n\nThe region was stirred. This is the apostolic anointing in action. It breaks hard ground open, establishes the kingdom, and directly engages demonic rulers. Anytime there is an extreme reaction, it is evidence of spiritual warfare. The region rose and responded to the ministry of Paul. His ministry was not affecting just the ground but the airspace above where demons traffic.\n\nThe prophetic anointing carries such a grace to labor alongside apostolic ministry. Prophets carry an anointing to tear down the false! They prophesy against deception and reveal the testimony of Jesus.\n\nThen the LORD put forth His hand and touched my mouth. And the LORD said to me, \"Now, I have put My words in your mouth. See, I have this day set you over the nations and over the kingdoms, to root out and to pull down, to destroy and to throw down, to build and to plant.\"\n\n\u2014JEREMIAH 1:9\u201310\n\nThe Lord placed His word in Jeremiah's mouth to expose, confront, and tear down demonic structures, lies, and concepts. Prophets are spiritual warriors who do not back down from the threats of the enemy. They have seen the hidden demon crouching in the corner. They see the lie and refuse to keep silent. They become powerful weapons in the arsenal of the Lord. One deception that often grips prophets and prophetic people is that they become overly warfare oriented without revealing Jesus. It is critical that prophets understand the full weight of the call and their responsibility to love and serve the body as well as tear down. If there is no building, it becomes highly problematic.\n\nO foolish Galatians! Who has bewitched you that you should not obey the truth? Before your eyes Jesus Christ was clearly portrayed among you as crucified.\n\n_\u2014_ GALATIANS 3:1\n\nReligious powers released a demonic enchantment over the Galatian church. Paul was speaking to them in a corrective fashion as their apostolic father. He could have used any language, but he chose to use the word _bewitched_. The purpose of this spell was to remove the believers from the liberty that had been established with the truth of the gospel and to replace truth with error, therefore shutting down the power and advancement of the kingdom.\n\nHow would this spell be broken? It needed to be addressed by the leaders and broken by the blood of the Lamb. Paul was exposing it in order to get them free. Exposure is warfare 101. You cannot conquer what you do not identify. Paul clearly marked the deception in order to bring the Galatians out of the snare of religious powers. This type of religious power moves to ensnare an entire group. It takes leadership within the group to renounce the authority of the enemy and break its power in the name of Jesus.\n\nMany regions, organizations, and ministries need bold, strong leaders who will unashamedly rise up and break the powers of the enemy. God always appoints leaders to usher groups into the promise. There is an anointing that flows as pictured in Psalm 133. The precious oil of Holy Spirit flowed down upon the beard of Aaron the priest and onto the people. That type of anointing breaks yokes, restores vision, and empowers people. Every spiritual relationship should unlock a flow.\n\nTHE CALL OF FIRST LOVE\n\nBut I have something against you, that you have abandoned the love you had at first. Remember therefore from where you have fallen. Repent, and do the works you did at first, or else I will come to you quickly and remove your candlestick from its place, unless you repent.\n\n\u2014REVELATION 2:4\u20135\n\nThe church at Ephesus had forgotten its primary calling. It had abandoned the pathway of first love. There is a love that is to be above every other. There is a lover who paid the highest price for us. Our truest and highest calling is to love Jesus and be loved by Him\u2014this is first love.\n\nThe entire mission of Christianity could be summed up simply like this: to be loved, to love God, and to love others. We are called to be living demonstrations of radical love, pursuit, and encounter. One of the deepest and most painful operations of the religious spirit is to trap us in a life of continual striving without truly experiencing the wonderful love and mercy of God.\n\nBut experiencing the wonderful love and mercy of God is our call! This is our passion\u2014to live lives of first-love pursuit and encounter. Jesus did not come and willingly offer His life to make us good slaves. He came to seek and save that which was lost. The Father sent Him on a divine rescue mission to restore us back to the family of God.\n\nFor the Son of Man has come to seek and to save that which was lost.\n\n\u2014LUKE 19:10\n\nThe mission of Jesus was to reposition us outside of the curse that resulted from the fall of Adam and to plant us in the family of God. He has delivered us from the power of darkness and has transferred us into the kingdom of His dear Son, in whom we have redemption through His blood, the forgiveness of sins.\n\n\u2014COLOSSIANS 1:13\u201314\n\nWe have been radically delivered from the domain of darkness. We have been translated into the kingdom of God as sons and daughters. We are to live above the ruling powers of hell. We are not to be bound by the schemes, plots, and lies of the enemy.\n\nThe religious spirit works to undo the reality of our new nature. When we were born again, we were essentially recreated. We have a new nature. We have a spirit that is made in the image and likeness of our Father. We are not limited. We are not under the realm of darkness or any of its regulations. We can live in the spirit realm. We can walk out the plans of God. We can live from the dimension of the glory of God.\n\nDo not lie one to another, since you have put off the old nature with its deeds, and have embraced the new nature, which is renewed in knowledge after the image of Him who created it.\n\n\u2014COLOSSIANS 3:9\u201310\n\nThe religious spirit wants to set a trap for us as it did for the Galatian church. It wants to bind us in defeat and strap us down with bondage. We must make a concerted effort to tap into our new nature by walking in the Spirit. There is a pathway that leads directly out of bondage and straight into the wonders of the kingdom. This is the first-love lifestyle! Pursuit becomes the norm, and encounter is a regular occurrence.\n\nTo walk in the new nature, we begin with the understanding that we were saved not for works alone but for relationship. God's highest desire for our lives is relationship with Him. In order to understand the practical dynamics of what this looks like, we must first lay a foundation of what a relationship is. It begins with a bond and is strengthened by investment.\n\nAll relationships have some type of bond. Family relationships have a bond of blood. Romantic relationships have a bond of attraction and affection. Business relationships have a bond of benefit and shared goals. Our relationship with the Father is based on a bond of unwavering love and commitment.\n\nWhile we were yet weak, in due time Christ died for the ungodly. Rarely for a righteous man will one die. Yet perhaps for a good man some would even dare to die. But God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us. How much more then, being now justified by His blood, shall we be saved from wrath through Him. For if while we were enemies, we were reconciled to God by the death of His Son, how much more, being reconciled, shall we be saved by His life. Furthermore, we also rejoice in God through our Lord Jesus Christ, through whom we have now received reconciliation.\n\n\u2014ROMANS 5:6\u201311\n\nGod is so good that He pursued us even when we had no interest in serving Him. Time and again humanity failed God, but He pursued us anyway. This is the foundation of our relationship with the Lord. He loves us with a deep covenant love. He pursued us. He ransomed us. He forgave us, and He is presently working on our behalf. He is our good Father! This is the starting point of our relationship with Him\u2014understanding His deep passion toward us.\n\nHe reconciled us. He brought us back into a position of sonship that Adam had forfeited. He paid it all for us. He provided His Son as a ransom for us. We begin our pursuit of God from a position of unfathomable love. We begin with the call of first love.\n\nTHE PURSUIT OF GOD\n\nEvery relationship moves beyond a bond into pursuit. Any relationship that loses pursuit also loses encounter. These two are vital components of relationship. Pursuit demands that we take practical steps motivated by love. In our relationship with the Father we must develop a lifestyle of continual pursuit. What does this look like? Let me lay out some very simple and practical steps to living free from the grasp of a religious spirit and remaining planted in a first-love lifestyle.\n\n\u2022 **Worship** \u2014This is one of the distinguishing marks of a son or daughter of God. We are born to be worshippers. Worship is the act of releasing our intimate affection to the Lord. It is the expression of our adoration. Worship is the engine of first love. The Greek word for worship literally means \"to blow kisses toward.\" Worship is the language of heavenly love! We are releasing our affection and adoration toward Jesus, and as we sit in His presence, we are receiving His love back to us.\n\nWorship is the constant act of surrender. It is not just a few fleeting moments; it is a lifestyle of surrender. David is one of the Bible's great heroes. His source of power was his radical love for God. Time and again he was the underdog. He was the reject and the misfit, but he found the presence of God and was never the same. Worship, radical adoration, will crown the misfit and the underdog, taking them from pit to palace!\n\nWe must develop a worship lifestyle. We lift our hands. We bow low before Him. We pour out our hearts. We worship with music. We sit in the quiet places, telling Him of our love. We surrender fully and deeply to Him. We feel His presence like waves rushing over us, encountering us, shifting us, refreshing us. This is worship! It is authentic, organic, and original. Make time for it, make space for it, and allow it to be a central focus of your life.\n\nBless the LORD, O my soul, and all that is within me, bless His holy name.\n\n\u2014PSALM 103:1\n\n\u2022 **Praise.** What is the distinction between praise and worship? In many ways they flow seamlessly together, but there are some key differences. To praise is to lift up. Praise is lifting up the name of Jesus, the power of God, and the promises of God above every other thing. When we praise, we are exuberantly lifting up the Lord. Praise holds the power to crush the weapons of the enemy. Praise will break heaviness.\n\nIsaiah 61:3 tells us to put on \"the garment of praise for the spirit of heaviness.\" Praise is an eruptive and exuberant expression of thanksgiving. When we praise God, we often begin from a place of remembering where and how He moved in our lives. This type of expression will create spiritual movement in your life. Every believer must be a praiser!\n\nPraise the LORD! Praise God in His sanctuary; praise Him in the firmament of His power! Praise Him for His mighty acts; praise Him according to His excellent greatness! Praise Him with the sound of the trumpet; praise Him with the lyre and harp! Praise Him with the tambourine and dancing; praise Him with stringed instruments and flute! Praise Him with loud cymbals; praise Him with the clanging cymbals! Let everything that has breath praise the LORD. Praise the LORD!\n\n\u2014PSALM 150:1\u20136\n\n\u2022 **Reading, studying, and meditating on the Word of God** \u2014The Word of God is a manual straight from the heart of God. Romans teaches us that as we renew our minds with the truth of God's Word, we position ourselves to fulfill the will of God (12:2). Simply put, this means we must learn to think differently. That is a process, not a one-time thing. All believers must become students of the Word of God. They must read it regularly, study it, and dig into it. They must also meditate on it, ponder it, and pray it out.\n\nIsaiah 55:11 says, \"So shall My word be that goes forth from My mouth; it shall not return to Me void, but it shall accomplish that which I please, and it shall prosper in the thing for which I sent it.\" The Word of God is our promise book. As we dive deep into the pages of the written Word, we are empowered to prosper and live from a place of victory. We can stand upon the promises of God and declare them over our lives. The Word will create strength, healing, and victory if we partner with it. We can receive it and decree it with great confidence. A journey into God's Word is an exploration of life-changing power. The Word of God is the voice of God in our lives. It is crafted to shape us and to plant us deep in the ways of our Father.\n\nFor the word of God is alive, and active, and sharper than any two-edged sword, piercing even to the division of soul and spirit, of joints and marrow, and able to judge the thoughts and intents of the heart.\n\n\u2014HEBREWS 4:12\n\n\u2022 **Prayer** \u2014Daily conversation, reflection, and communication with God is imperative to a vibrant first-love lifestyle. Volumes have been written about the various facets of prayer. There are so many pathways upon which we can travel into the heart of God. The primary thing is to make encountering and seeking God a priority. We need quiet times of prayer. We need times of governing prayer, of binding and loosing. We need times to lean in, be still, and listen. We need times of praying in the Spirit, building our faith, and unraveling the mysteries. Prayer was one of Jesus's most powerful weapons. When they were all looking for Him, He was alone, outside of the city pursuing the Father. This is our model. James 5:16 tells us, \"The effective, fervent prayer of a righteous man accomplishes much.\" A life framed by prayer is one of powerful destiny fulfilled.\n\nIn the morning, rising up a great while before sunrise, He went out and departed to a solitary place. And there He prayed.\n\n\u2014MARK 1:35\n\nSEEK HIM\n\nA first-love lifestyle, free from the grip of religious deception, is one of pursuit. There are practical steps to build a life of ongoing relationship with God. When you pursue, you will find Him. His power will show up. The men and women who do extreme exploits in a generation are those who have found His presence, His power, and His majesty. A first-love lifestyle creates a reservoir of power. Faith produces earth-shaking kingdom works.\n\nSo faith by itself, if it has no works, is dead.\n\n\u2014JAMES 2:17\n\nThis is the result of pursuit: the kingdom begins to manifest in your life, wonders are unlocked, and the supernatural comes bursting forth. This is what the religious spirit works overtime to stop. It does not want Christians operating in the power of the gospel with boldness, strength, and authority. It is impossible to hang out with Jesus and not manifest His power.\n\nAs you plunge headlong into the love of God, a deep well of His affection and nature will rise in your life. Love produces grace toward others. As God's love for you and toward you is unraveled, it becomes much easier to have grace and mercy for others when they fall short. The religious spirit has no grace with people. It uses rules and regulations to place shackles of bondage on people, but the love of God releases the grace and mercy of God.\n\nBe of the same mind toward one another. Do not be haughty, but associate with the lowly. Do not pretend to be wiser than you are.\n\n\u2014ROMANS 12:16\n\nLove produces personal freedom. As you discover how much God loves you, it sets you free from the lies, torment, accusation, and weapons of the enemy. The love of God acts as a powerful tool to enforce the revelation of freedom in Christ. The love of God empowers your identity as a son or daughter of God. As you live a first-love lifestyle, you discover who you are! This breaks the chains of bondage and releases your heart to dream and believe big. The love of God creates a God-confidence in your life. You become convinced that God is ready, willing, and able to do all that He has promised. The impossible now seems to be within your grasp through the power of the name, the Word, and the blood of Jesus.\n\nA life of first love plants you in a supernatural lifestyle. You begin to believe and expect the miracles of God. Your heart soars to believe for the miraculous. You step out of mundane, ordinary living and into the realm of the miraculous and extraordinary.\n\nThe goal of the religious spirit is to trap a believer into lifeless cycles in which he or she is unable to enjoy the privilege of intimacy with the Father. The work of Jesus on Calvary was total and absolute atonement. He paid the price for every human being who will receive Him. There is unfathomable freedom in the power of His blood.\n\nFreedom is not a single moment but an ongoing journey. We move from faith to faith, from glory to glory. By the power of Holy Spirit, blinders are removed, and truth is made manifest. We have been positioned in Jesus in heavenly places and are called to rule and reign.\n\nWhen we receive Jesus, we are made new. We receive a new nature that empowers us to live a life of the Spirit of God. Jesus's sacrifice gave us full access to the kingdom, to rule and reign with Him as sons and daughters of God. We can lay down the mentality of servanthood without sonship. We are empowered to rule and reign with Christ. We conquer and defeat the work of the religious spirit in our personal lives as we pursue the Lord, renew our minds, and take our places as sons and daughters, recognizing that we were born to live wildly and gleefully free!\n\nPOINTS TO CONSIDER\n\n\u00bb Are you falling more in love with Jesus every day?\n\n\u00bb Do you spend time with Him daily?\n\n\u00bb Do you have a revelation of all that Jesus did for you?\n\n\u00bb Do you honor Jesus by walking in the reality of all that He has secured for you?\n\nWHERE TO BEGIN\n\n\u00bb Spend time with Jesus, just talking with Him and listening. (Always have something to record what He says to you.)\n\n\u00bb Meditate on the goodness of God, on the nature of God, and on all the promises for you.\n\n\u00bb Read the Word of God for deeper revelation on the kingdom.\n\n\u00bb Be vigilant about counteracting the lies of the religious spirit with the truth of the Word of God. Don't let the religious spirit put you and keep you in bondage.\n\nPRAYER\n\n_Lord, give me a deeper revelation of Jesus and the price He paid for me. Show me how to walk in freedom in every area of my life. Give me understanding of reigning on this earth and of my identity as a son or daughter of God. I decree that I am free from sickness and disease, I have the mind of Christ, I am prosperous spirit, soul, and body, and I walk in all that the blood has purchased for me. In Jesus's name, amen._\nChapter 10\n\nENFORCING THE KINGDOM: LIVING FREE\n\nTHE MINISTRY OF Jesus brings deliverance. As He went forth in the earth, He broke the power of the devil.\n\nWhen Jesus entered Peter's house, He saw his wife's mother, lying sick with a fever. He touched her hand, and the fever left her. And she rose and served them. When the evening came, they brought to Him many who were possessed with demons. And He cast out the spirits with His word, and healed all who were sick.\n\n_\u2014_ MATTHEW 8:14\u201316\n\nEverywhere Jesus went, people were delivered. The church has the same bold call to freedom in this hour. Yet as we look at the church today, most people either do not believe in casting out devils or have never been taught how to do it, and many think it is mere foolishness. Yet while they are asleep, the enemy advances.\n\nWe can no longer have a Pollyanna attitude and simply pretend the enemy is not real. But we also do not want to overemphasize his power. Though it is important for us to understand his schemes and know how to thwart them, we must remember that we are victorious and triumphant children of the Most High.\n\nSomeone once asked me, \"Do you look for a devil behind every bush?\" My answer: \"No, but I know what to do when he shows up!\" We can live free in our minds, free in our bodies, free in our spiritual lives, and free in every area! The total price has been paid.\n\nNO LONGER CAPTIVE\n\nIf ever there has been a time for churches, ministries, and believers to be full of anointing, strength, and power, it is now! The world needs another great revival, but it is going to take a church on fire for that to happen. We must break the grip of these foul anti-anointing spirits and ride the mighty winds of God!\n\nFor our fight is not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, and against spiritual forces of evil in the heavenly places.\n\n\u2014EPHESIANS 6:12\n\nSpiritual warfare is a part of our existence as believers. There are battles, attacks, and traps. We are not fighting with human beings but being resisted by unseen powers. Paul is teaching in these verses that there will be conflict. There will be opposition. Developing a winning strategy for war is critical. We cannot and must not leave our destinies to change. We cannot afford to shun the responsibility that we have as believers to engage the enemy and enforce the authority of the kingdom.\n\nThere are people who question the need for teaching on demon powers, bondages, and deception. If you were the devil and you really wanted to infiltrate culture, take over society, and bind up the church, what would be your master plan of deception? Convincing people that you are either not real or not doing anything would be a pretty good start. This is exactly what he does. He works overtime to strip revelation away from believers so that they will not pray in the Holy Spirit and move in the power of God. Their impotence allows him to operate subtly and effectively behind the scenes.\n\nEffective warfare strategy begins with spiritual education. This is enforced time after time in the Bible. Paul admonished Timothy to be a good student.\n\nStudy to show yourself approved by God, a workman who need not be ashamed, rightly dividing the word of truth.\n\n\u20142 TIMOTHY 2:15\n\nLearning the tactics of the enemy, the nature of demon powers, and the authority of the believer equips you for masterful and victorious warfare. Educating yourself prepares you for the task at hand.\n\nYou must also realize that demons prefer the cover of darkness. The devil functions best when he is unseen and hidden. He releases wicked schemes under the radar. Battle strategy number one is exposure! You can never overcome what you fail to identify. Discerning what type of demonic power you are facing helps identify its tactics, operations, and schemes. This also helps you know how to combat it.\n\nTypically there are four points to break through a demonic attack. First, you discern the attack and identify the demon. Second, you enter into prayer to get wisdom and instruction. Third, you rise in authority to break the attack. You exercise the dominion that Jesus has given to you as a child of God. You boldly tell the devil no! You break his power and assignment over your life. You serve a spiritual eviction notice. This is what deliverance really is! It is evicting the enemy and sweeping the house clean.\n\nFinally you renew your mind and develop your faith to stay free. It is one matter to get free, but you must realize that the devil is persistent. If a demon has been operating in and around you for years and then you evict it, it will look for a way to come back. Jesus taught us this principle. If people are not willing to do some follow-up work, then it is better not even to minister deliverance to them because they can end up worse off.\n\nWhen an unclean spirit goes out of a man, it passes through dry places seeking rest, but finds none. Then it says, \"I will return to my house from which I came.\" And when it comes, it finds it empty, swept, and put in order. Then it goes and brings with itself seven other spirits more evil than itself, and they enter and dwell there. And the last state of that man is worse than the first. So shall it be also with this evil generation.\n\n_\u2014_ MATTHEW 12:43\u201345\n\nWe must recognize that deliverance is often a process and not a singular event. If people refuse to build their faith, study the Word, renew their minds, and fight the fight of faith, they can receive deliverance ministry and end up worse.\n\nGO TO THE POWER SOURCE\n\nI remember, a number of years ago as I was leading a church in a very religious territory, a severely bound man came to me for help. He told me that he had been molested as a child and had continual struggles. You could look at him and see the presence of an unclean spirit. It had attached itself to every part of his life. He was in desperate need of spiritual breakthrough and strong deliverance ministry.\n\nI asked him where he went to church, and he told me. In that region there were few places that believed in and practiced deliverance. He was attending a nice church that had nice one-hour meetings with no power or authority. You may be happy to attend a nice church; you may love the music, the short services, and all the nice, predictable messages\u2014until your family gets in trouble. Then you want a radical church that heals the sick, casts out demons, and believes in miracles.\n\nI asked the man if he had shared this issue with his pastor. He said he had and that his pastor had laughed and said that they did not believe in demons at that church. This is a problem. If you have demonic oppression and you are attending a nice church that refuses to deal with demons, you can easily sink deeper into spiritual trouble. This man was desperate and in major trouble, but he was in a place that refused to help him. The reality of demons pushed them beyond their comfort zone.\n\nYou must understand that deliverance is neither pretty nor convenient, but it is absolutely necessary. The church you go to could be the difference between life and death for you.\n\nIt is easy to be extremely zealous, see a need, and just start casting out spirits, but in reality, if the person involved is not willing to do some follow-up work, you are doing that person a disservice. I have learned this the hard way. Demonic oppression creates strongholds in the thought life that act as gateways into a person. Those thoughts are connected to desires. The thoughts must be dealt with, the desires must be resisted, and the flesh must be put under. (Fasting is a great deliverance tool.)\n\nThe enemy will use thoughts, emotions, feelings, and desires to try and creep back in. Many people just want it all dealt with at an altar with no effort on their part. This is lazy faith! James taught us that faith takes some elbow grease. There must be some effort exerted. Not only do we have to break the foothold of the enemy, but we also must resist his attempts to come back.\n\nTherefore submit yourselves to God. Resist the devil, and he will flee from you.\n\n\u2014JAMES 4:7\n\nAfter receiving deliverance, the journey continues. It is imperative that you resist\u2014stand against and firmly and repeatedly say no. It will take the prayers and agreement of others to keep the cord broken. Sometimes you must let go of some relationships and change your surroundings. You may have to stop watching certain programs and listening to certain music to slam the gateway shut and keep it that way.\n\nThe devil will lie to you and tell you that you are not free because you are having a tempting thought or desire. This is a heinous lie intended to overwhelm you and get you to give in to a spirit of defeat. You must recognize this lie and aggressively rise against it. You are not a captive to your mind, your body, or your appetites. You are in charge! You can use your authority to make these things line up with the Word of God.\n\nNo temptation has taken you except what is common to man. God is faithful, and He will not permit you to be tempted above what you can endure, but will with the temptation also make a way to escape, that you may be able to bear it.\n\n\u20141 CORINTHIANS 10:13\n\nThe word _temptation_ means \"a strong urge or desire.\" When a demon is removed, there can be root systems in the thought life. These thoughts attempt to give way to desires. These desires need to be bridled by the Word of God and your new nature. You must tap into the realm of the spirit and subdue the desire. The enemy provides temptation. Don't let the fact that you are being tempted cause you to question the freedom that belongs to you. The devil wants to use a fleeting thought or desire to convince you that you are bound. Do not buy his lies! Keep your confession strong and your mind rooted in the Word of God.\n\nThere is a powerful connection between what is in your heart and your mouth. When you believe something, you will think on it, and it will be planted deep inside of your thought life. You will then speak it out of your mouth. As you speak what you believe, you are releasing words of life or death in the realm of the spirit.\n\nDeath and life are in the power of the tongue, and those who love it will eat its fruit.\n\n\u2014PROVERBS 18:21\n\nOne vital key to living and staying free is to make speaking the Word of God part of your prayer life. Refuse to allow negative words of fear to dominate your mind and mouth. When the enemy screams at your mind that you are not free, boldly tell him that you are free and bind his lies up in Jesus's name! Dive deep into the Word of God, and find answers for your situation. Become a student of the freedom that you are seeking, and then pray, ponder, and meditate on those verses. Speak them out of your mouth. Call your mind free! Call your body free! Call your spiritual life free! Call your family free! Call your business, ministry, and financial life free! Prophesy the Word of God over your life by declaring what has already been written. Your mouth becomes a defense against the enemy.\n\nDecreeing the victory and breaking the power of the enemy are a crucial part of your personal freedom. Here are prayers and confessions for the various areas I have covered:\n\nBREAKING JEZEBEL ATTACKS\n\nIntimidation\n\n_Thank You, Lord, that You said in Proverbs 28:1 that the righteous are as bold as a lion! I break all intimidation from me in Jesus's name! I bind fear. I bind timidity in Jesus's name. I break the power of Jezebel's intimidation off my life in the name of Jesus. I command the lies of Jezebel to be broken off my mind. You foul Jezebel spirit, I command you to go, in Jesus's name. I will not listen to you. I will not surrender to you. I am free in the name of Jesus. Amen._\n\nRebellion\n\n_Lord, I thank You that I can and do fully surrender to You. You said in James 4:7 to submit myself to You and resist the devil. I come before You now, bowing my life before You. I offer myself as an offering to You. I give all of me to You. I repent for opening any doors of rebellion. I repent for partnering with rebellious thoughts or activities. I ask for Your mercy and forgiveness over my life in Jesus's name. I speak to every foul operation of rebellion, and I command it to go, in Jesus's name! I break the hold of Jezebel's rebellion over my life in the name of Jesus. I confess that I am free, that I honor authority and submit to all godly leaders, and that I live a life of godly surrender and submission in Jesus's name._\n\nWord curses\n\n_Thank You, Lord, that I am blessed coming in and blessed going out, according to Deuteronomy 28. No curse can prosper against my life! I release the power of the blood of Jesus over me in Jesus's name. I break every word curse of Jezebel over my life. Every demonic decree over me is broken. Every false accusation is exposed and proved a lie. No lie will stand against me. No decree of witchcraft against me will prosper, in the mighty name of Jesus. I break the power of Jezebel over my life in Jesus's name! I say that I am free and the Word of the Lord is prevailing in my life in Jesus's name. Amen._\n\nConfusion, manipulation, and lies\n\n_Thank You, Lord, that I walk in perfect peace, according to Isaiah 26:3. My mind is fixed upon You, Lord, and Your shalom is abundant in my life. I break every spirit and power of confusion sent from hell against me in the authority of Jesus's name. I break the power of lies and confusion from the Jezebel spirit. I break manipulative powers over my life in Jesus's name! I say that I am free from guilt, shame, and control. I refuse to manipulate others and strive to get my own way. I command all powers of manipulation to be broken over my life in Jesus's name. The Jezebel spirit has no authority over my life. I command every form of Jezebel witchcraft and attack to be broken in Jesus's name. Amen._\n\nStrange fire\n\n_Thank You, Lord, that I am consumed by Your holy fire in Jesus's name, according to Hebrews 12:29. The fire of purity prevails in my life. The fire of revival is burning in me. The fire of awakening is rich in my life. The fire of deliverance and power is strong in me. I break every attack of strange fire and sorcery over my life. No strange fire can enter my life, in Jesus's name. I break the power of Jezebel's witchcraft over my life in the mighty name of Jesus and command it to go right now. I loose the freedom and purity of the Lord Jesus Christ in every area of my life. I say that I am blessed and free in the name of Jesus. Amen._\n\nSeduction\n\n_Thank You, Lord, that I am pure in my thought life. Thank You, Lord, that I am sexually pure. I break all thoughts and temptations of lust in the name of Jesus. I break every spirit of perversion off my life and command them to go now in Jesus's name. I am not seduced or drawn away by the Jezebel spirit. I release the holiness and purity of the Lord over my life. Lord, You said in 1 Thessalonians 4 that I should be sanctified and avoid all immorality. I call for purity in my life. I consecrate my mind and body to You in the name of Jesus. Lust cannot operate in me. Immorality cannot prevail in my life in Jesus's name. I release the power and freedom of Jesus in my life right now. Amen._\n\nBREAKING PYTHON ATTACKS\n\nPhysical attacks\n\n_Thank You, Lord, that I am healed and free. I break every attack of choking, congestion, and breathing problems in the name of Jesus! I say that I am healed and free. According to Job 33:4, \"The Spirit of God has made me, and the breath of the Almighty has given me life.\" I thank You, Lord, that my breathing is normal! You foul python spirit, I command you in the name of Jesus to take your hands off my body. I command you to loose my lungs, throat, and breathing. Go now, in Jesus's name! I say I am healed and free by the blood of Jesus Christ. Thank You, Father, for Your power flowing over my body now in Jesus's name. Amen._\n\nConfusion and heaviness\n\n_I come against confusion, in Jesus's name! I break the power of the python spirit over my thought life. I release supernatural peace in the name of Jesus. I boldly confess that I have the mind of Christ, according to Philippians 2:5. My thoughts are clear. My thoughts are filled with the knowledge of God. I also break heaviness and confess that I am filled with joy. Python spirit, I break your grip over my thoughts and emotions in the name of Jesus! I command you to go right now, in Jesus's name. I say that I am free. I say that I have the joy of the Lord. I say that I am blessed now in Jesus's name. Amen._\n\nFalse utterances and false prophetic words\n\n_Father, I thank You for purity in every area of my life in Jesus's name! You said in Your Word that the spirit of prophecy is the testimony of Jesus_ (Rev. 19:10) _. I break every false prophetic word over my life in Jesus's name! I pull down every misleading and destructive utterance. I bind up every wrong decree, and I release the power of the blood of Jesus over my mind. I thank You, Lord, for cleansing my prophetic life so that I hear Your voice and align with Your will. I command every lie of python over my life to be broken now in Jesus's name! I command every false prophetic spirit to go now in Jesus's name. I thank You, Lord, that I am free in Jesus's name. Amen._\n\nFlattery\n\n_I break every flattering lie that has been spoken over me. In James 4:10 You said, \"Humble yourselves in the sight of the Lord, and He will lift you up.\" I come before You now, humbly asking for Your direction and guidance in every area of my life. I thank You, Lord, that my heart is fully surrendered to You in Jesus's name. I command every flattering decree of python to be broken over my life in Jesus's name. I break every deceptive word in Jesus's name. I command the power of the python spirit over me to be broken right now in the mighty name of Jesus. Amen._\n\nStalking\n\n_Thank You, Father, that according to Psalm 37:23, my steps are ordered of the Lord. You have provided_ _the right steps for me to take in every area of my life. I am confident in Your leadership and guidance. I am protected and loved by You as I follow Your leading. I break the stalking spirit of python over my life in Jesus's name! I am not harassed, followed, or irritated by the python spirit. Python spirit, I command you to go now, in Jesus's name! I confess that I am free. My steps are blessed, and my pathway, secure in the name of Jesus. Amen._\n\nLies and deception\n\n_Thank You, Lord, that I am led by Your Spirit, according to Romans 8:14. The Holy Spirit lives in me. The Holy Spirit is my teacher. I am not deceived. Deception cannot prevail in my life. I break the power of deception and deceiving spirits in the name of Jesus! I break lies and false thoughts off my life. I command the power of every lie to be broken in the mighty name of Jesus. I confess that the anointing lives in my belly, according to 1 John 2:20. The Spirit of truth lives in me, and I am led into all truth. I am not deceived. I break free from the grip of python in the authority of Jesus's name. I command the spirit of python to go from every area of my life now in Jesus's name! Spirit of python, I cast you out right now! Leave in the authority of the Lord Jesus. I am free in Jesus's name. Amen._\n\nBREAKING RELIGIOUS SPIRIT ATTACKS\n\nComplaining\n\n_Lord, I choose to meditate on Your goodness in my life. I am so thankful for every blessing, every miracle,_ _every breakthrough\u2014even the little things. I confess that You are my Father. You are my healer. You are my provider. You are my everything. You said in Philippians 2:14 that I should do all things without murmuring or disputing. Thank You, Father, that my mouth is filled with praise and my heart is filled with thanks. I rebuke the religious spirit and command its power over my life to be broken in the name of Jesus. I cast out the religious spirit now and break it off me in the mighty name of Jesus. Amen._\n\nOffense\n\n_Thank You, Lord, that I walk in love. You said in 1 Corinthians 13 that love is patient and kind. I confess that I am patient with others and I am kind. I do not keep track of wrongs. I do not hold grudges. I am free from offense. I break the power of offense off my life in the name of Jesus. I choose to walk in forgiveness toward others. I choose to release all pain and wounds. Father, I thank You for Your nature, Your love, and Your mercy operating in me toward others. I release all hurts, and I declare Your love is washing over me now in Jesus's name._\n\nCritical attitude\n\n_Thank You, Lord, that You said in 1 Corinthians 13 that love never fails! I say that I walk in love and mercy with people. I say that I see Your love for their lives. I say that I see the potential in people. I see the victory in their lives. I am not critical, in the name of Jesus. I break every critical attitude and thought process over my life. I refuse to partner with the religious_ _spirit and have a negative attitude. My eyes see as Christ would see. My mouth speaks as Christ would speak. I walk by faith and not by sight in the name of Jesus! I am free in Jesus's name. Amen._\n\nCondemnation\n\n_Thank You, Lord, that I am free from condemnation as a child of God, according to Romans 8. I break every lie that would try to condemn me or accuse me in the name of Jesus. There is no condemnation in Christ Jesus! The Spirit of life has made me free. I am free in my mind! I am free in my spirit! I am free in my body! I walk in Your grace. I receive Your promises. I receive Your goodness and Your love in my life. I break the weight, the lies, and the hindrance of condemnation off my life in the name of Jesus. I command condemnation to go in Jesus's name! I break every religious lie and mentality off my life and boldly declare that I am blessed, loved, forgiven, accepted, and free in the name of Jesus. Amen._\n\nDryness\n\n_Thank You, Lord, for the rivers of living water in my belly, according to John 7:38. I call forth Holy Spirit rivers in my life. I break the dry climate that Jezebel and religion would attempt to establish in my spiritual life. I release the rivers of refreshing. I release the rivers of joy. I release the rivers of healing. I release the rivers of mercy. I release the rivers of restoration. You said that You would anoint me with fresh oil. I declare fresh Holy Spirit oil running over every area of my life in the name of Jesus! Thank You, Father,_ _for the outpouring of Your spirit in my life. I break and bind the weight of the religious spirit and every religious attack, and I cast them out of my life in the name of Jesus! I command them to go now in Jesus's name. I call forth a spiritual climate shift in my life in the name of Jesus. Amen._\n\nBondage\n\n_Father, I thank You that Jesus paid the price for freedom in every area of my life. You said in John 8:36 that he whom the Son sets free is free indeed. I claim absolute freedom from all forms of religious bondage. I break heaviness and bondage off my thought life in Jesus's name. I break bondage off my spiritual life in Jesus's name. I release the power of God over me. I decree total freedom through the blood of Jesus. I am not bound; I am free. I command the religious spirit to go now, in the name of Jesus\u2014I cast you out and break your power over my life. As I read the Word of God, I say that I read it as a son or daughter and not a slave. I say that my mind is being renewed and my thoughts are lining up with Your promises. I release freedom over every area of my life in Jesus's name. Amen._\n\nAs you break the power of these demonic spirits over your life, you are invoking the rights and privileges afforded you by the shed blood of Jesus Christ. Authority must be exercised. The enemy will always attempt to overstep and push the limits. Establishing yourself in the revelation of your identity and freedom powerfully positions you to live without the evil influence of the #ToxicTrio.\nNOTES\n\nINTRODUCTION\n\n. _Merriam-Webster_ , s.v. \"conspire,\" accessed October 30, 2017, <https:\/\/www.merriam-webster.com\/dictionary\/conspire>.\n\nCHAPTER 1\n\nTHE REALITY OF THE UNSEEN REALM\n\n. _Merriam-Webster_ , s.v. \"dwell,\" accessed October 20, 2017, <https:\/\/www.merriam-webster.com\/dictionary\/dwell>.\n\n. John Eckhardt, _Prayers That Rout Demons_ (Lake Mary, FL: Charisma House, 2008), 1.\n\n. Blue Letter Bible, s.v. \" _energe\u014d_ ,\" accessed October 20, 2017, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G1754&t=KJV.\n\n. _HELPS Word-Studies_ , \"1754. _energ\u00e9\u014d_ ,\" Bible Hub, accessed October 20, 2017, <http:\/\/biblehub.com\/str\/greek\/1754.htm>.\n\n. James W. Goll, _The Seer Expanded Edition: The Prophetic Power of Visions, Dreams, and Open Heavens_ (Shippensburg, PA: Destiny Image Publishers, 2012), chapter 2, <https:\/\/books.google.com\/books?isbn=0768488567>.\n\nCHAPTER 2\n\nTHE POWER OF THE ANOINTING\n\n. Benny Hinn, _Operating in the Anointing_ (Bookmark Publishing, 2017), Lesson 1, <https:\/\/bennyhinnbiblestudy.org\/course\/operating-in-the-anointing\/>.\n\nCHAPTER 3\n\nJEZEBEL, THE SPIRITUAL ASSASSIN\n\n. _Merriam-Webster_ , s.v. \"rebellion,\" accessed October 23, 2017, <https:\/\/www.merriam-webster.com\/dictionary\/rebellion>.\n\nCHAPTER 4\n\nJEZEBEL'S SEDUCTION, SCHEMES, AND STRATEGIES\n\n. Blue Letter Bible, s.v. \" _tith\u0113mi_ ,\" accessed October 23, 2017, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G5087&t=KJV.\n\n. Blue Letter Bible, s.v. \" _pr\u014dton_ ,\" accessed October 23, 2017, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G4412&t=KJV.\n\n. Blue Letter Bible, s.v. \" _arch\u0113_ ,\" accessed October 23, 2017, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G746&t=KJV.\n\n. Blue Letter Bible, s.v. \" _exousia_ ,\" accessed October 23, 2017, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G1849&t=KJV.\n\n. _Merriam-Webster_ , s.v. \"seduce,\" accessed October 23, 2017, <https:\/\/www.merriam-webster.com\/dictionary\/seduce>.\n\n. Jennifer LeClaire, _Jezebel's Puppets_ (Lake Mary, FL: Charisma House, 2016), 41.\n\nCHAPTER 5\n\nCASTING DOWN THE DEMON QUEEN JEZEBEL\n\n. \"Jezebel Meaning,\" Abarim Publications, last updated September 1, 2017, <http:\/\/www.abarim-publications.com\/Meaning\/Jezebel.html#.We5AsGhSwdU>.\n\n. \"Jezebel No. 1,\" _All the Women of the Bible_ , Bible Gateway, accessed October 23, 2017, <https:\/\/www.biblegateway.com\/resources\/all-women-bible\/Jezebel-No-1>.\n\n. John Paul Jackson, _Unmasking the Jezebel Spirit_ (Flower Mound, TX: Streams Ministries International, 2002), chapter 2, <https:\/\/books.google.com\/books?isbn=0985863811>.\n\n. Blue Letter Bible, s.v. \" _apostolos_ ,\" accessed October 23, 2017, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G652&t=KJV.\n\nCHAPTER 6\n\nPYTHON, THE HEAT-SEEKING DECEIVER\n\n. Ruth Ward Heflin, _Revival Glory_ (Morrisville, NC: lulu.com, 2013), 154, <https:\/\/www.amazon.com\/Revival-Glory-Ruth-Ward-Heflin\/dp\/1304256162>.\n\n. Bible Hub, s.v. \" _ruach_ ,\" accessed October 24, 2017, <http:\/\/biblehub.com\/hebrew\/7307.htm>.\n\n. Blue Letter Bible, s.v. \" _pyth\u014dn_ ,\" accessed October 24, 2017, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G4436&t=KJV.\n\n. \"Apollo,\" _Encyclopedia Mythica_ , last modified January 31, 2004, accessed January 3, 2018, <http:\/\/www.pantheon.org\/articles\/a\/apollo.html>.\n\n. See, for example, Euripides, _Andromache_ , line 1106, <https:\/\/books.google.com\/books?isbn=0856687707>.\n\n. Kevin Kleint, \"Introducing the Oracle at Delphi,\" HonorOfKings.org, June 23, 2017, <https:\/\/www.honorofkings.org\/python-spirit-oracle-delphi\/>.\n\n. Jennifer LeClaire, _The Spiritual Warfare Battle Plan_ (Lake Mary, FL: Charisma House, 2017), 40.\n\nCHAPTER 8\n\nTHE DNA OF THE RELIGIOUS SPIRIT\n\n. _Merriam-Webster_ , s.v. \"pervert,\" accessed October 24, 2017, <https:\/\/www.merriam-webster.com\/dictionary\/pervert>.\n\n. A. A. Allen, _The Price of God's Miracle Working Power_ (Memphis, TN: Bottom of the Hill Publishing, 2012).\n\n. C. Peter Wagner, \"The Corporate Spirit of Religion,\" in _Freedom From the Religious Spirit_ (Bloomington, MN: Chosen Books, 2014), <https:\/\/books.google.com\/books?isbn=1441269029>.\n\nCHAPTER 9\n\nFIRST-LOVE LIFESTYLE: FREEDOM FROM THE RELIGIOUS SPIRIT\n\n. _Merriam-Webster_ , s.v. \"breach,\" accessed November 8, 2017, <https:\/\/www.merriam-webster.com\/dictionary\/breach>.\n\n. Blue Letter Bible, s.v. \" _proskyne\u014d_ ,\" accessed November 29, 2017, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G4352&t=KJV.\n\nCHAPTER 10\n\nENFORCING THE KINGDOM: LIVING FREE\n\n. _Merriam-Webster_ , s.v. \"temptation,\" accessed October 25, 2017, <https:\/\/www.merriam-webster.com\/dictionary\/temptation>.\n\n# Table of Contents\n\n  1. COVER PAGE\n  2. TITLE PAGE\n  3. COPYRIGHT PAGE\n  4. CONTENTS\n  5. ACKNOWLEDGMENTS\n  6. FOREWORD BY JOHN ECKHARDT\n  7. INTRODUCTION\n  8. CHAPTER 1 THE REALITY OF THE UNSEEN REALM\n  9. CHAPTER 2 THE POWER OF THE ANOINTING\n  10. CHAPTER 3 JEZEBEL, THE SPIRITUAL ASSASSIN\n  11. CHAPTER 4 JEZEBEL'S SEDUCTION, SCHEMES, AND STRATEGIES\n  12. CHAPTER 5 CASTING DOWN THE DEMON QUEEN JEZEBEL\n  13. CHAPTER 6 PYTHON, THE HEAT-SEEKING DECEIVER\n  14. CHAPTER 7 BREAKING PYTHON'S GRIP\n  15. CHAPTER 8 THE DNA OF THE RELIGIOUS SPIRIT\n  16. CHAPTER 9 FIRST-LOVE LIFESTYLE: FREEDOM FROM THE RELIGIOUS SPIRIT\n  17. CHAPTER 10 ENFORCING THE KINGDOM: LIVING FREE\n  18. NOTES\n\n## List of Pages\n\n  1. C\n  2. i\n  3. ii\n  4. iii\n  5. iv\n  6. v\n  7. vi\n  8. vii\n  9. viii\n  10. ix\n  11. x\n  12. xi\n  13. xii\n  14. xiii\n  15. xiv\n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n\"Anxiety is part of the 'broken brain' syndrome that millions of Americans suffer from. Conventional medicine can mask the symptoms, but it doesn't tackle the root cause. That's what the total mind-body system of qigong does. In this groundbreaking workbook, Master Kam Chuen Lam shows us how to use our own life-force energy to lift our minds and bodies to a new level of wellness. Open this book to any page and you'll find it's one of the most practical, helpful, and friendly guidebooks to your own well-being you can imagine.\"\n\n\u2014 **Mark Hyman, MD** , author of the #1 _New York Times_ bestseller, _The Blood Sugar Solution_\n\n\"Master Kam Chuen Lam's workbook is a valuable resource for anyone dealing with anxiety, worry, or stress. The ancient healing exercises he shares from decades of experience are presented in a format that is easy to understand, learn, and practice. Having worked with some of these exercises I can personally say they cultivate more peace, calm, and body awareness. This is a book I will be recommending to many of my clients for years to come!\"\n\n\u2014 **Tahir Bhatti, MD** , wellness physician, director of Integrative Holistic Services, and associate clinical professor in the department of psychology at the University of California, San Diego\n\n\"In a world where everyone experiences stress and worry, what a gift it is to read _The Qigong Workbook for Anxiety_. Master Kam Chuen Lam introduces readers to the ancient tradition of qigong in a friendly and accessible way. He teaches simple yet potent methods for relaxing both physical and emotional tension. His instructions are revolutionary in that he encourages us not to reject feelings of anxiety. Instead, by practicing these qigong exercises we become aware of our greater energy field and learn how to regulate our nervous system even when tension arises. This book is for anyone who wants to tap into their inner strength and live a calmer, healthier life.\"\n\n\u2014 **Amy Conway, MPH** , senior managerial consultant for Kaiser Permanente in Northern California\n\n\"Having worked for over thirty years in a busy and demanding dental practice, I am no stranger to stress at work. I can, without hesitation, attribute my current well-being largely to my daily practice and enjoyment of qigong. Being calm as a practitioner, I am able to bring harmony to the way I run my practice and patients enjoy the peaceful atmosphere. I strongly believe that Master Kam Chuen Lam's unique lineage, wisdom, and extensive knowledge [make this book] a must-have for all busy and stressed professionals who also want to remain truly healthy whilst having to endure daily challenges at work and home.\"\n\n\u2014 **Thye Yeoh** , qigong student and dental practitioner in the UK\n\n\"Do not let the simplicity of the exercises in this book fool you. ... I work as the compliance director of a major financial firm in London\u2014a highly stressful job\u2014and I can vouch from my own experience for the profundity of these exercises and the enormous wisdom of Master Kam Chuen Lam's teaching.\"\n\n\u2014 **Charles Beck** , qigong student and director of compliance, London, UK\n\n\"True health is more than exercise and good nutrition. It includes peace of mind. Those who possess it glow and seem to have access to a wellspring of pure energy. If you've ever wondered how to become such a person, qigong Master Kam Chuen Lam can guide you there.\"\n\n\u2014 **Susan Piver** , founder of the Open Heart Project\n\n\"Qigong combines two approaches to treating anxiety that are known to work: exercise and mindfulness. This highly accessible and readable workbook should help many more people enjoy the benefits of qigong.\"\n\n\u2014 **Elizabeth Murray** , general practitioner and professor of emergency health and primary care at the University of London, UK\n\n\"This book comes at the right time. More than ever, employees and managers are struggling to keep up with the ever-increasing speed and constantly rising demands of business life. The ensuing stress and burnout is literally killing people. Western approaches to dealing with the resulting issues have reached their limits. Master Kam Chuen Lam's workbook is addressing the root cause\u2014anxiety\u2014in a new way. It is based on age-old Chinese wisdom and Lam's own life's work devoted to the study of human energy. I highly recommend it to everyone who wants to be successful in the long run.\"\n\n\u2014 **Thomas Apfel** , corporate account manager at IBM\n\n\"Master Kam Chuen Lam's new book about dealing with anxiety is a revelation on its own. During my daily interaction with dental patients, I come across a big percentage of anxiety cases. I have been aiming to incorporate Master Kam Chuen Lam's relaxation techniques during treatment. As a result, patients often relax so much that they're telling us when they get off the dental chair that they feel as if they had a good rest! It is without a doubt, that in this book, various daily issues are addressed in an effective manner. Conventional Western medicine has no ammunition to combat stress or anxiety for everyday life or work. Look no further, Master Kam Chuen Lam's new book provides you with the essential elements to survive today's hectic lifestyle and emerge unscathed with simple techniques. It's the first time I've come across such an informative and to-the-point read.... It is a beacon of hope in today's jungle.\"\n\n\u2014 **Maria Maliderou, DDS, MSc** , qigong student and dentist based in London, UK\n\n\"Stand, be still, and master anxiety. Master Kam Chuen Lam's highly-engaging _The Qigong Workbook for Anxiety_ is a gift for anyone struggling with fear.\"\n\n\u2014 **Gary Lopes Cunha, LMFT, LPCC** , outpatient psychiatry clinician for Kaiser Permanente in Northern California\n\n\"Master Kam Chuen Lam's qigong exercises are like a warm internal shower that relaxes, yet energizes. They seem a perfect antidote to the anxieties of life and aging. Getting older myself and working with elders, as I do, I can attest to their powerful benefits. This workbook is a wonderful guide and support for new and experienced practitioners alike.\"\n\n\u2014 **Vicki Giella** , qigong participant and former manager of the Senior HelpLine, a telephone assistance service for elders, their families, and caregivers\n\nPublisher's Note\n\n_This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering psychological, financial, legal, or other professional services. If expert assistance or counseling is needed, the services of a competent professional should be sought._\n\nDistributed in Canada by Raincoast Books\n\nCopyright \u00a9 2014 Kam Chuen Lam\n\nNew Harbinger Publications, Inc.\n\n5674 Shattuck Avenue\n\nOakland, CA 94609\n\nwww.newharbinger.com\n\nCover design by Sara Christian\n\nAcquired by Jess O'Brien\n\nEdited by Jennifer Eastman\n\nAll Rights Reserved\n\n* * *\n\nLibrary of Congress Cataloging-in-Publication Data\n\nLam, Kam Chuen.\n\nThe qigong workbook for anxiety : powerful energy practices to rebalance your nervous system and free yourself from fear \/ Kam Chuen Lam ; foreword by Grandmaster Yu Yongnian.\n\npages cm\n\nSummary: \"Anxiety is an age-old problem. It exists in all cultures, and each has its own methods for treating it. For readers who suffer from anxiety but have found little relief from modern western treatments, the ancient practice of qigong--which can be understood as the cultivation of internal energy--may offer relief. In The Qigong Workbook for Anxiety, world-renowned and respected qigong master Kam Chuen Lam presents his first workbook for overcoming anxiety problems utilizing traditional Chinese energy-focusing exercises. By following these step-by-step exercises, readers will learn to transform feelings of anxiety into resilience and inner strength. The workbook also contains helpful illustrations\"-- Provided by publisher.\n\nIncludes bibliographical references.\n\nISBN 978-1-60882-949-1 (paperback) -- ISBN 978-1-60882-950-7 (pdf e-book) -- ISBN 978-1-60882-951-4 (epub) 1. Anxiety. 2. Qi gong. 3. Exercise therapy. I. Title.\n\nBF575.A6.L36 2014\n\n613.7'1489--dc23\n\n2014012473\n\n# Contents\n\nForeword\n\nAcknowledgments\n\nIntroduction\n\nPart 1 Energy, Anxiety, and Inner Strength\n\nComing Up for Air\n\nSitting Up\n\nThe Golden Ball\n\nResting the Golden Ball\n\nRolling the Golden Ball\n\nYour Inner Reservoir\n\nStarting Wu Ji\n\nCalming Your Nervous System\n\nStanding and Rolling the Golden Ball\n\nGathering Your Energy\n\nPart 2 Your Personal Energy Field\n\nArm Swinging\n\nOpening the Curtains\n\nTensing and Relaxing\n\nWorking with a Wall\n\nClenching and Relaxing\n\nSpreading and Relaxing\n\nStanding and Opening\n\nCalming the Mind and Body\n\nResting While Standing Still\n\nLearning to Stay Relaxed\n\nPart 3 Developing Your Inner Strength\n\nTaking Root\n\nUsing the Rolling Pin\n\nHolding the Golden Ball at Your Belly\n\nExpanding the Golden Ball\n\nGolden Ball Energy Stance\n\nComing Back\n\nWorking with Your Breath\n\nOverwhelmed, Lying Down\n\nIn the Middle of the Night\n\nPart 4 Using the Power of Your Senses\n\nLooking into a Glass of Water\n\nListening to the Garden\n\nScenting the Sweet Smell\n\nTaking a Cup of Tea\n\nFeeling What Touches Us\n\nRefreshment\n\nPart 5 Working with the Energy Around You\n\nGiant Strides\n\nBubbling Spring\n\nWaves on the Rocks\n\nFearless Rider\n\nCrushing the Flanks\n\nHolding Two Forces\n\nPracticing in Nature\n\nThe Two Poles\n\nNeck and Face Self-Massage\n\nSoothing Your Back\n\nEasing Headaches\n\nEnergy Circuit\n\nThe Lineage\n\nAbout the Author\n\nHealing and Training with Master Lam\n\nBooks and Videos with Master Lam\n\nReferences\n\nNotes on the Author and Foreword Writer\n\n# Dedication\n\nWith profound gratitude for the countless masters\n\nwho discovered the secrets of human energy\n\nand shared their knowledge of this art,\n\nfor my beloved late master, Yu Yongnian,\n\nand for my very dear wife and sons,\n\nI dedicate this book and its healing power\n\nto human well being and a world at peace.\n\n# Foreword\n\nBy Professor Yu Yongnian\n\nHonorary member of the Council of the Association of Qigong Science of the People's Republic of China, former adviser to the American-Chinese Qigong Association, and honorary chairperson of the Da Cheng Chuan Zhan Zhuang Chi Kung Research Groups (Europe).\n\nAnxiety is a problem everywhere in the world today. In the course of my lifetime, I have seen the world speed up. Everything is affected. Speed has changed the nature of people's work, travel, communications, and almost every field of life. Even their family lives have changed. This acceleration has also affected the human psyche. Our nervous systems feel the constant impact of the instability and violence of life around us. Millions of people are suffering from tremendous tension. It affects their health as individuals, and it has a serious effect on our societies as well.\n\nThe intensity of what we are experiencing is new, but anxiety itself is not.\n\nThe earliest texts of the natural scientists of my country, dating back thousands of years, examined this problem and talked about calming the mind. Perhaps the most famous is the _Tao Te Ching_ of the great master Laozi, who wrote:\n\n_Rest your mind..._\n\n_Do not tinkle like slivers of jade_\n\n_Or bang around like stone shards..._\n\n_From stillness and tranquility comes the order of the universe._\n\nLaozi's writings include some of the earliest known references to qigong, the art of cultivating the internal energy of the human being. I have studied this art for over seventy years. I had the privilege of being a student, in my early twenties, of the Grand Master Wang Xiang Zhai. He brought this art out of the realms of secrecy into the light of public teaching. He taught the rare qigong system known as _zhan zhuang_. It is the most powerful of all the systems for cultivating human energy. But it involves no external movement at all. This makes it unique.\n\nPracticing zhan zhuang qigong changed my life. At the time, I was a young dentist who suffered from a great deal of tension. I experienced it in my upper back as I treated my patients. All that changed after I started learning this art. Under my master's guidance, I went deeper. I began to see the inner significance and power of this art. I saw the transformation that its healing and calming power could have on people's lives.\n\nDuring the many decades I have practiced zhan zhuang qigong, I have also seen it change the lives of countless people who have been my students. Some people may think that, since this is part of China's cultural heritage, it is only helpful to Chinese people. Not at all. I have had the good fortune to teach in other countries in Asia, as well as in America and Europe. My experience has shown me that zhan zhuang qigong is a precious treasure in the heritage of all humanity.\n\nI am therefore delighted that Kam Chuen Lam, my disciple, has written this book to show people how they can use this ancient art to help them with their anxiety. Together, we bridge East and West. I studied Western medicine in my youth, becoming a professor of dentistry. Kam Chuen Lam studied traditional Chinese herbalism and bonesetting. We have used our different backgrounds to create a pathway for people everywhere to study qigong, which we have in common.\n\nHe first came to study with me in Beijing. After that, I went to London to supervise the production of his first book, _The Way of Energy._ It introduced the zhan zhuang system of qigong to the West. One of my favorite photos was taken on that trip. It shows me standing with one of the royal guards in the city.\n\nThis year, a large delegation of Kam Chuen Lam's students from nearly twenty countries around the world came to Beijing to show me their progress. I was delighted to be part of a huge energy circle with them. As I felt the current of their energy, which I was part of myself, I felt our circle was a powerful symbol. It made me appreciate not only the progress of those students, but also the progress of our whole system of zhan zhuang qigong. It has come a long way since I helped introduce it into hospitals in China in the 1950s. At that time, we saw how helpful it was in the treatment of chronic diseases. Now we know that it is helpful for people in all conditions of health.\n\n_The Qigong Workbook for Anxiety_ will open up a new chapter. It shows, step by step, how this ancient art can be used in our own age for one of the most common disorders that people are now suffering from\u2014the pain of the heart and mind caused by anxiety.\n\n\u2014Professor Yu Yongnian\n\nBeijing, 2013\n\nAge 94\n\nMaster Lam\n\n# Acknowledgments\n\nThis workbook has a long history, and many people have made it possible. I would like to express particular gratitude to my grand master, Wang Xiang Zhai, who courageously decided that the treasures of this tradition, which were passed on as a secret transmission over the centuries, should be made public.\n\nMy own master, Professor Yu Yongnian, graciously accepted me as a student\u2014even though he was taking a personal risk in doing so in the dangerous days of the Cultural Revolution in China, when ancient arts were under threat. He was a constant and powerful force in my life; and to the extent that I have been able to experience the depths of this art, it is thanks to his deep practice, immense skill, and great kindness. He wrote the foreword to this book just before his death in October 2013. As my senior students wrote in a funeral tribute to him, \"We have lost not only our grand master, but a noble human being.\"\n\nMy wife, Kaisin, has stood by me through all the years in which I have studied this art and endeavored to bring it to the world beyond China. Without her, what I have done would not have been possible. I would also like to thank my sons, Tin Yuan, Tin Yu, and Tin Hun, for their support, understanding, and unending help.\n\nLastly, I would like to thank all my students who have made a sincere effort to learn this art and who have remained with me over these years. In particular, I would like to thank my devoted student Richard Reoch, who has helped me with the planning and writing of many of my books, and who has been my constant companion in my efforts to bridge East and West.\n\nAuspiciously, we were introduced to New Harbinger Publications and their acquisitions editor, Jess O'Brien, who had, unknown to us, been reading and studying my books for over a decade. I am deeply grateful for New Harbinger's willingness to help me bring this healing wisdom into the world in this form.\n\nQi\n\nThis is the Chinese character for \"qi,\" pronounced \"chee.\" It represents the invisible, all-pervasive, and palpable life force of the universe. The top of the ideogram represents a cooking pot. Underneath it are four strokes that symbolize fire. The pot is filled with boiling water and steam is rising from it.\n\nJust as steam is capable of powering engines and other devices, so the power of qi enables us to do anything we want. It is like pressure\u2014air pressure, water pressure, wind pressure. In the human being, it takes the form of personal power, present throughout our being and visible in the photographs that scientists are now capable of taking of our aura.\n\nThe power of qi is capable of breaking through the blockages in our energetic pathways, such as those that we experience as the paralysis of anxiety or the congealed stagnation of fear.\n\n# Introduction\n\nThis workbook offers anyone who suffers from anxiety a step-by-step way to transform that agitated energy into resilient inner strength. You are introduced to the ancient art of harmonizing your own internal energy and drawing strength from the energy around you. This art is known as \"qigong\" (pronounced \"chee gung\"), which means \"internal energy work.\" Qigong is part of the world's oldest continuous system of natural medicine, developed in China over centuries.\n\n## Part 1: Energy, Anxiety, and Inner Strength\n\nPart 1 introduces you to a different way of thinking about yourself. In qigong, you are seen as a complete field of energy\u2014similar to the current thinking of atomic physics. The energy of your whole being is what is studied and strengthened.\n\nQigong is based on the truth that each person is unique and to be respected. Your feelings, attitudes, and behaviors\u2014including their deepest psychological patterns\u2014are part of your total energy field. Fearful and difficult feelings are not regarded as enemies but as part of the whole. Working with your complete field of energy, rather than splitting it up, is at the heart of this system of well-being.\n\n## Part 2: Your Personal Energy Field\n\nYou are first introduced to your energy field and are given simple exercises that help you get a feel for it. Anxiety, worries, and difficulties of all sorts make us tense, and when we are tense, our ability to feel what is happening in our bodies is blocked. This makes it very difficult to actually feel how tense we are. But after a little tension release, our natural sensitivity begins to be restored. Those natural feelings are what we need to remember and reacquaint ourselves with. It will help you learn how to use your bodily experience to develop a mental \"imprint\" of tension release. You can use the charts in this section of the workbook to help you track your personal experience.\n\n## Part 3: Developing Your Inner Strength\n\nOnce you start to feel that you are working with your energy in a different way, not resisting moments of anxiety and other strong emotions but embracing them within the larger flow of your body's energy field, you learn how to strengthen and cultivate the natural energy you possess. This increases your stamina, resilience, and inner strength. What previously made you panic is now experienced as an energy blip passing through the far larger, vibrant field of your total being. The gentle exercises and postures described in this part of the book are surprisingly simple and can be done without any equipment at all. Charts in this part of the book help you keep track of your daily energy training and the way you are using your new awareness of energy to work with your emotions from the perspective of inner strength.\n\n## Part 4: Using the Power of Your Senses\n\nOften, when we are suddenly gripped by anxiety, we have no time to reflect, meditate, or do calm breathing. How can we use the immediate energy of the world around us to be more grounded and less swept away? In part 4, you are introduced to the centuries-old tradition of using the power of sight, sound, smell, touch, and taste to influence your experience in subtle and profound ways. Our sensory experience is also part of the play and flow of our energy\u2014and a gateway to connecting with the constant movement of energy around us.\n\nIn this section, with the help of step-by-step instructions and charts, you learn simple methods that help balance and harmonize your mental, emotional, and physical experience. The methods form a path that strengthens and increases the flexibility of your whole nervous system. You are encouraged to do this in very personal ways\u2014to become the world's leading expert on your own experience.\n\n## Part 5: Working with the Energy Around You\n\nAnxiety can have the effect of closing us down. We can feel claustrophobic and alone with our agitation. No matter how we feel, we do not exist in isolation. Each of us exists in a vast, constantly dynamic field of energy: the world around us. We experience many different environments every day\u2014home, office, and the natural world\u2014all with their own energies, all influencing us in different ways. In this final part of the workbook, you learn ways of understanding and using these energies\u2014and also what you can do to protect yourself from harmful energies, which are common in this age of high-energy technology. You will also learn simple methods for developing and sharing positive energy with others, as well as how families and friends can help generate vibrant and healthy energy fields.\n\n## The Lineage\n\nThe art of qigong is the result of human inquiry, experimentation, and practice stretching over centuries. What I am sharing with you has been handed down in secret\u2014literally from person to person\u2014as masters passed on their wisdom to their students. It is only in modern times that it has become public. In passing on an art like this, it has always been important to know that the transmission is authentic. For this reason, I would like you to know the story of my lineage, and I hope you will find time to read these pages. Knowing this will be important to you\u2014especially if someone asks you about qigong, where it came from, and why you are practicing it.\n\n## Before You Begin\n\nYou will be working with your own energy field\u2014that makes it very personal. It will help if you pay close attention to your experience as you try the various exercises and suggestions. To help you do this, charts located throughout the workbook prompt you to think about and record what you are doing and how you are feeling. This way, you can be your own natural scientist, examining your own experience and deciding what is working best for you.\n\nMany people, when they start qigong, worry about whether they are doing the right thing, whether they are making progress, or (worst of all) whether they might be making a serious mistake. Guess what\u2014that makes them anxious. Don't beat yourself up. All the exercises and suggestions in this book have been developed and tested over centuries, and I have only included things that, in my experience, have proved helpful to my own students\u2014who come from all over the world, all backgrounds, and all ages.\n\nIt's natural to wonder if you are making \"progress,\" but actually, this idea is not really part of the qigong tradition. What happens when you practice qigong is very natural and almost imperceptible, the way a plant grows, day by day. If you feel impatient, uncertain, or frustrated, because you can't see or feel any progress, please note that down as part of your experience. Your natural energy will help you ride it all the better if you are fully aware of what's happening on your journey.\n\nSome people like to read books from the back to the front. So you might be reading this introduction last! That's all right. I have tried to sprinkle this kind of advice throughout the workbook. You do not have to do the exercises and suggestions in this book in a strict order. I suggest you look first at part 1 and follow the exercises and suggestions there first. But if you have already tried some of the practices for your senses from part 4 or some exercises from another part of the book, please continue to do that, working with whatever has the best effect on you. You can add the exercises and suggestions from the earlier parts of the book\u2014they will contribute to the power of whatever else you are doing.\n\nIf you want to develop a qigong routine, you can find one outlined in part 3. Following it is entirely up to you. You can also simply experiment with whatever you wish from this book and try to incorporate it into your lifestyle. If you do decide to follow a qigong routine, normally I recommend doing it in the morning. If you do it before breakfast, it is a good idea to drink a little warm water (not tea or coffee) before you start. If you have breakfast first, you should allow a short gap before you start your training. If you do your training at night, it is best to do it in a room with a light on. Preferably, there should be some source of fresh air in the room. If it is not possible to open a window, at least try not to practice in a room where the air is stale.\n\nDifferent people have different sensations when they start to practice qigong. It's common to feel warm, even hot. You might find yourself shaking a little\u2014sometimes a lot\u2014as your energy wakes up. Sometimes it's hard to sit or stand still, especially if you are always on the go. Those sensations are part of the practice. Whatever you experience is the result of the deep power of qigong.\n\n## Your Personal Responsibility\n\nIt is important that you understand that the advice I am offering in this book is a contribution toward your personal training and development. I have done everything in my ability to ensure that this book will be beneficial to your health and well-being. But I cannot possibly know every reader individually, and therefore I cannot adapt the instructions for each person.\n\nIt is therefore important that you understand that, owing to causes completely beyond my control, you might experience difficulties that you might feel are somehow connected with your qigong practice. People have been using my books for over twenty years, and no one has ever reported experiencing this. Nevertheless, please take personal responsibility and check that there is no medical reason to prevent you from undertaking the practices in this book.\n\n## What to Do if You Have a Problem Using Qigong\n\nIf at any time in your qigong practice you feel heat or pressure rushing to your head, start to have a headache, or feel dizzy or faint, you should immediately do the practice \"Gathering Your Energy\" (located in part 1) three times, slowly and carefully. You can do it three more times if you want to. Finish with a qi massage to the face and neck (located in part 5). Then completely relax and rest, breathing naturally.\n\n## A Further Word of Welcome\n\nNow that we have gone over all these useful tips in preparation for using this book, I would like to welcome you as you begin to explore this ancient art. The contemplation and inner work are personal and unique to each person. It is your own journey, working with the energy within you and around you. This art has been handed down from master to student in an unbroken succession of person-to-person transmission for over two thousand years. Through the pages of this book, I am delighted to extend this human wisdom now to you.\n\n## A Quick Check Before You Start\n\nWe'll be working with anxiety using the mind-body connection. The physical effects of anxiety in your body affect your mind\u2014your thoughts and emotions\u2014just as your thoughts and emotions have a direct effect on your body. We will be reading the physical signs throughout this workbook, so it will be useful to catalogue them now. But please don't mistake this for a physical check-up. In the centuries-old tradition of qigong, mind and body are inseparable, and we use each one to help the other. Please take a minute or two to record how you are feeling in this chart. You will want to come back to this chart later and see what you wrote.\n\nThe Bridge\n\nThe character for \"bridge\" literally means \"arch\" and is pronounced \"kiew.\" Its complexity expresses the dynamic and multifaceted quality of a bridge\u2014the quality of all that is involved in crossing a gap.\n\nAll bridges are complex. They require a combination of materials to take the strain of those who cross them. They also need flexibility. Whether made of bamboo, wood, metal, or stone, bridges need to be able to yield. A bridge allows communication\u2014allowing people and ideas to move from one place to another. A bridge is also always open, offering itself; anyone can pass over a bridge.\n\nA conversation can be a bridge. A person can be a bridge. An ambassador is a bridge. This book is a bridge.\nPart 1\n\nEnergy, Anxiety, and Inner Strength\n\n## \"Engaging, Not Avoiding\"\n\nOne of my students in California is a counselor for a major health organization that treats millions of people throughout the United States. In addition to being a health professional in the Western tradition, he has been studying internal energy work with me. We often talk about the tremendous health problems that people of all walks of life are faced with day after day in our modern world.\n\nHe has encouraged me to offer what I can from my years of studying, practicing, and teaching qigong, because he believes this kind of total energy work is part of what is desperately needed in these difficult times. \"Your tradition is a lineage of healer-warriors,\" he said. \"You teach the mastery of anxiety through the Way of Energy. It is possible to live a vibrant and meaningful life even with anxiety and in the midst of all the challenges we face.\"\n\nHe told me that the ancient tradition of qigong was very much in line with what many health-care professionals are now thinking. \"We need a perspective that emphasizes mindfulness, not avoidance,\" he told me. \"Avoidance in our day and age is packaged as feeling better, avoiding internal experience. This approach is reinforced with pharmacy ads promising happiness. The anxious usually need energy, but more importantly, they need a willingness to engage their worst fears if they are to have a chance of mastering them. Many have limited lives, because they want to 'feel better'\u2014by not experiencing anxiety. Paradoxically, this attitude keeps them chronically anxious in the long term.\"\n\nIn the ancient tradition of qigong, which is a way of working with all forms of energy, trying to avoid our actual experience is not considered healthy. Whatever we are experiencing is part of the total energetic field, both around us and in us. The more we are open to it, the more power we can draw from it.\n\nThus this book offers what may seem a different approach to working with anxiety. It works with the full energy spectrum of the human being, including all our experience, as well as the vast energy field that surrounds us at all times.\n\n## The Energy of Anxiety\u2014Bridging Two Cultures\n\nAnxiety is an age-old phenomenon. All human beings experience it. It takes place in all cultures. All medical and health-care systems offer ways of helping us deal with it. At the same time, it is hard to define. It is an elusive vibration that takes many forms.\n\nIt is evoked in one of the oldest classics of world culture, the _Tao Te Ching_ by Laozi:\n\n_Hesitant, like crossing a river in winter\u2014_\n\n_Cautious, like fearing four neighbors._\n\nIn contemporary Western culture, which has become the dominant model for much of the world, anxiety is often seen as a condition that can be treated in specific ways, partly through therapy and partly through medication. This is a relatively new way of trying to deal with anxiety.\n\nIn the older, classical Chinese model of health care, anxiety is seen to be part of a person's overall energetic pattern. Thus, the well-being of the whole person is what is studied and strengthened. The qigong tradition of cultivating internal energy is devoted to this approach. \"Qi\" means \"energy\"; \"gong\" means \"work.\"\n\nSome practitioners believe these two approaches are poles apart. In fact, they can be bridged. There is nothing in this book that requires anyone to give up whatever medication, therapy, or self-help methods they are using to cope with their anxiety. Also, the energy work offered in this book can be used whether you have a diagnosed anxiety disorder or simply struggle with varying levels of day-to-day anxiety.\n\nQigong does not treat your feelings\u2014even your most fearful and difficult feelings\u2014as enemies. This centuries-old tradition is based on respecting a person's feelings, attitudes, and behaviors\u2014including their deepest psychological patterns\u2014as part of their total energy field. Working with that total field of energy, rather than splitting it up, is at the heart of this system of well-being.\n\nThis is where ancient wisdom and the most contemporary scientific studies meet. Both focus on the totality of being. Both emphasize the reality of each person's direct, personal experience. The practice of qigong presented in this workbook is completely compatible with those therapeutic practices that empower individuals to relate with physical and emotional pain in ways that see these as powerful aspects of living a full life.\n\nInterestingly, the word \"anxiety\" in the English language comes from a root word meaning \"I cannot breathe\" or \"I'm choking.\" One of the translations for the Chinese word \"qi\" is \"breath.\" Often, people having a panic attack are told to take several deep breaths. Imagine what it would be like if we were able to breathe in that relaxed, calming way each moment of our lives.\n\nHuman Being\n\nIn classical Chinese, a single character of two strokes is sometimes used to represent a person. This is the uppermost of these two characters. It represents a person walking. Encoded within these two brushstrokes is the deep understanding that life is a process. It is always in motion. It is always changing. It is always shifting, in the same way that our weight moves from foot to foot. Whenever we see this character, it tells us that life is moving forward.\n\nBelow it is the character for \"body.\" Taken together, these two ideograms express the notion of \"human being.\" In the classical Chinese tradition, it is considered a rare and precious opportunity to be born as a human and have both a human mind and a human body. We believe this potential is a gift. It gives us the ability to accomplish many wonderful things. Whatever storms we encounter, whatever difficulties we face, we believe we should make the best possible use of this extraordinary opportunity.\n\n## Understanding Your Energy Field\n\nEvery human being is a field of energy. Energy doesn't simply flow through us\u2014we _are_ energy. This was discovered by humanity's earliest natural scientists\u2014the ancient Chinese physicians who studied the mysteries of the human body and mind. It has also been discovered by the modern scientists who reveal the wonders of our subatomic existence.\n\nWhatever we experience occurs within that field of energy. Whether we experience anxiety, fear, and pain or deep well-being, joy, and freedom from pain, it is all an experience of energy.\n\nIn classical Chinese medicine\u2014humanity's oldest continuous tradition of health care\u2014one of the principal causes of pain and dysfunction is tension. We tend to think of anxiety and tension as a problem of mental well-being, but to the earliest Chinese physicians, mind and body were so intimately connected that mental tension could be locked into the body at its most subtle level.\n\nTheir findings have been validated by modern neurological research. We now know that tension shows up at the cellular level, obstructing the flow of oxygen and other nutrients, as well as preventing the elimination of dead or infected organisms. The inevitable result of anxiety and tension is pain, disorder, and disease.\n\nInner Relaxation\n\nInner relaxation is therefore not a luxury. It is the key to health. Understanding this, China's earliest natural scientists and medical practitioners worked patiently to discover systems of health care that would reduce the effects of anxiety and promote inner healing.\n\nReferences to these systems appear in the oldest and possibly most influential text in the history of medicine, _The Yellow Emperor's Classic of Internal Medicine_. It dates back more than two thousand years, and its wisdom certainly goes back even farther than that. It includes an early reference to the system of cultivating energy described in this book: the practice of \"standing still without changing\" originally described by the Taoist master Laozi. _The Yellow Emperor's Classic_ , which was written much later, includes this passage:\n\n_The Emperor Huang Ti said:_\n\n_I have heard that in ancient times there were Spiritual Beings. Their knowledge was vast and their understanding was profound._\n\n_They stood between heaven and earth, connecting the universe;_\n\n_They understood and were able to control both yin and yang, the two fundamental principles of nature._\n\n_They inhaled the vital essence of life;_\n\n_They remained unmoving in their spirit;_\n\n_Their muscles and flesh were as one._\n\n_This is the Tao, the Way, which you are seeking._\n\nThis describes the essence of the mind-body practices in this book, the way in which apparently simple positions and gestures regulate the flow of life-giving energy that circulates within and around the human body and its nervous system at all times.\n\nMy own master in Beijing, Professor Yu Yongnian (pronounced \"Yoo Yongnyan\"), who died at the age of ninety-four while this book was in production, was fond of the writings of Laozi and liked to quote that passage from the Yellow Emperor. Talking to me about what it feels like to do these mind-body practices, he once said, \"As you stand, you breathe in not only the air, but the vital essence of life, real energy. Real energy is what you are inhaling.\"\n\nOur Energy Flow\n\nThere is an interesting relationship between the impact of this type of exercise on our physiology and its effect on our nervous system. Tests show that qigong has a deep, calming effect, while heightening our alertness. Practitioners of qigong tend to breathe more deeply, resulting in increased oxygen consumption. Their heart rate is slower but more powerful, resulting in increased stroke volume. They test well for alertness, but with negligible muscle tension. Those who regularly practice this type of exercise can attest to their improved powers of concentration, coordination, and inner balance.\n\nSeen through the lens of the qigong tradition, these are all signs of the unobstructed flow of the energy that enables life to flourish. If the flow of energy is blocked, then, like stagnant water, energy putrefies. What blocks the flow of qi within our bodies is tension\u2014the subtle effect of mental strain on our muscles and nerves. Qigong practice relieves this tension and clears the blockages.\n\nWith the blockages cleared, the heightened flow of energy automatically strengthens and calms our nervous system, because this inner relaxation and power is our natural state. What many people call a \"normal life\" is, in fact, the anxiety and discomfort we experience as a result of living with diminished and obstructed energy.\n\nA very low level of energy leaves us feeling drained of vitality, but we are also susceptible to short rushes of intense energy. We feel unstable. We often wake up feeling like we haven't slept. We feel hassled by family life and work. Sometimes we push ourselves late into the night. Underneath it all is constant anxiety. We feel vulnerable, often protecting ourselves with aggression toward others. It is almost like the vulnerability we experience if our immune system is weakened. Our emotional life shows the same patterns. We are vulnerable to inner turbulence. We are susceptible to panic and despair. All of this is the effect of weak, unreliable energy.\n\nAlthough you may recognize some aspects of that picture in yourself, it is probably not an accurate portrait. Your energy level is probably not that low. Much of the time you probably function well and feel fit. But since you are interested in this book, there are probably some days when you have an underlying sense that all is not well. You may feel that your energy fluctuates and that sometimes you lack stamina. You may be easily irritated at times, stressed and upset. This points to a level of energy that, while manageable at times, is irregular at others. It might be like a stream making its way through difficult terrain. It can be blocked by sudden shifts in the landscape or debris. On an emotional level, you may at times feel overwhelmed, frustrated, and agitated.\n\nA fuller and more stable energy flow is less likely to be obstructed. Like a river, the current is constantly flowing. We have greater staying power and are more resilient. We still experience the full range of emotions\u2014in fact, we are likely to be even more alive in our emotional life\u2014but the ups and downs of anxiety, fear, and other agitated states do not overwhelm us. They are part of the current. Opening up to that experience and raising your natural energy to that level is the possibility that this book offers you.\n\n## Natural Growth\n\nAll authentic growth takes time. So does healing and the process of deep strengthening. It is like giving birth.\n\nIn the more than thirty years that I have been teaching these arts and treating people in the West, I have always had to tell people that nature takes time to form, nourish, and give birth to new life. I tell my students, \"You can't make a plant grow by tugging on it every day. You simply put it in good soil, give it just enough water and light, and let it grow. If you do that, it will grow naturally. That is its nature.\"\n\nWe are all living at a time in history when the world seems to have sped up a lot, and there is a general expectation that everything can happen quickly, even overnight. So it's possible that some people have started reading and working with this book in the hope that it might quickly dispel all of the anxieties that they are having.\n\nIn my experience, some of the exercises and suggestions in this book can have almost immediate effects. That is not guaranteed, of course, because everyone is different, and the conditions in which people live and work vary widely. But even if you experience some immediate relief from the symptoms of your anxiety, that is not the point of this book\u2014or the profound wisdom of the qigong tradition.\n\nThis book and the practice of qigong are about the cultivation and strengthening of the energetic core of your being. This is why my grand master, Wang Xiang Zhai (pronounced \"Wang Shang Jai\"; his story is told later, in the section \"The Lineage\"), told his students:\n\n_Preserving the heavenly wisdom, and maintaining the state of quiescence,_\n\n_You are ready to act in response to all possible situations._\n\nThis energy work is hidden, unseen, and it takes place in the depth of your being. It is like the growth of a baby within a mother's body. Therefore, I suggest that you work with this book over a period of nine months, and in the opening and closing parts of this book, I have included charts you can use to pause and reflect on your overall experience of using the exercises over the course of that period. I hope this will help you care for and nurture deep, authentic growth within your being.\n\nThe next pages introduce you to some fundamental practices that will help you see the beneficial impact of apparently simple stretching, breathing, and calming exercises on your nervous system. Please don't be put off by their simplicity. They put you in touch with the deep simplicity and power of your own energy as a human being.\n\n## Coming Up for Air\n\n\"I'm drowning in it!\" I hear people saying that all the time. They may be drowning in their work and the pressure of deadlines or in the overwhelming demands and conflicts in their lives. Or, like the student whose story I tell in part 3, \"I Am Always Tired and Anxious,\" they might dream of themselves or others drowning. Such is the power that anxiety can exert over us. Just like someone who is sinking in water and running out of breath, we need to come up for air.\n\nIn the art of qigong there is a simple way to come up for air. You can do it sitting or standing; however, unless you are unable to stand, I recommend standing up to do this. So, if you are sitting as you read this page and really want to try coming up for air, stand up to do it.\n\nIt is best to stand with your feet shoulder-width apart. Now breathe in for a silent count of six seconds, opening your arms to your sides. Breathe in through your nose only, if you can.\n\nAs you are breathing in, your posture has the feeling of welcoming someone or being delighted to see an old friend coming toward you. Your palms are open. Look forward at the world.\n\nYour chest naturally expands a little as your lungs fill with air and your arms open outward. You can see that when I do this, I bend backward very slightly and look upward a little. Please don't overdo this\u2014do it just enough to experience the feeling of opening up and breathing in.\n\nSometimes, if you are very anxious or tense, your upper body, shoulders, and neck can feel stiff, so this movement could feel difficult the first few times you make it. This is natural\u2014a sign of how much you will benefit from \"coming up for air.\"\n\nAfter breathing in for a silent count of six, breathe out, exhaling through your nose and mouth at the same time, and let your arms relax forward. It is good to feel your breath naturally leaving your torso through both your nose and mouth when you exhale. This clears stale air from your system and naturally enhances the depth of your next inhalation.\n\nYour upper body relaxes so that you are very slightly bent over. Relax into this until you have exhaled completely. Your arms and hands hang loosely at your sides, possibly just a little forward, almost as if they are falling off. You feel a sense of natural release in your chest as your muscles relax. Your head can tilt a little forward too.\n\nComing up for air once is good. It can save the life of someone who is drowning. I recommend you do this six times, without rushing through it.\n\nThere is no need to hold the breath or exaggerate the movement. This is a very natural and subtle way of changing the energetic pattern of your being, right on the spot. See the effect for yourself.\n\n## \"I Have Always Been a Worrier\"\n\nOne of my students is an accountant. When he first came to see me, he was clearly worried about something. I didn't ask him about that, however, and he simply joined my classes and started learning qigong.\n\nAs I got to know him better, he opened up a bit. He told me, \"I have always been a worrier. I found out some time ago that often the things that I worried about were quite trivial. But they consumed most of my mind. My worrying often hampered me from doing something about the actual problem. The only break I had from this cycle of worry was when I went to work, where I had other responsibilities placed on me, like meeting deadlines or producing accounts to a deadline. But then I worried about them as well.\"\n\nIt seemed from what he told me that he worried about almost everything in his life. One day he told me that the things that worried him stretched from problems with his car to concerns about his own health and the health of his family. But, he told me, things had started to change.\n\n\"Once I started doing qigong, things began to change,\" he said. \"While I was standing without moving, I began to see that the thoughts that were causing me anxiety were like a broken record. They just went round and round. I soon realized that I could quite easily have other thoughts. I could think about good times and places. This had a profound effect on me. After my qigong practice, I felt the things that were troubling me were not as insurmountable. In fact, they often seemed insignificant. When I was practicing qigong, I was a different person\u2014without all the problems.\"\n\nThere was a period of time when I didn't see him much. I found out that there had been a serious illness in his family. His sister had been diagnosed with pancreatic cancer and was given eighteen months to live. \"My qigong practice was particularly useful then,\" he told me. \"I used to see her every day. My daily qigong practice helped me stop worrying about her and accept the reality of the situation. That helped me, and I think it helped everyone else.\"\n\nHe said, \"This is how I look at situations now. If they cause me concern and I can do something about them, then, of course, I do what I can. Otherwise, I am able to accept what is happening.\"\n\n## Sitting Up\n\nThe energy patterns we associate with anxiety have several forms. They can shift from one to the other, creating whirlpools of energy that leave us often feeling quite scattered. When people get to this state, they often say, \"I need to collect myself.\" Whether you are feeling that or not, it is always helpful to gather your energy. It naturally reorganizes itself if it has a chance to settle\u2014even for a short time.\n\nSitting up, in the midst of everything, for a minute or so is one way to do this. It is simple but extremely powerful. Anxiety and anxiety-related energy patterns often cause energy to get disturbed and blocked in the upper body and head. This contributes to confusion, increased tension, and headaches. The simple practice of sitting up helps move that energy lower in the body, where it can be more naturally rebalanced. As you can see from the relaxed position shown here, you do not need to hold yourself in a rigid position.\n\nIf possible, sit on a chair that supports your back, which will enable energy to flow freely and relieve your torso and inner organs of pressure. Since the spinal cord runs between the brain and the base of the spine, having your back supported and relaxed helps the flow of energy.\n\nThe difference between simply straightening the spine for a minute and the qigong practice you are going to do here is that you will hold this relaxed, upright position for between one and two minutes.\n\nYou breathe naturally while sitting up. Your eyes are open, although you can half-close them if you wish. Your hands can rest anywhere. To begin with, you are simply straightening the spine in a simple, natural way and remaining in that position for a short period of time.\n\nYou can think anything you want. Do not try to constrain the natural movement of the mind\u2014that only increases tension.\n\nAfter you have practiced the first stage of sitting up, the next stage is to sit with your ankles crossed. Some people find this more comfortable, and you can feel that it produces a different sensation in your lower body and abdomen.\n\nAlthough you will have your feet flat on the floor again in the third stage, this intermediate step is very helpful for lower body relaxation and for helping you settle into the sitting posture.\n\nKeep your spine as upright and relaxed as possible, without strain. You can relax your chest with the following simple method. First, breathe in deeply, and then breathe out through your nose. Don't force the air out, but exhale as fully as possible. As you do this, let your chest sink down a little. Your shoulders will drop a little. After this first complete breath, breathe naturally. Your hands can rest loosely in your lap, on your thighs, or in any position that is comfortable.\n\nThis is a moment of simply sitting still. It provides a healthy pause in your day and is part of the subtle process of cultivating your natural energy. When you feel ready, move on to the third stage.\n\nIf you are comfortable with the previous two sitting postures, you can try this slightly more advanced stage. You might be tempted to go straight to this posture; you can, but I recommend following the three stages of sitting up. To some people, these subtle adjustments may seem like nothing at all. It can be hard to understand why they are important.\n\nIn this third position, your knees are wider apart. This improves the flow of energy throughout the whole body and from the earth up into the body (earth energy is stabilizing).\n\nYour open hands rest comfortably on your thighs. This helps relax your hands, arms, and upper body, which improves the flow of energy from your head, through your torso, and to your lower body and back. Your eyes are open or half-closed.\n\nSitting like this for a minute or two, up to a maximum of five minutes, is very helpful for stabilizing your energy.\n\nDo not confuse this practice with any form of meditation. You are not asked to concentrate on anything, to breathe in any particular way, to stop thinking, or to do anything in particular with your thoughts or emotions. If you feel anxious while doing this, please just allow that to happen while sitting in this relaxed way. This, in itself, is strengthening and stabilizing for your energy. Your anxiety will learn to trust that.\n\nIf you are feeling particularly anxious, you can place a glass of water (preferably one made of green or blue glass) on a low table slightly in front of you and gaze gently at that. Further suggestions about the use of water in this kind of exercise are included in the practice \"Looking into a Glass of Water\" in part 4.\n\nThe Golden Ball\n\nThe Chinese understanding of the human body is based on a deep connection with the natural power and patterns of our internal energy. These two characters point to an aspect that seems mysterious to many outsiders. The top character has four strokes of the brush and means \"internal\" or \"inner\"; the lower one, also with four strokes, is best translated as \"ball.\"\n\nIn the Taoist tradition, this is known as the \"golden ball.\" \"Golden\" here means \"not easy to find.\" It is rich. It is rare. The golden ball is also sometimes known as the \"dragon ball.\"\n\nThe golden ball is one of the main fields of study for qigong masters and students. It develops within us as we cultivate our inner strength. As we work with the golden ball, our nervous system is strengthened until it is able to embrace all disturbances, including anxiety, shock, and fear.\n\n## The Golden Ball\n\nThere are several ways you can imagine the golden ball, all of which are part of the mind-body work of the qigong tradition. One way is to imagine that your body is surrounded by a sphere of golden light. This sounds fanciful, but this ancient, poetic idea expresses a phenomenon that you can now test with technology that measures the strength of your aura\u2014the energetic field of the human body that extends slightly beyond the surface of the skin. We experience an aspect of this energetic \"glow\" when we feel the heat of a person next to us, even when he or she is not actually touching us.\n\nThere are other energetic patterns that the qigong tradition refers to when speaking about the golden ball. For example, there is a very strong concentration of energy in the lower abdomen, below the belly button. In classical Chinese medicine, this is known as the _dantien (tan tien)_ , pronounced \"dan dyen.\" It is sometimes called the \"sea of qi,\" but is more literally translated as the \"field of elixir.\" This can be thought of as a golden ball. A skilled practitioner learns how to develop the energy of this golden ball and can do this both internally and externally.\n\nThe stronger the energy field of the golden ball in your dantien, the higher the overall energy level of your whole being. A higher energy level is capable of accommodating a wide variation of energy patterns associated with our physical, mental, and emotional states. Our energy becomes like a great river, with many currents and countercurrents borne along with its flow.\n\nOne way of working with the golden ball is to imagine that it is a physical ball\u2014light, like a balloon, or heavy, like a bowling ball\u2014that we hold in a relaxed way between our open hands. We can hold it and move it as if it were actually there. After a while, it surprises many people to find that they are starting to feel the presence of energy between their palms. Not everyone feels it right away, of course\u2014everyone's experience is different. \"What's that?\" people ask me, when they start to feel their hands tingling a little, and they are worried that their imagination is getting the better of them. \"It's your golden ball,\" I tell them.\n\n## Resting the Golden Ball\n\nYou can practice resting the golden ball in your lap while sitting. This adds power to the stabilizing benefit of the sitting practice you did earlier, \"Sitting Up.\"\n\nSit comfortably and be aware of your body weight resting on a chair or stool. Rest your back against the chair to support your spine. Let the muscles of your torso relax. Let your arms hang loosely by your sides. This will help you drop and relax your shoulders.\n\nSlowly spread your elbows, moving them away from your body, as if small balloons were inflating between your elbows and your sides.\n\nThen slowly move your hands in front of your lower abdomen, as if gently taking hold of a large golden ball in front of your belly.\n\nIt is extremely important to keep your chest completely relaxed as you do this. Do not struggle to hold your arms up with the muscles of your upper torso. Your elbows and shoulders are completely relaxed as you hold the ball in front of your belly.\n\nThen lower your hands so that the backs of your hands rest lightly on your thighs, so the golden ball is gently resting in your lap.\n\nTry not to lower your head, either to look down or because the relaxation of your shoulders makes you lean forward. Remain relaxed, looking forward. Breathe naturally.\n\nAs you imagine your hands holding the golden ball, your palms and fingers naturally develop a soft curve, as if molding themselves to the surface of the large ball. Your fingers are not held tightly together; they are gently separated, spreading across the surface of the golden ball.\n\nKeep your chest and shoulders relaxed to avoid any strain in your upper body. Your qi will flow more freely if your upper body is relaxed.\n\nSometimes when people start to relax like this, they have a tendency to let their hands and fingers droop and to close their eyes. Please try to remain in the original position, even if you feel a little drowsy.\n\nYour eyes are open or half-closed. If you don't feel like looking straight ahead, you can let your gaze rest on the floor about six feet in front of you.\n\nYou are holding the golden ball in your lap all this time. This means that your palms and fingers are in place, gently open and curved along the surface of the ball. Do not close them too much (as if holding a baseball or an egg) or open them too much (so much that a large ball would roll off your lap). This subtle work with the golden ball helps the overall circulation of energy in your body, settling any disturbances in its flow.\n\n## \"Worrying About the Person in the White Coat\"\n\nYou might be asking yourself if it is possible that simply sitting still and resting an imaginary golden ball on your lap could actually have an effect on your anxiety. One of my senior students recently shared with me an account of the usefulness of \"Resting the Golden Ball.\" She used to work as a nurse in a major London hospital and now leads qigong classes for older people.\n\n\"It's well-known that people get anxious when they visit a health professional,\" she told me. \"They start worrying about the person in the white coat. They might have worrying symptoms. They may be going to receive test results and are fearful about the diagnosis. Even if they are going for a routine check, they may be worried that they won't pass it.\"\n\nShe told me that one of her students had experienced this kind of anxiety. She was eighty-five years old and had managed to remain healthy, active, and independent. But in the last year, her husband had died, and one of her daughters had moved to Australia. So she was experiencing major changes in her life. The last thing she wanted to have at that point was a health problem. But she was diagnosed with high blood pressure and was asked to come back for a further test. My student told me what happened:\n\nI could see she was visibly worried. I reminded her that the qigong postures she was learning in class could be done sitting down and could help relax her. At the next class, she reported back to me. While sitting in the waiting room, she decided to put her qigong training to the test. She sat up comfortably, placing her feet flat on the floor. Without making a fuss, she practiced \"Resting the Golden Ball.\" She held her hands in front of her belly, holding the invisible ball, with her hands gently positioned on her thighs. She decided to stay like that until she was called in for her appointment.\n\nWhen she eventually went in to see the doctor, she felt told me she felt remarkably relaxed and calm. She had her blood pressure test\u2014it was lower than it had ever been on any of her previous visits. The medical professionals were extremely surprised, and, as you can imagine, she was delighted. Now she tells me she is doing the seated posture whenever she feels anxious\u2014and she is feeling much more relaxed about everything, including going back to the doctor.\n\n## Rolling the Golden Ball\n\nHaving rested the golden ball on your lap for a little while, you can now advance to the next energy practice.\n\nLift the golden ball up, and hold it in front of your chest for a moment. Then slowly move the golden ball upward and away from your chest\u2014outward\u2014in a gentle curve. Complete the circle by bringing the ball down and in toward your belly.\n\nKeep your elbows loosely bent, so they move naturally with the movement of your hands. Breathe out as the large ball circles up and outward. Breathe in as the ball comes down and in toward your belly.\n\nAs you roll the ball away from you and breathe out, imagine your energy is sending away any anxiety, pain, and other difficulties. You are using the movement and your mental intention to create a flow of energy that unblocks and releases the inhibiting patterns created by fear, anxiety, and uneasiness. This motion feels steady and full, strong and heavy.\n\nIf you can, make at least a dozen circles with the golden ball.\n\n## \"Several Hundred Peanuts\"\n\nWhen I was a young man, I wanted to join the police. This was in Hong Kong when it was still under British rule; the police force was known as the Royal Hong Kong Constabulary. To be accepted, I first had to graduate from the cadet college.\n\nOne day, when I was going about my duties at the cadet college, my commanding officer came up to me, very angry with me. He accused me of something I hadn't done, which made me extremely upset. Because I didn't apologize, he threatened to dismiss me. It was nearly the end of my dream of joining the police. Luckily, he let me off with a warning.\n\nI was furious. I had been wrongly accused, but if I had talked back, I would certainly have been kicked out of the college. I went to see my master, from whom I was learning martial arts. He was in his eighties and was a very great teacher. I told him my story and asked him what to do. I was angry, because I felt I would now be on a list of badly behaved cadets in the college.\n\nMuch to my amazement, after listening to me, my master gave me a sack of unshelled peanuts. He told me to sit down and start throwing them into a huge pot some distance in front of me. \"Throw your anger away,\" he said. \"Then I'll tell you what to do.\"\n\nThere must have been several hundred peanuts in the sack. When I had thrown them all into the pot, he made me tea. Then he asked me, \"What do you think you should do?\"\n\nFor a moment, my mind was blank. I didn't know what he was talking about, and when he reminded me, the incident no longer seemed to bother me. He had used my body to calm my mind.\n\n## Your Inner Reservoir\n\nOur internal energy naturally gathers in a reservoir deep inside our lower abdomens. As we go about our busy lives, most of us are unaware of this inner source of strength, and it can become depleted. When that happens, we can feel anxious.\n\nQigong gives you a way of replenishing your energy, as well as developing your inner reservoir and preparing it for fresh energy. I recommend that you do this practice after any of the previous exercises. It is also helpful at any time, no matter what you are doing. It is a wonderfully calming exercise that soothes your nerves and restores your energy.\n\nWith a relaxed, slow motion, bring your right hand to rest over your lower abdomen. Rest your entire palm and fingers there, just below your navel.\n\nThen slowly bring your left hand to rest on top of your right hand. Rest your entire palm and fingers on the back of your right hand.\n\nIt's important to make this movement slowly, and to try to avoid hunching your shoulders or tensing your chest. Otherwise there will still be some tension in your upper body. You can prevent this by first letting your arms hang loosely by your sides, then slowly moving your forearms and hands up toward your belly without raising your elbows.\n\nFeel the sensations in your belly and hands as you sit still. Your hands are calm over your belly. Breathe naturally.\n\nRest like this for a minute.\n\n## Your Experience, Day or Night\n\nYou have now been introduced to several exercises that you can experiment with at any time of the day or night, whenever you are feeling anxious, or when you just want to take a break or refresh your energy. You can use the following chart to make notes on your experience. There are no right or wrong experiences; everyone's sensations are different. Sometimes you feel no sensations at all. It might be useful to review your responses in the chart \"A Quick Check\" at the end of the introduction. If you find this helpful, you can continue to keep notes for yourself on a separate notepad.\n\nWu Ji\n\nIt takes many strokes of the brush to write these two ancient characters, yet they represent the deepest simplicity of the universe. Their meaning is mysterious and hard to fathom, but they also are the name of one of the most practical of all exercises in the world of qigong.\n\nThe phrase is \"wu ji,\" pronounced \"woo chee\" (and sometimes spelled \"wu qi\"). \"Wu\"\u2014the upper character\u2014symbolizes what exists before anything emerges from nonexistence. Just like the clear and infinite sky that can give birth to endless clouds and abundant rain.\n\nBeneath it is \"ji,\" which means \"end.\" Not \"end\" in the sense of \"final\" but in the sense of that which is never reached. Taken together, these two timeless characters are symbols for an endless circle that contains and gives birth to everything.\n\n## Wu Ji: The Position of Primal Energy\n\nThere are said to be thousands of styles of qigong. All are forms of internal energy work. Some involve elegant and graceful movements. Some are based on the movements of animals and birds. All aim to cultivate the energy of the human being, and some of them also teach the practitioner how to work with the energy of the universe around us.\n\nAmong these thousands of possibilities\u2014handed down from parents to children over the centuries, and secretly from masters to their students\u2014there is one method that is distinguished from the rest, _zhan zhuang_ qigong (pronounced \"jam jong chee gung\"). It literally means \"standing like a tree energy work.\" It is famed for its most remarkable characteristic: it involves no movement.\n\nThis is the mysterious practice described in the verses of the _Tao Te Ching_ , one of humanity's greatest literary treasures (and said to be one of the most widely translated texts in history).\n\n_Standing alone and unchanging,_\n\n_One can observe every mystery,_\n\n_Present at every moment and ceaselessly continuing\u2014_\n\n_This is the gateway to indescribable miracles._\n\nMy late master, Professor Yu Yongnian, whose story is told later in \"The Lineage,\" was the world's leading authority on zhan zhuang qigong. In a lecture in London in 1998\u2014on one of his rare trips outside China\u2014Professor Yu said:\n\nThe history of health care in China stretches back into the distant past. As elsewhere in the Far East, a tradition has developed of taking care of one's own body and mind....In the West, although material life has developed enormously, there has not been a corresponding development in taking care of the human being. As a result, in today's society, people face enormous pressures both at work and at home. These create depression and many other mental and emotional difficulties. What people need is a way of looking after themselves, a way of taking care of their mental and physical well-being. What is needed is a method of self-healing....What is real relaxation? How would we know how to do that? Simply sleeping is not full relaxation. Nor is normal exercise, which deals almost entirely with our muscles. What is needed is a method of looking after ourselves that also relaxes our minds at the same time.\n\nThat method is the system that Professor Yu taught me\u2014zhan zhuang qigong, \"Standing Like a Tree.\"\n\nThe fundamental practice of wu ji is to align yourself, normally in a standing position, so that your body is centered, well-balanced, and upright\u2014and to remain in that position, without moving, while you progressively relax all your muscle groups, until the flow of energy within you and around you gives you what might be called a complete energy bath. In the following pages, you are introduced to the opening phases of practicing wu ji.\n\nIn my experience, this is a wonderful practice for both calming and strengthening the nervous system. Students of mine who have experienced high levels of anxiety, panic, and fear have found that this ancient\u2014and apparently simple\u2014practice transformed their experience. I have included a number of their stories throughout this book.\n\nAlmost nothing seems more natural than simply standing still, but not everyone can do this, strange as that might seem. We might be familiar with the military posture of standing at attention, but that does not normally have an internal relaxation practice associated with it. There is also postural adjustment that normally takes place when people practice wu ji. It can take time to see clearly what adjustments might be needed and then to allow them to take place naturally as you stand in the position. Some people also find standing still very startling, since there is normally so much commotion in their lives. For these reasons, I would like to take you through a step-by-step approach to wu ji, which will begin in this first part of the book and will deepen later.\n\n## Starting Wu Ji\n\nTo begin your practice of wu ji, you will be using a wall or something similar to support your back and help you relax. Many people unconsciously hold a lot of tension in their upper body musculature, because of the stress and anxiety in their lives. So using a support at the base of the spine will make a big difference in the depth of your relaxation.\n\nFind something solid to lean against, such as a wall. In the kitchen, you can stand with your lower back against the fridge or a counter. Or you may have a piece of furniture that works well for you.\n\nStand with your feet shoulder-width apart. Rest your backside gently against the wall or other support behind you.\n\nYour shoulders and chest are relaxed. Your breathing is natural. Stand still, fully at rest with your eyes open.\n\nYour feet take the weight of your body, like the base of a great pyramid, and the wall or counter you are leaning against stabilizes you.\n\nThis photo shows the position from the front. Your arms are loosely relaxed by your sides.\n\nAt this stage in your practice, you are still resting your backside against a support. You may feel that you are tilted slightly forward as a result of doing this, particularly if you're using a wall as a support, but using the support for your back is helpful for the initial\u2014and essential\u2014inner relaxation process. Later you will progress to doing the full practice without the support, but at this stage it is important to become thoroughly familiar with the inner work without being distracted by instability in your back.\n\nAs you stand, please go through the progressive relaxation described just below, moving downward through your body.\n\nBegin with a couple of minutes, if you can manage that. Then increase, day by day, until you are able to stand for five minutes or more, relaxed and motionless.\n\n## Calming Your Nervous System\n\nStanding still in wu ji helps calm your nervous system. This process of relaxation is vital to your health and well-being. If your nervous system is agitated, the flow of energy throughout your body is seriously impeded. As you stand in wu ji, follow the following inner relaxation sequence.\n\nInner relaxation is the secret of all health and well-being. As you practice the postures and movements in this book, your inner work is to relax\u2014whether you are standing still or moving. You can then carry this inner relaxation with you throughout your day, whatever you are doing.\n\nInner Release\n\nBring your attention to the area of your eyes. Release any tension you may be holding in the skin or muscles. If your eyes feel tired, imagine you are releasing the pressure in your eyeballs. Then bring your attention to your jaw and make sure you are not clenching your teeth. Relax your tongue.\n\nLet this sense of relaxation flow down your neck and over your shoulders. Allow each shoulder to sink, releasing any tightness. Your arms will naturally drop slightly as your relax. Release tension in your chest by breathing out gently.\n\nImagine you are standing under a warm shower. As the water cascades down your back, feel it washing away any tension in your back, as if dried mud were being softened and washed away by the water. Feel the weight of your body sinking down through the soles of your feet.\n\nFinally, imagine a fine golden cord supporting you from the top of your head. It stretches up into the heavens, gently suspending you. Remain still, repeating this inner relaxation as you stand.\n\n## \"Like a Mountain with the Winds Howling Past\"\n\nOne of my students is an independent management consultant. A few years ago he contracted pneumonia and lost more than twenty pounds in twelve days. He was off work for three months. But when he recovered and went back to work, he found he didn't have his earlier strength.\n\nHe has now been with me for several years and recently said, \"My strength and immunity have increased to a level far better than average for my age. I had a weak lower back. This would often cause problems, including a very stiff upper back. Over the years, these have improved enormously....I genuinely feel as if, while I am getting older, my spine is getting younger.\"\n\nI asked him about his internal energy and what effect it has on his high-pressure work in the business community.\n\nHe said, \"Energetically, I have plenty of energy, focus, and sharpness. When it comes to anxiety, crises come and go. I went through a lengthy divorce. A company that went bankrupt owed me a great deal of money. It was an extremely difficult time for me. No one would choose to have those experiences. It has made a considerable difference to have some point every day when I could be like a mountain with the winds howling past\u2014calm and unmoved. That tranquility in the eye of a storm has made a huge difference.\"\n\nThis student of mine also practices meditation. I asked him how he feels about qigong from that perspective. \"The open-awareness of this qigong practice has allowed me to become more open and less self-orientated, as it were,\" he said. \"There can be a huge divide between a state of calm you might experience in retreat and then your daily life. This divide does not arise the way we practice qigong. There is no split to be healed.\"\n\n## Standing and Rolling the Golden Ball\n\nYou were introduced earlier to the golden ball, including the exercise of rolling the golden ball while seated. Now that you have also been introduced to wu ji, you combine these two practices in this new exercise.\n\nBegin by standing in wu ji. Imagine you are holding the golden ball in front of your chest, between your palms. Stand still for a moment before beginning the next movement.\n\nSlowly circle the golden ball up and away from you, so that it moves in an arc outward, away from your chest. Complete the circle by lowering and bringing it in toward your belly. Keep your elbows loose, so that they move fluidly with the large circle of the golden ball.\n\nBreathe out as the golden ball arcs up and outward. Breathe in as the ball comes down and in toward your abdomen. As you move the ball through the space in front of you, it feels strong and heavy.\n\nImagine that with each exhalation and outward arc of the ball, you are expelling any agitation and disturbing energy from your body.\n\nTry making up to thirty circles with the golden ball.\n\n## Gathering Your Energy\n\nThis practice is an essential aspect of qigong exercise. It is a method of collecting your energy, directing it into its natural reservoir in the lower abdomen, and sealing it there. You can use it by itself, as a way of calming and centering your energy, or you can use it at the end of another exercise or a period of practice. It can also be used as an emergency treatment if you feel uncomfortable at any point in your qigong practice. In fact, you can use it as an on-the-spot method for getting some ground under your feet if you are overcome by strong emotion, experience intense pressure, or find yourself in a very difficult situation.\n\nStand with your feet shoulder-width apart. To begin, your arms are at your sides.\n\nOpen your hands and raise your arms out to the sides with your open palms facing forward. Continue to raise your arms until your hands start to come toward each other above your head. Breathe in as your arms circle up.\n\nBring your hands together so that your left palm is on top of the back of your right hand. This makes your arms into a complete circle, with your palms overlapped and facing downward. Your hands are directly over your head.\n\nThen move your hands forward so that they are poised to come down in front of your face and upper body. Slowly lower your hands down in front of your body. You feel as if you are pressing a large ball down into water. Breathe out as your arms press down.\n\nThe movement continues until your palms are in front of your belly. You can let your hands gently slide down, touching your torso, if you wish.\n\nMake three large circles like this, slowly and calmly.\n\nThen calmly bring your palms to rest over your abdomen. Your right hand is against your belly, your left hand on top of it.\n\nRest in stillness, breathing naturally for a minute.\n\n## \"To My Surprise\"\n\nSoon after we were married, my wife and I went from Hong Kong to London on our honeymoon. We planned to stay only a short time, but we have ended up living there for more than thirty years. Some of my earliest students are still coming to see me. One of them, a lawyer, held a senior position in a large legal department for a national professional body in the United Kingdom.\n\nJust after he retired, he told me what he had been going through. \"As the date of my retirement approached, I was waking up at four in the morning, worrying about the big changes in income and daily activity. I was often unable to get back to sleep.\"\n\nThis student is quite elegant and a real gentleman. He was honest about himself: \"I am by temperament quite an anxious person. My fight-or-flight reaction has a low threshold and is easily triggered, causing mild panic and anxiety. This can be accompanied by thoughts of worst-case scenarios. They take on a life of their own.\"\n\nI knew he was being honest with me when he spoke about his fears of retirement. \"The accumulating sleep deficit and anxiety became self-perpetuating,\" he told me.\n\nHe wanted me to know what he did, because he had applied lessons he learned from studying qigong with me. \"I found that if I got out of bed and stood in wu ji and then one other qigong posture for ten to fifteen minutes, my breathing became deeper and slower automatically. This led to a virtuous cycle, whereby my anxious thoughts and feelings diminished without any conscious effort on my part.\"\n\nHe said, \"I had always thought that practicing qigong would make me feel wide awake. To my surprise, I found that on returning to bed, I felt calm and relaxed, and I fell asleep without much difficulty. I found the transition to retirement much easier than I feared\u2014and I now have much more free time to practice qigong and enjoy life!\"\n\n## Experiencing More Energy\n\nIn this part of the book, you have been introduced to the standing practice of wu ji and the progressive relaxation sequence that accompanies it. You have also learned standing and rolling the golden ball, and the practice of gathering your energy. These take you to a different level of energy work and have a more profound effect on your nervous system. You can use the following chart to make a note of your experience. It is all the more important at this stage to know that there are no right or wrong experiences; everyone's sensations are different. If you find these notes helpful, you can continue to keep notes for yourself on a separate notepad.\n\nInner Strength\n\nThese two Chinese characters stand for \"inner strength.\" Literally, they mean \"heart strength\" or \"experience held in the human heart.\" They express a feeling that is close to what we mean by \"confidence,\" but they point to something deeper than that: not confidence in something, but confidence in anything.\n\nFor example, in the face of difficulty, you are not shocked. You do not feel helpless or worried. It is as if you have encountered this problem many times before and are thoroughly familiar with it. You already hold that experience in your heart. And not only are you familiar with this particular difficulty, but you also recollect what to do about it. Hence, you are able to draw on \"experience held in the human heart.\" It is a hidden, personal treasure.\n\n## Inner Strength\n\n\"When I stand,\" said my grand master Wang Xiang Zhai, \"the earth is in my hands. The universe is in my mind.\"\n\n\"You are free,\" he told his students. \"You are a great fire. If anything comes toward you, it will be consumed in the fire. If it does not approach the fire, it will not be burned. You are merely the fire. You remain where you are, content to be alight.\"\n\n\"You are the sea,\" he told them. \"Whatever anyone gives you, you can take. They can also take from you anything they want. The sea is vast; it can give up anything and still remain the sea. Like the sea, you are endless and unceasing. This is true freedom.\"\n\nIn Chinese culture, we have the tradition\u2014which once also flourished in the West\u2014of apprenticeship. Students learn from their masters by serving them; following their instructions; doing whatever is asked; and slowly, slowly being shaped by the wisdom and experience of the teachers. The teachers, in turn, are obliged to devote themselves wholeheartedly\u2014for the sake of their art\u2014to shaping the apprentices, and fashioning them into future masters. I have tried to keep that tradition alive, even though I am living in a very different culture now. Thus, I feel very happy to share with you these precious words of my grand master, hoping that you will grasp their profound meaning.\n\nIf you are suffering from tremendous anxiety and perhaps other emotions that overwhelm you at times, you might have turned to this book out of despair. I offer these words to you as a source of help. If you are willing to follow the simple instructions and practices in this book\u2014to sit and stand, following the guidance I share with you\u2014and perhaps, from time to time, to read the remarkable words of my grand master, then stamina, resilience, and great strength will naturally arise in your being. You cannot generate that strength through willpower and wishful thinking alone. But if you follow with care the practices in this book, you will be able, day by day, to create the fertile conditions in your body and mind that will make it possible for that inner strength to blossom within you, silently and beautifully.\n\n## Growing from Within\u2014A Nine-Month Journey\n\nPlease take a few moments to reflect on your current experience. You will be using your answers here as a point of reference nine months from now, when you complete the \"Inner Growth\" worksheet at the end of this workbook. The purpose of these two charts is to help your inner development as you use this workbook and experiment with qigong. Please remember that every person is unique. There is no set pattern and no target. Qigong will help you ride the energy of all your experience. The following questions are meant for your personal, private use. Please treat them as an intimate conversation with yourself.\n\n## Frequently Asked Questions\n\n**Why does sitting and standing still in the various positions make you feel refreshed and energized? If it generates high levels of energy, why does it calm the nervous system?**\n\nThese are questions I am often asked about qigong. The basis of understanding qigong is an appreciation of energy. From the earliest natural scientists of Chinese civilization down to the quantum physicists of our own era, there has been a tradition of understanding all phenomena as expressions of energy. In Chinese, this energy\u2014or life force\u2014is known as \"qi.\" It is the basis of human life. It is none other than the vast force field that gives birth to and sustains everything in the universe\u2014visible and invisible. The practice of qigong helps you feel this wonderful world of energy. You learn how to work with it and relax into it. It is like learning to float in a great river.\n\n**Do I need to follow a strict sequence of exercises?**\n\nNo. Please feel free to experiment with the various exercises and suggestions in part 1. Everyone is different. Some people will make an immediate connection with an exercise; others won't. Some days you feel like trying one particular exercise; the next day you may not feel that way. There is a daily routine you can follow if you wish; you can find it in part 3. Many people like to add this routine to their day, since it only takes about ten minutes. But please feel free to make your own experiments with the practices and recommendations I am offering you.\n\n**What happens if I feel worse?**\n\nSometimes when we start \"Coming Up for Air\" and begin letting go of accumulated tension, the mind and heart can start to race, like a wild horse let loose in a field. This is normal. If you feel it is too much, follow \"Gathering Your Energy\" in part 1.\n\n**I'm feeling good. Can I skip ahead?**\n\nYes. This book is designed so that you can use it however you wish. It does not follow a strict, linear sequence. However, sometimes the very nature of anxiety keeps us skipping along, and we are unable to settle. Please notice this. When you find an exercise or a suggestion that you are attracted to, then please continue to do it regularly. You can add others as well, as additional experiments, slowly building up your repertoire. If you ever feel you are losing your way, come back to the very beginning and start \"Coming Up for Air.\"\n\n## Opening Up, Relaxing, and Gathering Your Energy\n\nAnxiety often makes us close up. We feel burdened, and we unconsciously constrict our muscles, particularly in the upper body. This puts pressure on our chests, and our breathing becomes shallow. That is why the very first qigong practice in this book is \"Coming Up for Air.\" This starts the process of opening up.\n\nAs you stretch your arms back, your chest expands. Your lungs open. You take in more oxygen. At the same time, you are stretching your whole upper body. Then, as you come back to the resting position, with your arms hanging slightly forward, there is a natural relaxation. That feeling of relaxation is the key to deeper qigong practice.\n\nSome people worry that they feel a little light-headed when they first start to do qigong exercises. This is _not_ a sign of illness or that you are doing anything wrong. As your qigong practice deepens, you will naturally develop a much stronger connection with the energy of the earth. This literally grounds you. At first, however, as your internal energy wakes up, you might feel it floating upward. You might feel a little dizzy or nauseous, or you might have a sudden rush of emotion. In that case, please follow the exercise \"Gathering Your Energy.\" This is what I call \"Qigong First Aid.\"\n\nEnergy Field\n\nThese two characters express in Chinese what we call an \"energy field.\" You will recognize the upper character as \"qi,\" \"energy.\" Underneath it is the character \"qung\" (pronounced \"chung\"), which represents an open space. It could be a football field, a large garden, or a great plain. It could simply be translated as \"area\" or \"field.\" This space is not closed, like a room or a house. It is open. Because it is open, it can expand\u2014it can grow. It can extend in all directions.\n\nEvery human being is an energy field. Our power, our capacity, and our potential are unbounded.\nPart 2\n\nYour Personal Energy Field\n\n## Health and Anxiety\n\nHuge numbers of people are seeking a method of relaxation that will enable them to function effectively in their busy lives. They feel a persistent level of anxiety and tension in almost everything they do and almost everywhere they go. The times we are living through are sometimes referred to as \"the age of anxiety.\" A student of mine showed me a magazine article that described people as \"all jumpy and edgy and short of breath.\" It said that physicians write over ninety-four million prescriptions for antianxiety medication annually in the United States alone (Miller 2012). The _New York Times_ recently featured a series of seventy essays on anxiety; the newspaper introduced the series by saying, \"Nearly one in five Americans suffers from anxiety. For many it is not a disorder, but part of the human condition\" (\"Anxiety\" 2013).\n\nPeople who suffer from anxiety have a wide range of symptoms, including restlessness and a feeling of being constantly on edge. They get tired easily. They have difficulty concentrating; their minds go blank with worry. They are frequently irritated. They experience body pains from persistent tension in their muscles. They have difficulty falling asleep; their sleep is disturbed, with them often waking up in the middle of the night; and in the morning they wake up feeling that they haven't slept.\n\nWe tend to think of tension merely as a problem of mental attitude. But to the early Chinese physicians, mind and body were so intimately connected that mental tension could be locked into the body at its most subtle level. Their findings have been validated by modern neurological research. We now know that tension shows up at the cellular level, obstructing the flow of oxygen and other nutrients as well as preventing the elimination of dead or infected organisms. The inevitable result of tension is pain, disorder, and disease. Inner relaxation is therefore not a luxury. It is a key to health.\n\nIn the tradition of classical Chinese medicine, tension is one of the most common causes of pain and dysfunction. Tension blocks the flow of vital energy, qi, in the human body. When our qi is flowing smoothly and without obstruction, we experience health. When it is blocked or diminished, we experience pain and ill health. We also experience this pain and _dis-_ ease in our attitudes, emotions, and thoughts. In fact, the two are so deeply interrelated as to be inseparable.\n\nThe exercises and practices in this book have been developed over many generations to balance muscular activity and relaxation so that your internal energy is strengthened. The overall effect on your nervous system and your mind is to calm you down, freeing you from debilitating anxiety.\n\nThe Benefits\n\nCirculation is a key determinant of our health. If the flow of blood, lymph, and other fluids is restricted, the body begins to degenerate. Our tissue loses its softness, dries out, and becomes hard. Joints stiffen and muscles ache. Our immune system is weakened as the flow of immune cells in the intercellular fluid slows and is blocked.\n\nAnxiety contributes to this degeneration. The many pressures of daily life put our whole system on constant alert, even in sleep. We see the results around us every day: hypertension, migraines, asthma, all manner of aches and pains, menstrual difficulties, depression, and heart attacks. We tend to store the effects of anxiety in our musculature\u2014as everyone with a stiff neck knows.\n\nThe gentle movements and postures of qigong slowly alternate contraction and release of all the major pairs of flexor and extensor muscles, as well as many of the smaller, deeper muscles in the body that normally we rarely use. This work warms and massages your internal body tissue, thereby releasing and encouraging the flow of blood, lymph, and intercellular fluid.\n\nThe inner relaxation usually helps our posture. This is especially noticeable in older people who take up qigong after a period of relative inactivity. The spinal muscles elongate. The ligaments become more supple and the movement of the spinal column eases. From the point of view of the flow of vital energy within the body, this has a profoundly beneficial effect on the central nervous system and our consciousness.\n\nMost people find that their breathing (which is easily affected and made shallow by anxiety) becomes deeper, slower, and more powerful as a result of qigong practice. Over time, I have noticed that their breathing rate can decrease from ten to twenty breaths per minute to six to eight per minute. Their breathing becomes smoother as their diaphragm becomes stronger. In hospitals in China, qigong has been used in the treatment of tuberculosis and chronic ailments such as bronchial asthma.\n\nDigestive processes also benefit from qigong. Anxiety affects the digestive system, with frequent complaints including gastric pain, ulcers, and bowel problems. In some cases, people who had bowel movements only once every three days (or even less frequently) have reported having daily bowel movements after doing even basic qigong exercises.\n\nMy master, Professor Yu Yongnian, took an interest in the effect of qigong on the cardiovascular system. In one of his experiments at Ten Lu (Railway) Hospital in Beijing, he measured blood counts of qigong practitioners (using the system described in this book) before and after their daily practice. He found that their hemoglobin levels increased, as did their production of both white and red cells. Their blood protein levels also increased significantly, resulting in increased absorption of oxygen from the lungs and improved circulation of the oxygen-rich blood to the rest of the organs.\n\nYou might wonder what all this has to do with levels of anxiety and, more particularly, with reducing or getting rid of anxiety. An old Latin saying provides an answer: \"A healthy mind in a healthy body.\" This was said, in good schools, to be the goal of educating the whole person. It also points to the interdependence between mind and body: the health of the body affects the health of the mind; the health of the mind affects the health of the body. Just take the example of Professor Yu's findings at his hospital. Imagine how our brain and nervous system would function if bathed in a nurturing flow of oxygen-rich blood. Wouldn't we feel alert, energized, and resilient? Imagine the contrast: the blood flow to the brain is constricted by muscular tension in the neck, and our shallow breathing means we have low levels of oxygen in our blood cells. Wouldn't this lead to reduced mental energy, increased worry, and a tendency to get stuck?\n\nYou may have tried many methods to lessen your anxiety. In the the following chart, you are asked, \"If you are already trying to control your anxiety, what have you found most useful, and why do you think this has helped you?\" It is useful to remind yourself of any methods that you have found useful, even if briefly. Nothing in this book counteracts any other method you might have found useful. You can practice these qigong exercises while receiving any other form of treatment or while using any other self-help method that offers relief from anxiety.\n\n## Using the Mind\n\nThe mind that is the anxious is same mind that can help us work with our anxiety. Our natural mental awareness is an ally. Although qigong may be different from other approaches you have tried, it has one important thing in common with most other methods: awareness. Your own awareness of what is happening to you makes a big difference in the effectiveness of what you do. Therefore, throughout this book there are charts to help you explore and keep track of your own experience.\n\nWhether you have started using the qigong practices in part 1 already or are dipping into different parts of this book looking for help, you may find it helpful to review your own experience of your anxiety. If you completed \"A Quick Check\" at the end of the introduction, you may find that your answers there will help you think about your responses to the questions below.\n\n## \"You Need to Train Your Spirit\"\n\nWhen I was very young, my father allowed me to begin training with a master in our city, Hong Kong. This was my first experience of the wisdom and power that was later to become the subject of my life's work and study. I learned many important lessons about anxiety, tension, and relaxation. My training was based on the principle that physical and mental health was part of the same discipline of working with energy.\n\nThe fundamental rhythm of all energy fields is the interplay of yin and yang\u2014the two interrelated energies in the famous Taoist symbol that weaves together light and dark. We need both energies to be whole, balanced, and healthy human beings, living a good life in the midst of the constantly shifting energies of human society and nature. Anxiety can be both yin and yang. Anxiety is yin when it undermines our ability to act. It is yang when it makes us frantic and tense. Like all energies, it can shift according to conditions. Qigong training cultivates the ability to work with yin and yang in all their forms.\n\nThe training I received from my master involved both yang and yin energies. First he taught me the martial arts. I learned some of the toughest forms, like those practiced by the famous Shaolin monks. One day I injured myself in a strenuous session. When I recovered, my master told me, \"Now you must learn tai chi.\" This is the graceful system of slow, gentle movements, known worldwide. What he said upset me. I loved the speed and force of what he had trained me in already.\n\nMy master could sense my resistance. \"You need to train your spirit,\" he said. My hard training had been developing my yang energy. Now I needed to cultivate my yin energy. During my years of teaching in the West, I have found that many people think that the gentleness of yin energy makes it weak. It is not. Indeed, in the _Tao Te Ching_ , Laozi points out that water is more powerful than rock.\n\nMy master knew that I was becoming too tense. My spirit was hardening. He knew that I needed to cultivate another aspect of my being. Later, during my philosophical studies, I thought back to how he trained me and realized that he was instructing me in how to understand and work with yin and yang.\n\n## Holding On and Letting Go\n\nProbably one of the most irritating things that anyone can say to an anxious person is \"calm down,\" \"relax,\" or \"just let go.\" Not only do those statements almost always make the person feel worse, they are actually based on an inaccurate way of understanding how human beings function.\n\nBeing alive is a constant process of holding on and letting go. These are not opposite states; they are two aspects of the amazing, dynamic process of being alive. In every human being, this process takes place simultaneously, in some way, in every function of the body and mind. Think about your breathing, or your heart beating, or the process of digesting food and drink. Think about how you go to sleep and then wake, the way you walk, the many ways in which you communicate. All involve holding on and letting go, woven together in the intimate fabric of life.\n\nSo too with anxiety and all our emotions. They are part of this continuous, unfolding rhythm of life. Understanding this lies at the heart of the qigong tradition and is beautifully expressed in the _Tao Te Ching_.\n\n_Sometimes breathing is hard, sometimes it comes easily;_\n\n_Sometimes there is strength, and sometimes weakness_\n\nThis passage reminds us that disturbances of all kinds are part of life. They are episodic in nature and are not the whole of life. Another truth is hidden within these simple words. All turbulence arises and subsides naturally, through its own organic processes. This is as true for our emotional life as it is for the weather. This knowledge is embodied in qigong. Thus, I do not ask you to force yourself into any mental state or reject anything about your experience. Instead, I am offering you a way of fully embracing\u2014even expanding\u2014the fullness of your experience.\n\nThe exercises in the following pages enable you to embody this. Please experiment with them and then use the charts provided to track your own experience.\n\n## Arm Swinging\n\nThis simple movement releases the tension so often associated with anxiety, and it increases your flow of energy.\n\nStand with your feet pointing straight forward, shoulder-width apart. Look straight ahead. Make sure that your head is not slumped forward or tilted backward. Relax your shoulders and let your hands hang loosely by your sides.\n\nSwing your arms loosely forward, with the backs of your hands facing forward, and your fingers gently spread apart. Let your hands swing up until they are level with your chest. Do not swing them higher than your shoulders.\n\nThen swing your arms back until they naturally stop behind you. Keep them slightly away from your body. Do not bring your hands together behind your back; they should remain the same distance apart as when they were swinging forward.\n\nLet the natural momentum of the swing carry your arms forward and then backward. Work toward a spontaneous and fluid swing. Build up to swinging your arms like this for up to three minutes, if you can.\n\n## What's Happening?\n\nHow can something so simple as swinging your arms help you with your anxiety? At first sight, it might seem impossible. That's especially true if you are someone who thinks that anxiety is a state of mind that has nothing to do with your body\u2014and certainly not your arms!\n\nBut perhaps you also know that going to the gym or going for a run is good not only for your physical fitness but also for your mind. It can clear your mind and improve your overall mental stamina. Qigong works on the same principle, except that in qigong we are working with an even deeper, more subtle approach.\n\nThe way you stand in the arm-swinging exercise, with your feet flat on the ground and shoulder-width apart, is a traditional way of connecting with the energy of earth\u2014the huge power source that nurtures all life on earth. Strong, sustaining energy comes up into our bodies through our contact with the earth. We know that intuitively when we go for a walk in nature or lie down on the grass.\n\nWhen you begin to swing your arms loosely, several things start to happen naturally in your body. First, the gentle swinging encourages your shoulders, chest, and neck to relax. You can experiment with this by hunching your shoulders and tensing your neck and chest while swinging your arms. You'll immediately find out how difficult it is\u2014you naturally want to loosen up. Loosening up these sensitive areas of the upper body is extremely important. It improves the flow of blood and lymph to and from the brain.\n\nAs your arms begin to swing, you feel sensations in your hands and fingers. Please try to keep your fingers open a little bit, so you can feel the air moving between them as they swing. This also helps encourage one the main effects of the swinging: the increased flow of blood all along the arms and hands, right through to the fingertips. That, in turn, stimulates increased circulation throughout the body.\n\nHow does that help with your anxiety? A combination of greater relaxation and increased blood circulation sends powerful signals to your whole nervous system. Anxiety often tenses and constricts your being, and with arm swinging, your body is sending the opposite signal to your nervous system\u2014relax and open up.\n\nFrom the classical Chinese point of view, there is also another reason. If you look at a chart of the meridians (energy pathways) that run through the body, you will see that all the main meridians have important terminal points in the hands and fingers. There's a saying in Chinese medicine: \"the qi flows with the blood.\" So the increased circulation caused by the swinging is also stimulating the flow of qi throughout all the meridians, clearing stagnation and increasing the movement of energy.\n\nDoes it work better if you swing your arms higher and faster? No! Here is the difference between qigong practice and a tough workout in the gym. For energy to flow smoothly and powerfully in your body and to calm and rejuvenate your nervous system, relaxation is essential. So, hidden within this seemingly simple, ancient tradition is the potent combination of power and relaxation.\n\nIt may surprise you that arm swinging is widely used as a morning exercise in schools, factories, and businesses throughout China. It rouses people's energy and generally seems to put them in a good mood to start the day. Also, people seem to appreciate doing it with others. There is something about the simplicity and the rising energy of the exercise that feels good to share with a friend, your family, or a large group of schoolmates or colleagues.\n\nEven more surprising to you, perhaps, is that this same exercise is used in traditional qigong hospitals in China for the treatment of serious illnesses, including some cancers, sometimes in conjunction with herbal treatments. In a hospital setting, arm swinging as a therapeutic treatment is done as a supervised exercise, since the patients devote themselves to the exercise for long periods of time.\n\nThe potency of the exercise lies in the subtle combination of the standing posture, which promotes a deep connection with the earth; the enhanced circulation of blood and qi caused by the arm swinging; and the inner relaxation, which releases internal blockages in the body's energy flow.\n\nPlease try it for yourself. There is a chart ahead to help you keep track of your experience at different times of the day and in different moods. Combining your exercise with the work of self-observation can deepen your own experience of this art.\n\n## \"I No Longer Feel Tired\"\n\nOne day a very charming lady came to see me at my clinic in London. She was middle aged and well dressed and had a dignified manner. It turned out that she was the head teacher of a good school in the city. I could tell immediately, from the way she walked, that she was suffering from postural pain. The way we hold our bodies is so often linked to anxiety.\n\n\"I have had this pain for years,\" she told me. \"I have been to countless doctors. I have tried Western medicine, Eastern medicine, and all manner of alternative therapies.\"\n\nIn the Chinese medical tradition, particularly in the art of qigong healing, pain is almost always a sign that the natural flow of energy in the body is being blocked. The blockage and the pain may persist for years, but it does not necessarily take all that long to unblock the energy and restore its natural flow. Of course, it may take time to overcome the habit of tensing up, but in my experience, it is almost always possible to make a difference.\n\nHere is where understanding the mind-body relationship is so important. Just as mental stress can produce tension and pain in the body, so too can we use bodywork to relieve the mind of its repetitive patterns of anxiety.\n\nI asked my patient, the head teacher, to stand up and to swing her arms gently to and fro. She began, but within a few minutes I could see her habit of tensing up. Her shoulders were hunched, her elbows were not loose, and the swing was forced.\n\nYears of caring, of being on constant alert, of helping others with their troubles\u2014all wonderful human qualities\u2014can have this effect on our nervous system if we are not careful. But because these patterns arise from a warm human heart, there is always a pathway back to a more natural state of being at ease, even in the midst of all the challenges we face.\n\nI corrected her arm movements and let her continue swinging.\n\n\"Ah,\" she said suddenly after a few minutes, \"I can feel it. I feel the relaxation in my tendons. They are no longer aching. My body feels comfortable.\"\n\nI thought she would stop at that point. Much to my surprise, she kept on going. And then she smiled and said, \"I no longer feel tired.\"\n\n## What Is Your Experience?\n\nPart 2 introduces you to the apparently simple exercise of arm swinging. This is something that you are welcome to try at any time of the day or night, whenever you are feeling anxious or want to take a break, or if you want to refresh your energy. Many people are skeptical about the benefits of this exercise, because it seems too simple. How can something so easy, almost childlike, make a difference? There is only one way to find out: try it.\n\nPlease use the following chart to keep track of your experience. Everyone's sensations are different. Similarly, the effects can be different on different days and at different times of day. If you find this helpful, you can continue to keep notes for yourself on a separate notepad.\n\n## Opening the Curtains\n\nHere is another apparently simple exercise that can help you in the midst of your anxiety. It is similar to the practice in part 1, \"Coming Up for Air,\" but it has an even more powerful effect on your inner organs and nervous system. It is a quick way of overcoming the constricting effects of anxiety, nervousness, and fear, which can cause us to tighten up internally.\n\nStand with your feet pointing straight forward, shoulder-width apart.\n\nImagine you are about to open heavy curtains that are drawn closed in front of a large window in front of you.\n\nRaise your hands to eye level, palms facing outward, as if placing them against the fabric of the heavy curtains. You are going to spread the curtains open, not pull them. Breathe in.\n\nAs you breathe out, use your open palms to spread the heavy curtains apart. Because they are heavy, you can't move them too quickly. Open the curtains until your arms are out to your sides at around a forty-five-degree angle.\n\nOnce you are familiar with this motion, you can make it more powerful. As you open the curtains, lean forward a little to put some pressure on the balls of your feet. This presses on one of the most powerful acupuncture points in your body, the _yongquan (yung chuan)_ point. It is a key power spot for your connection with the energy of the earth.\n\nThen bring your hands back to the starting position. As you do this, stop leaning forward, stand flat on your feet, and breathe in. Repeat the movement three, six, or nine times. Then slowly lower your arms.\n\n## Your Opening Experience\n\n\"Opening the Curtains\" works with the energy pathways that run from the _yongquan_ point in the middle of each foot up to your fingertips. You many experience new sensations as you do this, and you may detect a change in your experience of anxiety or other disturbing emotions. You may find the following chart a helpful way to keep track of any effect this practice has on you, physically and mentally _._\n\nPlease remember that these experiments are personal to you. If you find this exercise helpful and want to work with it over time, you can continue to keep notes for yourself on a separate notepad.\n\n## \"You Don't Know How Tense You Are\"\n\nOne day my master said to me, \"You don't know how tense you are.\" I was shocked and even a little resentful to hear him say that. I was a devoted student, still very youthful, doing my best to follow his instructions and training hard in the martial arts. A lot of the fast, repetitive exercises that have come down to us over the centuries, like swinging your arms rapidly in large circles, are designed to loosen up your joints and really free up your energy. We started our classes with vigorous warm-ups like this. The looser our joints, the more power in our punches. Even when we were sparring with our fellow students, our master would call out, over and over again, \"Relax! Relax!\"\n\nOn the day that he cautioned me about how tense I was, he came over and stood beside me and told me to do something new. \"Clench your fists very tightly,\" he said, making his own hands into two tight fists. \"Squeeze more tightly. Really squeeze hard. Now twist your fists in toward your body. Don't stop squeezing. Harder. Harder.\"\n\nMy fingers were getting sore from the pressure. The tendons in my forearms were really feeling the stretch. My wrists were starting to ache. My hands were getting hot. Then quietly he said, \"Release.\" He opened his hands, and I opened mine.\n\nI still remember the rush of energy. And I also remember that I was suddenly more relaxed than I had ever been. Oddly enough, he had showed me how to relax by asking me to tense up. Now I was learning that I could go much further than I had thought possible.\n\n## Tensing and Relaxing\n\nYou are now going to experiment with an approach to dealing with inner tension and anxiety that is patterned on the story \"You Don't Know How Tense You Are.\" The point of that story and the point of this exercise is that we might be in such a state of constant muscular and nervous tension that we have forgotten what it feels like to be relaxed. We may think we are relaxing, but we don't know what that would really feel like. In this sequence we deliberately increase the tension in our arm muscles and then let go so that our nerves begin to experience the feeling of relaxation. The alternation helps our nervous system remember another way of being.\n\nThere are two forms of this method of tensing and relaxing that you will learn. The first one you will do while sitting or standing still. With the second, you will learn to add a powerful movement to it. Both forms involve a very powerful twist of both arms. You feel the effect coiling up both arms into your chest and shoulders, and you feel the corresponding release of tension as you uncoil.\n\nBegin with your arms by your sides. Then gently bring your hands toward your abdomen, as if you are holding a large ball in front of your belly. Your elbows should be held away from your torso, allowing space under your arms. Relax your chest and shoulders. Breathe in.\n\nThen rotate your forearms inward so that the backs of your hands face each other. Your palms are turned outward, facing away from the sides of your body. Your fingers are pointing away from you. Breathe out as you do this.\n\nAs your forearms turn inward, you may feel a slight twist in your upper arms. Pause for a second or two, holding this position, feeling the twist. You feel it coiling up your arms, into your chest and shoulders. Do not breathe for a second\u2014 wait to breathe in on the next movement.\n\nNow, breathe in and let your arms return to the starting position, with your hands holding the imaginary ball in front of your belly. You feel the release of tension as you uncoil.\n\nIf you are doing this sitting down, you can rest your hands in your lap for a moment before doing the movement again. Breathe naturally for a moment. Repeat this sequence up to three times, if you wish.\n\nOnce you have become familiar with this exercise, you can add a further element. When you have twisted your forearms inward so that your palms are facing away from your body, squeeze your hands into fists as tightly as you can. Breathe in as you do this. Hold tight for a few seconds.\n\nThen breathe out as you unclench your fists. Your forearms are still turned inward. Then, breathing in, let your arms return to the starting position, holding the imaginary ball in front of your chest.\n\n## Working with a Wall\n\nOnce you have tried \"Tensing and Relaxing,\" you can add movement to it. You will be working with a wall, bouncing off it as you make the same twisting movements with your arms.\n\nThere are many different methods of tensing and relaxing, but they all work on the same principle. In the classical Chinese tradition, the alternation of tensing and relaxing is a perfect example of the powerful interaction between yin and yang. Tensing up involves yang energy\u2014strong, focused, and hard. Relaxing involves yin energy\u2014yielding, flexible, and soft.\n\nIn this exercise, the process of tensing and relaxing is rapid. It is a quick way of sending signals to the nervous system. The sudden and repeated interaction of yin and yang has the power to interrupt any patterns of anxiety, fear, panic, or obsession that may have taken hold of the mind.\n\nTo begin, stand with your back a few inches away from a wall. You will be bouncing your upper back against the wall, so check your position to make sure that you can do that. You can try rocking backward and forward on the soles of your feet to make sure you feel balanced and stable when your upper back touches the wall and when you lean a little forward. Most people need to be quite close to the wall to be able to do this.\n\nHold the large imaginary ball in front of your belly, as you did in \"Tensing and Relaxing.\" Your elbows are held away from your torso, allowing space under your arms.\n\nNow, let your body fall backward until your upper back bounces against the wall. As soon as your upper back touches the wall, there is a natural recoil that propels you forward away from the wall.\n\nAs you bounce forward, rotate your forearms inward so that the backs of your hands face each other, as you did in \"Tensing and Relaxing.\" Breathe out as you bounce forward, twisting your arms.\n\nWhen your body has come naturally forward, let your arms uncoil, so that you are holding the imaginary ball. Breathe in as you do this.\n\nRepeat the movement three, six, or nine times. Do not exaggerate the movement. You are not slamming your back against the wall. You are bouncing with a natural rhythm.\n\nAs you become familiar with the movement, the exercise will be a natural process of rapid tensing and relaxing, synchronized with your breathing. Your breath will come in short, shallow bursts in time with the bouncing.\n\nYou can record your experience in the workbook charts. In particular, if you decide to do this exercise when you are feeling particularly anxious, try noticing your different states of mind before, during, and after this vigorous form of tensing and relaxing.\n\n## Clenching and Relaxing\n\nEveryone reacts differently to qigong exercises. Some people find that a slower process of tensing and relaxing is more helpful than a rapid exercise. Please don't make the mistake of thinking that one method is better for your than the other. Don't worry that being able to do these exercises either faster or slower indicates something about your physical health or the state of your nerves. Each person is different. You need to experiment to see what works best for you.\n\nHere is a slower method you can try, using your hands to clench and relax, synchronized with your breathing.\n\nSit or stand upright. This automatically releases pressure on your internal organs. Let your arms hang loosely by your sides for a second. Relax your shoulders.\n\nSlowly open your elbows outward, away from your body. The movement is led from the elbows, keeping the shoulders still. Your arms rest in an open, relaxed curve on either side of your body.\n\nBreathe in. Then make your hands into fists and clench them tightly while you hold your breath. Hold tightly for a few seconds.\n\nOpen your hands as you breathe out, and lower your hands to your sides. Breathe naturally. Repeat up to three times, if you wish.\n\n## Spreading and Relaxing\n\nThis exercise is the reverse of \"Clenching and Relaxing.\" Here you spread your fingers wide apart, feeling the powerful stretch, and then let them relax.\n\nLike all of these relaxation techniques, you can do this standing or sitting. If you work at a desk, this exercise can be easily adapted to your needs: you can conclude by resting your hands on your desk, rather than your lap, if that is more convenient for you.\n\nBegin by raising your arms to chest level as if holding a large imaginary ball between your arms. Imagine the weight of your arms being supported by balloons under your arms.\n\nThen turn your hands so that your thumbs point toward each other. Your palms face downward and your fingers point away from your body. Breathe in.\n\nThen, as you breathe out, stretch your fingers and thumbs as wide apart as possible. The stretch should be as vigorous as possible, combined with a feeling that each finger is also extending forward.\n\nKeep your fingers and thumbs spread apart for a few seconds, and hold your breath. Relax the spread and breathe out while gently lowering your hands to your lap or by your sides. Breathe naturally. Repeat up to three times if you wish.\n\n## \"Locked Up in Anxiety\"\n\nOften, when people come to my classes or come to my clinic for treatment, one of the first things that I notice about them is that they are very tense. I can see the tension in their faces, the way they hold their bodies, and the way they move.\n\nI usually give them some simple exercises to do. I tell them the movements are a warm-up, but actually, this is a way of trying to help them relax a little. Some of those exercises, like arm swinging, are in this book. But some people are so tense that even after these \"warm-ups,\" their muscles are still tight. I used to tell them to relax, but that didn't work either. I realized that they had become so accustomed to being tense that they had no way of knowing how tense they actually were. They were literally locked up in anxiety.\n\nI decided to try a method I learned from one of my masters. I asked them to tense up. For example, I would ask them to clench their fists until their knuckles went white. When I saw they could do this, I would tell them, \"Squeeze harder; squeeze really tight!\" They would put all their effort into this, even getting short of breath. When I saw they couldn't stand it any longer, suddenly I would tell them to relax and breathe out.\n\nI could see the look in their eyes. The contrast between the extreme clenching of the fists and the sudden release of that tension gave them an experience they had not had for years. In that moment, they caught a glimpse of what it might be like to relax.\n\nNot only did they begin to let go of tension in their bodies, but that physical sensation began to affect their minds. Their minds were following their bodies. They could feel inner pressure being released. The process of tensing and relaxing was unlocking their anxiety.\n\n## Tracking Tension Release\n\nIt's important to become familiar with how your body and nervous system feel when they experience a moment of relief from the constant tension that accompanies anxiety. Even though we have these moments, they are easily swept away by the current of anxiety.\n\nYou can use your body to help your nervous system experience and develop a sustainable memory of tension release. After suffering from anxiety for a long time, many people have to relearn what it feels like to be relaxed.\n\nEach day, for one week, take a few moments immediately after swinging your arms or using any of the tensing and relaxing exercises in this part. Use the following chart to note how you feel in each of these nine areas of your body: shoulders, neck, mouth, eyes, elbows, wrists, fingers, shoulder blades, and lower back.\n\nYou do not have to be making any particular kind of progress each day. Just take a moment to feel each of these areas of your body after you have completed the exercise. Stand still, see how you feel in each area, and jot down a word or two that expresses the feeling.\n\nSome people wonder why I don't ask them to keep track of how they feel before doing these exercises and then make a comparison afterward. There's a good reason for this. Anxiety makes us tense up, and when we are tense, our ability to feel what is happening in our bodies is blocked. So it is very difficult to actually feel how tense we are. But after a little tension release, our natural sensitivity begins to be restored. _That's_ what we need to remember and reacquaint ourselves with.\n\n## Standing and Opening\n\nThis posture is the next stage in your qigong training. There is a subtle difference between this position and \"Starting Wu Ji.\" If you follow the instructions carefully, you will start to notice the different effect it has on your internal energy and any feelings of anxiety you may have.\n\nBegin by standing in wu ji. Your feet are shoulder-width apart, with your knees gently unlocked, and your feet pointing straight forward. Your arms are loosely by your sides, with your hands gently opened.\n\nKeeping your shoulders and chest completely relaxed, slowly open your elbows out to the sides. The movement is gentle. You feel as if you are softly curving your arms away from your body.\n\nAs you stand in this position, you feel a space opening up between your chest and your upper arms. There is more space under your arms. Imagining a small balloon under each arm that supports the weight of your upper arm will help you relax the muscles in your chest, shoulders, and upper arms. Breathe naturally through your nose.\n\nTry remaining in this position, without moving, for a minute or two. Try not to move, allowing your internal energy to find its way throughout your body as it rests in stillness.\n\n## Calming the Mind and Body\n\nNow we can go deeper with the practice of wu ji, which you were introduced to in part 1. This practice of \"Standing Like a Tree\" works on your mind in two ways: by calming the mind and strengthening it. But to strengthen the mind, you first have to calm the mind. Therefore, your training begins with calming the mind.\n\nThis process is helpful for working with anxiety, because a calm mind is less likely to panic than an agitated one. In addition, the qigong exercises in this book will help you develop your mental power to the point that you can encourage your body to relax precisely at the moment that you are experiencing high levels of stress.\n\nThe inner process of calming the mind begins, in the qigong tradition, with the body. We learn to use the nerves to send signals to our central nervous system. There are nerve endings almost everywhere in our bodies\u2014in our organs, our muscles, and our skin. If we have a physical shock, like an accident, those nerves send emergency signals to our central nervous system, which swings into action. Often it responds with the famous fight-or-flight reaction. This is a good example of how our bodies, through their network of nerves, affect our central command systems. We all feel the impact of strong emotions and thoughts on our cardiovascular system, muscles, and internal organs. This interactive energetic system is what makes qigong work\u2014why we can use the body to calm the mind.\n\nThe very first instructions you received on standing still in part 1 included \"Calming Your Nervous System,\" a progressive relaxation technique that starts from the eyes and moves carefully down through the body so that your musculature slowly releases the tension it normally carries all day and night (even during sleep).\n\nAfter relaxing your face, you carry on an internal conversation with yourself. You invite your muscles to relax, group by group: _I want to relax in the area of my neck._ You tune in to the physical sensations, looking for tension, then feeling the muscle fibers relax. Sometimes you might get distracted, in which case you talk yourself silently through the process: _Now I am relaxing my neck_.... _I am feeling it relax_.... _Now it is relaxing more_.... _I can move on._ You can support the relaxation of your upper body by continuing the process down through your hips, your thighs, your knees, your ankles, the soles of your feet, and your toes.\n\nOnce you have completed a full downward sequence of relaxation, start again, beginning again with your eyes. Notice if you have tensed up in the interval. Use that awareness of tension to guide yourself in your relaxation practice. You can do this over and over again while you are standing.\n\nI have noticed that this aspect of inner relaxation while practicing qigong is often overlooked\u2014perhaps because it is invisible. This is the deep inner work that the great masters who practiced this art were able to accomplish. They were not training themselves in physical endurance. They were transforming themselves through profound energy work.\n\n## Resting While Standing Still\n\nDespite all your efforts to relax while standing still, you may feel you are becoming tired and tense. One way of relieving this is to reposition your hands so that the backs of your wrists rest in the small of your back, just above your hip bones. Stand like this for a few moments, letting your shoulders completely relax and your wrists sink into the soft flesh in the small of your back.\n\nThis is an excellent way of relieving tension in your arms and shoulders while still benefiting from the standing position and allowing a smooth and unobstructed flow of energy through your body. It is an excellent position to use any time you feel anxious, tense, or tired.\n\nStanding deepens our connection with the ever-present energy of the earth. Whether we are in a garden or on the top floor of an office building, we are still able to connect with the potent energy of the earth\u2014the pull of gravity and the earth's magnetism. This energy is always available to us, naturally influencing our state of mind, our emotions, and our inner stability.\n\n## Learning to Stay Relaxed\n\nAs you work with tension release, using the sensitivity of the body as your guide, you are also increasing your mind's ability to focus on relaxation, experience it, and appreciate it. In this way, although you are exerting yourself mentally and physically, your mind is learning to be active but less agitated. This is not something everyone learns quickly. If you are having trouble relaxing, here are some additional methods used in qigong practice that you may find helpful.\n\nWhile you are standing, you almost always have thoughts, images, and feelings passing through your mind. You hear and see the world around you. You may become even more aware of your mental activity. You may have some anxiety. Try making a mental note of what you are experiencing: _Now I am feeling anxious. Now I'm worrying about going to the bank._ You don't need to stop thinking and feeling in order to calm down and get over your anxiety. The mind is naturally active. Let these experiences run their natural course. If you feel that your qigong is having no effect at all, you can say to yourself, _This is a very anxious person standing still and continuing to feel anxious._ And continue to stand still. If you experience a lot of anxiety while standing, try listening to a soothing piece of music while you follow the progressive relaxation sequence down through your body.\n\n## Inner Powers\n\nIn the classical Chinese tradition, a simple smile is said to have a beautiful effect on your whole nervous system. It is said to release a nectar-like substance that bathes all your internal organs.\n\nTry this for yourself. Once you are standing like a tree, bring your attention to your face. Let a gentle smile form. Feel the effect on your eyes, cheeks, and lips. Try using your natural sensitivity to see if you can feel any internal effect on your musculature and organs as you smile gently.\n\nAt an international gathering of my students, Madame Wang Yuk Fong, the daughter of Grand Master Wang Xiang Zhai, told them to smile while holding the zhan zhuang qigong postures. \"This is the 'inner laughter,' \" she said. \"That inner happiness will continue through your whole life. The more you stand, the more comfortable you feel. Everything looks very soft, relaxed, and at ease. Yet there is immense power inside you, which\u2014channeled by the mind\u2014expresses itself in movement and power.\"\n\n## \"I Never Shook So Much in My Life\"\n\nOne of my students travels a great deal. One of his trips took him to the Far East, where he was invited to demonstrate the art of tai chi, which he studies with me. It would be part of a large charity event to raise funds for the victims of recent floods. He asked my permission, because he would be demonstrating the form that bears my name\u2014Lam Style Tai Chi. I was happy to support the flood victims and said he could go ahead, but I warned him, \"There will be about eight thousand people watching you.\"\n\nWhen he returned, he told me that the arena had been completely packed. Luckily, as an international guest, he had a VIP seat at the front and was thrilled to see many masters demonstrating their arts. After a couple of hours, the organizers told him that his turn was coming up, and he went to the locker room to change.\n\n\"Then something happened to me that I never experienced before in my life,\" he told me. \"I was completely overcome with intense anxiety and nervousness. I was all alone in that changing room, trembling. I never shook so much in my life. At that point, it was going to be totally impossible for me even to walk out into the arena, let alone demonstrate.\"\n\nHe said, \"Then, in the midst of that panic, I remembered the qigong you taught me, but I was so anxious, I couldn't remember anything other than the starting position. I placed my feet shoulder-width apart and stood still, holding the golden ball. I just stood there, shaking, until I noticed that something had begun to change. The extreme shaking started to ease off. My breathing was less shallow; it slowed down a bit. The panic was passing. I was tired, but my mind was clearing.\"\n\n\"So, then what did you do?\" I asked him.\n\nHe replied, \"I heard the announcer call my name. I walked slowly into the arena and did the Lam Style form.\"\n\n\"How did it go?\" I asked him.\n\n\"I did my best for you,\" he said, humbly.\n\n## Your Standing Experience\n\nYou have now been introduced to the qigong practice of \"Standing and Opening.\" This takes you to a deeper level of wu ji, with fresh suggestions on how to calm your mind and body while doing this. As you probably discovered from standing still like this even for a couple of minutes, the body and mind have many sensations. In this age of constant movement, agitation, and distraction, we are not accustomed to simply standing still. Your experience of doing this is important, and I encourage you to take note of what goes on inside you\u2014your physical sensations and all your many thoughts and emotions. Most people experience resistance and discomfort at first, because this is so new. Please keep track of that too. It is a very important part of the overall process of cultivating your internal energy, strength, and resilience. You can use the following chart for your personal notes.\n\n## \"Much More Freedom\"\n\nOne of my students is a music teacher and a professional pianist. She has been studying music most of her life and gives public performances quite frequently. In addition to being a student of qigong, she is also adept in the Alexander Technique of postural training and teaches that system. What she told me is of value to everyone who suffers from anxiety\u2014not just pianists or other performers.\n\nIt takes a lot of energy to teach or perform in public. I have several students who do this and professionals who have come to me for treatment. Part of the challenge they face is that students, audiences, and crowds tend to draw on the energetic strength of the teacher or performer. Being the focus of other people's attention simply sucks out one's own energy. In addition to that, there is the anxiety or nervousness that people naturally feel when having to perform in front of others.\n\n\"My breathing used to be much more tense,\" my student, the pianist, told me. \"Practicing qigong has had an enormously beneficial effect. I have noticed that my breathing is much freer when playing. I am therefore less anxious and less dependent on adrenaline and muscle tension when performing in front of an audience.\"\n\nShe said, \"The practice of qigong is very grounding. Pianists often practice for long hours with shoulder, neck, and back tension. Learning to hold qigong postures opens energy pathways and reduces muscle tension. As a performer, I've experienced much more freedom in mind and body. I was a very nervous and tense performer in my early career. I now feel much lighter (and even cheerful!) when performing. Being calmer and more centered as a result of doing this work also means that I experience a different sense of time and space.\"\n\nWhen I asked her about the relationship of anxiety to her ability to perform, she said, \"It's clear. Ask any performer. Anxiety often stops us from responding in the moment and playing with a sense of ease and flow. As a performer, I can say that qigong is invaluable. Standing and sitting in the different energy postures would be a boon to any instrumentalist.\"\n\n## Frequently Asked Questions\n\n**What do I do if I feel anxious while doing these exercises?**\n\nIf you are prone to anxiety, then you know from personal experience that you can feel anxious at any moment, under almost any conditions, and that often it may not even be clear what you are anxious about. Qigong will help you work with the energy of your anxiety and make you much more capable of holding all such experiences in a far stronger field of personal energy.\n\nThere are three things you can do if, while doing any of these exercises, you feel anxious:\n\n  * Continue with the exercise, whether it is one of the stationary postures or one of the movements. Simply allow yourself to be anxious while doing the practice. This is deep training for your nervous system.\n  * Stop what you are doing and try the exercise \"Working with a Wall,\" gently bouncing with your upper back off a wall while exhaling. Do this three, six, or nine times, depending on how you are feeling. Most people find that the effect of this practice is to clear their mind and break the grip that powerful emotions can have on our being. It is a nonviolent way of \"bringing you around.\"\n  * Experiment with an instruction that Grand Master Wang Xiang Zhai gave to his students. He told them to imagine that they were like a great ocean or a great fire. A great ocean has an extraordinary power to absorb and purify. A great fire burns up whatever comes in contact with it. Let your anxiety or any other personal discomfort simply be absorbed and purified in the ocean of your being, or let it enter the healing fire of your energy.\n\n**Is there a risk that I could feel even more anxious while standing still?**\n\nThis can happen, but it is not likely that your anxiety is actually increasing. You are simply noticing it more clearly. It is a little like what happens when you clean your home: you notice all kinds of things that were there all along. If you have this experience, then I recommend you follow the advice I have given in answer to the first question above.\n\n**Can I overdo these tension release exercises?**\n\nSome people's anxiety can lead them to overdo things. It is hard to overdo these tension and release exercises, because there is a natural wisdom in the body. You get tired and your muscles hurt if you overexert yourself. In my experience, you can trust this automatic safety valve in the body.\n\n## Running Out of Anxiety\n\nThe energy of anxiety naturally exhausts itself. This is referred to, in poetic fashion, in the _Tao Te Ching_.\n\n_High winds do not last all morning._\n\n_Heavy rain does not last all day._\n\nIt doesn't always feel like it's going to happen, but there comes a point when the intensity of any strong emotion diminishes. The tensing and relaxing exercises I show you in this part of the book naturally support that process of lessening your anxiety. This means, in my experience, that it is almost impossible to hurt yourself doing these exercises. In fact, what happens is that your muscles tire and you naturally stop. But something else happens internally. Your nervous system is automatically involved in the physical alternation between tension and relaxation. Energetically, you could say that the muscular work acts like a suction pump. It draws the energetic whirlpool of your anxiety into the rhythm of tensing and relaxing, which helps to dissipate it.\n\nThe White Cloud Carving\n\nThis mysterious rock carving is often called the \"Chart of the Inner Landscape\" (the _Neijing tu_ ). It stands on the grounds of the White Cloud Taoist Temple in Beijing. It is said to be based on a scroll found on Songshan, one of the five sacred mountains of China.\n\nThe carving shows a side view of the human body. The head is shown as nine mountain peaks. The eyes are the sun and the moon. The spinal cord takes the form of a powerful river. You can see a sacred pagoda in the region of the throat and lungs. There are bright stars in the area of the heart, a grove of trees in the belly, and a burning cauldron in the lower abdomen.\n\nIt is a complete way of understanding the fundamental structures and processes of a human being.\nPart 3\n\nDeveloping Your Inner Strength\n\n## Human Energy Patterns\n\nThe evocative images of the White Cloud Carving may seem fanciful. They appear to bear no relationship to charts of the human body that are familiar to most of us today. Even if the imaginative drawings were to correspond to the various organs of the body, there seems to be a relationship between them in this model that is different from what we imagine when we think of ourselves as modern people. It is striking that there are multiple sources of powerful activity, and it is not completely clear who is in charge: the mountain sages, the weaver, the oxherd, or the figures tending the cauldron? And what has the greatest influence: the stars, the mountain peaks, the burning fire, the long waterway, or the interplay of yin and yang?\n\nOne way of understanding this centuries-old carving\u2014which is a foundational text for practitioners of acupuncture and other systems of Chinese medicine\u2014is to look carefully at how it depicts the human being as an interactive, energetic system, involving many different power spots within an overall complex of forces, profoundly influenced by the natural world and the galaxies.\n\nThis tradition of experiencing and working with the total energy of the human being is the basis of the qigong practices in this book. Instead of a separation between mind, spirit, and body, all elements of our being are understood as energetic forces. Their patterns and rhythms exert powerful influences on each other and on our entire being. Thus, properly cultivated, their collective energy can be used for the benefit of the whole person.\n\nIt is helpful to understand this\u2014or to accept that it offers an alternative way of understanding the human being\u2014in order to make sense of the value of qigong in dealing with phenomena such as anxiety and fear. Qigong works with patterns of energy in and around the body. Often, like many forms of energy, such as electricity and magnetism, human energy and its movements are invisible. What we experience is its effect.\n\nThere are five key aspects of the energetic structure of the human body used in the qigong system. The body is organized around a central column or channel. This runs from the top of the head to the bottom of the torso. Blockages anywhere along this channel seriously disrupt the flow of energy throughout your system. Sitting and standing with your back naturally upright helps prevent or clear such blockages.\n\nAt the top of the central channel is the uppermost point of the head. In Chinese medicine, this point is known as _bai hui (ba hui)_ , \"the hundred meetings.\" It is sometimes called \"the thousand-petaled lotus.\" This is a particularly sensitive point, through which a person's energy field connects with the energy that surrounds them.\n\nAt the other end of the channel is the lowermost point of the torso. This is known as the _huiyin (hui yin)_ point, \"the meeting of yin.\" This is a powerful point of connection with the supportive energy of the earth, especially when a person is seated.\n\nIn the lower abdomen, just below the navel and a short distance into the body, is the energy reservoir known as the dantien, the \"sea of qi.\" It is the principal area of energy storage within the entire energetic system of the body.\n\nThe human being, however, does not exist in isolation. We are at the hub of the energetic forces that surround us at all times. You can imagine that we are standing in the circle of a limitless sphere. It represents the vast and limitless field of energy we call the universe. This circle is not closed or limited. Traditionally, a circle is used in Chinese calligraphy to show open space. It is the symbol of limitless possibilities and all possible configurations of energy, visible and invisible.\n\nBelow us is a long curve. It is the uppermost arc of another sphere, one below us\u2014the earth, the vast storehouse of energy that supports all life-forms on our planet and is our traveling home in space.\n\nSince each human being is an energy field, and since energy attracts energy, we act as magnets for the energy that surrounds us. Constant streams of energy come to us and pass through us. The energy from the cosmos tends to be more variable and unstable\u2014it is associated with creativity. The energy from the earth tends to be more constant and stable\u2014it is associated with power. Your qigong practice enables you to harmonize these two forces within your being.\n\nIt is within these spheres of energy that we experience life\u2014all that happens to us and the many emotions that are constantly arising like swirling weather patterns. Anxiety is like that. So is fear, thought, and the energy of worrying. Like all energetic patterns, our emotions have great power and great potential. They are part of the total energy system that we learn to work with as our qigong deepens. That's what we'll be working on in this part of the book.\n\nEarth\n\nThis calligraphy is the artistic expression of one of the most profound and defining terms in the whole of Chinese culture\u2014the Tao. Normally, \"Tao\" is translated as \"the Way.\" In the _Tao Te Ching_ , the term \"Tao\" also denotes \"earth,\" meaning everything that surrounds us. Nothing in our environment or our experience is excluded. The earth is all embracing. It gives rise to everything in life\u2014birth, pain, all that we encounter, death, and renewal. As we say in Chinese, it includes \"this way\" and \"that way.\"\n\nThe central image is of a head on top of a body. Underneath, the long brushstroke depicts a long, simple raft in an ever-flowing stream. To the left is a boatman, plying his craft through the earth's waterways.\n\n## Taking Root\n\nLet's take your wu ji practice to a deeper level. You do this by developing your connection with the earth. You can do this whether you are standing directly on the earth in a garden or park or on the floor in your home, even if you are in a high-rise apartment.\n\nBegin by standing in the position \"Starting Wu Ji,\" described in part 1. Please follow the relaxation sequence \"Calming Your Nervous System\" in part 1.\n\nAfter standing in this position for a couple of minutes, let your mind rest its attention on the soles of your feet. Imagine that you are like a tree with its roots reaching down into the earth. (It may help if someone reads this page to you as you practice.)\n\nA mighty tree is deeply rooted in the earth. Its foundations are unseen. It draws its power from the soil in which it grows. Rising upward to the heavens, the tree's great mass is still. Countless creatures move across its surface, but the sturdy trunk is calm. It is silent and unmoved, filled with energy.\n\nFrom its tiniest roots deep in the earth to the buds and blossoms far above, the hidden power of the tree is moving, day and night, and season after season. Feeling wind and rain and stretching toward the light, the veins of its innumerable leaves breathe freely in the energy of space.\n\nYou are growing in this stillness. Beneath you is the earth, a sphere of power, fertile and immense. It sustains all living beings. As you sink and soften, inwardly relaxed, the earth's great power gives you energy.\n\nAbove you, expanding without limit, is the galaxy in which we live. You are standing in a universe of vast, immeasurable power. As you stand, relaxed and vertical, your brain, your senses, and your vital organs open fully, like blossoms in the light. This is the experience we call \"Standing Like a Tree.\" It is deep training for your whole being\u2014your mind, your heart, your nerves, and your internal organs. Outwardly, you are unmoving. You are growing within.\n\nLike the play of water and light in the leaves, like birds making their nests in the branches, like small animals making their way up and down the trunk, all manner of experiences may come to you as you stand. The tree may sway in storms and strong winds, amid thunder and lightning, but it draws strength from its roots. Let your deep training embrace whatever thoughts and feelings, itches or anxieties arise. Don't move, don't scratch. Just stand like a tree, rooted in the earth.\n\n## Finding Your Center\n\nAnxiety and other forms of tension often cause us to be off-center. Over the years, we become accustomed to holding our bodies in slightly crooked or tilted postures. The result is that when we stand still, we may notice that we are not properly aligned.\n\nThat is why it is very helpful to check to see if you are standing straight while practicing wu ji or any of the other standing practices in this part of the book. As you practice, your inner gyroscope will adjust to the correct, healthy posture.\n\nYou can check your posture yourself in a mirror. You can also ask a friend or partner to look carefully at your posture from all sides. Check that you are evenly balanced on your feet, without leaning to one side. Check that your navel is in line with the central point at the top of your head. If you drew a line from the top of your head straight down to the ground, it would pass through your navel and touch the ground exactly at the middle point between your feet.\n\nMake sure that your weight is spread evenly on your feet\u2014that you are not leaning forward or backward. Relax your knees. You can check to see if they are relaxed by locking them tightly and then letting the muscles naturally unlock.\n\nAs you stand, you can imagine that the distribution of weight in your body is like that of a pyramid. Your feet are the base. Your head is the peak. The base, spread firmly on the earth, is wide, dark, and heavy. The peak, pointing toward the sky, is upright and light. The feeling of heaviness in your feet comes from connecting with the earth's energy. The lighter sensation in your head is your own awareness of contact with the energy of space.\n\nAs you realign yourself in this way, the energetic and structural readjustment of your body has a corresponding effect on your nervous system. The emotional patterns that have been locked into long-standing physical distortions\u2014or have caused the temporary stiffness and tension of a particular day\u2014are freed up as your energy moves without obstruction.\n\nMy grand master, Wang Xiang Zhai, told his students, \"Begin by standing still. This is the practice that refines the flow of energy throughout the human body. Zhan zhuang transforms the weak into the strong and makes the awkward agile. Stand without moving\u2014each cell will work and grow. Your blood will move at full capacity and bring your vital functions into harmony. You stand in stillness, apparently inert. Within your being, you are filled with strength.\"\n\n## Using the Rolling Pin\n\nIf you do a lot of work with a computer, you may have an antistatic mat under your feet at your desk. This is a mat that is normally connected to a grounding wire or an electrical socket to absorb static electricity, which might otherwise damage electronic components. These mats are also helpful in keeping you\u2014the computer user\u2014grounded, as static electricity flows through you down into the mat and is dispersed.\n\nThis is a contemporary application of an age-old principle, one that has always been part of the qigong tradition. We are creatures of the earth\u2014we need its energy, and we also need its protection.\n\nEarth energy and our connection with the earth also help dissipate the agitated and erratic emotional energy that we experience as worries, anxieties, and annoyances. In everyday conversation, we call this \"settling down.\"\n\nThe earth nurtures and supports all the life-forms we know. The earth's energy bursts forth in spring. It blossoms fully in summer. It yields a rich harvest in autumn. In winter, it regenerates. This energy is always available to us, naturally influencing our state of mind, our emotions, and our inner stability.\n\nWhether we are standing in a garden or on the top floor of an office building, we are still able to connect with the potent energy of the earth. We are always affected by the energetic pull of gravity and the earth's magnetism.\n\nOur standing practice automatically strengthens our connection with the earth.\n\nIf you are familiar with acupuncture or reflexology, you know that there is an extremely potent and sensitive point, just behind the ball of the foot, on the central axis of the foot. In Chinese, this is known as the _yongquan_ point, the \"bubbling spring.\" It is our principal energetic connection point to the earth. Massaging or pressing this point draws agitated energy in our nervous system downward and helps dissipate it.\n\nOne way you can do this is to place a rolling pin, or a pen, pencil, paintbrush, or chopstick under your foot. Roll it backward and forward for a few minutes under the ball and middle of your sole. You can do this sitting or standing. It is not only a refreshing massage. It grounds you, and you'll see the effect on your nervous system.\n\n## Holding the Golden Ball at Your Belly\n\nWorking with the golden ball in this posture takes your practice of qigong to a more advanced level. It helps your spine decompress, which is beneficial to your central nervous system. The posture also stimulates the circulation of blood and qi throughout the body. Both factors aid our resilience when we are affected by difficult situations, anxiety, and fear.\n\nBegin by standing in wu ji. Your feet should be shoulder-width apart, with your toes pointing forward.\n\nLower your bottom slightly, about one inch\u2014as if you were about to begin sitting down on a high stool behind you.\n\nYour knees will bend slightly, but do not let them come forward over your toes. You will feel your lower back relaxing, as if you were actually sitting on the high stool. Relax your belly.\n\nLet your hands gently rise up in front of your hips and position them as if you were holding a large golden ball in front of your belly. As you do this, relax your palms. They are gently curved, as if resting on the round surface of the golden ball.\n\nBreathe naturally. To begin with, try remaining in this relaxed position for two minutes.\n\n## Easing Your Way Through Tension\n\nMany people experience a lot of tension in the very center of their body. The constant anxiety, worry, and nervous pressure they feel causes them to be stiff in their lower back, hips, intestines, and pelvis. This anxious stiffness cuts off the flow of energy, splitting the body into two. The symptoms are lower backache; hip pain; digestive disorders, including constipation; and groin problems. You could say this is one of the bodily imprints of our anxiety.\n\nBy relaxing in this position, holding the golden ball in front of your belly, you may find that these problems ease. As this part of the body relaxes, it has a corresponding effect on the invisible patterns of nervousness, anxiety, and fear.\n\nWhen holding the golden ball in front of the belly, many people hunch their shoulders and strain their chest muscles. One way to eliminate this unnecessary tension is to imagine that there is a strap running around behind your neck and stretching down to your wrists. The strap takes all the weight of your arms and, at the same time, has the effect of lowering your shoulders and relaxing your chest.\n\nSome people like to do this exercise while actually holding a beach ball or balloon, just to get a clear idea of what it feels like and to get the position correct. This is perfectly fine if it helps you.\n\nIt is good to feel that there is something expansive and spacious about the position\u2014so please don't hold the ball too close to you.\n\nIt also helps to imagine that you have just had a cup of your favorite warm drink, and its warmth is spreading through your abdomen. You could also imagine the relaxing feeling of a back massage or of a heating pad warming your belly or your lower back. This is the sensation that naturally develops when you hold the golden ball in front of your belly. It is the way your inner energy naturally begins to flow when you allow yourself to relax even for a couple of moments.\n\n## Expanding the Golden Ball\n\nIf you practice holding the golden ball at your belly for some time, you may feel that it is starting to expand. Sometimes your hands naturally begin to open wider. This is a natural effect of the increased flow of energy.\n\nThese sensations point you in the direction of the next practice with the golden ball. You can advance to this next position after you feel you are thoroughly familiar with holding the golden ball at your belly and can do it in a relaxed way for several minutes.\n\nSlowly let your hands open in front of you as if the golden ball was growing wider and wider.\n\nYour elbows remain slightly bent, just as they were when you were holding the ball in front of your belly.\n\nYour palms remain gently curved around the surface of the ball. Your fingers remain gently spread apart.\n\nYour hands should remain the same distance in front of your body as they were when you were holding the ball in front of your belly. They are just moving apart sideways, and they continue to move apart until your fingertips are pointing toward the earth\u2014as you can see in this photo.\n\nYour shoulders are relaxed, and your chest is relaxed. You continue to rest your backside on a high stool, just as you were before. Breathe naturally.\n\nAs you stand in this position, the golden ball no longer feels like something you are holding. You feel that in this expansive position, you yourself _are_ the golden ball.\n\n## \"Three Teacups\"\n\nIn the classical culture of China\u2014and much of the East\u2014there is a story about three teacups. It may help you as you explore the unfamiliar practices in this book and wonder, _Is this actually helping with my anxiety?_\n\nThe story goes like this. A master is teaching his students. During a break, he arranges for them all to drink tea together. At a certain point, he places three teacups in a row. One is upside down. One is half full of tea. One is empty. He picks up the teapot and pours out the tea in three equal amounts. The tea spills all over the upside-down cup and runs over the table. The second cup, being already half full, can only hold a little more tea, and soon tea is spilling over onto the table. The third cup is empty. The master is able to fill it completely.\n\nThis is not a story about how much information a student can take in. It is about the state of mind of a student who wishes to learn from a master. The master is not simply adding data to the student's mind; the master is transforming the student's being.\n\nThe first cup symbolizes the student who is resistant to change. The second cup represents the skeptical student, who takes only from the master what he or she chooses, based what he or she already knows\u2014possibly from another teacher.\n\nThe third cup, empty of tea, depicts the being of the student who is completely willing to follow the master's instructions, not out of blind stupidity, but because he or she is open to profound change\u2014and thus to transformative learning.\n\nThis attitude of the empty cup does not mean that you should suspend your critical judgment. That is why there are so many charts in this book to help you keep track of your direct, personal experience. But it does mean that second-guessing yourself or the instructions can get in the way of simply doing the practice.\n\nSome students have a long-standing habit of mind, and they might be deeply reluctant to let go of it. Their brains might resist the possibility of change, no matter how much they want to be free of their old patterns. In their case, emptying the teacup just long enough to follow the teacher's instructions could be the first step to that freedom.\n\n## Golden Ball Energy Stance\n\nOnce you are comfortable holding the golden ball at your belly for several minutes, you can experiment with this more advanced posture. It involves a slightly different stance and a different distribution of your weight. This posture is marvelous for increasing the flow of energy generated by the golden ball practice.\n\nBegin holding the golden ball at your belly with your weight evenly distributed over both your feet.\n\nShift your weight to your left foot and place your right foot slightly forward and to the right. Point it slightly away from you, toward the right.\n\nNow adjust your weight so that you feel that 60 percent of your weight is on your left foot and 40 percent is on your right foot.\n\nYour knees are relaxed, but not bent. Breathe naturally and stand still in this position for a minute.\n\nThen return to your original position with your feet shoulder-width apart and with your weight spread evenly over both feet. Relax in this position for a moment, still holding the golden ball at your belly. If you find it difficult to continue holding the ball, you can slowly lower your arms to hang loosely by your sides.\n\nThen, move back into the Golden Ball Energy Stance, but this time with your left foot a little forward and pointing slightly toward the left. Place 60 percent of your weight on your right foot, 40 percent on your left. Breathe naturally, and stand still in this position for a minute.\n\nYour arms are still holding the golden ball in a relaxed way in front of your belly. Your palms are gently curved around the surface of the golden ball, with your fingers slightly spread apart.\n\nAlthough you have changed the position of your feet and shifted your weight, you are still facing forward. Your head is upright. You are still standing straight, holding the golden ball at your belly, but just with more of your weight being borne by your rear leg and foot.\n\nAdvancing in This Practice\n\nThis stance increases the flow of energy throughout your body. It also strengthens your legs and deepens your connection with earth energy. Once you are able to hold this position for several minutes, you can take the practice to a more advanced level.\n\nThe first level is simply holding the position for a longer time. See if you can increase the period of standing to five minutes on each side, separated by an interval of two minutes in which you stand with your weight evenly distributed.\n\nOnce you have reached that level, try changing the distribution of your weight so that 70 percent of it is on your rear foot and 30 percent on your front foot. This is more demanding, but it takes you to a higher level of practice. You will likely need to start with shorter periods of time on each side. Then gradually increase to five minutes on each side, separated by an interval of two minutes with your weight evenly distributed.\n\n## Deepening Your Experience\n\nIn this part of the book, you are being introduced to the deepening practices of connecting with earth energy, taking root, and holding the golden ball at your belly. These are part of the methods used in the qigong tradition to simultaneously develop a person's energy field and produce a state of deep inner relaxation.\n\nPlease use the following chart to keep track of your experience. How you respond to these practices will be unique to you. We all have different emotional, physical, and mental responses. The effects can be different on different days and at different times of day. If you find these notes helpful, you can continue to keep notes for yourself on a separate notepad.\n\nWake Up\n\nThe first of the ideograms in this calligraphy means \"suddenly,\" and the second, \"understand.\" We might translate it as \"Suddenly, I understand!\" or \"Wake-up call!\"\n\nThese brushstrokes express a critical moment, one we've all experienced. We might have read a sentence many times. One day, we read it with fresh eyes; we clap our hands and say, \"Oh, now I get it!\"\n\nWe can also understand this as the moment when we \"come to our senses,\" awakening from a dream or an illusion.\n\nWe are floundering in the quicksand of anxiety, fear, and despair. We need to be rescued. \"Wake up!\" is the help that the energy of qigong offers us.\n\n## \"I Learned This from My Father\"\n\nOne day when I was young I saw that my father was very angry. I didn't know what had upset him. Suddenly, there was a startling _crack_ as he clapped his hands together very powerfully. I was shocked and immediately cried out, \"What are you doing?\"\n\n\"I'm waking myself up!\" he told me, smiling.\n\n\"But you are angry and now you are hurting yourself,\" I protested. I was so worried for him.\n\n\"No, my son,\" he said. \"I _was_ angry. I could have hurt others. Now everyone is safe; no one has been hurt.\"\n\nOver the years, I came to understand his words\u2014and what he was doing. It is easy to get swept away on a strong current of thoughts and feelings. We lose ourselves, and we lose track of what's actually happening around us. We are obsessed with something that happened in the past, or we are frantically agitated about something that might happen in the future. We are no longer present. Such moments in which we lose our way can do great harm. They make us forgetful and unaware. We are prone to accidents. We can be flooded with feelings of fear and aggression. Many people who have committed acts of violence speak of being lost to themselves in that moment. That's the mental state my father was waking himself up from. The single clap of his hands discharged it. That's what he meant when he said, \"Now everyone is safe; no one has been hurt.\"\n\nI had a student once who was so angry that he feared he would take out his rage on other people. He was obsessed with computer car games and realized he wanted to kill the imaginary people in the other cars on the screen. He was terrified that his violent obsession would take hold of him when he was driving in traffic. I told him the story of my father, and he understood immediately. Later, he thanked me. He had used this simple method several times. \"You saved me\u2014and you saved lives.\"\n\nThis way of waking ourselves up is not the same as self-wounding or being angry with ourselves. This is more like the way some people slap their thighs and shout \"I got it!\" when they have just solved a difficult problem. It's a way of bringing ourselves back from a daydream\u2014or a waking nightmare of any overwhelming emotion, including anxiety.\n\n## Coming Back\n\nWhen we are overtaken by worry, anxiety, or fear, it can be like being trapped in a nightmare. Sometimes we get lost like that when we are sleeping, and we suddenly wake up in panic. In qigong, getting lost like this is said to result from an imbalance of yin and yang. The soothing effect of qigong practice helps rebalance these energies. If you experience this difficulty at night, please see the sections that follow: \"Overwhelmed, Lying Down,\" \"I'm Always Tired and Nervous,\" and \"In the Middle of the Night.\"\n\nBut many people get overwhelmed and end up \"lost\" during the day. They are awake, but their minds have been captured by a waking nightmare. Our nervous system instinctively knows that we need to wake ourselves up, but on a conscious level, we don't know what to do. The lesson my father taught to me when he clapped demonstrates something you can try.\n\nThere are many moments like this, when you intuitively recognize that you need to \"come back.\" Your thoughts might be spinning around and around, trapped in a circle. You might be getting angrier and angrier at someone. Your body might be completely tense with worry\u2014wracked by chest, neck, or back pain.\n\nIf you want to break the cycle, clap your hands once, sharply, right in front of you. Notice the effect. It will be different for every person. It is like giving yourself a little energy shock. For a split second, your mind clears.\n\nIf you are in an office or another environment where it will be too disturbing to others if you clap your hands, use the back of one hand (your hand can be open or loosely curled\u2014like you are holding an egg) to slap the palm of your other hand, which is quieter. It might hurt for a second or two, but the shock brings you back from your nightmare or paralysis.\n\nYou are not punishing yourself or trying to hurt yourself. You are just using your energy to clear your mind for a second and come back into the present moment.\n\nIf you don't want to be seen doing this\u2014or if it hurts too much to clap or slap your hands\u2014try using the outside of your thigh. Curl your hand, as if you're holding an egg. Then strike your thigh with the side of your curled hand like you were cracking open the shell of a boiled egg. There is an important meridian (energy pathway) that runs up the outside of the thigh, normally in line with the outside seam of your trousers or jeans. A quick slap or blow on the surface stimulates the energy flow in the meridian, sending a helpful \"wake up\" jolt to your whole system. You can try this while sitting at your desk. Normally this hardly makes a sound, and no one will see it.\n\n## Working with Your Breath\n\n\"Take a deep breath,\" people sometimes say to a friend or coworker who is in a panic, hoping to help him or her calm down. But what do you do if you are in a panic yourself and you want to cool down before things get out of hand?\n\nThere are ways you can work with your breathing that can be of long-term benefit\u2014and give you some on-the-spot relief. Feel free to experiment with these. Everyone is unique and responds differently.\n\nDantien Breathing\n\nMany people breathe by raising and opening their chest cavity. The energy of their breathing is in their upper body. If they are experiencing anxiety and are told to breathe deeply, they will tend to hunch their shoulders and tense their upper body and neck. This has a short-term effect, but it is not as deeply relaxing as it could be.\n\nIn the qigong tradition, the focus of our breathing is the dantien in the lower abdomen (see \"The Golden Ball\" in part 1). It is possible to reorient our breathing using the following simple exercise. It can be done standing, sitting, or lying down. Place your right palm over your abdomen, just below your navel; rest your left palm on the back of your right hand.\n\nAs you inhale through your nose, allow your abdomen to expand gently, as if the breath was coming all the way down into it. When you breathe out, let the exhalation start from your abdomen. To begin with, draw your belly in lightly, so that you feel it is gently squeezing the air out of your lungs. If you practice this over time, your breathing will return to the original deep pattern you were born with.\n\nDouble Breathing\n\nIf you feel the need to boost your energy vitality in the midst of a difficult situation or a fit of anxiety, try this special breathing. Breathe in twice through your nose; that is, make two short, quick inhalations in succession. Then breathe out through your mouth. You feel you are blowing stale air out of your body.\n\nWhen you inhale twice quickly, you are taking in double the oxygen you normally inhale. And when you exhale, blowing stale air out of your body, you are dispelling a larger volume of carbon dioxide from your lungs than you normally do. In the classical Chinese way of understanding the body, this is described as taking in double the qi and then clearing stale qi from your energy field.\n\nTry this double breathing six times. Don't keep on doing it: it is only for a quick boost. If you then want to continue with another practice, do some of the standing qigong exercises in this book.\n\n## Overwhelmed, Lying Down\n\nMany people who suffer from anxiety feel constantly exhausted. The nervous effort of coping with anxiety makes us want to lie down and rest. But often we don't find resting easy\u2014the mind continues to spin, and we end up simply carrying on worrying while lying down. You may find the following gentle qigong practice helpful when you are tired and want to lie down or when you are lying down but still feeling anxious. It both relaxes and energizes you. This is also a wonderful way to start your day.\n\nLie flat on your back, with your head resting on the floor or the bed. Your arms are to your sides, slightly away from your body, with your hands positioned so that your palms are facing upward. Please sink into this position for a minute.\n\nThen, keeping your heels on the floor or bed, stretch your toes up so that your feet are at right angles with your legs. You'll feel a little stretch in the backs of your legs.\n\nThen lift your head slightly off the floor or the bed and look down toward your raised toes. Keep your gaze on your feet while you breathe in and hold your breath for a few seconds. Then breathe out and completely relax. Allow your feet to return to their original, relaxed position, and let your head sink back down to rest on the floor or bed. Repeat this six times\u2014breathing in as you move into the position, out as you relax.\n\nOnce you are familiar with this basic movement, you can add a little energy work to further increase the flow of energy throughout your body. Imagine an energy circuit running from the ball of one foot to the other. From the ball of your left foot, let the energy travel up to the top of your head and then back down to the ball of your right foot.\n\nAt first, you will need to imagine this energy circuit. Gradually, you will feel it. Try to keep your mind attentive to this energy flow while you are gazing down at your feet, holding your breath. When you breathe out, lower your head, and relax, you may feel the energy continuing to flow, but there is no need to concentrate on it.\n\nThen scan your entire body to see how you are feeling. You may feel like you want to get up. You may want to sleep. If you want to repeat this exercise, please wait a few minutes before repeating it, and you should only do it a total of three times in any one day. If you have enough energy to want to keep doing qigong exercises, then I recommend you get up and do some of the standing practices in this book.\n\nIf you have trouble falling asleep, you can use this practice when you get into bed. If you suffer from insomnia, you can try this in the middle of the night. For many people, it has an almost immediate effect. But if you are still having trouble getting back to sleep, there is a more powerful method in the pages that follow, called \"In the Middle of the Night.\"\n\n## \"I Am Always Tired and Nervous\"\n\nOne of my students in London is a real estate agent. He told me, \"I am always tired and nervous.\" The nature of his work is mentally strenuous and physically demanding.\n\nSeveral years ago he woke up in the middle of the night, which was common for him at the time. He was always on edge and couldn't sleep through the night. He turned on the television and saw my show, _Stand Still\u2014Be Fit_. The next week, he came to see me, and he has continued with me ever since.\n\nHe still has a lot of pressure in his job. Qigong does not magically get rid of anything that could cause him anxiety. It is really about helping him have good energy that, like a horse carrying a rider, can support him no matter what he is going through. Recently he told me about a nightmare he had had.\n\nI believe this internal energy work helps even my subconscious. I dreamed that I was chased by zombies into a huge barn. The dream was so real; it was frightening. I was screaming and running faster and faster. Suddenly I saw a gap in the top of the wall near the ceiling. I took an enormous, superhuman jump, squeezed through the gap, landed on open ground, and was safe. But my dream continued, and in it the whole town was swept away by a huge tidal wave. People were drowning and screaming. Properties crumpled down. The wind was howling. Trees were blown down. My entire house, except the roof, was submerged under water. Yet I was safe, as I was squatting on top of the roof in a sea of water. I woke up. I was dry. And do you know why? It is because my body energy was strong and warm. At some level, I was aware of this, and even in that nightmare, I could not fall into the water. I believe this is the effect of this practice, deep inside me.\n\n## In the Middle of the Night\n\nYou may suffer from sleeplessness, either having trouble getting to sleep or waking up in the middle of the night with your mind racing. Some people are completely overtaken by anxieties. They know they need a good night's sleep, but they just can't get it. I have found that many people are helped by this simple exercise that can be done lying in bed or on the floor. It is a variation on the exercise \"Overwhelmed, Lying Down.\"\n\nLie flat on your back, your arms resting by your sides. Your head is resting on the floor or the bed. Keeping your heels on the floor or bed, pull your toes up so that your feet are at right angles to your legs. You feel the stretch in the backs of your legs.\n\nThen turn your palms toward the sides of your thighs. Clench your fists. Pull your fists in toward your wrists so that they curl toward you, and you feel a strong stretch in your wrists. Squeeze your fists as strongly as you can.\n\nAs you do this, lift your head slightly off the floor or the bed and look down toward your toes. Breathe in and hold your breath for a few seconds.\n\nBreathe out and completely relax. Let your feet return to their original, relaxed position, let your hands relax, and let your head sink back down.\n\nRepeat this six times\u2014breathing in as you move into the position, out as you relax.\n\n## Waking Up Your Experience\n\nAt this point in part 3 you have been experimenting with practices that aim to wake you up or bring you back from daydreams, nightmares, or difficult situations, often involving high levels of anxiety. These including \"Coming Back,\" \"Dantien Breathing,\" \"Double Breathing,\" and practices you can do while lying down or experiencing sleeplessness.\n\nPlease use the following chart to keep track of your experience. The effects can vary on different days and at different times of day. If you find this helpful, continue to keep notes for yourself on a separate notepad.\n\nFirst, please reflect on your overall experience of trying the \"wake up\" practices below from this part of the book. Have you noticed a general effect on your energy levels when you try these \"wake up\" practices?\n\nNow, please reflect on your experience of the individual practices listed below.\n\n## \"Their Eyes Become Brighter\"\n\nOne of my students is a former cardiac and intensive care nurse who now teaches zhan zhuang to senior citizens in London. She is a former senior administrator for Amnesty International.\n\nShe told me, \"I have reserves of energy and resilience that enable me to carry on even when feeling tired. These exercises have affected me mentally. Learning to stand still has a calming effect; it gives me more self-confidence and has reduced my tendency to be hypercritical, of myself and others. I used to suffer from nonspecific anxiety, which I experience far less now. Qigong has also helped me to understand and tolerate pain.\"\n\nI asked her about the experience of the senior citizens whom she now teaches.\n\nMany of them are over sixty years of age, and many have some form of disability. Apart from the physical benefits of qigong, they report that the practice has a calming and relaxing effect, and that it helps them to sleep\u2014they sometimes visualize doing these exercises or standing or sitting still while lying in bed. It reduces their underlying anxiety, which they often experience as muscular tension. As they become more familiar with their body, they learn to use only the muscles needed and to let go of the others.\n\nWhat I see in all my classes is the way, over time, the students' posture improves\u2014they are standing or sitting with a stronger back. But the effect is more than physical. Their eyes become brighter.\n\nSo many of them say that qigong has boosted their self-confidence and reduced the level of fear they might feel. One gentleman told me it felt that this had opened up a whole new world to him and had a profound effect on his spiritual life.\n\n## Your Daily Routine\n\nIf you would like to develop and strengthen your qigong practice, I highly recommend that you experiment with a short daily sequence. How you start your day has a deep influence on your physical, mental, and emotional state throughout the day's events. A short morning routine will help clear your internal energy pathways, so that your energy is naturally able to absorb any anxiety, fear, or other strong feelings you might have during the day.\n\nYou could experiment with the following qigong energy-strengthening sequence before you get carried away with the day's worries\u2014even before you reach for your phone or your computer. Of course, you may have many thoughts and worries while you do this, but just let all those thoughts and feelings arise as you do your qigong. The following sequence, drawn from the practices in this book, will take you less than ten minutes a day.\n\nNow you are ready to start your day.\n\n## Frequently Asked Questions\n\n**What do I do if I can't stand still in wu ji or hold the golden ball long enough?**\n\nPlease don't overdo any of these practices. You are not in a competition\u2014either with yourself or with others. Please don't strain yourself. At the beginning, you will experience many different sensations. Some people get impatient. Some people get bored. Some people experience aches and pains. Some people feel nothing. This is all part of the way qigong works on your nervous system. Set yourself a clear target for your practice. But if you feel you need to stop sooner, that's fine. You need to pay attention to your own experience and be guided by your own natural wisdom.\n\n**Is it normal to shake while doing qigong?**\n\nShaking is perfectly normal. As you hold the positions, your muscles slowly tire. The parts of the muscles that you normally use may reach the limits of their endurance and call on the remaining parts of the muscles to help. This is the point\u2014a stage beyond tension\u2014at which you may begin to tremble. The shaking may then become more vigorous. That's natural. Don't try to stop shaking. You will gradually find yourself shaking less and less. For long periods, you may not shake at all. Relax in the position as much as you can. If you relax as much as you can, you will have taken your body and mind to a new level of development.\n\n**Is something wrong if one side of my body feels different than the other?**\n\nNothing at all. As you stand, one leg may feel longer than the other. One finger can be hot; another, cold. In the qigong tradition, these are signs that your internal energy is beginning to rebalance itself. At first you notice imbalances of which you may have been completely unaware. Normally, your body naturally rebalances itself. The initial experience of asymmetry is a sign that your training is working.\n\n**Is something wrong if my feet feel odd?**\n\nThis is natural. Our lives are so busy that we rarely have time to be still. When we sit or stand still for a little while, our nervous systems react. Some people find that their feet feel hot when they first stand in qigong postures. Sometimes they get pins and needles. These are the natural sensations of your feet bearing the relaxed weight of your body and making a powerful connection to the energy of the earth.\n\n**Does qigong affect my digestion?**\n\nQigong improves the flow of energy throughout your entire body, including all your internal organs. Sometimes people are surprised to find their stomachs rumbling or to experience burping and flatulence when they are doing these exercises. This is all part of qigong waking up your system. I recommend that you drink a glass of warm water before doing qigong training, especially if you start a daily routine. Never overeat. Chew your food thoroughly before swallowing. That helps your digestion\u2014whether you are doing qigong or not.\n\n## Adding Another String to Your Bow\n\nA lot of people take their anxiety personally. Because it is disturbing and makes them feel so uncomfortable, they tend to think that they might be doing something wrong\u2014they attribute it to a personal weakness or failure. But there is a completely different way of thinking about it. When I talked with my own master about his life experience\u2014he died recently in his mid-nineties\u2014he talked about the power of stillness. He pointed out how little we have of that nowadays and how much the world had sped up around us. If you look at our modern lifestyle, you could say that we live in a culture of anxiety. This is constantly affecting our experiences, our attitudes, and our nervous system.\n\nWe are all looking for ways to cope with this constant tide of anxiety. So please don't think that it is somehow your fault that you feel this way. The question is how best to help our nervous systems cope with the circumstances in which we find ourselves. I like to think that learning qigong adds another string to our bow. It gives us one more option in life. If it helps us to calm down and cope better with our own experience of anxiety, then it will also have a positive effect on everyone we come in contact with. They will experience your energy like a light breeze on a hot day.\n\nSenses\n\nThese thirteen strokes form the character \"yi.\" This comes close to representing what we mean by the word \"senses.\" \"Yi\" embraces the dimension of thought, feeling, and emotion in the human person.\n\nThe notion of \"yi\" is central to the Chinese tradition of meditation. According to this tradition's understanding of consciousness, awareness is like a movie screen. It receives the passing imprint of whatever is projected onto it. The screen itself is not changed. It is ready to receive whatever is projected onto it. It remains fresh, like a sheet of white paper on which calligraphy can be drawn.\n\nFear, anxiety, and all forms of mental disturbance are like these projections on a screen. Whatever we experience through our senses is also displayed on the screen. If we are aware of the evanescent, yet potent, nature of the senses, we can use our senses to tame and vanquish our demons.\nPart 4\n\nUsing the Power of Your Senses\n\n## Using the Power of Your Senses\n\nWe can use the power and sensitivity of our senses to reorient and rebalance our nervous system. We can draw on their power to influence our mental activity and any related physical symptoms. We can use them to help us work with even the most deep-seated patterns conditioned by anxiety, fear, and other emotional states.\n\nSome people find this hard to imagine. They are so used to experiencing recurrent and troubling states of mind that they believe these are immutable. \"That's just the way I am,\" people tell me.\n\nThere is also a widespread assumption that the body and mind are somehow distinct entities\u2014two different systems. They may be connected and working in parallel, but they are separate. This has long been reflected in some medical systems, with different methodologies of diagnosis and care for the mind and for the body. Since the senses are associated with physical sensation\u2014and thus the body\u2014the notion that they could have a powerful, even integrated, influence on our mental state runs counter to this deep-seated assumption of separateness.\n\nThese days, around the world, the study and treatment of the mind and the body are coming ever closer together. A more integrated medical model is emerging. Interestingly, the way we talk about our mental states in casual conversation already points to an intuitive awareness of this unity. We frequently rely on metaphors from our bodies and our senses to describe our elusive emotions: \"I'm so angry I can't see straight.\" \"I'm all knotted up with anxiety.\" \"I feel uptight.\" \"I'm under such pressure.\" \"I'm stewing in this.\" \"I feel so raw.\" \"I'm so touched by your love.\" \"Your grief is palpable.\"\n\nIn the qigong tradition, the so-called mind-body split does not need to be healed. It never existed. The human being is understood to be a single, complex field of energy. This is very similar to the latest studies and models developed in the field of quantum physics. It could be said that some energetic patterns are more dense and others are more ethereal, but there is no separation between mind and matter. Therefore, we are a living totality, which is constantly manifesting and influenced by innumerable tendencies\u2014both those that arise within our own field of energy and those that influence us from the immeasurable energetic field of the universe.\n\nBoth our senses and our mental states are part of this vibrating field of awareness. A single thought can cause our hearts to skip a beat. A single breath of fresh air can open our minds. This is the fundamental understanding on which the experiments and suggestions in this part of the book are based. They are not theoretical notions or new age fantasies. They are based on centuries of careful study, meticulous attention to detail, and the deep care of the qigong tradition.\n\nThe simple suggestions in the following pages may seem lightweight. For example, to suggest that burning a stick of incense could ease the paralysis of fear may seem nonsensical\u2014and might even feel insulting to the person who is caught in the icy grip of terror. But from an energetic point of view, pain and mental suffering is almost inevitably associated with blocked energy\u2014in the same way that ice impedes the flow of a great river. There is often a congealed, stagnant quality to the experience of being distressed, overwhelmed, and in despair. Like anything that is frozen, these mental states respond instinctively to warmth, expansiveness, and a widened horizon of awareness. This is so natural that we don't even need incense. We experience it every time we breathe the fresh air of the countryside, hear the sound of a nearby stream, and see birds taking wing in the sunlight.\n\nUnderstanding this and being able to use the fundamental energy of our being in this way is a path to human empowerment. It allows us to use all the faculties and energies of our being and to make a healthy relationship with our own energy and the natural energy of the living world around us.\n\nSight\n\nThe Chinese character \"ngan\" represents \"sight\" or, more correctly, the process of seeing. The square on the left-hand side contains two dots, representing our eyes. In the classical Chinese tradition, seeing is understood to be a two-way process. It involves the mind's receptivity to light entering the eyes and transmitting its energetic messages to the consciousness. It also involves the radiant power of the mind reaching out beyond itself\u2014through the eyes\u2014to the world around us.\n\nIn the qigong tradition, the eye is a gateway for the energy that surrounds us. It is like a telescopic dish receiving the signals of the galaxy.\n\n## Looking into a Glass of Water\n\nWhether you are currently experiencing anxiety or are in a situation that might produce anxiety, you can experiment with using your visual sense to affect your nervous system. As with any experiment, you do not have to believe it will work. You can adopt the approach of a scientist, carefully examining your experience to see the results. It is important to remember that in the qigong tradition, your entire being is a field of energy. Therefore, whatever happens to any part of your body or mind has an influence on your entire energy field. It is like adding a drop of watercolor paint to a glass of water\u2014the pigment diffuses throughout the liquid.\n\nIn the calligraphy for \"senses\" that opens this part of the book, you are introduced to the notion that the mind, like a sheet of white calligraphy paper, absorbs and reflects back whatever is painted upon it. Thus, while we may feel that our anxieties or fears operate at a deep level of the mind (whereas our sensory experiences are superficial), our senses are influencing the mind, like the strokes of a brush on paper. Therefore, we can use the power of our sensory experience to draw the mind in any direction, at any moment. Please bear this in mind as you experiment with these practices.\n\nWe can use the sight of water in numerous ways to literally keep our stream of thought fluid. The colors blue and green, which are reminiscent of the sea, have a similar effect. These can also be combined. If you have a blue- or green-tinted glass, fill it with water and place it on a table near you, so you can gaze at it comfortably. Rest your attention on the glass of water for a minute or so. Feel free to let any thoughts or feelings arise in your mind and heart. We are not trying to combat or suppress your natural feelings. We are simply letting the water and the blue or green color register in your visual field of consciousness.\n\nThe great qigong master Wang Xiang Zhai advised his students to imagine that they themselves were like a vast ocean. The ocean's immense and deep waters are home to inconceivable life-forms. Its waters have unknown depths. It is capable of accepting, absorbing, and purifying whatever enters it. In the following chart, you can record your experience of using a glass of water to imagine yourself this way.\n\nHere is another experiment. In Chinese businesses, you often see decorative fish tanks, which are kept (in homes as well as places of work) because of their impact on the environment. They have a beneficial effect on everyone who spends time there. If you have an aquarium yourself, you can experiment with standing or sitting a short distance from it and gazing at the life taking place within the tank. You may find it especially helpful to pay attention to one fish in particular. Observe it closely, follow its movements, watch the subtle patterns of its breathing and the delicate movements of its fins as it glides or floats in the water. If you wish to deepen this practice, see if you can identify with the fish or imagine what it might be like to be this supple being moving and floating in the waters of the tank. Allow yourself to feel that you yourself are that very fish.\n\nYou might be quite skeptical about whether this would have any effect on you, particularly if you are a highly intelligent, rational, and productive person, coping with many demands on you. You might say to yourself, _How on earth will staring at a guppy make a difference when I should have left for the office ten minutes ago?_ I can't answer that question for you. I can only suggest that you give it a try.\n\nYou can also experiment with the soothing power of candlelight. It is not accidental that people in virtually all cultures enjoy the subtle influence of candlelight. We may associate it with romance, but it is by no means limited to that. If you are working at home, having a meal, spending time with friends or family, or even rushing about frantically, see what happens if you light a candle. You don't have to stop and stare at it. Its light will influence the environment around you, even if you barely have time to notice it. But since this is an experiment, see if you can notice the subtle soothing and warming influence of this silent beacon.\n\n## Working with Sight\n\nUse the following chart to record your impressions of the experiments for using the power of sight to influence your state of mind. Often their impact is subtle, but you may notice their influence in your body, your physical movements, your personal habits, the way you speak, and the things you notice in your environment.\n\nSound\n\nThis simple character is \"yih.\" It depicts the human ear. Like the eyes, our ears are a bridge between our nervous system and the world around us.\n\nWe tend to think of the ear as an organ of hearing, but it is more than that. Deep within its structure is the delicate mechanism that governs balance. In some medical systems, this balance is thought of as a physical process. But in the tradition of classical Chinese medicine, balance is both inner and outer, spiritual and physical. This book is about both, using the power of the body to influence the workings of the mind. True balance is being able not only to stand upright but also to absorb in one's being the shock of the unexpected and the destabilization of anxiety.\n\n## Listening to the Garden\n\nWe all know that sounds have a profound influence on our mind and nervous system. Everybody is familiar with the effect of changing the soundtrack behind film footage. If we are shown a group of people running down the street accompanied by tinkling piano music, our minds\u2014without any verbal explanation\u2014jump to the conclusion that this is a happy scene. If we see exactly the same footage accompanied by the slow, sonorous notes of a cello, most of us instantly believe that we are about to watch something dreadful happen. Similarly, if someone screams\u2014even without words\u2014we are likely to be startled and upset. If we walk into a room where soft, flowing music is being played, we are likely to feel our bodies relaxing.\n\nObviously, we cannot control everything we hear, especially living in an age of constant noise: machines beeping or humming, phones ringing or vibrating, cars honking. But we can influence our sonic field in ways that relax, nurture, and strengthen our nervous system. Sometimes we just have to pay attention to sounds that are all around us\u2014but may not be those we notice most.\n\nWhen you awaken in the morning, before turning on the radio, checking your smartphone, or jumping out of bed, listen for the sound of the sparrows or other small birds that might be outside your window, twittering in a nearby garden. Particularly if you have had a stressful night, slept fitfully, or are waking up to a stressful day, I recommend the experiment of letting the music of these songbirds, however faint or distant, into the space of your mind.\n\nListening to water, in its many forms, can also have a potent effect on our nervous system. This is why many people place a small fountain somewhere in their home and increasingly in their offices or work cubicles.\n\nYou can also experiment with music. Wherever I teach qigong, I always try to have a recording of soft instrumental music playing. This influences the nervous system of my students, calming their minds and relaxing them so that they are more able to benefit from the power of qigong practice. A very helpful experiment is to listen to different types of music as you practice the qigong exercises in parts 1, 2, and 3 of this book. Notice the influence of different kinds of music. Try to see which music has the deepest and most helpful effect on your practice.\n\nMany people like to listen to their favorite music as they work, an example of the ancient truth about the influence of sound that has made its way into the modern world.\n\n## Working with Sound\n\nYou can use the following chart to record your impressions of the experiments for using the power of sound to influence your state of mind. The impact of this experiment with your senses is subtle. You may notice its influence in your body, your physical movements, your personal habits, the way you speak, and how you feel about what takes place around you.\n\n## \"The Reset Button of My Mind\"\n\nOne of my students works for the local governing council of a district in London. For almost as long as I have known her, she has cycled to and from her work and to our qigong classes. Recently she told me about an incident that made a deep impression on her.\n\nI was cycling home from work one day and felt very upset and anxious about a number of problems that I was having at work. In fact, I wasn't paying much attention to the road or the cars as I cycled along. There were many things that I was worrying about. My mind was completely absorbed in my difficulties.\n\nSuddenly, I heard the sound of a bird. Even though I was cycling in traffic, somehow the call of that bird penetrated everything else, and I heard it very, very clearly. It was so startling that I pulled over to the side of the road, got off my bike, and stood on the pavement looking up into the trees to see if I could spot the bird.\n\nI was overcome with the wonder of the world. I must have stood there for quite a few moments, looking up into the trees and listening to the bird's song.\n\nI remember this incident vividly, because when I got back on my bike and carried on cycling home, my mood had completely changed. Before, I was weighed down with my worries, anxieties, and all the things that were going wrong. But when I got back on my bike, I felt so much lighter in my spirit. I didn't even go back to thinking about the problems I was so anxious about.\n\nIt's not that those problems weren't serious. They were real difficulties I was having at work. The difference was my attitude or, perhaps you could say, my inner approach. Before, it was as if my mind was being crushed by what I was facing. I was in a tunnel. Afterward, my mind felt refreshed, no longer dragged down. Later on, I had a lot more clarity and some really good ideas about what I could do at work the next day. It was as if hearing the sound of that bird somehow pressed the reset button of my mind.\n\nIt wasn't just that she heard the sound of the bird. She paid attention to it, stopped, and listened.\n\nSmell\n\nThis noble character is \"bei\"; it depicts the human nose. This is not easy to see, because the character comprises many different elements\u2014the human person, feeling, two legs, and two hands.\n\nThis complexity, almost like a portrait of the entire human being, reflects the incredible way in which smell is capable of affecting our entire being. It is said that of all our senses, smell has the most immediate effect on our brain. A single odor can affect our entire body\u2014a perfume enchants us; a stench makes us shudder.\n\nKnowing this, we can use the power of smell to affect our state of mind, sedate pain, and awaken our sleeping spirit.\n\n## Scenting the Sweet Smell\n\nSmells have a very powerful effect on our nervous system. In this day and age, we may think of smells as merely atmospheric background to our experience, but they are constantly at work on us. They can affect our entire central nervous system. They can seduce us. They can jolt us awake. They can heal us. They can cause collapse. In traditional Chinese medicine, if a person was severely ill and too weak to eat, he or she would be subjected to strong, putrid odors\u2014so repulsive that the person would vomit. The effect was to jump-start the digestive system. It was an extreme measure, but it could save a life. Thus, even a terrible smell can be used to good effect on our mental and physical state. But don't worry, I'm not going to suggest that you try that!\n\nThere are some simple and obvious things you can experiment with. In the Far East\u2014and increasingly in the West\u2014many people use incense to create a warm and inviting feeling in their homes. Many people associate incense with religious ritual, but it can be used in a completely secular way. If you are free from smoke allergies, try burning some sandalwood or another incense in the room where you work or at home or when you are spending time with friends or family. There is now a wide range of fragrances available. You can experiment with different scents.\n\nIf you don't like incense, try aromatic oils\u2014there is a wonderful variety now available. Or you can try an aromatic candle. These are used to dispel stale and offensive odors in the home, but they can also be used simply to diffuse a gentle scent throughout the environment. If you have trouble sleeping, try the well-known method of using a lavender sachet or eye pillows filled with lavender.\n\nAs the calligraphy for \"smell\" suggests, the power of our olfactory system is so strong that it affects our entire central nervous system, more than any other of our senses. So while your anxiety or fear may seem overwhelming, impenetrable, or unshakeable, it is not necessarily immune to the subtle, healing influence of a sweet smell.\n\n## Working with Smell\n\nYou can use the following chart to record your impressions of using the power of smell to influence your state of mind. Often, the impact of these experiments is very subtle, but you may notice their impact in your body, your physical movements, your personal habits, the way you speak, and how you experience things around you. Remember that even an unpleasant experience can shift your state of mind.\n\nTaste\n\nIf you look carefully at the brushstrokes of this character, \"tsih,\" you will see at the bottom an open mouth. Above it is a long horizontal line, which represents the outstretched arms of a human being. It is like a child spreading his or her hands on a table and reaching forward to grasp a bowl of soup.\n\nEvery taste has a subtle\u2014and sometimes not so subtle\u2014effect on the inner core of our being. Indeed, the mouth plays an extremely important role in the Chinese medical diagnostic system. There are three main methods of diagnosis: taking the pulse, examining the eyes, and studying the tongue. Meticulous observation of the tongue is said to be the closest we can get\u2014without invasive procedures\u2014to observing the internal organs of the body. A skilled practitioner can read not only the state of the body but also our emotions in the subtle patterns of our tongue.\n\n## Taking a Cup of Tea\n\nIn Chinese culture, we say you never forget the food you had at home as a child. The texture, aroma, and taste of that food remains imprinted in our memories for the rest of our lives. It is not only true of our childhood. Taste has a lingering effect on us. Our memories of important moments of our lives often include a recollection of something we tasted at that time.\n\nIn this modern era, when so many people experience stress and persistent anxiety in their lives, it is not surprising that we also witness an increase in the consumption of \"junk food.\" We seek comfort and emotional release in the tantalizing flavors that are carefully blended in these foods, even though they have low nutritional value. The fact that junk food is of poor quality should not distract us from an age-old truth\u2014the taste of something sweet calms the spirit. For example, in Chinese cuisine, we prepare sweet rice, sometimes known as \"sticky rice.\" It has both a savory and a sweet taste and is normally served wrapped in a lotus leaf. In traditional Chinese medicine, sweet flavors are beneficial to the spleen and the earth energy of the body. A delicious taste is said to make the mouth calm and happy. The sensation is received by the brain and transmitted to the central nervous system. Similarly, if we want to wake up, there is nothing like chili powder or a curry. Our energy surges just from the aroma, even before we experience the taste.\n\nSo what should you do if you want to use the power of taste in an intelligent way and not succumb to the unhealthy eating habits of the modern world? There is an ancient tradition\u2014famous in China, but also enjoyed throughout the rest of the world\u2014that I highly recommend: drinking tea. Drinking a cup of tea is a way of engaging all our senses in a sophisticated way. There's the shimmering surface of the tea in the cup, the aroma that arises from the steaming tea, the delightful sound of the tea being poured from the pot into the cup, the warmth of the cup in your hands, and the taste of the tea itself. Every culture has its own version of a tea ceremony. Whether it is in a Middle Eastern market, a stately home in Europe, a Japanese teahouse, an Indian garden, or a Western restaurant, tea works its magic.\n\nIn many parts of the world, a good cup of coffee\u2014carefully brewed from fine-quality beans\u2014has taken the place of tea. What's important to remember, whether your choice of beverage is tea, coffee, or anything else, is that there is more to this than sipping or gulping down a drink. It's a gathering together of many, many elements of the lives we lead together, the richness of the natural world, and our appreciation of being alive.\n\nThe next time anxiety gets the better of you, I suggest a moment of world culture\u2014the age-old cup of tea, a good cup of coffee, or whatever you would love to taste.\n\n## Working with Taste\n\nYou can use the following chart to record your impressions of the suggestions and experiments for using the power of taste to influence your state of mind. Often the impact of these experiments is very subtle, but you may notice their impact in your body, your personal habits, the way you speak, and how you relate to other people.\n\nTouch\n\nYou may recognize this character. We saw it in the calligraphy for \"human being.\" It means \"body\" and can be understood as \"touch.\" From the topknot at the uppermost tip of this ideogram through the ends of the brushstrokes that depict the legs and feet, it tells us that the entire human body is exquisitely sensitive.\n\nIt is not only the physical body that is capable of feeling. We are blessed with a nervous system that reaches throughout our entire physical structure and is endlessly responsive to the most delicate of influences.\n\nSometimes, especially when we are anxious, distressed, or fearful, this sensitivity can be mistaken for weakness; but it can also be one of our greatest strengths. It all depends on how we use it.\n\n## Feeling What Touches Us\n\nThe calligraphy for \"touch\" shows the whole body from head to toe. This was the classical, poetic way of expressing the tactile receptivity of our bodies. Oddly enough, we often underestimate the power of touch, yet we are constantly experiencing it across the entire surface of our skin.\n\nWe feel the air and the atmosphere around us on our head, hands, and other exposed parts of our bodies. We feel our clothes brushing against us, from the shoes or sandals on our feet to the collars of our shirts. We feel the objects we hold in our hands and often greet each other with a handshake, a gentle touch, or a hug. When we cry, we feel the tears on our cheeks and cover our eyes with our hands. We embrace someone we love. When we shower, we feel water cascading over our faces and bodies. A massage rejuvenates and relaxes us. This entire nonverbal field of experience is constantly sending messages to our nervous system, affecting our emotions and shaping our experience of life.\n\nIf you are anxious or overwhelmed by the pace of life, there are three ways in which the power of touch may help you find a greater sense of equilibrium, nurture, and inner delight. The first is to ask yourself how you feel. Normally, we think only of our mental and emotional state: \"I feel upset,\" \"I feel overwhelmed,\" \"I feel afraid.\" But while you are experiencing those emotions, your entire body is also sensing the environment around you, feeling the clothes you are wearing, the phone in your hand, the cup from which you are drinking. Try taking a moment to extend your consciousness to this entire field of sensation and notice the effect.\n\nPay attention to the clothes you are wearing. Do they feel delightful and comforting to your skin? They don't have to be expensive or elegant. Next time you choose something to wear, don't consider only how it looks. Ask yourself how it feels, because you may be spending the rest of your day in contact with it.\n\nWhenever we experience a sudden shock, like losing our balance or getting bad news, we instinctively reach for something to hold. We can use this innate body wisdom to help us. It's not accidental or a sign of human weakness that progressive clinics and health-care centers often have soft toys on the chairs and sofas of their waiting rooms. An adult holding a teddy bear does not indicate infantile regression. We all feel better when we can touch and hold something that instinctively comforts us. Similarly, many overworked office workers like to use a rubber stress ball. This helps not just because of the effect of the tensing and relaxing, but also because of the simple experience of holding something. You can be as creative as you like in experimenting with this.\n\n## Working with Touch\n\nYou can use the following chart to record your impressions of the suggestions and experiments for using the power of touch to influence your state of mind. Often the impact of these experiments is very subtle, and you may notice their impact in your body, your personal habits, the way you speak, and how you relate to other people.\n\n## Refreshment\n\nAll our senses come into play when we have a meal, a snack, or a cup of tea or coffee with family or a friend. It's an honorable tradition. It meets our deep human need for refreshment, both nourishing us and revitalizing our spirit.\n\nOften, when people are at work, they \"grab a quick coffee\" or eat a sandwich while working. This is part of our fast-food environment. The problem is that while we are just \"getting a bite to eat,\" we are not really refreshing ourselves.\n\nMore and more people these days eat their meals alone. This is affecting family life and reducing simple human contact. That essential contact was part of the ancient tradition of taking tea together\u2014a tradition that still lives on in some parts of the world.\n\nWhat was important then\u2014and is still true today\u2014is the full sensory experience. That's why I recommend turning a meal or a drink with a friend or family member into an experience that encompasses all of your senses, if possible. For example, when making coffee with a filter, smell the freshly ground coffee. Fill the filter and place it in the glass bowl. Watch the coffee dripping though. Savor the wonderful aroma and the rich taste and warmth of the coffee as you share it with a friend. Your mind will calm down. You heart will warm. This is not time wasted. It is life refreshed.\n\n## Frequently Asked Questions\n\n**How can working with the senses help with strong feelings of anxiety, even panic? How can these simple methods help calm the nervous system?**\n\nWe tend to think that our emotional experiences are more deeply ingrained in our beings than superficial sensory experiences _._ We associate our emotions with the mind and our sensory experiences with the body. In the qigong tradition, there is virtually no distinction between mind and body. Both are regarded as aspects of a single field of energy. Our energy field can be influenced just as easily and powerfully by our sensory experiences as by our emotions. However we direct our attention shapes our experience.\n\n**Do I need to follow these suggestions in any particular order?**\n\nNo. The suggestions in this part of the book offer you a rich menu of possibilities. Everyone's experience of working with his or her senses will be different, just as it will be different for an individual on different days and at different times of the day. Please feel completely free to try any or all of these suggestions. Don't force yourself to work with something you don't want to. Enjoy experimenting with something that interests you.\n\n**Could experimenting with this kind of work increase my sensations of anxiety or fear?**\n\nWhenever you experiment with something new, there is an element of the unknown. If you have a long-standing habit of discomfort with uncertainty or fear of the unpredictable, then even the slightest change can bring up disturbing emotions. That can make these experiments all the more potent and helpful to you. For example, if you feel overwhelmed by deadlines and are worried you might lose your job by spending a minute listening to songbirds in the early morning, that might actually be the most powerful and beneficial moment in which to do exactly that.\n\n**Can I combine this use of the senses with the other exercises in this book?**\n\nYes. All the suggestions for working with the senses can be combined with your practice of the qigong postures and movements. You can listen to music while standing still. You can feel your clothes and the air around you as you practice the movements. You can experiment with the scent of an aromatic oil. You can have a cup of tea or coffee before and after.\n\n## The Unseen Power of the Mind\n\nIn the many years in which I have taught in the West and treated people with a wide range of illnesses and injuries, I have noticed how skeptical they can be. Once I treated a gentleman who was suffering from back pain. He had been to many doctors and tried many expensive methods. He was lying on the floor when I treated him. After a few minutes of qigong massage, I told him he could stand up. He refused to believe me. I had to get his wife to help me convince him that his back pain was gone and that he could get up from the floor. He stood up and was able to move freely, without pain. Two days later, he called to say that he had booked himself into a clinic for a scan to find out whether my treatment had actually worked. He was going to spend thousands on it. I asked him, \"Are you going to do this because you are in pain?\" \"No,\" he said, \"because I am no longer in pain.\"\n\nThis skepticism affects our view of the world, but most importantly, our view of our own minds. We tend to have a fixed\u2014and rather negative\u2014view of the mind and of ourselves. How is it possible, we might ask, that we could use sight, sound, smell, taste, and touch to ease persistent anxiety? Is it simply a way of temporarily distracting us from our ongoing experience of pain?\n\nThe truth is that when we use our senses to their full extent, we are doing far more than distracting ourselves. We are using the deep resources of the mind\u2014indeed, the full power of our entire energy system. Our senses are amazingly powerful. We can learn to tap into that power and use it for transformative, personal healing.\n\nWhat makes this possible is something that is only now being validated by modern science, but which has been known to the qigong tradition for centuries. Physicians now refer to what is termed \"the neuroplasticity of the brain.\" Put simply, it means that our brain activity, our thought patterns, and our sense of who we are are all far more fluid and flexible than we ordinarily think. The mind is a renewable resource. Its power is its freshness. We can tap into that freshness\u2014instantly. That's what our senses are constantly inviting us to do.\n\nThe World Around Us\n\nIn the classical tradition, the four characters in this calligraphy are, from top to bottom: \"heaven,\" \"human,\" \"join,\" and \"one.\" This would conventionally be translated as \"heaven and humanity joined together.\" However, there is a deeper way of understanding this.\n\nThe character for \"heaven\" is made up of four simple strokes, which represent, within the overall ideogram itself, \"heaven,\" \"earth,\" and \"human being.\" Thus, compressed within the very first movements of the brush is an expression of the totality and inseparability of all existence. It is already joined together.\n\nNowadays we might say \"galaxy\" rather than \"heaven.\" That also would convey both the breathtaking complexity and the beautiful simplicity expressed by this brushwork.\nPart 5\n\nWorking with the Energy Around You\n\n## A Shimmering Net\n\nThe universe is a vast field of energy. The entire cosmos is a shimmering net of constant communication. Vibrations move through this net at inconceivable speed. Although our senses seem to perceive a world of separate forms, whatever we experience is, in fact, a pattern of energy.\n\nTo talk about the cosmos as an energetic spectrum is now commonplace in Western physics. But it was a fundamental tenet of Chinese science well before this millennium. The Chinese philosopher and naturalist Lu Yen wrote, \"All substance and form is energy. It is yin and yang, the motion of the sun, moon and stars, everything that emerges and dissolves. It is the clouds, the mist, the fog, and moisture. The heart of all living beings, all growth and development is energy.\"\n\nThe original Chinese phrase that expressed our relationship to the rest of the universe was _chih chung_ , sometimes translated as \"magnetism.\" Just over two thousand years ago, in what is known as the Later Han Empire, natural scientists in China were exploring energetic attractions of all sorts. They established specific magnetic polarities, and their work has been described as \"the greatest Chinese contribution to physics\" (Needham, Wang, and Robinson 1962, 229). By the Tang Dynasty (618\u2013906 CE), a compass was in use.\n\nBut the way in which _chih chung_ was being explored took it far beyond the kind of magnetism in a fridge magnet. The meticulous observations of these early pioneers, who were also forming the basis of Chinese medicine, revealed that the human body also had magnetic properties. Latter-day scientists confirmed this through the presence of iron in the blood. We now know that our bodies are part of and are affected by the earth's magnetic field and that each of us has an energetic field extending beyond the visible limits of the body.\n\nIn these and many other ways, the energetic imprint of each human being is determined by the configuration of the force field of the universe at any one moment. This is why part 5 of the book opens with a calligraphy whose meaning is \"the world around us.\" Is that world separate from us? Not at all. More accurately, we could say that we are that world\u2014or that it is us. That is what is so beautifully and simply expressed by the four brushstrokes of the opening character of the calligraphy, which means \"heaven\" or \"galaxy.\" Compressed within those strokes is the profound realization that heaven, earth, and humanity are one. We ourselves are the near and distant galaxies and the celestial rhythms that embrace us. This is the inner experience of qigong.\n\n## \"This Immense, Living Field\"\n\nMy sons are always showing me fascinating items they have found on the Internet. One day they showed me images from the Hubble Telescope, the most powerful telescope yet developed. Astronomers had trained it on a patch of sky that seemed completely empty, with no visible planets or stars. They said this patch of dark sky was no larger than a grain of sand held out at arm's length.\n\nWhat the scientists found in that little point in space was the light from over three thousand galaxies. The telescope was receiving photons that had been traveling for more than thirteen billion years to reach Earth. The astronomers said that what they saw was \"one of the most profound and humbling images in all of human history.\" Each one of those galaxies, which showed up on the telescope's receptor as little dots of light, contained hundreds of billions of stars. Later, they found another patch with ten thousand galaxies. They called it the Ultra Deep Field (\"The Hubble Ultra Deep Field in 3D\" 2009).\n\nAs I watched these images, they called to mind some of the oldest texts in the spiritual legacy of the East: magnificent Buddhist sutras describing infinite world systems with radiant light streaming throughout the cosmos. These poetic descriptions are now seen to be remarkably consistent with the unending discoveries of modern science (Ricard and Trinh 2001).\n\nSometimes, contemplating the vastness of the universe can make us feel anxious. We may feel small and of no value compared to the cosmos. It can fill us with loneliness and dread, reminding us of our fragility. But there is another way of looking at this. Not one in which we feel intimidated by the universe, but one in which we feel part of this immense, living field of sparkling energy.\n\nIn the Taoist tradition, which I have studied extensively, the human body is understood to be a small galaxy. The whole point of the inner practices of that tradition, which gave birth to qigong, is to connect each of our small galaxies with the rest of the universe, with the immense energy of the incalculable number of galaxies that surround us.\n\nThat is part of what we are able to do through our qigong practice. We train our own energy field in order to strengthen our sensitivity to and our connection with the vast energy field in which we live.\n\n## Environmental Energy\n\nPlaces and environments are also fields of energy. You can sense this. Everyone has feelings about particular rooms and buildings. Those feelings are more than passing emotional states. You may be feeling anxious or suspicious about a location for a good reason. Our natural sensitivity tells us about the energetic qualities of locations. One place may feel welcoming, secure, or inspiring; another may feel threatening. Often, the feelings are health warnings. You just know a certain place is not a good one for you to be in.\n\nIn every moment of our lives we are picking up these kinds of messages. This shows that we are profoundly interconnected with our environments and that the energies that surround us shape our lives and our futures. Some messages warn us that if we sit in an exposed area, we may be approached from the rear without warning. Some tell us that our sleep may be disturbed if there is an open window over our bed. Others tell us where it feels best to have an intimate conversation.\n\nWorking with our own energy and the energy of our lives is the common experience of all humanity as we learn how best to live on this earth. It is our common home in space. Yet every person, every family, every home, every office, every day, and every place is different. Understanding and feeling this enables us to work with the energy of each situation as a fresh experience\u2014and therefore with a spirit that is alert, accommodating, and spontaneous.\n\nQigong heightens our ability to sense the world around us. We become more alert, but we also feel more emotionally sensitive. You may be surprised by new feelings as you look at your home, your neighborhood, and your office with increased sensitivity. Another level of your natural intelligence is waking up.\n\nKeen observation is important. That is why this is a workbook, with many charts that encourage you to reflect on your personal experience of qigong practice. There are questions about your physical and emotional experience, and many open-ended questions that simply invite you to ask yourself how you are reacting to the practices and exercises. This, too, is part of developing your sensitivity and appreciating it.\n\nAs that awareness develops on a personal level, it develops correspondingly on an environmental level. It's like having a new pair of glasses. You see so many things so much more clearly. You also become more curious about your own experience, about others, and about the amazing world around us. And that curiosity may lead to new and startling insights.\n\n## Home Energies\n\nWhether we live in an apartment or a large house, whether we spend most of our time at home or travel frequently, the energy around us has an impact on our lives. Although our home is meant to be a place of nurture and support, many people can feel anxious or subtly disturbed at home. And it is well known that moving to a new home is a cause of high anxiety for many people.\n\nFrom the point of view of our nervous system, there is one room in particular that has a pronounced effect\u2014the bedroom\u2014though other rooms have an impact as well. If we live in a one-room apartment, this effect is found in the part of the room where we sleep.\n\nMost of us spend at least a third of our lives asleep in our bedrooms. This part of our lives is extremely important to us. We need to sleep well\u2014deep, refreshing sleep. We need to dream\u2014clear, regenerative dreams.\n\nThe bedroom has other roles as well. For many people, the bedroom is also one of the places of greatest intimacy in our lives. It is a place of privacy, and it is also a refuge. It is often here that we come for protection, for security and peace. We can \"sleep\" on our problems. We can curl up and turn away from the stress of the day\u2014switching off the conscious activities of the brain and allowing the unconscious mind free rein.\n\nWe need to protect these precious hours of sleep and recuperation. If the bedroom is not a place of harmonious living, the consequences for our home lives can be devastating\u2014and how often we have seen that devastation exported emotionally into the rest of our lives.\n\nYou want to have a bedroom that is as safe as possible, where your privacy is ensured, where the risk of intruders is minimal, and where the risk of accident is reduced. These qualities are all the more important to you when it is dark and you are asleep.\n\nIf you are buying or renting a new place to live, you should pay careful attention to the internal relationship between the bedroom and the rest of the space. Since the energies from the rest of the home enter the bedroom principally through the bedroom door, it is important to be particular about the areas of the home onto which the door opens. This may not seem all that important to you compared with other factors in choosing a new home\u2014and the options you have may be limited\u2014but remember that you will likely be spending such a great deal of your time in your bedroom that you will be profoundly affected by the energies that enter and circulate in that room.\n\nIf you are choosing a new home, it will be helpful to know a few basic elements from an energetic perspective. Energies can easily flow from one room to another if there is a direct line of movement through the doorways. In some homes, the entrance to the bedroom is directly in line with the front door. You are less likely to sleep soundly in such a home. Your nervous system will be most relaxed if your bedroom is an interior, private place. In other homes, the entrance to the bedroom is directly in line with the bathroom or kitchen. In those instances, it is necessary to make arrangements that will keep the bedroom free from odors, fumes, and noxious energies.\n\nIt may not always be possible to have a separate bedroom with a door. You may live in an open-plan home or a studio apartment, leaving you without a protected, quiet place of repose. If that's your situation, try to set up your living space so that the sleeping area is clearly defined and protected at night by a barrier, such as a bookcase or a screen.\n\nQigong at Home\n\nWhether or not you are able to arrange your home taking all this into account, there is still one other thing that you can do: develop a daily qigong routine. It will make a difference in the quality of your life, both at home and away. Qigong is compatible with any other activity, like exercise or sports, and any sort of mind training, like meditation. Practicing a qigong routine every morning is like putting fresh batteries in your system before you start your daily activities. There is a sample routine in part 3.\n\nBecause you, your home, and anyone else who lives there are all fields of energy, the different energy patterns influence each other in a mutual, interdependent manner. We all know this from the simple fact that if one person is in a bad mood or acting in a disturbing way, he or she can upset everyone else and change the atmosphere in the home. Similarly, if you are regularly cultivating your life-enhancing energy and simultaneously calming your mind and nervous system, you will not only be less anxious and more emotionally resilient yourself, you will also have a beneficial effect on anyone you live with and the entire feel of your home.\n\n## Where You Work\n\nThe location of a shop, office, or factory is just as important to the well-being of the people who work there as the location of your home is to you and your family. Even if you don't have a choice about the location of your job, it is good to know if you are being exposed to helpful or potentially harmful energy there.\n\nUrban environments can affect our nervous systems in numerous ways, which makes it all the more of a challenge to live healthily. It is not just our individual temperament that causes us to be anxious, experience high levels of stress, and have difficulty sleeping. These patterns can be heavily influenced by our environment.\n\nIf you work in a shop, a bank, or a restaurant, it is likely to be on a busy street or in a mall, grouped together with similar businesses to attract as many customers as possible. This means you will be exposed to a substantial volume of moving, volatile energy. You will inevitably be affected by that turbulence.\n\nIf you are an office worker, on the other hand, it is likely that you will not be exposed to the same kind of bustling energy that you would inside a busy store. In fact, if you are looking for an office location for yourself, it is best to look for a quiet corner in a busy area. Often, this means a side street adjacent to a major intersection or on the upper floors above the shops at street level. Although most office workers cannot choose the location where they work, there are still ways you can use your qigong practice to help you counteract the difficult conditions in which you may be working. We will look at some special features of office life in the next section, \"Office Qigong.\"\n\nIf you work in a factory, you will know that they are, generally speaking, noisy places, and many generate fumes and waste. They need space, room to expand, and, usually, some open ground for off-site storage and transport. Since factories and light industrial complexes tend to be located in areas specially zoned for this kind of industry, your factory will likely be near other ones. The impact of the factories in the area is collective, which means that attention to health and safety and to each person's individual well-being is all the more important.\n\nBeing exposed to short periods of disturbed energy from a location is not necessarily a problem, even if it happens every day, as long as we are just dropping in. For example, the energy at a downtown intersection will be particularly disturbed. It might be a good location for stores that want to attract casual passersby, but if we were to try to live or work in that area, we would need to pay special attention to creating our own strong, protective energy field.\n\nOffice Qigong\n\nMost modern offices network their computers so that employees can readily share information, use e-mail, and access the Internet. Computers are connected to the network by cable or Wi-Fi system. Cables usually run under the floor; a Wi-Fi system is all-pervasive.\n\nAll electrical items have electrical and magnetic fields. In the workplace, that includes the wiring for power, phone and network cables, fluorescent lighting, computers, and air conditioning units, as well as the large number of invisible electronic networks whose signals penetrate everywhere. Everyone in the office is surrounded by these fields. Some of these fields may be of low power, but even these add up to an environment of energetic disturbance that can negatively affect the health of people who work in it for sustained periods of time.\n\nIn the classical system of understanding energy patterns in which I was trained, electricity and electronic equipment like computers, cell phones, and personal sound systems are considered fire energy. Like a fire, the energy shoots upward and is radiant and powerful. It is warming, transformative, and communicative. It can also cause overheating, burning, and dryness; it can be all-consuming.\n\nIn traditional Chinese medicine, a serious imbalance of fire energy affects the heart and disturbs the nervous system. The results of prolonged exposure to the upward force and intense power of fire energy can affect your concentration. It can lead to headaches, anxiety, and loss of sexual appetite. You find yourself nervous, jumpy, and unable to relax.\n\nIf you work in this type of office, you will always be using up your own energy as your system tries to cope with the excess fire energy that envelops you. This, in turn, weakens your immune system and therefore your resistance to disease. If you find yourself spending long hours working in this type of environment and your employer is not alert to this problem, you should develop your own ways of coping with the effects of these disturbing energy fields.\n\nMany practices offered in this book are helpful in counteracting the effects of fire energy. In part 3 you are introduced to the importance of connecting with the power of earth\u2014\"Taking Root\" and the other practices. Remember that you can literally \"ground\" yourself by using these practices, the way electrical equipment is grounded.\n\nAll the suggestions for using the power of your senses in part 4 can be helpful in an office environment. For example, the practice of gazing into a blue- or green-tinted glass filled with water can be easily adapted for discreet use at your desk. If you rest your eyes on the glass from time to time, you will notice the effect this apparently simple practice has on your nervous system.\n\nTake as many breaks as possible, to get out of the disturbing energy fields of the office. Literally go outside during breaks. Stand in the open air. This is a perfect time to practice the exercise \"Coming Up for Air\" in part 1. In meetings or at your desk, you can usually find a time when you can discreetly practice \"Resting the Golden Ball,\" or a tensing and relaxing exercise like \"Clenching and Relaxing.\"\n\nLike many of my students, you may find that regular morning qigong sessions make a huge difference, setting you up for the day. Over time, it strengthens your immune system and creates a strong field of energy that can protect you from the disturbed vibrations of your work environment. You can begin with \"Your Daily Routine\" in part 3.\n\nThe problems caused by working at a computer keyboard are, by now, well known. These are associated with tension; in extreme cases, the strain leads to damage of the nerves in the wrist. In traditional Chinese medicine, the condition is relieved by massage, the application of herbs, and special exercises. To help prevent or relieve tension and pain in the hands, you can use the following simple qigong exercise for a few minutes before starting work at the keyboard or if you start to feel tired while working.\n\n  * Begin by sitting or standing for two or three minutes in the position \"Holding the Golden Ball at Your Belly\" (in part 2).\n  * Then raise your hands to elbow height and turn them so that your palms are turned down toward the floor. Your fingers are pointing straight ahead.\n  * Slowly raise your thumbs as far as you can without moving your other fingers. Hold for one second. Then lower them, pressing down as far as you can. Hold for one second. Then relax your thumbs.\n  * Now repeat this for each of the fingers on your hands\u2014trying to raise and lower them in the same way, as far as you can without moving your other fingers.\n\nThese finger exercises not only help to remove the strain in your hands and improve the circulation of blood in your fingers, but are excellent stimulation for your brain. You will notice the effect that doing this has on your worries, anxieties, or other preoccupations.\n\n## Traveling Qigong\n\nMany people feel anxious when traveling. Of course, there is also the excitement and arousal of traveling, but for many people, the panic of getting packed and off to the station or airport on time is very unsettling.\n\nThere are different ways you can use qigong when traveling. First, there is a way of working with your mind that may help you. Then, over these next few pages I show you a sequence of six exercises that will greatly help your energy, especially if you are traveling in cramped conditions or on an airplane.\n\nPlease take a look at the calligraphy at the beginning of the section \"Earth\" in part 3. It shows the Chinese character for the Tao. There is a long single brushstroke at the bottom that represents a raft, the sort of flat, simple craft used for centuries for navigating waterways\u2014and still used today in some parts of the world. Just above it, at the left, is a small brushstroke that represents a person guiding the raft in the water. Picture yourself as that person. Your raft is floating in the currents of the water.\n\nAs you imagine this, you might close your eyes and feel that you are riding the current. You might sway a little as you do this. Try to feel that all the energy around you\u2014the movements of your car or bus, the sounds and smells of the station or airport, the commotion of your family, the turbulence of the plane\u2014are simply what you encounter as you balance on the raft, plying your oar in the water.\n\nIf you are not attracted to the image of a raft on water, you can imagine that you are being carried along in an old-fashioned bicycle taxi through the countryside. You feel the movement of the bike, the unevenness of the road, the air brushing against your face, and the feeling of the person who is riding the bike carrying you with them as you cycle along together.\n\nWhile these images may be most useful when you are traveling, you can also use them at work or in other situations where the environment is somewhat unstable and makes you feel uneasy.\n\n## Giant Strides\n\nThis is the first of six qigong exercises that you can do while sitting to rouse your energy, calm your mind, and improve your circulation and well-being. I originally developed them for use on an airplane, but you can use them on any other long journey as well, in a bus or on a train.\n\nSome seats make us slump so that our back is curved and our chest compressed. Try to sit up as best you can, but don't strain yourself too much.\n\nSlowly lift one foot off the ground. As you raise your foot, stretch your toes up toward you as fully as possible. Breathe in as you raise your foot. Then slowly lower your foot as you breathe out.\n\nBreathe in as you raise the other foot, again raising your toes up toward you as fully as possible.\n\nLift your feet eight times. Wait ten seconds. Then repeat the sequence two more times.\n\nAs you do this, imagine that you are taking huge strides over hills and mountains, like a giant. The movement is slow and powerful, your immense body covering miles with every step.\n\n## Bubbling Spring\n\nThis second traveling exercise focuses on one of the most potent acupuncture points on your body. Remain sitting comfortably in your seat, with your back as upright as possible.\n\nRaise the heel of one of your feet, keeping the ball of your foot firmly on the floor. Then press the ball of your foot down into the floor. Use your calf muscles to get the pressure going straight down into the ball of your foot.\n\nBreathe out as you press the ball of your foot down for a couple of seconds. Then relax, release the pressure on the ball of your foot, and breathe in. Lower your heel.\n\nThen repeat with your other foot. Press and relax eight times, first one foot, then the other. Wait ten seconds. Then repeat the sequence two more times.\n\nAs you press down, you are activating the _yongquan_ point on the sole of your foot, normally known as the \"bubbling spring.\" This action sharply increases the flow of qi throughout your system, and the work done by your calf muscles speeds blood circulation.\n\n## Waves on the Rocks\n\nIn this, the third of the traveling exercises, try to sit up in your seat so that you do not strain your upper back. Place both your hands on your knees, so that your palms and fingers cover the entire area around the joints. Your fingers should curve over your knees so that your fingertips are well below your kneecaps.\n\nSlowly rub your hands in smooth circles around the whole of your kneecaps. Your hands should circle around your knees starting with the inward aspect of each knee and circling around the front of the knee. The massage should be firm, but do not grip your knees.\n\nMake thirty circles around the knees. If you have knee problems, do this massage as many times as you wish. Breathe naturally as you do it.\n\nTo get the right feeling for this exercise, imagine that your knees are large rocks at the edge of the sea. Your hands flow around them just like waves that roll in, cover the rocks, wash completely around them, and then recede. Use your fingertips to move firmly in and around the ridges and spaces below your kneecaps.\n\nWhen you have finished, relax for a moment, with your hands resting on your knees or thighs.\n\n## Fearless Rider\n\nIn the fourth traveling exercise, begin by sitting in a resting position, with your back comfortably against your seat back. Your feet are flat on the floor.\n\nImagine that you are sitting in a saddle on the back of a powerful stallion. You are about to leap forward. As you feel the horse about to jump, you pull yourself forward fearlessly to ride upward with him. This forward pull takes place in the muscles of your buttocks, pelvis, and thighs. The shift in your hips automatically moves your back forward so that it is no longer touching the chair. As this happens, squeeze the muscles of your buttocks and thighs so that they are as taut as possible and you are fully sitting up. Breathe out as you come forward. Keep your upper body completely relaxed. .\n\nHold this forward position with the muscles of your buttocks and thighs tensed for a few seconds, without breathing in. Then gently release the pressure, completely relax, and sit back as you breathe in.\n\nSit up like this eight times. Wait ten seconds. Then repeat the sequence two more times\n\n## Crushing the Flanks\n\nIn the fifth traveling exercise, you are still riding the stallion. It is flying like the wind. To remain in position on its magnificent back, you need to grip its flanks tightly between your knees. As you crush the flanks, you imagine the full power of the horse carrying you with it.\n\nSome people, when traveling, experience tremendous anxiety because they feel trapped, helpless, and at the mercy of whatever is going on in the airplane or on the train or bus. Their feelings of powerlessness affect their overall energy. This exercise can help shift those feelings, as it boosts your qi.\n\nSit forward, if you can. Fold both your hands into loose fists and turn them palms downward. Bring your fists together so that the thumbs and index fingers of each hand touch each other. Place your fists together between your knees.\n\nSqueeze your knees together, pressing inward on your fists. Hold the squeeze for up to thirty seconds if you can. Breathe naturally as you do this.\n\nThen relax. Wait ten seconds. Then repeat the sequence two more times.\n\n## Holding Two Forces\n\nFor the final traveling exercise, sit up as straight as you can in your seat. Slowly breathe out, letting your chest sink inward a little. Relax your shoulders.\n\nThen, as you breathe in, bring your hands together in front of your chest and clasp them together.\n\nOpen your elbows outward so that your forearms form a right angle with your hands. Your palms are now vertical, pressing against each other.\n\nNow press your palms firmly toward each other as you breathe out. Keep pressing for thirty seconds while breathing naturally. Alternatively, you may prefer to do this exercise by pressing your palms together for ten seconds while making one long exhalation.\n\nThen slowly relax, keeping your arms in place. Breathe in slowly. Then repeat the pressing and relaxing, with synchronized breathing, two more times.\n\nYour arms are like two forces relentlessly opposing each other. You feel the pressure they exert on each other, but your mind and body remain relaxed, so that the power of the two forces is calmly held by the entire field of your energy.\n\n## Experiencing Your Environments\n\nAt the beginning of part 5, you were introduced to ways in which you can experience and work with the energy of your home and work environments. This is a different approach to working with energy, but since you spend a great deal of time in your home and work environments, they have a sustained effect on you, your health, and your inner life.\n\nPlease use the following chart to record anything you notice about your home or workplace. You can also list any changes you have made or are planning to try. Remember that everyone's experience is different, so you need to pay careful attention to what you are sensing yourself. If you would like to have a checklist of sensations to help you with your personal review, you can use the \"Quick Check\" chart at the end of the introduction. You can use this chart for notes. If you find this helpful, you can continue to keep notes for yourself on a separate notepad.\n\nThe Way of Nature\n\nNature is a living process of constant birth, death, and renewal. It is a process of perpetual rebalancing. It is never static, never the same. It is constantly coming to a beginning, over and over.\n\nThis is the deep truth embedded in the four characters of this calligraphy. At the very bottom, there are tiny brushstrokes. These represent fire\u2014the cycle of creation and destruction through which life forever renews itself, as a forest does after a fire.\n\nTaken together, the four characters of the calligraphy can be translated, \"The method of the Tao is the process of following nature.\" Thus, its message is one of profound relaxation, the antidote to anxiety. You may be experiencing anxiety now, but this is not a permanent state, nor is it your nature. You will cycle through this, like life, again and again, always changing.\n\n## Practicing in Nature\n\nAs you make progress in your qigong training, you will become sensitive to the flow of energy within you and to the energy that surrounds you. You may be drawn to practice outdoors\u2014in a garden, on a balcony, or in a local park. This is the way qigong has been practiced for centuries. You will feel refreshed by the air around you. However, it is not advisable to practice outdoors when it is windy or raining, and you shouldn't stand facing into direct sunlight, though sunlight on your back is beneficial. If you find it difficult to go outdoors for any reason, try to have a source of fresh air in the room where you do your exercises.\n\nYou can enhance your sensitivity to the natural world and derive immense benefit from it by learning to practice with a tree.\n\nThis photo was taken when I went to Muir Woods, the marvelous redwood forest in northern California. The power of the natural energy in the woodland is wonderfully invigorating.\n\nTrees are among the great powerhouses of nature, and the energy they radiate is extremely powerful. The presence of trees is a tremendous support for your practice, stabilizing and calming you with their protective power.\n\nIf you would like to try this practice, look for a tree that exudes a sense of strength, power, and well-being. It need not be a huge tree, but you should have the feeling that its energy is healthy.\n\nStand a few feet away from the tree in any of the qigong positions. For this practice (which is slightly different than what you see in the photo), please make sure you face the tree.\n\nImagine that there is a circuit of energy that extends from the crown of the tree down to the top of your head. It continues through your body down to your feet, then down into the earth, where it connects with the roots of the tree. The circuit continues back up through the tree until it begins to circle down through the top of your head again.\n\nAs you imagine the tree's energy coming in through the top of your head, breathe in. As it goes from your feet back into the tree, breathe out.\n\nAs you stand, allow your mind to rest on this continuous circuit. Experienced practitioners report feeling the mutual cycle of energy between them and the tree as they relax into the experience.\n\nYou can do any or all of the qigong practices in this book outdoors. If you start your day with a little routine, like the one at the end of part 3, doing all or part of your routine outdoors can give your energy a real boost and help your nervous system recover from any overnight worries, sleeplessness, or strain.\n\nYour Senses and Your Mind in Nature\n\nPracticing qigong outdoors is an excellent opportunity to use your senses and your mind to help you with your anxiety. Feel the air on your face, listen to the sounds of the trees, smell the grass and the flowers. Breathe freely in a relaxed way.\n\nIf there is a stream in the park or woodland, you can spend a few moments looking at the running water. If you see a stone in the water, observe how the water flows around and past it in an unending current. Try thinking of yourself as a smooth stone, standing in the current of all your anxieties and difficulties. They flow around you as you remain unmoving in the stream. Try to form a clear image of the water flowing past the stone in your memory, so you can use it later. Being a stone in the water doesn't mean that you become unfeeling or cold as a person. It simply means that you take your place in the flowing current of your life but are not swept away by it.\n\nDon't confuse this outdoor practice with meditation. You can talk to your friends or anyone you meet in the park. If you like to sing to yourself, please do. But please take this opportunity not to use headphones to listen to music or anything else. This is a time to appreciate nature just as it is.\n\n## Your Natural Experience\n\nYou have been taking your qigong practice out into nature. Outdoor practice enables you to feel more expansive, to do the exercises in the open air and to spend time making a connection with trees\u2014some of nature's most powerful energy transformers. You can do other qigong practices outdoors as well. In the chart below are some suggestions for practices that you can try outdoors. Follow your own preferences.\n\nIt is very helpful to keep track of your experience of doing qigong. You are the only person who can feel what is happening as you do this and how it is affecting your anxiety patterns. If you would like to have a checklist of sensations to help you with your personal review, you can use the \"Quick Check\" chart at the end of the introduction. If you find this helpful, you can continue to keep notes for yourself on a separate notepad.\n\n## The Two Poles\n\nAs you develop your practice of qigong, your hands become increasingly powerful conduits of energy. They are like the two poles of a magnetic field\u2014at any one time, one of these poles is positive, the other is negative. As your sensitivity increases and your qi becomes more and more powerful, you can feel the invisible current running between your palms.\n\nThere are numerous healing applications of this current. Your hands can be used in exactly the same way that magnets are used in magnet therapy. By placing your hands at different points on the body, the current passes through the body from one palm to the other.\n\nBefore performing the qigong massage techniques described on the following pages, you need to ensure that there is a strong flow of qi to your hands.\n\nStand in the position \"Holding the Golden Ball at Your Belly\" for at least five minutes. If possible, lower yourself down a little farther than usual. Your upper body should be relaxed. Spread your fingers amply apart.\n\nWhen you have completed your standing, rub your palms rapidly together for at least thirty seconds to warm them. Clap your hands together about a dozen times to bring your inner qi to the surface of your palms. You are then ready to work.\n\nYou will be using your hands like two magnetic poles of yin and yang energy. There are many ways of using this energy, including as first aid in some situations. If you have an accident, such as bumping some part of your body or stumbling and wrenching your ankle, you can apply your palms on either side of the affected area. This calms the nervous reaction and steadies the entire area, and\u2014depending on what has happened to you\u2014you may well find that it can relieve your pain and help reduce swelling.\n\nYou can imagine that the part of your body that is held between your hands is a small planet, like our own earth. The energy field between your hands is like the energy of the galaxy that surrounds the earth and sustains its biosphere. You are using the flow of your qi as a field of warmth, support, and nurture.\n\nIn our tradition, this use of qi is often referred to as \"natural medicine.\" The great Taoist sage Zhuangzi (also transliterated \"Chuang Tse\"), who lived about 2,500 years ago, wrote about this: \"The three treasures of vitality, energy, and spirit experience a daily flourishing of life and fill the whole body, so that great medicine can be expected to be produced naturally.\"\n\nThis \"natural medicine\" is one of the accomplishments of qigong practice. As you mature in your practice, the quality and power of your inner energy changes. What might have been a somewhat weak, irregular, unstable flow of energy becomes fuller, more constant, and unending\u2014less like a stream, more like a broad river. This enhanced flow of energy can be used for some forms of self-healing and helping others.\n\nI have done this kind of healing massage for most of my working life. I began with a clinic in Hong Kong after my training as a traditional herbalist and bonesetter, and I have since treated countless people in my centers in London and Piedmont, California. I have trained a number of my senior students in this method as well. They like to report their experiences to me, especially if they have been able to use their qi successfully in emergencies. One of them told me of such a success.\n\nI was walking on an uneven pavement and suddenly stumbled. My foot slipped off a loose paving stone, and my foot twisted to one side. I landed sharply on the side of my foot, pulling and twisting my ankle at the same time. The shock went right through my body. I immediately stopped and began rubbing my palms together. When they were warm, I knelt down and held my ankle between them. Then, after a little while, I gave the whole area a good, strong massage with my palms. I wasn't sure what the effect would be. When I stood up, I found I could place all my weight on the affected leg. Then I slowly moved my ankle. It was tender. I took a few steps and felt I could move it and continue walking. When I looked at my ankle that evening, there was no swelling.\n\nIf you have not practiced zhan zhuang qigong regularly, have been practicing incorrectly, or have only been practicing for a relatively short period of time, you should not expect to achieve miraculous results. But if you practice regularly and experience the sensations of the energy current between your palms, you can use your qi in this way and in the applications described in the following pages.\n\n## Neck and Face Self-Massage\n\nYou can use your energy for healing and calming self-massage. Your nervous system will respond wonderfully to a soothing qi massage of your neck and face, which will remove tension in your neck and bring radiant energy to your face.\n\nFirst, place your hands together and rub them together vigorously for a little while. Try to make sure that your fingers and fingertips are also rubbing against each other.\n\nThis automatically brings more qi to your hands. Most people experience the effect immediately: your hands and fingers become warm. Even if you do not experience increased warmth, your energy is there.\n\nThen rub your hands back and forth over the back of your neck. This helps to clear blockages and stiffness that impede the flow of energy to the brain and nervous system.\n\nYou can also give yourself a little \"qi facial\" this way. Gently rub your hands over your eyes, ears, and face, as if washing yourself. Try feeling the effect of holding your palms over your closed eyes for a few seconds\u2014and then over your ears.\n\n## Soothing Your Back\n\nAnxiety and other inner torment is one of the most common causes of backache. This painful condition affects millions of people. If you suffer from it, you may be able to give yourself some relief with a qigong massage. This should not be mistaken for a cure for back pain, especially if you are suffering from an injury to your back, a pinched nerve, or a chronic condition. But the warming flow of qi will help soothe you, lower your level of anxiety, and offer you some relief.\n\nFirst rub your palms together to stimulate your qi. Curl your fingers into open fists, as if you were holding an egg in each hand.\n\nPlace your open fists in the small of your back so that the curve between your thumb and your index finger rests in the soft flesh just above your hips. You are not using the whole of your hand, just the curve between your thumb and index finger. Then use your open fists to massage those two spots in your back for as long as you wish.\n\n## Easing Headaches\n\nThere are many causes of headaches. Some are the result of accidents and injuries, which require specialized treatment. But many ordinary headaches are caused by a blockage of qi in the area of the neck and shoulders, which affects the supply of blood and oxygen to the brain. You can use the following sequence for relief.\n\nBegin by stimulating the flow of qi to your palms using the \"Two Poles\" method.\n\nThen work with the tips of two or three fingers on each temple. Make smooth circles around the area of the temples, pressing gently inward. Make thirty-six small circles.\n\nThen run your fingers along both sides of the head, from the temples to behind the ears and down to the base of the skull at the back of your neck. Repeat this flowing motion six times.\n\nThen work the tips of two or three fingers on either side of the back of the neck, below the base of the skull. Make thirty-six small circles, pressing gently inward as you move. Finish by \"washing\" your head and face with firm, smooth, circular movements for about a minute.\n\n## Energy Circuit\n\nAnother way to work with the living energy of the world around you is to create an energy circuit with a partner. I recommend you do this with a family member or a good friend\u2014someone with whom you have a warm, supportive connection already. It is not necessary for both of you to have experience with qigong, but one of you should at least be experimenting with the practices in this book. If both of you are doing some daily qigong practice, of course, the effect will be even stronger.\n\nStand facing each other. It is best if your feet are shoulder-width apart. Move your arms so that they are positioned at a 45-degree angle to the side, level with your waist. The palm of your right hand faces down; the palm of your left hand is turned upward.\n\nHold your right hand over the other person's left hand, with your palm facing down over your partner's palm. Your two palms are facing each other. Leave some space in between your palms, as if you were holding an orange between them.\n\nYour energy will naturally create a circuit through your hands. Breathe naturally. Just stand for a few minutes like this, breathing naturally and relaxing.\n\nMany people ask me exactly what path the energy takes. You can think of it traveling from the palm of the first person's right hand (which is facing downward) into the palm of the other person's left hand. The right hand is passing energy into the left and the left is taking it. The energy simultaneously makes a circuit through your two bodies and continues through the connection of the other two hands.\n\nYou can also work with this energy circuit sitting down. Simply arrange yourselves so that you are facing each other and create the circuit with your outstretched palms. If one person is unable to stand up for any reason, that person can remain seated while the other stands.\n\nIf you have trouble figuring out the hand positions, please look carefully at the two hands in the foreground of the picture, which shows the correct position with one person's right hand on top, palm facing down, and the other's left hand underneath with the palm turned upward. The photo also shows the space between the hands. If you get one set of your palms properly aligned, then you can simply copy that (in reverse position) for the other set of palms. The photo also shows that the fingers of your hands are gently spread apart, but they are not tense.\n\nThere are many ways we can use and benefit from an energy circuit like this. If you are feeling anxious, just ask your partner to stand with you for a minute or two. In my experience with many students, even those two minutes can be extremely helpful. You will find that the energy of anxiety is neutralized.\n\nWhat happens if both people are feeling anxious? Your energy circuit will carry your anxiety along with it. The way energy flows, the weaker, most disturbed anxiety energy will flow downward toward the earth and dissipate into the ground. That will leave a strong circuit of supportive energy flowing between you.\n\n## Experiencing Energy Work\n\nHave you tried the various forms of energy work that you were introduced to in this part of the book? These include energy massages (\"The Two Poles\"), as well as working with another person to create a shared energy field (\"Energy Circuit\"). It's important to pay attention to the effect of these practices, so that you can learn from them and develop your ability.\n\nIt can also be wonderful to ask your partners about their experience, so there is space on this chart for that too. Remember that everyone's experience is unique\u2014not better or worse than anyone else's. If you find this helpful, you can continue to keep notes for yourself on a separate notepad.\n\n## Inner Growth\u2014A Nine-Month Review\n\nPlease come back to the following chart on the date that you entered in the box \"date nine months from now\" in the \"Growing from Within\" chart at the end of part 1. Answer these questions without looking back to see what answers you gave nine months ago. Then you can compare your answers and reflect on your experience.\n\nIt doesn't matter if you have maintained a regular qigong practice or followed any of the recommendations in this book. Life is a process of constant change, and it is always helpful to take a few minutes to review your experience. Please remember that every person is unique. These charts are meant for your personal, private use. Please treat them like an intimate conversation with yourself.\n\n## Frequently Asked Questions\n\n**Are there any times or conditions in which it is not good to practice qigong outdoors?**\n\nIt is best to practice qigong outdoors during the day. If you are an early riser and want to start practicing before dawn, that is fine, especially if you are able to arrange to include the sunrise in the period of your practice. Normally, you should try practice with sunlight striking your back. Sun energy connects with an important acupuncture point in the small of your back, known as _mingmen (ming men)_ , the \"gate of fire.\" Facing strong sunlight can make you squint and tense up, thus impeding the flow of energy. It is best not to practice outdoors if it is misty, foggy, or rainy. Damp conditions can also impede the flow of energy.\n\n**What happens if dogs or other animals see a person \"standing like a tree\"?**\n\nAnimals have a natural attraction to strong energy. Dogs and other pets, whether at home or outdoors, may take an interest in a person doing qigong. If they know you, they may wish to lie down near you or be in the area where you are practicing. If they don't know you, they might run over to you to check you out. If a dog you don't know is barking or jumping up and down, it is best to remain still and not make any sudden movements, in case this is startling. If you want to move away, lower your arms to a normal position and walk calmly and slowly on. Once when I was in a park with one of my students, a squirrel ran up his leg and sat for a while on his shoulder. We had a good laugh about how he was doing quite well at standing like a tree.\n\n**What should I do if there are no changes I can make to the conditions in my home or workplace?**\n\nSometimes it is simply impossible to make physical changes to your environment. In that case, it is important to realize that regular, consistent practice of qigong creates a strong energetic field that not only helps you internally, but also acts as a sphere of protection around you. This is sometimes referred to as the \"aura\" and is what can be seen using bioelectrography, also known as Kirlian photography, which picks up the electrical field around a person. In the section on your workplace (\"Where You Work,\" in part 5), I give specific suggestions to help you cope with the intense energy of many contemporary offices.\n\n**Can you pick up other people's illnesses by practicing qigong or exchanging energy with them?**\n\nIf you are in the same room as a person with a cough or a cold, you can pick that up through the normal route of physical infection. That's natural and can happen whether you are practicing qigong with them or not. But the practice of qigong has a positive effect on your immune system and helps you bounce back faster if you do have any kind of illness or injury. So the most important thing is to try to establish a good daily foundation of refreshing your energy. If you do that, you will be at less risk of becoming ill, no matter how you come in contact with illnesses.\n\n## Does This Really Work?\n\nThe only person who can answer this question is you. Personal experience is the basis of the qigong tradition. Qigong has always been handed down from person to person. And the reason it has been passed on is that one person after another found that it really worked for them. But if you don't find it helpful, or if you don't see a way of fitting it into your life, please don't worry. There are other methods of working with energy, and since energy is all around us, you will never be without it.\n\nWhat I have found is that people from all walks of life can find ways to fit a little qigong into their lives, and they find it helpful. That is why I have told their stories throughout the book. You may have noticed that my students come from a range of backgrounds: teaching, medical care, law, business, real estate, computing, international relations, and community service\u2014and add to that list my own early days as a police officer.\n\nSome of my students are artists. They too put qigong to use in their work. One is an opera singer. She told me, \"Zhan zhuang qigong has extended my capacity. The change is fundamental: you can hear the difference. I am stronger, breathe deeper, and project my voice farther. Zhan zhuang changed my awareness of my inner space. I couldn't have learned this just from taking singing lessons.\"\n\nI am not asking you to believe me. I am offering you the possibility of trying it out for yourself. Then you can decide whether it works or not.\n\nA Living Fire\n\nIf you want to keep a fire alive, you need to ensure that there are burning embers that will stay alight until dawn. That is the meaning of the first of these four characters.\n\nThe second character represents the fire itself.\n\nThe last two characters express the idea of \"passing the baton.\" In a relay race each runner carries the baton and then carefully hands it on to the next teammate.\n\nRead together, these four characters describe the notion of \"lineage.\" A lineage is a living fire. Like the Olympic flame, it is carefully kept alight and then is passed from hand to hand. That is clear from the last three characters. The first one, however, adds a warning. You can admire a fire that is already burning and decide to start your own. But that is a different fire. It is not the same as keeping the power of the original fire alive.\n\n# The Lineage\n\nThe practice of cultivating human energy has been passed down from master to student in a lineage that stretches back some twenty-seven centuries. Like all forms of energy, it can be used in a variety of ways. From those earliest days, there has always been a distinctive stream of this tradition that has used energy to relax and nurture the human being, both mentally and physically. My own master was fond of recounting the thousands of years of history through which the art of qigong has developed. It is thanks to those many generations of masters and students that it is possible for me to share with you today the benefits that qigong offers for alleviating anxiety.\n\nTwo of the earliest books about qigong are a medical treatise and one of China's most famed books of philosophy. _The Yellow Emperor's Classic of Internal Medicine_ ( _Huang Ti Nei Ching_ ), the world's most influential medical text, appeared some 2,400 years ago. In it, the court physician tells the emperor: \"The sages were tranquilly content with nothingness and the true vital force accompanied them always. Their vital spirit was preserved within.\"\n\nThe _Tao Te Ching_ , by Laozi\u2014one of the most widely read and influential books in the course of human civilization\u2014says, \"By standing alone and unchanging, you will find that everything comes to you and the energy of the cosmos will never be exhausted.\" \"Standing alone and unchanging\" was Laozi's way of describing the practice through which we have come to understand the full power of the universe.\n\nThe cultivation of internal energy was developed as part of Taoism. It included the practice of remaining completely still in fixed positions. The Taoist philosopher Zhuangzi tells us that the sages of old were \"still and unmoved....Their breathing came deep and silent.\" Their \"minds were free from all disturbance...forgetting everything.\" They were \"open to everything and forgot all fear of death.\" In the text, a disciple tells his master, \"I am making progress.\" \"What do you mean?\" asks the master. \"I sit and forget everything...becoming one with the great void in which there is no obstruction.\"\n\nBuddhist thought and practice also influenced the development of the qigong tradition. The development of \"one-pointedness\" of mind (that is, the ability of the mind to be clearly focused) in the Buddhist tradition made a major contribution to the cultivation and application of qi in the human energy system.\n\nThe practices of qigong were passed down through the centuries in a lineage that was largely kept secret. It was not until the twentieth century that this heritage was shared openly in China and brought to the West.\n\nThe person who unlocked these secrets for the modern world was Grand Master Wang Xiang Zhai. Born in 1885 in the Shenxian district of Hebei province, he suffered from poor health as a child. His father was determined to improve his son's physical condition and sent him for training under his uncle, Master Guo Yun Sin, who lived in his village. From Master Guo he learned the secret discipline of zhan zhuang qigong. After his master's death, the young Wang spent ten years traveling throughout China, meeting and studying under the great masters of his day.\n\nBy the mid-1940s, he was ready to teach and pass on the fruits of what he had learned and moved to Beijing, where he was soon recognized as a master of extraordinary wisdom and prowess. His accomplishment was vast. From his humble origins as a boy suffering from asthma that left him severely weakened, he became a pivotal figure in the qigong tradition, bringing many of its most profound secrets out from centuries of secrecy into the modern world.\n\nIn the course of his studies, Grand Master Wang delved deeply into the spiritual heritage of Chinese culture, immersing himself in the wisdom of the Taoist and Buddhist traditions. Grand Master Wang composed poems to guide his many students on the path of qigong practice\u2014attempting to convey in words the inner experience of this art.\n\n_You who wish to master this art, begin by standing still._\n\n_Breathe deeply, undisturbed, like air among the clouds._\n\n_Your spirit and your bones are being tempered in this forge._\n\n_Hold still, unwavering._\n\n_This wisdom will suffuse your being, ceaselessly._\n\nGrand Master Wang broke new ground by challenging the traditional secrecy of qigong instruction. He taught openly. \"Knowledge should not be hidden away like a secret,\" he said. \"It belongs to all humanity.\"\n\nOne of the earliest students of Grand Master Wang was a young man who had studied Western medicine and was serving as a dentist at the Ten Lu (Railway) Hospital. His name was Yu Yongnian.\n\nAfter nine years of study and practice under Grand Master Wang, Yu began introducing aspects of zhan zhuang qigong as treatment for chronic diseases at his hospital. His initial successes were such that a major medical conference was held at the Beijing Shoudong San Hospital three years later, in 1956, to introduce the zhan zhuang system to hospitals throughout China.\n\nAfter the Cultural Revolution, Professor Yu published groundbreaking books on zhan zhuang. The first edition of _Zhan Zhuang for Health_ came out in February 1982 (Beijing: Educational Publishers). By April a second edition of 120,000 copies was issued, and by 1987 a further 294,500 copies had been printed. A limited edition of his next book _, Application of Zhan Zhuang for Health_ , was published in Beijing in 1989, and in the same year a further work on zhan zhuang qigong was issued by Cosmos Books in Hong Kong.\n\nProfessor Yu became the world's leading authority on zhan zhuang qigong, was on China's National Qigong Research Council, was a former consultant to the American-Chinese Qigong Research Group, and was honorary chairperson of the Da Cheng Chuan Zhan Zhuang Chi Kung Research Groups (Europe).\n\n## \"Taking Tea with My Master\"\n\nAlmost as soon as I was introduced to the art of qigong, I realized that it was far more powerful than anything I had ever learned before. I wanted to go directly to the source and learn as much as I could. The world's leading authority on the zhan zhuang tradition of qigong was a master in Beijing, Professor Yu Yongnian. I wanted to learn directly from him.\n\nSadly, this was at the time of the Cultural Revolution in China. It was a very dangerous period. Since I was from Hong Kong (still a British colony at that time), I would be viewed as a foreigner, and traveling to Mainland China to visit Professor Yu could create difficulties for him. So I wrote to him; luckily, this was possible. After we had exchanged many polite letters, I told him I wanted to study with him. He replied that I should come to take tea with him at his home.\n\nEven though having a foreign visitor could place him in danger, he welcomed me. He served me tea in the traditional way, and we had a polite conversation. Then it was time for me to leave. As I left his home, he said nothing about whether I could be his student, but he told me, \"Please take tea with me again.\"\n\nI ended up visiting him three times, and only at the end of our third visit did he say he would start sharing some ideas with me. When I returned the next time, he said to me, \"Now you will need to start all over again at the very beginning.\" In this way, I became his devoted student. Teaching me was not a decision he took lightly\u2014that's why he had wanted to see me again and again. He wanted to observe me, listen to the way I spoke, watch the way I handled a teacup, get a feeling of my own attitude toward him, and examine what I had already learned. At the same time, we were strengthening the connection between us\u2014the invisible channel that would carry centuries of wisdom and experience forward to the future.\n\nI owe him a huge debt of gratitude, because he taught me in a very deep and meaningful way. What he transmitted was something that can only be passed on through the power of lineage.\n\nLineage is how a truly authentic art is handed down and kept alive, transmitted in a very personal way. That is why I want you, dear reader, to know about the centuries of human learning through which this art has been developed and passed on. Of course, reading a book is not the same as learning directly from a lineage holder. But now that you have started to take an interest in this art, if you read and respect the stories, photos, and statements in these pages\u2014which I have put together to give you a sense of our shared history\u2014I hope you will have a feeling of what it might be like to start taking tea with a master.\n\nI first heard about Professor Yu in the 1980s. It was not possible for me to meet him\u2014I could only write to him, because China was still in the grip of the Cultural Revolution. Later, I was finally able to travel to Beijing and was accepted as one of his students. Even though I now live and teach in the West, I continue to respect the classical tradition of the master-student relationship and, until his death in 2013, returned regularly to China to practice with Professor Yu and learn from his lifetime of experience in this art.\n\nThis calligraphy is a gift that I received from Professor Yu. A literal translation is \"Follow the Tao, day by day, less and less.\" It means that as we continue practicing qigong correctly every day, we gradually experience less and less of whatever troubles us. Less anxiety, less tension, less pain. It is like polishing something precious\u2014slowly the accumulated dust and dirt are removed and the original quality is revealed.\n\nYou may sometimes see \"qigong\" spelled \"chi kung,\" though the pronunciation is always the same. So why the difference in spelling? It reflects two ways of transliterating Chinese into English. \"Chi kung\" is the spelling using the Wade-Giles system of transliteration, which was developed by two British diplomats serving in China in the nineteenth century. Wade-Giles was widely in use throughout most of the twentieth century. Most of the Chinese masters who preserved the classical Chinese arts during the often catastrophic turbulence of that century used the Wade-Giles system when writing their teachings in English. I grew up using that system, and all my books have used it. Nowadays, however, the newer pinyin system of transliteration is used in China, having been adopted by the government. This spelling is much more common in the United States, and so I have used it in this book. To help readers familiar with the earlier system of transliteration, I have included that in parentheses on first reference. I have kept most of the titles of works or names that used the earlier system in that form.\n\nIf You Are Opening the Book Here and Looking for Help in a Hurry\n\nPeople tell me that they are drowning. They may be drowning in their work, the pressure of deadlines, or the overwhelming demands and conflicts in their lives. They need to come up for air.\n\nThere's a powerful qigong practice for this. It's called \"Coming Up for Air.\" You can try it for yourself: there are photos and clear instructions in part 1 of this book.\n\n\"Qigong,\" pronounced \"chee gung,\" literally means \"internal energy work.\" The system has been in use for twenty-seven centuries and is part of China's medical heritage. It helps the flow of vital energy in the body.\n\nDon't worry about the theory. You can read that later. This book is designed for practical use. You can pick out anything you would like to try. It has many illustrations and is designed to be readable at a glance, so you can rest the book beside you and follow the instructions. Many of the exercises in this book are designed to address particular problems.\n\nIf you start worrying first thing in the morning\u2014before you even get out of bed\u2014or if you suffer from insomnia, go to the exercises, \"Overwhelmed, Lying Down\" and \"In the Middle of the Night\" in part 3. You can do these deep practices lying on your back, in bed or on the floor, in the morning or at night.\n\nIf you suffer from panic attacks or constant anxiety that makes you tense up all the time, go to \"Tensing and Relaxing\" in part 2. You can also try \"Clenching and Relaxing.\" You clench and stretch your hands, using your muscles to trigger a relaxation effect in your nervous system, similar to using a stress ball.\n\nIf you want to find out in depth how to use this ancient health system to handle your anxiety and develop your inner strength, I recommend you read the introduction and then start with the exercises in part 1. You can use the charts to keep track of how you are doing.\n\nIf, however, you're eager to jump right in, just do that. You can open the book at any page and try whatever exercise pops up there. They are all designed to help you work with your anxiety.\n\n## \"Pervading Anxiety\"\n\nOne of my students is a commercial program manager. When he first came to see me, he told me, \"I am experiencing pervasive anxiety.\" He had done some martial arts when he was younger and decided some kind of internal energy practice might help him with his inner distress. Later, he told me how he was doing.\n\nMy anxiety has decreased substantially. Once I started learning from you, even when my anxiety arises I am more able to just observe it, which allows it to pass more easily, without me becoming embroiled in a futile battle against fear.\n\nI find that through practicing qigong, I am able to reach a state of peace, calm, and serenity that is unobtainable by any other means of meditation, relaxation, or even sleep. This is after having tried various meditation and relaxation techniques over a period of around six years.\n\nAfter practicing qigong, it feels like I have just had a shower. People often comment on how fresh and relaxed I look, which does not happen if I have not practiced for a while!\n\nYou can pass through the whole cycle of emotions, from resistance, anger, fear, pain, and elation....You learn to let go and observe the experiences and sensations, which is good practice and training for the many obstacles that appear in daily life.\n\nQigong is different from many other systems, because you are not \"trying\" to do anything. In other systems, you often have to try to send energy here or there, or you focus on your breathing. This often either leads to the energy only being in your head or causes tension from the \"trying.\" The zhan zhuang system seems to be more natural, as it seems to work on the basis that if you relax, the energy does the work for you, and your mind and body eventually end up where they need to be.\n\nEven after practicing for a short period of time, it is possible to quickly and easily obtain a state of consciousness where all of your worries disappear while strengthening your body and spirit. Over time, this sense of well-being also spreads into your daily life.\n\n# About the Author\n\nMaster Lam has helped millions on the road to health. He has devoted his life to the study of qigong and the other healing arts of Chinese culture. From an early age he learned under some of the greatest masters of his time, in Hong Kong, Taiwan, and China. He has been teaching in the West since 1975, opening up the secrets of Chinese health care through his books, videos, workshops, and classes.\n\nMaster Lam is a lineage holder of the classical zhan zhuang qigong tradition established by Grand Master Wang Xiang Zhai. This is widely known as \"Standing Like a Tree,\" an art Master Lam first introduced to the West in his groundbreaking book _The Way of Energy_. It is this time-tested system for regenerative health that lies at the heart of this workbook.\n\nMaster Lam is also the founder of Lam-style Tai Chi Chuan, developed to enable people in the West to learn and practice the principal foundational movements of tai chi chuan.\n\nBorn in Hong Kong at the end of the Second World War, Master Lam first studied under masters such as Lung Tse-Cheung, a disciple of Grand Master Hu Yue-Chang, known throughout China as \"the king of Iron Palm.\" He was trained in the arts of Choy Lee Fut, Northern Shaolin Kung Fu, and Iron Palm, as well as tai chi.\n\nHe then studied Chinese medicine, becoming a qualified bonesetter and herbalist, and opened a school and clinic in Hong Kong. At this stage in his career, he was introduced to a master trained in the tradition of Grand Master Wang Xiang Zhai, who had opened up the world of zhan zhuang qigong to the public. Master Lam went on to study in Beijing under Professor Yu Yongnian, a grand master of the art of zhan zhuang. Professor Yu, who recently died, was the world's leading authority on this system of qigong; his books are widely used in China.\n\nMaster Lam first settled in London in the mid-1970s. He became the first person officially recognized by the Inner London Educational Authority to teach the art of tai chi, thus opening up this field for formal acceptance in education. He opened a clinic in London and established both the Lam Association of Cultural Arts and the Zhan Zhuang Chi Kung Research Group (Europe), with members in nearly a dozen countries. During this time, he demonstrated the power of his art by using qigong to heal himself after a near-fatal accident and a life-threatening illness.\n\nIn 2009, Master Lam explored teaching and offering healing treatments in the United States. He settled in Piedmont, California. Now he divides his time between Europe and North America, giving private teaching, public classes and workshops, and professional seminars, as well as offering KTL Therapy, a synthesis of all his life experience named for the members of his family who work with him to provide it.\n\nMaster Lam is the author of a range of books on tai chi, qigong, and feng shui. Among his most widely read works, published in over a dozen languages, are _Step-by-Step Tai Chi_ and his qigong trilogy, _The Way of Energy_ , _The Way of Healing_ , and _The Way of Power_. He is well-known as the presenter of the ten-part Channel Four series _Stand Still\u2014Be Fit_ , which you can watch on Master Lam's website (see the section \"Master Lam on Film\" for more).\n\n# Healing and Training with Master Lam\n\nMaster Lam offers personal training, public classes, workshops, and professional seminars. He also offers qigong healing treatments. He is based in Piedmont, California, and in London, England. He travels widely throughout the year.\n\nIt is important to understand that the practices and recommendations in this book are based on a lifetime of study and practice. You are more than welcome to experiment with them yourself. Using this book, while beneficial to you personally, does not constitute an authorization to teach qigong to others, to present yourself as a student of Master Lam, or to publish this information in books or on websites. If you would like to take your training further or explore how to share it with others who might benefit, please contact Master Lam to discuss this and find out about the training and worksops he offers.\n\n## Healing Treatments\n\nMaster Lam has a lifetime of experience in healing a wide range of disorders. Through his own years of research and experience, Master Lam now offers a qigong healing system known as KTL Therapy. It is based on accumulated research into the healing properties of countless herbs, the physical structures of the body, and the pathways through which our internal energy moves. This system has proven to be effective in helping people with many illnesses and injuries. Testimonials are published on his website, <http:\/\/www.lamkamchuen.org>. The site includes a list of conditions that can be helped by this therapy. These include dorsalgia (back pain), whiplash, sciatica, joint dislocation, accidents and sports injuries, and neck and shoulder pain, as well as hip, knee, and ankle pain.\n\nIf you would like to make an appointment for treatment with Master Lam, please contact him. His phone number in the United States is 1-510-666-7889. His international headquarters is in London, England: +44 207 261 9049. His e-mail address is hq@lamassociation.org.\n\n## Personal Training\n\nIf you would like to study privately with Master Lam, please contact him for an appointment. His phone number in the United States is 1-510-666-7889. His international headquarters is in London, England: +44 207 261 9049. His e-mail address is hq@lamassociation.org.\n\n## Classes and Workshops\n\nMaster Lam offers public classes in Piedmont, California. For information about current classes, please visit his website, <http:\/\/www.lamkamchuen.com>. Master Lam is willing to arrange, on request, workshops on qigong for specific conditions and illnesses. If you are interested in arranging a workshop for Master Lam anywhere in the world, please send a written proposal to hq@lamassociation.org.\n\n## Professional Seminars\n\nIf you are a health professional interested in the application of qigong in the care of people suffering from anxiety or other conditions, Master Lam is more than willing to discuss offering specialized professional seminars. If you are interested in arranging a professional seminar anywhere in the world with Master Lam, please send a written proposal to hq@lamassociation.org.\n\n## Websites\n\nMaster Lam's websites include:\n\n<http:\/\/www.lamhealing.com>\n\nThis is the main website, where you will find details of the healing treatments that Master Lam offers using qigong and the other health systems in which he has been trained and that he has practiced throughout his life.\n\n<http:\/\/www.lamkamchuen.org>\n\nThis is the principal website for the classes and workshops that Master Lam and his sons offer in the United States. It also provides helpful background on the healing tradition they preserve.\n\n<http:\/\/www.lamkamchuen.com>\n\nThis is Master Lam's international website, which has extensive details on his art, his books and films, his classes, and his healing work worldwide.\n\n<http:\/\/www.lamassociation.org>\n\nThis is the website of the Lam Association of Cultural Art, Europe. This is Master Lam's European hub. It includes detailed information about Master Lam and his activities in European countries as well as links to his qualified instructors.\n\n## _Inner Strength_\n\n### Anxiety Workshops and Personal Consultations with Master Lam\n\n_Physicians write over ninety-four million prescriptions for anti-anxiety medication annually in the United States alone.*_\n\nWhether you are taking medication, or receiving therapy for anxiety\u2014or looking for a totally different treatment plan\u2014the centuries-old qigong system of internal energy strengthening can help you. It offers a different, time-tested method for counteracting the destabilizing effects of anxiety. It builds your internal reservoir of inner strength while allowing you to continue with any other form of treatment you wish.\n\nInternational qigong authority Master Lam offers workshops and personal consultations on request.\n\nIf you would like to arrange a workshop or personal consultation with Master Lam, please contact him. His phone number in the United States is 1-510-666-7889. His international headquarters is in London, England: +44 207 261 9049. His e-mail address is hq@lamassociation.org.\n\n* Miller, Lisa. 2012. \"Listening to Xanax.\" New York, March 18. http:\/\/nymag.com\/news \/features\/xanax-2012-3\/.\n\n# Books and Videos with Master Lam\n\nThrough his books and videos Master Lam has introduced and taught a range of Chinese cultural arts in the West. His works have sold over a million copies, in seventeen languages. For copies of his books, please consult your local or online bookseller. For videos and DVDs, please contact Master Lam directly at http:\/\/www .lamkamchuen.org or e-mail hq@lamassociation.org.\n\n_The Way of Energy_\n\nA comprehensive qigong manual, the first book in Master Lam's trilogy. This is the groundbreaking work that introduced the zhan zhuang system of qigong to the Western world. This book provides an understanding of the human energy system, including practical exercises to help harness internal energy and stimulate blood circulation.\n\n_The Way of Healing_\n\nThe second book in this trilogy presents a simple program of qigong and tai chi exercises to gently improve the inner energy flow, which is vital to good health. The routines are suitable for all ages and levels of fitness, and there are special modifications for people with disabilities and those who are in poor health.\n\n_The Way of Power_\n\nThe third part of the trilogy introduces _Da Cheng Chuan_ (\"the great accomplishment\"). This is the most powerful of all the martial arts. The book features classical qigong exercises, but also movements drawn from this extraordinary martial art.\n\n_Everyday Chi Kung_\n\nA program of seated qigong to fit in your day, at work, home, or play. Offering a fifteen-minute-a-day routine, this is the first book to focus on seated qigong.\n\n_Walking Chi Kung_\n\nA manual that focuses on qigong in motion and includes over fifteen different walking steps and systems.\n\n_Step-by-Step Tai Chi_\n\nThe leading beginner's guide to this ancient art, featuring clear step-by-step instructions and color photographs of the basic tai chi movements.\n\n_Tai Chi for Staying Young_\n\nA simple exercise program for all ages, featuring careful, step-by-step descriptions of movements and exercises that can be performed as a daily routine at home or in any other convenient location.\n\n_The Feng Shui Handbook_\n\nThis guide presents the philosophy of this art and shows how to use feng shui to bring harmonious energy into the home or office, thus promoting both health and general well-being.\n\n_The Personal Feng Shui Manual_\n\nA practical guide to creating your individual feng shui chart and direction finder to create harmony in everyday situations.\n\n_The Feng Shui Kitchen_\n\nThe energy of food; feng shui essentials; preparing and cooking; and recipes for the four seasons.\n\n_The Way of Tea_\n\nA detailed guide to the art of making and enjoying this delicious, healthy, and complex drink.\n\n## Master Lam on Film\n\n_Stand Still\u2014Be Fit_\n\nA ten-part television series that presents step-by-step qigong that stimulates the internal body, enhancing physical energy and promoting health and well-being. The complete series is available online at http:\/\/www.lamkamchuen.org\/Simple_Exercise\/Simple _Exercise.html.\n\n_The Way of Power_\n\nFilmed in China and London, it contains nearly fifty minutes of unique demonstrations by Master Lam, Professor Yu, Madame Wang Yuk Fong (daughter of Grand Master Wang Xiang Zhai), and students. It also includes a unique demonstration of internal power.\n\n_Golden Ball Tai Chi_\n\nClassical movement compiled by Master Lam for his students. Strengthens your body, clears your mind, and lifts your spirit. Filmed on locations with Master Lam teaching local citizens, it includes simple, rejuvenating exercises that can be done by people who need to sit or lie down.>\n\n# References\n\n\"Anxiety.\" 2013. _Opinionator_ (blog). _New York Times._ <http:\/\/opinionator.blogs.nytimes.com\/category\/anxiety\/>.\n\n\"The Hubble Ultra Deep Field in 3D.\" YouTube video, 4:17, posted by \"Deep Astronomy.\" August 2, 2009. <http:\/\/www.youtube.com\/watch?v=oAVjF_7ensg>.\n\nMiller, Lisa. 2012. \"Listening to Xanax.\" _New York,_ March 18. http:\/\/nymag.com\/news \/features\/xanax-2012-3\/.\n\nNeedham, Joseph, Ling Wang, and Kenneth Robinson. 1962. _Science and Civilisation in China._ Vol. 4, _Physics and Physical Technology._ Cambridge: Cambridge University Press.\n\nRicard, Matthieu, and Xuan Thuan Trinh. 2001. _The Quantum and the Lotus: A Journey to the Frontiers Where Science and Buddhism Meet._ New York: Crown.\n\n**Master Kam Chuen Lam** has been a traditional Chinese healer for over forty years. He is also an internationally respected authority in the Chinese health systems of qigong and tai chi. He has produced over ten titles in these subjects and they have been sold across the world in over seventeen languages.\n\n**Grandmaster Yu Yongnian** wrote the foreword to this workbook months before his death at age 94. He was the world's leading authority on the rare qigong system known as \"standing like a tree.\" As a medical doctor trained in Western medicine, he brought the health benefits of this ancient system into hospitals and clinics throughout China. He was a member of China's National Qigong Research Council and was a consultant to the American-Chinese Qigong Research Group. He was also a consultant to the Da Cheng Chuan Zhan Zhuang Chi Kung Research Groups (Europe).\n\nProfessor Yu travelled and taught in both Asian and European countries. He supervised the production of Master Kam Chuen Lam's first book, _The Way of Energy,_ and appears in both _The Way of Power_ and its companion video in a rare demonstration of his teaching methods and power.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nChuck Greif and the Online Distributed Proofreading Team\nat http:\/\/www.pgdp.net (This file was produced from images\navailable at The Internet Archive)\n\n\n\n\n\n\n\n\n           [Illustration: Faithfully yours Charles Dickens]\n\n\n\n\n                            PEARL-FISHING.\n\n                            CHOICE STORIES,\n\n                                 FROM\n\n                       Dickens' Household Words.\n\n                            SECOND SERIES.\n\n                                AUBURN:\n                        ALDEN, BEARDSLEY & CO.\n                              ROCHESTER:\n                        WANZER, BEARDSLEY & CO.\n                                 1854.\n\n      ENTERED, according to Act of Congress, in the year 1854, by\n\n                        ALDEN, BEARDSLEY & CO.,\n\nIn the Clerk's Office of the District Court of the United States for the\n                    Northern District of New York.\n\n                           THOMAS B. SMITH,\n                     STEREOTYPER AND ELECTROTYPER,\n                       216 William Street, N. Y.\n\n\n\n\nThe Publisher's Notice.\n\n\nThe large demand for the _First Series_ of this publication, has\nconfirmed the publishers in their opinion of its worth and its\nadaptability to meet the wants and tastes of the reading public, and\ninduced them to issue, in rapid succession, the present volume, which\nwill be found not less interesting and worthy of attention.\n\nThe publishers also announce their intention of continuing this series,\nwhich has been received with so much public favor.\n\nJune, 1854.\n\n\n\n\nContents.\n\n\n                                             PAGE\n\n  I.--THE YOUNG ADVOCATE                        7\n\n II.--THE LAST OF A LONG LINE                  33\n\nIII.--THE GENTLEMAN BEGGAR                    107\n\n IV.--EVIL IS WROUGHT BY WANT OF THOUGHT      130\n\n  V.--BED                                     167\n\n VI.--THE HOME OF WOODRUFFE THE GARDENER      184\n\n VII.--THE WATER-DROPS                        287\n\nVIII.--AN EXCELLENT OPPORTUNITY               325\n\n\n\n\nI.\n\nThe Young Advocate.\n\n\nAntoine de Chaulieu was the son of a poor gentleman of Normandy, with a\nlong genealogy, a short rent-roll, and a large family. Jacques Rollet\nwas the son of a brewer, who did not know who his grandfather was; but\nhe had a long purse and only two children. As these youths flourished in\nthe early days of liberty, equality, and fraternity, and were near\nneighbors, they naturally hated each other. Their enmity commenced at\nschool, where the delicate and refined De Chaulieu being the only\ngentilhomme among the scholars, was the favorite of the master (who was\na bit of an aristocrat in his heart) although he was about the worst\ndressed boy in the establishment, and never had a sou to spend; while\nJacques Rollet, sturdy and rough, with smart clothes and plenty of\nmoney, got flogged six days in the week, ostensibly for being stupid and\nnot learning his lessons--which, indeed, he did not--but, in reality,\nfor constantly quarrelling with and insulting De Chaulieu, who had not\nstrength to cope with him. When they left the academy, the feud\ncontinued in all its vigor, and was fostered by a thousand little\ncircumstances arising out of the state of the times, till a separation\nensued in consequence of an aunt of Antoine de Chaulieu's undertaking\nthe expense of sending him to Paris to study the law, and of maintaining\nhim there during the necessary period.\n\nWith the progress of events came some degree of reaction in favor of\nbirth and nobility, and then Antoine, who had passed for the bar, began\nto hold up his head and endeavored to push his fortunes; but fate seemed\nagainst him. He felt certain that if he possessed any gift in the world\nit was that of eloquence, but he could get no cause to plead; and his\naunt dying inopportunely, first his resources failed, and then his\nhealth. He had no sooner returned to his home, than, to complicate his\ndifficulties completely, he fell in love with Mademoiselle Natalie de\nBellefonds, who had just returned from Paris, where she had been\ncompleting her education. To expiate on the perfections of Mademoiselle\nNatalie, would be a waste of ink and paper; it is sufficient to say that\nshe really was a very charming girl, with a fortune which, though not\nlarge, would have been a most desirable acquisition to De Chaulieu, who\nhad nothing. Neither was the fair Natalie indisposed to listen to his\naddresses; but her father could not be expected to countenance the suit\nof a gentleman, however well-born, who had not a ten-sous piece in the\nworld, and whose prospects were a blank.\n\nWhile the ambitious and love-sick young barrister was thus pining in\nunwelcome obscurity, his old acquaintance, Jacques Rollet, had been\nacquiring an undesirable notoriety. There was nothing really bad in\nJacques' disposition, but having been bred up a democrat, with a hatred\nof the nobility, he could not easily accommodate his rough humor to\ntreat them with civility when it was no longer safe to insult them. The\nliberties he allowed himself whenever circumstances brought him into\ncontact with the higher classes of society, had led him into many\nscrapes, out of which his father's money had one way or another released\nhim; but that source of safety had now failed. Old Rollet having been\ntoo busy with the affairs of the nation to attend to his business, had\ndied insolvent, leaving his son with nothing but his own wits to help\nhim out of future difficulties, and it was not long before their\nexercise was called for. Claudine Rollet, his sister, who was a very\npretty girl, had attracted the attention of Mademoiselle de Bellefonds'\nbrother, Alphonso; and as he paid her more attention than from such a\nquarter was agreeable to Jacques, the young men had more than one\nquarrel on the subject, on which occasions they had each,\ncharacteristically, given vent to their enmity, the one in contemptuous\nmonosyllables, and the other in a volley of insulting words. But\nClaudine had another lover more nearly of her own condition of life;\nthis was Claperon, the deputy-governor of the Rouen jail, with whom she\nmade acquaintance during one or two compulsory visits paid by her\nbrother to that functionary; but Claudine, who was a bit of a coquette,\nthough she did not altogether reject his suit, gave him little\nencouragement, so that betwixt hopes, and fears, and doubts, and\njealousies, poor Claperon led a very uneasy kind of life.\n\nAffairs had been for some time in this position, when, one fine morning,\nAlphonse de Bellefonds was not to be found in his chamber when his\nservant went to call him; neither had his bed been slept in. He had been\nobserved to go out rather late on the preceding evening, but whether or\nnot he had returned, nobody could tell. He had not appeared at supper,\nbut that was too ordinary an event to awaken suspicion; and little alarm\nwas excited till several hours had elapsed, when inquiries were\ninstituted and a search commenced, which terminated in the discovery of\nhis body, a good deal mangled, lying at the bottom of a pond which had\nbelonged to the old brewery. Before any investigations had been made,\nevery person had jumped to the conclusion that the young man had been\nmurdered, and that Jacques Rollet was the assassin. There was a strong\npresumption in favor of that opinion, which further perquisitions tended\nto confirm. Only the day before, Jacques had been heard to threaten M.\nde Bellefonds with speedy vengeance. On the fatal evening, Alphonse and\nClaudine had been seen together in the neighborhood of the now\ndismantled brewery; and as Jacques, betwixt poverty and democracy, was\nin bad odor with the prudent and respectable part of society, it was not\neasy for him to bring witnesses to character, or prove an\nunexceptionable alibi. As for the Bellefonds and De Chaulieus, and the\naristocracy in general, they entertained no doubt of his guilt; and\nfinally, the magistrates coming to the same opinion, Jacques Rollet was\ncommitted for trial, and as a testimony of good will Antoine de\nChaulieu was selected by the injured family to conduct the prosecution.\n\nHere, at last, was the opportunity he had sighed for! So interesting a\ncase, too, furnishing such ample occasion for passion, pathos,\nindignation! And how eminently fortunate that the speech which he set\nhimself with ardor to prepare, would be delivered in the presence of the\nfather and brother of his mistress, and perhaps of the lady herself! The\nevidence against Jacques, it is true, was altogether presumptive; there\nwas no proof whatever that he had committed the crime; and for his own\npart he stoutly denied it. But Antoine de Chaulieu entertained no doubt\nof his guilt, and his speech was certainly well calculated to carry\nconviction into the bosom of others. It was of the highest importance to\nhis own reputation that he should procure a verdict, and he confidently\nassured the afflicted and enraged family of the victim that their\nvengeance should be satisfied. Under these circumstances could anything\nbe more unwelcome than a piece of intelligence that was privately\nconveyed to him late on the evening before the trial was to come on,\nwhich tended strongly to exculpate the prisoner, without indicating any\nother person as the criminal. Here was an opportunity lost. The first\nstep of the ladder on which he was to rise to fame, fortune, and a wife,\nwas slipping from under his feet!\n\nOf course, so interesting a trial was anticipated with great eagerness\nby the public, and the court was crowded with all the beauty and fashion\nof Rouen. Though Jacques Rollet persisted in asserting his innocence,\nfounding his defence chiefly on circumstances which were strongly\ncorroborated by the information that had reached De Chaulieu the\npreceding evening,--he was convicted.\n\nIn spite of the very strong doubts he privately entertained respecting\nthe justice of the verdict, even De Chaulieu himself, in the first flush\nof success, amid a crowd of congratulating friends, and the approving\nsmiles of his mistress, felt gratified and happy; his speech had, for\nthe time being, not only convinced others, but himself; warmed with his\nown eloquence, he believed what he said. But when the glow was over, and\nhe found himself alone, he did not feel so comfortable. A latent doubt\nof Rollet's guilt now burnt strongly in his mind, and he felt that the\nblood of the innocent would be on his head. It is true there was yet\ntime to save the life of the prisoner, but to admit Jacques innocent,\nwas to take the glory out of his own speech, and turn the sting of his\nargument against himself. Besides, if he produced the witness who had\nsecretly given him the information, he should be self-condemned, for he\ncould not conceal that he had been aware of the circumstance before the\ntrial.\n\nMatters having gone so far, therefore, it was necessary that Jacques\nRollet should die; so the affair took its course; and early one morning\nthe guillotine was erected in the court-yard of the jail, three\ncriminals ascended the scaffold, and three heads fell into the basket,\nwhich were presently afterwards, with the trunks that had been attached\nto them, buried in a corner of the cemetery.\n\nAntoine de Chaulieu was now fairly started in his career, and his\nsuccess was as rapid as the first step towards it had been tardy. He\ntook a pretty apartment in the H\u00f4tel de Marb\u0153uf, Rue Grange-Bateli\u00e8re,\nand in a short time was looked upon as one of the most rising young\nadvocates in Paris. His success in one line brought him success in\nanother; he was soon a favorite in society, and an object of interest to\nspeculating mothers; but his affections still adhered to his old love\nNatalie de Bellefonds, whose family now gave their assent to the\nmatch--at least, prospectively--a circumstance which furnished such an\nadditional incentive to his exertions, that in about two years from the\ndate of his first brilliant speech, he was in a sufficiently flourishing\ncondition to offer the young lady a suitable home. In anticipation of\nthe happy event, he engaged and furnished a suite of apartments in the\nRue du Helder; and as it was necessary that the bride should come to\nParis to provide her trousseau, it was agreed that the wedding should\ntake place there, instead of at Bellefonds, as had been first\nprojected; an arrangement the more desirable, that a press of business\nrendered M. de Chaulieu's absence from Paris inconvenient.\n\nBrides and bridegrooms in France, except of the very high classes, are\nnot much in the habit of making those honeymoon excursions so universal\nin this country. A day spent in visiting Versailles, or St. Cloud, or\neven the public places of the city, is generally all that precedes the\nsettling down into the habits of daily life. In the present instance St.\nDenis was selected, from the circumstance of Natalie having a younger\nsister at school there; and also because she had a particular desire to\nsee the Abbey.\n\nThe wedding was to take place on a Thursday; and on the Wednesday\nevening, having spent some hours most agreeably with Natalie, Antoine de\nChaulieu returned to spend his last night in his bachelor apartments.\nHis wardrobe and other small possessions, had already been packed up and\nsent to his future home; and there was nothing left in his room now, but\nhis new wedding suit, which he inspected with considerable satisfaction\nbefore he undressed and lay down to sleep. Sleep, however, was somewhat\nslow to visit him; and the clock had struck _one_ before he closed his\neyes. When he opened them again, it was broad day-light; and his first\nthought was, had he overslept himself? He sat up in bed to look at the\nclock which was exactly opposite, and as he did so, in the large mirror\nover the fire-place, he perceived a figure standing behind him. As the\ndilated eyes met his own, he saw it was the face of Jacques Rollet.\nOvercome with horror he sunk back on his pillow, and it was some minutes\nbefore he ventured to look again in that direction; when he did so, the\nfigure had disappeared.\n\nThe sudden revulsion of feeling such a vision was calculated to occasion\nin a man elate with joy, may be conceived! For some time after the death\nof his former foe, he had been visited by not unfrequent twinges of\nconscience; but of late, borne along by success, and the hurry of\nParisian life, these unpleasant remembrances had grown rarer, till at\nlength they had faded away altogether. Nothing had been further from his\nthoughts than Jacques Rollet, when he closed his eyes on the preceding\nnight, nor when he opened them to that sun which was to shine on what he\nexpected to be the happiest day of his life! Where were the high-strung\nnerves now! The elastic frame! The bounding heart!\n\nHeavily and slowly he arose from his bed, for it was time to do so; and\nwith a trembling hand and quivering knees, he went through the processes\nof the toilet, gashing his cheek with the razor, and spilling the water\nover his well-polished boots. When he was dressed, scarcely venturing to\ncast a glance in the mirror as he passed it, he quitted the room and\ndescended the stairs, taking the key of the door with him for the\npurpose of leaving it with the porter; the man, however, being absent,\nhe laid it on the table in his lodge, and with a relaxed and languid\nstep proceeded on his way to the church, where presently arrived the\nfair Natalie and her friends. How difficult it was now to look happy\nwith that pallid face and extinguished eye!\n\n\"How pale you are! Has anything happened? You are surely ill?\" were the\nexclamations that met him on all sides. He tried to carry it off as well\nas he could, but felt that the movements he would have wished to appear\nalert were only convulsive; and that the smiles with which he attempted\nto relax his features, were but distorted grimaces. However, the church\nwas not the place for further inquiries; and while Natalie gently\npressed his hand in token of sympathy, they advanced to the altar, and\nthe ceremony was performed; after which they stepped into the carriages\nwaiting at the door, and drove to the apartments of Madame de\nBellefonds, where an elegant _d\u00e9jeuner_ was prepared.\n\n\"What ails you, my dear husband?\" inquired Natalie, as soon as they were\nalone.\n\n\"Nothing, love,\" he replied; \"nothing, I assure you, but a restless\nnight and a little overwork, in order that I might have to-day free to\nenjoy my happiness!\"\n\n\"Are you quite sure? Is there nothing else?\"\n\n\"Nothing, indeed; and pray don't take notice of it, it only makes me\nworse!\"\n\nNatalie was not deceived, but she saw that what he said was true; notice\nmade him worse; so she contented herself with observing him quietly, and\nsaying nothing; but, as he _felt_ she was observing him, she might\nalmost better have spoken; words are often less embarrassing things than\ntoo curious eyes.\n\nWhen they reached Madame de Bellefonds' he had the same sort of\nquestioning and scrutiny to undergo, till he grew quite impatient under\nit, and betrayed a degree of temper altogether unusual with him. Then\neverybody looked astonished; some whispered their remarks, and others\nexpressed them by their wondering eyes, till his brow knit, and his\npallid cheeks became flushed with anger. Neither could he divert\nattention by eating; his parched mouth would not allow him to swallow\nanything but liquids, of which, however, he indulged in copious\nlibations; and it was an exceeding relief to him when the carriage,\nwhich was to convey them to St. Denis, being announced, furnished an\nexcuse for hastily leaving the table. Looking at his watch, he declared\nit was late; and Natalie, who saw how eager he was to be gone, threw her\nshawl over her shoulders, and bidding her friends _good morning_, they\nhurried away.\n\nIt was a fine sunny day in June; and as they drove along the crowded\nboulevards, and through the Porte St. Denis, the young bride and\nbridegroom, to avoid each others eyes, affected to be gazing out of the\nwindows; but when they reached that part of the road where there was\nnothing but trees on each side, they felt it necessary to draw in their\nheads, and make an attempt at conversation, De Chaulieu put his arm\nround his wife's waist, and tried to rouse himself from his depression;\nbut it had by this time so reacted upon her, that she could not respond\nto his efforts, and thus the conversation languished, till both felt\nglad when they reached their destination, which would, at all events,\nfurnish them something to talk about.\n\nHaving quitted the carriage, and ordered a dinner at the H\u00f4tel de\nl'Abbaye, the young couple proceeded to visit Mademoiselle Hortense de\nBellefonds, who was overjoyed to see her sister and new brother-in-law,\nand doubly so when she found that they had obtained permission to take\nher out to spend the afternoon with them. As there is little to be seen\nat St. Denis but the Abbey, on quitting that part of it devoted to\neducation, they proceeded to visit the church, with its various objects\nof interest; and as De Chaulieu's thoughts were now forced into another\ndirection, his cheerfulness began insensibly to return. Natalie looked\nso beautiful, too, and the affection betwixt the two young sisters was\nso pleasant to behold! And they spent a couple of hours wandering about\nwith Hortense, who was almost as well informed as the Suisse, till the\nbrazen doors were open which admitted them to the Royal vault.\nSatisfied, at length, with what they had seen, they began to think of\nreturning to the inn, the more especially as De Chaulieu, who had not\neaten a morsel of food since the previous evening, owned to being\nhungry; so they directed their steps to the door, lingering here and\nthere as they went, to inspect a monument or a painting, when, happening\nto turn his head aside to see if his wife, who had stopt to take a last\nlook at the tomb of King Dagobert, was following, he beheld with horror\nthe face of Jacques Rollet appearing from behind a column! At the same\ninstant his wife joined him, and took his arm, inquiring if he was not\nvery much delighted with what he had seen. He attempted to say yes, but\nthe word would not be forced out; and staggering out of the door, he\nalleged that a sudden faintness had overcome him.\n\nThey conducted him to the H\u00f4tel, but Natalie now became seriously\nalarmed; and well she might. His complexion looked ghastly, his limbs\nshook, and his features bore an expression of indescribable horror and\nanguish. What could be the meaning of so extraordinary a change in the\ngay, witty, prosperous De Chaulieu, who, till that morning, seemed not\nto have a care in the world? For, plead illness as he might, she felt\ncertain, from the expression of his features, that his sufferings were\nnot of the body but of the mind; and, unable to imagine any reason for\nsuch extraordinary manifestations, of which she had never before seen a\nsymptom, but a sudden aversion to herself, and regret for the step he\nhad taken, her pride took the alarm, and, concealing the distress she\nreally felt, she began to assume a haughty and reserved manner towards\nhim, which he naturally interpreted into an evidence of anger and\ncontempt. The dinner was placed upon the table, but De Chaulieu's\nappetite, of which he had lately boasted, was quite gone, nor was his\nwife better able to eat. The young sister alone did justice to the\nrepast; but although the bridegroom could not eat, he could swallow\nchampagne in such copious draughts, that ere long the terror and remorse\nthat the apparition of Jacques Rollet had awakened in his breast were\ndrowned in intoxication. Amazed and indignant, poor Natalie sat\nsilently observing this elect of her heart, till overcome with\ndisappointment and grief, she quitted the room with her sister, and\nretired to another apartment, where she gave free vent to her feelings\nin tears.\n\nAfter passing a couple of hours in confidences and lamentations, they\nrecollected that the hours of liberty granted, as an especial favor, to\nMademoiselle Hortense, had expired; but ashamed to exhibit her husband\nin his present condition to the eyes of strangers, Natalie prepared to\nre-conduct her to the _Maison Royale_ herself. Looking into the\ndining-room as they passed, they saw De Chaulieu lying on a sofa fast\nasleep, in which state he continued when his wife returned. At length,\nhowever, the driver of their carriage begged to know if Monsieur and\nMadame were ready to return to Paris, and it became necessary to arouse\nhim. The transitory effects of the champagne had now subsided; but when\nDe Chaulieu recollected what had happened, nothing could exceed his\nshame and mortification. So engrossing indeed were these sensations\nthat they quite overpowered his previous one, and, in his present\nvexation, he, for the moment, forgot his fears. He knelt at his wife's\nfeet, begged her pardon a thousand times, swore that he adored her, and\ndeclared that the illness and the effect of the wine had been purely the\nconsequences of fasting and over-work. It was not the easiest thing in\nthe world to re-assure a woman whose pride, affection, and taste, had\nbeen so severely wounded; but Natalie tried to believe, or to appear to\ndo so, and a sort of reconciliation ensued, not quite sincere on the\npart of the wife, and very humbling on the part of the husband. Under\nthese circumstances it was impossible that he should recover his spirits\nor facility of manner; his gaiety was forced, his tenderness\nconstrained; his heart was heavy within him; and ever and anon the\nsource whence all this disappointment and woe had sprung would recur to\nhis perplexed and tortured mind.\n\nThus mutually pained and distrustful, they returned to Paris, which they\nreached about nine o'clock. In spite of her depression, Natalie, who\nhad not seen her new apartments, felt some curiosity about them, whilst\nDe Chaulieu anticipated a triumph in exhibiting the elegant home he had\nprepared for her. With some alacrity, therefore, they stepped out of the\ncarriage, the gates of the H\u00f4tel were thrown open, the _concierge_ rang\nthe bell which announced to the servants that their master and mistress\nhad arrived, and whilst these domestics appeared above, holding lights\nover the balustrades, Natalie, followed by her husband, ascended the\nstairs. But when they reached the landing-place of the first flight,\nthey saw the figure of a man standing in a corner as if to make way for\nthem; the flash from above fell upon his face, and again Antoine de\nChaulieu recognized the features of Jacques Rollet!\n\nFrom the circumstance of his wife's preceding him, the figure was not\nobserved by De Chaulieu till he was lifting his foot to place it on the\ntop stair: the sudden shock caused him to miss the step, and, without\nuttering a sound, he fell back, and never stopped till he reached the\nstones at the bottom. The screams of Natalie brought the concierge from\nbelow and the maids from above, and an attempt was made to raise the\nunfortunate man from the ground; but with cries of anguish he besought\nthem to desist.\n\n\"Let me,\" he said, \"die here! What a fearful vengeance is thine! Oh,\nNatalie, Natalie!\" he exclaimed to his wife, who was kneeling beside\nhim, \"to win fame, and fortune, and yourself, I committed a dreadful\ncrime! With lying words I argued away the life of a fellow-creature,\nwhom, whilst I uttered them, I half believed to be innocent; and now,\nwhen I have attained all I desired, and reached the summit of my hopes,\nthe Almighty has sent him back upon the earth to blast me with the\nsight. Three times this day--three times this day! Again! again!\"--and\nas he spoke, his wild and dilated eyes fixed themselves on one of the\nindividuals that surrounded him.\n\n\"He is delirious,\" said they.\n\n\"No,\" said the stranger! \"What he says is true enough,--at least in\npart;\" and bending over the expiring man, he added, \"May Heaven forgive\nyou, Antoine de Chaulieu! I was not executed; one who well knew my\ninnocence saved my life. I may name him, for he is beyond the reach of\nthe law now,--it was Claperon, the jailor, who loved Claudine, and had\nhimself killed Alphonse de Bellefonds from jealousy. An unfortunate\nwretch had been several years in the jail for a murder committed during\nthe frenzy of a fit of insanity. Long confinement had reduced him to\nidiocy. To save my life Claperon substituted the senseless being for me,\non the scaffold, and he was executed in my stead. He has quitted the\ncountry, and I have been a vagabond on the face of the earth ever since\nthat time. At length I obtained, through the assistance of my sister,\nthe situation of concierge in the H\u00f4tel Marb\u0153uf, in the Rue\nGrange-Bateli\u00e8re. I entered on my new place yesterday evening, and was\ndesired to awaken the gentleman on the third floor at seven o'clock.\nWhen I entered the room to do so, you were asleep, but before I had time\nto speak you awoke, and I recognized your features in the glass.\nKnowing that I could not vindicate my innocence if you chose to seize\nme, I fled, and seeing an omnibus starting for St. Denis, I got on it\nwith a vague idea of getting on to Calais, and crossing the Channel to\nEngland. But having only a franc or two in my pocket, or indeed in the\nworld, I did not know how to procure the means of going forward; and\nwhilst I was lounging about the place, forming first one plan and then\nanother, I saw you in the church, and concluding you were in pursuit of\nme, I thought the best way of eluding your vigilance was to make my way\nback to Paris as fast as I could; so I set off instantly, and walked all\nthe way; but having no money to pay my night's lodging, I came here to\nborrow a couple of livres of my sister Claudine, who lives in the fifth\nstory.\"\n\n\"Thank Heaven!\" exclaimed the dying man; \"that sin is off my soul!\nNatalie, dear wife, farewell! Forgive! forgive all!\"\n\nThese were the last words he uttered; the priest, who had been summoned\nin haste, held up the cross before his failing sight; a few strong\nconvulsions shook the poor bruised and mangled frame; and then all was\nstill.\n\nAnd thus ended the Young Advocate's Wedding Day.\n\n\n\n\nII.\n\nThe Last of a Long Line.\n\n\nSir Roger Rockville of Rockville was the last of a very long line. It\nextended from the Norman Conquest to the present century. His first\nknown ancestor came over with William, and must have been a man of some\nmark, either of bone and sinew, or of brain, for he obtained what the\nAmericans would call a prime location. As his name does not occur in the\nRoll of Battle Abbey, he was, of course, not of a very high Norman\nextraction; but he had done enough, it seems, in the way of knocking\ndown Saxons, to place himself on a considerable eminence in this\nkingdom. The centre of his domains was conspicuous far over the country,\nthrough a high range of rock overhanging one of the sweetest rivers in\nEngland. On one hand lay a vast tract of rich marsh land, capable, as\nsociety advanced, of being converted into meadows; and on the other, as\nextensive moorlands, finely undulating, and abounding with woods and\ndeer.\n\nHere the original Sir Roger built his castle on the summit of the range\nof rock, with huts for his followers; and became known directly all over\nthe country of Sir Roger de Rockville, or Sir Roger of the hamlet on the\nRock. Sir Roger, no doubt, was a mighty hunter before the lord of the\nfeudal district: it is certain that his descendants were. For\ngenerations they led a jolly life at Rockville, and were always ready to\nexchange the excitement of the chase for a bit of the civil war. Without\nthat the country would have grown dull, and ale and venison lost their\nflavor. There was no gay London in those days, and a good brisk skirmish\nwith their neighbors in helm and hauberk was the way of spending their\nseason. It was their parliamentary debate, and was necessary to stir\ntheir blood. Protection and Free Trade were as much the great topics of\ninterest as they are now, only they did not trouble themselves so much\nabout Corn-bills. Their bills were of good steel, and their protective\nmeasures were arrows a cloth-yard long. Protection meant a good suit of\nmail; and a castle with its duly prescribed moats, bastions,\nportcullises, and donjon keep. Free Trade was a lively inroad into the\nneighboring baron's lands, and the importation thence of goodly herds\nand flocks. Foreign cattle for home consumption was as _striking an\narticle_ in their markets as in ours, only the blows were expended on\none another's heads, instead of the heads of foreign bullocks--that is,\nbullocks from over the Welch or Scotch Marches, as from beyond the next\nbrook.\n\nThus lived the Rockvilles for ages. In all the iron combats of those\niron times they took care to have their quota. Whether it was Stephen\nagainst Matilda, or Richard against his father, or John against the\nbarons; whether it were York or Lancaster, or Tudor or Stuart. The\nRockvilles were to be found in the _m\u00eal\u00e9e_, and winning power and\nlands. So long as it required only stalwart frames and stout blows, no\nfamily cut a more conspicuous figure. The Rockvilles were at Bosworth\nField. The Rockvilles fought in Ireland under Elizabeth. The Rockvilles\nwere staunch defenders of the cause in the war of Charles I. with his\nParliament. The Rockvilles even fought for James II. at the Boyne, when\nthree-fourths of the most loyal of the English nobility and gentry had\ndeserted him in disgust and indignation. But from that hour they had\nbeen less conspicuous.\n\nThe opposition to the successful party, that of William of Orange, of\ncourse brought them into disgrace; and though they were never molested\non that account, they retired to their estate, and found it convenient\nto be as unobtrusive as possible. Thenceforward you heard no more of the\nRockvilles in the national annals. They became only of consequence in\ntheir own district. They acted as magistrates. They served as high\nsheriffs. They were a substantial county family, and nothing more.\nEducation and civilization advanced; a wider and very different field of\naction and ambition opened upon the aristocracy of England. Our fleets\nand armies abroad, our legislature at home, law and the church,\npresented brilliant paths to the ambition of those thirsting for\ndistinction, and the good things that follow it. But somehow the\nRockvilles did not expand with this expansion. So long as it required\nonly a figure of six feet high, broad shoulders, and a strong arm, they\nwere a great and conspicuous race. But when the head became the member\nmost in request, they ceased to go ahead. Younger sons, it is true,\nserved in army and in navy, and filled the family pulpit, but they\nproduced no generals, no admirals, no arch-bishops. The Rockvilles of\nRockville were very conservative, very exclusive, and very stereotype.\nOther families grew poor, and enriched themselves again by marrying\nplebeian heiresses. New families grew up out of plebeian blood into\ngreatness, and intermingled the vigor of their fresh earth with the\nattenuated aristocratic soil. Men of family became great lawyers, great\nstatesmen, great prelates, and even great poets and philosophers. The\nRockvilles remained high, proud, bigoted, and _born\u00e9_.\n\nThe Rockvilles married Rockvilles, or their first cousins, the\nCraigvilles, simply to prevent property going out of the family. They\nkept the property together. They did not lose an acre, and they were a\nfine, tall, solemn race--and nothing more. What ailed them?\n\nIf you saw Sir Roger Rockville,--for there was an eternal Sir\nRoger--filling his office of high sheriff,--he had a very fine carriage,\nand a very fine retinue in the most approved and splendid of antique\ncostumes;--if you saw him sitting on the bench at quarter sessions, he\nwas a tall, stately, and solemn man. If you saw Lady Rockville shopping,\nin her handsome carriage, with very handsomely attired servants; saw her\nat the county ball, or on the race-stand, she was a tall, aristocratic,\nand stately lady. That was in the last generation--the present could\nboast of no Lady Rockville.\n\nGreat outward respect was shown to the Rockvilles on account of the\nlength of their descent, and the breadth of their acres. They were\nalways, when any stranger asked about them, declared, with a serious and\nimportant air, to be a very ancient, honorable, and substantial family.\n\"Oh! a great family are the Rockvilles, a very great family.\"\n\nBut if you came to close quarters with the members of this great and\nhighly distinguished family, you soon found yourself fundamentally\nastonished; you had a sensation come over you, as if you were trying,\nlike Moses, to draw water from a rock without his delegated power. There\nwas a goodly outside of things before you, but nothing came of it. You\ntalked, hoping to get talking in return, but you got little more than\n\"noes\" and \"yeses,\" and \"oh! indeeds!\" and \"reallys,\" and sometimes not\neven that, but a certain look of aristocratic dignity or dignification,\nthat was meant to serve for all answers. There was a sort of resting on\naristocratic oars or \"sculls,\" that were not to be too vulgarly\nhandled. There was a feeling impressed on you, that eight hundred years\nof descent and ten thousand a-year in landed income did not trouble\nthemselves with the trifling things that gave distinction to lesser\npeople--such as literature, fine arts, politics, and general knowledge.\nThese were very well for those who had nothing else to pride themselves\non, but for the Rockvilles--oh! certainly they were by no means\nrequisite.\n\nIn fact, you found yourself, with a little variation, in the predicament\nof Cowper's people,\n\n        ---- who spent their lives\n    In dropping buckets into empty wells,\n    And _growing tired_ of drawing nothing up.\n\nWho hasn't often come across these \"dry wells\" of society; solemn gulfs\nout of which you can pump nothing up? You know them; they are at your\nelbow every day in large and brilliant companies, and defy the best\nsucking bucket ever invented to extract anything from them. But the\nRockvilles were each and all of this adust description. It was a family\nfeature, and they seemed, if either, rather proud of it. They must be\nso; for proud they were, amazing proud; and they had nothing besides to\nbe proud of, except their acres, and their ancestors.\n\nBut the fact was, they could not help it. It was become organic. They\nhad acted the justice of peace, maintained the constitution against\nupstarts and manufacturers, signed warrants, supported the church and\nthe house of correction, committed poachers, and then rested on the\ndignity of their ancestors for so many generations, that their skulls,\nbrains, constitutions, and nervous systems, were all so completely\nmoulded into that shape and baked into that mould, that a Rockville\nwould be a Rockville to the end of time, if God and Nature would have\nallowed it. But such things wear out. The American Indians and the\nAustralian nations wear out; they are not progressive, and as Nature\nabhors a vacuum, she does not forget the vacuum wherever it may be,\nwhether in a hot desert, or in a cold and stately Rockville;--a very\nancient, honorable, and substantial family, that lies fallow till the\nthinking faculty literally dies out.\n\nFor several generations there had been symptoms of decay about the\nRockville family. Not in its property, that was as large as ever; not in\ntheir personal stature and physical aspect. The Rockvilles continued, as\nthey always had been, a tall and not bad-looking family. But they grew\ngradually less prolific. For a hundred and fifty years past there had\nseldom been more than two, or at most three, children. There had\ngenerally been an heir to the estate, and another to the family pulpit,\nand sometimes a daughter married to some neighboring squire. But Sir\nRoger's father had been an only child, and Sir Roger himself was an only\nchild. The danger of extinction to the family, apparent as it was, had\nnever induced Sir Roger to marry. At the time that we are turning our\nattention upon him, he had reached the mature age of sixty. Nobody\nbelieved that Sir Roger now would marry; he was the last, and likely to\nbe, of his line.\n\nIt is worth while here to take a glance at Sir Roger and his estate.\nThey bore a strange contrast. The one bore all the signs of progress,\nthe other of a stereotyped feudality. The estate which in the days of\nthe first Sir Roger de Rockville had been half morass and half\nwilderness, was now cultivated to the pitch of British agricultural\nscience. The marshlands beyond the river were one splendid expanse of\nrichest meadows, yielding a rental of four solid pounds per acre. Over\nhill and dale on this side for miles, where formerly ran wild deer, and\ngrew wild woodlands or furze-bushes, now lay excellent farms and\nhamlets, and along the ridge of the ancient cliffs rose the most\nmagnificent woods. Woods, too, clothed the steep-hill sides, and swept\ndown to the noble river, their very boughs hanging far out over its\nclear and rapid waters. In the midst of these fine woods stood Rockville\nHall, the family seat of the Rockvilles. It reared its old brick walls\nabove the towering mass of elms, and travellers at a distance recognized\nit for what it was, the mansion of an ancient and wealthy family.\n\nThe progress of England in arts, science, commerce, and manufacture,\nhad carried Sir Roger's estate along with it. It was full of active and\nmoneyed farmers, and flourished under modern influences. How lucky it\nwould have been for the Rockville family had it done the same.\n\nBut amid this estate there was Sir Roger solitary, and the last of the\nline. He had grown well enough--there was nothing stunted about him, so\nfar as you could see on the surface. In stature, he exceeded six feet.\nHis colossal elms could not boast of a properer relative growth. He was\nas large a landlord, and as tall a justice of the peace, as you could\ndesire; but, unfortunately, he was, after all, only the shell of a man.\nLike many of his veteran elms, there was a very fine stem, only it was\nhollow. There was a man, just with the rather awkward deficiency of a\nsoul.\n\nAnd it were no difficult task to explain, either, how this had come\nabout. The Rockvilles saw plainly enough the necessity of manuring their\nlands, but they scorned the very idea of manuring their family. What!\nthat most ancient, honorable, and substantial family, suffer any of the\ncommon earth of humanity to gather about its roots! The Rockvilles were\nso careful of their good blood, that they never allied it to any but\nblood as pure and inane as their own. Their elms flourished in the\nrotten earth of plebeian accumulations, and their acres produced large\ncrops of corn from the sewage of towns and fat sinks, but the Rockvilles\nthemselves took especial care that no vulgar vigor from the rich heap of\nordinary human nature should infuse a new force of intellect into their\nrace. The Rockvilles needed nothing; they had all that an ancient,\nhonorable, and substantial family could need. The Rockvilles had no need\nto study at school--why should they? They did not want to get on. The\nRockvilles did not aspire to distinction for talent in the world--why\nshould they? They had a large estate. So the Rockville soul, unused from\ngeneration to generation, grew--\n\n    Fine by degrees and _spiritually_ less,\n\ntill it tapered off into nothing.\n\nLook at the last of a long line in the midst of his fine estate. Tall he\nwas, with a stoop in his shoulders, and a bowing of his head on one\nside, as if he had been accustomed to stand under the low boughs of his\nwoods, and peer after intruders. And that was precisely the fact. His\nfeatures were thin and sharp; his nose prominent and keen in its\ncharacter; his eyes small, black, and peering like a mole's, or a hungry\nswine's. Sir Roger was still oracular on the bench, after consulting his\nclerk, a good lawyer,--and looked up to by the neighboring squires in\nelection matters, for he was an unswerving tory. You never heard of a\nrational thing that he had said in the whole course of his life; but\nthat mattered little, he was a gentleman of solemn aspect, of stately\ngait, and of a very ancient family.\n\nWith ten thousand a-year, and his rental rising, he was still, however,\na man of overwhelming cares. What mattered a fine estate if all the\nworld was against him? And Sir Roger firmly believed that he stood in\nthat predicament. He had grown up to regard the world as full of little\nbesides upstarts, radicals, manufacturers, and poachers. All were\nbanded, in his belief, against the landed interest. It demanded all the\nenergy of his very small faculties to defend himself and the world\nagainst them.\n\nUnfortunately for his peace, a large manufacturing town had sprung up\nwithin a couple of miles of him. He could see its red-brick walls, and\nits red-tiled roofs, and its tall smoke-vomiting chimneys, growing and\nextending over the <DW72>s beyond the river. It was to him the most\nirritating sight in the world; for what were all those swarming weavers\nand spinners but arrant radicals, upstarts, sworn foes of the ancient\ninstitutions and the landed interests of England? Sir Roger had passed\nthrough many a desperate conflict with them for the return of members to\nparliament. They brought forward men that were utter wormwood to all his\nfeelings, and they paid no more respect to him and his friends on such\noccasions than they did to the meanest creature living. Reverence for\nancient blood did not exist in that plebeian and rapidly multiplying\ntribe. There were master manufacturers there actually that looked and\ntalked as big as himself, and _entre nous_, a vast deal more cleverly.\nThe people talked of rights and franchises, and freedom of speech and of\nconscience, in a way that was really frightful. Then they were given\nmost inveterately to running out in whole and everlasting crowds on\nSundays and holidays into the fields and woods; and as there was no part\nof the neighborhood half so pleasant as the groves and river banks of\nRockville, they came swarming up there in crowds that were enough to\ndrive any man of acres frantic.\n\nUnluckily, there were roads all about Rockville; foot roads, and high\nroads, and bridle roads. There was a road up the river side, all the way\nto Rockville woods, and when it reached them, it divided like a fork,\nand one prong or footpath led straight up a magnificent grove of a mile\nlong, ending close to the hall; and another ran all along the river\nside, under the hills and branches of the wood.\n\nOh, delicious were these woods! In the river there were islands, which\nwere covered in summer with the greenest grass, and the freshest of\nwillows, and the clear waters rushed around them in the most inviting\nmanner imaginable. And there were numbers of people extremely ready to\naccept this delectable invitation of these waters. There they came in\nfine weather, and as these islands were only separated from the\nmain-land by a little and very shallow stream, it was delightful for\nlovers to get across--with laughter, and treading on stepping stones,\nand slipping off the stepping-stones up to the ankles into the cool\nbrook, and pretty screams, and fresh laughter, and then landing on those\nsunny, and to them really enchanted, islands. And then came fishermen,\nsolitary fishermen, and fishermen in rows; fishermen lying in the\nflowery grass, with fragrant meadow-sweet and honey-breathing clover all\nabout their ears; and fishermen standing in file, as if they were\ndetermined to clear all the river of fish in one day. And there were\nother lovers, and troops of loiterers, and shouting roysterers, going\nalong under the boughs of the wood, and following the turns of that most\ncompanionable of rivers. And there were boats going up and down; boats\nfull of young people, all holiday finery and mirth, and boats with\nduck-hunters and other, to Sir Roger, detestable marauders, with guns\nand dogs, and great bottles of beer. In the fine grove, on summer days,\nthere might be found hundreds of people. There were pic-nic parties,\nfathers and mothers with whole families of children, and a grand\npromenade of the delighted artisans and their wives or sweethearts.\n\nIn the times prior to the sudden growth of the neighboring town, Great\nStockington, and to the simultaneous development of the love-of-nature\nprinciple in the Stockingtonians, nothing had been thought of all these\nroads. The roads were well enough till they led to these inroads. Then\nSir Roger aroused himself. This must be changed. The roads must be\nstopped. Nothing was easier to his fancy. His fellow-justices, Sir\nBenjamin Bullockshed and Squire Sheepshank, had asked his aid to stop\nthe like nuisances, and it had been done at once. So Sir Roger put up\nnotices all about, that the roads were to be stopped by an Order of\nSession, and these notices were signed, as required by law, by their\nworships of Bullockshed and Sheepshank. But Sir Roger soon found that it\nwas one thing to stop a road leading from One-man-Town to Lonely Lodge,\nand another to attempt to stop those from Great Stockington to\nRockville.\n\nOn the very first Sunday after the exhibition of those notice-boards,\nthere was a ferment in the grove of Rockville, as if all the bees in the\ncounty were swarming there, with all the wasps and Hornets to boot.\nGreat crowds were collected before each of these obnoxious placards, and\nthe amount of curses vomited forth against them was really shocking for\nany day, but more especially for a Sunday. Presently there was a rush at\nthem; they were torn down, and simultaneously pitched into the river.\nThere were great crowds swarming all about Rockville all that day, and\nwith looks so defiant that Sir Roger more than once contemplated\nsending off for the Yeoman Cavalry to defend his house, which he\nseriously thought in danger.\n\nBut so far from being intimidated from proceeding, this demonstration\nonly made Sir Roger the more determined. To have so desperate and\nirreverent a population coming about his house and woods, now presented\nitself in a much more formidable aspect than ever. So, next day, not\nonly were the placards once more hoisted, but rewards offered for the\ndiscovery of the offenders, attended with all the maledictions of the\ninsulted majesty of the law. No notice was taken of this, but the whole\nof Great Stockington was in a buzz and an agitation. There were posters\nplastered all over the walls of the town, four times as large as Sir\nRoger's notices, in this style:--\n\n\"Englishmen! your dearest rights are menaced! The Woods of Rockville,\nyour ancient, rightful, and enchanting resorts, are to be closed to you.\nStockingtonians! the eyes of the world are upon you. 'Awake! arise! or\nbe forever fallen!' England expects every man to do his duty! And your\nduty is to resist and defy the grasping soil-lords, to seize on your\nancient Patrimony!\"\n\n\"Patrimony! Ancient and rightful resort of Rockville!\" Sir Roger was\nastounded at the audacity of this upstart, plebeian race. \"What! they\nactually claimed Rockville, the heritage of a hundred successive\nRockvilles, as their own. Sir Roger determined to carry it to the\nSessions; and at the Sessions was a magnificent muster of all his\nfriends. There was Sir Roger himself in the chair; and on either\nhand, a prodigious row of county squirearchy. There was Sir\nBenjamin Bullockshed, and Sir Thomas Tenterhook, and all the\nsquires,--Sheepshank, Ramsbottom, Turnbull, Otterbrook, and Swagsides.\nThe Clerk of the Session read the notice for the closing of all the\nfootpaths through the woods of Rockville, and declared that this notice\nhad been duly, and for the required period publicly posted. The\nStockingtonians protested by their able lawyer Daredeville, against any\norder for the closing of these ancient woods--the inestimable property\nof the public.\n\n\"Property of the public!\" exclaimed Sir Roger. \"Property of the public!\"\nechoed the multitudinous voices of indignant Bullocksheds, Tenterhooks,\nand Ramsbottoms. \"Why, Sir, do you dispute the right of Sir Roger\nRockville to his own estate?\"\n\n\"By no means;\" replied the undaunted Daredeville; \"the estate of\nRockville is unquestionably the property of the honorable baronet, Sir\nRoger Rockville; but the roads through it are the as unquestionable\nproperty of the public.\"\n\nThe whole bench looked at itself; that is, at each other, in wrathful\nastonishment. The swelling in the diaphragms of the squires Otterbrook,\nTurnbull, and Swagsides, and all the rest of the worshipful row, was too\nbig to admit of utterance. Only Sir Roger himself burst forth with an\nabrupt--\n\n\"Impudent fellows! But I'll see them ---- first!\"\n\n\"Grant the order!\" said Sir Benjamin Bullockshed; and the whole bench\nnodded assent. The able lawyer Daredeville retired with a pleasant\nsmile. He saw an agreeable prospect of plenty of grist to his mill. Sir\nRoger was rich, and so was Great Stockington. He rubbed his hands, not\nin the least like a man defeated, and thought to himself, \"Let them go\nat it--all right.\"\n\nThe next day the placards on the Rockville estate were changed for\nothers bearing \"STOPPED BY ORDER OF SESSIONS!\" and alongside of them\nwere huge carefully painted boards, denouncing on all trespassers\nprosecutions according to law. The same evening came a prodigious\ninvasion of Stockingtonians--tore all the boards and placards down, and\ncarried them on their shoulders to Great Stockington, singing as they\nwent, \"See, the Conquering Heroes come!\" They set them up in the centre\nof the Stockington market-place, and burnt them, along with an effigy of\nSir Roger Rockville.\n\nThat was grist at once to the mill of the able lawyer Daredeville. He\nlooked on, and rubbed his hands. Warrants were speedily issued by the\nBaronets of Bullockshed and Tenterhook, for the apprehension of the\nindividuals who had been seen carrying off the notice-boards, for\nlarceny, and against a number of others for trespass. There was plenty\nof work for Daredeville and his brethren of the robe; but it all ended,\nafter the flying about of sundry mandamuses and assize trials, in Sir\nRoger finding that though Rockville was his, the roads through it were\nthe public's.\n\nAs Sir Roger drove homeward from the assize, which finally settled the\nquestion of these footpaths, he heard the bells in all the steeples of\nGreat Stockington burst forth with a grand peal of triumph. He closed\nfast the windows of his fine old carriage, and sunk into a corner; but\nhe could not drown the intolerable sound. \"But,\" said he, \"I'll stop\ntheir pic-nic-ing. I'll stop their fishing. I'll have hold of them for\ntrespassing and poaching!\" There was war henceforth between Rockville\nand Great Stockington.\n\nOn the very next Sunday there came literally thousands of the jubilant\nStockingtonians to Rockville. They had brought baskets, and were for\ndining, and drinking success to all footpaths. But in the great grove\nthere were keepers, and watchers, who warned them to keep the path, that\nnarrow well-worn line up the middle of the grove. \"What! were they not\nto sit on the grass?\"--\"No!\"--\"What! were they not to pic-nic?\"--\"No!\nnot there!\"\n\nThe Stockingtonians felt a sudden damp on their spirits. But the river\nbank! The cry was. \"To the river bank! There they _would_ pic-nic.\" The\ncrowd rushed away down the wood, but on the river bank they found a\nwhole regiment of watchers, who pointed again to the narrow line of\nfootpath, and told them not to trespass beyond it. But the islands! they\nwent over to the islands. But there too were Sir Roger's forces, who\nwarned them back! There was no road there--- all found there would be\ntrespassers, and be duly punished.\n\nThe Stockingtonians discovered that their triumph was not quite so\ncomplete as they had flattered themselves. The footpaths were theirs,\nbut that was all. Their ancient license was at an end. If they came\nthere, there was no more fishing; if they came in crowds, there was no\nmore pic-nic-ing; if they walked through the woods in numbers, they must\nkeep to Indian file, or they were summoned before the county magistrates\nfor trespass, and were soundly fined; and not even the able Daredeville\nwould undertake to defend them.\n\nThe Stockingtonians were chop-fallen, but they were angry and dogged;\nand they thronged up to the village and the front of the hall. They\nfilled the little inn in the hamlet--they went by scores, and roving all\nover the churchyard, read epitaphs\n\n    That teach the rustic moralists to die,\n\nbut don't teach them to give up their old indulgences very\ngood-humoredly. They went and sat in rows on the old churchyard wall,\nopposite to the very windows of the irate Sir Roger. They felt\nthemselves beaten, and Sir Roger felt himself beaten. True, he could\ncoerce them to the keeping of the footpaths--but, then, they had the\nfootpaths! True, thought the Stockingtonians, we have the footpaths, but\nthen the pic-nic-ing, and the fishing, and the islands! The\nStockingtonians were full of sullen wrath, and Sir Roger was--oh, most\nexpressive old Saxon phrase--HAIRSORE! Yes, he was one universal wound\nof vexation and jealousy of his rights. Every hair in his body was like\na pin sticking into him. Come within a dozen yards of him; nay, at the\nmost, blow on him, and he was excruciated--you rubbed his sensitive\nhairs at a furlong's distance.\n\nThe next Sunday the people found the churchyard locked up, except during\nservice, when beadles walked there, and desired them not to loiter and\ndisturb the congregation, closing the gates, and showing them out like a\nflock of sheep the moment the service was over. This was fuel to the\nalready boiling blood of Stockington. The week following, what was their\nastonishment to find the much frequented inn gone! it was actually gone!\nnot a trace of it; but the spot where it had stood for ages, turfed,\nplanted with young spruce trees, and fenced off with post and rail! The\nexasperated people now launched forth an immensity of fulminations\nagainst the churl Sir Roger, and a certain number of them resolved to\ncome and seat themselves in the street of the hamlet and there dine; but\na terrific thunderstorm, which seemed in league with Sir Roger, soon\nrouted them, drenched them through, and on attempting to seek shelter in\nthe cottages, the poor people said they were very sorry, but it was as\nmuch as their holdings were worth, and they dare not admit them.\n\nSir Roger had triumphed! It was all over with the old delightful days at\nRockville. There was an end of pic-nic-ing, of fishing, and of roving in\nthe islands. One sturdy disciple of Izaak Walton, indeed, dared to fling\na line from the banks of Rockville grove, but Sir Roger came upon him\nand endeavored to seize him. The man coolly walked into the middle of\nthe river, and, without a word, continued his fishing.\n\n\"Get out there!\" exclaimed Sir Roger, \"that is still on my property.\"\nThe man walked through the river to the other bank, where he knew that\nthe land was rented by a farmer. \"Give over,\" shouted Sir Roger, \"I tell\nyou the water is mine.\"\n\n\"Then,\" said the fellow, \"bottle it up, and be hanged to you! Don't you\nsee it is running away to Stockington?\"\n\nThere was bad blood between Rockville and Stockington forever.\nStockington was incensed, and Sir Roger was hairsore.\n\nA new nuisance sprung up. The people of Stockington looked on the\ncottagers of Rockville as sunk in deepest darkness under such a man as\nSir Roger and his cousin the vicar. They could not pic-nic, but they\nthought they could hold a camp-meeting; they could not fish for roach,\nbut they thought they might for souls. Accordingly there assembled\ncrowds of Stockingtonians on the green of Rockville, with a chair and a\ntable, and a preacher with his head bound in a red handkerchief; and\nsoon there was a sound of hymns, and a zealous call to come out of the\ndarkness of the spiritual Babylon. But this was more than Sir Roger\ncould bear; he rushed forth with all his servants, keepers, and\ncottagers, overthrew the table, and routing the assembly, chased them to\nthe boundary of his estate.\n\nThe discomfited Stockingtonians now fulminated awful judgments on the\nunhappy Sir Roger, as a persecutor and a malignant. They dared not enter\nagain on his parish, but they came to the very verge of it, and held\nweekly meetings on the highway, in which they sang and declaimed as\nloudly as possible, that the winds might bear their voices to Sir\nRoger's ears.\n\nTo such a position was now reduced the last of the long line of\nRockville. The spirit of a policeman had taken possession of him. He had\nkeepers and watchers out on all sides, but that did not satisfy him. He\nwas perpetually haunted with the idea that poachers were after his game,\nthat trespassers were in his woods. His whole life was now spent in\nstealing to and fro in his fields and plantations, and prowling along\nhis river side. He lurked under hedges, and watched for long hours\nunder the forest trees. If any one had a curiosity to see Sir Roger,\nthey had only to enter his fields by the wood side, and wander a few\nyards from the path, and he was almost sure to spring out over the\nhedge, and in angry tones demand their name and address. The descendant\nof the chivalrous and steel-clad De Rockvilles was sunk into a restless\nspy on his own ample property. There was but one idea in his\nmind--encroachment. It was destitute of all other furniture but the\nmusty technicalities of warrants and commitments. There was a stealthy\nand skulking manner in everything that he did. He went to church on\nSundays, but it was no longer by the grand iron gate opposite to his\nhouse, that stood generally with a large spider's web woven over the\nlock, and several others in different corners of the fine iron tracery,\nbearing evidence of the long period since it had been opened. How\ndifferent to the time when the Sir Roger and Lady of Rockville had had\nthese gates thrown wide on a Sunday morning, and, with all their train\nof household servants at their back, with true antique dignity, marched\nwith much proud humility into the house of God. Now, Sir Roger--the\nsolitary, suspicious, undignified Sir Roger, the keeper and policeman of\nhis own property--stole in at a little side gate from his paddock, and\nback the same way, wondering all the time whether there was not somebody\nin his pheasant preserves, or Sunday trespassers in his grove.\n\nIf you entered his house, it gave you as cheerless a feeling as its\nowner. There was the conservatory, so splendid with rich plants and\nflowers in his mother's time--now a dusty receptacle of hampers, broken\nhand-glasses, and garden tools. These tools could never be used, for the\ngardens were grown wild. Tall grass grew in the walks, and the huge\nunpruned shrubs disputed the passage with you. In the wood above the\ngardens, reached by several flights of fine, but now moss-grown, steps,\nthere stood a pavilion, once clearly very beautiful. It was now damp and\nruinous--its walls covered with greenness and crawling insects. It was\na great lurking-place of Sir Roger when on the watch for poachers.\n\nThe line of the Rockvilles was evidently running fast out. It had\nreached the extremity of imbecility and contempt--it must soon reach its\nclose.\n\nSir Roger used to make his regular annual visit to town; but of late,\nwhen there, he had wandered restlessly about the streets, peeping into\nthe shop-windows; and if it rained, standing under entries for hours\ntogether, till it was gone over. The habit of lurking and peering about,\nwas upon him; and his feet bore him instinctively into those narrow and\ncrowded alleys where swarm the poachers of the city--the trespassers and\nanglers in the game preserves and streams of humanity. He had lost all\npleasures in his club; the most exciting themes of political life\nretained no piquancy for him. His old friends ceased to find any\npleasure in him. He was become the driest of all dry wells. Poachers,\nand anglers, and Methodists, haunted the wretched purlieus of his fast\nfading-out mind, and he resolved to go to town no more. His whole\nnature was centred in his woods. He was forever on the watch; and when\nat Rockville again, if he heard a door clap when in bed, he thought it a\ngun in his woods, and started up, and was out with his keepers.\n\nOf what value was that magnificent estate to him?--those superb woods;\nthose finely-hanging cliffs; that clear and _riant_ river coming\ntravelling on, and taking a noble sweep below his windows,--that\nglorious expanse of neat verdant meadows stretching almost to\nStockington, and enlivened by numerous herds of the most beautiful\ncattle--those old farms and shady lanes overhung with hazel and wild\nrose; the glittering brook, and the songs of woodland birds--what were\nthey to that worn-out old man, that victim of the delusive doctrine of\nblood, of the man-trap of an hereditary name?\n\nThere the poet could come, and feel the presence of divinity in that\nnoble scene, and hear sublime whispers in the trees, and create new\nheavens and earths from the glorious chaos of nature around him, and in\none short hour live an empyrean of celestial life and love. There could\ncome the very humblest children of the plebeian town, and feel a throb\nof exquisite delight pervade their bosoms at the sight of the very\nflowers on the sod, and see heaven in the infinite blue above them. And\npoor Sir Roger, the holder, but not the possessor of all, walked only in\na region of sterility, with no sublimer ideas than poachers and\ntrespassers--no more rational enjoyment than the brute indulgence of\nhunting like a ferret, and seizing his fellow-men like a bull-dog. He\nwas a specimen of human nature degenerated, retrograded from the divine\nto the bestial, through the long-operating influences of false notions\nand institutions, continued beyond their time. He had only the soul of a\nkeeper. Had he been only a keeper, he had been a much happier man.\n\nHis time was at hand. The severity which he had long dealt out towards\nall sorts of offenders made him the object of the deepest vengeance. In\na lonely hollow of his woods, watching at midnight with two of his men,\nthere came a sturdy knot of poachers. An affray ensued. The men\nperceived that their old enemy, Sir Roger, was there; and the blow of a\nhedge-stake stretched him on the earth. His keepers fled--and thus\nignominiously terminated the long line of the Rockvilles. Sir Roger was\nthe last of his line, but not of his class. There is a feudal art of\nsinking, which requires no study; and the Rockvilles are but one family\namong thousands who have perished in its practice.\n\n       *       *       *       *       *\n\nIn Great Stockington there lived a race of paupers. From the year of the\n42d of Elizabeth, or 1601, down to the present generation, this race\nmaintained an uninterrupted descent. They were a steady and unbroken\nline of paupers, as the parish books testify. From generation to\ngeneration their demands on the parish funds stand recorded. There were\nno _lacun\u00e6_ in their career; there never failed an heir to these\nfamilies; fed on the bread of idleness and legal provision, these people\nnourished, increased, and multiplied. Sometimes compelled to work for\nthe weekly dole which they received, they never acquired a taste for\nlabor, or lost the taste for the bread for which they did not labor.\nThese paupers regarded this maintenance by no means as a disgrace. They\nclaimed it as a right,--as their patrimony. They contended that\none-third of the property of the Church had been given by benevolent\nindividuals for the support of the poor, and that what the Reformation\nwrongfully deprived them of, the great enactment of Elizabeth\nrightfully--and only rightfully--restored.\n\nThose who imagine that all paupers merely claimed parish relief because\nthe law ordained it, commit a great error. There were numbers who were\nhereditary paupers, and that on a tradition carefully handed down, that\nthey were only manfully claiming their own. They traced their claims\nfrom the most ancient feudal times, when the lord was as much bound to\nmaintain his villein in gross, as the villein was to work for the lord.\nThese paupers were, in fact, or claimed to be, the original _adscripti\ngleb\u00e6_, and to have as much a claim to parish support as the landed\nproprietor had to his land. For this reason, in the old Catholic times,\nafter they had escaped from villenage by running away and remaining\nabsent from their hundred for a year and a day, dwelling for that period\nin a walled town, these people were among the most diligent attendants\nat the Abbey doors, and when the Abbeys were dissolved, were, no doubt,\namong the most daring of these thieves, vagabonds, and sturdy rogues,\nwho, after the Robin Hood fashion, beset the highways and solitary farms\nof England, and claimed their black mail in a very unceremonious style.\nIt was out of this class that Henry VIII. hanged his seventy-two\nthousand during his reign, and, as it is said, without appearing\nmaterially to diminish their number.\n\nThat they continued to \"increase, multiply, and replenish the earth,\"\noverflowing all bounds, overpowering by mere populousness all the severe\nlaws against them of whipping, burning in the hand, in the forehead or\nthe breast, and hanging, and filling the whole country with alarm, is\nevident by the very act itself of Elizabeth.\n\nAmong these hereditary paupers who, as we have said, were found in\nStockington, there was a family of the name of Deg. This family had\nnever failed to demand and enjoy what it held to be its share of its\nancient inheritance. It appeared from the parish records, that they had\npractised in different periods the crafts of shoe-making, tailoring, and\nchimney-sweeping; but since the invention of the stocking frame, they\nhad, one and all of them, followed the profession of stocking-weavers,\nor as they were there called, stockingers. This was a trade which\nrequired no extreme exertion of the physical or intellectual powers. To\nsit in a frame, and throw the arms to and fro, was a thing that might\neither be carried to a degree of extreme diligence, or be let down into\na mere apology for idleness. An \"idle stockinger\" was there no very\nuncommon phrase, and the Degs were always classed under that head.\nNothing could be more admirably adapted than this trade for building a\nplan of parish relief upon. The Degs did not pretend to be absolutely\nwithout work, or the parish authorities would soon have set them to some\nreal labor,--a thing that they particularly recoiled from, having a very\nold adage in the family, that \"hard work was enough to kill a man.\" The\nDegs were seldom, therefore, out of work, but they did not get enough to\nmeet and tie. They had but little work if times were bad, and if they\nwere good, they had large families and sickly wives or children. Be\ntimes what they would, therefore, the Degs were due and successful\nattendants at the parish pay-table. Nay, so much was this a matter of\ncourse, that they came at length not even to trouble themselves to\nreceive their pay, but sent their young children for it; and it was duly\npaid. Did any parish officer, indeed, turn restive, and decline to pay a\nDeg, he soon found himself summoned before a magistrate, and such pleas\nof sickness, want of work, and poor earnings brought up, that he most\nlikely got a sharp rebuke from the benevolent but uninquiring\nmagistrate, and acquired a character for hard-heartedness that stuck to\nhim.\n\nSo parish overseers learned to let the Degs alone; and their children\nregularly brought up to receive the parish money for their parents, were\nimpatient as they grew up to receive it for themselves. Marriages in the\nDeg family were consequently very early, and there were plenty of\ninstances of married Degs claiming parish relief under the age of\ntwenty, on the plea of being the parent of two children. One such\nprecocious individual being asked by a rather verdant officer why he had\nmarried before he was able to maintain a family, replied, in much\nastonishment, that he had married in order to maintain himself by parish\nassistance. That he never had been able to maintain himself by his\nlabor, nor ever expected to do it; his only hope, therefore, lay in\nmarrying and becoming the father of two children, to which patriarchal\nrank he had now attained, and demanded his \"pay.\"\n\nThus had lived and flourished the Degs on their ancient patrimony, the\nparish, for upwards of two hundred years. Nay, we have no doubt whatever\nthat, if it could have been traced, they had enjoyed an ancestry of\npaupers as long as the pedigree of Sir Roger Rockville himself. In the\ndays of the most perfect villenage, they had, doubtless, eaten the bread\nof idleness, and claimed it as a right. They were numerous, improvident,\nragged in dress, and fond of an ale-house and of gossip. Like the blood\nof Sir Roger, their blood had become peculiar through a long persistence\nof the same circumstances. It was become pure pauper blood. The Degs\nmarried, if not entirely among Degs, yet among the same class. None but\na pauper would dream of marrying a Deg. The Degs, therefore, were in\nconstitution, in mind, in habit, and in inclination, paupers. But a pure\nand unmixed class of this kind does not die out like an aristocratic\nstereotype. It increases and multiplies. The lower the grade, the more\nprolific, as is sometimes seen on a large and even national scale. The\nDegs threatened, therefore, to become a most formidable clan in the\nlower purlieus of Stockington, but luckily there is so much virtue even\nin evils, that one, not rarely cures another. War, the great evil,\ncleared the town of Degs.\n\nFond of idleness, of indulgence, of money easily got, and as easily\nspent, the Degs were rapidly drained off by recruiting parties during\nthe last war. The young men enlisted, and were marched away; the young\nwomen married soldiers that were quartered in the town from time to\ntime, and marched away with them. There were, eventually, none of the\nonce numerous Degs left except a few old people, whom death was sure to\ndraft off at no distant period with his regiment of the line which has\nno end. Parish overseers, magistrates, and master manufacturers,\nfelicitated themselves at this unhoped-for deliverance from the ancient\nfamily of the Degs.\n\nBut one cold, clear winter evening, the east wind piping its sharp\nsibilant ditty in the bare-shorn hedges, and poking his sharp fingers\ninto the sides of well broad-clothed men by way of passing jest, Mr.\nSpires, a great manufacturer of Stockington, driving in his gig some\nseven miles from the town, passed a poor woman with a stout child on her\nback. The large ruddy-looking man in the prime of life, and in the\ngreat-coat and thick-worsted gloves of a wealthy traveller, cast a\nglance at the wretched creature trudging heavily on, expecting a pitiful\nappeal to his sensibilities, and thinking it a bore to have to pull off\na glove and dive into his pocket for a copper; but to his surprise there\nwas no demand, only a low curtsey, and the glimpse of a face of singular\nhonesty of expression, and of excessive weariness.\n\nSpires was a man of warm feelings; he looked earnestly at the woman, and\nthought he had never seen such a picture of fatigue in his life. He\npulled up and said,\n\n\"You seem very tired, my good woman.\"\n\n\"Awfully tired, sir.\"\n\n\"And are you going far to-night?\"\n\n\"To Great Stockington, sir, if God give me strength.\"\n\n\"To Stockington!\" exclaimed Mr. Spires. \"Why you seem ready to drop.\nYou'll never reach it. You'd better stop at the next village.\"\n\n\"Ay, sir, it's easy stopping for those that have money.\"\n\n\"And you've none, eh?\"\n\n\"As God lives, sir, I've a sixpence, and that's all.\"\n\nMr. Spires put his hand in his pocket, and held out to her the next\ninstant half-a-crown.\n\n\"There stop, poor thing--make yourself comfortable--it's quite out of\nthe question to reach Stockington. But stay--are your friends living in\nStockington--what are you?\"\n\n\"A poor soldier's widow, sir. And may God Almighty bless you!\" said the\npoor woman, taking the money, the tears standing in her large brown eyes\nas she curtsied very low.\n\n\"A soldier's widow,\" said Mr. Spires. She had touched the softest place\nin the manufacturer's heart, for he was a very loyal man, and vehement\nchampion of his country's honor in the war. \"So young,\" said he, \"how\ndid you lose your husband?\"\n\n\"He fell, sir,\" said the poor woman; but she could get no further; she\nsuddenly caught up the corner of her gray cloak, covered her face with\nit, and burst into an excess of grief.\n\nThe manufacturer felt as if he had hit the woman a blow by his careless\nquestion; he sat watching her for a moment in silence, and then said,\n\"Come, get into the gig, my poor woman; come, I must see you to\nStockington.\"\n\nThe poor woman dried her tears, and heavily climbed into the gig,\nexpressing her gratitude in a very touching and modest manner. Spires\nbuttoned the apron over her, and taking a look at the child, said in a\ncheerful tone to comfort her, \"Bless me, but that is a fine thumping\nfellow, though. I don't wonder you are tired, carrying such a load.\"\n\nThe poor woman pressed the stout child, apparently two years old, to her\nbreast, as if she felt it a great blessing and no load: the gig drove\nrapidly on.\n\nPresently Mr. Spires resumed his conversation.\n\n\"So you are from Stockington?\"\n\n\"No, sir; my husband was.\"\n\n\"So: what was his name?\"\n\n\"John Deg, sir.\"\n\n\"Deg?\" said Mr. Spires. \"Deg, did you say?\"\n\n\"Yes, sir.\"\n\nThe manufacturer seemed to hitch himself off towards his own side of the\ngig, gave another look at her, and was silent. The poor woman was\nsomewhat astonished at his look and movement, and was silent too.\n\nAfter awhile Mr. Spires said again, \"And do you hope to find friends in\nStockington? Had you none where you came from?\"\n\n\"None, sir, none in the world!\" said the poor woman, and again her\nfeelings seemed too strong for her. At length she added, \"I was in\nservice, sir, at Poole, in Dorsetshire, when I married; my mother only\nwas living, and while I was away with my husband, she died. When--when\nthe news came from abroad--that--when I was a widow, sir, I went back to\nmy native place, and the parish officers said I must go to my husband's\nparish, lest I and my child should become troublesome.\"\n\n\"You asked relief of them?\"\n\n\"Never; oh, God knows, never! My family have never asked a penny of a\nparish. They would die first, and so would I, sir; but they said I might\ndo it, and I had better go to my husband's parish at once--and they\noffered me money to go.\"\n\n\"And you took it, of course?\"\n\n\"No, sir; I had a little money, which I had earned by washing and\nlaundering, and I sold most of my things, as I could not carry them, and\ncame off. I felt hurt, sir; my heart rose against the treatment of the\nparish, and I thought I should be better among my husband's\nfriends--and my child would, if anything happened to me; I had no\nfriends of my own.\"\n\nMr. Spires looked at the woman in silence. \"Did your husband tell you\nanything of his friends? What sort of a man was he?\"\n\n\"Oh, he was a gay young fellow, rather, sir; but not bad to me. He\nalways said his friends were well off in Stockington.\"\n\n\"He did!\" said the manufacturer, with a great stare, and as if bolting\nthe words from his heart in a large gust of wonder.\n\nThe poor woman again looked at him with a strange look. The manufacturer\nwhistled to himself, and giving his horse a smart cut with the whip,\ndrove on faster than ever. The night was fast settling down; it was\nnumbing cold; a gray fog rose from the river as they thundered over the\nold bridge; and tall engine chimneys, and black smoky houses loomed\nthrough the dusk before them. They were at Stockington.\n\nAs they slackened their pace up a hill at the entrance of the town, Mr.\nSpires again opened his mouth.\n\n\"I should be sorry to hurt your feelings, Mrs. Deg,\" he said, \"but I\nhave my fears that you are coming to this place with false expectations.\nI fear your husband did not give you the truest possible account of his\nfamily here.\"\n\n\"Oh, Sir! What--what is it?\" exclaimed the poor woman; \"in God's name,\ntell me!\"\n\n\"Why, nothing more than this,\" said the manufacturer, \"that there are\nvery few of the Degs left here. They are old, and on the parish, and can\ndo nothing for you.\"\n\nThe poor woman gave a deep sigh, and was silent.\n\n\"But don't be cast down,\" said Mr. Spires. He would not tell her what a\npauper family it really was, for he saw that she was a very feeling\nwoman, and he thought she would learn that soon enough. He felt that her\nhusband had from vanity given her a false account of his connections;\nand he was really sorry for her.\n\n\"Don't be cast down,\" he went on, \"you can wash and iron, you say; you\nare young and strong; those are your friends. Depend on them, and\nthey'll be better friends to you than any other.\"\n\nThe poor woman was silent, leaning her head down on her slumbering\nchild, and crying to herself; and thus they drove on, through many long\nand narrow streets, with gas flaring from the shops, but with few people\nin the streets, and these hurrying shivering along the payment, so\nintense was the cold. Anon they stopped at a large pair of gates; the\nmanufacturer rung a bell, which he could reach from his gig, and the\ngates presently were flung open, and they drove into a spacious yard,\nwith a large handsome house, having a bright lamp burning before it, on\none side of the yard, and tall warehouses on the other.\n\n\"Show this poor woman and her child to Mrs. Craddock's, James,\" said Mr.\nSpires, \"and tell Mrs. Craddock to make them very comfortable; and if\nyou will come to my warehouse to-morrow,\" added he, addressing the poor\nwoman, \"perhaps I can be of some use to you.\"\n\nThe poor woman poured out her heartfelt thanks, and following the old\nman servant, soon disappeared, hobbling over the pebbly pavement with\nher living load, stiffened almost to stone by her fatigue and her cold\nride.\n\nWe must not pursue too minutely our narrative. Mrs. Deg was engaged to\ndo the washing and getting up of Mr. Spire's linen, and the manner in\nwhich she executed her task insured her recommendations to all their\nfriends. Mrs. Deg was at once in full employ. She occupied a neat house\nin a yard near the meadows below the town, and in those meadows she\nmight be seen spreading out her clothes to whiten on the grass, attended\nby her stout little boy. In the same yard lived a shoemaker, who had two\nor three children of about the same age as Mrs. Deg's child. The\nchildren, as time went on, became play-fellows. Little Simon might be\nsaid to have the free run of the shoemaker's house, and he was the more\nattracted thither by the shoemaker's birds, and by his flute, on which\nhe often played after his work was done.\n\nMrs. Deg took a great friendship for this shoemaker; and he and his\nwife, a quiet, kind-hearted woman, were almost all the acquaintances\nthat she cultivated. She had found out her husband's parents, but they\nwere not of a description that at all pleased her. They were old and\ninfirm, but they were of the true pauper breed, a sort of person, whom\nMrs. Deg had been taught to avoid and to despise. They looked on her as\na sort of second parish, and insisted that she should come and live with\nthem, and help to maintain them out of her earnings. But Mrs. Deg would\nrather her little boy had died than have been familiarized with the\nspirit and habits of those old people. Despise them she struggled hard\nnot to do, and she agreed to allow them sufficient to maintain them on\ncondition that they desisted from any further application to the parish.\nIt would be a long and disgusting story to recount all the troubles,\nannoyance, and querulous complaints, and even bitter accusations that\nshe received from these connections, whom she could never satisfy; but\nshe considered it one of her crosses in her life, and patiently bore it,\nseeing that they suffered no real want, so long as they lived, which was\nfor years; but she would never allow her little Simon to be with them\nalone.\n\nThe shoemaker neighbor was a stout protection to her against the greedy\ndemands of these old people, and of others of the old Degs, and also\nagainst another class of inconvenient visitors, namely, suitors, who saw\nin Mrs. Deg a neat and comely young woman with a flourishing business,\nand a neat and soon well-furnished house, a very desirable acquisition.\nBut Mrs. Deg had resolved never again to marry, but to live for her boy,\nand she kept her resolve in firmness and gentleness.\n\nThe shoemaker often took walks in the extensive town meadows to gather\ngroundsell and plantain for his canaries and gorse-linnets, and little\nSimon Deg delighted to accompany him with his own children. There\nWilliam Watson, the shoemaker, used to point out to the children the\nbeauty of the flowers, the insects, and other objects of nature; and\nwhile he sate on a stile and read in a little old book of poetry, as he\noften used to do, the children sate on the summer grass, and enjoyed\nthemselves in a variety of plays.\n\nThe effect of these walks, and the shoemaker's conversation on little\nSimon Deg was such as never wore out of him through his whole life, and\nsoon led him to astonish the shoemaker by his extraordinary conduct. He\nmanifested the utmost uneasiness at their treading on the flowers in the\ngrass; he would burst with tears if they persisted in it; and when asked\nwhy, he said they were so beautiful, and that they must enjoy the\nsunshine, and be very unhappy to die. The shoemaker was amazed, but\nindulged the lad's fancy. One day he thought to give him a great treat,\nand when they were out in the meadows, he drew from under his coat, a\nbow and arrow, and shot the arrow high up in the air. He expected to see\nhim in an ecstacy of delight; his own children clapped their hands in\ntransport, but Simon stood silent, and as if awestruck. \"Shall I send up\nanother?\" asked the shoemaker.\n\n\"No, no,\" exclaimed the child, imploringly. \"You say God lives up there,\nand he mayn't like it.\"\n\nThe shoemaker laughed, but presently he said, as if to himself, \"There\nis too much imagination there. There will be a poet, if we don't take\ncare.\"\n\nThe shoemaker offered to teach Simon to read, and to solidify his mind,\nas he termed it, by arithmetic, and then to teach him to work at his\ntrade. His mother was very glad; and thought shoemaking would be a good\ntrade for the boy; and that with Mr. Watson she should have him always\nnear her. He was growing now a great lad, and was especially strong, and\nof a frank and daring habit. He was especially indignant at any act of\noppression of the weak by the strong, and not seldom got into trouble by\nhis championship of the injured in such cases amongst the boys of the\nneighborhood.\n\nHe was now about twelve years of age; when, going one day with a basket\nof clothes on his head to Mr. Spires's for his mother, he was noticed by\nMr. Spires himself from his counting-house window. The great war was\nraging; there was much distress amongst the manufacturers; and the\npeople were suffering and exasperated against their masters. Mr. Spires,\nas a staunch tory, and supporter of the war, was particularly obnoxious\nto the work-people, who uttered violent threats against him. For this\nreason his premises were strictly guarded, and at the entrance of his\nyard, just within the gates, was chained a huge and fierce mastiff, his\nchain allowing him to approach near enough to intimidate any stranger,\nthough not to reach him. The dog knew the people who came regularly\nabout, and seemed not to notice them, but on the entrance of a stranger,\nhe rose up, barked fiercely, and came to the length of his chain. This\nalways drew the attention of the porter, if he were away from his box,\nand few persons dared to pass till he came.\n\nSimon Deg was advancing with the basket of clean linen on his head,\nwhen the dog rushed out, and barking loudly, came exactly opposite to\nhim, within a few feet. The boy, a good deal startled at first, reared\nhimself with his back against the wall, but at a glance perceiving that\nthe dog was at the length of his tether, he seemed to enjoy his\nsituation, and stood smiling at the furious animal, and lifting his\nbasket with both hands above his head, nodded to him, as if to say,\n\"Well, old boy, you'd like to eat me, wouldn't you?\"\n\nMr. Spires, who sate near his counting-house window at his books, was\nstruck with the bold and handsome bearing of the boy, and said to a\nclerk, \"What boy is that?\"\n\n\"It is Jenny Deg's,\" was the answer.\n\n\"Ha! that boy! Zounds! how boys do grow! What that's the child that\nJenny Deg was carrying when she came to Stockington; and what a strong,\nhandsome, bright-looking fellow he is now!\"\n\nAs the boy was returning, Mr. Spires call him to the counting-house\ndoor, and put some questions to him as to what he was doing and\nlearning, and so on. Simon, taking off his cap with much respect,\nanswered in such a clear and modest way, and with a voice that had so\nmuch feeling and natural music in it, that the worthy manufacturer was\ngreatly taken with him.\n\n\"That's no Deg,\" said he, when he again entered the counting-house, \"not\na bit of it. He's all Goodrick, or whatever his mother's name was, every\ninch of him.\"\n\nThe consequences of that interview was, that Simon Deg was very soon\nafter perched on a stool in Mr. Spires' counting-house, where he\ncontinued till he was twenty-two. Mr. Spires had no son, only a single\ndaughter; and such were Simon Deg's talents, attention to business, and\ngenial disposition, that at that age Mr. Spires gave him a share in the\nconcern. He was himself now getting less fond of exertion than he had\nbeen, and placed the most implicit reliance on Simon's judgment and\ngeneral management. Yet no two men could be more unlike in their\nopinions beyond the circle of trade. Mr. Spires was a staunch tory of\nthe staunch old school. He was for Church and King, and for things\nremaining forever as they had been. Simon, on the other hand, had\nliberal and reforming notions. He was for the improvement of the people,\nand their admission to many privileges. Mr. Spires was, therefore, liked\nby the leading men of the place, and disliked by the people. Simon's\nestimation was precisely in the opposite direction. But this did not\ndisturb their friendship; it required another disturbing cause--and that\ncame.\n\nSimon Deg and the daughter of Mr. Spires, grew attached to each other;\nand, as the father had thought Simon worthy of becoming a partner in the\nbusiness, neither of the young people deemed that he would object to a\npartnership of a more domestic description. But here they made a\ntremendous mistake. No sooner was such a proposal hinted it, than Mr.\nSpires burst forth with the fury of all the winds from the bag of\nUlysses.\n\n\"What! a Deg aspire to the hand of the sole heiress of the enormously\nopulent Spires?\"\n\nThe very thought almost cut the proud manufacturer off with an\napoplexy. The ghosts of a thousand paupers rose up before him, and he\nwas black in the face. It was only by a prompt and bold application of\nleeches and lancet, that the life of the great man was saved. But there\nwas an end of all further friendship between himself and the expectant\nSimon. He insisted that he should withdraw from the concern, and it was\ndone. Simon, who felt his own dignity deeply wounded too, for dignity he\nhad, though the last of a long line of paupers--his own dignity, not his\nancestors'--took silently, yet not unrespectfully, his share--a good,\nround sum, and entered another house of business.\n\nFor several years there appeared to be a feud and a bitterness between\nthe former friends; yet it showed itself in no other manner than by a\ncareful avoidance of each other. The continental war came to an end; the\nmanufacturing distress increased exceedingly. There came troublous\ntimes, and a fierce warfare of politics. Great Stockington was torn\nasunder by rival parties. On one side stood pre-eminent, Mr. Spires; on\nthe other towered conspicuously, Simon Deg. Simon was grown rich, and\nextremely popular. He was on all occasions the advocate of the people.\nHe said that he had sprung from, and was one of them. He had bought a\nlarge tract of land on one side of the town; and intensely fond of the\ncountry and flowers himself, he had divided this into gardens, built\nlittle summer-houses in them, and let them to the artisans. In his\nfactory he had introduced order, cleanliness, and ventilation. He had\nset up a school for the children in the evenings, with a reading-room\nand conversation-room for the work-people, and encouraged them to bring\ntheir families there, and enjoy music, books, and lectures. Accordingly,\nhe was the idol of the people, and the horror of the old school of the\nmanufacturers.\n\n\"A pretty upstart and demagogue I've nurtured,\" said Mr. Spires often to\nhis wife and daughter, who only sighed and were silent.\n\nThen came a furious election. The town, for a fortnight, more resembled\nthe worst corner of Tartarus than a Christian borough. Drunkenness,\nriot, pumping on one another, spencering one another, all sorts of\nviolence and abuse ruled and raged till the blood of all Stockington was\nat boiling heat. In the midst of the tempest were everywhere seen,\nranged on the opposite sides, Mr. Spires, now old and immensely\ncorpulent, and Simon Deg, active, buoyant, zealous, and popular beyond\nmeasure. But popular though, he still was, the other and old tory side\ntriumphed. The people were exasperated to madness; and when the chairing\nof the successful candidate commenced, there was a terrific attack made\non the procession by the defeated party. Down went the chair, and the\nnew member, glad to escape into an inn, saw his friends mercilessly\nassailed by the populace. There was a tremendous tempest of sticks,\nbrick-bats, paving-stones, and rotten eggs. In the midst of this, Simon\nDeg, and a number of his friends, standing at the upper window of a\nhotel, saw Mr. Spires knocked down, and trampled on by the crowd. In an\ninstant, and, before his friends had missed him from among them, Simon\nDeg was then darting through the raging mass, cleaving his way with a\nsurprising vigor, and gesticulating, and no doubt shouting vehemently to\nthe rioters, though his voice was lost in the din. In the next moment\nhis hat was knocked off, and himself appeared in imminent danger; but,\nanother moment, and there was a pause, and a group of people were\nbearing somebody from the frantic mob into a neighboring shop. It was\nSimon Deg, assisting in the rescue of his old friend and benefactor, Mr.\nSpires.\n\nMr. Spires was a good deal bruised, and wonderfully confounded and\nbewildered by his fall. His clothes were one mass of mud, and his face\nwas bleeding copiously; but when he had had a good draught of water, and\nhis face washed, and had time to recover himself, it was found that he\nhad received no serious injury.\n\n\"They had like to have done for me though,\" said he.\n\n\"Yes, and who saved you?\" asked a gentleman.\n\n\"Ay, who was it? who was it?\" asked the really warm-hearted\nmanufacturer; \"let me know? I owe him my life.\"\n\n\"There he is!\" said several gentlemen, at the same instant pushing\nforward Simon Deg.\n\n\"What, Simon!\" said Mr. Spires, starting to his feet. \"Was it thee, my\nboy?\" He did more, he stretched out his hand; the young man clasped it\neagerly, and the two stood silent, and with a heartfelt emotion, which\nblended all the past into forgetfulness, and the future into a union\nmore sacred than esteem.\n\nA week hence and Simon Deg was the son-in-law of Mr. Spires. Though Mr.\nSpires had misunderstood Simon, and Simon had borne the aspect of\nopposition to his old friend in defence of conscientious principle, the\nwife and daughter of the manufacturer had always understood him, and\nsecretly looked forward to some day of recognition and re-union.\n\nSimon Deg was now the richest man in Stockington. His mother was still\nliving to enjoy his elevation. She had been his excellent and wise\nhouse-keeper, and she continued to occupy that post still.\n\nTwenty-five years afterwards, when the worthy old Spires was dead, and\nSimon Deg had himself two sons attained to manhood; when he had five\ntimes been mayor of Stockington, and had been knighted on the\npresentation of a loyal address; still his mother was living to see it;\nand William Watson, the shoe-maker, was acting as the sort of orderly at\nSir Simon's chief manufactory. He occupied the Lodge, and walked about,\nand saw that all was safe, and moving on as it should do.\n\nIt was amazing how the most plebeian name of Simon Peg had slid, under\nthe hands of the Heralds, into the really aristocratical one of Sir\nSimon Degge. They had traced him up a collateral kinship, spite of his\nown consciousness, to a baronet of the same name of the county of\nStafford, and had given him a coat of arms that was really astonishing.\n\nIt was some years before this that Sir Roger Rockville breathed his\nlast. His title and estate had fallen into litigation. Owing to two\ngenerations having passed without any issue of the Rockville family\nexcept the one son and heir, the claims, though numerous, were so\nmingled with obscuring circumstances, and so equally balanced, that the\nlawyers raised quibbles and difficulties enough to keep the property in\nChancery, till they had not only consumed all the ready money and\nrental, but had made frightful inroads into the estate itself. To save\nthe remnant, the contending parties came to a compromise. A neighboring\nsquire, whose grandfather had married a Rockville, was allowed to secure\nthe title, on condition that the rest carried off the residuum of the\nestate. The woods and lands of Rockville were announced for sale!\n\nIt was at this juncture that old William Watson reminded Sir Simon Degge\nof a conversation in the great grove of Rockville, which they had held\nat the time that Sir Roger was endeavoring to drive the people thence.\n\"What a divine pleasure might this man enjoy,\" said Simon Deg to his\nhumble friend, \"if he had a heart capable of letting others enjoy\nthemselves.\"\n\n\"But we talk without the estate,\" said William Watson, \"what might we do\nif we were tried with it?\"\n\nSir Simon was silent for a moment; then observed that there was sound\nphilosophy in William Watson's remark. He said no more, but went away;\nand the next day announced to the astonished old man that he had\npurchased the groves and the whole ancient estate of Rockville!\n\nSir Simon Degge, the last of a long line of paupers, was become the\npossessor of the noble estate of Sir Roger Rockville of Rockville, the\nlast of a long line of aristocrats!\n\nThe following summer, when the hay was lying in fragrant cocks in the\ngreat meadows of Rockville, and on the little islands in the river, Sir\nSimon Degge, Baronet of Rockville,--for such was now his title--through\nthe suggestion of a great lawyer, formerly recorder of the borough of\nStockington to the crown--held a grand f\u00eate on the occasion of his\ncoming to reside at Rockville Hall, henceforth the family seat of the\nDegges. His house and gardens had been restored to the most consummate\norder. For years Sir Simon had been a great purchaser of works of art\nand literature, paintings, statuary, books, and articles of antiquity,\nincluding rich armor and precious works in ivory and gold.\n\nFirst and foremost he gave a great banquet to his wealthy friends, and\nno man with a million and a half is without them--and in abundance. In\nthe second place, he gave a substantial dinner to all his tenantry, from\nthe wealthy farmer of five hundred acres to the tenant of a cottage. On\nthis occasion he said, \"Game is a great subject of heart-burning, and of\ngreat injustice to the country. It was the bane of my predecessor; let\nus take care it is not ours. Let every man kill the game on the land\nthat he rents--then he will not destroy it utterly, nor allow it to grow\ninto a nuisance. I am fond of a gun myself, but I trust to find enough\nfor my propensity to the chase in my own fields and woods--if I\noccasionally extend my pursuit across the lands of my tenants, it shall\nnot be to carry off the first-fruits of their feeding, and I shall still\nhold the enjoyment as a favor.\"\n\nWe need not say that this speech was applauded most vociferously.\nThirdly, and lastly, he gave a grand entertainment to all his\nwork-people, both of the town and the country. His house and gardens\nwere thrown open to the inspection of the whole assembled company. The\ndelighted crowd admired immensely the pictures and the pleasant gardens.\nOn the lawn, lying between the great grove and the hall, an enormous\ntent was pitched, or rather a vast canvas canopy erected, open on all\nsides, in which was laid a charming banquet; a military band from\nStockington barracks playing during the time. Here Sir Simon made a\nspeech as rapturously received as that to the farmers. It was to the\neffect, that all the old privileges of wandering in the grove, and\nangling, and boating on the river, were restored. The inn was already\nrebuilt in a handsome Elizabethan style, larger than before, and to\nprevent it ever becoming a fane of intemperance, he had there posted as\nlandlord, he hoped for many years to come, his old friend and\nbenefactor, William Watson. William Watson should protect the inn from\nriot, and they themselves the groves and river banks from injury.\n\nLong and loud were the applauses which this announcement occasioned. The\nyoung people turned out upon the green for a dance, and in the evening,\nafter an excellent tea--the whole company descended the river to\nStockington in boats and barges decorated with boughs and flowers, and\nsinging a song made by William Watson for the occasion, called \"The\nHealth of Sir Simon, last and first of his Line.\"\n\nYears have rolled on. The groves and river banks and islands of\nRockville are still greatly frequented, but are never known to be\ninjured: poachers are never known there, for four reasons. First, nobody\nwould like to annoy the good Sir Simon; secondly, game is not very\nnumerous there; thirdly, there is no fun in killing it where there is\nno resistance; and fourthly, it is vastly more abundant in other\nproprietors' demesnes, and _it is_ fun to kill it there, where it is\njealously watched, and there is a chance of a good spree with the\nkeepers.\n\nAnd with what different feelings does the good Sir Simon look down from\nhis lofty eyrie, over the princely expanse of meadows, and over the\nglittering river, and over the stately woods to where Great Stockington\nstill stretches farther and farther its red brick walls, its red-tiled\nroofs, and its tall smoke-vomiting chimneys. There he sees no haunts of\ncrowded enemies to himself or any man. No upstarts, nor envious\nopponents, but a vast family of human beings, all toiling for the good\nof their families and their country. All advancing, some faster, some\nslower, to a better education, a better social condition, a better\nconception of the principles of art and commerce, and a clearer\nrecognition of their rights and their duties, and a more cheering faith\nin the upward tendency of humanity.\n\nLooking on this interesting scene from his distant and quiet home, Sir\nSimon sees what blessings flow--and how deeply he feels them in his own\ncase--from a free circulation, not only of trade, but of human\nrelations. How this corrects the mischiefs, moral and physical, of false\nsystems and rusty prejudices;--and he ponders on schemes of no ordinary\nbeauty and beneficence yet to reach his beloved town through them. He\nsees lecture halls and academies, means of sanitary purification, and\ndelicious recreation, in which baths, wash-houses, and airy homes figure\nlargely; while public walks extend all round the great industrial hive,\nincluding wood, hills, meadow, and river in their circuit of many miles.\nThere he lived and labored; there live and labor his sons; and there he\ntrusts his family will continue to live and labor to all future\ngenerations; never retiring to the fatal indolence of wealth, but aiding\nonwards its active and ever-expanding beneficence.\n\nLong may the good Sir Simon live and labor to realize these views. But\nalready in a green corner of the pleasant churchyard of Rockville may\nbe read this inscription on a marble headstone:--\"Sacred to the Memory\nof Jane Deg, the mother of Sir Simon Degge, Bart., of Rockville. This\nstone is erected in honor of the best of Mothers by the most grateful of\nsons.\"\n\n\n\n\nIII.\n\nThe Gentleman Beggar\n\nAN ATTORNEY'S STORY.\n\n\nOne morning, about five years ago, I called by appointment on Mr. John\nBalance, the fashionable pawnbroker, to accompany him to Liverpool, in\npursuit for a Levanting customer,--for Balance, in addition to pawning,\ndoes a little business in the sixty per cent. line. It rained in\ntorrents when the cab stopped at the passage which leads past the\npawning boxes to his private door. The cabman rang twice, and at length\nBalance appeared, looming through the mist and rain in the entry,\nilluminated by his perpetual cigar. As I eyed him rather impatiently,\nremembering that trains wait for no man, something like a hairy dog, or\na bundle of rags, rose up at his feet, and barred his passage for a\nmoment. Then Balance cried out with an exclamation, in answer apparently\nto a something I could not hear, \"What, man alive!--slept in the\npassage!--there, take that, and get some breakfast for Heaven's sake!\"\nSo saying, he jumped into the \"Hansom,\" and we bowled away at ten miles\nan hour, just catching the Express as the doors of the station were\nclosing. My curiosity was full set,--for although Balance can be free\nwith his money, it is not exactly to beggars that his generosity is\nusually displayed; so when comfortably ensconced in a _coup\u00e9_, I\nfinished with--\n\n\"You are liberal with your money this morning; pray, how often do you\ngive silver to street cadgers?--because I shall know now what walk to\ntake when flats and sharps leave off buying law.\"\n\nBalance, who would have made an excellent parson if he had not been bred\nto a case-hardening trade, and has still a soft bit left in his heart\nthat is always fighting with his hard head, did not smile at all, but\nlooked as grim as if squeezing a lemon into his Saturday night's punch.\nHe answered slowly, \"A cadger--yes; a beggar--a miserable wretch, he is\nnow; but let me tell you, Master David, that that miserable bundle of\nrags was born and bred a gentleman; the son of a nobleman, the husband\nof an heiress, and has sat and dined at tables where you and I, Master\nDavid, are only allowed to view the plate by favor of the butler. I have\nlent him thousands, and been well paid. The last thing I had from him\nwas his court suit; and I hold now his bill for one hundred pounds that\nwill be paid, I expect, when he dies.\"\n\n\"Why, what nonsense you are talking! you must be dreaming this morning.\nHowever, we are alone, I'll light a weed, in defiance of Railway law,\nyou shall spin that yarn; for, true or untrue, it will fill up the time\nto Liverpool.\"\n\n\"As for yarn,\" replied Balance, \"the whole story is short enough; and as\nfor truth, that you may easily find out if you like to take the\ntrouble. I thought the poor wretch was dead, and I own it put me out\nmeeting him this morning, for I had a curious dream last night.\"\n\n\"Oh, hang your dreams! Tell us about this gentleman beggar that bleeds\nyou of half-crowns--that melts the heart even of a pawnbroker!\"\n\n\"Well, then, that beggar is the illegitimate son of the late Marquis of\nHoopborough by a Spanish lady of rank. He received a first-rate\neducation, and was brought up in his father's house. At a very early age\nhe obtained an appointment in a public office, was presented by the\nmarquis at court, and received into the first society, where his\nhandsome person and agreeable manners made him a great favorite. Soon\nafter coming of age, he married the daughter of Sir E. Bumper, who\nbrought him a very handsome fortune, which was strictly settled on\nherself. They lived in splendid style, kept several carriages, a house\nin town, and a place in the country. For some reason or other, idleness,\nor to plead his lady's pride he said, he resigned his appointment. His\nfather died, and left him nothing; indeed, he seemed at that time very\nhandsomely provided for.\n\n\"Very soon Mr. and Mrs. Molinos Fitz-Roy began to disagree. She was\ncold, correct--he was hot and random. He was quite dependent on her, and\nshe made him feel it. When he began to get into debt, he came to me. At\nlength some shocking quarrel occurred; some case of jealousy on the\nwife's side, not without reason, I believe; and the end of it was Mr.\nFitz-Roy was turned out of doors. The house was his wife's, the\nfurniture was his wife's, and the fortune was his wife's--he was, in\nfact, her pensioner. He left with a few hundred pounds ready money, and\nsome personal jewellery, and went to a hotel. On these and credit he\nlived. Being illegitimate, he had no relations; being a fool, when he\nspent his money he lost his friends. The world took his wife's part,\nwhen they found she had the fortune, and the only parties who interfered\nwere her relatives, who did their best to make the quarrel incurable. To\ncrown all, one night he was run over by a cab, was carried to a\nhospital, and lay there for months, and was during several weeks of the\ntime unconscious. A message to the wife, by the hands of one of his\ndebauched companions, sent by a humane surgeon, obtained an intimation\nthat 'if he died, Mr. Croak, the undertaker to the family, had orders to\nsee to the funeral,' and that Mrs. Molinos was on the point of starting\nfor the Continent, not to return for some years. When Fitz-Roy was\ndischarged, he came to me limping on two sticks, to pawn his court suit,\nand told me his story. I was really sorry for the fellow, such a\nhandsome, thoroughbred-looking man. He was going then into the west\nsomewhere, to try to hunt out a friend. 'What to do, Balance,' he said,\n'I don't know. I can't dig, and unless somebody will make me their\ngamekeeper, I must starve or beg, as my Jezebel bade me when we parted!'\n\n\"I lost sight of Molinos for a long time, and when I next came upon him\nit was in the Rookery of Westminster, in a low lodging-house, where I\nwas searching with an officer for stolen goods. He was pointed out to\nme as the 'gentleman cadger,' because he was so free with his money when\n'in luck.' He recognized me, but turned away then. I have since seen\nhim, and relieved him more than once, although he never asks for\nanything. How he lives, Heaven knows. Without money, without friends,\nwithout useful education of any kind, he tramps the country, as you saw\nhim, perhaps doing a little hop-picking or hay-making, in season, only\nhappy when he obtains the means to get drunk. I have heard through the\nkitchen whispers that you know come to me, that he is entitled to some\nproperty; and I expect if he were to die his wife would pay the hundred\npound bill I hold; at any rate, what I have told you I know to be true,\nand the bundle of rags I relieved just now is known in every thieves'\nlodging in England as the 'gentleman cadger.'\"\n\nThis story produced an impression on me,--I am fond of speculation, and\nlike the excitement of a legal hunt as much as some do a fox-chase. A\ngentleman a beggar, a wife rolling in wealth, rumors of unknown\nproperty due to the husband; it seemed as if there were pickings for me\namidst this carrion of pauperism.\n\nBefore returning from Liverpool, I had purchased the gentleman beggar's\nacceptance from Balance. I then inserted in the \"Times\" the following\nadvertisement: \"_Horatio Molinos Fitz-Roy_.--If this gentleman will\napply to David Discount, Esq., Solicitor, St. James's, he will hear of\nsomething to his advantage. Any person furnishing Mr. F.'s correct\naddress, shall receive 1_l._ 1_s._ reward. He was last seen,\" &c. Within\ntwenty-four hours I had ample proof of the wide circulation of the\n\"Times.\" My office was besieged with beggars of every degree, men and\nwomen, lame and blind, Irish, Scotch, and English, some on crutches,\nsome in bowls, some in go-carts. They all knew him as \"the gentleman,\"\nand I must do the regular fraternity of tramps the justice to say that\nnot one would answer a question until he made certain that I meant the\n\"gentleman\" no harm.\n\nOne evening, about three weeks after the appearance of the\nadvertisement, my clerk announced \"another beggar.\" There came in an old\nman leaning upon a staff, clad in a soldier's great coat all patched and\ntorn, with a battered hat, from under which a mass of tangled hair fell\nover his shoulders and half concealed his face. The beggar, in a weak,\nwheezy, hesitating tone, said, \"You have advertised for Molinos\nFitz-Roy. I hope you don't mean him any harm; he is sunk, I think, too\nlow for enmity now; and surely no one would sport with such misery as\nhis.\" These last words were uttered in a sort of piteous whisper.\n\nI answered quickly, \"Heaven forbid I should sport with misery; I mean\nand hope to do him good, as well as myself.\"\n\n\"Then, Sir, I am Molinos Fitz-Roy!\"\n\nWhile we were conversing candles had been brought in. I have not very\ntender nerves--my head would not agree with them--but I own I started\nand shuddered when I saw and knew that the wretched creature before me\nwas under thirty years of age and once a gentleman. Sharp, aquiline\nfeatures, reduced to literal skin and bone, were begrimed and covered\nwith dry fair hair; the white teeth of the half-open mouth chattered\nwith eagerness, and made more hideous the foul pallor of the rest of the\ncountenance. As he stood leaning on a staff half bent, his long, yellow\nbony fingers clasped over the crutch-head of his stick, he was indeed a\npicture of misery, famine, squalor, and premature age, too horrible to\ndwell upon. I made him sit down, sent for some refreshment, which he\ndevoured like a ghoul, and set to work to unravel his story. It was\ndifficult to keep him to the point; but with pains I learned what\nconvinced me that he was entitled to some property, whether great or\nsmall there was no evidence. On parting, I said \"Now Mr. F., you must\nstay in town while I make proper inquiries. What allowance will be\nenough to keep you comfortably?\"\n\nHe answered humbly after much pressing, \"Would you think ten shillings\ntoo much!\"\n\nI don't like, if I do those things at all, to do them shabbily, so I\nsaid, \"Come every Saturday and you shall have a pound.\" He was profuse\nin thanks of course, as all such men are as long as distress lasts.\n\nI had previously learned that my ragged client's wife was in England,\nliving in a splendid house in Hyde Park Gardens, under her maiden name.\nOn the following day the Earl of Owing called upon me, wanting five\nthousand pounds by five o'clock the same evening. It was a case of life\nor death with him, so I made my terms and took advantage of his pressure\nto execute a _coup de main_. I proposed that he should drive me home to\nreceive the money, calling at Mrs. Molinos in Hyde Park Gardens, on our\nway. I knew that the coronet and liveries of his father, the Marquis,\nwould ensure me an audience with Mrs. Molinos Fitz-Roy.\n\nMy scheme answered. I was introduced into the lady's presence. She was,\nand probably is, a very stately, handsome woman, with a pale complexion,\nhigh solid forehead, regular features, thin, pinched, self-satisfied\nmouth. My interview was very short. I plunged into the middle of the\naffair, but had scarcely mentioned the word husband, when she\ninterrupted me with \"I presume you have lent this profligate person\nmoney, and want me to pay you.\" She paused, and then said, \"He shall not\nhave a farthing.\" As she spoke, her white face became scarlet.\n\n\"But, Madam, the man is starving. I have strong reasons for believing he\nis entitled to property, and if you refuse any assistance, I must take\nother measures.\" She rang the bell, wrote something rapidly on a card;\nand, as the footman appeared, pushed it towards me across the table,\nwith the air of touching a toad, saying, \"There, Sir, is the address of\nmy solicitors; apply to them if you think you have any claim. Robert,\nshow the person out, and take care he is not admitted again.\"\n\nSo far I had effected nothing; and, to tell the truth, felt rather\ncrest-fallen under the influence of that grand manner peculiar to\ncertain great ladies and to all great actresses.\n\nMy next visit was to the attorneys Messrs'. Leasem and Fashun, of\nLincoln's Inn Square, and there I was at home. I had had dealings with\nthe firm before. They are agents for half the aristocracy, who always\nrun in crowds like sheep after the same wine-merchants, the same\narchitects, the same horse-dealers, and the same law-agents. It may be\ndoubted whether the quality of law and land management they get on this\nprinciple is quite equal to their wine and horses. At any rate, my\nfriends of Lincoln's Inn, like others of the same class, are\ndistinguished by their courteous manners, deliberate proceedings,\ninnocence of legal technicalities, long credit and heavy charges.\nLeasem, the elder partner, wears powder and a huge bunch of seals, lives\nin Queen Square, drives a brougham, gives the dinners and does the\ncordial department. He is so strict in performing the latter duty, that\nhe once addressed a poacher who had shot a Duke's keeper, as \"my dear\ncreature,\" although he afterwards hung him.\n\nFashun has chambers in St. James Street, drives a cab, wears a tip, and\ndoes the grand haha style.\n\nMy business lay with Leasem. The interviews and letters passing were\nnumerous. However, it came at last to the following dialogue:--\n\n\"Well, my dear Mr. Discount,\" began Mr. Leasem, who hates me like\npoison. \"I'm really very sorry for that poor dear Molinos--knew his\nfather well; a great man, a perfect gentleman; but you know what women\nare, eh, Mr. Discount? My client won't advance a shilling, she knows it\nwould only be wasted in low dissipation. Now don't you think (this was\nsaid very insinuatingly)--don't you think he had better be sent to the\nwork-house; very comfortable accommodation there, I can assure you--meat\ntwice a week, and excellent soup; and then, Mr. D., we might consider\nabout allowing you something for that bill.\"\n\n\"Mr. Leasem, can you reconcile it to your conscience to make such an\narrangement. Here's a wife rolling in luxury, and a husband starving!\"\n\n\"No, Mr. Discount, not starving; there is the work-house, as I observed\nbefore; besides, allow me to suggest that these appeals to feeling are\nquite unprofessional--quite unprofessional.\"\n\n\"But, Mr. Leasem, touching this property which the poor man is entitled\nto.\"\n\n\"Why, there again, Mr. D., you must excuse me; you really must. I don't\nsay he is, I don't say he is not. If you know he is entitled to\nproperty, I am sure you know how to proceed; the law is open to you, Mr.\nDiscount--the law is open; and a man of your talent will know how to use\nit.\"\n\n\"Then, Mr. Leasem, you mean that I must in order to right this starving\nman, file a Bill of Discovery to extract from you the particulars of his\nrights. You have the Marriage Settlement, and all the information, and\nyou decline to allow a pension, or afford any information; the man is to\nstarve, or go to the work-house?\"\n\n\"Why, Mr. D., you are so quick and violent, it really is not\nprofessional; but you see (here a subdued smile of triumph), it has been\ndecided that a solicitor is not bound to afford such information as you\nask, to the injury of his client.\"\n\n\"Then you mean that this poor Molinos may rot and starve, while you\nkeep secret from him, at his wife's request, his title to an income, and\nthat the Court of Chancery will back you in this iniquity?\"\n\nI kept repeating the word \"starve,\" because I saw it made my respectable\nopponent wince. \"Well, then, just listen to me. I know that in the happy\nstate of our equity law, Chancery can't help my client; but I have\nanother plan; I shall go hence to my office, issue a writ, and take your\nclient's husband in execution--as soon as he is lodged in jail, I shall\nfile his schedule in the Insolvent Court, and when he comes up for his\ndischarge, I shall put you in the witness-box, and examine you on oath,\n'touching any property of which you know the insolvent to be possessed,'\nand where will be your privileged communications then?\"\n\nThe respectable Leasem's face lengthened in a twinkling, his comfortable\nconfident air vanished, he ceased twiddling his gold chain, and at\nlength he muttered, \"Suppose we pay the debt?\"\n\n\"Why, then, I'll arrest him the day after for another.\"\n\n\"But, my dear Mr. Discount, surely such, conduct would not be quite\nrespectable?\"\n\n\"That's my business; my client has been wronged, I am determined to\nright him, and when the aristocratic firm of Leasem and Fashun takes\nrefuge, according to the custom of respectable repudiators, in the cool\narbors of the Court of Chancery, why, a mere bill-discounting attorney,\nlike David Discount, need not hesitate about cutting a bludgeon out of\nthe Insolvent Court.\"\n\n\"Well, well, Mr. D., you are so warm--so fiery; we must deliberate, we\nmust consult. You will give me until the day after to-morrow, and then\nwe'll write you our final determination; in the, meantime send us copy\nof your authority to act for Mr. Molinos Fitz-Roy.\"\n\nOf course I lost no time in getting the gentleman beggar to sign a\nproper letter.\n\nOn the appointed day came a communication with the L. and F. seal,\nwhich I opened not without unprofessional eagerness. It was as follows:\n\n    \"_In re Molinos Fitz-Roy and Another._\n\n     \"Sir,--In answer to your application on behalf of Mr. Molinos\n     Fitz-Roy, we beg to inform you that under the administration of a\n     paternal aunt who died intestate, your client is entitled to two\n     thousand five hundred pounds eight shillings and sixpence, Three\n     per Cents; one thousand five hundred pounds nineteen shillings and\n     fourpence, Three per Cents Reduced; one thousand pounds, Long\n     Annuities; five hundred pounds, Bank Stock; three thousand five\n     hundred pounds, India Stock, besides other securities, making up\n     about ten thousand pounds, which we are prepared to transfer over\n     to Mr. Molinos Fitz-Roy's direction forthwith.\"\n\nHere was a windfall! It quite took away my breath.\n\nAt dusk came my gentleman beggar, and what puzzled me was how to break\nthe news to him. Being very much overwhelmed with business that day, I\nhad not much time for consideration. He came in rather better dressed\nthan when I first saw him, with only a week's beard on his chin; but, as\nusual, not quite sober. Six weeks had elapsed since our first interview.\nHe was still the humble, trembling, low-voiced creature, I first knew\nhim.\n\nAfter a prelude, I said, \"I find, Mr. F., you are entitled to something;\npray, what do you mean to give me in addition to my bill for obtaining\nit?\" He answered rapidly, \"Oh, take half; if there is one hundred\npounds, take half; if there is five hundred pounds, take half.\"\n\n\"No, no, Mr. F., I don't do business in that way, I shall be satisfied\nwith ten per cent.\"\n\nIt was so settled. I then led him out into the street, impelled to tell\nthe news, yet dreading the effect; not daring to make the revelation in\nmy office for fear of a scene.\n\nI began hesitatingly, \"Mr. Fitz-Roy, I am happy to say that I find you\nare entitled to.... ten thousand pounds!\"\n\n\"Ten thousand pounds!\" he echoed. \"Ten thousand pounds!\" he shrieked.\n\"Ten thousand pounds!\" he yelled, seizing my arm violently.\n\n\"You are a brick,---- Here, cab! cab!\" Several drove up--the shout might\nhave been heard a mile off. He jumped in the first.\n\n\"Where to?\" said the driver.\n\n\"To a tailor's, you rascal!\"\n\n\"Ten thousand pounds! ha, ha, ha!\" he repeated hysterically, when in the\ncab; and every moment grasping my arm. Presently he subsided, looked me\nstraight in the face, and muttered with agonizing fervor, \"What a jolly\nbrick you are!\"\n\nThe tailor, the hosier, the bootmaker, the hairdresser, were in turn\nvisited by this poor pagan of externals. As by degrees under their hands\nhe emerged from the beggar to the gentleman, his spirits rose; his eyes\nbrightened; he walked erect, but always nervously grasping my arm;\nfearing, apparently, to lose sight of me for a moment, lest his fortune\nshould vanish with me. The impatient pride with which he gave his order\nto the astonished tradesman for the finest and best of everything, and\nthe amazed air of the fashionable hairdresser when he presented his\nmatted locks and stubble chin to be \"cut and shaved,\" may be _acted_--it\ncannot be described.\n\nBy the time the external transformation was complete, and I sat down in\na _caf\u00e9_ in the Haymarket opposite a haggard but handsome\nthoroughbred-looking man, whose air, with the exception of the wild eyes\nand deeply-browned face, did not differ from the stereotyped men about\ntown sitting around us, Mr. Molinos Fitz-Roy had already almost\nforgotten the past; he bullied the waiter, and criticized the wine, as\nif he had done nothing else but dine and drink and scold there all the\ndays of his life.\n\nOnce he wished to drink my health, and would have proclaimed his whole\nstory to the coffee-room assembly in a raving style. When I left he\nalmost wept in terror at the idea of losing sight of me. But, allowing\nfor these ebullitions--the natural result of such a whirl of events--he\nwas wonderfully calm and self-possessed.\n\nThe next day his first care was to distribute fifty pounds among his\nfriends the cadgers, at a house of call in Westminster, and formally to\ndissolve his connection with them; those present undertaking for the\n\"fraternity,\" that for the future he should never be noticed by them in\npublic or private.\n\nI cannot follow his career much further. Adversity had taught him\nnothing. He was soon again surrounded by the well-bred vampires who had\nforgotten him when penniless; but they amused him, and that was enough.\nThe ten thousand pounds were rapidly melting when he invited me to a\ngrand dinner at Richmond, which included a dozen of the most agreeable,\ngood-looking, well-dressed dandies of London, interspersed with a\ndisplay of pretty butterfly bonnets. We dined deliciously, and drank as\nmen do of iced wines in the dog-days--looking down from Richmond Hill.\n\nOne of the pink bonnets crowned Fitz-Roy with a wreath of flowers; he\nlooked--less the intellect--as handsome as Alcibiades. Intensely excited\nand flushed, he rose with a champagne glass in his hand to propose my\nhealth.\n\nThe oratorical powers of his father had not descended on him. Jerking\nout sentences by spasms, at length he said, \"I was a beggar--I am a\ngentleman--thanks to this----\"\n\nHere he leaned on my shoulder heavily a moment, and then fell back. We\nraised him, loosened his neckcloth--\n\n\"Fainted!\" said the ladies--\n\n\"Drunk!\" said the gentlemen--\n\nHe was _dead_!\n\n\n\n\nIV.\n\n\"Evil is Wrought by Want of Thought.\"\n\n\n\"It must come some day; and come when it will, it will be hard to do, so\nwe had best go at once, Sally. I shall have more trouble with Miss\nIsabel than you will with Miss Laura; for I am twice the favorite you\nare.\"\n\nSo said Fanny to her cousin, who had just turned to descend the\nstaircase of Aldington Hall, where they had both lived since they were\nalmost children, in attendance on the two daughters of the old baronet,\nwho were near their own ages, and had always treated them with great\nkindness.\n\n\"I am not sure of that,\" replied Sally, \"for Miss Laura is so seldom put\nout, that when once she is vexed, she will be hard to comfort; and I am\nsure, Fanny, she loves me every bit as well as Miss Isabel does you,\nthough it is her way to be so quiet. I dare say she will cry when I say\nI must go; but then John would be like to cry too, if I put him off\nlonger.\"\n\nThis consideration restored Sally's courage, and she proceeded with\nFanny to the gallery into which the rooms of their young mistresses\nopened; but here Fanny's heart failed her; and, stopping short, she\nsaid,\n\n\"Suppose we tell them to wait awhile longer, as the young ladies are\ngoing to travel. We might as well see the world first, and marry in a\nyear or two. But still,\" added she, after a pause, \"I could not find it\nin my heart to say so to Thomas; and I promised him to speak to-day.\"\n\nEach cousin then knocked at the door of her mistress. Laura was not in\nher room, and Sally went to seek her below stairs; but Isabel called to\nFanny to go in.\n\nFanny obeyed, and walking forward a few steps, faltered out, with many\nblushes, that as young Thomas had kept company with her for nearly a\ntwelvemonth, and had taken and furnished a little cottage, and begged\nhard to take her home to it; she was sorry to say, that if Miss Isabel\nwould give her leave, she wished to give warning and to go from her\nservice in a month.\n\nFanny's most sanguine wishes or fears, must have been surpassed by the\nburst of surprise and grief that followed her modest statement. Isabel\nreproached her; refused to take her warning; declared she would never\nsee her again if she left the Hall, and that rather than be served by\nany but her dear Fanny, she would wait upon herself all her life. Fanny\nexpostulated, and told her mistress that, foreseeing her unwillingness\nto lose her, she had already put Thomas off several months; and that at\nlast, to gain further delay, she had run the risk of appearing selfish,\nby refusing to marry him till he had furnished a whole cottage for her.\nThis, she said he had--by working late and early--accomplished in a\nsurprisingly short time, and had the day before claimed the reward of\nhis industry. \"And now, Miss,\" added she, \"he gets quite pale, and\nbegins to believe I do not love him, and yet I do, better than all the\nworld, and could not find it in my heart to vex him, and make him look\nsad again. Yesterday he seemed so happy, when I promised to be his wife\nin a month.\" Here Fanny burst into tears. Her sobs softened Isabel, who\nconsented to let her go; and after talking over her plans, became as\nenthusiastic in promoting, as she had at first been, in opposing them.\nThomas was to take Fanny over to see the cottage, that evening, and\nIsabel, in the warmth of her heart, promised to accompany them. Fanny\nthanked her with a curtesy, and thought how pleased she ought to be at\nsuch condescension in her young mistress, but could not help fearing\nthat her sweetheart would not half appreciate the favor.\n\nAfter receiving many promises of friendship and assistance, Fanny\nhastened to report to Sally the success of her negotiation. Sally was\nsitting in their little bedroom, thoughtful, and almost sad. She\nlistened to Fanny's account; and replied in answer to her questions\nconcerning Miss Laura's way of taking her warning, \"I am afraid, Fanny,\nyou were right in thinking yourself the greatest favorite, for Miss\nLaura seemed almost pleased at my news; she took me by the hand, and\nsaid, 'I am very glad to hear you are to marry such a good young man as\nevery one acknowledges John Maythorn to be, and you may depend upon my\nbeing always ready to help you, if you want assistance.' She then said a\ndeal about my having lived with her six years, and not having once\ndispleased her, and told me that master had promised my mother and yours\ntoo, that his young ladies should see after us all our lives. This was\nvery kind, to be sure; but then Miss Isabel promised you presents\nwhether you wanted assistance or not, and is to give you a silk gown and\na white ribbon for the wedding, and is to go over to the cottage with\nyou; now Miss Laura did not say a word of any such thing.\"\n\nFanny tried to comfort her cousin by saying it was Miss Laura's quiet\nway; but she could not help secretly rejoicing that her own mistress\nwas so generous and affectionate.\n\nIn the evening the two sweethearts came to lead their future wives to\nthe cottages, which were near each other, and at about a mile from the\nHall. John had a happy walk. He learned from Sally that he was to \"take\nher home\" in a month, and was so pleased at the news, that he could\nscarcely be happier when she bustled about, exclaiming at every new\nsight in the pretty bright little cottage. The tea-caddy, the cupboard\nof china, and a large cat, each called forth a fresh burst of joy. Sally\nthought everything \"the prettiest she had ever seen;\" and when John made\nher sit in the arm-chair and put her feet on the fender, as if she were\nalready mistress of the cottage, she burst into sobs of joy. We will not\npause to tell how her sobs were stopped, nor what promises of unchanging\nkindness, were made in that bright little kitchen; but we may safely\naffirm that Sally and John were happier than they had ever been in their\nlives, and that old Mrs. Maythorn, who was keeping the cottage for\nSally, felt all her fondest wishes were fulfilled as she saw the two\nlovers depart.\n\nFanny and Thomas, who had left them at the cottage door, walked on to\ntheir own future home, quite overwhelmed by the honor Miss Isabel was\nconferring on them by walking at their side.\n\n\"You see, Miss,\" said Thomas, as he turned the key of his cottage-door,\n\"there is nothing to speak of here, only such things as are necessary,\nand all of the plainest; but it will do well enough for us poor folks;\"\nand as he threw open the door, he found to his surprise that what had\nseemed to him yesterday so pretty and neat, now looked indeed \"all of\nthe plainest.\" The very carpet, and metal teapot, which he had intended\nas surprises for Fanny, he was now ashamed of pointing out to her, and\nhe apologized to Isabel for the coarse quality of the former, telling\nher it was only to serve till he could get a better.\n\n\"Yes,\" answered she, \"this is not half good enough for my little Fanny,\nshe must have a real Brussels carpet. I will send her one. I will make\nyour cottage so pretty, Fanny, you shall have a nice china tea set, not\nthese common little things, and I will give you some curtains for the\nwindow.\" Thomas blushed as this deficiency was pointed out. \"Why, Miss,\"\nsaid he, \"I meant to have trained the rose tree over the window, I\nthought that would be shady, and sweet in the summer, and in the winter,\nwhy, we should want all the day-light; but then to be sure, curtains\nwill be much better.\"\n\n\"Yes, Thomas,\" replied the young lady, \"and warm in the winter; you\ncould not be comfortable with a few bare rose stalks before your window,\nwhen the snow was on the ground.\" This had not occurred to Thomas, who\nnow said faintly, \"Oh, no, Miss,\" and felt that curtains were\nindispensable to comfort.\n\nSimilar deficiencies or short-comings were discovered everywhere, so\nthat even Fanny, who would at first be pleased with all she saw, in\nspite of the numerous defects that seemed to exist everywhere, gradually\ngrew silent and ashamed of her cottage. She did her utmost to conceal\nfrom Thomas how entirely she agreed with her mistress, and as this\ngenerous young lady finished every remark, by saying \"I will get you\none,\" or \"I will send you another,\" she felt that all would be right\nbefore long.\n\nAs Thomas closed the door, he wondered how in his wish to please Fanny\nhe could have deceived himself so completely as to the merits of his\ncottage and furniture; but he too comforted himself by remembering how\nhis kind patroness was to remedy all the defects; \"though,\" thought he,\n\"I should have liked better to have done it all well myself.\"\n\nThe lady and the two lovers walked homewards, almost without speaking,\ntill they overtook John and Sally, who were whispering and laughing,\ntalking of their cottage, Mrs. Maythorn's joy at seeing them happy,\ntheir future plans for themselves and her, and all in so confused a way,\nthat though twenty new subjects were started and discussed, none came to\nand conclusion, but that John and Sally loved each other and were very,\nvery happy.\n\n\"What ails you, Thomas?\" said John. \"Has any one robbed your house? I\ntold you it was not safe to leave it,\" but seeing Miss Isabel, he\ntouched his hat and fell back to where Fanny was talking to her cousin.\nIsabel, however, left them that she might take a short cut through the\npark, while they went round by the road.\n\nAt the end of the walk, Sally was half inclined to be dissatisfied with\nher furniture, so much had Fanny boasted of the improvements that were\nto be made in her own, but she could not get rid of the first impression\nit had made on her, and in a few days she quite forgot the want of\ncurtains and carpet, and could only remember the happy time when she sat\nin the arm-chair with her foot on the fender.\n\nAs the month drew to a close, the two sisters made presents to their\nmaids. Laura gave Sally a merino dress, a large piece of linen, a cellar\nfull of coals, and a five pound note. Isabel gave Fanny a silk gown\nthat cost three guineas, a beautiful white bonnet ribbon, a small\nchimney glass (for which she kindly went into debt), three left-off\nmuslin dresses, a painting done by her own hand, in a handsome gilt\nframe, and a beautiful knitted purse. Besides all this, she told Fanny\nit was still her intention to get the other things she had promised for\nthe cottage, as soon as she had paid for the chimney glass. \"I am very\nsorry,\" she said, \"that just now I am so poor, for unfortunately, as you\nknow, I have had to pay for those large music volumes I ordered when I\nwas in London, and which after all I never used. It always happens that\nI am poor when I want to make presents.\"\n\nFanny stopped her mistress with abundant thanks for the beautiful things\nshe had already given her. \"I am sure, Miss,\" said she, \"I shall\nscarcely dare wear these dresses, they look so lady-like and fine; Sally\nwill seem quite strange by me. And this purse too, Miss; I never saw\nanything so smart.\"\n\nIsabel was quite satisfied that she had eclipsed her sister in the\nnumber and value of her gifts, but she still assured Fanny she had but\nmade a beginning. Large and generous indeed, were this young lady's\nintentions.\n\nOn the wedding morning Isabel rose early and dressed herself without\nassistance, then crossing to the room of the two cousins, she entered\nwithout knocking. Sally was gone, and Fanny lay sleeping alone.\n\n\"How pretty she is!\" said Isabel to herself. \"She ought to be dressed\nlike a lady to-day. I will see to it;\" then glancing proudly at the silk\ngown, which was laid out with all the other articles of dress, ready for\nthe coming ceremony, her heart swelled with consciousness of her own\ngenerosity. \"I have done nothing yet,\" continued she; \"she has been with\nme nearly six years, and always pleased me entirely, then papa promised\nher mother that he should befriend her as long as we both lived, and he\nhas charged us both to do our utmost for our brides. Laura has bought\nSally a shawl, I ought to give one too--what is this common thing?\nFanny! Fanny! wake up. I am come to be your maid to-day, for you shall\nbe mistress on your wedding morning and have a lady to dress you. What\nis this shawl? It will not do with a silk dress, wait a minute,\" and off\nshe darted, leaving Fanny sitting up and rubbing her eyes trying to\nremember what her young mistress had said. Before she was quite\nconscious, Isabel returned with a Norfolk shawl of fine texture and\ndesign, but somewhat soiled. \"There,\" said she, throwing it across the\nsilk gown, \"those go much better together. I will give it you, Fanny.\"\n\n\"Thank you, Miss,\" said Fanny, in a tone of hesitation; \"but--but\nsuppose, Miss, I was to wear Thomas' shawl just to-day, as he gave it me\nfor the wedding, and John got Sally one like it--I think, Miss--don't\nyou think, Miss, it might seem unkind to wear any other just to-day?\"\n\n\"Why, it is just to-day I want to make you look like a lady, Fanny; no,\nno, you must not put on that white cotton-looking shawl with a silk\ndress, and this ribbon,\" said Isabel, taking up the bonnet, proudly.\nFanny looked sad, but the young mistress did not see this, for she was\nexamining the white silk gloves that lay beside the bonnet. \"These,\"\nthought she, \"are not quite right, they look servantish, but my kid\ngloves would not fit her, besides, I have none clean, and it is well,\nperhaps, that she should have a few things to mark her rank. Yes, they\nwill do.\"\n\nThere was so much confusion between the lady's offering help, and the\nmaid's modestly refusing it, that the toilette was long in completing.\nAt last, however, Isabel was in ecstasies. \"Look,\" said she, \"how the\nbonnet becomes you! and the Norfolk shawl, too, no one would think you\nwere only a lady's-maid, Fanny. Stop, I will get a ribbon for your\nthroat.\" Off she flew, and was back again in five minutes. \"But what is\nthat for, Fanny? Are you afraid it will rain this bright morning?\"\n\nFanny had, in Isabel's absence, folded Thomas' shawl, and hung it across\nher arm. \"I thought, Miss,\" answered she, blushing, \"that I might just\ncarry it to show Thomas that I did not forget his present, or think it\ntoo homely to go to church with me.\"\n\n\"Impossible,\" said Isabel, who, to do her justice, we must state, was\nfar too much excited to suspect that she was making Fanny uncomfortable;\n\"you will spoil all. There, put the shawl away,--that's right, you look\nperfect. Go down to your bridegroom, I hear his voice in the hall, I\nwill not come too, though I should like above all things to see his\nsurprise, but I should spoil your meeting, and I am the last person in\nthe world to do anything so selfish. One thing more, Fanny: I shall give\nyou two guineas, that you may spend three or four days at L----, by the\nsea-side; no one goes home directly, you would find it very dull to\nsettle down at once in your cottage; tell Thomas so.\" Isabel then\nretired to her room, wishing heartily that she could part with half her\nprettiest things, that she might heap more favors on the interesting\nlittle bride.\n\nLaura's first thought that morning had also been of the little orphan,\nwho had served her so long and faithfully, and whom her father had\ncommended to her special care. She, too, had risen early, but without\ndressing herself, she went across to Sally. Sally was asleep, with the\ntraces of tears on her cheeks; Laura looked at her for a few moments,\nand remembered how, when both were too young to understand the\ndistinction of rank, they had been almost play-mates; she wiped from her\nown eyes a little moisture that dimmed them, then putting her hand\ngently on Sally's shoulder, she said, \"Wake, Sally, I call you early\nthat you may have plenty of time to dress me first and yourself\nafterwards. I know you would not like to miss waiting on me, or to do it\nhurriedly for the last time. You have been crying, Sally, do not color\nabout it, I should think ill of you if you were not sorry to leave us,\nyou cannot feel the parting more than I do. I dare say I shall have hard\nwork to keep dry eyes all day, but we must do our best, Sally, for it\nwill not for John to think I grudge you to him, or that you like me\nbetter than you do him.\"\n\n\"Oh, no, Miss!\" replied Sally, who felt at that moment that she could\nscarcely love any one better than her kind mistress. \"Still John will\nnot be hard upon me for a few tears,\" added she, putting the sheet to\nher eyes.\n\n\"Come, come, Sally, this will not do, jump up and dress yourself\nquickly, that you may be ready to brush my hair when I return from the\ndressing-room; you must do it well to day, for you know I am not yet\nsuited with a maid, and do it myself to-morrow.\"\n\nThis roused Sally, who dressed in great haste and was soon at her post.\nLaura asked her many questions about her plans for the future, and found\nwith pleasure that most things had been well considered and arranged.\n\"There is only one thing, Miss,\" said Sally, in conclusion, \"that we are\nsorry for, and it is that we cannot offer old Mrs. Maythorn a home. She\nhas no child but John, and will sadly feel his leaving her.\"\n\n\"But why cannot she live with you and work as she does now, so as to pay\nyou for what she costs?\"\n\n\"Why, Miss, where she is she works about the house for her board, and\ndoes a trifle outdoors besides, that gets her clothing. John says it\nmakes him feel quite cowardly, as it were, to see his old mother working\nat scrubbing and scouring, making her poor back ache, when he is so\nyoung and strong; yet we scarcely know if we could undertake for her\naltogether. I wish we could.\"\n\n\"How much would it cost you?\"\n\n\"A matter of four shillings a week; besides, we must get a bed and\nbedding. That we could put up in the kitchen, if we bought it to shut up\nin the day-time, and, as John says, Mrs. Maythorn would help us nicely\nwhen we get some little ones. But it would cost a deal of money to begin\nand go on with.\"\n\n\"I will think of this for you, Sally. It would be easy for me to give\nyou four shillings a week now, but I may not always be able to do it. I\nmay marry a poor man, or one who will not allow me to spend my money as\nI please, and were Mrs. Maythorn to give up her present employments,\nshe would not be able to get them back again three or four years hence,\nnor would she, at her age, be able to meet with others; and if you would\nfind it difficult to keep her now, you would much more when you have a\nlittle family; so we must do nothing hastily. I will consult papa; he\nwill tell me directly whether I shall be right in promising you the four\nshillings a week. If I do promise it, you may depend on always having\nit.\"\n\n\"Oh, thank you, thank you, Miss, for the thought: I will tell John\ndirectly I see him; the very hope will fill him with joy.\"\n\n\"No,\" said Laura, \"do not tell him yet, Sally, for you would be sorry to\ndisappoint him afterwards, if I could not undertake it. Wait a day or\ntwo, and I will give you an answer; or, if possible, it shall be sooner.\nNow, thank you for the nice brushing: I will put up my hair while you go\nand dress; it is getting late. If you require assistance, and Fanny is\nnot in your room, tap at my door, for I shall be pleased to help you\nto-day.\"\n\nLaura was not called in; but when she thought the toilette must be\nnearly completed, went to Sally with the shawl which she had bought for\nher the day before. As she entered, Sally was folding the white one John\nhad given her. \"I have brought you a shawl,\" said Laura, \"which I want\nyou to wear to-day; it is much handsomer than that you are folding. See,\ndo you like it?\"\n\n\"Yes, Miss,\" said Sally, \"it is a very good one, I see,\" and she began\nto re-fold the other; but Laura noticed the expression of disappointment\nwith which she made the change, and taking up the plain shawl, said, \"I\ndo not know whether this does not suit your neat muslin dress better\nthan mine. Did you buy it yourself, Sally?\"\n\n\"No, Miss, it was John's present; but I will put on yours this morning,\nif you please, Miss, and I can wear John's any day.\"\n\n\"No, no,\" replied Laura, \"you must put on John's to-day. It matters but\nlittle to me when you wear mine, so long as it does you good service;\nbut John will feel hurt if you cast his present aside on your\nwedding-day, because some one else has given you a shawl worth a few\nshillings more.\" So Laura put the white shawl on the shoulders of Sally,\nwho valued it more than the finest Cashmere in the world.\n\nAs Sally went down stairs, she saw Fanny in tears on the landing. \"I\ncannot think how it is,\" answered she, in reply to Sally's questioning,\n\"but just on this day, when I thought to feel so happy, I am quite low.\nMiss Isabel has been so kind, she has dressed me, and quite flustered me\nwith her attentions. See what nice things she has given me--this\nshawl--though for that matter, I'd rather have worn Thomas's. Oh, how\nnice you look. Dear, so neat and becoming your station, and with John's\nshawl, too, but then Miss Laura has made you no present.\"\n\n\"Yes, a good shawl and a promise besides, but I well tell you about that\nanother time. Let us go in now, they must be waiting for us.\"\n\nFanny felt so awkward in her fine clothes, that she could scarcely be\nprevailed on to encounter the gaze of the servants; but her\ngood-natured cousin promising to explain that all her dress was given\nand chosen by her mistress, she at last went into the hall. Sally's\nexplanation was only heard by a few of the party, and as Fanny, in\ntrying to conceal herself from the gaze of the astonished villagers,\nslunk behind old Mrs. Maythorn, she had the mortification of hearing her\nsay to John, in the loud whisper peculiar to deaf people, \"I am so glad,\nJohn, the neat one is yours; I should be quite frightened to see you\ntake such a fine lady as Fanny to the altar; it makes me sorry for\nThomas to see her begin so smart.\"\n\nWhen the ceremony was over, the party returned to the Hall, where a\nhospitable meal had been provided for all the villagers of good\ncharacter who chose to partake of it. It was a merry party, for even\nFanny, when every one had seen her finery long enough to forget it,\nforgot it herself. Thomas was very good-natured about the shawl, and\ndelighted at the prospect of spending a few days at L----. He and Fanny\ntalked of the boat-excursions they would have, the shells they would\ngather for a grotto in their garden, and the long rambles they would\ntake by the sea-side, till they wondered how ever they could have been\ncontented with the prospect of going to their cottage at once.\n\nAs the pony-chaise which the good baronet had lent for the day, drove up\nto take the bridal party to L----, for John and Sally were also to spend\none day there, the two young ladies came to take leave of their\n_prot\u00e9g\u00e9es_. Laura said, \"Good-bye, Sally, I have consulted papa, and\nwill undertake to allow you four shillings a week as long as Mrs.\nMaythorn lives. Here is a sovereign towards expenses; you will not, I am\nsure, mind changing your five pound note for the rest.\"\n\nIsabel said, \"Good-bye, Fanny. I am very, very sorry to disappoint you\nof your treat at L----, but I intended to have borrowed the two pounds\nof Miss Laura, and I find she cannot lend them to me. Never mind, I am\nsure you will be happy enough in your little cottage. I never saw such a\nsweet little place as it is.\" So the bridal party drove away.\n\nIn less than a week the cousins were established in their new abode.\nSally settled and happy; but Fanny, unsettled, always expected the new\ncarpet, the china tea-set, and the various other alterations that Isabel\nhad suggested and promised to make. The young lady, however, was\nunfortunate with her money. At one time she lost a bank-note; at\nanother, just as she was counting out money for the Brussels carpet, the\nnew maid entered to tell her that sundry articles of dress were \"past\nmending,\" and must be immediately replaced. One thing after another\nnipped her generous intentions in the bud, and at last she was obliged\nto set out for her long-expected journey to France without having done\nmore towards the fulfilment of her promises than call frequently on\nFanny, to remind her that all her present arrangements were temporary,\nand that she should shortly have almost everything new.\n\n\"Good-bye, Fanny,\" said she at parting; \"I shall often write to you,\nand send you money. I will not make any distinct promise, for I dare say\nI shall be able to do more than I should like to say now.\"\n\nLaura had given Sally a great many useful things for her cottage, but\nmade no promise at parting. She said, \"Be sure you write to me, Sally,\nfrom time to time, to say how you are going on, and tell me if you want\nhelp.\"\n\nWhen Isabel was gone, Fanny saw that she must accustom herself to her\ncottage as it was, and banish from her mind the idea of the\nlong-anticipated improvements. It was, however, no easy task. The window\nonce regarded as bare and comfortless still seemed so, in spite of\nFanny's reasoning that it was no worse than Sally's, which always looked\ncheerful and pretty. To be sure, John, who did not think of getting\ncurtains, had trained a honeysuckle over it, still that made but little\nshow at present. The carpet, too, so long regarded as a coarse temporary\nthing, never regained the beauty it first had to the eye of Thomas, as\nhe laid it down the evening before he took Fanny to the cottage; and\nFanny could never forget, as she arranged her tea-things, that Miss\nIsabel had called them \"common little things;\" so of all the other\npieces of furniture that the young lady had remarked upon. Sally's house\nwas, in reality, more homely than her cousin's, yet as she had never\nentertained a wish that it should be better, and as Laura had been\npleased with all its arrangements, she bustled about it with perfect\nsatisfaction; and even to Fanny it seemed replete with the comfort her\nown had always wanted.\n\nAt the end of three months Isabel enclosed an order for three pounds to\nFanny, desiring her to get a Brussels carpet, and if there was a\nsufficient remainder, to replace the tea seat.\n\n\"I would rather,\" said Fanny to her cousin, \"put up with the old carpet\nand china, and get a roll of fine flannel, some coals, an extra blanket\nor two, and a cradle for the little one that's coming, for it will be\ncold weather when I am put to bed; but I suppose as Miss Isabel has set\nher mind on the carpet and china, I must get them.\"\n\nA week or two after John was invited, with his wife and mother, to drink\ntea from Fanny's new china. It was very pretty, so was the carpet, and\nso was Fanny making tea, elated with showing her new wealth.\n\n\"Is not Miss Isabel generous?\" asked she, as she held the milk-pot to be\nadmired.\n\n\"I sometimes wish Miss Laura had as much money to spare,\" replied Sally;\n\"for she lets me lay it out as I please, and I could get a number of\nthings for three guineas.\"\n\n\"Fie, Sally,\" said her husband; \"are not three shillings to spend as one\npleases, better than three guineas laid out to please some one else?\"\n\n\"Nonsense, John,\" said Fanny, pettishly; \"how can a carpet for my\nkitchen be bought to please any one but me?\"\n\n\"John isn't far wrong either,\" answered her husband; \"but the carpet is\nvery handsome, and does please you and me too, now it is here.\"\n\nTime passed on, and Fanny gave birth to a little girl. Isabel stood\nsponsor for her by proxy, sending her an embroidered cloak and lace cap,\nand desiring that she should be called by her own name. Little Bella was\nvery sickly, and as her mother had not been able to procure her good\nwarm clothing, or lay in a large stock of firing, she suffered greatly\nfrom cold during the severe winter that followed her birth. The spring\nand summer did not bring her better health; and as Fanny always\nattributed her delicacy to the want of proper warmth in her infancy, she\ntook a great dislike to the Brussels carpet, which now lay in a roll\nbehind a large chest, having been long ago taken up as a piece of\ninconvenient luxury in a kitchen. \"I wish you could find a corner for it\nin your cottage, Sally,\" said she, \"for I never catch a sight of it\nwithout worrying myself to think how much flannels and coals I might\nhave bought with the money it cost.\"\n\nLaura frequently sent Sally small presents of money, but Isabel, though\nnot so regular as her sister, surprised every one by the splendor of\nher presents, when they did come. As Bella entered her second year she\nreceived from her godmother a beautiful little carriage, which Thomas\nsaid must have \"spoilt a five-pound note.\" This was Isabel's last gift,\nfor it was at about this time that she accepted an offer from a French\ncount, and became so absorbed in her own affairs, that she forgot Fanny\nand Bella too. Poor Bella grew more and more sickly every month; the\napothecary ordered her beef tea, arrowroot, and other strengthening\ndiet, but work was slack with Thomas, and it was with difficulty that he\ncould procure her the commonest food. \"I am sure,\" said Fanny to her\ncousin, as little Bella was whining on her knee, \"that if only Miss\nIsabel were here, she would set us all right. She never could bear to\nsee even a stranger in distress.\"\n\n\"I wish,\" said Thomas, \"that great folks would think a little of what\nthey don't see. I'll lay anything Miss Isabel gives away a deal of\nmoney, more than enough to save our little one, to a set of French\nimpostors that cry after her in the street, and yet, when she knows our\nchild is ill, she never cares, because she can't see it grow thin, or\nhear it cry.\"\n\n\"For shame, Thomas,\" said his wife, \"do not speak so rudely of the young\nlady. Have you forgotten the pretty carriage she sent Bella, and how\npleased we were when it came?\"\n\n\"I don't mean any harm,\" answered her husband; \"only it strikes me that\nMiss was pleased to buy the carriage because it was pretty, and seemed a\ngreat thing to send us, and that she wouldn't have cared a straw to give\nus a little cash, that would have served us every bit as well.\"\n\n\"I never heard you so ungrateful, Thomas. Of course she wouldn't,\nbecause she wished to please us.\"\n\n\"Or herself, as John said; but maybe I am wrong; only it goes to my\nheart to see the child want food while there is a filagree carriage in\nthe yard that cost more than would keep her for six months.\"\n\n\"Well, cheer up,\" said Sally; \"Miss Laura will be coming home soon, and\nI'll lay anything she won't let Bella die of want.\"\n\n\"I'm afraid she won't think of giving to me, Sally,\" said Fanny\ndespondingly; \"I was never her maid, you know.\"\n\n\"You wouldn't fear, if you knew Miss Laura as I do, Fanny; she never\ncares who she helps so long as the person is deserving, and in want. She\nhas no pride of that sort.\"\n\nIsabel's marriage was put off, and Laura's return, consequently,\npostponed. As Bella grew worse every day, and yet no help came, the\nunselfish Sally wrote to her patroness, telling her of poor Fanny's\ndistress, and begging her either to send her help, or speak on her\nbehalf to her sister.\n\nIsabel was dressing for a party when Laura showed her Sally's letter.\n\"Poor Fanny,\" said she, \"I wish I had known it before I bought this\nwreath. I have, absolutely, not a half-franc in the world. Will you buy\nthe wreath of me at half-price, it has not even been taken from its\nbox.\"\n\n\"I do not want it,\" said Laura, \"but I will lend you some money.\"\n\n\"No, I cannot borrow more,\" said her sister despondingly. \"I owe you\nalready for the flowers, the brooch, the bill you paid yesterday, and I\nknow not what else besides; but I will tell Eug\u00e8ne there is a poor\nEnglishwoman in distress, I am sure he will send her something.\"\n\nEug\u00e8ne gave her a five-franc piece.\n\nIt was late one frosty evening when Sally ran across to her cousin's\ncottage, delighted to be the bearer of the long hoped-for letter. Fanny\nwas sitting on the fender before a small fire, hugging her darling to\nher breast, and breathing on its little face to make the air warmer.\n\"I'm afraid,\" said she, in answer to Sally's inquiries, \"that the child\nwon't be here long;\" and she wiped away a few hot tears that had forced\ntheir way as she sat listening to the low moans of the little sufferer.\n\n\"But I have good news for you,\" said her cousin, cheerfully. \"Here is a\nletter from Miss Isabel at last. I would not tell you before, but I\nwrote to Miss Laura, saying how you were expecting every week to be put\nto bed again, and how Bella was wasting away, and see, I was right about\nher, she has sent you a sovereign, and her sister's letter, no-doubt,\ncontains a pretty sum.\"\n\nFanny started up, and could scarcely breathe as she broke the seal. What\nwas her disappointment on seeing an order for five shillings!\n\n\"I am very sorry, my good Fanny,\" said Isabel, \"that just now I have no\nmoney. A charitable gentleman sends you five shillings, and as soon as I\npossibly can, I will let you have a large sum. I have not yet paid for\nthe carriage I sent you, and as the bill has been given me several\ntimes, I must discharge it before I send away more money. I hope that by\nthis time, little Bella is better.\"\n\nFanny laid her child upon the bed, and putting her face by its side,\nshed bitter tears. Sally did not speak, and so both remained till Thomas\ncame in from his work. Fanny would have hidden the letter from him, but\nhe saw and seized it in a moment.\n\n\"Five guineas for a carriage, and five shillings for a child's life,\"\nsaid he with a sneer, as he laid it down. \"Do not look for the large\nsum, Fanny, you won't get it; but I will work hard, and bury the child\ndecently.\"\n\nFanny felt no inclination to defend her mistress. For the first time, it\noccurred to her that Thomas and John might be right in their judgment of\nher. She raised Bella, as Thomas, who had been twisting up the money\norder, was about to throw it in the fire. He caught a sight of the\nchild's wan face, and, advancing to the bed, said, in a softened tone,\n\"Do you know father, pretty one?\" and as Bella smiled faintly, he added,\n\"I will do anything for your sake. Here, Fanny, take the money, and get\nthe child something nourishing.\"\n\nBella seemed to revive from getting better food; and the apothecary held\nout great hope of her ultimate recovery, if the improved diet could be\ncontinued; but expenses fell heavily on Thomas, Fanny was put to bed\nwith a fine strong little boy, and, although Sally and Mrs. Maythorn\ndevoted themselves to her and Bella, the anxiety she suffered from\nbeing separated from her invalid child, added to her former constant\nuneasiness, and want of proper food, brought on a fever that threatened\nher life. In a few days she became quite delirious. During this time\nIsabel was married, and Laura returned to England.\n\nWhen Fanny regained her consciousness she was in the dark, but she could\nsee some one standing by the window. On her speaking the person advanced\nto her side. \"Do not be startled to find me here,\" said a sweet soft\nvoice. \"Sally has watched by your side for three nights, and when I came\nthis evening she looked so ill that I insisted on her going to bed;\nthen, as we could find no one on whose care and watchfulness we could\ndepend, I took her place. You have been in a sound sleep. Dr. Hart said\nyou would wake up much better. Are you better?\"\n\n\"Yes, ma'am, a deal better; but where am I, and who is it with me?\"\n\n\"You are in your own pretty cottage, and Miss Laura is with you. You\nexpected me home, did you not?\"\n\n\"Oh, thank God; who sent you, dear Miss Laura? How is--but may-be I had\nbest not ask just while I am so weak. Is the dear boy well?\"\n\n\"Yes, quite well; and Bella is much better. I have sent her for a few\ndays to L----, with Mrs. Maythorn; the sea air will do her good.\"\n\n\"Oh, thank you--thank you--dear young lady, for the thought. I seem so\nbound up in that dear child, that nothing could comfort me for her loss.\nHow good and kind you are, Miss--you do all so well and so quietly!\"\n\n\"Yes, Fanny, dear,\" said Thomas, coming from behind the curtain and\nstooping to kiss his wife. \"Miss Laura has saved you and Bella, and me,\ntoo, for I couldn't have lived if you had died; and has found me work;\nand all without making one great present, or doing anything one could\nspeak about. I'll tell you what it is, wife, dear, Miss Isabel does all\nfor the best, but it is just as she feels at the moment. Now Miss\nLaura--if I may be so bold to speak, Miss--Miss Laura does not give to\nplease her own feelings, but to do good. I can't say it well, but do you\nsay it for me, Miss; I want Fanny to know the right words, to teach the\nlittle ones by-and-by. You know what I wish to say, Miss Laura.\"\n\n\"Yes, Thomas,\" said Laura, blushing, \"but I do not say you are right.\nYou mean, I think, that my sister acts from impulse, and I from\nprinciple. Is that it?\"\n\n\"I suppose that's it, Miss,\" said Thomas, considering, and apparently\nnot quite satisfied.\n\n\"You have no harder meaning, I am sure,\" said Laura, quietly, \"because I\nlove my sister very much.\"\n\n\"Certainly not, Miss,\" returned Thomas. \"But, myself, if I may take the\nliberty of gratefully saying so, I prefer to be acted to on principle,\nand think it a good deal better than impulse.\"\n\n\n\n\nV.\n\nBed.\n\n    \"Oh, Sleep! it is a gentle thing,\n      Beloved from pole to pole!\"\n\n\nWas the heart's cry of the Ancient Mariner at the recollection of the\nblessed moment when the fearful curse of life in death fell off him, and\nthe heavenly sleep first \"slid into his soul.\" \"Blessings on sleep!\"\nsaid honest Sancho Panza: \"it wraps one all around like a mantle!\"--a\nmantle for the weary human frame, lined softly, as with the down of the\neider-duck, and redolent of the soothing odors of the poppy. The fabled\ncave of Sleep was in the land of Darkness. No ray of the sun, or moon,\nor stars, ever broke upon that night without a dawn. The breath of\nsomniferous flowers floated in on the still air from the grotto's\nmouth. Black curtains hung round the ever-sleeping god; the Dreams stood\naround his couch; Silence kept watch at the portals. Take the winged\nDreams from the picture, and what is left? The sleep of matter.\n\nThe dreams that come floating through our sleep, and fill the dormitory\nwith visions of love or terror--what are they? Random freaks of the\nfancy? Or is sleep but one long dream, of which we see only fragments,\nand remember still less? Who shall explain the mystery of that loosening\nof the soul and body, of which night after night whispers to us, but\nwhich day after day is unthought of? Reverie, sleep, trance--such are\nthe stages between the world of man and the world of spirits. Dreaming\nbut deepens as we advance. Reverie deepens into the dreams of\nsleep--sleep into trance--trance borders on death. As the soul retires\nfrom the outer senses, as it escapes from the trammels of the flesh, it\nlives with increased power within. Spirit grows more spirit-like as\nmatter slumbers. We can follow the development up to the last stage.\nWhat is beyond?\n\n    \"And in that _sleep of death_, what dreams may come!\"\n\nsays Hamlet--pausing on the brink of eternity, and vainly striving to\nscan the inscrutable. Trance is an awful counterpart of sleep and\ndeath--mysterious in itself, appalling in its hazards. Day after day\nnoise has been hushed in the dormitory--month after month it has seen a\nhuman frame grow weaker and weaker, wanner, more deathlike, till the\nhues of the grave  the face of the living. And now he lies,\nmotionless, pulseless, breathless. It is not sleep--is it death?\n\nLeigh Hunt is said to have perpetrated a very bad pun connected with the\ndormitory, and which made Charles Lamb laugh immoderately. Going home\ntogether late one night, the latter repeated the well-known proverb, \"A\nhome's a home, however homely.\" \"Ay,\" added Hunt, \"and a bed's a bed,\nhowever _bedly_.\" It is a strange thing, a bed. Somebody has called it a\nbundle of paradoxes; we go to it reluctantly, and leave it with regret.\nOnce within the downy precincts of the four posts, how loth we are to\nmake our exodus into the wilderness of life. We are as enamored of our\ncurtained dwelling as if it were the land of Goshen or the cave of\nCirce. And how many fervent vows have those dumb posts heard broken!\nevery fresh perjury rising to join its cloud of hovering fellows, each\nmorning weighing heavier and heavier on our sluggard eyelids. A caustic\nproverb says,--we are all \"good risers at night;\" but woe's me for our\nagility in the morning. It is a failing of our species, ever ready to\nbreak out in all of us, and in some only vanquished after a struggle\npainful as the sundering of bone and marrow. The Great Frederic of\nPrussia found it easier, in after life, to rout the French and\nAustrians, than in youth to resist the seductions of sleep. After many\nsingle-handed attempts at reformation, he had at last to call to his\nassistance an old domestic, whom he charged, on pain of dismissal, to\npull him out of bed every morning at two o'clock. The plan succeeded,\nas it deserved to succeed. All men of action are impressed with the\nimportance of early rising. \"When you begin to turn in bed, its time to\nturn out,\" says the old Duke; and we believe his practice has been in\naccordance with his precept. Literary men--among whom, as Bulwer says, a\ncertain indolence seems almost constitutional--are not so clear upon\nthis point: they are divided between Night and Morning, though the best\nauthorities seem in favor of the latter. Early rising is the best\n_elixir vit\u00e6_: it is the only lengthener of life that man has ever\ndevised. By its aid the great Buffon was able to spend half a\ncentury--an ordinary lifetime--at his desk; and yet had time to be the\nmost modish of all the philosophers who then graced the gay metropolis\nof France.\n\nSleep is a treasure and a pleasure; and, as you love it, guard it\nwarily. Over-indulgence is ever suicidal, and destroys the pleasure it\nmeans to gratify. The natural times for our lying down and rising up are\nplain enough. Nature teaches us, and unsophisticated mankind followed\nher. Singing birds and opening flowers hail the sunrise, and the hush of\ngroves and the closed eyelids of the parterre mark his setting. But \"man\nhath sought out many inventions.\" We prolong our days into the depths of\nnight, and our nights into the splendor of day. It is a strange result\nof civilization! It is not merely occasioned by that thirst for varied\namusement which characterizes an advanced stage of society--it is not\nthat theatres, balls, dancing, masquerades, require an artificial light,\nfor all these are or have been equally enjoyed elsewhere beneath the eye\nof day. What is the cause, we really are not philosopher enough to say;\nbut the prevalence of the habit must have given no little pungency to\nhonest Benjamin Franklin's joke, when, one summer, he announced to the\nParisians as a great discovery--that the sun rose each morning at four\no'clock; and that, whereas, they burnt no end of candles by sitting up\nat night, they might rise in the morning and have light for nothing.\nFranklin's \"discovery,\" we dare say, produced a laugh at the time, and\nthings went on as before. Indeed, so universal is this artificial\ndivision of day and night, and so interwoven with it are the social\nhabits, that we shudder at the very idea of returning to the natural\norder of things. A Robespierre could not carry through so stupendous a\nrevolution. Nothing less than an avatar of Siva the destroyer--Siva\nwith his hundred arms, turning off as many gas-pipes, and replenishing\nhis necklace of human skulls by decapitating the leading\nconservatives--could have any chance of success; and, ten to one, with\nour gassy splendors, and seducing glitter, we should convert that pagan\ndevil ere half his work was done.\n\nBut of all the inventions which perverse ingenuity has sought out, the\nmost incongruous, the most heretical against both nature and art, is\nreading in bed. Turning rest into labor, learning into ridicule. A man\nhad better be up. He is spoiling two most excellent things by attempting\nto join them. Study and sleep--how incongruous! It is an idle coupling\nof opposites, and shocks a sensible man as much as if he were to meet\nin the woods the apparition of a winged elephant. Only fancy an elderly\nor middle-aged man (for youth is generally orthodox on this point),\nsitting up in bed, spectacles on his nose, a Kilmarnock on his head, and\nhis flannel jacket round his shivering shoulders,--doing what? Reading?\nIt may be so--but he winks so often, possibly from the glare of the\ncandle, and the glasses now and then slip so far down on his nose, and\nhis hand now and then holds the volume so unsteadily, that if he himself\ndidn't assure us to the contrary, we should suppose him half asleep. We\nare sure it must be a great relief to him when the neglected book at\nlast tumbles out of bed to such a distance that he cannot recover it.\n\nNevertheless, we have heard this extraordinary custom excused on the no\nless extraordinary ground of its being a soporific. For those who\nrequire such things, Marryat gives a much simpler recipe--namely, to\nmentally repeat any scraps of poetry you can recollect; if your own, so\nmuch the better. The monks of old, in a similar emergency, used to\nrepeat the seven Penitential Psalms. Either of these plans, we doubt\nnot, will be found equally efficacious, if one is able to use them--if\nanxiety of mind does not divert him from his task, or the lassitude of\nillness disable him for attempting it. Sleep, alas! is at times fickle\nand coy; and, like most sublunary friends, forsakes us when most wanted.\nReading in that repertory of many curious things, the \"Book of the\nFarm,\" we one day met with the statement that \"a pillow of hops will\nensure sleep to a patient in a delirious fever when every other\nexpedient fails.\" We made a note of it. Heaven forbid that the recipe\nshould ever be needed for us or ours! but the words struck a chord of\nsympathy in our heart with such poor sufferers, and we saddened with the\ndread of that awful visitation. The fever of delirium! when incoherent\nwords wander on the lips of genius; when the sufferer stares strangely\nand vacantly on his ministering friends, or starts with freezing horror\nfrom the arms of familiar love! Ah! what a dread tenant has the\ndormitory then. No food taken for the body, no sleep for the brain! a\nhuman being surging with diabolic strength against his keepers--a human\nframe gifted with superhuman vigor only the more rapidly to destroy\nitself! Less fearful to the eye, but more harrowing to the soul, is the\ndormitory whose walls enclose the sleepless victim of Remorse. No\npoppies or mandragora for him! His malady ends only with the fever of\nlife. _Ends?_ Grief, anxiety, \"the thousand several ills that flesh is\nheir to,\" pass away before the lapse of time or the soothings of love,\nand sleep once more folds its dove-like wings above the couch.\n\n\"If there be a regal solitude,\" says Charles Lamb, \"it is a bed. How the\npatient lords it there; what caprices he acts without control! How\nking-like he sways his pillow,--tumbling and tossing, and shifting, and\nlowering, and thumping, and flatting, and moulding it to the\never-varying requisitions of his throbbing temples. He changes _sides_\noftener than a politician. Now he lies full-length, then half-length,\nobliquely, transversely, head and feet quite across the bed; and none\naccuses him of tergiversation. Within the four curtains he is absolute.\nThey are his Mare Clausum. How sickness enlarges the dimensions of a\nman's self to himself! He is his own exclusive object. Supreme\nselfishness is inculcated on him as his only duty. 'Tis the two Tables\nof the Law to him. He has nothing to think of but how to get well. What\npasses out of doors or within them, so he hear not the jarring of them,\naffects him not.\"\n\nIn this climate a sight of the sun is prized; but we love to see it most\nfrom bed. A dormitory fronting the east, therefore, so that the early\nsunbeams may rouse us to the dewy beauties of morning, we love. Let\nthere also be festooned roses without the window, that on opening it the\nperfume may pervade the realms of bed. Our night-bower should be\nsimple--neat as a fairy's cell, and ever perfumed with the sweet air of\nheaven. It is not a place for showy things, or costly. As fire is the\npresiding genius in other rooms, so let water, symbol of purity, be in\nthe ascendant here; water, fresh and unturbid as the thoughts that here\nmake their home--water, to wash away the dust and sweat of a weary\nworld. Let no _fracas_ disturb the quiet of the dormitory. We go there\nfor repose. Our tasks and our cares are left outside, only to be put on\nagain with our hat and shoes in the morning. It is an asylum from the\nbustle of life--it is the inner shrine of our household gods--and should\nbe respected accordingly. We never entered during the ordinary process\nof bed-making--pillows tossed here, blankets and sheets pitched hither\nand thither in wildest confusion, chairs and pitchers in the middle of\nthe floor, feathers and dust everywhere--without a jarring sense that\nsacrilege was going on, and that the _genius loci_ had departed. Rude\nhands were profaning the home of our slumbers!\n\nA sense of security pervades the dormitory. A healthy man in bed is free\nfrom everything but dreams, and once in a life-time, or after adjudging\nthe Cheese Premium at an Agricultural Show--the nightmare. We once heard\na worthy gentleman, blessed with a very large family of daughters,\ndeclare he had no peace in his house except in bed. There we feel as if\nin a city of refuge, secure alike from the brawls of earth and the\nstorms of heaven. Lightning, say old ladies, won't come through,\nblankets. Even tigers, says Humboldt, \"will not attack a man in his\nhammock.\" Hitting a man when he's down is stigmatized as villanous all\nthe world over; and lions will rather sit with an empty stomach for\nhours than touch a man before he awakes. Tricks upon a sleeper! Oh,\nvillanous! Every perpetrator of such unutterable treachery should be put\nbeyond the pale of society. The First of April should have no place in\nthe calendar of the dormitory. We would have the maxim, \"Let sleeping\ndogs lie,\" extended to the human race. And an angry dog, certainly, is a\nman roused needlessly from his slumbers. What an outcry we Northmen\nraised against the introduction of Greenwich time, which defrauded us of\nfifteen minutes' sleep in the morning; and how indiscriminate the\nobjurgations lavished upon printers' devils! Of all sinners against the\nnocturnal comfort of literary men, these imps are the foremost; and\npossibly it was from their malpractices in such matters that they first\nacquired their diabolic cognomen.\n\nThe nightcap is not an elegant head-dress, but its comfort is\nundeniable. It is a diadem of night; and what tranquillity follows our\nself-coronation! It is priceless as the invisible cap of Fortunatus;\nand, viewless beneath its folds, our cares cannot find us out. It is\ngraceless. Well; what then? It is not meant for the garish eye of day,\nnor for the quizzing glass of our fellow-men, or of the ridiculing race\nof women; neither does it outrage any taste for the beautiful in the\nhappy sleeper himself. We speak as bachelors, to whom the pleasures of a\nmanifold existence are unknown. Possibly the \u00e6sthetics of night are not\nuncared for when a man has another self to please, and when a pair of\nlovely eyes are fixed admiringly on his upper story; but such is the\nselfishness of human nature, that we suspect this abnegation of comfort\nwill not long survive the honeymoon. The French, ever enamored of\neffect, and who, we verily believe, even _sleep_, \"_pos\u00e9_,\" sometimes\nsubstitute the many- silken handkerchief for the graceless\n\"_bonnet-de-nuit_.\" But all such substitutes are less comfortable and\nmore troublesome; and of all irritating things, the most irritable is a\ncomplex operation in undressing. \u00c6sthetics at night, and for the weary!\nNo, no. The weary man frets at every extra button or superfluous knot,\nhe counts impatiently every second that keeps him from his couch, and\nflies to the arms of sleep as to those of his mistress. Nevertheless,\nFrench novelette writers make a great outcry against nightcaps. We\nremember an instance. A husband--rather good-looking fellow--suspects\nthat his wife is beginning to have too tender thoughts towards a\nglossy-ringletted Lothario who is then staying with them. So, having\naccidentally discovered that Lothario slept in a huge peeked nightcowl,\nand knowing that ridicule would prove the most effectual disenchanter,\nhe fastened a string to his guest's bell, and passed it into his own\nroom.\n\nAt the dead of night, when all were fast asleep, suddenly Lothario's\nbell rang furiously. Upstarted the lady--\"their guest must be\nill;\"--and, accompanied by her husband, elegantly coiffed in a turbaned\nsilk handkerchief, she entered the room whence the alarum had sounded.\nThey find Lothario sitting up in bed--his cowl rising pyramid-fashion, a\nfool's cap all but the bells--bewildered and in ludicrous consternation\nat being surprised thus by the fair Angelica; and, unable to conceal his\nchagrin, he completes his discomfiture by bursting out in wrathful abuse\nof his laughing host for so betraying his weakness for nightcaps.\n\nThe Poetry of the Dormitory! It is an inviting but too delicate a\nsubject for our rough hands. Do not the very words call up a vision? By\nthe light of the stars we see a lovely head resting on a downy pillow;\nthe bloom of the rose is on that young cheek, and the half-parted lips\nmurmur as in a dream: \"Edward!\" Love is lying light at her heart, and\nits fairy wand is showing her visions. May her dreams be happy!\n\"Edward!\" Was it a sigh that followed that gentle invocation? What would\nthe youth give to hear that murmur,--to gaze like yonder stars on his\nslumbering love. Hush! are the morning-stars singing together--a lullaby\nto soothe the dreamer? A low dulcet strain floats in through the window;\nand soon, mingling with the breathings of the lute, the voice of youth.\nThe harmony penetrates through the slumbering senses to the dreamer's\nheart; and ere the golden curls are lifted from the pillow, she is\nconscious of all. The serenade begins anew. What does she hear?\n\n    \"Stars of the summer night!\n         Far in yon azure deeps,\n     Hide, hide your golden light!\n         She sleeps!\n     My lady sleeps!\n         Sleeps!\n\n     Dreams of the summer night!\n         Tell her her lover keeps\n     Watch! while in slumbers light\n         She sleeps\n     My lady sleeps!\n         Sleeps!\"\n\n\n\n\nVI.\n\nThe Home of Woodruffe the Gardener.\n\n\nI.\n\n\"How pleased the boy looks, to be sure!\" observed Woodruffe to his wife,\nas his son Allan caught up little Moss (as Maurice had chosen to call\nhimself before he could speak plain) and made him jump from the top of\nthe drawers upon the chair, and then from the chair to the ground. \"He\nis making all that racket just because he is so pleased he does not know\nwhat to do with himself.\"\n\n\"I suppose he will forgive Fleming now for carrying off Abby,\" said the\nmother. \"I say, Allan, what do you think now of Abby marrying away from\nus?\"\n\n\"Why, I think it's a very good thing. You know she never told me that we\nshould go and live where she lived, and in such a pretty place, too,\nwhere I may have a garden of my own, and see what I can make of it--all\nfresh from the beginning, as father says.\"\n\n\"You are to try your hand at the business, I know,\" replied the mother,\n\"but I never heard your father, nor any one else, say that the place was\na pretty one. I did not think new railway stations had been pretty\nplaces at all.\"\n\n\"It sounds so to him, naturally,\" interposed Woodruffe. \"He hears of a\nsouth aspect, and a <DW72> to the north for shelter, and the town seen\nfar off; and that sounds all very pleasant. And then, there is the\nthought of the journey, and the change, and the fun of getting the\nground all into nice order, and, best of all, the seeing his sister so\nsoon again. Youth is the time for hope and joy, you know, love.\"\n\nAnd Woodruffe began to whistle, and stepped forward to take his turn at\njumping Moss, whom he carried in one flight from the top of the drawers\nto the floor. Mrs. Woodruffe smiled, as she thought that youth was not\nthe only season, with some people, for hope and joy.\n\nHer husband, always disposed to look on the bright side, was\nparticularly happy this evening. The lease of his market-garden ground\nwas just expiring. He had prospered on it; and would have desired\nnothing better than to live by it as long as he lived at all. He desired\nthis so much that he would not believe a word of what people had been\nsaying for two years past, that his ground would be wanted by his\nlandlord on the expiration of the lease, and that it would not be let\nagain. His wife had long foreseen this; but not till the last moment\nwould he do what she thought should have been done long before--offer to\nbuy the ground. At the ordinary price of land, he could accomplish the\npurchase of it; but when he found his landlord unwilling to sell, he bid\nhigher and higher, till his wife was so alarmed at the rashness, that\nshe was glad when a prospect of entire removal opened. Woodruffe was\nsure that he could have paid off all he offered at the end of a few\nyears; but his partner thought it would have been a heavy burden on\ntheir minds, and a sad waste of money; and she was therefore, in her\nheart, obliged to the landlord for persisting in his refusal to sell.\n\nWhen that was settled, Woodruffe became suddenly sure that he could pick\nup an acre or two of land somewhere not far off. But he was mistaken;\nand, if he had not been mistaken, market-gardening was no longer the\nprofitable business it had been, when it enabled him to lay by something\nevery year. By the opening of a railway, the townspeople, a few miles\noff, got themselves better supplied with vegetables from another\nquarter. It was this which put it into the son-in-law's head to propose\nthe removal of the family into Staffordshire, where he held a small\nappointment on a railway. Land might be had at a low rent near the\nlittle country station where his business lay; and the railway brought\nwithin twenty minutes' distance a town where there must be a\nconsiderable demand for garden produce. The place was in a raw state at\npresent; and there were so few houses, that, if there had been a choice\nof time, the Flemings would rather have put off the coming of the family\ntill some of the cottages already planned had been built; but the\nWoodruffes must remove in September, and all parties agreed that they\nshould not mind a little crowding for a few months. Fleming's cottage\nwas to hold them all till some chance of more accommodation should\noffer.\n\n\"I'll tell you what,\" said Woodruffe, after standing for some time, half\nwhistling and thinking, with that expression on his face which his wife\nhad long learned to be afraid of, \"I'll write to-morrow--let's see--I\nmay as well do it to-night;\" and he looked round for paper and ink.\n\"I'll write to Fleming, and get him to buy the land for me at once.\"\n\n\"Before you see it?\" said his wife, looking up from her stocking\nmending.\n\n\"Yes. I know all about it, as much as if I were standing on it this\nmoment; and I am sick of this work--of being turned out just when I had\nmade the most of a place, and got attached to it. I'll make a sure thing\nof it this time, and not have such a pull at my heart-strings again. And\nthe land will be cheaper now than later; and we shall go to work upon it\nwith such heart, if it is our own! Eh?\"\n\n\"Certainly, if we find, after seeing it, that we like it as well as we\nexpect. I would just wait till then.\"\n\n\"As well as we expect! Why, bless my soul! don't we know all about it?\nIt is not any land-agent or interested person, that has described it to\nus; but our own daughter and her husband; and do not they know what we\nwant? The quantity at my own choice; the aspect capital; plenty of water\n(only too much, indeed); the soil anything but poor, and sand and marl\nwithin reach to reduce the stiffness; and manure at command, all along\nthe railway, from half-a-dozen towns; and osier-beds at hand (within my\nown bounds if I like) giving all manner of convenience for fencing, and\nbinding, and covering! Why, what would you have?\"\n\n\"It sounds very pleasant, certainly.\"\n\n\"Then, how can you make objections? I can't think where you look, to\nfind any objections?\"\n\n\"I see none now, and I only want to be sure that we shall find none when\nwe arrive.\"\n\n\"Well! I do call that unreasonable! To expect to find any place on earth\naltogether unobjectionable! I wonder what objection could be so great as\nbeing turned out of one after another, just as we have got them into\norder. Here comes our girl. Well, Becky, I see how you like the news!\nNow, would not you like it better still if we were going to a place of\nour own, where we should not be under any landlord's whims? We should\nhave to work, you know, one and all. But we would get the land properly\nmanured, and have a cottage of our own in time; would not we? Will you\nundertake the pigs, Becky?\"\n\n\"Yes, father; and there are many things I can do in the garden too. I am\nold and strong, now; and I can do much more than I have ever done here.\"\n\n\"Aye; if the land was our own,\" said Woodruffe, with a glance at his\nwife. She said no more, but was presently up stairs putting Moss to bed.\nShe knew, from long experience, how matters would go. After a restless\nnight, Woodruffe spoke no more of buying the land without seeing it; and\nhe twice said, in a meditative, rather than a communicative way, that he\nbelieved it would take as much capital as he had to remove his family,\nand get his new land into fit condition for spring crops.\n\n\nII.\n\n\"You may look out now for the place. Look out for our new garden. We are\njust there now,\" said Woodruffe to the children as the whistle sounded,\nand the train was approaching the station. It had been a glorious autumn\nday from the beginning; and for the last hour, while the beauty of the\nlight on fields and trees and water had been growing more striking, the\nchildren, tired with the novelty of all that they had seen since\nmorning, had been dropping asleep. They roused up suddenly enough at the\nnews that they were reaching their new home; and thrust their heads to\nthe windows, eagerly asking on which side they were to look for their\ngarden. It was on the south, the left-hand side; but it might have been\nanywhere, for what they could see of it. Below the embankment was\nsomething like a sheet of gray water, spreading far away.\n\n\"It is going to be a foggy night,\" observed Woodruffe. The children\nlooked into the air for the fog, which had always, in their experience,\narrived by that way from the sea. The sky was all a clear blue, except\nwhere a pale green and a faint blush of pink streaked the west. A large\nplanet beamed clear and bright; and the air was so transparent that the\nvery leaves on the trees might almost be counted. Yet could nothing be\nseen below for the gray mist which was rising, from moment to moment.\n\nFleming met them as they alighted; but he could not stay till he had\nseen to the other passengers. His wife was there. She had been a\nmerry-hearted girl; and now, still so young, as to look as girlish as\never, she seemed even merrier than ever. She did not look strong, but\nshe had hardly thrown off what she called \"a little touch of the ague;\"\nand she declared herself perfectly well when the wind was anywhere but\nin the wrong quarter. Allan wondered how the wind could go wrong. He had\nnever heard of such a thing before. He had known the wind too high, when\nit did mischief among his father's fruit trees; but it had never\noccurred to him that it was not free to come and go whence and whither\nit would, without blame or objection.\n\n\"Come--come home,\" exclaimed Mrs. Fleming, \"Never mind about your bags\nand boxes! My husband will take care of them. Let me show you the way\nhome.\"\n\nShe let go the hands of the young brothers, and loaded them, and then\nherself, with parcels, that they might not think they were going to lose\neverything, as she said; and then tripped on before to show the way. The\nway was down steps, from the highest of which two or three chimney-tops\nmight be seen piercing the mist which hid everything else. Down, down,\ndown went the party, by so many steps that little Moss began to totter\nunder his bundle.\n\n\"How low this place lies!\" observed the mother.\n\n\"Why, yes;\" replied Mrs. Fleming. \"And yet I don't know. I believe it is\nrather that the railway runs high.\"\n\n\"Yes, yes; that is it,\" said Woodruffe. \"What an embankment this is! If\nthis is to shelter my garden to the north--\"\n\n\"Yes, yes, it is. I knew you would like it,\" exclaimed Mrs. Fleming. \"I\nsaid you would be delighted. I only wish you could see your ground at\nonce; but it seems rather foggy, and I suppose we must wait till the\nmorning. Here we are at home.\"\n\nThe travellers were rather surprised to see how very small a house this\n\"home\" was. Though called a cottage, it had not the look of one. It was\nof a red brick, dingy, though evidently new; and, to all appearance, it\nconsisted of merely a room below, and one above. On walking round it,\nhowever, a sloping roof in two directions gave a hint of further\naccommodation.\n\nWhen the whole party had entered, and Mrs. Fleming had kissed them all\nround, her glance at her mother asked, as plainly as any words, \"Is not\nthis a pleasant room?\"\n\n\"A pretty room, indeed, my dear,\" was the mother's reply, \"and as nicely\nfurnished as one could wish.\"\n\nShe did not say anything of the rust which her quick eye perceived on\nthe fire-irons and the door-key, or of the damp which stained the walls\njust above the skirting-board. There was nothing amiss with the ceiling,\nor the higher parts of the wall,--so it might be an accident.\n\n\"But, my dear,\" asked the mother, seeing how sleepy Moss looked, \"Where\nare you going to put us all? If we crowd you out of all comfort, I shall\nbe sorry we came so soon.\"\n\nAs Mrs. Fleming led the way upstairs, she reminded her family of their\nagreement not to mind a little crowding for a time. If her mother\nthought there was not room for all the newly arrived in this chamber,\nthey could fit out a corner for Allan in the place where she and her\nhusband were to sleep.\n\n\"All of us in this room?\" exclaimed Becky.\n\n\"Yes, Becky; why not? Here, you see, is a curtain between your bed and\nthe large one; and your bed is large enough to let little Moss sleep\nwith you. And here is a morsel of a bed for Allan in the other corner;\nand I have another curtain ready to shut it in.\"\n\n\"But,\" said Becky, who was going on to object. Her mother stopped her by\na sign.\n\n\"Or,\" continued Mrs. Fleming, \"if you like to let Allan and his bed and\ncurtain come down to our place, you will have plenty of room here; much\nmore than my neighbors have, for the most part. How it will be when the\nnew cottages are built, I don't know. We think them too small for new\nhouses; but, meantime, there are the Brookes sleeping seven in a room no\nbigger than this, and the Vines six in one much smaller.\"\n\n\"How do they manage, now?\" asked the mother. \"In case of illness, say;\nand how do they wash and dress?\"\n\n\"Ah! that is the worst part of it. I don't think the boys wash\nthemselves--what we should call washing--for weeks together; or at least\nonly on Saturday nights. So they slip their clothes on in two minutes;\nand then their mother and sisters can get up. But there is the pump\nbelow for Allan, and he can wash as much as he pleases.\"\n\nIt was not till the next day that Mrs. Woodruffe knew--and then it was\nAllan who told her--that the pump was actually in the very place where\nthe Flemings slept,--close by their bed. The Flemings were, in truth,\nsleeping in an outhouse, where the floor was of brick, the swill-tub\nstood in one corner, the coals were heaped in another, and the light\ncame in from a square hole high up, which had never till now been\nglazed. Plenty of air rushed in under the door, and yet some more\nbetween the tiles,--there being no plaster beneath them. As soon as Mrs.\nWoodruffe had been informed of this, and had stepped in, while her\ndaughter's back was turned, to make her own observations, she went out\nby herself for a walk,--so long a walk, that it was several hours before\nshe reappeared, heated and somewhat depressed. She had roamed the\ncountry round, in search of lodgings; and finding none,--finding no\noccupier who really could possibly spare a room on any terms,--she had\nreturned convinced that, serious as the expense would be, she and her\nfamily ought to settle themselves in the nearest town,--her husband\ngoing to his business daily by the third-class train, till a dwelling\ncould be provided for them on the spot.\n\nWhen she returned, the children were on the watch for her; and little\nMoss had strong hopes that she would not know him. He had a great cap\nof rushes on his head, with a heavy bulrush for a feather; he was stuck\nall over with water-flags and bulrushes, and carried a long osier wand,\nwherewith to flog all those who did not admire him enough in his new\nstyle of dress. The children were clamorous for their mother to come\ndown, and see the nice places where they got these new playthings; and\nshe would have gone, but that their father came up, and decreed it\notherwise. She was heated and tired, he said; and he would not have her\ngo till she was easy and comfortable enough to see things in the best\nlight.\n\nHer impression was that her husband was, more or less (and she did not\nknow why), disappointed; but he did not say so. He would not hear of\ngoing off to the town, being sure that some place would turn up\nsoon,--some place where they might put their heads at night; and the\nFlemings should be no losers by having their company by day. Their\nboarding all together, if the sleeping could but be managed, would be a\nhelp to the young people,--a help which it was pleasant to him, as a\nfather, to be able to give them. He said nothing about the land that was\nnot in praise of it. Its quality was excellent; or would be when it had\ngood treatment. It would take some time and trouble to get it in\norder,--so much that it would never do to live at a distance from it.\nBesides, no trains that would suit him ran at the proper hours; so there\nwas an end of it. They must all rough it a little for a time, and expect\ntheir reward afterwards.\n\nThere was nothing that Woodruffe was so hard to please in as the time\nwhen he should take his wife to see the ground. It was close at hand;\nyet he hindered her going in the morning, and again after their early\ndinner. He was anxious that she should not be prejudiced, or take a\ndislike at first; and in the morning, the fog was so thick that\neverything looked dank and dreary; and in the middle of the day, when a\nwarm autumn sun had dissolved the mists, there certainly was a most\ndisagreeable smell hanging about. It was not gone at sunset; but by\nthat time Mrs. Woodruffe was impatient, and she appeared--Allan showing\nher the way--just when her husband was scraping his feet upon his spade,\nafter a hard day of digging.\n\n\"There, now!\" said he, good-humoredly, striking his spade into the\nground, \"Fleming said you would be down before we were ready for you;\nand here you are!--Yes, ready for you. There are some planks coming, to\nkeep your feet out of the wet among all this clay.\"\n\n\"And yours, too, I hope,\" said the wife. \"I don't mind such wet, after\nrain, as you have been accustomed to; but to stand in a puddle like this\nis a very different thing.\"\n\n\"Yes--so 'tis. But we'll have the planks; and they will serve for\nrunning the wheelbarrow, too. It is too much for Allan, or any boy, to\nrun the barrow in such a soil as this. We'll have the planks first; and\nthen we'll drain, and drain, and get rare spring crops.\"\n\n\"What have they given you this artificial pond for,\" asked the wife, \"if\nyou must drain so much?\"\n\n\"That is no pond. All the way along here, on both sides the railway,\nthere is the mischief of these pits. They dig out the clay for bricks,\nand then leave the places--pits like this, some of them six feet deep.\nThe railways have done a deal of good for the poor man, and will do a\ngreat deal more yet; but, at present, this one has left those pits.\"\n\n\"I hope Moss will not fall into one. They are very dangerous,\" declared\nthe mother, looking about for the child.\n\n\"He is safe enough there, among the osiers,\" said the father. \"He has\nlost his heart outright to the osiers. However, I mean to drain and fill\nup this pit, when I find a good out-fall; and then we will have all high\nand dry, and safe for the children. I don't care so much for the pit as\nfor the ditches there. Don't you notice the bad smell?\"\n\n\"Yes, indeed, that struck me the first night.\"\n\n\"I have been inquiring to-day, and I find there is one acre in twenty\nhereabouts occupied with foul ditches like that. And then the overflow\nfrom them and the pits, spoils many an acre more. There is a stretch of\nwater-flags and bulrushes, and nasty coarse grass and rushes, nothing\nbut a swamp, where the ground is naturally as good as this; and, look\nhere! Fleming was rather out, I tell him, when he wrote that I might\ngraze a pony on the pasture below, whenever I have a market-cart. I ask\nhim if he expects me to water it here.\"\n\nSo saying, Woodruffe led the way to one of the ditches which, instead of\nfences, bounded his land; and, moving the mass of weed with a stick,\nshowed the water beneath, covered with a whitish bubbling scum, the\nsmell of which was insufferable.\n\n\"There is plenty of manure there,\" said Woodruffe; \"that is the only\nthing that can be said for it. We'll make manure of it, and sweep out\nthe ditch, and deepen it, and narrow it, and not use up so many feet of\ngood ground for a ditch that does nothing but poison us. A fence is\nbetter than a ditch any day. I'll have a fence, and still save ten feet\nof ground, the whole way down.\"\n\n\"There is a great deal to do here,\" observed the wife.\n\n\"And good reward when it is done,\" Woodruffe replied. \"If I can fall in\nwith a stout laborer, he and Allan and I can get our spring crops\nprepared for; and I expect they will prove the goodness of the soil.\nThere is Fleming. Supper is ready, I suppose.\"\n\nThe children were called, but both were so wet and dirty that it took\ntwice as long as usual to make them fit to sit at table; and apologies\nwere made for keeping supper waiting. The grave half-hour before Moss's\nbed-time was occupied with the most solemn piece of instruction he had\never had in his life. His father carried him up to the railway, and made\nhim understand the danger of playing there. He was never to play there.\nHis father would go up with him once a day, and let him see a train\npass; and this was the only time he was ever to mount the steps, except\nby express leave. Moss was put to bed in silence, with his father's\ndeep, grave voice, sounding in his ears.\n\n\"He will not forget it,\" declared his father. \"He will give us no\ntrouble about the railway. The next thing is the pit. Allan, I expect\nyou to see that he does not fall into the pit. In time, we shall teach\nhim to take care of himself; but you must remember, meanwhile, that the\npit is six feet deep--deeper than I am high; and that the edge is the\nsame clay that you slipped on so often this morning.\"\n\n\"Yes, father,\" said Allan, looking as grave as if power of life and\ndeath were in his hands.\n\n\nIII.\n\nOne fine morning in the next spring, there was more stir and\ncheerfulness about the Woodruffes' dwelling than there had been of late.\nThe winter had been somewhat dreary; and now the spring was anxious; for\nWoodruffe's business was not, as yet, doing very well. His hope, when he\nbought his pony and cart, was to dispatch by railway to the town the\nbest of his produce, and sell the commoner part in the country\nneighborhood, sending his cart round within the reach of a few miles. As\nit turned out, he had nothing yet to send to the town, and his agent\nthere was vexed and displeased. No radishes, onions, early salads, or\nrhubarb were ready, and it would be some time yet before they were.\n\n\"I am sure I have done everything I could,\" said Woodruffe to Fleming,\nas they both lent a hand to put the pony into the cart. \"Nobody can say\nthat I have not made drains enough, or that they are not deep enough;\nyet the frost has taken such a hold that one would think we were living\nin the north of Scotland, instead of in Staffordshire.\"\n\n\"It has not been a severe season either,\" observed Fleming.\n\n\"There's the vexation,\" replied Woodruffe. \"If it had been a season\nwhich set us at defiance, and made all sufferers alike, one must just\nsubmit to a loss, and go on again, like one's neighbors. But, you see,\nI am cut out, as my agent says, from the market. Everybody else has\nspring vegetables there, as usual. It is no use telling him that I never\nfailed before. But I know what it is. It is yonder great ditch that does\nthe mischief.\"\n\n\"Why, we have nothing to do with that.\"\n\n\"That is the very reason. If it was mine or yours, do you think I should\nnot have taken it in hand long ago? All my draining goes for little\nwhile that shallow ditch keeps my ground a continual sop. It is all\nuneven along the bottom;--not the same depth for three feet together\nanywhere, and not deep enough by two feet in any part. So there it is,\nchoked up and putrid; and, after an hour or two of rain, my garden gets\nsuch a soaking that the next frost is destruction.\"\n\n\"I will speak about it again,\" said Fleming. \"We must have it set right\nbefore next winter.\"\n\n\"I think we have seen enough of the uselessness of speaking,\" replied\nWoodruffe, gloomily. \"If we tease the gentry any more, they may punish\nyou for it. I would show them my mind by being off,--throwing up my\nbargain at all costs, if I had not put so much into the ground that I\nhave nothing left to move away with.\"\n\n\"Don't be afraid for me,\" said Fleming, cheerfully. \"It was chiefly my\ndoing that you came here, and I must try my utmost to obtain fair\nconditions for you. We must remember that the benefit of your outlay has\nall to come.\"\n\n\"Yes; I can't say we have got much of it yet.\"\n\n\"By next winter,\" continued Fleming, \"your privet hedges and screens\nwill have grown up into some use against the frost; and your own\ndrainage----. Come, come, Allan, my boy! be off! It is getting late.\"\n\nAllan seemed to be idling, re-arranging his bunches of small radishes,\nand little bundles of rhubarb, in their clean baskets, and improving the\nstick with which he was to drive; but he pleaded that he was waiting for\nMoss, and for the parcel which his mother was getting ready for Becky.\n\n\"Ah I my poor little girl!\" said Woodruffe. \"Give my love to her, and\ntell her it will be a happy day when we can send for her to come home\nagain. Be sure you observe particularly, to tell us how she looks; and,\nmind, if she fancies anything in the cart,--any radishes, or whatever\nelse, because it comes out of our garden, be sure you give it her. I\nwish I was going myself with the cart, for the sake of seeing Becky; but\nI must go to work. Here have I been all the while, waiting to see you\noff. Ah! here they come! you may always have notice now of who is coming\nby that child's crying.\"\n\n\"O, father! not always!\" exclaimed Allan.\n\n\"Far too often, I'm sure. I never knew a child grow so fractious. I am\nsaying, my dear,\" to his wife, who now appeared with her parcel, and\nMoss in his best hat, \"that boy is the most fractious child we ever had;\nand he is getting too old for that to begin now. How can you spoil him\nso?\"\n\n\"I am not aware,\" said Mrs. Woodruffe, her eyes filling with tears,\n\"that I treat him differently from the rest; but the child is not well.\nHis chilblains tease him terribly, and I wish there may be nothing\nworse.\"\n\n\"Warm weather will soon cure the chilblains, and then I hope we shall\nsee an end of the fretting.--Now, leave off crying this minute, Moss, or\nyou don't go. You don't see me cry with my rheumatism, and that is worse\nthan chilblains, I can tell you.\"\n\nMoss tried to stifle his sobs, while his mother put more straw into the\ncart for him, and cautioned Allan to be careful of him, for it really\nseemed as if the child was tender all over. Allan seemed to succeed best\nas comforter. He gave Moss the stick to wield, and showed him how to\nmake believe to whip the pony, so that before they turned the corner,\nMoss was wholly engrossed with what he called driving.\n\n\"Yes, yes,\" said Woodruffe, as he turned away to go to the garden,\n\"Allan is the one to manage, him. He can take as good care of him as any\nwoman without spoiling him!\"\n\nMrs. Woodruffe submitted to this in silence; but with the feeling that\nshe did not deserve it.\n\nBecky had had no notice of this visit from her brothers; but no such\nvisit could take her by surprise; for she was thinking of her family all\nday long, every day, and fancying she should see them whichever way she\nturned. It was not her natural destination to be a servant in a\nfarm-house; she had never expected it,--never been prepared for it. She\nwas as willing to work as any girl could be; and her help in the\ngardening was beyond what most women are capable of; but it was a bitter\nthing to her to go among strangers, and toil for them, when she knew\nthat she was wanted at home by father and mother, and brothers, and just\nat present by her sister too; for Mrs. Fleming's confinement was to\nhappen this spring. The reason why Becky was not at home while so much\nwanted there was, that there really was no accommodation for her. The\nplan of sleeping all huddled together as they were at first would not\ndo. The girl herself could not endure it; and her parents felt that she\nmust be got out at any sacrifice. They had inquired diligently till they\nfound a place for her in a farm-house, where the good wife promised\nprotection, and care, and kindness; and fulfilled her promise to the\nbest of her power.\n\n\"I hope they do well by you here, Becky,\" asked Allan, when the surprise\ncaused by his driving up with a dash had subsided, and everybody had\nretired, to leave Becky with her brothers for the few minutes they could\nstay. \"I hope they are kind to you here.\"\n\n\"O, yes,--very kind. And I am sure you ought to say so to father and\nmother.\"\n\nBecky had jumped into the cart, and had her arms round Moss, and her\nhead on his shoulder. Raising her head, and with her eyes filling as she\nspoke, she inquired anxiously how the new cottages went on, and when\nfather and mother were to have a home of their own again. She owned, but\ndid not wish her father and mother to hear of it, that she did not like\nbeing among such rough people as the farm servants. She did not like\nsome of the behavior that she saw; and, still less, such talk as she was\nobliged to overhear. When _would_ a cottage be ready for them?\n\n\"Why, the new cottages would soon be getting on now,\" Allan said; but he\ndidn't know, nobody fancied the look of them. He saw them just after the\nfoundations were laid; and the enclosed parts were like a clay-puddle.\nHe did not see how they were ever to be improved; for the curse of wet\nseemed to be on them, as upon everything about the Station. Fleming's\ncottage was the best he had seen, after all, if only it was twice as\nlarge. If anything could be done to make the new cottages what cottages\nshould be, it would be done: for everybody agreed that the railway\ngentlemen desired to do the best for their people, and to set an example\nin that respect; but it was beyond anybody's power to make wet clay as\nhealthy as warm gravel. Unless they could go to work first to dry the\nsoil, it seemed a hopeless sort of affair.\n\n\"But, I say, Becky,\" pursued Allan, \"you know about my garden--that\nfather gave me a garden of my own.\"\n\nBecky's head was turned quite away; and she did not look round, when she\nreplied,\n\n\"Yes; I remember. How does your garden get on?\"\n\nThere was something in her voice which made her brother lean over and\nlook into her face; and, as he expected, tears were running down her\ncheeks.\n\n\"There now!\" said he, whipping the back of the cart with his stick;\n\"something must be done, if you can't get on here.\"\n\n\"O! I can get on. Be sure you don't tell mother that I can't get on, or\nanything about it.\"\n\n\"You look healthy, to be sure.\"\n\n\"To be sure I am. Don't say any more about it. Tell me about your\ngarden.\"\n\n\"Well: I am trying what I can make of it, after I have done working with\nfather. But it takes a long time to bring it round.\"\n\n\"What! is the wet there, too?\"\n\n\"Lord, yes! The wet was beyond everything at first. I could not leave\nthe spade in the ground ten minutes, if father called me, but the water\nwas standing in the hole when I went back again. It is not so bad now,\nsince I made a drain to join upon father's principal one; and father\ngave me some sand, and plenty of manure; but it seems to us that manure\ndoes little good. It won't sink in when the ground is so wet.\"\n\n\"Well, there will be the summer next, and that will dry up your garden.\"\n\n\"Yes. People say the smells are dreadful in hot weather, though. But we\nseem to get used to that. I thought it sickly work, just after we came,\ngoing down to get osiers, and digging near the big ditch that is our\nplague now: but somehow, it does not strike me now as it did then,\nthough Fleming says it is getting worse every warm day. But come--I must\nbe off. What will you help yourself to? And don't forget your parcel.\"\n\nBecky's great anxiety was to know when her brothers would come again. O!\nvery often, she was assured--oftener and oftener as the vegetables came\nforward; whenever there were either too many or too few to send to the\ntown by rail.\n\nAfter Becky had jumped down, the farmer and one of the men were seen to\nbe contemplating the pony.\n\n\"What have you been giving your pony lately?\" asked the farmer of Allan.\n\"I ask as a friend, having some experience of this part of the country.\nHave you been letting him graze?\"\n\n\"Yes, in the bit of meadow that we have leave for. There is a good deal\nof grass there, now. He has been grazing there these three weeks.\"\n\n\"On the meadow where the osier beds are? Ay! I knew it by the look of\nhim. Tell your father that if he does not take care, his pony will have\nthe staggers in no time. An acquaintance of mine grazed some cattle\nthere once; and in a week or two, they were all feverish, so that the\nbutcher refused them on any terms; and I have seen more than one horse\nin the staggers, after grazing in marshes of that sort.\"\n\n\"There is fine thick grass there, and plenty of it,\" said Allan, who did\nnot like that anybody but themselves should criticise their new place\nand plans.\n\n\"Ay, ay; I know,\" replied the farmer. \"But if you try to make hay of\nthat grass, you'll be surprised to find how long it takes to make, and\nhow like wool it comes out at last. It is a coarse grass, with no\nstrength in it; and it must be a stronger beast than this that will bear\nfeeding on it. Just do you tell your father what I say, that's all; and\nthen he can do as he pleases; but I would take a different way with that\npony, without loss of time, if it was mine.\"\n\nAllan did not much like taking this sort of message to his father, who\nwas not altogether so easy to please as he used to be. If anything vexed\nhim ever so little, he always began to complain of his rheumatism--and\nhe now complained of his rheumatism many times in a day. It was managed,\nhowever, by tacking a little piece of amusement and pride upon it. Moss\nwas taught, all the way as they went home, after selling their\nvegetables, how much everything sold for; and he was to deliver the\nmoney to his father, and go through his lesson as gravely as any big\nman. It succeeded very well. Everybody laughed. Woodruffe called the\nchild his little man-of-business; gave him a penny out of the money he\nbrought; and when he found that the child did not like jumping as he\nused to do, carried him up to the railway to listen for the whistle, and\nsee the afternoon train come up, and stop a minute, and go on again.\n\n\nIV.\n\nFleming did what he could to find fair play for his father-in-law. He\nspoke to one and another--to the officers of the railway, and to the\nowners of neighboring plots of ground, about the bad drainage, which was\ninjuring everybody; but he could not learn that anything was likely to\nbe done. The ditch--the great evil of all--had always been there, he\nwas told, and people never used to complain of it. When Fleming pointed\nout that it was at first a comparatively deep ditch, and that it grew\nshallower every year from the accumulations formed by its uneven bottom,\nthere were some who admitted that it might be as well to clean it out;\nyet nobody set about it. And it was truly a more difficult affair now\nthan it would have been at an earlier time. If the ditch was shallower,\nit was much wider. It had once been twelve feet wide, and it was now\neighteen. When any drain had been flowing into it, or after a rainy day,\nthe contents spread through and over the soil on each side, and softened\nit, and then the next time any horse or cow came to drink, the whole\nbank was made a perfect bog; for the poor animals, however thirsty,\ntried twenty places to find water that they could drink before going\naway in despair. Such was the bar in the way of poor Woodruffe's success\nwith his ground. Before the end of summer his patience was nearly worn\nout. During a showery and gleamy May and a pleasant June, he had gone\non as prosperously as he could expect under the circumstances; and he\nconfidently anticipated that a seasonable July and August would set him\nup. But he had had no previous experience of the peculiarities of\nill-drained land; and the hot July and August, from which he hoped so\nmuch, did him terrible mischief. The drought which would have merely\ndried and pulverized a well-drained soil, leaving it free to profit much\nby small waterings, baked the overcharged soil of Woodruffe's garden\ninto hard hot masses of clay, amidst which his produce died off faster\nand faster every day, even though he and all his family wore out their\nstrength with constant watering. He did hope, he said, that he should\nhave been spared drought at least; but it seemed as if he was to have\nevery plague in turn; and the drought seemed, at the time, to be the\nworst of all.\n\nOne day Fleming saw a welcome face in one of the carriages; Mr. Nelson,\na director of the railway, who was looking along the line to see how\nmatters went. Though Mr. Nelson was not exactly the one, of all the\ndirectors, whom Fleming would have chosen to appeal to, he saw that the\nopportunity must not be lost; and he entreated him to alight, and stay\nfor the next train.\n\n\"Eh! what!\" said Mr. Nelson; \"what can you want with us here? A station\nlike this! Why, one has to put on spectacles to see it!\"\n\n\"If you would come down, sir, I should be glad to show you....\"\n\n\"Well; I suppose I must.\"\n\nAs they were standing on the little platform, and the train was growing\nsmaller in the distance, Fleming proceeded to business. He told of the\nserious complaints that were made for a distance of a few miles on\neither hand, of the clay pits left by the railway brickmakers, to fill\nwith stagnant waters.\n\n\"Pho! pho! Is that what you want to say?\" replied Mr. Nelson. \"You need\nnot have stopped me just to tell me that. We hear of those pits all\nalong the line. We are sick of hearing of them.\"\n\n\"That does not mend the matter in this place,\" observed Fleming. \"I\nspeak freely, sir, but I think it my duty to say that something must be\ndone. I heard a few days ago, more than the people hereabouts\nknow,--much more than I shall tell them--of the fever that has settled\non particular points of our line; and I now assure you, sir, that if the\nfever once gets a hold in this place, I believe it may carry us all off\nbefore anything can be done. Sir, there is not one of us within half a\nmile of the station that has a wholesome dwelling.\"\n\n\"Pho! pho! you are a croaker,\" declared Mr. Nelson. \"Never saw such a\ndismal fellow! Why, you will die of fright, if ever you die of\nanything.\"\n\n\"Then, sir, will you have the goodness to walk round with me, and see\nfor yourself what you think of things. It is not only for myself and my\nfamily that I speak. In an evil day I induced my wife's family to settle\nhere, and....\"\n\n\"Ay! that is a nice garden,\" observed Mr. Nelson, as Fleming pointed to\nWoodruffe's land. \"You are a croaker, Fleming. I declare I think the\nplace is much improved since I saw it last. People would not come and\nsettle here if the place was like what you say.\"\n\nInstead of arguing the matter, Fleming led the way down the long flight\nof steps. He was aware that leading the gentleman among bad smells and\nover shoes in a foul bog would have more effect than any argument was\never known to have on his contradictious spirit.\n\n\"You should have seen worse things than these, and then you would not be\nso discontented,\" observed Mr. Nelson, striking his stick upon the\nhard-baked soil, all intersected with cracks. \"I have seen such a soil\nas this in Spain, some days after a battle, when there were scores of\nfingers and toes sticking up out of the cracks. What would you say to\nthat?--eh?\"\n\n\"We may have a chance of seeing that here,\" replied Fleming; \"if the\nplague comes, and comes too fast for the coffin-makers,--a thing which\nhas happened more than once in England, I believe.\"\n\nMr. Nelson stopped to laugh; but he certainly attended more to business\nas he went on; and Fleming, who knew something of his ways, had hopes\nthat if he could only keep his own temper, this visit of the director\nmight not be without good results.\n\nIn passing through Woodruffe's garden, very nice management was\nnecessary. Woodruffe was at work there, charged with ire against railway\ndirectors and landed proprietors, whom, amidst the pangs of his\nrheumatism, he regarded as the poisoners of his land and the bane of his\nfortunes; while, on the other hand, Mr. Nelson, who had certainly never\nbeen a market gardener, criticized and ridiculed everything that met his\neye. What was the use of such a tool-house as that?--big enough for a\nhouse for them all. What was the use of such low fences?--of such high\nscreens?--of making the walks so wide?--sheer waste?--of making the beds\nso long one way, and so narrow another?--of planting or sowing this and\nthat?--things that nobody wanted. Woodruffe had pushed back his hat in\npreparation for a defiant reply, when Fleming caught his eye, and, by a\ngood-tempered smile, conveyed to him that they had an oddity to deal\nwith. Allan, who had begun by listening reverently, was now looking from\none to another in great perplexity.\n\n\"What is that boy here for, staring like a dunce? Why don't you send him\nto school? You neglect a parent's duty if you don't send him to school.\"\n\nWoodruffe answered by a smile of contempt, walked away, and went to work\nat a distance.\n\n\"That boy is very well taught,\" Fleming said, quietly. \"He is a great\nreader, and will soon be fit to keep his father's accounts.\"\n\n\"What does he stare in that manner for, then? I took him for a dunce.\"\n\n\"He is not accustomed to hear his father called in question, either as a\ngardener or a parent.\"\n\n\"Pho! pho! I might as well have waited, though, till he was out of\nhearing. Well, is this all you have to show me? I think you make a great\nfuss about nothing.\"\n\n\"Will you walk this way?\" said Fleming, turning down towards the osier\nbeds, without any compassion for the gentleman's boots or olfactory\nnerves. For a long while Mr. Nelson affected to admire the reed, and\nwater-flags, and marsh-blossoms, declared the decayed vegetation to be\npeat soil, very fine peat, which the ladies would be glad of for their\nheaths in the flower-garden,--and thought there must be good fowling\nhere in winter. Fleming quietly turned over the so-called peat with a\nstick, letting it be seen that it was a mere dung-heap of decayed\nrushes, and wished Mr. Nelson would come in the fowling season, and see\nwhat the place was like.\n\n\"The children are merry enough, however,\" observed the gentleman. \"They\ncan laugh here, much as in other places. I advise you to take a lesson\nfrom them, Fleming. Now, don't you teach them to croak.\"\n\nThe laughter sounded from the direction of the old brick-ground; and\nthither they now turned. Two little boys were on the brink of a pit, so\nintent on watching a rat in the water and on pelting it with stones,\nthat they did not see that anybody was coming to disturb them. In answer\nto Mr. Nelson's question, whether they were vagrants, and why vagrants\nwere permitted there, Fleming answered that the younger one--the\npale-faced one--was his little brother-in-law; the other--\n\n\"Ay, now, you will be telling me next that the pale face is the fault of\nthis place.\"\n\n\"It certainly is,\" said Fleming. \"That child was chubby enough when he\ncame.\"\n\n\"Pho, pho! a puny little wretch as ever I saw--puny from its birth, I\nhave no doubt of it. And who is the other--a gypsy?\"\n\n\"He looks like it,\" replied Fleming. On being questioned, Moss told that\nthe boy lived near, and he had often played with him lately. Yes, he\nlived near, just beyond those trees; not in a house, only a sort of\nhouse the people had made for themselves. Mr. Nelson liked to lecture\nvagrants, even more than other people; so Moss was required to show the\nway, and his dark-skinned playfellow was not allowed to skulk behind.\n\nMoss led his party on, over the tufty hay- grass, skipping from\nbunch to bunch of rushes, round the osier-beds, and at last straight\nthrough a clump of elders, behind whose screen now appeared the house,\nas Moss had called it, which the gypsies had made for themselves. It was\nthe tilt of a wagon, serving as a tent. Nobody was visible but a woman,\ncrouching under the shadow of the tent, to screen from the sun that\nwhich was lying across her lap.\n\n\"What is that that she's nursing? Lord bless me! Can that be a child?\"\nexclaimed Mr. Nelson.\n\n\"A child in the fever,\" replied Fleming.\n\n\"Lord bless me!--to see legs and arms hang down like that!\" exclaimed\nthe gentleman; and he forthwith gave the woman a lecture on her method\nof nursing--scolded her for letting the child get a fever--for not\nputting it to bed--for not getting a doctor to it--for being a gypsy,\nand living under an alder clump. He then proceeded to inquire whether\nshe had anybody else in the tent, where her husband was, whether he\nlived by thieving, how they would all like being transported, whether\nshe did not think her children would all be hanged, and so on. At first,\nthe woman tried a facetious and wheedling tone, then a whimpering one,\nand, finally, a scolding one. The last answered well. Mr. Nelson found\nthat a man, to say nothing of a gentleman, has no chance with a woman\nwith a sore heart in her breast, and a sick child in her lap, when once\nhe has driven her to her weapon of the tongue. He said afterwards, that\nhe had once gone to Billingsgate, on purpose to set two fisherwomen\nquarrelling, that he might see what it was like. The scene had fulfilled\nall his expectations; but he now declared that it could not compare with\nthis exhibition behind the alders. He stood a long while, first trying\nto overpower the woman's voice; and, when that seemed hopeless, poking\nabout among the rushes with his stick, and finally, staring in the\nwoman's face, in a mood between consternation and amusement;--thus he\nstood, waiting till the torrent should intermit; but there was no sign\nof intermission; and when the sick child began to move and rouse itself,\nand look at the strangers, as if braced by the vigor of its mother's\ntongue, the prospect of an end seemed farther off than ever. Mr. Nelson\nshrugged his shoulders, signed to his companions, and walked away\nthrough the alders. The woman was not silent because they were out of\nsight. Her voice waxed shriller as it followed them, and died away only\nin the distance. Moss was grasping Fleming's hand with all his might\nwhen Mr. Nelson spoke to him, and shook his stick at him, asking him how\nhe came to play with such people, and saying that if ever he heard him\nlearning to scold like that woman, he would beat him with that stick; so\nMoss vowed he never would.\n\nWhen the train was in sight by which Mr. Nelson was to depart, he turned\nto Fleming, with the most careless air imaginable, saying, \"Have you\nany medicine in your house?--any bark?\"\n\n\"Not any. But I will send for some.\"\n\n\"Ay, do. Or,--no--I will send you some. See if you can't get these\npeople housed somewhere, so that they may not sleep in the swamp. I\ndon't mean in any of your houses, but in a barn, or some such place. If\nthe physic comes before the doctor, get somebody to dose the child. And\ndon't fancy you are all going to die of the fever. That is the way to\nmake yourselves ill; and it is all nonsense, too, I dare say.\"\n\n\"Do you like that gentleman?\" asked Moss, sapiently, when the train was\nwhirling Mr. Nelson out of sight. \"Because I don't--not at all.\"\n\n\"I believe he is kinder than he means, Moss. He need not be so rough;\nbut I know he does kind things sometimes.\"\n\n\"But, do you like him?\"\n\n\"No, I can't say I do.\"\n\nBefore many hours were over, Fleming was sorry that he had admitted\nthis, even to himself; and for many days after he was occasionally\nheard telling Moss what a good gentleman Mr. Nelson was, for all his\nroughness of manners. With the utmost speed, before it would have been\nthought possible, arrived a surgeon from the next town, with medicines,\nand the news that he was to come every day while there was any fear of\nfever. The gypsies were to have been cared for; but they were gone. The\nmarks of their fire and a few stray feathers which showed that a fowl\nhad been plucked, alone told where they had encamped. A neighbor, who\nloved her poultry yard, was heard to say that the sick child would not\ndie for want of chicken broth, she would be bound; and the nearest\nfarmer asked if they had left any potato-peels and turnip tops for his\npig. He thought that was the least they could do after making their\nfamous gypsy stew (a capital dish, it was said) from his vegetables.\nThey were gone; and if they had not left fever behind, they might be\nforgiven, for the sake of the benefit of taking themselves off. After\nthe search for the gypsies was over, there was still an unusual stir\nabout the place. One and another stranger appeared and examined the low\ngrounds, and sent for one and another of the neighboring proprietors,\nwhether farmer, or builder, or gardener, or laborer; for every one who\nowned or rented a yard of land on the borders of the great ditch, or\nanywhere near the clay-pits or osier-beds.\n\nIt was the opinion of the few residents near the Station that something\nwould be done to improve the place before another year; and everybody\nsaid that it must be Mr. Nelson's doings, and that it was a thousand\npities that he did not come earlier, before the fever had crept thus far\nalong the line.\n\n\nV.\n\nFor some months past, Becky had believed without a doubt, that the day\nof her return home would be the very happiest day of her life. She was\ntoo young to know yet that it is not for us to settle which of our days\nshall be happy ones, nor what events shall yield us joy. The promise had\nnot been kept that she should return when her father and mother removed\ninto the new cottage. She had been told that there really was not, even\nnow, decent room for them all; and that they must at least wait till the\nhot weather was completely over before they crowded the chamber, as they\nhad hitherto done. And then, when autumn came on, and the creeping mists\nfrom the low grounds hung round the place from sunset till after\nbreakfast the next day, the mother delayed sending for her daughter,\nunwilling that she should lose the look of health which she alone now,\nof all the family, exhibited. Fleming and his wife and babe prospered\nbetter than the others. The young man's business lay on the high ground,\nat the top of the embankment. He was there all day while Mr. Woodruffe\nand Allan were below, among the ditches and the late and early fogs.\nMrs. Fleming was young and strong, full of spirit and happiness; and so\nfar fortified against the attacks of disease, as a merry heart\nstrengthens nerve and bone and muscle, and invigorates all the vital\npowers. In regard to her family, her father's hopeful spirit seemed to\nhave passed into her. While he was becoming permanently discouraged, she\nwas always assured that everything would come right next year. The time\nhad arrived for her power of hope to be tested to the utmost. One day\nthis autumn, she admitted that Becky must be sent for. She did not\nforget, however, to charge Allan to be cheerful, and make the best of\nthings, and not frighten Becky by the way.\n\nIt was now the end of October. Some of the days were balmy\nelsewhere--the afternoons ruddy; the leaves crisp beneath the tread; the\nsquirrel busy after the nuts in the wood; the pheasants splendid among\nthe dry ferns in the brake, the sportsman warm and thirsty in his\nexploring among the stubble. In the evenings the dwellers in country\nhouses called one another out upon the grass, to see how bright the\nstars were, and how softly the moonlight slept upon the woods. While it\nwas thus in one place, in another, and not far off, all was dank, dim,\ndreary and unwholesome; with but little sun, and no moon or stars; all\nchill, and no glow; no stray perfumes, the last of the year, but sickly\nscents coming on the steam from below. Thus it was about Fleming's\nhouse, this latter end of October, when he saw but little of his wife,\nbecause she was nursing her mother in the fever, and when he tried to\namuse himself with his young baby at meal-times (awkward nurse as he\nwas) to relieve his wife of the charge for the little time he could be\nat home. When the baby cried, and when he saw his Abby look wearied, he\ndid wish, now and then, that Becky was at home; but he was patient, and\nhelpful, and as cheerful as he could be, till the day which settled the\nmatter. On that morning he felt strangely weak, barely able to mount the\nsteps to the station. During the morning, several people told him he\nlooked ill; and one person did more. The porter sent a message to the\nnext large Station that somebody must be sent immediately to fill\nFleming's place, in case of his being too ill to work. Somebody came;\nand before that, Fleming was in bed--certainly down in the fever. His\nwife was now wanted at home; and Becky must come to her mother.\n\nThough Becky asked questions all the way home, and Allan answered them\nas truthfully as he knew how, she was not prepared for what she\nfound--her father aged and bent, always in pain, more or less, and far\nless furnished with plans and hopes than she had ever known him; Moss,\nfretful and sickly, and her mother unable to turn herself in her bed.\nNobody mentioned death. The surgeon who came daily, and told Becky\nexactly what to do, said nothing of anybody dying of the fever, while\nWoodruffe was continually talking of things that were to be done when\nhis wife got well again. It was sad, and sometimes alarming, to hear the\nstrange things that Mrs. Woodruffe said in the evenings when she was\ndelirious; but if Abby stepped in at such times, she did not think much\nof it, did not look upon it as any sign of danger; and was only thankful\nthat her husband had no delirium. His head was always clear, she said,\nthough he was very weak. Becky never doubted, after this, that her\nmother was the most severely ill of the two; and she was thunderstruck\nwhen she heard one morning the surgeon's answers to her father's\nquestions about Fleming. He certainly considered it a bad case; he would\nnot say that he could not get through; but he must say it was contrary\nto his expectation. When Becky saw her father's face as he turned away\nand went out, she believed his heart was broken.\n\n\"But I thought,\" said she to the surgeon, \"I thought my mother was most\nill of the two.\"\n\n\"I don't know that,\" was the reply, \"but she is very ill. We are doing\nthe best we can. You are, I am sure,\" he said, kindly; \"and we must hope\non, and do our best till a change comes. The wisest of us do not know\nwhat changes may come. But I could not keep your father in ignorance of\nwhat may happen in the other house.\"\n\nNo appearances alarmed Abby. Because there was no delirium, she\napprehended no danger. Even when the fatal twitchings came, the arm\ntwitching as it lay upon the coverlid, she did not know it was a symptom\nof anything. As she nursed her husband perfectly well, and could not\nhave been made more prudent and watchful by any warning, she had no\nwarning. Her cheerfulness was encouraged, for her infant's sake, as well\nas for her husband's and her own. Some thought that her husband knew his\nown case. A word or two,--now a gesture, and now a look,--persuaded the\nsurgeon and Woodruffe that he was aware that he was going. His small\naffairs were always kept settled; he had probably no directions to give;\nand his tenderness for his wife showed itself in his enjoying her\ncheerfulness to the last. When, as soon as it was light, one December\nmorning, Moss was sent to ask if Abby could possibly come for a few\nminutes, because mother was worse, he found his sister alone, looking at\nthe floor, her hands on her lap, though her baby was fidgetting in its\ncradle. Fleming's face was covered, and he lay so still that Moss, who\nhad never seen death, felt sure that all was over. The boy hardly knew\nwhat to do; and his sister seemed not to hear what he said. The thought\nof his mother,--that Abby's going might help or save her,--moved him to\nact. He kissed Abby, and said she must please go to mother; and he took\nthe baby out of the cradle, and wrapped it up, and put it into its\nmother's arms; and fetched Abby's bonnet, and took her cloak down from\nits peg, and opened the door for her, saying, that he would stay and\ntake care of everything. His sister went without a word; and, as soon as\nhe had closed the door behind her, Moss sank down on his knees before\nthe chair where she had been sitting, and hid his face there till some\none came for him,--to see his mother once more before she died.\n\nAs the two coffins were carried out, to be conveyed to the churchyard\ntogether, Mr. Nelson, who had often been backward and forward during the\nlast six weeks, observed to the surgeon that the death of such a man as\nFleming was a dreadful loss.\n\n\"It is that sort of men that the fever cuts off,\" said the surgeon.\n\"The strong man, in the prime of life, at his best period, one may say,\nfor himself and for society, is taken away,--leaving wife and child\nhelpless and forlorn. That is the ravage that the fever makes.\"\n\n\"Well; would not people tell you that it is our duty to submit?\" asked\nMr. Nelson, who could not help showing some emotion by voice and\ncountenance.\n\n\"Submit!\" said the surgeon. \"That depends on what the people mean who\nuse the word. If you or I were ill of the fever, we must resign\nourselves, as cheerfully as we could. But if you ask me whether we\nshould submit to see more of our neighbors cut off by fever as these\nhave been, I can only ask in return, whose doing it is that they are\nliving in a swamp, and whether that is to go on? Who dug the clay pits?\nWho let that ditch run abroad, and make a filthy bog? Are you going to\ncharge that upon Providence and talk of submitting to the consequences?\nIf so, that is not my religion.\"\n\n\"No, no. There is no religion in that,\" replied Mr. Nelson, for once\nagreeing in what was said to him. \"It must be looked to.\"\n\n\"It must,\" said the surgeon, as decidedly as if he had been a railway\ndirector, or king and parliament in one.\n\n\nVI.\n\n\"I wonder whether there is a more forlorn family in England than we are\nnow,\" said Woodruffe, as he sat among his children, a few hours after\nthe funeral.\n\nHis children were glad to hear him speak, however gloomy might be his\ntone. His silence had been so terrible that nothing that he could say\ncould so weigh upon their hearts. His words, however, brought out his\nwidowed daughter's tears again. She was sewing--her infant lying in her\nlap. As her tears fell upon its face, it moved and cried. Becky came and\ntook it up, and spoke cheerfully to it. The cheerfulness seemed to be\nthe worst of all. Poor Abby laid her forehead to the back of her chair,\nand sobbed as if her heart would break.\n\n\"Ay, Abby,\" said her father, \"your heart is breaking, and mine too. You\nand I can go to our rest, like those that have gone before us; but I\nhave to think of what will become of these young things.\"\n\n\"Yes, father,\" said Becky gently, but with a tone of remonstrance, \"you\nmust endeavor to live, and not make up your mind to dying, because life\nhas grown heavy and sad.\"\n\n\"My dear, I am ill--very ill. It is not merely that life is grown\nintolerable to me. I am sure I could not live long in such misery of\nmind; but I am breaking up fast.\"\n\nThe young people looked at each other in dismay. There was something\nworse than the grief conveyed by their father's words in the hopeless\ndaring--the despair--of his tone when he ventured to say that life was\nunendurable.\n\nBecky had the child on one arm; with the other hand she took down her\nfather's plaid from its peg, and put it round his rheumatic shoulders,\nwhispering in his ear a few words about desiring that God's will should\nbe done.\n\n\"My dear,\" he replied, \"it was I who taught you that lesson when you\nwere a child on my knee, and it would be strange if I forgot it when I\nwant so much any comfort that I can get. But I don't believe (and if you\nask the clergyman, he will tell you that he does not believe) that it is\nGod's will that we, or any other people, should be thrust into a swamp\nlike this, scarcely fit for the rats and the frogs to live in. It is\nman's doing, not God's, that the fever makes such havoc as it has made\nwith us. The fever does not lay waste healthy places.\"\n\n\"Then why are we here?\" Allan ventured to say. \"Father, let us go.\"\n\n\"Go! I wonder how or where! I can't go, or let any of you go. I have not\na pound in the world to spend in moving, or in finding new employment.\nAnd if I had, who would employ me? Who would not laugh at a crippled old\nman asking for work and wages?\"\n\n\"Then, father, we must see what we can do here, and you must not forbid\nus to say 'God's will be done!' If we cannot go away, it must be His\nwill that we should stay and have as much hope and courage as we can.\"\n\nWoodruffe threw himself back in his chair. It was too much to expect\nthat he would immediately rally; but he let the young people confer, and\nplan, and cheer each other.\n\nThe first thing to be done, they agreed, was to move hither, whenever\nthe dismal rain would permit it, all Abby's furniture that could not be\ndisposed of to her husband's successors. It would fit up the lower room.\nAnd Allan and Becky settled how the things could stand so as to make it\nat once a bed-room and sitting-room. If, as Abby had said, she meant to\ntry to get some scholars, and keep a little school, room must be left to\nseat the children.\n\n\"Keep a school?\" exclaimed Woodruffe, looking round at Abby.\n\n\"Yes, father,\" said Abby, raising her head. \"That seems to be a thing\nthat I can do; and it will be good for me to have something to do. Becky\nis the stoutest of us all, and....\"\n\n\"I wonder how long that will last,\" groaned the father.\n\n\"I am quite stout now,\" said Becky; \"and I am the one to help Allan with\nthe garden. Allan and I will work under your direction, father, while\nyour rheumatism lasts; and....\"\n\n\"And what am I to do?\" asked Moss, pushing himself in.\n\n\"You shall fetch and carry the tools,\" said Becky; \"that is, when the\nweather is fine, and when your chilblains are not very bad. And you\nshall be bird-boy when the sowing season comes on.\"\n\n\"And we are going to put up a pent-house for you, in one corner, you\nknow, Moss,\" said his brother. \"And we will make it so that there shall\nbe room for a fire in it, where father and you may warm yourselves, and\nalways have dry shoes ready.\"\n\n\"I wonder what our shoe leather will have cost us by the time the spring\ncomes,\" observed Woodruffe. \"There is not a place where we ever have to\ntake the cart or the barrow that is not all mire and ruts; not a path in\nthe whole garden that I call a decent one. Our shoes are all pulled to\npieces; while the frost, or the fog, or something or other, prevents our\ngetting any real work done. The waste is dreadful. Nothing should have\nmade me take a garden where none but summer crops are to be had, if I\ncould have foreseen such a thing. I never saw such a thing\nbefore,--never--as market-gardening without winter and spring crops.\nNever heard of such a thing!\"\n\nBecky glanced towards Allan, to see if he had nothing to propose. If\nthey could neither mend the place nor leave it, it did seem a hard case.\nAllan was looking into the fire, musing. When Moss announced that the\nrain was over, Allan started, and said he must be fetching some of\nAbby's things down, if it was fair. Becky really meant to help him; but\nshe also wanted opportunity for consultation, as to whether it could\nreally be God's will that they should neither be able to mend their\ncondition nor to escape from it. As they mounted the long flight of\nsteps, they saw Mr. Nelson issue from the Station, looking about him to\nascertain if the rain was over, and take his stand on the embankment,\nfollowed by a gentleman who had a roll of paper in his hand. As they\nstood, the one was seen to point with his stick, and the other with his\nroll of paper, this way and that. Allan set off in that direction,\nsaying to his sister, as he went,\n\n\"Don't you come. That gentleman is so rude, he will make you cry. Yes, I\nmust go, and I won't get angry; I won't indeed. He may find as much\nfault as he pleases; I must show him how the water is standing in our\nfurrows.\"\n\n\"Hallo! what do you want here?\" was Mr. Nelson's greeting, when, after a\nminute or two, he saw Allan looking and listening. \"What business have\nyou here, hearkening to what we are saying?\"\n\n\"I wanted to know whether anything is going to be done below there. I\nthought, if you wished it, I could tell you something about it.\"\n\n\"You! what, a dainty little fellow like you?--a fellow that wears his\nSunday clothes on a Tuesday, and a rainy Tuesday too! You must get\nworking clothes and work.\"\n\n\"I shall work to-morrow, Sir. My mother and my brother-in-law were\nburied to-day.\"\n\n\"Lord bless me! You should have told me that. How should I know that\nunless you told me?\" He proceeded in a much gentler tone, however,\nmerely remonstrating with Allan for letting the wet stand in the\nfurrows, in such a way as would spoil any garden. Allan had a good ally,\nall the while, in the stranger, who seemed to understand everything\nbefore it was explained. The gentleman was, in fact, an agricultural\nsurveyor--one who could tell, when looking abroad from a height, what\nwas swamp and what meadow; where there was a clean drain, and where an\nuneven ditch; where the soil was likely to be watered, and where flooded\nby the winter rains; where genially warmed, and where fatally baked by\nthe summer's sun. He had seen, before Allan pointed it out, how the\ngreat ditch cut across between the cultivated grounds and the little\nriver into which those grounds should be drained; but he could not know,\ntill told by Allan, who were the proprietors and occupiers of the\nparcels of land lying on either side the ditch. Mr. Nelson knew little\nor nothing under this head, though he contradicted the lad every minute;\nwas sure such an one did not live here, nor another there; told him he\nwas confusing Mr. Smith and Mr. Brown; did not believe a word of Mr.\nTaylor having bought yonder meadow, or Mrs. Scott now renting that\nfield. All the while, the surveyor went on setting down the names as\nAllan told them; and then observed that they were not so many but that\nthey might combine, if they would, to drain their properties, if they\ncould be relieved of the obstruction of the ditch--if the surveyor of\nhighways would see that the ditch were taken in hand. Mr. Nelson\npronounced that there should be no difficulty about the ditch, if the\nrest could be managed; and then, after a few whispered words between the\ngentlemen, Allan was asked first, whether he was sure that he knew where\nevery person lived whose name was down in the surveyor's book; and next,\nwhether he would act as guide to-morrow. For a moment he thought he\nshould be wanted to move Abby's things; but, remembering the vast\nimportance of the plan which seemed now to be fairly growing under his\neye, he replied that he would go; he should be happy to make it his\nday's work to help, ever so little, towards what he wished above\neverything in the world.\n\n\"What makes you in such a hurry to suppose we want to get a day's work\nout of you for nothing?\" asked Mr. Nelson. He thrust half-a-crown into\nthe lad's waistcoat pocket, saying that he must give it back again, if\nhe led the gentleman wrong. The gentleman had no time to go running\nabout the country on a fool's errand; Allan must mind that. As Allan\ntouched his hat, and ran down the steps, Mr. Nelson observed that boys\nwith good hearts did not fly about in that way, as if they were merry,\non the day of their mother's funeral.\n\n\"Perhaps he is rather thinking of saving his father,\" observed the\nsurveyor.\n\n\"Well; save as many of them as you can. They seem all going to pot as it\nis.\"\n\nWhen Allan burst in, carrying nothing of Abby's, but having a little\ncolor in his cheeks for once, his father sat up in his chair, the baby\nsuddenly stopped crying, and Moss asked where he had been. At first, his\nfather disappointed him by being listless--first refusing to believe\nanything good, and then saying that any good that could happen now was\ntoo late; and Abby could not help crying all the more because this was\nnot thought about a year sooner. It was her poor husband that had made\nthe stir; and now they were going to take his advice the very day that\nhe was laid in his grave. They all tried to comfort her, and said how\nnatural it was that she should feel it so; yet, amidst all their\nsympathy, they could not help being cheered that something was to be\ndone at last.\n\nBy degrees, and not slow degrees, Woodruffe became animated. It was\nsurprising how many things he desired Allan to be sure not to forget to\npoint out to the surveyor, and to urge upon those he was to visit. At\nlast he said he would go himself. It was a very serious business, and he\nought to make an effort to have it done properly. It was a great effort,\nbut he would make it. Not rheumatism, nor anything else, should keep him\nat home. Allan was glad at heart to see such signs of energy in his\nfather, though he might feel some natural disappointment at being left\nat home, and some perplexity as to what, in that case, he ought to do\nabout the half-crown, if Mr. Nelson should be gone home. The morning\nsettled this, however. The surveyor was in his gig. If Allan could hang\non, or keep up with it, it would be very well, as he would be wanted to\nopen the gates, and to lead the way in places too wet for his father,\nwho was not worth such a pair of patent waterproof tall boots as the\nsurveyor had on.\n\nThe circuit was not a very wide one; yet it was dark before they got\nhome. There are always difficulties in arrangements which require\ncombined action. Here there were different levels in the land, and\ndifferent tempers and views among the occupiers. Mr. Brown had heard\nnothing about the matter, and could not be hurried till he saw occasion.\nMr. Taylor liked his field best, wet--would not have it drier on any\naccount, for fear of the summer sun. When assured that drought took no\nhold on well-dried land in comparison with wet land, he shook with\nlaughter, and asked if they expected him to believe that. Mrs. Scott,\nwhose combination with two others was essential to the drainage of three\nportions, would wait another year. They must go on without her; and\nafter another year, she would see what she would do. Another had\ndrained his land in his own way long ago, and did not expect that\nanybody would ask him to put his spade into another man's land, or to\nlet any other man put his spade into his. These were all the\nobstructions. Everybody else was willing, or at least, not obstructive.\nBy clever management, it was thought that the parties concerned could\nmake an island of Mrs. Scott and her field, and win over Mr. Brown by\nthe time he was wanted, and show Mr. Taylor that, as his field could no\nlonger be as wet as it had been, he might as well try the opposite\ncondition--they promising to flood his field as often and as thoroughly\nas he pleased, if he found it the worse for being drained. They could\nnot obtain all they wished, where everybody was not as wise as could be\nwished; but so much was agreed upon as made the experienced surveyor\nthink that the rest would follow; enough, already, to set more laborers\nto work than the place could furnish. Two or three stout men were sent\nfrom a distance; and when they had once cut a clear descent from the\nditch to the river, and had sunk the ditch to seven feet deep, and made\nthe bottom even, and narrowed it to three feet, it was a curious thing\nto see how ready the neighbors became to unite their drains with it. It\nused to be said, that here--however it might be elsewhere--the winter\nwas no time for digging; but that must have meant that no winter-digging\nwould bring a spring crop; and that therefore it was useless. Now, the\nsound of the spade never ceased for the rest of the winter; and the\nlaborers thought it the best winter they had ever known for constant\nwork. Those who employed the labor hoped it would answer--found it\nexpensive--must trust it was all right, and would yield a profit by and\nby. As for the Woodruffes, they were too poor to employ laborers. But\nsome little hope had entered their hearts again, and brought strength,\nnot only to their hearts, but to their very limbs. They worked like\npeople beginning the world. As poor Abby could keep the house and sew,\nwhile attending to her little school, Becky did the lighter parts (and\nsome which were far from light) of the garden work, finding easy tasks\nfor Moss; and Allan worked like a man at the drains. They had been\ncalled good drains before; but now, there was an outfall for deeper\nones; and deeper they must be made. Moreover, a strong rivalry arose\namong the neighbors about their respective portions of the combined\ndrainage; and under the stimulus of ambition, Woodruffe recovered his\nspirits and the use of his limbs wonderfully. He suffered cruelly from\nhis rheumatism; and in the evenings felt as if he could never more lift\na spade; yet, not the less was he at work again in the morning, and so\nsanguine as to the improvement of his ground, that it was necessary to\nremind him, when calculating his gains, that it would take two years, at\nleast, to prove the effects of his present labors.\n\n\nVII.\n\nIt was observed by Woodruffe's family, during one week of spring of the\nnext year, that he was very absent. He was not in low spirits, but\nabsorbed in thought, and much devoted to making calculations with pencil\nand paper. At last, out it came, one morning at breakfast.\n\n\"I wonder how we should all like to have Harry Hardiman to work with us\nagain?\"\n\nEvery one looked up. Harry! where was Harry? Was he here? Was he coming?\n\n\"Why, I will tell you what I have been thinking,\" said their father. \"I\nhave thought long and carefully, and I believe I have made up my mind to\nsend for Harry, to come and work for us as he used to do. We have not\nlabor enough on the ground. Two stout men to the acre is the smallest\nallowance for trying what could be made of the place.\"\n\n\"That is what Taylor and Brown are employing now on the best part of\ntheir land,\" said Allan; \"that is, when they can get the labor. There is\nsuch difference between that and one man to four or five acres, as there\nwas before, that they can't always get the labor.\"\n\n\"Just so; and therefore,\" continued Woodruffe, \"I am thinking of sending\nfor Harry. Our old neighborhood was not prosperous when we left it, and\nI fancy it cannot have improved since; and Harry might be glad to follow\nhis master to a thriving neighborhood; and he is such a careful fellow\nthat I dare say he has money for the journey,--even if he has a wife by\nthis time, as I suppose he has.\"\n\nMoss looked most pleased, where all were pleased, at the idea of seeing\nHarry again. His remembrance of Harry was of a tall young man, who used\nto carry him on his shoulders, and wheel him in the empty water-barrel,\nand sometimes offer to dip him into it when it was full, and show him\nhow to dig in the sand-heap with his little wooden spade.\n\n\"Your rent, to be sure, is much lower than in the old place,\" observed\nAbby.\n\n\"Why, we must not build upon that,\" replied the father; \"rent is rising\nhere, and will rise. My landlord was considerate in lowering mine to \u00a33\nper acre, when he saw how impossible it was to make it answer; and he\nsays he shall not ask more yet on account of the labor I laid out at the\ntime of the drainage. But when I have partly repaid myself, the rent\nwill rise to \u00a35; and, in fact, I have made my calculations in regard to\nHarry's coming at a higher rent than that.\"\n\n\"Higher than that?\"\n\n\"Yes; I should not be surprised if I found myself paying, as\nmarket-gardeners near London do, ten pounds per acre before I die.\"\n\n\"Or rather, to let the ground to me for that, father,\" said Allan, \"when\nit is your own property, and you are tired of work, and disposed to turn\nit over to me. I will pay you ten pounds per acre then, and let you have\nall the cabbages you can eat besides. It is capital land, and that is\nthe truth. Come--shall that be a bargain?\"\n\nWoodruffe smiled, and said he owed a duty to Allan. He did not like to\nsee him so hard worked as to be unable to take due care of his own\ncorner of the garden;--unable to enter fairly into the competition for\nthe prizes at the Horticultural Show in the summer. Becky now, too,\nought to be spared from all but occasional help in the garden. Above\nall, the ground was now in such an improving state that it would be\nwaste not to bestow due labor upon it. Put in the spade where you would,\nthe soil was loose and well-aired as needs be: the manure penetrated it\nthoroughly; the frost and heat pulverized, instead of binding it; and\nthe crops were succeeding each other so fast, that the year would be a\nvery profitable one.\n\n\"Where will Harry live, if he comes?\" asked Abby.\n\n\"We must get another cottage added to the new row. Easily done! Cottages\nso healthy as these new ones pay well. Good rents are offered for\nthem,--to save doctors' bills and loss of time from sickness;--and, when\nonce a system of house-drainage is set a-going, it costs scarcely more\nin adding a cottage to a group, to make it all right, than to run it up\nupon solid clay as used to be the way here. Well, I have a good mind to\nwrite to Harry to-day. What do you think of it,--all of you?\"\n\nFortified by the opinion of all his children, Mr. Woodruffe wrote to\nHarry. Meantime, Allan and Becky went to cut the vegetables that were\nfor sale that day; and Moss delighted himself in running after and\ncatching the pony in the meadow below. The pony was not very easily\ncaught, for it was full of spirit. Instead of the woolly insipid grass\nthat it used to crop, and which seemed to give it only fever and no\nnourishment, it now fed on sweet fresh grass, which had no sour stagnant\nwater soaking its roots. The pony was so full of play this morning that\nMoss could not get hold of it. Though much stronger than a year ago, he\nwas not yet anything like so robust as a boy of his age should be; and\nhe was growing heated, and perhaps a little angry, as the pony galloped\noff towards some distant trees, when a boy started up behind a bush,\ncaught the halter, brought the pony round with a twitch, and led him to\nMoss. Moss fancied he had seen the boy before, and then his white teeth\nreminded Moss of one thing after another.\n\n\"I came for some marsh plants,\" said the boy. \"You and I got plenty once\nsomewhere hereabouts, but I cannot find them now.\"\n\n\"You will not find any now. We have no marsh now.\"\n\nThe stranger said he dared not go back without them; mother wanted them\nbadly. She would not believe him if he said he could not find any. There\nwere plenty about two miles off, along the railway, among the clay-pits,\nhe was told; but none nearer. The boy wanted to know where the clay-pits\nhereabouts were. He could not find one of them.\n\n\"I will show you one of them,\" said Moss; \"the one where you and I used\nto hunt rats.\" And, leading the pony, he showed his old gypsy\nplay-fellow all the improvements, beginning with the great ditch,--now\ninvisible from being covered in. While it was open, he said, it used to\nget choked, and the sides were plastered after rain, and soon became\ngrass-grown, so that it was found worth while to cover it in; and now it\nwould want little looking to for years to come. As for the clay-pit,\nwhere the rats used to pop in and out,--it was now a manure-pit, covered\nin. There was a drain into it from the pony's stable and from the\npig-styes; and it was near enough to the garden to receive the refuse\nand sweepings. A heavy lid, with a ring in the middle, covered the pit,\nso that nobody could fall in in the dark, and no smell could get out.\nMoss begged the boy to come a little further, and he would show him his\nown flower-bed; and when the boy was there, he was shown everything\nelse: what a cart-load of vegetables lay cut for sale; and what an arbor\nhad been made of the pent-house under which Moss used to take shelter,\nwhen he could do nothing better than keep off the birds; and how fine\nthe ducks were,--the five ducks that were so serviceable in eating off\nthe slugs; and what a comfortable nest had been made for them to lay\ntheir eggs in, beside the water-tank in the corner; and what a variety\nof scarecrows the family had invented,--each having one, to try which\nwould frighten the sparrows most. While Moss was telling how difficult\nit was to deal with the sparrows, because they could not be frightened\nfor more than three days by any kind of scarecrow, he heard Allan\ncalling him, in a tone of vexation, at being kept waiting so long. In an\ninstant the stranger boy was off,--leaping the gate, and flying along\nthe meadow till he was hidden behind a hedge.\n\nTwo or three days after this one of the ducks was missing. The last time\nthat the five had been seen together was when Moss was showing them to\nhis visitor. The morning after Moss finally gave up hope, the glass of\nAllan's hotbed was found broken, and in the midst of the bed itself was\na deep foot-track, crushing the cucumber plants, and, with them,\nAllan's hopes of a cucumber prize at the Horticultural Exhibition in the\nsummer. On more examination, more mischief was discovered, some cabbages\nhad been stolen, and another duck was missing. In the midst of the\ngeneral concern, Woodruffe burst out a-laughing. It struck him that the\nchief of the scarecrows had changed his hat; and so he had. The old\nstraw hat which used to flap in the wind so serviceably was gone, and in\nits stead appeared a helmet,--a saucepan full of holes, battered and\nsplit, but still fit to be a helmet to a scarecrow.\n\n\"I could swear to the old hat,\" observed Woodruffe, \"if I should have\nthe luck to see it on anybody's head.\"\n\n\"And so could I,\" said Becky, \"for I mended it,--bound it with black\nbehind, and green before, because I had not green ribbon enough. But\nnobody would wear it before our eyes.\"\n\n\"That is why I suspect there are strangers hovering about. We must\nwatch.\"\n\nNow Moss, for the first time, bethought himself of the boy he had\nbrought in from the meadow; and now, for the first time, he told his\nfamily of that encounter.\n\n\"I never saw such a simpleton,\" his father declared. \"There, go along\nand work! Now, don't cry, but hold up like a man and work.\"\n\nMoss did cry; he could not help it; but he worked too. He would fain\nhave been one of the watchers, moreover; but his father said he was too\nyoung. For two nights he was ordered to bed, when Allan took his dark\nlantern, and went down to the pent-house; the first night accompanied by\nhis father, and the next by Harry Hardiman, who had come on the first\nsummons. By the third evening, Moss was so miserable that his sisters\ninterceded for him, and he was allowed to go down with his old friend\nHarry.\n\nIt was a starlight night, without a moon. The low country lay dim, but\nunobscured by mist. After a single remark on the fineness of the night,\nHarry was silent. Silence was their first business. They stole round the\nfence as if they had been thieves themselves, listened for some time\nbefore they let themselves in at the gate, passed quickly in, and locked\nthe gate (the lock of which had been well oiled), went behind every\nscreen, and along every path, to be sure that no one was there, and\nfinally, perceiving that the remaining ducks were safe, settled\nthemselves in the darkness of the pent-house.\n\nThere they sat, hour after hour, listening. If there had been no sound,\nperhaps they could not have borne the effort; but the sense was relieved\nby the bark of a dog at a distance, and then by the hoot of the owl that\nwas known to have done them good service in mousing, many a time; and\nonce, by the passage of a train on the railway above. When these were\nall over, poor Moss had much ado to keep awake, and at last his head\nsank on Harry's shoulder, and he forgot where he was, and everything\nelse in the world. He was awakened by Harry's moving, and then\nwhispering quite into his ear:--\n\n\"Sit you still. I hear somebody yonder. No--sit you still. I won't go\nfar--not out of call; but I must get between them and the gate.\"\n\nWith his lantern under his coat, Harry stole forth, and Moss stood up,\nall alone in the darkness and stillness. He could hear his heart beat,\nbut nothing else, till footsteps on the path came nearer and nearer.\nThey came quite up; they came in, actually into the arbor; and then the\nducks were certainly fluttering. In an instant more, there was a gleam\nof light upon the white plumage of the ducks, and then light enough to\nshow that this was the gypsy boy, with a dark lantern hung round his\nneck, and, at the same moment, to show the gypsy boy that Moss was\nthere. The two boys stood, face to face, motionless from utter\namazement, and the ducks had scuttled and waddled away before they\nrecovered themselves. Then, Moss flew at him in a glorious passion, at\nonce of rage and fear.\n\n\"Leave him to me, Moss,\" cried Harry, casting light upon the scene from\nhis lantern, while he collared the thief with the other hand. \"Let go,\nI say, Moss. There, now we'll go round and be sure whether there is any\none else in the garden, and then we'll lodge this young rogue where he\nwill be safe.\"\n\nNobody was there, and they went home in the dawn, locked up the thief in\nthe shed, and slept through what remained of the night.\n\nIt was about Mr. Nelson's usual time for coming down the line; and it\nwas observed that he now always stopped at this station till the next\ntrain passed,--probably because it was a pleasure to him to look upon\nthe improvement of the place. It was no surprise therefore to Woodruffe\nto see him standing on the embankment after breakfast; and it was\nnatural that Mr. Nelson should be immediately told that the gypsies were\nhere again, and how one of them was caught thieving.\n\n\"Thieving! So you found some of your property upon him, did you!\"\n\n\"Why, no. I thought myself that it was a pity that Moss did not let him\nalone till he had laid hold of a duck or something.\"\n\n\"Pho! pho! don't tell me you can punish the boy for theft, when you\ncan't prove that he stole anything. Give him a whipping, and let him\ngo.\"\n\n\"With all my heart. It will save me much trouble to finish off the\nmatter so.\"\n\nMr. Nelson seemed to have some curiosity about the business; for he\naccompanied Woodruffe to the shed. The boy seemed to feel no awe of the\ngreat man whom he supposed to be a magistrate, and when asked whether he\nfelt none, he giggled and said \"No;\" he had seen the gentleman more\nafraid of his mother than anybody ever was of him, he fancied. On this,\na thought struck Mr. Nelson. He would now have his advantage of the\ngypsy woman, and might enjoy, at the same time, an opportunity of\nstudying human nature under stress--a thing he liked, when the stress\nwas not too severe. So he passed a decree on the spot that, it being now\nnine o'clock, the boy should remain shut up without food till noon, when\nhe should be severely flogged, and driven from the neighborhood; and\nwith this pleasant prospect before him, the young rogue remained,\nwhistling ostentatiously, while his enemies locked the door upon him.\n\n\"Did you hear him shoot the bolt?\" asked Woodruffe. \"If he holds to\nthat, I don't know how I shall get at him at noon.\"\n\n\"There, now, what fools people are! Why did you not take out the bolt? A\npretty constable you would make! Come--come this way. I am going to find\nthe gypsy-tent again. You are wondering that I am not afraid of the\nwoman, I see; but, you observe, I have a hold over her this time. What\ndo you mean by allowing those children to gather about your door? You\nought not to permit it.\"\n\n\"They are only the scholars. Don't you see them going in? My daughter\nkeeps a little school, you know, since her husband's death.\"\n\n\"Ah, poor thing! poor thing!\" said Mr. Nelson, as Abby appeared on the\nthreshold, calling the children in.\n\nMr. Nelson always contrived to see some one or more of the family when\nhe visited the station; but it so happened, that he had never entered\nthe door of their dwelling. Perhaps he was not himself fully conscious\nof the reason. It was, that he could not bear to see Abby's young face\nwithin the widow's cap, and to be thus reminded that hers was a case of\ncruel wrong; that if the most ordinary thought and care had been used in\npreparing the place for human habitation, her husband might be living\nnow, and she the happy creature that she would never be again.\n\nOn his way to the gypsies, Mr. Nelson saw some things that pleased him\nin his heart, though he found fault with them all. What business had\nWoodruffe with an additional man in his garden? It could not possibly\nanswer. If it did not, the fellow must be sent away again. He must not\nburden the parish. The occupiers here seemed all alike. Such a fancy for\nnew labor! One, two, six men at work on the land within sight at that\nmoment, over and above what there used to be! It must be looked to.\nHumph! he could get to the alders dryshod now; but that was owing\nsolely to the warmth of the spring. It was nonsense to attribute\neverything to drainage. Drainage was a good thing; but fine weather was\nbetter.\n\nThe gypsy-tent was found behind the alders as before, but no longer in a\nswamp. The woman was sitting on the ground at the entrance as before,\nbut not now with a fevered child laid across her knees. She was weaving\na basket.\n\n\"Oh, I see,\" said Woodruffe, \"this is the way our osiers go.\"\n\n\"You have not many to lose, now-a-days,\" said the woman.\n\n\"You are welcome to all the rushes you can find,\" said Woodruffe; \"but\nwhere is your son?\"\n\nSome change of countenance was seen in the woman; but she answered\ncarelessly that the children were playing yonder.\n\n\"The one I mean is not there,\" said Woodruffe, \"We have him safe--caught\nhim stealing my ducks.\"\n\nShe called the boy a villain--disowned him, and so forth; but when she\nfound the case a hopeless one, she did not, and, therefore, probably\ncould not scold--that is, anybody but herself and her husband. She\ncursed herself for coming into this silly place, where now no good was\nto be got. When she was brought to the right point of perplexity about\nwhat to do, seeing that it would not do to stay, and being unable to go\nwhile her boy was in durance, she was told that his punishment should be\nsummary, though severe, if she would answer frankly certain questions.\nWhen she had once begun giving her confidence, she seemed to enjoy the\nlicense. When her husband came up, he looked as if he only waited for\nthe departure of his visitors to give his wife the same amount of\nthrashing that her son was awaiting elsewhere. She vowed that they would\nnever pitch their tent here again. It used to be the best station in\ntheir whole round--the fogs were so thick! From sunset to long after\nsunrise, it had been as good as a winter night, for going where they\npleased without fear of prying eyes. There was not a poultry-yard or\npig-stye within a couple of miles round, where they could not creep up\nthrough the fog. And they escaped the blame, too; for the swamp and\nditches used to harbor so much vermin, that the gypsies were not always\nsuspected, as they were now. Till lately, people shut themselves into\ntheir homes, or the men went to the public-house in the chill evenings;\nand there was little fear of meeting any one. But now that the fogs were\ngone, people were out in their gardens, on these fine evenings, and\nthere were men in the meadows, returning from fishing; for they could\nangle now, when their work was done, without the fear of catching an\nague in the marsh as they went home.\n\nMr. Nelson used vigorously his last opportunity of lecturing these\npeople. He had it all his own way, for the humility of the gypsies was\nedifying. Woodruffe fancied he saw some finger-talk passing, the while,\nthough the gypsies never looked at each other, or raised their eyes from\nthe ground. Woodruffe had to remind the Director that the whistle of the\nnext train would soon be heard; and this brought the lecture to an\nabrupt conclusion. On his finishing off with, \"I expect, therefore, that\nyou will remember my advice, and never show your face here again, and\nthat you will take to a proper course of life in future, and bring up\nyour son to honest industry;\" the woman, with a countenance of grief,\nseized one hand and covered it with kisses, and the husband took the\nother hand and pressed it to his breast.\n\n\"We must make haste,\" observed Mr. Nelson, as he led the way quickly\nback; \"but I think I have made some impression upon them. You see now\nthe right way to treat these people. I don't think you will see them\nhere again.\"\n\n\"I don't think we shall.\"\n\nAs he reached the steps the whistle was heard, and Mr. Nelson could only\nwave his hand to Woodruffe, rush up the embankment, and throw himself\npanting into a carriage. Only just in time!\n\nBy an evening train he re-appeared. When thirty miles off, he had wanted\nhis purse, and it was gone. It had no doubt paid for the gypsies' final\ngratitude.\n\nOf course, a sufficient force was immediately sent to the alder clump;\nbut there was nothing there but some charred sticks, and some clean pork\nbones, this time, instead of feathers of fowls, and a cabbage leaf or\ntwo. The boy had had his whipping at noon, after a conference with his\nlittle brother at the keyhole, which had caused him to withdraw the\nbolt, and offer no resistance. Considering his cries and groans, he had\nrun off with surprising agility, and was now, no doubt, far away.\n\n\nVIII.\n\nThe gypsies came no more. The fogs came no more. The fever came no more,\nat least, in such a form as to threaten the general safety. Where it\nstill lingered, it was about those only who deserved it,--in any small\nfarm-house, where the dung-yard was too near the house; and in some\ncottage where the slatternly inmates did not mind a green puddle or\nchoked ditch within reach of their noses. More dwellings arose, as the\nfertility of the land increased, and invited a higher kind of tillage;\nand among the prettiest of them was one which stood in the corner,--the\nmost sunny corner,--of Woodruffe's paddock. Harry Hardiman and his wife\nand child lived there, and the cottage was Woodruffe's property.\n\nYet Woodruffe's rent had been raised, and pretty rapidly. He was now\npaying eight pounds per acre for his garden-ground, and half that for\nwhat was out of the limits of the garden. He did not complain of it; for\nhe was making money fast. His skill and industry deserved this; but\nskill and industry could not have availed without opportunity. His\nground once allowed to show what it was worth, he treated it well; and\nit answered well to the treatment. By the railway he obtained what\nmanure he wanted from the town; and he sent it back by the railway to\ntown in the form of crisp celery and salads, wholesome potatoes and\ngreens, luscious strawberries, and sweet and early peas. He knew that a\nSurrey gardener had made his ground yield a profit of two hundred and\ntwenty pounds per acre. He thought that, with his inferior market, he\nshould do well to make his yield one hundred and fifty pounds per acre;\nand this, by close perseverance, he attained. He could have done it more\neasily if he had enjoyed good health; but he never enjoyed good health\nagain. His rheumatism had fixed itself too firmly to be entirely\nremoved; and for many days in the year, he was compelled to remain\nwithin doors, or to saunter about in the sun, seeing his boys and Harry\nat work, but unable to help them.\n\nFrom the time that Allan's work became worth wages in addition to his\nsubsistence, his father let him rent half a rood of the garden-ground\nfor three years, saying--\n\n\"I limit it to three years, my boy, because that term is long enough for\nyou to show what you can do. After three years, I shall not be able to\nspare the ground at any rent. If you fail, you have no business to rent\nground. If you succeed, you will have money in your pocket wherewith to\nhire land elsewhere. Now you have to show us what you can do.\"\n\n\"Yes, father,\" was Allan's short but sufficient reply.\n\nIt was observed by the family that, from this time forward, Allan's eye\nwas on every plot of ground in the neighborhood which could, by\npossibility, ever be offered for hire; yet did his attention never\nwander from that which was already under his hand. And that which was so\ngreat an object to him became a sort of pursuit to the whole family.\nMoss guarded Allan's frames, and made more and more prodigious\nscarecrows. Their father gave his very best advice. Becky, who was no\nlonger allowed, as a regular thing, to work in the garden, found many a\nspare half-hour for hoeing and weeding, and trimming and tying up, in\nAllan's beds; and Abby found, as she sat in her little school, that she\ncould make nets for his fruit trees. It was thus no wonder that, when a\ncertain July day in the second year arrived, the whole household was in\na state of excitement, because it was a sort of crisis in Allan's\naffairs.\n\nThough breakfast was early that morning, Becky and Allan and Moss were\nspruce in their best clothes. A hamper stood at the door, and Allan was\npacking in another, which had no lid, two or three flower-pots, which\npresented a glorious show of blossom. Abby was putting a new ribbon on\nher sister's straw bonnet; and Harry was in waiting to carry up the\nhampers to the station. It was the day of the Horticultural Show at the\ntown. Woodruffe had been too unwell to think of going till this morning;\nbut now the sight of the preparations, and the prospect of a warm day,\ninspired him, and he thought he would go. At last he went, and they were\ngone. Abby never went up to the station; nobody ever asked her to go\nthere, not even her own child, who perhaps had not thought of the\npossibility of it. But when the train was starting, she stood at the\nupper window with her child, and held him so that he might lean out, and\nsee the last carriage disappear as it swept round the curve. After that\nthe day seemed long, though Harry came up at the dinner-hour to say what\nhe thought of the great gooseberry in particular, and of everything else\nthat Allan had carried with him. It was holiday time, and there was no\nschool to fill up the day. Before the evening, the child became\nrestless, and Abby fell into low spirits, as she was apt to do when left\nlong alone; so that Harry stopped suddenly at the door when he was\nrushing in to announce that the train was within sight.\n\n\"Shall I take the child, Miss?\" said Harry. (He always called her\n\"Miss.\") \"I will carry him---- But, sure, here they come! Here comes\nMoss,--ready to roll down the steps! My opinion is that there's a\nprize.\"\n\nMoss was called back by a voice which everybody obeyed. Allan should\nhimself tell his sister the fortune of the day, their father said.\n\nThere were two prizes, one of which was for the wonderful plate of\ngooseberries; and at this news Harry nodded, and declared himself\nanything but surprised. If that gooseberry had not carried the day,\nthere would have been partiality in the judges, that was all; and nobody\ncould suppose such a thing as that. Yet Harry could have told, if put\nupon his honor, that he was rather disappointed that everything that\nAllan carried had not gained a prize. When he mentioned one or two, his\nmaster told him he was unreasonable; and he supposed he was.\n\nAllan laid down upon the table, for his sister's full assurance, his\nsovereign, and his half-sovereign, and his tickets. She turned away\nrather abruptly, and seemed to be looking whether the kettle was near\nboiling for tea. Her father went up to her; and on his first whispered\nwords, the sob broke forth which made all look round.\n\n\"I was thinking of one, too, my dear, that I wish was here at this\nmoment. I can feel for you, my dear.\"\n\n\"But you don't know--you don't know--you never knew----.\" She could not\ngo on.\n\n\"What don't I know, my dear?\"\n\n\"That he constantly blamed himself for saying anything to bring you\nhere. He said you had never prospered from the hour you came, and\nnow----\"\n\nAnd now Woodruffe could not speak, as the past came fresh upon him. In a\nfew moments, however, he rallied, saying,\n\n\"But we must consider Allan. He must not think that his success makes us\nsad.\"\n\nAllan declared that it was not about gaining the prizes that he was\nchiefly glad. It was because it was now proved what a fair field he had\nbefore him. There was nothing that might not be done with such a soil as\nthey had to deal with now.\n\nHarry was quite of this opinion. There were more and more people set to\nwork upon the soil all about them; and the more it was worked the more\nit yielded. He never saw a place of so much promise. And if it had a\nbad name in regard to healthiness, he was sure that was unfair,--or no\nlonger fair. He and his were full of health and happiness, as they hoped\nto see everybody else in time; and, for his part, if he had all England\nbefore him, or the whole world, to choose a place to live in, he would\nchoose the very place he was in, and the very cottage, and the very\nground to work on that had produced such a gooseberry and such\nstrawberries as he had seen that day.\n\n\n\n\nVII.\n\nThe Water-Drops.\n\nA FAIRY TALE.\n\n\nI.\n\nTHE SUITORS OF CIRRHA, AND THE YOUNG LADY; WITH A REFERENCE TO HER PAPA.\n\nFar in the west there is a land mountainous, and bright of hue, wherein\nthe rivers run with liquid light; the soil is all of yellow gold; the\ngrass and foliage are of resplendent crimson; where the atmosphere is\npartly of a soft green tint, and partly azure. Sometimes on summer\nevenings we see this land, and then, because our ignorance must refer\nall things that we see, to something that we know, we say it is a mass\nof clouds made beautiful by sunset colors. We account for it by\nprinciples of Meteorology. The fact has been omitted from the works of\nKaemtz or Daniell; but, notwithstanding this neglect, it is well known\nin many nurseries, that the bright land we speak of, is a world\ninhabited by fairies. Few among fairies take more interest in man's\naffairs than the good Cloud Country People; this truth is established by\nthe story I am now about to tell.\n\nNot long ago there were great revels held one evening in the palace of\nKing Cumulus, the monarch of the western country. Cirrha, the daughter\nof the king, was to elect her future husband from a multitude of\nsuitors. Cirrha was a maiden delicate and pure, with a skin white as\nunfallen snow; but colder than the snow her heart had seemed to all who\nsought for her affections. When Cirrha floated gracefully and slowly\nthrough her father's hall, many a little cloud would start up presently\nto tread where she had trodden. The winds also pursued her; and even men\nlooked up admiringly whenever she stepped forth into their sky. To be\nsure they called her Mackerel and Cat's Tail, just as they call her\nfather Ball of Cotton; for the race of man is a coarse race, and calling\nbad names appears to be a great part of its business here below.\n\nBefore the revels were concluded, the King ordered a quiet little wind\nto run among the guests, and bid them all come close to him and to his\ndaughter. Then he spoke to them as follows:--\n\n\"Worthy friends! there are among you many suitors to my daughter Cirrha,\nwho is pledged this evening to choose a husband. She bids me tell you\nthat she loves you all; but since it is desirable that this our royal\nhouse be strengthened by a fit alliance with some foreign power, she has\nresolved to take as husband one of those guests who have come hither\nfrom the principality of Nimbus.\" Now, Nimbus is that country, not\nseldom visible from some parts of our earth, which we have called the\nRain-Cloud. \"The subjects of the Prince of Nimbus,\" Cumulus continued,\n\"are a dark race, it is true, but they are famed for their beneficence.\"\n\nTwo winds, at this point, raised between themselves a great disturbance,\nso that there arose a universal cry that somebody should turn them out.\nWith much trouble they were driven out from the assembly; thereupon,\nquite mad with jealousy and disappointment, they went howling off to\nsea, where they played pool-billiards with a fleet of ships, and so\nforgot their sorrow.\n\nKing Cumulus resumed his speech, and said that he was addressing\nhimself, now, especially to those of his good friends who came from\nNimbus. \"To-night, let them retire to rest, and early the next morning\nlet each of them go down to Earth; whichever of them should be found on\ntheir return to have been engaged below in the most useful service to\nthe race of man, that son of Nimbus should be Cirrha's husband.\"\n\nCumulus, having said this, put a white nightcap on his head, which was\nthe signal for a general retirement. The golden ground of his dominions\nwas covered for the night, as well as the crimson trees, with cotton. So\nthe whole kingdom was put properly to bed. Late in the night the moon\ngot up, and threw over King Cumulus a silver counterpane.\n\n\nII.\n\nTHE ADVENTURES OF NEBULUS AND NUBIS.\n\nThe suitors of the Princess Cirrha, who returned to Nimbus, were a-foot\nquite early the next morning, and petitioned their good-natured Prince\nto waft them over London. They had agreed among themselves, that by\ndescending there, where men were densely congregated, they should have a\ngreater chance of doing service to the human race. Therefore the\nRain-Cloud floated over the great City of the World, and, as it passed\nat sundry points, the suitors came down upon rain-drops to perform their\ndestined labor. Where each might happen to alight depended almost wholly\nupon accident; so that their adventures were but little better than a\nlottery for Cirrha's hand. One, who had been the most magniloquent among\nthem all, fell with his pride upon the patched umbrella of an early\nbreakfast woman, and from thence was shaken off into a puddle. He was\nsplashed up presently, mingled with soil, upon the corduroys of a\nlaborer, who stopped for breakfast on his way to work. From thence,\nevaporating, he returned crest-fallen to the Land of Clouds.\n\nAmong the suitors there were two kind-hearted fairies, Nebulus and\nNubis, closely bound by friendship to each other. While they were in\nconversation, Nebulus, who suddenly observed that they were passing over\nsome unhappy region, dropped, with a hope that he might bless it. Nubis\npassed on, and presently alighted on the surface of the Thames.\n\nThe district which had wounded the kind heart of Nebulus was in a part\nof Bermondsey, called Jacob's Island. The fairy fell into a ditch; out\nof this, however, he was taken by a woman, who carried him to her own\nhome, among other ditch-water, within a pail. Nebulus abandoned himself\nto complete despair, for what claim could he now establish on the hand\nof Cirrha? The miserable plight of the poor fairy we may gather from a\ndescription given by a son of man of the sad place to which he had\ndescended. \"In this Island may be seen, at any time of the day, women\ndipping water, with pails attached by ropes to the backs of the houses,\nfrom a foul fetid ditch, its banks coated with a compound of mud and\nfilth, and strewed with offal and carrion; the water to be used for\nevery purpose, culinary ones not excepted; although close to the place\nwhence it is drawn, filth and refuse of various kinds are plentifully\nshowered into it from the outhouses of the wooden houses overhanging its\ncurrent, or rather slow and sluggish stream; their posts or supporters\nrotten, decayed, and in many instances broken, and the filth dropping\ninto the water, to be seen by any passer by. During the summer, crowds\nof boys bathe in the putrid ditches, where they must come in contact\nwith abominations highly injurious.\"[1]\n\nSo Nebulus was carried in a pail out of the ditch to a poor woman's\nhome, and put into a battered saucepan with some other water. Thence,\nafter boiling, he was poured into an earthen tea-pot over some stuff of\nwretched flavor, said to be tea. Now, thought the fairy, after all, I\nmay give pleasure at the breakfast of these wretched people. He pictured\nto himself a scene of love as preface to a day of squalid toil, but he\nexperienced a second disappointment. The woman took him to another room\nof which the atmosphere was noisome; there he saw that he was destined\nfor the comfort of a man and his two children, prostrate upon the floor\nbeneath a heap of rags. These three were sick; the woman swore at them,\nand Nebulus shrunk down into the bottom of the tea-pot. Even the thirst\nof fever could not tolerate too much of its contents, so Nebulus, after\na little time, was carried out and thrown into a heap of filth upon the\ngutter.\n\nNubis, in the meantime, had commenced his day with hope of a more\nfortunate career. On falling first into the Thames he had been much\nannoyed by various pollutions, and been surprised to find, on kissing a\nfew neighbor drops, that their lips tasted inky. This was caused, they\nsaid, by chalk pervading the whole river in the proportion of sixteen\ngrains to the gallon. That was what made their water inky to the taste\nof those who were accustomed to much purer draughts. \"It makes,\" they\nexplained, \"our river-water hard, according to man's phrase; so hard as\nto entail on multitudes who use it, some disease, with much expense and\ntrouble.\"\n\n\"But all the mud and filth,\" said Nubis, \"surely no man drinks that?\"\n\n\"No,\" laughed the River-Drops, \"not all of it. Much of the water used in\nLondon passes through filters, and a filter suffers no mud or any\nimpurity to pass, except what is dissolved. The chalk is dissolved, and\nthere is filth and putrid gas dissolved.\"\n\n\"That is a bad business,\" said Nubis, who already felt his own drops\nexercising that absorbent power for which water is so famous, and\nincorporating in their substance matters that the Rain-Cloud never\nknew.\n\nPresently Nubis found himself entangled in a current, by which he was\nsucked through a long pipe into a meeting of Water-Drops, all summoned\nfrom the Thames. He himself passed through a filter, was received into a\nreservoir, and, having asked the way of friendly neighbors, worked for\nhimself with small delay a passage through the mainpipe into London.\n\nBewildered by his long, dark journey underground, Nubia at length saw\nlight, and presently dashed forth out of a tap into a pitcher. He saw\nthat there was fixed under the tap a water-butt, but into this he did\nnot fall. A crowd of women holding pitchers, saucepans, pails, were\nchattering and screaming over him, and the anxiety of all appeared to be\nto catch the water as it ran out of the tap, before it came into the tub\nor cistern. Nubis rejoiced that his good fortune brought him to a\ndistrict in which it might become his privilege to bless the poor, and\nhis eye sparkled as his mistress, with many rests upon the way, carried\nher pitcher and a heavy pail upstairs. She placed both vessels, full of\nwater, underneath her bed, and then went out again for more, carrying a\nbasin and a fish-kettle. Nubis pitied the poor creature, heartily\nwishing that he could have poured out of a tap into the room itself to\nsave the time and labor of his mistress.\n\nThe pitcher wherein the good fairy lurked, remained under the bed\nthrough the remainder of that day, and during the next night, the room\nbeing, for the whole time, closely tenanted. Long before morning, Nubis\nfelt that his own drops and all the water near him had lost their\ndelightful coolness, and had been busily absorbing smells and vapors\nfrom the close apartment. In the morning, when the husband dipped a\nteacup in the pitcher, Nubis readily ran into it, glad to escape from\nhis unwholesome prison. The man putting the water to his lips, found it\nso warm and repulsive, that, in a pet, he flung it from the window, and\nit fell into the water-butt beneath.\n\nThe water-butt was of the common sort, described thus by a member of the\nhuman race:--\"Generally speaking, the wood becomes decomposed and\ncovered with fungi; and indeed, I can best describe their condition by\nterming them filthy.\" This water-butt was placed under the same shed\nwith a neglected cesspool, from which the water--ever absorbing--had\nabsorbed pollution. It contained a kitten among other trifles. \"How many\npeople have to drink out of this butt?\" asked Nubis. \"Really I cannot\ntell you,\" said a neighbor Drop. \"Once I was in a butt in Bethnal Green,\ntwenty-one inches across, and a foot deep, which was to supply\nforty-eight families.[2] People store for themselves, and when they know\nhow dirty these tubs are, they should not use them.\" \"But the labor of\ndragging water home, the impossibility of taking home abundance, the\npollution of keeping it in dwelling-rooms and under beds.\" \"Oh, yes,\"\nsaid the other Drop; \"all very true. Besides, our water is not of a sort\nto keep. In this tub there is quite a microscopic vegetable garden, so I\nheard a doctor say who yesterday came hither with a party to inspect\nthe district. One of them said he had a still used only for distilling\nwater, and that one day, by chance, the bottoms of a series of\ndistillations boiled to dryness. Thereupon, the dry mass became heated\nto the decomposing point, and sent abroad a stench plain to the dullest\nnose as the peculiar stench of decomposed organic matter. It infected,\nhe said, the produce of many distillations afterwards.\"[3] \"I tell you\nwhat,\" said Nubis, \"water may come down into this town innocent enough,\nbut it's no easy matter for it to remain good among so many causes of\ncorruption. Heigho!\" Then he began to dream of Princess Cirrha and the\nworthy Prince of Nimbus, until he was aroused by a great tumult. It was\nan uproar caused by drunken men. \"Why are those men so?\" said Nubis to\nhis friend. \"I don't know,\" said the Water-Drop, \"but I saw many people\nin that way last night, and I have seen them so at Bethnal Green.\" A\nwoman pulled her husband by, with loud reproaches for his visits to the\nbeer-shop. \"Why,\" cried the man, with a great oath, \"where would you\nhave me go for drink?\" Then, with another oath, he kicked the water-butt\nin passing--\"You would not have me to go there!\" All the bystanders\nlaughed approvingly, and Nubis bade adieu to his ambition for the hand\nof Cirrha.\n\n\nIII.\n\n     NEPHELO GOES INTO POLITE SOCIETY, AND THEN INTO A DUNGEON--HIS\n     ESCAPE, RECAPTURE, AND HIS PERILOUS ASCENT INTO THE SKY, SURROUNDED\n     BY A BLAZE OF FIRE.\n\nNephelo was a light-hearted subject of the Prince of Nimbus. It is he\nwho often floats, when the whole cloud is dark, as a white vapor on the\nsurface. For love of Cirrha, he came down behind a team of rain-drops\nand leaped into the cistern of a handsome house at the west end of\nLondon.\n\nNephelo found the water in the cistern greatly vexed at riotous\nbehavior on the part of a large number of animalcules. He was told that\nWater-Drops had been compelled to come into that place, after undergoing\nmany hardships, and had unavoidably brought with them germs of these\nannoying creatures. Time and place favoring, nothing could hinder them\nfrom coming into life; the cistern was their cradle, although many of\nthem were already anything but babes. Hereupon, Nephelo himself was\ndashed at by an ugly little fellow like a dragon; but an uglier little\nfellow, who might be a small Saint George, pounced at the dragon, and\nthe heart of the poor fairy was the scene of contest.\n\nAfter awhile there was an arrival of fresh water from a pipe, the flow\nof which stirred up the anger of some decomposing growth which lined the\nsides and bottom of the cistern. So there was a good deal of confusion\ncaused, and it was some time before all parties settled down into their\nproper places.\n\n\"The sun is very hot,\" said Nephelo. \"We all seem to be getting very\nwarm.\" \"Yes, indeed,\" said a Lady-Drop; \"it's not like the cool\nCloud-Country. I have been poisoned in the Thames, half filtered, and\nmade frowzy by standing, this July weather, in an open reservoir. I've\ntravelled in pipes laid too near the surface to be cool, and now am\nspoiling here. I know if water is not cold it can't be pleasant.\" \"Ah,\"\nsaid an old Drop, with a small eel in one of his eyes; \"I don't wonder\nat hearing tell that men drink wine, and tea, and beer.\" \"Talking of\nbeer,\" said another, \"is it a fact that we're of no use to the brewers?\nOur character's so bad, they can't rely on us for cooling the worts, and\nso sink wells, in order to brew all the year round with water cold\nenough to suit their purposes.\" \"I know nothing of beer,\" said Nephelo;\n\"but I know that if the gentlemen and ladies in this cistern were as\ncold as they could wish to be, there wouldn't be so much decomposition\ngoing on among them.\" \"Your turn in, sir,\" said a polite Drop, and\nNephelo leaped nimbly through the place of exit into a china jug placed\nready to receive him. He was conveyed across a handsome kitchen by a\ncook, who declared her opinion that the morning's rain had caused the\ndrains to smell uncommonly. Nephelo then was thrown into a kettle.\n\nBoiling is to an unclean Water-Drop, like scratching to a bear, a\npleasant operation. It gets rid of the little animals by which it had\nbeen bitten, and throws down some of the impurity with which it had been\nsoiled. So, after boiling, water becomes more pure, but it is, at the\nsame time, more greedy than ever to absorb extraneous matter. Therefore,\nthe sons of men who boil their vitiated water ought to keep it covered\nafterwards, and if they wish to drink it cold, should lose no time in\ndoing so. Nephelo and his friends within the kettle danced with delight\nunder the boiling process. Chattering pleasantly together, they compared\nnotes of their adventures upon earth, discussed the politics of\nCloud-Land, and although it took them nearly twice as long to boil as it\nwould have done had there been no carbonate of lime about them, they\nwere quite sorry when the time was come for them to part. Nephelo then,\nwith many others, was poured out into an urn. So he was taken to the\ndrawing-room, a hot iron having, in a friendly manner, been put down his\nback, to keep him boiling.\n\nOut of the urn into the tea-pot; out of the tea-pot into the slop-basin;\nNephelo had only time to remark a matron tea-maker, young ladies\nknitting, and a good-looking young gentleman upon his legs, laying the\nlaw down with a tea-spoon, before he (the fairy, not the gentleman) was\nsmothered with a plate of muffins. From so much of the conversation as\nNephelo could catch, filtered through muffin, it appeared that they were\ntalking about tea.\n\n\"It's all very well for you to say, mother, that you're confident you\nmake tea very good, but I ask--no, there I see you put six spoonfuls in\nfor five of us. Mother, if this were not hard water--(here there was a\nnoise as of a spoon hammering upon the iron)--two spoonfuls less would\nmake tea of a better flavor and of equal strength. Now, there are three\nhundred and sixty-five times and a quarter tea-times in the year ----\"\n\n\"And how many spoonfuls, brother, to the quarter of a tea-time?\"\n\n\"Maria, you've no head for figures. I say nothing of the tea consumed at\nbreakfast. Multiply----\"\n\n\"My dear boy, you have left school; no one asks you to multiply. Hand me\nthe muffin.\"\n\nNephelo, released, was unable to look about him, owing to the high walls\nof the slop-basin which surrounded him on every side. The room was\nfilled with pleasant sunset light, but Nephelo soon saw the coming\nshadow of the muffin-plate, and all was dark directly afterwards.\n\n\"Take cooking, mother. M. Soyer[4] says you can't boil many vegetables\nproperly in London water. Greens won't be greens; French beans are\ntinged with yellow, and peas shrivel. It don't open the pores of meat,\nand make it succulent, as softer water does. M. Soyer believes that the\ntrue flavor of meat cannot be extracted with hard water. Bread does not\nrise so well when made with it. Horses----\"\n\n\"My dear boy, M. Soyer don't cook horses.\"\n\n\"Horses, Dr. Playfair tells us, sheep, and pigeons, will refuse hard\nwater if they can get it soft, though from the muddiest pool.\nRace-horses, when carried to a place where the water is notoriously\nhard, have a supply of softer water carried with them to preserve their\ngood condition. Not to speak of gripes, hard water will assuredly\nproduce what people call a staring coat.\"\n\n\"Ah, no doubt, then, it was London water that created Mr. Blossomley's\nblue swallow-tail.\"\n\n\"Maria, you make nonsense out of everything. When you are Mrs.\nBlossomley----\"\n\n\"Now pass my cup.\"\n\nThere was a pause and a clatter. Presently the muffin-plate was lifted,\nand four times in succession there were black dregs thrown into the face\nof Nephelo. After the perpetration of these insults he was once again\ncondemned to darkness.\n\n\"When you are Mrs. Blossomley, Maria,\" so the voice went on, \"when you\nare Mrs. Blossomley, you will appreciate what I am now going to tell you\nabout washerwomen.\"\n\n\"Couldn't you postpone it, dear, until I am able to appreciate it. You\npromised to take us to Rachel to-night.\"\n\n\"Ah!\" said another girlish voice, \"you'll not escape. We dress at seven.\nUntil then--for the next twelve minutes you may speak. Bore on, we will\nendure.\"\n\n\"As for you, Catherine, Maria teaches you, I see, to chatter. But if\nMrs. B. would object to the reception of a patent mangle as a wedding\npresent from her brother, she had better hear him now. Washerwoman's\nwork is not a thing to overlook, I tell you. Before a shirt is worn out,\nthere will have been spent upon it five times its intrinsic value in the\nwashing-tub. The washing of clothes costs more, by a great deal, than\nthe clothes themselves. The yearly cost of washing to a household of the\nmiddle class amounts, on the average, to about a third part of the\nrental, or a twelfth part of the total income. Among the poor, the\naverage expense of washing will more probably be half the rental if they\nwash at home, but not more than a fourth of it if they employ the Model\nWash-houses. The weekly cost of washing to a poor man averages certainly\nnot less than fourpence halfpenny. Small tradesmen, driven to economize\nin linen, spend perhaps not more than ninepence; in the middle and upper\nclasses, the cost weekly varies from a shilling to five shillings for\neach person, and amounts very often to a larger sum. On these grounds,\nMr. Bullar, Honorary Secretary to the Association for Promoting Baths\nand Wash-houses, estimates the washing expenditure of London at a\nshilling a week for each inhabitant, or, for the whole, five millions of\npounds yearly. Professor Clark--\"\n\n\"My dear Professor Tom, you have consumed four of your twelve minutes.\"\n\n\"Professor Clark judges from such estimates as can be furnished by the\ntrade, that the consumption of soap in London is fifteen pounds to each\nperson per annum--twice as much as is employed in other parts of\nEngland. That quantity of soap costs six-and-eightpence; water, per\nhead, costs half as much, or three-and-fourpence; or each man's soap and\nwater costs throughout London, on an average, ten shillings for twelve\nmonths. If the hardness of the water be diminished, there is a\ndiminution in the want of soap. For every grain of carbonate of lime\ndissolved in each gallon of any water, Mr. Donaldson declares two ounces\nof soap more for a hundred gallons of that water are required. Every\nsuch grain is called a degree of hardness. Water of five degrees of\nhardness requires, for example, two ounces of soap; water of eight\ndegrees of hardness, then will need fifteen; and water of sixteen\ndegrees, will demand thirty-two. Sixteen degrees, Maria, is the hardness\nof Thames water--of the water, mother, which has poached upon your\ntea-caddy. You see, then, that when we pay for the soap we use at the\nrate of six-and-eightpence each, since the unusual hardness of our water\ncauses us to use a double quantity, every man in London pays at an\naverage rate of three-and-fourpence a year his tax for a hard water\nthrough the cost of soap alone.\"\n\n\"Now you must finish in five minutes, brother Tom.\"\n\n\"But soap is not the only matter that concerns the washerwoman and her\ncustomers. There is labor, also, and the wear and tear; there is a\ndouble amount of destruction to our linen, involved in the double time\nof rubbing and the double soaping, which hard water compels washerwomen\nto employ. So that, when all things have been duly reckoned up in our\naccount, we find that the outlay caused by the necessities for washing\nlinen in a town supplied like London with exceedingly hard water, is\nfour times greater than it would be if soft water were employed. The\ncost of washing, as I told you, has been estimated at five millions\na-year. So that, if these calculations be correct, more than three\nmillions of money, nearly four millions, is the amount filched yearly\nfrom the Londoners by their hard water through the wash-tub only. To\nthat sum, Mrs. Blossomley, being of a respectable family and very\npartial to clean linen, will contribute of course much more than her\naverage proportion.\"\n\n\"Well, Mr. Orator, I was not listening to all you said, but what I heard\nI do think much exaggerated.\"\n\n\"I take it, sister, from the Government Report; oblige me by believing\nhalf of it, and still the case is strong. It is quite time for people to\nbe stirring.\"\n\n\"So it is, I declare. Your twelve minutes are spent, and we will always\nbe ready for the play. If you talk there of water, I will shriek.\"\n\nHere there arose a chatter which Nephelo found to be about matters that,\nunlike the water topic, did not at all interest himself. There was a\nrustle and a movement; and a creaking noise approached the drawing-room,\nwhich Nephelo discovered presently to be caused by papa's boots as he\nmarched upstairs after his post-prandial slumberings. There was more\ntalk uninteresting to the fairy; Nephelo, therefore, became drowsy; his\ndrowsiness might at the same time have been aggravated by the close\nconfinement he experienced in an unwholesome atmosphere beneath the\nmuffin-plate. He was aroused by a great clattering; this the maid caused\nwho was carrying him down stairs upon a tray with all the other\ntea-things.\n\nFrom a sweet dream of nuptials with Cirrha, Nephelo was awakened to the\npainful consciousness that he had not yet succeeded in effecting any\ngreat good for the human race; he had but rinsed a teapot. With a faint\nimpulse of hope the desponding fairy noticed that the slop-basin in\nwhich he sat was lifted from the tray, in a few minutes after the tray\nhad been deposited upon the kitchen-dresser. Pity poor Nephelo! By a\nremorseless scullery-maid he was dashed rudely from the basin into a\ntrough of stone, from which he tumbled through a hole placed there on\npurpose to engulf him,--tumbled through into a horrible abyss.\n\nThis abyss was a long dungeon running from back to front beneath the\nhouse, built of bricks--rotten now, and saturated with moisture. Some of\nthe bricks had fallen in, or crumbled into nothingness; and Nephelo saw\nthat the soil without the dungeon was quite wet. The dungeon-floor was\ncoated with pollutions, travelled over by a sluggish shallow stream,\nwith which the fairy floated. The whole dungeon's atmosphere was foul\nand poisonous. Nephelo found now what those exhalations were which rose\nthrough every opening in the house, through vent-holes and the\nburrowings of rats; for rats and other venom tenanted this noisome den.\nThis was the pestilential gallery called by the good people of the\nhouse, their drain. A trap door at one end confined the fairy in this\nplace with other Water-Drops, until there should be collected a\nsufficient body of them to negotiate successfully for egress.\n\nThe object of this door was to prevent the ingress of much more foul\nmatter from without; and its misfortune was, that in so doing it\nnecessarily pent up a concentrated putrid gas within. At length Nephelo\nescaped; but, alas! it was from a Newgate to a Bastille--from the drain\ninto the sewer. This was a long-vaulted prison running near the surface\nunderneath the street. Shaken by the passage overhead of carriages, not\na few bricks had fallen in; and Nephelo hurrying forward, wholly\npossessed by the one thought--could he escape?--fell presently into a\ntrap. An oyster-shell had fixed itself upright between two bricks\nunevenly jointed together; much solid filth had grown around it; and in\nthis Nephelo was caught. Here he remained for a whole month, during\nwhich time he saw many floods of water pass him, leaving himself with a\nvast quantity of obstinate encrusted filth unmoved. At the month's end\nthere came some men to scrape, and sweep, and cleanse; then with a\nsudden flow of water, Nephelo was forced along, and presently, with a\nlarge number of emancipated foulnesses, received his discharge from\nprison, and was let loose upon the River Thames.\n\nNephelo struck against a very dirty Drop.\n\n\"Keep off, will you?\" the Drop exclaimed. \"You are not fit to touch a\nperson, sewer-bird.\"\n\n\"Why, where are you from, my sweet gentleman?\"\n\n\"Oh! I? I've had a turn through some Model Drains. Tubular drains they\ncall 'em. Look at me; isn't that clear?\"\n\n\"There's nothing clear about you,\" replied Nephelo. \"What do you mean by\nModel Drains?\"\n\n\"I mean I've come from Upper George Street through a twelve-inch pipe\nfour or five times faster than one travels over an old sewer-bed;\ntravelled express, no stoppage.\"\n\n\"Indeed!\"\n\n\"Yes. Impermeable, earthenware, tubular pipes, accurately dove-tailed. I\ncome from an experimental district. When it's all settled, there's to be\nwater on at high pressure everywhere, and an earthenware drain pipe\nunder every tap, a tube of no more than the necessary size. Then these\nlittle pipes are to run down the earth; and there's not to be a great\nbrick drain running underneath each house into the street; the pipes run\ninto a larger tube of earthenware that is to be laid at the backs of\nall the houses; these tubes run into larger ones, but none of them very\nmonstrous; and so that there is a constant flow, like circulation of the\nblood; and all the pipes are to run at last into one large conduit,\nwhich is to run out of town with all the sewage matter and discharge so\nfar down the Thames, that no return tide ever can bring it back to\nLondon. Some is to go branching off into the fields to be manure.\"\n\n\"Humph!\" said Nephelo. \"You profess to be very clever. How do you know\nall this?\"\n\n\"Know? Bless you, I'm a regular old Thames Drop. I've been in the\ncisterns, in the tumblers, down the sewers, in the river, up the pipes,\nin the reservoirs, in the cisterns, in the teapots, down the sewers, in\nthe river, up the pipes, in the reservoirs, in the cisterns, in the\nsaucepans, down the sewers, in the Thames--\"\n\n\"Hold! Stop there now!\" said Nephelo. \"Well, so you have heard a great\ndeal in your lifetime. You've had some adventures, doubtless?\"\n\n\"I believe you,\" said the Cockney-Drop. \"The worst was when I was pumped\nonce as fresh water into Rotherhithe. That place is below high-water\nmark; so are Bermondsey and St. George's, Southwark. Newington, St.\nOlave's, Westminster, and Lambeth, are but little better. Well, you\nknow, drains of the old sort always leak, and there's a great deal more\nwater poured into London than the Londoners have stowage room for, so\nthe water in low districts can't pass off at high water, and there's a\nprecious flood. We sopped the ground at Rotherhithe, but I thought I\nnever should escape again.\"\n\n\"Will the new pipes make any difference to that?\"\n\n\"Yes; so I am led to understand. They are to be laid with a regular\nfall, to pass the water off, which, being constant, will be never in\nexcess. The fall will be to a point of course below the water level, and\nat a convenient place the contents of these drains are to be pumped up\ninto the main sewer. Horrible deal of death caused, Sir, by the damp in\nthose low districts. One man in thirty-seven died of cholera in\nRotherhithe last year, when in Clerkenwell, at sixty-three feet above\nhigh water, there died but one in five hundred and thirty. The\nproportion held throughout.\"\n\n\"Ah, by the bye, you have heard, of course, complainings of the quality\nof water. Will the Londoners sink wells for themselves?\"\n\n\"Wells! What a child you are! Just from the clouds, I see. Wells in a\nlarge town get horribly polluted. They propose to consolidate and\nimprove two of the best Thames Water Companies, the Grand Junction and\nVauxhall, for the supply of London, until their great scheme can be\nintroduced; and to maintain them afterwards as a reserve guard in case\ntheir great scheme shouldn't prove so triumphant as they think it will\nbe.\"\n\n\"What is this great scheme, I should like to know?\"\n\n\"Why, they talk of fetching rain-water from a tract of heath between\nBagshot and Farnham. The rain there soaks through a thin crust of\ngrowing herbage, which is the only perfect filter, chemical as well as\nmechanical--the living rootlets extract more than we can, where impurity\nexists. Then, Sir, the rain runs into a large bed of silicious sand,\nplaced over marl; below the marl there is silicious sand again--Ah, I\nperceive you are not geological.\"\n\n\"Go on.\"\n\n\"The sand, washed by the rains of ages, holds the water without soiling\nit more than a glass tumbler would, and the Londoners say that in this\nway, by making artificial channels and a big reservoir, they can collect\ntwenty-eight thousand gallons a day of water nearly pure. They require\nforty thousand gallons, and propose to get the rest in the same\nneighborhood from tributaries of the River Wey, not quite so pure, but\nonly half as hard as Thames water, and unpolluted.\"\n\n\"How is it to get to London?\"\n\n\"Through a covered aqueduct. Covered for coolness' sake, and\ncleanliness. Then it is to be distributed through earthenware pipes,\nlaid rather deep, again for coolness' sake in the first instance, but\nfor cleanliness as well. The water is to come in at high pressure, and\nrun in iron or lead pipes up every house, scale every wall. There is to\nbe a tap in every room, and under every tap there is to be the entrance\nto a drain-pipe. Where water supply ends, drainage begins. They are to\nbe the two halves of a single system. Furthermore, there are to be\nnumbers of plugs opening in every street, and streets and courts are to\nbe washed out every morning, or every other morning, as the traffic may\nrequire, with hose and jet. The Great Metropolis mustn't be dirty, or be\ncontent with rubbing a finger here and there over its dirt. It is to\nhave its face washed every morning, just before the hours of business.\nThe water at high pressure is to set people's invention at work upon the\nintroduction of hydraulic apparatus for cranes, et c\u00e6tera, which now\ncause much hand labor and are scarcely worth steam-power. Furthermore--\"\n\n\"My dear friend,\" cried Nephelo, \"you are too clever. More than half of\nwhat you say is unintelligible to me.\"\n\n\"But the grand point,\" continued the garrulous Thames drop, \"is the\nexpense. The saving of cisterns, ball-cocks, plumbers' bills, expansive\nsewer-works, constant repairs, hand labor, street-sweeping, soap, tea,\nlinen, fuel, steam-boilers now damaged by incrustation, boards,\nsalaries, doctors' bills, time, parish rates--\"\n\nThe catalogue was never ended, for the busy Drop was suddenly entangled\namong hair upon the corpse of a dead cat, which fate also the fairy\nnarrowly escaped, to be in the next minute sucked up as Nubis had been\nsucked, through pipes into a reservoir. Weary with the incessant\nchattering of his conceited friend, whose pride he trusted that a night\nwith puss might humble, Nephelo now lurked silent in a corner. In a\ndreamy state he floated with the current underground, and was half\nsleeping in a pipe under some London street, when a great noise of\ntrampling overhead, mingled with cries, awakened him.\n\n\"What is the matter now?\" the fairy cried.\n\n\"A fire, no doubt, to judge by the noise,\" said a neighbor quietly.\nNephelo panted now with triumph. Cirrha was before his eyes. Now he\ncould benefit the race of man.\n\n\"Let us get out,\" cried Nephelo; \"let us assist in running to the\nrescue.\"\n\n\"Don't be impatient,\" said a drowsy Drop. \"We can't get out of here till\nthey have found the Company's turncock, and then he must go to this plug\nand that plug in one street, and another, before we are turned off.\"\n\n\"In the meantime the fire--\"\n\n\"Will burn the house down. Help in five minutes would save a house. Now\nthe luckiest man will seldom have his premises attended to in less than\ntwenty.\"\n\nNephelo thought here was another topic for his gossip in the Thames. The\nplugs talked of with a constant water-supply would take the sting out of\nthe Fire-Fiend.\n\nPresently among confused movements, confused sounds, amid a rush of\nwater, Nephelo burst into the light--into the vivid light of a great\nfire that leapt and roared as Nephelo was dashed against it! Through the\nred flames and the black smoke in a burst of steam, the fairy reascended\nhopeless to the clouds.\n\n\nIV.\n\nRASCALLY CONDUCT OF THE PRINCE OF NIMBUS.\n\nThe Prince of Nimbus, whose good-nature we have celebrated, was not good\nfor nothing. Having graciously permitted all the suitors of the Princess\nCirrha to go down to earth and labor for her hand, he took advantage of\ntheir absence, and, having the coast clear, importuned the daughter of\nKing Cumulus with his own addresses. Cirrha was not disposed to listen\nto them, but the rogue her father was ambitious. He desired to make a\ngood alliance, and that object was better gained by intermarriage with a\nprince than with a subject. \"There will be an uproar,\" said the old\nman, \"when those fellows down below come back. They will look black and\nno doubt storm a little, but we'll have our royal marriage\nnotwithstandstanding.\" So the Prince of Nimbus married Cirrha, and\nNephelo arrived at the court of King Cumulus one evening during the\ncelebration of the bridal feast. His wrath was seen on earth in many\nparts of England in the shape of a great thunderstorm on the 16th of\nJuly. The adventures of the other suitors, they being thus cheated of\ntheir object, need not be detailed. As each returns he will be made\nacquainted with the scandalous fraud practised by the Prince of Nimbus,\nand this being the state of politics in Cloud-Land at the moment when we\ngo to press, we may fairly expect to witness five or six more\nthunderstorms before next winter. Each suitor, as he returns and finds\nhow shamefully he has been cheated, will create a great disturbance; and\nno wonder. Conduct so rascally as that of the Prince of Nimbus is enough\nto fill the clouds with uproar.\n\n\n\n\nVIII.\n\nAn Excellent Opportunity.\n\n\nIn one of the dirtiest and most gloomy streets leading to the Rue Saint\nDenis, in Paris, there stands a tall and ancient house, the lower\nportion of which is a large mercer's shop. This establishment is held to\nbe one of the very best in the neighborhood, and has for many years\nbelonged to an individual on whom we will bestow the name of Ramin.\n\nAbout ten years ago, Monsieur Ramin was a jovial red-faced man of forty,\nwho joked his customers into purchasing his goods, flattered the pretty\n_grisettes_ outrageously, and now and then gave them a Sunday treat at\nthe barrier, as the cheapest way of securing their custom. Some people\nthought him a careless, good-natured fellow, and wondered how, with his\noff-hand ways, he contrived to make money so fast, but those who knew\nhim well saw that he was one of those who \"never lost an opportunity.\"\nOthers declared that Monsieur Ramin's own definition of his character\nwas, that he was a \"_bon enfant_,\" and that \"it was all luck.\" He\nshrugged his shoulders and laughed when people hinted at his deep\nscheming in making, and his skill in taking advantage of Excellent\nOpportunities.\n\nHe was sitting in his gloomy parlor one fine morning in Spring,\nbreakfasting from a dark liquid honored with the name of onion soup,\nglancing at the newspaper, and keeping a vigilant look on the shop\nthrough the open door, when his old servant Catherine suddenly\nobserved:--\n\n\"I suppose you know Monsieur Bonelle has come to live in the vacant\napartment on the fourth floor?\"\n\n\"What!\" exclaimed Monsieur Ramin in a loud key.\n\nCatherine repeated her statement, to which her master listened in total\nsilence.\n\n\"Well!\" he said, at length, in his most careless tones, \"what about the\nold fellow?\" and he once more resumed his triple occupation of reading,\neating, and watching.\n\n\"Why,\" continued Catherine, \"they say he is nearly dying, and that his\nhousekeeper, Marguerite, vowed he could never get up stairs alive. It\ntook two men to carry him up; and when he was at length quiet in bed,\nMarguerite went down to the porter's lodge and sobbed there a whole\nhour, saying, 'Her poor master had the gout, the rheumatics, and a bad\nasthma; that though he had been got up stairs, he would never come down\nagain alive; that if she could only get him to confess his sins and make\nhis will, she would not mind it so much; but that when she spoke of the\nlawyer or the priest, he blasphemed at her like a heathen, and declared\nhe would live to bury her and everybody else.'\"\n\nMonsieur Ramin heard Catherine with great attention, forgot to finish\nhis soup, and remained for five minutes in profound rumination, without\nso much as perceiving two customers who had entered the shop and were\nwaiting to be served. When aroused, he was heard to exclaim:\n\n\"What an excellent opportunity!\"\n\nMonsieur Bonelle had been Ramin's predecessor. The succession of the\nlatter to the shop was a mystery. No one ever knew how it was that this\nyoung and poor assistant managed to replace his patron. Some said that\nhe had detected Monsieur Bonelle in frauds which he threatened to\nexpose, unless the business were given up to him as the price of his\nsilence; others averred that, having drawn a prize in the lottery, he\nhad resolved to set up a fierce opposition over the way, and that\nMonsieur Bonelle, having obtained a hint of his intentions, had thought\nit most prudent to accept the trifling sum his clerk offered, and avoid\na ruinous competition. Some charitable souls--moved no doubt by Monsieur\nBonelle's misfortune--endeavored to console and pump him; but all they\ncould get from him was the bitter exclamation, \"To think I should have\nbeen duped by _him_!\" For Ramin had the art, though then a mere youth,\nto pass himself off on his master as an innocent provincial lad. Those\nwho sought an explanation from the new mercer, were still more\nunsuccessful. \"My good old master,\" he said in his jovial way, \"felt in\nneed of repose, and so I obligingly relieved him of all business and\nbotheration.\"\n\nYears passed away; Ramin prospered, and neither thought nor heard of his\n\"good old master.\" The house, of which he tenanted the lower portion,\nwas offered for sale; he had long coveted it, and had almost concluded\nan agreement with the actual owner, when Monsieur Bonelle unexpectedly\nstepped in at the eleventh hour, and by offering a trifle more secured\nthe bargain. The rage and mortification of Monsieur Ramin were extreme.\nHe could not understand how Bonelle, whom he had thought ruined, had\nscraped up so large a sum; his lease was out, and he now felt himself at\nthe mercy of the man he had so much injured. But either Monsieur Bonelle\nwas free from vindictive feelings, or those feelings did not blind him\nto the expediency of keeping a good tenant; for though he raised the\nrent, until Monsieur Ramin groaned inwardly, he did not refuse to renew\nthe lease. They had met at that period; but never since.\n\n\"Well, Catherine,\" observed Monsieur Ramin to his old servant on the\nfollowing morning, \"how is that good Monsieur Bonelle getting on?\"\n\n\"I dare say you feel very uneasy about him,\" she replied with a sneer.\n\nMonsieur Ramin looked up and frowned.\n\n\"Catherine,\" said he, dryly, \"you will have the goodness, in the first\nplace, not to make impertinent remarks; in the second place, you will\noblige me by going up stairs to inquire after the health of Monsieur\nBonelle, and say that I sent you.\"\n\nCatherine grumbled and obeyed. Her master was in the shop, when she\nreturned in a few minutes, and delivered with evident satisfaction the\nfollowing gracious message:\n\n\"Monsieur Bonelle desires his compliments to you, and declines to state\nhow he is; he will also thank you to attend to your own shop, and not to\ntrouble yourself about his health.\"\n\n\"How does he look?\" asked Monsieur Ramin with the most perfect\ncomposure.\n\n\"I caught a glimpse of him, and he appears to me to be rapidly preparing\nfor the good offices of the undertaker.\"\n\nMonsieur Ramin smiled, rubbed his hands, and joked merrily with a\ndark-eyed grisette, who was cheapening some ribbon for her cap. That\ngirl made an excellent bargain that day.\n\nTowards dusk the mercer left the shop to the care of his attendant, and\nsoftly stole up to the fourth story. In answer to his gentle ring, a\nlittle old woman opened the door, and giving him a rapid look, said\nbriefly,\n\n\"Monsieur is inexorable; he won't see any doctor whatever.\"\n\nShe was going to shut the door in his face, when Ramin quickly\ninterposed, under his breath, with, \"_I_ am not a doctor.\"\n\nShe looked at him from head to foot.\n\n\"Are you a lawyer?\"\n\n\"Nothing of the sort, my good lady.\"\n\n\"Well, then, are you a priest?\"\n\n\"I may almost say, quite the reverse.\"\n\n\"Indeed, you must go away; master sees no one.\"\n\nOnce more she would have shut the door, but Ramin prevented her.\n\n\"My good lady,\" said he, in his most insinuating tones, \"it is true I am\nneither a lawyer, a doctor, nor a priest. I am an old friend, a very old\nfriend of your excellent master; I have come to see good Monsieur\nBonelle in his present affliction.\"\n\nMarguerite did not answer, but allowed him to enter, and closed the door\nbehind him. He was going to pass from the narrow and gloomy ante-chamber\ninto an inner room--whence now proceeded a sound of loud coughing--when\nthe old woman laid her hand on his arm, and raising herself on tip-toe\nto reach his ear, whispered:\n\n\"For Heaven's sake, sir, since you are his friend, do talk to him; do\ntell him to make his will, and hint something about a soul to be saved,\nand all that sort of a thing: do, sir!\"\n\nMonsieur Ramin nodded and winked in a way that said \"I will.\" He proved,\nhowever, his prudence by not speaking aloud, for a voice from within\nsharply exclaimed,\n\n\"Marguerite, you are talking to some one. Marguerite, I will see neither\ndoctor nor lawyer; and if any meddling priest dare--\"\n\n\"It is only an old friend, sir;\" interrupted Marguerite, opening the\ninner door.\n\nHer master, on looking up, perceived the red face of Monsieur Ramin\npeeping over the old woman's shoulder, and irefully cried out,\n\n\"How dare you bring that fellow here? And you, sir, how dare you come?\"\n\n\"My good old friend, there are feelings,\" said Ramin, spreading his\nfingers over the left pocket of his waistcoat,--\"there are feelings,\" he\nrepeated, \"that cannot be subdued. One such feeling brought me here.\nThe fact is, I am a good-natured, easy fellow, and I never bear malice.\nI never forget an old friend, but love to forget old differences when I\nfind one party in affliction.\"\n\nHe drew a chair forward as he spoke, and composedly seated himself\nopposite to his late master.\n\nMonsieur Bonelle was a thin old man, with a pale sharp face, and keen\nfeatures. At first, he eyed his visitor from the depths of his vast\narm-chair; but, as if not satisfied with this distant view, he bent\nforward, and laying both hands on his thin knees, he looked up into\nRamin's face with a fixed and piercing gaze. He had not, however, the\npower of disconcerting his guest.\n\n\"What did you come here for?\" he at length asked.\n\n\"Merely to have the extreme satisfaction of seeing how you are, my good\nold friend. Nothing more.\"\n\n\"Well, look at me--and then go.\"\n\nNothing could be so discouraging; but this was an Excellent Opportunity,\nand when Monsieur Ramin _had_ an excellent opportunity in view, his\npertinacity was invincible. Being now resolved to stay, it was not in\nMonsieur Bonelle's power to banish him. At the same time, he had tact\nenough to render his presence agreeable. He knew that his coarse and\nboisterous wit had often delighted Monsieur Bonelle of old, and he now\nexerted himself so successfully as to betray the old man two or three\ntimes into hearty laughter.\n\n\"Ramin,\" said he, at length, laying his thin hand on the arm of his\nguest, and peering with his keen glance into the mercer's purple face,\n\"you are a funny fellow, but I know you; you cannot make me believe you\nhave called just to see how I am, and to amuse me. Come, be candid for\nonce; what do you want?\"\n\nRamin threw himself back in his chair, and laughed blandly, as much as\nto say, \"_Can_ you suspect me?\"\n\n\"I have no shop now out of which you can wheedle me,\" continued the old\nman; \"and surely you are not such a fool as to come to me for money.\"\n\n\"Money?\" repeated the draper, as if his host had mentioned something he\nnever dreamt of. \"Oh no!\"\n\nRamin saw it would not do to broach the subject he had really come\nabout, too abruptly, now that suspicion seemed so wide awake--_the_\nopportunity had not arrived.\n\n\"There is something up, Ramin, I know; I see it in the twinkle of your\neye; but you can't deceive me again.\"\n\n\"Deceive _you_?\" said the jolly schemer, shaking his head reverentially.\n\"Deceive a man of your penetration and depth? Impossible! The bare\nsupposition is flattery. My dear friend,\" he continued, soothingly, \"I\ndid not dream of such a thing. The fact is, Bonelle, though they call me\na jovial, careless, rattling dog, I have a conscience; and, somehow, I\nhave never felt quite easy about the way in which I became your\nsuccessor downstairs. It _was_ rather sharp practice, I admit.\"\n\nBonelle seemed to relent.\n\n\"Now for it,\" said the Opportunity-hunter to himself--\"By-the-by,\"\n(speaking aloud,) \"this house must be a great trouble to you in your\npresent weak state? Two of your lodgers have lately gone away without\npaying--a great nuisance, especially to an invalid.\"\n\n\"I tell you I'm as sound as a colt.\"\n\n\"At all events, the whole concern must be a great bother to you. If I\nwere you, I would sell the house.\"\n\n\"And if I were _you_,\" returned the landlord, dryly, \"I would buy it--\"\n\n\"Precisely,\" interrupted the tenant, eagerly.\n\n\"That is, if you could get it. Phoo! I knew you were after something.\nWill you give eighty thousand francs for it?\" abruptly asked Monsieur\nBonelle.\n\n\"Eighty thousand francs!\" echoed Ramin. \"Do you take me for Louis\nPhilippe or the Bank of France?\"\n\n\"Then, we'll say no more about it--are you not afraid of leaving your\nshop so long?\"\n\nRamin returned to the charge, heedless of the hint to depart. \"The fact\nis, my good old friend, ready money is not my strong point just now. But\nif you wish very much to be relieved of the concern, what say you to a\nlife annuity? I could manage that.\"\n\nMonsieur Bonelle gave a short, dry, churchyard cough, and looked as if\nhis life were not worth an hour's purchase. \"You think yourself\nimmensely clever, I dare say,\" he said. \"They have persuaded you that I\nam dying. Stuff! I shall bury you yet.\"\n\nThe mercer glanced at the thin fragile frame, and exclaimed to himself,\n\"Deluded old gentleman!\" \"My dear Bonelle,\" he continued, aloud, \"I know\nwell the strength of your admirable constitution; but allow me to\nobserve that you neglect yourself too much. Now, suppose a good sensible\ndoctor--\"\n\n\"Will you pay him?\" interrogated Bonelle sharply.\n\n\"Most willingly,\" replied Ramin, with an eagerness that made the old man\nsmile. \"As to the annuity, since the subject annoys you, we will talk of\nit some other time.\"\n\n\"After you have heard the doctor's report,\" sneered Bonelle.\n\nThe mercer gave him a stealthy glance, which the old man's keen look\nimmediately detected. Neither could repress a smile; these good souls\nunderstood one another perfectly, and Ramin saw that this was not the\nExcellent Opportunity he desired, and departed.\n\nThe next day Ramin sent a neighboring medical man, and heard it was his\nopinion that if Bonelle held on for three months longer, it would be a\nmiracle. Delightful news!\n\nSeveral days elapsed, and although very anxious, Ramin assumed a\ncareless air, and did not call upon his landlord, or take any notice of\nhim. At the end of the week old Marguerite entered the shop to make a\ntrifling purchase.\n\n\"And how are we getting on up-stairs?\" negligently asked Monsieur Ramin.\n\n\"Worse and worse, my good Sir,\" she sighed. \"We have rheumatic pains,\nwhich make us often use expressions the reverse of Christian-like, and\nyet nothing can induce us to see either the lawyer or the priest; the\ngout is getting nearer to our stomach every day, and still we go on\ntalking about the strength of our constitution. Oh, Sir, if you have any\ninfluence with us, do, pray do, tell us how wicked it is to die without\nmaking one's will or confessing one's sins.\"\n\n\"I shall go up this very evening,\" ambiguously replied Monsieur Ramin.\n\nHe kept his promise, and found Monsieur Bonelle in bed, groaning with\npain, and in the worst of tempers.\n\n\"What poisoning doctor did you send?\" he asked, with an ireful glance;\n\"I want no doctor, I am not ill; I will not follow his prescription; he\nforbade me to eat; I _will_ eat.\"\n\n\"He is a very clever man,\" said the visitor. \"He told me that never in\nthe whole course of his experience has he met with what he called so\nmuch 'resisting power' as exists in your frame. He asked me if you were\nnot of a long-lived race.\"\n\n\"That is as people may judge,\" replied Monsieur Bonelle. \"All I can say\nis, that my grandfather died at ninety, and my father at eighty-six.\"\n\n\"The doctor owned that you had a wonderfully strong constitution.\"\n\n\"Who said I hadn't?\" exclaimed the invalid feebly.\n\n\"You may rely on it, you would preserve your health better if you had\nnot the trouble of these vexatious lodgers. Have you thought about the\nlife annuity?\" said Ramin as carelessly as he could, considering how\nnear the matter was to his hopes and wishes.\n\n\"Why, I have scruples,\" returned Bonelle, coughing. \"I do not wish to\ntake you in. My longevity would be the ruin of you.\"\n\n\"To meet that difficulty,\" quickly replied the mercer, \"we can reduce\nthe interest.\"\n\n\"But I must have high interest,\" placidly returned Monsieur Bonelle.\n\nRamin, on hearing this, burst into a loud fit of laughter, called\nMonsieur Bonelle a sly old fox, gave him a poke in the ribs, which made\nthe old man cough for five minutes, and then proposed that they should\ntalk it over some other day. The mercer left Monsieur Bonelle in the act\nof protesting that he felt as strong as a man of forty.\n\nMonsieur Ramin felt in no hurry to conclude the proposed agreement. \"The\nlater one begins to pay, the better,\" he said, as he descended the\nstairs.\n\nDays passed on, and the negotiation made no way. It struck the observant\ntradesman that all was not right. Old Marguerite several times refused\nto admit him, declaring her master was asleep; there was something\nmysterious and forbidding in her manner that seemed to Monsieur Ramin\nvery ominous. At length a sudden thought occurred to him; the\nhousekeeper--wishing to become her master's heir--had heard his scheme\nand opposed it. On the very day that he arrived at this conclusion, he\nmet a lawyer, with whom he had formerly had some transactions, coming\ndown the staircase. The sight sent a chill through the mercer's\ncommercial heart, and a presentiment--one of those presentiments that\nseldom deceive--told him it was too late. He had, however, the fortitude\nto abstain from visiting Monsieur Bonelle until evening came; when he\nwent up, resolved to see him in spite of all Marguerite might urge. The\ndoor was half-open, and the old housekeeper stood talking on the landing\nto a middle-aged man in a dark cassock.\n\n\"It is all over! The old witch has got the priests at him,\" thought\nRamin, inwardly groaning at his own folly in allowing himself to be\nforestalled.\n\n\"You cannot see Monsieur to-night,\" sharply said Marguerite, as he\nattempted to pass her.\n\n\"Alas! is my excellent friend so very ill?\" asked Ramin, in a mournful\ntone.\n\n\"Sir,\" eagerly said the clergyman, catching him by the button of his\ncoat, \"if you are indeed the friend of that unhappy man, do seek to\nbring him into a more suitable frame of mind. I have seen many dying\nmen, but never so much obstinacy, never such infatuated belief in the\nduration of life.\"\n\n\"Then you think he really _is_ dying?\" asked Ramin; and, in spite of the\nmelancholy accent he endeavored to assume, there was something so\npeculiar in his tone, that the priest looked at him very fixedly as he\nslowly replied,\n\n\"Yes, Sir, I think he is.\"\n\n\"Ah!\" was all Monsieur Ramin said; and as the clergyman had now relaxed\nhis hold of the button, Ramin passed in spite of the remonstrances of\nMarguerite, who rushed after the priest. He found Monsieur Bonelle still\nin bed in a towering rage.\n\n\"Oh! Ramin, my friend,\" he groaned, \"never take a housekeeper, and never\nlet her know you have any property. They are harpies, Ramin,--harpies!\nsuch a day as I have had; first, the lawyer, who comes to write down 'my\nlast testamentary dispositions,' as he calls them; then the priest, who\ngently hints that I am a dying man. Oh, what a day!\"\n\n\"And _did_ you make your will, my excellent friend?\" softly asked\nMonsieur Ramin, with a keen look.\n\n\"Make my will?\" indignantly exclaimed the old man; \"make my will? what\ndo you mean, Sir? do you mean to say I am dying?\"\n\n\"Heaven forbid!\" piously ejaculated Ramin.\n\n\"Then why do you ask me if I have been making my will?\" angrily resumed\nthe old man. He then began to be extremely abusive.\n\nWhen money was in the way, Monsieur Ramin, though otherwise of a violent\ntemper, had the meekness of a lamb. He bore the treatment of his host\nwith the meekest patience, and having first locked the door so as to\nmake sure that Marguerite would not interrupt them, he watched Monsieur\nBonelle attentively, and satisfied himself that the Excellent\nOpportunity he had been ardently longing for had arrived. \"He is going\nfast,\" he thought; \"and unless I settle the agreement to-night, and get\nit drawn up and signed to-morrow, it will be too late.\"\n\n\"My dear friend,\" he at length said aloud, on perceiving that the old\ngentleman had fairly exhausted himself and was lying panting on his\nback, \"you are indeed a lamentable instance of the lengths to which the\ngreedy lust of lucre will carry our poor human nature. It is really\ndistressing to see Marguerite, a faithful, attached servant, suddenly\nconverted into a tormenting harpy by the prospect of a legacy! Lawyers\nand priests flock around you like birds of prey, drawn hither by the\nscent of gold! Oh, the miseries of having delicate health combined with\na sound constitution and large property!\"\n\n\"Ramin,\" groaned the old man, looking inquiringly into his visitor's\nface, \"you are again going to talk to me about that annuity--I know you\nare!\"\n\n\"My excellent friend, it is merely to deliver you from a painful\nposition.\"\n\n\"I am sure, Ramin, you think in your soul I am dying,\" whimpered\nMonsieur Bonelle.\n\n\"Absurd, my dear Sir. Dying? I will prove to you that you have never\nbeen in better health. In the first place you feel no pain.\"\n\n\"Excepting from rheumatism,\" groaned Monsieur Bonelle.\n\n\"Rheumatism! who ever died of rheumatism? and if that be all--\"\n\n\"No, it is not all,\" interrupted the old man with great irritability;\n\"what would you say to the gout getting higher and higher up every\nday?\"\n\n\"The gout is rather disagreeable, but if there is nothing else--\"\n\n\"Yes, there is something else,\" sharply said Monsieur Bonelle. \"There is\nan asthma that will scarcely let me breathe, and a racking pain in my\nhead that does not allow me a moment's ease. But if you think I am\ndying, Ramin, you are quite mistaken.\"\n\n\"No doubt, my dear friend, no doubt; but in the meanwhile, suppose we\ntalk of this annuity. Shall we say one thousand francs a year.\"\n\n\"What?\" asked Bonelle, looking at him very fixedly.\n\n\"My dear friend, I mistook; I meant two thousand francs per annum,\"\nhurriedly rejoined Ramin.\n\nMonsieur Bonelle closed his eyes, and appeared to fall into a gentle\nslumber. The mercer coughed; the sick man never moved.\n\n\"Monsieur Bonelle.\"\n\nNo reply.\n\n\"My excellent friend.\"\n\nUtter silence.\n\n\"Are you asleep?\"\n\nA long pause.\n\n\"Well, then, what do you say to three thousand?\"\n\nMonsieur Bonelle opened his eyes.\n\n\"Ramin,\" said he, sententiously, \"you are a fool; the house brings me in\nfour thousand as it is.\"\n\nThis was quite false, and the mercer knew it; but he had his own reasons\nfor wishing to seem to believe it true.\n\n\"Good Heavens!\" said he, with an air of great innocence, \"who could have\nthought it, and the lodgers constantly running away. Four thousand?\nWell, then, you shall have four thousand.\"\n\nMonsieur Bonelle shut his eyes once more, and murmured \"The mere\nrental--nonsense!\" He then folded his hands on his breast, and appeared\nto compose himself to sleep.\n\n\"Oh, what a sharp man of business he is!\" Ramin said, admiringly; but\nfor once omnipotent flattery failed in its effect; \"So acute!\" continued\nhe, with a stealthy glance at the old man, who remained perfectly\nunmoved. \"I see you will insist upon making it the other five hundred\nfrancs.\"\n\nMonsieur Ramin said this as if five thousand five hundred francs had\nalready been mentioned, and was the very summit of Monsieur Bonelle's\nambition. But the ruse failed in its effect; the sick man never so much\nas stirred.\n\n\"But, my dear friend,\" urged Monsieur Ramin in a tone of feeling\nremonstrance, \"there is such a thing as being too sharp, too acute. How\ncan you expect that I shall give you more when your constitution is so\ngood, and you are to be such a long liver?\"\n\n\"Yes, but I may be carried off one of those days,\" quietly observed the\nold man, evidently wishing to turn the chance of his own death to\naccount.\n\n\"Indeed, and I hope so,\" muttered the mercer, who was getting very\nill-tempered.\n\n\"You see,\" soothingly continued Bonelle, \"you are so good a man of\nbusiness, Ramin, that you will double the actual value of the house in\nno time. I am a quiet, easy person, indifferent to money; otherwise this\nhouse would now bring me in eight thousand at the very least.\"\n\n\"Eight thousand!\" indignantly exclaimed the mercer. \"Monsieur Bonelle,\nyou have no conscience. Come now, my dear friend, do be reasonable. Six\nthousand francs a-year (I don't mind saying six) is really a very\nhandsome income for a man of your quiet habits. Come, be reasonable.\"\nBut Monsieur Bonelle turned a deaf ear to reason, and closed his eyes\nonce more. What between opening and shutting them for the next quarter\nof an hour, he at length induced Monsieur Ramin to offer him seven\nthousand francs.\n\n\"Very well, Ramin, agreed,\" he quietly said; \"you have made an\nunconscionable bargain.\" To this succeeded a violent fit of coughing.\n\nAs Ramin unlocked the door to leave, he found old Marguerite, who had\nbeen listening all the time, ready to assail him with a torrent of\nwhispered abuse for duping her \"poor dear innocent old master into such\na bargain.\" The mercer bore it all very patiently; he could make\nallowances for her excited feelings, and only rubbed his hands and bade\nher a jovial good evening.\n\nThe agreement was signed on the following day, to the indignation of old\nMarguerite, and the mutual satisfaction of the parties concerned.\n\nEvery one admired the luck and shrewdness of Ramin, for the old man\nevery day was reported worse; and it was clear to all that the first\nquarter of the annuity would never be paid. Marguerite, in her wrath,\ntold the story as a grievance to every one; people listened, shook their\nheads, and pronounced Monsieur Ramin to be a deuced clever fellow.\n\nA month elapsed. As Ramin was coming down one morning from the attics,\nwhere he had been giving notice to a poor widow who had failed in paying\nher rent, he heard a light step on the stairs. Presently a sprightly\ngentleman, in buoyant health and spirits, wearing the form of Monsieur\nBonelle, appeared. Ramin stood aghast.\n\n\"Well, Ramin,\" gaily said the old man, \"how are you getting on? Have you\nbeen tormenting the poor widow up stairs? Why, man, we must live and let\nlive!\"\n\n\"Monsieur Bonelle,\" said the mercer, in a hollow tone, \"may I ask where\nare your rheumatics?\"\n\n\"Gone, my dear friend,--gone.\"\n\n\"And the gout that was creeping higher and higher every day,\" exclaimed\nMonsieur Ramin, in a voice of anguish.\n\n\"It went lower and lower, till it disappeared altogether,\" composedly\nreplied Bonelle.\n\n\"And your asthma----\"\n\n\"The asthma remains, but asthmatic people are proverbially long-lived.\nIt is, I have been told, the only complaint that Methuselah was troubled\nwith.\" With this, Bonelle opened his door, shut it, and disappeared.\n\nRamin was transfixed on the stairs; petrified with intense\ndisappointment, and a powerful sense of having been duped. When\ndiscovered, he stared vacantly, and raved about an Excellent Opportunity\nof taking his revenge.\n\nThe wonderful cure was the talk of the neighborhood, whenever Monsieur\nBonelle appeared in the streets, jauntily flourishing his cane. In the\nfirst frenzy of his despair, Ramin refused to pay; he accused every one\nof having been in a plot to deceive him; he turned off Catherine and\nexpelled his porter; he publicly accused the lawyer and priest of\nconspiracy; brought an action against the doctor, and lost it. He had\nanother brought against him for violently assaulting Marguerite, in\nwhich he was cast in heavy damages. Monsieur Bonelle did not trouble\nhimself with useless remonstrances, but, when his annuity was refused,\nemployed such good legal arguments as the exasperated mercer could not\npossibly resist.\n\nTen years have elapsed, and MM. Ramin and Bonelle still live on. For a\nhouse which would have been dear at fifty thousand francs, the draper\nhas already handed over seventy thousand.\n\nThe once red-faced, jovial Ramin, is now a pale, haggard man, of sour\ntemper and aspect. To add to his anguish, he sees the old man thrive on\nthat money which it breaks his heart to give. Old Marguerite takes a\nmalicious pleasure in giving him an exact account of their good cheer,\nand in asking him if he does not think Monsieur looks better and better\nevery day. Of one part of this torment Ramin might get rid, by giving\nhis old master notice to quit, and no longer having him in his house.\nBut this he cannot do; he has a secret fear that Bonelle would take some\nExcellent Opportunity of dying without his knowledge, and giving some\nother person an Excellent Opportunity of personating him, and receiving\nthe money in his stead.\n\nThe last accounts of the victim of Excellent Opportunities represent him\nas being gradually worn down with disappointment. There seems every\nprobability of his being the first to leave the world; for Bonelle is\nheartier than ever.\n\n                               THE END.\n\n                *       *       *       *       *\n\n              =Popular Work! Twelfth Thousand Now Ready!=\n\n                      LEWIE, OR THE BENDED TWIG.\n\n                           BY COUSIN CICELY,\n\n              Author of \"Silver Lake Stories,\" etc., etc.\n\n              =One Volume 12mo.,            Price $1.00.=\n\n  ALDEN, BEARDSLEY & CO., AUBURN, N. Y.,       }\n  WANZER, BEARDSLEY & CO., ROCHESTER, N. Y.,   } _Publishers_.\n\n    \"Mother! thy gentle hand hath mighty power,\n    For thou alone may'st train, and guide, and mould\n    Plants that shall blossom, with an odor sweet,\n    Or, like the cursed fig-tree, wither, and become\n    Vile cumberers of the ground.\"\n\nBrief Extracts from Notices of the Press.\n\n* * * A tale which deserves to rank with \"The Wide, Wide World.\" It is\nwritten with graphic power, and full of interest.--_Hartford Repub._\n\n* * * Her writings are equal to the best. She is a second Fanny\nFern.--_Palmyra Democrat._\n\n* * * It is recommended by its excellent moral tone and its wholesome\npractical inculcations.--_N. Y. Tribune._\n\n* * * Full of grace and charm, its style and vivacity make it a most\namusing work. For the intellectual and thinking, it has a deeper lesson,\nand while it thrills the heart, bids parents beware of that weakness\nwhich prepares in infancy the misery of man. \"Lewie\" is one of the most\npopular books now before the public, and needs no puffing, as it is\nselling by thousands.--_N. Y. Day Book._\n\n* * * The moral of the book is inestimable. The writer cannot fail to be\ngood, as she so faithfully portrays the evils which owe their origin to\nthe criminal neglect of proper parental discipline.--_Hunt's Merchants'\nMagazine._\n\n* * * The plot is full of dramatic interest, yet entirely free from\nextravagance; the incidents grow out of the main plot easily and\nnaturally, while the sentiment is healthy and unaffected. Commend us to\nmore writers like Cousin Cicely--books which we can see in the hands of\nour young people without uneasiness. Books which interest by picturing\nlife as it is, instead of giving us galvanized society.--_National\nDemocrat._\n\n* * * A touching and impressive story, unaffected in style and effective\nin plot.--_N. Y. Evangelist._\n\n* * * The story of the Governess, contained in this volume, is one of\nrare interest.--_Highland Eagle._\n\n* * * The story is a charming one--the most affecting we ever\nread.--_Jersey Shore Republican._\n\n* * * \"Cousin Cicely\" is just the person to portray family scenes.\n\n* * * This story will be profitable reading.--_Daily Capital City Fact,\nColumbus, Ohio._\n\n* * * The contents of the work are of the first order, and\nunexceptionable.--_Hartford Daily Times._\n\n       *       *       *       *       *\n\nFOOTNOTES:\n\n [1] Report of Mr. Bowie on the cause of Cholera in Bermondsey.\n\n [2] Report of Dr. Gavin.\n\n [3] Evidence of Mr. J. T. Cooper, Practical Chemist.\n\n [4] Evidence before the Board of Health.\n\n       *       *       *       *       *\n\nTypographical errors corrected by the etext transcriber:\n\ngave him little encouagement=> gave him little encouagement {pg 11}\n\nwhere an elegaut _d\u00e9jeuner_=> where an elegant _d\u00e9jeuner_ {pg 20}\n\nwo had sprung=> woe had sprung {pg 27}\n\nagaint the barons=> against the barons {pg 35}\n\nThs spirit of a policeman=> The spirit of a policeman {pg 62}\n\nthree feet together anwhere=> three feet together anywhere {pg 207}\n\nNepho now lurked=> Nephelo now lurked {pg 321}\n\ncried Nepho=> cried Nephelo {pg 322}\n\nyou are are not such a fool=> you are not such a fool {pg 334}\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Pearl-Fishing; Choice Stories from\nDickens' Household Words; Second S, by Charles Dickens\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by hekula03, David E. Brown, 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\n  _Chap-Books\n  and\n  Folk-Lore Tracts._\n\n  _Edited by\n  G. L. Gomme, F.S.A.\n  and\n  H. B. Wheatley, F.S.A._\n\n  _First Series._\n\n  I.\n\n\n\n\n  THE HISTORY\n  OF\n  THOMAS HICKATHRIFT.\n\n\n  PRINTED FROM\n  THE EARLIEST EXTANT COPIES,\n\n  AND EDITED, WITH AN INTRODUCTION,\n  BY\n  GEORGE LAURENCE GOMME, F.S.A.\n\n\n  LONDON:\n  PRINTED FOR THE VILLON SOCIETY.\n\n  1885.\n\n\n\n\nIntroduction.\n\n\nThere seems to be some considerable reason for believing that the hero\nof this story was a reality. The story tells us that he lived in the\nmarsh of the Isle of Ely, and that he became \"a brewer's man\" at Lyn,\nand traded to Wisbeach. This little piece of geographical evidence\nenables us to fix the story as belonging to the great Fen District,\nwhich occupied the north of Cambridgeshire and Norfolk.\n\nThe antiquary Thomas Hearne has gone so far as to identify the hero\nof tradition with a doughty knight of the Crusaders. Writing in the\n_Quarterly Review_ (vol. xxi. p. 102), Sir Francis Palgrave says:--\n\n\"Mr. Thomas Hickathrift, afterwards Sir Thomas Hickathrift, Knight,\nis praised by Mr. Thomas Hearne as a 'famous champion.' The honest\nantiquary has identified this well-known knight with the far less\ncelebrated Sir Frederick de Tylney, Baron of Tylney in Norfolk, the\nancestor of the Tylney family, who was killed at Acon, in Syria, in\nthe reign of Richard Coeur de Lion. Hycophric, or Hycothrift, as the\nmister-wight observes, being probably a corruption of Frederick. This\nhappy exertion of etymological acumen is not wholly due to Hearne, who\nonly adopted a hint given by Mr. Philip le Neve, whilome of the College\nof Arms.\"\n\nThere does not seem to be the slightest evidence for Hearne's\nidentification any more than there is for his philological conclusions,\nand we may pass over this for other and more reliable information.\n\nWe must first of all turn to the story itself, as it has come down\nto us in its chapbook form. It is divided into two parts. The first\npart of the story is the earliest; the second part being evidently a\nprinter's or a chapman's addition. Our reprint of the former is taken\nfrom the copy in the Pepysian Library at Magdalene College, Cambridge,\nand which was printed probably about 1660-1690; the latter is taken\nfrom the British Museum copy, the date of which, according to the\nMuseum authorities, is 1780.\n\nIn trying to ascertain something as to the date of the story apart from\nthat of its printed version, it will therefore be necessary to put out\nof consideration the second portion. This has been written by some\none well acquainted with the original first part, and with the spirit\nof the story; but in spite of this there is undoubted evidence of its\nliterary origin at a date later than the first part. But turning to the\nfirst part there are two expressions in this early Pepysian version\nwhich have not been repeated in the later editions--those of the\neighteenth century; and these two expressions appear to me to indicate\na date _after which_ the story could not have been originated. On\npage 1 we read that Tom Hickathrift dwelt \"in the _marsh_ of the Isle\nof Ely.\" In the earliest British Museum copy this appears as \"in the\n_parish_ of the Isle of Ely.\" Again, on page 11 Tom is described as\nlaying out the giant's estate, \"some of which he gave to the poor\nfor their common, and the rest he made pastures of and divided the\nmost part into _good ground_, to maintain him and his old mother Jane\nHickathrift.\" In the earliest British Museum copy the expression\n\"good ground\" is displaced by \"tillage.\" Now it is clear from these\ncurious transposition of words in the earliest and latest editions that\nsomething had been going on to change the nature of the country. The\neighteenth-century people did not know the \"marsh\" of Ely, so they read\n\"parish\": they did not know the meaning of \"good ground\" so they read\n\"tillage.\" And hence it is clear that at the printing of this earliest\nversion the fen lands of Cambridge and Norfolk had not yet been\ndrained; there was still \"marsh land\" which was being made into \"good\nland.\"\n\nBut I think there is evidence in this printed chap-book version of the\nstory which tells us that it was taken from a traditional version. Let\nany one take the trouble to read aloud the first part, and he will at\nonce perceive that there is a ring and a cadence given to the voice by\nthe wording of the story, and particularly by the curious punctuation,\nwhich at once reminds us of a narrative from word of mouth. And besides\nthis there is some little evidence of phonetic spelling, just such as\nmight have been expected from the first printer taking the story from\nthe lips of one of the Fen-country peasantry.\n\nNow this internal evidence of the once _viva-voce_ existence of\nthe printed legend of Tom Hickathrift has a direct bearing upon\nthe question as to the date of the earliest printed version. The\ncolloquialisms are so few, and the rhythm, though marked and definite,\nis occasionally so halting and approaches so nearly a literary form,\nthat we are forced to observe that the earliest printed edition now\nknown is certainly not the earliest version printed. There are too\nfew phoneticisms and dialect words to make it probable that the print\nin the Pepysian collection is the one directly derived from popular\ntradition. As the various printers in the eighteenth century altered\nwords and sentences here and there, as different editions were issued,\nso did the seventeenth-century printers; and therefore it is necessary\nto push the date of the printed version farther back than we can hope\nto ascertain by direct evidence. There is no reason why there should\nnot have been a sixteenth-century printed version, and to this period I\nam inclined to allocate the earliest appearance of the story in print.\n\nAnd then prior to the printed version was the popular version with\nits almost endless life, perhaps reaching back to that vague period\nindicated in the opening words of the story, \"in the reign before\nWilliam the Conqueror.\" Already internal evidence has, it is suggested,\npointed to a popular unwritten tradition of Tom Hickathrift's life\nand exploits. But we must ask now, Is there, or was there, any\ntradition among the peasantry of Lyn and its neighbourhood about Thomas\nHickathrift? And, if so, how far does this popular tradition reach\nback, and how far does it tally with the chap-book version? Again,\nis this popular tradition independent of the chap-book story, or has\nit been generated from the printed book? To answer these questions\nproperly we must closely examine all the evidence available as to the\nexistence and form of this popular tradition.\n\nTurning first of all to the historian of Norfolk, Blomefield,[A]\nwriting in 1808, gives us the following account:--\n\n\"The town of Tilney gives name to a famous common called Tilney Smeeth,\nwhereon 30,000 or more large Marshland sheep and the great cattle of\nseven towns to which it belongs are constantly said to feed. Of this\nplain of Smeeth there is a tradition, _which the common people retain_,\nthat in old time the inhabitants of these towns [Tilney, Terrington,\nClenchwarton, Islington, Walpole, West Walton, Walsoken, and Emneth]\nhad a contest with the lords of the manors about the bounds and limits\nof it, when one _Hickifric_, a person of great stature and courage,\nassisting the said inhabitants in their rights of common, took an\naxle-tree from a cart-wheel, instead of a sword, and the wheel for a\nshield or buckler, and thus armed soon repelled the invaders. And for\nproof of this notable exploit they to this day show, says Sir William\nDugdale [Dugd. _Hist. of Imbanking_, &c. p. 244; Weever's _Fun. Mon._\np. 866], a large grave-stone near the east end of the chancel in\nTilney churchyard, whereon the form of a cross is so cut or carved as\nthat the upper part thereof (wherewith the carver hath adorned it)\nbeing circular, they will therefore needs have it to be the gravestone\nof _Hickifric_, and to be as a memorial of his gallantry. The stone\ncoffin, which stands now out of the ground in Tilney churchyard, on\nthe north side of the church, will not receive a person above six\nfeet in length, and this is shown as belonging formerly to the giant\n_Hickifric_. The cross said to be a representation of the cart-wheel\nis a cross patt\u00e9e, on the summit of a staff, which staff is styled an\naxle-tree. Such crosses patt\u00e9e on the head of a staff were emblems or\ntokens that some Knight Templar was therein interred, and many such are\nto be seen at this day in old churches.\"\n\nNow the reference to Sir William Dugdale is misleading, because, as\nwill be seen by the following quotation, the position of the hero is\naltered in Dugdale's version of the legend from that of a popular\nleader to the tyrant lord himself:--\"Of this plain I may not omit\na tradition which the common people thereabouts have, viz., that in\nold time the inhabitants of the neighbouring villages had a fierce\ncontest with one Hickifric (the then owner of it) touching the bounds\nthereof, which grew so hot that at length it came to blows; and that\nHickifric, being a person of extraordinary stature and courage, took an\naxletree from a cart instead of a sword, and the wheel for his buckler,\nand, being so armed, most stoutly repelled those bold invaders: for\nfurther testimony of which notable exploit they to this day show a\nlarge gravestone near the east end of the chancel in Tilney churchyard,\nwhereupon the form of a cross is so cut as that the upper part thereof\nby reason of the flourishes (wherewith the carver hath adorned it)\nsheweth to be somewhat circular, which they will, therefore, needs have\nto be the wheel and the shaft the axletree.\" This version, taken from\nDugdale's _History of Imbanking_, 1772, p. 244, though differing in\nform, at all events serves to carry us back to 1662, the date when Sir\nWilliam Dugdale's _History_ was first published.\n\nBut the local tradition can be carried further back than 1662,\nbecause the learned Sir Henry Spelman, in his _Icenia sive Norfolciae\nDescriptio Topographica_, p. 138, and written about 1640, says, when\nspeaking of Tilney, in Marshland Hundred: \"Hic se expandit insignis\narea qu\u00e6 \u00e0 planicie nuncupatur Tylney-smelth, pinguis adeo et luxurians\nut Paduana pascua videatur superasse.... Tuentur eam indigen\u00e6 velut\nAras et Focos, fabellamque recitant longa petitam vetustate de\nHikifrico (nescio quo) Haii illius instar in Scotorum Chronicis, qui\nCivium suorum dedignatus fuga, Aratrum quod agebat, solvit; arreptoque\nTemone furibundus insiliit in hostes victoriamque ademit exultantibus.\nSic cum de agri istius finibus acriter olim dimicatum esset inter fundi\nDominun et Villarum Incolas, nec valerent hi ad versus eum consistere;\nredeuntibus occurrit Hikifricus, axemque excutiens a curru quem agebat,\neo vice Gladii usus; Rot\u00e2, Clypei; invasores repulit ad ipsos quibus\nnunc funguntur terminos. Ostendunt in c\u00e6meterio Tilniensi, Sepulcrum\nsui pugilis, Axem cum Rota insculptum exhibens.\"\n\nA still earlier version is to be found recorded by Weever in 1631. The\nfull quotation is as follows: \"Tylney Smeeth, so called of a smooth\nplaine or common thereunto adioyning.... In the Churchyard is a ridg'd\nAltar, Tombe, or sepulchre of a wondrous antique fashion, vpon which\nan axell-tree and a cart wheele are insculped. Vnder this Funerall\nMonument the Towne dwellers say that one Hikifricke lies interred; of\nwhom (_as it hath gone by tradition from father to the sonne_) they\nthus likewise report: How that vpon a time (no man knowes how long\nsince) there happened a great quarrell betwixt the Lord of this land or\nground and the inhabitants of the foresaid seuen villages, about the\nmeere-marks, limits, or bondaries of this fruitfull feeding place; the\nmatter came to a battell or skirmish, in which the said Inhabitants\nbeing not able to resist the landlord and his forces began to giue\nbacke; Hikifricke, driuing his cart along and perceiuing that his\nneighbours were fainthearted, and ready to take flight, he shooke the\nAxell tree from the cart which he vsed instead of a sword, and tooke\none of the cart-wheeles which he held as a buckler; with these weapons\nhe set vpon the Common aduersaries or aduersaries of the Common,\nencouraged his neighbours to go forward, and fight valiantly in defence\nof their liberties; who being animated by his manly prowesse, they\ntooke heart to grasse, as the prouerbe is, insomuch that they chased\nthe Landlord and his companie to the vtmost verge of the said Common;\nwhich from that time they haue quietly enioyed to this very day. The\nAxell-tree and cart-wheele are cut and figured in diuers places of\nthe Church and Church windowes, which makes the story, you must needs\nsay, more probable. This relation doth in many parts parallell with\nthat of one Hay, a strong braue spirited Scottish Plowman, who vpon\na set battell of Scots against the Danes, being working at the same\ntime in the next field, and seeing some of his countreymen to flie\nfrom that hote encounter, caught vp an oxe yoke (Bo\u00ebthius saith, a\nPlough-beame), with which (after some exhortation that they should\nnot bee faint-hearted) he beate the said straglers backe againe to\nthe maine Army, where he with his two sonnes (who tooke likewise such\nweapons as came next to their hands) renewed the charge so furiously\nthat they quite discomfited the enemy, obtaining the glory of the day\nand victory for their drad Lord and Soueraigne Kenneth the third,\nKing of Scotland; and this happened in the yeare 942, the second of\nthe King's raigne. This you may reade at large in the _History of\nScotland_, thus abridged by Camden as followeth.\"--Weever's _Funerall\nMonuments_, 1631, pp. 866-867.\n\nAnd Sir Francis Palgrave, quoting the legend from Spelman,\nobserves,--\"From the most remote antiquity the fables and achievements\nof Hickifric have been obstinately credited by the inhabitants of the\ntownship of Tylney. Hickifric is venerated by them as the assertor of\nthe rights and liberties of their ancestors. The 'monstrous giant'\nwho guarded the marsh was in truth no other than the tyrannical lord\nof the manor who attempted to keep his copyholders out of the common\nfield, Tylney Smeeth; but who was driven away with his retainers by the\nprowess of Tom armed only with his axletree and cart-wheel.\"[B] This\ndoes not appear to me to put the case too strongly. A tradition told\nso readily and believed so generally in the middle of the seventeenth\ncentury must have had a strong vitality in it only to be obtained by\nage.\n\nLet us now turn to the other side, namely, the existence of a\ntraditional version in modern days, because it is important to note\nthat the printing of a chapbook version need not have disturbed the\nfull current of traditional thought. In a note Sir Francis Palgrave\nseems to imply that the story was still extant without the aid of\nprinted literature. He writes:\n\n\"A Norfolk antiquary has had the goodness to procure for us an\nauthentic report of the present state of Tom's sepulchre. It is a\nstone soros, of the usual shape and dimensions; the sculptured lid or\ncover no longer exists. It must have been entire about fifty years ago,\nfor when we were good _Gaffer Crane would rehearse Tom's achievements_,\nand tell us that he had cut out the moss which filled up the\ninscription with his penknife, but he could not read the letters.\"[C]\n\nAnd Clare, in his _Village Minstrel_, tells us that:--\n\n  \"Here Lubin listen'd with awestruck surprise,\n  When Hickathrift's great strength has met his ear;\n  How he kill'd giants as they were but flies,\n  And lifted trees as one would a spear,\n  Though not much bigger than his fellows were;\n  He knew no troubles waggoners have known,\n  Of getting stall'd and such disasters drear;\n  Up he'd chuck sacks as we would hurl a stone,\n  And draw whole loads of grain unaided and alone.\"\n\nAnd this view as to the existence still of a traditional form of the\nstory is almost borne out by what the country people only recently\nhad to say relative to a monument in that part of the country over\nwhich Sir William Dugdale travelled, and of which he has left us such\na valuable memorial in his _History of Imbanking_. A writer in the\nJournal of the Archaeological Association (vol. xxv. p. 11) says:--\"A\nmound close to the Smeeth Road Station, between Lynn and Wisbech, is\ncalled the Giant's Grave, and the inhabitants relate that there lie the\nremains of the great giant slain by Hickathrift with the cart wheel\nand axletree. A cross was erected upon it, and is to be seen in the\nneighbouring churchyard of Torrington St. John's, bearing the singular\nname of Hickathrift's Candlestick.\"\n\nIt appears, then, that the following may be considered the chief\nevidence which we have obtained about the existence of the story:--\n\n  That a chapbook or literary form of the story has existed from the\n  sixteenth century;\n\n  That a traditional story existed quite independently of the literary\n  story in the seventeenth century;\n\n  That a traditional story exists at the present time, or until very\n  recently;\n\nAnd knowing what folk-lore has to say about the long life of\ntraditions, about their constant repetition age after age, it is not, I\nventure to think, too much to conclude that a story which can be shown\nby evidence to have lived on from mouth to mouth for two centuries\nis capable of going back to an almost endless antiquity for its true\noriginal.\n\nLet us now consider what may be the origin of this story. There is\none theory as to this which has gained the authority of Sir Francis\nPalgrave. The pranks which Tom performed \"must be noticed,\" says Sir\nFrancis, \"as being correctly Scandinavian.\" He then goes on to say,\n\"Similar were the achievements of the great Northern champion Grettir,\nwhen he kept geese upon the common, as told in his Saga. Tom's youth\nretraces the tales of the prowess of the youthful Siegfried detailed\nin the Niblunga Saga and in the book of Heroes. It appears from Hearne\nthat the supposed axle-tree, with the superincumbent wheel, was\nrepresented on 'Hycothrift's' grave-stone in Tylney churchyard in the\nshape of a cross. This is the form in which all the Runic monuments\nrepresent the celebrated hammer or thunderbolt of the son of Odin,\nwhich shattered the skulls and scattered the brains of so many luckless\ngiants. How far this surmise may be supported by Tom's skill and\nstrength in throwing the hammer we will not pretend to decide.\"[D]\n\nNow this takes the story entirely out of the simple category of local\nEnglish tradition, and places it at once among those grand mythic tales\nwhich belong to the study of comparative mythology and which take us\nback to the earliest of man's thought and belief. In order to test this\ntheory let us have before us the passages in Tom Hickathrift's history\nwhich might be said to bear it out, and then let us compare them with\nthe stories of Grettir.\n\nThe analysis of the story based upon the plan laid down by the\nFolk-Lore Society is as follows:--\n\n  (1.) Tom's parents are nobodies, \"a poor man and day labourer\" being\n  his father.\n\n  (2.) Tom was obstinate as a boy.\n\n  (3.) Loses his father, and at first does not help his mother, but\n  sits in the chimney corner.\n\n  (4.) Is of great height and size.\n\n  (5.) Strength is unknown until he shows it.\n\n  (6.) Commits many pranks, among which is the throwing \"a hammer five\n  or six furlongs off into a river.\"\n\n  (7.) Kills a giant with a club, Tom using axletree and wheel for his\n  shield and buckler.\n\n  (8.) Takes possession of the giant's territory and lives there.\n\n  (9.) Commits more pranks, \"kicks a football right away.\"\n\n  (10.) Escapes from four thieves and despoils them.\n\n  (11.) Is defeated by a tinker.\n\nIt will not be necessary to analyse the whole of the stories to which\nwe are referred for the mythic parallels of Tom Hickathrift; but I will\ntake out the items corresponding to those tabulated above. In the story\nof \"Grettir the Strong\" we have the following incidents:--\n\n  (1.) Grettir's father \"had his homestead and farm land.\"\n\n  (2.) Grettir was obstinate as a boy (does nothing on board ship.)\n\n  (3.) Plays pranks upon his father, and returns from attending the\n  horses to the fire-side (Iceland).\n\n  (4.) Is short, though strong, and big of body.\n\n  (5.) He had not skill to turn his great strength to account.\n\n  (6.) He wrestles with other lads, and commits many pranks, flings a\n  rock from its place.\n\n  (7.) Wrestles with Karr, the barrow dweller; and\n\n  (8.) Takes possession of Karr's weapons and wealth.\n\n  (9.) Fights with and conquers robbers.\n\nNow it cannot be denied that there is a great similarity in the thread\nof these two stories. Norfolk, the colony of the Northmen of old, may\nwell have retained its ancient tradition until the moving incidents\nof English economic history brought about the weaving of it into the\nactual life that was pressing round men's thoughts. It would thus leave\nout the great mass of detail in the old northern tradition, and retain\njust sufficient to fit in with the new requirements; and in this way it\nappears to me we have the present form of the story of Tom Hickathrift,\nits ancient Scandinavian outline, its more modern English application.\nNow it is curious to note that the cart-wheel plays a not unimportant\npart in English folk-lore as a representative of old runic faith. Sir\nHenry Ellis, in his edition of Brand's _Popular Antiquities_ (vol. i.\np. 298), has collected together some instances of this; and whatever\ncauses may have led to this survival there is nothing to prevent us\nfrom looking upon the wheel and axle in the story of Tom Hickathrift as\na part and parcel of the same survival.\n\nThere now remains to notice one or two points of interest outside the\nnarrative of the story itself. Of curious expressions we have--\n\n  fitted (p. 3), to pay any one out, to revenge one's self;\n\n  buttle of straw (p. 3);\n\n  shift (p. 3), to support, to make shift. See Davies's _Supplementary\n  Glossary_, _sub voce_ \"make-shift,\" \"shiftful\";\n\n  bone-fires (p. 11). See Ellis's _Brand's Popular Antiquities_, vol.\n  i. p. 300, note;\n\n  cocksure (p. 14), quite sure.\n\nOf proverbs there are--\n\n  to win the horse or lose the saddle (p. 8);\n\n  to make hay while the sun did shine (p. 10).\n\nOf games there are mentioned--\n\n  cudgells (p. 4);\n\n  wrestling (p. 4);\n\n  throwing the hammer (p. 4);\n\n  football (p. 13);\n\n  bear-baiting (p. 13).\n\nIt will be observed that the spelling of the name in the Pepysian copy\nis specially divided thus--Hic-ka-thrift; and though it seems probable\nthat some good reason must be assigned to this, I cannot find out\npoints of importance. But about the dubbing him Mr. (p. 7) or Master,\nas it would be in full, there is something of great interest to point\nout. This was formerly a distinct title. In Harrison's _Description\nof England_ we read, \"Who soeuer studieth the lawes of the realme,\nwho so abideth in the vniuersitie, or prefesseth physicke and the\nliberall sciences, or beside his seruice in the roome of a capteine\nin the warres can liue without manuell labour, and thereto is able\nand will beare the post, charge, and countenance of a gentleman, he\nshall be called master, which is the title that men giue to esquiers\nand gentlemen and reputed for gentlemen.\"--Harrison's _Description\nof England_, 1577 (edited by F. J. Furnivall for the New Shakspere\nSociety, 1877), p. 129.\n\nOf yeomen he says, \"And albeit they be not called master as gentlemen\nare, or sir as to knights apperteineth, but onelie John and Thomas,\"\n&c. (p. 134): and of \"the third and last sort,\" \"named the yeomanrie,\"\nhe adds, \"that they be not called masters and gentlemen, but goodmen,\nas goodman Smith, goodman Coot, goodman Cornell, goodman Mascall,\ngoodman Cockswet,\" &c. (p. 137).\n\nMr. Furnivall's note (p. 123) is as follows:--\"_Every Begger almost is\ncalled Maister._--See Lancelot's 'MAISTER Launcelet' in the _Merchant\nof Venice_, II. ii. 51, and the extract illustrating it from Sir Thomas\nSmith's _Commonwealth of England_, bk. I. ch. 20 (founded on Harrison,\ni. 133, 137), which I printed in _New Sh. Soc.'s Trans._ 1877-9, p.\n103-4. Also Shakspere getting his 'yeoman' father arms, and making him\na 'gentleman' in 1596.--(Leopold Shakspere, Introduction, p. ciii.).\"\nWe thus get still further indication of the early date of the story,\nthe significance of the title \"Master\" having died out during the\nseventeenth century.\n\nThe following is a bibliographical list of some of the editions, many\nothers having been printed from the beginning of this century:--\n\n  (1.) The history of Thomas Hickathrift. Printed for the booksellers.\n  London [1790.] 12mo. pp. 24.\n\n  Cap. i. Of his birth, parentage, and education, ii. How Thomas\n  Hickathrift's strength came to be known, iii. How Tom came to be\n  a Brewer's man; and how he came to kill a giant, and at last was\n  Mr. Hickathrift. iv. How Tom kept a pack of hounds; his kicking a\n  football quite away; also how he had like to have been robbed by four\n  thieves, and how he escaped.\n\n  (2.) The Pleasant and delightful history of Thomas Hickathrift.\n  Whitehaven: printed by Ann Dunn, Market Place [1780], pp. 24.\n\n  (3.) The History of Thomas Hickathrift. Printed in Aldermary\n  Churchyard, London. [1790.] 12mo. Part the first, pp. 24.\n\n  Similar contents to No. 1, with addition of cap. v. Tom meets with a\n  Tinker, and of the battle they fought.\n\n  (4.) The most pleasant and delightful history of Thomas Hickathrift.\n  J. Terraby, printer, Market Place, Hull. [1825.] 2 parts. 12mo. pp.\n  24; 24.\n\n  Same as No. 1. Second part, cap. i. How Tom Hickathrift and the\n  Tinker conquered ten thousand rebels. ii. How Tom Hickathrift and the\n  Tinker were sent for up to court, and of their kind entertainment.\n  iii. How Tom, after his mother's death, went a-wooing, and of the\n  trick he served a gallant who affronted him. iv. How Tom served two\n  troopers whom this spark had hired to beset him. v. Tom, going to be\n  married, was set upon by one and twenty ruffians, and the havock he\n  made. vi. Tom made a feast for all the poor widows in the adjacent\n  houses, and how he served an old woman who stole a silver cup at the\n  same time. vii. How Sir Thomas Hickathrift and his lady were sent\n  for up to court, and of what happened at that time. viii. How Tom\n  was made Governor of the East Angles, now called Thanet, and of the\n  wonderful achievement he performed there. ix. How the Tinker, hearing\n  of Tom's fame, went to be his partner, and how he was unfortunately\n  slain by a lion.\n\n  (5.) The history of Thomas Hickathrift. Printed for the Travelling\n  Stationers. 12mo. pp. 24.\n\n  Same as No. 3.\n\n[Illustration]\n\n\n\n\n  THE\n  PLEASANT HISTORY\n  OF\n  THOMAS HIC-KA-THRIFT.\n\n  [Illustration]\n\n  _Printed by J. M. for W. Thackeray and T. Passinger._\n\n\n\n\n  THE\n  CONTENTS.\n\n\n  What honour Tom came unto.\n\n  How Tom Hic-ka-thrift's strength came to be known.\n\n  How Tom came to be a Brewers man, and how he came to kill a gyant,\n  and at last was Mr. Hic-ka-thrift.\n\n  How Tom kept a pack of hounds and kickt a foot-ball quite away, and\n  how he had like to have been robbed with four thieves, and how Tom\n  escaped.\n\n\n\n\n[Illustration]\n\n\n\n\n  THE\n  PLEASANT HISTORY\n  OF\n  THOMAS HIC-KA-THRIFT,\n\n\n_His Birth and Parentage, and the true manner of his performing many\nmanly acts, and how he killed a gyant. Young man, here thou mayest\nbehold what honour Tom came unto._\n\n  And if that thou dost buy this Book,\n  Be sure that thou dost in it look,\n  And read it o're, then thou wilt say,\n  Thy money is not thrown away.\n\nIn the reign before William the conqueror, I have read in ancient\nhistories that there dwelt a man in the marsh of the Isle of Ely, in\nthe county of Cambridge, whose name was Thomas Hic-ka-thrift, a poor\nman and day labourer, yet he was a very stout man, and able to perform\ntwo days works instead of one: He having one son and no more children\nin the world, he called him by his own name, Thomas Hickathrift. This\nold man put his son to good learning, but he would take none, for he\nwas as we call them now in this age, none of the wisest sort, but\nsomething soft, and had no docility at all in him.\n\nGod calling this old man his father out of the world, his mother being\ntender of him, and maintained him by her hand labour as well as she\ncould; he being sloathful and not willing to work to get a penny for\nhis living, but all his delight was to be in the chimney corner, and\nwould eat as much at one time as might very well serve four or five\nordinary men; for he was in length when he was but ten years of age\nabout eight foot, and in thickness five foot, and his hand was like\nunto a shoulder of mutton, and in all parts from top to toe he was like\na monster, and yet his great strength was not known.\n\n[Illustration]\n\n\n_How Tom Hic-ka-thrifts strength came to be known, the which if you\nplease but to read will give you full satisfaction._\n\nThe first time that his strength was known was by his mothers going to\na rich farmer's house (she being but a poor woman) to desire a buttle\nof straw to shift herself and her son Thomas. The farmer being an\nhonest charitable man, bid her take what she would. She going home to\nher son Tom, said, I pray thee go to such a place and fetch me a buttle\nof straw, I have asked him leave. He swore a great oath he would not\ngo: nay, prithee, Tom go, said his old mother. He swore again he would\nnot go, unless she would borrow him a cart rope. She being willing to\nplease him, because she would have some straw, went and borrowed him a\ncart rope to his desire.\n\nHe taking it went his way; so coming to the farmer's house, the master\nwas in the barn, and two men a thrashing. Said Tom, I am come for a\nbuttle of straw. Tom, said the master, take as much as thou canst\ncarry. He laid down his cart rope, and began to make his buttle;\nbut said they, Tom, thy rope is to short, and jeer'd poor Tom, but\nhe fitted the man well for it: for he made his buttle, and when he\nhad made it, there was supposed to be a load of straw in it, of two\nthousand weight. But said they, what a great fool art thou, thou canst\nnot carry the tith on't? but Tom took the buttle and flung it on his\nshoulder, and made no more of it then we do of an hundred weight, to\nthe great admiration of master and men.\n\nTom Hic-ka-thrift's strength being known in the town, then they would\nnot let him any longer lie basking by the fire in the chimney corner,\nevery one would be hiring him to work; they seeing him to have so much\nstrength, told him that it was a shame for him to live such a lazy\ncourse of life, and to lie idle day after day, as he did. So Tom seeing\nthem bait at him in such a manner as they did, he went first to one\nwork then to another; but at length came a man to Tom and desired him\nto go with him unto the wood, for he had a tree to bring home, and he\nwould content him. So Tom went with him, and he took with him four men\nbeside; but when they came to the wood, they set the cart by the tree\nand began to draw it up with pullies; but Tom seeing them not able to\nlift it up, said, stand away, you fools, and takes the tree and sets it\non one end, and lays it in the cart. Now, says he, see what a man can\ndo. Marry, it is true, said they. So when they had done coming through\nthe wood they met the woodman, Tom asked him for a stick to make his\nmother a fire with. I, said the wood-man, take one what thou canst\ncarry. So Tom espyed a tree bigger then was in the cart, and lays it on\nhis shoulder, and goes home with it as fast as the cart and six horses\ncould draw it. This was the second time that Tom's strength was known.\n\nSo when Tom began to know that he had more strength then twenty men\nhad, he then began to be merry with men and very tractable, and would\nrun, or go, or jump, and took great delight to be amongst company,\nand to go to fairs and meetings, and to see sports and pastimes. So\ngoing to a feast, the young men were all met, some to cudgels, some\nto wrastling, some throwing the hammer, and the like; so Tom stood\na little to see their sport, and at last goes to them that were a\nthrowing the hammer; and standing a little by to behold their manlike\nsport, at last he takes the hammer in his hand to feel the weight of\nit, and bid them stand out of the way, for he would throw it as far\nas he could. I, said the smith, and jeer'd poor Tom, you'l threw it a\ngreat way I'le warrant you; but Tom took the hammer and flung it; and\nthere was a river about five or six furlungs off, and flung it into\nthat; so when he had done, he bid the smith go fetch his hammer again,\nand laught the smith to scorn; but when Tom had done that, he would go\nto wrastling, though he had no more skill than an ass had, but what he\ndid by strength; yet he flung all that came, for if once he laid hold\nthey were gone. Some he would throw over his head, some he would lay\ndown slyly, and how he pleased; he would not lock nor strike at their\nheels, but flung them two or three yards from him, ready to break their\nnecks asunder; so that none at last durst go into the ring to wrastle\nwith him, for they took him to be some devil that was come amongst\nthem; so Tom's fame was spread more in the country.\n\n[Illustration]\n\n\n_How Tom came to be a Brewer's man; and how he came to kill a Giant,\nand at last was Mr. Hic-ka-thrift._\n\nTom's fame being spread abroad in the country, there was not a man\ndurst give Tom an angry word for he was something foolhardy, and he did\nnot care what he did at them; so that those that knew him would not in\nthe least displease him. But at length there was a brewer at Lyn, that\nwanted a good lusty man to carry his beer in the marsh and to Wisbech;\nso hearing of Tom went to hire him, but Tom seemed coy and would not\nbe his man, until his mother and friends did perswade him, and his\nmaster intreated him; and likewise promised him a new suit of clothes\nand cloath him from top to toe; and besides he should eat and drink of\nthe best. So Tom at last yielded to be his man, and his master told him\nhow far he should go; for you are to understand there was a monstrous\nGyant, who kept some part of the marsh, and none durst go that way; for\nif they did he would keep them or kill them, or else he would make bond\nslaves of them.\n\nBut to come to Tom and his master, that he did more work in one day\nthen all his men would do in three; so that his master, seeing him so\ntractable, and to look so well after his business, made him his head\nman to go into the marsh, to carry beer by himself, for he needed no\nman with him. So Tom went every day to Wisbich, which was a very great\njourney, for it was twenty mile the road way.\n\nTom going so long that wearisome journey, and finding that way which\nthe Gyant kept was nearer by half, and Tom having gotten more strength\nby half then before by being so well kept, and drinking so much strong\nale as he did; one day he was going to Wisbich, and without saying\nanything to his master or to any of his fellow servants, he was\nresolved to make the nearest way to be a road or lose his life, to win\nthe horse, or lose the saddle; to kill or be killed; if he met with\nthe Gyant; and with this resolution he goes the nearest way with his\ncart, flinging open the gates for his cart and horses to go through;\nbut at last the Gyant spying him, and seeing him to be so bold, thought\nto prevent him, and came intending to take his beer for a prize, but\nTom cared not a fart for him, and the Gyant he met Tom like a lyon,\nas though he would have swallowed him. Sirrah, said he, who gave you\nauthority to come this way? Do you not know that I make all stand in\nfear of my sight, and you like a rogue must come and fling my gates\nopen at your pleasure! How dare you presume to do this? Are you so\ncareless of your life? Do you not care what you do? I'le make thee an\nexample for all rogues under the sun; dost thou not see how many heads\nhang upon yonder tree that have offended my law! But thy head shall\nhang higher then all the rest for an example. But Tom made him answer,\nA turd in your teeth for your news, for you shall not find me like one\nof them. No, said the Gyant, why thou art but a fool, dost thou come to\nfight with such a one as I am, and bring no weapon to defend thyself\nwithal? Said Tom, I have a weapon here will make you to know you are a\ntraytorly rogue. I, sirrah, said the Gyant, and took that word in high\ndisdain, that Tom should call him a traytorly rogue, and with that he\nran into his cave to fetch his great club, intending to dash out Tom's\nbrains at the first blow.\n\nTom knew not what to do for a weapon, for he knew his whip would do but\nlittle good against such a monstrous beast as he was, for he was in\nlength twelve foot, and six foot about the waste; but while the Gyant\nwent for his club, Tom bethought himself of a very good weapon, for he\nmakes no more ado, but takes his cart and turns it upside down, and\ntakes the axletree and the wheel for his shield and buckler, and very\ngood weapons they were in such time of need.\n\nThe Giant coming out again, began to stare at Tom, to see him take the\nwheel in one hand and the axle tree in the other to defend himself\nwith. O! said the Gyant, you are like to do great service with those\nweapons; I have here a twig, said the Gyant, that will beat thee and\nthy wheel and axle tree at once unto the ground; that which the Gyant\ncalled a twig was as thick as some mill posts are, but Tom was not\ndaunted for all that, for he saw there was but one way to kill or be\nkilled; so the Giant made at Tom with such a vehement force that he\nmade Tom's wheel crack again, and Tom lent the Gyant another as good,\nfor he took him such a weighty blow on the side of the head that he\nmade the Gyant reel again. What, said Tom, are you drunk with my strong\nbeer already.\n\nThe Gyant recovering laid on Tom most sad blows; but still as they came\nTom kept them off with his wheel so that he had no hurt at all. Tom\nplyed his work so well, and laid such huge blows at the Giant, that\nthe sweat and blood together ran down his face, and he being fat and\nfoggy, and fighting so long, was almost tired out, asked Tom to let\nhim drink a little, and then he would fight with him again. No, said\nTom, my mother did not teach me that wit; who's a fool then? Tom seeing\nthe Gyant begin to be weary, and finding him to fail in his blows, he\nthought best to make hay while the sun did shine, for he laid on so\nfast as though he had been mad, till he had brought the Gyant to the\nground. The Gyant seeing himself down, and Tom laying so hard on him,\nroared in a most sad condition, and prayed him not to take away his\nlife and he would do anything for him, and yield himself to him and be\nhis servant; but Tom having no more mercy on him than a dog of a bear,\nlaid still at the Gyant 'till he had laid him for dead, and when he had\ndone he cut off his head and went into his cave, and there he found\ngreat store of silver and gold which made his heart to leap. But when\nhe had done, he loaded his cart and went to Wisbich and delivered his\nbeer; and coming home to his master, he told it to him; but his master\nwas so overjoy'd at the news that he would not believe him till he had\nseen; and getting up the next day he and his master went to see if he\nspoke true or no, and most of the town of Lyn. But when they came to\nthe place and found the Gyant dead, he shewed them where the head was,\nand what silver and gold there was in the cave, all of them leapt for\njoy, for the Gyant was a great enemy to all the country.\n\nThis news was spread all up and down the country how Tom Hic-ka-thrift\nhad kill'd the Gyant, and well was he that could run or go to see the\nGyant and the cave; then all the folks made bonefires for joy; and\nTom was a better man respected than before. And Tom took possession\nof the cave by consent of the country, and everyone said that he did\ndeserve twice as much more. So Tom pulled down the cave and built him\na brave house where the cave stood; all the ground that the Gyant kept\nby force and strength, some he gave to the poor for their common, and\nthe rest he made pastures of and divided the most part into good ground\nto maintain him and his old mother Jane Hic-ka-thrift. And Tom's fame\nwas spread both far and near throughout the country; and then it was no\nlonger Tom, but Mr. Hickathrift, so that he was now the chiefest man\namongst them, for the people feared Tom's anger as much as they did the\nGyant before. So Tom kept men and maids, and lived most bravely; and\nhe made him a park to keep deer in; and by his house, which is a town,\nhe built a famous church and gave it the name of St. James' Church,\nbecause he killed the Gyant on that day, which is so to this hour and\never will be; and many more good deeds he did which is too tedious to\nwrite in this column, but to tell the chief I shall do my endeavour.\n\n[Illustration]\n\n\n_How Tom kept a pack of Hounds; and kickt a Foot-ball quite away;\nand how he had like to have been robbed by Four Thieves, and how he\nescaped._\n\nTom having so much about him and not used to it could hardly tell how\nfor to dispose of it, but yet he did use a means to do it, for he kept\na pack of hounds, and men to hunt with him; and who but Tom then. So\nhe took such delight in sport that he would go far and near to any\nmeetings, as cudgel-play, bear-baiting, football play, and the like.\nBut as Tom was riding one day, he seeing a company at football play he\nlighted off his horse to see that rare sport, for they were playing\nfor a wager; but Tom was a stranger there and none did know him there;\nbut Tom soon spoiled their sport, for he meeting the football took\nit such a kick that they never found their ball no more; they could\nsee it fly, but whither none could tell, nor to what place; they all\nwondered at it, and began to quarrel with Tom, but some of them got\nnothing by it, for Tom gets a spar which belonged to a house that was\nblown down and all that stood in his way he either killed or knocked\ndown, so that all the country was up in arms to take Tom, but all in\nvain, for he manfully made way wherever he came. So when he was gone\nfrom them, and was going homeward, he chanced to be somewhat late in\nthe evening. On the road there met him four lusty rogues that had\nbeen robbing of passengers that way, and none could escape them, for\nthey robbed all they met, both rich and poor. They thought when they\nmet Tom they should get a good prize, they perceiving he was alone,\nmade them cocksure of his money, but they were mistaken, for he got a\nprize by them. When they met with Tom they straight bid him stand and\ndeliver. What, said Tom, what should I deliver? Your money, sirrah,\nsaid they. But, said Tom, you shall give me better words for it first,\nand be better armed too. Come, come, said they, we do not come hither\nto prate, but we come for money, and money we will have before you stir\nfrom this place. I, said Tom, is it so? Nay then, said he, get it, and\ntake it.\n\nSo one of them made at him, but he presently unarmed him, and took away\nhis sword which was made of good trusty steel, and smote so hard at the\nothers that they began to set spurs to their horses and begone, but\nhe soon stayed their journey, one of them having a portmantle behind\nhim, Tom perceiving it to be money fought with more courage then he\ndid before, till at the last he had killed two of the four, and the\nother two he wounded most grievously that they cryed for quarter. So\nwith much intreating he gave them quarter, but he took all their money\nwhich was two hundred pounds to bear his charges home. So when Tom\ncame home he told them how he had served the football players and the\nfour thieves which caused a laugh from his old mother, and to refresh\nhimself went to see how all things did, and what his men had done since\nhe went from home. And going to the forest he wandred up and down, and\nat last met with a lusty tinker that had a good staff on his shoulder\nand a great dog to carry his bag and tools. So Tom asked the tinker\nfrom whence he came, and whither he was going, for that was no highway.\nBut the tinker being a sturdy fellow bid him go look, and what was that\nto him, but fools must be meddling. No, says Tom, but I'le make you\nto know before you and I part it is to me. I, said the tinker, I have\nbeen these three long years and have not had one combat with any man.\nI have challenged many a man but none durst make me answer; I think,\nsaid he, they be all cowards in this country, but I hear there is a man\nin this country which is called Tom Hickathrift that killed a gyant;\nhim I would fain see, said the tinker, to have one combat with him. I,\nsaid Tom, but methinks, said he, it might be master with you; I am the\nman, said he, what have you to say to me? Why verily, said the tinker,\nI am glad we are so happily met together, that we may have one single\ncombat. Sure, said Tom, you do but jest. Marry, said the tinker, I am\nin earnest. A match, said Tom. 'Tis done, said the tinker. But, said\nTom, will you give me leave to get me a twig? I, said the tinker, hang\nhim that will fight with a man unarmed, I scorn that.\n\nSo Tom steps to the gate and takes one of the rails for his staff; so\nto it they fell, the Tinker at Tom, and Tom at the Tinker, like two\ngiants they laid on at each other. The Tinker had a leathern coat on,\nand at every blow Tom gave the Tinker, his coat roar'd again, yet the\nTinker did not give way to Tom an inch. But Tom gave the Tinker a blow\non the side of the head, which felled the Tinker. Now, Tinker, where\nare you? said Tom.\n\nBut the Tinker being a nimble fellow, leapt up again, and gave Tom a\nblow, made him reel again, and followed his blows, and took Tom on the\nother side which made Tom's neck crack again. So Tom flung down his\nweapon and yielded the Tinker the better on't, and took him home to his\nhouse, where I shall leave Tom and the Tinker till they be recovered of\ntheir sad wounds and bruises.\n\n\nFINIS.\n\n[Illustration]\n\n\n\n\n[Illustration]\n\n\n\n\nTHE PLEASANT HISTORY\n\nOF\n\nTHOMAS HICKATHRIFT.\n\n_THE SECOND PART._\n\n\n\n\nCHAP. I.\n\n\n_Tom Hickathrift and the Tinker conquered and overcame three thousand\nrebellious subjects._\n\nIn and about the Isle of Ely, many disaffected persons, to the number\nof ten thousand and upwards drew themselves up in a body, presuming to\ncontend for their pretended ancient Rights and Liberties, insomuch that\nthe Gentry and civil Magistrates of the Country was in great danger;\nat which time the Sheriff, by night, privately got into the house of\nThomas Hickathrift, as a secure place of refuge, in so eminent a time\nof danger: where before Thomas Hickathrift, he laid open the villainous\nintent of this headstrong giddy-brained multitude, Mr. Sheriff, quoth\nTom, what service my brother, meaning the Tinker, and I can perform,\nshall not be wanting. This said, in the morning by day-break, with\ntrusty clubs they both went forth, desiring the Sheriff to be their\nguide, in conducting them to the place of the rebels' rendezvous,\nwhen they came there Tom and the Tinker marched up to the head of\nthe multitude, and demanded of them the reason why they disturbed\nthe Government? To which they answered with a loud cry, Our will's\nour law; and by that alone will we be governed. Nay quoth Tom if it\nbe so, these trusty clubs are our weapons, and by them you shall be\nchastised. Which words were no sooner out of his mouth but the Tinker\nand he put themselves both together in the midst of the throng and with\ntheir clubs beat the multitude down, trampling them under their feet\nevery blow which they struck laid twenty or thirty sprawling before\nthem. Nay, remarkable it was, the Tinker struck a tall man just upon\nthe nape of the neck, with that force that his head flew off, and was\ncarried violently fourteen foot from him, where it knockt down one of\ntheir chief ringleaders; Tom on the other hand still pressing forward,\ntill by an unfortunate blow he broke his club; yet he was not in the\nleast dismay'd; for he presently seized upon a lusty stout rawbon'd\nmiller, and made use of him for a weapon, till at length they clear'd\nthe field; so that there was not one found that dare lift up a hand\nagainst them, having run into holes and corners to hide themselves\nshortly after some of their heads were taken and made public examples\nof justice, the rest being pardoned at the humble request of Thomas\nHickathrift and the Tinker.\n\n[Illustration]\n\n\n\n\nCHAP. II.\n\n\n_Tom Hickathrift and the Tinker was sent for to Court and of their kind\nentertainment there, etc._\n\nThe King being truly informed of the faithful services perform'd by\nthese his loving Subjects, Thomas Hickathrift and the Tinker, he was\npleased to send for them to his Palace, where a Royal banquet was\nprepared for their entertainment, most of the Nobility being present.\nNow after the banquet was over, the King said unto all that were\nthere, these are my trusty and well-beloved subjects, men of approved\ncourage and valour, they are the men that overcame and conquer'd ten\nthousand which were got together to disturb the peace of my realm;\naccording to the character that hath been given to Tho. Hickathrift\nand Henry Nonesuch, persons here present, they cannot be matcht in any\nother kingdom in the world; were it possible to have an army of twenty\nthousand such as these, I dare venture to act the part of Alexander\nthe Great over again: yet in the meanwhile, as a proof of my Royal\nfavour, kneel down and receive the antient order of knighthood, Mr.\nHickathrift, which was instantly perform'd. And as for Henry Nonesuch,\nI will settle upon him, as a reward for his great service, the sum of\nForty Shillings a year, during life. Which said, the King withdrew, and\nSir Thomas Hickathrift and Henry Nonesuch the tinker, returned home,\nattended by many persons of quality, some miles from the Court. But to\nthe great grief of Sir Thomas, at his return from the Court, he found\nhis aged Mother drawing to her end, who in a few days after died and\nwas buried in the Isle of Ely.\n\n[Illustration]\n\n\n\n\nCHAP. III.\n\n\n_Tom after his old Mother's death went a wooing; and how he served a\nyoung Gallant who affronted him before his Mistress._\n\nTom's mother being dead, and he left alone in a large and spacious\nhouse, he found himself strange and uncouth, therefore he began to\nconsider with himself that it would not be amiss to seek out for a\nwife; and hearing of a young rich widow, not far from Cambridge, to her\nhe went, and made his addresses: and at the first coming she seem'd to\nshew him much favour and countenance; but between this and his coming\nagain, she had given some entertainment to a more genteel and airy\nspark, who happened likewise to come while honest Tom was there the\nsecond time; he look'd wishfully at Tom, and he star'd as wishfully\nat him again; at last the young spark began with abuseful language to\naffront Tom, telling him he was such a great lubberly whelp, adding\nthat such a one as he should not pretend to make love to a Lady, as he\nwas but a Brewer's servant. Scoundrel quoth Tom better words should\nbecome you, and if you do not mend your manners, you shall not fail\nto feel my sharp correction. At which the young Spark challenged him\nforth into the back yard; for, as he said, he did not question but to\nmake a fool of Tom in a trice. Into the yard they both walk together,\nthe young spark with a naked sword, and Tom with neither stick nor\nstaff in his hand, nor any other weapon. What says the spark, have you\nnothing to defend yourself? well I shall the sooner dispatch you. Which\nsaid, he ran furiously forward, making a pass at Tom, which he put by,\nand then wheeling round to his backside, Tom gave him such a swinging\nkick on the breech, which sent the spark like a Crow up into the Air,\nfrom whence he fell upon the ridge of a thatcht house, and then came\ndown into a large fish pond, and had been certainly drown'd if it had\nnot been for a poor shepherd who was walking that way, and seeing\nhim float upon the water, dragged him out with his hook, and home he\nreturned like a drowned Rat; while Tom enjoy'd the kind embraces of his\nfair Mistress.\n\n[Illustration]\n\n\n\n\nCHAP. IV.\n\n\n_Tom served two Troopers, whom the young Spark had hired to beset him,\netc._\n\nThis young galland being tormented in his mind to think how Tom had\nconquered and sham'd him before his Mistress, he was now resolved for\nspeedy revenge; and knowing that he was not able to coap with a man of\nTom's strength and activity, he therefore hired two lusty Troopers,\nwell mounted, to lie in ambush in a thicket which Tom had to pass\nthrough from his home to the young lady, and accordingly they attempted\nto set upon him: How now quoth Tom Rascals, what would you be at? Are\nyou indeed weary of the world, that you so unadvisably set upon one who\nis able to crush you in like a Cucumber; the Troopers laughing at him,\nsaid, that they were not to be daunted at his high words, High words\nquoth Tom, no I will come to action; and with that he run in between\nthese armed Troopers, catching them under his arm. Horse and Men, with\nas much ease as if they had been but a couple of Baker's babbins,\nsteering his course with them hastily towards his own home, and, as\nhe pass'd thro' a meadow, in which there was many Haymakers at work,\nthe poor distressed troopers crying out, Stop him stop him he runs\naway with two of the King's troopers. The hay-makers laught heartily\nto see how Tom hugged them along; ever and anon he upbraided them for\ntheir baseness; declared that he would make minced meat of them to feed\nthe Crows and Jackdaws about his house and habitation. This was such\na dreadful lecture to them, that the poor rogues begg'd that he would\nbe merciful, and spare their lives, and they would discover the whole\nplot, and who was the person that employ'd them; which accordingly they\ndid, and gain'd favour in the sight of Tom, who pardon'd them upon\npromise that they would never be concern'd in such a villainous action\nfor the time to come.\n\n[Illustration]\n\n\n\n\nCHAP. V.\n\n\n_Tom going to be married, was set upon by one-and-twenty Ruffians in\nArmour, and of the havock he made amongst them, etc._\n\nIn regard Tom had been hinder'd by these troopers, he delay'd his\nvisit to his Lady till next day, and then coming to her, gave her a\nfull account of what had happen'd; she was pleased at heart at this\nwonderful relation, knowing it was safe for a woman to marry with a\nman who was able to defend her against all assaults whatsoever, and\nsuch a one she found Tom to be. The day of marriage was accordingly\nappointed, friends and relations invited. Yet secret malice which is\nnever satisfied without sweet revenge, had like to have prevented the\nsolemnity for having three miles to go to church, where they were\nto be married, the aforesaid Gentleman had provided a second time\nRuffians in armour to the number of twenty-one, he himself being then\npresent either to destroy the life of Tom, or put them into strange\nconsternation; however thus it happened, in a lonesome place they\nbolted out upon them, making their first assault upon Tom, and with\na Speer gave him a slight wound, at which his love and the rest of\nthe women shrieked and cry'd like persons out of their wits, Tom\nendeavour'd all that he could to pacify them, saying, stand you still\nand I will show you pleasant sport. And with that he catch'd a back\nsword from the side of a Gentleman in his own company, with which he\nso bravely behaved himself that at every stroke he cut off a joint,\nloth he was to touch the life of any, but aiming at their legs and\narms, he lopt them off so fast, that in less than a quarter of an hour,\nthere was not one in the company but what had lost a limb, the green\ngrass being stained with their purple gore, and the ground strew'd\nwith legs and arms, as 'tis with tiles from the tops of the Houses\nafter a dreadful storm. His Love and the rest of the company standing\nall the while as joyful spectators, laughing one at another, saying,\nWhat a company of <DW36>s has he made, as it were in the twinkling of\nan eye! Yes, quoth Tom, I believe that for every drop of blood that I\nlost, I have made the Rascals pay me a limb as a just tribute. This\ndone, he stept to a Farmer's hard by, and hired there a servant giving\nhim twenty shillings to carry these <DW36>s home to their respective\nhabitations in his dung-cart; and then did he hasten with his love to\nthe church, where they were married, and then returned home, where they\nwere heartily merry with their friends, after their fierce and dreadful\nencounter.\n\n\n\n\nCHAP. VI.\n\n\n_Tom made a feast for all the poor Widows in the adjacent towns; and\nhow he served an old Woman who stole a Silver Cup at the same time,\netc._\n\nNow Tom being married, he made a plentiful feast, to which he invited\nall the poor widows in four or five parishes for the sake of his\nmother, which he had lately buried, this feast was kept in his own\nhouse, with all manner of varieties that the country could afford for\nthe space of four days, in honour likewise of the four victories which\nhe lately obtain'd. Now when the time of feasting was ended, a Silver\nCup was missing, and being ask'd about it, they every one deny'd they\nknew any thing of it. At length it was agreed that they should all\nstand the search, which they did, and the Cup was found upon a certain\nold woman, named the Widow Stumbelow; then was all the rest in a rage,\nsome was for hanging her, others were for chopping the old woman in\npieces, for her ingratitude to such a generous soul as Sir Thomas\nHickathrift; but he entreated them all to be quiet, saying they should\nnot murder a poor old Woman, for he would appoint a punishment for her\nhimself; which was this; he bor'd a hole thro' her nose and tying a\nstring therein, then order'd her to be stript stark naked, commanding\nthe rest of the old women to stick a candle in her fundament, and lead\nher by the nose thro' all the streets and lanes in Cambridge, which\ncomical sight caused a general laughter. This done, she had her cloaths\nrestor'd her again, and so was acquitted.\n\n[Illustration]\n\n\n\n\nCHAP. VII.\n\n\n_Sir Thomas Hickathrift and his Lady was sent for to Court, and of what\nhappened._\n\nThe tydings of Tom's wedding was soon nois'd in the Court, so that the\nKing sent them a royal invitation to the end he might see his Lady,\nthey immediately went, and were received with all demonstrations of\nJoy and Triumph. But while they were in their mirth, a dreadful cry\napproached the Court which proved to be the Commons of Kent who were\ncome thither to complain of a dreadful Giant that was landed in one of\nthe Islands: And brought with him abundance of Bears and young Lyons,\nlikewise a dreadful Dragon on which he himself rid, which monster and\nravenous beasts had frighted all the inhabitants out of the Island.\nMoreover they said if speedy course was not taken to suppress them\nin time, they might over-run the whole land. The King hearing this\ndreadful relation was a little startled, yet he perswaded them to\nreturn home and make the best defence they could for themselves at\npresent, assuring them that he should not forget them, and so they\ndeparted.\n\n\n\n\nCHAP. VIII.\n\n\n_Thomas Hickathrift was made Governor of the Island of the East Angles,\nnow called Thanet, and of the wonderful Achievements he performed\nthere._\n\nThe King hearing the aforesaid dreadful Tydings, immediately sate\nin Council to consider what was to be done for the overcoming this\nmonstrous Giant, and barbarous savage Lyons and Bears that with him had\ninvaded his Princely territories. At length it was agreed upon that\nThomas Hickathrift was the most likeliest man in the whole kingdom,\nfor undertaking of so dangerous an enterprise; he being not only a\nfortunate man of great strength, but likewise a true and trusty subject\none that was always ready and willing to do his King and country\nservice, for which reason it was thought necessary to make him Governor\nof the aforesaid Island; which place of trust and honour, he readily\nreceived, and accordingly he forthwith went down with his wife and\nfamily, to take possession of the same, attended with a hundred Knights\nand Gentlemen, who conducted him to the entrance of the Island which he\nwas to govern. A castle in those days there was, in which he was to\ntake up his head-quarters, the same being situated with that advantage\nthat he could view the Island for several miles upon occasion; the\nKnights and Gentlemen at last taking their leave of him, wish'd him all\nhappy success and prosperity. Many days he had not been there before it\nwas his fortune to behold the monstrous Giant mounted upon a dreadful\nDragon, bearing upon his shoulder a club of Iron, having but one eye,\nthe which was placed in his forehead, and larger in compass than a\nbarber's bason, and seem'd to appear like a flaming fire; his visage\nwas dreadful, grim and tawny; the hair of his head hanging down his\nback and shoulders, like snakes of a prodigious length; the bristles of\nhis Beard like rusty wire: And lifting up his blare eye, he happened to\ndiscover Sir Thomas Hickathrift, who was looking upon him from one of\nhis windows of the castle; the Giant then began to knit his brow, and\nbreath forth threatening words to the Governor, who indeed was a little\nsurpriz'd at the approach of so monstrous a brute; the Giant finding\nthat Tom did not make much haste down to meet him, he alighted from the\nback of the dragon, and chained the same to an Oak tree, then marching\nfuriously to the castle, setting his broad shoulder against a corner of\nthe stone walls, as if he intended to overthrow the whole building at\nonce which Tom perceiving, said is this the game you would be at; faith\nI shall spoil your sport, for I have a delicate tool to pick your teeth\nwithal; then taking his two handed sword of five foot long, a weapon\nwhich the King had given him to govern with taking this I say, down he\nwent, and flinging open the Gates, he there found the Giant, who by an\nunfortunate slip in his thrusting was fallen all along, where he lay\nand could not help himself. What, quoth Tom, do you come here to take\nup your Lodging? This is not to be suffer'd, and with that he ran his\nlong broad sword in betwixt the monstrous Giant's brawny Buttocks and\nout at his Belly, which made the monstrous Brute give such a terrible\ngroan that it seemed like roaring thunder, making the very neighbouring\ntrees to tremble; and then Tom pulling out his sword again, at six or\nseven blows he separated his head from his unconscionable trunk, which\nhead, when it was off, seemed like the root of a mighty Oak. Then\nturning to the Dragon, which was all this while chain'd to a tree,\nwithout any farther discourse, with four blows with his two-handed\nsword, he cut off his head also. This fortunate adventure being over,\nhe sent immediately for a team of horses and a waggon: which he loaded\nwith these heads, and then summoning all the Constables in the Country\nfor a guard, sent them to Court, with a promise to his Majesty, that he\nwould rid the whole Island likewise of Bears and Lyons before he left\nit.\n\n[Illustration]\n\n\n\n\nCHAP. IX.\n\n\n_The Tinker hearing of Tom s Fame went down to be Partner in his\nEnterprize; and how he was unfortunately slain by a Lyon._\n\nTom's victories rang so long, that they reach'd the ears of his old\nacquaintance, the Tinker, who, desirous of honour, resolved to go down\nand visit Tom in his new government, when coming there he met with kind\nand loving entertainment, for they were very joyful to see one another:\nNow after three or four days' enjoyment of one another's company, Tom\ntold the Tinker that he must needs go forth in search after wild Bears\nand Lyons, in order to rout them out of the Island. Well, quoth the\nTinker, I would gladly take my fortune with you, hoping that I may be\nserviceable to you, upon occasion. Well, quoth Tom, with all my heart,\nfor I must needs acknowledge I shall be right glad of your company.\nThis said, they both went forward, Tom with his two handed sword, and\nthe Tinker with his long pike-staff. Now after they had travelled\nabout four or five hours, it was their fortune to light of the whole\nknot of wild beasts together, being in number fourteen, of which six\nof them were Bears, the other eight young Lyons now when they had\nfastened their eyes upon Tom and the Tinker, these ravenous beasts\nbegan to roar and run furiously, as if they would have devoured them\nat a mouthful; but Tom and the Tinker stood side by side with their\nbacks against an Oak, and as the Lyons and Bears came within their\nreach, Tom with his long sword clove their heads asunder till they were\nall destroyed, saving one lyon, who, seeing the rest of his Fellows\nslain, was endeavouring to make his escape: now the Tinker being\nsomewhat too venturous, ran too hastily after him, and having given\nthe Lyon one blow, he turn'd upon him again, seizing him by the throat\nwith that violence, that the poor Tinker fell dead to the Ground; Tom\nHickathrift, seeing this, gave the Lyon such a blow that it ended his\nlife.\n\nNow was his joy mingled with sorrow for tho' he had cleared the Island\nof those ravenous savage beasts, yet his grief was intolerable for\nthe loss of his old friend. Home he returned to his lady, where in\ntoken of Joy for the wonderful success which he had in his dangerous\nenterprizes, he made a very noble and splendid Feast, to which he\ninvited most of his best Friends and Acquaintance, to whom he made the\nfollowing Promise.\n\n  My Friends while I have Strength to stand,\n    most manfully I will pursue\n  All Dangers, till I clear this Land,\n    of Lions, Bears, and Tygers too;\n  This you'll find true, or I'm to blame,\n    let it remain upon Record;\n  Tom Hickathrift's most glorious fame,\n    who never yet has broke his word\n  The Man who does his Country bless,\n    shall merrit much from this fair land;\n  He who relieved them in Distress,\n    His Fame upon Record shall stand:\n  And you my Friends who hear me now,\n    let honest Tom, for ever dwell,\n  Within your Minds and Thoughts I trow,\n    since he has pleas'd you all so well.\n\n\nFINIS.\n\n[Illustration]\n\n\n\n\nFOOTNOTES:\n\n\n[A] Blomefield's _History of Norfolk_, vol. ix. pp. 79-80; the same\nstory is related by Chambers in his _History of Norfolk_, vol. i. p.\n370. The parishes of W. and N. Lynn, though lying in marshland, are\nexcluded from any right of pasturage on the Smeth Common.\n\n[B] _Quarterly Review_, vol. xxi. p. 103.\n\n[C] _Quarterly Review_, vol. xxi. p. 102, note.\n\n[D] _Quarterly Review_, vol. xxi. pp. 102-103.\n\n\n\n\nTRANSCRIBER'S NOTES:\n\n\n  Italicized text is surrounded by underscores: _italics_.\n\n  Obvious typographical errors have been corrected.\n\n  Inconsistencies in hyphenation have been retained from the original.\n\n  Archaic or alternate spelling has been retained from the original.\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Chap-Books and Folk-Lore Tracts, Vol.\n1 (of 5), by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n_Also by Olivia St. Claire_\n\n203 WAYS TO DRIVE  \nA MAN WILD IN BED\n\nCopyright \u00a9 1996 by Sakkarra, Inc.\n\nAll rights reserved. No part of this book may be  \nreproduced or transmitted in any form or by  \nany means, electronic or mechanical, including  \nphotocopying, recording, or by any information  \nstorage and retrieval system, without permission in  \nwriting from the publisher.\n\nPublished by Harmony Books, New York, New York.  \nMember of Crown Publishing Group.\n\nRandom House, Inc. New York, Toronto, London, Sydney, Auckland\n\nwww.randomhouse.com\n\nHARMONY BOOKS is a registered trademark and the Harmony Books colophon is a trademark of Random House, Inc.\n\nDesign by John Fontana with Linda Kocur\n\nLibrary of Congress Cataloging-in-Publication Data is available upon request.\n\neISBN: 978-0-307-83320-4\n\nv3.1\n_To the sensual goddesses of history and heaven\u2014especially_\n\n_Aphrodite, Cleopatra, the Cloud Damsels, the Cosmic Yoni, the_\n\n_Djanggawo sisters, Hathor, Inanna, Lilith, Mother Earth_ ,\n\n_Nofretari, Oshun, Pele, Sekhmet, Shakti, the Snake Priestess_ ,\n\n_White Buffalo Woman, Wild Woman, Xochiquetzal, the_\n\n_Yoginis\u2014and to the Sex Goddess in every woman_\n\n# Acknowledgments\n\nMy deepest gratitude to all the women and men who so graciously shared their most intimate thoughts and feelings with me\u2014we have all learned so much from you. Heartfelt thanks as well to the extraordinary women who so lavishly gave me their support, advice, assistance, and love during the writing of this book\u2014Christine, Claudia, Deborah, Jane, Karen Marie, Marianne, Pamela, Paula, Sandy, and Tovi\u2014and to those who have so expertly guided me in birthing it into the world\u2014Shaye, Paula, Shima, and Miko.\n\n# Contents\n\n_Cover_\n\n_Other Books by This Author_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\n_Acknowledgments_\n\n_1_ awakening your inner sex goddess\n\n_2_ creating your sensual garden\n\n_3_ self-pleasuring\u2014the key to the kingdom\n\n_4_ awakening aphrodite's body\n\n_5_ the mirror of sensual transformation\n\n_6_ g(oddess)-spots and love nectars\n\n_7_ fantasies from the cloud damsels\n\n_8_ the secrets of divine orgasm\n\n_9_ cleopatra's pleasure toys\u2014electric, plastic, furry, and human\n\n_10_ the goddess with her consort\n\n_11_ daily goddessizers\n\nappendix: the inner sex goddess scrolls\n\n# 1  \nawakening your inner sex goddess\n\n_A woman learns to be a good lover from her own heart and from the heart of her being. She does not need to learn it from a man. She learns it from her inner goddess_.\n\n_\u2014 Elaine Kittredge, Masterpiece Sex_\n\nevery woman has an Inner Sex Goddess. She lives in your heart, your body, and your senses. She is the part of you that feels aroused by reading erotic literature or watching sexy movies. She is the part of you that skips a heartbeat when an attractive man in tight jeans walks by. She is the part of you that feels coy and flirtatious when you put on a clingy dress; after all, it was _her_ idea to wear the dress in the first place. Your Inner Sex Goddess is the one who bought this book and who tingles with anticipation at the thought of becoming, at last, her most gloriously seductive self.\n\nWhether you experience her as a feeling (like a buttery inner warmth), as one of the many roles you play in life (such as career woman, mother, friend, wife, party organizer), or as a distinct personality (like your inner child, inner critic, or inner male and female), your Inner Sex Goddess is a strong, important, and natural part of who you really are. And getting to know her is really getting to know your deepest self\u2014your beliefs and values, your fears and dreams, and the very personal poetry of your body. The truly wonderful thing about your Inner Sex Goddess is that, without instruction from any person, book, workshop, or man, she already knows how to be fabulously sexy and alluring. In her bones, she knows how to _feel_ sexy, even when you're tired and cranky, bored, or simply at a loss for inspiration. She knows exactly what to do with her hands when you are making love; she doesn't wonder if it's better to stroke his back or tickle his ear because, with unerring instinct, she automatically does the very thing that will most stimulate herself and her lover.\n\nIn fact, your Inner Sex Goddess is a boundless source of sexual energy, know-how, and charisma. Unashamed, self-confident, and regally proud of her sexuality, she understands that feeling and being sexy is a natural and beautiful way for a woman to nourish her spirit and learn to love the ecstasies of her sensation-rich body. Revealing herself to you with every swish of your hair or sway of your hip, your Inner Sex Goddess revels in the textures of life and the world of the senses\u2014and the more you revel in those textures and senses, the better you will get to know her; the freer she will be to enjoy pleasuring herself and her lover; and the more lovely, bold, and exciting she will become.\n\nIt is she who lives in the vivid sensitivities of your body. She is there when you appreciate the look of candlelight on the curve of your breast, the smell of a gardenia in your hair, the taste of wine on your lips and tongue, the feel of sun or silk on your skin, the sound of sultry jazz or samba music, the luxury of a hot bubble bath or a full-body massage, the delicious flushes and tingles of self-pleasuring or lovemaking. These physical sensations are doorways to your rich inner world of lush romance and hot libido.\n\nIn fact, we women have a natural gift for combining romance, love, and sex into one magical art form. Even our enticingly soft and curvy bodies seem to be specially designed for exquisite pleasure. Did you know, for instance, that we females are unique in having a highly sensitive organ whose _only_ purpose is sexual delight?\n\nBut even though we have all these natural erotic gifts, no one has schooled us in the art of being in tune with and expressing our inherent sexuality. So we have felt awkward, confused, and even cut off from one of the most precious and vital parts of ourselves. This is not the case in many indigenous cultures\u2014for instance, Native American, tribal African, Tantric Indian, and some ancient Far Eastern societies, to name a few. Placing a high priority on developing sexual joy and proficiency in a sacred, loving way, these traditions often use specially trained sexual mentors to instruct and advise young people on how to appreciate the beauty and majesty of their bodies, as well as how to bring the highest sensual pleasure to their future partners\u2014and to themselves. Naturally in touch with their Inner Sex Gods and Goddesses, people in these cultures believe sexuality is a divine gift and learn to treat their bodies with the utmost respect and reverence. We would be wise to learn from them.\n\n## the natural textures of life\n\nAncient indigenous cultures who were living closer to the earth and practicing rituals tied to the cycles of nature were more in tune with the senses and textures of life than our modern \"civilized\" society. Their uncomplicated existence and natural belief in the divine nature of plants, animals, elements, and life itself allowed them to be more intimately acquainted with the sensuality that dwelt within them\u2014and left them with practically no inhibitions. To them, sexuality was a natural part of creation, and their sex organs were a chalice of the sacred energy of life. For example, one Stone Age cave painting shows a male figure holding a bow, his penis connected by an unbroken \"power line\" to the genitals of a female figure, suggesting that the female ritually empowered the man to go out for the hunt\u2014archetypal roles that these early peoples acted out in daily life. In ancient India, men and women transformed themselves into gods and goddesses\u2014meditating, anointing themselves, drinking sacred potions, muttering mantras over their bodies, and having ritualized sex\u2014for the purpose of achieving enlightenment. Other prepatriarchal religions honored fertility, creation, and sexuality in their ceremonies as different aspects of the same divine force. The pagan rites of Beltane, for example, involved ritual or wildly abandoned sexual activity as the symbolic cause of the earth's fertility in spring, and the solstice celebrations of planting and reaping the crops as a gift from the sensual, abundant body of the earth as goddess.\n\nThe women of all these cultures innately understood that the power and divine sexuality of the feminine principle was inherent not only in the earth, the elements, the animals, and the goddesses their people worshiped but also in their own sex organs and instincts, in the smells and tastes of their fleshy bodies, and in the natural textures of their lives. They and their individual Inner Sex Goddesses lived interchangeably with each other on a daily basis, moving smoothly and powerfully from conjugal bed to cooking fire to sexual fertility rite to tribal council to wheat harvest to sacramental childbirth and back again. Though we have centuries of acculturation and stifling moral convention to conquer, we too can become the natural sex goddesses these women were born and raised to be. When we come to know, revere, and love our bodies and their pleasures, as they did, we will no longer see our sexuality as something separate from our everyday selves. For with knowledge, reverence, and love come power, respect, responsibility, control\u2014and the natural manifestation of each woman's own brand of lusty and graceful sexual divinity.\n\nFinding your particular version of vampiness means exploring the hitherto unknown territories of your body, your senses, and your hot-blooded imagination. Your Inner Sex Goddess waits there in the intoxicating realm of self-knowledge and self-pleasuring. In our quest to know her, then, we'll be investigating many of the different avenues of pleasure through which you may find and unleash her, including:\n\n\u2022 Transforming reading, writing, breathing, walking to work, and wearing clothes into aphrodisiacs for your senses\n\n\u2022 Reveling in creative autoeroticism and self-pleasuring\n\n\u2022 Having revealing inner dialogues with your goddess self\n\n\u2022 Exploring and eroticizing every square inch of your body\n\n\u2022 Inventing and enjoying wanton fantasies, dreams, and visions\n\n\u2022 Discovering new G-spots and lovemaking talents\n\n\u2022 Having lots of earthshaking orgasms\n\n\u2022 Playing with erotic toys, clothes, and props\n\n\u2022 Making wild passionate love to your man\n\nBut before we venture onto the superhighway of rich physical sensation, let's start by exploring some simple but very fruitful _mental_ pathways to your sensual kingdom. Your sexuality is contained not only in your five physical senses but also in the inner senses of your mind and heart, and the journey to your Inner Sex Goddess begins not with your lover or some esoteric technique but within the private realm of your fantasies, heartfelt desires, daydreams, and self-created images about who you are. Experiment with the following ideas and see which ones resonate for you.\n\n## daily dialog\n\nThroughout the day make a habit of asking your Inner Sex Goddess how she would do things. Just pretend that there is some creature inside you who knows how to be appropriately and naturally sexy, and have a conversation with her. Ask her, \"How would you dress this morning? walk down the street? talk to that man? move your hips? dance? put on your makeup? wear your hair? How would you make yourself _feel_ sexy? In what ways would you pleasure yourself? What toys would you use? How would you respond? Which man would you choose? How would you make love to him? Where would you put your hand?\" Maybe she would add a silk scarf to the conservative outfit you put on. Maybe she would walk to work through the sensation-stimulating park rather than taking the boring bus. Perhaps she would tango in the kitchen instead of going to the gym. Follow her advice and see where it takes you.\n\n## journal\n\nStart keeping a journal for your Inner Sex Goddess. Slip into her mind and heart and jot down what you see through her eyes, hear through her ears, feel through her emotions, think through her thoughts, and do through her body. Let her write about what she'd like to do, how she'd like you to relate to her, how she feels about herself, and how she'd like to express herself. Ask her questions about your sexuality and let her write the answers for you.\n\nWhen I did this one time, my Inner Sex Goddess wrote down that if you view any sensual act not as something you do but as something you receive, as she does, it takes on a whole new meaning. For instance, when you kiss your lover, don't think of what you are doing to him with your lips, but instead focus on receiving the softness of _his_ lips, receiving the smooth texture of the skin on his neck, receiving the hard round energy of his fingers as you suck on them. She said to close your eyes and concentrate on drinking in your lover's very essence. As much as I have attuned myself to the many nuances of lovemaking, I never would have thought about kissing quite this way. But keeping a journal for my Inner Sex Goddess brought out a wonderful new perspective that I have since eagerly incorporated into my love life. Find out what unique insights await you when you visit with your inner sexual divinity.\n\n## bodyknowing\n\nA particularly sensual woman I know says, \"For me, being in touch with my sexuality means deliberately cultivating a sort of magnetic, earthy feeling. I focus on being fully in my body and using all of its instincts and sensations, really feeling the electromagnetic energy pulsing around me. I imagine that I am a wild creature of the earth, a wolf woman, and knower of the moon mysteries. At those times I feel like an untamed, exotic animal using my bodyknowing to move sensuously through the natural world. It is then that my Inner Sex Goddess seems most alive.\"\n\nI recommend reading Clarissa Pinkola Est\u00e9s's superb book _Women Who Run with the Wolves_ to get an even deeper feel for the wildish woman inside you, so closely akin to your Inner Sex Goddess. Both these inner archetypes\u2014sensual, electric, and instinctual\u2014are characterized by their strong sense of bodyknowing.\n\n## inner movies\n\nUse the screen of your imagination to conjure up your Inner Sex Goddess. You may be surprised at who or what appears. Get comfortable. Close your eyes, breathe deeply. Relax and go inside yourself. Imagine a movie screen in your mind's eye and ask your Inner Sex Goddess to reveal herself on it. She may appear full-blown right away or slowly with a foot first and then an ankle, a leg, and so on. What does she look like? How does she dress and move? Is she a robed ancient queen or a contemporary woman in business suit over leopard-print panties? Observe her activities. Does she do different things than you do? What kind of perfume does she dab on? How does she communicate with men, verbally and nonverbally? What does she do in your workplace? at home? at school? in the neighborhood? What does she call herself\u2014maybe she prefers Angelique to your usual Anne. Ask her how you can be more in touch with her, in a safe, elegant, and appropriate way\u2014and how you can do this quickly and easily when need be. Request that she come to you in your dreams and teach you. Ask her to be with you when you make love with your partner and when you pleasure yourself. Thank her for revealing herself to you and give her a gift of gratitude. Let her give you a symbolic present and take careful note of what it is, for she will surely be communicating something special to you with this inner gift.\n\nYour Inner Sex Goddess may change her appearance from day to day, too. Sometimes mine appears as a beautiful naked Greek goddess and sometimes as the Great Cosmic Yoni (in the Tantric tradition, _yoni_ means \"vulva\"), who is the source of ecstasy, sensuality, and womanness. She has given me all sorts of symbolic presents, including a gold key (the key to any sense or experience I might want to open), a marble sculpture of a breast (representing the female qualities of nurturing and sensual beauty), and a leopard-skin G-string to remind me to be raunchy and have fun. Pat, a business executive friend of mine, says that her Inner Sex Goddess looks like Catwoman and is always admonishing her to slink around more and lick all the flavors of life. Another friend tells me that her Inner Sex Goddess looks like Marilyn Monroe, calls herself Lolita, and often shows up when she's making love to point out hidden erogenous zones on her man's body by kissing them with her red, pouty lips. My friend Georgia says that her Inner Sex Goddess is more like a colored ball of energy\u2014sometimes pink, sometimes turquoise\u2014that floats around Georgia's body making tingly sensations and whispering sexual secrets in her dreams.\n\n## mythological journeys\n\nAs the brilliant modern-day philosopher Joseph Campbell asserted, myths are often the key to our innermost selves, and mythological characters are the archetypes of our inner personalities. Exploring the tales of some of the mythological goddesses of sensuality can lead you to new discoveries about your own sexual persona. Of the scores of magnificent erotic deities, here are just a few to consider. Read about them, contemplate their qualities, gaze at their images, copy their poses, bring their accoutrements into your everyday life, or have inner conversations with them. Call on them just before you make love. Let them seep into your consciousness and connect you to your innermost sexual self.\n\nAphrodite is the Greek goddess of love, sensuality, and passionate relationships. It is she who graces your life when you fall madly in love. As the only goddess portrayed nude in Greek art, Aphrodite is proud of her beautiful nakedness and potent sexuality. In fact, in one representation of her as Aphrodite Kallipygos (Aphrodite of the Lovely Backside), she coyly draws up her robe to admire her own round fanny. The rose and lily, symbolizing the vagina, are her beloved flowers. Try draping yourself in a Grecian-style robe, even if only mentally, and lean back to gaze at your own luscious derriere. Meditate on the ripe power of this fetchingly plump piece of human art, potent enough to drive some otherwise sane men to a state of crazed desire. In emulating Aphrodite, you become a regal feminine beauty with the magical transforming power of erotic love nestled in every angle, wiggle, and curve of your body.\n\nIn the Semitic tradition, Lilith was the first woman, created at the same time as Adam. But when Adam tried to have sex with her, she grew wings and flew out of paradise to live in the desert, later becoming the bringer of agriculture and the protector of children. God then created the more compliant Eve. Let Lilith's fierce drive for sexual equality, independence, and wildness teach you about yours. Imagine yourself taking wing to escape a brand of sexuality you no longer desire and creating a powerful new one fashioned after your personal tastes and talents.\n\nOshun is the African Yoruba goddess of the river, sensuality, and creativity. As her waters dance sinuously through the countryside, she adorns herself with bangly stone bracelets, the beautiful clothing of the landscape, and the perfumes of the wildflowers. Also presiding over the arts, gold, and love, she provides a sensual healing balm to all who dip into her liquid body. She is completely self-contained, self-aware, and lovingly self-conscious of her natural erotic beauty as she glows in the moon's silvery romantic light. Mentally immerse yourself in her fluid sensuality and surge, ripple, and murmur like the river. Adorn yourself with flowers, flowing fabrics, and gold bracelets that jingle. Revel in your self-contained, liquid eroticism.\n\nIn the Mayan tradition, Xochiquetzal (Precious Flower) is the female deity of flowers, beauty, pleasure, and sexuality. Always joyous and free in her sensuality, she is sometimes represented as a butterfly, and her ancient worshipers often dress as hummingbirds and butterflies to dance around her rose-covered image. Merlin Stone, author of _When God Was a Woman_ , says that \"when a woman felt the pleasures of her body, it brought special joy to Xochiquetzal.\" Imagine dancing amid rose petals, hummingbird feathers, and butterfly wings as a form of ritual adoration. Worship your divine femininity by pleasuring your own fleshy rosebuds.\n\nShakti is a form of the Hindu Great Goddess and symbolizes the energizing life force of the universe. She is often represented as a snake, uncoiling sinuously up the human spine, awakening the life-giving, sensual energies within. Through worship of this divine goddess\u2014which included adoring a woman's body as the goddess's human form and having sacred sex with her\u2014male Tantric practitioners hoped to reach enlightenment. Slither like a snake and feel how this undulating movement awakens your sexual instincts. Sense the creative power of the universe uncoiling in your womb, ready to give birth to new life, new ideas, and new inner enravishments. Worship your own body as the physical manifestation of all that is divinely female\u2014beautiful, powerful, innately enlightened, life-nurturing, and sensual. You are the embodiment of sacred union.\n\nI also recommend that you read Hallie Iglehart Austen's inspired and gorgeously illustrated book _The Heart of the Goddess_ for poetic and practical journeys with scores of other goddesses from many cultures and time periods.\n\n## sex initiatrix\n\nAnother friend of mine likes to fantasize that she, as the officially recognized Sexiest Woman in the World, is teaching an inexperienced lover the art of pleasing a woman. Or she is a tribal sex shaman who is initiating a young boy into manhood during a sacred puberty ceremony. Or she is Sharon Stone, reducing otherwise strong men to butter simply by crossing her legs or dancing provocatively with another woman. Whether she indulges in these fantasies while standing in line at the grocery or when she is pleasuring herself or her lover, she finds it a rapid and surefire way to make her sexiest inner self come alive. And it's no wonder, for initiating virginal or emotionally unawakened men into the intoxicating mysteries of the female body is a powerful ritual we women have been performing for countless centuries. Try it and see if it doesn't awaken magic in you.\n\n## the heart of beauty\n\nThe Hawaiian shamans have a lovely way of connecting with their inner graces. They suggest finding something beautiful, like a rose or a pretty stone or a piece of yellow silk. Focus on its beauty, singling out and appreciating all the various things that make it lovely. Find the qualities it shares with you. For instance, notice the way the open petals of the rose resemble your pretty vaginal lips. Observe that the stone has many different sides, facets, and colors, like your multifaceted personality. Revel in the cool smoothness of the silk, like your satiny skin. The more you focus on the textures and senses of beauty, the more your subconscious starts to bring it into your life, the more you see it everywhere, and the more your mind and body begin to mimic these innate qualities of your Inner Sex Goddess.\n\n## the temple of erotic love\n\nOne of the most potent pathways to your inner sexual sanctum leads directly through the Jade Gate, as the Chinese call it\u2014your vaginal opening. Though Far Eastern and Indian Tantric cultures have bestowed upon the female sexual organs poetic names like the Grotto of the White Tiger, the Honey Pot, and the Valley of Joy, modern Western culture often views them as something unclean to be hidden and ashamed of. At best, \"women see their sex organs as an absence rather than a presence,\" laments a well-known Western gynecologist. But ancient and indigenous societies have long honored the vulva in particular as the primary source of life, pleasure, and spiritual development. Identifying it with the lotus\u2014flower of perfection and eternity, and the great void from which all existence arises\u2014the people of south India rubbed carved wooden yonis (vulvae) as a form of worship. Statues and paintings from South America, Africa, ninth-century Europe, and twentieth-century Australia too show goddesses displaying their sacred vulvae with authority and pride as the source of fertility, inspiration, and protection. Many of these statues appear over doorways as a reminder that the vagina is the literal gateway to life. Numerous churches and castles in the British Isles of the ninth century were thought to be sanctified by sheela-na-gigs, stone carvings of playful, smiling women with their vulvae spread wide. These peoples wisely believed, as did the tribal wizards of New Guinea, that \"All magic radiates from [the vagina] as fingers do from a hand.\"\n\nThat kind of reverence, respect, and love for this sacred part of a woman is an essential key to releasing the Inner Sex Goddess locked inside. So, like a female Indiana Jones on a quest for your true sexuality, you must transform what for you may be a mysterious void into a sacred temple of erotic love.\n\nThere are several ways to undertake this quest. You could make a list of all the wonderful things you can think of about your vulva\u2014for example, as the birth canal, it's the gateway to life; men go absolutely bananas over it; it provides the highest sexual ecstasy; its many amazing talents include vast expansion, tight contraction, and copious lubrication; it is cleverly hidden; its lips are provocative protrusions or delicate wings, whatever the case may be. Make a long, sensually detailed list. Or you could practice the following meditation.\n\nFirst, honor the entrance to your temple of erotic love by rubbing oils or powder on it. Place your hands over your pubic bone and vulva, feel the powerful energy that emanates from this cave of creation. As you allow the muscles in your stomach, genitals, and anus to relax, imagine that the warmth and love in your hands is being pulled into your vagina, illuminating the dark spaces and making the walls glitter with a jewellike brilliance. See this inner space being transformed into a vaulted temple, an elaborately decorated mosque, a holy shrine, a vast cathedral, or a shaman's magical cave. This is the magnificent entrance to the mysterious worlds of sexual ecstasy, sacred union, and the beginnings of life. A male friend once told me, \"When you give a woman oral love, your head is cradled at the source of woman, the source of life. It's a sacred place where you can worship, and the divinity you are worshiping gives you instant feedback and the most otherworldly sensations.\"\n\nAllow yourself to luxuriate in your hallowed temple of erotic love. Explore its inner recesses, discovering the sacred sensual secrets hidden there. You may want to imagine that you are one of many priestesses studying your art together or that you are performing some sexual healing for a man who has petitioned the temple goddess for rejuvenation. Learn from these activities. Ask the deity of this temple, your Inner Sex Goddess, to cleanse you of any negative feelings you may have about this sacred space and to fill it instead with reverence and love. Inquire how you may come to know her better and help her be a larger presence in your life. Just for a moment imagine that the goddess magically embodies herself in you and that you have all her knowledge, majesty, charisma, and heavenly experience. Promise her that you will always honor her temple and request that she be instantly and fully available to you whenever you merely think of this part of you. Thank her for inviting you here and gently take leave of your temple of erotic love for now. Rest in the knowledge that it and your Inner Sex Goddess are always there to nourish, support, and eroticize you.\n\n# 2  \ncreating your sensual garden\n\n_Because I am a goddess I will come to your heaven, and lie on your cloud.... Let my hovel be our heaven, let my bed be our cloud, On which we are united in perfect, heavenly bliss_.\n\n_\u2014 Erotic poetry of ancient Egypt, from Sacred Sex, Robert Bates_\n\nthe essence of being a goddess is that you transform even the most mundane events and surroundings into a celestial garden of delights by your very existence. Cultivated and in full bloom, your inner garden grows to encompass the world around you, expanding your capacity for pleasure, luring divine consorts to your domain and elevating the quality of life and lovemaking, for you and for your lover, to sublime heights. It is your Inner Sex Goddess, who, by the simple act of viewing the entire world as a sensual paradise, creates this erotic garden, first in your own life and then by divine osmosis in the lives of those around you.\n\nNow that you recognize your inherent, natural potential as a goddess, your first divine act must be to create the sensual garden that sustains and enriches your inner life and provides a perfumed playground for all who enter it. Start by tasting the ambrosia of life, reaching out and grabbing the sensuous feeling of wind in your hair or silk on your skin, savoring the magic of a striking piece of art, reveling in the mystery of a blossoming flower, sinking into the romance of a poetic phrase or a lover's sigh. These simple yet profound acts of transformation are your natural birthright as a woman.\n\nWhile men have the gift of robust action, women have the gift of powerful perception. Sensually, men tend to focus on their genitals, whereas women are more holistic, more interested in arousing their entire bodies and their feelings. A man often wants to get right to the physical pleasure, while a woman knows how to heighten sensitivity by first creating the mood of romance. She knows that stimulating your \"romance hormones\" will create the context for the magic that can come from stimulating your sex hormones. In fact, many women find that if they pleasure themselves and their lovers only in a physical way, they feel bored, dishonored, and cut off from their deep feelings and true Sex Goddess nature.\n\nAt the same time, though, women intuitively understand that the physical senses are the very doorways to inner eroticism\u2014the openings through which magic passes from the outer to the inner world and back again, the alchemical tubing that transforms sunsets, cantatas, and freshly baked bread into the hot pheromones of love. To a woman's Inner Sex Goddess, the whole world is an aphrodisiac. It's simply a matter of noticing it. And a woman becomes a Sex Goddess not by virtue of a perfect body, a bag of exotic sexual tricks, or a Valentino-like lover but by virtue of her ability to abandon herself to the electrically sensual qualities of everything around and within her and to pamper herself with them.\n\nSo if you want to make your Inner Sex Goddess a larger part of your life, use all of your senses as doorways between the exterior world and the magical interior kingdom where she dwells. See, hear, taste, smell, and feel her in every fire flame, zesty orange peel, and cat's purr. Leave room for your efficient, taking-care-of-business self, yes; she's important too. But right now, it's your sultry inner seductress that we're showering with attention. Deliberately create occasions for her to come out and play. Pamper, treat, and romance her all the time. She will turn your everyday life into a garden of sensual delights, bring your spicy inner flavors to the surface, and galvanize your man to new erotic ecstasy.\n\n## sowing the seeds of sensuality in your garden\n\n_1_. Send yourself flowers. Fresh, fragrant flowers not only make you feel beautiful and adored but also provide a pretty-to-look-at and heavenly-to-smell prop for your bedroom, or for your bathroom, kitchen, or living room\u2014wherever you decide to take your self-love tryst. Get all of your senses involved with the flowers\u2014arrange them in a pretty vase and place them in a different spot every day; inhale their fragrance deep inside you; caress your skin with their velvety petals; garnish your food with them. Let your Inner Sex Goddess select the bouquet for its sensual appeal\u2014two blood-red roses, one lushly fragrant gardenia, several exotic and silky tiger lilies. Get luxurious!\n\n_2_. Give yourself perfume. Your favorite kind. Something that smells wickedly sexy. Spray your sheets with it. Dab it behind your ears, between your beautiful breasts, along your soft inner thighs. Smell your luscious self. One hedonistic friend of mine likes to apply a musky scent between her toes and then massage them to bring out the aroma. She says it makes her feel pampered.\n\n_3_. Have plenty of romance-making candles around\u2014tapers for elegance, votives to create a soft glow, big fat ones to help you recall phallic encounters, aromatic ones to awaken your earthy nature. Light them in the bathroom when you luxuriate in the tub. Fill your bedroom with many flames of love. Look at your gorgeous body in the mirror in shimmering candlelight. Fantasize that you're performing a magic sex ritual by candle flame and the light of the moon. Become aware of the incandescent quality of your Inner Sex Goddess. Glow.\n\n_4_. Prepare a candlelight dinner for one. Set a romantic table with flowers, special dinnerware, candles, a stunning centerpiece. Indulge in aphrodisiacal appetizers such as oysters, avocados, mangoes, figs, or other juicy fruits. Savor an entr\u00e9e that you can eat with your fingers, and lick them frequently. Indulge in a sinfully rich and gooey dessert. Relish your food. Inhale its aroma. Lick it. Tear it apart with your fingers. Taste it slowly. Fruits are lovely to rub on your body and gently, slowly lick off. Don't do the dishes. Go directly to bed with yourself.\n\n_5_. Treat yourself to a bottle of good wine. Red wine makes me feel much more sensuous than white or pink, but follow your own sensual urges here. A wine with a good nose will add another texture. Pour it into a delicate wineglass. Savor it. Let it slide warmly over your tongue and down your throat. Pour it on yourself and lick it off. Allow yourself to slip into the relaxed, uninhibited playground of Dionysus, the god of wine. Become his goddess consort.\n\n_6_. Humanistic psychologist Abraham Maslow stated that the two most common triggers for peak experiences are music and sex. How about music and sex together? Put on some evocative music and listen to it with your whole body. My Inner Sex Goddess loves cool, sultry jazz, but other women I know prefer hot, driving salsa, romantic Mathis or Vandross, lush and elegant Ravel or Chopin, dreamy Debussy, the fireworks of Tchaikovsky, earthy blues, electric, pulsing rock, or ethereal flute music. (See the \"Inner Sex Goddess Scrolls\" at the back of this book for lots more specific musical suggestions.) Play whatever turns you on at the moment. Close your eyes and slip inside the sound. Let it loosen your pelvis and shake your shoulders. Slink around in it. Sing or hum. Feel your voice vibrate your chest, throat, and face. See the colors the music creates behind your eyes. Taste its brassy, peppery, or creamy essences. Exult in the emotions it provides. Melt into the harmony of the spheres.\n\n_7_. Dance. Naked. When there's no one else around to criticize your technique, your Inner Sex Goddess is free to undulate, writhe, slither, rock 'n' roll, hula, fandango, samba, or even strip to her heart's content. Close your eyes and surrender to the mood of your movement. Or dance in front of a mirror and see how sexy you look with breasts jiggling and hips swaying. Dance _with_ the mirror; press up against it and feel its cool slickness against your skin. Make a hot, moist patch on it with your breath and flatten your nipple against it. Perform the dance of the seven veils with a big filmy scarf or a fringed shawl. Be provocative. Not only is this fun, loosening, limbering, and hormone-stimulating, it's also great practice for the sexy striptease you're going to do for your man someday soon.\n\nIf you feel totally uncoordinated or sexless when you dance, then try almost any other form of physical movement\u2014yoga, aerobics, stretching to music, walking, tai chi, swimming, running. I have a shy, pixielike friend who is actually embarrassed to go to the gym because she gets overly excited by the movements of her thighs while using the StairMaster!\n\n_8_. Give yourself some sexy lingerie. I love to wear lacy camisoles under my sweaters, blouses, and blazers, letting enough show so that it looks very feminine and demure but at the same time revealing and sexy as hell. My Inner Sex Goddess simply adores peekaboo lace. My friend Jennifer, on the other hand, always wears satin panties and matching plunging bras under her accounting manager pinstripe suits. She says it helps her remember she's a shameless hussy. Let your Inner Sex Goddess investigate and supplement your wardrobe. Try sleeping in a silky teddy instead of your regular oversized T-shirt. Be aware of how these textures glide and flutter over your skin. Admire yourself in a mirror often. Pose like a Victoria's Secret model. _Feel_ like one. Buy and wear\u2014if only for yourself\u2014something that makes you look and feel like an expensive call girl. My friend Lisa, a harried mother of twin three-year-olds, says the easiest way for her to feel sexy on command is to don her see-through merry widow with lacy garters and ribbons that untie in strategic places. When she wears it with lace-top stockings and five-inch heels, she says she looks and feels deadly. Sometimes she adds her G-string silk panties. Of course, she often sports these duds for her man, but just as often she wears them only for herself. They make her feel expensively sexy. And they become exciting props for self- or partner-lovemaking.\n\n_9_. Dress sensuously all the time. This does not mean you have to look overtly sexual; obviously that's not always appropriate. It means that you wear things that feel sensuous and soft; fabrics and shapes that heighten your awareness of your skin and movements. Even with your most severe business suit or your sportiest tennis togs, you can add at least one little thing, seen or unseen, that has a sensate quality to it. Silk, satin, cashmere, lace, Lycra, pearls, filmy scarves, fur, velvet, earrings that dangle against your neck, red lipstick, a push-up bra, see-through panties, a flowing skirt that billows against your bare skin, lots of bracelets, a snake armband worn on the upper arm, a wide choker, short tight skirts, silk or lace T-shirts, anything leather against your skin. Go braless. Go pantyless. Definitely go slipless, unless it's a sexy one.\n\nIt's amazing how sexy you will feel when you go without panties, particularly when you're wearing a long, flowing skirt that swishes when you walk; you're all covered up, but you feel so deliciously _exposed_. Whether your sensuous attire is demurely hidden from public view, proudly flaunted for your lover, or unabashedly admired in your own mirror, your Inner Sex Goddess will be softly vibrating and ready for further attention.\n\n_10_. Soak in a hot, scented bubble bath. Spend hours at it. Read a steamy novel. Tease yourself with a soft skin brush. Bask in the glowing candlelight. Dream. Fantasize. Croon to yourself. Paint your toenails. Smooth the bubbles over your face, neck, arms, breasts, tummy, thighs, genitals. Arch your feet coquettishly and tickle your glistening toes. Massage your scalp as you wash your hair. Let the water drain while you're still in the tub. Or finish off with a cold shower. Pat yourself dry with the biggest, fluffiest towel you own. Or don't use a towel at all, and move directly into a thick terry-cloth robe. Or smooth perfumed body lotion over every inch of your skin. Float into your warm bedroom, and if there's a man in it, wake him up seductively. If not, sizzle to your own inner music.\n\n_11_. Read something erotic. Curl up with a good story by Ana\u00efs Nin and identify with the hot French coquettes she writes about. Transport yourself to another time and place where anything sexy, a little kinky, and very feminine goes. Like the women in the stories, allow yourself to be carried away by your passions until you feel that familiar warm glow begin to burn inside you. Then let your hands take it from there. There's lots of other good erotic literature out there too, something to fuel every fire, from the seductive S&M stories of A. N. Roquelaure to the taboo and poetic writing of Marco Vassi to anthologies like _Deep Down: The New Sensual Writing about Women_ and _TheGates of Paradise_. There's always Nancy Friday's _My Secret Garden_ as well as many other collections of women's fantasies and even some of the letters and stories in magazines like _Playboy_ , _Penthouse_ , and _True Romance_. (See the \"Inner Sex Goddess Scrolls\" at the back of this book for a more complete list.) Or maybe a hot love scene from your favorite romance or trashy novel will light your inner fire. One highly organized friend of mine makes a copy of every good love scene she finds and puts it in a hot-pink file folder that she keeps locked in her nightstand drawer. Some of them are pretty dog-eared by now, but they never fail to arouse her passionate nature. Find your own favorites and keep them handy for reading to your Inner Sex Goddess or your real-life sex god.\n\n_12_. Write your own erotica. Even the feel and sound of pen on paper as you write can be a sensual delight. Release your erotic feelings, hopes, and dreams onto the paper. Let your pen take flight, and follow its fanciful wanderings. Express how you felt when you last made love; recall the sensations that stirred you emotionally, sink into the feelings, and let them pour onto the paper. Invent a wildly romantic interlude in the forest, and write down exactly what you'd like to have happen. Describe a piece of music or a part of your body in sensual detail. Imagine your ideal lover and give him written instructions on how to drive you wild in bed. Tell the story of your bizarre sexual experiences on another planet. Pen an imagined page from the journal of Fanny Hill, Cleopatra, or Oshun, the river goddess. Write intensely brief poetry, cool factual journalism, or hot turgid prose. Create a passionate romance novel or a porn comic book. Try on different writing styles to suit your mood or to create a new one. Since no one will be grading this paper, really let yourself go. Don't be afraid to dig down deep inside you and reveal your heart's secret erotic desires.\n\nA single real estate broker I know told me she once spent a lone day at the beach feeling the blazing sun on her skin and fantasizing about a hot lover who would sneak up and inflame her body even further with steamy kisses. Later, feeling compelled to pour her sensual feelings onto paper, she wrote a rhymeless poem describing her fantasy love fest. She said that concentrating on finding the right words helped her relive and enhance the experience, and rereading it now and then recaptures all those sizzling skin sensations and simmering passions. Her poem turned out to be one of her most treasured personal turn-ons.\n\n_13_. Ogle sexy pictures. Your Inner Sex Goddess is much more visually erotic than you might imagine, so give yourself permission to open the optic channels to your libido. She loves those bare-chested men in Calvin Klein ads, lovers in the ecstatic embraces and wild positions depicted so beautifully in Oriental woodcuts and Indian miniatures, pictures of luscious babes who look rather like you in men's magazines, beefcake postcards (one woman I know collects a few eye-poppers in every town she visits), hunky-man calendars (my calendar for this year happens to be \"The Men of Hawaii,\" and are they hot!), nude photos you've taken of your lover, the sexually explicit drawings found in _The Joy of Sex_ and other sex instruction books, museum postcard reproductions of beautifully sculpted nude Greek statues, the cover of a steamy novel, magazine photos of movie stars in action. Actually, some of the most erotic images I've ever seen are photographs or paintings of nature\u2014Georgia O'Keeffe's paintings of flowers that look like women's vulvae; beautifully rendered still lifes where the soft fruit or flowers resemble swollen lips, rounded breasts, or plump behinds; photos of dense redwood forests exuding damp earthiness and phallic sensuality; drawings of curvy, openly inviting shells glistening with moisture.\n\nCollect your personal preferences and keep them at the ready in your erotic file folder. Drool and fantasize to your libido's content. Imagine you are _in_ the picture, and follow your bliss.\n\n_14_. Watch X-rated videos. Yes, real women watch porn flicks. In fact, your Inner Sex Goddess can literally be transfigured by visual eroticism. One of the best descriptions I've ever heard of this phenomenon was reported by Sallie Tisdale in her fabulous book _Talk Dirty toMe:_ \"Watching my first adult movie, watching for the first time a man penetrate another woman, was like leaving my body all at once. I was outside my body, watching, because she on the screen above me _was_ me; and then I was back in my body very much indeed. My lust was aroused as surely and uncontrollably by the sight of sex as hunger can be roused by the smell of food.\"\n\nAdult videos are much improved these days, and often you don't even have to fast-forward to get past all the boring or stupid stuff. For sleek erotic atmosphere with gorgeous men and women, try Andrew Blake's films. Candida Royale and her Femme Productions put out many sexy films that are more romantic and woman-centered. These and many other modern hard-, soft-, and medium-core porn videos (see \"The Inner Sex Goddess Scrolls\" for a more complete list) provide a great tool for expanding your sexual horizons and acting out vicariously what you might never do in real life. Whether you're into fantasizing about vampires, whips, bisexuality, group scenes, or just plain sizzling sex and plenty of it, there's a film made just for you. Your Inner Sex Goddess can also learn some interesting new tricks to try by yourself or with a partner. As my fiftyish friend Judy said when she loaned me one of her favorites, \"I thought the film was trash, but it made me laugh, think, and swoon all at the same time. I couldn't wait to plug in my vibrator.\" I recommend that you (1) read the package carefully to get a feel for how kinky, outlandish, or romantic the film is; (2) trade videos and personal reviews of them with friends; (3) watch alone at first; and (4) watch with a vibrator in your lap.\n\n_15_. Collect, admire, and play with erotic objects. Joanna, my plain but profoundly charismatic soul sister, likes to collect phallic objects, for instance. Her favorite is a two-inch-long crystal penis, every detail true to life. Obviously too small to do anything with, this toy is just for her Inner Sex Goddess to fantasize over. I have a stunning satinwood statue of a man arching over a woman's shoulder to kiss her breast; I love to gaze at it and stroke it. My pleasingly plump friend Marsha collects beautifully tasteful representations of women's pelvises and derrieres; they make her feel good about her own. Another friend likes lots of velvet pillows for looking, touching, and rubbing against sensitive body parts. Some women find that certain wild animals and the coiled power they represent arouse sexual thoughts\u2014panthers, snakes, and stallions, for instance\u2014and they enjoy having pictures, statuettes, or stuffed toys of these creatures around to inspire, admire, and pet. Pretty feathers, and even feathery plants, can be a sensual delight for the eyes and skin. Having a fetish for erotic collectibles not only results in some fabulous toys for your lovemaking lair but also keeps your sensual self alive and tantalized all the time, always on the lookout for a new addition to your secret sensitizing collection.\n\n_16_. Aroma is another powerful stimulus for your Inner Sex Goddess. In fact, the ancient Egyptians, believing that fragrant oils were created by the gods, used scents to evoke in themselves the joyous sensuality of Hathor or the powerful regenerative qualities of Isis, for instance. Unleash your inner goddesses by inhaling deeply of _all_ the \"essential oils\" of life\u2014ocean air, baking bread, freshly washed hair, damp earth, baby skin, lemon peel, the musky aroma of sex, and especially flowers and their liquid essences. The lush smell of white ginger instantly unlocks all my voluptuous instincts, so I always return from Hawaii with ginger sachets to tuck in my drawers and pillows. Try placing a drop of your favorite essential oil on a bedside lightbulb or on melting wax as it drips down the side of a candle. Spray gardenia perfume in your linen closet and later unfold the fragrance onto your bed. Dip a cotton ball in jasmine oil and nestle it between the mounds of your breasts. Bathe in lavender-scented water. Saturate a lacy white handkerchief with ylang-ylang oil, wear it next to your skin for a while, and then carelessly leave it lying around in one of your man's favorite haunts.\n\n_17_. Burn incense. A room filled with the heady aroma of musk, rose, jasmine, ylang-ylang, or amber, one of my favorites, somehow oozes sensuality. Like the ancient Tantricas, use the pungent smoke of your favorite incense to activate and eroticize your chakras (energy centers). Hold a stick of burning incense near your pubic bone and then move it slowly up the front of your body all the way to the top of your head, pausing for a few moments at your genitals, diaphragm, heart, and throat. Keep the incense about two inches away from your skin. Somehow this raises your sexual temperature and infuses your body and mind with heat, preparing you for an unusually sensual experience.\n\n_18_. Develop an eye for the erotic qualities of nature. Women are especially talented at seeing and feeling the sensuous beauty in a vivid sunset, a purple craggy mountain at dawn, frothy ocean waves, emerald green rolling hills, tall grasses rippling in the wind, a certain iridescence of light and shade in the forest. Feel the thickness or sweetness of the air at the beach, in the mountains, in a forest glade, in the steamy jungle. Smell and taste it. Gardeners know that massaging wet dirt can be a sensuous experience. Caress the tender petals of a flower. Eat a blade of grass. Close your eyes and let the sound of crashing ocean waves wash over your body. Trickle hot sand over your feet and legs. Press your body against the rough bark of a tree. Walk barefoot in the grass. Feel with your eyes as you watch a horse run, a cat stretch, a bird soar, a snake slither, a butterfly float, a tiger spring, or a dragonfly mate.\n\nIf you live in the city and can't get away, take your Inner Sex Goddess to a museum or art gallery and let her drink in the sensual images of nature depicted in fine works of art. Luxuriate in Monet's sumptuous gardens or John Constable's overpowering, detailed landscapes. Absorb the subtleties of light or merge with the stillness in a photograph. Become intimate with Ansel Adams and his stunning black-and-white adorations of canyons and deserts. Imagine yourself into the setting of a van Gogh and let your animal senses run wild.\n\n_19_. Mine your past for experiences and sensations that touch off interior sparks. Old love letters, postcards, snapshots, prom mementos, valentines, love songs, discarded clothing of his and yours, and even photos of a magnificent beach you went to by yourself are a living treasure chest of evocative sense triggers. Close your eyes and submerge yourself completely in the sights, sounds, and smells associated with that memory. Feel his arms around your waist as you danced to that old song; taste the souvlaki and baklava from that romantic trip to the Greek islands; drink in the smell of his skin when he wore that old flannel shirt. Vividly recall the rapid beating of your heart and the warm tingle under your skin that burned these moments into the very fibers of your body. Your Inner Sex Goddess lives in those vibrant sense memories.\n\n_20_. As a little girl, you knew the power of your mother's lipstick and high heels to transform you into a sexy, grown-up chick; or a clown's hat to bring out your silly oaf side; or a bandanna and a stick for a horse to turn you into a Wild West cowgirl. This stuff still works, even on your fuddy-duddy adult self, because Your Inner Sex Goddess adores the game of dress-up. Try on any persona that excites your erotic imagination and act the role to the hilt in front of your private mirror. Adorn yourself like a tart: net stockings, garter belt showing beneath short tight skirt, five-inch heels, brightly rouged cheeks, and teased hair. Wriggle your behind, whip your handbag around, expose your push-up bra and crotchless panties, and stand disdainfully arched against the doorway to attract imaginary customers. Dress up like a harem girl, with floating sheer scarves, beaded vest, lots of jangling jewelry, ankle bracelets, heavy Cleopatra eyeliner, and face veil. Pull the scarves provocatively across your breasts and through your legs; with head down, peek out from behind your veil with inviting eyes; let your scarves and ropes of jewelry fly out from your body as you twirl, dance, and undulate. Or be a demure geisha and don a kimono, wide silk sash, chopsticks in your piled-up hair, and the whitest makeup you can find. Geishas consider the nape of the neck very erotic, and they extend their white makeup around to the back of the neck, making a long, pointed pattern with it in the space where their kimono collar falls away. Step mincingly about the room, flutter your fan in strategic places, purse your lips, and enticingly caress your brazenly exposed neck and ankles. Or transform yourself into a hula dancer by baring your breasts, tying a sarong or big scarf low around your hips, and arranging flowers and leaves in your hair. Sway your hips, raise your arms gracefully to one side, and perform the sensuous _ami_ movement by rotating your hips around in one big continuous circle. Slip into the memory of how you looked and felt, or take your harem scarf or your net stockings to bed with you, the next time you want to entice your Inner Sex Goddess or your consort lover.\n\n_21_. Try the practice of vivification. The Native Americans believed that everything\u2014rocks, trees, streams, animals, clouds\u2014had a living spirit, and they addressed each one with the same respect, awe, and love they would give to any divine being. When you become conscious of every flower, shell, fig, painting, or piece of music as a living essence of sensuality, its vital spark will come alive for you, revealing all its rich colors, secret smells, and soul-tingling textures. And in getting to know this voluptuous being, you will be transformed. As Joseph Campbell said, \"The [self] that sees a 'thou' is not the same [self] that sees an 'it.' \"\n\n## your body: the temple within the garden\n\nWe don't need to attend a sex seminar, travel to an exotic locale, or search out the perfect lover to open wide the doors of our true sensuality. All we have to do is be fully present right here in the sacred temple of our own bodies. As the Boston Women's Health Collective has so brilliantly put it, \"Our Bodies _Are_ Ourselves\" (emphasis mine). Through the senses of our bodies we know the world and ourselves. Through our bodies we _know_. We know what we want and need. We know what we feel. We know what to do. We know who we are. We know ourselves as women. In fact, the poet Adrienne Rich has written that women \"think through the body.\"\n\nBy becoming more aware of our bodies, bringing our attention to each part as we observe or touch it, we tap into the deep wisdom contained in these our miraculous sensate morsels of flesh. We begin to learn the language of our instincts, which direct and guide us when we make love. All too often, though, we let our heads take the lead in bed and leave our wise bodily instincts behind. This separates us from our full sensuality and leaves us feeling unfulfilled, as though something were still lacking. It is! What's missing is the full experience of the senses we are trying so hard to stimulate, and the powerful sensate wisdom of our bodies. So often we criticize, ignore, or try to change our bodies. But without them, how could we have the wonder that is sex and sensuality?\n\nYour Inner Sex Goddess knows that in order to be a great lover, with your man or with yourself, to enjoy lovemaking and discover new sexual delights every day, you have to be fully present in your body. You have to revere it as the sacred temple of your sensuality and consciously inhabit all of its rooms, hallways, and turrets. So when you want to tempt the voluptuousness out of your bones or turn your lover into hot mush, start by taking the time to consciously inhabit the temple of your body. Turn your mental switch from outer visual mode to inner sensate mode. Really _feel_ the sensations of your skin. John Gray, author of _Men Are from Mars, Women Are from Venus_ , says that women have more \"cuddling hormones\" than men and that the skin all over their bodies is ten times more sensitive than men's, as sensitive as the skin on a man's penis. So take advantage of those extra pleasure sensors. Concentrate on the tingles, the tickles, the heat rising and falling, the waves of pleasure. Luxuriate in your body. Here are some things you can do to increase your awareness and turn on your body senses.\n\n_22_. Rub your hands together rapidly for about ten seconds before you touch your body. This (1) brings your awareness to your hands and the physicalness of your body, (2) sensitizes your hands, (3) makes them nice and warm, (4) creates energy that you will definitely feel when you touch your hands to your body. Watch out for tingles!\n\n_23_. Close your eyes and take several slow, deep breaths. Feel the breath traveling to every corner of your body. Let it tingle. Imagine that you are breathing in through your breasts and nipples. Let them become filled with heat and energy. Focus intently on the subtle sensations and a new sense of presence in your breasts. Then imagine breathing in through your vagina. Again, really concentrate your attention there. Feel its fullness, warmth, and energy.\n\n_24_. Another sensitizing breath technique that comes from the Tantric tradition is called the Breath of Fire. Sit up straight and pant very fast and very hard. Make a hissing sound as you inhale and exhale. Continue this for about fifteen seconds (you can work up to two minutes, but take it slowly at first). Then inhale very deeply, hold your breath for a count of five, and exhale fully and slowly. As its name suggests, this technique will ignite your sexual fires; and if you want to direct them to a specific place in your body, you can do so simply by focusing your attention there as you perform your final slow exhalation. Feel the hot inner breath actually traveling to your favorite erotic hot spot.\n\n_25_. During rhythmic activities like running, rowing, singing, and making love, people's breathing will often synchronize. If you consciously model your breathing pattern after the excited panting or deep moaning rhythms of peak sex, you will find your body and mind responding with the physical sensations of arousal, even releasing titillating pheromones.\n\n_26_. Contract every muscle in your body as tightly as you can for five seconds. All at the same time, clench your fists, hunch your shoulders, curl your toes, pull in your stomach, tighten your leg and arm muscles, scrunch your eyelids shut. Contract everything. Then release your tension completely for five seconds. Do this five times. Finally, relax, relax, relax. Let go. See what happens.\n\n_27_. Lick yourself all over. Start by sliding your tongue across the tips of your fingers. Then travel down to lap your palm, encircle your wrist with a wet tongue, lick the back of your hand and your knuckles, trace a tongue line down your forearm with occasional nibbles along the way, and flick your tongue in the hollow of your elbow. Continue languorously over the rest of your body. When you are with your lover, ask him to join you by licking and biting your other arm.\n\n_28_. Sensitize your head by massaging your scalp or by shampooing your hair while imagining that a lover is doing it, brushing your hair for a long time, pulling slowly on your earlobes, tracing circles on your scalp with your fingernails, pressing the hollows beside your eyes and behind your ears, tapping your head all over with your fingertips, rubbing your scalp with a rough towel. Close your eyes and let these physical tingles awaken the sensual you.\n\n_29_. Bathe in the hot sun\u2014nude, if possible. Sunlight actually increases the level of the hormone androgen, the female counterpart of testosterone, filling your body with delicious liquid desire and bringing your Inner Sex Goddess to life.\n\n_30_. My friend Gloria has developed a simple but elegant way to be deeply in touch with the sensations of her body. She says that when her erotic feelings first start to blossom, the bottoms of her feet start to tingle. She then focuses on moving that tingle up to her vagina, or wherever she wants it, and she moves and breathes in ways that keep that prickle of excitement growing, no matter what else may follow. Whatever your body's calling cards of arousal are\u2014tingling feet, melting chest muscles, burning palms, soft tongue\u2014focus on those sensations and visualize them expanding into your fingers, breasts, pelvis, toes, head, wherever. Feel them actually traveling along your nerve pathways, glowing, heating, and thrilling your entire body.\n\n_31_. Rub your skin with fur, velvet, chamois, felt, honey, olive oil, cold cream, rough facial cleansing cream, suede, cold cans of soda, fresh porcini (wild mushrooms whose firm silkiness feels like the skin on the tip of your man's penis). Stand in the rain, run through the wind, dip your hands in snow, submerge yourself in a cool, bubbling brook. Lie naked on a rug, a wooden floor, the sand, a bed of leaves. Ask your Inner Sex Goddess what her favorite skin treat is and indulge yourself with it.\n\n_32_. Sensuously apply perfumed lotion to your entire body. Don't forget the backs of your knees, your underarms, the nape of your neck. Really focus on the physical sensations as you do this\u2014the place where you are rubbing, your hands as they caress your skin, the heady aroma. Be aware of every one of the millions of nerve endings all over your skin.\n\n_33_. Give yourself a massage. It's a good idea to trim your nails first so you can use your fingertips without getting stabbed. Take your time and be very sensuous about it. Slather yourself with oil, scented or unscented. As you move your hands over your body, focus your attention on the delicious sensations that are exchanged between fingers and skin, each experiencing its own distinct pleasures. Start at your feet and slowly work your way up your legs and torso, leaving your genitals for last. Or begin with your face and scalp and work your way down. Spend lots of time on your fingers and hands because they are extremely sensitive to touch and because they take quite a beating during your busy day. Use long, sensuous strokes that go deep into the skin, or work in soothing circular motions. Large, two-handed circles over the tummy are especially nice. Let your hands discover all those little knots of tension, and knead them out. Cup your hands over your eyes, breasts, and pubic bone. Caress, hold, and cradle your body the way a lover would.\n\n_34_. As you are performing number 32 or 33, do the following body awareness exercise. If you have started with your foot, for instance, close your eyes and \"listen\" to the sensate messages your foot is sending you. How does it feel? Does it want stronger or gentler pressure? Lift your hands for a moment and be aware that the foot you have stroked probably feels lighter and more vibrant than your other foot. As you massage your foot, tell it how beautiful and strong it is, and thank it for supporting you so well. Let your foot drink in all this pleasure. Continue in the same manner with every part of your body.\n\n# 3  \nself-pleasuring\u2014the key to the kingdom\n\n_We human beings are able to make love more frequently and sensuously than perhaps any other animal. Yet we are often disappointed after lovemaking. Why? Because most of us are like owners of a precious Stradivarius violin that we have never learned to play_.\n\n_\u2014 Jolan Chang, The Tao of the Loving Couple_\n\nas a sacred temple, your body is the divine instrument of pleasures fit for the gods. And before you can use it to make beautiful music with your partner, you have to become a virtuoso on your own Stradivarius. In order to seduce others with your heavenly melodies, you must first seduce yourself by drawing out your deep, wild sexuality, turning on the magnetic firelight of your body, and inflaming your own delicious passion. Even the gods and goddesses of ancient Egypt obeyed this basic rule, for among them it is said that the sun brought the world into existence by first pleasuring itself. After that, it was the female sky who, by lowering her beautiful potent body over the male earth, created the union that produced life.\n\nAutoeroticism has long been practiced as a natural and elegant way of honoring the glorious gift of a body and communing with one's inner divinity\u2014from primitive cave drawings of masturbation to the erotic dildos found in ancient Egyptian tombs, to a Buddhist temple in Orissa that boasts beautiful bas-reliefs of men and women stroking themselves. Even today's sex therapists extol the benefits of autoeroticism as a vital tool for self-knowledge and relationship-enhancement, professing that sexual virtuosity with a lover is attained by first becoming a sexual virtuoso with yourself. Many ordinary women bear this out; they have told me that, despite the fact that their first experience with lovemaking was merely okay or even disappointing, they later learned to reach sexual heights and become extraordinary lovers by focusing on sensualizing their own bodies. Such was the case with me.\n\n## my story\n\nI had a very strict and sheltered upbringing that did not include getting information about sex, although I always suspected that something mysterious, powerful, and potentially dangerous or enjoyable (both equally discouraged) lurked beneath the currents of everyday life. During high school I necked and petted with my boyfriends, but genital contact was so out of the question that I didn't even think about it. Then, finally, at age nineteen, I had intercourse for the first time\u2014a somewhat boring encounter with a steady boyfriend that produced the required bloody sheet, but not much in the way of passion, excitement, or romance. Subsequent boyfriends helped me see that sex could be sexier than had been apparent in the first encounter, but my eyes weren't truly opened to the sensuality and power I owned for myself (and with which I could light all sorts of flashing sparks in men) until several years later when I bought a vibrator.\n\nA trusted friend had raved about the fun she was having with her mechanical instrument of pleasure, and I wanted to explore too. Suddenly I could concentrate fully on my own body and the delicious inner and outer feelings I was having without having a man's wants and needs to worry about. And because the vibrator was so intense and indefatigable, and because it would stay exactly where I wanted it to, I found myself making a discovery: \"Oh, _that's_ what an orgasm is!\" Over the next several months I explored every inch of my body and learned to love my hips, little toes, breasts, inner elbows, genitals, and earlobes for the voluptuous and powerful way they could make me feel. I gave up feeling embarrassed or ashamed about any of my body's shortcomings, and even about the fact that it could give me such pleasure. In fact, learning about my own personal poetry gave me greater self-esteem, zest, and natural charm. And the more I enjoyed my own body, the hotter and sexier I felt, the better lover I became, and the more men seemed to be attracted to me. That's why, when women say to me,\n\n\"I have a really great lover. Why would I want to play with myself?\"\n\n\"If I masturbate, it means I can't get a lover and have to settle for second best.\"\n\n\"I already know how to massage my hot button. Who needs anything else?\"\n\n\"My man feels too threatened by my vibrator.\"\n\n\"I masturbate every once in a while, but it's kinda boring.\"\n\n\"I'm afraid I'll get so aroused by women's bodies I'll become a lesbian.\"\n\n\"Isn't that really more for men?\"\n\n\"I could never do _that!\"_\n\n\"I don't really know how.\"\n\n\"I'm not oversexed!\"...\n\nI say that you can settle for good sex or even pretty great sex if you want to, but I believe that self-loving is one of the most vital keys to your inner kingdom of abandoned, expert, and rapturous sexuality. It's certainly one of the easiest, fastest, and most reliable ways to unleash your Inner Sex Goddess. And if you want some sensible, virtuous reasons to spend precious time and energy on what may seem like frivolous, threatening, or wickedly self-indulgent pleasures, here they are.\n\n### Reason # 1\n\nSelf-Pleasuring Is the Single Most Important Thing You Can Do to Become a Truly Great Lover.\n\nBy focusing on your own total fulfillment, you gain access to the complete you\u2014the savage you, the loving you, the lusty trampy you, the giving you, the unbridled you, the inventive and adventurous you, the magnetic you, the Aphrodite you, the instinctive wild you, the Sex Goddess you. You discover a whole new way of being a woman.\n\nYou have the keys to your body, your physical and emotional sensations, and your most lascivious libido. Self-pleasuring allows you to find out firsthand:\n\n\u2022 What most turns you on\u2014or off\n\n\u2022 What makes you feel less inhibited\n\n\u2022 What feels good real slow and gentle or very hard and fast\n\n\u2022 How long you like to be stimulated\n\n\u2022 How sensitive your nipples and clitoris are\n\n\u2022 Where your G-spot is and how it likes to be rubbed\n\n\u2022 The best way for you to reach orgasm\n\n\u2022 How to extend your sexual capacity into multiple orgasmic bliss\n\nBy becoming familiar with your sexual organs and the pleasure they can provide, you discover just how beautiful and powerful they really are; your genitals may even become your favorite feature. I know a woman who feels that her pink flowerlike inner vaginal lips are one of her strongest assets. She says men are fascinated with her large fleshy petals and want to stroke and kiss them endlessly!\n\nWhen you discover and awaken all of your personal hot spots and habits, when you find out for yourself what the tight contractions of your vagina feel like to a man, when you see how sexy you look with hips undulating, nipples erect, and in the full throes of orgasm, you have a visceral experience of the high-wattage sexual power that is yours to command, enjoy, and give. You _know_ you are a deliciously dangerous weapon of love.\n\nAnd when you're in firm control of all your rich sexual assets and weapons, needing nothing outside yourself, yet wanting and commanding the whole world of sensuality, you are prepared to fully inflame your man's body, his mind, and that most powerful erogenous zone of all\u2014his imagination.\n\n### Reason # 2\n\nSelf-Pleasuring Makes You Feel Sexy.\n\nIn _203 Ways to Drive a Man Wild In Bed_ , I explained that the first secret of great sex is \" _Feel_ sexy, and you will _be_ sexy,\" One of the fastest, surest, and most enjoyable ways to summon up that sense of power and sensuality is to titillate your body and mind with some self-loving.\n\nThat's because thinking about sex makes you feel sexy; preparing to have sex makes you feel sexy; knowing and appreciating your body makes you feel sexy; loving your genitals makes you feel sexy; seeing how beautiful you look when you are flushed with excitement makes you feel sexy; orgasm makes you feel sexy. And stimulating your sex hormones makes you feel sexy; that is, after all, their physiological function\u2014to engorge your womb with blood, to start your vaginal juices flowing, to activate the sensation-rich nerve endings in your genitals and breasts, to stiffen your nipples, to flush your skin with extra blood and energy. In fact, the electromagnetic energy field surrounding your body begins to overflow with sexual energy and potency.\n\nIn fact, self-pleasuring actually expands your capacity for pleasure. The more pleasure you get, the more you want. The more you learn to enjoy, the more you can handle. It's like when you finally try the Flying Teacup ride at the carnival: you get off feeling giddy and exhilarated. _Now_ you are ready to take on the Giant Killer Roller Coaster! Building the pleasure of sex and sensuality is similar. All of your senses come alive, expand, and reach enthusiastically for more.\n\nAnd most important, being in touch with your Inner Sex Goddess makes you feel sexy. She is the repository of all your sensual drives, feelings, and knowledge. When you respect and love her and give her free rein to express herself, she will reward you with the riches of your own natural sexual power and artistry. Your Inner Sex Goddess _knows_ how to feel and be sexy. She _is_ sex.\n\n### Reason # 3\n\nSelf-Pleasuring Puts You in Touch with Your Body.\n\nWe ignore our bodies most of the time. We're too busy organizing complex work and social schedules, we're devising five-year plans and household budgets, we're weighing the pros and cons of buying a new refrigerator or a new dress, we're making decisions and formulating opinions, we're determining if a new man is safe, mature, and stable enough to go to bed with. We even make physical exercise a rigorous mental obsession, calculating our heart rates and personal best scores. We tend to view the input and sensations of our bodies as less important, less reliable, and less real than those of our minds. But this separation of mind and body can be terribly self-destructive.\n\nI find that when I neglect the pleasures of self-loving I become more critical of my body. I see it as a separate entity with cottage cheese thighs and wrinkles around the eyes. I lose a nurturing awareness of the rhythms and cycles of my body and of nature. I don't know and love myself as well. I start feeling powerless and depressed.\n\nBut a luxurious session of self-play can quickly restore my feelings of oneness with body and soul. It can do the same for you. By focusing your attention on the sensations of your skin, muscles, and seemingly countless erogenous zones and by feeling your womanness intensely, you come to know your body again and love all of its pleasures. You may find you actually like your fleshy thighs and crinkly eyes; they too are full of sweet sensations. Your body, mind, and soul begin resonating together once again. You feel healthier, earthier, more in tune with yourself, more human. Your body lets you in on its inherent playfulness and love of dancing, exercising, and making love. You reclaim the joy of being alive, and you feel glad to have your sensuous female body, which was made for pleasure.\n\n### Reason # 4\n\nSelf-Pleasuring Strengthens and Increases the Flexibility of Your Love Muscle.\n\nIn case you don't already know it, the pubococcygeus (PC) is your primary love muscle. It's the one you use to stop the flow of urine, and if it's well trained, you can use it to massage a man's penis to ecstasy. It is also the muscle in which you experience most of the contractions of orgasm, and if you keep it well toned, it helps you have bigger, better, and more frequent orgasms. Just flexing it can make you feel sexy and hot. That's why having a strong and flexible PC muscle is crucial to great sex, whether you're flying in tandem or solo.\n\nThe increased blood supply and toning contractions of orgasm vitalize your PC muscle. It feels good if it's exercised and strengthened, and then it pays you back by giving you better climaxes and more control of your partner's pleasure too. And of course the easiest way to give yourself plenty of lovely toning orgasms\u2014unless your man is available for hot sex twenty-four hours a day\u2014is by self-pleasuring.\n\n### Reason # 5\n\nSelf-Pleasuring Teaches You How to Have Orgasms\u2014Lots of Them\u2014Easily and Quickly.\n\nMany women have never had an orgasm with or without a partner. Many are not sure if they have or not. Many women fake it just to keep the peace or to avoid hurting his feelings. According to recent surveys, only about 30 percent of women have orgasms through intercourse alone, but 80 to 99 percent achieve climax with masturbation. This is astonishing. It's completely normal and very common _not_ to have an orgasm through intercourse all by itself. Most women need additional stimulation to experience their full sexual potential and enjoyment, but all the while they think they're supposed to be climaxing all over the place just from their lover's penis. While some women do, most do not. The truth is that once you learn how and where your body needs to be stimulated in order to have an orgasm, you can start having them any time you want to, easily and quickly\u2014during intercourse, while masturbating with your partner, and through self-pleasuring on your own.\n\nA woman's body is an amazing pleasure machine, capable of trillions of orgasms, big ones and small ones. In fact, according to Dr. Andrew Stanway in _The Art of Sensual Loving_ , \"any one woman's total sexual output through her life (intercourse plus self-produced orgasms) is probably more than a man's.\" It seems that a woman's capacity for sexual gratification is nearly limitless. And the more orgasms you have, the more your body learns to respond to all types of stimulation so that it is easier to have more and more wonderful climaxes.\n\nThe best way to teach your body to have easy orgasms is through self-pleasuring. You can try out all kinds of fun things in complete privacy, without having to worry if you're taking too much time, looking silly, or doing something incredibly weird. If you find it takes twenty minutes of constant clitoral massage to make you come, then you've discovered a great secret. If you learn that pulling on your nipples or massaging your anus while you thrust your finger in and out of your love tunnel throws you over the edge, then you've found the key to unlocking your orgasmic bliss. Chances are, you are not going to discover these things while having sex with your man. You'll be too embarrassed to explore yourself or too busy worrying about his pleasure, or you won't have enough time because he's already had his orgasm and gone to sleep. Stop denying yourself! Use your hands to teach your body how to have its ultimate pleasure.\n\n### Reason # 6\n\nSelf-Pleasuring Puts You in Control of Your Own Pleasure.\n\nWhen you know your body and its sexual responses really well, _you_ can control whether, when, and how to have your orgasm. You know just what it takes to create warm liquid feelings throughout your entire body and to bring on those delicious orgasmic contractions. Then you can guide your man to help you have _multiple_ orgasms, if that's your desire. The point is that you know exactly what you want and need, and you know precisely how to achieve it. You are your own mistress.\n\nAnd then there are times when you don't have a man in your life. Fortunately that doesn't mean you have to give up being your Sex Goddess self, because while you are self-pleasuring you can fantasize the most outrageous encounters, imagining every single touch, kiss, and stroke being done exactly the way you want it, for as long as you want it, as tenderly or roughly as you want it, by whomever you desire. Your imaginary sex partner anticipates your every sexual wish and never tires of titillating you all the way to the outside edge of your pleasure envelope. I've had several lovely liaisons with a fantasy Mel Gibson, for instance, where he licked me all over for three hours, massaged and kissed my breasts just the way I like it for about an hour and a half, and then made hard and fast love to me for forty-five minutes without stopping till we both collapsed in luscious exhaustion.\n\nAnd even when you do have a man in your life, he will sometimes be away, ill, or just not into it. Again, your own loving hands are always there to do your bidding. Sometimes your man may be just plain tired out from having given you ten star-burst orgasms in a row, but you want eleven that night. On those occasions you always have the ability to continue riding the waves on your own. In other words, you are not at the mercy of a partner's sexual whims, skill, or presence to enjoy your own ecstasies.\n\n### Reason # 7\n\nSelf-Pleasuring Leads to Self-Confidence in All Areas of Life.\n\nWhen you know how to turn yourself on and realize that you can do so anytime without depending on anyone else, you will gain a tremendous sense of self-control and self-confidence. You feel surer of yourself, more deserving, and more powerful.\n\nThis exhilarating sense of potency naturally spreads to other areas of your life. Learning to take responsibility for your own sensual enjoyment gives you the confidence to take the same initiative elsewhere. The aliveness and power you feel as a self-created Sex Goddess help you know that you are a fascinating and limitlessly capable creature, regardless of the role you are playing at the moment. My banker friend Charlotte says, \"Now that I'm in touch with my full range of feelings and experiences, I am more complete, more integrated. I know the amazing things I'm capable of in all areas of my life. And I like myself for it.\"\n\nAccording to research cited by Lonnie Barbach in _For Yourself_ , \"consistently orgasmic women tend to describe themselves as contented, good-natured, insightful, self-confident, independent, realistic, strong, capable, and understanding while non-orgasmic women tend to describe themselves as bitter, despondent, dissatisfied, distrustful, fussy, immature, inhibited, prejudiced, and sulky.\" Gee, I'd rather be around a frequently orgasmic woman. Wouldn't you? And so would your husband or lover, your children, your co-workers, and your friends. You become an asset to the rest of the world when you are happy, strong, and fulfilled. And you know it.\n\n### Reason # 8\n\nSelf-Pleasuring Is the Key to Overcoming Sexual Guilt and Self-Repression.\n\nThe more time you spend getting to know your body\u2014feeling comfortable with it and all the wonderful things it can do for you and seeing how beautiful it is at rest and during all stages of excitement\u2014the more you come to understand that sexual pleasure is a natural and especially wonderful benefit of living in a body. Our bodies love to be stroked and caressed; they thrive on it. In fact, if deprived of physical affection they can become ill.\n\nFinding out that self-pleasuring is a natural and beautiful function, something that makes you feel great about yourself and more loving to those around you, frees you from all that guilt you may have been carrying around. It breaks down the walls of inhibition and helps you let go of the fears and musty old taboos that have been keeping you from fully enjoying your natural Sex Goddesshood. And once you do release whatever sexual guilt is lurking in all those dark inner corners, you will be able to express your sexuality and your true self more freely\u2014 _with_ your lucky lover. You can start enjoying sex more and regretting it less.\n\n### Reason # 9\n\nSelf-Pleasuring Promotes Health, Well-Being, and Energy.\n\nYour sex muscles are not isolated. To paraphrase an old song, \"The sex bone's connected to the thigh bone; the thigh bone's connected to the back bone,\" and so forth. Healthy, well-toned pelvic and vaginal muscles lead to healthy, well-toned stomach muscles, leg muscles, and on and on. To say nothing of the increased flow of blood and energy that permeates your entire body when you indulge in a vigorous self-pleasuring workout. Exercise makes for healthy, happy muscles. It also boosts your energy production and releases endorphins that create a sense of well-being. In fact, according to Dr. Deepak Chopra, a chemical called Interleukin-2 floods the body during any positive exhilarating activity\u2014and I think most of us will agree that sizzling self-play qualifies as such an activity. Interleukin-2 is a powerful anticancer neuropeptide, the body's chemical equivalent to a feeling of joy. So for a quick mental and physical pick-me-up that will also build long-term health\u2014some women say it even makes them feel younger\u2014indulge in your favorite form of self-play.\n\n### Reason # 10\n\nSelf-Pleasuring Reduces Stress and Tension.\n\nThe intense muscular and emotional release of selfinduced orgasm provides quite a tonic for anxiety, tense muscles, job stress, traffic jam tension, or the frustration of a bad hair day. Stress and tension restrict the flow of blood and energy through your body and cause blocks to creativity, good health, and joyful living. Like a good massage, self-pleasuring can soothe and release these tensions with the simple touch of a finger. My friend Diane, who has a very stressful job, says, \"I sometimes masturbate every day if it's a tough work week. I don't know how I'd manage the stress caused by intense days filled with frantic meetings and impossible deadlines without it!\"\n\n### Reason # 11\n\nSelf-Pleasuring Helps You Sleep.\n\nOn those nights when your mind is racing, or you're all keyed up from some intense conversation or event, or worry is keeping your eyes glued to the ceiling, you can invite the Sandman into your bed by indulging in some stress-releasing self-massage. It has been scientifically proven that giving yourself a lovely orgasm relaxes your muscles and tunes your brain waves to alpha (the brain waves of dreaming sleep and creativity) or even theta (deep sleep). And it's a \"sleeping pill\" that has no side effects and will leave you energized, not stupefied, in the morning.\n\n### Reason # 12\n\nYou Don't Have to Stop Feeling Good When You or Your Sex Organs Are in a Delicate Condition.\n\nSuppose that you have an infection, you've recently had non-womb-related surgery, you are in the last stages of pregnancy, you just gave birth, or you are having your period. These are times when it might not be comfortable, desirable, or safe to indulge in love play with a partner. But during such times, you usually need _more_ loving, not less. During those times, you could really use the recuperative magic of sensual bliss. Fortunately, you can always give yourself just the right gentle caresses and love strokes to soothe your soul and keep you in life-affirming touch with your Inner Sex Goddess.\n\n### Reason # 13\n\nFor Older Women, Self-Pleasuring Increases Vaginal Lubrication and Lessens the Discomfort of Dryness.\n\nThis is true not just when you are pleasuring yourself, but all the time. By keeping your vaginal glands active, before, during, and after the menopausal years, you are encouraging them to produce more lubrication regularly. So you'll be juicier for your man too. When you keep your vaginal muscles contracting and those hormones secreting, you'll stay sexually active and in touch with your Inner Sex Goddess.\n\n### Reason # 14\n\nSelf-Pleasuring Provides Relief from Menstrual Cramps.\n\nOne of the benefits of having nicely toned and frequently orgasmic vaginal muscles is that they are stronger, more elastic, and much better able to handle the inner movements associated with your menstrual period. The spasms that release no-longer-needed blood from your uterus are therefore less intense, jarring, and painful. And these contractions don't have to enlist the support of surrounding stomach and back muscles to do their job. Most women who make self-pleasuring a favorite habit find that they experience much less pelvic cramping and greatly diminished or no menstrual backaches. A nice side effect, no?\n\n### Reason # 15\n\nSelf-Pleasuring Enhances Lovemaking with Your Partner.\n\nIf you are hot, you can't help but light his fire.\n\nIf you have a strong PC muscle to flutter on your lucky man's love organ, and if your body is willing and eager to have lots of deep orgasms, you are going to bring new vigor to his body and sexual imagination.\n\nIf you know all your secret erogenous zones and personal turn-ons, you can be much more confident and creative as a lover. You can gently guide him to massage your inner lips counterclockwise, if that's what makes you hot, and let him feel the power of bringing you to a teeth-chattering climax. You can teach him all sorts of sizzling new sex tricks. And if you think your man doesn't really want you to be so brazenly aggressive, think again. According to a book of sex surveys published in 1991, _Do You Do It with the Lights On?_ , 91 percent of men\u2014and a whopping 98 percent of men age thirty to fifty-four\u2014not only wanted their partners to take a more active role in sexual play, they wished them to choose the position, place, and procedure.\n\nIf you are sexually self-sufficient, you are freer to express this wanton sexuality with your man. That way he doesn't feel so pressured to perform or to puzzle out how to make you fly higher than anybody else has. He doesn't have to spend sleepless nights wondering if he's man enough to make you have the multiple orgasms he's read women secretly expect. You can seductively show him how. Your relaxed self-confidence and euphoria gently rub off on him and magically bring out his natural savage aptitudes and talents.\n\nIf you learn how to be deeply intimate with yourself, you can be more intimate with your partner. Likewise, your increased openness encourages him to be more intimate with you. And so that elusive soul sharing that we call intimacy blossoms and grows in your most treasured relationship. A lover once told me, \"When you are open with me about your feelings and show me your vulnerable, out-of-control side, I feel I can be open and vulnerable with you. And that makes me feel good.\"\n\nIf you pleasure yourself regularly, you will feel sexier. You will _be_ sexier. You will want to make love to your man more often, and you will do so with more gusto and expertise. He will think he has died and gone to heaven.\n\nAnd last but certainly not least, once you get comfortable with and good at pleasuring yourself, you can put on a really hot, provocative show for your lover. Men _love_ to watch women give themselves pleasure. Whether you tie him down and make him watch you for hours or simply wet your finger and slip it quickly inside yourself while staring into his eyes during foreplay, he'll be bewitched. Every true Sex Goddess has this mouth-watering act in her sexual repertoire. (More details on self-pleasuring for and with your partner in Chapter 10.)\n\n### Reason # 16\n\nIn the Age of HIV, Self-Pleasuring Is the Ultimate in Safe Sex.\n\nObviously, when you have sex only with yourself, you are in no danger of catching sexually transmitted diseases from someone else. No small concern these days.\n\nBut even in partner sex, masturbating with and for each other can be a delightful way to stay safe. Using your hands, vibrators, dildos, and a variety of other sex toys allows you to exchange a lot of loving and fun sexual energy without exchanging any body fluids. And even without body fluids, you can have the sensual moistness of hot sex if you add playful fruit\u2014cucumbers, zucchini, peaches, mushed-up bananas, and so forth\u2014to your repertoire as well. (More details in Chapter 9.)\n\n### Reason # 17\n\nThe Joy of Self-Pleasuring Is Always Available.\n\nMorning, evening, and in-between times, at the office, while traveling, at the beach, on an airplane, when you're bored, when you look or feel too grungy to be seen by another human being, when he's sleeping or watching a football game, between the time he forgets the anniversary of your first kiss and the time when you forgive him, between lovers, after a divorce, when you have the jitters, when you're feeling lonely, when you're feeling sexy and there's no one else around, when you find yourself alone with an X-rated movie or a picture of a man with muscles to die for, whenever wild and crazy thoughts come bouncing unbidden into your head\u2014you can always pleasure yourself and be your own Inner Sex Goddess anytime.\n\nOne friend recalls that when she was on her way to see her lover after a three-month separation, she became so excited thinking about his waiting erection that she slipped into an airport bathroom during a stopover, hid herself in a tiny stall, and hastily created her own urgent and explosive, if necessarily silent, orgasm. Thank Goddess, your hands always travel with you!\n\n### Reason # 18\n\nSelf-Pleasuring Is One of the Very Best Ways to Unleash Your Inner Sex Goddess.\n\nGoing all the way back to the ancient Sumerian goddess Inanna, divine and human women have innately understood that self-loving is the magical open sesame to the inner realms of power. In fact, Inanna is thought of as the patron goddess of self-pleasuring because she was in a constant state of ecstatic union with herself, and that inner union gave birth to the fertility of all living things. As a matter of fact, self-love _is_ the creative state, because when you _know_ that you are new each day and that you are a beautiful and perfect temple of pleasure, your benevolent, loving power gives life to all those around you.\n\nAlthough there are many roads\u2014mental, emotional, and spiritual\u2014to the kingdom of your Inner Sex Goddess, the physical path of sensual awareness and self-pleasuring is usually the most direct route simply because it is so tangible, it is easily accessible, and it fairly bursts with the power of all the five senses. And you will find that if you perform any of what you might consider the mechanical or physically impersonal acts of self-pleasuring with self-love you will evoke the very mental, emotional, and even spiritual feelings of love and affection you usually have only with an intimate mate. Not only is this more fun, stimulating, and satisfying, but it also expands your self-esteem and enhances your ability to communicate love to yourself and your partner\u2014and to receive it from him. And this is what being a true Sex Goddess is all about.\n\n# 4  \nawakening aphrodite's body\n\n_Round her delicate throat and her silvery breasts they fastened necklaces of gold which they, the gold-filleted Hours, wear themselves when they go to the lovely dances of the gods in their father's house_.\n\n_\u2014 first Homeric \"Hymn to Aphrodite,\" translated by Jules Cashford_\n\neven Helen of Troy, a woman so gorgeous and enticing that a war was fought over her, bowed to the awesome beauty and sensual power of Aphrodite, Queen of Heaven and Goddess of the Dawn. Despite one of Aphrodite's temporary disguises as an old woman, Helen recognized the Golden One by the \"sweet throat of the goddess and her desirable breasts and her eyes that were full of shining.\" Most often portrayed nude, the goddess of love was completely unself-conscious about her body and its sexuality, often flaunting it by decorating herself only with golden ornaments. In fact, her unembarrassed naked sensuality and her skill at pleasuring drew countless lovesick men to her like a magnet. Aphrodite was a woman who knew her own erogenous zones and how to activate them to excite herself and her lovers. She truly understood that Woman embodies all the senses in their exalted form and that pleasuring her beautiful body is a natural way for a woman to nourish her erotic soul.\n\nAwakening to the Aphrodite within you, then, means learning how to delight in the pleasures of your own flesh and opening your body to the rich sensations and wanton feelings locked inside. And your own hands, mythically adorned with Aphrodite's brazenly worn gold rings and bracelets, are the keys. So use your golden goddess fingers to love and awaken your body, exploring all its curvy contours, discovering and intensifying its unique pleasure points, and transform it, like Aphrodite's own body, into a sumptuous temple of erotic delights in which you and your man can worship regularly.\n\nIt's natural at first to feel some inhibitions about manually pleasing yourself in every wild way you can possibly think of, but try to let those impediments go. Be as bold and as reverent as Aphrodite. If you do feel shy about jumping right in to try _all_ of the following suggestions, that's okay. Just start out with what feels comfortable and fun to you. All you may want to do just now is read about them. Don't be surprised, though, if you soon find yourself feeling tempted to try a thing or two, getting more and more adventurous, and even making up your own kinky list of self-pleasuring scenarios. Try to give yourself and your body the benefit of the doubt by crossing over your embarrassment threshold just a little bit. After all, there's no one around to judge you or find out just how much fun you're having. Sip some wine. Relax. Let go. Let your hair down. Brush it out. Like Aphrodite, throw caution, inhibition, and self-consciousness to the wind.\n\n## aphrodite's boudoir\n\nIt will be much more tempting for your Inner Sex Goddess to come out and play if you create a loving and sensual nest in which she can feel comfortable, safe, and uninhibited. So before you begin your self-play sessions, take the time to create a bedroom Aphrodite would have been proud of. Take the phone off the hook or use the answering machine. Lock the door, post a sign, or do whatever you have to do to ensure privacy. Make sure the room is warm and that you have something to drink. Gather together all the things you might want or need\u2014oils, a towel, a large mirror, a handheld mirror, perfume, wine, sensual foods, music, lingerie, costumes, scarves, sensual fabrics, pillows, erotic literature, sexy pictures, X-rated videos, special erotic objects, vibrators, dildos, and other sex toys of your choice. (More on vibrators and other toys in Chapter 9.) You won't use all of these things every time, but have them handy and gather whichever items you want for that particular session _before_ you start. Dress the bed in pretty, fresh linens. Artfully arrange flowers, candles, incense, and evocative paintings or photos. Really focus on creating a special, sensual, and freeing environment for your self-loving ritual. Doing so will make you feel special, sensual, and free.\n\nYou can even use your lingerie to add a touch of sass to the interior design of your Aphrodite-like bower. Hang a see-through negligee over the bedpost. Dangle a lacy garter belt from the knob of your dresser. Let several long strings of pearls cascade from one corner of a mirror, and float a silky teddy from the other. Drape lace-topped stockings over a chair. Adorn your pillow with a feather boa and your bedside lamp with a filmy scarf. Display your long satin gloves on the nightstand. Dressing your boudoir in lingerie and jewels will transform it into a sensual palace that beckons your Inner Sex Goddess to appear in its midst\u2014 _and_ will be a real turn on to your lover.\n\n## the breasts beautiful\n\nAphrodite, like any clever Sex Goddess, certainly knew how to make the most of her feminine assets. So she is depicted, in many statues and paintings, winsomely exposing two of a woman's most naturally glorious and seductive attributes, her breasts. Wildly arousing to most red-blooded men and exquisitely sensitive to your own or a lover's touch, your golden orbs provide a softly feminine place to begin your erotic self-arousal, especially since there is a direct nerve connection between the nipples and the clitoris. Although the intensity of breast and nipple sensation is different for each individual Sex Goddess, Masters and Johnson, the definitive scientific sex researchers, found that about 1 percent of the women they studied were actually capable of going all the way to orgasm simply by caressing their own breasts. But even for the rest of us, breast and nipple massage can provide some highly erotic sensations, and it will usually start your libido sizzling and your divine vaginal nectars flowing.\n\n_35_. Begin by using the feathery touch of your nails or fingertips to stroke the delicate skin on the side of your torso and under your arms. Slide your fingers down across your stomach and make circles all over your belly and midsection as you raise your hands, and your temperature, up to your swelling breasts. Repeat several times and let yourself float on the hypnotically languorous sensations.\n\n_36_. Slide your palm in circular motions around the whole area of your breast. Press firmly. Do one breast at a time or use both hands for the two together. Keep this up even after your nipple has become erect. Go faster, slower, harder. Cup as much of each creamy globe in your hand as possible and move them both around. Pull them out. Press them in. Jiggle them. Squeeze them. Imagine that your lover is caressing them. Continue until your breasts feel very warm and tingly.\n\n_37_. One day, as I leaned over the side of the bed to retrieve the bottle of massage oil I had dropped, I discovered that my breasts felt more sensitive when they were hanging down. I later learned that this is because of increased blood supply to the nipples. So by all means try this leaning-over position. Besides feeling lushly full and tender, your breasts are freer to move as well. Shake them. Give them love taps. Push them together. Massage as in number 36.\n\n_38_. Massage your nipples. Press on them. Roll them gently between your fingers. Use firmer and firmer pressure. Pinch them. Pull them and hold, or pull and release completely in very rapid strokes. My friend Carrie, a computer specialist with no current lover but plenty of libido, says she likes to hold on to her nipples and shake her breasts with them. Try this for the intensity of the feelings and the visually exciting stimulation.\n\n_39_. Use one hand to encircle your breast and push it up and out, as a push-up bra would do. This pulls the nipple taut. With your other hand, roll, pinch, and pull the nipple. Wet your fingers and play with it. Rub your fingers back and forth rapidly. Use the pad of your index finger to rotate your nipple in small circles. Flick your fingers or fingernails across each bright berry. Know that your man would be fascinated to watch these maneuvers. Keep all of this up way past the point at which you feel the urge to move to your pubic area.\n\n_40_. In one very elegant sex movie, I watched with fascination as a woman licked her own nipples. Even though she was rather small-breasted, she was able to accomplish this delightful trick by bringing her breast close to her mouth with her hand, bending her neck down, and using the _bottom_ of her tongue to reach the nipple. Close your eyes and feel the lapping of your lover's tongue.\n\n_41_. Rub your breasts and nipples with silk, velvet, fur, feathers, and even rougher textures like wool, terry cloth, or a very soft brush.\n\n_42_. Write your partner's name or some sizzling words of love across your chest with your fingertips, spray-on perfume, or hot red lipstick. Use your nipples to dot the i's or make lovely curlicues on appropriate letters.\n\n_43_. Pull on long, tight leather gloves and play with your breasts and nipples.\n\n_44_. One friend of mine suggests circling the nipples with an ice cube. She says, \"It makes my nipples stand up immediately and feel all wet and fiery.\" Her husband often requests the pleasure of watching her do this, and he usually jumps right in to lick the watery drips from her upturned buds. Apparently it makes him \"feel all wet and fiery\" too.\n\n_45_. Dab whipped cream or honey on your nipples, or dip them in wine. Lick it off\u2014or let _him_ lick it off.\n\n_46_. Massage your nipples with Vicks VapoRub, Ben-Gay, or some other cream that makes your body heat rise.\n\n_47_. One day I got the idea from an old photograph of a French courtesan to apply rouge to my nipples. While this is a makeup trick you'll want to secretly expose to your lover, try it on your own, too, and see how ripe and enticing your nipples will look. Let the color come off as you massage your pretty rosebuds. Reapply as desired.\n\n_48_. One bawdy corporate comptroller I know told me that she likes to put her accounting pencils to better use in the privacy of her own boudoir. Pinching her nipples taut, she rubs the rounded pencil erasers, softened from use on her daytime ledgers, against her sensitized areolae.\n\n_49_. Put on a pretty bra or swimsuit top, but pull the material away so that your nipples and most of your breasts are exposed. Fondle your velvet tips. Now pull the supporting bottom part of the bra up over your breasts so that it rests above the breasts and pushes them down. Again, this puts more pressure into the nipple and sensitizes it. Massage, pull, and pinch your nipples. You can, of course, dress up in any sexy lingerie\u2014camisole, teddy, merry widow, bra with the nipples cut out\u2014and do the same thing. There's no reason you shouldn't look just as gorgeous for yourself as you do for your lover, and besides, your breasts will be sensitized by the extra exquisite pressure. As one young nursing student I know says, \"The cool silk and low-cut lace bodice of my favorite camisole do wonderful things with my nipples, especially when I wear it under my uniform.\"\n\n_50_. Adorn your fingers with rings before you begin your breast play. Depending on the rings, you may want to turn them around so the showy parts are on the insides of your fingers. They look great, and you can tease your nipples with their different textures as you massage.\n\n_51_. Use a vibrator on your nipples. Put on one of the ridged or pronged attachments and hold it steady or move it back and forth across the areolae. Circular motions are good, too. My friend Brigitta, a pale Nordic beauty in a still-lusty twenty-year marriage, says she prefers to use a Swedish massager that fits on the back of her hand. She straps it on and lets her vibrating hand and fingers loose on her sensitive breast skin. Sometimes she shares the delicious vibes with her man. (More on vibrators in Chapters 9 and .)\n\n_52_. Get in the bathtub and use your shower massager to wash your nipples with sensation. This is a favorite with several friends of mine, who say it reminds them of being nubile young girls in the shower after gym class, giggling over all the exciting new sensations they were just discovering then. It makes me feel like a Polynesian beauty, standing naked in a steamy tropical waterfall. Discover your own liquid fantasy.\n\n_53_. Watch yourself in a mirror as you do any or all of the above. Marvel at the hypnotic beauty of your golden orbs, radiating like the Aphrodite within you\u2014to please your own aesthetic senses or titillate your favorite man.\n\n## aphrodite's private foreplay\n\nAs the Goddess of Love, Aphrodite knew instinctively that the flower of her desire lay resting lightly in her clitoral rosebud and that it had to be treated with the utmost delicacy in order to blossom to its full beauty. While no two buds of love are exactly the same, we modern-day Aphrodites know from all-too-frequent experience that most don't like to be attacked by a too-eager hand. They like to be slowly warmed up first. Hence, your Inner Sex Goddess has gifted you with the instinctive skill of your own brand of private foreplay for the clitoris. Use it for solo as well as partner sex.\n\nWhile your breast play will already have begun to unfold the petals of your clitoral sensitivity, it's still a good idea to go gently and slowly when approaching direct stimulation of this sensitive area. In fact, some women never touch the glans of their clitoris directly when self-pleasuring; it's just too tender. Instead, they concentrate on the pubic mound. And even though most women do require at least some direct stimulation of the clitoris to have an orgasm, it's still best to start by stimulating the surrounding areas.\n\n_54_. You might want to begin by stroking your inner thighs and buttocks. Remember to be present in your body, as we learned to do in the body-consciousness exercises in Chapter 2. Really feel all of the luscious sensations you are creating. As you reach the genital area, use a finger or two to gently massage around your anus. (Make sure you are freshly bathed.) Softly rub the perineum (between anus and vulva) with back-and-forth or circular motions. Give it a few gentle love taps. Concentrate on building a fire in this lower portion of your genitals.\n\n_55_. Most women enjoy the comforting feeling of cupping their entire pubic mound in one hand. Press and massage gently in circles, from side to side, or back and forth. Knead it like bread. Shake it gently. Recall the feeling of your man's pelvis pumping against this fleshy mound.\n\n_56_. Still holding your pubis tenderly like a ripe fruit, press your outer vaginal lips together. Then caress them gently up and down so that they begin to open up and expose the moist inner lips. Continue this up-and-down motion, occasionally reaching farther down to include your perineum in one long sensuous stroke over it and your inner lips.\n\n_57_. You may find it intoxicating to pay some special attention to your delicate perineum. Try sitting with feet flat, knees bent, and legs spread. Rotate your pelvis forward so you can reach it easily. With the touch of a butterfly's wings, caress your inner thighs and allow your fingers to cross over your perineum. Tease this dainty area with your fingernails and inscribe wings, flowers, and stars there. Brush it lightly with a feather or cosmetic brush. Wet your finger and make maddeningly delicious circles. Imagine how provocative this would look to a Peeping Tom.\n\n_58_. My friend Jenny, a shy librarian with a hot lover who is cleverly disguised as an ordinary mailman, says that when self-pleasuring she loves to vibrate her pubic mound with her whole hand. She suggests laying one finger on each vaginal lip and vibrating them, too. You can use your other hand to vibrate your perineum and\/or anus simultaneously if you choose. Jenny admits that she has taught this trick to her favorite mail carrier too.\n\n_59_. According to Oriental erotic texts, a woman's upper lip is directly connected to her clitoris. While massaging your pubis or vaginal lips, try biting your upper lip, or run the fingernails of your other hand over it, to send tingly signals to the rosebud you're about to touch.\n\n_60_. If the upper lip connection is not a powerful one for you, try moistening your lips with Vaseline or a lovely supermoist lip gel. Lightly rub your fingers over your glistening lips. Slip a finger between them.\n\n_61_. Recline languorously on a pillow and let your legs fall open. Rest the palm of your hand on your pubic area but without touching your clitoris. Exerting a firm but light pressure, use your hand to move the skin of your pubis around in slow circles over the underlying tissue and bone. Occasionally release your hand and lightly pat your beautiful mound of pleasure. Alternate light patting with deep circling until moans escape your lips. After observing their women get all hot and bothered by doing this, men seem to feel especially clever when they \"invent\" this foreplay technique to use on you themselves. So be sure to let your lover catch you at it.\n\n_62_. While holding your vaginal area, move your hips around in circles. Undulate like a snake. Or do some pelvic thrusts \u00e0 la Elvis. Keep your hand loose enough so that your pelvic motions cause it to slide lightly around your moving mound. I have a very modest friend who claims that pretending she's a rock star and flagrantly thrusting her hips around frees her wilder, lustier self to come out of hiding. So, with her hand suggestively placed over her naked sex, this is how she begins every self-pleasuring session. When you try this, you may want to imagine that you are getting private lessons in hip swiveling from the King himself.\n\n## the ripe rosebud\n\nThe lovely little organ whose only function is to give pleasure originally got its name from Aphrodite's compatriots, the ancient Greeks. Wise in the ways of both loving and philosophy, they understood that this tiny knob was the key to a woman's orgasmic pleasure. So from their word for \"key,\" _kleis_ they coined the term _kleitoris_. Just as every door key is different from all others, every woman's clitoris looks and behaves differently and has highly individual preferences for stimulation. These preferences can range from no direct touch at all to indirect pressure with only the palm of the hand to intense and deep finger massage directly on the glans. So there is no \"right\" way to pleasure your ripe rosebud. There is only the way that feels good, comfortable, and sensuous to you. And that may change from day to day or from month to month, depending on your emotional state, your physical condition, and your monthly cycle. So consult your Inner Sex Goddess and be adventurous yet sensitive to the responses of your own body. Here are some suggestions to try out as you take the time to deepen the sensual bond between you and your own pleasure organ.\n\n_63_. Lubrication is not only the key to a good relationship with your clitoris, but it also extends the sensations over larger parts of your vaginal area. So before you caress this delicate bud directly, make sure your fingers are moistened with your own saliva, the juices from your vagina, K-Y jelly, or oil. You can use baby oil, coconut oil, vegetable oil, or a good massage oil. Olive oil is probably a little too thick, but sunflower, safflower, and grapeseed are good choices. Scented massage oils add another dimension of sensuality; the flower essences, musk, and ylang-ylang are particularly potent for sexual play. Just make sure the oil you use contains no alcohol, which can irritate the delicate mucous membranes. And avoid Vaseline too; because it's not water soluble it can be uncomfortable and unhealthy if it gets in your vagina or urethra.\n\nExperiment with different lubes for different occasions. Your own juices are always available, completely natural, and very earthy and sexy. Oils, which are slipperier, are good for intense friction and hand gliding. Scented oils add luscious aromatic vapors and can catalyze some exotic fantasies. The choice is deliciously yours.\n\n_64_. You may want to begin by using your entire hand\u2014heel, palm, and fingers closed together. Cup your hand over your pubic mound, touching your clitoris lightly, and inscribe circles around and over it. Press with the palm or heel of your hand. Then try using your overhanging fingers. Keep the circular motions going as you explore the sensations arising from varying pressures and parts of the hand. Most women find that sustained repetitive movements are the most satisfying. Also lovely is imagining that your lover's hand or penis is caressing you tenderly there.\n\n_65_. Karol, a sexy secretary I know, says she likes to rub her clitoris indirectly through its shaft. You can stroke on the right side if you are right-handed and on the left if you are left-handed. Switch sides when you feel the urge; this technique can provide an entirely different sensation, and it can renew sensitivity when one side gets \"tired.\"\n\n_66_. Experiment with various kinds of touch and rhythms\u2014light feathery pats, gentle massage, firm steady pressure, intense rubbing, playful love slaps, fast circles, slow back-and-forth movements, languid strokes alternating with quick taps, rolling motions, pinches, vibrating with your hands or fingers. When you find a touch that really lights your fire that day, stay with it for a while. As you explore your own key rhythms and touches, sense the wetness seeping into your loins.\n\n_67_. If you can stand it, try massaging your rosebud firmly with just one finger. Play around with circling, side-to-side, and rolling motions. Create a vibrating motion with your finger as you press down hard. Rock your pelvis up to meet your trembling digit as you would to reach for a lover's touch. Add another finger and probe some more. If the pleasure gets too agonizingly intense, stop momentarily and massage your thighs or belly instead.\n\n_68_. Like my friend Bess, a forty-something banker, you might like to try taking hold of the shaft of your clitoris with thumb and forefinger. Roll it between the fingers. Pull on it. Move it up and down. Squeeze. Release. Roll it again. Try using two different fingers, or the fingers of your two hands. Bess says, \"I need a lot of stimulation, and somehow this direct, forceful handling really gets my body humming.\"\n\n_69_. Contract your PC muscles rhythmically while caressing your clitoris. Squeeze and release the muscles around your anus. These tightening movements heighten the sensitivity of your clitoral nerves. Let your contractions roll back and forth between the two areas as you continue rubbing your love button. Test varying rhythms. Do it to rock music.\n\n_70_. I always wondered why the women in X-rated movies frequently use both hands to stretch their outer lips wide before provocatively stroking themselves. Often we think that their immodest actions are meant only to provide better sight lines or are done purely for the benefit of some debauched man. But I've found that, as usual, women are much smarter than that. After all, men don't invent these unusual and creative moves. Women do. Because, yes, they drive men crazy, but more important, they feel great. Through personal experimentation, I've learned that spreading your love lips open the way these women do makes the sensitive skin of your clitoris feel deliciously exposed and taut. Try it yourself, and then stimulate your lovely pleasure organ any way you choose. Imagine that _you_ are a raunchy porn star\u2014and your man is salivating over every frame of the movie.\n\n_71_. A luscious thing to do while gloriously outspread is to thump your clitoris lightly and quickly with your middle finger or the side of your hand. (Wouldn't a slap from your lover's penis be lovely too!) Alternate with hard circular massaging.\n\n_72_. Sara, a dietitian with six kids, says that whenever she can steal a moment to herself, one of her favorite techniques is to stroke down toward her clit from the top of her pubic mound, not quite touching its surface or sometimes gliding lightly over the top of it. Adapting a technique she learned from her deliciously randy husband, she occasionally slides her fingers down farther to gather some juice from her vagina and then circles her wet fingertips around her sensitized bud. Try it. You'll like it too.\n\n_73_. You can learn a lot from watching a man play with himself. Because the clitoris is closely akin to the penis, you can adapt almost any male masturbation method to the caressing of your love bud. For instance, try holding and stroking your clitoris up and down as he would his penis. Or slide from bottom to tip with the fingers of one hand, and then switch to the other hand. Do this quickly and with upstrokes only. Improvise with anything that works for him.\n\n_74_. While caressing your clitoris, squeeze or rub your vaginal lips. Using only one hand will keep the rhythms and movements in concert, but you could also try using two hands to circle clockwise with one while the other moves counterclockwise. Or one hand could administer gentle blows while the other presses and massages deeply.\n\n_75_. Pull gently on your pubic hair while teasing your clitoris. For me, this always brings back arousing memories of a wonderful man who loved to reach around in front of me and hold my pubic hair while he rubbed his penis against my clitoris from behind.\n\n_76_. Tug on a nipple in rhythm with your clitoral strokes. Roll both between your fingers. Massage in tandem circles as your legs fall invitingly open for an imaginary lover.\n\n_77_. In one sex class I attended, the instructor asserted that \"any good clitoral massage should include the urethra which is actually part of the G-spot and is therefore highly sensitive to erotic pleasuring.\" At the next class most of the women, including me, excitedly attested to the truth of this theory. In the name of scientific research, test it out for yourself.\n\n_78_. Tickle the underside of your love bud with one finger while stroking, patting, rubbing, or flicking the side or top of the clitoral hood with another. This intensely erotic trick can make you climb the walls, so apply it judiciously.\n\n_79_. As you lovingly stroke your clitoris, be aware of how the sensations change. At the beginning it will be tender and want a light touch. But as your excitement grows and you get closer to orgasm, your rosebud will swell and become harder and stiffer, much as a man's penis does. At this point, press harder, circle or stroke faster, vibrate more intensely. Thrust your pelvis against your hand. Arch your back and pant, moan, or scream.\n\n_80_. My friend Rosie reminds me of a plum\u2014small, round, and sweet, but tart at the same time. Very frank about the fact that she's been self-pleasuring regularly since the age of five, she says, \"I love to massage my soft spot [the perineum, between the vagina and anus] with the fingertips of both hands. Then I smooth my fingers up between my vaginal lips, collecting some wetness so my fingers can slide more easily, then I let them slide over my clit, down outside the outer lips, and back to my soft spot. Each time I reach my clit, I give it a little extra push, squeeze, or roll. After about three minutes of this, I'm really shaking. But then I circle my clit and soft spot with both hands at the same time, which makes me feel unbearably sexy, until I have an orgasm that vibrates my whole body.\"\n\n_81_. While you rub your pleasure organ, try bearing down as you do when you have a bowel movement. This pushes the entire pelvic floor down and brings more pressure into the clitoris, making it exquisitely sensitive and trembly. Many women report that this action alone can bring on an orgasm very quickly.\n\n_82_. Don some of your laciest, sexiest panties and massage your clitoris through them. Pull the two sides together at the crotch so that you have a thin, taut ribbon of fabric to pull against your pulsing love bud. Since this is as exciting to feel as it is to watch in the mirror, keep it in mind for a visual turn-on when you're in bed with your man.\n\n_83_. A technique I modified from an Oriental love text is really meant for pleasuring a man's \"Jade Stalk.\" But there's no reason your \"Rosy Stem\" shouldn't enjoy it too. Hold your little stem between your fingers and stroke it in rhythm to the throbbing of your heart. Then give it one fast up-and-down stroke, and return to the slow, sensuous stroking for about ten seconds. Then two rapid up-and-down strokes and back to the slow stuff for another ten seconds. Increase to three fast, and back to slow, and so on.\n\n_84_. After your love button has become engorged with excitement, try holding it between your fingers while you use the palm of your other hand, or your flattened fingers, to massage lightly in circles over the tip. Fantasize, if you wish, that it's really the pulsing head of your man's penis that's rubbing you there.\n\n## the voluptuous vagina\n\nAphrodite wasn't the only one who was hip to the benefits of loving her own body. Since ancient times, Taoist medical practitioners and Tantric initiates have taught that to have a whole and healthy body and mind, you must fully love your genitals and the sexual energy harbored there. When you come to know and love your vagina as the mysterious cavelike home of the sacred life force, you will find yourself developing a new, more charismatic relationship with your femininity, your sexuality, and your man.\n\nWhile some women prefer to explore the voluptuousness of their vaginas first before moving on to the clitoris\u2014feeling that the dainty bud is just too sensitive in the beginning\u2014others feel that starting with clitoral massage warms up the vagina's fewer nerve endings so that it becomes more eager for erotic touch. But whatever your personal preference, it's delicious to pleasure both erogenous zones at the same time. You can practice the following techniques while you are massaging your pearly bud or try them all by themselves. Mix and match to your libido's content.\n\n_85_. Stroke and massage all around your vaginal opening. Run your finger up and down over the juicy slit, gently opening it wider and wider. Pull the lips apart and caress the moist skin just inside the opening. Many women find that the wider their vaginal opening is stretched, the more acute the sensations. Relish this feeling of vulnerable exposure.\n\n_86_. Dip one or two fingers into your voluptuously wet furrow. In addition to keeping your fingers lubricated for clitoral stroking, this adds another level of sensation. (Short fingernails are a prerequisite.) Press upward to feel the back of your clitoral membrane and explore its moist sensitivities from this other side. If you curl your hand, you can keep the knuckles of your free fingers pressed against the outside of your clitoris at the same time.\n\n_87_. Slide one or more fingers in and out of your vagina just as you like your lover's penis to do. Slant your hand and sweep your fingers around to feel the luscious friction from all angles and against all surfaces of your vaginal canal. Stretch your lips wide. Go fast. Go slow. Tease yourself unmercifully.\n\n_88_. A very sultry woman I know says she likes to insert just one finger about two inches deep. Then she crooks her finger and tickles first the upper and then the lower walls of her vaginal barrel. It's \"exquisite torture,\" says she.\n\n_89_. While performing 87 and 88, use the thumb of the same hand to rub your clitoris or pull on your inner lips. This is much like the tug of your lover's penis as it dips in and out between all those sensitive tissues.\n\n_90_. Adapting a cooking technique for stuffing chickens, I discovered that it feels heavenly to slip your thumb inside your vagina and then glide it in circular motions against your other fingers, massaging the thin, wet membrane of your vaginal wall between them. Try out your own inner recipe.\n\n_91_. When your vaginal tissues are engorged with sexual juices and excitement, their sensitivity to touch and pressure is magnified. One inch of thumb can feel like nine inches of a man. So if you simply anchor your thumb tip inside and press it down toward your spine, you can mobilize your fingers to simultaneously tease your clitoris with circular massage or up-and-down sweeps.\n\n_92_. Plunge into the depths of your coral abyss with two or three fingers and rotate them as if you were stirring up a scrumptious batter. Or spread your fingers out like a scissors, as far apart as you can. Open and release several times.\n\n_93_. Adorn your index finger with a big flashy ring and thrust it in and out of your steamy vagina. The X-rated movie stars who love to perform this act for the camera know that it adds elegant beauty to the scene and feels fantastic, too. Show this one to your lover.\n\n_94_. Experiment with different body positions. Kneel, lie on your stomach, stand in front of a mirror. Or try lying on your back with feet braced against the headboard or wall. This, and kneeling with your legs spread, tenses the pelvic muscles and pulls your vagina taut to heighten sensation.\n\n_95_. When her husband is away, my friend Karen says she uses one of those standing pillows with arm rests as a substitute. She snuggles into its \"embrace\" or props herself against her stand-in hubby's open \"legs\" while she diddles herself just the way he would.\n\n_96_. With the flat of your hand, try stretching the soft skin of your pubic mound upward toward your belly. Hold it taut while you slide the fingers of your other hand quickly in and out of your tightened opening. Conjure up your favorite fantasy.\n\n_97_. Spread open your outer lips and use your thumb to gently strum the inner ones. Play this tune for your man and his drum major's baton.\n\n_98_. My friend Gail, a brilliant graphic artist with fiery red hair and tons of freckles and boyfriends, graciously offered to share one of her favorite maneuvers. She says she loves to insert two fingers into her vagina and press her clitoris with two fingers of the other hand. Then she rubs the two sets of fingers together through all the sensitive skin between. I've found you can also do this with only one hand by inserting your middle finger and using your thumb for the clitoris. What a divine massage!\n\n_99_. Hold open your vaginal lips with the fingertips of both hands. Knead the flower-petal skin in circular motions, down toward the anus, around toward the inner thighs, and up toward the clitoris. Keep going around and around till you can't stand it anymore.\n\n_100_. When a Hawaiian friend of mine mentioned that she likes to go \"swimming with the dolphins\" when she pleasures herself, I couldn't wait to hear the details. She says she lays her hand flat against the entire vaginal area. Then she lets the tip of her middle finger tease her vaginal opening and occasionally slide in and out. She alternates pressing hard against the top and undersides of her pubic mound with the heel and ball of her palm. Then she rolls and undulates her whole hand in wavelike movements, the way dolphins swim. Or you can let your middle finger lie against your clitoris with the fingertip just inside your vagina; then circle, roll, and undulate your hand like a sinuous creature of the deep.\n\n_101_. While thrusting your fingers in and out like a lover's hard shaft, massage your inner thighs, belly, derriere, or nipples with your other hand. All of these sensitive tissues are erotically connected and will respond quite intensely to tender caresses at any location. You rarely get to experience these delightful combinations with a partner because it's just too awkward; so take the opportunity to treat yourself to this extra loving.\n\n## the love muscle\n\nThe PC muscle, the one you squeeze to stop the flow of urine, is one of the greatest love secrets in Aphrodite's body. Just by twitching it, any ordinary goddess can give and receive ambrosial delights worthy of the heights of Mount Olympus. A well-toned PC muscle will greatly enhance your pleasure during self-loving as well as while making love with a partner, and it will drive most men wild when flexed against their penis. When strong and healthy, this muscle is highly sensitive to vaginal stimulation and is actually able to bring his thrusting penis closer to your G-spot. It makes for increased lubrication, stronger clitoral sensations, more intense and more frequent climaxes, and even multiple orgasms. How does all this magic take place?\n\nLying about one inch below the surface of the skin, the PC muscle runs from the pubic bone to the tailbone, firmly supporting the anus and internal sex organs. It may be as thin as half an inch or as thick as two inches. In addition to cradling your organs of pleasure, the love muscle is also home to the pudendal nerve, which detects feeling around the clitoris, the vaginal lips and opening, and the anus. This nerve sends signals to the brain that say \"This feels good; send blood and energy here,\" and it later receives brain signals that start the pleasurable contractions of orgasm. After every contraction of your PC muscle, whether you flex it deliberately or an orgasm does it for you, blood rushes into the vagina, making the tissue swell and darken and increasing lubrication. Some women describe this sensation as a melting or \"I love you\" feeling. In fact, besides the actual nerve connections between the PC muscle and the brain, there seems to be some extrasensory bond between the love muscle and the heart that, when activated, makes you feel soft, fuzzy, and warm inside.\n\nThe miraculous abilities and pleasures of a well-trained PC muscle are extolled in the ancient love texts of the Oriental and Tantric traditions, too. They describe women with such a high degree of PC control that they were able to pick up small objects and thrust them out with great force, using only their vaginal muscles. Imagine what they could have done with a man's penis! But the real beauty of such dexterity is that they were able to provide their own intense vaginal and clitoral stimulation\u2014without a penis, dildo, vibrator, pillow, hand, or finger. Their agile vaginas were completely self-contained pleasure units. We too can share in this special female magic simply by exercising our PC muscles regularly. And you will find that these sexual calisthenics are a major turn-on in and of themselves.\n\n_102_. The fastest and easiest way to get started enjoying your PC muscle is to simply flex it quickly about twenty-five times\u2014anytime, anywhere; no one will know. The sensation will be more intense if you put your attention on your clitoris or vaginal opening while you are squeezing. This will strongly focus your awareness, energy, and excitement. In fact, some women say they can reach a climax from this exercise alone. At the very least, it will bring your Inner Sex Goddess immediately to attention.\n\n_103_. When your muscles have something to resist their squeezing pressure, they can develop even faster, and you can get a better idea of just how strongly you are flexing. Insert a finger, a clean carrot, or a dildo into your vagina, squeeze and hold it for five seconds. Release and squeeze again. After several of these, do some quick clenches. Swoosh the inserted object around to provide differing levels of pressure and to excite your PC muscle into more intense flexing. See what it feels like to push the muscles _out_ as well; you can accomplish this by pressing down as you do when you have a bowel movement. If you close your eyes and focus on your moist vaginal walls sliding against the object, you will find a warm, pleasurable sensation building up inside you. Let your libido and your Inner Sex Goddess take it from there.\n\n_104_. Here's a Tantric PC exercise that always makes me feel like a volcano boiling, oozing, and erupting. You may want to perform this one to music, as I do. Stand with your feet apart at shoulder width and begin slowly squeezing your PC muscle. Breathe in deeply with each squeeze, and exhale with each release. Feel the volcanic heat coming up the insides of your thighs and into your womb. Keeping your feet firmly planted, rock your hips back and forth sensuously\u2014rock forward as you inhale and squeeze, rock back as you exhale and release. Feel the heat building in your loins. After a bit, move your hips around in circles as you continue flexing and releasing. Imagine that you are a hula dancer or a belly dancer; sway your pelvis provocatively and perform expertly sexual internal dances. Feel the volcanic heat radiating from your pelvis as you release a low \"ahhh\" with each exhalation and PC relaxation. As you continue undulating your hips and flexing your love muscle, imagine that the hot energy you are generating rises up through your belly, breast, throat, and face to the top of your head, where it showers you with tingly liquid ecstasy. Slowly return to earth whenever you're ready.\n\n_105_. The ancient Chinese Taoists developed a complex set of stone egg exercises designed to strengthen and tone the vaginal muscles. For a time, these exercises were taught only to the Empress and the Emperor's concubines in the royal palace, supposedly so they could better please the Emperor. And I'm sure they did. But these wise women knew that they were secretly enjoying themselves, enhancing their health, and even prolonging their youth. If, like these multitalented royal ladies, you'd like to increase your sexual energy and expertise, here's a simplified version of the egg techniques.\n\nSelect a stone or a wooden egg about one inch in diameter from a gem store or a New Age shop. Keep it scrupulously clean with soap and water. Always perform the egg exercises while standing, feet apart at shoulder width, and warm yourself up first with some self-massage. Insert the egg into your vagina, large end first, using your own juices or some oil to lubricate it. Hold the egg inside you by clenching the muscles at the opening of your vagina. When you first start practicing, simply feel your internal muscles contracting against the smooth egg. Get used to holding it inside you and feeling how it moves.\n\nAs you become more expert, you can move on to more advanced techniques such as the following. Inhale as you squeeze the muscles at the other end of your vagina, deep inside. Then begin to move the egg up and down, and from side to side, with the muscles in the middle section of your vagina. This takes practice and dexterity, but the sense of passionate control you will experience is worth the effort. Even if you just learn to keep the egg snug inside, you can't help but revel in its smooth sensuality and the delicious feeling of fullness it provides. Try different size eggs, too\u2014bigger egg, more fullness; smaller egg, more exercise and control. Eventually you can train your vaginal muscles to draw the egg in all by themselves and expel it forcefully. Your vaginal expertise, whether applied to an egg, a finger, or a penis, is limited only by your imagination.\n\n## the bewitching behind\n\nAphrodite Kallipygos (Aphrodite of the Lovely Backside) captivated scores of lovers simply by undraping her fleshy fanny. She truly understood that a woman's behind has the power to ignite her own inner fires as well as to bewitch almost any man. You'll begin to understand the fanny's electric power if you use your erotic imagination to recall how delicious it feels when you rub yours up against your man's body while lying in the spoon position, or when he cups your luscious cheeks in his hands as you make love. I didn't really catch on to this myself until about the twentieth time my lover, plunging into me from behind, moaned in the throes of ecstasy, \"Your ass looks so fine when it quivers, and it feels so good bouncing up against me.\" Meanwhile I actually _was_ quivering, with even more intense pleasure than he! So complete the awakening of your Aphrodite's body by discovering, with your own gentle hands and fingers, the unexpected pleasures of derriere and anal massage, a sensual treat fit for a goddess.\n\n_106_. While you are pleasuring your vagina or your clitoris, smooth your hands over the silky curves of your behind. Use feathery fingertips or firm palm pressure; squeeze gently or hard; scrape your nails across the delicate skin; it's especially sensitive because it is rarely exposed to sun and air. Rub the rough lace or cool silk of a camisole against your fanny. Don your elbow-length gloves or furry mittens and massage your rounded curves and the deep valley between them. Experiment with wet hands, ringed fingers, naked knuckles, and rolling forearms. Have fun with feathers.\n\n_107_. Diane, an otherwise very buttoned-down attorney friend of mine, says she likes to rub her behind against the cool sheets of her bed the whole time she is pleasuring herself in other ways. Sometimes she places a pillow between derriere and bed to add a new sensation and a higher elevation. Or she lays her backside against the rough wool of the blanket and undulates with the motions of her vaginal self-play. Occasionally she even stands with her back against the wall\u2014she has textured wallpaper, but even smooth paint will do\u2014and slides up and down while she uses her hands to caress the front of her body. She loves to find different textures to stimulate her bottom during self-play and often sits in various locations to produce interesting new effects\u2014smooth, hard wooden chairs; rocking chairs; plump velvet couches; canvas deck chairs; rope hammocks; fabric-covered office chairs that roll around; desks; kitchen countertops; toilet seats; bathtubs, dry or with a thin layer of water; rugs and carpets; stairs; grass; sand; the rough bark of a tree limb; a swing; a large basket\u2014the possibilities are truly limitless.\n\n_108_. The anus is one of the most erotically sensitive spots on your body, and if you are freshly bathed, it's just as clean as the rest of you. So don't be squeamish about discovering its pleasures. Just remember that the secret to enjoying anal play is lubrication\u2014either your own saliva or a good massage oil. Start out by simply wetting your index finger and pressing or massaging all around and over the outside of the opening. Keep your mind and body focused on the erotic by simultaneously massaging your nipples, clitoris, or vagina. As you enjoy these luscious sensations, let yourself relax and feel good about this often outcast part of your body. Indulge in ecstatic sense memories of your lover's tickling caresses to your behind. Breathe deeply and consciously relax your sphincter muscles. Then gently and slowly work your finger inside your anus and keep massaging. Circular or come-hither motions are best at first; save the in-and-out movements for later when you're more accustomed to the feeling and more relaxed. Discover your personal internal hot spots by trying different pressures, motions, and locations. Many women find an especially delicious place to rub is on the vagina side of the anal canal, which is really the other side of the magical G-spot. In fact, this is how many otherwise nonorgasmic women finally achieve a shuddering climax. But even as a simple accompaniment to the main melody of clitoral or vaginal self-pleasuring, anal massage is a delight you won't want to miss.\n\n_109_. An especially deep and earthy feeling of ecstasy can be achieved by massaging\u2014from both sides\u2014that thin, silken spot that lies inside between the vaginal and anal canals. You can do this by inserting the finger of one hand in the vulva and the finger of the other hand into the anus, then rubbing them together. It's probably easiest to accomplish this while lying on your back with legs drawn up to your chest. One inventive friend of mine says she likes to leave a hand free for breast or clitoral massage at the same time. So she simply uses her thumb inside the vagina and slides the pinky of the same hand into her anal opening, bringing thumb and pinky together around the delicate internal tissues. Considering the possibilities for variation here, discovering your own style of anal play\u2014and later revealing them to your panting lover\u2014can be at least half the fun.\n\n# 5  \nthe mirror of sensual transformation\n\n_Woman is the creator of the universe, the universe is her form.... Whatever form she takes, whether the form of a man or a woman, is the superior form. In woman is the form of all things, of all that lives and moves in the world. There is no jewel rarer than woman, no condition superior to that of a woman_.\n\n_\u2014 Saktisangama Tantra_\n\nPriestesses of the Tantric tradition, sometimes called yoginis, took great pride in the firepower of their bodies and sexuality to burn up obsolete ideas and beliefs, transforming them into jewels of expanded sensual perception and wisdom. Intimately familiar with the physical and esoteric function of every part of their bodies, they deliberately adopted certain postures and palpated special body points that enhanced their sensual energy flow. They had the ability to see the true nature of their resplendent physical forms reflected in a quiet pool, a looking glass, or a lover's eyes.\n\nBy looking into the yogini's Mirror of Sensual Transformation and seeing yourself as a magnificent creature made for love, you too will be able to shed worn-out ideas about puffy tummies and sagging chins, throw radiant light on heretofore hidden and mysterious places, and come to truly understand that \"in woman is the form of all things.\" If you approach this anatomical exploration as an exhilarating journey of rediscovery, you will no doubt have provocative revelations about yourself and your sexuality\u2014revelations that will make you an unusually creative bed partner. In fact, sensuality experts as diverse as ancient Tantrics and modern-day sex therapists _require_ expertise in this kind of self-pleasuring investigation before moving on to any other kind of sexual teachings. It's an eye-opening fount of knowledge and an autoerotic turn-on. So grab your Inner Sex Goddess, light a few candles, and make a special ritual out of discovering the divine firepower of your own sumptuous body.\n\n_110_. Set up a full-length mirror and a good-sized hand mirror in a private, quiet room. Lock the door and make sure you will have at least an hour and a half completely to yourself. Then start out by creating a soft sensual mood\u2014after all, you are about to seduce the sacred inner you. Take a scented bath. Light some candles and incense. Put on sultry music. Fix your hair in a pretty style and dab on some musky perfume or body lotion. Don your sexiest lingerie and pose before the mirror, proudly erect with hands on hips, as the yoginis do, or reclining provocatively like the centerfold models in _Playboy_ magazine. Then seductively strip down to complete nakedness, or perhaps leave on a string of pearls or a thin chain around your waist, or a flower in your hair. Think of all this as preparation for your lover. Just as Cleopatra's attendants would have readied her for romance with scented milk baths and opulent adornments, so should you carefully prepare for a loving session with yourself. Crawl into the translucent, hot skin of your Inner Sex Goddess and enjoy.\n\nThen sit or stand about ten inches away from the full-length mirror and stare gently into your eyes. Relax, breathe deeply, and imagine your eyes becoming like soft jelly. Look into your soul. The Tantrics recommend eyegazing like this for about five to ten minutes, longer if you can. Really let yourself trespass on the inner you\u2014sultry, deeply sensual, romantic, tender, and loving.\n\nBegin to explore your body with your hands. Look at yourself from all angles\u2014front, back, side, standing, kneeling, sitting with legs apart and together. Do not criticize. If you have to, imagine that you are from another planet where lumpy thighs are considered exquisitely sexual. Examine every nook and cranny with the handheld mirror, praising and patting each delicious body part. Many women find that looking at their own bodies this way turns them on. Admire and caress your entire body, except the genitals, for at least forty-five minutes.\n\nNow lovingly examine your pubic area in both the full-length and handheld mirrors. If you have a makeup mirror with a magnifying side, use that, too. Observe the different colors, shapes, and textures. Label each part by touching it gently. Notice first the contour of your upper pubis. Does it slope invitingly down to your vulva, or is it a high, provocative mound? Is your pubic hair a dark, thick bush or more like wispy blond grass? Does it form a triangle? a bold band? a small circlet peeping over the edge? How would someone who was mesmerized by it, like an adoring lover, describe this part of you?\n\nAs you slide your fingers farther down and along the hair-covered outer lips of your vulva (labia majora), be aware of their spongy or pillowy texture. See if yours are puffy, flat, long, or curved. Would you describe their color as burnt sienna? dusky rose? amorous amber? Regarded by some as the female equivalent of a man's scrotum, these lips contain glands that, when aroused, release a film of sweat that chemically stimulates and attracts the opposite sex. Did you know they were so enticing?\n\nWhen you stroke your hairless inner lips (labia minora), notice their smooth, moist slickness. They too contain glands that produce lubricating juices. Feel, smell, and even taste these fluids when they arrive. Try to characterize the size, shape, and texture of your inner lips, which vary greatly from woman to woman. Some women have described theirs as having \"the feel of satin sheets,\" \"caviar coloring,\" \"edges like a doily,\" or \"the clean lines of a Jaguar racing car.\" Do yours remind you of a flower? the inside of a peach? folded velvet? a sea creature? a fleshy heart? a yin-yang symbol? Is the left lip bigger or smaller than the right one? Do the inner lips stay tucked demurely inside the outer ones, or do they protrude or droop temptingly? Notice how the texture, moisture, and warmth of both your inner and outer lips change as you excite them, engorging with blood, thickening, and darkening.\n\nThen slide on up to the clitoris, or the Jewel Terrace, as it's lovingly known in the Orient. Slip back the hood of skin that covers it and run your finger along the shaft. Although it is much like a man's penis, the clever little clitoris is unique\u2014no organ whose _sole_ function is to stimulate and elevate erotic pleasure exists on a man's body. The most sensitive of your sexual organs, it is often the _key_ (remember that's the literal translation of \"clitoris\" in Greek) to sensual ecstasy. That's because it is crammed full of tender nerve endings, more than any other part of the vulva. Here too are intensely sensitive microscopic fibers called Pacinian corpuscles, jammed even closer together in the tip of the clitoris, making this apex a keenly acute receptor of sharp pleasure\u2014or pain, if too roughly handled. Gently experiment with differing strokes and amounts of pressure on your clitoris, touching it directly\u2014with a wet finger is usually best\u2014and indirectly through the surrounding tissues, up and down the shaft, across the tip, circling the whole area. How does it look and feel as it receives these different types of loving attention?\n\nUsually about 3\/4 inch to 1 1\/4 inches long, each woman's clitoris will have a size, shape, and character all its own. Some women have described theirs as a \"tiny dewdrop,\" a \"tight, pulsing rosebud,\" or a \"big flashy cultured pearl.\" How would you describe your \"love diamond\"? In an Aboriginal creation myth, the two Djanggawo sisters had clitorises so long that they formed grooves on the earth as they walked, and the well-endowed sisters had to rest their clitorises on their thighs in order to have access to their vaginas. What is the relationship of your clitoris to the vaginal opening? Is it a mere finger's width away, or more like a wrist's width distant? It may be that women whose clitorises are very near to their vaginas are the ones able to have orgasms through intercourse alone.\n\nAs you press and massage your love bud, notice the tiny ridge of tissue that runs underneath. Not just the little pebble you see on the outside, the clitoris is actually a much larger organ that lies partially buried beneath the surface. In actuality, the clitoral system is at least 30 percent larger inside your body than it is on the outside, so you can experience flashes of burning pleasure when it is stimulated from inside your vagina too. The clitoral shaft is composed of two rods of spongy tissue that fill with blood during sexual arousal. (It should be enlarging and becoming more sensitive and \"erect\" right now as you touch it.) These two rods bend backward and connect with the pelvic bones on either side of the vagina, while another muscular rod extends from the inner part of the shaft, splits in two, and surrounds the vaginal opening. During arousal, these muscles contract and keep blood from leaving the clitoris, helping to keep it enlarged and tugging gently on it as well. Try stroking your vaginal lips as you watch in the mirror what happens when the pressure on these connected tissues sends thrilling messages to your clitoral bud. Imagine that your lover's penis, rubbing in the vicinity, will do the same.\n\nAs you massage your inner lips, notice how your vagina naturally opens to your finger. Approach this portal in a sacred way, as do the people of the Sepik River in New Guinea. They enter and exit their ceremonial houses by crawling between the outspread legs of carved figures of the sacred feminine form. Slip through the entrance of your inner cave to feel the creamy warmth within, and the differences in texture and sensation between all the sides of its elastic walls. Some sexologists assert that the first third of the vagina is more sensitive to touch than the remainder because more nerve endings are located there, and that the deeper two-thirds is more responsive to pressure and stretching. Many women feel differently. Slide your fingers deep inside and see what's true for you.\n\nFeel the power of your PC muscles as you alternately squeeze and push them against your fingers. Watch your muscles contract in the mirror. Imagine how this milking action would feel to a penis. Explore the thick folds of your vaginal walls, some running horizontally and some vertically, and the milky secretions they produce. Feel, smell, and taste this love juice.\n\nDelight in your vagina's miraculous ability to compress or expand to accommodate the smallest tampon, the largest erect penis, and even the birth of a baby. See if you can get your fingers far enough inside to feel the way the upper two-thirds of your vagina opens up like a sultan's tent, and the lower third swells to create a warm, spongy, soft area of tissue that will eventually grip a penis tightly. Experience the particular way your magical inner cave lifts and opens wide as you become more aroused, pulling your vaginal lips apart, sensitizing and flooding them, too, with blood and moisture.\n\nIn the Quodoushka tradition, a Cherokee equivalent of Tantra, the depth of the vagina and its characteristics are thought to be important clues to a woman's entire sexual temperament. In Harley Swiftdeer's _Quodoushka Manual_ , five female genital anatomy types are described, including Dancing Woman and Sheep Woman. With a vagina of average depth, small inner lips, and a high, small clitoris that springs up from beneath its hood like a dancer, Dancing Woman prefers a lover with a short, thin penis ( _tipili_ , or \"sacred snake\"). Sheep Woman, on the other hand, has a deeper vagina, larger, thicker vulval lips, and a lower, more hidden clitoris. She produces copious lubrication and can climax only if she feels a heart connection to her partner. What is the nature of your \"feathered flying serpent\" ( _tupuli_ ), as the Cherokees call it?\n\nLook in the mirror now to see how gorgeous your vulva appears when excited with love, and how much it has changed since you started this scintillating adventure. Reconsider your vagina not as an unattractive and unknown void but as a divine portal to the most secret parts of the inner you, a doorway to your Inner Sex Goddess, a Vermilion Gate, as the Orientals call it, through which to receive your lover in the most royal fashion. Praise and perfume this wondrous, sacred part of yourself.\n\nSince there are few things more beautiful than a woman aroused, gaze in the mirror now and see how gorgeous your whole body is\u2014how pink your luscious lips have become, how your eyes have dilated and your skin flushed. Admire the way your breasts have swollen and your nipples have become harder and darker, erect with desire. Your clitoris, too, is erect and pulsating. Your vaginal lips may have swollen up to three times their normal size, and they may have darkened to a deep wine or a bright red. Your toes, feet, or stomach muscles may be fluttering or quivering in a high state of turbulent excitation. Notice how your body not only _looks_ different but also _smells_ and _moves_ differently. Take in the whole image of you in full bloom\u2014glorious, sensual, and powerful. Tell your reflection how radiant, delicate, sexy, graceful, glowing, shapely, and devastating you look. Bask in your beauty and thank yourself, your body, and your Inner Sex Goddess for revealing their wonderful secrets to you.\n\n_111_. Finish off this mirrored self-play by continuing to arouse and ravish yourself almost to the point of orgasm\u2014at least three times and maybe up to fifteen times. With each buildup to and retreat from the point of no return, your excitement will reach higher and higher levels. Then finally let yourself climax, watching your orgasm in the mirror, and float in the dreamy sensations for an eternity.\n\n_112_. A very saucy friend of mine, who actually appeared in a _Playboy_ video of over-forrty-and-still-fabulous women, suggests the following alternative way to discover your own body: perform all of the activities in this chapter not in front of a mirror but before a video camera. If you don't own one, you may be surprised to find out how inexpensive they are to rent. Watching your own sexy, self-loving video at a later time can be not only immensely informative but also a titillating turn-on\u2014for you and maybe for a fortunate partner!\n\n# 6  \ng(oddess)-spots and love nectars\n\n_Woman has sex organs just about everywhere. She experiences pleasure almost everywhere_.\n\n_\u2014 Luce Irigaray, This Sex Which Is Not One_\n\nis there really such a thing as a G-spot? Does every woman have one? Will I be able to pinpoint mine? Can it give me a different type of orgasm? Yes, yes, yes, and _yes!_ In fact, the G-spot is one of your Inner Sex Goddess's favorite treasures. She knows where yours is, and she's been eagerly waiting to initiate you and your lover into the hidden ecstasies of its potent orgasms and love nectars.\n\nAlthough the ancient Romans, Japanese, Chinese, Tantric Indians, and indigenous cultures throughout history were intimately familiar with this special sweet spot inside a woman, modern society seemed to lose touch with its existence until 1950, when a German gynecologist named Ernst Grafenberg researched and wrote about it. His findings caused quite a stir among the medical and scientific crowd, but not too many other people knew about them. Then thirty years later a trio of psychologists and sex therapists expanded Grafenberg's work into a book and decided to name this ancient and utterly female body part after him\u2014the _man_ who supposedly discovered it! So it became known as the G-spot, although it does go by other, more accurate and interesting names. One woman I know affectionately calls it her love button. The Federation of Feminist Women's Health Centers refers to it, very clinically, as the urethral sponge. In Panama they speak of this savory spot as _la bella loca_ , which literally means \"the beautiful crazy.\" I like to think of it as a woman's goddess spot. Whatever it's called, this lovely hidden site is a pleasuring bonanza.\n\nIn fact, many sex researchers believe that learning to have a G-spot climax is an important key to achieving multiple orgasms. That's because after you've had one vaginal or clitoral orgasm, the extreme sensitivity of those areas will make you want to stimulate other less overwrought areas of your genital pleasure zone. Meanwhile, some doctors and psychologists are still haggling over whether women have vaginal orgasms, clitoral orgasms, or some combination of the two, or whether there even _Is_ a vaginal orgasm. We should do them a favor and clue them in to the fact that women have _all kinds_ of orgasms, including clitoral, vaginal, G-spot, and breast! The beauty of the female body is that it's sensitive to orgasmic stimulation all over\u2014and, yes, the G-spot is _especially_ sensitive to the erotic touch, and can give you an entirely different type of orgasm than you've ever had before. It may be more subtle and somehow deeper than the ones you're used to, but maybe not; maybe your G-spot orgasms will be more electrifying or tinglier or more wavelike. Your own goddess spot mystery waits there for you to discover.\n\n## what and where is the g-spot?\n\nEveryone has a pet theory. According to Grafenberg, the G-spot is an area in the vagina composed of the paraurethral glands and ducts, a complex network of blood vessels, nerve endings, and the tissue surrounding the neck of the bladder. The apparent function of this network is to act as a buffer between your lover's penis and your urethra by swelling and filling with blood during sexual excitement. But many doctors say the G-spot is merely a small cluster of nerve endings and blood vessels, like a little bean deep inside the vagina. And some feminist sex researchers claim it covers a lot more territory\u2014that, in fact, it encompasses the entire complex of tissues around the urethral opening, all the way up the long bladder tube, and all around the vaginal canal.\n\nThe somewhat esoteric quality and imprecise location of any given G-spot may account for the fact that some women and their mates believe they don't have one; having been told it has to be located in one specific area, they conclude that a lack of special sensitivity there means the G-spot doesn't exist for them. How unfortunate, especially since, as we've seen, doctors and sex researchers can't even agree on where this one specific spot is! Some say it's about two inches inside the vagina, others say it's above and in back of the pubic bone, and still others say it's much deeper than that, almost at the end of the vaginal canal.\n\nMy Inner Sex Goddess and I have our own personal theory. Apparently there are very tender nerve endings associated with the urethra, at the opening and all along the bladder neck, probably designed to protect the very important function of waste removal. Since the bladder neck runs parallel to the vaginal canal, on the side toward your stomach, you can feel this tender bladder tissue through your vaginal walls. These urethral nerve endings are supersensitive in various spots along the way; and the exact location of these spots is _different for every woman_. In other words, every woman has her own personal G-spot\u2014or even two or three G-spots!\u2014that will look, feel, and locate itself in a manner completely unique to her. Just like everything else about a woman's sexuality, your G-spot is a highly individualized treat.\n\nBut most doctors and sexologists do seem to agree that the main function of the G-spot is to help you achieve a high degree of sexual intoxication! When you are drunk with rapture, just imagine what an uninhibited and wanton hussy you become, how much more intensely you feel even the barest touch, how much easier and more delicious it is to abandon everything to your feeling of pleasure and that of your lover. This is the special bounty of your G-spot.\n\nHow, then, do you find your special sweet spot or spots? Call on your Inner Sex Goddess to help you; she will raise your level of attunement to the subtler sensations of this area. The other helpful trick is to keep in mind the G-spot's second main function\u2014to swell during sexual excitement for protection of the urethra and bladder. When you are aroused, the G-spot will rapidly become engorged, get harder, and may even develop well-defined edges, making it much easier to locate.\n\nYou can take the following steps to find your goddess spot:\n\n_Step 1_. Get yourself into a state of arousal.\n\n_Step 2_. Assume a seated or squatting position, like sitting on the toilet, for instance. It's almost impossible to find your sweet spots while lying on your back because gravity tends to pull your internal organs down and away from the vaginal entrance. (Remember this when you are trying to feel your G-spot during intercourse!)\n\n_Step 3_. Lubricate your finger and first massage gently around the urethral opening, sometimes called the meatus, not around the vaginal opening. See if and how sensitive this external area is for you. A whole new experience opened up for me when I discovered this as one of my personal goddess spots, and I couldn't understand why I'd never thought of it before. But it may not be yours, so explore carefully and with an open mind.\n\n_Step 4_. Still making sure your finger is lubricated, insert it gently into your vagina. Letting yourself be guided by your instincts, use firm upward pressure to slowly investigate along the front of your internal vaginal wall. (Some women find that pressing down at the same time on the outside of the abdomen with the other hand helps pinpoint areas of special sensitivity.) Because the bladder tube runs on the other side of the front vaginal wall, you'll find your own G-spot somewhere along this path. One woman I know said she found her special spot when she felt that she was touching \"an inverted clitoris, one that was inside my vagina, and much more pleasurable for me than my clitoris.\" But anywhere you feel a slight jolt, a momentary urge to urinate, a small, rather hard lump, or a warm, deep sensual feeling is _it_. If you don't find a G-spot right away, or even after you do, don't stop searching. You may have more than one!\n\n_Step 5_. A commonly found G-spot lies about half an inch inside the vagina at about eight o'clock or four o'clock (if the top of your vagina is twelve o'clock). One woman I know found her special spot \"in front and a little to right of center. When I touch it, or when my lover's penis does, it feels pleasurable and kind of liquidy.\"\n\n_Step 6_. Be sure to try the area just above your pubic bone. As you probe inside the vagina, you first encounter the lumpy Skene's gland just inside the opening. Then there's the hard area of the pubic bone itself. Beyond it lies a smoother area leading back to the cervix. From there, come back toward the pubic bone a little and hook your finger in back of it where there's a small hollow. Wag your finger right and left. Many women find their G-spot in this area; and it is the place to which most scientists refer. But if, like me, you aren't especially sensitive right there, don't stop searching yet!\n\n_Step 7_. Keep tracing your finger along the front vaginal wall, as far as you can reach. Stroke and circle with your finger. Deep inside the vagina, almost to the end, is another very common sweet spot. I especially notice mine here when my lover's penis is thrusting deep into me from behind, but it's difficult to reach with my own finger. So don't give up too soon in trying to find a possible deep sweet spot. You may want to insert a wet finger or dildo and duplicate the angle of rear-entry intercourse to fully explore the path of your goddess spot.\n\n_Step 8_. Notice and remember where your sensitive spots are and how they felt when stimulated. As I've mentioned, you may have one or more G-spots; some of them may be larger than others; you may be hotly sensitive to G-spot stimulation or just mildly warmed by it; upon contact with your G-spot you may or may not have an immediate orgasm, and it may feel sharper or softer than those that are clitorally stimulated. One friend of mine describes her G-spot climaxes as \"sudden, sharp, and explosive,\" while another says, \"It feels like a warm, deep wave spreading throughout my whole body.\" I don't always have an orgasm when I rub my deeper G-spot, but when I do have one, it's like a slow, pulsing release. On the other hand, massaging the sweet spot at my outer urethral opening can make me feel as if I've been shot out of a cannon. Your experience will be different still.\n\nHere's what Diane, my buttoned-down attorney friend, says about her G-spot explorations: \"At first I had trouble finding my G-spot because I had this set idea about where it was supposed to be. And it just wasn't there. Then one time when Tom was making love to me from behind and we were lying on our sides, he angled into some corner of me that felt really good. The next day I pressed around inside by myself and found it again\u2014a sort of mushy, earthy place that made me feel as if I were melting down. When I told Tom about it, he probed around with his penis until I started shuddering. This made him really hot, and he just kept thrusting into that place. Somehow this spot seemed connected to a wild animal inside me, and once it got loose, I started bucking and screaming for Tom to 'fuck me hard!' That night we made love for hours with a ferocity and urgency I'd forgotten existed in our relationship.\"\n\n## love nectar\n\nWhen you massage your goddess spot, you may find yourself ejaculating a small amount of clear fluid from your urethra. Despite appearances, this is not urine. It is not the color of urine, it does not smell or taste like urine, nor does it stain like urine. Furthermore, it has now been scientifically tested to prove it is not urine. This sweet juice is your love nectar, and it falls under the special province of Xochiquetzal, Mayan goddess of beauty, sexuality, and flowers. Sometimes represented as a butterfly, Xochiquetzal symbolizes the sacrament of sipping transformational ambrosia\u2014as the butterfly sucks the nectar of the flower, so the lover drinks the honeyed liquid of his woman's vulva. Never be embarrassed by it; it is a lovely female ejaculation that announces the presence of your very enjoyable orgasms.\n\nGrafenberg observed that with women, as well as with men, the ejaculation that occurs at orgasm involves fluid being ejected in gushes through the urethra and that this happens most often when the G-spot is being stimulated. Since we know that sweet spots are really urethral nerve endings and tissues this really makes sense. It also makes sense that G-spot ejaculation is intimately related to the strength of the PC muscle, the one that controls the flow of urine. The stronger your PC muscle is, the more likely you are to emit love nectar. Some women claim they ejaculate every time they make love; others say it happens only occasionally. Some detect a cyclical pattern, possibly related to the phases of the moon. And many women never ejaculate at all and have just as good a time. So whatever your orgasmic flow experience, don't think it strange.\n\nIn fact, the female production of love nectar has been recognized and even reverently cultivated for centuries. Aristotle observed that women expelled fluid during orgasm. So did the famous second-century Greek physician Galen. And in the seventeenth century, Dutch anatomist Regnier de Graaf described the \"female prostate\" and \"female semen\" in detail. The Bataro tribe of Uganda was known to have a custom called _kachapati_ , or \"spray the wall,\" in which older women taught younger ones how to ejaculate. And in some ancient and modern Tantric teachings, sexual intercourse is said to have the specific esoteric function of stimulating the flow of female fluid, which they call _rajas_. The men of this Tantric school even make a ritual out of collecting the precious juice on a leaf, adding it to a bowl of water, offering it to the deity presiding over the ritual, and then drinking it themselves.\n\nThe taste of this orgasmic fluid has been variously described as very sweet, tangy, bitter, or tart. Not only is every woman's flavor different from that of others, but the taste of any given woman's _rajas_ may change depending on her mood and the time of month. I was once lucky enough to have a lover who delighted in what he called the \"sweet 'n' spicy\" flavor of orgasmic juice. He loved to stimulate me to produce this nectar so that he could drink it directly from my body.\n\nSo never try to inhibit the natural flow of your precious love fluids. Remember, the Tantrics worshiped them. Try seeing yourself as they did\u2014as the embodiment of the divine female principle, fountain of life and pleasure, source of the magical elixirs of physical and spiritual transformation\u2014at the very least, a Sex Goddess who's so hot she overflows with the sweet nectar of love.\n\n## the goddess spot pleasure dance\n\nNow that you've located your particular sweet spot and learned that it's perfectly okay to secrete special love juices, let's explore the best ways to incorporate the G-spot into your self-pleasuring adventures. You may find a direct shortcut to the temple of your Inner Sex Goddess.\n\n_113_. It's usually best to save the G-spot for later in your pleasuring session, maybe even after you've had your initial orgasm, so that it's swollen, sensitized, and easier to find and excite. Massage, stroke, tickle, and fondle your sweet spot as you would your vaginal lips or clitoris. You will probably want to use heavier pressure on your goddess spot than you would on your clitoris, and you will probably feel the sensation deeper inside your body. Abandon yourself to its earthier tones of pleasure. You can perform this erotic dance all by itself, or you can add clitoral, nipple, or navel and tummy massage to your private party. Or simply stroke yourself all over and dream about Antonio Banderas. With the G-spot, prolonged stimulation is usually the best way to rouse your Inner Sex Goddess from her sensual slumbers.\n\n_114_. The _Kama Sutra_ suggests rubbing your G-spot with the middle or third finger. The ancient erotic text considers the index finger too heavy and aggressive for this delicate pleasure. Of course the middle finger is often the longest one, too\u2014making it easier to reach your G-spot. You may also want to experiment gently with dildos, vibrators, and peeled carrots to provide a longer reach.\n\n_115_. Positioning is important. Remembering that lying on your back is not the best way to find your sweet spot, you will probably correctly surmise that it's also not the best way to stimulate it. Try kneeling, squatting, or sitting on your heels with your knees apart. The positions you get into for rear-entry sex are good, too. Sandy, a friend who has created the most sumptuous bathroom I've ever seen, likes to sit on the edge of her bathtub, knees spread wide, and feet in the tub. She says this position provides a great angle for finding and rubbing the somewhat elusive love button, and she can keep her fingers wet at the tap.\n\n_116_. Get on your hands and knees and use your fingers to imitate a thrusting penis bumping up against your G-spot from behind. Make sure your fingers are well lubricated and the nails trimmed. You can achieve the best angle by sliding your fingers around behind your leg and fanny, instead of approaching from the front. The rear entry and the downward motion, along with maybe a lusty \"being taken\" fantasy, add up to an unusually exhilarating experience.\n\n_117_. If you want to learn to make love nectar, or to do so more at will, here's how. First increase your PC muscle strength and control by conscientiously doing about one hundred quick squeezes a day for at least a month. Then, in one of your self-loving sessions, get really hot and ready by giving yourself several regular orgasms with your favorite method of clitoral and vaginal stimulation. You should feel as if you're on an orgasmic plateau. Now stimulate your goddess spot with your fingers or a vibrator; if you have a sweet spot on the outside at your urethral opening, this is usually the best one to start with. Here's the key: push out and down with your vaginal muscles, rather like the way you would push with a bowel movement or when giving birth. Keep pushing as you rub, and do whatever it is that makes you so excited you feel like shaking or exploding.\n\nDavid, a middle-aged accountant friend of mine, told me about the first time his wife favored him with her love juice: \"First I got her incredibly hot by licking her lovely pussy until she came three times. Then I slid my two fingers just inside her opening and circled around the rough area there. She began to moan and cry 'Don't stop!' which really turned me on too. Pretty soon her vagina was pumping my fingers, almost shoving them out. And then she squirted the most delicious warm liquid all over my hand. I loved it!\" The important things to take note of here are her high level of arousal, the sweet-spot massage, and the shoving that she did. These are the keys to making a successful and exciting brew of goddess love nectar, either by yourself or with your man. Happy trails!\n\n# 7  \nfantasies from the cloud damsels\n\n_People walk down the street and see someone they are attracted to and they can undress that person right in their minds and perform all kinds of sexual acts on them in their fantasies. They go to bed at night and generate fantastic pornographic images in their sleep. This is just something that happens. It seems to be a natural activity_.\n\n_\u2014 James Hillman, former director, Zurich Jung Institute_\n\nfantasizing is as natural as walking, sleeping, and breathing. In fact, we can't seem to stop doing it. And why would we want to? The exciting sensory images that come unbidden from our subconscious, from the heart of our most intimate erotic reservoirs, are as necessary to our lives as the bread we eat, yet as naughtily delicious as chocolate candy. They make our skin tingle, our hearts flutter, and our deep inner caves moisten. And like magic carpets, they transport us to enchanted realms of raging desire, perfect love, and unashamed carnality.\n\nThe goings-on in our heads\u2014our attitudes, beliefs, personal rules and regulations, wishes and dreams\u2014are at the very source of who we really are. They keep us posted on who we think we are now, who we'd like to be, who we'd like to \"try on\" for five minutes, and who we'd like our partners to be. And they determine who we _will_ be, if we so desire. Erotic, irrational, and completely carefree, this private world is ours alone\u2014unassailable by anyone else's opinions or needs, undemanding of anyone else's presence or good graces, unfathomable to anyone else's linear everyday thinking. Our personally fabricated mythology constitutes an intimate inner fortress and playground that gives us strength, joy, and freedom. For even if we can't quite manage to let ourselves go stark raving wild in our jobs, relationships, and bedchambers, our fantasies give us the opportunity to visit a land where letting it all hang out is deliciously de rigueur.\n\nThese exotic passports to otherwise forbidden erotic territory are issued by the ethereal Cloud Damsels who float through our imagination. Erotic nature spirits who live in a lofty dimension parallel to human reality, the Cloud Damsels, known as Apsaras in Hindu and Buddhist mythology, are sacred beings of both earth and sky. Originally human women but now goddesses (like you!), their special gift is the ability to combine sensuous flesh with divinity, sexuality with spirituality, fantasy with reality. With their fluid, sweet-smelling bodies, they may take one human lover or hundreds of godlike consorts, effortlessly combining the earthy sensuality of one with the impossibly wild and divine imaginings of the other, and vice versa. Tall, ornate headdresses sprout from their temples like the abundant fruitfulness of their erotic imagination. They delight in spicing up any and all sexual encounters with a tidbit of juicy fantasy, elevating the mundane to the glorious, human sweat to divine ambrosia, and solo sex to a magnificent garden richly populated with amorous Greek Adonises, dashing and swarthy sheikhs, and insatiable young sexpots of the male, female, animal, and extraterrestrial persuasion.\n\nThe Cloud Damsels, as close companions to your Inner Sex Goddess, live in the most uninhibited and uncensored part of your mind. There they are free to reflect the vivid images of your raciest thoughts, your most intimate desires, and your most deeply hidden pleasure stimulators without having to bother with the complications of real life, real people, or real consequences. They give you the freedom to indulge in any outrageous scenario you care to\u2014being penetrated by a huge black stallion, making oral love to another woman, having wild sex with Bj\u00f6rn Borg on Wimbledon's center court as thousands of horny onlookers applaud\u2014while still keeping your marriage, your self-esteem, your emotions, and your pocket-book intact. What a divine luxury!\n\nWhile the five physical senses are potent conveyances to the realm of your Inner Sex Goddess, fantasizing can be thought of as the sixth, vitalizing sense. Under the purview of the otherworldly Cloud Damsels, it's the sense that enhances all the others, opening up even more diverse avenues for your unfettered sexuality, and, as Nancy Friday says in _My Secret Garden_ , \"taking one further, faster in the direction in which the unashamed unconscious already knows it wants to go.\" Your creative and erotic imagination is the sense that can intuit, interpret, and invent intangibles\u2014like romance, atmosphere, mystery, intoxicating danger, forbidden delights, and irrational, all-consuming lust\u2014all the things that stimulate our particularly female hormones. For we women are creatures of rhapsody and enigma, more stimulated by emotion than by action, more aroused by the romance of sex than by the mechanics of sex, and more unloosed by imagination than by facts, formulas, and graphic realities. We come fully alive in our fantastical imaginings.\n\nThat's why your Inner Sex Goddess recommends investing _every_ sensual activity with erotic imagery\u2014exploratory sessions with your body, sybaritic warm-ups for sex, partner lovemaking, and especially self-pleasuring, where it's just you and your unbridled imagination. As the Cloud Damsels in you know, all of these fleshly pleasures can be even further glorified by healthy dollops of divinely inspired phantasm. And in this rarefied atmosphere of magical reality, _anything_ goes. There is nothing to be ashamed of, for your daydreams, reveries, and mental inventions simply reflect your unique, and completely normal, sexual personality\u2014uncensored, rare, courageous, and exquisitely precious. You don't have to act them out, tell them to anyone, or even figure them out. On the other hand, you can do all of those things if it would please and excite you to do so. Your flights of fancy are your private possessions to invent, modify, reveal, conceal, discard, repeat, and indulge in as you will.\n\n## having a pleasant flight\n\nWhen you allow yourself to travel to the vast open spaces of fantasyland, you will find your sensual life taking on a new radiance. Women who give free rein to their Cloud Damsel visions report that their experience of sex is more intense, more ecstatic, and more full-bodied than they ever thought possible. In fact, many women say they can achieve orgasm by fantasy alone, with no physical stimulation at all. The key is to surrender completely to the irrational brilliance of your own erotic mind and to let your fantasies lead you, not the other way around. Here are some tips for making the most of your fantasy flights:\n\n\u2022 Impose no limits. While few of us would actually have sex in the middle of Times Square, fantasizing about it can be delicious, healthy fun. So don't label yourself a pervert or nymphomaniac just because you like to dream about being trussed up, blindfolded, and ravaged by twenty-seven Mongol warriors. On the contrary, be as outrageous and outlandish as you possibly can. Enjoy the rare opportunity to explore the impossible and to step into someone else's skin\u2014whether it be Venus de Milo, Irma la Douce, a nymphomaniac mermaid, a tightly laced Victorian damsel aching to have her bodice ripped, a primitive cavewoman, a creature from _Star Wars_ , or even a male truck driver with \"horns.\" Whatever your weirdest fantasy, there are at least a thousand completely normal women out there whose kinky castle-building is even more over-the-top than yours. So let your unfettered hallucinations bring out all of your muskiest natural perfumes, your most salacious proclivities, and your wildest primal instincts. Make no judgments. Take no prisoners or guilt trips. These private tours are safe within the boundaries of your head and heart.\n\n\u2022 I think it's best _not_ to share your fantasies with your mate. He might feel threatened, intimidated, outraged, or even betrayed, especially if his delicate male ego is called into question. Worse yet, he might criticize or belittle your fanciful efforts, puncturing a gigantic hole in your fantasy balloon. But most important, if you know someone else will be privy to your secret exotic dreams, you will not feel free to indulge in the really juicy, borderline ones\u2014the most liberating and fun visions of all!\n\nThe exceptions to this are those fantasies that you specifically want to turn into reality and that you feel are safe, nonthreatening, and practical enough to do so. I once conjured up a vision of taking my lover to the opera, both of us dressed to the nines, and slipping off for a stand-up quickie in some dark corner while the onstage drama raged on. The elegance, furtiveness, and danger of this make-believe scenario really appealed to my naughty Inner Sex Goddess\u2014and to my man, when I shared it with him. For months it was a joint fantasy that we used to titillate each other. Then one night at the theater, he got up in the middle of the first act and led me to a vestibule, where, behind a swinging door, we pounded into each other until we heard the audience applaud and begin to leave the theater proper. Just in the nick of time, we refastened our clothes and nonchalantly emerged from behind the door to join the exiting audience for intermission. It was fantastic, but then I had to think up a new outrageous daydream!\n\n\u2022 Fantasize at every opportunity\u2014while commuting on the train, lying in the sun, ironing tablecloths, enduring a meeting, fixing dinner, observing a flower, drifting off to sleep, showering, flying on a plane, walking to work, and especially when looking at and touching your body or that of your lover. Fantasize while you are preparing for a sensual encounter, making love to your man, or indulging in your favorite self-pleasuring activities. The more you flex your fantasy muscles, the more alive you will feel, the more abandoned and creative a lover you will be, and the better you will know your Inner Sex Goddess, for it is in the realm of your vivid imagination that she lives, breathes, and grows.\n\n\u2022 Use props. Make your fantasies as real as possible by adding a dash of tactile reality\u2014a filmy scarf for your harem sojourn (while in Egypt I bought scads of sheer belly-dancing scarves with dangling beadwork just for that purpose); a furry tail for your Catwoman reverie; a length of rope or a blindfold for your \"rape\" fantasies; mesh stockings and a brief, frilly apron for the French maid in you; sandalwood or musk incense for imaginary Tantric rituals; juicy fruits for fantasy love with Tom Jones. And of course always have your favorite toys readily available\u2014vibrators, videos, scented massage oil, ben wa balls, and so forth. Your castles in the air become much more erotically solid when draped with the stuff of reality.\n\n\u2022 Let your erotic visions speak a different language. In fantasies, men don't have \"penises\"; they flaunt swollen \"cocks.\" You no longer own a \"bosom\" and \"genitals\"; now you've got creamy \"tits\" and a \"juicy pussy.\" And horny hunks don't swarm around to have \"intercourse\" with you; they are panting to \"fuck\" you silly. For many of us, the brazen language of sex can instantly trigger your most wanton instincts and transform your fantasies from bland into sizzling. Foreign words, too, can add just the right touch of exotic romance when you are daydreaming about sexy Italian counts or rough-hewn blond Vikings. Phrases such as _\"Baise-moi, ch\u00e9ri!\" \"Bums Mich Schneller, Liebling!\"_ and _\"Chiavami, tesoro!\"_ are just the ticket to exciting foreign fantasylands. Learn to savor these powerfully evocative words. Whisper them to yourself in the dark of night. Chant them to yourself. Use them to catapult otherwise casual musings into the realm of hot, depraved fantasy masterpieces.\n\n## going to the source\n\nI am in the African bush with an untamed lover. We roll in the leaves and wallow in the mud like languorous primitive beasts, and then he washes me off in a cool green river. We glide through the water on the back of an alligator, and when the creature tries to reach around and devour me, my wild lover saves me from its cruel jaws with the power of his muscled arms. He proudly parades me through the village, naked on the back of a giant wrinkled elephant. In his hut he leaves me bare but wraps himself up in white robes so that I can see only his dark smoldering eyes as they travel hotly over my quivering nakedness. Then he parts his robes for me so I can touch his hot, hard body, and I kneel down to take his swollen member in my mouth. Caressing my head as I suck him, he brings me under his robes, where it's even hotter than under the African sun. He pours his honeyed come down the front of my body and rubs it into my breasts so that I'll bear his scent. Slowly we sink down onto the rough mat, and he enters me from behind, biting my neck and growling softly like a wild animal. His huge shaft plunges hard against the very deepest part of my woman's well, and soon I am panting in uncontrolled ecstasy. Hours pass as he keeps me teetering on the brink of breathless release. Finally, possessing me completely by reaching around to put one hand on my throbbing bud and the other on a burning hot nipple, he thrusts even more powerfully into me, snarling like a great tiger\u2014and I scream in the ecstatic agony of my body's surrender. Later he bathes me under a cascading waterfall and lies with me on the banks of the river, while we listen to the hyenas laugh.\n\nOne night, while out dancing in a tight red dress, I am suddenly grabbed by three men who blindfold me and throw me into the back of a van. Days later, awaking from the drugs they've given me, I find myself standing on the auction block of a white slave market. I am covered from head to toe with robes; only my eyes are visible. The auctioneer says, \"The bidding will start at one million rupees for this exceptionally fine specimen!\" Suddenly he rips away my robes, exposing my entire naked body; the audience gasps in excited approval. Several customers, men dressed in long robes and turbans, come up and bounce my breasts between their hands, pinch my nipples, and grab my bottom. One man in a blue sash even rubs my clit and thrusts an incredibly long finger deep inside me, pumping it in and out to observe my reaction. Though I should feel humiliated and angry, I find myself feeling haughty and sexy instead, because I can see that every one of these men has a huge erection jutting out under his robe. I start gyrating my hips, squeezing my own tits, and massaging my pussy. I look right into the eyes of the man with the blue sash and lick my lips suggestively. I take my fingers out of my wet sex and stick them in his mouth. He starts sucking them hungrily and whips out his iron rod to get at it with his hands. All the other men do the same, so that all I see are these gorgeous shafts aching and throbbing for me. I am turned on tremendously, and I keep undulating and fucking myself with my fingers until all of those men spurt their come all over my body\u2014all except the man in the blue sash: he sticks his stone-hard member straight up my pussy and fucks me hard, fast, and deep, right there in front of everybody, and I explode in uncontrollable spasms. I select him as my lucky owner and lead him off to demand more fucking.\n\nThough some women quite easily and prolifically conjure up elaborate visions like the ones above to accompany their self-pleasuring interludes, many of us are just as happy dreaming up much simpler scenarios\u2014like \"What would my next-door neighbor look like without his clothes? I'll bet his stomach muscles ripple, all the way down to his...\" or \"I'd love to rub my oiled-up breasts all over my husband's body.\" That's okay. Fantasies can run the gamut from simple musings about real men and situations to complicated visions of exotic lands, strangely sexed people, and bizarre or impossible copulations. Almost anything you see, read, hear about, remember, or experience can become a rich source of voluptuous inner visions for you. The world, and especially the sensualized domain that your Inner Sex Goddess perceives, is full of juicy ideas. So never feel stumped as to where and how to come up with really tasty fantasy material. Simply tune your otherwise rational brain in to the outlandishly opulent wavelength of your personal Cloud Damsels and view the world through their erotic-tinted glasses. Allow yourself to recognize a sensual image when you see one, relish it, take a mental picture of it, write about it in your journal, free-associate it with any people, places, or things that float across your mind. Let your imagination range freely over completely new territory. Then, when next you want to embody your Inner Sex Goddess\u2014whether you are self-pleasuring, making love to your man, or standing in line at the bank\u2014flip through your mental catalog of self-produced fantasy flicks and plug in the hottest one.\n\n_118_. Recall a man, situation, or previous lovemaking session that made your love button shiver. In your mind, savor the sounds, smells, and textures that went with it. Embellish them. In your imagination let them evolve to an extreme pleasure point. Here's an example: \"I had a lover once who had the most beautiful smooth brown skin. Just the smell of it drove me insane with lust. Whenever I want to get hot, I remember that musky scent and the feel of his skin against my chest. Sometimes I get to swimming in that smell and imagine that it's pouring in through my nose, mouth, and vagina to fill every corner of my body. It turns my insides to warm honey.\" And another: \"My boyfriend and I used to jump on his motorcycle and go for wild moonlight rides. Remembering the feel of that vibrating hunk of metal and my boyfriend's ass between my legs makes me tremble even now. It works especially well in tandem with my vibrator.\"\n\n_119_. Read steamy novels. Become one of the characters and immerse yourself in the hottest scenes\u2014clothes, food, toys, locales, the whole megillah. Some of those anonymous Victorian porn books can be delightfully kinky\u2014with wide-eyed, pantalooned maidens turning into sex-crazed, riding crop\u2013wielding mistresses who love to watch their brothers impale every servant in the house. Or become a different sex. \"Henry Miller's books really get my juices flowing,\" a friend told me. \"I love to put on my husband's clothes\u2014when he's not around, of course\u2014and pretend I'm one of Miller's cocky male heroes, throwing around words like 'prick' and 'pussy' and whipping out my huge member to fuck every woman who crosses my path. It's fun to wear the shoe on the other foot for a change.\"\n\n_120_. Peruse adult magazines. The photos, especially in _Penthouse_ , are often meant to be realistic fantasies. A ten-page layout of two women playing with ice dildos supplied me with raw fantasy material for months. Don't forget to read the stories and the letters from other readers too. \"A man wrote in about secretly watching his wife shed all her inhibitions and have pornographic sex with two hunky gardeners, and for some reason that lit all my fuses,\" my friend Mary told me. \"Improvising my own version, I have great fun imagining my husband spying on me as I have my way with the virile young guys who clean our pool. My favorite part is when I sit on the lap of one, his love-stick buried deep inside me and my back to him, while the other guy slurps on my clit. This is just too much for my husband, and he has to come out of hiding and join in by sucking my tits. Of course I would never do this in real life, but that's what makes it even more fun to fantasize about!\"\n\n_121_. Walk in nature. When you see a waterfall, fancy that you are standing under it with a long-haired Polynesian lover, feeling the water and his hands on your breasts. Let the intense mossy green of a forest return home with you as a mental talisman that evokes vivid color, moist earthy feelings, and romantic medieval imagery. See mountains as breasts to suckle, trees and skyscrapers as mighty shafts to penetrate you, rock clefts as mouths to kiss and be kissed by. Let a mud puddle conjure up visions of rolling in the mud with a crocodile, a wolf, or a primitive tribal lover. Use these natural spices to flavor your solo or partner lovemaking. One nature lover I know told me this fantasy. \"All of a sudden this log became a platform on which I could display myself to hundreds of men. In my mind, I lay back on it, nude, the log between my legs so that I was spread-eagled, my sex open wide. I could imagine the bark prickling my bare back and ass as I massaged my juicy opening, and all these men with huge naked erections stared at me hungrily. But no one was allowed to touch me because I was the sacred white goddess of the forest, and they had to worship me from afar.\"\n\n_122_. Become another woman\u2014someone you've read about, seen in a movie, or heard of in a song. \"I have different fantasy personalities depending on the activity,\" a woman told me. \"For pleasuring myself, I mentally play the role of Salome, whom I see as someone unashamed of touching her body and with lots of marvelous veils and things to play with. With my husband, I imagine that I am Xaviera Hollander, the Happy Hooker, flaunting my bold sexuality and giving him everything a man could possibly dream of. And when I just want to relieve the boredom of ironing, I invent a scene where I'm 'Brown Sugar' from that Rolling Stones song. She's a saucy little tart who drives all kinds of rock stars wild.\"\n\n_123_. When you see an attractive man, imagine what he looks and feels like naked. What outrageous things would he do to you? What naughty acts would you perform for him? How would his penis be shaped? What color is his pubic hair? Does he have a tattoo on his right inner thigh? Would he like to shave your mound? How does the inside of his mouth taste? Would he prefer you in white cotton, red lace, or black leather? Does he cry out when he comes? Are his nipples sensitive? his earlobes? his eyelids? Would he have a penchant for licking your cleavage? Would you like to lick his? The next time you play with your vibrator, imagine it's his hands that are giving you those sudden shocks of pleasure.\n\n_124_. Read sex manuals. Pick out some of the kinkier suggestions that you might never have the nerve to do in real life\u2014like having your man whip you lovingly, wrapping his penis with pearls, or inviting a third person into your love-play\u2014and fantasize about them. Devise your own variations. Pretend that you are being photographed or drawn for the book's illustrations. As you fill your mind with these vivid images, let them stir you into self-pleasuring action. \"I've always read about watching yourself have sex in an overhead mirror, but I'd be too embarrassed to do this with my husband. So sometimes when we make love, or especially when I'm by myself, I imagine that I can see the reflection of his tidy little rear bouncing up and down on top of me. It gives me the most amazing tingles,\" a friend told me.\n\n_125_. Be alert for accidental triggers to your erotic imagination. Wrong numbers, incorrectly addressed mail, switched luggage or tickets, accidental grocery cart collisions, misinterpreted words or names, and unusually striking newspaper articles can all carry the seeds of a great fantasy garden. I used to get occasional wrong-number phone calls from people asking for Natasha. Eventually, one deeply disappointed caller told me that Natasha was a dominatrix who had always had this phone number, and maybe I'd like to get paid for \"punishing\" him instead of just insisting that this \"bad boy\" had reached the wrong number! I hung up on him and got my phone number changed, but in the meantime I had great fun fantasizing about rudely making appointments with these men\u2014how they would love the verbal abuse!\u2014putting on hip boots and chains, and satisfying their every desire for erotic leather punishment.\n\n_126_. Let music shape your feelings into images. Brazilian music or reggae can take you to hot tropical climes where small bikinis, dark sultry men, and devil-may-care attitudes have free rein. The heavy drumbeat of hard rock may stir your primitive instincts. Baroque sonatas may turn you into a bird flying free of all society's limitations or a guitar being plucked by a man's strong fingers. \"I don't fantasize about people or events but more about feelings,\" Carol told me. \"I can put on a piece of music, concentrate on a particular instrument or melody line, and use that to focus my sensations and feelings into one electric-blue chord that flows throughout my body as I want it to. This chord feels hot, cold, and throbbing all at the same time, and it kind of beams into my hands, breasts, and genitals as the music trips along. It intensifies everything that I'm doing to myself, and if I focus it in my clitoris, it always gives me an orgasm.\"\n\n_127_. Advertisements are a great source of fantasy these days. How about those Calvin Klein ads showing the naked torso of a perfectly sculpted man on the sand? Imagine rolling in the waves with him or feeling the bulge in his jeans. The ads for Guess jeans often depict a hot western scenario or a coffee-shop t\u00eate-\u00e0-t\u00eate with gorgeous men and women lolling about. You could walk right into one of those scenes and get branded by your favorite cowboy. TV commercials for perfume are usually designed specifically to evoke romantic fantasies: Liz Taylor seductively bringing luck to some classy dude in a casino; a pouty-lipped woman challenging you that \"not every woman can wear Red,\" apparently because it's so hot and bold; some Fabio look-alike galloping up on a horse to scoop you up and away to cloudland. And then there's the next-door neighbor who comes to borrow coffee. My friend Teresa fantasizes, \"That sexy hunk rings my chimes all right. When he comes to ask for coffee, I invite him in, and while I'm reaching up to get the tin off the shelf, he comes up behind me, pins me against the cupboard, raises my skirt, and screws the living daylights out of me. Then I undo his shirt, put honey on his nipples, and lick it off, and somehow we forget all about the coffee.\"\n\n_128_. Delve into your dreams for fantasy scenarios. As messages from the subconscious, dreams can give us clues to the \"forbidden\" desires and fears we otherwise keep hidden from ourselves, and in fantasy it's safe to explore them. I once had a powerfully erotic dream about the very ordinary man who was my boss at the time. Although in real life I liked and respected him, there had been no particular sexual chemistry between us\u2014and it would have been a career disaster to get romantically involved with him. But in my nocturnal imagining, I let myself see the suave and sexy Don Juan personality that he probably never revealed even to himself and that lit a huge bonfire in my libido. Intentionally refraining from any real-life dalliance with him, I undressed him with my eyes and made dangerously abandoned imaginary love to him in the hallway for months. It made going to work such a treat!\n\nYou can also program your dreams to work things out for you. A friend told me she'd been having mental flashes of two of her female colleagues licking and biting her all over. She did not want to explore the idea of making love to a woman in real life, but these images were so persistent that she decided to tackle the idea in her dreams. After about five nights of falling to sleep deliberately thinking about these women, she finally had a dream in which the three of them made an intensely erotic porn movie together, stroking, sucking, tweaking, and kissing each other in every possible entangled combination. She enjoyed the dream so much, and it felt so natural to her, that she treasured it as a favorite fantasy for years. Even though she still had no desire for same-sex love in her daily life, she allowed herself to relish the beauty of woman in her dreams and fantasies.\n\n_129_. Assign yourself a role\u2014any role\u2014and see where it takes you.\n\n\u2022 You're a librarian who just can't help sneaking hunky young college students in between the shelves for some real education.\n\n\u2022 You're a sports reporter, and when you find yourself in the football players' locker room, you tear off all your clothes and massage yourself to orgasm while the entire team cheers.\n\n\u2022 You're a teenager on your first date, making out for hours, then finally letting him feel your breast, his hot, desperate panting loud in your ear.\n\n\u2022 You're stranded on a desert isle with only the wild animals for loving company.\n\n\u2022 You're a harem girl, pampered, perfumed, bathed, and licked by hundreds of beautiful lusty women every day and chosen to suck the sultan's golden dagger every night.\n\n\u2022 You're a porn film star and have to watch hugely endowed men and beautifully molded women having sex constantly; then you let three men and two women suck, fuck, and bite you for ninety-seven camera takes.\n\n\u2022 You're a nineteenth-century French maid whose master often finds it necessary to spank you for dereliction of duty.\n\n\u2022 You're a Tantrica in an ancient Oriental garden pungent with the smell of orange blossoms and jasmine, and an Indian god with twelve arms smooths every inch of your body with his twelve hands while making transcendental love to you.\n\n\u2022 You're an executive at a meeting, and one of your colleagues has slipped under the conference table to lick you between the legs.\n\nYou're a hostage who is tied down, forced to perform cunnilingus on six women, spanked with the thick penises of ten huge men, and then taken from behind by each one in turn.\n\n\u2022 You're Lady Godiva, riding bareback and bareassed through town with only your long tresses to cover your voluptuous charms.\n\n\u2022 You have a blind date, and you're getting dressed to go meet a new man for the first time, dreaming up all his sensational qualities and odd quirks, anticipating the flirtatious eye signals and dinner conversation, imagining all the salacious things he might try to (and you might let him) do to you.\n\n\u2022 You're a despotic ruler whose male subjects must pay tribute to you by sucking your nipples and then rubbing their come all over your body.\n\n\u2022 You're a glamorous gambler at a casino in Monaco, and a visiting Italian prince follows you around the casino all night and finally whisks you off to his yacht for a moonlight tryst.\n\n\u2022 You're a waitress at an elegant restaurant who occasionally slips into the men's room and has sex with whoever is in there, no matter how many or how few.\n\n\u2022 You're a bee who has ecstatic orgasms by flying deep into the well of fragrant, brilliantly colored flowers and rolling gleefully around in their soft golden pollen.\n\n## cloud damsel visions\n\nWhile most fantasies are purely for lustful fun, a certain kind of imagining is meant for more serious business\u2014the business of designing reality the way you want it to be. Whether that means making changes in the way your partner feels and behaves, making changes in the way you feel and behave, or calling a new lover to you, these visions are fantasies with a powerful purpose. Indulge in them often enough, and you will find your real life starting to take the shape of your utopian dreams. It's as if you and your Cloud Damsels sit around a white witches' magic cauldron: you throw in mental pictures of what you want, together you brew it up with imagination, and the resulting steam carries your desires to their intended destination in reality, infusing the person or thing with the smoky essence of your erotic vision. More potent than ordinary visualizations, these magnetic mirages carry all the power of your fervid physical desire, hotly charged emotion, and most unbridled imagination. Your inner voices will prompt you when it's appropriate to use them.\n\n_130_. Imagine summoning a lover to you. When my friend Vivian wants to bring a new man into her life or make a long-distance connection with a current one, she merely thinks intensely of that person while she pleasures her body. She feels herself calling to him energetically and sees the man consciously receiving the connection. Vivian says this seems to work even better for her when done in the shower because her pleasure is more intense and because she envisions the steam from her very hot, wet body floating off with the powerful erotic magic of her dream. When I tried this once, I not only had a deliciously entertaining shower, but the lover I'd been thinking of phoned me from three thousand miles away immediately after I left the bathroom!\n\n_131_. Design beautiful new feelings about your own body that will translate into bewitching behavior. Because I believe it's so important for women to learn to love and admire their genitals, I often recommend they try autoerotic play enhanced with the following fantasy: Envisage your mate or imaginary lover ardently licking the lips of your sex and moaning, \"I love your gorgeous pussy. It tastes sooooo good and makes me so hard.\" As your passion mounts, he kisses you even more intensely and groans, \"Give me more of your delicious nectar. I love to lap it up.\" Embroider on this basic scenario as you wish and allow your hands or vibrator to act it out. In addition to being delectable self-pleasuring fun, this vision helps you (1) to know in your bones what a magical, sexy jewel your vulva is; (2) to radiate physical heat from your sex organs to those of your lover; and (3) to invent elegantly lewd pelvic moves.\n\n_132_. Fantasize your mate into the Valentino-like lover you desire. Suppose your man, though he means well, just doesn't spend enough time on foreplay. You've tried retraining him with erotic massage, seductive self-pleasuring shows, and tender talk, but you've had only moderate success. Time for voluptuous visions. In a private pleasuring session, imagine that your once-reluctant lover is now slowly teasing you in all the sweetly agonizing ways you crave. In vivid detail, see him lovingly massage your toes; feel his slow, sensuous caress on your inner thigh; smell the scent of his fevered skin as he bites your nipple; hear the passionate endearments he whispers in your ear; taste his tangy tongue on the corners of your mouth. Simultaneously, inflame all these hot spots with your own loving hands, building your excitement to a feverish heat. The combination of the powerful creative juices of sex, your steaming emotions, and your fierce intention will drive this blazing message straight into your lover's heart.\n\n# 8  \nthe secrets of divine orgasm\n\n_And then it happened like a miracle, this pulsation of pleasure unequalled by the most exalted musicians, the summits of perfection in art or science or wars, unequalled by the most regal beauties of nature, this pleasure which transformed the body into a high tower of fireworks gradually exploding into fountains of delight through the senses_.\n\n_\u2014 Ana\u00efs Nin_\n\nWe women know that orgasm is not a _goal_ to be achieved at all costs, like a touchdown or a home run. In fact, sometimes we don't even care if we have an orgasm because we enjoy the sweet ecstasy of the _process_ of lovemaking so much. Yet that pleasure \"unequalled by the most regal beauties of nature\" is a divine gift to which we are most definitely entitled, and which your Inner Sex Goddess most dearly wishes to bestow upon you. In fact, the blissful waves of orgasm are vital food for the goddesses within your body, strengthening all your subtle energy centers and nourishing your physical and mental capabilities as well. The ecstatic pulsations of your orgasm also constitute one of the most deeply satisfying gifts you can give your lover. Few things turn a man on more than knowing he has solved the eternal mystery, found the elusive Grail, and sent your body into uncontrolled spasms of rapture\u2014which then cascade all over _his_ body.\n\nWe Sex Goddesses have this divine frenzy well within our power\u2014to receive and to give. The research of Masters and Johnson indicates that women's capacity for orgasm far exceeds that of men. And the more we exercise this ability, the more sensitive and responsive to orgasmic stimuli we become, achieving even greater heights of climactic ecstasy, capacity, and skill. In fact, Mary Jane Sherfey in _The Nature and Evolution of Female Sexuality_ speculates that without the restraints of societal and internal taboos, women might be freed to behave more naturally, in a similar fashion to our close relatives the higher primates who copulate, and probably reach orgasm, up to fifty times a day when in heat. How happy and healthy they and their mates must be!\n\nNot too far removed from this state of natural, sexually vibrant well-being are the islanders of Mangaia in central Polynesia. Believed to be the most orgasmically advanced society in the world, they consider female orgasm not an indulgence but a necessity. At puberty, Mangaian males are taught how to stimulate women to maximum sexual pleasure, and the man who fails to give his woman at least two orgasms every time they make love loses his status in the island's society. Sadly, our culture is not so advanced.\n\nBut actually it's much better to retain control of your own body and its pleasures anyway, to realize that nobody _gives_ you orgasms. You create them yourself by knowing how your body reaches its peak and then directing your feelings and activities in a way that _allows_ an orgasm to occur. This command of simultaneous control and abandon is the special talent and gift of your Inner Sex Goddess and of the many sex goddesses, ancient and modern, who have discovered the secrets of first-time, dependably regular, teeth-chattering, and multiple orgasm and have honed their skills into a fine art. Let's listen in on one of their powwows.\n\nOnce upon a time a group of ancient priestesses and modern-day sex goddesses sat around the campfire trying to figure out how to induce the Goddess Who Makes You Shudder Inside to visit them. They knew she could be elusive and that she required praise and petting that met certain very high standards. But these women had clues and secrets that they whispered to each other over the fire. \"Try rubbing very fast.\" \"Let yourself go.\" \"Maybe worshiping a cucumber will make her come,\" they said. And lo, the mysterious seeds of their shared female wisdom eventually bore fruit, the Goddess Who Makes You Shudder Inside appeared full bloom in their midst, and they and their lucky mates lived ecstatically ever after. Each in her turn had revealed a secret....\n\n### Secret # 1\n\n_Know and Love Your Sexual Self\u2014Regularly_.\n\nThis, of course, is what you've been doing throughout this book\u2014exploring your body and finding not its faults but all of its sexy hollows and curves; discovering your personal hot spots, desires, and sexual style; becoming intimate with your Inner Sex Goddess; loving your sensuality, your wild erotic thoughts, and your genitals; learning how to pleasure yourself, and thereby your lover. As one woman said, \"I never had an orgasm until I gathered up the courage to look at my vagina, decided it was beautiful and lovable, and finally got comfortable touching myself and feeling a little bit out of control. That's when all heaven broke loose.\"\n\nA woman who loves her sexuality and regularly pleasures herself to orgasm has a high degree of sexual self-confidence. She wants to express herself sexually more than the usual woman and possesses a higher-than-average sensual thermostat. She likes her body and thinks of herself as lovable and valuable. She expects to delight herself and her lover more than the ordinary woman would; and she most often does. Her self-pleasure is its own reward. Yet it's because she's so hot that she gets the best lovers and the best loving. \"When I realized that _I_ am the one, not a man, who brings sexuality into my life, I felt as if I had been reborn. My body, now truly my own, seemed very precious and powerful. It showed me how to have _real_ orgasms.\"\n\n### Secret # 2\n\n_Surrender_.\n\nWilhelm Reich, the psychoanalyst who wrote _The Function of the Orgasm_ , described climax as \"the ability to surrender to the flow of sexual energy without any inhibition.\" Abandoning yourself to your hot-blooded feelings is essential if you want to be carried over the orgasmic edge, for the very essence of orgasm is letting go, relinquishing control, being lost in the moment, submitting to your body's imperative. You must ask your critical, control-freak self to step aside and let your voluptuous feelings show the way. One very wise woman said, \"You can't _force_ yourself to orgasm; you have to relax, get in the flow, and let your body do its own thing.\" And yet another has said, \"I get myself as excited as possible and then just let go. I kind of surrender to my sensations.\"\n\n### Secret # 3\n\n_Focus_.\n\nEvery orgasmic woman I know says she intoxicates herself to orgasm by throwing everything else out of her mind and concentrating only on her body and its heady feelings of pleasure. At that moment nothing else exists. \"I empty my mind and merge with my feelings,\" one woman told me. \"When I touch my breasts, I concentrate fiercely on the tips of my nipples. When I stroke my vagina, I am so focused there that I _become_ my vagina, pulsing and contracting.\" This ability to loose yourself from the bonds of normal reality and focus totally on the altered reality of voluptuous sensation is what allows many women to achieve extraordinary or multiple orgasmic peaks. \"In my private world, only the tip of my clitoris exists. But at that moment it feels bigger than my whole body, as if I'm one giant clitoris rippling on waves of vibration.\"\n\n### Secret # 4\n\n_Get the Right Kind of Clitoral Stimulation\u2014and Plenty of It_.\n\nWhether you like it direct or indirect, fast or slow, hard or easy, with water or your hand or a vibrator, find out what drives your clitoris crazy\u2014and stick with it until your body can't stand the agonizing pleasure anymore and has to climax to relieve the tension. One friend says, \"My best orgasms happen when I spread my vaginal lips apart and let the faucet in the tub run full blast right on my clit. The faucet is fabulous and can last a lot longer than I can.\" Another self-educated woman chimes in, \"I like to move a vibrator up and down over my clitoris for about ten minutes nonstop. Then I just hold it steady and push it in until I explode.\" Still another says, \"I can accelerate my orgasm by holding the sides of my clitoral hood between my finger and thumb and gently massaging it while I slide a wet finger over the tip of the clitoris itself. Fireworks!\"\n\n### Secret # 5\n\n_Try Your Goddess Spot_.\n\nWherever it is, your G-spot is _your_ area of special sensitivity, so capitalize on it. Three different sex goddesses say: \"I arch my back so my fingers can reach a real deep sexy place inside me. Then just a couple of strokes and I go off like a cannon.\" \"I can come several times if I put a dildo just inside my vagina and rub it up and down quickly over that rough spot I love.\" \"My G-spot is outside over my urethra. If I massage it in circles, orgasmic waves start right away.\"\n\n### Secret # 6\n\n_Flex Your PC Muscle_.\n\nContracting the PC muscle greatly increases sensation and adds enough additional stimulation to bring many women to the threshold of orgasm. One wild maiden confides, \"Tightening makes the feelings stronger. When I get close to climaxing, I squeeze hard and hold my vaginal walls together till the contractions stop.\" And another says, \"After I have an orgasm, I use my PC to mimic the contractions until I come again.\"\n\n### Secret # 7\n\n_Watch Yourself in the Mirror_.\n\nMany women are tantalized into orgasm by seeing what their own buildup of sexual tension looks like and watching how their body moves with mild or intense orgasms. One says, \"When I see my vaginal lips swelling, opening up like a hothouse flower and undulating real sexy-like, I start quivering inside and sort of roll into an internal shudder.\" Another autoerotic voyeur declares, \"My nipples get very dark and big, as if someone's been kissing and biting them for hours. And then my whole body seems to get pink and light up. Watching this sends me right over the edge.\"\n\n### Secret # 8\n\n_Use a Vibrator_.\n\nThis secret is especially helpful if you've never had an orgasm. But even if you have, new sensual peaks await you, as the following story illustrates. \"A friend told me about the fun she was having with her vibrator, and I wanted to explore too. It was great because I could direct the intense pulsing movements exactly where I wanted them and keep them going for a really long time. I felt that I was building up to some sort of explosion or precipice, and I remember thinking, Could this be dangerous? Soon I felt as if I'd reached the top of some mountain and then fallen off into a blissful state of delicious inner pulsing and uncontrollable contractions that I had never dreamed my body was capable of.\" (See Chapter 9 for details on using vibrators.)\n\n### Secret # 9\n\n_Fantasize_.\n\nThe most erotic organ in the body is your brain. It is the epicenter of your sexual earthquakes. And as you know from the previous chapter, your personal Cloud Damsels are intimately familiar with the wilds of your interior erotic terrain. In the blink of an eye, they can create at least ten hot fantasies that will expertly touch off any number of private landslides. So do as more than 80 percent of orgasmic women do: fire up your imagination to bring on a shuddering orgasm. Here's what just one of them said, \"I like to get on all fours with a vibrator beneath me on a pillow and imagine that I am being taken doggie-style. This wonderful little fantasy brings on multiple orgasms and makes me really hot for more loving from my real-life man.\" Another creative vamp shares this fantasy: \"I lie flat on my back, insert three wet fingers, and fantasize that an Arab prince has whisked me off to his tent and laid me down on a long table covered with luscious exotic foods. The women of his harem start licking me all over like a piece of fruit, and the prince himself slips his tongue deep inside me. That always makes me come right away!\"\n\n### Secret # 10\n\n_Play a Role_.\n\nSometimes playing the part of a sex kitten, a dominatrix, or an innocent young girl can free the wildly orgasmic you from the bonds that restrain her. Dress the part and really act it out. Three playful Sarah Bernhardts confide: \"When I wear my red lace bra and see-through panties, I start slinging my hips around and getting naughty. I feel like a high-class call girl capable of firing off a hundred orgasms.\" \"I like to dress up like Lolita, with red pouty lips and little schoolgirl dresses. Then I do that 'shy Di' look. Sexy and sweet at the same time, I feel it's okay to do anything, including innocently losing control of my body and, 'Oh my! I've just had an orgasm!' \" \"I _pretend_ that I'm having an incredibly intense orgasm. I shout and fling myself around in a frenzy. My body doesn't seem to know the difference and actually comes to a climax.\"\n\n### Secret # 11\n\n_Vividly Picture a Body Part_.\n\nI used to think I was the only oddball who did this, but it turns out that many other women find visualizing body parts extra stimulating too. I usually imagine a close-up view of my breasts, nipples extended and quivering. Somehow this heightens their sensitivity and sends intense orgasm signals to my vagina. Other sensual visualizers share their secrets: \"I get an image of what my clitoris must look like to a man, from a straight-on view. It seems huge, red, and wet. Then if I just touch it with my fingertip, I have an orgasm immediately.\" \"I see my vaginal walls swelling and filling with blood, and the back of my vagina opening up to accommodate this huge dildo I have. I see the dildo in there, too, thrusting against the walls and forcing them to open even more. The power of this image really gets me, for some reason, and I then imagine I'm actually watching the spasms of these swollen walls as they release all the energy they've been holding.\"\n\n### Secret # 12\n\n_Keep Your Rhythm and Pressure Steady_.\n\nThis is a biggie. Almost every woman I know says the trouble with trying to have an orgasm with a man is that he's likely to change the exquisite thing he's doing to you just when you've reached a level of unbearable sexual tension. At that point, it's crucial to _not change a thing_. \"Yes!\" a friend of mine agreed. \"It must be exactly the same rhythm, motion, pressure, speed, everything for the last twenty or thirty seconds. If not, the whole thing is ruined!\" Another smart lady capitalizes on this essential element: \"After I come, I just keep up the same steady, hard stimulation even though at first it feels like I can't stand to be touched. Pretty soon that intensity goes away and I just keep on riding the waves of pleasure into multiple-orgasmic bliss.\"\n\n### Secret # 13\n\n_Tease Yourself_.\n\nYou can intensify and prolong your orgasm by building tension to a peak but stopping before you go over the edge. Lighten your touch or move to a different area while the level of arousal drops, then rises again. The more times you do this, the greater will be your release in the end. \"I love to tease myself into a powerful orgasm by moving my vibrator down from my clit to my vaginal opening when I'm almost ready to come. I hold it there for a while and hang suspended until all pulse stops racing and then move it back to my clit,\" my friend Joan said. \"I do this about ten times before I finally let myself come; then I just explode.\" Another shameless self-tease confesses, \"If I take my time, build up slowly, then let myself come down again, maybe just rubbing my breasts between peaks, I get this almost unbearable feeling of wanting to jump out of my skin after about five times.\"\n\n### Secret # 14\n\n_Experiment with Breathing Patterns_.\n\nEvery woman seems to have a different way of using her breath to enhance her orgasmic capabilities. Explore your breathing repertoire until you find just the right lung action for you. \"I make sure that I have an orgasm every day, because it helps relieve the stress of my high-power job,\" a colleague told me. \"I find that breathing deeply is very important for me, no matter what I'm doing, but especially when I'm pleasuring myself. I just breathe very deeply and imagine that the air is being breathed in and out of my vagina. It seems to make my insides tingle just enough to bring me off beautifully.\"\n\nOther heavy breathers say: \"I've found that if I hold my breath, it helps me over the plateau.\" \"Panting in short, rhythmic gasps really increases the sensation in my whole vulval area and makes me come faster. In fact, if I keep up these pants after my first orgasm and rub myself more gently, the hypersensitivity in my clit disappears and I can go into another sexual buildup and another orgasm\u2014even five or six more.\"\n\n### Secret # 15\n\n_Make Sounds_.\n\nWithholding noises can stifle your orgasmic feelings, while letting them out frees you to express what you feel inside. Here's what several vocal women have to say: \"If I don't scream like a banshee, I can't come.\" \"When I moan, I get more into my feelings. I don't know if I could have an orgasm without letting these sexual urges out.\" \"I actually talk to myself about how sexy I look and how good it feels. I say things I would tell a man, like, 'Ooooo, it feels so good when you suck my tits.' It makes me really hot, and then I just can't keep from coming.\"\n\n### Secret # 16\n\n_Change or Combine Types of Stimulation_.\n\nOnce you have an orgasm, try changing your type or place of stimulation to encourage another one. Or combine several styles at once. One voracious lady says, \"My first orgasm comes from fingering my clitoris. But my second one usually arrives only if I plunge a dildo real deep into my vagina and pump it hard.\"\n\nMany women say that if they simply change positions, with their fingers or vibrator approaching from a different angle, they can induce more orgasms. Another woman gluttonous for pleasure says, \"After I come once, I just add on another erogenous zone. I keep my fingers in my vagina, but also rub my bottom or twirl my nipples to make more climaxes.\"\n\n### Secret # 17\n\n_Push_.\n\nBearing down or pushing seems to trigger orgasm for many women. Sara says, \"When I want to have an orgasm, I just push down, as I did when I had my baby. The extra pressure sends me flying.\" Another multiply orgasmic lady says, \"After my first climax, my clitoris is less responsive to touch, so I create more sensation by pushing hard against my fingers to bring on another one.\" And another affirms, \"Bearing down on the dildo makes all my inner skin feel exposed and extra-sensitive. I come right away like that, several times.\"\n\n### Secret # 18\n\n_Learn to Distinguish Between Clitoral and Vaginal Orgasms_.\n\nVaginal orgasms are quite different from clitoral ones, and you may not realize you're having one because your contractions are not the intense muscular spasms of clitoral climax but the deeper, more oceanlike waves of vaginal orgasm, and this may set you on an orgasmic plateau that leads to other things.\n\n\"When I have a vaginal orgasm, my whole pelvis, and sometimes my entire body, bounces and shakes,\" Rachel explained. \"It happens when I use my dildo to rub the very back of my uterus, deep inside. After one of these, I'm satisfied, but then I crave lots more sex, mainly with a real live cock. So my husband loves it when I please myself this way.\" One vibrating vixen says, \"I love to make my clit come first with Jerry the Vibrator. Then I set him aside and use my fingers to caress just inside my vulva until it starts undulating inside. Then if I touch my G-spot, this delicious liquid starts flowing that feels like it comes from the middle of my being.\"\n\n### Secret # 19\n\n_Ride Your State of Mind Beyond Orgasm_.\n\nOne cosmic seductress shared these provocative thoughts on going beyond orgasm: \"To me, orgasm is like the electric charge of putting your finger in a socket\u2014an exciting feeling but a little crude. I find that if I get my head out of the way, feel completely safe, and let go of all my inhibitions and fears, I can go beyond that violent charge into a delicious hum, a state of bliss that can go on for quite a long time. When I'm completely in touch with my Inner Sex Goddess, she knows what to do with my body and leads the way. But you have to be willing to unite completely\u2014not with a partner but with yourself.\"\n\nThe Taoist masters, too, have always regarded the feeling of oneness achieved during and following orgasmic ecstasy as the most easily accessible mystical experience. So if you approach pleasuring and orgasm not just as a physical release but as a spiritual adventure, you too may find that a mystical place even beyond orgasm lies deliciously within your self-loving grasp.\n\n# 9  \ncleopatra's pleasure toys\u2014electric, plastic, furry, and human\n\n_In retrospect, I can see there were times I came close to romantic love with my vibrator. I solved the problem of possessiveness by buying several and lending them to friends_.\n\n_\u2014 Betty Dodson, Sex for One_\n\nalthough it is said that, on many nights, Cleopatra would summon hundreds of her soldiers to take their individual turn satisfying and being satisfied by her seemingly endless capacity for sexual pleasuring, once she met Mark Antony she became a one-man woman. And while he was off fighting wars and acquiring new territory for their kingdom, she would slake her robust sexual appetite with a wide assortment of inventive and luxurious toys, from dildos fashioned of smooth wood or stone to gold-chain pleasure halters, papyrus reed ticklers, and carefully selected clusters of grapes. Aside from quenching a queen's lusty thirst, Cleopatra's pleasure toys also kept the jewel of her libido sparkling, piqued and developed her prodigious imagination, and enhanced her magnetically irresistible sex appeal. Her eyes literally smoldered with passionate invitation when Antony returned from campaigning, and she wantonly initiated him into whatever new and secret seductions she had invented during their long separation. She was a woman profoundly in touch with her Inner Sex Goddess.\n\nAs a queen, Cleopatra knew that gold and land were her most necessary allies. But as a woman, she believed that a finely carved dildo was truly a girl's best friend. Much more charming and fun companions on a solitary evening, they're much more affordable and easier to procure too. Today's vibrating dildos and modern pulsing massagers provide a reliable source of high orgasmic pleasure that can't be beat\u2014in fact, using a vibrator is how I discovered what an orgasm was in the first place. My friend Sally says she used to need an hour and a half of hand stimulation before she could have an orgasm, but with her trusty electric toy she can reach a shuddering climax in ten minutes or less if she wants to. Joan, a very focused and intense woman in her frisky sixties, reports that she needs up to two hours of rapid pulsation directly on her clitoris to reach her peak. For her, a vibrator provides the consistent, long-lasting stimulation her body wants and needs but that neither she nor her husband has the stamina to maintain by hand. Well-known sex therapist Pauline Abrams says, \"It takes longer for me to come with a vibrator, because I can't control the feedback system as well as with my own hand,\" but most women find that the concentrated, unflagging stimulation provided by an electric pleasure toy makes for easier, faster, and often more intense orgasms than they can count on with any other method.\n\nA vibrator works by quivering and shaking the millions of nerve endings in the skin of your genitals. It can trigger at least a million more sensors than the most skillful hand or penis, meaning that an intensely high level of arousal, if not ecstatic orgasm, is inevitable. But don't worry that you'll become addicted. In fact, quite the opposite is usually the case. Most women find that when they learn the full extent of their erotic capacity by self-pleasuring with vibrators and other sex toys, they are at last able to transfer these satiating talents and sensations to other styles of lovemaking, whether on their own or with their partners.\n\nJust like sex goddesses, vibrators come in many styles, shapes, and sizes. Like me, you may choose to acquire an entire inventory of vibrating toys to suit your different sensual moods. Or, like Joan, you may prefer the simplicity of one perfect pleasure wand, always impeccably attuned to your particular cravings. Whichever your predilection, you can buy most types of vibrators in an ordinary drugstore because they are basically massagers for the whole body. For other types, such as dildo vibrators, you may need to visit your local sex store or browse through the many tasteful sex toy catalogs currently available. Once acquired, you can prolong your vibrator's life and the quality of service it gives you by storing it in a cool, dry place\u2014I keep mine in a cloth bag, snuggled in a drawer\u2014and wiping it clean after every use with a cloth moistened with warm water or alcohol. Never immerse your vibrator-massager in water.\n\n## swedish-style scalp massagers\n\nHeld on by straps around the palm, these massagers are small vibrating boxes that sit on the back of your hand. They make your whole hand vibrate gently. Although the pulsations they produce are not as intense as wand or coiloperated vibrators, the advantage of Swedish-style massagers is the skin-to-skin contact they permit and the versatility of being able to transport vibrations to any body area that your eager hand and fingers can reach. So for a lovely silky sort of vibratory experience, strap on your Swedish massager and make your hand flow sensually over your body like that of one of Cleopatra's handmaidens in the milk bath.\n\n_133_. Glide over your entire body with your electrified hand, delving into every crevice and cranny with this quivering sensation. When you vibrate your skin, you stimulate the flow of blood to that area, a marvelous health and beauty treatment for the entire body. And you will discover that areas previously not so erotically sensitive will awaken to new sensual thrills at your vibrating touch\u2014your scalp, lower back, fingers, and toes, for example. So skate lightly over the skin, hold firmly in one spot, or massage as you go. Tingle your soft lips, earlobes, and nipples. Massage your tummy and inner thighs. Tickle the bottoms of your feet, your inner elbows, and the curve of your fanny. Try performing the Mirror of Sensual Transformation ritual with a vibrating hand and see how marvelously different it feels.\n\n_134_. Vibrate your vulva. Cup your entire pubic mound in your pulsating palm; slowly press in and feel the deep waves ripple throughout your entire body. Let your fingers rest lightly on your clitoris and vaginal lips. Massage your rosebud and moist vaginal opening with your softly buzzing hand. Squeeze them gently. Insert one vibrating fingertip inside your vagina. Just hold it there while you tingle; then rotate it and slide it in and out. Sneak up on your G-spot and make it throb. With your new quivering touch, you can add a fresh dimension to what you did in numbers 35 through 132.\n\n_135_. If you didn't become an immediate fan of anal self-play in Chapter 4, try it with your _electrified_ fingers. You may discover, as did my friend Sharon, that the delicate tissues there, richly endowed with sensual nerve endings, literally invite vibrations. She says, \"I love to strap on my Swedish massager and lay the tip of my middle finger against the opening in my behind. The vibrations feel sooooo good, and somehow primal. At the same time I rub my clit and labia with the other hand. It's kind of like being rocked in an unbelievably sexy cradle.\"\n\n## electric wand and coil-operated vibrators\n\nAs used by most modern sex goddesses, these vibrators are great for super stimulation of the genital area, especially the clitoris, thereby quickly propelling you into the blissful throes of first-time orgasms, nectar-producing orgasms, and even multiple orgasms. As such, they are highly recommended by Aphrodite, the Cloud Damsels, the sensual descendants of Cleopatra, and 99.7 percent of all Inner Sex Goddesses as an especially fine way to develop deep communion with them. And the more you express your Inner Sex Goddess self, the deeper and more electric is your communion with your lover. So pick out the vibrator type best suited to you, and aim for the stars.\n\nA wand vibrator is composed of a tennis-ball-sized vibrating head attached to a foot-long phallic handle. One of the oldest and most popular types, this magic wand generates a strong penetrating vibration, diffused over the rounded rubber head, and it's very easy to maneuver. It can be easily stationed between you and your lover, providing both of you with powerful vibes and free hands.\n\nCoil-operated vibrators have smaller handles, usually about seven inches long, and a vibrating metal plug on the side to which you can fasten a fun assortment of vinyl attachments\u2014rounded, pronged, and cuplike. They generate more focused, super-fast vibrations, are relatively lightweight and quiet, and have the advantage of variety over the vibrating head. Both wand and coil-operated vibrators usually have two speeds of operation, which basically translate to \"intense\" and \" _really_ intense.\" So handle these toys with care, but do handle them.\n\n_136_. Since wand and coil-operated vibrators were really made to be body massagers, you can and should use them to voluptuously tingle the skin all over your body. Employ the longer wand vibrators to pleasure places you can't normally reach with your hands. Massage the middle of your back, for instance, while you twirl your nipples with your free hand, or extend the wand down to tickle the bottoms of both feet while you rub your clitoris with your free hand. Play with the attachments of a coil-operated model on different parts of your body\u2014perhaps the suction cup over a toe, the pronged number to scratch your back, or the small rounded one to slip into your navel. Or simply give yourself an all-over vibrating massage before you move on to more sexual endeavors.\n\n_137_. Although the possible combinations of body parts and vibrator attachments are tantalizingly endless, here are some suggestions for each tingling attachment that Cleopatra would have loved to have in her collection. The dome of the suction cup is perfect for crowning your clitoris with euphoric pulsations; it fits over and around your little Jewel of the Nile without touching it directly. Try using the pronged attachment to simulate the grazing of Mark Antony's teeth on your milk-wet nipples. The small rounded tip is excellent for penetrating _in_ to sensitive places, like a smoothed Nile pebble having an earthquake inside your moist interior caves. And the ringed diffuser spreads the tantalizing tingles over the fleshy mounds of your honeyed breast or your jasmine-perfumed pubis; and, if pressed down firmly, it sends caressing pulsations deep into your very bones.\n\n_138_. Nipples and vibrators go well together. The round ball of a wand vibrator can impart a wonderful tingly feeling when smoothed over or pressed deep into a rosy areola. Cupping a nipple with the domed attachment of a coil-operated model might remind you of a small, sucking mouth. For a sharper, more intense sensation, pass either the pronged or the spot massager lightly over your nipples; this can send darts of excitement to your lower regions, warming them up for further self-play.\n\n_139_. When you feel ready to move the vibrator to your genitals, you may want to soften the intensity of its pulsations at first. Too strong a pressure or movement can kill your building passion. Try placing a folded towel, a pillow, or a piece of clothing, like maybe your sexy panties or teddy, between your delicate vaginal or clitoral tissues and the head of the vibrator. Or encase the whole vibrator in a thick cotton sock. This will help diffuse the vibrations, and the different texture against your skin will add a thrill all its own.\n\n_140_. Massage your outer and inner love lips with the vibrator. Long, slow strokes bring out the best and hottest levels of sensation. At some point stop and hold the head maddeningly steady against your urethral opening or your vaginal opening. Let the constant, steady pulsations melt into you and spread a throbbing glow throughout your whole body.\n\n_141_. You know how maddeningly exciting it is when your lover teases you by completely removing his penis from your vulva, pausing, and then resuming his amorous assault. Do the same with your vibrator by continually raising it away from your body and bringing it back again.\n\n_142_. Try inserting your stone egg (from the PC muscle exercises) or some ben wa balls into your vagina and then holding the vibrator against your vaginal opening. The egg or balls will rattle around deliciously.\n\n_143_. One sexy editor I know says she likes to rest a finger on her clitoris and then vibrate her finger with the rim of the suction cup attachment. Of course, this finger arrangement can also travel farther down across your inner lips to create a throbbing vulval massage.\n\n_144_. Though this is a delicate undertaking, most women use their vibrators to bestow otherwise impossible raptures upon their yearning clits. Techniques for doing this are as varied as the women who invent and relish them. Some stir their wands over and around the clitoral tip in slow, lazy circles. Others rub up and down or back and forth. They may alternate these movements with simply holding the vibrator lightly in place or rhythmically pulsing it in and out like a slow, gentle love tap. One young and blissfully married swim instructor I know says she prefers to plant her vibrator firmly over her clitoris and then very slowly increase the urgency by pressing it in closer and closer until the vibrator is mushing her clitoris into her pubic mound and the entire area is throbbing wildly. Then she switches the vibrator control to high!\n\n_145_. While vibrating your clitoris in the manner of your choice, trail your moistened fingers over the soft furrow between your labia, circle around your vaginal opening, and slip inside. Rotate, tickle, or thrust appropriately.\n\n_146_. A favored method of my friend Beverly, a bookish professor of philosophy, is to vibrate her clitoris while manually massaging her nipples, and then alternate with the opposite procedure\u2014vibrating her trembling tips while manually massaging her love bud. For Beverly, as for many women, there is a direct and powerful connection between these two hot spots that is electrified by the simultaneous pulsing and pulling actions. She says it's a great turn-on for her to watch this in the mirror too\u2014as it would be for any lucky lover, by the way.\n\n_147_. You can simulate the skin-to-skin pulsations of a Swedish scalp massager by resting your wand or coiloperated vibrator on the back of your hand as you stroke, knead, and slap with your fingers or palm. With a little imagination, these strokes could feel like the delicious ministrations of your man's mouth, hand, or \"sacred snake.\"\n\n_148_. Insert a fingertip to pinpoint your G-spot and then vibrate it by placing your magic wand on the part of your finger, knuckle, or hand that remains outside.\n\n_149_. My next-door neighbor, an avid sensualist, suggests laying your vibrator, head up, on a pillow, kneeling or squatting over it, and touching your genitals to its pulsing head as lightly or firmly, as fast or slow, as you desire. Pump your hips, \u00e0 la Elvis, over the humming surface.\n\n_150_. Free your hands for erotic self-exploration by creatively engineering a harness to hold your vibrator in place against your labia or clitoris. Tie a long scarf under your pelvis and vibrator, and over your shoulder. Two scarves crisscrossed around your hips can work even better. Or don the bottom half of your bikini swimsuit and insert the vibrator head between cloth and skin.\n\n_151_. I've found that if you hold the vibrator lightly against your love bud and use the middle finger of your other hand to tickle and caress the outside of your vaginal opening and the area just inside it, the voluptuous sensations and images of a lover's penis rubbing against your moist vulva are magically and meltingly evoked.\n\n_152_. One day over crullers and coffee my friend Marion confessed to me, \"I love to rent steamy sex videos and watch them with a vibrator in my lap. I copy every thing they do in the movie, whether it involves somebody's hand or tongue or even a man's penis, except that I'm doing it with my vibrator. I rub it all over my face, breasts, genitals, and derriere. Whatever they do, I get turned on by watching it and feeling my own vibrations at the same time.\" A great way to double your pleasure!\n\n_153_. Using either a wand vibrator or one of the smooth attachments on a coil-operated vibrator, slip the oscillating head around to your anal opening and massage gently. Don't try to insert it. With your free hand, rub your swollen labia and\/or clitoris. Close your eyes and float on a dreamy, cradling cloud, or imagine your lover's tongue there.\n\n_154_. Even though your vibrator is a wonderfully powerful love machine, don't make it do all the work. Move your body against it, gyrate your hips, swivel your fanny into it, undulate over and around its pulsing head. From a prone position with knees bent, thrust your pelvis up off the bed, tighten your tummy and PC muscle, and grind against the massager. Kneel or stand with legs outstretched and do the lambada or the limbo with your mechanical partner. Bend over and slip the vibrator between your legs to massage your tender thighs and labial mound; grip it with your thighs and squeeze, as if it's another phallus you know and love.\n\n_155_. Get personal. Plenty of women have told me they have pet names for their vibrators\u2014Sam, Vic the Vibrator, Thor (the god of thunder!), Valentino, Reginald Rutabaga III, and so on. The woman who dubbed her vibrator Valentino often plays flamenco music and wears one of those black gaucho hats with a chin strap when she has a rendezvous with her electrically powered lover.\n\n_156_. Many sex toy shops and mail-order catalogs carry a variety of vibrator attachments specifically designed for sexual use\u2014including \"cliticklers\" (perfect for focusing vibrations on one spot) and G-spotters (curved to tuck against your labia or to insert for vaginal vibration). There's even one shaped like a two-pronged twig that tingles your clitoris and your inner vaginal walls simultaneously or pulsates deliciously inside your anus. Sometimes half the fun of these attachments is shopping for them in the first place and imagining the dazzling variety of uses to which you can put them. So don't be too shy to indulge in this harmlessly pleasurable pursuit\u2014or to actually cavort with these naughty little toys either!\n\n_157_. Okay, say your vibrator has brought you to a thrilling orgasmic peak, and say your love bud is now supersensitive and shy of further vibratory touch, so you think the fun is over. Only your Inner Sex Goddess knows for sure, but you could probably go on to several more orgasms and reach places in your deep, hot sexuality you never knew existed. If you simply relax, take a few deep belly breaths, and let your hand rest between the vibe and you while you gently rotate against it, you can sustain the lovely orgasmic plateau and allow your clitoris to rejuvenate itself for further, heightened responses. This is when your Inner Sex Goddess _really_ comes alive.\n\n## dildos and battery vibrators\n\nDerived from the Italian word _diletto_ , which means \"delight,\" dildos are penislike objects used for delightful sensual pleasuring. Battery-operated vibrators are simply dildos that vibrate. Since you do basically the same things with both devices, I've lumped them together here for convenience\u2014after all, it's their penislike quality that matters.\n\nAnd that's of course what mattered to Cleopatra when she had her wood and stone dildos especially carved to order. We don't know for sure, but since clay dildos were found in Egyptian tombs around that same time, we can only imagine what elaborate pleasure wands the Queen of the Nile might have had prepared for her eternal resting place. We do know that Cleopatra treasured her phallic possessions as practical magic for keeping her ripe, vitalizing sensuality within easy reach. The Egyptian temptress intuitively understood that sense memories are stored all over the body, and the touch of phallus to vulva can spark off internal nerve messages that evoke lust and love.\n\nThroughout the ages, people of all cultures have unabashedly used carved phalluses as delightful keys to unlock deep sexual feelings. The people of ancient times actually considered dildos sacred. Originally worshiped as fertility gods, they were believed to house the magical potency of life itself. In India, holy lingams made of stone, metal, or ivory, representing the god Siva, were used to deflower virgins as a sacred rite of passage before marriage. And in ancient Rome, it was a statue of the phallus god Mutunus Tutunus who presented his staggering equipment for affianced young girls to ceremonially straddle.\n\nIn ancient Japan and China, a wife would often have a replica of her husband's penis made of tortoiseshell, horn, or wood, with his name beautifully inscribed on it. Greatly cared for and kept in a specially made box, it was a much-venerated likeness of his actual form that could be doted upon and used for self-satisfaction during the prolonged absence of a loved and loving mate.\n\nIn the harems of Arabia and the social and sacred events of Greece, dildos continued to develop new and more interesting forms, and around A.D. 500 a nice touch was added: dildos began to be made from sealing wax so they would absorb body heat. By the 1100s, the clever French had invented long red rubber instruments called _consolateurs_ that held milk or other liquids that could be pumped out to simulate ejaculation.\n\nToday both vibrating and nonvibrating dildos come in an astonishing variety of shapes, sizes, and colors, perfect for each libidinous lady and her every sensual mood. Vibrating dildos made from plastic, vinyl, or rubber are relatively inexpensive and portable, and they produce vibrations gentler than the wand or coil-operated vibrators. Some are plain hard cylinders while others approximate the softer feel and luscious look of a real penis. Take your pick. I have a delightfully mischievous friend who loves to give the plain cylindrical kind\u2014not very lifelike, but effective\u2014as first-time vibrator gifts. Wearing a catlike grin, she told me, \"They're nonthreatening, affordable, and always appreciated.\"\n\nRegular dildos are equally insertable, portable, and affordable\u2014they just don't vibrate. What they do is provide a delicious feeling of fullness in the vagina (or rectum); the movement is up to you. They too are made from rubber or plastic, but some are fashioned from a very lifelike silicone, and some even sport little protrusions that look and feel like testicles.\n\nBecause they come in a wide variety of shapes, sizes, and colors, buying a regular or a vibrating dildo can be a mind-boggling experience. I've seen them as small as 4 1\/4 inches long and an inch in diameter, and as big as 10 inches long by 2 1\/4 inches. They come in every color, from flesh to black to turquoise, and in shapes ranging from plain cylinders to ducks, whales, and dolphins to lifelike models of a famous porn star's equipment. Relying on your own personal tastes, you shouldn't have too much trouble selecting a color and shape to suit you, but size can be deceiving. Some women choose a dildo too small because they underestimate the expanded scope of an aroused vagina, while others find that their eyes are much bigger than their sexual appetites. I recommend experimenting with vegetables first\u2014zucchini, cucumbers, and carrots\u2014to determine the perfect fit for you. Then let your Inner Sex Goddess be your guide.\n\n_158_. First things first. Always lubricate your dildo before use because, far from being innately slippery, rubber and plastic can be somewhat sticky and can even pinch or abrade your sensitive tissues. But the oiling-up process can be a scintillating project in itself. Like Cleopatra, you might want to wet your dildo with your mouth by sucking on it lustily, as you would your man's penis. When I attended a class in the art of fellatio, I was delighted to discover that the instructor actually got turned on by using a lifelike rubber dildo to demonstrate oral technique. This practice is not at all weird or perverted; it's sensual as all get-out. In fact, I often have hot oral sex with my dildo while I pleasure myself in other ways at the same time. Try it. The other lubing option is to rub your dildo with oil in the same sensuous way you would massage your lover's equipment. Again, with a little imagination, this procedure can be an entrancing turn-on for you.\n\n_159_. Asking what women like to do with dildos is like asking \"What do women want?\" Basically, everything. Your repertoire should be limited only by your imagination. With your vibrating dildo, try everything you like to do with your regular vibrator. Use your nonvibrating dildo to perform some version of whatever you enjoy doing with your hands and fingers. Make either kind do all the lovely things to you that your man's love tool does, and all the things you _wish_ it could do. It's your dream penis to do with\u2014and be done unto with\u2014as you please. Look at it. Fondle it. Lick it. Screw it. Make it last for hours. Experiment with wild new ways of making love that you can shock your lucky man with later.\n\n_160_. Here's what my friend Charlene, a freckled, fresh-faced mom, likes to do: \"First I smooth the oiled head of my dildo up and down my vaginal lips. Then I press it in a little deeper so it snuggles in between them. Still moving it forward and back, I bring the dildo up to massage my clitoris, then down to nuzzle it barely inside my opening\u2014just enough so that the lip of the dildo head catches on the rim of my vagina. After I pull it in and out a few times, I start the whole routine over again, keeping myself from pushing the entire length deep inside as long as I can\u2014because as soon as I do, I have the most fantastic orgasm.\"\n\n_161_. Create a welcoming space for your dildo by fingering your clitoris long enough to get your juices flowing. Still massaging your delicate rosebud, dip the dildo into your musky lubricating fluids and slip it slowly inside you. Then slide it in and out, matching the cadence of your thrusts with your clitoral strokes and firing off all those lovely inner sex neurons. Keeping the two motions going, quicken your pace until you are thrusting and rubbing at lightning speed. See how far out into the galaxy you can take yourself.\n\n_162_. As you are thrusting, you might want to get your PC muscle into the rhythm too, squeezing it around the plunging dildo so that you can feel its texture against your sensitive vaginal walls. Remember that contracting your inner muscles also pulls on your clitoris, so you'll get a double whammy effect with these sensations. And it's great practice for milking your man's hot member.\n\n_163_. Try inserting your dildo as deep as possible and just holding it there. If it's a vibrator, flick on the switch after it's in place and let the vibrations radiate throughout your entire vaginal cavity. With a nonvibrating dildo, just focus on the luscious feeling of fullness. Let your other hand caress your body erotically as you fashion daydreams of cradling the love-swollen penis of the sexiest man alive.\n\n_164_. I know a woman who prefers to insert her dildo at oblique angles so that it stretches and pulls at her vaginal opening yet penetrates very little. She says, \"This way, it hits all those special spots, inside and out, that drive me crazy, and I feel always on the edge, craving more.\"\n\n_165_. Many women like to alternate deep, powerful plunges that pound into the inner recesses of their love cave with shallow, almost playful thrusts that excite the tighter entryway section. You can even let the dildo slip out completely sometimes, experiencing a lovely tug against your inner lips and vaginal rim each time you reinsert it. Show your lover how to do this with his flesh-and-blood phallus.\n\n_166_. Experiment with varying leg positions to achieve different effects: knees together or spread wide; sitting, reclining, or kneeling; standing in front of a full-length mirror; on your side with legs drawn up; on all fours; in a chair with your legs draped over the arms; face down while imagining being taken by a Mongol warrior. My manicurist says she likes to half recline with one knee up and the other flat on the bed. Then she brings her dildo in from behind, leaving plenty of room for the other hand to massage her clitoris or outer lips. This also brings the tip of the dildo in contact with the front wall of her vagina, where she undoubtedly has a goddess spot or two.\n\n_167_. A great position, if you can manage it, is lying on your back, knees spread wide, feet together and drawn up close to your bottom. Hold the dildo inside you and maneuver it with your feet. All the muscles around your vagina and its lips are being stimulated by the stretch of your legs as well as the exertions of your feet. And this lightly athletic marvel leaves both hands free for erotic exploration to your nipples and clitoris or other erotic venues. If there were a Sexual Olympics, this would surely be a hotly contested event!\n\n_168_. If you have a dildo with a suction cup base, try sticking it to the floor; then squat over it with legs spread wide and bounce up and down wildly. You can create very deep penetration and play with your breasts at the same time. Do your man the biggest favor of his life and let him watch you next time.\n\n_169_. Mary Ann, whose husband is often away scouting locations, has become quite chummy with her vibrating dildo. She says it makes her even hotter for Bill when she stretches her love lips wide open with the fingers of one hand and plunges her dildo in and out with the other. She comments, \"Not only does this feel great and remind me of my lover, but it's incredible to watch in the mirror too.\"\n\n_170_. As you tease, thrust, and rub with your dildo, letting yourself get more and more rapidly out of control, you may find it amusing to deliberately slow the pace by plunging your dildo in as deep as possible and just letting it rest there. Massage your clitoris very languidly and gently until your breathing is back to normal. Then speed up again, bringing yourself to a peak three or four times before you finally let the overpowering spasms carry you over the edge.\n\n_171_. Sex therapist Pauline Abrams says, \"I have my best orgasms if I put a dildo inside me so that it touches my cervix and then use the vibrator. That makes my orgasms much stronger.\" You can use the vibrator to steam up your clitoris and, while doing slow, sensuous penetration, gyrate on the dildo and vibrator at the same time. Or occasionally move the vibrator to the base of the dildo to send fiery pulsations rocketing up inside you. Another idea is to hold the dildo inside you by squeezing your legs together so that one hand is free for the vibrator and the other is available to tease your nipples or to perform some other delight.\n\n_172_. Insert your dildo about halfway and revolve it around and around the neck of your womb as if you were stirring up a delicious batter of passion cookies. This is especially intoxicating when done with a vibrating dildo turned up to full speed.\n\n_173_. Adorn your dildo with a ribbed or contoured condom to create a subtle but stimulating variation on the sensations it produces. It's a great way to practice safe fantasy sex with your favorite outrageously endowed porn star.\n\n_174_. Accessorize. Don a hat, scarf, gloves, shirt pulled open to expose your breasts, garter belt, and stockings under a tight dress, earrings in your pubic hair, temporary tattoos\u2014anything you might not feel brazen enough to wear for your lover. The sight of yourself in the mirror, coquettishly attired, with a lovely plump phallus sliding in and out of you, may give you the salacious ideas you've always wished you had.\n\n_175_. For a delightful tease, skim a vibrating dildo back and forth over your perineum for five to ten minutes before allowing yourself to push it into the deep well of your vagina.\n\n_176_. In some sex toy stores and catalogs, you can purchase hollow dildos containing a rubber or latex tube that's open at one end. You fill the tube with warm water or cream, or take a tip from the Italians and use egg whites or even fish eggs. Then you thrust the dildo in and out in your favorite manner and eject your fantasy lover's \"vital essence\" at the moment of your climax.\n\n_177_. Though this may be not to everyone's taste, I just have to pass along what seems like an exotically stimulating idea from a very adventurous friend of mine. Says she, \"I like that vibrating feeling in my behind; it seems to go all the way up through my vagina and stomach. That's why I often slide my vibrator dildo gently into my rear and smooth my fingers around the stretched opening there. Usually I just lie back and let the feeling build, but sometimes I play with my clitoris or the inside of my vagina. If I slip my fingers inside, I can feel the vibrations through the skin of my vagina, especially if I twirl the vibrator dildo around in that direction. It's a real deep, earthy sensation that makes me shudder and shake like a volcano.\"\n\n_178_. And here's one more from yet another lusty adventuress: \"I think two dildos are better than one, and I like to come up with creative ways to use both my small penislike vibrator and my big lifelike dildo. First I make the little one slippery by sliding it into my cozy slit. Then I slip it into the hole in my bottom and switch it on. Next I glide my Superman-size dildo ever so slowly into my vagina, push it in and out, and churn it around in circles. I imagine that some burly, sweaty construction worker is having his way with me and being real rough. This somehow blasts all my inhibitions right out of the water, and I pound against him like a wild woman. And once I've got all that going, I still have one hand free to stroke my clit. I haven't yet tried using another vibrator for my clitoris, but that's next on my list!\"\n\n## cleo's grab bag\n\nWith the power, ingenuity, and resources to have any sensual toy imaginable created for her, the precocious Queen of the Nile collected quite a cache of inventive pleasuring instruments. But her favorites were always her dildos and the simple little things like ibis feathers, lion fur, and of course milk baths. These pleasures she indulged in for both solo and partner lovemaking. While vibrators and dildos are at the top of almost every modern Sex Goddess's toy list, there is a veritable cornucopia of other scintillating playthings you too can use to coax delectable pleasures from your silky, sexy body\u2014and that of your lover. Though some of the suggestions that follow may bend the boundaries of your self-pleasuring comfort zone, put on your sultry Cleopatra crown and let her try them out for you. You may be surprised at your exotically expanded capacity for pleasure.\n\n_179_. Ancient Indian texts suggest that natural substances make the best penetration toys; they recommend radishes, mangoes, and gourds as well as the usual bananas, carrots, cucumbers, and zucchini. (One zany advertising writer says, \"Thank God for zucchini\u2014it has ridges! But I always practice safe vegetable sex and use condoms to avoid contact with pesticides.\") Fascinatingly, the ancients also suggested a \"reed made soft with oil\" and the stalk of a plant which, when soaked in hot water, swells up and takes on an \"agreeable warmth and texture.\" Mmm, sounds yummy. Of course, if they were around today they might also advocate using pickles (with their titillating knobs and crevices), Popsicles (deliciously icy, and fun to eat afterward), corn on the cob (lube it well first!), or even hot dogs (wet, slippery, flexible, cold, and sort of lifelike in color). Consult your Inner Sex Goddess on any richly inventive ideas she may have.\n\n_180_. An artist friend of mine suggests sculpting your vegetable of choice to size, or to an interesting shape, with a potato parer, leaving enough skin at the bottom so it won't slip out of your fervid grasp. She says she goes into another world when she puts on her headphones, inserts her artistically molded eggplant, and lets her regular vibrator tickle her clitoris unmercifully. When designing your vegetable penis, don't trim too close to the center or it will go limp.\n\n_181_. Almost any smooth object shaped like a cylinder can be used as a dildo\u2014like candles, for instance. Just remember that your internal tissues are delicate; so don't use anything that's glass or metal or that has rough edges or sharp corners. Be as discriminating about your stand-in lovers as you are about your flesh-and-blood men.\n\n_182_. Ben wa balls can provide an exhilarating internal massage, especially if you activate them with a vibrator (as suggested in 142) or by walking, dancing, or exercising while they jiggle around. These are especially handy for an ultimately unattainable fantasy\u2014having a man's penis massaging your vaginal walls all day. You could also insert a vibrating dildo in your bottom and point it toward the ben wa balls just on the other side of the sensitive dividing tissues.\n\n_183_. The Japanese have a slightly different version of their own ben wa balls\u2014called rin-no-tama balls. One of these small brass spheres is hollow and the other contains a tiny heavy metal ball or quicksilver, so that the balls constantly roll against each other. You insert them in your vagina\u2014put the empty ball in first\u2014and secure them with a tampon. Any pelvic movement causes them to shimmy around and tickle your insides mercilessly.\n\n_184_. Most sex toy stores or catalogs carry a little gem of an item called a Kegelcisor. Made of heavy brass, it's a small barbell for exercising your PC muscle (doing Kegel exercises), based on the concept that it's easier to build up strength and tone if you have to work against a resistive device. It's about 6 and 3\/4 inches long and has three small spheres, one at either end and one in the middle, that are 1\/4 inch to 1 and 1\/8 inches in diameter. This fancy fitness tool builds love muscles to grasp your man with, and it can be a uniquely gratifying dildo as well.\n\n_185_. Ice is a fabulous toy. Rub a frosty ice cube across your nipples, over your vaginal lips and clitoris, and inside the walls of your womb. You may prefer \"warm\" ice that has been sitting out for some time or the cold shock of ice cubes fresh from the freezer. Place an ice cube inside your vagina and let it melt as you massage your vulva, or use a vibrator on your labia to blend an internal \"cherry\" daiquiri. You can make a chilling dildo by freezing a cucumber or by very carefully filling an extra-sturdy and\/or large condom with water and popping it in the freezer. A few hours later cut away the condom and _voil\u00e0!_ \u2014an ice penis to play with.\n\n_186_. A truly marvelous toy for sale at most sex stores is a Venus Butterfly massager. This tiny vibrating pad, which lies over your clitoris and straps on around your thighs, allows you to enjoy continuous titillation on your love bud while your hands are completely free to roam over the rest of your hot bod.\n\n_187_. Of course, it's good to have some playthings for the rest of your body too. Taking an inventory of the self-pleasuring treasure troves of several sensuous friends (nightstand drawers, shoe boxes under the bed, even safe-deposit boxes), I discovered the following collection of simple little toys kept handy to make the skin tingle all over the body: a small piece of rabbit fur, ostrich and peacock feathers, a feather duster, silk scarves, the satin from a discarded camisole, a lace doily, black leather gloves, a small rubber ball, wool mittens, the ratty pronged attachment from a long-abandoned vibrator, a satin-smooth wooden egg, a collection of thimbles for each fingertip, cloth flowers, a soft cuddly teddy bear, an old woven place mat, elbow-length cotton gloves, part of a beaded car-seat cover, felt squares, a terry washcloth, a large fluffy makeup brush, the ruffle from a taffeta petticoat, pieces of Velcro, and the long Cleopatra wig from an old Halloween costume. Whether you employ these skin-sational toys by your own private self or invite your lover to join in, they provide easy and fun inducements to coax your Inner Sex Goddess out of hiding.\n\n_188_. Nobody talks about it, but millions of ordinary, virtuous, and sane women find that a little loving bondage provides a hot, intense, and daring addition to their self-pleasuring repertoire. If you've wondered about it but are a little squeamish, self-play can be a wonderfully private, fun, and safe outlet for your budding bondage fantasies\u2014and a special delight for your sizzling Cleopatra-esque personality. While sex toy stores have all kinds of fancy handcuffs, whips, clamps, blindfolds, and chains, there are perfectly wonderful substitutes for all of these wicked toys right in the sanctity of your own palace. Scarves, stockings, bras, and panties make fabulous handcuffs and blindfolds; choker necklaces, ribbons, or long strands of pearls can become decorative slave collars; belts, towels, dustpans, and Ping-Pong paddles can be used for sensuous self-spankings. One friend of mine, who is particularly partial to nipple stimulation, says that clamping her nipples with padded clothespins creates a delicious teasing pinch while leaving her hands free for other delightfully depraved activities. (If you don't have the padded variety, use cotton balls or scarves as a cushion.) Reading a book like _The Story of 0_ can give you plenty more inventive ideas for playing at submission while providing an erotic state of mind perfect for exploring the sweet tortures of passion.\n\n_189_. Another unexpected pleasure can come from playing with toys especially made to fit into your bottom. Depending on your personal preferences, you can insert small dildos, carrots, or cucumbers to provide an exquisite feeling of fullness, or larger specially made anal plugs that stretch, and thereby titillate, your tight opening. Anything you insert into your nether regions should be completely smooth and seamless with a flared base to keep it from slipping all the way inside. And always use lots of lubrication. Once your fanny toy is comfortably and safely in place, you have both hands free to roam over other erotic territories. One woman who enjoys euphoric anal sex with her partner told me that, when alone, she likes to circle her fingers around the rim where her rear sex toy emerges while she massages her clitoris with the other hand. Another lusty lady says, \"When I really want to come fast, I slide this special rubber plug inside my bottom and then stroke the inside of my vagina with my left hand. This turns my vaginal walls to warm pudding, and the plug makes me feel so wide and extended that my body just opens and comes.\"\n\n_190_. Water as a toy offers a fount of sensuous possibilities. Sit in front of the hot, powerful jet stream in a whirlpool bath. Use a bidet, or simply pour warm water over your genitals. Lie in the tub with your vulva directly under the faucet and turn the water on hard, perhaps switching from hot to cold to hot. Remove the head from your shower and let the extra weight of the water falling from a height feel like a thousand tiny fingers rapidly running over your genitals. Marilyn Chambers, star of the classic porn video _Behind the Green Door_ , says, \"What's really good is a hose in your bathtub. You just die, I swear. Just hook a hose to the faucet and take the spray attachment off. You put it on your clit, then you turn the water up and down\u2014hot or cold\u2014and you go through the ceiling. It's the most wrenching orgasm you will ever have in your life.\"\n\n_191_. Then there's a shower massager. The kind you can move around by hand is the only kind to have, of course. Dial the setting to any and every type of spray that takes your fancy\u2014diffuse, pulsing, hard, soft. Wash your inner thighs and navel with hot, tingling liquid. Rub the hard spray directly on your nipples. Spurt pulsing water against your clitoris. Open your labia lips and let the warm liquid pour like cream all over your sensitive membranes. Focus the stream with your hand, and finger yourself within the gushing current. Let hot pulses cascade down the cleavage of your bottom. Spray your entire genital area at different angles and from varying distances, in circles, in long sensuous strokes. Shake it over your throbbing bud and thrust the massager rapidly closer and farther away from your body. Let your Inner Sex Goddess splash around like a wild creature of the deep.\n\n_192_. I'll bet you've never thought of your douching equipment as a toy, but many women have told me they find it extra-stimulating to pleasure themselves while douching. They say the warm water spraying inside and then dripping over their fingers as they rub their clitoris drenches them in ecstasy\u2014and makes them think of penises bursting with love juice. Even if you don't wish to partake of this watery rapture, you can always use the dry attachments as delightfully different dildos.\n\n_193_. Brightly colored ribbons, small soft flowers, and decorative hair combs can be used to adorn your pubic hair as a playful masterpiece of sensual art. If you want to go a little further, try shaving your genitalia (an erotic sensation in itself), admiring your innocently exposed elegance, and then applying a pretty, feminine temporary tattoo. Follow up these aesthetics with master strokes of pleasuring massage to your decorated pubis. Reveal your artwork to your man, if you dare.\n\n# 10  \nthe goddess with her consort\n\n_You must perceive what you are through knowing yourself and your pleasures; for only then can you give the pleasure you seek and accept the pleasure given to you_.\n\n_\u2014 Ashley Thirlby, Tantra_\n\nnow that you've awakened all the erotic centers of your body and mind and learned to bring yourself via a thousand different paths to the highest peaks of sexual ecstasy, it's time to invite your consort into the game. Of course you've been making love to your man all along, and he's been reaping the benefits of your rapidly developing Sex Goddesshood, but have you shared your hottest autoerotic expertise and passion with him? Have you, for instance, thought about tying up your partner and making him watch while you undulate, shiver, and throb to your own inner rhythms? Well, if not, now's the time to knock his socks off.\n\nGoddess vamp that you are, however, even you may at first feel a little shy about revealing your intimate moves and feelings so boldly. But there are several good reasons for overcoming any lingering bashfulness.\n\n_Reason # 1_. According to recent research, many couples have their best sex when they engage in self-pleasuring or mutual self-pleasuring prior to or during intercourse. It adds variety (a crucial element in any long-term relationship), spice, mutual trust, and vulnerability, and it deepens intimacy immeasurably. In _The Art of Sensual Loving_ , Dr. Andrew Stanway says, \"The value of masturbation cannot be overestimated in any loving relationship.\"\n\n_Reason # 2_. You can turn your lover's flame up to extra hot with a simple twist of your nipple or stroke of your sexual triangle. Why do you think there's so much female masturbation seen in X-rated videos? Because men are intensely visual and begin to drool helplessly at the sight of a woman in the throes of her own passion. The fact that you are lusty and confident enough to allow him this intimate and thrilling sight of _you_ is extra frosting on his erotic cake. A man once told me, \"When a woman takes on the full power of her sexuality by loving her own body, and _lets me see it_ , she empowers me to be my sexiest self and takes us to places neither one of us has been before.\"\n\n_Reason # 3_. It takes the passivity out of being the receiving partner. Even in partner sex, you must still be responsible for your own passion, arousal, and orgasmic pleasure. So it's fortifying to know that instead of simply having to respond and follow where your man leads, you can add some flashy steps of your own. And through some magical process of sexual osmosis, it turns out that taking the turns _you_ like makes the dance much more exciting for both of you.\n\n_Reason # 4_. It's the best way for your man to learn your sexual needs and desires. You can show him, for example, that when you are touched lightly your body reaches out for more sensation, and that it contracts away if pressed too hard. You can let him see that when your nipples are twirled at just the right angle, your vulva opens up for him like a sweet, honey-dripping flower. Men want to know that their tremendous sexual prowess is propelling their partners to the heights of ecstasy. But never forget that, most of the time, they are completely mystified about what _really_ turns women on. How provocatively refreshing it is, then, to be shown in such a nonthreatening and salaciously seductive way _exactly_ how to drive you wild in bed!\n\n_Reason # 5_. You can learn hot new tricks from your man too. Not long after I vibrated my way to sexual self-rediscovery, I met Michael, a dedicated and natural sensualist. Sometimes I brought my vibrator to bed with us, but what Michael really wanted to see was me stroking myself between the legs, swelling, undulating, and oozing delicious, slippery moisture. He used to love watching me give myself an orgasm with my fingers and often said it was the most erotic thing he'd ever seen. I loved to watch him, too. In fact, that's how I learned most of what I know about exciting a man's penis. By watching him pleasure his own, I found out, for example, about that \"one slow, five fast strokes\" routine that drives most men insane.\n\nClearly, self-pleasuring with a partner can provide an electrifying boost to any couple's love life. But whether you discuss its lusty virtues with your mate beforehand\u2014to feel him out, prepare, and excite him\u2014or make it a sizzling surprise, it's probably best to start with a small, unintimidating sample of your self-pleasuring power. Over the next several days or lovemaking sessions, take your cues from each other, slowly getting used to your new erotic freedom, before you gradually crescendo to a stirring display of autoerotic fireworks. You'll probably both feel more comfortable that way.\n\n_194_. Sometime during foreplay look him straight in the eye as you suck the tip of your forefinger and slide it seductively over your lips. Then use your wet fingertip to tease his lips and mouth or to give your nipples a slippery pinch, or insert it smoothly into your vagina. Although his eyes will be riveted to your actions, keep him physically involved too by using your other hand to massage his body somewhere.\n\n_195_. As a variation on number 194, start by brazenly wetting your finger in _his_ mouth before trailing it over, under, and into your body.\n\n_196_. Cup your breasts in your hands and present them to him. Bring your nipples to his mouth, lift his hand to them, or play provocatively with them yourself. Deliberately twirl, tweak, pull, and pinch in the ways that make you hot, so he'll learn something while he's getting an erection.\n\n_197_. Provocatively slather oil on your breasts and then take your pleasure on his body, rubbing your slick, tingly nipples roughly or gently all over his face, neck, chest, nipples, stomach, legs, buns, testicles, and penis. Slide his toes, knees, flattened hand, and finally his hardened penis between your pearly globes. Let your deeply satisfied moans inflame him even further.\n\n_198_. As a variation on the previous suggestion, do the same with your oiled pubic hair and vulva or (a personal favorite of my lover's) with your well-lubricated bottom.\n\n_199_. Give your man a tactile and sensual thrill by asking him to lick your fingers while you massage your hottest erogenous zones.\n\n_200_. Take photos of yourself massaging your body and send them to him, insert them in the folds of his newspaper, or leave them propped up on his pillow.\n\n_201_. Some women prefer to blindfold themselves when first self-pleasuring for their lovers. Says my shy friend Milly, \"I can be in my own private world and feel free to do all sorts of naughty things I'd be too embarrassed to do otherwise. Sometimes I even keep the blindfold on after we start making love because the feeling of not knowing what's coming next and the heightened sense of touch are so exciting.\" In fact, not being under the scrutiny of your gaze can be freeing for your partner too.\n\n_202_. You might try supplementing what your man is doing to you with _his_ hands. A sensuous publishing exec I know says, \"When my husband has his fingers in my vagina, it feels more intimate if I stroke his hand with mine. Sometimes I rub my clitoris or massage my outer lips against his fingers. We have fun getting our fingers all tangled up together.\"\n\n_203_. Another opportunity for partial privacy is when your lips are locked together in a passionate kiss and your eyes are closed or busy staring into each other's orbs. This is a good time to stroke his back with one hand and massage your breasts or vulva with the other. He just makes you so _hot!_\n\n_204_. Incorporate any of the sense-awakening activities from Chapter 2 into your partner self-pleasuring. Take a bath with him and run the water and your soapy hands over your genitals and his. Dance for him while seductively undulating your hands over your private places. Use flowers, feathers, or fur to stroke yourself, and him, provocatively. Pour wine over your breasts or genitals, rub it in, lick your fingers, and press your wine-soaked parts against him. Smooth ice over your nipples and into the narrow glove of your sex, then let him lick off the cold drippings.\n\n_205_. One closet vamp I know likes to use her lover's body as a self-stimulating tool. She says, \"I rub my vulva against his knee, foot, or fanny cheeks. Or I tickle my breast with his chest hairs or pinch a nipple between his toes. Sometimes I insert his big toe in my vagina and use it like a dildo. Of course I'm fingering my clit the whole time, too, and looking him straight in the eye.\"\n\n_206_. Actually, one of the best self-pleasuring toys you'll ever have is your man's very own love rod. Use it, slippery with your lubrication but firmly ensconced in your caressing hand, to trace circles around your clitoris; to tease your inner lips and coral opening; to tickle your nipples; to massage your facial muscles; and, keeping it decidedly under the control of your maddening grasp, to inflame the inner walls of your vagina.\n\n_207_. Several sex toys have been specifically designed to transform your lover's penis into an even more exciting self-pleasuring tool for you. The fact that you show up with one of them in the bedroom will ignite his lustful imagination even before you start. Latex or rubber rings with beads in them or soft prongs all around can be slipped down to the base of your man's organ. The snug fit increases his erection, and the beads or prongs rub deliciously against your vaginal lips and clitoris as he thrusts in and out of you. The beaded ones can also be made smaller to fit around the base of his penis head, thus providing you with an outrageous internal massage. French ticklers, or rubber sleeves that fit over the penis and sport nubs all over or feathery plumes on the end, also tease your insides and may quickly stimulate you into a frenzied vaginal orgasm. How potent he'll feel then!\n\n_208_. My friend Liana's husband loves to watch her do the vacuuming while she's dressed in her leather corset or lace teddy. To further relieve the boredom of this mundane task, Liana occasionally reaches down to massage her exposed love nest. And of course it just so happens that she has to bend over then to examine a speck of dust on the carpet, giving her husband a spectacular view. Apparently she sometimes rubs the humming vacuum cleaner wand between her thighs, too. I always wondered how she got her husband to help with the housework!\n\n_209_. Men love to see a woman spread her sex lips wide while she plays with herself, but women often feel hesitant to do this because they think it must look grotesque. On the contrary, men find the sight intensely erotic and beautiful. One lover of mine watched fascinated as he moaned over and over, \"You're so pretty, so pretty. Yes, open that flower for me.\" Soon he simply had to join in.\n\n_210_. My friend Teresa, who's been ecstatically married to the same man for thirty years, says she likes to cup, lift, and massage her fanny while her man is on top during intercourse. In addition to providing a deeper angle of entry, this adds rear stimulation that excites her immensely. And when she wants to come, she can easily slip her finger into her anus and start the spasms rolling.\n\n_211_. When he is taking you from behind, massage your breasts or reach down to rub your love bud. Again, you may want to include him in the massage by occasionally stroking his thrusting penis while you're in the neighborhood.\n\n_212_. You can, of course, pleasure yourself from any lovemaking position, but spoon fashion is a particularly good one because your hands and the front of your body are completely free and, as you don't face each other, any embarrassment can easily be hidden. Even the mystery of what your man can't see you doing but can feel in your undulating response is a veiled enticement for him. Anyway, most men are very aroused when a woman gets so excited, presumably by him, that she just has to touch herself erotically. So if you have any doubts, you may want to try the spoon position as a seductive starter.\n\n_213_. Speaking of lovemaking positions, when you are lying on your stomach and he's taking you from behind, use the friction of the sheet or pillow to graze against your nipples and clitoris. Your deepened undulations will turn up the heat in his body as well as yours.\n\n_214_. Sometimes it's easier to abandon your inhibitions when you dress up like a tart. As a birthday present for my man, I once donned a red lace merry widow with garters and stockings (no panties!) and let him ogle me as I stood up on the bed to dance, wriggle, and seductively stroke all my exposed parts. Later he raved that it was the best birthday present he ever received.\n\n_215_. Tie him up and make him watch you give yourself an orgasm. Concentrate on your arousal, letting him see just what kind of stroking, pressure, speed, and positions best turn you on. When you've had your first climax, untie him and allow him to join in the fun of bringing you to another.\n\n_216_. Occasionally remove your wet finger from your glistening, musky depths and put it in his mouth to suck.\n\n_217_. A bank teller I know says, \"My lover gets wildly excited when I kneel over his chest, pinning down his arms with my legs, and then rub my clit and vagina. He says he loves to see my aroused sex and the ecstasy on my face, feel my tensed muscles, hear my panting, and inhale my scent. He usually has just enough arm movement to stroke his penis and time his orgasm with mine.\"\n\n_218_. When you are on top during intercourse, give him an additional show and yourself greater pleasure by sitting up straight and massaging your flushed breasts. You may want to occasionally wet your finger in your mouth or vagina and moisten your stiffened nipple with it. Or lean back and stroke your clitoris as you keep thrusting on him.\n\n_219_. When you feel especially bold, try kneeling over his face as you stroke your vaginal lips. He'll love the view; and you can occasionally allow him to lick you.\n\n_220_. One bawdy friend of mine shares the following: \"Nothing makes my boyfriend hotter than when I stroke his shaft to get him hard and start his pre-ejaculate fluids going. I take some on my fingers and put it in my mouth or massage my nipples or clit with it. Another thing I love to do is rub myself all over with his come juice. It's a great lubricant for self-massage, and it drives him crazy to watch me do this.\"\n\n_221_. Self-pleasuring is a lovely accompaniment to oral sex as well. When he has his tongue buried in your dewy depths, you can get so carried away that you abandon restraint to massage your clitoris or caress your thighs and breasts. Or you can get the urge as you're kissing him too. As one very ladylike matron told me, \"I feel so primal when I lick my husband's penis that a different me comes out. I don't feel shy at all about rubbing myself at the same time and even imitating with my fingers what I'd like him to be doing inside me. I always give him a good view of this, too, because it makes him really excited.\"\n\n_222_. When you have both become more comfortable with this type of love-play, you should encourage your man to pleasure himself for you as well. It's highly erotic to watch each other simultaneously and to respond, even from a small distance, to the other's building arousal and orgasm. You can lie next to or opposite each other, legs perhaps entwined, and feast your eyes and senses for hours\u2014or you can get a little more aggressive about it. One new bride, who was obviously getting her marriage off to a flaming start, told me, \"I'll never forget the excitement of watching my lover, kneeling between my outspread legs as I masturbated for him, stroking his engorged penis to orgasm right above me. Neither one of us could take our eyes off the spectacle of the other's sex in full, pulsing bloom. It was magnificent!\" When you do this, try to retain some shred of your rational mind to observe how he is specially treating his love organ\u2014that way, you'll know just how to make it your own later.\n\n_223_. Self-pleasuring with your lover by phone can also be quite titillating. My friend Sara, who carried on a long-distance romance for over a year, says, \"I used to talk to Jeff with one hand on the receiver and the other between my legs. I'd give him a blow-by-blow description of what I was doing, panting and moaning into the phone, or even putting the mouthpiece down below so he could hear the wet sounds I was making. Sometimes I'd start masturbating first and then call his answering machine as I got excited enough to come\u2014and there'd just be these long drawn-out 'Ooooooo's' for a sexy message. He never looked at another woman the whole time he was gone.\"\n\n_224_. The best way to overcome any jealousy your man might feel about your vibrator or dildo is to bring it to bed with you and let him play too. Share with him the lusty creature you become when a vibrator is throbbing you to orgasm, and be sure to pet him a little at the same time. Then use your toy gently on him. Let him see how delicious it looks when you tease yourself with a dildo, telling him you always imagine it's his lovely thick limb, and invite him to thrust the dildo powerfully inside you while his own equipment is free to cradle elsewhere\u2014like in your cleavage, the cleft of your bottom, or between your buttery love lips. Or put on sexy lingerie and give him his own personal X-rated show, with you as the female lead gyrating in flagrant abandon with the male star\u2014your vibrating dildo. If you always involve him and let him see more of your passion than he'd normally be privileged to view, your toys will become not a threat but favorite tools for joint self-pleasures.\n\n_225_. My friend Catherine has a very clever suggestion for including your man in the vibrator action. She says that when her husband is making love to her from behind, she often presses a vibrator to her thighs, pelvic mound, or clitoris. Not only does this raise her passion to fever heat, but it also transmits tingling pulsations to her lucky fellow's erection right through the walls of her throbbing flesh. Or when they are face-to-face, she simply lodges a small battery-powered vibrating dildo between her genitals and his. That way, each time he thrusts, they both get a powerful quivering jolt.\n\n_226_. Remember all those suggestions in Chapter 9 for using a vibrator on the back of your hand while your fingers tantalize your clitoris, vaginal walls, or G-spot? Well, simply substitute your lover's hand for your own, slipping his fingers seductively between the vibrator and your trembling flesh. He'll love participating firsthand in your pulsing, swelling excitement.\n\n_227_. With or without a vibrator, you might drive your man wildest of all by bringing yourself to the frenzied peak just before orgasm while he watches in panting fascination, then stopping completely, looking at him with fever in your eyes, and moaning, \"Come fuck me _now!_ \"\n\n# 11  \ndaily goddessizers\n\n_She learned to be at home in her body, to be her own best friend, her favourite lover._\n\n_\u2014Tee Corrinne,  \nDreams of the Woman Who Loved Sex_\n\n\"I make time for self-pleasuring every day, even if I have to lock myself in the bathroom, skip lunch, or tell my mate to give me an hour by myself. If I didn't, I'd be stressed out at work, a boring lover, and a crabby witch to my friends and family. They deserve better than that, and so do I.\"\n\n\u2014 Anne, executive secretary\n\n\"My sexuality makes me feel alive, important, and good about myself.\"\n\n\u2014 Sandy, housewife and mother of six children\n\n\"It makes me feel great to know that we women can be many different kinds of people, professional as well as sexy.\"\n\n\u2014 Jennifer, computer specialist\n\n\"The more time I spend getting in touch with my inner sensuality, the better my day seems to go. It gets my sex hormones flying, and then men are drawn to me like magnets.\"\n\n\u2014 Lelia, retired attorney\n\nlike riding a bicycle, the knack of unleashing your autoerotic Sex Goddess is not lost if you don't do it all the time. But I've found that the greatest benefits of keeping her active in your life\u2014enhanced health, well-being, and energy; stress reduction; charismatic self-confidence; hot lovemaking expertise; and constant sexual radiance\u2014are made fully available only if you keep your self-pleasuring thermostat on high with regular practice. Smart, sexy women, like the ones quoted above, pleasure themselves and their Inner Sex Goddess\u2014even if only in some small, five-minute way\u2014every single day. Besides innately understanding that self-pleasure is necessary for their physical, emotional, and mental well-being, they feel it's a precious and delightful gift they love to unwrap as often as possible.\n\nLike them, you should bestow upon yourself the same love and attention you would naturally give any romantic relationship. That's what keeps love affairs (and lovers) healthy, hot, and juicy. A love affair with yourself is no different. So schedule time for sensuality; delegate household chores to children, mate, or cleaning person; give up an hour at the health club; skip part of a gossip session (giving up gossip completely is of course unthinkable); knock a half hour off your shopping; do whatever you have to do to establish your self-pleasuring as a priority. Never forget that you _deserve_ to live in a perpetually fresh garden of ecstatic delights. And, as a reigning Sex Goddess, your glorious kingdom extends to those you love and care for.\n\n## daily goddessizers\n\nEven if you are the busiest woman on the planet, I'm sure you can find time for at least one of these quick and easy goddessizers every day. They'll keep your sensual motor purring and your Inner Sex Goddess glowing.\n\n\u2022 In your morning shower, let the water run over your beautiful genitals. As you wash, massage your clitoris and dip your soapy fingers into your vagina.\n\n\u2022 As you towel off, rub the rough terry cloth against your soft vulva.\n\n\u2022 When you apply body lotion, massage your nipples for an extra minute.\n\n\u2022 Admire your naked body as you dress. Preen, pose, and play.\n\n\u2022 Whenever you go to the bathroom, touch your vulva lovingly.\n\n\u2022 Move through your whole day being turned on by sounds, smells, beautiful scenes, awareness of your sexual juices, the sight of someone's full mouth, provocative ads, a glimpse of the moon.\n\n\u2022 Caress a tree on the way to work.\n\n\u2022 Sensitize your clitoris by lightly stroking it for ten minutes every day.\n\n\u2022 Roll in fur.\n\n\u2022 Squeeze and rub your thighs together, especially when you're on the gym machines.\n\n\u2022 Exercise naked.\n\n\u2022 Flex your PC muscle whenever you think of it; no one at the meeting will know.\n\n\u2022 Stand in the wind, close your eyes, open your palms, inhale its freshness.\n\n\u2022 Flash yourself in the mirror.\n\n\u2022 Invent a hot fantasy while riding the bus to work, standing in line at the bank, or under the hair dryer at the beauty salon.\n\n\u2022 Choose your lunch menu based on color and fragrance. Admire, lick, and smell your food.\n\n\u2022 Give your Inner Sex Goddess a mental hug at least once every day. Like you, she thrives on attention, love, and praise. Ask her advice on what to wear.\n\n\u2022 Indulge in a chocolate-covered ice-cream bar. Rub it on your lips and anywhere else you can get away with.\n\n\u2022 Whenever you look in a mirror, tell yourself, \"I love you. You are intelligent, powerful, sexy, beautiful, and irresistible.\" Blow yourself a kiss if no one's looking.\n\n\u2022 While talking on the phone, massage your face, neck, and breasts.\n\n\u2022 Occasionally bring your attention to your _hara_ (located about two inches below your navel), considered by martial arts specialists to be the body's center of gravity and power. This snaps you immediately into the present moment and the world of the senses. Float there as you experience your own sensuality, remaining motionless or stroking yourself luxuriously.\n\n\u2022 Dance the hula as you file or dust.\n\n\u2022 Self-pleasure in the \"exotic\" locations you visit every day\u2014the ladies' room at work, a telephone booth, a restaurant bathroom stall, someone else's backyard, or an empty meeting room.\n\n\u2022 On the commuter train, feel the heavy vibrations come up through your feet and into your legs, thighs, and pelvis. Let your body hum.\n\n\u2022 Punctuate your housecleaning with short vibrator sessions\u2014ten minutes fully clothed on the couch with the vibrator, ten minutes vacuuming, ten minutes back on the couch, ten minutes dusting, and so on.\n\n\u2022 Wash dishes in the nude.\n\n\u2022 Doodle pictures of body parts and other sensuous things. Write in your journal about all of the erotic things you've seen, smelled, touched, heard, and done today.\n\n\u2022 In the grocery store, fondle the cucumbers and zucchini.\n\n\u2022 Treat your self-pleasuring as a brief meditation. Close your eyes and drift in the alpha brain waves produced by using a vibrator on your erogenous zones. Tune in to your sensual intuition, daydreams, and creative juices.\n\n\u2022 Lying in bed at night, massage and love your body. Go to sleep thinking of yourself as a sensual wonderland, a famous temptress, or the answer to your man's erotic dreams.\n\n## for special, languorous occasions\n\nWhen you do have time for a longer, more luxurious visit with your Inner Sex Goddess, treat yourself to:\n\n\u2022 an hour-long hot scented bath with candles, music, and wine\n\n\u2022 an extended version of any of the \"Seeds of Sensuality\" activities from Chapter 2\n\n\u2022 the hottest and longest fantasy or role-play you can dream up, complete with toys, clothes, and props\n\n\u2022 the ritual of the Mirror of Sensual Transformation, with or without a vibrator\n\n\u2022 a two-hour teasing session in which you bring yourself to the peak at least fifteen times before you finally allow yourself to explode into one earth-shattering orgasm\n\n\u2022 a two-hour teasing session in which you give yourself fifteen orgasms\n\n\u2022 an X-rated video accompanied by candles, incense, wine, and a vibrator\n\n\u2022 a lazy afternoon with a fantasy or real life love slave\n\n\u2022 a languorous self-massage with aromatherapy oils or in a hot bath\n\n\u2022 two hours of reading or writing erotic literature, massaging your breasts and genitals the whole time\n\nFor an extra treat, try this lovely Tantric sensual meditation:\n\nStimulate yourself to a peak of excitement, then close your eyes and focus on your _relaxed_ breathing. Imagine yourself to be _in_ your perineum or clitoris. Feel a sensation, heat, tingle, shape, color, or sound there. As you continue pleasuring yourself, breathe energy into this area and let your special sensation expand as you exhale. After a few moments move your imagined location high up into your womb and feel a different sensation there. Repeat the breathing and expanding. Next, travel to your navel with the same process, and then to your heart. As you fully experience the warmth and love in your heart, finally let yourself have an orgasm that washes deliciously over your entire body. Bask for a while in your sensual Nirvana before reentering the everyday world.\n\nYour Inner Sex Goddess brings pleasure, confidence, and joy not only to you but, through you, to all those in your life. A whole world of women in such full sensual flower would be a beautiful garden indeed. So, like all of the mythological and real-life sex goddesses you've read about in this book, don't be afraid to get really decadent and outrageous. On the contrary, be proud of your multifaceted, lusty creativity. Remind yourself every day that you are a luscious, highly erotic woman\u2014a true Sex Goddess divinely designed for pleasure and loving.\n\nFreely radiate your resplendent sensuality, and the whole world will be transformed into your divine kingdom of love.\n\n# Appendix\n\n## the inner sex goddess scrolls\n\n_books, magazines, music, and videos to get you in the mood_\n\nOf the millions of erotic books, magazines, videos, and pieces of music out there, here are just a few that have been highly recommended by the Inner Sex Goddesses of many ultra-sensual women (including my own) especially for self-arousal. Try them out and see which ones resonate for you. If you find a particular work that really brings your Aphrodite self out to play, explore the other pieces that author, composer, or director has done too. Visit your local sex toy store or browse the pages of home pleasure toy catalog or adult magazines for further choices. Shop the erotica section of your favorite bookstore. Ask the clerk at the video or music store for recommendations. Share suggestions with your friends. Keep your eyes and ears open and your Inner Sex Goddess antennae alert; once you start looking, you'll find that all sorts of wonderful erotica shows up naturally within your Sensual Domain. Happy trolling!\n\n#### Literature\n\n_Delta of Venus_ by Ana\u00efs Nin (actually _anything_ by Nin is fabulously erotic)\n\n_Fanny Hill_ by John Cleland\n\n_Lady Chatterley's Lover_ and _Women in Love_ by D. H. Lawrence\n\n_Cher\u00ed_ by Colette\n\n_Emmanuelle_ by Emmanuelle Arsan\n\n_The Story of O_ by Pauline Re\u00e1ge\n\n_The Pearl_ by a collection of Victorian authors\n\n_Plaisir d'Amour_ by Anne-Marie Villefranche\n\n_The Pleasures of Loving_ , compiled by Maren Sell\n\nThe Black Cat books, any of them, published by Grove Press\n\n_Tropic of Cancer_ by Henry Miller\n\n_The Naked Lunch_ by William Burroughs\n\n_The Claiming of Sleeping Beauty_ , _Beauty's Punishment_ , and _Beauty's Release_ by A. N. Roquelaure\n\n_Exit to Eden_ by Anne Rice\n\n_The Erotic Comedies_ , or anything else, by Marco Vassi\n\n_The Gates of Paradise_ , edited by Alberto Manguel\n\n_Lust_ , edited by John and Kirsten Miller\n\n_My Secret Garden_ and _Forbidden Flowers_ by Nancy Friday\n\n_Pleasures: Women Write Erotica_ by Lonnie Barbach, Ph.D.\n\n_Endless Love_ by Scott Spencer\n\n_The Intimate Kiss_ by Gershon Legman\n\n\"Song of Solomon\" from the Bible\n\n_Erotic Poems_ , edited by Peter Washington\n\n_The Book of Eros_ , edited by Lily Pond and Richard Russo\n\n#### Art Books\n\n_The Art of Arousal_ by Dr. Ruth Westheimer\n\n_The Erotic Edge_ by Lonnie Barbach, Ph.D.\n\n_The Erotic Arts_ by Peter Webb\n\n_Erotic Art of the East_ and _Primitive Erotic Art_ by Philip Rowson\n\n#### Adult Magazines\n\n_Playgirl_\n\n_Woman on Top_\n\n_Playboy_\n\n_Penthouse_ (great for the pictorials with people making love)\n\n_Penthouse Letters_ ( _hot_ letters from readers)\n\n_Yellow Silk_ (collections of erotic writing)\n\n_Tantra_\n\n_For Women_\n\n_Paramour_\n\n#### Classical Music\n\n_Hungarian Rhapsody #2_ by Liszt\n\n_Bolero_ by Ravel\n\n_The 1812 Overture_ by Tchaikovsky\n\n_Scheherazade_ by Rimsky-Korsakov\n\n_Symphony No. 9_ (\"Ode to Joy\") by Beethoven\n\n_Canon in D Major_ by Pachelbel\n\n_Romeo and Juliet_ by Prokofiev\n\n_Meditation_ by Mischa Maisky, cello\n\n_The Double Life of Veronique_ soundtrack by Krzysztof Kieslowsky\n\n_Carmina Burana_ by Orff\n\n_The Firebird_ by Stravinsky\n\n_Suite No. 3 in D Major_ by Bach\n\n\"Love Death\" aria from _Tristan und Isolde_ by Wagner\n\n\"Celeste Aida\" from _Aida_ by Verdi\n\n\"O soave fanciulla\" from _La Boh\u00e8me_ by Puccini\n\n\"Tu, tu, amore tu?\" from _Manon Lescaut_ by Puccini\n\n_Carmen_ by Bizet\n\n#### Modern Music\n\n\"Layla\" (original or \"unplugged\" version) by Eric Clapton\n\n\"Brown Sugar,\" \"Let's Spend the Night Together,\" or countless other hits by the Rolling Stones\n\n\"Fields of Gold\" by Sting\n\n\"Lay, Lady, Lay\" by Bob Dylan\n\n\"Knights in White Satin\" by the Moody Blues\n\n\"Black Magic Woman\" by Santana\n\n\"Baby, Baby\" by Smokey Robinson\n\n\"Fire\" by Jimi Hendrix\n\n\"Stir It Up\" by Bob Marley\n\n\"Love Me Tender\" by Elvis Presley\n\n_Inside the Taj Mahal_ (flute) by Paul Horn\n\n\"Dream Lover\" by Mariah Carey\n\n\"Crazy for You\" by Madonna\n\n\"Caribbean Queen\" by Billy Ocean\n\n_Rhythm of the Saints_ by Paul Simon\n\n\"(You Make Me Feel Like) A Natural Woman\" by Aretha Franklin\n\n\"Fly Me to the Moon\" by Frank Sinatra\n\n\"Unforgettable\" by Natalie and Nat King Cole\n\n\"Sweetest Taboo\" by Sade\n\n_CrazySexyCool_ by TLC\n\n\"Purple Rain\" by Prince\n\n\"Sex\" by Morphine\n\n\"Funky Blues\" by Charlie Parker\n\n\"Fooled Around and Fell in Love\" by Elvin Bishop\n\n_Taj Mahal_ by Taj Mahal\n\n_Compact Jazz_ by Antonio Carlos Jobim\n\n_Nouveau Flamenco_ by Ottmar Liebert\n\n_Dreams_ by Kitaro\n\n_Enigma_ by Enigma\n\n_Chariots of Fire_ soundtrack by Vangelis\n\n#### Adult Videos\n\n_Behind the Green Door_\n\n_Delta of Venus_\n\n_The Seven Seductions_\n\n_Pandora's Mirror_\n\n_Taboo American Style_\n\n_Night Trips_ , _House of Dreams_ , _Hidden Obsessions_ (and _any_ other films directed by Andrew Blake\u2014he's one of the best)\n\n_Three Daughters_ (and any other films put out by Candida Royale and her Femme Productions; they are specifically directed toward women)\n\n_The Licorice Quartet_ (and anything else directed by Harry Paris)\n\n_Boiling Point_ , _Cat House_ , _The Girl with the Heart-Shaped Tattoo_ (and any other films directed by Toni English, a woman)\n\n_Better Sex_ (an instructional series)\n\n_Red Shoe Diaries_ (the half-hour TV series from Showtime)\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n## ANDREW GRAHAM-DIXON\n\n## Caravaggio\n\n#### A Life Sacred and Profane\n\nPENGUIN BOOKS\n\n## Contents\n\n_List of Illustrations and Illustration Acknowledgements_\n\n_Maps: Milan,_ c. _1590_ _;_ _Rome,_ c. _1600_ _;_ _Valletta,_ c. _1607_ _;_ _Naples,_ c. _1610_ _;_ _Italy,_ c. _1610_\n\n_Preface and Acknowledgements_\n\nPart One: Milan, 1571\u201392\n\nPart Two: Rome, 1592\u20135\n\nPart Three: Rome, 1595\u20139\n\nPart Four: Rome, 1599\u20131606\n\nPart Five: The Alban Hills, Naples, Malta, Sicily, Naples, Porto Ercole, 1606\u201310\n\n_Epilogue_\n\n_Illustrations_\n\n_Notes_\n\n_Further Reading_\n\n##### ABOUT THE AUTHOR\n\nFor more than twenty-five years, Andrew Graham-Dixon has published a weekly column on art, first in the _Independent_ and more recently, the _Sunday Telegraph_. He has written a number of acclaimed books, including _A History of British Art_ and _Renaissance_ , and is twice winner of the Hawthornden Prize, Britain's top prize for writing about art. He is one of the leading figures in broadcasting in the UK, having presented seven major television series on art for the BBC.\nPENGUIN BOOKS\n\n##### CARAVAGGIO\n\n'The author's insights into the individual paintings are remarkable, but what really gives the book its pep is the author's picking apart of the scanty evidence about Caravaggio's private life, the prostitutes and friends who people his paintings, and the seedy world of late-Renaissance Rome through which he swaggered' Andrew Holgate, _Sunday Times_ , Books of the Year\n\n'Remarkable... Sheds fresh light on the painter whose hot temper was as renowned as his work... uncovers details on pivotal events during the artist's turbulent career' Roya Nikkhah, _Sunday Telegraph_\n\n'Highly readable... thorough and elegant... _Caravaggio_ has done the artist proud' Ian Thomson, _Spectator_\n\n'Andrew Graham-Dixon's absorbing biography leaves no stone unturned... he is an entertaining art historian' _Economist_\n\n'Graham-Dixon's excellent new book... the complexity of Caravaggio's character as well as his art is vividly evoked by Graham-Dixon throughout this book... Graham-Dixon's biography will surely quickly establish itself as the outstanding introduction to Caravaggio's life and art' Brian Allen, _Standpoint_\n\n'Andrew Graham-Dixon brings the bad-boy genius of the seventeenth century to life as vividly as if he were one of today's pop stars. His book is scholarly, perceptive and very well written' John Richardson\n\n'An entertaining read... an engaging and well-written account of Caravaggio's life' Sheila McTighe, _Art Quarterly_\n\n'A beautifully paced narrative... its tremendous narrative drive... steers the story fluently and plausibly between the pitfalls of academic dryness and overdone speculation. As a human story it is ultimately tragic' Charles Nicholl, _Sunday Times_\n_To the memory of my mother, Sue, who first awoke my love of reading, writing and looking at art_\n\n## List of Illustrations\n\n_All works are by Michelangelo Merisi da Caravaggio (1571\u20131610), unless otherwise stated._\n\n1. _Portrait of Caravaggio_ by Ottavio Leoni (1578\u20131630), _c_. 1621\u20135 but clearly based on a life drawing originally done in _c._ 1595\u20131606, Biblioteca Marucelliana, Florence, Italy. Photo: Scala, Florence. Drawing: black chalk heightened with white on blue paper, 9.2 x 6.4 in. (23.4 x 16.3 cm).\n\n2. The _sacro monte_ at Varallo, Italy. Copyright _\u00a9_ Sabine Tilly.\n\n3. The _sacro monte_ at Varallo, Italy. Copyright _\u00a9_ Sabine Tilly.\n\n4. _The Lamentation_ by Guido Mazzoni (1450\u20131518), 1492, Church of Sant'Anna dei Lombardi, Naples, Italy. akg-images\/De Agostini Picture Library. Terracotta.\n\n5. _Mary Magdalen_ by Donatello ( _c._ 1386\u20131466), 1453\u20135, Museo dell'Opera del Duomo, Florence, Italy\/The Bridgeman Art Library. Gilt wood, height 74 in. (188 cm).\n\n6. _Portrait of Carlo Borromeo_ by Carlo Dolci (1616\u201386), Palazzo Pitti, Florence, Italy\/The Bridgeman Art Library. Oil on canvas (octagonal), 36.8 x 30.3 in. (93.5 x 77 cm).\n\n7. _The Adoration of the Shepherds_ by Simone Peterzano (1540\u201396), 1578\u201382, Charterhouse of Garegnano, Milan, Italy\/Alinari\/The Bridgeman Art Library. Fresco.\n\n8. _Venus and Anchises_ , The Farnese Gallery (detail) by Annibale Carracci (1560\u20131609), begun 1597, Palazzo Farnese, Rome, Italy\/The Bridgeman Art Library. Fresco.\n\n9. _Perseus and Andromeda_ by Giuseppe Cesari (1568\u20131640), 1602, Kunsthistorisches Museum, Vienna, Austria\/The Bridgeman Art Library. Oil on Canvas, 20.4 x 15 in. (51.8 x 38.2 cm).\n\n10. _Boy Peeling a Fruit_ , _c._ 1592\u20133, The Royal Collection \u00a9 2009, Her Majesty Queen Elizabeth II. Oil on canvas, 24 x 19 in. (61 x 48.3 cm).\n\n11. _Boy with a Basket of Fruit_ , _c._ 1593\u20134. Galleria Borghese, Rome, Italy\/Alinari\/The Bridgeman Art Library. Oil on Canvas, 27.6 x 26.4 in. (70 x 67 cm).\n\n12. _Boy Bitten by a Lizard_ , _c._ 1595, National Gallery, London, UK. \u00a9 The National Gallery, 2010. Oil on canvas, 26 x 19.5 in. (66 x 49.5 cm).\n\n13. _Boy Bitten by a Crayfish_ by Sofonisba Anguissola (1532\u20131625), _c._ 1554, Museo di Capodimonte, Gabinetto dei Disegni, inv. 1030, Naples, Italy. Photo: Scala, Florence, courtesy of the Ministero Beni e Att. Culturali. Drawing.\n\n14. _Self-Portrait as Bacchus_ , _c._ 1593\u20134, Galleria Borghese, Rome, Italy\/Lauros\/The Bridgeman Art Library. Oil on Canvas, 26.4 x 20.9 in. (67 x 53 cm).\n\n15. _Bacchus and Ariadne_ by Titian (Tiziano Vecellio) ( _c._ 1488\u20131576), 1520\u201323, National Gallery, London, UK. \u00a9 The National Gallery, 2010. Oil on canvas, 69.5 x 75.2 in. (176.5 x 191 cm).\n\n16. _The Cardsharps_ , 1595, Kimbell Art Museum, Fort Worth, Texas, USA\/Art Resource, NY\/Scala, Florence. Oil on canvas, 37  x 51  in. (94.2 x 30.9 cm).\n\n17. _The Gypsy Fortune-Teller_ , 1595, Pinacoteca Capitolina, Palazzo Conservatori, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 59.5 x 45.7 in. (151.2 x 116 cm).\n\n18. _Portrait of Cardinal Francesco Maria del Monte_ by Ottavio Leoni (1578\u20131603), 1616, Museum Purchase, Collection of the John and Mable Ringling Museum of Art, The State Art Museum of Florida, a division of Florida State University, USA. Drawing: black chalk heightened with white on blue paper, 9 x 6\u00bd in. (22.9 x 16.5 cm).\n\n19. _The Palazzo Madama_ by Giuseppe Vasi (1710\u201382). Engraving.\n\n20. _Medusa_ , _c._ 1598, Galleria degli Uffizi, Florence, Italy. Photo: Scala, Florence, courtesy of the Ministero Beni e Att. Culturali. Oil on canvas attached to wood, diameter 22 in (55.5 cm).\n\n21. _The Musicians_ , _c._ 1595\u20136, Metropolitan Museum of Art\/Art Resource\/Scala, Florence, Italy. Oil on canvas, 36\u00bc x 46  in. (92.1 x 118.4 cm).\n\n22. _Le Concert Champ\u00eatre_ by Titian (Tiziano Vecellio) ( _c._ 1488\u20131576), _c._ 1510, Louvre, Paris, France\/Giraudon\/The Bridgeman Art Library. Oil on canvas, 41.3 x 53.9 in. (105 x 137 cm).\n\n23. _Concert_ by Callisto Piazza (1500\u20131561), 1520, The Philadelphia Museum of Art, Philadelphia, USA\/Art Resource\/Scala, Florence. John G. Johnson Collection, 1917. Oil on panel 35  x 35\u00be in (90.5 x 90.8 cm).\n\n24. _The Lute Player_ , _c._ 1596, Hermitage, St Petersburg, Russia\/The Bridgeman Art Library. Oil on canvas, 37 x 46.9 in. (94 x 119 cm).\n\n25. _Basket of Fruit_ , _c._ 1596\u20138, Ambrosiana, Milan, Italy\/The Bridgeman Art Library. Oil on canvas, 12.2 x 18.5 in. (31 x 47 cm).\n\n26. _The Penitent Magdalen_ , _c._ 1595\u20136, Galleria Doria Pamphilj, Rome, Italy\/Alinari\/The Bridgeman Art Library. Oil on canvas, 48.2 x 38.8 in. (122.5 x 98.5 cm).\n\n27. _St Francis of Assisi in Ecstasy_ , _c._ 1596, Wadsworth Atheneum Museum of Art, Hartford, CT, USA\/Art Resource NY\/Scala, Florence. The Ella Gallup Sumner and Mary Catlin Sumner Collection Fund, 1943.222. Oil on canvas, 36  x 50\u00bc in. (92.5 x 127.8 cm).\n\n28. _The Rest on the Flight to Egypt_ , _c._ 1595\u20136, Galleria Doria Pamphilj, Rome, Italy\/Alinari\/The Bridgeman Art Library. Oil on canvas, 53.3 x 65.6 in. (135.5 x 166.5 cm).\n\n29. _The Judgement of Hercules_ by Annibale Carracci (1560\u20131609), _c._ 1595\u20136, Museo e Gallerie Nazionali di Capodimonte, Naples, Italy\/The Bridgeman Art Library. Oil on canvas 65.7 x 93.3 in. (167 x 237 cm).\n\n30. _Jupiter, Neptune and Pluto_ , _c._ 1598\u20139, Casino dell'Aurora, Villa Boncompagni Ludovisi, Rome, Italy\/akg-images\/Andrea Jemolo. Ceiling painting in oil, 124.4 x 59.8 in. (316 x 152 cm).\n\n31. _Bacchus_ , _c._ 1597\u20138, Galleria degli Uffizi, Florence, Italy\/Alinari\/The Bridgeman Art Library. Oil on canvas, 36.6 x 33.5 in. (93 x 85 cm).\n\n32. _Martha and Mary Magdalen_ , _c._ 1598, Detroit Institute of Arts, USA\/Gift of the Kresge Foundation and Mrs Edsel B. Ford\/The Bridgeman Art Library. Oil and tempera on canvas, 38.5 x 52.2 in. (97.8 x 132.7 cm).\n\n33. _St Catherine_ , 1599, Museo Thyssen-Bornemisza. Photo: Scala, Florence. Oil on canvas, 68.1 x 52.4 in. (173 x 133 cm).\n\n34. _Portrait of Fillide Melandroni_ , _c._ 1598, Gem\u00e4ldegalerie des Kaiser-Friedrich Museum, Berlin, Germany, inv. 356 (destroyed in 1945). Photo: Scala, Florence\/BPK, Bildagentur f\u00fcr Kunst, Kultur und Geschichte, Berlin, Germany. Oil on canvas, 26 x 20.9 in. (66 x 53 cm).\n\n35. _Judith and Holofernes_ , 1598\u20139, Palazzo Barberini, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 57.1 x 76.8 in. (145 x 195 cm).\n\n36. _The Entombment_ , 1604, Vatican Museums and Galleries, Vatican City, Italy\/The Bridgeman Art Library. Oil on canvas, 118.1 x 79.9 in. (300 x 203 cm).\n\n37. _Piet\u00e0_ by Michelangelo Buonarroti (1475\u20131564), 1499, Basilica di San Pietro, Vatican, Rome, Italy\/The Bridgeman Art Library. Marble, 68.5 x 79.8 in. (174 x 195 cm).\n\n38. _The Calling of St Matthew_ , 1599\u20131600, Contarelli Chapel (left-hand-side wall), San Luigi dei Francesi, Rome, Italy\/The Bridgeman Art Library. Oil on panel, 126.8 x 133.9 in. (322 x 340 cm).\n\n39. _The Creation of Adam_ (detail, post-restoration) by Michelangelo Buonarroti (1475\u20131564), 1511\u201312, Sistine Chapel, Vatican, Italy. Fresco, 110.2 x 224.4 in. (280 x 570 cm).\n\n40. _The Martyrdom of St Matthew_ , 1599\u20131600, Contarelli Chapel (right-hand-side wall), San Luigi dei Francesi, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 127.2 x 135 in. (323 x 343 cm).\n\n41. _The Conversion of St Paul_ (first version), 1600\u20131601, Palazzo Odescalchi, Collection of Principessa Nicoletta Odescalchi, Rome, Italy. Photo: Scala, Florence. Oil on cypress, 90.6 x 68.9 in. (230 x 175 cm).\n\n42. _The Crucifixion of St Peter_ , 1600\u20131601, Cerasi Chapel, Santa Maria del Popolo, Rome, Italy\/Alinari\/The Bridgeman Art Library. Oil on canvas, 90.6 x 68.9 in. (230 x 175 cm).\n\n43. The Cerasi Chapel (front view), Santa Maria del Popolo, Rome, Italy. \u00a9 Archivo Fotografico, Soprintendenza Speciale per il Patrimonio Storico Artistico ed Etnoantropologico e per il Polo Museale della Citt\u00e0 di Roma.\n\n44. _The Conversion of St Paul_ (second version), 1600\u20131601, Cerasi Chapel, Santa Maria del Popolo, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 90.6 x 68.9 in. (230 x 175 cm).\n\n45. _The Supper at Emmaus_ , 1601, National Gallery, London, UK. \u00a9 The National Gallery, 2010. Oil on tempera on canvas, 55.5 x 77.4 in. (141 x 196.2 cm).\n\n46. _St_ _John the Baptist_ , 1602, Musei Capitolini, Rome, Italy. Photo: Scala, Florence. Oil on canvas, 50.8 x 37.4 in. (129 x 95 cm).\n\n47. _Ignudo_ (detail, post-restoration) by Michelangelo Buonarroti (1475\u20131564), 1511\u201312, Sistine Chapel, Vatican, Italy\/Bridgeman Art Library. Fresco.\n\n48. _The Betrayal of Christ_ , 1602, National Gallery of Ireland, Dublin. With kind permission of the Jesuit community, who acknowledge the generosity of Dr Marie Lee Wilson. Oil on canvas, 52.6 x 66.7 in. (133.5 x 169.5 cm).\n\n49. _Street Scene_ by Francesco Villamena (1564\u20131624), 1601, British Museum, London, UK. \u00a9 The Trustees of the British Museum. Engraving, 13.6 x 19.5 in. (34.5 x 49.5 cm).\n\n50. _St Matthew and the Angel_ (first version), 1602, Gem\u00e4ldegalerie des Kaiser-Friedrich Museum, Berlin, Germany, inv. 365 (destroyed in 1945). Photo: Scala, Florence\/BPK, Bildagentur f\u00fcr Kunst, Kultur und Geschichte, Berlin, Germany. Oil on canvas, 87.8 x 72 in. (223 x 183 cm).\n\n51. _St Matthew and the Angel_ (second version), 1602, Contarelli Chapel, San Luigi dei Francesi, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 116 x 76 \u00be in. (295 x 195 cm).\n\n52. _The Death of the Virgin_ , 1606, Louvre, Paris, France\/The Bridgeman Art Library. Oil on canvas, 145\u00bc x 96\u00bd in. (369 x 245 cm).\n\n53. _The Death of the Virgin_ by Carlo Saraceni (1579\u20131620), 1610, Church of Santa Maria della Scala, Rome, Italy. Photo: Scala, Florence\/Fondo Edifici di Culto- Min. dell'Interno. Oil on canvas, 180.7 x 107.5 in. (459 x 273 cm).\n\n54. _Doubting Thomas_ , _c._ 1603, Schloss Sanssouci, Potsdam, Brandenburg, Germany\/Alinari\/The Bridgeman Art Library. Oil on canvas, 42.1 x 57.5 in. (107 x 146 cm).\n\n55. _The Sacrifice of Isaac_ , 1603, Galleria degli Uffizi, Florence, Italy\/Alinari\/The Bridgeman Art Library. Oil on canvas, 40.9 x 53.1 in. (104 x 135 cm).\n\n56. _Omnia vincit amor_ , 1602, Gem\u00e4ldegalerie, Staatliche Museen, Berlin, Germany\/Alinari\/The Bridgeman Art Library. Oil on canvas, 61.4 x 44.5 in. (156 x 113 cm).\n\n57. _Divine Love_ by Giovanni Baglione (1566\u20131644), 1602, Palazzo Barberini, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 94.5 x 56.3 in. (240 x 143 cm).\n\n58. Study for _The Resurrection_ by Giovanni Baglione (1566\u20131644), _c._ 1602\u20133, Louvre, Paris, France\/Lauros\/Giraudon\/The Bridgeman Art Library. Oil on canvas, 33.8 x 22.2 in. (86 x 56.5 cm).\n\n59. _St Jerome Writing_ , _c._ 1605, Galleria Borghese, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 44.8 x 62.8 in. (113.8 x 159.5 cm).\n\n60. _St Francis in Meditation_ , _c._ 1603, Palazzo Barberini, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 50.5 x 37.2 in. (128.2 x 94.4 cm).\n\n61. _St John the Baptist_ , _c._ 1604, Nelson Art Gallery, Kansas City, USA. Photo: Scala, Florence. Oil on canvas, 68 x 52 in. (172.7 x 132 cm).\n\n62. _The Madonna of the Rosary_ , _c._ 1603\u20134, Kunsthistorisches Museum, Vienna, Austria. \u00a9 Mondadori Electa\/The Bridgeman Art Library. Oil on canvas, 143.5 x 98.2 in. (364.5 x 249.5 cm).\n\n63. _The Madonna of Loreto_ , 1604, Cavalletti Chapel, Church Sant'Agostino, Rome, Italy. Photo: akg-images\/Electa. Oil on canvas, 102.4 x 59.1 in. (260 x 150 cm).\n\n64. _The Madonna of the Palafrenieri_ , 1605\u20136, Galleria Borghese\/The Bridgeman Art Library. Oil on canvas, 115 x 83.1 in. (292 x 211 cm).\n\n65. The contract for _The Madonna of the Palafrenieri._ Archivio dell'Arciconfraternita di Sant'Anna dei Palafreneiri, Cod. 843 v.sch.n 62. As reproduced and cited in Maurizio Marini, _Michelangelo Merisi da Caravaggio 'pictor praestantissimus'_ (second edition, Rome 1979), p. 56.\n\n66. View of Zagarolo. Photo: Arsetfuror.\n\n67. _The Supper at Emmaus_ , 1606, Pinacoteca di Brera, Milan, Italy\/The Bridgeman Art Library. Oil on canvas, 55.5 x 68.9 in. (141 x 175 cm).\n\n68. _David with the Head of Goliath_ , 1606, Galleria Borghese, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 49.2 x 39.8 in. (125 x 101 cm).\n\n69. _Sleeping Cupid_ , 1608, Palazzo Pitti, Florence, Italy\/The Bridgeman Art Library. Oil on canvas, 28.3 x 41.3 in. (72 x 105 cm).\n\n70. _The Seven Acts of Mercy_ , 1606, Pio Monte della Misericordia, Naples, Italy\/The Bridgeman Art Library. Oil on canvas, 153.5 x 102.4 in. (390 x 260 cm).\n\n71. _Roman Charity_ (black and white photo, detail) by Pierino del Vaga (Pietro Buonaccorsi) (1501\u201347), _c._ 1528\u201334, Palazzo Doria Pamphili, Genoa, Italy\/Alinari\/The Bridgeman Art Library. Oil on canvas.\n\n72. _The Flagellation_ , 1607, Museo e Gallerie Nazionali di Capodimonte, Naples, Italy\/The Bridgeman Art Library. Oil on canvas, 112.6 x 83.9 in. (286 x 213 cm).\n\n73. _The Crucifixion of St Andrew_ , 1607, Cleveland Museum of Art. Leonard C. Hanna, Jr, Fund 1976.2. Oil on canvas, 79.7 x 60.1 in. (202.5 x 152.7 cm).\n\n74. _St Jerome Writing_ , 1607, Co-Cathedral of St John, Valletta, Malta. Photo: Scala, Florence. Oil on canvas, 46.1 x 61.8 in. (117 x 157 cm).\n\n75. _Portrait of Fra Antonio Martelli_ , 1607, Palazzo Pitti, Florence, Italy\/The Bridgeman Art Library. Oil on canvas, 46.7 x 37.6 in. (118.5 x 95.5 cm).\n\n76. _Portrait of Alof de Wignacourt, with His Pageboy_ , 1607, Louvre, Paris, France\/Peter Willi\/The Bridgeman Art Library. Oil on canvas, 76.4 x 52.8 in. (194 x 134 cm).\n\n77. _The Resurrection of Lazarus_ , 1609, Museo Regionale, Messina, Sicily, Italy\/Lauros\/Giraudon\/The Bridgeman Art Library. Oil on canvas, 149.6 x 108.3 in. (380 x 275 cm).\n\n78. _The Adoration of the Shepherds_ , 1609, Museo Regionale, Messina, Sicily, Italy\/Lauros\/Giraudon\/The Bridgeman Art Library. Oil on canvas, 124 x 83 in. (314 x 211 cm).\n\n79. _The Beheading of St John_ , 1608, Co-Cathedral of St John, Valletta, Malta\/The Bridgeman Art Library. Oil on Canvas, 142.1 x 204.7 in. (361 x 520 cm).\n\n80. _St John the Baptist_ , 1610, Galleria Borghese, Rome, Italy\/The Bridgeman Art Library. Oil on canvas, 62.6 x 48.8 in. (159 x 124 cm).\n\n81. _The Adoration of the Shepherds_ (black and white photo), 1609, Oratory of San Lorenzo, Palermo, Italy\/Alinari\/The Bridgeman Art Library. Stolen in 1969. Oil on canvas, 105.5 x 77.6 in. (268 x 197 cm).\n\n82. _The Burial of St Lucy_ , 1608, Basilica di Santa Lucia al Sepolcro, Syracuse, Italy. Photo: Scala, Florence. Oil on canvas, 160.6 x 118.1 in. (408 x 300 cm).\n\n83. _The Denial of St Peter_ , 1610, Metropolitan Museum of Art, New York, USA\/Art Resource\/Scala, Florence. Gift of Herman and Lila Shickman, and Purchase, Lila Acheson Wallace Gift, 1997. Oil on canvas, 37 x 49.4 in. (94 x 125.4 cm).\n\n84. _The Martyrdom of St Ursula_ , 1610, Intesa Sanpaolo Collection, the Gallery of Palazzo Zevallos Stigliano, Naples. Oil on canvas, 56.3 x 70.9 in. (143 x 180 cm).\n\n85. _Experiment on a Bird in the Air Pump_ by Joseph Wright of Derby (1734\u201397), 1768, National Gallery, London, UK\/The Bridgeman Art Library. Oil on canvas, 72 x 96 in. (182.9 x 243.9 cm).\n\n86. _The Raft of the Medusa_ by Theodore G\u00e9ricault (1791\u20131824), 1819, Louvre, Paris, France\/The Bridgeman Art Library. Oil on canvas, 193.3 x 281.9 in. (491 x 716 cm).\n\n87. Still from _Mean Streets_ , directed by Martin Scorsese (1942\u2013 ), 1973, Taplin-Perry-Scorsese\/The Kobal Collection.\n\nText illustration, p. 325: Sketch of Caravaggio's sword and dagger made by the police officer who arrested him on the evening of 28 May 1605. As reproduced and cited in Maurizio Marini, _Michelangelo Merisi da Caravaggio 'pictor praestantissimus'_ (second edition, Rome, 1979), p. 54.\n\nText illustration, p. 328: _A Knight of Malta being Defrocked_ by Wolfgang Kilian (1581\u20131662), from C. von Osterhausen, Eigentlicher und gr\u00fcndlicher (Bericht, Augsburg, 1650), no. II. Photo: Zentralinstitut f\u00fcr Kuntsgeschichte. Engraving, 5 x 3 in. (12.7 x 7.5 cm).\n\n### Map 1\n\n### Milan, _c_. 1590\n\n### Map 2\n\n### Rome, _c_. 1600\n\n### Map 3\n\n### Valletta, _c_. 1607\n\n### Map 4\n\n### Naples, _c_. 1610\n\n### Map 5\n\n### Italy, _c_. 1610\n\n## Preface and Acknowledgements\n\nThis book has taken me a shamingly long time to write, more than ten years in total. My excuse is that I have had a lot of other things to do at the same time. For the first five of those ten years I was responsible for two weekly articles for the _Sunday Telegraph_ (latterly reduced to one, to make life workable); in 2007 I had to stop work on _Caravaggio_ almost completely to finish a book about Michelangelo's paintings in the Sistine Chapel; and throughout the past decade I have spent at least five months of every year writing and presenting various television series about the history of art for the BBC.\n\nWhile often frustrating, the many delays and interruptions have, overall, worked to the book's advantage. Had I delivered my manuscript more quickly, I might have caused my miraculously patient and long-suffering publisher, Stuart Proffitt, considerably less stress. But I would not have been able to take advantage of numerous recent archival discoveries \u2013 a set of remarkable finds that cumulatively have transformed our knowledge of Caravaggio, particularly of his later years. Because those discoveries have emerged piecemeal, often in out-of-the-way academic journals or private publications, I have found myself in the unusual and fortunate position of writing about one of the greatest artists ever to have lived fully four centuries after his death, yet able to draw on fresh and important documentary material unavailable to previous biographers.\n\nAs a result, I believe I have been able to shed light on aspects of Caravaggio's life that have until now remained shrouded in mystery to all except the scholars most closely involved \u2013 including the painter's sexuality, the circumstances that led him to commit the murder of 1606 that cast such a long shadow over the rest of his life, and the events surrounding his imprisonment on the island of Malta. In addition I publish here for the first time some hitherto overlooked descriptions of the Osteria del Cerriglio, the establishment in Naples where he was badly assaulted near the end of his life in a vendetta attack. By returning to other previously discovered documents I believe I have also been able to offer a convincing solution to the riddle of how Caravaggio met his death in the summer of 1610.\n\nMy principal focus throughout is on the artist's paintings. I dwell on them at length because they are the main reason to be interested in Caravaggio, notwithstanding the tempestuous drama of his life. Attentive readers will notice that I am less generous in my attributions than many other scholars of Caravaggio's work: I prefer to be too rigorous than over-inclusive. It may be assumed that if I do not mention a particular picture, for example the frequently proposed _Narcissus_ from the Barberini Collection, it is because I am not satisfied that Caravaggio painted it. The main exception to this is _The Annunciation_ in the Mus\u00e9e des Beaux-Arts in Nancy, which is indeed a Caravaggio, but one so badly damaged as not to be worth discussing here.\n\nI have incurred many debts in writing this book, above all to the community of scholars whose researches have yielded so much new information over the past half-century or so, especially in recent years. I am deeply grateful to Sandro Corradini for helping to guide me through the labyrinth of Rome's criminal archive and for sharing the fruits of his twenty years and more of research there. Maurizio Marini took me on a memorable tour of Caravaggio's old haunts in the artist's quarter of the city and made interesting suggestions, which I have developed, about the significance of damage done to the ceiling of a particular room in a house in the present-day Vicolo del Divino Amore. Maurizio Calvesi generously communicated his insights into the painter's 'pauperist' religious orientation, and the role that members of the Colonna family may have played in the various events of his life. In Naples, Vincenzo Pacelli showed me his archival discoveries concerning Caravaggio's last painting, _The Martyrdom of St Ursula_ , and shared some speculations about the painter's final days.\n\nMy thanks are also due to Peter Robb, who met me in Naples and sent me on what proved to be anything but a wild-goose chase on the island of Malta. On Malta itself I profited from conversations with Fr John Azzopardi and Keith Sciberras, who have between them shed much light on Caravaggio's ill-fated attempt to join the Order of the Knights of St John. John T. Spike, who received me at his home in Florence, allowed me to see an advance copy of the CD-ROM catalogue and bibliography that accompanied his monograph on Caravaggio: an invaluable guide to the vast literature on the artist. My old friend Mary Hersov, former Head of Exhibitions at the National Gallery in London, has talked and walked Caravaggio with me far beyond the call of duty.\n\nHelen Langdon, whose own biography of Caravaggio appeared in 1998, has also been extremely supportive throughout the writing of this book. In particular, she generously allowed me to profit from the time-consuming work that she put into combing through Riccardo Bassani and Fiora Bellini's sporadically fascinating but deeply flawed book of 1994, _Caravaggio assassino_ \u2013 the curate's egg of recent Caravaggio studies \u2013 sifting the true not only from the false but also from the outright invented. Helen also set me straight at a particular crossroads in my research into the painter's second and final stay in Naples, for which I am very grateful.\n\nI have not spoken to Sir Denis Mahon in the course of writing my book, but, like everyone engaged in serious study of Caravaggio, I have benefited enormously from his pioneering work. The shades of Walter Friedlaender and Roberto Longhi have given me much assistance along the way, as has that of my old tutor at the Courtauld Institute, Michael Kitson, whose wisdom I sought to absorb along with the smoke of many amiably shared packets of cigarettes. I have drawn rather more lateral inspiration from the work of John Michael Montias, whose _Vermeer and His Milieu_ of 1989 is a truly remarkable work. The shape of my own book has been certainly influenced by his, as well as by a meeting with Montias at his home in New Haven in the autumn of 2001. Without laying any claim to Montias's eminence as an archival scholar, I have myself tried to spin a 'web of social history', to use his phrase \u2013 to convey, through an account of one man's life and milieu, some sense of an entire lost world, in this case the civilization of Italy at the end of the sixteenth century and the beginning of the seventeenth. Charles Nicholl's books about Marlowe and Shakespeare, _The Reckoning_ and _The Lodger_ , have been among my other touchstones.\n\nWriting about Caravaggio has been an intellectual challenge, but it has also been an adventure, one which has led me into some fairly unusual situations. With John Azzopardi's generous help and the loan of a slightly rickety ladder, I have inspected the stone well, or _guva_ , in which Caravaggio was imprisoned on Malta (I can now laugh at the practical joke of his pretending to lock me in and leave me there, although it seemed less funny at the time). I have duelled (after a fashion) with master-swordsman Renzo Musumeci Greco in his Roman fencing school, in an attempt to understand the sort of manoeuvres that might result in the emasculation of a man during a swordfight. I have walked along the quays of the old port at Valletta with the Maltese naval historian Joseph Sciberras, to learn about transport by _felucca_ in Caravaggio's time. I have been allowed to inspect the book of the dead in the parish of Porto Ercole by local historian Giuseppe La Fauci. I have spent some happy hours poring over Caravaggio reproductions with the film director Martin Scorsese, who generously gave his time to open my eyes to the artist's importance for modern cinema. To these and all the others who have gone out of their way to help me \u2013 the boy who lowered that ladder down the _guva_ on Malta, the sacristan who got the keys to the church of Santa Lucia in Syracuse, the librarians and archivists in London, Rome, Naples, Milan and Malta who found so many books and documents \u2013 a heartfelt thank you.\n\nCloser to home, I would like to thank my producer Silvia Sacco for devising a schedule for my television and other work that made the seemingly impossible possible. Without her constant encouragement, moral support and ruthless deadline-setting, I really might never have written the book at all. Without the help of my researchers, I would certainly never have been able to finish it. Opher Mansour did a first-rate job of translating Corradini's essential anthology of archival documents, _Materiali per un processo_ , from a mixture of legalistic Latin and often difficult sixteenth- and seventeenth-century Italian vernacular slang. Opher also allowed me to read his enlightening doctoral thesis about censorship in Caravaggio's Rome and unearthed several eyewitness accounts of the plague that ravaged Milan in the mid 1570s. In the very final stages of the book, Nicholas Stone Villani took time away from his own thesis to travel to Italy on my behalf, where he found out the seedy truth about the Osteria del Cerriglio. My principal researcher throughout has been Eug\u00e9nie Aperghis-van Nispen tot Sevenaer, who has been unfailingly helpful, resourceful and thorough in carrying out what must sometimes have seemed a daunting series of tasks. She also did the picture research for the book and secured the reproduction permissions. While running her marathon, Eug\u00e9nie was ably assisted by Kasja Berg, who on more than one occasion responded to my plaintive demands for particular texts or documents with exemplary calm and efficiency. My mother and father, far more knowledgeable about music than I ever will be, kindly brought their considerable erudition to bear on Caravaggio's early paintings of musicians and lute-players, greatly to my advantage.\n\nI will always remain affectionately grateful to Roger Parsons, with whom I first began to explore the complexities of Caravaggio's world such an absurdly long time ago. Stuart Proffitt has made extremely valuable suggestions concerning style, structure and approach. Donna Poppy, my copy editor, has improved my original manuscript immeasurably with her rigorous and unsleeping eye for sense, proportion, perspective and detail. Finally I would like to thank my wife, Sabine, who must have read this book ten times while I was writing it once, for contributing so many emendations, corrections and indeed fresh ideas \u2013 and also my whole family, for helping me to keep my sanity and managing to keep their own while enduring the difficult birth of this long-gestated child.\n\n_London, February 2010_\n\n## PART ONE  \nMilan, 1571\u201392\n\n#### DARKNESS AND LIGHT\n\nCaravaggio's art is made from darkness and light. His pictures present spotlit moments of extreme and often agonized human experience. A man is decapitated in his bedchamber, blood spurting from a deep gash in his neck. A man is assassinated on the high altar of a church. A woman is shot in the stomach with a bow and arrow at point-blank range. Caravaggio's images freeze time but also seem to hover on the brink of their own disappearance. Faces are brightly illuminated. Details emerge from darkness with such uncanny clarity that they might be hallucinations. Yet always the shadows encroach, the pools of blackness that threaten to obliterate all. Looking at his pictures is like looking at the world by flashes of lightning.\n\nCaravaggio's life is like his art, a series of lightning flashes in the darkest of nights. He is a man who can never be known in full because almost all that he did, said and thought is lost in the irrecoverable past. He was one of the most electrifyingly original artists ever to have lived, yet we have only one solitary sentence from him on the subject of painting \u2013 the sincerity of which is, in any case, questionable, since it was elicited from him when he was under interrogation for the capital crime of libel.\n\nMuch of what is known about him has been discovered in the criminal archives of his time. The majority of his recorded acts \u2013 apart from those involved in painting \u2013 are crimes and misdemeanours. When Caravaggio emerges from the obscurity of the past he does so, like the characters in his own paintings, as a man _in extremis_.\n\nHe lived much of his life as a fugitive, and that is how he is preserved in history \u2013 a man on the run, heading for the hills, keeping to the shadows. But he is caught, now and again, by the sweeping beam of a searchlight. Each glimpse is different. He appears in many guises, moods and predicaments. Caravaggio throws stones at the house of his landlady and sings ribald songs outside her window. He has a fight with a waiter about the dressing on a plate of artichokes. He taunts a rival with graphic sexual insults. He attacks a man in the street. He kills a man in a swordfight. He and a gang of other men inflict grievous bodily harm on a Knight of Justice on the island of Malta. He is himself attacked by four armed men in the street outside a low-life tavern in Naples. His life is a series of intriguing and vivid tableaux \u2013 scenes that abruptly switch, as in the plays of his English contemporary William Shakespeare, from comedy to tragedy, from low farce to high drama.\n\nAnyone attempting a biography of Caravaggio must play the detective as well as the art historian. The facts are rarely straightforward and the patterns of intention that lie behind them often obscure. The artist's life can easily seem merely chaotic, the rise and fall of an incurable hot-head, a man so governed by passion that his actions unfold without rhyme or reason (this was, for centuries, the prevailing view of him). But there is a logic to it all and, with hindsight, a tragic inevitability. Despite the many black holes and discontinuities in the shadowplay of Caravaggio's life, certain structures of belief and certain habits of behaviour run through all that he did and all that he painted. The evidence has to be decoded using guesswork, intuition, speculation and above all a sense of historical imagination \u2013 a willingness to delve as deeply as possible into the codes and values that lie behind the words and deeds of a far distant past.\n\nA lot has been made of Caravaggio's presumed homosexuality, which has in more than one previous account of his life been presented as the single key that explains everything, both the power of his art and the misfortunes of his life. There is no absolute proof of it, only strong circumstantial evidence and much rumour. The balance of probability suggests that Caravaggio did indeed have sexual relations with men. But he certainly had female lovers. Throughout the years that he spent in Rome he kept close company with a number of prostitutes.\n\nThe truth is that Caravaggio was as uneasy in his relationships as he was in most other aspects of life. He likely slept with men. He did sleep with women. But he settled with no one. From a very young age, and with good cause, he suffered from a deep sense of abandonment. If any one thing lay behind the erratic behaviour that doomed him to an early death, it was the tragedy that befell him and his family when he was still just a little boy. The idea that he was an early martyr to the drives of an unconventional sexuality is an anachronistic fiction.\n\nTo understand the emotions that drove him and the experiences that most deeply shaped him, it is necessary to begin where he was born: in the town of Caravaggio, in Lombardy, from which he would later take his name. He lived both there and in the nearby city of Milan for the first twenty-one years of his life. His youth is the least documented period of his existence \u2013 the darkest time, in every sense, of this life of light and darkness. But in its shadows may be found some of the most important clues to the formation of his turbulent personality.\n\n#### FACTS AND FICTIONS\n\nThere are three early biographies of Caravaggio. All were composed after his death, and each is unreliable for different reasons. The first was written during the second decade of the seventeenth century by Giulio Mancini, a physician from Siena who met Caravaggio in Rome, probably in about 1592, and who knew him well between 1595 and 1600. The second was published in 1642 by Giovanni Baglione, a rival painter who had competed and quarrelled with Caravaggio during his years in Rome, in particular between 1601 and 1606, on one occasion suing him for libel in response to some scabrous verses, on another going so far as to accuse him of hiring paid assassins to kill him. The third was written, three decades later, by an antiquarian and art theorist named Giovanni Pietro Bellori, who had never known Caravaggio and who based his own account on those of the two earlier authors.\n\nMancini is sporadically informative but frustratingly brief. Baglione is more circumstantial and surprisingly objective, given that he was writing the life of a man whom he suspected of having plotted to murder him. As a rule of thumb, Baglione is the most trustworthy early source. His biography has been shown to be extremely accurate in its presentation of the bare facts. Many later discoveries of original documents concerning Caravaggio have simply confirmed the truth of his original account. Baglione is only really unreliable in his smug, moralizing conclusions, which are plainly coloured by _Schadenfreude_. This is particularly evident in the mean-spirited passages that tell the story of Caravaggio's various falls from grace.\n\nBellori wrote his life of Caravaggio considerably later. It was published in 1672, more than sixty years after the painter's death. Bellori plainly drew much of his material from Baglione. But he did glean some new facts. He also went to much trouble to see the painter's works _in situ_. He was seduced by their power and their drama, and fascinated by the novelty of Caravaggio's technique. Bellori wrote about the painter's art with far greater sensitivity than either Mancini or Baglione. Yet he was also fundamentally appalled by it. Caravaggio's vivid capturing of poverty and violence \u2013 his depictions of Christ and the Virgin Mary as barefoot paupers, his bloodily realistic portrayals of Christian martyrdom \u2013 went directly against Bellori's own most cherished beliefs. Bellori upheld the academic principle that art should not represent the world as it is, but as it should be, sweetened and idealized. So although he responded instinctively to Caravaggio's captivating realism, he felt bound to condemn him all the more strongly for it. Bellori crystallized what would remain for centuries the standard academic objection to the painter's work:\n\n> Repudiating all other rules, [Caravaggio] considered the highest achievement not to be bound to art. For this innovation he was greatly acclaimed, and many talented artists seemed compelled to follow him... Such praise caused Caravaggio to appreciate himself alone, and he claimed to be the only faithful imitator of nature. Nevertheless, he lacked _invenzione_ , decorum, _disegno_ [draughtsmanship], or any knowledge of the science of painting. The moment the model was taken from him, his hand and his mind became empty.\n\nBellori went on to say that 'Just as certain herbs produce both beneficial medicine and most pernicious poison, in the same way, though he produced some good, Caravaggio has been most harmful and wrought havoc with every ornament and good tradition of painting.' In other words, the painter might have had a gift for mimicking reality, but there was no depth to him. If Bellori were to be believed, he was little more than a machine for producing optically convincing images \u2013 a kind of human camera, with his workshop a prototypical photographer's studio, long before the invention of photography itself. In this way was the myth of Caravaggio as an untutored, thoughtless virtuoso, the master of a debased and pernicious brand of naturalism, attached like an anchor to his posthumous reputation. In fact, he was an extremely thoughtful, inventive painter, a close and careful reader of the texts that he was called to dramatize and to embody in the form of images. But how and where he got his education remains unknown, partly because his three biographers have so little to say about his early life.\n\n#### MODEST ORIGINS, NOBLE CONNECTIONS\n\nCaravaggio was born three years after the publication of the second, revised edition of Giorgio Vasari's pioneering anthology of artists' biographies, _The Lives of the Most Excellent Painters, Sculptors and Architects_. Vasari's book was the model on which later writers such as Baglione and Bellori based their own collections of artists' lives. In it, he confirmed and sought to extend a great rise in the status of artists within the Italian peninsula during the period now known \u2013 also largely thanks to Vasari's efforts \u2013 as the Renaissance. Previously the profession of art had been ranked low because it involved work with the hands and was therefore classed as a form of manual labour, a craft rather than a liberal art. But implicit throughout Vasari's thousand and more pages is the belief that the greatest artists deserve to be ranked with poets and philosophers as men of true genius, rightful companions of kings and princes.\n\nAs well as raising the reputation of his own profession, Vasari established certain formulae for writing the life of an artist. Particularly famous painters and sculptors, such as Giotto or Michelangelo, are established as miraculous prodigies from an early age: the brilliance of Giotto, for example, is said to have been discovered by the older artist Cimabue, who came upon the young man when he was still a callow shepherd and found him drawing perfectly upon a stone. But no such uplifting fables are attached to the youth of Caravaggio by his biographers. Mancini compresses his early life to just two sentences, and Baglione to a paragraph. Bellori has a tale to tell about the young Caravaggio, but it runs counter to the kind of prodigy stories favoured by Vasari because it is designed to stress the artist's principal failing, as Bellori saw it \u2013 his supposed _lack_ of intellect, which meant that his work could never rise from mere craft.\n\nBellori's story tells of Caravaggio's origins as the son of an artisan. Since the painter 'was employed in Milan with his father, a mason, it happened that he prepared glue for some painters who were painting frescoes and, led on by the desire to paint, he remained with them, applying himself totally to painting. He continued in this activity four or five years...' Bellori may have meant to imply that this imitative, unreflective training predisposed Caravaggio to his great mistake \u2013 that of recognizing 'no other master than the model, without selecting from the best forms of nature'. His moral is certainly blunt: once a craftsman, always a craftsman.\n\nThe story is not exactly true but like many stories about Caravaggio it contains elements of the truth. He could never have been employed in tasks such as preparing glue or plaster for his father, because his father died when Caravaggio was only five years old. But the record shows that Fermo Merisi was indeed a mason. This might suggest that the artist's origins were, as Bellori implies, rooted in the humble world of the artisan. But the sources hint at a more complicated truth. There is room for ambiguity because Fermo Merisi's job of mason could encompass different ways of working with stone, and possibly even the vocation of architect.\n\nBaglione's brief account broadly agrees with that of Bellori \u2013 he simply says that the artist, 'born in Caravaggio in Lombardy, was the son of a mason, quite well off'. But Mancini makes the artist's background sound considerably grander. According to him, 'He was born in Caravaggio of honourable citizens since his father was majordomo and architect to the Marchese di Caravaggio.' Mancini may have got the gist of his account from the artist himself, in particular the idea that Caravaggio was of better than merely common birth. A number of incidents in the painter's later life indicate that he believed that he came from good stock, and deserved respect on account of that. It is important to establish the truth, because Caravaggio's elevated sense of his own status would lie at the root of many of his future troubles.\n\nMost of the known facts about Caravaggio's youth were published by the scholar Mina Cinotti in 1983. One of the more revealing documents to emerge from her research records the wedding of the artists' parents. On 14 January 1571 Fermo Merisi married a woman called Lucia Aratori. Fermo was born in about 1540 and was a widower, with a daughter named Margherita by his first marriage. Lucia was some ten years younger than him and had not been married before. Fermo was recorded as resident in Milan, but the marriage took place in the town of Caravaggio, where both his bride and the rest of his family lived. It would have been an unexceptional wedding had it not been for the presence, among the witnesses, of the Marchese Francesco I Sforza di Caravaggio. The marchese was a member of one of the leading noble families of Italy, the Sforza, who were former lords of Milan. His wife, the young Marchesa di Caravaggio, was from the enormously powerful Colonna family. These were the most important people in the neighbourhood.\n\nThe presence of nobility at the nuptials of the Merisi family turns out to have had precious little to do with Caravaggio's father. Fermo Merisi was just an ordinary stonemason, perhaps reasonably well off but with no great social pretensions. He was certainly _not_ an architect. In a number of documents relating to him he is referred to as a _mastro_ , designating him as a qualified artisan with the right to set up his own workshop and hire apprentices. He ran this modest business in Milan. His probate inventory lists 'some old iron mason's tools', but does not include any books or instruments that would indicate a knowledge of the theoretical aspects of architecture. His retention of an independent workshop makes it unlikely that he was in the direct employ of the Marchese di Caravaggio. Caravaggio's paternal grandfather, Bernardino Merisi, was himself no higher up the social scale. He too had run a small business. He was a wine merchant and vintner based at the family home in Porta Seriola, in the north-east quarter of Caravaggio.\n\nThere were in fact close links between Caravaggio's family and the noble Colonna dynasty, but all on the side of the painter's mother. Her father, Giovan Giacomo Aratori, was an _agrimensor_ , or 'surveyor', whose job it was to help resolve disputes over land ownership. He was also involved in buying and selling land. His work brought him directly into contact with the Colonna, who owned much property in the region. Whereas Caravaggio's father and paternal grandfather worked with their hands, Giovan Giacomo was a professional rather than an artisan. His work required more literacy than that of a mason, as well as a knowledge of geometry and arithmetic. In 1570, a year before the birth of his grandson, the future painter, he was made a member of the college of land surveyors of the Duchy of Milan.\n\nGiovan Giacomo Aratori also played his part in the religious life of Caravaggio. The most celebrated event in the history of this sleepy little agricultural town had occurred in 1432, when a peasant girl working in the fields was reputed to have had a vision of the Virgin Mary. According to legend a freshwater spring had miraculously gushed from the spot where she experienced her vision, and a shrine had been subsequently erected to the honour of the wonder-working 'Madonna della Fontana'. By the second half of the sixteenth century, the shrine of Santa Maria della Fontana had become the most significant religious institution in Caravaggio. It was administered by a body of _scolari_ , to which Giovan Giacomo was elected at various times from the mid 1560s onwards.\n\nIn addition, he held important positions in the local _comune_ , as councillor, treasurer and emissary to the Spanish authorities (the Duchy of Milan, including the town of Caravaggio, was at that time part of the vast Habsburg empire, controlled by Philip II of Spain from the Escorial, his palace and monastery outside Madrid). Giovan Giacomo's many responsibilities meant that he was a familiar figure among the local nobility. He acted directly as an agent for the Marchese Francesco Sforza I di Caravaggio, served as a legal witness for the Sforza family and collected rents on their behalf. Some documents connect him directly to the marchese, others to the marchese's wife, Costanza Colonna.\n\nThere were yet more intimate links between the Colonna family and the Aratori clan. Giovan Giacomo's daughter Margherita, Caravaggio's maternal aunt, was wet-nurse to the Sforza children. She lived in the Colonna household for many years and breastfed Costanza Colonna's sons, including the future adventurer and sometime militant Knight of the Order of St John, Fabrizio Sforza Colonna. In 1584, as a reward for her service, Costanza gave Margherita a small estate in Fara d'Adda, near the town of Caravaggio. As late as 1601 Margherita was still in regular touch with the marchesa, writing letters to her in Rome \u2013 at a time when Caravaggio, elsewhere in the city, was receiving some of his most important commissions.\n\nCostanza Colonna would be called on many times by Caravaggio. Always she would respond. She would be a constant support to him in times of crisis, giving him shelter when he was on the run and shielding him when he was under sentence of death. Yet, unlike any of his other noble allies or protectors, she would never try to acquire a painting by his hand. All the evidence suggests that she genuinely cared for him, perhaps even loved him as a child of her own. Her influence and that of her family, with its tentacular network of feudal and familial alliances, reaching right across the Italian peninsula, can be sensed throughout Caravaggio's life _\u2013_ but especially during his later and more troubled years.\n\nSocial class, in particular questions of 'nobility' and 'virtue', would be at issue in many of Caravaggio's future disputes and quarrels. These were matters of intense debate in medieval and Renaissance Italy. In northern Europe the aristocracy took its own pre-eminence for granted and assumed that nobility was a quality that could only truly inhere in those fortunate enough to be born into the upper, landed classes. There, a nobleman was easily identified: a man of virtue and pure blood, who had the right to bear arms in the service of his monarch, who was a skilled swordsman and horseman and would never dirty his hands with trade. In Italy the situation was more ambiguous, because Italian society was more fluid and its ruling elites more diverse, made up of imperial knights, communal knights, magnates and other types of feudal lord. It was also an increasingly urbanized society, and that too led to the blurring of social distinctions. From the second half of the fourteenth century onwards, urban patriciates sought to tighten their hold on government. The men who made up those bodies, which included merchants, moneylenders, textile manufacturers and other drivers of early capitalism, were themselves intensely class conscious. They founded their own dynasties, staked their own claims to _nobilit\u00e0_ \u2013 so much so that the very term itself became, in Italy, shifting and unstable. As early as the fourteenth century, writers ranging from the poet Dante to medieval jurists had struggled to define the concept. Legal definitions based purely on titles conferred by the monarchy or the church were countered by those who preferred to regard nobility as a moral quality to which, in theory, almost anyone could aspire.\n\nWhat position did Caravaggio's maternal grandfather occupy within this world of subtly shaded social distinctions? Giovan Giacomo Aratori is referred to in the documents of the time as _signor_ , _messer_ or _dominus_. While his social status was certainly higher than that of anyone on the Merisi side of Caravaggio's family, neither he nor his descendants possessed any actual titles. He was a member of what might be called the upper, professional bourgeoisie, while the likes of Bernardino and Fermo Merisi belonged to the petty, trade bourgeoisie. Mancini's statement that Caravaggio was born into a family of 'very honourable citizens' \u2013 _cittadini_ is the word he uses in Italian \u2013 was entirely accurate.\n\nBut in the small world of Caravaggio, where the artist spent much of his youth, his status may to him have seemed grander than that. As we have seen, his maternal grandfather was a highly respected man, but other factors may have conspired to make him feel that both he and his family were blessed by aristocratic favour. Maybe Costanza Colonna showed particular favours or kindnesses to Caravaggio's mother, Lucia, sister to her own children's wet-nurse. Lucia's early years of motherhood were hard indeed, marked by bereavement and loss. Costanza Colonna too had suffered a difficult time during the early years of her marriage to Francesco I Sforza. She had been married off, as the custom then was among the nobility, at the age of thirteen. The duties of a wife had at first been abhorrent to her, so much so that she had at one point threatened suicide. Did Costanza Colonna feel a particular sympathy for Lucia and her young children during the harsh years of their early upbringing? It is impossible to know for sure, but she certainly took a particular interest in Caravaggio's wellbeing later in his life. Perhaps the date of his birth had something to do with it too, because as far as anyone in Christendom was concerned \u2013 but especially a Colonna \u2013 he was born at an auspicious time.\n\n#### THE ANGEL WITH SWORD AND SHIELD\n\nCaravaggio grew up as Michelangelo Merisi. It was an evocative name for a future artist \u2013 the same Christian name as that of the most famous Italian sculptor and painter of all, Michelangelo Buonarrotti, who had died just seven years earlier. But Caravaggio's parents did not have that in mind when they named their son. They called him Michelangelo for reasons of faith and superstition. He came into the world on 29 September 1571. His parents named him after the Archangel Michael, whose feast day it was.\n\nThis was a charged and momentous time in the history of Christendom. Throughout the 1550s and 1560s the Christian powers of the western Mediterranean were threatened by the forces of Islam \u2013 led first by the Ottoman Sultan Suleiman 'The Magnificent', and then by his successor, Selim II. The bitter and bloody conflict between Muslim and Christian reached a climax at exactly the moment of Caravaggio's birth. In 1570\u201371 Christian Cyprus, a strategically vital island fortress long controlled by the Venetians, fell into Ottoman hands. The garrison stationed at Famagusta, the last Christian stronghold in Cyprus, fought bravely before being forced to surrender. The survivors of the siege were cruelly massacred. Churches and cathedrals were converted into mosques, their stained glass smashed, their paintings and sculptures destroyed, their belltowers turned into minarets. Pope Pius V was appalled not only by the atrocity and its immediate consequences, but also by the possibility that the Ottomans might gain control over the principal trade routes of the Mediterranean. He joined forces with the Venetians, and together the allies sought additional support wherever they could find it. Missions were sent to Spain, to Portugal and to all the independent states of Italy. The princely families of southern Europe rallied together and thousands of soldiers were pressed into service. The result was no mere political alliance, but a self-styled Holy League for the defence of Christendom against the infidel.\n\nUnder the command of Don Juan of Austria, illegitimate brother of Philip II of Spain, a vast fleet of galleys \u2013 most of them constructed, in record time, within the great dockyard-cum-factory production-line that was Venice's Arsenale \u2013 set out to humble the Turkish navy. Eight days after Caravaggio's birth, on 7 October 1571, the two sides met in the Greek Gulf of Corinth, then known in the west as the Gulf of Lepanto. The result was the last great sea battle fought between galley-rowed ships. Both sides suffered heavy casualties. Eight thousand Christians died, and many more Turks. But, while the fleet of the Holy League survived the battle all but intact, the Ottoman fleet was destroyed and its commander-in-chief killed. One of the heroes of the battle was the commander of the papal forces, Marcantonio Colonna, father to Costanza Colonna, father-in-law to Francesco I Sforza, who had been witness at the wedding of Fermo Merisi and Lucia Aratori. After the victory, the pope declared that the Virgin Mary herself had interceded with God on behalf of the Holy League. Henceforward, the day of the victory was to be remembered as the Feast of Our Lady of Victory. Marian cults across Catholic Europe received a huge boost to their popularity. In Venice the day was declared a permanent _festum solemnis,_ to be marked every year by a procession led by the doge, and by celebratory masses. All across Italy, churches were built in honour of Santa Maria della Vittoria. Devotion to the Rosary reached a new pitch of intensity.\n\nThe Battle of Lepanto was a triumph to salve the wounds of a Christian world that had been sundered by the Reformation some half a century before. The Protestant king of faraway Scotland, James VI, was so carried away by the news that he wrote an epic poem to celebrate the great Catholic victory (though he felt compelled to add a prefatory disclaimer that Don Juan of Austria, hero of his verses, should still be regarded as 'a foreign papist bastard'). Meanwhile, Costanza Colonna's father, Marcantonio, made his triumphal entry into Rome. He rode into the city on a white horse, a modern-day Mark Antony stealing the glory of the caesars of old. But he also had the decorum to temper that show of pride with a spectacular display of humility. Having processed in triumph, he exchanged the armour of victory for rags and set forth on a pilgrimage to give thanks to Our Lady of Loreto.\n\nMichelangelo Merisi had been born on a day full of promise for zealous Christians, whose world was under threat. Archangel Michael had been the guardian angel of the Hebrew nation, and was associated with the protection of the faithful from harm. He had also been adopted, in Christian times, as the principal saint of the Church Militant. In depictions of the Last Judgement, he weighs the souls of the blessed and the damned, separating good from evil. In such paintings he is commonly shown wearing chain-mail and armed with a sword and shield, symbols of the archangel's ancient association with knights and crusades, and holy wars against the infidel.\n\nMichelangelo was a fitting Christian name for any child within the sphere of the Colonna family, defenders of the faith and warriors against heresy \u2013 but all the more so in the case of a child born not just on the saint's name day, but on the eve of a great battle between Christian and Muslim in which the head of the Colonna family himself would take a leading role. When victory at the Battle of Lepanto followed within just over a week of his birth, the hopes and prayers attendant on his baptism were answered. Perhaps he was thought of as a child who had brought good luck. Perhaps that was another reason why, despite his difficult personality and frequent lapses into criminal behaviour, Costanza Colonna would always stand by him.\n\n#### TOWN AND CITY\n\nThe artist's early life was divided between the town of Caravaggio and the city of Milan. The contrast between the two could hardly have been greater. Set in the fertile plains of Lombardy, Caravaggio was a quiet place, architecturally undistinguished, which had once been a Roman outpost. The activities of the town revolved around agriculture, which was vital to the booming prosperity of the region. Since the later Middle Ages the entire area had been intensively developed. Irrigation channels, networks of stream and canal that still criss-cross the fields today, had been systematically introduced. Better understanding of crop rotation had transformed the area into a prime producer of cereals. Large plantations of mulberry trees were grown as feed for silkworms, silk being the essential raw material for Milan's booming textile industry. The people of Caravaggio lived and worked by the rhythms of nature. They were known for their phlegmatic character, their solid business sense and their piety, the symbol of which was, from the 1580s onwards, the construction of the great shrine dedicated to Santa Maria della Fontana. Caravaggio was tranquil bordering on dull, a place where it felt as though nothing much had happened for a hundred years and more.\n\nMilan, the great city, two hours' ride away, had a population of 100,000, much the same as that of London or Paris at the time. Milan was noise and bustle, trade and industry, a populous and prosperous city \u2013 the place where Fermo Merisi, Caravaggio's father, went to work each day with his mason's tools of iron. It was a city known for the skill of its stone-workers and the ingenuity of its sword-makers. Milanese armour, Milanese swords and Milanese daggers were renowned as the finest in Italy. The men of the city were famous for their swordsmanship, a skill at which Caravaggio would come to excel.\n\nThe men of Milan were also known for their singular reluctance to marry. 'In Italy marryage is indeede a yoke, and that not easy, but so grevious, as brethren no where better agreeing, yet contend among themselves to be free from marryage.' Distrust of matrimony was common enough in sixteenth- and seventeenth-century Italy, especially among the upper classes, to have provoked many such comments from visitors. Italian humanists, including Petrarch and Leonbattista Alberti, had railed against marriage as a distraction to the intellect and a potential cause of economic ruin. Nowhere was the misogynistic cult of celibacy stronger than in Lombardy. It did not necessarily entail sexual abstinence, merely a refusal to be yoked to any single woman. The rate of celibacy among the Milanese aristocracy reached unprecedentedly high levels in the second half of the seventeenth century, so much so that it has been calculated that more than fifty per cent of all high-born males in the city never married at all. Caravaggio would never marry either, although it is impossible to establish whether this was another example of the painter imitating aristocratic _mores_ , or simply the result of his restless temperament.\n\nThe traveller Thomas Coryat visited Milan in 1608, by which time Caravaggio was long gone from the city. But the Englishman's vivid account, published in 1611 under the title _Coryat's Crudities_ , describes the metropolis much as it had been when Caravaggio was young. Coryat noted Milan's conspicuous opulence, and the many luxury trades that thrived there: 'No City of Italy is furnished with more manuary arts than this. Their embroderers are very singular workemen, who worke much in gold and silver. Their cutlers that make hilts are more exquisite in the art than any that I ever saw. Of these two trades there is a great multitude in the city: Also silkemen do abound here, which are esteemed so good that they are not inferiour to any of the Christian world.'\n\nHe grouped the city's armourers and sword-makers together with its embroiderers and silk-workers, perhaps implying that all were working in different branches of the Milanese fashion industry. The ability to fight was certainly just as important, to a young man out to impress, as the clothes that he wore. Swordsmanship was part of that intangible code of pseudo-chivalric skills and values encompassed by the Italian words _virt\u00f9_ and _nobilit\u00e0_ \u2013 although in Caravaggio's Italy it was never easy to tell whether a young man's aspirations to virtuous nobility were rooted in fact or fantasy.\n\nCoryat was also struck by the number of churches in Milan and impressed by the city's close links with some of the most dynamic figures of early Christianity. He visited the church of St Ambrose, where the relics of Ambrose himself, Bishop of Milan in the fourth century, were preserved. He seems not to have visited Santa Maria delle Grazie, the Dominican friary for which Leonardo da Vinci, more than a hundred years before, had painted his famous _Last Supper_ (another English traveller, Fynes Moryson, who visited Milan in 1618, noted that 'in this Monastery... in the place where the Friers eate, the supper of our lord is painted with wonderfull art'). But Coryat did pay a trip to Milan's cathedral, 'an exceeding glorious and beautifull Church, as fair if not fairer then the Cathedrall Church of Amiens', where he witnessed 'one of the nayles wherewith Christ was crucified, as they affirme'. He then climbed the cathedral tower to get a view of the whole city and the plains beyond, the little town of Caravaggio somewhere in their midst. As he did so, looking out and beyond the city's nine great gates, he encompassed the whole world of the artist's childhood:\n\n> There I observed the huge suburbs, which are as bigge as many a faire towne, and compassed about with ditches of water: there also I beheld a great part of Italy, together with the lofty Apennines; and they shewed me which way Rome, Venice, Naples, Florence, Genoa, Ravenna &c. lay. The territory of Lombardy, which I contemplated round about from this tower, was so pleasant an object to mine eyes, being replenished with such unspeakable variety of all things, both for profite and pleasure, that it seemeth to me to be the very Elysian fieldes, so much decantated and celebrated by the verses of Poets, or the Tempe or Paradise of the world. For it is the fairest plaine, extended about two hundred miles in length that ever I saw, or ever shall if I should travell over the whole habitable world: insomuch that I said to my selfe that this country was fitter to be an habitation for the immortall Gods than for mortall men.\n\nMilan was built on a circular plan. At the centre of the circle stood the massive, intimidating Castello Sforzesco. This daunting structure had originally been built as a palace for the mighty Sforza dynasty. It is an epitome of the Renaissance architecture of tyranny, with its dark and towering walls covered with diamond rustication, like the studs on a knuckleduster. When Milan came under Habsburg control in the 1530s, the patronage of the Sforza came to an end. Caravaggio knew the building as the fortress from which the city's Spanish governors nervously ruled, ever on the lookout for insurrectionists within and heretics without. Hostile, watchful suspicion was a frame of mind deeply ingrained in the Spanish rulers of Milan. They knew how important it was that they maintained their grip on the city. Whoever controlled Milan controlled the overland route from Italy to the rest of Europe. Milan would be on the front line in the event of any attempted invasion from the Protestant north, so it needed its heavy fortifications. In Caravaggio's day, the principal danger was thought to lie in France and, to a lesser degree, Switzerland. For all the animosity between the Milanese and their occupiers \u2013 caused to a large extent by taxes and grain levies on the local people to feed the Spanish troops \u2013 the religious and political interests of Catholic Spain and Catholic Italy were as one in the face of such perceived threats from the Protestant north.\n\nMilan had been a strategically important city since the days of the Roman empire, when Julius Caesar and Pompey had at different times made their residence there. Unlike the caesars of the past, Philip II of Spain did not go so far as to live in Milan. But he jealously guarded his power over the city. He had inherited both his crown and his empire from his father, Emperor Charles V. There is no more vivid document of the larger world of _realpolitik_ , of the patterns of religious and political division that fractured Europe in Caravaggio's time, than the long 'Instruction' that the ailing Charles V dictated, towards the end of his life, for the benefit of his son. He warned his heir to watch out for the dangers lying in wait throughout the vast range of his territories, from Spain to Naples, from the Netherlands all the way across to Germany and Austria. He gave particular emphasis to the importance of keeping a grip on Milan, a key military outpost at the crossroads of Europe:\n\n> Leave German affairs, as I now do, to my brother Ferdinand, but maintain contacts there, for your vigilance must be alert throughout all the possessions of our house. The most constant threat comes from France. Their kings have been and are bound to us by treaties, but remember that they are not true to their undertakings and only keep to their word when they are too poor to go to war... Keep a good guard on our northern borders with France, and maintain a fleet of galleys in the Mediterranean as a warning both to the Turks and to the French. We need to maintain good relations with Genoa because of its port, so take good care for this.\n> \n> In the north-east I have strengthened Flanders against France by my annexation of Guelders, Utrecht and Frisia. Still, you must keep money on hand there in case there is need for a sudden mobilisation; the inhabitants are reasonably loyal to us, but do not relax your watchfulness... I have settled the affairs of Savoy somewhat to the detriment of our ally the Duke, but do not help him to recover the lands occupied by the French even if they are his by right. That could give the French an excuse to press south again against our Milan and if that happens our links with Genoa and Florence and our rule in Naples and Sicily could all be put at risk.\n\nStill Charles V continued, spinning out a web of complex alliances and counter-alliances, seeking to pass on to his son his own, pragmatically paranoid brand of statecraft:\n\n> Further to Italy: do not trust the Pope, who neither honours his word nor has the general interests of Christianity at heart; keep an eye on any strengthening of the Duke of Ferrara's family relationships with the French; Venice is unlikely to form any close attachment to France, Florence is much indebted to our support of the Duke and is safe, but be watchful of Lucca and Siena. Above all, keep Milan and Naples well garrisoned with troops regularly paid to keep them loyal to us. As for the rest, remember that the Swiss covet part of our Franche-Comte; keep on good terms with England but, given the Pope's resentment against that country, very warily; with Scotland, you need have little to do.\n\nCharles V gave Philip II that advice in 1548. By the 1570s relations between Spain and the papacy had somewhat improved, but Europe remained the same fractious place described in the emperor's world-weary anatomy of the continent's political and religious divisions. And Milan, important enough to get two mentions in his long memorandum, remained vital to Spanish interests. Charles V always regarded the city as 'the key to Italy', and his son Philip II never deviated from that view. To lose Milan would not only expose the whole of Spanish rule in southern Italy to danger; it would also separate Spain from its territories in the Low Countries. The caution of Milan's Spanish rulers was exacerbated by their knowledge that the city's defence was in the hands of no more than around 5,000 soldiers. Any hint of trouble \u2013 the merest suggestion that the French were fomenting revolt in Genoa, the chance appearance of a group of gypsies from Venice \u2013 and a state of emergency was liable to be declared.\n\nOn the surface, the city where Caravaggio spent much of his youth was run as it had been during the era of Sforza rule. The Duchy of Milan might have become a vassal of Spain, but its bureaucratic apparatus remained unchanged and the same magistracies held the reins of power. The most significant difference was that the Consiglio Segreto, or 'secret council', that had once advised the Sforza dukes now reported to the Spanish governor. The Senate continued to exercise supreme judicial and administrative authority in the city, but was obliged to do so with a careful eye to Spanish interests. The members of the Senate were jurists drawn from the Milanese patriciate, men with a strong sense of Milanese legal traditions, whereas the governor of the city was one of the highest representatives of the Spanish sovereign, who was naturally disposed to act in accordance with Spain's larger strategic aims. Milanese politics was a balancing act, a fragile equilibrium of occupier and occupied.\n\nInstitutional continuity under Spanish rule was mirrored by a continuity of approach to the balance of secular and religious powers. The Sforza had pursued a gradually consistent policy of strengthening civil authority and weakening that of the Church. One of their main aims had been to establish control over ecclesiastical nominations in the Duchy of Milan, so that those whom they considered politically undesirable, or outright hostile, could be excluded from powerful positions such as that of bishop. Under Spanish rule, this strategy was pursued to the point where many other traditional powers of the Church were usurped by the state. Frequently, it was the civil, rather than the religious, authority that tried those accused of heresy, that took responsibility for discipline in the duchy's convents and monasteries, and that assumed the right to punish clerical abuses. This naturally reinforced Spanish power over all areas of life in Milan, but, though its aim was to limit ecclesiastical powers and privileges, it was never intended to weaken the Catholic faith itself.\n\nIn his instructions to his viceroys and governors, the fervently devout Philip II constantly stressed that the defence of Catholicism was his absolute priority. He had inherited a medieval Spanish conception of his role as monarch, according to which his first duty was _servicio de Dios_. He was brought up to believe that as king he had been singled out as the instrument of divine will. So, by a self-perpetuatingly circular logic, his policies were held to be those decreed by God and those best calculated to advance the holy mission of Catholicism. Spain's cause _was_ the cause of God; and this was true even if Spanish policies clashed directly with those of the supreme ecclesiastical authority, the pope. That was exactly what happened in Milan during the years immediately before and after the birth of Caravaggio. Other circumstances besides conspired to create a mood of incendiary religious fervour, often bordering on hysteria, in the city where the artist spent his formative years.\n\n#### CARLO BORROMEO\n\nThe dominant figure in Milan during Caravaggio's youth was not a Spaniard but an Italian. Carlo Borromeo was a dour and deeply pious man with a fierce sense of mission. He became Archbishop of Milan in 1565. He saw the city as the world itself in microcosm, a place teetering on the brink of damnation, teeming with sinners to be converted and souls to be saved. Like the ascetic Dominican friar Savonarola, who had preached in Florence almost a hundred years before, Borromeo galvanized the Milan of Caravaggio's childhood into regular paroxysms of mass repentance. His appearance, gaunt, hollow-cheeked, charismatically severe, was itself symbolic: a visible sign, like the rags adopted four centuries earlier by St Francis of Assisi, that Borromeo had renounced wealth and privilege to follow directly in the footsteps of Christ and his apostles.\n\nAlthough he would become one of the most radical reformers of the Catholic faith and way of life, he had first been pressed into the service of the Church by the forces of old-fashioned nepotism. His uncle, Pope Pius IV, appointed him to the position of his own private secretary and elevated him to the rank of cardinal when Borromeo was still barely in his twenties (and despite the fact that he had received no theological training). Yet he soon justified that favouritism. A skilled negotiator, he played a vital part at the end of the Council of Trent. This was the hugely significant nineteenth Ecumenical Council of the Roman Catholic Church \u2013 and was, in essence, the Catholic Church's concerted response to the multiple challenges to its authority posed by the Protestant Reformation.\n\nIt was at the Council of Trent that the Catholic Church reaffirmed the importance of the sacraments and the role of the priesthood; that it insisted on the importance of good works as well as of faith, in contradiction of Martin Luther's belief in 'justification by faith alone'; that it pronounced its own interpretation of the Bible final, branding any Christian with the temerity to substitute his or her own interpretations a heretic; and that it reaffirmed a multitude of Catholic practices that had been criticized by reformers in the north, such as pilgrimage and the veneration of saints and their relics. These were the basic principles that would underpin the Counter-Reformation, as it became known, the Catholic riposte to Protestant reformers. Yet such was the mood of contention surrounding the questions under debate, in which nothing less than the future of the Catholic Church was at stake, that there were many times when it seemed as though agreement might never be reached. First summoned in 1537, the council was concluded only in 1562\u20133. Carlo Borromeo was one of the men who saved it, at the last, from breaking down altogether.\n\nHe was a hard worker, who rarely slept for more than five hours and often went without food due to the permanent backlog of papal business requiring his attention. But to those who did not know about his punishing regime he may well have seemed like just another corrupt cardinal-nephew, the latest in a long line of such self-serving placemen. The pope appointed him to a dazzling array of positions, including the Protector of Portugal, of Lower Germany and of the seven Catholic cantons of Switzerland; as well as Protector of the Carmelites, Franciscans, Humiliati, Canons Regular of the Holy Cross at Coimbra and of the orders of St John and Christ in Portugal. He was also the absentee abbot of a number of monastic foundations. From these sources and his family estates he derived an annual income of around 50,000 scudi, a princely sum in mid sixteenth-century Italy. He was a keen and energetic huntsman who spent lavishly on his horses and hounds, and equally lavishly on his household, which he turned into a magnificent manifestation of his innate asceticism and sobriety: he kept one hundred and fifty servants and retainers, all dressed from head to toe in a uniform of funereal black velvet.\n\nBorromeo in his youth was a volatile combination of pride and piety, but it would take a personal tragedy to transform him into one of the most fervent and inventively radical priests of the Counter-Reformation. His elder brother, Federico Borromeo, died suddenly in 1562. Carlo, who was administrator of the pope's native diocese, the Archbishopric of Milan, was widely expected to give up his career in the Church, renounce his vows of piety and continue the family line by marrying in order to father a son and heir. Instead, he concluded that all man's earthly hopes and aspirations amounted to no more than a handful of dust. He gave up the trappings of wealth, sacked the majority of his household staff and forbade those remaining in his service to wear garments of silk or to indulge in any other luxuries. He took holy orders and briefly considered retreating from the world altogether, to a monastery. Eventually he decided that it was his role in the divine plan to revive and reform the Roman Catholic Church \u2013 and he set about the task with the evangelical zeal of a man convinced that he had God on his side.\n\nIt was only after the death of his brother that Carlo Borromeo's influence would really be felt in Milan. In 1565 he was consecrated Archbishop of Milan. He signalled his intentions by making his triumphal entry into the city wearing archbishop's robes, rather than dressed as a cardinal. It was his way of indicating that he came with his own sense of duty and purpose, not as the mere servant of papal Rome. He was determined to make the city and its provinces into the crucible for an extraordinary socio-religious experiment. Under his steely control and watchful gaze, the 900,000 souls of the Duchy of Milan were to be systematically indoctrinated in the ways of his own, deeply ascetic brand of piety. What he attempted was nothing less than a form of forced mass conversion to what he saw as the real and true tenets of Christian faith.\n\nThe archbishop had a darkly pessimistic view of human nature. He passionately opposed the doctrine of free will favoured by so many Protestants, and by some within his own church. To him, the idea that man had a God-given ability to choose between good and evil was a pernicious lie. He had a revealing disagreement with another prominent figure in the Counter-Reformation Catholic Church, the Bolognese theologian Gabriele Paleotti. Paleotti argued that 'Since God created human volition free and its own arbiter, it can be forced by no chains, but only sparked with the help of God's grace.' In Borromeo's bleaker view, human nature is 'already tainted by sin' \u2013 the Original Sin of Adam and Eve \u2013 and 'is by itself so inclined to evil that we easily neglect and forget to do good'. Borromeo's stern conclusion was that 'we need help and stimulants to live well, and always someone to remind us of it.' What that sentence portended, for the people of Milan, was a systematic attempt to change their way of life and transform their habits of thought. Borromeo saw himself as a spiritual successor to St Ambrose. Just as Ambrose had defied the Roman emperor, so Borromeo challenged the Spanish governors of Milan with the aim of asserting his own authority as the city's spiritual leader.\n\nOne of his first acts was to reassert the ancient right of the Archbishop of Milan to maintain a private army. Borromeo's so-called _famiglia armata_ , or 'armed family', which was a corps of armed men drawn from his own household, became a key weapon in his fight to reform what he regarded as the rotten state of the city. The archbishop claimed wide-ranging powers, so that those suspected of any offence that he judged to touch on public morality \u2013 such as heresy, blasphemy or sodomy \u2013 were liable to receive the not so tender attentions of his 'family'. He revived the defunct civil and criminal tribunals of the archiepiscopal curia, and reopened ancient prisons for the confinement of those found guilty in Milan's ecclesiastical courts. Borromeo's insistence on the unrestricted use of his _famiglia armata,_ and his extension of ecclesiastical authority into areas of life long regulated by the secular courts and justice system, led to numerous clashes with the city's Spanish rulers. At the climax of one particularly acrimonious row over jurisdiction, Borromeo went so far as to excommunicate the Spanish governor, the Marqu\u00e9s de Requesens. When Requesens retaliated, Borromeo himself was nearly exiled from the city.\n\nBorromeo saw to it that the Spanish Inquisition, which operated in Spain's offshore Italian possessions, Sardinia and Sicily, was excluded from Milan. He was able to clip and curtail Spanish power in a number of such ways, largely because he had strong support in Rome. He removed jurisdiction over alleged religious crimes from Milan's Spanish rulers, insisting that he himself should be the final judge in all such cases. But he also slowly won the grudging respect of the pious Philip II \u2013 to whom he explained, in a long and persuasive letter, that his aim was not to usurp Spanish power but to strengthen the Roman Catholic Church.\n\nFor two decades, throughout the formative period of Caravaggio's life, the archbishop pushed through a multitude of Church reforms intended to control the hearts, souls and minds of the people at large, measures that ranged from the introduction of new confessionals to the segregation of men and women in church. He made no secret of his distaste for self-styled elite clerical associations such as the Theatines or the Barnabites. He actually suppressed one of the orders nominally under his protection, the Humiliati, on the grounds that its membership was restricted to a corrupt, self-serving clique of aristocrats. He was promptly shot in the back at close range by a disgruntled Humiliatus, but survived the attack unharmed \u2013 an escape subsequently attributed, by his hagiographers, to divine intervention.\n\nBorromeo was an obsessive regulator and centralizer, and he did his best to turn his clergy into a spiritual equivalent of the _famiglia armata_ \u2013 a body of Christian soldiers animated by a single purpose and method. All priests with a pastoral duty were obliged to preach every Sunday and feast day. Reports were gathered from every parish in the diocese and underperforming priests were summoned to the archbishop's presence to practise their skills in front of him (they also had to leave a written copy of their sermon behind for his perusal). Not least because Milan was on the frontier with partly Protestant Switzerland, Borromeo was determined to turn his diocese into a shining demonstration of revitalized Roman Catholicism \u2013 a beacon to those who had erred, the brilliance of which might persuade them to mend their ways. He built new churches by the score and trained up an army of new priests to spread the word of God to their congregations. He founded diocesan seminaries and many schools. By the time of his death some 40,000 children in the diocese of Milan were being educated at any one time, an unprecedentedly high proportion of the juvenile population. Caravaggio's own family would be directly involved in the Borromean spiritual experiment: his only brother became a priest. No one in Milan and its surroundings was left untouched by the archbishop's plans for spiritual revival.\n\nBorromeo saw sinfulness everywhere and envisaged his priests as an army of spiritual stormtroopers taking the battle to the devil. No detail was too small to escape his eye, especially in the design of churches, which he saw as machines for the purification of an evil world. He wrote entire volumes of instructions about the minutiae of ecclesiastical architecture, the so-called _Instructiones_ , in which he pronounced on matters ranging from the precise amount of space church architects should allow for each member of the congregation ('one cubit and eight ounces square' exactly, four square feet in modern terms) to the appropriate scale and decoration of the entrances: 'the middle doorway must be distinguished by its width and ornamented with sculptures of lions... to represent the Temple of Solomon and the vigilance of bishops.' Borromeo was particularly concerned to ensure that men and women be separated from one another in church. He devised movable screens to be erected between groups of male and female worshippers, to prevent them exchanging glances with one another \u2013 often, in Borromeo's view, the first occasion for sin. He also sought to set strict controls on the clothes worn by worshippers, especially female worshippers, whom he berated for coming to church dressed for seduction, as if they were going to Carnival rather than participating in a holy ritual.\n\nThe mass of directives issued from the archiepiscopal palace of Milan must have occasionally wearied even the most conscientious of the Milanese clergymen whose task it was to enforce them. Here for example is the archbishop on the intricacies of the holy-water stoup, its placement, its design, its necessary accessories:\n\n> [The holy-water stoup] should not be put outside, but rather inside the church, accessible to those who enter and at their right hand, if possible. One font should be placed on the side where men enter and another... where the women enter. These [stoups] should not be near the wall but distant from it in proportion to the space that is there. They should be supported... on a small pillar, or some type of base on which nothing profane appears. There should be a sprinkler on a small metal chain hanging from the rim... it should not terminate with a sponge rather than bristles. It may terminate with a sponge only if it is enclosed in a silver, tin, or brass-perforated knob that has bristles on the outside.\n\nThe bristles about which Borromeo was so particular symbolized the cleansing branches of hyssop that purify the souls of the faithful in the ancient biblical psalm (Psalm 50): 'Sprinkle me with hyssop and I shall be cleansed; wash me and I shall be made whiter than snow.'\n\nMany years later, near the end of his life, when Caravaggio was a fugitive from justice in the Sicilian town of Messina, he was offered some holy water in a little church. The story is told by Francesco Susinno, a Messinese writer of artists' lives. It suggests that Caravaggio may still have carried with him some sardonic remembrance of the obsessive concern with the purification of souls that had coloured so much of his childhood and youth in Milan: 'One day he went into the church of the Madonna of Pilero with certain gentlemen, and the politest of them stepped forward to offer him some holy water. Caravaggio asked him what it was for, and was told 'to cancel venial sins'. 'Then it is no use,' he said. 'Because mine are all mortal.' That terse remark captures the darkest of the painter's moods \u2013 the sullen conviction that nothing, certainly no holy water, could ever wash _his_ soul clean or whiten the stain of _his_ sins.\n\n#### 'EGO TE ABSOLVO'\n\nFor Carlo Borromeo, confession was the Church's greatest weapon in the war on sin and evil. His highest priority was to regularize and control the administering of the sacrament of penance \u2013 which he believed could be used not only to mould the individual conscience but to redesign society. In Borromean Milan the hearing of confession was restricted to trained teams of diocesan confessors, who were allowed to operate only under direct licence from the archbishop himself. Each confessor was obliged to attend weekly classes to hone his confessional technique and receive the latest instructions from Borromeo. The archbishop told his confessors that they were even more important than the parish priests when it came to the saving of souls; he told them that they 'have the souls in their hands' and 'speak to Jerusalem's heart'.\n\nIn 1566 a new Roman Catechism had been composed, under Borromeo's supervision, in which the sacrament of penance had been described as 'the fortress of Christian virtue'. It had preserved the Roman Catholic Church from the attacks of the devil and his heretical minions, and it was to be considered responsible for 'whatever today's Church has preserved in holiness, piety and religiosity'. Borromeo went to great lengths to ensure that 'the fortress of virtue' remained pure of carnality or corruption. He popularized a new article of furniture for the administering of the sacrament, the confessional box, to create a physical separation between confessor and penitent \u2013 and thus avert any danger of unclean thoughts polluting their necessarily intimate relationship. It placed the confessor in his own kind of indoors fortress, making him invisible to the penitent and, it was hoped, immune to temptations and blandishments.\n\nThe archbishop's suspicious view of human nature extended to his own priests and confessors. In the late 1570s, when a woodworker named Rizzardo Taurini was commissioned to build five confessionals for the new Jesuit church of San Fedele in Milan, he provoked Borromeo's rage by fractionally curtailing one of the partitions at the bottom of the standard double sentry-box design. The Jesuit provost of San Fedele recalled the archbishop's outraged protest \u2013 'the confessor can easily touch the woman's feet with his own.' Borromeo repeated the objection several times, to the evident exasperation of the provost, who found the archbishop's insistence on the moral dangers inherent in a proximity between two people's feet more than faintly absurd. 'He greatly insisted on this,' the provost remembered, 'as if lust enters the body through one's shoes, and he is unaware that in his confessionals the woman's mouth is close to the confessor's ear.' The Jesuit knew a truth that Borromeo did not want to acknowledge: no matter how strong the grilles and walls of any confessional box, nothing could absolutely prevent priests and penitents from harbouring feelings for one another. The partitions intended to separate man from woman might even enhance the illict thrill of such emotions. This exchange between the worldly provost and the archbishop reveals the paranoid fear of sinfulness \u2013 and the corresponding desire to close off almost every avenue of human sensuality \u2013 that lay at the heart of Borromean piety.\n\nBorromeo believed that confession was nothing less than an instrument, given to him by God, to purify the world. The sacrament of penance already gave the confessor a fearsome weapon for the discipline of each soul \u2013 the power to grant or withold absolution. But Borromeo enhanced that power by putting checks in place to ensure that penance was true and not merely a matter of words and assurances. He insisted that confessors make enquiries about their penitents with their parish priests. Those priests in turn were instructed to tell confessors of any conditions that might disqualify a particular penitent from absolution \u2013 adultery, for example, or cohabitation outside wedlock. If absolution were not granted, the penitent would soon find himself or herself before the episcopal magistrates, and under the threat of imprisonment.\n\nBorromeo also ordered his confessors to interrogate their penitents for any knowledge they might have of heretics, or of anyone harbouring prohibited books. This cast the net wide, since such was the repressive cultural effect of the Counter-Reformation that the list of banned books \u2013 the Index \u2013 included many of the works now considered part of every Italian's intellectual heritage: Boccaccio's _Decameron_ , the poetry of Petrarch and Ariosto, the political theory of Machiavelli, to name just some. Anyone who made too public a display of owning any of those books was likely to find themselves given away to the authorities.\n\nThe situation in Milan was not unique, in that harsh measures against heresy were being taken in cities all across Italy, but it was extreme. For example, when the Roman Inquisition recommended that Catholic confessors should encourage their penitents to inform on heretics, Borromeo applied the condition with particular severity. In Milan any confessor too fastidious to enquire about heresy was summarily excommunicated; and if a penitent _did_ confess to knowledge of heretical activities, he or she was immediately sent to higher authorities to give further information about these enemies of the faith \u2013 to supply names and addresses, to give details of what they had done or might be planning to do. Only then might the penitent return to their confessor to hear the consoling words, ' _Ego te absolvo_.'\n\nBorromeo did not entirely succeed in his efforts to turn Milan into a model Tridentine police state. Even though he had the firm support of the pope and, eventually, Philip II of Spain, some of his attempts to redesign the Milanese way of life met with angry resistance. He tried in vain to ban dancing on feast days and Sundays, and in 1579 he even attempted to kill off the exuberantly joyful pre-Lenten tradition of Carnival. To the disgust of many Milanese citizens, he prohibited all jousts, tournaments, masquerades, plays and dances, and declared the automatic excommunication of all those participating in or attending such spectacles. Borromeo regarded Carnival as the devil's work, a dissolute rite lodged like a parasite at the beginning of the holy season. His prohibition was backed up by the threat that his confessors would exploit their information-gathering networks to identity those who had taken part in the celebrations. But this time he had gone too far. Borromeo's attempt to shut down the festive life of the city was met with panic by the civic authorities and rage by the people. Rome and Spain both had to intervene to prevent a popular uprising and Borromeo was forced, reluctantly, to acknowledge that there were limits to his power. In the end, the Milanese were content to fall short of spiritual perfection.\n\n#### THE POWER OF THE IMAGE\n\nFor all his inflexibility, Borromeo was an immensely charismatic and transformative individual. He changed his world, and has been rightly remembered as one of the most dynamic figures in the history of the Roman Catholic Church. In the words of Ludwig von Pastor, author of _The History of the Popes_ , 'he stands as a milestone in the history of the Church, at the boundary line between two epochs, the dying Renaissance and the triumphant Catholic reform.' As the first resident Archbishop of Milan for nearly a century, he cast a giant shadow over the city throughout the 1570s and early 1580s.\n\nThere is good reason to believe that the acts and ideas of Carlo Borromeo played a profound part in the formation of Caravaggio \u2013 an artist whose greatest gift would be an unprecedentedly stark and vivid naturalism, deeply attuned to the ideals of Counter-Reformation piety that permeated the city of his youth. Borromeo embodied more than just a particularly direct and messianic form of piety. His faith was rooted in an intense, spectacularly visual imagination. Borromeo's way of believing in Christ \u2013 which involved putting Christ at the centre of his life not in an abstract way, but as actually as possible \u2013 depended essentially on a process of mental projection identical to that required in painting pictures.\n\nIn early life Borromeo had been strongly influenced by the founder of the Jesuits, St Ignatius Loyola. He had read and followed Loyola's _Spiritual Exercises_ , a work that placed great emphasis on the role of visualization in Christian meditation. Loyola specifically advised his readers to visualize Christ's sufferings, insisting that the necessary prelude to any deep and serious meditation on Christ's Life and Passion was a mental process that he termed 'composition, seeing the place'. What that involved was, in effect, a kind of internalized version of the act of painting itself: 'In contemplation or meditation on visual things, as in contemplating Christ our Lord, who is visible, composition will be to see by the eye of the imagination a physical place where that thing is found which I wish to contemplate. By a physical place I mean, for example, a temple or mountain, where Jesus Christ, or Our Lady is found, according to that which I wish to contemplate.'\n\nThe Ignatian belief in the power of visualization carried within it the implication that if worshippers can see the image of Christ in their mind's eye, then they can empathize with his sufferings all the more fully \u2013 opening themselves to that emotional involvement which leads to the deeper forms of meditative experience. But the idea was not new to Loyola. It goes back to the Middle Ages, and finds especially powerful expression in the writings of the early followers of St Francis of Assisi. A good example is an early Franciscan tract entitled the _Little Book on the Meditation on the Passion of Christ Divided According to Seven Hours of the Day_ , which describes the exercise thus: 'It is necessary that when you concentrate on these things in your contemplation, you do so as if you were actually present at the very time when he suffered. And in grieving you should regard yourself as if you had our Lord suffering before your very eyes, and that he was present to receive your prayers.'\n\nA late thirteenth-century guide to prayer entitled _Meditations on the Life of Christ_ , probably written by a Franciscan friar from Tuscany, vividly demonstrates how such practices developed. The process involved ever more complicated and detailed visualizations, so that a succession of almost cinematic images would follow, one after another, in the mind:\n\n> reflect on the benignity of the Lord in having to sustain persecution so soon and in such a way... He was carried to Egypt by the very young and tender mother, and by the aged saintly Joseph, along wild roads, obscure, rocky and difficult, through woods and uninhabited places \u2013 a very long journey. It is said that couriers would take thirteen or fifteen days; for them it was perhaps two months or longer. They are also said to have gone by way of the desert, which the children of Israel traversed and in which they stayed forty years. How did they carry food with them? And where did they rest and stay the night? Very seldom did they find a house in that desert. Have pity on them, for it was a very difficult, great and long exertion for them as well as for the Child Jesus. Accompany them and help to carry the Child and serve them in every way you can... Here there comes a beautiful and pious, compassionate meditation... These and other things about the boy Jesus you can contemplate. I have given you the occasion and you can enlarge it and follow it as you please.\n\nThe rise of this form of popular devotion was closely linked to the development of painting, from the thirteenth to the fifteenth centuries and beyond. Throughout Western European Christendom, and especially in Italy and the Low Countries, artists competed with each other to create convincing illusions of actual presence, developing new techniques such as mathematically calculated perspective to paint ever more convincing images of the life and sufferings of Christ. Painters made their pictures as realistic as they could in order to assist worshippers in their own acts of mental picture-building. The common goal was to summon up the events described in the New Testament as vividly as possible, so that devout Christians might imagine themselves present as actors at the scene \u2013 mourning the dead Christ, for example, or helping to tend him as an infant on the Holy Family's flight to Egypt, as the author of _Meditations on the Life of Christ_ had written. Religious painting and religious meditation were, in fact, branches of the same activity.\n\nBut by the middle years of the sixteenth century the relationship between art and religious contemplation in Italy had become less straightforward. In the more sophisticated artists' circles, the idea of appealing to the popular devotional imagination with images of painstaking realism was regarded with disdain. Art became seen instead as an idealized, generalized language for the expression of higher thought. Michelangelo, the outstanding painter-sculptor of the High Renaissance, deliberately distanced himself from the pious naturalism of earlier religious painting, which he associated above all with the oil painting traditions of Flanders: 'They paint in Flanders,' he contemptuously remarked in the 1540s, 'only to deceive the external eye, things that gladden you and of which you cannot speak ill. Their painting is of stuffs, bricks and mortar, the grass of the fields, the shadows of trees, and bridges and rivers, which they call landscapes, and little figures here and there. And all this, though it may appear good to some eyes, is in truth done without _reason_ , without symmetry or proportion, without care in selecting or rejecting.' It was a form of painting, he concluded, fit only for 'young women, monks or nuns, or certain noble persons who have no ear for true harmony'.\n\nBut Michelangelo's subtle, poetically allusive and metaphorical ideal of art seemed, to many in the Roman Catholic Church, to be increasingly out of tune with the times. His use of the idealized nude figure was considered scandalous and his famous cycle of paintings for the Sistine Chapel ceiling was systematically censored in the late 1550s with the addition of a multitude of decorously placed fig leaves. Religious art was a highly controversial subject. Protestant reformers had attacked religious images altogether, on the grounds that they violated the Second Commandment ('Thou shalt not make to thyself any graven image'). The Catholic clerics who assembled at the Council of Trent had their own counter-argument, based on centuries of Church tradition. They resoundingly defended religious paintings and statues as divinely ordained tools for transmitting the messages of the Bible to the illiterate poor. But, at the same time, they acknowledged that many religious artists had forgotten their fundamental role to aid and assist devotion. It seemed clear to most of the leading formulators of Counter-Reformation policy that artists had become so caught up in abstruse ideas, so concerned to demonstrate their own ingenuity and originality, that they had forgotten the humility required of them as servants of the will of God. Not only was the Sistine Chapel ceiling censored, but the Venetian artist Paolo Veronese was publicly castigated for including all kinds of irrelevant details in a painting of _The Last Supper_. The Venetian Inquisition, which called Veronese to account for himself, was outraged by the presence in that picture of parrots, dwarfs, buffoons and, worst of all, Germans (regarded with detestation throughout Italy ever since Charles V's army led by Lutheran _Landsknechts_ had sacked Rome in 1527). The painter was forced to find an ingenious solution to the dilemma, which he did by changing both title and subject: Veronese's _Last Supper_ became instead a depiction of _Christ in the House of Levi_.\n\nSuch developments marked a great shift in attitudes. During the Renaissance religious artists had come to believe that, within fairly loose constraints of Christian orthodoxy, they were free to interpret and depict the stories of the Bible as they liked. As a result of the Counter-Reformation, the Roman Catholic Church set new and stringent limits on the presumed freedom of artists. The principal aim of this policy was to replace the Renaissance cult of freedom and originality with the ideals of artistic duty and responsibility. The second half of the sixteenth century witnessed a widespread call to order \u2013 a movement intended to take religious art back to the values of an earlier and supposedly purer time. Carlo Borromeo was at its forefront. As well as containing recommendations of every kind about church architecture and decoration, his _Instructiones_ set out his views on art with typical forthrightness. No animals or other distracting details should be included, unless actually mentioned in the biblical text that the artist had been instructed to illustrate. In the seventeenth chapter of his book, devoted to the correct representation of sacred events, Borromeo determined the appropriate fines and punishments for artists who failed to meet the strictest standards of decorum. In Milan, errant artists as well as heretics were liable to come to the attentions of the archbishop's _famiglia armata_. No painter could be in any doubt about what was required of him. Images should be clear and direct. It was the job of art simply to educate spectators and move them to penance.\n\nBorromeo's influence on art in his native Milan is well documented. Simone Peterzano, the feeble late Mannerist painter with whom Caravaggio would sign a contract of apprenticeship, developed a sparer and more austere style in direct response to Borromeo's pronouncements. The archbishop himself owned a collection of paintings that, to judge by its contents, he is likely to have used in his meditations. According to an inventory of 1618, these included an _Adoration of the Magi_ by Titian, an _Agony in the Garden_ by Antonio Campi and an _Annunciation to the Shepherds_ by Jacopo Bassano (all three paintings can be seen, today, in Milan's Pinacoteca Ambrosiana). Such works reflect his taste for the art of Venice and the Veneto, and his marked preference for small-scale devotional pictures. But the most intriguing aspect of Borromeo's taste, for the student of Caravaggio, is his implicit rejection of high art in favour of more traditional, popular visual representations aimed squarely at the promotion of mass piety. Within five years of becoming Archbishop of Milan, he had sold his entire personal collection of art and given the proceeds to charity. This was an act consistent with his personal asceticism, indicating that Borromeo shared the widespread belief \u2013 propounded by supporters and opponents of the Reformation alike \u2013 that money spent on 'dead' images of Christ, i.e. paintings, could be yet better spent on Christ's 'living' images, namely the real flesh-and-blood poor.\n\n#### THE SACRED MOUNTAIN\n\nBorromeo was not against religious art _per se_ , but he had forceful likes and dislikes. There was a powerfully retrospective cast to his thought. He believed that the best solution to the problems of the modern Catholic Church lay in a return to the past. As a corollary to that, he favoured popular spectacle over the intellectual abstractions of supposedly sophisticated 'High' Renaissance art. Long after he had sold his own paintings, Borromeo continued to sponsor and support particular forms of popular Christian visual spectacle \u2013 events and phenomena that were literally 'vulgar', in the sense of being aimed directly at the _vulgus_ , the crowd, the general mass of people. Borromeo himself staged numerous theatrical performances of his own extreme ideal of Christian faith. In times of trouble or pestilence for the city, he would march barefoot through the city of Milan with thousands of his supporters, all in sackcloth and ashes. Such processions might themselves be seen as a form of choreographed visual art.\n\nBorromeo's theatricality was another reflection of his belief in the value of constantly remembering and re-enacting the life of Christ \u2013 whether actually or in the mind's eye. It was deliberately unsophisticated, direct and immediate, and that was part of its point. Borromeo was intentionally attempting to revive the emotive methods of the itinerant medieval friars such as Francis of Asissi and his followers. The teachings of Francis had unleashed a flood of early Renaissance painting on the walls of churches throughout fourteenth- and fifteenth-century Italy, clear images bringing Christ's message to the poor. But Francis had also helped to found yet more popular and rabble-rousing forms of artistic expression \u2013 not only penitential processions of the kind imitated by Borromeo, the pious medieval equivalent of performance art, but also a very particular type of folkish _mise-en-sc\u00e8ne_ in which painted statues were arranged to conjure up events from the Bible. The first and most widely copied example of this was the crib that Francis created at the monastery of Greccio for the Christmas of 1223: a three-dimensional mock-up of the Nativity, complete with painted carvings of Mary, Joseph and the infant Christ, it was all done, in his words, 'to bring home to the people of Greccio what the birth of Christ at Bethlehem was like'.\n\nFrancis's innovation of celebrating the Nativity with the creation of a crib proliferated and mutated. Over the following centuries it produced other, far more elaborate traditions of folk art, including the so-called _sacro monte_ , or 'sacred mountain'. It was here that several of the most vital elements of popular piety \u2013 including the practice of empathetic visualization of the life of Christ, the ideal of religious meditation and a much enlarged version of the sculptural arrangement devised by St Francis in the crib \u2013 all came together in a single carefully orchestrated experience.\n\nThe earliest _sacro monte_ came into being at the end of the fifteenth century when a Franciscan friar named Bernardino Caimi decided to re-create the sites of Christ's life and passion \u2013 ranging from Bethlehem to Nazareth, from Gethsemane to Mount Sion \u2013 in the mountains above the town of Varallo, in what is today the Piedmont region. Caimi received papal permission and support for his plan, which involved the construction of numerous chapels linked by mountain paths. Each chapel was to contain polychrome figures acting out the stories of the Bible. Eventually a total of forty-five such chapels were built, allowing pilgrims who climbed up to them to travel even further in spirit \u2013 journeying all the way from Original Sin, where they would encounter Adam and Eve tempted by the serpent, to Golgotha, 'the place of the skull', where Christ was crucified. Carlo Borromeo spent the last days of his life on just such a pilgrimage, ascending the mountain at Varallo and praying day and night among its painted figures.\n\nSomewhat decayed, and much restored during the intervening centuries, these sculptures remain _in situ_ today. Some of the figures are carved; others are formed from terracotta or stuffed fabric. The effect is inconsistent but full of lively touches of naturalism, somewhere between sculpture and waxwork theatre. The chapel of the Massacre of the Innocents is particularly vivid and gruesome, with its goitred executioner and grieving mothers, its floor strewn with dismembered babies. The _sacro monte_ took the kind of interior, spiritual journey advocated for centuries in manuals of prayer and meditation, and turned it into an actual, physical itinerary, with suitably moving or horrifying scenes for the traveller moving up the mountain to witness at each new point of arrival. The sacred mountain gave a palpable form and structure to the instructions contained in devotional handbooks such as the fifteenth-century Venetian text _The Garden of Prayer_ \u2013 books that, like the Franciscan prayer manuals before them and Ignatius's _Spiritual Exercises_ afterwards, counselled the worshipper to summon up a chain of places and images as vividly as possible within the space of the mind:\n\n> The better to impress the story of the Passion on your mind, and to memorize each action of it more easily, it is helpful and necessary to fix the places and people in your mind: a city, for example, which will be the city of Jerusalem \u2013 taking for this purpose a city that is well known to you. In this city find the principal places in which all the episodes of the Passion would have taken place \u2013 for instance, a palace with the supper-room where Christ had the Last Supper with the Disciples, and the house of Anne, and that of Caiaphas, with the place where Jesus was taken in the night, and the room where he was brought before Caiaphas and mocked and beaten... And then too you must shape in your mind some people, people well known to you, to represent for you the people involved in the Passion \u2013 the person of Jesus himself, the Virgin, Saint Peter, Saint John the Evangelist, Saint Mary Magdalen, Anne, Caiaphas, Pilate, Judas and the others, every one of which you will fashion in your mind.\n\nThe sacred mountain was designed to ease the process of devotional visualization. The worshipper must make the physical effort of ascending from one chapel to another, but once inside each space he or she would find that the job of visualization had already been accomplished. The images at Varallo were begun by the artist Gaudenzio Ferrari in the late fifteenth century, but they were ultimately destined to be created, re-created and continually restored in a centuries-long collaborative process involving generations of sculptors, craftsmen and architects. What those images did was, precisely, to re-create scenes from the Bible as if enacted by 'people well known to you'.\n\nThe most skilfully carved and painted of the figures have a shocking actuality about them. This is not art that seeks to idealize or generalize life; it is art that aspires to the condition of a simulacrum of life itself. Collectively, the chapels of the _sacro monte_ exemplified an ancient, pious fairground form of realism \u2013 a type of art that has in general been overlooked or avoided by most art historians precisely because of its naked and self-conscious 'vulgarity'. Yet the art of the _sacro monte_ also had strong roots in traditions of high artistic realism going back to the start of the Renaissance. This was a tradition that had produced the sculpturally immediate, emotionally vivid and highly theatrical fourteenth-century paintings of Giotto \u2013 which themselves had strong links with certain forms of sacred drama, miracle plays and the like, promoted by the Franciscans and other orders of mendicant friars; as well as the startlingly lifelike fifteenth-century sculptures of Donatello, creations such as the _Mary Magdalen_ or the _Habakkuk_ , which struck his contemporaries as so eerily imbued with human presence that he was even suspected of necromancy.\n\nThis tradition of the work of art as, essentially, a speaking likeness intended to bring the Bible to life was displaced during the later Renaissance \u2013 or, at least, it was so transformed by the values and imperatives of the High Renaissance, of Michelangelo and Raphael and the Mannerists who came after them, that its original, uncanny effects were greatly diminished. Yet it continued to thrive away from the perceived centres of art such as Rome or Florence. In Emilia-Romagna and throughout Lombardy, unsettlingly realistic groups of figures were created by a school of sculptors working in the malleable and highly expressive medium of terracotta. Their art is still insufficiently appreciated, but a sculptor such as Guido Mazzoni from Modena, whose breathtakingly emotive works can still be seen in churches across northern Italy \u2013 and indeed as far south as Naples \u2013 deserves to be ranked alongside any of his better-known contemporaries. The traditions of the highly realized terracotta sculpture, and of the _sacro monte_ , played a crucial role in shaping the imagination of pious Italians in the fifteenth, sixteenth and seventeenth centuries.\n\nBoth traditions were also deeply influential on Caravaggio. Caravaggio's mature paintings, such as _The Crucifixion of St Peter_ and _The Conversion of St Paul_ , are blatantly rooted in the traditions of popular pious realism that produced the sculptures of the sacred mountain and the freestanding groups created by Mazzoni and other such masters. So clear and direct is the connection, so manifest the visual resemblance, that it might even be said that his principal strategy as a religious artist was to translate the effects of these two particular branches of theatrical sculpture into the painting of his time. The way in which he paints the wrinkled faces and bodies of his protagonists has its exact parallel in the wizened physiognomies conjured from clay by the masters of terracotta sculpture in Lombardy and Emilia-Romagna \u2013 so much so that some of the older faces in his painting might almost have been copied direct from sources in terracotta sculpture.\n\nCaravaggio's fondness for going into gruesome, visceral detail \u2013 his depiction, for example, of the gouts of blood that spurt from the decapitated tyrant's head in _Judith and Holofernes \u2013_ also vividly testifies to the affinity between his art and the rowdy, bloody, popular spectacle of much _sacro monte_ imagery. But even more telling is his constant habit of framing and composing scenes as though confined within a single, small, contained, theatrical space. There is very little landscape in Caravaggio, very little feel of the open air. The scenes he depicts are mostly to be imagined taking place indoors. He habitually collapses the immensity of the world to the confines of a room in which he can control the action and rigorously limit the cast of actors \u2013 a space analogous to the densely packed, theatrical spaces devised by the creators of the popular, pious, sculptural _mises-en-sc\u00e8ne_.\n\nTo say this is to deny neither Caravaggio's virtuosity nor his powers of invention. The way in which he adapted the conventions of popular sculpture to painting, the way in which he made them thoroughly _pictorial_ \u2013 above all through his use of light and shade \u2013 was so original that it gave painters nothing less than a whole new grammar and vocabulary. The very idea of looking _back_ , past the etiolated late Mannerism of his day, past the art of the High Renaissance, to vivid and robust traditions of popular religious sculpture \u2013 that too was a profoundly original move. It ran directly counter to the prevailing aesthetic orthodoxy of late Renaissance thought, as expressed by Giorgio Vasari in his _Lives of the Artists \u2013_ namely, the belief that art should continually evolve and progress, and that it was the duty of every artist to increase that forward momentum, whether he did so incrementally or through some great leap of innovation.\n\nThere was admittedly room for a degree of retrospection within Vasari's fundamentally teleological view of history. As he tells the tale of a great rebirth, or _Rinascita,_ in Italian art, Vasari allows that the earliest masters, such as Giotto or the Pisano family, had been obliged to look back to the art of classical antiquity to rekindle the painting and sculpture of their own day. In articulating that belief, he was adapting a commonplace of the poet Petrarch's thought \u2013 the idea that the classical past was a 'golden age' that could be recovered and eventually surpassed thanks to the efforts of humanist scholarship \u2013 and applying it to the discipline of art history. Because Vasari was the very first art historian, his notion of art as essentially progressive has proved peculiarly persistent. But the idea that an Italian artist born in the sixteenth century might have thought it worthwhile to look back past the art of Michelangelo and Raphael, not to the classical world but to the art of the earlier Renaissance, and even to the popular arts of the Middle Ages \u2013 that would have shocked and bewildered Vasari. He would have regarded such a preference, for the old and the popular over the new and sophisticated, absurd and perverse. Yet Caravaggio exhibited and proclaimed just such a preference. It might even be said that he was the first self-conscious primitivist in the entire history of post-classical Western art. The force, imagination and ingenuity with which he asserted his position made his art seem all the more iconoclastic and persuasive \u2013 compelling the admiration of subsequent painters as diverse as Rembrandt and Rubens, Ribera and Zurbar\u00e1n.\n\nHow was it that Caravaggio came to formulate his aggressively retrograde strategy? The most plausible solution would seem to lie in the painter's roots in Borromean Milan. Caravaggio's reinvention of devotional religious painting along the lines of a direct, theatrical, visceral, popular art would take place not in Milan but in Rome, and it would happen more than a decade after Borromeo's death. But it would represent, none the less, a strikingly faithful translation, into the field of art, of the imperatives of Carlo Borromeo's piety. In matters of piety Borromeo was a dramatist, a populist and \u2013 for all his innovations in church architecture and furnishing \u2013 a primitivist. The religious art that Caravaggio was destined to create could hardly have been more closely aligned to the beliefs and sensibilities of the charismatic Archbishop of Milan.\n\nThere are certainly elements of Caravaggio's religious painting of which Borromeo would not have approved. The full-breasted figure of the Virgin Mary in Caravaggio's _The Madonna of the Palafrenieri_ would doubtless have offended his sense of decorum. He would have been disturbed altogether by the painter's intense sensuality, by his feel for the flesh and blood of the human body and by his sensitivity to the suggestions implicit in the least exchange of glances. Yet even here, Borromeo may have exerted a subtle influence on the painter. Caravaggio paints with a strong and unmistakable sense of the perils and the powers of looking. His pictures both embody and evoke an acute and piercing gaze. Caravaggio sees what he sees with such intensity \u2013 even if it is only an image in his mind's eye, an image conjured from the imagination \u2013 that he makes seeing itself seem a compulsive and potentially fraught act. It is as if he feels at every moment that to see is also to possess and, potentially, to _be_ possessed. This is why Caravaggio's paintings have a destructive effect on pictures by other artists hung anywhere near them in art galleries. They exert such a sensually charged, magnetic attraction that they seem almost as though backlit, or somehow illuminated from within, while the pictures around them \u2013 even those of great artists, whether Rembrandt or Poussin or Vel\u00e0zquez \u2013 appear by comparison to recede, to retreat from the gaze.\n\nPablo Picasso was another artist whose pictures project such deep intensity of looking that they have an obliterating effect on other works of art. In his case, the phenomenon seems to have been linked to a powerfully distinctive way of seeing learned from the culture in which he grew up \u2013 the male-dominated and intensely Catholic society of late nineteenth-century Andalusia, where they even had a phrase for this kind of looking, the _mirada fuerte_ (which literally means 'strong gazing'). It has been succinctly characterized by the historian David Gilmore: 'When the Andalusian fixes a thing with a stare, he grasps it. His eyes are fingers holding and probing... the sexual element is present also... The light of the eyes is highly erotic... In a culture where the sexes are segregated to the point of mutual invisibility, the eye becomes the erogenous zone par excellence...' The explanation for Caravaggio's own intensity of looking may also lie in the distinctive milieu of his youth, the milieu of Borromean Milan \u2013 a place where, just as in Picasso's Andalusia, rigorous attempts were made to keep men and women apart.\n\nIf the dangers inherent in the sense of touch disturbed Borromeo, he was yet more disturbed by the opportunities for corruption furnished by the sense of sight. In the words of the _Methodus Confessionis_ \u2013 the sixteenth-century confessor's manual that Borromeo himself recommended to his Milanese clergy \u2013 sight was described as the most dangerous of all the senses precisely because it was superior to the rest and had the ability to 'incite man to many sins'. In a sermon delivered in the Lombard town of Lecco on 2 July 1583, Borromeo went even further. Reflecting on the murky biblical tale of the rape of Dinah (Genesis 34), he argued that the origins of all such sexual crimes lay within the sense of sight. Dinah was at fault for the rape, he decreed, because she had allowed herself to be seen and had underestimated what can happen when men are given sight of the flesh that they sinfully covet. The eyes, he pronounced, 'are like two gates to the castle of our body. So when they are in the Devil's control, he is also the master of our heart, and can introduce into our soul whatever he wants... consequently, since the eyes can introduce great mischief into the soul, they are to be guarded with the utmost diligence. For \"death has come up through our windows\" [Jeremiah 9:21]: therefore we have to keep them shut.'\n\nIt was remarkable advice to give to any congregation. To avoid sin, close your eyes. Take Borromeo's views one step further and the logical thing to do would be to inflict blindness systematically on the whole Christian community to keep it pure of lust and other evil thoughts. He was also of course perfectly capable of taking exactly the opposite line, especially when it came to images of the kind to be encountered on the sacred mountain at Varallo. There, Carlo Borromeo would counsel the faithful to open their eyes as wide as possible \u2013 as he did, in the days before his death \u2013 and drink in the spectacle of Christ's suffering. For all his contradictions, one thing is certain. At the centre of the paranoid and extremist edifice of Borromeo's religious thought there was a profound, superstitious belief that the sense of sight was the most direct route to the soul.\n\nThe evidence of his art suggests that Caravaggio was profoundly shaped by the insistent, manic ocularity of Borromean piety. He would surely have been a very different artist had he not been exposed to the very particular nexus of Milanese attitudes that linked seeing with guilt-ridden sensuality on the one hand, and salvation on the other. His pictures certainly speak of an intense sensitivity to every aspect of visual experience. So too do his notorious arguments and quarrels. Nearly all of the disagreements that would mark and mar Caravaggio's life would turn on a glance taken amiss, a perceived slight or insult implying a potential loss of face. When he looked at people, nothing missed his attention and sometimes his sensitivity may have fooled him into seeing things that really were not there. When people looked back at Caravaggio, they did so at their peril.\n\n#### PLAGUE\n\nIn the summer of 1576, when Caravaggio was almost five years old, the city of Milan was struck by an outbreak of bubonic plague. A census taken at the end of that year in the Milanese parish of Santa Maria della Passerella records the presence there of Fermo Merisi and his wife Lucia. Also listed were Fermo's daughter by his first marriage, Margherita, as well as Giovan Battista Merisi, Caravaggio's brother, four years old at the time. Mysteriously, there is no mention of Caravaggio himself, or of his two-year-old sister, Caterina, or of his still younger sibling, Giovan Pietro.\n\nIt is possible that by November or December they had already been evacuated from the city to the relative safety of the countryside, although it is not clear why they alone should have been sent away, leaving the equally vulnerable Giovan Battista at risk. Perhaps the censor missed their presence in the household; perhaps they were not there on the day that he came; perhaps Fermo and Lucia hurriedly managed to conceal at least some of their children when the censor visited, making it easier to evacuate them, unnoticed, at a later date. The movement of people and goods was strictly controlled in time of plague, and almost as soon as the contagion had become apparent, in August, Carlo Borromeo had issued edicts prohibiting anyone from leaving the city. At the end of October, as the disease appeared to abate, this quarantine was briefly lifted \u2013 although even then only a select group of wealthier families was allowed to leave. Is it possible that Caravaggio's parents took advantage of their contacts with the Colonna family to secure safe passage for their children, away from Milan, at that time? All that the historical record shows, for sure, is that the whole family had moved to their home in Caravaggio by the autumn of 1577 at the latest. Whatever the subsequent course of events, there is no reason to believe that the artist was not living with his parents in Milan when the outbreak struck in August of 1576. So he is likely to have witnessed much of the horror of the epidemic, especially during its early months, at first hand.\n\nThe symptoms of bubonic plague ( _yersinia pestis_ ) are grim and unmistakable. On infection, plague bacteria swiftly multiply in the sufferer's lymphatic system, affecting tonsils, adenoids, spleen and thymus. Within a day or two, the victim suffers fevers, chills and headaches. Vomiting and diarrhoea follow. But the most decisive sign of plague, the true mark of death, is the appearance of the so-called 'buboes' \u2013 swellings caused by internal bleeding that appear in the neck, groin and armpits, at the point of the lymph nodes, oozing both blood and pus. Damage then quickly spreads throughout the victim's underlying tissue, until the whole body is covered in dark, purplish blotches. The majority of sufferers die within about four days of contracting the disease.\n\nTo the young Caravaggio and his contemporaries, the plague was a visitation, a mysterious curse, like a torture from the bowels of hell inflicted on the living. What could be more appalling than this death by spontaneous internal putrefaction, this sudden consuming of the body from within? Bubonic plague was carried by rats and transmitted to human beings by fleas that jumped from those rats, but no one in Caravaggio's world was aware of that fact. The concept of infection was understood and so was the importance of quarantine, so that houses known to be plague-ridden were locked from the outside, their inhabitants forbidden to leave until the contagion had passed (when all those trapped within were usually found dead). But, although the precise cause of the disease was unknown, human intuition had groped towards an understanding that the illness might in some way be linked to hygiene. Public proclamations issued at regular intervals by Milan's health office in the late 1570s give no fewer than seven accounts of the true cause of the plague, and although they differ widely it is striking how many of them involve rumours about unclean bedding or clothes \u2013 which no doubt did, indeed, harbour the fleas that carried the disease. The plague was said to have begun in Venice, originating with Jews trading in used household goods; to have been brought from Mantua by a Jew who had come from that city to Milan selling \u2013 again \u2013 used household goods; to have been caused by a citizen of Mantua, a man carrying the disease who had spent the night in a Milanese inn and thereby infected the bedding on which he had slept; to have been transmitted by a dirty shirt, worn by a traveller and unwisely confiscated by an innkeeper in part settlement of his bill; and so on.\n\nAs well as carrying inklings of the truth about the actual mechanisms of infectious transmission, these stories vividly demonstrate the extent to which plague was liable to stir up a hornets' nest of prejudice. There was a long, ignoble history of such accusations. During the fourteenth century, when all of Europe suffered unprecedented mortality rates from outbreaks of the Black Death, as plague became known, the rumour was put about that Jews were deliberately spreading the disease. The idea of a _pestis manufacta_ , a plague-inducing substance secretly manufactured by the enemies of Christianity, took hold of the popular imagination. In the lower Rhineland and parts of Provence, many Jews suspected of such terrorist acts had been interrogated and tortured, and when confessions had been extorted from them entire Jewish communities had been systematically liquidated. There was no such response to the Milanese plague of the 1570s, but there was a recurrence of rumours about the role of _untori_ \u2013 unguent-spreaders \u2013 in creating the epidemic. According to a Jesuit eyewitness, Paolo Bisciola, 'it is said that there were certain men who went about touching the walls, gates, and streets with artificial unguents, which opinion many affirmed through the discovery one morning that almost all of the gates and _cadenazzi_ of the Corsa di Porta Nuova _,_ had been smeared, and the walls in various places had been soiled by unguents.' The presumed culprits, this time, were not the Jews but the Spanish. For a while, the people of Milan convinced themselves that their hated rulers were to blame for the evil that afflicted them. The Spanish governor of Milan felt compelled to pass legislation that forbade anyone from repeating the accusation \u2013 which only made the Milanese population even more jittery.\n\nIn truth, the Spanish government _was_ partly to blame, although not in the way so luridly imagined. The plague had actually been brought from Sicily to Milan by the entourage of Don Juan of Austria, illegitimate brother of Philip II of Spain and hero of Lepanto: the group had arrived in the city in August 1576, with several of its members already close to death. The inner circle of the Milanese Senate knew this, and so did members of the city's Health Tribunal. The plethora of alternative official explanations was a smokescreen created partly to protect the reputation of the Spanish royal family and preserve the _status quo_. The dirty laundry, the flea-infested bedding, really _was_ theirs; but this had to kept from the people at large or riots might ensue.\n\nAny risk of the truth coming out was dispelled when Carlo Borromeo involved himself in the situation. From his perspective, the plague was a God-given opportunity to force home his severe spiritual message; and, as far as he was concerned, there was absolutely no ambiguity about its source. It had nothing to do with bedlinen or shirts or phantom _untori_ going around wiping poison on gateposts. It certainly had nothing to do with the Spanish or the visit of Don Juan. Its source was human sin. The Milanese had neglected their souls, had confessed too infrequently, had debauched themselves at Carnival and had indulged in luxuries. The plague was being visited upon them by a vengeful God, and even if it seemed terrible it was truly a blessing in disguise \u2013 a call to universal repentance that could not be ignored.\n\nBorromeo was so prominent a figure in the events of the plague of 1576\u20138 that it would eventually be nicknamed after him \u2013 remembered forever as _la peste di San Carlo_. It brought out, in equal measure, his extreme piety and his fondness for the exercise of bureaucratic control. In the first two months of the epidemic, when as many as 10,000 people are thought to have died, the city almost fell into a state of anarchy. There was inadequate provision for the disposal of so many corpses and during 'the terrible September' of 1576 \u2013 the month of Caravaggio's fifth birthday \u2013 carts heaped high with bodies rolled along the cobbled streets of the town at all hours of the day and night. Piles of half-naked cadavers were left in open view. The much feared _monatti_ , or 'gravediggers', public health officials whose responsibility it was to collect the dead and purge houses of disease, were said to be running amok. Dark tales abounded of the _monatti_ pillaging the houses they were supposed to make safe, and raping the few female survivors they found there.\n\nFaced with a city descending into nightmare, Borromeo requested and duly received a brief from Pope Gregory XIII giving him full authority to redirect all the energies of his clergy to the alleviation of the plague. He mobilized his private army and summoned all the priests and monks of his diocese to a vast congregation. To each was assigned a different task. No more corpses were to be left outside. Every victim was to be given a proper burial, 'with crosses and lights'. Borromeo also organized partial quarantines, especially for women, whom he regarded not only as more likely to occasion sin but as the primary carriers of plague (because, he said, they talked so much and constantly visited each other's houses). On his orders, many of the women of the city were confined within their homes for long periods of time or held in purpose-built isolation cabins. Such measures were not always entirely effective: because women continued to live in terror of the _monatti_ , they often failed to disclose the presence of plague or the existence of the dead, with the result that the disease continued to spread, albeit behind closed doors.\n\nBorromeo's most successful strategy against the disease was the reopening of Milan's _lazzaretto_ , one of the first great penitentiaries for the plague built in Italy. This was a vast moated structure that had originally been constructed during the late fifteenth century on the orders of Francesco Sforza, following an outbreak of plague in 1483\u20135. The _lazzaretto_ \u2013 so named after Lazarus, who had been raised from the dead by Jesus and who was often portrayed in art as a plague victim, spotted with buboes \u2013 had been left empty for more than fifty years. But it served Borromeo's purposes well. Into its compound of 288 rooms were squeezed nearly all of Milan's homeless and destitute (a nearby monastery was also conscripted to accommodate any overspill, confining the spread of the disease to great effect). In its regimented spaces, those who remained healthy could be segregated from those who fell ill. The bodies of the deceased were kept in plague pits at the centre of the compound, to be taken away at regular intervals for mass burial. Death was being bureaucratically managed, to Borromeo's satisfaction. Most important of all, the whole itinerant population of the Milanese diocese was now being held in one place.\n\nThe man whom Borromeo installed in charge of the _lazzaretto_ was a Franciscan friar named Fra Paolo Bellintano, who later published a detailed account of his methods. Under his stewardship, the _lazzaretto_ became a kind of centralized fortress for the management of plague. He employed a team of _sbirri_ \u2013 policemen, or constables, recruited like himself from within the Franciscan order \u2013 to maintain discipline within the _lazzaretto_ itself, and enforce their own standards on those working with plague victims in the city at large. The _sbirri_ were, he wrote, an essential part of his tactical plan:\n\n> Many needs arose during the day which could not be remedied without these. I dare to say that without them Milan would have become a den of thieves. How would I have been able to hold back so many _monatti_ who cleaned out the houses without fear of justice? Decrees and proclamations could have been made without end, and they would not have feared a thing. But they saw that almost every day I had people whipped, birched, imprisoned, scourged, tied to a column, and that I imposed other punishments besides. And they did not want to become familiar with all this [i.e., suffer the same fate themselves].\n\nAs a result, he noted with tight-lipped satisfaction, the behaviour of the _monatti_ was soon much improved.\n\nBellintano's account of the methods applied within the _lazzaretto_ graphically demonstrates how the terrors of the plague were used to scourge the collective Milanese soul. The people were being punished for their attachment to the pleasures of Carnival, yet Bellintano himself takes a positively carnivalesque relish in the ways that he and his _sbirri_ devised to ensure their penitence. He tells the story of how, one night, the inmates put on a secret dance to raise their spirits. One of his fellow Franciscans, Fra Andrea, got wind of the party and determined to puncture the festive mood. He went to the plague pit in the middle of the _lazzaretto_ and retrieved the bloated corpse of an old woman. As he heaved her on to his shoulders, a great belch of air was expelled from her swollen guts. Unperturbed, he told her to keep quiet and get ready for a dance. He went to the room where the inmates were dancing and asked to join their party. When they opened the door, he threw the dead body into their midst, shouting out 'Let her dance too!' There then followed a brief sermon, after which, Bellintano drily observes, 'the dance ended.'\n\nPublic counterparts to such dark, private theatricals were the processions organized by Carlo Borromeo. Believing that the only way to salvation was passionate identification with the suffering body of the Lord, he staged a series of re-enactments of Christ's journey to Calvary. In October 1576 he announced three days of fasting and ordered that Milan's most prized relic, a nail said to come from the True Cross on which Christ had been crucified, be taken out of the Cathedral:\n\n> His holiness performed the three processions dressed _di mestitia_ , with a large rope around his neck, barefoot and hooded, dragging his clothes on the ground, and with a large Crucifix in his arms. And on the Sabbath he carried the Holy Nail in procession, supplicating God by the merits of His Most Holy Passion to turn away the ire he had conceived against this people, and use them with mercy. He went in the same habit, and manner, as the previous days, but was also accompanied by about a thousand flagellants, who beat themselves continuously, causing great pity in whoever saw them. All the portable relics of the city were also carried in procession that day. But that which most moved the people to tears, penitence and dolour was [the sight of] the illustrious Cardinal in such sad and mournful dress, that great black cross on which he carried the Holy Nail, the blood that was seen to issue from his feet. When the procession was finished he preached a public sermon almost three hours long, with such spirit and fervour that he was like another St Paul. I believe there were few who did not weep. When the days of prayer were over he carried it all through the city once more, especially in those places where the disease was worst. On this journey he spilt much blood from his feet, and was accompanied by barefoot priests and monks, with ropes around their necks.\n\nThe author of this eyewitness description, the Jesuit Paolo Bisciola, describes how huge crowds were exhorted to join in these mass demonstrations of faith. He also notes, without apparent irony, that 'on these occasions, the plague grew very much.' Bisciola's account is also interesting for its visual detail. He says that Borromeo ordered temporary altars, lit with candles, to be set up throughout the streets of the city, 'so that to walk in the streets was like walking in church'. As autumn advanced, and as the nights drew in, the city seemed ablaze with 'the lights of piety and religion'. On a multitude of outdoor altars 'there burned a great quantity of candles and much incense'. Flame and shadow: Milan had become a city of chiaroscuro.\n\nThe artist was five when the plague reached its height, and only a year older when it reached into his life and tore his family to pieces. The sequence of events is charted in a series of documents from the archives. On 20 October 1577 the death was recorded, in Caravaggio, of the artist's father, Fermo Merisi, his paternal grandfather, Bernardino Merisi, and his (unnamed) grandmother. The document states that they died within a day of each other. An earlier document, of 17 August 1577, indicates that Caravaggio's uncle Pietro had died earlier in the year. The document in question is a claim to Pietro's estate, which indicates that he died without having made a will, that is, unexpectedly. It also provides the only explicit link to the plague, stating that Pietro lived in Milan but was in Caravaggio because of the epidemic sweeping the city.\n\nThe archive contains no description of the events that took place in the Merisi household in late October 1577. It was not a notary's job to paint pictures of human suffering. But the art of Caravaggio's maturity would be saturated in the ineradicable memory of night terrors. It would be an art of paroxysm and abandonment, filled with images of turmoil in dark places. Towards the end of his life, working in Sicily \u2013 the place where Milan's great plague had started \u2013 he would paint a huge altarpiece of the _The Resurrection of Lazarus_. He himself chose the subject of the picture. It would be a meditation on death and salvation \u2013 a work that, though shaded by ambiguity, has a miraculous story to tell. But nothing could change the story of Caravaggio's own early years. No miracle had raised his father, his grandfather, his uncle, from the dead. By the age of six, Caravaggio had lost almost every male member of his family.\n\n#### THE BAD APPRENTICE\n\nBy 18 February 1578 the plague finally abated. A fifth of the population of the diocese of Milan was dead and everyone else was trying to reassemble their lives. On that day Caravaggio's mother, Lucia, signed a document in which she assumed legal guardianship of all four of her children. It shows that the family was now resident in Caravaggio.\n\nNearly a year later, on 21 January 1579, another document shows that Fermo Merisi, her late husband, had died intestate. It apportions his estate and parts of his parents' estate too. They had also died intestate, on the same night as him, creating a tangle of legal complications. There had evidently been some dispute between Lucia and Fermo's three half-brothers \u2013 a quarrel about who was to get what. Arbitration was necessary. The largest property at issue, Bernardino's house and land in Porta Seriola (together with his business premises), went to the half-brothers. In exchange Lucia and her children were relieved of Fermo's debts, which amounted to 1,737 lire. They were also granted four modest landholdings, which the document goes on to list:\n\n  1. 7 _pertiche_ of land in Canigio Nuovo (which is divided, with three quarters going to Fermo's brothers, the rest to Lucia's children);\n  2. land and a vineyard in Rovere, 8 _pertiche_ ;\n  3. 2\u00bd _pertiche_ of land on the road to Calenzano;\n  4. an orchard outside Porta Prato, 1 _pertica_.\n\nOne _pertica_ , in Lombardy at that time, was the equivalent of approximately 6,500 square feet. So altogether the Merisi family, Fermo's wife and her four children ended up with roughly 18.5 _pertiche_. That is 120,250 square feet, just under three acres. The land's value amounted to about 3,000 imperial lire, or 500 gold scudi: not a lot of money.\n\nNothing is known about Caravaggio's early education but he certainly received one. An inventory of his possessions made several years afterwards, when he was a fully fledged painter working in Rome, reveals that he owned several books. The pictures that he would paint, later in life, are without doubt the works of a questing, curious and literate mind. His brother Giovan Battista, who was destined for the Church, would later study at the Jesuit Collegio Romano in Rome. The Jesuits were among the most intellectually demanding of the religious orders, so Caravaggio's brother must have been given at least the rudiments of an education in classical and Italian literature. Numerous grammar schools had been set up across the diocese of Milan at the instigation of Carlo Borromeo, who believed that educated souls were less likely to stray into temptation. Caravaggio's brother probably attended one such school, which makes it more likely that Caravaggio did too.\n\nBy 1583 Giovan Battista Merisi had decided that he was destined for the Church. He was following in the footsteps of his father's brother, Ludovico, who was a priest. By 1584, it seems, Caravaggio had decided to become a painter. On 6 April of that year, at the age of thirteen, he signed a contract of apprenticeship with Simone Peterzano. The contract was signed in Milan, where Peterzano had his workshop, and it spelled out the nature of Caravaggio's commitment to his _dominus_ , or 'Master', and described what he was to expect in return:\n\n> The said Michelangelo will stay and live with the said Master, Simone, to learn the art of painting for the next four years beginning from today, and that the said Michelangelo will train in that art night and day, according to the custom of the said art, well and faithfully, and that he will commit no deceit or fraud upon the goods of the said Master, Simone.\n> \n> The said Master, Simone, is required and obliged to support the said Michelangelo in his house and workshop, and instruct him in that art all that he can, so that at the end of the four years he will be qualified and expert in the said art, and know how to work for himself. The said Michelangelo is required to give and pay to the said Master, Simone, for his recompense, twenty-four gold scudi at the rate of six imperial lire to the scudo, to be paid in advance every six months by the said Michelangelo to the said Master, Simone, of which he now receives ten scudi in advance payment, of which Michelangelo promises to pay the remainder.\n\nThese were not exactly standard terms. Caravaggio and his family had to pay Peterzano 24 gold scudi each year of the apprenticeship, six months in advance \u2013 a total of 96 scudi. Payment for apprenticeships was not an invariable part of such contracts, in that the apprentice's labour was regarded as recompense to the Master for his tuition. When the Master also provided board and lodging, as in Caravaggio's case, some payment from the apprentice was customary, but Peterzano's fee on this occasion was unusually high. For example, when the painter Gerolamo Lomazzo had been apprenticed in Milan in 1556, he had been required to pay just 8 gold scudi a year. Peterzano's only other known apprentice, Francesco Alicati, was actually _paid_ 24 scudi a year for his contributions in the workshop. The implication is that Alicati already had some skills in painting, whereas Caravaggio had none.\n\nSimone Peterzano was an eclectic and mediocre artist who was originally from Bergamo but preferred to stress his links with Venice, where he may have been trained. He claimed to be a disciple of Titian, the most celebrated painter of Renaissance Venice, and sometimes even signed his pictures _titiani alumnus_ , 'pupil of Titian'. A number of contemporary sources refer to him as Simone Veneziano. The most extensive surviving example of his art is to be found in the presbytery of the Certosa di Garegnano, north-west of Milan. There, he and his workshop painted a monumental fresco cycle depicting scenes from the life of Christ. Work was begun in 1578 and finished in 1582, so the resulting pictures are a reasonable guide to Peterzano's style as it was when he took Caravaggio on as his apprentice just two years later. It is a flaccid, bloodless late variant of Mannerism, exemplified by _The Adoration of the Shepherds_ at Garegnano \u2013 an exercise in saccharine piety, complete with a cast of lumpen shepherds whose decorously draped forms, in various postures, were perhaps meant to demonstrate virtuosity but only reveal Peterzano's inadequacies as a painter of the human anatomy. At the centre of the picture a sober and dignified Joseph, the sole convincing figure, is joined by a slack-jawed, pinheaded Mary. Both kneel in adoration of a mannequin baby Jesus, while unconvincing angels circle overhead.\n\nWhat Peterzano's fresco cycle communicates more vividly than anything else is his determination not to cause offence. His pictures embody the Tridentine timidity that infected so much Italian painting in the years that immediately followed the Counter-Reformation. Before he had begun work on the Garegnano fresco cycle, the artist had been made to sign a contract obliging him to follow the new rules of decorum laid down by the Council of Trent: 'All the human figures, and above all the saints, should be executed with the greatest honesty and gravity, and there should not appear torsos, nor other limbs or parts of the body, and every action, gesture, clothes, attitude and drapery of the saints should be most honest, modest and full of divine gravity and majesty.' Peterzano was careful to follow these instructions \u2013 all the more careful, no doubt, because Carlo Borromeo himself was known to visit the charterhouse at Garegnano to practise the spiritual exercises. It might be said that he painted according to the negative principles of Borromean piety, in the sense that his overriding priority was to avoid courting controversy or violating decorum. It would be Caravaggio's genius to express that same piety in boldly positive terms, to create an art of agonized humility and bleeding flesh that would stir up controversy wherever it was seen. In short, there is no trace of a debt to Peterzano's work in the art of Caravaggio's maturity. Were it not for the existence of the actual contract of apprenticeship, there would be no reason whatever to connect the two men.\n\nSo what did Caravaggio learn during his apprenticeship? It might be supposed \u2013 it is the conventional view \u2013 that he received a traditional grounding in the techniques of Renaissance painting. In other words, he learned to prepare and grind colours; he learned how to draw; and he learned how to paint in _buon fresco_ , the 'true fresco' technique, like Peterzano himself. But Caravaggio never painted a fresco and no single drawing exists by his hand. X-rays of his oil paintings show that he did not even use preparatory drawings on the canvas, as a guide for the brush. In other words, there is almost no resemblance between his daringly improvisatory techniques and those that would have been taught in the studio of an artist such as the safe, dull and cautious Simone Peterzano.\n\nIt seems that something must have gone awry during Caravaggio's apprenticeship. He was a painter of extraordinary innate talent, a unique virtuoso when it came to conjuring the illusion of three-dimensional reality within the two dimensions of painting. Yet his earliest known works, while forceful, are relatively gauche and crude. Those pictures were done after 1592 and they were done in Rome. If someone with his gifts really _had_ applied himself to the study of art in Milan for four whole years from 1584 to 1588 \u2013 working 'day and night', as the contract says \u2013 he should have been far better than he actually was by then. The breakneck pace of Caravaggio's subsequent acceleration, from uncertain beginnings to full-blown mastery, begs further questions. Was it perhaps only in the early 1590s that he first took painting seriously? Is it possible that he began his career with the merest smattering of an education, and taught himself most of what he knew about painting on the job? Could it be that he spent much of his presumed apprenticeship playing truant?\n\nThe hypothesis has the virtue of helping to explain Caravaggio's extreme technical originality. It is easier, in some ways, for a man to reinvent painting if he has almost nothing in the way of conventionally ingrained techniques to impede him. His contemporaries described him as a difficult young man who liked to settle disagreements with violence and who was prone to disappear for days on end. There is no reason to believe that he was anything but an unruly teenager. Even if he did absorb some of the rudiments of art, he is unlikely to have been a model student. What evidence we have suggests that he was probably a very bad one.\n\n#### 'THEY COMMITTED A MURDER'\n\nThe bare bones of the archive \u2013 and they are pretty bare, for this part of the painter's story \u2013 indicate that these were difficult years for the whole Merisi clan. On 25 August 1584 the richest and most influential member of the family, Giovan Giacomo Aratori, went to his grave. On 7 June 1588 Caravaggio's youngest brother, Giovan Giacomo, died of unknown causes. By this time the painter's apprenticeship, such as it was, had finished. He was back in Caravaggio by 25 September 1589, to sell a parcel of land. He was not quite eighteen, so the sale could take place only with his mother's permission, which she gave. There were more land sales the following year. On 30 May and 20 June 1590 Caravaggio and his brother, the future priest Giovan Battista, parted with all of their remaining property in Canigio Nuovo, 'to clear debts [accumulated] by them or their mother, or by the said Michelangelo by entering into a contract'. The brothers had no option, it seems, but to eat into their rapidly dwindling capital.\n\nThere is a hint of trouble too in the fact that legal responsibility for this sale had been suddenly passed from the brothers' mother, Lucia, to their uncle, the priest Ludovico Merisi, who was their next closest relative. Was Lucia ill or incapacitated in some way? It seems so. On 29 October 1590 she made her will, bequeathing her entire property in equal proportions to her three surviving children. Exactly a month after that she died.\n\nThere were two more sales and then \u2013 on 11 May 1592 \u2013 the final division of Lucia's estate between Caravaggio and his two siblings. Giovan Battista got some land and the family's two houses in Porta Folceria. Caterina got some land as well as an undertaking from Giovan Battista that he would pay her dowry of 200 lire. Caravaggio was excused from any obligation to either of them and took nothing except the cash from one last land sale. It looks like the behaviour of a man who wanted to cut all ties with his past. Not long after the division of the property, he would leave Caravaggio and Milan, never to return.\n\nBy the middle of 1592 he had raised altogether 1,957 imperial lire from the family's capital \u2013 the equivalent of 600 gold scudi, or about six times the cost of his apprenticeship. By the end of the same year, he would have run through it all. No one knows what he did with the money, just as no one knows exactly what he was doing with his life during and after his apprenticeship. He was twenty-one years old in 1592. By the same age his namesake, Michelangelo Buonnarotti, had established himself as one of the leading artists in Italy. Yet, as far as anyone has been able to establish, Michelangelo Merisi, soon to be known as Caravaggio, had not even painted a picture.\n\nAll this suggests he was not so much a slow developer as a reluctant one. Perhaps he did not even want to be an artist. Perhaps he explored other possibilities, such as becoming a mercenary or soldier of fortune. He was good with a sword, and the alacrity with which he would later jump at the chance to become a Knight of Malta suggests he may always have nurtured romantic fantasies about becoming a knight at arms. Another distinct possibility is that he had got into bad company and was just living it up during these years, with no thought to the future \u2013 until the money finally ran out. Children often define themselves in opposition to one another, and the fact that Caravaggio's brother was chosen for the priesthood is in itself suggestive. If Giovan Battista was the good little boy, maybe Caravaggio had taken the role of the rogue. It would not have been surprising. He had grown up with barely a single close male role model. In fact almost all the men closest to him \u2013 the men who might have controlled him, helped him, shown him how to live \u2013 had died of the plague.\n\nThere was no shortage of opportunities for getting into trouble in Milan. Carlo Borromeo was not just flourishing his priestly rhetoric when he called it a city of sin. Milan had a reputation as a violent place, infested with vagabonds, conmen, pimps and whores. Street crime was rife and the murder rate soared during the 1580s and 1590s. The Spanish governor was constantly issuing proclamations about the need to clean up the city and offering rewards for the capture of bandits, muggers and murderers. There are passages in the writings of Caravaggio's biographers which suggest that he got involved \u2013 and got out of his depth \u2013 in this dangerous Milanese underworld. Bellori baldly states that 'being disturbed and contentious, because of certain quarrels he fled from Milan.' But that is not the last word from him on the matter, because on the front page of his copy of Baglione's biography of Caravaggio (still preserved in the Vatican Library), he wrote a further note, just as bald but more informative: 'he ground colours in Milan and learned to colour and because he had killed one of his companions he fled the country.'\n\nBaglione, who seems to have known nothing about Caravaggio's life in Milan, is silent on the subject. Mancini, in _his_ life, was keen to tell the version of the story that Caravaggio himself wanted the world to believe, sweeping any suggestions of ill-doing under the carpet of a single brisk sentence: 'At a young age he studied diligently for four to six years in Milan, though now and then he would do some outrageous thing because of his hot nature and high spirits.'\n\nMancini too left some scribbled, marginal mutterings that throw more light into this dark corner. There is a manuscript copy of Mancini's life of Caravaggio in the Marciana Library in Venice that contains a number of barely legible lines of disconnected prose: 'They committed a murder. Prostitute tough guy gentleman. Tough guy hurts gentleman prostitute slashes insult into the skin with knife. Policeman killed. They wanted to know what the accomplices... He was in prison for a year and then he wanted to see his property sold. In prison he didn't confess he came to Rome and said no more about it.'\n\nIt is with this gnomic, fragmented record \u2013 this mangled account of mysterious skulduggery and impenetrable misdeeds \u2013 that Caravaggio's life in Milan comes to a close.\n\n## PART TWO  \nRome, 1592\u20135\n\n#### VIOLENT TIMES\n\n> 'Whore, bitch, tart! I throw a bowl of shit in your face! Go on, fuck yourself with a horsewhip! I'll stick the handle of my paintbrush up your arse!'\n\nThese are the words of an artist scorned, addressed to a courtesan who refused to sleep with him. They are preserved in a deposition in the State Archives of Rome for 1602. The man was before the magistrates for abuse and physical assault. As well as insulting and beating her, he had actually knifed the woman. She had been badly injured, cut deeply to the face. The facial wound was an example of a _sfregio_ , a slash with the blade inflicted as a mark of shame \u2013 doubly damaging to a courtesan, whose face was her fortune.\n\nThere are many such tales in the annals of the lives of the artists who thrived, floundered or failed in Counter-Reformation Rome. Here is another example.\n\nAn artist catches his mistress in the company of his own younger brother, an assistant in his workshop. He pursues his brother to St Peter's, where they are busy on a commission, and breaks two of his ribs with a crowbar. He then tries to kill him with his sword, but the brother escapes and seeks sanctuary in a church. Meanwhile, the artist sends his servant to the house of his offending mistress, with instructions to give her a _sfregio_. He finds her in bed and slashes her face with a razor.\n\nNeither of these stories directly concerns Caravaggio. The first is about a now forgotten painter whose misdemeanours took place at the start of the seventeenth century. The second involves the flamboyant sculptor Gianlorenzo Bernini, who caught his brother _in flagrante_ with his mistress in 1638. More than thirty years and a gulf of talent separated the two artists but they behaved in an identically hot-headed way. Both men acted in the heat of the moment, spurred on by a slight to their honour \u2013 a loss of face punished, with terrible literalness, by cuts and slashes to the actual faces of their victims.\n\nDuring his own fourteen years in Rome, Caravaggio would become embroiled in more than his fair share of assaults, disputes and bloody vendettas. He was a violent man, but it is important to remember that he lived in a violent world. Throughout seventeenth-century Italy \u2013 throughout seventeenth-century Europe \u2013 an inflammatory code of honour prevailed. The _fama_ of an individual, by which was meant not only his fame or reputation but also his good name, was paramount. Any insult to it had to be paid for, and the price was often blood. Caravaggio went to greater extremes than his contemporaries, in life as in art. He was no angel, even if he had been named after one. He had a hot temper and was forever spoiling for a fight. But he was not the freak or absolute exception that he has often been painted to be \u2013 both by his enemies and by those who have claimed to idolize him.\n\n#### IN ROME\n\nHaving cut all ties with his family, the artist travelled to Rome in the autumn of 1592. Bellori says that Caravaggio went there via Venice, 'where he came to enjoy the colours of Giorgione, which he then imitated'. A brush with Venetian art at this formative moment in his life seems likely, although Bellori overplays Caravaggio's indebtedness to Giorgione. Giorgione's work had inspired Titian, the most celebrated painter of Renaissance Venice. But that axis of Venetian painting \u2013 rich, brightly coloured, with a strong sense of paint as eloquent, material _stuff_ , to be pushed about with the fingers as well as manipulated by the brush \u2013 did not hold the young Caravaggio's attention. Aside from his innate sense of pictorial drama, he would have little in common with either of those great masters. Only in the paintings of his very last years would he move towards the impressionistic manner of Titian's later work.\n\nIf any Venetian painter touched him to the core it was Jacopo Tintoretto. Tintoretto's brooding, monumental religious canvases, full of dramatic contrasts of light and dark \u2013 lightning strikes of supernatural illumination that shiver like spiritual electricity \u2013 are the only late sixteenth-century Italian paintings to prophesy elements of Caravaggio's own mature style. Simone Peterzano, who liked to think of himself as a painter in the Venetian mould, may well have inspired his unruly apprentice to visit the city. If so, he contributed to the final eclipse of whatever dim influence his own art might have had on Caravaggio's imagination.\n\nThe trip to Venice remains hypothetical, but highly plausible. According to such a version of events, the young Caravaggio arrives in Rome with his memory full of vast, dark pictures teeming with images of humanity _in extremis_. This helps to make sense of his subsequent development. The preferred, monumental scale of his work as well as the ambition behind it; the extreme sense of light and dark; even the distinctive, low-toned palette that Caravaggio would make his own \u2013 where could he have got the first glimmerings of all this, if not from Venice, and Tintoretto?\n\nFor the next fourteen years Caravaggio would be at the heart of Roman Catholic Christendom, achieving fame and notoriety in equal measure. The most vivid late sixteenth-century account of the city was written by the French essayist and philosopher Michel de Montaigne, who spent several months there in late 1580 and early 1581. That was ten years before Caravaggio's time. But the city Montaigne described was, by and large, the city that Caravaggio knew.\n\nMontaigne was immediately struck by the ugliness and poverty of the surrounding countryside: 'The approaches to Rome, almost everywhere, look uncultivated and barren, either for want of soil, or, what I consider more likely, because this city has hardly any labourers and men who live by the work of their hands.' The few labourers who _were_ to be encountered in the fields tended to be migrant workers, from the mountains of northern Italy: 'When I came here I found on the way many groups of villagers who came from the Grisons and Savoy to earn something in the season by labouring in the vineyards and the gardens; and they told me that every year this was their source of income.'\n\nRome was a city of migrants. Its shifting population was drawn from every corner of the Christian world \u2013 priests seeking preferment, pilgrims seeking salvation, courtesans seeking riches. 'It is the most universal city in the world,' proclaimed Montaigne, 'a place where strangeness and differences of nationality are considered least; for by its nature it is a city pieced together out of foreigners; everyone is as if at home.'\n\nIt was also a suspicious city. On arrival, Montaigne's baggage was seized. The books in his travelling library were meticulously inspected by Rome's customs officials. They were looking for forbidden texts, for evidence of heresy, and, although they found little to concern them, Montaigne was struck by the severity of their regulations: 'the rules were so extraordinary here that the book of hours of Our Lady, because it was of Paris, not of Rome, was suspect to them, and also the books of certain German doctors of theology against the heretics, because in combating them they made mention of their errors.' Much to Montaigne's annoyance, the authorities confiscated a book 'on the histories of the Swiss, translated into French, solely because the translator \u2013 whose name, however, is not given \u2013 is a heretic... it is a marvel how well they know the men of our countries.'\n\nMore than half a century had passed since the Lutheran troops of Emperor Charles V sacked the city in 1527. But Rome had still not recovered. Thousands had died during the Sack and many others had abandoned their homes. Montaigne was struck by the contrast between the splendour of the papal court \u2013 'remarkable houses and gardens of the cardinals... palaces divided into numerous apartments, one leading to another' \u2013 and the squalid, neglected condition of so much of the rest of the city.\n\nRelics and reminders of ancient Rome were everywhere, so that 'in many places we were walking on the tops of entire houses... in truth, almost everywhere, you walk on the top of old walls which the rain and the coach ruts uncover.' But so mangled were the tangible remains of the classical past that Montaigne felt the totality of its destruction more keenly than anything else: 'those who said that one at least saw the ruins of Rome said too much, for the ruins of so awesome a machine would bring more honour and reverence to its memory: this was nothing but its sepulchre. The world, hostile to its long domination, had first broken and shattered all the parts of this wonderful body; and because, even though quite dead, overthrown, and disfigured, it still terrified the world, the world had buried its very ruin.'\n\nLike any other migrant worker from the north, Caravaggio would have entered the city through the Porta del Popolo, into the Piazza del Popolo. In those days the great square was flanked on its northern side by the church and monastery of Santa Maria del Popolo, and on the south by a line of ordinary houses. Attempts had recently been made to aggrandize this main avenue of entry to the city. In 1587 an obelisk had been erected in the middle of the square. A marble fountain had been added too, but still the piazza was anything but grand. A traveller just arrived might get something to eat from one of the fritter vendors at the foot of the obelisk. He might sit with his back to the stump of a classical column \u2013 there were several protruding from the ground, like broken teeth \u2013 to munch his snack. There was a drinking trough nearby, used by farmers bringing pigs and goats to market, and a watering place where women did their laundry in the open air.\n\nMuch of Rome was still down at heel, as it had been when Montaigne visited. But by the time of Caravaggio's arrival the city was in the throes of a great transformation. In the spring of 1585 a devout Franciscan, Felice Peretti, Cardinal of Montalto, had been elected Pope Sixtus V. Energized by the same sense of mission as the formidable Carlo Borromeo \u2013 with whom he collaborated on an edition of the writings of St Ambrose \u2013 he set out to rebuild Rome both spiritually and physically. The edicts of the Counter-Reformation, handed down at the Council of Trent, were to be scrupulously observed. The fabric of the city itself had to be transformed into the visible symbol of a triumphantly reaffirmed Catholicism.\n\nUnder Sixtus V and his immediate successors, the appearance of Rome was dramatically altered. Seven grand new radial avenues were created to link the seven principal Christian basilicas and to ease the passage of pilgrims through the city. Many of the ancient Christian sites of Rome \u2013 including the catacombs, the tombs of the early martyrs \u2013 were excavated and restored. The dome of St Peter's, begun by Bramante nearly a century earlier, continued by Antonio da Sangallo and finally redesigned by 'the divine' Michelangelo, had at last been completed. Within a year of Caravaggio's arrival a gleaming ball topped by a golden cross had been mounted above its lantern.\n\nAs if to justify Montaigne's assertion that ancient Rome still 'terrified the world', its vestiges were yet more thoroughly subjected to Christian zeal. Prominent remains of antiquity were appropriated \u2013 moved, transformed, sometimes defaced and demolished \u2013 to demonstrate the eternal triumph of a resurgent Catholic Church over paganism and heresy alike. Sixtus V's principal architect, Domenico Fontana, transported a vast obelisk from the Circus of Nero to the square of St Peter's. Inscriptions were added to its base, declaring that a monument erected to the impious cults of the ancient gods had been brought to 'the threshold of the apostles' and consecrated to 'the undefeated cross'. The old Renaissance spirit of admiration for the art and literature of the classical past began to be regarded with a distrust that bordered on outright hostility.\n\nThe same, severe repudiation of pagan antiquity had been expressed by one of the most prominent commissions of Sixtus's predecessor, Gregory XIII: Tommaso Laureti's painting _The Triumph of Christianity_ \u2013 a fresco decoration for the Sala di Costantino in the Vatican Palace, completed in the mid 1580s. In a chilly atrium, a statue of Mercury lies shattered at the foot of an image of Christ on the Cross. The fragments of stone that symbolize the destruction of the ancient gods \u2013 hand, torso, decapitated head \u2013 have been placed in the foreground, at the start of a brutally insistent single-point perspective scheme. The vanishing point of the picture is like a black hole, where all energy converges. The painter rushes the eye from pagan idol to redeeming Christ and beyond \u2013 to a glimmering avenue of architectural mystery that stands, by implication, for the ineffable mystery of the one true faith.\n\n#### CLEMENT VIII\n\nCaravaggio arrived in Rome some seven or eight months after the election of a new pope. Clement VIII was determined to carry on the work begun by his predecessors, albeit in a somewhat less militant style. He was a shrewd, cautious and deeply pious man, whose pontificate was marked by a relaxation of hostility towards the culture and mythology of antiquity. In the private sphere, at least, it became permissible to commission paintings on profane subjects from the artists of the city. So it was that during the 1590s the Bolognese painter Annibale Carracci covered the ceiling of the Palazzo Farnese \u2013 the palace of Cardinal Odoardo Farnese, one of the richest men in all Italy \u2013 with a dizzying cornucopia of nudes re-enacting the loves of the gods on earth, in the air and in the water. There had been nothing like this joyful celebration of Eros in Rome since the Renaissance.\n\nClement VIII had been elected, on 30 January 1592, on the strength of his supposed moderation. In practice, he would tread a fine line between political pragmatism and Counter-Reformation zeal. He could be ruthless in the suppression of heresy and dissent, so the Rome that Caravaggio knew could hardly be described as a haven of creative and intellectual freedom. It was under Clement's pontificate that the speculative mystic Giordano Bruno \u2013 who believed in a thousand different worlds spinning through space, but denied the existence of God \u2013 was burned at the stake in 1600. Clement was not actively hostile to Philip II of Spain, but he set out to emancipate the papacy from what he perceived as undue Spanish influence. Rival French and Spanish factions lobbied tirelessly for influence in Rome, and at times their disagreements spilled over into street fights and public brawls. Clement steered a skilful middle course. He cultivated closer relations with France, acknowledging the legitimacy of Henri IV's claims to the throne and thus paving the way for the French king to renounce Protestantism and return to the Catholic fold. He then brokered the peace of Vervins of 1598, which effected a rapprochement between Henri IV and Philip II.\n\nThe French king's conversion was a tremendous coup for the Catholic Church and would come to be considered the greatest triumph of Clement's reign. (The pope also courted James I of England, whose queen, Anne of Denmark, was already a convert to Catholicism, but to no avail.) At home he did his utmost to restrict the powers of the aristocracy, reining in the feudal barons of the Papal States at every opportunity. In 1597, when the venerable Este dynasty failed due to the lack of a male heir, he promptly claimed title to the family's fiefdom of Ferrara and incorporated it into the papal states. Clement revised the Vulgate, promulgated a new edition of the _Index librorum prohibitorum_ and took his duties as Bishop of Rome no less seriously than his role as supreme pontiff. He curbed prostitution, introduced a general ban on the carrying of weapons in public, outlawed duelling, made libel a capital offence and sought to enforce the strict celibacy of his clergy. The papal _sbirri_ , the constabulary, were a vital tool in his control of the city. They were the equivalent of the Bishop of Milan's _famiglia armata_ , but even more numerous. They were given wide-ranging powers, including the power to stop and search anyone suspected of heresy, of bearing arms or of being out after curfew without good cause. They did much of their work at night and were known for the dark cloaks that they wore to conceal themselves as they tailed their suspects, or paid unannounced visits to the houses of witnesses and potential informers.\n\nPunishment, by contrast, took place in broad daylight. Death by execution was a grim public spectacle, a theatre of retribution designed to instil fear and the spirit of penitence into all who witnessed it. In 1581 Montaigne had observed the last moments of 'a famous robber and bandit captain' by the name of Catena:\n\n> they carry in front of the criminal a big crucifix covered with a black curtain, and on foot go a large number of men dressed and masked in linen, who, they say, are gentlemen and other prominent people of Rome who devote themselves to this service of accompanying criminals led to execution and the bodies of the dead; and there is a brotherhood of them. There are two of these, or monks dressed and masked in the same way, who attend the criminal on the cart and preach to him; and one of them continually holds before his face a picture on which is the portrait of Our Lord, and has him kiss it incessantly. At the gallows, which is a beam between two supports, they still kept this picture against his face until he was launched. He made an ordinary death, without movement or word; he was a dark man of thirty or thereabouts...\n\nAfter the criminal's death, his body was cut into pieces. At this point, Montaigne notes, 'Jesuits or others get up on some high spot and shout to the people, one in this direction, the other in that, and preach to them to make them take in this example.' Such executions were still very much part of life in the Rome that Caravaggio knew.\n\nReligious observance was not a matter of choice. At Easter everyone living in Rome was obliged to take Communion and procure a ticket of evidence from the priest who administered the sacrament. Procuring the ticket \u2013 proof of orthodoxy, and necessary to pass muster with the police \u2013 was itself part of a system of surveillance and involved a separate visit to the priest, who was obliged to write down the name and address of each communicant. But he also had to write down other details, noting for example who lived where and with whom, and listing their servants. It was, in effect, an annual census. It is because Counter-Reformation Rome was such an intensely controlled society that so much is known about those who lived there.\n\nAs in the Milan of Caravaggio's youth, great importance was attached to the question of what people should see, or be allowed to see. In a world where even the death of a criminal could be orchestrated as a grisly spectacle, religious art was inevitably subject to all kinds of supervisions. At the very start of his pontificate (1592\u20131605, therefore coinciding almost exactly with Caravaggio's years in Rome) Clement proved particularly keen to establish himself as a ruthless enforcer of the doctrines laid down by the Council of Trent. On 8 June 1592, some four months after his election, he issued the papal Bull _Speculatores domus Israel_ , declaring a 'Visitation' of all churches of the city of Rome. The clergy would be inspected and so would the fabric and decoration of their churches, including works of art.\n\nIt was to be no comprehensive survey. The churches Clement actually visited are listed, in order, in the so-called Secret Vatican Archives (Archivio Segreto Vaticano). He started at the top, with St Peter's itself on 3 July 1592. He then went on to Santa Maria Maggiore, followed by San Giovanni in Laterano. By the time the Visitation had finally petered out, four years later, only twenty-eight churches had been covered. The reason was not dilatoriness but Clement's meticulous attention to detail. He insisted on visiting every church himself, and interrogating any suspect members of his clergy personally. Even though he surrounded himself with an entourage of four cardinals and three bishops \u2013 including Audwyn Lewis, the Bishop of Cassano, a Welsh Catholic who had left England in 1579 \u2013 the work of inspection was painfully slow. Its eventual abandonment may be taken as further proof of Clement's common sense. Although deeply concerned for the well-being of the Church, Clement was not a man in the same obsessive mould as Carlo Borromeo. He stopped, perhaps knowing that his point had been made. The mere threat of the Visitation had reminded the Roman clergy to pay close attention to the works of art in their churches, and to use their powers of censorship if necessary. Caravaggio's career would be directly affected as a result. Several times during his years in Rome he would experience the humiliation of having a painting intended for the altar of a Roman church rejected on the grounds of indecency or impropriety.\n\n#### IN THE ARTISTS' QUARTER\n\nLike his immediate predecessors, the new pope was determined both to stabilize the foundations of the Catholic Church and to reassert the Eternal City as the radiant centre of Christendom. The beauty of Rome's churches must compel faith and crush heresy. That is why the city was filled with artists. Painters, sculptors and architects throughout Italy, and further afield, knew that there was more than enough work to go round in Rome. As Florence had been during the fifteenth century, and as Paris would be at the peak of Louis XIV's power, Rome under Clement VIII was the artistic capital of Europe. The artists of the city were so numerous \u2013 at a rough estimate, there were 2,000 of them, out of Rome's total population of around 100,000 \u2013 that they had their own quarter.\n\nThis was an area of a little more than two square miles situated, roughly, between the Piazza del Popolo and the Piazza di Spagna. Artists tended to arrive in groups \u2013 whether from Naples or Bologna, Lombardy or Emilia-Romagna, Flanders or France \u2013 and to board together to save money; it was not unusual for two or three to share a room, using the space both as bedroom and workshop. Rome could be stiflingly hot, which placed a premium on the lower, cooler floors of rental accommodation. But that suited the traditionally hard-up community of painters, who preferred the less expensive upper floors anyway, because there was more light there to paint by. They particularly favoured houses backing on to the Pincio, the hill perched above the Piazza del Popolo.\n\nDifferent national groups of artists ran in gangs and swapped racial insults with their rivals. There were stereotypes to fit all. Germans were crude, the Flemish were drunks, and the French were violent thugs hiding behind a veneer of fake refinement. The Italians themselves, according to the exiled English earl encountered in Rome by the hero of Thomas Nashe's novel of 1594, _The Unfortunate Traveller_ , were addicted to 'the art of epicurising, the art of whoring, the art of poysoning, the art of sodomitry'.\n\nAn Englishman like Nashe grouped all Italians together, but for them the matter of national belonging was less clear cut. Italians had some sense of communal identity but an even sharper feeling for the distinctions that set them apart from one another. The Bolognese were known to hate the Tuscans, while most Romans treated Sicilians as if they were little better than peasants. Neapolitans were said to be obsessive about horsemanship. The Milanese, as we have seen, were famously keen swordsmen, and naturally unruly \u2013 although 'Lombards' as a group were often stereotyped as sluggish provincials, heavy of mind and body thanks to their rustic diet.\n\nIn 1589 the writer Giovanni Botero went so far as to propose a north\u2013south fissure in the Italian temperament: 'Those who live in northern countries but not in the extreme north, are bold but lack cunning; southerners on the other hand are cunning but not bold... They are as the lion and the fox; whereas the northerner is slow and consistent in his actions, cheerful and subject to Bacchus, the southerner is impetuous and volatile, melancholy and subject to Venus...' The personality of Caravaggio would be hard to locate on this particular map. He does not fit either profile, and in fact he would make his sense of his own singularity the subject of one of his earliest paintings \u2013 a self-portrait _as_ Bacchus, but a Bacchus who is suffering and full of melancholy.\n\n#### CITY OF MEN, CITY OF WHORES\n\nThe chronology of Caravaggio's early years is impossible to establish with precision, although we can work out that he changed his address frequently \u2013 ten times or so between 1592 and 1595. The houses changed but the milieu was always the same: the dark network of alleyways clustered around the Piazza del Popolo; the narrow streets bordering the Palazzo Firenze, home of the Medici's ambassadors to Rome; the Piazza Navona, which had once been the stadium of Domitian and still preserved its ancient double-horseshoe outline; the Campo de' Fiori, the marketplace.\n\nThis artists' quarter was a dangerous area of the city. Fights were common and fists were not the only weapons used. In an attempt to deter armed violence, the papal police made a public example of anyone caught wielding a _pugnello_ , the short-handled dagger that was so often Exhibit A in the cases brought before Rome's criminal tribunals. At the corner of Via del Corso and Via dei Greci, in full shaming view of the city whose laws he had violated, the arrested suspect would be subjected to the _strappado_ , an excruciating form of rope torture. The victim's hands were tied behind his back, with another loop of rope passed beneath his joined arms. He was then hauled into the air and left dangling for half an hour, the full weight of his body gradually pulling his arms further and further back and behind him. The inevitable result was dislocation of both shoulders. Victims eventually recovered but they did not forget the pain in a hurry. A painter subjected to the _strappado_ could not work for weeks.\n\nAt night Caravaggio, his friends and his enemies shared the streets with the city's prostitutes. Rome's whores and courtesans had long been one of the sights of the city. In the early 1580s Montaigne had noted a craze for open-topped carriages especially adapted for the purpose of erotic ogling: 'One preacher's joke was that we turned our coaches into astrolabes... To tell the truth, the greatest profit that is derived from this is to see the ladies at the windows, and notably the courtesans, who show themselves at their Venetian blinds with such treacherous artfulness that I have often marvelled how they tantalise our eyes as they do; and often, having... obtained admission, I wondered at how much more beautiful they appeared to be than they really were...'\n\nBy Caravaggio's time the prostitutes were so numerous that they had been coralled by papal edict into their own enclosure by the Tiber, the Ortaccio di Ripetta \u2013 a name which joked that the place was a kind of reverse Eden, since the literal meaning of _ortaccio_ was 'evil garden'. But they would escape after dark to ply their trade in the ill-lit streets around the Piazza del Popolo. They were an embarrassment to the authorities because their very presence in such great numbers represented a blatant betrayal of Christian ethics at the very heart of the Catholic world.\n\nRome was not just an overwhelmingly male city; it was a city full of young and unattached men competing desperately with one another for favours. The city's whores were a much needed outlet for the accumulated sexual energy of this male-dominated, testosterone-fuelled society. But they were also, often, an occasion for violence in themselves. Some girls offered certain services for free to clients whom they liked, which could easily breed resentment. For an artist, the service might be posing naked for a picture (painting the nude model was officially illegal and this was one way of getting around the rules). But if the girls' pimps discovered such an arrangement, there was generally trouble.\n\nThe young artists who came to Rome to make a name for themselves lived on top of each other, competed for the same work, drank in the same taverns, frequented the same restaurants and bought their materials \u2013 paints, canvases, stretchers \u2013 from the same artists' supplier. His name was Antinoro Bertucci and he had a street stall on the Corso. Painters and sculptors of all nationalities would meet there, to buy what they needed for the next day's work and to discuss the latest gossip and news \u2013 to find out about workshop vacancies, to learn who was in and who was out with this or that influential prelate or cardinal. A visit to Antinoro's in the evening was also a good way to dodge the curfew regulations, because the paint-seller kept a fire burning at all hours. Going out for heat and light was a legitimate excuse for breaking curfew, so if a gathering should be interrupted by the police everyone would say they had gone to Antinoro for that reason. 'We were at Antinoro's because our fire had gone out' is a phrase that recurs in the witness statements of Rome's artists.\n\nThe ultimate ambition of every artist was the same: namely to work for the cardinals closest to the pope, to secure the most important commissions and win lasting fame \u2013 along with the money and security that went with it. The rules of the game were by no means straightforward. Everybody knew of mediocre artists who had been promoted above their abilities, and of deserving painters who had been overlooked. In a world where competition and rivalry were intense, resentment flourished easily, so Antinoro's was a rumour factory as well as an artists' supply shop. Stories of sabotage abounded \u2013 of the scaffolding that collapsed in the night, of the painter whose rival poisoned his colours with acid so that all his blues turned green in a matter of days. Once an artist got a reputation for bad luck or inefficiency, once it got around that he had 'the evil eye', his commissions would soon dry up.\n\nThough they disagree in some details, taken together Caravaggio's early biographers paint a convincing picture of a young man struggling to find his way in a harsh and unfamiliar world. During these first years in Rome he had contacts to ease his passage into the city: his uncle, Ludovico Merisi, the priest, was living there in 1591\u20132, Costanza Colonna in late 1592. But he seems to have fended mostly for himself, moving restlessly from one studio to another in search of employment \u2013 and quite possibly instruction too.\n\nBaglione reports that 'in the beginning he settled down with a Sicilian painter who had a shop full of crude works of art.' Bellori, in his marginal notes to Baglione's life, gives the Sicilian painter a name of sorts \u2013 Lorenzo Siciliano, whose line of work was painting crude bust-length heads for general sale. According to Bellori, in Lorenzo's workshop 'Caravaggio painted heads for a groat apiece and produced three a day.' The 'heads' in question may have been portraits of the famous men of the past, a subject in vogue among collectors of art since the middle years of the fifteenth century. The mercenary soldier and intellectual Federigo da Montefeltro, ruler of Urbino, had turned his private study into a gallery of such pictures, encompassing figures as various as Cicero and St Thomas Aquinas. If Caravaggio did paint his own versions of such subjects, none is known.\n\nIt was while staying with Lorenzo Siciliano that Caravaggio met an ambitious but unpredictable young Sicilian artist called Mario Minniti. According to the eighteenth-century biographer Francesco Susinno, who had access to sources now lost or destroyed, the young Minniti had been forced to flee Syracuse, via Malta, to get away from unspecified troubles. Arriving in Rome, he lodged with a hack painter from Sicily, in whose studio he befriended Caravaggio. Susinno implies that they were united in their dissatisfaction with the type of work which 'that coarse artisan' demanded of them, and dreamed together of rising to greater things. They became close friends, Minniti even posing for the impoverished Caravaggio \u2013 who could ill afford a model \u2013 on several occasions. His moon-shaped, mumpish face can be recognized in a number of early works. Minniti would also prove to be a useful contact much further along the pitted track of Caravaggio's life.\n\nAccording to Mancini, during this period Caravaggio also lodged with a beneficed priest of St Peter's named Pandolfo Pucci, from the town of Recanati. He may have been introduced through his connections with the Colonna: the priest was household steward to a member of the Peretti family, and the Peretti and the Colonna were close. Mancini says that Pucci gave the artist a room and allowed him to paint there in exchange for domestic chores. The deal was not to the painter's satisfaction, and not only because he was not the sort of man to take pleasure in doing the housework: 'Worse, he was given nothing but salad to eat in the evening, which served as appetizer, entr\u00e9e and dessert \u2013 as the corporal says, as accompaniment and toothpick. After a few months he left with little recompense, calling his benefactor and master \"Monsignor Salad\".'\n\nMancini records that during his time with the parsimonious Pucci, Caravaggio 'painted some copies of devotional images', which the priest took home to Recanati with him when he left Rome in 1600. No trace of them has been found. Lost too is 'a portrait of an innkeeper who had given him lodgings' as well as another, unnamed portrait mentioned by the biographer. But the young Caravaggio's painting of a 'boy who is peeling a pear with a knife', also mentioned by Mancini, has perhaps survived.\n\nThere are at least ten versions of a similar subject, all showing the same rudimentary composition. The painting in the British Royal Collection, which is thought to have been acquired by Charles II, is conceivably the original picture described by Mancini. It was already recorded in the James II inventory of the collection as a work by 'Michael Angelo', which indicates that it was regarded as an autograph Caravaggio as early as the seventeenth century. An adolescent boy in a white shirt sits at a table on which various fruits, including cherries, peaches and nectarines, are scattered. The boy's shirt is spotlessly white, his hands unblemished, details suggesting that he is of noble birth. He peels not a pear but a green Seville or Bergamot orange, a bitter fruit. Perhaps this symbolizes his determination to choose the path of virtue, to avoid the sweeter temptations that life has to offer, or perhaps it is an emblem of the disappointments and difficulties that lie ahead even for a boy like this, blessed by wealth and fortune \u2013 a characteristically sour note for Caravaggio to have struck. But the mood of the painting is anything but ominous, so it might be unwise to burden it with too much hidden meaning. If it is indeed an autograph work, this undistinguished genre picture confirms just how little progress Caravaggio had made as a painter by the early 1590s. The handling is crude, the boy's expression wooden. Only in the extreme contrast of light and shade \u2013 the whiteness of the shirt, the depth of the shadows \u2013 can some presage of Caravaggio's later work be discerned.\n\n#### THE BROTHERS CESARI\n\nFor the rest of his early time in Rome, Caravaggio appears to have been very much out on his own. Having left Monsignor Salad to his greens, he probably spent some time in the studio of a Sienese painter called Antiveduto Gramatica. Gramatica was an artist of limited gifts about whom little is known save for the fact that his father had a questionable sense of humour. After predicting his son's premature birth, he registered his prescience by giving him his joke of a name \u2013 _antiveduto_ , meaning 'foreseen'.\n\nCaravaggio may have entered his studio in early 1593. In the same year Gramatica became a member of the guild of painters, the Accademia di San Luca. He was prolific, turning out small-scale devotional pictures, portraits and copies of portraits by the score. Particularly popular were his copies of a series of _Famous Men_ then at the Villa Medici. Caravaggio, whose work for Lorenzo Siciliano had probably included similar pictures, may have painted his own copies of the Villa Medici 'heads' while in Gramatica's studio. If so, it is possible that he now came to the attention of his future protector, the Medici cardinal Francesco Maria del Monte, for the first time.\n\nThe next studio in which Caravaggio found work was more exalted. It was that of Giuseppe Cesari, otherwise known as the Cavaliere d'Arpino, one of the most prominent artists in Rome in the 1590s. Giuseppe Cesari was only three years older than Caravaggio but far more successful. He was from a family of artists: his father, Muzio, was a painter. His brother, Bernardino, was his chief assistant and may also have acted as workshop manager. When he was a boy, Giuseppe had shown such precocious gifts as a draughtsman that his mother had taken him to Rome. At the age of just thirteen he had found work as a colour mixer for Niccol\u00f2 Circignani, who was then directing the decoration of the Vatican Loggie for Pope Gregory XIII. He soon graduated to the painting team and made enough of a name for himself to win several important independent commissions in Rome during the later 1580s. Following the death of his most influential patron, Cardinal Farnese, in 1589, he accepted an invitation to carry out a series of paintings for the Certosa di San Martino in Naples, including a monumental canvas, _The Crucifixion_. He returned to Rome in 1591 and on Clement VIII's accession became the pope's leading painter.\n\nCesari's art was a limp but occasionally elegant hybrid of High Renaissance and Mannerist styles. As a painter of religious subjects, he answered the Counter-Reformation call for clarity, grace and decorum. But in his smaller, erotically charged mythological pictures, he experimented with complicated poses and frequently arcane symbolism. He painted the rape of Europa and the judgement of Paris, as well as the naked Diana and her companions surprised by the huntsman Actaeon. A composition of Perseus swooping from the skies to rescue an alluring and unusually languid Andromeda from the clutches of a diminutive lapdog of a dragon proved especially popular; several versions survive.\n\nCaravaggio probably began working for the Cesari brothers in the middle of 1593. It was an eventful time for his new employers. In late 1592 Bernardino Cesari had been sentenced to death for associating with known bandits and had run away to Naples. But by May of the following year he was back in Rome, having secured a papal pardon thanks to the intervention of Cardinal Paolo Emilio Sfondrato, one of several powerful patrons of the Cesari workshop. Giuseppe now needed all the help he could get. The studio was busy: the papal treasurer, Bernardo Olgiati, had commissioned the decoration of an entire chapel in the church of Santa Prassede. Figures of the prophets, sibyls and doctors of the Church were required, as well as a _Resurrection_ and a monumental _Ascension_. There was also a commission to decorate the vault of the Contarelli Chapel, in the church of San Luigi dei Francesi, with small scenes set into an intricate stuccowork design. Six years later Caravaggio would win the commission to paint two large canvases for the walls of the very same chapel \u2013 works that would instantly establish him as the most original religious artist of his time, and forever overshadow the earlier contributions of the Cesari workshop. But in 1593 he was just another apprentice painter from Lombardy with everything to prove.\n\nThe sources say that Caravaggio was employed to paint 'flowers and fruit'. Artists from northern Italy had recently begun to work in the relatively new field of still life painting. It was a genre of secular art \u2013 albeit frequently with undertones of religious meaning \u2013 that had its roots in post-Reformation Flanders and Holland. But its popularity had begun to spread southwards across Europe during the late sixteenth century. The Spanish played a part in disseminating this new taste, buying works of art in their northern territories and taking them to the cities that they controlled in Italy. It is likely that Caravaggio had seen Dutch or Flemish still life paintings during his time in Milan, and perhaps that was why Giuseppe Cesari marked him out for the same line of work. It is likely that he would have contributed the decorative festoons of frescoes by the Cesari workshop, as well as painting canvases for direct sale to private clients. Cesari also bought and sold Dutch and Flemish cabinet pictures, which suggests that he was well aware of the new taste among Roman collectors for the novelties of landscape and still life painting. But the supply of such pictures from the north was inevitably limited. Who better than an artist from Lombardy \u2013 a man from fertile Caravaggio, surrounded by the orchards that supplied Milan \u2013 to create homegrown depictions of flowers, fruit and vegetables?\n\nThere may have been an element of condescension about Cesari's decision to channel Caravaggio's energies towards still life \u2013 a trace of dismissive preconception about painters from the north. It was not unknown for Lombard artists to be caricatured as rustic provincials, country bumpkins of painting, unacquainted with the grand traditions of Renaissance art. Artists were ranked according to a strict hierarchy determined by subject matter. At the top were those who specialized in paintings of the human figure shown in heroic or significant action \u2013 paintings from the Bible or mythology. Lower down the scale came portraits, then paintings of animals. Then came another relatively new genre, the landscape, followed last and least by the humble still life. Such distinctions mattered a lot, especially to a man as touchy and as self-conscious about his own status as Caravaggio.\n\nThere are strong indications that he resented the lowly nature of the work that he was given to do in the Cesari studio. Bellori suggests that he already had the ambition to work in the higher reaches of art, but had to take whatever employment he was offered simply to survive:\n\n> Since models, without which he did not know how to paint, were too expensive, he did not earn enough to pay his expenses. Michele was therefore forced by necessity to work for Cavaliere Giuseppe d'Arpino, who had him paint flowers and fruit, which he imitated so well that from then on they began to attain that greater beauty that we love today. He painted a vase of flowers with the transparencies of the water and glass and the reflections of a window of the room, rendering flowers sprinkled with the freshest dewdrops; and he painted other excellent pictures of similar imitations. But he worked reluctantly at these things and felt deeper regret at not being able to paint figures.\n\nShorthand notes in the manuscript of Mancini's biography yield several tantalizing glimpses of Caravaggio in the Cesari workshop. They are obscure and hard to interpret, but suggest that the painter's relationship with his employers was fraught. At first they rescue him, but then they let him down or betray him in some unspecified way. There is a reference to Caravaggio in poor and ragged clothing. Then Bernardino Cesari takes him into the 'Torretta', which was the name of the building in which the Cesari workshop was housed. He is put up on a straw mattress on a raised platform, presumably some kind of minstrels' gallery in one of the rooms. He boards there, in all, for eight months. But at a certain point something bad happens, although Mancini's notes do not say what that something was. Giuseppe Cesari is a witness: 'Giuseppe sees and is petrified and in order to distract him makes him retreat and flee so he does not appear.' Following this nameless act and its enigmatic aftermath, the Cesari brothers seem to feel that Caravaggio's presence in their workshop has to be concealed. Mention is made of 'C. G.', short for Giuseppe Cesari, painting a picture of St Joseph on which Caravaggio perhaps collaborates; but wherever this takes place, Cesari is very keen that Caravaggio should not be seen. Then Caravaggio gets kicked by a horse so badly that his leg swells alarmingly, but a surgeon is not called because he still must not be seen by anyone. A Sicilian friend who owns or runs a shop \u2013 more likely to be Lorenzo Siciliano than Mario Minniti \u2013 takes him to the hospital of the Consolazione. The Cesari brothers never go to visit him and he never goes back to them.\n\nMancini's finished life of Caravaggio is much less circumstantial than his notes, except on the subject of the artist's stay in the hospital of the Consolazione. During his convalescence there, Caravaggio is said to have painted 'many pictures for the prior, who brought them to Seville, his home'. Reversing the chronology suggested by his jottings, Mancini then asserts that it was _after_ his illness that the painter 'stayed with Cavaliere Giuseppe'. But whatever the precise sequence of events, a clear enough picture emerges of Caravaggio during this time of evident hardship. He is proud and touchy. He is growing in ambition, but increasingly disgruntled. He has not lost his knack for getting into trouble. He grudgingly performs his duties as a still life painter.\n\n#### SELF-PORTRAIT AS BACCHUS, BOY WITH A BASKET OF FRUIT\n\nNone of the still lives that Caravaggio painted while he was with the Cavaliere d'Arpino seem to have survived. But there are two early pictures with a provenance that places them in the Cesari workshop. Both contain carefully worked still life elements, although neither is a pure still life painting: _Boy with a Basket of Fruit_ and the so-called _Sick Bacchus_ , or _Self-Portrait as Bacchus_.\n\nThey must have been done for Giuseppe Cesari in 1593\u20134, because both were still in his possession as late as 1607, the year when he unwisely clashed with the covetous papal nephew Scipione Borghese. Borghese was an avid art collector, as well as a great admirer of Caravaggio's work, who had long had his eye on Giuseppe Cesari's considerable backroom stock. He made an insultingly low offer for the pictures, and when Cesari had the temerity to refuse, Borghese used his influence to have the troublesome painter-dealer arrested on trumped-up charges. He then appropriated Cesari's entire collection of 105 paintings. The two pictures by Caravaggio have been in the Borghese collection ever since \u2013 and may still be seen in the Galleria Borghese today.\n\nThey are unusual works, with more than a hint of awkwardness about them, especially the _Self-Portrait as Bacchus_ , which two of Caravaggio's early biographers found sufficiently memorable to single out from the rest of his juvenilia. Mancini refers to 'a beautiful Bacchus who was beardless', while Baglione mentions 'a Bacchus with different bunches of grapes, painted with great care but a bit dry in style'. 'Dry' seems a more appropriate epithet than 'beautiful'.\n\nThe ancient god of wine and mystic revelry holds a bunch of white grapes in his right hand. At the same time he brings up his shadowed left hand to clasp and even crush them, as in a wine-press. The bloom on the grapes, which dusts them with a layer of whiteness and dulls the reflected light caught in their opalescent skins, is echoed by the dry and whitish lips of the god himself. His pallour is an enigma, which the dark pools of his eyes \u2013 mocking and mysterious \u2013 do nothing to dispel. The still life that lies before him has an unsettling pathos. Two ripe, furred peaches lie beside a bunch of purple grapes on a forbiddingly cold and otherwise bare ledge of stone. Vine leaves trail off into darkness.\n\nThe title frequently used in modern times, _Sick Bacchus_ , is a legacy of the Italian art historian Roberto Longhi. Longhi believed that Caravaggio painted it as an allegorical self-portrait just after his discharge from the hospital of the Consolazione. Whether the work alludes to the artist's illness is open to question, but it is certainly a self-portrait. Baglione groups it with a number of other, long-since vanished 'portraits of himself in the mirror'. The distorted right shoulder of the figure, so close to the picture plane as to seem almost touchable, may reflect the painter's use of a faintly convex mirror. The effect is at once intimate and disconcerting. The promise of a close relationship is held out by the figure's proximity, but denied by the cool evasiveness in his eyes. His right leg, so lost in semi-darkness as to have become little more than a blur, is half raised, which suggests that he could be about to get up. Sensual gratification is the half-promised gift that he brings. But he might disappear at any moment, leaving behind just darkness and the taste of ashes, not of wine.\n\nWhy would Caravaggio have painted himself like this? What might he have meant by it? The notion that he intended the work as a record of his own illness is ingenious, but there is a better and simpler explanation for the artist's liverish complexion. The picture is set at night, the time for Bacchic revelry. The light that flares so brightly on the figure's shoulder, giving his face its greenish cast, is simply the light of the moon.\n\nThe significance of this hypnotizing self-portrait is best sought in its symbolism, although that is anything but straightforward. In one sense Bacchus is an apt alter ego for an artist, because according to his legend he is subject to fits of divine inspiration. Caravaggio was not the first painter to associate himself with the god of wine. In Borromean Milan, the city of his upbringing, a group of painters, including the well-known artist and theorist Gian Paolo Lomazzo, had formed a mock-academy dedicated to the cult of Bacchus. The young Caravaggio's appropriation of the same Bacchic symbolism may have been his way of announcing his strong sense of his own capabilities, in which case there may have been an element of personal manifesto involved in the play-acting. It is tempting to imagine that he painted this truculent picture to show Giuseppe Cesari that he could be much more than a hack studio assistant.\n\nWhile Bacchus symbolizes inpiration, he also stands for disorder, anarchy, an unruly surrender to the senses. He is passion, opposed to the reason embodied by Apollo. He is the enemy of civilization, capable of laying waste to an entire society: in Euripides' tragedy _The Bacchae_ he destroys Thebes by luring its people into the mountains to join in his revels. The city's outraged king, Pentheus, is torn limb from limb by the god's intoxicated female followers, the Bacchantes. Pentheus' mother, Agave, is at their forefront, bearing her son's head aloft in triumph. In her ecstasy she sees him as a lion, fit to be slaughtered.\n\nThe madness and the maenadism associated with the myth had been painted most memorably \u2013 and most disconcertingly \u2013 by Titian in his celebrated _Bacchus and Ariadne_ , now in London's National Gallery. As Bacchus leaps down from his chariot to join the mortal woman with whom he has suddenly fallen in love, his rowdy mob continues with its orgy. The god's followers include the fat Silenus, drunk beyond coherence, and a young satyr with glazed eyes who drags behind him, as if it were a toy, the severed head of a sacrificed calf. It had been Titian's achievement to distil the violence and weirdness of the Bacchic cults to a single image. He had conjured up a Renaissance equivalent to the frenzy described in Catullus' famous 64th poem \u2013 which was, almost certainly, one of his principal sources:\n\n> Bacchus was rushing up and down with his dancing band of satyrs... looking for you, Ariadne. Some of them were waving thyrsi with covered points, some were tossing about the limbs of a mangled steer, some were girding themselves with writhing serpents; some were bearing in solemn procession dark mysteries enclosed in caskets, mysteries which the profane desire in vain to hear. Others were beating tambourines with uplifted hands, or were raising sharp ringings from cymbals of rounded bronze...\n\nAll this is relegated to the background of Caravaggio's self-portrait, which, in its dryness, restraint and small scale, is a world away from Titian's seductively orgiastic mythology. But it is there by implication. The violence that impends, the rending of the flesh, the drunkenness, the cannibalism \u2013 these things lurk in the teasing expression on the painter's face. Might he have actually painted the picture behind his master's back? Could it have been an act of truancy from the demeaning drudgery of the pure still life painting to which he had been assigned in the Cesari workshop? It has a sorceror's apprentice feel to it, with its hints of illicit goings on, after dark and away from prying eyes. By the light of the moon, the young painter dares to dress up as a god of misrule.\n\n_Boy with a Basket of Fruit_ is a fresher, brighter painting. But there is maybe more to this work too than at first meets the eye. The viewer is confronted by a blushing, smooth-skinned adolescent, with dark curly hair and an expression of amorous intensity on his face. On the admittedly slender evidence of a later self-portrait by Mario Minniti, it is possible that this was one of the pictures for which Caravaggio persuaded his new Sicilian friend to model. The boy carries a woven basket filled to overflowing with fruit \u2013 a cornucopia by comparison with the mere pair of peaches and the solitary bunch of grapes perched before the figure of Caravaggio-as-Bacchus. The basket contains four bunches of grapes, one red, two black and one green, as well as three apples, a peach and a pair of medlars. A pomegranate, split open to reveal its purple seeds, and four figs, two green and two black \u2013 the latter so ripe that they too have split to disclose the yellow and purple flesh within \u2013 also appear.\n\nThe picture has been interpreted in a number of sharply differing ways. It is plainly a kind of demonstration piece, painted to exhibit the young Caravaggio's skill in depicting not only fruits and foliage, but also the human face and form. Some writers have regarded it as a straightforward genre painting, a portrait of a handsome young fruit-seller plying his trade. Others claim to detect echoes of classical literature \u2013 in particular, the fables of Pliny the Elder, whose _Natural History_ is the principal source of information about the painters of antiquity. Pliny's encyclopedic book contains several stories and parables intended to demonstrate the heights of virtuosity reached by the artists of ancient Greece, as they competed to create an art of perfectly deceptive illusionism:\n\n> The contemporaries and rivals of Zeuxis were Timanthes, Androcydes, Eupompus, Parrhasius. This last, it is recorded, entered into a competition with Zeuxis. Zeuxis produced a picture of grapes so dexterously represented that birds began to fly down to eat from the painted vine. Whereupon Parrhasius designed so lifelike a picture of a curtain that Zeuxis, proud of the verdict of the birds, requested that the curtain should now be drawn back and the picture displayed. When he realised his mistake, with a modesty that did him honour, he yielded up the palm, saying that whereas he had managed to deceive only birds, Parrhasius had deceived an artist.\n\nPliny adds that, as a riposte to Parrhasius, Zeuxis also painted a picture of a child holding grapes. Once more the birds tried to eat the fruit, but this time Zeuxis felt he had failed. He disconsolately pointed out that if his picture had been perfectly lifelike, the birds would have been too frightened by the painted boy to peck at the painted grapes in his hands.\n\nIt was not uncommon for Italian artists of the fifteenth and sixteenth centuries to create their own versions of lost paintings from the classical past. So perhaps Caravaggio's choice of subject was intended to evoke that same picture of a child holding grapes by Zeuxis \u2013 and, indeed, to surpass it. No birds would ever dare to pick at the fruit in _this_ basket. The blushing boy, whose tunic has slipped off his shoulder, is tremblingly alive. There is a slight awkwardness in the handling of his anatomy \u2013 an uncertainty in the juncture of his collarbone and right shoulder, which seems as a result unnaturally enlarged \u2013 but he is a compelling presence none the less. While the basket of fruit advertises Caravaggio's ability to capture different tones, textures and colours, the figure of the boy demonstrates a yet rarer gift: the ability to suggest human emotion. Those ardent, intently gazing eyes are filled with longing, even love. This striking intensity of feeling is inconsistent with the notion that the picture is simply a genre painting, a snapshot of daily life. Neither can it be readily explained by reference to the classical past.\n\nHow should we think about this remarkable face? Those who subscribe to the romantic myth of Caravaggio as a social and sexual outsider, boldly expressing the love that dares not speak its name, are obliged to twist the fruit-bearer's expression of amorous yearning into the come-hither eyelash-flutterings of a rent boy. Howard Hibbard's biography of Caravaggio, published in 1983, contains a brief but exemplary statement of this line of argument: 'There is a soliciting aspect to this picture, and since some of Caravaggio's other paintings of the 1590s are apparently homosexual in implication, we may read at least unconscious elements of this kind into the _Boy with a Basket_ , whose fruits have various potentially symbolic meanings.'\n\nAlthough Hibbard's interpretation is, I believe, thoroughly misguided, it contains an element of truth. There _is_ a link between the figure's mood of sensual abandon and the luscious fruits that he bears, many of which \u2013 especially the figs, apples and pomegranate \u2013 had ancient sexual connotations. But the explanation for that lies not in the artist's supposedly devil-may-care determination to flaunt his homosexuality. It lies in the words of an ancient Persian love poem, absorbed long ago into the Judaeo-Christian tradition and known as the Song of Songs or the Song of Solomon, the most flagrantly erotic text in all of the Old Testament. It takes the form of a poetic dialogue between two lovers, the Bride and the Groom, who express their feelings for one another in imagery of a rich and fecund natural world.\n\nThe Groom compares his beloved to a garden: 'A garden inclosed is my sister, my spouse; a spring shut up, a fountain sealed. Thy plants are an orchard of pomegranates, with pleasant fruits...' (4:12\u201313). For her part, the Bride describes the Groom as 'white and ruddy, the chiefest among ten thousand. His head is as the most fine gold, his locks are bushy, and black as a raven... His mouth is most sweet: yea, he is altogether lovely. This is my beloved, and this is my friend, O daughters of Jerusalem' (5:10\u201316). Finally, the Groom describes the fruition of their desires: 'How fair and how pleasant art thou, O love, for delights! This thy stature is like to a palm tree, and thy breasts to clusters of grapes. I said, I will go up to the palm tree, I will take hold of the boughs thereof; now also thy breasts shall be as clusters of the vine, and the smell of thy nose like apples; and the roof of thy mouth like the best wine for my beloved, that goeth down sweetly, causing the lips of those that are asleep to speak' (7:6\u20139).\n\nThe iconography of Caravaggio's painting is extremely close to that of the Song of Songs. The boy's basket is filled with the fruits described in the poem, while the boy himself has all the attributes of the Groom, with his ruddy cheeks, his hair 'as black as a raven'. So tender and languorous is his gaze that he might readily be imagined actually reciting the verses of the Song of Songs to his beloved. His lips are parted, as if to speak or sing.\n\nThe Song of Songs was a controversial religious text among Christians and Jews alike precisely because of its profound eroticism. In the first century AD one of the rabbis to argue most passionately for its inclusion in Jewish scripture, as the 'Holy of Holies', also condemned the secular practice of singing it in banqueting halls, which suggests that sacred interpretation of the text had long been shadowed by suspicion of its sensuality. By the time Caravaggio painted his _Boy witha Basket_, in the late sixteenth century, Christian Church fathers had spent considerably more than a millennium teasing out what they had come to see as the redemptive symbolism of the poem's tale of love. The Groom's passion for the Bride was held to express Jesus Christ's boundless love for his holy mother, Mary. The metaphor of the Bride as an 'inclosed garden' was easily transformed into a symbol of Mary's virginity.\n\nBut, to judge by the remarks of St Teresa of Avila, who wrote her own commentary on the Song of Songs in 1573, such forms of allegorical interpretation were not always easily understood by congregations in the world of the Catholic Counter-Reformation. As she noted, bawdy laughter at the sexual connotations of the poem's language might easily interrupt even the most solemn, sacerdotal reflections on the Song of Songs: 'Indeed, I recall hearing a priest... preach a very admirable sermon, most of which was an explanation of those loving delights with which the bride communed with God. And there was so much laughter, and what he said was so poorly taken, that I was shocked.'\n\nCaravaggio's painting, like that priest's sermon, has also provoked ribald comments and has inevitably been susceptible to erotic interpretation. That ambiguity has perhaps always been part of its meaning. To borrow a phrase applied to Caravaggio's work as a whole by his contemporary Cardinal Ottavio Paravicino, it is a picture that seems poised 'between the sacred and the profane' \u2013 in this case, concealing a devout message within an apparently profane, secular subject. To those who would be blind to its spiritual dimensions, the painting was designed to remain a merely enchanting parade of sensual delights \u2013 a picture of a boy with a puzzlingly languorous expression on his face, carrying a basket of fruit. But to those who knew how to see through the sensual surface, the boy reveals himself as the Groom in the Song of Songs and therefore as the type of the young Jesus Christ, an image at once of love and vulnerability.\n\nHe is bare from the shoulder, not only because he is rapt in symbolic love for his divine mother, but also in anticipation of his crucifixion, the sacrificial gift of love he bears to all humanity. The shadows that flicker on the wall behind him, set against the light that illuminates his face, are shadows of death from which his own image, and with it the promise of eternal life, radiantly emerges. The same Christian message, that eternal life can be salvaged from the jaws of death, lurks in his basket of fruit. Withered, worm-eaten leaves of the vine contrast with ripened bunches of grapes. From death, once more, shall come life. The fading foliage is decay, transience, the passing of all things here on earth. The grapes are wine, the wine of the Eucharist that is the sacrificial blood of Christ. The picture offers not only a gift but a stark contrast of alternatives. What will you have? Death or life? Darkness or light?\n\n#### BOY BITTEN BY A LIZARD\n\nCaravaggio's _Self-Portrait as Bacchus_ and _Boy with a Basket of Fruit_ are subtle and ambitious paintings, not the work of a painter likely to be satisfied with long hours and low pay working as another artist's fruit and flower specialist. They corroborate Bellori's assertion that Caravaggio 'worked reluctantly' at whatever hack work was assigned to him and 'felt deeper regret at not being able to paint figures'.\n\nWith the _Bacchus_ , Caravaggio asks to be taken seriously, to be recognized as a painter not only of inspiration and intelligence but of something more than that. The picture announces Caravaggio's spirit of unruly unpredictability, and shows for the first time the face of a man quite capable of overthrowing the tired artistic conventions of his time. With the _Boy with a Basket_ , he demands to be regarded as better than a mere still life painter, and expresses the hope that one day \u2013 one day soon \u2013 he might be allowed to try his hand at devotional pictures.\n\nEven this early in his career, at a time when so much of his life and personality are obscure, certain things are clear. Caravaggio wants to paint the human figure and he wants to treat what, for his contemporaries, are the deepest and most serious subjects \u2013 the great Christian themes of salvation and damnation. His art is both sensually and intellectually seductive. It is carefully calculated to appeal to the more discerning and well-educated type of Roman patron \u2013 someone likely to be high up in the hierarchy of the Roman Church, keenly attuned to the subtle devotional symbolism of a picture such as the _Boy witha Basket_, or to respond to a secular, mythological painting like the _Self-Portrait as Bacchus_.\n\nSo it is no coincidence that the young Caravaggio should have gravitated towards the company of churchmen. The more he could infiltrate the higher circles of the Roman clergy, the more likely he would be to win meaningful patronage. At first he had stayed with the unsatisfactory Pandolfo Pucci, 'Monsignor Salad'. Around the beginning of 1595, after eight months in the Cesari workshop and a spell in hospital, he lodged once more with a man of the cloth. According to Mancini, the struggling Caravaggio found support from a certain 'Monsignor Fatin Petrigiani, who gave him the comfort of a room in which to live'.\n\nThere were no fond farewells to Giuseppe Cesari. Whether Caravaggio left Cesari's employ before or after the murky events that led to his hospitalization, they parted on bad terms. Whatever the personal reasons for the bad blood between them, professional jealousy also probably played a part. There are hints, in Mancini's manuscript notes, that Cesari deliberately attempted to hold his talented young apprentice back for fear of being outshone. The perennially abrasive Caravaggio was a born innovator who had little time for the art of most of his contemporaries (he would later say as much in one of his several appearances before the Roman magistrates), so likely regarded the fey late Mannerism of Cesari's mature style with naked contempt. Cesari's prestige among the most influential Roman collectors and patrons can only have made Caravaggio's own position all the more galling to him. Being studio assistant was bad enough, but being studio assistant to an overrated mediocrity must have been more than his pride could stand.\n\nAccusations of arrogance echo through the early biographies of Caravaggio. 'Michelangelo Merisi was a satirical and proud man,' writes Baglione; 'at times he would speak badly of the painters of the past, and also of the present, no matter how distinguished they were, because he thought that he alone had surpassed all the other artists in his profession.' Bellori explicitly says that pride drove Caravaggio to leave Cesari and strike out on his own. At this point Bellori introduces another character into the narrative, a well-known painter of amusing bizzareries named Prospero Orsi, who suddenly appears as the rebellious Caravaggio's sidekick, egging him on to rebellion and independence: 'When he met Prospero, a painter of grotesques, he took the opportunity to leave Giuseppe in order to compete with him for the glory of painting. Then he began to paint according to his own inclinations; not only ignoring but even despising the superb statuary of antiquity and the famous paintings of Raphael, he considered nature to be the only subject fit for his brush.'\n\nBellori treats Caravaggio's rejection of Cesari as if it had been the publication of a manifesto. In his eyes, Caravaggio had not just turned away from one man's influence; he had repudiated the entire classical and Renaissance canon and abandoned those principles of selection and idealization on which all truly great and lasting works of art must be founded. He describes it as an act of foolhardy hubris. 'As a result, when he was shown the most famous statues of Phidias and Glykon, in order that he might use them as models, his only answer was to point towards a crowd of people, saying that nature had given him an abundance of masters.'\n\nThe idea that anyone would have taken the time to call the young Caravaggio's attention to the sculptures of classical antiquity is probably fanciful. The neatness of his rejoinder strengthens the suspicion that this is parable, rather than fact (Bellori in effect admits as much when he concedes that 'A similar story is told about the painter Eupompus'). Yet the fiction is revealing because it contains, in a nutshell, the academic artist's innate distrust of Caravaggio's startling naturalism. The painter is cast as gifted but fatally proud, a man bent on dragging art down into the gutter \u2013 leading it towards the mere unthinking replication of reality. The same attitude, softened by time but equally misguided, lies behind more recent attempts to expose the presumed trickery behind Caravaggio's art \u2013 the suggestion that the painter must have used some kind of lens to achieve his effects, or the hypothesis that it was all (literally) done with mirrors. The one grain of truth in Bellori's account may lie in what it has to say about the sheer strength of early audience response, favourable or otherwise, to the seductively lifelike qualities of Caravaggio's paintings.\n\nHaving left the Cesari studio, Caravaggio certainly needed to sell his paintings. His stay with Monsignor Petrigiani may not have lasted long. Baglione says that, soon after leaving Cesari, Caravaggio 'tried to live by himself' and that he painted some self-portraits at this time, lacking the funds to hire a model. 'He also painted a boy bitten by a lizard emerging from flowers and fruits; you could almost hear the boy scream, and it was all done meticulously.'\n\nThere are two extant versions of this subject, one in the Roberto Longhi Foundation in Florence, the other in the National Gallery, London. Technical analysis, as well as its slightly more crude and direct style, suggests that the Longhi picture was created first, probably in late 1594 or at the start of 1595. The handling of the drapery is more assured in the London picture, which also points to a slightly later date. Yet the very existence of this second, slightly more sophisticated variation indicates that Caravaggio had scored enough of a success with his original version to create a market for replicas.\n\nOnce again, Caravaggio paints a single figure in an interior lit by raking light. But this time he animates the figure, having him actively recoil in pain and, as Baglione says, utter an almost audible scream. The painter emphasizes the effects caused by his use of a single light source, pushing the contrast between light and dark to an unprecedented degree.\n\nThe subject is a moment of compressed drama. A young man has been unpleasantly surprised during what should have been a quiet moment of unalloyed pleasure. Reaching out towards the selection of fruit laid out on the table before him \u2013 two bright red cherries, some figs and some grapes are visible \u2013 he finds that he himself is being bitten, by a creature that has been lurking unseen. The animal, a lizard, buries its fangs into the fleshy part of his middle finger. The boy's face, startled and flushed with the sudden consciousness of pain, is strongly illuminated. His bare shoulder and tensed right hand, from which the lizard still dangles, are thrown into sharp relief.\n\nThere is a slightly clumsily painted pink rose behind the boy's ear, while the artist has also included a vase on the table in front of him, which is three quarters full of water and contains another rose and some stalks of flowering jasmine. Light slows and thickens to a texture like that of milk in the depths of the water. Reflections play in the convex surface of the vase, and two drops of condensation trickle down its fatly curved side. This is a piece of painting that evokes Giorgio Vasari's description of a work by the young Leonardo da Vinci, the most famous painter to have worked in Caravaggio's home town of Milan \u2013 a picture of the Virgin 'in which, besides the marvellous vividness, he had imitated the dewdrops so that the picture seemed more real than life'.\n\nThe exquisite still life is a naked demonstration of skill \u2013 a reminder that when Caravaggio painted it he was working for the open market and therefore, in a sense, crying out his wares. He included the detail to impress his mastery of certain virtuoso techniques in oil painting on his prospective Roman audience \u2013 ways of painting the reflection and refraction of light, of capturing the precise wetness and viscosity of a drop of sweat, a drop of water or a drop of blood, which could make the practice of art seem almost like a form of magic. Despite Vasari's encomium to Leonardo, such skills were primarily associated with artists from Flanders. Jan van Eyck had been the first Renaissance master to master them, followed by Rogier van der Weyden, Hans Memlinc and others. Caravaggio's inclusion of such effects in his own work advertised his roots in Lombardy, in northern Italy, where Flemish art was better known than in the rest of the peninsula. His handling of the vase and its reflections indicate that he was familiar with the work of later Flemish masters such as Jan Bruegel.\n\nBut the still life detail _is_ only a detail, a grace note in a picture designed primarily as a vehicle for the depiction of a human being gripped by sudden, strong emotion. Contrary to Bellori's assertion that Caravaggio turned away from all artistic tradition to pursue an art rooted solely in study from life, the figure of the boy is extremely sculptural. He was painted from a model, but he also evokes that very tradition of classical statuary which, according to Bellori, Caravaggio despised. The most obvious precedent for the boy who screams in pain was the celebrated classical statue of _Laoco\u00f6n_ and his sons, wrapped in the coils of snakes, which had been excavated in Rome less than a hundred years earlier. Even the lizard may have been inspired by a classical sculpture, namely the so-called _Apollo Sauroctonus_ , or _Lizard Apollo_ , which is now in the Louvre but was probably in Rome in Caravaggio's time. The reptile climbing up a tree trunk in that sculpture is shown from the same, sharply profiled angle \u2013 seen as if from above \u2013 as Caravaggio's lizard.\n\nThe German art historian, painter and engraver Joachim von Sandrart, who travelled widely in Italy between 1628 and 1635, gave _BoyBitten by a Lizard_ a prominent place in the short account that he wrote of Caravaggio's early years. To judge by its tone, he must have spoken to artists or collectors who still remembered the picture's thrilling impact from some forty years before: 'In the beginning, he painted many faces and half-length figures in a sharp, dry manner. One of these is that of a child with a basket of flowers and fruit, from which a lizard emerges, biting the hand of the child who begins to cry bitterly, so that it is marvellous to look at and it caused his reputation to increase notably throughout Rome.'\n\nSandrart mistakenly refers to the picture's vase of flowers as a basket. Perhaps he confused its still life, in his memory, with that in the earlier _Boy with a Basket_. But his report vividly demonstrates the extent to which the exploits of the young Caravaggio were still remembered, still talked about, in Rome even as late as the 1630s. His informants, whoever they were, also gave him to understand its startling combination of emotional intensity and artistic naturalism as a gauntlet thrown down at the feet of Giuseppe d'Arpino and his followers: 'Because Arpino generally painted large works in fresco, which does not in itself have the same strength of colour or the intrinsic truth of oil colours, and because Caravaggio was very excellent in the latter, he offered Giuseppe and many others a challenge which resulted in endless quarrels. This brought them to swords' points...' Sandrart also tells the story that Caravaggio painted a picture in the Roman church of San Lorenzo in Damaso, next to an altarpiece by Giuseppe d'Arpino, in which 'he represented a nude giant who sticks out his tongue at Giuseppe's work as if he wished to ridicule it.' The tale of the nude giant with the mischievous tongue is certainly apocryphal, nor is there any other evidence to suggest that Caravaggio and Giuseppe Cesari, the Cavaliere d'Arpino, ever came to blows (if they had, Cesari might never have lived to a ripe old age). But there is perhaps a glimmer of fire behind all the smoke. In Sandrart's telling of the story, Caravaggio becomes far more than a disgruntled studio assistant with the nerve to walk out on his boss. He becomes a rival, someone who turns away from his former master's style and subject matter because he has his own ideas.\n\nThe most original aspect of _Boy Bitten by a Lizard_ is the fact that it depicts an ordinary person \u2013 someone distinguished by no particular signs of rank or status \u2013 in the grip of a strong emotion. One of the few known precedents for this lay in late sixteenth-century Bolognese art. Giorgio Vasari tells of a female painter called Sofonisba Anguissola, originally from Cremona, who created a drawing for Tommaso de' Cavalieri \u2013 once a close friend of the great Michelangelo \u2013 in which she depicted 'a little girl laughing at a boy who is weeping because one of the cray-fish out of a basket full of them, which she has placed in front of him, is biting his finger; and there is nothing more graceful to be seen than that drawing, or more true to nature'.\n\nVasari's story about another work of art may shed light on the meaning of Caravaggio's. The introduction of a faintly malevolent laughing girl complicates the story of a boy bitten by surprise. Perhaps Anguissola may have intended some playful allusion to the hazards of adult love that lie in store for every child. This, in turn, may begin to suggest the symbolic intentions that lay behind Caravaggio's own choice of the theme. Is the presence of erotic temptation implied in his _Boy Bitten by a Lizard_? There is reason to think so.\n\nThere is an air of abandonment about the boy, imparted both by his languid state of undress and by the rose in his hair. Roses are traditional emblems of romantic love, but the other blossoms present in the picture add a less innocent note to its symbolism. Jasmine was a traditional symbol of desire (Caravaggio would include the same flower in his later portrait of a well-known Roman courtesan). The boy's clothing, such as it is, a wispy piece of white drapery, might be no more than a twisted bedsheet. He who reaches for cherries and apples has grasped at sexual temptation. Now he is receiving his just reward. A sexual subtext lurks, as the lizard had done, in that pile of luscious fruit. The animal is zoologically inaccurate \u2013 real lizards have no teeth \u2013 but charged with metaphorical potency. A toothless reptile has been transformed into the very image of the _vagina dentata_.\n\nIt would have required no great ingenuity on the part of Caravaggio's contemporaries to unlock his meaning. In the sign language of the Italian street \u2013 symbolism in its most vivid, popular form \u2013 the bitten finger represented the wounded phallus. The English diarist John Evelyn witnessed a quarrel between two boatmen in seventeenth-century Genoa, at the end of which one of them 'put his finger in his mouth and almost bit it off by the joynt, shewing it to his antagonist as an assurance to him of some bloodie revenge'. The threat on that occasion, as Evelyn euphemistically hints, was castration. A different fate can be understood to lie in store for Caravaggio's decadent young man: in Rome, city of courtesans, the reward for promiscuity was venereal disease. 'The French disease', they called it in Italy (although the French themselves preferred to think of it as 'the Pox of Naples').\n\n_Boy Bitten by a Lizard_ is a _vanitas_ painting, a reflection on the pitfalls that await those who give themselves up to the pleasures of the flesh. It is a work of art that functions in a way exactly analagous to the action which it depicts. An apparently innocuous image, full of sweet fruit and lingering sensual detail, hides the sourest of morals. The message of the picture might seem unnecessarily severe, but it should be remembered that Caravaggio's target audience was the higher Roman clergy. They needed the alibi of moral reflection to enjoy \u2013 let alone purchase \u2013 a picture such as this.\n\n#### GYPSIES AND ROGUES AND A CARDINAL SNARED\n\nFor all his ingenuity, Caravaggio did not enjoy immediate success with _Boy Bitten by a Lizard_. According to Mancini, the painter was forced to sell the work for next to nothing. In Baglione's yet bleaker telling of the story, Caravaggio failed to find a buyer for any of the pictures that he painted after leaving the Cesari workshop: 'He was unable to to sell these works, and in a short time he found himself without money and poorly dressed.'\n\nDesperate for money, the artist went to the picture-dealers of Rome. According to Baglione, 'some charitable gentlemen expert in the profession came to his aid, and finally Maestro Valentino, a dealer in paintings at San Luigi dei Francesi, managed to sell a few.' This 'Maestro Valentino' was actually Costantino Spata, who did indeed have a shop in the piazza bordering San Luigi dei Francesi, the so-called 'church of the French'. He befriended Caravaggio and Prospero Orsi. He sold their pictures on commission and was seen drinking with them on several occasions.\n\nCostantino Spata played a vital role in Caravaggio's career. It was through him that the painter came to the attention of one of his most important supporters, his principal patron during his early years in Rome. Baglione tells the story in a few words: 'This was the means by which he met Cardinal del Monte, an art lover, who invited him to his home.' Cardinal del Monte would nurture Caravaggio through the next few crucial years of his life. Not only would he house, clothe and protect him, but he would introduce him to a circle of the most powerful and influential collectors in Rome, and negotiate the difficult waters of higher Church patronage on his behalf.\n\nDel Monte, the 'art lover', whose palace was just around the corner from the Piazza di San Luigi, was one of Spata's clients. Did the dealer and the painter think up a deliberate strategy to get the cardinal's attention? Did Spata even advise Caravaggio on _what_ to paint, helping to bait the hook that would land the big fish? Certainly, the work that Caravaggio created for his new dealer to try to sell was markedly different from anything he had painted before.\n\nThe two pictures with which Caravaggio and Spata successfully tempted del Monte, _The Gypsy Fortune-Teller_ and _The Cardsharps_ , still exist. The first is to be found in the Capitoline Museum in Rome (a later and even finer version of the same composition, painted for a friend of del Monte, is in the Louvre). The second is in the Kimbell Art Museum in Fort Worth, Texas. Between them, they mark a radical new departure for Caravaggio, and indeed they are among the most innovative pictures created anywhere in Europe in the late sixteenth century.\n\nEach painting shows a scene of trickery and deceit, enacted by half-length figures. 'Genre picture' was the less than satisfactory term eventually settled upon by art historians to describe such works. But the genre picture in this vein did not exist until Caravaggio invented it. Although there had been shadowy precedents for such work, in prints and drawings and in marginal details of paintings about other things, _The Gypsy Fortune-Teller_ and _The Cardsharps_ introduced a new concept to art: the low-life drama. Hung together in a single room in del Monte's Roman palace, their influence was soon felt far and wide. The taste for such pictures grew rapidly and spread across all of Europe. Caravaggio's tricksters spawned a whole world of painted rogues, created by a multitude of artists including Bartolomeo Manfredi in Italy, Rembrandt in Holland and Georges de La Tour in France.\n\nThe differing dimensions of the two canvases suggest that they were not painted as a pair, although both are offspring of the same idea. In _The Gypsy Fortune-Teller_ , a sharply dressed young man with a sword at his hip has fallen under the spell of a smiling young Romany traveller. She fixes him with an intense and slightly nervy stare. He returns her hypnotic gaze with a dreamy, half-lost expression of his own. Shadows play on the dun-coloured wall behind the two figures. The precise nature of the action was explained by Mancini: 'I do not think I have seen a more graceful and expressive figure than the Gipsy who foretells good fortune to a young man... he shows the Gipsy's slyness with a false smile as she takes off the ring of the young man, who shows his naivete and the effects of his amorous response to the beauty of the little Gipsy who foretells his fortune and steals his ring.' Under the pretence of reading the young man's palm, the streetwise confidence trickster is actually robbing him.\n\n_The Cardsharps_ plays a variation on the same theme, a gentleman fooled out of his money. The scene is a gambling den, in which we encounter the second of Caravaggio's fresh-faced, rich young men, playing a game of cards. He is dressed in sumptuous black silk over a lace-trimmed shirt \u2013 sleek finery that has drawn the attention of not one but two urban predators. The yellow-and-black stripes of their costumes suggest the image of a pair of wasps buzzing around a honeytrap. Some honey has already been extracted, to judge by the detail of a backgammon board, pushed to the edge of the gaming table. Having failed at one game, the young gentleman is trying to win back his losses at another. His optimism is undimmed, to judge by the half-smile that plays on his lips. But he cannot possibly win. The young cheat sitting opposite him has a choice of extra cards tucked into his belt behind his back. The other peeks over the young gentleman's shoulder and signals in code to his partner in crime, letting him know exactly what will be required to ensure a winning hand.\n\nThe older of the two conmen, with his holed, threadbare glove and black cloak \u2013 perfect for melting into the unlit gloom of Rome's streets by night \u2013 is the spying accomplice described in numerous books and pamphlets of the time. There was a thriving literature devoted to the tricks of the street and, in particular, the devices of the card cheat. A popular Italian treatise on gambling, entitled _The Book on Games of Chance_ ( _Liber de ludo aleae_ ), was written by the mathematicican, astronomer and failed card-player Gerolamo Cardano (1501\u201376). Countless other texts listed the various techniques used by cheats at the gaming table. One of the most widely read, a work first published in England in 1552 under the title _A Manifest Detection of the Most Vile and Detestable Uses of Dice-play, and Other Practices Like the Same_ , contains a more or less exact description of the ruse played out in Caravaggio's _Cardsharps_ : 'Of this fraternity there be that called helpers, which commonly haunt taverns or ale-houses, and cometh in as men not acquainted with none in the company, but spying them at any game will bid them God-speed and God-be-at-their-game, and will so place himself that he will show his fellow by signs and tokens, without speech commonly, but sometime with far-fetched words, what cards he hath in his hand, and how he may play against him. And those between them both getteth money out of the other's purse.'\n\nThere is a narrative and symbolic affinity between the two pictures painted for Cardinal del Monte and the work that immediately preceded them, _Boy Bitten by a Lizard_. All three tell of a man undone by his own vices, of youth suddenly clouded by the prospect of disease, loss or debt \u2013 a pattern that the artist had perhaps experienced during his own youth in Milan. But what made _The Gypsy Fortune-Teller_ and _The Cardsharps_ so startlingly original was their unprecedentedly close focus on the world of the street and the gambling den.\n\nThe subject matter of these paintings was highly topical. Counter-Reformation Rome was a city in which all manner of thieves, rogues and scoundrels thronged. Their presence was a symptom of social crisis. Recurrent plague not only destroyed lives, but ravaged economies in the cities and states where it struck. The number of displaced and unemployed people had grown alarmingly during Caravaggio's lifetime. Sixteenth-century Italy had also been racked by an almost constant state of war, resulting in a large, permanently uprooted population of mercenaries. When they had money such men gambled, drank and whored. If the recruiting officer did not call, they were liable to turn to crime. In Rome, ever a magnet for the poor in need of alms, they often masqueraded as pilgrims.\n\nA vivid picture of this confusing world, where scurrility often dressed itself in the clothes of virtuous indigence, is painted by a set of notarial documents dated February 1595 (probably only a year or so before Cardinal del Monte purchased _The Gypsy Fortune-Teller_ and _The Cardsharps_ ), which record the interrogation of a young man in the prison on the Sistine Bridge, in the centre of Rome. 'I am called Pompeo,' the boy declared. 'I was born in Trevi near Spoleto, I am about 16 years old, I have no occupation, I was arrested by your men in the church of S. Giacomo degli Spagnoli, because I was begging for alms during Mass.' When he was asked if he knew anything about other beggars in the city, and whether they formed a single sect, or many, he gave the following answer: 'Sir, among us poor beggars there are different confraternities [ _compagnie_ , a word normally used to refer to religious confraternities]... the first is called the Confraternity of the Grencetti, those who, while they are begging for alms in the churches in a crowd, cut purses... The second is called the Confraternity of the Sbasiti, and includes those who pretend to be ill and lie on the ground as if they were dying and keep groaning and demanding alms. The third is called the Baroni, who are healthy and upright, and are sturdy beggars who do not want to work.' By the time he had finished, Pompeo had listed no fewer than nineteen groups of fraudulent beggars. They included the Formigotti, who pretended to be discharged soldiers; the Rabrunati, who faked epilepsy by eating soap and then foaming at the mouth; and the Pistolfi, who posed as priests to extract 'donations' from their victims.\n\nThe same set of documents also contains the testimony of another man claiming familiarity with different groups of criminals at work in Rome in the 1590s. His name was Girolamo, and to Pompeo's nineteen categories of villain he added seventeen more, including the Marmotti, who affected to have been struck dumb, and the Spillatori, who \u2013 like Caravaggio's _Cardsharps_ \u2013 sought out the gullible in taverns and inns and cheated them out of their money using marked cards and loaded dice. Girolamo disagreed with Pompeo's (presumably) ironic comparison of such groups of criminal specialists with religious confraternities. 'They are not confraternities [ _compagnie_ ] but crafts [ _arti_ ], like shoemakers, goldsmiths and so on.'\n\nThe exact status of these texts is questionable, and the precision with which they reflect actual criminal activity in Caravaggio's Rome is open to debate. Peter Burke, who translated and republished them, notes that 'these documents cannot now be found in the Roman archives, and are best known though a copy made a few years later, which used to be in the former Imperial Library in Berlin', with 'the rather literary title of the \"delightful examination\" of rogues, \" _Il dilettevole essamine de' guidoni, furfanti o calchi_ \" '.\n\nIt is likely that the documents are altered transcripts of reported speech; and almost certain that they have been liberally 'improved' by the late sixteenth-century writer who gave them their present form. Many of the practices that they record, such as the use of soap to feign epilepsy, were (and still are) used by real fraudsters. They are part of an oral history of fraudulence that certainly has a basis in fact. But other elements seem to have been exaggerated. Analysis of the certain testimonies of known criminals that still _are_ to be found in the Roman archives presents a less well-organized and significantly less colourful picture of criminal activity \u2013 gangs of crooks robbing at random when the opportunity presents itself, fencing stolen goods to the city's Jewish pedlars and immediately eating and drinking away the proceeds.\n\nFor all that, the tales told by 'Pompeo' and 'Girolamo' contain their own kind of truth, halfway between fact and fiction. What these stories reveal above all is a particular set of stereotypes about the seamy side of life in Rome, in which a certain sector of the city's elite wished to believe: that criminals were so well organized as to constitute a dark, mirror-version of normal society, complete with 'fraternities' or 'guilds' of particular ill-doers; that they had, in effect, created a kind of inverted world of their own, a _mondo alla rovescia_ , or 'world upside down' (a phrase of the time which anticipates the modern term 'underworld'); that any apparently needy beggar might easily turn out to be a crook.\n\nThere was something of a craze for the classification of rogues in Caravaggio's Italy. Tommaso Garzoni's _La piazza universale di tutte le professioni del mondo_ ( _Universal Marketplace of the World_ ), a compendium published in 1585, had listed seventeen types of false beggar, including five that appear in the accounts of 'Pompeo' and 'Girolamo'. There is strong evidence that such reports tended to multiply at times of genuine social crisis, especially when the existing structures of poor relief were being put under intense pressure by plague or famine. There were several occasions during the 1580s and 1590s \u2013 the plague of Caravaggio's childhood, in 1576\u20137, being a prime example \u2013 when the sheer number of beggars and indigent people at large threatened to overwhelm the ability of the Italian states to function. Tough measures against itinerant beggars and 'vagabonds' were periodically introduced throughout the late sixteenth century: in Florence in 1576, in Milan the following year, in Genoa in 1582, in Palermo in 1590 and in Rome throughout the 1590s. The stories that circulated about 'fraudulent beggars', the growing literature devoted to their typology and taxonomy, were partly a reflection of reality and partly a reflection of deep anxiety within the governing classes of society. They multiplied in much the same way as stories about supposed illegal immigration or welfare fraud multiply in right-wing newspapers today, at times when the economy or social services come under strain.\n\nThe medieval Christian attitude to the poor had been essentially supportive. Every poor person was to be seen as the living image of the impoverished Christ himself, and helped accordingly. St Francis of Assisi had gone so far as to declare himself married to 'Lady Poverty'. But by the late sixteenth century such attitudes had undergone a sea change. In many states \u2013 including the papal states \u2013 the poor were viewed with increasing distrust and hostility. In some places they were simply driven out by edict. Elsewhere the great _lazzaretti_ , the plague hospitals of the Middle Ages and Renaissance, were converted into poor houses where the indigent were coralled and forced to do menial work.\n\nThe ruling and religious elites of the day were bitterly divided by the issue. The more authoritarian wing of the Catholic Church favoured rigorous means of social control and repression. But there were also those more sympathetic to the ancient medieval view \u2013 orders such as the Franciscans and Jesuits \u2013 who continued to plead for sympathy with the poor. There was, in other words, a form of right-wing \/ left-wing split in Caravaggio's Rome over the treatment of poverty. One way to view the growing literature devoted to 'roguery' \u2013 whether the writings of Garzoni or the 'improved' trial transcripts attributed to 'Pompeo' and 'Girolamo' \u2013 is as propaganda for the right.\n\nAll this helps to clarify the most important questions that need to be asked about these two pivotal works in the painter's career. Can the _Gypsy Fortune-Teller_ and _Cardsharps_ be seen as a plea for sympathy for those afflicted by poverty? Or are they simply a translation into painting of the sinister mechanisms of state control \u2013 a _visual_ means ( _pace_ Burke) 'of legitimating the repressive measures' taken against those living on the margins of society?\n\nCaravaggio's pictures were certainly not painted for political ends. They were created to amuse and entertain an art-loving cardinal. But it may be significant that del Monte, who dared to hang these startlingly novel pictures of low-life characters on the wall of his palace, was a Medici supporter who publicly shared the Medici's known sympathy for the 'pauperist' views \u2013 as they have been termed \u2013 of Filippo Neri and the Oratorians. In other words, he was a man who stood to the left in Rome's divide on the issue of the poor.\n\nCaravaggio was undoubtedly familiar with the overwhelmingly negative picture of the rogue or trickster presented in so much of the moralizing literature of his time. His depiction of the _Cardsharps_ is so close to the accounts of card cheats in texts such as _A Manifest Detection_ , or the seventeenth-century judge Antonio Maria Cospi's book of advice to magistrates, _Il giudice criminalista_ (1643), that he may actually have consulted such works when planning his composition. Cospi's section on the marking of cards is a virtual gloss on the action in _The Cardsharps_ :\n\n> I have seen those who have marked the edge of the corner of the card with ink, who bend with the right hand a suit towards the narrower part, and bend the other suit at the same angle, but on the longer side... Many other observations could be made, not all of which could be foreseen or imagined, but this is enough to awaken the mind of the magistrate, should some suspect cards come into his hands, to observe or discover if there is some other mark on them. This is as much as I can say as regards the eye. There are also those who know the cards by touch, and these make a little hole with a needle that stands out in relief on the underside of the cards. According to the place where they feel this slight relief, they know which card it is that goes to their opponent or that they take from themselves. Others play with thick cards with such thick colours that they have a certain relief. They keep the tip of the middle finger of the right hand well shaven, so that the skin there is very sensitive, and on touching the card with that finger, they sense those colours and know which card is underneath.\n\nThis explains why the cheat's accomplice in Caravaggio's picture has two prominent holes in his glove. The glove has not been worn by use. Its stitching has been unpicked, so that the trained and sensitive middle finger and thumb of the sharper can do their work. But, despite such finely observed details, it would be a mistake to assume that Caravaggio's picture is an overtly moralizing work of art. The artist might have drawn on texts such as Cospi's book for magistrates, but he himself reserved judgement.\n\nDespite Caravaggio's vaunted reputation for realism, he emphatically refused to present his image of cardsharping as a slice of reprehensible reality. This is no snapshot from the scene of a crime. It is a piece of lively, intriguing theatre. The gestures of the crooks \u2013 especially the pantomimic semaphore of the accomplice's hand signal \u2013 plainly come from the world of drama. In an actual gambling den, such overt gesticulation would soon be discovered. But imagine _The Cardsharps_ as a scene from a play, performed for an audience happy to suspend disbelief, to enjoy the sense of superiority that comes from knowing that _they_ can see everything that the gulled cardplayer is blind to \u2013 and the exaggerated body language of the figures makes perfect sense.\n\n_The Cardsharps_ plays on the threat of _il mondo alla rovescia_ , a world turned upside down, where wily guttersnipes win and aristocrats lose. But its message is not morally straightforward. The rich young man will no doubt return to his palace at the end of the game. The scruffy _bravi_ robbing him blind \u2013 who may indeed be out-of-work mercenaries, to judge by the younger cheat's sword \u2013 will no doubt drink their gains away and end up back in the gutter. But for this brief moment they are victorious.\n\nCaravaggio's painting is ambiguous, but contains a hint of where his sympathy \u2013 or, at least, his empathy \u2013 might lie. He paints the young gull with a form of smooth indifference, as a softly generalized figure of aristocratic insouciance. By contrast, the cheats themselves are live, lithe, fascinating. The older man's concentration is absolute, and touched by a sense of desperation. The younger conman, gazing with fixity at his prey, is as tense and alert as a feral cat. Caravaggio paints his desperadoes like a man who feels _with_ them, if not necessarily for them. He understands the deep seriousness of their desire to work their trick, to carry out their strategy without a hitch. When he painted the picture his own predicament was not altogether dissimilar to theirs.\n\nThe trickster in _The Gypsy Fortune-Teller_ belonged to an even more reviled class of 'vagabond' than the cardsharps. For Cospi, gypsies were the lowest of the low because their habits of thievery were innate rather than learned. His entry on them in _Il giudice criminalista_ is an undisguised racist diatribe \u2013 a nearly hysterical expression of burgeoning hatred for a people initially welcomed to Italy, in the early fifteenth century, as refugees and pilgrims:\n\n> They are thieves by nature, descended from Cus son of Ham, cursed son of Noah... They still feel this paternal curse, wandering dispersed around the world without being able to find a homeland or other permanent place... They sell their own sons for food... They come from the region between Egypt and Ethiopia and wander through the world, erecting their tents outside cities in fields and highways. They make deception, changes and prognostication from the lines of the hand, and earn their living by these amusing frauds... Like beasts, they consider marriage to their own sisters legitimate... The women steal chickens, and while they pretend to tell one's fortune by the signs of the hand, rob the peasants and steal the women's purses and handkerchiefs.\n\nEnglish attitudes could be just as virulent. In Thomas Dekker's pamphlet of 1608, _Lanthorne and Candle-light_ , gypsies are described as: 'a people more scattered than the Jewes and more hated: beggarly in apparell, barbarous in condition, beastly in behaviour and bloudy if they meete advantage. A man that sees them would sweare they all had the yellow Jawndis, or that they were Tawny Moores bastardes...'\n\nBut when Caravaggio painted his _Gypsy Fortune-Teller_ he went against the grain of such crude stereotypes. His gypsy is a thief, for sure, but she is a far cry from the subhuman monster of Cospi and Dekker. She is a beautiful enchantress, an exotic swindler who steals her victim's heart as surely as she pilfers the ring from the hand he drowsily surrenders to her. Like Caravaggio's cardsharps, she has stepped into his painting straight from the theatre \u2013 and emphatically not from the pages of judges or journalists seeking to control a perceived social menace.\n\nThree years before Caravaggio painted the picture, Cesare Ripa published an enormously influential guide to the symbolism of the post-Renaissance world, entitled the _Iconologia_. The book is a description, as its title page says, of 'diverse Images of Virtues, Vices, Affections, Human Passions, Arts, Disciplines, Humours, Elements, Celestial Bodies, Provinces of Italy, Rivers and every region of the world'. The gypsy appears twice in Ripa's encyclopedia of imagery, each time as a woman. On the one hand, she is an emblem of poverty, shown 'with a twisted neck' in the act of begging for alms: 'poverty is represented in the guise of a Gipsy,' explains Ripa, 'because a poorer folk than this is not to be found; for they have neither property nor nobility nor taste, nor hope of anything that can give a particle of that happiness that is the aim of political life.' But she is also a symbol of comedy, of light-hearted resilience to the blows of fortune.\n\nUnder this aspect, Ripa notes, 'her dress should be of various colours; in her right hand she should carry the horn which is used as a musical instrument; in her left hand she should have a mask, and she should wear socks on her feet. The diversity of colours signifies the varied and diverse actions dealt with by this sort of poetry, which delight the eye of the mind no less than variety of colours the eye of the body through their expression of the accidents of human life, of virtues, vices and worldly conditions, as found in every quality and kind of people, except those of princely blood.' Here Ripa is distantly following the classical theory of theatrical genres propounded by Aristotle in his _Poetics_. Aristotle's distinction between tragedy and comedy, much parroted by the more prescriptive literary theorists of sixteenth-century Italy, held that tragedy should focus on the actions of the elite \u2013 kings and princes \u2013 while comedy should concern itself with the behaviour of those at the very bottom of the social heap.\n\nCaravaggio's _Gypsy Fortune-Teller_ is smiling poverty personified. But she is no mere emblem. With her turban-like headscarf and long cloak she is, in fact, dressed precisely as a real gypsy in late sixteenth-century Italy. Cesare Vecellio's _Habiti Antichi et Moderni_ , or _Costumes Past and Present_ , published in 1590, contains a description that tallies more or less exactly with Caravaggio's painting. Vecellio notes that gypsy women 'bind a cloak of woollen cloth over the shoulder, passing it under the arm, and it is long enough to reach down to their feet'. The cloak, known in Italian as a _schiavina_ , is defined by the writer as 'a long garment of coarse wool, worn by gypsies and hermits'.\n\nThe ancestry of Caravaggio's beautiful gypsy can be clearly traced \u2013 in accordance with the assumptions behind Ripa's _Iconologia_ \u2013 to the world of comic theatre. The gypsy was a stock figure in the performances of Italian Commedia dell'Arte, the popular acting companies of the sixteenth century \u2013 so much so that the name _zingaresche_ , derived from _zingara_ , or 'gypsy', was given to a whole range of comic theatrical productions. A series of French prints known as the _Recueil Fossard_ documents the performances of an Italian Commedia dell'Arte troupe given in France in the late sixteenth century. One of those prints, depicting the encounter of the brazen whore 'Peronne' with the louche aristocrat 'Julien le Debauche', bears a striking resemblance to Caravaggio's own _Gypsy Fortune-Teller_.\n\nThe connections between Caravaggio's painting and the theatre do not stop there. One of the more celebrated late sixteenth-century performances of a _zingarescha_ can in fact be traced directly to the milieu of Cardinal del Monte. In 1589, when the cardinal's Medici patron, the Grand Duke of Tuscany, married Christine of Lorraine, a theatrical festival was staged to celebrate their union. A play entitled _La Pellegrina_ was performed, together with six extravagant intermedii involving all kinds of elaborate stage machinery and sets representing both a fiery hell and a cloud-capped Mount Olympus. Shortly after this, according to the diary of an eyewitness, the grand duke invited the Comici Gelosi \u2013 one of the leading Commedia dell'Arte troupes \u2013 to 'act a comedy of their own choice'. The two leading ladies, Isabella Andreini and Vittoria Piissimi,\n\n> nearly came to blows, for Vittoria wanted to act _Zingara_ and the other wished to perform her Pazzia, entitled _La Pazzia d'Isabella_ [ _The Madnessof Isabella_] \u2013 given that Vittoria's favourite part is Zingara and Isabel's La Pazzia. However they finally agreed that the first piece to be acted would be _Zingara_ , and that _La Pazzia_ would be given another time. And so they performed the said _Zingara_ with the same _Intermezzi_ as were prepared for the great play; and indeed whoever has not heard Vittoria perform _Zingara_ has neither seen nor heard something marvellous, and certainly all were very satisfied with the play.\n\nCardinal del Monte attended this actual performance, and it is quite conceivable that Caravaggio himself had first-hand experience of the Gelosi on stage. The company, which was the most prestigious in Counter-Reformation Italy, had close links with his home city of Milan, where its first recorded performance took place in 1568, just three years before the painter's birth. During its forty-year existence it often played there and in the other principal towns of northern Italy: Florence, Ferrara, Genoa, Mantua and Venice.\n\nIt is possible that Caravaggio's painting was actually inspired by the memory or repute of Vittoria Piissimi's celebrated performance as the _Zingara_. In Artemio Giancarli's comedy, written in 1545, the gypsy plays the role of kidnapper, temptress and wily thief. But those who saw Piissimi in the role remembered her, above all, as the temptress: 'a beautiful sorceress of love, she entices the hearts of a thousand lovers with her words; a sweet siren, she enchants with smooth incantations the souls of her devout spectators.' Caravaggio's contemporaries praised his own, painted gypsy in strikingly similar language. In Mancini's eyes, she might have been 'false' and 'sly', but above all she was beguilingly beautiful \u2013 the most 'graceful and expressive figure', indeed, that he had ever seen in art. The painter's friend, the poet Gaspare Murtola, went even further. In Murtola's madrigal in praise of Caravaggio's painting, the gypsy is not only an enchantress, she is also the painter's alter ego. Just as she deceives her fresh-faced admirer, so Caravaggio beguiles the world with the freshness and the beauty of his art:\n\n_Non so qual si piu maga_ ,\n\n_O la donna, che fingi_ ,\n\n_O tu che la dipingi_\n\nI don't know who is the greater magician,\n\nThe woman, who deceives,\n\nOr you, who paint her\n\nThe poet rhymes _fingi_ with _dipingi_ , cheating and painting. So Caravaggio is not merely the painter of rogues, crooks and the enchantresses of the street. He is the painter _as_ vagabond. And suddenly all of his subtle counterfeiting has paid off. His illusions have worked their magic, his paintings have been sold \u2013 and he has been invited to live in the house of a cardinal. It is the autumn of 1595 and he is twenty-four years old.\n\n## PART THREE  \nRome, 1595\u20139\n\n#### FRANCESCO MARIA BOURBON DEL MONTE\n\nCaravaggio's patron looks out at posterity from a vivid drawing by the printmaker, painter and master-draughtsman Ottavio Leoni. He has kind but piercing eyes and a fully receded hairline. His thin lips and slightly weak mouth are disguised, not altogether successfully, by a wispy salt-and-pepper beard. Cardinal Francesco Maria Bourbon del Monte was approaching seventy when he sat for the likeness; twenty years had passed since he had taken Caravaggio into his home. But he was still the same inquisitive, thoughtful man whom the painter had known. His epitaph would stress above all that he had always done his best to support 'the good arts'.\n\n_Ars longa, vita brevis_. The picture was done, in a single sitting of perhaps half an hour, in black chalk with white highlights on fine-grained paper the colour of a hazy blue sky. The cardinal seems to endure the ordeal of keeping still with patience and forbearance: unlike many powerful men, he does not frown and fidget his way through a sitting. There is a mixture of worldliness, compassion and curiosity in his gaze. The finishing touch is a tricorn hat, rendered in dense cross-hatching, perched on the smooth dome of his forehead. It makes him look a little bit like a chess piece come to life.\n\nFrancesco Maria del Monte may have been the first father-figure in Caravaggio's life. Giovanni Baglione, terse as ever, described the artist's time with del Monte as a rare idyll in his otherwise troubled existence. 'In these quarters Michelangelo was given room and board, and soon he felt stimulated and confident.' Stimulated and confident: such adjectives were not often applied to Caravaggio by people who actually knew him. This is the only passage in Baglione's biography of Caravaggio where he appears as anything other than mad, bad and dangerous to know. We can sense the painter's genuine relief at having found, at last, a refuge from the storms of his early life.\n\nWhen Caravaggio met him, del Monte was in his late forties, one of the younger and more energetic cardinals. But, unlike most of those elected to the curia, he was neither particularly rich nor especially aristocratic. He owed his position to a combination of solid family connections, considerable charm and \u2013 so jealous contemporaries muttered \u2013 outrageous good fortune. Del Monte had been born in Venice, on the Fondaco dei Turchi, in 1549. It is a measure of his family's importance to the city that the great Venetian painter Titian attended his baptism. So too did the notorious poet, pamphleteer and pornographer Pietro Aretino, a man who might be said to have embodied the deepest contradictions of his age. On the one hand, he encouraged Pope Paul IV to fig-leaf the genitalia in Michelangelo's frescoes for the Sistine Chapel; on the other, he wrote such works as _Tales of Nuns_ , _Wives and Courtesans_ , the opening scene of which involves numerous nuns, their lubricious mother superior and a copious supply of glass dildos. Also present at the ceremony was the less colourful but widely celebrated architect Jacopo Sansovino.\n\nDespite the pomp that attended his baptism, del Monte would not actually be brought up in Venice. Del Monte's father, Renieri, is known to have been in the service of the dukes of Urbino. Ever since the days of the fifteenth-century soldier-intellectual, Federigo da Montefeltro, the rulers of Urbino had hired out their services as battle-hardened mercenaries to the highest bidder. The Duke of Urbino who employed del Monte's father was particularly active on behalf of the Venetians during the years from 1539 to 1552. Since Renieri went by the title of 'colonel', it seems likely that he was a soldier, who had won the respect of the Venetians by fighting their enemies.\n\nDel Monte was decidedly not a military man, but a student of law and humanities. He and his elder brother, Guidobaldo \u2013 later to become a distinguished mathematician and the author of a treatise on perspective \u2013 were educated at the courts of the della Rovere family in Pesaro and Urbino. They also studied at Padua, long established as a centre of humanist learning, which was where Prince Francesco Maria della Rovere himself received his education. Del Monte had been named in honour of Prince Francesco Maria. But he later switched allegiances and eventually travelled to Rome, in 1572, in the service of a Sforza cardinal.\n\nDel Monte switched allegiances again in the early 1570s. He won the patronage of Cardinal Ferdinando de' Medici, younger son of Grand Duke Cosimo I, ruler of Florence and Tuscany. Groomed for the Church from an early age, Ferdinando had been made cardinal when he was just fourteen years old. He was a patron of music as well as a discerning art collector, who adorned the gardens of the Villa Medici with ancient Roman sculptures. Del Monte worked for many years as Ferdinando's secretary and assistant. By the mid 1580s he had become his closest confidant. Then, in 1587, both men's lives were transformed by news of a dramatic series of events in the Medici stronghold of Florence. Ferdinando's elder brother, Grand Duke Francesco I, had died of a mysterious illness. The duke's wife had succumbed to the same ailment. In Florence, with its long and murky political history of plot and counter-plot, foul play was inevitably suspected. With Medici power in the balance, Ferdinando felt compelled to renounce his vows and return to Tuscany. He became grand duke, and del Monte his right-hand man. A contemporary witness described the atmosphere at court in the immediate aftermath of Ferdinando's accession. The new grand duke would dine alone, allowing no one save his trusted adviser to share 'his most secret thoughts'. He considered del Monte a kindred spirit, the source added, because he was 'knowledgeable in literature and other learned subjects'.\n\nFerdinando's resignation from the curia left the ruling family of Florence without a voice in Rome. So in 1588 the new Grand Duke of Florence used his influence with Pope Sixtus V to have del Monte appointed in his place. Del Monte would remain a cardinal for almost forty years, reporting to his Florentine master on the twists and turns of papal politics and promoting the interests of Tuscany whenever he could. His umbilical connection to Florence, and to the court of Ferdinando de' Medici, would have numerous consequences for Caravaggio's career.\n\nThe Medici had strong links with the pauperist wing of Counter-Reformation Catholicism. During his years in Rome, Ferdinando had been close to the charismatic churchman Filippo Neri, the dominant religious personality in the city during the second half of the sixteenth century and founder of an order of secular priests known as the Congregation of the Oratory. His style of teaching was informal and direct, inspired by a desire to return to the simplest and most direct forms of Christian belief. He preferred discussion to sermonizing, improvisation to the set text, and had a knockabout, down-to-earth sense of humour. Despite a profound difference in temperaments, Neri was greatly admired by Carlo Borromeo, who on several occasions in the 1560s and 1570s protected him from accusations of heresy. One of Neri's ideals was pilgrimage, which he interpreted as a model for the Christian life itself, as a journey travelled in prayer. As well as the Oratory, he founded the Archconfraternity of the Most Holy Trinity of the Pilgrims and Convalescents \u2013 the Santissima Trinit\u00e0 dei Pellegrini \u2013 to care for the poor and the sick, and especially the many thousands of destitute pilgrims who travelled to Rome during Jubilee years. The Pellegrini would eventually have their own church, a somewhat severe building constructed between 1587 and 1597 and designed by Martino Longhi the Elder (the father, as it turned out, to one of Caravaggio's most turbulent companions). But much of its energies were devoted simply to health care. For several years Ferdinando de' Medici served honourably as protector of its hospital. Probably as a result of his connections with del Monte and the Medici, Caravaggio would himself develop close links with Neri's Archconfraternity of Pilgrims, and with the Order of the Oratory. Two of his most impressive altarpieces, _The Madonna of Loreto_ and _The Entombment of Christ_ , would result.\n\nThe political alliances of the Medici would also shape the development of Caravaggio's painting. Throughout Cardinal del Monte's long Roman career, but especially during the early years, when he was closest to Caravaggio, the balance of European power was delicately poised between Spain and France. Like other members of the family before him, Ferdinando de' Medici favoured France. He married the Valois princess Christine of Lorraine, the marriage celebrated with great pomp and ceremony in 1589. Just over a decade later the Medici's links with France would become closer still. Ferdinando's niece, Marie de' Medici, would marry Henri IV and become Queen of France. That union could never have taken place without Henri IV's acceptance into the Church of Rome. So throughout the early 1590s del Monte's overriding concern, and the Medici's greatest goal, was to ensure that Henri IV's conversion from Protestantism went ahead as planned. Del Monte's diplomatic style was subtle and self-effacing, but effective. In 1593, when the long-hoped-for event occurred, the Medici cardinal could congratulate himself on having played his part in one of the decisive political events of the age. Clement VIII was deeply grateful for del Monte's help in winning the French king back to the Catholic faith. The cardinal's position within the curia was strengthened as a result.\n\nIt was no coincidence that Caravaggio's first major religious commission, secured for him by del Monte \u2013 'his cardinal', as the jealous Baglione put it \u2013 would be for San Luigi dei Francesi, the church of the French in Rome. The paintings would be completed in 1600, the year of Henri IV's marriage to Marie de' Medici. The first of them, _The Calling of St Matthew_ , showing the saint roused from spiritual slumber by the coming of Christ, probably alluded to the conversion of the French king. When del Monte looked at the picture, he could reflect on his finest hour as a servant of the Medici, and France.\n\nThe cardinal had two official residences in Rome, the Palazzo Firenze, near the old Roman harbour of the Ripetta, and the Palazzo Madama, around the corner from San Luigi dei Francesi. The Palazzo Madama was where he chose to live, and where he gave Caravaggio room and board \u2013 presumably on one of the attic floors of the palace, in the servants' quarters. The painter's new surroundings were visible proof of his sudden change of fortunes, a far cry from his mean lodgings with Monsignor Insalata and a world away from the ramshackle platform on which he had been compelled to sleep in the Cesari workshop. The Palazzo Madama was an imposing building in the heart of Rome, its broad fa\u00e7ade emblazoned with the famous Medici coat of arms, a shield decorated with six round balls \u2013 often, apocryphally, said to symbolize pills, but actually emblematic of bezants, or coins, in allusion to the family's origins as moneylenders. The state rooms of the palace were richly decorated with tapestries and oriental carpets, as well as a small but choice selection of classical sculptures and other hallowed relics of the distant Roman past. These included the most celebrated cameo-glass vessel to have survived from antiquity, the so-called Portland Vase.\n\nDel Monte was forever buying and selling works of art, antiquities, precious stones, sculptures and curiosities. He kept a sharp lookout for anything that might interest his Medici patrons. In 1607 he excitedly reported his acquisition of some fragments of clothing, discovered on the Appian Way, that had once belonged to a Roman consul alive at the time of First Punic War. He was sending them to the grand duke as a gift, he wrote, so that he could study 'the weaving of those times' (as ruler of Florence, a city at the centre of the Italian textile trade, Ferdinando could reasonably be assumed to take an interest in such a find).\n\nThe cardinal was an insatiable accumulator of all kinds of things, but above all he accumulated paintings. His collection included allegories and narrative pictures as well as a number of still lives \u2013 and, of course, Caravaggio's _Cardsharps_ and _Gypsy Fortune-Teller_ , those pioneering experiments in the painting of contemporary rogues and tricksters. Inventories show that at his death del Monte owned around 600 paintings, enough to furnish an entire museum. He possessed copies of celebrated pictures by masters of central Italian painting such as Leonardo da Vinci, Michelangelo and Raphael. But he was also drawn to the very different traditions of Venetian painting, owning no fewer than five pictures attributed \u2013 possibly with more optimism than accuracy \u2013 to Titian. The names of Palma Vecchio and Jacopo Bassano also figure in the lists, alongside that of Giorgione, an artist to whom Caravaggio was often compared in his youth.\n\nThe Venetian tradition valued _colore_ above _disegno_ , emphasizing the primacy of colour rather than design \u2013 whereas for the great painters of central Italy, the Tuscan\u2013Roman axis of art for which Giorgio Vasari was such a vocal and persuasive spokesman, drawing was the foundation stone of all excellence. Caravaggio seems to have had almost no interest at all in theories of art. But he shared the Venetian preference for working on canvas, rather than in the medium of fresco. In the ages-old debate about the relative merits of _disegno_ and _colore_ he might have sided with the Venetians. Not a single independent drawing survives by Caravaggio's hand. Even X-rays of his finished work have failed to yield anything resembling a conventional underdrawing.\n\nThe nature of the collections in the Palazzo Madama may have reflected the cardinal's roots in Urbino. Like Federigo da Montefeltro, whose _studiolo_ was lined with portraits of famous men, del Monte made a point of collecting images of those whom he admired. By far the greatest part of his collection was made up of portraits, a pantheon of intellectual and spiritual heroes. A late inventory refers to '277 pictures without frames... of various popes, emperors, cardinals and dukes and other illustrious men and some women'. In addition, the collection contained 67 paintings of saints. These too were portraits of a kind \u2013 images of those individuals from sacred history whom the cardinal especially venerated.\n\nThe breadth of del Monte's interests was reflected not only in the various rooms of his palace, which contained a well-stocked library and an extensive collection of scientific instruments, but also in the wide circle of his acquaintances. Del Monte collected remarkable men in real life as well as in art. He knew writers, bibliophiles and collectors of rare manuscripts. He knew musicians and composers. He knew alchemists, astronomers and others working on the ill-defined border between medieval belief and modern enquiry.\n\nInspired by the researches of his own brother, Guidobaldo, del Monte took a lively interest in scientific discovery. He was an early and enthusiastic supporter of Galileo, and played a crucial role in the astronomer's career by helping him to secure the patronage of the Medici. Without the powerful support and protection of Florence's ruling dynasty, some of Galileo's most important work might never have been done. The Palazzo Madama contained a tangible symbol of the scientist's gratitude: the gift of a telescope. Del Monte's views on the controversial hypothesis of a heliocentric universe are unrecorded. But he might have agreed with Cesare Baronio, a prominent member of Filippo Neri's Order of the Oratory, who famously remarked that 'the scriptures teach us how to move to heaven, not how the heavens move.'\n\nDel Monte's interest in science extended to experiment as well as study. He dabbled in alchemy and had a fully equipped laboratory in the Palazzo Madama. Within a year of Caravaggio's arrival in his household, the cardinal acquired a third residence, a country retreat at Porta Pinciana, up in the hills above the western edge of the city, not far from the Villa Borghese. Here del Monte established a pharmaceutical distillery. The distillation of drugs, whether from plants, metals or other substances, was something of a fad in the elite circles of Roman society at the time. In his _Epistolae medicinales_ , the Sicilian physician Pietro Castelli (1590\u20131661) noted that the apothecaries of the day worked not only in their own shops but also in the private households of virtuosi. The efficacy of the resulting cures could be questionable. The German taxonomist and doctor Johannes Faber publicly boasted that the celebrated 'Cardinal Dal [ _sic_ ] Monte' had given him the recipe for a highly effective drug made from the meat of a poisonous snake. But he did not specify whether he had actually put the medicine to the test. Another of del Monte's supposed remedies was rumoured to have killed a man.\n\nFaber's story suggests that the cardinal took an interest in the activities of the Hospital of Santo Spirito, one of Rome's largest and most important institutions for the care of the poor and the sick. Faber was himself a physician of the hospital, which, in his estimate, provided more than 12,000 people with food, shelter and medical care every year. Del Monte was friendly with another doctor working at the Santo Spirito, Giulio Mancini, Caravaggio's future biographer. Mancini, born of humble Sienese parents, had trained in medicine in the city of Padua. He got his job at the hospital in 1595 and probably met the painter in del Monte's house in that same year. That would explain why Mancini knew so much more than the other early biographers about Caravaggio's dark deeds in Milan and his very first years in Rome.\n\nThe Hospital of Santo Spirito was closely connected to the papal court. A post there was often the prelude to a successful career in medicine at the highest level, and so it proved in the case of Mancini, who eventually rose to become physician by appointment to the pope. But Mancini's commitment to the relief of the poor seems to have been genuine, rather than just place seeking. When he died he left his considerable fortune to be distributed among the impoverished students of his native Siena. He was known for his unconventional behaviour and beliefs: a French obituarist wrote that Mancini was an amateur astrologer and _un Grand Ath\u00e9_ , 'a great atheist'. Del Monte probably befriended him because of his reputation as an experimental chemist and connoisseur of art. The two men seem to have shared an essentially philanthropic approach to life.\n\nDel Monte was a philanthropist but he was certainly no firebrand of Counter-Reformation piety. In one of his letters he describes an evening spent gambling at the game of hazard at the Farnese Palace, in the company of the cardinal-nephew Pietro Aldobrandini. Having lost heavily \u2013 'I more than he', del Monte noted ruefully \u2013 both men finished the evening in the company of a pair of courtesans, listening to music. Overall, the pattern of del Monte's friendships and alliances suggests that he was a worldly, benevolent, diplomatic, curious, open-minded and socially adept man, with a rare sensitivity to genius in other people and a strong sense of Christian charity.\n\nBut a considerably more negative picture of him was painted by his contemporary Dirck van Amayden, who composed the principal early biography of del Monte. Amayden's text, which has had a definite influence on the cardinal's posthumous reputation, is so hostile that it amounts to a thinly veiled character assassination. The author's method was a form of devious insinuation. This involved the recounting of various scurrilous tales about the cardinal, followed by half-hearted protestations on the part of the author to the effect that such dreadful things \u2013 surely \u2013 could not have been true.\n\nThe pattern is set by Amayden's discussion of Ferdinando de' Medici's patronage of del Monte. He begins with the phrase 'It is said', eternal refrain of the unreliable reporter. In this case, 'it is said' that del Monte wormed his way into Ferdinando's affection by arranging clandestine trysts between the young Medici buck and the wife of another man, 'the bride of one Cesarino'. The double calumny is followed up instantly by a sly denial of which Shakespeare's Iago would have been proud: 'this nevertheless I would not believe, knowing perfect friendship arises out of virtue, not vice.'\n\nNot only is Amayden's del Monte an accomplice to adultery. He also turns out, at the end of the biography, to be a closet homosexual with a particular fondness for young men. The author's explosion of this bombshell inevitably casts something of a shadow over his earlier, flatly dutiful assertion that del Monte was an intellectually enlightened patron of the arts and sciences who 'was very liberal to painters, chemists and similar'. The scandalized reader is naturally inclined to wonder whether the cardinal might have asked for certain favours, from certain young men, in return for his support. The very last paragraph of Amayden's life of del Monte addresses the question in the author's characteristic style and leaves the matter open:\n\n> He was of unusual sweetness of behaviour, and loved to be familiar with youths, not, however, for a criminal reason, but from natural sociability. This is presumably connected with the fact that he prudently hid it before Urban was elected. When Urban was made pope he threw off all restrictions; in the longed-for reign he indulged his inclination openly, and, though aged and almost blind, more a trunk than a man and therefore incapable of allure, a young man of short stature got a benefice from him.\n\nThe image of del Monte in old age as an absurd and enfeebled pederast is hardly flattering. It is also, in all probability, a fiction. Amayden spent his life in the service of Spain, promoting the cause of the mighty Habsburgs with unwavering constancy and taking every opportunity to blacken the names of their enemies. The pro-French faction at the papal court was anathema to him and he had a professionally ingrained hatred of the Medici. So this was a man with every motive to slander the memory of del Monte, whose curial vote had always been cast in favour of the Medici and their French allies. Amayden's story about del Monte belatedly coming out of the closet in 1623, on the accession of Urban VIII, should also be read as a slander aimed at the pope himself. The subtext is that Urban's reign was so licentious that every sin suddenly dared to show its face. This too fits with the hispanophile Amayden's anti-French agenda, since Urban VIII had shown great favour to Cardinal Richelieu, Governor of France under Louis XIV.\n\nDespite its implausibility, Amayden's text has insidiously shaped the legend of Caravaggio. It has fostered a deeply fanciful view of del Monte's household as a louche pleasure palace, subversively lodged at the heart of Catholic Rome. Through the rooms of this imaginary Palazzo Madama passes a parade of freethinkers and sexual outsiders, mostly exquisite young men. The shadowy figure of del Monte, libertine masquerading in a cardinal's robes, looks on with a mischievous twinkle in his eye. Viewed through the lens of this seductive fantasy, many of the pictures that Caravaggio painted during his sojourn with del Monte are correspondingly distorted. They become thrillingly decadent and disappointingly flimsy at one and the same time \u2013 mild exhalations of homoerotic yearning, shot through with an abiding spirit of perversity.\n\nThe best corrective to Amayden's sweet-tongued libel is an eyewitness description of Cardinal del Monte's household as it actually was in the 1590s, published for the first time in 1991. The recipient of the description was Grand Duke Ferdinando de' Medici. Its author was a musician and gentleman called Emilio de' Cavalieri. It presents a thoroughly believable, down-to-earth picture of the milieu in which Caravaggio, in his mid twenties, found himself:\n\n> Del Monte amazes me in regard to spending that he can live on what he has and do it so honourably. It is true that for his clothing he doesn't spend a _giulio_ ; he has had only one livery made; his coach is also the first he has had; he makes the best of what he has; he has bought himself a carriage and with this he keeps himself; the mouths he feeds in all don't amount to fifty; he doesn't keep horses or gentlemen but his servants are treated well and given good meals \u2013 all that is seen through your highness's favour of a beautiful home, [the fitting out and decoration of] which is now finished; as a cardinal of Rome, he formally receives at table in the morning with his silverware; and he is courted by more Romans than cardinals for his great trafficking, which is all honest, with his metalworkers; and his antechamber is always filled with people; there are no high-ranking clergy. The reason for this is that he is not involved in important transactions and those that come do so only to visit... I have made this speech so that you will know the truth...\n\nThis account is just as partisan in its way as that of Amayden (Cavalieri was a close friend of the cardinal and would later name him as one of his executors), but it has the disorderly ring of truth about it. Del Monte was anything but well off by the standards of most Roman cardinals. His residences were Medici property, not his own. His recorded income was approximately 12,000 ducats a year, by no means a great deal of money for a man in his position, so Cavalieri's sympathetic description of the household's thrifty but somewhat threadbare imitation of late Renaissance courtly pomp tallies well with the known facts. The reference to del Monte's 'trafficking... with all his metalworkers' suggests necessary financial dealings conducted on the side. The insistence that 'he is not involved in important transactions' with the pope or his fellow grand clerics may have been meant to reassure the grand duke that del Monte was sticking purely to Medici-approved business.\n\nCavalieri was from an old Roman family associated both with the arts and with artists. The legendary Michelangelo had been close to his father, Tommaso de' Cavalieri, and had given him a highly finished presentation drawing \u2013 possibly _The Rape of Ganymede_ of 1532, now in the Royal Collection at Windsor \u2013 as a token of his affections. Emilio himself was a composer and impresario, principally employed at the Medici court from the late 1580s as master of ceremonies for the elaborate entertainments known as _intermedii_ , dramatizations of myth and legend, set to music.\n\nDel Monte and Emilio de' Cavalieri probably met when the latter was orchestrating the unusually lavish spectacles that marked Grand Duke Ferdinando's marriage to Christine of Lorraine in 1589. By the early 1590s they had become firm friends. Del Monte was in Florence in 1595 to see a production of Cavalieri's _Gioco della Cieca_ , an early experiment in musical drama inspired by antiquity. Over the next ten years the composer often visited the cardinal in Rome. Cavalieri's letters back to Florence are a valuable source of information about del Monte's deep immersion in the musical culture of his time, illuminating his tastes and responses to the music that moved him. They also shed some oblique shafts of light on the very first picture that Caravaggio painted for his new benefactor: a compellingly ambiguous depiction of a group of musical performers, about to give a concert.\n\n#### 'IF MUSIC BE THE FOOD OF LOVE'\n\nPainted around the end of 1595, _The Musicians_ is one of the artist's most puzzlingly unorthodox creations. Four young men wearing classical drapery have been crowded into an airless interior. The central figure meets the spectator's gaze with a languorous, distracted look, absent-mindedly fingering the strings of the lute that he cradles in the crook of his right arm. Behind him, a dark-haired boy holding a barely visible cornetto \u2013 a hybrid instrument of the Renaissance, with a trumpet-like mouthpiece and the fingerholes of a recorder \u2013 looks up in a way that suggests the troupe has been disturbed while rehearsing.\n\nA third young man, pressed so close to the foreground that he might almost be on the point of falling out of the picture, studies a sheaf of music. He is presumably the singer, and therefore the star attraction. But he looks as though he is still learning his song and his back is conspicuously turned to the viewer. He wears his costume carelessly, as if he knows that nothing much is going to happen for a little while yet. The folds of white cloth in which he is draped have fallen off his shoulder and become ruched up under the purple silk bow meant to hold them in place, leaving him almost naked from the waist up. The somewhat ragged group is completed by a curly-haired boy, sitting to the lutenist's right, who has a pair of Cupid's wings strapped to his back and a quiver full of arrows hanging at his right shoulder. But firing darts of love is plainly the last thing on his mind. He looks down and helps himself to some grapes, as much out of boredom as hunger.\n\nThe picture is not in good condition, having suffered considerable damage during the two hundred years that it spent in obscurity after disappearing into a series of unknown collections in the early eighteenth century. The violin and the page of music in the foreground have been largely reconstructed by modern restorers; the lute has lost its strings. But the work's fundamental originality and oddity remain undimmed, despite considerable areas of paint loss.\n\n_The Musicians_ was clearly one of Caravaggio's better known early pictures, because both Bellori and Baglione mention it specifically. Baglione says that 'For Cardinal del Monte he painted a Concert of Youths from nature, very well.' Bellori describes it in the same terms: 'the Concert of Youths portrayed from life in half figures'. The young man with the cornetto, at the back, resembles Caravaggio himself, while the lutenist may be his friend Mario Minniti. But the composition as a whole radiates an air of contrivance. It resembles a frieze or bas-relief, rendered in paint. The four boys are so similar in aspect and demeanour that they might be clones of each other. The suspicion lingers that they were all based on the same figure, depicted from different angles and then collaged together to form a single composition. Perhaps when Baglione and Bellori talked of Caravaggio portraying from life and painting from nature they were not talking about the artist's processes \u2013 the use of models, and so on \u2013 but trying to capture the distinctive _mood_ of his picture. For all its artifice, it does have a certain clumsy lifelikeness. And that is precisely what made it so different from most earlier paintings of similar subjects.\n\nBy the late sixteenth century there was a long-established tradition of so-called 'concert' pictures. The genre had originated in Venice, and in its early form it is exemplified by the so-called _Le Concert Champ\u00eatre_ of around 1510, now in the Louvre. Once thought to have been painted by Giorgione but now generally attributed to Titian, it is a tender and lyrical fantasy. A young man in fine clothes strums at his lute while conversing with a shepherd. Two naked women are present alongside them, one filling a glass jug with water, the other breaking off from playing on her recorder to listen to the two men's conversation. The action takes place outside, in a golden, idealized landscape based loosely on that of the Veneto itself. The precise meaning of the _Concert Champ\u00eatre_ (if it has one) is open to debate, but the allegorical thrust of Titian's dreamlike vision is clear enough. It has its roots in the ancient, classical fantasy of pastoral retreat. The city sophisticate retreats to nature and finds there a world as pure as the clearest spring water, and a harmony as sweet as that of the most beautiful music. In Arcadia, he retunes the strings of his very being.\n\nAlongside this idealizing tradition of musical picture, there was another and more prosaic sort of painting that showed singers and musicians in mid performance. Sometimes such works were enlivened by touches of bawdy humour. In Callisto Piazza's _Concert_ of circa 1525, a group of performers is crowded into a shallow space, together with a single, male member of their audience. The most prominent of the musicians, a woman playing a lute, wears a low-cut bodice and has a coquettish expression on her face. Her admirer, who has evidently been enjoying the performance in more ways than one, wheels to face the viewer of the painting with a knowing look in his eye. The artist has furnished him with a phallic prop in the form of a sheathed dagger, fastened at his hip, which points towards the girl at an angle carefully calculated to indicate just what he has in mind.\n\nCaravaggio's _Musicians_ cannot easily be squeezed into the existing tradition of sixteenth-century musical paintings. It is certainly not a pastoral in the Venetian mode. Nor does it depict an actual performance, showing instead the preparations for one. There was no precedent for this. The presence of the boy with wings has prompted speculation that the picture might have been intended as an allegory of Music and Love. But that offers no real explanation for Caravaggio's most obvious departure from convention. Why should he have chosen to depict this rather ramshackle scene of musicians rehearsing?\n\nSolutions to the enigma may be found in the unusually broad and experimental musical tastes of his patron. Cardinal del Monte was actively involved in music at the papal court throughout the 1590s. Clement VIII put him in charge of a far-reaching reform of liturgical music, and he served as Protector of the Sistine Choir. Music was also an essential part of life at his various residences. In one of his letters back to Florence, Emilio de' Cavalieri gives a richly evocative description of an impromptu concert that took place one day in 1602 at del Monte's country house at Porta Pinciana. The admired soprano Vittoria Archilei was the surprise guest at an afternoon party, together with her husband and accompanist. Also present were cardinals Paravicino and Acquaviva, who had ostensibly come to see a vineyard in the grounds of del Monte's estate. Archilei was prevailed on to sing. She stunned her small audience with the naked emotion of her performance \u2013 so much so that even Cavalieri, who had helped to train her famously expressive voice, was surprised. He reported that because she was 'in a wild mood and singing in a vaulted room, I have never heard her in more beautiful voice. She gave so much satisfaction that Acquaviva said to me: \"I for shame did not weep.\" Paravicino said he never thought such refinement was possible. They are both musicians.'\n\nSuch expressions of dumbfounded pleasure go beyond the courtly formulas of polite approval. Archilei had clearly given an unusually affecting performance, but that is not the only explanation for the strength of response she received. Its surprise lay essentially in the fact that she sang on her own, in public, to the simplest of instrumental accompaniments. By the early 1600s, medieval polyphony \u2013 many voices singing different lines of music simultaneously \u2013 had been the overwhelmingly dominant mode of music for centuries. Monody, in which a single melodic line is carried by a solitary singer, was still relatively uncommon in concert performance. The solo voice accompanied by the solo instrument was unfamiliar, arresting. As the rapturous response to Archilei's singing shows, its potential was only just being developed.\n\nThe polyphonic and monodic modes are at opposite ends of music's emotional spectrum. Polyphony subsumes the individual voice within a choral harmony, reflecting the desire to conjure up an essentially otherworldly sound, such as the singing of the angelic host. Words are hard to distinguish in the layers of polyphonic singing. Syntax dissolves and sense is sacrificed for an effect of transcendence. By contrast, monody puts precise meaning and specific human emotions at the heart of music. The single melodic line, the solo voice, is easily understood. To follow its meanderings is to follow the contours of feeling expressed by words and music together (the theme of Vittoria Archilei's song would, almost certainly, have been unrequited love). It might be said that while polyphony aspires to heaven, monody expresses man.\n\n'The solo voice contains all the purity of music, and style and melody are studied and appreciated more carefully when one's ears are not distracted by more than one voice.' Baldassare Castiglione's _Book of the Courtier_ of 1528 shows that the fashion for the solo voice had roots in an earlier period of the Renaissance. Further proof of this lies in the fact that the Flemish composer Adrian Willaert, choirmaster at St Mark's Basilica in Venice in the 1530s, had rearranged a number of polyphonic madrigals so that they could be sung for the solo voice. What seems to have been most strikingly new in the more experimental singing of Caravaggio's time, seventy years later, was its strong emphasis on vocal expression. This was characterized by the development of the _stile rappresentativo_ , a style of monodic singing that followed the natural accents and rhythms of spoken language. It was an innovation that transformed the performance of choral music, and the style in which Vittoria Archilei would have sung.\n\nEmilio de' Cavalieri was himself a composer at the forefront of this shift in musical sensibilities. He understood exactly what was going on in del Monte's house that afternoon in 1602. What he describes, very precisely, is the shock experienced by the listeners as they encounter raw feeling through the medium of music. On this particular occasion, the already unfamiliar experience is amplified by the wildness of the singer's own mood and the cavernous acoustic of a high-ceilinged room. The audience of cardinals Acquaviva and Paravicino is genuinely astonished, and Cavalieri's parting shot \u2013 'they are both musicians' \u2013 is meant to underscore the sheer novelty of the performance. These men are experienced listeners and practitioners; they know music very well; but they have never heard music quite like _this_.\n\nThe origins of the musical transformation epitomized by Archileo's performance were (and still are) debated. A group of Florentine musicians active in the 1570s and 1580s had built a whole philosophy around the doctrine of a return to monody. For them, this was an extension of the Renaissance ideal of reviving the modes of classical antiquity. Their spokesman had been the humanist author Vincenzo Galilei \u2013 father of the scientist and astronomer \u2013 who was partly inspired by the mistaken belief that the drama of ancient Greece had been sung rather than spoken. Galilei argued in favour of the perceived simplicity and emotional directness of ancient monody, conjuring the romantic vision of a world in which singers might reclaim the fabled powers attributed to Orpheus. He urged that song and drama should be reunited once more, to tell the stories of ancient legend and move the hearts of men. The ideas expressed in Galilei's _Dialogue on Ancient and Modern Music_ , of 1581, would have profound implications for music in Italy and beyond. New and ever more nakedly emotional songs for the solo voice would be written and performed. The _stile rappresentativo_ would triumph and the abstract patterns and harmonies of medieval polyphony fall out of fashion. In the music of court entertainment, individual performers separated from the chorus to sing passionate songs of love and death. Such songs would become known as arias, as the old genre of _intermedii_ metamorphosed into a new, startlingly dramatic art form that became known, simply, as 'opera'.\n\nMany different musicians and composers claimed a hand in these changes. Cardinal del Monte's friend Emilio de' Cavalieri was prominent among them. Cavalieri was thoroughly disgusted when his rival, the singer and composer Giulio Caccini, took credit for inventing the _stile rappresentativo_. 'Everyone knows _I_ am the inventor of this style,' Cavalieri angrily countered in a letter of 1600. Posterity has sided with Caccini in that particular argument, partly because of his especially close association with the circle of Vincenzo Galilei. But Cavalieri's other big claim, to have written the very first opera, has been more widely accepted. The work in question, a musical drama in three acts entitled _La Rappresentazione di Anima e di Corpo_ , was composed in 1600 \u2013 at the height of his friendship with Caravaggio's cardinal, and just two years before his sudden death from an unknown illness.\n\nCardinal del Monte was, then, rather more than a mere amateur of music: through his various musical activities and associations he had assisted at the birth of momentous changes in both composition and performance. By supporting a pioneer like Cavalieri, by hosting events like Vittoria Archilei's remarkable concert, by reordering the priorities of liturgical music at the papal court and by subtly altering the style of the Sistine Chapel Choir itself to favour the expressive qualities of the human voice brought out by the _stile rappresantativo_ \u2013 by doing so much, del Monte had placed himself at the forefront of musical experimentation at the turn of the seventeenth century.\n\nDel Monte was also friendly with the nobleman and banker Vincenzo Giustiniani, whose palace was directly opposite the Palazzo Madama. Giustiniani was a fellow musical enthusiast, who in 1628 would write _A Discourse on Music_ describing the so-called musical _camerino_ \u2013 a purpose-built private chamber, 'nobly decorated with paintings made for the sole purpose' of setting the right mood and tone for intimate musical performances. One of del Monte's first acts on moving to the Palazzo Madama had been (in his own words) to 'reserve a room for Harpsichords, Guitars, a Chitarrone and other instruments'. By the late 1620s such rooms were a familiar sight in the palaces of the Roman aristocracy. But in the early 1590s, when del Monte had created his own _camerino_ at the Palazzo Madama, he had been setting a new trend. To judge by the inventory made after his death, it must have been a headily atmospheric space \u2013 a cross between a private concert chamber and a miniature museum on the theme of music. Del Monte's _camerino_ contained no fewer than thirty-seven musical instruments, not including the 'chest where the viols are'. On its walls hung four pictures, all of which were listed, simply, as _una musica_ , 'a scene of music'. One of these was Caravaggio's _Musicians_.\n\nSo why did the painter depart from all the known conventions of the so-called 'concert picture' and depict his musicians as an ensemble of the blatantly unready? Further clues lie in Emilio de' Cavalieri's letters, which show (among much else) that Cardinal del Monte was extremely interested in the technical aspects of singing and performing. One of the most promising singers in his household was a Spanish castrato named Pedro Montoya, to whom Cavalieri gave six singing lessons, some of which del Monte himself must have attended: 'The Cardinal del Monte was amazed because he [Montoya] can already sing to the same standard as Onofrio [probably Onofrio Gualfreducci, a gifted castrato attached to the household of Cardinal Montalto] and if he does not cause trouble, within a month he will surpass Onofrio.'\n\nCaravaggio's painting moodily evokes the milieu of del Monte's household \u2013 a laboratory of musical experiment and innovation, where performers rehearsed under the tutelage of the cardinal and his friends, and where the expressive, classically inspired _stile rappresantativo_ was taken to new extremes. The space into which Caravaggio's four boys have been crammed evokes the cluttered intimacy of the _camerino_ itself. Dressed in their makeshift _all'antica_ costumes, they are preparing to take part in a piece of musical theatre of just the kind favoured and supported by del Monte. A single voice will be accompanied by only two instruments, in emulation of that imagined golden age when the songs of Orpheus were heard. The theme of the piece is the intoxicating effect of music on those who are in love. The song studied by the boy with his back to the viewer is no longer decipherable, but it probably expressed some variant of the sentiments voiced by Shakespeare's Count Orsino in _Twelfth Night_ : 'If music be the food of love, play on.'\n\nBy painting a rehearsal rather than a performance, Caravaggio went behind the scenes of the traditional concert picture. He showed the long hours of preparation and the artifice that made possible the final, polished performance. In doing so, he paid subtle tribute to the active role del Monte himself played in the musical culture of his time. Once hung in the room that the cardinal had consecrated to music, the picture conjures up a scene in which his own, animating presence is forever awaited. It is the picture of a process that depends on the energies of the patron himself. Only when the cardinal arrives can the final preparations be completed, and the concert begin.\n\n#### _THE LUTE PLAYER_ AND THE _BASKET OF FRUIT_\n\nThe second of Caravaggio's musical paintings, _The Lute Player_ , was commissioned by del Monte's friend Vincenzo Giustiniani and probably painted around 1596. An effeminate young man plucks at the strings of a lute while gazing out at the viewer with an expression of such soulfulness that his eyes seem to be brimming with tears. Two musical part-books and a violin lie on the table before him beside some scattered fruit and a glass carafe full of flowers. The scene is lit by a bright, diagonal shaft of light that casts strong shadows.\n\nThe wistful singer has sometimes been taken for a girl. Bellori, for example, described the figure as 'a woman in a blouse playing a lute with the sheet music in front of her'. But the 1638 inventory of Giustiniani's collection unambiguously listed the work as 'a half-length figure of a youth who plays the lute, with diverse flowers and fruits and music books... from the hand of Michelangelo da Caravaggio'. There would seem little reason to doubt its accuracy. The singer's face is androgynous but the shirt, open almost to the waist, reveals no sign of a cleavage.\n\nIt is possible that Caravaggio's _Lute Player_ is an idealized portrait of del Monte's promising but potentially troublesome castrato, Pedro Montoya. Montoya joined the Sistine Chapel Choir in 1592 and left in 1600, so he was almost certainly in del Monte's household when the picture was painted. The soft, hairless skin and slightly swollen face of Caravaggio's lutenist are consistent with the hormonal side-effects of castration. There may be a glancing allusion to the pitch of the boy's voice in the part-books that lie on the table before him. The five-staved sheets of an open part-book reveal a number of madrigals. Beneath lies another part-book, prominently marked 'Bassus'. It is closed, perhaps the painter's way of indicating that this particular singer never would be capable of hitting the low notes.\n\nCastrati were much in favour in Rome in the years around 1600. Their rise coincided with that of the professional female singer, and both reflected the new taste for piercingly emotional music arranged for the single voice. In his _Discourse on Music_ , Vincenzo Giustiniani noted that 'the famous Vittoria Archilei' had established 'the true method of singing of women', adding that it applied equally well to sopranos singing in falsetto and the castrati of the Sistine Chapel choir. The castrato voice was valued for its sweetness and sensuality, as well as for its clarity of enunciation.\n\nCastrati were encouraged to learn musical instruments so that they might accompany themselves. Such tuition is likely to have been part of the regime in del Monte's household. The cardinal himself played the Spanish guitar, and it is possible that Caravaggio learned the same instrument while living there. A deposition lodged against him by his landlady in 1605 includes the complaint that he came to her house late at night with a group of friends, playing the selfsame instrument and singing lewd songs, and a later inventory of his possessions lists one.\n\nThe singer in the _Lute Player_ is anything but raucous. He opens his mouth 'not more than is necessary to converse with friends', as a contemporary singing manual advised those performing chamber music of this kind. The picture is in such good condition that the sheet music open on the table is still legible: four madrigals by the Flemish composer Jacques Arcadelt ( _c_. 1505\u201368): ' _Chi potra dir_ ', ' _Se la dura durezza_ ', ' _Voi sapete_ ' and ' _Vostra fui_ '. Their texts are a compendium of the conventions of the courtly love tradition, shot through with plaintive simile and metaphor \u2013 beauty that blinds like the sun, ardent fires of passion, cold unyielding marble of a proud woman's heart. ' _Chi potra dir_ ' is representative:\n\nWho can express what sweetness I taste\n\nIn gazing on that proud light of my lady\n\nThat shames the celestial sphere?\n\nNot I, who am unable to find within myself\n\nThe proper words,\n\nSo that, looking on her beautiful face and mien,\n\nSo as not to see less well\n\nI would deign to lose together both life and light.\n\nThe amorous mood of the song is conveyed by the singer's passionate, voluptuous expression. The beam of light that rakes the room, illuminating the boy's face with its flash of radiance, may be Caravaggio's own metaphor for 'that proud light of my lady'. The melancholy poetry of a song has been translated into the texture of painting.\n\nThe prominent still life may have been intended to enhance the bittersweet mood. Faded flowers traditionally symbolized the transience of life and love. Baglione singled them out for particular praise, focusing on 'the carafe of flowers filled with water, in which we see clearly the reflection of a window and other objects in the room, while on the petals of the flowers there are dewdrops imitated most exquisitely'. The flowers have indeed been depicted with meticulous care, each one sharply individuated. But they pose a puzzle because neither they, nor the fruit, can possibly have been painted by Caravaggio himself. The handling is very different in this part of the painting, much harder in the outlines, with a pernicketiness in the finish that is quite alien to his style. The vase of flowers strikes an especially discordant note. The enamelled blooms are piled high in a merely decorative profusion. They have none of the weight, none of the mute and insistent singularity, of things seen and painted by Caravaggio. It is conceivable that the fruit and flowers were added by the Netherlandish painter Jan Bruegel (1568\u20131625), who was in Rome in the mid 1590s. The second son of Pieter Bruegel the Elder, Jan Bruegel was a favourite of Carlo Borromeo's cousin Cardinal Federico Borromeo, a friend of del Monte who lived close to the Palazzo Madama from 1597 to 1601. With Borromeo's encouragement, Bruegel would later become a specialist painter of flowers in vases. Given that he was certainly in Caravaggio's circle and in Rome at the right time, he is a plausible candidate for authorship of _The Lute Player_ 's mysterious bouquet.\n\nJan Bruegel's patron, Federico Borromeo, was one of the first collectors of still life painting. It was for him that Caravaggio painted his only pure example of the genre, the _Basket of Fruit_ , now in the Ambrosiana Library in Milan. Created some time in the later 1590s, it is among the very first autonomous still life pictures, a muffled explosion of morbidity and metaphysical aspiration, and another testament to Caravaggio's extreme originality.\n\nThe _Basket of Fruit_ was elaborated from the earlier _Boy with a Basket of Fruit_. What it shows, essentially, is the basket without the boy. The weight of the painter's attention on his apparently straightforward theme is palpable. The basket on this occasion contains a considerably reduced ration of produce. There are figs, an apple, a quince, a peach, a pear and four bunches of bloom-clouded grapes. These are the fruits of late summer, ripe to succulence, but also on the turn. There is a dark wormhole in the red-streaked cheek of the apple. Two of the grapes at the apex of the uppermost bunch have shrivelled to raisins. There are black spots as well as bright pearls of dew on some of the foliage. A parched vine leaf has turned thinner than paper, while the peach leaves have curled and dried to dark, perforated twists. The basket of woven straw has been placed on the most minimally suggested of ledges. It overhangs the edge a little, a fact that the painter indicates with the smallest crescent of black shadow. The transience of nature is linked to precariousness. Entropy and the fear of falling are connected in Caravaggio's mind.\n\nThe background of the painting is a golden void, reminiscent of a cream-coloured wall in the sun. An early inventory of Federico Borromeo's collection shows that it was kept unframed. This may have been deliberate: it may have been hung high on a wall the same colour as the ground, to emphasize its _trompe-l'\u00e6il_ effect and make it yet more of a tantalus. In any case, the blank background had been a characteristic peculiarity of Caravaggio's work from the very start of his career. It is the hallmark of an artist utterly uninterested in extraneous detail. For Caravaggio, making images is a way of focusing the mind. To paint something is to isolate it for the purposes of contemplation.\n\nThe total isolation of forms in the _Basket of Fruit_ might have lent the image a quality of objectivity, akin to that of the contemporary specimen studies in the albums compiled for Cassiano dal Pozzo collectively known as 'The Paper Museum'. But in Caravaggio's hands it has the opposite effect. The objects of his concern have been removed from the world of the merely mundane. The possibility of transfiguration seems to linger in the stillness that surrounds them. The picture implicitly contrasts death with hope. From withered grapes comes wine, just as from the dead body of Christ flowed the blood of salvation. The fruits and leaves are haloed by the golden light, which seems to emanate from several sources. The light gives modelling to some and reduces others to sharp and ragged silhouettes. There is a measure and a rhythm to the arrangement of objects and shapes, which creates the sense that there must be more going on than meets the eye. Leaves droop and curl with an exquisite sense of placement, fastidiously arranged in such a way as to hang clear in the golden air.\n\nCaravaggio exhausted the genre of still life \u2013 for himself, at least \u2013 in the act of painting his only example of it. Roberto Longhi, searching for words to express the uniqueness of the painting, called it 'a humble biological drama'. It is precisely the picture's dramatic quality that makes it so unusual, and so powerful. The _vanitas_ connotations of bruised fruit and the eucharistic implications of grapes come together in the painted basket. But the accumulation of symbols is not enough for Caravaggio, who prefers to express his meaning dynamically. His idea of still life painting is not an assemblage of objects but a theatre of forms. The huddled windfall fruit have a corporeal solidity. The grapes overspill. The foliage is uncannily anthropomorphized. The most daring detail is the single tendril of vine that reaches into the picture from an unexplained point of origin outside its right-hand edge. The blackened silhouettes of leaves hanging from that single, knubbled stalk might almost be hands stretching. They are figures of death or desperation reaching towards the light and life. The work is, in the end, only a still life painting, but it is filled with the same energies, the same sense of agony and paroxysm, as Caravaggio's greatest religious pictures.\n\nThe origins of the _Basket of Fruit_ remain obscure. It might have been commissioned directly by Federico Borromeo when he lived in Rome, between April 1597 and May 1601. Or it might have been a gift from Cardinal del Monte. In a letter to Borromeo of 1596, del Monte thanks him for certain presents received and expresses his intention to reciprocate with a gift of 'paintings and clocks'. He says he is sorry for the delay, caused by the fact that 'I am dealing with persons with whom I have to arm myself with patience' \u2013 which sounds like a reference to the temperamental Caravaggio.\n\nOpinion is also divided on the question of what Borromeo thought of the painting. He singled it out in an intriguing passage in his _Musaeum_ , part of a tract entitled ' _De Pictura Sacra'_ , 'Of Sacred Painting', written in 1618: 'Of not little value is a basket containing flowers in lively tints. It was made by Michelangelo da Caravaggio who acquired a great name in Rome. I would have liked to place another similar basket nearby, but no other having attained the incomparable beauty and excellence of this, it remained alone.' Borromeo's reference to flowers instead of fruit has called into question the sincerity of his admiration for the painting. If he had truly loved it, surely he would never have made such an elementary mistake. But it was not uncommon for a churchman to refer to the mystic grapes of Christian belief as _flores vineae_ , the 'flowers of the vine' \u2013 a phrase inspired by the Song of Songs. Rather than indicating Borromeo's indifference to the picture, the passage may actually reveal his awareness of its deeper connotations.\n\n#### PAINTINGS FOR PRAYER AND DEVOTION\n\nFederico Borromeo's ownership of the _Basket of Fruit_ suggests that with del Monte's support Caravaggio's circle of patrons and collectors soon widened. As well as painting secular subjects, such as _The Musicians_ and _The Lute Player_ for del Monte himself, the artist created a number of private devotional works \u2013 images of the saints, and the Holy Family, intended as aids to prayer and meditation. These too were subtle and original works that did much to enhance his steadily growing reputation for independence of thought and style.\n\nTwo of these religious pictures, _The Penitent Magdalen_ and _The Rest on the Flight to Egypt_ , have been together in the Pamphili Collection in Rome ever since they were first recorded there in an inventory of 1652. Their earlier history is not known for certain but it is likely that their first owner was Donna Olimpia Aldobrandini, whose heir married Prince Camillo Pamphili in 1647, taking all the family pictures with her. Olimpia Aldobrandini was the niece of Cardinal Pietro Aldobrandini, with whom Cardinal del Monte played hazard and frequented the company of courtesans, so she would have been well placed to buy works from Caravaggio in the later 1590s; these two pictures appear to have been part of the unsold stock that the painter brought with him when he moved to the Palazzo Madama. Mancini, the most reliable source of information about the early years, says that both were painted at around the same time as _The Gypsy Fortune-Teller_ , when Caravaggio was living in the house of Monsignor Fatin Petrigiani. That would place them in 1595\u20136, a date consistent with their lightness of palette and slightly soft style.\n\nBellori thought _The Penitent Magdalen_ was a shockingly unorthodox work of art and described it as one of the most extreme examples of Caravaggio's obsession with reproducing raw and unmediated reality:\n\n> Since Caravaggio aspired only to the glory of colour, so that complexion, skin, blood and natural surfaces might appear real, he directed his eye and work solely to that end, leaving aside all the other aspects of art. Therefore, in order to find figure types and to compose them, when he came upon someone in town who pleased him he made no attempt to improve on the creations of Nature. He painted a girl drying her hair, seated on a little chair with her hands in her lap. He portrayed her in a room, adding a small ointment jar, jewels and gems on the floor, pretending that she is the Magdalen. She holds her head a little to one side, and her cheek, neck and breast are rendered in pure, simple, and true colours, enhanced by the simplicity of the whole figure, with her arms covered by a blouse and her yellow gown drawn up to her knees over a white underskirt of flowered damask. We have described this figure in detail in order to show his naturalistic style and the way in which he imitates truthful colouring by using only a few hues.\n\nNeither for the first nor the last time in his life of Caravaggio, Bellori's criticism was mitigated by grudging admiration. Compelled by his own academic dogma to dismiss the work, he none the less responded instinctively to its vivid style and unusual composition.\n\nMary Magdalen was one of the most popular saints of the Counter-Reformation Catholic Church, which placed a heavy emphasis on the moral responsibilities of each and every believer. She was held up as a shining example of penance and conversion, a beacon for all those languishing in darkness and sin. According to the most prominent part of her legend, she was a prostitute who repented, and 'the woman whom Jesus loved'. The biblical Mary Magdalen was the woman from whose mouth 'seven devils' were exorcised in the gospel of St Luke. But tradition also identified her with Mary, the sister of Martha and Lazarus of Bethany, and with the unnamed repentant sinner blessed by Christ for washing his feet with her tears, drying them with her hair and anointing them with oil (Luke 7:37\u201350). In the Eastern Church these figures were regarded as separate individuals, but in the West all three were merged into Mary Magdalen. Her fable was further embroidered during the Middle Ages, when she was said to have travelled from the holy land to the South of France, where she 'went into the desert and dwelt there thirty years without knowing of any man or woman'.\n\nIn the mid fifteenth century the Florentine sculptor Donatello had carved a harrowingly ascetic sculpture of Mary Magdalen in the wilderness, portraying her as a gap-toothed hermit with withered flesh, wasting away in a hair shirt. But by Caravaggio's time, Donatello's raw immediacy of imagining had fallen out of fashion. The image of Mary Magdalen, epitome of the repentant prostitute, had become fossilized by convention into two basic types: either she appeared as a beautiful nude in a landscape, decorously draped by tresses of her own luxuriant hair, praying before a crucifix and skull; or she appeared as a demure aristocrat reading a book indoors.\n\nCaravaggio rejected both of those stereotypes to create his own, highly distinctive image of Mary Magdalen. The hands of the figure are boneless and the anatomy of her chest and neck unconvincing, but the painter's conception of the subject is impressively original and characteristically dramatic. Caravaggio placed the girl who modelled for the painting on a chair so low that her knees must have been only inches off the ground. As a result, she is seen from above, almost as in a compressed version of a bird's-eye view, so that at first sight it is not quite clear whether she is sitting or lying outflung on the ground. It may have been this extremely unusual perspective, so alien to the pictorial conventions of the time, that led Bellori to think of the work as a perverse exercise in purely optical painting. But it has a poetic point to it. By seating Mary Magdalen so low, Caravaggio emphasizes her humility \u2013 the etymology of which, derived from the Latin _humus_ , or 'ground', itself expresses the idea of abasement.\n\nCaravaggio's Magdalen is no emblem, but a person in turmoil. She sits in darkness, but above her an abstract wedge of light intrudes, as if to dramatize the light of Christ entering her soul. The painter depicts her in the immediate aftermath of her conversion \u2013 the moment just after Christ says, 'Thy faith has saved thee; go in peace.' A single tear trickles down the side of her nose. She has torn off her gold and her jewels and scattered them on the ground. The glass jar beside her, three quarters full, contains the same unguent with which she anointed Christ's feet. It is echoed by the shape of a vase in the damask of her dress, a visual pun that may have its own significance. She is herself like a vessel that has been filled with the spirit of the Lord. Her closed eyes suggest the idea that she is looking within, perhaps even experiencing the transports of mystic vision. She looks as though she is cradling an imaginary child. Perhaps she is thinking about her namesake, the Virgin Mary, and reflecting on the mystery of the mother whose divine child is foredoomed to die.\n\nWhen Caravaggio reimagined the Magdalen in this way, when he thought about the heart of her story and asked himself how to bring that story to life, he was doing just what Carlo Borromeo had urged the preachers of post-Tridentine Milan to do. The true precedent for his painting lay in images formed from words, rather than paint or stone. There is nothing like his _Penitent Magdalen_ in the visual art of late sixteenth-century Italy, but turn to the sermons of the time and close parallels can be found. The following, for example, is a description of Mary Magdalen in the passionate transports of her conversion given in a sermon by Francesco Panigarola, a preacher closely associated with Borromeo: 'now she retraces her steps, her legs stagger, now she starts to take off all her vain ornaments, now to utter cruel invectives against indecent womankind, now to disparage the beauty of the eyes, now to groan, exclaiming, \"Oh roof, why don't you fall down and crush me?\" Casting down her necklaces and jewels, shaking her tresses, violently wringing her hands, she trembled...' Caravaggio's picture is like a still image pulled from the flow of such thoughts. It would always be his practice as a religious painter to rethink sacred story as living drama. Perhaps, when he did so, he often began by remembering the images conjured up in the sermons of his childhood.\n\n_The Rest on the Flight to Egypt_ , always closely associated with _The Penitent Magdalen_ , must have been painted at around the same time. Caravaggio employed the same young model, a redhead, in both cases. In the second picture she has metamorphosed from distraught courtesan to exhausted Madonna. She sits cross-legged on the ground, cradling the infant Christ in her lap. The baby sleeps peacefully but the mother's rest is more fitful; Mary's head lolls, her cheek resting on the crown of the infant Christ's head. Her brow is furrowed. There is some hesitancy in Caravaggio's painting here. He has trouble articulating the junctures of chin and neck, neck and shoulder, and her limp hands are only a little more convincing than those of the _Penitent Magdalen_. But there is great tenderness, none the less, in the artist's idea of mother and child. The heavy-headed Mary is a refugee huddled with her baby, snatching a moment of rest while she can. Bellori, that keen if reluctant admirer of Caravaggio's humanity, noted the poignancy of 'the Madonna who, with her head inclined, sleeps with her baby at her breast'.\n\nOn the other side of the painting sits a wizened, greybeard St Joseph. Wrapped in folds of heavy brown cloth, he has the weather-worn face of a working man. He rubs one of his bare feet with the other in a way that suggests he is feeling the cold. His head is placed almost disconcertingly close to that of the ass, which stands patiently behind him. The objects beside him, a bundle wrapped in green striped cloth and a flagon of wine sealed with a twist of paper, speak of the family's hurried displacement. Caravaggio's Holy Family is very much in hiding. They nestle close together, within a bower sheltered by undergrowth. Like illegal immigrants seeking to avoid detection, they have made themselves small and unobtrusive.\n\nThe painter's biblical source was the Book of Matthew, 2:12\u201315, which recounts the events immediately preceding Herod's massacre of the innocents: 'the angel of the Lord appeareth to Joseph in a dream, saying, Arise, and take the young child and his mother, and flee into Egypt, and be thou there until I bring thee word: for Herod will seek the young child to destroy him.' Between Mary cradling the Christ child and the figure of Joseph, Caravaggio has included the angel of the Lord \u2013 a smooth-skinned adolescent boy, scantily draped, seen from behind, face in half-profile. The only upright form in the painting, the figure resembles a pillar of divine light against the dark clouds that hover over the landscape in the background. The angel plays a violin while Joseph obligingly holds up a musical part-book.\n\nThe music being played has been identified. It is the four-voiced _Quam pulchra es et quam decora_ , composed by Noel Bauldewyn ( _c_. 1480\u20131520) to a medieval text drawn in patchwork fashion from the verses of the Song of Songs. The angelic music solemnizes the spiritual union of Caravaggio's sleepy mother and child, so they are to be understood as the true husband and wife in the scene. Joseph earnestly contemplates the angel, as if straining to understand the mystical significance of the heavenly vision. But, like the slow and faithful ass with which he has been paired, he does not fully grasp the elusive meaning of the notes that fill the air.\n\nHis incomprehension might well have been echoed by many of Caravaggio's contemporaries. Because the painter rendered the notes in the musical part-book without the actual words from the Song of Songs, a vital part of the iconography of his painting was encrypted from the outset. _The Rest on the Flight to Egypt_ has an air of secrecy about it. In many respects it is a touchingly direct dramatization of a biblical story. But it is also occluded, a painting of different levels and layers designed to speak fully only to those who have been initiated into its mysteries.\n\nThe standing angel is one of the young Caravaggio's most haunting inventions. There was nothing in the Bible or any of the Christian apocrypha to suggest the playing of heavenly music during the Holy Family's flight to Egypt. A music-making guardian angel did appear in medieval miracle plays telling the story. The painter may have seen dramas of this kind, which were traditionally staged at Christmas, and perhaps he intended to evoke popular sacred theatre. But Caravaggio's precise visual source for the angel was far removed from the world of medieval piety. He lifted the figure directly from _The Judgement of Hercules_ , a mythological picture of 1596 painted for one of the ceilings of the Palazzo Farnese by Annibale Carracci. That painting, now in the Museo di Capodimonte, Naples, shows the mythical character Hercules choosing between two female figures personifying Vice and Virtue. It was the scantily clad figure of Vice that Caravaggio daringly chose to transform into his own half-naked angel. Caravaggio's career would be marked by bitter rivalry with a number of other painters, including Annibale Carracci. So it seems likely that he was deliberately courting comparisons between his own work and Carracci's _Judgement of Hercules_ , which was painted only a few months earlier.\n\nAppropriation is Caravaggio's pretext for a virtuoso display of his own powers. The thief turns out to be a magician. Annibale's figure is a heavy, sculpturally draped figment of the late Renaissance, a being abstracted from reality into the realm of art. Caravaggio's angelic boy is a type of ideal beauty, but he has been brought down to earth. His feet touch the dark soil, his slender legs shift to transfer his weight to his left side, his curly hair is tousled by the wind. Even his wings, evidently modelled on those of a pigeon, announce Caravaggio's distinctive attachment to actuality. An unnecessary curl of surplus string coils from one of the pegs of the angel's violin, a final grace note of captivating realism. All this serves to emphasize the gulf between Carracci's disembodied spirit of sensuality and Caravaggio's fully realized angel. But the most daring trope of inversion is the transformation of the figure's essential meaning. An embodiment of temptation has been recast as an angel. Vice has been sanctified. The profane is invested with sacred meaning, just as it is in the Songs of Songs. Like the half-concealed text at its heart, _The Rest on the Flight to Egypt_ is charged with erotic feeling. The alluring and mysterious angel, sensuality and divinity intertwined, splits the picture like a bolt of lightning.\n\nShortly after painting _The Penitent Magdalen_ and _The Rest on the Flight to Egypt_ , Caravaggio created perhaps the most daring of all his early devotional pictures. _St Francis of Assisi in Ecstasy_ is now in the Wadsworth Atheneum, Connecticut. The painting is a nocturne, set in a landscape of gloomy indistinctness lit by the first distant gleams of dawn. The painter directs a focusing beam of divine light on to the form of the ecstatic saint, as he swoons into the arms of his guardian angel.\n\nScholars disagree about who commissioned the picture from Caravaggio. The two candidates are Ottavio Costa, a rich banker, and del Monte himself. There is a case to be made for either, and even for both. A picture of St Francis by Caravaggio is recorded in Costa's will of 1605, while 'A _St Francis in Ecstasy_ by Michel Agnolo [ _sic_ ] da Caravaggio with an adorned gold frame of four palmi' was sold by del Monte's heir in 1627. It is not known if these were one and the same picture, or two versions of the same composition. Cardinal del Monte's Christian name was Francesco, and as we have seen he had a particular fondness for images of the saints. So, though the complications of its early history may never be fully unpicked, the picture now in Connecticut may well have hung at one time in the Palazzo Madama \u2013 perhaps not far from Caravaggio's earlier painting of a musical rehearsal. There is a striking resemblance between the boy with Cupid's wings in _The Musicians_ and the angel cradling the ecstatic saint in the later picture. The same model probably posed for both figures.\n\n_St Francis of Assisi in Ecstasy_ is a crucial painting in Caravaggio's early development. It announces the stark tenebrism that would become the hallmark of the painter's revolutionary style \u2013 that 'boldly dark and black colouring,' in Bellori's words, 'which he used abundantly to give relief to the forms'. It also displays for the first time Caravaggio's lifelong fascination for the strongest and most intense strains of Counter-Reformation Catholic spirituality. It expresses the idea of a transfiguring love of Christ, a love so deep that it becomes a form of mystic self-annihilation.\n\nIn Italy during the later years of the sixteenth century, the thirteenth-century Francis was regarded as the saint to emulate above all others. The Counter-Reformation Church looked back to the Middle Ages as a time of powerfully simple piety, uncomplicated by divisive theological speculation. Francis had practised an emotive, theatrical form of preaching, which spoke to the feelings rather than the intellect. He encouraged his followers to venerate nature as God's blessed creation, and held that the only way to follow Christ's message was to live it out in daily life. In prayer, he sought to visualize the events of the New Testament. In public, he would act them out, turning his own body into the living image of Jesus Christ. In sackcloth and ashes, haltered like a beast, he would re-enact the humiliations of the journey on the road to Calvary. The central event of Francis's life was itself a miracle of empathetic identification. One day the saint focused his prayers so strongly on the image of Christ that the wounds of Crucifixion were miraculously branded on his own body. The idea of the stigmata was in turn burned into the Christian folk memory, becoming the ultimate symbol of the power of prayer and visualization to lead the believer towards God. This is the theme of Caravaggio's picture, and it became a guiding principle of his art. All his religious paintings would be re-enactments or reimaginings, closely akin to the vivid theatricality of Franciscan devotion.\n\nThe miracle of the stigmata is most fully described in St Bonaventure's mid thirteenth-century _Life of St Francis._ Bonaventure relates that the saint went up to Mount La Verna, an isolated mountain at the centre of the Apennines, with one of his followers, Brother Leo, to pray and fast. While he was absorbed in devotion to Christ, a seraph with six flaming wings appeared to him in the sky. As the seraph came nearer, Francis saw the figure of a man crucified between its wings:\n\n> He marvelled exceedingly at the appearance of a vision so unfathomable, knowing that the infirmity of the Passion doth in no wise accord with the immortality of a Seraphic spirit. At length he understood therefrom, the Lord revealing it unto him, that this vision had been thus presented unto his gaze by the divine providence, that the friend of Christ might have foreknowledge that he was to be wholly transformed into the likeness of Christ crucified, not by martyrdom of body, but by enkindling of heart. Accordingly, as the vision disappeared, it left in his heart a wondrous glow, but on his flesh also it imprinted a no less wondrous likeness of its tokens. For forthwith there began to appear in his hands and feet the marks of the nails, even as he had just beheld them in that Figure of the Crucified... The right side, moreover, was \u2013 as if it had been pierced by a lance.\n\nSt Bonaventure makes a minute distinction here. The Lord makes it known to Francis that he will be transformed into 'the likeness of Christ crucified' not by the mortification of his flesh but by the inner burning of love in his heart \u2013 'not by martyrdom of body, but by enkindling of heart'. It is love, not pain, that transfigures the human being in search of God. Once Francis understands this, the seraph disappears. A wound appears in Francis's heart at that very moment; immediately afterwards the marks of the stigmata appear on the saint's hands and feet.\n\nCaravaggio's composition indicates not only that he had read St Bonaventure's _Life of St Francis_ , but that he intended to dramatize the crucial moment in the story \u2013 the moment when, as the seraph disappears, the wound appears in the saint's heart. As Caravaggio's saint swoons backwards, he reaches involuntarily with his right hand towards a rent in his habit where a wound in his side has already started bleeding. There are no signs of stigmata in his hands or his feet, and there is no seraph in the sky. There was no precedent, in depictions of St Francis, for including the wound in his side and omitting the others. There was no precedent, either, for the compassionate kneeling angel who cradles the saint in his arms. Caravaggio was also the first artist to depict the saint lying down at the moment of his stigmatization.\n\nIn every sense \u2013 style, iconography, drama \u2013 the painting broke new ground. It certainly gives the lie to the slander that Caravaggio was an untutored Lombard realist, bent solely on dazzling with the mimetic brilliance of his art. _St Francis of Assisi in Ecstasy_ is a picture full of subtle, poetic reflections on the deeper meaning of Francis's transfiguring moment of communion with Christ. Barely visible in the gloom behind the angel and the saint, a little group of shepherds is gathered round a campfire, one of them pointing excitedly to the heavens. Here, Caravaggio consciously echoes the traditional imagery of Christ's Nativity \u2013 which had itself often been painted as a nocturne \u2013 as if to imply that at the moment of his swooning ecstasy Francis really has been reborn in the image of Christ himself. Implicit in this brief moment of death-in-life is, therefore, a second birth for Francis, marking out his destiny to live only in and for Christ \u2013 to live as _alter Christus_ , or 'another Christ', as his legend had it.\n\nThe group of saint and angel echoes another tradition of Christian art, recalling images of the dead Christ cradled in the arms of his mother, the Virgin Mary. Caravaggio's angel is taller than the figure of St Francis, which has sometimes been put down to the painter's youthful clumsiness, but this too is actually a poetic device, enhancing the pathos of the saint's helpless body: in many images of the Virgin mourning Christ's death, including Michelangelo's celebrated marble _Piet\u00e0_ in St Peter's, the mother's supporting body is much larger than that of her lifeless son. This echo of Christ's death and lamentation may also have been meant to express the idea that it was through the act of meditating on Christ's passion that Francis brought the miracle of his own transfiguration upon himself. By thinking about the dead Christ, he achieved the state of electrifying empathy that summoned forth the blessed vision of the seraph.\n\nFrancis was a figure from the relatively recent past. His legend was treasured by Catholics, not least because it seemed such a tangible demonstration of the continuing presence of a miracle-working God in the real and actual world. Protestants disapproved of the veneration of saints and their relics, arguing that too much worship had been displaced from its proper focus on God alone. The destruction of shrines and the suppression of pilgrimage in countries of the Protestant north was, in part, an attempt to stem this perceived haemorrhaging of holiness from the divine centre to the apocryphal margin. But in Catholic Italy, it was feared that such theological purism might rob the world altogether of its Christian magic. To abandon the images and relics of the saints, together with the rituals associated with their veneration, might create the sense of a terminally disenchanted present, cut adrift from the sacred past.\n\nMany of the religious initiatives of the Counter-Reformation addressed this nexus of fear, desire and belief. One of the challenges that the Catholic Church set itself in Caravaggio's time was that of demonstrating that the old and the modern Christian worlds were not distinct and separate eras but formed, instead, a single unbroken continuum. The very fabric of Rome, where so many of the dramas of Christian history had been played out, was itself interrogated for evidence of this. The discovery of the catacombs, burial places of the earliest generations of Roman Christians, led to a boom in the field of what might be called sacred archaeology. The seventeenth-century Bishop of Vaison, Joseph Maria Suarez, examined the mosaics of ancient Rome from a Christian perspective. Antonio Bosio's study of the city's buried architectural bones, _Roma Sotterranea_ , was posthumously published in 1632. Another scholar, Antonio Gallonio, spent years studying the instruments and reconstructing the methods of early Christian martyrdom, publishing the results in 1591 as _The Tortures and Torments of the Christian Martyrs_. Gallonio gave the different chapters of his book gorily circumstantial titles \u2013 'Of the Wheel, the Pulley and the Press as Instruments of Torture', or 'Of Instruments Wherewith the Heathen Were Used to Tear the Flesh of Christ's Faithful Servants, to wit Iron Claws, Hooks and Currycombs' \u2013 and his text proved immensely popular. The more lavishly illustrated editions bear vivid witness to the author's underlying ambition, that of making the holy deaths of venerable memory seem as gruesomely fresh as yesterday's executions.\n\nThe past was not to be thought of as the past. The age of miracles and martyrdoms was not another time, dead and buried, the passing of which was to be mourned; it was part of the present. To go on pilgrimage, as Catholics were encouraged to do, was to reaffirm precisely that belief, because to travel to a holy site was to move through time as well as through space \u2013 journeying back through the centuries, in mind and spirit, to relive the events of sacred history as if they were taking place in the here and now. Ascanio Donguidi, Augustinian Canon Regular of San Giovanni in Laterano, one of the principal pilgrimage churches of Rome, published a guidebook for prospective pilgrims in 1600. On approaching St Peter's, he advised,\n\n> You will greatly enjoy thinking about your visit to all the Saints whose relics are kept in that Church. Imagine yourself having found the saints present and alive... O with how much great devotion and fervour and joy of heart you would go into Saint Peter's, if you truly believed to find him... sitting in his Pontifical Throne, or how you would hurry, even run, if you imagined being able to find present [and alive] in said Church where they are buried, all the Holy Martyrs, Popes, Confessors and Holy Virgins. And [it is not only a figment of your imagination] because it is true that you are going to visit them, and it is very true that they are living a life of glory. And they will listen to your petitions, prayers and supplications and will present them to God.\n\nNot only were the old saints to be venerated, to be conversed with as if they were still alive. It was to be known that their miracles were being actively repeated in the contemporary world, in the lives of new saints elect. Ecstasies of empathetic love akin to those of Francis loomed large in the lives of sixteenth-century penitents, priests and charismatic nuns. St Teresa of Avila's memoirs, published in 1588, famously told of an angel coming to her when she was deep in prayer and piercing her breast with an arrow of divine love: 'It is not bodily pain, but spiritual, though the body has a share in it \u2013 indeed, a great share, so sweet are the colloquies of love which pass between the soul and God that if anyone thinks I am lying I beseech God, in his goodness, to give him the same experience.'\n\nCaravaggio's painting was intended to prompt reflections on more than the stigmatization of Francis alone. _St Francis of Assisi in Ecstasy_ embodied an ideal of transfiguring Christian love, exemplified not only by St Francis, and by St Teresa, but also \u2013 still closer to Roman hearts \u2013 by St Filippo Neri, founder of the Oratory. Like Teresa, Neri had been fascinated both by the legends of a saint such as Francis of Assisi, and by the power and authority of the primitive Church. In early life he prayed continually in the catacombs of Rome's most ancient saints of all; and it was in the Catacomb of San Sebastiano that he experienced his own ecstasy and his own divine wound of love:\n\n> In 1544, just before the feast of Pentecost, Philip, while still a layman, was praying to the Holy Ghost in the Catacomb of San Sebastiano, when he seemed to see a globe of fire which entered his mouth and sank down into his heart. At the same time he felt a fire of love which seemed to be a positive physical heat, so that he had to throw himself on the ground and bare his breast to cool it. When he rose he was seized with a violent trembling, accompanied by an extraordinary sense of joy, and putting his hand to his heart, felt there a swelling as big as a man's fist... At the same time there began that palpitation of the heart which lasted throughout his life, and made itself felt particularly when he was praying, hearing confession, saying Mass, or giving communion, or when he was speaking on some subject which stirred his emotions. So violent was this palpitation that it was described by those who knew him best as being like the blows of a hammer, while the trembling it caused was such as to shake his chair, his bed, or sometimes the whole room. Yet, when he pressed his penitents to his heart they felt an extraordinary consolation...\n\nSuch modern stories of saintly ecstasy were well known to those who commissioned and paid for Caravaggio's early devotional pictures. Neri had confided the tale of his blissful ordeal by divine fire to none other than Cardinal Federico Borromeo, owner of Caravaggio's _Basket of Fruit_. Near the end of his life, Cardinal del Monte delivered a _laudatio_ of St Teresa, on the occasion of her canonization.\n\nCaravaggio's strong and unusual emphasis on the love that burned within Francis's heart expressly evoked the parallels between his legend and those of the modern saints. The sacred past is projected into the present. The holy light that shone on Francis might still shine on anyone with eyes to see. _St Francis of Assisi in Ecstasy_ is more than an illustration of an episode in the life of a saint. The picture offers a consoling dream of transfiguration, a condition of oneness with Christ to which anyone might aspire. That is the significance of its most striking detail, the creased and eyes-closed face of the ecstatic saint. Caravaggio painted St Francis as a real, flesh-and-blood human being, a man with sharply defined features, someone who might be easily recognized, even on the dark streets of Rome by night. Not only that \u2013 he gave the saint his own face. It is an extraordinary statement of self-identification, but one that would be more than justified by Caravaggio's subsequent religious works. No other painter of his time would do more to revive and proclaim the solemn, ascetic sense of humility at the heart of the Franciscan ideal.\n\n#### BETWEEN SACRED AND PROFANE\n\nThe sacred and the profane are inextricably intertwined in Caravaggio's early work. Pictures of apparently mundane subjects are depth-charged with spiritual yearning, while flashes of intoxicating eroticism dart from pictures of the saints or the Holy Family. The painter dreams of angelic beauty, but can only embody it as one beautiful boy after another. The sensual and sexual appeal of such youthful, smooth-skinned figures as the coquettish, music-playing angel in _The Rest on the Flight_ , or the angelic ministrant to St Francis, has been taken as evidence of the painter's homosexuality. The truth is not straightforward. Caravaggio was capable of being aroused by the physical presence of other men. He could not have painted such figures in the way that he did if that were not so. But he was equally attracted to women, as certain other paintings from the late 1590s, such as the transfixing _St Catherine of Alexandria_ , plainly demonstrate. Insofar as the art reveals the man, Caravaggio's painting suggests an ambiguous sexual personality. On the evidence of his paintings he was neither heterosexual nor homosexual, terms that are in any case anachronistic when applied to his world. He was omnisexual.\n\nThe devouring way in which Caravaggio looked at the world made it all but impossible for him to paint idealized forms. There is a quality of seemingly involuntary vividness in many of the details of his paintings \u2013 a quality that, increasingly, he learned to control and to manipulate. This both intrigued and fascinated his contemporaries, and brought a dangerous unpredictability to his pictures. Something base and ordinary might suddenly seem touched by a miracle; a holy mystery could shade into figments snatched from an erotic daydream. Caravaggio's early work is beguiling, in part, because it is so ambiguous and metamorphic. It expresses the truancies of the painter's imagination and allows space for the unregulated responses of the viewer's wandering eye. It speaks of piety but makes concessions to the impious mind, guiltily mingling the pleasures of the world with a genuine sense of devotion.\n\nCaravaggio's more sophisticated patrons were attuned to such subtle ambiguities. The best evidence for this is an intriguing correspondence between a cleric from Vicenza named Paolo Gualdo and Cardinal Ottavio Paravicino. Paravicino, who had been present at Vittoria Archileo's concert, was a friend of del Monte and one of three Roman cardinals certainly linked with Filippo Neri's Oratory. Gualdo was a scholarly cleric with strong connections to the humanist culture of the Veneto in general and Vicenza in particular. He was a friend of the poet Tasso and wrote a biography of the architect Palladio. He was also a lover of painting, who had tried and failed to obtain a picture by Caravaggio.\n\nIn one of his letters to Paravicino, Gualdo harks back to that disappointment. Referring to himself in the third person, and writing in a spirit of knowingly ironical self-deprecation, he casts himself in the role of a simple impecunious man of the cloth, motivated by philanthropy as well as the love of art, whose overtures to Caravaggio have been unfairly rebuffed: 'the good priest has a certain discernment when it comes to painting, but not very many jewels to fund his fancy, so this seemed a good occasion to help a _galant'huomo_ of the art of painting, and in the process obtain some graceful little picture.'\n\nThe allusion to Caravaggio as a _galant'huomo_ suggests not only his pre-eminence as a painter but also a degree of social pretension. The term, which was used interchangeably with _valent'huomo_ , signified a virtuoso or a man of especial expertise in his chosen field. But it also carried associations of worth and, by extension, honour. Gualdo's letter was written in 1603, but, given that he refers to a small picture, and elsewhere mentions the name of Cardinal del Monte, it seems reasonable to believe that he had the painter's work of the mid to late 1590s in mind.\n\nA yet more interesting letter about Caravaggio was written by Paravicino to Gualdo in August 1603. It is a teasing text, composed in courtly riddles and insinuations, which takes the form of an imaginary encounter between the phantom of Caravaggio and a caricatured version of Paravicino's friend, Gualdo himself, the cleric from Vicenza:\n\n> Michelangelo da Caravaggio, excellent Painter, says that he came as a shade or spirit to Vicenza, and met a _galant'huomo_ who loves paintings and who asked him wondrous many questions. He describes, but does not paint with his brush, a priest with the air of a solemnly reformed cleric, a man who, if he did not speak, would appear to be a Theatine. [The Theatines were a Counter-Reformation order of clerics noted for their asceticism and moral severity.] But when he does open his mouth he touches on every topic, and does so in a spirit of gallantry. It seems to me that he has a tincture of all the sciences, says Caravaggio, but since I lack the necessary expertise myself I cannot touch the marrow of his actual knowledge. He describes himself as extremely keen to have something painted, one minute speaking of various churches, the next of having some beautiful work painted for his lordship the Bishop of Padua. But Caravaggio would have made for him some painting that would have been in that middle area, between the sacred, and the profane \u2013 a kind of picture that he would not have wanted to see from a distance...\n\nThe aim of the letter is to tease Gualdo and to puncture, with the lightest of touches, his holier-than-thou pretensions \u2013 which had themselves been expressed in a spirit of ironic self-parody. Caravaggio and his art are merely the tools employed to that ludic end. But, for all its cryptic circumlocutions, Paravicino's letter reveals much about the risky pleasures enjoyed by keen-eyed connoisseurs of Caravaggio's painting.\n\nThe whole passage turns on the play between appearance and reality. The figure of Gualdo seems at first to be a severe and utterly correct Counter-Reformation cleric, but he then discourses with gallantry on every subject under the sun, showing that he has a more restless mind than first appearances had suggested. He is not necessarily irreligious, although he thinks about more than religion alone. But even this second Gualdo, _galant'huomo_ himself of art and all the sciences, may not be everything that he seems, since the fictional Caravaggio of the letter confesses that he himself lacks the wherewithal to judge the true extent of his knowledge.\n\nA similar contrast between seeming and being is drawn in the second part of Paravicino's tale, about the imaginary commissioning of a picture. Gualdo says that he is thinking of a painting to be given to a church, or to his superior, the Bishop of Padua. But Caravaggio sees through the smokescreen of Gualdo's request and understands what would _really_ please him. He decides not to paint an altarpiece, a monumental and unimpeachably pious type of painting. That is because a public work of art, designed to be seen and read from a distance, would not suit a man of Gualdo's personality. Instead, Caravaggio will paint for him something very different \u2013 a work that might appear devout but will also appeal to a taste for profane pleasures. It will be a picture for private contemplation, 'not one that he would have wanted to see from a distance', because it would yield its secrets and pleasures only when viewed at close quarters. Such a picture, it is archly implied, would be the exact complement to Gualdo, because it would be just as slippery as the man himself. For the priest who is not entirely priestly, for the man who is not all he seems, Paravicino has found the perfect gift: a work by Caravaggio.\n\n#### _BACCHUS_ AND _HEAD OF THE MEDUSA_\n\nNone of the artist's pictures are more teasingly poised between the sacred and the profane than the _Bacchus_ that he painted sometime in 1597 or 1598 for his patron Cardinal del Monte. This later Bacchus is very different from the _Self-Portrait as Bacchus \u2013_ the malingering reveller, impersonated by the artist himself, painted during his apprentice days of discontent. The model, this time, seems to have been Caravaggio's Sicilian friend Mario Minniti. He is a swarthy, ruddy-cheeked, well-fed god of wine, crowned with a wreath of grapes and vine leaves. An air of Dionysian mystery still clings to him, but he is very much the Greek god in his Roman incarnation. Wearing a toga, he lounges on a _triclinium_ , as the ancient Romans did when feasting.\n\nThere is a decanter on the table in front of him, two thirds full of a wine so darkly crimson that it looks almost black. There are bubbles at its surface and its level is askew, a minute touch of realism that makes the moment captured in the painting seem ever more fleeting. The wine is still swinging in the heavy bowl of the decanter. The boy-god has just set it down, after pouring a glassful of the liquid into the fine-stemmed Venetian goblet that he holds, delicately, in his left hand. He offers the wine to the viewer of the painting. His expression is gently quizzical, his half-raised eyebrow both invitation and challenge: unriddle me if you can.\n\nThe _Bacchus_ is a sophisticated, courtly work of art, calculated to catch the eye and then hold it. It is an enigma embodied as a rich store of captivating details. Viewed from a certain perspective, the picture seems ripe with sensuality, bordering on outright lubricity. The barely draped boy might be no more than an elaborately wrapped sexual gift. Does he himself not hint at that possibility, with the suggestive play of his right hand in the knot of black ribbon that binds his clothes?\n\nThat would be the profane approach to the picture. But there is space for a devout approach too. There is another way of undoing that knot. Bacchus is the god of wine and of autumnal fruitfulness, and in keeping with that Caravaggio has given him another of his overflowing baskets of fruit. The black grapes have never seemed so lustrous, the figs so ripe. But the foliage once more is withered, the apple worm-eaten, the quince and the plum bruised. The pomegranate has split and collapsed, disgorging its fleshy seeds. Once more, a sense of eucharistic implication hovers in the still air. Summer has become autumn and the sere leaves at the basket's edge are the presage of death to come. But there is hope here too: the transcendent promise of eternal life is contained in the glass of wine held so carefully by the boy-god \u2013 and with such precise metaphorical intent \u2013 directly above the basket of decaying fruit.\n\nAccording to the Neoplatonic thought of the Renaissance, classical myth was alive with shadowy anticipations of Christian truth. The legend of Dionysus, who died to be reborn, was regarded as a pagan prophecy of the coming of Christ. So it was that the figure of Dionysus\/Bacchus became associated with the Saviour himself. Caravaggio's Bacchus has sad, solemn eyes. Those aware of his Christian aspect might also have noted how the toga that drapes him so loosely also resembles a winding sheet. The wine that he offers is the wine of his blood, an allusion lightly pointed by the heart-shaped shadow, angled towards the figure's heart, cast by the decanter. The apparent promise of physical delight has been transfigured, changed to a metaphysical gift.\n\nThe picture plays on the deceptive nature of appearances, yet also flaunts the very deceptions that brought it into being. As he had done in _The Musicians_ , Caravaggio allows the viewer to peer behind the scene of his own artifice. The model's face and hands are sunburned, to indicate that he is someone who has to go out into the world and earn a living under the harsh noonday sun. There are crescents of black dirt under the fingernails of his left hand. His Roman bed of repose has been created by the expedient of draping white sheets over a somewhat grubby cushion decorated with blue ticking, part of which shows through. This is not _really_ Bacchus, but a young man playing his part.\n\nThe _Bacchus_ soon found its way into the collections of the Medici in Florence. It is likely that del Monte specifically commissioned it as a gift for the grand duke. But the present does not seem to have gone down well. The god with dirty fingernails and sunburned skin may have struck the Medici as a joke in poor taste, or perhaps they were scandalized by the picture's close focus on the sensual body of a half-naked boy. Either way, the picture disappeared from view as soon as it entered the Medici collections. When it finally resurfaced, some four hundred years later, in a basement storeroom of the Uffizi Galleries, not only had it never been catalogued, it had never even been framed.\n\nIn 1598 del Monte gave another painting by Caravaggio to his Medici protector. Baglione writes that the artist created 'a head of a terrifying Medusa with vipers for hair placed on a shield, which the Cardinal sent as a gift to Ferdinando, Grand Duke of Tuscany'. Unlike the Bacchus, the _Medusa_ was enthusiastically received and prominently displayed in the Medici collections. It is one of Caravaggio's most startling inventions. Painted on to a circular piece of canvas stretched over a convex shield of poplar wood, the picture conjures up the legendary monster at the instant when she breathes her last. In Greek myth, the serpent-haired Medusa turned all who gazed upon her to stone, until the hero Perseus, looking only at her reflection in his brightly polished shield, cut off her head. In Caravaggio's painting, thick jets of blood spurt from the horrible creature's neck, which has been neatly severed just below the jaw. Her eyes stare and her mouth opens in a soundless scream. The snakes of her hair coil convulsively, each writhing in its own separate corkscrew agony of death.\n\nThe dying monster with arrestingly masculine features is yet another of the artist's self-portraits. ' _Item_ : a convex mirror', reads one of the entries in an inventory of Caravaggio's possessions. The distortions of the painter's face as it appears in the _Medusa_ indicate that he used a convex mirror to paint it. As in a convex reflection, the cheeks and forehead have been slightly broadened and elongated. Caravaggio deepens the game further by making his own convex reflection, painted on to a convex shield, look as though it is actually _concave_. The shadow cast by the Medusa's head creates the illusion of a curved circular surface scalloped away from the viewer, like a shallow bowl.\n\nCaravaggio treated the commission as a pretext for the display of his own special skills and techniques \u2013 so much so that the picture might almost be regarded as his own emblem, or _impresa_. Just as the face is the painter's own, studied from life, the snakes too were painted from actual, wriggling specimens. It is a mark of Caravaggio's pragmatism that the snakes are not vipers, but watersnakes of a type commonly found in the Tiber. He must have asked a fisherman to net some for him.\n\nJust as Perseus had slain the snake-haired Gorgon, Caravaggio set out to vanquish every other artist to have attempted the subject. The _Medusa_ is a work of such flourish and bravado that it has the look of a painting submitted for a prize. Giorgio Vasari had argued that without the intense spirit of competition between Florentine artists there could have been no Italian Renaissance. His _Lives of the Artists_ is full of accounts of such rivalry, and tales of actual contests that had taken place between artists in earlier times \u2013 for example, the story of Ghiberti and Brunelleschi competing for the commission to create a set of bronze doors for the Baptistry in Florence, or that of Michelangelo and Leonardo da Vinci working, side by side, on two enormous battle paintings for the council chamber of the city's town hall. By commissioning the _Medusa_ , Cardinal del Monte was consciously arranging another such competition. He was pitting Caravaggio against the celebrated Leonardo himself: not only was Leonardo's _Medusa_ one of his most famously idiosyncratic creations, it also happened to be in Florence, in the collection of the Medici. The work is lost now, surviving only in the form of a vivid account in Vasari's life of Leonardo. The story begins with Leonardo's father asking him to paint something on a shield of fig wood:\n\n> And afterwards, having given it a coat of gesso, and having prepared it in his own way, he began to think about what he could paint upon it, that might be able to terrify all who should come upon it, producing the same effect as once did the head of the Medusa. For this purpose, then, Leonardo carried to a room of his own into which no one entered save himself alone, lizards great and small, crickets, serpents, butterflies, grasshoppers, bats, and other strange kinds of suchlike animals, out of the number of which, variously put together, he formed a great ugly creature, most horrible and terrifying, which emitted a poisonous breath and turned the air to flame; and he made it come out of a dark and jagged rock, belching forth venom from its open throat, fire from its eyes, and smoke from its nostrils, in so strange a fashion that it appeared altogether a monstrous and horrible thing; and so long did he labour over making it, that the stench of the dead animals in that room was past bearing, but Leonardo did not notice it, so great was the love that he bore towards art.\n\nCaravaggio also studied live animals in the process of creating his own monster, but otherwise his _Medusa_ could hardly have been more different to that described by Vasari. Leonardo's painting sounds complicated and full of circumstantial detail, conjuring up rocks and crags, a theatrical entrance on the part of the monster, and even the air itself thick with fire and smoke, just the sort of picture that mirrored his restless mind. By contrast, the brilliance of Caravaggio's _Medusa_ reflects the painter's remorseless pursuit of a realist conceit. Leonardo had painted a picture of the Medusa that seemed wittily appropriate as the decoration of a shield. Caravaggio did something bolder and conceptually far more pure. He created a painting that sought to transcend painting and become the very thing that it depicts. His _Medusa_ is not a painting _of_ a shield, or at least it pretends not to be. It pretends to _be_ the shield itself, held in Perseus's hand at the very instant when he has killed the Medusa. It is a painting meant to be admired at close quarters, passed round from hand to hand. To look at the picture thus would be to _become_ the conquering hero himself \u2013 to gaze, through his eyes, at the reflection of the Medusa, as she in turn watches herself die, in her own reflection, in the shield's mirror.\n\nThe best way to grasp the true nature of Caravaggio's illusion \u2013 to complete the circle of gazes demanded by the painter's conceit \u2013 would indeed be actually to hold it. Did Ferdinando de' Medici do just that, and smile at the ingenuity of Caravaggio's idea? Certainly, the sense that this was not a picture like other pictures, a picture simply to be hung on a wall, persisted among later generations of the Medici. An inventory of the family's armoury from 1631 reveals that it was displayed as part of a suit of armour arranged to look like a standing knight at arms. It was brandished by the figure, in fact, just like a real shield.\n\nCaravaggio's _Medusa_ was designed to transform its owner into Perseus himself. To give such a picture to a Medici was to pay him a comfortingly familiar compliment. The Perseus myth had been assimilated into the mythology of Medici power in the middle years of the sixteenth century, when the family had assumed absolute control over what had once been the Florentine republic. Benvenuto Cellini's chillingly persuasive, larger-than-life bronze of _Perseus_ , brandishing a scimitar and holding up the Medusa's head, was a public symbol of Medici might \u2013 a vivid demonstration of exactly what would happen to anyone with the temerity to resist Medici rule. Caravaggio's _Medusa_ , reviving those old associations with the lightest of touches, is a clever piece of praise as well as a virtuoso work of art.\n\nBut its biggest compliment of all is paid implicitly to the painter himself. He it is who personifies the Medusa, the monster who might be defeated but whose magical powers, none the less, loom larger than anything else in her legend. With eyes wide open and mouth agape, the painter takes on her role and in doing so claims for himself her dark powers of enchantment. Whomsoever the Medusa looks at, she freezes, preserving them forever in a single, charged instant of being. From the flux of life she takes a moment and makes it last for all time. That is what Caravaggio does too. Her magic is his magic, a petrifying art.\n\n#### IN THE LABORATORY OF THE ALCHEMIST\n\nSometime around 1599 del Monte invited Caravaggio to his villa near the Porta Pinciana and commissioned him to decorate the ceiling of the Tesoretto, a narrow, rectangular room next to the distillery where the cardinal conducted his alchemical experiments. A hidden, private space, it is reminiscent of the _studiolo_ of Francesco de' Medici in the Palazzo Vecchio in Florence, a chamber like a jewellery box, which had been richly decorated in the Mannerist style by Giorgio Vasari and his assistants in the late 1560s. Francesco de' Medici himself appears in one of those paintings, in the character of an alchemist. Although Caravaggio did not actually paint Cardinal del Monte surrounded by his phials and retorts, he did create a kind of portrait of the alchemically inclined mind. _Jupiter, Neptune and Pluto_ is a wall painting but it was executed in the unusual and fugitive medium of oil on plaster, which strengthens the suspicion that Caravaggio had never learned to paint in fresco, despite his supposed apprenticeship to Peterzano. The picture, which is still _in situ_ and in surprisingly good condition, was first described by Bellori:\n\n> In Rome in the Ludovisi Gardens near the Porta Pinciana, they attribute to Caravaggio the _Jupiter, Neptune and Pluto_ in the casino of Cardinal del Monte, who was interested in chemical medicines and adorned the small room of his laboratory, associating those gods with the elements and with the globe of the world placed in their midst. It has been said that Caravaggio, reproached for not understanding either planes or perspective, placed the figures in such a position that they appear to be seen from sharply below, so as to vie with the most difficult foreshortenings...\n\nFor the first and last time Caravaggio flirted with out-and-out Mannerism. The picture's primary function might almost be, as Bellori insinuates, to demonstrate difficulties triumphantly overcome. The plunging perspective is of a type known as _di sotto in s\u00f9_ , literally meaning 'of above, from below', executed here with light-hearted bravado. Jupiter, mounted on an eagle, reaches a hand into the translucent celestial sphere at the centre of the ceiling's painted sky. The frowning figure of Neptune, mounted on a rearing seahorse, is yet another of Caravaggio's self-portraits. The most dramatically foreshortened figure is that of Pluto, whose carefully painted penis is uncircumcised and surrounded by a dark bush of pubic hair. The Mannerist painter Giulio Romano had painted a similarly vivid _di sotto in s\u00f9_ depiction of male genitalia \u2013 the undercarriage of a flying charioteer \u2013 in his mid sixteenth-century decorations of the Palazzo del Te in Mantua. Caravaggio's bawdy fantasy of airborne larking about belongs squarely in the same tradition.\n\nThere is an allegorical alibi for the emphatic phallus. The overarching theme of the painting is the procreative role of the three elements. From their seminal confluence, everything in the known universe depends. The picture reflects a particular twist in sixteenth-century alchemical theory. During the middle years of the century, the cardplaying astronomer Gerolamo Cardano had proposed a revision of the ancient Aristotelian belief in the four elements of Fire, Air, Water and Earth. Cardano argued that fire should not properly be regarded as an element, thereby reducing their number to three. Caravaggio followed this refinement, presumably advised by del Monte.\n\nMichelangelo had made dramatic use of _di sotto in s\u00f9_ perspective for his depictions of _God Separating Light and Darkness, God Creating the Sun and Moon_ and _God Calling Forth Life from the Waters_ on the ceiling of the Sistine Chapel. The subject of Caravaggio's painting for del Monte is, in essence, a profane version of the same story, told at the start of the Book of Genesis. Caravaggio's use of the same device may have been his way of mischievously pointing up the parallel between the most famous cycle of religious frescoes in all of Rome and his own, rather more playful ceiling decoration.\n\nCaravaggio was always highly responsive to circumstance and milieu. Throughout his life, his art would be deeply coloured by the different social, political and religious environments that he encountered. Entering the circle of Cardinal del Monte, living in his palace, absorbing his ideas, listening to his musicians, looking at his art collections \u2013 those experiences are all clearly reflected in Caravaggio's paintings of the late 1590s. His work becomes more sophisticated, and more intellectually rarefied. Certain details, such as the exquisite wine glass held up by Bacchus, with its delicately blown stem and the whirlpool patterning of its shallow bowl, express his palpable delight in a previously unknown world of beauty and luxury.\n\nThe work of this period is also marked by a spirit of experiment. The artist is trying out new ideas and striving to impress, so much so that he occasionally paints against the grain of his own dark and intense personality. The mythical Mannerist comedy of _Jupiter, Neptune and Pluto_ would not be repeated. But the fact that Caravaggio was prepared to undertake a commission so alien to his own sensibility demonstrates his determination to succeed.\n\nAway from his painting room, and away from the company of Cardinal del Monte, Caravaggio was still the same turbulent young man who had committed nameless misdeeds in Milan. Those who knew him at this time thought of him as a person split asunder, a man who contrived to live two opposing lives. Karel van Mander, a Dutch painter in Rome, described him as a piece of living chiaroscuro:\n\n> There is... a certain Michelangelo of Caravaggio who is doing remarkable things in Rome... he... has risen from poverty through his industry and by tackling and accepting everything with farsightedness and courage, as some people do who refuse to be held down through timidity or through lack of courage but who advance themselves candidly and fearlessly and who boldly pursue gain \u2013 a procedure which, if it is taken in honesty, in a proper manner, and with discretion, deserves no censure. For Fortuna will offer herself by no means frequently of her own accord; at times we must try her, prod her, and urge her...\n> \n> But again there is beside the grain the chaff, to wit that he does not pursue his studies steadfastly so that after a fortnight's work he will swagger about for a month or two with his sword at his side and with a servant following him, from one ball-court to the next, ever ready to engage in a fight or argument, with the result that it is impossible to get along with him. All of which is wholly incompatible with our Art. For certainly Mars and Minerva have never been the best of friends. Yet as regards his painting, it is such that it is very pleasing in an exceedingly handsome manner, an example for our young artists to follow...\n\nCaravaggio lived his life as if there were only Carnival and Lent, with nothing in between. His pictures are the legacy of his lenten days. To encounter his carnivalesque alter ego it is necessary to consult the records preserved in the labyrinthine vaults of Rome's Archivio di Stato: a paper city within the city of stone, made up of witness statements and accusations. Through smokescreens of rumour, denial and furtive insinuation, this other figure comes clearly into view, accompanied by his friends, his lovers and his enemies.\n\n#### PAINTERS, SWORDSMEN AND WHORES\n\nOn 11 and 12 July 1597 three men were summoned to appear before the Tribunal of the Governor of Rome and interrogated in connection with a case of assault. It was literally a cloak-and-dagger affair. A young man called Pietropaolo, apprentice to a barber-surgeon, had been hurt in a fight on the Via della Scrofa. Following the incident, he had been detained in prison because he refused to reveal the identity of his attacker. Violent crime was on the increase in Rome and steps were being taken to limit the unlicensed bearing of arms. Pietropaolo's silence irked the authorities. They were also interested in the discovery of a black cloak near the scene of the crime. Someone had handed it into the barber's shop where the apprentice worked. Who was that person? Might he be able to shed light on the matter?\n\nThe investigators would soon discover that the man who found the cloak was Michelangelo da Caravaggio, artist in the service of Cardinal del Monte. Because he had friends in high places, they decided not to call him until their preliminary enquiries were complete. Instead, they called two of the friends with whom he had been seen on the evening of the fracas, namely Prospero Orsi, the painter of grotesques, and the picture-dealer Costantino Spata. In the event, Caravaggio himself never would be asked to testify in connection with the case. But the words of those who were summoned give us vivid glimpses of the life that he led, at night, in the streets of Rome.\n\nIt is with the testimony of Pietropaolo's employer, a barber-surgeon called Luca, that the case records begin. 'I am a barber, and I practise the profession of barber here by the church of Sant'Agostino,' Luca told the court. What did he know about the article of clothing that had been handed into his shop? Luca answered that a _ferraiuolo_ \u2013 a black cloak, fastened with iron hooks \u2013 had been given to his apprentice, Pietropaolo, on the night in question. Luca himself was busy having dinner with his father and some others, but later Pietropaolo showed him the cloak and told him that a certain painter had brought it. 'He told me his name, but I can't remember it.' When pressed to say if he knew the painter in question, Luca said that certainly he did: 'One time he came into my shop to be spruced up, and another time he came to have a wound dressed... he had been in an argument with one of the grooms of the Giustiniani or Pinello families.'\n\nWhen the case investigator expressed scepticism that Luca could not remember the man's name, he protested that it was the truth: 'Really, sir, I do not recall it.' Then, as if ashamed of his own memory lapse, he gave a startlingly precise physical description of the painter whose injuries he had treated, and whose beard he had trimmed. All is carefully transcribed in the spidery handwriting of a court notary. Suddenly here is Caravaggio, caught in the flashbulb glare of a barber's memory: 'This painter is a stocky young man, about twenty or twenty-five years old, with a thin black beard, thick eyebrows and black eyes, who goes dressed all in black, in a rather disorderly fashion, wearing black hose that is a little bit threadbare, and who has a thick head of hair, long over his forehead.'\n\nReturning to the matter of the lost-and-found cloak, Luca remembered Pietropaolo telling him that another man had been present when it was handed in \u2013 'a certain Costantino, who buys and sells paintings and whose shop is attached to the Madonella next to San Luigi dei Francesi'. All this happened shortly after Pietropaolo had been assaulted, and the fact that he was looking after the shop suggests that he cannot have been very badly injured. Luca made no mention of his apprentice's wounds. Instead, he stressed that the cloak had never belonged to his apprentice in the first place. He probably hoped the whole investigation would come to nothing, so that normal business could be resumed.\n\nCostantino Spata was the next witness called by the court. 'I am a seller of old paintings and I have a shop at San Luigi,' he declared. He had been there for four years, since 1593, always on the same premises. He lived over the shop with his wife, Caterina Gori, and their four children, two adolescent girls and two little boys. The boys went to school, to the ' _Letteratura_ '. Costantino had no assistant and he did not employ members of his family in the business, so when he went out the shop was closed.\n\nHaving established these particulars, the investigator asked him to recall the events of the previous Tuesday evening, the night of the fracas. Costantino said that, having been in his shop all day, he closed up at sunset, when the _Ave Maria_ sounded. Just as he did so, two painters whom he knew walked by.\n\n> One of them was Monsignor Michelangelo from Caravaggio who is the painter of Cardinal del Monte, and lives in the house of the said cardinal, and the other was a painter called Prospero, who comes from I know not where but lives near Monsignor Barberini over a boarding house... he is of small stature, with a little black beard, and is around twenty-five to twenty-eight years old. They asked me if I had eaten and I said yes but they said they hadn't eaten and wanted to go to dinner at the Tavern of the Wolf [' _all'hostaria della Lupa_ '], where we all went together, and I stopped there with them while they ate.\n\nAfter dinner the three companions left the tavern. Moments later, the trouble erupted. 'We all heard someone coming towards us from the Piazza San Luigi, yelling out and saying \"ahi, ahi\".' Caravaggio and Prospero headed off in the direction of Sant'Agostino, while Costantino hurried homewards. As he was walking, a running man passed him. For the rest of his testimony, Costantino stonewalled the investigators. He never really saw the man who was in such a hurry. He could not judge his height or make out how he was dressed. He could not even tell if he had a cloak, or if he was wearing a hat on his head. He did not see if either of the painters had picked up a cloak. He did not have his glasses with him, and without his glasses he could not see very well. Besides, it was dark.\n\nProspero Orsi, the last witness to be called, corroborated Costantino's story. He also went into more detail about Caravaggio's decidedly marginal involvement in the evening's events. Half an hour before the sounding of the _Ave Maria_ , Prospero recalled, Caravaggio had come round to his place. They had gone out to eat. After dinner, they were walking along the Via della Scrofa when they heard shouts coming from the Piazza San Luigi \u2013 'screams and laments, someone saying \"ohime, ohime\" and other words'. But because they were still some way off, because it was getting dark and there was very little street lighting, Prospero could not really see what was going on. Moments later a man sprinted past him. What did he look like, this man? 'Sir, I cannot say who that man was. I didn't see his face, I didn't see his clothes, because he passed like a shadow.'\n\nAfter the man ran past them, Prospero and Caravaggio carried on walking in the direction of the Pantheon. They came across a black cloak lying on the ground. Prospero didn't touch the cloak, so he could not say of what material it was made. Caravaggio picked it up and said he would give it to a neighbour. With those words, he turned around and went back to the corner of Sant'Agostino and gave the cloak to a young man at the barber's shop there. 'I don't know the young man's name because I don't go to that barber's shop,' Prospero added. The two artists wandered back towards San Luigi dei Francesi, where they bumped into Costantino again. He was closing his shop for the night. Prospero parted with Caravaggio at the Palazzo Madama and went home. The investigator enquired whether any of the parties involved had been bearing arms. 'Costantino and I were not carrying weapons of any kind,' Prospero said. But Caravaggio was wearing a sword. 'He is the only one to carry a sword, because he is in the service of Cardinal del Monte. Before he used to carry it by day. Now he only carries it sometimes when he goes out at night.'\n\nCaravaggio's remark that he would give the cloak to 'a neighbour', and his immediate decision to hand it in at Luca's shop, indicate that he (unlike his conveniently short-sighted and confused friends) had immediately recognized the running man. Caravaggio knew that it was Pietropaolo, because he himself went to the barber-surgeon's shop on the corner of Sant'Agostino. He assumed the cloak was his and took it straight to the apprentice's place of work \u2013 only to find that Pietropaolo had run so fast that he was there to receive it himself.\n\nThe enquiry was dropped and the case was closed, unsolved. It was a trivial matter. But the testimonies of those involved, fragmented and confused, reveal much about Caravaggio and the milieu in which he moved. He might have gone up in the world, but he had not forgotten his old friends. He still kept company with Prospero Orsi, who had pushed him to leave the Cesari workshop, and with Costantino Spata, the hard-pressed picture-seller with many mouths to feed. Taking advantage of his newfound status as member of a cardinal's household, Caravaggio was now carrying a sword openly in the streets of the city. He was not afraid to use it. He had been to the barber-surgeon's at least once, to have his wounds dressed, following a fight with a groom attached to another noble Roman household. History does not relate whether the groom's injuries were worse than his.\n\nThe barber-surgeon's account of Caravaggio's physical appearance closely matches descriptions of the painter in other early sources. Bellori, echoing Vasari's idea that artists resemble their own work, wrote that 'Caravaggio's style corresponded to his physiognomy and appearance; he had a dark complexion and dark eyes, and his eyebrows and hair were black; this colouring was naturally reflected in his paintings... driven by his own nature, he retreated to the dark style that is connected to his disturbed and contentious temperament.' Bellori's Caravaggio is the epitome of the melancholy artist, born under the sign of Saturn: dark looks, dark temperament, dark art. But the evidence of the criminal archive suggests a more literal explanation for Caravaggio's sartorial style. People who went dressed in dark colours did so to avoid detection, especially at night. To describe a man as someone who wore black when the _Ave Maria_ sounded was to mark him out as a trouble-maker. Like the dark cloak that may or may not have belonged to Pietropaolo, Caravaggio's black clothes were a form of urban camouflage, designed to enable him to disappear into the poorly lit streets of the city at night. He pursued the same strategy on the streets as he did in the studio. In life as in art he hid what he wanted to hide in the shadows.\n\nBellori also confirmed Luca's description of the 'disorderly' and 'threadbare' state of Caravaggio's clothing. But he added an interesting twist to it. 'We cannot fail to mention his behaviour and his choice of clothes, since he wore only the finest materials and princely velvets; but once he put on a suit of clothes he changed only when it had fallen to rags.' Caravaggio was one of those who liked to play the gentleman, to put on airs and graces and set himself above the people at large \u2013 even though his actual status remained highly ambiguous.\n\nSo the dark, threadbare Caravaggio was wearing a sword by night in 1597. He was evidently still doing so in spring of the following year: on 4 May 1598 he was arrested for bearing arms in a public place at eleven o'clock at night. A certain lieutenant Bartolomeo, attached to the Bargello of Rome, reported that 'I encountered Michelangelo da Caravaggio between the Piazza Navona and the Piazza Madama, carrying a sword, without a licence, and also a pair of compasses, so I took him and put him in prison at the Tor di Nona.' The following day, after a night in jail, Caravaggio was in court. Asked to account for his actions, he replied with righteous indignation. 'I was arrested last night... because I was carrying a sword. I carry the sword by right because I am Painter to Cardinal del Monte. I am in his service and live in his house. I am entered on his household payroll.' Unusually, the notary transcribing the day's evidence capitalized the word 'painter', perhaps in an attempt to capture the self-important stress that Caravaggio had placed on the word. Because he had such powerful contacts, the magistrate had no choice but to release him. The case was dismissed.\n\nJoachim von Sandrart, a German etcher and art historian, wrote a brief account of Caravaggio's life that contains a short but telling paragraph about the painter's years in Rome. According to Sandrart, Caravaggio liked to go about 'in the company of his young friends, mostly brash, swaggering fellows \u2013 painters and swordsmen \u2013 who lived by the motto _nec spec_ , _nec metu_ , \"without hope, without fear\".' A number of those friends can be identified from contemporary sources. Prospero Orsi and Costantino Spata seem to have been among his more peaceable companions. Caravaggio's Sicilian friend, the painter and occasional model Mario Minniti, must also be included. He was no stranger to violence but he lacked Caravaggio's positive relish for it. According to his biographer, Francesco Susinno, Minniti was close to Caravaggio during his early years in Rome but eventually abandoned him; 'he settled down and got married, because he found the turbulent adventures of his friend too much to stomach.'\n\nOrazio Gentileschi was among the more volatile 'painters and swordsmen' with whom Caravaggio kept company. Orazio was a gifted artist in his own right, but a difficult man with a short temper, whose reputation was under a cloud. In 1615 Grand Duke Cosimo II of Tuscany \u2013 son and heir to Cardinal del Monte's patron, Ferdinando de' Medici \u2013 considered bringing Orazio to Florence as an artist and asked his agent in Rome, Piero Guicciardini, to file a report on the painter's character. The resulting reference was less than favourable: 'he is a person of such strange manners and way of life and such temper that one can neither get on nor deal with him.'\n\nSome ascribed the fiery temperament of painters to the toxic qualities of the materials that they used. Lead white and vermilion were particularly poisonous. The mere touch or smell of either might cause a variety of symptoms including depression, anxiety and increased aggressiveness. Those suffering from 'Painter's Colic', as it was called, also tended to drink heavily. While wine alleviated some symptoms, it exacerbated others, and was itself the catalyst for innumerable brawls and scraps in the artist's quarter. Most of the people around Caravaggio were involved in some kind of violent incident at one time or another. Orazio Gentileschi's own daughter, Artemisia, herself a painter, was raped by another artist. His name was Agostino Tassi. The enraged Orazio subsequently accused Tassi of 'repeatedly deflowering' his daughter. Artemisia's matter-of-fact testimony, given at Tassi's trial on 9 May 1611, gives a very clear view of the real violence that stalked the lives of so many of Caravaggio's contemporaries.\n\nArtemisia told the court that she was with her sister, Tuzia, when the attack took place:\n\n> After midday dinner the weather was wet and I was painting a portrait of one of Tuzia's children for my own pleasure, when Agostino came by. He was able to get inside because there were masons in the house and they had left the door open. When he found me painting he said, 'Not so much painting, not so much painting,' and he took the palette and brushes out of my hand and threw them around, saying to Tuzia, 'Get out of here.' When I said to Tuzia to stay and not leave me with him as I had motioned to her before, she said, 'I don't want to stay and argue, I want to be off.' Before she left Agostino put his head on my breast and after she had gone he took me by the hand and said, 'Let us walk about a bit because I hate to be sitting down.' As we walked up and down the room some two or three times I told him that I felt unwell and that I thought I had the fever, and he answered, 'I have more fever than you,' and after walking up and down two or three times, each time going past my bedroom door, when we came to the door of the bedroom he pushed me inside and locked it.\n> \n> Once it was locked, he pushed me on to the edge of the bed with one hand on my breast, and he put one of his knees between my thighs so that I could not close them, and he lifted my clothes, doing so with much difficulty. He placed one of his hands with a handkerchief over my throat and my mouth so that I could not scream... and with his member pointed at my vagina he began to push it into me, having first put both his knees between my legs. I felt a terrible burning and it hurt me very much, but because of the gag on my mouth I could not cry out, though I tried as best as I was able to scream and call Tuzia. And I scratched his face and pulled his hair and before he could put it inside me again I grabbed his member so tightly that I even removed a piece of flesh. But none of this deterred him and he continued what he was intent upon, staying on top of me for a long time and keeping his member inside my vagina. And after he had finished his business he got off me.\n> \n> Seeing myself free I went to the table drawer and took out a knife and moved towards Agostino saying, 'I want to kill you with this knife because you have dishonoured me.' And he opened his tunic saying, 'Here I am,' and I threw the knife at him; he shielded himself otherwise I would have hurt him and might easily have killed him. The outcome was that I wounded him slightly on the chest and he bled little because I had scarcely pierced him with the point of the knife. Then the said Agostino fastened his tunic and I was weeping and lamenting the wrong he had done me and to pacify me he said, 'Give me your hand and I promise to marry you as soon as I am out of the mess I am in.' He also said to me, 'I warn you that when I take you as a wife I want no foolishness,' and I answered him: 'I think you can see whether there is foolishness.'\n\nOne of the most persistent trouble-makers in Caravaggio's own immediate circle was the architect Onorio Longhi. He was just two years older than the painter and they had much in common. Onorio was from Viggi\u00f9, near Varese in Lombardy, close to where Caravaggio himself had been brought up. Longhi's family had links with the Colonna dynasty. Onorio's father, Martino, also an architect, had been called to Rome to work for the Colonna family. After Martino's death in 1591, Onorio took on the family architectural practice, overseeing the completion of the church of Santa Maria in Vallicella in Rome, among other significant commissions. In between, he found plenty of time to get into trouble with the law. He was constantly in court, charged with disturbing the peace and a variety of other offences. G. P. Caffarelli, whose four-volume _The Families of Rome_ was written between 1603 and 1615, described Onorio as 'a little lawless leader for the youth' (' _e un poco scapo scelerato per la gioventu_ '). He certainly seems to have been the ringleader of the group or gang that included Caravaggio. The architect and the painter were often seen together. Longhi was brash and talkative and generally took the lead, while Caravaggio tended to be more taciturn and veiled.\n\n'I am a gentleman... and I don't care about anything, I just go to eat and drink.' That was how Onorio defiantly introduced himself in a witness statement given in court on 4 May 1595. Like Caravaggio, he was all the more dangerous because he had connections with a powerful household. As a _servitore_ of the Colonna family, he too could defy the general ban on carrying weapons issued by Pope Clement VIII. 'I don't carry a sword by day or by night,' Longhi declared in a witness statement of 1595. 'Instead my servant, who accompanies me, carries it.' Two years earlier, in 1593, a prostitute named Margarita Fanella had testified that Onorio was armed at least part of the time \u2013 'sometimes yes and sometimes no I have seen him carry it [his sword] in the street when he goes around with other gentlemen.' Also according to Margarita, 'he has a little blonde beard, that he grows quite thick... he goes dressed in rich dark velvet.' It comes as no surprise to learn that he too was one of those who dressed in black after the sounding of the _Ave Maria_.\n\nOne cause for his many court appearances was a festering dispute with his brother Stefano, which broke out into actual violence on more than one occasion. Onorio had been away when their father had died, and he believed that Stefano had failed to pay over his full share of the inheritance. In 1599 Onorio succeeded in having Stefano imprisoned for four or five months for non-payment of the debt. In the autumn of 1600 Stefano issued a counter-suit, in which he claimed that on 7 July 1598 Onorio had come to his lodgings with three armed companions, possibly including Caravaggio, and had tried to break down the door, shouting, 'You cuckolded thief, I want you to die by these hands!'\n\nThat was by no means the only accusation levelled against Onorio. The investigators who looked into Stefano's counter-suit of 1600 also examined evidence linking him to a number of other unsavoury incidents. They were particularly interested in a long-standing grievance between Onorio and a certain widow called Felice Sillano. Under questioning, Onorio told them that he knew her and that she was a respectable woman. But he denied that he had once tried to beat her door down while shouting, 'You whore, you slut, you coward!' That interrogation itself went back to an older, unresolved case. Another set of transcripts, from 1599, reveal that Felicita Silano ( _sic_ ) had already sued Onorio \u2013 together with 'Claudio, the stone-cutter' \u2013 for threatening behaviour:\n\n> It is now two nights ago that the said Onorio came to my doors saying: 'Open, you baggage and slag.' Having left, he came back and wanted to kick down the door, threatening me with further insults. He said that if I spoke he would beat me over the head with his sword. This same Claudio came and did the same thing five or six months ago, inciting others, and particularly Onorio, to cause trouble at my door. I am not a woman to put up with such treatment, and therefore I am now bringing the matter before the court.\n\nThe record of the outcome does not survive; probably, the case petered out.\n\nOne of Longhi's biographers said of him that 'he was naturally bizarre and had a head that smoked.' The quarrel with his brother and the dispute with Felice Sillano were a fraction of his misdemeanours, which ranged from the extremely serious to the utterly trivial. He would be present when Caravaggio committed murder. But he was just as likely to be found getting into fights with passers-by who made the mistake of bumping into him in the street, or shouting insults at tradesmen for showing him insufficient respect. Case transcripts for 1595 record a scuffle in a cake-maker's shop, after he had gone in to buy 'a certain type of soft white meringue'.\n\nThe violent words and deeds of Caravaggio and his contemporaries may now seem random and chaotic. In fact, the behaviour recorded in Rome's criminal archive conformed to a particular set of codes. To attack a woman's house, to threaten to break down her doors, was also to insult her honour, because, according to the mores of the time, people's dwellings represented the occupants themselves. A common way to cast aspersions on someone's name was to commit the crime of _deturpatio_ : daubing paint \u2013 or sometimes excrement \u2013 on to the doors or windows of their home. In such cases the front of the house metaphorically represented the face. But the locked home also represented the human body, secure in virtue, hence Artemisia's emphasis, in her account of being raped, on the carelessness of the masons who had been working on her house. By insisting that it was they, not she, who had left the door open, she was emphasizing that Agostino's violation of her home and her body was none of her doing.\n\nEven an act as apparently gratuitous as Longhi's attack on a maker of meringues had its own significance. It was Longhi's way of asserting that he was a _valent'huomo_ , a cut above the common herd. Deportment books of the time encouraged gentlemen and aristocrats to cultivate a deliberate air of insolence towards the lower orders. In 1616 Giovanni Bonifacio published a manual on deportment entitled _L'arte de' cenni_ , in which he argued that gesture and facial expression themselves amounted to a language as complex as that of human speech. Bonifacio devoted fifty-eight sections to the eyes alone, itemizing different types of wink and squint, differentiating between the promise of a raised eyebrow and the threat of a frown. He also went into some detail about the significance of the out-thrust elbow. To walk with arms akimbo gave the impression of strength, he noted, it being the demeanour of someone prepared to thrust through the mass of ordinary people, to 'push their way through crowds'. Anthony Van Dyck captured this modishly insouciant arrogance in his portraits of swaggering English aristocrats at the court of Charles I. Van Dyck's portraits were painted in London in the 1630s and 1640s, but they evoke an aggressive air of sharp-edged hauteur that had been fashionable among 'gentlemen' throughout Europe for decades \u2013 the same style of threatening superiority that was being aped by men such as Onorio Longhi, and indeed Caravaggio himself, in Rome at the turn of the seventeenth century. They too were men who walked around with their elbows \u2013 and often their swords \u2013 out-thrust.\n\nIt would however be a mistake to regard Longhi as a simple thug. He was a man of learning, a poet as well as an architect, who used his literary skill to curry favour with the great and the good. When Ferdinando de' Medici's first son was born, Onorio marked the occasion with a witty poem full of satirical jibes against the Spanish. He associated with writers and musicians, as well as the likes of Caravaggio. Men working in the liberal professions, who could dream of rising through the social hierarchy, were attracted to the mock-chivalric ethos of his circle. Some of his companions were genuinely high-born, the sons of Rome's leading families. Others were simply out-of-work soldiers. Most lived around the Campidoglio, between the Piazza dei Sant'Apostoli and the Piazza Montanara.\n\nBut Longhi was also undoubtedly dangerous. Perhaps the most telling detail in descriptions of him in the archive is the fact that he often went around the streets of Rome on horseback, as if he were a knight and his servant were his page. Onorio and Caravaggio and those who ran with them \u2013 or against them \u2013 did not just copy the clothes and the manners of the aristocracy. They behaved like modern, debased versions of the 'veray parfit gentil knights' of the old romance tradition. Instead of wandering through the forests of Arthurian legend, doing battle with monsters and saving damsels in distress, they frequented the streets and taverns of Rome, picking fights with pimps and vying for the favours of whores.\n\nThis topsy-turvy translation of courtly manners and codes of honour, from high-flown literature to the most ordinary milieux of modern life, was by no means restricted to Italy. It became a _leitmotif_ of seventeenth-century prose, poetry and drama across Europe. The misapplication of chivalric codes of honour to the circumstances of modern life is the great theme \u2013 the essential running joke \u2013 around which the whole of Cervantes's picaresque novel _Don Quixote_ revolves. The Don might be aged, Sancho Panza fat and ridiculous, and they might inhabit a gentler and more absurd world than the Rome of Pope Clement VIII; but their escapades are none the less close parodies of the scrapes and adventures in which Caravaggio, Onorio Longhi and their companions were habitually embroiled. A man with a barber-surgeon's basin on his head is mistaken for a rival knight, and beaten by Don Quixote until he bleeds. The action takes place on a dusty road in provincial Spain \u2013 but events very much like it took place every day in Caravaggio's Rome.\n\nTommaso Garzoni's _Piazza universale_ of 1585 includes a vivid characterization of Rome's ragged army of self-appointed knights errant, marauding through the city:\n\n> Every day, every hour, every moment, they talk of nothing but killing, cutting off legs, breaking arms, smashing somebody's spine... For study, they have nothing other than the thought of killing this or that person; for purpose, nothing more than to avenge the wrongs that they have taken to heart; for favour, nothing more than serving their friends by butchering enemies...\n\nThe author goes on to describe a day in the lives of such 'dregs-of-knaves... scum-of-scoundrels'. Armed with an improbably extensive arsenal of weaponry, they sally forth into the city to do their worst. Despite, perhaps, some element of caricature, Garzoni paints a convincing picture of Caravaggio's life, as it must have been, during his Carnival months, when he would 'swagger about... with his sword at his side... ever ready to engage in a fight or argument'.\n\nThe day begins in the piazza and ends, more often than not, in the whorehouse:\n\n> In the morning they get out of bed and straightaway pull on their hose, put jacket and breastplate on their back, hats on their heads, gauntlets or hunting gloves on their hands, sword and dagger at their side, arquebus in a bag and its iron balls in their breeches. And thus, armed like St George, they swagger out of the house, make a circuit of the piazza and, with four companions, make themselves master of the field...\n> \n> Then they go in a group to walk about the district, bullying everyone they meet, demanding the right of way, and with their plumes, whether black or white, they flutter fearlessly about, so that they will be taken for the boldest swordsmen on earth. Then they stop on a street corner and here, drawn up in a circle, they make fun of whoever passes, and mockingly salute whomever they like with their hats, deride the farmers, poke fun at the masters, and stop their servants by force... They also make it their custom to go out in the piazza and, as ruffians, stop to look at the peasant-girls and the countrywomen, whom they harass... Then they go to where the whores and procuresses are found: there they play a bit with Laura, strut about with Betta, and mess around with Rosa. With Cieca they have an argument, pinching a pair of clogs and taking away her shoes, or giving her some slaps on the head, pinching her buttocks, biting her breasts, and making her howl like a wretched bitch. On the way home they meet some other _bravi_ , by whom they are punished as they deserve...\n\nThere is no written evidence to place Caravaggio in the company of any particular Laura or Betta, any Rosa or Cieca, but he was certainly friendly with a number of prostitutes, some of whom modelled for him. His favourite was a dark-eyed girl destined to become one of Rome's most famous courtesans. Caravaggio painted her portrait, perhaps in exchange for favours received, which she would bequeath to her richest lover and patron, a Florentine nobleman named Giulio Strozzi. It later passed into the collection of Cardinal del Monte's friend Vincenzo Giustiniani, in whose collection it was catalogued as 'a courtesan called Fillide', and eventually the work entered the collections of the Kaiser-Friedrich Museum in Berlin. It was destroyed during the Second World War, but black-and-white photographs survive. They show a smouldering beauty, in understated finery, with a look of wary self-possession about her. She could be a sister to Edouard Manet's hard-faced nineteenth-century whore _Olympia_. She clutches a posy of jasmine blossoms, symbol of erotic love, to her breast.\n\n'Fillide' was Fillide Melandroni. The literary land of Arcadia was peopled with pure and innocent shepherdesses named Fillide (or Phyllida, as the name is often anglicized), but this was her real name. Someone, once, had dreamed of a bright future for her \u2013 her father perhaps, whose own name, Enea, conjured up the epic deeds of Aeneas recounted by the Roman poet Virgil.\n\nFillide was born in Siena, but in early adolescence she was uprooted to Rome. Her father had died when she was still young, so money was scarce. Her aunt Pietra was already waiting on tables in a taverna in the city; perhaps it was she who encouraged the family to move there in search of better prospects. Fillide made the journey with her mother, Cinzia, and her brother, Silvio. They arrived on a rainy day in February 1593, less than a year after the young Caravaggio had first come to the city. They shared their coach with the Bianchini family. Sibilla Bianchini had a son called Matteo and two daughters, Alessandra and Anna. Anna was the same age as Fillide.\n\nThe two families lodged together in the same house on Via dell'Armata. Close by was the church of Santa Caterina, patron saint of their home town, Siena. Not long after arriving in Rome, the two mothers, Cinzia and Sibilla, put their daughters to work as prostitutes. In April 1594 the two girls were arrested together for being out after curfew, on suspicion of soliciting. The investigating magistrates called them 'Donna Anna' and 'Donna Fillide', which made them sound more grown-up than they really were. They were fourteen and thirteen years old, respectively.\n\nCaravaggio painted his portrait of Fillide around 1598, by which time she was seventeen or eighteen, and when she and Anna Bianchini were still going around together. According to a witness statement made in that same year, Anna was 'smaller rather than bigger' and had 'long red hair'. There is an outside possibility that she was the girl who modelled for _The Penitent Magdalen_ and the sleeping Virgin Mary in _The Rest on the Flight to Egypt_.\n\nFillide first appears in a devotional painting of about 1598, _Martha and Mary Magdalen_ , now in the collection of the Detroit Institute of Arts. The picture is badly damaged and of questionable quality: the brightly illuminated Magdalen is puffy and distorted and clumsily portrayed, giving her a pop-eyed appearance, but it is certainly by Caravaggio.\n\nThe painter shows the moment when the Magdalen, urged on by her sister Martha, forswears her life of harlotry. The shadowy Martha, viewed in half-profile, is another figure who may possibly have been modelled by Fillide's friend, Anna Bianchini. Like preachers of the time, Martha counts on her fingers the reasons to repent. But her sister has already decided to devote herself to God. Fillide as Mary Magdalen once again holds a flower to the bodice of her scarlet silk dress. This time it is not perfumed jasmine but orange blossom, symbol of purity. A gap-toothed ivory comb and a precariously propped-up convex mirror \u2013 probably the same convex mirror in which Caravaggio studied his own distorted features to paint the screaming _Medusa_ \u2013 here symbolize worldly vanities renounced. The mirror also evokes the prophecy of St Paul: 'Now we see through a glass darkly, but then we shall see face to face' (1 Corinthians 13:12). Her mind has turned away from the things of this world, and towards the next.\n\nThe ring on the wedding finger of her left hand symbolizes her decision to embrace chastity and become a bride of Christ. The finger sticks out at an angle, as though dislocated or nerve-damaged. Another figure by Caravaggio for which Fillide modelled, that of the slightly later _St Catherine_ , suffers from the same slight deformity of the same finger: Fillide must have had a damaged hand.\n\nThe painter could easily have disguised or corrected the flaw, but he chose to preserve it on both canvases. Why? The most likely explanation is that he intended it as an advertisement of his militant naturalism. To animate the old stories of Christianity, to make them seem as though taking place in the present day, he had developed his own unique method: he would systematically restage the sacred dramas, using real, flesh-and blood people, and paint the results. The crooked finger was there to call attention to the most distinctive aspect of his practices: his reliance on the study of actual models carefully posed in the stage setting of the studio.\n\n_Martha and Mary Magdalen_ was probably painted for Olimpia Aldobrandini. The picture's subject may have reflected Olimpia's own charitable activities. Gregory Martin, an English Catholic priest who was in Rome in the 1580s, observed that a group of noblewomen had formed an association for the reform of whores. Regardless of their own safety, they would go into the Ortaccio di Ripetta, the 'evil garden', and plead with prostitutes to mend their evil ways: 'honest and wise matronnes of Rome... match them selves with the famous or rather infamous and notorious sinful wemen of the citie, such as sometime Marie Magdalen was, and so by their wordes and behaviour and promises and liberality towards them, they winne them to honest life, and by Gods merciful hand working with exceeding charity they plucke them out of the deepe pitte of dayly fornication, as it were raysing dead stinking carcasses out of their graves.' Perhaps Olimpia Aldobrandini was one of these modern-day Marthas. The reformed prostitutes were known as _convertite_ , or convertites. 'These be therefore so called,' noted Martin, 'bycause they are converted from their naughty life, and of common whores and harlots made good Christian wemen...'\n\n#### 'I'LL GET YOU NEXT TIME'\n\nThe young Fillide Melandroni could pose as the Magdalen, but actually it seems that repentance was the last thing on her mind. During the years when Caravaggio knew and painted her, she was often in trouble with the authorities, and not solely for prostitution. On 4 December 1600 a Roman court investigated an accusation of assault made against her and another courtesan, Tella Brunora. The litigant was a third prostitute, Prudenza Zacchia, who lived directly 'behind the monastery for the Convertites of the city'. All three of the women involved worked for the same pimp, a man called Ranuccio Tomassoni from Terni. Caffarelli, in _The Families of Rome_ , described Ranuccio as 'a good young man, and good in his conduct', but others disagreed. Caravaggio, for one, would become his deadly enemy.\n\nRanuccio belonged to a family of soldiers and mercenaries with long-established links to the Farnese dynasty. One of Ranuccio's brothers, Giovan Francesco, had served with honour under the general of the pontifical army, Giovan Francesco Aldobrandini. Another brother, Alessandro, had fought in Flanders. Ranuccio himself never saw military service, although he often carried a sword in the evenings. His excuse was that he was in the service of Cardinal Cinzio Passeri Aldobrandini. But his real job was running an unruly gang of prostitutes. If any of his girls' clients turned nasty, it was as well to be armed. As well as taking a cut of their earnings, he took payment in kind from those he favoured. This sometimes caused trouble, as an investigation of December 1600 revealed.\n\nPrudenza Zacchia was called first by the court. She had recently been charged with throwing a brick at an agent of the Governor of Rome, but this time she was the injured party. She claimed that Fillide and Tella had been conducting a vendetta against her:\n\n> Your Honour should know that yesterday evening, at about the first hour of the night, the two accused came to look for me, and not finding me, they gave my mother several kicks, and went out. Nothing else happened yesterday evening. This morning I was in the house of Ranuccio, who lives at the Rotonda [i.e. next to the Pantheon]. The said Fillide, the accused, came to this said house and attacked me with a knife, which she had in her hand, and was restrained by Ranuccio. She came at me in every way, and gave me many blows, and tore off a lot of my hair. Then she left...\n> \n> Later I was in a downstairs room in my own house, when both of the accused arrived and entered the house by force. In coming in they gave my mother, who was at the door, a shove. The said Fillide came at me with a knife to disfigure me, and she hauled me up by the mouth to give me a scar. I defended myself with my hand, which she cut on the wrist and wounded me, as your Honour can see... and as soon as they saw that I was bleeding, they went with God. Later they went out again to have a go at me, and if they hadn't been restrained by certain gentlemen... they would have gone at me again. Then the said Fillide came up to the window, and started to taunt me, saying that she wanted to scar me all over. I am making a complaint about this...\n\nOther testimony fleshed out the story. Geronimo Mattei told the court that he had been warming himself by the fire downstairs in Ranuccio's house earlier that morning. 'Ranuccio was in bed together with a woman called Prudenza Zacchia... and a woman called Fillide came into the house and ran upstairs to the said Ranuccio and Prudenza, and as soon as she saw the said Prudenza she began saying, \"Ah, you slag, you baggage, there you are!\" and at the same time she ran to the table and took a knife and went to the said Prudenza, saying, \"Whore, I'm going to scar you everywhere.\" '\n\nGeronimo intervened and took the knife from Fillide, but he could not prevent her from attacking Prudenza again. This time, 'she tore lots of hair from her head.' Later on, he told the court, 'I was passing by the said Prudenza's house, just behind the monastery of the Convertites, when Prudenza called out to me and showed me a wound in her hand, saying that Fillide and Tella had attacked her in her house, and that Fillide had taken a knife to scar her, and had wounded her in the hand...' It is not hard to imagine how Fillide might have acquired the bent and damaged finger of her own left hand.\n\nThere was one more witness, a man called Cesare Pontoni. He was a close friend of Ranuccio and Giovan Francesco Tomassoni, and often testified in cases where they were involved. He had witnessed only the last incident in this serial fracas. Cesare told the court that he was walking down the street when he saw Fillide shouting at Prudenza. Fillide was at the window of Tella's house, which was opposite where Prudenza lived. Prudenza was standing in her own doorway. 'You dirty whore!' Fillide screamed. 'I hurt your hand, when I wanted to stab you in the mouth, but I'll get you next time!' Moments later Fillide advanced on Prudenza with a stone in her hand, yelling, 'You dirty whore! I want to cut you! I want to cut you!'\n\nThe interpretation of the case is clear enough. Prudenza and Fillide were vying for Ranuccio Tomassoni's affections. Amongst the elaborate rituals of insult and injury, the crucial terms in the court documents are _sfregio_ and the related verb _sfregiare_. Literally, a _sfregio_ was a facial scar, but in the honour code of the time it also carried the figurative meaning of a serious affront to a person's reputation. When Fillide said, repeatedly, that she wanted to cut Prudenza in the face, she was expressing a desire to dishonour and shame her. She uttered her threats publicly because she wanted her intentions to be known in the public arena of the street \u2013 the theatre in which reputation was made and harmed. Prudenza repeated those threats in court for the same reason. To accuse someone of the intention to inflict a _sfregio_ was to alert the law to a potentially serious offence.\n\nIn the event, the many threats of wounds to Prudenza's face seem not to have actually been carried out, perhaps because Fillide's main aim was to frighten her rival. If she had actually cut her in the mouth, or sliced off her nose \u2013 a not unheard-of tactic in the more extreme revenge assaults \u2013 Prudenza would have become damaged goods. That would not have pleased Ranuccio. The impression that emerges from the testimony is that, for all her apparent wildness, Fillide knew what she was doing and remained in control throughout. Probably because nobody was seriously hurt, the case seems to have come to nothing.\n\nCaravaggio's name does not appear in the trial transcripts involving Fillide, Tella, Prudenza and Ranuccio, so these documents shed little direct light on the painter's future quarrels with the pimp. But they shed a good deal on the murky world in which both men moved. Ranuccio's contacts and alliances may also be significant. His family's patrons, the Farnese, were supporters of Spain against France, so the Tomassoni clan was closely connected with Rome's pro-Spanish faction. This was true of Ranuccio's friends too. Politics could have been one cause of bad blood between him and Caravaggio. But the artist's relationship with Fillide may have been another: a mere painter was hardly a desirable client for Ranuccio's most beautiful courtesan.\n\n#### PAINTING FILLIDE\n\nFillide had been miscast as the virtuous heroine of _Martha and Mary Magdalen_ , which is perhaps why that picture seems less than completely convincing. On two other occasions Caravaggio painted her more as the historical record suggests she truly was \u2013 tough, passionate, with a capacity for violence. Even though she only appears in devotional pictures, her presence in them tips the balance of his art from sacred to profane.\n\nIn 1598 or 1599 Caravaggio painted a startlingly sado-erotic _Judith and Holofernes_ , with Fillide in the leading role. Like that of David and Goliath, the biblical story of Judith was a parable of underdog virtue triumphing over tyrannical might: the Jewish heroine of the tale seduces the ruthless Assyrian general and then slays him, with his own sword, in his tent. It was a subject that had been treated by many celebrated artists. Michelangelo had depicted Judith and her maidservant elegantly bearing aloft the tyrant's severed head, as his corpse writhed in darkness, in one of the four paintings at the corners of the Sistine Chapel ceiling. The great Florentine Renaissance sculptor Donatello had created a famous bronze _Judith_ in which the heroine hacks implacably at the neck of her victim. But even Donatello's stark and visceral image pales by comparison with Caravaggio's clinically violent conception of the subject.\n\nOnce again, the painter brought a scene from the biblical past into the world of his own time, but never before had he done so with such brutal, shocking immediacy. Sanctified execution in an Assyrian tent has become murder in a Roman whorehouse. The bearded Holofernes, lying naked on the crumpled sheets of a prostitute's bed, is a client who has made a terrible mistake. He wakes up to realize that he is about to die. Fillide pulls on his hair with her left hand, not only to expose his neck but to stretch the flesh taut so that it will part more easily under the blade. In her right hand, she holds the oriental scimitar \u2013 Caravaggio's one concession to historical accuracy \u2013 with which she has just managed to sever her victim's jugular. She frowns with grim concentration, as he screams his last, and as the blood begins to spray from the mortal wound in bright red jets. A theatrical swag of dark red drapery hovers directly above the act of murder.\n\nCaravaggio has imagined the whole scene as a fantastically extreme version of the kind of violent incidents in which he and his companions were often embroiled. 'I want to cut you! I want to cut you!' Fillide would yell at her rival Prudenza. Here, the threat is fully carried out. The heroine's grizzled maidservant, readying herself to bag up the bloody trophy of a severed head, reinforces the impression that the action is indeed taking place in a darkened brothel somewhere in Rome. She is the stock figure of the procuress, the whore's wizened partner in corruption. Caravaggio adds a sexual frisson to the thrill of bloody violence: beneath the diaphanous fabric of her tight-fitting bodice, Fillide's nipples are visibly erect. It is the sort of detail that Cardinal Paravicino may have had in mind when he made his remark about pictures that he 'would not have wanted to see from a distance'.\n\n_Judith and Holofernes_ divided Caravaggio's contemporaries. Annibale Carracci's succinct condemnation of the work encapsulated the reservations of all those who found Caravaggio's realism rude and indecorous. 'When pressed to speak his opinion on a _Judith_ by Caravaggio, he replied \"I don't know what to say except that it is too natural.\" ' Artemisia Gentileschi, by contrast, was fascinated by it. During the second decade of the seventeenth century, she made a name for herself by painting numerous versions of the same subject in a darkly tenebristic style directly modelled on Caravaggio's own. She gave an idiosyncratic twist to the theme by using it to take public revenge on the man who had raped her, painting herself as the sword-wielding heroine and Agostino Tassi as her victim.\n\nIn 1599 Fillide Melandroni appears once more in Caravaggio's art, as _St Catherine of Alexandria_. The masterpiece of his early career, it is another picture that simmers with violent sexuality. Less shocking than the _Judith and Holofernes_ , but equally striking, it encapsulates the intense and powerfully inverted eroticism with which the Counter-Reformation Church infused the idea of martyrdom. The haloed saint is isolated in a bare, featureless, dark room illuminated by a single light source coming from the left. She kneels on a red damasked cushion and wears splendid robes of purple, to indicate her royal birth. The mood is intimate, suspenseful. She holds the viewer's gaze.\n\nThe saint is alone with the attributes decreed by her legend. A martyr's palm lies crosswise on the floor at her feet. Beside her is the spiked wheel on which the Roman emperor Maxentius had intended to break her body, painted from an ordinary Roman cartwheel of coarse-grained oak. (A section of it is broken, because God sent a thunderbolt to shatter it before it could be used on the saint.) The actual instrument of her death was a sword. Caravaggio, with the expertise of a swordsman, has furnished Fillide with a weapon appropriate to her sex \u2013 a light, thin, perfectly deadly rapier. He did not have such a sword himself, so he borrowed one. The hilt is so intricate that it must have been painted from a real example.\n\nThe picture's subject is a yearning for death so strong that it resembles sexual desire. The saint leans towards the wheel and its vicious spikes of grey steel as if leaning towards a lover. A fold of extraneous drapery has wrapped itself around the longest and darkest of the wheel's spikes. She caresses the pommel of the sword and runs a finger lovingly along its blood-groove. Death by the sword is her consummation. To be penetrated by its steel is to be married, forever, to Christ. Her face is flushed, her eyes excited.\n\nThe composition is austere, the forms monumental, the paint handled with a subtle brilliance. The soft-focus depiction of the muted drapery around the wheel-spike anticipates the work of Vel\u00e0zquez, and in fact Caravaggio would rarely repeat such levels of virtuosity. But it is not hard to see why some of the artist's contemporaries might have been troubled by such a picture. Was it really a picture of St Catherine, rapt in the joyful embrace of death? Or was it just a picture of a sexy modern girl, with some studio props, alone in a room? In truth, it was both. Caravaggio's technique opened his art to ambiguity because it exposed the painter himself directly to reality. His responses inevitably coloured every image that he created, whatever its mythical construction might be. Caravaggio could turn Fillide into Mary Magdalen, into Judith, into St Catherine, but the transformation could never be absolute. After all, it was Fillide that he saw in the room, Fillide with her damaged hand, breathing softly and looking back at him, with her wide appraising eyes, as she tried to hold the pose.\n\nBy the end of the 1590s Caravaggio had invented a new style and a new approach to painting, and in the three pictures for which Fillide posed he arrived at something like a fixed, settled method. In some early works he had used a light ground, like other painters from Lombardy. But in these later paintings he used a dark ground and worked from dark to light, a technique that he may have seen for the first time in the art of Tintoretto. It suited him in a number of ways. A dark ground enabled him to focus only on the essentials of a scene, as he imagined it. Dark paint creates an illusion of deep shadow around the principal forms and therefore also does away with the need to paint background detail: Bellori, in his biography of the painter, noted that Caravaggio 'left the ground visible in the half-tones', meaning that in places he could model form simply by leaving the canvas in the unpainted state in which it had been prepared. (The technique is visible, for example, in the frame of the mirror in _Martha and Mary Magdalen_.) Caravaggio was fond of short-cuts and liked to work quickly, which suggests another reason behind his extreme tenebrism: quite apart from their expressive effect, pools of darkness, like visible ground, simply mean that there is less to paint.\n\nCaravaggio's habitual impatience is manifest too in his frequent practice of working wet-in-wet rather than waiting for each layer of oil paint to dry. He was unique among the painters of his time in making no preparatory drawings for his pictures, preferring to block out his compositions directly on the primed canvas. Having posed his models, he often marked the exact positions of heads and other contours by making light incisions in the base layer of paint, presumably so that he could reset the models' positions after every break in the work. No other artist of his time used such incisions. Caravaggio's exceptional working procedure argues strongly for the hypothesis that he learned little from his master, Peterzano, and was largely self-taught.\n\nCaravaggio did not draw because his method of composition was essentially theatrical \u2013 proto-cinematic, it might be said, because lighting was also involved. He composed by staging scenes, or fragments of scenes, that he knitted together, collage-like, on his canvas, using shadow to mask the joins. The scenes involved objects, models, props. Fillide knelt on a real purple cushion, leaned against a real wheel and held a real sword while Caravaggio painted her. Sometimes, not surprisingly, the absence of preparatory drawing led him to make a mistake in posing his models: halfway through _Judith and Holofernes_ , he realized that a head half severed would look more detached from the neck and trunk of the body than the head of his very alive model. X-rays show that he painted over the first head of Holofernes, reposed the man and painted him again to achieve the necessary degree of grisly separation.\n\nCaravaggio's method also involved setting lights, or at least controlling illumination in some way. Joachim von Sandrart gave a short description of his technique, saying that 'as he wished to effect a more perfect roundness and natural relief, he regularly made use of gloomy vaults or other dark rooms which had one small source of light from above; so that the darkness, by means of strong shadows, might leave power to the light falling upon the model, and thus produce an effect of high relief.' There is evidence of it in _Martha and Mary Magdalen_. The brilliant square of light reflected in the surface of the convex mirror is Caravaggio's 'source of light from above' made visible on the canvas. Bellori noted that Caravaggio began to work in this way at around the time that he painted the _St Catherine_ and other pictures close to it in date, in 1598\u20139. His pictures from these final years of the sixteenth century 'have a darker colour', he observed, 'as Michele [ _sic_ ] had already begun to darken the darks'.\n\nBellori went on to give his own account of how Caravaggio achieved his famously extreme contrasts of light and dark:\n\n> the colouring he was introducing was not as sweet and delicate as before, but became boldly dark and black, which he used abundantly to give relief to the forms. He went so far in this style that he never showed any of his figures in open daylight, but instead found a way to place them in the darkness of a closed room, placing a lamp high so that the light would fall straight down, revealing the principal part of the body and leaving the rest in shadow so as to produce a powerful contrast of light and dark. The painters then in Rome were greatly taken by this novelty, and the young ones particularly gathered around him, praised him as the unique imitator of nature, and looked on his work as miracles.\n\n#### A BROTHERLESS MAN\n\nMany of Caravaggio's pictures of the later 1590s look like demonstration pieces. Each new work shows a new difficulty overcome. But with the _St Catherine_ of 1599 Caravaggio had reached a higher level of mastery and assurance. He painted the picture for del Monte, who had a special devotion to the martyr, probably because she was the patron saint of scholars. Perhaps this was the work that persuaded the cardinal that his proteg\u00e9 was ready for bigger commissions.\n\nIt was at this otherwise propitious time, for reasons unknown, that Caravaggio finally severed all links with his family. In 1594 his sister Caterina had married a 'maestro Bartolommeo Vinizzoni' \u2013 'maestro' indicating that he was an artisan of some kind. Caravaggio had not attended the wedding. He had also been avoiding his brother, Giovan Battista, the priest, who was in Rome studying moral theology with the Jesuits from the autumn of 1596 to the winter of 1599. Before Giovan Battista went back to Lombardy, to be ordained as a subdeacon in the province of Bergamo, he decided to call on Caravaggio. Giulio Mancini tells the strange story of what happened when the two brothers met:\n\n> Caravaggio had an only brother, a priest, a man of letters and of high morals who, when he heard of his brother's fame, wanted to see him and, filled with brotherly love, arrived in Rome. He knew that his brother was staying with Cardinal del Monte, and being aware of his brother's eccentricities, he thought it best to speak first to the Cardinal, and to explain everything to him, which he did. He was well received by the Cardinal, who told him to return in three days. He did so. In the meantime, the Cardinal called Michelangelo and asked him if he had any relatives; he answered that he did not. Unwilling to believe that the priest would tell him a lie about a matter that could not be checked, and that would do him no good, he asked among Caravaggio's compatriots whether he had any brothers, and who they were, and so discovered it was Caravaggio who had lied. After three days the priest returned and was received by the Cardinal, who sent for Michelangelo. At the sight of his brother he declared that he did not know him and that he was not his brother. So, in the presence of the Cardinal, the poor priest said tenderly: 'Brother, I have come from far away to see you, and thus I have fulfilled my desire; as you know, in my situation, thank God, I do not need you for myself or for my children, but rather for your own children if God will do you good as I will pray to His Divine Majesty during my services, as will be done by your sister in her chaste and virginal prayers.' But Michelangelo was not moved by his brother's ardent and stimulating words of love, and so the good priest left without even a goodbye.\n\nMancini neither comments on the story nor explains it in any way. But the structure of his telling, which is like a fable, may contain clues about what he believed was going on. Three times Caravaggio is asked to recognize his brother, the priest, and three times he refuses him. Like St Peter denying Christ three times 'before the cock crows twice' (Mark 14:66\u20138), Caravaggio denies his brother, himself Christ's servant on earth. The implication is that religion, somehow, lay at the heart of the matter. Was Caravaggio ashamed to look his pious brother in the eye? Mancini may have thought so.\n\nOttavio Leoni drew a portrait of Caravaggio at around this time. He has the dark, dishevelled hair and bushy eyebrows described by Luca, the barber-surgeon. But it is his expression that seems most striking. His mouth is set and sullen. There is determination and truculence in his eyes, but there is sadness there too \u2013 a look of profound loneliness, and abandonment.\n\n## PART FOUR  \nRome, 1599\u20131606\n\n#### THE ST MATTHEW CHAPEL\n\nFor the priests at San Luigi dei Francesi, the Contarelli Chapel had been nothing but trouble. For years the chapel \u2013 the fifth one along on the left, in the national church of the French \u2013 had been little better than a building site. Not only did it make the church look bad, the priests complained, it was giving Rome's French community a bad name.\n\nThe saga had begun in 1565, when a French cardinal named Mathieu Cointrel (or Matteo Contarelli, as his name was Italianized) had paid a considerable sum to acquire the chapel, where he intended to be buried. Contarelli had already given generously during the construction of San Luigi dei Francesi, footing the bill for its fine marble fa\u00e7ade, designed by Giacomo della Porta. But despite his best efforts his own chapel was still all but bare of decoration when he died in 1585.\n\nThe cardinal himself had contracted Girolamo Muziano, a competent but unexceptional painter, to paint frescoes on its two lateral walls and decorate its vault. Muziano had prevaricated for years, only to renege on the commission with almost nothing painted. In 1587 the executor of Contarelli's will, Virgilio Crescenzi, had commissioned a marble altarpiece from a Flemish sculptor, Jacques Cobaert. Crescenzi had also persuaded Giuseppe Cesari to fresco the walls and ceiling. Cesari had completed the frescoes on the vault by 1593, when Caravaggio was part of his studio, but he never got round to the rest because he was deluged by other assignments, including several from the pope himself. Meanwhile the sculptor, Cobaert, was said to be working away, although there was nothing to show for it. Thrilled by the importance of the commission, but paralysed by self-doubt, he toiled for years on what he hoped would be his _magnum opus_. His contract was renewed in 1596, yet as the end of the sixteenth century approached there was still no sign that he would ever actually deliver the work. Those close to him remarked that Cobaert was becoming ever more paranoid and secretive.\n\nIn 1597 the patience of the long-suffering priests had finally snapped. With the Jubilee year of 1600 fast approaching, they had sent a petition to the pope:\n\n> Most Blessed Father, the French community of the Church of San Luigi in Rome... humbly represents that the chapel... founded in this church by the late Cardinal... and provided by him with one hundred gold scudi per annum for two chaplains, has been closed for more than twenty-five years and is at present still closed. And if Your Holiness does not bring His authority to bear in the matter, there is a danger that the chapel will never be completed, because Signor Abbate Giacomo Crescenzi, the executor of the will of the above-named Cardinal since the retirement of his father Virgilio Crescenzi... has not finished it and excuses himself on the grounds of difficulties with the sculptor, the painter and other things. Thus the soul of the deceased has been cheated of its masses and the church of San Luigi similarly cheated of the endowment which was destined for the chapel. All of this is a discredit to the divine service and a shame for the community, and it leads people to believe that the neglect is the fault of the community when they see the chapel continually boarded-up and closed while various other churches in Rome are constructed from their foundations up... the heirs and sons of the Crescenzi, accumulating [revenues] year after year and day after day, have bought many and various offices in the Cancelleria, real estate and other things without doing anything which relates to the will of the testator and without even having anniversary services said for the soul of the deceased...\n\nAs a result of this tirade, Clement VIII ordered the Crescenzi to surrender Contarelli's legacy and entrusted responsibility for the chapel to the governing body of the Fabbrica di San Pietro \u2013 the works office of St Peter's. Giuseppe Cesari was approached again and asked to finish what he had begun, but he pleaded overwork. Del Monte, whose palace was directly opposite the church, followed these developments carefully. Del Monte was friendly with the Crescenzi family. He busied himself behind the scenes, pulled the right strings and somehow won the commission for Caravaggio, an artist as yet untried in the public arena of large-scale religious painting. 'With the support of his Cardinal he got the commission for the Contarelli chapel in San Luigi dei Francesi,' Baglione noted, with a touch of bitterness. On 23 July 1599 Caravaggio signed a contract with the two rectors of the church in which he undertook to complete the side panels for the chapel by the end of the year for a fee of 400 scudi.\n\nIt was a daunting challenge for a young and relatively inexperienced artist. So far Caravaggio had never painted a picture with more than four figures in it. None of his previous canvases had been more than four or five feet across. Suddenly, he would have to produce two monumental paintings, each more than ten feet in width and almost the same in height. He had, it is true, painted a number of devotional pictures, but he was known principally as a painter of genre scenes with a talent for still life. Now he was being invited to create complex religious narrative paintings. It was a chance to compete with the greatest artists of the past. But if it went wrong, Caravaggio's failure would also be very public.\n\nThe subjects of the two lateral pictures for the Contarelli Chapel had been prescribed by Cardinal Cointrel himself. He had wanted his burial chapel to be dedicated to St Matthew, his name saint, and so the two pictures on either side of the altar were to tell stories from the apostle's life. The painting on the left was to show Matthew, the tax collector, being summoned by Christ. That on the right was to show the saint's glorious martyrdom at the hands of a pagan assassin. Cointrel had also had very particular ideas about how these scenes might be depicted, which are reflected in the unusually circumstantial wording of an attachment to one of the contracts for the painting of the chapel:\n\n> For the St Matthew Chapel... At the right side of the altar, that is, on the side of the gospel, there is to be a painting 17 palmi high and 14 palmi long in which is painted the same St Matthew in a store or large room used for tax collection with various items pertaining to such an office, with a counter such as tax collectors use, with books, and monies that have been received, or as shall seem best. From this counter St Matthew, dressed as a practitioner of his trade would be, should rise in order to follow Our Lord who passes along the street with his disciples and calls him to the apostolate; and the attitude of St Matthew should show the painter's skill, as should also the rest. On the left side, that is, of the epistle, there should be another painting of the same height and length as above in which is painted a long wide space in the form of a temple, with an altar raised up on the top of three, four, or five steps: where St Matthew dressed in vestments to celebrate the mass is killed by the hands of soldiers and it might be more artistic to show the moment of being killed, where he is wounded and already fallen, or falling but not yet dead, while in the temple there are many men, women, young and old people, and children, mostly in different attitudes of prayer, and dressed according to their station and nobility, and benches, carpets, and other furnishings, most of them terrified by the event, others appalled, and still others filled with compassion.\n\nThe level of detail in these instructions shows how carefully painters had to tread in Rome at the end of the sixteenth century. In the event Caravaggio took artistic licence, but he remained faithful to the spirit of the patron's recommendations. None of the documentary sources specifies the medium in which the works were to be carried out, although such was the pre-eminence of fresco in the traditions of Rome's Christian art that its use was probably assumed. But frescoes must be painted _in situ_ , the pigment applied directly to a fast-drying layer of wet plaster. The technique would have required Caravaggio to depart from his studio practice \u2013 the painting of live models posed in carefully controlled light conditions. Reluctant to abandon the procedures that had already won him admirers, Caravaggio purchased two large canvases and set to work in his usual way.\n\nTrue to his method of transposing the biblical past to the present day, Caravaggio imagined _The Calling of St Matthew_ taking place in a dingy room somewhere in modern Rome. Christ and St Peter have just entered the dim and plainly furnished office of Matthew, the tax-gatherer. Here they encounter five men, grouped around a table set close to a bare wall relieved only by a single window. The window's shutter is open, but little light penetrates through its four dull panes, which are made not of glass but of oilskin held in place by crossed strings. There are coins on the table as well as a moneybag, an open account book and an inkwell from which the stem of a quill protrudes. A transaction is taking place.\n\nAt the far end of the table, a young man sitting in a savonarola chair is absorbed in calculation. He is the taxpayer, who has settled his dues and is now receiving a small amount of change. His shoulders are hunched as he counts the meagre handful of coins before him and prepares to draw the money in. Directly behind him, a bespectacled old man wearing a fur-trimmed coat peers down at the table, as if to check that all the sums have been done correctly. Next to them sits Matthew himself, accompanied by his page, a round-faced boy who leans with friendly familiarity on his master's shoulder. Caravaggio's Sicilian friend, the painter Mario Minniti, posed for this figure. Opposite sits another young man in fine page's livery, his striped black and white sleeves flashing and shimmering in the half-dark of the room. He is presumably the taxpayer's minder. Tradition has it that this figure was also modelled by a painter, Lionello Spada. But the identification may be apocryphal, a flight of fancy inspired perhaps by the sword \u2013 _spada_ , in Italian \u2013 that he wears at his side.\n\nThe biblical account of Matthew's election to the apostolate is terse in the extreme: 'And as Jesus passed forth from thence, he saw a man, named Matthew, sitting at the receipt of custom: and he saith unto him, Follow me. And he arose, and followed him' (Matthew 9:9). Caravaggio has painted the moment at the heart of this truncated narrative, when Christ has just spoken his simple two-word command. Matthew, at once astonished and compelled, points to his own chest as he gazes up into the eyes of the Saviour. There is incredulity in his expression and a question frozen on his lips: 'Who, me?' He continues, absent-mindedly, to count out one last coin of the taxpayer's change, but he knows in which direction his destiny is taking him. He braces his legs, preparing to stand up and step into his new existence. The command is irresistible, its outcome inevitable. Christ fixes the tax-collector with a hypnotizingly intense stare. Even as he reaches out towards Matthew, he has already begun to leave the room. His bare feet, half hidden in deep shadow, are turned away from the company of men back towards the outside world. In a moment he will have left, taking his new apostle with him. All has been done that needed to be done.\n\nMatthew and his companions, grouped around the coin-strewn table, might almost be gaming in the tavern of the _Cardsharps_ , painted for Cardinal del Monte five years earlier. The disconsolate man paying his taxes and raking back a pile of change looks like a gambler who has just won an annoyingly small pot. Indeed that was exactly what the seventeenth-century writer Joachim von Sandrart took him for, years after the picture was painted. 'Christ is represented in a dark room,' he wrote, 'which he has entered with two of His followers and finds the tax collector Matthew in the company of a gang of rogues with whom he is playing cards and dice, and sitting about drinking. Matthew, as if afraid, conceals the cards in one hand and places the other on his breast; in his face he reveals that alarm and shame which is the result of his feeling that he is unworthy to be called to the Apostolate by Christ. One of the other men takes his money from the table by sweeping it with one hand into the other, and attempts to sneak away; all of which seems true to life and nature itself.'\n\nSandrart plainly failed to give the painting his full attention, but none the less his misinterpretation evokes a mood that Caravaggio intended to create. The tax-gatherer's office, with its basement gloom and its cast of mercenary characters, is a convincingly seedy den of iniquity. Christ brings light into this darkness, just as he brings illumination and divine purpose to Matthew's dreary, money-grubbing existence. The picture's main light source is high and to the right, to suggest daylight flooding in from above, perhaps through an open door and down a flight of stairs. It flashes on to the face of Matthew, along a diagonal parallel with the line traced by Christ's golden halo and his outstretched, spotlit, beckoning hand. It is the light of ordinary mundane reality, yet it is also the light of God.\n\n_The Calling of St Matthew_ is built on contrasts, and not only the contrast of light and shade. Whereas Matthew and his companions are dressed in foppish modern finery, Christ and the solemn, reproving figure of St Peter go barefoot and wear simple, timeless robes. They belong to a different time and place, and a different moral and spiritual universe. They might be an apparition or a dream, projected from the distant sacred past into a profane Roman present.\n\nWith _The Calling of St Matthew_ Caravaggio was staking his claim to a place in the great Italian tradition of monumental religious painting. He had the confidence to weave an overt reference to that tradition into the very fabric of his picture. The hand that Christ holds out to Matthew is a direct paraphrase of one of the most celebrated images of Michelangelo's Sistine Chapel ceiling, a detail appropriated from _The Creation of Adam_ in which the animating finger of God reaches towards the languid hand of the first man. Yet it is the hand of Adam, not God, that Caravaggio has chosen to give to his own solemnly beckoning figure of Christ. This apparent homage to Michelangelo is actually a statement of Caravaggio's independence of thought, and the detail adds a subtly appropriate layer of meaning to the picture. Caravaggio's Christ becomes a second Adam, made in God's image but purged of sin, calling Matthew to his redemption: 'For as in Adam all die, even so in Christ shall all be made alive' (1 Corinthians 15:22).\n\nThe hand of Christ is not the only such allusion in the painting. The grouping of figures around the table has been calculated to resemble a profane version of the Last Supper. The young man counting his change, oblivious to the call of Christ, clutches a bag of money in his shadowed left hand. He is like Judas with his fifty pieces of silver. It is from the company of worldly Judases, to that of Christ the Saviour, that Matthew has been called.\n\nThe painting is poetical and metaphorical, although the piety of which it speaks is harsh, direct and forbidding. It also has a haunting quality, the character of a personal renunciation: in _The Calling of St Matthew_ , Caravaggio revisited the world of his own early genre paintings, but only to consign that world to darkness. The picture was the artist's first public demonstration of his formidable naturalism, but it is less like a depiction of real life than a dream of escaping reality altogether, of being called away from a life of vice, suddenly and inexplicably, and summoned into the presence of God. Did Caravaggio himself dream of being chosen like this \u2013 of being rescued from his own unruly, imperfect nature?\n\nThe painter fought long and hard with the second of his pictures for the Contarelli Chapel, _The Martyrdom of St Matthew_. His struggle became common knowledge among the gossiping artists of Rome. Bellori, writing seventy years afterwards, knew enough to declare that Caravaggio 'did it over twice', a claim confirmed when the picture was examined during conservation in 1966. X-ray photographs reveal the painter's aborted first composition, in which the bearded martyr stood before the altar, hands outstretched to protect himself from the assault of three armed men. As one of the assassins prepared to attack, another strode in from the side, sword at the ready. A third stepped in with his back to the viewer, as if entering the scene from directly in front of the picture. Shocked spectators looked on. In this initial attempt, the figures were considerably smaller than in the completed version. The architecture, square columns and pilasters with heavy cornicing, was correspondingly more prominent. The painter was perhaps struggling to depict the scene as it had been described in Contarelli's instructions: 'a long wide space in the form of a temple... where St Matthew dressed in vestments to celebrate the mass is killed by the hands of soldiers'.\n\nUnhappy with his first effort, Caravaggio painted it out and rethought his approach. Now he was determined to give his composition a focal point, to make an image that would be at once more monumental and more dynamic. He made two fundamental changes to the composition that had displeased him: he greatly increased the scale of the individual figures, and he reduced the number of executioners from three to one. The story of murder and martyrdom was in this way compressed to a single brutal act. In the finished work a snarling youth wields a sword over the prone figure of Matthew, who lies at the base of a simple stone altar. The saint, whose chasuble is splashed with blood, has already been wounded. The assassin grasps him by the wrist, subduing Matthew at the same time as turning his body towards him, the better to administer the _coup de gr\u00e2ce_. In the first version of the painting the killers had been represented as athletic youths stripped to the waist. In the final version the single assassin is nude save for a loincloth. A high, raking light falls on the scene, catching his pale skin and accentuating his musculature. It also catches Matthew's white vestments and his helpless upturned face.\n\nCaravaggio's final composition resembles a centrifuge, with peripheral forms and figures seeming to fly off in all directions, driven away by the violence at the centre. On the right-hand side a statuesque altar-boy screams, open-mouthed. To the left several onlookers recoil, including two men in shadow. One raises his hands in a gesture of instinctive shock and revulsion, while the other simply stares, transfixed. Behind them two _bravi_ , one armed with a sword and wearing a plumed hat, look back as they prepare to flee the scene. Two more distant figures, isolated against the darkness, have already taken flight. One is shown in half-profile while the other is shown in full-face, picked out by a sudden shaft of light. He turns back to stare at the killing, his eyes full of sadness, regret, guilt. His features are unmistakably those of Caravaggio himself.\n\nThe painter's treatment of the foreground was long regarded as a puzzle. To the right-hand side, two nearly nude figures huddle together on a fold of striped blanket. Opposite them, another reclining nude supports his weight on both hands while dangling his right leg into a dark area of void space. The near-nudity of Matthew's assassin might be explained by the fact that he is pagan, but why should these other figures be half naked in church at the celebration of Mass? It has been argued that they are no more than an expedient compositional device, and that their function is essentially to swell the small crowd of witnesses. A more plausible explanation, advanced by Giovanni Urbani in his report on the cleaning of both Contarelli Chapel pictures in 1966, is that the nudes should be regarded as recent converts to Christianity who are about to be baptized. The evidence suggests that his hypothesis is correct.\n\nThe principal source for the story of Matthew's martyrdom was the popular compendium of saints' lives known as _The Golden Legend_. There it is told that Matthew travelled to Ethiopia, where he converted many people to Christianity. His followers built him a church, where he baptized the king and queen and their daughter, Ephigenia, who entered a religious order. Matthew's martyrdom was the work of the king's successor, Hirtacus, who came to the throne determined to marry Ephigenia. When Matthew counselled her to remain a bride of Christ, Hirtacus ordered that the troublesome priest be killed. The story emphasizes that Matthew's martyrdom was the direct consequence of his missionary zeal: in earlier Italian pictorial narratives of the saint's life, the image of his killing was often immediately preceded by a depiction of him baptizing new converts.\n\nConversion and baptism were themes highly appropriate to the national church of the French, whose own king had himself been so recently converted to Catholicism. The setting of Caravaggio's painting is a baptismal chapel, with steps leading down from the altar to a lustral pool, around the edges of which the naked converts have gathered. The significance of the painting's architecture was long unrecognized, for the simple reason that hardly any such baptismal chapels have survived. But they were once a common sight in Italian churches, especially in the north. In Rome, where baptism by aspersion was the general practice, stepped pools were not necessary. But in Milan, where they practised the Ambrosian rite of baptism by full bodily immersion, such chapels contained a deep pool at the base of the altar. The liturgically precise Archbishop of Milan, Carlo Borromeo, writing in his _Instructiones fabricae et supellectilis ecclesiasticae_ , described an arrangement that closely corresponds to the setting of _The Martyrdom of St Matthew_ : 'a baptistery should be in the centre of the chapel. It should be eleven cubits wide and deep enough so that the descent to it from the floor of the chapel consists of at least three steps. By the descent and moderate depth it should resemble a sepulchre.' It seems that Caravaggio painted the kind of baptismal chapel that he remembered from his childhood in Milan.\n\nBorromeo's instruction that the baptistry should resemble a sepulchre reflects the Christian belief that baptism and death are closely connected: to be baptized is to enter a new life in Christ, and to die is also to embark on the journey to a new existence \u2013 eternal life among the blessed. Baptism and death by martyrdom were even more intimately linked in Christian theology, in part because of the belief that the wound in Christ's side had flowed with water as well as blood during the Crucifixion. The early Church father, Tertullian, commented that this was 'to make us, in like manner, called by water, chosen by blood. These two baptisms He sent out from the wound of his pierced side in order that they who believed in his blood might be bathed with the water; they who had been bathed in the water might likewise drink the blood.'\n\nThese ideas are woven together in _The Martyrdom of St Matthew_. The artist has imagined Matthew, missionary to the heathen, being murdered in the very act of conducting Mass during the sacrament of baptism. As he dies his blood flows into the baptismal pool. In this detail, murder is sanctified to a holy rite. The saint's death is a baptism of blood, a rebirth into immortality. Above, invisible both to the assassin and to the saint's shocked congregation, an angel perched on a heavy outcrop of cloud descends to thrust the palm of martyrdom into Matthew's open right hand.\n\nCaravaggio's hard-won solution to the challenge of the picture combined theological subtlety with dramatic immediacy and narrative plausibility. The murderer, all but naked like the circle of converts awaiting baptism, has sprung up from their midst. He turns out to have been a pagan in disguise, lurking among the ranks of the faithful.\n\nThe principal visual inspiration for _The Martyrdom of St Matthew_ is often said to have been Titian's famous _St Peter Martyr_ , which had been painted for an altar in the Venetian church of SS Giovanni e Paolo. That work must have been in Caravaggio's mind as he devised his own image of martyrdom, since his fallen saint and executioner are undeniably close to the same figures in Titian's composition. Yet the effect of his crowded tableau of figures is closer still to the three-dimensional sculptural _mises-en-sc\u00e8ne_ of popular religious art in his native Lombardy. _The Martyrdom of St Matthew_ resembles nothing so much as the chapel sculptures of the sacred mountain at Varallo, near Milan \u2013 in particular, perhaps, the many figures assembled in a frozen re-enactment of _The Massacre of the Innocents_. Caravaggio's picture is like a partial, spotlit memory of that crowd scene.\n\nCaravaggio's self-portrait as one of the fleeing onlookers in the _Martyrdom_ is partly a kind of signature, in line with well-established Renaissance convention. A hundred years earlier Luca Signorelli had included himself as a solemn witness at the end of the world, in his fresco cycle of scenes from the Book of Revelation in Orvieto Cathedral. Caravaggio too is a witness. Including himself in the scene may have been his way of proclaiming that he really did see it all unfold, just like this, in his mind's eye. But there is perhaps more to it: he is not only an observer, but also a participant, a furtive accessory to the dreadful act. Like the converts in the foreground of the painting, he has stripped naked to be baptized; unlike them, he has gathered his blanket around him and taken to his heels. The self-portrait, in this instance, reads like a _mea culpa_. If Caravaggio had actually been there, he suggests, he would have had no more courage than anyone else. He would have fled like the others, leaving the martyr to his fate. According to the logic of his own narrative, he remains unbaptized and therefore outside the circle of the blessed. He is a man running away, out of the church and into the street.\n\n#### A BACK-STABBING, AND OTHER MISADVENTURES\n\nOn 4 July 1600 the painter received a final payment of 50 scudi for _The Calling_ and _The Martyrdom of St Matthew_. The two pictures were complete by that date, but may not have been set into the walls of the chapel until the autumn. Only in December did the carpenter employed to do the work submit his bill:\n\n> for lining the two pictures which are on either side of the chapel of Cardinal Contarelli, which are both 14\u00bd palms broad and 15 palms long; for fixing the laths in the wall so that the boards may be nailed, for putting three [laths] for each picture... and for dividing the fir-wood boards, by 50, all my own material \u2013 it amounts to 20 scudi 20 baiocchi. For making the frames of the said pictures of my own white-poplar timber \u2013 it amounts to 20 scudi and 20 baiocchi.\n\nThe pictures for the Contarelli Chapel were compellingly original public works of art. At a stroke they brought Caravaggio's new style of painting to a much broader public. His matchless sense of drama and his use of extreme contrasts of light and dark would prove intoxicatingly influential. The painting of such seventeenth-century masters as Rembrandt in Holland, Georges de La Tour in France, Ribera in Spain, even the work of much later Romantic artists such as G\u00e9ricault and Delacroix, all are inconceivable without the pictorial revolution first unleashed by Caravaggio in his two pictures of scenes from the life of St Matthew. It is no exaggeration to say that they decisively changed the tradition of European art. But in their own time, they were controversial.\n\nCaravaggio's rival, imitator and future biographer Giovanni Baglione went to see the pictures as soon as they were installed. His account of the visit conveys the impact of Caravaggio's work on those who first saw it. But it also hints at the jealousies aroused by the sudden rise to fame of a previously little-known painter from Lombardy. Baglione went to see the pictures with Federico Zuccaro, the president of Rome's art academy, the Accademia di San Luca. Sixty years old, Zuccaro was an _\u00e9minence grise_ who aspired to the mantle of Michelangelo while painting late Mannerist monstrosities. He claimed to be unimpressed by Caravaggio's work, as Baglione reported with evident pleasure:\n\n> This commission with the paintings done after life... made Caravaggio famous, and the paintings were excessively praised by evil people. When Federico Zuccaro came to see this picture, while I was there, he exclaimed: 'What is all the fuss about?' and after having studied the entire work carefully, added: 'I do not see anything here other than the idea of Giorgione in the picture of the saint when Christ calls him to the Apostolate'; and, sneering, astonished by such commotion, he turned his back and left.\n\nAt first sight Zuccaro's response seems as puzzling as it is petty. Caravaggio's monumental, tenebristic _Calling of St Matthew_ has little in common with the works of Giorgione, painter of _The Tempest_ , _The Sleeping Venus_ and the _Three Ages of Man_. But Zuccaro's phrase, 'the idea of Giorgione', suggests that he meant to invoke the Venetian master first and foremost as a stereotype \u2013 the embodiment of a particular approach to painting. It is by no means certain that the crusty academician was familiar with Giorgione's actual works. But he certainly knew Giorgio Vasari's life of Giorgione, which had emphasized the painter's absolute dependence on the evidence of his own eyes. According to Vasari, Giorgione 'would never represent anything in his works without copying it from life'. Vasari was a partisan of the Tuscan\u2013Roman approach to art, with its strong emphasis on idealized forms, usually realized in the medium of fresco; his portrayal of Giorgione as a slavish naturalist was part of a systematic damning of the great Venetian oil painters with faint praise. So in declaring that Caravaggio was merely another Giorgione, Zuccaro was tarring him with the same brush: the comment was shorthand for saying that Caravaggio had no faculty of invention or imagination, that he was a painter who brought everything down to the level of mundane actual life, even the sacred mysteries. The curmudgeonly and conservative Zuccaro may well have been genuinely disturbed by the painter's decision to depict Matthew in his tax office as if he were a character in a low-life genre scene. If so, he would not be the last to take offence at Caravaggio's perceived sins against decorum.\n\nIn certain circles of the Roman art world Caravaggio would always be seen as an unwelcome outsider. Not only did Zuccaro criticize him for being an empty-headed naturalist, but he also implied that Caravaggio was polluting the pure and noble traditions of Roman painting with a seditious foreign idea \u2013 'the idea of Giorgione' \u2013 brought in from Venice. Caravaggio's dark and monumental oil paintings would certainly have looked extremely Venetian in the chapel of a Roman church in 1600, because only in Venice, where dampness and humidity discouraged fresco painting, was it common to see such large works of religious art carried out in oil on canvas. Caravaggio's painting must have seemed truly foreign, alien.\n\nCaravaggio may not have been unduly concerned by Zuccaro's dislike for his work, but it was hardly a good omen. The Accademia di San Luca was an influential organization that could play an important role in a painter's career. He seems to have tried to be a part of it. A few years earlier he had been one of 105 artists to participate in the religious devotion known as the Forty Hours, annual celebrations in honour of St Luke. At that time he was not yet a member of the academy, although there is evidence to suggest that he may have joined some time after 1600. He was, however, never admitted to its inner circle.\n\nThe Contarelli paintings divided opinion, but they instantly established Caravaggio as one of the leading painters of the city. However, there is no sign that success mellowed him. His life on the streets of the city was more turbulent than ever. At some point during the winter of 1600 \u2013 the precise date is unknown \u2013 he clashed with one of Rome's many unemployed mercenaries. Both men drew their swords. The painter outfought the soldier, who retired hurt. Caravaggio's friend, the notoriously hot-headed Onorio Longhi, was also involved. The injured man prosecuted and the legal document that records the affair also notes that it was settled out of court. Caravaggio must have compensated the man for his injuries:\n\n> In favour of Michelangelo Caravaggio, summoned and prosecuted for a sword wound which he had inflicted on the hand of Flavio Canonico, a former sergeant of the guards at Castel Sant'Angelo, with the complicity of Onorio Longhi, without danger to life, but with a permanent scar... the most Illustrious and Reverent Lord, the Governor [of Rome], in view of the accord and reconciliation obtained from the aforesaid Flavio who was the injured party, ordered that the lawsuit... and all other documents existing against the aforesaid [Caravaggio] for the above mentioned cause shall be cancelled and annulled and that the same [Caravaggio] shall not be molested any further on the ground of the aforesaid incident...\n\nFlavio Canonico was not Caravaggio's only victim that winter. On 19 November 1600 the painter was charged with a nocturnal assault on a young art student named Girolamo Spampa from Montepulciano in Tuscany. This is what Spampa told the court:\n\n> You should know that last Friday night, three hours after nightfall, while returning from the Academy [the Accademia di San Luca], where I had been studying, when I got to the Via della Scrofa \u2013 Messer Orazio Bianchi was with me \u2013 and I was knocking at the candlemaker's door to get some candles, the defendant came up with a stick and began to beat me. He gave me a good many blows. I defended myself as best I could, shouting: 'Ah, traitor, is that a way to act!' Some butchers arrived with lights, and then Michelangelo drew his sword and made a thrust at me, which I parried with my cloak, in which he made a gash, as you can see, and then fled. Then I recognized him, whereas previously I had not been able to recognize him.\n\nSpampa's description of the attack was confirmed by his companion, Orazio Bianchi, who gave his own town of origin as Lyon in France; he was the moderately accomplished religious painter Horace Le Blanc. Le Blanc's finest hour would come in 1622, long after his return from Italy, when he was commissioned to design sets for the triumphal entry of Louis XIII and Anne of Austria into Lyon. His decorously idealized paintings, and the pattern of his later career, when he served for years as master of Lyon's guild of painters, identify him as a pillar of the academic establishment. Although he was only about twenty years old in 1600, he was already a member of the Accademia di San Luca.\n\nCaravaggio's attack on the industrious young Spampa and his aesthetically conservative friend was not a spur-of-the-moment fracas: it was a premeditated assault that reeked of vendetta. Caravaggio had clearly lain in wait for the young student, tailing him through the dark Roman streets as he made his way home from the Accademia di San Luca. Revenge attacks of this sort were often carefully calculated. The convention was that the punishment should fit the crime. Is it possible that Spampa, keen to nail his colours to the mast of the academy, had been parroting Federico Zuccaro's criticisms of the Contarelli Chapel pictures? Had Caravaggio been tipped off by one of his own friends and allies? If so, his response had a certain brutal logic to it. Spampa had been guilty of back-stabbing. So Caravaggio attacked him from behind.\n\nThe case went no further, perhaps because of lack of evidence, perhaps because Cardinal del Monte intervened on Caravaggio's behalf. But there were other incidents besides, including one sighting of the painter that suggests he himself had been on the receiving end of a beating.\n\nIn late October 1600 trouble had broken out again between Caravaggio's friend Onorio Longhi and Onorio's brother Stefano. The pair were still arguing over their contested inheritance. Stefano had charged Onorio with assault and threatening behaviour. In the course of a three-day investigation of his grievances and accusations, the court looked into a number of incidents in which Onorio had been involved. Caravaggio's presence is mentioned. The evidence is confused and fragmentary, but presents a vivid picture of the painter's life on the streets of Rome in the first year of the new century.\n\nDuring the investigation, Onorio Longhi was asked to cast his mind back to an altercation involving himself, Stefano and others that had taken place earlier that year. Under cross-examination, he conjured up the vibrancy of Rome in the high summer in holiday mood, packed with men watching sport and spoiling for a fight:\n\n> Yes, sir, if I remember correctly, in July I was at the French tennis court at Santa Lucia della Tinta to see a match between two fencers, one of whom is called Cencio Abruzzese and the other is a Bolognese, whose name I don't know. After I had seen the two fencers fight, I went on to Piazza Navona, where some people were playing ball. I went up to watch them play. I met Vicenzo da Ascoli, a fencer, Livio Freta, who had been the judge of the said combat, Fulvio Scocimarro da Riete and Geronimo Roncalli, a merchant; there was one other with them, who they say is from Terni, but I don't know his name or anything else. They asked me what I thought about the fight between the two fencers, and who struck the most blows. I told them in my opinion Cencio had struck the most; then the man from Terni suggested that I hadn't seen well, or that I didn't understand much. I told him that he had gambled away ten scudi on the first hit, and in response the said Stefano made a mistake and threw a punch at me, then he put his hand to his sword. For my honour and defence I put my hand to my sword too. We threw so many punches that I don't know who was hit, because we were separated by many, and I was alone. But according to what I heard from the Duke of Acquasparta, who brokered a peace between us, he told me that he was hurt a little in the hand.\n\nThe magistrate then told Onorio that he was not asking about that fight, but another one that had taken place on the Via della Scrofa near the harbour of the Ripetta, the whore's part of town. It was a brawl that had started because someone had called out 'Testicles for a penny' (a double-edged provocation, since the Italian word for testicles, _coglione_ , could also signify a moron or imbecile). Longhi remembered that fracas too and described it:\n\n> Sir, I was walking down the street with some friends of mine. We were talking among ourselves and I said to them that bollocks were one a penny. Someone happened to be passing, accompanied by a certain painter whom I didn't know at all. He took it as meant for himself and told me not to speak to him like that, saying that he ate bollocks like me fried. We went at each other with our fists, and were separated. Then I went off on my own business, because after the fist-fight, those two took up stones to throw, but I didn't throw back because we had been separated.\n\nLonghi insisted that nothing else happened and that no one else came to blows, but the magistrate continued with his cross-examination:\n\n> _Who else was present at the scene?_\n> \n> With him, that is, the one who came to blows with me, was one Marco Tullio, a painter, and with me was Michelangelo Merisi, the painter, who separated us.\n> \n> _Was Caravaggio armed at the time?_\n> \n> At the time Messer Michelangelo was convalescing, so he had his sword carried by a boy. This boy had the sword and was with him when the fight occurred, but Messer Michelangelo never took it out of the scabbard.\n> \n> _Did anyone else take up the scabbard and throw it, and if so, at whom?_\n> \n> When Messer Michelangelo was separating us, my adversary drew the scabbard to himself. I don't know what he did with it then and whether he threw it at me or not.\n> \n> _Did Michelangelo have the sword in its scabbard? Why was the scabbard thrown?_\n> \n> I don't know about that, because Messer Michelangelo was so ill he could barely stand, and when he saw the sword without its sheath he went off about his own business.\n\nSuddenly the magistrate changed tack, perhaps revealing his main reason for enquiring into this otherwise apparently trivial affair. Longhi was asked to give the name of the man from Terni, the one with whom he had fought. He replied that he was not sure, but that he had heard that the man was called Luca Ciancarotta. At this point, the cross-examiner abruptly brought up the name of Ranuccio Tomassoni, also from Terni.\n\n> _Had Longhi ever had an argument with the said Ranuccio? If so, when had he quarrelled with him, and over what?_\n> \n> No, sir, I've never had any words with Ranuccio Tomassoni from Terni. Even if he is a little related to the said Stefano, he is my friend and we've never had any disagreements in the past.\n> \n> _Had he ever tried to attack Ranuccio, alone or in company?_\n> \n> No, sir, no such thing, because, as I said, Ranuccio is my friend. We ate together only a few days ago. And I've never had any arguments or attacked him.\n\nA swordfight on a tennis court; the painter barely able to walk, 'convalescing' probably not from illness but from injuries sustained in some fight or other; an argument involving testicles; a rivalry with a group of men from Terni. These events would soon enough be replayed \u2013 low farce turning to tragedy \u2013 in the lives of Caravaggio and Ranuccio Tomassoni. Stirrings of the trouble that lay ahead between the two men can be sensed behind the evasive testimony of Onorio Longhi.\n\nLonghi did know Ranuccio Tomassoni well enough to be on first-name terms, as he had claimed. Just two weeks after the investigations of late October 1600, he was up before the magistrates again. The case at hand was his alleged assault of Felice Sillano, which by then had rumbled on for more than two years. Part of the investigation turned on whether a particular witness could possibly have recognized Longhi at night. When challenged, the witness turned to him and said 'I know you by your voice, because I've heard you talking with Messer Ranuccio at the Rotonda [the Pantheon], and seen you playing tennis in the Vicole de' Pantani.' Longhi may once have been on good terms with the philandering, tennis-playing pimp. But their friendship had soured by the summer of 1600, perhaps because of Caravaggio's relationship with Fillide Melandroni, or perhaps because Ranuccio had taken the side of Stefano Longhi in the brothers' long-running battle over their inheritance.\n\nIn the archives of the tribunal of the Governor of Rome is another illuminating document, in effect an early seventeenth-century restraining order. On 17 November 1600 a sculptor called Hippolito Butio, of Milan, gave his pledge that Longhi would neither attack, nor cause to be attacked, a whole host of people. The list included the long-aggrieved Felice Sillano as well as Stefano Longhi and Flavio Canonici ( _sic_ ), whose hand Caravaggio had marked with 'a permanent scar', and Ranuccio Tomassoni.\n\nThere is a strong sense, in all this, of battle lines being drawn. A dangerous pattern of alliances was forming, a web of personal and patriotic rivalries. Caravaggio and Onorio Longhi stand on one side of the street, while Tomassoni and his henchmen from Terni gather on the other.\n\n#### TWO PAINTINGS FOR TIBERIO CERASI\n\nMeanwhile, in the autumn of 1600 Caravaggio had been offered another important commission. Two more lateral pictures were required, this time on the subjects of _The Conversion of St Paul_ and _The Crucifixion of St Peter_ , for a chapel that had been acquired by Monsignor Tiberio Cerasi in the Church of Santa Maria del Popolo. It was another exceptional opportunity for the artist to excel on a public stage. The Augustinian foundation of Santa Maria del Popolo, at the northern edge of Rome, marked the start of one of the principal routes of pilgrimage through the city. Caravaggio was well aware that over the years millions of pilgrims would see his depictions of Peter and Paul, the beloved Princes of the Apostles. With the help of Cardinal del Monte, he was becoming famous.\n\nCaravaggio's new patron, Monsignor Tiberio Cerasi, was a rich man. Born in 1544, he had made his fortune practising law at the papal court. Since 1596 he had been Treasurer-General to the Apostolic Chamber, responsible for authorizing papal expenditure. During the same period Cardinal Vincenzo Giustiniani was the Depositary-General, whose job it was to receive and distribute the funds. The two men were often brought together by their work. It may have been Giustiniani, close friend of Cardinal del Monte and owner of _The_ _Lute Player_ , who first suggested that Cerasi employ Caravaggio to paint two pictures for his burial chapel in Santa Maria del Popolo.\n\nCaravaggio may also have been favoured by the religious order that owned the church, the Augustinian Friars of the Congregation of Lombardy. They were from his corner of Italy, and would have had every chance to admire his new pictures in San Luigi dei Francesi, which was only a short walk from their door. St Augustine himself had regarded human beings as essentially helpless recipients of divine mercy, measured out according to the inscrutable logic of a predestined universe, so an Augustinian community may have been impressed by the painter's _The Calling of St Matthew_ , which shows a sudden, inexplicable shining of divine grace into the life of a sinner.\n\nCardinal Giustiniani's involvement in the commission is confirmed by the contract for the new work. He is described in the role of banker, making the first payment to Caravaggio on Cerasi's behalf. The document is dated 24 September 1600:\n\n> Michael Angelo [ _sic_ ] Merisi da Caravaggio... outstanding painter of the city, contracts with Tiberio Cerasi to paint two pictures on cypress wood, each with a length of ten Roman palmi and a width of eight, representing the Conversion of St Paul and the Martyrdom of St Peter, for delivery within eight months, with all figures, persons, and ornaments which seem fit to the painter, to the satisfaction of his Lordship. The painter shall also be obliged to submit specimens and designs of the figures and other objects with which according to his invention and genius he intends to beautify the said mystery and martyrdom. This promise the said painter has made for an honorarium and price of 400 scudi in cash... [having received] 50 scudi in the form of a money order directed to the Most Illustrious Vincenzo Giustiniani... For all this the parties have pledged themselves... They have renounced to the right of appeal, in perfect consent and have taken their oaths respectively: the Prelate according to the custom of his rank, by touching his breast; Messer Michel Angelo [ _sic_ ], by touching the Bible...\n\nTiberio Cerasi had only acquired the burial chapel in Santa Maria del Popolo in July, barely more than two months earlier, and clearly wanted to avoid the kinds of delay that had plagued the decoration of the Contarelli Chapel. He may also have suspected that he did not have long to live. Two years earlier he had signed his last testament, in which he declared the Hospital of Santa Maria Consolazione to be the _erede universale_ and residuary legatee of his will. His father, Stefano Cerasi, had worked there as a physician, and Tiberio had always kept close ties with the institution. In his will, Cerasi wrote that his love for the hospital was greater than his poor bequest could convey. He was a man with his mind on the next world, determined to be in credit when the final reckoning came. He would die while work was still in progress on the paintings for his chapel, but the results would surely have pleased him. Caravaggio's dark and solemn style was well suited to his penitential mood.\n\nSaints Peter and Paul were deeply revered in Rome. Their heads were reputedly preserved in St John Lateran, their bodies buried before the high altar of St Peter's. It was believed that they had both been martyred in the city on the same day, baptizing the Roman Church with their blood. Because they were regarded as 'the founders of the Apostolic See', the stories of their lives were often presented together, but it was unusual to see depictions of the particular episodes prescribed by Cerasi placed side by side. The conversion of Paul was not usually paired with Peter's martyrdom, but with Peter receiving the keys from Christ.\n\nThere was one notable precedent for the chosen arrangement. In the 1540s, in the Pauline Chapel, next to the Sistine Chapel, Michelangelo had painted his last pair of monumental frescoes, a _Conversion of St Paul_ and a _Martyrdom of St Peter_ for Pope Paul III. They are forbiddingly gloomy pictures, part of Michelangelo's long retreat from ideal beauty. By commissioning Caravaggio to repeat the same juxtaposition of themes in his own burial chapel, Cerasi was implicitly setting him in competition with the ghost of the most celebrated Renaissance artist of all.\n\nTo this challenge, the patron added another. In addition to Caravaggio's lateral panels of Paul and Peter, he commissioned a painted altarpiece for his chapel. The artist chosen was the Bolognese painter Annibale Carracci, who had recently completed the breathtaking cycle of mythological paintings for the ceiling of the Palazzo Farnese. The subject for his new work was to be _The Assumption of the Virgin_. Carracci and Caravaggio were the two most talented painters in Rome. Cerasi had secured the services of both, beginning what he must have hoped would be a thrilling battle for pre-eminence.\n\nCarracci was the senior of the two, some fifteen years older than Caravaggio. Before starting work, he looked at his rival's new pictures in the nearby church of the French and probably assumed that Caravaggio would repeat the pattern of his _Calling_ and _Martyrdom of St Matthew_ : making the past seem present, painting from carefully posed models, using intense contrasts of light and shade. During the course of his considerably longer career, Carracci himself had flirted with similar methods and devices. But now his sense of competition pushed him to the opposite extreme.\n\nPainting _The Assumption of the Virgin_ , Carracci reverted to the pure, sweet style of the High Renaissance. He brightened and softened his colours and ruthlessly eliminated any hint of real life. Swathed in drapery the colour of a summer's sky, arms outspread, an expression of beatific serenity on her perfectly round face, Carracci's Virgin Mary rises from the tomb like an ecstatic doll. Her feet rest on a cushion of winged cherubim's heads, while a decorous cast of bearded apostles has been arranged, below her, in various standard poses of politely expressed wonderment. The painting is airless and spaceless, all its figures pushed up to the picture plane as if to a sheet of glass. There is no suggestion of the sacred erupting into the world of the everyday. It is a dream of pure transcendence.\n\nCarracci's picture is a point-by-point refutation of all Caravaggio's innovations in the Contarelli Chapel. Harking back to the serenity of Raphael's middle style, it is an insistently retrograde work of art \u2013 a doctrinaire assertion of the importance of _disegno_ , in the sense both of drawing and of idealized composition. But it also anticipates the swooning, aerially propelled visions of the incipient Baroque \u2013 the style that in Italy at least would for a time triumph over Caravaggio's harsh brand of pious naturalism. _The Assumption of the Virgin_ is a reminder of the powerful tides of taste against which Caravaggio was swimming.\n\nAnnibale Carracci delivered his work on time, to the approval of Tiberio Cerasi, and it was duly installed on the altar of the chapel in Santa Maria del Popolo. Caravaggio struggled with his own commission. Working on panels of cypress wood, as the contract had stipulated, was very different from his usual practice of working on canvas. Oil paint does not penetrate panel in the same way that it works its way into the weft of a canvas. The resulting surface is more reflective, with more emphatically succulent colours and shadows that do not recede as fully into darkness. According to Baglione, Caravaggio persevered with the two panel pictures, but they 'were painted in a different style' and 'did not please the patron'. Cerasi rejected them and the artist had no choice but to start again, this time in his preferred medium of oil on canvas. His abortive first efforts were sold on, Baglione added, to Cardinal Sannesio.\n\nOnly one of these unsatisfactory compositions survives. The attribution to Caravaggio has sometimes been questioned, but is now generally accepted. The dimensions of the panel are very close to those of the two oil paintings that the painter eventually completed for the Cerasi Chapel. Allowing for the painter's use of an unfamiliar support, the style is convincing. The model for the angel reappears in at least one of Caravaggio's later works.\n\nIt is not hard to see why Tiberio Cerasi rejected this first _Conversion of St Paul_. The composition is a clutter and a jumble. As the bearded Paul squirms on the ground, shielding his eyes from the dazzling celestial vision, his horse rears up and foams at the mouth. The saint's aged retainer, clutching a shield decorated with a crescent moon and wearing an absurdly elaborate plumed helmet, resembles a baffled spear-carrier in a comic opera. Hearing a noise like a thunderclap, but seeing nothing, he brandishes his weapon at thin air. The young, bearded Christ descends from the heavens, reaching down to the stricken Paul with a gesture of grave compassion. He and the angel accompanying him lean awkwardly across a snapped branch of a laurel, like a pair of parachutists stuck in a tree. This cumbersome arrangement probably reflects some actual studio contrivance. A ladder and a length of rope may have been used to help the models assume their poses for this part of the _mise-en-sc\u00e8ne_.\n\nOnce again, the artist's memories of the popular religious art of Lombardy are much in evidence. The picture bears a strong resemblance to some of the more overcrowded scenes enacted by the busy mannequin figures in the chapels of the _sacro monte_ tradition. The painting is more upright than the Contarelli Chapel canvases, yet Caravaggio has tried to squeeze almost as much dramatic action into the narrower compass allowed for by the cramped dimensions of the Cerasi Chapel. As a result, the forms and figures seem bizarrely compressed, with heaven and earth forced into a weird and unconvincing proximity.\n\nDuring his early struggles with the Cerasi Chapel commission, Caravaggio was handicapped by an apparent inability to get away from the famous prototype of Michelangelo's restless and turbulent _Conversion of St Paul_ in the Pauline Chapel. The rearing horse and reeling saint, the figure of Christ descending from the heavens, arm outstretched \u2013 he borrowed and adapted all these elements from Michelangelo's far larger and more densely populated painting, as if he were setting out to create a condensed version of the earlier work. It was only when Cerasi rejected the painting out of hand that Caravaggio reconsidered and found a diametrically different solution. For his second _Conversion of St Paul_ , he went back to basics. He returned to oil on canvas and went back to the biblical source of the story, to find a new way of getting to its heart and bringing it to life.\n\nThe tale of Paul's conversion is told in the Acts of the Apostles. The Roman citizen Saul of Tarsus, the future St Paul, was travelling to Damascus with letters of authority to persecute the Christians. A harsh and ruthless man, 'breathing out threatenings and slaughter against the disciples of the Lord', he was abruptly stopped in his tracks by a miracle:\n\n> And as he journeyed; he came near Damascus: and suddenly there shined round about him a light from heaven: And he fell to the earth, and heard a voice saying unto him, Saul, Saul, why persecutest thou me? And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest... And he trembling and astonished said, Lord, what wilt thou have me to do? And the Lord said unto him, Arise, and go into the city, and it shall be told thee what thou must do. And the men which journeyed with him stood speechless, hearing a voice, but seeing no man. And Saul arose from the earth; and when his eyes were opened, he saw no man: but they led him by the hand, and brought him into Damascus. And he was three days without sight, and neither did eat nor drink' (Acts 9:1\u20139).\n\nCaravaggio's own revelation, on rereading these words, may have been as sudden. The story is essentially a parable played out between the twin poles of his own art \u2013 a tale of light and darkness. Lost in the shadows of evil and ignorance, a vicious man is suddenly bathed by the light of God and his soul is washed clean. In the moment of his ecstatic vision the divine light enters him, invades and permeates his whole being. Filled with this inner illumination, the light of truth and faith, he becomes blind to the mundane world about him. There are striking parallels with the story of Matthew called by Christ as Caravaggio had imagined it in the Contarelli Chapel. But this time the metaphor of illumination, which the painter had brought to the bare text of Matthew's gospel, is there in the biblical account itself: 'there shined about him a light from heaven'. In that phrase, he found his key to understanding the nature of Paul's conversion. Turning his back on the tumult and drama conventionally associated with Paul's conversion, Caravaggio created a picture of unprecedented calm. Gone are the creakingly theatrical figures of Christ and the angel, replaced by a spectral radiance that is the light of God. There is no noise, no clamour, no comedy of misapprehension here \u2013 just simple ignorance contrasted with miraculous divine illumination, an irresistible tide of light that floods the saint and changes him forever.\n\nPaul's retainer stands quietly to one side, lost in his thoughts and half lost in the shadows. A hard-faced balding man with a furrowed brow, he tends with calm solicitude to the horse from which his master has fallen. Below, almost beneath the animal's hooves, the figure of Paul lies on his back with his eyes closed like a man dreaming of his lover. His arms are open wide, embracing the light that envelops him, filling him with truth and wisdom and humanity. He is considerably younger than the wizened, bearded Paul of the rejected version. This Paul is very much the tough Roman soldier described in the Acts of the Apostles \u2013 a hard-bodied athlete with a granite jaw who has suddenly been melted by the love of God. His sword lies by his side, resting in folds of red drapery as if to symbolize the rivers of Christian blood that he had meant to shed when he set out for Damascus.\n\nIn the moment of Paul's ecstasy, the world is brought to a standstill. A physical journey has turned into a spiritual odyssey. Caravaggio's decision to purge the story of visible narrative was brave and unorthodox, but expressive. Bellori, missing the point with perfect eloquence, described the picture as 'the Conversion of St Paul, in which the history is completely without action'. On the contrary: the action has been completely internalized, so that we see or sense it unfolding within Paul's soul. He is being moulded by the light that models his figure with its soft and gentle rays. In the chiaroscuro that plays along the length of his outstretched left arm, in the shafts woven through the tips of his fingers, in the gleams reflected in the dull sheen of his fingernails, light itself becomes palpable \u2013 something he feels, accepts, draws into the depths of his body.\n\nThis is a painting to be understood intuitively, instinctively. It is not an intellectual picture, nor one that shows any interest in beauty as conventionally understood. It is designed to speak not to the rich or theologically learned but to the poor \u2013 to roughshod peasants and sunburned labourers, ordinary people who had made the long pilgrimage south to Rome and found themselves, at last, inside the city walls. The composition is dominated by the solid, heavy form of the patiently standing horse, lifting a heavy hoof so as not to tread on the prone body of its master. The animal is no thoroughbred, but a stocky piebald beast of burden. Caravaggio paints the weight and density of its powerful flank. He paints the animal's patience and loyalty. He even conjures up a feeling of the heat that emanates from its slow, heavy body \u2013 in rural parts, in the little town where he had been brought up, poor people kept their livestock in their homes in the winter months to keep themselves warm. This is an essential part of the picture's plainspeaking intimacy. It is like a hearth, inviting cold bodies to gather round and warm themselves in the act of devotion.\n\nThe horse evoked other folk memories too. Like the benign ox and ass in traditional depictions and plays of the Nativity, the animal standing quietly in the dark recalls the manger in which Christ was born. Seen through half-closed eyes, the animal's groom might almost be St Joseph. The association adds another level of meaning to the scene. In the moment of his conversion Paul is helpless yet blessed, bathed by the light of God, just as Christ was in his infancy.\n\nBehind all this is the old idea of the _Imitatio Christi_ , which was central to the ethics of the old pauperist orders such as the Franciscans. To understand Christ's message is to become like him, to follow in his footsteps \u2013 to undergo a profound, internal metamorphosis. At the instant of his inner rebirth as a Christian, Paul mystically experiences the whole life of Christ, its beginning and its end. He becomes, in his own mind, both Christ the blessed child and Christ the doomed adult, sacrificed to save mankind. In the movements of his body are reflected the motions of his soul. He reaches his arms out like a baby. As he does so, his gesture mimes the Crucifixion.\n\nThe theological justification for pairing St Paul's conversion with St Peter's martyrdom was the belief that each event represented a mystical death. At his conversion, Paul dies to the world to be reborn in Christ; at his martyrdom, Peter literally dies, to meet his rewards in heaven. Such symmetry is implicit in the relationship between Caravaggio's two paintings. The prone body of Paul, cruciform in a gesture of spiritual empathy, is echoed by the actually crucified body of Peter.\n\nAccording to legend, Peter insisted that he be crucified upside down because he felt unworthy to die the same death as Christ. In _The Crucifixion of St Peter_ , Caravaggio shows him already nailed to the cross, defiantly half rearing up as his executioners toil to raise him into place. He exhales against the pain, stomach muscles tensing, and looks away out of the picture. His eyes are fixed on the actual chapel's altar, as if to stress that death by martyrdom is another form of participation in the rite of the Mass. Even as his own blood is shed, he trusts that he will be saved by the flesh and blood of Christ. The rock in the foreground is the symbol of his hard, enduring faith, cornerstone of the Church itself: 'thou art Peter, and upon this rock I shall build my church; and the gates of hell shall not prevail against it' (Matthew 16:18\u201319).\n\nThe action takes place in some dim corner of a nocturnal world lit only by the flash of God's grace. The light falls on Peter and the straining figures of his three executioners, but the martyr alone is alive to its message of salvation. The others grunt and sweat under the burden of his weight, grimly immersing themselves in the practical business of hoisting up a human body nailed to a cross. They look as though they are trying not to think about what they are actually doing \u2013 or pretending to themselves that it might be some more innocent and straightforward task, such as erecting a fence-post, or heaving the joist of a house into place.\n\nThe executioners are insensitive to the point of insentience, blind to the mystical significance of the death they so callously arrange. Their figures are pushed up so close to the front edge of the picture that they seem almost to spill out into the real world. Like _The Conversion of St Paul_ , _The Crucifixion of St Peter_ is a painting aimed squarely at poor and ordinary people. It is a challenge as well as a call to conscience: viewers are brought into its space and invited to take the place of Peter's executioners, at least in the mind's eye \u2013 to make good their failings, to show compassion and mercy, to open up to the light of God.\n\nThe Renaissance scholar and connoisseur Bernard Berenson, who published a short and rather tetchy book about Caravaggio in 1951, was offended by the aggressive directness of the picture:\n\n> But for the noble Titianesque head of the victim, the rest is a study in the raising of a heavy weight without the aid of machinery. Of the chief performers, the one who acts as crane and the other as booster, we see the back of one and the buttocks of the other. We do not see their faces. No need. They are mere mechanisms. Hard to conceive a more dehumanized treatment of the subject. No doubt the arrangement of the four figures as crossed diagonals taking up the entire canvas was a happy thought...\n\nThe executioners were certainly intended to shock. The presence of these coarsely posed, unmistakably low-brow figures underscored Caravaggio's total rejection of High Renaissance and Mannerist elegance. This is all the more apparent in the Cerasi Chapel, where Annibale Carracci's large and centrally placed altarpiece perfectly embodies the traditions to which Caravaggio's work is so brutally opposed. Carracci had sought to pre-empt his rival by creating a work designed to reassert the values of idealized beauty, splendid colour and lofty transcendence. In doing so, he may have hoped to sow seeds of self-doubt in Caravaggio's mind. But the younger painter was only spurred on to a more blatant statement of his own, very different priorities. In place of Carracci's emotionless splendour of effect he offered up his own spare, low-toned, militantly 'poor' art. Carracci had used rich colours, colours that literally embodied wealth and magnificence, like the celestial blue of the Madonna's cloak, painted in the costly medium of ultramarine. In stark contrast, Caravaggio kept rigorously to a palette of humble, ordinary, cheap colours: the earth colours, ochre and umber, carbon black, lead white, verdigris. The use of costly ultramarine was actually specified by Cerasi, who doubtless wanted posterity to know that no expense had been spared. But Caravaggio used the colour in such a way as to reject its rich associations. The dying Peter's robe, lying in a heap in the bottom corner of the _Martyrdom_ , has been painted in murkily shadowed ultramarine. As Bellori noted, Caravaggio avoided more brilliant vermilions and blues, and even when he did use them generally 'toned them down'.\n\nThe lives of Christ and his followers were neither rich nor splendid. Their deaths were brutal. Caravaggio insists on these home truths in every detail of the Cerasi Chapel paintings, whether it be the glint of the crouching executioner's spade or the black dirt so deeply ingrained in the upturned heel and ball of his left foot. Like Carlo Borromeo preaching in rags, the art of Caravaggio expressed an aggressively harsh piety. With _The Conversion of St Paul_ and _The Crucifixion of St Peter,_ he took his uncompromisingly severe style of painting to an ascetic extreme. As a parting gesture to his rival, as if to stress the depth of his disdain for Carracci's brand of vapid magnificence, Caravaggio contrived a cunning insult: the rump of St Paul's proletarian carthorse is pointedly turned towards Carracci's _Assumption of the Virgin_.\n\n#### IN THE HOUSE OF THE MATTEI\n\nCaravaggio finished his two lateral paintings for the Cerasi Chapel towards the end of 1601. Earlier in the year he had left the household of Cardinal del Monte to accept the hospitality of another powerful figure in the Roman Curia, Cardinal Girolamo Mattei.\n\nThe Mattei were powerful. They lived in a honeycomb complex of houses and palaces built over the ruins of the ancient Roman Teatro di Balbo, in the heavily populated district of Sant'Angelo, between the Tiber and the Campidoglio. The adjoining residences of the various branches of the family formed an entire block, known as the Isola dei Mattei. At its centre, looking out across the Piazza Mattei, was the massive Palazzo Mattei, home to Cardinal Girolamo.\n\nCaravaggio moved there some time before 14 June 1601, when he gave his address on agreeing a contract for an altarpiece of _The Death of the Virgin_ , for the church of Santa Maria della Scala in the district of Trastevere in Rome: he is described as 'Michelangelo Merisi from Caravaggio, painter of the city, living in the palace of the illustrious and reverend lord cardinal Mattei'. The terms of the contract allowed him twelve months to paint the new altarpiece. He would complete the work eventually, but not until long after that deadline had passed.\n\nCaravaggio probably remained in the household of the Mattei family until at least the beginning of 1603. His precise movements are hard to track following his departure from the household of Cardinal del Monte, who was himself friendly with the Mattei and may have been instrumental in the artist's move. Caravaggio's change of address should not be seen as marking a break between him and del Monte. The painter continued to rely on his old protector for support. On the evening of 11 October 1601 Caravaggio was stopped for carrying arms without a licence in the district of the Campo Marzio. The policeman who made the arrest reported that the painter 'insisted that he was on the household roll of the Cardinal del Monte, and because he did not have a licence and I did not know if it was true, I took him to prison at the Tor di Nona.' Nothing came of it and the painter was soon released, probably with del Monte's help. For his part the Medici cardinal seems to have remained on good terms with his prot\u00e9g\u00e9, continuing to make allowances for his erratic behaviour.\n\nCardinal Girolamo Mattei was one of three brothers. Although he was not the eldest, his elevated position in the Roman curia meant that it was he who lived in the principal family palace. He was a member of the strictest order of Franciscan friars, the Observants. Cardinal Mattei was noted for his dislike of conspicuous display and may have influenced Caravaggio's turn towards a harsh and simplified language of Christian painting in 1601. The pictures for the Cerasi Chapel, so stark and ascetic, were finished when Caravaggio was living in the Palazzo Mattei.\n\nGirolamo's two brothers, Ciriaco and Asdrubale, shared a house close to the Palazzo Mattei. Ciriaco was a year older than the cardinal, while Asdrubale was ten years younger. Both men had added to their considerable inherited wealth by marrying advantageously. They were known as enthusiastic collectors of ancient Roman sculpture and as connoisseurs of contemporary art. The family account books show that it was they, rather than their brother the cardinal, who commissioned paintings from Caravaggio. For Asdrubale he created a painting of _St Sebastian_ that has long since been lost. For Ciriaco he painted no fewer than three gallery paintings on sacred themes, all of which survive.\n\nThe archives of the Mattei family show that Caravaggio was paid by Ciriaco at the start of January 1602 for 'A painting of Our Lord Breaking Bread'. This is _The Supper at Emmaus,_ now in the National Gallery in London. The painting tells the biblical story of the risen Christ, days after the crucifixion, sharing a meal with two of his astonished followers. According to the gospel of Luke, at first they did not recognize him: 'And then it came to pass, as he sat at meat with them, he took bread, and blessed it, and brake, and gave to them. And their eyes were opened, and they knew him; and he vanished out of their sight' (Luke 24:30\u201331). Caravaggio paints the moment just before the vanishing. Dressed in robes of red and white, colours that symbolize his triumphant resurrection, Jesus reveals his identity with a gesture of gentle benediction. In the act of blessing the square and solid loaves of bread, he both confirms that he has indeed risen from the dead and affirms his own bodily presence in the Eucharist. The claws of a scrawny boiled chicken, pathetic image of mortality, are contrasted with the life-giving hands of Christ. A simple meal has become a sacrament.\n\nThe Bible says that the village of Emmaus 'was from Jerusalem about threescore furlongs', but Caravaggio imagined a place much closer to home. His _Supper at Emmaus_ is served up in a rough Roman tavern, the kind of place where the painter would meet his friends and start arguments with his enemies. As the Saviour announces himself, a hardbitten innkeeper looks on with an expression of uncomprehending suspicion, as if he might be wondering whether this pale, plump-faced young man and his ragged companions will be able to pay their bill.\n\nMeanwhile the two disciples are frozen in the throes of astonished, dawning recognition. One has his back to us. As he prepares to lever himself upright, his hands are braced on the arms of the same savonarola chair that Caravaggio had used in _The Calling of St Matthew_. At the point of his bony elbow, there is a small rent in his rough green tunic, through which his white undershirt shows. The other disciple, who wears a pilgrim's shell on his mantle, spreads his arms as wide as he can, measuring the extent of his amazement like an angler demonstrating the size of a fish that got away. His gesture also mirrors the Crucifixion, as if to shape the question springing to his mind. How is it possible that a man whom he so recently saw nailed to the cross, a bleeding corpse, should live and breathe and speak once more?\n\nThe hands of Christ and the wondering apostle seem to reach out of the painting, through the membrane that separates illusion from reality. The effect is worked through skilful foreshortenings of perspective. The apostle's outspread arms plot the whole depth of the picture. His right hand, half lost in the darkness, seems blurred by movement. His other hand, so close to the picture plane as to seem almost touchable, is sharply in focus. From the tip of Christ's thumb, back along the dappled sleeve of his red shirt to his shoulder, his arm is a piece of art that measures distance, in graded lights and darks, with such illusory precision that it is almost impossible to look at the painting and believe it truly flat.\n\nYet Caravaggio's intense realism is also, on this occasion, shot through with a strong sense of the uncanny. It is as if the painter has asked himself a series of direct, straightforward questions about the story that he was given to depict. What happens to the world when a miracle takes place? How might it be possible to tell, should the risen Christ suddenly come among us? What do things actually look like at such moments? _The Supper at Emmaus_ contains Caravaggio's answers to those questions.\n\nThe idea that divine visitations are inevitably accompanied by thunderclaps and clouds of angels is dismissed as naive and childish. Caravaggio, himself so keen-eyed and attentive to every last nuance of visual experience, imagines the process to be subtler than that. God is light, so he announces his presence among men in the elusive forms of a shadowplay. The innkeeper cannot see it, but by standing where he does he casts a shadow on the wall that gives Christ a dark but unmistakable halo. Below, a basket of fruit is balanced precariously on the leading edge of the table. It is the same basket that Caravaggio had painted for Federico Borromeo, and its contents are nearly the same too \u2013 a worm-eaten apple, a pomegranate and fig, withered grapes and trailing vine leaves, embodying decay but also symbolizing the hope of Christian redemption. The fruit and the teetering basket cast a second meaningful shadow, this one shaped like the tail of a fish, the ancient mnemonic sign for Christ used by his earliest followers. Caravaggio's painting suggests that those who would prefer to be saved, rather than damned, might do well to pay attention to such details. Even those in the presence of a miracle might easily miss it.\n\nBellori unwisely chose to single out _The Supper at Emmaus_ as an example of the painter's thoughtless literalism and lack of decorum: 'in addition to the vulgar conception of the two Apostles and of the Lord who is shown young and without a beard, the innkeeper wears a cap, and on the table is a dish of grapes, figs and pomegranates out of season. Just as certain herbs produce both beneficial medicine and most pernicious poison, in the same way, though he produced some good, Caravaggio has been most harmful and wrought havoc with every ornament and good tradition of painting...' The biographer concluded this little homily with the reflection that many other painters had been bewitched by the 'error and darkness' of Caravaggio's painting, 'until Annibale Carracci came to enlighten their minds and restore beauty to the imagination of nature'.\n\nBellori's misreading of _The Supper at Emmaus_ does at least have the virtue of highlighting some of the picture's most effective devices. The writer found his eye drawn to Caravaggio's wicker basket of fruit, so beautifully painted, only to complain that the fruits within were 'unseasonal'. He clearly felt they should have been the fruits of Easter, the time of Christ's crucifixion. Guilty of the very literalism for which he blamed Caravaggio, Bellori was oblivious to the symbolic meanings concealed within the basket of fruit, and completely blind to the significant shape of its shadow.\n\nHe was also perturbed by the disrespectful figure of the innkeeper, who wears his cap in the presence of Christ. But this is no mere oversight, or vulgar lapse, on the part of the painter; it is a detail essential to his telling of the story. The innkeeper fails to doff his cap because he does not realize whom he serves. He remains in darkness, even though a miracle is taking place before his eyes. In Caravaggio's interpretation, the story of the meal at Emmaus becomes a parable about those who see and those who do not.\n\nBellori disliked the evident poverty of the two disciples and can almost be heard tut-tutting over that prominent torn sleeve. More telling is his other complaint, about Caravaggio's depiction of Christ as 'young and without a beard'. The painter's decision to depart from the traditional image of a solemn, bearded Christ \u2013 such as he had recently painted in _The Calling of St Matthew_ \u2013 was certainly unusual. But once again, it is essential to his understanding of the story as a tale of hard-won recognition.\n\nThe principal source for the story of the Supper at Emmaus is the gospel of Luke, Chapter 24, but there is also a fleeting reference to it in Chapter 16 of the gospel of Mark: 'After that he appeared in another form unto two of them, as they walked, and went into the country.' Caravaggio seized on the three words, 'in another form'. They are the only explanation given in the Bible for the apostles' failure to recognize Christ. Risen from the dead, he took on a different physical appearance. It seems that Caravaggio's inspiration for the picture's main idea \u2013 the idea of an unobvious miracle, a miracle that men must struggle to see \u2013 had its origins in a careful reading of the Bible.\n\nBack in the 1540s Michelangelo had placed a similarly controversial, young and beardless Christ at the centre of his _Last Judgement_ , on the altar wall of the Sistine Chapel. The image was meant to evoke the early traditions of Christian Rome, where Christ had often been depicted in the guise of the sun god, Apollo. There are strong echoes of Michelangelo's Apollonian Christ, judging all mankind at the end of the world, in Caravaggio's own figure of Christ in _The Supper at Emmaus_. In the fresco of _The Last Judgement_ , Christ's left arm is turned against the seething mass of the damned, while with his right he beckons the blessed up into heaven. Caravaggio appropriated those same gestures, adapting them with surprisingly little modification for his own figure's act of blessing the bread. It is another formal echo charged with spiritual meaning. Christ's appearance to his two disciples at Emmaus prefigures his final appearance to the whole human race on the day of judgement.\n\nTwo more payments were made to Caravaggio by Ciriaco Mattei in 1602, one in July, the other in December. These were for a painting which has been plausibly identified with the _St John the Baptist_ now in the Capitoline Museum in Rome. Once again, Caravaggio treated his appointed subject in an unusual and idiosyncratic way. The saint, who is shown during his legendary retreat into the desert, appears without several of his usual attributes. He carries neither a cross nor a banderole. The lamb of God who usually accompanies him has metamorphosed into a sheep with horns. He embraces the animal, which nuzzles his cheek. It was conventional to depict St John as a haggard ascetic in animal furs, but Caravaggio presents him as a cheerfully smiling, ruddy-cheeked adolescent. Most unusual of all, he is stark naked. The boy reclines on a scrap of fur, but his discarded clothes lie around him in a heap.\n\nThe picture is so unconventional that even its very subject has been called into question. As early as 1620 the author of a guidebook to the Mattei collection gave the work a mythological title, referring to it as a _Pastor Friso_ , which identified the naked young man as a pagan shepherd. A number of subsequent scholars have taken that attribution seriously. Others have argued that Caravaggio intended to depict the biblical Isaac, son of Abraham, stripped for sacrifice and rejoicing after his sudden stay of execution. None of these hypotheses has much merit. Ciriaco Mattei presented the picture to his son, Giovanni Battista Mattei, whose name saint it certainly depicts and for whom it was almost certainly intended from the outset. An inventory of his possessions drawn up in 1616 refers to 'A painting of San Gio: Battista with his Lamb by the hand of Caravaggio', and it is safe to assume that the picture's owner knew its true subject. When Giovanni Battista made his will, seven years later, he gave instructions that the painting 'of St John the Baptist by Caravaggio' be left to none other than Cardinal Francesco del Monte. This implies that the Mattei family felt an abiding sense of obligation to del Monte for releasing Caravaggio into their service.\n\nAlthough its subject is easily established, the work is still intriguingly unusual. Why did Caravaggio paint John the Baptist in this strange, splay-legged pose? Why is the figure smiling so enigmatically? Why, above all, is he nude? Part of the answer to those questions lies in the art of the immediate past.\n\nDuring the early years of the seventeenth century, when Caravaggio was forging his style and making his reputation, he gave a great deal of thought to the works of Michelangelo. He had been born just seven years after the death of 'the divine Michelangelo', as Vasari had called him. Like every ambitious painter of his generation, he would have regarded Michelangelo's works as a summit of excellence. And as if to force such comparisons upon him, Michelangelo also happened to be his own namesake. Caravaggio had already been invited to compete with the older artist by the choice of subjects for the Cerasi Chapel. In that case, he had asserted his independence from his predecessor by reconceiving his two canonically Michelangelesque themes in a radically un-Michelangelesque manner. But in other works of the period, he complicated the game of rivalry and homage. _The Supper at Emmaus_ , with its Michelangelesque Christ, is just one of several instances. The Capitoline _St John the Baptist_ is another.\n\nThe picture is a variation on the theme of Michelangelo's _ignudi_ , the idealized male nudes which frame the nine great narrative paintings telling stories from the Book of Genesis on the Sistine Chapel ceiling. Michelangelo's male nudes are the only non-Christian elements in the whole of his scheme. They had been included as a compliment to Pope Julius II, who commissioned him to paint the ceiling: they bear festoons of oak leaves and acorns, emblems of the pope's family name, della Rovere. Collectively, they symbolize the idea of a golden age described in the writings of antiquity, the conceit behind them being that the reign of Julius amounted to another such blessed period in the lives of men. But by the second half of the sixteenth century the _ignudi_ had become controversial. Their nudity was deemed unbecoming, their pagan symbolism judged suspect, and a painter called Daniele da Volterra was hired to fig-leaf their genitalia.\n\nThe pose of Caravaggio's smiling _St John the Baptist_ has been directly borrowed from one of the four _ignudi_ who frame _The Sacrifice of Noah_ on the Sistine Chapel ceiling. The seventeenth-century writer who believed the painting to be an image of a pagan shepherd was probably responding, unconsciously, to its neo-pagan source in the art of Michelangelo, and in this sense the Capitoline _St John_ is another of Caravaggio's pictures on the borderline between 'the sacred and profane', in Cardinal Paravicino's phrase. But the true subtlety of the work lies in its double inversion of the famous but controversial prototype that inspired it.\n\nWhereas Michelangelo's nudes collectively represent a languorously beautiful ideal, an imaginary museum of male beauty raised up to the vault of heaven, Caravaggio has clearly painted a picture of a real, flesh-and-blood boy. The fact that the model has been posed just like an _ignudo_ emphasizes the gulf between Michelangelo's idealizing aesthetic and Caravaggio's countervailing realism. The flesh of Michelangelo's nudes is chiselled, marmoreally perfect. Caravaggio's adolescent saint is slight and skinny. His ribcage shows through the light-dappled flesh of his side and there is dirt under his toenails. He is an _ignudo_ brought down to earth, but not in a spirit of homage. The echo is there to assert Caravaggio's difference, to make it unavoidable.\n\nCaravaggio has also reversed the sense of Michelangelo's nudes in the act of appropriating their form. Those who have seen the Capitoline _St John_ as a daringly sexy depiction of a Christian saint, laughing provocatively as he turns to face the viewer, miss the point of the picture entirely. The truth is that Caravaggio has taken Michelangelo's notoriously pagan imagery, a classically phrased compliment paid to a pope, and fully reclaimed it for Christianity. His _ignudo_ is no sleepy, sensual emblem of a vanished golden age, but an ecstatic prophet bathed in the light of divine revelation. The naked, rejoicing boy embraces the animal by his side because it has been sent to him by God to show him what will come to pass. He sees in it the destiny of Christ the saviour, with whose fate his own is intertwined, and whom he will one day baptize.\n\nThe painter's decision to give the animal horns is unusual, but underscores the significance of the scene. It recalls the image of a sacrificial ram, and may also have been inspired by a detail of the Sistine Chapel ceiling: only a few feet from the _ignudo_ whose pose is closest to that of Caravaggio's _St John_ , Michelangelo had painted a ram being prepared for slaughter in _The Sacrifice of Moses_. Those who have misinterpreted Caravaggio's picture as an image of Isaac delivered from sacrifice are in one way simply overreacting to a genuine element of the painter's intended meaning. He meant to emphasize the idea of sacrifice by giving the sheep horns, but the sacrifice he had in mind was not that of Isaac but of Christ himself.\n\nIn the upper-right-hand corner, barely visible in the shadows, a small detail clarifies the picture's iconography: the foliage of a vine, symbolizing grapes and the wine of the Eucharist. The sacrificial sheep and the vineleaves are the outward signs of the saint's inner contemplation. In his mind's eye, he is looking into the future, seeing Christ's blessed death and the salvation of mankind. That is the reason for the smile on his face. It is the beatific smile of a mystic, a seer.\n\nThe last of the three pictures commissioned from Caravaggio by Ciriaco Mattei was _The Betrayal of Christ_. It was paid for on 2 January 1603, and probably painted just a few weeks or months before. The picture has had an eventful history. It remained in the Palazzo Mattei in Rome for nearly two centuries, after which it disappeared into the obscurity of a Scottish private collection. In 1990 it was rediscovered in the possession of the Irish Jesuit Fathers of the house of St Ignatius in Dublin, who placed it on indefinite loan to the National Gallery of Ireland.\n\nThe story of Christ's betrayal by Judas is told in all four gospels. Caravaggio followed certain stage directions, but ignored others. According to Matthew, at the time appointed for Christ's arrest, 'Judas, one of the twelve, came, and with him a great multitude with swords and staves, from the chief priests and elders of the people.' (Matthew 26:47). John adds a detail crucial to Caravaggio's nocturnal conception of the scene, equipping Judas and his men 'with lanterns and torches' (John 18:3). Mark gives the most economical account of the treacherous kiss given to Christ by Judas: 'And he that betrayed him had given them a token, saying, Whomsoever I shall kiss, that same is he; take him, and lead him away safely. And as soon as he was come, he goeth straightway to him, and saith, Master, master; and kissed him. And they laid their hands on him, and took him' (Mark 14:44\u20136). Only in Luke could Caravaggio have found the idea that Christ showed his foreknowledge of Judas's treachery by flinching at the kiss: 'Judas drew near unto Jesus to kiss him. But Jesus said unto him, Judas, betrayest thou the Son of Man with a kiss?' (Luke 23:48).\n\nDespite his academic prejudices, Bellori often responded instinctively to the sheer humanity and psychological depth of Caravaggio's painted dramas. His is by some distance the most eloquent early account of the picture: 'Judas is shown after the kiss with his hand on the Lord's shoulder; a soldier in full armour extends his arms and his ironclad hand towards the chest of the Lord, who stands still, patiently and humbly, his hands crossed before him, as John runs away behind with outstretched arms. Caravaggio rendered the rusty armour of the soldier accurately with head and face covered by a helmet, his profile partially visible. Behind him a lantern is raised and we see the heads of two other armed men.'\n\nEarlier artists had often envisaged the betrayal as a chaotic crowd scene, confusing the eye with a multitude of soldiers and panicking disciples. Caravaggio's new technique of emphatic chiaroscuro was the perfect editing device for avoiding such unnecessary complications. He used it here as a ruthless means of exclusion, spotlighting the figures at the very centre of the drama and casting everything else into deepest shadow. In his interpretation, the whole story becomes an elemental conflict between good and evil, innocence and malignity. The pale, delicate, emotionally sensitive face of Christ is set hard against the brutish, sunburned face of Judas. There is great sorrow, mingled with resignation, in Christ's half-closed eyes. In the moment of betrayal, Judas seems to lament the fatal move he has only just made. He stares into space like a man possessed, as if he is already haunted by the guilt that will soon drive him to suicide.\n\nTo this drama of juxtaposed faces, the painter has added a subplot of hands. Judas reaches out to grasp Christ with his left hand, the sinister side. Christ instinctively shrinks from the clutching embrace. Below, isolated and emphasized by a bright pool of light, Christ's own hands are clasped in a gesture of great pathos. His fingers are entwined, palms pushed away, in a movement that speaks at once of regret and acceptance of his fate. It is a detail that suggests the influence of Cardinal Mattei: the Franciscans placed the concept of _Abnegatio_ , the complete denial of self and dedication to others, at the centre of their teaching. According to the Franciscan ethic of the _Imitatio Christi_ , 'the imitation of Christ', his calm acceptance of cruelty and torture was a constant source of wonder and inspiration. Caravaggio's _Betrayal of Christ_ is one of the most powerfully moving images of that Christian ideal. It is a work that allies Caravaggio, once again, with the deepest strains of severity in Counter-Reformation spirituality.\n\nBut there is more to the eloquently compressed composition of the picture than the figures of Christ and Judas alone. To the left, just behind Christ, a terrified disciple runs away into the night. Two soldiers approach from the other side to make the arrest. They are grim and impersonal, as lacking in compassion as the stolid executioners in _The Crucifixion of St Peter_. They stand for the implacable forces unleashed by the act of betrayal. Just enough of the face of the soldier on the right is visible, beneath his helmet, to reveal that Caravaggio used the same model who had sat for the open-armed disciple in _The Supper at Emmaus_. This would have been all the more evident when both pictures hung in Ciriaco Mattei's palace. The effect must have been slightly disconcerting, like watching the same actor playing utterly contrasting roles simultaneously. Did Caravaggio do it on purpose, to demonstrate the versatility of his method? The other soldier is so obscured by the burnished steel of his helmet that almost nothing can be seen of his features. He is faceless as well as pitiless.\n\nThe encounter of Judas and Christ is charged with feeling, an exchange through which guilt and saddened acquiescence flow. Those emotions are amplified by the fleeing disciple, whose red cloak billows excitedly above the heads of Christ and Judas, linking his form so closely to theirs that he seems less like a person in his own right than like their psychic emanation \u2013 a scream forced out into the night sky by their inner turmoil. But the soldiers feel nothing and they show no capacity for feeling. They are all murderous efficiency, armoured against compassion. Christ, Judas and the disciple are beings of yielding flesh; the soldiers seem made of the very steel that they wear. Bellori was particularly struck by the horrid contrast between dark metal and soft human tissue: 'a soldier in full armour extends his arms and his ironclad hand towards the chest of the Lord.'\n\nCaravaggio took the idea for this vivid distinction from a woodcut of Christ's arrest in the garden of Gethsemane by the German Renaissance artist Albrecht D\u00fcrer. In D\u00fcrer's image the soldier's chain-mailed arm, which calls to mind the scales of an armadillo, reaches across the kissing Judas to the vulnerable figure of Christ with exactly the same gesture. In his own picture, Caravaggio sharpened that ugly juxtaposition and made it even more shocking. The soldier's black armour, jointed at shoulder and elbow, looks like the carapace of some gigantic insect. His black hand points into Christ's neck like the sting of a scorpion about to pierce its prey.\n\nCaravaggio was in the habit of pillaging prints and engravings for compositional ideas during the early years of the seventeenth century. He probably kept a stock of such images, ready catalysts for his imagination. D\u00fcrer's woodcut was not his only source for _The Betrayal_. There is another borrowing in the composition, this time from an engraving actually commissioned by the man for whom the picture was destined, Ciriaco Mattei himself. In 1601 Mattei had asked the engraver Francesco Villamena to commemorate a particularly brutal Roman street battle between members of the city's pro-French and pro-Spanish factions. From this bruising scene of affray, Caravaggio took one detail: that of the crescent-shaped, billowing cloak that connects the fleeing disciple to the figures of Christ and Judas. Once again, the painter makes the detail an organic part of his own, very different composition. The actual borrowing is less interesting than the fact that Caravaggio should have been thinking about a scene of modern-day violence in Rome when he devised the composition of his _Betrayal_. It seems that he envisaged the scene, from the start, as just the kind of fracas with which he was personally familiar \u2013 a nocturnal scuffle, with figures crying out in the dark Roman night as the _sbirri_ pounce on their man.\n\nAt the back right of the scene, set slightly apart from its principal action yet straining to witness it, the painter included his own self-portrait. He holds up a lantern with his right hand, a gesture which some have seen as a proud flaunting of his new and very particular method \u2013 a demonstration of exactly that technical revolution in studio lighting which Bellori attributed to Caravaggio when he spoke of him 'placing a lamp high so that the light would fall straight down, revealing the principal part of the body and leaving the rest in shadow so as to produce a powerful contrast of light and dark'. The trouble with this is that the light that Caravaggio holds up is actually rather feeble. It fails to illuminate anything much except for the painter's own face and intent expression. The light that falls across Christ's saddened face and clenched hands, the light that gleams in the metal armour of the soldiers, comes from somewhere else. It is fluid and cannot be easily located.\n\nCaravaggio knew very well that studio lighting, the trick for which he was fast becoming famous, can only ever be a device. What counts is what is done with it. The dim paper lantern in Caravaggio's hand may indeed be the emblem of his method. But its ineffectiveness is significant. The painter brings light to the scene, but in a symbolic rather than a literal sense. The real light-source is his imagination.\n\n#### DIRTY FEET\n\nCaravaggio was increasingly well known in Rome \u2013 the volatile, aggressive painter with the lightning-strike style, who made stories from the Bible look as though they were taking place right here, right now. He was also becoming famous as the painter of feet: the overlapped feet of Joseph, feeling the chill during the flight to Egypt; Christ's bare feet, on the stone flags of Matthew's counting house; the nail-pierced feet of St Peter and the dirt-ingrained feet of his coldly indifferent executioner.\n\nFeet were controversial. Here is a passage from a book about the Christian saints and martyrs, written by the theologian Niccol\u00f2 Lorini del Monte in the second decade of the seventeenth century:\n\n> In sum, feet may be taken by the holy Church as symbolising the poor and humble. In addition to the authority of the Holy Fathers, there are also fundamental reasons [for this], because the poor may be called the feet of the Church, and to caress the feet is to caress the poor of Christ. Because if the feet are the last and most lowly part of the human body, the poor and humble are the last part, and those who hold the last place in the Church. The poor carry the heaviest burdens of the world. They are exposed to all the blows, bumps and knocks... And if the feet are a member [of the body], then nude and uncovered they lack rosiness, and it is not a shameful thing to discover them: the poor servants of Christ were not guilty and had nothing of which to be ashamed... But yet, as they are vile the poor may be humiliated; nevertheless, God has honoured them so much that he has willed the greatest Heroes of the world serve them, and caress them, and fatten them up with servitude and earthly sustenance.\n\nLorini's paean to feet was part of a discussion of the life of a popular thirteenth-century saint, Elizabeth of Hungary. The daughter of a king, she was famous for receiving the poor into the apartments of her palace, where she fed them, dressed their wounds and washed their feet as Christ himself had washed the feet of his disciples. Francis of Assisi, whose followers called him _alter Christus,_ 'another Christ', had also abased himself at the feet of the poor and needy. In Caravaggio's time wealthy members of certain religious confraternities emulated such venerable examples \u2013 clothing, feeding and washing the feet of poor pilgrims coming to Rome. To do so was quite literally to embrace humility, to lower the proud self to the ground in emulation of Christ. The Latin root of _humilitas_ is the word _humus_ , meaning 'ground'. The word 'humble' is part of the same linguistic family. To honour the foot is to honour the lowest part of the human body, and implicitly to humble the self in the sight of God.\n\nWhen Caravaggio painted the saints and martyrs with bare feet, he was firmly allying himself with the pauperist wing of the Catholic Church. Not only was he explicitly welcoming the poor into his pictures, making them feel part of the same impoverished family as that of Christ and his followers, he was also implicitly calling on the rich to follow the example of those such as St Francis, the merchant's son, and Elizabeth of Hungary, the princess, who had given away all their worldly possessions to minister to the poor. The message would not always be well received.\n\nThe painter was kept busy by other commissions as well as by the demands of the Mattei family during the first three years of the century. Early in 1602, several months before painting _The Betrayal of Christ_ , he had learned that he was required once more at the church of San Luigi dei Francesi. Although more than a year had passed since Caravaggio had finished the lateral canvases for the Contarelli Chapel, the completion of the whole decorative scheme had been delayed by the prevarications of Jacob Cobaert. At the end of January 1602 the tardy Flemish sculptor finally delivered his marble altarpiece of Matthew and the angel, still partially incomplete. It was instantly rejected by the increasingly irritable and fractious coalition of Mathieu Cointrel's executors. Just eight days later Caravaggio was asked to replace the sculpted altarpiece with a painting of the same subject. Matthew was to be shown writing his gospel. The contract specified that he must be depicted taking dictation from an angel; those were the only figures required. It was a clear brief, but its execution would prove to be far from straightforward and Caravaggio would end up having to paint two versions of the picture. The root of the problem would be his depiction of the saint's feet.\n\nCaravaggio's first _Matthew and the Angel_ for the Contarelli Chapel eventually passed to the Kaiser-Friedrich Museum in Berlin. Like the lost portrait of Fillide, it was destroyed by fire during the Second World War, but a record of its appearance is preserved in black-and-white photographs. Possibly because he knew that his picture was replacing a marble altarpiece, the painter created a powerfully sculptural composition. Matthew and his attendant angel, a tender winged boy who guides the saint's writing hand, form a single monumental group. The evangelist sits with his body twisted effortfully around the great book in his lap. His shoulders are hunched, his neck arched forward so that he can peer at the text. The gleaming white pages of the book and the dark jerkin that he wears obscure and interrupt much of his anatomy. His body is reduced to its component elements: balding, bearded head on a bull neck; gnarled hands and forearm; bare legs and heavy feet; toes thrust almost into the viewer's face. This Matthew is an aggressively inelegant, proletarian figure, conceived along the lines of St Peter in the Cerasi Chapel and very different from the pale-skinned tax-gatherer or the heroic fallen priest depicted in Caravaggio's earlier pictures for the chapel. The suggestion is that he is both writing and reading for the first time, like a peasant made suddenly and miraculously literate.\n\nThe gospel of Matthew was at the centre of a controversy between Catholics and Protestants. In the fourth century, St Jerome had asserted that Matthew wrote in Hebrew. But at the start of the sixteenth century the humanist author Erasmus had questioned whether the received version had really been translated from a Hebrew original. This raised the possibility that the biblical book of Matthew was based on a later, corrupt version of the text \u2013 posing a grave threat to the authority of the Church itself. In 1537 a Protestant Hebraist named Sebastian Munster published his own translation of a Jewish manuscript that he claimed was the true text of Matthew's gospel, and which differed from the received version in numerous places. Caravaggio was certainly aware of this: the words in the book on Matthew's lap are written in Hebrew, and he has been careful to ensure that they exactly mirror the sense of the received version approved by the Catholic Church.\n\nBecause Matthew has just started writing his gospel, the painter shows its opening lines: 'The book of the generations of Jesus Christ, the son of David, the son of Abraham.' Matthew, aided by the angel, is about to finish the next phrase, 'Abraham begat', which marks the start of the gospel's tracing of the lineage of Christ. As the bloodline leading to the salvation of mankind is announced, Matthew stares in wonder.\n\nAccording to St Jerome, Matthew was the first of the apostles to write his gospel. By the time Caravaggio painted his picture, this had become part of Catholic tradition. He alludes to it by implication through his emphatic use of chiaroscuro. As the wizened, sunburned figure of Matthew receives the very first divinely inspired Christian text, he is bathed in light. Through him, the whole world will be illuminated. As so often during this phase of his career, Caravaggio defines his own art by contrast with that of Michelangelo. Once more, he has the Sistine Chapel in mind, specifically the vast, sculptural figures of the prophets who sit enthroned at the level of the pendentive arches. Michelangelo's monumental figures, like Caravaggio's Matthew, are shown in the spasms of divine revelation, reading or writing the prophecies vouchsafed to them by God. Also like Caravaggio's Matthew, they are barefoot, and often accompanied by inspiring angelic figures.\n\nBut Caravaggio evokes the comparison with Michelangelo's prophets only to offer his own, opposed conception of divine inspiration. His _St Matthew_ perfectly reverses all of the properties of the Michelangelesque figure of the prophet. Michelangelo's prophets are nobly idealized figures, decorously draped, but Caravaggio's Matthew is an ordinary, imperfect human being in working clothes that leave his arms and legs bare. Michelangelo depicts troubled intellectuals, straining to grasp God's veiled meanings, but Caravaggio's sainted peasant is a simple man stunned by the directness of his revelation. Whereas Michelangelo's prophets sit on carved thrones of marble, Caravaggio's apostle sits on a simple wooden chair, the same savonarola chair already used for the _Calling of Matthew_ and the _Supper at Emmaus_.\n\nPerhaps the most touching aspect of the painting is the intimacy of the relationship between the stooped saint and the tender young angel, whose wings enfold the whole scene in a hushed embrace. The angel is God's messenger but also the embodiment of Christian love \u2013 a love so generous it encompasses even those as ragged and gnarled as the cross-legged, doltish St Matthew. The contrast between the two figures is the contrast between extreme youth and encroaching old age. Frailty is being overcome, an old man is being made young by the teachings of a child, which are the teachings of Christ himself, and the writing of the first word of the first gospel marks the very instant when the Old Testament is being replaced by the New.\n\nDespite or more likely because of its brusque singularity Caravaggio's picture 'pleased nobody', according to Baglione. The _St Matthew_ was rejected as soon as it was delivered. Bellori gave the fullest account of events: 'Here something happened that greatly upset Caravaggio with respect to his reputation. After he had finished the central picture of St Matthew and installed it on the altar, the priests took it down, saying that the figure with its legs crossed and its feet rudely exposed to the public had neither decorum nor the appearance of a saint.' That was, of course, precisely Caravaggio's point: Christ and his followers looked a lot more like beggars than cardinals. But the decision of Mathieu Cointrel's executors \u2013 who included Fran\u00e7ois Cointrel, his nephew and heir \u2013 was final. Saving Caravaggio's blushes, Vincenzo Giustiniani took the painting of _St Matthew_ for his own collection. According to Bellori, Giustiniani also prevailed on the congregation of San Luigi dei Francesi to allow the painter to try again.\n\nThe resulting picture, his second version of _St Matthew and the Angel_ , was accepted without demur. It remains on the altar of the chapel. The character of the painting, and indeed the very fact that it was commissioned at all, suggests that those in charge of the commission had few doubts about the painter's ability. As far as they were concerned, it was merely his taste, and the tenor of his piety, that was suspect: if he was given the right instruction, these could easily be amended.\n\nThe second _St Matthew_ suggests that Bellori's account of the reasons for the rejection of the first was correct. Matthew the shockingly illiterate peasant has suddenly been turned into Matthew the dignified, grey-haired sage. This scholar-saint kneels at his desk, quill pen at the ready. He is draped in red robes and has been equipped with an expression of dignified attentiveness. Rather than guiding his uncertain hand, the angel now counts off the verses as he dictates them. The pages of the book are no longer visible, but since the angel has got to the index finger of his left hand \u2013 number two, in the gestural rhetoric of the time, since Italians counted the number one with their thumbs \u2013 it seems that he has once more got to the start of the second verse, and Abraham's begetting of Christ's lineage. The angel's airborne arrival from behind Matthew closely echoes the composition of Tintoretto's _Virgin Appearing to St Jerome_ , which Caravaggio may have seen in Venice. There is no suggestion of intimacy here. A message is not vouchsafed tenderly as an act of love, but handed down from on high as an emanation of divine authority.\n\nCaravaggio's second _St Matthew and the Angel_ is a much diluted, dutifully toned-down version of his original idea. Matthew's poverty and humility are not rudely proclaimed, but politely whispered. The most tellingly emphatic of the painter's several adjustments relate to the apostle's feet. They are shown in profile rather than thrust towards the viewer, still bare but unlikely to offend anybody.\n\nFor the first but not the last time, Caravaggio's work had been censored. His sin when painting the first _St Matthew_ had been to make holy poverty and humility unpalatably real. On this occasion his embarrassment was spared by Vincenzo Giustiniani, but Giustiniani's purchase of the first _St Matthew_ itself created a paradox. A work of art expressly designed to articulate ideals of popular piety, to appeal to the broadest possible audience, had been deemed unsuitable for mass consumption. Instead, the picture had found a home in the collection of a noted connoisseur. The implication was that there was something dangerous, even seditious, about Caravaggio's emphatically humble vision of the origins of Christianity. In a prominent church, such an intoxicatingly powerful painting might serve as a rallying cry. It might have an influence. Its visual language might help shape the visual language of the Counter-Reformation Catholic Church. But confined to the collection of a rich man, it became something much less potent: an interesting work of art, an experiment in a new style, but altogether too strange and adventurous for anyone but a sophisticate and his friends to appreciate.\n\n#### LONG LIVE _V_\n\nVincenzo Giustiniani was a torchbearer for Caravaggio's intensely ascetic religious art. He had probably helped the painter to win the commission to paint the lateral canvases in the Cerasi Chapel, and if Bellori is to believed it was he who persuaded the executors of Mathieu Cointrel to allow Caravaggio a second attempt at the _St Matthew_ altarpiece for the Contarelli Chapel. Now, probably around the start of 1603, Giustiniani commissioned one of Caravaggio's most uncompromisingly pauperist depictions of Christ and his disciples, _Doubting Thomas_ , and when it was finished he displayed it prominently in the huge Palazzo Giustiniani, opposite San Luigi dei Francesi, on Via Crescenzi. Its presence there, in addition to that of the first _St Matthew_ , meant that the Giustiniani collection was fast becoming an advertisement for Caravaggio's new approach to devotional art. Vincenzo Giustiniani must have hoped that with his support the artist would eventually win papal favour, obtain great public commissions and become one of those painters who transform the depiction of Christian belief. Without him, many of Caravaggio's most remarkable pictures might never have been created.\n\n_Doubting Thomas_ is a raw picture about the palpable proving of faith. Joachim von Sandrart, impressed by the graphic realism of the work, described it as a picture of 'Christ, in whose holy wounds, Thomas, in the presence of the other apostles, is putting his finger. By means of good painting and modelling he was able to show on the faces of all those present such an expression of astonishment and naturalness of skin and flesh that in comparison all other pictures seemed to be coloured paper.' Sandrart was responding not only to the flesh wound in Christ's side but also to the wrinkled and wizened skin of Thomas and his companions. Thin and papery with age, perhaps it was this that elicited his thoughts about other pictures looking like coloured paper. Caravaggio, once the painter of withered autumn fruits, was becoming increasingly the painter of withered human beings, battered by age and poverty.\n\nLike _The Supper at Emmaus_ , _Doubting Thomas_ was inspired by a legend of the risen Christ. The painter's source was a passage from the gospel of John:\n\n> when the doors were shut where the disciples were assembled for fear of the Jews, came Jesus and stood in the midst, and saith unto them, Peace be with you. And when he had so said, he shewed unto them his hands and his side. Then were the disciples glad, when they saw the Lord... But Thomas, one of the twelve, called Didymus, was not with them when Jesus came. The other disciples therefore said unto him, We have seen the Lord. But he said unto them, Except I shall see in his hands the print of the nails, and thrust my hand into his side, I will not believe. And after eight days again his disciples were within, and Thomas with them: then came Jesus, the doors being shut, and stood in the midst, and said, Peace be unto you. Then saith he to Thomas, Reach hither thy finger, and behold my hands; and reach hither thy hand, and thrust it into my side: and be not faithless, but believing. And Thomas answered and said unto him, My Lord and my God. Jesus saith unto him, Thomas, because thou hast seen me, thou hast believed: blessed are they that have not seen, and yet have believed' (John 20:20\u201329).\n\nCaravaggio chose a half-length frieze-like composition and a close-up view, further excluding all extraneous detail with his usual blanket of shadow. The story is distilled to its essence. Four faces, arranged in the configuration of a diamond, bear mute witness to the miracle of the Resurrection. Christ gently accepts the indignity of being surgically investigated by his sceptical follower. Holding aside the folds of his burial sheet, he guides Thomas's hand towards him and draws the disciple's forefinger into his open wound. Two fellow-disciples crowd round, eyes fixed on the clinical probing of divine flesh. Christ too looks down, as though assisting at his own autopsy. The place where finger meets wound is a different kind of vanishing point, achieved without the calculations of perspective. All converges at the place where the miracle is proved to be true, and the metaphysical and the empirical meet.\n\nThomas and his fellow apostles are men in the same mould as the first _St Matthew and the Angel_ , earnest, ordinary, with heavily lined brows and sunburned faces. Thomas's sleeve needs restitching at the shoulder. An otherwordly radiance floods the scene, illumination the herald of revelation. Cured of his doubt, Thomas himself looks not at the wound in Christ's side but instead to the light.\n\nThere is also, once more, the suggestion of a reminiscence of Michelangelo's paintings for the Sistine Chapel. Thomas's reaching gesture is another of Caravaggio's inverted variations on the fingertip-touching act of generation at the centre of Michelangelo's _Creation of Adam._ In the act of touching Christ, Thomas is born again in unquestioning faith.\n\nGiustiniani commissioned one other, very different picture from Caravaggio. It is the single, stunning exception to the prevailingly solemn body of work produced by the artist during these middle years of his career. _Omnia vincit amor_ , or _Love Conquers All_ , was painted in the summer of 1602. The most nakedly libidinous of the painter's secular mythological works, it is a mischievously joyful celebration of Eros \u2013 a laughing proclamation of the power of sexual love.\n\nA smooth-skinned, naked young Cupid, far removed from the wizened saints of Caravaggio's devotional pictures, confronts the viewer with a puckish smile. The figure half sits and half stands, one leg raised and bent at an angle of almost ninety degrees to his body. Awkwardly perched on a table draped with a white sheet, he occupies an interior cluttered with the stuff of intellectual, artistic, military and political endeavour: men may fight and dream, create and aspire, but in the end love will always triumph over all. The picture's symbolism is concisely explained in a 1638 inventory of the Giustiniani collection, where it is listed as 'A painting of a smiling Cupid, in the act of disparaging the world'. The question of whether it may have had other and deeper meanings, both for Caravaggio and for his patron, Vincenzo Giustiniani, has been hotly contested.\n\nThe pubescent boy who modelled for the picture also modelled for the _St John the Baptist_ , painted by Caravaggio for Ciriaco Mattei at around the same time. Like that other picture, _Omnia vincit amor_ is a variation on Michelangelo's _ignudi_ in the Sistine Chapel, although the effect on this occasion could hardly be more different. The _St John_ is a sanctified version of a Michelangelo nude, spiritually transformed by Christian revelation. This dazzling Cupid in an airless room is devoid of conscience or piety. He embodies a triumphant, amoral, vibrant sexuality.\n\nThe figure's pose carries echoes not only of the _ignudi_ , but also of famous sculptures, one by Michelangelo, the marble figure of _Victory_ , and one by Donatello, the celebrated bronze _David_ , the first freestanding image of the male nude since antiquity, a work charged with homoerotic overtones. Caravaggio's sexy adolescent is the extrovert alter ego of Donatello's veiled, ambiguous, naked young man. The most sensually explicit detail of Donatello's bronze is a feather, a plume from the felled Goliath's helmet, that tickles David's inner thigh. The same motif is repeated in Caravaggio's painting, but here the boy's leg is brushed by the tip of one of his own Cupid's wings.\n\nJoachim von Sandrart, a guest of Vincenzo Giustiniani between 1629 and 1635, reported that the marchese prized the _Omnia vincit amor_ above all the other works in his collection. He gave an admiring, if not altogether accurate, description of the picture itself, and an arresting account of the manner in which it was originally displayed:\n\n> Caravaggio painted for the Marchese Giustiniani a life-size Cupid as a boy of about 12 years old, seated on a globe, and raising his bow in his right hand. On his left are various instruments, a book for studies, a laurel wreath. The Cupid has the brown wings of an eagle. Everything is accurately and clearly designed with bright colours and a three-dimensionality that approximates reality. This painting was among 120 others in a gallery of the most celebrated artists. But, I recall, it was covered with a curtain of dark green silk, and was shown last, after all the others, to avoid eclipsing the other works.\n\nThe objects strewn at Cupid's feet and by his side form a dispersed but uniquely haunting still life: a hallucination of things. They allude to the arts, sciences and letters. A compass and triangle, representing architecture as well as geometry, are prominent in the left foreground. A violin and a lute, rendered in extreme foreshortening, are propped on a musical part-book. A manuscript, emblem of literary ambition, lies open and abandoned on the floor. A laurel wreath has been dropped on to an empty cuirass and other scattered pieces of armour, of the same dark steel as that worn by the sinister soldier in _The Betrayal of Christ_. These signs of military glory undone are complemented by the crown and sceptre obscurely nestling in the dishevelled sheets near Cupid's raised calf. Poking out from behind his right thigh is the rim of a celestial globe, blue with gold stars. Astronomy too has been laid low by Cupid, who holds up two arrows \u2013 not his bow, as Sandrart had asserted \u2013 to symbolize his triumph over all the works and schemes of industrious but easily tempted humanity.\n\nThe objects in the painting may have been selected to reflect Vincenzo Giustiniani's own interests and family history. He was an author and a well-known musical amateur with a keen interest in astrology. The Giustiniani also had an illustrious military and political history. According to one ingenious (but incorrect) interpretation, the picture is not even intended to show love's triumph over all worldly endeavours. Instead it is a celebration of Vincenzo Giustiniani's many accomplishments, a Neoplatonic allegory of the passion propelling him and his family to so many different forms of excellence. But if art and culture really were being celebrated, why would their remnants litter the floor like bric-a-brac?\n\nSandrart's remark about the patron keeping the picture until the end of a tour through his house \u2013 saving the best until last \u2013 is suggestive. Having shown his guests his splendid palace, his collections of classical statuary, his musical _camerino_ , his pictures by the great masters of Italian art, Giustiniani would show them _this_ \u2013 an allegory of all hubris, creative and intellectual, brought low at the feet of love. An elegant gesture of knowing self-deprecation was surely intended. As rich and influential as he was, as accomplished in the arts, letters and sciences, even he still had to concede \u2013 with a graceful smile, of course \u2013 that there was a limit to his powers. Before love, all must give way.\n\nBut _Omnia vincit amor_ was more than just an excuse for that graceful flourish of rhetoric. The picture is arrestingly littered with letter _v_ 's. The majuscule in the musical part-book is a _v_. The set square is arranged in the form of a _v_. The compasses form an upside down _v_. The violin and lute fall across each other to form a _v_. The crown and sceptre shape a _v_. So does the curiously awkward arrangement of the Cupid's splayed legs. His wings echo the shape too. They are eagle's wings, which also formed part of the Giustiniani family crest. All these _v_ 's are also implicated in an orgiastic series of sexual consummations. The set square pushes at the furled circle formed by the part-book's leaves. The compass straddles the set square. The bow of the violin has slid over the neck of the instrument. The sceptre phallically pierces the circle of the crown. Even the white sheet on which the boy rests has contrived to fold itself, at the point just below Cupid's phallus, into the shape of the female sex.\n\nThe phrase _omnia vincit amor_ is taken from Virgil's _Eclogues_ , where it is followed by the line _et nos cedamus amori_ : 'Love conquers all; let us lovers all yield to it.' In Caravaggio's painting, the objects of art and culture have not merely been conquered by love, they have given themselves up to passion. The picture buzzes and pulsates with libidinous energy. It is a mythology shot through with a raucously erotic and life-affirming sense of comedy, a fantasy of learning and knowledge suddenly caught up in the throes of sexual self-abandonment.\n\nBut why would Caravaggio have painted such a picture for a Roman nobleman? And why would a man such as Vincenzo Giustiniani have wanted one? There was in fact a long tradition of such erotically charged mythologies in Italian painting. They were usually created on the occasion of family weddings. The earliest examples were painted on the panels of the wedding chests traditionally given by groom to bride in fifteenth-century Tuscany. By the end of the fifteenth century the mythological love painting had emancipated itself from the decoration of wedding chests to become an independent art form. Botticelli's _Birth of Venus_ is the most famous example. The goddess of love rises from the sea and steps on to dry land. As she does, a cloak is readied by Venus's handmaiden to wrap her perfect body. A particular bunched fold of that cloak, close to Venus's face, has been painted by Botticelli to resemble the female sex and within it a tiny leaf is folded, an emblem of fertility. There are shades, here, of Caravaggio's vulva-like twist of sheet in _Omnia vincit amor_ \u2013 and an anticipation of its meaning. _The Birth of Venus_ was painted as a gift to a Medici bride. Its message was unambiguous. Like the virgin Venus leaving the sea and arriving on earth, the bride was about to leave her former chastity behind and embark on married life. As she did so, the painting was offered to her as a prayer for the future fertility of the union.\n\nThis tradition was still alive in Caravaggio's time, although by then it had mutated into yet more spectacular forms \u2013 none more so than the mythological-erotic ceiling painted by Annibale Carracci for the ceiling of the Farnese Palace between 1597 and 1601. A deliberately pagan parody of Michelangelo's Sistine Chapel ceiling, Carracci's Farnese decorations constitute a vast panorama of the loves of the gods, a comical riot of the sexual indiscretions of Jupiter, Juno and a veritable horde of other, amorously inflamed deities. The overarching theme of the ceiling is _Omnia vincit amor_. The work was painted, just like the wedding chests of Tuscan tradition, just like Botticelli's _Birth of Venus_ , to celebrate a wedding.\n\nIt is highly likely that Caravaggio's own _Omnia vincit amor_ was commissioned on the occasion of a wedding in the Giustiniani family. Whether it commemorates a particular event or not, its essential meaning is clear. Vincenzo was a family name among the Giustiniani, so all those orgiastically active _v_ 's in the picture may be taken to stand both for the man who commissioned the work and for his many heirs and descendants. Long live the Giustiniani, the picture priapically exclaims: long may they prosper, and long may they procreate.\n\n#### THE BLACK WINGS OF ENVY\n\nGiovanni Baglione and his friends were not happy to see the painter from Lombardy doing so well. Forty years later, when he came to write his short biography of Caravaggio, Baglione still seethed with a sense of injustice when he thought of his rival winning a string of commissions from Vincenzo Giustiniani and Ciriaco Mattei. As far as Baglione was concerned, Caravaggio's patrons had been fooled by nothing more than clever publicity: 'The Marchese had been put into this frame of mind by Prosperino delle Grottesche, Caravaggio's henchman [Prospero Orsi, the painter of grotesques]... Moreover, Signor Ciriaco Mattei succumbed to the propaganda... Thus Caravaggio pocketed from this gentleman many hundreds of scudi.'\n\nKarel van Mander's _Schilderboek_ of 1604 includes some pithy remarks on the rivalries that divided Rome's competing factions of artists during the early years of the seventeenth century. According to van Mander, Clement VIII and his papal court commissioned so many new works that they stirred up a frenzy of competition among painters and sculptors: 'a new ardour is kindled; lean Envy secretly begins to flap her black wings and everyone strives to do his best to gain the coveted prize.'\n\nThe dark wings of Caravaggio's Cupid certainly fanned the flames of Giovanni Baglione's envy. Infuriated by the acclaim with which Caravaggio's _Omnia vincit amor_ had been received \u2013 the curtain of green silk later described by Sandrart proof of its status as the _coup de th\u00e9\u00e2tre_ of Vincenzo Giustiniani's whole collection \u2013 Baglione responded with an act of provocation. On 29 August 1602 he brought a new work to the artists' exhibition held annually in the courtyard of San Giovanni Decollato. Caravaggio was not taking part in the show, but his friend and follower Orazio Gentileschi did have a picture on display. Baglione's painting was an attack on both of them, as Gentileschi would later explain under cross-examination: 'there certainly is some rivalry among us. When I hung a picture of _St Michael the Archangel_ in San Giovanni dei Fiorentini [a slip of the tongue; the exhibition took place in San Giovanni Decollato], Baglione showed up and hung one of his opposite, a _Divine Love_ that he had done to vie with an _Earthly Love_ by Michelangelo da Caravaggio.'\n\nThe full title of Baglione's painting, now in the Gem\u00e4ldegalerie in Berlin, is _Divine Love Overcoming Earthly Love, the World, the Flesh and the Devil_. It is no masterpiece but it is a clever and vicious painting, carefully calculated to try to wound Caravaggio and cloud his reputation. Inverting both the moral and the message of _Omnia vincit amor_ , Baglione shows love conquered by virtue. A saint in armour subdues a cowed and cowering Cupid, while the devil skulks in darkness to one side. Although armed with a thunderbolt rather than shield and sword, the figure of _Divine Love_ evokes traditional images of Archangel Michael trampling Satan underfoot \u2013 a detail that can only have been intended to sharpen Baglione's satire. Caravaggio's own holy namesake is shown exorcizing the erotic and demonic spirits of Caravaggio's art. The resemblance to St Michael may also have been meant as a sideswipe at Gentileschi and his own 'picture of _St Michael the Archangel_ '. But Caravaggio was the primary target. Baglione's parody is completed by the emphatically Caravaggesque lighting that flashes across the vengeful angel and the prone, flaccid form of Cupid below.\n\nNot content with satirizing Caravaggio and his art, Baglione even dared to offer his own picture to Vincenzo Giustiniani's brother, Cardinal Benedetto Giustiniani. Yet more galling, it was accepted, and Baglione given the traditional painter's reward of a gold chain. Gentileschi, who would tell this part of the story too in court, was distinctly unamused by the whole affair. He tried to get his own back by telling Baglione that his avenging angel should have been naked and childlike, perhaps deliberately misunderstanding the figure's pointed and satirical resemblance to the Archangel Michael: '[Baglione] had dedicated _Divine Love_ to Cardinal Giustiniani and although said picture was not liked as much as the one by Michelangelo \u2013 all the same, that Cardinal gave him a neck chain. That painting had many flaws. I told him he had done a grown-up man in armour who should have been nude and putto, so he did another that was completely nude.'\n\nThat last remark is only half true. Baglione did paint another _Divine Love_ , the version now in the Palazzo Barberini in Rome, but he completely ignored Gentileschi's advice. The second and equally grown-up angel is certainly no naked putto. He still wears armour, albeit modified this time to a breastplate and some wisps of drapery. The most important change has been made elsewhere, at the bottom of the picture. The devil, previously shown lurking in obscurity, now wheels round with an expression of startled guilt on his face. Despite his staring eyes, his fangs and his pointed ears, he is unmistakably a portrait in caricature of Caravaggio, caught _in flagrante_ with a flushed and furtive Cupid. Baglione's second _Divine Love_ went beyond satire. It was a visual accusation of sodomy.\n\nBaglione repeated that charge verbally, and in public. He and his friends talked openly about Caravaggio keeping company with a _bardassa_ \u2013 vulgar Italian slang derived from a Turkish word for a young man who took the female part in sexual encounters with other men. Rome's artists gossiped, so people may have begun to look at Caravaggio's _Omnia vincit amor_ in a different light. The identity of the boy who had modelled for Cupid was known. He was Cecco di Caravaggio, who prepared the artist's paint and his canvases. If Baglione was to be believed, not only was he Caravaggio's assistant and model, he was also his catamite.\n\nHalf a century later, the story was still current. In about 1650 an English artist called Richard Symonds was shown round the Giustiniani collection in Rome. He made notes on the pictures, writing down any anecdotes that struck him. He obviously spent a while in front of _Omnia vincit amor_. The _custode_ told him that it was one of the most precious pictures in the collection, that it had cost 300 scudi, and that both the Cardinal of Savoy and a member of the Crescenzi family had offered 2,000 ducats for it. But the punchline of the spiel, no doubt regularly trotted out to tourists visiting the palace, was the scandalous relationship between artist and model. Here is a full transcription of the entry in Symonds's travel journal:\n\n> Cupido di Caravaggio \/ Card di Savoya profe. \/ 2 milia duboli p[er] \/ il Cupido di Caravaggio \/ Costo 3 cento scudi. \/ Checco del Caravaggio tis \/ calld among the painters \/ twas his boy \u2013 \/ haire darke, 2 wings \/ raie, compasses lute \/ violin & armes & laurel \/ Monsr Crechy vuolle dare \/ 2 milia dubole \/ Twas the body & face \/ of his owne boy or servant \/ that laid with him.\n\nFrom the readiness with which the story was believed and then accepted into local legend, it seems that nobody had been particularly surprised to hear about Caravaggio's alleged homosexual proclivities. He was known to be an impetuous man who followed his passions. He kept company with whores and courtesans, such as Fillide Melandroni, and on the evidence of his paintings he was equally alive to the physical charms of men. Caravaggio and Francesco Boneri, alias Cecco, were close: Cecco stayed with him even after he was obliged to leave Rome in 1606. There is a good chance that the rumours were true and that Caravaggio did indeed have a sexual as well as a working relationship with 'his owne boy or servant'.\n\nWhatever the reality, Baglione's accusations were damaging and dangerous. Sodomy was a capital crime in Clement VIII's Rome, and though the authorities were unlikely to investigate the well-connected Caravaggio's sexual behaviour, as long as he was reasonably discreet, the potential harm to his name and prospects was immense. Once an artist had been smeared as a pederast, his work was smeared too. People were liable to stop taking it seriously, seeing it only through the lens of its creator's presumed sexual aberration. This had happened half a century before, notoriously, to an artist named Giovanni Antonio Bazzi. Bazzi had offended the famous chronicler of artists' lives, Giorgio Vasari, who had taken his revenge in print: 'since he always had about him boys and beardless youths, whom he loved more than was decent, he acquired the by-name of Sodoma.' This was all a pure fabrication on Vasari's part, first started perhaps by Bazzi's rival, the Sienese painter Domenico Beccafumi. But the mud stuck, and to this day the artist is known as Il Sodoma, 'the Sodomite'.\n\nCaravaggio was deeply sensitive about his reputation. He never knowingly allowed the least slight to go unpunished. His nocturnal assault on Girolamo Spampa, the art student from Montepulciano, was proof of that. Spampa had probably been recycling Federico Zuccaro's criticisms of Caravaggio's Contarelli Chapel paintings, which he had most likely heard from Baglione in the first place. Either way, Baglione must have been aware of the beating that Caravaggio had given Spampa, which had taken place just eighteen months before. He knew that his satires and smears would not be forgotten. Sooner or later, Caravaggio would retaliate.\n\nThe autumn and winter of 1602 passed without incident, as Caravaggio bided his time. Only in spring of the following year did he give vent to his simmering anger. He was stung into retaliation by the unveiling of Baglione's largest work yet, _The Resurrection_ , an altarpiece for the principal Jesuit church in Rome, the Ges\u00f9. That work is now lost, but, to judge by Baglione's preparatory study in the Louvre, it was a clumsy and grandiose essay in the same proto-Baroque idiom as Carracci's _Assumption of the Virgin_. In the sketch Christ stands heavily on a stage-flat cloud as angelic choirs hymn his heavenward ascent. Below on earth, one of the soldiers guarding the tomb gets drowsily to his feet, while others snooze or look on in laboured poses of amazement.\n\nBaglione could carry off the mock sublimity of a parody like the _Divine Love_. But when he strove for effects of awe-inspiring transcendence, he was undone both by his lack of skill and by the essentially prosaic nature of his imagination. His shortcomings can only have been magnified by the scale of the altarpiece for the Ges\u00f9: eight metres high and nearly five across. The picture was in its allotted place by Passiontide 1603, but kept under wraps until Easter Sunday itself, the day of the Resurrection. It seems never to have been much loved. Caravaggio and his friends set the tone for its reception by poking fun at it from the moment it was unveiled. There would be no protests when it was quietly removed from the church towards the end of the seventeenth century, following alterations to the transept altars.\n\nCaravaggio observed (and probably helped to orchestrate) the picture's unfavourable reception with rancorous pleasure. He was already annoyed that Baglione had been given such a prestigious assignment \u2013 and all the more irritated because he suspected that Baglione had won the job through the ruse of offering his satire, the _Divine Love_ , to Benedetto Giustiniani. Cardinal Giustiniani was a Jesuit and had probably intervened with the general of the order, Claudio Acquaviva, to obtain the commission for Baglione. So when his rival produced his monumental flop, Caravaggio decided that it was the moment to take his revenge. What better time to kick a man than when he is down?\n\nShortly after Easter Sunday 1603 a couple of newly composed satirical poems caused something of a sensation in the artists' quarter of Rome. Copies were passed round. Impromptu recitals were held. The verses were aimed at 'Gioan Bagaglia' or 'Gian Coglione', 'John Baggage' or 'Johnny Testicle'. They were not the most ingenious nicknames for Giovanni Baglione, but they were effective. One of the poems also included a swipe at 'Mao', the alias of Tommaso Salini, who was a minor still life painter and Baglione's closest associate.\n\nThe first poem is crude and makeshift, a mock-sonnet with all the subtlety of a punch in the face:\n\n_Gioan Bagaglia tu no[n] sai un ah_\n\n_le tue pitture sono pituresse_\n\n_volo vedere con esse_\n\n_ch[e] non guadagnarai_\n\n_mai una patacca_\n\n_Ch[e] di cotanto panno_\n\n_da farti un paro di bragesse_\n\n_ch[e] ad ognun mostrarai_\n\n_quel ch[e] fa la cacca_\n\n_portela adunque_\n\n_i tuoi disegni e cartoni_\n\n_ch[e] tu ai fatto a Andrea pizzicarolo_\n\n_o veramente forbete ne il culo_\n\n_o alla moglie di Mao turegli la potta_\n\n_ch[e] libelli con quel suo cazzon da mulo pi\u00f9 non la fotte_\n\n_perdonami dipintore se io non ti adulo_\n\n_ch[e] della collana ch[e] tu porti indegno sei_\n\n_et della pittura vituperio_.\n\nJohn Baggage you don't even know\n\nThat your pictures are mere woman's-work\n\nI want to see\n\nThat you won't even earn a counterfeit penny from them\n\nBecause with as much canvas\n\nAs it would take to make yourself a pair of breeches\n\nYou can show everyone\n\nWhat shit truly is\n\nTherefore take\n\nYour drawings and cartoons\n\nThat you have made, to Andrea the grocer's shop\n\n[so he can wrap fruit and veg in them]\n\nOr wipe your arse with them\n\nOr stuff them up the cunt of Mao's wife\n\nBecause he isn't fucking her anymore with his donkey cock\n\nPray pardon me, painter, if I do not worship you\n\nBecause you don't merit that chain you wear round your neck\n\nAnd your painting deserves only vituperation.\n\nBenedetto Giustiniani's award of a gold chain to Baglione evidently still rankled. Rubens, Van Dyck and Rembrandt would all paint themselves wearing chains of gold, symbols of accomplishment and courtly patronage. It was a mark of intellectual distinction, a sign of honour, but it had been conferred on Baglione for painting a picture that explicitly _dishonoured_ Caravaggio.\n\nThe second poem was rather more carefully constructed, in regular hendecasyllabic lines. Its attacks on Baglione were slightly less sexually graphic, at least until the last line:\n\n_Gian Coglione senza dubio dir si puole_\n\n_quel ch[e] biasimar si mette altrui_\n\n_ch[e] pu\u00f2 cento anni esser mastro di lui._\n\n_Nella pittura intendo la mia prole_\n\n_poi ch[e] pittor si vol chiamar colui_\n\n_Ch[e] no[n] pu\u00f2 star p[er] macinar con lui._\n\n_I color no[n] ha mastro nel numero_\n\n_si sfaciatamente nominar si vole_\n\n_si sa pur il proverbio ch[e] si dice_\n\n_ch[e] chi lodar si vole si maledice_\n\n_Io no[n] son uso lavarmi la bocca_\n\n_ne meno di inalzar quel ch[e] no[n] merta_\n\n_come fa l'idol suo ch[e] \u00e8 cosa certa._\n\n_Se io metterme volessi a ragionar_\n\n_delle [s... re] fatte da questui_\n\n_no[n] bastarian interi un mese o dui._\n\n_Vieni un po' qua tu ch'[e] vo' biasimare_\n\n_l'altrui pitture et sai pur ch[e] le tue_\n\n_si stano in casa tua a' chiodi ancora_\n\n_vergogna[n]doti tu mostrarle fuora._\n\n_Infatti i' vo' l'impresa aba[n]donare_\n\n_ch[e] sento ch[e] mi abonda tal materia_\n\n_massime s'intrassi n[e] la catena_\n\n_d'oro ch[e] al collo indegnamente porta_\n\n_ch[e] credo certo [meglio] se io non erro_\n\n_a pi\u00e8 gle ne staria una di ferro._\n\n_Di tutto quel che ha detto con passione_\n\n_per certo gli \u00e8 p[er] ch\u00e9 credo beuto_\n\n_avesse certo come \u00e8 suo do\u00f9to_\n\n_altrime[n]te ei saria un becco fotuto._\n\nCall him Johnny Bollock,\n\nthis man who sets about criticizing another man\n\nwho could be his master for a hundred years.\n\nI mean in my beloved art of painting,\n\nbecause he would like to call himself a painter\n\nalthough he's not even fit to grind colours for that other man.\n\nUsing colours isn't as easy as one two three,\n\neven if he shamelessly wants to pretend it's like that.\n\nEveryone knows the truth of the proverb:\n\nMen like to attack those whom they should really be praising.\n\nI'm not one for washing my mouth out,\n\nnor for exalting someone who doesn't merit it,\n\nas he praises his false idol.\n\nIf I wanted to start describing\n\nthe pathetic things this man has done\n\na whole month wouldn't be enough, nor even two.\n\nCome here for a moment, you who like to criticize\n\nthe paintings of another,\n\neven though you know that your own\n\nare still in your house\n\nbecause you're ashamed to show them to anyone.\n\nIn fact I'm going to stop this humiliation in a minute\n\nbecause I've just got too much material to work with\n\nespecially if I start on that necklace\n\nof gold which you so undeservedly wear round your neck\n\nbecause I believe, if I'm not mistaken, that you should\n\nreally have an iron one attached to your ankle.\n\nAs for all that [Johnny Bollock] has said with such passion,\n\nwell, it can only be because he's drunk, in my opinion,\n\nas he ought to be,\n\notherwise he'd just be a fucked-over cuckold.\n\nThere was a long and carnivalesque tradition of colourful insult in Rome, embodied by the battered ancient statue known as _Pasquino_ , which stood at the corner of Palazzo Braschi to the western side of Piazza Navona. It had long been the custom to attach squibs, satires, scurrilous pieces of graffiti and other outbursts of defamatory rage to the wall next to the statue, under the cover of darkness. There was a collective noun for these libels: _pasquinate_ , or 'pasquinades' _._ Caravaggio, a familiar sight in the Piazza Navona, sword strapped to his side, may well have attached the verses attacking Baglione to the so-called 'speaking statue'. In any case it must have been fairly obvious to anyone who knew about the trouble between the two men that Caravaggio was behind this poetry of scabrous ridicule.\n\nThe timing of his attack was unwise. During the early years of the seventeenth century there was a fierce crackdown on libel in Rome, in direct response to the widespread unrest that had followed a notorious trial and public execution. In the summer of 1599 a beautiful young noblewoman called Beatrice Cenci had been sentenced to death for murdering her tyrannical and incestuous father. Her mother, Lucrezia, and her two brothers, Giacomo and Olimpio, had also been convicted as accomplices to the crime. Appeals for clemency were turned down by the pope, and on 11 September of the same year the execution had taken place on a temporary scaffold erected on the Ponte Sant'Angelo. Lucrezia and Beatrice were publicly beheaded, while Giacomo had his flesh torn piece by piece from his body with red-hot pincers. Olimpio was spared because he was still a minor. But he was forced to watch the suffering of the others and roused with cold water on the many occasions when he fainted. A vast crowd was present, including perhaps Caravaggio: artists were encouraged to witness executions so that they might imagine the pains of the Christian martyrs all the more sharply. The sympathies of the people were firmly with the Cenci. It was widely believed that the execution was no more than a judicial murder, designed to enrich Clement VIII and other members of the Aldobrandini family. The pope did not care for justice, it was said. He was just looking for an excuse to sequester the Cenci's estates. There was rioting and seven died in the crush.\n\nThe notoriously tough Governor of Rome, Ferrante Taverna, had personally supervised the extraction of Beatrice Cenci's confession. In the immediate aftermath of her trial and execution, he clamped down brutally on the circulation of seditious rumours. At the end of 1599 he issued a decree, _Contro detrattori della fama, & honor' d'altri in lettre d'avisi, versi, prose, o altrimenti_, 'to curb the audacity of those... who use their pernicious tongues, in writing newsletters to various parts, filling their papers with lies and calumnies'. Tough penalties would be imposed on anyone who 'defamed, and detracted from honour and reputation... under the guise of cleverly written poems, and witty epigrams, or libellous prose, & pasquinades'. A man found guilty of the most serious form of libel could expect a sentence of from seven years to life rowing in the papal galleys. Many of those convicted requested beheading instead.\n\nWithin a matter of months of Caravaggio's poems starting to circulate, Baglione and Tommaso Salini \u2013 Mao, as he was known \u2013 decided to take the case to court on a charge of criminal libel. They prepared their evidence with care. Salini cultivated the acquaintance of a painter who was close to Caravaggio, Filippo Trisegni. In a show of friendship, he lent Trisegni various studio props, including a helmet, and promised to teach him how to paint cast shadows. Eventually Salini managed to wheedle a copy of the first poem out of Trisegni. He then persuaded him to write the second one out for him in longhand. Armed with Exhibit A and Exhibit B in the case for the prosecution, Baglione and Salini struck back at their enemies. On 28 August, Baglione lodged a complaint with the Governor of Rome about some _libelli famosi_ , or 'famous libels'. The accused were Onorio Longhi, Caravaggio, Orazio Gentileschi and the hapless Filippo Trisegni. Baglione produced his manuscripts of the offending poems, incriminatingly written out in the hand of Caravaggio's known friend Trisegni, as he addressed his deposition to Judge Alfonso Tomassino, judicial representative of Governor Taverna:\n\n> You should know that I am a painter by profession and have been practising this profession here in Rome for a good many years. Now it happens that I have gone and painted a picture of the resurrection of Our Lord for the Father General of the Company of Jesus, which is in a chapel of the church of Ges\u00f9. When they found out about the said picture, which was this past Easter, the said accused were envious because they intended, I mean, the said Michelangelo intended to do it himself. So this Michelangelo out of envy, as I said, and the said Onorio Longhi and Orazio, his friends and followers, have gone round speaking ill of me and reproaching my work. And, in particular, they have done some verses that dishonour and insult me. They gave these round and circulated them among many different people, these being the ones I'm showing you, which I had from the painter Messer Tommaso Salini. He told me he got them from Filippo Trisegni, also a painter, and that a part of the said verses were written by Filippo in his presence, being those that begin 'John Baggage' and end 'your painting deserves only vituperation' and that the others are those on this quarter page that begin 'Johnny Bollock' and end 'otherwise he'd just be a fucked-over cuckold'. So I take action against the above-named and any others that have assisted, or in whatever way were aware of and found guilty of this fact, asking that action be taken against them as justice requires, since the above-named accused have always persecuted, emulated, and envied me seeing that my works are held in higher esteem than theirs.\n\nImmediately afterwards Mao Salini made his own deposition, fleshing out Baglione's account of events with more detail. Salini particularly relished telling the story of how he had trapped Filippo Trisegni into helping him obtain copies of the incriminating poems:\n\n> I was taking a stroll around with the said Filippo asking him what painters were saying about the picture that the said Giovanni had done for the church of Ges\u00f9. He told me that Michelangelo of Caravaggio, Onorio Longhi, and Orazio Gentileschi, all three painters, had put together some verses against the said Giovanni, and against me since I am his friend, concerning the said painting. And so, a few days later, the said Filippo with fine words gave me a paper with several verses written against the said Giovanni, which were on a quarter-page. Then he told me that the said Orazio had written them together with Ottavio Padovano [a nickname for Ottavio Leoni], likewise a painter, and that Ludovico Bresciano, also a painter, was going around distributing them to numerous painters. In particular, he had given them to one Mario, in like manner a painter, who lives in Via del Corso.\n> \n> Then \u2013 I don't know how many days later \u2013 the said Filippo came to my house one day towards evening to see a painting. After showing it to him, I begged him to tell me a little about the sonnet that he had given me a while back. That was before he gave me the said verses that I mentioned above. Then he told me that he had already given it to me once, and when I said that I had misplaced it he finally wrote the sonnet out for me there in my house on half a page, which if I don't remember badly begins 'John Baggage', telling me that the said Michelangelo and Onorio had written it, and that he had received it from a catamite [ _bardassa_ ] of Onorio and Michelangelo called Giovanni Battista who lives behind the Banchi.\n> \n> What's more, he told me that the said Michelangelo, knowing that this Filippo had been handed the said sonnets, had warned him to be careful that these sonnets didn't fall into the hands of the said Giovanni or in my hands because trouble would be caused, and that certain young men had done them at his home for their pleasure, telling me also that a certain Bartolomeo, servant of the said Michelangelo, was going around distributing these sonnets to whoever wanted one, and that he had also written others.\n\nWhat Salini so carefully described was exactly what the new libel law had been brought in to eradicate: a systematic attempt to blacken a man's reputation. The cast of villains was impressive. First there was the naive accomplice, Trisegni, with his 'fine words', taking a malicious pleasure in the whole affair while unwittingly playing the part of a patsy. Then there was the network by which the slanders were distributed. It included the Lombard artist called Ludovico from Brescia and his friend Mario \u2013 possibly Caravaggio's old friend from Sicily, Mario Minniti \u2013 who took the poems from one artist's studio to the next. An alleged catamite and a servant of doubtful morals were also involved. Finally, at the centre of the conspiracy, were Caravaggio and his shady friends. They wrote the poems at Caravaggio's home, which at this time was the Palazzo Mattei, and they did so out of pure malice, 'for their pleasure'. After Salini's testimony, the judge had little choice but to prosecute.\n\nLess than two weeks later, the black-cloaked _sbirri_ swooped. On 11 September 1603 they took Filippo Trisegni while he was having lunch at home in the Via della Croce. Caravaggio was seized in the Piazza Navona the same day. Less than twenty-four hours later, they arrested Orazio Gentileschi at his quarters in Via Paolina. They also searched Gentileschi's rooms and removed certain letters and sonnets, to be used as evidence. The only defendant to escape arrest was Onorio Longhi. He had left town, presumably after a tip-off.\n\nThe accused were kept apart, in order to prevent them getting their respective stories straight before the trial. Gentileschi was held at the crumbling Corte Savella. Caravaggio and Trisegni were kept in solitary confinement at the Tor di Nona. Their dark, single cells were on the first floor. Nearby was another prison known as the _galeotta_ , so called because it housed convicts who had already been condemned to row in the papal galleys. It was a vivid reminder of the fate awaiting Caravaggio and his associates if Giovanni Baglione had his way.\n\nFilippo Trisegni was the first of the accused to give evidence. He began by pretending that he barely knew Mao Salini: 'I know a painter called Tommaso but I don't know his surname. He lives near me on Via della Croce. I usually call him Mao and I believe he's from Rome.' But there was an immediate surprise in store for him. Salini had provided the court with a note written to him by Trisegni, requesting the loan of an iron helmet. The judge produced the note, which surely implied that Trisegni and his neighbour were closer than he had just said. He had written the note, had he not? Wrongfooted, Trisegni backtracked hurriedly. He could not deny that the note was in his handwriting. On reflection, he added, 'the said Tommaso is a very good friend of mine, who lent me chalks and anything else I needed.'\n\nIt was not an auspicious beginning. But Trisegni managed to recover his composure when questions were asked \u2013 as he must have known they would be \u2013 about certain scurrilous verses. His responses were teasing and ambiguous, clearly intended to deflect attention away from himself, as well as from Caravaggio and the other suspects. He implied that a mysterious man called Gregorio Rotolanti had commissioned the verses from a yet more mysterious and unnamed man, a student of physics or logic \u2013 Trisegni was not sure which. Here is the main body of his testimony:\n\n> I heard some insulting poems about the said Tommaso told by Gregorio Rotolanti, and since Tommaso is my friend I pretended to like them and begged the said Gregorio to give me a copy. And so I went to his home and he let me copy some verses but I don't remember exactly what they said. The wife of the said Tommaso was mentioned in it and he was called Mao and it said 'Your pictures are woman's work' or something similar and I think they were on a quarter-page. After I copied them I went to see the said Tommaso and told him that he ought to be aware that while he was going around speaking badly of other people's paintings people were speaking badly of him too. Then I told him I wanted to show him something that had been written against him and so I showed him that poem. I gave it to him and [Mao] pressed me to tell him who had given it to me and who had written it, but I never wanted to tell him because I didn't want to cause trouble.\n> \n> Yet he pressed me and named a lot of people and especially Michelangelo of Caravaggio, Bartolomeo, who was Michelangelo's servant, and Orazio Gentileschi, a man from Parma called Ludovico Parmigianino, another called Francesco Scarpellino, and he asked me if the person could be one of them. I said to him 'It could be any one of them \u2013 it might be one of them but I don't want to tell you his name.' I had been waiting for him to teach me how to do figures in cast shadows and then I would have told him. But he never taught me, and so I didn't tell him.\n> \n> But he did beg me to tell him if I heard anything else about him. And so I spoke to the said Rotolanti, who said that he had another poem against the said Tommaso. Then on the following day we met on Via della Croce and went into the apothecary, where he let me copy some more verses that I think began with, 'Johnny Bollock'. It was in a nice style and well written. But I don't know much about verse and out of several verses I made one. Rotolanti didn't want to give me the original because he said he wanted to learn it by heart. I asked him who had written them and he answered that a young man had written it, a student of logic or maybe physics, a _valent'huomo_ , who was really good at it and would write a sonnet or two for a woman for me if I so desired. He told me he was a graduate and wrote exquisite verse. So I returned to see Tommaso about fifteen or twenty days after the first visit and showed him this other poem and left it with him. It was on a quarter-page filled with small handwriting from one side to the other. He told me that he had lost the first one and asked me if I would be kind enough to give him a copy. Since I had learned almost all of it by heart I made him a copy there at his home the way I remembered it.\n\nTrisegni's testimony directly contradicted Salini's deposition to the court. The man whom he claimed had given him the verses, Gregorio Rotolanti, now became a key witness. But he was never called to testify. Perhaps he had gone into hiding, like Onorio Longhi. Or maybe Trisegni had simply invented Rotolanti and his story about the student of physics or logic with a talent for versification.\n\nGentileschi was next to be questioned by the court. The main purpose was to identify the handwriting of the various documents that the _sbirri_ had confiscated at his home. These included a letter to which some playful verses were attached, as well as four sonnets written on a single piece of paper. After having admitted that 'I know how to write but not very correctly', Gentileschi denied that any of the verses produced in evidence were written in his hand. He said that the eight lines of verse attached to the letter were written by a friend of his called Lodovico. As for the four sonnets, 'I say that about six to eight days ago one Giovanni Maggi, an engraver and painter that lives in Vicolo dei Bergamaschi, gave them to me. He dabbles in these things but I don't really know if it's his handwriting. But it could be his handwriting.'\n\nOn the next day, 13 September, the magistrate arranged a confrontation between 'Mao' \u2013 Tommaso Salini \u2013 and Filippo Trisegni. This was common practice when two witnesses had given opposing testimonies. Neither man changed his story. Salini repeated his assertion that Trisegni had listed the names of those who had written the two poems. He stressed once again that Trisegni had told him that he been given some of the verses by the _bardassa_ , the catamite 'who lives behind the Banchi'. But Trisegni denied everything and maintained that Salini was lying.\n\nLater the same day Caravaggio was called to give evidence. The notary, Decio Cambio, made his usual careful report of the proceedings, but the painter gave him less work than any of the other witnesses. He was haughty and taciturn. He gave his answers grudgingly and kept them short. At one point he asked, with evident irritation, how much more he would have to listen to, as if to suggest that the enquiry was just an elaborate waste of his time. He declared, improbably, that he had never heard of any scurrilous poems written about Giovanni Baglione and Mao Salini. He claimed that he had not even spoken to Orazio Gentileschi for three years. But he made no pretence of his utter contempt for Baglione's work. That was the one subject on which, dangerously, he became almost effusive. But perhaps the most noteworthy part of his testimony is his grumpy, one-line definition of a good painter.\n\nHere is the full court record of his interrogation:\n\n> _Before His Excellency the Illustrious Alfonso Tomassino assistant examining magistrate and myself... Michelangelo Merisi of Caravaggio interrogated under oath by the magistrate._\n> \n> _Asked how, under what circumstances, and for what reason he was incarcerated._\n> \n> _Answered_ :\n> \n> I was arrested the other day in Piazza Navona but I don't know what the reason and circumstances are.\n> \n> _Asked what profession he exercises_.\n> \n> My profession is painting.\n> \n> _Asked if he knew and knows some painters in Rome and which ones._\n> \n> _Answered_ :\n> \n> I think I know almost all the painters in Rome starting with the _valent'huomini_. I know Gioseffe, Caracci, Zucchero [ _sic_ , for Zuccaro], Pomarancio, Gentileschi, Prospero, Giovanni Andrea, Giovanni Baglione, Gismondo and Giorgio Todesco, Tempesta, and others.\n> \n> _Asked if all the painters named above are his friends and are all (as they are commonly called) valent'huomini._\n> \n> _Answered_ :\n> \n> Almost all the painters I have listed above are my friends, but not all of them are _valent'huomini_.\n> \n> _Asked to identify what he means by the word 'valent'huomo'._\n> \n> _Answered_ :\n> \n> By the term ' _valent'huomo_ ' I mean he who knows how to do well, that is, he who knows how to do his art well. So in painting a _valent'huomo_ is one who knows how to paint well and imitate natural objects well.\n> \n> _Asked to identify which of them are his friends and which are his enemies_.\n> \n> _Answered_ :\n> \n> Of those I listed above neither Gioseffe, nor Giovanni Baglione, nor Gentileschi, nor Giorgio Todesco are my friends because they don't speak to me. The others all speak and converse with me.\n> \n> _Asked to specify which of the above-named he considers and holds to be valent'huomini (as they are commonly called) and those which he does not consider as such_.\n> \n> _Answered_ :\n> \n> Of the painters I listed above as good painters, Gioseffe, Zucchero, Pomarancio and Annibale Carracci; as for the others I don't consider them _valent'huomini_.\n> \n> _Asked if he knows if the above-named painters are judged respectively to be good or bad by other painters, just as he judges and reputes them to be_.\n> \n> _Answered_ :\n> \n> _Valent'huomini_ are those who are well versed in painting and will judge as good and bad painters those that I judge myself to be good and bad. But those that are bad and ignorant painters will judge as good painters the ignorant just as they are.\n> \n> _Asked if he knows of any person or painter who praises, considers and holds to be a good and virtuous painter one of those painters named above that he esteems is not a good painter_.\n> \n> _Answered_ :\n> \n> I don't know of any painter that praises and considers a good painter any of those painters that I don't esteem to be good painters.\n> \n> _Then_ :\n> \n> I have gone and forgotten to tell you that Antonio Tempesta \u2013 he too is a _valent'huomo_.\n> \n> _Asked in particular if any other painter has praised and praises the said painter Giovanni Baglione and who he might be._\n> \n> _Answered_ :\n> \n> I don't know of any painter who thinks that Giovanni Baglione is a good painter.\n> \n> _Asked if he has seen any of the works of the said Giovanni Baglione and which ones._\n> \n> _Answered_ :\n> \n> I've seen almost all the works of Giovanni Baglione, that is, the big chapel at the Madonna dell'Orto, a painting in San Giovanni Laterano, and recently _The Resurrection of Christ_ at the Ges\u00f9.\n> \n> _Asked how he judges the above-named painting The Resurrection, and if he knows if it was praised or reproached by other painters._\n> \n> _Answered:_\n> \n> I don't like that painting there at the Ges\u00f9 because it's a bungle. I think it's the worst he's done and I haven't heard any painter praise the said painting. Of all the painters I've spoken to no one liked it.\n> \n> _Then_ :\n> \n> Except that it is praised by someone who's always with him, who's called his guardian angel, who was there praising it when it was unveiled. They call him 'Mao'.\n> \n> _Asked to identify with whom or with which painters he saw that painting The Resurrection._\n> \n> _Answered:_\n> \n> Prospero and Giovanni Andrea. And I saw it on other occasions when I went to the Ges\u00f9, but I don't remember if any other painters were with me.\n> \n> _Then_ :\n> \n> What am I going to have to hear next about this matter?\n> \n> _Asked if he knows if the said Mao is a painter and if he has done any paintings and if he has seen any of them._\n> \n> _Answered:_\n> \n> Maybe he dabbles in art and still daubs, but I've never seen any work of this Mao.\n> \n> _Asked if he has met and knows Onorio Longhi and Ottavio Padovano_.\n> \n> _Answered_ :\n> \n> I know Onorio Longhi, who's very much a friend of mine, and I also know Ottavio Padovano.\n> \n> _Then_ :\n> \n> But I've never spoken to Ottavio Padovano.\n> \n> _Asked if he has ever had a conversation with the said Onorio Longhi and Orazio Gentileschi concerning the said painting The Resurrection._\n> \n> _Answered:_\n> \n> I've never spoken to Onorio Longhi about the said painting of the Resurrection by Baglione, and it's been more than three years since I've spoken to Gentileschi.\n> \n> _Asked if he knows the painter Lodovico Bresciano and Mario, also a painter._\n> \n> _Answered:_\n> \n> I know a Lodovico Bresciano and a Mario, both painters. This Mario once stayed with me and it's been three years since he left and I haven't spoken to him since. As for Lodovico I've never spoken to him.\n> \n> _Asked if he knows someone called Bartolomeo, who was once a servant of his, and his whereabouts._\n> \n> _Answered:_\n> \n> I know Bartolomeo. He was once my servant who went two months ago to the Castello del Soderino.\n> \n> _Asked if he knows someone called Giovanni Battista, a young man who lives behind the Banchi._\n> \n> _Answered:_\n> \n> I don't know any young man called Giovanni Battista or any young man who lives behind the Banchi.\n> \n> _Asked if he knows how to write verse in the vulgar tongue._\n> \n> _Answered:_\n> \n> Your Excellency, no. I don't dabble in verse either in the vulgar tongue, or in Latin.\n> \n> _Asked if he has ever heard of a poem or composition written in the vulgar tongue in which the said Giovanni Baglione was mentioned._\n> \n> _Answered:_\n> \n> I have never heard it in verse or in prose, in the vulgar tongue, or in Latin.\n> \n> _Asked if he has ever heard of a poem or composition written in the vulgar tongue in which the said Giovanni Baglione was mentioned._\n> \n> _Answered:_\n> \n> I have never heard in verse or in prose, in the vulgar tongue or in Latin, or in any other form, anything where mention of the said Giovanni Baglione was made.\n> \n> _And to his Excellency who said that according to the Bar he was aware that mention had been made of this Giovanni Baglione, and also of the said Mao, in some verses in the vulgar tongue._\n> \n> _Answered:_\n> \n> Never have I received information that mention has been made of Giovanni Baglione or of the said Mao in verse in the vulgar tongue.\n\nCaravaggio's definition of a good painter as 'one who knows how to paint well and imitate natural objects well' is almost comically prosaic. Perhaps he meant it as a deliberately provocative, no-nonsense assertion of his own notoriously direct naturalistic approach. But it is equally likely that he was just playing dumb. Caravaggio knew very well that there was more to painting than the mere reproduction of appearances. But it was not in his interests to appear before the court as an intellectual. After all, intellectuals were the kind of people who might write poetry in their spare time.\n\nCaravaggio's eventual list of _valent'huomini_ was influenced by calculation. Most of those chosen were conservative and academically minded painters. None were his friends, least of all Annibale Carracci, with whom he had crossed paintbrushes in the Cerasi Chapel. He included Federico Zuccaro, who had insulted Caravaggio's Contarelli Chapel canvases in the presence of Giovanni Baglione. Zuccaro was president of the Academy, so no wonder Caravaggio wanted the court to think he thought well of him. This was not merely false magnanimity: it was a shrewd attempted alignment with respectability.\n\nShortly after Caravaggio gave his evidence, Baglione decided to concentrate his attack on Gentileschi. He came back to the court that afternoon with another exhibit for the prosecution. It was an angry letter to Baglione from Gentileschi, written earlier that summer. Baglione had been on a pilgrimage to the shrine at Loreto and Gentileschi had asked him to bring him back some silver figurines of the Madonna. Baglione had given him two figures, but they were in lead, which Gentileschi had taken as a slight. Having explained the background to the court, Baglione then produced the letter:\n\n> To Giovanni the painter,\n> \n> I am not returning your Madonna figurines as you deserve but will keep them for the devotion they represent. However, I consider you a man with just about enough courage to buy them in lead. Your other actions have shown everyone all the riches you are made of and I don't give a hoot about you.\n> \n> I'd like you to do me a favour by hanging some offal on that chain you wear around your neck as an ornament to match your worth. I told you that if you sent me one in silver I would pay you for it. I would never under any circumstances send one in lead to a courteous gentleman, like the ones you see worn on hats.\n> \n> And with this I take leave of you and return your friendship and who is saying this to you cannot be a blackguard.\n\nThis was conclusive proof of Gentileschi's enmity. The robust style of the letter also showed that he had misled the court when he said he could barely string a sentence together on paper. Most incriminating of all, however, was the reference to Baglione's chain. 'In the sonnets written against me he mentions a neck chain saying I ought to wear an iron chain instead,' Baglione told the court. 'In this note he also speaks about the neck chain saying I should hang some offal in place of the chain. I steadfastly insist that it must have been him...'\n\nThe next day, as a final throw of the dice, the examining magistrate recalled Orazio Gentileschi. If he could be made to crack under cross-examination, the case would suddenly be thrown wide open.\n\nAt first the magistrate lulled Gentileschi into a false sense of security, asking him a series of questions about other artists, which Gentileschi parried with ease. It was at this point in the trial that he blithely volunteered the story about Baglione bringing his picture of _Divine Love_ to the annual artists' exhibition, to compete with his own _St Michael the Archangel_. He presumably wanted to demonstrate that he had nothing to fear from a discussion of the rivalry between them. He was happy to admit to the occasional disagreement with Baglione, but he also took care to distance himself from Caravaggio, complaining that both men had a habit of looking down on him:\n\n> I haven't spoken to the said Giovanni Baglione since the St Michael affair and especially because he expects me to raise my hat to him on the street and I expect him to raise his hat to me. Even Caravaggio, who's a friend of mine, expects me to salute him, and although both of them are my friends, there's nothing more between us. It must be six or eight months since I've spoken to Caravaggio, although he did send round to my house for a Capuchin's robe and a pair of wings I lent him. It must be about ten days since he sent them back.\n\nGradually the magistrate turned the conversation towards the incriminating note, although for the time being he kept the document itself up his sleeve. Did Gentileschi ever remember Baglione leaving Rome? Had Baglione ever given anything to him? The accused stumbled slowly but surely into the trap. Gentileschi told the court that Baglione had gone to Loreto and that he had brought him back some lead figurines 'like the ones that are worn on hats'. He had hoped for silver ones, but he had thanked Baglione graciously none the less. What about a note? Had he written a note? Gentileschi pretended to struggle to remember. Baglione had written to him first, as he recalled, saying that he had heard that Gentileschi was complaining about the figurines. He had felt obliged to reply. 'I answered him in a note that I held devotion dear, that I was surprised that he would write these things to me, that I had thanked him in the presence of several people... and that he shouldn't think I was interested in all that silver nonsense.'\n\nAfter a little more sparring the magistrate suddenly produced the note itself, to Gentileschi's evident consternation. At first he tried to deny that he had written the note, then realized that the handwriting was so evidently his that he had better own up to it. But when confronted with the line about hanging offal on Baglione's chain, he panicked and half-heartedly denied authorship of the letter once again. Drowning in his own inconsistency, all he could find to cling to was an implausible insinuation that the letter was a forgery. His testimony rapidly descended into incoherence. 'It seems to me,' he stammered, 'yet it doesn't, that I wrote about offal and chains, but the handwriting looks like mine. I recognize this letter by my handwriting. It's true I wrote about someone who had done something bad and that he face up to it but I don't think I've written about chains and offal.'\n\nAt this point the notary reported, ' _He got confused_.' Again and again the magistrate pressed him for the truth about the letter, but Gentileschi just went round and round in ever-decreasing circles: 'it doesn't seem to be in my hand, but I know I haven't written this note in this manner: it is like my handwriting but I don't know of having written these things... the handwriting looks like mine, but I don't think I've written this letter in this way, but it is my handwriting...' It was not quite a confession, but he must have signed such a testimony with a heavy heart.\n\nBut just as the case seemed to have swung decisively against them, the defendants were reprieved. Someone must have told the Governor of Rome to call off his hounds, because on 25 September 1603 Caravaggio was suddenly released from prison. He was bailed under guarantee from the French ambassador, which strongly suggests that Cardinal del Monte, friend to the Medici and to France, had engineered his release. The condition of bail was that he 'was not to leave his habitual residence without written permission... at the risk of being condemned to the punishment of galley slave'. He was also obliged to make himself available for a further hearing in a month's time. In another document of the same date, Ainolfo Bardi, Count of Vernio, undertook to ensure that Caravaggio would offend 'neither the life nor the honour' of either 'Giovanni Baglione, painter' or 'Tommaso, alias Mau [ _sic_ ]'.\n\nIn the event, there were no further hearings and the case was dropped. But that was not quite the end of the affair. By November 1603 Caravaggio's friend Onorio Longhi was back in Rome. He wanted revenge for the ordeal of the lawsuit and tried to pick a fight with Baglione and Salini. This time, Longhi was the one who ended up in court, arrested for threatening behaviour by a _sbirro_ who signed himself 'Tullio, assistant to the head of police'. The events leading up to the arrest were described by Salini in his deposition to the court:\n\n> I was in the church of Minerva together with my friend Messer Giovanni Baglione. We wished to hear Mass and while we were waiting I saw Onorio Longhi who was standing in front of us staring at me saying something very softly with his mouth that couldn't be heard. Then he beckoned me with his head and I went over and asked him what he was calling me for. He started to say, 'I'd like to make you swing from a wooden scaffold, you fucking grass.' To which I answered that he was insulting me so in church but that outside he wouldn't have dared say such a thing. Then, raising his voice, Onorio told me to come outside and said I was a fucker and a grass if I didn't and that I should come out and he'd be waiting for me.\n> \n> He immediately went out of the rear door of the Minerva and picked up a stone saying, 'Come out, you scum you grass.' Then I told him he was lying through his teeth and that he should put down the brick or we'd be uneven. And then the said Messer Giovanni Baglione came out and held me back and Onorio began to say, 'There are two of you', and a companion of the said Onorio, a procurator from Truffia who lives in Montecitorio, turned towards Messer Giovanni and seeing him with a dagger said, 'Put the dagger down.' Onorio, likewise, said 'Get the dagger off him',... and the said procurator approached him [and punched the said Messer Giovanni in the chest ]... and all at once Onorio threw the brick at Messer Giovanni, which hit his hat but didn't hurt him. Then he turned towards me, but having a stone in my hand I told him to stop or I'd knock him down. All the same he came towards me saying, 'You fucking grass', and so I called him a liar and entered the church and he went off with the said procurator.\n\nShortly after this, Salini added, Longhi challenged him to a swordfight again, 'near the door of the friars' cloister'. When he refused to rise to the bait, Longhi went to wait at Salini's home on the corner of Via della Croce. When Baglione and Salini got there, he shouted at them to arm themselves, but still they refused to fight with him. Salini went to see his tailor and Baglione went into 'the baker's shop that sells bread, wine and charcoal'. (Were it not for this aside in Salini's testimony, it might never have been known that bakers in the artists' quarter of seventeenth-century Rome sold charcoal from their bread-ovens to painters.)\n\nThe tale of provocation ends there, with Salini trying on shirts and Baglione buying artist's materials and the enraged Longhi left yelling in the street. The story as Salini had told it was confirmed by another eyewitness, Lazzaro Visca, a barber. True to his profession, he added one small tonsorial detail to the picture: the infuriated Longhi had a red beard.\n\n#### A DAGGER, A PAIR OF EARRINGS, A WORN-OUT BELT\n\nOnorio Longhi left the custody of the _sbirri_ with a caution as the enmity between rival factions of artists rumbled on. Caravaggio had not been involved in the latest fracas, if only because he was away from Rome at the time. He had left the city shortly after the collapse of the libel trial, to research a new picture he had been asked to paint.\n\nIn early September 1603 the heirs of Ermete Cavalletti had acquired a chapel in the church of Sant'Agostino. Caravaggio was in prison at the time, but immediately after his release Ermete's widow, Orinzia Cavalletti, commissioned him to paint an altarpiece depicting the Madonna of Loreto. Cavalletti's late husband had been particularly devoted to the cult of Loreto, a small town in the Marches, east of Rome, which was home to the fabled Holy House of the Virgin Mary, and during the last year of his life he had organized a pilgrimage to the Holy House, then one of the principal pilgrimage sites in the Roman Catholic world. Caravaggio decided to familiarize himself with the shrine and its legend before starting work on the altarpiece, so he too followed the pilgrims' trail. Orazio Gentileschi may or may not have asked him to pick up a few silver figurines when he got there.\n\nWhile he was on the road, Caravaggio accepted another commission, this time to paint an altarpiece for the Capuchin church of Santa Maria di Costantinopoli in the town of Tolentino. The picture is lost and was nearly forgotten altogether, but in the late nineteenth century a scholar working in the town's municipal archives stumbled on a letter about it. Dated 2 January 1604, it was sent to the priors of the town by a nobleman from Tolentino, Lancilotto Mauruzi, who was living in Rome at the time. He congratulated them on securing Caravaggio's services and wanted them to know that he was 'a most excellent painter, of great worth, in fact the best in Rome today'. He pleaded with them to treat the artist well, because if he created one of his 'extraordinary' paintings for Tolentino it would forever bring honour to the town. The fate of the work is unknown, but it must have disappeared some time after 1772, when the author of a local guidebook rapturously described it as 'a singular and precious production of Knight Michelangelo Amerigi da Caravaggio, in which, in his strong and dark manner, he depicted St Isidora Agricola piercing a tree with a spike and miraculously bringing forth a fountain: the figure is so natural, seeming to be alive, such is the delicacy of the flesh tones... there are also other figures, transfixed by the sight of the miracle: in fact one could say it it is a miracle of art, so true to life does it seem.'\n\nCaravaggio went on to Loreto, for how long we do not know, but he was back in Rome by the beginning of 1604. In the aftermath of the libel trial, his life became increasingly unsettled. Having lodged for several years with powerful patrons, first del Monte and then the Mattei family, he was now living in rented accommodation. Not long after his release from prison he had moved into a small two-storey house in the Vicolo dei Santa Cecilia e Biagio (now the Vicolo del Divino Amore), a narrow curved street flanked on one side by the walls of the Palazzo Firenze. The palace was part of the machinery of Medici power in Rome, serving among other things as the post office for Tuscany, and del Monte was often required to be there in the course of his duties. It seems that despite his changed circumstances the painter was still keen to stay close to his protector.\n\nUnlike many Roman streets, the Vicolo was, and still is, unshaded by any large church or monastery. In high summer the sun beats down on to the roofs and through the windows of the houses there, creating dramatically Caravaggesque effects of light and shade. The painter's new home had a cellar and a small courtyard garden with its own well. His landlady, Prudentia Bruni, charged him the modest sum of 40 scudi a year in rent. His reasons for moving are not known. Perhaps he wanted privacy. He was living alone with his one assistant, according to a communion census carried out for the parish of San Nicola dei Prefetti. The document in question, part of a _Status animarum_ , or report on 'the state of souls', records that 'Michelangelo, painter' had taken communion as required, together with 'Francesco, his servant' \u2013 Cecco, the model for _Omnia vincit amor_ , the source of all those rumours about Caravaggio's homosexuality that would still be current half a century later. There must have been talk, but Caravaggio was paying no attention to it.\n\nAn inventory of Caravaggio's household contents was drawn up when his tenancy ended abruptly in August 1605, an event that will be described below. The inventorist noted the objects in the order in which he encountered them, so the list of things also describes a sequence of rooms. Like many of the documents concerning Caravaggio, the inventory is fascinating but tantalizing. Reading it is like leafing through a dossier of arbitrarily cropped and framed snapshots of the things a man once owned \u2013 the furniture he sat on, the weapons with which he fought, the books he read, the tools he used. But all the photographs are just a little out of focus. Crucial details are missing and there is no one to fill in the gaps:\n\n> This is the inventory of all the personal property of the painter Michelangelo from Caravaggio... First, a kitchen-dresser made of white poplar-wood, with three compartments and an alder frame, containing eleven pieces of glassware, namely glasses, carafes and flasks covered in straw, a plate, two salt-cellars, three spoons, a carving board and a bowl, and on the above-mentioned dresser two brass candlesticks, another plate, two small knives and three terracotta vases. _Item_ a water jug. Two stools. _Item_ a red table with two drawers. _Item_ a couple of bedside tables. A picture. _Item_ a small chest covered with black leather, containing a pair of ragged breeches and a jacket. A guitar, a violin. A dagger, a pair of earrings, a worn-out belt and a door-leaf. _Item_ a rather big table. _Item_ two old chairs and a small broom. _Item_ two swords, and two hand daggers. _Item_ a pair of green breeches. _Item_ a mattress. _Item_ a shield. _Item_ a blanket. _Item_ a foldaway bed for servants. _Item_ a bed with two posts. _Item_ a chamber-pot. _Item_ a stool. _Item_ an old chest. _Item_ a majolica basin. _Item_ another chest containing twelve books. _Item_ two large pictures to paint. _Item_ a chest containing certain rags. _Item_ three stools. _Item_ a large mirror. _Item_ a convex mirror. _Item_ three smaller pictures. _Item_ a small three-legged table. _Item_ three large stretchers. _Item_ a large picture on wood. _Item_ an ebony chest containing a knife. _Item_ two bedside tables. _Item_ a tall wooden tripod. _Item_ a small cart with some papers with colours. _Item_ a halberd. _Item_ two more stretchers.\n\n'Once he put on a suit of clothes he changed only when it had fallen to rags.' Bellori's remark, repeated in one or two other early sources, finds confirmation in the ragged breeches and jacket and the worn-out belt. There is little in the way of cooking equipment, which suggests that neither the painter nor his apprentice spent a great deal of time in the kitchen. It is not surprising to learn that Caravaggio kept books, and a pity that the notary did not list their titles (though if they _had_ been identified, a multitude of over-ingenious iconographic hypotheses would have been unleashed on the painter's work). At any rate, their presence gives the lie to the old academic caricature of Caravaggio as an unlettered copyist of appearances. But we cannot tell anything conclusive from what is here about his relationship with his servant: all the references to beds and bedding occur in one place, so it can be inferred that the two men either shared sleeping quarters, or slept in adjoining rooms. But Cecco certainly had his own separate bed, or mattress.\n\nThe convex mirror is likely to be the one in which Caravaggio had studied his own, distorted self-portrait while painting the _Medusa_. It is probably the same object that can still be seen, propped up on the table, in the _Martha and Mary Magdalen_ in Detroit. The mention of a 'large mirror' is more mysterious: mirrors were extremely expensive in the early seventeenth century, and, given Caravaggio's often-remarked disregard for his own appearance, it is unlikely he would have owned one for reasons of vanity. It was most likely another tool of his studio, which he used to bounce or reflect light, much like a modern cinematographer. The 'large painting on wood' could have been one of the two botched first attempts at the Cerasi Chapel commission.\n\nOne thing is clear from the inventory: Caravaggio was living modestly. For all Baglione's allegations of 'the many hundreds of scudi' that he had 'pocketed' from Ciriaco Mattei, there was not much sign of suddenly acquired wealth in the meagre possessions assembled in the little house on the Vicolo dei Santa Cecilia e Biagio. Caravaggio was a very well known painter by early 1604, but his future was by no means assured. When Clement VIII commissioned a series of altarpieces for St Peter's in these, the last years of his pontificate, Caravaggio was not among the artists approached.\n\nPart of the problem was almost certainly his personality. By now he would have been notorious as a proud and difficult man. Baglione and his clique would have been only too happy to reinforce that impression. But there were other, more powerful forces working against Caravaggio. The Catholic Church was moving decisively away from the severe Counter-Reformation piety embodied so powerfully by his work. The religious attitudes that he had grown up with in Milan were falling increasingly out of favour among those in positions of power. Carlo Borromeo's belief that the princes of the Church should clothe themselves in humility and model their lives on those of Christ's own poor disciples was falling terminally out of fashion. Poverty and the poor were there to be controlled, regulated, put in their place. In parallel, the idea that Christian art should exalt poverty was regarded as increasingly eccentric and distasteful by senior churchmen, from the pope downwards. It was the function of art to hymn the majesty of God in his heaven \u2013 and therefore to bathe the papal court and the upper hierarchies of the Church in the reflected glory of that higher, celestial court. Like the art of Caravaggio, the art favoured by a newly triumphalist Church was aimed at the poor as well as the rich. But its approach was very different. It did not welcome the poor and the meek or make them feel that they, ultimately, were the inheritors of the earth. It was there to awe, daunt and stupefy them, to impress them with visions of a force so powerful it could not be resisted \u2013 and must, therefore, be obeyed.\n\nFor all his sensitivity and genius, there could ultimately be no place in this new Baroque sensibility for an artist such as Caravaggio. If anything, his art was becoming even more pared down, more severe, with the passage of time. This pattern of development, begun with the Cerasi Chapel paintings, had continued throughout 1603. In his evidence given at the libel trial, Orazio Gentileschi had remembered Caravaggio returning 'a Capuchin's robe and a pair of wings'. Cecco had worn the wings when he sat for _Omnia vincit amor_. The Capuchin robe, sacred uniform of the Franciscan order, had been worn by the rather older model whom Caravaggio cast in another picture of the same period, _St Francis in Meditation_ , now in the Galleria Nazionale d'Arte Antica, Palazzo Barberini, Rome. Nothing is known about why, or for whom, this ascetically morbid picture was painted. The saint kneels alone in darkness beside the simple wooden crucifix that assists his meditations. His torn, patched cloak is the symbol of his piety and of his utter disdain for the things of this world. He holds a skull in his hands, staring deep into the sockets where eyes once were. He is shown lost in contemplation of his own mortality, and of the eternal life that awaits him thanks to Christ's crucifixion at Golgotha, 'the place of the skull'. But the idea of the picture is stronger than its execution. The folds of the drapery have been cursorily painted and the penitential pose seems artificial. Caravaggio had always composed with a strong sense of theatre, but here his work tips over into theatricality.\n\nAccording to Gentileschi, Caravaggio had returned the borrowed habit at around the beginning of September 1603, so probably the _St Francis_ was completed shortly before then. _The Sacrifice of Isaac_ is another picture from this time. Less penitentially gloomy, it clothes an Old Testament legend in the same robes of holy poverty. The lined and bald Abraham is cousin to Caravaggio's earlier pauper saints and strongly resembles the uppermost disciple in his _Doubting Thomas_. A simple man of simple faith, he steels himself to do God's bidding, holding his screaming son down as if the boy \u2013 modelled by Cecco in yet another guise \u2013 were just one more lamb brought to slaughter.\n\n_The Sacrifice of Isaac_ had been the subject of a famous competition in fifteenth-century Florence between Lorenzo Ghiberti and Filippo Brunelleschi, for the commission of a cast-bronze pair of doors for the city Baptistry. Each artist had produced a single bas-relief in bronze, which between them became the two most famous Renaissance treatments of the subject. The eventual winner, Ghiberti, had devised a composition of great elegance and delicacy, setting the scene in a gracefully abbreviated landscape and lending a balletic quality even to the murderous gesture of the Old Testament prophet. Brunelleschi had produced a much brusquer and more violent interpretation of the story, focusing on the action itself, where Abraham plunges the dagger dramatically towards his son even as the angel stays his hand. Brunelleschi lost the competition, but unsurprisingly it was with his vivid, violent, essentially late medieval view of the story that Caravaggio identified. Caravaggio's greatest Dutch admirer, Rembrandt, would give a yet more brutal emphasis to the drama, by showing Abraham's smothering hand clamped over the face of his helpless son.\n\n_The Sacrifice of Isaac_ is also notable for Caravaggio's very last view of landscape: an idyllic glimpse of the Roman _campagna_ complete with winding path, avenue of cypresses, a country villa and a distant monastery silhouetted against a fragment of summer sky. From 1604 onwards, his painted world shrinks in on itself, and even events set outside look as though they are taking place in a darkened theatre. Middle tones almost disappear. Increasingly, there is only darkness and light.\n\n#### A PLATE OF ARTICHOKES AND OTHER STORIES\n\nCaravaggio would receive just three commissions for large-scale public religious paintings between 1603 and 1606. As the spareness and solemnity of his work became increasingly out of step with the times, he was forced to watch from the sidelines as lesser painters overtook him in the unstable hierarchy of Roman patronage. In reaction, he became ever more aggressive. An eighteenth-century writer of artists' lives, Filippo Baldinucci, recounts the story of Caravaggio's jealous attack on a Florentine painter called Domenico Passignano. Passignano's sin had been to secure the coveted commission to paint an altarpiece for St Peter's. One day his assistant was alone with the unfinished picture, which was still curtained off from public scrutiny, when Caravaggio went to take a look. Showing 'no respect for place or person', he drew his sword, cut a slit in the fabric and poked his head through the hole. His assessment of the work in progress was predictably caustic: 'As bad as I thought, from a painter like him'. The message was presumably passed on to Passignano by his startled apprentice. The legend was soon absorbed into the collective memory of the Italian artist.\n\nCaravaggio was picking fights with other people too. On 24 April 1604 he got into an argument with a waiter at one of his local restaurants, the Osteria del Moro, or 'Tavern of the Blackamoor'. In the course of an altercation concerning artichokes, he smashed a plate against the man's face. Stopping only to have his wound dressed at the barber-surgeon's, the waiter took his grievance straight to court. He gave his name as Pietro de Fossaccia and declared that he was originally from Lago Maggiore. This is his testimony against Caravaggio:\n\n> At about seventeen hours [half past twelve] the above-named defendant with two other men was eating in the Tavern of the Blackamoor, near the Church of the Magdalen, where I am employed as a waiter. I had brought him eight cooked artichokes, to wit, four cooked in butter and four in oil, and the said defendant asked me which were done in butter and which in oil. I replied: 'Smell them, and you will easily know which are cooked in butter and which in oil.' Thereupon, he flew into a rage and without further words seized an earthen plate and flung it in my face. It hit me here in the left cheek, wounding me slightly. Then he got up and snatched the sword of one of his companions, which was lying on the table, perhaps with intent to strike me. But I got away from him, and came here to the office to file a complaint.\n\nA copyist called Pietro Antonio de Madii, from Piacenza, had also eaten at the Tavern of the Blackamoor that lunchtime. He was called as an eyewitness. He partially corroborated the waiter's story. But, in recalling the exact words that had been exchanged, he shed new light on the incident. The verbal precision of his evidence may have reflected the habits of his work, as a transcriber of others' words:\n\n> I was dining at the Tavern of the Blackamoor. On the other side of the room there was Michelangelo da Caravaggio, the painter. I heard him ask whether the artichokes were done in oil or butter, they being all in one plate. The waiter said: 'I don't know', and picked one up and and put it to his nose. Michelangelo took it badly and sprang to his feet in a rage, saying: 'It seems to me, you fucked-over cuckold, that you think you're speaking to some kind of vulgar provincial [ _barone_ ].' And he seized the plate and threw it at the waiter's face. I did not see Michelangelo grasp the sword to threaten the waiter.\n\nCaravaggio was being touchy about status again. The fight may have been sparked by a question about butter and olive oil, but the argument was really about something else. The painter was accusing the waiter of a quasi-racist insult. The Romans were proud of their olive oil \u2013 Montaigne had remarked on its quality when he visited the city \u2013 and scorned northern Italians for lacking the discrimination to appreciate its fine but faintly bitter taste. Lombards were easily caricatured as cowherds from far-off plains and mountains, who thought a meal was not a real meal unless it was dripping with butter and cheese. The painter accused the waiter of taking him for a _barone_ , which has been imperfectly translated above as 'vulgar provincial'. Its literal meaning is 'baron', but used ironically, in the language of demotic insult, it means the opposite \u2013 a low parody of an aristocrat, somebody from the sticks who thinks he has taste but actually has none.\n\nCaravaggio responded to the implied slight by hurling a plate at the man's mouth. It was an impetuous act, but as usual the painter had put thought into his violence. The punishment mirrored the perceived crime: 'You think I have no taste? Taste this.' The insult that accompanied the assault, _becco fottuto_ , or 'fucked-over cuckold', was the same phrase that had been used at the end of the poem addressed to 'Johnny Bollock'. Caravaggio must have used it fairly frequently.\n\nDespite only three major commissions in these troubled years, Caravaggio certainly had work to do. His altarpiece for _The Death of the Virgin_ was years overdue and he still had the picture of the Madonna of Loreto to paint. But he seems to have found it increasingly difficult to concentrate for prolonged periods. In the past he had been a conscientious respecter of deadlines, but now he had a growing reputation for unreliability. It was with the Caravaggio of these years in mind that Karel van Mander wrote his comment about the painter spending a month in the streets for every two weeks in the studio, swaggering about with his sword at his side, 'with a servant following him, from one ball-court to the next, ever ready to engage in a fight or argument'.\n\nIt was probably in the summer of 1604, between fights, that Caravaggio painted the hauntingly intense _St John the Baptist_ now in the Nelson-Atkins Museum in Kansas City. The picture was almost certainly painted for the Genoese banker Ottavio Costa. There is an early copy in the church of the Oratory of the Confraternity at Conscente, in Liguria, which was a fief of the Costa dynasty. The family had paid for the building of the church, so it may be that Caravaggio's painting was originally destined for its high altar, and subsequently replaced by the copy for reasons unknown. Perhaps Ottavio Costa was so impressed by the work when he saw it that he decided to keep it for his art collection in Rome.\n\nThe picture is very different to the _St John the Baptist_ painted for Ciriaco Mattei a couple of years before. As in the earlier painting, the saint occupies an unusually lush desert wilderness. Dock leaves grow in profusion at his feet. But he is no longer an ecstatic, laughing boy. He has become a melancholy adolescent, glowering in his solitude. Clothed in animal furs and swathed in folds of blood-red drapery, he clutches a simple reed cross for solace as he broods on the errors and miseries of mankind. The chiaroscuro is eerily extreme: there is a pale cast to the light, which is possibly intended to evoke moonbeams, but the contrasts are so strong and the shadows so deep that the boy looks as though lit by a flash of lightning. This dark but glowing painting is one of Caravaggio's most spectacular creations. It is also a reticent and introverted work \u2013 a vision of a saint who looks away, to one side, rather than meeting the beholder's eye. This second _St John_ is moodily withdrawn, lost in his own world-despising thoughts. The picture might almost be a portrait of Caravaggio's own dark state of mind, his gloomy hostility and growing sense of isolation during this period of his life.\n\nOnly one other painting by Caravaggio can be securely dated to 1604. It is _The Entombment_ , a large and ambitious altarpiece for the Oratorian church in Rome, Santa Maria in Vallicella, a few hundred yards west of the Piazza Navona, close to where the Tiber snakes around the Vatican. He finished it some time shortly before 1 September, when the picture is described as 'new' in a document recording that it had been paid for by a man called Girolamo Vittrice. Girolamo commissioned the work for the burial chapel of his uncle, Pietro, who had died in 1600. Like many of the painter's most important patrons, the Vittrice family was closely connected with Filippo Neri's Oratory and therefore directly allied with the emphatically populist, pauperist wing of the Roman Church. Pietro Vittrice had been particularly close to Filippo Neri himself and had strongly supported the core values of the Oratory, with its stress on the importance of charitable works, its antipathy to elaborate ritual and its ambition to revive the simple and direct faith associated with the early Church.\n\nCaravaggio's monumental and dramatic altarpiece for Vittrice's burial chapel was immediately recognized as one of his most accomplished paintings. Baglione baldly stated that 'this is said to be his best work', a judgement that was echoed rather more circumstantially by Bellori:\n\n> One of the best works by Caravaggio is the _Deposition of Christ_ [ _sic_ ] in the Chiesa Nuova of the Oratorians which has received well-deserved praise. The figures in the painting are placed on a stone in the opening of the sepulchre. In the centre Nicodemus supports the Sacred Body under the knees, embracing it, and as the hips are lowered, the legs jut out. On the other side St John places one arm under the shoulder of the Redeemer whose face is upturned and his breast deathly pale; one arm hangs down with the sheet and all the nude parts are drawn forcefully and faithfully from nature. Behind Nicodemus are seen the mourning Marys, one with her arms upraised, another with her veil raised to her eyes, and the third looking at the Lord.\n\nCaravaggio had Michelangelo in mind again when he created _The Entombment_. Pietro Vittrice's burial chapel was dedicated to the Piet\u00e0, the solitary lamentation of Mary over the dead Christ. Caravaggio deliberately harked back to one of the most hallowed images of that earlier event in the story, namely Michelangelo's marble _Piet\u00e0_ in St Peter's. The limp right arm of Caravaggio's dead Christ, with its prominent veins, is a direct paraphrase in paint of the same element in Michelangelo's composition. The flesh of the arm gently bulges over the supporting hand of St John, just as it does over the hand of the Virgin Mary in the marble _Piet\u00e0_. But in Caravaggio's painting, John's hand inadvertently opens the wound in Christ's side. For the pathos and poetry of Michelangelo's sculpture, in which Mary mourns the man she once cradled as a child, Caravaggio substitutes his own intense morbidity. Caravaggio's dead Christ is punishingly unidealized. He truly is the Word made flesh: a dead man, a real corpse weighing heavily on those who struggle to lay him to rest. John strains not to drop the sacred burden. Nicodemus stoops awkwardly as he clasps the body around the knees in a bear-hug, locking his right fist like a clamp around his left forearm.\n\nOnce more, the painter emphasizes the bare feet of Christ and his disciples. Nicodemus's feet, so firmly planted on the tomb slab by the heavy load of the corpse, are veined and creased at the ankle. Christ's feet dangle limply in space. Such details could be controversial elsewhere, but the Oratorians' sense of decorum was evidently undisturbed by Caravaggio's insistence on holy poverty. Christ's drapery has been given strong emphasis, shining with particular force in the darkness of Calvary. His winding sheet dangles below the tomb slab, touching the leaves of a plant \u2013 a juxtaposition perhaps meant to symbolize the hope of new life brought even to the darkness of the grave. Pietro Vittrice had especially venerated the Holy Shroud of Turin, fabled as the winding sheet in which Christ had been interred.\n\n_The Entombment_ is a powerfully sculptural painting. It alludes to Michelangelo's _Piet\u00e0_ but ultimately looks back again to the polychrome _mises-en-sc\u00e8ne_ of the _sacri monti_ and the vivid terracotta sculptures of northern Italian tradition. The figures are tightly grouped, each responding to the tragedy of death in a different way. Caravaggio's Madonna, who has been given a wimple so that she resembles a nun, gazes solemnly at the dead body of her son. The other two female figures are more overtly expressive. Mary Magdalen, eyes uptilted in a trance of sorrow, raises her hands to the heavens. The third Mary bows her head and weeps. Both these figures were modelled by Fillide Melandroni. She had dropped out of Caravaggio's art for a while, but was clearly still part of his life. Her continued presence in Rome at around this time is confirmed by a communion census report of Easter 1603, in which she is recorded as living with her brother and aunt from Siena in the parish of Santa Maria del Popolo. By now she was twenty-two years old.\n\nThere is a hint of theatricality about the pair of figures modelled by Fillide, which jars with Caravaggio's prevailing rhetoric of brutal realism. Perhaps he was trying to sweeten the bitter pill of his art, at least to a degree. There would be hints of compromise in one or two other Roman pictures of the period, a sign perhaps that his confidence had been knocked by rejection. In fact the whole composition of _The Entombment_ has a slightly staged and artificial feel to it, although this may reflect a specific aspect of the image's meaning. There was a tradition of painting Christ's entombment not as a dramatic event but as a moment of votive stasis \u2013 a presentation of his sacrifical body both to the congregation of the church and, symbolically, to all mankind. Caravaggio's Nicodemus looks directly out of the picture as he and John seem, indeed, to hold Christ's body up to view, and Bellori's description of the painting repeatedly refers to the image of Christ as 'the Sacred Body'. This element of the composition may well have been created to complement the actual liturgy of the Mass, as the painting originally hung directly above the altar in the chapel: at the moment when the priest elevated the host, the actual flesh of Christ, he was obliged by the design of Caravaggio's composition to align it with the painted body of Christ.\n\n#### STONES THROWN AND A DOOR DEFACED\n\nOn 4 June 1604 Caravaggio was sentenced for his attack on the waiter, along with several people convicted of unrelated offences. The other men and women in the dock included a furrier, a launderer and a recent convert to Christianity from Judaism. Their various crimes and punishments are detailed in the judicial Latin of the document, but whereas the entry under Caravaggio's name gives the nature of his offence \u2013 wounding a man under the left eye with an earthenware plate \u2013 it fails to specify the penalty. Perhaps he was again let off with a warning, thanks to his powerful friends.\n\nOn 19 October he was back in prison at the Tor di Nona. This time he and some friends were accused of throwing stones at some police officers. The alleged incident had taken place two days earlier, at 9.30 in the evening, on the Via dei Greci. His fellow defendants were Ottaviano Gabrielli, a bookseller; Alessandro Tonti of Civitanova, a perfume-maker; and Pietro Paolo Martinelli, a courier to the pope. Their testimonies are contradictory, but they give at least a piecemeal picture of the incident.\n\nCaravaggio had eaten with Martinelli and Gabrielli that evening at the Osteria della Torretta, the 'Tavern of the Little Tower'. After dinner they had decided to walk to the Piazza del Popolo. They were halfway there when they were arrested for throwing stones. 'We were arrested because a stone had been thrown and they wanted me to tell who had thrown it, whereas I didn't know,' Caravaggio testified. 'I told the constables, \"Go and look for the man who threw the stone, and no more abusive words.\" ' In his version of events, he had been walking along with his old friend Onorio Longhi, as well as the bookseller Gabrielli and someone else whose name he did not know. They had stopped to chat in the street to a girl called Menicuccia \u2013 a nickname for the courtesan Menica Calvi \u2013 when he heard stones flying through the air. He was under the impression that the stones had been thrown at his friends. He insisted that he was just an innocent bystander.\n\nThe pope's courier, Martinelli, distanced himself from the whole affair. He claimed to have walked on ahead with another friend. Ottaviano Gabrielli denied having been present at dinner with the others. He asserted that he was on his way to meet the girlfriend of a friend of his when he got caught up in the evening's events. Gabrielli admitted having been in prison once before, on suspicion of selling prohibited books. On the evening in question, he said, he had witnessed Caravaggio's arrest but had not been involved in the events leading up to it. As the _sbirri_ had taken Caravaggio away, the painter had appealed to Gabrielli for help. The bookseller recalled his precise words: 'Go to the house of the Illustrious Cardinal del Monte and speak to Monsignor the Cardinal del Monte, or to his majordomo... and go to the house of Signora Olimpia Aldobrandini.' Gabrielli had taken the message but had been arrested himself later.\n\nFor his part, the perfume-maker said that he had had nothing to do with any of it. He had never been in prison before, he said. It was unfair. He had only been going for a walk on the Via del Babuino. During his one night in jail, he had clearly been struck by Caravaggio's confidence in his powerful patrons. The perfumier remembered the painter saying, 'Whatever happens, I'll be out tomorrow.'\n\nIt is clear from Caravaggio's testimony that this had been only one of several run-ins with the police in the autumn of 1604. He was accused of using offensive language to the _sbirro_ who had arrested him, Corporal Malanno. Caravaggio answered that the said Malanno had a grudge against him. The policeman was hostile and insulting whenever he bumped into him, the painter complained, but he stoutly denied having called the arresting officer a 'cocksucker' on the night in question.\n\nA month later Caravaggio was stopped again, late at night, walking along a narrow conduit called the Chiavica del Bufalo. The arresting officer filed his report on 18 November 1604:\n\n> Five hours after nightfall at the Conduit of the Bufalo, Michelangelo da Caravaggio, who was carrying a sword and dagger, was halted by my men. When asked if he had a licence, he answered, 'Yes,' and presented it, and so he was dismissed, and I told him he could leave, and said, 'Goodnight, sir.' He replied loudly, 'You can stick it up your arse,' and so I arrested him, since I did not wish to bear such a thing. I ordered my men to take him, and when he was bound he said, 'You and everyone with you can shove it up your arses.' And so I put him in the jail of the Tor di Nona.\n\nAs always the evidence is fragmentary, but what we have of it at the end of 1604 strongly suggests a life going awry. Caravaggio is living in rented accomodation with only his apprentice Cecco for company. He has several commissions but works at them in sporadic bursts. He flares up at the merest hint of an insult. He goes looking for trouble late at night and even manages to pick a fight with the police when they are on the point of letting him go.\n\nWinter came and went with little sign of Caravaggio doing much in the way of work: no pictures from his hand are known from these months. His former rival, Annibale Carracci, had fallen into a deep melancholy after the completion of the Farnese Gallery, so deep that it prevented him from working altogether. In the terminology of the time, Caravaggio was choleric rather than melancholic, but he too seems to have been afflicted by some form of painter's block. By early 1605 his debts had begun to mount up. His rent was in arrears. His landlady, Prudentia Bruni, kept sending him reminders that he studiously ignored.\n\nMeanwhile the city was in a state of political flux. Clement VIII took to his bed in February 1605 and died on 3 March after a short illness. The supporters of the French faction in Rome rejoiced when Alessandro de' Medici was elected as Pope Leo XI, but he was frail and old and soon after his election he too died, on 27 April. Rome was a turbulent city at the best of times, but it was doubly unstable whenever the papal throne was empty. During this interregnum normal government was effectively suspended. According to long tradition, a blanket amnesty was given to the inmates of the city's jails. The felons celebrated their newfound freedom with predictable exuberance. The regular civic authorities tried to maintain their grip on the population, but their jurisdiction was frequently contested at such times by the _caporioni_ , the heads of the city districts.\n\nThree days after the death of the short-lived Medici pope Leo XI, simmering rivalry between the French and Spanish factions broke out into open street warfare. Soon there was a full-blown riot, with fighting spilling over from the Piazza della Trinit\u00e0 into the Via dei Condotti. The _bargello_ of Rome, the city's principal law enforcement officer, tried to restore order with his troops. But he was met, with equal force, by Giovan Francesco Tomassoni, _caporione_ of the Campo Marzio district, his brothers Ranuccio and Alessandro, and their own ragtag militia. The ensuing argument revolved around jurisdiction over prisoners. The _bargello_ wanted to take a number of men into custody, whereas the Tomassoni demanded that the men be handed over to them, for reasons that are not clear: they were either allies, whom the pro-Spanish Tomassoni intended to set free, or they were enemies, sought for the darker purposes of retribution.\n\nThe three Tomassoni brothers ended up in court over the incident. The outcome is unknown, but the testimony given by two eyewitnesses paints a vivid picture of the upheavals in Rome. The first to be called was Lieutenant Antonio Crepella, an officer under the command of the _bargello_ who had been assigned to patrol duties on the day in question:\n\n> Sir, I was with the _bargello_ of Rome, who was leading the entire constabulary, and we were walking slowly around Rome. When we were in Piazza Trinit\u00e0 we saw a large crowd of people towards Via de' Condotti, who were quarrelling and had their swords out. So we hurried there and the people, when they saw us, ran off in all directions. Chasing after them, we caught seven or eight. We then led them off as prisoners towards Tor di Nona by order of the _bargello_... and when we were in the Piazza of Cardinal Borghese [Piazza S. Eustachio, in front of the Palazzo Borghese] the _caporione_ of Campo Marzio, Captain Francesco Tommasoni da Terni, appeared in front of us, along with his brother Ranuccio, and another brother whose name I don't know, but who is older than them, with a great crowd of people who were all from the militia, which Francesco captains.\n> \n> He and his brothers were all three armed with swords, daggers and prohibited pistols. Some of the militiamen were armed with arquebuses, some with halberds, and some with other weapons. Their captain Francesco said to me, 'Hey! What prisoners are these?' I told him they were prisoners who had been fighting in Piazza della Trinit\u00e0, where there had been people hurt, and someone may have been killed. He replied that I should stop, and that he wanted the prisoners himself, and that I should hand them over to him, because they were taken in his _rione_ , and that he wanted to know what was going on, because it was up to him to account for these things. I replied: 'Captain Francesco. Don't get in my way. Let me go, and talk with Captain Girolamo, who is on his way. Don't make [trouble]. These prisoners fought with us, and we can't hand them over to you. Let us take them to prison, then go and talk to the Governor, and get satisfaction.\n> \n> The said Captain Francesco answered me: 'I want you to leave them with us,' and put his hand to the pistol he was carrying, and his brothers also took their pistols, saying, 'Leave them here! Leave them here, or we'll cut you all into pieces, you fucking pricks!' and their militiamen shouted, 'To arms! To arms! Beat the drum!' One of them pointed his halberd at my chest, saying: 'Get away from here! What are you doing here? Get out of here!' Finally, having surrounded us, the said Francesco, his brothers and the militiamen took the prisoners away from the _sbirri_ and led them away themselves. Then they let us go, and I came straight here to the office to give my account.\n\nThe next, unnamed witness, another officer under the command of the _bargello_ , was asked to identify the participants. He could only indicate Francesco and his brothers. His account differs little from that of the lieutenant, until he comes to the aftermath of the event:\n\n> And while we were waiting [at the 'office' where they had gone to report the incident] the _caporione_ Captain Francesco Tomassoni came and said, 'Go to the Heavens. The prisoners are mine.' And we said, 'Take them,' but also that he would have to give us an account of it. Then he said, 'Do us a favour, take us to the militia [Tomassoni's own headquarters].' And so we escorted him to his house in Piazza S. Lorenzo in Lucina. When we were inside we told him to make a list of the prisoners with their names and surnames, and someone who was dressed in long clothes began to write it. While they were being written down, I told him to make out the receipt saying that we had consigned them to him, and he replied: 'I don't want to make out a receipt, etc.' Ranuccio, the _caporione_ 's brother, came towards me and said: 'I'll talk with my brother Gian Francesco, who's here in a house where the prisoners are, etc.' Then the _caporione_ and his people forced us to halt, saying, 'Stop there! Stop there!' and putting their hands to their swords and pointing them at us _sbirri_ and prisoners. I know two of them, one of whom was Captain Ranuccio, and another one, an old man who is his relative.\n\nGiovanni Baglione was also a _caporione_ during the period of the two Vacant Sees. His area of jurisdiction was the district of Castello. Perhaps his civic duties brought him into contact with the Tomassoni clan. Baglione would later describe Ranuccio Tomassoni as an honourable young man, which suggests that they may have been friends. Honourable or not, Ranuccio and his family were certainly well connected in Rome. When his brother Alessandro died later in the year of an unspecified illness, he was accorded the signal honour of burial in the Pantheon.\n\nOn 29 May 1605 Camillo Borghese was elected as Pope Paul V. The new Borghese pope, considerably less severe than his predecessor, allowed the revival of the traditional nepotism of the papal court, ensuring that his nephew Scipione was elected to the cardinalate. The papal nephew loved food and art in equal measure and would soon become an acquisitive collector of Caravaggio's pictures. But private and public domains were very different. The official religious style of the Borghese papacy would be far removed from Caravaggio's simplicity and austerity. For major commissions, the graceful manner of an artist such as Guido Reni was preferred. The ground was being prepared for the soaring majesty of the full-blown Baroque style.\n\nOn the eve of Paul V's coronation, Caravaggio was back in jail. He had been stopped yet again for bearing arms. When he failed to produce a licence for his weapons, he was taken to prison \u2013 not the Tor di Nona this time but the governor's jail. The name of the arresting officer was Captain Pino. His testimony was brief:\n\n> Last night about seven hours after nightfall [3 a.m], as I was on patrol with my constables at Sant'Ambrogio on the Corso, there came a man by the name of Michelangelo, wearing a sword and dagger. Stopped and asked whether he had a licence to carry the said weapons, he said he had not. I had him arrested and brought to jail, and I now make my report, as is my duty, that he may be punished according to justice.\n\nIn the margin of his report, Captain Pino drew a little sketch of the offending sword and dagger. The questioning of Caravaggio followed. The court notary took down his responses and made a note of the outcome:\n\n> I was seized on the street of the Corso in front of the Church of Sant'Ambrogio. It may have been eight hours after nightfall [4 a.m.] for it was light, and I was seized because I had a sword and dagger.\n> \n> I have no written licence to carry a sword and dagger. However, the Governor of Rome had given oral orders to the captain and his corporal to let me carry them. I have no other licence.\n> \n> _He recognized the weapons taken from him by the constables_.\n> \n> _He was allowed to go at large, with three days' time to prepare his report_.\n\nJust what Caravaggio had been doing in the middle of the night is anyone's guess. He is unlikely to have been up to much good. Six weeks later, on 19 July 1605, he was back in the Tor di Nona, having been cautioned for the crime of _deturpatio portae_ , or defacing doors. A woman called Laura della Vecchia and her daughter Isabella lodged the complaint.\n\n_Deturpatio_ was a specific legal term that can be translated as 'house-scorning'. It was invariably a response to a perceived slight or injury. House-scorners generally operated in the dead of night, when they were less likely to be disturbed by the police. They often made a lot of noise, shouting insults or singing lewd songs as a prelude to the vengeful assault itself. Then they would throw stones, damaging shutters and blinds. Sometimes they would also hurl animal bladders filled with blood or ink to leave other visible marks of shame. Excrement was often smeared on to doors and door handles. Doodles were drawn, scurrilous graffiti in the shape of erect phalluses or cuckold's horns.\n\nThe charges levelled at Caravaggio by Laura della Vecchia and her daughter do not specify which of these methods the painter had employed. The wording of the complaint against him suggests that the worst damage was done to the door of the house. That may in itself suggest the nature of the painter's grievance. House-scorning was an almost exclusively male activity, and the most common perpetrators were men whose amorous attentions had been rejected by women. Had Isabella della Vecchia led Caravaggio on in some way, only to change her mind? Had Laura della Vecchia shut the door of her house \u2013 and therefore, metaphorically, the door of her daughter's chastity \u2013 against the infuriated painter? Or perhaps Isabella was just one of the many whores with whom the abrasive Caravaggio mingled, and quarrelled. There is reason to believe that sex, in some form, lay at the root of the argument. As the spring turned to summer in the troubled year of 1605, even the painter's relationships with women were going badly.\n\n#### A CACKLING OF GEESE\n\nCaravaggio did manage to start work on at least one picture in the heat of the Roman summer of 1605: _The Madonna of Loreto_ , commissioned for the Cavalletti Chapel in the Roman church of Sant'Agostino some eighteen months earlier. Caravaggio's painting was clearly shaped by his experience of visiting Loreto and its Holy House, which was said to have flown miraculously from Nazareth to Italy in the Middle Ages, eventually touching down in Loreto one night in December 1294. Protestants, predictably, dismissed the cult of Loreto as a sham. Even the credulity of many devout Catholics was strained by a legend according to which the childhood home of Jesus Christ himself had been aerially projected, by the force of miracle, from Nazareth to an obscure wood in the eastern Marches of Italy. The popularity of the shrine was sustained by its dramatic popular appeal, and by the persuasive rhetoric of its promoters. Louis Richeome's influential tract _Le Pelerin de Lorette_ was originally published in French in 1604, the year before Caravaggio painted his picture. Soon translated into Latin, Italian and a number of European vernacular languages, Richeome's text was a bestseller that brought thousands more pilgrims to the doors of the Holy House.\n\nRicheome placed great emphasis on the miracle of the Incarnation and eloquently made the case for regarding Loreto as the holiest of all holy shrines. The following passage is taken from _The Pilgrime of Loreto_ , the English translation of his book:\n\n> when we shall have reckoned up by name, the most renowned places of all the world, as well out of profane Writers, as out of the sacred Scriptures, the Chamber of _Loreto_ exceedeth them all in this condition, in having been the closet, where the marriage of the Sonne of God with our humane Nature was celebrated in the B. Virgin's womb, the most high and mysterious worke, that the holy Trinity maker of all things, did ever accomplish; for therein God was made man; the Creator, a creature; the supreme cause, an effect; the Word, flesh; the spirit did take a body; the first is become last, and Alpha, Omega...\n\nBy Caravaggio's time there were two basic conventions for depicting the shrine of Loreto. The Madonna and child might be shown sitting on the roof of the Holy House, as they had been said to do during its magic carpet-like flight from Nazareth to Italy. Or the Madonna might simply be shown standing, holding the Christ child, in a pose derived from an ancient cult statue said to have been carved by St Luke himself that was housed on the altar of the shrine.\n\nDeparting from the limited conventions of existing Lauretan imagery, Caravaggio depicted two poor modern pilgrims kneeling at the entrance to the famous shrine. They are husband and wife, or perhaps mother and son. They have come in all humility, as every pilgrim was advised to do, to pray to the Queen of Heaven. Their feet are bare and dirty, their clothing begrimed, patched and poor. They have been rewarded for their honest piety and their weeks on the pilgrimage trail with a vision. The Virgin has chosen to appear to them, in the very doorway of the Holy House of Loreto itself. The infant Christ appears with his blessed mother, clasped in her arms, a finger of his right hand raised in the gesture of benediction. Haloed by a filigree circle of gold, Mary cranes her neck towards the pilgrims, as if to make sure that she catches every last word of their prayers.\n\nIn Caravaggio's time, it was the custom for pilgrims to enter Loreto barefoot, wearing simple clothes. Their immediate destination was the simple dwelling of the Holy House itself, which, like the modest barn of Francis of Assisi's first church, had been shoehorned into a splendid marble architectural casing, itself contained within the vast nave of a later cathedral. Once arrived, the pilgrims were to circle the holy dwelling three times, on their bare knees. Having made this slow crawl towards the hope of salvation, they were finally allowed to enter the shrine.\n\nAll this is the implied prelude to Caravaggio's gentle fantasy of a painting. The work is a _tour de force_ of naked religious populism: spare to the point of banality, blatant in its appeal to the masses. The gratification that it offers is instant, the idea that it embodies too good to be true. It is the realization, in art, of every pilgrim's dream. At the end of the barefoot, knee-scraping journey, a vision. The door to the Holy House has become the door to Heaven itself. The two weary pilgrims are greeted by the Virgin and Child and implicitly welcomed towards another, better place. They will have no further need of their walking sticks, now they have come this far.\n\nSuch is the sheer directness of its appeal to popular piety, _The Madonna of Loreto_ has often been regarded as something of an embarrassment \u2013 a saccharine, sentimental picture, the only work in Caravaggio's entire _\u0153uvre_ with something of the chocolate-box about it. But in its time it was unusual and daring. No artist had ever given such prominence, in a major religious altarpiece, to two such nakedly proletarian figures as the pair of kneeling pilgrims.\n\nThere was an old tradition of including portraits of men and women who had paid for certain altarpieces within the work themselves. Such donor portraits, as they have become known, often place the kneeling figures of such pious benefactors to either side of the Virgin and Child. They are included within the scene, yet they are also apart from it, witnesses rather than participants. In _The Madonna of Loreto_ , Caravaggio turned this convention on its head, first by making the kneeling figures central to the sacred story (the story's catalyst, even, since it is their faith that has called forth the vision of the merciful Madonna and child), and secondly by depicting them not as wealthy donors but as poor pilgrims who have circled the shrine at Loreto three times on their bare knees. The man's filthy naked feet, turned towards the viewer, emphasize this shockingly complete inversion of an old pictorial tradition.\n\nWhat might the true donors of the picture, the Cavalletti, have made of all this? Might they not have been disconcerted by Caravaggio's substitution of their images by those of the two poor pilgrims? It would only have required a relatively minor adjustment to the picture for the normal proprieties to be observed. He could easily have painted the standing Madonna and Child with the kneeling figures of Ermete and Orinzia Cavalletti to either side, in the manner of traditional donor portraits. Yet he did not, and no such alteration was asked of him.\n\nErmete Cavalletti was of course dead by the time Caravaggio finished _The Madonna of Loreto_. But he would most likely have approved of the painter's innovations. Ermete's dedication to the Santissima Trinit\u00e0 dei Pellegrini is proven: as a member of that lay confraternity, he, a rich man, had abased himself in imitation of Christ and washed the feet of poor pilgrims. Caravaggio's painting no less dramatically asserted the pauperist values of that institution. In fact the painter might be said to have repeated that act of self-abnegation, on Cavalletti's behalf, by putting poor pilgrims in place of his rich patrons. The replacement may even imply a kind of wishful metamorphosis, with the kneeling pilgrims as metaphorical portraits of Ermete and Orinzia Cavalletti themselves \u2013 transformed, through their humility of heart, into honorary members of the blessed poor.\n\nWhether that too was part of Caravaggio's meaning, there is every indication that the family approved wholeheartedly of his picture. Not only was it accepted without demur and without alterations, but Orinzia Cavalletti arranged for her own burial beneath the floor of the same chapel.\n\nOnce again, Caravaggio had painted a monumental altarpiece aimed squarely at the poor and the hungry. The location of the church for which he painted the picture was also part of its message and part of its significance. With the completion of _The Madonna of Loreto_ , Caravaggio now had major works on display in two of the most frequently visited churches on the principal pilgrimage axis through northern Rome. Every year wave after wave of pilgrims would enter the city from the north at the Porta del Popolo. Immediately on their left was the church of Santa Maria del Popolo, for which Caravaggio had painted _The Conversion of St Paul_ and _The Crucifixion of St Peter_. The main pilgrimage route from there towards St Peter's then led directly along the Via di Ripetta and its continuation, the Via della Scrofa, to the corner of the Via dei Coronari. Turning right on to that street, in the direction of the Tiber and the Ponte Sant'Angelo, the pilgrim would find himself in one of the most congested thoroughfares in all Rome. The church of Sant'Agostino lay at the start of the Via dei Coronari, so named after its multitude of Rosary-makers' shops, thronged by pious tourists buying Rosaries and other devotional souvenirs of their visit to the Eternal City. Caravaggio knew that he was guaranteed a vast audience of the pious and the humble by virtue of Sant'Agostino's prominent place on the city's Christian itinerary. To the pilgrims who entered the church and walked into the Cavalletti Chapel, he offered a perfected mirror image of their own travels, one in which they could see themselves reaching the wished-for end of every pilgrim's journey.\n\nIt was this direct appeal by Caravaggio to the poor, and the central role he gave them in his theatre of Christianity, that most shocked his critics. Writing from the perspective of the later seventeenth century, when the pauperist ideals of the early Counter-Reformation lay in ruins, Bellori cast Caravaggio in the role of a seditious revolutionary. With pictures such as _The Madonna of Loreto_ he had opened a Pandora's Box of vulgarity: 'Now began the imitation of common and vulgar things, seeking out filth and deformity, as some popular artists do assiduously... The costumes they paint consist of stockings, breeches, and big caps, and in their figures they pay attention only to wrinkles, defects of the skin and exterior, depicting knotted fingers and limbs disfigured by disease.'\n\nBellori's disgust for Caravaggio's 'popular' art, his lazar-house realism, was echoed by Giovanni Baglione. Unlike Bellori, Baglione was a contemporary of Caravaggio, and had gone to see the picture soon after it was installed. His predictable dislike of the work was only intensified by the huge crowds that it drew: 'In the first chapel on the left in the church of Sant'Agostino, he painted the Madonna of Loreto from life with two pilgrims; one of them has muddy feet and the other wears a soiled and torn cap; and because of this pettiness in the details of a grand painting the public made a great fuss over it.'\n\nThe word Baglione used for public was _popolani_ , which specifically denoted the lower classes: peasants, _hoi polloi_. To convey the kind of fuss they made over the picture, he used _schiamazzo_ , which means a din of chattering, but can also be used to describe the cackling of geese.\n\nBellori and Baglione represented the values of the academy, of idealized classical style. But they spoke not only for a particular notion of decorum in art: they spoke also for power and for wealth, and for forms of religious art that spoke down to, rather than for, the mass of Christian believers. Caravaggio had not painted _The Madonna of Loreto_ for them. He had painted it for the _popolani_ , and whether they cackled like geese or not, the _popolani_ took it to their hearts. Not for nothing is the picture commonly known by its 'popular' title \u2013 which is, simply, _The Madonna of the Pilgrims_.\n\n#### LENA WHO STAYS ON HER FEET IN THE PIAZZA NAVONA\n\nPrecisely when Caravaggio finished and delivered the altarpiece to Sant'Agostino is unknown. It may not have been until the autumn of 1605, or even later: he was probably still working on the picture at the end of July, but could have done no work on it at all in August, because for the whole of that month he was again in trouble with the law.\n\nOn 29 July 1605 a junior notary called Mariano Pasqualone accused Caravaggio of assault and grievous bodily harm. The young man arrived, still bleeding, in the legal offices of a certain Paolo Spada, where a clerk of the criminal court took his statement under oath:\n\n> I am here in the office because I have been assaulted by Michelangelo da Caravaggio, the painter, as I am going to relate. As Messer Galeazzo and I \u2013 it may have been about one hour after nightfall [8.30 p.m.] \u2013 were strolling in Piazza Navona in front of the palace of the Spanish ambassador, I suddenly felt a blow on the back of my head. I fell to the ground at once and realized that I had been wounded in the head by what I believe to have been the stroke of a sword. As you can see, I have a wound on the side of my head. Thereupon, the aggressor fled.\n> \n> I didn't see who wounded me, but I never had disputes with anybody but the said Michelangelo. A few nights ago he and I had words on the Corso on account of a woman called Lena who is to be found standing at the Piazza Navona, past the palace, or rather the main door of the palace, of Messer Sertorio Teofilo. She is Michelangelo's woman. Please, excuse me quickly, that I may dress my wounds.\n\nAfter Pasqualone's departure from the office, his companion, Galeazzo Roccasecca, who gave his profession as a writer of apostolic letters, added his own witness statement:\n\n> I saw a man with an unsheathed weapon in his hand. It looked like a sword or a small pistol. He turned round at once and made three jumps and then turned towards the palace of the Illustrious Cardinal del Monte, which was nearby down the little street where we were. He wore a black cloak on one shoulder only. I said to Messer Mariano, 'What is it? What is it?' and he replied to me, 'I have been assassinated and I am wounded.' I saw that he had a wound in the head and he said, 'I am assassinated... it could not have been anyone other than Michelangelo da Caravaggio.' And that is the truth.\n\nSome seventy years later Giambattista Passeri wrote a long and circumstantial account of what might have been behind the trouble between Caravaggio and Pasqualone. Passeri was a painter, poet and author of artists' lives, who had clearly been told some version of the story while he was doing his research in the artists' studios of Baroque Rome. Having applied a liberal coat of literary polish to the original anecdote, he included it, as an entertaining diversion, in the first edition of his life of the painter Guercino:\n\n> In the first chapel to the left of the entrance in S. Agostino, Caravaggio painted the Holy Virgin with the Child in her arms and two pilgrims adoring her. At that time he lived in the House of the Eight Corners, in one of the little streets behind the Mausoleum of Augustus. Nearby lived a lady with her young daughter, who was not at all unattractive; they were poor but honest people. Michelangelo wished to have the young girl as a model for the Mother of God which he was to paint in this work, and he succeeded in this by offering them a sum of money which was large enough, considering their poverty, to enable him to carry out his wish.\n> \n> This girl was being courted by a young man who was a notary by profession and who had asked the mother for her daughter's hand in marriage. However, he had always received a negative answer because this simple and naive woman was unwilling to give her daughter to a notary since, as she said, all notaries are surely bound for damnation. The young man was indignant at this refusal, but he nevertheless did not lose track of his beloved. Thus he found out that she frequently went to the house of Caravaggio and remained there for long periods of time posing for him.\n> \n> Full of jealousy and totally enraged, he contrived to meet the mother one day and said to her, 'My good woman, you're so scrupulous and such a good guardian, and here your lovely daughter, whom you refused to let me marry, goes to this miserable painter so that he can do anything he likes with her. Really, you have made a wise choice and one which is worthy of your class, refusing to let her marry a man like me so that you can make her the concubine of this scoundrel. Now you can just keep her and I hope it will do you a lot of good.' Then turning his back, he left her confused and completely upset.\n> \n> It seemed to this lady that she had inadvertently done the wrong thing by taking her daughter to Michelangelo, even though she had done so in perfectly good faith, and it also seemed that this notary had good reason, at least from his point of view, for treating her so badly. She went immediately to Caravaggio in tears and complained about what had happened on his account. He smiled bitterly at this accusation and asked her who it was that had so unjustly mistreated her. From her description he easily recognized him as a person whom he frequently met in the street. He consoled the lady with gentle words and sent her home.\n> \n> He was upset by this incident and, being by nature irritable and violent, the next morning he put a hatchet under his coat and went out to look for this young man. This being Wednesday, market day, he carried the affair right into Piazza Navona, just when a fair was being held there. It took place in front of the church of S. Giacomo degli Spagnoli, near the Triton Fountain. He went up to the notary and gave him such a terrific blow on the head with the hatchet that he fell to the ground unconscious and covered with his own blood. And Michelangelo said, 'Now learn to behave yourself if you don't know how.' After this misdeed he took refuge at San Luigi dei Francesi and remained there for a long time. Fortunately for Caravaggio, the notary did not die from the blow, even though he was unconscious and for a long time remained ill. It was some years before they settled their feud and the indemnity.\n\nIn several striking respects, Passeri's account is impressively close to the witness statements given in the immediate aftermath of the attack. Passeri gets Pasqualone's profession right, correctly specifies the place where the attack occurred, describes it as a surprise assault on an unarmed man. So even though his 'poor but honest' female characters sound like the heroines of a fairy story, Passeri's assertion that the young lady at the centre of the dispute was Caravaggio's model is worth taking seriously.\n\nPasseri's description of the two women as virtuous creatures from the world of fable also marks the one serious fault-line between his account and the original witness statements. Although brief, Pasqualone's description of the mysterious Lena suggests that she was actually a prostitute. Two very precise phrases in his testimony may shed light not only on this specific incident, but also on the whole vexed question of Caravaggio's nocturnal existence \u2013 the other life that he pursued, so vigorously, in the shadows of the city.\n\nPasqualone's exact words in Italian when first describing the girl were ' _una donna chiamata Lena che sta in piedi a Piazza Navona_ ' \u2013 literally, 'a woman called Lena who stays on her feet in the Piazza Navona'. This way of describing a woman who can always be found standing in a certain place carries an insinuation; the phrase is still current Italian slang for a streetwalker, a whore. There was in fact a known prostitute called Lena Antognetti working in the area at around this time, who was arrested in Piazza Catinara, the present-day Piazza Cairoli, opposite the church of San Carlo ai Catinari, on the night of 1 November 1604. She was apparently on her way home but was stopped for being out after curfew.\n\nThe young notary's second remark about Lena is even more intriguing. He baldly says ' _e donna di Michelangelo_ ', which literally means 'she is Caravaggio's woman', but implies a particular form of possession. Pasqualone pointedly does _not_ describe Lena as ' _la donna di Michelangelo_ ' but as ' _donna di Michelangelo_ ': not ' _the_ woman of Caravaggio' but simply 'woman of Caravaggio', a phrase that objectifies her and carries the suggestion that she is one of several such women. Pasqualone might simply have been saying that Lena was one of several prostitutes frequented by Caravaggio, but it is also possible that he meant to imply that she was one of several prostitutes _controlled_ by Caravaggio \u2013 and that the painter, therefore, was a part-time pimp.\n\nPasqualone's remarks offer an explanation for much of Caravaggio's seemingly random nocturnal escapades and unpredictable behaviour. His life becomes no less violent, but more logical. Caravaggio certainly used whores as models. He painted Fillide Melandroni and, quite possibly, Fillide's friend Anna Bianchini. He painted Lena the streetwalker, and according to Giulio Mancini, who knew Caravaggio well, at least one other prostitute modelled for him in Rome. Perhaps he and his friends just happened to know a lot of whores and courtesans \u2013 after all, such women tended to move in the same circles and live in the same places as painters, sculptors and architects. But it is conceivable that there was more to it than that. Caravaggio needed women to model for him, so rather than be at the mercy of pimps for a reliable supply of girls, why not secure his own small team of whores? He would get free use of female models, which was by no means otherwise easy to arrange. He would not be beholden to anybody, which always made him uneasy. He would earn a bit of extra money on the side, and there would have been some free sex. For their part, the prostitutes would get their own livelihood, and a formidable protector.\n\nCaravaggio used his contacts in high places to ensure that he could carry a sword and a dagger with impunity wherever he went. If he could not always actually produce a licence, he could usually count on Cardinal del Monte, or his majordomo, to get him out of trouble. Maybe one of the reasons Caravaggio went about armed the whole time was that when he was out on the street, he was also out on duty, looking out for 'his' women.\n\nMany of the known incidents involving him lend at least circumstantial support to the hypothesis. What kind of argument over a whore could have led to a sudden, brutal assault of the kind perpetrated by Caravaggio on Pasqualone? The romantic answer is that both men were in love with the girl. But if that were so, a duel would have been the solution. It would have been a matter of honour, whereas Caravaggio treated his victim with a calculated show of contempt. If, on the other hand, Lena was one of Caravaggio's prostitutes, the shameful attack from behind becomes easier to explain. Pasqualone was perhaps a client who had not paid, or had mistreated the girl in some way, so Caravaggio took his revenge publicly, sending out a clear message to anyone who might be watching.\n\nSeen in this light, many of the smaller or more puzzling details to emerge from the painter's criminal record suddenly come into sharper focus. He is often seen out and about, carrying a sword, in the small hours of the morning. He attacks the house of two women, who have annoyed him in some way that seems to relate to sex. On the evening of the stone-throwing, he stops in the street to chat with Menicuccia, a whore whom he clearly knows well. All of this is consistent with the behaviour of a pimp.\n\nThe enmity between Caravaggio and Ranuccio Tomassoni, soon to reach its climax, may have been in some way territorial: certainly Tomassoni was a pimp himself. Caravaggio painted one of Tomassoni's girls, Fillide Melandroni, and, having got her to model for him, perhaps he also tried to persuade her to work for him.\n\n#### THE CASE OF THE DAMAGED CEILING\n\nAt the end of July 1605, concurrently accused of the assault on Pasqualone and the _deturpatio_ of Laura and Isabella della Vecchia, Caravaggio skipped bail and fled to the coastal city of Genoa. He probably took letters of introduction with him from some of his patrons and protectors in Rome. Ottavio Costa and Vincenzo Giustiniani, both enthusiastic collectors of Caravaggio's work, had strong links with the city, as above all did the Colonna family, his protectors since boyhood. The Marchesa Costanza Colonna was living in Rome, at the Palazzo Colonna, between 1600 and 1605. The Colonna family had intermarried with one of the great Genoese families, the Doria. As soon as Caravaggio got to Genoa, he sought out one of the marchesa's relations, Prince Marcantonio Doria, who, although it came to nothing, offered him a prestigious commission.\n\nCaravaggio spent the best part of a month away. On three separate occasions between 3 and 19 August, a Roman court notary reported his failure to attend hearings in the case brought by Laura della Vecchia. Repeated summonses were addressed to him and he was eventually fined for contempt. Meanwhile, his movements were being carefully tracked by Fabio Masetti, an agent in Rome working for Cesare d'Este, Duke of Modena.\n\nMasetti was keeping a close watch on Caravaggio in the summer of 1605 because he was trying to get a picture out of him. Earlier in the year Cesare d'Este had conceived the idea of staging another pictorial competition between Caravaggio and Annibale Carracci. This rematch of the Cerasi Chapel contest of 1601 was to have been staged for a 'Chapel of the Madonna' in the newly renovated ducal castle in Modena. The idea was that each artist should paint a scene from the life of the Virgin. Carracci was to paint the altarpiece, Caravaggio a single canvas for one of the side walls. But there were problems from the outset. Carracci was crippled by depression, and Caravaggio had good reason to dislike the terms of the commission: his picture was to be considerably smaller than his rival's, and in March 1605 another of the duke's agents in Rome, Attilio Ruggieri, reported that Caravaggio was trying to wriggle out of the assignment altogether, truculently remarking that the duke would be better off hiring a miniaturist to paint such small figures. Caravaggio's fee was to be just 50 or 60 scudi, compared to the 200 offered to Carracci.\n\nIn the end he had stuck with it, and by the summer responsibility for handling the commission had passed from Attilio Ruggieri to Masetti, whose letters from Rome to Modena are a paper trail of mounting frustration. On 17 August, Masetti told the duke that it was impossible to force a painting out of the depressive Carracci, saying that there was nothing for it but to accommodate the painter's strange 'humour'. Meanwhile, he added, 'Caravaggio is in contempt of court, and is to be found in Genoa.' On 20 August, Masetti reported that Carracci continued to be utterly intractable, but that efforts were being made to secure Caravaggio's return to Rome: 'a settlement is now being negotiated for Caravaggio; as soon as it's concluded, I'll be on his back.' Perhaps del Monte was still busy on Caravaggio's behalf \u2013 Galeazzo Roccasecca had hinted as much in his testimony for Pasqualone at the end of July.\n\nMasetti's next letter, of 24 August, shows that the Este agent was now trying to put pressure on Caravaggio through del Monte:\n\n> When I heard that Caravaggio had appeared in Rome in hope of a settlement, I petitioned Cardinal del Monte to command him to despatch Your Highness's painting, which he promised me would be ready quickly, though one can't rely on [Caravaggio]. It is said that he is funny in the head [' _e uno cervello stravantissimo_ '; literally, 'he is a very extravagant brain'] and also that Prince Doria sought to have him paint a loggia for him [in Genoa] and wanted to give him 6,000 scudi for it, but that he didn't want to accept, though he had almost promised. It occurred to me to sound out whether, under these circumstances of his non-attendance [in Rome], he would be happy to move there [to Modena], where he could have given every satisfaction to Your Highness. But seeing that he is so unstable I have done no more.\n\nCaravaggio's refusal of a prince's ransom for the small task of decorating a loggia struck Masetti as typically capricious. But the painter had never learned to work in fresco, so he could not have accepted the commission even if he had wanted to. Besides, he had business to attend to in Rome. Apart from anything else, he needed to arrange some more modelling sessions with Lena and finish off the overdue _Madonna of Loreto_.\n\nCaravaggio was indeed back in Rome a week or so before the end of the month. On 26 August he signed a judicial peace with Mariano Pasqualone. Damages had probably been paid, although the legal conventions governing such documents made it sound like a gentleman's agreement: 'the above-named parties, exhorted and persuaded by mutual friends, determined to make peace as befits good Christians...'\n\nIn exchange for a pardon from the Governor of Rome, Caravaggio put his name to the declaration:\n\n> I, Michelangelo Merisi, having been insulted by Messer Mariano, clerk of the Vicar's Court, as he would not wear a sword in the daytime, resolved to strike him wherever I should meet him. One night, having come upon him accompanied by another man and having perfectly recognized his face, I struck him. I am very sorry for what I did, and if I had not done it yet, I would not do it. I beg him for his forgiveness and peace, and I regard the said Messer Mariano with a sword in his hand as a man fit to stand his ground against me or anybody else. I, Michelangelo Merisi, do affirm all the above.\n\nPasqualone's lawyers must have insisted on some of the more humiliating phrases in this fulsome apology. Did Caravaggio sign it through gritted teeth? Or did he simply regard it, phlegmatically, as a means to an end? Assault with a lethal weapon was a serious crime. He had been let off lightly. Intriguingly, the judicial peace was actually signed at the Palazzo Quirinale, in the antechamber of the papal nephew, Scipione Borghese. It is possible that the new Borghese cardinal had helped to arrange Caravaggio's truce with Pasqualone. It was at around this time that Caravaggio's darkly penitential depiction of _St Jerome Writing_ entered Scipione Borghese's collection. Perhaps the work was a gift, in recognition of a favour received. It is a strikingly sombre painting. The wizened and emaciated figure of Jerome sits in semi-darkness, writing in a great book. His deeply shadowed face and the bald dome of his head are modelled so severely, in chiaroscuro, as to resemble the skull that lies on the desk before him as a _memento mori_. It is a morbid visual rhyme.\n\nGetting on with his life turned out to be more difficult than Caravaggio might have hoped. While he was away in Genoa, his infuriated landlady had taken advantage of his absence to seize his possessions in lieu of rent, and change the locks on the house in Vicolo dei Santa Cecilia e Biagio. This occasioned the inventory of its contents, which was made on 26 August (see p. 271). 'Two large pictures to paint' were included on the list. Perhaps one of them was the half-painted canvas of _The Madonna of Loreto_ , still in his studio but now frustratingly inaccessible. The other, most probably, was the incomplete and long-overdue _Death of the Virgin_ , which had been commissioned by the jurist Laerzio Cherubini, to serve as the altarpiece for a chapel that he had acquired in the church of Santa Maria della Scala in Rome, all the way back in 1601.\n\nCaravaggio decided to vent his rage on the landlady who had locked him out with another _deturpatio_. Just four days after he had signed his peace with Mariano Pasqualone, he was being prosecuted yet again, for throwing stones at her windows. The attack took place in the small hours of 1 September and later that day Prudentia Bruni was airing her grievances in court:\n\n> Last night at about the fifth hour [1 a.m.], the said Michelangelo came and threw so many stones at the shutters of my windows that he broke them all down one side, as Your Lordship sees.\n> \n> [The notary adds: _then she showed the wooden shutter broken in one part, and also some stones that were in the said window, which was noted down as evidence_.]\n> \n> And a little after this he came back with some others, playing a guitar; they stopped on the corner of the alley, and he talked with his companions, but I couldn't hear the exact words they were saying.\n> \n> The said Michelangelo did this because he rents a house of mine, which is beside my [own] house. Some days ago he wounded a notary of the Vicariate and left. I was owed rent for six [the notary adds: _correction, four_ ] months, and he had broken a ceiling of mine in the said house, so I had obtained a mandate... to take the things that were left in the house, giving a security in the form of the deposit, which I did. Because of this he broke my shutters in order to spite me. There were three others in company with him. So I am making this complaint, and demand that they be punished in conformity with justice...\n\nFour days later the magistrate examined two of the landlady's neighbours, a woman called Francesca Bartoli and a lady called Lucretia, who was the widow of a certain Ferdinando, from Perugia. Each separately denied that she had seen or heard a thing on the night in question. But as Caravaggio's recent neighbours, they of course knew what he was like.\n\nPrudentia Bruni had referred to a damaged ceiling in the rented house. Might it have been caused by the painter's unorthodox working methods? Caravaggio's proto-cinematographic fondness for powerfully directed downlighting must have involved some ingenious studio set-ups. As we have seen, Bellori wrote that he placed 'a lamp high so that the light would fall straight down'. Sandrart echoed Bellori's remark, saying that Caravaggio liked to work in a dark space with a single source of light from above. In practice this might have involved a powerful flame, perhaps a torch made of pitch, the light from which might have been directed by the use of one of the mirrors in the studio. That would certainly have been enough to char the studio ceiling. Or Caravaggio may simply have taken advantage of the powerful, raking sunlight beating down on the roofs of the houses in his street. Perhaps he had got the effect he wanted just by blacking out his windows and smashing a hole in the ceiling to let the sunlight in.\n\nThe painter's latest prosecution did nothing to lighten the mood of Cesare d'Este's agent. On 7 September he sent yet another gloomy missive to his master in Modena. Neither of the two required pictures had even been started. There was no hope of getting anything from Annibale Carracci, who was completely incapable of working because of his depression. As for Caravaggio: 'last Saturday, his contempt of court for wounds given to a notary was settled, but now there's some other affair he's involved in.'\n\nFive weeks later, on 12 October, in a desperate bid to stir Caravaggio into action, Masetti gave him an advance of 12 scudi. By that time the painter had managed to find a new place to live, and had probably retrieved his possessions from Prudentia Bruni (though we do not know how the case between them was settled). He had moved into the house of Andrea Ruffetti, a lawyer with an interest in art and literature, in the Piazza Colonna, almost next door to the palace of Caravaggio's very first protector, the Marchesa di Colonna.\n\nThe new lodging brought about no more ordered a life. Towards the second half of October, Caravaggio was injured in a fight with a person or persons unknown. The affair was serious enough to warrant investigation. But when the law came calling on him, the bedridden painter was decidedly incommunicative. The investigating officer's report is dated 24 October 1605:\n\n> I, the notary of warrants, etc., visited the painter Michelangelo Caravaggio who was lying in bed in the house of Sr Andrea Ruffetti in Piazza Colonna, wounded in the throat and left ear. Because of the bandages placed on it, this wound could barely be seen, but it is noted here. He was sworn to tell the truth and interrogated by me as to where, by whom and for what reason he was wounded. He replied: 'I wounded myself with my own sword when I fell down these stairs, I don't know where it was and there was no one there.' Although I exhorted him several times to tell the truth, he replied, 'I can say no more.' And I got no other response from him.\n\nThroughout the winter, the Este agent continued to chase Caravaggio for the duke's painting. On 5 November 1605 he reported that 'Caravaggio says the picture is almost ready and that he needs money; I replied that once the appointed thing has been done there will be no want of the money.' But by 16 November, Masetti had capitulated to Caravaggio's demands. On that day he noted paying the artist another 20 scudi 'because the painting will definitely be finished by this coming weekend'.\n\nBut it was not ready by the coming weekend. Nor the next. Nor the one after. After a long silence Fabio Masetti wrote one more exasperated letter on 18 January 1606: 'I have given Caravaggio 32 scudi for this thing. He goes red when he sees me.' Cesare d'Este never would get his painting.\n\n#### 'SO MUCH TROUBLE'\n\nCaravaggio is unlikely to have felt any regret over the unfinished and most probably unstarted picture for the ducal palace of Modena, or any embarrassment at taking money for it. By the beginning of 1606 he had already begun work on a far more prestigious commission. He had finally been asked to paint an altarpiece for St Peter's, the central church of Catholic Christendom.\n\nThe wheels of papal influence had turned in the painter's favour. Scipione Borghese, pleased with his new picture _St Jerome_ Writing, had praised Caravaggio to the pope. According to Bellori, the papal nephew personally 'introduced Caravaggio to Pope Paul V, whom he portrayed seated, and by whom he was well rewarded'. That portrait is lost, although a hamfisted copy of it still survives in the collections of the Palazzo Borghese. With papal favour came papal preferment. Paul V had plans for St Peter's. Caravaggio now became part of them.\n\nIn September 1605 the new pope had ordered the final demolition of the ancient nave of Old St Peter's, which still survived beneath the great dome of the new cathedral begun by Bramante and completed by Michelangelo. Seven altars lost in the destruction of the old basilica were moved to the new transept. One of these was owned by the Confraternity of the Palafrenieri, or 'papal grooms', whose patron saint was Anne. By the end of October the members of the confraternity had resolved to commission an altarpiece for their 'altar of St Anne in St Peter's'. Within a month they had been steered towards Caravaggio.\n\nOn 1 December 1605 Antonio Tirelli, deacon, gave the painter a down payment of 25 scudi. Caravaggio would receive only another 50 scudi in total, a low fee for such an important work, but the painter was in no position to bargain. He was recently evicted, deeply in debt, scarred by a swordfight and in trouble, yet again, with the law. The commission must have seemed like a God-given chance for him to paint his way out of trouble. The altarpiece was finished and delivered in less than four months.\n\n_The Madonna of the Palafrenieri_ , sometimes known as _The Madonna of the Serpent_ , is an unsettling picture. Monumental in scale, almost ten feet tall and more than six across, it shows three figures in a tall room, absorbed in a confrontation with pure evil. The Virgin and the infant Christ together crush the head of a serpent beneath their feet. As the foul creature writhes in its death agonies, St Anne, frail and bent by age, looks on in solemn contemplation. By God's grace, the devil is defeated.\n\nRaven-haired Lena, ' _donna di Caravaggio_ ', was required once again to play the part of the Virgin. Wearing a coral-coloured dress with a deep d\u00e9colletage, she leans to support her son as he steps forward, his foot upon hers, hers upon the snake. He is a curly-haired, red-headed boy of about four years old. Were it not for the presence of the animal, they might just be mother and son playing a game of walk-on-my-feet as grandmother watches.\n\nThe mood of the picture is still and strange. There is no sense of drama, because instead of telling a story Caravaggio was obliged to embody an allegory. The result is like an image from an emblem book staged as a _tableau vivant_ by flesh-and-blood human beings. The voluptuously full-breasted Virgin holds her smooth-skinned son under his arms. St Anne, half lost in the shadows, has corded sinews around her neck and collarbone, while the skin of her lined face looks as dry as autumn leaves.\n\nThe theme prescribed for the picture was calculated to make a specific theological point. Its origin lay in a much debated passage in the biblical Book of Genesis, in which God curses the serpent that has tempted Eve to eat from the Tree of Knowledge: 'I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel' (Genesis 3:15). There was a long tradition of regarding this as a prophetic reference to the Virgin Mary, the so-called 'Second Eve'. By giving birth to Jesus Christ, she would redeem mankind from Original Sin and undo the evil done by the treacherous snake in the garden of Eden \u2013 bruising the head of the serpent, as had been predicted in Genesis. But Protestants, suspicious of the cult of the Virgin Mary and concerned that it detracted from the proper worship of Jesus Christ, disputed this interpretation. Martin Luther declared that Christ, and Christ alone, could redeem mankind. The Catholic Church had reaffirmed its position, in a papal Bull of 1569 that proclaimed 'the Virgin crushed the head of the serpent with the aid of him to whom she had given birth.'\n\nCaravaggio's picture was intended to translate word into image, to embody this article of Catholic faith as a vivid picture that all could understand. He was careful, at every point, to emphasize the underlying significance of his allegory. The serpent writhes in uneven, broken coils, while Christ forms a perfect circle with the thumb and forefinger of his outstretched left hand, a circle mirrored by the floating haloes of his mother and grandmother. The serpent is death. Christ is eternal life, perfection incarnate. The humble figure of Anne is there not only because her presence was decreed by the spiritual allegiance of the Confraternity of Palafrenieri, but to reinforce the idea of the Immaculate Conception, the belief that her daughter, Mary, was preserved from Original Sin. In Caravaggio's painting, Anne is shrouded in darkness, while her daughter is bathed in light. That is because Anne's virtue, great as it had been, was only a dim prefiguration of Mary's radiance.\n\nThe Virgin and Christ child seem tense and alert, as they concertedly crush the serpent. Caravaggio himself seems to have approached his task of painting in a mood of wary circumspection, deliberately curbing the aggressive side of his originality and softening the rougher edges of his style: there were no horses' rumps here, no grimy feet thrust in the face of the viewer, no red rags to conventional piety or decorum, or so he hoped. He did his utmost to produce an unimpeachably correct endorsement of the Marian orthodoxy laid down by the Counter-Reformation Church \u2013 which can only have made what happened next all the more painful. The story of the picture's reception is told in three prosaic documents in the archives of the Palafrenieri.\n\nOn 8 April, Caravaggio delivered the painting and gave a certificate to the deacon of the confraternity. It is the only known example of a statement by the artist written in his own handwriting: 'I, Michelangelo da Caravaggio, am content and satisfied with the picture that I have painted for the Company of St Anne, in faith I have written and underwritten this 8th day of April 1606.' On 14 April the picture was put on display in the confraternity's chapel. The record shows that 1 scudo was paid on that day to a carpenter named Pierfrancesco, for installing the painting on the altar of St Anne in St Peter's. On 16 April, two days later, the picture was removed. Orders were given that it should be taken away and stored in Sant'Anna dei Palafrenieri, the church of the papal grooms. The confraternity records 'payment to two porters to carry the painting of St Anne from St Peter's to their church'. On a spring morning, Caravaggio's monumental altarpiece was loaded into a mule-driver's cart and drawn slowly along the cobbled streets of the city.\n\nTo a painter so sensitive about his honour, the humiliation of this sudden reverse must have been deeply wounding. Why was the painting rejected? At around the time when Caravaggio delivered his picture, a dispute had arisen over the Palafrenieri's rights to the altar in St Peter's, but even after its resolution in early May the Palafrenieri made it clear that they still did not want the painting. By the middle of June 1606 they had sold it to Scipione Borghese for 100 scudi. The arrangement suited both parties: the Palafrenieri disposed of the picture that displeased them, clearing a small but tidy profit on their original outlay, and Borghese acquired a new work by the painter he most admired at a knockdown price.\n\nIt is possible that the Palafrenieri had simply taken exception to Caravaggio's portrayal of their beloved patron, St Anne, as a withered old lady thrown into deep shadow. The theology behind it allowed her to represent all the ancient generations before the coming of Christ, who had lived in darkness \u2013 but the Palafrenieri still might not have liked the overall effect. Their main objections, however, probably centred on the portrayal of the other two figures. The most plausible account of the picture's rejection is given by Bellori, who baldly states that it was taken out of St Peter's 'because of the offensive portrayal of the Virgin with the nude Christ child'.\n\nThe infant Christ's nudity may have been thought improper but a wisp of drapery could easily have been added. It was surely Caravaggio's embodiment of the Virgin as Lena in a low-cut dress that really caused the difficulty. Appealing once more to the mass of ordinary Catholics \u2013 and especially women, among whom the cult of Mary was strongest \u2013 Caravaggio had painted her as the kind of mother with whom real mothers might identify. He had stressed her tenderness, leaning down over the child with gentle solicitude, but in the process he had revealed quite a lot of her cleavage. It is not difficult to see why such a voluptuous Virgin Mary might have caused misgivings. Cardinal Gabriele Paleotti, whose _Discourse on Sacred and Profane Images_ was a widely consulted book of rules for the Counter-Reformation artist and patron, wrote that a picture of the Madonna with even the slightest hint of lasciviousness made him 'sick to my stomach'.\n\nThere is no reason to doubt Caravaggio's pious intentions. He probably called attention to the Madonna's full breasts to stress her maternal aspect and posed her as he did to impart the touching awkwardness of actual life. But, if so, he miscalulated. The woman in red, leaning forward with her skirts hitched up, was just too real to be allowed into St Peter's. In the eyes of the Counter-Reformation Church, Mary was pure and perfect, the Queen of Heaven. Caravaggio's Mary was just not like that. As Roberto Longhi memorably remarked, she might almost be 'a peasant woman killing a viper in a barn'.\n\nThe hellfire Dominican preacher Savonarola had once declared that artists should depict the Virgin as a pauper, not a queen, but that had been in Florence a hundred years before Caravaggio's time, when collective repentance was in the air along with smoke from Savonarola's bonfires of the vanities. In Rome under Pope Paul V such views were not widely held. The Borghese papacy was characterized by a return to pomp and magnificence, a decisive rejection of the austerity that had marked the age of Carlo Borromeo and, to a lesser degree, that of Clement VIII. Caravaggio, whose approach to religious painting had been shaped so powerfully by Borromean ideals of piety, found himself in a difficult position. In one sense his work was favoured by those in power. But in another and more important sense it was deemed entirely unacceptable.\n\nScipione Borghese, who ended up buying _The Madonna of the Palafrenieri_ , clearly liked Caravaggio's powerfully dramatic style. But he did so as a connoisseur. As a cardinal, he looked at it differently. Caravaggio may have had friends among the elite clerics of the Borghese papacy, but they were not prepared to put the weight of the Church behind his visions of holy poverty. The rejection of his altarpiece for St Peter's, and its acquisition for the Borghese collection, fundamentally altered its nature as a work of art. It was secularized, and in the process was also neutered.\n\nThe same thing had happened to him once before, in 1602, when Vincenzo Giustiniani had stepped in to buy the first _St Matthew_. But on that occasion the picture had been for a burial chapel in the church of the French and Caravaggio had been invited to paint another version. This time the picture was for St Peter's and he was given no second chance. It was a watershed in his career. Thereafter he became an increasingly isolated figure \u2013 an artist whose work would be tolerated, even admired, in private, or at the provincial margins of the Catholic world, but not at its centre.\n\nDespite this enormous setback Caravaggio refused to change his approach. Shortly after delivering _The Madonna of the Palafrenieri_ , he finally completed his long overdue altarpiece of _The Death of the Virgin_. This huge and deeply moving picture is stark evidence of the painter's reluctance to compromise, and of his moral resilience.\n\nNever before in the history of Christian painting had Mary, mother of God, been made to seem as poor and frail and vulnerable as this. Wearing a simple red dress, unlaced at the bodice to make her more comfortable in her last moments, she lies stretched out on the makeshift bier of a plank of wood. She looks shockingly dead. The apostles have gathered around her lifeless form, to pay their last respects. They are grave and serious men in the winter of their lives, each expressing pain and sorrow in his own different way. Those nearest the body are the most convulsed by grief. One man cries and rubs at his tears. Another covers his eyes and holds himself by the throat as if to choke off his own sorrow. Two others stare intently at her prone body, as if rapt in contemplation of the miracle that once grew within this mortal flesh.\n\nCaravaggio suggests that the Virgin's own last thoughts had been of that miracle, and that even now she might be dreaming of it. Her right hand rests gently on her own slightly swollen stomach, remembering the sacred baby that once grew in the blessed womb. Standing slightly to one side, St John the Evangelist, his head propped on one hand, is the picture of melancholy reflection. Mary Magdalen sits shuddering with grief on a chair pulled right up to Mary's bed. She must have been the last person to hold the dead woman's hand. As some crowd round the body, others must wait. At the back of the room, more men can be seen, talking quietly among themselves or simply thinking their own grave thoughts. Perhaps they have just come in. Once more, Caravaggio evokes the messiness of actual life. People have always mourned their dead like this, and always will.\n\n_The Death of the Virgin_ is the most bleakly mundane of Caravaggio's sacred dramas, the deathbed scene of a poor and ordinary woman. It drew another of Longhi's pithy metaphors: 'a scene from a night refuge', he called it. The Virgin's dwelling is certainly poor and humble, with its rough plastered wall and simple ceiling of coffered wood. Her feet, bare like those of the apostles, poke out straight and stiff from the folds of her dress. There is perhaps a hint that _rigor mortis_ has begun to set in. The copper basin on the floor of the room adds a final note of pathos. The body of the Virgin, too, is an empty vessel, and there is little hint of transcendence.\n\nThere is a stratagem behind the painting's apparent mood of hopeless bereavement: it invites the viewer into the darkness and doubt of death. It even dares to suggest \u2013 the deepest fear of all, in an age of faith \u2013 that perhaps this meagre life is all that there is. But peer into the gloom and all is not as it seems. Just as he had done in _The Supper at Emmaus_ , with its mystical shadowplay, Caravaggio weaves a sense of the miraculous into hard and ordinary reality. The signs of salvation have to be looked for, even if at first sight they appear to be lacking. The Virgin's face is much younger than those of the apostles, which indicates that she has been spared by God the ravages of age. The thinnest of haloes, shining in the dark air, encircles her head. Above her a great swag of drapery hangs from the ceiling of the room. Literally, it is the canopy of the Virgin's bed, but spiritually it is a sign from above. Its colour relates to her body, while its form tells the story of her soul. It is being drawn upwards, whirled to heaven by unseen energies.\n\nThe church of Santa Maria della Scala, for which the painting was intended, belonged to the order of the so-called Discalced Carmelites, the shoeless Carmelites. This may have encouraged Caravaggio to believe that his uncompromisingly severe depiction of the Virgin and apostles as shoeless paupers might find favour there. But he was once again disappointed. No sooner was his painting delivered than he learned that it too had been rejected.\n\nGiulio Mancini watched the whole situation unfold and even took the trouble to talk to the Fathers of the Carmelite order about why they had rejected the picture. In his biography of Caravaggio, he baldly states that 'the fathers of that church had it removed because Caravaggio portrayed a courtesan as the Virgin.' Had they simply got wind of the fact that the painter had modelled his Madonna on a prostitute, and found it scandalous? Caravaggio would certainly not have publicized his method, since the practice had been explicitly condemned in Cardinal Paleotti's _Discourse on Sacred and Profane Images_. Maybe one of the Carmelite fathers simply recognized the girl in the painting as a local streetwalker, or perhaps one of the painter's enemies helpfully pointed it out to them. She is not Lena, who had modelled for _The Madonna of Loreto_ and _The Madonna of the Palafrenieri_. Mancini seems to have known her identity, although he does not give her a name. In marginal notes to the manuscript of his life of Caravaggio, he elaborates tantalizingly on the bare bones of the story: 'the fathers rejected it because he had painted, in the person of the Madonna, the portrait of a courtesan whom he loved \u2013 and had done so very exactly, without religious devotion.' It is impossible to establish the true nature of Caravaggio's relationship with the girl \u2013 lover, pimp or simply employer.\n\nBut the model's identity cannot have been the sole reason for the rejection of the painting. One painted face can easily be substituted for another, a detail that could have been altered in less than a day's work. It seems it was Caravaggio's fundamental approach to the subject \u2013 essentially, his blunt portrayal of the Virgin as an actual dead woman \u2013 that the fathers could not bear. In the autumn of 1606 Mancini talked to the Carmelite fathers and subsequently wrote a letter to his brother in Siena in which he alluded to the picture being 'compromised by its lasciviousness and lack of decorum'. Later in the same document he reiterated that it was 'well made but without decorum or invention or cleanliness'. To say a picture had been created 'without invention' was shorthand for saying that it had been painted from reality rather than the imagination. The other two objections, about cleanliness and decorum, were versions of the same criticism. This was the heart of the fathers' objections. The Madonna had been made to look dirty and indecorous. She had been made to look real.\n\nThe best evidence for this is the picture that eventually ended up on the altar of the church. Having sacked Caravaggio, the church fathers passed the commission on to an artist called Carlo Saraceni. Taking his cue from images of the Virgin as the Queen of Heaven, such as Annibale Carracci's _Assumption of the Virgin_ in the Cerasi Chapel, he depicted an ecstatic Mary being translated to heaven at the moment of her death. But even that was not a sufficiently happy ending for the Madonna. The Carmelites of Santa Maria della Scala wanted a choir of angels to waft her to heaven, so Saraceni had to cook up a second version of his own sweet confection, adding a topping of cherubs. His picture, finally completed in 1610, can still be seen in the church today. Caravaggio's painting is in the Louvre.\n\nComing so soon after the rejection of his altarpiece for St Peter's, this second disappointment must have cut Caravaggio to the quick. Looking back on it years later, Mancini wondered if the refusal of _The Death of the Virgin_ might not have been the tilting point of the painter's whole life. 'Perhaps consequently Caravaggio suffered so much trouble,' he wrote. It is just an aside, but it should not be taken lightly. Mancini was there at the time. He had seen what happened next. In the immediate aftermath of the two rejections, Caravaggio committed the darkest of his many crimes, the crime that would blight the rest of his life. He killed a man.\n\n#### DEATH ON A TENNIS COURT\n\nFor several years Caravaggio and Ranuccio Tomassoni had been heading obscurely towards their final confrontation on the streets of Rome. Exactly what happened, and why, has been the subject of much speculation, but one thing is certain. On 28 May 1606 Caravaggio killed his enemy in a swordfight.\n\nThe earliest account of the murder is contained in a document in the Roman archives, which dates to the day of the killing itself. It was a Sunday, and the anonymous author saw Caravaggio's crime as part of a sinister pattern, as rowdy festivities across the city threatened to spiral dangerously out of control:\n\n> The celebrations began for the [anniversary of the] coronation of the Pope... towards evening at Ripa Grande there were celebrations and fighting with boats. In the midst of the festivity and the contest, someone gave somebody else a knock, and was stabbed to death. In Campo Marzio the same evening the painter Michelangelo Caravaggio wounded and killed Ranuccio da Terni with a sword-thrust through the thigh; he had barely confessed before he died, and was buried in the Rotonda [the Pantheon] the next morning. After that his brother, Captain Giovan Francesco, unsheathing his sword, killed another soldier (formerly a captain) of the Castel Sant'Angelo. The above-mentioned Giovan Francesco, Michelangelo and one other were also wounded in the same quarrel.\n\nUntil quite recently the only known accounts of the murder were those given by Caravaggio's three principal biographers. They were written long after the event itself, and give only the sketchiest sense of what might really have happened, but each contains vestiges of a complicated truth.\n\nMancini insinuates, as we have seen, that at the time of the killing Caravaggio was even touchier than usual because he had been upset by the rejection of _The Death of the Virgin_. He also implies that the painter was provoked, and he places the perenially hot-headed Onorio Longhi at the scene of the crime: 'Finally, as a result of certain events he almost lost his life, and in defending himself Caravaggio killed his foe with the help of his friend Onorio Longhi and was forced to leave Rome.'\n\nBaglione moralized the murder, describing it as the predictable outcome of Caravaggio's innate criminality. He also explained its cause. An argument over a tennis match had got out of hand:\n\n> Michelangelo was quite a quarrelsome individual, and sometimes he looked for a chance to break his neck or jeopardise the life of another. Often he was found in the company of men who, like himself, were also belligerent. And finally he confronted Ranuccio Tomassoni, a very polite young man, over some disagreement over a tennis match. They argued and ended up fighting. Ranuccio fell to the ground after Michelangelo had wounded him in the thigh and then killed him. Everyone who was involved in this affair fled Rome...\n\nBellori echoed Baglione's account, adding a colourful account of the fight itself: 'during a tennis match with a young friend of his, they began hitting each other with their rackets. At the end he drew his sword, killed the young man, and was also wounded himself.'\n\nThe idea that the fight was in some way connected to a game is seemingly confirmed by two _avvisi_ , small booklets that were the rudimentary forerunners of the modern newspaper. They were sold on the streets of the city, especially around the statue of _Pasquino_ , to the cry of ' _Nove e Avvisi!_ ', or 'News and Notices!'\n\nOne of these _avvisi_ , written on 3 June 1606, six days after the murder, establishes the scene of the crime. It also confirms the involvement of the two other men who had been mentioned in the very first report of 28 May. According to the _avviso_ of 3 June, Ranuccio Tomassoni's brother, the former soldier and _caporione_ Giovan Francesco Tomassoni, had indeed joined the fight, drawing his sword on another soldier. But this _avviso_ contradicts the earlier document's statement that the other man had been killed, saying instead that he had been seriously wounded and was now in prison awaiting trial. It also provides his name, and specifies that he was a companion of Caravaggio:\n\n> because of a game near the palace of the Grand Duke [i.e. the Palazzo Firenze] an argument arose between the son of the late Colonel Lucantoni da Terni, and Michelangelo da Caravaggio, the famous painter; Tomassoni was killed by a blow given to him while, retreating, he fell on the ground. Then his brother, Captain Giovan Francesco, and Petronio the Bolognese, Caravaggio's companion, entered the fray; Giovan Francesco seriously wounded Captain Petronio, and wounded Caravaggio in the head. Caravaggio saved himself by running away, and Petronio was put in prison, where he remains.\n\nThis would appear to confirm Baglione's account of an argument over a tennis match. The _avviso_ mentions a game near the Grand Duke of Tuscany's palace. There were indeed tennis courts directly opposite the Palazzo Firenze: although they have long since disappeared, the street where they once stood is still the Via di Pallacorda, i.e. 'Tennis Street'.\n\nThe other _avviso_ that mentions the murder was written on 31 May 1606. It does not name Caravaggio's wounded companion, simply describing him as a Bolognese captain serving in the papal fortress of the Castel Sant'Angelo. It confirms that he had been wounded rather than killed, and had now been put in prison. This report also blames the fight on a game, on which money had been wagered. But it also makes the fight itself sound more like an outbreak of gang warfare than a chance fracas. A total of eight people are now said to have been involved, two bands of four:\n\n> On the aforesaid Sunday night a serious quarrel took place in the Campo Marzio, with four men on either side. The leader of one side was Ranuccio of Terni, who died immediately after a long fight; and of the other Michelangelo da Caravaggio, a painter of some renown in our day, who reportedly received a wound, but his whereabouts [are] not known. Severely wounded, however, and taken to prison, was one of his companions whom they call the Captain, from Bologna, and who was a soldier of Castel Sant'Angelo. The incident is alleged to have been caused by a dispute over a game involving 10 scudi which the dead man had won from the painter.\n\nA number of other documents found in the Roman archives confirm many elements of the accounts given in the two _avvisi_. On 29 May 1606, the notary responsible for the registry of births and deaths in the parish of San Lorenzo in Lucina recorded that Ranuccio Tomassoni had been murdered in the Via della Scrofa. Since the fatal blow had actually been struck on a tennis court in the nearby Via di Pallacorda, this reference must be to Tomassoni's place of death \u2013 presumably at the shop of a barber-surgeon, who was unable to stem the flow of blood from the stricken man's wounds. The mortal thigh wound mentioned by several sources is consistent with this. Caravaggio must have caught Tomassoni high in the leg, near the groin, severing or at least seriously rupturing the femoral artery. It is very difficult to stop the bleeding from such injuries, which make the tying of an effective tourniquet all but impossible. Tomassoni would have died quickly, as the sources indicate, but it is unlikely that he would have had time to confess, as the author of the first report of 28 May optimistically suggested.\n\nWhile Ranuccio Tomassoni's companions were taking him and his brother Giovan Francesco to the barber-surgeon's in the Via della Scrofa, Caravaggio's friends were tending to Captain Petronio Toppa from Bologna. They took him to another barber-surgeon, a man called Pompeo Navagna, who treated him for a cut in his left arm so deep that seven pieces of bone had to be removed before it could be dressed. He had eight stab wounds in his left thigh, one in his left shin, and another in his left heel. Taken altogether, Navagna concluded, these were life-threatening injuries, and despite them Toppa had subsequently been taken to the prison of Tor di Nona for questioning.\n\nMeanwhile, Fabio Masetti was still keeping his eye on Caravaggio and reporting the latest developments back to Cesare d'Este in Modena. In a letter of 31 May he confirmed that Caravaggio had been wounded, and that he had fled Rome. According to Masetti's spies, the painter was on his way to Tuscany, a logical destination, given his links with Cardinal del Monte and the Medici. Masetti even found cause for a certain grim optimism in this sudden turn of events: 'The painter Caravaggio has left Rome badly wounded, having killed a man who provoked him on Sunday evening. I am told that he is heading in the direction of Florence, and perhaps will also come to Modena, where he will give satisfaction by making as many paintings as are wanted.'\n\nOn the same day, another letter was written by another representative of the Este in Rome, Pellegrino Bertacchi. He too had heard that a game of tennis had been the cause of all the trouble: 'the fight was over the question of a penalty, while we were playing at racquets, near the [palace] of the Ambassador of the Grand Duke [i.e the Palazzo Firenze].' He had also heard that the painter 'lay down his head, mortally wounded' and that 'two others were dead.' Clearly all kinds of wild rumours were flying about.\n\nBut a month later some of the smoke had cleared and the _sbirri_ had begun to get to the bottom of the whole murky business. As the _avviso_ of 31 May had stated, eight men had been involved. By the end of June the authorities had established the names of everyone on Ranuccio Tomassoni's side. He had been accompanied by his two brothers-in-law, Ignazio and Federigo Giugoli, as well as by his brother Giovan Francesco. Between 28 June and 8 July, summonses were issued to all three, instructing them to appear before the court and remain resident at their customary addresses. Caravaggio's partners in crime were Petronio Toppa, another Bolognese soldier by the name of Corporal Paulo Aldato and \u2013 just as Mancini would later report \u2013 his old friend Onorio Longhi. There was no need to call Toppa, who was in jail already, still recovering from his injuries. No one seemed to know anything much about Paulo Aldato, save for the fact that he had only one eye. So just two further summonses were sent, to Caravaggio and Longhi.\n\nCaravaggio, by then long gone, would never appear in court to answer the charges against him. But, as Baglione would later write, everyone else involved in the affair had run away from Rome too \u2013 everyone else except for the unfortunate Petronio Toppa. Onorio Longhi had fled to Milan. Giovan Francesco Tomassoni was nowhere to be found. Nor were the Giugoli brothers, whose father, Flaminio, paid caution money to the court on their behalf on 27 July.\n\nThe continued absence of so many of the participants casts considerable doubt on the story that the fight between Caravaggio and Ranuccio Tomassoni had been sparked by an argument over a tennis match. If that had been so, why would at least three apparently innocent bystanders, namely Longhi and the two Giugoli brothers, have defied court orders and gone into hiding? It made no sense.\n\nThe known facts of the case point to a very different explanation of the fight. The pattern of the evening's events could hardly be clearer. Four men on one side, four on the other: two combatants, two seconds, four witnesses. An encounter on a tennis court, a flat field that was often also used as a fencing arena \u2013 as on the day back in 1600, when Onorio Longhi watched a fencing match take place on the French tennis court at Santa Lucia della Tinta. The fight between Caravaggio and Ranuccio Tomassoni was no chance row. It was a prearranged duel. The stories about a tennis match, a bet, a disputed call \u2013 they were all fabrications, tall tales put about by the participants themselves to hide what had really happened. It was an expedient pretence: duelling was illegal in papal Rome, and punishable by death.\n\nBy the end of June, when the first summonses were issued, Judge Angelo Turchi and his fellow investigators had rumbled the cover-up of a tennis match. By the second week of July, even some of the participants had given up pretending that it had been anything other than a duel. On 11 July 1606, a notary recorded Mario and Giovan Francesco Tomassoni's acknowledgment of the writ served against Giovan Francesco. Writing in judicial Latin, he recorded their joint undertaking to do nothing in breach of the peace, in effect a vow not to take the law into their own hands \u2013 there were perhaps concerns that a vendetta might develop. He also recorded their plea for the conclusion of the investigation into Giovan Francesco, in 'distant parts'. But the most crucial elements of this document are a couple of scraps of vocabulary. Not once, but twice, Mario and Giovan Francesco referred to the dispute between Caravaggio and their late brother as 'a duel'.\n\nBy the start of August, Petronio Toppa was well enough to undergo questioning. On 6 August 1606 Toppa called two witnesses in his defence. The first was Captain Francesco Pioveno, of Vicenza, who testified that he had known the Bolognese soldier for about twelve years. He gave a ringing endorsement of his former comrade in arms: 'Captain Petronio, who's been in the wars and has been a soldier with me in these two garrisons, in Lucca and Rome... I've known him as a soldier, and I've always considered him an honourable soldier.'\n\nThe second witness was Francesco fu Menici of Lucca. He gave his profession as a gentleman's valet, although before that he had been a soldier. He had known Petronio Toppa for about eight years. They had fought together in the Hungarian campaigns of the 1590s. Unlike the first witness, Francesco Pioveno, Menici had been in the vicinity of the tennis court on the evening of Ranuccio Tomassoni's death. He said he had not seen the fight itself, but he gave an account of the prelude to it.\n\nOn the night in question, Menici said, he had seen his friend Petronio sitting\n\n> in front of the Florentine ambassador's, in front of the tennis court. He was with another Bolognese, who had only one eye, but I don't know his name. I think they call him Paulo, but I don't know his name and I don't want to say what I don't know [for sure]... I don't remember exactly when the argument was, but it was a Sunday, and it could be about a month and a half or two months ago. I wasn't present at the fight and I didn't see who was in it or what happened. I passed through there because I was coming alone from the French ambassador's house, and in passing I saw Captain Petronio and said to him, 'At your service' [this was possibly a deliberate irony, given Menici's profession as a valet] and he returned the greeting and said, 'Where are you going?' I replied, 'I'm going home,' and he replied that he wanted me to wait, because he was waiting to perform a service, and that afterwards he would come too, but he didn't tell me what service he wanted to do, and I replied that I couldn't wait and was in a hurry.\n> \n> So I left and went towards Campo Marzio, and the man who was with him left him and came with me up to Piazza Campo Marzio past the Manescalco, and said that he wanted to go and see a whore of his nearby. Then I went home. I don't know if that man without an eye, who came with me up to Campo Marzio, returned from there to Captain Petronio. When I passed the captain I saw that there were others around him, besides the one I said, and they were armed with swords...\n\nToppa's own testimony and his eventual fate are unknown. But the evidence given by the second witness called in his defence confirms that the contest between Caravaggio and Tomassoni was indeed a duel.\n\nThanks to the nature of Rome's judicial processes there exist, in addition, four statements by those who saw and acted in the duel itself. Apart from Petronio Toppa, all the men involved had run away from Rome immediately after the swordfight. Having been subsequently summoned, to no avail, all were presumed guilty and sentenced to mandatory exile. Over the following months and years, each sought to bargain the terms of a return. As they did so, they were obliged to account for their actions on the night in question. The resulting evidence is patchily informative, but it does at least clarify the circumstances in which the second swordfight, between Giovan Francesco Tomassoni and Petronio Toppa \u2013 the duel within the duel \u2013 had started.\n\nRanuccio Tomassoni's brothers-in-law, Ignazio and Giovan Federico Giugoli, revealed little as they submitted to the due process of law. In their petition for an end to their exile, they admitted that they had been present at the fight in which their kinsman had been killed, but said no more than that. The reason they gave for wanting to return to Rome was that their father, Flaminio Giugoli, who had paid the caution money for them, had died while they were away. They needed to sort out his affairs, or the family would fall into ruin.\n\nOnorio Longhi, from his native Milan, protested his total innocence in the killing. He had witnessed the fight but asserted that he had been there simply to keep the peace (hardly likely, given his record of inflammatory remarks, provocative behaviour and incitement to assault). He too said little of any substance about the duel itself and he finished, like the Giugoli brothers, by invoking his family:\n\n> Onorio Longhi in all humility declares to Your Holiness that in 1606 he was banished from Rome, as can be seen in the trial records of the Tribunal of the Governor of that city... because he was present at the murder committed by Michelangelo da Caravaggio on the person of Ranuccio Tomassoni, in which deed the speaker was not at fault. On the contrary, he accompanied Caravaggio as his well-wisher, so that no disorder should occur and to exhort him to make peace, as God and his conscience are witnesses. Thus, he has remained conscious of his own innocence and has obtained peace, and at this time is retained in Milan in the service of His Imperial Majesty and desires to return to his fatherland and to his wife and five children in order to serve the Holy Church and Your Holiness.\n\nGiovan Francesco Tomassoni's plea for his own exile to be revoked was more informative. He acknowledged having intervened in the fight between his brother and Caravaggio, and accounted for his own actions in some detail:\n\n> When the speaker saw his brother injured, bleeding and thrown to the ground, [any] obligations to keep the peace or pledges not to offend were entirely dissolved. He and the said Michelangelo standing beside each other, he wounded him [Caravaggio] in the head with a sword and would perhaps have killed him in the presence of others, save that the aforementioned Captain Petronio, and others, were present. The said Captain Petronio defended Caravaggio with a naked blade, and he [Giovan Francesco] wounded him several times.\n\nFrom this patchwork of biographies, letters, _avvisi_ and witness statements, a clear picture of the fight can now be established.\n\nBecause it is a matter of honour between Caravaggio and his long-standing enemy, they must be allowed to settle it alone. According to custom, the duellists' seconds must promise not to intervene, while Onorio Longhi, the one-eyed Bolognese soldier Paulo Aldato and the Giugoli brothers must undertake to attend simply as witnesses. Once these formalities about 'keeping the peace' have been agreed, the duel can be arranged.\n\nOn the evening chosen for the settling of scores, the air is thick with foreboding. Toppa, the painter's appointed second, is ready and waiting in front of the tennis court. As the evening wears on, he is joined by one man and then another. Having briefly wandered off, the one-eyed soldier from Bologna returns to complete the group. Everyone concerned is trying hard to look casual, but they emanate a powerful sense of menace none the less. All are armed with swords, not a tennis ball or racket in sight. Somewhere nearby, Ranuccio Tomassoni is meeting his brother and his two brothers-in-law. As nightfall approaches, the vendetta is about to be settled.\n\nThe duel does not last long. Real swordfights are short and sharp, nothing like modern fencing matches. Tomassoni and Caravaggio are wearing no helmets or body armour, because that would have made their story about an argument over a tennis match completely implausible. They use the full width of the court, fighting in a channel formed by the two lines of their witnesses and seconds. At the climax of the duel, Caravaggio seizes the initiative and the tiring Ranuccio Tomassoni stumbles in his retreat. Caravaggio lunges at the groin of his fallen opponent, piercing his femoral artery. Blood spurts in jets from the wound. Caravaggio withdraws his sword and prepares to strike again, but at this moment Giovan Francesco Tomassoni steps out of line to help his 'injured, bleeding' brother. As luck would have it, the rhythms of the fight have placed him right next to Caravaggio at this critical moment. He draws his sword in a flash and strikes the painter in the head, preventing him from inflicting further damage on the stricken Ranuccio. Seeing this violation of 'the peace or pledges not to offend', Petronio Toppa draws his sword and saves Caravaggio's life, at grave danger to his own. As he and Giovan Francesco engage, Onorio Longhi and the one-eyed Bolognese intervene to prevent further injury on both sides.\n\nMeanwhile, Federico and Ignazio Giugoli do what they can to help their brother-in-law. Caravaggio, stunned by his injury, can fight no more. At this point the carnage stops and everyone disperses into the twilit streets. As Ranuccio's friends carry his ominously still body towards the barber-surgeon's on the Via della Scrofa, they unconsciously re-enact Caravaggio's great altarpiece of _The Entombment_ in the nearby Chiesa Nuova \u2013 solemn depiction of men struggling under the weight of a heavy corpse, stilled mirror-image of the scene in the street outside.\n\n#### BANDO CAPITALE\n\nNone of the witnesses said anything to the investigating magistrate about the causes of the duel. Who had challenged whom, and why?\n\nThere are possible answers to those questions too in the extensive but partial dossier of archival evidence about the killing. Fabio Masetti, in his letter to Cesare d'Este of 31 May, had said that Caravaggio had 'killed a man who provoked him', which implies that it was Tomassoni who had challenged the painter. Mancini, the only other source with anything to say on the matter, spoke of Caravaggio 'defending himself', which also suggests that Tomassoni was the initiator of the fight.\n\nThe most telling clue to the nature of Tomassoni's grievance may lie in the identity of his chosen witnesses. He chose his brother, a soldier, as his second, and the other men there were his two brothers-in-law. It may be that Tomassoni challenged Caravaggio over a question of family honour \u2013 a question, specifically, involving the honour of Ranuccio Tomassoni's wife. Her name was Lavinia. The painter had already stolen or tried to steal one of the pimp's whores, Fillide Melandroni. Had he compounded that insult by starting something with Lavinia Tomassoni too? If so, that would have made Ranuccio a cuckold \u2013 a _becco fotuto_ , to use one of the painter's favourite phrases. Since nothing has been found in the archives to connect Caravaggio directly with Lavinia Tomassoni, a verbal provocation is more likely to have been the cause of the trouble. Perhaps Caravaggio had heard that Lavinia Tomassoni was being unfaithful to her husband, and had taunted him by repeating that rumour to his face. We may never know exactly what lay behind the duel but some kind of insult concerning Lavinia is the most likely explanation. This would explain why Tomassoni wanted Lavinia's brothers watching when he exacted his revenge \u2013 or so he hoped \u2013 on the troublesome painter who had dared to cast aspersions on his wife.\n\nWhatever the precise truth, another chain of archival evidence suggests that Lavinia Tomassoni was hardly a model wife and mother. Less than a fortnight after Tomassoni's death, arrangements were set in train for a close friend of the family, the lawyer Cesare Pontoni, to look after the couple's only daughter. Her name was Felicita Plautilla Tomassoni, and she was still only a baby. Tomassoni's widow, Lavinia, made the excuse that she was too young to bring up a child on her own. She said that she wanted to remarry (which she did, as another document in the archive reveals, within less than a year). Lavinia's mother-in-law, Tomassoni's mother, claimed that she was too old to look after the little girl. On 10 October 1606 the legal guardianship of Felicita was ratified and she became Pontoni's ward. Only one other document has been found concerning her, the will of a relation who died on 17 August 1615. This reveals that the girl had by then dropped her first name, Felicita, which means 'happiness' in Italian, and had entered a nunnery. The document refers to her as Sr Plautilla of the convent of S. Silvestro in Urbe.\n\nOne other detail suggests that the cause of the fight may have been some kind of sexual insult. Ranuccio Tomassoni bled to death from the femoral artery. Caravaggio had struck him a low blow, aiming perhaps at the groin and missing by just a fraction. Was the artist using his sword as if it were a paintbrush, attempting to mark out the most graphic of sexual insults on the body of his enemy? Wounds were meaningful, as Fillide Melandroni had graphically indicated when threatening to cut the face of her own love-rival, Prudenza Zacchia. A cut to the face was a _sfregio_ , but it was by no means the only form of symbolic, premeditated injury that vengeful Italians inflicted upon their enemies.\n\nThe practice was sufficiently common to be mirrored in the provisions of the law, where widely differing penalties were specified for different forms of revenge wound. The fourteenth-century statutes of Florence set a fine of 50 lire for the loss of members including the foot, hand, tongue and eye \u2013 but for both eyes the penalty was much steeper. The mid sixteenth-century statutes of Caravaggio's native Lombardy valued teeth at 50 lire apiece, and set a fine of 500 lire for the amputation of a hand. The loss of a single testicle was assessed at the same rate as four teeth or a tongue, namely 200 lire. The penalty for castration was understandably more than double that, at 500 lire. It is entirely possible that Caravaggio was not actually trying to kill Ranuccio Tomassoni, but attempting to make mincemeat of his testicles with a duelling sword.\n\nLike many other aspects of the Roman judicial system, sentencing was irregular and inconsistent. All those involved in the duel were sentenced to exile, but the precise sentences are not known and can only be guessed at by the different dates on which the various convicted men petitioned for their return. On this evidence, it seems that those on Caravaggio's side were dealt with more harshly than the supporters of Tomassoni: Giovan Francesco Tomassoni's plea to return, duly granted, was filed on 9 December 1606, and that of the Giugoli brothers less than two years later, but Onorio Longhi felt able to seek his own return only in the spring of 1611.\n\nThe most serious penalty was reserved for Caravaggio. As well as being sentenced to indefinite exile from Rome, he was condemned as a murderer and made subject to a _bando capitale_ , a 'capital sentence'. This meant that anyone in the papal states had the right to kill him with impunity; indeed there was a bounty for anyone who did so. The phrase meant exactly what was indicated by the etymology of its second word, derived from the Latin _caput_. To claim the reward, it would not be necessary to produce the painter's body. His severed head would suffice.\n\nCaravaggio's sword and dagger, drawn by a policeman (see p. 287).\n\n## PART FIVE  \nThe Alban Hills, Naples, Malta, Sicily, Naples, Porto Ercole, 1606\u201310\n\nA Knight of Malta being Defrocked by Wolfgang Kilian (detail). The ceremony of _privatio habitus_ took place directly beneath Caravaggio's altarpiece, which can just be made out here at the far end of the oratory.\n\n#### ON THE RUN\n\nCaravaggio had been seriously injured in the swordfight, but he needed to get out of Rome quickly. After having his wounds dressed, he returned briefly to his lodgings in the house of the lawyer Andrea Ruffetti to gather a few necessary possessions \u2013 clothes, painting materials, whatever money he could lay his hands on. But it was not safe to stay at Ruffetti's overnight because the _sbirri_ knew to look for him there. So, accompanied by Cecco, Caravaggio went to the neighbouring Palazzo Colonna and threw himself on the mercy of his family's first protectress, Marchesa Costanza.\n\nBleeding, bedraggled, wild-eyed with adrenalin, he confessed to the murder and asked for her help. Despite the seriousness of his crime, she gave it. For all his sins, perhaps in her eyes he was still the lucky child \u2013 the boy whose birth on the feast day of the Archangel Michael had once seemed like such a good omen, and whose very name was like a prayer invoking her father's famous victory at the Battle of Lepanto. Caravaggio and Cecco were given a bed for the night. Early next morning the injured painter and his boy left the palace in the marchesa's coach and four, bound for the safe haven of a remote Colonna fiefdom in the Alban Hills. As the carriage clattered through the streets of the city, its blinds were firmly drawn.\n\nNo actual account of Caravaggio's flight from the city survives, so all of the above is speculation. But something much like it must have happened. The fact of the matter is that within a day or two of the murder the painter had indeed been spirited out of Rome, deep into Colonna territory.\n\nHe probably went first to Zagarolo, and moved between there and Palestrina, both small towns owned and controlled by the Colonna, some twenty miles from Rome and suitably off the beaten track. The Colonna Palace at Zagarolo still looms today over thickly wooded hills, a forbidding fortress in a wild and remote landscape. It is exactly the sort of place where a man in fear of his life might choose to take refuge. Mancini stated categorically that this was, indeed, Caravaggio's first stopping place on his flight from Rome: 'He first reached Zagarolo, where he was secretly housed by the prince.' Bellori echoed Mancini, adding the detail that the painter was being pursued: 'Fleeing from Rome, without money and being followed, he found refuge in Zagarolo under the protection of Duke Marzio Colonna.'\n\nWe do not have much information about his first few months of exile. All three early biographies refer to a picture of Mary Magdalen that the artist supposedly painted while he was in the Alban Hills, but it has never come to light. Mancini and Bellori also mention a _Supper at Emmaus_ from the same period, which does survive. A solemn and introverted work, it now hangs in the Pinacoteca di Brera in Milan. This second _Supper at Emmaus_ is strikingly different from the painting on the same subject in London, created just five years earlier. It seems to announce a change within Caravaggio himself, and certainly marks the transformation of his style. With this troubled picture begins the last phase of the painter's life and work.\n\nThe older Caravaggio would often revisit themes and motifs that he had painted before. But only on this occasion did he rework the entire composition of a picture from earlier in his career. The second painting is almost identical in size to the first, the figures are the same scale, and virtually repeated \u2013 with the addition, in the Brera version, of the innkeeper's wizened wife, waiting to serve a rack of lamb. The tablecloth and the Turkish rug over which it has been laid are almost the same. But in this later painting it is as if someone has turned off the lights, so deep are the shadows.\n\nChrist is no longer the beardless youth of five years before, the Apollonian judge calmly looking forward to the end of time. He is the conventional type of Jesus, with light beard and shoulder-length hair, but pushed to the point of exhaustion. He is a pained and troubled figure, a Man of Sorrows who has suffered much and struggles even to raise his hand, poised just inches above the table, in the revelatory gesture of blessing.\n\nThe first _Supper at Emmaus_ , a spotlit drama of sudden recognition, has blurred to an image from a dream. The theatre of Caravaggio's early Roman painting has contracted to a space that seems more like the inside of the artist's mind, a space of memory or mental projection. That some of the figures in the picture almost certainly _were_ painted from memory, rather than from life, enhances the effect. The innkeeper resembles the innkeeper in the earlier version of the picture, but seen at one remove or through half-closed eyes. The figure of his wife, so beaten down by existence, was surely based on Caravaggio's recollections of the old woman who had recently modelled for St Anne in the _Madonna of the Palafrenieri_.\n\nThe lamb on the dish that she holds is a scrap of meat so shrivelled and inconspicuous that it barely performs the iconographic task required of it \u2013 the scantest of allusions to Christ's sacrifice and the death of all flesh. Even the still life on the table has been reduced to a bare, eucharistic minimum, just some broken loaves and a chipped majolica jug. Darkness surrounds the huddled figures seated at this simple meal, but there are no shades of transcendence here: no halo is cast on the wall behind Christ, no pattern of the divine is suggested in the shadows that fall on the drab white tablecloth. The invigorating light of a miraculous dawn has weakened to the feeble gleams of the end of the day.\n\nThe paint has been applied thinly and the colours muted to earths and ochres. The faces of all the figures are less sharply differentiated than they are in any of Caravaggio's earlier pictures. Many of the technical departures of the artist's later work are related to his circumstances: he stops painting from models, in all but a few cases, because he has no time to find them or money to pay them, and he paints quickly because he has to move on.\n\nThe disciple sitting with his back to the viewer is seen only as a silhouette, a few stray curls of his tousled hair picked out against the folds of Christ's blue-green robe. His hands express surprise but also uncertainty, as if he cannot quite believe the miracle to which he is a witness; his right hand is a shadowy form set against shadows, light glimmering in the interstices between the fingers. This recalls similar passages in the darkest paintings of Tintoretto, and indeed an eloquent Venetian indistinctness enters Caravaggio's painting at this time.\n\nThe wiry, sunburned disciple seated to Christ's left is fiercely intent. He grips the table hard with both hands, grasping for a sense of reality. Can it really be true? Can he really be performing this one last miracle? Can he really have come back from the dead? The tendons in his neck stand out as he stares with desperate intensity at the half-lit face of Christ. But there is a stoop in his shoulders, a defeated weariness about him, suggesting that in his heart of hearts he still cannot quite bring himself to believe in the miracle, that it would be too much to hope for. Christ himself seems about to disappear into the surrounding blackness of the inn \u2013 as indeed he does, in the gospel of Luke, at the very moment when the disciples realize who he is. This dark and occluded picture has the quality of a confession. How much harder Caravaggio now finds it to see the possibility of salvation.\n\nOne other painting survives from the early months of Caravaggio's exile, a depiction of _David with the Head of Goliath_ , which has traditionally been misdated to the end of his life. Its subject is David's well-known act of giant-slaying, recounted in the Old Testament (1 Samuel 17:48\u201351): a familiar story, but treated by Caravaggio in a strikingly unfamiliar way. The sombre young hero is intriguingly unexultant in his moment of triumph. He holds his grisly prize at arm's-length, staring down almost absently at the trails of blood still pouring from the severed neck of his vanquished foe. Mild disgust is mingled, in David's complicated and contemplative expression, with gentle sadness. Cecco modelled for this figure, as he had for the exuberant _Omnia vincit amor_. But what a change has come over him. He looks older, more drawn. His brow is furrowed and there are bags under his eyes. Exile and flight had taken their toll on him too.\n\nThe disembodied head of Goliath seems still to be screaming, in an extension of his death agony. Light glints on his irregular row of front teeth and is reflected in the wetness of his lower lip. The extreme tenebrism of Caravaggio's technique isolates these few charged details, distilling the drama to a compelling vignette while casting everything extraneous into darkness. Having reduced the story to an apparent bare minimum of incident, the painter deepens the meaning of his picture by deftly weaving in other layers of association. David's earlier act of hurling his shot at the giant's head is subtly implied by the way his white shirt has been looped through his belt to shape a kind of sling. His vulnerably naked torso and softly compassionate, almost Christ-like expression hint at the larger perspective of theological meaning in which the slaying of Goliath was to be understood.\n\nDavid evokes the youthful Christ, because the story of David slaying Goliath was often seen as an Old Testament prefiguration of Christ subduing Satan. The inscription on the blade of the sword held by David spells out the letters 'H.OC.S'. This is the acronym of a phrase from St Augustine's commentary on Psalm 33, in which he remarks that 'As David overcame Goliath, this is Christ who kills the Devil.' The Latin phrase used by Augustine is _humilitas occidit superbiam_ : 'humility kills pride'.\n\nThe most insistent of the picture's meanings is carried by its most blatant detail. The death's head of Goliath is a self-portrait, a depiction of Caravaggio himself _in extremis_. There is a terrible, ambiguous intensity behind his fixed stare. He seems, horribly, half dead and half alive, his right eye glazed over and closing while his left eye is still bright with outrage and pain. He is like one of the damned souls glimpsed by Dante in the _Inferno_ , an outcast moaning forever in torment.\n\nThe last piece in the puzzle of this haunting picture is furnished by the identity of its intended owner. Caravaggio painted the _David and Goliath_ for Scipione Borghese, papal nephew and the chief administrator of papal justice \u2013 the man who, more than any other, had the power of life and death over Caravaggio himself. The _David and Goliath_ was Caravaggio's darkly ingenious plea to the one man who could save him: his way of saying that Borghese was welcome to have his head in a painting, if only he would let him keep it in real life.\n\nThanks to the help of his protectors, Caravaggio was able to despatch his pictures to Rome. The second _Supper at Emmaus_ was sold to the banker Ottavio Costa; the _David and Goliath_ , a precious gift to Scipione Borghese, may have arrived in Rome in the same Colonna carriage. It was likely to have been well appreciated: the papal nephew already owned Caravaggio's severe _St Jerome_ _Writing_ , which the painter had probably also given to him as a gift, in exchange for helping to resolve the affair of his assault on the lawyer Pasqualone; and within less than a year he would sequester the entire art collection of the unfortunate Giuseppe Cesari, largely in order to get his hands on two early pictures by Caravaggio, the _Boy with a Basket of Fruit_ and the moonlit _Self-Portrait as Bacchus_.\n\nNews of the _David and Goliath_ 's arrival at the Palazzo Borghese was kept quiet. Scipione Borghese did not actually hang the picture for several years, perhaps because he did not want the artist's death's-head petition to him to be too widely known. But as the summer of 1606 turned to autumn, it seems that he was indeed working behind the scenes on Caravaggio's behalf. News of the painter's presence in the Alban Hills had spread to Rome, where it was rumoured that there were plans for Caravaggio to make a swift return. On 23 September the Este agent Fabio Masetti wrote to his masters in Mantua that 'Caravaggio, having committed the murder previously reported, is staying at Pagliano with the plan of coming back soon. I will get repayment from him of the 32 scudi...'\n\nBut though Caravaggio had support in Rome, he also had enemies. Bellori's brief comment about his 'being followed' on his flight from the city is a reminder that certain people were determined to see him brought to justice. The Tomassoni clan may have sent men after him. Within Rome itself their voices must have been raised against an early pardon for Caravaggio. Whatever deal was being brokered on his behalf, by the end of September it had fallen through and the painter had resigned himself to a lengthy period of exile.\n\nCaravaggio probably used the proceeds of the sale of the second _Supper to Emmaus_ to pay his way to Naples. Certainly by early October he was living and working there, where he felt safe enough to show his face in public. But the fear of reprisals stayed with him. He was careful to remain under the protection of the Colonna family, who maintained a powerful presence in the city. With their help, he would attempt to repair his damaged and disordered life.\n\n#### IN THE CITY OF BEGGARS\n\nNaples at the start of the seventeenth century was the largest city in southern Europe. Its population was 300,000, three times that of Rome, and would soon grow to half a million. Founded by the Greeks in ancient times, and built around the crescent of a natural bay, Naples had always been a port town. Its lifeblood was maritime commerce. Although Muslim corsairs and Barbary pirates continued to make predatory forays from their bases along the African coast, the seas had become safer for Neapolitan traders since the victory of the Christians over the Turks at Lepanto. Ships from Naples travelled to Flanders, Holland, England and Germany, as well as to Sicily, Spain and northern Africa.\n\nThe sharp-eyed English traveller George Sandys visited Naples in 1611, just after Caravaggio's time there. He was impressed by the sheer range of foods, fabrics and other materials on sale in the city's many markets:\n\n> The concourse of sundry nations to this haven, doth adde an overabundance to their native plenty. _Apulia_ sends them almonds, oyle, honey, cattell, and cheese. _Calabria..._ silke, figges, sugar, excellent wines, minerals, and matter for the building of ships. _Sicilia_ releeveth them with corne, if at any time their own soile prove ungrateful... _Africa_ furnisheth them with skinnes; _Spaine_ with cloth and gold; _Elba_ with steele and iron; and we with our countries commodities: so that nothing is wanting.\n\nThe city's traders dealt not only in goods but also in people: there were 10,000 slaves within the Neapolitan population.\n\nAccording to the phlegmatic and worldly Giulio Cesare Capaccio, long-time secretary of the city's administration, Naples was living proof that industry rather than piety was the key to a city's prosperity. 'It is not fate or the stars that determine the greatness of cities,' he proclaimed, 'but commerce and the concourse of people as in Antwerp, Lisbon, Seville, Paris, and Naples.' In his drily patriotic book about the city, the _Guida de' forestieri_ , Capaccio anticipated the later Romantic adage 'See Naples and die', asserting that 'there is nobody who does not desire to see it, and who does not desire to die here. Naples is the whole world.' That world included distinct Neapolitan communities of Pisans, Catalans, Ragusans, Germans, Flemings and French. The French and the Ragusans had their own consulates in the city. So too did the English, who ran the city's textile trade.\n\nLike the painter's home province of Milan, Naples was under Spanish rule. The city was the capital of the so-called Kingdom of the Two Sicilies, another part of the immense Spanish empire, which had passed from Philip II to his son, Philip III, in 1598. Travellers approaching from the sea were impressed by the scale and density of the town. Tier after tier of buildings rose up from the half-moon of the bay's shoreline, stretching into the hills and towards distant Mount Vesuvius, smoking ominously on the horizon. The seaward limits of Naples, like all its boundaries, were marked by high walls of stone. Massive fortifications dominated both skyline and waterfront, embodying Spanish naval and military might. Naples had three castles: the Castel Sant'Elmo, built in the shape of a six-pointed star on the top of the hill above the centre of the city; the Castel Nuovo, which stood beside the shore and was home to the Spanish viceroy; and the Castel dell'Ovo at the south-east corner of the city, so named after the egg-shaped rock on which it was perched.\n\nNaples was a bastion of Habsburg rule over the southern Mediterranean. An army of Spanish soldiers was stationed in its garrisons, a navy of Spanish galleons moored in its harbour. The policy of the city's rulers was driven by two overriding aims: to safeguard the territories of the Spanish empire and subjugate the Neapolitan aristocracy to the will of the Spanish monarch. Under a succession of sternly autocratic viceroys, those aims had been ruthlessly pursued. The old structures of Neapolitan society had been systematically eroded, as the aristocracy, who had been a thorn in the side of Neapolitan rulers for centuries, were stripped of their powers and forced to renounce their ancient rule as despots on their rural estates. Most had been persuaded to leave their fiefdoms in the countryside and move to Naples itself, where they were compensated for the loss of real power with the sybaritic rewards of life at the court of the Spanish viceroy. The Italian historian Benedetto Croce encapsulated their decline in a single, acerbic sentence of his _History of the Kingdom of Naples_ : 'Idleness, luxury, rivalry in conspicuous display, the construction of huge palaces, the attendance of large numbers of servants, the abandonment of family and the frequenting of courtesans (a custom copied, apparently, from the Spaniards) led the baronial families, in the course of a few generations, to ruin.'\n\nAs the power of the barons dwindled, a new class of professionals and entrepreneurs flourished: lawyers, tax advisers, importers and exporters of grain, moneylenders, traders in luxury goods. Many came from Genoa, others from Tuscany, traditional breeding ground of merchants and financiers. Regardless of background, those involved in trade and finance were routinely referred to as 'Jews' by the habitually anti-Semitic Neapolitans. The actual Jewish population had been decimated by a systematic campaign of expulsions begun a hundred years before.\n\nThe rich dressed in the Spanish manner and travelled through the streets in carriages or covered litters. George Sandys remarked that there were as many litter-bearers touting for work on the streets of Naples as there were boatmen on the busy wharves of London. But the city's most striking feature was its ubiquitous crowd of beggars and paupers. In every street and in every alley thronged a seething, jostling mass of the poor. 'Nowhere in the world,' wrote Capaccio, 'is there anything so obtrusive and undisciplined, the result of the mixture and confusion of so many races... miserable, beggarly and mercenary folk of a kind such as to undermine the wisest constitution of the best of republics, the dregs of humanity, who have been at the bottom of all the tumult and uprisings in the city and cannot be restrained otherwise than by the gallows.' He likened the Neapolitan crowd to a constant swarm of insects. Wherever he went, he heard 'a murmuring... as if it were the buzzing of bees'.\n\nDespite the city's prosperity, there was work for only a fraction of its ever-growing population. Every day, every week, every year, an unstoppable flood of rural migrants poured into its already close-packed mesh of streets. They came to escape the harshness and uncertainty of life on the land, where petty banditry was rife and where the failure of one crop could doom an entire family to starvation. Their plight had been further exacerbated by new and punitive royal taxes, exacted by the Spanish from the rural peasantry, who abandoned their smallholdings in droves.\n\nAs its population climbed inexorably, Naples became caught in a vicious circle that made mass unemployment and grim poverty inevitable facts of life there. The authorities lived in continual fear of social unrest, with good reason. There had been brief, bloody rebellions in 1508 and 1547. To avert the threat of revolution, the viceregal government guaranteed food and provisions even in times of scarcity or famine. Grain was stockpiled in vast quantities to ensure that corn and bread would always be available, at state-controlled prices, to all the inhabitants of the city. Such measures had the inevitable effect of attracting yet more immigrants, thus exacerbating the very crisis the government had intended to alleviate.\n\nIn a vain attempt to check the city's growth, the authorities introduced restrictive building ordinances, which prohibited the construction of new dwellings outside the city walls; the intention was to stem the tide of immigrants by the simple expedient of depriving them of anywhere to live. But workers from the countryside continued to flow into Naples, so the new regulations simply meant that living conditions became ever more cramped. It has been estimated that some 21,000 people were squeezed into every square mile of the city.\n\nEven the physical appearance of the population was transformed by this wrenching demographic shift. Pressure on the food supply meant that for the majority pasta replaced vegetables and fruit as the staple diet. Despite the best efforts of the government, many people lived in a state of permanent semi-starvation. Neapolitans became shorter in height and notably more prone to the illnesses and deformities caused by malnutrition: goitres in the throat, rotten teeth, rickets and scurvy. The ragged and homeless were themselves seen as a kind of disease afflicting the body politic. The poorest citizens were known as the _lazzari_. The term literally means 'lepers', but in Spanish-controlled Naples it was used to encompass an entire social underclass, a subproletariat of the destitute. At night they huddled under market stalls, in courtyards, beneath porticoes, anywhere shelter could be found. By day they sought refuge in churches or took to the streets to beg. They were everywhere, complained Capaccio, clogging the very arteries of his city. 'Nothing is more difficult than getting about in Naples, wherever I go and at whatever time.'\n\nThe chronic shortage of housing was made yet more acute by the city's many churches and monasteries, by the grand scale of its civic buildings and by the determination of the authorities to maintain large areas of park and orchard in the urban centre. Because space was so precious, it was rigorously exploited. Houses in Naples commonly rose to six storeys, twice as tall as those in any other Italian city. The streets were narrow and they were still arranged in the same tight grid-plan formation that had been laid down by the Greek founders of the settlement more than two thousand years earlier. The centre of the city was dark. Overshadowed by unbroken lines of tall buildings, its congested lanes and alleys were rarely penetrated by direct sunlight. Despite the sunshine of southern Italy, most daily life took place in deep shadow, in a form of civic space not unlike the bottom of a well.\n\n#### THE SEVEN ACTS OF MERCY\n\nLittle is known about Caravaggio's first visit to Naples. The archives of the city have not even yielded his address. He may have stayed on the Via Toledo, in the palace of Luigi Carafa Colonna, Costanza Colonna's nephew. But it is more likely that he and Cecco were given rooms in Costanza's own residence at Chiaia, a grand fortified block of a building on the edge of town, close to the sea. He is securely documented as having stayed there during his second visit to Naples, three years later.\n\nAccording to Bellori, Caravaggio was deluged with work from the moment he arrived in the city, 'since his style and reputation were already known'. Within days of his arrival he had been commissioned by Niccol\u00f2 Radolovich, a rich grain merchant from Ragusa, to paint a large altarpiece of 'the Madonna and child, surrounded by choirs of angels, with St Dominic and Francis embracing below, with St Nicholas on the right and St Vitus on the left'.  On 6 October he received 200 ducats in advance payment and later the same day opened an account at the Banca di Sant'Eligio, where he deposited the money. Radolovich wanted his picture as soon as possible: the contract specified that the altarpiece was to be delivered by December.\n\nThe Radolovich altarpiece has been lost, if it ever existed. None of the artist's early biographers mention the picture, so perhaps it was never painted. Might Caravaggio have had second thoughts about taking on the type of stiff, static and rather old-fashioned composition prescribed by the contract? The Virgin Mary wafted to heaven in clouds of cherubim: hardly a subject to bring out the best in him. Less than three weeks after agreeing the deal with Radolovich, Caravaggio cashed a money order for 150 ducats drawn on his new bank account. Perhaps he took the money out to give his client a refund.\n\nAt around the same time, late October or early November, he took on a more prestigious commission: to paint a monumental picture for the high altar of a new church in the heart of Naples. The church was the Chiesa del Pio Monte della Misericordia, close to the cathedral, on the corner of the Via dei Tribunali and the narrow Vico dei Zuroli. The subject of the altarpiece was to be the Seven Acts of Mercy, the good works encouraged by the Christian spirit of charity, such as feeding the hungry and giving shelter to pilgrims. It was a topical theme in Naples, where the plight of the poor was so brutally visible.\n\nThe pauperist strain of Counter-Reformation piety, to which Caravaggio had given such uncompromising expression in his Roman altarpieces, was especially strong in southern Italy. The Pio Monte della Misericordia was a lay confraternity devoted to the care of the sick and the needy, an institution at the front line of attempts to alleviate the urban crisis gripping seventeenth-century Naples. It had been founded in 1601 by seven idealistic young noblemen who were dissatisfied with the narrowness and superficiality of life at the court of the Spanish viceroy. Moved by the plight of the _lazzari_ , they would meet every Friday at the Hospital of the Incurables, 'to serve and succour those poor invalids with food and sweetmeats'. As their confraternity grew and flourished, they broadened its activities to encompass all seven of the traditional Christian acts of mercy. They also built a church. It had been consecrated in the middle of September 1606, a mere fortnight before Caravaggio's serendipitous arrival in Naples.\n\nThe original statute of the Pio Monte had been written in 1603. The document placed great emphasis on the practice of 'corporal mercy', by which was meant hands-on charity, as opposed to the spiritual offering of prayer. It also expressed the confraternity's fiercely independent spirit, insisting on its freedom from ecclesiastical control: 'finally we wish that our Monte be not subject to the ordinary [i.e. the Archbishop of Naples], but that the workings of the Monte be autonomous and free from the jurisdiction of this ordinary.' The papal authorities made the concession, although they insisted on keeping it secret for fear of setting an undesirable precedent.\n\nCaravaggio's new patrons were powerful and persuasive men, with deep pockets. They offered him 400 ducats, twice the fee that had been proposed for the Radolovich altarpiece. They were evidently determined to get their man. Caravaggio had come to Naples in their time of need, at the exact moment when they were looking for a painter to give permanent visual expression to their sense of charitable mission.\n\nThe prime mover of the commission was probably Giovanni Battista Manso, the Marchese di Villa, one of the seven founding members of the Pio Monte. Manso was interested in the arts, especially poetry. He was a patron of Giambattista Marino, a poet famous for his restless and unruly nature, who had himself struck up a friendship of sorts with Caravaggio in Rome, and had possibly cast an eye over the scurrilous verses addressed to 'John Baggage'.\n\nManso was sharp and open-minded, with a keen and speculative intelligence. He was a friend of Galileo and regularly visited Tommaso Campanella, freethinking cleric and author of _The City of the Sun_ , during his 27-year imprisonment by the Inquisition. Manso was also friendly with Costanza Colonna's nephew, Luigi Carafa Colonna. Together, in 1611, they would found the Accademia degli Oziosi, one of the leading literary academies of Naples. Manso liked to entertain poets and other writers at his villa in coastal Puteoli, a place he fondly described in his biography of the poet Torquato Tasso: 'on a most beautiful sea-shore... a beautiful house somewhat elevated above all the others and encompassed all around by very beautiful gardens'. Many years later Manso would play host to the English poet John Milton on his visit to Naples. Milton described him in a Latin epigraph as 'a very noble and authoritative man'.\n\nTolerant of outsiders and misfits, interested in intellectual innovators, close to the Colonna family \u2013 all this indicates that Manso is likely to have been well disposed to Caravaggio. He was first and foremost an author, a connoisseur of literature rather than painting, but this too points to his involvement in the commission, which seems to have reflected a very literary conception of the subject of the acts of mercy. All seven acts were to be depicted on a single canvas, together with the figure of the Madonna della Misericordia, the 'Virgin of Pity', descending from heaven to give her blessing. Caravaggio would rise to the challenge of this busily elaborate iconography with one of the most compellingly humane pictures of the seventeenth century.\n\nThe painter was deeply responsive to the different worlds through which his stormy life would take him. He had an unerringly keen sense of milieu, a sharp eye for all that sets one place apart from another, whether architecture or mood, the quality of light or the quality of human behaviour. That responsiveness was one of the foundations of his art. It was the means by which he made holy legend seem real and true to those who looked at his pictures, embodied in a painted world that looked and felt like _their_ world. When in Rome, he had brought the Bible and its stories to Rome. When he moved to Naples, he shifted his visions of the sacred past there too.\n\n_The Seven Acts of Mercy_ is set at the bottom of the crowded well of a Neapolitan street corner. It is night-time, but the street is full of people. In the foreground a beggar half kneels and half crouches, light flaring off his pale naked back. His skin is stretched tightly across his shoulderblades, over the curve of his vertebrae and the cage of his ribs. A young man in silk and velvet clothing, wearing a feather in his cap, looks down at the half-naked pauper with an expression of troubled compassion. They are just two in the midst of a throng. Beside them, an innkeeper gives the nod to a sad-faced pilgrim and a sunburned man looks skyward with pained relief as he slakes his thirst with a trickle of water.\n\nTo their right, someone really has seen Naples and died. The corpse is being carried away. Only the dirt-ingrained soles of the cadaver's feet are visible. The face of the dead person's pallbearer is lost in deep shadow. Behind, a swarthy and bearded sexton in plain white vestments is reciting the funeral office. There is a flickering, mobile quality to the light, especially where it falls on the folds of the priest's cassock, which has an almost phosphorescent glitter. Its source is the pair of candles that the priest holds aloft, a torch against the blackness of night. A more mysterious light also falls from above, its source hidden.\n\nSmoke rises from the coarse tallow and the priest chants in a deep, melancholy voice. Next to the departing corpse, a dull-eyed woman bares her breast and gives succour to an old man through the grille of his prison cell. Above, a contemplative Virgin Mary cradles her son and looks down on the scene. The Madonna and child are wrapped in the embrace of two intertwined angels.\n\n'For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in: Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me.' The different groups in Caravaggio's painting represent the different forms of charity listed in the gospel according to Matthew (25:36\u20137). To the six biblical acts of mercy the medieval Church had added one more: the burial of the dead. It was traditional to represent each of the acts separately. But, having been asked to combine them all in one picture, Caravaggio turned an apparent handicap to his own advantage. For a dark and desperately overcrowded town, he created a dark and desperately overcrowded altarpiece.\n\nWith the exception of the burial of the dead, which is implicitly set in the present, each of the acts of mercy is enacted by a figure from history or legend. The sunburned man with a desperate need for water is Samson, whose thirst was miraculously quenched from the jawbone of an ass (Judges 15:18\u201319). The bearded traveller sheltered by the stolid innkeeper is Christ the pilgrim. The young _bravo_ with a plumed hat, who evokes bittersweet memories of the finely dressed ne'er-do-wells in Caravaggio's first Roman pictures, is a representation of St Martin of Tours. He has drawn his sword to cut his cloak in half, as the medieval saint had done, to clothe a pauper, in the most frequently recounted episode of his life. The unclothed wretch at the saint's feet has already been given his piece of cloth, which the pauper grasps in his left hand as if to begin covering his nakedness. The blade of St Martin's sword glints in the darkness to the left of the beggar's face. Half lost in the shadows, virtually under the innkeeper's feet, another curly-headed figure squats with his hands clasped in supplication. The solemn and melancholy saint may be about to give away the second half of his cloak to this second beggar. He gives his charity to two people and is presumably meant to embody two different acts of mercy, not only clothing the naked but visiting the sick. Of all the charitable figures, he has been placed closest to the picture plane, and therefore closest to the congregation in the church of the Pio Monte. Literally, he represents St Martin, but he is also an alter ego for the seven young Neapolitan noblemen who had founded the Pio Monte. Like them, he is an aristocrat helping those who have been struck down by sickness and poverty.\n\nThe two figures at the right, the half-undressed woman and the greying old man behind the bars of his cell, are drawn from the legends of ancient Rome. They also embody twin acts of charity, namely feeding the hungry and visiting prisoners in jail. The imprisoned Cimon was starving to death when his daughter, Pero, came to him and nourished him with milk from her breast. The subject was known as the _Caritas Romana_ , 'Roman Charity', and seen as a classical prefiguration of the Christian spirit of mercy. Since Caravaggio had visited the Palazzo Doria, in Genoa, on his brief flight to the city in the summer of 1605, he must have known Pierino del Vaga's fey Mannerist version of the same subject, in which an elegantly dressed young woman in a windswept cloak smuggles her left breast through the grille of a prison cell with a gesture of improbable grace. Caravaggio's interpretation is harsher, darker, gratingly realistic. Looking around her furtively, as if wary of detection, the dark-haired young woman performs her act of mercy with a troubled and anxious air. The old man who suckles at her breast has been reduced by his plight to a second infancy. Her dress is folded up under his chin like a bib. Two viscous drops of milk are caught in the strands of his beard.\n\n_The Seven Acts of Mercy_ is a picture that collapses time and space, drawing the whole world and all the world's history into its dark centre. Classical antiquity, the Old Testament, the New Testament, the Middle Ages and the present day \u2013 every epoch is symbolically represented in the different episodes that crowd the canvas. 'Naples is the whole world,' Capaccio wrote, and in Caravaggio's painting a corner of the city has been transformed into precisely that. This one dark street, this scene of desperation and pain and death, is the painter's microcosm for the brutality of existence itself. Briefly, it has been blessed and transfigured, made other than cruel experience normally proves. Here, the thirsty drink, the homeless are given shelter, and a sword is used not to kill a man but to put clothing on his back.\n\nThe embracing angels, themselves a celestial vision of fraternal love, descend earthwards in a rush, bearing the Madonna and Child with them. The leading angel's hand reaches down and into the world of fallen humanity \u2013 the highest reaching towards the lowest, the hand of the angel extended towards its visual rhyme, the left hand of the wretch at the very bottom of the painting, itself pressed down on the hard and unyielding ground. But a gulf of darkness and confusion separates the angel from the wretch. In that darkness there is space for the shadow of a doubt.\n\nThe tumbling angels and Madonna of Mercy are unusually heavy and corporeal, so emphatically realized that the wings of one angel cast the clearest of shadows on the prison wall. Yet the sense of hectic, jostling movement that ripples through the entire composition has the effect of making everything in it seem unsettlingly provisional. At any moment the celestial vision might disappear, the lights that flare gutter and go out, and the world plunge back into impenetrable night.\n\n#### THE MECHANICS OF EVIL\n\nCaravaggio painted the monumental altarpiece at breakneck speed, in little more than seven weeks. He received the balance of his fee on 9 January 1607, by which time the painting was probably installed on the high altar of the church of the Pio Monte. The confraternity soon came to see it as one of their greatest treasures. At a group of meetings held in the summer of 1613 the congregation decided that the painting could never be sold at any price. By then, several offers of 2,000 scudi or more \u2013 five times the original fee for the work \u2013 had already been turned down. One of the would-be purchasers was the Spanish poet Juan de Tassis y Peralta, Conde de Villamediana, but he was forced to content himself with a copy painted from Caravaggio's original. In the 1650s, when the complex of the confraternity's buildings was remodelled, a new centrally planned Baroque church was created with the specific aim of giving Caravaggio's altarpiece yet more prominence, space and light \u2013 a rare instance of an entire building being constructed around a single picture.\n\nThe _Seven Acts_ guaranteed further commissions and more work for Caravaggio. Sometime in the early months of 1607 he agreed to paint another altarpiece, on the subject of Christ's flagellation, for a chapel within the courtyard of a Dominican monastery in Naples. The picture was finished by 11 May 1607, when a final payment of 250 ducats was made. It has remained in Naples ever since, although it is no longer in the chapel for which it was commissioned, but in the Museo di Capodimonte.\n\nWith _The Flagellation of Christ_ , Caravaggio resumed his old rivalry with Michelangelo. The most celebrated earlier version of the subject had been for the Roman church of San Pietro in Montorio, painted by Sebastiano del Piombo but to Michelangelo's designs. Sebastiano's High Renaissance Christ is sorrowful but withdrawn. He is an idealized victim enduring the blows of a group of animated, mildly grotesque tormentors, in the setting of a grand apsidal chapel supported by marble columns with finely carved Corinthian capitals.\n\nCaravaggio took the same basic composition but made it his own by giving yet more emphasis to the cruelty and suffering implicit in the subject. He moved the viewer much closer to the grim act of torture, enlarging the figures and narrowing the complex architecture of the earlier painting to the truncated shaft of a single pillar in a darkened space. To that shadowy pillar, a reduced cast of torturers strive to bind the spotlit figure of Christ. Naked save for a loincloth and a crown of thorns, he is a strikingly statuesque figure. Just like the Christ of Michelangelo and Sebastiano, he might almost be a sculpture come to life. But he is more beaten down, more nakedly vulnerable. His exhaustion is conveyed by the line of his neck, the way he has wearily allowed the weight of his head to sag on to his shoulder. Too tired to hold himself upright, he has stumbled forward from the base of the pillar.\n\nResponding to their victim's state of collapse with angry determination, two of his tormentors are kicking and yanking him back into place. The torturer at the right, whose face is half hidden by shadow, is tightening the cords with which Christ's arms are bound. The man on the left is pulling his hair to straighten his body for the first blows. He snarls bestially, brandishing a makeshift whip in his other hand.\n\nA third torturer kneels at Christ's feet, binding a sheaf of twigs into a flay. He goes about his work with care, only looking up to see how soon the work of flagellation need begin. Just as he had done in _The Crucifixion of St Peter_ for the Cerasi Chapel, Caravaggio focused on the grim mechanics of evil. The kneeling man's shadowed profile is shown in silhouette against Christ's left thigh and bright white loincloth. Placing such emphasis on the proximity of one man's body to another is Caravaggio's way of heightening the horror of the scene. Torture is a misbegotten form of physical intimacy.\n\nHis new audience was impressed but also startled by Caravaggio's intense and troubling realism. The shock of their initial reaction can still be sensed in an account of _The Flagellation_ , written more than a hundred years later, by the Neapolitan art historian Bernardo de Dominici: 'This work when it was shown to the public attracted much attention, in particular the figure of Christ which was taken from a common and not a noble model as is necessary for the representation of God made Man: everyone, from the amateurs to the professors, was shocked by his new manner: the use of deep and terrible shadows, the truth of the nakedness, the cold light without reflections.'\n\nApart from his irrelevant complaint about the supposedly ignoble Christ \u2013 actually one of the painter's most gracefully sculptural figures \u2013 de Dominici's remarks epitomize the Neapolitan response to Caravaggio's art. Pictures such as the _Seven Acts_ and _The Flagellation_ were greeted with stunned admiration, bordering on bewilderment. They created a sensation and transformed Neapolitan painting virtually overnight. Caravaggio's extreme chiaroscuro and his brutal sense of reality were the catalyst for the birth of a new school of tenebristic painting in Naples. And through this city at the crossroads between Italian and Spanish art, Caravaggio's starkly powerful new style was transmitted to Spain itself. There it would have an even deeper transformative effect on native traditions. The work of the greatest Spanish religious painters of the seventeenth century, Ribera and Zurbar\u00e1n, is unimaginable without the influence of Caravaggio. The gruesome particularity of Baroque Spain's polychrome statuary, so bloodily realistic in its conjurings of saints martyred and Christ crucified, is also deeply Caravaggesque in spirit. The painter's years of exile and displacement are reflected, obliquely, in the westward spread of his influence.\n\nTwo further altarpieces survive from this period. _The Crucifixion of St Andrew_ , which now hangs in the Cleveland Museum of Art, may have been even more directly responsible than the _Seven Acts_ or _The Flagellation_ for the dissemination of Caravaggio's influence in Spain. Bellori records that the picture was acquired by the Spanish viceroy in Naples, Don Juan Alonso de Pimentel y Herrera, Conde de Benavente, and taken by him to Valladolid on his return home to Spain in 1610. Its presence is confirmed by an entry in an inventory of the contents of the palace of the counts of Benavente drawn up in 1653, where it is described as 'a large painting of a nude St Andrew when he is being put on the cross with three executioners and a woman, with an ebony frame' and attributed, in a marginal annotation, to 'micael angel caraballo[ _sic_ ]'.\n\n_The Crucifixion of St Andrew_ was almost certainly commissioned directly from Caravaggio by the Conde de Benavente himself. The viceroy had a special devotion to the saint, having played a significant role in the early seventeenth-century renovation of the crypt of St Andrew in the cathedral of Amalfi. In 1610, the year of his departure from Naples, he is reported to have made a special pilgrimage to Amalfi, 'moved by devotion to visit the tomb of St Andrew'. It seems highly probable that he commissioned Caravaggio's painting as an aid to his own prayers, and that it was destined from the start for the private chapel of his palace in Spain.\n\nThe picture is a harsh and daringly abbreviated depiction of a withered old man dying the cruel death of a martyr. Its true subject is not actually 'St Andrew when he is being put on the cross', as the writer of that Spanish inventory understandably assumed, but the miracle that occurred when his would-be executioners attempted to take him off it. According to _The Golden Legend_ , the saint met his death in Patras, in Greece, after incurring the wrath of the Roman proconsul Aegeas. To prolong his agony, Aegeas ordered that Andrew be tied rather than nailed to the cross. For two days he hung there in the scorching sun, continuing to preach his forbidden Christian message to a crowd of twenty thousand. On the third day, the people grew restless and threatened Aegeas with death unless he put an end to the sufferings of 'an old man full of gentleness and piety'. But the saint prayed to God to be allowed to die on the cross, just as Christ had done. When Roman soldiers tried to unbind him, 'they could not touch him, for instantly their arms fell back powerless... a dazzling light came down from Heaven and enveloped him... and when the light vanished, he breathed forth his soul.'\n\nThis is the moment that Caravaggio chose to depict. As the flash of divine light fades, the old man stops breathing and his eyes begin to roll up into his head. This is the parody of a deathbed scene, with the dying man forced to expire, against nature, in an upright position. His livid yellow skin is stretched tight across his ribcage. Wizened and pathetically shrunken, he exhales his last breath. The painter captures that moment when a man does indeed give up the ghost, when he suddenly becomes strange and unfamiliar, no longer like _himself,_ as the life slackens out of him and death takes over his mouth, his eyes, his limbs, twisting them into unfamiliar forms.\n\nMight the painter have observed his model, this very man, at the moment of his death? Might he have used his contacts in the Pio Monte della Misericordia to gain access to the Hospital of the Incurables \u2013 not to 'serve and succour' the terminally ill, but to paint one of their number? It is an image that reeks of mortality. The dead man's face and neck are sunburned, the rest of his emaciated body pale. He looks just like an actual human being at the end of an actual hard life, a malnourished _lazzaro_ who has swapped the hardships of the land for the brutality of the city.\n\nThe odour of the geriatric ward hovers too about the figure of the old woman in the bottom-left-hand corner of the painting. Sun-scorched like the saint himself, with a face heavily lined and wrinkled and a goitre in the neck, she frowns with fellow feeling. Her strong and sad eyes are full of pity. Gazing up towards the dying martyr, she plays the part of a chorus of one, standing in for the twenty thousand who had listened as Andrew preached.\n\nThe party of Roman soldiers sent to untether the saint from his cross has also been reduced to a solitary figure, a man teetering on a ladder. He struggles to free his arms from the invisible force that has paralysed them. As he does so, he arches away from the saint. The two bodies perform a kind of dance, its symmetry shaping a contrast between life and death. One is curved in tension, balanced against the possibility of a fall. The other is curved involuntarily, by the sideways sag of its own dead weight. Below, the lightly bearded figure of Aegeas looks up wonderingly at the miracle. His armour gleams darkly, evoking memories of the malign armoured soldier in Caravaggio's _Betrayal of Christ_. Two other figures loiter, their faces obscured by darkness. Landscape and sky have been reduced to a cursory smear.\n\n#### CARAVAGGIO AND RUBENS\n\nThe other picture by Caravaggio to surface during his first visit to Naples was a large altarpiece of _The Madonna of the Rosary_. It was first mentioned by Frans Pourbus the Younger, a painter at the court of Mantua who was in Naples in the autumn of 1607. He had seen it for sale along with another painting by Caravaggio, a _Judith andHolofernes_ that has since disappeared. On 15 September he wrote to his master, Vincenzo I Gonzaga, the Duke of Mantua, to inform him that 'I have seen here two most beautiful paintings from the hand of Michelangelo da Caravaggio. One is a _Rosary_ and was made as an altarpiece; it is 18 palmi high and they are asking no less than 400 ducats for it. The other is a painting of medium size with half figures and is a _Judith and Holofernes_ ; they will not let it go for less than 300 ducats. I did not want to make an offer because I did not know the intentions of Your Highness; however, they have promised not to let the painting go until they have been informed of the wishes of Your Highness.'\n\nIn the same letter Pourbus implied that the picture had been painted by Caravaggio in Naples, but its tight, dry, highly finished style is utterly at odds with the brusquely abbreviated technique that characterizes the known works of the first Neapolitan period, such as the _Seven Acts_ or _The Crucifixion of St Andrew_. Poised and somewhat theatrical, with its crowd of anxious paupers clustered in supplication at the feet of the Virgin and Child, _The Madonna of the Rosary_ must have been painted considerably earlier in Caravaggio's career. Figures and forms are clearly delineated, the play of light and shade in the drapery far more sharply defined than such passages in any of the painter's later works. The red drapery bunched above the head of Mary is more precisely described even than the similar swag of red cloth in Caravaggio's last Roman altarpiece, _The Death of the Virgin_.\n\nCaravaggio was an artist compelled to be true to himself, incapable of stretching to the ventriloquistic impersonation of his own earlier manners. As he grew older, his style moved inexorably towards simplification, abbreviation, occlusion. _The Madonna of the Rosary_ is closest in spirit and appearance to _The Madonna of Loreto_ and _The Entombment_. It breathes the same air of unclouded popular piety as those pictures of 1604.\n\nGiven the prominence of St Dominic in the legend of the Rosary, this large and imposing work was perhaps commissioned as the altarpiece of a Dominican church somewhere in or near Rome. Its appearance in Naples can best be explained by another of the rejections that occurred so frequently during the painter's years in Rome. The naked and conspicuously dirty feet of the kneeling paupers in the foreground were probably to blame. Caravaggio may have taken the large and valuable canvas with him when he fled from Rome after the killing of Ranuccio Tomassoni.\n\nWorship of the Rosary had begun in Italy in the early years of the twelfth century. According to tradition, the Virgin appeared in a vision to St Dominic one night in 1208, holding a string of beads in her hand. She showed him how to use the beads in prayer and instructed him to preach the technique to Christians everywhere. Each bead represented a different mystery in the life of the Virgin or of Christ. As the worshipper moved the beads along the string, one by one, he or she was to visualize one particular mystery at a time, to bring it forth in the mind's eye and focus devotion upon it, while reciting the _Ave Maria_ and the _Pater Noster_. Protestants disapproved of the Rosary, but during the second half of the sixteenth century the cult went from strength to strength. At a time when the Church was actively seeking to strengthen its hold on the mass of ordinary believers, the distribution of Rosary beads was recognized as a cheap and effective way of encouraging prayer and piety at every level of society.\n\nCaravaggio stressed the inclusive nature of the cult by giving great prominence to the huddled crowd of the poor, reaching out in unison for the strings of beads held out by Dominic in both hands. In most depictions of the subject, the saint is shown himself receiving the Rosary from the Madonna. But here she acts the part of a heavenly overseer, supervising its distribution to the people. Supporting a plump Christ child on her knee, she points down towards the bottom-left-hand corner of the picture, in the direction of a mother who, like herself, is accompanied by her young son. The Virgin seems gently concerned that they should not be forgotten in the clamouring press of people.\n\nOn the other side from St Dominic stands St Peter Martyr, distinguished by the head-wound of his martyrdom. He was a Dominican friar, who had been killed by a stone thrown by a heretic. Accompanied by another darkly cowled and inscrutable member of the order, he gestures towards the Madonna and child and looks out at the viewer with a yearning, soulful expression on his face.\n\nOne other figure also looks at us. With the bearing and demeanour of an aristocrat, dressed in black and wearing a fine lace ruff, he kneels at Dominic's elbow and stares meaningfully out from the picture. He is presumably its donor, the man for whose chapel it had been commissioned in the first place. Who is he? There is perhaps a clue in the painting's composition. He has been aligned with a massive fluted column. The column was a symbol of the Colonna dynasty, strongly associated with the Madonna of the Rosary ever since the Battle of Lepanto, at which Costanza's father Marcantonio had played a pivotal role. In Rome in the early 1570s, Filippo Neri had attributed victory there to the prayers of the faithful to the Madonna of the Rosary. Various names have been proposed for the donor, including that of Don Marzio Colonna, who had sheltered Caravaggio after his flight to the Alban Hills. But since the commission of the painting remains entirely undocumented, his precise identity remains a mystery.\n\nFor whatever reason, the picture had disappointed the man in the lace ruff. Whoever he was, he had turned it down, and so it appeared on the open market in Naples in the autumn of 1607. By that time it was in the hands of two minor painter-dealers, Abraham Vinck and Louis Finson, whose stock also contained the now lost _Judith and Holofernes_. It is not clear whether they had purchased these works or whether they were selling them on commission on the painter's behalf. It was Finson and Vinck who told Frans Pourbus that _The Madonna of the Rosary_ would cost the Duke of Mantua 400 ducats, which happens to be exactly in line with the fee Caravaggio had received for the _Seven Acts_. But the deal must have fallen through, because in the end the dealers themselves kept hold of the painting. Finson subsequently took it to Aix-en-Provence and then Antwerp, where he died in 1617. _The Madonna of the Rosary_ would eventually become part of the royal collection of the Viennese Habsburgs, and is nowadays to be seen in Vienna's Kunsthistorisches Museum. It is a key picture in the transmission of Caravaggio's style to northern Europe. But one particular chapter in its history introduces the crucial role that would be played in that process by another great artist.\n\nIn 1620 or shortly afterwards the picture was bought from the heirs of Finson and Vinck and donated to the principal Dominican church in Antwerp by a group of painters and connoisseurs. It would remain there for more than a hundred and fifty years. The archives of the Dominican Fathers of Antwerp record that the most celebrated Flemish painter of the seventeenth century, Peter Paul Rubens, was a driving force behind the bequest: 'The large painting... now in the chapel above the altar, is a work of Michelangelo da Caravaggio and was given by various art lovers, including among others, Rubens, Bruegel, Van Bael, Cooymans. Seeing that they could acquire this extraordinary great work of art for a good price, they bought it out of affection for the chapel and to have in Antwerp a rare art work...'\n\nRubens was a middle-aged man in the early 1620s, but he had been deeply impressed by Caravaggio's work from the start of his career. Like many other artists from northern Europe, he had travelled to Rome in his youth, forty years before, to study the art of classical antiquity and the Renaissance. While he was there he had been struck, as if by the force of revelation, by Caravaggio's Roman altarpieces. The violence and drama of the works of Rubens's early maturity, such as _The Massacre of the Innocents_ , would be deeply touched by Caravaggio's influence. Through Rubens, that influence would be transmitted to Flanders and Holland, where an entire school known simply as 'the Caravaggisti' would come into being. The development of Rembrandt's subtle, shadowy realism would be part of the same story, which can ultimately be traced all the way back to the time of Caravaggio's first visit to Naples, and Rubens's early encounter with his pictures in Rome. In 1607, within just a few months of Pourbus's negotiations on behalf of the Gonzaga to buy _The Madonna of the Rosary_ \u2013 the very picture that Rubens, over a decade later, would help donate to the Dominicans in Antwerp \u2013 Rubens himself was acting for the Duke of Mantua in regard to another painting by Caravaggio that was for sale on the open market in Rome.\n\nThe picture in question was _The Death of the Virgin_ , another rejected altarpiece from Caravaggio's Roman years. Laerzio Cherubini, who had commissioned the work only to reject it in the summer of 1606, wanted to recoup his outlay. He had put it on the market in January 1607 and it had been snapped up by Caravaggio's future biographer, Giulio Mancini. The fragmentary evidence of Mancini's correspondence suggests that he paid 200 scudi for it, and that he intended to sell it to an unnamed purchaser in his home town of Siena. His letters to his brother, who was helping him with the negotiations for the sale there, show that he was concerned that Caravaggio's indecorous depiction of the Virgin might cause a stir. 'Someone knowledgeable will reprove us, but as it is for the service of God and the embellishment of the city, I will pay no attention to complaints.'\n\nBut by the middle of February, Mancini was considering other options. The Duke of Mantua, one of Rubens's most valued patrons, had shown interest in buying the picture. His agent in Rome, Giovanni Magno, had opened negotiations with Mancini and was taking advice about the painting from, among others, Rubens. It seems likely that it may have been Rubens's idea to acquire the picture for Mantua in the first place.\n\nOn 17 February, Magno wrote a cautiously encouraging letter to the duke's secretary, Annibale Chieppio, about the potential acquisition. While he himself found _The Death of the Virgin_ rather difficult and unpalatable, it had been greatly praised by the experts and connoisseurs: 'Last Sunday I saw the painting by Caravaggio, proposed by Signor Peter Paul Rubens who, when he saw it again, was still more satisfied by it... It pleased me to a degree corresponding to the concordant judgement of the professionals. However, because people of little experience desire some pleasure to the eyes, I was more impressed by the testimony of the others than by my own feeling which is not sufficient to understand well certain occult artificialities which place this picture in such high esteem. The painter, however, is one of the most famous for the collectors of modern things in Rome, and the picture is held to be one of the best paintings he has ever made. Thus, presumption is in favour of this painting in many respects, and really one can observe in it certain very exquisite parts...'\n\nIn Magno's next letter, of just a week later, he told the duke's secretary that the price for the painting had been agreed by Rubens at 280 scudi. Mancini would make a profit of 80 scudi on the deal. He was content with that, and at this point the prospective purchaser in Siena disappears from the story. By the end of March, Magno was writing to confirm that he had taken possession of the picture on behalf of the Duke of Mantua.\n\nWithin a week, the painters of Rome had heard about the purchase and were clamouring to be allowed a sight of Caravaggio's painting before it left the city. It had been removed so quickly from its intended altar in Santa Maria della Scala in the summer of 1606, just before the murder of Ranuccio Tomassoni, that almost no one had had a chance to view the work. On 7 April 1607 Magno reported to his masters in Mantua that\n\n> I found it necessary, in order to gratify the painters' guild, to let the purchased picture be seen all week long. Many of the most famous painters have been flocking there with a good deal of curiosity because this picture was the talk of the town, but scarcely anybody had been allowed to see it. It has certainly been a great satisfaction to me to let it be enjoyed by the public because it has been commended for the exceptional art with which it was done. It will be forwarded next week.\n\nIn the event, despatch of the painting was delayed because Rubens wanted to be sure that it survived the journey. On 14 April, Magno wrote to say that 'The purchased picture is at Sr Peter Paul Rubens's disposal, ready to be forwarded. But he, in order to preserve it from injuries, is having I know not what sort of case constructed, which will necessarily delay the shipment until after the holidays.' By May, _The Death of the Virgin_ had reached Mantua.\n\n#### FLIGHT FROM NAPLES\n\nClearly Caravaggio was held in deep regard by his fellow painters in Rome, despite being under capital sentence for murder. But he still had his enemies there. All had not been forgotten and forgiven. There were those who claimed that Caravaggio was still up to no good, still making trouble in the city even though he was in exile from it. Within days of his arrival in Naples, he had been accused, _in absentia_ , of another attempted murder back in Rome. It was said that an assassin named Carlo Piomontese, working to Caravaggio's orders, had tried to kill a man who was on his way into church to hear Mass. Carlo Piemontese was a painter, a man also known as _il Bodello_ , a nickname for sodomites. The victim of his alleged assault was none other than Caravaggio's old adversary Giovanni Baglione.\n\nBaglione's accusations are to be found in a series of depositions recorded by a notary in a Roman court of law at the beginning of November 1606:\n\n> Last Sunday at the 14th or 15th hour I was walking to mass at Trinit\u00e0 de' Monti. I was alone, and wearing a sword and a cape. I was walking down the stairs toward the Medici gardens when, as I set foot on the last step the said Carlo, who was hiding behind a pilaster on the stairs, attacked me with an unsheathed sword and struck me a blow that hit me on the shoulder, and tore my cloak and coat, as Your Lordship can see when I show you here... [ _Then I, the notary, saw a black cloak with a cut on the left shoulder, and a coat with a similar cut_ ] Then he aimed a blow at my head, which struck me on the arm with the flat of the sword. Seeing myself attacked in this way I put my hand to my sword also. In grasping it he wounded me in the said right hand, as you can see... [ _Then I, the notary, saw a little scar on the index of the right hand_ ] Then we exchanged some blows and my sword broke, because I think that he was wearing a breastplate, or something else of iron. Then some people came up, and we separated.\n\nUntil this attack, things had gone well for Baglione in the autumn of 1606. In September he had been knighted as a Cavaliere di Cristo. In October he had received the further honour of being voted _principe_ , or 'head', of the Accademia di San Luca. Baglione believed that his success in the elections for that post had provoked the attempt on his life. Three weeks before the attack, he said in his evidence, Carlo Piemontese had come to the academy and attempted to disrupt the vote: 'As he was not one of the Congregation, was under twenty years of age, and had no reason at all for being admitted, I told him that he should go outside until the _principe_ had been chosen. He answered me that he was a painter like the others, and as he was there already he wanted to stay, but he did it in such a way that he was not balloted, and did not vote in the creation of said _principe_ , and nothing else occurred.'\n\nThat earlier incident had passed off without violence, but Baglione believed his election had continued to gnaw at Carlo Piemontese. The would-be assassin was friendly with two other painters, Carlo Saraceni and Orazio Borgianni, who were themselves close to Caravaggio. Baglione believed that the three of them had formed a cabal, to block his campaign and ensure that a member of Caravaggio's faction be elected instead. When their plans were foiled, they resorted to violence. He knew this, he said, because on the day of the vote for _principe_ , his groom had seen Saraceni and Borgianni standing outside the Accademia with Carlo Piemontese, stirring him up into a frenzy. He had also been told \u2013 although he did not say by whom \u2013 that the mastermind of the whole plot was Caravaggio himself:\n\n> My servant told me that, while he was outside holding my horse, there came out the aforementioned Carlo, Orazio and Carlo Veneziano, and that they incited the said Carlo by saying 'that prick' and other insulting words [about me]. They did this because they wish me ill, now and in the past, and are adherents of Caravaggio, who is my enemy. I heard that he gave them something, and someone else another thing, and told them to kill me, and to bring the news to Caravaggio, who would give them a fine reward.\n\nThe final judgement of the case is unknown. Two of the accused, Saraceni and Borgianni, made unusually large donations to the Accademia di San Luca on St Luke's feast day in the following year, which suggests that the affair may have been settled out of court. Baglione's accusation was potentially very damaging for the absent Caravaggio: at the moment of his arrival in Naples, just as he was taking what he hoped would be the first steps on the way to a pardon, his name was once more associated with violence and murderous intent.\n\nBut behind the scenes it seems efforts were being made on Caravaggio's behalf by the Colonna and his other allies. In May 1607, some six months after the assault on Baglione and just as Rubens was packing up _The Death of the Virgin_ for transport to Mantua, it was again rumoured that he would soon be returning to Rome. The Este agent, Fabio Masetti, still fretting about the 32 scudi he had advanced to him sixteen months earlier, had remained alert for new developments. Reporting back to Modena from Rome on 26 May, he sounded a distinctly hopeful note: 'It has not been possible to recover the money because of a homicide committed by the said painter, on account of which he has been banished. However, as the said homicide was accidental and the painter was badly wounded too, a reprieve is being negotiated and a pardon is hoped for. So, when he is back, I shall not fail to recover the said 32 scudi.'\n\n'The said homicide was accidental' and 'the painter was badly wounded too' \u2013 Masetti was no doubt repeating the same arguments, perhaps even the very same phrases, that were being used in Caravaggio's defence. By the start of June, Masetti was sufficiently optimistic to let the painter himself know that he would be waiting for him when he got back to Rome. 'I have written a letter to Caravaggio the painter for the restitution of the 32 scudi,' he informed his superiors in Modena, 'although it was not the first one, and the other time he failed to send a reply.'\n\nYet again the hapless Masetti was to be disappointed. Caravaggio, painting _The Crucifixion of St Andrew_ , once more failed to reply. By the end of June, unpredictable as ever, he had left Naples by boat and was travelling ever further away from the city of Rome. His destination was the island of Malta, southernmost bastion of the Christian faith against Turks and Corsairs, and home to the military Order of the Knights of St John.\n\nJust why Caravaggio took the extraordinary decision to go to Malta is one of the many puzzles of his later years. Piecemeal clues in the historical archive suggest that he went in the hope of finding freedom and forgiveness. He appears to have believed that by allying himself with Malta's formidable militia of warrior knights he might win permanent redemption for his crimes. But in the tough world of the Christian soldier he would be undone, once again, by his own volatility.\n\n#### THE FRIARS OF WAR\n\nTo become a Knight Hospitaller of the Sovereign and Military Order of St John of Jerusalem, Rhodes and Malta, Custodian of the Poor of Jesus Christ and Servant of the Sick, was to join one of the most venerable and powerful organizations in the Christian world. The order's roots lay deep in the medieval past, when the religious zeal of the pilgrim and the aristocratic ethos of chivalry became closely interwoven. In the year 1070 a group of noblemen from Amalfi, in Italy, founded a hospital in the city of Jerusalem to care for fellow Christians weakened by the long pilgrimage to the Holy Land. After the First Crusade, and following the capture of the Holy City, they were formally constituted as a nursing and military order. 'The Friars of War', as they became known, were dedicated both to the service of the sick and to the defence of the Christian faith against the threat of Islam.\n\nWithin less than a century, the Knights of St John had established a vast network of hospitals and fortifications along the pilgrim routes leading from Europe to Jerusalem. Over the next two hundred years, they developed into a formidable army of aristocratic Christian warriors, building and defending a long chain of castles to safeguard the land frontiers of the Holy Land, from Asia Minor to Egypt. The Knights of St John were the crack troops of Christendom, but they also bore the brunt of wave after wave of attacks from the armies of Islam. By the end of the thirteenth century they had been made to relinquish almost all of their hard-won possessions. When the Christians were finally forced out of the Holy Land altogether, the knights were the last to leave, finally defeated at the Siege of Acre in 1291.\n\nThe history of the order over the next three centuries would be no less bloody and no less embattled. The knights found a new home on the Greek island of Rhodes, a strategically vital maritime base at the crossroads of Europe, Asia and Africa. Having captured the island, they fortified it and set about creating a fleet of fighting ships. From that time on, they were no longer an army of Christian footsoldiers but a naval force. From their base on Rhodes they mounted raids on Turkish shipping and vulnerable coastal settlements, taking slaves and capturing hostages for ransom.\n\nIn the Islamic world they were regarded as brutal and pitiless marauders. In their monastic uniform of black robes, proudly emblazoned at the chest with a white eight-pointed cross, the Knights of St John represented a militantly aggressive form of Christianity. Their activities inevitably attracted reprisals. In 1480 a Turkish fleet laid siege to Rhodes, only to be repelled with crippling losses. Forty years later, in 1522, Sultan Suleiman the Magnificent once more sent a flotilla of ships to conquer Rhodes. After six months of attack and counter-attack, the knights were finally defeated and expelled from the island.\n\nIn 1530 the Holy Roman Emperor Charles V gave them another new home, on Malta, part of his Two Kingdoms of Sicily. His motives were part religious, part strategic. Charles V wanted to protect the southern flank of Europe and ultimately Rome itself against the might of Islam. His reasoning was that if anyone could hold Malta, the Knights of St John could. He ceded it to them in exchange for an annual tribute of a single falcon.\n\nThirty-five years later, in 1565, the Turks once more laid siege to an island garrisoned by their most hated adversaries. The Siege of Malta lasted for months and would forever be remembered, both for the ferocity of the fighting and for the atrocities committed on both sides. The official historian of the Order of St John, Giacomo Bosio, included a harrowing account of it in his three-volume _Dell'Istoria della sacra religione_ , the last part of which was published in 1602, five years before Caravaggio arrived on Malta. At the height of the siege, Bosio recounted, having captured Fort St Elmo, the Turks proceeded to massacre their Christian captives. The day allotted for the killing was 24 June, the feast day of St John and therefore one of the two most holy days of the year for members of the order (the other being 29 August, the day that marked the saint's decapitation at the whim of Salome). Making grim play of the significance of the date, the Turks turned the killing itself into an obscene parody of a Christian religious festival: 'All the cadavers which by their clothing could be recognized as knights or men of importance were gathered up; and it was ordered that they be stripped nude, decapitated, and that their hands be severed. Then, out of disresepect for the Holy Cross and to make sport of the knights' military overgarments, on each corpse four huge incisions were made with scimitars, making the sign of the Cross on both the fronts and the backs.'\n\nOn a later occasion, in a similar spirit of vengeful parody, the Turks crucified a number of headless knights' corpses and floated them into the harbour at Birgu. Bosio wrote that 'after having had them lashed to various pieces of wood with their arms spread apart so as to form, similarly, the sign of the Cross, and bound in such a manner as to make one body tow the other in a long chain, they were then tossed into the sea. The water, it was thought, would carry them and this truly horrible spectacle over to our brethren at Birgu, and it in fact did so.' The intention was to strike terror into the last remnants of Malta's Christian garrison. But the Grand Master of the Order of St John, a doughty Frenchman named Jean de la Valette, responded to the Turks' flotilla of death with a fusillade of his own. He ordered his Turkish captives to be decapitated and had their heads fired from cannons at the Turkish soldiers occupying Fort St Elmo. In the end the knights held firm, despite their crippling losses, and the Turks were forced to withdraw. By the end of the siege just 50 Knights of St John survived. More than 7,000 defenders had lost their lives, but the last great effort of the Ottoman Turks to seize control of the western Mediterranean had been successfully repulsed.\n\nOver the decades that followed there was a surge of new recruits to the Order of St John, lured to Malta by the dream of emulating the exploits of the heroes of 1565. Just six years after the siege, that other famous Christian victory, at the Battle of Lepanto, had fanned the flames of such enthusiasm yet further. Hundreds of young noblemen from the leading families of Europe travelled to Malta to seek knighthood, honour and glory. They wanted to fight, and if necessary die a martyr's death, at the front line of conflict with the forces of Islam.\n\nAccording to Bellori, Caravaggio too nurtured the dream of becoming a Knight of St John. He was 'eager to receive the Cross of Malta', in the words of the biographer. But why? His art had electrified Naples. According to sources in Rome, steady progress was being made in the negotiation of his pardon. It would seem like an odd moment to travel yet further south, to a barren and rocky island at the farthest frontier of Christendom. It is possible that the fantasy of becoming a knight had long been with him \u2013 after all, he was a keen and talented swordsman, who had been brought up in the immediate aftermath of the Battle of Lepanto. Or perhaps he still felt vulnerable to attack or apprehension by a bounty hunter, aware of the price on his head. By papal dispensation the Knights of St John were above the law, subject to their own unique legal code. In Malta, Caravaggio would be safe. Furthermore, if he could win a knighthood he would, _de facto_ , have gained pardon for his crimes.\n\nBut there was probably more to it than that. Caravaggio had always been extremely touchy about status. At his trial for libel, he had contemptuously dismissed the rank and file of Rome's artists by saying that hardly any of them deserved the title of _valent'huomo_ , literally, a 'worthy man'. Caravaggio took pride in his own worth. The poems attacking Giovanni Baglione, in which he certainly had had a hand, made much of the gold chain awarded to his rival. The perceived injustice of the honour clearly rankled with Caravaggio as much, if not more, than anything Baglione had actually said or done. But by the summer of 1607, nearly a year after the murder of Tomassoni, Baglione had just been knighted and his stock had risen yet further. Caravaggio, by contrast, was still a fugitive from justice. Even if he were pardoned and allowed to return, he would be going back to Rome as a man in disgrace. But to return, himself, with a knighthood \u2013 and not just an honorific papal knighthood but a knighthood in the Order of St John, proudly wearing the eight-pointed cross on his chest \u2013 that would be very different. If he could manage that, he could face his rivals down.\n\nJoachim von Sandrart tells an undoubtedly apocryphal tale about the cause of Caravaggio's decision to go to Malta, which, for all its evident fancifulness, may contain a kernel of truth. In Sandrart's story, Caravaggio's former employer Giuseppe Cesari, on horseback, passes him one day in the streets of Rome. Caravaggio challenges Cesari to a duel and tells him to dismount from his horse so that they can fight. But he is rebuffed:\n\n> Giuseppe answered... that it was not fitting for a knight, named by the Pope, to duel with someone who was not a knight. With this politely cutting answer, he wounded Caravaggio more than he might have with his sword, for this talk so stunned and confused Caravaggio that he immediately (as he did not intend to defer the matter) sold all his belongings to the Jews for whatever he could get, and set out for Malta and the Grand Master with the purpose of soon himself becoming a knight...\n\nThe tale is clearly a fiction, because Caravaggio was nowhere near Rome when he decided to go to Malta. But it has the ring of psychological truth. The unpalatable thought of lesser painters being dubbed knight may well have impelled him on his Maltese adventure.\n\nMalta was not, however, a place where someone could simply turn up unannounced. The whole island was a fortress, and security was tight. No one was allowed in from the mainland without a passport and papers prepared by the order's network of receivers. The receiver in Naples was a high-ranking official named Giovanni Andrea Capeci. Capeci would have had to gain approval from the Grand Master of the order on Malta itself before completing the necessary paperwork, and such permissions, especially for a fugitive from papal justice, were no simple matter. One of Caravaggio's friends in high places would have been needed to broker the arrangement with the Knights of Malta. Who helped him? There are a number of possibilities, because several people in the painter's network of patrons and protectors turn out to have had links with the Order of St John.\n\nIn the summer of 1607, at exactly the same time as Caravaggio chose to go to the island, two cousins of the noble Giustiniani family \u2013 avid collectors of Caravaggio's work in Rome \u2013 were on their way to Malta to offer the Grand Master a family property in Venosa, near Naples, as a naval base for the knights on the mainland. Perhaps they were prevailed on to put in a good word for the talented artist with a criminal record.\n\nOttavio Costa, the banker who had recently bought Caravaggio's second _Supper at Emmaus_ while the artist was in hiding in the Alban Hills, also had connections with the Knights of the Order of St John. His wife's uncle was Ippolito Malaspina, an illustrious member of the heroic old guard of Malta, and something of a living legend. A veteran of the great Siege of 1565, Malaspina had gone on to captain one of the Maltese galleys at the Battle of Lepanto, in the year of Caravaggio's birth. In 1603 he had been appointed commander of the papal fleet, as a result of which he temporarily delegated his responsibilities on Malta and moved to Rome for two years \u2013 years during which Caravaggio painted some of his most highly acclaimed Roman altarpieces. Malaspina would certainly have known of the painter's work and may even have met him. By the summer of 1607 he had been reappointed to a number of senior posts in the order, including Prior of Naples. He was very close to the Grand Master himself, a Frenchman named Alof de Wignacourt, having played an important part in Wignacourt's election in 1601. The possibility that Malaspina's advocacy might have been instrumental in Caravaggio's acceptance on Malta is strengthened by the fact that one of the first pictures the artist painted when he got there \u2013 another depiction of _St Jerome Writing_ \u2013 was done for Malaspina himself: the Malaspina family crest is prominently painted into the right-hand edge of the canvas.\n\nThis was not the total of Caravaggio's contacts with the upper echelons of the order. The idea of going to Malta, to seek redemption for crimes committed, almost certainly emanated from his most constant guardians and protectors, the Colonna dynasty. A prominent member of the Colonna family had recently done exactly the same thing himself.\n\nIn 1602 Costanza Colonna's second son, Fabrizio Sforza Colonna, had been convicted of crimes considered so shameful that their precise nature was left unmentioned in the reports of the day. Following his arrest, he was taken to Rome and imprisoned while the pope considered his case. Costanza Colonna pleaded for mercy on her son's behalf. In deference to his rank, the pope decided to give the noble prisoner a second chance. He was sent to Malta in 'privileged exile', on condition that he remain on the island for at least three years, placing himself at the service of the Christian faith. By 1605 this black sheep of the Colonna family was deemed to have expiated his sins, and had been elected co-Prior of Venice, a post that he shared with his uncle, Ascanio Colonna. The following year he was made a member of the governing Venerable Council of the order and elevated to the rank of General of the Galleys. There could hardly have been a better way for the grandson of Marcantonio Colonna, hero of Lepanto, to complete his return from disgrace and exile.\n\nA less exalted version of the same process of redemption seems to have been planned for Caravaggio. Costanza Colonna, who had seen things go so well for her son on Malta, may well have been the driving force behind the whole scheme. She had long taken a virtually maternal interest in Michelangelo Merisi da Caravaggio, who was close to the same age as her own Fabrizio. What had worked for one difficult young man might work for the other.\n\nA number of recently discovered documents place Fabrizio and Costanza Colonna in Naples in the summer of 1607. In fact they both arrived in the city just a matter of days before Caravaggio embarked for Malta. It has also emerged that he made the journey to the island in one of a flotilla of galleys commanded by none other than Fabrizio Sforza Colonna himself.\n\nOn his first voyage as General of the Galleys, Fabrizio Sforza Colonna had travelled to Barcelona to take delivery of a new flagship and a large number of slaves and convicts donated to the order by the Spanish crown. Discovering that the new Spanish flagship was poorly constructed, he had a replacement fitted out in the shipbuilding port of Marseilles. By the early summer of 1607 he was back in Italian waters, collecting his mother, Costanza, from the Torre del Greco near Naples, the spectacular seaside residence of the princes of Stigliano. The two of them carried on to Naples itself, where the final arrangements for Caravaggio's journey to Malta would soon be concluded.\n\nSo it was that on 25 June 1607, bearing with him the good wishes of his protectress, Caravaggio embarked for the island fortress of Malta. It is not known whether his faithful assistant and rumoured lover, Cecco, accompanied him. Probably, he did not: Cecco appears in no more of Caravaggio's paintings after this date.\n\n#### THE ISLE OF ST JOHN\n\nThe voyage to Malta was fraught with tension. The flotilla's first stop was Messina, in Sicily, where Fabrizio Sforza Colonna received an urgent warning from Grand Master de Wignacourt. Seven large galleys from the Barbary Coast had just been sighted in the waters off Gozo, Malta's sister island. Five of them had disembarked soldiers and mounted an unsuccessful attack on the order's garrison there. Wignacourt suspected that the enemy had received intelligence about the imminent arrival of the flotilla from Naples and intended to engage them in battle. He was concerned about 'the advantage that the enemy has because of the larger number of vessels and because our galleys are burdened and with provisions in tow'.\n\nBy the beginning of July, the enemy vessels were still in Maltese waters, so Fabrizio Sforza Colonna continued to delay his departure for the last leg of the journey. Meanwhile, Grand Master Wignacourt sent a frigate from Malta to reinforce the flotilla. On 10 or 11 July the galleys of the order left Sicily. All on board would have been in a state of alert, fully armed for combat. In the event, the journey passed without incident. On 12 July, in the fierce heat of midsummer, Caravaggio arrived in the harbour of Malta's capital city, Valletta.\n\nTo a man in search of renewal and redemption, it must have been an inspiring sight. An entirely new city, built of honey-coloured limestone that glowed pink in the sun, Valletta had been constructed at breakneck speed in just forty years. After the turmoil of the Great Siege, the knights realized that they had to fortify the narrow headland known as the Xiberras Promontory, which connected the island's two principal harbours. The construction of the new capital by an army of slaves, on the steepest incline of the headland, had been an immense undertaking but once complete it meant that the knights' principal garrison was all but impregnable. It was named in honour of Jean de la Valette, Grand Master during the siege. The pope's best military engineer, Francesco Laparelli, was responsible for the plan. The sheer stone fortifications of the citadel rose directly from the craggy outcrop of the island itself, with the sea acting as a moat on both sides.\n\nWithin its walls, Valletta was laid out on the Renaissance model of the ideal city. The principal architect responsible for the buildings was Girolamo Cassar, who was from Malta but had studied in Rome. His palaces and churches were designed to reflect the knights' ideals of Christian sobriety and military discipline, with long, severe fa\u00e7ades of rusticated stone. The streets were laid out in a grid, with nine thoroughfares running across the peninsula and twelve running from top to bottom. Their strict geometry was softened by gardens and fountains, providing shade and water. Getting from the harbour end of Valletta, up the steep hill to the centre of town, and to the grand Cathedral of St John, was hard work even for the fittest. (Centuries later, the club-footed English poet Byron would bid farewell to Malta with the words 'adieu, ye cursed streets of stairs'.)\n\nApproaching Malta for the first time, Caravaggio was surrounded by symbols of the island's fierce rule of law. On the first promontory on the left of the harbour was the forbidding spectacle of a gallows. Within the harbour itself, prominent on the left-hand side, was the Castel Sant'Angelo, where many of the most famous events of the siege had taken place. By the time of Caravaggio's arrival, it had become a prison for disorderly knights. Another hallowed site from the recent Maltese past was the Castel Sant'Elmo, where so many members of the order had lost their lives in 1565. A late sixteenth-century German visitor to Malta, Hieronymus Megiser, noted that some of the rocks there were still visibly sprinkled with gore. The stains were pointed out with pride by his Maltese hosts, as the glorious blood of Christian martyrs.\n\nMalta was a remote and harsh place, rocky and sun-parched, unlike anywhere Caravaggio had ever known. But it was also fertile, having been famous since antiquity for the quality of its cotton \u2013 Cicero had had his clothes made on Malta \u2013 as well as for the sweetness of its honey and its bounteous quantities of almonds, olives, figs and dates. As Megiser noted, the island encompassed two utterly distinct societies, 'Malta Africana' and 'Malta Europeana'. The world of the indigenous islanders had remained unchanged for centuries. Its people were dark-skinned, spoke a language incomprehensible to Europeans and lived in humble settlements much like the tribal villages of nearby coastal north Africa. Cosmopolitan Valletta was utterly different, a flammable blend of extreme Christian piety, simmering military aggression and barely contained sexual dissipation.\n\nTo the English poet and adventurer George Sandys, who unknowingly followed in Caravaggio's footsteps four years later, the two Maltas were indeed worlds apart:\n\n> The _Malteses_ are little lesse tawnie than the _Moores_ , especially those of the country, who go halfe-clad, and are indeed a miserable people: but the Citizens are altogether Frenchified; the Great Maister, and major parts of the Knights being _French_ men. The women wear long blacke stoles, wherewith they cover their faces (for it is a great reproch to be seen otherwise) who converse not with men, and are guarded according to the manner of Italy. But the jealous are better secured, by the number of allowed curtizans (for the most part _Grecians_ ) who sit playing in their doores on instruments; and with the art of their eyes inveagle these continent by vow, but contrary in practise, as if chastitie were only violated by marriage. They here stir early and late, in regard of the immoderate heat, and sleep at noone day.\n\nIt is not known where Caravaggio lived during his time on the island. Prospective knights on their first tours of duty were given accommodation in the auberge belonging to their particular Langue, or country. Altogether there were eight Langues, of Italy, Provence, Auvergne, England, France, Aragon, Castille and Germany. The Italian auberge, with its long fa\u00e7ade decorated with trophies and escutcheons, was close to the main city gate of San Giorgio. But Caravaggio is unlikely to have lodged there with the other Italian novices, because when he first arrived no one other than his Colonna protectors seems to have known of his plan to be elevated to a knighthood. It appears from the archive that his desire for the Cross of St John was not communicated to the highest levels of the order until the winter of 1607. So he probably lived in the household of Fabrizio Sforza Colonna, at least during the first months of his stay.\n\nCaravaggio was swiftly made aware of the sharp divide between public morality and the private behaviour of the knights and their companions. On 14 July, two days after his arrival on the island, a welcome party was thrown for him and a number of other new arrivals by a Sicilian knight named Giacomo Marchese. Marchese was overheard joking about a Greek painter who kept two wives. But for at least one of the other guests, it was no laughing matter. Judge Paolo Cassar, a doctor of civil and canon law, promptly denounced the unnamed painter to the Inquisition. On 26 July, Caravaggio was called by the Inquisitor, Leonetto Corbiaro, to answer questions about the identity of the alleged bigamist. He answered with his customary reticence, learned in the courts of Rome:\n\n> About that which you ask me, most Reverend sir, I know nothing except that in the house of the Knight Fra Giacomo de Marchese there was staying a Greek painter who arrived with the galleys, but about the rest I have nothing to say concerning the said knight nor about anything which concerns the Holy Office [of the Inquisition] also I know nothing of the name of the said painter nor of which country he claims as his homeland.\n\nThe case fizzled out, but it was a clear sign of how easy it could be to get into trouble with the law on Malta. Even more fearsome than the Inquisition was the Grand Master, Alof de Wignacourt, whose rule on the island was absolute. 'This man is a Pickard borne,' Sandys would write, 'about the age of sixtie, and hath governed eight years. His name and title, _The illustrious and most reverent Prince my Lord Frier Alosius of Wignian-court, Greate Maister of the Hospitall ofSainte Johns of Jerusalem: Prince of Malta and Goza_ [ _sic_ ]. For albeit a Frier (as the rest of the knights) yet is he an absolute Soveraigne, and is bravely attended on by a number of gallant young gentlemen.' Like all Knights of Malta, Wignacourt was bound by vows of poverty and celibacy. But he lived in grand style none the less, in the Grand Master's Palace, an elegant building constructed around a courtyard garden, with walls frescoed with scenes of the Great Siege by a minor Italian artist named Matteo Perez d'Aleccio \u2013 who, like Caravaggio, had fled to Malta after getting into trouble in Rome. Wignacourt surrounded himself with young page boys, the flower of the European aristocracy. On his death, he bequeathed to the order more than 200 slaves and a fortune in ransom money.\n\nAs supreme authority on Malta, Wignacourt was answerable only to the pope. He presided over the Venerable Council of the order, composed of the eight Conventual Bailiffs \u2013 one for each Langue \u2013 and the Grand Priors. The Venerable Council framed the order's statutes. Wignacourt was also in charge of the Criminal Council, which had the often demanding job of ensuring that those statutes were obeyed. As the leader of an all-male elite fighting unit, especially trained in privateering, pillage and kidnap, one of his main priorities was simply to preserve order. This was by no means easy, and a blind eye was diplomatically turned to certain habitual misdemeanours. Wignacourt made no attempt to close down Valletta's many brothels: in 1581, when one of his predecessors had attempted to eliminate prostitution on Malta, the result had been a full-scale riot. But other offences were dealt with swiftly and ruthlessly, on a sliding scale of punishment.\n\nThe list of prohibitions and mandatory penalties is itself a testament to the difficulty of maintaining order among several hundred proud Knights of Malta. Punishment for the offence of being incorrectly dressed, without the eight-pointed cross of the order, was the 'quarantaine', which insisted that the miscreant be confined to his auberge for forty days, during which time he was to fast in penitence and submit to regular public floggings by the vice-prior in the conventual church. Repetition of the same offence brought a three-month prison sentence. The penalty for rowdy behaviour inside the auberges was deprivation of seniority within the hierarchy of the knights. Insults traded between brother knights in the Grand Master's presence meant the loss of three years' seniority. More serious crimes were punished by defrocking, the permanent deprivation of a knight's habit. This was the penalty ordained for a variety of offences, including assault on a fellow knight, heresy, apostasy, theft, duelling and the abandonment of comrades in battle. If a knight killed in anger, he was sentenced to a traditional Maltese death. The procedure was described by George Sandys: 'If one of them be convicted of a capitall crime, he is first publicly disgraced in the Church of Saint John where he received his Knight-hood, then strangled and thrown into the sea at night-time.'\n\nIt would be harder to enter the brotherhood of Malta than Caravaggio had perhaps imagined. Knights of Justice were the elite of the order, from whose ranks the Grand Crosses who sat on the Venerable Council were drawn, but to be considered for such a knighthood the candidate had to be able to prove unbroken noble lineage of two hundred years. Below Knights of Justice came Knights of Grace, but they too had to prove a high degree of nobility. Considering his humble origins, Caravaggio could only aspire to the still lower Knighthood of Magistral Obedience, which was reserved for men of merit \u2013 _valent'huomini_ , to use his own favoured terminology \u2013 and awarded at the discretion of the Grand Master. But just before Caravaggio's arrival on the island, Wignacourt had introduced a statute putting an end to the conferment of such knighthoods. He had grown irritated by the number of applicants for them and felt they were cheapening the status of the brotherhood as a whole. Honorific knighthoods were viewed as being open to corruption, tradeable awards akin to a form of simony. On his travels in Palestine, Sandys was contemptuously amused by the sight of an apothecary from Aleppo being dubbed a knight in exchange for hard cash.\n\nIf Caravaggio were to become a Knight of Malta, special arrangements would have to be made by the Grand Master himself. Little is known about the artist's activities during his first several months on the island. But on the evidence of three pictures that he painted during the second half of 1607, he was working hard to impress those in the upper echelons of the order. With each new commission, he moved closer to the centre of power.\n\n#### SIGNED IN BLOOD\n\nSoon after establishing a workshop on Malta, Caravaggio painted a sombre devotional picture for Ippolito Malaspina. _St Jerome Writing_ had the same subject as the picture painted for the papal nephew, Scipione Borghese, two years earlier. The image of the saint, this time, is less a generic old man and more the portrait of an actual individual. Spotlit in the gloom of his study, Jerome has wispy grey hair scraped across the sunburned crown of his head, deep wrinkles, a slightly cauliflowered right ear \u2013 emphasized by the raking light \u2013 and a dimpled, beak-like nose. In contrast with his coppery, weathered face, his bare torso is pale and white. His physique is lean, although the skin at his ribs and belly has begun to sag with age.\n\nThe saint looks down at the pages of the book in which he is writing. In his right hand he holds a quill, in his left an inkpot. On the desk before him lie three symbolic objects: a stone the colour of a bruise \u2013 the stone with which, according to his legend, he used to beat his breast; a tip-tilted skull, eyes gaping and teeth glinting; and a crucifix on which a stretched figurine of the agonized Christ is represented in shadowy foreshortening.\n\nTo the saint's right, his red cardinal's hat hangs from a rudimentary peg. All else is in shadow. Semi-nude, swathed up to the waist in a sheet of red drapery, Jerome the scholar-saint looks more like a military man sitting up in bed before first light, writing out the orders of the day. The sinews at the juncture of neck and shoulderblade are taut with nervous energy. Did Caravaggio model him on Malaspina himself? One of Wignacourt's closest advisers, Malaspina had been away from Malta for four years, and had returned on the same flotilla that had brought the artist to the island. Now in his late sixties, he had chosen to rededicate himself to the Order of St John, and to God. Caravaggio's picture was perhaps intended to commemorate that decision.\n\nWith its skilful foreshortenings, dramatic light and shade and compellingly lifelike depiction of dignified old age, the picture was a virtuoso performance and a demonstration of just what Caravaggio could do for the Knights of the Order of St John. Malaspina would eventually bequeath the _St Jerome Writing_ to the chapel of the Italian Langue \u2013 it now hangs in the co-cathedral of St John in Valletta, having survived a heist in 1985, during which it was cut out of its frame with a Stanley knife \u2013 but originally he hung it in his house. Because Malaspina was in Wignacourt's immediate circle, the painting would soon have been seen by all the right people. More commissions followed.\n\nIn the autumn or winter of 1607 Caravaggio was approached to paint the likeness of one of the most senior and distinguished Knights of Malta, Fra Antonio Martelli. The picture, which now hangs in the Pitti Palace in Florence, is one of the most impressive of all seventeenth-century portraits. This depiction of an obdurate and forceful man, in lean old age, rheumy eyes gazing off into the distance, anticipates the mature portraiture of Rembrandt by some half a century. In the darkness of Caravaggio's Maltese studio, the air feels dense with thought. The old warrior, mouth set in an expression of habitual determination, looks out and away \u2013 but it is really as if he is looking _within_ , sifting his own memories and remembering his old battles. His left hand rests on the pommel of his sword, a swiftly painted tangle of finely wrought metal, while in his right hand he holds a string of Rosary beads. These are the twin attributes of the Friar of War, dedicated at once to God and the profession of arms. Caravaggio has painted the hands so cursorily they seem unfinished. It was the sitter's face that fascinated him.\n\nMartelli was seventy-four years old when Caravaggio painted him and had been a Knight of Malta for almost fifty, a doughty veteran of the Great Siege and numerous other battles and engagements. He had served, for many years, as Ferdinando I de' Medici's _consigliere di guerra_ , his 'councillor of war'. He was a canny and gifted diplomat, the scion of an ancient Florentine family that had made the mistake of opposing the Medici earlier in the sixteenth century, and had paid the price. Martelli had undone the damage and re-established the influence of his family in the Grand Duchy of Florence. In the portrait he is dressed for winter, in a thick black monastic habit with a white long-sleeved shirt underneath. A large, eight-pointed cross of the Order of St John gleams on his chest in the half-dark. The habit marks him out as a Grand Cross of the order, the highest rank to which an elite Knight of Justice could rise. Martelli was indeed a member of the Venerable Council, an ally of Wignacourt and Malaspina. He had been appointed the order's Prior of Messina in 1606, and was regarded in Medici circles as something of a phenomenon: as late as 1618, when Martelli was in his eighty-fifth year, the Grand Duke of Florence would appoint him General of Artillery in Tuscany. There is a strong sense of his formidable character in Caravaggio's portrait, even a hint of hero-worship about it. As a child, thanks to the plague, he had known few men on whom to model himself. A man like Martelli may have awoken in him not only admiration, but something resembling the emotion of filial respect.\n\nBy November or December, Caravaggio was painting the Grand Master himself. According to Bellori he depicted Wignacourt both seated and standing with a page, but only the second picture survives. It can be seen in the Louvre. The painting is somewhat damaged but remains none the less an impressive essay in a type of grand manner portrait originally pioneered by Titian in his work for the Spanish royal family more than half a century earlier. It is a more old-fashioned, stiff, conservative work of art than the portrait of Martelli, which may be a reflection of Alof de Wignacourt's unbending sense of his own importance. Instead of a monastic habit the Grand Master wears an ornate suit of mid sixteenth-century armour, a deliberate anachronism intended to evoke the glorious past, and in particular the heroism of the Great Siege. Bellori says the portrait was actually hung in the knights' armoury, where the similarly ornate armour of Jean de la Valette was also displayed, which must have reinforced the association. He holds the baton of high office in his gauntleted hands.\n\nWignacourt had a prominent wart on the left side of his nose, which Caravaggio has been careful to shroud in shadow. The Grand Master looks off to one side, but his look, unlike that of Martelli, is neither introspective nor retrospective. He looks to the future with necessary vigilance, the guardian of Christendom's frontline with Islam. While Wignacourt strikes a slightly creaking pose of authority, his adolescent pageboy enters the scene, stage right, with an expression of cool wariness in his wide and curious eyes. He holds the Grand Master's helmet against his downy cheek, allowing its plumes to caress the side of his face. The pageboy's identity is unknown, but a possible candidate is Alessandro Costa, son of Caravaggio's patron Ottavio Costa. He had travelled to Malta on the same flotilla as the artist, entering Wignacourt's entourage of pages on his arrival. Within the conventions of state portraiture, he represents innocent youth, in contrast with Wignacourt's wise old age. But his presence also adds an irregular, unexpected frisson of eroticism to the scene. Caravaggio's evident interest in the boy threatens to unbalance the composition.\n\nBy all accounts, Wignacourt was delighted by the portrait. The artist's biographers are unanimous in asserting that Caravaggio received the cross of the Order of St John as a direct reward for the Grand Master's portrait. Wignacourt may even have discussed the matter with him during sittings for the picture. If so, he would have made Caravaggio aware that there was a considerable stumbling block to the conferment of the honour. Having abolished the Knighthood of Magistral Obedience, the only kind for which Caravaggio might have been eligible, Wignacourt would have to appeal directly to Pope Paul V for permission to reinstate it: he was obliged to seek papal support whenever going beyond the letter of the order's statutes. That is exactly what he did.\n\nOn 29 December 1607 the Grand Master wrote to his ambassador in Rome, Francesco Lomellini. He briefed Lomellini on his desire to bestow knighthoods of Magistral Obedience on Caravaggio and one other person. He did not name the painter, simply referring to him as 'a person of great virtues, honourable, and respectful', and explaining that he wanted to give him a knighthood in order so as 'not to lose' him \u2013 ' _per non perderlo_ '. Wignacourt wrote separately to the historian of the Order of St John, Giacomo Bosio, who was then in Rome, urging him also to lobby for papal favour in the matter. In addition, Wignacourt's secretary, Francesco dell'Antella, wrote in support of the petition to grant the two knighthoods. The message was duly passed on to the pope himself:\n\n> Most Holy Father, the Grand Master of the [Order of the] Hospital of St John of Jerusalem wishes to honour some persons who have shown virtue and merit and have a desire and devotion to dedicate themselves to his service and that of the [Order of the] Hospital and does not have at the present moment any more suitable way of doing so; he therefore humbly begs Your Holiness to deign to grant to him, by a Brief, the authority and power for one time only to decorate with the habit of a Magistral Knight two persons favoured by him and to be nominated by him; despite the fact that one of the two had once committed homicide in a brawl and despite that it is prohibited by the Chapter General of the Order that the habit of a Magistral Knight can be conceded any further. He begs to receive this request as a very special favour, because of the great desire he holds to honour such persons who have shown virtue and merit. And may the Lord preserve you for a long time.\n\nThe request was granted at once. Papal permission was given in a missive of 7 February 1608, spelling out that 'It has pleased the Most Holy Father to approve for Aloph de Wignacourt Grand Master of the [Order of the] Hospital of St John of Jerusalem authority to present the habit of a Magistral Knight to two persons favoured by him despite the fact that one of the two committed homicide in a brawl.' On 15 February the letter reached Malta. Wignacourt had secured for Caravaggio his much coveted knighthood.\n\nThere were two conditions. Like any other novice, he could not be dubbed knight until he had spent a full year on the island, so he would have to stay until mid July to receive the honour. He would also have to pay a tribute known as the _passaggio_ before he could be allowed to enter the brotherhood. Being a fugitive from justice, Caravaggio had little money, but Wignacourt had a solution to that as well. The Oratory of St John, attached to the co-cathedral of St John in Valletta, had only recently been completed. It was one of the most important buildings in Malta's new capital. But it had no altarpiece. If Caravaggio would supply one, the picture would be accepted in lieu of his _passaggio_.\n\nThe subject specified for the work was _The Beheading of St John_ , which also meant a deadline before the end of summer. Wignacourt wanted to unveil the work on the Feast Day of the Decollation of St John \u2013 the day that marked his beheading \u2013 which was 29 August. Ideally the artist would have finished the work by July, so he could receive his knighthood exactly a year after his arrival on the island.\n\nIt is intriguing that Wignacourt should have omitted Caravaggio's name from his letter to the pope. Perhaps he had been tipped off that there were those in Rome who would lobby against the petition if they knew that it was to benefit Caravaggio; or perhaps he worried that Paul V might himself object, because conferment of a Maltese knighthood automatically commuted a capital sentence to one of exile. The reference to a man 'who had once committed homicide in a brawl' was a smokescreen: the phrase made it hard for anyone in Rome to connect the candidate for a Maltese knighthood with Caravaggio, who had killed a man not in a brawl but in a premeditated duel, which was a very different matter. It is possible that Caravaggio himself had lied about the murder to Wignacourt \u2013 duelling was banned on Malta and greatly frowned upon by the Grand Crosses of the order. Whether he did so or not, the Grand Master himself was being economical with the truth. It seems he was absolutely determined that the deal should go through.\n\nWignacourt was a dynamic and formidable Grand Master, with great aspirations for Malta. Caravaggio's portrait shows him as the proud absolute ruler of a brand-new city, founded on the monastic ideals of Christian chivalry. But he must have been conscious to a degree that Valletta was something of an artistic desert. Wignacourt had tried once before to address that deficiency, attempting but failing to lure an unnamed Florentine painter to Malta in 1606. the Grand Master knew that it would be hard to tempt any truly sought-after artist to faraway, provincial, sun-baked Malta, in the shadow of the threat of Islam. But now fate had brought Caravaggio to the island. He had even come of his own accord.\n\nThe Grand Master's ambitions and the painter's needs might appear to have dovetailed perfectly: Wignacourt would get his great altarpiece, while Caravaggio would get his knighthood, and the death sentence that had hung over him for nearly two years would be lifted. But the painter may not have understood the true nature of the deal being dangled in front of him. That phrase in Wignacourt's first letter to his Roman ambassador Lomellini, in which he talks of knighting Caravaggio in order 'not to lose him', is telling. Not to lose him carries a further implication, which might seem obvious but has gone largely overlooked: to keep him. By giving Caravaggio a knighthood, Wignacourt would automatically acquire the power to do just that. Under the statutes of the order no Knight of Malta was allowed to leave the island, even for a day, without the Grand Master's permission. For Caravaggio, his knighthood was a short-cut back to Rome. But there is nothing to indicate that Wignacourt viewed it like that at all. It was just as likely his way of laying a trap. Having got a great painter to Malta, why should he ever let him go?\n\nThere was no reason why any of this would have dawned on Caravaggio until his actual arming as a Knight of Magistral Obedience. Only then need he be informed about the extent of the obedience required of him. Meanwhile, in the spring and summer of 1608, he concentrated on planning and painting the largest altarpiece of his entire career, _The Beheading of St John_. The daunting scale of the work, which was to be over ten feet high and more than fifteen feet across, meant that he probably had to change workshop. In addition, models would have to be found and a few necessary props sourced: a butcher's knife, a gilded plate, a sheepskin and a length of rope.\n\nThe story of St John's martyrdom is told in the New Testament books of Matthew (14:3\u201312) and Mark (6:17\u201328). King Herod had thrown John into prison because he had dared to reprimand him for his illicit marriage to Herodias. Herod's consort plotted with her daughter, Salome, to bring about John's execution. At the king's birthday feast, Salome danced so seductively for Herod that he granted her anything she desired. She asked for the head of John the Baptist. An executioner beheaded the saint in his prison. The severed head was laid on a platter and given to Salome at the feast.\n\nThere were two main conventions for artists painting John's martyrdom. Either they depicted the moment when the dish was served up to a gloating Salome, or they depicted the instant before the beheading, with the executioner poised to strike. Caravaggio painted his own version of the latter subject, but imagined something even darker taking place. The scene is set in the gloomy courtyard of an oppressively harsh prison, beside a gateway built of massively heavy stone quoins and a barred window at which two prisoners huddle pathetically for a glimpse of the killing. The executioner is another in the line of Caravaggio's impassive, workmanlike killers. He leans over the body of his victim, whose hands are trussed cruelly behind his back. The executioner has laid down his sword, the cold steel of its blade glinting on the dull earth.\n\nShockingly he has made a botch of the job, cutting deep into the saint's neck, deep enough to sever the jugular, but leaving the head still attached to the trunk. Now he reaches behind him for the sharpened knife in the scabbard at his belt, which he needs to cut the last flap of flesh connecting John's head to his body. He grabs the saint by his hair so that he can get at the place he needs to with his knife. He might be a butcher working at his slab.\n\nDoes the saint still live? His pale face seems animated, as if he were in his last death agonies, recoiling from the gurgled, choking rush of his own blood. In the frozen world of Caravaggio's painting, he must wait forever for the _coup de gr\u00e2ce_. A swathe of red drapery has been thrown carelessly across his otherwise naked body. This sudden shock of colour in the prison gloom emphasizes the atrocious nature of what is taking place. It is like a pictogram or symbol of bloodletting in the dark. The martyr lies on a sheepskin, which symbolically makes of him a blessed Christian lamb, brought to sacrifice. The painter has contrived to pick out the martyr's naked left foot with a stray shaft of light. Surrounded by pools of darkness, placed next to some twisting coils of rope, it is almost like a still life detail \u2013 separate from the rest of the scene and yet emblematic of the poor and painfully solitary death which the saint endures.\n\nThe novices of the Order of St John listened to sermons and received instruction in the oratory for which Caravaggio's painting was destined. The place was both a school for the martyrs of the future and a burial ground for the martyrs of the past \u2013 the bones of the knights who had died at the Great Siege were interred beneath its stone-flagged floor. Within the oratory, novices were trained in the hard ways of the Knights of Malta and made to understand that they too might have to face death in a distant land at the hands of unbelievers. Caravaggio's altarpiece was designed to make sure that they could be under no illusions about what that might mean. A martyr's death brought the reward of eternal glory with the saints in heaven, but there would be nothing glorious about the death itself. It could be a death much like this one, a sordid act of butchery in a dark and lonely place. The picture is like a catechism, an asking of questions. Are you sure you have it in you to be a Knight of the Order of St John? Are you ready to die? To die like this?\n\nNext to the executioner, underscoring Caravaggio's transposition of John's legend to a cruel present, stands the figure of a Turkish jailor with heavy black keys dangling at his belt. He directs operations with an air of weary impatience, pointing unnecessarily at the richly chased and gilded plate onto which the severed head must be placed. Beside him stands an old woman with her head in her hands, distraught at the spectacle of the martyrdom. She is another version of the goitrous peasant woman gazing piteously at the crucified body of the saint in _The Crucifixion of St Andrew_. She stands for Christian pity and prayer. The main group of five figures is completed by that of the serving girl who has been sent to collect the head. Her pose has an eloquent woodenness about it. She is trying her best to carry out a task that appals her, affecting a mechanical workaday demeanour that the expression on her face belies. She stares fixedly down at the plate in her hands, pursing her lips like somebody desperately stifling the impulse to puke.\n\nWhat she cannot bear to look at is the spurting of the saint's blood from the deep gash in his nearly severed neck. It is so thick that it resembles a skein of red wool laid on the ground. Beneath the main pool of coagulating gore, there lie some thinner threads of blood. Anyone looking closely at the picture sees that they have been made to spell out the letters of Caravaggio's own name: 'F. Michelangelo'. Inscribed in the blood of St John the Baptist himself, this is the only example of the artist's signature. He had never signed a painting before, and would never do so again.\n\nThis boldly idiosyncratic gesture has been subjected to a variety of anachronistic modern interpretations. It has been read, for example, as the veiled retrospective confession of Caravaggio, the murderer; and as a proto-Freudian token of his fetishistic obsession with violence and death. But the true meaning of the signature in blood is clear and unambiguous. The key to it lies in a tradition of Christian symbolism to which Caravaggio had already alluded earlier in his career. Years before, when painting _The Martyrdom of St Matthew_ for the Contarelli Chapel in Rome, he had evoked the ancient link between martyrdom and baptism by having Matthew's blood flow into a baptismal pool. The blood signature alluded to the same association, although its meaning was subtly different. In _The Martyrdom of Matthew_ , it was Matthew and Matthew alone who had been reborn into immortality through his own martyrdom. In _The Beheading of St John_ , it is not only the martyr who gains eternal life. Caravaggio himself has been symbolically reborn, through his acceptance into the ranks of those dedicated to the martyred John the Baptist.\n\n_The Beheading of St John_ was Caravaggio's gift to the Knights of Malta, a due paid in lieu of his _passaggio_ into the Order of St John. Its completion, therefore, marked his entry into the brotherhood of knights. Hence that prominent 'F' before his name. It stands for 'Fra', or 'Brother', the official prefix of any Knight of St John. The artist's signature, written in John the Baptist's blood, was a public proclamation. It was Caravaggio's way of asserting that his own mortal sin, the murderous letting of a man's blood, had been washed away by the blood of his new patron saint. Now he could return to Rome, not as a criminal but as a proud Christian soldier.\n\n#### COMPETING WITH MICHELANGELO\n\nCaravaggio must have added his signature to the work some time shortly after 14 July 1608; because it was on that date, exactly a year and two days after his arrival on Malta, that he was invested with the habit of a Knight of Magistral Obedience and given the title 'Fra Michelangelo Merisi'. The address given by the Grand Master at the ceremony of the investiture can only have increased the artist's pleasure in his newfound status. In the Bull of his reception, Wignacourt went so far as to compare Caravaggio with Apelles of Kos, the celebrated painter of ancient Greece:\n\n> Whereas it behooves the leaders and rulers of commonweals to prove their benevolence by advancing men, not only on account of their noble birth but also on account of their art and science whatever it may be, so that human talent, hopeful of obtaining reward and honour, might apply itself to praiseworthy studies:\n> \n> And whereas the Honorable Michael Angelo, born in the town Carraca, in the vernacular called Caravaggio, in Lombardy, having been called to this city, burning with zeal for the order, has communicated to us his fervent wish to be adorned with our habit and insignia.\n> \n> Therefore, as we wish to gratify the desire of this excellent painter, so that our island of Malta, and our order, may at last glory in this adopted disciple and citizen with no less pride than the island of Kos (also within our jurisdiction) extols her Apelles; and that, should we compare him to more recent artists of our age, we may not afterwards be envious of the artistic excellence of any other outstanding man of equally important name and brush... and as we wish to comply with the pious wish of the aforesaid Michael Angelo, we receive and admit him, by the grace of God almighty and by a papal authorization especially granted to us for the purpose, to the rank of Brethren and Knights known as Brethren and Knights of Obedience...\n\nThis pretty sequence of tributes pays the greatest compliment of all to Wignacourt himself, because if Caravaggio is a new Apelles, the Grand Master is by implication a second Alexander the Great. The author of the encomium, with its flowery phrases and polished rhetoric, was almost certainly Wignacourt's erudite secretary, Francesco dell'Antella, who had also played a part in lobbying for Caravaggio's knighthood some months earlier. Dell'Antella was a learned Florentine, who for a Knight of Malta prided himself on his sophistication and classical learning. He also took an unusually strong interest in art. He was himself a gifted amateur draughtsman, who produced an impressively detailed drawing of Valletta as one of the illustrations for his friend Giacomo Bosio's history of the Order of St John. Dell'Antella would eventually become an official member of the Accademia del Disegno in Florence, the first Italian art academy, founded by Giorgio Vasari in the mid sixteenth century. Like Caravaggio, dell'Antella was a proud and stormy man with a tendency to violence. He had even killed Wignacourt's own nephew in a swordfight, but had been magnanimously forgiven by the Grand Master on the grounds that he had been unjustly provoked. This similarity of temperament and history may have given the two men some affinity.\n\nPerhaps as a gesture of gratitude to dell'Antella, perhaps to commission, Caravaggio painted a wry and learned cabinet picture for him, _Sleeping Cupid_. The mischievous and malign child-god lies sleeping, one wing folded beneath him, the other reduced to the barely perceptible rim of a feathery arc. In his left hand he limply holds a bow, of Indo-Persian design, and a feathered dart of love. A dim light illuminates the scene, suggestive of the first glimmers of dawn. The picture is a darker, drowsier, dreamlike version of the _Omnia vincit amor_ painted in 1602 for Vincenzo Giustiniani. This time the Cupid is not an adolescent boy but a child, with the plump, fleshy body and heavy, lolling head of a toddler.\n\nThe painting is close in spirit to a poem about a statue of sleeping Cupid in Giambattista Marino's _La_ _Galeria_ , an anthology of verse inspired by works of art both real and imaginary. Marino was a close contemporary of Caravaggio, and had been a friend of his in Rome, so it is possible that the painter had the poet's verses in mind when he painted his picture.\n\nMarino begins by warning the prospective visitor to his poetical museum against waking the image of the sleeping child:\n\n_Gu\u00e0rdati Peregrino_\n\n_non gli andar si vicino,_\n\n_nol desar, prega, ch'egli_\n\n_dorma in eterno pur, n\u00e9 mai si svegli._\n\n_Se tu'l sonno tenace_\n\n_rompi al fanciul sagace,_\n\n_desto il vedrai pi\u00f9 forte_\n\n_trattar quell'armi, ond'\u00e8_\n\n_e peggior che Morte_.\n\nLook out, Pilgrim\n\ndo not get so close,\n\ndo not rouse him, pray that he\n\nsleeps forever and never wakes up.\n\nIf you break the clever boy's sleep,\n\nright away you will see him yet more strongly\n\ntake up those weapons that make him\n\nworse than death.\n\nMarino's sleeping child is lost in cruel dreams of deceptions, massacres and sufferings. Dawn is breaking and he will soon awake to visit more miseries of love on his countless victims. The poem ends with a question, and a joking reminder that the subject of all these fears and fantasies is after all merely a work of art:\n\n_Qual tu ti sia, che 'l miri,_\n\n_temi non vivi e spiri?_\n\n_Stendi securo il passo:_\n\n_toccal pur, scherza teco, egli \u00e8 di sasso_.\n\nWhoever you are, who gaze upon him,\n\ndo you fear lest he live and breathe?\n\nLengthen your stride with confidence, do not tiptoe,\n\nTouch him even \u2013 I was teasing you \u2013 he is made of stone.\n\nCaravaggio's painting also plays teasingly on the boundary between art and reality. The sleeping boy is an image, but of a disconcertingly lifelike kind. His teeth can be seen glinting behind his half-closed lips. The abandon with which his head is thrown back and the look of absorption on his face powerfully conjure the illusion of a real child caught up in a vivid dream. But there are other ways of looking at this picture too. Like Marino's poem, Caravaggio's painting looks back knowingly to the world of antiquity. Not only does it evoke the myth of Cupid; it also calls to mind the many ancient Greek legends about images of art so deceptively convincing that they seemed real \u2013 the painted grapes of Xeuxis which, as Pliny the Elder relates, fooled the birds into pecking at them, or the statue of a woman infused with such love by the sculptor Pygmalion that she actually came to life and stepped down off her pedestal. In painting the _Sleeping Cupid_ , Caravaggio was making his own contribution to the imaginary art gallery of the classical past \u2013 and living up to the classical compliment that had so recently been paid to him at his investiture as a Knight of Malta. He had been dubbed the new Apelles, and now he was wittily acting the part.\n\nThere was yet another layer of allusion for the learned Francesco dell'Antella to enjoy as he contemplated his new possession, this time to a more recent work of art. The subject of Cupid asleep was famously associated with Caravaggio's namesake, Michelangelo Buonarroti. When Michelangelo was young, he had created a sculpture of a _Sleeping Cupid_ so perfectly classical in spirit that he was able to pass it off as a genuine antique work of art. Giorgio Vasari tells the story:\n\n> [He] set himself to make from another piece of marble a Cupid that was sleeping, of the size of life. This, when finished, was shown... to Lorenzo di Pier Francesco [Medici] as a beautiful thing, and he, having pronounced the same judgement, said to Michelangelo: 'If you were to bury it under ground and then send it to Rome treated in such a manner as to make it look old, I am certain that it would pass for an antique, and you would thus obtain much more for it than by selling it here.' It is said that Michelangelo handled it in such a manner as to make it appear an antique; nor is there any reason to marvel at that, seeing as he had genius enough to do it, and even more.\n\nMichelangelo Buonarroti had created a sculpture of Cupid to rival the masterpieces of antiquity. Now Michelangelo Merisi was vying with him by emulating that very act of classical emulation. Implicit in the gesture was the Renaissance conceit of the _paragone_ , a contest between different art forms. Michelangelo, the sculptor, had given tangible form to his _Sleeping Cupid_. Caravaggio, the painter, could not do that. But he could create a greater illusion of flesh and blood, and he could use his mastery of chiaroscuro to evoke the light of approaching dawn.\n\nThe _Sleeping Cupid_ is essentially a _jeu d'esprit_. But it is also a vitally important painting for the understanding of Caravaggio's work as a whole, demonstrating his high degree of erudition and establishing beyond any doubt an explicit spirit of competition with Michelangelo, which had seemed at least implicit in so many of his Roman paintings, from _The Calling of St Matthew_ onwards. Francesco dell'Antella made the comparison overtly when he went out of his way to show the picture to a great-nephew of Michelangelo named Francesco Buonarroti, who was also a Knight of Malta; and he then actually sent the work to Florence in the hope that the most celebrated member of the modern Buonarroti family, Fra Francesco's brother, the poet and dramatist Michelangelo the Younger, would give his opinion of it. Michelangelo the Younger clearly did see this rivalrous homage to his great-uncle's marble Cupid, because on 24 April 1610, dell'Antella wrote to him to say: 'I value now more than before my Cupid, after hearing the praise of your Lordship for which I kiss your hand.'\n\n#### APELLES IN PRISON\n\nThe perennially spiky Caravaggio was celebrated on Malta. In his own estimation he had always been a _valent'huomo_. Now he was truly being treated like one. Alof de Wignacourt was delighted with the painter's work for the order. According to Bellori, he was so impressed by the enormous new altarpiece for the Oratory of St John that 'as a reward, beside the honour of the Cross, the Grand Master put a gold chain around Caravaggio's neck, and made him a gift of two slaves, along with other signs of esteem and appreciation for his work'. Finally, Caravaggio had got his own gold chain.\n\nIt may only have been at this moment of apparent happiness and prosperity that the full implications of being a Knight of Malta finally dawned on him. Not only was he bound to the island by the Grand Master's whim, but obliged to live in strict observance of the statutes of the order. Sexual indiscretions were liable to be tolerated as long as they were committed out of the public eye, but any other disorderly conduct would be ruthlessly dealt with under the knights' code of law. That meant no shouting or trading of insults, no fighting, no duelling with swords. For a man like Caravaggio, that was never going to be easy, especially in a town like Valletta. The city teemed with proud young noblemen from the different national Langues, intensely conscious of the most minute differences in rank and status. As Alof de Wignacourt himself remarked in a letter to the pope, 'It is impossible, in a place where so many are devoted to the profession of arms, and where so much importance is given to points of honour, that there should not be numerous fights and brawls.'\n\nCostanza Colonna and her son, Fabrizio Sforza Colonna, must have known that they were taking a calculated risk when they sent Caravaggio to Malta. Their hope must have been that the ruthless military discipline of the Order of St John would persuade him to keep his temper under control. Everything went to plan for a while, as Caravaggio painted for the central figures of the Maltese establishment. But the gamble did not pay off. The painter's pride in his knighthood came before his greatest fall from grace. Caravaggio's character had always been a volatile compound, an uneasy blend of Lenten piety and the raucous spirit of Carnival. This was never more true than on Malta.\n\nIt is impossible to know what triggered the outburst that undid him. Perhaps it was his shocked realization that Wignacourt indeed wanted 'not to lose him' \u2013 to chain him to the island, perhaps not forever, but for several more years. Whatever the cause, just weeks after admission to the Order of St John, Caravaggio lashed out against its authority. In the space of a few hours he went from hero to villain.\n\nThe early biographers are vague on the subject of what went wrong for Caravaggio on Malta. Mancini does not even mention the incident. According to Baglione, who was better informed, the cause of the trouble was an argument with a Knight of Justice: 'In Malta Michelangelo had a dispute with a Knight of Justice and somehow insulted him.' Knights of Justice were higher ranking than mere Knights of Magistral Obedience. So Baglione may have meant to imply an argument over status, which was just the sort of 'point of honour' liable to cause the frequent fights between brothers of the order mentioned by Wignacourt in his letter to the pope.\n\nBellori's account is similar to Baglione's, except that in his version of the story Caravaggio's mercurial nature is the driving force behind the calamity. Like the hero of a Greek tragedy, he is a man ruinously undone by a fatal flaw of character: 'He lived in Malta in dignity and abundance. But suddenly, because of his tormented nature, he lost his prosperity and the support of the Grand Master. On account of an ill-considered quarrel with a noble knight, he was jailed and reduced to a state of misery and fear.'\n\nCaravaggio certainly committed an offence on Malta, one serious enough to merit imprisonment. But for centuries the exact nature of that offence remained a mystery. Generations of historians combed the archives in Malta, where the great books of statutes, crimes and punishments are still preserved in the library of the Order of St John, but to little effect. One of the volumes stored there revealed much about the aftermath of Caravaggio's crime, but nothing about the crime itself. Tantalizingly, a number of adjacent pages in that same book had been systematically and deliberately painted out with a thick layer of opaque pigment.\n\nIt was only in 2002, after the Maltese scholar Keith Sciberras had taken the initiative of X-raying some of those obscured pages, that the truth was revealed. The painter had indeed become embroiled in an altercation with 'a noble knight', just as Bellori had indicated. Baglione turned out to have been right too. The aggrieved party was indeed a Knight of Justice, Fra Giovanni Rodomonte Roero, the Conte della Vezza. He was seriously injured in the incident.\n\nOne of several documents thus uncovered was a report of the preliminary results of an enquiry ordered by Grand Master Wignacourt and the Venerable Council on 19 August 1608. The purpose of that enquiry was to outline the events of a 'tumult' that had taken place the night before. The incident had involved several knights, some of whom had smashed open the door of the residence of the Organist of the Conventual Church of St John, Fra Prospero Coppini.\n\nAs a result of that preliminary enquiry, a criminal commission was set up to investigate the incident in more detail. The three investigators were Fra Philiberto de Matha, Fra Giovanni Gomes de Azevedo and Antonio Turrensi. They established that a brawl, involving seven knights altogether, had broken out in the house of Fra Coppini, but since he himself had not taken part, he was absolved. The commission found that Fra Giovanni Rodomonte Roero, the Conte della Vezza, had been the victim of an assault by six aggressors, including Caravaggio.\n\nThe artist's companions on the night in question included two senior figures in the Maltese hierarchy, both Knights of Justice like the victim, Roero. One was Fra Giulio Accarigi, who was originally from Siena but who had been a Knight of Malta since 1585. He had a reputation for violence and a criminal record to match, having spent two months in detention for assault in 1595 and a further two years in jail some ten years later. The other Knight of Justice involved was Fra Giovanni Battista Scaravello, from Turin, who had arrived on Malta in 1602 and had entered the Order of St John two years after that.\n\nTwo young novices were also implicated: Francesco Benzi, who had come to the island in 1606; and Giovanni Pecci, from Siena, who had arrived on Malta within a day of Caravaggio himself, on 13 July 1607. Both men would have known the painter as a fellow novice. One of the conditions of entry to the order was a rigorous programme of training in the selfsame Oratory of St John \u2013 also known as the Oratory of the Novices \u2013 for which the artist had painted his altarpiece of _The Beheading of St John_. Benzi and Pecci would have prepared for their knighthoods alongside Caravaggio.\n\nNo eyewitness description of the fight has been found in the Maltese archives, so the parts played by those involved have to be deduced from the punishments each received. Accarigi and Scaravello seem to have taken minor roles. Each would eventually be given six months in jail, a relatively mild sentence in the harsh context of Maltese justice (although it is also possible that they were let off lightly on account of their rank). Benzi and Pecci were condemned to two and four years in prison respectively. The main culprits appear to have been Caravaggio himself and a certain Fra Giovanni Pietro de Ponte, who was a deacon of the church, and another frequent offender. De Ponte was identified by the criminal commission as a prime mover of the assault. On the night of the fight he had been carrying a small pistol referred to as a _sclopo ad rotas_. It was a bullet or bullets from the _sclopo_ that had inflicted serious wounds on Roero. De Ponte would be defrocked, deprived of his habit and denied forever his status as a Knight of Malta. Caravaggio was never sentenced for his part in the assault, for reasons that will become clear. But his crime was clearly deemed to be at least as serious as that of de Ponte, because the first report of the criminal commission into the case recommended that these two \u2013 but none of the others involved \u2013 be arrested immediately.\n\nThat report was submitted to the Venerable Council, whose members included Alof de Wignacourt, Fabrizio Sforza Colonna, Antonio Martelli and Ippolito Malaspina, on 27 August 1608. Present too would have been Wignacourt's secretary, Francesco dell'Antella, for whom Caravaggio had recently painted the _Sleeping Cupid_. But Malta's strict code of discipline, and the seriousness of the assault on the Conte della Vezza, would have given the artist's patrons and supporters no choice but to order his immediate arrest. On 28 August 1608 Caravaggio was seized and imprisoned within the forbidding precincts of the Castel Sant'Angelo.\n\nThe timing of his offence could not have been more perversely precise. Caravaggio had managed to get himself thrown into jail on the eve of one of the most important days in the calendar of the Knights of Malta: 29 August was the Feast of the Decollato, the day on which the order gathered in the Oratory of St John to remember the decapitation of its patron saint. In 1608, it was also the day Wignacourt had chosen to unveil Caravaggio's monumental altarpiece of _The Beheading of St John_. But instead of attending in his knight's robes, the painter now languished in an underground cell.\n\nThe 'tumult' cast a long shadow over the celebrations of the feast of the Decollato. To make matters worse a dispute had arisen between the confraternity responsible for arranging those celebrations, the Compagnia di San Giovanni Decollato, and the musicians of the Conventual Church \u2013 including, coincidentally, Fra Prospero Coppini, the organist whose door Caravaggio had helped to kick in. The musicians were unhappy about their pay and most of them went on strike, so that on the feast day itself neither Vespers nor the solemn Mass was sung in the oratory before Caravaggio's picture. The unveiling for which Wignacourt had planned so carefully could hardly have gone more badly wrong.\n\n#### 'A ROTTEN AND DISEASED LIMB'\n\nCaravaggio spent the entire month of September detained in the _guva_ , an underground cell cut directly into the rock of the Castel Sant'Angelo. It is a bell-shaped chamber, eleven feet deep, sealed by a heavy trap door, and reserved for knights who had been guilty of serious offences. The traces of their presence remain in the form of several melancholy graffiti, one of which records the last-known words of a sixteenth-century Scottish Knight of Malta, one John Sandilands: 'imprisoned forever, victim of evil triumphing over good \u2013 so much for friendship'.\n\nCaravaggio's own thoughts were less mournful and more pragmatic. Few had ever broken out of the Castel Sant'Angelo, while escape from the _guva_ itself was unheard of, but he was determined to do so. Even if he could scale the walls of the rock-cut cell, he would then need to climb the ramparts of the castle itself. After that he would have to lower himself down a sheer 200-foot precipice to the sea. To do all this he would need help.\n\nGetting off Malta itself would pose a whole tangle of other problems. Caravaggio would need a boat, skippered by a brave and corruptible captain. But the boat would be unable to collect him at the bottom of the castle cliff, because the only way to the open sea from there lay through the narrow opening of Valletta's Grand Harbour. Any vessel attempting to escape by that route would certainly have been spotted by the order's patrols. The journey would have to be made from one of the island's many small bays, and by night, to avoid detection. This meant that Caravaggio would have to swim round the promontory on which the Castel Sant'Angelo stood, then make his way to a quieter part of the island by foot, to wait for the vessel skippered by his accomplice. From there, the most logical destination would be Sicily, the nearest part of the mainland, some sixteen hours away with a favourable headwind.\n\nSomehow, Caravaggio did indeed manage all of this. By the end of October 1608 he was in the Sicilian port town of Syracuse, some sixty miles from Malta. Bellori describes the artist's great escape in a single terse sentence: 'In order to free himself he was exposed to grave danger, but he managed to scale the prison walls at night and to flee unrecognized to Sicily, with such speed that no one could catch him.' Baglione adds that a rope ladder was used in Caravaggio's escape, but neither writer makes any suggestions about who might have helped his getaway. He must have had help from someone on the inside in the Castel Sant'Angelo, but who that someone was remains a mystery.\n\nCaravaggio was officially declared missing on 6 October, when\n\n> was heard the complaint of Lord Brother Hieronymus Varays, Procurator for the Treasury of the Order, made against brother Michael Angelo Marresi [ _sic_ ] de Caravaggio who while detained in the prison of the Castle of St Angelo fled from it without permission of the most illustrious and most exalted Lord the Grand Master and departed secretly from the district, against the form of the Statute 13 concerning prohibitions and penalties: the most Illustrious and most exalted Lord the Grand Master and Venerable Council commissioned the Lord Brothers Joanni Honoret and Bladius Suarez that, through the agency of the Master Shield-Bearer, they should see that all due diligence is shown in finding out the said Brother Michael Angelo and in summoning him to appear, and should gather information about his flight...\n\nThere is a strong implication here that an expeditionary force was sent to recapture Caravaggio and render him up to the Maltese court to face sentence both for the assault on the Conte della Vezza and for his defiance in fleeing the island. The Grand Master was known to be extremely severe on knights who transgressed Statute 13 of the order's legal code by leaving Malta without his permission. He insisted that all fugitives be returned to Valletta at once, preferably in secrecy. It was his normal practice in such cases to write to all the order's receivers in the major cities and ports of Europe to demand the immediate detainment of the renegade knight. Yet seven weeks later Caravaggio was still at large in Sicily, having evaded whatever attempts had been made to rearrest him. On 27 November, his trial on Malta went ahead in his absence. The Venerable Council determined that he had escaped from prison using ropes. It decided to disgrace him and deprive him of his habit. At the same time, the council heard and passed judgement in the case of the August assault. Four of the six guilty knights were sentenced to jail terms, while the church deacon, Giovanni Pietro de Ponte, was to be defrocked like Caravaggio.\n\nAccording to Maltese custom, criminal trials and ceremonial punishments were carried out in the Oratory of St John, where Caravaggio's _Beheading of St John_ now hung directly over the main altar. So four days after the trial, on 1 December 1608, the ritual defrocking known as the _privatio habitus_ took place in that very room. The archive records that 'a General Assembly was summoned of the Venerable Bailiffs, the Priors, Preceptors and Brothers in the Church and Oratory of St John our patron, at the sound of the bell, according to the ancient and praiseworthy custom of the Holy Order of St John of Jerusalem... the information inspected and carefully read out against Michelangelo Merisi de Caravaggio...'\n\nWolfgang Kilian's mid seventeenth-century engraving of a criminal trial on Malta conjures up the scene of Caravaggio's _privatio habitus_. On either side of the Oratory of St John sit the massed ranks of the order's Grand Crosses. In December 1608 they would have included not only the artist's most prominent patrons such as Antonio Martelli, but also many other veterans of the great sea and land battles of recent European history \u2013 survivors of the Great Siege, of Lepanto, perhaps even the Spanish Armada. Before this assembly of heroes, Caravaggio's greatest humiliation was to take place.\n\nIn Kilian's engraving (see p. 328), the Grand Master sits, just as Alof de Wignacourt would have done, at the near end of the church. At the far end, the guilty knight kneels, directly beneath Caravaggio's depiction of St John's decapitation. Because Caravaggio was to be defrocked _in absentia_ , a wooden stool draped with the habit of a Knight of Magistral Obedience would have been placed at the altar end of the church \u2013 more or less directly beneath Caravaggio's own signature, flowing in blood from the neck of John the Baptist.\n\nBefore the conclusion of the ceremony, there was one last formality to be gone through: 'The Lord Shield-Bearer... repeated in a loud voice in the Public Assembly so that the said Brother Michael Angelo de Caravaggio being personally summoned once, twice, thrice and a fourth time, an abundant notice, did not appear nor as yet doth he appear...' The oratory fell silent for the brief, necessary moment of Caravaggio's inevitable non-appearance. Then the robe of a Knight of Malta, so proudly but so briefly worn by him, was stripped from the stool by Grand Master Wignacourt himself, and the last damning words were written in the record: 'the said Brother Michael Angelo de Caravaggio was in the Public Assembly by the hands of the Reverend Lord President deprived of his habit, and expelled and thrust forth like a rotten and diseased limb from our Order and Society.'\n\n#### _THE BURIAL OF ST LUCY_ AND A BLACK DOG CALLED CROW\n\nCaravaggio was on the run for the second time in his life. His destination was the port town of Syracuse at the western edge of Sicily, where his old fellow apprentice, Mario Minniti, had established a thriving studio. Minniti had contacts in the town Senate. If they could be persuaded to look favourably on Caravaggio, they had the power to protect him from Maltese law. He had never been in more trouble than now. This time he had managed to alienate his entire network of supporters, not only the Colonna and their allies, who had manoeuvred to get him to Malta, but also the formidable Alof de Wignacourt and his army of knights. Caravaggio desperately needed some new friends in high places.\n\nThere is evidence that he took a deliberately circuitous route, landing at one of the island's smaller and more southerly ports, such as Pozzallo or Scicli, before working his way north-east. En route, he stayed in the little town of Caltagirone, some sixty miles inland from Syracuse. A recently rediscovered eighteenth-century document records that Caravaggio was seen visiting a church there, Santa Maria di Ges\u00f9. He was impressed by the beauty of a sixteenth-century marble Madonna by Antonello Gagini on one of its altars. 'Whoever wants her more beautiful, should go to heaven,' he reportedly said. Caravaggio was continuing to measure himself, as he had done throughout his life, against the standard of Michelangelo and his school: Gagini had been one of Michelangelo's most gifted pupils, and was said to have assisted the sculptor on his final version of the tomb of Pope Julius II, in the Roman church of San Pietro in Vincoli.\n\nAs the painter made his way from Caltagirone to Syracuse, he found himself once more within a realm ruled distantly by Philip III of Spain. The island had been praised for its warm climate and natural abundance since antiquity, but under the Spanish the majority of its people suffered great privations and hardship. Part of the reason was Spain's own economic crisis, caused by the sudden dwindling of its vast revenues from the silver mines of Latin America, under the pressure of competition from other European nations. A succession of Spanish viceroys in Sicily were encouraged to bleed it of its natural resources. The people became poorer as their rulers enriched themselves, concealing the true nature of this unequal transaction behind the grandest of architectural fa\u00e7ades. During the late sixteenth and early seventeenth centuries, cities such as Syracuse, Messina and Palermo became stage-sets for the performance of the rituals of absolute Spanish power. Splendid new churches and palaces were built in an extravagant local version of the Baroque style. Grand axial routes were ruthlessly cut through the fabric of Sicily's medieval cityscapes, distracting attention from the miserable lot of the poor, and allowing the rich to move serenely through each city without ever seeing its warren of slums.\n\nTravelling to Syracuse by land from the southern tip of Sicily was the best way of avoiding the pursuing knights, but Caravaggio exposed himself to other risks. Such was the discontent with Spanish rule that by the early years of the seventeenth century much of the island's interior had degenerated into lawlessness, with many regions at the mercy of competing clans of _banditti_. The Spanish authorities had retaliated against these roaming gangs with a degree of success, but travel in the rural hinterlands of Sicily was still considered dangerous by George Sandys in 1615: 'This Vice-Roy hath well purged the country of Bandities, by pardoning of one for the bringing in or death of another: who did exceedingly, and yet do too much infest it. Besides, the upland inhabitants are so inhospitable to strangers, that betweene them both there is no travelling by land without a strong guard; whom rob and murder whomsoever they can conveniently lay hold on.' Despite the dangers, Caravaggio made it to Syracuse safely around the middle of October 1608.\n\nThe main source of information about Caravaggio's activities in Sicily is a manuscript of 1724 entitled _The Lives of the Messinese Painters_ , written by a priest and amateur painter called Francesco Susinno. Susinno's sources were in the painters' studios of Sicily, where memories of Caravaggio's unprecedentedly emotive style of painting and perplexing personality were still strong more than a century after his death. In Susinno's words, Caravaggio 'was welcomed by his friend and colleague in the study of painting, Mario Minniti, a painter from Syracuse, from whom he received all the kindness that such a gentleman could extend to him. Minniti himself implored the Senate of that city to employ Caravaggio in some way so that he could have the chance to enjoy his friend for some time and be able to evaluate the greatness of Michelangelo, for he had heard that people considered him to be the best painter in Italy.'\n\nA commission from the Senate would mean protection from the Knights of Malta. The knights maintained an active presence in Syracuse, but so long as he was working for them, the city's fathers would look after him. Once again, Caravaggio's predicament would be his patrons' opportunity. Once again, he would be given the chance to paint his way out of trouble.\n\nThe timing of his arrival in Syracuse could hardly have been more opportune. Previously strained relations between the religious authorities and the Senate had improved in the early years of the seventeenth century, as a result of which the city had embarked on a vigorous programme of renovating its churches and monasteries, commissioning new altarpieces and boosting the cults of local saints.\n\nOne of the most actively venerated of those saints was the fourth-century martyr St Lucy, a native of Syracuse said to have met her end during the persecutions of the Emperor Diocletian. In the severe climate of the Counter-Reformation, with so many in the Roman Catholic faith calling for a return to the simple piety of the early Church, the cults of the ancient Christian martyrs were resurgent. A statue of St Lucy had already been placed on the ramparts of Syracuse, and the Senate had agreed to finance the creation of a costly silver reliquary to house some of her supposed remains. Not long before Caravaggio's arrival in the city, the authorities had also decided to restore the church most closely associated with her, the medieval basilica of Santa Lucia al Sepolcro. The church lay outside the city walls, having been built directly above the ancient Christian catacombs where, according to her legend, the virgin martyr had been interred. A local archaeologist and historian, Vincenzo Mirabella, had made a study of the site, re-emphasizing its significance in the sacred history of 'Syracuse the Faithful'. The newly restored church would need a painting telling the story of Lucy's martyrdom for its main altar. Who better to create it than 'the best painter in Italy'? The altarpiece that he produced can still be admired in the Franciscan church outside the old city walls.\n\nThe subject given to Caravaggio was _The Burial of St Lucy_ , the final episode in the story of her martyrdom and one seldom painted. There was a logic behind the choice. The civic and religious authorities of Syracuse were especially keen to reinforce the local cult of St Lucy, which had been severely damaged by the theft of the saint's remains during the Middle Ages. Rival cults of Lucy had been established elsewhere, most notably in Venice, where ownership of her relics was now being claimed. The authorities wanted Caravaggio's picture to remind the world that wherever her bones might have been taken, Syracuse was the original site of Lucy's miracles and the place where she had first been laid to rest.\n\nThe fullest account of her martyrdom is to be found in _The Golden Legend_. Lucy, a virgin of noble lineage, born in Syracuse, resolved to imitate the poverty and humility of Christ. She swore a vow of chastity and steadily gave away her possessions to the poor. Her former husband-to-be, a non-believer, failed at first to realize that she had converted to Christianity and suspected her of courting another man with her wealth. When she informed him that she was now a bride of Christ, he denounced her to a Roman judge named Paschasius. The judge punished the saint by giving her over to 'the ribalds of the town', instructing them 'to defoul her, and labour her so much till she be dead'. But when they came to take her away to a brothel, they could not move her. Paschasius sent for reinforcements, even for a team of oxen, but 'she abode always still as a mountain, without moving.' Then the judge ordered a great fire to be lit around her immobile form, commanding his torturers to pour boiling oil and resin on her. Throughout her ordeal, she prayed to God, so infuriating her tormentors that they thrust a sword straight through her throat. Even then she did not die, or even budge an inch, until Holy Communion was offered to her: 'Lucy never removed from the place where she was hurt with the sword, and died not till the priest came and brought the blessed body of our Lord Jesu Christ. And as soon as she had received the blessed sacrament she rendered up and gave her soul to God, thanking and praising him of all his goodness. In that same place is a church edified in the name of her...'\n\nCaravaggio painted the moment just after St Lucy received communion and died. Having breathed out her soul, she lies on a bare expanse of ground. Her body is small, crumpled, pathetic. Her right arm is outstretched, the foreshortened hand reaching out like the hand of a beggar asking for charity. The other arm rests on her belly. Lucy's skin is pallid with death, her mouth slack. Her head lolls helplessly back. There is a deep gash in her neck, but no sign of burns. Her frail body is framed by the hulking figures of two gravediggers, stooping to plunge their spades into the earth. The contrast between the slight saint and the giants who have come to bury her is extreme and disconcerting. This was Caravaggio's way of continuing to suggest the brutality of Lucy's martyrdom \u2013 the death of a young woman at the hands of thuggish men \u2013 even in the moment of her burial.\n\nThe bull-necked, crop-haired gravedigger to the right, whose tightly draped buttocks have been given such rude prominence, is loutishly absorbed in his task. Were he to stand up, he would tower over all the other figures in the painting. He is a man-mountain, at least ten feet tall. His workmate is similarly gigantic but more aware of his surroundings. Veins bulge in the left forearm and right wrist, but, as he bends to dig, he loses concentration on the job in hand. He seems transfixed by the figure of the bishop, to the right of the scene, whose blessing hand is picked out by a ray of illumination. The toiling worker has suddenly become aware of the momentous, sacred nature of the ritual in which he is taking part. According to her legend, in her last moments Lucy had expressed the hope that her martyrdom might convert some of her tormentors to Christianity. In the figure of the second gravedigger, that hope is about to be realized. He has looked up, and seen the light.\n\nThe picture has been much abraded by time, but its power and originality are undimmed. The strong illusion of reality is all the more impressive given the painter's bold distortions of scale and perspective. Clustered behind the body of the saint, a small group of mourners has gathered. Their faces and bodies are seen in fragments, occluded by shadow and the body of the first gravedigger, forming a collage of griefstruck response. To the right of the priest's blessing hand, a man's face has been abbreviated to little more than a furrowed brow and a pair of staring, anxious eyes; this may be a daringly cropped self-portrait. To the left, three more saddened faces appear. A bald man prepares to wipe tears from his eyes. Beside him, the much repainted figure of a woman stares impassively into space, while, between them, their bearded companion seems frozen by melancholy. At the feet of these three figures, Caravaggio has included a kneeling reprise of the old woman with her face in her hands from his recent altarpiece of _The Beheading of St John_.\n\nThe mourners are completed by a young man draped in a snaking length of red cloth, and a veiled elderly woman. His hands are clasped in front of him, at waist height, while hers are held up to her cheek. Both stare down with intense grief at the dead body on the ground in front of them. These last two mourners have been taken directly from the conventions of earlier Renaissance painting, in which Mary and the Apostle John were depicted in exactly the same postures of grief. Caravaggio's use of this deliberate archaism evokes the Crucifixion and emphasizes the martyr's emulation of Christ. The wound in her neck and her outstretched left arm reinforce this chain of associations.\n\nThe picture has a hallucinogenic quality, the feeling again of a work painted from memory rather than from models. Just as the old woman with her head in her hands has been borrowed from Caravaggio's earlier _Beheading of St John_ , the sinister figure in armour standing next to the blessing bishop is another version of Aegeas in _The Crucifixion of St Andrew_ , painted in Naples in 1607. More than ever, Caravaggio's painting evokes the old, folk traditions of Italian polychrome sculpture. Lucy and those who mourn, bury and bless her could almost be mannequins of wax, dressed in real clothing and given real hair. The tall, bare room in which the burial takes place evokes the simple chapels of the old _sacro monte_ , where stories of the life of Christ are told through assemblages of straining and struggling figures very much like these.\n\nThe artist may have drawn on older memories too. When plague had struck Milan in 1576 he would have seen many hasty burials and ragged funeral processions. The picture's iconography is ingeniously suggestive of hope and redemption, but its mood is overwhelmingly bleak. No angel descends to hymn the martyr's soul to heaven. Almost half the painting is dark bare stone, wall and arch both isolating and seeming to press down on the figures crowded around the dead body. The snapshot immediacy of the image, with its extremely innovative effects of cropping and occlusion, is suggestive of alienation and abandonment. Not until the middle of the nineteenth century, when Manet and Degas began to crop and cut their images in the name of capturing 'modern life', would there be anything to rival Caravaggio's weird dislocations in _The Burial of St Lucy_. The bishop and soldier, pushed to one side by the scything asymmetry of the composition, remain deeply ambiguous. Officially they stand for good and evil, light and dark, the compassionate Christian as opposed to the ruthless pagan response to a martyr's death. But they have been so brusquely sidelined by the artist that it is tempting to wonder whether he was opening the way to another view of what they might embody. Church and state stand by, united in their ineffectiveness, as yet another innocent goes to the grave.\n\nPerhaps with the priorities of the Senate of Syracuse in mind, Caravaggio also included a strong visual reference to the saint's original place of burial in his picture. The church of Santa Lucia was built directly on top of the city's ancient Christian catacombs, where according to legend her body had first been put into the ground. The high, arched interior in which he set _The Burial of St Lucy_ was directly based on those actual catacombs, which he had visited for himself; in this way, he perpetuated the act of interment linking Lucy to the city, creating an illusion that made it look as though her body was forever about to be entombed beneath the church itself. However dark its mood, Caravaggio's painting was a brilliant stroke of propaganda for the city.\n\nHe started work on the altarpiece soon after arriving in Syracuse, probably at around the beginning of November. He must have worked extremely fast. The altarpiece was monumental in scale, the largest work of his Sicilian period and one of the largest pictures he had ever undertaken. Yet it was finished by the start of the following month, comfortably in time for St Lucy's feast day on 13 December. 'The big canvas came out so well that it became famous,' wrote Susinno; 'the idea behind it was so good that there are many copies in Messina and in other cities of the Regno.'\n\nThe sources indicate that Caravaggio was in a state of nervous anxiety during much of this period. During his first weeks in Sicily, the galleys of the Order of St John were a constant and highly visible presence around the harbours of the island, including that of Syracuse. Susinno records that 'Caravaggio was very distracted, restless, indifferent to his own existence: many a time he would go to bed fully dressed, with his dagger (from which he was never separated) at his side... Even when dressed ordinarily he was always armed, so that he looked more like a swordsman than a painter.' Baglione tells us that it was around this time that he acquired 'a black dog that was trained to play various tricks, which he enjoyed immensely'. The animal was probably not just for amusement and company. Caravaggio gave it the ill-omened name of Corvo \u2013 'Crow' \u2013 a raucous creature with an aggressive personality to match.\n\n#### THE TYRANT'S EAR\n\nCaravaggio may have painted _The Burial of St Lucy_ in Mario Minniti's substantial workshop. It was quite unlike any of his own modest studios. In pursuit of success and respectability, Minniti had reinvented himself as a gentleman-painter, employing an army of assistants to transfer his compositions from paper to canvas and adding only the finishing touches himself. So busily productive was Minniti's workshop that quality inevitably suffered. 'Many weak paintings by him can be seen around,' wrote Susinno. 'If he had contented himself with just a few public works he would have been as celebrated as Caravaggio himself.'\n\nCaravaggio made at least one other acquaintance in Syracuse, with whom he went on a foray to see some of the sights of the town: Vincenzo Mirabella, antiquarian, mathematician and archaeologist. Caravaggio may have sought his help in researching the Christian catacombs of Syracuse, to give _The Burial of St Lucy_ an authentically antique setting: the catacombs were one of Mirabella's specialities, and he would include a lengthy account of them in his book, the _Dichiarazioni della pianta delle antiche Siracuse_ , published in 1613.\n\nElsewhere in the same book, Mirabella would tell of how he took Caravaggio to see another of the oldest sites of Syracuse, a huge grotto said to have been used as a prison by the ancient tyrant Dionysus. According to local folklore, Dionysus had ordered a deep and narrow slit to be cut into the roof of this 'speaking cave', so named because of its extraordinary acoustic qualities, which amplified noise in such a way as to make the least sound perfectly audible. At the cave's single entrance, the tyrant built a great gate, so that he could confine his prisoners within. On the hilltop above the cave, perched directly over the slit cut into its apex, he placed the house of his jailor. While his captives languished hundreds of feet below, Dionysus could eavesdrop on their every word. He could hear their admissions of guilt, learn their plans, discover the names of their friends and allies.\n\nAfter explaining all this to Caravaggio, Mirabella was struck by the acuteness of the painter's response. 'I remember,' he wrote, 'when I took Michelangelo da Caravaggio, that singular painter of our times, to see that prison. And he, considering its strength, and showing his unique genius as an imitator of natural things, said: \"Don't you see how the tyrant, in order to create a vessel that would make all things audible, looked no further for a model than that which nature had made herself to produce the self-same effect. So he made this prison in the likeness of an ear.\" Which observation, not having been noticed before, but then being known and studied afterwards, has doubly amazed the most curious minds.' To this day, the great cave \u2013 now part of the Archaeological Park of Syracuse \u2013 continues to be known as 'The Ear of Dionysus'.\n\nJudging by the portrait that serves as frontispiece to Mirabella's book, its author was a dapper and fashion-conscious gentleman with a self-consciously quizzical stare. He waxed his extravagantly long handlebar moustache and favoured the Spanish style of dress, wearing a high-necked lace ruff over a dark, finely embroidered shirt. But Mirabella was also highly respected by some of the best minds in early seventeenth-century Italy. A year after the publication of his book, he would be enrolled in Rome's foremost scientific society, the Accademia de' Lincei \u2013 named for that sharp-eyed animal, the lynx \u2013 after his friend Federico Cesi wrote a letter supporting his application to the great astronomer Galileo Galilei. In Cesi's words to Galileo, Mirabella was 'a knight from Syracuse, noble by birth and very rich, learned in Greek and Latin, man of letters and most erudite in Mathematics and primarily in the theory of Music, in which he is greatly esteemed and admired by his proposer. He has already published a worthy volume on the Antiquity of his birthplace with diligent description of the same...' Mirabella subsequently became a friend and correspondent of Galileo himself. They exchanged letters about 'spots on the sun' and the astronomer lent 'the knight from Syracuse' telescope lenses on more than one occasion.\n\nMirabella was impressed by the empirical tenor of Caravaggio's thought, and by his evident interest both in acoustics and in the mechanism of the human ear. Their exchange gives a rare glimpse of Caravaggio not as a violent criminal, nor as a probable lover of young men and whores, but as an intellectual and sophisticate. This was the same man who, in Rome, had moved in a circle of speculative thinkers and connoisseurs such as Giulio Mancini and Cardinal del Monte \u2013 himself another correspondent of Galileo's \u2013 and who had befriended poets such as Giambattista Marino.\n\nBut Caravaggio's remarks about 'The Ear of Dionysius' seem also to have reflected his increasingly apprehensive state of mind. The tyrant's prison grotto was a potent image of his own contracting world \u2013 a 'speaking cave' where every movement was monitored by spies, every remark overheard by eavesdroppers. Behind the logic of his observation lurked a paranoid fear of surveillance and recapture.\n\n#### LAZARUS RISING\n\nTypically, the most strenuous efforts to recapture an errant Knight of Malta were made in the period leading up to his trial, and in theory the ceremony of the _privatio habitus_ diminished the urgency of Wignacourt's campaign to get the painter back to Valletta. But even if Caravaggio was aware of that, he did not feel safe in Syracuse. Despite the success of _The Burial of St Lucy_ , Susinno records that 'the unquiet nature of Michelangelo, which loved to wander the earth, soon after led him to leave the home of his friend Minniti'. He departed more or less immediately after the work was finished, not even waiting to see it unveiled. By 6 December, a full week before the Feast of St Lucy, he was in the nearby city of Messina. There, he showed no sign of lowering his guard. He continued to wear his dagger and sword in public and to sleep fully clothed. Susinno wrote that 'his spirit was more disturbed than the sea of Messina with its raging currents that sometimes rise and sometimes fall.'\n\nWith a population of 100,000, Messina was as large and vibrant a city as Rome itself. North of Syracuse, it was separated by the narrowest of straits from Calabria on the mainland. Its port was one of the busiest in the Mediterranean. Messina was a town at the junction of east and west, Africa and Italy, and another centre of the thriving European slave trade. George Sandys described it as a stylish but dangerous place:\n\n> the meanest artificers wife is clothed in silke: whereof an infinite quantity is made by the worme... The Gentlemen put their monies into the common table (for which the Citie stands bound) and receive it againe upon their bils, according to their uses. For they dare not venture to keepe it in their houses, so ordinarily broken into by theeves (as are the shops and ware-houses) for all their crosse-bard windows, iron doores, locks, bolts, and barres on the inside: wherein, and in their private revenges, no night doth passe without murder. Every evening they solace themselves along the Marine (a place left throughout betweene the Citie walls and the haven) the men on horsebacke, and the women in large Carosses, being drawne with the slowest procession. There is to be seene the pride and beauties of the Citie. There have they their play-houses, where the parts of men are acted by women, and too naturally passioned, which they forbeare not to frequent upon Sundayes...\n\nCaravaggio's situation was awkward and fraught with danger. The galleys of Malta were in the waters of Messina throughout the last months of 1608. Not only that, but some time before 4 November Fra Antonio Martelli had taken up residence as the order's prior of the city. He is unlikely to have looked on Caravaggio's transgressions with a fond and forgiving paternal eye, but his ability to move openly against the painter was compromised because in the winter of that year the order was in litigation with the Senate of Messina, a state of affairs that continued for the duration of Caravaggio's stay in the city.\n\nAccording to Susinno, Caravaggio's fame had preceded him. He must have won the favour of the Senate, because his services were immediately in demand: 'The new reputation of Caravaggio appealed to the sympathetic people of Messina, who always favoured strangers, and the impressive excellence of such a man was such that they wanted him to stay, and they gave him commissions.' Emboldened by Fra Martelli's relative impotence, Caravaggio even had the cheek to present himself to his eager new clients as a fully fledged Knight of Malta. When the first of his Messinese altarpieces was consigned for delivery, the relevant document referred to the work as being 'by the hand of fr. Michelangeli Caravagio [ _sic_ ] Knight of the Order of Jerusalem'. Caravaggio was well aware that violation of the order's thirteenth statute meant inevitable expulsion, so he cannot simply have been acting in ignorance of his own disgrace.\n\nThere are suggestions that the painter still hoped to win a pardon from Wignacourt. According to Bellori, 'hoping to placate the Grand Master, Caravaggio sent to him as a present the half-figure of Herodias with the head of St John the Baptist in a basin' \u2013 a work that can tentatively be identified with a painting of a similar subject now in the National Gallery, London. But by continuing to pose as a Knight of Malta, he must have damaged his cause further. The stony-faced Fra Antonio Martelli is unlikely to have been impressed and reported back, no doubt, to the Grand Master. Alof de Wignacourt's desire to have Caravaggio forcibly extradited from Sicily can only have been strengthened by his insulting masquerade.\n\nAs Susinno suggested, the painter was soon hard at work for his new Messinese patrons. On 6 December a wealthy merchant named Giovan Battista de' Lazzari entered into an agreement to build and decorate the central chapel of the church of the Padri Crociferi, the 'cross-bearing fathers', a confraternity of hospitallers devoted to caring for the sick. Determined to obtain the services of Caravaggio, the de' Lazzari family offered him a huge sum of money to paint the principal altarpiece for their new chapel, more than three times the fee he might have expected for an equivalent commission when he was at the peak of his fame in Rome. The proposed title was _The Madonna, St John the Baptist and Other Saints_ , which suggests a rather static image of the type that would come to be known as a _Sacra Conversazione_ , or 'sacred conversation'. Caravaggio must have disliked the subject, because in a play on his patrons' family name he made the counter-suggestion that they commission him to paint _The Resurrection of Lazarus_ instead. The de' Lazzari accepted the proposal, and some time around the start of 1609 Caravaggio started work.\n\nIn Rome at the height of the Renaissance it had not been unknown for a famous artist to alter the terms of a commission. Michelangelo had famously plucked up the courage to tell Pope Julius II that his initial plan for the Sistine Chapel ceiling was 'a poor thing', replacing the pope's proposal of twelve apostles in a field of classical decoration with his own vastly more ambitious scheme of illustrations to the Book of Genesis. But in the provincial artistic milieu of Messina, Caravaggio's assertion of independence was still being talked about a hundred years later. Susinno was even more struck by it than he was by the huge fee that the painter was paid:\n\n> When some wealthy members of the house of Lazzaro wished to build a new chapel for the church of the Padri Crociferi, they commissioned Caravaggio to paint a large canvas and agreed to pay the sum of 1,000 scudi. Caravaggio conceived the Resurrection of Lazarus, alluding to their family. Those noblemen were greatly satisfied, and the artist was given free rein to fulfil his creative fantasy. It is commendable to give liberty to great artists to operate at their own will, instead of tying both their hands when they are ordered to execute a certain work in this or that manner or form.\n\nWhy, apart from this play on his patrons' name, did Caravaggio want to paint the story of Lazarus raised from the dead? It was a subject rarely depicted since the very early Renaissance. Following the conventions of Byzantine art, Giotto and Duccio had painted Lazarus, plague-spotted, rising from his tomb still wrapped in his grave clothes. In Caravaggio's later work, there is a powerful thrust towards both the subject matter and the style of much earlier Christian art. The only artist before him to have deliberately regressed in a comparable way had been \u2013 again \u2013 Michelangelo. With the creation of the _Rondanini Piet\u00e0,_ late in life, Michelangelo had plunged his art back to the angular and ascetic forms of Gothic carving. Caravaggio's _The Resurrection of Lazarus_ makes a similarly unorthodox statement of primitivist intent.\n\nLight and dark, which Caravaggio had previously manipulated in the service of a beguilingly deceptive optical realism, now serve an altogether different purpose. Their function is simply to amplify meaning and feeling \u2013 to reduce, to pare away, to lose or annihilate everything irrelevant to the essentials of the story he wants to tell. Nine tenths of the painting is bitumen black, a great pit of darkness in which the action unfolds. It is the darkness of death. To the left, the deeply shadowed figure of Christ enters the sepulchre of Lazarus and with a gesture of his right hand bids the dead man to awake: 'Lazarus, come forth' (John 11:43). Around his shadowed form a gaggle of bystanders can be seen craning their necks for a view of the impending miracle.\n\nBelow Christ's beckoning hand, two swarthy and sunburned labourers lift the dead man's tombstone, while another raises the corpse from the grave. As the last of the three workmen stumbles forward, cradling the exhumed body, panic, disgust and wonder are mingled together on his face. Lazarus is an emaciated and green-tinged corpse, just plucked from the tomb and seemingly reluctant to wake from the sleep of death. Light streams into the sepulchre from behind Christ, flowing along the line of his outstretched right arm towards the right hand of Lazarus, which reaches as if involuntarily towards the source of illumination. His left hand reaches down, towards the bony litter of the sepulchre, a human skull and thighbone gleaming softly in the low light. He is caught between life and death, suspended at the very moment of his animation.\n\nThe parable of Lazarus was traditionally regarded as a miracle performed by Christ in prefiguration of his own crucifixion. In raising Lazarus from the tomb, Christ saved him from sin and death, just as by dying on the Cross, he would save mankind from Original Sin and open the way to salvation. Caravaggio was certainly aware of the theological parallel, since he has arranged Lazarus's body in the same configuration as that of Christ on the Cross. Lazarus's two sisters, Martha and Mary, gather around him like the mourners of Christ at the moment of his deposition. The detail of Martha's face, pressed so close to that of her reviving brother, was taken directly by Caravaggio from an ancient Christian prototype of the _Mater Dolorosa_ , the Virgin Mary mourning the death of her son. Cheek pressed to cheek, eyes to mouth, mouth to eyes, the motif of two faces interlocked like pieces in a jigsaw puzzle had long been a standard trope of Byzantine painting. It was used, for example, by the twelfth-century master who painted the fresco of the _Lamentation_ in the church of St Pantaleimon in modern-day Macedonia \u2013 and from Byzantium the device entered Italian painting in the thirteenth and fourteenth centuries. Caravaggio probably took the motif from an Italian source, and it is not impossible that he saw it in a Byzantine icon in Sicily.\n\nJust as he had done in _The Burial of St Lucy_ , where the two central mourners derive from early Renaissance images of the Crucifixion, Caravaggio introduced a deliberate archaism from a much earlier tradition of art into _The Resurrection of Lazarus._ In both cases, he did so to evoke a parallel between the subject in hand and the Crucifixion of Christ. Perhaps he meant the gestures as acts of humility, a renunciation of his own illusionistic virtuosity, a penitential clipping of his own Icarus wings. He had always been an austere painter, a painter for, and of, holy poverty, but never more so than now. In emulation of Cardinal Borromeo, who had counselled a return to the austere values of the ancient Church, Caravaggio formulated his own modern version of a purged and primitive style. There is almost no colour in these works, almost no sense of space, just twisted groupings of figures arranged frieze-like in the convulsions of sorrow, melancholy or agonized bewilderment.\n\nLazarus was traditionally believed to have died of the plague. Hence the Italian word for a plague house, _lazzaretto_ (the slang word for the Neapolitan poor, _lazzari_ , shares the same etymology). Once again, as in _The Burial of St Lucy_ , Caravaggio had painted a scene like many he must have witnessed during the darkest years of his childhood in Milan \u2013 a group of people gathered around a grave, lit by what seems like guttering torchlight. He had set himself the challenge of redeeming those memories of death and desperation, of transfiguring them into representations of the miraculous. In this, he cannot be said to have entirely succeeded.\n\nFor all his efforts, what is expressed in this last and darkest flowering of Caravaggio's art is anything but a simple and straightforward sense of piety. The shadowed figure of Christ in _The Resurrection of Lazarus_ is another figment of the painter's memory, a second version of the statuesque Christ beckoning Matthew from shadow into light in the painter's very first large religious painting, _The Calling of St Matthew_ in the Contarelli Chapel. But so shadowy is the Saviour's form that he might be missed altogether by the inattentive viewer. That kind of uncertainty, whether fully intended or not, has subversively worked its way to the very heart of the picture. Lazarus is suspended between death and life, extinction and salvation. As one hand reaches towards the light, the other extends down towards the tomb. His eyes are sightless, his body gripped still by _rigor mortis_. Will he truly be saved? All is still in the balance.\n\nWhereas light flooded into Caravaggio's earlier religious paintings, here the illumination struggles to penetrate the gloom. The whole painting conveys a sense of just how hard it is truly to see \u2013 and perhaps believe in \u2013 salvation. Both men holding up Lazarus's tombstone look back with bewildered expressions at Christ, squinting and blinking in confusion. Above them, Caravaggio has included his own self-portrait. He gazes out of the picture, staring directly at the invisible source of light pulsing into the sepulchre, a look of yearning desperation on his face.\n\n#### ADORATION, DESOLATION\n\nCaravaggio's Sicilian biographer told colourful tales about how he painted _The Resurrection of Lazarus_. According to Susinno, Caravaggio asked for a room in the hospital run by the confraternity of the Padri Crociferi, hired some workmen, and arranged a grisly modelling class:\n\n> in order to give the central figure of Lazarus a naturalistic flavour he asked to have a corpse dug up that was already in a state of decomposition, and had it placed in the arms of the workmen who, however, were unable to stand the foul odour and wanted to give up their work. Caravaggio, with his usual fury, raised his dagger and jumped on them, and as a result those unlucky men were forced to continue their job and nearly die, like those miserable creatures who were condemned by the impious Maxentius to die tied to corpses. Likewise Caravaggio's picturesque room could in some fashion be called the slaughterhouse of the same tyrant.\n\nThis is surely a parable invented to illustrate the painter's reputedly excessive attachment to naturalism. Lazarus cannot have been modelled from a corpse in a state of decomposition, because his body is shown in the involuntary stasis of _rigor mortis_ ; that the painter could find the corpse of a real man who just happened to have died in a cruciform pose is less than plausible. The figure is an invention, although it seems likely enough that the three workmen were painted from local models, since their faces are unfamiliar from the rest of Caravaggio's work, and they do have the ungainly actuality of real individuals. Susinno tells an equally tall tale about a lost first version of the picture, which the painter supposedly slashed to ribbons with his dagger when a member of the de' Lazzari family had the temerity to criticize one or two elements. It is a story designed to perfect the caricature of the painter as a wild man of art, deranged by his own passions.\n\nThe biographer is more illuminating when he turns to the third and last of Caravaggio's surviving Sicilian altarpieces, _The Adoration of the Shepherds_. Long neglected, because so far off the beaten track, it is one of the most startlingly direct, wrenchingly emotional religious paintings of the seventeenth century. A sombre and profoundly personal work, it is the last great painting of Caravaggio's traumatic life. Susinno, who responded to it with heartfelt sincerity, believed it to be in fact the greatest of all his works:\n\n> In this canvas he represented the Nativity with life-size figures, and this in my opinion is the best of all his paintings, because here this great naturalist abandoned his sketchy, allusive style and demonstrated his naturalism once more without the use of bold shadows... This one great work of art would have been enough for Caravaggio's glory for centuries to come, because here he removed himself completely from dryness and from exceedingly dark tones. Instead, on the ground is a basket with carpentry tools alluding to St Joseph's trade. Above, on the right, the Virgin is seen stretched out on the ground, looking at the Christ child wrapped in cloth and caressing him. She is leaning on a haystack, behind which those animals are grazing; on the left side, at the foot of the Virgin, St Joseph is seated in attractive drapery, deep in thought. Nearby the three shepherds adore the newborn child; the first one has a staff in his hand and is dressed in a white garment, the second with his two hands joined in prayer shows a bare shoulder that looks like living flesh, and finally the third one looks on admiringly, his bold head painted marvellously. The rest of the canvas consists of a black background with rough wood that constitutes the shed. Indeed, the background was higher, and it was necessary to cut off a large section in order to fit the canvas into the chapel.\n\nCaravaggio's _Adoration of the Shepherds_ is the most tragic of nativities. Mary has just given birth to her tiny, swaddled child. She slumps, exhausted from her labours, not against a haystack as Susinno mistakenly said, but against the side of a manger. Behind her, in the half dark, the biblical ox and ass stand patient and impassive. Mary is the _Madonna del Parto_ \u2013 the 'Madonna of childbirth' \u2013 and also the Madonna of humility. She reclines on the bare earth of the cattleshed, strewn with strands of straw that catch the light like threads of gold. Those pieces of light are her only riches, but her eyes are closed to them. She is a refugee mother, utterly alone in the dark with her defenceless child.\n\nJoseph is singled out by his halo, but he is not with her \u2013 he is with the shepherds, part of her audience. None of the men are truly with her, and there is no sense that any of them can help her. The bald shepherd closest to Mary and her child reaches out to touch them. But his hand is kept back from actually making contact, as if by an invisible force. The men are suspended in an eternal agony of empathy. Their faces radiate compassion and helplessness. What can be done? Does the world really have to be like this?\n\nOnce again, Caravaggio in Sicily reached back to the oldest popular traditions of Christian art. The motif of the tiny baby, crawling on his mother's body, pressing his face to hers and reaching to touch her with his little hand, is drawn from Byzantine art. The whole scene has been conceived as another of the painter's assemblages of sculpturally realized figures, but this time what is evoked is not the sacred mountain with its chapels, but the tradition of the Christmas crib, begun by St Francis at the monastery of Greccio in the chill winter of 1223.\n\nIt is no coincidence that _The Adoration of the Shepherds_ was painted for one of Messina's Franciscan churches. Such was the depth of the friars' attachment to Caravaggio's painting that they later fought tooth and nail to keep it. 'At various times princes have been attracted by this Nativity and sought to take it away,' wrote Susinno, 'but they were unable to do so because the Capuchin Fathers made an appeal to the Senate, which in those days was more important, and its authority made them realize that those Fathers were its only custodians. As a result the picture remained in Messina, and I can affirm in truth that this unique work is the most masterly painting by Caravaggio.'\n\nAll of Caravaggio's great Sicilian pictures reach back \u2013 back to the oldest and most direct forms of Christian art, and back to his own oldest and most painful memories. Whether he was conscious of it or not, _The Adoration of the Shepherds_ is an uncanny allegory of his own emergence into the dark world of Milan under plague back in the 1570s \u2013 born to a mother soon to be bereaved, born to be abandoned by all save her. That is why the men in the picture look on but cannot touch, like dreams or ghosts. They see the mother and child's abandonment, but can do nothing to assuage it. They are hardly in the same place, but in another shadowland. Iconographically, the gnarled and saddened men are Joseph and the shepherds. Emotionally, they are Caravaggio's father, his uncles, his grandfather \u2013 all the men in the family that he might have had, but lost. Caravaggio's own father's tools had been those of a simple stonemason. Here they are replaced by the equally humble tools of the carpenter, placed with such desolation to the other side of Mary.\n\nThe set square, the saw, the adze, the white rag, lie there unused, a _memento mori_ , oblique memorial to an ordinary man who left an extraordinary child to fend for himself. This is Caravaggio's last still life. These are among his last truly meaningful, eloquent brushstrokes. The picture is almost unbearable.\n\n#### 'LIKE A CRIMINAL ESCAPING FROM HIS GUARDS'\n\nIt is hard to know what Caravaggio did during his time in Messina, other than paint. Susinno says that he paraded himself as a heretic: 'Apart from his profession, Caravaggio also went about questioning our holy religion, for which he was accused of being a disbeliever...' But he also tells that dark story of the painter's visit to a Messinese church, where he refused holy water on the grounds that it was only good for washing away venial sins. 'Mine are all mortal' were Caravaggio's words, hardly those of a man untroubled by questions of salvation or damnation. Regrettably, there is no hard evidence about his beliefs. In religion, as in so much else, Caravaggio was perhaps a man divided \u2013 torn between doubt and faith, angry rebellion and sullen obedience.\n\nHe stayed longer in Messina than he had in Syracuse. He had won the favour of the Senate \u2013 which commissioned and paid for the _Adoration_ , according to Susinno \u2013 and perhaps that added to his sense of security. But his behaviour remained erratic. 'He used to have his meal on a slab of wood, and instead of using a tablecloth, most of the time he would eat on an old portrait canvas; he was foolish and crazy, more cannot be said.'\n\nSusinno's weirdest story about Caravaggio concerns his alleged sexual interest in a group of adolescent schoolboys who used to play near the dry docks at the eastern end of Messina. It is an unusual anecdote in the context of the Sicilian author's _Lives_ , which are not otherwise salacious:\n\n> He used to disappear during holy days to follow a certain grammar teacher called Don Carlo Pepe, who escorted his pupils for recreation to the arsenal. There galleys used to be built... Michele went to observe the positions of those playful boys and to form his inventions. But the teacher became suspicious and wanted to know why he was always around. The question so disturbed the painter, and he became so irate and furious... that he wounded the poor man on the head. For this action he was forced to leave Messina. In short, wherever he went he would leave the mark of madness.'\n\nHaving seemingly implied that the schoolteacher was accusing Caravaggio of an indecent interest in his pupils, Susinno himself asserts that the painter's real motive for following the boys was artistic. The priest-biographer ends up by writing off the whole incident as yet another instance of Caravaggio's mental instability. But because of its very oddity and untidiness, the story has the reek of truth. Caravaggio was hunted, haunted and lonely in Messina. It is by no means inconceivable that he should have sought companionship, even sexual solace, in the company of young men. Susinno's anecdote might even help to explain one of the most enigmatic and homoerotic paintings of Caravaggio's Sicilian period, his last depiction of _St John the Baptist_ , now to be seen in the Borghese Gallery. Was this one of Don Pepe's pupils? Did Caravaggio persuade him to model for him, and perhaps more?\n\nPlaced in a cursory wilderness, landscape lost in shadow, accompanied by a cursory lamb of God, the boy reclines on a swag of red drapery and fixes the viewer with a sullen, sultry, knowing gaze. Is this really John the Baptist, prophet and seer, possessor of secret knowledge, or a swarthy Sicilian boy, older than his years and conscious of his sexual appeal? The artist still had the picture with him when he died: it was in the inventory of his last effects, which suggests that it was not painted to order but on impulse.\n\nCaravaggio had evaded capture, first in Syracuse, then in Messina. He may have been forced to leave Messina because of the fracas with the schoolmaster, but he was probably planning to leave anyway because he suspected that his enemies were closing in on him. According to Bellori, 'misfortune did not abandon Michele, and fear hunted him from place to place. Consequently he hurried across Sicily and from Messina went to Palermo, where he painted another Nativity for the Oratorio of San Lorenzo... The Virgin is shown adoring her newborn child, with St Francis, St Lawrence, the seated St Joseph, and above an angel in the air. The lights are diffused among shadows in the darkness.'\n\nFrom this moment on in the painter's story, the light is diffused among a great many shadows. But certain facts are clear. As Bellori said, Caravaggio left Messina for Palermo, sometime around the height of summer 1609. Once again, he painted an altarpiece for the Franciscans, this time for an oratory in the possession of a confraternity known as the Compagnia di San Francesco. Perhaps in deference to the sensibilities of its members he painted a rather sweeter version of the heartbreakingly bare _Adoration_ in Messina. The Virgin is still weary, still seated on the ground, but without the same sense of desolation and isolation. Comparison between the two works is no longer possible, since the Palermo version was allegedly stolen by order of a Sicilian Mafia boss in 1969, and has never been recovered.\n\nCaravaggio did not stay long in Palermo. Within two months, at most, he was on the move again. By the middle of September 1609 he was back in Naples. Baglione says he left because 'his enemy was chasing him'. Bellori agrees: 'he no longer felt safe in Sicily, and so he departed the island and sailed back to Naples, where he thought he would stay until he got word of his pardon allowing him to return to Rome.'\n\nOn his return to Naples, Caravaggio stayed in the Colonna Palace at Chiaia. With its vast terraced gardens, close to the sea, it was an idyllic retreat from the cares of life, with the added bonus of thick walls. The fact that Caravaggio had evidently been accepted back into the Colonna fold suggests not only that Marchesa Costanza had forgiven him, yet again, but also that she had negotiated some kind of truce with Alof de Wignacourt and the Knights of Malta. Her own son, Fabrizio Sforza Colonna, remained in post as admiral of the Grand Master's galleys. He owed both his liberty and the rescue of his reputation to Wignacourt. In such circumstances, it would have been inconceivable for Costanza Colonna to have protected a known fugitive from the order. Whatever it involved, a deal must have been struck on Caravaggio's behalf. He would presumably have been required to send some paintings to the Grand Master, as well as putting an end to the absurd pretence that he was still a Knight of Magistral Obedience.\n\nNews that Caravaggio was back in Naples soon got around and offers of work followed. 'In Sant'Anna de' Lombardi he painted the Resurrection,' Bellori wrote. The picture does not survive, because the chapel that once housed it in the Neapolitan church of the Lombards was destroyed by an earthquake at the turn of the nineteenth century. But documents and eyewitness accounts confirm that Caravaggio, himself a Lombard by origin, did indeed paint a large altarpiece of _The Resurrection of Christ_ for Sant'Anna. To judge by the praise heaped on it, it was a strange and morbidly enthralling picture, the lost masterpiece of Caravaggio's later years.\n\nCaravaggio's patron, Alfonso Fenaroli, had obtained the rights to the third chapel on the left side of the church on 24 December 1607, six months after the painter had left for Malta at the end of his first stay in Naples. Fenaroli must have commissioned the new altarpiece as soon as the artist arrived back from Palermo, probably sometime around the beginning of September 1609. Working in the abbreviated and fluent style of his Sicilian altarpieces, Caravaggio finished it before the end of the following month. Nearly a hundred and fifty years later, the travelling French connoisseur Charles-Nicolas Cochin was bowled over by it. By then the picture had darkened with age, and the identity of its creator had been forgotten. Cochin had every reason to pass it over, but it seemed so bizarrely original, so memorable and so sinister, that it drew him in:\n\n> In the third chapel on the left, one sees a painting representing the resurrection of Jesus Christ. It is a singular invention, Christ is not even shown rising into the air and he walks past the sentries [who guard the holy sepulchre]. All of which gives a low idea of him, and makes him look like a criminal escaping from his guards. Also, he has been given the character of a scrawny suffering man. From a purely pictorial point of view the composition is really beautiful and the style is strong and felt with great taste. It is much blackened. No one knows the name of the artist. This piece is beautiful.\n\nOnly in these words is the vivid ghost of a great painting preserved. Cochin was unaware of the painter's identity, his flight from Rome, his escape from Malta, his restless peregrinations through Sicily \u2013 yet still, purely from the power of his work, he sensed the depth of Caravaggio's unease. The painter had made Christ 'look like a criminal escaping from his guards'. Just as he had done in his haunting Sicilian pictures, Caravaggio was putting his own memories and emotions at the heart of his work. Whatever he set out to paint \u2013 the death of a martyr, the infancy of Christ or his resurrection \u2013 he always ended up painting himself.\n\n#### THE KNIGHT'S REVENGE\n\nHis work had never been bleaker or more emotionally naked. But in the autumn of 1609 Caravaggio had some grounds for optimism. Alof de Wignacourt seems to have been appeased, which lifted the threat of sudden rendition to Malta, and negotiations were reopened for the papal pardon that would allow him to return to Rome at last. Emboldened, perhaps, by the sense that his fortunes were about to change, Caravaggio fatally let down his guard. He paid an ill-advised visit to the Osteria del Cerriglio, a Neapolitan tavern frequented by artists and poets and much celebrated in the popular literature of the time.\n\nThe Cerriglio was located in a narrow alleyway behind the Neapolitan church of Santa Maria La Nova. There are a number of theories about the original meaning of its name, which may have derived from the _cierro_ , local slang for the long forelock worn by the cut-throats who were often to be seen there; from the merry appearance ( _cera_ ) of those who had enjoyed its hospitality; from an oak forest ( _cerrillo_ ) that had once stood nearby; from the _cerilleros_ , the vagabonds and wastrels who caroused at the tavern; or simply from the name of its owner. It was famously a place where the wine flowed more freely than water, but a number of hitherto overlooked documents reveal that the Osteria del Cerriglio was also notorious as a brothel. Giulio Cesare Cortese, an exact contemporary of Caravaggio, wrote a mock-epic poem entitled _La conquista del Cerriglio_ , in which the imaginary moment of the tavern's foundation is marked by 'huge orgies'. Real orgies took place there too: another of the painter's contemporaries, Giambattista Basile, called it 'that place where the courtesans \/ wallowed \/ in front of disapproving passersby \/ stripping the gullible to the bone', while yet another poet of the period, Giovan Battista del Tufo, added the detail that 'moreover, for gentlemen, \/ There is a door for entering secretly.' The Cerriglio was especially popular among men seeking sex with other men, to judge by the insinuation in Basile's description of it as a place 'where Bacchus reigns and Venus is shunned'.\n\nThe nineteenth-century Neapolitan poet, playwright and historian Salvatore di Giacomo, whose work on the underworld of seventeenth-century Naples has been largely forgotten, unearthed several incriminating references to the tavern in the archives of the city. 'The Cerriglio was not wholly frequented by well-mannered individuals, and the innkeeper would often turn a blind eye if not turn his back altogether,' he wrote in his pioneering study of 1899, _Prostitution in Naples_. Elsewhere, di Giacomo described just what 'gentlemen' such as Caravaggio might find when they walked though the brothel's discreetly concealed door and entered its upper rooms: 'These rooms nowadays would be called _higher chambers_. Since the end of the 16th century, by which time the Cerriglio was already famous, they had made up a separate quarter [of the tavern]... in one of these little rooms, in circa 1671, a slave was caught practising what are nowadays referred to as certain psychopathic sexual acts, which were thought of in less scientific terms in the seventeenth century and punishable with beheading.' The only sexual act punishable by beheading was sodomy. The Cerriglio clearly catered for a wide range of sexual appetites.\n\nCaravaggio's problems arose when he tried to leave the tavern. He had been followed there by a group of armed men, who waited for him in the street outside as he took his pleasure within. As soon as he walked out of the door, they ambushed him. On 24 October 1609, a Roman newspaper included the following notice: 'Word has been received from Naples that Caravaggio, the famous painter, has been murdered. Others say disfigured.' The rumour of his death turned out to have been exaggerated. He had not been killed, but he had been severely injured.\n\nWithin days of the publication of the newspaper report, Caravaggio's old friend and biographer, Giulio Mancini, put out his own antennae. Mancini did not yet know the full truth, but what he did know filled him with anxiety. He wrote to his brother Deifebo in Siena: 'It's said that Michelangelo da Caravaggio has been assaulted by 4 in Napoli and the witnesses say he has been given a facial scar. If so it would be a sin and is [the next word, which begins with a _d_ but is illegible, could be 'disturbing' or 'a disgrace'] to everybody. Let God make it not so.'\n\nMancini wrote that Caravaggio had been _sfregiato_ , cut on the face, which in the honour code of the day was an injury inflicted to avenge an insult to reputation. The same word had been used by the writer of the Roman news report. It lends both brief accounts of the assault a grim specificity, and explains the other detail gleaned by Mancini: that Caravaggio had been attacked by a group of four men. This was no drunken fracas but a premeditated act, a vendetta attack ruthlessly executed: three men to hold him down, one man to cut the marks of shame into his face.\n\nYears later, the painter's biographers gave their own terse versions of what had happened. They were unanimous on two points. It was a coldblooded attack \u2013 a hit \u2013 and it was perpetrated by a man or a group of men from Malta.\n\nBaglione's report of the assault at the Cerriglio follows seamlessly from his account of Caravaggio's incarceration on Malta and his subsequent escape. It is clear that Baglione believed the two episodes were linked as surely as cause and effect:\n\n> In Malta, Caravaggio had a dispute with a Knight of Justice and in some way affronted him. For this he was thrown into prison. But he escaped at night by means of a rope ladder and fled to the island of Sicily. In Palermo he executed several works, but because he was still being pursued by his enemy he had to return to Naples. There his enemy finally caught up with him and he was so severely slashed in the face that he was almost unrecognisable.\n\nBellori, writing considerably later than Baglione, thought the cause of the assault lay elsewhere. In his account it was not the revenge attack of an insulted Knight of Justice, but a mission carried out by implication on the orders of Alof de Wignacourt:\n\n> [Caravaggio] felt that it was no longer safe to remain in Sicily and so he left the island and sailed back to Naples, intending to remain there until he received news of his pardon so that he could return to Rome. At the same time seeking to regain the favour of the Grand Master of Malta, he sent him as a gift a half-length figure of Herodias with the head of St John the Baptist in a basin. These attentions availed him nothing, for stopping one day in the doorway of the Osteria del Cerriglio he found himself surrounded by several armed men who manhandled him and slashed his face.\n\nFrancesco Susinno, writing still later, but from a position considerably closer to the events on Malta and Sicily, leaned towards Baglione's version of events: ''The fugitive arrived in Palermo, and in that city also left excellent works of art. From there he moved again to Naples, chased there by his angered antagonist, and was badly wounded on the face.'\n\nTo these counterposed explanations of the attack may be added one other possibility: that its origins lay not in Malta but in Rome, and that it was carried out either by or on behalf of the aggrieved relations of the late Ranuccio Tomassoni. There is no suggestion that this was the case in any of the early biographies, nor in any contemporary source. In fact there is no hard evidence of any kind to support the hypothesis. But the theory has been advocated by at least one influential scholar of Caravaggio's life and work in recent years.\n\nA great deal of archival research has been done on Caravaggio over the past half-century. Many new discoveries have been made, and it is striking how in almost every case the historical facts have tended to confirm the accounts of one or other of Caravaggio's early biographers. Baglione has generally proved to be more accurate than Bellori, which is not surprising: he was part of Caravaggio's own circle, and although the two men were enemies they took more than a passing interest in each other's activities. Baglione knew who Caravaggio's friends and allies were in Rome, and understood the complicated and violent codes of honour by which he lived and died, whereas Bellori was simply baffled by them. A fairly straightforward process of elimination establishes Baglione's account of the assault in the Osteria del Cerriglio as the most credible explanation of the whole dark business.\n\nThe modern suggestion that Ranuccio Tomassoni's relations were the aggressors lacks merit on the grounds of chronology, geography and logic. The attack in the Cerriglio took place more than three years after Caravaggio had murdered Tomassoni. Even if it is assumed that the Tomassoni clan was still bent on revenge, which in this case would have been a dish served very cold indeed, it is unlikely that they would have attempted an attack on the painter in distant Naples: far better to wait until his heralded return to Rome, where they could watch his movements and plan their strike with a greater certainty of success. The most powerful argument against their involvement is the nature of the wounding Caravaggio suffered. He had been cut in the face. In the language of vendetta, the _sfregio_ was punishment for an insult to honour and reputation. But the painter had murdered Tomassoni, not merely insulted him. An eye for an eye: if the Tomassoni had been behind the assault in Naples, Caravaggio would have been killed, not disfigured.\n\nBellori's suggestion that Alof de Wignacourt ordered the attack is equally illogical. Caravaggio had not personally insulted Wignacourt, nor had he attacked his reputation. True, he had defied the Grand Master's authority. But the appropriate punishment for that was extradition back to Malta. The facial wounding of an errant knight at a house of ill repute was not something Wignacourt would have sanctioned. His involvement seems even less likely, given that at the time of the attack Caravaggio was living in the household of the mother of Wignacourt's admiral of the fleet. The Grand Master was ruthless but he was also intensely pragmatic. If he had wanted satisfaction from Caravaggio, he would have taken it in the form of pictures.\n\nBaglione's account, to which the Sicilian biographer Susinno subsequently gave his imprimatur, is the only one entirely consistent with the known facts of the case. It has the cold logic of vendetta, stressing the symmetry between insult given and punishment received, even in the author's choice of words. Baglione says Caravaggio had 'affronted' the Knight of Justice on Malta, a usage that etymologically conjoins insult with the notion of a metaphorical loss of face ( _affronto_ , the word used by Baglione, has the same root as _fronte_ , Italian for 'forehead'). In revenge, Caravaggio's enemy literalized that same insult, slashing him in the face.\n\nThat enemy was, we now know, Giovanni Rodomonte Roero, the Conte della Vezza. We also know that he left Malta shortly after Caravaggio's escape from the island. That too is consistent with Baglione's assertion that the painter was slowly but surely tracked by his enemy, who followed him to Sicily from Malta and finally caught up with him at the Osteria del Cerriglio. Since the facts to have emerged from the Maltese archive tally so exactly with the arc of Baglione's narrative, it is only logical to believe that the rest of his account is also correct. He asked the right questions of the right people, and he established the truth: it was indeed a vendetta, begun in Malta and finished in Naples.\n\nWhatever the painter had said or done to him on the night of the fracas in Malta, Roero had been left with a burning sense of grievance. Maltese Knights of Justice were not known for their propensity to forgive and forget. The Conte della Vezza was evidently proud and mercilessly persistent. He had a team of accomplices. This was the man who hunted Caravaggio down, who stood over him as he struggled, who cut his face.\n\nAfter exacting his bloody revenge, Roero vanished from historical view. That too seems to have been part of his plan. He may have been helped by friends within the Maltese judiciary. Shortly after the revenge attack, all details of Caravaggio's crime on Malta were carefully painted out of the archive there by an unknown hand. In this way, the artist's name was obliterated from the great book of crimes and punishments. So too was the name of his victim and assailant. Having got his revenge, Roero meticulously covered his traces. Even Baglione, who plainly knew so much, never discovered the name of Caravaggio's assailant.\n\n#### TWO LAST PAINTINGS\n\nCaravaggio seems never to have fully recovered from the attack at the Osteria del Cerriglio. Crippled and perhaps partially blinded by his injuries, he went into the limbo of a long convalescence. On Christmas Day 1609, two months after the assault, Mancini's correspondence with his brother Deifebo communicated a solitary scrap of inconclusive rumour: 'It's said that Caravaggio is near here, well looked after, also that he wants to return to Rome soon, and that he has powerful help.' Negotiations for a papal pardon may have been progressing, but in truth Caravaggio was nowhere near Rome. Mancini had been misinformed. The painter was in Naples, presumably at the Colonna Palace at Chiaia, fighting for his life. He would remain there for at least six months.\n\nMancini's letter apart, from October 1609 until May 1610 there is a striking absence of evidence about Caravaggio's activities. He apparently does nothing, says nothing. The archive falls silent, like a cardiograph flatlining. It then flickers briefly, but only twice. Each flicker takes the form of a painting.\n\nThe seriousness of Caravaggio's injuries is shockingly apparent in _The Denial of St Peter_ , a melancholic and withdrawn devotional work painted some time in the summer of 1610. It is a terminally raw and ragged thing \u2013 an image snatched from the pit of darkest adversity, painted by a man who could barely hold a brush. The stark and pared down style evolved in Sicily has been appallingly coarsened. Three figures, two men and a single woman, confront one another in the shallowest of spaces. The conception is subtle, the composition strikingly original and the mood bitterly sad. But such is the uncertainty of the handling that the whole image looks disconcertingly unfocused. It is still recognizably a Caravaggio, but the brushwork is so broad, the definition of forms so unsure, that the painter seems to have fallen prey to some form of essential tremor, an uncontrollable shaking of the hands, as well as perhaps to damage of the eyes.\n\nThe story that the painting illustrates is told in all four books of the New Testament. According to the gospels, Christ prophesied that his disciple Peter would deny him three times before the cock had crowed twice. On the day of Christ's arrest in the garden of Gethsemane, Peter followed his master into the courtyard of the high priest Caiaphas. He waited there as Christ was tormented by his accusers: 'And some began to spit on him, and to cover his face, and to buffet him, and to say unto him, Prophesy: and the servants did strike him with the palms of their hands. And as Peter was beneath in the palace, there cometh one of the maids of the high priest: And when she saw Peter warming himself, she looked upon him, and said, And thou also wast with Jesus of Nazareth. But he denied, saying, I know not, neither understand I what thou sayest.' Twice more, Peter was asked if he knew Jesus, and each time he gave the same answer: 'And the second time the cock crew. And Peter called to mind the word that Jesus said unto him, Before the cock crow twice, thou shalt deny me thrice. And when he thought thereon, he wept' (Mark 14:65\u201372).\n\nCaravaggio has combined elements from all three denials in a single image. Behind the figures, a reddish-brown smudge and some scattered flecks of brighter pigment suggest the fire by which Peter warms himself, damp logs spitting sparks into the air. On the left, his face entirely in shadow, stands one of Caiaphas's guards. He looks like a dim memory of the malign soldier in the much earlier _Betrayal of Christ_ , which had shown the moment directly before Peter's threefold denial. This soldier's red shirtsleeve is indicated in a few summary strokes of red paint with swiftly dashed-in highlights. A wedge-shaped piece of light fragments and disperses in the darkness of his armour. His face and hands are a blur. Beside him, a single girl stands in for both maids challenging Peter. She stares intently at the soldier while pointing at Peter with a half-sketched hand.\n\nThe most eloquent figure in the picture is Peter himself, his bald head creased with lines and his face carrying an expression of deep, glassy-eyed self-recrimination. He points both of his own hands towards himself, as if to complete the triple accusation. He denies Christ and hates himself in the same moment. A tear wells out of a corner of his half-hidden right eye. He is the embodiment of saddened guilt, a man who knows he has done wrong and can hardly bear to confront himself.\n\nAgainst the odds, it is a moving and powerful image. Caravaggio has drawn on all his long-practised ingenuity. But his strategies are those of evasion. Crop the figures to extreme close-up, to avoid problems of anatomical articulation. Arrange the faces at odd or oblique angles, to obviate the need for accurate depictions of human physiognomy. Smother any awkward areas in blankets of shadow. Wherever gleams of illumination do pierce the darkness, they reveal the imprecision of the painter's touch. His draughtsmanship, the way he draws with the brush, has collapsed altogether. Peter's hands are like flesh-coloured mittens, his left thumb so botched it resembles the claw of an animal. Light flaring in darkness had once been Caravaggio's signature, the source of all his pictorial magic. Now it exposes his illness and incapacity, and shows us how that magic has evaporated.\n\nOnly one other painting survives by Caravaggio's hand. Darker still than _The Denial of Peter_ , and yet more abbreviated in style, _The Martyrdom of St Ursula_ is his last picture. Once more, a group of fragmentary figures has been arranged in a frieze-like composition within the shallowest of shadowy spaces. There is almost no light at all, and very little sense of scene or background, save for some shadowy drapery intended perhaps to signify the inside of a tent. It is a picture so entirely lacking in the connective tissue of illusion that it is like language without conjunctions or prepositions: killer's face, hands; shocked woman's eyes; victim stunned; two men watching.\n\nThe painting's subject is drawn from the life of St Ursula, as recounted in _The Golden Legend_. A chaste princess led 11,000 virgins on an ill-fated pilgrimage through Germany:\n\n> And then all these virgins came... to Cologne, and found that it was besieged with the Huns. And when the Huns saw them they began to run upon them with a great cry, and enraged like wolves on sheep, and slew all this great multitude. And when they were all beheaded, they came to the blessed Ursula, and the prince of them, seeing her beauty, so marvellous, was abashed, and began to comfort her on the death of the virgins, and promised to her to take her to his wife. And when she had refused him and despised him, he shot at her an arrow, and pierced her through the body, and so accomplished her martyrdom...\n\nThe convention was to paint a vast crowd scene, an orgy of death. Caravaggio did the opposite. He envisaged the scene of Ursula's martyrdom as a horribly intimate ritual wounding. The murderous Hun, who seems horrified by the result of his own actions, has just shot Ursula at point-blank range in the stomach. The victim of a sexual insult \u2013 'she had refused him and despised him' \u2013 responds by subjecting the woman who had scorned him to a vile parody of pregnancy. Her swollen belly has been impregnated by the tip of an imperfectly painted arrow. She looks down with an expression of quiet surprise as blood spurts from the entry point, making a gesture with her hands that suggests she wants to part the flesh of her stomach still further. She is about to give birth to her own death.\n\nThree others complete the group. Ursula's shocked maidservant hovers like a ghost between the killer and her mistress. In her left hand she holds the pole of a Christian banner, while with her right she reaches, too late, for the Hun's bow. A soldier in black armour, shown in half-profile, approaches to catch the martyr should she swoon or fall. Directly behind Ursula's stooped white mask of a face, another ghoulish face stares sightlessly into space. It is as if she has grown a second head. This is the last of all Caravaggio's self-portraits.\n\nWhat did he mean by this strange, haunting device? To suggest his own sympathy for the martyr, his wish to die like her? Or was he painting his realization that he was actually dying \u2013 and dying, like her, from a revenge wound inflicted at close quarters? His mouth is half open, as though to suggest that he is gasping, that he feels the arrow piercing his flesh too. Had Caravaggio turned the whole scene into a proxy for his own traumatic ordeal at the Osteria del Cerriglio? The assassin has the weatherbeaten face of a warrior. Is he too a portrait, an image dredged up from painter's worst memories?\n\nThere are no answers to these questions. With the completion of the picture, darkness closes in on Caravaggio.\n\n#### THE BOATMAN'S STORY\n\nCaravaggio painted his last picture for Prince Marcantonio Doria of Genoa, who had probably sheltered him when he briefly fled Rome in the summer of 1605 after assaulting the notary Mariano Pasqualone. The prince, who had once tried to commission an entire fresco cycle from Caravaggio, had to content himself with a single canvas. He probably chose the subject of _The Martyrdom of St Ursula_ in honour of his beloved stepdaughter Ursula, who like her namesake had committed herself to a lifetime of chastity by taking religious vows.\n\nA small comedy of errors attended the delivery of the painting. It had been finished by 10 May 1610. But the very next day Doria's procurator in Naples, Lanfranco Massa, apologized to his master for having nearly ruined it: 'I thought to send you the painting of Saint Ursula this week, but to be sure that it was dry, I put it in the sun yesterday, and this instead caused the thick varnish which Caravaggio put on to liquefy; I want to obtain Caravaggio's opinion on how to do it so as not to harm it. Signor Damiano has seen it and was amazed, like all the others who saw it...'\n\nIt took more than two weeks to put the picture right, but by the end of May it was ready for despatch from Naples to Genoa. On 27 May, Massa wrote to Prince Doria: 'I am sending with P. Alessandro Caramano on his boat a long box inside of which is the painting of _The Martyrdom of St Ursula_ , carefully packed, for which [you] will be required to pay 50 soldi in conformance with the shipper's policy.'\n\nThe correspondence is completed by a shipper's manifest, dated the same day: 'Sr Lanfranco Massa has loaded in the name of God and of good fortune in the present port of Naples onto the _felucca_ named _Santa Maria di Porto Salvo_ , owned by Alessandro Caramano, a box containing the painting of _The Martyrdom of St Ursula_ , made by the hand of Michel'Angelo [ _sic_ ] Caravaggio, very well packed, in order to consign it in the same condition upon arrival in Genoa to Sr Marcantonio Doria who will pay two and one-half libri of that money should God carry it safely.' The manifest was signed by a certain Antonio Feraro, 'on command of the above stated Alessandro Caramano who does not know how to write'.\n\nGod and the illiterate boatman indeed carried the painting safely to distant Genoa. According to a note of receipt in the margin of Massa's second letter, it arrived on 18 June, precisely three weeks after it had left Naples. Three weeks after _that_ , sometime around the second week of July, Caravaggio himself embarked on a _felucca_ travelling from Naples to Rome. He left from the Colonna Palace at Chiaia. He had three paintings with him, two of St John the Baptist and one of Mary Magdalen.\n\nThe timing of Caravaggio's departure suggests that he waited for Alessandro Caramano to return before leaving for Rome himself. He probably wanted to use Caramano for his own journey too. He was a trusted skipper, whose _felucca_ had a sufficiently large hold to carry bulky pictures packed in wooden boxes. Caravaggio was very ill and no doubt more than a little apprehensive, so it made sense for him to choose a boatman whom he knew. But this time, the 'St Mary of the Safe Harbour' did not bring good luck.\n\nAccording to Bellori, the painter felt confident to return to Rome because he had 'by then obtained his freedom from the pope through the intercession of Cardinal Gonzaga'. The recently appointed Cardinal Ferdinando Gonzaga was the son of Vincenzo I Gonzaga, who had purchased Caravaggio's _Death of the Virgin_. The family may have hoped eventually to obtain more pictures from the artist in return for their support. But the young cardinal seems not to have dealt with the pope directly, approaching him instead through the papal nephew, Scipione Borghese. Borghese was already the proud owner of the artist's first version of _St Jerome Writing_ as well as his _David with the Head of Goliath_. Insatiable collector that he was, Borghese agreed to help obtain Caravaggio's pardon, but only if the artist gave him his entire stock of unsold pictures as soon as he got to Rome.\n\nA Roman _avviso_ of late July supports Bellori's report that Caravaggio had been granted his long-awaited pardon for the murder of Ranuccio Tomassoni some time before he left Naples, saying that the painter was travelling to Rome 'because His Holiness had lifted the _bando capitale_ which he was under'. But Baglione was not so sure: he makes it sound as though negotiations were still continuing, even as Caravaggio set out from Naples. The painter was travelling 'on the word of Cardinal Gonzaga, who was arranging his pardon from Pope Paul V'. If the pardon had not yet been officially agreed, that may help to explain why things would go so badly wrong for him on his journey to Rome.\n\nEach writer put a slightly different slant on what happened next.\n\nIn Baglione's telling, it became the parable of a fittingly miserable death, brought on by the painter's own impetuosity and the burning July sun:\n\n> When Caravaggio went ashore he was suddenly arrested. He was held for two days in prison and when he was released, the _felucca_ was no longer to be found. This made him furious and in his desperation he started out along the beach in the cruel July sun, trying to catch sight of the vessel which was carrying his belongings. Finally he reached a village on the shore and was put to bed with a malignant fever. He was completely abandoned and within a few days he died miserably \u2013 indeed, just as he had lived.\n\nBellori gives a broadly similar account, although he emphasizes that Caravaggio was still in agony from the injuries he had received in the vendetta attack. He also embellishes the painter's detention on landing, turning it into a case of mistaken identity. The idea should not be taken too seriously, since Bellori probably just got it from misreading the phrase 'suddenly arrested' in Baglione's considerably earlier account:\n\n> he boarded a _felucca_ , and, suffering the bitterest pain, he started out for Rome... When he went ashore the Spanish guard arrested him by mistake, taking him for another Cavaliere, and held him prisoner. Although he was soon released, the _felucca_ which was carrying him and his possessions was no longer to be found. Thus in a state of anxiety and desperation he ran along the beach in the full heat of the summer sun, and when he reached Porto Ercole, he collapsed and was seized with a malignant fever. He died within a few days at about forty years of age...\n\nMancini gives much less detail. Wrongly and a bit strangely, since he knew better, he has Caravaggio leaving for Rome from Malta. He also omits the story of Caravaggio's imprisonment, release and desperate pursuit of the _felucca_ , but agrees with Baglione and Bellori that the painter died at Porto Ercole:\n\n> He left with the hope of being pardoned and went to Civita Vecchia [Porto Ercole, to be precise, according to a marginal note in the text], where, stricken with a malignant fever, he died miserably and without care, at the height of his glory, being about thirty-five or forty years of age. He was buried nearby.\n\nCertain elements of the early biographers' accounts of Caravaggio's death are questionable. He certainly did not travel to Porto Ercole on foot, for example. But in essence they got the facts right. What they said happened was, more or less, what actually happened.\n\nThe true sequence of events has been confirmed and fleshed out by two contemporary newspaper reports and a remarkable letter found in the state archive of Naples.\n\nOn 28 July a Roman _avviso_ reported that 'There has been news of the death of Michelangelo Caravaggio, the famous painter, excellent in colouring and in drawing from nature, following his illness in Port' Ercole.' Three days later, another Roman _avviso_ confirmed the news, adding the detail that he had died at Porto Ercole 'while he was coming from Naples to Rome, having obtained the lifting of the death sentence he was under'. The speed with which these reports appeared suggests that the writers may have received their information direct from Porto Ercole itself, which was a day's fast ride from Rome.\n\nBut the papal nephew, Scipione Borghese, had heard the news even quicker than the _avviso_ writers. He knew that Caravaggio was dead as early as 23 July. But although he got his information with lightning speed, it was not entirely reliable, because Borghese's source wrongly told him that the painter had died not at Porto Ercole but on the little island of Procida, a day's sail west of Naples. Presumably for that reason Borghese immediately wrote to the papal nuncio in Naples, Deodato Gentile, Bishop of Caserta, urgently demanding more intelligence. He wanted to know what had happened to poor Caravaggio. Even more pressingly, he wanted to know what had happened to the paintings in the dead man's luggage. As far as Borghese was concerned, they were now his property.\n\nThe papal nuncio in Naples was indeed able to tell Scipione Borghese what had happened. His response to the papal nephew anticipates the accounts of Caravaggio's death given by both Baglione and Bellori by many years. Yet it turns out to match their descriptions of what happened so exactly, albeit with more detail, that it was in all probability their main source of information in the first place. What Gentile told Borghese became common knowledge in Rome.\n\nGentile's letter was dated 29 July. He began by acknowledging receipt of Borghese's request for information, which had reached him on 24 July. He confessed that Caravaggio's death was 'completely new' to him. But he had made enquiries and found answers to the papal nephew's questions. He gave Borghese the full story of the painter's death, as he now understood it:\n\n> Poor Caravaggio did not die at Procida, but at Port' Ercole, because having arrived with the _felucca_ , in which he went to Palo, he was incarcerated by the captain there. In the uproar, the _felucca_ went back out into the open sea and returned to Naples. Caravaggio stayed in prison, then freed himself by paying over a huge sum of money, and perhaps on foot reached Port' Ercole by land, where, falling ill, he departed this life.\n> \n> On its return, the _felucca_ brought the things he'd left behind to the house of the lady Marchesa of Caravaggio, who lives at Chiaia, and from where Caravaggio himself had left. I immediately made sure the pictures were there, and found that there are no more than three, the two St Johns and the Magdalen, and they are in the above-mentioned house of the lady Marchesa, to whom I have sent [a message] straight away to ask that they be well looked after, so they are not ruined before they can be seen, or come into anyone's hands, since they were intended [for Your Lordship], and it is necessary to negotiate on Your Lordship's behalf with the heirs and creditors of the said Caravaggio and give them honest satisfaction.\n\nDeodato Gentile signed off with a promise to make sure that the paintings would end up in 'the hands of Your Most Illustrious Lordship'.\n\nDespite this letter's dispassionate clarity, and despite the fact that it was written within days of the events that it describes, all kinds of arcane conspiracy theories about Caravaggio's death continue to proliferate. He is said to have been the victim of a plot involving the Knights of Malta, or Costanza Colonna, or the pope himself \u2013 or all of them, acting fiendishly in concert. He is said to have been ambushed at sea, his body cut in pieces and dumped underwater in a sack. The proponents of such theories invariably claim that the information gathered by Deodato Gentile was nothing more than a smokescreen of falsehood and fabrication \u2013 a tall tale to cover up a murder. But there is no real reason to doubt the report that Gentile carefully filed to Scipione Borghese, who was not only one of the most powerful men in Italy but the head of the papal system of justice. Attempting to deceive such a man would have been foolhardy, and probably futile.\n\nIn truth, the supposed mystery of Caravaggio's death is nothing of the kind. Conspiracy theories are a distraction. Caravaggio's true fate was dark and dramatic enough to need no elaboration or reinvention. His last journey can now be clearly reconstructed, the cause of his death understood.\n\nThis is what happened.\n\nHoping that his pardon had been arranged, the painter set out from Naples to Rome on or around 9 July 1610. He left in a _felucca_ , probably the _Santa Maria di Porto Salvo_ , with his three paintings stowed in the hold. He is unlikely to have been the only passenger. A _felucca_ was a two-masted boat with square-set sails and a spitsail, which could be rowed if winds were unfavourable. It was crewed by between six and eight men, and to hire one was expensive. The usual practice was for a skipper to wait until he had two or more customers going in the same direction before beginning a journey. It is probable that Caravaggio had a travelling companion who was going to Porto Ercole, or the boatman had a delivery to make there. Either way, Caravaggio knew that Porto Ercole was the boat's final destination.\n\nAbout a week after setting sail, the _felucca_ carrying Caravaggio and his paintings docked at Palo, a high-security fort manned by a Spanish garrison, some twenty miles west of Rome. It was not the most common landing place for travellers to Rome, especially travellers wanting to arrive discreetly. But Palo was a centre for the distribution and transportation of goods and materials, as well as a fortress. It made sense for Caravaggio to land there because with his three heavy paintings in large boxes he needed a horse-drawn carriage or cart to complete his journey.\n\nWhen he got to Palo, however, something went badly wrong. His papers may have been out of order, or perhaps he just made a remark the captain of the garrison did not like. Whatever it was, before his luggage could be unloaded, he was taken away to a holding cell. 'In all the uproar,' Deodato Gentile told Scipione Borghese, 'the _felucca_ went back out into the open sea and returned to Naples.' Gentile's wording suggests a fracas, with Caravaggio resisting arrest, shouting and perhaps characteristically trying to draw his sword as he was forcibly restrained.\n\nAvoiding further involvement in the scuffle, the skipper put out to sea again. He would indeed return to Naples, but not immediately. First, with his other customer, or to make his other delivery, he had to get to Porto Ercole, some fifty miles north and, depending on the wind, about a couple of days' journey by sea further from Rome, and from Naples.\n\nMeanwhile, Caravaggio was forced to cool his heels in jail. The cause of his imprisonment may have been trivial, because he was allowed to buy his way out. At this point all the accounts become a little vague, or fanciful, suggesting there were no witnesses to what happened next. According to Deodato Gentile, Caravaggio, 'perhaps on foot, reached Porto Ercole by land'. Baglione elaborated that speculation into the maddened run of a desperate man along a parched coastline in the height of summer: 'in his desperation he started out along the beach in the cruel July sun, trying to catch sight of the vessel which was carrying his belongings.' The story clearly appealed to Bellori, who repeated it.\n\nBut it is obviously false. Assuming that Caravaggio got out of jail within a day of his arrest, he left Palo on 16 or 17 July. Scipione Borghese knew that he was dead by 23 July, which means that he actually died on 21 July at the very latest, and probably earlier. In other words, his journey from Palo to Porto Ercole can only have taken a few days, probably just a couple. But the distance between the two places is some fifty miles. In high summer, a man convalescing from serious injuries would have struggled to make that journey on foot in less than four or five days.\n\nCaravaggio may have been desperate, but he was not mad. Throughout his life he had shown a cool head in tight situations. It suited Baglione's purposes to invent the story of the enraged pursuit, because it paved the way for his smug ending \u2013 'he died miserably \u2013 indeed, just as he had lived' \u2013 but the truth is that Caravaggio had to catch up with the boat because it was carrying the paintings that were the price of his compact with Scipione Borghese. If he did not, he could not return to Rome. He knew from conversations with the skipper, or with his travelling companion, that the boat had gone to Porto Ercole. Given the 'uproar' that had accompanied his arrest at Palo, he certainly could not count on the boat returning there with his possessions. So he had to go to find it.\n\nPalo was a staging post, so even though the boat had a head start, he could easily get to Porto Ercole first. He would simply have to ride post along the coastal delivery route. With a change of horse, he might cover the whole distance in a single day. It would have been exhausting, but it was no insane race against fate. It was the logical thing to do. He left Palo probably on 16 or 17 July, and a day later arrived in Porto Ercole, another small coastal settlement, manned by a Spanish garrison. But the stress of his arrest at Palo, and the effort of getting to Porto Ercole to recover his paintings, finally broke him. In Porto Ercole, probably on 18 or 19 July, Caravaggio died.\n\nThe boat carrying his paintings arrived almost simultaneously, perhaps shortly afterwards. The skipper and crew soon learned the news of Caravaggio's illness and death. The painter was buried, hurriedly and without ceremony. In the heat of summer a body would decay quickly, so there could be no delay. Since he died alone, without relatives or friends to care for him, he was placed in an unmarked grave. His death was not recorded in the parish records. This has been regarded as a sinister omission by the conspiracy theorists. But there is an unsinister explanation for it. Porto Ercole's only priest was in dispute with the town fathers and on strike at the time. No deaths were recorded there in the summer of 1610.\n\nThe boat carrying Caravaggio's possessions could do no more than return to Naples. Presumably the vessel left immediately. It was certainly back in Naples by 29 July, when Deodato Gentile reported to Scipione Borghese that Caravaggio's paintings had been returned to Costanza Colonna's palace.\n\nThe exact cause of Caravaggio's death is unknown. Deodato Gentile, writing just over a week after the event, simply said that he fell ill and departed life. On the evidence of the agonized self-portrait in _The Martyrdom of St Ursula_ , and the shakiness of the hand that painted it, he was already unwell when he set out for Rome. The stress of his arrest, and the frantic ride to Porto Ercole in the extreme heat of July, was more than a man in his condition could take. Heat exhaustion, or perhaps a heart attack, may have been what finally killed him.\n\nOne question remains. Where did Deodato Gentile get his information? What was his source for all this close detail about the final journey and strange, sad death of a sick man trying to reach Rome from Naples? Whatever it was, it was also the source on which the later biographers drew when they elaborated their own accounts of the painter's death. No one added anything meaningful to it, except red herrings like Bellori's mistaken arrest or Baglione's headlong footrace along the coast.\n\nIt might be thought that Gentile had put his feelers out in Porto Ercole, where the death had taken place, or had sent for information to Palo, where Caravaggio had been arrested. But he could have done neither of those things: the dates of his correspondence with Scipione Borghese preclude it. Borghese wrote to Gentile on 23 July and Gentile received the letter the following day. He replied to Borghese just five days later, on 29 July. It was two or three days by boat to Palo, the same again to Porto Ercole. By horse, even riding post, it would have taken at least four days in each direction, since it is two hundred miles from Naples to Porto Ercole. A week, more likely ten days, would have been needed to get there, make enquiries, and then report back. So Gentile must have found his information in Naples. Who could he have spoken to? Who would have known all this?\n\nOnly one person could have told the papal nuncio about what happened when Caravaggio landed at Palo. Only one person could have told him about the painter's death in Porto Ercole. That person was the boatman, who had just returned to Naples with the dead painter's belongings. His crew had accompanied him, but it was the owner of the boat whom Deodato Gentile would have brought in for questioning. The whole story must have been his testimony. Hence the use of nautical terminology \u2013 'the _felucca_ went back out into the open sea', he had said, _alto mare_ 158 \u2013 as well as the ship's-eye perspective of the entire account. Hence too the vagueness after Caravaggio is arrested and the boat pulls off: that was the moment when the boatman lost sight of the painter.\n\nThe interview would have been short and to the point. The boatman was being accused of nothing and had nothing to hide. He had no reason to be evasive, so he simply told the truth as best he could.\n\nWhere did you take Michelangelo Merisi? Palo, the garrison. What happened there? Some kind of trouble. They arrested him. There was a real uproar, so it was best to take the boat on to Porto Ercole.\n\nHow did the painter get to Porto Ercole? The skipper does not know, so he shrugs and makes a guess, not thinking properly about the distances involved \u2013 'perhaps on foot'.\n\nWhat happened at Porto Ercole? He is not sure about that either, probably because the painter had died before he got there. But he does know that Caravaggio had fallen ill, and had died at that place. It had probably only just happened when the boatman arrived. He may even have been asked to identify the body, so they could bury it as soon as possible.\n\nWhat about the paintings? Of course, he knows all about them. They are back at the Marchesa of Caravaggio's house, the palace at Chiaia, the one at the edge of the city, facing the bay. He had returned them just the day before. That is where he had taken them from in the first place, when the poor man had hired him.\n\nDeodato Gentile could have had all this information second-hand from Costanza Colonna herself, because she must have quizzed the skipper of the _felucca_ when he came back to her house in Naples, with the pictures but without Caravaggio. But he did not. Gentile makes it clear in his letter that he had _not_ spoken to her, that he had only sent a message telling her to keep Caravaggio's pictures safe at all costs. Gentile's source can indeed only have been the captain of the _felucca_ himself \u2013 the skipper, in all probability, of the _Santa Maria di Porto Salvo_.\n\nCaravaggio appears for the first time as a flesh-and-blood human being in the documentary records through the fleeting testimony of a Roman barber-surgeon named Luca. The painter had been 'a stocky young man, about twenty or twenty-five years old, with a thin black beard, thick eyebrows and black eyes, who goes dressed all in black, in a rather disorderly fashion, wearing black hose that is a little bit threadbare, and who has a thick head of hair, long over his forehead'. That was in 1597. Less than thirteen years later, wounded and worn down, our last glimpses of Caravaggio are through the testimony of a humble boatman, Alessandro Caramano. Like Luca the barber's apprentice, Alessandro was just an ordinary man. He could not read and he could not write. But he could tell the truth about what he had seen with his own eyes.\n\nCaravaggio had lived much of his life close to the margins of society, surrounded by poor and ordinary people. He painted them, staging the stories of the Bible with their bodies and their faces. He painted _for_ them and from their perspective. In the end he died among them and was buried among them, in an unmarked grave. He was thirty-eight years old.\n\n#### AFTERMATH\n\nIn Naples, Rome and Malta, people in high places briefly lamented the passing of 'poor Caravaggio'. Then they got into an unseemly scramble for his last few paintings.\n\nHaving been told by the boatman that the three pictures in the painter's luggage had been deposited with Costanza Colonna, Deodato Gentile had immediately written to her claiming them on Scipione Borghese's behalf. But he had been too late. The Knights of Malta had also found out about Caravaggio's death. On the very day that Gentile wrote to Costanza Colonna, the local prior of the Knights of Malta barged his way into her palace and forcibly confiscated the pictures. Caravaggio had been dead for only ten days, but an unholy row was already brewing over his last things.\n\nOn 31 July 1610, Gentile reported back to Borghese in Rome: 'Most Illustrious and Reverend Sir... The Marchesa of Caravaggio has informed me that the paintings of Caravaggio are no longer in her house, but have been sequestrated by the Prior of Capua... said prior is claiming that Caravaggio was a serving brother in his religious order, and that therefore all the spoils are his to take. The Marchesa says that this is all folly and vanity, and the prior is not right. I will do my best to find where they are kept, and use all diligence to secure them in the name of Your Illustrious Lordship...'\n\nOn the death of any Knight of Malta, his possessions indeed automatically reverted to the order. Suddenly it suited Wignacourt and his prior to pretend that Caravaggio's defrocking had never taken place, and that he had still been a Knight of Magistral Obedience when he died. But the marchesa, who knew very well that Caravaggio had been stripped of his knighthood, saw straight through this rather crude gambit. Hence her audible disgust for the prior and his men, turning up at her house and taking the paintings, as if she were a bankrupt and they were the bailiffs \u2013 it was indeed all _vanit\u00e0_.\n\nDeodato Gentile concluded his letter of 31 July by advising Borghese to write to Don Pedro Fern\u00e1ndez de Castro, Conde de Lemos, who had recently taken over the post of Spanish viceroy in Naples. Don Pedro was the most powerful man in the city. Borghese followed Gentile's advice, informing the viceroy of the prior's false claims and appealing to him for help. But the wheels of Spanish diplomacy moved painfully slowly. It was mid August before Don Pedro swung, rather confusedly, into action. He told the impudent Prior of Capua that it was no good pretending that Caravaggio had died a Knight of Malta, and that he would have to surrender all claims to the pictures. But the Spanish viceroy had evidently failed to realize exactly what had happened. Somehow or other he had got it into his head that the paintings were still being argued over in Porto Ercole, some two hundred miles north. So he fired off a peremptory letter to the head of the Spanish garrisons in Tuscany, together with an inventory listing the works of art that he particularly wanted to secure:\n\n> Honoured Sir, I have been informed that the painter Michael Angelo di Caravaggio has died at Port' Ercole and that you have in your possession all his property, especially the items indicated in the inventory which accompanies this letter, the property having been taken over as a spolium under the pretext that the deceased was a member of the Order of St John, and that it belonged to the Prior of Capua who has declared that he has no right to this spolium inasmuch as the deceased was not a Knight of Malta; and thus I charge you that as soon as you receive this letter you send me the aforesaid property by the first _felucca_ available, and especially the painting of _St John the Baptist_ , and if by chance it has been disposed of or removed from the property for whatever reason, you shall endeavour by all means to see that it is found and recovered in order to send it well packed with the other property and deliver it here to the proper authority, and you shall carry this out unconditionally, informing me of the receipt of this letter. From my desk, Naples, August 19, 1610.\n\nIt would be another five months before anything more was heard about the paintings. By then two of them had disappeared altogether, perhaps into the hands of Caravaggio's creditors, perhaps to Malta. The only work of art that anyone could locate for sure was a _St John_ , which turned out to be the picture of the saint as an olive-skinned Sicilian boy painted at around the time Caravaggio had left Messina for Palermo. By the winter of 1610 it had found its way into the house of the Spanish viceroy, who seems to have become singularly reluctant to give it up.\n\nOn 12 December the beleaguered Bishop of Caserta, Deodato Gentile, was finally able to report further developments to Scipione Borghese. He apologized for still not having despatched the _St John_ , which his lordship 'must have given up for lost', and explained the reasons. The viceroy had wanted to have a copy made of the painting for his own collection. In addition, there had been obscure problems with Caravaggio's inheritors and creditors \u2013 this part of the document is barely legible \u2013 and since he had left many debts there were people who had needed to be satisfied. Gentile promised to press the matter and obtain the painting. Only in August of the following year did the papal nuncio finally manage to prise it from the grip of the Spanish viceroy and send it, at long last, to Rome. He apologized that it had been slightly damaged in all the toing and froing. The picture has remained in the Borghese collection ever since.\n\n#### DOING JESUS LIKE CARAVAGGIO\n\nThe messy story of what happened to Caravaggio's last paintings is also a microcosm of his afterlife, and a parable illustrating his singularity as a painter. He had always been an outsider, a troublemaker, a difficult and dangerous man. Yet his art was so compelling, so original, so unforgettable, that people were simply transfixed by it. They fought to look at it, gathering in their hundreds every time a new altarpiece was unveiled, and they fought to acquire it, even though everything else about Caravaggio \u2013 his terseness, his weird dress sense, his violence, his sexual reputation, his unerring gift for getting into trouble \u2013 seemed so disconcerting and strange.\n\nCaravaggio was not only the most disturbed but also the most unconventional of the truly great painters of the Italian tradition. His whole career ran counter to type, defiantly contradicting the patterns of training, patronage and even the actual practice of painting that were expected of a successful artist. It is clear that during his obscure early years, something went awry during his supposed apprenticeship to the Milanese painter Peterzano. Essentially, Caravaggio taught himself to paint. He may have picked up technical tips and clues in places like Giuseppe Cesari's studio, but his basic method was empirical. He looked at the way light falls, and at the way people behave. The fact that he was obliged to invent himself may partly explain his deep originality. The advantage of not having been taught was that he had nothing to unlearn.\n\nOnce he had begun to find his own way, Caravaggio painted with such force, such a stunning sense of drama, such a deep sense of humanity, that prestigious commissions flooded towards him. The simple truth is that he was a far greater painter than any of his contemporaries. But, despite winning the support of Cardinal del Monte, and despite his network of protectors within the Colonna family, he never found a secure place in the hierarchies of power and patronage. He painted as if the rich and the powerful were his enemies, as if he really did believe that the meek deserved to inherit the earth. Ultimately, he acted in the same way too. Only once in his life did he come close to achieving a truly settled position, a respected place among men of real power and influence, and that was on Malta. But almost as soon as he had been knighted, he managed to have himself thrown into jail. With hindsight it looks like a complete act of self-sabotage, as if he could not bear the thought of truly belonging and of walking the corridors of power.\n\nCaravaggio was also unique among the great Italian painters in _how_ he went about painting. He had no studio in anything like the conventional sense. He had the odd boy to help him, Cecco in particular, but essentially he painted all by himself. He did not draw. He never established a workshop with specialist assistants to help with the painting of drapery or landscape, as other artists did. He gathered around himself no real circle of pupils, and there were no acolytes to spread the word, no one to disseminate his methods and his beliefs. There were no portfolios of his drawings to pass around. There was nothing except his pictures themselves, and there were not very many of those because he had died so young. Under the circumstances, the vast impact of his work is all the more remarkable.\n\nFor more than a century and a half after his death, the classicizing critics of Europe's academic art tradition made a concerted and resolute attempt to blacken his name. According to their beliefs, much influenced by the strains of Neoplatonist philosophy, it was art's duty to present an idealized version of reality, and not \u2013 as Caravaggio was held to have done \u2013 merely to represent the real world in all its unregenerate ugliness. Bellori was the arch-exponent of the anti-Caravaggist movement in academic thought, but there were many others, notably the Spanish painter and author Vicente Carducho, who demonized Caravaggio as an anti-Christ of art, the antithesis to his saintly predecessor and namesake, the 'divine' Michelangelo. So influential was the rhetoric of Caravaggio's posthumous enemies that the great French seventeenth-century painter Poussin was persuaded that he had been 'sent into the world to destroy painting'.\n\nDespite the sustained drive to denigrate and marginalize his work, Caravaggio's paintings were too profound and affecting to be suppressed. Gradually but inexorably, his dramatic sense of composition, his strikingly stark handling of light and dark and his sheer rawness of feeling worked themselves into the DNA of Western art. During the years immediately after his death, hardly a single important painter escaped his influence. Rubens, Vel\u00e0zquez and Pietro da Cortona all echoed his compositions or copied his devices and traits. Within a generation, entire schools of so-called Caravaggisti established themselves in both Italy and the Netherlands. Partly perhaps because of the location of the French Academy in Rome, at the top of the Spanish Steps, and within easy walking distance of so many of his most important altarpieces, he would have an especially powerful impact on French art. His influence can be detected in the work of such widely differing French painters as Valentin de Boulogne and Georges de La Tour. There was a particularly strong resurgence of interest in his art during the Neoclassical and Romantic periods. In England, Joseph Wright of Derby's _Experiment on a Bird in the Air Pump_ of 1768 transformed the scientific demonstration of the effects of a vacuum on a living creature into a hushed modern version of a miracle as painted by Caravaggio. In France, the self-appointed painter to the Revolution, Jacques-Louis David, painted the dead Marat slumped in his bath as if he were one of Caravaggio's spotlit martyrs, and in 1819 Theodore G\u00e9ricault conceived arguably the first great masterpiece of French Romanticism, _The Raft of the Medusa_ , as a modern, secularized version of an altarpiece by Caravaggio.\n\nTowards the end of the nineteenth century Caravaggio's work did fall somewhat out of fashion. His paintings attracted relatively little attention from those pioneering the still embryonic discipline of art history, whose attentions were biased by the market. (The purpose of much early art historical research was to establish the provenance and therefore the value of pictures coming to auction, but, since nearly all of Caravaggio's major pictures were immovable altarpieces, very few of his works ever came up for sale.) Neither did his pictures seem especially interesting to painters of the early Modern period, such as C\u00e9zanne or, later, the Cubists and Futurists, because it was their stated ambition to flatten, distort and destroy the conventions of post-Renaissance illusionist painting. Caravaggio was too much of an 'optical' painter for their taste. They preferred the so-called Italian 'primitives', painters such as Giotto and Duccio, whose disregard for conventional perspective seemed closer to a Modernist aesthetic. They might have been interested in Caravaggio's late Sicilian pictures, which responded to powerful strains of primitivism in Counter-Reformation thought, but those paintings had fallen into neglect and were all but unknown by the early twentieth century. It is symbolic of this one period of genuine neglect that the young Picasso, for all his magpie eclecticism and positively Oedipal obsession with the art of the past, never showed the slightest interest in reworking or pastiching the art of Caravaggio. It was only when Picasso grew older that his attitude changed. In 1937, while working on _Guernica_ , his agonized frieze of suffering inspired by the horrors of the Spanish Civil War, he told Salvador Dal\u00ed that he wanted the horse at the centre of the painting to have the same presence as the horse in Caravaggio's _Conversion of St Paul_ : 'I want it to be so realistic \u2013 just like in Caravaggio \u2013 that you can smell the sweat.'\n\nCaravaggio's reputation was decisively rehabilitated for the twentieth century by the gifted and eloquent Italian art historian Roberto Longhi, who put on an extremely influential retrospective of the painter's work in 1951. Since then Caravaggio has become perhaps the most widely popular of all the Old Masters. In many respects he is the perfect painter for an age pruriently obsessed with the lurid private lives of famous people. His fame has never been greater, and his private life was nothing if not lurid. His many sins and misdemeanours, his irregularities and eccentricities, so long used to blacken his name, have now made him a posthumous celebrity. But the deeper pull is still that of his art.\n\nSince Longhi staged his ground-breaking exhibition, Caravaggio's influence has continued to spread. But his work seems to have been less of an inspiration to those ploughing the increasingly conceptualist fields of fine art than to those working with photography and film. One of the few painters to have had a profound impact on disciplines other than painting itself, he may fairly be considered as a pioneer of modern cinematography. Pier Paolo Pasolini, who made some of the most powerful Italian films of the 1960s, was profoundly influenced by Caravaggio's sense of light, by his narrative directness, and by his casting of poor and ordinary working people in leading roles. Martin Scorsese, one of the most gifted American directors of the last forty years, has been disarmingly explicit about the depth of his own admiration for Caravaggio. He was introduced to the painter's work in the late 1960s by screenwriter Paul Schrader when they were working on _Taxi Driver_ , his film about a vigilante killer taking on New York's underworld of drug dealers and whores. He sees Caravaggio very much with the eyes of someone looking for things he can use, borrow, adapt. In Scorsese's words, the long tradition of Caravaggio as a true artist's artist is both reincarnated and refreshed. It is worth quoting him at length:\n\n> I was instantly taken by the power of the pictures, the power of the compositions, the action in the frames, the way he designed the composition and the subject matter... there was no doubt it could be taken into cinema because of the use of light and shadow, the chiaroscuro effect...\n> \n> Initially I related to the paintings because of the moment that he chose to illuminate in the story. _The Conversion of Paul_ , _Judith Beheading Holofernes_ : he was choosing a moment that was not the absolute moment of the beginning of the action, it's during the action, in a way. You sort of come upon the scene midway and you're immersed in it. It was very different from the composition of the paintings that preceded it, the Renaissance paintings. It was like modern staging in film. It was as if we had just come in the middle of scene and it was all happening. It was so powerful and direct. It was startling, really. He would have been a great film-maker, there's no doubt about it. I thought, I can use this too...\n> \n> So then he was there. He sort of pervaded the entirety of the bar sequences in another film I made around then, _Mean Streets_. There's no doubt about that. He was there in the way I wanted the camera movement, the choice of how to stage a scene. It's basically people sitting in bars, people at tables, people getting up, that sort of thing. _The Calling of Matthew_ , but in New York! Making films with street people was what it was really about, like he made paintings with them. They weren't like the usual models from the Renaissance. They were people who were really living life. That's why it played into my mind in _Mean Streets_...\n> \n> Then that extended into a much later film, _The Temptation of Christ_. Why couldn't we have people who lived on the street play apostles? They had been fishermen, Jesus was a carpenter. Caravaggio takes the Virgin Mary and has a prostitute play the Virgin Mary. She's a woman and the Virgin Mary's a woman. It's shocking and provocative. It doesn't judge the person. It doesn't make judgement on the prostitute when making her the Virgin and this is something very powerful and compassionate...\n> \n> So in doing _The Last Temptation of Christ_ the idea was that Jesus was going to be Jesus Christ on Eighth Avenue and 49th Street in New York, where we shot _Taxi Driver_ those years ago. It hasn't changed much since then, it's a little better now, but really you might as well be in a den of iniquity most of the time. It was quite a place, especially at three and four in the morning. This is where Jesus would go. He wouldn't be hanging out on Park Avenue in New York. He'd be in the street with the crack addicts and the prostitutes. The idea was to do Jesus like Caravaggio.\n\n1. _Portrait of Caravaggio_ by Ottavio Leoni. A Roman barber named Luca described the painter just as he appears here: 'a stocky young man, with a thin black beard, thick eyebrows and black eyes... dressed all in black'.\n\n2\/3. The _sacro monte_ at Varallo, in modern Piedmont. In a series of chapels linked by mountain paths, polychrome figures play out stories from the Bible in vivid and often bloody _mises-en-sc\u00e8ne_.\n\n4. _The Lamentation_ by Guido Mazzoni. Caravaggio was influenced by this intense form of realism. He knew Mazzoni's work well and painted a picture for the Neapolitan church that housed this very sculpture.\n\n5. _Mary Magdalen_ by Donatello. Donatello's disconcertingly lifelike sculptures may also have shaped Caravaggio's imagination.\n\n6. _Carlo Borromeo_ (detail) by Carlo Dolci. Dour and ascetic, Borromeo was the dominant force in Milan during Caravaggio's formative years. He would remain legendary in the city for centuries.\n\n7. _The Adoration of the Shepherds_ (detail) by Simone Peterzano. The young Caravaggio signed a contract of apprenticeship with Peterzano, but just what he learned from this feeble disciple of Titian is not clear.\n\n8. The Farnese Gallery by Annibale Carracci (detail). For his Farnese patrons, Carracci daringly revived pagan and erotic mythology in Counter-Reformation Rome.\n\n9. _Perseus and Andromeda_ by Giuseppe Cesari. The young Caravaggio spent some unhappy months as an assistant to Cesari, who used him to paint fruit and flowers.\n\n10. _Boy Peeling a Fruit_. This may be one of Caravaggio's earliest surviving pictures. If so, it shows how little progress he had made as a painter by his early twenties.\n\n11. _Boy with a Basket of Fruit_ : 'a blushing, smooth-skinned adolescent, with dark curly hair and an expression of amorous intensity on his face' who may represent the Groom in the biblical Song of Songs.\n\n12. _Boy Bitten by a Lizard_. In the language of the Italian street the bitten finger represented the wounded phallus.\n\n13. _Boy Bitten by a Crayfish_ by Sofonisba Anguissola.\n\n14. _Self-Portrait as Bacchus_. 'The picture has a sorceror's apprentice feel to it, with hints of illicit goings on...'\n\n15. _Bacchus and Ariadne_ by Titian _._\n\n16. _The Cardsharps._ A young aristocrat is cheated by two feral and predatory conmen.\n\n17. _The Gypsy Fortune-Teller_. 'A sweet siren, she enchants with smooth incantations.'\n\n18. _Cardinal Francesco Maria del Monte_ by Ottavio Leoni. Friend to writers, musicians, artists and scientists, del Monte was Caravaggio's first patron.\n\n19. The Palazzo Madama in Rome, where Caravaggio lived for several years under del Monte's protection.\n\n20. _Medusa_. 'Whomsoever she looks at, she freezes. From the flux of life she takes a moment and makes it last for all time. That is what Caravaggio does too. Her magic is his magic, a petrifying art.'\n\n21. _The Musicians_. By painting a rehearsal instead of an actual performance, Caravaggio went behind the scenes of the traditional concert picture.\n\n22. _Le Concert Champ\u00eatre_ by Titian. Music in the pastoral mode.\n\n23. _Concert_ by Callisto Piazza. Music as erotic stimulus.\n\n24. _The Lute Player_. This may be Pedro Montoya, a castrato favoured by del Monte. His swollen cheeks are consistent with the hormonal effects of castration.\n\n25. _Basket of Fruit_. Caravaggio's only surviving pure still life painting.\n\n26. _The Penitent Magdalen_. In a paroxysm of repentance, she has torn off her gold and her jewels and scattered them on the ground. By seating the Magdalen so low, Caravaggio emphasized her humility.\n\n27. _St Francis of Assisi in Ecstasy_. 'The picture offers a consoling dream of transfiguration, a condition of oneness with Christ.' It seems to be a self-portrait.\n\n28. _The Rest on the Flight to Egypt_. A scantily draped adolescent angel stands between Mary and Joseph. Caravaggio borrowed the figure from a composition by Annibale Carracci (below).\n\n29. _The Judgement of Hercules_ by Annibale Carracci.\n\n30. _Jupiter, Neptune and Pluto_. Decorations for the ceiling of del Monte's alchemical laboratory, in his house near the Porta Pinciana.\n\n31. _Bacchus_. The model for Caravaggio's second depiction of the god of wine was probably his Sicilian friend and fellow painter, Mario Minniti.\n\n32. _Martha and Mary Magdalen_. Fillide Melandroni, not so penitent courtesan, modelled for the Magdalen. Her friend Anna Bianchini possibly sat for the figure of Mary's sister Martha.\n\n33. _St Catherine_. Fillide also posed for this depiction of the Christian martyr Catherine. 'She leans towards the wheel and its vicious spikes of grey steel as if leaning towards a lover.'\n\n34. _Portrait of Fillide Melandroni_. She kept this portrait with her until her death. It later passed into the German national collections and was destroyed by fire during the Second World War.\n\n35. _Judith and Holofernes_. Fillide once more, frowning with concentration as she plays the part of the Old Testament biblical heroine beheading a tyrant.\n\n36. _The Entombment_. This stark, strong composition was much admired by the people of Rome, whom Giovanni Baglione described as 'chattering like geese' at its unveiling.\n\n37. _Piet\u00e0_ by Michelangelo. Caravaggio adapted the limp right arm of Michelangelo's dead Christ for his own version of the same figure in _The Entombment_ (above).\n\n38. _The Calling of St Matthew_. 'And as Jesus passed forth from thence, he saw a man, named Matthew, sitting at the receipt of custom: and he saith unto him, Follow me. And he arose, and followed him' (Matthew 9).\n\n39. _The Creation of Adam_ by Michelangelo (detail). Caravaggio modelled the hand of his beckoning Christ (above) on the hand of Michelangelo's Adam.\n\n40. _The Martyrdom of St Matthew_. Caravaggio included his self-portrait among those fleeing from the scene of the crime. He looks back as if to regret his failure to help the stricken Matthew.\n\n41. _The Conversion of St Paul_ (first version). One of Caravaggio's few failures, it was immediately rejected by the patron, Tiberio Cerasi. The reeling servant looks like a spear-carrier in a bad comic opera.\n\n42. _The Crucifixion of St Peter_. Peter insisted that he be crucified upside down, because he felt unworthy of the same death as Christ. His unfeeling executioners impassively hoist him aloft.\n\n43. The Cerasi Chapel (front view). ( _above the altar_ ) Annibale Carracci's _Assumption of the Virgin_. ( _left_ ) _The Crucifixion of St Peter_. ( _right_ ) _The Conversion of St Paul_ , with horse's rump aimed at Carracci's Virgin Mary.\n\n44. _The Conversion of St Paul_ (second version). Rejecting the tumult and drama of his first, unsuccessful treatment of the subject (Plate 41), Caravaggio internalized the action so that we sense it unfolding within Paul's soul.\n\n45. _The Supper at Emmaus_. As the risen Christ reveals himself to the two disciples, they react with gestures of awe and astonishment.\n\n46. _St John the Baptist_. Caravaggio's workshop assistant, Cecco, modelled for this depiction of the saint in the wilderness as a smiling and ecstatic young boy.\n\n47. _Ignudo_ by Michelangelo. The male nudes on the Sistine Chapel ceiling evoke the golden age of classical myth. Caravaggio appropriated motifs from Michelangelo's work throughout his life.\n\n48. _The Betrayal of Christ_. 'The pale, delicate, emotionally sensitive face of Christ is set hard against the brutish, sunburned face of Judas.'\n\n49. _Street Scene_ by Francesco Villamena. A notorious battle between Rome's pro-French and pro-Spanish factions. Caravaggio took the billowing cloak for his own nocturnal street scene, _The Betrayal of Christ_.\n\n50. _St Matthew and the Angel_ (first version). Caravaggio's first, rejected version of the altarpiece. Objections were raised to its lack of decorum. During the Second World War, it met the same fate as the lost _Portrait of Fillide Melandroni_ (Plate 34).\n\n51. _St Matthew and the Angel_ (second version). More refined but less forceful, the saint has metamorphosed from illiterate bumpkin to dignified sage. The picture still hangs over the altar in the Contarelli Chapel today.\n\n52. _The Death of the Virgin_ , in which a prostitute modelled for the Virgin. The picture was rejected by the fathers of Santa Maria della Scala, the last straw that may have triggered Caravaggio to commit murder.\n\n53. _The Death of the Virgin_ by Carlo Saraceni, the picture that replaced Caravaggio's rejected altarpiece (above).\n\n54. _Doubting Thomas_. 'Then saith he to Thomas, Reach hither thy finger, and behold my hands; and reach hither thy hand, and thrust it into my side: and be not faithless, but believing' (John 20).\n\n55. _The Sacrifice of Isaac_. Abraham holds his squealing son down as if the boy were a lamb brought to slaughter. These are the last glimpses of landscape in Caravaggio's work.\n\n56. _Omnia vincit amor_. 'Love conquers all.' Cupid was modelled by Cecco. An English visitor to Rome was told that ''Twas the body & face \/ of [Caravaggio's] owne boy or servant \/ that laid with him.'\n\n57. _Divine Love_ by Giovanni Baglione. The avenging angel triumphs over the devil, who has been caught _in flagrante_ with his young catamite. The sodomitic Satan on the left is a libellous caricature of Caravaggio.\n\n58. Study for _The Resurrection_ by Giovanni Baglione. This study preserves the composition of Baglione's lost altarpiece for the Ges\u00f9, which Caravaggio mocked openly: 'It's a bungle... the worst he has done.'\n\n59. _St Jerome Writing_. A penitentially solemn picture. It may have been a gift to Scipione Borghese, papal nephew, for helping Caravaggio obtain a pardon for violent assault in the summer of 1605.\n\n60. _St Francis in Meditation_. The saint is lost in contemplation of his own mortality, and of Christ's crucifixion at Golgotha, 'the place of the skull'.\n\n61. _St John the Baptist_. A world away from the earlier St John modelled by Cecco. This glowering adolescent 'might almost be a portrait of Caravaggio's own dark state of mind' during his later years in Rome.\n\n62. _The Madonna of the Rosary_. This altarpiece was greatly admired by Peter Paul Rubens, one of a group of connoisseurs who bought the picture for a prominent church in Antwerp in 1651.\n\n63. _The Madonna of Loreto_. Two humble pilgrims to Loreto, with patched clothes and dirt-ingrained feet, are granted a miraculous vision of the Madonna and child.\n\n64. _The Madonna of the Palafrenieri_. The Virgin Mary and the infant Christ crush the serpent Satan as St Anne looks on. The picture was turned out of St Peter's, probably because of the Madonna's full cleavage.\n\n65. The contract for the ill-fated _Madonna of the Palafrenieri._ His blood signature aside (Plate 80), this is the sole surviving example of Caravaggio's handwriting.\n\n66. View of Zagarolo in the Alban Hills outside Rome. Palazzo Colonna in Zagarolo was Caravaggio's first hide-out after the murder of Ranuccio Tomassoni.\n\n67. _The Supper at Emmaus_. A strikingly different interpretation of a subject Caravaggio had treated so subtly five years earlier (Plate 45). After the murder his style became increasingly bleak, dark and morbid.\n\n68. _David with the Head of Goliath_. Cecco as David, Caravaggio as the severed head of Goliath. Often misdated to the end of the painter's life, but actually painted in 1606 as a homicide's plea for clemency.\n\n69. _Sleeping Cupid_. Painted in Malta for a Florentine humanist, this picture was inspired by a celebrated sculpture of the same subject by Michelangelo.\n\n70. _The Seven Acts of Mercy_. 'This one dark street, scene of desperation and pain and death, is the painter's microcosm for the brutality of existence itself.'\n\n71. _Roman Charity_ (detail) by Pierino del Vaga.\n\n72. _The Flagellation_. Torture as a misbegotten act of intimacy.\n\n73. _The Crucifixion of St Andrew_. The painting does not show Andrew being bound to the cross, as some have thought, but the moment of his death. His former tormentors strive but fail to release him.\n\n74. _St Jerome Writing_. Painted for one of the most senior Knights of Malta, a virtuoso demonstration of Caravaggio's gifts to the artist's new circle of patrons in the Order of St John.\n\n75. _Portrait of Fra Antonio Martelli_. 'This depiction of an obdurate and forceful man, in lean old age, rheumy eyes gazing off into the distance, anticipates the mature portraiture of Rembrandt by some half a century.'\n\n76. _Portrait of Alof de Wignacourt, with His Pageboy_. Instead of a monastic habit, the Grand Master wears a suit of sixteenth-century armour, evoking the heroic defence of Malta at the great Siege of 1565.\n\n77. _The Resurrection of Lazarus._ 'Lazarus, come forth' (John 11). As Christ bids him to rise from the grave, Lazarus seems reluctant to wake from death.\n\n78. _The Adoration of the Shepherds_. The most tragic of nativities: Mary is a refugee mother utterly alone in the dark with her defenceless child.\n\n79. _The Beheading of St John_. In the blood gushing from the saint's neck, Caravaggio signed his name. It is his only signature on a painting.\n\n80. _St John the Baptist_. Caravaggio's last known depiction of the saint, who was modelled on this occasion by a swarthy and sun-tanned Sicilian adolescent.\n\n81. _The Adoration of the Shepherds_. Painted for a Franciscan confraternity in Palermo, the last of Caravaggio's Sicilian altarpieces was allegedly stolen by order of a Mafia boss in 1969. It has not been recovered.\n\n82. _The Burial of St Lucy_. 'The picture's iconography is ingeniously suggestive of hope... but its mood is overwhelmingly bleak. Almost half the painting is dark bare stone.'\n\n83. _The Denial of St Peter_. One of just two paintings that can be dated to after the painter's face was slashed in late 1609. The seriousness of his injuries is shockingly apparent in the work's ragged, fumbling style.\n\n84. _The Martyrdom of St Ursula_. Caravaggio's very last work contains his last self-portrait. Behind Ursula's bowed head, the painter's face appears. He seems to groan, as he stares sightlessly into space.\n\nCaravaggio's influence reached forward to the Enlightenment, continued into the Romantic period and has infiltrated the DNA of modern cinema.\n\n85. _Experiment on a Bird in the Air Pump_ by Joseph Wright of Derby.\n\n86. _The Raft of the Medusa_ by Theodore G\u00e9ricault.\n\n87. Still from _Mean Streets_ , directed by Martin Scorsese.\n\n## Epilogue\n\nCaravaggio's contemporaries would doubtless have been amazed by the extent of his posthumous fame. Few of those who knew him could ever have imagined that he and his work would survive so far into the future, that he would be remembered so long after they had all been forgotten.\n\nBut it was true. Hardly any of the artists with whom Caravaggio had been close made any mark at all on posterity. His Sicilian friend Mario Minniti lived into his sixties, turning out quantities of mediocre altarpieces and making himself a small fortune, but no great reputation, in Messina. His old assistant Cecco Boneri established something of a career for himself as Cecco del Michelangelo, but soon slipped into near-total obscurity. The hot-headed architect Onorio Longhi, who had been his second in the duel, returned to Rome a year or so after Caravaggio's death, only to die himself five years later of syphilis.\n\nCaravaggio's old enemy Giovanni Baglione lived long and prospered, winning numerous grand commissions from popes as well as princes and aristocrats. When he died he was nearly eighty, a Knight of Christ and a wealthy man. But he too would soon be forgotten \u2013 or at least remembered mostly for being Caravaggio's adversary and biographer. Orazio Gentileschi, who had once laughed along with Caravaggio at 'Johnny Baggage', was the only one of his close acquaintances to amount to much as an artist. A painter of considerable power and invention, he ended his career as a court painter to Charles I, dying in London in his mid seventies in 1639 just a few years before the start of the English Civil War. Orazio's daughter, Artemisia, who had been raped by Agostino Tassi, also became a gifted and successful painter in her own right.\n\nAnd what of those with whom Caravaggio drank and dined, quarrelled and fought? What of those whom he loved and hated? What of the waiter with the cut face, the sharp-eyed barber-surgeon, the disgruntled notary? What of all the pimps and soldiers and the boys and girls who lived by selling their bodies? Some survived in his paintings, whether as villains or martyrs, torturers or apostles. Most disappeared without trace. But one fragment has survived: the last will and testament of Fillide Melandroni.\n\nFillide had been Caravaggio's first model. She was the disconcertingly sexy _St Catherine_ , as well as the girl holding the flower up to her breast and gazing out with a smouldering, coquettish stare in the portrait he had painted of her in 1598. She was Fillide the courtesan, who perhaps won the heart and certainly lightened the purse of the Florentine aristocrat Giulio Strozzi.\n\nIn the summer of 1618, Fillide was thirty-seven or thirty-eight years old, close to the same age as Caravaggio when he died. She was still living in Rome, but by now she had her own house. She had clearly gone up in the world. But she was mortally ill, perhaps with the same form of venereal disease that had cut short the life of Caravaggio's friend Onorio Longhi. On 3 July she died.\n\nSoon after, an inventory was made of her now considerable possessions. Her main reception room was decorated with gilded leather panels. At its centre stood a table covered with a Turkish carpet, and around the table there were eight leather-covered chairs. In the bedroom she had a large gilded four-poster bed with a green taffeta canopy and a chest containing some lengths of luxury fabric. She had books, vases, plants, an inkwell of silvered copper, a pearl necklace, twenty gold buttons and two gold pendants with pearls.\n\nOn 19 November her estate was settled and division was made of her goods. The will that she had made four years earlier was read out. It seems that she was happy for all her property to be sold and the proceeds parcelled out, in specified fractions, to her chosen legatees. But she wanted one particular object to go to one particular individual: 'Item: she states and declares that she has in her house a painting or portrait by the hand of Michelangelo da Caravaggio that belongs to Giulio Strozzi. She wishes it to be restored and consigned to Sr Giulio.'\n\nFillide's portrait by Caravaggio, the picture that would be consumed four centuries later by the flames of the Second World War, was the most precious thing that she had. She wanted it to go to Strozzi, her protector, who had allowed her to keep it for so long. Perhaps she liked the thought of being with him, in surrogate, after she died. Perhaps she still loved him.\n\nFrom the inventory of her possessions and the terms of her will we may think that Fillide was not quite the same woman she had been when Caravaggio knew her. Once, she had shamelessly touted for business as a prostitute in the very shadow of the monastery of the Convertites, the religious foundation for the reform of prostitutes, and had assaulted her rival, Prudenza, in her house directly next door to it, screaming as she did so: 'You dirty whore! I want to cut you! I want to cut you!' Now, as well as the portrait by Caravaggio, her house contained three small devotional paintings: of the Nativity, of the Virgin Mary and of the Penitent Magdalen, the prostitute who mended her ways. Her will specified that she wanted to be buried in her parish church. At the end, as death approached, she left several legacies to religious institutions dedicated to the Virgin, so that Masses would be said for her soul after she had died, and a fifth of her entire legacy to the Convertites. The bequest was stipulated in the penultimate clause in her will, set down by the notary in black and white.\n\nBut who knows what Fillide really felt, or what she really believed. Like the dark-haired painter she had once known, she moved in that uncertain realm, 'between the sacred and the profane'.\n\n## Notes\n\n##### PART ONE: MILAN, 1571\u201392\n\n1. See Helen Langdon (ed.), _The Lives of Caravaggio_ (London, 2005), pp. 89, 81.\n\n2. The very structure of Bellori's _Lives of the Modern Painters, Sculptors and Architects_ consigns Caravaggio to darkness. In arranging the engraved portraits that illustrated his book, Bellori made sure that the artists whom he truly valued be given dignifying attributes such as books or paintbrushes to hold. So for example Nicolas Poussin, one of Bellori's heroes, holds a book fixed with a fine clasp and gazes out with an expression of grave calm on his face. Caravaggio, by contrast, has his hand on the hilt of a sword and glares nervously sideways with the furtive and guilty eyes of a criminal. One of just twelve artists singled out for inclusion, he has been allowed his place at the table of art history. But he sits on the wrong side, a Judas among the true apostles. For an arresting interpretation of some of the fictional elements of the early biographies see 'Caravaggio's Deaths' by Philip Sohm, _Art Bulletin_ , vol. 84, no. 3 (Sept. 2002), p. 452.\n\n3. See Helen Langdon, _The Lives of Caravaggio_ , p. 57.\n\n4. Ibid., p. 41.\n\n5. Ibid., p. 27.\n\n6. See in particular M. Cinotti, _Novita sul Caravaggio_ (Milan, 1983).\n\n7. For the importance of Caravaggio's maternal relations and their contacts, Giacomo Berra, 'Il Giovane Michelangelo Merisi da Caravaggio: la sua famiglia e la scelta dell'ars pingendi', _Paragone_ , vol. 53 (2002), pp. 40\u2013128, is the invaluable source.\n\n8. See Richard A. Goldthwaite, _Wealth and the Demand for Art in Italy 1300\u20131600_ (Baltimore, 1993): 'Generally speaking... the argument got shifted from the nature of nobility to the behaviour of the noble; and along the way, most of the essential elements of the traditional definition \u2013 arms, service, virtue, blood, economic activities \u2013 were qualified.' So many different ideas were 'bandied about the concept', the writer adds, 'that one could have it just about any way he wanted it'.\n\n9. See Diarmaid MacCulloch, _Reformation: Europe's House Divided 1490\u20131700_ (London, 2003), pp. 330\u201332.\n\n10. See C. Hughes (ed.), _Shakespeare's Europe: Unpublished Chapters of Fynes Moryson's Itinery_ (New York, 1967), p. 49; see also D. E. Zanetti, 'The Patriziato of Milan from the Domination of Spain to the Unification of Italy: An Outline of the Social and Demographic History', _Social History_ , no. 6 (Oct. 1977), pp. 745\u201360.\n\n11. See D. E. Zanetti, 'The Patriziato of Milan', pp. 750\u201352.\n\n12. See Thomas Coryate, _Coryat's Crudities_ (London, 1611), p. 102.\n\n13. See 'Instrucciones de Carlos-Quinto a Don Felipe su hijo', in C. Weiss (ed.), _Papiers d'Etat du Cardinal de Grenvelle_ , vol. 3 (Paris, 1842), pp. 267\u2013318. My attention was brought to this document by John Hale, who cites it in his _The Civilisation of Europe in the Renaissance_ (London, 1993), pp. 95\u20136.\n\n14. See Agostino Borromeo, 'Archbishop Carlo Borromeo and the Ecclesiastical Policy of Philip II in the State of Milan', in John M. Headley and John B. Tomaro (eds.), _San Carlo Borromeo: Catholic Reform and Ecclesiastical Politics in the Second Half of the Sixteenth Century_ (Washington, London and Toronto, 1988), pp. 85\u2013111.\n\n15. See Ludwig von Pastor, _The History of the Popes_ (London, 1951), vol. 15, p. 108.\n\n16. See Wietse de Boer, _The Conquest of the Soul: Confession, Discipline and Public Order in Counter-Reformation Milan_ (Leiden, Boston and Cologne, 2001), p.73; Paolo Prodi, 'San Carlo Borromeo e il Cardinale Gabriele Paleotti: due vescovi della Riforma Cattolica', _Critica Storica_ , 3 (1964), pp. 135\u201351.\n\n17. See Agostino Borromeo, 'Archbishop Carlo Borromeo'.\n\n18. See Diarmaid MacCulloch, _Reformation: Europe's House Divided_ , pp. 411\u201312.\n\n19. See E. Cecilia Voelker, 'Borromeo's Influence on Sacred Art and Architecture', in John M. Headley and John B. Tomaro (eds.), _San Carlo Borromeo_ , pp. 173\u201387.\n\n20. Ibid., p. 178.\n\n21. See Wietse de Boer, _The Conquest of the Soul_ , p. 43.\n\n22. Ibid.\n\n23. Ibid., p.122.\n\n24. See Diarmaid MacCulloch, _Reformation: Europe's House Divided_ , pp. 406\u20137.\n\n25. See Wietse de Boer, _The Conquest of the Soul_ , pp. 62\u20133: 'Confessors thus became quite literally law enforcement officers, who were to use their privileged access to the soul to assist in the application of church law. Having dispensed with such matters, they turned to the confession proper. But they continued to wear their uniforms as agents of discipline, constantly weighing the need to deny absolution to those considered unwilling to mend their sinful ways... if obstinacy was undeniable, the refusal of absolution was to be no empty threat. The Milanese confessor was to display the same combination of holy zeal and legal spirit that was characteristic of his bishop.'\n\n26. See Ludwig von Pastor, _The History of the Popes_ (London, 1930), vol. 19, p. 108.\n\n27. See David Freedberg, _The Power of Images: Studies in the History and Theory of Response_ (Chicago and London, 1989), p. 179. Freedberg's excellent account of traditions of visualization in Christian meditation gives passing mention to Borromeo (but not Caravaggio).\n\n28. Ibid., p. 171.\n\n29. Ibid., p. 168.\n\n30. See Michael Baxandall, _Painting and Experience in Fifteenth Century Italy: A Primer in the Social History of Pictorial Style_ (Oxford, 1972), p. 45: 'The painter was a professional visualizer of the holy stories. What we now easily forget is that each of his pious public was liable to be an amateur in the same line, practised in spiritual exercises that demanded a high level of visualization of, at least, the central episodes of the lives of Christ and Mary.'\n\n31. Cited in Roger Fry, 'Flemish Art at Burlington House. I', Burlington magazine 50, 287 (Feb. 1927), p. 68.\n\n32. See Michael Baxandall, _Painting and Experience in Fifteenth Century Italy_ , p. 46.\n\n33. See David Gilmore, _Aggression and Community: Paradoxes of Andalusian Culture_ (New Haven, 1987), p. 161.\n\n34. See Wietse de Boer, _The Conquest of the Soul_ , p. 113.\n\n35. Ibid., p. 114.\n\n36. See M. Cinotti, _I pittori bergamaschi_ (Bergamo, 1983), p. 235.\n\n37. Ibid. Giovan Pietro, who is first mentioned in a document of 1578, died in childhood.\n\n38. See Ann G. Carmichael, 'The Last Past Plague: The Uses of Memory in Renaissance Epidemics', _Journal of the History of Medicine and Allied Sciences_ , vol. 53, no. 2, (Apr.1998), p. 143.\n\n39. Ibid., p. 137.\n\n40. See Paolo Bisciola, _Relatione verissima del progresso della peste di Milano, qual principio nel mese d'agosto 1576_ (Ancona and Bologna, 1577). The translation here is that of Ann G. Carmichael in 'The Last Past Plague'.\n\n41. See Ann G. Carmichael, 'The Last Past Plague', pp. 137, 141.\n\n42. See Paolo Bisciola, _Relatione verissima del progresso della peste di Milano_.\n\n43. See Fra Paolo Bellintano, _I due Bellintani da Sal\u00f2 et il dialogo della pesta di Fra Paolo_ , F. Odorici (ed.), in Francesco Colombo (ed.), _Raccolta di cronisti e documenti storici lombardi inediti_ (Milan, 1857), vol. 2, p. 296.\n\n44. See ibid.; the story is singled out in Ann G. Carmichael, 'The Last Past Plague'.\n\n45. See Paolo Bisciola, _Relatione verissima del progresso della peste di Milano_.\n\n46. See M. Cinotti, , _I pittori bergamaschi_ , p. 203.\n\n47. For this document and the division of land, see ibid., pp. 235, 250, 206.\n\n48. See Giacomo Berra, 'Il Giovane Michelangelo Merisi da Caravaggio'.\n\n49. See _The Age of Caravaggio_ , Royal Academy exhibition catalogue (London, 1985), p. 73.\n\n50. The contract is quoted in M. Gregori, (ed.), _Gli affreschi della Certosa di Garegnano_ (Turin, 1973), p. 10; I have used the translation offered in Helen Langdon, _Caravaggio: A Life_ (London, 1998), p. 24.\n\n51. Ibid., p. 57.\n\n52. See Walter Friedlaender, _Caravaggio Studies_ (Princeton, 1955), p. 233.\n\n53. See Helen Langdon, _The Lives of Caravaggio_ , pp. 89, 27.\n\n54. The passage was first detected and deciphered by the art historian Maurizio Calvesi, author of a book aptly entitled _Le realt\u00e0 del Caravaggio_ ( _The Realities of Caravaggio_ ) (Turin, 1990). I am grateful to him for sharing his insights with me.\n\n##### PART TWO ROME, 1592\u20135\n\n1. ' _Bugiaronaccia poltrona puttana de tio te voglio tirare una pignatta de merda sul mostaccio... fatti fottere dal boia e ho in culo te con quanti n'hai_ ': my attention was called to this passage by Alexandra Lapierre, who very kindly allowed me to examine her personal collection of transcripts from criminal archives concerning the activities of artists in late sixteenth- and early seventeenth-century Rome. She quotes the document, in a slightly different translation, in her historical novel _Artemisia_ (London, 2000), p. 16, where it appears in the mouth of Agostino Tassi, a protagonist in her story \u2013 artistic licence, because it was actually uttered by another, now long-forgotten painter. The original document is dated 1602. She specifies its location in a note to her book; see pp. 369\u201370.\n\n2. See James Fenton, 'Bernini at Harvard \/ Chicago Baroque', in _Leonardo's Nephew_ (London, 1998), for a concise retelling of the story, which is rehearsed at fuller length in Charles Avery, _Bernini: Genius of the Baroque_ (London, 1997).\n\n3. See Helen Langdon, _The Lives of Caravaggio_ , p. 57.\n\n4. See _The Complete Works of Montaigne_ , D. Frame (trs.) (London, 1958), p. 1,163.\n\n5. Ibid., p. 1,164.\n\n6. Ibid., p. 1,172.\n\n7. Ibid., p. 1,143.\n\n8. Ibid., pp. 1,142, 1,150.\n\n9. Ibid., p. 1,142.\n\n10. Ibid., p. 1,150.\n\n11. See Helen Langdon, _Caravaggio: A Life_ , p. 34; and Walter Friedlaender, _Caravaggio Studies_ , p. 59.\n\n12. See _The Complete Works of Montaigne_ , p. 1,148.\n\n13. I am grateful to Opher Mansour for allowing me to read his unpublished doctoral dissertation for the Courtauld Institute in London, 'Offensive Images: Censure and Censorship in Rome under Clement VIII 1592\u20131605', from which this information about Clement's Visitation is drawn.\n\n14. This figure necessarily involves guesswork, but, given the sheer amount of artistic activity in Rome at the time, and given the size of many painters' and sculptors' workshops, it is likely to be on the low side.\n\n15. Quoted in John Hale, _The Civilisation of Europe in the Renaissance_ , p. 53.\n\n16. See Giovanni Botero, ' _The Reason of State' and 'The Greatness of Cities'_ , trans. by Robert Peterson 1606, P. J. and D. P. Waley (trs.) (London, 1956), p. 38.\n\n17. See _The Complete Works of Montaigne_ p. 1,168.\n\n18. My thanks again to Alexandra Lapierre for guiding me through the history of the artists' quarter and for sharing the fruits of her own research so generously in conversation.\n\n19. See Helen Langdon, _The Lives of Caravaggio_ , p. 41.\n\n20. Ibid., p. 27.\n\n21. The suggestion is made by Bellori in notes written while he was preparing his life of Caravaggio.\n\n22. See Helen Langdon, _The Lives of Caravaggio_ , p. 58.\n\n23. See Giulio Mancini, _Considerazioni sulla pittura_ , vol. 1 (Rome, 1956), pp. 226\u20137.\n\n24. Ibid., p. 226.\n\n25. See Helen Langdon, _The Lives of Caravaggio_ , p. 41. I am grateful to John T. Spike for the suggestion \u2013 very plausible, I think \u2013 that the picture is a nocturne.\n\n26. See for example the entries in _Caravaggio\u2013Rembrandt_ , Rijksmueum exhibition catalogue (Amsterdam, 2006), and _The Age of Caravaggio_ , Royal Academy exhibition catalogue _._\n\n27. See Pliny, _Natural History_ , Book 35, 64\u20136.\n\n28. I am indebted to Maurizio Calvesi for this suggestion, made to me in conversation in September 2001. See Maurizio Calvesi, _Le realt\u00e0 del Caravaggio_ , and for an English language version of his interpretation see his _Caravaggio_ (Florence, 1998), pp. 26\u20137.\n\n29. The rabbi's name was Akiva. See Carl W. Ernst, _Interpreting the Song of Songs: The Paradox of Spiritual and Sensual Love_ for a helpful guide through the theological intricacies of the centuries-long tradition of exegesis (www.unc.edu\/-cernst\/articles\/sosintro.htm, 28 Oct. 2008).\n\n30. St Teresa of Avila, 'Meditation on the Song of Songs', _The Collected Works of St Teresa of Avila_ , vol. 2, Kieran Kavanaugh, OCD, and Otilio Rodriguez, OCD (trs.) (Washington, DC, 1980).\n\n31. See Helen Langdon, _The Lives of Caravaggio_ , p. 28.\n\n32. Ibid., p. 49.\n\n33. Ibid., p. 41.\n\n34. It was painted on a light grey ground like a number of Caravaggio's earliest works, whereas the National Gallery picture was painted on a warmish ground, which accords with the painter's practice from around 1596.\n\n35. All quotations from Sandrart taken from the translation given in Walter Friedlaender, _Caravaggio Studies_ , pp. 263\u20136.\n\n36. See Giorgio Vasari, _Lives of the Painters, Sculptors and Architects_ , Gaston du C. de Vere (trs.), David Ekserdjian (ed.), vol. 1 (London, 1996), p. 860.\n\n37. Cited in Peter Burke, _The Historical Anthropology of Early Modern Italy_ (Cambridge, 1987), p. 98.\n\n38. See Helen Langdon, _The Lives of Caravaggio_ , p. 42.\n\n39. See Helen Langdon, _Caravaggio: A Life_ , p. 78. Langdon writes (and lectures) particularly well about Caravaggio's pictures of rogues. The idea that the cardsharps are rather like wasps in human clothing \u2013 see below \u2013 I owe to her.\n\n40. See Helen Langdon, _The Lives of Caravaggio_ , p. 42.\n\n41. Mancini, cited in Howard Hibbard, _Caravaggio_ (London, 1983), p. 350 (he was writing about the later version, a picture that he particularly loved, but his remarks are equally applicable to the painting owned by del Monte).\n\n42. Cited in Todd P. Olson, 'The Street has Its Masters: Caravaggio and the Socially Marginal', in _Caravaggio: Realism, Rebellion, Deception_ , Genevieve Warwick (ed.) (Delaware, 2006), p. 76.\n\n43. The quotations reprinted here have been extracted from the essay 'Perceiving a Counter-Culture', in Peter Burke, _The Historical Anthropology of Early Modern Italy_ , pp. 63\u201375. My summary of the different types of beggar is an abridged version of Burke's.\n\n44. Ibid., pp. 65\u201371. My discussion of poverty, religion and politics throughout this section of the book owes a great deal to Burke's lucid analysis.\n\n45. See Antonio Maria Cospi, _Il giudice criminalista_ , pp. 374\u20137.\n\n46. Ibid.\n\n47. Cited in John F. Moffitt, 'Caravaggio and the Gypsies', _Paragone_ , vol. 53 (2002), p. 141.\n\n48. Cited in D. J. Gordon, 'Gypsies as Emblems of Comedy and Poverty', _Journal of the Gypsy Lore Society_ , vol. 23 (1944), pp. 39\u201342.\n\n49. Ibid.\n\n50. See John F. Moffitt, 'Caravaggio and the Gypsies', p. 134.\n\n51. Giuseppe Pavoni, Di\u00e1rio, 1589, pp. 29\u201330, cited in Kenneth Richards and Laura Richards, _The Commedia dell'Arte: A Documentary History_ (Oxford, 1990), p. 74.\n\n52. Ibid., p. 60.\n\n53. Tommaso Garzoni, quoted in ibid., pp. 221\u20132.\n\n##### PART THREE: ROME, 1595\u20139\n\n1. See Helen Langdon, _The Lives of Caravaggio_ , p. 42.\n\n2. See Creighton Gilbert, _Caravaggio and His Two Cardinals_ (Pennsylvania, 1995), p. 116.\n\n3. See Helen Langdon, _Caravaggio: A Life_ , p. 79.\n\n4. See Zbgniew Wazbinski, _Il Cardinale Francesco Maria del Monte 1549\u20131626_ (Florence, 1994), p. 77, cited in Helen Langdon, _Caravaggio: A Life_ , p. 81.\n\n5. Decorated with languorous, graceful figures and a flying putto, it is now one of the treasures of the British Museum.\n\n6. He might have appreciated the French Romantic painter Delacroix's slashing cut through that particular Gordian knot: the observation that a painter's every brushstroke necessarily incorporated the act of drawing.\n\n7. See Helen Langdon, _Caravaggio: A Life_ , p. 96.\n\n8. For del Monte, Mancini, health care and alchemy, see Silvia De Renzi, ' \"A Fountain for the Thirsty\" and a Bank for the Pope: Charity, Conflicts and Medical Careers at the Hospital of Santo Spirito in Seventeenth-Century Rome', in _Health Care and Poor Relief in Counter-Reformation Europe_ , Ole Peter Grell, Andrew Cunningham and Jon Arrizabalaga (eds.) (London, 1999), pp. 102\u201331.\n\n9. De Renzi's scholarly study of the hospital (see above) concludes, ambiguously, that 'Reasons to apply for a job at the Santo Spirito could be various: a somewhat difficult-to-detect religious and moral commitment, and the more evident search for a prestigious position, were interwoven.'\n\n10. See Creighton Gilbert, _Caravaggio and His Two Cardinals_ , p. 205.\n\n11. Ibid. Gilbert has done all scholars of Caravaggio and del Monte a service by so thoroughly exposing Amayden's untrustworthiness as a biographer.\n\n12. The letter in question was discovered in the Florentine State Archives by the scholar Franca Trinchieri Camiz, who published it for the first time in 1991. See Franca Trinchieri Camiz, 'Music and Painting in Cardinal del Monte's Household', _Metropolitan Museum Journal_ , no. 26 (Hartford, 1991).\n\n13. See Franca Trinchieri Camiz, 'La \"musica\" nei quadri di Caravaggio', _Caravaggio. Nuove riflessioni, Quaderni di Palazzo Venezia_ , vol. 6 (Rome, 1991).\n\n14. See Keith Christiansen, _A Caravaggio Rediscovered: The Lute Player_ (New York, 1990).\n\n15. See Franca Trinchieri Camiz, 'La \"musica\" nei quadri di Caravaggio'.\n\n16. See Claude V. Palisca, 'Musical Asides in the Diplomatic Correspondence of Emilio de' Cavalieri', _Musical Quarterly_ , vol. 49, no. 3 (July 1963), p. 346.\n\n17. See Keith Christiansen, _A Caravaggio Rediscovered: The Lute Player_ , p. 26.\n\n18. See the entry on Emilio de' Cavalieri in _The Grove Dictionary of Music_ (Oxford, 2003).\n\n19. See Zbgniew Wazbinski, _Il Cardinale Francesco Maria del Monte_ , pp. 137\u20138.\n\n20. See Creighton Gilbert, _Caravaggio and His Two Cardinals_ , p. 116.\n\n21. See Keith Christiansen, _A Caravaggio Rediscovered: The Lute Player_ , p. 46.\n\n22. See Helen Langdon, _The Lives of Caravaggio_ , p. 63.\n\n23. See Keith Christiansen, _A Caravaggio Rediscovered: The Lute Player_ , p. 32.\n\n24. See Franca Trinchieri Camiz, 'Music and Painting in Cardinal del Monte's Household', p. 220.\n\n25. As Franca Trinchieri Camiz remarks, in 'Music and Painting in Cardinal del Monte's Household': 'the voice was well suited for solo performance because of its greater capacity for proper phrasing, which allowed the expression of the strong emotions in fashion during this period'(p. 221).\n\n26. Ibid., p. 218; for a counter-example, see the very different, open-mouthed singers, accompanied by lutes and polyphonically hymning the infant Christ, in Piero della Francesca's _Nativity_ in the National Gallery, London.\n\n27. See Colin Slim, 'Musical Inscriptions in Paintings by Caravaggio and His Followers', in _Music and Context_ , A. Shapiro (ed.) (Cambridge, Mass., 1995).\n\n28. See Keith Christiansen, _A Caravaggio Rediscovered: The Lute Player_ , p. 90. The translation given is that of Louis E. Lord.\n\n29. The late seventeenth-century writer Pietro Paolo Bosca actually referred to it as a 'tantalus'. See P. P. Bosca, _De origine et statu Bibliothecae Ambrosianae_ (Milan, 1672), p. 126. Cited by John T. Spike, _Caravaggio_ (New York, 2001), in his CD-ROM catalogue entry on the _Basket of Fruit_.\n\n30. Cited by John T. Spike, _Caravaggio_ , in his CD-ROM catalogue entry on the _Basket of Fruit_.\n\n31. My thanks to Maurizio Calvesi for this observation.\n\n32. It is a fair assumption that the two pictures have the same history. So to trace one is, in effect, to trace both. _The Rest on the Flight to Egypt_ is linked to Olimpia Aldobrandini by an inventory of her collection compiled in 1611, which mentions 'A large painting of the Madonna's Flight into Egypt in a frame', albeit without naming the artist. The hypothesis that this is a reference to Caravaggio's painting is strengthened by circumstantial evidence. An inventory of 1622, listing pictures in the Villa Aldobrandini in Frascati, mentions 'A large painting on canvas of a Madonna embracing the child and a Saint Joseph... copy of Caravaggio'. The presence of this copy in one of the other residences of Olimpia Aldobrandini's family suggests that the original was indeed in her possession. The inventory reference is cited in John T. Spike's CD-ROM catalogue entry on _The Rest on the Flight to Egypt_.\n\n33. See Helen Langdon, _The Lives of Caravaggio_ , p. 28.\n\n34. See Bernard Aikema, 'Titian's _Mary Magdalen_ in the Palazzo Pitti: An Ambiguous Painting and Its Critics', _Journal of the Warburg and Courtauld Institutes_ , vol. 57 (1994), p. 58.\n\n35. See Colin Slim, 'Musical Inscriptions in Paintings by Caravaggio and His Followers'.\n\n36. See John T. Spike in his CD-ROM catalogue entry on the _St Francis of Assisi in Ecstasy_.\n\n37. See Helen Langdon, _The Lives of Caravaggio_ , p. 63.\n\n38. St Bonaventure's _Legenda maior_ was one of the most readily available literary sources for painters working in the post-Tridentine period. It was the official biography of the saint, written in 1262. Bonaventure derived much of his information from the very first life of Francis, written by Thomas of Celano in _c_. 1230, just four years after the saint's death. See Pamela Askew, 'The Angelic Consolation of St Francis of Assisi in Post-Tridentine Italian Painting', _Journal of the Warburg and Courtauld Institutes_ , vol. 32 (1969), pp. 280\u2013386.\n\n39. Cited in Pamela M. Jones, 'The Place of Poverty in Seicento Rome: Bare Feet, Humility and the Pilgrimage of Life in Caravaggio's _Madonna of Loreto_ ( _c_. 1605\u20136) in the Church of S. Agostino', in _Altarpieces and Their Viewers in the Churches of Rome from Caravaggio to Guido Reni_ (Aldershot, 2008), p. 107.\n\n40. _The Life of Teresa of Jesus: The Autobiography of St Teresa of Avila_ , E. Allison Peers (trs.) (New York, 2004), Chapter 29.\n\n41. Quoted in Radleigh Addington, _The Idea of the Oratory_ (London, 1966), p. 3.\n\n42. This is a confident assertion based on comparisons with known portraits of Caravaggio, but not a documented fact.\n\n43. For the correspondence between Paravicino and Gualdo, see G. Cozzi, 'Intorno al Cardinale Ottavio Paravicino, a Monsignor Paolo Gualdo e a Michelangelo da Caravaggio', _Rivista storica italiana_ , vol. 73 (1961), pp. 36\u201368. I am indebted to Opher Mansour, who allowed me to see his translations of, and commentaries on, these letters, in his unpublished doctoral thesis submitted to the Courtauld Institute: 'Art, Offensive Images: Censure and Censorship in Rome under Clement VIII 1592\u20131605' (London, 2003).\n\n44. It is often said that there is a hidden self-portrait reflected in the carafe \u2013 see, for example, Langdon, _Caravaggio: A Life_ , p. 151. I have inspected the painting under high magnification and there is no such self-portrait in it.\n\n45. See Helen Langdon, _The Lives of Caravaggio_ , p. 43.\n\n46. See n. 41 above.\n\n47. See Giorgio Vasari, _Lives of the Painters, Sculptors and Architects_ , vol. 1, p. 629.\n\n48. See Helen Langdon, _The Lives of Caravaggio_ , p. 93.\n\n49. He did so, perhaps, because there was an established connection between that particular artistic style and alchemy. See my comments on Francesco de' Medici's _studiolo_ , on p. 159 above.\n\n50. Again, see n. 41 above. The resemblance to Ottavio Leoni's portrait of Caravaggio is, in my opinion, incontrovertible in the _Jupiter, Neptune and Pluto_. The identification with Francis is a little less certain but I am still confident that the saint is a self-portrait.\n\n51. See Walter Friedlaender, _Caravaggio Studies_ , p. 260.\n\n52. Sandro Corradini discovered the case. With Maurizio Marini, he subsequently published the transcripts in full, together with a useful interpretative essay. See Sandro Corradini and Maurizio Marini, 'The Earliest Account of Caravaggio in Rome', _Burlington Magazine_ , vol. 40, no. 1,138 (Jan. 1998), pp. 25\u20138.\n\n53. The building still stands in Rome today. It is still a barber's shop!\n\n54. See Helen Langdon, _The Lives of Caravaggio_ , p. 92.\n\n55. See Fiora Bellini, 'Tre documenti inediti per Michelangelo da Caravaggio', _Prospettiva_ , no. 65 (Jan. 1992), pp. 70\u201371.\n\n56. See Walter Friedlaender, _Caravaggio Studies_ , pp. 263\u20135.\n\n57. See Francesco Susinno, _Le vite de' pittori messinesi e di altri che fiorirono in Messina_ , V. Martinelli (ed.) (Florence, 1960), p. 117.\n\n58. See Rudolf and Margot Wittkower, _Born Under Saturn_ (New York, 1963), p. 198. Orazio Gentileschi eventually prospered in France and Genoa in the 1620s. He was called to London in 1626 to become a painter at the court of King Charles I, who rewarded him with a generous stipend.\n\n59. See G. P. Caffarelli, 'Famiglie romane', Biblioteca Angelica MS 1638, cc. 88r\u2013v; cited (reliably) in Riccardo Bassani and Fiora Bellini, _Caravaggio assassino_ (Rome, 1994), p. 13, n. 20.\n\n60. ASR, Tribunale criminale del Senatore (TCS), reg. 1438, testimony of Onorio Longhi, 4 May 1595, cc. 20v\u201322v.\n\n61. Ibid.\n\n62. Ibid., reg. 444, testimony of Margherita Fannella, 4 May 1595.\n\n63. Cited in Sandro Corradini, _Materiali per un processo_ (Rome, 1993), document 15, 25\u20137 Oct., deposition by Stefano Longhi and others.\n\n64. Ibid.\n\n65. See Rudolf and Margot Wittkower, _Born Under Saturn_ , p. 196.\n\n66. See L. Pascoli, _Vite de' pittori, scultori ed architetti moderni_ (Rome, 1730), vol. 2, pp. 512\u201313.\n\n67. See ASR, TCS, reg. 1438, testimony of Onorio Longhi, 4 May 1595, cc. 20v\u201322v.\n\n68. See Christopher Breward, 'Fashioning the Modern Self: Clothing, Cavaliers and Identity in Van Dyck's London', in _Van Dyck and Britain_ , Karen Hearn (ed.) (London, 2009), pp. 34\u20135.\n\n69. See Tommaso Garzoni, _La piazza universale di tutte le professioni del mondo_ (Rome, 1996), pp. 1,263\u201383, for the following quotations.\n\n70. ASR, Tribunale del governatore (TCG), reg. 483, witness statement of Anna Bianchini, 22 Apr. 1594, c. 144v; cited (reliably) in Riccardo Bassani and Fiora Bellini, _Caravaggio assassino_ , p. 74, n. 5.\n\n71. ASR, Archivio Sforza Cesarini, s. xii, b. 1b, filza 1, interrogationes et testes 1596\u20137, cc. n.n; cited (reliably) in Riccardo Bassani and Fiora Bellini, _Caravaggio assassino_ , p. 53, n. 5.\n\n72. Inventories show that she owned a painting of the Magdalen repenting by Caravaggio. The banker Ottavio Costa also owned a version of the same subject. Scholarly opinion is divided about who originally owned the Detroit painting but the balance of evidence currently available favours Olimpia Aldobrandini.\n\n73. See Gregory Martin, _Roma sancta_ , George Bruner Parks (ed.) (Rome, 1969), p. 143.\n\n74. For Ranuccio Tomassoni, see Riccardo Bassani and Fiora Bellini, _Caravaggio assassino_ , pp. 55\u201373.\n\n75. For all the following testimonies see Sandro Corradini, _Materiali per un processo_ , document 17.\n\n76. Such figures appear with great frequency in northern European genre painting, especially Dutch art. Her tough, harsh face was probably modelled on a male Roman portrait bust.\n\n77. See Carlo Cesare Malvasia, _Le vite de' pittori bolognese_ , edition of 1678 (Bologna, 1841), vol. 1, p. 344.\n\n78. His methods might be described as a kind of empirical Tintorettism, in the sense that they are the techniques a painter might evolve if he wanted to emulate Tintoretto but had never been trained in Tintoretto's actual methods \u2013 which were rather different, and certainly involved drawing.\n\n79. See Walter Friedlaender, _Caravaggio Studies_ , p. 264.\n\n80. See Helen Langdon, _The Lives of Caravaggio_ , p. 64. The idea that Caravaggio also used some kind of lens or camera obscura is a red herring. Caravaggio had plenty of enemies who would have no doubt taken pleasure in exposing him as a cheat, but no such device is mentioned by any of the early writers. Nor does anything like it appear in the only known inventory of his possessions.\n\n81. Ibid., p. 33.\n\n##### PART FOUR: ROME, 1599\u20131606\n\n1. The document is reprinted and translated in Walter Friedlaender, _Caravaggio Studies_ , p. 297.\n\n2. See Herwarth R\u00f6ttgen, _Il Caravaggio: ricerche e interpretazioni_ (Rome, 1974), pp. 20\u201321; the translation is from John T. Spike, _Caravaggio_. The contract in question is the one signed by Giuseppe Cesari on 27 May 1591. As Cesari's successor, it seems highly probable that Caravaggio would have been made aware of Contarelli's wishes.\n\n3. Quoted in Walter Friedlaender, _Caravaggio Studies_ , p. 265.\n\n4. See Helen Langdon, _The Lives of Caravaggio_ , p. 69.\n\n5. See for example Catherine Puglisi, _Caravaggio_ (London, 1998), pp. 157\u201360.\n\n6. See G. Urbani, 'Il restauro delle tele del Caravaggio in S. Luigi dei Francesi', _Bollettino dell'Istituto Centrale del Restauro_ , vol. 17 (1966).\n\n7. See Franca Trinchieri Camiz, 'Death and Rebirth in Caravaggio's _Martyrdom of St Matthew_ ', _Artibus et Historiae_ , vol. 11, no. 22 (1990), pp. 89\u2013105.\n\n8. See E. Cecilia Voelker, Charles Borromeo's 'Instructiones fabricae et supellectilis ecclesiasticae', translation with commentary, dissertation, Syracuse University, 1977, pp. 250\u201351.\n\n9. See Anti-Nicene Christian Library, _Translations of the Writings of the Fathers Down to AD 325_. _Volume 9: The Writings of Tertullian_ , I, 25. Cited in 'Death and Rebirth in Caravaggio's _Martyrdom of St Matthew_ '.\n\n10. Titian's painting is lost, destroyed by fire, but its design can still be studied from prints.\n\n11. See Walter Friedlaender, _Caravaggio Studies_ , p. 300.\n\n12. See, in particular, his thunderously inept contributions to the fresco cycle begun by Giorgio Vasari in the dome of Florence cathedral.\n\n13. See Helen Langdon, _The Lives of Caravaggio_ , p. 45. The translation here gives 'the style of Giorgione', which I have changed to 'idea' because the Italian word Baglione used was _pensiero_.\n\n14. See Giorgio Vasari, _Lives of the Painters, Sculptors and Architects_ , vol. 1, p. 641.\n\n15. San Luigi dei Francesi was open to such innovations from outside. When Caravaggio accepted his commission, it was already one of the few churches in Rome to have a great Venetian canvas \u2013 by Jacopo Bassano \u2013 above its high altar.\n\n16. See Helen Langdon, _Caravaggio: A Life_ , p. 75.\n\n17. See Sandro Corradini, _Materiali per un processo_ , document 21, 7 Feb. 1601.\n\n18. See Walter Friedlaender, _Caravaggio Studies_ , pp. 269\u201370.\n\n19. All the material from the investigation of Onorio Longhi in Oct. 1600, discussed below, is from Sandro Corradini, _Materiali per un processo_ , document 15.\n\n20. See above, p. 74.\n\n21. Ibid., document 16, 20 Jan., deposition by Stefano Longhi and others.\n\n22. Ibid., document 18.\n\n23. Cited in John T. Spike, _Caravaggio_ , in his CD-ROM catalogue entries for _The Conversion of St Paul_ and _The Crucifixion of St Peter_ ; and Walter Friedlaender, _Caravaggio Studies_ , pp. 302\u20133.\n\n24. See Denis Mahon, 'Egregius in Urbe Pictor: Caravaggio Revisited', _Burlington Magazine_ , vol. 93, no. 580 (July 1951), p. 226.\n\n25. Caravaggio was familiar with the place too. He had convalesced in the Hospital of Santa Maria Consolazione in 1592\u20133, after being kicked by a horse.\n\n26. Sixtus V; see Helen Langdon, _Caravaggio: A Life_ , p. 181.\n\n27. Quoted in John T. Spike, _Caravaggio_ , p. 106.\n\n28. See Helen Langdon, _The Lives of Caravaggio_ , p. 88, where Bellori says that 'Caravaggio did not use cinnabar reds or azure blues in his figures; and if he occasionally did use them, he toned them down, saying they were poisonous colours.'\n\n29. See Fiora Bellini, 'Tre documenti per Michelangelo da Caravaggio', pp. 70\u201371.\n\n30. See Helen Langdon, _The Lives of Caravaggio_ , p. 91.\n\n31. See Charles Scribner III, 'In Alia Effigie: Caravaggio's London _Supper at Emmaus_ ', _Art Bulletin_ , vol. 59, no. 3 (Sept. 1977), pp. 375\u201382, for an illuminating account of the youthful Christ and his theological significance.\n\n32. The author's name was Gaspare Celio, whose book was published in Naples in 1638. He described the picture as 'a Pastor Friso, in oil, by Michelangelo da Caravaggio'. See the entry in John T. Spike, Caravaggio, CD-ROM catalogue entry no. 29.\n\n33. See Conrad Rudolph and Steven F. Ostrow, 'Isaac Laughing: Caravaggio, Non-Traditional Imagery and Traditional Identification', _Art History_ , vol. 24, no. 5 (Nov. 2001), pp. 646\u201381. The article advances the theory that the painter meant to depict Isaac instead of St John. It also contains a very good summary of the hard documentary evidence that disproves its own argument.\n\n34. 'Un quadro di San Gio: Battista col suo Agnello di mano del Caravaggio', cited in ibid., p. 649.\n\n35. He described it as 'di San Giovanni Battista del Caravaggio'; cited in ibid.\n\n36. See Helen Langdon, _The Lives of Caravaggio_ , p. 72.\n\n37. See Sergio Benedetti, 'Caravaggio's Taking of Christ: A Masterpiece Rediscovered', _Burlington Magazine_ , vol. 135, no. 1,088 (Nov. 1993), p. 740.\n\n38. See Niccol\u00f2 Lorini del Monte, _Elogii delle piu principali S. Donne del sagro calendario, e martirologio romano_ (Florence, 1617), p. 316. My attention was called to this passage by Pamela M. Jones's enlightening study of the pauperist context of Caravaggio's Rome in her book _Altarpieces and Their Viewers in the Churches of Rome from Caravaggio to Guido Reni_ ; see pp. 75ff. in particular.\n\n39. Cointrel's nephew and heir, Fran\u00e7ois, took possession of Cobaert's dull and stolid sculpture, eventually having it completed by another artist and placed in a chapel in SS Trinit\u00e0 dei Pellegrini, where he himself would eventually be buried.\n\n40. See Irving Lavin, 'Divine Inspiration in Caravaggio's Two St Matthews', _Art Bulletin_ , vol. 56, no. 1 (Mar. 1974), pp. 59\u201381.\n\n41. See Helen Langdon, _The Lives of Caravaggio_ , p. 45 for Baglione's remark, p. 66 for Bellori's.\n\n42. Bellori's bald statement that the _Doubting Thomas_ was painted for 'the Marchese Vincenzo Giustiniani' is supported by most of the available evidence. Giustiniani certainly owned the picture by 1606, because in the summer of that year he wrote a letter comparing his own, original _Doubting Thomas_ by Caravaggio to a copy in Genoa. Baglione asserted that the _Doubting Thomas_ was painted for Ciriaco Mattei, but this is probably a rare slip of the pen on his part. He may have confused the picture with _The Betrayal of Christ_ , which certainly was painted for Ciriaco Mattei and which, oddly, Baglione does not mention at all. In summary, there is a remote possibility that the _Doubting Thomas_ was painted for Ciriaco Mattei, then later acquired by Vincenzo Giustiniani. But the balance of probability favours a direct commission from Giustiniani himself. For a good analysis of the arguments and a precis of the relevant documents, see John T. Spike, _Caravaggio_ , CD-ROM catalogue entry for _Doubting Thomas_.\n\n43. See Walter Friedlaender, _Caravaggio Studies_ , p. 264.\n\n44. Inventory of 9 Feb. 1638; see John T. Spike, _Caravaggio_ , CD-ROM catalogue entry for _Omnia vincit amor._\n\n45. The resemblance to Michelangelo's _Victory_ was first noted by Walter Friedlaender, _Caravaggio Studies_ , p. 93.\n\n46. See Joachim von Sandrart, _L'Accademia Todesca della archittetura, scultura e pittura_... (Nuremberg, 1675). Quoted by Robert Enggass, 'L'Amore Giustiniani del Caravaggio', _Palatino_ , vol. 11 (1967), pp. 13\u201319. This translation is from John T. Spike, _Caravaggio_.\n\n47. The idea is advanced by Robert Enggass in the article cited in the previous note above. If this hypothesis is to be believed, Cupid does not trample the arts and sciences underfoot, but inspires them to flourish in the Giustiniani household. Such an interpretation is, however, flatly contradicted by the Giustiniani inventory of 1638, describing 'Cupid disparaging the world'. It is also at odds with the purely visual evidence of the painting. In particular, the discarded shell of an empty suit of armour cannot possibly have been intended by the painter as a compliment to the military prowess of his patron. Nor can Caravaggio's impishly provocative, full-frontally nude Cupid be plausibly transmuted into a Neoplatonic emblem of the Earthly Love that sparks man to Divine Creativity.\n\n48. For an earlier conversation inspired by a painting of Cupid between the Venetian collector Gabriel Vendramin and the connoisseur Anton Francesco Doni, see Catherine Whistler, 'Titian's _Triumph of Love_ ', _Burlington Magazine_ , vol. 151, no. 1,277 (Aug. 2009), n. 19, in which the author cites Doni's _I marmi_ (Venice, 1552), vol. 3, fols. 40\u201341: ' _e fra l'altro mi mostr\u00f2 un leone con un Cupido sopra. E qui discorremo molto della bella invenzione, e lodassi ultimamente in questo, che l'amore doma ogni gran ferocit\u00e0 e terribilit\u00e0 \u00e0 persone_.'\n\n49. The Courtauld Galleries in London contain a particularly good example of two such chests in their original condition. As well as being embellished with complex narrative paintings about love, drawn from classical mythology, they are decorated with split pomegranates spilling their seeds, a kind of symbolic prayer for fertile married union.\n\n50. See Charles Dempsey, ' \"Et nos cedamus amori\": Observations on the Farnese Gallery', _Art Bulletin_ , vol. 50, no. 4 (Dec. 1968), pp. 363\u201374.\n\n51. See Helen Langdon, _The Lives of Caravaggio_ , pp. 45\u20136.\n\n52. See Karel van Mander, _Het Schilderboek_ (Haarlem, 1604), cited in Beverly Louise Brown, 'The Black Wings of Envy: Competition, Rivalry and _Paragone_ ', in _The Genius of Rome_ , Royal Academy exhibition catalogue, p. 251.\n\n53. See Gianni Papi's essay 'Cecco del Caravaggio', in _Come dipingeva il Caravaggio: atti della giornata di studio_ , Mina Gregori (ed.) (Milan, 1996).\n\n54. This transcription was made from the original MSS of Symonds's travel journal by John Gash, who published it in the _Burlington Magazine_ , vol. 140, no. 1,138 (Jan. 1998), pp. 41\u20132.\n\n55. See Giorgio Vasari, _Lives of the Painters, Sculptors and Architects_ , vol. 2, p. 418.\n\n56. See Maryvelma Smith O'Neil's entertainingly revisionist study, _Giovanni Baglione: Artistic Reputation in Baroque Rome_ (Cambridge, 2002), p. 17. I am indebted to her lucid account of the libels and their consequences, although not convinced by her suggestion that Baglione was an injured innocent in the affair.\n\n57. These transcriptions of the poems are taken from Anthony Colantuono, 'Caravaggio's Literary Culture', in _Caravaggio, Realism Rebellion, Reception_ , Genevieve Warwick (ed.) (Newark, 2006), p. 58.\n\n58. Percy Bysshe Shelley's verse play _The Cenci_ was inspired by these events.\n\n59. See Maryvelma Smith O'Neil, _Giovanni Baglione: Artistic Reputation in Baroque Rome_ , p. 13.\n\n60. The libel trial documents were first published in full in G. A. Dell'Acqua and M. Cinotti, _Il Caravaggio e il sue grandi opere da S. Luigi dei Francesi_ (Milan, 1971), pp. 153\u20137. The translation offered here is by Don Var Green and can be found in full in Maryvelma Smith O'Neil, _Giovanni Baglione: Artistic Reputation in Baroque Rome_ , pp. 337\u201362. I have made a couple of slight alterations, to match my own translation of the two poems at the centre of the case, and in one or two instances have preserved the original Italian usages.\n\n61. The document is printed in full in Maryvelma Smith O'Neil, _Giovanni Baglione: Artistic Reputation in Baroque Rome_ , pp. 357\u20138.\n\n62. See Sandro Corradini, _Materiali per un processo_ , document 26.\n\n63. See Maryvelma Smith O'Neil, _Giovanni Baglione: Artistic Reputation in Baroque Rome_ , pp. 358\u201362.\n\n64. Salini added the detail about the punch in the chest in a slightly later piece of testimony; I have inserted it here for the sake of clarity.\n\n65. The document is reprinted in full in Maurizio Marini, _Michelangelo Merisi da Caravaggio 'pictor praestantissimus'_ (second edition, Rome 1979), p. 472.\n\n66. See Tullio Lazzari, _Ascoli in prospettiva_ (Ascoli, 1722), p. 40.\n\n67. The document is dated 6 June 1605. It is quoted, and photographically reproduced, in Maurizio Marini, _Michelangelo Merisi da Caravaggio 'pictor praestantissimus'_ , p. 53.\n\n68. See Sandro Corradini, _Materiali per un processo_ , document 57. The translation is from Catherine Puglisi, _Caravaggio_ , p. 420.\n\n69. Nowadays many people have books they do not read, but books were so expensive in Caravaggio's time that ownership of a volume can be taken as an indication of familiarity with its contents.\n\n70. See Helen Langdon, _Caravaggio: A Life_ , p. 279.\n\n71. See Walter Friedlaender, _Caravaggio Studies_ , p. 280.\n\n72. Ibid., p. 260.\n\n73. Ibid., p. 249.\n\n74. The translation is from ibid., p. 281; the fullest transcription of these documents is in G. A. Dell'Acqua and M. Cinotti, _Il Caravaggio e il sue grandi opere da S. Luigi dei Francesi_ , p. 158.\n\n75. G. A. Dell'Acqua and M. Cinotti, _Il Caravaggio e il sue grandi opere da S. Luigi dei Francesi_ , p. 158.\n\n76. See Walter Friedlaender, _Caravaggio Studies_ , p. 281.\n\n77. These officials were drawn from the lay population and elected to their posts by the noble families of the city. Hence they reflected the factionalism and competing dynastic ambitions that existed at the highest level of Roman society. During the so-called Vacant See, the interregnum between one pope's death and another's election \u2013 but only at that time \u2013 the _caporioni_ were allowed to act as judges in the districts under their control. Trouble often ensued during these periods. See Laurie Nussdorfer, 'The Politics of Space in Early Modern Rome', _Memoirs of the American Academy in Rome_ , vol. 42 (1997), pp. 161\u201386.\n\n78. All this testimony is in Sandro Corradini, _Materiali per un processo_ , document 41.\n\n79. Ibid., document 47.\n\n80. See Walter Friedlaender, _Caravaggio Studies_ , p. 282.\n\n81. The term 'house-scorning' was coined by Elizabeth S. Cohen. The discussion that follows is heavily indebted to her pioneering work in the field of seventeenth-century social history, especially the essay 'Honour and Gender in the Streets of Early Modern Rome', _Journal of Interdisciplinary History_ , vol. 22, no. 4 (Spring 1992), pp. 597\u2013625.\n\n82. Louis Richeome, _The Pilgrime of Loreto_ , facsimile of the 1629 edition, _English Recusant Literature 1558\u20131640_ , vol. 285, D. M. Rogers (ed.) (London, 1976), p. 33.\n\n83. Thousands of pilgrims visited Loreto every year and their experience was carefully orchestrated. The pilgrimage diaries of the Santissima Trinit\u00e0 dei Pellegrini, founded by Filippo Neri and supported by the patrons who paid for Caravaggio's _Madonna of Loreto_ , the Cavalletti family, contain much information about the structure of a visit to Loreto. They strongly suggest that the painter wanted his picture to evoke an actual pilgrimage.\n\n84. The placement of Caravaggio's works within the geography of Rome has received relatively scant consideration. Pamela Jones's essay, 'The Place of Poverty in Seicento Rome', included in _Altarpieces and Their Viewers_ , contains a penetrating analysis of the significance of the geographical locations of some of Caravaggio's works.\n\n85. See Helen Langdon, _The Lives of Caravaggio_ , p. 90.\n\n86. Ibid., p. 46.\n\n87. See Walter Friedlaender, _Caravaggio Studies_ , p. 284.\n\n88. See G. A. Dell'Acqua and M. Cinotti, _Il Caravaggio e il sue grandi opere da S. Luigi dei Francesi_ , p. 158.\n\n89. It was left out of later editions.\n\n90. See Jacob Hess, 'Nuovo Contributo alla vita del Caravaggio', _Bolletino d'Arte_ , anno 26, ser. 3 (July 1932), pp. 42\u20134.\n\n91. Rome's criminal archives include a report written by the constable who arrested her. See Sandro Corradini, _Materiali per un processo_ , document 38.\n\n92. If this is so (which is certainly possible), he would have been using the phrase in the same straightforward sense as the one-eyed Bolognese corporal, possibly called Paulo Aldato, who appears to say something similar in a later criminal action involving Caravaggio. Aldato (if that was his name) is reported as saying that he wanted to visit ' _una sua puttana_ ' \u2013 one of his prostitutes \u2013 on a street nearby. There is no implication that Aldato was a pimp. See Sandro Corradini, _Materiali per un processo_ , document 101.\n\n93. He would later claim that he had tried to challenge Pasqualone to a fair and open fight, but probably only to put his own actions in a better light.\n\n94. See Sandro Corradini, _Materiali per un processo_ , documents 48\u201352, 54.\n\n95. Giuliana Marcolini, 'Cesare d'Este, Caravaggio, e Annibale Carracci: una duca, due pittori e una _committenza_ \"a mal termine\" ', in _Sovrane passioni: studi sul collezionismo estense_ , Jadranka Bentini (ed.) (Milan, 1998), pp. 23\u20134. Ruggieri's letter reporting Caravaggio's riposte was dated 2 Mar. 1605.\n\n96. Had it not been for the discovery of Masetti's correspondence, the details of Caravaggio's trip to Genoa would have remained unknown. See Sandro Corradini, _Materiali per un processo_ , document 53.\n\n97. Ibid., document 55.\n\n98. Ibid., document 56.\n\n99. See Walter Friedlaender, _Caravaggio Studies_ , p. 285.\n\n100. Ibid.\n\n101. See Sandro Corradini, _Materiali per un processo_ , document 58.\n\n102. Ibid., document 59. This is a slightly free translation; Masetti uses the phrase ' _un' altra questione_ ', meaning 'another question'.\n\n103. Ibid., document 67.\n\n104. Ibid., document 68.\n\n105. Ibid., document 71.\n\n106. Carracci did eventually deliver his own picture for the duke, thought to be identical to the painter's _The Birth of the Virgin_ now in the Louvre.\n\n107. See Helen Langdon, _The Lives of Caravaggio_ , pp. 73\u20134.\n\n108. See Luigi Spezzaferro, 'La pala dei Palafrenieri', _Colloquio_ (1974), which reprints the documents from the archive of the confraternity associated with the commission.\n\n109. Ibid.; the translation is given in John T. Spike, _Caravaggio_ , where the painting appears as entry no. 48. The same is true for the two documents that follow. For a reproduction of this document in Caravaggio's handwriting, see illustration no. 65.\n\n110. See Helen Langdon, _The Lives of Caravaggio_ , p. 90.\n\n111. See Gabriele Paleotti, 'Discorso intorno alle imagini sacre e profane', in _Trattati d'arte del Cinquecento fra Manierismo a Controriforma_ , vol. 2, P. Barocchi and P. Barocchi (eds.) (Bari, 1961), p. 370.\n\n112. See Roberto Longhi, _Opere complete_ (Florence, 1968), vol. 4, p. 58.\n\n113. The date of _The Death of the Virgin_ is disputed, but there are compelling reasons to place its completion close to the very end of Caravaggio's Roman period, i.e. around May 1606. Before the discovery of the contract for the painting, of 14 June 1601, the work was dated 1606 by most art historians on purely stylistic grounds. There seems little reason to reverse that view simply because of the discovery of the contract. It was common for paintings to be delivered late, sometimes years late (witness the travails of poor Fabio Masetti). The picture is certainly much closer in its _facture_ , palette and mood to Caravaggio's later, post-Roman works than it is to such paintings of 1601\u20132 as _The Supper at Emmaus_. In my opinion, it was finished directly after the _Madonna of the Palafrenieri_ , since it is painted in the looser, freer style of that picture's right half \u2013 the half containing St Anne \u2013 which directly prefigures the style of the artist's last years. As a compromise solution some experts have chosen to date the painting to 1604, but this seems perverse, bearing in mind both the picture's appearance and the existing documentary evidence. The first detailed reference to the picture occurs in a letter by Giulio Mancini, dated 14 Oct. 1606, in a context strongly suggestive of the picture having been finished just a matter of months earlier. Another reference to it from around the same time occurs in the correspondence of an agent working for the Duke of Mantua, who noted that the painters of Rome were complaining that they had not yet been able to see the painting. If it really had been finished as early as 1604, it would seem strange indeed that Caravaggio's friends and rivals had still not seen it all of two years later. In addition, in his biography of the painter Mancini explicitly connects its rejection with 'the trouble' that ruined Caravaggio's life, i.e., the killing of Ranuccio Tomassoni. It would therefore seem logical to assume that it was the very last picture the artist painted before his flight from Rome.\n\n114. Saints should never be given the recognizable features of 'persons of ill repute', Paleotti had written. Gabriele Paleotti, 'Discorso intorno alle imagini sacre e profane', p. 360. I am obliged to Opher Mansour for pointing out both these references to me.\n\n115. See Giulio Mancini, _Considerazioni sulla pittura_ , vol. 1, pp. 120, 132; see also Walter Friedlaender, _Caravaggio Studies_ , p. 195.\n\n116. See Michele Maccherini, 'Caravaggio nel carteggio familiare di Giulio Mancini' in _Prospettiva_ , vol. 86 (1997), pp. 71\u201392.\n\n117. See Sandro Corradini, _Materiali per un processo_ , document 78. It is not clear what type of document this is; perhaps a journal.\n\n118. See Helen Langdon, _The Lives of Caravaggio_ , pp. 29\u201331.\n\n119. Ibid., p. 52.\n\n120. Ibid., p. 76.\n\n121. See Peter Burke, 'Rome as Center of Information and Communication for the Catholic World 1550\u20131650', in _From Rome to Eternity: Catholicism and the Arts in Italy, c. 1550\u20131650_ , Pamela M. Jones and Thomas Worcester (eds.) (Leiden, Boston and Cologne, 2002), p. 259.\n\n122. See Sandro Corradini, _Materiali per un processo_ , document 81.\n\n123. The tennis courts were all destroyed in a fire during the eighteenth century. The site is now occupied by an underground car park. I am grateful to Maurizio Marini for showing me its exact whereabouts.\n\n124. See Sandro Corradini, _Materiali per un processo_ , document 85; the translation given here is from Walter Friedlaender, _Caravaggio Studies_ , p. 286.\n\n125. Ibid., document 82.\n\n126. His report on the man's injuries can still be consulted in the 'Barbitonsores' section of the Roman State Archives. This document confirms that Caravaggio's ally in the fight had indeed been badly wounded. See ibid., document 80.\n\n127. Ibid., document 83.\n\n128. Ibid., document 84.\n\n129. Ibid.document 95: ' _initi duelli cum Michelangelo de Caravaggio... ac pro presenti duello_ '.\n\n130. For the document discussed below, see ibid., document 101.\n\n131. Ibid., documents 163, 164.\n\n132. Ibid., document 145.\n\n133. Ibid., document 111.\n\n134. I am grateful to Sandro Corradini for talking me through this series of archival documents, which remain unpublished. For another precis of their contents, see Helen Langdon, _Caravaggio: A Life_ , p. 313.\n\n135. See Sandro Corradini, _Materiali per un processo_ , document 109. Pontoni, who was a lawyer, also appears in document 17, testifying in the case of Fillide's knife attack on Prudenza Zacchia.\n\n136. Ibid., document 151.\n\n137. See Romolo Caggese (ed.), _Statuti della reppublica fiorentina. Volume 2: Statuto del podest\u00e0 del anno 1325_ (Florence, 1921); and _Volumen statutorem civitatis Maceratae_ , facsimile reprint of the 1553 edition, Arnaldo Forni (ed.) ([n.p., n.d.]). I am indebted to Elizabeth S. Cohen and Thomas S. Cohen for allowing me to read their essay ' _Sfregio_ : Facial Mutilation as Expressive Act' when it was still in draft form. It was that essay that called my attention to the legal penalties cited in the statute books noted above.\n\n##### PART FIVE: THE ALBAN HILLS, NAPLES, MALTA, SICILY, NAPLES, PORTO ERCOLE, 1606\u201310\n\n1. See Helen Langdon, _The Lives of Caravaggio_ , pp.31, 76.\n\n2. A picture in a private Roman collection has been put forward several times as a candidate, but it is so clumsy and sentimental that it cannot possibly have been painted by Caravaggio.\n\n3. No documents relating to this work survive. It has been romantically placed at the end of Caravaggio's life \u2013 in the quatercentenary exhibition in Rome in 2010 it was once more dated to 1610 \u2013 but it was not among the pictures listed as being on the boat with him when he travelled to Rome for the last time in July of that year, and it is besides painted in a style quite different from that of Caravaggio's last-documented picture, _The Martyrdom of St Ursula_ , now in the boardroom of the Banco di Napoli. Given that the style of the _David with the Head of Goliath_ is so close to that of _The Seven Acts of Mercy_ of 1606\u20137 in Naples \u2013 compare, for example, the handling of light in striated drapery in both pictures \u2013 and given that it indeed entered the Borghese collection (it can still be seen in the Villa Borghese in Rome), I believe that Caravaggio painted it expressly for Scipione Borghese to try to secure a pardon for his crimes. The identification of the severed head of Goliath as a self-portrait has been universally accepted, on the basis of visual comparison with Ottavio Leoni's portrait of Caravaggio in the Uffizi, and with other known self-portraits that occur within Caravaggio's _\u0153uvre_.\n\n4. Mancini is the source for this information: See Helen Langdon, _The Lives of Caravaggio_ , p. 31.\n\n5. It was only in 1613 that he ordered a frame to be made for it, according to a Borghese palace inventory: see the CD-ROM catalogue entry on the painting in John T. Spike, _Caravaggio_.\n\n6. Sandro Corradini, _Materiali per un processo_ , document 106, 23 Sept. 1606.\n\n7. See George Sandys, _A Relation of a Journey_ (London, 1615), pp. 253\u20134.\n\n8. Cited by Jeanne Chenault Porter in 'Reflections of the Golden Age: The Visitor's Account of Naples', in _Parthenope's Splendor: Art of the Golden Agein Naples_, published as Papers in Art History from the Pennsylvania State University, vol. 7, Jeanne Chenault Porter and Susan Scott Munshower (eds.) (Pennsylvania, 1993), p. 11.\n\n9. See Giuseppe Galasso, 'Society in Naples in the Seicento', in _Painting in Naples 1606\u20131705: From Caravaggio to Giordano_ , catalogue to the exhibition at the Royal Academy of Arts, London, Clovis Whitfield and Jane Martineau (eds.) (London 1982), p. 28.\n\n10. It is probable that the open-weave Neapolitan canvases on which Caravaggio would paint some of his greatest pictures were of English origin: see Clovis Whitfield, 'Seicento Naples', in _Painting in Naples 1606\u20131705_ , p. 19.\n\n11. Benedetto Croce, _History of the Kingdom of Naples_ , Frances Frenaye (trs.), H. Stuart Hughes (ed.) (Chicago, 1970), p. 116. Croce's text was first published as _Storia del regno di Napoli_ (Bari, 1925).\n\n12. Quoted in ibid., p. 120.\n\n13. Quoted in Giuseppe Galasso, 'Society in Naples in the Seicento', in _Painting in Naples 1606\u20131705_ , p. 25.\n\n14. See ibid., _passim_.\n\n15. See ibid., p. 25.\n\n16. See Helen Langdon, _The Lives of Caravaggio_ , pp. 76\u20137.\n\n17. For the details and documents concerning this commission, see Vincenzo Pacelli, 'New Documents concerning Caravaggio in Naples', _Burlington Magazine_ , vol. 119, no. 897 (Dec. 1977), pp. 819\u201329; and Vincenzo Pacelli, _Caravaggio: Le sette opere di misericordia_ (Salerno, 1984), p. 102.\n\n18. The quote is taken from the manuscript of C. De Lellis, _Aggiunta alla Napoli sacra del d'Engenio_ _1654\u201389_ , cited in Vincenzo Pacelli, _Caravaggio: Le sette opere di misericordia_ , p. 12.\n\n19. See Ferdinando Bologna, 'Caravaggio: The Final Years', in _Caravaggio: The Final Years_ , exhibition catalogue, the National Gallery (London, 2005), p. 22.\n\n20. Tiberio del Pezzo was the member of the confraternity who signed the documents authorizing payment to Caravaggio, but since he was only a deputy his role is likely to have been marginal. For the documents concerning this commission, see Vincenzo Pacelli, _Caravaggio: Le sette opere di misericordia_ , p. 102.\n\n21. See pp. 250\u201353, above. Caravaggio painted Marino's portrait in 1600 or 1601. It does not survive.\n\n22. See Estelle Haan, _From Academia to Amicitia: Milton's Latin Writings and the Italian Academies_ (Philadelphia, 1998), p. 122.\n\n23. Ibid., p. 119.\n\n24. These documents are usefully summarized in John T. Spike, _Caravaggio_ , in the CD-ROM catalogue entry on the picture.\n\n25. See Helen Langdon, _The Lives of Caravaggio_ , p. 77, where Bellori states that 'he was commissioned to do the _Flagellation of Christ at the Column_ in the Di Franco Chapel of the church of San Domenico Maggiore.' There is another, half-length depiction of _The Flagellation_ in Rouen that many scholars believe to be an autograph Caravaggio, but I am not convinced by it. Two other versions of the subject, one in Lucca and the other in a Swiss private collection, were published respectively by Roberto Longhi and Denis Mahon in the 1950s. I am not convinced by those paintings either.\n\n26. See Vincenzo Pacelli, 'New Documents concerning Caravaggio in Naples', p. 820.\n\n27. See Bernardo de Dominici, _Vite de' pittori, scultori ed architetti napoletani_ (Naples, 1742\u20133), pp. 275\u20136. Cited in John T. Spike, _Caravaggio_ , in his entry on _The Flagellation._\n\n28. This could be said to bring full circle that fruitful interplay between painting and sculpture already embodied by Caravaggio's own work. He himself had been powerfully influenced by the polychrome statuary of Lombardy and the _sacri monti_.\n\n29. See Ann Tzeutschler Lurie and Denis Mahon, 'Caravaggio's _Crucifixion of St Andrew_ from Valladolid', _Bulletin of the Cleveland Museum of Arts_ , vol. 64 (Jan. 1977), pp. 3\u201324. The picture had reportedly found its way to a convent in Spain by 1972, and was on sale in the art market in Switzerland a year later; it was purchased by the Cleveland Museum of Art through the L. C. Hanna Jr Bequest in 1976. See the CD-ROM catalogue entry on the painting in John T. Spike, _Caravaggio._\n\n30. See _Caravaggio: The Final Years_ , exhibition catalogue, p. 109. Keith Christiansen's entry on this particular painting also contains an outstandingly lucid account of the wider issues surrounding the much debated chronology of Caravaggio's later pictures.\n\n31. The quotation is taken from _The Golden Legend_ of Jacobus de Voragine, translated and adapted from the Latin by Granger Ryan and Helmut Ripperberger (New York, 1969), p. 13.\n\n32. John Varriano, in his _Caravaggio: The Art of Realism_ (Pennsylvania, 2003), notes that 'goiters are known to be geographically linked to mountainous places and were especially common in the region around Naples, the site where the earliest research on the disease was conducted.'\n\n33. See Walter Friedlaender, _Caravaggio Studies_ , p. 314.\n\n34. It was taken there by Finson, who by that time had assumed sole ownership of the work. He subsequently bequeathed it to his friend and business partner, Vinck.\n\n35. The picture was either sold or given to Emperor Josef II of Austria when he visited Antwerp in 1781.\n\n36. See the CD-ROM catalogue entry on the painting in John T. Spike, _Caravaggio_ , for the relevant documents.\n\n37. For Mancini's correspondence in connection with the sale of _The Death of the Virgin_ , see Michele Maccherini, 'Caravaggio nel carteggio familiare di Giulio Mancini', _Prospettiva_ , vol. 86 (1997), pp. 71\u201392.\n\n38. For Magno's correspondence with Chieppio, see Walter Friedlaender, _Caravaggio Studies_ , pp. 308\u201310.\n\n39. See Sandro Corradini, _Materiali per un processo_ , document 110, 2\u20134 Nov. 1606.\n\n40. See Maryvelma Smith O'Neil, _Giovanni Baglione: Artistic Reputation in Baroque Rome_ , pp. 166\u20137.\n\n41. Walter Friedlaender, _Caravaggio Studies_ , p. 313.\n\n42. Ibid., p. 313.\n\n43. Giacomo Bosio, _Dell'istoria della sacra religione_ (Rome, 1594\u20131602), vol. 3, p. 574. My attention was called to this and the following quotation by David M. Stone's article 'The Context of Caravaggio's _Beheading of St John_ in Malta', _Burlington Magazine_ , vol. 139, no. 1,128 (Mar. 1997), pp. 161\u201370.\n\n44. Ibid., vol. 3, p. 574.\n\n45. Helen Langdon, _The Lives of Caravaggio_ , p. 79.\n\n46. See Walter Friedlaender, _Caravaggio Studies_ , p. 266.\n\n47. Just a year before, at the start of 1606, Capeci had helped to organize another painter's journey to Malta. He had provided the artist in question, an unnamed Florentine, with canvases and pigments. He had also arranged his passage to the island via Messina in Sicily. For unknown reasons the painter from Florence never actually made it on to the island, but the episode strongly suggests that Capeci was involved in Caravaggio's transfer to Malta as well. See Keith Sciberras and David Stone, _Caravaggio: Art, Knighthood and Malta_ (Malta, 2006), p. 22.\n\n48. See Maurizio Calvesi, _Le realt\u00e0 del Caravaggio_ , pp. 132\u20133; Keith Sciberras and David Stone, _Caravaggio: Art, Knighthood and Malta_ , p. 20; Helen Langdon, _Caravaggio: A Life_ , p. 346.\n\n49. Much of this sequence of events was established by the discovery of a long-overlooked letter of June 1607 in the Farnese deposit in the Naples State Archives, written by Alessandro Boccabarile, agent for Duke Ranuccio Farnese: 'Eight days ago five galleys of the Religion of Malta arrived here, from Provence, under the command of the Prior of Venice, brother of the Marquis of Caravaggio [i.e., Fabrizio Sforza Colonna]. He brought his mother, who was staying at the Torre del Greco with the Prince of Stigliano... The aforesaid galleys will leave for Malta once the Feast of St John is over, taking with them two unequipped galleys newly made in Provence, and above all slaves...' This material was discovered by Antonio Ernesto Denunzio and published in the catalogue to the National Gallery's _Late Caravaggio_ exhibition, p. 49.\n\n50. Keith Sciberras's second chapter in _Caravaggio: Art, Knightood and Malta_ , entitled 'Virtuosity Honoured, Chivalry Disgraced', is an invaluable source of information about Caravaggio's time on Malta, much of it recently unearthed by Sciberras himself in the Maltese archives. For the documents concerning the journey to Malta, see p. 22.\n\n51. See George Sandys, _A Relation of a Journey_ , p. 234.\n\n52. For these documents, see John Azzopardi's contribution to the catalogue _The Church of St John in Valletta 1578\u201398 and the Earliest Record of Caravaggio in Malta_ , Fr John Azzopardi (ed.) (Malta, 1978).\n\n53. See George Sandys, _A Relation of a Journey_ , p. 230.\n\n54. See Helen Langdon, _Caravaggio: A Life_ , p. 345.\n\n55. See Commander Denis Calman, _Knights of Durance_ (Malta, 1963), p. 12.\n\n56. See George Sandys, _A Relation of a Journey_ , p. 230.\n\n57. Martelli did not take up his position until 1608, when a Medici agent reported his arrival in the Sicilian port town: 'yesterday the galleys arrived here from Malta and with them Prior Martelli, who if much aged remains in excellent health.' For this quotation, see Keith Sciberras and David Stone, _Caravaggio: Art, Knighthood and Malta_ , p. 89. Stone's account of Caravaggio's portrait of Martelli is clear and perceptive. The most informative essay on Martelli's life and career is by John Gash: 'The Identity of Caravaggio's Knight of Malta', _Burlington Magazine_ , vol. 139, no. 1,128 (Mar. 1997) pp. 156\u201360. Some authors continue to question the picture's attribution to Caravaggio, others to doubt that it represents Martelli. But a mid seventeenth-century inscription in a Medici collection inventory records the name of the painting's sitter as Antonio Martelli and I see no reason to doubt that. He was a celebrated man and the painting probably hung in the Vasari corridor alongside other depictions of worthies and notables treasured by the Medici, which makes it all the more likely that the inventorist would have got his name right. It used to be thought that Martelli could not have been painted by Caravaggio on Malta in 1607\u20138 because of his appointment to the Priory of Messina in 1606, so the archival evidence showing that he did not actually leave for Messina until the autumn of 1608 is important. Last but not least, the nonpareil moral and intellectual force of the painting, its abbreviated style, its depth of chiaroscuro, even such details as the slightly blocky impasto highlights in the prominent sunburned ear of the sitter \u2013 all scream out late Caravaggio. I cannot see who else could possibly have painted the picture.\n\n58. For these documents, see Fr John Azzopardi, 'Documentary Sources on Caravaggio's Stay in Malta', in _Caravaggio in Malta_ , Philip Farrugia Randon (ed.) (Malta, 1986), pp. 45\u201356; and Stefania Macioce, 'Caravaggio a Malta e i suoi referenti', in _Storia dell'Arte_ , vol. 81 (1994), pp. 207\u20138. Helen Langdon, in conversation with me, has expressed second thoughts about whether this document actually refers to Caravaggio. She points out that Knights of Malta were so universally prone to violence that the reference to a homicide committed does not necessarily point the finger at Caravaggio alone as the intended recipient of one of the two knighthoods for which papal approval was being requested. However, given the scarcity of Knighthoods of Magistral Obedience awarded by Wignacourt \u2013 indeed, so great was his reluctance to award such knighthoods at all that he had all but abolished them \u2013 it seems highly unlikely that he gave two in the same year to men who had committed murder. In my opinion, the man mentioned in the document, and Caravaggio, are beyond all reasonable doubt the same person.\n\n59. See n. 47 above.\n\n60. I am indebted to Keith Sciberras for explaining this crucial sequence of points to me, in conversations on Malta in 2001.\n\n61. It has been suggested that he may have painted the work _in situ_ , in the Oratory of St John itself, but I think that is implausible on the grounds that the light in that space would have been so far from ideal, even in the summer months. There is no absolute proof either way, but I think it more likely that he found a space elsewhere and adapted it accordingly.\n\n62. Some writers have identified her with Salome, others with Salome's mother, Herodias, Herod's consort. But she is dressed in the clothes of a serving wench. Everything about the way in which Caravaggio painted her indicates that she is meant to be seen as a member of the chorus, not as a leading player in the drama.\n\n63. The inscription has occasionally been thought to imply the phrase ' _fecit_ Caravaggio', 'Caravaggio made this', rather than 'Fra Michelangelo'. But the fact that it was his reception painting into the Order of St John argues compellingly for the latter as the true reading.\n\n64. Cited in John T. Spike, _Caravaggio_ , pp. 209\u201310. Fr John Azzopardi published a photograph of the document, with transcription and translation, in 'Documentary Sources on Caravaggio's Stay in Malta', pp. 55\u20136.\n\n65. I am grateful to John T. Spike for pointing out to me dell'Antella's probable authorship of the Bull, and for teasing out the implication that by praising Caravaggio as Apelles, it offers even higher praise to Wignacourt as his patron.\n\n66. For dell'Antella's life and personality, see Helen Langdon, _Caravaggio: A Life_ , p. 354; and Keith Sciberras and David Stone, _Caravaggio: Art, Knighthood and Malta_ , p. 80.\n\n67. I am indebted to Elizabeth Cropper's enlightening article about the links between Marino and Caravaggio: 'The Petrifying Art: Marino's Poetry and Caravaggio', _Metropolitan Museum Journal_ , vol. 26 (1991), pp. 193\u2013212. For the connections between Marino's poem about sleeping Cupid and Caravaggio's painting, see pp. 199\u2013200. The quotations are taken from Giambattista Marino, _La Galeria_ , Marzio Pieri (ed.) (Padua, 1979), vol. 1, pp. 273\u20137. The translations given here are my own very slightly adapted versions of those given in Cropper's article.\n\n68. See Giorgio Vasari, _Lives of the Painters, Sculptors and Architects_ , vol. 2, p. 650.\n\n69. Letter dated 24 Apr. 1610. See David Stone, 'In Praise of Caravaggio's _Sleeping Cupid_ : New Documents for Francesco dell'Antella in Malta and Florence', _Melita historica_ , vol. 12, pp. 165\u201377.\n\n70. See Helen Langdon, _The Lives of Caravaggio_ , p. 80.\n\n71. See fn. 33 to Keith Sciberras's second chapter in _Caravaggio: Art, Knightood and Malta_ , for the full quotation in Italian. The translation given here is my own.\n\n72. See Walter Friedlaender, _Caravaggio Studies_ , p. 292. The translation given there is more faithful than that in Helen Langdon, _The Lives of Caravaggio_ , pp. 52\u20133. Baglione's wording is important.\n\n73. See Helen Langdon, _The Lives of Caravaggio_ , p. 80.\n\n74. I owe a debt of gratitude to Fr John Azzopardi for showing me around the Maltese archive, and allowing me to examine for myself the documents \u2013 both legible and obliterated \u2013 relating to Caravaggio's crime and punishment on Malta. Keith Sciberras, who also generously shared much information with me on my visits to the island, first published the results of his X-ray examinations under the title ' \"Frater Michael Angelus in tumultu\": The Cause of Caravaggio's Imprisonment in Malta', _Burlington Magazine_ , vol. 144, no. 1,189 (Apr. 2002), pp. 229\u201332. My account of the events surrounding Caravaggio's crime is, inevitably, hugely dependent on his pioneering research.\n\n75. According to Malta's Liber Conciliorum for 1608\u201310, less than two years after the 'tumult' involving Caravaggio, De Ponte was sentenced for two months for fighting ' _cum levi sanguinis effusione_ ' with a certain Fra Francesco Sarsale. See Keith Sciberras, ' \"Frater Michael Angelus in tumultu\": The Cause of Caravaggio's Imprisonment in Malta', fn. 37.\n\n76. I am grateful again to Fr John Azzopardi for helping to find the ladder and letting me into the _guva_. Keith Sciberras doubts that Caravaggio would have been kept in the _guva_ , arguing that he would most probably have been detained in one of Castel Sant'Angelo's semi-open prisons. But, given Wignacourt's stated desire 'not to lose him', expressed in the petition to the pope for Caravaggio's knighthood, I share Fr Azzopardi's view that he would indeed have been confined in the _guva_ , which was after all the most high-security of the island's jails. It may also be worth noting that, according to a long oral tradition on Malta, the _guva_ was Caravaggio's place of imprisonment.\n\n77. Helen Langdon, _The Lives of Caravaggio_ , p. 81.\n\n78. See Faith Ashford, 'Caravaggio's Stay in Malta', _Burlington Magazine for Connoisseurs_ , vol. 67, no. 391 (Oct. 1935), pp. 168\u201374.\n\n79. See Keith Sciberras and David Stone, _Caravaggio: Art, Knightood and Malta_ , p. 34.\n\n80. See Faith Ashford, 'Caravaggio's Stay in Malta', p. 174.\n\n81. Ibid.\n\n82. The document was published by A. Spadaro, 'Il percorso smarrito e l'importante inedito: la presenza del pittore a Caltagirone', in _Foglio d'Arte_ , vol. 8, no. 2 (1984\u20135), pp. 6\u20137; I was alerted to it by Gioacchino Barbera and Donatella Spagnolo's essay 'From _The Burial of St Lucy_ to the Scenes of the Passion: Caravaggio in Syracuse and Messina' in the catalogue to the exhibition _Caravaggio: The Final Years_ , pp. 80\u201387.\n\n83. See George Sandys, _A Relation of a Journey_ , p. 234.\n\n84. Susinno's manuscript containing biographies of artists to have worked in Sicily, and particularly Messina, was first published by Valentino Martelli in Florence in 1960. Susinno's life of Caravaggio, which was included in that manuscript, was usefully reprinted and translated in Howard Hibbard, _Caravaggio_ : see p. 381. Minniti had lived a chequered life since returning to his native Sicily in about 1604, at one point having been forced to seek sanctuary in the Carmelite monastery at Syracuse 'for a homicide casually committed' \u2013 for which see Francesco Susinno, _Le vite dei pittori messinesi_ of 1724, Valentino Martelli (ed.) (Florence, 1960), p. 117. By the autumn of 1608 Minniti had long since redeemed himself, by painting numerous altarpieces for the religious institutions of Syracuse and Messina. He often worked for the Franciscans, which indicates that he had a close relationship with the order. His sister, Maria, was a Capuchin tertiary. See Gioacchino Barbera and Donatella Spagnolo, 'From _The Burial of St Lucy_ to the Scenes of the Passion: Caravaggio in Syracuse and Messina' in the catalogue to the exhibition _Caravaggio: The Final Years_ , p. 81.\n\n85. See Susinno in Howard Hibbard, _Caravaggio_ , p. 381; in fact, the church and its adjacent monastery were not assigned to the Minorite friars of the Franciscan order until 1618. But they had been lobbying to have the site restored and given to them for many years, so they are also likely to have had a strong say in the choosing of Caravaggio. The Franciscans were the poorest of the poor orders. They are likely to have been highly sympathetic to an artist whose work so aggressively insisted on the poverty of Christ and his early followers. Franciscan involvement also supports Susinno's account of the part played by Minniti in winning the commission for Caravaggio.\n\n86. St Lucy's name was derived from the Latin word _lux_ , meaning 'light', a fact that had not been lost on the early Church fathers. St Ambrose, in his commentaries on her martyrdom, noted that 'In Lucy is said the way of light.'\n\n87. Susinno as reprinted in Howard Hibbard, _Caravaggio_ , from which the translation used here derives. See p. 381.\n\n88. See Keith Sciberras and David Stone, _Caravaggio: Art, Knighthood and Malta_ , pp. 35\u20136.\n\n89. Susinno as reprinted in Howard Hibbard, _Caravaggio_ , from which the translation used here derives. See p. 386.\n\n90. Ibid. The name of the dog is disclosed by Giovanni Baglione in a comic aside in a passage from his life of Caravaggio's follower Carlo Saraceni. See Giovanni Baglione, _Le vite de' pittori, scultori, architetti, dal pontificato di Gregorio XIII del 1572, fino a' tempi di Papa Urbano VIII_ _nel 1642_ (Rome, 1642), p. 147.\n\n91. See Francesco Susinno, _Le vite dei pittori messinesi_ , p. 119.\n\n92. See Vincenzo Mirabella, _Dichiarazioni della pianta delle antiche Siracuse, e d'alcune scelte medaglie d'esse e de' principi che quelle possedettero_ (Naples, 1613), p. 89. The whole passage is quoted in Italian in Maurizio Marini, _Michelangelo Merisi da Caravaggio 'pictor praestantissimus'_ , p. 100. The translation given here is my own.\n\n93. See Ferdinando Bologna, 'Caravaggio: The Final Years', in _Caravaggio: The Final Years_ , p. 32.\n\n94. See Keith Sciberras and David Stone, _Caravaggio: Art, Knighthood and Malta_ , pp. 36\u20137.\n\n95. See George Sandys, _A Relation of a Journey_ , pp. 245\u20136.\n\n96. Susinno as reprinted in Howard Hibbard, _Caravaggio_ , from which the translation used here derives. See p. 382.\n\n97. The documents recording this commission are now lost, presumed destroyed in the catastrophic earthquake that struck Messina in 1908. Before their destruction, they were transcribed and published. See V. Sacc\u00e0, 'Michelangelo da Caravaggio pittore. Studi e ricerche', in _Archivio storico messinese_ , vol. 7 (Messina, 1906), p. 58, and vol. 8 (Messina, 1907), p. 78.\n\n98. There is proof positive that he was familiar with the knights' book of statutes in his Maltese altarpiece, _The Beheading of St John_. The image of the prison, with inmates, is clearly drawn from one of the illustrations in the order's book of statutes. See David M. Stone, 'The Context of Caravaggio's _Beheading of St John_ in Malta', _Burlington Magazine_ , vol. 139, no. 1,128 (Mar. 1997), pp. 161\u201370. It should also be noted that the document of consignment in which he is referred to as a Knight of Malta is dated June 1609, a full seven months after his expulsion from the order. It therefore seems highly unrealistic to argue that he did not know about his expulsion.\n\n99. See Helen Langdon, _The Lives of Caravaggio_ , p. 84. Caravaggio had made a similar plea for his head to Scipione Borghese just after the murder. The London picture is much weaker than the Borghese _David and Goliath_ , however. It is not Herodias (or Salome) with the head, but a female servant.\n\n100. See V. Sacc\u00e0, 'Michelangelo da Caravaggio pittore. Studi e ricerche'. Caravaggio's name is not mentioned in the document of 6 Dec., but, given his strong association with the poor orders and charitable ministries, and given Susinno's remark that he left Messina soon after completing _The Burial of St Lucy_ , which must have been ready by her feast day on 13 Dec., it is a reasonable assumption that Giovan Battista de' Lazzari had Caravaggio in mind from the start. Indeed, he may have been spurred to make his undertaking by the very opportunity that Caravaggio's arrival provided. I take the document of 6 Dec. as a _terminus ante quem_ for Caravaggio's arrival in Messina from Syracuse.\n\n101. Susinno as reprinted in Howard Hibbard, _Caravaggio_ , from which the translation used here derives. See p. 382.\n\n102. As George Sandys noted, the Eastern faith was tolerated in Sicily: 'Their religion is Romish yet there are not so few as ten thousand who are of the tollerated Greeke church.' See George Sandys, _A Relation of a Journey_ , p. 238.\n\n103. Susinno as reprinted in Howard Hibbard, _Caravaggio_ , from which the translation used here derives. See p. 384.\n\n104. Ibid. See p. 385.\n\n105. It can be found in countless icons of the Virgin and Child, one of the most famous examples being Russia's most sacred icon, _Our Lady of Vladimir_ , which was painted in Constantinople in the eleventh century and taken to Kiev a hundred years later to mark the conversion to Christianity of the peoples of Russia. Caravaggio will have been familiar with the motif from icons in Sicily, or from the rich traditions of Italo-Byzantine painting of the thirteenth and fourteenth centuries, examples of which were to be seen all over the Italian peninsula.\n\n106. Susinno as reprinted in Howard Hibbard, _Caravaggio_ , from which the translation used here derives. See p. 385.\n\n107. Ibid. See p. 386.\n\n108. Ibid.\n\n109. Ibid.\n\n110. Helen Langdon, _The Lives of Caravaggio_ , p. 84.\n\n111. I arrive at this date by common sense. We know that Caravaggio was seriously wounded by a gang of assailants in Naples in late Oct. 1609, as will be explained below, pp. 415\u201320. He was very badly injured indeed. The only two paintings that can be dated to after that time, _The Denial of Peter_ and _The Martyrdom of St Ursula_ , are so radically unlike his Sicilian paintings that the difference can only logically be explained by incapacity and illness. We also know that Caravaggio painted a large altarpiece for the Fenaroli Chapel in Sant'Anna de' Lombardi during his second and last stay in Naples, i.e. after arriving there from Palermo in 1609. He cannot have painted it on his first visit to the city, because the patron had only acquired rights to the chapel on 24 Dec. 1607, when Caravaggio had already left Naples for Malta. In my opinion, it is clear from the visual evidence of _The Denial of Peter_ and _The Martyrdom of St Ursula_ that when he painted those works Caravaggio could barely wield a brush. On the empirical evidence of the pictures, his eyesight had been damaged as well as possibly his nervous system. It is therefore inconceivable that he could have painted any kind of large and ambitious altarpiece after the assault of late Oct. 1609. In other words, he must have painted the Fenaroli altarpiece in Naples before the wounding took place. Assuming he worked flat out, and assuming it was commissoned from him the moment he disembarked from Palermo, he still would have needed at least four to six weeks to paint it. Therefore, he must have been back in Naples from Sicily some four to six weeks before the wounding of late Oct. On that basis, I set a date some time around the first week of September for his return to Naples.\n\n112. See Helen Langdon, _The Lives of Caravaggio_ , p. 53.\n\n113. Ibid., p. 84.\n\n114. I make this assumption because we know for sure that Caravaggio _left_ Naples from the Colonna Palace at the end of his second stay in the city, in July 1610: that fact is documented. Given that the early sources all say he went to Naples from Palermo because he was in fear of pursuit, it seems logical to suppose that he was at the Colonna Palace at Chiaia throughout his time there in 1609\u201310, under the protection of the Marchesa Costanza Colonna.\n\n115. See Helen Langdon, _The Lives of Caravaggio_ , p. 77.\n\n116. For the reasoning behind these assertions concerning the date of the lost _Resurrection_ , see n. 111 above.\n\n117. See Charles-Nicolas Cochin, _Voyage d'Italie_... (Paris, 1758), vol. 1, pp. 171\u20132; the passage is quoted in Maurizio Marini, _Michelangelo Merisi da Caravaggio 'pictor praestantissimus'_ , p. 568.\n\n118. ' _orgie siffatte_ ': for a useful summary of the poem, see Giuseppe Ferrari, _Opuscoli politici e letterari_ (Naples, 1852), p. 462. For the poem in full, see Giulio Cesare Cortese, _Opere_ (Naples, 1666), 6 vols.\n\n119. See Giambattista Basile, 'Talia, overo lo Cerriglio', _Egloca III_ , _Le Muse Napolitane_ , in _Collezione di tutti i poemi in lingue napoletane_ , tome 21, vol. 2 (Naples, 1788), p. 267: ' _Lloco le Cortesciane_ \/ _Fanno lo sguazzatorio_ : \/ _E all'_ _uocchie de corrive,_ \/ _A spesa de perdente_ \/ _Ne sporpano tant' ossa..._ '\n\n120. As cited in Salvatore di Giacomo, _La prostituzione in Napoli nei secoli XV, XVI e XVII: documenti inediti_ (Naples, 1899), p. 82.\n\n121. See Giambattista Basile, 'Talia, overo lo Cerriglio', p. 257. Basile's exact phrase is ' _dove trionfa Bacco, dove se scarfa Venere_ ': _se scarfa_ is Neapolitan dialect, which I translate as 'is shunned', having taken specialist advice from Nicholas Stone Villani, who kindly consulted a number of experts in historical Neapolitan usage on my behalf.\n\n122. See Salvatore di Giacomo, _La prostituzione in Napoli_ , p. 83.\n\n123. Ibid., p. 119.\n\n124. See Walter Friedlaender, _Caravaggio Studies_ , p. 292.\n\n125. See Michele Maccherini, 'Caravaggio nel carteggio familiare di Giulio Mancini', p. 83.\n\n126. See p. 63 and p. 180, above.\n\n127. This translation is broadly that given by Friedlaender in _Caravaggio Studies_ , p. 236. I use the word 'affronted' instead of 'insulted', because it is closer to Baglione's usage in Italian, _affronto_ , which I believe itself carries an implied meaning, as will be explained on p. 420, below.\n\n128. This translation is again broadly that given by Friedlaender in _Caravaggio Studies_ , p. 251. I have corrected Friedlaender's mistranscription of 'Herodias' as 'Salome'.\n\n129. Susinno in Howard Hibbard, _Caravaggio_ , p. 386.\n\n130. Most notably Maurizio Marini. I am grateful to him for sharing his views with me during the course of a fascinating two days of excursions and peregrinations in Caravaggio's Rome in the autumn of 2001. I should add that when Marini expressed his view that the Tomassoni might have been responsible for the attack, Keith Sciberras had yet to publish the facts of Caravaggio's crime on Malta, which tilt the balance very much towards Malta as the source of the attack; in other words, Marini was not in possession of all the facts when we spoke.\n\n131. See Keith Sciberras's second chapter in _Caravaggio: Art, Knightood and Malta_ , fn. 49.\n\n132. Keith Sciberras, who made the discovery of Caravaggio's crime on Malta, had to X-ray the book to get at the documents. In his account of his discovery, he notes that the records of the crime were covered over not long after they had been inscribed, i.e the coverings-over date from the early seventeenth-century, consistent with the idea that they might represent a cover-up arranged by Roero himself. See Keith Sciberras, ' \"Frater Michael Angelus in tumultu\": The Cause of Caravaggio's Imprisonment in Malta', pp. 229\u201332.\n\n133. See Michele Maccherini, 'Caravaggio nel carteggio familiare di Giulio Mancini', p. 83.\n\n134. Only one other work in Caravaggio's entire known _\u0153uvre_ is painted in the sadly attenuated post-assault style of his last year, and that is _The Martyrdom of St Ursula_ , discussed below, pp. 423\u20134. That picture is securely datable on the basis of original documents concerning its consignment. These two works are utterly distinct in style, and they clearly show a tragic falling off in the painter's manual dexterity that can only be accounted for by his injuries.\n\n135. For all the documents concerning _The Martyrdom of St Ursula_ , see Vincenzo Pacelli, 'Caravaggio 1610: la \"Sant'Orsola confitta dal tiranno\" per Marcantonio Doria', _Prospettiva_ , vol. 23 (Oct. 1980), pp. 24\u201330. They are helpfully translated in John T. Spike, _Caravaggio_ , in the CD-ROM catalogue entry on the picture.\n\n136. The date of his departure can be inferred from the journey time by sea from Naples to Palo, where he tried to go on land with his things \u2013 roughly seven days \u2013 and the date of his death, which must have occurred sometime between 18 and 21 July 1610.\n\n137. See Helen Langdon, _The Lives of Caravaggio_ , p. 85.\n\n138. The details of the deal emerge in Deodato Gentile's letter to Borghese about Caravaggio's death, of 29 July 1610; see below, p. 429.\n\n139. See Walter Friedlaender, _Caravaggio Studies_ , p. 292.\n\n140. See Baglione's original Italian, as reprinted in Walter Friedlaender, _Caravaggio Studies_ , p. 233. I have given my own translation.\n\n141. Baglione uses the words _in cambio_ , literally, 'in change', a phrase that has frequently been misleadingly translated as 'mistakenly', on the assumption that Baglione meant to imply that Caravaggio was arrested 'in exchange' (so to speak) of someone else. But in sixteenth- and early seventeenth-century Italian usage, _in cambio_ is most frequently a phrase of emphasis with little actual meaning, carrying more or less the same thrust as 'in fact'. It can also sometimes imply the idea of a swift change, in which case the English word 'suddenly' is a good equivalent. Baglione probably meant it in this latter sense. The modern mistranslations take their cue from Bellori, who clearly based his own account of Caravaggio's death on that of his predecessor, Baglione. He himself seems to have misunderstood Baglione's use of _in cambio_ , amplifying it into his own tale of a case of mistaken arrest \u2013 as will be discussed below, p. 427.\n\n142. Apart from the word 'suddenly' \u2013 explained in the note above \u2013 I have used the translation given in Walter Friedlaender, _Caravaggio Studies_ , p. 236.\n\n143. See n. 141 above.\n\n144. I have used the translation given in Walter Friedlaender, _Caravaggio Studies_ , pp. 251\u20132.\n\n145. Ibid., p. 258.\n\n146. See Sandro Corradini, _Materiali per un processo_ , document 138, 28 July 1610.\n\n147. Ibid., document 140, 31 July 1610.\n\n148. Much has been made of this reference to Procida, and many a paranoid theory has been erected on its shaky foundations. But whoever told Borghese that Caravaggio died there may just have been making a logical guess based on the knowledge that the painter had left from Naples. Boats from there hitting bad weather often took refuge in Procida. The English traveller George Sandys had exactly that experience when he left Naples to travel to Rome a few years after Caravaggio: he got caught in a storm and ended up making an unscheduled visit to the island before continuing on to Rome, via Nettuno.\n\n149. Borghese's side of the correspondence has been lost. But the content of his letter of 23 July can be inferred from Gentile's letters back to him, which do still survive. All these documents were discovered by Vincenzo Pacelli, through brilliant sleuth work in the Neapolitan archives. They are conveniently brought together with much other archive material in Sandro Corradini, _Materiali per un processo_.\n\n150. The courier must have travelled post haste, changing horses as he rode, since Naples is a little over 120 miles from Rome.\n\n151. See Sandro Corradini, _Materiali per un Processo_ , Document 139, 29 July 1610.\n\n152. See, in particular, the fanciful closing sections of Peter Robb's quasi-biography of the painter, _M_ (Sydney, 1998), in which the pope, Costanza Colonna and the Knights of Malta are held to have conspired to have Caravaggio assassinated. Vincenzo Pacelli, who located Gentile's correspondence with Borghese in the first place, also believes in a plot. In his view, the Knights of Malta, Costanza Colonna and Scipione Borghese conspired to have Caravaggio killed. He told me so in conversation in 2001. The plotters' motive, according to Pacelli, was their shared belief that Caravaggio had not only become an atheist but that he was using his most important commissions to profess, as it were, in subtle code, his heretical non-belief in God. The prime example given to me by Pacelli of a supposedly atheist painting infiltrated by Caravaggio into a Catholic church was _The Seven Acts of Mercy_. I am grateful to him for sharing his theories with me, but I have to say that I find them implausible.\n\n153. I am indebted to the Maltese naval historian Joseph Scibberas for explaining how transport by _felucca_ really worked in early seventeenth-century Italy.\n\n154. When George Sandys went there from Naples a few years later, he went via the much sleepier port of Nettuno (see n. 148, above) to avoid detection as an Englishman and a Protestant.\n\n155. I noticed this on a visit to Palo in 2001. The old fortress is still in existence, although nowadays it is a luxury hotel patronized by prominent Italian politicians, playboys and their supermodel girlfriends. The insignia of the old postal service can still be seen on the wall.\n\n156. Besides, to make the journey on foot would have been to defeat its very purpose, which was to get to Porto Ercole preferably before or, at worst, at the same time as the boat. If Caravaggio had arrived four or five days after leaving Palo, the boat would already have got to Porto Ercole, unloaded and left. So if a horse or horses had not been available, there would have been no point in his even attempting the journey.\n\n157. He refused to record any of the names of the dead for the entire year of 1610. I am grateful to Giuseppe La Fauci for showing me the book of the dead for the relevant period in the archives of the town, and for explaining the absence of records for the year in question. The death certificate that was 'found' in 2001 in Porto Ercole, a separate piece of paper with Caravaggio's name on it, is entirely inconsistent with the manner in which deaths were conventionally noted down in Porto Ercole \u2013 i.e., as entries in the book of deaths. I am sure that document is a forgery.\n\n158. In Langdon, _Caravaggio: A Life_ , p. 388, the author wrongly takes this phrase to mean 'high seas', as in tall waves, and suggests that a storm was brewing and the sea was swelling, which forced the boat to pull away from shore. However _alto mare_ does not mean that; it simply means 'the open sea'.\n\n159. See Sandro Corradini, _Materiali per un processo_ , document 141, 31 July 1610.\n\n160. See Walter Friedlaender, _Caravaggio Studies_ , p. 293.\n\n161. See Sandro Corradini, _Materiali per un processo_ , document 144, 10 Dec. 1610.\n\n162. Quoted in Carlton Lake, _In Quest of Dal\u00ed_ (Michigan, 1969), p. 46.\n\n163. Martin Scorsese's remarks have been directly transcribed from his conversations with the author in December 2005, which included an interview filmed for and subsequently transmitted by _The Culture Show_ (BBC Television, directed by David Shulman).\n\n## Further Reading\n\nDetailed references to nearly all of the many sources I have consulted in writing this book will be found in the Notes (see pp. 447\u2013 above). What follows here is a short list of texts that I would recommend to any non-specialist reader wishing to pursue an interest in Caravaggio and his world.\n\nWietse de Boer, _The Conquest of the Soul: Confession, Discipline and Public Order in Counter-Reformation Milan_ (Leiden, Boston and Cologne, 2001). A fascinating, highly detailed account of the religious milieu created in Milan by Archbishop Carlo Borromeo during Caravaggio's youth.\n\nMaurizio Calvesi, _Le realt\u00e0 del Caravaggio_ (Turin, 1990). For those who can read Italian, this is highly recommended. A broad-ranging, discursive study of many aspects of Caravaggio's life and work, full of biographical insights and intuitions that have been borne out, to a remarkable extent, by later documentary finds. Also worth reading for Calvesi's many ingenious iconographical interpretations of the pictures.\n\nSandro Corradini, _Caravaggio: materiali per un processo_ (Rome, 1993). Hard work, requiring a mastery of demotic Italian as it was spoken in Caravaggio's day, as well the ability to read the judicial Latin used by the notaries of the time. Hard to get hold of too, since it was published in a tiny edition. But I cannot omit it from this list. Containing the fruits of more than two decades of privately conducted research in the archives of Rome, Corradini's book is the essential anthology of documents concerning the darker and more violent aspects of Caravaggio's life.\n\nWalter Friedlaender, _Caravaggio Studies_ (Princeton, 1955). Pioneering study of the painter's life and work, superseded in some respects by the research of later scholars, but still remarkably fresh, and full of wise and heartfelt responses to the individual paintings. Contains useful translations of numerous primary documents as well as dual-language versions of Mancini, Baglione and Bellori's biographies, and is supplemented by the short biographical remarks written by Karel van Mander and Joachim von Sandrart, also both in the original and in English translation.\n\nHoward Hibbard, _Caravaggio_ (London, 1983). Deeply unreliable on the facts of the painter's life, but still worth reading for some of the author's interpretations of the pictures themselves. Also contains a useful translation of Francesco Susinno's eighteenth-century life of Caravaggio.\n\nHelen Langdon (ed.), _The Lives of Caravaggio_ (London, 2005). A handy, pocket-sized edition of the three principal early biographies of the artist, by Giulio Mancini, Giovanni Baglione and Pietro Bellori.\n\nHelen Langdon, _Caravaggio: A Life_ (London, 1998). Much the best twentieth-century biography of Caravaggio, outstanding in particular on the painter's years in Rome. The last few chapters should be read with care, however, because some of Langdon's assumptions and conclusions concerning Caravaggio's later life have been overtaken by subsequent archival discoveries.\n\nRoberto Longhi (ed. Giovanni Previtali), _Caravaggio_ (Rome, 1982). For those who can read Italian, these are collected writings on the artist by arguably the greatest and certainly the most influential Caravaggio scholar of the twentieth century. Full of pithy, down-to-earth descriptions of the paintings.\n\nDiarmaid MacCulloch, _Reformation: Europe's House Divided 1490\u20131700_ (London, 2003). A brilliant, panoramic overview of the religious history of the period.\n\nMaurizio Marini, _Caravaggio. Michelangelo Merisi da Caravaggio, 'pictor praestantissimus'_ (Rome, 1989). Another for the reader of Italian. Marini's book draws together a great number of the primary documents relating to Caravaggio's paintings, and reprints many of them in facsimile.\n\nJohn T. Spike, _Caravaggio_ (New York, 2001). Somewhat marred by the author's attributional optimism (so much so that both the front and back covers of the book show pictures that Caravaggio never painted!). But the accompanying CD-ROM catalogue is tremendously useful, bringing together just about all the primary sources for all of the major paintings. In his catalogue Spike also furnishes remarkably long and comprehensive lists of bibliographical references for every single one of Caravaggio's paintings \u2013 a monumental endeavour, making his catalogue indispensable to anyone wanting to explore any particular work in great depth and detail.\nPENGUIN BOOKS\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 (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, Auckland 0632, New Zealand   \n(a division of Pearson New Zealand Ltd)\n\nPenguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa\n\nPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England\n\nwww.penguin.com\n\nFirst published by Allen Lane 2010\n\nPublished in Penguin Books 2011\n\nCopyright \u00a9 Andrew Graham-Dixon, 2010\n\nMaps copyright \u00a9 Alan Gilliland, 2010\n\nThe moral right of the author has been asserted\n\nThe acknowledgements on pp. ix\u2013xii constitute an extension of this copyright page.\n\nEvery effort has been made to trace and contact copyright holders. If there are any inadvertent omissions, we ask that you contact the publisher.\n\nAll rights reserved\n\nWithout limiting the rights under copyright reserved above, no 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 written permission of both the copyright owner and the above publisher of this book\n\nISBN: 978-0-14-196294-8\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Chris Curnow, Joseph Cooper, Christian\nBoissonnas, The Internet Archive for some images and the\nOnline Distributed Proofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\n  BIRDS AND NATURE.\n\n  ILLUSTRATED BY COLOR PHOTOGRAPHY.\n\n  VOL. VIII.       SEPTEMBER, 1900.       NO. 2.\n\n\n\n\n  CONTENTS.\n\n\n                                                         Page\n  SEPTEMBER.                                               49\n  THE MALLOWS.                                             50\n  EAGLE LORE.                                              53\n  THE SNOWDROP'S PHILOSOPHY.                               55\n  THE GLADNESS OF NATURE.                                  56\n  FLOWERS AND THEIR INVITED GUESTS.                        59\n  THE ASTERS.                                              62\n  SCHOOL GARDENS.                                          65\n  THE FLICKER'S MISTAKE.                                   66\n  TIGER-LILIES.                                            68\n  FLOWERS IN THE CRANNIED WALL.                            68\n  THE WILD YELLOW LILY.                                    71\n  WHAT DO WE OWE THE BIRDS?                                72\n  TO THE VESPER BIRD.                                      73\n  THE VESPER SPARROW.                                      74\n  THE WORSHIP OF NATURE.                                   77\n  BIRD-STUDY.                                              78\n  THE OREGON JUNCO.                                        80\n  THE CALICO BASS.                                         83\n  THE GROWTH AND VARIATION OF FISH.                        84\n  THE ORIGIN OF THE FISH.                                  90\n  THE BANANA.                                              95\n\n\n\n\nSEPTEMBER.\n\n\n    The golden-rod is yellow;\n      The corn is turning brown;\n    The trees in apple orchards\n      With fruit are bending down.\n\n    The gentian's bluest fringes\n      Are curling in the sun;\n    In dusty pods the milkweed\n      Its hidden silk has spun.\n\n    The sedges flaunt their harvest,\n      In every meadow nook;\n    And asters by the brook-side\n      Make asters in the brook.\n\n    From dewy lanes at morning\n      The grapes' sweet odors rise;\n    At noon the roads all flutter\n      With yellow butterflies.\n\n    By all these lovely tokens\n      September days are here,\n    With summer's best of weather,\n      And autumn's best of cheer.\n                                        Helen Hunt Jackson.\n\nCopyright, 1900, by A. W. Mumford.\n\n\n\n\nTHE MALLOWS.\n\n\nA number of interesting plants are found grouped under the name of\nthe Mallow Family (Malvaceae). They are the common Mallow, a weed of\nwaysides and cultivated grounds; the Indian Mallow or Velvet-leaf, with\nits large velvety leaves and yellow flowers, a visitor from India which\nhas escaped from cultivation and become a pest in corn and grain fields\nand waste places; the Musk Mallow, which has also escaped from our\ngardens; the Marsh-Mallow, the root of which abounds in a mucilage that\nis extensively used in the manufacture of confections; the Hollyhock of\nour gardens, which was originally a native of China and the beautiful\nRose-Mallow of our illustration.\n\nThe Mallow Family includes about eight hundred species which are widely\ndistributed in the temperate and tropical countries. The technical name\nis from a Greek word having reference to the soothing effect produced\nby many of the species, when applied to wounded surfaces.\n\nAll are herbs. Most of those found in the United States have been\nintroduced from Europe and Asia. Only a very few are native, and no one\nof these is very common.\n\nThe flowers and fruits are all similar in structure to that of the\ncommon hollyhock.\n\nThe disk-like fruits of the common round leafed Mallow of our dooryards\nare often called \"cheeses\" by the children and are frequently gathered\nand eaten by them. The cotton plant, one of our most important economic\nplants, is also closely related to the Mallow. The Cotton of commerce\nis the woolly hair of the seeds of this plant which is a native of\nnearly all tropical countries and is cultivated in temperate regions.\n\nThe beautiful Rose-Mallow has its home in the brackish marshes of the\nAtlantic sea coast. It is also occasionally found on the marshy borders\nof lakes and rivers of the interior.\n\nThe plants grow to the height of from three to eight feet. The leaves\nare egg-shaped and the lower ones are three-lobed. The under side of\nthe leaves is covered with fine and soft whitish hairs.\n\nThe flowers, produced in August and September, are large, varying from\nfour to eight inches in diameter, and may be solitary or clustered\nat the top of the stem. The color of the petals is usually a light\nrose-pink, but occasionally white, with or without crimson at their\nbases.\n\nNeltje Blanchan in \"Nature's Garden\" speaks of this beautiful plant as\nfollows:\n\n\"Stately ranks of these magnificent flowers, growing among the tall\nsedges and 'cat-tails' of the marshes, make the most insensate traveler\nexclaim at their amazing loveliness. To reach them one must don rubber\nboots and risk sudden seats in the slippery ooze; nevertheless, with\nspade in hand to give one support, it is well worth while to seek them\nout and dig up some roots to transplant to the garden. Here, strange to\nsay, without salt soil or more water than the average garden receives\nfrom showers and hose, this handsomest of our wild flowers soon makes\nitself delightfully at home under cultivation.\"\n\n  [Illustration: SWAMP ROSE-MALLOW.\n                 (_Hibiscus Moscheutos._)\n                 FROM \"NATURE'S GARDEN\"\n                 COPYRIGHT 1900, BY\n                 DOUBLEDAY, PAGE & COMPANY.]\n\n\n\n\nEAGLE LORE.\n\nCURIOUS STORIES OF THE OLD-TIME FAITH IN THE \"KING OF THE FEATHERED\nTRIBES.\"\n\n\nBirds were trusted, honored and made the symbols of wisdom and power\nin the old time, and they have not, at least in their emblematical\nsignification, been neglected in modern times. The eagle, in\nparticular, is exalted to a high and potential distinction. On the\nbanner of a hundred States he is displayed as a conquering symbol and\nfloats to-day over many a fair realm where Rome's imperial standard\nnever penetrated.\n\nThe eagle has always been considered a royal bird, and was a favorite\nwith the poets. They called him king of the air and made him bear the\nthunderbolts of Jove. Euripides tells us that \"the birds in general are\nthe messengers of the gods, but the eagle is king, and interpreter of\nthe great deity Jupiter.\"\n\nThe eagle figures in the early legends of all people. When the ancient\nAztecs, the mound-builders of the Mississippi Valley, were moving\nsouthward under Mexi, their king, their god, Vitziputzli, whose image\nwas borne in a tabernacle made of reeds and placed in the center of the\nencampment whenever they halted, directed them to settle where they\nshould find an eagle sitting on a fig-tree growing out of a rock in a\nlake. After a series of wanderings and adventures that do not shrink\nfrom comparison with the most extravagant legends of the heroic ages\nof antiquity, they at last beheld perched on a shrub in the midst of\nthe lake of Tenochtitlan a royal eagle with a serpent in his talons and\nhis broad wings opened to the rising sun. They hailed the auspicious\nomen and laid the foundation of their capital by sinking piles into the\nshallows. This legend is commemorated by the device of the eagle and\nthe cactus, which forms the arms of the modern Mexican Republic.\n\nA goose, it is said, saved Rome once upon a time, but it was an\neagle that directed the selection of the ancient Byzantium--now\nConstantinople--as the capital of the Eastern Empire. The site of\nancient Troy had been settled upon by Constantine, and the engineers\nwere engaged in surveying the plan of the city, when an eagle swooped\ndown, seized the measuring line, flew away with it and dropped it at\nByzantium. At any rate, this was the story told to the soldiers and\nmarines, in order to reconcile them to the change of plan, which they\nmight otherwise have deemed an unfavorable omen, though the splendid\nsituation of the new capital and its long prosperity, prove how\nadmirably sagacious was the choice of its founder.\n\nIn the reign of Ancus Martius, King of Rome, a wealthy man, whose name\nwas Tarquin, came to that city from one of the Etruscan States. Sitting\nbeside his wife in his chariot, as he approached the gates of Rome, an\neagle, it is said, plucked his cap from his head, flew up in the air,\nand then, returning, placed it on his head again. Not a few suspect\nthat the eagle was a tame one and had been taught to perform this\ntrick. If so, however, the apparent prodigy lost none of its effect in\nthe popular belief, and Tarquin succeeded Ancus as King of Rome. The\neagle's head on the Roman sceptre, and later on its standard, took its\norigin from this occurrence.\n\nPlutarch, in his life of Theseus, relates that when Cymon was sent by\nthe Athenians to procure the bones of that hero, who had long before\nbeen buried in Scyros, to reinter them in his former capital, he found\ngreat difficulty in ascertaining the burial place of the ancient\nmonarch. While prosecuting his search, however, he chanced to observe\nan eagle that had alighted on a small elevation and was trying with\nhis beak and claws to break the sod. Considering this a fortunate\nomen, they explored the place and discovered the coffin of a man of\nextraordinary size, with a lance of brass and a sword lying by it.\nThese relics were conveyed to Athens amid great rejoicing, where they\nfound a resting place in the famous temple of Theseus, whose ruins are\nstill in existence.\n\nThe old historians state that the Greek poet Aeschylus lost his life\nthrough an eagle's mistaking his bald head for a rock and dropping a\ntortoise upon it in order to break the shell of his amphibious prey,\nbut which broke, instead, the poet's skull. That an eagle, proverbially\nthe keenest-sighted of created things, should mistake a man's head\nfor a stone is absurd beyond the necessity of comment. The story is\nprobably intended for an allegory, showing how stupidity can overwhelm\ngenius, or a dull criticism smash a lively poet.\n\nIn A. D. 431 there was war between the Emperor Theodosius II. and\nGenseric the Vandal, and Marcian, the general of the former, was taken\nprisoner. The unfortunate captive was doomed to death. At the place\nof execution an eagle alighted on his head and sat there some time\nundismayed by the tumult around it. Upon seeing this, and believing\nthat the captive was destined for some exalted fortune, Genseric\npardoned him and sent him home. About eighteen years afterwards\nTheodosius died, and, as his sister had married Marcian, the latter\nbecame Emperor of Constantinople.\n\nDuring the wars between the Christians and the Moors, of Spain, a\nSpanish knight engaged in combat with a gigantic Moslem. The conflict\nremained undecided for a long time, but at last the Spaniard began to\nlose ground. At this juncture an eagle, swooping from above, flew into\nthe face of the Moorish giant, and, taking advantage of this sudden and\nmiraculous intervention, the Spanish champion plunged his sword into\nthe heart of his antagonist, thus winning the battle.\n\nRudolph, count of Hapsburg, one morning was looking out of his castle\nwindow upon the surrounding country, and while thus engaged noticed an\neagle circling strangely above a certain place in the forest. Taking\nsome men at arms he proceeded to the spot, where he found a beautiful\nand high-born lady held captive by a band of robbers. He rescued her\nand afterwards married her. When a new emperor was wanted in Germany he\nobtained the election through the influence of his wife's relatives. In\nthis romantic fashion began the glory of the present reigning house of\nAustria.\n\nI have alluded to the prominence of eagles in the arms of nations and\nindividuals. The famous ensign of the Roman legions verified the text\nof Scripture when, in referring to the eagle, Job says: \"Where the\nslain are there is she,\" for the Roman bird flew over nearly the whole\nknown world and delighted in destruction and in threatening it. The\nByzantine Caesars sported a double-headed eagle to indicate that they\nwere lords of both the Eastern and the Western world. The Russians\nadopted the symbol from those princes. About four hundred years ago a\nlady, who claimed to be the heir of the Byzantine Emperor, married Ivan\nIII., Czar of Russia, who, therefore, assumed the Greek arms, which may\npossibly be restored again to Constantinople by Russian arms.\n\nThe United States chose for her emblem the same imperial and triumphant\nbird. Some have considered it as not altogether an appropriate device\nfor our republican government. Students of natural history have\nobserved that the eagle is mean and cowardly. He lives, moreover, a\nlife of rapine, plundering birds that are bolder and more industrious\nthan himself. This is rather a bad character for our national bird.\n\nThe ancients would probably be horrified at such a criticism of their\nroyal bird, and, after all, it is not surprising that they held him\nin such reverence. These people of the long ago had no books nor\nnewspapers, but they were proficient students in the book of nature.\nBy them the birds were accounted prophets, and by their varied flights\nthey foretold future events and regulated the movements and enterprises\nof nations.\n\nWe call the wisdom of birds instinct, but they considered it divine\nintelligence. Nor was it strange that they should take them for the\ninterpreters of fate, seeing that in many things the birds were wiser\nthan themselves, for they seemed to have a knowledge of the future\nthat was denied to man.\n\nWe have some idea of how these people regarded the movements of the\nbirds from one of the ancient Greek writers, who, in a play entitled\n\"The Birds,\" makes them give the following account of themselves: \"We\npoint out to man the work of each season. When the crow takes his\nflight across the Mediterranean it is seed-time--time for the pilot\nto season his timber. The kite tells you when you ought to shear your\nsheep; the swallow shows you when you ought to sell your watch-coats,\nand buy light dresses for the summer. We birds are the hinge of\neverything you do. We regulate your merchandise, your eating and\ndrinking, and your marriages.\"\n\nThis Greek play-writer probably voiced the sentiments of the majority\nof the people, who had implicit faith in what they called \"the\nprophecies of the birds;\" and it is not surprising that they endowed\nthe eagle--the king of the feathered tribes--with almost supernatural\nwisdom.\n                                        Phebe Westcott Humphrey.\n\n\n\n\nTHE SNOWDROP'S PHILOSOPHY.\n\n\n    \"I should think you'd lose heart in this frosty air,\"\n    Said a sparrow one day to a snowdrop fair.\n    \"You're almost hidden down there in the snow,\n    And I see you shiver whene'er the winds blow.\n    If I were you I wouldn't bloom\n    If I couldn't grow with the roses in June.\n    What right have they any more than you,\n    To live in the summer when skies are blue\n    And bright with sunshine the whole long day?\n    They have it easy enough, I must say;\n    But you're so meekly quiet and white,\n    You're afraid to speak up when you have the right.\"\n\n    \"But, my dear,\" said the snowdrop, \"can't you see\n    That summer can do very well without me?\n    My place is to blossom right here in the snow,\n    No matter where the roses grow.\n    It's lovely to be a summer flower,\n    But I am content to do all in my power\n    To sweeten the gloom of this wintry day,\n    And be brave if the sky is so cold and gray.\n    I cannot be helpful by being sad;\n    I have my work and that makes me glad\n    To bloom my fairest and grow my best,\n    And let kind nature do all the rest.\"\n                                        Wildea Wood.\n\n\n\n\nTHE GLADNESS OF NATURE.\n\n\n    Is this a time to be cloudy and sad,\n      When our mother Nature laughs around,\n    When even the deep blue heavens look glad,\n      And gladness breathes from the blossoming ground?\n\n    There are notes of joy from the hang-bird and wren,\n      And the gossip of swallows through all the sky;\n    The ground-squirrel gaily chirps by his den,\n      And the wilding-bee hums merrily by.\n\n    The clouds are at play in the azure space,\n      And their shadows at play on the bright green vale,\n    And here they stretch to frolic chase,\n      And there they roll on the easy gale.\n\n    There's a dance of leaves in that aspen bower;\n      There's a titter of winds in that beechen tree;\n    There's a smile on the fruit, and a smile on the flower,\n      And a laugh from the brook that runs to the sea.\n\n    And look at the broad-faced sun, how he smiles\n      On the dewy earth that smiles in his ray;\n    On the leaping waters and gay young isles--\n      Ay, look, and he'll smile thy gloom away!\n                                        William Cullen Bryant.\n\n  [Illustration: LADY'S SLIPPER.\n                 (Cypripedium hirsutum.)\n                 COPYRIGHT 1900, BY\n                 A. W. MUMFORD, CHICAGO.]\n\n\n\n\nFLOWERS AND THEIR INVITED GUESTS.\n\n\nIt must be taken for granted in this paper that the reader has such\nknowledge of the parts of the flower as could be obtained from the\npaper on \"A Typical Flower,\" printed in the June number.\n\nWhen flowers first appeared it became necessary to secure the transfer\nof the pollen grains to the stigmas. This was necessary in order that\nthe ovule might be developed into a seed containing a young plant or\nembryo. At first the currents of air were selected as the agents of\nthis pollen transfer, and the flowers were adapted to what is known\nas wind-pollination. As the wind is an inanimate agent any transfer\nby it is largely a matter of chance. In order to increase the chances\nof successful pollination it was necessary for pollen to be developed\nin enormous quantities, so that it might fall like rain. In this way\nstigmas would be reached, but at the same time an enormous amount\nof pollen would be wasted. The evergreens are good illustrations of\nwind-pollinated plants, and their showers of pollen are very familiar\nto those who live near pine forests. When these showers come down in\nunaccustomed regions they are often spoken of as \"showers of sulphur,\"\nand the local newspapers are full of accounts of the mysterious\nsubstance.\n\nIn wind-pollinated plants not only must the pollen be excessively\nabundant, but it must also be very light and dry. Sometimes the\nbuoyancy is increased by the development of wings on the pollen grains,\nas in the case of pines. This habit of pollination is found not only\namong the evergreens, but also among many important families of the\nhigher plants, as in the ordinary forest trees, the grasses, etc.\n\nWhen the higher forms appeared, however, flowers of a different\ncharacter gave evidence that a new type of pollination was being\ndevised. Instead of the old wasteful method, insects were called in to\nact as agents of the transfer. By securing an animate agent there is\na definiteness in the pollination and a saving in pollen production\nwhich is quite in contrast with the wind method. It must not be\nsupposed that all flowers have learned to use insects with equal skill,\nfor many of them may be said to be clumsy in their arrangements. On the\nother hand, certain families have reached a high degree of organization\nin this regard, and arrange for insect visits with a skill and\ncompleteness of organization which is astonishing.\n\nIn order to secure visits from insects, so that pollination may be\neffected, flowers have been compelled to do several things. In the\nfirst place, they must provide an attractive food. This has taken two\nprominent forms, namely, nectar and pollen. There are insects, such\nas butterflies, which are not only attracted by the nectar, but whose\nmouth parts have only been adapted for sucking up a liquid. There\nare other insects, however, like the bees, wasps, etc., which are\nable to take the more substantial pollen as food. Accordingly insects\nwhich visit flowers may be roughly divided into the two classes,\nnectar-feeders and pollen-feeders.\n\nIn the second place, the flower must notify the insect in some way that\nthe food is present. This is done primarily by the odors which flowers\ngive off. It must not be supposed that odors which are sensible to\nus are the only ones sensible to insects, for in general their sense\nof smell is far keener than ours. It is also probably true that the\ndisplay of color, which is so conspicuously associated with flowers, is\nan attraction to insects, although this has become somewhat doubtful\nlately by the discovery that certain insects which were thought to be\nattracted by color have proved to be color blind. At present, however,\nwe have no reason to suppose that color is not associated in some\nprominent way with the visits of insects.\n\nIt should be noticed, also, that two kinds of pollination are possible.\nThe pollen may be transferred to the stigma of its own flower, or it\nmay be carried to the stigma in some other flower, and this other\nflower may be some distance away. The former method may be called\nself-pollination, the latter cross-pollination. It seems evident that\nflowers in general have made every effort to secure cross-pollination.\nThis would seem to imply that it is a better method for some reason,\nalthough we may not be able to explain why. Apparently, however, while\nflowers in general have tried to secure cross-pollination, they have\nnot entirely abandoned the chances of self-pollination, so that if\none should fail the other may be used. In this way it will be found\nthat a great many plants have two kinds of flowers, the ordinary showy\nkind, and in addition to them inconspicuous flowers which are never\nseen except by those acquainted with their presence. For example, in\nthe common violet, in addition to those flowers with which everyone\nis familiar, others are developed which are concealed by the cluster\nof leaves, which never open, but which are able to produce very well\ndeveloped seeds.\n\nWith nectar and pollen provided as food, and with odor and color\nnotifying the insects of their presence, it remains to be noted that\nthe suitable insects are those which fly. A creeping insect is of no\navail in the work of pollination, since the pollen will be rubbed from\nits body as it crawls from one flower to the next. How the flowers ward\noff the visits of creeping insects, which are attracted as well as the\nflying ones to the food provided, will be described in a subsequent\npaper.\n\nA good illustration of the workings of insect pollination may be found\nin the sweet pea, or in any member of the pea family. The flower has a\nrough resemblance to a butterfly, whose projecting body is represented\nby a structure like the keel of a boat. In this keel is a cluster of\nstamens, and also the pistil with its stigma at the top. While lying in\nthis keel the stamens shed their pollen upon the style, which usually\nhas hairs or some sticky surface to receive it. Accordingly the style\nbears the stigma on top and masses of pollen stuck to its sides below.\nAn insect being attracted to such a flower naturally lands upon the\nkeel as upon a shelf, with its head toward the center of the flower,\nwhere the nectar is deposited. If the insect is heavy enough the\nweight of its body pushes down the keel, but the contained style is\nanchored, so that it seems to dart out, and strikes the insect's body,\nfirst with the stigma at the tip, and then glancing along rubs its\nside against the body of the insect. The insect flies away with pollen\nrubbed upon its body, and when it goes through the same performance at\nanother flower, the new stigma strikes it first and gets some of the\npollen, and then some more pollen is smeared on, and so the pollen is\ncarried from one flower to the stigma of another flower. It is easy to\nsee the effect of the weight of a heavy insect by pressing down the\nkeel with a pencil, when the style will be seen to dart forth at the\ntip.\n\nPerhaps one of the most common ways of securing pollination is that in\nwhich the pollen and stigma are not ready at the same time in the same\nflower. The pollen may be ready to shed, but the stigma is not ready to\nreceive, or the reverse may be true. This would seem very effective in\npreventing self-pollination. Illustrations of this kind are exceedingly\nnumerous, but perhaps as common a one as any is furnished by the great\nfireweed, Epilobium. It has a conspicuous purple flower, and if a patch\nof the plants be examined the flowers will be found in two conditions.\nIn one set the cluster of stamens will be found projecting straight\nout from the flower, while the style with its stigma is turned back\nout of the way under the flower. In the other set the stamens, having\nshed their pollen, are turned back behind the flower, while the style\nhas straightened up, and the mature stigma holds the same position\nthat the anthers did the day before. An insect, in visiting such a\ngroup, therefore, may fly straight towards a flower whose stamens are\nprojecting and shedding, and its body will be dusted with the pollen.\nIf it now flies to a flower which is a little older, whose stamens\nare out of the way, but whose style is projecting, its body carrying\nthe pollen will strike the stigma. In this way the pollen is very\neffectively transferred from one flower to another.\n\nIt would be impossible to give any adequate account of the subject of\ninsect-pollination in general, as it is an immense subject with an\never-increasing literature. Every kind of flower has its own particular\nway of solving the problem, so that the subject will never be completed\nuntil all flowers have been questioned and their answers obtained.\n\nAny account, however brief, should not omit mention of the orchids,\nwhich in the matter of insect-pollination have reached the highest\ndegree of organization. So detailed are their adaptations that\neach kind of flower is adapted to a particular kind of insect. The\naccounts given of the various ways in which orchids attract insects\nand secure pollination really surpass belief, until one has actually\nobserved some of the plants and their insects at work. Any greenhouse\nfurnishes abundant examples of orchids, and our illustration represents\none of the most common of our native orchids, the ordinary yellow\nLady-slipper. In most orchid flowers there is a long tubular spur, at\nthe bottom of which the nectar is found, which is to be reached by\nlong probosces, such as can be found only in moths and butterflies. In\nLady-slippers, however, there is a different arrangement. The flowers\nhave a conspicuous pouch in which the nectar is secreted, and a flap\noverhangs the opening of the pouch. Behind the flap are the two pollen\nmasses, between which is the stigmatic surface. A bee crowds itself\naway into the pouch and becomes imprisoned, and may frequently be found\nbuzzing about uneasily. The nectar is in the bottom of the pouch, and\nafter feeding the bee moves toward the opening overhung by the flap,\nand rubs itself against the stigma and then against the anthers,\nreceiving the pollen on its back. A visit to another flower will result\nin rubbing some of the pollen upon the stigma, and in receiving more\npollen for another flower.\n\nOne of the most remarkable cases of insect-pollination is that shown\nby the ordinary Yucca, which is pollinated by a small moth, the plant\nand the moth being very dependent upon one another. The flowers of\nYucca occur in very large prominent clusters, and hang like bells. In\neach bell-shaped flower there are six hanging stamens, and a central\novary ribbed lengthwise like a melon. At the tip of the ovary is a\nfunnel-shaped opening, which is the stigma. During the day the moth\nhides quietly in the recesses of the flower, but at dusk she becomes\nvery active. She travels down the stamens, and, resting on the open\nanthers, scrapes out the somewhat sticky pollen with her front legs.\nHolding the little mass of pollen she runs up on the ovary, stands\nastride of one of the furrows, pierces through the wall with her\novipositor, and deposits an egg in an ovule. After depositing several\neggs, she runs to the apex of the ovary and begins to crowd the mass\nof pollen she has collected into the funnel-like stigma. These actions\nare repeated several times, until many eggs are deposited and repeated\npollination has been effected. As a result of all this, the flower is\npollinated and seeds are formed, which develop abundant nourishment for\nthe moth larvae, whose eggs had been laid in the ovule. Just how the\ninsect learned that this behavior on her part would secure food for her\nyoung is hard to imagine.\n\nIn studying any flower there are three questions that should be asked:\n(1) How does it hinder self-pollination?; (2) How does it secure\ncross-pollination?; (3) How does it discourage the visits of unsuitable\ninsects?\n                                         John Merle Coulter.\n\n\n\n\nTHE ASTERS.\n\n\nThe mythical origin of the Asters is set forth in an old Greek story,\nwhich states that after the gods had abandoned the earth, because of\nthe crimes and dissensions that came with the Brazen Age, Astraea,\nthe goddess of innocence and purity, alone remained, endeavoring to\nredeem the degenerate race of mortals. She, too, finally left, and\nbecame known among the stars as the constellation Virgo, or the Virgin.\nAfter the wrath of Jupiter had been appeased by the destruction of the\nearth by water, Virgo, noticing that the summit of Mount Parnassus had\nalone escaped the flood, planted there a seed, whose flowers should\nreflect the azure hue of her new home and whose heart should typify the\nGolden Age that some day will come again to mankind. This plant, Virgo\ndestined as a symbol of her mission of purity and so she gave it her\nearly name, Astraea or Aster. That the plants might bloom for all races\nof men, Zephyrus, the lover of Flora, queen of the flowers, took the\nseeds and distributed them throughout the earth from polar snows to the\nsun-kissed lands of the equator. Hence it is that the Aster, in some of\nits varied forms, is found in all countries, over two hundred and fifty\nspecies being known to botanists. Although the plant is cosmopolitan,\nit is essentially an American form, one hundred and fifty of the total\nknown species belonging to North America. Of the balance, Russia claims\ntwenty, Europe ten and Canada sixty or seventy.\n\nIt seems as though Nature, after the first blush of spring, relaxed her\nefforts for a supreme endeavor towards the close of the floral season.\nThen she assumes her festal robes and the woodlands and fields become\ngorgeous with the purple of the Asters, the gold of the sunflowers and\ngolden-rod, with here and there the cardinal and blue of the lobelias.\n\nAmong all this symphony of color, no plant is more lavish of its\ncharms than the New England Aster (Aster Novae Anglae). Botanically\nconsidered, the Asters belong to the Compositae, a family of plants\nincluding from ten to twelve thousand species and characterized by\nlarge numbers of flowers, crowded together into single heads, each\nof which gives the impression of a single flower. What appear to be\npetals, are known as ray flowers and give the characteristic color, as\nthe purple, blue or white of the Aster or the yellow of the Sunflower.\nThese rays consist of flowers, whose petals have been joined together\nand spread out flat, the points of the petals usually appearing on\nthe end of the ray. In the case of the Asters, the ray flowers, which\noccur in a single row, are pistillate or have a pistil and no stamens\nand hence are capable of producing seeds. The center or disk flowers\nare tubular, yellow in color and perfect, containing both stamens and\npistils. The heads are surrounded by an involucre, having leaf-like\ntips and are variously massed or branched along the stems of the plant.\n\nWith few exceptions, the Asters are perennial, coming up each year from\nthe old underground portions and flowering in autumn. They vary in\nheight from a few inches to eight feet or more, but in the case of the\nNew England Aster, the completed growth is generally from two to seven\nor eight feet. This species has a stout and somewhat hairy stem clothed\nwith many leaves which are pointed, have entire edges and a clasping\nbase. The ray flowers in the common form are purple, but in the two\nvarieties of the species, they are rose-purple or white.\n\nThe plant derives its name from the fact that its general distribution\nin the Eastern States together with the beauty of its flowers gained\nit an early recognition among the pioneers of New England, where it\nsoon became a favorite. The statement is made that it was the chosen\nflower of John Alden and Priscilla and, on many occasions, old books,\nhanded down from revolutionary days, have been found to contain dried\nspecimens of the flowers.\n\n  [Illustration: NEW ENGLAND ASTER.\n                 (_Aster Novae-Angliae._)]\n\n  [Illustration: LATE PURPLE ASTER.\n                 (_Aster patens._)\n                 FROM \"NATURE'S GARDEN.\"\n                 COPYRIGHT 1900, BY\n                 DOUBLEDAY, PAGE & COMPANY.]\n\nThe Late Purple Aster (Aster patens) while not an uncommon form,\nis one of the most beautiful of all the Asters. The rays are long and\nshowy, in color purplish-blue or deep violet. The plants attain a\nheight of from one to three feet, the stems having rigid, bristly hairs\nand the leaves, which are entire, have a clasping base.\n\nThe Asters have been highly considered from very early times. Virgil\nstates that the flowers were used to decorate the altars of the gods\nand the ancients placed great faith in the efficacy of the leaves as a\ncharm against serpents. The American Indians have always prized these\nplants as a cure for skin diseases, calling them the bee flower, as\nthey supposed that the frequent visits of honey bees, concentrated in\nthe Asters the virtues of many other forms of flowers.\n                                        Charles S. Raddin.\n\n\n\n\nSCHOOL GARDENS.\n\n\nThere is nothing more desolate than the average surroundings of the\npublic school, and it would be cheerful news to learn that the recent\npamphlet brought out by the United States Department of Agriculture\nupon the School Gardens of the Rhine might bring about a reform in\nthis direction. Attention is called to the matter by a writer in the\nOutlook, who finds the pamphlet highly suggestive. Says the writer: \"It\nis a common experience to enter from an absolutely barren schoolyard\ninto a schoolroom decorated with botanical and natural history charts,\nand to find these charts and text-books are the only mediums used for\nteaching these branches of the natural sciences. The pamphlet above\nnamed shows the practical application of the schoolroom work. The\ngrounds are cultivated entirely by the pupils, two hours' work per week\nbeing compulsory. The result is that the community life is affected.\nThe farms and gardens are cultivated with new knowledge; the boys\nand girls work in the home grounds with greatly increased interest.\nDestructive insects and disease are watched for. The products of the\nfarms and gardens in this district bring the best prices, because\nthey are handled with care and intelligence. The first requisite for\nsuch work is such practical knowledge as will make success possible.\nThe introduction of the school garden into this country is entirely\nfeasible. It would create a new avenue of employment for the students\nin our agricultural colleges and experiment stations; it will make\nanother avenue for the use of the knowledge collected by our Department\nof Agriculture. Our township system would make a practical division\nfor the control of one agricultural supervisor and instructor.\"--The\nWestern Journal of Education.\n\n\n\n\nTHE FLICKER'S MISTAKE.\n\n\n\"My dear,\" said Mrs. Flicker, one bright day, as Mr. Flicker came\nflying home in high feather, \"we have made a mistake--a horrible\nmistake.\"\n\nNow, Mr. Flicker was a very polite bird, but he was so used to his\nwife's little peccadilloes that, though sometimes he listened patiently\nto her tale of woe, at other times he just tossed his head, absolutely\nwithout fear of what man might do to him. On this particular day the\nwarblers were whistling and flashing in and out of willow trees across\nthe stream, the wild grape and strawberry and the sweet clover made\nthe air fragrant, the sun shone out gaily from a cloudless sky, far\nand wide on the earth lay greens upon greens, and overhead stretched\nheaven's blue--a June day--why should Mr. Flicker fear? With Mrs.\nFlicker it was different; she had laid the eggs, she had patiently kept\nthem warm; she was now watching her little baby Flickers jealously;\nwhat wonder that she grew morbid and fearful, and exaggerated every\nsmall annoyance! Mr. Flicker saw now that she was trembling with\nexcitement, as she said again, \"We have made a horrible mistake.\"\n\n\"What about?\" asked he.\n\n\"Do you know,\" she said, solemnly, \"what kind of a tree this is in\nwhich we have put our nest?\"\n\n\"A very good tree, indeed,\" said Mr. Flicker, bristling, for he had\nselected the tree; \"a remarkably fine tree, with this hollow limb in\nthe midst of so much foliage.\"\n\n\"But, my dear, it is a cherry tree.\"\n\n\"So much the better,\" said the gay Mr. Flicker; \"most birds like cherry\ntrees.\"\n\n\"Yes, and boys like cherry trees!\"\n\n\"Well, and what of that?\"\n\nIt will plainly be seen that Mr. Flicker was no logician, but then, he\ncould fly far, far away toward the heavenly blue, while logicians--the\nvery wisest of them--\"on their feet must go plodding and walking.\"\n\n\"What of that!\" mocked Mrs. Flicker, nervously. \"Well, there have been\nboys in this tree this very morning, picking cherries, and I am worn\nout with fluttering and fussing and calling, to attract their attention\nfrom the nest.\"\n\nMr. Flicker thought he knew boys, and while he might be considered a\nfair and generous-minded bird in most things, it is a lamentable fact\nthat he never could quite understand why Nature in her infinite wisdom\nhad thought it necessary to produce anything so incongruous as a boy.\nBut, as has been said, Mr. Flicker's reasoning powers were limited.\nHe was sober now--boys always sobered him. But after all, he had the\nspirit and digestion of a bird, and even the fussy Mrs. Flicker fussed\nonly in a bird-like manner. So they talked it over and hoped for the\nbest, especially as the babies showed signs of the greatest precocity\nand bade fair to fly away in a few days and be safe from harm.\n\nThe next day as Mr. Flicker was returning from his favorite ant-hill,\nhe was startled by the frightened screams of his wife, and for some\ntime after he reached the nest she could do nothing but scream and\ncry and hop distractedly from branch to branch. Mr. Flicker followed\nher about and tried to comfort her, though he felt that this was no\nimaginary grievance.\n\n\"What is it, my love; what is it?\" he begged softly.\n\n\"Go look in the nest,\" said she.\n\nHe flew to the nest, and then his cries and shrieks rose above hers,\nand they hopped from branch to branch like demented bird-folk. Mr.\nFlicker, when quite himself, was gay and trustful and debonair, but\nhe was, besides all this, a proud and natural parent, and when he\nfound that one of his precious babies was missing, his grief, though\nloud, was sincere. Mrs. Flicker told him how a dreadful, hideous boy,\nwith frightful sprawling legs and arms had climbed the tree to pick\ncherries--how he had found the nest in spite of all that she could\ndo--how he had pushed his long arm down into the hollow limb and taken\nout and examined one baby after another, and had then run off with\none, putting the others back in the nest.\n\n\"Oh, help! help!\" suddenly cried poor Mrs. Flicker, \"here they come\nagain! They will take all the others. What shall we do?\"\n\nMr. Flicker looked, and, true enough, there they were, coming over\nthe hill through the orchard--two boys, and another. The agonized\ncries sounded through all the trees, coming not so much from the\nFlickers themselves as from the friendly cat-birds and robins and cedar\nwaxwings and sparrows who, forgetting the slights they had received\nfrom the Flickers, joined in a noble effort to attract the attention\nof the intruders and keep them away from the cherry tree. On they\ncame, however, paying not the slightest heed to the medley of cries\nabout them--two boys and a gray insignificant person who seemed to be\ndirecting the cruel expedition. Straight to the cherry tree they made\ntheir way, up went the sprawling boy, and before the crazy birds could\ntell what had happened, the three were making their way back through\nthe orchard again. The cat-birds followed them and the others kept up\ntheir cries for some time afterward.\n\nAt first Mrs. Flicker refused to return to her empty nest, but as night\ncame on she grew calmer and decided not to abandon her home. She knew\nshe could lay more eggs and raise another family, but she would not\nbelieve that there could ever again be such brave and beautiful babies\nas her stolen ones. As she at last came to the nest, she heard a soft\nlittle familiar call, and peeping in--lo! there were the babies just\nas she had left them except that the stolen one had been returned and\nlay cuddled safe and warm beside the others! There was a happy Flicker\nfamily in the old cherry tree that night.\n\nNot long after this the cherries disappeared, and the baby Flickers,\none by one, took their flying lessons and flew away on their own strong\nwings. Then the nest was molested no more. And when the banks of the\ncreek were bright with golden-rod and asters, and the milkweed pods\nwere bursting, the Flickers started on their southern journey. Of\ncourse the next summer is a long way off, and no one can tell what may\nhappen. But it might be that even if the Flickers cannot forgive, they\ncan forget--which is the better, after all, if you can do but one. And\nwhen the April days come round again, remembering only the fragrant\nair and the fat ant-hills of the orchard, they may return again to the\ncherry tree. Who knows?\n                                        Nell Kimberly McElhone.\n\n\n\n\nTIGER-LILIES.\n\n\n    I like not lady-slippers,\n    Nor yet the sweet-pea blossoms,\n    Nor yet the flaky roses,\n      Red, or white as snow;\n    I like the chaliced lilies,\n    The heavy Eastern lilies,\n    The gorgeous tiger-lilies,\n      That in our gardens grow.\n\n    For they are tall and slender;\n    Their mouths are dashed with carmine;\n    And when the wind sweeps by them,\n      On their emerald stalks\n    They bend so proud and graceful--\n    They are Circassian women,\n    The favorites of the Sultan,\n      Adown our garden walks!\n\n    And when the rain is falling,\n    I sit beside the window\n    And watch them glow and glisten,\n      How they burn and glow!\n    O, for the burning lilies,\n    The tender Eastern lilies,\n    The gorgeous tiger-lilies,\n      That in our garden grow!\n                                        Thomas Bailey Aldrich.\n\n\nFLOWERS IN THE CRANNIED WALL.\n\n    Flower in the crannied wall,\n    I pluck you out of the crannies;\n    Hold you here, root and all, in my hand,\n    Little flower--but if I could understand\n    What you are, root and all, and all in all,\n    I should know what God and man is.\n                                        Alfred Tennyson.\n\n\n\n\n  [Illustration: WILD YELLOW OR CANADIAN LILY.\n                 (_Lilium Canadense._)\n                 FROM \"NATURE'S GARDEN.\"\n                 COPYRIGHT 1900, BY\n                 DOUBLEDAY, PAGE & COMPANY.]\n\nTHE WILD YELLOW LILY.\n\n\nAmong our common wild flowers, that quickly attract the attention of\nthe observer is the Yellow Lily (Lilium canadense). Its home is in the\nswamps, the wet meadows and fields of Canada and the United States,\neast of the Missouri river. It is also called the Canada, the Field and\nthe Meadow Lily.\n\nThis plant, with about forty-five sister species--all beautiful,\nbelongs to the genus Lilium. All are natives of the Northern Hemisphere\nand are found distributed around the world. About sixteen species are\nnatives of the United States. The flowers vary in color. Some are red,\nothers white or yellow and some are more or less mottled.\n\nNo plants are more frequently mentioned in Ancient Myths and by the\nclassical poets. Though the white lily (Lilium candidum) was, even\nbefore the time of Homer, known as a garden flower, yet the earliest\ndescriptions of the lilies found in cultivation were written by Gerard\nin the year 1597.\n\nIt is thought by some that the \"lilies of the field,\" spoken of in the\nseventh chapter of Matthew, are the red lily described by Pliny. The\nwhite lilies have long been considered the symbol of purity and were\noften used by the great masters in the pictures of the Annunciation,\nin which they were represented as held by the Angel Gabriel. How\nappropriate is the white lily, with its glossy and pure white petals\nfor the decoration of Easter time!\n\nThe slender stalk of the Yellow Lily arises from a scaly bulbous and\nthickened underground stem, growing to a height of from two to five\nfeet. The leaves are narrow and lance-shaped, from two to six inches\nin length and usually attached in whorls of from three to eight. Each\nstalk bears from one to fifteen flowers, the ground color of which is\nyellow or reddish with brownish spots toward the base of each division,\nwhich are six in number and are spreading and gracefully arched. The\nflowers, appearing in June, July and August, are nodding and vary in\nlength from two to four inches. The fruit pods are oblong, large, and\nbear numerous seeds.\n\nClosely related to the plant of our illustration, and at times closely\nresembling it, is the beautiful Turk's Cap Lily (Lilium superbum).\nThis species is wonderfully prolific in the production of flowers,\nsometimes bearing forty or more on a single stalk. It is one of the\ntallest of the lilies, and frequently the marshes of the eastern states\nare transformed by its presence into striking masses of color, orange,\norange-yellow or red.\n\n\n\n\nWHAT DO WE OWE THE BIRDS?\n\n\nThe answer to this question needs to be presented from two distinctly\ndifferent points of view--the commercial and the esthetic. In\npresenting the commercial point of view it will be necessary to ignore\nthe use of any bird as an article of food, because we are now speaking\nof the living bird. Likewise it will be necessary to ignore the side\nwhich might be presented by the millinery trade, because that, too, has\nto do with the dead bird. We shall have occasion to present the general\nsubject of the demands of fashion at a later time. This paper, then, is\nconcerned only with our debt to the living bird.\n\nIn the June number of Birds and Nature some general remarks were made\nabout what the birds eat. In this paper it will be necessary to go more\ninto particulars in order to get clearly before us just wherein our\ndebt lies.\n\nFirst of all, we owe our physical comfort to the birds, because they\ncheck the increase in insect life. The mosquito and gnat, the horse\nfly and common housefly would soon rival the plagues of Egypt were the\nbirds to disappear. If anyone doubts this let him go into the Cascade\nmountains where the scarcity of the birds gives great liberties to the\n\"deer flies.\" And they take all liberties without so much as a \"thank\nyou, I guess I will!\"\n\nWe owe our fruits largely to the birds. This statement anyone may prove\nby simple experiment. First drive the birds from your garden because\nyou think they are eating the buds and blossoms, instead of the insects\nwhich sting the buds. You will be rewarded with a scanty and stunted\nfruit crop. Next conclude that you won't get fruit anyway, and so let\nthe birds do as they please. You will be pretty sure to harvest a\nfairly good crop at least. Lastly, encourage the birds to visit your\ngarden and orchard in their northward passage, as well as during the\nsummer season. Build nesting boxes for the swallows, wrens and martins.\nPlant a mulberry tree for the fruit-loving robins and cat-birds. Now\nyour fruit and garden are returning an abundant yield of the best\ngrade. If the birds take a little for themselves have they not earned\nit? There is enough and to spare.\n\nWe owe corn and other grains largely to the birds, because they help\nto keep in check the insects which attack the cereals. During the\ngrasshopper plagues very many birds feed upon the grasshoppers which\ndo not usually touch grasshoppers. Probably chief among our grain\nfield helpers is the Bronzed Grackle, who is so much in disfavor for\nthe ravages he makes upon those same fields when the corn is in the\nroasting-ear stage. But he earns far more than he eats. The birds of\nprey destroy vast numbers of the little rodents which help themselves\ntoo freely to the planted grains.\n\nWe owe the preservation of the remnant of our forests, and all our\ntrees and bushes largely to the birds who eat the insects which attack\nthe trees and bushes. The woodpeckers are after the insect which is\ndestroying the tree, not after the life of the tree.\n\nSpace would fail us to speak of the debt we owe to all the birds. There\nare the scavenger water-birds--gulls, terns and the like--the scavenger\nland birds--the vultures--the ducks, geese and swans, who check the\nencroachments of vegetable life upon our streams, ponds and lakes; the\nherons, cranes, rails, coots, gallinules and shore-birds, which feed\nupon the water and mud-inhabiting insects and other small animals; the\nsparrows and grouse, which destroy vast quantities of the seeds of\nharmful plants. In short, the only birds about whose usefulness there\nis any doubt are the English Sparrow, Crow, Blue Jay and four of the\nhawks. These are far too few for us to condemn all birds.\n\nWe cannot afford to overlook the esthetic side of this question. How\nmuch of our pleasure and happiness do we owe the birds directly for\ntheir intensely busy lives, the neatness and beauty of their dress,\nthe perpetual joy of their songs? Can you imagine a world without\nbirds? Are the returned warmth and the green vegetation all that make\nthe summer months more pleasant than the winter season? Rob the tropics\nof their birds and you rob them of their heart. Pasadena, California,\nis a bird paradise, but take away its mocking birds, its orioles, its\ntowhees, its gorgeous humming birds, and the many other birds which\nenliven every lawn, and you have taken away one of its chief charms.\n\nBut it is not simply that we are entertained by the birds, nor even\nthat we are pleased with their neatness and beauty. Where their lives\ntouch ours we feel an uplifting influence. We are better fitted for the\nservice which it is our privilege to render to the world by the touch\nof the bird life. Our horizon is broadened beyond the self-interest,\nthe egoistic, to the altruistic conception of life. We cannot live in\nthe presence of these creatures so full of life without being spurred\nto more earnest effort ourselves. When we fail to see in the world of\nnature about us what it is our privilege to see we are losing that much\nof life. Let us open our eyes to all the influences that may shape our\nlives toward best living.\n                                        Lynds Jones.\n\n\n\n\nTO THE VESPER BIRD.\n\n\n    Sweet bird of twilight wake in me\n    Bright memories of melody\n    Outpoured from every nesting-tree\n        At early morning gray.\n    O sing that I may ponder on\n    The songs away with noontide gone,\n    Ere shadows troop across the lawn\n        And voices die away.\n    Long have I waited wistfully;\n    And lest thy gift unheeded be,\n    Lo, now my gardens are for thee,\n        Thou truant all the day!\n                                        Frank English.\n\n\n\n\nTHE VESPER SPARROW.\n\n\nIn the fields, the pastures and along the roadsides of the Eastern\nUnited States and the British Provinces may be found the unobtrusive\nVesper Sparrow (Poocaetes gramineus). It is also known by other\nnames such as the Bay-winged Bunting or Sparrow, the Grass-Finch and\nsometimes, though incorrectly, it is called the Field Sparrow. The\nlatter name should only be applied to one of the Chipping Sparrows\n(Spizella pusilla).\n\nThe characteristics of the male and the female are the same. The\nexposed part of the outer and the tip of the second tail feathers are\nwhite. This character is very marked as the bird alights. The feathers\nof the underside of the body are usually yellowish-white and the tops\nof the wings are a light chestnut-brown. It does not seem to shun one's\npresence, but will run along the side of the road, a short distance\nahead, occasionally stopping for observation.\n\nThe Vesper Sparrow builds its nest on the ground without reference\nto any special plant protection except that of grass and other low\nherbage. The eggs are usually four in number, the general color of\nwhich is light gray marked, in a variable manner, by dull reddish-brown\nspots or blotches.\n\nWhen frightened from her nest the mother-bird will endeavor to attract\nthe attention of the intruder by slowly flying away and occasionally\nfeigning injury by falling.\n\nMr. John Burroughs, in his little book, \"Wake Robin,\" writes in an\nadmirable manner of the song and habits of this little bird. He says:\n\"Have you heard the song of the Field-Sparrow? If you have lived in a\npastoral country, with broad upland pastures, you could hardly have\nmissed him. Wilson, I believe, calls him the Grass-Finch, and was\nevidently unacquainted with his powers of song. The two white lateral\nquills of his tail, and his habit of running and skulking a few yards\nin advance of you as you walk through the fields, are sufficient to\nidentify him. Not in meadows or orchards, but in high, breezy pasture\ngrounds, will you look for him. His song is most noticeable after\nsundown, when other birds are silent, for which reason he has been\naptly called the Vesper Sparrow. The farmer following his team from the\nfield at dusk catches his sweetest strain. His song is not so brisk and\nvaried as that of the Song-Sparrow, being softer and wilder, sweeter\nand more plaintive. Add the best parts of the lay of the latter to\nthe sweet vibrating chant of the Wood Sparrow (Spizella pusilla), and\nyou have the evening hymn of the Vesper-bird--the poet of the plain\nunadorned pastures. Go to those broad, smooth, uplying fields, where\nthe cattle and sheep are grazing, and sit down on one of the warm,\nclean stones, and listen to this song. On every side, near and remote,\nfrom out the short grass which the herds are cropping, the strain\nrises. Two or three long, silver notes of rest and peace, ending in\nsome subdued trills or quavers, constitute each separate song. Often\nyou will catch only one or two of the bars, the breeze having blown the\nminor part away. Such unambitious, unconscious melody! It is one of\nthe most characteristic sounds in Nature. The grass, the stones, the\nstubble, the furrow, the quiet herds, and the warm twilight among the\nhills, are all subtly expressed in this song; this is what they are at\nleast capable of.\"\n\n  [Illustration: VESPER SPARROW.\n                 (Poocaetes gramineus.)\n                 1\/2 Life-size.\n                 FROM COL. CHI. ACAD. SCIENCES.\n                 COPYRIGHT 1900, BY\n                 A. W. MUMFORD, CHICAGO.]\n\n\n\n\nTHE WORSHIP OF NATURE.\n\n\n    The ocean looketh up to heaven\n      As 'twere a living thing;\n    The homage of its waves is given\n      In ceaseless worshiping.\n\n    They kneel upon the sloping sand,\n      As bends the human knee,\n    A beautiful and tireless band,\n      The priesthood of the sea!\n\n    They pour the glittering treasures out\n      Which in the deep have birth,\n    And chant their awful hymns about\n      The watching hills of earth.\n\n    The green earth sends its incense up\n      From every mountain-shrine,\n    From every flower and dewy cup\n      That greeteth the sunshine.\n\n    The mists are lifted from the rills,\n      Like the white wing of prayer:\n    They lean above the ancient hills\n      As doing homage there.\n\n    The forest-tops are lowly cast\n      O'er breezy hill and glen,\n    As if a prayerful spirit pass'd\n      On nature as on men.\n\n    The clouds weep o'er the fallen world,\n      E'en as repentant love;\n    Ere, to the blessed breeze unfurl'd,\n      They fade in light above.\n\n    The sky is as a temple's arch,\n      The blue and wavy air\n    Is glorious with the spirit-march\n      Of messengers at prayer.\n\n    The gentle moon, the kindling sun,\n      The many stars are given,\n    As shrines to burn earth's incense on,\n      The altar-fires of Heaven!\n                                        John Greenleaf Whittier.\n\n\n\n\nBIRD-STUDY.\n\n\nTo be intimate with Nature is as important to the investigator as the\nability to technically classify the things found therein.\n\nIn this connection we copy, by permission, the words of Olive Thorne\nMiller, from the \"School Room Methods and Nature Study:\"\n\n     \"Recognizing a bird on sight or hearing, knowing his nest and\n     eggs, when he arrives in the spring, and when he departs in the\n     fall, does not by any means imply that one is acquainted with the\n     bird himself. All these facts are easily acquired; they have been\n     set down in the books these many years.\n\n     But whoso really desires to know the little being so beautifully\n     enshrined; to see his home ways with his mate and little ones; to\n     find out his personal habits; his likes and dislikes; his tastes;\n     his disposition; in a word his personality, for him is something\n     very different from book study. He must go into the field and\n     observe for himself; for well as we may know our common birds\n     by sight, glibly as we can explain their anatomy, give their\n     scientific names, and their place in our classification, of their\n     lives and habits we are in almost total ignorance.\n\n     This is a field of inquiry as fascinating as it is fresh and\n     unexplored. Nothing but the greed of collecting and the passion\n     for classifying, could so long have blinded men to the charm of\n     studying life instead of death, the individual instead of the\n     skin. And this is the beautiful work left for us to do, to make\n     the world acquainted with the lives of our little brothers in\n     feathers.\n\n     For this work are needed, patience that knows no fatigue,\n     accuracy of observation, enthusiasm that scorns such trifles\n     as wet feet, torn garments, insect bites and stings, burning\n     sun or blistering wind, and above all--lacking which all else\n     is useless--truthfulness that will report correctly, without\n     exaggeration or coloring. To one possessing these qualities a\n     whole world of delight is open.\n\n     Nor is this world so difficult to enter as it seems at first.\n     Science--whose help is needed--has, to be sure, shrouded itself\n     in technicalities, buried its facts under scientific terms, and\n     hidden its names in a dead language. But all this, which perhaps\n     was necessary, can be got over. With a little courage, and some\n     perseverance, this bristling array of difficulties may be broken\n     through, and the charming goldfinch be as lovely and bewitching\n     under the name of Spinus tristis, as of thistle bird, or yellow\n     bird.\n\n     How shall we go to work? This is the first question always.\n     Let me give you a few hints: Some fine morning dress yourself\n     in modest-hued array, dull olive of medium shade best; discard\n     all conspicuous details of costume; take off ribbons and\n     veils, and all fluttering things; reject the spring hat with\n     its eccentricities of flowers, fruits, feathers, or general\n     fluffiness, and put on a plain shade hat, as near the color of\n     the dress as possible; leave parasol, bag or basket and book all\n     at home. Slip into a flat pocket on the outside of your gown or\n     coat, a small note book with sharpened pencil attached to it,\n     and suspend by narrow ribbon around the neck, so that it will\n     hang above the waist and be ready for instant use, an opera glass\n     without its case. On your left arm carry a light folding camp\n     stool--and start out.\n\n     Bid adieu to your friends, and go alone, for the temple of Nature\n     can never be entered in crowds, nor even in pairs. Turn your\n     steps to the best place you know of; an old orchard, a grove with\n     underbrush near a house, a ravine, a swamp, or the edge of woods.\n     Walk slowly and leisurely along, with little noise of footsteps,\n     and without swinging arms.\n\n     Arrived in your chosen spot look sharply around for the flitting\n     forms of the birds. When you see one, stop at once; quietly slip\n     your stool off your arm and sit upon it, with as little motion as\n     possible. If you place it against a tree trunk, to furnish a back,\n     you can be comfortable in that one position an hour without moving.\n\n     Now slowly raise your opera glass to your eyes, adjust the focus\n     to bring the bird clearly before you, and proceed to study him.\n     First you want his description so that you can name him. Look\n     very carefully at him, his size and shape, his coloring above and\n     below, his peculiar markings, the shape of his tail at the end,\n     and the color and shape of the beak. As you settle one point write\n     it in your note book, which you have quietly drawn out of its\n     pocket.\n\n     His description recorded, proceed to note his manners; whether\n     quiet or restless, whether he jerks his tail, or his head; walks\n     or hops. See what he is doing; picking up insects, digging them\n     from bark or ground, seeking them among flowers or leaves, or\n     whether he is eating seeds from the grass or weeds. Sit there as\n     long as that bird is in sight, and note down everything he does,\n     even his calls and his song as it sounds to you.\n\n     When you go home take your manual and look for a description that\n     matches yours. This is where troubles begin, not only the obscure\n     scientific terms, and the Latin names, but the knowing where in\n     that big book to start. You will be helped by observing what the\n     bird ate. If he hammered on the bark and picked his food from tree\n     trunk or limb, look among the woodpeckers; if he flew out, made a\n     turn or two and back to his perch seek him among the fly-catchers;\n     if he was eating seeds, look among the finches; and so on.\n\n     When by a little work you have passed this Rubicon--where so many\n     turn back discouraged--you will reap your reward, success. Having\n     persevered, and named your bird without help, you will feel a new\n     pleasure in his acquaintance, as if he belonged to you, and you\n     will never forget him.\n\n     Then go out and make acquaintance with another. You will find\n     him easier to identify, and as you will become familiar with its\n     idiosyncrasies the manual will lose its terrors for you.\n\n     Of course all this trouble will be avoided if you begin with the\n     study of scientific ornithology. But in that case you are in\n     danger of becoming absorbed in the science, and getting to care\n     more for the dry bones and the dead skin, than for the living\n     bird, and thus adding one more to the ornithologists, and taking\n     one from the students of life.\"\n\n\n\n\nTHE OREGON JUNCO.\n\n\nResidents of the Atlantic, Middle, Southern and Middle Western States\nare, doubtless, well acquainted with the slate-colored Junco. This\nlittle feathered specimen is more familiarly known as \"Snowbird.\"\n\nThe Oregon Junco (\"Junco hyemalis var. Oregonus\") is a sub-species, and\nis found throughout the Pacific coast region from California to Sitka.\nIt is, by no means, confined exclusively to Oregon. Its darkest-hued\nplumage makes the bird very conspicuous when the ground is covered with\na soft and spotless mantle of snow.\n\nThe sooty-black head, flesh-colored bill and white breast, sharply\ncontrast in color. On the sides are pinkish colored feathers; the back\nis rufous-brown and the two outer tail feathers pure white, showing\nwhen the bird flies. In western Oregon it is a winter visitant,\narriving with the first cool days of autumn.\n\nAs winter approaches these snowbirds become more plentiful, hopping\nabout in the small bushes in quest of food. A great deal of pleasure\nand interest may be found in studying these birds, especially when the\nground is covered with snow. By casting bread crumbs on the snow, the\nlittle fellows flock around, and are easily tamed. In winter their\nonly note is a sort of chirp, sometimes uttered several times in quick\nsuccession when alarmed. With the warm days of spring they begin their\nsong, sometimes many singing at once, and soon the majority disappear\nto a higher altitude to breed.\n\nThe Oregon Junco builds its nest in hollows in the ground under low\nbushes. The nest is constructed flush with the surface and in holes\namong the roots of bushes and trees, and under woodpiles. Usually,\nthe nest is made of dry grasses rather loosely placed together, with\na lining of cowhair, and contains four and sometimes five handsome\ngreenish-white eggs, spotted and wreathed with purple.--J. Mayne\nBaltimore.\n\n       *       *       *       *       *\n\nOlive Thorne Miller, in her fascinating little book, \"The First Book\nof Birds,\" speaking of how the birds work for us, says: \"Chickadees\nlike to eat the eggs of cankerworms; and for a single meal one of\nthese tiny birds will eat two hundred and fifty eggs, and he will take\nseveral meals a day. Now, cankerworms destroy our apples. When they get\ninto an orchard in force, it looks, as Miss Merriam says, as if it had\nbeen burned over. Robins, cat-birds, and shrikes, and several others,\nlike to eat cutworms, which destroy grass and other plants. As many as\nthree hundred of them have been found in the stomach of a robin, of\ncourse for one meal. Ants are very troublesome in many ways, and three\nthousand of them have been taken from the stomach of one flicker.\"\n\nWhy kill these birds that are so useful to us and so beautify nature?\nMany others are just as useful and some that occasionally do damage\namply repay us in other ways.\n\n\n\n\n  [Illustration: THE CALICO BASS.\n                 (Pomoxys sparoides.)\n                 COPYRIGHT 1900, BY\n                 A. W. MUMFORD, CHICAGO.]\n\nTHE CALICO BASS.\n\n\nThe Calico Bass (Pomoxys sparoides) is so called because of the mottled\nand variegated coloring of the body and fins. It is also called the\nStrawberry Bass, the Grass Bass, the Bitter Head, the Lamp-lighter and\nthe Barfish.\n\nIt is abundant in all the lakes and ponds of the region of the Great\nLakes and the upper Mississippi river, where it shows a preference for\nquiet, cool and clear water and grass covered bottoms.\n\nThe Calico Bass is closely related to the Crappie (Pomoxys annularis)\nof the lower Mississippi valley. It is, however, seldom seen where the\nCrappie is abundant, as the latter prefers muddy sloughs and bayous and\nis not found as far north as the former.\n\nThe body of the Calico Bass is elongated, is much compressed and of a\nbright, silvery olive-green color. The sides and fins are mottled with\na darker green or brownish-green, the blotches being gathered into\nirregular bunches. The vertical fins also have markings in the form of\na network surrounding paler spots. The mouth is large and oblique. The\nusual length of the adult is about twelve inches.\n\nThe Calico Bass obtains its food largely from the lower forms of animal\nlife, such as crustaceans, worms and insects.\n\nIt is said that \"from the fact that it thrives well in slow-moving\nwaters, it deserves the favorable consideration of owners of large\nmill ponds, where there is a steady flow of water, as it requires very\nlittle care, except the first planting of it in waters suitable to its\nnature. It is not averse to an occasional minnow, but is not regarded\nas peculiarly aggressive, though provided by nature with an armature\nthat enables it to defend itself against all comers.\"\n\n       *       *       *       *       *\n\n    \"Forthwith the sounds and seas, each creek and bay\n    With fry innumerable swarm, and shoals\n    Of fish, that with their fins and shining scales\n    Glide under the green wave, in sculls that oft\n    Bank the mid sea: part single, or with mate,\n    Graze, the seaweed their pasture, and through groves\n    Of coral stray, or, sporting with quick glance,\n    Show to the sun their waved coats, dropt with gold.\"\n                                        --Milton, \"Paradise Lost.\"\n\n\n\n\nTHE GROWTH AND VARIATION OF FISH.\n\n\nHow can you tell the age of a fish? This question is often asked and\njust so often is the answer unsatisfactory.\n\nA fish is a cold-blooded animal; that is, his temperature is nearly\nthe same as that of the water in which he lives. His circulation is\nsluggish and his appetite is a variable quantity. He has the capacity\nto take in large quantities of food at one meal and properly assimilate\nit; on the other hand he is able to fast for weeks at a time. He has\nhis own notions about eating, and it is quite impossible to induce him\nto change them, and all this has considerable influence on his rate\nof growth. It is out of the question to expect him to grow when he is\nfasting; on the other hand he must draw on the fat he has stored up in\nhis body to furnish him energy for his muscular movements and to carry\non the ordinary functions of nutrition. The fish here has an advantage\nover the warm-blooded animals, for he does not need to generate heat\nto keep his body at a constant temperature. The amount of food often\neaten at one time is quite remarkable. I remember once of taking nearly\none pound of sunfish from the stomach of a Large-mouthed Black Bass.\nThis does not indicate that a bass must eat such meals three times each\nday, it only shows his capacity to make use of a large quantity of food\nwhen it is abundant and his stomach feels the need of it. A trout is a\ngood feeder; his stomach and mouth are large, much in size like that\nof the black bass. From experiments conducted at Neosho, Missouri, by\nMr. Page, he found that a young trout did best on a daily ration of\nsolid food equal to about seventy-five per cent of its weight. On this\namount the trout would reach an average length of six inches in one\nyear. The average amount of solid food consumed daily by a man is from\none and one-half to two per cent of his weight, or more than twice that\nconsumed by our active, growing young trout. As mentioned before, the\ntrout is relieved from generating heat to keep his body at a constant\ntemperature, and at one usually much higher than the medium in which he\nlives.\n\nAs an example of the ability of fishes to go for some time without\neating, we need only mention our Pacific salmon. There are five species\nof these large fishes on the Pacific coast. In the early spring (April)\nmany of the largest species, the Chinook, start up the Columbia river\nfor the purpose of spawning. They reach the headwaters of the Columbia\nin Idaho early in September. During this journey they eat nothing. We\nknow they do not eat, for of the thousands caught each year for the\ncanneries none are found with food in their stomachs; besides, this\norgan has become much shrunken. If they did eat on this journey there\nwould not, I believe, be enough animal and plant life in the Columbia\nto furnish each salmon with more than one meal. Now many of them make\nthe journey against a strong current for more than one thousand miles,\nand reach an elevation of about eight thousand feet above the sea.\nWhen they leave the ocean they are in excellent condition, by the time\nthey have reached their journey's end they are thin and haggard, their\nvitality is so reduced that soon after spawning they die--literally\ndie of starvation. Their eggs hatch during the winter. By the next\nwinter the young salmon are from four to five inches in length, and by\nthe following fall or early winter they go to the sea, having reached\nan average length of about ten inches. After leaving the fresh water,\nwhich only afforded them a scant subsistence for nearly two years,\nthe generous ocean gives them plenty of sea room and an abundance of\nfood, which in a few years prepares them to repeat the long journey\nof their parents. We are, in case of most fishes, ignorant of their\nlife histories, as we are of the salmon's. We know the average rate of\ngrowth of the salmon for the first two years, but we know nothing more\nof them until they return to fresh water to spawn.\n\nI mentioned that trout in the Neosho Fish Hatchery grew, under\nfavorable circumstances, to a length of six inches in one year. It must\nnot be taken for granted, however, that trout six inches in length are\none year old. In their native streams, in cooler regions, they will not\noften attain this length in two or more years.\n\nIn general we do not find large fishes in small bodies of water;\nneither do we find the fish in our small aquaria growing at an alarming\nrate. The fish disdains to outgrow his surroundings; he may feel his\nimportance, and consider himself in many ways superior to the other\nfishes in the pond with him, but he will not permit himself to grow to\nsuch a size as to make the question of securing a living a difficult or\nirksome one.\n\nFishes spawn but once each year, and the time and length of the\nspawning season is not the same for all species. With some species\nthe season is short, while with others it may extend through three\nor more months. In the latter case those produced the first part of\nthe spawning season are at the end of six months much larger than\nthose which appear at the close. It is therefore evident that the\nfishes of any single brood by the end of the year will vary greatly in\nsize, often to such an extent that the broods of one season cannot be\nseparated from those of the preceding season; especially is this true\nof our smaller species. Mr. Moenkhaus, in making a study of the two\nspecies of darters, the Sand Darter or \"Johnny,\" and the Log Perch,\nfound by collecting a large, miscellaneous lot of these fishes, from\na given locality, that it was possible to separate them in groups\naccording to size of one, two or three years of age, which indicates a\nquite uniform rate of growth for these two species.\n\nMr. Voris collected a miscellaneous lot of over five hundred specimens\nof the Blunt-nosed Minnow from Turkey Lake in Indiana, varying from\none to three inches in length. These, when separated as far as\npossible, according to sizes, did not fall into distinct groups of\ndifferent ages. In my own collecting and study of fresh water fishes\nI have always been impressed with the difficulty of recognizing the\nage of fishes, except that the smallest taken was considered to be\nthe product of the preceding spawning season. Here is an interesting\nquestion to which but little attention has been given. Any one will\nfind much interest in studying the rate of growth of fishes under\ndifferent circumstances. We know that the rate of growth is in no way\nuniform, as is the case with our warm-blooded animals. We also know\nthat among fishes there is no uniform adult size, as there is in case\nof warm-blooded animals (birds and mammals). In general, we cannot\nspeak of a fish as being full-grown; at the same time there seems to be\na limit of size for each species in each body of water, beyond which\nonly a few go. The Chinook salmon we mentioned reach an average weight\nof twenty to thirty pounds, although individuals are occasionally taken\nof forty, sixty or even one hundred pounds weight. These large fishes\nare by no means common, the other species of salmon never attain the\nsize of the Chinook.\n\nThere is an interesting family of fishes in our fresh waters known as\nMinnows; these fishes are too small and too full of bones to become\na favorite for the table. They are the most helpless of all our\nfresh water fishes, being soft, and, as they are slow swimmers, they\nbecome an easy prey to larger fishes, and form a large part of their\nfood supply. They have been constantly driven into smaller streams\nand shallow water, until they have become exceedingly dwarfed. Their\nonly use in the economy of fish life seems to be to assimilate small\norganisms, converting them into such shape that they can be taken by\nthe larger fishes. Now the Minnows of all the United States east of the\nRockies are small and, except in case of a few species, they are less\nthan six inches in length. The predatory fishes, such as the Sunfishes\nand Perches, Pike and Pickerel, are their worst enemies. In the Rocky\nMountains there are none of these fishes, and many minnows there grow\nto a length of two feet or more. The only enemy of importance they\nhave is the trout, but the minnow finds a more congenial climate in\nthe larger bodies of water, too warm for the trout. The struggle for\nexistence has been a severe one, especially so in our streams where\nspecies of fish are the more numerous. It has greatly limited the\ngrowth of most species beyond an average size, and is in many places\nresponsible for the fact that often a species may become dwarfed in\ncertain bodies of water. In the Salmon river in Idaho it was not an\nuncommon thing to catch trout of three or four pounds weight. In the\nsmaller tributaries and in the smaller mountain lakes it was unusual\nto catch one weighing over one-half pound, the average being less than\none-fourth pound. I have no doubt that many of those from the small\nlakes of one-half pound were as old as the large ones taken from the\nSalmon river.\n\nFish eat and grow very irregularly. The average size of individuals,\nwhich we would ordinarily call adults, for some species, is different\nin different bodies of water. Their growth is influenced largely by the\nsize and depth of the body of water in which they live, also by its\ntemperature and the amount of suitable food it contains. The value or\nextent of each of these influences is imperfectly understood.\n\nThe forms of fishes are very numerous. Some are extremely long and\nslender, as many of the species of Eels, Pipe-fishes and the like,\nwhile others are extremely short, like Sunfish of the ocean. Others,\nlike the Trunk Fishes, are nearly equal in all dimensions. The average\nform and the one which best suits our idea of a fish, is the Black\nBass, or other fishes of similar pattern. To know the advantages of\nthese forms one must study the fishes in their native element. The\npeculiar forms which many species take are the most noticeable in those\nfound in the tropics. The struggle for existence there is the most\nsevere, and it seems as if each species had labored to take on some\npeculiar form which would assist most in its preservation. In this\nrespect color also plays an important factor. It is in the tropics and\namong the many species of corals that we find the most highly-colored\nfishes.\n\nMany fishes have the power to change their color, and this they can do\nin a very short time. The flounders are a peculiar family, the young\nwhen born are symmetrical. Early in life they take on the habit of\ntheir parents and lie on one side, the eye on the underside disdains to\nlook downwards and so begins to move toward the other side. The bones\nof the head suit themselves to this change and soon our flounder has\nboth eyes on the same side of the head. The upper side is colored much\nto resemble sand, and the under side becomes nearly white. The flounder\nprotects himself by covering his body, except the eyes, with sand.\nFlounders live on sandy bottoms, some in shallow water, while others\nare found in deepest parts of the ocean. If flounders are placed in an\naquarium and arranged so the light can fall on the under side of their\nbodies, this, too, becomes dark, much like the other side.\n\nIt is interesting to study the habits of fishes in a small aquarium,\nand to especially notice their ability to change color, and how rapidly\nthey do it. So many persons seem to be saturated with the idea that an\naquarium must have in it one or more gold fish. This seems to me to be\na mistake when our streams contain so many species suitable for the\naquarium which are far more handsome than the gold fish, and which,\nif you give them half a chance, will teach you something of interest.\nMr. Ford, of Berwyn, Illinois, has a small aquarium, in his house, in\nwhich he keeps from fifteen to twenty-six species of native fishes.\nAmong these are several species of Darters, the most beautifully\ncolored and the most interesting of all our fresh-water forms. Then\nthere are Minnows, Suckers, Catfishes, Sunfishes, the Pike, Mud\nMinnow, Top Minnow, and so on. To one who would know fishes, any one\nof these species is more desirable than gold fish. The study of fishes\nin an aquarium, such as the one possessed by Mr. Ford, is extremely\ninteresting. They will teach you much about their habits, besides\ngiving you many lessons showing their ability to change color and adapt\nthemselves to their surroundings.\n\n  [Illustration: A MOUNTAIN LAKE.\n                 CHICAGO:\n                 A. W. MUMFORD PUBLISHER.]\n\nThe Blind Fishes, which inhabit caves in this country, are very\ninteresting. They have lost their color, if they ever had any, being\nwhite. In many the eyes have become so degenerated as to be entirely of\nno service when the fish is in the light. The head is furnished with\ntactile organs, which enables them to feel their way in the dark. In\nfact, they are well adapted for the life they lead. Dr. Eigenmann tells\nus that Blind Fishes were not accidentally swept into caves or driven\nthere by their enemies, \"but entered them deliberately and avoided\ncoming out into the light.\" In other words, they preferred \"darkness\nrather than light.\" Having simplified eyes and highly developed sense\norgans, they were able to live in the dark. The many ages they have\nlived in the caves has better fitted them for their existence in total\ndarkness. The Blind Fishes were not always blind, but have become so\nbecause of their own preferences.\n\nThe readers are, if they will only study fishes, sure to find them\nextremely interesting. There are a wonderful variety of fishes, each\nwell adapted for the life it leads. You will find them in the brooks,\ncreeks, rivers and lakes or ocean, wherever you happen to be, and you\nare sure to be highly repaid for all the study or attention you may\ngive them.\n                                        Seth E. Meek.\n\n\n\n\nTHE ORIGIN OF THE FISH.\n\nA BIRD-FISH STORY.\n\n\nOnce upon a time, and that was in the long ago, there lived a Koko-bird\nalong the forest shores of the Boozoo river. I am not quite certain in\nwhat country this river is but I believe it is somewhere in Gazazuland.\nIt does not matter much where it is or was, but of one thing I am\nabsolutely certain, and that is that the river did exist, else how\ncould the bird have lived along its shores? Now this bird was quite\nbeautiful, could sing quite well, and could fly quite gracefully;\naccomplishments which all of the other birds of the community willingly\nadmitted, but the Koko-bird was very boastful. In a loud, arrogant\nvoice he would proclaim himself the handsomest, the most musical and\nthe most graceful of all the feathered tribe. At first his neighbors\ntried to ignore these boasts, hoping that the Koko-bird would in time\nlearn better manners, but he did not; on the contrary, he became more\nboastful every day, in fact every minute, so that his presence became\nalmost unbearable, causing great mental irritation and a feeling of\nnausea in those who were obliged to listen to him. A bird committee\nwas therefore appointed to obtain an audience with the Golden Eagle,\nwho was then the ruler of all the birds, and petition his majesty to\nconvene the bird council in order that suitable punishment might be\nmeted out to the boastful Koko. The very next day the meeting was\ncalled by special and very swift bird messengers. The Koko-bird was\nbrought a prisoner before the king of the birds, the bird council and a\nvast concourse of birds from far and near, who had come to witness the\ntrial. In a measured and stentorian voice the king asked the following\nquestions of the culprit:\n\n\"Are you the handsomest of birds?\"\n\n\"I am,\" replied the Koko-bird.\n\n\"Are you the best singer among birds?\"\n\n\"I am,\" again replied the boastful bird.\n\n\"Are you the most graceful and the highest flyer among birds?\"\n\n\"I am,\" replied the braggart for the third time.\n\nThe king of birds then flapped his right wing and there came forth\nthe gorgeous bird of paradise, with the beautiful and wonderful\ntail feathers and crown, at the sight of which the members of the\nbird council individually and collectively flapped their wings in\nadmiration. The eagle once more turned to the Koko-bird and in a\nterrible voice demanded:\n\n\"Are you still the handsomest among birds? Heed well your answer.\"\n\nThe Koko-bird gave one sidelong squint at the beautiful bird and said:\n\n\"I am,\" in a very indifferent tone of voice; whereat the assembled\nbirds were astonished.\n\nThe king of birds then flapped his left wing and there came forth a\nnightingale which began to sing so sweetly that some of the listeners\nfell from their perches out of sheer ecstasy and they would have been\nhurt by the fall had they not caught themselves in the air by means of\ntheir wings. Even the king of birds was greatly moved, for he was seen\nto brush a tear from his right eye before he turned to the Koko-bird\nand spoke in a thunderous voice:\n\n\"You have heard this marvelous singer. Are you still the best vocalist\namong birds? Heed well your answer.\"\n\nThe Koko-bird merely yawned and said, \"I am,\" and again the birds were\ngreatly astonished.\n\nThe king of birds now nodded his head and there arose out of the\nmultitude of birds a blue crane, whose home was near the Gingago\nriver in farthest India. Its wings moved in even, silent, graceful\nundulations. It gradually rose higher and higher. All of the birds,\nwith the exception of the Koko-bird, watched it spellbound until it\nappeared a mere speck in the distance. The Koko-bird gave one glance\nat the high flyer, then curled one foot up in his feathers, shook his\nhead, closed his eyes and dozed peacefully.\n\nFor the third time the king of birds turned toward the Koko-bird and\nspoke in a voice even more terrible than on previous occasions.\n\n\"Are you the most graceful and highest flyer among birds? Answer me\nquick and heed well your answer.\"\n\nThe Koko-bird merely opened one eye and said sleepily, \"I am,\" whereat\nthe vast concourse of birds were astonished for the third time. Some\nopened their bills in amazement at such unheard-of audacity; others\nhooted and screamed, clamorously, demanding that the wicked Koko be\nseverely dealt with.\n\nThe king of birds now flapped both wings to demand silence and\nattention. Those who had their bills open closed them with a snap and\nthe clamorous ones became perfectly quiet. The king then turned toward\nthe council and spoke in an even, stentorian voice, as follows:\n\n\"Gentlemen birds of the council. The prisoner, otherwise known as the\nKoko-bird, stands before you, self-accused and self-condemned. I commit\nhim to your judgment. Let his punishment be as severe as the bird law\nwill permit.\"\n\nThe bird council then adjourned to the large council tree where they\nremained in closed session for one hour. They then returned to the bird\nassembly and the leader thus addressed the king of birds:\n\n\"Your majesty, the grand council of this bird assemblage, convened by\nyou, find the prisoner guilty and fix upon the following punishment:\n\n\"1. Because of his boast that he is the handsomest of birds his tail\nand wing feathers shall be pulled out and all other feathers shall be\nshorn close.\n\n\"2. Because of his boast that he is the best singer among birds he\nshall be struck dumb.\n\n\"3. Because of his boast that he is the most graceful and highest\nflyer among birds, he shall forever be prevented from moving in the\natmosphere in which we move.\"\n\nNo sooner had the speaker finished when the handsome feathers of the\nKoko-bird disappeared. This so surprised Koko that he actually awoke\nfrom his slumber. He tried to say, \"Well! well! what has happened,\" but\ncould not utter a sound. The king of birds now flew away, which was\nthe signal for the adjournment of the assembly, for, you see, their\nwork was done. All of the birds began to depart for their respective\nhome trees, but before doing so each one said something sarcastic\nor insulting, hoping to humiliate the forsaken culprit. This merely\nannoyed Koko a little. He tried to retaliate by boldly declaring that\nhe was the handsomest, the most musical and the most graceful of all\nbirds, as he had often done before, but he could not, for had not the\ncouncil decreed that he be \"struck dumb?\" He tried to catch the little\nsparrow, who, by his derisive twitterings, annoyed him even more than\nthe vulture, by his coarse insults, but his wings would not carry him.\nHe merely succeeded in falling into the Boozoo river.\n\n\"Now I shall be drowned,\" he thought, for you remember he could not\ntalk. But behold! he did not drown; by means of his featherless wings\nand tail he could swim beautifully on top of the water as well as in\nit. His body feathers being gone, they did not become water-soaked and\ngive him the snuffles, a severe cold, or perhaps pneumonia. Koko was\nastonished to find that water, which he had formerly feared, was not\nbad at all. He could drink whenever he wanted to without having to\nstand at the edge of the river bank, as he formerly did and get his\nfeet all mud. In time his wings and feet became fins and the feather\nstumps became scales; in other words, the erstwhile boastful Koko\nbecame a fish.\n\nThe Koko-fish (for so must the Koko-bird be called now), would have\nlived in the Boozoo river peacefully had not an owl noticed him one day.\n\n\"O, ho! What is this?\" said the wise one, blinking both eyes. \"Such a\ncreature was never seen before. I must investigate closer.\" So saying\nhe flew to a lower limb and looked hard at Koko. Koko, in turn, stared\nat the owl out of one eye; he did not wink or blink but simply stared\nand said nothing.\n\n\"By my wisdom,\" said the owl, \"if this isn't Koko. I know him by his\neye. Well! well! what may not happen next?\"\n\nThat night the wise owl repaired in all haste to Urtzook in Tartary,\nwhere the bird council was again in session, and reported his wonderful\nfind, whereat the king of birds and all present were greatly astonished\nfor the fourth time. They expressed a fear that Koko would some day\nleave his watery element and return to them. The king turned to the\nwise one and said:\n\n\"How know you that the creature which you beheld in the limpid waters\nof the Boozoo is the erstwhile Koko? and let me remind you, heed well\nyour answer.\"\n\n\"Uh! how do I know, indeed,\" replied the owl, \"by his eye, by his cold\nstare.\"\n\n\"Our enemy, the Boa, also hath an eye with a cold stare; is he\ntherefore also a metamorphosed Koko? Again heed well your answer,\"\ncontinued the king in a somewhat sarcastic tone.\n\nThe owl winked and blinked, adjusted his spectacles and made answer.\n\n\"The undeniable evidence that the creature referred to is the\nmetamorphosed Koko-bird is as follows: All the wise birds of your\nkingdom, including your humble servant, have searched far and near and\nhave found no Koko-bird. We, ahem, I, have found this creature with the\ncold stare; therefore, it follows that this staring, scaly, wingless\nand featherless creature must be the metamorphosed Koko-bird, for how\ncould it be otherwise?\"\n\nAll doubt vanished at such display of wisdom and the king of birds at\nonce dispatched the Flipflap bird to the banks of the Boozoo river,\ninstructing him to keep a sharp lookout on the now scaly Koko and to\ndrive him back into the water should he attempt to leave it. Even\nto this day the guardian of fish may be seen perched upon a stump,\nclosely watching the rippling waters. As soon as one of the finny tribe\napproaches near the surface he makes a dash for it, compelling it to\nreturn with all speed. For his faithful services the Flipflap bird has\nbeen dubbed Kingfisher, which is a much nicer name.\n                                        Albert Schneider.\n\n\n\n\n  [Illustration: BANANA.\n                 2\/3 Natural-size.\n                 COPYRIGHT 1900. BY\n                 A. W. MUMFORD, CHICAGO.]\n\nTHE BANANA.\n\n\nTall and stately, capped by a gracefully arched group of leaves and a\nnodding spike of numerous flowers, the banana is noted alike for its\nbeauty, its nourishing fruit and its many qualities of economic value.\nSome one has said, \"The banana is the queen among ornamental herbs, and\nthe household god of the laborer's cottage.\"\n\nTo him who dwells in the tropics the banana is as wheat and rice are to\nthe inhabitants of more temperate regions.\n\nNearly all the authorities on the distribution of plants believe the\nbanana to be a native of Asia and that it was not found in the New\nWorld previous to its introduction by man. An argument which strongly\nsupports this theory is the lack of native names for the plant in\nMexico and in South America. It was mentioned by the early Latin and\nGreek writers, but seems to have been unknown to the ancient Egyptians.\n\nBotanical authorities quite generally agree that the numerous varieties\nof our common banana are produced from Musa sapientum. The generic\nname, Musa, is by some claimed to have its origin in the Arabic word\nMoux, their name for this group of plants. Others claim that the name\nwas given in honor of Antonius Musa, a physician who cured Augustus\nCaesar of a disease that had been considered incurable. The specific\nname has its origin in the myth that the groves of the banana plant\nwere used by the sages or wise men (sapientes) of India for their\ncouncils and for rest, they also partaking of the fruit.\n\nAnother species of the genus Musa is called paradisiaca from the\nmythical story that it was the forbidden fruit of Paradise. The common\nname of this species is the plantain and by many it is considered\nthe parent of the numerous varieties in cultivation in Asia and the\nadjacent islands and also in the New World. Many eminent authorities\nbelieve that both the banana and the plantain, with the numerous\nvarieties of each, are the same species.\n\nThe banana plant is herbaceous and dies down to the ground after\nfruiting. The true stem is underground and perennial, sending up new\nshoots each season, which grow rapidly and in a few months bear ripened\nfruit.\n\nThe stalk that bears the flowers grows to a height of from fifteen to\ntwenty feet and is surrounded by the sheathing bases of the leaves. The\nflower cluster or spike is terminal and from two to four feet in length\nand nodding. The oblong leaves are dark green in color, from five to\nten feet in length, and from one to two feet in width. The beautifully\narching leaves and the pendulous cluster of flowers or fruits forms an\nattractive foliage and makes the plant a noted ornament for the garden.\n\nThe many varieties of both the banana and plantain, which vary in\ntaste, color, form and size, are very widely distributed throughout the\nworld, being usually found in a zone bounded by 38 degrees North and 38\ndegrees South latitude. It is said that a single plant will produce, on\nthe average, in one year three bunches of fruit weighing fifty or more\npounds. The amount of labor required in its cultivation is very small,\nespecially in the older plantations.\n\nThe number of bananas on a single stalk of the ordinary variety varies\nfrom about one hundred to two hundred, with an average of about one\nhundred and thirty. When a plantation is fully developed growth is so\nrapid and so constant that ripe bunches of fruit may be gathered each\nweek.\n\nFor the best results a good, fertile soil is required. It is\ninteresting to note that but little moisture is needed, for the plants\nattract water, either from the air or the waters deep under ground, and\nthe surface of the ground is always moist even in a time of unusual\ndrought.\n\nThe stalk that bears the heavy bunch of fruit, occasionally weighing as\nmuch as eighty pounds, may be easily cut down by a single stroke of a\nscythe or a machete.\n\nUnder cultivation the fruit seldom produces perfect seeds, but if\ndeveloped in a state of nature it is said that they will mature and\nthat many varieties are produced.\n\nThe banana is frequently used in coffee plantations to make the\nnecessary shade for the young coffee plants and at the same time it\nyields an income while the planter is waiting for the production of the\ncoffee berry.\n\nNatives of the tropics have found the leaves a cool and useful\nthatching for the roofs of their huts.\n\nThe unripe fruits contain a large percentage of starch and the pulp,\nwhen dried and reduced to a powder, makes an excellent and nutritious\nflour or meal. The ripe fruit contains about twenty per cent of starch,\nthe remainder having been changed into sugar during the process of\nripening. Even intoxicating drinks are made by the Africans from the\njuice, known as \"banana beer\" and \"banana wine.\" It is not the fruit\nalone that is used as food, as also the pith, the top of the flower\ncluster and the young and tender shoots delight the taste and nourish\nthe body.\n\nThe economic value of the fibers of some of the species was known to\nthe Chinese and Japanese from remote times. The fiber obtained from\nthe leaves of both the banana and the plantain are valuable in the\nmanufacture of paper and fabrics of various kinds.\n\nOne of the most interesting and valuable of the species of Musa is the\nWild Plantain (Musa textilis) of the Philippine Islands. The fiber\nobtained from this plant is the Manila or Cebu hemp of commerce, which\nis used, in this country, mainly for the manufacture of binding twines,\ncordage and mats. In France the finer fibers are quite extensively used\nfor the manufacture of fine veils, crapes, hats, delicate underclothing\nand many other articles of apparel. The natives of the Philippines call\nthis fiber Abaca. It is called Manila because most of the fiber is\nexported from the seaport of that name. We are told that \"Manila hemp\nbegan to be used extensively in this country, in Salem and Boston, in\n1824 to 1827.\"\n\nProbably the most peculiar of all the species is the Chinese banana\n(Musa Cavendishii), which is extensively cultivated in China and\nthroughout the South Sea Islands. It is a dwarf, the plant seldom\nattaining a height of more than six feet. It is robust and yields a\ngreat harvest of fruit, a single bunch bearing from two hundred to\nthree hundred bananas, the flavor of which is excellent.\n\nThe opposite of the Chinese form is the Abyssinian (Musa ensete), which\nmay be called the giant plantain. It attains a height of thirty or more\nfeet and the leaves are sometimes twenty feet long by three feet wide.\nThe fruit is pulpless and dry, but the inner part of the stalk and the\nyoung stalks are boiled and used for food. It is without doubt the most\nhandsome species of this wonderfully useful and beautiful group of\nplants.\n                                        William Kerr Higley.\n\n\n\n\n  +----------------------------------------------------------------- +\n  | Transcriber's Note:                                              |\n  |                                                                  |\n  | Minor typographical errors have been corrected without note.     |\n  |                                                                  |\n  | Punctuation and spelling were made consistent when a predominant |\n  | form was found in this book; otherwise they were not changed.    |\n  |                                                                  |\n  | Ambiguous hyphens at the ends of lines were retained.            |\n  |                                                                  |\n  | Italicized words are surrounded by underline characters,         |\n  | _like this_.                                                     |\n  |                                                                  |\n  | The Contents table was added by the transcriber.                 |\n  +------------------------------------------------------------------+\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Birds and Nature, Vol. VIII, No. 2,\nSeptember 1900, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nAll rights reserved.  \nPublished in the United States by Ten Speed Press, an imprint of the Crown Publishing Group, a division of Random House, Inc., New York.  \nwww.crownpublishing.com  \nwww.tenspeed.com\n\nTen Speed Press and the Ten Speed Press colophon are registered trademarks of Random House, Inc.\n\nCopyright \u00a9 2011 by Sunset Publishing Corporation\n\nFor photograph copyright information, see Credits.\n\nLibrary of Congress Cataloging-in-Publication Data is on file with the publisher\n\neISBN: 978-1-60774-059-9\n\nv3.1\n\n# Contents\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\nINTRODUCTION\n\nThe One-Block Garden Warm-Season Planting Plan and Timeline\n\nThe One-Block Garden Cool-Season Planting Plan and Timeline\n\nESSENTIAL GARDENING GUIDELINES\n\nBefore You Plant\n\nHow to Plant\n\nHow to Water\n\nHow to Fertilize\n\n**SUMMER**\n\nTHE STORY OF OUR SUMMER FEAST\n\nGARDEN\n\nSummer Garden Plan\n\nPROJECTS\n\nHow to Raise Honeybees (and Make Honey)\n\nHow to Make Cheese (Part I)\n\nFresh Chive Cheese\n\nOregano Queso Blanco\n\nHow to Make Beer\n\nSummer Wheat Beer\n\nBelgian Abbey Ale\n\nHow to Raise Chickens\n\nRECIPES\n\nSkillet-Roasted Edamame\n\nDeviled Cucumber Cups\n\nTempura Squash Blossoms\n\nTomato and Herb Salad with Fresh Chive Cheese\n\nWatermelon-Chile Salad\n\nPurslane-Cucumber Salad\n\nCr\u00e8me Fra\u00eeche\n\nCorn Soup with Roasted Poblanos and Zucchini Blossoms\n\n'Trombetta' Zucchini and Its Flowers\n\nOven-Baked Steak Fries with Green Chile Mayonnaise\n\nRosemary Potatoes Anna\n\nGrilled Summer Succotash\n\nPattypan Squash with Eggs\n\nWhole-Wheat Pizzas\n\nFresh Chive Cheese and 'Sweet Million' Tomato Sauce\n\nCorn and Zucchini\n\nCherry Tomato, Ricotta, Mint, and Chile\n\nPotato, Onion, and Gouda\n\n'Sweet Million' Tomato Sauce\n\nSummer Lemongrass Custards\n\nWatermelon, Cantaloupe, or Honeydew Sorbet\n\nPeppermint-Lemongrass Tisanes\n\n**Summer Preserved**\n\nCanned Heirloom Tomatoes\n\nDried Herbs\n\nDried Chiles\n\nDried Corn\n\nRoasted Poblanos for the Freezer\n\nSlow-Roasted Tomatoes for the Freezer\n\n**FALL**\n\nTHE STORY OF OUR FALL FEAST\n\nGARDEN\n\nFall Garden Plan\n\nPROJECTS\n\nHow to Make Wine\n\nSyrah\n\nChardonnay\n\nHow to Make Vinegar\n\nHow to Make Cheese (Part II)\n\nRicotta\n\nFromage Blanc\n\nFeta\n\nGouda\n\nHow to Grow Mushrooms\n\nHow to Make Olive Oil\n\nRECIPES\n\nFresh Apple Cider and Applesauce\n\nPickled Cocktail Mushrooms and Onions\n\nQuinoa Bites with Walnut Romesco\n\nCreamy Flageolet Dip with Red Pepper Sticks\n\nRoasted Spiced Butternut Squash Seeds\n\nRoasted Tomato-Fennel Soup\n\nHerb Vegetable Broth\n\nButternut Squash and 'Cipollini' Onion Soup\n\nGrilled Radicchio and Fennel Salad with Apples and Toasted Walnuts\n\n'Scarlet Emperor' Rago\u00fbt\n\nStuffed Poblanos with Red Pepper Sauce\n\nQuinoa Huaraches with Egg and Parsley Salad\n\nButternut Squash Gnocchi with Chard and Sage Brown Butter\n\nAncho Chile\u2013Sauced Noodles with Shiitakes and Butternut Squash\n\nMexican Skillet Eggs\n\nCreamy Scrambled Eggs with Oyster Mushrooms\n\nWhole-Wheat Rosemary Shortbreads\n\nWalnut-Honey Crisps\n\nHoney Ice Cream\n\nLemon-Thyme Ice Cream Sandwiches\n\nApple Cheese Puff\n\nButternut Squash Compote with Honey and Toasted Walnuts\n\nHomemade Butter and Buttermilk\n\nLast-Minute Pineapple Guava Preserves\n\n**WINTER**\n\nTHE STORY OF OUR WINTER FEAST\n\nGARDEN\n\nWinter Garden Plan\n\nPROJECTS\n\nHow to Make Salt\n\nHow to Make Escargots (from Your Own Garden Snails)\n\nGarlic Butter Escargots\n\nHow to Make Mead (Honey Wine)\n\nRECIPES\n\nArugula and Red Butterhead Lettuce Salad with Tangerines and Hard-Cooked Eggs\n\nWheat Berry Ciabatta\n\nEgg Cloud ( _Nuvolone_ )\n\nFeatherlight Pancakes\n\nEgg and Gouda Crepes\n\nKale Colcannon\n\nVegetable Shepherd's Pie\n\nRicotta Manicotti\n\nWinter Vegetable Chowder\n\nWheat Berry \"Risotto\" with Roasted Tomatoes and Broccoli Rabe\n\nBraised Winter Greens with Preserved Lemons and Red Chile\n\nCaramelized Tangerine and Ricotta Tart\n\nTangerine Honey Flan\n\nSticky Chewy Tangerine Marmalade\n\nPreserved Lemons\n\n**SPRING**\n\nTHE STORY OF OUR SPRING FEAST\n\nGARDEN\n\nSpring Garden Plan\n\nPROJECTS\n\nHow to Own (or Co-Own) a Dairy Cow\n\nHow to Make Tea\n\nRECIPES\n\nRadishes, Fresh Homemade Butter, and Salt\n\nFavas and Ricotta on Buttermilk Crackers\n\nGouda Goug\u00e8res\n\nCarrot and Beet Chips\n\nGarden Borscht\n\nMesclun Salad with Spring Beets and Dill\n\nNasturtium Salad with Omelet Ribbons\n\nWhole-Leaf Radish and Herb Salad\n\nGrilled Carrot Salad\n\nRoasted Beets and Tops with Tarragon\n\nFresh Pickled Beets (and Eggs)\n\nFive Ways with Fresh Eggs\n\nFava Leaf and Parsley Quiche\n\nStrawberries with Fromage Blanc and Lemon Honey\n\nStrawberry Crepes\n\nStrawberry Cr\u00e8me Fra\u00eeche Sherbet\n\nStrawberry Lemonade\n\n**A Spring Tea Party (because we could)**\n\nScone Tarts: Lemon, Strawberry, and Tangerine\n\nClotted Cream\n\nLemon Curd\n\nStrawberry Oven Jam\n\nTea Sandwich Trio\n\nEgg Nasturtium\n\nGouda Arugula\n\nGreen Onion\u2013Parsley\n\nWhole-Wheat Honey Sandwich Bread\n\nEPILOGUE\n\nACKNOWLEDGMENTS\n\nTHE ONE-BLOCK TEAM\n\nAPPENDIX: Regional Planting and Harvesting Timelines for Your Own One-Block Feasts\n\nINDEX\n\nPhoto Credits\n\n_The garden at Sunset_\n\n# Introduction\n\nIt's the start of the day at Sunset magazine. On the way to the coffee machine, I pass Kimberley and Margaret. Kimberley is an imaging specialist; she color-corrects the cascade of photographs that streams into our office every month. Margaret is our production coordinator, adjusting every detail of the layouts until the pages are ready for publication. But right now, Kimberley and Margaret are beekeepers. They're both in bulky white jumpsuits, with veiled helmets tucked under their arms, heading out to inspect our hives.\n\nAs for me, I'm carrying a white bucket filled with leftover spinach, fennel trimmings, and chile seeds. With my coffee in the other hand, I go out to see the girls\u2014the squawking hens in Sunset's test garden who dive-bomb the bucket as soon as I set it down. I collect a couple of still-warm eggs from the nest box and go back inside, past the egg sign-up sheet with its list of colleagues waiting their turn for eggs, past the big crocks of vinegar with their bracingly tangy smell, and into the kitchen to put away the eggs. It makes me happy to know that our under-the-counter fridge holds wheels of Gouda that we made a few weeks ago from the milk of our own cow, that the latest batch of beer is percolating in a corner, and that the shelves in the pantry are lined with bottles of our own olive oil and jars of honey from the hives.\n\nAlthough it may sound like we're running a farm here, we aren't. Our main business is putting out a monthly lifestyle magazine about the West. But the Sunset office is a little different from other magazine offices.\n\nIt was designed by its longtime former owners, Bill and Mel Lane, to be what Bill liked to call \"a laboratory of Western living.\" On its five acres, landscape architect Thomas Church planted extensive gardens that start with the cacti of the Southwest and arc over to the ferns and firs of Washington and Oregon, with California redwoods in between. The Lanes installed a test garden, too, for trialing the newest flowers, fruits, and vegetables, so we could tell our readers which varieties to count on. The building itself, constructed in the early 1950s by architect Cliff May, embodied the then-new idea of outdoor living, with sliding glass doors to bring in the view, tile floors that flowed from inner rooms to outer patios, and courtyards with shady overhangs and comfortable seating. Many of the home stories in the magazine over the years have focused on bringing the outdoors in and vice versa. We've built patios in our parking lots, put together an adobe oven, and made all kinds of garden structures\u2014from tool sheds to benches to raised beds\u2014and published the plans. In the sand volleyball court we used to have, we once pit-roasted a pig, Hawaiian style. It ended in a visit from the fire department, but in a laboratory, not all experiments go smoothly.\n\nIn the kitchen, we made a point of celebrating Western foods like Meyer lemons and abalone, starting when some of them were barely known outside the West. (One of my favorite bits of display type, from a 1970 article: \"It's endlessly versatile. It's a dip, a sauce, a dressing, a spread. It's _guacamole_.\") We've been fans of local eating for a long time. So we were thrilled when, several years ago, it became a bona fide movement: People everywhere were seeking out ingredients close to home, grown by farmers they could meet or at least learn about. It was civilized and humane, healthy in the largest sense of the word, and it produced great-tasting food.\n\nAs we saw people (even restaurants) setting food-gathering boundaries for themselves, experimenting with 150-mile, 100-mile, and even 50-mile diets, we looked around at our living laboratory\u2014which occupies a full city block\u2014and thought, why don't we try a one-block diet? It would take local eating to its logical conclusion, since you can't get much more local than your own back yard. Starting in May, we'd grow everything we needed in our organic garden for an end-of-summer feast in August. (Our planting plans and timelines are included here.)\n\nTo make it interesting, we'd raise more than just plants. We would figure out how to make absolutely everything we needed for a well-rounded meal, from protein to cooking fat to seasonings to sweetener to even wine\u2014the kinds of ingredients you would typically buy at the store, even if you're a dedicated locavore.\n\nAlmost before we knew it, we had formed Team Chicken, Team Olive Oil, and Team Bee to explore exactly how to produce the eggs, oil, and honey that we wanted. Several other teams materialized\u2014you'll read about them in the first chapter\u2014and we became like an old-fashioned neighborhood of do-it-yourselfers heading for the best block party ever.\n\nAfter a little more thought, we decided that the block should be reduced to a medium- to large-size backyard, so that it would be easier for our readers to replicate all that we were planning. We began describing our adventures on a blog, called One-Block Diet (http:\/\/oneblockdiet.sunset.com), where we recorded what worked and what didn't, the practical tips we learned from experts, and the fun we had.\n\nBy the time we sat down to our summer feast in August, the project had assumed a larger life. Various strands of it were still evolving: for instance, the Syrah we'd made was nowhere near ready to drink (it was still aging); our cheeses were tasty, but we wanted to explore making different and more ambitious kinds; and we thought it would be fascinating to turn our honey into mead. We found ourselves not only continuing the projects we'd started, but coming up with new ones and planning menus for the seasons ahead. And pretty soon it all added up to this book, with extra recipes developed along the way.\n\nHowever, we ended up doing our research in fits and starts, skipping to winter and then to fall and back to spring. For the sake of seasonal continuity and flow, and so that our plan is helpful to anyone following it, I've condensed some of our experiences and adjusted dates where it made sense.\n\nWhen we started the project, we knew nothing about the time-honored skills we were trying to explore, apart from gardening and cooking. We had no idea how to make wine, or keep bees, or milk a cow. We are people with full-time office jobs and home lives, and yet we found that all of these things are completely possible and extremely rewarding.\n\nAlthough this book is as comprehensive as we could possibly make it, with recipes that use only what we grew or raised ourselves, we designed it to be a resource that anyone\u2014backyard garden or no\u2014could dip into and still really enjoy. All the recipes work with store-bought or farmers' market ingredients, for instance. And even if you choose just one project, like homemade vinegar, you will feel the undeniable pride that is part of what makes food from scratch so rewarding.\n\nBut if you want to dive in deeper, say, and collaborate with your neighbors so that one of you raises chickens and another bees, and maybe the woman on the corner grows tomatoes and corn, and your buddy across the street makes the beer\u2014then just imagine how great your block parties could be.\n\n**HOW TO USE THIS BOOK**\n\nAlthough there's a lot of advice and information in this book, we designed it to be used by all kinds of readers and meet different kinds of needs.\n\nMake a Fast Recipe\n\nSay it's a Monday night after work, and you want a quick recipe for that nice farmers' market produce you just bought. You'll find dozens of choices here\u2014actually, more than half of the recipes in the book take less than an hour.\n\nPlant a Vegetable or Two\n\nOr, try growing some of that produce yourself. We've provided planting and harvesting advice for every fruit or vegetable in the book, in the sections titled \"Garden.\" We also have a short section on gardening basics.\n\nGrow a Whole Feast\u2014or All Four\n\nIf you want to grow and cook an entire seasonal feast, like we did, you can decide which menu sounds appealing (every chapter begins with one) and then follow that menu's planting plan, which is included in the garden section. The recipes we cooked for our feasts\u2014plus more dishes we developed using that season's ingredients\u2014are at the end of the chapter. Or if you're an expert gardener who enjoys a challenge, check out the warm-season and cool-season planting plans that will take you through a whole year of one-block feasts\u2014and some really interesting plants, too.\n\nMake Food from Scratch\n\nMaybe you love the idea of from-scratch cooking and want to make as much of your own food as possible. Our project guides can help you make cheese, olive oil, vinegar, beer (from wheat, barley, and hops), and wine; and even salt from seawater. These are true adventures, and will change the way you look at food. You'll be surprised by how easy some of them are.\n\nRaise Food-Producing Animals\n\nOnce, \"A Chicken in Every Pot\" was a countrywide dream; now it seems to be a chicken (or flock) in every backyard. If you're curious about getting laying hens, read our guide on this page. We've also included guides to raising honeybees and a dairy cow.\n\nOr Just Read\n\nAnd, if you just want to plop on the sofa and read a story, we wrote one for you. The chapter introductions describe our successes and failures as we gardened, cooked, and otherwise worked our way toward making the book you have in your hands.\n\nLastly, if you see something you especially like in these pages (or omissions or errors), or want to share your stories about raising your own food, let us know! Come comment on our facebook page for this book, www.facebook.com\/SunsetMagazine, or stop by our blog about this project, <http:\/\/oneblockdiet.sunset.com>, whose readers have helped shape and improve everything we've done.\n\n## THE ONE BLOCK GARDEN:  \n _Warm-Season Planting Plan_\n\nThe illustrations and timelines on this page and immediately following show every plant we raised for our one-block feasts, with dimensions indicating how much space we allotted for each. Some plants from the warm-season garden (like the citrus trees, most of the herbs, the fennel, and the butternut squash) remain in the cool-season garden plan, either because we used them during the cool season or because we relied on them year-round.\n\nPlanting and Harvesting Timeline for Northern California\n\nThe timeline below shows when we planted and harvested all the crops for our four seasonal one-block feasts in Menlo Park, California; the information on specific crops in the garden section of each chapter reflects the general guidelines for each plant. If you live outside Northern California, turn to this page for other region-specific planting and harvesting timelines. To learn how to produce your own transplants (seedlings) from seed, see Starting Plants from Seed.\n\n10 feet\n\nKey\n\nColors indicate the seasonal feasts that use these crops and in which chapter you'll find growing instructions for them:\n\n  = summer (see this page)\n\n  = fall (see this page)\n\n  = winter (see this page)\n\n  = spring (see this page)\n\n  = seed\n\n  = transplant\n\n  = plant (gallon-can size or larger)\n\n  = bare-root\n\n  = start any time of year\n\n### MARCH\n\n**PLANT**\n\n**25.** Chives\n\n**21.** Lemon tree (any time in mild climates, but best in spring after last frost)\n\n**19.** Marjoram (and year-round)\n\n**7.** Nasturtiums\n\n**18.** Oregano (and year-round)\n\n**20.** Parsley (and for fall and spring feasts; succession sow; replant in cool season)\n\n**10.** Peppermint\n\n**3.** **11.** Potatoes (from seed potatoes; plant in towers or trenches)\n\n**14.** Rosemary (and year-round)\n\n**15.** Tangerine tree\n\n**16.** Tarragon, French\n\n**17.** Thyme (and year-round)\n\n**PROJECTS**\n\nBEER (for spring feast)\n\nBEES (through April)\n\n### APRIL\n\n**PLANT**\n\n**26.** Basil (in batches, every 2 weeks through August)\n\n**2.** Beans (through August)\n\n**5.** Chiles (and for fall feast; through May)\n\n**30.** Corn (in batches, every 2 weeks through August)\n\n**22.** Cucumbers\n\n**24.** Edamame (in batches, every 2 weeks through May)\n\n**13.** Hops (through May)\n\n**23.** Lemongrass\n\n**12.** Pattypan squash, 'Bennings Green Tint' (in May)\n\n**12.** Pattypan squash, 'Sunburst' (in May)\n\n**5.** Peppers (through August)\n\n**8.** Pineapple guava\n\n**6.** Purslane\n\n**4.** Quinoa\n\n**28.** Tomatoes (through May)\n\n**32.** Zucchini\n\n**PROJECTS**\n\n**9.** TEA\n\nCOW\n\n### MAY\n\n**PLANT**\n\n**31.** Butternut squash (through June)\n\n**29.** Melons (through June)\n\n**1.** Florence fennel (through August)\n\n**HARVEST**\n\n  All crops for spring feast\n\n### JUNE\n\n**HARVEST**\n\n  Barley (planted in November)\n\n  Onions, 'Spanish White' (planted in October) and let cure\n\n  Potatoes\n\n  Wheat (planted in January)\n\n**PROJECTS**\n\nBEER (for summer feast)\n\nCHEESE\n\n### JULY\n\n**HARVEST**\n\n  Garlic (planted in October) and let cure\n\n  Onions, cippolini (planted in February) and let cure\n\n### AUGUST\n\n**HARVEST**\n\n  All other crops for summer feast\n\n**PROJECTS**\n\nCHICKENS (buy chicks; raise indoors until fully feathered, 8 to 10 weeks)\n\n## THE ONE BLOCK GARDEN:  \n _Cool-Season Planting Plan_\n\nPlanting and Harvesting Timeline for Northern California\n\n10 feet\n\nKey\n\nCrops not numbered or mentioned are carryovers from the warm-season planting plan. Colors indicate the seasonal feasts that use these crops:\n\n  = summer (see this page)\n\n  = fall (see this page)\n\n  = winter (see this page)\n\n  = spring (see this page)\n\n  = seed\n\n  = transplant\n\n  = plant (gallon-can size or larger)\n\n  = bare-root\n\n  = start any time of year\n\n### SEPTEMBER\n\n**PLANT**\n\n(If using seeds for crops below, start them in August)\n\n**18.** Arugula (also for spring feast; succession sow through March)\n\n**9.** Broccoli rabe\n\n**10.** Broccoli romanesco\n\n**16.** Cabbage, Savoy (through October)\n\n**11.** Cauliflower\n\n**14.** Kale, Tuscan\n\n**15.** Kale, Curly-Leafed\n\n**17.** Lettuce, Red Butterhead\n\n**12.** Mustard (every few weeks through December)\n\n**20.** Radicchio\n\n**13.** Swiss chard (and for winter feast; through January)\n\n**19.** Thyme\n\n**HARVEST**\n\n**26.** Hop flowers and let dry\n\n### LATE SEPTEMBER\u2013EARLY OCTOBER\n\n**PLANT**\n\n**23.** Parsley (also for spring and summer feasts; succession sow February through March)\n\n**22.** Sage\n\n**HARVEST**\n\n**1.** Quinoa\n\n**PROJECTS**\n\nWINE (harvest and crush grapes)\n\n### OCTOBER\n\n**PLANT**\n\n**2.** Fava beans\n\n**27.** Garlic (late October to early November, from cloves)\n\n**25.** Onions, 'Spanish white'   or sets\n\n**HARVEST**\n\n  All crops for fall feast\n\n**PROJECTS**\n\nMUSHROOMS (order logs  ; plant morels)\n\n### NOVEMBER\n\n**PLANT**\n\n**1.** Barley (after harvesting quinoa; through February)\n\n**PROJECTS**\n\nOLIVE OIL\n\nVINEGAR\n\n### JANUARY\n\n**PLANT**\n\n**24.** Strawberries (into February)\n\n**28.** Wheat\n\n**PROJECTS**\n\nSALT\n\nESCARGOTS\n\nMEAD\n\n### FEBRUARY\n\n**PLANT**\n\n**4.** Beets (through August)\n\n**3.** Carrots\n\n**7.** Chervil (through March)\n\n**8.** Dill\n\n**21.** Green onions\n\n**6.** Mesclun (through mid-March for late spring harvest)\n\n**5.** Radishes (through June)\n\n**26.** Onions, cippolini (very end of February)     or sets\n\n**HARVEST**\n\n  All crops for winter feast\n\n# Essential Gardening Guidelines\n\nThe following information is bare-bones basic for organic gardening. If you would like to learn about mulching (to keep soil moist and squelch weed growth), ways to protect young plants (with row covers, netting, or fencing), training and staking plants, and other details about organic fruit and vegetable gardening, see our _Western Garden Book of Edibles_ (Sunset Publishing Corporation, 2010). For specifics on how to grow each plant we raised for our feasts, see the garden guides at the beginnings of the first four chapters in this book. Because growing conditions vary by region (and even within neighborhoods), the best source for local planting information is your county's agricultural cooperative extension office; find yours at www.csrees.usda.gov\/Extension\/index.html.\n\n## BEFORE YOU PLANT\n\nIt is extremely tempting to put your seeds and seedlings in the ground right away. But if you want them to flourish and feed you well, you must create a hospitable environment for them first. The two most important elements are sun and soil.\n\n### Sun: More Is Better\n\nMost vegetables, herbs, and fruits need 6 to 8 hours of full sun daily. Plant them in a spot that is not shaded by buildings, trees, or shrubs. Shadows lengthen in fall and winter to the north of tall structures, so use a compass to figure out where north lies, and make sure any tall plants or structures to the south won't cast shadows over your growing bounty.\n\n### Soil: The First Ingredient\n\nRich, well-drained soil gives your plants the edge they need to produce abundant, flavorful harvests with minimal attention. If your garden is organic, as ours is, keeping the soil healthy is key to plant health, too.\n\n#### ASSESSING YOUR SOIL\n\nHow do you know whether your soil is good, just okay, or a plant-killing disaster? Most soils fall more or less into one of these three categories.\n\n**Loam** This is the good stuff, dark, rich, crumbly, and wonderful-smelling, especially when slightly damp. It drains well but doesn't dry out too fast, can hold nutrients longer than other soil types, and contains enough air for healthy root growth. Give it the squeeze test: Thoroughly wet a patch of soil and let it dry out for a day. Then, pick up a handful and squeeze it firmly in your fist. If it is slightly crumbly but still holds a loose ball, it's loam.\n\n**Clay (heavy) soils** These are made up of very small particles that pack together tightly, producing a compact mass with microscopic pore spaces (the areas between soil particles). Because water and nutrients percolate slowly through the tiny spaces, drainage usually takes a while. It's not easy for roots to penetrate clay soil, and during long rainy spells (or if overwatered), the soil stays wet, sometimes to the point of causing root rot. When subjected to the squeeze test, heavy soil forms a tight, slippery ball.\n\n**Sandy (light) soils** At the other end of the spectrum are soils with large, irregularly rounded particles and large spaces between them that allow water and nutrients to drain away freely. Plants growing in sand are unlikely to suffer root rot, but you need to water them more often to keep their roots moist. The frequent watering leaches nutrients away, so you'll have to fertilize more often, too. Try the squeeze test: If the soil feels gritty, doesn't hold its shape, and crumbles in your hand, it's sandy.\n\n#### FIXING YOUR SOIL\n\n**Compost** This \"gardener's gold\" is an easy, natural way to create good soil. Plenty of compost loosens clay soils and improves drainage. Added to sandy soils, it increases moisture retention by wedging into the large pore spaces between the soil particles. To learn how to make it yourself, see Cultivating Compost, below. You can also buy bagged compost at nurseries.\n\nA few days before planting, dampen the soil slightly to make it easier to dig. With a spading fork (also known as a digging or garden fork), dig to a depth of about 10 inches, breaking up clods of earth and removing any stones, weeds, and debris as you go.\n\nSpread a hefty layer of compost (3 to 6 inches) over the soil and fork it in, along with granular controlled-release fertilizer (see How to Fertilize). Then level the bed with a rake, breaking up any remaining clods. Water well and let the improved soil settle.\n\nEach time you replant a bed with new crops, you'll first need to rejuvenate the soil by repeating these steps, but adding only 1 to 2 inches of compost instead of 3 to 6 inches. Your soil will get progressively fluffier and richer.\n\nCULTIVATING COMPOST\n\nIn the Sunset test garden, we have two side-by-side wooden-slat bins that look like big open crates. We also like the three-bin system often used for composting, which can handle a greater amount of material, but our garden doesn't have room. One of our two bins holds a \"hot\" pile and the other a \"cold\" pile. The \"hot\" pile you build all at once, and you get finished compost in about 3 months. The \"cold\" pile, which you build gradually, breaks down more slowly. Having the two piles gives us a way to deal with both a whole load of raw materials at once and a smaller, steadier stream from the garden and kitchen.\n\nBuild a Two-Bin Compost System\n\n**1. Set up bins** You can use pretty much any kind of bin\u2014wooden, plastic, or chicken wire shaped around rebar\u2014set directly on the soil. That way, beneficial fungi, bacteria, and worms have a better chance of finding their way into the compost, and the bins keep the piles from getting messy. Ours are 3 feet square (the minimum size for efficient \"hot\" composting), about 3 feet tall, and\/or open on the top and one end for easy pile turning.\n\n**2. Add materials** Any compost pile has four main ingredients: **browns** (carbon), including dried grass, dried leaves, woody stems, straw, and eggshells, with the biggest pieces no larger than 6 inches; **greens** (nitrogen), including animal manure (not dog or cat), fresh grass, fresh leaves and other fresh plant parts (stalks, roots), coffee grounds and kitchen scraps (no fat or meat), and coffee grounds (with the biggest pieces no larger than 6 inches); **water,** usually with rainfall and condensation providing all that is needed to keep the pile as moist as a wrung-out sponge; and **air,** which reaches the compost through the open top and front of the bin. **To keep out of your compost:** Weeds with seedheads, cat and dog feces, disease- or insect-infested plants, meat scraps, and copious amounts of other food scraps (keep the food scraps to less than 50 percent of the greens; otherwise, they will rot and attract flies and rodents). Feed both piles a good range of greens and browns. Treat your compost like your diet\u2014diversity is best.\n\n**3. Build and tend the hot pile** Alternate 6-inch-thick layers of browns and greens, watering each layer thoroughly and ending with a brown layer. This is a great place to toss the still-green summer crops you ripped out at the end of the season.  \nThe pile will heat up from the inside over the next week or so. Then use a spading fork to \"turn\" (that is, mix) the pile, and keep turning it once a week. We drag all of it out onto a tarp to turn it and then pile it back in the bin.  \nYou'll have finished compost in about 3 months. It's ready when the majority of the contents smell and look like lush, earthy, dark brown soil. It might need to be sifted if harder-to-break-down additions (like branches) haven't decomposed.\n\n**4. Build the cold pile** You worry less about the ratio of browns to greens in a cold pile. Just toss in whatever might be coming from the garden or kitchen. It decomposes from the bottom up and no turning or any other maintenance is necessary: If it starts to smell, add more browns. It's ready when it turns a rich, dark brown; smells earthy; and has the look and feel of coffee grounds, which can take from 6 to 12 months.\n\nOur two-bin setup is not a perfect system. The space isn't big enough to hold all of the garden scraps, and can't produce enough compost to feed all of our new plantings, so we end up buying bagged compost from the garden store.\n\nBut it's a start, and we have one thing working strongly in our favor: readily available chicken excrement from our coop to add to both piles. This amendment, high in nitrogen, jump-starts any pile's breakdown. You need to let it age in the compost pile for at least a month, however, before you can use the compost on plants. Otherwise, it will burn them.\n\nWe get immense satisfaction out of shoveling the chicken poop into the compost bin, and a satisfaction just as great from putting finished compost back out into the planting beds. We feel like we're helping along one of nature's most miraculous systems: using \"waste\" to become food once again. Not everyone would get such pleasure out of shoveling crap. But we like seeing the loop close, so we wouldn't have it any other way.\n\n _\u2014Johanna Silver_\n\n_The \"hot\" pile_\n\n## HOW TO PLANT\n\nOnce you have prepared your soil and let it settle for a few days, your garden is ready to be planted. You can buy seeds and start your own plants, sow seeds directly in the ground, or buy transplants (seedlings) from a nursery. With seeds, you can order exactly what you want from a mail-order company if your local nursery doesn't carry it. Seek out All America Selections (AAS) seeds whenever possible. They have been judged as winners for their stellar performance in trials across the country by the AAS, a nonprofit organization whose mission is to promote new garden-seed varieties. Nursery transplants are more expensive, and the choice is more limited, but they can be planted immediately.\n\n### Starting Plants from Seed\n\nWhen you start your plants from seed, you can experiment with more varieties than you ever knew existed. (Many of our one-block crops are only available as seeds.) It's also a fascinating science project and an exercise in patience and care.\n\nSeeds are dormant until placed in a moist medium (such as potting mix) that encourages germination. We start ours in cell packs left over from buying seedlings, and either keep them in our garden's greenhouse until the seedlings are up before planting them outdoors, or sow them directly into the ground. Seed starting flats or peat pots, available at nurseries and home-supply stores, work too, and you don't need a greenhouse\u2014a bright indoor windowsill will work fine.\n\nWhen do you start seeds? For most warm-season vegetables, seeds should be sown in early spring (except in hot desert climates), so the plants are ready to set out in the garden when the weather has warmed up. For cool-season crops, you want the seedlings ready to plant in early spring or in fall. Each packet will give you exact guidelines for when to start that particular seed. The label will specify any special care the seedlings might need, too, like a heating mat set underneath the cell pack to speed germination. The packet will also tell you whether the plant does best sown directly in the ground (certain crops, like carrots, prefer not to be transplanted). For tips on direct sowing, see opposite.\n\n#### FOUR SEED-STARTING SECRETS\n\n**1. Use a light, fluffy soil mix** Make sure you use potting mix instead of planting soil or compost. Potting mixes or soils contain all the necessary ingredients (perlite, vermiculite, peat moss) to stay light and fluffy and drain properly. Dampen the soil before you sow your seeds.\n\n**2. Plant more seeds than you need** Whether sown directly in the ground or into cell packs, it's a good idea to plant double the amount of seeds you think you need. In the ground, this means to drop two seeds, instead of one seed, into each hole. In cell packs, sow eight tomato seeds if you want four tomato plants. That way, you'll have backups in case any of the seeds fail to germinate. You can always carefully separate the starts, pot up the extras, and pass them along to friends or coworkers.\n\n**3. Plant shallow** Sow seeds twice as deep as the seed is wide. This means barely under the surface for tiny seeds like lettuce or carrots and about \u00bc inch deep for crops such as squash. And you don't need to pat the seeds into the soil. You can just sprinkle soil on top of them; it really is okay.\n\n_Shallow-furrow sowing_\n\n**4. Water thoroughly** Once you have sown your seeds, sprinkle them gently but thoroughly with water. Be extremely diligent about keeping the seeds moist until they germinate (sprout).\n\n#### PLANTING SEEDLINGS (TRANSPLANTS)\n\n**1. Move the seedlings to a halfway house** Once the seedlings have developed their first true leaves, transplant each one to its own pot (a 4-inch-deep pot is ideal) before hardening it off and putting it into the ground. Or, you can keep the seedlings in their cell packs until the second set of true leaves appear, then harden them off and transplant directly into the ground. If you opt to transplant the seedlings to pots, here is how to do it:\n\n\u2022 Fill the pots with potting mix, moisten the mix, and let it drain.\n\n\u2022 Remove each seedling from its cell pack by gently squeezing the sides of the pack with one hand, turning the pack on its side, and then cradling the soil ball between two fingers of your other hand as you tip the seedling out.\n\n\u2022 Poke a hole in the moistened potting mix. Carefully lift a seedling and its root-ball, keeping your fingers under it for support, and ease it into the hole in the potting mix. Firm the mix around the base of the seedling.\n\n\u2022 Once all the seedlings are transplanted, water them immediately, then set the pots in bright light in the greenhouse or indoors near a window (but keep them out of direct sunlight for a few days).\n\n\u2022 Feed the seedlings weekly with an all-purpose ferti\u00adlizer sold for starting seeds or with a liquid fertilizer diluted to half-strength. When they sprout a couple more leaves, they are ready to be hardened off.\n\n**2. \"Harden off\" the seedlings** The seedlings will need to be gradually acclimated to the outdoors before they are planted permanently outside. This process, called hardening off, prevents transplant shock and will help your seedlings thrive once they are in the ground. The seed packets will tell you when to begin the process.\n\nOver the course of a week or so, take the 4-inch pots of seedlings (or the seedlings in cell packs if you have decided against the halfway house) outdoors for several hours each day, steadily increasing the amount of time until you are leaving them outdoors all night. Bring your babies indoors, though, should the temperature suddenly spike or drop. If you buy your transplants at a nursery, they are already hardened off and can be planted directly in the garden (see next step).\n\n**3. Transplant the plants into the garden** Dig a hole the same depth as the 4-inch pot and an inch or two wider. Ease the plant out of the pot. Try not to disturb the roots, unless the root-ball is very tight. If that's the case, tease apart the roots a little to encourage them to grow out, rather than around and around in a coil. Place each plant in its hole so that the top of the root-ball is even with the soil surface. (Tomatoes are an exception; they are planted more deeply.) Firm the soil around the roots, then water with a gentle flow that won't disturb the soil or the roots.\n\n#### SOWING SEEDS DIRECTLY IN THE GARDEN\n\nMany vegetables grow best if you plant the seeds in the garden from the start. These include root crops (carrots, beets, radishes, turnips, and parsnips), as well as corn, peas, and beans.\n\nSow the seeds after the soil has warmed up to at least 60\u00b0F. You can check the temperature with an instant-read kitchen thermometer or a soil thermometer, available at garden shops. Read the seed packet (or the plant entries in this book) for more information on when to sow each crop. Make sure that the soil is loose and finely crumbled, so the tiny seedlings can push through. Refer to the plant entries or to the seed packet for the proper planting depth; seeds planted too deep won't sprout. As with seeds you start indoors, plant them twice as deep as the seed is wide.\n\nTo be sure you get enough plants, sow a lot of seeds close together. Then, when the seedlings are 1 to 2 inches tall, thin the seedlings to their proper spacing (specified in the plant entries and on seed packets).\n\nYou can sow three different ways:\n\n**_Shallow furrows_** This is the most common way to sow. Use a trowel or the corner of a hoe to make a furrow the correct depth for the seeds you are planting. Sow the seeds evenly, then pat the soil gently over them. To make straight rows, stretch a string between two stakes and plant beneath it. Or, lay a board on the soil and plant along its edge.\n\n**_Clusters_** This is a traditional way to grow sprawling plants like squash and melons, because it gives them more room. Sow five or six seeds in a circle and pat soil over them. Check the plant entries or the seed packet for how far apart to space the circles.\n\n**_Broadcast_** Scattering seeds in wide bands instead of planting in rows is more space efficient for smaller crops, such as lettuces, carrots, or radishes. Scatter the seeds evenly over the soil. Cover by scattering soil over the seeds or by raking gently, first in one direction and then perpendicular to it.\n\nAfter sowing, water the planted area with a fine mist, being careful not to dislodge the seeds. You want to keep the soil moist but not soggy, and also prevent it from forming a crust (so that the germinating seed won't bang its head against a ceiling). In hot weather, covering the soil with damp burlap helps retain moisture. Be sure to remove the burlap as soon as the seeds begin to sprout. If you plan to water with basins or furrows, dig them before you sow your seeds.\n\nHOW TO STORE LEFTOVER SEEDS\n\nUsually a packet gives you more seeds than you can use. If the leftover seeds are stored airtight in a cool, dark place, they will last for at least another year, especially if they were fresh when you bought them. (Check the date on the packet before buying.)\n\nTo store the seeds, put them (in their packets for easy identification) in mason jars, resealable heavy plastic bags, or sturdy plastic containers, and stick them in the freezer. If you're not certain whether your seeds are viable, wrap them in a moistened paper towel, pop them in a resealable plastic bag, and put the bag in the sun. If they sprout in the time specified on the packet, they're good for planting.\n\n## HOW TO WATER\n\nEvery garden has different needs depending on the weather, the soil, the age and variety of the plant(s), and a host of other factors, so figuring out the best way to water is an ongoing learning process. What follows are general guidelines. For more specific information on the watering requirements of a particular crop, see the plant entries in each season.\n\n### Frequency\n\nWater new transplants at least once a day in 60\u00b0 to 70\u00b0F weather and up to two or three times a day when it's hot and windy. Your goal is to keep the soil moist but not soggy. As the plants grow and their roots reach deeper, you can water less frequently. In general, vegetables that are flowering or beginning to set fruit, form heads, or develop edible roots need to be watered more often than older plants.\n\nYour soil texture also influences how often you need to water. Clay soils need less frequent watering than loam soils, and loam soils can be watered less frequently than sandy soils. (For more on soil types, see this page.)\n\n### Amount\n\nOnce the plants are established, water them deeply enough to moisten the entire root zone. This encourages roots to grow down farther. Deeper roots have access to more moisture, which allows the plants to go longer between waterings. Frequent shallow sprinklings are inefficient because they encourage shallow root growth, which leaves plants subject to stress from heat and drying winds. To check water penetration in your soil, dig a hole with a trowel after watering.\n\n### Ways to Water\n\nWe used the following methods in our one-block garden:\n\n**By hand with a sprinkler hose or watering can** Using a sprinkler nozzle on the end of a hose or a watering can with a sprinkler snout is useful for newly seeded beds, new transplants, and container plants because you can apply the water gently and put it exactly where it's needed. This method also rinses away dust and discourages certain pests (especially spider mites). Sprinkling has some disadvantages, as well. It wastes water through evaporation, particularly in windy weather. It may also encourage leaf diseases on some crops, especially in humid climates. The best time to water by hand is in the morning (leaves will dry off during the day) when the air is still.\n\n**By hand, flooding a basin or furrow** Basins, which are used to water large fruiting plants such as our lemon tree, tea bushes, and pineapple guava tree, are doughnut-shaped depressions in the soil surrounding the plants. As the plants grow, you need to expand the basins. Furrows, or shallow ditches, dug near smaller plants that are growing in rows work well on level ground. Broad, shallow furrows are generally better than deep, narrow ones: The wider the furrow, the wider the root area you can soak, since water moves primarily downward rather than sideways. To avoid damaging the roots, dig basins and furrows before you set out plants or sow seeds. Furrows are great because they are a fast, efficient way to reach the plants' roots.\n\n**Soaker hoses** These are long tubes made of perforated or porous plastic or rubber with hose fittings at one end. When you attach a soaker to a hose and turn on the water supply, water seeps or sprinkles from the soaker along its entire length. Soakers are ideal for irrigating rows of vegetables. To water beds, snake the soaker back and forth around the plants. To water trees, coil a soaker around the outer edges of the root zone. Soakers water efficiently and evenly, but you have to be there to turn on the hose.\n\n**Drip irrigation** Drip systems deliver water slowly by drip emitters that you attach to plastic tubing yourself, or by emitter lines, tubes with factory-installed emitters spaced at regular intervals. Emitters can be adjusted so that the water is applied directly over plant roots, reducing your water use. Because most of the surface is not moistened, drip systems also cut down on weed growth. Drip-irrigation kits and supplies are sold by agricultural-supply stores, retail nurseries, garden centers, and mail-order suppliers.\n\n_Drip irrigation_\n\n## HOW TO FERTILIZE\n\nFirst, you'll need to figure out the label. All fertilizer labels list the percentage by weight of three primary nutrients, in this order: nitrogen (N), phosphorus (P), and potassium (K). Fertilizers labeled \"general purpose\" or \"all-purpose\" are complete fertilizers, meaning they contain all three nutrients. For example, a fertilizer labeled 10-10-10 contains 10 percent nitrogen, 10 percent phosphorus, and 10 percent potassium. A fertilizer labeled 20-10-5 contains more nitrogen (N) than the other nutrients, so it's sometimes called a high-nitrogen fertilizer. Complete fertilizers can be liquid, granular, or pellets.\n\n### Choosing Fertilizers\n\nLight feeders such as lettuce, radishes, and tomatoes that have been planted in rich soil well amended with compost rarely, if ever, need supplemental fertilizer. But many other crops, including melons and onions, need extra food to thrive. Where fertilizer is called for in this book, we most often use the types listed below.\n\n**Fish emulsion** This creamy, brown, sometimes fishy-smelling liquid fertilizer is made from whole fish, or oils and by-products of processed fish. Dilute it with water according to label directions and apply it with a watering can. (We also use it as a foliar spray for an added boost.) It is fast acting, slightly acidic, and relatively mild.\n\n**Granular organic blend** Dr. Earth granular fertilizer for tomatoes, vegetables, and herbs is a 5-7-3 formulation that combines fish bonemeal, kelp meal, alfalfa meal, soft rock phosphate, seaweed extract, potassium sulfate, and more. It is usually added during the growing season.\n\n**Controlled-release fertilizer** This is our one exception to organic gardening. We sometimes use it for containerized fruit trees, which tend to be heavy feeders. The beadlike granules release nutrients gradually\u2014the rate depends on temperature, weather, and moisture\u2014over several months. Scatter and dig the granules into freshly prepared soil at the rate of about 3 tablespoons for every 4 square feet.\n\n_\u2014Kathleen N. Brenzel and Johanna Silver_\n\n_The Summer Garden_\n\n#  SUMMER\n\n_The One-Block Summer Feast_\n\nSkillet-Roasted Edamame  \nDeviled Cucumber Cups\n\nCorn Soup with Roasted Poblanos and Zucchini Blossoms\n\nTomato and Herb Salad with Fresh Chive Cheese  \nRosemary Potatoes Anna  \nPattypan Squash with Eggs\n\nWatermelon, Cantaloupe, or Honeydew Sorbets  \nPeppermint-Lemongrass Tisane\n\nChardonnay and Summer Wheat Beer\n\nIt all started with the menu. In the Sunset kitchen in mid-May, we cooks dreamed about the end-of-summer dishes we wanted to make: Peppery arugula salads with a rainbow of ripe tomatoes in oranges and yellows, greens and purples, reds and pinks. A platter of avocados and oranges with paper-thin red onions. Sweet corn on the cob, definitely. Ripe figs, because we had spotted a vine growing out back.\n\nWith a tentative menu drawn up, we sat down with the garden department and got a reality check. Arugula, a cool-season crop, would wilt in our summer heat. Our fig vine had been pruned so severely the year before that it probably wouldn't bear much fruit\u2014not enough to plan on, anyway. We had no avocado trees, and even if we bought some young ones, they would take several years to produce.\n\nThere were consolations, however. We could grow good tomatoes, though they would be on the small side in our cool climate. Corn would not have the savory depth that it does in the Midwest, but it would be sweet and juicy. 'Yukon Gold' potatoes would be no problem. \"How about zucchini?\" suggested Lauren Swezey, our garden projects editor. \"Zucchini does really well here.\" Privately, I was crushed\u2014zucchini is just about the most boring summer vegetable I can think of. But then Lauren described a wondrous variety called 'Trombetta di Albenga'. She took out a seed packet with a picture on it. \"It curves like a trombone,\" she said. \"And it's sweet and a little crunchy. Completely delicious.\" We were sold.\n\nOver the next few days, we settled on a cooking fat (not peanut oil, because peanuts need a southern climate, or corn oil, because three cups would require about sixty pounds of corn, and we wanted to eat our corn). What Sunset did have were twenty-one olive trees, planted all around the property as landscaping back in the 1950s. They were loaded with fruit, and surely it wouldn't be too hard to figure out how to press it.\n\nFor seasoning, we would plant chiles, lemons, and potent summer herbs. And, because we lived close to the Pacific, it seemed worth trying to make some salt from seawater.\n\nWhat would we do for protein? Our menu sounded good, but gossamery. We asked ourselves what we were collectively capable of, and it did not include raising meat animals. Eggs and cheese seemed more doable. We could keep chickens right in the garden, and as for the milk for cheese, the closest dairy would do. We didn't dream (then) that we might someday have a cow.\n\nFor dessert, we'd need a sweetener, and honey seemed like the natural solution. Why not try keeping some bees? Plus, all those pollinators would help our crops produce.\n\nWithin a couple of weeks, we'd finalized our menu. I wandered out into the garden to imagine how it might all look. A pair of grapevines caught my eye. What if we made wine? Our little vines wouldn't supply enough grapes, but maybe we could find a vineyard nearby. Wine editor Sara Schneider loved the idea and agreed to launch Team Wine. In the meantime, Rick LaFrentz, our head gardener, volunteered to lead Team Beer. He had brewed at home using kits, and wanted to try planting barley, wheat, and hops to make beer from the ground up. It was intriguingly medieval of him.\n\nOur made-from-scratch project had not even started, and here we were, \"importing\" wine grapes and milk and ocean water. But we would transform the imports into foods that would be wholly our own: grapes into wine, milk into cheese, water into salt.\n\nItalians have a lovely word for the locally grown produce in their farmers' markets: _nostrani_ \u2014\"ours.\" It usually sells out first because it's often the best. That's exactly what this summer dinner would be, from start to finish. Ours.\n\nI walked around the office to see if any of my other colleagues wanted to join the project. Erika Ehmsen, our stately, calm copy chief, knew where we might get the hundreds of pounds of grapes we needed: Thomas Fogarty Winery, in the nearby Santa Cruz Mountains. Her dad and Dr. Fogarty, a cardiologist, had worked together, and she'd visited as a child and had always wanted to go back. She signed up for Team Wine. Researcher Elizabeth Jardina and art director Jim McCann liked the idea of chickens. Garden associate editor Julie Chai, whose heritage is Korean and German, wanted to join Team Vinegar (formed in the wake of Team Wine). \"I have kimchi on one side and sauerkraut on the other,\" she told me. \"I love all things fermented.\" Margaret Sloan, production coordi\u00adnator, and Kimberley Burch, imaging specialist, were drawn to Team Bee. Margaret had always thought bees were mysterious and fascinating, and Kimberley, concerned about colony collapse disorder, mainly wanted to help increase the bee population.\n\nBy early June, we had started researching our various projects and were ready to roll. The fact that we had no experience with any of them, except cooking and gardening and a bit of beer making, did not dim our enthusiasm a bit.\n\nThat was a good thing, because it would take us about a year and a half to create everything we needed for our summer feast. The wine grapes wouldn't be ready until October; then they'd need months to ferment and mature. The olives ripened in November. And we could not harvest the wheat and barley until the following summer.\n\nAt least we could get the chickens going. I personally yearned for fabulous-looking breeds like Silkies, whose feathers are as soft as kitten fur, or Polish, whose feather crests make them look like Tina Turner. Fancy chickens are not always the best layers, though. In the end, we just got what Half Moon Bay Feed & Fuel happened to have on the day we visited, which were tiny chicks from breeds known to lay well: Ameraucanas (exciting, since they would lay blue- and green-shelled eggs), Rhode Island Reds, and Buff Orpingtons.\n\nWe bought two of each, and fussed over them like crazy, hanging a heat lamp over their wire cage\u2014which we put in a storage shed behind our main office building\u2014and visiting them every couple of hours. The runt, a Rhode Island Red we called Ruby, immediately developed a condition that the feed store had warned us about: pasty butt. Untreated, it can block a chick's digestion and be lethal. This we handled as instructed by the feed store, gingerly applying cotton balls soaked in warm water to the tiniest butt imaginable. Ruby survived.\n\nAs our chickens grew that fall, we plunged into wine making. Fogarty Winery agreed to sell us both Syrah grapes (a relatively forgiving variety, so we would have a chance of making drinkable wine) and Chardonnay juice (since pressing white grapes is much trickier than pressing the red). On a golden October morning, we drove out to the winery and picked five hundred pounds of small, luscious Syrah grapes from Fat Buck Ridge, a vineyard with sweeping views to the west. It was surprisingly fast\u2014a couple of hours of snipping and we had all the grapes we could handle. We had a quick picnic in the vineyard, with a bottle of Fogarty Syrah from that very ridge\u2014\"the prototype, the goal... the competition,\" Sara joked later.\n\n_Among the Syrah vines at Fat Buck Ridge, in the Santa Cruz Mountains_\n\nWe had to crush the grapes as soon as possible. Waiting for us back at Sunset, with a giant pile of equipment, was Dan Brenzel, husband of our garden editor, Kathy Brenzel, and a serious home winemaker himself. His crusher-destemmer was entirely capable of handling our harvest. But a couple of us had seen the grape-stomping episode of _I Love Lucy_ at an impressionable age, so we did some foot crushing, too, there in the Sunset parking lot. We laughed so hard we practically fell out of the garbage cans we were using for crushing. As Erika pointed out, it was sort of like a StairMaster. Set in quicksand.\n\nThe next several days were tense for Team Wine. Sara woke up at 3:00 a.m. worried that our grape pulp (called must), which was soaking under the eaves in a big vat to give the juice as much flavor and color as possible, might be spoiling. So we added yeast to kick off the fermentation and crowd out any unwelcome microorganisms. (We'd already added yeast to our Chardonnay juice.)\n\nAnd at first everything went like it should. The yeast attacked the sugars and belched carbon dioxide gas, which made the pulp gurgle and foam as though it were alive. A warm, thick, seething cap of skins and seeds formed on the surface, and we punched it down a few times a day so it could deliver its goodness to the juice. But the sugar level wasn't dropping fast enough. This meant we risked a \"stuck fermentation\" that could lead to spoiled wine. We decided to move our vats into the warm building, hoping to prod the yeast into gobbling more sugar. It worked, and the sugar eventually hit zero. We had gotten through the hardest part.\n\nNow we eased the Syrah into a slower, secondary fermentation by sprinkling in powdery, freeze-dried malolactic bacteria. It was as easy as adding salt to a stew. A few days later\u2014once the bacteria had a chance to settle in\u2014we pressed our wine off its skins and seeds. This involved using Dan's homestyle basket press, which looked not unlike a toy rocket. We poured the inky slurry into the press and set up a tag team running buckets from the press's spigot over to several waiting carboys (big glass jugs). Then, once we'd popped airlocks into the tops of the carboys, we left our beast alone to be tamed by beneficial bacteria and the gentling hand of time.\n\nMeanwhile, our chicks had become fully feathered young hens and were ready for their coop in the test garden. The henhouse itself, which opened right into a fully enclosed yard\u2014to keep out raccoons and other chicken hunters\u2014was a grand sloping-roofed structure given to us by Wine Country Coops, in Napa. The hens took to it as though to the manor born, hopping up to its comfortable perches and exploring the nest boxes. Pretty soon, their personalities\u2014yes, they do have them, it turns out\u2014began to show: Rhode Island Red Carmelita was the boss lady, the most aggressive and the shiniest feathered. (Elizabeth worried that she might be a rooster.) Ruby was a talker, croaking insistently whenever we came to visit, as though cussing us out for something. Alana, slightly shy, was named for our managing editor, Alan Phinney, because her lustrous head feathers resembled his hair (he is a good sport about this). Her fellow Ameraucana, Ophelia, was much more outgoing and instead of clucking produced a sort of low foghorn tootle. The two Buff Orpingtons, Honey and Charlotte\u2014both big, fluffy, and blonde\u2014were the most inclined to jump on your lap for a pat. They weighed as much as house cats and liked to be stroked under their wings, where the down was softest. It made them burble happily, as did an ear of corn or a bulb of fennel thrown down for them to peck. And the burbling of a happy chicken is a very sweet sound.\n\nEverything was going well. Then, in late October, disaster struck. Olives began pelting the ground weeks before we had expected to harvest them. Not only that, the fruit looked shrunken and deformed. We sent some off to The Olive Press, up in Sonoma, for advice, and were told that we had the worst infestation of olive fruit fly maggots that the store's owner had ever seen.\n\nApparently, you can press olives with a small percentage of maggots\u2014although you might end up with a flavor flaw known in the olive oil trade as \"grubby\" (seriously). Our fruit, though, was riddled with them. We could try treating the trees in the future, but this year the grubs had won.\n\nWe badly wanted olive oil, and we really wanted to learn how to pick and press olives. Fortunately, Valencia Creek Farms, in the Santa Cruz foothills, agreed to sell us eight hundred pounds of olives, which we could pick ourselves. So we rented a U-Haul and drove off to the foothills again.\n\nAt the orchard, we picked big, fat green Ascolanos. The rest of our order had been harvested for us earlier that day, and although we felt a little sheepish about this, we were also very grateful when we realized that it would have taken our five Team Olive members about eleven hours to pick what we needed.\n\n_Ascolano olives at Valencia Creek Farms_\n\nAs with grapes, speed is of the essence when pressing olives. The second they are off the trees, they start to degrade. We loaded up our twenty-two crates and drove south, to Pietra Santa Winery, near Hollister, and its olive press. The olives were destemmed and washed, and then ground by three massive stones into a pinkish, brownish paste that looked exactly like chopped liver.\n\nWe tasted the olive oil right from the spigot. It was grass green, flecked with bits of olive, and extremely fresh and bright. The mild climate of Santa Cruz had tamed the olives, so the oil was not nearly as peppery as a typical \"three-cough\" Tuscan oil. We were absurdly and instantly proud of it. Our hens were not laying yet and our wine was still aging, but we had our olive oil and it felt great.\n\nWe were now in January, and with every passing day of the new year, we expected eggs. Chickens usually start laying when they are between four and six months old, and ours were past the five-month mark. Team Chicken was beside itself with anticipation, especially when the hens started showing the classic signs of being on the verge: fat pink combs and a tendency to suddenly crouch in front of you, tail up high. This is called the egg squat, and the chicken hopes you are a rooster bent on mating. We stroked the squatters from head to tail, which is supposed to relax them and help the egg form\u2014but still, no eggs! Elizabeth caught herself actually commanding the chickens to lay. Team leader Jim put a marble egg paperweight in the nest box to encourage them. We checked the nest boxes four or five times a day. And then, finally, in late January, on a quiet Sunday, our Ameraucana Ophelia laid a small and exquisite blue egg.\n\n_Our first egg (behind the decoy)_\n\nOver the next several days, Ophelia gave us two more eggs\u2014enough for a group taste. The yolks were deep orange-yellow and the whites so firm they actually were hard to break up with a fork. I scrambled them very slowly, and we all stood around and had small bites. They were velvety and voluptuous, with a rich, round flavor that made ordinary eggs taste like imitations.\n\nOne by one, the other chickens stepped into the nest box. Alana laid a light green egg the shape of a kiwifruit, with splotches of darker green. Carmelita, after half an hour of squawking and cackling and thrashing, produced a brown egg with a dent in the bottom (we figured out later that she was standing up when she laid it). Ruby's egg was brown too, but tiny and freckled. The Buff Orpingtons brought up the rear with creamy beige eggs. Pretty soon we were getting five or six eggs a day, and no two were ever the same.\n\nOur next project took us up into the hills again, but this time north, to Sonoma, and the home of cookbook author Paula Wolfert. Paula is famous for her meticulous research and had just finished a major investigation of vinegar. She gave Team Vinegar a start-to-finish lesson, complete with emphatic advice: \"If it smells like furniture polish, throw it out!\" We left with jars of her precious vinegar \"mother,\" the live starter culture for vinegar, which she had gotten from a friend who had brought it over from France. It was probably at least forty years old. The slippery, gelatinous bits of this venerable mother would convert part of our stash of Syrah into the best wine vinegar ever. Or, so we hoped.\n\nWe fed the mothers with cautious dribbles and then healthy pours of Syrah, and in a matter of weeks, we had some incredible vinegar. It was exhilaratingly strong, fruity, and complex, and left deep purple stains on whatever it touched. \"It knocked my socks off,\" said Kathy Brenzel, Sunset's garden editor and the team leader. \"I could've sipped it from a wineglass and been happy.\" This was nothing like grocery-store red-wine vinegar. We actually had to add water to it, just to tame its jets.\n\nBy now we were well into spring, a good time to get started on some cheese. We knew mozzarella would be tremendous with the big tomato and herb salad we had planned, but we also knew that making it was kind of difficult and also possibly painful (you have to stretch the cheese while it's boiling hot). Instead, Team Cheese bought a copy of Ricki Carroll's _Home Cheese Making_ and started experimenting with recipes.\n\nWe had only a handful of ingredients: organic milk, \"imported\" from the excellent Straus Family Creamery, just north of San Francisco; lemons and herbs from our garden; and sea salt, which recipe editor Amy Machnak had figured out how to make from pure Pacific Ocean water. (Her fianc\u00e9, Andrew, a diver, had hauled in forty gallons of it for her.) So it's not surprising that the recipes did not turn out exactly as Carroll described. Plus, we lost control of our heat, overwhisked the curds, and used too much lemon juice. But in the end we produced two simple cheeses (and have since fine-tuned the recipes): a crumbly, salty, oregano-flecked white cheese we could sprinkle into soup, and a firm, sliceable log flavored with chives, which we planned to use in the salad. We felt pleased with ourselves. \"I never thought you could really _make_ cheese,\" said features editor Christine Ciarmello, who had moved west the month before, from Florida. \"I've just come from the land of fast food, and here I've been responsible for the birth of this little cheese.\"\n\nSoon after, in May, when the weather grew soft and warm, we planted just about everything else we hoped to eat. Test-garden coordinator Ryan Casey, who was starting his own small farm in Half Moon Bay, turned the soil in our backyard-size plot with a digging fork until it was as loose and light as bread crumbs and then raked in a rich, dark layer of compost. He planted the little seedlings lightning fast, scooping out each hole with one tanned, bare hand and plopping the plant in with the other, then sliding soil around the stem. For seeds, he dropped them into furrows with an upswing of his hand and then pinched the furrows closed on the downswing. His grace and economy of motion were mesmerizing.\n\nThe warmth meant flowers, too, and nectar flow, the food supply for bees. It was time to set up our hives. We decided to get what are called nucs\u2014brand-new little colonies, each with a queen, bees, and brood (bee larvae)\u2014rather than packaged bees, which would take longer to get established. But we were nervous. Already all the equipment seemed overwhelming, and we were a little scared of the bees themselves.\n\nMaster beekeeper Randy Oliver had nucs to sell, and he offered to give us a crash course in beekeeping, too. Over the phone, he told me, \"You guys give me two hours and I'll shift your paradigm.\" In his lush, pine-scented bee yard, in the Sierra foothills, that is exactly what happened. He showed us that if we moved calmly among the bees, as though practicing tai chi, they would ignore us completely. He taught us how to hold the frames to catch the sunlight, so we could see the tiny eggs\u2014each smaller than a grain of rice\u2014to make sure the queen was laying. Bare hands, he said, are better for working with bees, because gloves can crush them. Then he carefully shook dozens of bees into our bare cupped hands. Holding them was electrifying. It felt like holding a mass of warm, vibrating air. No stings. By the end of the afternoon, we were completely smitten, and drove home with two hives of gentle, golden bees\u2014two living bee worlds\u2014in the back of our rented truck.\n\n_First beekeeping lesson_\n\nIn the meantime, a minor crisis had erupted in the chicken coop. Honey, our sweetest chicken, was being terrorized by the other hens. They would peck her little pink comb till it bled, and she'd hide in the nest box all day, not eating, not drinking. We tried all kinds of things to disrupt the pecking order, such as squirting the attackers with water guns, painting Honey's poor chewed comb with nasty-tasting liquid baby soap, and yelling \"No!\" repeatedly. Nothing worked. Honey was wasting away, her keel bone poking out like the prow of a ghost ship.\n\nFinally, we built her a separate enclosure within the coop, with her own little nest box. Elizabeth called it the Honeydome. Unfortunately, the minute we put her in it she flew out to join her oppressors. Elizabeth claimed Honey had \"a crazy form of chicken Stockholm syndrome.\" To block another escape, we extended the chicken-wire fence to the ceiling of the coop. But Honey just pressed herself against the mesh, and the other hens pecked away at her. We were at wit's end. And then, miraculously, the others started to leave her alone, and we put her back in the yard.\n\nThat was handy, because we suddenly had a new resident for the Honeydome: a half-grown chick we named Nugget, because it had been found (implausibly) in a McDonalds parking lot. The family that found the chick had taken her to their vet's office, where one of our cooks, there with her cat, heard the story and asked us to please adopt the little thing. It was too good a story to resist, so Nugget joined the flock, settling down in the Honeydome. Eventually the chickens would accept her (we sure hoped it was a her), and we would integrate them.\n\nAll was now calm on the chicken front, but our bees having a bumpy ride. Betty and Veronica\u2014the names we had given the hives\u2014had been attacked by ants right after they arrived. Scrub jays ate the bees in midair. Then varroa mites, horrible ticklike parasites that suck bees' blood and spread disease, invaded the hives. We did everything we could to help, throwing rocks at the jays, setting out ant bait, putting the hive legs in water to deter the ants, and dusting the bees with powdered sugar (which makes the mites lose their grip and fall off, and prompts the bees to clean themselves, knocking off more mites). We doused the hives with Apiguard, a foul-smelling mite-fighting gel, taking the risk that it might make any honey-in-progress taste like Listerine. Although it didn't get rid of the mites, it put a nice dent in the population.\n\nAs we tried to protect our poor bees, we explored the hives, too, learning to spot the queens, with their elongated abdomens, and the big-eyed, square-bottomed drones, whose main purpose in life was to mate with a virgin queen. All the rest were worker bees, and they each progress through a series of jobs, which we started to recognize. Guard bees stood sentry at the edge of the landing board and on the tops of the frames. House bees stayed in the hive, taking care of the queen and feeding the larvae. Mortuary bees carried the bodies of their dead sisters out of the hive. Foragers brought back food\u2014pollen and nectar\u2014and as they landed, we could see bright balls of pollen packed into pouches on their fuzzy back legs. On hot days, the house bees would stand on the landing board (and inside, too) and fan with their tiny wings, ventilating the hive. It was hypnotic and comforting to watch this orderly, intelligent, collective universe go about its business. Kimberley and Margaret became so fascinated that they started stashing their beekeeping outfits next to their desks, so they could get to the bees faster.\n\nRyan had put in drip irrigation in June, and the garden took off like something in a fairytale. The melon vines darted sideways, carpeting their beds. The pattpyan squash plants sprouted leaves as big as baby elephant ears, with bright yellow flowers underneath and nubbins of baby squashes. One day in early July, Ryan dipped a hand under the potato bed and pulled out a marble-size 'Yukon Gold'; they were coming along nicely. The much-anticipated 'Trombetta di Albenga' zucchini twirled up their arched trellis. The corn was already past my knees. Everywhere, bees zipped and buzzed and buried their shiny golden bodies in the blossoms.\n\n_'Trombetta' zucchini shooting out tendrils_\n\nWe were not having superb luck with our wheat and barley, however, which we'd planted back in the winter. Squirrels pillaged the wheat, and we worried the barley would be next. As insurance, Team Beer started up a batch of wheat beer using a kit, and the funky-malty smell of wort\u2014the sweet liquid that would ferment into beer\u2014hung in the kitchen as thick as fog. We poured it into the primary fermenter and slapped on an airlock to let carbon dioxide escape, and the yeast turned the wort into a bubbling mass that made the airlock dance for days.\n\nOur chickens had been healthy and strong, popping out eggs like Pez dispensers. Then champion layer Ophelia developed a strange, puffy chest. It was a hard lump about the size of a tennis ball. Elizabeth did some quick research and figured out that Ophelia's crop\u2014a little sac where a chicken's food sits and moistens before traveling down toward its two stomachs\u2014was stuffed with some kind of indigestible mass. Impacted crop, it was called, and it was serious. A chicken could die from it.\n\nWe turned Ophelia upside down and massaged her. This was ineffective and stressful. Our next option: carefully slice her open with a sterilized X-acto knife and remove the mass (apparently many chicken owners do this successfully), or take her to the vet. Worried we would botch the backyard surgery, we went to the vet, who extracted a giant wad of soggy straw from Ophelia's chest. She had, it seemed, been munching on the flooring of the coop. Before she returned, we removed every scrap of straw from the coop, and replaced the straw with largish hunks of pine bark. No chicken could possibly eat those.\n\nIt was August, and our first feast was now just a few weeks away. We did not have honey yet, but we had plenty of eggs, good olive oil, and our first beer (from the kit; we'd save our homegrown grain for round two). We had not yet bottled the Chardonnay, but we could pour some of it into bottles just for the occasion. (The Syrah wouldn't be ready for at least another six months.) Our own crops were finally ripe enough for us to see how they worked in the recipes we'd developed. It was the moment we cooks had been waiting for.\n\nWe brought in armloads of vegetables, so fresh they still seemed to be growing. The ears of corn were plump and firm, with tassels as fragrant as grass, and not sticky and brown the way they often are at the store and even the farmers' market. When we sliced kernels off the ears, sweet, milky juices ran down the knife\u2014they hadn't had time to thicken into starch. The cucumbers, picked small, were crisp and mild, without a hint of bitterness. We had grown edamame (soybeans) to see how they would do, and the pods were unexpectedly fat and fuzzy, with creamy beans inside. The sun-warmed tomatoes tasted like candy. Everything we had planned for the menu would work just fine.\n\nDays before the dinner, we got our final ingredient. Margaret and Kimberley rushed into the kitchen with four dripping frames of golden honey laid out on sheet pans. With pastry scrapers, we cut the combs from the frames, sending thick, shiny, voluptuous rivers of honey streaming into the bowls beneath. Margaret said she wanted to lick her fingers and sing like Pooh Bear. Kimberley was so happy she was practically quivering. I couldn't stop eating the honeycomb: It was soft and supple, and gushed honey with every chew.\n\nOn a hot late-summer evening, we set our table in the garden, a few feet from the plants that we had so carefully tended. We started with smoky, skillet-roasted edamame\u2014just soybeans and salt\u2014and cucumber cups with deviled-egg centers, courtesy of the flock. Our wheat beer had nice fizz and flavor. The Chardonnay was tart, clean, and incredibly refreshing, like biting into a cold green apple.\n\nNext came corn soup, cold and smooth, each bowl sprinkled with crunchy diced trombetta zucchini, spicy chopped poblanos, and crumbled oregano cheese, with a zucchini blossom laid on top. For the main course, we ate golden wedges of potato cake; pattypan squashes, each cupping a baked egg; and our tomato salad, with homemade chive cheese and the most labor-intensive vinaigrette we had ever made, even though it was just salt, olive oil, red-wine vinegar, and a drop or two of honey. We finished with sorbets made from nothing but melons and honey, and little glasses of warm peppermint-lemongrass \"tea.\"\n\n_Our summer feast_\n\nIt had taken us a year and a half to make this dinner. And in only a year and a half, we had learned to make wine, beer, cheese, olive oil, salt, and vinegar; grown thirty different herbs, fruits, and vegetables; successfully raised chickens and bees; and come to count on one another in ways we had not before. We would never again take any of these foods for granted. And we were all hungry for more. Fall lay ahead; we could live a whole new menu.\n\n# THE SUMMER GARDEN\n\nJuicy tomatoes and supersweet corn are warm-season edibles, meaning they need warm soil and air temperatures to grow steadily and produce crops. We grew these, and other heat lovers for our summertime feast, in a spot that gets full sun for at least six hours per day. The edamames and squashes are really easy to grow from seed. Other crops, including tomatoes, are easier and bear more plentifully if grown from nursery transplants.\n\n_Potatoes grown in \"towers\" made of reed screens_\n\n10 feet\n\n**SUMMER GARDEN PLAN**\n\n1. Potatoes (towers)\n\n2. Barley\n\n3. Chiles\n\n4. Purslane\n\n5. Peppermint\n\n6. Potatoes (trench)\n\n7. 'Spanish White' onions\n\n8. Hops\n\n9. Garlic\n\n10. Wheat\n\n11. Rosemary\n\n12. 'Benning's Green Tint' pattypan squash\n\n13. Cucumbers\n\n14. Lemongrass\n\n15. Edamame\n\n16. Chives\n\n17. Basil\n\n18. 'Sunburst' pattypan squash\n\n19. Tomatoes\n\n20. Melons\n\n21. Corn\n\n22. Parsley\n\n23. Thyme\n\n24. Oregano\n\n25. Marjoram\n\n26. 'Trombetta di Albenga' zucchini\n\n27. Lemon tree\n\n### Barley 'Lacey'\n\nThis high-yield, low-protein, six-row malting-type barley was developed for the brewing industry. We planted it in a 5-by-10-foot bed, with high hopes for harvesting the 9 pounds of barley we needed for making beer. Unfortunately, squirrels found our patch and ripped up a bunch of plants\u2014we think to use for building their nests. We ended up with just 2 pounds of barley!\n\n**Best Site** Full sun and well-drained soil.\n\n**Days to Harvest** 90 to 120 days from seed.\n\n**Planting and Care** Plant in November or December in moist, not wet, soil. Sow seeds 1 to 2 inches deep and about 3 inches apart. Space rows 7 inches apart. Water regularly.\n\n**How to Harvest** The following summer, when the stalks are thoroughly dry and golden brown, clip them off a couple of inches above the ground. For details about processing the barley, see the directions for making Belgian Abbey Ale.\n\n**Seed Source** Howe Seeds, www.howeseeds.com.\n\n### Basil 'Genovese'\n\nWe are constantly using the tender, spicy-sweet green leaves for flavoring tomato and Italian dishes. They make wonderful pesto and give a lift to salads. 'Genovese' is easy to grow from both seed and transplants and bees love it.\n\n**Best Site** Full sun and well-drained soil enriched with compost.\n\n**Days to Harvest** 88 days from seed. For the best harvest, allow the seedlings to at least double in size before picking any leaves, and then pick them sparingly, pinching tips to induce bushiness, until the plant fills out.\n\n**Planting and Care** Sow seeds in batches every two weeks from midspring through early- to mid\u00adsummer. Plant in rows 18 inches apart, placing seeds 2 to 3 inches apart in each row and \u00bc inch deep. Once seedlings sprout, thin to 12 inches apart. If starting from seedlings, plant 12 inches apart. Water as needed to keep soil evenly moist.\n\n**How to Harvest** Pick leaves whenever you need them. To keep leaves coming, pinch off branch tips and flowers. To harvest lots of leaves for making pesto, cut plants back by half (they will regrow). Or, plant seedlings in succession every month and harvest the entire plant.\n\n**Seed Source** Stokes Seeds, www.stokeseeds.com.\n\n### Chile, Poblano\n\nA staple in Mexican cooking, especially for chiles rellenos, these fleshy, broad-shouldered, mildly spicy peppers grow on multistemmed plants about 2 feet tall. Often called pasillas, especially in California, they are 3 to 6 inches long. Flowers appear 50 days after sowing seed, followed by fruits that ripen from dark green to red. When dried, poblanos are known as anchos and have a deep, fruity taste.\n\n**Best Site** Full sun and well-drained soil amply amended with compost.\n\n**Days to Harvest** 100 days from seed.\n\n**Planting and Care** Plant nursery seedlings when soil is warm in spring, placing the plants 18 to 24 inches apart. In containers, one or two plants are plenty. To start from seed, sow seeds indoors 6 to 8 weeks before the last spring frost. Let the soil dry out between waterings, and cut back on water as the chiles mature to concentrate their flavor.\n\n**How to Harvest** Poblanos are almost always harvested green; their flavor mellows as they ripen. Snip pods off with scissors or pruners.\n\n**Seed Source** Tomato Growers Supply Co., www.tomatogrowers.com.\n\n### Chile, Serrano\n\nAbout as long as your index finger, serranos start out green, then turn red as they mature. Hotter than jalape\u00f1os, they are extremely versatile in the kitchen, where they are added to everything from guacamole to stews. Plant heights vary from 1\u00bd to 5 feet tall; each plant produces about 50 pods.\n\n**Best Site** Same as for Chile, Poblano, above.\n\n**Days to Harvest** 75 to 80 days from seed.\n\n**Planting and Care** Same as for Chile, Poblano, above.\n\n**How to Harvest** Harvest serranos in the green or red stage. They are usually harvested green, when their heat has a lively, almost citrusy zing, but they are just as good red, when their flavor turns warm and ripe.\n\n**Seed Source** Same as for Chile, Poblano, above.\n\n### Chive\n\nChives are related to onions and garlic. They grow in delicate-looking clusters of narrow, slender green leaves that are loaded with flavor\u2014like a green onion, but more intense. In midspring, they produce round clusters of lilac-pink blossoms that look like puffballs. The flowers are edible and are pretty in green salads.\n\n**Best Site** Full sun or partial shade and fairly rich, well-drained soil.\n\n**Days to Harvest** Leaves can be harvested any time, once the plants begin to fill out.\n\n**Planting and Care** Grow from seed in spring or from nursery plants. Sow seeds \u215b to \u00bc inch deep and 8 to 12 inches apart. Space nursery plants 8 to 12 inches apart. Water regularly to keep soil evenly moist.\n\n**How to Harvest** Chives will send up pickable leaves from early summer until first frost. If you start from seed, harvest lightly in the first year, snipping stems near the base from the outside of the clump. For a more concentrated flavor, cut flowers while still in bud.\n\n**Seed Source** Renee's Garden, www.reneesgarden.com.\n\n### Corn 'Honey Select'\n\nTriple-sweet types like this one combine great corn flavor with especially sweet, tender kernels. 'Honey Select' hybrid sweet corn is so good you can even eat it uncooked, fresh off the stalk. If you are grilling, boiling, or steaming it, however, cook it for no more than a few minutes to preserve its juiciness and crunch.\n\n**Best Site** Full sun and moist, deeply cultivated, well-drained soil. Fertilize and cultivate prior to planting.\n\n**Days to Harvest** 80 days from seed.\n\n**Planting and Care** Start in spring after soil has warmed in moist, not wet, soil. Plant in blocks of at least 16 plants, in 4 rows of 4 plants each, planting in batches every 2 weeks. Sow seeds 4 to 6 inches apart and 1 to 2 inches deep in rows 2 to 3 feet apart. Thin to 8 to 12 inches apart when seedlings are 6 inches tall. Water regularly and deeply.\n\n**How to Harvest** When silks start to turn brown (about 3 weeks after they first appear), pull back the husk and try popping a kernel with your thumbnail. If the juice is milky, it's time to pick. (Clear juice means the corn isn't ripe yet; pasty means it's overripe.)\n\n**Seed Sources** Ed Hume Seeds, www.humeseeds.com; Park Seed, www.parkseed.com.\n\n_'Honey Select' corn_\n\n### Cucumber 'Diva'\n\nA vining type that reaches 5 to 6 feet tall and 1 to 2 feet wide, 'Diva' is very productive. Its cukes are sweet and crunchy, with no hint of bitterness. They also have no spikes on the skin, common in other varieties, which makes harvesting them a pleasure. We tried to trellis the plants, but they still flopped over and required tying. In the end, our best method was to plant them in hills or rows and let them ramble over the ground.\n\n**Best Site** Full sun and moist, well-drained soil.\n\n**Days to Harvest** 58 days from seed; 32 days from seedlings.\n\n**Planting and Care** Sow seeds in spring after the last frost date, or start seeds indoors 4 weeks before the last frost date. Space rows 3 to 6 feet apart, then sow seeds 8 to 12 inches apart and 1 inch deep, in clusters of 2 or 3 seeds. When seedlings are a few inches tall, thin them to a single seedling per cluster. Alternatively, you can grow plants in hills; make soil mounds 4 to 6 feet apart, and sow seeds 1 inch deep in each hill, in clusters of 4 to 6 seeds. When seedlings are a few inches tall, thin them to 2 or 3 per hill. Water cukes regularly through the growing season, especially when the fruits begin to swell.\n\n**How to Harvest** For cucumbers of supreme crunch and sweetness, harvest them while they are still young and no more than 6 to 8 inches long. To keep new cukes coming, harvest 3 to 4 times a week.\n\n**Seed Source** Park Seed, www.parkseed.com\n\n### Edamame 'Sayamusume'\n\nSoybeans are nutty, buttery, and high in protein, and we grew them to add substance to our summer table. We chose 'Sayamusume' for its plump beans; its sturdy, well-filled pods; and its high yield. Despite its productivity, we didn't grow nearly enough to satisfy our appetite. We would have welcomed double or even triple the harvest we got from 10 plants\u2014especially because 'Sayamusume' freezes well (we defrosted some in winter, and they were delicious).\n\n**Best Site** Full sun and well-drained soil.\n\n**Days to Harvest** 85 days from seed.\n\n**Planting and Care** Plant in spring after weather warms and night temperatures are consistently above 50\u00baF. Moisten the soil thoroughly before planting. Sow seeds 1 inch deep and 3 inches apart in rows 2 feet apart, and do not water again until seedlings have emerged; once growth starts, keep soil moist. Soybeans ripen all at once, so sow them in batches every few weeks for a continuous crop. When seedlings are several inches tall, thin to final spacing of 6 inches apart.\n\n**How to Harvest** Pluck off pods when they are nice and swollen (that means the beans have reached full size) but are still green.\n\n**Seed Source** Renee's Garden, www.reneesgarden.com.\n\n_'Sayamusume' Edamame_\n\n### Garlic 'Spanish Roja'\n\nThis heirloom \"hard-necked\" variety (meaning it develops large outside cloves and no inner cloves) is a Rocambole type known for its purple-blushed skin. The large, crisp, juicy cloves peel easily. Their flavor is pungent, though not bitingly so, and they turn buttery soft when cooked.\n\n**Best Site** Full sun and rich, well-drained soil (where soils are poor, grow it in raised beds).\n\n**Days to Harvest** Set out cloves in fall for an early-summer harvest.\n\n**Planting and Care** Plant cloves with pointed tops up, placing them 1 inch deep, 4 to 8 inches apart, in rows 16 to 18 inches apart. In cold-winter climates, mulch heavily at the onset of winter to help prevent soil heaving, which can force the cloves out of the ground. Irrigate evenly and weed regularly or mulch. Rake back mulch in spring and pinch off any blossoms that develop. Once the leaf tips start to turn yellowish brown, stop watering.\n\n**How to Harvest** When green leaves begin to turn yellow around July, stop irrigating for about 2 weeks, then carefully lift the bulbs out of the ground with a garden fork. Pulling by hand may crack the bulbs and decrease shelf life. Hang the bulbs in bundles to dry in a dry, well-ventilated area until skins are papery, about 3 weeks. Brush off the dirt, cut off most of the roots, and clip off the tops to 1 inch (unless you want to braid them). Store bulbs in mesh bags or paper bags in a cool (around 50\u00b0F), airy place out of direct sunlight for 4 to 6 months.\n\n**Clove Sources** Irish Eyes Garden Seeds, www.gardencityseeds.net; and Peaceful Valley Farm & Garden Supply, www.groworganic.com.\n\n### Hops\n\nThe pretty little cone-shaped flowers grow on a mammoth climbing vine 12 to 18 feet tall, often much taller. They provide a key flavoring ingredient in beer, which is why we grew them. We chose 'Centennial' for its slightly bitter edge and 'Nugget' for its spicy and herbal notes. We lost most of the 'Centennial' to some sort of dieback. But 'Nugget' thrived, and donated the hops for making our beer. Our total bounty, from the single 'Nugget' rhizome we planted, was about 1 pound, and when you consider that you need only 2 to 3 ounces of hops to brew a 5-gallon batch of beer, that's a lot\u2014more than enough for a party!\n\n**Best Site** Full sun and light, well-drained soil. You must have a strong trellis for the vine to climb (for details on our trellis, see the directions on this page). Plant your hops at the base\u2014and the vines will race right up and attach themselves.\n\n**Days to Harvest** 150 days from rhizome.\n\n**Planting and Care** Plant in spring once the last frost date has passed. Hops of the same variety can be planted 3 feet apart; space different kinds at least 5 feet apart to keep them from becoming entangled. Plant the rhizomes vertically, about 2 inches deep, with the bud pointing up. Mulch the soil and keep it moist. Hops need watering three times per week to almost daily when weather is hot, windy, and dry.\n\n**How to Harvest** Flowers will be ready to harvest between July and September, though your first year's yield will probably be tiny. Harvest the flower cones when they turn pale green and the yellow lupulin glands are visible underneath the bracts. When squeezed, the cones should feel slightly papery and have a pronounced odor: 'Nugget' is woodsy, and 'Centennial' has a citrusy, floral smell.\n\n**Rhizome Sources** Freshops, www.freshops.com; Nichols Garden Nursery, www.nicholsgardennursery.com.\n\n_'Nugget' hops_\n\n### Lemon 'Eureka'\n\nA good all-around lemon tree, 'Eureka' yields fruits that are large, tart, juicy, and versatile. The tree is aptly named: It bears fruit year-round in mild climates.\n\n**Best Site** Full sun and rich, fast-draining soil. Before planting, dig in a 4- to-6-inch layer of compost to a depth of 1 foot.\n\n**Days to Harvest** Depends on the age of the plant you buy. Fruit production generally begins when a tree is 3 or 4 years old. Ours was 5 years old when we started the project.\n\n**Planting and Care** Plant in spring after all danger of frost is past. Remove any fruit from the tree before planting from a nursery container. Add a 2-inch layer of mulch to help keep moisture in the soil and to prevent weeds. If your area is hot, wrap trunks of the newly planted trees with paper bands (available from nurseries) to prevent sunburn. Water consistently (twice a week in normal summer weather), and apply citrus fertilizer several times during the March to October growing season.\n\n**How to Harvest** When fruit feels heavy in your hand, looks fully formed, and is yellow (not green), pick it and try it. Citrus won't ripen off the tree, so you need to sample to know when to pick. Ripe lemons can hang on the tree for months and be fine, but if they get puffy, they are too old.\n\n**Plant Sources** Sold as grafted plants in 2- to 15-gallon cans in almost every nursery; Four Winds Growers, www.fourwindsgrowers.com.\n\n_'Eureka' lemon_\n\n### Lemongrass\n\nIntensely aromatic and citrusy, this grasslike plant is a key ingredient in many Southeast Asian cuisines. We used it to make a wonderful end-of-dinner drink and a hauntingly delicious dessert custard.\n\n**Best Site** Full sun and rich, well-drained soil.\n\n**Days to Harvest** The flavor is best when the bulbous leaf base swells to about \u00bd inch in diameter. Plants grow fastest in tropical weather, so our stalks had relatively slender bases. It took 4 months for ours to produce harvestable stems.\n\n**Planting and Care** Buy a potted plant and repot in a larger pot or in the ground in spring once all danger of frost is past. Eventually it will grow into a multistem clump 3 to 4 feet tall and equally wide. During the growing season, water well and feed monthly with half-strength fish emulsion. Lemongrass is sensitive to frost, so in winter, move potted plants to a bright spot indoors or cover in freezing weather.\n\n**How to Harvest** Push an outside stem to the side, then twist and pull it off. Cut off and discard the leaves (or twist into knots and steep in boiling water for tea).\n\n**Seedling Source** Nichols Garden Nursery, www.nicholsgardennursery.com.\n\n### Marjoram, Sweet\n\nSweet marjoram (known just as \"marjoram\" in the culinary world) bears a strong resemblance to its kin, oregano. But its little leaves are softer and finer textured than those of oregano, and the plant has a tingly, spicy flavor, like a blend of mint, cloves, and cinnamon. We grow it along gravel paths and as a pretty edging for our vegetable beds.\n\n**Best Site** Full sun and soil that drains especially well, or in a container indoors on a sunny windowsill.\n\n**Days to Harvest** About 70 days (leaves can be picked any time once plants are established and growing).\n\n**Planting and Care** Buy nursery plants in spring and plant them 9 inches apart. Keep the soil moist, not soggy, until the plants are established, then water less. Mature plants thrive on little water. Or, to plant from seed, sow in spring after the soil has warmed. Till the soil well, rake it smooth, then sow the seeds thinly and cover them lightly. Keep the seedbed moist and well weeded while seedlings are young.\n\n**How to Harvest** Pinch off leaves. For a stronger flavor, cut stems when they are in bud (the blossoms are edible, too).\n\n**Seed Source** Renee's Garden, www.reneesgarden.com.\n\n### Melons\n\nWe grew luscious, fragrant, cream-colored 'Sharlyn' melon, which tastes like a cross between a honeydew and a cantaloupe; seedless 'Sugar Baby' watermelon, a small, crisp, juicy melon that won't take up an entire shelf in the fridge; and deep orange, meltingly soft 'Ambrosia' cantaloupe (cantaloupes are also known as muskmelons). All these melons need plenty of heat in order to sweeten.\n\n**Best Site** Full sun and rich, well-drained soil.\n\n**Days to Harvest** 85 days from seed.\n\n**Planting and Care** Plant outdoors in spring after the soil has warmed to at least 60\u00b0F (raised beds warm up faster than flat ground). Mix a 2- to 3-inch layer of compost into the soil. Plant seeds 1 inch deep and 4 inches apart, in rows 4 to 6 feet apart. As seedlings grow, thin them to 10 inches apart. Use drip irrigation or a soaker hose to avoid wetting the foliage, and water often. When the melons reach full size but are not yet ripe, cut back on watering to keep the fruits from splitting and to concentrate their flavors.\n\n**How to Harvest** When fully ripe, cantaloupes slip off the vine easily. 'Sharlyn', a honeydew cross, is ready when the fruit has a strong, fruity fragrance and there's an orange tone to the skin beneath the tan-colored netting. For watermelons, wait for the tendril next to the stem to wither and the \"resting spot\" on the underside of the melon to turn creamy yellow. Avoid twisting the melons as you check them. It can damage the stem and thus their nutrient supply.\n\n**Seed Source** Seeds of Change, www.seedsofchange.com.\n\n### Onion 'Spanish White'\n\nLarge and evenly spherical, with good steady onion flavor, 'Spanish White' won't turn sweet when cooked. We planted in fall with high hopes for a good crop. But by the following April, our plants weren't looking so good. The bulbs didn't swell; many just produced flowers. We wondered whether temperatures or erratic watering was the cause. By summer, half our plants had not produced bulbs. Luckily, the rest of them finally matured.\n\n**Best Site** Full sun (or light shade) and fine-textured (well-raked), loose, rich, well-drained soil.\n\n**Days to Harvest** 150 says from seed; 100 days from sets.\n\n**Planting and Care** Plant from seeds or sets (miniature, dormant onions). Sow seeds when the soil has warmed to at least 35\u00b0F and preferably 50\u00b0F, setting them about \u00bc inches deep in the soil, in rows 15 to 18 inches apart. Thin seedlings to 4 or 5 inches apart. Put out sets in spring, 4 to 6 weeks before the last frost date. Sets are easiest and produce quick results, but they may bolt into flowers, rendering the onion inedible. Select the smallest sets available, because they are less likely to bolt than large ones. To plant, push the sets under the soil, aligning their pointed ends with the soil level; space 4 to 5 inches apart. Water plants\u2014whether from sets or seeds\u2014regularly when green tops are growing. Weed regularly but carefully, because bulbs are easily damaged.\n\n**How to Harvest** Water less frequently toward the end of the growing cycle. When 25 to 50 percent of the foliage in your onion patch has fallen over, the onions are nearly ready. Stop watering, bend the rest of the green tops to the ground, and let the bulbs harden and cure in the soil for 3 weeks before you pull them. Store the bulbs in a cool, dry spot.\n\n**Seed\/Sets Source** Gurney's Seed & Nursery, www.gurneys.com.\n\n### Oregano, Italian\n\nA familiar seasoning for bean dishes, pastas, and vegetables, Italian oregano ( _Origanum_ x _majoricum_ ) grows about 30 inches tall, producing tiny, pointy green leaves with an intense, spicy flavor. It is used in Mexican recipes, too, which means we relied on it a lot in our one-block cooking.\n\n**Best Site** Full sun and soil that drains especially well, or in a container indoors on a sunny windowsill.\n\n**Days to Harvest** Leaves can be picked any time once plants start to fill out.\n\n**Planting and Care** Buy nursery plants in spring and plant them 18 inches apart. Keep the soil moist, not soggy, until the plants are established, then water less. Mature plants thrive on little water.\n\n**How to Harvest** Pinch off leaves any time; for a stronger flavor, cut stems when they are in bud (the flowers are edible).\n\n**Seedling Source** Mountain Valley Growers, www.mountainvalleygrowers.com.\n\n### Parsley 'Gigante Italian'\n\nMost chefs prefer to cook with Italian flat-leafed parsley ( _Petroselenium crispum neapolitanum_ ), which has a more robust flavor than the curly-leafed types once commonly used as garnishes. Flat-leafed parsley is also very easy to grow in a pot or in the ground. We grew 'Gigante Italian', which has large, crisp, mellow-tasting leaves that we used all kinds of ways, including whole in salads.\n\n**Best Site** Full sun or part shade (afternoon shade in the hottest climates) and rich, well-drained soil that's been well amended with compost.\n\n**Days to Harvest** 80 days from seed, or, from seedlings, any time after plants have filled out.\n\n**Planting and Care** Start seeds in the spring when the weather is settled but still cool (or in fall in hottest desert climates). Scatter seeds thinly over the soil, then cover them with \u00bc inch of soil. Parsley tends to germinate unevenly over several weeks, but don't be discouraged! Just keep the soil evenly moist. When the seedlings have a few fully formed leaves, thin them to 18 to 24 inches apart. Water regularly through the growing season and feed plants occasionally with liquid fish emulsion. If you would rather take the easy route, plant seedlings in containers (at least 12 inches wide and 10 inches deep) or in the ground.\n\n**How to Harvest** Start picking leaves when the plants are 8 to 10 inches tall and relatively full. Harvest entire stems from the base to encourage growth.\n\n**Seed\/Seedling Source** Renee's Garden, www.reneesgarden.com.\n\n_'Gigante Italian' parsley_\n\n### Pattypan Squash\n\nWe grew disk-shaped, scalloped-edged 'Benning's Green Tint' (lime green) and 'Sunburst' (brilliant yellow) for their tender skins, delicate flavor, and few seeds. Plus, they are small and perfect for stuffing.\n\n**Best Site** Full sun and rich, well-drained soil.\n\n**Days to Harvest** 50 days from seed.\n\n**Planting and Care** Sow seeds indoors in a warm place in spring 1 week or so before the last frost date, or wait until the soil has warmed (at least 2 weeks after the last frost date) and sow seeds outdoors. To sow outdoors, plant seeds 12 inches apart and 2 to 3 inches deep in rows 3 to 5 feet apart. Once seedlings have several sets of leaves, thin them to 2 to 4 feet apart. Water regularly and deeply. Avoid splashing water on leaves, stems, and flowers.\n\n**How to Harvest** Cut stems close to the fruits with a sharp knife (wear gloves; those stems are prickly). Unless you need larger squashes for stuffing, harvest them when they are 2 to 3 inches across\u2014they taste best as tender \"babies.\" Regular harvesting keeps the plants producing through September.\n\n**Seed Sources** For 'Benning's Green Tint', Baker Creek Heirloom Seeds, www.rareseeds.com, and Southern Exposure Seed Exchange, www.southernexposure.com; for 'Sunburst', Park Seed, www.parkseed.com.\n\n_'Sunburst' pattypan squash_\n\n### Peppermint\n\nPeppermint has dark, slender, smooth-edged leaves and purplish stems. We love it for its almost shocking mintiness\u2014it makes the best cup of mint tea ever. Like all mints, it spreads rapidly by underground stems and can be invasive in a kitchen plot. To keep it in bounds, we always grow it in a pot and set the pot atop paving (it can even root through the drainhole!).\n\n**Best Site** A large container filled with potting soil, in a sunny or partially shaded location\u2014mint is not fussy.\n\n**Days to Harvest** Leaves can be picked any time.\n\n**Planting and Care** Set out young nursery plants in spring, spaced at least 6 inches apart, in containers. For best growth, keep the soil moist. After 3 years, plants become bare in the center, so you need either to replace them or to dig them up, divide each plant, and replant a few of the pieces with roots.\n\n**How to Harvest** Snip sprigs of new growth.\n\n**Seedling Source** Mountain Valley Growers, www.mountainvalleygrowers.com.\n\n### Potato 'Yukon Gold'\n\nThe best all-purpose potato, this spud has buttery, sweet flesh that is equally suited to boiling and baking. We hung onto our harvest for as long as we could, using it into the winter. A pretty plant with lightly fragrant flowers, the potato is a cinch to grow. But knowing when the spuds are ripe for digging is another matter. Our best advice: Watch for foliage to die back, then pull one up.\n\n**Best Site** Full sun and fertile, fast-draining soil high in organic matter and with a pH on the acid side (below 5.5).\n\n**Days to Harvest** 90 to 120 days after planting for fully mature potatoes. You can dig potatoes when they are small and tender (when plants begin to bloom), or when fully mature (when plants die down). For long-term storage, dig only mature potatoes. To store potatoes, see Our Makeshift Root Cellar.\n\n**Planting and Care** Plant seed potatoes in spring 4 to 6 weeks before the last frost date, around mid-March in our area. Two days before planting, cut them into chunks about 1\u00bd inches square, each with two \"eyes\" from which sprouts will emerge, and leave them to dry (callus) in a dry area, either inside or outside, to help prevent rotting.\n\n**How to plant in furrows** Dig furrows 6 to 8 inches wide and 4 inches deep, spacing them 2\u00bd to 3 feet apart. (Closer spacing will result in higher yields of smaller potatoes.) Then set the chunks 12 to 18 inches apart in the furrows and cover with 2 inches of soil. After sprouts emerge, add another 2 inches of soil, leaving the foliage tips exposed. As the vines grow, continue adding soil, mounding until you have ridges 4 inches high and 18 inches wide. The soil cover keeps the tubers at the right temperature, with the right amount of moisture. It also protects them from the sun, which can raise levels of solanine, a toxic compound. (The sun also turns potato skins green, which is usually a good indicator of toxin development.)\n\nKeep soil uniformly moist during the growing season. Weed and watch for pests.\n\nWhen most of the foliage has turned yellowish brown, water the plants for the last time, wait for 14 days, and then cut away the vines. This \"sets,\" or hardens, potato skins so they won't peel or bruise easily. If you dig them up without hardening them, the skins will be as fragile as tissue paper and the potatoes will be moist, sweet, and highly perishable. Harvest these \"fresh\" potatoes carefully and cook them the same day or the next.\n\n**How to plant in towers** (See photo.) To save space and make potatoes a snap to harvest, plant them in towers made of wire and reed screening (find materials in a hardware or home supply store). Loosen the soil with a spading fork, then rake it smooth. Using a trowel, trace a circle about 3 feet in diameter in the prepared soil, then place seed potato pieces (see Planting and Care) about 5 inches apart within the circle. Bend a piece of welded wire into a cylinder 3 feet in diameter and 3 to 4 feet tall; secure the cut ends together with wire. Put the \"cage\" on top of the soil circle around your newly planted potato sets. Wrap a piece of reed screening (we bought a 12-foot privacy screen and cut it to fit) around the wire cage, then tie its ends together with twine. Pour a 3-inch layer of compost and another 3 inches of straw into the cylinder. Hand-water the potato sets every other day or so.\n\nOnce foliage appears (be patient; ours didn't show up for several weeks), continue adding compost and straw, leaving about 6 inches of green stem and leaves on top exposed to sunlight. Nearing the tower's top, just add straw.\n\n**How to Harvest**\n\n**How to harvest furrows** Within 5 to 7 days of cutting away the vines, preferably when the weather is cool and overcast, dig up the plants carefully with a spading fork, keeping it 10 inches away from the plants to avoid injuring the potatoes. Lift each plant gently, shake off the loose soil, and pull the potatoes from the vines. Gather them in burlap bags or baskets and keep out of strong sunlight.\n\nIf you are planning to store the potatoes long-term, it is vital that the air temperature be cool when you harvest, so do it early in the morning (see Our Makeshift Root Cellar).\n\n**How to harvest new potatoes in towers** After the plants flower, open and remove the reed screening, then poke your hand through the welded wire and feel around for potatoes. Pick some small, tender, thin-skinned \"new\" potatoes. Replace the screening.\n\n**How to harvest mature potatoes in towers** After the plants flower, withhold water for 2 to 3 weeks, or until the entire plant dies back. Then pull up the towers. All the potatoes will spill out in a perfect cascade\u2014no digging required, and no spade damage to the potatoes.\n\n**Seed Potato Sources** The Cook's Garden, www.cooksgarden.com; Gurney's Seed & Nursery, www.gurneys.com; Henry Fields Seed & Nursery, www.henryfields.com; Irish Eyes Garden Seeds, www.irisheyesgardenseeds.com; Wood Prairie Farm, www.woodprairie.com.\n\n_'Yukon Gold' potatoes_\n\n### Purslane\n\nOkay, we know what you're thinking: Why would anyone in their right mind want to plant a weed in their summer garden\u2014especially when it seems to pop out of the soil all by itself? Here's why: Purslane's rounded, succulent leaves add a citrusy zip and juicy texture to salads and stir-fries, and they're high in vitamin C. The little yellow-flowered green plant ( _Portulaca oleracea_ ) is eaten with relish in many countries, including France (where it's called _pourpier_ ), Mexico ( _verdolaga),_ Holland ( _postelein_ ), and Turkey ( _semizotu_ ). We used it in a delectable Turkish-style salad with garlic, chiles, and mint.\n\n**Best Site** Full sun and loose soil that's been enriched with compost.\n\n**Days to Harvest** 60 to 70 days from seed.\n\n**Planting and Care** Sow seeds after last frost date (in our area, April, when weather warms up to 68\u00b0F or more, is a good time). Sow the seeds \u00bc inch deep in warm soil and rake the seeds in lightly. Keep the bed well watered. Seeds germinate quickly (7 to 10 days).\n\n**How to Harvest** Once plants are growing well (June to October), pick tender tips and shoots as you need them. To preserve juiciness, harvest in early morning when temperatures are cool.\n\n**Seed Source** Seeds of Change, www.seedsofchange.com.\n\n_Purslane_\n\n### Rosemary 'Tuscan Blue'\n\nWe like this plant, sometimes sold under the name 'Blue Spires', for its sturdy good looks\u2014it's an upright variety with broad leaves and dark blue flowers; its piney, intense, uplifting fragrance when you rub it between your fingers; and its versatility in the kitchen. Plus, the plant is tough, standing up to wind, salt spray, heat, and poor soil.\n\n**Best Site** Full sun and soil that drains especially well, though it will tolerate poor, dry soil.\n\n**Days to Harvest** Leaves can be picked any time.\n\n**Planting and Care** Set out small nursery plants in early spring (the typical kitchen garden will need only a few of these vigorous plants), spacing them at least 24 inches apart. Water the plants regularly until they are established, then only enough to keep them from drying out. Rosemary often fails in soggy soil.\n\n**How to Harvest** Snip branch tips with scissors or pruners.\n\n**Sources** Plants are widely sold at nurseries.\n\n_'Tuscan Blue' rosemary_\n\n### Thyme, French\n\nLike rosemary and oregano, thyme grows well in our Mediterranean climate, and we constantly make use of its tiny, aromatic leaves in our cooking. Plus, it looks gorgeous in the garden, clustered at the end of a row and nearly always shooting up at least a few pretty little blossoms. Our bees like it, too.\n\n**Best Site** Full sun and light, well-drained soil, though it will tolerate poor, dry soil.\n\n**Days to Harvest** Seedlings appear 14 to 21 days after sowing, and need at least a month to fill out before you start picking. Once mature, leaves can be picked any time.\n\n**Planting and Care** Set out small nursery plants in early spring, 8 to 12 inches apart. Water regularly until the plants are established, then only to keep the plants from drying out completely. Or, to plant from seed, wait until the soil has warmed, then sow thinly \u00bd inch deep in a fine-textured (well-raked) bed and cover the seed lightly. Keep the seedbed moist and well weeded while seedlings are young.\n\n**How to Harvest** Snip branch tips (the flowers are edible, too).\n\n**Seed\/Seedling Sources** Seedlings are sold in almost every nursery, or order seeds from Renee's Garden, www.reneesgarden.com.\n\n### Tomatoes\n\nWe grew five different tomatoes: deep yellow 'Sungold' cherry tomatoes, for their fabulous flavor and juiciness; prolific 'Sweet Million' red cherry tomatoes; dependable red 'Early Girl'; citrusy, tart 'Green Zebra'; tender, yellow-and-red-streaked 'Marvel Stripe'; and succulent, magenta-purple 'Brandywine'. We loved them all. But we grazed most often on 'Sungold', popping the juicy, sun-warmed fruits into our mouths like candy, right off the vine.\n\n**Best Site** Full sun (at least 8 hours a day) and fertile, well-drained soil rich in organic matter.\n\n**Days to Harvest** 50 to 90 days after setting out plants.\n\n**Planting and Care** Plant seedlings as soon as the danger of frost has passed. Remove the lowest leaves, then plant them deeply, burying at least half of the stem to encourage vigorous root growth. Space the seedlings widely, at least 24 inches apart, and support them with sturdy stakes or wire cages to help prevent rot. If starting from seed, sow indoors in a warm place 5 to 7 weeks before setting them out (at least 10 days after the last frost). Water often enough to keep soil damp but not soggy, and keep water off the leaves. Tomatoes need lots of fertilizer. In rich soils that have been well amended with compost, they don't need anything. But if you have unamended soil, dig a single application of controlled-release fertilizer into the soil at planting time. Or feed plants lightly with a dilute liquid fish emulsion every two weeks from blossom set until harvest.\n\n**How to Harvest** Snip fruit from the stem with scissors or pruners or gently pull by hand. Tomatoes ripen best on the vine. If your plants still have green tomatoes hanging on them after the weather starts turning cold, pick them from the vine and bring them indoors to ripen.\n\n**Seed\/Seedling Sources** For seeds, Seeds of Change, www.seedsofchange.com, or Renee's Garden, www.reneesgarden.com; for seedlings, Laurel's Heirloom Tomato Plants, www.heirloomtomatoplants.com.\n\n_'Early Girl' tomatoes_\n\n### Wheat\n\nInitially, we grew wheat for brewing our beer: The carbohydrates in wheat convert to sugars when the grain is malted (sprouted), then, during fermentation, the sugars convert to alcohol. Once our one-block project progressed, we realized we could not have a year's worth of menus without a loaf of bread, crackers, or a crust of any kind. So, we chose a soft white wheat, which produces golden \"berries\" with a nutty, sweetish flavor.\n\n**Best Site** Full sun and well-tilled, good-draining soil with plenty of compost mixed in. Till in blood meal and bonemeal (about 1 pound each per 100 square feet) along with the compost.\n\n**Days to Harvest** 130 to 150 days from seed.\n\n**Planting and Care** In January, scatter seeds over prepared soil (about 6 handfuls, or \u00bc pound, per 100 square feet), then rake seeds in so they are buried to a depth about three times their diameter. After planting, soak soil thoroughly, then water as needed to keep moist (every week in dry weather; we tend to get lots of rain in January, so we didn't need to water at all). Weed regularly. When the wheat starts turning brown, stop watering; when your tooth can no longer dent the berries, the crop is ready to harvest.\n\n**How to Harvest** Snip the seedheads from the stalks with scissors or pruners. For details about processing the wheat, see How to Make Beer.\n\n**Seed Sources** Homegrown Harvest, www.homegrownharvest.com; for 'Sonora' soft white wheat, contact Pie Ranch, www.pieranch.org; Full Belly Farm, www.fullbellyfarm.com; EatWell Farm, www.eatwell.com); or Whole Grain Connection, www.sustainablegrains.org.\n\n### Zucchini 'Trombetta di Albenga'\n\nProbably the most captivating plant of our summer menu, 'Trombetta di Albenga' is an extremely vigorous climbing vine with big, fan-shaped leaves and huge, pale green zucchini that can curve like trombones. They taste sweet and mild, stay crunchy even when cooked, and all the seeds are concentrated in the bulbous end, so most of what you get is seed free. We grew ours over an arbor (one plant on each side), so we could look up at the pretty squashes as they ripened and to give their voluptuous shapes room to dangle.\n\n**Best Site** Full sun and rich soil amended with aged manure or compost.\n\n**Days to Harvest** 85 days from seed.\n\n**Planting and Care** Sow seeds indoors in a warm place in spring 1 week or so before the last frost date. Or, wait until the soil has warmed, at least 2 weeks after the last frost date, and sow seeds outdoors. Sow 3 or 4 seeds 1 to 2 inches deep about 4 inches from a vertical support (you will want to save space by growing this vigorous vine on a sturdy trellis, fence, or stakes). Thin to the strongest seedling for each stake or 12 to 24 inches apart. Water deeply.\n\n**How to Harvest** The fruits are best when they are 10 to 12 inches long, though they are usually still tender up to 30 inches. Use pruners to cut the fruits\u2014with a bit of stem attached\u2014from the vine. Pick flowers in the morning, snipping them off the vine when they're cool and fresh and still open; eat that day if possible and no later than the next. Keep in the refrigerator in a plastic bag until ready to use.\n\n**Seed Source** Renee's Garden, www.reneesgarden.com.\n\n_'Trombetta di Albenga' zucchini_\n\n_\u2014Kathleen N. Brenzel_\n\nTHE \"IMPORTS\"\n\n**We went outside our garden for these local foods:** milk, whole and unhomogenized (for cream, and for cheese); olives (for olive oil); seawater (for salt); wine grapes (for wine); brewing kit (as a backup for our beer crops); mother of vinegar (for vinegar)\n\n# THE SUMMER PROJECTS\n\n##    \nHOW TO RAISE  \nHONEYBEES  \n(AND MAKE HONEY)\n\nWe began our bee adventure in part because we needed honey to sweeten our summer feast. But we also did it for the love of bees.\n\nYou have probably heard the statistic: bees, traveling from plant to plant and pollinating as they go, are responsible for producing about one-third of our country's food supply. Unfortunately for them and for us, they have been dying in huge numbers over the past several years. No one has conclusively figured out what causes colony collapse disorder, as it has been termed.\n\nWe figured that by raising bees, we would contribute, in at least a small way, to the overall population of bees. We also thought it would be easy. Plus, we have plenty of plants for them to browse, and conditions that are relatively benign (good weather, lots of well-watered suburban gardens). Life should be great for bees at Sunset.\n\nNow, having seen our bees battle ants, bluejays, wasps, buckeye poisoning (the pollen is toxic to bees), varroa mites, tracheal mites (probably), hive beetles, wax moths, and American foulbrood, we understand just what these hardworking little creatures are up against (for a running diary of our life with bees, see <http:\/\/oneblockdiet.sunset.com\/team_bee>). And it's deepened our appreciation for what marvelous, interesting creatures they are (see Amazing Bees).\n\nThe stings? Well, working in the hives is not the killer-bee kind of drama you might see in a late-night horror movie. Bees only sting in self-defense and when they are protecting the hive. If you have calm, docile bees, and move slowly as you work the hives, you can hold them in your bare hands. Even with care, of course, we have been stung a few times, and for the most part it was no big deal.\n\nRaising bees has been and continues to be an incredible experience. We will never taste honey again without a sense of gratitude.\n\n###    \n2 COLONIES OF HONEYBEES\n\nBoth of our hives are filled with European honeybees ( _Apis mellifera ligustica_ ), known for their gentleness and productivity. We recommend asking your neighbors whether they'd be comfortable with hives next door (promise them honey!) before you actually go out and buy bees. Also, check with your city to see if local regulations allow you to keep bees.\n\n#### WHAT TO USE\n\nUnless otherwise noted, our beekeeping supplies came from Dadant & Sons (www.dadant.com or 217\/847-3324). We also recommend Mann Lake Ltd. (www.mannlakeltd.com or 800\/880-7694) for beekeeping equipment. Prices do not include shipping.\n\n##### 2 MASTER POLLINATOR KITS\n\nEach kit ($310 each) includes:\n\n**_1 telescoping outer cover with galvanized metal top and an inner cover_** The inner cover helps insulate the box. The outer cover provides weather protection. Used together, they prevent the bees from gluing the hive shut with propolis (a sticky substance they collect from plants to bond parts of the hive) and wax.\n\n**_2 hive bodies_** These are the \"brood\" boxes, deep boxes that the bees fill with brood (bee larvae), honey, and pollen. Each box has 10 frames loaded with Plasticell foundation (a thin sheet of plastic molded into hexagonal cells and coated with beeswax to make it easier for the bees to build the honeycomb).\n\n**_1 bottom board_** We did not use this, preferring the \"Country Rubes\" bottom board, which helps you screen for mites.\n\n**_1 metal hive tool_** This 10-inch-long tool is used to pry apart frames and to scrape off propolis and extra comb.\n\n**_1 bee brush_** A gentle sweep with this brush removes bees from the surfaces of combs and frames.\n\n**_1 smoker_** A small metal can with bellows that puff smoke when the fuel inside is lit. Smoke calms the bees, making working in the hive easier for the beekeeper.\n\n**_2 pounds smoker fuel_** We have had the best luck with KwikStart Smoker pellets (about $8 for 100) and cotton fuel (about $2) from Mann Lake Ltd. We have also used eucalyptus leaves, dead grass, pine needles\u2014anything that will provide long-lasting, cool smoke.\n\n_Preparing the smoker_\n\n**_1 entrance feeder_** Warned that these encourage robbing by intruder bees, we made our own feeders (see Other Equipment).\n\n**_1 zipper veil suit_** We chose large, to fit the largest member of Team Bee, but it became hazardous to the smallest members of our team, because the loose pant legs allowed bees to crawl into the suit. We ultimately bought suits that fit Team Bee regulars.\n\n**_1 plastic helmet and veil that zips to the suit_** A zipped-on veil keeps bees from sneaking under the helmet.\n\n**_1 pair leather beekeeping gloves_** The gloves come in small, medium, large, and extra large. Most of the women on Team Bee found small to be the best size.\n\n**_2 books_** These guides proved invaluable: _The Hive and the Honey Bee,_ edited by Joe M. Graham (Dadant & Sons, 1992); and _First Lessons in Beekeeping,_ by Keith S. Delaplane (Dadant & Sons, 2007).\n\n**_A year's subscription to the American Bee Journal_**\n\n**_1 medications brochure_** We found sound advice for dealing with bee diseases and problems in these pages.\n\n##### NOT INCLUDED IN THE KIT\n\n**_2 queen excluders, one for each hive_** An excluder is a screen with a mesh small enough to prevent the queen (larger than the rest of the bees) from crawling up into the \"honey super\" (a box designated for honey collection) and laying eggs there. About $11.50 each.\n\n**_2 \"Country Rubes\" bottom boards, one for each hive_** A bottom board is a small box on which the hive rests. Designed to help manage varroa mites (see Pest Control), it has an open, screened bottom and a removable plastic board. $39 each from www.countryrubes.com.\n\n**_2 honey supers (unassembled), one for each hive_** A honey super is a box that is smaller than the brood box and is set on top of it. It is normally used with a queen excluder to get clean honey. Each measures 6\u215d inches tall, with frames. $22.75 each.\n\n**_20 Plasticell foundations_** Plastic sheets, 5\u00bd inches by 16\u00be inches, used for filling the frames in the supers. About $0.75 each.\n\n**_2 drone frames, one for each hive_** These are used to combat varroa mites (see Pest Control). $3 each.\n\n**_3 vented helmets_** Much more comfortable than the plastic helmets from Dadant. A helmet paired with a round veil (see below)\u2014plus a white, long-sleeved shirt, light-colored pants, and closed shoes\u2014works fine as a basic beekeeping outfit, with gloves for heavy or prolonged work or for days when the bees are grumpy. About $14 each from Mann Lake Ltd.\n\n**_3 round veils with string tie-ons_** These are not quite as secure as zip-on veils, but they are fine for working with gentle bees like ours, and you can put them on faster. About $12 each from Mann Lake Ltd.\n\n##### BEES\n\nYou can get bees three ways:\n\n**_Packaged bees and caged queen_** It takes time to build up the colony this way, but it's the least-expensive choice. You can usually order packaged bees through your local beekeepers' guild. Preorder as early as the fall and certainly no later than early spring, as bees are only available for a short time in spring. About $65.\n\n_Our caged queen, Califia (with packaged bees)_\n\n**_Nuc (short for \"nucleus\")_** A nuc is a young hive, usually covering no more than 5 frames of comb, with a newly laying queen. Starting this way helps you get a jump on honey production. Buy from a reputable beekeeper to avoid getting diseased equipment or sick bees. We ordered two nucs from master beekeeper Randy Oliver and drove to his location in Grass Valley, California, to pick them up. $90 for each nuc, queen included; www.scientificbeekeeping.com.\n\n**_Well-established swarms or colonies_** Large colonies can be daunting if you've never kept bees before, and beginning beekeepers shouldn't try to capture a swarm. Leave that to a more experienced beekeeper (contact your local beekeepers' guild to find such a person), and perhaps he or she will help you start a hive with the captured swarm.\n\n##### OTHER EQUIPMENT\n\n**_2 hive stands, one for each hive_** The stands raise the hives off the ground. Made from scrap two-by-fours, each is 16\u00bc inches wide, 20 inches long, and 1 foot tall and has an open top. We painted our stands with white latex paint.\n\n**_8 sturdy square plastic food containers_** 32-ounce size. About $4 for 4 containers at a grocery store.\n\n**_Terro ant bait_** About $7 for a package of 3 units at a nursery, garden center, or hardware store.\n\n**_Apiguard_** For treating varroa mites. About $31 (enough for 10 individual treatments).\n\n**_Formic acid_** Also for treating varroa mites. About $40 (enough for 10 treatments) from Mann Lake Ltd.\n\n**_2 feeder tops, one for each hive_** These boards hold upended jars that dispense sugar syrup down into the hive to feed the bees.  \nTo make a board, trim a thin piece of plywood to fit on top of the hive. Then cut a hole with a diameter slightly smaller than that of the feeder jar lid (see below). Put the board on top of the hive. Set a filled jar over the hole, lid down, so that it completely covers the opening.\n\n**_2 (1-quart) glass jars and lids_** Punch several small holes in each lid with a nail to create feeder lids, fill the jars with sugar syrup, and then screw the lids on the jars. Any large clean glass jar will do.\n\n**_10 cups (about 4\u00bd pounds) granulated sugar_** You need this sugar to make the sugar syrup for the feeder jars. About $7.50 for a 5-pound bag at a grocery store.\n\n**_Powdered sugar_** Used to control mites. You will need 1 cup per brood box per hive per week for as long as you are dusting for mites (see Pest Control). About $2 for 2 pounds at a grocery store.\n\n**_Wooden spoons_** About $2 each at a cookware store.\n\n**_Bench scraper_** Also known as a pastry scraper. From $8 online or at a cookware store.\n\n**_Large glass bowl_** About $9 online or at a cookware store.\n\n**_Cheesecloth_** About $4.50 for 2 square yards at a hardware store or cookware store.\n\n**_Large stainless-steel strainer_** About $25 for a good sturdy one (we like OXO brand) at a cookware store.\n\n**_Honey jars and lids_** About $9 per 24-count box of 3-ounce hexagonal jars and about $6 per 12-count box of 6-ounce hexagonal jars from Mann Lake Ltd. (We used 5 boxes of the small jars and 2 boxes of the larger jars to bottle 31 pounds of honey.)\n\n**_5-gallon food-grade plastic bucket with 1\u00bd_** ** _-inch honey gate_** Bubbles and foam rise to the surface and the pure honey settles to the bottom, where it can be drawn from the honey gate (a kind of faucet). About $19 from Mann Lake Ltd.\n\n#### HOW TO DO IT\n\n**1. Choose the location** Bees need four things. First, they need **sun** , or afternoon shade if your weather is hot. Second, they must have access to fresh **water** near the hive. We used a large plant saucer with stones in the center for the bees to land on and refreshed the water every day. A shallow bubble fountain would work well, too. Third, the hive must be **protected from wind** , which can blow rain (or snow) into the hive, making it harder for the bees to keep the hive warm. Finally, bees need **privacy**. Don't put the hives near high-traffic areas, play areas, swimming pools, or pet areas. Give each hive plenty of space\u201450 feet from high-traffic areas is ideal, but if space is limited, position the hive so the entrance is near a tall fence or hedge. This will force their flight path overhead to minimize contact with people and pets. And screening them from view will keep bees and people happy.\n\n**2. Prepare the location** Hives should face south, if possible, and they need to be kept off the ground to protect them from dampness and critters. After clearing the brush and leveling the ground, we poured a cement pad to make care easier.\n\n**3. Install the bees** Spring, when blooming flowers furnish a food supply, is the time to put your bees in their hives. Once you've chosen how to buy them, the best bet is to rely on your source for installation instructions.\n\nHere is what happened when we picked up our bees from Randy Oliver at his property in Grass Valley: He gave us an introductory class in beekeeping, showing us how to use the hive tool and the smoker, handle bees, and check for eggs, brood (larvae), and queen\u2014all vital signs of a colony's health. Randy loaded 5 frames of his gentle hybrid bees and a queen into each of our two brood boxes and sealed the openings by stuffing them with our beekeeping gloves. We used ratchet straps to secure the boxes in the back of our truck. When we got back to Sunset, we positioned the brood boxes in their designated locations and removed the gloves from the entrances. One hive we named Betty; the other, Veronica.\n\n_Hives Veronica and Betty on a sunny day_\n\n**4. Feed the bees** Young colonies have a lot of work to do\u2014storing pollen and nectar, sealing all the cracks and seams in their new home, and taking care of the queen and new brood. To make their adjustment easier, we fed them a \"nectar.\" Here is how to make it: Dissolve equal parts granulated sugar and water and use to fill the quart jars. Top with the feeder lids and invert the jars into the holes. The lids should not drip; they should be barely moist. The bees will drink what they need from the lids.\n\nIn the beginning, our nucs drank about three-fourths of a quart jar per day. Over the next 3 weeks or so, it tapered off to the point where we realized sugar water was no longer necessary. The bees were finding their nutrition in flowers. Plus, sugar water makes for insipid honey and should not be continued if it is not needed.\n\n**5. Inspect the hives inside and out** Much of beekeeping is simple observation and response. If you are a novice beekeeper, inspect the hive about once a week for a couple of months so that you can learn. Once you feel comfortable, adjust your routine to every two weeks. Make sure the outside of the hive is clean and free of bee poop, the landing board is free of litter, and there are no ants on the hive. Open the hives and check frames for larvae and eggs (on warm days only). If the queen is healthy, you will see plenty of larvae in various stages of development. If you don't see evidence of a healthy queen, consult an expert. Your local beekeeping guild is a good source.\n\nUltimately, the less often you inspect the hive, the better for its health. Opening the hives and thoroughly checking them requires smoking to keep the bees calm. This stresses the bees and it takes them about a day to recover. As you learn more, you will find you won't need to pull many frames to know what is going on inside. And you will figure out a lot simply by observing the bees as they come and go from the hive.\n\n_Comb from our top-bar hive_\n\n**6. Check regularly for pests and diseases** Varroa mites are the pest most typically found in hives. Left unchecked, they can cripple and eventually kill the hive (see Pest Control, for hints about checking for mites and mite control). Other pests you need to watch for include the small hive beetle and the wax moth. Diseases you need to be on the lookout for are American and European foulbrood. Early intervention can often mean the difference between a healthy hive and a dead hive.\n\n**7. Expand the hive when necessary** Start with one deep hive body-brood box. When the bees have filled it with 7 or 8 frames of bees and brood, top it with a second brood box. Let the bees build up brood cells in the second brood box, too. When the second brood box is well filled (7 or 8 frames of bees), top it with a queen excluder, if you choose to use one, and, finally, the honey super (the box from which you will collect most of your honey).\n\n##### PEST CONTROL\n\nBees are like flying balls of delicate spun sugar filled with honey. Everything wants to eat them. Here are three of the worst pests we battled, and the tactics we used.\n\n**Ants** Argentine ants can kill a hive by robbing honey and eating the brood. We couldn't spray to kill the ants, since that would also kill the bees. We tried Terro ant bait\u2014little containers filled with boric acid mixed with a sweet substance ants like\u2014with some success. In the end, we were most successful with a physical barrier. We placed each leg of the hive stands in plastic tubs filled with water that the ants could not cross.\n\n**Small hive beetles** Hive beetle larvae will eat all parts of the hive, including the baby bees. We kill the beetles on site, and have been experimenting with traps like AJs Beetle Eater ($5.25) from Dadant.\n\n**Varroa mites** The most damaging pests a beekeeper has to deal with are these mites, as they threaten the survival of a hive once they become established. They suck the blood of adult bees and lay their eggs in brood cells, where their larvae feed off bee babies, infecting them with viruses and weakening and even killing them. To save their bees, beekeepers use a variety of methods:\n\n**1. Monitoring** A 24-hour count of a natural mite fall will give you a good idea of a hive's infestation. Coat the bottom of your Country Rube board with petroleum jelly or cooking spray (to trap the mites), slide it into the lower part of the bottom board, wait for 24 hours, and then pull it out and count the mites. Anything more than 10 mites per brood box indicates you have a problem.\n\n**2. Sugar dusting** The powdered sugar method lets you both count the mites and control them. Sift powdered sugar, 1 cup per brood box, over the tops of the frames and brush it into the hive. The powdered sugar makes the mites lose their grip on the bees and fall off; plus the bees groom the sugar off their bodies, dislodging more mites. Again, use the bottom board to capture the fallen mites. You should not see more than a few mites 10 minutes after dusting. If there are more, you have a problem.\n\n_Brushing powdered sugar into a hive_\n\n**3. Mite trapping** Drone frames will also help trap varroa mites. These frames are designed to encourage bees to make drone comb cells, which are larger than worker comb cells. Since varroa mites prefer drone brood 10 to 1, the drone comb makes a great mite trap. Just before the drones hatch (24 days after the eggs were laid), destroy the drone comb (you can freeze it and return it to the hive, or simply cut it out), and replace the drone frame for the next cycle. (Since our queens have already mated and have a lifetime's supply of sperm inside of them, they do not need the drones in order to reproduce.)\n\n**4. Apiguard** A gel infused with thymol, made from the oils of thyme plants. It works well, but it makes the honey stored during the treatment taste like mouthwash.\n\n**5. Formic acid** More toxic than thymol, formic acid kills the mites by gassing them. It makes the honey inedible for humans, so it is applied in the fall and winter, when the nectar flow is slow or stopped. You need to wear a respirator when applying it.\n\nFor more information on mite control, see the sources listed under Helpful Information, opposite.\n\n##### HONEY COLLECTION\n\nWe were lucky to collect honey the first summer. Typically, during the first year the bees build up their hive, and if they overwinter well, you can begin harvesting in the late spring or early summer of the second year.\n\nThree months after bringing our bees home, we had 4 frames packed with honey, each weighing about 8 pounds. Lacking a professional extractor, we used the following low-tech method.\n\n**1. Cut and crush** Using the bench scraper, we cut the honey\u2014wax and all\u2014off the foundation into a bowl, balancing the frame on a wooden spoon set across the bowl like a bridge. Then we used a wooden spoon to crush the honey and wax in the bowl.\n\n_Cutting the honeycomb off the foundation_\n\n_Crushing the comb_\n\n**2. Straining and settling** We poured this slurry of wax and honey through a double layer of cheesecloth and the stainless-steel strainer into our food-grade plastic bucket. Then we left it to drain and settle for a couple of days (bubbles and foam rose to the surface).\n\n**3. Bottling** We covered the floor with newspapers and got our jars ready. Then we loosened the honey gate (the stopper at the bottom of the bucket) to release the honey into each jar. In went the honey, on went the lids. It was as simple as that. From 4 full frames of honeycomb, we reaped 12 pounds, 10 ounces of honey. We rinsed the leftover wax and froze it. Later, we rendered the wax in a solar wax melter and used it for craft projects like lip balm and hand salve (for more information, see <http:\/\/oneblockdiet.sunset.com\/team_bee>.) We had a second surprise harvest later in the summer, bringing our total to about 31 pounds of pure, fragrant honey.\n\nAMAZING BEES\n\nWorker bees\u2014all females\u2014work hard their whole lives. That's not true for the males, known as drones. They lounge about stealing honey until it is time for them to fly out and look for a queen to mate with. But only a day or two after a working girl pulls herself from her six-sided cell, she begins contributing to the health of the hive.\n\nAs they age, worker bees take on a series of jobs (although tasks sometimes overlap, depending on the needs of the hive). At five to ten to days, they are house bees, cleaning the hive (bees are very clean), tending the brood (baby bees), taking care of the queen, and unloading nectar from the foraging bees and storing it in the wax combs. (The foragers take in the rest of the groceries\u2014the pollen\u2014themselves.) As they age, house bees start taking on other tasks, like guarding the hive and ventilating it by fanning their tiny wings across the comb. At two to three weeks, they become foragers, flying up to three miles away to find suitable blooming plants. Even as foragers they will help build comb, ventilate the hive, and protect it from honey-hungry animals. Bee colonies are adaptable, and every worker can switch jobs when needed.\n\nIn the summer, the worker bees work so hard they just wear out after about thirty-five to forty days. The queen, no slacker herself, lays up to 1,500 eggs a day during the spring and summer brood buildup to provide a constant supply of new workers. In the winter, bees live longer\u2014140 days or more\u2014keeping the hive and queen warm, dry, and safe until spring, when the season begins again.\n\n_Drones (in center)_\n\n_Workers_\n\n_Queen (in center)_\n\n#### HELPFUL INFORMATION\n\n##### GENERAL WEB SITES\n\n\u2022 www.scientificbeekeeping.com\n\n\u2022 www.sanmateobee.org\n\n\u2022 www.sfbee.org\n\n\u2022 www.beesource.com\n\n\u2022 www.beginningbeekeeping.com\n\n\u2022 www.caes.uga.edu\/departments\/ent\/bees\/beekeeping.html\n\n##### WEB SITES ABOUT HOW TO CONTROL VARROA MITES\n\n\u2022 www.caes.uga.edu\/departments\/ent\/bees\/disorders\/honey-bee-parasites.html#varroa\n\n\u2022 www.countryrubes.com\/information\/informationandpictures.html\n\n\u2022 www.scientificbeekeeping.com\n\n\u2022 www.dadant.com\/Apiguard-Howtouse_003.htm\n\n##### BOOKS AND PUBLICATIONS\n\n\u2022 _The Hive and the Honey Bee_ , edited by Joe M. Graham (Dadant & Sons, 1992).\n\n\u2022 _First Lessons in Beekeeping_ , by Keith S. Delaplane (Dadant & Sons, 2007).\n\n\u2022 _The Backyard Beekeeper_ , by Kim Flottum (Quarry Publishing Group, 2010).\n\n\u2022 _American Bee Journal_ , $24.95 per year (monthly publication); 51 South 2nd Street, Hamilton, IL 62341, 217\/847-3324.\n\n\u2022 _Bee Culture: The Magazine of American Beekeeping_ , $25 per year (monthly publication); 623 West Liberty Street, Medina, OH 44256, 800\/289-7668.\n\n_\u2014Margaret Sloan and Kimberley Burch_\n\n##    \nHOW TO MAKE  \nCHEESE (PART I)\n\nMilk needs very little encouragement to become cheese. Add heat and coagulant (like vinegar or lemon juice) and curds separate from whey to create, tah-dah, cheese.\n\nOur first cheeses, for our summer feast, couldn't have been simpler, because we restricted ourselves to what we could get from our garden: lemons for coagulation and herbs for flavoring, plus salt, which we made from seawater, and milk, from a Bay Area dairy (eventually we found our own cow). But even with our supersimple cheeses, the results were slightly different depending on how we adjusted the variables: temperature, time, amounts, and technique.\n\n#### BEFORE YOU START\n\n**_Sanitize_** All kinds of bacteria love the warm, nourishing medium of milk. To keep bad bacteria from messing with the good, scrub your work surface with antibacterial soap. Boil utensils\u2014ladle, whisk, measuring cups, and so on\u2014for 10 minutes in an 8-quart pot before you use them. Wash everything else with antibacterial soap and let air-dry. After you finish making cheese, wash your equipment and let it air-dry.\n\n**_Check your thermometer_** If the recipe calls for a dairy thermometer, check that it is calibrated correctly, or you might ruin your cheese. It's easy to check. As you're sterilizing it (submerging only the stem) with your other utensils, check the readout: It should be 212\u00b0F, the temperature of the boiling water. If it isn't, hold the face and adjust the pointer by turning the washer underneath the face with small pliers.\n\n###    \nFRESH CHIVE CHEESE\n\nMAKES 18 ounces (7-inch log; 28 servings)\n\nTIME about 2\u00bd hours, plus about 2 hours draining and chilling time\n\nThis recipe is based on one for _chenna_ , an Indian cheese, in Ricki Carroll's _Home Cheese Making_. Making it involves kneading the curds into a satiny, moist \"dough\" that you then roll into a log, chill, and slice to use. We added it to our Tomato and Herb Salad.\n\n#### WHAT TO USE\n\n**_Milk_** Lacking a cow, goats, or sheep here at Sunset, we had to go outside our one-block parameter for milk. (Later, we bought a dairy cow.)  \nBoth pasteurized and raw whole milk will work in any of our cheese recipes. The only milk that will not work is ultra-pasteurized milk; the extreme heat of the pasteurizing process (at or above 280\u00b0F for 2 seconds) damages proteins in the milk and makes it harder for curds to form. Ultra-pasteurized milk and cream also have an unpleasant \"cooked\" flavor.  \nWe used cream-top organic whole milk from Straus Family Creamery (www.strausfamilycreamery.com), near Tomales Bay in Marin County. It's sold in San Francisco Bay Area stores, starting at about $4.50 per gallon; 1 gallon yields about 1 pound cheese.\n\n**_Lemons_** We used plain old 'Eureka' lemons\u2014in our case, from our garden\u2014squeezed and the juice strained.\n\n**_Sea salt_** Salt is key for flavor development. It also extracts moisture from the curd and helps preserve the cheese. We used salt extracted from seawater (see How to Make Salt). Any good fine sea salt will work.\n\n**_Herbs_** You can use any herbs that appeal to you, from parsley to tarragon to mint. We chose chives, because we like their sharp, oniony flavor and emerald color against the white cheese.\n\n**_8-quart heavy-bottomed stainless-steel pot_** For heating the milk. From $60 at a cookware shop.\n\n**_Large colander_**\n\n**_Cheesecloth_** Lining the colander with a finely woven cheesecloth prevents any small curds from tumbling through the holes. About $4.50 for 2 square yards at hardware stores or cookware shops.\n\n**_Stainless-steel perforated cheese ladle_** A cheese ladle has a large, flat surface that stirs milk and curds efficiently; you can also use a large slotted spoon. About $7.\n\n**_Dessert or salad plate_**\n\n**_5-pound weight_** Anything will do, from a barbell to heavy cans. We used round kitchen-scale weights.\n\n**_Stand mixer with dough hook attachment (optional)_**\n\n#### HOW TO DO IT\n\n1 gallon whole milk, pasteurized or raw\n\n\u00bd cup freshly squeezed lemon juice (4 to 5 large lemons)\n\n2 tablespoons coarsely snipped fresh chives\n\n\u00be teaspoon fine sea salt\n\n**1. Separate the curds and whey** Pour the milk into the 8-quart pot and bring to a gentle boil over medium-high heat, stirring often to prevent scorching (this will take about 30 minutes, so have a book handy). As soon as the milk boils, remove from the heat and slowly drizzle in the lemon juice while stirring slowly and gently with a rubber spatula. Keep stirring until solid white curds separate from greenish white, translucent liquid whey. This happens within seconds. If the curds have separated but the whey is still milky instead of clear, return the pot to low heat and cook until the whey is clear, then remove from the heat. Let sit until the curds have settled below the whey, about 15 minutes.\n\n**2. Drain the curds** While the curds are settling, line the colander with a double thickness of cheesecloth and set it in the sink. Ladle the curds into the colander and rinse gently with lukewarm water for 5 seconds. Gather the cheesecloth into a knot over the curds and twist gently to squeeze out some of the liquid (the \"bundle\" should still be dripping after you have done this). To save the whey left in the pot for making ricotta, strain it through a cheesecloth-lined strainer into a container and chill until ready to use (up to 1 day).\n\n**3. Press the curds** Place the dessert plate on the cheesecloth-wrapped curds and top with the 5-pound weight. Let drain for 45 minutes. At this point, the curds may still be dripping a bit, but that's okay. You want some moisture in the cheese.\n\n**4. Knead the curds** Unwrap the curds and put them in the bowl of the stand mixer fitted with the dough hook. Add the chives and salt and beat on low speed until the cheese looks silky, 10 to 12 minutes. It should be as moist and smooth as cream cheese. Or, combine the curds, chives, and salt in a large bowl and knead by hand for about the same amount of time. The hand-kneaded cheese will be denser and more tightly grained than the machine-beaten cheese.\n\n**5. Shape and store the cheese** Roll the cheese into a log about 2 inches in diameter. Wrap in waxed paper and then plastic wrap. Refrigerate until cold and firm, at least 1 hour, before using. It will keep for up to 3 days.\n\n_PER \u00bc-INCH SLICE: 86 cal., 49% (42 cal.) from fat; 4.6 g protein; 4.7 g fat (2.9 g sat.); 6.9 g carbo (0 g fiber); 103 mg sodium; 20 mg chol._\n\n_Sanitizing the equipment_\n\n_Field trip to Cowgirl Creamery_\n\n_Tasting curds in the Sunset kitchen_\n\n###    \nOREGANO QUESO BLANCO\n\nMAKES about 2 cups (10\u00bd ounces)\n\nTIME about 2 hours\n\nCrumbly and mild, this cheese reminds us of fresh Mexican _queso blanco_ , or \"white cheese,\" which is often also called _queso fresco_. It is based on the recipe for lemon cheese in Ricki Carroll's _Home Cheese Making_ , but we more than doubled the lemon juice of the original, and we whisked the curds before draining them. The result: tart, moist, fluffy small curds that are good on soups, salads, or pizza. We sprinkled the cheese on our corn soup.\n\n#### WHAT TO USE\n\nSame as for Fresh Chive Cheese.\n\n#### HOW TO DO IT\n\n2 quarts whole milk, pasteurized or raw\n\n\u00bd cup plus 1 tablespoon freshly squeezed lemon juice (4 to 5 large lemons)\n\n1\u00bdtablespoons fresh oregano leaves, minced\n\n1 teaspoon fine sea salt\n\n**1. Separate the curds and whey** Pour the milk into the pot and bring just to boiling over medium-high heat, stirring often to prevent scorching. As soon as the milk looks as though it is about to boil, remove the pot from the heat and drizzle in the lemon juice while whisking briskly. Cover and let sit for 10 minutes.\n\n**2. Drain the curds** Meanwhile, line the colander with a double thickness of cheesecloth and set the colander in the sink. Pour in the curds. Tie two opposite corners of the cheesecloth into a knot over the curds, then repeat with the other two corners. Hang the cheesecloth sack from the sink faucet until the curds have stopped draining, 1 to 2 hours.\n\n**3. Flavor and store the cheese** Transfer the curds to a bowl and mix in the oregano and salt with your fingers, breaking up the curds into small grains. Cover tightly and refrigerate. It will keep for up to 1 week.\n\n_PER TABLESPOON: 38 cal., 47% (18 cal.) from fat; 2 g protein; 2 g fat (1.3 g sat.); 3.2 g carbo (0 g fiber); 91 mg sodium; 8.5 mg chol._\n\n#### HELPFUL INFORMATION\n\n##### WEB SITE\n\n\u2022 The Pacific Northwest Cheese Project, www.pnwcheese.typepad.com.\n\n##### CHEESE-MAKING SUPPLIES\n\n\u2022 The Beverage People (wine-, beer-, and cheese-making supplies), www.thebeveragepeople.com or 800\/544-1867.\n\n\u2022 Hoegger Supply Company, hoeggergoatsupply.com or 800\/221-4628.\n\n\u2022 New England Cheesemaking Supply, www.cheesemaking.com or 413-397-2012.\n\n##### BOOKS AND DVDS\n\n\u2022 _Home Cheesemaking: Recipes for 75 Homemade Cheeses,_ by Ricki Carroll (Storey Books, 2002). First published in 1982, this little book has been reissued three times and has guided many a beginner to professional success.\n\n\u2022 _The Cheesemaker's Manual_ , by Margaret Peters-Morris (Glengarry Cheesemaking Inc., 2003). From a guru of the cheese-making world and the owner of Glengarry Cheesemaking Inc, in Lancaster, Ontario (Canada), this concise, well-organized book is a bit dry but packed with helpful information for home cheese makers, plus about fifty recipes for soft, hard, and washed-rind cheeses. Glengarry also sells a DVD, _Home Cheesemaking with Margaret Morris_ , in which Ms. Morris calmly and cheerfully demonstrates how to make feta, Gouda, Camembert, and Cheddar (about $38).\n\n##    \nHOW TO MAKE  \nBEER\n\n_Barley_\n\nMaking home-brewed beer doesn't take long. You can make ale, the easiest type, in only four to six weeks. Also, you can play around with all sorts of ratios and roasts to make exactly the kind of beer you like. The biggest challenge is becoming familiar with the process and with the key pieces of equipment.\n\nMost home brewers use brewing kits to make beer, and that's how we started, too. For our second batch in the spring, we made beer totally from scratch\u2014as in, we used the wheat and barley that we had planted, threshed, winnowed, and malted (sprouted) ourselves, and grew hop flowers (called cones) for flavoring and preserving the brew. We were in a little over our heads, but we had expert advice and fun doing it.\n\nWe recommend that you start with a kit and either brew with a friend who has made beer before or take a class. Once you're comfortable with the process, progress to partial mash brewing (making beer from malted grain steeped in malt extract; we plan to do this next, and will write about it on our Team Beer blog, <http:\/\/oneblockdiet.com\/team_beer>). Then try making all-grain brews, first from pre-malted grains, then from grain you malt yourself, and ultimately\u2014if you're so inclined\u2014from grain you grow, for a beer that is as personal as it gets.\n\n###    \nSUMMER WHEAT BEER (from a kit)\n\nMAKES 48 (12-ounce) bottles\n\nTIME about 1 month\n\nFresh and light, yet flavorful, this beer went well with the round, ripe flavors of our summer feast.\n\n#### WHAT TO USE\n\nAll of our materials, unless otherwise mentioned, came from William's Brewing Company, 2594 Nicholson Street, San Leandro, California; www.williamsbrewing.com or 800\/759-6025.\n\n**_Honey wheat brewing kit_** A beginner-level kit contains one 6-pound pouch wheat extract, one 2-pound pouch blackberry honey, 2 ounces flavoring hop pellets (pulverized and compressed hop cones), 1 ounce aromatic hops, one 125-milliliter packet liquid yeast, and 1 packet corn sugar (fine, white and powdery; also called priming sugar) for carbonation. About $36.\n\n**_Saniclean_** Safe for septic systems and does not stain; good for sanitizing your chiller, fermenters, ladle, tubing, and bottles. Use 2 ounces for 5 gallons of water. About $13 for a 1-quart bottle.\n\n**_7-gallon boiling pot_** Also known as a brewing pot or brew kettle. A large, sturdy stockpot, available at any cookware or housewares store, will do. Or, you can use the 8-gallon boiling pot with spigot that we used for our Belgian Abbey Ale.\n\n**_Medium plastic ladle_**\n\n**_Long-handled plastic stirring spoon_**\n\n**_Immersion chiller or plate chiller_** A coil of copper tubing, the immersion chiller is placed in the hot wort (the sweet liquid, made from malted grains, that will become beer) and then flushed with cold water to cool the wort rapidly. You can also put the pot in a sink full of cold water and change the water when it gets hot, but the chiller works much faster. The plate chiller, also known as a Shirron heat exchanger, is a slender metal box that brings in the cold tap water and the hot wort at the same time through separate tubes, instantly chilling the wort to about 70\u00b0F. The chiller costs about $55, the heat exchanger about $87.\n\n**_Brewer's thermometer_** Indispensable for measuring the temperature of the new wort. If the wort is too hot, it will destroy the yeast. Brewer's Edge makes a thermometer with a large, easy-to-read face and a 12-inch probe stem. About $12.\n\n**_Primary fermenter_** Also called a siphonless fermenter, this 7-gallon food-grade plastic bucket with a spigot is used for the fermenting stage. We've also used a 5-gallon carboy, but this is cheaper. About $30.\n\n**_Strainer_** Choose a sturdy one that fits into the top of the primary fermenter. About $15.\n\n**_Hydrometer_** A hydrometer measures the specific gravity (density) of the wort relative to water. Dissolved sugars make up the density of the wort, and the sugars are what ferment into alcohol, so the hydrometer measurement is a good indicator of the potential alcohol level of your beer. The MT300 hydrometer from www.morebeer.com has a potential alcohol scale. About $6.\n\n**_Hydrometer jar_** A plastic tube with a flat base, sized to fit the hydrometer; used for holding the beer sample. $4.75 from www.morebeer.com.\n\n**_Rubber stopper_** This almost fully seals the opening at the top of the primary fermenter. A small perforation in the middle accommodates the airlock (see below). $1.00 to $1.50, depending on size.\n\n**_Airlock_** You fill this small plastic cylinder with water and insert it into the rubber stopper. This keeps bacteria from entering the new wort. It also allows the carbon dioxide in the fermenting beer to find a way out, so your container doesn't explode. About $1.\n\n**_Large, dark cloth_**\n\n**_Secondary fermenter_** Also known as a priming tank, this second 7-gallon ** _,_** food-grade plastic bucket with a spigot is similar to the primary fermenter, but the spigot is placed lower on the bucket. This bucket is also used for bottling; it includes a small length of tubing, one end of which attaches to the spigot. The other end fills the bottles. About $20.\n\n**_3-foot length food-grade vinyl tubing_** Clear vinyl, with a \u215c-inch interior diameter, used to siphon the beer from one vessel to another. About $0.30 per foot at a home wine-making or plumbing-supply store.\n\n**_Bottles_** You'll need 48 pry-top bottles (screw-top bottles are harder to seal) in dark green or brown glass (sunlight shining through clear glass can stimulate the growth of bacteria). We scrounged ours from various sources, namely friends, family, and colleagues. Free.\n\n**_Jet bottle washer_** This fits on any outdoor hose thread faucet, such as those on a garage sink. About $12.\n\n**_Bottle tree_** Invert your newly washed and sterilized bottles on this multipronged \"tree\" to dry. We like the 81-bottle model. About $30.\n\n**_Beer caps_** A pack of 320 pry-type caps costs about $6.\n\n**_Capper_** We like the easy-to-use Emily capper; $14.\n\n**_Or, get a complete kit_** William's Brewing also offers a \"home brewery,\" which includes most of the tools above (excluded are the bottles, caps, and boiling pot) and the ingredients for your beer of choice (from $110). The company also throws in a beginner's home-brewing book and a DVD guide.\n\n#### HOW TO DO IT\n\nBREWING (about 3 hours, plus up to 3 days for the yeast to swell)\n\n**1. Get the yeast going** Pop the puffy center of your liquid-yeast packet and shake the packet. That bubble contained your yeast, and surrounding the bubble is wort for the yeast to feed on and multiply. Let the packet swell; this takes at least a day and up to three. Do not start the beer until the yeast packet is at least 1\u00bd inches thick.\n\n**2. Prepare your boiling pot** Sterilize it with Saniclean solution, rinse with water, and fill the pot with 5 gallons of water. Add 2 teaspoons gypsum to the water to harden it.\n\n**3. Mix the wheat extract with the hot water** Cover the pot and bring the water to a full boil over high heat. (That much water can take a long time to boil, so don't be in a hurry.) When the water boils, turn off the heat and squeeze the wheat extract into the water. Use the ladle to scoop hot water into the pouch to remove the residual extract, and then use your stirring spoon to mix the extract with the water. Now the mixture is considered wort.\n\n**4. Bring the wort to a boil** When the extract is well mixed, cover the pot partially and bring the wort to a boil over high heat. Then uncover and keep the wort at a rolling boil. Be careful not to let it boil over the top of the pot, because it is a very sticky mess to clean up.\n\n**5. Add the flavoring hops** Five minutes after the wort has begun to boil, add the flavoring hops. Boil for 45 minutes.\n\n**6. Add the honey** Squeeze in the honey as you did the wheat extract, ladling some of the hot wort into the pouch to loosen any honey residue. Boil for 3 minutes.\n\n**7. Add the aromatic hops** Stir in the aromatic hops and boil for 2 minutes more. Remove the wort from the heat.\n\n**8. Chill the wort** Use the immersion chiller or plate chiller, or put the pot in a cold-water bath in the sink, to cool down the wort to the yeast's ideal fermentation temperature. The yeast packet will note the temperature; ours listed a range of between 58\u00b0 and 74\u00b0F. Depending on which method you use, the chilling process can take anywhere from a few minutes (the plate chiller) to 30 minutes (immersion chiller) to more than 1 hour (the sink).\n\n**9. Start fermenting** Pour the wort through the strainer into the primary fermenter, add the yeast, and give the wort a good stir with the stirring spoon.\n\n**10. Measure the original specific gravity (density) of the wort** Insert the hydrometer into the hydrometer jar, drizzle in enough beer from the fermenter's spigot to make the hydrometer float, and read off the original specific gravity (OG) and the potential alcohol. Make a note of the OG. Although the precise figure isn't as important when you're using a kit, it will tell you how dense the wort is and will give you a way to measure the progress of the fermentation (as the sugars in the wort convert to alcohol, the mixture will become less dense).\n\n**11. Set up the beer for primary fermentation** Put the fermenter lid on and insert the rubber stopper into the hole in the lid. Insert the airlock into the stopper, and then fill the airlock one-third full with water. Leaving only the airlock exposed, wrap the fermenter in the dark cloth. This blocks sunlight, which can stimulate the growth of bacteria. Keep the wort at the yeast's ideal fermentation temperature (see step 8). You can use an electric heating pad under the fermenter if the room gets too chilly.\n\n##### FERMENTATION (2 to 4 weeks)\n\n**1. Primary fermentation** After a day or two, the wort should start to ferment. As the carbon dioxide escapes, the airlock will start to sputter, bubble, and hiss vigorously. Once it has calmed down, generally within a few days, the primary fermentation is complete.\n\n**2. Secondary fermentation and racking** Let the beer keep on fermenting for the time specified by the kit. In our case, this was 12 more days (for a total fermentation time of 16 days). Even though the airlock may not be sputtering, carbon dioxide is still forming and quietly seeping out through the airlock.\n\nMeanwhile, the dead yeast cells are slowly filtering to the bottom, creating sediment (or trub). Moving the beer off the trub into a clean container (racking) helps to clarify it as it ferments. To rack your beer, set the primary fermenter on a countertop and the secondary fermenter on the floor below. Remove the airlock from the primary fermenter. Work one end of the 3-foot vinyl tubing onto the spigot of the primary fermenter, put the other end in the open secondary fermenter, and let the beer flow down. Shut off the spigot when you see sediment starting to enter the tube. Remove the vinyl tubing, snap the lid on the secondary fermenter, and seal it with the stopper and airlock. Wrap it up in the dark cloth to keep out light, and re-wrap after subsequent rackings (below).\n\nWhen another layer of trub forms on the bottom of the secondary fermenter, usually in about a week, rack the beer again the same way, this time returning it to the primary fermenter.\n\nAt the end of this secondary period, check the final gravity again with your hydrometer (see Brewing, step 10). Our kit specified a reading of 1.017 or less. If the reading is higher than that, let the beer ferment for another few days.\n\n**3. Clean your bottles** At least a day before you think you will be bottling, start getting the bottles ready. Mix up a bucketful of Saniclean solution and put a batch of bottles in it for a few minutes, making sure the solution fills them completely. Then empty out the sanitizer back into the bucket and rinse the bottles with hot water using the jet bottle washer. Invert the bottles onto the bottle tree to drain. Sanitize the rest of the bottles the same way.\n\n##### BOTTLING DAY (about 3 hours)\n\n**1. Carbonate the brew** If the beer is not already in the secondary fermenter, siphon it in there. Stir in the corn sugar, which the surviving yeast will gobble up, releasing carbon dioxide\u2014in other words, carbonation.\n\n_Carbonating_\n\n**2. Bottle the beer** Dunk the bottles in Saniclean solution and drain them on your bottle tree. Put the end of the secondary fermenter's tubing into your first empty bottle, holding it below the fermenter. Open the fermenter's valve and fill to within 1 inch of the neck bottom.\n\n**3. Cap the bottles** Put a cap on the bottle, place the capper over it, and push down on the capper's arms to seal.\n\n##### CARBONATION IN BOTTLE (1 to 2 weeks)\n\n**1. Let the beer rest** Store in a cardboard box at room temperature (68\u00b0F or above) for a week. Then try a bottle: If it's fizzy, chill the beer and drink as soon as possible. If it is not fizzy enough, let it go for another week. When you like it, chill it.\n\nThis beer tastes best when it is freshly made. Ours kept well in the refrigerator for months, but the fizz and flavor were at their peak right after bottling.\n\n###    \nBELGIAN ABBEY ALE (from barley, wheat, and hops)\n\nMAKES 48 (12-ounce) bottles\n\nTIME 1\u00bd years\n\n_Belgian Abbey Ale_\n\nOur aim was to make a combination barley-wheat beer, totally from scratch.\n\n#### WHAT TO USE\n\nAll-grain brewing requires a few more pieces of equipment than kit-brewing does, but if you have already brewed using a kit, you can use the same equipment and just add to it. All of our materials, unless otherwise mentioned, came from nearby William's Brewing Company, 2594 Nicholson Street, San Leandro, California; www.williamsbrewing.com or 800\/759-6025.\n\n**_2 hop rhizomes ('Nugget' and 'Centennial') for planting_** The rhizome, the main stem of the hop vine, is sold cut into small pieces that look like little brown sticks.\n\n**_3 pounds barley seed for planting, to yield 9 pounds malted, prepped barley_** Barley is the main grain used in beer. We planted 'Lacey', a malting type (versus a cereal type) developed for brewing. About $5 per pound from Howe Seeds, www.howeseeds.com.  \nOur total yield was about 2\u00bd pounds of grain, after winnowing and threshing (squirrels carried away most of the crop). The total yield after malting (sprouting) was 2 pounds. We needed about 9 pounds malted, prepped barley to make one 5-gallon batch of beer, so we had to supplement with 7 pounds store-bought malted barley. If we do this again, we'll grow more barley, as insurance. About $2 per pound for whole malted grain, $2.25 per pound for crushed malted grain.\n\n**_2 pounds wheat seed for planting, to yield 6 pounds malted, prepped wheat_** We planted organic soft white wheat. It is high in carbohydrates, which convert to sugars and then alcohol. About $5 per 2-pound bag from Homegrown Harvest, www.homegrownharvest.com.  \nThose same marauding squirrels caused us to lose a lot of wheat. Our total harvest was about 1\u00bd pounds. Our total yield after malting (sprouting) was 1 pound. We needed 6 pounds for our 5-gallon batch of beer, so, as with the barley, we had to supplement with 5 pounds of store-bought malted seed (wheat malt). About $2 per pound for whole wheat malt, $2.25 per pound for crushed wheat malt.\n\n**_Trellis for growing hop vines_** Hop vines can reach upward of 30 feet, so we had to get creative. The lower part was a 5-by-10-foot piece of rigid \u00bc-inch wire mesh. With deck screws, we attached four wooden blocks to a section of our garden fence (one block for each corner of mesh), screwed a lag hook into each block, and fitted the mesh over the hooks. Above the trellis and about 3 feet below the top of the fence, we attached two 2-inch-square wooden poles, each 8 feet tall. We attached them with deck screws to the fence and spaced them 10 feet apart. We ran several lengths of electrical wire between the poles, so the hops could grow across as well as up. The trellis cost $60 at a scrap metal shop. Everything else came from Home Depot: 6-inch-square wooden blocks (cut from an 8-foot board), about $8.50; 8-inch lag hooks, about $1.50 each; 8-foot poles, about $5 each; outdoor deck screws, about $8.75 for a 1-pound box of 75 screws; and uncoated electrical wire, from $0.25 per foot.\n\n**_Hop-drying frame_** We built a 2-by-4-foot wooden frame from two-by-fours, and then nailed a piece of fine wire mesh to the bottom, sized to fit the frame. About $15 for all supplies at Home Depot.\n\n**_Vacuum food sealer_** We vacuum-sealed our dried hops for storage. (You can also store your hops in two heavy-duty resealable plastic bags, one set inside the other.) Seal-A-Meal VS230 Vacuum Food Sealer, about $71 from www.amazon.com.\n\n**_Several ordinary kitchen colanders_** for separately draining the soaked barley and wheat.\n\n**_Grain mill_** The Barley Crusher MaltMill, with a 15-pound hopper, for use with a standard \u215c-inch power screwdriver. $138 from www.barleycrusher.com.\n\n**_Abbey Ale yeast_** Fruity and rich, this distinctive yeast is used in two of the world's six Trappist breweries. Like many Belgian yeasts, it is capable of continuing to ferment even when alcohol levels are high (high alcohol kills many yeasts). We had a lot of malted grain in this recipe (that is, potential sugars and therefore alcohol), so we thought it would be the right choice. About $6 for a 1-ounce tube from www.morebeer.com.\n\n**_Corn sugar_** Also known as priming sugar, this fine, white, powdery sugar (sometimes available granulated) feeds the yeast, which unleashes the carbon dioxide that carbonates the beer. About $2 per pound.\n\n**_Food-grade gypsum_** A blend of powdered calcium and sulfur, gypsum is used to harden the water. Hard water makes for clearer beer by helping coagulate malt proteins and encouraging the sedimentation of yeast. You will need only 1 teaspoon gypsum for each 5-gallon batch. About $3 for an 8-ounce bag.\n\n**_8-gallon boiling pot with spigot_** This big, stainless-steel vessel is also known as a brewing pot or brew kettle. About $85.\n\n**_5-gallon all-grain mash tun_** A Rubbermaid beverage cooler with a spigot attached and a fine-mesh screen laid over the bottom to keep the grains from clogging the spigot. About $125 from www.morebeer.com.\n\n**_5-gallon all-grain hot liquor tank_** A Rubbermaid beverage cooler with a spigot attached. About $48 from www.morebeer.com.\n\n**_Sparger_** Also known as a sparge tip, this plastic water dispenser has a central tube and four arms to keep it balanced atop the mash tun. Its job is to dribble water through the tube and arms very slowly and evenly into the mashed grain to rinse off residual sugar. About $25 from www.morebeer.com.\n\n**_Two 3-foot lengths food-grade vinyl tubing_** Made of clear vinyl, and with a \u215c-inch interior diameter, the tubing is used to connect the hot liquor tank to the sparger and to let the wort flow from the mash tun into the boiling pot. It is also used later to siphon the beer into bottles. About $0.30 per foot online or at a home wine-making or plumbing-supply store.\n\n**_Immersion chiller or plate chiller, Saniclean, stirring spoon, brewer's thermometer, primary fermenter, strainer, hydrometer, hydrometer jar, rubber stopper, airlock, secondary fermenter, bottles, jet bottle washer, bottle tree, beer caps, capper_** See the What to Use list for Summer Wheat Beer.\n\n**_Or, get an All-Grain System kit_** The kit includes the mash tun, hot liquor tank, and sparger, plus tubing. $219 from www.sanfranciscobrewcraft.com; $235 from www.morebeer.com.\n\n#### HOW TO DO IT\n\n##### GROWING THE CROPS\n\n**1. Plant the hops** In April, 18 months before we would savor our brew, we planted the rhizome pieces (we knew that the first season would yield few, if any, flower cones, so we had to plan for a second-season harvest). Because of the hop vine's prodigious size and abundance, we planted the pieces about 6 feet apart, right beneath the metal trellis that would support their fast-growing shoots (see photo).\n\n**2. Plant the barley** In November (for Northern California; for other regions, see planting calendars), plant the seed and care for the crop as directed on here.\n\n**3. Plant the wheat** In January (for Northern California; for other regions, see planting calendars), plant the seed and care for the crop as directed on here.\n\n##### HARVESTING, THRESHING, AND WINNOWING THE BARLEY AND WHEAT\n\n**1. Harvest the barley** After the barley forms seeds in the spring, wait for the stalks to dry; they should be golden brown. If the weather is hot, this won't take long (we harvested in mid-May). Use hand pruners to clip a couple of inches below the seedheads.\n\n**2. Dry the barley** Lay a tarp out indoors and spread the freshly harvested stalks in a single layer on the tarp. When the color of the barley changes from golden brown to grayish brown, it is fully dried. This takes about a month, depending on the heat and humidity.\n\n**3. Harvest the wheat** Like the barley, the wheat seedheads should dry on the stalk to a golden brown before harvesting. We harvested our wheat a bit early because our crop was being devoured by squirrels. Harvest the wheat the same way you harvested the barley.\n\n**4. Dry the wheat** Dry it the same way you dried the barley. It will take about the same amount of time.\n\n**5. Thresh and winnow the barley and the wheat** Keeping the barley and wheat separate, thresh (loosen the grains from the stalks and the papery casings surrounding each grain, called chaff) and winnow (separate the seeds from the stalks and chaff) them. Each member of Team Beer tried a different tactic, but most ended up laboriously hand-peeling the casings from the grains and then either blowing off the chaff or soaking the grain and skimming the debris.\n\nWhat we should have done, we've since learned, is thresh by banging the seedheads against the inside of a clean metal trash can to get them off the stalk. Then we should have stuffed the loosened seeds into a sturdy, thickly woven pillowcase or sack (clean of course) and walked in place on it, twisting our feet over the grain to rub off those papery bracts. As for winnowing, it's apparently easiest done in front of a floor fan turned to the lowest setting. You put a large sheet pan (or maybe a sheet itself) on the ground and then let loose handfuls of grain above the fan's windstream. The grains drop and the chaff flies away.\n\nIf we grow grain again, we'll try these methods.\n\n##### HARVESTING AND DRYING THE HOPS\n\n**1. Harvest the hops** 'Centennial' succumbed to fungus and bugs, so we were left with 'Nugget'. By September, the resiny balls called lupulins, found at the base of the flower cones, had turned golden yellow and were quite bitter, a sign of ripeness. (The lupulins, along with alpha acids in the flower, contribute a spicy, herbal, pleasingly bitter note to beer.) Harvesting was easy: We just snipped the cones from the vine with scissors.\n\n_Hop rhizomes_\n\n_Hop harvest_\n\n_'Nugget' hops_\n\n**2. Dry the hops** We spread a carpet of newspapers on the floor and set our hop-drying frame on top, with a couple of bricks under each corner so air could circulate underneath. We placed the hop cones on the screen, spreading them out in a single layer. After about a week of drying, the hops were still green, but sprang back when pressed\u2014a sign that they were dry enough. We kept small vacuum-sealed bags in the freezer. For our Belgian Abbey Ale, we used 3 ounces of hops.\n\n##### MALTING THE GRAIN (at least 6 days)\n\nThis involves converting the barley and the wheat from a starch storage unit (a seed) to a slightly sweet, crunchy grain that can be further converted, through the brewing process, into the fermentable sugary solution known as wort (pronounced \"wert\").\n\n_Malted grain_\n\n**1. Soak the grain** Put the 2 pounds homegrown barley and 1 pound homegrown wheat seeds in separate bowls. Add water to cover by at least 2 inches, then leave the seeds to soak for 8 hours. Drain them separately in colanders, and let them dry in the colanders for 8 hours (if they are drenched for longer than 8 hours at a time, they will be deprived of oxygen and will die). Soak the seeds separately again the same way for another 8 hours. At this point, the seeds should have begun to swell and turn white, a sign that they are about to germinate (sprout). If they haven't, drain, dry, and soak them again the same way until they appear about to germinate.\n\n**2. Sprout the grain** Line large rimmed baking sheets with paper towels. Keeping the wheat and barley separate, spread the swelled seeds in a single layer on the lined baking sheets. The paper towels absorb moisture and so maintain humidity, which is essential to helping the seeds germinate.\n\nPut the pans in large, dark garbage bags and tie the ends closed. Let the seeds sit at room temperature until they sprout. Barley seed takes 4 to 7 days; wheat seed takes about 3 days. Note that each seed will put out root hairs (each finer than a root) and shoots. The shoots are much larger than the root hairs, and each seed has just one shoot but several root hairs. When the shoot has grown to one-half to three-fourths the length of the seed, stop the germination by baking the seeds.\n\n**3. Bake the grain** Preheat the oven to between 100\u00b0 and 125\u00b0F (if your oven has a pilot light, just turning the pilot light on may bring the oven to the perfect temperature). Remove the baking sheets from the plastic bags and slip them into the oven. Bake the seeds for about 1 hour, then take them out of the oven. Remove the towels from the pan (you will yank off some roots and shoots, too, but that's fine), spread the seeds evenly on the pans, and bake the seeds for 24 hours. Move them around with a spatula every 30 minutes for the first 4 to 6 hours, to help them dry evenly.\n\nWhen the seeds are fully dried, bite into one. If it is rock hard, it is still a starch storage unit and won't make beer. Put the seeds back in the oven for another 30 minutes and test again. When they are crunchy and slightly sweet, the germination has converted the starch to sugar, and you have malt\u2014the key to making beer.\n\n**4. Clean the grain** One last step in the malting process: rub the fine dried root material off the grains. This will give you a clearer beer.\n\n##### BREWING DAY (about 3 hours)\n\nWe brewed under the guidance of our friend Chuck Schwalbach, who is married to Sunset's manufacturing manager, Diane Schwalbach. Chuck, a product designer for Apple, really knows his way around all-grain home brewing and helped us each step of the way. We highly recommend that you make your first all-grain batch with a knowledgeable friend, or that you take a class.\n\nHoping to create a particularly fresh-flavored beer, we decided to roast and grind the grain the same day we brewed the beer.\n\n**1. Roast the grain** Light 20 to 30 charcoal briquettes in the fire grate of your grill and let them burn until they are covered with ash. Push them into two small piles on either side of the grate. Put 8 ounces of the malted barley in an aluminum pie pan and place the pan on the cooking grate over the cleared space. Roast the grain, shaking the pie pan often, until about two-thirds of the grains are black. The timing will depend on the intensity of the fire; plan on about 20 minutes. Don't roast more than 1 pound of grain for a 5-gallon batch of beer, or your beer may turn out grainy and bitter.\n\n**2. Grind the grain** Pour all of the grain\u2014malted wheat, malted barley, and roasted barley\u2014into the hopper of the grain mill, turn on the mill, and let it rip.\n\n**3. Prepare your boiling pot** Sterilize your boiling pot with Saniclean solution. Rinse off the sanitizer with clean water.\n\n**4.** **Mash (hot-soak) the malted grain** The key here is to mash the grain at just the right temperature for the grain's starches to convert to sugar. Chuck measured the temperature of our grain and calculated that we'd need the mash water to be 172\u00b0F when poured into the mash pot. As soon as the water and grain mingled, the temperature would drop to 152\u00b0F. The mash conversion temperature can range from 150\u00b0 to 158\u00b0F, depending on the style of beer you want. (For more on this, see www.howtobrew.com, an excellent in-depth online brewing guide from beer guru John Palmer; refer to Sections 2 and 3 to learn about all-grain brewing.)\n\nFill the boiling pot with 13 quarts water, add 1 teaspoon gypsum, set the pot on the stove top, and heat the water to 180\u00b0F (measure it with your brewer's thermometer). Meanwhile, set your mash tun (the cooler fitted with the mesh bottom) on a nearby countertop. Attach one of your vinyl tubes to the spigot of the boiling pot, put the other end in the mash tun, and open the spigot to let the hot water flow into the mash tun. Then let the water in the mash tun cool to 172\u00b0F. (It's easier to start with hotter water and cool it down than have to pour the water back into the boiling pot and heat it up again.)\n\nPour the ground grain into the mash tun. Using the stirring paddle, mix well. The temperature of the mash should immediately decrease to 152\u00b0F. Let the mash rest for 1\u00bd hours.\n\n_Adding grain to the mash tun_\n\n**5. Sparge (rinse) the mash** About 30 minutes before the mash has finished resting, start heating up 4\u00bd gallons of water in the boiling pot. Bring it to a boil to evaporate the chlorine in the water.\n\nRemove the pot from the heat and let the water cool to 172\u00b0F. Transfer the hot water to the hot liquor tank (the cooler without the mesh bottom) and elevate the tank at least 1 foot above the mash tun. (We hoisted the tank onto the top of a wall oven, for example.) Set the sparger over the mash tun.\n\nPlace the now-empty boiling pot at least 1 foot below the mash tun. Attach one end of a length of vinyl tubing to the spigot of the mash tun, and put the other end in the empty boiling pot.\n\nWhen the water registers 172\u00b0F (you can add ice to cool it down quicker), it is cool enough so that it won't leach tannins from the grain husks into your beer. Attach a second vinyl tube to the spigot of the hot liquor tank and the other end to the top of the sparger. Open up the spigot and let the water flow down through the sparger into the mash. (The point of sparging is to rinse off the residual sugar that adheres to the grain.)\n\nAt the same time, open up the spigot of the mash tun and let the sweet brown liquid\u2014the wort\u2014flow into the boiling pot. One hand will be monitoring the hot water flow from above, into the sparger, and the other hand will be monitoring the flow from the mash tun into the boiling pot below. You want the inflow to be at about the same pace as the outflow. Try to keep the grain covered by 1 to 2 inches of water, so that when the water drizzles down from the sparger arms, it hits water and does not displace the grain at all or create channels, which can cause uneven rinsing. Properly done, sparging should take about 30 minutes for a 5-gallon batch of wort.\n\n_Sparging_\n\n**6. Boil the wort and add the hops** Measure the wort to be sure you have 5 gallons. If you don't, add water to make up the difference. Bring the wort to a boil over high heat and boil for 1\u00bd hours.\n\nAfter the wort has boiled for 30 minutes, add 1 ounce hops. At the 1-hour mark, add another 1 ounce hops; at the 1\u00bc-hour mark, add \u00bd ounce hops; and at the end of 1\u00bd hours of boiling the wort, add the last \u00bd ounce hops. The early additions will give the beer bitterness (the hops release alpha acids, which contribute more bitterness the longer they boil), and the later additions will add aroma.\n\n**7. Chill the wort** Use the immersion chiller or plate chiller, or put the pot in a cold-water bath in the sink, to cool down the wort to 68\u00baF, the ideal fermentation temperature for Abbey Ale yeast (the temperature will be specified on the yeast label). Depending on which method you use, the chilling process can take anywhere from a few minutes (the plate chiller) to 30 minutes (the immersion chiller) to an hour or more (the sink).\n\n_Chilling (with a plate chiller)_\n\n**8. Start fermenting** Pour the cooled wort through the strainer (to remove the spent hops) into the primary fermenter, add the yeast, and give the wort a good stir with the stirring spoon.\n\n**9. Measure the wort's original specific gravity** Insert the hydrometer in the hydrometer jar, drizzle in enough beer from the fermenter's spigot to make the hydrometer float, and read off the original specific gravity (OG) and the potential alcohol. Make a note of the OG. For this beer, our OG was 1.062 degrees; the corresponding readout of potential alcohol, marked on the hydrometer, was about 7.84 percent. The OG tells you how dense the wort is and will give you a way to measure the progress of the fermentation (as the sugars in the wort convert to alcohol, the mixture will get less dense).\n\n_Measuring original gravity_\n\n**10. Set the beer up for primary fermentation** Put the fermenter lid on and insert the rubber stopper into the hole in the lid. Insert the airlock into the stopper, then fill the airlock one-third full with water. Leaving only the airlock exposed, wrap the fermenter with the dark cloth. This blocks sunlight, which can stimulate the growth of bacteria. Keep the wort at 68\u00b0F, the ideal fermentation temperature for this yeast. You can use an electric heating pad if the room gets too chilly.\n\n##### FERMENTATION (2 to 4 weeks)\n\nFollow the fermentation instructions for Summer Wheat Beer, but try to keep the beer at around 68\u00b0F. In Step 2 (secondary fermentation and racking), let the beer settle for 3 weeks. The final specific gravity for this beer should be 1.021 (and the corresponding alcohol content should be 7.65 percent). Surprisingly, our final specific gravity reading was 1 (the specific gravity of water), an indication that something had gone very wrong.\n\n##### BOTTLING DAY (about 3 hours)\n\nFollow the bottling instructions for Summer Wheat Beer.\n\n##### CARBONATION IN BOTTLE (at least 3 weeks)\n\n**Let the beer rest** Store in a dark cardboard box at room temperature (68\u00b0F or above) for 3 weeks. Then try a bottle: If it's fizzy, chill the beer and drink it as soon as possible. If it is not fizzy enough, let it go for another week or two, then try it again. When you like it, chill it.\n\nAfter all our work, the Belgian Abbey Ale was, unfortunately, kind of funky. The final specific gravity did not lie. We think the brew got contaminated at some point during primary fermentation. Or, possibly our yeast was bad. We hope we have better luck next time.\n\n#### HELPFUL INFORMATION\n\n\u2022 John Palmer's easygoing yet detailed introduction to making beer at home: www.howtobrew.com.\n\n\u2022 _The Complete Joy of Home Brewing,_ by Charlie Papszian (Harper Paperbacks, 2003).\n\n\u2022 American Homebrewers' Association, www.homebrewersassociation.org.\n\n\u2022 Product questions section on William's Brewing Company site, www.williamsbrewing.com.\n\n\u2022 _The Home Brewer's Answer Book,_ by Ashton Lewis, (Storey Publishing, 2007) columnist for _Brew Your Own_ magazine, www.byo.com.\n\n\u2022 Helpful and lively forum for makers of AG (all-grain) brews on www.homebrewtalk.com, www.realbeer.com, and www.onebeer.net.\n\n\u2022 Forums on www.forums.morebeer.com.\n\n_\u2014Rick LaFrentz_\n\n##    \nHOW TO RAISE  \nCHICKENS\n\nWhat could be more idyllic than a flock of hens happily clucking in your backyard? We got six baby chicks in August 2007 and raised them to provide eggs, not meat. (We wanted protein we wouldn't have to kill.) They yielded a side benefit, too: their droppings, which make great fertilizer for our garden (see Cultivating Compost for more on this). Plus, chickens are unexpectedly entertaining.\n\nThey're not like other pets, however. For one thing, you probably eat others of their species. Some people who raise backyard chickens find they lose interest in eating meat. Others, after observing their chickens' behavior and getting to know them, decide that they feel okay about chickens as meat.\n\nIn our experience, chickens do not have the emotional range of cats or dogs. If you are looking for a cuddly creature with whom you will have an emotional relationship, chickens may not be the most rewarding option.\n\nThis is not to say that we don't like our chickens. We enjoy watching them scratch around, and we like feeding them treats. We feel the obligation to make them safe and comfortable and healthy. And we like their eggs a lot.\n\n_Carmelita (at left) and Ruby_\n\n#### BEFORE YOU START\n\nBefore you get your baby chicks (adorable balls of fluff), evaluate your space and your lifestyle.\n\n**Does your city allow you to keep chickens?** Every city has its own rules. Our municipality (Menlo Park, California) lets residents keep hens, but not roosters. That's fairly common in cities: Many have no problem with a few quiet hens (usually classified as pets), but they ban noisy roosters. Check your local regulations before picking out your flock. It would be dreadful to get them and then have to get rid of them.\n\n**Do you have space?** Each chicken should have at least 10 square feet to run around in, plus 4 square feet of house.\n\n**Can you keep them safe?** Making their digs secure is extremely important, especially at night. Chickens are prey, and they sleep so soundly that they seem unconscious\u2014morsels waiting to be devoured. They are vulnerable to attack by raccoons, skunks, foxes, weasels, and other predators. Raccoons are particularly nasty, and they're particularly clever about using their little nasty hands to get into your coop. Also, keep in mind that your other pets (cats, dogs) may be predators. (Read more about chicken coops here.)\n\n**What will do you with them when they stop laying eggs?** Hens lay best in the first year, and may lay sporadically for four or five years, but they can live for eight years (or longer). After their prime egg-laying years are over, will you be happy to keep caring for them?\n\n**What will you do if one gets injured or sick?** Before you acquire your flock, and regardless of your philosophical feelings about whether your chickens are pets, you must make sure you have a plan for what to do in case something happens to one of them. If you choose to take your chickens to a vet, locate an avian vet in your area in advance, preferably one familiar with chicken health problems.\n\nNot everyone who raises chickens chooses to treat them when they get sick. Some people will euthanize a sick or injured animal; others have a friend, family member, or neighbor willing to do it in an emergency.\n\n**Can you afford it?** You'll be getting good fresh eggs for free, but setting up a coop can cost a few hundred dollars, depending on the materials you use, and chicken food is an ongoing expense. If you decide to visit the vet, those bills can add up quickly.\n\n_One of our baby Buff Orpingtons_\n\n###    \nONE FLOCK OF 6 CHICKENS\n\n2 Ameraucanas (Ophelia lays blue eggs, Alana green)\n\n2 Buff Orpingtons (Honey and Charlotte; buff-colored eggs)\n\n2 Rhode Island Reds (Carmelita and Ruby; deep brown to bronze eggs)\n\n#### WHAT TO USE\n\n**_The coop_** Most of the work in raising chickens is setting up their space. Chickens need a box to lay their eggs in (plan on one or two boxes for half a dozen hens), a secure house with a roost for them to sleep on, and a place to run around and do their chicken thing. \"The coop\" refers to both the house and the enclosed run where chickens scratch around and spend the day.  \nWe got our 4-by-6-foot henhouse from Wine Country Coops, whose houses are both luxurious (glass windows! high-quality wood! beautiful construction!) and pricey ($1,300 and up; www.winecountrycoops.com or 707\/829 8405). Ours is built on stilts with an open-mesh floor, which lets droppings fall through to the ground below. You can get a less posh chicken house at a feed store, or you can build one yourself. Find inspiration at www.backyardchickens.com.\n\n**_The yard_** Allow 10 square feet per hen. (You can give your chickens less room if you let them range freely outside their yard. We don't, because they like to eat the seedlings in our garden.) We marked the boundaries of our yard with wooden posts; stretched chicken wire between the posts and buried it 12 inches into the ground to keep digging predators at bay; and then topped the yard with a corrugated, translucent plastic roof to allow light in but keep the chickens dry in foul weather. (Note: If raccoons are a problem in your area, you should use \u00bd-inch hardware cloth instead of chicken wire, since raccoons can reach right through the wire and grab; they can also pull it apart if sufficiently motivated.) The door of our chicken house opens right into the yard, so the coop is completely enclosed and protected. The yard also has a human-size door\u2014made of two-by-fours, more chicken wire, and a hinge\u2014so that we can easily get in and out to feed and visit the chickens. (About $200 in supplies from a hardware store.)\n\nUnless otherwise mentioned, we purchased everything below from a local feed store, Half Moon Bay Feed & Fuel (650\/726-4814). McMurray Hatchery sells a starter kit for about $65 (www.mcmurrayhatchery.com). Many supplies can be purchased from a pet store, such as Petco (www.petco.com) or Petsmart (www.petsmart.com).\n\nEGGS: QUESTIONS AND ANSWERS\n\n**Q:** Do you need a rooster for eggs?\n\n**A:** Chickens lay eggs with or without the presence of roosters. Since we don't have a rooster (they are not allowed in Menlo Park), the eggs are not fertilized, so they will never hatch into chicks. (This is just as well, because we're not in the market for more chickens.) If we did have a rooster, the eggs would be fertilized while they were, ahem, still in the hen. The rooster doesn't do anything to eggs once they have been laid.\n\n**Q:** Do colored eggs taste different?\n\n**A:** Different varieties of chickens lay different colors of eggs. Our Ameraucanas lay blue and green eggs; the other four lay brown eggs of varying shades. All of them taste exactly the same, and wonderful.\n\n**Q:** How many eggs do you get a day?\n\n**A:** Chickens usually lay one egg a day. When they get to be two or three years old, this drops off, and the eggs become more sporadic. Many chickens on large industrial farms lay two eggs a day.\n\n##### FOR CHICKS\n\n**_A warm indoor location_** We used a storage shed on the Sunset grounds.\n\n**_Heat lamp, reflector, bulb, and clamp_** Until they have feathers, chicks need to be kept very warm. About $20.\n\n**_Wire cage or enclosure_** This houses the chicks for the first few weeks of their lives. It is a few feet in length and the heat lamp hangs above it. We prefer the type with a door at the top, because it makes reaching in easier. You can use a cardboard box, but it's more difficult to clean and may not be escape-proof. From $50.\n\n**_1-gallon plastic water fount_** A narrow lip keeps the chicks from tumbling into the water. Buy one with a wide base so it won't tip easily. $5 to $10.\n\n**_Chick feeder_** Make sure your feeder has a top, or the chicks will scratch their food right out of it. About $5.\n\n**_Chick starter and grower food_** A finely ground, high-protein (20 to 22 percent) mixture of grains. This is what a chick should eat until it is 18 weeks old. (Some brands sell separate starter and grower crumbles; in that case, feed the chicks starter until they are 8 weeks old, then switch to the grower food until they start laying.) Medicated chick starter helps stave off the nasty parasitic infection called coccidiosis; alternatively, you can buy chicks that have been vaccinated for coccidiosis. Don't do both. $30 to $40 for a 50-pound sack of organic starter and grower food.\n\n**_Electrolyte powder_** Dissolve this in the chicks' water according to package instructions. It will shore up their frail systems with nutrients, and is especially important to use if the chicks are sick or stressed from traveling. Stop adding it when the chicks are 10 weeks old. $8 for an 8-ounce packet; a single packet is enough to raise at least a dozen chicks.\n\n##### FOR ADULT CHICKENS\n\n**_5-gallon water and food dispensers_** Galvanized steel dispensers hang from the roof of the coop and provide the chickens with a steady source of food and water. $40 to $50 each.\n\n**_Large covered trash can_** If rats are a problem in your area, stow the food dispenser in a trash can at night. Make sure the can has a sturdy lid that snaps closed securely. About $15 from a home supply store.\n\n**_Layer pellets_** These pellets have less protein and a coarser formulation than chick starter. Begin feeding them to your hens at 18 weeks or when they start laying. This is the chickens' main food for life. If you like, you can give it to them in crumble form instead, but the pellets are less messy. We order Purina organic layer pellets through our closest pet-supply store, San Mateo Pet Supply (www.sanmateopet.com or 650\/365-6738). About $40 for a 50-pound sack.\n\n**_Coarse-ground oyster shells_** Strew a couple of handfuls on the floor of the yard a few times a week for your chickens to peck up. The crushed shells supply calcium to strengthen eggshells, which can otherwise be weak and rubbery. $12 for 50 pounds.\n\n**_Treats from the garden and the kitchen_** Chickens love leafy greens, chile seeds, weeds picked from the garden (especially anything in the dandelion family and wild grasses), plain yogurt, leftover bread and cooked pasta, apple cores, overripe strawberries and other fruits, and insects and worms they find in the dirt. If your hens aren't roaming all over the garden during the day, though, they can become overweight. This means that you need to restrict treats for cooped chickens to greens and maybe a little fruit, with the occasional dish of yogurt. Give them only as much as they'll eat within a 15-minute period.\n\nFOODS TO AVOID\n\nEggshells (can encourage them to peck their own eggs), citrus (weakens shells), onions and garlic (can impart a funny flavor to eggs), and legumes (some uncooked beans are toxic for birds). We avoided giving them meat so as not to attract rats\u2014and because we think it's creepy\u2014but chickens will eat just about anything.\n\n#### HOW TO DO IT\n\n**1. Buy the chicks** Some companies will mail you chicks (newly hatched, they can survive for 2 days without food or water), but for beginning chicken raisers, we recommend buying them at a local feed store. Not only is it reassuring to pick up your cheeping chicks in person, but feed stores also have knowledgeable employees who can be valuable resources. Plus, mail-order shipping can be hard on chicks, and getting a package that contains a dead chick or two can be traumatic for first timers.\n\nWe got our hens as sexed 2- and 7-day-old chicks from Half Moon Bay Feed & Fuel (650\/726-4814). \"Sexed\" means the store's staff were pretty sure that they were girls. They came vaccinated for Marek's disease, a fatal poultry disease, and cost $4.50 each. We chose them because they were the kind the feed store happened to have that day. That said, Rhode Island Reds are good, old-fashioned layers, as are Buff Orpingtons. Our Ameraucanas proved to be fine layers, too.\n\n**2. Raise the chicks indoors** Keeping the chicks alive for the first few weeks was a fun kind of bustle. We kept them in a warm shed in their wire cage, with the heat lamp suspended above, until they started to feather out. We visited them several times a day, making sure that the heat lamp was the right distance from the cage to maintain the ideal temperature (90\u00b0 to 100\u00b0F for the first few weeks, and then decreased by 5 degrees per week after that). We made sure that food and water were always available, and we cleaned the cage every few days. Also, we gently picked them up to get them used to people. It worked: As adults, they are friendly and easy to handle.\n\n**3. Move the flock outdoors** At 5 weeks, the chicks were fully feathered, the sign that they were ready to leave the nest, so to speak. It was September and balmy when we took them out to their coop in our test garden. During the cold months, we kept the heat lamp inside their house for extra warmth at night. This was probably unnecessary; chickens huddle together for warmth, and supplemental heat really isn't needed unless nighttime temperatures go below freezing.\n\n**4. Encourage and collect eggs** Most chickens will begin to lay when they are somewhere between 18 and 24 weeks, depending on the chicken and the weather. Moderate warmth and 14 hours of daylight encourage laying. We put a marble egg in the nest box when the girls were of age, hoping it would give them the right idea (a \"decoy\" is supposed to help). Our first egg, a long-awaited event, appeared in January, courtesy of our Ameraucana Ophelia. Over the next several weeks, the other hens followed.\n\n**5. Routine maintenance** Adolescent and adult chickens don't require much care. Feeding and watering, cleaning the coop, and collecting the eggs pretty much sums up what you need to do.\n\nThe hens must be fed daily, and clean water must always be available. If a chicken goes thirsty for even a few hours, she may temporarily stop laying.\n\nWhatever you use as flooring in your coop must be scooped out and replaced about once a month, more often if it gets smelly faster. We initially used straw as bedding and on the floor of the coop, but we don't recommend it. Ophelia ate it and got it knotted up in her crop. Now we use pine bark, which is sturdy enough not to blow away, but easy for the hens to move around so they can dig in the dirt. We hose out the house about four times a year, and we compost the droppings.\n\nWe collect the eggs every day, not only because we want to eat them, but also because egg buildup can encourage broodiness (a condition in which a hen refuses to get off the nest in the hope of hatching chicks).\n\nBeyond this basic care, we recommend daily visits because it's enjoyable to go see the ladies. Plus, bringing them treats and petting and picking them up helps them get used to human presence and makes it easier to handle them if you've got a sick chicken you need to check out.\n\nHOW A CHICKEN LAYS AN EGG\n\n**\"The acme of food packaging.\" \u2014Alan Davidson, _The Oxford Companion to Food_**\n\nEvery day, our chickens deliver eggs to us\u2014anywhere between two and six, depending on the weather (in winter, they slow down). Each egg is slightly different in shape, in color, and in thickness of shell.\n\nIf you have a day job, it's hard to catch a chicken in the act of laying an egg. The thing pops out in less than a minute. Within the chicken, it takes only about a day for the egg to form.\n\nHere's how it works.\n\nEvery female chick is born with thousands of undeveloped yolks, or ova, grouped together near the middle of her backbone in a large cluster, the ovary.\n\nWhen a hen is old enough to lay, these ova begin to mature. At any given time within an adult hen's body, her ova are in various stages of development: Some are tiny yellow bubbles, others are the size of marbles. Every twenty-four to twenty-six hours, one ripens fully and is released into the instant grip of the waiting oviduct. As the egg moves down this tube, sperm from the rooster, if there is any, attempts to penetrate the egg; most eggs proceed down the oviduct unfertilized.\n\nNext, the egg is coated with layers of gel-like albumen (the egg white) and wrapped in a thin, translucent, antimicrobial membrane. At this point, the egg is only about half its final size. Time elapsed: about four hours.\n\nThe soft, shell-less egg then moves into the thickest part of the oviduct, the shell gland (uterus). For five hours or so, water and salts pump up the albumen to full size. Now a floating cloud of calcite (calcium carbonate) begins to swirl around the egg. Precisely spaced protein points on the membrane's surface attract the calcite particles, which build up in geometric columns to form a thin, smooth crystal: the shell.\n\nThe shell-encased egg is just centimeters from the exit. The oviduct releases a coating that both blocks bacteria and colors the egg\u2014blue, green, brown, or white\u2014which seems like a lovely bit of primping on nature's part just before the egg emerges.\n\nFinally, the hen gets an urge and climbs into the nest box. After a bit of heaving, out pops the egg. There it is, protein rich, marvelous, ready to go. We take it away to eat, and the hen starts making another one, as if it were the most ordinary thing in the world.\n\n _\u2014Margo True_\n\n_Ophelia_\n\n#### HELPFUL INFORMATION\n\n\u2022 **Your local feed store:** Often, this will be the place that supplied your chicks, and someone on the premises will know quite a bit about chicken care. At our store, we were given an informative how-to-raise-chicks handout and told about a condition we hadn't known of until that moment: pasty butt. This is basically a clogged vent, which can be deadly for a little chick. Pasty butt is easily resolved with cotton balls dipped in warm water and gently applied to get the crud off.\n\n\u2022 **The forum atwww.backyardchickens.com:** No matter what weird problem you're having, someone else has had it first. If you're looking for practical advice and real anecdotes, this is the best resource on the Web. Sifting through the active message board will get you up to speed on what to worry about, what not to worry about, and what to do next.\n\n##### BOOKS\n\nEvery month, a few more books on chicken raising seem to hit the shelves. These are some of our favorites.\n\n\u2022 _Creating Your Backyard Farm_ , by Nicki Trench (Cico Books, 2010). Good introduction to keeping hens, including how to deal with health problems.\n\n\u2022 _The Backyard Homestead_ , edited by Carleen Madigan (Storey Publishing, 2009). A substantial section on raising chickens offers great-looking coop designs. This is a terrific all-around raise-your-own-food book.\n\n\u2022 _The Chicken Health Handbook_ , by Gail Damerow (Storey Publishing, 1994). Packed with technical (and often grisly or terrifying) information, but also extremely useful; reading it will help you prevent sickness in your flock.\n\n\u2022 _The Joy of Keeping Chickens: The Ultimate Guide to Raising Poultry for Fun or Profit_ , by Jennifer Megyesi (Skyhorse Publishing, 2009). A heartfelt guide to the pleasures of the backyard flock by the owner of Fat Rooster Farm, in Royalton, Vermont. With artful photographs of chickens and humans by Geoff Hansen.\n\n\u2022 _Keep Chickens! Tending Small Flocks in Cities, Suburbs, and Other Small Spaces_ , by Barbara Kilarski (Storey Publishing, 2003). Geared toward absolute beginners and inspiringly enthusiastic.\n\n\u2022 _Keeping Chickens with Ashley English_ , by Ashley English (Lark Books, 2010). A slim volume with helpful advice on topics such as how to choose a good layer, plus interviews with chicken keepers.\n\n\u2022 _Minnie Rose Lovgreen's Recipe for Raising Chickens_ , by Minnie Rose Lovgreen (NW Trillium Press, 2009). This friendly little book, dictated by the eighty-six-year-old Minnie Rose to her friend Nancy Rekow in 1974, reads as though Minnie herself were sitting at your kitchen table, giving you advice.\n\n\u2022 _Raising Chickens for Dummies_ , by Kimberly Willis with Rob Ludlow (Wiley Publishing, 2009). Willis, a poultry breeder, and Ludlow, owner of the excellent Web site www.backyardchickens.com, include a section on how to butcher and process chickens.\n\n\u2022 _Storey's Guide to Raising Chickens_ , by Gail Damerow (Storey Publishing, 1995). Technical and thorough, it's great for those who have had some experience and are looking for more extensive information.\n\n_\u2014Elizabeth Jardina_\n\n_Ruby the chicken_\n\n_Our henhouse arrives (top); Honey, brooding (bottom right)_\n\n# THE SUMMER RECIPES\n\n##    \n  SKILLET-ROASTED EDAMAME\n\nSmoky, salty, and addictive, these soybeans were inspired by a recipe for griddle-roasted peas in Niloufer Ichaporia King's _My Bombay Kitchen: Traditional and Modern Parsi Home Cooking_ (University of California Press, 2007). To eat them, pop the peas out of the pod with your fingers\u2014or, if you like the salt and char, with your teeth.\n\nMAKES about 4 cups\n\nTIME about 10 minutes\n\n1 pound fresh or frozen edamame (soybeans) in the pod\n\nFine sea salt\n\n1. If using fresh edamame, bring a large pot of salted water to a boil. Add the edamame and boil for 2 minutes. Drain and let cool. If using frozen edamame, skip this step.\n\n2. Heat a large, heavy skillet or a wok over high heat. Toss in half of the edamame and cook, turning with tongs, until blackened in several spots, 5 to 8 minutes. Pour into a bowl. Repeat with the remaining edamame.\n\n3. Toss the edamame with salt to taste.\n\n_PER \u00bd-CUP SERVING 65 cal., 38% (25 cal.) from fat; 6 g protein; 2.8 g fat (0 g sat.); 5 g carbo (2.8 g fiber); 18 mg sodium; 0 mg chol._\n\n##    \n  DEVILED CUCUMBER CUPS\n\nThese pretty little bites show off the bright color of eggs from pasture-raised chickens. If you want the cups to have fluted edges, you will need two round, fluted biscuit cutters, each at least \u00be inch high: one 1\u00bd inches in diameter, for the narrower parts of the cucumber, and the other 2 inches in diameter, for the fatter sections. A melon baller is useful, too, for removing the seeds.\n\nMAKES 24 cucumber cups, or 8 servings\n\nTIME about 1 hour\n\n3 large eggs\n\nIce cubes\n\n4 cucumbers, each at least 7 inches long\n\n2\u00bd tablespoons Green Chile Mayonnaise\n\n2\u00bd tablespoons minced red onion, rinsed\n\nAbout 1 teaspoon freshly squeezed lemon juice\n\n\u00bd teaspoon fine sea salt, plus more to finish\n\n1 to 2 red or green serrano chiles, seeded and slivered, for garnish\n\n1. Put the eggs in a small saucepan and fill with water to cover by 1 inch. Bring to a boil; immediately reduce the heat to a simmer and cook, uncovered, 10 minutes. With a slotted spoon, transfer the eggs to a bowl of water and ice cubes and let cool for 1 minute, then crack all over and put back in the ice water for 5 minutes to help loosen the shells. Lift the eggs out of the water, peel, and pat dry.\n\n2. Meanwhile, cut the cucumbers crosswise into 24 rounds, each \u00be inch thick (you will have some cucumber left over). For fluted edges, cut out fluted rounds, using a biscuit cutter in the size that works best (see headnote). Using a melon baller or small spoon, scoop the seeds from the center of each cucumber round, leaving at least a \u00bc-inch-thick layer of cucumber on the bottom and sides. Set the rounds on a baking sheet.\n\n3. Shred the eggs with a razor-sharp grater (such as a Microplane) or against the finest slatted holes of a box grater, and put in a bowl. Add the mayonnaise, onion, lemon juice, and salt. Mash the mixture together with a fork.\n\n4. Fill the cucumber cups with the egg mixture, mounding it slightly. Sprinkle each filled cup with salt and a few drops of lemon juice. Top each with a sliver of serrano.\n\n**MAKE AHEAD** You can make both the filling and the cucumber cups up to 1 day ahead and cover and refrigerate them separately. Blot the cups dry just before filling.\n\n_PER CUCUMBER CUP 24 cal., 58% (14 cal.) from fat; 1.1 g protein; 1.7 g fat (0.35 g sat.); 1.4 g carbo (0.36 g fiber); 61 mg sodium; 29 mg chol._\n\n##    \nTEMPURA SQUASH BLOSSOMS\n\nWe came up with this recipe as a way to control the crazy productivity of our pattypan squash plants: If you eat the blossoms, the squashes can't form. The dish is just as tasty and works just as easily with zucchini blossoms.\n\nMAKES 4 to 6 servings\n\nTIME about 20 minutes\n\nExtra virgin olive oil for deep-frying\n\n\u00bd cup plus 2 tablespoons Summer Wheat Beer or other wheat beer\n\n\u00bd cup 'Sonora' white whole-wheat flour,* whole-wheat pastry flour, or a mix of all-purpose flour and cake flour\n\n\u00bc teaspoon fine sea salt, plus more for sprinkling\n\n12 small pattypan squash blossoms or blossoms from 'Trombetta di Albenga' or other zucchini, about 3 inches long\n\n\u00bc cup Fresh Chive Cheese or fresh goat cheese mixed with 2 teaspoons minced chives\n\n1. Pour the oil to a depth of 2 inches into a medium saucepan and heat to 350\u00b0 to 375\u00b0F on a deep-frying thermometer. Meanwhile, in a small bowl set in a larger bowl of ice and cold water, whisk together the beer, flour, and salt, stopping when the batter is almost but not quite smooth.\n\n2. Brush any debris or insects off the blossoms (check inside, too). Gently open the petals and stuff each blossom toward its base with about 1 teaspoon cheese, depending on the size of the blossom. Leave enough room for the petals to close easily over the cheese.\n\n3. Dip a few stuffed blossoms in the batter to coat. One at a time, lift the blossoms from the batter, allowing the excess batter to drip back into the bowl, and slip them into the hot oil. Fry the blossoms, turning them occasionally, until golden and crisp, about 1 minute. Transfer with a slotted spoon to paper towels to drain. Repeat with the remaining blossoms, dipping and frying only a few at a time. Sprinkle the blossoms with salt, if you like, and serve immediately.\n\n_PER 3-BLOSSOM SERVING 146 cal., 58% (85 cal.) from fat; 4.1 g protein; 9.8 g fat (2.6 g sat.); 12 g carbo (1.3 g fiber); 152 mg sodium; 11 mg chol._\n\n* _We used finely milled 'Sonora' wheat flour because it makes a delicate, crisp batter. SeeThe Allure of Local Wheat for more on local flours._\n\n  TOMATO AND HERB SALAD WITH FRESH CHIVE CHEESE\n\nThis is the salad to make with dead-ripe tomatoes from your backyard. The surprise is the easy homemade cheese.\n\nMAKES 8 servings\n\nTIME about 20 minutes\n\n2 pounds assorted vine-ripened tomatoes\n\n12 to 14 slices Fresh Chive Cheese or fresh mozzarella\n\n1 teaspoon fine sea salt, plus more to finish\n\n\u00bd teaspoon honey\n\n1\u00bd tablespoons Syrah Vinegar or other red-wine vinegar\n\n\u00bc cup extra virgin olive oil\n\n2 or 3 sprigs marjoram, plus flowers if any\n\n10 to 12 fresh basil leaves, plus flowers if any\n\n1. Slice large tomatoes; leave small tomatoes whole, or cut into wedges or halves. Arrange on a platter with the cheese.\n\n2. In a small bowl, whisk together the salt, honey, and vinegar until the salt has dissolved; whisk in the oil. Spoon the dressing over the tomatoes and cheese.\n\n3. Strip the leaves and flowers from the marjoram sprigs onto the salad. With kitchen scissors, snip the basil leaves and flowers over the tomatoes. Sprinkle with salt.\n\n_PER SERVING 227 cal., 59% (135 cal.) from fat; 8.5 g protein; 15 g fat (5.8 g sat.); 17 g carbo (1.5 g fiber); 424 mg sodium; 32 mg chol._\n\n##    \nWATERMELON-CHILE SALAD\n\nIf you have ever had a cold slab of watermelon sprinkled with salt, chile, and lime on the streets of a Mexican town, you'll recognize the roots of this salad.\n\nMAKES 4 to 6 servings\n\nTIME about 15 minutes\n\n4 teaspoons freshly squeezed lemon juice\n\n\u00bd teaspoon fine sea salt\n\n2 to 2\u00bd teaspoons minced green serrano chile\n\n2 teaspoons finely chopped fresh chives\n\n2 tablespoons extra virgin olive oil\n\n6 cups cubed watermelon (1-inch cubes)\n\n\u2153 cup loosely packed small flat-leaf parsley leaves (chop coarsely if large)\n\n1. In a large bowl, whisk together the lemon juice, salt, chile, chives, and oil to make a dressing.\n\n2. Add the watermelon and parsley to the dressing and toss gently to coat evenly.\n\n_PER SERVING 86 cal., 49% (42 cal.) from fat; 1.1 g protein; 4.9 g fat (0.7 g sat.); 11 g carbo (0.79 g fiber); 191 mg sodium; 0 mg chol._\n\n##    \nPURSLANE-CUCUMBER SALAD\n\nWhat better way to control the weeds in your garden than to eat them? And you can definitely eat purslane. It is pleasantly lemony and its fat little leaves have a juicy crunch. This Turkish-style salad is one of our favorite ways to get rid of purslane.\n\nMAKES 4 to 6 servings\n\nTIME about 20 minutes\n\n1 large cucumber, about 7 inches long\n\n2\u00bd cups lightly packed purslane or watercress sprigs\n\n2 teaspoons finely chopped fresh peppermint\n\n1 clove garlic\n\n\u00bc teaspoon fine sea salt\n\n2 tablespoons Cr\u00e8me Fra\u00eeche (recipe follows)\n\n\u00bd teaspoon crushed dried red serrano chile or red chile flakes\n\n3 tablespoons extra virgin olive oil\n\n1 tablespoon freshly squeezed lemon juice\n\n1. Peel the cucumber, halve lengthwise, and remove the seeds. Cut into \u00bd-inch cubes and put in a serving bowl. Add the purslane and peppermint.\n\n2. Mince the garlic, sprinkle it with the salt, and mash to a paste with the flat side of a big chef's knife. In a small bowl, whisk together the garlic paste, cr\u00e8me fra\u00eeche, chile, oil, and lemon juice to make a dressing.\n\n3. Pour the dressing over the salad and toss to coat evenly.\n\n_PER SERVING 90 cal., 87% (78 cal.) from fat; .86 g protein; 9 g fat (2.2 g sat.); 2.5 g carbo (0.4 g fiber); 107 mg sodium; 4.2 mg chol._\n\n  CR\u00c8ME FRA\u00ceCHE\n\nIn France, this luxuriously thick, tangy, slightly nutty-tasting cream is often made with unpasteurized cream, and naturally occurring bacteria do the work of thickening. If you are using pasteurized cream, you have to add a fermenting agent to the milk to get it to thicken. Even so, homemade cr\u00e8me fra\u00eeche is wonderful stuff, and less expensive than store-bought. There are many ways to make it. This is our favorite.\n\nMAKES about 1 cup\n\nTIME about 5 minutes, plus at least 8 hours to ripen\n\n1 cup heavy cream\n\n2 tablespoons buttermilk\n\n1. Pour the cream and buttermilk into a small bowl and stir to combine. Cover tightly with plastic wrap and let sit at room temperature until thickened, at least 8 hours or up to overnight.\n\n2. Stir well, re-cover, and refrigerate. It will keep for up to 1 week.\n\n_PER TABLESPOON 53 cal., 94% (50 cal.) from fat; 0.4 g protein; 5.6 g fat (3.5 g sat.); 0.5 g carbo (0 g fiber); 7.4 mg sodium; 21 mg chol._\n\n##    \n  CORN SOUP WITH ROASTED POBLANOS AND ZUCCHINI BLOSSOMS\n\nThe essence of sweet, just-picked summer corn, this soup lends itself to a gardenful of garnishes. It is equally good hot or cold.\n\nMAKES 8 servings\n\nTIME about 1 hour\n\n3 large poblano chiles\n\n10 freshly picked ears corn,* any variety\n\n2 tablespoons olive oil\n\n1 large white onion, chopped\n\n1 to 2 green serrano chiles, finely chopped\n\n2 teaspoons fine sea salt\n\n4 large cloves garlic, minced\n\n1 'Trombetta di Albenga' zucchini (7-inch-long piece) or 1 medium regular zucchini, seeded and cut into \u00bc-inch dice (about 1 cup)\n\n8 'Trombetta di Albenga' zucchini blossoms or other zucchini blossoms, brushed clean and halved lengthwise\n\n1 cup Oregano Queso Blanco, crumbled store-bought queso fresco,** or crumbled mild feta, homemade or store-bought\n\n_Corn Soup with Roasted Poblanos and Zucchini Blossoms_\n\n1. Preheat the broiler. Put the poblanos on a rimmed baking sheet. Broil about 4 inches from the heat source, turning as needed, until blackened all over, at least 15 minutes. Drape the pan with a sheet of aluminum foil and let the chiles stand until cool enough to handle. Gently pull off and discard the blackened skins. Stem and seed the chiles, then cut into \u00bd-inch dice.\n\n2. Meanwhile, remove the husks and silk from the corn. Working with 1 ear at a time, stand it on one end in a deep, wide bowl. Using a sharp knife and starting at the top of the ear, cut off the kernels.***\n\n3. Heat the oil in a large, wide pot over medium heat and add the onion, serranos, and salt. Cook, stirring occasionally, until the onion and chiles have softened, about 5 minutes. Stir in the garlic and corn kernels and cook for 3 minutes more.\n\n4. Pour just enough water into the pot to cover the corn. Cover, bring to a simmer over medium heat, and cook for 15 minutes. Remove from the heat and let cool slightly.\n\n5. While the corn mixture is cooking, put the diced zucchini in a small saucepan with salted water just to cover. Bring to a simmer and cook until tender-crisp, about 2 minutes. Drain.\n\n6. In batches, whirl the corn mixture in a blender until very smooth. Strain through a fine-mesh strainer into a bowl, mashing with the back of a spoon to press out as much liquid as possible. Thin the soup with water if you like. To serve the soup hot, return it to the pot and reheat gently (don't boil). To serve the soup cold, cover and chill for at least 2 hours. Just before serving, season with salt if necessary.\n\n7. Divide the soup among 8 bowls. Top each bowl with 2 tablespoons diced poblanos, 2 tablespoons diced zucchini, and 2 zucchini blossom halves. Sprinkle each serving with 1 tablespoon cheese, and pass the remaining cheese at the table.\n\n**MAKE AHEAD** You can make the soup through step 6 up to 2 days ahead, cover its components seperately, and refrigerate.\n\n_PER SERVING 238 cal., 34% (80 cal.) from fat; 9.2 g protein; 8.9 g fat (3.2 g sat.); 36 g carbo (4.3 g fiber); 695 mg sodium; 17 mg chol._\n\n* _Just-picked corn is much juicier than corn that has been sitting around for a while, so if you are using older, chewier, starchier corn, you may need to add more water to the pot when simmering the corn._\n\n** _Queso blanco, also known as queso fresco, is a mild, fresh white Mexican cheese with a moist, crumbly texture._\n\n*** _Save the corn cobs for your chickens. They'll love them._\n\n##    \n'TROMBETTA' ZUCCHINI AND ITS FLOWERS\n\nOur 'Trombetta' zucchini plants produced big, curving, crisp, pale green zucchini and enormous blossoms, both of which worked perfectly for this recipe. But you can use any kind of zucchini\u2014or squash, actually\u2014and its flowers for this dish. If you have two big frying pans, use them both to cook all the zucchini at once. Otherwise, cook the dish in two batches. It does not need to be served piping hot.\n\nMAKES 4 to 6 servings\n\nTIME about 20 minutes\n\n2 tablespoons extra virgin olive oil\n\n2 pounds 'Trombetta di Albenga' or other zucchini, cut on the diagonal into \u00bd-inch-thick slices\n\n1 teaspoon fine sea salt\n\n2\u00bd teaspoons minced fresh marjoram\n\n30 'Sungold' or other yellow cherry tomatoes\n\n4 cloves garlic, thinly sliced\n\n2 to 3 teaspoons freshly squeezed lemon juice\n\n8 to 10 'Trombetta di Albenga' or other zucchini blossoms\n\n1. Heat 1 tablespoon of the oil in a large nonstick frying pan over medium heat. Add half of the zucchini slices and cook, without stirring, until the slices on the bottom begin to brown, 2 to 3 minutes. Sprinkle with \u00bd teaspoon of the salt and 1 teaspoon of the marjoram. Using a thin spatula, turn the zucchini so the bottom slices are on top.\n\n2. Add half of the tomatoes and half of the garlic and cook, stirring gently every now and then, until most of the zucchini slices have browned a little, about 10 minutes, and the tomatoes are hot. Transfer to a bowl.\n\n3. Return the pan to medium heat and heat the remaining 1 tablespoon oil. Add the remaining zucchini and cook the same way with the remaining salt, tomatoes, garlic, and 1 teaspoon marjoram. Return the first batch of zucchini to the pan and sprinkle with lemon juice to taste. Tear all but 3 blossoms into petals and fold into the zucchini.\n\n4. Turn the zucchini out onto a platter or into a wide serving bowl and sprinkle with the remaining \u00bd teaspoon marjoram. Tear the remaining 3 blossoms into petals and scatter over top.\n\n_PER 1-CUP SERVING 84 cal., 56% (47 cal.) from fat; 4.6 g protein; 5.4 g fat (0.81 g sat.); 7.6 g carbo (2.1 g fiber); 398 mg sodium; 0 mg chol._\n\n##    \nOVEN-BAKED STEAK FRIES WITH GREEN CHILE MAYONNAISE\n\nWe didn't have steak, but managed to eat plenty of these regardless, especially dunked in the mayo\u2014and washed down with cold, fizzy homemade beer.\n\nMAKES 4 servings\n\nTIME about 45 minutes\n\n1 pound 'Yukon Gold' potatoes, cut into \u00bd-inch-thick wedges\n\n2 tablespoons extra virgin olive oil\n\n\u00bd teaspoon fine sea salt\n\nGreen Chile Mayonnaise (recipe follows)\n\n1. Preheat the oven to 425\u00b0F. Place the potato wedges on a rimmed baking sheet, drizzle with the oil, and sprinkle with the salt. Toss the potato wedges to coat evenly.\n\n2. Roast, turning the wedges halfway through, until golden and crisp, about 35 minutes.\n\n3. Transfer the fries to a platter, spreading them out evenly (they will stay crisp longer on a platter than heaped in a bowl), and season with more salt, if you like. Serve with the mayonnaise.\n\n_PER SERVING 153 cal., 39% (60 cal.) from fat; 2.7 g protein; 7 g fat (1 g sat.); 20 g carbo (1.3 g fiber); 287 mg sodium; 0 mg chol._\n\n###   GREEN CHILE MAYONNAISE\n\nA fiesty spread that's good on sandwiches, too. Have ingredients at room temperature to help the mayonnaise thicken. (If the egg is straight from the refrigerator, put it in warm water for several minutes before you use it to help it warm up.) Seed the chile if you want less kick. You can also leave out the garlic and serrano entirely for a mild mayo; see here.\n\nMAKES About 1 cup\n\nTIME 5 minutes\n\n1 large egg\n\n2 garlic cloves, minced\n\n\u00bd to 1 red or green serrano chile, finely chopped\n\n1\u00bd to 2 teaspoons fine sea salt\n\n2 tablespoons lemon juice, plus more to taste\n\n1 cup extra-virgin olive oil\n\nIn a blender, whirl together the egg, garlic, chile, 1\u00bd tea\u00adspoons of the salt, and 2 tablespoons lemon juice until the mixture thickens slightly, about 1 minute. With the motor running, add the oil through the top of the blender in a slow, steady stream, whirling until all of it has been incorporated and the mayonnaise is thick. Whirl in more salt and lemon juice to taste, if you like.\n\n**MAKE AHEAD:** Up to 4 days, covered and chilled.\n\n_PER TABLESPOON 87 cal., 95% (83 cal.) from fat; 0.55 g protein; 9.7 g fat (1.5 g sat.); 0.4 g carbo (0.02 g fiber); 52 mg sodium; 18 mg chol._\n\n##    \n  ROSEMARY POTATOES ANNA\n\nA thin, golden brown \"cake\" of sliced potatoes cooked in a heavy skillet, potatoes Anna typically calls for butter. We used our own olive oil instead. This recipe works best with potatoes that you have allowed to cure (see here). If they are freshly dug, they will be too moist. (All potatoes in grocery stores and most potatoes in farmers' markets are cured.)\n\nMAKES 8 servings\n\nTIME about 1 hour\n\n3 pounds 'Yukon Gold' potatoes\n\n1 tablespoon extra virgin olive oil\n\n\u00bd teaspoon fine sea salt\n\n2 teaspoons minced fresh rosemary\n\n1. Preheat the oven to 375\u00b0F. Peel the potatoes and cut into \u215b-inch-thick slices.\n\n2. In a 10-inch ovenproof nonstick frying pan, heat 2 teaspoons of the oil over medium-high heat. When the oil is hot, arrange about one-fifth of the potatoes in the pan in concentric circles, starting from the outer edge, covering the bottom completely. The potatoes should sizzle when you put them down on the hot pan. If they don't, wait for the pan to get hot enough. Mix together the salt and rosemary in a small bowl, and sprinkle the potatoes in the pan with about one-fifth of the mixture. When the potatoes begin to brown around the edges, reduce the heat to low. Top the first layer of potatoes with 4 more layers, arranging them the same way and sprinkling each layer with the salt mixture. Before you sprinkle the final layer, brush the remaining 1 teaspoon oil over the potatoes.\n\n3. With a heatproof spatula, gently pull back the edge of the layered potatoes to see if the bottom layer is starting to brown. If it is, transfer the pan to the oven and bake until the potatoes are tender when pierced with a fork, about 45 minutes. If it isn't, increase the heat to medium-high and cook until the bottom layer begins to brown, then transfer the pan to the oven. Check the potatoes occasionally, and if they start to look dry or begin to curl, cover the pan with aluminum foil.\n\n4. Remove from the oven. Run the spatula between the layered potatoes and the pan sides. Invert a flat, round plate or platter over the pan and invert the pan and the plate together, releasing the potatoes onto the plate. Serve hot or warm, cut into 8 wedges with a serrated knife.\n\n**MAKE AHEAD** The finished dish can be kept warm in a 200\u00b0F oven for up to 2 hours.\n\n_PER SERVING 141 cal., 11% (15 cal.) from fat; 3.6 g protein; 1.7 g fat (0.2 g sat.); 27 g carbo (1.8 g fiber); 154 mg sodium; 0 mg chol._\n\n##    \nGRILLED SUMMER SUCCOTASH\n\nSuccotash usually involves a mixture of boiled corn, lima beans, and bell peppers. We had the corn, no problem. Instead of limas, we used edamame. And our poblano chiles stood in for the bells. Then, because every summer needs a grilled salad, we grilled the corn and chiles\u2014and threw in a few ripe cherry tomatoes for sweetness at the last minute.\n\nMAKES about 5 cups, or 4 to 6 servings\n\nTIME about 45 minutes\n\n4 large ears corn\n\n\u00bc cup extra virgin olive oil, plus more for brushing\n\n\u00be teaspoon fine sea salt, plus more for sprinkling\n\n3 poblano chiles\n\n1 cup shelled fresh or frozen edamame (soybeans)\n\n2 cloves garlic\n\n2 teaspoons freshly squeezed lemon juice\n\n1\u00bd cups 'Sweet Million' or other red cherry tomatoes, cut in half\n\nAbout \u00bd cup heavy cream (optional)\n\n1. Prepare a grill for medium-hot cooking (350\u00b0 to 450\u00b0F; you should be able to hold your hand above the cooking grate for only 5 to 7 seconds). Strip the husks and silk from the corn, brush the ears with oil, and sprinkle with salt.\n\n2. Arrange the corn and chiles on the cooking grate. Grill the corn, turning often, until lightly browned, 10 to 15 minutes. Grill the chiles, turning as needed, until completely blackened on all sides, about the same amount of time.\n\n3. Meanwhile, bring a pot of well-salted water to a boil. Add the edamame, bring back to a boil, and cook for 2\u00bd minutes if fresh or 4 minutes if frozen. Drain and set aside.\n\n4. Mince the garlic, sprinkle it with the \u00be teaspoon salt, and mash to a paste with the flat side of a big chef's knife. In a small bowl, whisk together the garlic paste, lemon juice, and the \u00bc cup olive oil to make a dressing.\n\n5. Let the corn and chiles cool until they can be handled. Then, stand 1 ear on end in a deep, wide bowl and, slicing downward, cut off the kernels.* Repeat with the remaining ears. You should have about 3 cups kernels. Peel the blackened skin from the poblanos. Holding each poblano over the bowl of corn kernels, slit the chile and let the juices drain into the bowl. On a work surface, stem and seed the chiles and cut into \u00bd-inch dice. Add the diced chiles to the corn.\n\n6. Add the tomatoes and edamame to the corn and chiles, drizzle with the vinaigrette, and toss together gently to coat evenly. Eat drizzled with cream if you like.\n\n_PER SERVING 217 cal., 49% (106 cal.) from fat; 6.6 g protein; 12 g fat (1.6 g sat.); 25 g carbo (5.4 g fiber); 296 mg sodium; 0 mg chol._\n\n* _Save the corn cobs for your chickens. They'll love them._\n\n##    \n  PATTYPAN SQUASH WITH EGGS\n\nHarvest or buy the squashes when they are 4 inches across. We found that this was the perfect size for holding a single large egg. Since our chickens had just started laying, their eggs tended to be small, so some of our squashes held two eggs. (If the egg won't quite fit, scoop out a little of the white with a spoon.)\n\nMAKES 8 servings\n\nTIME about 1 hour\n\n8 pattypan squashes, yellow or green or a combination, each 4 inches across\n\nAbout 3 tablespoons extra virgin olive oil\n\n\u00be teaspoon fine sea salt, plus more for sprinkling\n\n1 white onion, finely chopped (about \u00becup)\n\n4 large cloves garlic, minced\n\n1\u00bc teaspoons minced fresh thyme\n\n8 large or 16 small eggs, at room temperature\n\n1. Preheat the oven to 375\u00b0F. Cut off the top (stem end) of each squash and reserve. Using a small spoon, scoop out and discard the flesh (or reserve for another use) from each squash, leaving a shell at least \u00bc inch thick. Brush the squashes and their tops all over with about 2 tablespoons of the olive oil, and set, cut side up for squashes and stem side up for tops, on a rimmed baking sheet.\n\n2. Bake until tender when pierced with a sharp knife, 20 to 30 minutes. The tops will cook more quickly, so check them after 15 minutes and remove them when they are done. When the shells are ready, remove from the oven and sprinkle the insides of the shells and the tops with salt. Set the tops aside; leave the shells on the baking sheet.\n\n3. While the squash shells are baking, heat the remaining 1 tablespoon oil in a large frying pan over medium-high heat. Add the onion and the \u00be teaspoon salt. Cook, stirring occasionally, until the onion is very soft and starting to brown, 8 to 10 minutes. Add the garlic and cook, stirring, until fragrant, about 30 seconds. Add the thyme and stir until combined. Keep hot.\n\n4. Divide the hot onion mixture among the squash shells, spreading it over the bottom and up the sides to form a thin coating. Crack 1 large egg or 2 small eggs into each squash and drizzle with a little more oil. Cover the baking sheet with aluminum foil and bake until the egg whites are firm but the yolks are still loose, 15 to 20 minutes.\n\n5. Sprinkle each egg with a pinch of salt and serve immediately, with the squash tops replaced or set to the side.\n\n**MAKE AHEAD** Bake the squash cups up to 4 hours ahead, cover, and keep at room temperature. Rewarm the cups in the oven before adding the onion mixture and the eggs.\n\n_PER SQUASH 165 cal., 48% (80 cal.) from fat; 10 g protein; 8.9 g fat (2.1 g sat.); 14 g carbo (3.9 g fiber); 366 mg sodium; 212 mg chol._\n\n##    \nWHOLE-WHEAT PIZZAS\n\nIf we had known more about wheat when we started our project, we would have grown high-protein hard red wheat for breads and pizzas in addition to our low-protein soft white wheat. The soft wheat was excellent for beer making and for pastries and crackers, but made gummy bread and crackerlike pizza dough. So, after some exasperating dough experiments, we found a great locally grown and milled hard wheat flour called 'Expresso' and used it for this recipe (see The Allure of Local Wheat for more on local flours). The crust is reasonably puffy, and it has a nutty, wholesome flavor that is great with toppings of fresh vegetables and cheeses.\n\nIf your pizza dough balls are fully proofed (puffy and ready to be stretched into pizza crusts), but you are not yet ready for them, punch them down and form them into tight little balls, then let them rise for an hour. They will puff up again and be good to go.\n\nWe tried to make a whole-wheat starter from just wheat and water to use as leavening, but we couldn't get it to work consistently. So, in the interest of creating a reliably delicious pizza dough, we ended up with packaged yeast. However, we'll keep exploring a way to grow a good starter on <http:\/\/oneblockdiet.sunset.com\/team_kitchen>.\n\nMAKES 6 pizzas, or 6 servings\n\nTIME about 1\u00bd hours\n\n1 package (2\u00bc teaspoons) active dry yeast\n\n1 teaspoon honey\n\n3\u00bd to 4 cups 'Expresso' whole-wheat flour* or other whole-wheat flour, plus more for sprinkling\n\n1\u00bd tablespoons extra virgin olive oil\n\n1 teaspoon fine sea salt\n\nTopping of choice (recipes follow)\n\n1. Put the yeast, honey, and 1\u00be cups warm water (100\u00b0 to 110\u00b0F) in the bowl of a stand mixer fitted with a dough hook. Let stand until the yeast dissolves and bubbles appear, 5 minutes. Add 1 cup flour and mix with the dough hook on low speed until combined. Gradually add 2\u00bd cups more flour and mix for 8 minutes to thoroughly combine.\n\n2. Add the oil and the salt and mix on low speed for 7 minutes more. The dough should feel elastic\u2014when you poke it, it should spring back\u2014and only slightly sticky. If it is quite sticky, mix in more flour, 1 tablespoon at a time, until the stickiness subsides.\n\n3. Turn the dough out onto a floured work surface and cut into 6 equal portions. Roll each portion into a tight ball and place on a well-floured tray. Dust the tops with flour.\n\n4. Cover the tray loosely with plastic wrap and a kitchen towel. Let the dough balls rise at warm room temperature until they are soft, pillowy, and full of air, 1 to 1\u00bd hours. (Alternatively, cover the tray as directed and refrigerate overnight. Let come to room temperature for at least 1 hour before continuing.)\n\n5. Place a pizza stone or baking sheet on the lowest rack of the oven and preheat the oven to 500\u00b0F (or as high as your oven will go) for at least 30 minutes.\n\n_Corn and Zucchini Pizza (bottom); \"Fresh Chive Cheese and 'Sweet Million' Pizza\" (top)_\n\n6. Working with 1 dough ball at a time (keep the others covered), set the ball on a well-floured pizza peel or baking sheet. Tap the center of the ball first to deflate it, then stretch and pat the dough out from the middle with your fingers into a 7- to 8-inch round, leaving the outer rim untouched (the dough will not be very stretchy, so be careful not to tear it). Add the toppings.\n\n7. With a quick thrust of the peel (set the tip of the peel or sheet on the stone first to make it easier), shove the pizza onto the stone. Bake until the crust is puffy and browned, 8 to 12 minutes, or follow baking directions in the topping recipe. Repeat with remaining dough balls and toppings.\n\n**PIZZA TOPPINGS**\n\nWe borrowed from future seasons to come up with these pizza toppings; in the summer, we had only the first one.\n\n### FRESH CHIVE CHEESE AND 'SWEET MILLION' TOMATO SAUCE\n\nParbake the pizza crusts until firm and dry but not browned, about 3 minutes. Spread with a thin layer of 'Sweet Million' Tomato Sauce (at right) or other pizza sauce. Top with thin slices of Fresh Chive Cheese or fresh mozzarella sprinkled with chopped chives, and bake until the crust is puffy and browned, 3 to 4 minutes more. Finish with a drizzle of extra virgin olive oil, a sprinkle of sea salt, and chopped fresh thyme.\n\n_PER \u00bc PIZZA (APPROX.) 202 cal., 45% (91 cal.) from fat; 7.5 g protein; 10 g fat (3.7 g sat.); 22 g carbo (2.6 g fiber); 425 mg sodium; 20 mg chol._\n\n### CORN AND ZUCCHINI\n\nHeat extra virgin olive oil in a frying pan over medium heat, add slivered white onion, diced zucchini, minced garlic, chopped fresh marjoram, and fine sea salt to taste and cook until tender. Mix in fresh corn kernels. Mix fromage blanc, homemade or store-bought, with enough heavy cream to create the consistency of softened cream cheese. Spread the cheese mixture onto the pizza crusts, spoon the zucchini-corn mixture evenly over the cheese layer, and top with a few more spoonfuls of the fromage blanc mixture. Bake as directed in step 7. Halfway through baking, sprinkle with Oregano Queso Blanco or crumbled feta, homemade or store-bought and mixed with minced oregano.\n\n_PER \u00bc PIZZA (APPROX.) 164 cal., 40% (66 cal.) from fat; 4.9 g protein; 7.5 g fat (2.7 g sat.); 21 g carbo (3.1 g fiber); 234 mg sodium; 13 mg chol._\n\n### CHERRY TOMATO, RICOTTA, MINT, AND CHILE\n\nTop pizza crusts with generous smears of ricotta cheese, homemade or store-bought, mixed with crushed dried serrano chiles or red chile flakes and a little fine sea salt. Cover with whole cherry tomatoes. Sprinkle with more chile flakes, if you like, and drizzle with extra virgin olive oil. Bake as directed in step 7. Finish with torn fresh peppermint leaves and a sprinkle of sea salt.\n\n_PER \u00bc PIZZA (APPROX.) 122 cal., 43% (53 cal.) from fat; 3.7 g protein; 6.1 g fat (1.4 g sat.); 15 g carbo (2.6 g fiber); 148 mg sodium; 4 mg chol._\n\n### POTATO, ONION, AND GOUDA\n\nYou won't need a baking stone or preheated baking sheet for this Roman-style pizza. For 6 pizzas, peel 1\u00be pounds 'Yukon Gold' potatoes and slice as thinly as possible on a mandoline. Place the slices in a very large bowl of well-salted water and let soak for at least 1 hour at room temperature (or several hours if chilled). Drain and pat dry, then toss with 2 cups finely chopped white onion and about \u00bd cup extra virgin olive oil.\n\nFor each pizza, oil a large rimmed baking sheet, then gently stretch and flatten each dough ball into a thin rectangle (about 10 by 7 inches). Brush with more oil, then cover the surface of the dough with no more than a double or triple layer of the potato-onion mixture.\n\nBake as directed in step 7 until the potatoes are just beginning to turn brown, about 10 minutes. Remove from the oven, sprinkle with shredded Gouda cheese, homemade or store-bought, and with chopped fresh rosemary. Return to the oven and bake until the crust is browned, the edges of the potato slices are crisp, and the cheese is bubbling, about 5 minutes more.\n\n_PER \u00bc PIZZA (APPROX.) 197 cal., 47% (92 cal.) from fat; 6 g protein; 11 g fat (2.8 g sat.); 21 g carbo (3 g fiber); 271 mg sodium; 11 mg chol._\n\n* _'Expresso' has a protein content of 12.3 percent, which we needed to give our pizza crust a good soft, chewy texture. SeeThe Allure of Local Wheat for more on local flours._\n\n##    \n'SWEET MILLION' TOMATO SAUCE\n\nWe created this recipe especially for our red 'Sweet Million' cherry tomatoes, but you can use any small, tender cherry or pear-shaped tomato. The sauce has a rustic character and nice crunch from the tomato seeds and skins.\n\nMAKES about 1\u00be cups\n\nTIME about 1 hour, 10 minutes\n\n1 quart 'Sweet Million' tomatoes\n\n1 tablespoon minced garlic\n\n1 tablespoon extra virgin olive oil\n\n1 teaspoon honey\n\n\u00bd teaspoon crushed dried red serrano or \u00e1rbol chile\n\n\u00bd teaspoon fine sea salt\n\n1 teaspoon chopped fresh basil\n\n1 teaspoon chopped fresh oregano\n\n1. Pulse the tomatoes in a food processor to chop coarsely. Heat the garlic and oil in a medium saucepan over medium heat, stirring, 1 minute. Stir in the tomatoes, honey, chile, and salt. Bring the mixture to a boil over high heat. Reduce the heat to low and simmer uncovered, stirring often, until very thick, 50 to 60 minutes.\n\n2. Stir in the basil and oregano and use right away, or cool, cover, and refrigerate.\n\n**MAKE AHEAD** Refrigerate, tightly covered, for up to 3 days. Or, pack into a sturdy plastic container or resealable plastic bags and freeze for up to 5 months.\n\n_PER \u00bc-CUP SERVING 42 cal., 50% (21 cal.) from fat; 0.78 g protein; 2.3 g fat (0.29 g sat.); 4.7 g carbo (1.1 g fiber); 164 mg sodium; 0 mg chol._\n\n##    \nSUMMER LEMONGRASS CUSTARDS\n\nSilky and delicate in texture, with a surprising little zing from the lemongrass, these custards are lovely on their own. But a pouf of whipped cream puts them over the top.\n\nMAKES 6 servings\n\nTIME about 1\u00bd hours, plus at least 1\u00bd hours to steep\n\n4 large stalks lemongrass\n\n2 cups whole milk\n\n2 large eggs\n\n\u2153 cup plus 2 teaspoons honey\n\n\u2153 cup heavy cream\n\n1. Trim the tough outer layers from the lemongrass stalks, then slice the core. In a small saucepan over medium-high heat, combine the milk and lemongrass, stirring, until small bubbles form, 5 minutes. Let cool, stirring often, for 20 minutes. Cover and refrigerate for at least 1\u00bd hours or up to overnight.\n\n2. Preheat the oven to 350\u00b0F. In a bowl, whisk together the eggs and the \u2153 cup honey until well blended. Strain the milk mixture into the egg mixture (discard the lemongrass) and whisk again. Divide the custard among six 5-ounce ramekins.\n\n3. Set the ramekins in a 9-by-13-inch baking dish. Pull out the oven rack halfway and put the baking dish on the rack. Pour boiling water into the baking dish to come halfway up the sides of the ramekins. Carefully slide the rack into place. Bake until the custards jiggle only slightly when ramekins are gently shaken, about 30 minutes.\n\n4. Using a wide metal spatula, transfer the ramekins to a cooling rack. Let cool for at least 15 minutes.\n\n5. In a bowl, whisk together the cream and the remaining 2 teaspoons honey until thick. Serve the custards warm or chilled, with spoonfuls of cream on top.\n\n**MAKE AHEAD** Cover tightly and refrigerate for up to 1 day.\n\n_PER SERVING: 184 cal., 45% (83 cal.) from fat; 5 g protein; 9.2 g fat (5.1 g sat.); 22 g carbo (0.04 g fiber); 64 mg sodium; 97 mg chol._\n\nHERB BLOSSOMS ON THE TABLE\n\nOne of the loveliest aspects of growing your own herbs is that you have access to their flowers, which you rarely see in grocery stores because they are so fragile. Even farmers' markets can have trouble keeping them fresh. When herbs are a few feet from your kitchen door, you can pick them minutes before you need them and scatter the blossoms over salads or soups. (They taste like the herbs themselves, only spicier.)\n\nBeyond cooking, the blossoms make unusual, fragrant bouquets for your table, and because they are short, they don't block the view of diners sitting across from each other. We like to put bunches of them, along with the leafy sprigs, in small glasses all over the table.\n\n  WATERMELON, CANTALOUPE, OR HONEYDEW SORBET\n\nThis sorbet is only as good as your melon, so use the best fruit you can find (or grow). We especially loved the flavor of our sweet, honeyed 'Ambrosia' cantaloupes in this sorbet, but the greenish 'Sharlyn' melons and 'Sugar Baby' watermelons were good too. The sorbet is best when eaten as soon as possible after freezing, because the lack of refined sugar makes it turn icy as it sits in the freezer. The good news is that any leftover sorbet makes an excellent granita: Turn it into a square baking pan, rake it with a fork until fluffy, cover, and freeze. To serve, let the granita sit at room temperature for 5 minutes, then rerake with the fork until fluffy.\n\nMAKES 1 quart, or 8 servings\n\nTIME about 1 hour, plus at least 20 minutes in an ice cream maker\n\n8 to 10 cups ripe cantaloupe, honeydew, or watermelon chunks (1-inch chunks)\n\n3 tablespoons honey\n\n1. Whirl the melon in a blender or food processor until smooth. Pour into a bowl, cover, and chill until cold.\n\n2. In a saucepan, bring \u00bd cup water to a boil. Remove from the heat and stir in the honey. Pour into a bowl and chill until cold. Strain into the melon puree.\n\n3. Freeze in an ice cream maker according to the manufacturer's directions. Serve immediately.\n\n_PER \u00bd-CUP SERVING 87 cal., 5% (4.5 cal.) from fat; 1.6 g protein; 0.5 g fat (0.1 g sat.); 22 g carbo (1.4 g fiber); 17 mg sodium; 0 mg chol._\n\n##    \nPEPPERMINT-LEMONGRASS TISANES\n\nHerbal infusions, or _tisanes_ (the French term), make a soothing, caffeine-free finish to dinner. We paired this zingy mentholated one with our melon sorbet (at left), and liked the way it brought out the ripe flavors of the fruit. This is an especially enjoyable tisane to make, because the peppermint leaves release an intoxicating aroma as you crush them. You can use ordinary spearmint, but it won't be nearly as fragrant.\n\nMAKES 8 servings\n\nTIME about 10 minutes\n\n12 stalks lemongrass\n\n6 cups loosely packed peppermint sprigs, plus 8 small sprigs or leaves\n\n2 tablespoons honey (optional)\n\n1. Bring 8 cups water to a boil in a medium pot over high heat. Meanwhile, peel 2 or 3 tough outer layers off each lemongrass stalk, then mash the core with a meat mallet or the bottom of a small, heavy frying pan. Cut the stalks into 2-inch lengths.\n\n2. Remove the boiling water from the heat, add the lemon-grass, cover, and steep for 2 minutes. Crush the 6 cups mint sprigs in your hands and add to the pot. Re-cover and let steep for 5 to 8 minutes.\n\n3. Pour through a strainer into a large, heatproof pitcher and stir in the honey. Divide among 8 small tea glasses or cups and top each serving with a mint sprig.\n\n**MAKE AHEAD** The tisane can be made through step 2 a day ahead; once strained, cover and refrigerate. Reheat to a simmer before serving (don't allow it to boil).\n\n_PER 1-CUP SERVING 61 cal., 15% (9 cal.) from fat; 4.5 g protein; 1 g fat (0.3 g sat.); 12 g carbo (fiber n\/a); 41 mg sodium; 0 mg chol._\n\n# SUMMER PRESERVED\n\n_Clockwise from top right: dried poblano (ancho) chiles; canned 'Marvel Stripe' tomatoes; dried corn; dried peppermint; dried rosemary; dried flageolet beans; dried serrano chiles_\n\nThroughout the summer, we put up some of what we'd grown, to make cooking more exciting for the rest of the year. Four tools were our best friends in these undertakings:\n\n**1. A boiling-water canner** You can use the canner for tomatoes and for jams and lots of other preserves. Use it once and it's no longer intimidating.\n\n**2. A dehydrator** Haul out that 1970s dehydrator (or go get a new one) and put it to work. It's the coolest and most efficient way to dry herbs and chiles, and you can use it to dessicate any fruit or vegetable into a fine snack for hiking or biking. The dehydrator is ridiculously simple to use: Cut up your food, spread it on the dehydrator tray, and plug in the dehydrator. During the next day or two, your food will dry in a gentle waft of warm, recirculating air, without a chance of molding or bug infestation.\n\n**3. The freezer** We forget that the freezer can be a second pantry for fruits and vegetables. The two main tips we can give you: Pack your food into doubled resealable plastic bags or an airtight sturdy plastic container to prevent freezer burn, and label the bag or container with the contents and the date. We always think we'll recognize a food down the road, and then find ourselves puzzling over dark, icy blobs, wondering what they are and how long they've been in the freezer.\n\n**4. A \"root cellar\"** This is in quotes because we don't actually _have_ a root cellar. Few Californians\u2014or anyone living in a warm place\u2014do. If you live in a cold climate and have a basement, the basement will work. For specifics on this, we recommend the excellent and venerable _Root Cellaring_ _,_ by Mike and Nancy Bubel (Storey Communications, Inc., 1979, 1991), and _The Complete Root Cellar Book_ , by Steve Maxwell and Jennifer Mackenzie (Robert Rose, 2010). To see what we used instead to store our potatoes (essential for year-round cooking), refer to Our Makeshift Root Cellar.\n\nWe wanted to preserve our onions and garlic, too, because they are also indispensable in year-round cooking, but unfortunately we grew the wrong varieties for long-term storage. So, after the last onions and garlic were eaten in the fall, we had to buy them for winter and spring, because frankly we can't cook without them. We will plant at least two varieties next time\u2014one each for \"keeping.\"\n\nOUR MAKESHIFT ROOT CELLAR\n\nWe planned how to store our spuds with help from Greg Lutovsky, who owns Irish Eyes Garden Seeds in Washington State and specializes in potatoes.\n\nFirst, choose unbruised, unblemished potatoes for storing. After harvest (see tips on how to harvest), let the potatoes cure, spread out in a single layer, at room temperature in a dark, well-ventilated place for about 2 weeks. This will toughen their skins and make them last longer.\n\nArrange the potatoes in single layers in slatted stackable boxes (we used clear acrylic letter trays from an office supply store). Slide the boxes into a clean, empty fridge set to between 40\u00b0 and 42\u00b0F. The beat-up old \"extra\" fridge some people have in their garages is great for this. (Don't store apples or alliums like garlic, onions, or shallots in the same space, adds Lutovsky, because all of them emit ethylene gas, which will make the potatoes sprout.) Unscrew the fridge's lightbulb, so the potatoes don't develop toxins (which they do if exposed to light).\n\nBesides cold temperatures and complete darkness, storage potatoes like extremely humid conditions\u2014in the neighborhood of 90 percent; otherwise, they shrivel. To boost moisture, put a couple of wide deep pans filled with water next to the potatoes. Also, remove the shelf that covers the crisper drawers and fill the drawers with water. To monitor the humidity and temperature inside the fridge, set a combination hygrometer (moisture meter) and thermometer on a shelf. It costs about $10 at a hardware store.\n\nThe last requirement: ventilation. Set a small desk fan on one of the fridge's shelves to keep the air circulating. Plug the fan's power cord into an electrical socket near or behind the fridge (the fridge shuts easily over the cord). Refill the water once a week or so. The potatoes should last for at least 3 months and, depending on variety and your maintenance of the storage conditions, up to 6.\n\n##    \nCANNED HEIRLOOM TOMATOES\n\nCan your own ripe tomatoes, and you'll thank yourself all winter long for bottling up the essence of summer. For this easy recipe, adapted from the \"USDA Complete Guide to Home Canning\" (www.uga.edu\/nchfp\/index.html), you just squish raw skinned tomatoes into jars. This cold-pack technique may cause the fruit and liquid to separate a bit during processing, but the results still taste delicious.\n\nWhen working with tomatoes, it's essential for food safety that you acidify them as noted in the recipe, and that you do not increase the amount of herbs or add any other ingredients. Look for canning jars, rings, and lids in a hardware store or a well-stocked grocery store.\n\nMAKES 6 to 7 quarts\n\nTIME about 3 hours\n\n17 pounds ripe red or yellow heirloom tomatoes\n\n14 tablespoons lemon juice*\n\n7 teaspoons fine sea salt (optional)\n\n7 sprigs thyme, each 3 to 4 inches long (optional)\n\n1. Fill a 20-quart boiling-water canner half full with water. Set the rack so it rests on the pan rim, cover, and bring the water to steaming over high heat. While the water is heating, wash 7 wide-mouthed quart jars, screw rings, and flat lids in hot, soapy water and rinse and drain. Nest the lids inside the rings, place in a medium saucepan, and cover with water. Heat until small bubbles form (do not boil). Remove the pan from the heat, cover the pan, and set aside until the lids and rings are needed. No need to dry before using.\n\n2. Meanwhile, peel the tomatoes: Fill a large saucepan three-fourths full of water and bring to a boil over high heat. Add only enough tomatoes to make a single layer and leave them in the water just until the skins split and will peel easily with a knife, 20 to 40 seconds. Using a slotted spoon, lift out the tomatoes and let cool until they can be handled. Working over a bowl to catch the juices, core the tomatoes, pull off the skins, and trim any browned areas. Repeat with the remaining tomatoes.\n\n3. Put 2 tablespoons lemon juice and 1 teaspoon salt into each jar. Add the tomatoes to the jars, cutting them first if needed to make them fit. Push down on the tomatoes to ensure a compact fit, making sure the tomatoes are covered with juice and leaving \u00bd-inch headspace. If the tomatoes are not covered with juice, add the juice from the bowl as needed to cover. Using the handle of a fork, poke 1 thyme sprig down the side of each jar. Release air bubbles by gently running a table knife around the inside of jars. Wipe the rim of each jar clean with a damp cloth. They must be clean for the lids to seal.\n\n4. With tongs, lift the rings and lids from the hot water. Center a flat lid on each jar, making sure the sealing compound is resting on the jar rim. Screw the metal rings on firmly, but do not overtighten.\n\n5. Put the jars on the canner rack and lower the rack into the water. The water should cover the jars by at least 1 inch. If necessary, add more hot water now and during processing. Cover the canner, bring the water to a boil, then boil for 1 hour and 25 minutes (add 5 minutes for every 3,000 feet in altitude above sea level). The jars may leak a little; this is okay.\n\n6. Turn off the heat and let the jars stand in the water in the canner for 5 minutes. Line a work surface with kitchen towels. Then, using the tongs and a hot pad, lift the rack with the jars onto the edge of the canner. With a jar lifter or hot pads, remove the jars from the rack and set them upright on the towels. Do not tighten the rings. Let the jars cool completely at room temperature. You may hear a satisfying \"ping\" as the lids form a vacuum seal.\n\n7. Press on the center of each lid. If it stays down, the jar is sealed; if it pops up, it isn't. You can still eat the tomatoes if the seal did not form, but you have to keep the jar refrigerated and finish its contents fairly quickly (within a week). Remove the rings. Wipe the jars and lids with a clean, damp cloth. Replace the rings, if you like.\n\n8. Label the jars and store in a cool, dark place for up to 1 year.\n\n_PER \u00bd-CUP SERVING 23 cal., 10% (2.4 cal.) from fat; 1.1 g protein; 0.26 g fat (0.04 g sat.); 5.2 g carbo (1.5 g fiber); 7.1 mg sodium; 0 mg chol._\n\n* _The USDA recommends that you acidify tomatoes with bottled ReaLemon lemon juice, which has a standardized acidity of pH 2.5, or that you use \u00bdtsp. citric acid (Fruit Fresh) per quart jar instead of lemon juice. We used juice from our own lemons, but tested the pH using acid test strips (0 to 2.5 pH range) fromwww.indigo.com to verify that the pH was 2.5._\n\n##    \nDRIED HERBS\n\nPreserving your herb harvest is as simple as hanging up bunches and waiting for them to air-dry, or, even simpler, spreading the sprigs on racks and drying them in a dehydrator for several hours. They will keep for a year.\n\n### TO AIR-DRY\n\nImmerse the herbs in a sinkful of water and give them a good rinse. Shake off as much water as you can (you may want to go outside for this). Tie handful-size bunches at the cut end with kitchen string, leaving some string hanging.\n\nSelect a spot away from the sun (which will darken the herbs) and dust to hang the bunches. We used thumbtacks to attach them by their strings to walls around the office. Also, leave enough space for air to flow freely around each bunch. The herbs are ready when the leaves are completely dry and crumbly, 1 to 2 weeks. Some herbs may darken; this will not affect the flavor.\n\nWorking over a large bowl, strip the leaves from the stalks, discarding the stalks and allowing the leaves to drop into the bowl. Sort through the bowl and remove any stems. Crumble the leaves with your hands. (Chives are a little more difficult to crumble; you may need to chop them.) Pack into airtight containers and store at room temperature in a dark place for up to 1 year.\n\n### TO DRY IN A DEHYDRATOR\n\nRinse the herbs and shake them dry in the same way as for air-drying. Break off and discard any large stems. Arrange the herbs in a single layer (do not crowd them, or they will take longer to dry) on one or more dehydrator trays, and dehydrate until dry and crumbly, 8 to 24 hours. Store as directed for air-dried herbs.\n\n##    \nDRIED CHILES\n\nFor spicing up our cooking all year, we relied heavily on dried serrano and poblano chiles, picked from our bushes when fully ripe and at their fruitiest. Rinse the chiles and pat them dry before using either of the following methods.\n\n### TO AIR-DRY\n\nUsing a needle and strong thread (fishing line works well, too), pierce red, fully ripe chiles through the base of the stem and space out along the length of your thread. (The beautiful New Mexican _ristra_ style\u2014chiles threaded so they overlap\u2014works well in that state's arid climate, but in more humid areas, the chiles rot.) Leave enough thread on either end to tie to shelving or some other support, so the chiles hang like a necklace. Hang the necklace in a well-ventilated spot out of the sun and let the chiles dry until they have withered. Serranos are ready when they are brittle, at least 2 weeks. Poblanos (called anchos in their dried state) should be taken down when they are evenly darkened, fully wrinkled, and leathery but still supple, 2 to 3 weeks. Put both types in airtight containers. They will keep, stored at room temperature, for up to several months.\n\n### TO DRY IN A DEHYDRATOR\n\nArrange rinsed, dried whole chiles in a single layer on one or more dehydrator trays. Because poblanos are thick, you may have to invert the tray above them for the first day to accommodate their girth. Once they have shrunk, you can stack the trays normally. Dehydrate the serranos until they are brittle, 1 to 2 days. The anchos should be leathery but still supple, which will take 2 days.\n\n##    \nDRIED CORN\n\nWe ate most of our corn fresh off the cob, but dried some of it for use later in the year (we put it into stuffed poblano chiles). Using a dehydrator is the simplest and fastest approach.\n\nRemove the husks and silk from the ears. Cut each ear into 2-inch lengths and set the pieces in a single layer on the dehydrator trays. Dehydrate until just dried, about 36 hours. Rub the dried kernels off the ears.\n\nTwo ears will yield about 1\u00bc cups fresh kernels or \u2153 cup dried kernels. Store the dried kernels in airtight containers at room temperature for up to several months.\n\n_PER \u00bc-CUP SERVING 150 cal., 12% (18 cal.) from fat; 4 g protein; 2 g fat (0 g sat.); 31 g carbo (6 g fiber); 15 mg sodium; 0 mg chol._\n\n##    \nROASTED POBLANOS FOR THE FREEZER\n\nWhen chiles are in season, we like to roast lots of them and stash them away in bags in the freezer. Then when we want to make chiles rellenos, a salsa, or a stew, we just pull out a bag. We freeze the chiles whole with the seeds, so they can be prepped in a number of different ways.\n\nMAKES 8 roasted chiles\n\nTIME about 45 minutes\n\n8 poblano chiles, about 2 pounds total\n\n1. Preheat the broiler. Put the chiles on a rimmed baking sheet lined with aluminum foil. Place in the broiler about 4 inches from the heat source and broil, turning as needed, until blackened all over, at least 15 minutes.\n\n2. Remove the poblanos from the broiler, cover with a kitchen towel, and let cool until they can be handled. Gently pull off and discard the blackened skins. Let the chiles cool completely, then arrange them in a single layer in a 1-gallon resealable plastic bag. Seal the bag closed, pressing out the air. Seal the first bag in a second bag.\n\n3. Freeze for up to 6 months.\n\n_PER CHILE 33 cal., 45% (15 cal.) from fat; 1.7 g protein; 0.17 g fat (0.02 g sat.); 7.8 g carbo (1.2 g fiber); 5.8 mg sodium; 0 mg chol._\n\n_Slow-Roasted Tomatoes for the Freezer_\n\n##    \nSLOW-ROASTED TOMATOES FOR THE FREEZER\n\nThese slow-roasted tomatoes freeze well and add deep, tomatoey flavor to any dish long after summer has gone. Any variety or size of tomatoes will work, but in our climate, tomatoes seldom get much bigger than 3 inches in diameter. Roast tomatoes of the same size together so they cook evenly.\n\nMAKES 8 cups\n\nTIME about 8 hours\n\n10 pounds tomatoes\n\n\u00bd cup extra virgin olive oil\n\n3 cloves garlic, finely chopped\n\n2 teaspoons fine sea salt\n\n\u00bc cup chopped fresh oregano\n\n1. Preheat the oven to 250\u00b0F. Core all tomatoes except cherry tomatoes. Cut small tomatoes in half, keep cherry tomatoes whole, and cut medium and large tomatoes into 1\u00bd-inch-thick wedges. Arrange the tomatoes, cut side up and packed tightly together, on rimmed nonreactive or aluminum foil\u2013lined baking sheets.\n\n2. In a small bowl, mix the oil and garlic, then drizzle over the tomatoes. Sprinkle with the salt and oregano.\n\n3. Roast the tomatoes, switching the pans to a different oven rack every 2 hours, until they have wrinkled and shrunk by more than half but are still slightly moist, 6 to 8 hours. If roasting cherry tomatoes, begin checking after 5 hours.\n\n4. Let the tomatoes cool completely, then transfer to a sturdy airtight container and store in the refrigerator for up to 1 week or in the freezer for up to 5 months.\n\n_PER \u00bc-CUP SERVING 54 cal., 59% (32 cal.) from fat; 1.2 g protein; 3.8 g fat (0.54 g sat.); 5.2 g carbo (1.6 g fiber); 147 mg sodium; 0 mg chol._\n\n# FALL\n\n_The One-Block Fall Feast_\n\nQuinoa Bites with Walnut Romesco  \nPickled Cocktail Mushrooms and Onions  \nCreamy Flageolet Dip with Red Pepper Sticks\n\nGrilled Radicchio and Fennel Salad with Apples and Toasted Walnuts  \nButternut Squash Gnocchi with Chard and Sage Brown Butter\n\n_Cheese platter:_ Gouda and feta with fresh honeycomb, walnuts, pineapple guava slices, Last-Minute Pineapple Guava Preserves, and Whole-Wheat Rosemary Shortbreads  \nHoney Ice Cream\n\nChardonnay\n\nOur cornstalks were dried out and toppling over. The lovely trombetta zucchini vine had just a few withered fruits left. The pattypans, so verdant and green a month before, had splotchy mildew all over their leaves, and we'd all eaten way too many squashes\u2014even the chickens were sick of them. It was time to move past summer.\n\nJohanna Silver, our new test-garden coordinator, waded into the pattypan patch one late August day and started vigorously ripping out the monster vines. A former city kid from Denver, she's a girl who loves the physical exercise of gardening, and she threw her whole body into it. By the end of the day, a big swath of the garden had been cleared, ready to be planted with new crops for our fall feast.\n\nThis time the Food and Garden teams had formed a plan well in advance. In went seeds and seedlings for radicchio and fennel, to be thrown on the grill for a salad; plenty of sage, because it is indispensable in fall cooking; Swiss chard, for the same reason; and squat, juicy little cipollini onions, as a backup for our already-harvested 'Spanish White' onions. (We had discovered, belatedly, that this variety\u2014and our sweet, juicy 'Spanish Roja' garlic\u2014weren't the right kind for storing, and had both of them stashed away in a cool, dry spot, hoping they would survive long enough to use for the dinner.) I tried to persuade Team Garden to plant more garlic, but it just couldn't be done. If it were started now, in late summer for a fall harvest, the cloves wouldn't form.\n\nSome crops, needing a longer spell in warm weather, had already been planted: Three kinds of beans, to be shelled and dried, and a butternut squash vine. Along with our cipollini onions, they would go into a big vegetable cassoulet-type dish, our main course. Johanna had the brilliant idea of planting quinoa, one of the fabled \"ancient grains\" of the Andes, and now it stood in a thick patch about five feet tall, with colossal yellow-orange seedheads. It was close to our grapevine, and looking at them together inspired our first course: quinoa-stuffed grape leaves! The quinoa would be cooked with raisins from the same grapevine, of course. This made us instantly want to have nuts, to add to the stuffing, but we had not a single nut tree on the grounds. Even if we bought a baby tree immediately, it would take years to bear fruit. Nuts seemed crucial to a fall feast, so we added walnuts to our list of \"imports\" and hoped we could soon plant a couple of our own trees.\n\nApples also seemed indispensable for a fall menu. We did have a dwarf apple tree, but it had pooped out the year before and was still standing around doing nothing. So apples went on the imports list, too.\n\nWe chose to use them in salad instead of in something sweet, because we had a grand plan for dessert. Fabulous, nectarlike Calimyrna figs grew on one straggly old vine near our test kitchen. We'd dry them at their peak, in September, and simmer them with our raisins in a Syrah syrup, until they turned into a dark, velvety, sophisticated compote, which we would lavish with dollops of homemade cr\u00e8me fra\u00eeche.\n\nWith our menu set and our garden growing, we figured we might as well take Team Cheese to the next level. After our bumbling experiments with making cheese from a book, we needed expert help. We found it at Bellwether Farms, in the rolling, green hills of Sonoma County. In Bellwether's warm, milk-scented cheese-making room, Liam Callahan showed us how to turn milk into ricotta, one of the glories of Italian gastronomy. We saw how gently he handled the curds to make them fat and fluffy, using nothing more than vinegar (although we would use lemon juice), salt, and heat. We spooned up the sweet new ricotta right out of the mold, and tried to commit its taste and texture to memory\u2014our cheese lodestar.\n\nAt Cowgirl Creamery, up in cool, foggy Marin county, co-owner Sue Conley and cheese maker Jonathan White explained how to make their award-winning fromage blanc, a luxuriously smooth, rich, spreadable cheese that can be used umpteen ways in cooking. It was a breakthrough for us. Mysterious ingredients that only real cheese makers seemed to understand, like cheese culture and rennet, were just ingredients, we realized. They were no harder to add to milk than cocoa, although it took longer to taste and see their full effects.\n\nThese lessons inspired us to go even further. We pored through home cheese-making catalogs and bought cheese molds, a few different types of cultures, our own little bottle of rennet, and even a small cheese press. In a fit of ambition, we decided we would do a full-out cheese course for our fall feast and make Gouda and feta. Each of these took several days to finish (and the Gouda then had to age for months). It took many tests to get them both right, which led to our coining a new term: cheese slave. Finally, we managed to produce cheeses that tasted surprisingly good, like actual, bona fide Gouda and feta, and our pride blocked out our pain.\n\nOur gorgeous eggs, on which we had come to be utterly dependent, would, of course, be a main ingredient in our fall feast. Then, suddenly, the chickens stopped laying. Not only that, they were shedding feathers like crazy. The infamous annual chicken molt had arrived.\n\nOnce a year, usually when the weather starts to turn cold, chickens drop their old feathers to make way for strong new ones, and their energies go into producing feathers, not eggs. This went on for several weeks, threatening to leave us eggless for our dinner. Plus, the chickens looked freaky, as though they had been mauled. So many feathers were strewn around the coop that Jim, our Team Chicken leader, said it looked like the scene of a pillow fight.\n\nWe had an even bigger chicken drama to deal with, though. Nugget, our adopted chick, had gotten nasty, nipping at us when we came into the coop. Nugget's red feathers had grown unusually long and glistening. Nugget had a suspicious upright stance.... We hadn't heard actual crowing yet, but Johanna, who had been in charge of chickens at a farm, said it out loud: \"Guys, Nugget is a dude.\"\n\nThat meant we had to get rid of him. Roosters are illegal in our city, as they are in many others, because they're noisy. And even if keeping Nugget hadn't been against the law, we knew he had to go. We couldn't handle an explosion of chicks.\n\nSome of us on Team Chicken voted for eating him, in time-honored chicken-raising fashion. Plus, as Elizabeth pointed out, we publish chicken-dinner recipes in the magazine all the time. \"If we're chicken eaters, why not eat this chicken?\" But others argued that we weren't exactly running a farm, that Nugget had a name, and we should find him a home.\n\nWe tried, we really did, but had zero luck. Finally, we decided to take him to the feed store where we had bought our chicks. We'd noticed cages of half-grown roosters there, and the clerk had said that sometimes people \"adopted\" them. Sure, it was a euphemism, but we had no choice. To the feed store he would go.\n\nI wrote about our decision on our one-block blog, and late one night, when Nugget was groggy and unsuspecting, my valiant boyfriend, Peter\u2014who'd dealt with roosters at his grandparents' farm\u2014and I crept into the coop and nudged Nugget into a big, sturdy box. We taped it and left him boxed up till dawn\u2014and his ride to the feed store.\n\nThat morning, I happened to check my e-mail before hopping in the car. A woman named Tina had sent me an urgent message. She had read my blog post and was aghast. Her e-mail sat atop a chain of other e-mails from bird rescuers all around the Bay Area, discussing how to save Nugget. It was amazing. Upshot: Tina urged me to call a number in San Juan Bautista, belonging to a person who would possibly take Nugget.\n\nSo I did, and the very next day, Nugget was off to an entirely different fate. He rode south to join an extended family of other rescuees: emus; chinchillas; a hedgehog; many, many rabbits; and about thirty other roosters who lived in a fenced-in orchard, eating bugs, and\u2014due apparently to the absence of hens\u2014not ripping one another to shreds. It was too good to be true.\n\nAnd it got even better. When Nugget's new owner extracted him from the box, he behaved so beautifully (possibly stunned by the car ride?) that she decided to put him with her own backyard flock of hens. So Nugget, twice rescued, ended up with a harem.\n\n_Nugget the rooster_\n\nWhile Nugget began his new, charmed life, our luck had turned sour in the garden. The perfectly ripe, green-gold Calimyrna figs had been gobbled up by squirrels, right through the bird netting. In other words, those cursed furballs ate our dessert. It is hard to dream about a dish for weeks on end and then have it chewed to bits in one day.\n\nThen the mother of one of our interns came for a tour of the garden. A discerning cook of Palestinian and Syrian background, she nibbled a grape leaf from our vine, since it was an ingredient she knew well\u2014and immediately spat it out. Apparently, we should have picked in spring, when the leaves were tender and sweet, rather than waiting till fall, when they were leathery and disgusting. Also, the grapes, which we had imagined would automatically turn into plump raisins, had molded. Good-bye, first course.\n\nTeam Kitchen had an emergency powwow. Instead of our figgy compote, we would use what we had lots of\u2014honey\u2014and make ice cream. Then, to replace our stuffed grape leaves, we decided to create a trio of appetizers\u2014which lacked only one ingredient: mushrooms.\n\nAnd so Team Mushroom was born. The easiest and quickest way to get a good variety of fungi is to order \"logs,\" which we soon discovered looked like misshapen loaves of bread. Mainly they were sawdust and bran, held together with threadlike mycelium, the main body of the mushroom.\n\nWithin a week of their arrival, our logs sprouted a miniforest of oyster, shiitake, and poofy, white pom pom mushrooms. It was immensely gratifying. \"When I have a kid and he needs a science project that works, I'm getting one of these,\" said Brianne McElhiney, assistant to the editor-in-chief. She became the main mushroom-log caretaker, and whenever she went on vacation, the logs would send up crazy, deformed shapes\u2014almost as if they sensed her absence.\n\nSpooky or no, mushrooms cast an undeniable spell. Once we began to learn about them, we felt as though we had gone down a rabbit hole and emerged in the marvelous world of fungi. Did you know that the largest organism on earth is a four-square-mile fungus in eastern Oregon? And that it is at least twenty-four hundred years old? These were the kinds of factoids we now sprinkled into party conversation.\n\nWhen we found morel spawn for sale, we couldn't resist. Morels are incomparably tasty and usually harvested wild. If we could grow them, well, that would be close to miraculous. Elaine Johnson, our associate food editor, serious mushroom-lover, and co-captain of Team Mushroom, guided the adventure. In a damp patch of soil near the redwood trees, we dug in the spongy white threads\u2014essentially the baby fungus, from which morels would soon sprout like crazy, we hoped. We would get morels in anywhere from three months to two years, said the seller. We fed the bed with a steady diet of fresh compost and Elaine tenderly showered it with ash from the pizza oven, and we waited. It's been six months, and so far _nada_ , but we are patient. Perhaps also a tiny bit obsessed.\n\n_Mushroom harvest (logs)_\n\n_Scooping ash for the morel bed_\n\nBesides the butternut squash, we had one other phenomenally successful crop: our 'Nugget' hop vine, which had shot right up its fourteen-foot trellis over the summer and then sideways, too. It hadn't been ready to pick for our summer beer, but was now loaded with pale green flowers. We picked pounds of the slightly sticky blossoms, dried them, and stashed them in the freezer until we were ready to make beer again.\n\nThe other components of our beer were wheat and barley. Earlier in the summer, we had harvested what was left of both after the squirrels had ripped through, and, since we had already made a backup beer, spread the stalks of grain on a tarp inside to fully dry. Now it was time to thresh, an ancient and picturesque-sounding activity that brings to mind charming villagers rhythmically beating golden sheaves of grain against stone floors.\n\nWe couldn't have been less picturesque if we had tried. To thresh properly, you have to convince each tiny grain to let go not only of the stalk, which isn't so hard, but also of the papery envelope\u2014the chaff\u2014that tightly encloses it. Nothing we did, not rubbing the grains together in a bowl or mashing them against the bottom of a perforated bucket, seemed to loosen the envelopes. (Then again, we hadn't actually gotten around to researching how to thresh; there was so little left, after the squirrels' rampage, that it seemed easier to just wing it.) Stephanie Dean, our test-kitchen coordinator, bent over her portion of wheat and barley at her desk, peeling the envelope from each seed. Then she took it home and talked her relatives into helping. It was torture.\n\nAs we were thinking that perhaps what we should have done was bag up the grain and whack it like a pi\u00f1ata, or maybe run it over with a car, we found out\u2014from a grain historian who threshes her own heritage wheat\u2014that we should have put the seedheads in a large, strong sack and then danced the twist on it with rubber-soled shoes. Seriously. If you don't have an actual threshing machine, the twist is the way to go. Afterward, the winnowing\u2014getting rid of the bits of envelope\u2014is relatively easy, and we had done it correctly, more or less: You let the grains fall into a bowl in a light breeze, and the breeze whisks away the chaff. The picturesque villager would have a broad woven winnowing basket, for tossing the grain. We had metal mixing bowls, but it still worked.\n\nIn the end, we had about three pounds of wheat and barley combined, ready to be stored until we made our next beer. If nothing else, it made us deeply appreciate the people who grow our grains and mill them into flour.\n\nCompared to the wheat and barley, the quinoa was a cinch to thresh. We just stripped the dusty dried seeds (blessedly unenclosed by papery envelopes) off the seedheads into bags. Then, on the advice of Seeds of Change, who sold us our quinoa to plant, we figured out a better way to winnow. In the test kitchen, we draped a big sheet over some chairs and a table to create a kind of giant catcher's mitt, positioned a small fan opposite it, and put a baking pan in front of the fan. Then we rubbed handfuls of the quinoa in midair above the baking sheet. The wind blew the chaff into the sheet and the heavier seeds fell onto the baking pan\u2014more or less. We had to winnow each batch a few times to get it completely clean, but it did work. The kitchen acquired a fine layer of quinoa dust, however. Next time, we'll do it outside.\n\n_Threshing quinoa_\n\nBy early October, well past the usual honey season, hive Veronica had produced nearly twenty more pounds of honey. This bonanza, after the twelve pounds we had collected in August, seemed remarkable, especially because new hives, just building up their colonies, often yield no honey at all their first year. The women of Team Bee were beaming as they carried in the heavy golden frames. Over the months, they had grown even more entranced by the bustling of the hives, by the way the bees took care of and communicated with one another\u2014licking each other clean, for instance, and clustering around the queen on cold days to keep her warm. \"I like the honey, but I like the bees better,\" said Margaret. Sometimes, when she was stressed, she would take not a coffee break but a bee break, and go out to sit with the hives for a while. Brianne, a new member of Team Bee, was so inspired that she set up her own hives at home. By this time, all three of them could point out the queen bees within a humming mass of workers in only a few seconds.\n\nKimberley deepened her bond with bees one day when she found one flying around inside the test kitchen. Not wanting any of our allergic coworkers to possibly be stung, she dipped a finger in honey\u2014we were still processing the harvest, so had plenty sitting around\u2014and held it out to the bee, who landed and began lapping it up. \"I could feel a little tickle. She didn't move at all, and I was able to walk her out.\" The girl hand-fed a _bee_.\n\nIt was mid-October now, and almost time for the feast. The hens had sprouted new feathers and resumed laying, so at least we had eggs. But we were having major problems with the main course. In the test kitchen, Amy Machnak had been doing her best to make a knockout vegetable cassoulet using all our pretty beans, plus mushrooms, peppers, squash, and onions. But no matter what she tried, it kept turning out gloppy. \"It's just a big pot of mess!\" she moaned. She abandoned cassoulet and experimented with individual potpies, draping a sort of biscuity wafer over each. They looked sweet, but did not taste exciting. She fooled around with a vegetable terrine next, but it didn't hold together. We realized we were trying to cram all our crops into a single dish, and we had to simplify. Time was running out.\n\nWith days to go, Amy focused on our garden savior, the butternut squash. From it she made a glowing orange gnocchi, lightened with the ricotta we had recently learned to make, and scattered with Swiss chard and the seeds from the squash, roasted till crunchy. The finishing touch: a drizzle of sage browned butter. It was delicious. Our menu had an anchor.\n\nThis time, we set the table in our outdoor kitchen, at dusk. Candles twinkled down the center, and the light from inside the building shone onto us as we stood around in chatty clusters, eating tiny quinoa cakes with a spicy red pepper sauce, pickled mushrooms and onions, and a creamy dip made from the flageolet beans. We had our crisp Chardonnay again, but we were still waiting for the Syrah. It had tasted wild and thorny and not completely drinkable when we'd sampled it a few days before. We were betting it would mellow by the time our next feast rolled around, though.\n\nIt felt so refreshing to be at the table not as seasoned coworkers who knew how to put out a magazine together, but as people who had collectively decided to make something happen, despite our not knowing how. Everything we ate\u2014the grilled radicchio and fennel salad, Amy's triumphant gnocchi, the go-for-broke cheese platter, the rich honey ice cream\u2014tasted great, but it meant so much more. Not that we were consciously thinking this (mostly we were just enjoying catching up with one another), but on some level we knew how much each bite had cost us, and also given us, and that made us happy.\n\n_Scarlet Emperor' beans, freshly opened (right), 2 hours after opening (center), and dried (left)_\n\n# THE FALL GARDEN\n\n10 feet\n\n**FALL GARDEN PLAN**\n\n1. Pineapple guava\n\n2. Fennel\n\n3. Quinoa\n\n4. Peppers and chiles\n\n5. Beans\n\n6. Parsley\n\n7. Chives\n\n8. Rosemary\n\n9. Radicchio\n\n10. Tomatoes\n\n11. Cippolini onions\n\n12. Swiss chard\n\n13. Sage\n\n14. Thyme\n\n15. Oregano\n\n16. Butternut squash\n\n17. Lemon tree\n\nAs with our summertime garden, most of the crops we planted for our fall menu thrive in a spot that gets full sun for at least 6 hours per day. Some plants, such as beans and winter squashes, are easily started from seed. Others, like peppers and tomatoes, fare better when started from nursery transplants. For planting tips, see here.\n\nALL BEANS\n\n**Best Site** Full sun and well-drained soil.\n\n**Planting and Care** Sow seeds as soon as the soil is warm. Heavy seed leaves must push through the soil, so be sure the soil is reasonably loose and open.\n\n_For vining types:_ Insert poles 1 to 2 feet apart in rows, and sow seeds 1 inch deep and 1 to 3 inches apart. Or, sow along a sunny wall, fence, or trellis and train vines on a web of light string supported by wire or heavy twine.\n\n_For bush types:_ Plant seeds 1 inch deep and 1 to 3 inches apart, allowing 2 to 3 feet between rows. With both types, moisten the soil thoroughly before planting, then do not water again until the seedlings have emerged. Keep the soil moist throughout thegrowing season. Fertilize after the plants are in active growth and again when the pods start to form, working a 5-10-10 fertilizer into the soil along the row.\n\n### Bean 'Cannellini'\n\nMost beans are frost-sensitive heat lovers, easy to grow from seed. Gardeners can choose from many types, but 'Cannellini' is one of our favorites. A mainstay in Italian cuisine, this pearly white bean, a cousin to the kidney bean, cooks up rich and meaty. It's wonderful in soups and stews, or all by itself with just a little olive oil and fresh herbs. The beans grow on bush plants to 26 inches tall.\n\n**Days to Harvest** 85 days from seed.\n\n**How to Harvest** Pick pods in the green \"wax\" stage to use fresh, and at the dried stage for shelling and storage. For dried beans (which was how we used them), let the pods completely dry out on the bush through October, then pick and shell.\n\n**Seed Source** Gourmet Seeds, www.gourmetseed.com.\n\n### Bean, Flageolet\n\nThis pretty little kidney-shaped, bush-type, pale green bean comes from southern France, where it is traditionally served with lamb. We used it for a creamy bean dip and in a vegetarian shepherd's pie.\n\n**Days to Harvest** 90 days from seed.\n\n**How to Harvest** Pick these shelling beans in fall after the pods have matured and the leaves have dried and dropped off. For dried beans (which was how we used them), let the pods completely dry out on the bush through October, then pick and shell.\n\n**Seed Source** Seed Savers Exchange, www.seedsaversexchange.com.\n\n### Bean 'Scarlet Emperor'\n\nA stunner in the garden, this vine grows to 8 to 10 feet tall. Its scarlet-orange flowers give way to velvety green pods that turn plump and juicy as they grow to 8 inches long. The beans themselves are large, like big limas, and a shocking lipstick pink color that quickly darkens to purple.\n\n**Days to Harvest** 75 days from seed.\n\n**How to Harvest** To eat fresh, harvest when beans are fully formed and starting to pull away from the pod (you'll see the bulge of the developing bean through the shell) but before pods start to change color and the beans inside start to rattle when you shake them. Remove the beans from the pods and seal them in plastic bags to store in the refrigerator for 10 to 14 days (you can also freeze them for longer storage). For dried beans (which was how we used them), let the pods completely dry out on the bush through October, then pick and shell.\n\n**Seed Source** Territorial Seed Company, www.territorialseed.com.\n\n### Butternut Squash\n\nLike all winter squashes, butternuts have a hard rind and a firm, dryish flesh. They are excellent for baking and store well, too. The plant pumps out delicious fruits on a tall vine, and it was by far our most successful fall crop. We loved how pretty it looked, rambling up a sturdy metal arbor over a gravel path, with most of the fruits dangling downward like pi\u00f1atas.\n\n**Best Site** Full sun and rich, well-drained soil. Give the vine plenty of room to sprawl, unless you are training it up a trellis or arbor, as we did.\n\n**Days to Harvest** 75 to 80 days from seed.\n\n**Planting and Care** When soil temperatures warm to about 50\u00b0F, sow seeds directly in the ground 1 inch deep, spacing them 2 to 4 feet apart in rows. Or plant seeds on mounds or hills of soil about 8 to 12 inches across and 4 to 6 feet apart, spacing seeds on each hill 4 inches apart. Once plants reach 4 inches tall, thin to 3 seedlings per hill. Water at planting time, then every 2 or 3 days until seeds germinate. As the plants grow, give the roots plenty of water, but keep the leaves dry to avoid mildew. Feed the plant regularly by working a complete fertilizer into the soil around the mounds or along rows.\n\n**How to Harvest** Allow a squash to ripen on the vine until it has thoroughly hardened, then cut it from the vine with 1 inch of stem attached.\n\n**Seed Source** Burpee, www.burpee.com.\n\n_Butternut squash_\n\n### Chiles\n\nWe picked the last of our summer garden poblanos and serranos for our October feast. See the Summer Garden.\n\n### Chive\n\nSee the Summer Garden.\n\n### Fennel, Florence or Finocchio\n\nCooks prefer this annual fennel _Foeniculum vulgare azoricum_ over the perennial common fennel ( _F. vulgare_ ) for its larger, thicker leafstalk bases. These bulbous bases, which have a celerylike crunch and a sweet, light anise flavor, can be steamed, saut\u00e9ed, baked, grilled, simmered in soups and stews, or eaten raw. The feathery leaves resemble those of dill, but have a coarser texture, and make good garnishes and seasonings. The seeds are a staple \"spice\" in our pantry.\n\n**Best Site** Full sun and well-drained soil rich in organic matter.\n\n**Days to Harvest** 90 to 100 frost-free days from seed.\n\n**Planting and Care** Sow seeds directly in the ground, \u00bc inch deep and 4 to 6 inches apart. Thin seedlings to 1 foot apart. Keep the soil moist until the first leaves appear. As they grow, water the plants regularly, but don't overwater. To make the bulb more tender, sweeter, and extra white, try \"blanching\" it: When the bulb is about the size of an egg, pile up compost around it.\n\n**How to Harvest** Snip off fronds as you need them. Cut whole stalks by slicing them off where they join the bulb. As for the bulb itself, wait for it to gain some girth: at least 3 inches end to end and a couple of inches wide (ours were most flavorful at about 5 inches wide). Then harvest by slicing horizontally through the bulb just above the root line. This leaves the base of the bulb behind, from which new sprouts will often shoot. Or, pull up the bulb, roots and all.  \nEven though Florence fennel is grown for its bulb rather than its seeds, you can still harvest seeds from it to use in cooking. They are less plump than seeds from common fennel, but they are still very flavorful, especially when fresh. To harvest the seeds, wait until the plant sends up its giant flower stalks (they can grow up to 6 feet tall) and then blossoms, sometime in fall. The flower clusters will eventually produce pale green seeds. When they are plump and swollen, cut the stalk from the plant (the seeds won't fall off), rinse off the aphids if you have them, and let the clusters dry on paper towel\u2013lined baking sheets inside at room temperature, uncovered, until the seeds are completely hard (4 to 5 days; bite into one to check). Each large cluster will yield about 1 teaspoon seeds, so just one plant will yield around \u00bc cup fennel seed. Store in a tightly closed glass jar; for maximum freshness, keep in the fridge.\n\n**Seed Source** Burpee, www.burpee.com.\n\n_Harvesting Florence fennel seeds_\n\n### Lemon 'Eureka'\n\nSee the Summer Garden.\n\n### Mushrooms\n\nSee How to Grow Mushrooms.\n\n### Onion, Cipollini\n\nThese 2- to 3-inch, flattish bulbing onions are Italian heirlooms\u2014so special they are usually listed in their own class in catalogs. They come in red, straw yellow, and white and have a mild, sweet flavor. You'll need patience to grow them, though; to form bulbs, they need long days (typical of northerly climates from Bakersfield, California, north to Alaska), and they take more than 4 months to mature. We think they are worth the wait. We used them to make a melt-in-your-mouth cocktail pickle and also added them to a rich bean rago\u00fbt.\n\n**Best Site** Full sun and warm, fast-draining soil that has been amended with compost.\n\n**Days to Harvest** About 105 days from seed.\n\n**Planting and Care** We planted sets as soon as the ground warmed up. Plant seeds a bit later in spring (March or April), \u00bd inch deep in rows 18 inches apart, then thin seedlings to 2 or 3 inches apart. Water regularly to keep soil moist just below the surface.\n\n**How to Harvest** Pull up the onions in late summer after the tops have died down, then store them in a cool, dry place.\n\n**Seed\/Seedling\/Set Sources** Gourmet Seed International, www.gourmetseed.com; Territorial Seed, www.territorialseed.com; Northwest Seed & Pet, www.nwseed.com.\n\n### Oregano, Italian\n\nSee the Summer Garden.\n\n### Parsley 'Gigante Italian'\n\nSee the Summer Garden.\n\n### Peppers 'Carmen' and 'Mariachi'\n\n**'Carmen'** An early-maturing Italian sweet pepper, 'Carmen' has horn-shaped, 6-inch-long fruits that taste great whether grilled or saut\u00e9ed. As they mature from green to red, they get even sweeter. That's when they are best sliced fresh for salads. The plants grow to 28 inches tall.\n\n**'Mariachi'** These cone-shaped beauties deliver a blast of color and a hint of heat\u2014spicy but only mildly hot. They change from creamy yellow to bright red all summer on plants 18 to 24 inches tall. The plants produce 4-inch fruits well into fall (although you can start harvesting them in the summer).\n\n**Best Site** Full sun and fertile, well-drained soil.\n\n**Days to Harvest** 65 days for 'Mariachi' to 75 days for 'Carmen' from seed.\n\n**Planting and Care** Sow seeds indoors 6 to 8 weeks before the last spring frost. Plant seedlings only after the soil is warm in spring (peppers are heat lovers). Place the seedlings 18 to 24 inches apart. One or two plants in a container (at least 14 inches wide and 12 inches deep) are plenty. Let the soil dry out between waterings, and cut back on water as the peppers mature to concentrate their flavor.\n\n**How to Harvest** Harvest at the red or green stage. Snip off pods with scissors or pruners.\n\n**Seed\/Seedling Sources** 'Carmen', Tomato Growers Supply Co., www.tomatogrowers.com; 'Mariachi', Burpee, www.burpee.com. Check your local nursery for seedlings of these varieties.\n\n_'Mariachi' pepper_\n\n### Pineapple guava\n\nThe hardiest of subtropical fruits, pineapple guava ( _Feijoa sellowiana_ ), also called feijoa, blooms in late spring or early summer. Showy, inch-wide white flowers with big tufts of red stamens are edible, too\u2014crisp, succulent, and sweet: Toss them into fruit salads or use them to top cakes and cupcakes. The oval, gray-green fruits that follow have dense, sweet, slightly grainy flesh that tastes a bit like pineapple. They grow on evergreen, multistemmed shrubs (10 to 15 feet tall and wide).\n\n**Best Site** Full sun and well-drained soil.\n\n**Days to Harvest** Fruits are ready to harvest 4 to 5\u00bd months after flowering in warm climates, 5 to 7 months after bloom in cooler areas. Plants can take 2 to 3 years to bear fruit.\n\n**Planting and Care** Set out nursery plants (gallon size or larger) in spring if you are in the mildest climates of California, Arizona, or Hawaii; give them room to spread and water regularly to start. Once established, pineapple guava needs only occasional watering, although fruiting is best with deep regular waterings. Feed lightly with a complete fertilizer such as 8-8-8 once every 2 months during the spring and summer growing season. The plant can take almost any amount of training (in late spring) to shape it as an espalier, screen, hedge, or small tree.\n\n**How to Harvest** To harvest lots of fruits at a time, wait until the first ones drop, then spread a tarp underneath the tree and give the trunk a shake. Repeat every few days. Or, gently squeeze a fruit to see whether it is ripe. If it gives to the touch, pick by hand. Ripe pineapple guavas, which are green (sometimes with a red blush) when mature, can be stored in the refrigerator for up to a month and frozen for up to a year.\n\n**Plant Sources** Look for plants in garden centers, or ask your nursery to order it for you from a wholesaler such as Monrovia Nursery, www.monrovia.com.\n\n### Quinoa 'Faro'\n\nGrown mostly for its edible seed, quinoa (say it: \"keen-wa\") has an unusually high protein content\u201416 to 23 percent\u2014and contains all eight essential amino acids. We grew 'Faro', a variety bred to thrive at sea level, and loved its statuesque height (nearly 5 feet) and its big, heavy, golden orange seedheads. Having gotten so little from our summer planting of wheat and barley, we wondered how much this next grain experiment would yield. But we're always willing to grow anything once. After threshing and winnowing, which was a pain but not nearly as bad as threshing the wheat and barley, we were rewarded with about 15 quarts of quinoa from just a couple of tablespoons of seed.\n\n**Best Site** Full sun and loose, well-drained soil.\n\n**Days to Harvest** 100 to 110 days from seed.\n\n**Planting and Care** Sow seeds in midspring when soil is warm (at least 65\u00b0F) by scattering them over soil that's been raked smooth, then watered lightly with a hose. Cover them lightly with \u00bc to \u00bd inch of soil. (We raised 54 plants in a 4-by-8-foot raised bed.) Keep the soil moist until the seeds germinate and the first two or three leaves appear, then let them get super-thirsty. You want to give them just 10 to 12 inches of water for the entire season.\n\n**How to Harvest** The crop is ready in September or so, when it looks dry, starts to flop over, and the seeds barely dent when you push into them with a fingernail. Cut off the seedheads, lay them flat indoors, and let them dry completely (including stems and leaves) until the stalks are brittle. This takes about 2 weeks. Next, roughly remove the seeds from the stems\u2014that's the threshing part\u2014and then comes the \"fun\" part: winnowing to separate the seeds from the rest of the particulate matter. It's a messy process, best done outdoors. Set up a small fan and put a tray in front of it to catch the seeds. Then rub the quinoa between your hands in front of the fan. The heavier seeds drop onto the tray, while the featherlight chaff blows away.\n\n**How to Rinse for Cooking** Quinoa seeds are coated with a bitter, soapy substance called saponin, which has to be rinsed off before cooking. (Store-bought quinoa is prerinsed, but still benefits from a dunk in water.) Because our sea-level quinoa was a particularly bitter variety, we were extra-thorough when it came to cleaning. We soaked the seeds for a few hours in warm water, changing the water often. Then we rinsed them several times, rubbing the grains between our fingers, until the water was no longer cloudy. We did big batches at once and thoroughly dried the seeds on kitchen towels before storing them in airtight containers in the freezer.\n\n**Seed Source** Seeds of Change, www.seedsofchange.com.\n\n_'Faro' quinoa (top); quinoa laid out to dry; then, threshing the seeds (bottom)_\n\n### Radicchio 'Palla Rossa Ashalim'\n\nThe radicchio we grew is really a red-leaved Italian chicory ( _Cichorium intybus_ ) that forms a lettucelike head. The stunning purple-red leaves with bright white veins taste sharp and spicy, with a hint of bitterness that adds a pleasing bite to mixed salads. Radicchio is a bit trickier to grow than lettuce: It takes slightly longer to mature, doesn't tolerate heavy frost (mild frost is fine, and gives it a wonderfully sweet flavor), and needs to be monitored, much like cabbage, to determine when to harvest.\n\n**Best Site** Full sun and compost-rich, well-drained soil.\n\n**Days to Harvest** 65 days from seed.\n\n**Planting and Care** Plant in midspring or early summer for a fall harvest; in mild winter climates, plant in fall for a winter harvest. Sow seeds \u00bc to \u00bd inch deep in rows spaced 18 inches apart, then thin seedlings to about 12 inches apart. Protect from frosts with floating row covers. Keep soil evenly moist during the growing season, and apply a 10-10-10 fertilizer after growth starts.\n\n**How to Harvest** When the heads start feeling firm to the touch, slice them from above the crowns with a sharp knife, then peel away the outer leaves to reveal the white-veined leaves inside. (If enough time remains before a frost, the crowns may resprout and develop second heads.) Heads are past their prime for harvest when they turn hard and their leaves turn tough and bitter.\n\n**Seed Source** Botanical Interests, www.botanicalinterests.com.\n\n_'Palla Rossa Ashalim' radicchio_\n\n### Rosemary 'Tuscan Blue'\n\nSee the Summer Garden.\n\n### Sage\n\nCommon sage ( _Salvia officinalis_ ) is a kitchen garden essential\u2014one we would never be without. You can use its soft, aromatic gray-green leaves fresh or dried to flavor everything from soups, stews, and egg dishes to vegetables and vinegar. A small perennial subshrub (1 to 3 feet tall), it comes in many forms; our current test-garden favorites include 'Icterina', whose leaves are edged with yellow, and 'Tricolor', whose new leaves are flushed with purple.\n\n**Best Site** Grow it in full sun in cool climates, afternoon shade in hot climates, in a spot with loose, well-drained soil.\n\n**Days to Harvest** Any time after the plants have filled out.\n\n**Planting and Care** Set out plants from nursery containers with the crown (the thickened area at the stem base) slightly above the surrounding soil. Space several at 12- to 24-inch intervals in garden beds (they make pretty edgings). Or, grow a single plant in a 16-inch-wide container. Water once a week or so (more in hot, windy weather) for the first year, then taper off to occasional irrigations (once a month during the warm season if leaves look droopy). Apply a granular 10-10-10 fertilizer once a year in spring when new growth begins.\n\n**How to Harvest** Pick leaves as needed.\n\n**Seedling Source** Mountain Valley Growers, www.mountainvalleygrowers.com.\n\n_Sage_\n\n### Swiss chard\n\nGrow this form of beet for its leaves and stalks rather than its roots. It's one of the easiest vegetables you can grow, and you can harvest leaves all summer (and, in mild climates, into fall and even spring, depending on when you plant). We grew both green-and-white chard and 'Bright Lights'\u2014with leaves ranging from green to burgundy and stalks in various shades of yellow, orange, pink, purple, and red.\n\n**Best Site** Full sun and fertile, well-drained soil.\n\n**Days to Harvest** About 60 days from seed.\n\n**Planting and Care** Sow the big, tan, crinkly seeds \u00bd inch deep and 2 inches apart in spaded soil any time from spring to early summer. Gardeners in mild winter climates (like us) can plant it in the late summer to early fall for harvest starting in the late fall through winter. As seedlings appear, thin them to 12 inches apart. Keep the soil evenly moist. Feed with a 10-10-10 fertilizer after plants are established and again 6 weeks later.\n\n**How to Harvest** When the plants are 12 to 18 inches tall, begin cutting the outer leaves, keeping the core of the plant intact to continue growing.\n\n**Seed Sources** Nichols Garden Nursery, www.nicholsgardennursery.com; Renee's Garden, www.reneesgarden.com; and Territorial Seed International, www.territorialseed.com.\n\n_Swiss chard_\n\n### Thyme, French\n\nSee the Summer Garden.\n\n### Tomatoes\n\nAt this point in the year, our summer garden was still producing 'Sungold', 'Sweet Million', and 'Brandywine' tomatoes. See the Summer Garden.\n\n_\u2014Kathleen N. Brenzel_\n\nTHE \"IMPORTS\"\n\n**We went outside our garden for these local foods:** milk, whole and unhomogenized (for cream, butter and buttermilk, and for cheese); walnuts, Honey Crisp apples, whole-wheat flour ('Sonora' and 'Expresso')\n\n_Crushing and destemming Syrah grapes_\n\n# THE FALL PROJECTS\n\n##    \nHOW TO MAKE  \nWINE\n\nWine making is like following a very large, very slow recipe, with strange and fascinating moments along the way. At the end of the process, with luck (as Team Wine seems to have had), you'll have pretty decent stuff to proudly call your own.\n\nIf you can, pick your own grapes. We considered quitting our day jobs after our October 2007 experience of harvesting dusty, juicy Syrah grapes in Thomas Fogarty Winery's remote and gorgeous Fat Buck Ridge Vineyard in California's Santa Cruz Mountains.\n\nThe home wine-making journey will get under your skin and give you a huge appreciation for the quality in the bottles you buy. Just remember to keep a record of everything you do to your wine and to sanitize everything every step of the way. Then celebrate the chance to use that high-school chemistry.\n\n_Fogarty Winery's Fat Buck Ridge_\n\n###    \nSYRAH\n\nOurs is an inky, deep purple wine. Blueberries and blackberries mingle with leather, smoke, and bacon. To make the journey worthwhile, you should make at least four cases (twelve bottles per case); we made sixteen cases.\n\n#### WHAT TO USE\n\n**_Clipboard, pencil, and log sheet_** You need to record your wine's progress, additions you make (such as yeast), and any preventative or preservative measures you take. We downloaded a free \"log chart\" from _WineMaker_ magazine (www.winemakermag.com).\n\n**_Wine grapes_** To make four cases of wine, you need about 125 pounds of grapes. Make sure you can get your hands on the wine grapes before you buy or rent any equipment. By midsummer, look into harvesting grapes in early fall at a U-pick vineyard (try El Dorado County, California's www.edc-farmtrails.org and search for \"wine grapes\"). PickYourOwn.org (www.pickyourown.org) lets you search for U-pick farms near you, with grape growers posting when vines are ready to harvest.  \nYou could put dibs on grapes in summer for fall delivery by mail or special truck shipment (both methods are fairly expensive). But if you live near a grower with excess grapes, you can load up for less by cruising by with your own van or truck (this is primarily an option in California, Oregon, and Washington). At MoreGrapes! (www.moregrapes.com), growers on the West Coast and beyond list available grapes, including Syrah from about $1 per pound. And many wine-making association Web sites dedicate a page to \"grapes available\" (check out the El Dorado Wine Grape Growers Association's comprehensive site: www.eldoradograpes.com).  \nA good first stop is to ask about grape options at your local home wine-making shop. To find one, go to www.winemakermag.com and click \"Resource Guide\" and then \"Supplier Directory.\" Many home-brew beer shops have wine contacts, too. They will likely refer you to small local growers or to largely Web-based enterprises like F. Colavita & Son (www.cawinegrapes.com) that cater to home winemakers.\n\n**_Potassium metabisulfite_** The industry-standard chemical for equipment sterilization and protecting finished wine from spoiling. Buy it in powdered form, not Campden tablets (for which you have to take the extra step of crushing). We used \u00bd teaspoon per 5-gallon carboy (water cooler\u2013style glass jugs) of Syrah. We sprinkled some into each piece of equipment we needed to sterilize, then added some water, sloshed it around every inner surface, and rinsed liberally. We also mixed some with water to create our airlock solution and to sterilize bottles before filling and corking them. About $5 per \u00bd pound (you shouldn't need more than this, for everything) online or at a home wine-making store.\n\n**_Large plastic garbage can_** For stomping grapes with your feet, after which you'll have to pick through the must (the newly crushed grapes) to remove the stems. About $10 at a hardware store or home-improvement center. Or, use a hand-cranked **_crusher-destemmer_** (like a large trough with a rotating screw in the middle), which gives you relatively stem-free must. Rent this from a home wine-making store; buying one will cost at least $450.\n\n**_5-gallon food-grade plastic buckets_** For schlepping grapes, must, and wine and for sanitizing tubing. About $7 at a hardware store.\n\n**_Two 32-gallon food-grade plastic drums with lids_** For your must's first (primary) fermentation. About $70, including lid, at a home wine-making store. Or, you can use two **_large plastic garbage cans_** (if you bought one for crushing and destemming, just pick through the must to remove the stems, then add yeast to start fermentation). About $10 each at a hardware store or home-improvement center.\n\n**_Cheesecloth_** For keeping bugs out of your fermenter. About $4.50 for 2 square yards at a hardware or cookware store.\n\n**_Garden hose and sprayer_** For making quick work of cleaning equipment at the end of a messy day. About $30 at a hardware store or home-improvement center.\n\n**_Wine yeast_** For starting your fermentation to convert your grape-juice must to wine. We used a strain called ICV-D80. About $1.50 for an 8-gram packet online or at a home wine-making store; for most wines, you need 1 gram of yeast per 1 gallon of must.\n\n**_Wooden or stainless-steel wine-cap punch-down tool_** For punching down the cap of skins and seeds, which gives red wine its beautiful color and contributes flavor and tannins. Or, with some basic lumber and tools, you can make a wooden \"plonker\" (for project directions, visit www.winemakermag.com\/component\/resource\/article\/107-build-3-winemaking-projects). At least $80 for a stainless-steel model at home wine-making stores.\n\n**_Hydrometer_** For measuring the sugar level of your fermenting must. About $6 from www.morebeer.com.\n\n**_Hydrometer jar_** A plastic tube for holding the wine sample you're measuring with the hydrometer. Some hydrometers come with their own jars. About $5 online or at a home wine-making store.\n\n**_Floating thermometer_** For taking the temperature of your fermenting must. About $10 online, or at a home wine-making store.\n\n**_Malolactic bacteria_** For prompting a secondary fermentation. $15 to $20 for 2.5 grams (enough for 66 gallons of wine) online or at a home wine-making store.\n\n**_Bladder press_** For pressing your young wine off the skins and seeds. Rent this at a home wine-making store; buying one will set you back at least $1,500. Or, you can use a **_basket press_**. Again, rent this at a home wine-making store; buying one will run at least $275.\n\n**_5-gallon glass carboys_** Surrogate barrels\u2014buy one for every 5 gallons of wine you are making, plus one extra for the racking stage. From $20 to $30 at a home wine-making or home-brewing shops.\n\n**_Airlocks_** For keeping oxygen and bugs out of your wine while allowing carbon dioxide to bubble out. Buy one for each carboy. About $1.50 each online or at a home wine-making store.\n\n**_Rubber stoppers_** Aka bungs with small holes in the center, for sealing your carboys while accommodating the airlocks. About $1.50 each online or at a home wine-making store.\n\n**_Two 3- to 5-quart nonreactive mixing bowls_** For sterilizing stoppers, tubing, and corks. Stainless steel is ideal (don't use aluminum or copper). Borrow them from your kitchen, or buy them for as low as $10 each online or at a cookware store.\n\n**_Wine thief_** A glass tube for extracting wine from a carboy for tasting and testing. From $25 online or at a home wine-making store. We resorted to a **_plastic turkey baster_**. From $5 online or at a cookware store.\n\n**_Plastic funnel_** For pouring wine into carboys. Choose an asymmetrical one with a high side to avoid splash back. From $3.50 online or at a home wine-making or cookware store.\n\n**_6- to 8-foot lengths of food-grade vinyl tubing_** For siphoning wine. We used two tubes with a \u215c-inch interior diameter: one could be siphoning while the other soaked in sanitizer. About 30\u00a2 per foot online or at a home wine-making or plumbing-supply store.\n\n**_Sturdy worktable_** For elevating the full carboys when racking. From $75 at a home-improvement center.\n\n**_Oak chips or cubes_** For barrel flavor substitute. We used small cubes\u201426 grams per 5-gallon carboy of Syrah. $10 to $30 per pound; buy online or at a home wine-making store.\n\n**_Digital kitchen scale_** For weighing the oak chips. From $30 at a cookware store.\n\n**_Wine bottles_** Reuse empty wine bottles (you'll need to scrub off the original labels and sanitize the bottles) or buy new ones (sanitize these, too). Consider a Rh\u00f4ne-style bottle (with sloping shoulders) if you want your Syrah to look authentic. From $15 per dozen at a home wine-making store; you'll need 26 standard (750-milliliter) wine bottles for each 5-gallon carboy of wine you're making.\n\n**_2 bottle rinsers and 1 \"bottle tree\" drying rack_** For sanitizing the insides of the bottles. You just slip a rinsed wine bottle over the sprayer nozzle and pump it up and down on the nozzle a few times, then drain each bottle on the drying rack, which is shaped like a tall, spiky tree trunk. Each bottle rinser runs about $20, and a drying rack starts at about $40; both can be purchased at a home wine-making or home-brew store.\n\n**_Manual bottle filler_** For controlling the bottle-filling process. You attach the plastic rod with a spring-loaded tip to your plastic tubing. About $6 online or at a home wine-making or home-brew store.\n\n**_Corks_** Price depends on quality. Natural cork is divided into several grades. Flor corks are the finest grade (about $150 for 250 corks), but we decided to go with similarly top-quality 1\u00be-inch \"overrun\" corks left over from bottling runs at Napa wineries (we paid $19 per 100 corks at MoreWine in Los Altos, California; www.morewinemaking.com or 800\/600-0033). Buy corks online or at a home wine-making store.\n\n**_Corker_** Rent a \"floor corker\" at a home wine-making store, or buy time on a vacuum corker at a home wine-making or home-brew store. Hand corkers are available and cost less, but they limit your ability to move between bottling elements. Buying a floor corker at a home wine-making store will cost at least $85 and more along the lines of $850 for a really sturdy one (the $85 version should be fine).\n\n**_Labels_** You can use a design-graphics program like Adobe Illustrator, or you can work with your local home wine-making or home-brew store to create a label on their software. Conveniently, we had designers down the hall.\n\n#### HOW TO DO IT\n\nWine is a living, changing thing, and your timeline will vary based on environmental factors like temperature, starting sugar level, and yeast health. Here is the timeline we followed. Consult the resources listed in Helpful Information for more general guidance.\n\n##### OCTOBER 4, 2007\n\n**Harvested 500 pounds of Syrah grapes** We picked our grapes at Thomas Fogarty Winery's Fat Buck Ridge Vineyard in the San Francisco Bay Area's Santa Cruz Mountains. After hauling the grapes back to our office, we crushed them, both by foot in sanitized garbage cans and with a hand-cranked crusher-destemmer. Then we put the destemmed must (crushed grapes, skins, seeds, and juice) into two food-grade plastic drums, covered each with cheesecloth and drum lid, and allowed the must to cold-soak (to extract color and flavor) for 4 days.\n\n_Pushing grapes into the crusher-destemmer_\n\n_Crushing by foot_\n\n_Shin-deep in grapes_\n\n##### OCTOBER 8\n\n**Added yeast** Following the package directions, we added the yeast to start primary fermentation.\n\n##### OCTOBER 12\u201328\n\n**Punched down the \"cap\"** Two or three times a day, we used our punch-down tool to push the cap of skins and seeds that continually formed on our fermenting must back down into the juice. The cap was thick and radiated heat, and we had to stand on a chair to get enough leverage to punch through it for the first few strokes.\n\nEvery afternoon during this period, we took two measurements: The wine's temperature (with the floating thermometer), to make sure the wine was warm and fairly constant (it hovered around 78\u00b0F), and the wine's sugar level (a Brix scale reading with the hydrometer), to check that the yeast continued to eat up the sugar, converting it into alcohol.\n\n##### OCTOBER 22\n\n**Started malolactic fermentation** This secondary, bacterial fermentation\u2014ML for short\u2014converts harsh malic acids into softer lactic ones. It's as simple as sprinkling the powdery bacteria into the wine (following the package directions, of course).\n\n##### OCTOBER 29\n\n**Pressed** **the wine off its pomace** The pomace is the skins, seeds, and other solids that form the cap. We used the bladder press, catching the \"free-run\" wine (what naturally spilled through the slats of the press and out through the spigot with only gravity's assistance). This was followed by the \"press-run\" wine, which was coaxed off the pomace by inflating the bladder of the press with a garden hose to 15 psi\u2014pounds (of pressure) per square inch.\n\nAs the wine streamed from the press, we ran a bucket brigade from the spigot over to sanitized 5-gallon carboys, each topped with a plastic funnel. Then we outfitted each carboy with an airlock filled about halfway with \"meta\" solution (2\u00bc teaspoons potassium metabisulfite dissolved in 1 quart water; we made our meta solution once and stored it in a sterilized gin bottle) and stuck the airlock in a bung to create a seal on top of the carboy, keeping oxygen (the enemy of wine) and bugs (we don't like to share, and we didn't want to risk bacteria from their tiny legs) out of the wine while allowing carbon dioxide to bubble out.\n\n_Pressing the Syrah_\n\n##### NOVEMBER 2007\u2013APRIL 2008\n\n**Periodically checked airlocks** Every two weeks or so, we checked the airlocks (refilling them with new meta solution if wine had burbled into the lock or if the meta level was looking low) and tasted our wine. In mid-February, we gave some wine samples to Fogarty winemaker Michael Martella to test pH and total acidity (TA) to make sure things were on track. Our pH was 3.8 (a little high; Martella thought it would come down), and our TA was .64 (which Martella said was fine).\n\nEach time we pulled wine out of a carboy with our wine thief (okay, our turkey baster) to sample it, we \"topped off\" the headspace (the air between the wine level and the airlock) with more Syrah (because, as mentioned above, oxygen is the enemy of wine). The extra Syrah came from smaller glass jugs, also topped with airlocks, that we had set aside for this purpose.\n\n##### APRIL 29, 2008\n\n**Racked the wine** We siphoned our wine off the lees (the spent yeast and other sediment that drops out as fermentation ends) into fresh carboys. Here's how: Put a full carboy on top of your worktable and position an empty, sterilized carboy on the floor under it. Remove the airlock and set it upright in a sterilized container to save for the carboy you are about to fill up. Put one end of the sterilized vinyl tubing in the full carboy (don't let the tubing end get too close to the lees at the bottom of the carboy), then gently suck the open end of the tube until wine is flowing toward it. (Be prepared to quickly pinch the open end and stop the wine with your index finger.) Put this open end into the clean carboy and let gravity do the rest of the work for you. Keep pushing the tubing in the full carboy lower and lower toward the lees to keep the wine flowing, but don't let it dip into the lees. Ask a friend or two to help you out for this step.\n\nOnce the wine was racked, we added oak chips for body and rounded flavor: 26 grams of small, medium-toast oak cubes in each 5-gallon carboy.\n\nWe also added \u00bd teaspoon powdered potassium metabisulfite per 5-gallon carboy to protect our finished wine from spoilage by microbes. We filled the headspace with more Syrah, and then topped the new carboys with the meta-filled airocks. Finally, we moved them into a cool, temperature-controlled environment (our basement, which is a steady 56\u00b0F).\n\n##### LATE FALL 2008 (A YEAR AFTER WE STARTED)\n\n**Created labels** In anticipation of bottling, we got our label design ready and printed it (see it here).\n\n##### JANUARY 30, 2009\n\n**Bottled our Syrah** With a drain and a hose-rigged spigot nearby for easy cleaning and cleanup, we dragged out the worktable; stainless-steel bowls for sterilizing our siphon tubing, bottle filler, and corks; two bottle rinsers; and our \"bottle tree\" drying rack.\n\n**_Sterilized bottles_** We whipped up a batch of bottle wash (4\u00bd teaspoons potassium metabisulfite dissolved in 1 gallon water) and split this solution between our two bottle rinsers. We gave each bottle a rinse with bottle wash: two pumps on one spring-loaded bottle rinser to wash away any dust, followed by two pumps on our second rinser to complete the sterilization. Then we placed each bottle onto the tree for 10 to 15 minutes of drying.\n\n**_Siphoned wine into bottles_** Here's how we did it: Stick one end of vinyl tubing into a carboy (as close to the remaining sediment as you dare), then put the other end in your mouth. Start sipping, as you did for racking, keeping a hand near to pinch off the tubing. Grab your bottle filler, and gently but firmly slide the rod into the tubing. As long as you keep the rod end lower than the carboy that's up on the table, your siphon is ready whenever you are.\n\nTake a bottle off the tree, place it at your feet, and poke the bottle filler into the empty bottle. Depress its tip against the bottom of the bottle, and the rod will let wine flow in. Fill the bottle until it is approaching full, pausing to let it foam as needed. When the wine crests the top of the bottle neck, gently lift the rod out.\n\n**_Corked the bottles_** Even though they're natural, corks still need sterilizing before being put in contact with wine (which remains alive, even when bottled). Prep another sanitizing solution (for corks, use \u00bd teaspoon potassium metabisulfite dissolved in 1 gallon water) in one of your stainless-steel bowls. To keep the corks from bobbing up in the solution, nestle a smaller nonreactive bowl on top of the corks.\n\nAfter 20 minutes, they're ready to use. Leave them bathing until you need them; just set the bowl at the base of the floor corker for easy access.\n\nPlace a full bottle on the spring-loaded platform of the corker, position a sanitized cork in the jaws of the \"compression chamber,\" or iris, then brace the corker with your foot and use your hands to pull down the handle. In one fluid motion, the cork gets squeezed on four sides and a metal rod comes down to plunge the compressed cork into the bottle.\n\nOnce in a bottle, the cork expands to create a tight seal. Place the bottles upright (cork side up) for 24 to 48 hours, then flip them upside down in a case or store them horizontally, either in a wine rack or by tilting a full case of bottles on its side, so the wine stays in contact with the cork.\n\n##### A FEW DAYS LATER\n\n**Labeled our Syrah** It was just a matter of lining the label up straight while pressing the label onto a clean, dry bottle.\n\nBut don't start pulling corks and sharing your wine with friends immediately. Your living wine is adapting to its new life in a bottle and could be experiencing bottle shock. Give your wine a few weeks to settle into its smaller confines, then uncork away.\n\n###    \nCHARDONNAY\n\nOur Chardonnay has a touch of creaminess to it, but with the vibrant, juicy acidity of crisp Granny Smith apples and none of the caramel and toast you would get from new oak barrels.\n\nIn terms of home wine making, the biggest differences between white and red wine are that for white, the juice is pressed off the skins before you start fermentation, and the fermentation takes place right in the glass carboys. Since pressing white grapes takes special equipment, we started with juice, not grapes.\n\n#### WHAT TO USE\n\nTo make Chardonnay, follow the What to Use list for Syrah. If you are starting with pressed juice, you won't need these items from the Syrah list:\n\n**_Large plastic garbage can or crusher-destemmer_**\n\n**_Food-grade plastic drum with lid or large plastic garbage can_**\n\n**_Cheesecloth_**\n\n**_Punch-down tool_**\n\n**_Bladder press or basket press_**\n\nYou'll also need:\n\n**_Juice (or wine grapes)_** In midsummer, you can put dibs on pressed juice. We bought 20 gallons of freshly pressed Chardonnay\u2014enough to make just over 100 standard (750-milliliter) bottles of wine\u2014from Thomas Fogarty Winery.\n\n**_Oak chips or cubes_** We used small cubes, adding a small handful per 5-gallon carboy of Chardonnay (we oaked only half of our 20 gallons). See our Syrah guide for where to buy.\n\n**_Rubber stoppers (bungs) without holes_** For \"stirring\" the lees (rolling the carboys to add character and complexity). You need one for each carboy. About $1.25 each online or at a home wine-making store.\n\n**_Duct tape_** For securing the stoppers when rolling the carboys. About $4 at a home-improvement center.\n\n**_Thick cardboard or a thick old rug_** For rolling the carboys.\n\n**_Wine bottles_** Consider a Burgundy-style bottle (sloping shoulders and a relatively wide base) if you want your Chardonnay to look authentic. See our Syrah bottle-buying advice for more tips and how to calculate the number of bottles you'll need.\n\n#### HOW TO DO IT\n\nThe process of Chardonnay making is much the same as for Syrah. Check our step-by-step guide for Syrah for more details on any of the phases described.\n\n##### OCTOBER 4, 2007\n\n**Brought home our Chardonnay juice** We split the 20 gallons among four 5-gallon carboys. Then we started fermentation by adding yeast to each carboy. Finally, we outfitted the carboys with airlocks filled about halfway with meta solution.\n\n_Yeast slurry_\n\n_Adding the slurry to Chardonnay juice_\n\n_Fitting airlocks into carboys_\n\n##### OCTOBER 11\n\n**Added oak chips** We put them in half of our carboys, placing just a small handful in each one.\n\n##### OCTOBER 16\n\n**Took two measurements** We noted both the wine's temperature and its sugar level.\n\n##### OCTOBER 17\n\n**Started malolactic fermentation.**\n\n##### MID-OCTOBER 2007 TO MARCH 2008\n\n**Periodically tasted** We sipped our Chardonnay over the months to see how it was evolving. Also, in mid-February, we gave some wine samples to Fogarty winemaker Michael Martella to test pH and total acidity (TA) to make sure things were on track. The pH was 3.2 (low, which means our acidity is high, which is an excellent thing, we think, for a Chardonnay). The TA was .68 (great).\n\nEach time we pulled wine out of a carboy, we had to \"top off\" the vacant headspace with more Chardonnay. We opened bottles of other West Coast Chardonnays for this purpose. Or we could have added marbles to the carboy.\n\n##### MARCH 3, 2008\n\n**Stirred the lees** We swapped each carboy's airlock and stopper for a stopper without a hole, duct-taped it down, and rolled each carboy on a thick rug until all of the sediment that was lodged on the bottom had been mixed back up into the wine. Then we added \u2153 teaspoon powdered potassium metabisulfite (swirled in a little Chardonnay and slightly heated in a microwave to dissolve the powder) per 5-gallon carboy to protect our wine. Finally, we retopped each carboy with a meta-filled airlock.\n\n##### MARCH 27\n\n**Rolled the carboys** We took our Chardonnay out on the rug again and retopped it with meta-filled airlocks.\n\n##### APRIL 29\n\n**Ditto.**\n\n##### MAY 13\n\n**Rolled the carboys** One last time, the carboys took a spin. Then we moved them to our basement.\n\n##### JULY 8\n\n**Racked the Chardonnay** We siphoned the wine off its lees it into fresh carboys. Then we added \u00bc teaspoon powdered potassium metabisulfite per 5-gallon carboy, filled the headspace with Chardonnay, and topped the new carboys with meta-filled airlocks.\n\n##### OCTOBER 1, 2008\n\n**Bottled, corked, and labeled** After letting our Chardonnay rest for nearly three months, we bottled, corked, and labeled it. Beyond wanting to let the wine settle, we were waiting for cooler weather so as not to expose the wine to the heat.\n\n_Corking Chardonnay_\n\n#### HELPFUL INFORMATION\n\n##### WEB SITES\n\n\u2022 For general and specific advice, plus free charts and sulfite calculator, check out www.winemakermag.com.\n\n\u2022 For general advice, visit www.grapestompers.com.\n\n##### BOOKS AND OTHER PUBLICATIONS\n\n\u2022 _The Way to Make Wine: How to Craft Superb Table Wines at Home_ by Sheridan Warrick (University of California Press, 2006).\n\n\u2022 _The Wine Maker's Answer Book: Solutions to Every Problem, Answers to Every Question_ by Alison Crowe (Storey Publishing, 2007).\n\n\u2022 _WineMaker Beginner's Guide_ (Battenkill Communications, no date).\n\n\u2022 _WineMaker_ magazine (available at home wine-making shops; find subscription information and many articles on www.winemakermag.com).\n\n##### HOME WINE-MAKING SHOPS\n\n\u2022 Find a home wine-making or home-brew shop near you: go to www.winemakermag.com and click \"Resource Guide\" and then \"Supplier Directory.\"\n\n\u2022 Our favorite local shop, MoreWine in Los Altos, California, also has an online shop (great for yeast, malolactic bacteria, and most equipment and materials); www.morewinemaking.com or 800\/600-0033.\n\n\u2022 An excellent Web-based store for equipment, materials, and kits is E.C. Kraus, www.eckraus.com or 800\/353-1906.\n\n_\u2014Erika Ehmsen_\n\n##    \nHOW TO MAKE  \nVINEGAR\n\nDelicious, fancy vinegars are easy to find in the United States. Go to a well-stocked grocery store and you can get everything from French Banyuls to Spanish sherry to Italian balsamic.\n\nWhat is harder to find is good ordinary red-wine vinegar. Most of what is available commercially for a couple of bucks a bottle is thin and flavorless. Some of it is laboratory-produced acetic acid, diluted with water and dressed up with coloring. Slightly better, though only marginally, is vinegar made using a speeded-up fermentation process (anywhere from one to three days). Traditional red-wine vinegar, left to ferment naturally on its own, takes about seventy-five days and results in a much richer texture and flavor.\n\nThe good news is that \"slow vinegar\" is easy to make at home, tastes wonderful, and is cheap to produce (it feeds on leftover wine). We have two crocks going in our kitchen that yield a constant supply for salad dressings, sauces, and gifts.\n\n_Decanting aged vinegar_\n\n###    \nSYRAH VINEGAR\n\nBecause we were already making Syrah wine, we decided to divert some of it to making vinegar. The result was a deep purple-red vinegar, intensely fruity and sharp and fresh.\n\n#### WHAT TO USE\n\nAvoid aluminum or iron kitchen tools for making vinegar, because they can impart off flavors. We used plastic, glass, and wood.\n\n**_A notebook and a pencil_** It's easy to lose track of when and how much you fed your vinegar, unless you record the dates and amounts. This is most important when the vinegar is newly establishing itself.\n\n**_A good mother_** In the world of vinegar, a \"mother\" is a live starter, similar to a sourdough starter for bread. It is home to acetic acid\u2013producing bacteria of the _Acetobacter_ genus that convert wine to vinegar. The mother will form a not-unpleasant and actually quite fascinating thin, somewhat firm gelatinous layer on the surface of your vinegar crock. This is a sign that the bacteria are alive and well and doing their work.  \nYou can either get your mother from a vinegar-making friend, as we did (see How to Do It), or buy it from a vinegar-supply shop, which more often than not will also be selling wine and\/or beer supplies (typically about $12.50 per 8-ounce jar).\n\n**_Leftover red wine_** We used our own Syrah, but you can use any decent, fruity red wine. It's a great use for that half bottle you didn't finish at dinner.\n\n**_1- to 1\u00bd-gallon fermenting container_** The best fermenting containers allow for a wide surface area and have an open top, so that the bacteria have enough oxygen, and are enclosed to keep out light, which slows down the bacteria's progress. A 1-gallon glass iced-tea jar can work (keep it in a cardboard box), but Mason jars are too narrow. We had our best results with 5-liter Italian demijohns enclosed in mesh holders ($37.50 each from Oak Barrel Winecraft, Inc., www.oakbarrel.com) and with gorgeous 1-gallon clay crocks ($80.00 each, from Clay Coyote Pottery, www.claycoyote.com).\n\n**_A simple countertop thermometer_** _Acetobacter_ bacteria prefer the 70\u00b0 to 90\u00b0F range.\n\n**_1-gallon aging container_** New vinegar is very sharp, almost feisty. To let your vinegar mellow, age it for at least a month in a separate, sterilized, lidded container after straining and pasteurizing.  \nClay crocks produce a fine, soft vinegar and were our favorite aging containers. French oak crocks are pricey (6-liter crock, $160 from Oak Barrel Winecraft), but they impart a lovely mellow toasty flavor to the vinegar and look stunning.\n\n**_Cheesecloth_** It keeps out fruit flies and dust, yet allows oxygen to reach the bacteria. About $4.50 for 2 square yards at a hardware or cookware store.\n\n**_1- to 1\u00bd-foot length of plastic tubing_** This is what you will use to feed your vinegar. $1 per foot at a hardware store.\n\n**_Plastic funnel_** Make sure the funnel is narrow enough to fit securely into your tubing and bottles. From $3.50 online and at a cookware store.\n\n**_Large plastic colander_** sized to fit the top of the pot, below. From $6 at a cookware store.\n\n**_Coffee filters_** About $3.50 for 40 filters at a grocery store.\n\n**_8-quart stainless-steel heavy-bottomed pot_** for pasteurizing the vinegar. From $60 at a cookware store.\n\n**_Candy or deep-fry thermometer_** with bracket so you can clip it to the side of the pot. From about $6 online or at a cookware store.\n\n**_Storage-serving bottles and corks_** The Olive Oil Source, www.oliveoilsource.com, sells bottles by the case, in various sizes; 200-milliliter bottles with corks are about $19.50 for a case of 12.\n\n**_21-quart boiling-water canner with canning rack_** Handy for sterilizing your bottles and for pasteurizing the vinegar. From $16 online and at a cookware store.\n\n**_Jar lifter_** For gripping steaming, slippery bottles, draining them, and positioning them for filling. Use it for canning and jam making, too. About $9 at a cookware store.\n\n#### HOW TO DO IT\n\n**1. Find the mother** We got our mother for free, from cookbook author and vinegar maven Paula Wolfert. Paula was given hers several years ago by her friend Abra Bennett, a food writer from Bainbridge Island, Washington. She thinks it's at least forty years old.\n\nPaula cut 5 playing card\u2013size pieces from a mother fished out of one of her crocks, slipped each piece into a small glass jar, and fed it with just enough diluted red wine (1 cup wine and \u00bd cup water) for it to travel safely back to our offices. \"Start small,\" she advised us. \"Swamp it, and it'll be dead.\"\n\n_A piece of the mother in its traveling jar_\n\n**2. Start the mother** Pour each jarful of mother into a fermenting container and add more diluted wine: 2 cups wine to 1 cup water. Cover the open top of each container with a double layer of cheesecloth and roll a rubber band down around the rim to keep it in place (or screw on a jar ring). Put the containers in a warm place out of direct sunlight.\n\n_Covering the vinegar with cheesecloth and a jar ring_\n\n**3. First feedings: a set of 3** Once the bacterial conversion has begun, which will be about 1\u00bd weeks after you have started the mother, you will need to start feeding your vinegar. Add 2\u00bd cups Syrah to each container three times over a period of 1\u00bd weeks.\n\n_Feeding the vinegar_\n\n_How to feed your vinegar:_ Sometime into the first-feeding period, a mother will form on the surface of your vinegar. To avoid \"swamping\" it, slide one end of the plastic tubing underneath the edge of the mother, then fit the funnel into the other end and pour in the wine. Work on a surface that's easy to wipe clean.\n\nYour vinegar is ready when it smells and tastes like vinegar. It is possible to do a titration test to figure out just how acidic the mixture is, but this is unnecessary unless you're using the vinegar for pickling food (you will need 4 percent acidity for pickling); for details, see Fresh Pickled Beets. We dipped clean plastic spoons into our vinegar, took thoughtful sips, and if it tasted like vinegar\u2014tart, sharp, strong, delicious\u2014we either pasteurized, aged, and bottled it (steps 5, 6, and 7), or continued feeding it (step 4) to increase the volume. If it tasted more like wine than vinegar, we left it alone until it was ready.\n\n_Warning sign:_ If your vinegar ever starts to smell like furniture polish, throw it away. It has been contaminated and can't be saved. Always taste a bit of the wine before you feed it to your vinegar.\n\n**4. Maintenance feedings** Once the _Acetobacter_ bacteria have established themselves vigorously, add 1 to 2 cups of wine every few weeks or so. When the vinegar is at a vinegary moment (taste it to see), you can pasteurize and age it.\n\nEvery now and then, the mother will wear out and sink to the bottom, and a new mother will take its place. This doesn't harm your vinegar, but after a while the old mothers will start to absorb the vinegar in the crock. Every month or so, using clean hands or plastic or wooden salad tongs, gently push aside the nice firm top mother, fish out any settled mothers, and throw them away.\n\n**5. Pasteurize the vinegar** This step ensures that a mother won't grow in the bottle or, if you decide to age your vinegar, the aging vessel. However, you can keep a mother from growing if you store the vinegar in the refrigerator (the bacteria go dormant). To age vinegar, though, you have to pasteurize or the mother will form again.\n\nFill the canner with hot water up to the first ring from the bottom, insert the canning rack upside-down (handles down), and bring it to a boil, covered. Meanwhile, line the colander with a few coffee filters and place it over the 8-quart pot. With a plastic ladle, scoop the vinegar into the lined colander. Homemade vinegar is exhilaratingly strong, and because it will evaporate as you heat it, you may want to add from 1 to 3 cups water to the pot. Keep tasting until it reaches the strength you like.\n\nPosition a clean candy thermometer in the vinegar and clip it to the side of the pot. Ease the pot into the boiling water, turn the heat off, and heat the vinegar to 155\u00b0F; hold it there for 30 minutes. If it climbs above 155\u00b0F, lift the pot out and let it sit near the stove until the temperature drops to 155\u00b0F. If it drops below, turn the burner on to give it a boost. You can now let your vinegar age for a while (step 6) or you can bottle it (step 7).\n\n**6. Age the vinegar** Let the pasteurized vinegar cool, then pour it into the aging vessel. Store in a dark, quiet, cool (50\u00b0 to 75\u00b0F) place. Taste periodically to see how the flavor is developing. After a couple of weeks, you should start to notice the mellowing. During aging, solids fall to the bottom of the vessel, so the vinegar not only mellows but also becomes beautifully bright and clear.\n\nWhen you like what you taste, sterilize a batch of bottles and bottle your vinegar, bearing in mind that some aging will continue in the bottle. You can also keep topping up your aging vessel with newly pasteurized vinegar for as long as you like.\n\n**7. Bottle the vinegar** Wash the bottles, corks, a plastic ladle, and a plastic funnel in hot, soapy water, then rinse. To sterilize the bottles, immerse them in the canner, cover with water by one inch, bring to a boil, and boil for 10 minutes. Remove the bottles carefully with a jar lifter (or tongs) and tip out the hot water. Using the ladle and funnel, fill each bottle with hot pasteurized vinegar or room-temperature aged vinegar, leaving \u00bd-inch headspace. Stopper the bottles with the clean corks.\n\nA bottle of unpasteurized vinegar is a form of insurance for vinegar making. If something goes wrong with your current batch of vinegar, your unpasteurized vinegar can be used to start a new one. A bottle of this \"live\" vinegar is also a great gift for friends interested in developing their own stash of flavorful homemade vinegar.\n\n#### HELPFUL INFORMATION\n\n\u2022 _Homemade Vinegar: Make Your Own with Mother of Vinegar_ , by Patrick and Carole Watkins (A-Printing Publishing, 1995).\n\n\u2022 _Making Vinegar at Home_ , by Frank Romanowski, Mark F. Larrow, and Gail Canon; revised by Mark F. Larrow (Beer & Winemaking Supplies, Inc., 1994).\n\n\u2022 _Vinegar: The User-Friendly Standard Text Reference & Guide to Appreciating, Making, and Enjoying Vinegar_, by Lawrence J. Diggs (Quiet Storm Trading Co., 1989; iUniverse, 2003).\n\n##    \nHOW TO MAKE  \nCHEESE (PART II)\n\nAs our one-block project continued, we kept making cheeses using just milk, lemon juice, salt, and herbs, but we also learned to use cheese cultures (freeze-dried packets of microorganisms that give acidity, flavor, and texture) and rennet (enzymes that help coagulate milk). Most of the world's great cheeses\u2014all the soft-ripened, oozy beauties and the long-aged hard cheeses\u2014depend on cultures for their identities, and many need rennet to form curds. We never seemed to make exactly the same cheese twice, although they were similar. And once we started playing around with presses and brines, the chances for variation only multiplied. So far, we have made six cheeses, with more to follow, which we'll write about on <http:\/\/oneblockdiet.sunset.com\/team_cheese>.\n\nAt this point, we like the fact that our cheeses are one of a kind. Their mutability helps us understand why some of our favorite artisanal cheeses differ from time to time. It also puts us in awe of the discipline, knowledge, and skill required to make a consistently great cheese.\n\n#### BEFORE YOU START\n\nTwo basic rules apply to all our cheese making: sanitizing all your equipment and making sure your thermometer is accurate. For details, see Before You Start. As for equipment for the four cheeses that follow, most of what you'll need is available at a good grocery store, cookware store, or hardware store. (We haven't included ordinary cookware like saucepans, baking sheets, or measuring cups in the lists, assuming those would already be in your kitchen.) For specialized cheesemaking tools and ingredients, we turned to the Beverage People, in Santa Rosa, California (www.thebeveragepeople.com or 800\/544-1867), unless otherwise mentioned.\n\n_Cheesemaking supplies_\n\n###    \nRICOTTA\n\nMAKES 1\u00bc to 1\u00bd cups (3\u00bd cups if you use all milk)\n\nTIME about 1\u00bd hours\n\nRicotta, which means \"recooked\" in Italian, is essentially a thrifty way to squeeze more cheese out of what another cheese left behind (usually with some milk added for richness). Bellwether Farms, in Sonoma, California, is famous for its sweet, light, delicious ricotta, and cheese maker Liam Callahan showed us how to make it at home. \"It takes a little bit of practice,\" he says, but the results are worth it.\n\n#### WHAT TO USE\n\n**_Fresh whey_** We used leftover whey from making Fresh Chive Cheese and Feta. We have also used whey from Fromage Blanc, but its acidity causes the curds to form almost as soon as the milk heats. Our Gouda does not yield enough whey. Use the whey right away or chill for up to 1 day.\n\n**_Milk, sea salt, and lemons_** For more on these ingredients, see How to Make Cheese (Part I). You can also make this recipe using 1 gallon whole milk instead of the whey-milk combination. It won't technically be ricotta, but it will taste similar, just less delicate.\n\n**_21-quart boiling-water canner with canning rack_** Large and lightweight, an aluminum canner filled with hot water makes a great water bath for your milk pot. The water bath helps keep the temperature steady and prevents the milk pot from scorching. From $16 at www.amazon.com.\n\n**_8-quart heavy-bottomed stainless-steel pot, colander, cheesecloth_** See the What to Use list for Fresh Chive Cheese for more about these tools.\n\n**_Dairy thermometer_** Unlike a candy thermometer, it measures low temperatures. Get the kind with a bracket so you can clip it to the side of the pot. About $25.\n\n**_Stainless-steel perforated cheese ladle_** You can use a slotted spoon, but a cheese ladle's large, flat surface makes stirring the milk and curds more efficient. $7.\n\n**_1 ricotta mold_** (4\u00bd inches wide; about $9 each; product code CH87). A small, open-weave basket made of food-grade plastic, it gives the cheese a pretty shape. Or, substitute the basket from a small salad spinner (8 inches in diameter or smaller works best).\n\n**_2 or 3 cookie or biscuit cutters without handles_** These are used to elevate the mold to let the whey drain freely (from $2 each at a cookware store). Or use small empty cans with the tops and bottoms removed.\n\n#### HOW TO DO IT\n\n10 cups fresh whey\n\n6 cups whole milk, pasteurized or raw (or 1 gallon whole milk if not using whey)\n\n2 teaspoons fine sea salt\n\n3\u00bd tablespoons freshly squeezed lemon juice if using whey (or 4\u00bd tablespoons if using only milk), from 2 to 3 large lemons\n\n**1. Heat the milk** Put the canner on the stove and insert the canning rack upside down (handles down). Fill the canner with water to 1 inch below the top ring. Cover the canner and bring the water to a boil over high heat. Meanwhile, fill a sink with cold water. This will be your cool-down spot should you need it.\n\nUncover the canner and set the empty 8-quart pot on the upturned rack (it will float a little). Pour the whey, milk (or milk only), and salt into the pot and insert the tip of the thermometer into the whey mixture. With the cheese ladle, stir the mixture 20 times with a gentle surface-to-bottom circular motion to evenly distribute the heat.\n\nRe-cover the canner and heat the mixture over high heat, undisturbed, until it registers 192\u00b0 to 194\u00b0F (the water will be boiling) on the thermometer, 30 to 40 minutes. Adjust the heat if necessary to maintain the temperature. If the mixture starts to get too hot, immerse the pot in the water-filled sink.\n\n_Heating milk for ricotta_\n\n**2. Acidify the milk** Slowly pour the lemon juice over the hot milk while stirring it with the cheese ladle. (The acid coaxes the remaining proteins in the whey to come together.) Stir 20 times with a gentle surface-to-bottom circular motion. Small curds will begin to form (they may have already). Re-cover the canner and leave the mixture undisturbed over high heat for 25 minutes to allow the curds to finish forming. The temperature should stay between 192\u00b0 and 194\u00b0F; check occasionally and adjust the heat as needed. Meanwhile, line the mold with a double thickness of cheesecloth, trimmed to hang slightly over the rim. Set cookie cutters in the stainless-steel bowl.\n\n**3. Drain the curds** Lift the cheese pot out of the water bath and put it next to the sink. Hold the mold over the sink and gently ladle the curds and whey into the mold. Allow the mold to drain until the whey flow slows to a trickle. This usually happens within a couple of minutes. Smooth the cheese in the mold so it's level, then set the mold on top of the cookie cutters in the bowl.\n\nCover the bowl with plastic wrap and chill right away to preserve the ricotta's delicate sweetness. Let the curds drain until there are no visible pockets of liquid (curds should still be moist). This will take up to 30 minutes. Your ricotta is ready to eat. You can either invert the mold onto a plate, or just spoon the cheese out of the mold.\n\n_Draining ricotta curds_\n\n**4. Store the cheese** Wrap the mold airtight with plastic wrap. Or, if you haven't used a mold, transfer the cheese to an airtight container. It keeps, chilled, for up to 4 days.\n\n_PER \u00bc-CUP SERVING 108 cal., 66% (71 cal.) from fat; 7 g protein; 8.1 g fat (5.2 g sat.); 2 g carbo (0 g fiber); 52 mg sodium; 32 mg chol._\n\n###    \nFROMAGE BLANC\n\nMAKES about 4 cups cheese and 10 cups whey\n\nTIME about 20 minutes, plus 16 to 18 hours to culture and drain\n\nSmooth, creamy, and delicately tangy, fromage blanc is a lot more sophisticated than its French name\u2014\"white cheese\"\u2014lets on. It's very versatile in the kitchen, and we've used it in a number of recipes in this book.\n\nSue Conley, co-owner of Cowgirl Creamery, in Petaluma and Point Reyes Station, California, taught us the Cowgirl way to make fromage blanc. She prefers to start with raw milk and use a gentle pasteurizing method, rather than the more intensive one typical for store-bought pasteurized milk. \"The less aggressive method maintains the integrity of the milk and allows it to form a strong curd,\" she explained. \"It also allows the culture to more fully express the flavors inherent in the milk as it ripens.\" (If you would like to use raw milk and gently pasteurize it yourself, see here.) If the milk is homogenized, the fat globules are much smaller and are difficult to retain in the curd; they escape into the whey, compromising the flavor, texture, and yield of the cheese. You can make a fine fromage blanc with pasteurized milk, but avoid homogenized milk if possible.\n\nFor specialized cheesemaking tools and ingredients, we turned to the Beverage People, in Santa Rosa, California (www.thebeveragepeople.com or 800\/544-1867), unless otherwise mentioned.\n\n#### WHAT TO USE\n\n**_Milk and sea salt_** For more on these ingredients, see the What to Use list for Fresh Chive Cheese.\n\n**_Farmhouse culture_** **_(MA4001)_**. Freeze-dried microscopic bacteria (it looks a lot like the freeze-dried yeast used for baking) that, once revived in warm milk, give the cheese both its flavor and texture\u2014and acidify the milk so that curds develop properly. Fromage blanc may be made with other cultures, but the Beverage People carries this one, and we liked the results. $10 per packet; each packet can make at least 12 batches of fromage blanc. Store it in the freezer in a resealable plastic bag.\n\n**_Calcium chloride (optional)_** A type of salt added as a liquid brine, calcium chloride helps firm up the curds (particularly useful with most commercially pasteurized and homogenized milk, which has a weaker structure than raw milk). Use it if your fromage blanc is not as thick as you'd like. $6.50 for a 120-milliliter (about 4-ounce) bottle.\n\n**_Vegetarian rennet_** A lab-created version of the natural enzymes that coagulate milk, this rennet is fine for any young cheese, but can give a bitter flavor to aged cheeses. That's why some cheese makers prefer using animal rennet, a group of enzymes derived from the stomachs of calves, lambs, or sheep, for making aged cheeses. $10 for a 60-milliliter (about 2-ounce) squeeze bottle.\n\n**_Cr\u00e8me fra\u00eeche_** This enhances the smoothness to the cheese and rounds out its flavor. See the recipe here.\n\n**_Dairy thermometer_** See the What to Use list for Fresh Chive Cheese.\n\n**_Stainless-steel perforated cheese ladle_** You can use a large slotted spoon you already have, but a cheese ladle has a large, flat surface that makes the stirring of the milk and curds more efficient. $7.\n\n**_Cheese bucket_** A 5-gallon, food-grade plastic bucket to capture whey. From about $7 at a housewares stores.\n\n#### HOW TO DO IT\n\n1 gallon whole milk, pasteurized or raw\n\n\u215b teaspoon farmhouse culture (MA4001)\n\n2 drops (0.2 milliliter) calcium chloride, if needed\n\n1 drop (0.1 milliliter) vegetarian rennet\n\n\u00bd cup cr\u00e8me fra\u00eeche, homemade or store-bought\n\n\u00be to 1 teaspoon fine sea salt\n\n**1. Heat the milk** Pour the milk into the 8-quart pot and insert the tip of the dairy thermometer. Heat the milk over medium-high heat until it registers 85\u00b0F on the thermometer, stirring often with the cheese ladle to prevent scorching. This takes about 5 minutes.\n\n**2. Add the culture** Remove the pot from the heat and remove the thermometer. Sprinkle the culture as evenly as possible over the milk and let sit for 10 minutes. Using the ladle, stir the milk 20 times to incorporate the culture, using a gentle, surface-to-bottom circular motion.\n\n**3. Add the calcium chloride and rennet** Dilute the calcium chloride in 2 tablespoons cool water. Pour the calcium chloride mixture evenly all over the milk and, using the ladle, stir the milk for 1 minute to get it moving in a circular direction. Dilute the rennet in 2 tablespoons cool water, pour it into the still-moving milk, and stir gently a couple of times in the same direction. (Rennet is extremely sensitive to trace minerals found in water and to agitation\u2014so don't dilute it until you're ready to add it, and incorporate it very gently.) Stir once in the opposite direction to stop the movement of the milk. Cover the pot with a double layer of cheesecloth and let rest overnight on the counter (the temperature can be anywhere between 60\u00b0 and 85\u00b0F). The milk will ripen and set during this time, creating curds (your future cheese, full of proteins and enzymes) and whey.\n\n_Sprinkling fromage blanc culture_\n\n**4. Drain the curds** Line the colander with a double thickness of cheesecloth and set the lined colander over the cheese bucket. Ladle the curds out of the pot into the colander. Drain the curds, scooping and turning them with a soup spoon every hour or so to help them dry evenly, until the mixture resembles thick sour cream. This will take 6 to 8 hours. The whey will drain into the bucket throughout this period. If you'd like to use it for making Ricotta, periodically transfer it to a bowl, cover, and refrigerate. Use the whey within a day.\n\n_Draining fromage blanc curds_\n\n**5. Add the cr\u00e8me fra\u00eeche and salt** When the cheese is ready, turn it into the large bowl and stir in the cr\u00e8me fra\u00eeche and the salt to taste. The cheese is now ready to eat. It will keep in an airtight container in the refrigerator for up to 1 week.\n\n_Adding cr\u00e8me fra\u00eeche and salt_\n\n_PER OUNCE (ABOUT 2 TABLESPOONS) 45 cal., 60% (27 cal.) from fat; 2 g protein; 3 g fat (2 g sat.); 2 g carbo (0 g fiber); 40 mg sodium; 15 mg chol._\n\n###    \nFETA\n\nMAKES 2 small (4\u00bd-inch) wheels (about 1\u00bd pounds total)\n\nTIME about 6 hours, plus overnight to drain, at least 3 hours to brine, and at least 1 day to dry\n\nTangy, crumbly feta can be made many different ways. We've based our method on one developed by Margaret Morris, a well-known Canadian cheese maker and owner of Glengarry Cheesemaking and Dairy Supply, who documented it on her DVD _Home Cheesemaking with Margaret Morris_ (see Helpful Information, for details). This recipe demonstrates the amazing ability of milk to maintain a stable low temperature. Once you heat it up to 85\u00b0F, it's likely to stay there for several hours.\n\nFor specialized cheesemaking tools and ingredients, we turned to the Beverage People, in Santa Rosa, California (www.thebeveragepeople.com or 800\/544-1867), unless otherwise mentioned.\n\n#### WHAT TO USE\n\n**_Milk and sea salt_** For more on these ingredients, see the What to Use list for Fresh Chive Cheese.\n\n**_Feta cheese culture (ChoozitMT1)_** This mixture of four different freeze-dried microscopic bacteria (it looks a lot like the freeze-dried yeast used for baking) is what gives the cheese both its flavor and texture; it also acidifies the milk so that curds develop properly. $13 per packet; each packet can culture about 6 gallons of milk, or 6 batches of cheese. Store it in the freezer in a resealable plastic bag.\n\n**_Lipase powder_** One of the enzymes found in a calf's stomach, lipase breaks down the fatty acids in the milk into pleasingly tangy compounds. It makes cow's milk taste more like sheep's or goat's milk, the more traditional milks for making feta. $11 for 26 grams (just under 1 ounce).\n\n**_Calcium chloride and vegetarian rennet_** For more on these ingredients, see the What to Use list for Fromage Blanc.\n\n**_21-quart boiling-water canner with canning rack, dairy thermometer, 2 ricotta molds_** See the What to Use list for Ricotta. You can also use a small salad spinner insert (see photos) if you don't have the molds.\n\n**_8-quart heavy-bottomed stainless-steel pot, colander, cheesecloth_** For more on these, see the What to Use list for Fresh Chive Cheese.\n\n**_Stainless-steel perforated cheese ladle_** See the What to Use list for Fromage Blanc.\n\n**_Wooden spoon with a round, narrow handle_**\n\n**_Thin metal spatula (at least 8 inches long)_**\n\n**_Cheese bucket_** A 5-gallon, food-grade plastic bucket to capture the whey. From about $7 at a houseware store.\n\n#### HOW TO DO IT\n\n1 gallon whole milk, pasteurized or raw\n\n\u00bc teaspoon feta cheese culture (ChoozitMT1)\n\n\u00bc teaspoon lipase powder\n\n\u00be teaspoon calcium chloride\n\n\u00bc teaspoon vegetarian rennet\n\n\u00bd cup fine sea salt\n\n1\u00bd teaspoons freshly squeezed lemon juice\n\n**1. Heat the milk** Put the canner on the stove and insert the canning rack upside down (handles down). Fill the canner with water up to the first ring from the bottom. Bring the water to a boil, covered, over high heat. Meanwhile, fill a sink with cold water. This will be your cool-down spot should you need it.\n\nTurn off the heat, uncover the canner, and set the empty 8-quart pot on the upturned rack (it may float a little). Pour the milk into the pot and insert the tip of the thermometer into the milk. Stir the milk with the cheese ladle 20 times in a gentle surface-to-bottom circular motion until it registers 85\u00baF. This will take only a couple of minutes.\n\nMake sure the milk stays between 85\u00b0 and 88\u00b0F for the duration of the cheese making. If it starts to climb above 86\u00b0F, lift the pot from the canner and set it in the sink of cold water until the temperature returns to 85\u00b0F; then return the pot to the water bath. If the temperature of the milk sinks, set the canner with the milk pot over medium-high heat, stirring gently every minute or so, until the milk once again registers 85\u00b0F.\n\n**2. Add the culture** Measure the feta culture into a measuring cup, add a big spoonful of warm milk from the pot, and stir to dissolve. Pour the mixture into the milk pot and, using the cheese ladle, stir 20 times with a gentle surface-to-bottom circular motion. Cover the canner and let the milk sit for 45 minutes to ripen.\n\n**3. Add the lipase, calcium chloride, and rennet** Put the lipase in a measuring cup, add a big spoonful of warm milk from the pot, and stir to dissolve. Pour the mixture into the milk and, using the ladle, stir 20 times with a gentle surface-to-bottom circular motion. Put \u00bd teaspoon of the calcium chloride in the measuring cup, add \u00bc cup cool water, and stir to dissolve. Add the mixture to the milk the same way you added the lipase mixture. Put the rennet in the measuring cup, add \u00bc cup cool water, stir to dissolve, and add to the milk the same way. Cover the canner and let the milk sit for 1\u00bd hours to set. Check every now and then to be sure the temperature stays at 85\u00b0F.\n\n**4. Check the curd set** Resting your hand on the edge of the pot, insert the tip of the wooden spoon handle at a 45\u00b0 angle into the middle of the pot and slowly lift it straight up. If the curd is ready, it will break cleanly over the rising handle. If it leaves lots of soft, milky residue on the handle, it is not. Re-cover, let rest for another 30 minutes, and check again. Repeat once more if the curds still haven't set. (If they are _still_ too soft, the milk may be weak or not properly acidified; or possibly the culture, lipase, or rennet is too old; or the temperature wasn't right. Drain the curds, add salt, and eat them like cottage cheese instead.)\n\n**5. Cut the curds** With the spatula, rapidly cut the curd into 1-inch slices, cutting all the way to the bottom of the pot. Then cut the curd again, perpendicular to the first direction, to create 1-inch cubes. Now slash the cubes on a diagonal, again spacing the cuts about 1 inch apart. The curds should look like soft tofu.\n\nLet the curds sit for 1 hour, covered, stirring every 10 minutes with the cheese ladle to break up large chunks (the curds should look like rough cottage cheese). Be sure the curds' temperature is still 85\u00b0F.\n\n**6. Drain the curds** Lift the stockpot from the canner and set it on the counter. Remove the thermometer. Line the colander with a double thickness of cheesecloth and set the lined colander over the cheese bucket. Drain the sink of water and put the bucket-colander setup in the empty sink. Put 1 mold in the lined colander and carefully ladle curds into the mold, filling it about halfway. When the mold is filled, transfer it to the rack set inside the baking sheet and fill the second mold. Even out the curds between the molds.\n\nEmpty the whey from the bucket (or save it for making ricotta cheese: put it in an airtight container, chill, and use within a day.) Let the cheeses drain for 30 minutes, then invert them into your clean hands and ease them back into the molds so that their bottoms are now facing up. Over the next 2 hours, invert them twice more; then let them drain overnight at room temperature, loosely covered with plastic wrap.\n\n_Feta\u2014our first attempts, using a salad spinner_\n\n**7. Brine the cheese** Meanwhile, in a saucepan, bring 4 cups water and \u2153 cup salt to a boil, stirring until the salt dissolves. Remove from the heat and let cool to room temperature. Stir in the remaining \u00bc teaspoon calcium chloride and the lemon juice to create a brine.\n\nUnmold the cheeses onto a clean work surface and cut into quarters. Put into a glass or ceramic bowl and pour in the brine. Let the cheeses pieces float in the brine, covered with plastic wrap and chilled, 1\u00bd hours. Turn the cheeses over and cure for 1\u00bd hours longer.\n\n**8. Dry the cheese** Transfer the cheese pieces to the cooling rack in the baking pan. Chill, turning them a couple of times a day, until their surfaces are dry and hard, for at least 1 day and up to 3. The feta is ready to eat.\n\n**9. Store the cheese** Make a storage brine with remaining 2\u00bd tablespoons sea salt dissolved in 4 cups water. Put the cheeses in an airtight container, pour in brine, and store, chilled, for up to 3 months. Whenever the brine starts forming a cloud on the surface, replace it with a fresh batch of brine. If you find the cheese too salty, soak it in cold water for 15 minutes before eating.\n\n_PER 2-OUNCE SERVING (about \u00bc cup crumbled) 150 cal., 73% (109 cal.) from fat; 8 g protein; 12 g fat (8.5 g sat.); 2 g carbo (0 g fiber); 632 mg sodium; 50 mg chol._\n\n###    \nGOUDA\n\nMAKES one 2-pound wheel\n\nTIME about 5 hours to set, cut, and wash the cheese; 6 to 8 hours to press; 3 to 4 days to dry; 4 hours to coat; and about 30 minutes to wax, plus between 2 and 6 months to age\n\nMild and creamy, Gouda gave us the \"melting cheese\" we needed for a number of different dishes in this book. As we did for our feta, we've based our method on one developed by Canadian cheese maker Margaret Morris, who calmly and cheerily demonstrates it on her _Home Cheesemaking with Margaret Morris_ DVD (see Helpful Information).\n\nFor specialized cheesemaking tools and ingredients, we turned to the Beverage People, in Santa Rosa, California (www.thebeveragepeople.com or 800\/544-1867), unless otherwise mentioned.\n\n#### WHAT TO USE\n\n**_Milk and sea salt_** For more on these ingredients, see the What to Use list for Fresh Chive Cheese.\n\n**_Annatto_** (sometimes called \"butter color\") A natural dye often used to color butter, annatto is made from the orange-red (achiote) seed. About $10 for 60 milliliters (about 2 ounces).\n\n**_Mesophilic type III culture_** A mixture of two strains of freeze-dried microscopic bacteria (it looks like the freeze-dried yeast for baking) that help create flavor and texture; it also acidifies the cheese so that curds develop properly. $11 for 10 grams, enough for 8 batches of cheese. Store in the freezer in a resealable plastic bag.\n\n**_Calcium chloride and vegetarian rennet_** For more on these ingredients, see the What to Use list for Fromage Blanc.\n\n**_21-quart boiling-water canner with canning rack, dairy thermometer, cheese bucket_** See the What to Use list for Ricotta.\n\n**_10-quart heavy-bottomed stockpot_** For setting the curd. About $60 at IKEA.\n\n**_Colander, cheesecloth, cheese ladle_** For more on these, see the What to Use list for Fresh Chive Cheese.\n\n**_Wooden spoon_** **_with long, round, narrow handle_**\n\n**_Stainless-steel wire whip_** The whip, designed specifically for cheese making, is 20 inches long and is sturdy enough to cut curds evenly. $8.25 from Hoegger Supply Company (www.hoeggergoatsupply.com or 800\/221-4628). A large standard wire whisk (not a balloon whisk) will also work, but not quite as well.\n\n**_2-pound cheese press_** Made of solid hardwood, it comes with a 1-pound press as part of the Hoegger's Deluxe Two Hoop Cheese Press. About $90 from Hoegger Supply Company (above). We used the 2-pound press instead of a pricier, single Gouda mold (the 1-kilogram Dutch Kadova mold, $82, from Glengarry Cheesemaking and Dairy Supply, www.glengarrycheesemaking.on.ca). The Hoegger mold works well, but produces a Gouda with sharp edges rather than the classic, round-shouldered shape.\n\n**_Cheese-press pressure scale_** This handy tool tells you how much pressure you're putting on the cheese. About $40 from Hoegger Supply Company (above).\n\n**_Cheese draining\/ripening mat_** A nylon grid that allows air to circulate underneath the cheese as it dries. $3.\n\n**_Cheese coating (cream wax)_** A liquid wax coating that you apply to your new cheese with a pastry brush; it quickly sets into a flexible rind that keeps the cheese moist and helps prevent mold. $19 for a 500-gram (17\u00bd-ounce) jar.\n\n**_New pastry brush_**\n\n**_Red cheese wax_** This paraffin-based wax is melted and used for dipping your new cheese into after you have sealed it with cheese coating (or, use your own beeswax, as we did; see above). About $6 per 1-pound block.\n\n**_Small deep saucepan_** At least 4 inches deep, for melting wax, and wide enough to dip the finished cheese.\n\n#### HOW TO DO IT\n\n2 gallons whole milk, pasteurized or raw\n\n2 drops annatto\n\n\u00bc teaspoon mesophilic type III culture\n\n\u00bd teaspoon calcium chloride\n\n\u00be teaspoon vegetarian rennet\n\n\u00bd cup fine sea salt\n\n**1. Heat the milk** Put the canner on the stove and insert the canning rack upside down (handles down). Fill the canner with water up to the first ring from the bottom. Bring the water to a boil, covered, over high heat. Meanwhile, fill a sink with cold water. This will be your cool-down spot should you need it.\n\nRemove the canner from the heat and set the stockpot on the upturned rack inside it. The water should come 2\u00bd inches up the sides of the stockpot; add more hot water as needed. (Some stockpots have handles that rest on the edge of the canner, preventing the pot from sitting fully on the rack.) Pour the milk into the pot and insert the tip of the thermometer into the milk. Stir the milk with the cheese ladle 20 times in a gentle surface-to-bottom circular motion to distribute the heat evenly. Watch for the thermometer to register 85\u00baF; this will take only a couple of minutes.\n\nThe milk should be held at about 85\u00baF for the duration of the cheese making. If it starts to climb above 86\u00b0F, lift the pot from the canner and set it in the sink of cold water until the temperature returns to 85\u00b0F; then return the pot to the water bath. If the temperature of the milk sinks, briefly set the canner with the milk pot over low heat until the milk again registers 85\u00b0F.\n\n**2. Add the annatto and culture** Add the annatto to \u00bc cup cool water and stir with a soup spoon to dissolve. Pour the mixture into the milk and, using the ladle, stir 20 times with a gentle surface-to-bottom circular motion. Immediately sprinkle the culture as evenly as possible over the milk and let rest for 10 minutes.\n\n**3. Add the calcium chloride and rennet** Place the calcium chloride in a measuring cup, add \u00bc cup cool water, and stir to dissolve. Pour the mixture into the milk and, using the ladle, stir 20 times in the same way as before. Place the rennet in a measuring cup, add \u00bc cup cool water, and stir to dissolve. Stir the mixture into the milk the same way you did the calcium chloride. Cover the canner and let the milk sit for 1 hour to set. Check the temperature occasionally to make sure it remains at 85\u00b0 F and correct if needed as described in step 1.\n\n**4. Check the curd set** Resting your hand on the edge of the pot, insert the tip of the wooden spoon handle at a 45\u00b0 angle into the middle of the pot and slowly lift it straight up. If the curd is ready, it will break cleanly over the rising handle. If it leaves lots of soft, milky residue on the handle, it is not. Re-cover, let rest for another 30 minutes, and check again. Repeat once more if the curds still haven't set. (If they are _still_ too soft, the milk may be weak or not properly acidified; or possibly the culture, lipase, or rennet is too old; or the temperature wasn't right. Drain the curds, add salt, and eat them like cottage cheese instead.)\n\n**5. Cut the curds** Drag the wire whip rapidly but evenly through the entire mass of curd once, breaking it into roughly \u00bd-inch bits. Repeat the motion, this time dragging perpendicular to the first direction. Then drag the whip through the curd on a diagonal.\n\nUsing the ladle, stir the curd for 5 to 10 minutes, using surface-to-bottom strokes in a circular motion. Let the curds settle into the whey for 5 minutes. Meanwhile, start heating 7 cups of water to 175\u00b0F in a saucepan.\n\n**6. Wash the curds** Lift the stockpot from the canner and set it on a work surface. Remove the thermometer. Line the colander with a double thickness of cheesecloth and set the lined colander over the cheese bucket. Drain the sink of water and put the bucket-colander setup in the empty sink.\n\nScoop out all the curds into the lined colander and let drain until the whey has almost stopped dripping. Pour the whey in to a large stainless-steel bowl. Return two-thirds of the whey and all of the curds to the pot.\n\nMeasure 3 cups of the 175\u00b0F water (you may need to heat the water a bit) and add to the curds and whey. Scoop the ladle down one side of the pot and lift the curds, jiggling the ladle to separate them. Using a surface-to-bottom motion, keep lifting and jiggling the curds until they begin to shrink, 10 to 15 minutes. Let the curds settle to the bottom of the pot, about 10 minutes.\n\n**7. Wash the curds a second time** Scoop out the whey down to the level of the curds and pour into the colander. Return any curds in the colander to the pot and discard the whey. Add enough of the 175\u00b0F water (2 to 3 cups) to the curds to heat them to 100\u00b0F, and stir and shake the curds with the ladle as before for about 10 minutes. Let the curds settle for another 10 minutes.\n\nScoop up a spoonful of the curds and squeeze them gently. If they're springy and moist, they're ready to be pressed. If they feel soft, let them sit in the hot water for several more minutes. When the curds are ready, pour them into the colander and let drain for 5 minutes.\n\n**8. Press the cheese** Arrange a double thickness of cheesecloth on the base of the press and place the 2-pound hoop on the base (or set a 2-pound Gouda mold into the press). Set the press in a rimmed baking pan. Scoop the curds into the press, packing them down as evenly as possible. Cover the curds with another double thickness of cheesecloth and set the follower (the inset top) of the mold over the curds. Screw down the wing nuts on either side of the hoop simultaneously to distribute the pressure evenly. Screw in the central pressure-gauge screw and tighten to 35 pounds of pressure.\n\nLeave the cheese in the press for 6 hours, checking the pressure every couple of hours and retightening the pressure-gauge screw as needed to register 35 pounds pressure. After 3 hours, turn the cheese over in the mold.\n\n**9. Brine the cheese** Meanwhile, in a medium saucepan, bring 4 cups water and the salt to a boil over high heat, stirring until the salt is mostly dissolved. Let the brine cool to room temperature, then slip the cheese out of the mold and into the brine. Cover and brine the cheese in the refrigerator for 12 hours.\n\n**10. Dry the cheese** Put the draining mat on a cake cooling rack and set both on the baking sheet. Remove the cheese from the brine, set it on the on draining mat, and let the cheese air-dry at room temperature, flipping it every several hours, until dry to the touch, 3 to 4 days.\n\n**11. Coat the cheese** Using the new pastry brush, paint the top and sides of the cheese with the cream wax. Let the wax dry completely, about 1 hour. Turn the cheese over, paint the bottom, and let dry completely, about 1 hour. Coat the cheese again the same way and let dry completely.\n\n_Applying cream wax to Gouda_\n\n**12. Wax the cheese** Line the small deep saucepan with heavy-duty foil. Put 1 pound cheese wax in the pan. Fill the roasting pan halfway with water and bring to a boil over medium-high heat. As it's heating, set the pan of wax in the roasting pan to melt. When the wax has melted, dip the cheese sideways into the wax to coat as completely as possible. Lift out, let harden (this takes only a couple of seconds), then dip the other side. Repeat at least twice, or until you have nice coverage. Set the cheese on the cooling rack (remove the mat first) to harden completely.\n\n**13. Age (ripen) the cheese** Ripen the cheese at 50\u00ba to 55\u00b0F and about 75 percent humidity for at least 2 months and up to 6 months. Make sure the area is well ventilated. You can repurpose an old refrigerator for greater control when ripening cheese. See Our Little Cheese Cave, opposite, for directions. The longer you ripen the cheese, the stronger in flavor and drier in texture it will be.\n\nOUR LITTLE CHEESE CAVE\n\nAlthough it would be very romantic if we had a cool, stone-lined cave dug into a mountain for aging cheeses, or even a modern cheese maker's walk-in cooler \"cave,\" our cheese cave is cheap and humble: a dorm fridge with its thermostat rejiggered to allow temperatures above 40\u00b0F. The living bacteria in cheeses are what ripen it, and they shut down when the temperature is too low. On the other hand, if the temperature shoots high, as room temperatures can, the cheeses may spoil; that's why they need to age in an atmosphere that can be controlled.\n\nTo override the fridge's normal temperature settings, we bought a Controller external thermostat (from Williams Brewing, www.williamsbrewing.com, about $60). It permits a range of temperatures from 37\u00b0 to 80\u00b0F, adjustable with a knob, and is easy to install. We also put in a tiny electric desk fan (from about $9 at hardware stores) for air circulation\u2014its cord snakes out through the fridge's door seal and into a nearby plug\u2014and shallow pans of water, for humidity. Lastly, we bought a combination thermometer and hygrometer (from about $9 online or at hardware stores), and set it next to our cheese so we could monitor the temperature and humidity.\n\n_PER 2-OUNCE SERVING 201 cal., 70% (140 cal.) from fat; 14 g protein; 15 g fat (9 g sat.); 1 g carbo (0 g fiber); 464 mg sodium; 64 mg chol._\n\n#### HELPFUL INFORMATION\n\nFor information on web sites, cheese-making supplies, and books and DVDs.\n\n_\u2014Margo True and Elaine Johnson_\n\n##    \nHOW TO GROW  \nMUSHROOMS\n\nWe took both the fast route, raising mushrooms from logs that you purchase as kits, and the slow, planting a morel patch in our garden. The kits are as easy as growing a houseplant. The process is fun and encourages the mad scientist in you as you watch the logs that you purchase as kits sprout weird, Dr. Seuss-like fungi. Home-cultivated mushrooms have a delicious, mild flavor and tender texture, and you can experiment with several varieties.\n\nMorels, on the other hand, are a long-term endeavor, with no guarantees that you will reap the rewards the first year (we are still waiting). But what rewards when they do come! Few treats compare with a panful of fragrant, earthy morels saut\u00e9ed in butter.\n\n###    \nSHIITAKE, OYSTER, AND POM POM BLANC MUSHROOMS (grown from a kit)\n\n_Shiitake mushrooms_\n\n_Oyster mushrooms_\n\n_Pom pom mushroom_\n\n#### WHAT TO USE\n\n**_Log kits_** Mushrooms are the fruit of a fungus whose body is the mycelium, a net of threadlike fibers. In a mushroom-log kit, the mycelium is distributed throughout a growing medium, such as sawdust mixed with rice bran. The log looks like a loaf of bread covered with a fine, soft mold. As the mushrooms grow, they sprout from the log. Although the kits come with instructions, they are fairly general, and we added a lot of extra notes based on our experiences.  \nFrom Far West Fungi (www.farwestfungi.com), we bought a shiitake minifarm kit ($19.50) and a tree oyster minifarm kit ($19.50). From Gourmet Mushrooms and Mushroom Products (www.gmushrooms.com), we purchased a shiitake kit ($20.00), plus kits for pom pom blanc mushrooms ($19.00) and brown oyster mushrooms ($17.00). Each kit comes with plastic bags, an important part of the mushroom-growing environment. The kits from Gourmet Mushrooms also include a sponge for the log to rest on, which allows the log to take up the moisture it needs without sitting in a pool of water.\n\n**_Extra materials_** Depending on the brand and type of kit, you may also need a spray bottle and a deep plate large enough to hold each log with a little water.\n\n#### HOW TO DO IT\n\n**1. Prepare the log's environment** For Far West Fungi kits, untie the twist tie and retie it at the top of the bag. Cut 10 to 20 (see package directions) small X-shaped holes in the upper part of the bag to allow the mushroom culture to breathe without drying out.\n\nFor Gourmet Mushrooms kits, leave the plastic covering on the logs. If shiitake logs have not started to sprout, they may need a pregrowing rest period of 7 to 12 days (see kit directions); ours didn't need this. Then rinse the log if the kit directs you to (it depends on the mushroom).\n\nNext, cut two X-shaped openings through the plastic covering over the oyster or pom pom blanc mushrooms and into the log. Each opening should be about 1 inch deep and on either side of the log near the top. Gouge a 1-inch-wide, 1-inch-deep hole in the top of the shiitake log with a table knife. Set the log on its sponge in a large, rimmed plate and add cool water to a depth of \u00bd to \u00be inch. Don't let standing water touch the log. Open the perforated plastic sleeve (or for shiitakes, inflate the bag by blowing into it), and put the sleeve over the log so the bottom of it rests inside the water dish. The top of the sleeve should be closed.\n\n**2. Find a good spot** Set the log indoors at room temperature or a bit cooler, away from the kitchen, which can foster mold growth (we learned this the hard way), and away from direct light and drafts.\n\n**3. Encourage growth** For Far West Fungi kits, lightly mist the mushrooms once or twice a day, but not enough to cause water to collect in the bottom of the bag. Open the bag a little as needed so the environment stays humid but not wet. Within a week or two, you'll begin to see tiny knobs, or \"pins\"\u2014the start of mushrooms\u2014on the log. When these are about 1 inch long or \u00bd inch across, remove the twist tie and open the bag slightly at the top to make room for the mushrooms to grow. Otherwise, you'll get strange, strangulated shapes.\n\nFor Gourmet Mushrooms kits _,_ add water as needed to the plate to keep the water about \u00bd inch deep. Every day until you see growth (2 to 21 days for pom poms or oysters, 5 to 9 days for shiitakes), open up the top of the plastic sleeve and either mist or drizzle a little water over the logs (for pom poms or oysters) or pour water into the hole in the top (for shiitakes).\n\n**4. Let the mushrooms mature** A few days after you see growth starting, the mushrooms will reach full size. As they grow, open up the sleeves a bit as needed to give them room, and continue to add moisture as before.\n\n**5. Harvest** Shiitakes are ready when the white cottony veil beneath the cap has fully broken away from the stem but the caps still curl under slightly. Harvest oyster mushrooms when the caps have grown but before their outer edges begin to curl up. Pom poms are ready when the heads have formed and are still firm and white. Cut mushrooms at the base with a sharp knife. Cook them, or enclose them in paper bags (not plastic ones, which can trap moisture and make the mushrooms slimy), and refrigerate. You may get anywhere from 8 ounces to 2 pounds mushrooms per log.\n\n_First fruits: shiitake and oyster mushrooms_\n\n**6. Cook** If consumed raw, these mushroom varieties can make some people ill.\n\n**7. Encourage additional mushrooms** You should get three or four more \"flushes\" of mushrooms that are a week to a month apart before the mycelium runs out of food. To encourage additional growth, clean off any remaining mushroom stems from the log with a knife. Wrap the log in paper towels (for Gourmet Mushrooms kits) or a plastic bag (for Far West Fungi kits) and let rest for about a week at room temperature for oysters and pom poms, 2 to 3 weeks for shiitakes. Then repeat the steps to grow more mushrooms, using a clean plastic bag. We had only sporadic production from our logs after the first harvest.\n\n**8. Deal with moldy logs** We had a little mold during our mushrooms' initial growth period, and simply scraped it off with a table knife, decreased the water, and opened the bags to let in more air. Between harvests, we got a lot of mold on a couple of the logs; we had some luck scraping it off with the knife, then cleaning the area with rubbing alcohol.\n\n**9. Recycle the logs** Once they have stopped producing, add them to your compost pile.\n\n###    \nMOREL MUSHROOMS (grown from spawn)\n\nIn the wild, morels grow in a number of different colors. We are raising common, or yellow, morels.\n\n#### WHAT TO USE\n\n**_Spawn_** This is mycelium in a wad of white, crumbly fiber that can be mixed straight into soil. As it grows, the mycelium spreads underground and produces mushrooms\u2014the fruit\u2014above ground when conditions are right. We bought our spawn from Gourmet Mushrooms and Mushroom Products (www.gmushrooms.com) for $30.\n\n**_Soil_** You need a 4- to 16-square-foot patch of ordinary, well-drained garden soil, in the ground or in a raised bed (in contact with the ground). The area should be shaded at least half the day, with less than 3 hours of direct sunlight, and protected from wind (you can build a windbreak if needed).\n\n**_Shovel_**\n\n**_Compostables_** Fresh materials from your kitchen, such as vegetable trimmings, coffee grounds, eggshells, and the like. We also added ash from a wood fire, as morels in the wild grow well in areas where there has been a forest fire or controlled burn.\n\n**_Water source_** Pick a spot within reach of a hose, or plan to make frequent trips with a watering can.\n\n#### HOW TO DO IT\n\n**1. Let the spawn rest** After the spawn arrives, it needs to sit in a cool (below 70\u00b0F), dark place for a couple of days. It will also keep in the refrigerator for up to 6 months.\n\n**2. Plant the spawn** Morels can be planted any time of year as long as the soil isn't frozen.\n\n**3. Prepare the soil** With a shovel, turn the soil to a depth of 6 to 12 inches to loosen it. Clear away any rocks or debris.\n\n_Turning the morel bed_\n\n**4. Scatter the spawn** Crumple the spawn over the prepared soil, then turn the soil under. Rake it smooth on top.\n\n_Raking it smooth_\n\n**5. Add compostables** About once a week, dig in a bowlful of fresh food scraps, 9 to 18 inches deep. Water the soil so it stays damp. Continue this for at least 3 months.\n\n**6. Wait** Now you need to let the soil rest so the mycelium can spread underground. Typically, it takes a full year before you'll see mushrooms, but depending on when you planted your spawn, you could get results in as soon as 2 months. As the soil rests, you may get plants sprouting from the compost. This is okay, and may even be helpful, as they will shade the area.\n\nIt's hard to predict exactly what month the mushrooms will pop above ground. They need the right conditions for a \"morel spring,\" which can be any time of year, though more often in spring. Typically they appear when there has been a period of heavy rain followed by a period of light to no rain for a week or two, with the weather changing to highs in the upper 60s and 70s. This can be January to April in areas with mild climates and May or June in areas with colder climates. Also, the spawn we used produces a warm-weather morel that comes up later than wild morels.\n\n**7. Water regularly** In warm weather, water your patch often enough to keep it damp.\n\n**8. Check for mushrooms** Our morels did not emerge in May, as we had hoped, and they haven't appeared in the months since. When we finally spot them, it will be a happy day. Young morels look like small, white toothpicks poking out of the soil. We'll probably need to give them a week or two to reach full size.\n\n**9. Harvest** We're planning to trim the mushrooms at the base without digging up the soil, which would disturb the mycelium. Gourmet Mushrooms tells us that the yield can vary from year to year\u2014sometimes none, and sometimes as many as 50, but the mycelium may keep producing for up to 15 years.\n\n**10. Cook** Since morels grow in dirt, they'll need a quick rinse to remove soil. They should only be eaten cooked. A word of caution: If the mushrooms you've picked don't look like morels, do not eat them until you've checked with an expert, such as someone at your local mycological society (see a couple of choices in Helpful Information, below).\n\n#### HELPFUL INFORMATION\n\n##### WEB SITES\n\n\u2022 \"6 Ways Mushrooms Can Save the World: Paul Stamets on TED.com.\" A dazzling 18-minute presentation from Stamets, an \"entrepreneurial mycologist,\" on the potential that mushrooms have to transform biotechnology. Watch and be awed by the not-so-humble 'shroom. <http:\/\/blog.ted.com\/2008\/05\/06\/paul_stamets>.\n\n\u2022 Fungi Perfecti LLC, www.fungi.com. A one-stop resource from Paul Stamets (see above) with indoor and outdoor mushroom kits, culinary mushroom products, dietary supplements, information for kids, recipes, articles, and books\u2014including Stamets' _Mycelium Running: How Mushrooms Can Help Save the World_ (Ten Speed Press, 2005).\n\n\u2022 The Great Morel, www.thegreatmorel.com. Interactive site with stories, recipes, and photos, from fellow 'shroomers around the world.\n\n##### BOOKS\n\n\u2022 _Growing Gourmet and Medicinal Mushrooms_ , by Paul Stamets (Ten Speed Press, 2000). An in-depth resource for growing and learning about mushrooms.\n\n##### ORGANIZATIONS\n\n\u2022 Cascade Mycological Society, www.cascademyco.org. Based in Eugene, Oregon, the group promotes wild mushroom education and identification, leads wild forays, and hosts an annual wild mushroom feast.\n\n\u2022 Mycological Society of San Francisco, www.mssf.org or 866\/807-7148. Gives information on how mushrooms grow, how to cultivate mushrooms, mushrooms in the wild, and recipes.\n\n_\u2014Elaine Johnson_\n\n##    \nHOW TO MAKE  \nOLIVE OIL\n\nIt was the sight of the olive trees all around Sunset that did it: tall, stately, and loaded with thousands of ripening olives. We needed some sort of cooking fat for our one-block feast, and after rejecting peanuts and corn for oil, we realized that our trees held the answer in their branches. Plus, we would be using a heretofore wasted resource\u2014every fall, olives rained down onto the ground and into the bushes, feeding only the birds and insects.\n\nUnfortunately, they were feeding the insects a little too well. Our olives, we learned, were thoroughly infested with the maggots of olive fruit flies. So we picked olives at a nearby, fruit-fly-free olive farm instead and drove them to a commercial olive press, where we had planned to press our olives anyway.\n\n_Harvesting olives in the Santa Cruz foothills_\n\nThe actual making of olive oil is instantly gratifying. Unlike with wine, there are no months of fermenting and aging. It all happens in a single day. The thick, jewel green new oil is ready to taste immediately, mere hours after it was fruit hanging on a tree. This new oil\u2014what the Italians call _olio nuovo_ \u2014is the biggest perk of crushing olives yourself, since it's difficult to find in stores. The other big perk: a stock of your own good oil to give as gifts and use whenever you like.\n\nWe are still holding out hope that by next harvest season we will be able to use our own olives, which we believe are a combination of Mission and Redding Picholine. In fact, as this book went to press, we had just started spraying our trees with an organic control called GF-120 that has been recently approved for backyard trees. For more on this, see Our Future Plans and see <http:\/\/oneblockdiet.sunset.com\/team_olive>, where we will post ongoing updates of Our Battle Against the Olive Fruit Fly.\n\n###    \nEXTRA VIRGIN OLIVE OIL\n\n20 gallons, pressed from a blend of six Italian olives: Maurino, Leccino, Frantoio, Pendolino, Ascolano, and Taggiasca\n\n#### WHAT TO USE\n\n**_Olives_** To make 20 gallons of olive oil, you'll need about 800 pounds of olives. Because many growers worry about liability these days, finding a farm where you can pick your own fruit is difficult. We got lucky and were kindly allowed to pick at a ranch, but it was a one-time-only situation. You can order olives by mail, however, through the Olive Growers Council of California (559\/734-1710 or adin@goldstate.net) and from La Conda Ranch (www.lacondaranch.com or 530\/824-5946). Call to place your order starting in early September. Prices vary depending on variety and growing conditions. As an example, last season, La Conda Ranch sold their olives for $1.75 per pound (plus shipping.) All of these olive farms are in California\u2014in fact, 98 percent of U.S. olive production is in the state.\n\n**_Olive press_** We didn't buy one. They are expensive (up-ward of $200,000!) and we were already spending money on the olives themselves. A separator and a washing system boost the bill even higher. The home-version First Press, available through the Olive Oil Source (see Helpful Information), is a relatively reasonable $2,790 and includes all the machinery you need. The advantage is that you can run just a few buckets of olives through it and get oil, and do it fresh from your trees. It is slow and labor-intensive, however.  \nWith a larger amount, at least a few hundred pounds, you can go to a communal mill: an olive-oil maker that rents its mill periodically to the public during harvesttime (and also takes care of the separating and washing). We took our olives to Pietra Santa Winery in Hollister, which makes its own superb olive oil ($525 minimum fee per ton or less; www.pietrasantawinery.com or 831\/636-1991). Frantoio Ristorante & Olive Oil in Mill Valley (www.frantoio.com or 415\/289-5777) will press a half ton for $250, which includes a $50 pomace disposal fee. If you want the skins and seeds yourself and can haul them away, you'll save $50. Frantoio will also press quantities less than a half ton, but they mix the olives with other small lots, so you'll end up with a blend. Other communal mills offer this small-lot service, too, on \"community press days.\" One is McCauley Brothers Olive Groves, in Brentwood (www.mccauleyolivegroves.com or 925\/754-6457); call for prices. For more communal presses and community press days, search the list of millers on the Olive Oil Source's website. It's usually about $450 per ton, depending on the mill, and less for smaller amounts.\n\n**_A cool, dark place_** Where you keep your oil as it settles and once it is bottled is important. Olive oil needs to be kept at temperatures between 60\u00b0 and 70\u00b0F, in as dark a place as possible. (Heat and light are the top degraders, along with oxygen.) We stored our olive oil in a temperature-controlled shed at Sunset, and laid big sheets of black plastic over the containers during the settling period to block the light completely.\n\n**_5-gallon food-grade plastic drums with spigot_** You need these for storing the new oil. They're about $15 each from the Olive Oil Source. We bought seven, figuring we could always use the extra three for the next harvest. Stainless-steel containers are ideal as they block out all light and oxygen and are beautiful. They are also expensive (from $515 for a 50-liter tank). Plastic drums admit a small amount of oxygen, but not enough to affect the oil as long as the drums are filled to the top. The drums do need to be scrupulously washed with a food-grade degreaser for reuse, because even a trace of old oil will contaminate your nice fresh batch.\n\n**_Power drill with a small bit_** You need to make a \"breather hole\" in each plastic drum when you are ready to bottle. From about $30 at a hardware store.\n\n**_Bottles and corks_** Dark green or amber glass is best, to prevent light from spoiling the oil. We chose small bottles (250-milliliter\/about 1-cup size) to give away as gifts and also because we could use up an opened bottle more quickly, with less chance of it oxidizing. Bottles sell out fast during harvest season in fall, so place your order well ahead of time. The Olive Oil Source sells bottles by the case, in various sizes; 200-milliliter bottles with corks in dark amber are about $19.50 for a case of 12. If you buy a minimum of 7 cases (over $100), the $20 handling fee is waived.\n\n**_Plastic funnels_** Get at least two, and make sure they're narrow enough to fit securely into your bottles. From $3.50 online or at a cookware store.\n\n**_Towels or rags_** For mopping up and wiping bottles.\n\n**_Sturdy worktable_** Where you will set the containers of olive oil for bottling, with their spigots hanging over the edge. From $75 at a home-improvement center.\n\n**_Plastic tarps_** As much as you'll try not to, you _will_ splatter, making a tarp under and around the worktable indispensable. If you care about your worktable top, spread one there, too. About $40 for a 10-by-100-foot tarp at a home-improvement center.\n\n**_Large aluminum pans_** Old turkey roasting pans or the like, placed beneath the spigots, work well for catching drips.\n\n**_Mallet or small hammer_** For tapping corks into bottles. From $6 at a hardware store.\n\n**_Hot, soapy water_** Mix up a batch in a bucket to use for cleaning slippery hands while bottling.\n\n#### HOW TO DO IT\n\nGiven that our own olive trees were infested with olive fruit fly maggots, we didn't grapple with harvesting them. If you have a backyard tree or several, and are interested in learning how to harvest olives, control the olive fruit fly, or care for your trees, see Helpful Information.\n\n##### NOVEMBER 29\n\n**Pick and press** We drove to Valencia Creek Farms in Aptos (www.valenciacreekfarms.com), about 50 miles south of Sunset, and hand-picked olives. We put them in 23 plastic crates, each of which measured \u00be by 16 by 9 inches and held 40 to 44 pounds of olives. Valencia Creek's Chris Banthien sells them for $1 per crate. Similar crates are available through the Fairfield, California, branch of MacroPlastics (www.macroplastics.com or 707\/437-1200).\n\nWe immediately drove 30 miles south to Pietra Santa Winery to press the olives. Our 800 pounds were mechanically washed and then crushed in an Italian stone press (three enormous rotating stone wheels weighing a total of 7,000 pounds). The mush was separated, filtered, and poured into the 5-gallon plastic drums we had brought with us.\n\n_Our brand-new oil_\n\n_The stone mill at Pietra Santa Winery_\n\n##### NOVEMBER 30\n\n**Settling** Olive particles in oil makes it spoil faster (although it does taste delicious when fresh), so we needed to let the solids fall to the bottom of the drums. We reserved a small portion to use as _olio nuovo_ , and then we put the drums in the Sunset prop shed and set them on their sides so they would be in the same position on bottling day, with the spigot facing down (if we stored them upright and then turned them, all the sediment would swirl back up into the oil). We draped them with black plastic to shut out light, and set the thermostat to 65\u00b0F to keep them properly cool.\n\n##### FEBRUARY 6\n\n**Bottle** After about 2 months of quiet resting, each drum had a layer of olive sludge on its bottom (settling usually happens within 30 to 45 days). It was time to bottle. Here's how:\n\n\u2022 **First, taste it** You don't want to go through all that work of bottling if your oil is no good! We tasted the oil as best we could to make sure it hadn't acquired any of the flaws that can downgrade an extra virgin olive oil to just plain olive oil: fustiness, mustiness, vinegariness, muddiness, or just plain rancidity. Phew\u2014it was still good. It was less \"green\" tasting and peppery than when just pressed, and the color had mellowed to a golden hue, but good.\n\n\u2022 **Divide the labor** Put 1 or 2 people to work filling the bottles, another pushing in the corks, and a fourth wiping the bottles free of oil and putting them in storage boxes (we reused the boxes that the bottles came in).\n\n\u2022 **Spread the tarp** Lay the tarp on the ground where you will be working. Set your worktable on top and cover it with a tarp, too, if it's a stainable surface.\n\n\u2022 **Set the drums on the table** Position each so the spigot is hanging over the edge. Drill a small \"breather hole\" into the tiny cap opposite the spigot to release pressure and enable flow. (Once the drum is empty and cleaned, it can be reused for storage as long as the cap's cover is in place; it comes with the drum.)\n\n\u2022 **Put the aluminum pans on the ground** Slip a pan under each spigot to catch drips.\n\n\u2022 **Fill and cork the bottles** Slip a funnel into the neck of the first bottle, stick it under the spigot, open up the valve, and let the oil pour in, stopping within \u00bd inch of the rim to leave room for the cork. Close the valve, push in the cork, clean off the bottle, and put the bottle in the box. Keep going until you've bottled up all the oil.\n\n_Bottling olive oil_\n\n##### APRIL 22\n\n**Store** We moved our boxes of oil down into the Sunset wine cellar, where it's always cool and dark.\n\n##### NOVEMBER\n\n**An anniversary taste** One year had passed since the harvest. Some bottles were starting to lose flavor and turn bitter once they were removed from their cool home in the cellar. Others stayed stable.\n\n##### FEBRUARY\n\n**A last taste** Although our remaining couple dozen bottles tasted passable, the oil was no longer as flavorful as it was in November. We relegated these final bottles to cooking, rather than saving them for drizzling over dishes before serving.\n\n#### OUR FUTURE PLANS\n\nWe have finally mounted a plan of attack against the olive fruit fly. By the time you read this, we hope to have executed the plan, harvested our olives, and pressed them into oil. (Check our blog to read all about it.)\n\n##### IDENTIFY WHICH TREES TO TREAT\n\n**Short, or at least thinned in the canopy** Olive trees for fruit production should be no more than 14 feet high, and\/or pruned so that they are not densely foliaged. Big, full trees like ours are ideal habitats for the fruit fly. Our trees were planted decades ago as ornamentals, however, so they are now 40 to 50 feet tall and beautiful, and are part of the landscaping at Sunset. Team Olive can't go hacking away at them. But we don't want them to stand around spreading disease, either (untended, infested public and backyard trees pose a major problem for olive growers in California). One avenue we are exploring for most of these statuesque lovelies is spraying them with a plant growth regulator called FruitStop, which goes on at blossom time and prevents the baby olives from forming. This way we'll still have pretty trees. No olives to pick, but no olive fruit flies to contend with either\u2014and we'll focus on treating the trees that we do hope to harvest.\n\nIn the meantime, we've decided to treat just four of our twenty-one olive trees\u2014the shortest (25 to 30 feet) and relatively best-pruned ones.\n\n_Infested olive tree at Sunset_\n\n**Relatively free of surrounding bushes and other plants** This makes harvesting and treating them easier and also means they're relatively pest-free (olives with larvae drop into bushes and can't be raked up; the larvae burrow into the ground and overwinter there).\n\n##### CHOOSE THE WEAPONS\n\n**Organic spray** The best organic control of the olive fruit fly is a spray called GF-120 NF Naturalyte, which contains a tiny amount of natural pesticide called Spinosad, plus a sticky, molasses-like substance (hydrolyzed protein). When you spray the stuff on the trees, it lands as gooey globs that act as bait. Ingesting GF-120 makes the insects' muscles contract, leading to paralysis\u2014and, within a day or two, death. It doesn't hurt birds (even if they eat the bugs). It can harm honeybees that land directly on the globs, but according to olive ranchers we spoke to who also raise bees, the bees would much rather go for blossoms than for the GF-120, and don't seem affected at all by the spraying. Just in case, spray early in the morning, before the bees are up.\n\nThe price of GF-120 NF Naturalyte ranges from $100 to $150 per gallon. For sources, see Helpful Information.\n\n**Hanging traps** Also known as olipe traps. We're painting liter-size plastic bottles yellow, filling them with water, cutting a \u00bc-inch hole in the top of each bottle, dropping 4 yeast tablets (available from Napa Valley Ag Supply, 707\/963-3495; $59 for 180 tablets) into each bottle, and then hanging the bottles. The flies love the yeast. They'll fly into the bottles and never come out. Traps will also be a great indicator of just how infested our trees are. We plan to check them (and dump out the dead flies) once a week.\n\n##### PICK UP FALLEN OLIVES\n\nOlive fruit flies really like the olives they find on the ground. And any larvae laid inside burrow into the dirt and live there all winter long, to hatch as flies in the spring. So we will have to be diligent about picking up fallen fruit from last year, to help keep the olive fly population down (it would help to get all the old fruit off the tree, too). We do, of course, still have day jobs, but it looks like Olive Pick-Up might be the new lunch hour.\n\n##### OUR WORRIES, OUR DOUBTS\n\nOkay, we'll be honest here. Our number-one worry: Is this all for naught? Our trees have been infested for _years, probably decades._ (The olive fruit fly was discovered in California in 1998, and in our complete ignorance that we even had a problem, we have never sprayed our trees. And, like we said, we are only treating four of our twenty-one trees. That means that seventeen trees are going untreated. We can't imagine that these relentless olive flies will be afraid to cross the street to where the treated trees are. The Spinosad spray attracts flies from up to a couple of miles away! As Bill Krueger, an olive-tree expert from UC Davis, told us, \"They're the superfly of fruit flies.\" Is this going to work? \"Well...,\" says Sunset's head gardener, Rick LaFrentz, \"I wouldn't bet on it. But you have to try. You have to try.\" So, we're giving it our best shot.\n\n##### HARVESTING\n\nHow we'll go about actually picking the olives from our tall trees is still somewhat up in the air. Ladders that tall, we're told, are pretty dangerous. It is more likely we'll stand on standard ladders and beat our trees with sticks. We hear that PVC pipes (a type of plastic, and the main kind of irrigation pipe buried underground) are good, too. Or, if we can get a hold of some mechanized rakes, that would be helpful. We'll definitely need tarps to spread out under the trees to catch the olives as they drop.\n\n##### PLANTING NEW TREES\n\nThis is an idea we've been tossing around: planting a few short, high-density olive trees on trellises. It's the latest trend in the industry, and UC Davis has an exhibition planting near its Olive Center. High-density trees are easier to manage, harvest, and\u2014best of all\u2014are more likely to hold their own against the olive fly.\n\n#### HELPFUL INFORMATION\n\n##### ORGANIZATIONS\n\n\u2022 The Olive Oil Source, based in Santa Ynez, California (805\/688-1014 or www.oliveoilsource.com), sells everything from books to mills to bottles and has a trove of resources and links on its Web site, including information on how to grow and harvest olives.\n\n\u2022 The new Olive Center at the University of California at Davis (www.olivecenter.ucdavis.edu) has horticultural information about raising olives for oil and for the table. Also, the university harvests its own landscape olive trees and makes good oil, which it sells in the campus bookstore and through its Web site for $10 per bottle.\n\n\u2022 The California Olive Oil Council (COOC) gives pointers and guidance to anyone interested in growing olives or learning about olive oil. From time to time, they hold informative olive oil\u2013tasting workshops open to the public. They also certify extra virgin olive oils via an expert tasting panel, following the Italian model. Look for the COOC label the next time you buy oil and you won't be disappointed (www.cooc.com or 888\/718-9830).\n\n##### BOOKS\n\n\u2022 _California's Olive Pioneers: Early Essays on Olives & Olive Oil_, foreword by Judith M. Taylor, MD (Robert Mondavi Institute for Wine and Food Science, 2009); available through the Olive Center (see opposite).\n\n\u2022 _Olive Oil: From Tree to Table_ by Peggy Knickerbocker (Chronicle Books, 1997). A gorgeous overview (photographs by Laurie Smith) of olive oil production around the world, with guidance on tasting and usage, plus recipes.\n\n\u2022 _Olive Production Manual_ , edited by Louise Ferguson and G. Steven Sibbett (A.N.R. Publications, 1994). A good introduction to olive growing.\n\n\u2022 _Olives: The Life and Lore of a Noble Fruit_ , by Mort Rosenblum (Absolute Press, 2000). A mesmerizing story of the olive and its deep, abiding roots in cultures around the Mediterranean, prompted by the author's quest to revive some ancient olive trees on his property in Provence.\n\n\u2022 _The New American Olive Oil: Profiles of Artisan Producers and 75 Recipes,_ by Fran Gage (Stewart, Tabori & Chang, 2009). Profiles of growers, with recipes, tips on choosing and storing extra virgin olive oils, and how to host an olive oil tasting.\n\n\u2022 _The Olive in California: History of an Immigrant Tree_ , by Judith M. Taylor, MD (Ten Speed Press, 2004). A richly detailed account of more than two hundred years of olive growing in the Golden State.\n\n##### OLIVE FRUIT FLY CONTROL\n\nThe olive fruit fly invaded California more than ten years ago and has ruined many an olive crop. Here are some resources to help you battle the pest:\n\n_Tyingolipe traps in one of our olive trees_\n\n\u2022 \"Controlling Olive Fruit Fly at Home,\" by Paul Vossen and Alexandra Kicenik Devarenne, two of California's leading olive fruit fly experts. cesonoma.ucdavis.edu\/files\/27230.pdf.\n\n\u2022 \"Olive Fly Control,\" The Olive Oil Source, Santa Ynez, California. www.oliveoilsource.com\/olive_fly.htm.\n\n\u2022 University of California Cooperative Extension. www.ucanr.org (enter \"olive fruit fly\" in search field).\n\n\u2022 \"Pruning: Topping Tall or Neglected Olive Trees,\" by Paul Vossen. www.ucce.ucdavis.edu\/files\/filelibrary\/2161\/36224.pdf.\n\n\u2022 Ernie's Pest Control, Orland, California; 530\/865-9829. Sells GF-120 NF Naturalyte; specializes in olive fruit fly control.\n\n\u2022 ISCA Technologies, www.iscatech.com. For yellow sticky traps.\n\n\u2022 Wilbur-Ellis, St. Helena, California; 707\/963-3495 (website is hard to navigate; it's easier to call). Sells GF-120 NF Naturalyte, yellow sticky traps, torula yeast tablets, and FruitStop.\n\n_\u2014Rachel Levin_\n\n_Butternut Squash and 'Cipollini' Onion Soup_\n\n# THE FALL RECIPES\n\n##    \nFRESH APPLE CIDER AND APPLESAUCE\n\nNot having a powerful juicer or a purpose-built press for creating cider from the apples, we came up with this twofer: a rather unorthodox way to make the cider, plus an applesauce bonus.\n\nMAKES 2\u00bd to 3 cups cider and 3\u00bd to 4 cups applesauce\n\nTIME about 1 hour\n\n8 or 9 large Honey Crisp or other favorite local apple (about 5 pounds total)\n\n1. Peel, quarter, and core the apples, then cut the quarters in half crosswise. Put half the apples in a food processor and add \u00be cup water. Whirl into a smooth puree.\n\n2. Pour the puree into a medium-mesh strainer set over a bowl, and press and rub with the back of a spoon to extract the cider. Transfer the puree to a saucepan.\n\n3. Process the remaining apples the same way, adding \u00be cup of the cider instead of water. Add the puree to the saucepan.\n\n4. Cover the pan and bring the puree to a gentle simmer over medium heat, stirring occasionally. Cook, stirring occasionally and reducing the heat if the sauce starts to stick, until very soft, 15 to 20 minutes.\n\n5. Serve the applesauce warm or at room temperature. Serve the cider hot or at room temperature.\n\n**MAKE AHEAD** Both the applesauce and the cider can be made up to 3 days in advance and refrigerated. Bring to room temperature before serving.\n\n_PER 1 CUP CIDER 120 cal., 0% from fat; 0 g protein; 30 g carbo (0 g fiber); 25 mg sodium; 0 mg chol._\n\n_PER \u00bd-CUP APPLESAUCE 51 cal., 0.02% (1.1 cal.) from fat; 0.2 g protein; 0.1 g fat (0 g sat.); 13.8 g carbo (1.3 g fiber); 2.4 mg sodium; 0 mg chol._\n\n  PICKLED COCKTAIL MUSHROOMS AND ONIONS\n\nThese tangy, spicy, thyme-scented pickles would be great with a martini.\n\nMAKES about 5 cups\n\nTIME about 40 minutes, plus at least 9 hours to cool and chill\n\n4 tablespoons olive oil\n\n4 cloves garlic, peeled and crushed\n\n2 cups Chardonnay\n\n6 to 8 sprigs thyme, plus a few small sprigs\n\n2 dried red serrano or \u00e1rbol chiles, finely chopped\n\n1 teaspoon fine sea salt\n\n\u00bd pound small cipollini onions, peeled\n\n\u00bd pound medium-size shiitake mushrooms, stemmed\n\n\u00bd pound medium-size oyster mushrooms, stemmed\n\n1 to 2 teaspoons freshly squeezed lemon juice\n\n1. In a large frying pan, heat 1 tablespoon of the oil over medium-high heat. Add the garlic and cook, stirring, for 1 minute. Add the wine, thyme (the amount depends on your tastes), chiles, and salt. Reduce the heat to low, cover partially, and simmer for 5 minutes.\n\n2. Add the onions, mushrooms, and the remaining 3 table-spoons oil. Cover and simmer until the mushrooms are barely tender, 10 to 15 minutes.\n\n3. Remove from the heat, pour into a heatproof bowl, and let cool completely. Remove and discard the thyme sprigs and stir in the lemon juice to taste. Cover and refrigerate for at least 8 hours or up to 1 week.\n\n4. To serve, bring to room temperature. Put the onions and mushrooms in a bowl, pour a little of the marinade over them, and add a few small thyme sprigs. Set out with toothpicks.\n\n_PER \u00bc-CUP SERVING 56 cal., 45% (25 cal.) from fat; 0.7 g protein; 2.8 g fat (0.4 g sat.); 3.3 g carbo (0.7 g fiber); 116 mg sodium; 0 mg chol._\n\n  QUINOA BITES WITH WALNUT ROMESCO\n\nWe liked these crisp-edged little nuggets so much that we made a bigger version as a main course.\n\nMAKES about 40 patties\n\nTIME about 1\u00bd hours, plus 30 minutes to cool\n\n1 cup quinoa*\n\n2 large eggs, lightly beaten\n\n\u00bc teaspoon fine sea salt\n\nAbout 4 tablespoons extra virgin olive oil\n\n1 cup Walnut Romesco (recipe follows)\n\nAbout 40 fresh flat-leaf parsley leaves\n\n1. Fill a large bowl halfway with water. Add the quinoa and rub the grains briefly between your palms in the water. Drain in a fine-mesh strainer, then combine with 2 cups water in a medium saucepan. Bring to a boil, reduce the heat to a simmer, and cook until translucent in the center and a white ring appears, about 15 minutes. Transfer to a bowl and let cool. Stir in the eggs and salt.\n\n2. In a large nonstick frying pan, heat 1 tablespoon of the oil over medium heat. Working in batches of 8 to 10, spoon 2 teaspoons of the quinoa mixture into the pan for each patty, spacing them about \u00bd inch apart. Cook until browned underneath, about 4 minutes. Turn the patties with a fork and cook until just browned on other side, about 3 minutes more. Transfer the patties to a plate lined with a paper towel to drain. Cook the remaining quinoa mixture the same way, adding more oil as needed.\n\n3. Arrange the patties on a serving plate and top each with about \u00bd teaspoon romesco and a parsley leaf.\n\n**MAKE AHEAD** You can prepare the patties through step 2 up to 4 hours ahead and leave them at room temperature. To serve, reheat them at 350\u00b0F until hot, 10 minutes.\n\n_PER 3-BITE SERVING 96 cal., 52% (50 cal.) from fat; 2.7 g protein; 5.8 g fat (1 g sat.); 8.6 g carbo (0.9 g fiber); 53.8 mg sodium; 33 mg chol._\n\n### WALNUT ROMESCO\n\nWe eat this Spanish-style sauce (traditionally made with almonds and usually tomatoes) on everything from potatoes to toast to green beans to eggs. It's as versatile as pesto.\n\nMAKES about 1 cup\n\nTIME about 30 minutes\n\n2 'Mariachi' or 'Carmen' peppers or red bell peppers\n\n4 tablespoons extra virgin olive oil\n\n4 dried red serrano or \u00e1rbol chiles\n\n\u00bd cup walnut pieces, lightly toasted\n\n2 small cloves garlic\n\n\u00bd teaspoon fine sea salt\n\n1 tablespoon red-wine vinegar\n\n1. Preheat the broiler. Rub the peppers with 1 tablespoon of the oil and put on a rimmed baking sheet. Broil, turning as needed, until completely charred on all sides, 12 to 15 minutes. During the last minute, add the chiles to the baking sheet to warm and soften. Let the peppers and chiles cool until they can be handled. Stem the chiles, and stem, seed, and peel the peppers.\n\n2. In a food processor, pulse the walnuts and garlic until coarsely chopped. Add the chiles, peppers, salt, vinegar, and the remaining oil and process until smooth.\n\n**MAKE AHEAD** Keeps, covered and chilled, at least 3 days.\n\n_PER 2-TABLESPOON SERVING 114 cal., 85% (98 cal.) from fat; 1.4 g protein; 11 g fat (1.4 g sat.); 3.3 g carbo (1.2 g fiber); 141 mg sodium; 0 mg chol._\n\n* _For instructions on how to rinse homegrown quinoa to rid it of bitterness, seehere._\n\n  CREAMY FLAGEOLET DIP WITH RED PEPPER STICKS\n\nWith their elegant, oblong shapes and soft green color, flageolet beans look like bits of pale jade. They are sweet and mild and make a great bean dip.\n\nMAKES about 2 cups\n\nTIME about 10 minutes\n\n2 cups drained cooked flageolet beans (see method)\n\n\u00bd cup cr\u00e8me fra\u00eeche, homemade or store-bought\n\n2 tablespoons extra virgin olive oil\n\n1 tablespoon freshly squeezed lemon juice\n\n1 teaspoon finely shredded lemon zest\n\n\u00be teaspoon fine sea salt\n\n2 tablespoons coarsely chopped fresh flat-leaf parsley\n\n'Mariachi' and 'Carmen' peppers or other sweet red peppers, seeded and cut into narrow strips\n\nIn a food processor, whirl together the beans, cr\u00e8me fra\u00eeche, oil, lemon juice and zest, and salt until smooth. Transfer to a bowl and stir in the parsley. Serve with pepper strips alongside the dip.\n\n_PER 3-TABLESPOON SERVING 190 cal., 35% (67 cal.) from fat; 9 g protein; 8 g fat (3 g sat.); 22 g carbo (9 g fiber); 177 mg sodium; 10 mg chol._\n\n  ROASTED SPICED BUTTERNUT SQUASH SEEDS\n\nNext time you cut up a butternut squash, save the seeds for this tasty snack. The roasted seeds are good sprinkled over gnocchi, soup, or salads, too.\n\nMAKES about \u00be cup\n\nTIME about 1 hour\n\n1 cup butternut squash seeds (from about 3 squashes), rinsed free of any squash flesh and patted dry (be patient; this takes a quite a while)\n\n1 tablespoon extra virgin olive oil\n\nAbout \u00bd teaspoon fine sea salt\n\n\u00bd teaspoon finely ground dried red serrano or \u00e1rbol chile\n\n1. Preheat the oven to 350\u00b0F. In a bowl, toss the seeds with the oil, salt, and ground chile until they are evenly coated. Spread the seeds on a rimmed baking sheet\n\n2. Roast the seeds, turning them often with a spatula, until lightly golden and crunchy, about 15 minutes. Let them cool, then season with more salt, if you like. Store in an airtight container at room temperature for up to 2 weeks.\n\n_PER TABLESPOON 34 cal., 55% (19 cal.) from fat; 1 g protein; 2.2 g fat (0.4 g sat.) 1 g carbo (000 g fiber); 230 mg sodium; 0 mg chol._\n\n##    \nROASTED TOMATO-FENNEL SOUP\n\nThe velvety richness of this soup comes not from cream, but from the slowly roasted vegetables themselves\u2014and plenty of good olive oil. We used the very last of our vine-ripened tomatoes for this soup, but you could easily use roasted tomatoes that you froze earlier in the season; just thaw and reheat in the broth, then blend in with the fennel and onion.\n\nMAKES about 6 cups or 4 servings\n\nTIME about 1\u00bd hours\n\n3 pounds ripe red tomatoes such as Brandywine, cored, halved, seeded, and cut into 1-inch wedges\n\n2 fennel bulbs, trimmed and cut into 1-inch wedges\n\n1 white onion, cut into slivers\n\n1\u00bc teaspoons fine sea salt\n\n1 teaspoon dried thyme\n\n\u00bc cup extra virgin olive oil, plus more for finishing\n\n3 cups Herb Vegetable Broth (recipe follows), heated to a simmer\n\n1. Preheat the oven to 425\u00b0F. Combine the tomatoes, fennel, and onion in a roasting pan. Sprinkle with the salt and thyme, drizzle with the oil, and toss to coat evenly. Spread the vegetables in a single layer in the pan and roast them, turning once halfway through cooking, until very tender and collapsed, about 1 hour.\n\n2. Remove from the oven, add the broth, and stir to scrape up any browned bits. Puree the soup in a blender.\n\n3. Serve the soup with a drizzle of olive oil.\n\n_PER 1\u00bd-CUP SERVING 277 cal., 56% (155 cal.) from fat; 4.3 g protein; 18 g fat (2.4 g sat.); 29 g carbo (7 g fiber); 1,404 mg sodium; 0 mg chol._\n\n##    \nHERB VEGETABLE BROTH\n\nUnlike some vegetable broths, which can be strong flavored, this one tastes light, sweet, and gentle.\n\nMAKES 4 to 4\u00bd quarts\n\nTIME about 2 hours\n\n\u00bc cup extra virgin olive oil\n\n2\u00bd pounds cipollini or white onions, loose peels removed and coarsely chopped\n\n2 cups coarsely chopped mushroom stems or whole mushrooms such as shiitake and\/or oyster\n\n20 large sprigs flat-leaf parsley\n\n16 large sprigs thyme\n\n10 fresh sage leaves\n\n2 sprigs rosemary, 6 inches each\n\n2 tablespoons fine sea salt\n\n1. In an 8- to 10-quart pot, heat the oil over high heat. Add the onions and mushrooms and saut\u00e9 until the vegetables start to brown, 10 minutes. Add the parsley, thyme, sage, rosemary, salt, and 5 quarts water. Cover and bring to a boil. Reduce the heat and simmer about 1\u00bc hours.\n\n2. Strain the broth through a colander set over another large pot. Discard the solids. For perfectly clear broth, strain again through a fine-mesh strainer.\n\n3. Use the broth right away, or let cool, cover, and refrigerate for up to 1 week or freeze for up to 6 months.\n\n_PER 1-CUP SERVING 49 cal., 63% (31 cal.) from fat; 0.6 g protein; 3.6 g fat (0.5 g sat.); 4.3 g carbo (0 g fiber); 842 mg sodium; 0 mg chol._\n\n##    \nBUTTERNUT SQUASH AND 'CIPOLLINI' ONION SOUP\n\nEvery cook's repertoire should include a good butternut squash soup. For this one, you roast a big batch of vegetables and puree most of them for the soup. A few choice pieces become the garnish, along with fried sage leaves and a drizzle of brown butter.\n\nMAKES about 2\u00bd quarts or 6 servings\n\nTIME about 2 hours\n\n4\u00bd pounds butternut squash (from 1 large or 2 medium)\n\n3 tablespoons plus \u00bd cup extra virgin olive oil\n\n1\u00bd tablespoons fresh thyme leaves\n\n1\u00bd teaspoons fine sea salt, plus more for sprinkling\n\n1\u2153 pounds small cipollini onions, peeled\n\n4 tablespoons unsalted butter\n\nAbout 18 small fresh sage leaves, patted dry\n\nAbout 7 cups Herb Vegetable Broth, heated until boiling\n\n1. Preheat the oven to 425\u00b0F. Cut the ends off the squash and discard. Cut the squash in half lengthwise, then scrape out the seeds (save the seeds for roasting, if you like). Using a vegetable peeler, peel the squash halves, then cut the flesh into 1\u00bd-inch chunks. Put the chunks on a large rimmed baking sheet and toss with 2 tablespoons of the oil, 1 tablespoon of the thyme, and 1 teaspoon of the salt. Spread in a single layer.\n\n2. Place the onions on a second rimmed baking sheet. Drizzle with 1 tablespoon of the oil and sprinkle with the remaining 1\u00bd teaspoons thyme and \u00bd teaspoon salt. Toss to coat evenly, then spread in a single layer.\n\n3. Roast the vegetables for 30 minutes. Using a wide spatula, turn over the onions and squash chunks and switch the pan positions. Continue to roast until the onions are very soft and golden brown all over, 10 to 15 minutes longer, and the squash is tender and beginning to brown, 20 to 25 minutes longer.\n\n4. While the vegetables are roasting, melt the butter in a small saucepan over medium-low heat. Heat, shaking the pan occasionally, until the butter has turned a dark gold, 5 to 6 minutes. Pour into a heatproof bowl and set aside.\n\n5. In another small saucepan, heat the remaining \u00bd cup oil over medium-high heat. Working in batches, add the sage leaves and cook, stirring occasionally, until their color fades to light olive, 20 to 30 seconds. With a slotted spoon, transfer to paper towels to drain. Sprinkle with salt.\n\n6. When the vegetables are ready, remove the pans from the oven. Set aside 12 small squash pieces and 6 small onions for garnish. Working in batches, puree the remaining roasted vegetables in a blender with 6 cups of the broth until very smooth, then pour into a large saucepan. Divide about 1 cup broth between the vegetable-roasting pans. Stir to scrape up any browned bits, then add to the saucepan with the soup. Heat the soup over medium-low heat until piping hot. Add enough additional broth to create a velvety, pourable soup.\n\n7. Ladle the soup into warmed bowls. Garnish each serving with the reserved whole onions, squash chunks, fried sage, and a drizzle of brown butter.\n\n_PER SERVING 493 cal., 66% (326 cal.) from fat; 5 g protein; 38 g fat (9 g sat.); 43 g carbo (12 g fiber); 1,609 mg sodium; 20 mg chol._\n\n  GRILLED RADICCHIO AND FENNEL SALAD WITH APPLES AND TOASTED WALNUTS\n\nOur radicchio took forever to ripen. It sat stubbornly in the ground, green as green cabbage, for months. Finally, tinges of red started to take hold. By the time we used it for this salad, its leaves were a beautiful streaky combination of green and red. It lasted well into winter, turning redder and sweeter after the first frost.\n\nMAKES 4 to 6 servings\n\nTIME about 1 hour\n\n3 small heads radicchio, about 1\u00bd pounds total\n\n2 large fennel bulbs with stalks and feathery fronds\n\n4 tablespoons extra virgin olive oil\n\nAbout 1\u00bd teaspoons fine sea salt\n\n\u00bc cup Syrah Vinegar or other fruity red-wine vinegar\n\n\u00bc lemon\n\n1 Honey Crisp or other firm, tart-sweet red apple\n\n\u00be cup coarsely chopped toasted walnuts\n\n1. Prepare a grill for medium-low heat (about 300\u00b0F; you should be able to hold your hand 5 inches above the cooking grate for only 7 to 9 seconds).\n\n2. Cut the radicchio heads lengthwise into quarters. Trim the stalks from the fennel bulbs and cut the bulbs crosswise into \u00bd-inch-thick slices. Snip enough fronds from the stalks to measure about \u00be cup loosely packed fronds. Slice the stalks very thinly and set aside.\n\n3. Put the radicchio quarters and fennel bulb slices on rimmed baking sheets and drizzle with 3 tablespoons of the oil, doing your best to keep their pieces together. Sprinkle lightly with \u00bd teaspoon of the salt.\n\n4. Arrange the radicchio and fennel bulb slices on the cooking grate. Grill the radicchio, turning as needed, until the edges are crisp and browned on all sides, 15 to 20 minutes. As they are ready, return them to the baking sheet. Grill the fennel slices, turning as needed until browned on both sides and very tender, 20 to 25 minutes. (If the grate bars are widely spaced, grill the fennel on a baking sheet.) Return the slices to the baking sheet.\n\n5. Meanwhile, in a small bowl, whisk together the remaining 1 teaspoon salt and the vinegar. Squeeze the lemon quarter into a medium-size bowl filled with water. Quarter the apple through the stem end and core with a melon baller or a knife. Slice the apple quarters thinly crosswise, and slip the slices into the lemon water.\n\n6. Cut away the core from each radicchio quarter and separate the leaves. Slice the outer leaves in half lengthwise. Cut the grilled fennel into bite-size pieces. Drain the apple slices. Put the radicchio, grilled fennel, apple slices, and about three-fourths of the sliced raw fennel stalks in a large bowl. Whisk the remaining 1 tablespoon oil into the vinegar-salt mixture to make a vinaigrette. Pour all but about 1 tablespoon of the vinaigrette over the salad and toss gently to coat evenly.\n\n7. Divide the salad among 4 to 6 plates. Top with the remaining sliced raw fennel and the walnuts, drizzle with the remaining vinaigrette, and garnish with the fennel fronds.\n\n_PER 2-CUP SERVING 251 cal., 68% (171 cal.) from fat; 5 g protein; 19 g fat (2 g sat.); 19 g carbo (6 g fiber); 640 mg sodium; 0 mg chol._\n\n##    \n'SCARLET EMPEROR' RAGO\u00dbT\n\n'Scarlet Emperor' is a particularly pretty variety of scarlet runner bean. The flowers are a deep lipstick pink and, in summer, pop out all over the vine\u2014which itself is a graceful, twirling, high-climbing plant. The bean inside the pod, when freshly picked, is the same color the flower was; then it quickly oxidizes to a shiny purple, then purple-black. This rago\u00fbt (a French style of stew) is simple yet incredibly flavorful, and we ate big bowls of it by itself with bread, or with braised greens.\n\nMAKES 4 to 6 servings as a side dish\n\nTIME about 1\u00bd hours\n\n2 tablespoons extra virgin olive oil, plus more for finishing\n\n12 cipollini onions, peeled and halved\n\n16 roasted tomato halves (see Slow-Roasted Tomatoes for the Freezer)\n\n1 tablespoon minced garlic\n\n2 cups drained cooked 'Cannellini' beans (method follows)\n\n2 cups drained cooked 'Scarlet Emperor' beans (method follows)\n\n1\u00bd teaspoons fine sea salt\n\n1 tablespoon chopped fresh rosemary\n\n1. Preheat the oven to 350\u00b0F. In a Dutch oven or other heavy pot, heat the oil over medium heat. Add the onions and cook, stirring often, until browned and softened, 5 to 6 minutes. Add the tomatoes and garlic and cook, stirring, until fragrant, 1 minute. Stir in the beans and salt. Pour in just enough water to cover the beans.\n\n2. Cover the pot, place in the oven, and bake for 30 minutes. Stir the rago\u00fbt, return the pot to the oven, uncovered, and cook for another 30 minutes to blend the flavors and reduce and thicken the liquid.\n\n3. Remove from the oven and stir in the rosemary. Spoon into warmed bowls, and top each serving with a drizzle of oil.\n\n_PER 1-CUP SERVING 238 cal., 24% (58 cal.) from fat; 12.5 g protein; 6.4 g fat (0.9 g sat.); 33 g carbo (6.5 g fiber); 619 mg sodium; 0 mg chol._\n\n### HOW TO COOK DRIED BEANS\n\nIn general, 1 pound (about 2\u00bc cups) dried beans yields 2 to 2\u00bd pounds (about 6 cups) cooked beans.\n\nIn a large pot, combine the dried beans with water to cover by 2 inches, cover, and bring to a boil over high heat. Reduce the heat to a simmer and cook, occasionally skimming off any foam from the surface, until the beans are just tender, 45 minutes to 1\u00bd hours. Add fine sea salt to taste (1 teaspoon for every 2 cups uncooked beans is about right) when the beans are still a little crunchy, about 15 minutes before they have finished cooking. For the most velvety texture, let the beans cool in their liquid. The beans can be stored, covered, in their liquid, in the refrigerator for up to 2 days and frozen (drained) for up to 1 month.\n\n_Stuffed Poblanos with Red Pepper Sauce_\n\n##    \nSTUFFED POBLANOS WITH RED PEPPER SAUCE\n\nWith their filling of quinoa and dried corn, these stuffed poblanos are both earthy and light. We used the last of the fresh poblanos in our garden for this recipe.\n\nMAKES 6 servings\n\nTIME about 1\u00bd hours\n\n\u2154 cup quinoa*\n\n\u00bc cup dried corn kernels**\n\n6 poblano chiles\n\n2 'Mariachi' or 'Carmen' peppers, or red bell peppers\n\n3 cloves garlic, unpeeled\n\n\u00bd white onion, unpeeled\n\n1 cup shredded Gouda cheese, homemade or store-bought\n\n1 large egg, lightly beaten\n\n2 tablespoons finely chopped fresh oregano\n\n1 teaspoon fine sea salt\n\n\u00bc cup extra virgin olive oil\n\n2 tablespoons freshly squeezed lemon juice\n\n1. Fill a large bowl halfway with cool water. Add the quinoa and briefly rub the grains between your palms in the water. Drain the quinoa in a fine-mesh strainer, transfer to a saucepan, add 2 cups water, and bring to a boil. Reduce the heat and simmer until translucent in the center and a white ring appears, about 15 minutes. Let cool for a few minutes, then fluff with a fork.\n\n2. In a small saucepan, combine the corn kernels with water just to cover. Bring to a boil, then reduce the heat and simmer until tender, about 2 minutes. Drain.\n\n3. Preheat the broiler. Place the poblanos on a rimmed baking sheet lined with aluminum foil. Broil, turning as needed, until completely charred on all sides, 15 to 20 minutes. Cover the poblanos with a kitchen towel and let cool until they can be handled. Reduce the oven temperature to 425\u00b0F.\n\n4. Put the red peppers, garlic, and the onion half, cut side down, on a rimmed baking sheet lined with foil. Roast until the peppers have blackened slightly, about 20 minutes. Let the vegetables cool until they can be handled. Leave the oven set at 425\u00baF.\n\n5. When the poblanos are cool enough to handle, peel off and discard the skins. Cut a lengthwise slit 2 to 3 inches long on one side of each poblano, and carefully scoop out the seeds with your fingers or a small spoon. Put the poblanos back on the baking sheet.\n\n6. In a bowl, mix together the quinoa, \u00bd cup of the cheese, the egg, the oregano, \u00bd teaspoon of the salt, and the corn. Carefully spoon the mixture into the poblanos, dividing it evenly. Cover the poblanos with foil.\n\n7. Bake the poblanos for 15 minutes. Uncover, sprinkle with the remaining cheese, and bake until the cheese has melted and has begun to turn golden, about 5 minutes.\n\n8. While the poblanos are baking, peel the roasted garlic and onion and stem and seed the red peppers. Put them all in a food processor; add the oil, lemon juice, and the remaining salt; and puree until smooth. If the puree is cool, warm it in a small saucepan over low heat.\n\n9. To serve, spoon the warm sauce into a serving dish and top with the hot stuffed poblanos.\n\n_PER POBLANO WITH \u00bc CUP SAUCE 286 cal., 52% (149 cal.) from fat; 10 g protein; 17 g fat (5 g sat.); 25 g carbo (4 g fiber); 548 mg sodium; 57 mg chol._\n\n* _For instructions on how to rinse homegrown quinoa to rid it of bitterness, seehere._\n\n** _We used corn from our summer garden that we dried ourselves (here). Feel free to use fresh corn kernels, if you like, increasing the amount to \u00bd cup._\n\n##    \nQUINOA HUARACHES WITH EGG AND PARSLEY SALAD\n\nIn Mexico, _huaraches_ is the word for sandals and also for the thick, oval tortilla-like cakes that resemble them. They are usually piled with lots of delicious toppings, and so are these.\n\nMAKES 6 servings\n\nTIME about 1\u00bd hours\n\n1 cup quinoa*\n\n9 large eggs\n\n\u00bd teaspoon fine sea salt\n\n4 tablespoons extra virgin olive oil\n\n2 cups drained cooked 'Cannellini' beans (see method), warmed\n\n2 cups Walnut Romesco (double recipe)\n\n2 cups fresh flat-leaf parsley leaves\n\n1 teaspoon freshly squeezed lemon juice\n\n\u00bc cup 1-inch-long fresh chive pieces\n\n1. Fill a large bowl halfway with cool water. Add the quinoa and briefly rub the grains between your palms in the water. Drain the quinoa in a fine-mesh strainer, transfer to a saucepan, add 2 cups water, and bring to a boil. Reduce the heat and simmer until translucent in the center and a white ring appears, about 15 minutes.\n\n2. Transfer the quinoa to a bowl and refrigerate 20 minutes. Meanwhile, put 6 eggs in another small saucepan and add water to cover. Bring to a boil over medium-high heat, remove from the heat, cover, and let stand for 15 minutes. Drain the eggs and rinse with cold water until cool. Peel the eggs and quarter lengthwise. Set aside.\n\n3. Lightly beat the remaining 3 eggs, add to the cooled quinoa with \u00bc teaspoon of the salt, and stir to combine.\n\n4. In a large nonstick frying pan, heat 1 tablespoon of the oil over medium heat. To make 2 _huaraches_ at a time, spoon 2 ladlefuls (about \u00bd cup) of the quinoa mixture into the oil for each _huarache_ and use a spoon or spatula to spread into \u00bd-inch-thick oval patties. Cook the patties, turning once, until firm and golden brown, about 6 minutes. Transfer each patty to a plate and repeat with the remaining quinoa mixture, adding 1 tablespoon oil to the pan for each batch.\n\n5. Top each patty with \u2153 cup beans. Then dollop 2 heaping tablespoons of the romesco onto each mound of beans. In a small bowl, gently mix together the parsley, the remaining 1 tablespoon oil, the lemon juice, the chives, the remaining \u00bc teaspoon salt, and the quartered eggs. Carefully set the salad mixture on top of the cakes.\n\n_PER SERVING 685 cal., 61% (421 cal.) from fat; 23 g protein; 48 g fat (7.6 g sat.); 47 g carbo (10 g fiber); 808 mg sodium; 322 mg chol._\n\n* _For instructions on how to rinse homegrown quinoa to rid it of bitterness, seehere._\n\n  BUTTERNUT SQUASH GNOCCHI WITH CHARD AND SAGE BROWN BUTTER\n\nUnlike many of the vegetables we grew for our fall dinner, our butternut squash thrived. They bulged by the dozen from the vine, so we made the most of them, as in this radiant main course.\n\nMAKES 4 to 6 servings\n\nTIME about 4 hours\n\n**GNOCCHI**\n\n3 pounds butternut squash (about 1 large)\n\n1\u00bd teaspoons extra virgin olive oil\n\n\u2154 cup ricotta cheese, homemade or store-bought\n\n1 large egg, lightly beaten\n\n1\u00bd tablespoons honey\n\n\u00be teaspoon fine sea salt\n\nAbout 3 cups sifted 'Expresso' whole-wheat flour* (to remove some of the coarse germ), plus more for sprinkling\n\n**SAGE BUTTER**\n\n\u00bd cup unsalted butter\n\n2\u00bd teaspoons freshly squeezed lemon juice\n\n2 tablespoons thinly sliced fresh sage leaves\n\n1 tablespoon extra virgin olive oil, plus more for oiling baking sheet\n\n1\u00bd teaspoons minced garlic\n\n9 to 10 ounces Swiss chard (about 1 bunch), stemmed and leaves chopped\n\nAbout \u00be cup Roasted Spiced Butternut Squash Seeds\n\n1. To start the gnocchi, preheat the oven to 350\u00b0F. Cut off the ends of the squash and discard. Cut the squash into large chunks, but do not peel. Scrape off the seeds and save them for roasting if you like. Rub the flesh sides of the squash chunks with the oil, set on a rimmed baking sheet, and cover with aluminum foil.\n\n2. Roast the squash until tender when pierced with a knife tip and starting to brown, 45 to 60 minutes. Remove from the oven and let cool, then scoop the flesh from the skins into a medium-mesh strainer set over a bowl. Let the squash drain of any excess liquid, about 15 minutes. Reserve the squash and liquid separately and throw away the skins.\n\n3. In a food processor, whirl the squash flesh until smooth. If it's dry and not moving freely, add 2 tablespoons of squash liquid. Continue adding the liquid 2 tablespoons at a time, until the squash has the consistency of velvety mashed potatoes. Set the puree aside.\n\n4. To make the sage butter, melt the butter in a large saucepan over high heat. Reduce the heat to medium and simmer until the butter darkens to a medium brown, about 10 minutes. Be careful not to let the butter get too dark. Remove from the heat and stir in the lemon juice and sage. Set the butter aside.\n\n5. To finish the gnocchi, in a large bowl, combine 2 cups squash puree (save the extra for another use), the ricotta, egg, honey, and salt, mixing well. Gently stir in 2 to 3 cups of the flour in \u00bd-cup increments, adding just enough flour to form a very soft dough. It should still be quite wet.\n\n6. Bring a large pot of salted water to a boil. Meanwhile, heavily dust a work surface with more sifted flour. Scoop about 1 cup of the dough onto the flour (the dough will be extremely soft) and sift a little more flour on top. Roll the dough with your hands into a rope 15 inches long. Cut the rope into 1-inch pieces, sprinkling the dough with more flour if it gets sticky. As the pieces are cut, transfer them to a rimmed baking sheet lightly dusted with sifted flour. Repeat with the remaining dough.\n\n7. Reduce the heat under the boiling water to medium-low. Working in batches of 15 to 20 pieces, carefully drop the gnocchi into the water and cook until they are firm and float to the surface, about 5 minutes. With a slotted spoon, transfer the gnocchi to a large colander to drain, then to a lightly oiled rimmed baking sheet. Cook and drain the remaining gnocchi the same way.\n\n8. To assemble the gnocchi, heat 1\u00bd teaspoons of the oil in a large nonstick frying pan over medium-high heat. Working in batches, add the gnocchi to the pan and cook, stirring often, until browned, 5 to 8 minutes. Transfer to a warmed serving dish.\n\n9. Add the remaining 1\u00bd teaspoons oil and the garlic to the same pan over medium-high heat and cook, stirring, until the garlic is fragrant, 2 minutes. Add the chard and cook, stirring occasionally, until softened, 5 minutes. Gently combine the chard with the gnocchi.\n\n10. Warm the reserved sage butter and drizzle half of it over the gnocchi. Sprinkle the gnocchi with the roasted squash seeds. Pass the remaining sage butter at the table.\n\n**MAKE AHEAD:** You can prepare the gnocchi through step 6, cover tightly, and refrigerate for up to 3 days. Or, you can freeze the gnocchi on the baking sheet until solid, then transfer them to resealable plastic bags for up to 3 weeks. Cook directly from the refrigerator or freezer.\n\n_PER 1\u00bd-CUP SERVING 733 cal., 49% (366 cal.) from fat; 22 g protein; 41 g fat (15.5 g sat.); 77 g carbo (13 g fiber); 594 mg sodium; 91 mg chol._\n\n* _We liked the nutty flavor of this freshly milled local flour, but any whole-wheat flour will work. SeeThe Allure of Local Wheat for more on local flours._\n\n##    \nANCHO CHILE\u2013SAUCED NOODLES WITH SHIITAKES AND BUTTERNUT SQUASH\n\nWe used a combination of flours to give these noodles a more supple, finer texture than they have when they are made from regular whole-wheat flour. If you like a rustic, hearty noodle, use just the regular whole-wheat flour.\n\nMAKES 4 to 6 servings\n\nTIME about 3 hours\n\n1 cup 'Sonora' whole-wheat flour* or whole-wheat pastry flour, plus more for sprinkling\n\n1\u00bc cups 'Expresso' whole-wheat flour* or other whole-wheat flour\n\n1\u00bd teaspoons fine sea salt\n\n3 large eggs plus 2 large egg yolks\n\n6 tablespoons extra virgin olive oil\n\n1 butternut squash, 2 to 2\u00bd pounds\n\n4 small ancho chiles (about 1 ounce total)\n\n\u00bc white onion\n\n3 cloves garlic, unpeeled\n\n1 pound shiitake mushrooms, stems removed and caps sliced 1 inch thick\n\n4 roasted tomato halves (about 2 tablespoons; see Slow-Roasted Tomatoes for the Freezer)\n\n\u00bd cup chopped fresh flat-leaf parsley\n\n1. To make the noodle dough, in a large bowl, whisk together the flours and \u00bd teaspoon of the salt. Form a well in the center and add the eggs and egg yolks to the well. Break up the eggs with a fork and mix in 1 tablespoon of the oil. Gradually mix the flour from the sides of the well into the eggs. Stir until a dough forms.\n\n2. Turn the dough out onto a well-floured work surface and sprinkle with a little more flour. Knead the dough, dusting it lightly with flour whenever it starts sticking to your hands, until it is smooth, 2 to 3 minutes. Cover with a damp kitchen towel and let rest for at least 30 minutes or up to 1\u00bd hours.\n\n3. Meanwhile, preheat the oven to 450\u00b0F. Cut off the ends of the squash and discard. Halve the squash lengthwise and scrape out the seeds. Rinse any flesh off the seeds and pat dry on a kitchen towel. Using a vegetable peeler, peel the squash halves, then cut them crosswise into \u00be-inch-thick slices.\n\n4. Start the ancho sauce: In a heavy frying pan (not nonstick), toast the chiles over medium-high heat, pressing down on them occasionally with tongs, until soft and pliable, 2 to 3 minutes. Transfer 3 of the chiles to a small heatproof bowl and pour 1\u00bc cups boiling water over them. Let soften for 20 minutes. Stem the remaining chile, break into pieces, and pulverize in a spice grinder or in a mortar with a pestle.\n\n5. Add the onion and garlic cloves to the same frying pan over medium-high heat and toast, turning occasionally, until browned in several spots, 10 minutes. Transfer to a small bowl. Add 1 tablespoon of the oil to the same pan over medium heat, add the squash seeds, sprinkle with \u00bd teaspoon of the salt, and toast, stirring, until crisp, 5 minutes. Pour into a bowl.\n\n6. Cook the vegetables: Arrange the squash slices in a single layer on a large rimmed baking sheet and arrange the mushrooms in a 9-by-13-inch baking dish. Sprinkle the squash slices and mushrooms with the ground chile, then sprinkle with the remaining \u00bd teaspoon salt. Add \u00bd cup water to the mushrooms, cover the dish tightly with aluminum foil, and braise in the oven until tender, about 30 minutes. Drizzle the squash slices with 2 tablespoons of the oil and roast, turning the slices halfway through cooking, until browned and tender, about 40 minutes. Set aside, covered.\n\n7. Finish the sauce: Stem the soaked chiles and put in a blender. Add \u00be cup of the chile soaking water and the toasted squash seeds and blend until smooth. Peel the toasted garlic and add to the blender along with the onion, tomatoes, and the remaining oil. Blend again until smooth.\n\n8. Divide the dough into 4 equal portions and pat each into a square \u00bd inch thick. Put 1 square on a floured work surface (keep the others covered with the towel), and roll out \u215b inch thick. Using a pizza cutter, cut the dough sheet into noodles \u00bc inch wide. Scoop up the noodles with a bench scraper or a large, wide spatula and lay them on a cooling rack. Repeat with the remaining dough. Let the noodles dry for at least 20 minutes or up to 1 day.\n\n9. Bring an 8- to 10-quart pot of salted water to a boil. Add the noodles to the boiling water, shaking off excess flour. Cook until chewy-tender, 3 to 4 minutes. Drain the noodles, reserving \u00bd cup of the cooking water, and pour into a large bowl.\n\n10. Add the ancho sauce to the noodles and toss to coat, adding as much of the reserved cooking water or remaining chile soaking water as needed to loosen the sauce a little. Add three-fourths of the parsley and toss again. Arrange the noodles on a warmed platter. Top with the roasted squash slices, the braised mushrooms, and the remaining parsley.\n\n_PER SERVING 431 cal., 40% (172 cal.) from fat; 14 g protein; 20 g fat (3.6 g sat.); 58 g carbo (13 g fiber); 627 mg sodium; 177 mg chol._\n\n* _We used finely milled 'Sonora' white whole-wheat flour, because it behaves like commercial whole-wheat pastry flour. For the regular whole-wheat flour, we used 'Expresso', a hard red winter wheat. SeeThe Allure of Local Wheat for more on local flours._\n\n##    \nMEXICAN SKILLET EGGS\n\nOur eggs seemed especially flavorful in this one-pan dinner, fortified with pureed 'Scarlet Emperor' beans (our substitute for black beans, because they have a similar starchy texture), intense oven-roasted tomatoes preserved from summer, and our last few fresh chiles. Once we ran out of fresh, we made it with frozen poblanos and dried serrano chiles.\n\nMAKES 4 to 6 servings\n\nTIME about 30 minutes\n\n2 tablespoons extra virgin olive oil\n\n\u00bd white onion, sliced\n\n2 red or green poblano chiles, stemmed, seeded, and sliced\n\n1 tablespoon minced garlic\n\n1 tablespoon minced red serrano chile\n\n1\u00bd teaspoons fine sea salt\n\n3 cups drained cooked 'Scarlet Emperor' beans (see method)\n\n8 roasted tomato halves (see Slow-Roasted Tomatoes for the Freezer), halved\n\n4 to 6 large eggs\n\n1 tablespoon fresh oregano leaves\n\n1. In a large nonstick frying pan, heat the oil over medium heat. Add the onion and poblanos and cook, stirring often, until starting to soften, about 4 minutes. Add the garlic, serrano chile, and \u00bd teaspoon of the salt and cook, stirring, until fragrant, about 1 minute. Set aside off the heat and cover to keep warm.\n\n2. In a small saucepan, combine the beans, \u00bd cup water, and the remaining 1 teaspoon salt over low heat and heat just until warm. Transfer to a food processor and puree until smooth and loose, adding more water as needed.\n\n3. Return the frying pan to medium heat, pour the bean puree over the vegetables, and heat until the mixture begins to simmer. Scatter the tomato halves over the beans. Use a spoon to make 4 or 6 wells, depending on how many people you are serving, and crack an egg into each well. Cover the pan and cook until the eggs are just set but still a little runny in the center, about 5 minutes. Sprinkle with the oregano and serve.\n\n_PER EGG WITH \u00bc CUP BEANS 217 cal., 42% (93 cal.) from fat; 13 g protein; 10.5 g fat (20 g sat.); 18 g carbo (2 g fiber); 649 mg sodium; 211 mg chol._\n\n##    \nCREAMY SCRAMBLED EGGS WITH OYSTER MUSHROOMS\n\nThis recipe is based on one by Niloufer Ichaporia King, a wonderful Bay Area cook and the author of _My Bombay Kitchen: Traditional and Modern Parsi Home Cooking_.\n\nMAKES 4 to 6 servings\n\nTIME about 30 minutes\n\n2 tablespoons unsalted butter, homemade or store-bought\n\n7 ounces oyster mushrooms, stems trimmed if woody and any large caps halved\n\n1\u00bd teaspoons fresh thyme leaves, minced\n\nFine sea salt\n\n2 cloves garlic, minced\n\n8 large eggs\n\n\u00bd cup whole milk\n\n1. In a nonstick frying pan, melt the butter over medium heat. Pour about half of the melted butter into a medium bowl and set aside. Add the mushrooms, thyme, and \u00bc teaspoon salt to the frying pan and cook, stirring every now and then, until the mushrooms have softened and are starting to brown, about 5 minutes. Add the garlic and cook, stirring, for 1 minute more.\n\n2. Add the eggs, milk, and a couple of pinches of salt to the bowl with the melted butter and whisk to blend. Pour the egg mixture into a medium saucepan and cook over low heat, stirring slowly with a rubber spatula and keeping contact with the bottom of the pan, until small, soft curds begin to form, 10 to 15 minutes.\n\n3. Stir in the mushrooms and keep cooking and stirring until the eggs are just set, about a minute longer.\n\n_PER SERVING 151 cal., 63% (95 cal.) from fat; 10 g protein; 11 g fat (5 g sat.); 4 g carbo (0.7 g fiber); 222 mg sodium; 299 mg chol._\n\n  WHOLE-WHEAT ROSEMARY SHORTBREADS\n\nWe wanted some kind of buttery cracker to go with our fall cheese platter. These melt-in-your-mouth shortbreads are too fragile to be topped or spread with anything, but they taste wonderful with cheese and fruit.\n\nMAKES about 12 shortbread squares\n\nTIME about 30 minutes, plus 1 hour to chill\n\n\u00be cup unsalted butter, homemade or store-bought, at room temperature\n\n1 teaspoon honey\n\n\u00bd teaspoon fine sea salt\n\n\u00bd teaspoon minced fresh rosemary\n\n1\u00bd cups 'Sonora' whole-wheat flour,* plus more for sprinkling\n\n1. In a medium bowl, using a mixer on medium speed, beat together the butter and honey until light and fluffy. Beat in the salt and rosemary. On low speed, gradually beat in the flour just until well combined.\n\n2. Line a rimless cookie sheet with waxed paper and scrape dough onto the paper. Sprinkle the dough lightly with flour. With a floured rolling pin, roll out the dough to about \u00bc inch thick. Top with another sheet of waxed paper and refrigerate for at least 1 hour or up to 2 days.\n\n3. Preheat the oven to 325\u00b0F. Using a 2-inch square cookie cutter (we used one with a scalloped edge), cut out as many squares as possible and place on 2 rimmed baking sheets, spacing them \u00bd inch apart. Gather up the dough scraps, reroll, cut out more squares, and add to the baking sheet. (If the dough gets too sticky to roll, chill and try again.)\n\n4. Bake until set but not browned, 10 to 12 minutes. Let cool completely on the baking sheets set on wire racks.\n\n**MAKE AHEAD** Shortbreads keep, stored airtight, at room temperature, for up to 1 week.\n\n_PER SHORTBREAD 159 cal., 66% (105 cal.) from fat; 2.4 g protein; 12 g fat (7 g sat.); 12 g carbo (2 g fiber); 176 mg sodium; 31 mg chol._\n\n* _We liked a medium-fine grind of this local flour, but regular whole-wheat flour works well, too. SeeThe Allure of Local Wheat for more on local flours._\n\n##    \nWALNUT-HONEY CRISPS\n\nWe started out trying to make _tuiles_ , thin, blond, curvy French cookies that shatter when you eat them. Our sole sweetener, though, was honey\u2014which is too moist to produce anything thin or crisp enough to hold a curve, and our wheat flour made the cookies dark. So these are _tuiles'_ flat, homey, but very tasty cousins.\n\nMAKES 10 cookies\n\nTIME about 30 minutes\n\n\u2153 cup unsalted butter\n\n\u00bc cup honey\n\n\u215b teaspoon fine sea salt\n\n\u00bc cup finely chopped toasted walnuts\n\n\u00bc cup 'Expresso' whole-wheat flour* or other whole-wheat flour\n\n1. Preheat the oven to 325\u00b0F. Line 2 rimmed baking sheets with parchment paper.\n\n2. In a small saucepan, heat the butter, honey, and salt over medium heat until boiling, stirring often. Remove from the heat and whisk in the walnuts and flour. Continue to whisk until the mixture is very smooth and thick and no fat dots appear on the surface, about 3 minutes.\n\n3. For each cookie, spoon 1 tablespoon batter onto the prepared sheets, spacing the spoonfuls about 3 inches apart (the batter will spread a lot).\n\n4. Bake the crisps until dark golden brown, about 12 minutes. Halfway through baking, switch the pan positions and rotate the pans back to front to ensure the cookies bake evenly. Let cool completely on the pans on wire racks. Store in an airtight container at room temperature for up to 1 day.\n\n_PER COOKIE 95 cal., 78% (74 cal.) from fat; 3.4 g protein; 8.3 g fat (1.3 g sat.); 1.8 g carbo (0.5 g fiber); 97 mg sodium; 0 mg chol._\n\n* _We liked the nutty flavor of this freshly milled local flour, but any whole-wheat flour will work. SeeThe Allure of Local Wheat for more on local flours._\n\n##    \n  HONEY ICE CREAM\n\nWe were knocked out by the intensely floral, seductive flavor of our honey all over again in this simple, lovely ice cream. Use your favorite honey when you make this.\n\nMAKES 4 cups\n\nTIME about 1 hour, plus 6 hours to freeze\n\n2 cups heavy cream\n\n1\u00bd cups whole milk\n\n4 large egg yolks\n\nAbout 1 cup honey\n\nPinch of fine sea salt, plus more for finishing\n\nIce cubes\n\n1. Pour the cream and milk into a medium saucepan and bring to a simmer over medium heat. Right before it comes to a simmer, in a medium heatproof bowl, whisk together the egg yolks, \u00be cup of the honey, and the salt.\n\n2. Immediately pour the cream and milk slowly into the yolk mixture while whisking constantly. Return the mixture to the pan and cook over medium heat, stirring constantly and adjusting the heat to prevent the mixture from boiling, until it begins to thicken, about 8 minutes.\n\n3. Fill a large bowl with ice cubes and water, and nest a medium bowl in the ice water. Strain the custard mixture through a fine-mesh strainer into the medium bowl. Let cool completely, stirring occasionally and replacing the ice if needed.\n\n4. Freeze in an ice cream maker according to the manufacturer's directions. Transfer to an airtight container and freeze until firm, at least 6 hours or up to 2 weeks.\n\n5. To serve, scoop ice cream into bowls. Drizzle with more honey and top with a sprinkle of salt.\n\n_PER \u00bd-CUP SERVING 390 cal., 60% (234 cal.) from fat; 4.1 g protein; 26 g fat (15.5 g sat.); 38.7 g carbo (0.1 g fiber); 328 mg sodium; 192 mg chol._\n\n##    \nLEMON-THYME ICE CREAM SANDWICHES\n\nMade with nutty whole-wheat flour and honey, the cookie part tastes like graham crackers, with the barest hint of thyme. The ice cream brings back summer, when we harvested the last of the heavy, dripping frames of honey.\n\nMAKES 30 ice cream sandwiches (60 cookies)\n\nTIME about 1 hour, plus about 1\u00be hours to chill\n\n1 cup butter, at room temperature\n\n\u00bd cup honey\n\nGrated zest of 1 lemon\n\n1 teaspoon minced fresh thyme\n\n2\u00bc cups whole-wheat pastry flour*\n\nAbout 1 pint Honey Ice Cream (opposite), frozen overnight\n\n1. In a large bowl, using a mixer on medium speed, beat together the butter, honey, lemon zest, and thyme until smooth. On low speed, gradually beat in the flour until a smooth dough forms.\n\n2. Divide the dough in half. Put each portion on a sheet of plastic wrap and pat into a rough log about 7 inches long. Move the logs, still on the plastic wrap, onto a rimmed baking sheet and freeze until firm enough to shape, about 20 minutes.\n\n3. Pat each log until it is smooth and reaches a uniform 1\u00bd inches in diameter, squeezing the dough together if it starts to crack. Wrap airtight in plastic wrap and freeze until firm enough to slice, 20 to 30 minutes.\n\n4. Preheat the oven to 325\u00b0F. Line 2 rimmed baking sheets with parchment paper. Unwrap the dough. Using a thin, sharp knife, cut the logs into \u00bc-inch-thick slices. Arrange the slices \u00bd inch apart on the baking sheets.\n\n5. Bake the cookies until the edges are golden brown, 12 to 15 minutes. Switch the pan positions and rotate the pans back to front about halfway through baking so the cookies bake evenly. Let cool on the pans on wire racks for 5 minutes. Transfer to the racks to cool completely.\n\n6. To make the sandwiches, put a baking sheet in the freezer until it is cold. Arrange 5 or 6 cookies, bottom side up, on a work surface. Put 1 tablespoon ice cream on each cookie, then top with a second cookie, bottom side down, and squish gently to spread the ice cream evenly. Put the finished sandwiches on the baking sheet in the freezer, then repeat with the remaining cookies and ice cream, working in batches. When all of the sandwiches are made, wrap them individually in plastic wrap and freeze until the ice cream is firm, about 1 hour.\n\n**MAKE AHEAD** Plain cookies can be stored in an airtight container at room temperature for up to 2 days or in the freezer for up to 1 month. Ice cream sandwiches will keep in the freezer for up to 1 week.\n\n_PER ICE CREAM SANDWICH 156 cal., 56% (87 cal.) from fat; 1.5 g protein; 9.7 g fat (6 g sat.); 17 g carbo (1.2 g fiber); 88 mg sodium; 42 mg chol._\n\n* _We used finely milled local 'Sonora' whole-wheat flour, but any whole-wheat pastry flour will work. SeeThe Allure of Local Wheat for more on local flours._\n\n_Apple Cheese Puff_\n\n##    \nAPPLE CHEESE PUFF\n\nThis pastry is all about layers: a thin, buttery crust, then a poufed circle of cream puff dough topped with creamy, sweetened cheese and caramelized apples. You can cook the apples a day ahead, but not the crust\u2014it gets leathery as it sits. This is a dessert best eaten in one go.\n\nMAKES one 7- to 8-inch pastry or 9 servings\n\nTIME about 1\u00bd hours\n\n\u00bd cup plus 3 tablespoons cold unsalted butter, homemade or store-bought\n\n1 cup 'Sonora' whole-wheat pastry flour*\n\n1 large egg plus 1 large egg white, lightly beaten\n\n4 Honey Crisp or other firm, tart-sweet apples\n\n9 tablespoons honey\n\n\u00be cup fromage blanc, homemade or store-bought**\n\n1. Preheat the oven to 350\u00b0F. Cut \u00bc cup of the butter into small cubes. Put \u00bd cup of the flour in a small bowl, scatter the butter cubes over the top, and mix the butter into the flour with a pastry blender or your fingertips until the mixture is the consistency of crushed crackers. Sprinkle in 1 tablespoon water and mix with a wooden spoon until the dough comes together. If the dough is not coming together, sprinkle in another 1 tablespoon water. Press the dough into a disk.\n\n2. Put the disk on an ungreased rimmed baking sheet and pat into a round 8 inches in diameter and \u215b inch thick.\n\n3. In a saucepan, bring \u00bc cup of the butter and \u00bd cup water to a rolling boil. Remove from the heat and quickly stir in the remaining \u00bd cup flour with a wooden spoon. Pour in the eggs and stir vigorously until smooth. With a wet soup spoon, spread the mixture over the dough disk, pushing it higher on the edge to form a narrow rim. Poke the center several times with a fork to keep the pastry from puffing up when it bakes.\n\n4. Bake the pastry until crisp and brown, 45 to 55 minutes. Let cool completely on the pan on a cooling rack.\n\n5. Meanwhile, make the toppings: Peel, halve, and core the apples, then slice \u00bc inch thick. In a large nonstick frying pan, melt the remaining 3 tablespoons butter over medium-high heat. Add the apples and cook, stirring occasionally, until they start to brown, about 8 minutes. Turn the heat to its lowest setting, cover, and cook the apples until they are very soft, about 20 minutes more. Add 6 tablespoons of the honey, raise the heat to medium-high, and cook, uncovered, until the apples are deep golden brown and most of the liquid has evaporated, about 20 minutes. Let cool until just warm.\n\n6. In a small bowl, whisk together the fromage blanc and the remaining 3 tablespoons honey. Spread the mixture on the cooled pastry and top with the warm apples.\n\n_PER SERVING 387 cal., 52% (200 cal.) from fat; 5.5 g protein; 22.5 g fat (13.8 g sat.); 43 g carbo (2.9 g fiber); 303 mg sodium; 91.5 mg chol._\n\n* _We liked this local finely milled soft whole-wheat flour for this recipe because its texture is great in pastries. Any whole-wheat pastry flour will work, however. SeeThe Allure of Local Wheat for more on local flours._\n\n** _Store-bought fromage blanc is often much drier than homemade fromage blanc. If you use it, stir in enough heavy cream (at least 3 tablespoons) to give it the consistency of softened cream cheese._\n\n##    \nBUTTERNUT SQUASH COMPOTE WITH HONEY AND TOASTED WALNUTS\n\nA quick glance at the ingredients here might give you an unpromising impression, but this dessert is a standout: as the squash bakes slowly in honey, it takes on a deeply floral, slightly caramelized note. Resist the temptation to take it out of the oven once it is cooked. When left in the oven to cool, the squash becomes infused with the flavor of the honey, and its texture turns velvety. The recipe is based on one in _The Slow Mediterranean Kitchen: Recipes for the Passionate Cook_ (Wiley, 2003) by Paula Wolfert.\n\nMAKES 4 to 6 servings\n\nTIME about 2 hours, plus about 3 hours to stand\n\n1 small butternut squash, about 1\u00be pounds\n\n1 cup honey\n\n1 teaspoon salted butter, homemade (at right) or store-bought\n\n\u00bd cup walnut halves\n\n\u00be cup cr\u00e8me fra\u00eeche, homemade or store-bought\n\n1. Preheat the oven to 300\u00b0F. Cut off the ends of the squash and discard. Cut the squash in half lengthwise, then scrape out the seeds (save the seeds for roasting, if you like). Using a vegetable peeler, peel the squash halves, then cut the flesh into 1\u00bd-inch chunks. You should have 3\u00bd to 4 cups.\n\n2. Put the squash chunks in a deep 1- to 1\u00bd-quart baking dish such as a souffl\u00e9 dish. Add the honey and stir to coat the squash chunks evenly. Cut a piece of parchment paper a few inches larger than the diameter of the dish. Crumple the parchment, moisten it with water, and then flatten it on a work surface. Cover the squash with the parchment, tucking it around the inside of the dish to fit snugly.\n\n3. Bake the squash, stirring every 30 minutes or so, until it is tender when pierced, 1\u00be to 2 hours. Turn off the oven and leave the squash inside for about 3 hours to finish cooking evenly.\n\n4. Meanwhile, in a small frying pan, melt the butter over medium heat. Add the walnuts and cook, stirring often, until the nuts are lightly browned, 3 to 4 minutes.\n\n5. Spoon the slightly warm squash and honey syrup into bowls. Garnish each serving with a spoonful of cr\u00e8me fra\u00eeche and a scattering of the walnuts.\n\n**MAKE AHEAD** The compote can be refrigerated for up to 2 days. Warm slightly in a microwave for about 1\u00bd minutes or in a 300\u00baF oven before serving.\n\n_PER SERVING 387 cal., 40% (156 cal.) from fat; 3.5 g protein; 17 g fat (8 g sat.); 60 g carbo (4 g fiber); 11 mg sodium; 27 mg chol._\n\n##    \nHOMEMADE BUTTER AND BUTTERMILK\n\nHere is your chance to ignore that rule about not overbeating cream and go straight for the clumps of butter. Unlike commercial butter, which has small amounts of culture added, homemade butter tastes extra sweet and fresh\u2014and so does the buttermilk that's left over. (We use it to make Wheat Berry Ciabatta.)\n\nMAKES 1 cup butter and \u00be cup sweet buttermilk\n\nTIME about 10 minutes\n\n2 cups heavy cream\n\nFine sea salt (optional)\n\n1. Whirl the cream in a food processor until it separates into buttermilk and clumps of butter that look like fluffy scrambled eggs. Then keep whirling until the butter forms bigger clumps. This takes 2 to 3 minutes total.\n\n2. Set a fine-mesh strainer over a bowl. Pour the butter-milk and butter into the strainer and let drain briefly. Squeeze the butter with your hands to extract the remaining buttermilk (it's okay if a little is left).\n\n3. Turn the butter into another bowl and stir in salt to taste, if you like. Both the butter and buttermilk will keep in the refrigerator for up to 1 week.\n\n_PER TABLESPOON BUTTER 101 cal., 100% (101 cal.) from fat; 0.1 g protein; 11.5 g fat (7.3 g sat.); 0 g carbo (0 g fiber); 1.5 mg sodium; 30.5 mg chol._\n\n_PER \u00bd CUP BUTTERMILK 75 cal., 20% (14.5 cal.) from fat; 6 g protein; 1.5 g fat (1.2 g sat.); 9 g carbo (0 g fiber); 193 mg sodium; 7.5 mg chol._\n\n  LAST-MINUTE PINEAPPLE GUAVA PRESERVES\n\nWe wanted some sort of fruity component for our cheese course, and in the wake of the Fig Disaster, we scoured the garden for fruit, any fruit at all\u2014and found some very ripe pineapple guavas still hanging on our tree.\n\nWe had not considered including guavas in the feast (we'd been focused on those figs), but suddenly they seemed perfect: We would make a classic Spanish-style membrillo (quince paste)\u2014so good with cheese\u2014only with pineapple guavas! Like quince, pineapple guavas are extremely fragrant and are used to make purees, pastes, and preserves. Also known as feijoas, they have a wonderful flavor that combines pineapple, pear, and strawberry, and their flesh is slightly grainy, like that of a Bosc pear.\n\nWe ended up with both a thick guava jam (shown in photo) and, in later tests, a firm paste that can be cut into pieces. Both were delicious, and went well with the Gouda and feta.\n\nMAKES about 2 cups\n\nTIME about 3 hours, plus 30 minutes to cool and 2 hours to chill\n\n2 pounds ripe pineapple guavas (fejoias)\n\n1 cup water\n\n1 cup honey\n\n1. Trim off the ends of the guavas, then chop coarsely, skins and all. Put the chunks in a large pot with the water. Cook over low heat, stirring occasionally, until the chunks are soft, about 1 hour.\n\n2. Preheat the oven to 250\u00b0F if making a paste. Transfer the cooked fruit to a food processor and pulse until smooth. Strain the mixture back into the pot and add the honey. Cook over medium-low heat, stirring frequently with a rubber spatula and scraping the bottom of the pot to prevent the mixture from sticking, until it is very thick and bubbling, the consistency of mashed potatoes, about 1 hour. The color goes from green to light brown. If you want a firm paste, continue with the next step. Otherwise, let the preserves cool, then chill them until ready to serve.\n\n3. Scrape the jam into a nonstick 9-inch square baking dish and bake, stirring occasionally, until dark brown and very thick, about 45 minutes. Remove the preserves from the oven, stir, and let cool 30 minutes. Chill until cold, then cut into pieces.\n\n**MAKE AHEAD** The preserves can be made up to 2 weeks in advance and stored, tightly covered, in the refrigerator.\n\n_PER 2-TABLESPOON SERVING 85 cal., 3% (3 cal.) from fat; 0.05 g protein; 0.3 g fat (0 g sat.); 22 g carbo (2.2 g fiber); 2 mg sodium; 0 mg chol._\n\n_Fall cheese platter:Gouda and feta with fresh honeycomb, walnuts, pineapple guava slices, Last-Minute Pineapple Guava Preserves, and Whole-Wheat Rosemary Shortbreads_\n\nTHE ALLURE OF LOCAL WHEAT\n\nAfter we ran out of our own wheat, we decided to try to find the next best thing: locally grown whole-wheat flour.\n\nAt a farmers' market near our office, we found a finely milled soft white variety called 'Sonora', grown and milled by Full Belly Farm in Yolo County, to the northeast of us. At Pie Ranch in Pescadero, on the coast to the south, we found more 'Sonora', with a medium-fine texture. Eatwell Farm\u2014in inland Solano County\u2014sold several flours at San Francisco's Ferry Plaza Farmers Market; we tried 'Expresso', a hard red type.\n\nWhen you're used to standard refined all-purpose flour, it's exciting to cook with flour made from nonstandard wheat varieties. They each have their own character and flavor, and you have to adjust your recipes to suit them. The 'Sonora' has a delicate, fresh taste, and, we discovered, works best as a pastry flour; it is very soft and doesn't develop the stretchiness needed for bread making. The very finely milled 'Sonora' sold by Full Belly Farm was great for tempura, which usually requires rice flour in order to be lacy and crisp. The protein content of 'Sonora' is typically 9 percent, about the same as pastry flour. The 'Expresso', at 12.3 percent protein, makes good, nutty-tasting bread with a fine rise. All three farms we bought from mill their grains every couple of weeks, so the flour is fresh and sweet\u2014another big difference between the little guys and the big manufacturers.\n\nModern Wheat\n\nAs we cooked with the flours, we found out more about them. 'Expresso', developed by Arizona breeder Kim Shantz in 2005, is a thoroughly modern \"inbred\" variety created to fight off a common wheat disease called stripe rust. It is mainly grown in the Sacramento Valley. Its parent, 'Express', came from an international breeding program in Mexico. Neither had roots in California. \"Breeders use varieties from around the world. You can't really regionalize any more,\" says Shantz. I asked him whether the fine flavor of 'Expresso' flour was making the wheat more popular. \"Big producers don't care so much about that,\" he said. \"They're looking at protein level and mixing qualities.\" (The term _mixing qualities_ refers to the strength of the dough and how long it can be mixed without breaking down.)\n\nNigel Walker, of Eatwell Farm, bought the 'Expresso' originally to feed to his chickens. Then he heard that it made good bread, so he milled some, and now he sells all he can supply. \"I know it's not an heirloom. But it makes really good baking flour and responds to organic methods,\" he says. Modern flours typically need heavy doses of herbicide to fight off weeds. But so far, Walker's 'Expresso' is doing fine on his organic farm.\n\nLandrace Wheat\n\n'Sonora' wheat comes from the other end of the wheat spectrum. It is a landrace variety, meaning it has adapted to its specific environment over many years. In the case of 'Sonora', that means centuries. Monica Spiller, the founder of Whole Grain Connection, in Mountain View, California, thinks its cultivation in California probably predates the Spanish missionaries. It is known to have existed in Sonora, in northern Mexico, since at least the early 1700s, hence its name. (The big white flour tortillas famous in that part of Mexico were once made with 'Sonora' wheat.)\n\nIn California, 'Sonora' thrived. Along with a few other wheat varieties, it made California the number one wheat producer in the country by the late 1800s. So much of it was grown in 1884\u2014nearly three million acres\u2014that it set the floor prices at the London International Wheat Exchange.\n\nUnfortunately, its decline began immediately afterward, with the rise of hard red wheat from the Midwest, and the invention of the roller mill. Hard red was not only superprolific, but it could be put through a roller mill that sheared off the germ and bran to produce refined white flour, for which the public had an insatiable appetite. 'Sonora' didn't work well in the roller mills\u2014it was best ground whole\u2014and, not being quite as white as refined white flour, it floundered. By 1910, California's total wheat acreage fell to six hundred thousand, and gradually modern hybrids replaced the older strains.\n\nWheat Revival\n\nIt is thanks in large part to Spiller that 'Sonora' is back on the scene. A former chemist, she has spent the past thirty years researching which varieties do best in our area, and 'Sonora', which she found preserved at the United States Department of Agriculture's Small Grains Collection in Idaho, tops the list. Unlike hybrids, which tend to get knocked out by evolving strains of stripe rust, 'Sonora' is enduringly resistant to the disease, requires no chemical fertilizing, and needs very little water once it gets going, which makes it great for drought-prone California and also Arizona. The next time we plant wheat, we'll probably give both 'Sonora' and 'Expresso' a try.\n\n**HELPFUL INFORMATION**\n\nWhole-wheat flour\n\n**Oregon Tilth** A good list of national producers who grow organic wheat (www.tilth.org\/producer-search\/producers).\n\n**Anson Mills, Columbia, South Carolina** Owner Glenn Roberts grows a range of heritage wheats for flour, including colonial-era 'Red May' (www.ansonmills.com).\n\n**Bob's Red Mill, Milwaukie, Oregon** The whole-grain, stone-ground giant gives guided tours of the mill. (www.bobsredmill.com)\n\n**Pie Ranch, Pescadero, California** Organic 'Sonora' wheat berries and flour. Sold at the farm stand in Pescadero, March through October; Pie Ranch's flour (as well as its produce) also goes to make pies and other baked goods at local sustainable food businesses such as Mission Pie in San Francisco and Companion Bakers in Santa Cruz. $3 per 1-pound bag (www.pieranch.org).\n\n**Full Belly Farm, Guinda, California** Organic 'Sonora' wheat flour and wheat berries sold at various Bay Area farmers' markets and some grocery stores. $3 per 1\u00bd-pound bag (www.fullbellyfarm.com).\n\n**Eatwell Farm, near Dixon, California** Organic 'Expresso' hard red wheat flour, as well as flour milled from several other varieties. Sold at San Francisco's Ferry Plaza Farmers Market on Saturdays. $1 per 1-pound bag wheat berries, $5 per 3-pound bag flour (www.eatwell.com).\n\n**Community Grains, Oakland, California** Oliveto restaurant co-owner Bob Klein has just launched a new business that sells freshly milled, locally grown wheat flours, as well as super-flavorful polentas, at specialty stores in the Bay Area. \"This is not health nut stuff,\" says Klein. \"This is stepping into full flavor.\" The Web site, www.communitygrains.com, offers terrific in-depth information about wheat.\n\nWheat seeds\n\n**Whole Grain Connection, Mountain View, California** Monica Spiller's Web site includes a history of wheat growing in America, recipes for bread made with barm (sourdough starter), and a report on field trials of various old wheat strains, plus a plan for how to bring them back into production. $1.25 per pound for seeds, plus packaging and shipping (www.sustainablegrains.org or 650\/938-2865).\n\n**Wheat berries from Pie Ranch, Full Belly, and Eatwell Farm** (see above, under whole-wheat flour) can be planted as seed. Wheat berries from the grocery store may not be viable, but theoretically could sprout, too.\n\nBooks\n\n_Homegrown Whole Grains_ , by Sara Pitzer (Storey Publishing, 2009). A charming, restfully designed, yet information-packed guide to growing your own, with helpful illustrations.\n\n_Professional Baking_ , by Wayne Gisslen (John Wiley & Sons, fifth edition, 2008). Although intended primarily for culinary-school students, this is an invaluable book for home bakers, too, and answers questions you didn't know you had.\n\n_Small-Scale Grain Raising_ , by Gene Logsdon (Chelsea Green Publishing Company, 2009). First printed in 1977, this book explains how to grow, thresh, winnow, grind, and store all kinds of grains at home, including wheat.\n\n_'Sonora' wheat_\n\n# WINTER\n\n**_The One-Block Winter Feast_**\n\nArugula and Red Butterhead Lettuce Salad with Tangerines and Hard-Cooked Eggs\n\nWinter Vegetable Chowder\n\nWheat Berry Ciabatta with Homemade Butter\n\nTangerine Honey Flan\n\nChardonnay, Syrah, and Mead\n\nBright, chilly weather with an occasional frost, followed by a few months of rain\u2014this is winter in the Bay Area. Even though it's nothing like the long, hunker-down, snow-blanketed hiatus of the Midwest or the Northeast, it's still a slow time that made us take stock of where we were and what we had.\n\nOur wines sat quietly aging in their carboys. The olive oil, stored in small dark bottles, was a little over a year old and starting to lose some of its oomph. But we had cases and cases of it, enough to last several more months if we kept it cool and dark. The vinegar thrived, despite our forgetting to feed it several times. Our hens were doing fine, laying eggs regularly, and they liked the heat lamp we'd turn on for them whenever the temperature hit freezing. They'd hang out in the henhouse all morning until the sun could warm up the yard. Their favorite new treat was maggoty olives. They hailed down from our trees daily, tiny weights on the collective conscience of Team Olive\u2014which hadn't yet found a practical and affordable solution for dealing with the infestation of olive fruit flies. But at least the chickens were happy. They scarfed up those black, wormy nuggets like party crashers at a caviar bar.\n\nWe were especially reluctant to treat our trees because we didn't know whether the spray would harm our bees. And the poor bees! As it was, they were battling a seemingly unending stream of pests and diseases. The latest two were the small hive beetle, which eats brood, bees, and honey; and nosema, which is sort of like bee flu. We dealt with the first by using traps and pouring a concrete slab under the hives so the beetles couldn't pupate in the ground beneath, and the second miraculously seemed to go away on its own.\n\nMites were a more persistent problem. We had already tried dusting the bees with powdered sugar (when they clean themselves, they knock off the mites) and using Apiguard, a mite-killing gel that the bees track all through the hive. Now it was time for pads soaked with formic acid, which gives off awful, powerful fumes that make your throat burn and your eyes water. It was a good thing that the women of Team Bee were utterly smitten with the hives\u2014with their uncanny collective intelligence, their hardworking drive, and their sheer power, which Margaret would sometimes describe as \"a tiger in a box.\" They would do whatever they could to help those hives survive.\n\nWhen we had finished surveying all of our projects, we felt pretty happy about them\u2014and the sheer fact that we'd managed to keep them going, given how busy our lives were. They provided great holiday presents for our friends and families: jars of honey, fragrant hand salve (from our own beeswax, which Team Bee had carefully melted in a homemade solar wax melter), and bottles of olive oil and vinegar and beer. Cheeses weren't added to the list of gifts, because they took so long and we had made so few. We labeled all our jars and bottles (see How to Make Your Own Labels) and it felt good to be giving gifts that represented more than what they were\u2014but were fine just _as_ they were, too.\n\n_Pouring beeswax hand salve_\n\nOur winter menu garden had been planned and mostly planted in the fall. This time around, we were going for a cozy supper of big food: A giant, bright winter salad of frilly escarole and crisp endive, with poached eggs and croutons from homemade bread (we had a bit of wheat carefully set aside for flour). Then we'd have a huge stew or a creamy hot chowder of winter greens and other vegetables, and a loaf of bread on the side with homemade butter. For dessert, Stephanie would make one of her specialties, a tender olive oil and orange cake. We told ourselves (blithely, as it turned out) that we would adapt the cake to use the ingredients on hand. We wouldn't make beer this time around, since at long last we would have our Syrah as well as Chardonnay\u2014and, in a surprise development, mead.\n\n_Our partly fermented mead, with airlock inserted_\n\nSooner or later, every beekeeper considers making mead, or honey wine. Team Bee had visited a local meadery in the fall and been completely inspired by what they had sipped. This was no cloying, supersweet Renaissance faire beverage. It was crisp and simple, with \"the luscious flavor of the honey shining through,\" Brianne said. Mead, as explained to them by the meadery's owner, seemed easy to make, too: Mix water, honey, and yeast. So that's exactly what the newly formed Team Mead did\u2014using some of the same equipment we had used for beer\u2014and now the result was fermenting in a glass carboy.\n\nThe winter garden was shaping up to be a beauty. It was laid out in a U shape facing the path, compact and tidy\u2014unlike our fall garden, which had no particular design other than to give the plants optimum sun exposure. The two arms of the U were meant to mirror one another, each planted with the same pleasing symmetry of greens, with brassicas like cauliflower, cabbage, and weird, beautiful broccoli romanesco also worked into the pattern. The garden department threw in some red butterhead lettuce, too (in our mild winter, it would do fine). Our hardy thyme and oregano beds neatly capped each end of the U.\n\nAmong the garden's many admirers were snails. As in just about every California garden, we had tons of them, thanks to some well-meaning immigrant in the 1800s who brought these edible gastropods (known as _petit gris_ in France) to the West thinking they could form a great new food supply\u2014and instead introduced a major agricultural pest. Whenever Johanna found them snacking on our pretty garden, she would mercilessly crunch them with her boot or feed them to the chickens, who loved them.\n\nThen one day, it occurred to her that they could fulfill their original nineteenth-century purpose, right here on our one-block table. She walked into the test kitchen and persuaded Amy, who is a lot less squeamish than your average cook (her father was a hunter), to form Team Escargot.\n\n_Tasting escargots_\n\nWithin a week, they had purged, fattened, and even _named_ about a dozen snails. Amy, not having had time to research snail cooking\u2014do you pound them like abalone and flash-cook? or braise like octopus?\u2014relied on French-style panache and threw Shelli, Chelle, Shelby, and the rest of the clan into a saut\u00e9 pan with our wine, butter, and herbs. We tasted bravely and regretted it. Turns out it takes more than a quick saut\u00e9 to remove mucus from a snail.\n\nBut Amy always rises to a challenge. Boiling, she read, would help deslime the snails. This produced lots of bilious green goo and some screaming from Amy, who refused to ever cook snails again: \"The French can have them!\" Several weeks later she got her mojo back, added a salting step to the process to reduce the slime, and made some incredibly tasty escargots. We did not add them to the menu, though\u2014making snails for the entire staff was too much to ask of any cook.\n\nSixteen months after we had picked our Syrah grapes, crushed and destemmed them, fretted over them as they cold-soaked, went through two separate fermentations, pressed them, and then figured out how to siphon off the lees, our wine was at last ready to be bottled. We had to make sure with a final taste test, and used a turkey baster to sip from our first carboy. The wine was more than just okay\u2014it was a flood of dark, delicious fruit. Cheers broke out all around. \"We did it!\" said Sara, and danced a little jig. She swirled, swished, and sipped, and got to analyze her own wine: brambles (good in Syrah) and blackberries, plus leafy tobacco, mocha, and black pepper.\n\n_Sampling the Syrah_\n\nWe tasted each of our seven carboys and were amazed by how different they were\u2014proof, as Sara said, that wine is alive, and how tiny molecular shifts along the way can affect the \"yum factor.\" The star carboys were definitely 1 and 5. Carboy 2 was good but with sweeter-seeming fruit, said Sara, and molasses character. Number 3 would need some age to smooth out its rough spots.\n\nThen we got down to work. The entire team pitched in, including Dan Brenzel, our wisecracking advisor and perpetual provider of home wine-making equipment. Using the same techniques from a couple of months earlier, when we bottled the Chardonnay, we siphoned the inky wine into the 135 bottles we had saved up and sanitized, and then used Dan's nifty floor corker to drive each cork home. We ended the day with purple hands, purple teeth, and high spirits.\n\nWe hadn't exactly been hibernating in winter, what with most of our projects needing tending of one kind or another. Then we really started to bustle: We launched Team Cow.\n\nFor months, I'd been dreaming of finding our own cow. We'd been \"importing\" excellent store-bought organic milk and cream from Straus Family Creamery, north of San Francisco, but getting milk from our own cow\u2014preferably a Jersey, which produces the richest milk in cowdom\u2014would take our dairy products to a whole new level. Imagine the butter, ice cream, and cheese we could make from her superfresh, unadulterated, unhomogenized milk! Plus, we'd get to know an actual cow, an animal once common in backyards and farms across the country. Now they're familiar to most of us only as a picture on a milk carton.\n\nWe already knew we couldn't keep a cow at Sunset (even though our city regulations allow it, amazingly enough). The twice-daily milkings weren't the main issue. In order to lactate, cows must get pregnant and give birth, and the impregnation part\u2014and then the delivery-of-calf part, and the dealing with calf (possibly many calves) afterward\u2014seemed a little intense for people with desk jobs. Sharing a cow with a knowledgeable owner seemed like the best solution.\n\nWe looked and looked for a cow. Once there were small dairies all over the San Francisco Peninsula, but those are gone now. Finally we found her in Pescadero, on the coast to the south, at Pie Ranch Farm. She was an adorable Jersey named Adelaide, with a thick, glossy coat and sparkling brown eyes. Her owners, Jered and Nancy Lawson, had already set up a share arrangement, and we would be one of several owners.\n\nUnfortunately, Adelaide got a staph infection in her udder right before our first visit. The Lawsons decided she had to dry up, which would help her heal, and then go through a whole new pregnancy (about nine months, just like humans) before her milk would be available. So Team Cow, with much regret, had to press the pause button.\n\nAt least our bees were on the mend. Both hives had lots of baby bees, enough to ensure that they would survive their mite infestation. Also, because mites prefer to lay their eggs in the larger drone cells rather than worker bee cells, we got in the habit of removing the drone frames every few weeks and freezing them. This killed the drone larvae along with the mites, but drones are relatively expendable, and, as Margaret remarked, \"beekeeping is not all sweetness and honey.\"\n\nThe garden had lived up to its promise and was a gorgeous, dense display of textures and colors\u2014reds, burgundies, jade, lime green. The snowy heads of cauliflower were like pom-poms. Broccoli rabe towered at the back, with sprays of spicy yellow blossoms. The broccoli romanesco could mesmerize you for minutes with its perfectly symmetrical, spiraling turrets, like something out of an Escher engraving. And with their curvy, frilly leaves, the chard and kale and lettuce looked like the feathers on Vegas showgirls.\n\nIt was almost heartbreaking to have to harvest it for our feast. We managed to stop thinking about how barren and forlorn it looked once we had carried our huge, luscious basketfuls into the kitchen and started cooking. What a pleasure it is to cook with vegetables that still have life-force in them! Everything was crisp, tight, and juicy.\n\nWe were missing the escarole (a seed shortage prevented us from ever planting it) and the endive (it never germinated), so we changed our salad to use what we did have: red butterhead lettuce and arugula. We hard-cooked the eggs instead of poaching them, because liquidy poached yolks, great on crisp endive and escarole, would have turned the tender lettuces into a sticky clump.\n\nFor bread to go with our chowder, we ground some of our wheat berries, first coarsely (and laboriously) in a sausage grinder, and then finely in tiny batches in an electric coffee grinder. (We probably should have gotten a home grain mill, but we had so little to grind.) We turned the flour into a couple of rough, rustic loaves of ciabatta; we could just imagine butter pooling and melting in its nooks and crannies. So Elaine made butter, using nothing but cream and a food processor.\n\n_We ground flour with a sausage grinder_\n\nI had hoped we'd be able to preserve some onions and a few garlic bulbs from the summer\u2014they are crucial kitchen workhorses, and we really needed them for this winter menu. But they had barely lasted through fall; we hadn't grown good storage varieties. We caved and added both to our winter \"imports\" list, telling ourselves we would grow storage types next year. Surprisingly, though, we did have a few remaining Yukon Gold potatoes, which we had experimentally stored in an old fridge. We had just enough for our winter chowder.\n\nGiven the chilly weather, we ate our winter feast indoors, at a couple of long tables in the kitchen. We poured our deep, dark Syrah along with our Chardonnay, and I think it's the first time I've ever made a toast _to_ wine as well as with it.\n\nI had worried that this dinner would somehow be lumpy and bland. But it was as vigorous and pretty as the garden it came from. The salad had bits of juicy tangerine and yellow egg scattered among its red and green leaves. We sprinkled the chowder with yellow broccoli rabe and blue rosemary flowers, and it looked and tasted great with hunks of the dark, shaggy ciabatta. Dessert actually _had_ started out lumpy and bland\u2014it's why I'd been worried. Several days earlier, when Stephanie tried to make her olive oil cake with whole-wheat flour and honey instead of the usual refined white flour, baking powder, and sugar, it cooked up into a dark, heavy mass. She ditched it and instead made a lovely big flan for the dinner, glistening with caramelized honey and bits of tangerine peel. It looked like a golden moon.\n\nI'd also wondered whether our winter menu would be substantial enough. When it's cold, you need some rib-sticking food. Thanks, though, to the cream in the chowder, the butter, the whole-grain bread, and all those hearty winter vegetables\u2014plus eggs, milk, and honey in the flan\u2014it had sturdiness as well as freshness.\n\nWe ended dinner with another toast, this time to (and with) our mead, poured into tiny frosted shot glasses. It was a baby, really not meant to be tasted until it had aged for at least a year, better two. Even so, it was surprisingly drinkable\u2014not \"like cough syrup,\" Brianne's biggest fear. Like so much else in our one-block project, it was living and evolving, and every taste in the months to come would teach us more.\n\nHOW TO MAKE YOUR OWN LABELS\n\nWe labeled nearly all of the foods we made, because it was fun and because we gave away lots of food as gifts, especially around the holidays. It's not difficult to produce your own labels. Here's how:\n\n**Create the design** Use a design\/graphics program like Adobe Illustrator, or work with your local home wine-making\/home-brew store to create a label on their software.\n\n**Print the labels** We laser-printed each design onto white Avery 5265 full-sheet labels (available at office-supply stores and www.officemax.com).\n\n**Cut the labels** On a self-healing mat (available at craft stores and www.dickblick.com), line up a metal ruler along the label's edge and use a craft knife (available at craft stores and www.dickblick.com) to cut out each label. (We also tried trimming labels with a paper cutter, but sticky bits gummed up the blade.)\n\n**Stick them on** Make sure the surface of the container to be labeled is clean and dry. Peel back one corner of the label and use that sticky spot to help you position the label on the surface. Then reach under the label and gently remove the backing with one hand; with the other, smooth down the label as you peel off the backing. Use a paper towel to do the smoothing, to keep the label clean. With wine, wait at least 48 hours after you bottle to start labeling; you want to give your wine some time to adapt to the bottle.\n\n# THE WINTER GARDEN\n\n10 feet\n\n**WINTER GARDEN PLAN**\n\n**1. Thyme**\n\n**2. Arugula**\n\n**3. Lettuce**\n\n**4. Savoy cabbage**\n\n**5. Mustard**\n\n**6. Broccoli rabe**\n\n**7. Broccoli romanesco**\n\n**8. Cauliflower**\n\n**9. Swiss chard**\n\n**10. Kale, Tuscan**\n\n**11. Kale, curly-leafed**\n\n**12. Oregano**\n\n**13. Marjoram**\n\n**14. Rosemary**\n\n**15. Lemon tree**\n\n**16. Tangerine tree**\n\nThe leafy greens and succulent cruciferous vegetables we raised for our winter menu grow best when air temperatures are cool. Yet they thrive in sunny locations (at least 6 hours of sun per day). Arugula is easy to grow from seeds, while other crops, including lettuce, yield plentifully from nursery plants. (For seed-starting and planting tips, see Essential Gardening Guidelines.) If you can, avoid planting in any \"frost pockets\"\u2014low areas that can get frost earlier than other parts of your garden.\n\n### Arugula\n\nArugula or rocket ( _roquette_ in French) has tender, deep green leaves that add a peppery bite to salads. Crops come fast: You can pick baby leaves in as little as 3 weeks. To prolong the harvest, sow in succession every 3 weeks.\n\n**Best Site** An open, sunny spot and well-drained soil.\n\n**Days to Harvest** 35 days from seed.\n\n**Planting and Care** Sow seeds during cool weather in ground that has been raked or hoed clean of weeds and clods. You can either broadcast (scatter) the seeds or sow thinly in rows, and cover lightly with \u00bc inch of soil. Water the plot lightly and often to bring up the seedlings, then regularly (once a week or so) as they grow. To speed them up, apply a high-nitrogen fertilizer after the first unformed leaves appear. Thin seedlings to about 6 inches apart (the thinnings are great in salads). Arugula also thrives in pots at least 6 inches deep and 14 inches wide.\n\n**How to Harvest** Pick the leaves as needed once plants are larger.\n\n**Seed Source** Burpee, www.burpee.com.\n\n_Arugula_\n\n### Broccoli Rabe\n\nBroccoli rabe, also called broccoli raab or rapini, is a choice cool-season crop to grow alongside cabbage and carrots. It resembles broccoli, but instead of producing one giant head, it grows many longer, smaller budding stalks that you can selectively harvest all winter and spring. The plant grows 12 to 15 inches tall, and usually resprouts from the stalks until hot weather settles in.\n\n**Best Site** Full sun, though it can tolerate some shade, and well-drained soil. Prefers cool weather.\n\n**Days to Harvest** 60 days from seeds.\n\n**Planting and Care** Plant in well-composted soil in early fall, sowing seeds 2 inches apart in rows 6 to 8 inches apart. When seedlings are 3 to 4 inches tall, thin them to 6 inches apart. Keep the soil evenly moist.\n\n**How to Harvest** Clip stalks when buds appear, or allow a few of their yellow flowers to bloom for splashes of garden color (the blooms are edible, too). Plants will keep shooting out new buds for an extended yield.\n\n**Seed Sources** (search under \"broccoli raab\") Johnny's Selected Seeds, www.johnnyseeds.com, and John Scheepers Kitchen Garden Seeds, www.kitchengardenseeds.com.\n\n### Broccoli Romanesco 'Veronica'\n\nThis variety's round heads, made up of chartreuse florets that spiral into little peaks, make it look like cauliflower from another planet. It has a texture and flavor similar to mild, sweet cauliflower, too. The florets turn a slightly deeper, more olive green when cooked, and can also be eaten raw as whole florets (good with dips) or sliced in salads.\n\n**Best Site** Grow it in loamy soil that's been well amended with compost, and make sure it gets full sun (though it can tolerate some shade). Broccoli Romanesco prefers cool weather.\n\n**Days to Harvest** 77 days from seed.\n\n**Planting and Care** In mild climates, plant in late summer or early fall for a winter harvest. (Plants develop the best heads when they mature in cool weather; in heat, the flower buds open prematurely.) Start seeds in flats or small pots indoors 6 weeks before transplanting into the garden. Before planting in the garden, work some well-aged compost and a granular complete fertilizer into the soil. Space the plants 18 to 24 inches apart, keep the soil evenly moist, and cover the seedlings with floating row covers (available at nurseries) until they're big and leafy enough to withstand cabbage worms.\n\n**How to Harvest** For the best quality, harvest the heads when they are firm and tight. As they get older, they open up and lose their crunch.\n\n**Seed Source** John Scheepers Kitchen Garden Seeds, www.kitchengardenseeds.com.\n\n_Broccoli Romanesco 'Veronica'_\n\n### Cabbage, Savoy 'Alcosa'\n\nThis curly-leaved Savoy cabbage forms tight heads that are ideal for closely spaced planting. Also, its leaves are dense, crinkled, and colorful\u2014blue-green outside, lighter green to creamy white inside\u2014and pretty in garden beds with other greens. We love this cabbage in stir-fries, because it retains its crunch, and in salads. It has an exceptionally mild, sweet flavor.\n\n**Best Site** Full sun in mild coastal areas, part shade in hot-summer interiors, and loose, fast-draining soil enriched with compost.\n\n**Days to Harvest** 72 days from seed.\n\n**Planting and Care** Time your planting so heads will mature in cooler weather before or after hot summer months. We started our plants from seeds, sowing them in flats in our greenhouse 4 weeks before planting the seedlings outdoors in garden beds in September. Space the seedlings 24 to 30 inches apart. Water the plants often enough to ensure they never wilt, and give them frequent light feedings of dilute liquid fish emulsion (follow package directions).\n\n**How to Harvest** When the heads feel firm and look well formed, cut them off at the base with pruners or loppers. The plants can take light frost, which sweetens their flavor, but heavy freezes can knock them down.\n\n**Seed Source** Johnny's Selected Seeds, www.johnnyseeds.com.\n\n_'Alcosa' Savoy cabbage_\n\n### Cauliflower 'Cassius'\n\nWe grew this variety because it produces round, creamy white heads 7 to 8 inches across\u2014the perfect size for a compact kitchen garden. Some varieties need to have their leaves tied together over the heads to keep the sun from discoloring them, but 'Cassius' is a self-blanching variety, so it didn't need all that fussing. Its flavor is mellow and sweet.\n\n**Best Site** Full sun and well-drained soil.\n\n**Days to Harvest** 65 to 75 days from seed.\n\n**Planting and Care** Start seeds in flats or small pots indoors 4 to 6 weeks before transplanting into a garden bed that has been well amended with compost. Space seedlings 18 to 24 inches apart. Keep soil moist until seedlings get growing, then water deeply and regularly, especially in dry weather. Feed once or twice with complete fertilizer before heads start to form.\n\n**How to Harvest** Use a sharp knife to slice the entire head from the mature stem. It's a bit of a dance to know when cauliflower is ready to harvest. The head should be firm and fully formed; harvest it before the curd (the white part) separates into flowers (check it regularly). Make the cut beneath the top set of leaves.\n\n**Seed Source** Territorial Seed Company, www.territorialseed.com.\n\n_'Cassius' cauliflower_\n\n### Kale, Curly-Leafed and Tuscan\n\nRich and nutrient-packed (high in vitamins A and C), kale is indispensable for winter soups and stews. We grew two varieties: 'Winterbor', whose curly-edged, blue-green leaves form low rosettes that stretch out to 2 feet, and 'Nero di Toscana', a Tuscan heirloom kale whose strappy, dark green leaves form statuesque, upright plants that reach 3 feet. (Tuscan kale is also often labeled lacinato or dinosaur kale.) Both kale types are ornamental and cold-hardy and add texture and color to the winter garden. Light frost sweetens their flavors.\n\n**Best Site** Full sun or light shade in mild climates, part shade in hotter inland areas.\n\n**Days to Harvest** 55 to 65 days for 'Winterbor' and 60 days for 'Nero di Toscana' from seed.\n\n**Planting and Care** Plant in September for a winter crop. Till the soil and mix in compost or aged manure before planting. Sow seeds in place and thin seedlings to 24 to 36 inches apart. Or, set out nursery seedlings at the same spacing. Keep the soil moist during the growing season.\n\n**How to Harvest** Remove leaves from the outside of the clusters as needed, or harvest the entire plant by pulling it up and cutting off the base.\n\n**Seed Source** John Scheepers Kitchen Garden Seeds, www.kitchengardenseeds.com.\n\n### Lemon 'Eureka'\n\nSee the Summer Garden.\n\n### Lettuce, Red Butterhead\n\nLettuce is one of the most satisfying vegetables you can grow: It's fast and easy from seed as long as you plant it at the right time of year (fall or spring in mild climates). We chose 'Marvel of Four Seasons' ('Merveille des Quatre Saisons'), a French heirloom variety whose loosely cupped green leaves are tinged with shades of ruby, rose-pink, and bronze.\n\n**Best Site** A sunny spot in cool areas, part shade where it is hot, and loose, well-drained soil.\n\n**Days to Harvest** 50 to 60 days from seed.\n\n**Planting and Care** Sow seeds about 4 inches apart in well-prepared soil, then barely cover the seeds with about \u00bc inch of soil. Thin seedlings to 12 inches apart. Spray the seedbed with water regularly until the seeds germinate, then water regularly to keep the soil moist. Feed once or twice during the growing season with dilute liquid fish emulsion.\n\n**How to Harvest** Pick leaves any time from the seedling (thinning) size on. Harvest after loose heads form.\n\n**Seed Sources** The Cook's Garden, www.cooksgarden.com, and John Scheepers Kitchen Garden Seeds, www.kitchengardenseeds.com.\n\n_Red Butterhead lettuce_\n\n### Marjoram, Sweet\n\nSee the Summer Garden.\n\n### Mushrooms\n\nSee How to Grow Mushrooms.\n\n### Mustard 'Green Wave'\n\nMustard comes in different colors and textures, but we like 'Green Wave' for its lime-green hue, ruffled edges, and pungent, peppery flavor. The plant looked so good in the garden that we neglected to harvest it and ended up with very big (and very spicy) leaves. Luckily, they were perfect when braised with other winter greens (see recipe).\n\n**Best Site** Full sun and well-drained soil.\n\n**Days to Harvest** 50 days from seed.\n\n**Planting and Care** We planted mustard in October for our winter dinner. You can also plant in early spring or late summer; the key is to avoid exposing the plants to too much heat, which causes premature bolting (setting of seed). Start seeds indoors or plant them directly in well-fertilized soil \u00bd inch deep, eventually thinning the seedlings to 6 to 12 inches apart. Water regularly, and feed with a complete fertilizer every 2 to 3 weeks.\n\n**How to Harvest** Pick outer leaves as needed, or let the plant reach about 6 inches tall, then snip it down to an inch above the soil (new leaves can shoot up again from the base).\n\n**Seed Source** Territorial Seed Company, www.territorialseed.com.\n\n_'Green Wave' mustard_\n\n### Oregano, Italian\n\nSee the Summer Garden.\n\n### Rosemary 'Tuscan Blue'\n\nSee the Summer Garden.\n\n### Swiss Chard\n\nSee the Fall Garden.\n\n### Tangerine 'Dancy'\n\n'Dancy' is the standard variety that appears in markets before Christmas: smallish and seedy, but with excellent, complex flavor. We picked from a tree that has been at Sunset for decades. Like all tangerines, it's a member of the mandarin family.\n\n**Best Site** Full sun and deep, fast-draining soil.\n\n**Days to Harvest** Plants start bearing fruits in early winter.\n\n**Planting and Care** Plant after spring frost but before summer heat. Dig in a 4- to 6-inch layer of compost before planting, then dig a hole twice as wide as the root-ball and mix a granular, controlled-release general-purpose fertilizer into the backfill. Or, plant in large containers (at least 18 inches in diameter) in fast-draining potting mix. Mulch with compost. Water established trees every other week, soaking the soil thoroughly. Feed several times during the growing season with a fertilizer formulated for citrus (follow package directions).\n\n**How to Harvest** Citrus fruits ripen only on the tree. Pluck one and taste for ripeness.\n\n**Plant Sources** Shop nurseries, or online at Four Winds Growers, www.fourwindsgrowers.com.\n\n### Thyme, French\n\nSee the Summer Garden.\n\n_\u2014Kathleen N. Brenzel and Johanna Silver_\n\nTHE \"IMPORTS\"\n\n**We went outside our garden for these local foods:** Milk, whole and unhomogenized (for butter and cheese); seawater (for salt); active dry yeast (for bread); sweet mead liquid yeast (for mead); whole-wheat flour and wheat berries ('Sonora' and 'Expresso'), garlic (ours ran out); 'Spanish White' onions (ditto); walnuts\n\n# THE WINTER PROJECTS\n\n##    \nHOW TO MAKE  \nSALT\n\nOf all the projects we attempted as part of our One-Block feasts, this may have been the most far-fetched. We had read of other people's efforts, most memorably Michael Pollan's in _The Omnivore's Dilemma_ , in which he scavenges water from trash-strewn wetlands and evaporates it on his stove top into brown salt that, he writes, \"tasted so metallic and so much like chemicals that it actually made me gag.\"\n\nBut we persisted because we knew we had to have seasoning for our dinner, and figured\u2014what with the San Francisco Bay to one side of us (the same bay that Pollan harvested, actually) and the Pacific Ocean on the other, we had some water to choose from. Also, the other raw materials that we had \"imported\" for our feast\u2014grapes for the wine, olives for the oil, and milk for the cheese\u2014at least were transformed from their natural state by our own hands. It would be copping out to just go buy salt.\n\nThe process proved surprisingly easy, and the yield was much higher than we had expected. And our salt looked pretty (pure white), smelled fresh, tasted exactly like the ocean, and made a fine seasoning for our feast. As to whether it was safe to consume or would behave like normal salt in cooking, we weren't sure. We were just happy that we had, in fact, made our own salt from local seawater.\n\n###    \nPACIFIC SEA SALT\n\nWe hauled drums filled with Pacific seawater to our Menlo Park offices and reduced several gallons to salt.\n\n#### WHAT TO USE\n\n**_Food-grade plastic drums_** We had a few clean 5-gallon food-grade plastic drums left over from making olive oil, so we used them to transport the brine. Any thoroughly washed bucket will do. Drums cost about $15 each, plus $20 handling fee if your order is less than $100, at The Olive Oil Source, www.oliveoilsource.com or 805\/688-1014.\n\n**_Coffee filters or a triple layer of cheesecloth_** For filtering the water. About $4.50 for 2 square yards at a hardware store or cookware store.\n\n**_Pyrex liquid-measuring cups_** To measure the hot brine.\n\n**_Stockpot_** For boiling the brine. From $60 online or at a cookware shop.\n\n**_Two Pyrex 9 by 13 inch baking dishes_** For baking and evaporating the brine. About $7 each at a cookware shop.\n\n**_Rigid metal spatula_** Once the salt crystals have formed, you have to scrape them off the pan.\n\n#### HOW TO DO IT\n\n**1. Collect seawater** After thinking about the nearest cleanest portion of the ocean, we settled on Bean Hollow State Beach near Pescadero, since no streams (which can carry toxins and other runoff) flow into the ocean there. We collected 40 gallons of the chilly Pacific in our plastic drums. Another time, we collected seawater near Fort Bragg, a small coastal town north of San Francisco. Both batches were processed the same way.\n\n**2. Filter** Back at the office, we poured a test batch of water through a coffee filter (or layered cheesecloth) into the stockpot.\n\n_Our Pacific sea salt_\n\n**3. Boil** We brought the filtered water to a rolling boil and held it there for 20 minutes to kill any noxious bacteria. We were aware that this would have no effect on other toxins or heavy metals. In fact, if they were present, the cooking might concentrate them.\n\n**4. Filter again** To remove any remaining particles, we filtered the water again, this time into heatproof measuring cups.\n\n**5. Bake** The first time we made salt, we used aluminum baking sheets, which gave us a large yield\u2014but the salt was grayish and tasted like aluminum. We then switched to Pyrex baking dishes and got salt as white as snow with no funny taste. Here's how: We poured the filtered water in 3-cup batches into the pans, then baked it in a 350\u00b0F oven for about 2\u00bd hours, checking often toward the end to make sure we didn't burn the crystals as they formed.\n\n_Sun evaporation as an alternative._ Feeling sheepish about the fossil fuel consumed by baking, we also tried putting the brine-filled baking sheets outside to evaporate in the sun. This didn't work very well, owing to spotty weather and to lots of leaves and other debris blowing into the exposed salt. We tried again, covering the pans with cheesecloth, and after several days, we had salt. The yield was much lower than what we got with the baking method, however.\n\n_Or stovetop boiling._ Another early attempt involved boiling filtered seawater until the liquid completely evaporated and salt lined the inside and bottom of the pan. This ended with a steamy kitchen and burning hot crystals of salt flying out of the pot like popcorn and hitting us on the arms. Plus, boiling down all that water and then running the fan to cool off the kitchen wasn't energy efficient, so baking became the best choice.\n\n**6. Scrape** At the end of baking, the crystals rimmed the pan, and we scraped them off with the metal spatula. The yield varied depending on the source of the brine: We got about 2 tablespoons good-tasting, snowy white salt from 3 cups brine collected at Pescadero. The same amount of brine from Fort Bragg yielded between \u2153 and \u00bd cup.\n\n_Our first stab at making salt_\n\n**7. Taste** We sent a sample of our Fort Bragg salt to salt expert Mark Bitterman, owner of the specialty shop The Meadow in Portland, Oregon. Here's what he said: \"[It] has a lovely opaque, cr\u00e8me fra\u00eeche color that comes from an abundance of the sea's trace minerals. The crystals are marvelously diverse in form, ranging from finely fringed micro-grains to layers of laminated flakes. And the flavor: a roiling oceanic assault on the senses\u2014only nice, like being caught in a wave of sun-warmed water, briny and bitter and butter-sweet.\" He further pointed out that, unlike commercial salt manufacturers, we had reduced all the available salts in our brine, each of which has a different salinity and flavor. The big guys tend to go for sodium chloride only, which provides most of the salt and precipitates first, because it is more efficient. \"What you guys have done is grab it all, including the calcium and magnesium salts. You've got a big dynamo of saltiness. It has everything in it.\" This, it seems, is the hallmark of artisanal salts. We are artisans!\n\n**8. Test** We spent some fruitless time contacting various clean-water agencies to ask about getting our water samples tested. One officer, with a branch of the Environmental Protection Agency, said to us, \"Why would anyone want to drink salt water?\" He had a point. So we moved on to the salt itself.\n\nWe then considered sending a salt sample to a food lab, but a heavy-metals screening would have cost $275 for each possible contaminant (and there are many), and a test for pesticides at least $100. Kind of pricey for seasoning, we thought. As Bitterman pointed out, salt is extremely dilute in the ocean\u2014and is not a bioaccumulator of toxins\u2014so the chances of it containing large amounts of bad stuff are probably not all that high.\n\nAt this point, we are happy to have produced salt that tastes good and, we hope, might be harmless.\n\n#### HELPFUL INFORMATION\n\nSalt is a fascinating subject. It has roughly fourteen thousand applications, from de-icing roads to food preservation, and was so valued in Roman times that soldiers were paid wages in it.\n\n##### BOOKS\n\n\u2022 _Neptune's Gift: A History of Common Salt_ , by Robert P. Multhauf (Johns Hopkins University Press, 1978).\n\n\u2022 _Salt: A World History_ , by Mark Kurlansky (Penguin Books, 2002).\n\n\u2022 _Salted: A Manifesto on the World's Most Essential Mineral, with Recipes_ , by Mark Bitterman (Ten Speed Press, 2010).\n\n##### OTHER RESOURCES\n\n\u2022 \"How to Make Salt from Sea-Water,\" by John Leconte (Governor and Council of South Carolina, 1862). An oddly engrossing pamphlet written at the request of the state's governor by a chemistry and physics professor at South Carolina College. At a time when the oceans were cleaner than they are today, Leconte spells out how to boil down seawater in 20-foot sheet-iron pans over wood fires. You'll find yourself engrossed in the details of sludging and soccage (crystallization). For the free download, visit www.docsouth.unc.edu\/imls\/lecontej\/leconte.html.\n\n\u2022 The Salt Institute, an association of salt producers, puts forth all sorts of information about salt on its Web site\u2014everything from stats on world production (China leads) to solution mining to nutrition (www.saltinstitute.org).\n\n_\u2014Amy Machnak_\n\n##    \nHOW TO MAKE  \nESCARGOTS  \n(FROM YOUR OWN GARDEN SNAILS)\n\nGrowing edibles inevitably means growing snails, especially because our garden is organic and hospitable to wildlife. We have a big snail population snacking on our carefully tended leafy vegetables. One day Johanna, our test-garden coordinator, suggested that we eat them. Eat them? As in\u2014 _escargots_?\n\nThat was definitely the French approach to dealing with snails in the garden. In fact, the French don't consider them a nuisance at all, but a reason to spend time with the family, walking with buckets through their yards\u2014even grassy fields\u2014in pursuit of their prize. They refer to it as \"hunting.\"\n\nBut were ours the type of snails that could be eaten? Maybe they produced toxins. And, most important, how would we make them taste like the French gourmet item served in overpriced restaurants?\n\nFirst, we experimented with saut\u00e9ing them, and discovered that preparing snails wasn't as simple as tossing them around in butter in a frying pan. They were, er, slimy. So we did some research and consulted many authorities on the cooking of France, including M. F. K. Fisher and Georgeanne Brennan, and reread our own article on escargots, published in 1988. All advised pretty much the same technique for cooking snails, with slight variations: Purge the snails (put them on a cornmeal diet in a closed dish for several days to clear their innards of any noxious stuff they may have been nibbling), boil them, extract them from their shells, and then use them in whatever recipe you have in mind.\n\nHaving now nabbed and cooked several batches of snails, let's just say that making escargots, though not difficult, is not for the squeamish. In fact, it reminds us of a sixth-grade science project. But the final outcome is delicious.\n\n###    \nESCARGOTS\n\nWe plucked our little shelled nuisances from the garden.\n\nHelix aspersa _in the one-block garden_\n\n#### WHAT TO USE\n\n**_About 12 live snails_** Choose the biggest snails possible. Unfortunately, we didn't seem to have any big snails. Ours were all small to medium.\n\n**_Bowl_** For collecting snails.\n\n**_Kitchen towel_** To drape over the snail-collecting bowl, to keep your catch from crawling out.\n\n**_Roasting pan_** A large, rimmed pan, such as a 10-by-14-inch roasting pan. About $10 at most cookware stores or online.\n\n**_About 2 cups yellow cornmeal_** For the purge. About $3 for 1 pound at most grocery stores.\n\n**_Several thyme sprigs_** For the purge.\n\n**_2 tablespoons fine sea salt_** For salting to remove slime, plus more for seasoning when saut\u00e9ing.\n\n**_Small bowl_** For salting the snails.\n\n**_Large saucepan_**\n\n**_\u00bd cup white wine_**\n\n**_Fresh herbs_**\n\n**_\u00bd small onion_**\n\n**_1 carrot_**\n\n**_2 celery stalks_**\n\n**_Garlic_**\n\n**_Flat-leaf parsley_**\n\n**_Lemon juice_**\n\n**_White wine_**\n\n**_Butter_**\n\n#### HOW TO DO IT\n\n**1. Choose wisely** Determine the species of snail you have; not all are edible. The most popular snail in France is the big, fat Burgundy snail, _Helix pomatia_. Ours are common brown garden snails, _Helix aspersa_ , also considered fine to eat. They were brought to this country, probably by the French, in the 1850s and are now a major pest in gardens everywhere. Make sure you hunt for snails in a garden free of pesticides. This isn't a problem for us, as Sunset has had a no-spray policy for decades.\n\n**2. Collect** We found the majority of our snails on the inside rims of potted plants, mostly edibles but occasionally flowers, too. They tend to like plants that give shade in the afternoon heat, such as agapanthus, but they also like to hang out in whichever plant they are eating, especially radicchio, lettuce, kale, cabbage, and strawberries. Just pull them off and put them in a basket or bowl, covered with a towel, for transporting to the kitchen.\n\n**3. Wash** Since snails crawl around in dirt, you need to make sure they are really clean. Rinse them in a colander under cold running water while rubbing them together.\n\n**4. Purge** This controlled feeding of the snails ensures that their digestive tracts are clean of impurities, toxins, and dirt. Put the snails into the roasting pan with a small dish of fresh water, about \u00bc cup yellow cornmeal poured into the corner of the pan, and a few sprigs of thyme from the garden. Cover the pan with a double layer of cheesecloth secured around the edge of the pan with rubber bands or tied string. Every other day, take the snails out of the pan, wash the pan, return the snails to the pan, and replace the cornmeal, the bowl of water, and the thyme. Do this for about 2 weeks. You'll know the purge is working when you see little yellow and green dried poop trails in the pan.\n\n**5. Salt** Put the snails in a small bowl and sprinkle them with the 2 tablespoons salt. This kills them and also helps to pull the slime from them. Cover the bowl with aluminum foil and poke a few holes in the foil. Let sit for about 12 hours.\n\nFill the bowl with water and then drain. Repeat as needed, rinsing off the salt and mucus each time, until the water runs clear.\n\n**6. Boil** In a large saucepan, bring 4 cups water and the \u00bd cup white wine to a boil. Add a few herb sprigs, such as thyme and flat-leaf parsley. Then chop the onion, carrot, and celery and add them to the pan. Simmer for 30 minutes to create a flavorful broth. Drop the snails into the broth and simmer them, skimming off any green, slimy foam that appears on the surface (this is the science-project part), until the meat releases easily from the shell when prodded with a toothpick, about 15 minutes (longer if you have larger snails). Lift the snails out with a slotted spoon, letting any slime and mucus drip off the spoon, and transfer to a plate. Then use the toothpick to pluck the snails from their shells. The standard instruction is to cut the tough \"foot\" off the snail. Our snails were so small that the cut was more like a scrape, but we did it anyway.\n\nYou can save the shells, dry them completely, and serve the cooked snails in them for a lovely presentation. Our shells were too small and not worth the trouble.\n\n**7. Cook: Garlic Butter Escargots** At this point, the snails can be used in any recipe you like (we've seen recipes for everything from baked escargots to spaghetti with escargots). We went the traditional French route and tossed them in a little garlic, parsley, lemon, white wine, and, of course, butter\u2014lots of melted butter. We ate them in a little dish with a side of warm crusty bread and were instantly transported to a sidewalk cafe on the Champs-\u00c9lys\u00e9es.\n\n#### HELPFUL INFORMATION\n\n##### BOOKS\n\n\u2022 _The Food and Flavors of Haute Provence_ , by Georgeanne Brennan (Chronicle Books, 1997).\n\n\u2022 \"Fifty Million Snails,\" from _Serve It Forth_ , by M. F. K. Fisher (Harper & Brothers, 1937; reprinted by North Point Press, 1989).\n\n##### OTHER RESOURCES\n\n\u2022 \"Snails as Food: Escargot\" by Robert Hawthorne (8-page pamphlet from the University of California, Davis; publication #2222; revised 1975; order through <http:\/\/anrcatalog.ucdavis.edu\/snails>)\n\n\u2022 \"The Snail Eaters,\" a reminiscence by Joyce Hanson on Leite's Culinaria, <http:\/\/leitesculinaria.com\/47375\/writings-snail-eaters.html>.\n\n\u2022 For general information about snails and about cooking them, visit www.en.wikipedia.org\/wiki\/Heliciculture#Turning_snails_into_escargots.\n\n_\u2014Amy Machnak_\n\n##    \nHOW TO MAKE  \nMead (HONEY WINE)\n\nMost people think of mead as an overly spiced and unbearably sweet beverage found only at Renaissance faires, but there is a whole world of delicious, easy-to-drink mead, both sweet and dry. Along with beer and wine, it is one of our most ancient drinks, thought to have been made as early as 7000 BCE. We tend to associate mead with the Vikings and the Celts of northern Europe, but it's been quaffed in many other places around the globe, including China, India, Greece, and Africa (it is still popular in Ethiopia, where it's called _tej_ ). In places where grapes could not be grown, mead offered a different way to make wine. Dozens of different styles of mead exist, from morat (made with mulberries) to cyser (honey and apple juice fermented together) to metheglin (a traditional Welsh brew involving herbs and spices).\n\nWe were inspired to have a crack at it ourselves after accompanying our local beekeepers' guild to Rabbit's Foot Meadery, in Sunnyvale, California. We chatted with the owner and acclaimed mead maker, Michael Faul, and realized that basic mead was not hard to make: Honey, water, and yeast are all it takes.\n\nAfter doing some research, we came up with a stream-lined recipe that is easy to re-create. We left out all spices and refrained from boiling the honey to preserve more of its character. This gentle treatment also retains more of the nutritional benefits of the raw honey.\n\nOur mead still needs a few years of aging to mellow fully, but even after eight months, it already tastes better to us than many of the professionally made meads we have tried. The flavor of the honey shines through, undisguised by fruit flavors or excessive spices. Follow Team Mead's blog (oneblockdiet.sunset.com\/team-mead) to watch our progress as we embark on future batches of dry mead and of melomel, mead flavored with fruit.\n\n###    \nTRUE MEAD\n\nSweet, crisp, unspiced honey wine, showcasing the floral and faint eucalyptus flavors of our one-block honey.\n\nMAKES 5 gallons\n\nTIME at least 6 months\n\n#### WHAT TO USE\n\nUnless otherwise noted, all of our materials came from More Beer in Los Altos, California (www.morebeer.com or 800\/600-0033).\n\n**_Saniclean_** For sanitizing all equipment, including containers, tubing, and bottles. Safe for septic systems and doesn't stain. Use 2 ounces for 5 gallons of water. About $13 for a 1-quart bottle from Williams Brewing Company, www.williamsbrewing.com.\n\n**_5-gallon plastic bucket_** For mixing up batches of Saniclean solution. About $7 at a hardware store.\n\n**_Water_** Room-temperature tap water is fine. If your water contains heavy chlorine or minerals, use bottled water (not distilled).\n\n**_6-gallon, food-grade plastic bucket with spigot_** Used for mixing the must and for fermenting and bottling. About $13. We actually used a 5-gallon glass carboy (see photo) since we happened to have a few on hand, but this is much cheaper.\n\n**_About 12 pounds honey_** From our own hives.\n\n**_Hydrometer_** A probe-like tool that measures the specific gravity (density) of the must relative to water. Dissolved sugars make up the density of the must, and sugars are what ferment into alcohol, so the beginning hydrometer measurement is a good indicator of your mead's potential alcohol level. (The higher the specific gravity, the higher the potential alcohol level.) The MT300 hydrometer has a potential alcohol scale included. About $6 from morebeer.com.\n\n**_Hydrometer jar_** A plastic tube with a flat base, sized to fit the hydrometer; for holding the must and mead samples. Some hydrometers come with their own jars. An 11\u00bd-inch jar is about $5.\n\n**_Yeast_** We used 1 vial of White Labs Sweet Mead liquid yeast. About $6.25.\n\n**_Stirring paddle or spoon_** Any sort of long-handled spoon or spatula will work.\n\n**_5-gallon glass carboy_** The main fermentation container. From $20 to $30 at a home wine-making or home-brewing shop.\n\n**_Rubber stopper_** You want a no. 7 stopper with a small hole (to accommodate the airlock) for sealing the opening at the top of the carboy. $1.50.\n\n**_Airlock_** You fill this small plastic cylinder with water and insert it into the rubber stopper on top of the carboy. This keeps bacteria and other airborne impurities from entering the mead, which is highly susceptible to contamination in its early stages. It also allows carbon dioxide to escape, rather than build up inside the carboy. About $1.25.\n\n**_Blanket or a large, dark cloth_** For wrapping the mead as it ferments. It blocks sunlight, which can stimulate the growth of bacteria.\n\n**_Racking cane_** This cane-shaped stiff plastic tube (\u00be inch in diameter by 21 inches long) attaches to the vinyl tubing (below) used to rack the must. $2.50.\n\n**_4-foot length of food-grade vinyl tubing_** Made of clear vinyl, and with a \u00be-inch interior diameter, the tubing is used to both rack the mead (siphon it off its sediment to another container) and bottle it. Cut it into a 3-foot section and a 1-foot section. About 30\u00a2 per foot online or at a home wine-making or plumbing-supply store.\n\n**_Bottles_** You'll need 48 pry-top bottles (screw-top bottles are harder to seal) in a dark green or brown glass (sunlight shining through clear glass can stimulate growth of bacteria). We got ours from friends, family, and colleagues. Free.\n\n**_Jet bottle washer_** This fits on any outdoor hose thread faucet, like those on an outdoor or garage sink. About $12.\n\n**_Bottle tree_** Invert your newly washed and sterilized bottles on this multipronged \"tree\" for easy drying of lots of bottles at once. We like the 81-bottle model from Williams Brewing Company (www.williamsbrewing.com or 800\/759-6025; it's the same one we used for our beer). About $30.\n\n**_Beer caps_** A pack of 50 pry-type caps costs about $1.50.\n\n**_Capper_** The only way to cap your bottles. We like the double-armed, easy-to-use Emily capper from Williams Brewing Company (above). About $14.\n\n#### HOW TO DO IT\n\n**1. Sterilize your equipment** Sterilize everything with a Saniclean solution in the 5-gallon plastic bucket. Rinse the equipment well and set on clean dish towels to dry.\n\n**2. Make the must (honey water)** Add 4\u00bd gallons tap water to the 6-gallon bucket. Add enough honey to give the must your target specific gravity, which predicts potential alcohol content. For example, for a potential alcohol content of 10.74 percent, the beginning specific gravity needs to be 1.08. To get a specific gravity reading, insert the hydrometer in its jar, drizzle in enough must from the bucket's spigot to make the hydrometer float, and read off the original specific gravity (OG). Make a note of the OG. If your hydrometer does not include a potential alcohol scale, you can use a handy online potential alcohol calculator (see Helpful Information) to figure out the corresponding specific gravity.\n\nThe thickness of the honey you use, not the amount, will determine the specific gravity of the must. We recommend first adding about 2 cups honey to the water, then stirring in more, \u00bd cup at a time, measuring the specific gravity with your hydrometer after each addition. Be sure there is enough liquid in the bucket to fill a 5-gallon carboy. If it is looking short, add more water and honey, paying close attention to the specific gravity.\n\n**3. Pitch the yeast** Add the entire vial of liquid yeast to the must and stir for 5 minutes.\n\n**4. Primary fermentation** Transfer the liquid from the bucket (through the spigot attached to the 1-foot tubing) into the clean glass carboy, leaving 2 inches of headspace at the top. Seal the carboy with the rubber stopper and insert the airlock into it. Store the carboy in a warm room (70\u00b0 to 80\u00b0F) and cover it with a blanket or dark cloth to keep out the light. Within a week, fermentation will begin.\n\nWhen the mead stops bubbling and the dead yeast particles have sunk to the bottom, it will be time to rack the mead. This typically takes between 1 and 3 months. (It took our mead 2\u00bd months.)\n\n**5. Secondary fermentation and racking** To rack your mead _,_ put the carboy on a counter and the plastic bucket with the spigot on the floor below. Remove the airlock and stopper from the carboy. Attach the 3-foot length of vinyl tubing to the racking cane and insert the tip of the cane into the carboy. Start gently sucking on the end of the tubing to get the mead flowing; when the mead begins to move down the tube, pinch the end, set it in the bucket, and let the mead flow down. Stop when you see that you're getting close to the sediment. Remove the siphoning tubing and racking cane. Clean and sterilize your carboy and siphon the mead back into the carboy (switch positions with the bucket). Replace the rubber stopper and airlock and store in a cool, dark place (60\u00ba to 70\u00baF) for 3 months. Re-rack the mead every 2 to 4 months until it is clear (this may take up to a year).\n\nWhen your mead is clear, it is ready to bottle. Take one more specific gravity reading. As the sugars in the must convert to alcohol, the mixture becomes less dense, and the reading goes down. The reading won't tell you what the alcohol content is. The point, rather, is to let you know whether fermentation has worked. Our specific gravity sank from 1.08 to .998, so we knew we had been successful.\n\n_Racking the mead_\n\n**6. Bottle the mead** Mix up a plastic bucketful of Saniclean, put a batch of bottles into the solution, and let them sit there for a few minutes. Make sure the solution fills the bottles completely. Then empty the solution out of the bottles back into the bucket and rinse out the bottles with hot water using your jet bottle washer. Be sure to rinse the lip and neck of the bottles, too, to wash off any excess sanitizer. Invert the bottles on the bottle tree to drain.\n\nTo bottle the mead, siphon the mead from the carboy into the 6-gallon bucket with the spigot. Set the bucket on a counter and attach the 1-foot length of vinyl tubing to the spigot. Put the other end of the tubing in the neck of your first empty bottle. Open the spigot and fill the bottle about halfway up the neck. Cap each bottle as soon as it's filled.\n\n**7. Cap the bottles** Put a cap on the bottle, place the capper over it, and push down on the capper's arms to seal.\n\n**8. Let the mead age** Your mead will be fine for drinking once you have bottled it, but we recommend you let it age in a cool, dark place (60\u00b0 to 70\u00b0F) for a minimum of 2 to 3 years. From everything we've heard, it only gets better.\n\n#### HELPFUL INFORMATION\n\n\u2022 For recipes, supplies, discussion boards, and information about the art of mead making, visit www.gotmead.com.\n\n\u2022 For a potential alcohol calculator, visit www.gotmead.com\/index.php?option=com_content&task=view&id=745&Itemid=16.\n\n\u2022 The Web site of the National Honey Board, www.honey.com, offers a handy backgrounder on mead, including recipes and a great resources guide, at www.honey.com\/images\/downloads\/makingmead.pdf#search='mead'.\n\n_\u2014Brianne McElhiney_\n\n_Arugula and Red Butterhead Lettuce Salad with Tangerines and Hard-Cooked Eggs_\n\n# THE WINTER RECIPES\n\n  ARUGULA AND RED BUTTERHEAD LETTUCE SALAD WITH TANGERINES AND HARD-COOKED EGGS\n\nWe wanted a hearty salad for our winter menu, with lots of different textures and flavors. The core idea was eggs on toast\u2014which we translated into wedges of hard-cooked egg and crunchy, garlicky croutons. The lettuces lighten everything up and the tangerines are nuggets of juicy sweetness.\n\nIf you use eggs from your own chickens, or are buying eggs from the farmers' market, let them sit in the fridge for at least a week before you cook them (if eggs are too fresh, they're hard to peel).\n\nMAKES 6 to 8 servings\n\nTIME about 1 hour\n\n6 to 8 large eggs (not super-fresh)\n\n2 teaspoons freshly squeezed tangerine juice\n\n\u00bd teaspoon finely grated tangerine zest\n\n\u00be teaspoon fine sea salt\n\n\u00bd cup extra virgin olive oil\n\n2 cloves garlic\n\n3 thin slices Wheat Berry Ciabatta, cut into \u00bd-inch cubes (about 1\u00bd cups)\n\n5 cups loosely packed arugula leaves\n\n6 cups loosely packed red butterhead lettuce leaves (about \u00bd small head)\n\n2 large or 4 small tangerines\n\n1. Preheat the oven to 400\u00b0F. Put the eggs in a small pot and cover with water by about 1 inch. Bring to a boil, immediately reduce the heat to a simmer, and cook, uncovered, for 10 minutes. Remove from the heat, immerse the eggs in ice water to cover, and let cool for 1 minute. On the countertop, crack each egg all over and return to the ice water for 5 minutes. Peel under cold water. Quarter eggs lengthwise and set aside.\n\n2. Meanwhile, in a small bowl, whisk together the tangerine juice, zest, and \u00bd teaspoon salt. Whisk in \u00bc cup of the oil to make a dressing and set aside. Mince the garlic, sprinkle it with the remaining \u00bc teaspoon salt, and mash to a paste with the flat side of a chef's knife.\n\n3. In a heatproof measuring cup, combine the remaining \u00bc cup olive oil and the garlic paste and microwave it for 10 seconds. Put the bread cubes on a rimmed baking sheet, drizzle with the garlic oil, and toss to coat evenly. Spread in a single layer and bake, stirring once or twice, until crisp, about 15 minutes. Set aside.\n\n4. Rinse the greens and dry twice in a salad spinner. Peel the tangerines and remove the thready white pith. Cut the tangerines crosswise, and separate into chunks, discarding any seeds.\n\n5. In a large bowl, toss the greens gently but thoroughly with just enough of the dressing to coat. Add the tangerines and the croutons and toss gently. Divide the salad among individual plates. Add a quartered egg to each and drizzle the eggs with a little more dressing.\n\n_PER 2-CUP SERVING 288 cal., 64% (184 cal.) from fat; 9.7 g protein; 21 g fat (3.9 g sat.); 17 g carbo (2.9 g fiber); 359 mg sodium; 212 mg chol._\n\n##    \n  WHEAT BERRY CIABATTA\n\nWe ate this shaggy-crumbed, chewy, steamy loaf slathered with homemade butter and sprinkled with sea salt. Although you will need to start it the day before, it requires very little effort that first day. We used sweet buttermilk left over from making Homemade Butter, and 'Expresso' (hard red) wheat berries and whole-wheat flour (for more on local flours, see the Allure of Local Wheat). Hard red wheat berries are the most common type sold in stores, and any brand will work; any regular whole-wheat flour will work, too.\n\nMAKES two 1\u00bd-pound loaves (about 22 slices per loaf)\n\nTIME 7 to 8 hours (mainly unattended time), plus overnight to chill starter\n\n1 cup wheat berries\n\n4\u00bd cups whole-wheat flour, plus more for dusting and sprinkling\n\n\u00be teaspoon active dry yeast\n\n\u00bc cup honey\n\n\u00bc cup Homemade Buttermilk, or whole milk\n\nAbout \u2153 cup extra virgin olive oil\n\n2 teaspoons fine sea salt\n\n3 cups ice cubes or ice-cold water\n\n1. In a saucepan, combine the wheat berries and 4\u00bd cups water and bring to a boil over high heat. Reduce the heat to a simmer, cover, and cook until the berries are very tender and at least half of them have burst, 1 to 1\u00bd hours. Remove from the heat and let cool in the pan to room temperature. Drain the berries; you should have about 3 cups. (They can be cooked up to 3 days in advance and stored, tightly covered, in the refrigerator.)\n\n2. In a bowl, stir together 1 cup of the flour, \u00bc teaspoon of the yeast, and 1 cup cold water to make the starter. Cover and refrigerate overnight.\n\n3. The next day, let the starter sit at room temperature for 1 hour; then scrape it into the bowl of a stand mixer fitted with a dough hook. Add the remaining 3\u00bd cups flour, 1 cup cold water, the honey, the buttermilk, and \u00bc cup of the oil. Knead the mixture on low speed until just combined but still shaggy, about 2 minutes. Turn off the mixer and let the dough sit for 45 minutes to allow the flour to absorb the moisture.\n\n4. Sprinkle the remaining \u00bd teaspoon yeast over the dough and knead on low speed until a smooth dough forms, about 4 minutes. Add the salt and knead for 1 minute more. Add the reserved wheat berries and continue to mix on low speed just until most of the berries are incorporated.\n\n5. Use about 1 tablespoon of the oil to grease a large deep baking dish. Transfer the dough to the dish along with any loose wheat berries and cover the pan with oiled plastic wrap. Let the dough sit in the dish in a warm spot until almost doubled in size, about 1\u00bd hours.\n\n6. Line 2 baking sheets with parchment paper and set aside. Turn the dough out onto a floured work surface and divide it in half. Flour your hands; then, using your palms, press each portion flat to release any air. Fold both long sides of each piece of dough into the center, forming 2 rectangular loaves each 10 to 12 inches long. Place each loaf seam side down on a baking sheet.\n\n7. Loosely cover the loaves with oiled plastic wrap and let them sit in a warm spot until doubled in size and very puffy, 1 to 1\u00bd hours.\n\n8. Set 2 oven racks in the oven with 4 inches between them. Put a large rimmed baking pan on the oven floor. Preheat the oven to 500\u00b0F.\n\n9. Peel the plastic wrap from the loaves, being careful not to tear their surfaces. Lightly sprinkle the loaves with whole-wheat flour and put them, still on the baking sheets, on the center racks in the oven. Before closing the oven door, pour about 3 cups ice cubes into the baking pan on the oven floor, then quickly close the oven door to trap steam. Reduce the oven temperature to 450\u00b0F.\n\n10. Bake the loaves, switching the positions of the baking sheets once, until an instant-read thermometer inserted into the center of a loaf registers 190\u00baF, 20 to 25 minutes.\n\n11. Transfer the loaves to racks and let cool to room temperature. You can eat the bread right away, but if you rip into a loaf while it's still hot, it will be gluey when it cools. We usually eat one hot and let the other one cool.\n\n_PER \u00bd-INCH SLICE 160 cal., 22% (35 cal.) from fat; 5.2 g protein; 4.1 g fat (0.61 g sat.); 28 g carbo (4 g fiber); 213 mg sodium; 1 mg chol._\n\n##    \nEGG CLOUD ( _NUVOLONE_ )\n\nA dish from the Italian Alps, _nuvolone_ \u2014the word means \"big cloud\"\u2014is like a deconstructed souffl\u00e9, with the whites piled in a fluffy peak above the liquid yolk. This recipe is based on one in Manuela Darling-Gansser's _Winter in the Alps: Food by the Fireside_. It's great for a weekend breakfast.\n\nIf you want to make multiple egg clouds, beat all the egg whites together, but put each yolk in a separate little bowl (or in its eggshell) until using, which makes it easier to keep them intact. Prep all of your ingredients before you whip the egg whites\u2014and reduce the cooking time to 9 minutes.\n\nMAKES 1 serving\n\nTIME about 20 minutes\n\n1 teaspoon unsalted butter, homemade or store-bought, plus more for greasing\n\n1 large egg, separated (keep the yolk intact)\n\nPinch of fine sea salt\n\n1\u00bd tablespoons coarsely shredded aged Gouda, homemade or store-bought\n\n2 teaspoons heavy cream\n\n\u00bc teaspoon dried or \u00bd teaspoon fresh minced marjoram\n\n1. Preheat the oven to 375\u00b0F. Generously butter a ramekin 3 inches in diameter and 1\u00bd inches deep.\n\n2. In a small bowl, beat the egg white until soft peaks form. Add the salt and beat into stiff peaks. Quickly and gently fold in 1 tablespoon of the Gouda. Spoon the beaten white into the ramekin, pushing it up the sides of the dish and leaving a well in the center.\n\n3. Ease the egg yolk into the well. Top the yolk with the cream, marjoram, butter, and the remaining \u00bd tablespoon Gouda. Smooth the white over the yolk and build it up into a frothy pile. Put the ramekin on a rimmed baking sheet.\n\n4. Bake until the top is pale gold and the white is set around the rim of the ramekin, 9 to 10 minutes. Inside, the yolk will be liquidy and the white will be cooked. Serve immediately.\n\n_PER SERVING 164 cal., 75% (123 cal.) from fat; 9.3 g protein; 14 g fat (7.1 g sat.); 1.1 g carbo (0.06 g fiber); 324 mg sodium; 241 mg chol._\n\n##    \nFEATHERLIGHT PANCAKES\n\nThese are based on a recipe _Sunset_ published years ago for German \"egg cakes\" ( _Eierkuchen_ ) from the Elk Cove Inn, in the Northern California coastal town of Elk. They are good with whipped cream and berries, too.\n\nMAKES 8 pancakes, or 4 servings\n\nTIME about 30 minutes\n\nUnsalted butter for cooking and serving\n\n1 cup whole-wheat pastry flour*\n\n\u00bd cup whole milk\n\n\u00bc teaspoon fine sea salt\n\n1 tablespoon honey\n\n6 large egg yolks\n\n4 large egg whites\n\n1 teaspoon freshly squeezed lemon juice\n\nWarmed honey for serving\n\n1. Preheat an electric griddle to 300\u00b0F or a nonstick frying pan over medium-low heat. Preheat the oven to 200\u00b0F. Butter a rimmed baking sheet.\n\n2. In a medium bowl, whisk together the flour, milk, salt, honey, and egg yolks until well blended. In a large bowl, beat the egg whites with the lemon juice until moist, soft peaks form. Gently fold the whites into the yolk mixture.\n\n3. When the griddle is ready, butter it and spoon on \u00bd-cup portions of the batter. Cook until golden on the bottom and slightly puffed, about 3 minutes. Flip and cook until golden on the other side and cooked through (break into one to check), 1 to 2 minutes more. As they are cooked, put the pancakes on the buttered baking sheet, cover with aluminum foil, and keep warm in the oven.\n\n4. Serve hot, with butter and honey.\n\n_PER 2-PANCAKE SERVING 267 cal., 37% (100 cal.) from fat; 12 g protein; 11 g fat (4.8 g sat.); 30 g carbo (4 g fiber); 241 mg sodium; 325 mg chol._\n\n* _We used finely milled local 'Sonora' soft white whole-wheat flour, which behaves like commercial whole-wheat pastry flour in recipes. SeeThe Allure of Local Wheat for more on local flours._\n\n##    \nEGG AND GOUDA CREPES\n\nSimple yet elegant, these egg-topped crepes make an excellent brunch or a light dinner. The nuttiness of the whole-wheat crepes pairs well with the butterscotch notes in the Gouda.\n\nMAKES 4 servings\n\nTIME about 35 minutes\n\n4 large Whole-Wheat Crepes, made in a 12-inch pan (recipe follows)\n\n1 tablespoon extra virgin olive oil\n\n\u00bd cup finely chopped white onion\n\n1 cup finely shredded Gouda, homemade or store-bought\n\n1 teaspoon unsalted butter, homemade or store-bought\n\n4 large eggs\n\nFine sea salt\n\nAbout \u00bd cup small arugula leaves\n\n1. Preheat the oven to 250\u00b0F and warm the crepes in the oven while it is heating. In a large nonstick frying pan, heat the oil over medium heat. Add the onion and cook, stirring often, until translucent, 4 to 5 minutes.\n\n2. Working with 1 crepe at a time, evenly spread about 1 heaping tablespoon of the onion in the center of the crepe. Sprinkle \u00bc cup of the cheese over the onion. Fold the sides of the crepe over the filling, overlapping to form a square, and press to seal. Put the crepe on a rimmed baking sheet, folded side down. Cover with aluminum foil and keep warm in the oven. Repeat with the remaining crepes, onion, and cheese.\n\n3. In the same pan, melt the butter over medium heat and swirl to coat the bottom. Crack the eggs into the pan, trying to keep them from touching. Cook, turning once, until the whites are set but the yolks are still runny.\n\n4. Place each crepe on a plate and top with an egg. Sprinkle each egg with a pinch of salt and then scatter a few arugula leaves on top.\n\n_PER CREPE 367 cal., 61% (224 cal.) from fat; 19 g protein; 25 g fat (12 g sat.); 17 g carbo (2.4 g fiber); 770 mg sodium; 342 mg chol._\n\n### WHOLE-WHEAT CREPES\n\nThese crepes have a delicately nutty flavor and supremely tender texture.\n\nMAKES 8 large (12-inch), 10 thickish medium (8-inch), or 16 small (6-inch) crepes\n\nTIME about 30 minutes\n\n1\u00bd cups whole milk\n\nAbout 3 tablespoons unsalted butter, homemade or store-bought, melted\n\n3 large eggs\n\n1 cup whole-wheat pastry flour*\n\n\u00bd teaspoon fine sea salt\n\n1. In a saucepan, combine the milk and 2 tablespoons of the butter and heat over low heat until warm.\n\n2. Put the eggs in a blender and pulse just to mix. Add the flour, salt, and \u00bc cup of the warm milk mixture and whirl to combine. With the blender running, slowly pour the remaining milk mixture into the blender through the lid opening. Strain the batter into a bowl.\n\n3. Heat a 12-inch, 8-inch, or 6-inch nonstick frying pan over medium heat. Brush the pan with a little of the remaining 1 tablespoon butter, then pour about \u2153 cup batter into the largest pan, \u00bc cup batter into the medium-size pan, or 2 tablespoons into the smallest pan and swirl to coat the bottom of the pan evenly. Cook, turning once, until set and starting to brown, about 2 minutes total. Transfer the crepe to a plate or a baking sheet. Repeat with the remaining batter, brushing the pan with more butter as needed and stacking the crepes as you go.\n\n**MAKE AHEAD** Crepes can be made up to 4 days ahead, wrapped in plastic wrap, and chilled, they can also be frozen, stored between layers of wax paper, for 1 month.\n\n_PER LARGE CREPE 148 cal., 48% (71 cal.) from fat; 5.3 g protein; 7.9 g fat (4.2 g sat.); 14 g carbo (2 g fiber); 187 mg sodium; 95 mg chol._\n\n_PER MEDIUM CREPE 118 cal., 48% (57 cal.) from fat; 4.3 g protein; 6.3 g fat (3.3 g sat.); 11 g carbo (1.6 g fiber); 149 mg sodium; 76 mg chol._\n\n_PER SMALL CREPE 74 cal., 47% (35 cal.) from fat; 2.7 g protein; 4 g fat (2.1 g sat.); 6.9 g carbo (1 g fiber); 93 mg sodium; 48 mg chol._\n\n* _We used finely milled local 'Sonora' soft white whole-wheat flour, which behaves like commercial whole-wheat pastry flour in recipes. SeeThe Allure of Local Wheat for more on local flours._\n\n##    \nKALE COLCANNON\n\nColcannon is one of the genius ways that the Irish have with potatoes\u2014mashing them up with milk, good butter, and cooked kale or cabbage. It is simple but delicious.\n\nMAKES 4 to 6 servings\n\nTIME about 40 minutes\n\n1\u00bd pounds Yukon Gold potatoes, unpeeled, cut into large, evenly sized chunks\n\n\u00bd pound Tuscan kale, tough ribs removed and leaves coarsely chopped\n\n\u2153 cup whole milk, or as needed, warmed\n\n2 tablespoons unsalted butter, homemade or store-bought, plus more for serving (optional)\n\n\u00be teaspoon fine sea salt\n\n1. In a saucepan, combine the potatoes with cold water to cover by 1 inch and bring to a boil over high heat. Reduce the heat and simmer, covered, until the potatoes are tender, about 20 minutes.\n\n2. Meanwhile, pour water to a depth of \u00bd inch in another saucepan, insert a steamer basket, cover, and bring to a boil over high heat. Add the kale to the basket and cook, covered, turning the kale occasionally with tongs, until tender, about 20 minutes.\n\n3. Drain the potatoes, return them to the pan, and add the milk, butter, and salt. Mash with a potato masher, keeping the potatoes slightly chunky. Add the kale and stir to combine. Stir in a little more milk if the mixture seems too thick.\n\n4. Serve the colcannon with a pat of butter if you like.\n\n_PER \u00be-CUP SERVING 145 cal., 27% (39 cal.) from fat; 3.8 g protein; 4.4 g fat (2.7 g sat.); 23 g carbo (1.7 g fiber); 309 mg sodium; 12 mg chol._\n\n##    \nVEGETABLE SHEPHERD'S PIE\n\nCreating a hearty shepherd's pie without meat was challenging, but meaty mushrooms and hearty greens and beans did the trick.\n\nMAKES 4 to 6 servings\n\nTIME about 1\u00be hours\n\n1\u00bd pounds Yukon Gold potatoes, peeled and quartered\n\n1 tablespoon plus 2 teaspoons extra virgin olive oil\n\n\u00bd white onion, cut into slivers\n\n1\u00bc teaspoons fine sea salt\n\n1 clove garlic, finely chopped\n\n2 tablespoons plus 2 teaspoons whole-wheat flour*\n\n\u00bc cup Chardonnay\n\n2 cups Herb Vegetable Broth\n\n\u00bc pound Tuscan kale, tough ribs removed and leaves chopped\n\n\u00bd pound Swiss chard, ribs removed and leaves chopped\n\n6 ounces oyster mushrooms, halved if large\n\n1 cup drained cooked flageolet beans (see method)\n\n2 tablespoons unsalted butter, homemade or store-bought\n\n3 tablespoons whole milk\n\n1 large egg plus 1 large egg yolk\n\n1. In a saucepan, combine the potatoes with cold water to cover by 1 inch and bring to a boil over high heat. Reduce the heat, cover, and simmer until the potatoes are tender, about 20 minutes. Drain the potatoes, return them to the pan, and cook over low heat, stirring occasionally, until dry and floury, about 5 minutes.\n\n2. While the potatoes are cooking, in a large pot, heat the oil over medium heat. Add the onion and \u00be teaspoon of the salt and cook, stirring often, until the onion is softened and lightly golden, about 7 minutes. Add the garlic and cook until fragrant, about 1 minute.\n\n3. Whisk the flour into the onion mixture and cook, whisking, for 1 to 2 minutes. Gradually whisk in the wine and continue whisking to break up any lumps. Then gradually whisk in the broth and bring the mixture to a simmer. Preheat the oven to 425\u00b0F.\n\n4. Add the kale and chard to the broth mixture, cover, and cook over medium heat until the greens are tender, about 10 minutes. Stir in the mushrooms and cook, covered, until the mushrooms are almost tender, about 6 minutes.\n\n5. Add the flageolets, return the mixture to a simmer, and cook until the consistency of thick but pourable gravy, about 5 minutes.\n\n6. Meanwhile, press the potatoes through a ricer into a bowl (or mash with a potato masher). Mash the butter and milk into the potatoes, then stir in the egg, egg yolk, and the remaining \u00bd teaspoon salt.\n\n7. Transfer the vegetable mixture to an 8-inch square baking pan, patting it into an even layer. Carefully spread the potato mixture on top, covering the vegetable mixture evenly. If you like, using a spoon, sculpt the top into freeform peaks or waves.\n\n8. Bake until the potatoes are lightly golden and slightly puffed, about 25 minutes. Let stand for 5 to 10 minutes to thicken and cool slightly before serving.\n\n_PER SERVING 248 cal., 40% (98 cal.) from fat; 10 g protein; 11 g fat (3.9 g sat.); 36 g carbo (6.8 g fiber); 924 mg sodium; 81 mg chol._\n\n* _We used 'Expresso' whole-wheat flour, a hard red winter wheat. SeeThe Allure of Local Wheat for more about local flours. Any regular whole-wheat flour will work in this recipe._\n\n##    \nRICOTTA MANICOTTI\n\nMaria Helm Sinskey, author of _The Vineyard Kitchen: Menus Inspired by the Seasons_ , makes her manicotti with crepes instead of dried pasta\u2014just like her great-grandmother did. Ethereal and rustic at the same time, they were the inspiration for this recipe.\n\nIf you prefer crisp-edged manicotti, put the filled crepes on an oiled rimmed baking sheet in pairs (no tomato sauce beneath), leaving about an inch of space between the pairs. Top with two-thirds of the tomato sauce (leave the ends of manicotti bare) and bake in a 450\u00b0F oven until the edges are nicely browned, about 10 minutes, rotating the pan back to front halfway through so the manicotti brown evenly.\n\nMAKES 10 manicotti or 4 to 6 servings\n\nTIME about 40 minutes\n\n3 cups 'Sweet Million' Tomato Sauce or other tomato-based pasta sauce\n\n\u00bc cup Chardonnay\n\n3\u00bd cups ricotta, homemade or store-bought\n\n\u00bd cup chopped fresh basil\n\n2 tablespoons freshly squeezed lemon juice\n\n1 teaspoon crushed dried red serrano or \u00e1rbol chile\n\n1 teaspoon fine sea salt\n\n10 medium Whole-Wheat Crepes, made in an 8-inch pan (they will be quite thick)\n\n\u00bd cup finely shredded Gouda, homemade or store-bought\n\n1. Preheat the oven to 450\u00b0F. In a small bowl, mix together the tomato sauce and wine. Spread some of the sauce in the bottom of a 9-by-13-inch baking dish.\n\n2. In a bowl, mix together the ricotta, basil, lemon juice, chile, and salt. Spoon about \u2153 cup of the filling over each crepe, leaving a \u00bd-inch border around the edges. Roll up the crepes and place them seam sides down in the baking dish. Spread the remaining tomato sauce evenly over the filled crepes.\n\n3. Bake the crepes until hot and bubbling, about 15 minutes. Sprinkle the Gouda evenly over the top and continue to bake until the cheese melts, about 2 minutes more.\n\n_PER MANICOTTO 343 cal., 57% (197 cal.) from fat; 17 g protein; 22 g fat (12 g sat.); 20 g carbo (3 g fiber); 691 mg sodium; 127 mg chol._\n\n_Winter Vegetable Chowder_\n\n  WINTER VEGETABLE CHOWDER\n\nCold rain hits the San Francisco Bay Area in January and February. A big bowl of this creamy chowder gives lasting, delicious warmth. To make it pretty, we sprinkled it with broccoli rabe and rosemary flowers.\n\nMAKES 12 cups, or 6 main-course servings\n\nTIME about 1\u00bd hours\n\n2 pounds 'Yukon Gold' potatoes, unpeeled, cut into \u00be-inch chunks (about 6 cups)\n\n1 head cauliflower, about 2\u00bd pounds, cut into florets\n\n1 head broccoli romanesco, about 1\u00bc pounds, cut into florets\n\n4 cups (about 1 bunch) chopped broccoli rabe plus any flowers\n\n3 tablespoons extra virgin olive oil, plus more for serving\n\n1 tablespoon fine sea salt\n\n3 tablespoons chopped garlic\n\n1 cup chopped white onion\n\n\u00bd cup heavy cream (optional)\n\nChopped fresh rosemary, plus any flowers, for garnish\n\n1. Preheat the oven to 350\u00b0F. Put half the potatoes on a rimmed baking sheet. Pick out 3 cups bite-size cauliflower florets and combine them with the broccoli romanesco and broccoli rabe on another rimmed baking sheet (set aside any broccoli rabe flowers). Chop the remaining cauliflower and set aside. Drizzle 1 tablespoon of the oil and sprinkle 1 teaspoon of the salt over each baking sheet.\n\n2. Roast the vegetables, stirring often, until golden brown and tender when pierced with a fork, about 20 minutes for the mixed vegetables and 25 minutes for the potatoes.\n\n3. Meanwhile, heat the remaining 1 tablespoon oil in a large pot over medium heat. Add the garlic and onion and cook, stirring often, until translucent, about 5 minutes. Add the remaining potatoes, the chopped cauliflower, and 6 cups water and bring to a simmer. Cover and cook until the vegetables are tender, about 15 minutes. Remove from the heat and let cool slightly.\n\n4. Working in batches, ladle about 2 cups of the liquid and vegetables into a blender and process until smooth. Pour through a fine-mesh strainer set over a large bowl, using the back of the ladle to push the soup through.\n\n5. Pour the puree back into the pot and bring to a simmer over low heat. Stir in the cream and the remaining 1 teaspoon salt, then add the roasted vegetables and heat through.\n\n6. Serve the chowder into deep soup bowls, drizzled with oil and sprinkled with chopped rosemary and broccoli rabe and rosemary flowers.\n\n_PER 2-CUP SERVING 273 cal., 23% (64 cal.) from fat; 11 g protein; 7.5 g fat (1.1 g sat.); 43 g carbo (5.3 g fiber); 1,205 mg sodium; 0 mg chol._\n\n##    \nWHEAT BERRY \"RISOTTO\" WITH ROASTED TOMATOES AND BROCCOLI RABE\n\nMany grains other than rice can be cooked like risotto, and wheat berries from soft winter wheat are one of them. As you stir them, they drink in hot liquid and their starchy outer hulls dissolve into delectable creaminess. We used local 'Sonora' soft white wheat berries from Pie Ranch in Pescadero, California. Despite the name, they are actually pale yellow, and have a lovely, fresh, slightly corny flavor. Hard red wheat berries are what you are likely to find in the grocery store, and while they won't become as creamy, their earthier, deeper flavor is good in this recipe, too.\n\nWe ate this as a main course, with a chunk of homemade feta and some hot bread on the side.\n\nMAKES 8 cups or 4 to 6 servings\n\nTIME about 3 hours\n\n2 cups wheat berries, preferably soft white wheat such as 'Sonora'\n\n5 tablespoons extra virgin olive oil\n\n1 bunch broccoli rabe, about 1 pound, tough ribs removed and leaves chopped into 1-inch pieces\n\n1 teaspoon fine sea salt\n\n4 large cloves garlic, finely chopped\n\n1 white onion, chopped (about 1\u00bd cups)\n\n\u00be cup Slow-Roasted Tomatoes for the Freezer, thawed; or sun-dried tomatoes packed in oil\n\n1 cup chopped walnuts\n\n2 tablespoons unsalted butter, homemade or store-bought\n\n1. In a large pot, combine the wheat berries and 2\u00bd quarts water, cover, and bring to a boil over high heat. Remove from the heat and let sit, covered, for 1 hour. Bring back to a boil, then reduce the heat to a simmer. If using soft white wheat berries, simmer until about half of the berries have just burst, about 30 minutes. If using hard red wheat berries, simmer until about half of the berries have burst wide open, 2 to 2\u00bd hours. Drain the wheat berries and set aside.\n\n2. In a large nonstick frying pan, heat 3 tablespoons of the oil over medium heat. Add the broccoli rabe and cook, stirring occasionally, until wilted, about 4 minutes. Sprinkle \u00bd teaspoon of the salt over the greens, cover, reduce the heat to low, and cook until just tender, about 15 minutes more. Add the garlic, raise the heat to medium, and cook, stirring occasionally, for 3 minutes. Remove from the heat, cover, and set aside.\n\n3. In a saucepan, bring 6 cups water to a simmer. Cover and keep at a simmer over low heat.\n\n4. In a heavy-bottomed 8-quart pot, heat the remaining 2 tablespoons oil over medium heat. Add the onion and cook, stirring occasionally, until translucent, about 10 minutes. Add the wheat berries and the remaining \u00bd teaspoon salt and cook, stirring often, until hot, about 3 minutes.\n\n5. Add \u00bd cup of the simmering water and cook, stirring, until completely absorbed by the wheat berries. Continue adding water, \u00bd cup at a time, stirring until each addition is absorbed before adding the next, until the berries look creamy, 15 to 30 minutes. You will have water left over. Keep the pot at a steady simmer throughout, reducing the heat to medium-low if the mixture starts to boil.\n\n6. Chop the tomatoes and add them and the cooked broccoli rabe to the wheat berries. Heat, stirring, until heated through, 2 to 4 minutes.\n\n7. Remove the pot from the heat and stir in the walnuts and butter. For a looser risotto, stir in \u00bd to 1 cup of the remaining simmering water. Serve immediately.\n\n_PER SERVING 535 cal., 52% (277 cal.) from fat; 15 g protein; 32 g fat (5.6 g sat.); 56 g carbo (11 g fiber); 490 mg sodium; 10 mg chol._\n\n##    \nBRAISED WINTER GREENS WITH PRESERVED LEMON AND RED CHILE\n\nIf you have a jar of preserved lemons and some dried chiles on hand, you can make these supremely satisfying greens quickly. We especially like the combination of sweet Savoy cabbage with mildly pungent mustard greens, but you can use any greens in your garden.\n\nMAKES 4 to 6 servings\n\nTIME about 25 minutes\n\n3 tablespoons extra virgin olive oil\n\n\u2154 cup finely chopped white onion\n\n2 tablespoons chopped preserved lemon peel\n\n\u00bc to \u00bd teaspoon crushed dried red serrano chiles or \u00e1rbol chiles\n\n3\u00bd quarts loosely packed whole leaves of mixed greens such as Savoy cabbage, mustard, Swiss chard, and\/or kale that have had the tough stems and ribs removed\n\nFine sea salt (optional)\n\n1. In a 5- to 6-quart saucepan, heat 1\u00bd tablespoons of the oil over medium heat. Add the onion and cook, stirring often, until translucent, 4 to 5 minutes. Stir in the preserved lemon and chile and cook, stirring, for about 1 minute.\n\n2. Add half the greens and \u00be cup water and raise the heat to medium-high. Cover the pan and cook until the greens have wilted, 2 to 3 minutes. Add the remaining greens and cook, covered, until all are wilted, about 5 minutes more. Stir, reduce the heat to low, and cook, covered, until the greens are tender, about 5 minutes more.\n\n3. If there is too much liquid in the pan, uncover and boil over high heat for a few minutes to evaporate most of it. Stir in the remaining 1\u00bd tablespoons oil and season to taste with salt.\n\n_PER \u2154-CUP SERVING 118 cal., 53% (63 cal.) from fat; 4.6 g protein; 7.3 g fat (1 g sat.); 13 g carbo (6.5 g fiber); 401 mg sodium; 0 mg chol._\n\n##    \nCARAMELIZED TANGERINE AND RICOTTA TART\n\nBellwether Farms, in Sonoma, California, produces some of America's finest ricotta, and head cheese maker Liam Callahan taught us how to make ricotta at home (see here). It is terrific in this tart, which was inspired by one that Liam and his mother, Cindy, serve to guests at Bellwether. Their version is based on a tart by chef Emeril Lagasse, who invented it using Bellwether cheeses.\n\nIf you make the rectangular tart, you'll have enough pastry dough, filling, and topping left over for a 4- to 5-inch mini tart. Instead of the caramelized tangerines, you can top the tart with any ripe, in-season fruit and glaze it with some warmed jam.\n\nMAKES 1 round 9- or 10-inch tart, or 1 rectangular 4\u00bd- by-14-inch tart plus 1 round 4- to 5-inch mini tart\n\nTIME about 4 hours, plus at least 2 hours to chill\n\n**TOPPING**\n\n\u2153 pounds small, firm tangerines (avoid the loose \"zipper-skin\" type)\n\n7 tablespoons honey\n\n1 cup freshly squeezed tangerine juice, plus more if needed\n\n**CRUST**\n\n1\u00bd cups 'Sonora' whole wheat flour* or whole-wheat pastry flour, plus more for dusting\n\n\u00bc teaspoon fine sea salt\n\n\u00bd cup cold unsalted butter, homemade or store-bought, cut into cubes\n\n1 tablespoon honey\n\n3 to 4 tablespoons ice water\n\n**FILLING**\n\n1 cup ricotta, homemade or store-bought\n\n\u00be cup fromage blanc, homemade or store-bought**\n\n\u00bd cup cr\u00e8me fra\u00eeche, homemade or store-bought\n\n2 tablespoons honey\n\n2 tablespoons Sticky Chewy Tangerine Marmalade or store-bought tangerine or orange marmalade\n\n1. To make the topping, preheat the oven to 275\u00b0F. Wash the tangerines well, but don't peel them. Slice them very thinly crosswise. Arrange half the slices in an 8-inch square baking dish, keeping them more or less in a single layer. Drizzle the slices with 1\u00bd tablespoons of the honey. Top with the remaining tangerine slices and drizzle with another 1\u00bd tablespoons honey. Pour the tangerine juice over the layered slices. It should almost cover the fruit; if it doesn't, add more as needed. Cover the dish with aluminum foil and cook until the peels are very soft, 2 to 2\u00bd hours.\n\n2. Meanwhile, make the dough for the crust. Sift the flour and salt into a food processor and whirl briefly to blend. Scatter the butter cubes over the top and pulse until the mixture looks like shaggy fresh bread crumbs. Drizzle the honey over the mixture. With the motor running, drizzle in the ice water, adding just enough for the dough to come together. Turn the dough out onto a work surface and press together into a rectangle or disk, depending on the shape of the tart you are making. Wrap the dough in plastic wrap and refrigerate for at least 1 hour or up to 2 days.\n\n3. While the dough is chilling, make the filling. In a bowl, beat the ricotta, fromage blanc, cr\u00e8me fra\u00eeche, and honey until smooth with a wooden spoon. Then beat in the marmalade. Cover and refrigerate until needed.\n\n4. When the tangerine slices are ready, drizzle with another 2 tablespoons honey and raise the oven temperature to 350\u00b0F. Roast the slices, uncovered, until they start to brown, about 50 minutes. Remove from the oven and drizzle with the remaining 2 tablespoons honey.\n\n5. While the tangerines are roasting, bake the crust. Lightly dust a work surface with flour. Unwrap the dough, set it on the floured surface, and let it warm up for 10 minutes. With a lightly floured rolling pin, roll out the dough about \u215b inch thick to fit a 9- or 10-inch round tart pan with a removable bottom or a 4-by-14-inch rectangular tart pan with a removable bottom. Using a pastry scraper or a thin metal spatula, scoot the dough off the surface and gently ease it into the tart pan. Trim the dough even with the pan edge. If you are making a rectangular tart, gather up the scraps and reroll to a line a 4- or 5-inch tartlet pan.\n\n6. Prick the bottom of the crust all over with a fork. Line the crust with parchment paper and fill with pie weights or dried beans. Slide the crust into the oven alongside the tangerines and bake until golden, 30 to 40 minutes. Remove from the oven, remove the pie weights and parchment, and bake the crust until deep golden, 10 to 15 minutes more. Let cool on a rack.\n\n7. Spread the filling in the cooled crust to within \u00bd inch of the rim. Arrange the warm or cooled tangerines over the filling and drizzle with any accumulated juices. Chill for at least 2 hours before serving.\n\n**MAKE AHEAD** The tart can be made up to 1 day ahead and refrigerated. You can also make the crust, filling, and topping up to 1 day ahead (keep the crust at room temperature; refrigerate the filling and topping), then assemble and chill for 2 hours before serving.\n\n_PER SLICE OF ROUND TART 342 cal., 44% (152 cal.) from fat; 6.5 g protein; 17 g fat (10 g sat.); 42 g carbo (3.8 g fiber); 236 mg sodium; 49 mg chol._\n\n* _We used local 'Sonora' soft white wheat flour because it behaves like commercial whole-wheat pastry flour. SeeThe Allure of Local Wheat for more about local flours._\n\n** _Store-bought fromage blanc is often much drier than homemade fromage blanc. If you use it, stir in enough heavy cream (at least 3 tablespoons) to give it the consistency of softened cream cheese._\n\n_Tangerine Honey Flan_\n\n##    \n  TANGERINE HONEY FLAN\n\nWithout baking soda or baking powder, we couldn't make cakes. But we could make this fantastic flan\u2014smooth, rich, cool, and creamy.\n\nMAKES 1 (9-inch) flan\n\nTIME about 2 hours, plus 2 hours to chill\n\n\u2153 cup plus \u00bd cup honey\n\n2\u00bd cups whole milk\n\n1 tablespoon finely shredded tangerine zest\n\n4 large eggs plus 2 large egg yolks\n\nThinly sliced tangerine peel for garnish (optional)\n\n1. Preheat the broiler and position the rack 5 inches from the heating element.\n\n2. Spread \u2153 cup honey in an ovenproof 9-inch quiche dish or pie pan.\n\n3. Broil the honey until bubbly and medium amber, 5 to 10 minutes. Watch closely to make sure it doesn't burn. Remove from the oven and swirl gently if unevenly colored, then let the honey cool (it will solidify and become fairly hard). Reduce the oven temperature to 325\u00b0F and reposition the rack in the center of the oven.\n\n4. In a small saucepan, combine the milk and tangerine zest and heat over medium-low heat until steaming. Meanwhile, whisk together the remaining \u00bd cup honey, the whole eggs, and the egg yolks in a bowl.\n\n5. Slowly drizzle the warm milk mixture into the egg mixture while whisking constantly. Strain the custard through a cheesecloth-lined strainer into a clean bowl. Put the quiche dish of honey in a roasting pan and pour the custard into the dish.\n\n6. Pull out the center oven rack halfway and put the roasting pan on the rack. Pour hot water into the pan to come halfway up the sides of the quiche dish. Carefully slide the rack into place. Bake until the custard jiggles only slightly when the dish is gently shaken, about 1 hour.\n\n7. Carefully remove the roasting pan from the oven. Let stand for 10 minutes, then transfer the custard dish to a rack to cool to room temperature.\n\n8. Cover the custard and chill for at least 2 hours or up to overnight. To serve, carefully run a knife along the inside edge of the dish to release the custard. Invert a large rimmed plate over the custard, carefully invert the dish and plate together, and then lift off the dish. Top with the tangerine peel.\n\n_PER SERVING (1\/12th OF FLAN) 135 cal., 27% (37 cal.) from fat; 4.2 g protein; 4.1 g fat (1.7 g sat.); 22 g carbo (0.1 g fiber); 48 mg sodium; 111 mg chol._\n\n##    \nSTICKY CHEWY TANGERINE MARMALADE\n\nThis Scottish-style marmalade is based on a recipe from my family's friend Mary Latker, whose relatives owned the R. & W. Scott jam factory in Carluke, near Glasgow. It's thick and delectably chewy, with a pleasantly bitter edge.\n\nThe recipe has three stages: an overnight soak, a first boil, and a second boil in small batches, with sugar added (we used honey instead). Preserves made this \"small-batch\" way have a better consistency and a fresher, brighter taste. You can double the recipe if you like, but you must still do the second boil in small batches.\n\nWe don't process the marmalade jars in boiling water, and have had no trouble (the jars seal properly and the marmalade keeps for several months at room temperature). That said, water-bath processing has the advantage of driving out every last bit of air from the jars, and marmalade or preserves that are processed this way keep their color longer. If you'd like to try it, follow the instructions that come with canning jars, or download directions from the National Center for Home Food Preservation at www.homefoodpreservation.com; look under Make Jam & Jelly, then go to Processing Jams and Jellies.\n\nMAKES 3 half-pint jars, plus a little extra\n\nTIME about 3 hours, plus overnight to soak\n\n2 pounds firm tangerines (9 to 15, depending on size)\n\n\u00bd lemon\n\n2\u00bc cups honey\n\n1. Wash 2 or 3 tangerines, then score their peels into quarters and remove them. Slice the peels into thin slivers (you should have about \u00bd cup) and put the slivers in a large, wide heavy-bottomed pot.\n\n2. Peel the remaining tangerines and remove most of the thready white pith from all the tangerines, including those from step 1. Separate the tangerines into segments, cut the segments in half crosswise, and remove any seeds. Put the segments in the pot.\n\n3. Wash the lemon half, then cut it, peel and all, into \u00bc-inch dice and remove any seeds. Add to the pot. Measure the contents of the pot. You should have about 5 cups (add a little extra peel or fruit if you are short.) Return the fruit to the pot and add 1\u00bd cups water for every 1 cup fruit. Cover and refrigerate overnight.\n\n4. The hard boil: Cover the pot and bring to a boil over high heat. Boil rapidly until the peels are very soft and cooked through, 25 to 30 minutes. The tangerine peels can toughen after seeming soft, so cook them well.\n\n5. Meanwhile, put 3 half-pint canning jars on a rack (or on several cookie cutters) in a stockpot and fill with hot water. Bring to a boil, then boil for 10 minutes to sterilize. Wash canning lids and rings for the jars with hot, soapy water.\n\n6. The small-batch boil: Measure the fruit pulp, including the liquid; you should have about 6 cups. Working in batches, put 2 cups pulp in a heavy-bottomed medium saucepan (plain steel, no dark coating inside; this helps you see the true color of the marmalade as it darkens). Stir in \u00be cup of the honey. You can do more than one batch at a time if you have enough equipment.\n\n7. Put a small plate in the freezer to chill. Bring the fruit mixture to a boil over medium-high heat and cook, uncovered, at a hard, foaming boil, stirring occasionally with a wooden spoon, until the mixture starts to thicken, about 20 minutes. To test, spoon a dollop onto the chilled plate. If the dollop has rounded \"shoulders\" instead of flattening out, and nearly holds its shape when you run a finger through it, the marmalade is ready. (It will thicken as it cools.) If it seems too thin, continue to boil, watching carefully (honey burns easily) and testing periodically the same way. If the marmalade begins to burn, reduce the heat and simmer until it is thick enough to set.\n\n8. Empty the jars of hot water and set them on a counter. Ladle in the marmalade to within \u00bc inch of the rims and skim any foam from the surface. Wipe the rim edges clean with a damp kitchen towel. Set the inner lids on the jars, then screw on the bands. Transfer the jars to a cooling rack until they self-seal (they give off an audible \"snick\" as the lid is sucked in toward the surface of the jam). Test a seal by pushing on the center of the lid. If it doesn't spring back, it is sealed.\n\n9. If you have not been making batches simultaneously, repeat steps 6, 7, and 8 with the remaining fruit pulp. The marmalade will keep in a cool, dark place for several months. If a jar did not seal properly, store it in the refrigerator and use within a month. If you have leftover marmalade, or if the final batch failed to make a full jar, store that in the refrigerator too.\n\n_PER TABLESPOON 56 cal., 1% (0.42 cal.) from fat; 0.18 g protein; 0.05 g fat (0.01 g sat.); 15 g carbo (0.41 g fiber); 0.95 mg sodium; 0 mg chol._\n\n##    \nPRESERVED LEMONS\n\nMost preserved lemons take a good month to cure. Freezing them overnight speeds up the softening of the lemons' cells, and they are ready in about a week. Slice or chop the peel and put in salads, cooked greens, or quinoa, or puree with cooked vegetables for soup\u2014carrot is especially good. We used 'Eureka' lemons, because it's what we grew; but if you have Meyer lemons, by all means use them.\n\nMAKES 2 cups\n\nTIME about 10 minutes, plus at least 1 week to stand\n\n5 to 7 lemons\n\n5 tablespoons fine sea salt\n\n1. Rinse the lemons well. Quarter 5 lemons lengthwise, put in a bowl, and sprinkle with the salt. Mix with your hands to coat the lemons evenly with the salt, then crush them with your hands or a potato masher until the juices seep out. Pack the lemons into a 1-pint widemouthed jar with a clamp lid. The lemon quarters should be immersed in the juice. If not, squeeze the juice from the remaining 2 lemons and add as needed to cover the lemon pieces.\n\n2. Cover the jar with plastic wrap, roll a rubber band down around the rim to hold the wrap in place, and freeze for at least 8 hours or up to overnight.\n\n3. Let the lemons thaw at room temperature. Remove the plastic wrap and close the jar lid. Let stand at room temperature until the peels have softened, 7 to 8 days. Invert the jar twice a day to make sure the lemon pieces are evenly moistened.\n\n4. The lemons are now ready to use. They can be stored in the refrigerator for up to 1 year (they will turn a deeper yellow). To use them, scrape the pulp from the peel and discard the pulp. If you want a less salty flavor, rinse the peel under cold running water. Cut the peel as directed in individual recipes.\n\n_PER \u00bc LEMON 3.8 cal., 0% from fat; 0 g protein; 0 g fat (0 g sat.); 1.3 g carbo (0.5 g fiber); 1,680 mg sodium; 0 mg chol._\n\n# SPRING\n\n_The One-Block Spring Feast_\n\nRadishes, Fresh Homemade Butter, and Salt  \nFavas and Ricotta on Buttermilk Crackers\n\nMesclun Salad with Spring Beets and Dill  \nGrilled Carrot Salad  \nFava Leaf and Parsley Quiche\n\nStrawberries with Fromage Blanc and Lemon Honey\n\nChardonnay, Belgian Abbey Ale, Strawberry Lemonade\n\nSpring starts early in the San Francisco Bay Area. In February, stone-fruit trees puff out in popcornlike balls of pink and white. With every passing week, new flowers open on trees and from bushes and beds: mock orange and rose and lilac vine, sending out a sweet blended perfume.\n\nAround the beginning of the month, our bees began pouring from the hives. We saw them everywhere, dancing over the flowers and herbs, their golden bodies glinting in the sun. Despite the pleasure of watching them and anticipating the honey to come, Team Bee braced itself. Spring is swarm season.\n\nSwarming typically happens because the hive is overcrowded. The crammed-in bees will raise a new queen (or several), and then she and up to half of the colony will fly away. It can seriously deplete the hive, and beekeepers try to prevent this by stacking on another box to give their bees more room.\n\nWe had added boxes to both hives a few weeks earlier, but it wasn't enough to keep Betty from swarming. One morning, we spotted a big, buzzing clump of bees up in a nearby tree\u2014and more bees cascading thickly down the front of Betty. Luckily, bees are gentle when swarming; their goal is to protect the queen (hidden in the buzzing ball) and find a new home, not attack, so even though we had rushed out without suits or veils, we had little to worry about. But we were helpless to intervene.\n\nTwo days later, Betty (now ex-Betty, most likely) swarmed again. An \"afterswarm\" is rare, but does happen when the bees raise more than one queen. The air was thick with darting, roaring bees, and Kimberley and Margaret stood in the middle of it, engulfed by the tornado. \"It felt like being encompassed by the Other,\" said Margaret later. \"You could feel their weird, humming, crackling energy.\" Even for those of us standing at the edge, it was an awesome sight. We watched for half an hour, and the bees kept flying crazily back and forth. By the end of day, they still hadn't managed to locate a new place to live and were pooled forlornly under the hive.\n\nThe next morning, Brianne found the swarm clustered on some empty clay pots nearby. She had just been to a lecture on swarm catching, and without hesitation gently slid an empty box under the bees while brushing them in with a bee brush (kind of like a duster). We gave the swarm to the head of our local beekeeper's guild. And we left the twice-swarmed hive alone for a while to settle down with, we hoped, a new resident queen.\n\nWe had been excited about all of our menus. But spring seemed especially wonderful, because the contrast between the dense, minerally, sulfurous vegetables of winter\u2014kales, cabbages, cauliflowers\u2014and the light, fresh crops of spring\u2014strawberries, peas, carrots\u2014was so dramatic. With the sun came sunny, sweet-tasting food.\n\nAfter our now-customary Food and Garden meeting to figure out what we would like to grow versus what we actually could grow, Johanna began the planting. She started the more tender herbs\u2014like dill (even though it's more of a summer herb, it would still grow in our spring), tarragon, and feathery chervil\u2014in the greenhouse, since the ground wasn't quite warm enough yet. She planted carrots, strawberries, and soft mixed lettuces that we would be able to snip when barely formed and toss into salads. Green onions went in, and radishes, and three kinds of beets. We craved peas, preferably English peas or sugar snaps, but if planted now, in March, they wouldn't be ready for a feast in May.\n\nThis was terribly disappointing. What is a spring menu without peas? Then we spotted some distinctly legume-like plants climbing up the side of the greenhouse. Johanna reluctantly revealed that they were fava beans, which she had planted specifically as a cover crop back in October to deliver nitrogen to a spot that needed it. Ah! Fava beans could be just as good, if not better, in our spring menu. Maybe the test garden could spare a few? \"Please let us eat some. They're so delicious,\" I wheedled. \"And we have no peas.\" Johanna graciously gave in. Team Kitchen lucked out, and next year, we will plan ahead for peas.\n\nChickens are tamer than bees, but they care less about the good of the collective. Honey\u2014the henpecked chicken for whom we had built her own separate protective enclosure the year before\u2014was again being tormented by the rest of the flock: more bleeding comb, more hiding in the nest box, serious wasting away of plump chicken body. We fretted. This could not go on.\n\nFinally, we realized that Honey was just being broody. This is common among chickens, and we felt a bit silly that it had taken so long for us to recognize it. Poor Honey was trying to hatch eggs; that was why she refused to leave the box, not because she was avoiding the sorority from hell down in the yard. However, non-brooding hens, we learned, do typically attack a brooder whenever she's off the nest. It's as though they're telling her to get back on the job.\n\nOur tactic was to haul Honey out of the nest box, plop her down next to the food, and stand guard long enough so she could eat and drink. She would puff up her feathers until she looked like a big yellow dandelion, and she'd make wet, squelchy clucks that sounded as though she were babbling. This completely enraged the other chickens. We'd fend them off with menacing foot thrusts while Honey chowed down. Within several days, she was back to her usual sleek self, and Team Chicken was able to relax.\n\nAt last, Team Beer attempted its most ambitious brew, using our own hops and our own agonizingly threshed and winnowed wheat and barley. Now we had to malt the grains\u2014that is, make them sprout, which naturally converts their starches to sugars. Malted grain, once ground up and soaked in warm water, produces wort, the sweet liquid that yeast feed on and convert to beer.\n\nTeam Beer's leader, Rick LaFrentz, is also Sunset's head gardener and has a knack for making seeds sprout. He soaked our wheat and barley for hours and then enclosed the seeds in plastic bags to keep them moist. Within days, fine little root hairs and shoots appeared. A bit of drying and rubbing off of roots and shoots, and the malted grain was ready.\n\nFor help on brewing day, we enlisted a friend with lots of experience in home brewing from grain: Chuck Schwalbach, husband of Diane, who works in manufacturing at Sunset. Chuck brought in some very useful tools, including a plate chiller, which cools the hot wort almost instantly. As the ripe smell of wort filled the kitchen, we hoped that the grains had fully converted their starches to sugars. Without sugars, our beer would bomb. So it was a big relief when, at the end of the day, Chuck measured the new brew's density and found that it had plenty of sugar: enough, he predicted, to give our beer 7 percent alcohol. We toasted our success with mugs of the grainy, sweet wort, tinged with bitterness from the hops.\n\nBeer wasn't the only thing we planned to brew for the spring feast. Months earlier, Chris Ryan, our executive editor and a dedicated tea drinker, had observed that our project lacked caffeine. We had assumed that growing coffee and tea would be impossible, that the plants needed to be shrouded in tropical mists. But no! Tea could survive our climate, apparently, if coddled. Back in the fall, Chris had tracked down some plants at a South Carolina nursery that were mature enough to yield leaves in spring, and sent away for three of them.\n\nThey survived the winter, and as soon as the weather warmed up, they began to unfurl small, shiny green leaves. It only took about a week and a half to wither, roll, ferment, and dry our minuscule harvest. To celebrate, it seemed fitting to have a tea party. We summoned up our inner Britons and made dainty tea sandwiches\u2014and, not having the ingredients for scones, little tarts filled with clotted cream and preserves.\n\n_Picking tea_\n\nTeam Cow was celebrating, too. We'd finally found a new cow. It had taken months of calling all over the area and beyond\u2014to 4-H clubs, farms and ranches, even backyard cow owners. Our cow lived about one hundred miles to the south, at Claravale Farm, a raw-milk dairy near Pinnacles National Monument. It was a beautiful place set in a remote, grassy valley framed by mountains, with chickens clucking in the bushes, a pistachio orchard, several century-old buildings from the town that once stood there, and a milking herd of fifty-five Jersey cows.\n\nRon Garthwaite and his partner, Collette Cassidy, let us pick out a young, good-looking, chocolate brown Jersey, No. 64. We named her Holly, after Hollister, the nearest big town, and arranged to buy her (she would continue to live at Claravale, though.). We got to milk her by hand, which is much harder than you might think: milk shoots sideways, down your sleeve, or refuses to come out at all. It is, however, extremely relaxing to put your shoulder against a big warm animal and (once you get the hang of it) rhythmically squirt milk from soft, stretchy teats into a bucket. Mainly, though, we used an individual milking machine, with tubes that attach to each teat and pipe the milk straight into an enclosed container. This is more efficient (the cow is milked in ten minutes, versus up to forty-five), more comfortable for the cow, and more sanitary, since the milk is never exposed to the air, flies, or whatever might happen to fall into an open bucket.\n\nAnd Holly's milk was a revelation. It tasted sweet and pure, with an indescribable lightness of texture and the slightest hint of grass. \"It hasn't been homogenized, pasteurized, standardized, or fortified,\" said Ron. He'd just described the milk we'd all taken for granted as the real thing\u2014until now.\n\n_Ron Garthwaite and Collette Cassidy with their Jerseys_\n\nAfter each visit, we took Holly's milk home in one-gallon canning jars that we buried in a giant cooler full of ice. At Sunset, we drank it as it was, from the jars. If we felt we hadn't kept the milk cold enough, or if anyone at Sunset requested it, we gently pasteurized some. So far, we've used the milk to make rich, delicious, sweet-tasting ricotta with a much higher yield than from store-bought milk\u2014doubtless because Jersey milk has more butterfat and proteins than the milk of any other breed. We'll be trying out all kinds of other cheeses, too, in the months ahead\u2014and yogurt, butter, and ice cream.\n\nBy March, our spring garden had sprouted in a neat rectangle of green leaves of various hues and textures. Johanna had broadcast the herb seeds\u2014meaning she scattered them the way nature would\u2014and sowed the other crops in rows, so we had a combination of soft masses and orderly lines. The carrots had been heavily seeded in case some refused to germinate, but most of them had sprouted, forming a dense patch. Johanna carefully pulled out more than half to make some growing space for the rest. The thinnings were delicious in salad.\n\nWhile out in the garden one day, we noticed that Alana, layer of pretty green eggs, was not her perky self. Her tail was drooping and she had a sad, lethargic look in her eyes, as though something very bad were going on internally.\n\nElizabeth read up on droopiness and lethargy. She concluded that it was most likely egg binding, which is when an egg gets stuck in the oviduct; it can be fatal if not fixed. The home remedy involves helping the chicken relax. So, following the standard advice, we massaged Alana's vent with mineral oil (not as bad as it sounds) and gave her a warm bath. We also massaged her stomach in the direction of the vent to coax the egg out, but felt nothing.\n\nAlana got worse and worse. After about five days, she was nearly unable to stand, and kept her eyes closed most of the time. By this point in our chicken keeping, we had decided we would limit veterinary care. We didn't want to be cruel to our chickens, but they were, after all, farm animals, and if they were in pain we would give them a quick, merciful end. Also, their diseases were often not treatable. That said, we did want to know whether Alana had something that could harm the rest of the flock.\n\nSo Elizabeth took Alana to an avian vet in nearby Mountain View. He was stumped, too, but he came up with many untreatable possibilities, from Marek's disease (a deadly contagious virus) to botulism to heavy-metal poisoning. \"Her prognosis doesn't look good,\" he told her. The choice was pretty clear. We decided to put her out of her misery and send her body to the state lab, to understand what had happened to her.\n\nThis was the first death in our one-block project. Okay, it was the first warm-blooded death\u2014we'd had hundreds, if not thousands, of dead bees. After investing a lot of time and concern in one chicken, Elizabeth found the death difficult, not only because she was present for it but also because she had made a real effort not to get attached to any of the hens. Plus, she had always tried to keep in mind that she (like the rest of us on Team Chicken) enjoyed a good chicken curry and that our own chickens were perfectly edible. So her sadness took her by surprise.\n\nWe each ended up feeling differently about Alana's end. Jim, an animal lover to his core, sensed even more acutely the innocence and vulnerability of animals that come under the care of humans. \"It lays superiority and responsibility at your feet, even when you don't want them,\" he said. Elizabeth said it gave her some insight into empathy and how it should be directed. \"The chickens are not pets. They're animals that have a job to do. I had to ask myself, how sad do I feel about this? And I had to control my empathy, which was an eye-opening experience. I might not give them human names next time.\" Other people on staff (not on Team Chicken, however) had no qualms about killing one of our flock, whether to end suffering or to put meat on the table. As for me and where I stood on the detachment scale, well, probably near Elizabeth. Mostly, I felt gratitude for that chicken and her beautiful, delicious eggs, which she had so generously produced day after day.\n\nSeveral weeks after Alana's body went off to the lab, we got the results back. She didn't die of anything infectious, but of kidney failure, and she had some inflammation around the heart. Also, a lot of internal fat. We were probably overfeeding the chickens. No more enchiladas, girls! We cut back on kitchen scraps and started giving them only greens, with a few other fruit and vegetable bits. And Johanna put together a movable garden enclosure for them out of chicken wire and rebar, so they could run around a little more and scarf up slugs at the same time. Unfortunately, we couldn't give them free rein of the garden because they would also eat our vegetables, which we were coaxing toward maturity for our spring feast.\n\nWe had had phenomenal luck getting our winter garden to ripen all at once. That was not going to happen with this one. The favas were ready now, in April (right on schedule for favas), and so were the radishes, but everything else needed a few more weeks of grow time. We weren't worried: Favas keep well layered between damp paper towels or newspaper, and we quickly replanted the radishes (they grow fast). We harvested twenty-five pounds of big, meaty pods from those fava vines in about half an hour and packed them away, along with their entirely edible greens, in the fridge.\n\nBeverage-wise, we were moving right on schedule. Our beer had taken a couple of days for fermentation to kick in, but when it did, it blew the airlock clean off the carboy. Now, after six weeks of racking it and letting it percolate, Team Beer filled and capped a grand total of fifty-one bottles. We tasted it expectantly. Even though it wasn't yet carbonated, it seemed balanced, with a nice graininess and fragrant hops. After all that work, we had probably made some decent beer.\n\nThree weeks later, when carbonation was complete, we popped a few caps for a group taste. Maybe we hadn't cleaned our bottles quite well enough, or maybe something had crept into the brew while we were bottling, but the beer was undeniably funky. At first we tried to deny it, saying things like, \"It has a zingy, citrusy edge,\" and \"Boy, is that blond.\" The more honest among us noticed flavors of plastic jug and bathroom cleanser, and then we all did. Team Beer drew on its inner Buddhist and tried to think about the journey, not the end.\n\nNothing we seemed to do could keep the mites from swarming all over our bees. At best, we kept them at bay. Our newest tactic was to bring in an alternative hive called a top-bar. So far, we'd been using traditional Langstroth hives, which have stacked boxes and frames with preexisting foundation on which the bees build honeycomb and brood cells. The top-bar is a long single-story box you build yourself, with strips of wood (the bars) running across the top. The bees build their own comb, as they do in the wild, and anchor it to the bars. The top-bar was reputed to reduce disease and pests, and we were willing to try anything.\n\nFor this hive, we installed bees for the first time. We had purchased our first two hives as nucs (new colonies complete with queens). We built this one from scratch. The bees had arrived by mail, in a very buzzy shoebox-size box, with the queen in a small, separate capsule inside. During the trip, she had been emitting the powerful pheromones that were gradually bonding the other bees to her, but without her own traveling compartment, they might have killed her first. We named her Califia, after the queen of the mythical island of California. She was a beautiful bee, the color of a ripe apricot.\n\nWe removed the cork from the bottom of the queen's capsule and stuffed in a marshmallow instead, which the other bees would chew through in a few days, giving the queen time to fully cast her scent-spell over her subjects before they released her. We hung the little cage inside the hive. Kimberley shook the box of bees over the open hive, and they fell in with the sound of rice pouring from a box. After a couple of hours, they were already out on scouting flights. Three weeks later, fourteen of the fifteen bars had comb descending from them in snowy white lobes. Our new bees were busy and capable. Now we hoped they would stay healthy, too.\n\nA couple of days before our spring dinner, we checked out the favas. The pods were fine, and enough of the greens had lasted to give us a filling for two big quiches, which together formed the centerpiece of our menu. Then we went into the garden with Johanna and harvested our crops, pulling up golden and red beets and dusty pink ones\u2014the amazing 'Chioggias' that, when sliced crosswise, looked like swirly peppermint lollipops. The carrots were just a matter of pulling, too, and all the herbs were easy\u2014we just snipped. Pull on a green onion, though, and it snaps off. You have to dig around each one and then tug to get the whole thing.\n\n_Harvesting fava beans_\n\nIn the kitchen, we tasted our vegetables so we would know how best to cook them. With a bowl of water for swishing and a knife for root trimming, we chomped straight from the basket. Everything was sweet and juicy, especially the radishes, which were as crisp and mild as apples, and the green onions, which had only a hint of heat sneaking in toward the end of the chew. The frilly little chervil had a clean, good flavor. French tarragon numbed our tongues with a hit of potent licorice, like it was supposed to. The strawberries were best of all\u2014dead ripe and supersweet. We were glad our plan for them was simple: We wouldn't get in their way.\n\nA breeze kicked up on the day of our feast, so we ate inside, next to sliding doors open to the garden. The beer was there for the curious, and we still had plenty of Chardonnay to sip as we stood around eating freshly baked buttermilk crackers with sweet ricotta, mint, and fava beans. We had a platter of radishes, too, with fresh butter and salt\u2014it was all they needed.\n\nWe covered the table with pink roses that Johanna had picked, and bottles of wine and pitchers of strawberry lemonade. Everything was served on platters: mesclun salad with paper-thin circles of red, golden, and lollipop-swirled beets; tall, custardy quiches laced with saut\u00e9ed fava leaves; and a warm salad of grilled carrots with tarragon. Elaine was in charge of dessert, and she had been experimenting with our strawberries for several days. In the end, she turned away from strawberry fromage blanc cheesecake (because we already had a crusty dish, the quiche) and strawberries simmered in wine syrup (it tasted wintry). Instead, she spooned homemade fromage blanc into bowls, added the strawberries, and drizzled warm lemon-infused honey on top. It was so simple, and it was just right.\n\nThe dinner looked like a garden in full bloom, and it tasted wonderful, too. And yet the real triumph\u2014as with each of the feasts that preceded it\u2014happened on the way to the table.\n\nWe had coaxed food from pure nature, and that had required forming a relationship with it. We'd been moved to tears and totally frustrated; we'd been overjoyed and awestruck. We'd seen into the microscopic heart of cheese, beer, and wine; worked with bees, chickens, and a cow; and made gardens that grew like green symphonies, each little plant playing its part, contributing its flavor and beauty to the whole. Our project had taken place, more or less, on one block, but what we learned from it went far beyond.\n\n# THE SPRING GARDEN\n\nTender greens, crisp green onions, and plump radishes are cool-season crops, meaning they grow best in cooler temperatures. Long periods of hot weather can cause them to turn bitter and to bolt (set seed) before they produce edible parts. In mild-winter climates, plant them in very early spring so they will mature before summer heat settles in, or in late summer for a fall or winter crop. Because growing conditions vary by region (and even within neighborhoods), check the planting times for your area \"One-Block Feasts Across the Country.\" Many of the crops we planted thrive in full sun in cooler areas, which means the planting location needs at least 6 hours of sun per day. In hot climates, give them part shade. Some crops, such as mesclun and radishes, are easy to grow from seeds. Others on our spring menu, including strawberries, are easier if you start with bare-root plants or nursery seedlings. For seed-starting and planting tips, see here.\n\n_10 feet_\n\n**SPRING GARDEN PLAN**\n\n1. Tea\n\n2. Nasturtiums\n\n3. Peppermint\n\n4. Chervil\n\n5. Dill\n\n6. Fava beans\n\n7. Carrots\n\n8. Beets\n\n9. Arugula\n\n10. Mesclun\n\n11. Radishes\n\n12. Strawberries\n\n13. Green onions\n\n14. Tarragon\n\n15. Thyme\n\n16. Parsley\n\n17. Lemon tree\n\n18. Tangerine tree\n\n### Arugula\n\nSee the Winter Garden.\n\n### Beets\n\nBest known for their edible roots, beets also send up leaves that are tasty in soups and salads if picked when they are young and tender. The plump, heart-shaped bulbs come in various colors, from red and deep plum to golden. We chose three heirlooms: 'Bull's Blood', with blood red foliage that adds color to salads; golden, which forms sweet, mild-tasting globes; and supersweet 'Chioggia', whose roots reveal alternating rings of white and pink when sliced.\n\n**Best Site** Full sun and fertile, well-drained soil enriched with compost.\n\n**Days to Harvest** 40 to 52 days for 'Bull's Blood', 55 days for golden, and 54 days for 'Chioggia' from seed.\n\n**Planting and Care** In mild climates, sow seeds in early spring or late summer so plants will mature in mild weather. Space them 1 inch apart, then cover with \u00bc inch of compost. (To prolong the harvest, sow at monthly intervals.) When the plants are small, thin them to 3 inches apart. Keep the soil evenly moist, and apply a dilute fish emulsion after the tops are up.\n\n**How to Harvest** Pull up the roots when they are about 3 inches wide (the bigger they grow, the woodier they get).\n\n**Seed Sources** Burpee, www.burpee.com, and Ed Hume Seeds, www.humeseeds.com.\n\n### Carrot 'Nantes'\n\nPulled straight from the earth and showered at the sink, homegrown carrots taste sweeter than any you can buy at the grocery store. But flavor, even among homegrown types, differs by variety. We grew 'Nantes', which forms blunt-tipped orange carrots about 7 inches long\u2014great for munching raw. They are also delicious roasted with other root vegetables or grilled with thyme (see recipe).\n\n**Best Site** Full sun with light, deep, well-drained soil free of clods and stones. Raised beds filled with planting mix are nearly perfect.\n\n**Days to Harvest** 70 days from seed.\n\n**Planting and Care** In mild-winter climates, sow seeds in early spring when the soil is warm (carrots are at their sweetest when the last few weeks of growth occurs in cool weather). Or, sow in early fall. Soak the bed, then scatter the seeds thinly on top, in rows 15 inches apart. Cover with \u00bc inch of compost to keep the soil surface from crusting. When the tops are 1 to 2 inches tall and have two or three leaves, thin the seedlings to 2 inches apart (eat the thinnings in salads or steam them in butter). Keep the soil evenly moist throughout the growing season. After the first thinning, work a narrow band of complete fertilizer into the soil about 2 inches beyond the row.\n\n**How to Harvest** Begin to harvest when the carrots are fully grown (about 7 inches long; pull one up to check) and the tops are green and full.\n\n**Seed Source** Burpee, www.burpee.com.\n\n### Chervil\n\nThis lacy annual herb\u2014one of French cooking's four classic fines herbes, along with tarragon, parsley, and chives\u2014resembles parsley, but is paler green and more delicate. It tastes like parsley, too, but with overtones of anise. We like to add the fragile whole leaves to salads, fold them into omelets, or float them on soups. As the plant matures, 1- to 2-foot-tall flower stems topped with white blossoms will rise above the low mound of ferny foliage. Cut those off\u2014ideally before they open\u2014to keep the tasty leaves coming (flowering causes the foliage to lose its flavor and aroma). Spring crops bolt (flower and set seed) quicker than fall crops, because they mature in warmer weather.\n\n**Best Site** Part shade and well-drained soil.\n\n**Days to Harvest** 6 to 8 weeks from seed.\n\n**Planting and Care** Plant seeds in early spring or in fall (where winters are mild), sowing them about \u00bc inch deep and 2 inches apart. When the seedlings are up (they germinate in 10 to 14 days), thin them to about 4 inches apart. Keep the soil evenly moist, and don't let the plants dry out completely between waterings or they will set seed. Feed every 3 weeks or so with liquid fish emulsion.\n\n**How to Harvest** Snip off tender young leaves as needed and use them immediately. The delicate flavor of chervil is volatile and can dissipate soon after harvest (add it to cooked dishes at the last minute).\n\n**Seed Sources** Renee's Garden, www.reneesgarden.com, and John Scheepers Kitchen Garden Seeds, www.kitchengardenseeds.com.\n\n### Dill\n\nBig, umbrella-like clusters of yellow flowers and soft, feathery foliage make dill as pretty as a spring wildflower. The blooms that poke above the stately 3- to 5-foot-tall plant provide nectar for butterflies and beneficial insects. But the most compelling reason to grow dill is the pungent aroma of its seeds and leaves, which can flavor so many dishes. You can use the seeds in pickling and in vinegar and the leaves to flavor sauces and soups.\n\n**Best Site** Full sun and well-drained soil; protect from the wind.\n\n**Days to Harvest** 40 to 50 days to harvest leaves, 85 to 105 days to harvest seed.\n\n**Planting and Care** In early spring, sow seeds directly in the ground, \u00bd inch deep and 1 to 2 inches apart in rows 6 inches apart. (Once the plants' deep tap roots get growing, transplanting will be difficult.) Thin the seedlings to 12 inches apart when they are 2 to 3 inches tall. Water regularly\u2014about once a week during dry periods. If you don't want plants to resow (which they will, vigorously), shear off the heads before they set seed, or watch the seeds carefully and harvest them promptly (see below).\n\n**How to Harvest** Snip off leaves as you need them. To collect the seeds, tie small paper bags or tightly woven cheesecloth (double or triple layers if necessary) over the seedheads when the seeds begin to turn brown, and leave the bags in place for a week or so. Then snip the stems below the bags, bring the heads into the kitchen, and shake the seeds out onto a tray. Store them in airtight containers.\n\n**Seed Source** Renee's Garden, www.reneesgarden.com.\n\n### Fava Bean\n\nNot a true bean, the fava is actually a giant vetch (an ancient type of legume). Unlike true beans, it is cold hardy, which means gardeners in mild climates can plant it in the fall for harvest in late winter or early spring (as we did). As long as the weather stays cool, the plants can last into midspring. You can cook and eat the immature pods like edible-pod peas, or let the pods hang on the plant to ripen into dry shelling beans. One caveat: Some people of Mediterranean, Asian, and African ancestry have an enzyme deficiency that can cause severe reactions to these beans and their pollen.\n\n**Best Site** A mild, sunny location with loose, fast-draining soil.\n\n**Days to Harvest** 65 days from seed.\n\n**Planting and Care** Sow seeds as soon as the soil is warm. Heavy leaves must push through the soil, so be sure the soil is loose and open. For a bush type of fava, which is what we grew, plant seeds 1 inch deep and 1 to 3 inches apart, allowing 24 to 36 inches between rows. Moisten the soil thoroughly before planting, then do not water again until seedlings have emerged. Keep the soil evenly moist throughout the growing season. Fertilize the soil after the plants are in active growth and again when the pods start to form, working a complete fertilizer into the soil along the row.\n\n**How to Harvest** To eat the pods, pick them when they are small (about 3 inches long), plump, and deep green. To eat the fresh beans inside (but not the pod), wait until the pod and beans are larger but are still bright green. For dry yellow shelling beans, wait until the pods have blackened and drooped.\n\n**Seed Source** John Scheepers Kitchen Garden Seeds, www.kitchengardenseeds.com.\n\n_Fava beans_\n\n### Green Onion 'White Lisbon'\n\nSometimes called scallions, green onions are either bulbing onions that you harvest young (before the bulbs grow), or bulbless bunching onions. We grew 'White Lisbon', a green bunching type with delicate stems and sweet, juicy tops.\n\n**Best Site** Full sun and loose, fertile, well-drained soil.\n\n**Days to Harvest** 60 to 65 days from seed.\n\n**Planting and Care** Plant green onions from seed or nursery seedlings. Sow seeds \u00bc inch deep and \u00bd inch apart in rows 12 to 15 inches apart. Plant seedlings 4 to 5 inches apart. Keep the soil evenly moist and feed plants regularly with a dilute liquid fish emulsion.\n\n**How to Harvest** Pull up green onions when the tops are 12 to 18 inches tall.\n\n**Seed Source** Botanical Interests, www.botanicalinterests.com.\n\n_'White Lisbon' green onions_\n\n### Lemon 'Eureka'\n\nSee the Summer Garden.\n\n### Mesclun\n\nThe word _mesclun_ , from a southern French word for \"mixture,\" refers to an assortment of greens picked when young and tender. Sweet red and green lettuces, piquant green arugula, spicy mustard, peppery red Komatsuna, tender Swiss chard leaves\u2014some or all might turn up in one of these colorful mixes. Succulent \"weeds\" such as purslane and tender m\u00e2che might be included, too. You can choose a mix that contains the greens you like.\n\n**Best Site** Full sun in fall, winter, and spring in mild climates, and well-drained soil enriched with plenty of compost.\n\n**Days to Harvest** 35 days from seed.\n\n**Planting and Care** Scatter seeds over the prepared seedbed in spring. To prolong the harvest, sow seeds every 2 or 3 weeks. Thin seedlings to about 3 inches apart. Keep the soil evenly moist, and apply half-strength liquid fish emulsion after the first fully formed leaves appear.\n\n**How to Harvest** Snip off the young plants above ground level. Most will produce new leaves for your next harvest.\n\n**Seed Source** Burpee, www.burpee.com.\n\n### Nasturtium\n\nRound, bright green leaves of this pretty annual ( _Tropaeolum majus_ ) have a refreshing herbal fragrance. When young and tender, they also have an appealing, dewy sweetness and a peppery flavor (as do the buds and flowers) that adds zip to salads. You can grow vining nasturtiums to trail over the ground or climb (to 6 feet), or you can plant bushy dwarf types that top out at 18 inches. Either way, they will reseed themselves: We still have offspring of 'Copper Sunset' popping up here and there, even though we haven't planted it for years.\n\n**Best Site** Full sun or part shade and well-drained or sandy soil.\n\n**Days to Harvest** 45 to 60 days from seed.\n\n**Planting and Care** For an early summer harvest, sow seeds in March or April, or for a spring harvest, sow in October in mild-winter or hot-summer areas. Plant the large seeds 10 to 12 inches part and 1 inch deep. Keep the soil well watered throughout the growing season. Dig a little compost into the soil before planting and your nasturtiums won't need fertilizer.\n\n**How to Harvest** Pick leaves and flowers as you need them. For the best flavor, harvest them early in the morning when temperatures are cool.\n\n**Seed Source** Renee's Garden, www.reneesgarden.com.\n\n### Parsley 'Gigante Italian'\n\nSee the Summer Garden.\n\n### Peppermint\n\nSee the Summer Garden.\n\n### Radish 'Easter Egg II'\n\nAmong the easiest vegetables to grow, radishes are also fast to mature. We especially like 'Easter Egg II', a feast for the eyes that blends pink, rose, purple, and white radishes in one colorful mix. All have crisp, juicy white flesh. The best part: You can harvest from 2 to 5 pounds per 10-foot row!\n\n**Best Site** Full sun in mild climates and part shade where it is hot; fast-draining soil that's been well amended with compost.\n\n**Days to Harvest** 25 days from seed.\n\n**Planting and Care** Sow seeds as soon as the soil is workable in spring, then at weekly intervals until warm weather approaches (plants go to seed when temperatures rise). In mild climates, you can also sow seeds at regular intervals in fall and winter. Plant the seeds \u00bd inch deep and 1 inch apart in rows 1 to 1\u00bd inches apart. Keep the soil evenly moist from seed to harvest. Ten days after planting, feed with a dry or liquid fertilizer (any kind), applying it alongside the rows.\n\n**How to Harvest** Pick as soon as radishes reach full size (about 3 weeks after sowing, as noted above, but longer for slower varieties). If you leave them in the ground too long, they will turn woody and too pungent to eat.\n\n**Seed Source** Burpee, www.burpee.com.\n\n### Strawberry 'Sequoia'\n\nNothing tastes sweeter than a plump, sun-warmed strawberry picked at the peak of ripeness. It is nature's best dessert\u2014divinely succulent and juicy, low in calories (about 50 per cup), and high in vitamin C. If you only have room for growing one crop, we suggest this one, since many commercial harvests are treated with chemicals.\n\nWe chose 'Sequoia', a locally adapted variety that bears its entire crop once a year. (It is called a \"June bearer,\" but we had berries in May.) The plants reach 6 to 8 inches tall and spread by runners about 12 inches across. The berries are large and luscious.\n\n**Best Site** Full sun and loose, well drained, well-amended soil.\n\n**Days to Harvest** 90 to 104 days from January-started bare-root plants.\n\n**Planting and Care** Set out bare-root plants in January or February (trim their roots back to about 6 inches and soak them in water for about 30 minutes before planting). You can also set out plants in late summer or fall for a spring harvest. Plant seedlings from small pots in late February or March. Plant with the crown (the point where the leaves come together at the stem base) slightly above the soil level (a buried crown will rot); the topmost roots should be about \u00bc inch below the soil. Mulch to deter weeds (we used straw, which did a good job of keeping ripening fruit clean). Keep birds from stealing your berries by covering the bed with netting or floating row covers. Water regularly (at least once a week) during the bearing season. Feed June-bearers twice a year\u2014lightly when new growth starts, and more heavily after fruiting. Give ever-bearing types regular light feedings throughout the growing season, and alpine strawberries once in early spring and once after fruiting begins. For all strawberries, use a complete fertilizer.\n\n**How to Harvest** After a fruit has colored up, pinch through the stem with your thumbnail to detach it.\n\n**Plant\/Seedling Sources** Bare-root plants and seedlings are sold at nurseries.\n\n_'Sequoia' strawberry_\n\n### Tangerine 'Dancy'\n\nSee the Winter Garden.\n\n### Tarragon, French\n\nThis sprawling, largely flowerless woody perennial ( _Artemisia dracunculus sativa_ ) has dark green, aromatic leaves with an anise flavor and grows to less than 2 feet tall. The leaves are a classic seasoning for chicken dishes, sauces, and vegetables. Be careful not to plant plain old Russian tarragon\u2014 _A. dracunculus_ \u2014by mistake; it has tougher, narrower leaves and a pungent, bitter flavor.\n\n**Best Site** Full sun and rich, well-drained soil.\n\n**Days to Harvest** Leaves can be harvested any time.\n\n**Planting and Care** In early spring, as soon as the soil starts to warm up, set plants about 24 inches apart in garden beds or put a single plant in an 8-inch-deep container. Water regularly and let the plants dry out between irrigations. Apply a complete fertilizer during spring growth and again after major harvests. To keep a fresh supply coming, cut plants almost at the ground during the growing season. Divide the plants every other year during early fall.\n\n**How to Harvest** Pinch off leaves.\n\n**Seedling Source** Mountain Valley Growers, www.mountainvalleygrowers.com.\n\n_French tarragon_\n\n### Tea\n\n_Camellia sinensis_ is the tropical shrub grown commercially in Asia for making green, black, and oolong teas. It is rare in the United States, and we couldn't find mature plants at nurseries or growers on the West Coast, so we ordered them from a source in North Carolina (Camellia Forest Nursery, www.camforest.com or 919\/968-0504). But if you are patient, you can order seedlings from the two sources listed at right.\n\nOur three pretty, round tea shrubs have leathery, dark green leaves and fragrant white fall blooms. Initially we planted them in big pots so we could move them under the eaves in winter (the plants are frost tender), but they weren't happy, so now they are in the ground (we will cover them when frost hits). The plants will eventually grow to 15 feet tall.\n\n**Best Site** Full sun, or light shade inland, and well-drained soil enriched with plenty of compost, or use potting soil. Protect from wind and from winter frost.\n\n**Days to Harvest** Plants must be at least 3 years old.\n\n**Planting and Care** Set the plant in the soil so its base is above soil level, then keep the roots cool by applying a \u00bd-inch-thick layer of mulch (keep it away from the stem's base). Young plants are best in light shade under tall trees. Water your tea plant regularly, at least weekly during the April to October growing season for the first few years; older plants can get by with less water. Feed occasionally with an acid plant food formulated especially for camellias.\n\n**WHEN TO HARVEST** In spring, pick the two uppermost new leaves and the new buds. To prepare them for making tea, see How to Make Tea.\n\n**SEEDLING SOURCES** Greer Gardens, www.greergardens.com, and Nichols Garden Nursery, www.nichols gardennursery.com.\n\n### Thyme, French\n\nSee the Summer Garden.\n\n_\u2014Kathleen N. Brenzel_\n\nTHE \"IMPORTS\"\n\n**We went outside our garden for these local foods:** Milk, whole and unhomogenized, from our own cow; active dry yeast; whole-wheat flour ('Sonora' and 'Expresso'), garlic (ours ran out); 'Spanish White' onions (ditto)\n\n_Ron Garthwaite and Holly, our cow, at Claravale Farm (top); Pouring Holly's milk into bottles at Claravale Farm (bottom right)_\n\n# THE SPRING PROJECTS\n\n##    \nHOW TO OWN (OR CO-OWN)  \nA DAIRY COW\n\nGetting a cow might seem like a large leap for us office-working suburbanites. (Like jumping over the moon, you say?) We had good reasons. Milk had been an integral part of our one-block feasts from the start, giving our vegetarian menus protein and making cooking more interesting; like eggs, milk is one of those building-block ingredients that open up all sorts of possibilities for recipes. We had been \"importing\" unhomogenized, pasteurized bottled milk from Straus Family Creamery (by buying it at the store). But if we could have a cow\u2014especially a Jersey, a breed that gives particularly rich, delicious milk\u2014we could make a key ingredient our own.\n\nWith a cow of our own, we'd see firsthand the fluctuations in milk from season to season. We'd be able to try fresh, unprocessed milk, a food that most of us, in our age of mass-produced dairy products, had never tasted\u2014even though humans have been consuming it since at least 5000 BCE. And we'd understand more about how this taken-for-granted substance is made.\n\nAlso, several of us happened to really like cows, the way some people like horses. We wanted to get to know one of these patient, generous animals whose relationship with humans has profoundly shaped our culture.\n\nTo our amazement, our city, Menlo Park, allows backyard cows. We imagined what it might be like to have a brown Jersey hanging around the office, contentedly grazing on Sunset's lawn. For many reasons, including our complete and utter lack of experience, we quickly decided our cow would be better off living on a farm and being taken care of by people who knew what they were doing\u2014and could teach us.\n\nThe rewards so far include learning to milk by hand (well, sort of); the pleasure of wandering around a beautiful and well-managed farm with healthy cows; the incomparable flavor of sweet, fresh Jersey milk; and the richness and nuance of the cheeses we've made. Because this is our newest project, what is written here reflects only our first steps. We'll be describing our continuing experiences on our blog, <http:\/\/oneblockdiet.sunset.com\/team-cow>.\n\n###    \nONE DAIRY COW\n\nHolly is an 800-pound, 2\u00bd-year-old Jersey. She gives between 5 and 6 gallons of milk a day.\n\n#### HOW TO DO IT\n\nYou can own a cow three ways:\n\n**1. Buy the cow and keep her at home** This is a serious commitment, involving twice-daily milkings and learning how to keep lactation going (by having your cow impregnated every 12 to 18 months; a cow has to have a calf in order to produce milk). It also involves figuring out what to do with the calves (sell or use them for beef? Occasionally keep a female calf to raise as another milk cow?). You also need to have enough space for a cow and her calf and be sure you can properly feed them (a lactating cow eats a lot) and keep them healthy (see Raising Dairy Cows for helpful resources that will tell you more). And a cow is expensive: A good adult dairy cow costs as much as $2,500, a calf around $500. That said, she will give you wonderful milk, enough to share, if you like; meat, if you choose (her calves); and manure for the garden and pasture. Keeping a single cow at home is also the safest way to have raw milk, because one cow is easier to keep clean than a herd, and you have more control over the milk.\n\nCheck with your city to see whether local rules let you keep a cow, and then start learning everything you can about cows, including differences between breeds. Jerseys are often suggested as ideal backyard cows because they are gentle and friendly, smaller than other breeds and thus easier to handle, and also produce richer milk but less of it\u2014a volume that is easier to deal with than the 14-gallon-a-day flow of, say, a Holstein. You might also consider sharing your cow with neighbors to offset the expenses and to divide up the milk.\n\nMost sources say that beginners should look for a cow pregnant for the second or third time. She'll be \"trained\" in milking, and you won't have to deal with the impregnation aspect of owning a cow for at least a year.\n\n**2. Co-own or share a cow who lives on a farm** Cow-sharing is most common in states where the sale of raw milk is banned, because it gives the owners legal access to their cow's fresh milk. The ownership fee, paid to the farmer, covers the cost of the cow's care and board. A cow-share contract can also be set up as a share in an entire herd. For more on how to find or set up a cow-share (or goat-share), see Helpful Information.\n\nPerhaps you are wondering whether cow-shares exist at regular (non-raw-milk) dairies. To the best of our knowledge, they don't, and for two reasons: Most commercial dairies aren't set up to capture the milk of a single cow. They are very large operations that pump milk straight from the entire herd into refrigerated trucks, which then drive the milk to a separate facility\u2014the creamery\u2014for pasteurizing and processing. Also, their contract with the creamery is usually exclusive, which means they can't do business with outside parties. In any case, it might not be a good idea to share a cow from one of these dairies. Since the milk is destined for pasteurization, the allowable pathogen count is higher than at a raw-milk dairy. In other words, it would be risky to drink the milk unless it was pasteurized.\n\n**3. Own a cow who lives on a farm** Like cow-sharing, this arrangement isn't common in California, because the sale of raw milk is legal in our state. However, Ron Garthwaite of Claravale Farm, a raw-milk dairy near Paicines, California, was willing to make a one-time arrangement with us to own one of his cows and board her at his farm. He has also agreed to manage her care and give us lessons in the art of cow keeping.\n\nAlthough we did not set out to find a cow at a raw-milk dairy, we've since learned that the legal requirements for cleanliness at such an operation are much more stringent than for a pasteurized-milk dairy, since the milk will be leaving the premises raw. We also see how healthy Ron's cows are and notice the care with which he handles them. His farm is inspected regularly and has never been cited. He doesn't pool the milk with that of any other herd, he supervises every step of its (minimal) production, and he distributes it only in California. All this reassures us that the milk we're getting is of the highest quality.\n\n#### WHAT TO USE\n\nClaravale Farm supplied most of the equipment we needed, as part of our boarding contract. We've included prices anyway, should you be interested in keeping a cow at home. The source, unless otherwise mentioned, is Nasco dairy supply, 800\/558-9595 or www.enasco.com\/farmandranch\/Dairy+Supplies.\n\n**_Teat Dip_** An iodine-based sanitizing solution for the teats. About $23 for a 1-gallon jug.\n\n**_Plastic teat dipper_** A cup, filled with teat dip, that makes dipping each teat easier. About $3.\n\n**_Individual cow milker_** Commercial dairies use a system of tubes to carry the milk of several (or more) cows at once into a large chilling tank. For one cow, you need a self-contained setup; this one includes a vacuum pump, an electronic pulsator, the \"claw\" (suction cups that attach to the teats), the tubing, and the milking pail. $1,749 from Hoegger Supply Co., 800\/221-4628 or www.hoeggergoatsupply.com\n\n**_Stainless-steel milk strainer_** Right after we collect the milk, we pour it through this to separate out any dirt. Includes a steel mesh strainer disk. $199.\n\n**_6\u00bd-inch round milk filter disks (non-gauze)_** Thin, paperlike disks that fit into the strainer, on top of the steel mesh strainer disk. $6.25 per box of 100.\n\n**_Aluminum milk pail or jug_** To hold the strained milk. $40.\n\n**_1-quart widemouthed canning jars_** We like this size for transporting milk, because the jars are small enough for the milk to cool quickly when immersed in ice water. Also, they are easy to clean, and the wide top allows us to skim the cream without effort. From about $18 for a case of 12 jars (enough for Holly's afternoon-milk yield of 2\u00bd to 3 gallons) at a well-stocked grocery store, hardware store, or cookware store.\n\n**_A big cooler (about 2\u00bd feet long), full of ice_** For rapid cooling and transporting of the milk in the trunk of our car. From about $75 for a 75-quart cooler at a hardware store.\n\n**_Dairy thermometer_** For keeping tabs on the temperature of the milk during transport, and to measure the temperature of the milk if you pasteurize. Get the kind that you can clip onto the side of the pot, and make sure it's accurate (see here for calibrating instructions). About $25 from the Beverage People, www.thebeveragepeople.com or 800\/544-1867.\n\n**_21-quart boiling-water canner with canning rack_** If you want to pasteurize (see opposite). When filled with hot water, the canner makes a great water bath for your milk pot. From $16 on www.amazon.com.\n\n**_8-gallon heavy stockpot_** If you opt to pasteurize, you'll need this, too.\n\n**_Stainless-steel perforated cheese ladle or large slotted spoon_** For pasteurizing. We like the large, flat surface of the cheese ladle because it's more efficient for stirring the milk. From $7 for the ladle at www.thebeveragepeople.com.\n\nHOW TO BATCH-PASTEURIZE MILK\n\n1. For equipment details, see opposite. Fill the stockpot with water and bring to a boil. Add the thermometer (submerging only the stem) and the ladle; boil for 10 minutes. Transfer the ladle and the thermometer to a sanitized work surface and pour the hot water into the boiling-water canner. Add enough water to fill to within 1 inch below the top ring, insert the canning rack upside down (handles down and folded under), cover, and bring water to a boil over high heat. Meanwhile, fill a sink with cold water (your emergency cool-down spot).\n\n2. Pour the milk into the stockpot. Attach the dairy thermometer to the side of the pot, and ease the pot into the hot-water bath until it rests on the canning rack. The water should come 2\u00bd inches up the sides of the pot; add more boiling water if you need to. Keep the burner on high.\n\n3. Stir the milk with the ladle 20 times, using a slow surface-to-bottom circular motion to distribute the heat evenly. Watch the thermometer; when it hits 140\u00b0F, turn off the burner and let the milk rise to 145\u00b0F. Hold it at 145\u00b0F or just above, covered, for 30 minutes, boosting with a blast of heat from the burner if necessary, or removing the pot from the canner if the temperature gets above 150\u00b0F (you can also put the pot in the sink to cool down rapidly).\n\n4. Add plenty of ice to the sink of water and immerse the milk pot in the ice water. Let the milk cool, stirring with a slow surface-to-bottom circular motion every now and then. If you're making cheese, let it cool to the temperature specified in the recipe, then use immediately. Otherwise, cool the milk to room temperature, then pour it into the cleaned canning jars and refrigerate.\n\n#### HOW TO DO IT\n\n**1. Find the cow** For our best general advice, see see here.\n\nCows are few and far between on the San Francisco Peninsula. Educational farms near us had cows, but they were already being milked to their limits. 4-H clubs focus on raising younger animals, not adult dairy cows. The backyard cow owners we tracked down had all decided to let their animals \"dry off\" for a while, meaning they weren't producing milk.\n\nWe finally found Claravale Farm, and Holly, through another farm (the owners had bought their cow from Claravale), and through a state brand inspector, who handles documentation when cows are sold and who pointed us in Claravale's direction. With the free assistance of the Farm-to-Consumer Legal Defense Fund (see Helpful Information: Cow-Shares), we drew up an initial bill of sale and a cow-boarding agreement. Holly's price was $2,500, but we sold most of her milk in advance back to Claravale, since we knew we wouldn't be able to use it and the Claravale owners definitely could. The cost per year worked out to be $270.\n\n**2. Milk the cow** She ambles into the milking barn, ideally 12 hours after the last milking, lowers her head to the trough, and starts chewing grain\u2014the treat that puts her in a good mood and encourages the \"letting down\" of her milk. Meanwhile, a stanchion holds her head in place just in case she tries to back out.\n\nBefore milking starts, we dip each of her four teats in a bright red iodine sanitizing solution. The first few squirts go onto the ground, to clear the ducts of any dirt and also the iodine once it has done its job. Then we wipe off the iodine with a clean damp paper towel.\n\nWe tried hand milking, and it's quite a bit harder than you might think. Each teat fills up with milk when the cow \"lets down.\" To keep it from shooting back up into the udder instead of into the pail, you have to close off the teat right where it meets the udder, using your thumb and forefinger; then you squeeze the milk in the teat down and out with your remaining fingers. Ron demonstrated on Holly and a thick stream of milk shot into the pail. When we tried, we got little dribbles. Well, room for improvement! The truth is that cows are usually milked with a milking machine, which gets the job done in a fraction of the time and keeps the milk cleaner than a set of hands squirting into an open pail. It hooks up to the cow by means of a \"claw,\" a configuration of four cylindrical suction cups, each of which attaches to a teat and rhythmically draws the milk off through pipes to a central chilling tank. So we could keep Holly's milk separate, Ron showed us how to attach a small, individual milking unit whose pipes end up in a portable canister. The claw goes on with a little _swooosh_ of pressurized air and, when the milk flow slows, pops off easily. After milking, we again dip each teat in sanitizer, and Holly saunters back out to pasture.\n\n_The \"claw\"_\n\n**3. Filter the milk** We immediately lug the milk canister to the cool milk-tank room, where we pour it into the strainer, set atop a sanitized steel jug. The milk is quite warm\u2014it's 102\u00b0F\u2014and sweet-smelling steam rises from it as we pour.\n\n_Filtering milk into a sanitized jug_\n\n**4. Bottle the milk** We've lined up our jars in a long stainless-steel sink. Now two of us lift the jug and pour the milk as accurately as possible into the jars, trying not to spill. Another Team Cow member screws the lids on the jars as soon as they are filled and plunges the jars into the ice-filled cooler. There's always one jar that ends up only partly filled. We insert the dairy thermometer into this jar and fit it into the cooler too, covering it with foil as best we can to prevent sloshing; it is our temperature gauge. Fresh milk should be cooled right away to at least 40\u00b0F. At Claravale, the pipes take the milk directly to the chiller tank, which drops its temperature to 36\u00b0F within minutes after it has been in the cow. We're not as fast, but we're not far behind.\n\n**5. Transport the milk (and taste)** We drive back to Sunset and put the milk in the refrigerator. Usually we have a glassful first, because it is absolutely delicious.\n\n**6. Store the milk** Chilled at 40\u00b0F or below, it stays fresh and sweet for a week or so. Whenever we haven't been certain that we've kept the milk cool enough on its long ride back to Menlo Park, we've pasteurized it using the low-heat, long-time \"batch\" method, which destroys any harmful organisms but interferes least with the milk's light, fresh flavor. Most commercial dairies use what is known as the high- temperature, short-time (HTST) method, which means that the milk is heated to 162\u00b0F (sometimes up to 171\u00b0F) for 15 seconds, or the ultrahigh temperature (UHT) method, which blasts the milk at or above 265\u00b0F for 2 seconds.\n\n**7. Cook with the milk** Jersey milk has more butterfat by weight than Holstein milk, the grocery-store standard, does (over 5 percent versus 3.6 percent), and more protein, too. This, and the fact that it's not homogenized or high-temperature pasteurized\u2014both of which alter the structure of the milk\u2014means it coagulates more readily and the cheese yield is higher. We made terrific, fluffy ricotta with Holly's milk, because the whey forms many more and fatter curds than whey from Holstein milk. When we made Gouda, though, it set so quickly and firmly that it turned into a sort of cheese superball. Our friends at Cowgirl Creamery, in Point Reyes, California (see How to Make Cheese), advised us to reduce the rennet by 40 percent when using Jersey milk. We'll try that next time.\n\nMILK: THE FRESH AND THE COOKED\n\nIn the world of food and drink, few controversies generate as much passionate outrage as the debate over raw milk. If you want to own a cow and drink her milk, you will have to decide where you stand. This is a simplified overview; to learn more, see Helpful Information.\n\nThe argument for fresh (raw)\n\nSupporters say raw milk is a pure, healthful food that is not only nutritious, but also capable of curing such chronic conditions as eczema, asthma, and even hepatitis. They also claim that pasteurization destroys vitamins and beneficial bacteria.\n\nThe argument for cooked (pasteurized)\n\nDetractors, which include the U.S. Department of Agriculture and the U.S. Food and Drug Administration, insist that raw milk should be universally avoided, and rules in many states ban the sale of it. They contend that pasteurization\u2014heating the milk to kill harmful microbes\u2014leaves the nutritional value of the milk intact.\n\nOur opinion so far\n\nRaw milk is not inherently harmful. It has to be invaded by bad bacteria to be unsafe, and if that doesn't happen, it's perfectly fine to drink. (Interestingly, if left to sour naturally, raw milk is inhospitable to those bad bacteria\u2014but we'd rather drink the milk when it's sweet.) That said, bacteria love raw milk, and can quickly contaminate it if the cow is diseased or dirty or the milk is mishandled. Many outbreaks of foodborne illness have been connected to raw milk. (As they have to a range of other foods, from hamburger to spinach.)\n\nUnfortunately, it seems pretty clear that our current system of mass production and distribution can't keep bacteria at bay. Raw milk needs to come from healthy cows, be checked vigilantly for pathogens, and be kept cold and very clean in order to be safe. It has to be handled as the fragile, fresh substance that it is, and requires a different and more careful (and more localized) way of selling.\n\nWe know our cow and how she is cared for, and how her milk is treated. We love how her milk tastes. So, when we know we've chilled it right and kept it cold, we drink it raw. And when we're in doubt, we pasteurize\u2014gently.\n\n#### HELPFUL INFORMATION\n\n##### BUYING DAIRY COWS\n\n\u2022 Your county's agricultural extension office; see www.csrees.usda.gov\/Extension\/index.html for a nationwide listing.\n\n\u2022 Small dairy farms, which periodically have cows for sale. Claravale Farm, in Paicines, California, where we bought our cow, Holly (and where we keep her), sells cows occasionally. Contact owner Ron Garthwaite through www.claravaledairy.com or 831\/628-3219.\n\n\u2022 Large-animal veterinarians often have clients who will sell a dairy cow.\n\n\u2022 Livestock brand inspectors in your state, who handle the paperwork whenever a cow is sold or moved, often know of cows for sale.\n\n##### RAISING DAIRY COWS\n\n\u2022 Your county's agricultural extension office may know of workshops near you. See www.csrees.usda.gov\/Extension\/index.html for a listing of offices nationwide.\n\n\u2022 _The Backyard Cow_ , by Ann Williams (Prism Press, 1979). A sweet, little British book that covers the basics of cow rearing.\n\n\u2022 _Chore Time_ , by Tim Wightman (Farm-to-Consumer Legal Defense Fund, 2008). Two DVDs take you through the ABCs of small-scale milk production.\n\n\u2022 _The Encyclopedia of Country Living_ (10th edition), by Carla Emery (Sasquatch Books, 2008). An invaluable resource for anyone interested in learning to live off the land, this personable, comprehensive guide (more than 900 pages long) covers everything from calculating harvest yields to stocking a fish pond to making your own furniture polish. The chapter on raising dairy cattle is an encouraging and friendly introduction, and it supplies a guide to breeds and where to find them, plus a handy list of questions to ask when buying a family milk cow.\n\n\u2022 _The Family Cow_ , by Dirk van Loon (Garden Way Publishing, 1976). A very good introduction to the dairy cow, complete with taking care of the land that feeds her and building fences to keep her in.\n\n\u2022 www.motherearthnews.com offers a mother lode of articles, many dating back to the 1970s but others current, on keeping cows. Search under \"Keep a Family Cow\" for the most comprehensive listing.\n\n\u2022 _Raw Milk Production Handbook_ , by Tim Wightman (Farm-to-Consumer Legal Defense Fund in association with the Weston A. Price Foundation, 2008). A beginner's guide to buying, raising, and milking a healthy, happy cow; it's the most current resource we've found.\n\n\u2022 _The Self-Sufficient Life and How to Live It: The Complete Back-to-Basics Guide_ , by John Seymour (DK Publishing, 2009). Written in 1976 by Britain's \"Father of Self-Sufficiency,\" this is an expanded edition. Like many DK books, it is beautifully illustrated, and the chapter on cows is inspirational.\n\n##### COW-SHARES\n\n\u2022 Farm-to-Consumer Legal Defense Fund, (703) 208-3276 or www.farmtoconsumer.org\/cow-shares.html\n\n\u2022 The Weston A. Price Foundation, a nonprofit devoted to promoting whole, unprocessed foods, has a list of state-by-state chapters that can help locate a share in your area, www.westonaprice.org\/chapters and also www.realmilk.com\/cowfarmshare.html.\n\n##### RAW MILK\n\n\u2022 www.realmilk.com (a branch of the Weston A. Price Foundation)\n\n\u2022 www.fda.gov\/Food\/ResourcesForYou\/Consumers\/ucm079516.htm\n\n\u2022 \"The Udder Truth,\" by Hannah Wallace; www.salon.com, January 19, 2007.\n\n\u2022 _Safe Handling: Consumers' Guide_ , by Peggy Beals (MI Fresh Milk Council, 2009). A guide to raw-milk handling from an advocate's perspective.\n\n\u2022 \"Should This Milk Be Legal?\" by Joe Drape; www.nytimes.com, August 8, 2007.\n\n\u2022 _The Untold Story of Milk\u2014The History, Politics and Science of Nature's Perfect Food: Raw Milk from Pasture-Fed Cows_ , by Ron Schmid, ND (NewTrends Publishing, 2009).\n\n##### MILK HISTORY AND SCIENCE\n\n\u2022 _Milk: The Surprising Story of Milk Through the Ages, with 120 Adventurous Recipes That Explore the Riches of Our First Food_ , by Anne Mendelson (Alfred A. Knopf, 2008). One of America's finest food historians offers a well-researched, beautifully written exploration of milk as used by cultures over the globe, from ancient times to the present day. The book includes an extensive, worldly, and fascinating collection of recipes.\n\n\u2022 _Nature's Perfect Food: How Milk Became America's Drink_ , by E. Melanie DuPuis (New York University Press, 2002). A history of milk in the United States that does a good job of puncturing myths.\n\n\u2022 _On Food and Cooking: The Science and Lore of the Kitchen_ , by Harold McGee (Scribner, 2004). Chapter 1, \"Milk and Dairy Products,\" is a compelling read that makes the science of milk a page-turner.\n\n##    \nHOW TO MAKE  \nTEA\n\nWith all due respect to Teams Salt and Escargot, Team Tea submits that attempting to grow, here in Menlo Park, _Camellia sinensis_ \u2014a crop that's happiest in, say, the rain forests of Yunnan or the hill stations of Darjeeling\u2014is _the_ most far-fetched One-Block project.\n\nAs far as we can tell, only two tea plantations are currently operating in the western United States, and neither of them is in California. What's more, these two farms (one is in Hawaii, the other is in Washington) produce only Japanese-style green teas, not black tea. (Although green and black teas all come from the same kind of plant, they are processed in different ways.) Since some of us on the team are hopelessly addicted to the black stuff, that's what we had our hearts set on.\n\nThe first hurdle was finding mature tea bushes and figuring out where they would grow best on the Sunset grounds. After that, the actual processing of the tea turned out to be pretty simple.\n\nWe are still in the early days of tea growing. So far, we've had one very small harvest from our three bushes, which resulted in a single pot of pale yellow, very delicate, distinctly tea-flavored brew. But we're looking forward to learning, and drinking, more.\n\n###    \nBLACK TEA\n\nOur first harvest: 1 heaping tablespoonful of dried tea leaves, rolled into needles like Asia's finest.\n\n#### WHAT TO USE\n\n**_Tea bushes_** When we first started looking for tea bushes, we thought we'd be able to buy them from Forest Farm Nursery up in Williams, Oregon (www.forestfarm.com). But the nursery only had seedlings at the time, and a bush generally needs to be 3 years old before you can start plucking its leaves. Further research led us to Google Answers and a reference to the Camellia Forest Nursery, in North Carolina, which sells both seedlings and mature tea plants. There are hundreds of subvarieties and cultivars of _C. sinensis_ , but we figured we could at least see which of the three main types\u2014large leaf (Assam bush), medium leaf (Java bush), and small leaf (China bush)\u2014does best in our climate, so we ordered one of each, in late fall. Kind of a splurge, as they were $50 each. Camellia Forest Nursery, www.camforest.com or 919\/968-0504.\n\n**_Rimmed baking sheet_** Perfect for all stages of the leaf processing. About $7 at a cookware store.\n\n**_Wine cellar or other cool, damp place_** We invaded the Sunset wine cellar, commandeering a side table where we left our tea leaves to oxidize. A regular cellar would work just as well, or even a garage. (But maybe park the car outside while your tea is oxidizing, to avoid adding the fine aroma of car exhaust.)\n\n**_Reliable oven thermometer_**\n\n**_Metal tin with an airtight lid_** Tea lasts much longer if kept in one of these. You can buy a fancy tea caddy, or you can reuse an old Peet's tin, which is what we do.\n\n#### HOW TO DO IT\n\n**1. Have the tea bushes inspected** One wrinkle we failed to foresee was that because our plants were coming from outside California, they needed to be inspected by our county's Office of Weights and Measures (California's title for its county agriculture departments) before we could put them outside. (They were also examined, en route, by California state inspectors, which added a few days to the estimated shipping time.) After a brief scare involving a mystery beetle (which turned out to be benign), the county guy lifted the quarantine. To find out whether your plants need to be inspected, contact your county's agriculture department.\n\n**2. Acclimate the plants** After being shipped cross-country, the bushes were looking bedraggled. Our garden-department friends helped move them out into the test garden and then repotted them in three lovely green ceramic pots, staking them, too. The people at Camellia Forest told us that in our region, we wouldn't start seeing new growth (or picking leaves) until April. With winter coming on, we decided to leave them in the sun for the time being.\n\n**3. Transplant the plants** By early spring, it was becoming clear that the bushes weren't loving the full sun, or being in pots: The leaves were spotty and blemished and the plants weren't thriving. We found a spot in dappled shade and dug three relatively shallow holes, maybe 16 inches deep. Like citrus trees, tea bushes don't like their crowns (where the stem meets the roots) covered with earth, so we were careful not to plant them too deep. According to what we had read, tea plants do best at higher elevations, where the summers are warm and wet and the winters dry and moderately cold\u2014the opposite of our weather. But we also had read that they can \"prosper surprisingly well in a range of adverse climatic conditions, tolerating dry summers and wet winters.\" And we had seen plenty of healthy, happy ornamental camellias growing in Northern California. So we crossed our fingers and waited. (For more on how to grow _C. sinensis_ , see the tea entry under the Spring Garden).\n\n_Newly transplated Camellia sinensis_\n\n_Transplanting the young tea bushes_\n\n**4. Harvest the leaves (April)** According to the people at Camellia Forest, we were supposed to pick 1 or 2 of the newly growing leaves _and_ the leaf bud on each stem. At this early stage of the bush's life, the idea is to leave a couple new leaves on the stem unpicked, so some growing gets done. Then, after a week or two, you can go back and pick from that stem again. We played it safe: On our first sweep, we ended up with only about 25 leaves or buds. (We had to combine the leaves from all three plants to get that total; someday, we're hoping to have enough leaves from each bush so that we can try making single-variety teas.) We picked them in the morning, having read that tea leaves picked in the afternoon can have unpleasantly higher levels of tannin.\n\n_Picking tender new leaves_\n\n**5. Wither the leaves** Basically, the first step in making black tea from our leaves was simple: Just spread the leaves out on the baking sheet for the rest of the day to let them wilt. Keep checking them\u2014if the room is too dry or you leave them too long, some of the smaller leaves might dry to a crisp. You want them sad and wilty looking, like salad that's been left out too long. For us, that took from about ten in the morning until six that night.\n\n**6. Roll the leaves** Roll the wilted leaves firmly, one at a time, between your fingers and thumbs. You're supposed to be bruising the leaves to release their enzymes, and you'll know you're doing it hard enough if you can feel the juices in the leaves being released. Then spread out the needle-shaped rolled leaves on the baking sheet.\n\n**7. Ferment the leaves** This just means letting the leaves sit for a few hours or days to oxidize until all the green color is gone and they're completely brown. We let our first batch sit for 3 full days in the wine cellar, and by the end of that time, there were still patches of green amid the brown. At that point, since it was a Friday, we moved on to the next step. With our second batch of leaves (which we picked a couple weeks later), we gave them 5 days. Even so, we still saw a few hints of green.\n\n**8. Dry the leaves** Although the tea looked plenty dry to us, we decided to follow Camellia Forest's instructions to dry the tea, still on its baking sheet, in a 250\u00b0F oven for 20 to 25 minutes. We let it cool thoroughly (for a few hours) and transferred the leaves carefully into our metal tin.\n\n**9. Brew the tea** The finished product, once we poured briskly boiling water into a teapot with our precious tablespoonful of tea leaves and let it steep for 5 minutes, was more delicate than we had hoped, though it definitely had a tealike scent and a tannic afterbite. At this point, milk would probably have overwhelmed it. For our next harvest, we'll try picking the leaves in the afternoon\u2014maybe that extra tannin will help add some body\u2014and we'll let them oxidize for however long it takes for the leaves to turn entirely brown. And Team Cow will provide the milk.\n\n#### HELPFUL INFORMATION\n\n##### BOOKS\n\n\u2022 In _For All the Tea in China: How England Stole the World's Favorite Drink and Changed History_ (Viking, 2010), journalist Sarah Rose tells the tale of how Robert Fortune (he was kind of a nineteenth-century botanical equivalent to Indiana Jones) smuggled tea plants from China to India on behalf of the British East India Company.\n\n\u2022 Sadly out of print but worth looking for used, _A Time for Tea_ (Knopf, 1991), by British writer Jason Goodwin, is a beautifully and quirkily written history of tea through the ages.\n\n\u2022 _The Tea Enthusiast's Handbook_ , Mary Lou Heiss and Robert J. Heiss (Ten Speed Press, 2010), gives a good overall introduction to the topic of tea, with in-depth descriptions of the most famous types.\n\n##### OTHER RESOURCES\n\n\u2022 This link to the University of Florida IFAS Extension page leads to growing information aimed, of course, at gardeners living in Florida, but has helpful advice even for those living elsewhere: www.edis.ifas.ufl.edu\/hs308.\n\n\u2022 Here is the link to the Google Answers page that led us to Camellia Forest Nursery and provided lots of useful (albeit undocumented) information: www.answers.google.com\/answers\/threadview\/id\/566527.html.\n\n\u2022 On Google Books, we found a scanned copy of a classic prewar work on tea production that we're looking forward to reading, called _Indian Tea: Its Culture and Manufacture of Tea_ , written by Claud Bald and published in Calcutta in 1903.\n\n_\u2014Christine Ryan_\n\n_Radishes, Fresh Homemade Butter, and Salt_\n\n# THE SPRING RECIPES\n\n  RADISHES, FRESH HOMEMADE BUTTER, AND SALT\n\nThis classic country French hors d'oeuvre could not be easier. And when both the radishes and butter are very fresh, it is delicious.\n\nMAKES 6 servings\n\nTIME about 5 minutes\n\n24 juicy, just-picked radishes\n\nUnsalted butter, homemade or store-bought\n\nFine sea salt\n\nCut the bigger radishes in half and leave smaller ones whole. Serve with butter, salt, and a butter knife or two.\n\n_PER SERVING (4 RADISHES AND \u00be TEASPOON BUTTER) 28 cal., 93% (26 cal.) from fat; 0.15 g protein; 2.9 g fat (1.8 g sat.); 0.61 g carbo (0.28 g fiber); 7.4 mg sodium; 7.6 mg chol._\n\n##    \n  FAVAS AND RICOTTA ON BUTTERMILK CRACKERS\n\nPlump fava beans and milky-sweet ricotta are as springy as spring gets. A little bit of mint adds an extra spark. We used true peppermint, which is more mentholated than regular grocery-store spearmint, but spearmint works, too (use a little more of it).\n\nUnless they are very small, when you can eat them pod and all, favas need to be shelled, and then each bean popped out of its tough skin. This takes a little time, but it is easy to do and strangely relaxing, like knitting. Do it while you're watching TV.\n\nMAKES 8 servings (16 crackers)\n\nTIME about 45 minutes\n\nFine sea salt\n\n3 pounds fava beans in the pod\n\n16 Buttermilk Crackers (recipe follows) or store-bought crackers\n\n1 cup ricotta cheese, homemade or store-bought\n\nAbout \u00bd cup finely sliced fresh peppermint leaves\n\nAbout \u00bc cup extra virgin olive oil\n\n1. Bring a small pot of salted water to a boil. Meanwhile, shell the fava beans (you will have about 2 cups beans). Boil the beans for 2 minutes. Drain into a colander and rinse with cold running water until cool.\n\n2. To peel the tough skin from each bean, tear it open at the bean's round end with your fingernail or a paring knife and then pop out the bean.\n\n3. Spread the crackers with a thin layer of ricotta and top with the mint and then the favas. Arrange on a platter, sprinkle with salt and drizzle with the oil.\n\n**MAKE AHEAD** You can prepare the favas up to 1 day ahead and store them in an airtight container in the refrigerator.\n\n_PER SERVING 496 cal., 45% (226 cal.) from fat; 22 g protein; 26 g fat (8.7 g sat.); 57 g carbo (4.4 g fiber); 1,213 mg sodium; 32 mg chol._\n\n### BUTTERMILK CRACKERS\n\nHomemade crackers are surprisingly easy to make and taste much fresher than the average store-bought cracker. If you have made your own butter, use the leftover buttermilk to make these crackers.\n\nMAKES 16 crackers, each 2 by 4 inches\n\nTIME about 40 minutes\n\n2 cups whole-wheat flour* or whole-wheat pastry flour, plus more for dusting\n\n2 teaspoons fine sea salt, plus more for sprinkling (optional)\n\n1\u00bd teaspoons crushed dried red serrano or \u00e1rbol chiles (about 2 chiles)\n\n4 tablespoons cold unsalted butter, homemade or store-bought, cut into small cubes\n\n\u2154 cup Homemade Buttermilk (154) or whole milk\n\n1 tablespoon honey\n\nAbout \u00bc cup extra-virgin olive oil\n\n1. Preheat oven to 375\u00b0F. In a food processor, whirl together the flour, salt, and chile. Add the butter and pulse until the mixture looks like fine cornmeal. Pour in the buttermilk and honey and pulse just until incorporated.\n\n2. Turn the dough out onto a lightly floured work surface, form into a ball, and then divide the ball in half. Set half of the dough aside and cover it with a damp kitchen towel to keep it from drying out.\n\n3. Pat the remaining dough half into a rough rectangle, dust with flour, and roll out into a paper-thin rectangle. Trim off any ragged edges. (You can save the scraps and reroll them, though the resulting crackers will be a little tough.) Gently roll the dough around the rolling pin, and then unroll it onto a rimmed baking sheet. Using the tip of a sharp knife, score it into eight 2-by-4-inch rectangles. Brush with half of the oil and sprinkle very lightly with salt. Repeat with the remaining dough half. Poke both dough sheets all over with a fork.\n\n4. Bake for 8 minutes, then switch the pan positions and rotate the pans back to front. Continue to bake until the dough is a pale brown with some slightly darker edges, about 3 minutes longer. Let cool on the pans on racks, then break into individual crackers along the scored lines.\n\n**MAKE AHEAD** The crackers will keep for up to 1 week in an airtight container at room temperature.\n\n_PER CRACKER 118 cal., 50% (59 cal.) from fat; 2.6 g protein; 6.8 g fat (2.4 g sat.); 13.3 g carbo (2 g fiber); 292 mg sodium; 8 mg chol._\n\n* _We used 'Sonora' soft white wheat flour because it's local and because we like its delicate flavor and slightly_ nubbly _texture in these crackers. But regular whole-wheat flour or whole-wheat pastry flour will work too. SeeThe Allure of Local Wheat for more on local flours._\n\n##    \nGOUDA GOUG\u00c8RES\n\nChoux pastry, made with just eggs, salt, butter, and flour, is the basis for all kinds of rich treats\u2014\u00e9clairs, profiteroles, cream puffs, and these tasty, cheesy morsels.\n\nMAKES 25 to 30 small puffs\n\nTIME about 40 minutes\n\n4 tablespoons unsalted butter, homemade or store-bought, cut into cubes\n\n\u00bc teaspoon fine sea salt\n\n\u00bd cup whole-wheat pastry flour* mixed with \u00bc cup whole-wheat flour\n\n2 large eggs\n\n2 ounces aged Gouda, homemade or store-bought, coarsely shredded (about \u00bd cup)\n\n\u00bd teaspoon minced fresh rosemary\n\n1. Preheat the oven to 425\u00b0F. Line 2 rimmed baking sheets with parchment paper.\n\n2. In a heavy-bottomed medium saucepan, combine the butter, salt, and \u00bd cup water and bring to a rolling boil over high heat. Remove from the heat, add the flour all at once, and vigorously stir with a wooden spoon until smooth and shiny. Stir in the eggs, one at a time, mixing in the first egg completely before adding the second one. Stir in the cheese and then the rosemary.\n\n3. Spoon the dough into a pastry bag fitted with a \u00bd-inch plain tip. Pipe the dough onto the prepared baking sheets in 1 tablespoon mounds, spacing them about 1 inch apart. Or, snip a \u00bd-inch hole in one corner of a resealable plastic bag, fill the bag with the dough, and use it to pipe the dough.\n\n4. Bake for 10 minutes, then switch pan positions and rotate the pans back to front. Reduce the heat to 300\u00b0F and continue to bake until the puffs are golden brown, 10 to 12 minutes more. Serve warm.\n\n**MAKE AHEAD** You can freeze the puffs for up to 2 weeks. Let cool, then put in a resealable plastic bag and freeze. To serve, reheat in a 350\u00b0F oven for 8 minutes.\n\n_PER GOUG\u00c8RE 36 cal., 61% (22 cal.) from fat; 1.2 g protein; 2.4 g fat (1.4 g sat.); 2.4 g carbo (0.4 g fiber); 39 mg sodium; 20 mg chol._\n\n* _We used freshly milled 'Sonora' soft white wheat flour because it is the closest local substitute for commercial whole-wheat pastry flour. SeeThe Allure of Local Wheat for more about local flours._\n\n##    \nCARROT AND BEET CHIPS\n\nWhy not go beyond the potato and make seasonal chips? We love how colorful and flavorful these are. We used extra virgin olive oil to stay within our One-Block pantry, but it would be a waste otherwise, since the oil is expensive and high heat destroys its nuances. Several other cheaper, more neutral-flavored oils, such as safflower or canola, have a higher smoke point and would be ideal.\n\nMAKES about 2 ounces carrot chips (2 cups) and about 1 ounce (1\u00bd cups) for each kind of beet chip\n\nTIME about 1 hour\n\n3 very large carrots, at least 1\u00bd inches in diameter, peeled and cut into 4-inch lengths\n\n3 beets of any variety, about 5 ounces each, trimmed and peeled\n\nExtra virgin olive oil for deep-frying\n\n1\u00bd to 2 teaspoons fine sea salt\n\n1. Using a mandoline or other handheld slicer, slice the carrots paper-thin lengthwise. Thinly slice the beets, slicing any red beets last so they don't stain the others.\n\n2. Pour the oil to a depth of \u00bd inch into a large frying pan and heat over medium-high heat to 375\u00b0F on a deep-frying thermometer. (Some olive oils have lower smoke points; if your oil starts smoking, reduce the heat until the thermometer registers 360\u00b0F).\n\n3. Working in small batches, fry the carrot chips first, then the beet chips, stirring occasionally with a slotted spoon, until they are crisp, 1 to 2 minutes. Transfer to a double thickness of paper towels laid over a paper bag to drain. Return the oil to 375\u00baF before frying each batch.\n\n4. Sprinkle the chips with salt while they're hot.\n\n**MAKE AHEAD** You can fry the chips up to 2 days ahead and store them in an airtight container at room temperature. If they get soggy, spread them in a single layer on a rimmed baking sheet and reheat in a 250\u00b0F oven for 10 minutes. Let them cool for at least 10 minutes in a single layer before serving.\n\n_PER \u00bc CUP CARROTS 51 cal., 61% (31 cal.) from fat; 0.47 g protein; 3.6 g fat (0.52 g sat.); 4.8 g carbo (1.4 g fiber); 401 mg sodium; 0 mg chol._\n\n_PER \u00bc CUP BEETS 50 cal., 62% (31 cal.) from fat; 0.76 g protein; 3.6 g fat (0.51 g sat.); 4.5 g carbo (1.3 g fiber); 403 mg sodium; 0 mg chol._\n\n##    \nGARDEN BORSCHT\n\nThis is a gentle, fabulous soup that owes its subtle grace to fennel seeds that we harvested from our plants at the end of their cycle, when they stood nearly six feet tall.\n\nMAKES 6 servings\n\nTIME about 1\u00bd hours\n\nAbout 2 pounds large beets, with tops intact\n\n1 tablespoon extra virgin olive oil\n\n1 white onion, chopped\n\n1 carrot, peeled and chopped\n\n1 clove garlic, minced\n\n1 tablespoon crushed fennel seeds (see see here for details on how to harvest)\n\n4 cups Herb Vegetable Broth\n\n\u00bc cup plus 4 teaspoons coarsely chopped fresh dill\n\n2 teaspoons fine sea salt\n\nCr\u00e9me fra\u00eeche, homemade or store-bought, for serving\n\n1. Trim the leafy tops from the beets and chop them, including the stems. Peel the beets, quarter them through the stem end, and then thinly slice each quarter crosswise. Set the tops and beets aside.\n\n2. In a large pot, heat the oil over medium heat. Add the onion, carrot, garlic, and fennel seeds and cook, stirring often, until the vegetables have softened, about 4 minutes. Add the beet tops, then pour in the broth, 3\u00bd cups water, and the \u00bc cup dill. Cover, bring to a simmer, and cook, stirring occasionally, until the beets have softened and the flavors have blended, about 1 hour.\n\n3. Season the soup with the salt, then serve dolloped with cr\u00e8me fra\u00eeche and some of the remaining dill.\n\n_PER SERVING 112 cal., 39% (44 cal.) from fat; 2.6 g protein; 5.1 g fat (0.73 g sat.); 16 g carbo (4.4 g fiber); 1,396 mg sodium; 0 mg chol._\n\n  MESCLUN SALAD WITH SPRING BEETS AND DILL\n\nLettuces are one of the most rewarding crops in the back-yard garden. They are easy to grow, and you can harvest the young, tender leaves right before dinner to make a salad that is as fresh as it gets.\n\nMAKES 4 to 6 generous servings\n\nTIME about 20 minutes\n\n\u00bd teaspoon finely shredded lemon zest\n\n2 tablespoons freshly squeezed lemon juice\n\n\u00bd teaspoon sea salt\n\n\u00bc cup extra virgin olive oil\n\n3 small to medium red and golden beets ('Chioggia', 'Bull's Blood', or other red beet, and golden), trimmed and peeled\n\n8 cups loosely packed mesclun (baby lettuces)\n\n\u00bd cup small sprigs dill\n\n1. In a small bowl, whisk together the lemon zest, lemon juice, and salt, then whisk in the oil.\n\n2. Slice the beets into paper-thin rounds, preferably with a mandoline or other handheld slicer. Slice red beets last so they don't stain the others. Put the red beet slices in a medium bowl and toss with 1 tablespoon of the dressing to coat evenly.\n\n3. In a large shallow bowl, toss together the mesclun, dill, and any light-colored beet slices. Drizzle on the remaining dressing and toss to coat evenly. Tuck in the red beet slices.\n\n_PER SERVING 110 cal., 74% (81 cal.) from fat; 1.8 g protein; 9.4 g fat (1.3 g sat.); 7.8 g carbo (2.3 g fiber); 230 mg sodium; 0 mg chol._\n\n##    \nNASTURTIUM SALAD WITH OMELET RIBBONS\n\nNasturtiums have been self-seeding at random in the Sunset garden for several years. They are at their peak in spring and early summer, blooming in billowy clusters of orange and yellow\u2014and were hard to ignore when we were snipping lettuces for this salad.\n\nMAKES 4 to 6 servings\n\nTIME about 20 minutes\n\n1 tablespoon freshly squeezed lemon juice\n\n1\u00bd teaspoons honey\n\n\u215b teaspoon plus \u00bc teaspoon fine sea salt\n\n2 tablespoons plus 1 teaspoon extra virgin olive oil\n\n3 large eggs\n\n3 cups loosely packed nasturtium leaves, stems removed\n\n3 cups loosely packed mesclun (baby lettuces)\n\n1 to 2 cups nasturtium blossoms, stems removed\n\n1. In a small bowl, whisk together the lemon juice, honey, and \u215b teaspoon salt. Whisk in 2 tablespoons of the oil.\n\n2. In a 10-inch nonstick frying pan, heat the remaining 1 teaspoon oil over medium heat. Meanwhile, in another small bowl, whisk together the eggs and the remaining \u00bc teaspoon salt until well blended.\n\n3. When the oil is hot, pour the eggs into the pan. As the eggs begin to set, lift the edge with a spatula and tilt the pan to let the uncooked mixture on top flow underneath. Work your way around the omelet, lifting one section of the edge at a time and tilting the pan, until the omelet is set underneath. Continue to cook until the top is only slightly moist, 1 to 2 minutes.\n\n4. Slide the omelet onto a heatproof plate. Invert the pan over the plate and, holding the pan and plate together with pot holders, flip them over together so the omelet falls back into the pan. Return the omelet to the heat until the underneath is just set, 1 to 2 minutes more. Slide the omelet onto a cutting board and cut into ribbons \u00bc inch wide.\n\n5. In a serving bowl, combine the nasturtium leaves, mesclun, and omelet ribbons and toss gently to mix. Drizzle on the dressing and toss again. Sprinkle with the nasturtium blossoms and serve.\n\n_PER SERVING 92 cal., 75% (69 cal.) from fat; 3.6 g protein; 8 g fat (1.6 g sat.); 2.4 g carbo (0.25 g fiber); 184 mg sodium; 106 mg chol._\n\n##    \nWHOLE-LEAF RADISH AND HERB SALAD\n\nInstead of lettuces, this salad has lots of just-picked herbs plus radish leaves, which have a lovely, fresh, \"green\" taste.\n\nMAKES 4 to 6 servings\n\nTIME about 20 minutes, plus 30 minutes to chill\n\n3 cups loosely packed small, tender radish leaves (or use larger, tender leaves torn into pieces)\n\n2 cups quartered radishes\n\n2 cups loosely packed small fresh flat-leaf parsley leaves\n\n\u00be cup loosely packed small sprigs dill\n\n\u00be cup loosely packed sprigs chervil\n\n\u00be cup loosely packed fresh tarragon leaves\n\n1 teaspoon Syrah Vinegar or 1\u00bd teaspoons other red-wine vinegar\n\n\u00bd teaspoon fine sea salt\n\n3 tablespoons extra-virgin olive oil\n\n1. In a large serving bowl, combine the radish leaves, radishes, parsley, dill, chervil, and tarragon. Cover and chill for about 30 minutes to crisp the leaves.\n\n2. In a small bowl, whisk together the vinegar and salt, then whisk in the oil to make a dressing. Drizzle the dressing over the salad and toss gently to coat evenly. Season with salt to taste.\n\n_PER SERVING 99 cal., 67% (66 cal.) from fat; 3 g protein; 7.7 g fat (1.1 g sat.); 6 g carbo (1.8 g fiber); 234 mg sodium; 0 mg chol._\n\n_Grilled Carrot Salad (bottom); Mesclun Salad with Spring Beets and Dill_\n\n##    \n  GRILLED CARROT SALAD\n\nWe grew 'Nantes' carrots, an exceptionally sweet and juicy French variety with rounded tips. You can often find them at farmers' markets, too.\n\nMAKES 6 servings\n\nTIME about 30 minutes, plus 1 hour to marinate\n\n\u2153 cup extra virgin olive oil\n\n3 tablespoons Syrah Vinegar or 4 tablespoons other red-wine vinegar\n\n\u00bd teaspoon fine sea salt\n\n3 tablespoons fresh thyme leaves\n\n1\u00bd pounds carrots\n\n3 green onions, ends trimmed and halved crosswise\n\n1. In a large bowl, whisk together the oil, vinegar, salt, and 2 tablespoons of the thyme leaves. Add the carrots and green onions, turn to coat evenly, and marinate at room temperature for about 1 hour.\n\n2. Prepare a grill for medium heat (350\u00b0 to 450\u00b0F; you should be able to hold your hand above the cooking grate for only 5 to 7 seconds).\n\n3. Lift the carrots out of the marinade onto the grill and cook, turning often with tongs, until charred on all sides and tender when poked with a paring knife, about 10 minutes for smaller carrots and 15 minutes for larger ones. About 5 minutes before the carrots have finished cooking, lift the green onions out of the marinade, reserving the marinade, and cook them, turning once, until charred and softened on both sides, about 5 minutes total. As the carrots and green onions are done, return them to the marinade.\n\n4. Transfer the carrots to a cutting board and cut each carrot on the diagonal into 3 equal pieces. Return the carrot pieces to the marinade and toss with the onions.\n\n5. Arrange the carrots and green onions on a large serving platter (discard the marinade) and sprinkle with the remaining 2 tablespoons thyme leaves.\n\n_PER SERVING 157 cal., 69% (108 cal.) from fat; 1.3 g protein; 13 g fat (1.8 g sat.); 12 g carbo (3.5 g fiber); 267 mg sodium; 0 mg chol._\n\n##    \nROASTED BEETS AND TOPS WITH TARRAGON\n\nWe especially like the deep, earthy sweetness of red beets in this recipe, but you can use any kind of beet.\n\nMAKES 4 to 6 servings\n\nTIME about 1\u00bc hours\n\nAbout 3 pounds large beets, with tops intact\n\n4 tablespoons extra virgin olive oil\n\n\u00bc cup fresh tarragon leaves\n\n\u00bc teaspoon fine sea salt\n\n1. Preheat the oven to 350\u00b0F. Trim the leafy tops, including stems, from the beets, rinse thoroughly, and set aside. Arrange the beets in a roasting pan just large enough to hold them in a single layer. Drizzle the beets with 1 tablespoon of the olive oil and cover the pan with aluminum foil.\n\n2. Bake the beets until tender when pierced with a knife, about 50 minutes. Set aside until cool enough to handle.\n\n3. While the beets are cooking, roughly chop the reserved beet tops, including the stems. Heat 1 tablespoon of the olive oil in a large pot over medium-low heat. Add the chopped beet tops and stems and pour in \u00be cup water. Cover and cook, stirring often, until they are wilted and tender, about 30 minutes.\n\n4. Peel the beets, cut into large chunks, and add them to the beet tops in the pot. Stir in the tarragon and salt.\n\n5. Turn the mixture into a serving dish and drizzle with the remaining 2 tablespoons oil.\n\n_PER SERVING 164 cal., 51% (84 cal.) from fat; 4.2 g protein; 9.7 g fat (1.4 g sat.); 18 g carbo (7 g fiber); 382 mg sodium; 0 mg chol._\n\n##    \nFRESH PICKLED BEETS\n\nThis super-easy small-batch pickle is stored in the refrigerator (no canning required). We used our homemade Syrah Vinegar, which gives the pickles a tart, fruity depth.\n\nIf you want to can your own pickles using homemade vinegar, you'll need a recipe designed for canning, because proportions of ingredients are important for safety; for details, see the \"USDA Complete Guide to Home Canning,\" (www.uga.edu\/nchfp\/index.html), and also follow their directions for water bath processing. Also, homemade vinegars can vary in acidity, so you'll need to measure the percent of acidity (acetic acid) in your vinegar to be sure it's the same as what's listed on the bottle of commercial vinegar called for in the canning recipe. You can do a quick check with a Country Acid Test Kit ($8.95) from the Beverage People (www.thebeveragepeople.com or 800\/544-1867); ask to have the instructions for titrating vinegar included.\n\nMAKES about 1\u00be cups beets, plus pickling liquid\n\nTIME about 1 hour, plus at least 3 hours to chill\n\n1 pound red beets\n\n3 tablespoons honey\n\n1\u00bd teaspoons fine sea salt\n\n\u2153 cup Syrah Vinegar or \u00bd cup other red-wine vinegar\n\n1. Trim the leafy tops off the beets, leaving about 1 inch of the stem intact. (Reserve the tops for another use.) Put the beets in a saucepan with just enough water to cover, cover the pan, and bring to a boil over high heat. Reduce the heat to a simmer and cook until the beets are just tender when pierced with the tip of a sharp knife, about 20 minutes.\n\n2. Drain the beets and let them cool until they can be handled. Peel, slice into \u00bd-inch-thick rounds or wedges, and put the pieces into a heatproof bowl.\n\n3. Meanwhile, in another saucepan, bring the honey, salt, and 2 cups water to a boil over high heat. Reduce the heat to a simmer and cook until the honey dissolves. Add the vinegar and stir well; pour over the beets.\n\n4. Let cool, then cover and chill for at least 3 hours to let the flavors develop.\n\n**MAKE AHEAD** The pickled beets will keep in the refrigerator for up to 2 weeks.\n\n_PER \u00bd-CUP SERVING 58 cal., 0% (1 cal.) from fat; 1.1 g protein; 0.11 g fat (0.02 g sat.); 14 g carbo (1.3 g fiber); 530 mg sodium; 0 mg chol._\n\n### PICKLED EGGS\n\nYou can also put a few peeled hard-cooked eggs in with your beets, if you like. Add the whole eggs to the bowl of beets before you pour in the hot vinegar mixture. After 2 to 3 days of pickling, the whites will have acquired a brilliant outer ring of fuchsia, the yolks will have remained a vivid yellow, and the eggs will taste pleasantly pickled. They are best eaten right away. After 4 days or more, they will start to get rubbery.\n\n_PER EGG 78 cal., 62% (48 cal.) from fat; 6.3 g protein; 5.3 g fat (1.6 g sat.); 0.56 g carbo (0 g fiber); 62 mg sodium; 212 mg chol._\n\n##    \nFIVE WAYS WITH FRESH EGGS\n\nWhen your backyard hens are just starting to lay eggs, it's nice to get to know them in the simplest possible recipes\u2014you'll immediately see (and taste) how different they are from industrial eggs. For the following methods, we are assuming that your eggs are standard large (2 ounces in the shell). If they are small or jumbo, adjust the timings by a minute in either direction.\n\n### 1. FRIED, OVER EASY\n\nUse your most slick-surfaced, well-seasoned frying pan. It should be large enough to allow a bit of space around each egg so you can turn them easily. Heat the pan over medium-low heat until hot, then add a bit of unsalted butter, homemade or store-bought. When the butter has foamed up and settled back and is just beginning to brown, crack an egg on your countertop, hold it close to the pan's surface, and open it into the pan. Repeat with another egg, being careful not to crowd them. Sprinkle with fine sea salt and let cook until the white is almost set (touch it to test). Gently turn each egg over (we like to use a spatula in each hand) and remove the pan from the heat. Let the eggs sit in the pan for 30 seconds, then serve.\n\n### 2. SCRAMBLED\n\nUse the same well-seasoned frying pan you would use for frying eggs, above, and heat it over medium heat until hot. Meanwhile, crack the eggs into a bowl and sprinkle them with fine sea salt (and fresh minced herbs, if you like), but don't mix them. Add some unsalted butter, homemade or store-bought, to the hot pan, let it foam up and settle back, and then pour in the eggs. When the whites are just starting to set, stir the eggs with a wooden spoon to mix the whites with the yolks. Keep stirring until they are almost but not quite as set as you like (they will continue to cook off the heat). Serve right away.\n\n### 3. POACHED\n\nFill a saucepan with enough water to cover the eggs you are cooking and bring to a gentle simmer. At the same time, pour water to a depth of about 1\u00bd inches into a large frying pan and bring to a gentle simmer. Gently lower the eggs, still in their shells, no more than two at a time, into the saucepan of water. Simmer for 15 seconds to set the outermost layer of egg white, then lift out. Crack 1 egg into a spouted measuring cup, then touch the spout to the surface of the simmering water in the frying pan and slip the egg into the water. Repeat with the remaining eggs. Cook the eggs until they look softly set, 2 to 3 minutes. Lift them out with a slotted spoon, slide them onto warmed plates, and serve right away.\n\n**MAKE AHEAD** You can poach the eggs as directed, then arrange them in a single layer in a glass or ceramic baking dish, cover the dish, and refrigerate for up to 2 days (yes, really). To reheat, pour water to a depth of 1\u00bd inches into a large frying pan and heat until steaming hot (but not quite boiling). Gently lower the eggs into the water and let stand until they feel hot to the touch, about 5 minutes. Lift out the eggs with a slotted spoon and serve right away.\n\n### 4. SOFT COOKED\n\nThis method produces tender but completely cooked whites and yolks that are still liquid in the center. They are not boiled, so we don't use this word to describe them. (To hard-cook eggs for use in salads or sandwiches, see here.) If the eggs are chilled, first bring them to room temperature, which helps keep them from cracking. Put the eggs into a saucepan just big enough to hold them, add warm water to cover by 1 inch, and let sit for about 5 minutes. Remove them from the water and bring the water to a boil. Gently lower the room-temperature eggs into the water, turning down the heat until it is at a barely bubbling simmer (a few bubbles rise from the bottom of the pan but don't break the surface). Simmer the eggs, uncovered, for 6 minutes. Remove from the water and let sit for 3 to 4 minutes before eating.\n\n### 5. OMELET\n\nHeat a 10-inch nonstick frying pan over medium heat. Whisk together 6 eggs, \u00bd teaspoon fine sea salt, and about 2 tablespoons minced fresh herbs (fines herbes\u2014a mix of chervil, parsley, chives, and tarragon\u2014is especially good). Add 2 teaspoons unsalted butter, homemade or store-bought, to the pan. Once the butter is foaming but before it starts to brown, pour in the eggs and reduce the heat to low. As the eggs begin to set, lift one edge with a spatula and tilt the pan to let the uncooked mixture on top flow underneath. Work your way around the omelet, lifting one section of the edge at a time and tilting the pan, until the omelet is set underneath. Cover and cook until the top is set but still quite moist, 1 to 2 minutes. If you like, spoon some ricotta, homemade or store-bought, over one-half of the omelet, and gently fold the other half on top. Slide onto a warmed plate and serve right away.\n\n**SERVES 3 TO 4.**\n\n_Whisking eggs for an omelet_\n\n_Fava Leaf and Parsley Quiche_\n\n  FAVA LEAF AND PARSLEY QUICHE\n\nWe like growing fava plants not only for their tasty beans, but also for their broad, abundant, easy-to-pick leaves, which have an earthy-sweet, faintly grassy flavor and can be cooked and eaten as you would spinach.\n\nWe borrowed our method for the quiche from a wonderful recipe in Thomas Keller's _Bouchon_.\n\nMAKES one 10-inch quiche, or 8 servings\n\nTIME about 1 hour, plus 1\u00bd hours to chill and about 1 hour to cool\n\n**CRUST**\n\n1\u00bd cups whole-wheat flour,* plus more for dusting\n\n\u00bd teaspoon fine sea salt\n\n\u00be cup cold unsalted butter, homemade or store-bought, cut into small cubes\n\n\u00bc cup ice water\n\n**FILLING**\n\n2 teaspoons extra virgin olive oil\n\n1\u00bd teaspoons minced garlic\n\n6 ounces baby fava or spinach leaves\n\n\u2153 cup chopped green onions\n\n1 cup fresh flat-leaf parsley leaves\n\n\u00bd teaspoon fine sea salt\n\n**CUSTARD**\n\n1\u00be cups whole milk\n\n1\u00be cups heavy cream\n\n5 large eggs\n\n1 teaspoon fine sea salt\n\n1. To make the dough for the crust, put the flour and salt in a food processor and whirl briefly to blend. Add the butter and pulse until the mixture looks like cornmeal. Add the ice water and pulse again until the dough comes together, about 30 seconds. Turn the dough out onto a work surface and press into a disk. Wrap in plastic wrap and chill for at least 1 hour and up to 1 day.\n\n2. Lightly dust a work surface with flour. Unwrap the dough, place it on the floured surface, and let it warm up for 15 minutes. With a floured rolling pin, roll out the dough into a 12-inch round. Roll the round gently around the rolling pin, then unroll over a 10-inch deep-dish pie pan. Press the dough evenly onto the bottom and up the sides of the pan, fold the overhang under itself to create a high edge on the pan rim, and crimp or flute the edge. Chill for at least 30 minutes. Meanwhile, position an oven rack on the lowest rung in the oven and preheat the oven to 325\u00b0F.\n\n3. Line the chilled crust with parchment paper and fill with pie weights or dried beans. Bake the crust on the bottom oven rack for 25 minutes. Remove the pie weights and parchment and continue to bake until the crust is slightly browned and looks dry, about 10 minutes more. Remove the crust from the oven and set it on a rimmed baking sheet. Raise the oven temperature to 375\u00b0F.\n\n4. While the oven is heating, make the filling. In a large frying pan, heat the oil over medium heat. Add the garlic, fava leaves, green onions, parsley, and salt and cook, stirring often, until the leaves are wilted, about 4 minutes. Transfer the mixture to a cutting board and chop the leaves coarsely, then arrange evenly over the bottom of the warm crust.\n\n5. To make the custard, heat the milk and cream in a large heavy-bottomed saucepan over high heat, just until the mixture begins to simmer. Remove from the heat. Put the eggs in a blender, add half of the hot milk mixture, and whirl for a few seconds to combine. Add the salt and the remaining hot milk mixture and whirl again just to combine. Pour the mixture into the crust, being careful to keep the greens evenly distributed.\n\n6. Bake the quiche on the lowest rack until slightly browned around the edges and beginning to puff in the center, about 25 minutes. Let cool to warm or room temperature before serving.\n\n_PER SERVING 512 cal., 75% (386 cal.) from fat; 11 g protein; 43 g fat (25 g sat.); 24 g carbo (4 g fiber); 688 mg sodium; 255 mg chol._\n\n* _We used freshly milled 'Sonora' soft white wheat flour because it is the closest local substitute for commercial whole-wheat pastry flour. SeeThe Allure of Local Wheat for more about local flours._\n\nARE BACKYARD EGGS BETTER FOR YOU?\n\nIn 2007, _Mother Earth News_ magazine, using an accredited, independent lab, tested eggs from fourteen different pasture-raised flocks around the United States for their nutritional content. Here is how the figures compare to data on commercial eggs, supplied by the U.S. Department of Agriculture.\n\nPastured eggs versus commercial eggs\n\n  * one-third less cholesterol\n  * one-third less saturated fat\n  * two-thirds more vitamin A\n  * two times more omega-3 fatty acids\n  * three times more vitamin E\n  * three times more beta-carotene\n  * at least three times more vitamin D\n\nThe _Mother Earth_ hens roamed around all day, pecking up green plants, seeds, insects, worms, and some grain supplement. Our Sunset flock doesn't have free range of the garden (they tend to eat our tender seedlings to the nub), so they can't qualify as \"pasture raised.\" We feed them organic layer pellets and greens from our kitchen, with occasional fruit and a few snails, and they do have the occasional (supervised) outing.\n\nWe were curious to see how eggs like ours\u2014from average, not-very-cooped-up but not-pastured-either hens\u2014would stack up nutritionally, since many backyard chicken owners have a similar setup. We sent off a dozen eggs to Columbia Food Laboratories in Corbett, Oregon, for testing, focusing on four nutrients: cholesterol and saturated fat (the \"bad\" stuff), and omega-3s and vitamin D (two of the \"good\" elements). Current research suggests that vitamin D in particular may have all kinds of health benefits, including helping build bones, boost muscle strength, and reduce inflammation.\n\nThe results? Unimpressive when it comes to cholesterol and omega-3s (we were quite sad about this, but science doesn't lie). We are happy to report, however, that our eggs have _half_ the saturated fat of commercial eggs\u2014even less than pastured!\u2014and _three times_ the amount of vitamin D. If you are interested in getting your eggs tested, contact Columbia Food Laboratories, www.columbiafoodlab.com. or 503\/695-2287.\n\n  STRAWBERRIES WITH FROMAGE BLANC AND LEMON HONEY\n\nAlthough technically a fresh cheese, fromage blanc tastes like a thicker, silkier version of sour cream, and is terrific with strawberries.\n\nMAKES 6 servings\n\nTIME about 30 minutes\n\n1\u00bc cups (about 9 ounces) fromage blanc, homemade or store-bought*\n\n\u00bc cup heavy cream\n\n2 pints strawberries, preferably small\n\n\u00bd cup honey\n\nFinely shredded zest of 1\u00bd lemons\n\n1. In a bowl, whisk together the fromage blanc and cream until very smooth. Divide the mixture among 6 shallow bowls.\n\n2. Set aside 6 to 12 small, perfect berries. Hull the remaining berries and halve or quarter lengthwise if large. Scatter the berries over the bowls of fromage blanc.\n\n3. In a glass measuring cup, combine the honey and lemon zest and microwave until beginning to bubble, about 30 seconds. Let cool about 5 minutes.\n\n4. Drizzle the warm honey over the berries and fromage blanc. Top each bowl with 1 or 2 of the reserved berries.\n\n_PER SERVING 263 cal., 39% (103 cal.) from fat; 5.3 g protein; 11 g fat (6.8 g sat.); 36 g carbo (1.6 g fiber); 351 mg sodium; 44 mg chol._\n\n* _Store-bought fromage blanc is often much drier than homemade fromage blanc. If you use it, stir in enough heavy cream (at least 3 tablespoons) to give it the consistency of softened cream cheese._\n\n##    \nSTRAWBERRY CREPES\n\nAlthough supremely easy, these breakfast crepes are elegant, too.\n\nMAKES 6 servings\n\nTIME about 25 minutes\n\n\u00bd cup unsalted butter, homemade or store-bought\n\n\u00bd cup honey\n\n4 cups hulled and quartered strawberries\n\n12 small Whole-Wheat Crepes, made in a 6-inch pan\n\n1. In a large frying pan, melt the butter over medium heat. Add the honey and strawberries and cook until the strawberries have begun to release their juices, about 2 minutes. Remove from the heat.\n\n2. Fold each crepe in half and then in half again to form a triangle. On each of 6 plates, arrange 2 triangles, overlapping them. Spoon the strawberries and their sauce over the crepes.\n\n_PER SERVING 475 cal., 51% (244 cal.) from fat; 8.9 g protein; 28 g fat (16 g sat.); 52 g carbo (5.1 g fiber); 284 mg sodium; 184 mg chol._\n\n##    \nSTRAWBERRY CR\u00c8ME FRA\u00ceCHE SHERBET\n\nRich, tangy cr\u00e8me fra\u00eeche takes the place of milk in this smooth, refreshing dessert.\n\nMAKES 4 cups\n\nTIME about 1 hour, plus time to freeze\n\n4 cups hulled and quartered strawberries\n\n\u00bd cup honey\n\n1 teaspoon freshly squeezed lemon juice\n\n\u215b teaspoon fine sea salt\n\nIce cubes\n\n\u00bd cup cr\u00e8me fra\u00eeche, homemade or store-bought\n\n1. In a large saucepan, combine the strawberries, honey, lemon juice, salt, and \u00bc cup water and cook over medium heat, stirring often, until the strawberries have softened and the mixture is simmering, about 20 minutes.\n\n2. Remove the strawberry mixture from the heat and puree it in batches in a blender.\n\n3. Fill a large bowl with ice cubes and water. Pour the puree through a fine-mesh strainer into a medium bowl, pressing the mixture with a spoon to extract as much liquid as possible. Discard the contents of the strainer. Set the bowl in the ice bath and let cool completely, stirring occasionally.\n\n4. Add the cr\u00e8me fra\u00eeche to the cold strawberry liquid and whisk until smooth. Freeze in an ice cream maker according to the manufacturer's directions. Eat right away, or transfer to an airtight container and freeze until completely firm, about 6 hours, before serving.\n\n**MAKE AHEAD** The sherbet can be stored in the freezer for up to 1 week.\n\n_PER \u00bd-CUP SERVING 144 cal., 36% (52 cal.) from fat; 1.1 g protein; 5.8 g fat (3.6 g sat.); 24 g carbo (1.6 g fiber); 38 mg sodium; 13 mg chol._\n\n##    \n  STRAWBERRY LEMONADE\n\nThis refreshing lemonade is just as much about the strawberries as it is about the lemons.\n\nMAKES 7\u00bd cups, or 6 servings\n\nTIME about 20 minutes\n\n2 pints strawberries, hulled\n\n\u00be cup honey\n\n1 cup freshly squeezed lemon juice (from 5 to 6 lemons)\n\nIce cubes for serving\n\n1. In a blender, puree the strawberries and honey. Pour through a fine-mesh strainer set over a bowl, pressing the mixture with a spoon to extract as much liquid as possible. Discard the contents of the strainer.\n\n2. Pour the strawberry juice into a pitcher and stir in the lemon juice and 4 cups water. Serve over ice.\n\n**MAKE AHEAD** The lemonade can be made up to 4 days ahead and refrigerated.\n\n_PER SERVING 83 cal., 3% (2.5 cal.) from fat; 0.84 g protein; 0.29 g fat (0.01 g sat.); 22 g carbo (2.1 g fiber); 1.9 mg sodium; 0 mg chol._\n\n# A SPRING TEA PARTY\n\n# (because we could)\n\nAs soon as we learned that we could grow tea in Menlo Park, actual _Camellia sinensis_ \u2014the civilizing sip of queens and commoners alike\u2014the idea of trying to make a tea party seized hold. Proper British afternoon tea is all about nuance and delicacy, however, and we had relatively few ingredients to pull it off.\n\nBut we were determined. In the end, the tea party food we created had a hearty note (there's no getting around that with sandwiches made with whole-wheat bread), but it mostly qualified as dainty. We added a bowl of freshly picked strawberries and some thick whipped cream sweetened with honey, and thoroughly enjoyed ourselves one spring afternoon, with our afternoon tea out in the garden.\n\n_Scone Tarts_\n\n##    \nSCONE TARTS\n\nWhat is afternoon tea without scones, clotted cream, and jam? With no baking powder or baking soda, we didn't even try to make scones, knowing they would be disastrous. In their place, we made mini tarts and filled them with scone toppings. They were buttery and delicious, crumbling as soon as we put them in our mouths.\n\nYou can use any shape of tart pan you like for this recipe, as long as it is small (1 to 2 inches in diameter). Even a mini-muffin pan will work. Also, this recipe can easily be doubled.\n\nMAKES about sixteen 2-inch tarts\n\nTIME about 1\u00bd hours, plus 1 hour to chill\n\n**TART CRUSTS**\n\n\u00be cup whole-wheat pastry flour*\n\n\u215b teaspoon fine sea salt\n\n4 tablespoons cold unsalted butter, homemade or store-bought, cut into small cubes\n\n2 teaspoons honey\n\n1 to 2 tablespoons ice water\n\n**FILLINGS**\n\nAbout \u00bc cup clotted cream, homemade or store-bought\n\nAbout 2 tablespoons Sticky Chewy Tangerine Marmalade or store-bought citrus marmalade\n\nAbout 2 tablespoons strawberry jam, homemade** or store-bought\n\nAbout 2 tablespoons lemon curd, homemade or store-bought\n\nA few tangerine segments, cut in half\n\n1 strawberry, hulled and sliced lengthwise paper-thin\n\nSeveral thin lemon zest strips\n\n1. To make the tart crusts, sift the flour and salt into a food processor (discarding any bran) and whirl briefly to blend. Add the butter and pulse until the mixture looks like shaggy fresh bread crumbs. Drizzle the honey over the mixture. Then, with the motor running, drizzle in the ice water, adding just enough for the dough to come together. Turn the dough out onto a work surface and press into a disk. Wrap in plastic wrap and refrigerate for at least 1 hour.\n\n2. Preheat the oven to 350\u00b0F. Lightly dust a work surface with flour. Remove the dough from the refrigerator, unwrap it, place it on the floured surface, and let it warm up for 10 minutes. For each 2-inch tart shell, pull off a ball of dough about 1\u00bd inches in diameter and put it in the bottom of a 2-inch tart pan. With floured thumbs, gently press the dough into the pan, rotating the pan as you press, until the dough is an even \u215b inch thick on the bottom and sides. (You can use a smaller pan, but reduce the amount of dough; it should form a layer no more than \u215b inch thick.) Trim the dough even with the pan rim by brushing outward across the rim with your fingers.\n\n3. Prick the bottom of each tart crust all over with a fork. If you have extra empty tart pans in the same size, fit the empty pans on top of the pastry-lined pans, to create slightly neater shells (they won't shrink as much when baked). Put the pans on a rimmed baking pan and bake until thoroughly golden, 10 to 15 minutes. Let cool completely in their pans, at least 15 minutes but no more than 1 hour (or they will start to stick).\n\n4. Invert each tart pan and squeeze gently to release the crust. Arrange the crusts on a large platter. Fill each crust partway with clotted cream, then fill to the top with the marmalade (warmed if too thick), strawberry jam, or lemon curd. Top each tart with a piece of tangerine, a strawberry slice, or a lemon zest strip.\n\n**MAKE AHEAD** Tart dough can be made up to 2 days ahead and refrigerated. Tart crusts can be baked up to 1 day in advance and stored in an airtight container at room temperature. The tarts can be assembled up to 1 hour before serving and kept at room temperature. (Do not refrigerate them or the crusts will become soggy.)\n\n_PER 2-INCH TART 69 cal., 57% (39 cal.) from fat; 0.63 g protein; 4.4 g fat (2.8 g sat.); 7.1 g carbo (0.75 g fiber); 32 mg sodium; 19 mg chol._\n\n* _We used finely milled 'Sonora' whole-wheat flour, which behaves like commercial whole-wheat pastry flour. SeeThe Allure of Local Wheat for more information on local flours._\n\n** _If using ourStrawberry Oven Jam recipe, sweeten to taste with honey before using._\n\n_Clockwise, from top right:Strawberry Oven Jam, Lemon Curd, and Clotted Cream_\n\n##    \nCLOTTED CREAM\n\nA star of the English tea tray, clotted cream is produced by slowly heating pure cream until it forms a devastatingly rich, thick layer on top, which is then scooped off for slathering onto warm scones.\n\nMAKES about 1\u00bc cups\n\nTIME 2 days\n\n5 cups heavy cream (not ultrapasteurized)\n\n1. Preheat the oven to 175\u00b0F. Pour the cream into a wide, heavy-bottomed pot, cover, and heat for at least 8 hours and up to 12 hours.\n\n2. Remove the pot from the oven, cover, and refrigerate the cream overnight (at least 8 hours).\n\n3. Using a wide, flat spoon, skim the thick layer of clotted cream from the chilled cream and place in a bowl or jar (reserve the remaining thin cream for another use). Cover and refrigerate until firm enough to spread, about 4 hours.\n\n**MAKE AHEAD** The cream will keep, refrigerated, for up to 1 week. It will get thicker the longer it sits.\n\n_PER TABLESPOON 70 cal., 97% (68 cal.) from fat; 0 g protein; 7.5 g fat (5 g sat.); 5 g carbo (0 g fiber); 2.5 mg sodium; 23 mg chol._\n\n##    \nLEMON CURD\n\nYou can use any citrus you like to make this rich, tangy spread. We spooned it into little tart crusts for our one-block tea party. It is also delicious blended with vanilla ice cream or used as a cake filling.\n\nMAKES about 1\u00bd cups\n\nTIME about 15 minutes\n\n\u00bd cup unsalted butter, homemade or store-bought\n\n\u00bd cup freshly squeezed lemon juice (from about 3 lemons)\n\n\u2154 cup honey\n\n8 large egg yolks\n\n1. In a heavy-bottomed medium saucepan, melt the butter with the lemon juice over medium-high heat. In a bowl, whisk together the honey and egg yolks until blended.\n\n2. Slowly whisk the hot lemon-butter mixture into the honey-egg mixture, \u00bd cup at a time. Pour the mixture back into the saucepan and cook over medium-high heat, whisking constantly, until the mixture is thick and just starts to boil, about 4 minutes.\n\n3. Remove from the heat, pour into a bowl or other container, and press plastic wrap directly onto the surface to prevent a skin from forming. Let cool, then refrigerate.\n\n**MAKE AHEAD** The curd will keep in the refrigerator for up 1 week.\n\n_PER TABLESPOON 82 cal., 57% (47 cal.) from fat; .99 g protein; 5.3 g fat (3 g sat.); 8.4 g carbo (0.04 g fiber); 3.7 mg sodium; 80 mg chol._\n\n##    \nSTRAWBERRY OVEN JAM\n\nMaking strawberry jam without sugar or commercial pectin is challenging. Honey tends to burn over high heat, resulting in a bitter jam. A slow-cooker yields a jam that is too liquidy. We kept at it and finally arrived at this easy method, which produces a not-too-sweet, fresh-tasting jam with a nice, spreadable consistency.\n\nMAKES about 1 cup\n\nTIME about 3 hours\n\n2 pints strawberries, hulled\n\n2 tablespoons honey, plus more to taste (optional)\n\n1 tablespoon freshly squeezed lemon juice\n\n1. Preheat the oven to 200\u00b0F. Combine the strawberries, honey, and lemon juice in a food processor and pulse 20 to 30 times to chop the berries, stopping to scrape down the sides of the work bowl as needed. Be careful you don't puree the berries.\n\n2. Spread the strawberry mixture in a thin, even layer on a rimmed baking sheet. Bake, scraping up and stirring with a flat, wide metal spatula every hour and then respreading into an even layer, until the jam is as thick as you like, 2 to 3 hours. It will continue to thicken slightly as it cools.\n\n3. Let cool, then transfer to an airtight container. Stir in more honey before serving if you want a sweeter jam.\n\n**MAKE AHEAD** The jam will keep in an airtight container in the refrigerator for up to 1 week or in the freezer for up to 3 months.\n\n_PER TABLESPOON 20 cal., 5% (1 cal.) from fat; 0.25 g protein; 0.11 g fat (0.01 g sat.); 5 g carbo (0.73 g fiber); 0.67 mg sodium; 0 mg chol._\n\n_Tea Sandwich Trio, clockwise from top: Egg-Nasturtium, Green Onion\u2013Parsley, and Gouda-Arugula_\n\n##    \nTEA SANDWICH TRIO\n\nThere is something particularly pleasing about sitting in your garden and eating dainty tea sandwiches filled with the garden itself. We made three kinds of sandwiches: hard-cooked egg and peppery nasturtium; Gouda with arugula; and, in a tribute to James Beard's famous onion sandwiches, green onion and parsley.\n\nMAKES 36 miniature sandwiches, or 6 servings\n\nTIME about 20 minutes, plus 30 minutes to chill\n\n18 very thin slices Whole-Wheat Honey Sandwich Bread\n\nEgg-Nasturtium Filling (recipe follows)\n\nGouda-Arugula Filling (recipe follows)\n\nGreen Onion\u2013Parsley Filling (recipe follows)\n\n1. Fill the sandwiches as directed in each filling recipe, using 6 slices of bread to make 3 large sandwiches with each filling. Chill the sandwiches until cold, at least 30 minutes.\n\n2. Trim the crusts from each sandwich and cut into 4 small squares, rectangles, or triangles. Arrange on a platter or individual plates to serve.\n\n**MAKE AHEAD** The sandwiches can be assembled up to 2 hours ahead, covered with a barely damp kitchen towel and plastic wrap, and chilled.\n\n_PER EGG-NASTURTIUM SANDWICH 60 cal., 65% (39 cal.) from fat; 1.7 g protein; 4.6 g fat (0.87 g sat.); 3.6 g carbo (0.52 g fiber); 109 mg sodium; 40 mg chol_\n\n_PER GOUDA-ARUGULA SANDWICH 46 cal., 57% (26 cal.) from fat; 1.8 g protein; 3 g fat (1.8 g sat.); 3.5 g carbo (0.53 g fiber); 74 mg sodium; 9.5 mg chol_\n\n_PER GREEN ONION\u2013PARSLEY SANDWICH 35 cal., 34% (12 cal.) from fat; 1.3 g protein; 1.3 g fat (0.75 g sat.); 4.6 g carbo (0.62 g fiber); 111 mg sodium; 4.7 mg chol_\n\n### EGG-NASTURTIUM FILLING\n\nMAKES enough for 12 miniature sandwiches\n\nTIME about 20 minutes\n\n2 large eggs, at least 1 week old (superfresh eggs are difficult to peel)\n\n3 tablespoons Green Chile Mayonnaise, made without garlic or chile, or regular mayonnaise flavored with a little freshly squeezed lemon juice\n\n3 tablespoons coarsely chopped nasturtium leaves, plus 6 whole leaves\n\nFine sea salt (optional)\n\n12 nasturtium blossoms\n\n1. Put the eggs in a small pot and cover with water by about 1 inch. Bring to a boil, immediately reduce the heat to a simmer, and cook, uncovered, for 10 minutes. Remove from the heat, immerse the eggs in ice water to cover, and let cool for 1 minute. Crack each egg all over on the counter and return to the ice water for 5 minutes.\n\n2. Peel the eggs under cold water and put them in a bowl. Chop them with a pastry blender or mash them with a fork. Stir in the mayonnaise and chopped nasturtium leaves and season with salt.\n\n3. To fill the sandwiches, spread half of the bread slices with the egg mixture, then top each one with 2 whole leaves and 4 blossoms. Top with the remaining bread slices.\n\n### GOUDA-ARUGULA FILLING\n\nMAKES enough for 12 miniature sandwiches\n\nTIME about 5 minutes\n\nAbout 1 \u00bd tablespoons salted butter, homemade or store-bought, softened\n\nAbout 2 ounces young or aged Gouda, homemade or store-bought, thinly sliced\n\n\u2153 cup loosely packed small arugula leaves\n\n1. To fill the sandwiches, lightly spread each bread slice on one side with butter. Layer the cheese and then the arugula on the buttered side of 3 bread slices.\n\n2. Top with the remaining bread slices, with the buttered side down.\n\n### GREEN ONION\u2013PARSLEY FILLING\n\nMAKES enough for 12 miniature sandwiches\n\nTIME about 10 minutes\n\n\u2153 cup fromage blanc, homemade or store-bought mixed with 3 tablespoons heavy cream\n\n3 tablespoons finely chopped green onions\n\n\u00bc cup finely chopped fresh flat-leaf parsley\n\n\u215b teaspoon fine sea salt\n\n1. In a small bowl, stir together the fromage blanc, green onions, parsley, and salt.\n\n2. Spread the mixture on 3 of the bread slices, dividing it evenly. Top with the remaining bread slices.\n\n##    \nWHOLE-WHEAT HONEY SANDWICH BREAD\n\nThis mild-tasting loaf tastes equally good with something savory, like cheese, or something sweet, like jam. It makes great toast, too.\n\nMAKES 2 loaves; each loaf yields about twenty \u00bd-inch-thick slices\n\nTIME about 2\u00bd hours\n\n2 packages (2 \u00bd teaspoons each) active dry yeast\n\n\u00bc cup honey\n\n6 \u00bd cups whole-wheat flour,* plus more for dusting\n\n2 tablespoons unsalted butter, homemade or store-bought, melted\n\n1 tablespoon fine sea salt\n\nAbout 1 teaspoon extra virgin olive oil\n\n1. Put the yeast, honey, 1 cup of the flour, and 3 cups warm water (100\u00b0 to 110\u00b0F) in the bowl of a stand mixer fitted with a dough hook. Let stand until the yeast dissolves and bubbles appear, about 5 minutes. Add the remaining 5\u00bd cups flour and mix with the dough hook on low speed just until the flour is no longer dry, about 1 minute. Add 1 tablespoon of the butter and the salt and mix on medium-low speed until the dough is smooth, begins to pull away from the sides of the bowl, and feels a bit tacky, 5 to 7 minutes.\n\n2. Use some of the oil to grease a large bowl. Transfer the dough to the bowl and turn the dough to coat it evenly on all sides with oil. Cover the bowl with plastic wrap and let sit in a warm spot until the dough is doubled in size, about 1 hour.\n\n3. Use the remaining 1 tablespoon butter to grease two 8\u00bd-by-4\u00bd-inch loaf pans. Dust each pan lightly with flour, tapping out the excess, and set the pans aside.\n\n4. Turn the dough out onto a lightly floured work surface and divide in half. Using the palm of your hand, flatten 1 piece into a rectangle, popping any air bubbles. Position the rectangle with a long side facing you. Fold the farthest edge toward you \u00bd inch at a time, alternating with folds of the shorter sides toward the center, until you reach the long side nearest you. With the heel of your hand, press all along the length of the seam to seal it. Place the loaf, seam side down, in a prepared pan. Repeat with remaining dough half and place in the second pan. Loosely cover each pan with lightly oiled plastic wrap and let them sit in a warm spot until the dough is doubled in size, 30 to 45 minutes. Meanwhile, preheat the oven to 350\u00b0F.\n\n5. Peel the plastic wrap from the pans, being careful not to tear the surface of the loaves. Bake until the loaves are browned and sound hollow when tapped on the bottom or sides (use pot holders to tip the loaf out of the pan, then tap), 45 to 50 minutes. Turn the loaves out onto racks and let cool completely before slicing.\n\n_PER SLICE 80 cal., 12% (9.6 cal.) from fat; 2.8 g protein; 1.1 g fat (0.45 g sat.); 16 g carbo (2.5 g fiber); 169 mg sodium; 1.5 mg chol._\n\n### WHOLE-WHEAT ROLLS\n\nFollow the recipe through step 2. Butter and flour two 12-cup standard muffin pans. Divide the dough into 24 equal pieces. Roll each piece into a ball and place a ball in each prepared cup. Cover the pans loosely with lightly oiled plastic wrap and let sit in a warm spot until the dough doubles in size, about 50 minutes. Bake the rolls in a 350\u00b0F oven until they are browned and sound hollow when tapped on the bottom (lift a roll out of the pan with a knife tip, then tap), about 15 minutes. Turn out onto racks to cool completely.\n\n_PER ROLL 134 cal., 10% (14 cal.) from fat; 4.8 g protein; 1.6 g fat (0.72 g sat.); 27 g carbo (4.2 g fiber); 282 mg sodium; 2.5 mg chol._\n\n* _We used 'Expresso' whole-wheat flour, but any regular whole-wheat flour will work. SeeThe Allure of Local Wheat for more information on local flours._\n\n# Epilogue\n\nWe haven't exactly had a smooth ride with this one-block project of ours. And yet, to a person we think it's been worth it.\n\nWe've had to do without so many ingredients we regularly use unthinkingly all the time, like baking soda and baking powder, cinnamon, black pepper, vanilla, chocolate, and coffee\u2014and sugar, which is much more versatile than honey because it doesn't burn as easily or ooze. Crops we were counting on failed or were destroyed by pests (most perniciously, our lovely olive trees). Our first prospective cow got mastitis, and one of our chickens died. And, most recently, Team Bee member Brianne had to be rushed to the emergency room with a suddenly developed allergy to bee stings. (She recovered well, and is undergoing desensitization therapy so she can work with the hives again.)\n\nAnd yet. All that we've done, successfully or not, has changed our whole outlook on food\u2014and on ourselves.\n\nNow that we've made beer, for instance, and understand more about all the ways in which it can go wrong, we're in awe of the artisans who create great beers like Chimay, time after time.\n\nWe have a deeper way of seeing and relating to the natural world. Margaret can tell the age of a bee by the condition of its wings. Brianne seems to spot mushrooms wherever she goes\u2014she just has an eye for them now. Jim is so gentle with the hens that even skittish Charlotte lets him pick her up. Erika says that one of the best rewards of making wine was \"getting to feel alive in a new way. We got to be part of nature\u2014and be low-tech, without our computers glaring at us.\" In the kitchen, we cooks look at plants differently now. We consider a vegetable's potential from top to bottom, beyond the parts normally considered edible. Fava leaves and radish greens are delicious, we've found, and so are crisp, juicy pineapple guava blossoms, and pod-on shelling beans if you eat them when they are very small.\n\nAnd finally, we cooks (okay, mainly me) have come to understand the gardener's point of view. At the beginning of this project, I thought that a garden could be planted to follow a menu. Now I know that a menu follows a garden. Nature always leads, and a smart cook learns how to dance.\n\nThat said, I'm not sure we would have had as much fun without a menu as our roadmap. It was the idea of a well-thought-out, well-cooked meal that drove us to figure out a cooking fat (olive oil) and sweetener (honey), seasoning (salt), and protein (eggs and cheese)\u2014and really everything else, too.\n\nAnd it was our projects that led to us getting to know one another. Our office is sort of like a neighborhood, in that most of the people here see each other nearly every day and say hello and maybe chat in the hall. But that's often where the conversation ends. Through collaborating on cheese and wine, bees and olive oil, and so on, we learned who likes to camp, whose kids play baseball, about musical tastes and books being read, and who tells a good joke\u2014all the small things that give texture to life. I'm not sure whether this has helped us in our professional work together, but I'd like to think so.\n\nOf course, we are all proud that we have managed to produce some excellent food, using skills more common a few generations ago. Beyond having had four thoroughly enjoyable dinners and plenty of good food in between, we have learned a lot, even though we are far from being experts in anything. We fling around terms like hydrometer and titration, rennet and rumen, sodium metabisulfite and GF-120 as though they were spare change. We're more capable now.\n\nThere's something else going on here, too. Part of what is compelling about these projects is that they're somewhat mysterious and unpredictable. They involve cooperating with other living beings, some human, some not, and most of them microscopic. There is no single way to succeed, so the projects have truly become our own, with results that reflect us and everything around us: the weather, the earth, our tastes and choices, our skill levels. And what we've produced\u2014the weird torpedo-shaped eggs and the carrots with crooks in them, the superfloral honey and the supersharp vinegar\u2014seems unique, too, and wonderful in a world that persists in prizing Red Delicious apples, all the same size, shape, and color.\n\nWhat we like most about our One-Block project is that it has connected us to nature itself, free of standardized packaging and predigested ways of seeing. It has given us the quirks and surprises we didn't even know we craved until we had them, and didn't know we'd lost until we'd found them.\n\nTHE ONE-BLOCK TEAM. _Back row, left to right:_ Erika Ehmsen (with wine), Julie Chai, Sara Schneider, Sarah Epstein, Sara Jamison, Rachel Levin, Alan Phinney, Elaine Johnson, Sheila Schmitz. _Middle row, left to right:_ Lauren Bonar Swezey, Christine Ryan, Sophie Egan, Katie Tamony, Vanessa Speckman, Kimberley Burch, Margaret Sloan, Stephanie Dean, Kathy Brenzel, Rick LaFrentz, and Erin Shitama. _Front row, left to right:_ Jim McCann, Brianne McElhiney, Margo True, Amy Machnak, Elizabeth Jardina, and Johanna Silver. _Not pictured:_ Ryan Casey, Christine Ciarmello, Dale Conour, Trina Enriquez, Barb Newton.\n\n# The One-Block Team\n\n**Kathy Brenzel \u2022 Garden Editor \u2022 Team Vinegar**\n\nEvery year, Kathy's husband makes wine, most recently using grapes from their neighbors' merlot, cabernet, and zinfandel vines. With so much wine aging in carboys and oak barrels in their garage, Kathy decided it was time to learn how to turn some into equally delicious vinegar for dressing salads.\n\n**Kimberley Burch \u2022 Imaging Specialist \u2022 Teams Bee, Mead, Olive, and Salt**\n\nWith the exception of a college year abroad in Wales, Kimberley has lived her entire life in the West, where _Sunset_ magazine was always around. A love of nature led her to become a photographer, and now she is in heaven as Sunset's imaging specialist, photographer for the one-block website, and \"Queen Bee\"\u2014leader of Team Bee.\n\n**Ryan Casey \u2022 Organic Farmer and Former Test Garden Coordinator \u2022 Team Garden**\n\nA true nature nerd and lover of good food, Ryan found the One-Block garden a perfect place to get his hands dirty, watch birds, and discover the vast amount of edible possibilities to grow. He now produces fresh vegetables, fruits, and flowers at Blue House Farm (www.bluehouseorganicfarm.com) in Pescadero, California.\n\n**Julie Chai \u2022 Associate Garden Editor \u2022 Teams Vinegar and Cheese**\n\nAs a member of Team Vinegar, she's dug deep into crocks of fermenting wine to pull out the aging \"mother\" with her bare hands, and has stirred pots of milk for hours as a member of Team Cheese. In coming years, she's hoping to find a way to create Team Chocolate.\n\n**Christine Ciarmello \u2022 Features Editor \u2022 Team Cheese**\n\nEver since she learned to churn butter as a Girl Scout, Christine has been fascinated with the chemistry of food. Team Cheese gave her a peek into the wonderful world of curds. The San Francisco\u2013based editor says her desert island food is definitely ricotta.\n\n**Dale Conour \u2022 Partner and Head of Strategy at Dial House brand strategy agency and Former Executive Editor \u2022 Teams Bee and Beer**\n\nAs fascinating as the bees were, Dale simply enjoyed the opportunity to wear a pith helmet. His beer-making experience deepened his respect for beer craftsmen everywhere, and he now always performs a silent salute to the brewmaster before imbibing. A native of the West, he lives in San Francisco.\n\n**Stephanie Dean \u2022 Test Kitchen Coordinator \u2022 Teams Beer, Kitchen, and Cow**\n\nBefore Sunset, Stephanie worked for chocolate and olive oil producers and in a food product development lab, and has a master's degree in nutrition. What she loved most about the One-Block project was how cooking with what's available forced us to be creative.\n\n**Sophie Egan \u2022 Researcher \u2022 Teams Tea and Cow**\n\nGetting close to one's food sources is thrilling to Sophie, a recent Stanford graduate and Seattle native. After learning that green and black tea all come from the same plant, she was excited to tackle the tea-making process. And she couldn't pass up Team Cow's offer to milk her first cow.\n\n**Erika Ehmsen \u2022 Copy Chief \u2022 Team Wine**\n\nThis comma jockey relished the chance to pursue perfection in a glass, eagerly blending _I Love Lucy_ \u2013inspired grape stomping with a high school affinity for chemistry. After tucking her young kids into bed, Erika sneaks off to her laptop to mince words for her next Team Wine post on the One-Block blog.\n\n**Trina Enriquez \u2022 Copy Editor \u2022 Team Olive**\n\nEven though Trina's dad grows a jungle of backyard fruit\u2014nectarines, persimmons, kiwis, avocados, pluots\u2014Trina herself took it all for granted as a kid. Living in the Bay Area and working at Sunset finally opened her eyes, though. Her dad still cures olives from one of his trees, a process that piqued her interest in Team Olive.\n\n**Sarah Epstein \u2022 Recipe Retester \u2022 Team Wine**\n\nSarah's background in agriculture and plant science fueled her curiosity about how wine happens, from planting and picking to sipping and pairing. She spends her days testing recipes in the Sunset kitchen, where she has used (and tasted) products from all the One-Block teams.\n\n**Elizabeth Jardina \u2022 Freelance Writer and Former Researcher \u2022 Team Chicken**\n\nElizabeth is a friend to chickens and a former vegetarian. A native Texan, she now lives in San Mateo in Merv Griffin's childhood home.\n\n**Sara Jamison \u2022 Former Photo Style Coordinator \u2022 Team Wine**\n\nBefore Sunset, Sara taught arts and crafts to elementary-school children. At Sunset, she transferred her artistic vision to the photo department, styling and propping sets and managing day-to-day details. She has always loved a good glass of wine, so when Team Wine started up, she was ripe for the picking.\n\n**Elaine Johnson \u2022 Associate Food Editor \u2022 Teams Mushroom, Kitchen, Cheese, Cow, and Bee**\n\nEver since she pruned her first lemon tree (badly) at age 10, Elaine has been fascinated with growing her own food. She took up canning and vegetable-gardening in college in Corvallis, Oregon, and still grows and preserves food year-round out of her home garden in Palo Alto, California.\n\n**Rick LaFrentz \u2022 Head Gardener \u2022 Team Beer**\n\nRick has worked in the gardens at Sunset for over 31 years. What he did before that he can't remember. He'd had some experience making beer (from extract), but it's been quite an education trying to make a beer completely from scratch.\n\n**Rachel Levin \u2022 Senior Editor \u2022 Team Olive**\n\nAs a travel editor-writer who loves to cover all things food, too, Rachel jumped at the chance to join a One-Block team. Team Chicken seemed like too much responsibility. Olives sounded easier (until she met the olive fly).\n\n**Amy Machnak \u2022 Recipe Editor \u2022 Teams Salt, Olive Oil, Vinegar, Kitchen, and Escargot**\n\nAmy grew up in the suburbs of Detroit, in a family of avid hunters and gardeners who instilled the foraging and farm-to-table philosophy in her long before it was trendy. A former pastry chef turned writer and devout Western enthusiast, she was giddy at the idea of making recipes from only what grew at Sunset, and often tells people at cocktail parties that she made salt.\n\n**James McCann \u2022 Art Director \u2022 Team Chicken**\n\nJim hooked up with Team Chicken because as a city boy, he's always had a hidden desire to be a farmer and know what it's truly like to live off the land. Plus, he claims he's never had enough straw in his hair, feathers in his mouth, or poop on his shoes.\n\n**Brianne McElhiney \u2022 Editorial Assistant \u2022 Teams Bee, Cheese, Mead, Mushroom, Chicken, and Cow**\n\nA native of the Bay Area, Brianne is no stranger to the innovative and forward-thinking lifestyle of the West. A graduate of Pepperdine University with a degree in business administration and a passion for food, she is balancing her enthusiasm for business with the adventurous and sustainable spirit of Western living\u2014a life made, not bought.\n\n**Barb Newton \u2022 President \u2022 Team Cow**\n\nBarb is a proud member of Team Cow. She came to Sunset from Wisconsin\u2014need she say more?\n\n**Alan Phinney \u2022 Managing Editor \u2022 Teams Wine and Beer**\n\nIn his 19-year stint at Sunset _,_ Alan has had his hand in covering many aspects of growing and gardening, and food and wine. Yet literally handling grapes and grain as they went from field to bottle has meant a whole new world of learning and appreciation.\n\n**Christine Ryan \u2022 Executive Editor \u2022 Team Tea**\n\nWondering whether true black tea could thrive in Northern California, Christine (a five-cup-a-day drinker) volunteered to head up Team Tea. Apart from her tea-fermenting duties, she serves as executive editor of the magazine; before coming to Sunset, she was abusing caffeine at _7\u00d77_ , _Travel Holiday_ , _Gourmet_ , and _European Travel & Life_ magazines.\n\n**Sheila Schmitz \u2022Sunset.com editor**\n\nSheila coached the One-Block team in the art of blogging, and she loves to grow tomatoes, peppers, cucumbers, lettuces, herbs, and fruit in her San Jose, California backyard. She keeps the garden imperfect to leave room for daydreaming and surprises.\n\n**Sara Schneider \u2022 Wine Editor \u2022 Team Wine**\n\nWriting about wine makes it irresistible to produce it if you get the chance\u2014to get inside this chemically alive, fascinating beverage. Heading up Team Wine was a true career highlight for this former English teacher, culinary school grad, and food editor.\n\n**Erin Shitama \u2022 Assistant to the President \u2022 Teams Bee and Mushroom**\n\nErin grew up foraging for wild berries and onions around her home on the eastern shore of Maryland, and was drawn to the locavore lifestyle long before she knew locavore was a word. She's happiest when she's on the road with only her passport and a backpack, sampling local food, language, and culture along the way.\n\n**Johanna Silver \u2022 Test Garden Coordinator \u2022 Teams Garden, Escargot, Chicken**\n\nJohanna came to Sunset with a background in organic farm apprenticeships and garden education. She's since added ornamental gardening to her list of loves while at the magazine, but participating in the One-Block project keeps her in touch with her edible roots\u2014er, beginnings.\n\n**Margaret Sloan \u2022 Production Coordinator \u2022 Team Bee**\n\nMargaret wears many hats at Sunset; she researches stories, illustrates maps, squares up page designs, and makes sure the magazine will print properly. She's been fascinated by bees ever since a swarm settled on a tree near her house years ago, but never dreamed she'd have a job that required her to wear a beekeeper's helmet, or that she'd learn to like bugs.\n\n**Vanessa Speckman \u2022 Photo & Imaging Assistant \u2022 Teams Bee and Mead**\n\nVanessa grew up in an Italian and Irish family with a hands-on kitchen, learning how to do everything from crush grapes and construct wine barrels with her _nonno_ to making good chicken broth. Harvesting honey at Sunset and understanding how vital bees are to the environment inspired Vanessa and her brother to make their own mead at home.\n\n**Lauren Bonar Swezey \u2022 Former Garden Special Projects Editor \u2022 Teams Garden, Wine, Chicken**\n\nAs manager of the Sunset test garden and long-time garden writer and editor, Lauren helped identify which vegetable varieties were best for Menlo Park's mild climate, and oversaw planting, care, and harvest. Her favorite part of the project? The excitement of witnessing the garden's daily transformation from bare plot into veritable vegetable factory, and then sampling the tasty harvest.\n\n**Katie Tamony \u2022 Editor-in-Chief \u2022 Team Mushroom**\n\nShe was so intrigued by the idea of \"making\" her own mushrooms\u2014a favorite food! Katie's usual job at Sunset is to direct the editorial work of our incredibly creative group in magazines, books, our web site, and television. And to greenlight projects like this!\n\n**Margo True \u2022 Food Editor \u2022 All teams, but especially Kitchen, Chicken, and Cow**\n\nWhat a journey it's been. What's next? How about Team Fish?\n\n# Acknowledgments\n\nFor almost every one of our projects, we relied on the advice of people wiser and more knowledgeable than us. These generous people helped steer us in the right direction:\n\n**Bees** Randy Oliver, of Scientific Beekeeping, for shifting our paradigm; master beekeepers Tom Vercoutere and Tina and Thomas Keller, for their help, support, and soothing influence in the face of panic; and the Beekeepers' Guild of San Mateo County, for patiently answering our questions and lending us a honey extractor.\n\n**Beer** Chuck Schwalbach, for sharing his extensive beer knowledge and vast array of brewing equipment, and for being so calm and organized.\n\n**Cheese** Sue Conley, Peg Smith, Jonathan White, and Maureen Cunnie at Cowgirl Creamery; and Liam Callahan and his mom, Cindy Callahan, at Bellwether Farms, for their inspiring cheesemaking lessons and followup advice.\n\n**Chickens** Jody Main, for a demystifying day with her hens.\n\n**Cow** Ron Garthwaite and Collette Cassidy, of Claravale Farms, who made Team Cow possible.\n\n**Garden** Renee Shepard, of Renee's Garden Seeds, for always encouraging us to start from seed; also Darryl Wong, of Free Wheelin' Farm, for being our on-call crop advisor.\n\n**Olives** Chris Banthien at Valencia Farms, where we picked olives, and Alessio Carli, olive oil and winemaker at Pietra Santa Winery, where we crushed them, as well as the winery's director of marketing, Jayme Nunn; Dan Flynn, for backyard tree harvesting advice; and Alexandra Devarenne, Bill Krueger, Frank Zalom, Paul Vossen, and Ernie Simpson, for their excellent tips on olive fruit fly control.\n\n**Salt** Mark Bitterman, \"selmelier\" at The Meadow, in Portland, Oregon, for his infectious enthusiasm and for helping us make better salt.\n\n**Vinegar** The incomparable Paula Wolfert, cookbook author, who taught us the ways of vinegar-growing and gave us pieces of her vinegar mother.\n\n**Wine** Thomas Fogarty Winery's Dr. Thomas Fogarty, Anne Krolczyk, and especially winemaker Michael Martella, who visited Sunset to check up on the wine; also Dan Brenzel, home winemaker and retired chemist, who kindly loaned us just about every scrap of equipment we needed\u2014and whose wisecracks still make us laugh.\n\nTo all of them, we owe a giant debt of gratitude. Many others contributed along the way, including my sweetheart, Peter Lang, who trapped Nugget the rooster and spent many hours tending chickens on weekends; Andrew Hash, who collected gallons of Pacific Ocean; and Hank Shaw, our friend at http:\/\/honest-food.net, who gave us pointers on cooking snails; and the readers of our blog, http:\/\/oneblockdiet.sunset.com, who have offered countless good suggestions and stories of their own food-from-scratch adventures.\n\nIn the kitchen, we salute our amazing team of retesters\u2014home cooks who come to Sunset and test our recipes with diligence and dedication to make sure they'll work for anyone. We are grateful for all their work on this book. Thanks also to Molly Watson, former Sunset recipe editor, for creating some of the recipes for our first one-block menu back in the summer of 2007.\n\nEvery now and then, we've needed to construct things like a hen yard or a concrete pad to shield our beehives from ants. Tony Soria and Dan Strack, whose talent and hard work keeps Sunset's office and grounds operating smoothly, have been our heroes and helped us out whenever we asked. Speaking of construction, we have possibly the prettiest henhouse in Menlo Park, built by James Stamp of Wine Country Coops; thank you, James.\n\nThe photos of our feasts were art directed by Sunset's creative director, Mia Daminato, whose graceful aesthetics set the tone for this book. Our luscious recipe shots are the work of staff photographer Tom Story and photo editor Sue Smith\u2014thank you so much\u2014and the garden shots were coordinated by photo editor Linda Peters with good-natured aplomb. A big thank you to photo director Yvonne Stender, for overseeing the photography and making sure it came in on time and on budget. Warm thanks to Sunset's imaging specialist, Kimberley Burch, for efficiently cataloguing hundreds of possible images for this book and adding to them with her own photography. You went above and beyond! Thanks too to Spencer Toy, for our fine group portrait and for the many other shots he contributed to these pages.\n\nAt Ten Speed Press, Melissa Moore was our champion editor and warm guiding hand, assisted with aplomb by Emily Timberlake. Nancy Austin and Katy Brown created the beautiful design. We much appreciated Sharon Silva and Linda Bouchard for their scrupulous attention to every word. And we are very thankful to Aaron Wehner, who instantly understood what we were doing and signed us on. Working with all of them has been a wonderful experience.\n\nAnd lastly, heartfelt thanks to Katie Tamony, our editor in chief at Sunset, for saying Yes to this quirky project and committing the time, resources, and enthusiastic support to help it blossom and grow\u2014and to Barb Newton, our president, who was always ready for the next adventure.\n\n# Appendix\n\n**Regional Planting Calendars for Your Own One-Block Feasts**\n\nThe planting and growing information in this book is for Northern California, where Sunset is located. Over the decades, we've defined a series of 32 climate zones for the entire West\u2014based on winter low temperatures as well as elevation, latitude, rainfall, and proximity to climate-altering mountains, oceans and lakes\u2014with the aim of helping our readers figure out what to plant in their areas and how best to care for those plants. This information is presented in detail in the _Sunset Western Garden Book_ (Sunset Publishing, Eighth Edition, 2007). We've applied that knowledge to the planting timelines that follow to help you grow our one-block crops no matter where in the West you live.\n\nWhereas our warm-season and cool-season planting timelines reflect how we planted crops in Northern California, the calendars here show you all the seasonal plantings and harvests possible in your area, to give you maximum choice for planning menus. The calendars begin in March simply for the sake of consistency.\n\n_Edamame sprout_\n\n**One-Block Feasts across the Country**\n\nIn certain regions of the country, planting times roughly resemble those in parts of the West. If you live in one of the following areas, consult the timeline suggested for your region, but make allowances for your different climate conditions.\n\n\u2022 **New England** (Connecticut, Massachusetts, Rhode Island). Check out our Mountain and Intermountain West timeline.\n\n\u2022 **Midwest** (Illinois, Iowa, Michigan, Wisconsin). Your spring planting dates are much the same as in the Pacific Northwest, but your summers are hotter and winters colder.\n\n\u2022 **Mid-Atlantic States** (Maryland, Delaware, Virginia). Your spring planting dates are much the same as in the Pacific Northwest, but your summer summers are hotter and winters colder.\n\n\u2022 **The Mid-South** (parts of Georgia, Alabama) Your region shares some planting dates with parts of California, but heat, humidity, and rainfall significantly shift the harvest and summer planting dates.\n\n\u2022 **A National Climate Map** For more precisely tailored information for your climate, see our map of 45 national climate zones at www.sunset.com\/usclimatezones. The accompanying descriptions will tell you when to plant warm-season crops and when to start cold-season crops in your area, so you can apply that information to growing the plants in this book; refer to the seasonal garden guides for specific information on planting and harvesting each crop. For a print copy of the national zones and much more on countrywide gardening, pick up a copy of the _Sunset National Garden Book_ (Sunset Publishing, 1997; available on www.amazon.com).\n\nBecause growing conditions can vary so much by region and even microregion, the best source for local planting information is your county's agricultural cooperative extension office; find yours at www.csrees.usda.gov\/Extension\/index.html. Local independent nurseries are excellent sources of advice, too.\n\n# SOUTHERN CALIFORNIA  \n(low elevations west of the mountains)\n\n**MARCH**\n\n**Plant**\n\nArugula (through Apr)\n\nBeets (through May)\n\nBroccoli rabe\n\nCarrots (except interior valleys)\n\nChervil\n\nDill\n\nFlorence fennel\n\nGreen onions (through Apr)\n\nLettuce (through Apr)\n\nMesclun (through Apr)\n\nMustard\n\nParsley (through Apr)\n\nPotatoes\n\nQuinoa\n\nRadishes (through Apr)\n\nSwiss chard (through May)\n\nTea\n\n**Harvest (Jan or Feb planting)**\n\nArugula\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nGreen onions\n\nKale\n\nLemons\n\nLettuce\n\nMesclun\n\nMustard\n\nParsley\n\nRadishes\n\n**_WARM SEASON_**\n\n**APRIL**\n\n**Plant**\n\nBasil (through Jun)\n\nBeans (through Jul)\n\nChiles\n\nCorn (through Jul)\n\nCucumbers (through May)\n\nEdamame (through Jul)\n\nHerbs (perennial)\u2014chives, marjoram, oregano, peppermint, rosemary, sage, French tarragon, thyme\n\nLemon tree\n\nLemongrass\n\nMelons, all kinds (through Jun)\n\nNasturtiums\n\nPeppers\n\nPineapple guava\n\nPurslane\n\nSquash, butternut\n\nSquash, summer (pattypan and zucchini)\n\nTangerine tree\n\nTomato\n\n**Harvest**\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nFava beans\n\nGreen onions\n\nHerbs (perennial)\n\nLemons\n\nLettuce\n\nMesclun\n\nMustard\n\nParsley\n\nRadicchio\n\nRadishes\n\nStrawberries\n\nSwiss chard\n\nTea\n\n**MAY**\n\n**Harvest**\n\nBarley\n\nBasil\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nChervil\n\nDill\n\nFava beans\n\nFlorence fennel\n\nGreen onions\n\nHerbs (perennial)\n\nLemons\n\nLettuce\n\nMesclun\n\nOnions, 'Spanish White' and cippolini\n\nParsley\n\nStrawberries\n\nSwiss chard\n\nWheat\n\n**JUNE**\n\n**Harvest**\n\nBarley\n\nBasil\n\nBeets\n\nCabbage\n\nCarrots\n\nChervil\n\nDill\n\nEdamame\n\nFlorence fennel\n\nGarlic\n\nHerbs (perennial)\n\nLemons\n\nLemongrass\n\nLettuce\n\nNasturtiums\n\nOnions, 'Spanish White' and cippolini\n\nParsley\n\nPotatoes\n\nPurslane\n\nQuinoa\n\nSquash, summer (pattypan and zucchini)\n\nSwiss chard\n\nWheat\n\n**JULY**\n\n**Plant**\n\nCorn\n\n**Harvest**\n\nBasil\n\nBeans\n\nCucumbers\n\nChiles\n\nDill\n\nEdamame\n\nFlorence fennel\n\nGarlic\n\nHerbs (perennial)\n\nLemons\n\nLemongrass\n\nMelons\n\nNasturtiums\n\nPeppers\n\nPotatoes\n\nPurslane\n\nQuinoa\n\nSquash, summer (pattypan and zucchini)\n\nTomatoes\n\n**_COOL SEASON_**\n\n**AUGUST**\n\n**Plant**\n\nBroccoli romanesco (through Oct)\n\nCabbage (through Oct)\n\nCauliflower\n\nLettuce (through Feb)\n\nPotatoes (through Sept)\n\n**Harvest**\n\nBasil\n\nBeans\n\nChiles\n\nCorn\n\nCucumbers\n\nEdamame\n\nHerbs (perennial)\n\nHops\n\nLemons\n\nLemongrass\n\nMelons\n\nNasturtiums\n\nPeppers\n\nPurslane\n\nSquash, butternut\n\nSquash, summer (pattypan and zucchini)\n\nTomatoes\n\n**SEPTEMBER**\n\n**Plant**\n\nArugula (through Feb)\n\nBeets (through Feb)\n\nBroccoli rabe (through Feb)\n\nCarrots\n\nChervil\n\nGreen onions\n\nKale (through Oct)\n\nMesclun (through Feb)\n\nMustard (through Oct)\n\nParsley\n\nRadicchio (through Feb)\n\nRadishes (through Feb)\n\nSwiss chard\n\n**Harvest**\n\nBeans\n\nChiles\n\nCorn\n\nCucumbers\n\nEdamame\n\nHerbs (perennial)\n\nHops\n\nLemons\n\nMelons\n\nMesclun\n\nPeppers\n\nRadishes\n\nSquash, butternut\n\nSquash, summer (pattypan and zucchini)\n\nTomatoes\n\n**OCTOBER**\n\n**Plant**\n\nBarley\n\nCarrots (except interior valleys)\n\nFava beans (through Nov)\n\nGreen onions\n\nOnions, 'Spanish White' and cippolini (through Nov)\n\nParsley (through Feb)\n\n**Harvest**\n\nArugula\n\nBeans\n\nBroccoli rabe\n\nChiles\n\nCorn\n\nEdamame\n\nHerbs (perennial)\n\nKale\n\nLemons\n\nLettuce\n\nMesclun\n\nMustard\n\nPeppers\n\nPineapple guava\n\nRadishes\n\nTomatoes\n\nSquash, butternut\n\n**NOVEMBER**\n\n**Plant**\n\nCabbage (except interior valleys through Jan)\n\nCarrots (except interior valleys through Dec)\n\nGarlic (through Dec)\n\nKale (except interior valleys through Jan)\n\nMustard (along coast through Dec)\n\nPotatoes\n\nWheat\n\n**Harvest**\n\nArugula\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nGreen onions\n\nHerbs (perennial)\n\nKale\n\nLemons\n\nLettuce\n\nMesclun\n\nMustard\n\nParsley\n\nPineapple guava\n\nPotatoes\n\nRadicchio\n\nRadishes\n\nSwiss chard\n\n**DECEMBER**\n\n**Harvest**\n\nArugula\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nChervil\n\nGreen onions\n\nHerbs (perennial)\n\nKale\n\nLemons\n\nLettuce\n\nMesclun\n\nMustard\n\nParsley\n\nPotatoes\n\nRadicchio\n\nRadishes\n\nSwiss chard\n\nTangerines\n\n**JANUARY**\n\n**Plant**\n\nBroccoli romanesco (except interior valleys through Feb)\n\nCauliflower (except interior valleys through Feb)\n\nHops (through Feb)\n\nMustard\n\nOnions, 'Spanish White' and cippolini (except along the coast)\n\nStrawberries (through Feb)\n\n**Harvest**\n\nArugula\n\nBeets\n\nCabbage\n\nCarrots (except interior valleys)\n\nChervil\n\nGreen onions\n\nHerbs (perennial)\n\nKale\n\nLemons\n\nLettuce\n\nMesclun\n\nMustard, along coast\n\nParsley\n\nRadicchio\n\nRadishes\n\nTangerines\n\n**FEBRUARY**\n\n**Plant**\n\nCabbage\n\nCarrots\n\nChervil\n\nDill\n\nFlorence fennel\n\nGreen onions\n\nMustard\n\nOnions, 'Spanish White' and cippolini (through Nov)\n\nPotatoes\n\nSwiss chard\n\n**Harvest**\n\nArugula\n\nBeets\n\nBroccoli rabe\n\nCabbage\n\nCarrots (except interior valleys)\n\nGreen onions\n\nHerbs (perennial)\n\nKale\n\nLemons\n\nLettuce\n\nMesclun\n\nMustard (along coast)\n\nParsley\n\nRadishes\n\nTangerines\n\n# SOUTHWEST DESERTS  \n(in New Mexico, Arizona, Utah, Nevada, and California)\n\nThis calendar has no warm or cool seasons. That's because growing seasons here are especially diverse. Instead, follow planting times for one of the four regional climate zones: zone 10, high desert; zone 11, intermediate deserts of California and southern Nevada; zone 12, Arizona-Sonoran desert; zone 13, low deserts of California and Arizona. If no zone is noted, the information pertains to all Southwest gardens.\n\n**MARCH**\n\n**Plant**\n\nBasil (zone 12)\n\nBeans (zone 12)\n\nBeets\n\nBroccoli rabe (zones 10, 11)\n\nBroccoli romanesco (zones 10, 11)\n\nCabbage (zones 10, 11)\n\nCarrots (zone 10, 11)\n\nCauliflower (zones 10, 11)\n\nChervil (zones 10, 11, 12)\n\nCorn (zones 12, 13)\n\nCucumbers (zone 13)\n\nDill (zone 11)\n\nEdamame (zone 13)\n\nFava Beans (zone 11)\n\nFlorence fennel (zone 11)\n\nHops (zone 10)\n\nKale (zones 10, 11)\n\nLettuce (zones 10, 12)\n\nMelons (zones 12, 13)\n\nMesclun\n\nMustard (zones 10, 11)\n\nOnions, 'Spanish White' and cippolini (zone 10)\n\nParsley (zones 10, 11)\n\nPotatoes (zone 11)\n\nPurslane (zone 13)\n\nRadicchio (zone 11)\n\nRadishes (zones 10, 12)\n\nStrawberries\n\nSquash, summer (pattypan and zucchini) (zones 12, 13)\n\nSwiss chard (zones 10, 12)\n\nTomato (zone 12, 13)\n\nSquash, butternut (zone 13)\n\n**Harvest**\n\nArugula (zone 12)\n\nBeets\n\nCarrots (zones 12, 13)\n\nFava Beans (zones 12, 13)\n\nFlorence fennel (zones 12, 13)\n\nGreen onions\n\nLettuce (zones 12, 13)\n\nMesclun\n\nParsley (zones 12, 13)\n\nQuinoa (zone 13)\n\nRadicchio (zone 12)\n\nRadishes (zones 12, 13)\n\nSwiss chard (zone 12)\n\n**APRIL**\n\n**Plant**\n\nBasil (zones 10, 11, 12; from mid-month)\n\nCarrots (zone 10, 11)\n\nChervil (zones 10, 11, 12)\n\nChiles\n\nCorn (zones 10, 11)\n\nCucumbers (zone 11; from mid-month, 13)\n\nDill (zones 10, 11)\n\nEdamame (zones 12, 13)\n\nFava Beans (zone 10)\n\nFlorence fennel (zones 10, 11)\n\nHerbs (perennial)\u2014chives, marjoram, oregano, peppermint, rosemary, sage, French tarragon (zone 10 only), thyme\n\nLemon tree (zones 12, 13; outside potted for wintering indoors elsewhere)\n\nLemongrass\n\nLettuce (zone 10),\n\nMesclun\n\nNasturtiums (zones 10, 11, 12)\n\nOnions, 'Spanish White' and cippolini (zone 10)\n\nParsley (zones 10, 11)\n\nPeppers\n\nPineapple guava (zones 12, 13)\n\nPotatoes, (zone 10)\n\nPurslane (zones 10, 11, 12)\n\nQuinoa (zones 10, 11, 12)\n\nRadicchio (zone 10)\n\nRadishes (zones 10, 11, 12)\n\nSquash, butternut (zone 10)\n\nSquash, summer (pattypan and zucchini) (zones 12, 13)\n\nTangerine tree (zones 12, 13; outside potted for wintering indoors elsewhere)\n\nTea (zone 12)\n\nTomato (zones 10, 11)\n\n**Harvest**\n\nBarley (zones 12, 13)\n\nBasil\n\nBeans\n\nBeets\n\nBroccoli rabe (zones 10, 11)\n\nCarrots (zones 12, 13\n\nCucumbers (zone 13)\n\nDill (zones 12, 13)\n\nFlorence fennel (zones 12, 13)\n\nGreen onions\n\nLettuce (zones 10, 11)\n\nMesclun\n\nOnions, 'Spanish White' and cippolini\n\nParsley (zones 12, 13)\n\nQuinoa (zone 13)\n\nSwiss chard (zones 10, 12)\n\n**MAY**\n\n**Plant**\n\nArugula (zone 12)\n\nBasil (zones 10, 11, 12)\n\nCarrots (zone 11)\n\nChiles\n\nCorn (zone 10)\n\nCucumbers (zone 10, 11),\n\nEdamame (zones 11, 12)\n\nMesclun (zones 10, 12)\n\nMelon (zones 10, 11)\n\nPeppers (zones 10, 12)\n\nSquash, butternut (zones 10, 11)\n\nSquash, summer (pattypan and zucchini) (zones 10, 11)\n\nTomato (zones 10, 11)\n\n**Harvest**\n\nBarley (zones 12, 13; late in month)\n\nBasil\n\nBeans\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCarrots\n\nChervil (zones 10, 11, 12)\n\nCorn (zone 13)\n\nCucumbers (zone 13)\n\nDill (zones 10, 11)\n\nFlorence fennel (zones 10, 11)\n\nGarlic\n\nGreen onions\n\nLettuce (zones 10, 11, 12)\n\nMesclun\n\nNasturtiums (zone 13)\n\nPotatoes (zone 13)\n\nOnions, 'Spanish White' and cippolini\n\nParsley (zones 10, 11)\n\nPurslane (zone 13)\n\nRadicchio (zones 10, 11)\n\nRadishes, zones 10, 11, 12\n\nStrawberries\n\nSwiss chard (zones 10, 12)\n\nTea\n\nWheat\n\n**JUNE**\n\n**Plant**\n\nCabbage (zone 10)\n\nCarrots (zone 11)\n\nCorn (zone 10)\n\nEdamame (zone 10)\n\nSquash, summer (pattypan and zucchini) (zones 10, 11)\n\n**Harvest**\n\nArugula (zone 12)\n\nBarley\n\nBasil\n\nBeans\n\nBeets (zones 10, 11)\n\nCarrots (zones 10, 11)\n\nChervil (zones 10, 11, 12)\n\nCorn (zones 12, 13)\n\nCucumbers\n\nDill (zones 10, 11)\n\nEdamame (zone 13)\n\nFlorence fennel (zones 10, 11)\n\nGreen onions\n\nHerbs (perennial)\n\nLemongrass\n\nLettuce (zone 10)\n\nMelons (zones 12, 13)\n\nMesclun (zone 10, 12)\n\nNasturtiums (zones 10, 11, 12)\n\nOnions, 'Spanish White' and cippolini\n\nParsley (zones 10, 11)\n\nPotatoes (zones 12, 13)\n\nPurslane\n\nRadicchio (zone 10)\n\nStrawberries\n\nSwiss chard (zone 10)\n\nTomatoes (zones 12, 13)\n\n**JULY**\n\n**Plant**\n\nBeans (zone 12; last half of month)\n\nBeets (zone 10)\n\nCabbage (zone 10)\n\nCarrots (zone 11)\n\nCorn (zones 10, 11, 12)\n\nMesclun (zone 10)\n\nRadicchio (zone 10)\n\nSquash, butternut (zones 12, 13)\n\n**Harvest**\n\nBasil\n\nCarrots (zone 11)\n\nChiles\n\nCorn (zones 10, 11)\n\nCucumbers (zones 10, 11)\n\nEdamame (zones 12, 13)\n\nFava beans\n\nGarlic\n\nHerbs (perennial)\n\nLemongrass\n\nMesclun (zones 10, 12)\n\nNasturtiums (zones 10, 11, 12)\n\nPeppers\n\nPotatoes (zone 11)\n\nPurslane\n\nQuinoa (zones 10, 11, 12)\n\nSquash, butternut (zone 13)\n\nSquash, summer (pattypan and zucchini) (zones 10, 11)\n\nTomatoes\n\n**AUGUST**\n\n**Plant**\n\nBeans (zones 12, 13)\n\nBeets (zones 10, 11)\n\nCarrots (zone 10)\n\nCorn (zones 12, 13)\n\nCucumbers (zones 12, 13)\n\nLettuce (zone 10)\n\nMesclun (zone 10)\n\nRadicchio (zone 11)\n\nRadishes (zones 10, 11, 12)\n\nSwiss chard (zone 10)\n\n**Harvest**\n\nCarrots (zone 11)\n\nChiles\n\nCorn (zone 10)\n\nEdamame (zones 11, 12)\n\nFava beans\n\nHerbs (perennial)\n\nHops\n\nLemons\n\nLemongrass\n\nMelons (zones 10, 11)\n\nMesclun (zone 10)\n\nPeppers\n\nPotatoes (zone 10)\n\nPurslane\n\nQuinoa (zones 10, 11, 12)\n\nSquash, butternut (zone 10)\n\nSquash, summer (pattypan and zucchini) (zones 10, 11)\n\nTomatoes (zones 10, 11, 12)\n\n**SEPTEMBER**\n\n**Plant**\n\nArugula (zones 10, 11, 12)\n\nBeets\n\nBroccoli romanesco\n\nCabbage\n\nCarrots (zones 12, 13)\n\nCauliflower\n\nChervil (zones 12, 13)\n\nChiles\n\nDill (zone 12)\n\nFlorence fennel (zone 12)\n\nKale\n\nLettuce (zones 10, 11, 12)\n\nMesclun\n\nMustard\n\nNasturtiums (zone 13)\n\nParsley (zone 12)\n\nPeppers\n\nRadicchio (zones 12, 13),\n\nRadishes (zones 11, 12, 13)\n\nSwiss chard (zones 12, 13)\n\n**Harvest**\n\nBeets\n\nCarrots (zone 11)\n\nCorn (zone 10)\n\nEdamame (zone 10, 11, 12)\n\nHerbs (perennial)\n\nLemons\n\nLemongrass\n\nMesclun\n\nRadishes (zones 10, 11, 12)\n\nSquash, butternut (zones 10, 11)\n\nSwiss chard (zone 10)\n\nTomatoes (zones 10, 11, 12)\n\n**OCTOBER**\n\n**Plant**\n\nArugula (zones 12, 13)\n\nBeets (zones 12, 13)\n\nBroccoli rabe (zones 12, 13)\n\nBroccoli romanesco (zones 12, 13)\n\nCabbage (zones 12, 13)\n\nCarrots (zones 10, 12, 13)\n\nCauliflower (zones 12, 13)\n\nDill (zones 12, 13)\n\nFava Beans (zones 12, 13)\n\nFlorence fennel (zones 12, 13)\n\nGarlic\n\nHerbs (perennial)\u2014rosemary, sage, thyme\n\nKale (zones 12, 13)\n\nLettuce (zones 12, 13)\n\nMesclun (zones 12, 13)\n\nMustard (zones 12, 13)\n\nParsley (zones 12, 13)\n\nRadicchio (zones 12, 13)\n\nRadishes (zones 11, 12, 13)\n\nSwiss chard (zones 12, 13)\n\n**Harvest**\n\nArugula (zones 10, 11, 12)\n\nBeans\n\nBeets\n\nCabbage\n\nChervil (zones 12, 13)\n\nCorn (zones 10, 11, 12)\n\nCucumbers (zones 12, 13)\n\nEdamame (zone 10)\n\nHerbs (perennial)\n\nLemons\n\nLettuce (zone 10)\n\nMesclun\n\nNasturtiums (zone 13)\n\nRadishes (zones 11, 12, 13)\n\nSquash, butternut (zones 12, 13)\n\nSwiss chard (zone 10)\n\n**NOVEMBER**\n\n**Plant**\n\nArugula (zones 12, 13)\n\nBarley (zones 12, 13)\n\nBeans\n\nBeets (zones 12, 13)\n\nBroccoli rabe (zones 12, 13)\n\nBroccoli romanesco (zones 12, 13)\n\nCabbage (zones 12, 13)\n\nCarrots (zones 12, 13)\n\nCauliflower (zones 12, 13)\n\nDill (zones 12, 13)\n\nFava Beans (zones 12, 13)\n\nFlorence fennel (zones 12, 13)\n\nGarlic\n\nKale (zones 12, 13)\n\nLettuce (zones 12, 13)\n\nMesclun (zones 12, 13)\n\nMustard, transplant (zones 12, 13)\n\nOnions, 'Spanish White' and cippolini\n\nParsley, seedling (zones 12, 13)\n\nQuinoa (zone 13)\n\nRadicchio (zone 12)\n\nRadishes (zones 12, 13)\n\nSwiss chard (zones 12, 13)\n\nWheat\n\n**Harvest**\n\nArugula\n\nBeets\n\nCorn (zones 12, 13)\n\nCucumbers (zones 12, 13)\n\nHerbs (perennial)\n\nLemons\n\nLettuce\n\nMesclun (zones 12, 13)\n\nNasturtiums (zone 13)\n\nPineapple guava\n\nRadicchio (zones 12, 13)\n\nRadishes (zones 11, 12, 13)\n\nSquash, butternut (zones 12, 13)\n\nSwiss chard (zones 12, 13)\n\nTangerines\n\n**DECEMBER**\n\n**Plant**\n\nArugula (zones 12, 13)\n\nBarley (zones 12, 13)\n\nBeets (zones 12, 13)\n\nCarrots (zones 12, 13)\n\nDill (zone 13)\n\nFlorence fennel (zone 13)\n\nLettuce (zones 12, 13)\n\nMesclun (zones 12, 13)\n\nOnions, 'Spanish White' and cippolini\n\nParsley (zones 12, 13)\n\nPotatoes (zone 13)\n\nRadicchio (zone 12)\n\nRadishes (zones 12, 13)\n\nSwiss chard (zones 12, 13)\n\n**Harvest**\n\nArugula\n\nBeets\n\nCarrots (zones 12, 13)\n\nDill (zones 12, 13)\n\nFlorence fennel (zones 12, 13)\n\nLemons\n\nLettuce (zones 12, 13)\n\nMesclun (zones 12, 13)\n\nParsley (zones 12, 13)\n\nRadicchio (zones 12, 13)\n\nRadishes (zones 12, 13)\n\nSwiss chard (zones 12, 13)\n\nTangerines\n\n**JANUARY**\n\n**Plant**\n\nArugula (zones 12, 13)\n\nBasil (zone 13)\n\nBeets (zones 12, 13)\n\nCarrots (zones 12, 13)\n\nDill (zone 13)\n\nFlorence fennel (zone 13)\n\nHops (zone 10)\n\nLettuce (zone 12)\n\nMesclun (zones 12, 13)\n\nParsley (zones 12, 13)\n\nPotatoes (zone 13)\n\nRadicchio (zone 12)\n\nRadishes (zones 12, 13)\n\nStrawberries (through Mar)\n\nSwiss chard (zone 12)\n\n**Harvest**\n\nArugula (zones 12, 13 through Feb)\n\nBeets\n\nBroccoli romanesco (zones 12, 13)\n\nCabbage (zones 12, 13)\n\nCarrots (zones 12, 13)\n\nCauliflower (zone 13)\n\nDill (zones 12, 13)\n\nFlorence fennel (zones 12, 13)\n\nGreen onions\n\nKale (zones 12, 13)\n\nLemons\n\nLettuce (zones 12, 13)\n\nMesclun (zones 12, 13)\n\nParsley (zones 12, 13)\n\nRadicchio (zones 12, 13)\n\nSwiss chard (zones 12, 13)\n\nTangerines\n\n**FEBRUARY**\n\n**Plant**\n\nBasil\n\nBeans (zone 13)\n\nBeets\n\nCarrots (zones 12, 13)\n\nCorn (zone 13)\n\nCucumbers (zone 13)\n\nDill (zone 13)\n\nFlorence fennel (zone 13)\n\nHop (zone 10)\n\nLettuce (zones 11, 12)\n\nMesclun (zones 11, 12)\n\nNasturtiums (zone 13)\n\nPotatoes (zones 12, 13)\n\nRadishes (zones 12, 13)\n\nSwiss chard (zones 10, 12)\n\n**Harvest**\n\nArugula (zones 12, 13)\n\nBeets\n\nCabbage (zones 12, 13)\n\nCarrots (zones 12, 13)\n\nCauliflowe (zones 12, 13)\n\nDill (zones 12, 13)\n\nFava Beans (zones 12, 13)\n\nFlorence fennel (zones 12, 13)\n\nGreen onions\n\nKale (zones 12, 13)\n\nLemons\n\nLettuce (zones 12, 13)\n\nMesclun (zones 11, 12, 13)\n\nParsley (zones 12, 13)\n\nRadicchio (zones 12, 13)\n\nRadishes (zones 12, 13)\n\nSwiss chard (zones 12, 13)\n\n# PACIFIC NORTHWEST  \n(Western Oregon, Western Washington, and Southwestern British Columbia)\n\n**MARCH**\n\n**Plant**\n\nArugula (through May)\n\nBeets (through May)\n\nBroccoli rabe (through Apr, warmest zones)\n\nBroccoli romanesco (through Apr, warmest zones)\n\nCabbage (through Jul)\n\nCarrots (through Jul)\n\nCauliflower (through May)\n\nChervil\n\nHops\n\nLettuce (through May)\n\nMesclun (through Jul)\n\nMustard (through Apr)\n\nOnion (through Jun) (Green onions only)\n\nParsley (through May)\n\nPotatoes (through May)\n\nRadicchio (through Jul)\n\nRadishes (through Apr)\n\nStrawberries\n\nSwiss chard (through May)\n\n**Harvest**\n\nHerbs (perennial)\n\nLemons\n\n**_WARM SEASON_**\n\n**APRIL**\n\n**Plant**\n\nDill (through Jun)\n\nFlorence fennel (through Jul)\n\nHerbs (perennial)\u2014chives, oregano, peppermint, rosemary, sage, French tarragon, thyme (thorough May)\n\nHops (through May)\n\nLemongrass\n\nMarjoram (through May)\n\nPineapple guava (warmest location)\n\nTomato (last half of month and through May)\n\n**Harvest**\n\nArugula\n\nCauliflower\n\nChives\n\nLemons\n\nMesclun\n\nMustard\n\n**MAY**\n\n**Plant**\n\nBasil (through Jul)\n\nBeans (through Jun)\n\nBroccoli rabe (cooler zones through Jun)\n\nBroccoli romanesco (cooler zones through Jun)\n\nChiles\n\nCorn\n\nCucumbers\n\nEdamame\n\nHoneydew melon\n\nLemon tree (through Jun; outside in summer, sunroom in winter)\n\nMelons\n\nNasturtiums (through Jun)\n\nPeppers\n\nPineapple guava\n\nPurslane (through Jun)\n\nQuinoa\n\nRadishes (except in hottest zones)\n\nSquash, butternut (through Jun)\n\nSquash, summer (pattypan and zucchini, through Jun)\n\nTangerine tree (through Jun; outside in summer, sunroom in winter)\n\nWatermelon\n\n**Harvest**\n\nArugula\n\nBeets\n\nBroccoli rabe\n\nCarrots\n\nCauliflower\n\nChervil\n\nFava Beans\n\nGreen onions\n\nHerbs (perennial)\n\nLemons\n\nLettuce\n\nMesclun mix\n\nMustard\n\nRadicchio\n\nRadishes\n\nWheat\n\n**JUNE**\n\n**Plant**\n\nBeets (cooler zones only)\n\nCabbage (cooler zones only through Jul)\n\nCauliflower (cooler zones, through Jul)\n\nFava Beans\n\nLettuce (cooler zones, through Jul)\n\nRadishes (cooler zones)\n\n**Harvest**\n\nArugula\n\nBasil\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nDill leaves\n\nGarlic\n\nGreen onions\n\nHerbs (perennial)\n\nLemongrass\n\nLemons\n\nLettuce\n\nMarjoram\n\nMesclun\n\nMustard\n\nParsley\n\nPotatoes\n\nRadicchio\n\nRadishes\n\nStrawberries\n\nSwiss chard\n\n**JULY**\n\n**Plant**\n\nArugula (late in the month)\n\nBeets (warmer zones)\n\nRadishes (except in hottest zones)\n\n**Harvest**\n\nBarley\n\nBasil\n\nBeans\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nCucumbers\n\nDill\n\nEdamame\n\nFennel\n\nGreen onions\n\nHerbs (perennial)\n\nLemongrass\n\nLemons\n\nLettuce\n\nMarjoram\n\nMesclun\n\nNasturtiums\n\nOnions, 'Spanish White' and cippolini\n\nParsley\n\nPotatoes\n\nPurslane\n\nRadicchio\n\nRadishes\n\nStrawberries\n\nSquash, summer (pattypan and zucchini)\n\nSwiss chard\n\nTomato\n\n**_COOL SEASON_**\n\n**AUGUST**\n\n**Plant**\n\nArugula (through Oct)\n\nBasil\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots (through Sept)\n\nCauliflower\n\nChervil (last half of month)\n\nKale\n\nLettuce (except in warmest zones)\n\nMesclun (through Sept)\n\nMustard (through Sept)\n\nOnions, 'Spanish White' and cippolini (through Sept)\n\nParsley\n\nRadicchio (through Nov)\n\nRadishes (except in warmest zones)\n\nSwiss chard\n\n**Harvest**\n\nBasil\n\nBeans\n\nBeets\n\nCabbage\n\nCarrots\n\nChiles\n\nCorn\n\nCucumbers\n\nDill leaves, seeds\n\nEdamame\n\nFennel bulbs\n\nGreen onions\n\nHerbs (perennial)\n\nHop flowers\n\nLemongrass\n\nLemons\n\nLettuce\n\nHoneydew melon\n\nMarjoram\n\nMelons\n\nMesclun\n\nNasturtiums\n\nOnions, 'Spanish White' and cippolini\n\nParsley\n\nPeppers\n\nPotatoes\n\nPurslane\n\nRadicchio\n\nRadishes\n\nStrawberries\n\nSquash, summer (pattypan and zucchini)\n\nSwiss chard\n\nTomato\n\nWatermelon\n\n**SEPTEMBER**\n\n**Plant**\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCauliflower\n\nChervil\n\nFava beans (through Oct)\n\nHerbs (perennial)\u2014rosemary, sage, thyme (through Oct)\n\nKale\n\nLettuce\n\nPurslane\n\nRadishes\n\n**Harvest**\n\nArugula\n\nBasil\n\nBeans\n\nBeets\n\nBroccoli rabe (late)\n\nBroccoli romanesco\n\nCabbage\n\nCauliflower\n\nChiles\n\nCorn\n\nCucumbers\n\nDill\n\nEdamame\n\nFennel\n\nGreen onions\n\nHerbs (perennial)\n\nHoneydew melon\n\nKale\n\nLemongrass\n\nLemons\n\nLettuce\n\nMarjoram\n\nMelons\n\nMesclun\n\nMustard\n\nParsley\n\nPeppers\n\nQuinoa\n\nRadicchio\n\nRadishes\n\nStrawberries\n\nSquash, butternut\n\nSquash, summer (pattypan and zucchini)\n\nTomato\n\nWatermelon\n\n**OCTOBER**\n\n**Plant**\n\nBarley (through Nov)\n\nGarlic (through Nov)\n\nWheat (through Dec)\n\n**Harvest**\n\nArugula\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nChervil\n\nDill\n\nFennel\n\nHerbs (perennial)\n\nKale\n\nLemons\n\nLettuce\n\nMarjoram\n\nMesclun\n\nMustard\n\nParsley\n\nPineapple guavas\n\nRadicchio\n\nRadishes\n\nSquash, butternut\n\nSwiss chard\n\nTomato\n\n**NOVEMBER**\n\n**Plant**\n\nFava Beans (first half of the month)\n\n**Harvest**\n\nArugula\n\nBroccoli romanesco\n\nCarrots\n\nChervil\n\nLemons\n\nMustard\n\nParsley\n\n**DECEMBER**\n\n**Harvest**\n\nArugula\n\nLemons\n\n**JANUARY**\n\n**Plant**\n\nHops (through Mar)\n\nStrawberries (through Feb)\n\n**Harvest**\n\nHerbs (perennial)\n\nLemons\n\n**FEBRUARY**\n\n**Harvest**\n\nHerbs (perennial)\n\nLemons\n\n# MOUNTAIN AND INTERMOUNTAIN WEST  \n(Mountainous areas of all Western states, including the Northwest)\n\n**MARCH**\n\n**Plant**\n\nBeets (through Apr in mildest areas)\n\nFava Beans (mildest areas)\n\nHops\n\nLettuce (in mildest areas, through Apr)\n\nMesclun (through Jun)\n\nMustard (mildest areas)\n\nOnions, 'Spanish White' and cippolini\n\nPotatoes (mildest areas)\n\nRadicchio\n\nRadishes (mildest areas, through Apr)\n\nStrawberries (through Apr)\n\nSwiss chard (milder areas through Jun)\n\n**Harvest**\n\nLemons\n\nTangerines\n\n**APRIL**\n\n**Plant**\n\nArugula\n\nBroccoli rabe (through Jun)\n\nBroccoli romanesco\n\nCabbage\n\nCarrots (through Jun)\n\nCauliflower (through May)\n\nChervil\n\nFava Beans\n\nHops\n\nOnions, 'Spanish White' and cippolini\n\nQuinoa (through May)\n\n**Harvest**\n\nLemons\n\nMesclun\n\nRadishes\n\nSwiss chard\n\nTangerines\n\n**_WARM SEASON_**\n\n**MAY**\n\n**Plant**\n\nBasil (through Jun)\n\nBeans (through Jun)\n\nBeets\n\nBroccoli romanesco (through Jun)\n\nCabbage (through Jun)\n\nChiles\n\nCorn\n\nCucumbers\n\nDill (through Jun)\n\nEdamame (through Jun)\n\nFlorence fennel\n\nHerbs (perennial)-chives, oregano, peppermint, sage, French tarragon, thyme\n\nLemon tree, (outdoors in frost-free season, sunroom in cold months)\n\nLemongrass\n\nLettuce (everywhere through Jun)\n\nMarjoram\n\nMelons\n\nMustard (coldest-winter areas)\n\nNasturtiums\n\nPeppers\n\nPotatoes (everywhere)\n\nPurslane\n\nRadishes (colder areas)\n\nRosemary (though Jun)\n\nSquash, butternut\n\nSquash, summer (pattypan and zucchini)\n\nTangerine tree (outdoors in frost-free season, sunroom in cold months)\n\nTomato (cold-winter areas)\n\nWatermelon\n\n**Harvest**\n\nArugula\n\nBeets\n\nBroccoli rabe\n\nGreen onions\n\nLemons\n\nLettuce\n\nMesclun\n\nRadicchio\n\nRadishes\n\nSwiss chard\n\nTangerines\n\n**JUNE**\n\n**Plant**\n\nCucumbers\n\nPotatoes (cold-winter areas)\n\nSwiss chard (cold-winter areas)\n\n**Harvest**\n\nBeans\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nFava Beans\n\nGarlic\n\nGreen onions\n\nLemons\n\nLettuce\n\nMesclun\n\nRadishes\n\nSwiss chard\n\nTangerines\n\n**JULY**\n\n**Plant**\n\nBeets (mild-summer areas)\n\nLettuce (except in hot-summer areas)\n\nMesclun (except in hot-summer areas)\n\n**Harvest**\n\nBasil\n\nBeans\n\nBeets\n\nBroccoli rabe\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nChervil\n\nChiles\n\nCorn\n\nCucumbers\n\nEdamame\n\nFava Beans\n\nGarlic\n\nGreen onions\n\nHerbs (perennial)\n\nHops\n\nLemons\n\nLemongrass\n\nLettuce\n\nMarjoram\n\nMesclun\n\nNasturtiumss\n\nPeppers\n\nPotatoes\n\nPurslane\n\nRosemary\n\nStrawberries\n\nSquash, summer (pattypan and zucchini)\n\nSwiss chard\n\nTangerines\n\nTomato\n\nWheat\n\n**_COOL SEASON_**\n\n**AUGUST**\n\n**Plant**\n\nArugula (except in hot-summer areas)\n\nLettuce (except in hot-summer areas)\n\nMesclun (except in hot-summer areas)\n\nRadicchio\n\n**Harvest**\n\nBarley\n\nBasil\n\nBeans\n\nBeets\n\nBroccoli romanesco\n\nCabbage\n\nCarrots\n\nCauliflower\n\nChervil\n\nChiles\n\nCorn\n\nCucumbers\n\nDill\n\nEdamame\n\nFava Beans\n\nFlorence fennel\n\nHerbs (perennial)\n\nHoneydew melons\n\nHops\n\nLemons\n\nLemongrass\n\nLettuce\n\nMarjoram\n\nMelons\n\nMesclun\n\nNasturtiums\n\nOnions, cippolini\n\nPepper\n\nPotatoes\n\nPurslane\n\nQuinoa\n\nRosemary\n\nSquash, summer (pattypan and zucchini)\n\nSwiss chard\n\nTangerines\n\nTomato\n\nWatermelons\n\n**SEPTEMBER**\n\n**Plant**\n\nBarley\n\nMesclun\n\nMustard (mildest areas)\n\nRadishes\n\nWheat (through Oct)\n\n**Harvest**\n\nArugula\n\nBasil\n\nBeans\n\nBeets\n\nCorn\n\nCucumbers\n\nChiles\n\nDill\n\nEdamame\n\nHerbs (perennial)\n\nLemons\n\nLemongrass\n\nLettuce\n\nMarjoram\n\nMelons\n\nMesclun\n\nNasturtiums\n\nOnions, 'Spanish White'\n\nPeppers\n\nPotatoes\n\nPurslane\n\nRosemary\n\nSquash, butternut\n\nSquash, summer (pattypan and zucchini)\n\nTangerines\n\nTomato\n\n**OCTOBER**\n\n**Plant**\n\nBarley\n\nGarlic (last half of month through Nov)\n\nRadishes (mild-winter areas)\n\n**Harvest**\n\nArugula\n\nHerbs (perennial)\n\nLemons\n\nLettuce\n\nMarjoram\n\nMesclun\n\nPotatoes\n\nRadishes\n\nRosemary\n\nSquash, butternut\n\nTangerines\n\n**NOVEMBER**\n\n**Harvest**\n\nLemons\n\nRadicchio\n\nRadishes\n\nTangerines\n\n**DECEMBER**\n\n**Harvest**\n\nLemons\n\nTangerines\n\n**JANUARY**\n\n**Harvest**\n\nLemons\n\nTangerines\n\n**FEBRUARY**\n\n**Plant**\n\nSwiss chard (in mildest areas)\n\n**Harvest**\n\nLemons\n\nTangerines\n\n# Index\n\n### A\n\nAle. _See_ Beer\n\nApples\n\nApple Cheese Puff\n\nFresh Apple Cider and Applesauce\n\nGrilled Radicchio and Fennel Salad with Apples and Toasted Walnuts\n\nArugula\n\nArugula and Red Butterhead Lettuce Salad with Tangerines and Hard-Cooked Eggs\n\ngrowing\n\nTea Sandwich Trio\n\n### B\n\nBarley\n\nBelgian Abbey Ale\n\ngrowing, 2.1, 3.1, 3.2\n\nthreshing\n\nBasil\n\ngrowing\n\nTomato and Herb Salad with Fresh Chive Cheese\n\nBeans. _See also_ Edamame; Fava beans\n\ncooking dried\n\nCreamy Flageolet Dip with Red Pepper Sticks\n\nGrilled Summer Succotash\n\ngrowing\n\nMexican Skillet Eggs\n\nQuinoa Huaraches with Egg and Parsley Salad\n\n'Scarlet Emperor' Rago\u00fbt\n\nVegetable Shepherd's Pie\n\nBeekeeping, 1.1, 1.2, 3.1, 6.1, 10.1, 10.2, 14.1, 14.2\n\nBeer\n\nBelgian Abbey Ale\n\nmaking, 1.1, 3.1, 14.1, 14.2\n\nSummer Wheat Beer\n\nBeets\n\nCarrot and Beet Chips\n\nFresh Pickled Beets\n\nGarden Borscht\n\ngrowing\n\nMesclun Salad with Spring Beets and Dill\n\nRoasted Beets and Tops with Tarragon\n\nBelgian Abbey Ale\n\nBeverages. _See also_ Beer; Tea; Wine\n\nFresh Apple Cider\n\nPeppermint-Lemongrass Tisanes\n\nStrawberry Lemonade\n\nTrue Mead\n\nBorscht, Garden\n\nBread\n\nTea Sandwich Trio\n\nWheat Berry Ciabatta\n\nWhole-Wheat Honey Sandwich Bread\n\nBroccoli rabe\n\ngrowing\n\nWheat Berry \"Risotto\" with Roasted Tomatoes and Broccoli Rabe\n\nWinter Vegetable Chowder\n\nBroccoli romanesco\n\ngrowing, 11.1\n\nWinter Vegetable Chowder\n\nBroth, Herb Vegetable\n\nButter\n\nmaking\n\nSage Butter\n\nButtermilk\n\nButtermilk Crackers\n\nmaking\n\nButternut squash\n\nAncho Chile\u2013Sauced Noodles with Shiitakes and Butternut Squash\n\nButternut Squash and Cipollini Onion Soup\n\nButternut Squash Compote with Honey and Toasted Walnuts\n\nButternut Squash Gnocchi with Chard and Sage Brown Butter\n\ngrowing\n\nRoasted Spiced Butternut Squash Seeds\n\n### C\n\nCabbage, Savoy\n\nBraised Winter Greens with Preserved Lemon and Red Chile\n\ngrowing\n\nCanning, 5.1, 5.2\n\nCantaloupe\n\nCantaloupe Sorbet\n\ngrowing\n\nCarrots\n\nCarrot and Beet Chips\n\nGrilled Carrot Salad\n\ngrowing\n\nCauliflower\n\ngrowing\n\nWinter Vegetable Chowder\n\nCheese\n\nApple Cheese Puff\n\nButternut Squash Gnocchi with Chard and Sage Brown Butter\n\nCaramelized Tangerine and Ricotta Tart\n\nCorn Soup with Roasted Poblanos and Zucchini Blossoms\n\nEgg and Gouda Crepes\n\nEgg Cloud\n\nFavas and Ricotta on Buttermilk Crackers\n\nfeta\n\nFresh Chive Cheese\n\n_fromage blanc_ , 8.1\n\nGouda\n\nGouda Goug\u00e8res\n\nmaking, 1.1, 3.1, 6.1, 8.1\n\nOregano Queso Blanco\n\nricotta\n\nRicotta Manicotti\n\nStrawberries with Fromage Blanc and Honey\n\nStuffed Poblanos with Red Pepper Sauce\n\nTea Sandwich Trio\n\nTempura Squash Blossoms\n\nTomato and Herb Salad with Fresh Chive Cheese\n\nWhole-Wheat Pizzas\n\nChervil\n\nChickens, raising, 1.1, 3.1, 6.1, 10.1, 14.1\n\nChiles\n\nAncho Chile\u2013Sauced Noodles with Shiitakes and Butternut Squash\n\nCorn Soup with Roasted Poblanos and Zucchini Blossoms\n\nDeviled Cucumber Cups\n\ndrying\n\nGreen Chile Mayonnaise\n\nGrilled Summer Succotash\n\ngrowing\n\nMexican Skillet Eggs\n\nRoasted Poblanos for the Freezer\n\nStuffed Poblanos with Red Pepper Sauce\n\nWalnut Romesco\n\nWatermelon-Chile Salad\n\nChips, Carrot and Beet\n\nChives\n\nFresh Chive Cheese\n\ngrowing, 2.1, 2.2\n\nChowder, Winter Vegetable\n\nCider, Fresh Apple\n\nClimate zones\n\nColcannon, Kale\n\nCompost, fm.1, fm.2\n\nCorn\n\nCorn Soup with Roasted Poblanos and Zucchini Blossoms\n\ndrying\n\nGrilled Summer Succotash\n\ngrowing\n\nStuffed Poblanos with Red Pepper Sauce\n\nWhole-Wheat Pizzas\n\nCows. _See also_ Milk\n\nmilking, 14.1, 16.1\n\nowning\/co-owning, 10.1, 14.1, 16.1, 16.2\n\nCrackers, Buttermilk\n\nCream\n\nclotted\n\ncr\u00e8me fra\u00eeche\n\nCrepes\n\nEgg and Gouda Crepes\n\nRicotta Manicotti\n\nStrawberry Crepes\n\nWhole-Wheat Crepes\n\nCucumbers\n\nDeviled Cucumber Cups\n\ngrowing\n\nPurslane-Cucumber Salad\n\nCustards, Summer Lemongrass\n\n### D\n\nDehydrating, 5.1, 5.2, 5.3\n\nDesserts\n\nApple Cheese Puff\n\nButternut Squash Compote with Honey and Toasted Walnuts\n\nCaramelized Tangerine and Ricotta Tart\n\nHoney Ice Cream\n\nLemon-Thyme Ice Cream Sandwiches\n\nStrawberries with Fromage Blanc and Honey\n\nStrawberry Cr\u00e8me Fra\u00eeche Sherbet\n\nSummer Lemongrass Custards\n\nTangerine Honey Flan\n\nWalnut-Honey Crisps\n\nWatermelon, Cantaloupe, or Honeydew Sorbet\n\nDill\n\nDrip irrigation\n\n### E\n\nEdamame\n\ngrowing\n\nSkillet-Roasted Edamame\n\nEggs. _See also_ Chickens, raising\n\nArugula and Red Butterhead Lettuce Salad with Tangerines and Hard-Cooked Eggs\n\nCreamy Scrambled Eggs with Oyster Mushrooms\n\nDeviled Cucumber Cups\n\nEgg and Gouda Crepes\n\nEgg Cloud\n\nFava Leaf and Parsley Quiche\n\nfried\n\nGreen Chile Mayonnaise\n\nLemon Curd\n\nMexican Skillet Eggs\n\nNasturtium Salad with Omelet Ribbons\n\nomelet\n\npastured vs. commercial\n\nPattypan Squash with Eggs\n\nPickled Eggs\n\npoached\n\nQuinoa Huaraches with Egg and Parsley Salad\n\nscrambled\n\nsoft cooked\n\nSummer Lemongrass Custards\n\nTangerine Honey Flan\n\nTea Sandwich Trio\n\ntesting\n\nEscargots, 10.1, 12.1\n\n### F\n\nFava beans\n\nFava Leaf and Parsley Quiche\n\nFavas and Ricotta on Buttermilk Crackers\n\ngrowing\n\nFennel\n\nGrilled Radicchio and Fennel Salad with Apples and Toasted Walnuts\n\ngrowing\n\nRoasted Tomato-Fennel Soup\n\nFertilizers\n\nFish emulsion\n\nFlan, Tangerine Honey\n\nFreezing, 5.1, 5.2\n\n### G\n\nGarden Borscht\n\nGardening. _See also_ _individual plants_\n\nclimate zones\n\nfertilizers\n\nplanting techniques\n\nsoil, fm.1, fm.2\n\nsun\n\ntimelines, fm.1, 18.1\n\nwatering\n\nGarlic\n\nGnocchi, Butternut Squash, with Chard and Sage Brown Butter\n\nGoug\u00e8res, Gouda\n\nGreen onions\n\ngrowing\n\nTea Sandwich Trio\n\n### H\n\nHerbs. _See also_ _individual herbs_\n\nbouquets of\n\ndrying\n\nHerb Vegetable Broth\n\nTomato and Herb Salad with Fresh Chive Cheese\n\nWhole-Leaf Radish and Herb Salad\n\nHoney. _See also_ Beekeeping\n\nApple Cheese Puff\n\nButternut Squash Compote with Honey and Toasted Walnuts\/a>\n\ncollecting, 1.1, 3.1, 6.1\n\nHoney Ice Cream\n\nStrawberries with Fromage Blanc and Honey\n\nSummer Wheat Beer\n\nTangerine Honey Flan\n\nWalnut-Honey Crisps\n\nWhole-Wheat Honey Sandwich Bread\n\nHoneydew melons\n\ngrowing\n\nHoneydew Sorbet\n\nHops\n\nBelgian Abbey Ale\n\ngrowing, 2.1, 3.1, 3.2, 3.3\n\nHuaraches, Quinoa, with Egg and Parsley Salad\n\n### I\n\nIce cream\n\nHoney Ice Cream\n\nLemon-Thyme Ice Cream Sandwiches\n\nIrrigation\n\n### J\n\nJams and preserves\n\nLast-Minute Pineapple Guava Preserves\n\nPreserved Lemons\n\nSticky Chewy Tangerine Marmalade\n\nStrawberry Oven Jam\n\n### K\n\nKale\n\nBraised Winter Greens with Preserved Lemon and Red Chile\n\ngrowing\n\nKale Colcannon\n\nVegetable Shepherd's Pie\n\n### L\n\nLabels\n\nLemongrass\n\ngrowing\n\nPeppermint-Lemongrass Tisanes\n\nSummer Lemongrass Custards\n\nLemons\n\ngrowing\n\nLemon Curd\n\nLemon-Thyme Ice Cream Sandwiches\n\nPreserved Lemons\n\nStrawberry Lemonade\n\nLettuce. _See also_ Mesclun\n\nArugula and Red Butterhead Lettuce Salad with Tangerines and Hard-Cooked Eggs\n\ngrowing\n\n### M\n\nManicotti, Ricotta\n\nMarjoram\n\ngrowing\n\nTomato and Herb Salad with Fresh Chive Cheese\n\nMarmalade, Sticky Chewy Tangerine\n\nMayonnaise, Green Chile\n\nMead, 10.1, 12.1\n\nMelons\n\ngrowing\n\nWatermelon, Cantaloupe, or Honeydew Sorbet\n\nWatermelon-Chile Salad\n\nMenus\n\nfall\n\nspring\n\nsummer\n\nwinter\n\nMesclun\n\ngrowing\n\nMesclun Salad with Spring Beets and Dill\n\nNasturtium Salad with Omelet Ribbons\n\nMexican Skillet Eggs\n\nMilk. _See also_ Buttermilk; Cows; Cream\n\nbottling\n\ncooking with\n\nfiltering\n\nhistory of\n\npasteurized, 16.1, 16.2\n\nraw, 16.1, 16.2\n\nstoring\n\nMushrooms\n\nAncho Chile\u2013Sauced Noodles with Shiitakes and Butternut Squash\n\nCreamy Scrambled Eggs with Oyster Mushrooms\n\ngrowing, 6.1, 8.1\n\nHerb Vegetable Broth\n\nPickled Cocktail Mushrooms and Onions\n\nVegetable Shepherd's Pie\n\nMustard\n\nBraised Winter Greens with Preserved Lemon and Red Chile\n\ngrowing\n\n### N\n\nNasturtiums\n\ngrowing\n\nNasturtium Salad with Omelet Ribbons\n\nTea Sandwich Trio\n\nNoodles, Ancho Chile\u2013Sauced, with Shiitakes and Butternut Squash\n\n_Nuvolone_ , 13.1\n\n### O\n\nOlives\n\nfruit flies, 8.1, 8.2\n\nharvesting\n\noil, 1.1, 8.1, 8.2\n\nOnions. _See also_ Green onions\n\nButternut Squash and Cipollini Onion Soup\n\ngrowing, 2.1, 7.1\n\nHerb Vegetable Broth\n\nPickled Cocktail Mushrooms and Onions\n\nWhole-Wheat Pizzas\n\nOregano\n\ngrowing\n\nOregano Queso Blanco\n\n### P\n\nPancakes, Featherlight\n\nParsley\n\nFava Leaf and Parsley Quiche\n\ngrowing\n\nQuinoa Huaraches with Egg and Parsley Salad\n\nTea Sandwich Trio\n\nWhole-Leaf Radish and Herb Salad\n\nPattypan squash\n\ngrowing\n\nPattypan Squash with Eggs\n\nTempura Squash Blossoms\n\nPeppermint\n\ngrowing\n\nPeppermint-Lemongrass Tisanes\n\nPeppers, sweet. _See also_ Chiles\n\nCreamy Flageolet Dip with Red Pepper Sticks\n\ngrowing\n\nStuffed Poblanos with Red Pepper Sauce\n\nWalnut Romesco\n\nPickles\n\nFresh Pickled Beets\n\nPickled Cocktail Mushrooms and Onions\n\nPickled Eggs\n\nPineapple guava\n\ngrowing\n\nLast-Minute Pineapple Guava Preserves\n\nPizzas\n\ntoppings for\n\nWhole-Wheat Pizzas\n\nPlanting. _See also_ _individual plants_\n\ntechniques\n\ntimelines, fm.1, 18.1\n\nPotatoes\n\ngrowing\n\nKale Colcannon\n\nOven-Baked Steak Fries with Green Chile Mayonnaise\n\nRosemary Potatoes Anna\n\nVegetable Shepherd's Pie\n\nWhole-Wheat Pizzas\n\nWinter Vegetable Chowder\n\nPreserves. _See_ Jams and preserves\n\nPurslane\n\ngrowing\n\nPurslane-Cucumber Salad\n\n### Q\n\nQuiche, Fava Leaf and Parsley\n\nQuinoa\n\ngrowing\n\nQuinoa Bites with Walnut Romesco\n\nQuinoa Huaraches with Egg and Parsley Salad\n\nrinsing\n\nStuffed Poblanos with Red Pepper Sauce\n\nthreshing\n\n### R\n\nRadicchio\n\nGrilled Radicchio and Fennel Salad with Apples and Toasted Walnuts\n\ngrowing\n\nRadishes\n\ngrowing\n\nRadishes, Fresh Homemade Butter, and Salt\n\nWhole-Leaf Radish and Herb Salad\n\nRolls, Whole-Wheat\n\nRoot cellars, 5.1, 5.2\n\nRosemary\n\ngrowing\n\nRosemary Potatoes Anna\n\nWhole-Wheat Rosemary Shortbreads\n\n### S\n\nSage\n\ngrowing\n\nSage Butter\n\nSalads\n\nArugula and Red Butterhead Lettuce Salad with Tangerines and Hard-Cooked Eggs\n\nGrilled Carrot Salad\n\nGrilled Radicchio and Fennel Salad with Apples and Toasted Walnuts\n\nMesclun Salad with Spring Beets and Dill\n\nNasturtium Salad with Omelet Ribbons\n\nPurslane-Cucumber Salad\n\nTomato and Herb Salad with Fresh Chive Cheese\n\nWatermelon-Chile Salad\n\nWhole-Leaf Radish and Herb Salad\n\nSalt\n\nSandwiches\n\nLemon-Thyme Ice Cream Sandwiches\n\nTea Sandwich Trio\n\nWhole-Wheat Honey Sandwich Bread\n\nSauces\n\nApplesauce\n\n'Sweet Million' Tomato Sauce\n\nWalnut Romesco\n\nScone Tarts, 18.1, 18.2\n\nSeeds\n\nRoasted Spiced Butternut Squash Seeds\n\nstarting from, fm.1, fm.2\n\nstoring\n\nShepherd's Pie, Vegetable\n\nSherbet, Strawberry Cr\u00e8me Fra\u00eeche\n\nShortbreads, Whole-Wheat Rosemary\n\nSnails. _See_ Escargots\n\nSoaker hoses\n\nSoil\n\nassessing, fm.1, fm.2\n\nfixing\n\nSorbet, Watermelon, Cantaloupe, or Honeydew\n\nSoups\n\nButternut Squash and Cipollini Onion Soup\n\nCorn Soup with Roasted Poblanos and Zucchini Blossoms\n\nGarden Borscht\n\nRoasted Tomato-Fennel Soup\n\nWinter Vegetable Chowder\n\nSquash. _See_ Butternut squash; Pattypan squash; Zucchini\n\nStrawberries\n\ngrowing\n\nStrawberries with Fromage Blanc and Honey\n\nStrawberry Cr\u00e8me Fra\u00eeche Sherbet\n\nStrawberry Crepes\n\nStrawberry Lemonade\n\nStrawberry Oven Jam\n\nSuccotash, Grilled Summer\n\nSummer Lemongrass Custards\n\nSummer Wheat Beer\n\nSun\n\nSwiss chard\n\nBraised Winter Greens with Preserved Lemon and Red Chile\n\nButternut Squash Gnocchi with Chard and Sage Brown Butter\n\ngrowing\n\nVegetable Shepherd's Pie\n\n### T\n\nTangerines\n\nArugula and Red Butterhead Lettuce Salad with Tangerines and Hard-Cooked Eggs\n\nCaramelized Tangerine and Ricotta Tart\n\ngrowing\n\nSticky Chewy Tangerine Marmalade\n\nTangerine Honey Flan\n\nTarragon\n\nTarts\n\nCaramelized Tangerine and Ricotta Tart\n\nScone Tarts\n\nTea\n\ngrowing, 14.1, 15.1\n\nmaking\n\nparty\n\nTea Sandwich Trio\n\nTempura Squash Blossoms\n\nThyme\n\nTimelines, fm.1, 18.1\n\nTisanes, Peppermint-Lemongrass\n\nTomatoes\n\nCanned Heirloom Tomatoes\n\nGrilled Summer Succotash\n\ngrowing\n\nMexican Skillet Eggs\n\nRicotta Manicotti\n\nRoasted Tomato-Fennel Soup\n\n'Scarlet Emperor' Rago\u00fbt\n\nSlow-Roasted Tomatoes for the Freezer\n\n'Sweet Million' Tomato Sauce\n\nTomato and Herb Salad with Fresh Chive Cheese\n\n'Trombetta' Zucchini and Its Flowers\n\nWheat Berry \"Risotto\" with Roasted Tomatoes and Broccoli Rabe\n\nWhole-Wheat Pizzas\n\nTransplants\n\n### V\n\nVegetables. _See also_ _individual vegetables_\n\nHerb Vegetable Broth\n\nVegetable Shepherd's Pie\n\nWinter Vegetable Chowder\n\nVinegar, 1.1, 8.1\n\n### W\n\nWalnuts\n\nButternut Squash Compote with Honey and Toasted Walnuts\n\nGrilled Radicchio and Fennel Salad with Apples and Toasted Walnuts\n\nWalnut-Honey Crisps\n\nWalnut Romesco\n\nWheat Berry \"Risotto\" with Roasted Tomatoes and Broccoli Rabe\n\nWatering\n\nWatermelon\n\ngrowing\n\nWatermelon-Chile Salad\n\nWatermelon Sorbet\n\nWheat\n\nBelgian Abbey Ale\n\nflour\n\ngrowing, 2.1, 3.1, 3.2\n\nlocal\n\nseeds\n\nSummer Wheat Beer\n\nthreshing\n\nWheat Berry Ciabatta\n\nWheat Berry \"Risotto\" with Roasted Tomatoes and Broccoli Rabe\n\nWhole-Wheat Crepes\n\nWhole-Wheat Honey Sandwich Bread\n\nWhole-Wheat Pizzas\n\nWhole-Wheat Rolls\n\nWhole-Wheat Rosemary Shortbreads\n\nWine\n\nChardonnay\n\nmaking, 1.1, 8.1, 10.1\n\nSyrah\n\nWinter Vegetable Chowder\n\n### Z\n\nZucchini\n\nCorn Soup with Roasted Poblanos and Zucchini Blossoms\n\ngrowing\n\nTempura Squash Blossoms\n\n'Trombetta' Zucchini and Its Flowers\n\nWhole-Wheat Pizzas\n\n# Credits\n\nAuthor photo \u00a9 2011 by Thomas J. Story\n\nFood\/location photography \u00a9 2011 by Thomas J. Story: FM.1, FM.2, FM.4, FM.5, 1.1, 1.8, 1.13, 1.14, 1.18, 1.19, 1.53, 1.54, 1.55, 1.56, 1.57, 1.58, 1.59, 1.60, 1.61, 1.62, 1.64, 1.65, 1.66, 2.1, 2.7, 2.8, 2.9, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, 2.60, 2.61, 2.66, 2.62, 2.63, 2.64, 2.65, 2.68, 3.8, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.21, 3.23, 3.24, 3.25, 3.26, 3.27, 3.28, 3.29, 3.30, 3.31, 3.32, 3.33, 3.34, 3.35, 4.1, 4.6, 4.8, 4.21, 4.22, 4.23, 4.24, 4.25, 4.25, 4.27, 4.28, 4.29, 4.30, 4.31, 4.32, 4.33, 4.34\n\nFood styling \u00a9 2011 by Karen Shinto: FM.1, 1.1, 1.8, 1.54, 1.55, 1.57, 1.58, 1.61, 1.64, 1.66, 2.35, 2.36, 2.37, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, 2.61, 2.66, 2.62, 2.63, 2.64\n\nFood styling \u00a9 2011 by Dan Becker: 1.56, 1.59, 1.60, 1.62, 2.9, 3.1, 3.23, 3.24, 3.25, 3.30, 3.32, 3.33, 4.23, 4.24, 4.25, 4.27, 4.31\n\nFood styling \u00a9 2011 by Randy Mon: 2.60, 3.26, 3.27, 3.28, 3.29, 3.31, 3.34, 4.1, 4.21, 4.22, 4.25, 4.29, 4.30, 4.32, 4.33, 4.34\n\nLocation photography \u00a9 2011 by Kimberley Burch: FM.6, FM.9, FM.11, 1.4, 1.21 (top left, bottom), 1.24, 1.27, 1.28, 1.29, 1.34, 1.36, 1.39, 2.14, 2.38, 2.42, 2.48, 2.49, 2.50, 3.4, 3.22, 4.3, 4.16, 4.20\n\nLocation photography \u00a9 2011 by Rob Brodman: FM.8, 1.11, 1.16, 1.20, 2.11, 2.15\n\nLocation photography \u00a9 2011 by E. Spencer Toy: FM.11, 1.2, 1.9, 1.22, 1.30, 1.48, 1.49, 1.51, 1.52 (bottom left, bottom right), 1.63, 1.68, 2.6, 2.10, 2.12 (bottom center, bottom right), 2.13, 2.16, 2.18, 2.19, 2.21, 2.20, 2.22, 2.23, 2.24, 2.25, 2.27, 2.28, 2.29, 3.3, 3.9, 3.17, 4.10, 4.17, 4.18, 4.19\n\nGarden plan illustrations \u00a9 2011 by Akiko Aoyagi Shurtleff\n\nThe following material is reprinted with permission:\n\nCourtesy Margo True: FM.3, 1.17, 1.26, 1.32, 1.38, 1.43, 1.44, 1.45, 1.46, 2.4, 2.30, 2.43, 2.46, 2.45, 2.53, 3.19, 4.4, 4.5, 4.7, 4.11 (top and bottom right), 4.13, 4.15\n\nCourtesy Johanna Silver: FM.7, 2.12 (top, bottom left), 3.20, 4.9\n\nCourtesy Sheila Schmitz: FM.10, 1.7, 1.10, 1.12, 1.15, 1.52 (top), 2.51, 2.52, 1.35\n\nCourtesy Elizabeth Jardina: 1.5, 1.47, 1.50, 2.2, 2.3, 3.2, 3.5\n\nCourtesy Erika Ehmsen: 1.3, 2.17, 2.26\n\nCourtesy Elaine Johnson: 1.6, 1.21 (top right), 4.2, 4.12, 4.14\n\nCourtesy Brianne McElhiney: 1.23, 1.31, 1.33, 2.5, 2.39, 2.40, 2.41, 2.44, 2.47, 4.11 (bottom left)\n\nCourtesy Alan Phinney: 1.37\n\nCourtesy Rick LaFrentz: 1.40, 1.41, 1.42\n\nCourtesy Native Seeds\/SEARCH: 2.67\n\nCourtesy Ron Ehmsen: 3.6\n\nCourtesy Doni Jackson: 3.7\n\nCourtesy Mark Bitterman: 3.18\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"RYDER\n\nA Men of Clifton Montana Novel\n\nBook 5\n\nSusan Fisher-Davis\nErotic Romance\n\nRyder\n\nCopyright \u00a9 2015 Susan Fisher-Davis\n\nFirst E-book Publication: October 2015\n\nCover design by Amy Valentini\n\nEdited by Amy Valentini\/Romancing Editorially\n\nProofread by Toby Schuler\n\nAll cover art copyright \u00a9 2015 by Susan Fisher-Davis\n\nALL RIGHTS RESERVED: This literary work may not be reproduced or transmitted in any form or by any means, including electronic or photographic reproduction, in whole or in part, without express written permission.\n\nAll characters and events in this book are fictitious. Any resemblance to actual persons living or dead is strictly coincidental.\n\nPUBLISHER: Blue Whiskey Publishing\n\nSusan Fisher-Davis\n\nwww.susanfisherdavisauthor.com\n\n# Table of Contents\n\nTitle Page\n\nCopyright Page\n\nDedication\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nEpilogue\n\nAbout the Author\n\n# Dedication\n\nA big Thank you to Kenolivier Gisbert for being Ryder.\n\n<https:\/\/www.facebook.com\/olivier.gisbert.7?fref=ts>\n\nTo Neuf Neuf Photography for the amazing shoot you did for me.\n\n<https:\/\/www.facebook.com\/NeufNeufPhotographer?fref=ts>\n\nYou both are amazing and I love you!\n\nTo my readers. I do this for you. Without you, I wouldn't be here.\n\n# Chapter One\n\nRyder Wolfe knew trouble. Most of the time, he knew how to handle it but as he turned to look in the direction which had drawn the attention of every other man in the place, he wondered if he'd met his match. Taking a deep breath, he spun on the bar stool, and set his elbows on the bar behind him. Like every man in the bar, his gaze settled on the angelic looking beauty standing in the doorway. Yeah, she was definitely trouble. She was tall, at least her long legs made her appear so. He couldn't really tell...yet. Blonde hair swirled around her face, falling onto her shoulders like a halo. She stood with her legs slightly spread. His gaze traveled over her, from the top of that beautiful blonde hair to a face that could stop traffic or at least cause a few accidents. Her nose was pert and below it, she had the most kissable mouth he'd ever seen. At thirty-three, he was a loner, but he loved women. Respected them, or at least tried to but he usually loved them then left them. He had run away from home when he turned sixteen so he grew up fast, and he grew up hard.\n\nOnce more, his gaze swept the room. His mouth twitched in irritation. The men in the bar looked like a bunch of hungry dogs waiting for a bone. A fight was coming. Ryder could feel it in his bones. He looked back to the woman. Finishing where he left off, his scrutiny settled on her chest. The top three buttons of her white blouse were unbuttoned just enough to tease a man. She had nice cleavage. Her stomach was flat too. The white blouse she wore was tucked into what had to be the tightest, shortest, red skirt, he'd ever seen. Any shorter, and she'd be arrested for indecent exposure. Ryder had to shake his head on that one. There wasn't anything indecent about it but it sure was inciting indecent thoughts.\n\nDown his eyes went to those long legs. He almost smiled. Always having been a leg man, he thought hers were great. The shoes on her feet had him biting the inside of his cheek to hold back a groan. Red stilettos. Good God, his mouth was watering. He felt his groin tighten. He spun back around on the stool and took a long pull on his beer. He shook his head. You never let a woman get to you. Don't start now.\n\nHe made the mistake of looking her way again. She was walking toward the bar. Her hips swayed just enough to draw attention, and no more. Ryder watched another man walk up to her. She graced him with a slight smile but moved past him. The guy looked like he'd just lost his best friend. Ryder covered his smile by taking another drink. She walked past him. He could smell her perfume. Light, not too strong. Just enough that it teased his nose, and made him want to bury his face in her neck. He watched her by way of the mirror behind the bar. She sat on a stool three down from his, but Ryder kept his eyes on the mirror.\n\nShe glanced up, and in the mirror, her eyes connected with his. He felt like someone had sucker-punched him. She really was the most beautiful woman he'd ever seen. He watched as Keith, the bartender, stepped up to ask what she wanted to drink. Ryder didn't hear her response, but Keith nodded and handed her a beer. In a bottle yet. She looked at it as if she didn't know what to do with it. Keith set a glass in front of her, bringing a smile to her face. She had a perfect, white smile. When she smiled, deep dimples appeared in her cheeks. Ryder couldn't stop looking at her mouth.\n\nRyder watched as she poured her beer into the glass. She was pouring it too fast, and it was going to be all foam. As the beer reacted as he predicted, the foam rising high above the top of the glass, she set the bottle on the bar top and leaned down. She quickly took a sip and Ryder did all he could not to laugh. He was sure he was the only one who saw the face she made when she tasted the beer. The smile left his face though when he saw her small pink tongue come out and lick at the foam on her upper lip. Squeezing his eyes shut against the image, Ryder turned around on the stool to watch the pool game going on behind him. He kept his eyes on the game, paying no more attention to the woman. He watched as Big Earl came strolling over to her and tried to squeeze in between two bar stools, but failed. Jesus! Didn't the big ox know how big he was? Ryder kept his eyes on the game, but Big Earl's voice carried over the noise in the bar.\n\n\"Hello, honey. Heaven must be missing an angel.\"\n\nRyder nearly choked on his beer. He found it hard to believe that any man would still use that sorry excuse of a pick-up line. He mentally shrugged. He wasn't getting involved. She wasn't his problem. He raised the bottle to his lips and took another sip, but almost spit it out when he heard her speak.\n\n\"That's nice of you to say. I'm looking for a man.\"\n\nBig Earl let out a loud laugh. \"Well, honey, you came to the right place.\" He leaned closer to her. \"I could be the man you're looking for.\"\n\nRyder turned back around and watched them in the mirror. He watched as the woman leaned forward as if she was going to whisper in Big Earl's ear. He listened closely, hoping she was going to be creative in her rejection of the jerk.\n\n\"You might be. Are you Ryder Wolfe?\" She spoke in a soft, smoky voice.\n\nRyder slammed the beer bottle down on the bar. He felt every set of eyes, in the bar, except hers, settle on him. He picked up his cowboy hat, from where he'd laid it on the bar, and slammed it on his head. He'd almost gotten to the door when Keith called his name.\n\nShit! He turned around slowly, and looked at Keith. When Keith didn't say anything, Ryder raised an eyebrow. He heard Keith clear his throat. Ryder narrowed his eyes at him, letting him know he'd done the wrong thing. No one bothered Ryder unless he wanted to be bothered. He was well-known as one mean son of a bitch when things bothered him. He was a tough man. He saw Keith swallow and move his eyes toward the blonde, who was now looking in his direction. Ryder shifted his gaze to her, and swore under his breath.\n\nRyder walked toward the woman. Every step felt like he was walking to the electric chair. He stopped in front of her. She slid off the stool, settling easily on those sexy shoes. She was tall in those heels but at six four, he was taller. The top of her head came to his mouth. Without the heels, she would barely reach his shoulder.\n\n\"You're Mr. Wolfe?\" she asked him, looking everywhere but at him.\n\n\"Who wants to know?\" Ryder almost kicked himself for using such a childish comeback. She didn't seem to notice.\n\n\"My name's Kelsey Sullivan.\" She stuck her hand out. He ignored it. Her brow furrowed in reaction to his obvious insult. She took a deep breath before continuing. \"Frank Sanders told me to contact you when I got to Clifton. He said you would help me.\"\n\nRyder was so angry, he was sure steam was coming out of his ears. Damn it, Frank! He took her elbow in his large hand, and led her to a table away from prying eyes. He nodded for her to sit down. When she did, he removed his hat, took a seat across from her, and stared at her over the small table. He couldn't seem to keep his eyes off her. She, on the other hand, wasn't having any problem keeping her eyes off him. She didn't seem to want to look him in the eye. She was nervous. He sat back in the chair and folded his arms over his chest. He was getting impatient.\n\n\"Why would Frank send you to me? I'm not a damn babysitter. I'll let Frank know that too.\"\n\n\"Babysitter?\" she asked. Even as she diverted her eyes, he knew she waited for his answer.\n\n\"Yes. Well, I don't have time for this.\" He told her with a wave of his hand. He stood up, and looked down at her. \"You might want to go home and see who else Frank can get to help you. I'm not that man.\" He spun on his heel and headed for the door.\n\nRyder was still fuming even after he reached the parking lot. Frank had no right to send someone to him. He was beside his truck before he knew it. He reached into the inside pocket of his coat, and pulled out his cell phone. He punched in Frank's number, cursing every time it rang. Finally, he heard Frank answer.\n\n\"What the hell were you thinking, sending someone here? I'm not a fucking babysitter, Frank, and God damn it, you know that!\"\n\n\"Are you done yelling at me?\" Ryder heard Frank ask in a calm voice. This made Ryder angrier. He opened his mouth to tell him he was just getting started when Frank continued to speak.\n\n\"I didn't know what else to do, Ryder. I've known Kelsey, and her family, for years. In fact, I'm her godfather. I've always told her to call me if she ever needed anything. Well, she does and she did. She needs to disappear until her stalker is caught\u2014again. They had him once but when he posted bail, he bolted. Your ranch is the best place I could think of.\"\n\nRyder swore under his breath. \"You didn't think to call me and ask first? I'm supposed to hide this...this woman for God knows how long. How long has he been on the run?\"\n\n\"A week. They've got a BOLO out on him. She's scared, Ryder,\" Frank said in a low gentle voice.\n\nRyder sighed. How could he turn down Frank's request? He owed Frank too much to refuse a favor since he and his wife had practically raised Ryder.\n\n\"But how long am I supposed to let her stay here?\" Ryder asked, and then listened as Frank went into the details. Ryder swore when Frank pulled the trump card.\n\n\"You owe me, Ryder. I swore I'd never call you on it but in this matter, I have to.\"\n\nRyder was royally pissed off and intended to let Frank know. He swore up a blue streak for the next few minutes and wouldn't be surprised if Frank had set the phone down while he did. Finally, Ryder took a breath as he listened to Frank continue.\n\n\"Look, I know you don't want her there but...Kelsey's my goddaughter. I've known her since she was born. Her parents are like family to Grace and me. You, of all people, know how we feel about family.\"\n\nRyder sighed. He knew Frank had a smile on his face even though he couldn't see it. Frank loved him like a son, and although he had never said so, Ryder loved him, and his wife, Grace. They had taken him in when he was sixteen. A lonely teenager on his own, with nowhere to go, no one to watch over him\u2014until Frank and Grace.\n\n\"I know I owe you. I'll take her to the ranch, but they need to find him. Fast.\" Ryder didn't wait to hear Frank's response, but instead hung up the phone and stuck it back into his pocket. He turned to head back into the bar when he saw Kelsey standing outside of the door, looking his way. He walked toward her, stopping a few feet from her. He put his hands on his hips, and looked at her. He saw tears in her eyes. He swore softly, and glanced away, then back to her. He didn't do well with emotional women.\n\n\"I spoke to Frank. I'll take you to my ranch until your stalker is found.\"\n\n\"Thank you. I hope you're right about that,\" she said softly. Ryder nodded, and motioned his head for her to follow him. He started walking toward his truck but when he didn't hear her behind him, he turned back around. He saw her walking from the bushes with a suitcase. He realized she'd hidden her suitcase there, rather than carrying it into the bar. She caught up to him, and smiled up at him. He frowned at her and walked away.\n\nBehind them, the bar doors swung open and Ryder heard Big Earl yell at them. \"Where the hell you think you're going with my woman, Wolfe?\"\n\nRyder hung his head and swore. He told Kelsey to get into the truck. He was grateful she did what she was told. Ryder watched as Big Earl stomped toward him with fire in his eyes. Ryder stood tall for sure. Big Earl wasn't Big Earl because he was tall, but rather because he was wide. Wider than a Mack truck, it seemed to most people. Not to Ryder. He folded his arms over his chest, and waited for the big ox to get to him. When Earl got close enough, he put his head down, and charged at Ryder. Ryder was ready and stepped aside when Earl got there. Earl fell to the ground face first. Getting up slowly, he glared at Ryder. He let out a growl when he saw Ryder smile. He stood in front of Ryder, and took a swing but he was ready. Ryder leaned back quickly, and then faster than lightning, his fist shot out, hitting Earl square on the chin. The big man went down like a ton of bricks, and didn't get back up. Ryder made sure he was all right, and then walked to his truck. He got in, rubbing the knuckles on his right hand. They were going to be sore tomorrow. He glanced over at Kelsey. She was just sitting there staring at him. He mumbled under his breath.\n\n\"What did you say?\" he heard her ask.\n\n\"I said I knew you were trouble the minute you walked into the bar.\" He started the truck, and after another look at her, shook his head, put the truck in gear, and headed home.\n\n****\n\nKelsey sat quietly in the truck looking out the front windshield. She nibbled on her bottom lip. He certainly wasn't much of a talker or very friendly. What had she gotten herself into? She supposed she had to trust this Ryder Wolfe. If Frank told her she could trust Ryder, then she would. She knew he wasn't happy about the situation, but she wasn't either. She wasn't happy about anything in her life right now.\n\nTwo months ago, she'd started receiving emails, and then it escalated to letters sent to her place of work. When she started receiving mail at her home too, she knew it was time to report it. The police found the man and arrested him but then he made bail, and ran. After she told her parents about it, they suggested she call Frank since he used to be in law enforcement. Frank sent her to the silent and disagreeable man beside her.\n\nKelsey sighed and looked out the windshield. She was shocked to see snow highlighted in the headlights. She glanced over to Ryder. His big hands were leisurely holding the steering wheel as if he were out for a Sunday afternoon drive. The snow didn't seem to bother him at all.\n\n\"It's snowing,\" she said, continuing to stare at him. He didn't take his eyes off the road when he spoke to her.\n\n\"Brilliant deduction. It's been snowing since we left the bar.\"\n\nKelsey's eyes widened. \"It's late September.\" Did he have to be such a smart-ass?\n\nRyder glanced at her then back to the road. \"You're in northern Montana. Of course, it's snowing. Where are you from?\"\n\n\"Georgia. Born and raised.\" She looked at him. \"Have you always lived in Montana?\"\n\nRyder shook his head, but didn't supply anything further. Silence filled the space once more. Fifteen minutes later, he slowed the truck and steered it onto a dirt road. The only light illuminating the road was coming from the headlights of the truck. He drove slowly until they arrived at a gate. There he stopped the truck and stepped out.\n\nKelsey watched him as he pushed the gate open, and then walked back to the truck and climbed in. He put the truck in gear and drove through the gate then stepped out, and repeated the action only closing the gate this time. After returning to the truck, he drove it further down the road until she could see lights up ahead.\n\nRyder steered the truck into an area in front of a log home, shut off the engine, and nodded for her to get out. She opened the door, and the scent of snow filled her nose, she carefully climbed down from the truck fully aware of her high heels, and then reached for her suitcase from the back seat. She looked up at the yard lights. Still amazed at seeing snow in September, she stood there watching the snowflakes dance through the light. It started snowing harder. Suddenly, Ryder was there beside her. She glanced up at him, his hat shading his eyes. He took the suitcase from her.\n\nWhen he started walking toward the house, she followed him to the back door where a light on the porch made the trek easy. Ryder opened the back door and reached inside to turn the kitchen light on. The warmth from the house came out to greet them. She hadn't realized how cold it had gotten and shivered a bit from the change in temperature. He motioned for her to go inside so she stepped into the house.\n\nKelsey glanced around the kitchen. The countertops were black marble. The appliances matched, including the smooth-top stove, with a built-in black microwave above it. The floor reflected the color scheme in its large square tiles. A kitchen table sat to the left of the entry door, under a window with white plantation blinds. Kelsey fell in love with it. She turned to tell him she thought the kitchen was beautiful, but meeting his gaze straight on made her speechless.\n\nShe found herself gazing up into moss green eyes lined with thick black lashes. Slowly, she took in this obstinate man's good looks as he removed his coat, from the top of his brown cowboy hat, sitting low on his forehead to his nose, which was straight and narrow. He had high cheekbones and a wonderful, sumptuous mouth. Stubble covered his lower face, neck, and jaw. His hair was black, pitch black actually, and touched the collar of his T-shirt. She watched him hang up his jacket on one of a row of hooks on a wall leading away from the kitchen, and she couldn't resist perusing his wide chest, the T-shirt stretched tight over it. Next, her eyes took in his flat stomach, which led to the snap of his jeans. Jeans, he seemed to fill out most impressively, covered long legs. On his feet were well-worn cowboy boots. The words to describe this man, which sprang into her mind, were cowboy, sexy, and dangerous. Kelsey lifted her gaze and met his stare. She found a tight-lipped expression.\n\n\"Do I pass?\" he asked her. Just then, two dogs came running into the kitchen and saved her from answering.\n\nRyder told them \"No\" in a voice that demanded obedience. Both dogs stopped and sat down in front of her. She smiled and put her hand out to them. They were bumping into each other trying to get closer but barely moved.\n\n\"What breed are they? They're so pretty.\" She glanced up at Ryder.\n\n\"Border Collies,\" he said as he walked to the back door and opened it to let them out. They both ran off the porch, and then disappeared around a small barn. Ryder closed the door, removed his hat, hung it on a hook alongside his jacket, and then disappeared into a small room off the kitchen. Kelsey took her coat off and followed him. They almost bumped into each other as he came out the door she was going in. She laughed softly and looked up at him. He was a big man. He took her coat from her and hung it on another hook near his things. No smile touched his lips as he turned to her.\n\n\"Are you hungry? I could make you some soup or something.\"\n\n\"I'm too tired to be hungry. I just want to go to bed.\" She felt a blush rush up her neck and into her cheeks. She had no idea why saying that made her blush but suddenly, she couldn't look at him.\n\n\"I'll show you to your room then. Follow me.\" Ryder left the kitchen. After a few mortifying seconds of wishing she could crawl into a hole, Kelsey followed him. You're making such an ass of yourself. Calm down. He's just a man. God, yes, but what a man!\n\nKelsey rolled her eyes at her own foolishness as she followed him through what appeared to be a foyer or maybe not, since he didn't turn any lights on until they reached a hallway. It was then she noted they had passed a darkened living room. He opened the first door on the left side of the hallway and turned a light on inside the room. He motioned for her to go in. He set the suitcase down just inside the door.\n\n\"You have your own bathroom. If you need more blankets, there are some in the linen closet. If you need anything, my room is the last one on the right.\" He frowned at her. \"How did you know to find me at Dewey's? The bar,\" he clarified.\n\n\"When I arrived at the bus station, I saw a Sheriff's Deputy and I asked him where I could get a taxi to take me to your house. He told me he'd just been by the bar and saw your truck there.\" She shrugged. \"So I took a taxi there.\"\n\nHe gave her a nod and softly closed the door behind him leaving her alone.\n\nKelsey stood in the center of the room and looked around. The walls were painted in a soft blue. There were white lace curtains on the window and a blue and white quilt on the bed. The chest of drawers was mahogany with a small TV sitting on top of it. Along a wall sat a large mahogany dresser and the soft carpet was light beige. It was a surprisingly nice room.\n\nShe walked to the nightstand beside the bed, turned on the small table lap, and then flipped off the ceiling light. Sighing, she lifted her suitcase up on the bed and opened it. She hoped she wasn't here very long or she'd have to get more clothing. She put her underwear in one of the drawers, and hung her blouses, and jeans in the roomy closet.\n\nMoving toward the bed, something outside in the yard caught her eye. She slowly moved to the window and looked out. It was Ryder. He was walking toward the barn. He had certainly made no bones about not wanting her in his home. Other than answering her question about the dogs and telling her where to get extra blankets, he hadn't said much. Geesh! What a motor mouth! She chuckled at the thought.\n\nKelsey took a seat on the bed and wondered what to do next. The fatigue of the past two months, the journey here, and not knowing what was going to happen was weighing heavy on her. It had been a long flight from Georgia to Montana, and now the time difference was starting to affect her. She counted on her fingers and realized that it was nine at night in Georgia, but only seven here. The bus ride from Butte to Clifton had been long and boring.\n\nWhen she heard the dogs barking, she stood up and looked out the window again. She watched in amusement as Ryder played with them. He was throwing a ball for them. He'd throw it as far as he could and the dogs would run after it. Both of the dogs, eager to please their master, tried carrying it back to him. She heard him laugh and her heart hit her stomach. He had a deep, rich laugh that made her smile. He glanced toward the window as if sensing she was there. She jumped back. She didn't want him to see she'd been watching him.\n\nHow long was she going to have to be here? With a man she knew nothing about...except that Frank Sanders trusted him. She sighed. She knew that was good enough for her. She loved Frank as if he were an uncle to her instead of a godfather. She even called him uncle. She knew he'd never put her in danger.\n\nKelsey fell back on the bed and placed her hands over her face. Please God, please let the stalker give up on me...please. She whispered the same prayer every day, hoping one day soon God would hear her, and give an answer.\n\n****\n\nThe next morning Kelsey awoke to what sounded like hammering. She was surprised at how hard she had slept. She expected to toss and turn since she was in a strange place but she must have been more exhausted than she thought. Climbing from the warmth of her bed, she looked out the window. There was a smattering of snow on the gravel driveway and the grass, but the sun was shining. Feeling energized, she hurried into the bathroom to shower. After dressing, she headed for the kitchen.\n\nWhen she came upon the living room, she stopped. Unlike the night before when it was dark, it was bright with sunlight and looked very welcoming. To her right was a cherry door with leaded glass. A large red overstuffed couch filled the room in front of her with its back to her. Two red recliners sat opposite of the couch. The stone fireplace had a large flat screen TV hanging above the mantel centered between the living room and kitchen. She loved how it was open on both sides. A window seat with large cushions sat at the bottom of a large picture window that took up most of the front wall. In the corner next to the window seat was a beautiful rocking chair made from cedar. The floor was dark cherry hardwood. The whole room was beautiful. A room she'd love spending a day in reading a good book.\n\nShe was preparing a pot of coffee in the kitchen when the back door opened. She turned to see Ryder enter with the dogs following behind. She could feel and smell the cold coming off them. The dogs greeted her with pants and wagging tails. Without a word, they sat down in front of her. She smiled and reached down to pet them. They put their paws up for her to shake. She laughed and looked at Ryder, who busied himself taking his hat and coat off. He hung his hat on a peg by the door, then walked into the laundry room and hung up his coat.\n\nShe watched as he walked back into the kitchen, rolling up the sleeves on his blue and black checkered flannel shirt. She could see he had a T-shirt on under it. His forearms were large and tanned. Dark hair covered his arms. She'd always thought that was sexy but when she took notice of his mussed hair as well, it only added to the sexiness that was Ryder Wolfe.\n\n\"I'm making coffee. Would you like some?\" she asked, watching him.\n\nRyder nodded his head as he washed his hands at the sink. As he reached for a paper towel, she noticed a small crescent shaped scar by his right eye. His eyes were even greener than she'd first thought. His lashes were long and thick. His hair so black that the ceiling lights made it appear to have a blue sheen. His eyebrows were thick and black. The man was, without a doubt, drop dead gorgeous, and sexier than sin.\n\nKelsey turned away from him and wanted to fan herself. Suddenly, the room seemed as if it was on fire. It was so hot! She was so into her own thoughts, she didn't realize he'd spoken until she saw him staring at her. She blushed and glanced away from him.\n\n\"Are you all right?\" Ryder asked her.\n\nShe nodded. \"I've just got a lot on my mind. I'm sorry. What did you ask me?\"\n\n\"I didn't ask you anything. I said I didn't know if you even liked coffee when I made the pot, so I didn't make any extra. I will tomorrow morning.\"\n\n\"What are the dogs' names?\" She blushed. What a lame question! Oh well, she was curious.\n\n\"Badger and Buttons,\" he said while pointing to each dog. \"I didn't name them. They had those names when I adopted them from the shelter.\"\n\nWonderful! A gorgeous man who also saves animals. Did he have any flaws? Dipping her eyes down to the fly of his jeans again, she had her answer. Nope. No flaws.\n\n****\n\nRyder pulled a chair out from the table and sat down. He watched her fuss over the coffee and saw her nod again. Was she nervous? Was she afraid of him? He mentally shook his head. No, she wasn't afraid, nervous perhaps, but she knew and trusted Frank. If Frank trusted him enough to watch over her and protect her, then she wouldn't be here if she didn't trust him. He moved his gaze over her as she poured a second cup of coffee. The night before, the bar had been dark and although he had appreciated her beauty, he hadn't wanted anything to do with her. Then she told him Frank sent her. He clenched his jaw at remembering. Damn it! He didn't need any of this. Frank knew that he didn't have time for this shit.\n\nHe sneaked a look back at Kelsey and allowed his eyes to peruse the view of her backside. He almost groaned. Jesus, those jeans hugged her ass like a second skin.\n\nHis dick twitched in appreciation so he leaned back in the chair, crossed his arms, and looked everywhere but at her. When she walked into his line of sight and set his coffee down in front of him, he caught a whiff of her perfume again. Just like last night, subtle, but not too strong. Still, enough to make him want to hold her close and breathe it in. He tried not to watch her as she sat down in the chair across from him. She spooned sugar into her cup but kept her eyes down as if she was determined not to look at him.\n\nRyder lifted his cup and took a sip of coffee. Thankfully, it was not scalding hot anymore, or he would have burned his mouth since he couldn't keep his gaze from going back to her. Her skin was flawless and his notice of freckles across her nose told him she wore no makeup. He liked that. Most women would put makeup on regardless of the situation, and especially around a man they didn't know. For what reason that was, he had no idea. He preferred a natural looking woman.\n\nHe took a deep breath. He knew he had to let her stay. He'd do it for Frank because he'd do just about anything Frank asked him to do. He let his breath out slowly and stood up, moving to the sink without glancing at her. After rinsing his cup, he set it aside, and walked to the laundry room where he'd left his coat. Returning, he didn't look in her direction but put his hat on, gave an abrupt nod, and headed out the door.\n\n# Chapter Two\n\nKelsey sat at the table for a long while after Ryder left. She wondered what she was going to do all day since he certainly gave her no suggestions. He barely spoke more words than were necessary. She stood, walking to the sink to rinse her cup. She stopped there and gazed out the window. She wondered what he did to keep busy. Did he have horses he had to take care of? Would he mind if she went out there? She wasn't about to go where she wasn't invited to find out. He just seemed so private. A real loner. Sighing, she returned to her bedroom, got her cell phone out, and called Frank.\n\n\"What kind of man do you have me staying with, Uncle Frank?\" She smiled as she heard him chuckle. She wasn't sure what the relationship between Ryder Wolfe and her godfather was but she needed to know. \"Who is he to you?\" she asked curiously.\n\nShe heard him take a deep breath and exhale. She listened as Frank told her about Ryder.\n\n\"He's had a rough life. He ran away from home when he was sixteen. His father...and I use that term loosely, used to smack him, and his mother, around. Ryder's mother stayed drunk most of the time. His father was a heavy drinker too, and a mean drunk. Ryder was smart to get away from them. I found him sleeping in our barn.\" Frank chuckled as if remembering back. \"He had a real bad attitude too, and didn't want help, from anyone. I was ready to call the cops, but Grace saw something in him that I didn't. He needed love. We found his parents and asked them to give him up to us. At first, they said they wouldn't do it unless we gave them a lot of money. Money! They actually wanted money instead of their son.\" Frank sighed. \"But once we told them we'd report them for child abuse, and not reporting him missing, they changed their mind. They said Ryder was nothing to them anyway. So we took him in and told him if he ran off, we'd report him for being a runaway. For months, he fought us every step of the way. Eventually, Grace got him to see that she loved him, cared about him, and it got through to him. She was the mother he never really had.\n\n\"It took a little longer for him to trust me. It finally happened when we went fishing when he was seventeen. He caught me with his hook when he cast out. I swore up a blue streak. Scared him to death, I guess. He thought I was going to hit him. Even at seventeen, already getting big and strong, he was scared. I had him take the hook out of my hand. The whole time, he had tears streaming down his face. When he finally removed it, he hung his head, as if waiting for me to start hitting him. I put my hand on his shoulder and told him I wasn't angry with him. He said, 'you sure were cussing a lot.' I laughed and told him, I guess I was...because it hurt! Ryder looked at me with the most amazing look on his face. He couldn't believe I could be that angry and not hit him. He gave me the biggest grin, and from that day on, we've been close. He calls us Mom and Dad now, unless he's angry with me, then I'm Frank. He went back to school and graduated. From there he went to college, and then returned to Clifton to raise Paint horses. His grandfather left him that ranch, but that's something I'll let him tell you about because it's not all cut and dry. I was hoping to get in touch with him before you arrived but Grace had an art show, and time slipped away from me.\"\n\n\"I'm all right with it. I'm not sure Ryder is though.\"\n\nFrank chuckled. \"He'll be fine. He'd do anything we ask of him.\"\n\n\"I'm just not sure how long I should stay here.\"\n\n\"Until I'm sure that SOB isn't looking for you. We'll just take it one day at a time. In the meantime, enjoy Montana. It's beautiful there.\"\n\n\"All right. I love you, and tell Aunt Grace I love her too.\"\n\nFeeling a little better about this enigmatic man, Ryder Wolfe, she gave a low sigh, put her phone away, and decided since she was obviously stuck here for a while, she'd make herself useful and fix dinner. Wondering what food was available, she returned to the kitchen.\n\n****\n\nRyder pounded the hammer as hard as he could to work out his frustrations. How the hell was he supposed to live with her, and for who knew how long? He didn't like having a woman in his home, let alone living here. No matter how temporary it was. He blew out a breath because he knew he had no choice. Frank and Grace asked him to do this and so he would\u2014for them. But damn, if Kelsey wasn't so fucking beautiful, it wouldn't be such a hardship. Her long hair looked so soft that he wanted to wrap it around his hand and pull it to his face. The hammer missed the nail and hit his thumb. Grimacing in pain, Ryder dropped the hammer and grabbed his thumb.\n\n\"Motherfucker!\"\n\nHe stuck his thumb into his mouth, trying to ease the pain, and make it feel less like it was going to explode like in the cartoons. That's what you get for not having your mind on your work, and on a woman instead, stupid. A damn woman! Shit!\n\nHis thumb was hurting. It throbbed something fierce. He shook his hand trying to alleviate the pain. but it didn't seem to help. He bent over, picked up the hammer, and put it away. There was no way he'd get any more work done since he wasn't able to concentrate on it anyway. He put his tools away, left the small barn, and entered the horse barn. Two of his ranch hands walked toward him.\n\n\"Hey, Ryder,\" Jay Rodriquez greeted him.\n\n\"How's it going, guys?\"\n\n\"It's been a pretty good day, boss,\" Gavin Hill told him.\n\n\"Great. If you're done, you guys can head on out.\" He nodded at them and started around them. \"Oh, wait. You'd best know, there's a woman here for a while...\" he started to explain but hesitated when both men raised their eyebrows. \"No, not for that, get your minds out of the gutter. Frank sent her here from Georgia. She has a stalker and he thought she'd be safer here until the guy was found again. The police had him once but he jumped bail.\"\n\n\"That stalker shit isn't something to take lightly. An ex-boyfriend stalked my sister and it got downright creepy,\" Jay said.\n\n\"Is she all right?\" Ryder asked.\n\n\"Yeah. But he tried to kidnap her. Luckily, the police caught up with them and helped her, but Jamie is almost afraid to step outside the house now. I know she'll get over it eventually but it was real bad for a while.\"\n\n\"I just want you both to keep an eye out for any strangers coming around. Although I'm not sure how anyone would find her here, but you never know.\"\n\n\"Have you told Sam?\" Gavin asked him.\n\n\"No, but that's a good idea. I just worry that the more people who know about her being here, the greater the chance of her stalker finding her. Then again, most folks around here I can trust to help keep an eye out. I'll give him a call though. Have a good evening,\" Ryder said waving them on their way, as he walked down the aisle of the barn.\n\nHe entered his office, pulled his cell phone from his pocket, and called Sam Garrett, the Sheriff of Clifton County, Montana. After explaining the situation to Sam, he did as Sam suggested, he called a select group of his friends to come to his house that night. He would introduce them to Kelsey and explain what was going on.\n\nAn hour later, he entered the house and almost fell to his knees from the smells coming from the kitchen. Kelsey stood at the stove. She glanced over her shoulder at him.\n\n\"I thought I'd earn my keep and make dinner. Hope you like fried chicken, corn, and mashed potatoes. Go on and take a shower. It should be ready by the time you're done.\"\n\nRyder stared at her for a few seconds until she raised her eyebrow at him. He nodded his head, removed his hat and coat, hung them up, and then headed toward the bathroom. It was going to be the fastest shower on record. He hadn't realized just how hungry he was until he entered the house.\n\n****\n\nKelsey was just setting the food on the table when he returned. His hair was still wet and she found herself tempted to comb her fingers through it.\n\n\"Do you need any help?\" he asked her, still tucking his shirt into the top of his belted jeans.\n\n\"Nope, all under control. Just have a seat.\"\n\nRyder pulled out a chair but sat only after waiting for her to take a seat. She put food on her plate, and then watched as Ryder piled his plate high. After he took a first bite of chicken, he moaned.\n\n\"Damn, this is good.\"\n\n\"Thank you. I love to cook so since you're letting me stay here, I'll fix dinner. Breakfast and lunch too, if you want.\"\n\n\"I get up before the sun, but lunch would be good. Would you mind cooking enough for the men too?\"\n\n\"No, not at all. How many?\"\n\n\"Six altogether. Is that a problem?\"\n\n\"Only if you don't have enough food.\" She smiled.\n\n\"There are two freezers filled to capacity in the mudroom.\"\n\nKelsey laughed. \"Okay, it won't be a problem then.\"\n\nRyder nodded as he continued to eat. \"A group of my friends are coming over tonight. I want them to meet you, and know what's going on. Some of them will understand well what you're going through, and with their help, and the help of my men, we'll have more eyes keeping watch for strangers.\"\n\n\"All right.\" Kelsey knew he was doing the right thing, but she had to wonder what his friends might think of her staying with Ryder.\n\n\"What's wrong?\" he asked her.\n\n\"What will they think of me staying here?\"\n\n\"Nothing. I plan to tell them the truth. They're not going to judge you, if that's what you think.\"\n\nKelsey huffed. \"You trust them?\"\n\n\"With my life.\"\n\n\"Good enough for me then.\"\n\nAn hour later, they sat in the living room watching TV when someone knocked on the back door. Ryder glanced at her, motioned for her to stay put, stood, and headed toward the kitchen. He came back in with three tall men and three women accompanying him.\n\n\"Kelsey, this is Jake, Gabe, and Wyatt Stone. This is Becca, Jake's wife, Emma, Gabe's wife, and Olivia, Wyatt's wife. Everyone, this is Kelsey Sullivan. Once everyone else arrives, I'll tell you what's going on.\"\n\nKelsey looked at each man, and almost groaned. They were all gorgeous. She could easily see the resemblance between them, the black hair, and dark eyes. Only Gabe had a mustache, but it seemed to suit him. Their wives were beautiful too. The men took seats on the chairs Ryder had brought in from the dining room. The women sat on the sofa beside her. They each smiled at her with curious expressions on their faces. She smiled at them and shrugged, making them laugh.\n\n\"I'm really curious here, Ryder,\" Olivia said.\n\n\"Why am I not surprised, Liv?\" Ryder teased.\n\nA few minutes later, two more couples entered. Ryder introduced them as Madilyn and Brody Morgan, and Trick and Kaylee Dillon.\n\n\"Where's Sam? He's usually on time,\" Ryder asked Brody.\n\n\"He pulled in just as we were coming in.\"\n\nA tall man entered and Kelsey gazed around the room. All of these men were gorgeous. She'd never seen so many hot men in one place, including the one wearing a badge who was the last to join them.\n\n\"Kelsey, this is Sam Garrett. He's the sheriff here in Clifton,\" Ryder told her.\n\n\"Hello.\" She shook his warm hand and hoped she didn't sound as breathless as she felt. Sam Garrett could arrest her anytime. She glanced over to Olivia and saw her mouth the words, \"hot, huh?\" making Kelsey bite back a giggle.\n\nOnce everyone had taken seats, Ryder explained what was going on. The women gasped and the men wore frowns.\n\n\"How long has this been going on, Kelsey?\" This question came from Sam.\n\n\"Two months before Frank sent me here.\"\n\n\"You had to be terrified,\" Emma said, reaching out to pat Kelsey's knee.\n\n\"I was. I still am...\"\n\n\"We'll all keep an eye out, Ryder. In this little town, we all know each other and this time of the year there should be no strangers,\" Wyatt said.\n\n\"Do people come here to ski?\" Kelsey asked.\n\n\"No. Clifton is an hour from the nearest resort. People usually don't stay that far away from the slopes since they like to get on them early,\" Ryder told her.\n\n\"If it was spring or summer, it would be different. We have loads of tourists here then. I own a bed and breakfast that does amazingly well and people come from all over. Strangers are all we see then. We'll be open only a few more weeks, and the people that are here now, we know,\" Becca said.\n\nKelsey nodded. \"I'm sorry to include you all in this.\"\n\n\"Don't worry about it,\" Sam said, smiling. He turned to Ryder. \"I think you should bring her to Evelyn and Stan's engagement party. We can introduce her and let folks know she's under the town's protection. That way everyone will keep an eye out for anything unusual or if someone comes around asking about her. I'll contact the police in Atlanta and have a picture of the perp faxed over.\"\n\nKelsey saw Ryder swear. \"I suppose it would only make sense to take her to the party. Everyone will eventually see her around anyway and letting them know what's going on can only help us, as long as it stays only with the town folks.\"\n\n\"You want to take me out in public?\" Kelsey asked.\n\n\"I think the more people who know you and why you're here, the more will be on the lookout for any strangers, and I'll pass the picture around too,\" Sam told her the same thing Ryder had.\n\n\"Whatever you say, you're the sheriff.\" Kelsey teased and smiled at him.\n\nSam grinned at her. \"Yes, ma'am, I am.\"\n\nShe glanced over at Ryder and saw him frowning at her. She stared back at him until he finally looked away. What was his problem? Was he jealous of Sam? Too bad. What woman in her right mind wouldn't do a little harmless flirting with Sam Garrett? In fact, if the other men were single, she'd flirt with them too. She loved gorgeous men and the room was full of them\u2014including Ryder. No matter how hot she found the others, there was something more about him that made him stand out. He had that brooding cowboy thing going on and it was damn sexy.\n\n\"Tell me how it all started,\" Sam said, in what she imagined was his investigative voice.\n\nKelsey sighed. \"It started with a message on my social media page. He asked me what hospital I worked at. I'm a nurse.\" She gazed up at Sam. \"I told him Georgia. I didn't want to tell him exactly where I was. We chatted for a while and I thought he was nice. I wasn't looking to hook up or anything, I was just being nice.\" She shook her head. \"I should have known better. He always seemed to pop in anytime I was online and we'd chat. I liked talking to him.\"\n\n\"Did the police find him through social media?\" Jake asked her.\n\n\"No. They found him when they discovered he was driving by my house every day. I didn't know I'd even met him before, until I saw him when he was arrested. He's the uncle of one of my patients. I started getting letters at the hospital. They'd be at the front desk with a note saying they were for me. The police couldn't get fingerprints from anything. He was smart. Next thing I know I'm getting emails from him. Telling me he wanted to be with me, and that he...he loved me. I suppose he got my email from my social media.\" At Ryder's grunt, she glared at him. \"I know it was stupid. You don't have to be rude about it.\"\n\n\"You think I'm being rude about it? I'm just surprised you're a nurse. You should know better than to do something like that. There are all kinds of idiots out there. As you now know.\" Ryder practically growled and came to his feet. Kelsey gasped.\n\nSam glared at him. \"That's enough, Ryder. We all make mistakes.\"\n\nRyder glared back at Sam. \"Yes, we do. but in this day and age, no woman should be talking with a man she doesn't know.\"\n\n\"Excuse me, but I'm living with a man I don't know,\" Kelsey shouted.\n\n\"It's not the same thing,\" Ryder muttered.\n\n\"Okay, let's all just calm down,\" Sam said. \"It's done and over with. We'll all keep an eye out. We'll do our best to keep you safe, Kelsey.\"\n\n\"Thank you. His name is Wayne Dowling,\" Kelsey said, quietly.\n\nOlivia put her arm around Kelsey. \"We'll take care of you.\"\n\nKelsey tried to smile. \"Thank you, Olivia. I appreciate it. It's nice that most of you will.\"\n\nRyder's lips flattened as she stared up at him. He spun away from her and returned to his seat in the recliner. She really wished she didn't have to stay here with him. She wished she could ask the others for haven with them but she didn't want to interfere in their lives. Besides, Frank had sent her to Ryder and she wouldn't want to upset Frank by leaving Ryder's ranch. He trusted Ryder.\n\n\"What about your job?\" Gabe asked her.\n\n\"I took a leave of absence. Thing is, I don't know how long they'll allow me to be away though. If I don't go back soon, I know I'll be replaced.\" It saddened her to think of not going back to the hospital. Would she get back in a reasonable amount of time or would they hire someone to take her place? She hated being a bother to Ryder and she knew, without a doubt, that she was. He didn't want her here anymore than she wanted to be here. She glanced over to him and saw him still staring at her. A sudden heat filled her face.\n\n****\n\nRyder wanted to shake her. What the fuck was she thinking? Talking with a man on social media like an immature kid who didn't know better. Sure, he was a stranger but Frank vouched for him, and she trusted Frank or she wouldn't be here. His hands tightened around the arms of the chair until he was sure his fingers would leave indents in the material. Did Frank know the entire story? Had Kelsey told him about talking with the stalker?\n\n\"Stop staring at me like you want to choke me, Ryder,\" Kelsey snapped at him.\n\n\"Because I do!\" He stood. \"What were you thinking?\"\n\n\"I wasn't, okay? Could we talk about this later?\" Kelsey glanced around the room, suddenly looking embarrassed.\n\n\"We're going to leave you two to talk. Bring her to the party Saturday, Ryder,\" Sam practically ordered. Ryder gave him a terse nod. Everyone said their goodbyes and left. Ryder continued to stare at her.\n\n\"Tell me what possessed you to talk with a stranger so much that he thinks he's in love with you?\"\n\n\"I guess I liked the attention,\" she whispered, as if she wanted to disappear.\n\n\"What?\"\n\nKelsey glared up at him. \"I liked the attention, all right?\"\n\n\"A woman who looks like you needed attention from a perfect stranger?\"\n\nShe shot up off the sofa. \"Just what does that mean? A woman who looks like me?\"\n\n\"It means, why would a beautiful woman need to seek out attention from a stranger?\"\n\n\"You think I'm beautiful?\" She sat back down.\n\nRyder snorted. \"Do you never look in the mirror? Of course, you're beautiful. That's why I don't get this at all.\" He ran his fingers through his hair.\n\n\"Thank you, but I know I'm not beautiful.\"\n\nHe stared at her, and then burst out laughing. \"Are you fishing?\"\n\n\"What? No. Of course not.\"\n\n\"Kelsey, I know a lot of beautiful women and trust me when I say you're beautiful.\" When she opened her mouth to speak, he put his hand up. \"Don't argue with me. I'm not the only one who thinks so. I saw the way Sam was looking at you.\" He began pacing the room.\n\nShe smiled at him, as if pleased at the idea that the sheriff might be attracted to her. \"Sam is very good-looking.\"\n\n\"Too much for his own good.\"\n\nKelsey laughed. \"Don't worry. He has nothing on you, Ryder Wolfe.\"\n\nRyder stopped pacing and turned to look at her. Was she being sarcastic? He suddenly hoped not. He grinned. \"Is that so?\"\n\n\"Yes,\" she said with a teasing smirk. Now he wasn't sure again.\n\nRyder stared at her, and then suddenly frowned. Was it possible she truly didn't think she was a beautiful woman? \"Did someone put you down at one time?\"\n\nShe cleared her throat. \"A man I was seeing. A teacher. I found out he was married and then he told me I wasn't the only one he was seeing because he couldn't stand to look at me.\" Her voice cracked and he knew those words had hurt her. He understood how words could cause as much pain as fists.\n\n\"And you believed the asshole?\"\n\n\"When you hear things enough, you begin to believe them. He never hesitated to tell me I was ugly.\"\n\n\"Why would he say things like that?\" Ryder walked toward her and stopped in front of her.\n\n\"I have a long scar on my left thigh from a car accident.\"\n\nRyder swore and turned from her. \"Son of a bitch. He drilled it into your head that you were ugly because of a scar on your leg?\" He looked back at her.\n\nKelsey nodded. \"It took a long time for me to even let him see it and once he did, he started telling me how people didn't like scars. It was a fact of life.\"\n\n\"I'd like to choke him with my bare hands.\" He squatted down in front of her and took her hand in his. \"Scars are a fact of life, but in no way should anyone make you feel ugly because of one. I have one here.\" He pointed to the one at the corner of his eye. \"And I have scars you can't see. It took me a long time to get past them, but I did. Don't let him keep making you feel ugly, Kelsey. You are one beautiful woman. I thought so the minute you walked into Dewey's.\"\n\nKelsey leaned forward and kissed the place alongside his eye where that scar lived. He froze at the touch of her lips. His eyes met her blue ones and nothing in this world could have stopped him from leaning forward, and pressing his lips to hers. Her gasp filtered across his lips right before she pulled his bottom lip between her teeth. Ryder growled and moved to his knees. He cupped her face in his hands and deepened the kiss, moving his tongue into her mouth to taste her. Her arms wrapped around his neck as she moaned into his mouth. Ryder moved to sit beside her on the sofa but as his body came full up against hers and his hands moved, one to cup her firm sweet bottom, while the other slid into her silky hair, he felt her tense. Suddenly, he remembered what they had just discussed. This other guy had made her feel ugly and self-conscious, and now he was pawing her like a rutting horse.\n\nRyder pulled back, breaking the kiss. He stared down into blue eyes heavy with desire and wished she understood just how incredibly gorgeous she was. He wanted to show her how beautiful he thought she was. He dipped his eyes and watched her tongue slide over the delectable lips he'd just tasted. How he wanted to kiss her there, and lower, and lower still until she begged him to take her. He felt his body react and knew he needed to stop this before it got out of hand.\n\nLeaning back and releasing Kelsey, he cleared his throat. \"I guess that was out of line. It's been a long day. I'm sorry.\"\n\nKelsey stared at him, her chest heaving drawing his eyes to her nipples reaching toward him through the thin cloth of her top. He glanced away.\n\n\"Yeah, I suppose it was. I shouldn't have kissed you.\"\n\nShe shouldn't have kissed me! Where the hell did that come from? Ryder was about to comment that he was the one to take things too far when she pulled herself off the sofa, straightened her clothes, and started toward the door.\n\n\"Goodnight,\" she said, throwing the word over her shoulder as she left the room.\n\n\"What the hell...\" Ryder mumbled at the empty room. He ran his hand over his face and took a deep breath willing his body to calm down. \"Being gentlemanly just doesn't pay off sometimes.\"\n\n# Chapter Three\n\nSaturday evening, Ryder and Kelsey rode in silence to the Clifton town hall. They'd barely spoken two words to each other in the three days that had passed since...the couch incident. That was how Kelsey thought of it. Still feeling embarrassed about kissing him, and for thinking even for a moment, he had actually been interested at all. He'd certainly pushed her away fast enough. He may have told her he thought she was beautiful, but what man turned away a beautiful woman? His tall, dark, and silent treatment hadn't helped either. In fact, she was getting angrier by the day because he made her feel foolish. Damn him.\n\nShe glanced over to him. He kept his eyes straight on the road ahead. He hadn't looked at her at all in the past few days. She didn't even ask him what she should wear for the party. She'd called Olivia instead, and asked her.\n\n\"Just jeans are fine. It's not fancy even if it is an engagement party,\" Olivia had told her.\n\nArmed with that advice, Kelsey wore skinny jeans tucked into Ugg boots and a blue sweater that matched her eyes. She knew the sweater brought out the color of her eyes, making them sparkle. She cleared her throat.\n\n\"Who are Evelyn and Stan?\"\n\nRyder flinched as if surprised by her voice. \"Stan Watson is the foreman at Becca's B and B. Evelyn Robinson is the cook. They're both in their sixties, I believe,\" he explained with a shrug, but didn't take his eyes off the road. \"They fell in love and decided to get married. They're a great couple. You'll like them.\"\n\n\"That's great that they're getting married. Love knows no age limits.\"\n\nRyder grunted and steered the truck into the parking lot. He found a spot easily enough and after parking it, climbed out. Not waiting for him, Kelsey opened her door and stepped down out of the cab. He stood at the front of the truck waiting on her. She stuck her chin up and walked past him, clenching her jaw when she heard him sigh. She hoped he wasn't going to embarrass her in front of his friends. She spun around to glare at him, half-expecting him to make some snide remark but he just grinned and stepped up next to her, taking her elbow in his hand.\n\n\"Try not to let everyone know you hate me,\" he whispered against her ear. Her hate him! She stared at him and confusion raced over her. Two can play this game.\n\n\"You're asking the impossible,\" Kelsey muttered, but smiled when he chuckled.\n\nThey entered the town hall and Ryder led her around the room introducing her to people, and somewhat, explaining the situation without giving too much detail. Once done, they joined their friends at the table. Kelsey pulled out a chair and sat down between Olivia and Emma. Ryder sat across from her beside Becca.\n\n\"You look great,\" Becca told her.\n\nKelsey smiled at her. \"Thank you. So do you.\"\n\nJake sat forward and put his arm around his wife. \"She always does.\" He kissed Becca's temple, making her blush. Kelsey laughed at how adorable they were.\n\n\"How are things going out at the Wolfe ranch?\"\n\n\"Good.\" Kelsey smiled, hoping her discomfort wasn't too apparent to the women.\n\n\"I'm sure it's not a hardship being with a man who looks like Ryder,\" Emma said from beside her.\n\nKelsey laughed. \"You live with a gorgeous man. All of you do.\"\n\n\"Yes, we do, but we also love looking at other gorgeous men.\" Olivia sighed. \"Like Sam.\" She nodded her head toward the door and Kelsey watched as the sexy sheriff entered the town hall.\n\n\"He is really hot,\" she said wondering if he would turn her away like Ryder had.\n\n\"I've said it before and I'll say it again, if I hadn't met Wyatt, I would have gone after Sam. That man is fucking sexy.\"\n\nAs Sam walked further into the room, all the women followed him with their eyes. Olivia turned around and fanned herself, making the women laugh.\n\n\"He certainly fills out a pair of jeans,\" Becca said.\n\n\"Front and back, honey,\" Emma added with a big grin.\n\n\"Oh yeah, for sure. That's ten pounds of sugar in a five pound sack,\" Olivia said and then sighed. The women all burst out laughing at her dramatics.\n\n\"They're talking about Sam,\" Wyatt said aloud, sounding somewhat annoyed.\n\n\"I figured as much,\" Jake said frowning.\n\n\"What?\" Olivia asked as she batted her eyelashes.\n\n\"It's just a damn good thing we love you,\" Gabe said and then winked at Emma.\n\nTrick and Kaylee along with Brody and Madilyn took seats at the table.\n\n\"What's everyone talking about?\" Kaylee asked.\n\n\"How the women are drooling over Sam,\" Jake told her.\n\nKaylee laughed. \"Sam is hot.\"\n\n\"What?\" Trick growled from beside her. Kelsey watched as Kaylee smiled at her husband and Trick winked at her.\n\nMadilyn laughed. \"He is. I don't think there's a woman in town who doesn't think so.\"\n\nThe men laughed. Wyatt shook his head. \"Why didn't you go out with him then, instead of Ryder?\"\n\nOlivia laughed. \"He never asked.\" She squealed when Wyatt pulled her onto his lap.\n\n\"You dated Ryder?\" Kelsey asked, suddenly feeling very inadequate since Olivia was the most beautiful woman she'd ever seen.\n\nRyder chuckled. \"I wouldn't call going out once dating.\"\n\n\"What happened that you didn't go out more?\"\n\nRyder looked at Kelsey. \"We just didn't feel it. Although I remember, Wyatt hated me for a while.\"\n\n\"A while? What makes you think I still don't? I can hold a grudge, you know.\" He grinned when Olivia whispered something in his ear. \"She's talking dirty to me, so no grudge held.\"\n\nRyder burst out laughing along with everyone else.\n\nAs the humor of the group, helped put her more at ease, Kelsey watched as a woman came up behind Ryder, put her arms around his neck, and leaned over his shoulder. He glanced over his shoulder at her.\n\n\"You didn't call me like you said you would, Ryder.\"\n\nRyder looked at Becca and winked. \"I thought you wouldn't want me to after the last time we spoke.\"\n\nThe woman laughed softly. \"You know better than that. When have I ever stayed mad at you?\" She glanced over at Kelsey, and for some reason, gave her a dirty look. \"Dance with me, Ryder...unless your date won't let you.\"\n\nRyder's gaze slid to Kelsey then back to Nicki. \"She's not my date, Nicki. Be nice.\"\n\nNicki sneered at Kelsey. \"Sorry.\"\n\nKelsey didn't think she sounded sorry. \"No problem. If I was his date, you sure as hell wouldn't be hanging all over him.\"\n\nOlivia nearly choked on her drink, and Becca placed her hand over her mouth. Everyone else laughed. Kelsey didn't know why she had said what she did but apparently, it didn't bother Ryder, because he was grinning at her.\n\nNicki pulled on his arm. He rolled his eyes, stood up, and started toward the dance floor. Nicki looked back at Kelsey.\n\n\"In your dreams, honey.\" Then she quickly caught up to Ryder.\n\n\"I can't believe I said that,\" Kelsey said with embarrassment, and put her hands on her cheeks.\n\nOlivia laughed. \"I loved it. Nicki Shaffer deserved it. She thinks she's the only woman Ryder's interested in.\" She shrugged. \"He dates a few women at a time. I seriously think he's one man who will never settle down.\"\n\nWyatt grunted. \"I thought I was that man for a while. It just takes the right woman.\" He looked at Olivia and winked. She put her head on his shoulder, and let out a big sigh, making everyone laugh.\n\n****\n\nRyder had his hands on Nicki's waist while her arms wrapped around his neck, her fingers tangling in his hair. She was gazing up at him.\n\n\"Who's that woman?\" she asked.\n\nRyder shrugged. \"Nobody important, she's staying with me for a while. Frank sent her to me for some protection. She seems nice enough. You, however, were being a bitch.\"\n\nNicki pouted at him. \"You know how I feel about you, Ryder. Do you think I need any more competition where you're concerned?\"\n\nRyder scowled. \"Nicki, you know how I feel. I'm not committed to you or anyone else.\"\n\n\"So, are you saying you're interested in her?\"\n\n\"I didn't say that but even if I was, it wouldn't be any of your business.\"\n\n\"Sometimes, I really hate you, Ryder. There's going to come a day when you want me and I might not be available anymore.\" Nicki stalked away from him as soon as the song ended.\n\nRyder returned to the table, wishing the night would end. Having two women angry with him for no apparent reason was more than enough. The band suddenly stopped playing, and Becca stood on the stage. Everyone stopped what he or she was doing to pay attention.\n\n\"We're here tonight to celebrate the love of Evelyn and Stan. It's about time you two did this. You've been dancing around each other for quite a while now.\" The crowd laughed. \"Seriously, we all love you and are so very, very happy for you both. It will be a beautiful wedding, and Jake and I are thrilled to death to have it at the B and B. Let's all raise our glasses to Evelyn and Stan.\"\n\nEveryone raised their glasses and toasted the couple, who smiled and leaned into each other. Becca walked straight to Evelyn and hugged her, and then Stan. Jake hugged Evelyn and shook Stan's hand then he led Becca back to the table.\n\nThe band started playing again to get back into the party mood. A slow song started and everyone left the table to dance, except for Kelsey and Ryder.\n\n\"Would you like to dance?\" Ryder asked her, not sure how she might answer.\n\n\"Oh, I don't know. Where's the she-devil you were dancing with earlier?\"\n\nRyder chuckled, almost with relief. \"I don't know. I'm sure she's around. Why? You're not afraid of her are you?\"\n\nKelsey snorted. \"Please. I could take her in a heartbeat.\"\n\nRyder laughed, suspecting she could at that. He stood, and put his hand out to her. She came around the table and accepted it. He led her to the dance floor and turned her in his arms. Kelsey had her hands on his shoulders and his were around her waist. Kelsey softly laughed. Ryder leaned his head back to look at her, raising his eyebrow.\n\n\"Don't look now, but the she-devil is shooting darts at us with her eyes.\"\n\nRyder huffed out a laugh. \"I'm sure she is.\"\n\nKelsey giggled. \"I just waved at her.\"\n\nRyder threw his head back and laughed. Kelsey laughed with him. He turned her so he could see just in time to watch Nicki stalk toward the bathroom.\n\n\"And you call her a she-devil. You're just wicked.\"\n\nKelsey laughed. \"You have no idea.\"\n\nRyder was about to try once more to apologize about the other night when Sam interrupted the band, and walked to the mic. \"Sorry to interrupt, folks, but it's snowing and accumulating kind of fast. Some of you may want to leave now.\"\n\n\"Are we leaving?\" Kelsey asked Ryder with a worried expression.\n\nHe nodded. \"I think we should. If it's getting bad here in town, it'll be worse on the outskirts.\" He led her back to the table and talked with Gabe and the others. He turned to Kelsey. \"Everyone is leaving. We're all going together in case there's any problem. We all live in the same direction, so it makes sense.\"\n\n\"It wasn't even snowing when we came in,\" Kelsey said as Ryder handed her coat over.\n\n\"It can happen fast here and pile up in no time. No one will take the chance that they can't get home. Everyone who lives out of town will probably leave.\" Ryder glanced around the room and watched as most of the people were getting their coats and hats on to leave.\n\nKelsey nodded. \"I hate it for Stan and Evelyn.\"\n\n\"They know how it is and I'm sure they're ready to go too. In fact, I think they're already gone.\" Ryder smiled as he pulled his coat on.\n\nThe group walked out to the parking lot together. Once on the road, they pulled the vehicles into a caravan, and soon the snow got heavier.\n\n\"I can't believe this,\" Kelsey said peering out through the windshield.\n\n\"Welcome to Montana.\" Ryder chuckled and kept his eyes on the truck in front of him.\n\nAs they drove on, each time one of the trucks pulled into a driveway, Ryder would blow the horn. His driveway was the last one so he saw everyone get home safely. Now if he could just concentrate on driving and not on the beautiful woman beside him. He tightened his grip on the steering wheel. He should have just kept his damn hands to himself the other night. He still didn't understand why she felt what she did was wrong. She'd given him a sweet touching kiss on his cheek and he'd taken it to pawing her and wanting to take it much further...much, much further. Touching her was not only wrong but also stupid. Frank wouldn't like what had almost happened. He trusted him to watch over her, and he was nearly as bad as the stalker.\n\nRyder slowed the truck to pull into his driveway and it slid on the snow-covered road. Kelsey let out a gasp and gripped the center console and suicide bar on the dash.\n\n\"We're fine, I got it,\" he said quietly and saw her nod out of the corner of his eye. He was glad he'd left the gate open, was his thought, as he drove through, and toward the house. He stopped the truck by the back door and shut the engine off. \"Are you all right?\"\n\n\"Yes. That was just a little scary.\" Kelsey looked at him with large round eyes. \"I've never driven in snow and now I'm glad I didn't have to. I would have panicked.\"\n\n\"Rule of thumb...never slam on the brakes. The wheels lock up, and the vehicle will slide.\"\n\n\"I'll never drive in it, so no worries.\" She smiled at him.\n\nTheir eyes met and held for longer than was necessary. Ryder cleared his throat, opened his door, and climbed out. He trudged through the snow, around the front of the truck, planning to help Kelsey but she met him there.\n\n\"Take my hand so you don't slip,\" Ryder told her as he held out his hand.\n\n\"If I fall, you're going down with me.\" She laughed even as she put her smaller hand in his.\n\n\"Wouldn't be the first time my ass has hit the snow. Come on.\"\n\nThey made their way through the quickly accumulating snow to the porch. The dogs barked from inside the house. Ryder opened the door and they ran out and off the porch. Once inside, Ryder helped her out of her coat, and then removed his own and his hat.\n\n\"It sure got cold really fast,\" Kelsey said rubbing her hands up and down her arms.\n\n\"Go take a hot bath. I think I'll watch some TV before I head to bed.\"\n\n\"All right.\" She seemed to hesitate as if she wanted to say something but then didn't. \"Goodnight, Ryder.\"\n\n\"'Night.\" He watched as she walked down the hallway and disappeared into her room. The closing of the door had him leaving out a breath he hadn't realized he'd been holding. He flopped down in the recliner and picked up the remote. He suspected it was going to be another fucking sleepless night.\n\n****\n\nAfter changing into sweatpants and a T-shirt, Kelsey fell back on the bed and sighed. She wanted Ryder. All night with him, sitting across from him, and then dancing with him had put her body on high alert. Watching him dance with Nicki, Kelsey had wanted to skin her alive. The woman had her hands all over him.\n\nShe punched the quilt-covered mattress, thinking about them together. Ryder slept with that woman and Kelsey knew it as well as she knew her own name. Bitch!\n\nStaring at the ceiling, she wondered how much longer she was going to have to be here. The more time she spent around Ryder, the harder it was going to be not to touch him or let him know how much she wanted him even though he wasn't interested. Why should he be? He has Nicki. He was probably disappointed he had to bring her home instead of being with the she-devil.\n\nKelsey was twenty-seven years old and never been married. She could count on one hand the number of men she'd had sex with and still have fingers left over. Daniel Ward had been the last man she'd been with and she really thought she was in love, until she found out he had a wife and three kids at home. Some of her friends had tried to tell her he was married, but she'd believed him when he said he was separated and getting a divorce. Believed him completely, until his wife came to her apartment and told her different. Stupid! Stupid! Stupid!\n\nDaniel even came to her apartment that very night to tell her he'd leave his wife but she shut the door in his face. Once a liar, always a liar. He'd fucked around on his wife, and Kelsey knew he'd do the same to her if she stayed with him. He'd made her feel such the fool when she found out he never intended to leave his wife. Damn men. The police had even checked into Daniel as the stalker, but it wasn't him. He'd moved on to greener pastures with a new young teacher at his school, or so she'd heard. The man couldn't keep it in his pants and his wife couldn't keep him at home.\n\nThe only other two men she'd been with had been one in high school and one in college. They had both been too immature. Inexperienced, neither knew anything about satisfying a woman. She didn't seem to have much luck where men were concerned.\n\nKelsey pushed off the bed and moved to the window where she watched the snow falling outside. Amazing. Snow so early in the year. She slipped her feet into her slippers and padded to the door. Opening it quietly, in case Ryder had gone to bed, she stepped into the hallway and strolled toward the kitchen. As she passed the living room, she saw the flickering glow of the television. She peeked into the room and realized Ryder was sitting in a recliner.\n\nQuickly retreating, she hurried to the kitchen for a snack. The light was on so she moved directly to the refrigerator and pulled the door open, peering inside. Suddenly, a barking sound came from outside. She started toward the back door but stopped when Ryder entered the kitchen. He stopped. He stared at her. His hair was slightly mussed as if he had been sleeping, and his shirt was open and the T-shirt beneath clung to his hard body.\n\nAnother bark sounded from the other side of the door. Ryder moved quickly to the door, opened it and the two dogs flew inside bringing the scent of cold and snow in with them. They shook, spraying cold wet droplets around the kitchen. Most of it showered Ryder's jeans but some hit Kelsey too, making her laugh and jump back a step.\n\n\"Go on, get in there,\" Ryder commanded with a chuckle, sending the dogs into the laundry room. He followed them.\n\nKelsey smiled. The dogs had certainly enjoyed themselves out in the snow. Like children, she thought. She returned to the refrigerator and grabbed an apple from the crisper, and a bottle of white wine from the door. She was pouring wine into a tumbler when Ryder returned to the kitchen.\n\n\"Want some?\" she asked, showing him the bottle.\n\n\"No, thanks, got a beer in the other room,\" Ryder answered and headed toward the living room.\n\nKelsey frowned as she watched him leave the room. Why did she even try? She took a crackling bite of the apple and chewed the tasty fruit, thinking about Ryder. Was she really so unappealing? Less appealing than that she-devil, who hung all over him at the party.\n\nShe took a mouthful of wine and swallowed hard. She bit into the apple again, feeling annoyance and perhaps anger build with each bite. When she finished, she slam dunked the core into the trash bin, picked up her glass of wine, took a fortifying gulp, and marched into the living room.\n\nRyder was sitting in the recliner, a bottle of beer in one hand, his arm resting on the arm of the chair. She walked into the room and took a seat on the sofa. He glanced at her but didn't say anything. She sipped at her wine and glanced at the TV. Some classic black and white movie was playing but the sound was low so she didn't really know what the actors were saying.\n\n\"So...you and Nicki,\" she began.\n\n\"There is no me and Nicki,\" Ryder said in a low voice before taking a drink of beer.\n\n\"I just figured...well, the way she was hanging on you and...\" Kelsey paused and looked down at the glass in her hand. \"Well, you weren't interested in me.\"\n\n\"Not interested in you,\" Ryder exclaimed, his eyes meeting hers in the dim light of the room. \"You're the one who told me it was a mistake.\"\n\n\"Yeah, well then, I guess it was.\" She finished the last of her wine and set the glass on the coffee table. \"I don't compare to the other women you know.\" She thought about how beautiful Olivia was and knew she was nothing like her, or even Nicki.\n\n****\n\nKelsey stood. Ryder knew she was once more thinking herself ugly thanks to the bastard who told her so while claiming to love her. She was so wrong about him not being interested in her though. He could have had Nicki tonight had he wanted, but he couldn't stop thinking about Kelsey. Holding her in his arms while dancing reminded his body just how soft and wonderful she felt when he kissed her the other night.\n\nWhen she moved to leave, he came to his feet. She stopped and looked at him with a curious expression. He moved forward until he stood in front of her. He heard her inhale as he smoothed her hair back from her beautiful face. Dipping his head, he touched his lips to hers and felt her tense for a moment. Here we go again!\n\nRyder hesitated in deepening the kiss and nearly stepped back when Kelsey snaked her arms around his shoulders and pressed her warm body against him. His body immediately reacted and his arms slid down to her waist and pulled her close. Lowering her onto the couch, he spread out alongside her while his tongue danced with hers. He heard her moan low in her throat and his body liked it.\n\nHer hands pushed his outer shirt off over his shoulders before sliding her fingers across his chest and down along his stomach to the bottom of his T-shirt. She lifted it over his head and pulled her lips from his. He watched as her eyes roamed over his chest and down his muscled stomach. He pushed her shirt up and groaned when he saw her beautiful breasts. The rosy nipples peaked in anticipation and arousal.\n\n\"You're beautiful, Ryder,\" she whispered on an exhalation.\n\nHe grinned with satisfaction and moved his mouth to her neck as his hand slipped inside her sweatpants to her very core. He kissed her and smiled against her lips.\n\n\"Commando?\" he said as he pulled them off her and tossed them onto the floor.\n\nKelsey blushed and nodded. \"Touch me, please...\" Her plea ended on a gasp when he moved his fingers between her wet, slick folds.\n\n****\n\nKelsey pressed her body against his hand as he inserted a finger and rubbed his thumb against her clitoris. His tongue lapped at her nipple, and then he sucked it into his mouth. What he was doing was driving her crazy. She'd never felt like this, and she wanted him inside her. Now. Kelsey threw her head back as her orgasm grabbed her. A low moan emerged from her before she cried out his name. Her body trembled as she came down from her high. She felt Ryder move his mouth up her neck to her lips, taking them in a deep kiss. He removed his hand from her still throbbing body, and lifted his lips from hers. She groaned as she watched him stick his finger into his mouth and sucked on it. She'd never seen anything so erotic in her life.\n\nShe wrapped her hand around his neck, pulled him down to her, and ran her tongue along his lips, making him groan. He took control of the kiss and deepened it, his tongue dipping deep inside her mouth. The growl that tore from him when she sucked on it made her feel very much in control. He moved his lips across her cheek to her ear.\n\n\"I want to fuck you,\" he whispered, his hot breath in her ear. \"I want to feel you clench around my cock when you come. I\u2014\" All of a sudden, he pulled away from her.\n\n\"What...?\"\n\n\"We can't do this, Kelsey,\" he said in a rough hoarse voice.\n\n\"It looks like we already are.\"\n\nRyder sat up and moved away from her. He cleared his throat and ran his hand around the back of his neck. \"Frank would kill me.\"\n\n\"Frank isn't here and besides, who would tell him?\"\n\n\"No one has to tell him. I'd know he'd be disappointed in me. I'm supposed to be protecting you, not fucking you.\"\n\nShe sat up and pulled her T-shirt down. \"Well, we certainly didn't get that far, did we?\" she said, suddenly feeling angry and humiliated. He'd used her just as Daniel had.\n\nRyder stared at her. \"No, we didn't, because I came to my senses.\"\n\n\"Was that before or after you were licking me off your finger?\" She grabbed her sweatpants from the floor and pulled them on. She stood and glared down at him. She saw his jaw clench, and a muscle ticked in his cheek.\n\n\"Go to bed,\" he growled suddenly.\n\n\"Are you coming with me? Oh. No. Wait. You're too afraid to be a man here and finish what you started.\"\n\n\"I got you off, didn't I?\" he shouted, noticeably angry.\n\nShe spun away from him and then turned back to him. \"You're right. At least I got off.\" Her eyes shifted to the bulge behind his fly. \"You'll have to take care of that by yourself.\"\n\nKelsey wanted to hit him, and demand he just admit he wanted her, but feared he didn't. She stormed off to her bedroom, slamming the door behind her.\n\n****\n\nRyder swore under his breath when he heard the door slam. What the fuck is wrong with you? He ran his fingers through his hair, and then down over his face. He could still smell her scent on his finger and that just made his cock swell even more. Son of a bitch!\n\nHe pulled on his shirts and stomped to the kitchen, grabbed his hat and coat, and walked out the back door. He trudged through the snow to the barn. Frank would have his balls in a sling if he even thought, for one minute, that Ryder would touch Kelsey.\n\nRyder walked into his office and closed the door. After taking his hat and coat off, he sat behind the desk and booted up the computer. Why he did, he had no idea. He just needed to get his mind off Kelsey. He snorted...like that would happen with his body still wanting her.\n\nHe groaned when he thought about how responsive she'd been. When he felt her inner muscles clenching around his finger, he just about came himself. He'd wanted to shove his jeans down and thrust into her hot, wet heat.\n\nHis hands grasped the edge of the desk, his knuckles turning white trying to push back his need to march back into the house, go to her room, throw her on the bed and fuck her till she screamed for more. He jerked when his cell phone rang. He picked it up and groaned when he saw who it was.\n\n\"Hey, Frank,\" he answered through a clenched jaw.\n\n\"How is everything, Ryder?\"\n\n\"Fine. Why?\"\n\nA silence greeted him for a few seconds. \"I was just wondering,\" Frank finally said.\n\nRyder cleared his throat. \"Everything is just fine.\"\n\n\"Good. I tried to call Kels, but she didn't answer. Is she all right?\"\n\nAll right? She was fucking fantastic was Ryder's thought, but he couldn't say that to Frank. \"She's probably in the shower.\"\n\nShit! Ryder didn't need that image in his head. His dick was hard enough to pound nails already.\n\nFrank chuckled in his ear. \"Okay, then. Tell her I called and I love her. I'll talk to you soon, son.\"\n\n\"All right, Dad.\" Ryder ended the call and walked to the window.\n\nThis was ridiculous. He wondered how much longer she was going to be here. He wasn't sure how much longer he could take having her here with him and it'd only been a few days. He slammed his fist on the wall alongside the window, and then swore at the pain. His knuckles hadn't healed from hitting Big Earl yet.\n\nRyder sighed, and returned to his seat behind the desk. He figured he could look over the books, and do whatever else came to mind until he headed back inside. Give his body some time to calm down. He couldn't avoid her forever. If only he could.\n\n****\n\nKelsey leaned her head back against the tub and closed her eyes. Damn him. He wanted her. She restlessly moved her legs. She was still on fire from his touch. No man had ever made her feel that way that quickly. She wanted him inside her and he'd moved away. Moved away! He got a fucking conscience and pulled back from her. Told her they couldn't do it. Damn him. God damn him! If he were standing in front of her, she'd slug him. She hoped his dick stayed hard all night. She snorted and laughed. Then she sobered when she remembered how she'd felt him hard against her thigh and how fantastic he'd felt. Long, thick, and hard. She jerked when a sudden knock sounded on the bathroom door. He called her name.\n\n\"Go away, Ryder.\"\n\n\"Are you all right?\"\n\n\"Sure. I'm not the one sexually frustrated right now.\" She placed her hand over her mouth and grinned maliciously when she heard him swear.\n\n\"I just wanted to tell you that Frank called. He said to tell you he loved you.\" He cleared his throat. \"I'm going to bed.\"\n\n\"Thank you...Ryder?\"\n\n\"Yes?\" Why did he suddenly sound hopeful? Was he? No! He'd made his opinion quite clear.\n\n\"Hope you and your hand have a nice night together,\" she yelled through the door, and then laughed when she heard him swearing as he walked away.\n\nDamn him anyway. Even Daniel hadn't gotten her to an orgasm like he had, and Ryder had only used his hand. She shivered against the warm water, feeling her body react at the memory. If he was that good with just his hand, she wondered what he could do with the rest of his body. She'd never felt that way before. She groaned when she remembered him sticking his finger into his mouth and sucking her essence from it. God! He was so fucking hot.\n\nShe wanted to see him...all of him. Every. Delectable. Inch. Of. Him. Because what she'd felt against her leg had been very impressive, and she had no doubt he knew what to do with it. Daniel had never touched her like that. He'd never done much in the foreplay area. He was too anxious to get inside her, and get off. She'd always been left wanting...craving for more, just as she was now.\n\nKelsey pulled the plug on the tub and stood up as the water began draining away like she wished the throbbing of her body would. The water ran down her body as she reached for a fluffy towel. After drying off, she pulled on a pair of lounge pants and a T-shirt. Entering her bedroom, she looked at the empty bed and sighed. She had a feeling it was going to be a long night. Again. After pulling the quilt back on the bed, she climbed beneath the covers and turned the bedside light off. Taking a deep breath, she blew it out and stared at the ceiling hoping sleep would come before daylight crept through the lace curtains.\n\n# Chapter Four\n\nSunlight played across her face, waking her. Kelsey stretched against the pillows and mattress before sitting up and glancing toward the window. Once again, she heard hammering. She threw back the covers, swung her legs over the side of the bed, and slipped her feet into her slippers. She pulled the quilt from the bed up around her shoulders and walked to the window. The sun reflected off the snow making it almost impossible to see at first. Once her eyes adjusted, she saw smoke pouring out of the chimney that jutted up from the smaller barn roof. She decided she was going out there today, so she quickly dressed and headed toward the kitchen. After getting a quick cup of coffee, she put on her boots, hat, and coat before heading outside. She let her eyes adjust to the brightness before walking down the steps. Someone had shoveled a path through the snow leading to the barns. Kelsey followed it and stopped at the door of the smaller barn. She smelled fresh sawdust. Taking a deep breath, she opened the door and stepped inside.\n\nShe let her eyes adjust again and once they did, she gazed around in awe. Wood furniture sat everywhere. She heard a saw running and walked toward the sound. She saw Ryder standing at a table saw with safety goggles on. She stepped in front of him and waved her hands so he'd see her. She watched him turn off the saw, raise his goggles up, and push them up on top of his head.\n\n\"So this is what you do in here? You make furniture?\" She was astounded at the beauty of the furniture sitting around. From bedroom suites to dining room tables and rows of rocking chairs. She realized the one that was in his living room was just like the ones sitting in there.\n\nHe grinned. \"What do you think?\"\n\nKelsey shook her head and walked toward the rockers. She ran her hand over the top of one of them. The surface was as smooth as glass. The cedar gleamed, and the lighter shade of wood running through the darker wood was beautiful. She sat in one of them and rocked it. She watched Ryder walking toward her. She gazed up at him when he stopped in front of her.\n\n\"I love this. I'd love to have one.\" She grinned.\n\nRyder took a seat in the rocker beside her. She glanced at him as she rocked the chair. Who knew this is what he did out in this barn? She glanced around at the other pieces in the space and when her eyes landed on an armoire, the beauty of the piece stole her breath. It was made from cherry.\n\nShe pushed out of the rocker and walked over to it. She ran her hands over it, and then opened the doors looking inside and inhaling the wonderful aroma. It was a truly beautiful piece of furniture. She turned around and looked at him with admiration.\n\n\"You made all of this? You made this?\" she asked caressing the fine wood.\n\n\"Yes, and most of them are sold already too. Just waiting for pick-up. One of my best customers owns a furniture store in Billings. He usually sends a big truck here to pick up a huge order. He came here last month to pick up an armoire I'd made for him to give to his wife for her birthday.\"\n\n\"My mom would love to go through here. She's a big furniture nut.\" She spun around and looked at him. \"You made the furniture in my bedroom, didn't you?\"\n\n\"I made all the furniture in my home, except the couch and recliners. Well, maybe a few other things.\" He shrugged and flashed a grin at her.\n\n\"But you have horses too?\"\n\n\"My main business is my ranch, the Paint horses I breed and sell. This is just a hobby that took off like wildfire.\"\n\n\"I can see why. It's all so beautiful.\" This was the most talkative he'd been since her arrival. Of course, they both failed to talk about the issue hanging in the air between them.\n\n\"Thank you. Look, I need to get back to work. I only work out here on the weekends and I have a customer coming tomorrow to pick up a bedroom suite.\"\n\n\"All right. I'll get out of your way. Did you have breakfast?\"\n\n\"Hours ago. I'd love some lunch though in about an hour. If you don't mind?\"\n\nShe smiled. \"No problem. I'll fix you something. Is there supposed to be more snow?\"\n\n\"Yes. The weatherman is calling for up to a foot more tonight. I'll have to make sure we have plenty of firewood and gas for the generator, in case the power goes out.\"\n\n\"Do you think it will?\"\n\nRyder shrugged and moved to stand by the table saw. \"We never know around here. It has before and it will again. Don't worry, we'll stay warm.\" He lowered his goggles and turned on the saw.\n\nKelsey nodded and left the barn to head back to the house. She looked around at all of the snow. She'd never seen so much snow, not in person anyway. It was beautiful. She glanced back to the barn when she reached the back door. She knew of one way to stay warm...with Ryder. Of course, all she had to do was think about the time on the couch and she was burning up. Damn. If only she could get him to admit he wanted her and to stop worrying about what Frank would think. Suddenly, she was hoping the power would go out tonight.\n\nAbout an hour later, Ryder entered the kitchen and took off his hat and coat. After hanging them up, he told he was going to make a fire, just in case and headed off to the living room. A short while later, he returned to the kitchen and washed his hands in the kitchen sink. He took a seat at the table and Kelsey set a hamburger in front of him.\n\n\"I hope that's all right. I didn't know how big of a lunch you'd want.\"\n\n\"It's just fine,\" Ryder told her as he spread mustard on the bun. He glanced up at her and raised an eyebrow. \"Aren't you eating?\"\n\n\"I already did.\"\n\nHe nodded, picked up the sandwich, and took a big bite. \"Umm...this is good. Different.\"\n\n\"I added some spices to the meat. Who wants a plain old hamburger?\"\n\n\"Not me, not anymore.\"\n\nKelsey laughed, appreciating how he was enjoying his lunch. \"I have steaks out for dinner. How do you like yours?\"\n\n\"Rare.\"\n\nKelsey made a face and gave a mock shudder. \"How in the world do you eat it like that? Isn't it still mooing?\"\n\n\"Only way to eat a steak, darlin', more than that and you cook the flavor out.\"\n\n\"Uncle Frank eats it that way too.\"\n\nRyder chuckled. \"Who do you think got me started on it that way?\"\n\n\"They were good to you, weren't they?\"\n\nHe stared at her for a few seconds. \"Yes, they were.\"\n\nIt was obvious he wasn't going to say anything more on the subject. He simply finished his burger and then stood. After putting his plate in the sink, he grabbed his hat and coat, and was gone.\n\nKelsey blew out a breath she hadn't realized she was holding. She was never sure what she could say without offending him. The man was a puzzle, but one she'd very much love to solve.\n\n****\n\nAfter working a few more hours on the furniture order, and Kelsey feeding him one of the best steak dinners he'd had in a long time, Ryder returned to the main barn to feed the horses. Since it was the weekend, he took care of it himself. He gave his men weekends off when possible. The only one around full-time was his manager, Cookie Bridges, and he rarely left the ranch. He and his wife, Charlene lived in a small house behind the barns.\n\nAs he walked down the aisle with the bucket of feed, the horses stuck their heads over their gates. He talked to each one as he fed them. He opened Tramp's gate and entered the stall. The horse butted his head against Ryder's chest.\n\n\"Hey, buddy. Are you hungry?\" Ryder chuckled when the horse bobbed his head up and down.\n\nTramp was one of Ryder's best studs. The horse had high stud fees, and it was because of this particular horse and his stud that his business thrived the way it did. People from across the states came to buy a Paint with Tramp's bloodlines. A five year old with black, brown, and white markings, he stood seventeen hands tall and had blue eyes. He was a gorgeous animal and Ryder loved him. The feeling was mutual. Tramp would let no one else near him if Ryder wasn't close by.\n\nAs Ryder poured the oats into the feed bucket, Tramp pushed him in the back. Ryder spun around and grinned. \"What the hell was that for? I'm pouring the food as fast as I can. You can't be that hungry.\"\n\nTramp took the brim of Ryder's hat in his teeth and pulled it from his head. He bobbed his head up and down with the hat between his teeth. When Ryder reached for it, Tramp turned his head away. Ryder put the bucket down and stared at the horse fighting to keep from laughing.\n\n\"You think you're such a smartass. Give me my hat or you're dog food.\" Tramp dropped the hat and butted Ryder against his chest. Ryder chuckled and patted Tramp's strong neck. \"You know better than to believe that of me, don't you, boy?\"\n\nOnce the feed was in the bucket, Tramp pushed Ryder out of the way and started eating. He ignored Ryder as he munched on the oats. \"You don't even acknowledge me now, do you?\"\n\nWhen the horse didn't raise his head, Ryder smiled, patted Tramp on the neck, picked up the other bucket, and left the stall. He wanted to get inside before it got much darker and the snow piled even higher.\n\nAs he checked to make sure everything was secure in the barn, he thought about Kelsey. She was hard to read, that one. Last night, she was angry, angry that he'd respected her and stopped things before they got out of control. She hadn't brought it up so he sure as hell wasn't going to, but then she'd complimented his work on the furniture and now she was cooking for him like he was something special. Did that mean she'd forgiven him, or was she just planning her revenge through kindness and compliments? Women. Impossible. Simply impossible.\n\nLeaving the barn, he pulled his collar up, whistled for the dogs that appeared from around the side of the barn, and together they hurried through the snow as it began falling faster and heavier. Just as he opened the back door and entered the kitchen, the lights flickered once. He stopped, took off his hat, and waited. Just as he shrugged out of his damp coat, the lights flickered again, and then went out. Damn!\n\n****\n\nKelsey jerked when the lights went out. It was early evening but with the snow, it seemed later. The kitchen went dark yet with the reflection off the snow, the yard seemed suddenly brighter. She watched as Ryder hung up his hat and coat. Then he turned toward her.\n\n\"I think you should get blankets and a pillow from your room, along with anything else you need, and put them in the living room. I don't know how long the power will stay out and I'd rather you get everything now then have to do it in the dark.\"\n\n\"All right.\" Kelsey headed down the hallway and pulled a pillow and blanket off the bed. She had just entered the living room and laid them on the sofa when the power suddenly came back on. She sighed in relief.\n\nRyder shrugged. \"We never know what might happen. I don't think it would be a bad idea to sleep in the living room tonight anyway. I have a generator, but since this early snow caught me off guard, I hadn't laid in any additional fuel for it, so I'm not sure how long it will keep the lights on.\" He took a seat in the recliner. \"Whatever you wear to bed, you'd better make sure it will keep you warm.\"\n\n\"I'll change into my sweatpants...\" She hesitated for a heartbeat remembering what had happened when she wore them last night. Heat flashed over her body thinking about Ryder stripping them off her legs. She refused to look at him and headed toward her bedroom.\n\nShit, shit, shit! She clenched her fists and hissed. Maybe he had forgotten how he'd humiliated her. She snorted. Yeah, about as much as she had. Her body shivered with sudden arousal remembering the way he had brought her body to life and now she might have to sleep next to him in the living room. Should she pray for the power to go off or stay on? Either way, she decided God was punishing her for some misdeeds she was unaware of having committed.\n\nKelsey quickly changed her clothes and walked slowly back to the living room. Ryder was adding wood to the fire and the room was feeling very warm and cozy. The blue and orange flames wrapped around the logs filling the room with the scent of burning wood. She took a seat on the sofa, pulling her legs up under her, and stared up at the TV, but she couldn't keep her eyes from seeking out Ryder. He took his seat in the recliner again, stretched his longs legs out in front of him, and crossed his booted feet at the ankles.\n\nGod! He was so gorgeous. That black hair needed a trim but it looked so sexy on him and the stubble on his face, neck, and jaw made her fingers itch to touch it. She'd love to scrape her fingernails over it. Her eyes traveled down his chest to the fly of his jeans and she had to bite her lip to keep from groaning as she remembered how he'd felt against her leg. Hard, long...she raised her eyes and saw him looking at her. She felt a flaming hot blush heat her face.\n\n\"You keep looking at me like that and I'll finish what I started last night, darlin',\" he said in a low tone of voice.\n\n\"Please\u2014\"\n\n\"Please? You begging now?\"\n\nShe was sure her mouth dropped open, but she saw he was trying not to grin. Was he baiting her to make her angry, or had he changed his mind? She gave him a half-smile. \"I don't think I'd need to beg,\" she said with a shrug and glanced back to the TV.\n\nRyder laughed. \"Damn right you wouldn't.\" He pulled his tall frame from the recliner and it came up right with a thud drawing her attention back to him. \"I better find some candles. I have a feeling we may need them.\"\n\n\"I can't believe this snow,\" Kelsey said as she gazed out the front window, grateful for the change in subject. The snow was coming down so heavily that it looked like a whiteout or what she thought a whiteout would be like since she'd never experienced one at home in Georgia. She just knew she couldn't see the driveway at all. \"Are your horses in?\"\n\n\"Yes, indeed. I always make sure the babies are in when it gets this bad.\"\n\n\"Babies?\" Kelsey grinned.\n\nRyder chuckled. \"Yep, those horses are my babies.\" He left the room.\n\nKelsey couldn't keep from watching the snow. Watching it fall was fascinating. She'd only seen snow like this on television or in the movies. Never in person. She glanced toward Ryder when he returned. \"Did you find candles?\"\n\n\"I did. Flashlights and a deck of cards too. If the power goes off, we'll most likely get bored so we can always play cards.\" He raised an eyebrow. \"Unless you've got some better ideas on how to pass the time?\" He laughed when she shook her head feeling the heat in her cheeks again. \"I need to check the generator. I'll be right back.\"\n\n\"Where is it?\"\n\n\"In the shed next to the house.\"\n\n\"You're going out in this?\" she asked in a shocked voice.\n\n\"I've done it before.\" He walked out of the room and headed to the kitchen. A few minutes later, she heard the back door open and close. His dogs ran from the back room to the kitchen.\n\nKelsey called to them and they ran to her, but kept whining. \"I'm sure he'll be fine. Maybe you two can sleep with me, if I need to keep warm, huh. Can you keep me warm, Badger? How about you, Buttons?\" She laughed when the dogs tried to push each other out of the way to get to her.\n\nShe heard the back door open and close then a sound like he was stomping his feet. Ryder walked into the living room rubbing his hands together. He wore a frown of frustration.\n\n\"Let's hope the power stays on. There's almost no fuel in the generator. I can't believe I didn't think to get any for it. Always be prepared. Son of a bitch,\" he muttered.\n\n\"The fire should keep us warm.\"\n\n\"Yeah, we do have plenty of wood. I'll keep it going all night. I'll even light the one in my bedroom.\"\n\n\"You have a fireplace in your bedroom?\"\n\n\"Yes. Why?\"\n\n\"Well, it's just...you know, so romantic.\"\n\nRyder stared at her and then burst out laughing. \"There's nothing romantic about it. I put one in there for just this reason...snow storms and power outages.\"\n\n\"You really know how to burst a woman's bubble, Ryder.\"\n\n\"Why? Because I'm not romantic?\"\n\n\"Yes.\"\n\n\"Well, if you're really game, darlin', we can sleep in there and I'll show you just how romantic I can be,\" he said with a dangerous waggle of his eyebrows.\n\nKelsey was about to answer when the lights went out again. She looked up at Ryder. \"I wonder how long it will be out this time.\"\n\nThe wind picked up outside, howling as it blew snow against the windows.\n\n\"I guess we'll find out.\"\n\n\"I can't believe it's snowing like this. It's only mid-October.\"\n\n\"Well, as I said before. Welcome to Montana.\"\n\n\"Should we go ahead and start a card game?\"\n\n\"Or we could make-out.\"\n\nShe could hear the laughter in his voice and threw a couch pillow at him. He dodged it and laughed.\n\n\"I'm not the one who threw ice water on things last night so be careful or I'll take you up on your offer.\" She picked the cards up and started shuffling them. She nearly laughed at the incredulous expression on his face. She suspected she'd shocked him.\n\n\"Poker?\" she asked him. \"Strip maybe?\"\n\n\"Now who's teasing?\" he countered.\n\n\"Don't throw out innuendoes if you can't handle getting them back, Ryder. You make me think you've only got sex on the brain.\" Ha! Like she didn't as well.\n\nHe burst out laughing. \"I'm a man, remember?\"\n\nOh, she remembered all right. How could she not? She looked at him from under her lashes. The firelight played over his hair. She'd love to take him up on that make-out session. She dealt the cards. Five card stud. They played for a few hours. She beat him one time. He was relentless and good.\n\n\"You do realize that if we'd been playing strip, you'd be naked by now.\" He winked at her, making her laugh.\n\n\"I can't believe we've been playing this long.\" She yawned. \"I'm getting sleepy now.\"\n\n\"Let's get the blankets ready and get some sleep.\"\n\nKelsey nodded and helped him get the blankets on the sofa and in the recliner. The fire was keeping the room warm but if it died, they'd both get cold.\n\n\"Don't worry. I'll keep the fire going,\" Ryder told her as if reading her mind.\n\n\"I know you will, but you'll need to sleep too.\"\n\n\"I will. Get some sleep, Kelsey. Maybe the power will come back on before the night is over.\"\n\nShe nodded and crawled between the blankets. She'd taken the couch and he lay back in his recliner. The fire felt good but she prayed the power came back on.\n\n****\n\nRyder pulled his boots off dropping them to the floor alongside the chair, and then lifted the lever on the recliner to lift the footrest and lower the chair's back. He put the pillow behind his head and tugged on the blanket to cover his chest. He glanced over at Kelsey, but she had her back to him with the blanket almost covering her entire head. All he could see was the top of her blonde head.\n\nHe wasn't sure how long he sat there staring at her but when she moaned in her sleep, his dick woke up. Christ! He threw the blankets off and stood to add wood to the fire, although he thought it was already hot as hell in the room. The damn woman was going to be the death of him. He should have told Frank he couldn't keep her in his home. She was too beautiful. Her body was fantastic and he walked around with a constant hard-on. A hard-on that was only going to go away once he had himself buried deep inside her. He groaned.\n\nAs quietly as he could, he laid the logs in the grate and moved them around with the poker.\n\n\"Ryder?\"\n\nHe glanced over his shoulder. \"What?\"\n\n\"The power is still off?\"\n\n\"Yes. Go back to sleep.\"\n\n\"What time is it?\"\n\nHe looked over to the grandfather clock in the corner. \"Three.\"\n\n\"Have you slept at all?\"\n\n\"Christ, Kelsey. Go to sleep,\" he growled at her.\n\n\"What are you so angry about?\"\n\nRyder stood and ran his fingers through his hair. \"I'm not angry.\"\n\nKelsey snorted. \"Yeah, okay...right.\"\n\nHe spun around and glared at her. \"Could you just let it go and go back to sleep?\"\n\n\"Maybe I'm not sleepy anymore.\"\n\n\"Holy shit! Fine then you can take care of the fire and I'll go to sleep.\" He watched her sit up.\n\n\"It's not the fire keeping you awake. You're suddenly in a piss ass mood for some reason.\"\n\nHe strode over to the couch and stood over her, his hands at his hips. \"You want to know why I'm in a piss ass mood. Because of you. All right? I can't sleep because I can't stop watching you sleep. I'm wide-awake because of you. You're making my dick hard. Happy now?\"\n\nKelsey sat there staring up at him just blinking those big blue eyes. Then she narrowed them at him, threw back her covers, and slowly stood, her eyes never wavering from his. \"Whose fault is that? Yours, Ryder. Not mine. I wasn't the one who put the brakes on things. That's all on you.\"\n\n\"You know why\u2014\"\n\n\"Bullshit, Ryder. We're adults. This has nothing to do with Frank. So grow some balls and man up.\" She placed her hands on his chest and shoved him. He stepped back from her.\n\n\"The hell it doesn't have anything to do with Frank. He trusted me to keep you safe, not take you to bed,\" Ryder shouted at her.\n\n\"But you want to,\" she shouted back.\n\nRyder clenched his hands into fists. \"Hell yes, I want to. I'm a man, damn it, and you're a beautiful woman.\"\n\nKelsey stepped closer to him, resting a hand on his chest, and speaking in a lower voice. \"Prove it.\"\n\n\"Prove what?\"\n\n\"That you're a man.\"\n\n\"Go back to sleep,\" he growled as he started to turn away from her.\n\n\"Pussy,\" she whispered the challenge bold in her tone.\n\nHe spun around to face her. \"What? What did you just call me?\"\n\n\"You heard me.\" Her chin went up.\n\n\"I don't think I heard you right. Say it again.\"\n\n\"I called you a...pussy. Want to hear me say it again? You. Are. A. Pussy.\"\n\nRyder reached out, grasped her arms, and pulled her up against him. His mouth slammed down on hers and he thrust his tongue into her mouth. She moaned and when he let go of her arms, she wrapped them around his neck. He groaned when she pushed her hips against him. He moved his hands, down along her waist to cup her butt and lift her up. Her legs locked around his hips, and she moved against his hard cock. If she didn't stop, he was going to come right there.\n\nWithout releasing her mouth, he carried her to his bedroom and to the bed, laying her in the middle of it. He stared down at her, at her already swollen mouth and wanted more. He had one knee on the bed, and his eyes on her, as he removed his shirt, and then lay down beside her. He traced his fingertips over her eyebrows, nose, and lips then down her neck to the top of her shirt. He leaned down to kiss her while he rolled her sweatshirt up, stopping only long enough to lift it over her head. He groaned when he saw her breasts. She had fantastic breasts. His mouth covered hers again, slipping his tongue into her mouth, while he cupped her breast in his hand and kneaded it gently. His cock was so hard he ached.\n\n\"God, Kelsey. You're so beautiful. You're pure perfection.\"\n\nRyder ran his tongue down her neck to her breasts where one at a time, he took a nipple into his mouth. Kelsey grabbed fistfuls of his hair in her hands pressing his face closer and he gave her what he wanted. He tugged on her nipple gently with his teeth, swirling his tongue around it. Her hands ran down across his back, raking her nails lightly against his skin until her hands reached his hips, slipping around front to the snap on his jeans. She opened it and slowly lowered the zipper. His hard cock pressed against the back of her fingers, through the cloth of his boxer briefs.\n\nHe lifted his head, his eyes meeting hers while his hand slid along the top edge of her sweatpants, slipping inside, and then into her panties. She squirmed against his touch, her eyelids fluttered shut, and she moaned. He lowered his mouth over her lips and brushed them in a soft caress.\n\n\"Do you want me to stop?\" he whispered against her lips.\n\nKelsey's eyes flew open and she drew back slightly. \"Are you friggin' nuts? No!\"\n\nRyder chuckled. \"I was hoping you'd say that.\" He pressed his lips to hers, kissing her deeper, tasting the sweetness of her mouth. His fingers moved into the folds of her sex. She was wet, making him moan. He almost shot off the bed when she wrapped her hand around his hard dick. She smiled up at him. \"You don't seem to be lacking in any areas, Ryder.\"\n\nHe grinned against her lips, and then pressed his lips to hers again. He felt her push his jeans down past his hips but he had no plans to let her take control. He broke the kiss and pressed his lips in quick kisses and the occasional lick of his tongue along her neck, down her chest and across her stomach, removing her sweatpants as he did.\n\nSnagging her red panties in his teeth, he pulled them down. She was panting and he heard her giggle when he used his teeth. He moved his tongue through her curls, stroking his tongue across her clitoris turning her giggle to a gasp. He pushed her panties down over her legs and off. He kissed his way up her legs, stopping at the scar on her outer left thigh. He pressed his lips to the mark, tracing his tongue against it before moving along to the sweet wet heaven where her legs come together.\n\nRyder flicked his tongue against her clitoris, hearing her moan, and then put his mouth over it to suck on it. She squirmed and lifted her hips, pressing her body against his face. He slowly inserted two fingers into her as he continued to suck her. He reveled in the feel of her hot body coming and clenching around his fingers right before she cried out his name. He pressed his cock against her leg before spreading her legs with his knees and settling between them. He was so ready to inch into her, but then he made himself stop.\n\n\"Don't stop now, Ryder,\" Kelsey whispered in pleading voice, cupping his face, and gazing into his eyes.\n\n\"I need a condom,\" he muttered. He pulled away long enough to retrieve a condom from the drawer in the night table by the bed. After ripping the packet open, he sheathed himself, and then settled between her thighs once more. He cupped her face, brushing hair away from her cheeks, and stared down into her flushed face. \"Are you sure?\"\n\n\"Yes, I've never been so sure of anything in my life,\" she said in a breathless voice then gasped and moaned as he began pressing into her. Her legs wrapped around his hips and she quickly matched his rhythm.\n\n\"You feel so fucking good,\" Ryder whispered against her ear. He smiled when he felt her tremble, and then pressed her mouth to his neck.\n\n\"Oh, God. Ryder, you feel amazing,\" she whispered in his ear then she caught his earlobe with her teeth, making him shudder.\n\nHe kissed her hard, moving his tongue into her mouth and growled against it when she sucked on it. He was breathless, his voice hoarse when he lifted his head. \"When you suck on my tongue, it shoots straight to my cock.\"\n\n\"I want to suck on that too,\" she whispered, and then gasped, her eyes wide as if in surprise. \"I've never said anything like that in my life.\"\n\nRyder chuckled and kissed her again. \"Say whatever you want. I love it and as far as doing that, I'll let you...later.\"\n\nKelsey laughed as she tightened her legs around his waist. He picked up the pace and moved into her harder and faster. Her inner muscles clenched around his cock and her head went back against the pillow as she screamed out his name. Ryder tried to hold out, but he couldn't. Feeling her squeezing him sent him over the edge. He growled out her name and came. Hard. It had never felt anything like this before. He collapsed on top of her with both of them trying to catch their breaths. Their bodies still pressed against each other were slick with sweat.\n\n\"So, that's what a mind-blowing orgasm feels like,\" Kelsey whispered breathlessly.\n\n\"It must be. Never felt like that for me before.\" Ryder rolled off her and lay beside her, closing his eyes against the way his body felt...like every ounce of energy had been sucked out of him.\n\n\"Really?\"\n\nIt took him a moment to answer. \"Yes. Really.\"\n\nCool air settled over his skin raising gooseflesh. He moved to sit on the side of the bed and removed the condom. It wasn't something he wanted her to see.\n\n\"Where are you going?\"\n\n\"I need to put more wood on the fires. I'll be right back.\" He stood, pulled on his jeans then headed out the door, dropping the condom in the trash as he left the room to go to the living room.\n\nThe fire was still going strong but he wanted to keep it going to keep the house warm. He added logs, stoked it to a blaze, and then carried some wood back to the bedroom. He crouched down, added logs to the fire, and used the poker to move them around until the fire was fully alive again. He glanced over his shoulder to see Kelsey sitting up and watching him. Her breasts gleamed in the firelight. He felt his dick start to swell again. Son of a bitch! She reached her arms out to him.\n\nHow could something that left him feeling so distasteful, feel so totally amazing?\n\n\"Come back to bed, Ryder.\" She was so beautiful, so alluring. He couldn't resist her even when he knew he should.\n\nHe pushed to his feet, removing his jeans as he moved toward the bed. His conscience told him, he should leave them on as a protective barrier but he resisted that thought knowing full well, the cloth wouldn't stop him. After climbing back into the bed, he gathered Kelsey close and wrapped his arms around her. She wrapped an arm around his waist and rested her cheek on his chest. He shifted making her groan in disappointment. But when he reached for the quilt at the bottom of the bed and pulled it over them, she smiled up at him.\n\n\"We need to get some sleep. It's late.\" He kissed the top of her head. He stared at the flames flickering shadows across the ceiling until he heard her breathing change. He knew she'd fallen asleep. Now, if he could just do the same.\n\n# Chapter Five\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nThe next morning, Kelsey awoke to an empty bed. She glanced around the room. The light from the fireplace hadn't been enough to allow her to see it last night, not that she was very interested at the time. It was definitely a man's room. A beautiful cedar dresser sat against the far wall, next to the fireplace, with a door between them leading to a bathroom. French doors leading to a covered deck and a closet sat on another wall. Her eyes traveled back to the fireplace. It was open on both sides. She threw the covers back and walked into the bathroom. Inside, she found a large Jacuzzi tub next to the fireplace and a stand-up shower against the opposite wall. She smiled with delight. A large skylight allowed sunshine through, lighting up the room. She sat on the side of the tub, fantasizing about lounging in it with Ryder, and desperately wanted to try it out. She'd never seen a fireplace in a bathroom before. It was so romantic. Sighing, she pushed any romantic thoughts from her mind, and decided to take a shower before getting some breakfast.\n\nAfter returning to her room for fresh clothes and dressing, she entered the kitchen to find Ryder standing at the stove frying bacon. She cleared her throat and then blushed when he turned toward her not knowing how he'd receive her this morning. He smiled at her and walked to her, cupped her face in his hands and kissed her lips.\n\n\"Good morning,\" he murmured against her lips.\n\n\"Good morning. You were gone when I woke.\"\n\nHe chuckled. \"I was starving. You made me work up quite an appetite.\"\n\n\"Oh sure. It was all me.\" She wrapped her arms around his neck and kissed him back.\n\n\"Go sit down. I'll make you some breakfast then I need to head to the barn.\"\n\n\"You're leaving me for the barn?\"\n\n\"I have horses to take care of, darlin'.\"\n\n\"What if I need taking care of?\" she asked with a teasing tilt of her head\n\nRyder groaned. \"I'll be back later.\"\n\n\"Okay, those horses must really be something if you'd rather spend the morning with them than me.\"\n\n\"Oh, they are. Dark brown eyes that look so sad when I leave them. Except Tramp. He has blue eyes.\"\n\n\"I can look sad,\" she told him frowning and pouting out her bottom lip.\n\nHe leaned down and nibbled on her lip. \"I promise I'll make it up to you. They need to be fed and the men didn't make it in.\"\n\n\"Did we get a lot of snow?\"\n\n\"A foot, but it will be gone by tomorrow. It's supposed to warm up.\"\n\nKelsey sighed. \"Darn...I guess that means no power loss and no snuggling.\"\n\n\"Really? Who made up that rule?\" He nibbled on her neck.\n\n\"I think your bacon is burning.\" She grinned.\n\n\"Shit.\" Ryder hurried back to the stove and saved the bacon. \"What kind of eggs do you like?\"\n\n\"Scrambled.\"\n\n\"I like mine over-easy\u2014for future reference.\"\n\n\"If you're going to abandon me at first light, you'll get them over-easy your head,\" she muttered.\n\nRyder burst out laughing. \"I'll make my own breakfast then. I'm usually up at three-thirty but someone had other ideas last night.\"\n\n\"Oh yeah, blame me. You happened to be up at three.\"\n\nHe waggled his eyebrows at her. \"I was up after that too.\"\n\nKelsey laughed. \"Yes, and I'm so glad you were. It was a first for me. I've never had an orgasm during sex before.\"\n\n\"What? Are you serious?\"\n\n\"Yep. Sad, huh?\"\n\n\"What the hell kind of men have you been with?\"\n\n\"Not very sexually skilled ones obviously, including the last one who I thought I was in love with. He was only interested in getting himself off.\"\n\n\"And you thought you loved him?\"\n\n\"I did. I figured if I had love then I could live with bad sex.\"\n\n\"No one should have to live with bad sex,\" Ryder muttered.\n\n\"I know that now. I'll never settle again.\" She pulled a chair out from the table and sat down. Ryder placed a plate in front of her and her stomach growled in reaction to the wonderful aromas.\n\n\"Sounds like you worked up an appetite too.\"\n\nKelsey grinned, feeling the heat rise in her cheeks. \"Honestly, I had kind of hoped to work up a greater appetite before getting up.\" She shrugged.\n\nShe heard Ryder groan. \"I won't be in the barn long and perhaps we'll work one up for lunch.\" He winked at her and grinned when she blushed yet again.\n\nShe was in so much trouble here. The man was way beyond hot and sexy. The things he'd done and whispered to her made her face heat up just thinking about them. Damn. It was getting hot in here. She smiled shyly at him, picked up her fork, and dug into breakfast. She moaned at the first taste and saw his eyebrows lift. Hoping to distract him from all the sex talk, she changed the subject.\n\n\"These are fantastic. You can cook breakfast for me anytime.\"\n\nRyder seemed surprised but then he grinned. \"Deal.\"\n\nThey finished their meal in relative silence. When he was finished, he stood then carried his plate to the sink. He glanced over his shoulder at her but then grabbed his coat and hat, putting them. \"I'll be back later.\"\n\n\"Can I come out there?\"\n\nHe looked startled by her question so he hesitated, but then grinned. \"Sure, if you want to, but its damn cold out there.\"\n\n\"I'm not some delicate flower, Ryder. I've never been in snow like this. I want to go outside.\"\n\nHe chuckled. \"Okay, No need to get testy. Get dressed, and come on out. I have to clear a path to the barn, so give me a few minutes.\"\n\n\"All right. Thank you.\" She smiled.\n\nHe shook his head as he opened the door. \"She wants to go outside,\" he muttered before closing the door behind him.\n\nKelsey laughed and then finished her breakfast. She'd clean up the kitchen before going out to join him. She hoped she packed her mittens. Her mom was supposed to be shipping her more clothes but she hadn't expected snow so soon after arriving here.\n\nKelsey sighed. She was hoping she wouldn't need more things but it looked like she was going to be here a while longer. Her mother told her that letters were still showing up at her apartment. As for emails, she hadn't received anymore but she wasn't sure if that was because her stalker feared they were too easy to trace or the blocking really worked. She couldn't help but wonder if her stalker could still find her and what would he do if he found her?\n\nEver since arriving in Clifton, she'd thought about going home. It was her first thought in the morning and her prayer at night. She wanted to go home. Or did she? After last night, she wasn't sure anymore. Being here with Ryder wasn't easy, and the man was hard to ignore.. He was gorgeous and sexy as sin, which was just bad for her. The most this could ever be was a fling...a lusty, mind-blowing sexy fling. She would go home someday. Right?\n\nHer insides quivered and she swore her heart skipped a beat thinking about last night. Sex had never been like that for her\u2014ever. Lord, what she had been missing all this time.\n\nRyder had pulled her from a deep sleep with his hands and mouth once more during the night and it had been just as mind-blowing. He could bring her to orgasm so easily with his mouth, his hands, and his body. Shit. Now she was horny. She had to stop lusting after him. He was right, it wasn't why she was here and it could only lead to disappointment in the end. Right?\n\nShe needed to get her coat on and go see the man in the barn. Hey cowboy, how about a roll in the hay? She laughed at the dirty thoughts rushing through her mind and even wondered if he'd go for it. She imagined kissing every inch of him and wanting to since, she hadn't gotten that chance yet. He had made it all about her last night. How she wanted time to explore that gorgeous body. The thought of running her fingers, and tongue, over his solid pecs and a six-pack stomach gave her chills and heat flashes at the same time. He said working the ranch and lifting bales of hay all day was more than enough workout to keep him in shape. You can say that again!\n\nKelsey quickly donned her coat and pulled a beanie over her head. Luckily, she did have her mittens. After putting them on, she stepped onto the porch. The cold hit her hard. She knew it would be cold but this snatched her breath away. Ryder had cleared a path that led directly to the barn. She hurried along the path and stepped inside the barn. She let her eyes adjust, and the warmth of the building washed over her, and then walked down the aisle. Horses stuck their heads over their stalls as if eager to see who the visitor was to their home. She rubbed their noses as she passed them, smiling at the noises they made in response. She saw Ryder come out of a room carrying a bucket. Her heart nearly stopped at the image he presented. He was the quintessential cowboy. She'd always been into cowboys and she was definitely into this one. He stopped when he saw her, smiled, and then walked toward her. He stopped in front of her.\n\n\"Hi, darlin',\" he said. \"I see you made your way here.\"\n\n\"Hi, yourself,\" she responded a bit disappointed he didn't kiss her. \"Yeah, the yellow brick road was easy to follow.\"\n\nHe chuckled and walked around her. \"Come on, I have to feed Tramp, and then we can go back to the house where it's much warmer.\"\n\n\"Why do we need to go to the house? We could always have a roll in the hay, as they say.\" She was trying not to laugh.\n\nRyder spun around, raised an eyebrow, and laughed. \"Have you ever had a roll in the hay? It pokes you.\"\n\n\"Oh, well I'd rather have you poke me,\" she told him with a grin.\n\nRyder burst out laughing and turned away.\n\nShe frowned at his lack of eagerness at her idea. She followed him through the barn watching him work. She couldn't keep her eyes off the way his jeans hugged his tight butt. Damn. She'd really hoped he'd be more in the mood. Maybe things looked different to him in the light of day.\n\nSome while later, Kelsey entered the kitchen with Ryder behind her. She was freezing. Georgia didn't do cold like Montana does. Her fingers felt numb, and she wasn't sure she if she could even smile the way her cheeks felt. She stood in the kitchen shivering.\n\n\"Take your coat off,\" Ryder said from behind her.\n\n\"No. I'm too cold.\"\n\nHe put his hands on her shoulders and leaned close. \"I'll have you warm in no time.\" The sound of that had her smiling as he turned her to face him and started unbuttoning her coat.\n\nShe kept her eyes on his face. He glanced up and their eyes met. She took her bottom lip between her teeth. She watched as he closed his eyes for a few seconds. When he opened them, she gasped at the heat in them.\n\n\"Ryder?\"\n\n\"I want you, Kelsey, honestly I do, but...,\" he paused, his fingers fell away from the front of her coat and he raked his fingers through his hair. He stepped away and pulled his coat off, hanging it on a hook alongside his hat. \"I'm going to get a fire going.\"\n\nKelsey stood there, silent, watching his back disappear into the living room. He wanted her, but... But what? She yanked off her mittens, hat, and then her coat, tossing them onto one of the kitchen chairs. Rubbing her hands to warm them, she walked into the living room and stood watching Ryder stoke the fire. Soon she was warm again, but she wasn't sure if it was from the blazing fire or the anger building inside her.\n\n****\n\nRyder could feel her eyes on him and knew what she expected of him since they had flirted wildly before he left to work in the barn. He hadn't meant to, he had intended to stay cool as usual but he wasn't used to having women stay overnight here and she looked so sexy sleeping in his bed. By the time she'd come in for breakfast, he thought he'd convinced himself it was only the one night. Nothing more. But then she was there, so beautiful and so sweet. She'd all but said he was the best she'd ever had. Hearing that made his body stand up and take notice again. When he left her, he was serious about coming back and working up an appetite for lunch and much more. It wasn't until he was in the cold light of the real world that things slapped him in the face. She was here for his protection. She could be gone tomorrow for all he knew and besides, he wasn't looking for anything permanent and Kelsey Sullivan had commitment written all over her.\n\n\"Warmer now,\" he asked standing and placing the poker in its stand. She was frowning at him and as he moved away to take a seat in his recliner, he silently begged her not to go there.\n\n\"Did I do something wrong?\"\n\nDamn. She was going there.\n\nHe shook his head and stared at the fire. \"Last night...\"\n\n\"Last night was amazing,\" Kelsey interjected coming to stand between him and the flames. \"You can't deny that it was.\"\n\n\"I'm not, darlin', but I am going to admit that it was a mistake.\"\n\nKelsey stepped back, her arms dropped to her sides and she turned to stare at the fire. He knew it wasn't what she wanted to hear but it was the truth. He might be a lot of things, but he was honest.\n\n\"Look, it was great, really great, but I'm not boyfriend material, Kelsey,\" he said rubbing his hand across the back of his neck.\n\nShe turned slowly, folded her arms across her chest, the action pushing her deliciously beautiful breasts high and making him all the more aware of them.\n\n\"Did I ask you to be my boyfriend, Ryder?\"\n\n\"No...but...well, I know how women get,\" he stumbled trying his best to crawl out of the hole he'd somehow dug for himself.\n\n\"Oh, you do, do you? Well, maybe I'm not like other women. Maybe I just like to have some fun. Maybe I'm not looking for a boyfriend,\" she exclaimed with a saucy toss of her silky blonde hair.\n\n\"It's just...well, you'll be leaving to go home once they find this guy and my home is here. So...there's just...\"\n\n\"Ryder, I get that you're uncomfortable with me here. Hell, I'll bet I'm the first woman to sleep in that bed with you...\" When he glanced down at her accusation, she laughed. \"I am, aren't I?\"\n\nHe nodded. \"This is my place. My sanctuary. I don't bring women here. My grandmother was the last truly good woman to live on this land...\"\n\nKelsey moved forward, perched on the edge of the coffee table, and reached a hand out. Her fingers gently stroked his hand and his eyes met hers. What was it about her that made him want to talk? He hated to talk.\n\n\"Frank told me a bit about how you came to live with him and Grace. He told me...about your parents.\"\n\nHe glanced at the fire. The thought of her knowing how unwanted he was by his own parents made him want to crawl into a hole and pull it in after him.\n\n\"You owe your parents nothing, they owe you. They brought you into this world and it was you that they owed a good life.\"\n\nHe looked at her and saw true compassion in her blue eyes. What she said was true and it wasn't the first time he'd heard that. Grace had told him that more than once until she'd won him over. He owed his parents nothing but he owed Frank and Grace the world for saving him.\n\n\"I loved my grandfather, much more than I loved my parents. He knew how they treated me and did everything he could to protect me, but he got sick and couldn't take care of me. I'd stay with him in the summertime, here on the ranch. I hated when school would start and I'd have to go back to them. I'm sure Frank told you how they treated me. They were drunks. All they did was lay around the house and make me wait on them. If I didn't do something they told me to do, I got a beating. The bigger I got, the worse they got. I wanted to live with my grandfather.\"\n\n\"Why didn't you or your grandfather report them?\"\n\n\"The same reason every child who's beaten doesn't. Embarrassment. My grandfather didn't report them because they threatened to take me away and he'd never see me again. They were ruthless enough to do it, and he knew it. The only ones who knew about how it was were my friends. The ones you met here. I made them promise not to tell anyone\u2014ever. To this day, no one mentions it.\" He paused, and she squeezed his hand.\n\n\"One of my friends, Riley Madison, lived through it with his own father. We were close. I tried to get him to run away with me, but he wouldn't go. He was afraid if he did and his father got him back, the beatings would be worse. Our parents were smart in the way they beat us. They never left a bruise where anyone could see it. If I had bruises and had to take gym, my mother would send a note in that I was sick and couldn't attend the class.\" He shook his head. \"I hated them both and I still do.\"\n\n\"They're still alive?\"\n\n\"Oh yeah, in fact, they call me almost monthly to ask for money. It will be a cold day in hell when I give them any. My grandfather died when I was fifteen. I was heartbroken. I loved him so much. I couldn't take it anymore so when I turned sixteen, I ran\u2014as far, and as long, as I could. I was heading for Florida and had made it as far as Georgia. I was so tired. I came upon a dirt road. Well, at least I thought it was a dirt road. It was Frank and Grace's driveway. I followed it until I came to a barn. I figured I'd just rest there for the night.\" He laughed. \"I was woken up by someone kicking my foot. I jumped up and tried to run, but Frank caught me by the collar. He told me he was taking me inside while he called the cops. But Grace wouldn't let him make the call. She told him to leave me alone and asked me if I was hungry. I told her no but when she put two ham sandwiches in front of me, I wolfed them down. She told Frank I was going to sleep on the couch. Of course, he told her if everything she owned was gone in the morning along with me, not to bitch to him. She winked at me and I knew right then, I wanted her to be my mom.\"\n\nKelsey reached over with her other hand and with both, she cupped his hand gently. She reminded him of Grace sometimes. \"They are both amazing people.\"\n\n\"I know, and I'm very fortunate to have them in my life. I call them Mom and Dad. They are my parents as far as I'm concerned.\"\n\nKelsey laughed. \"Frank told me about taking you fishing.\"\n\nRyder chuckled. \"I couldn't believe anyone could be that angry and not want to hit something, or someone. He's the one who talked me into trying my hand at building furniture. I had so much anger in me, and he said to take it out on something with a hammer. I thought he was joking but once I started doing just that, I began to love it.\"\n\n\"So how did you go from making furniture to having this ranch and breeding horses?\"\n\n\"My grandfather left me this ranch when he died, but I couldn't take possession until I turned twenty-one. Thing is, I didn't come back until I was twenty-three. I was traveling for two years and hard to track down. The lawyer finally contacted Frank. It seemed he never knew I'd been living with Frank and Grace. My parents were living here on the ranch in the original house. The house they burned to the ground by one of them falling asleep with a cigarette in their hand. They both got out but everything was lost. When I came back to the ranch, they had an RV hooked up, and they were living out of it. I told them to get off the ranch but they refused, so I called the sheriff and had them evicted. They weren't happy but they went. My grandfather had left me a lot of money too. He knew I dreamed of owning a Paint horse ranch.\"\n\nRyder took a deep breath. \"He left me a letter, telling me how proud he was of me and to follow my dream. I did. I started going to auctions and buying horses. I bought a few horses from Becca when she decided to turn the house into a bed and breakfast. She needed gentle riding horses, so I took the Paints off her hands. When I bought Tramp, I knew he was the perfect stud. I was right too. People come from across the states to have their mares bred with him. I make a ton of money on stud fees alone. Jake, Gabe, and Wyatt helped me get things established. They raise and sell Quarter horses along with training them. Because of them, I was well on my way.\"\n\n\"They seem like great friends. I like all your friends.\"\n\nRyder narrowed his eyes at her. \"Is that so?\"\n\n\"Yes. It's too bad Trick is married. He is really hot and don't even get me started on the sheriff.\" She squealed when he suddenly pulled her onto his lap. She wrapped her arm around his shoulders and smiled down at him.\n\nHe felt better telling her his past. He'd never shared any of it with a woman before.\n\n\"Trick would never leave Kaylee, so no hope for you there.\"\n\n\"Oh, okay. What about Sam?\"\n\n\"You're really pushing it, darlin'.\" Ryder laughed. \"Sam is seeing someone, so leave him alone. Christ, we all grew up with the girls, and then women, chasing after Wyatt, Trick, and Sam.\"\n\n\"Oh yeah, like the rest of you are chopped liver.\"\n\nRyder burst out laughing. \"We had our share, but it got to be where it was a competition. When we were in our twenties, we were terrible with the girls at least until Sam took off to college, then the playing field got a little easier. When Trick married Kaylee, it was down to Wyatt. Then he left for the Marines. Finally, the rest of us had a chance.\"\n\n\"Oh please. I don't believe that for a minute.\"\n\nRyder grinned. \"No?\"\n\n\"Nope.\" Kelsey brushed a kiss along his lips, and then frowned. \"What kind of name is Trick?\"\n\n\"His real name is Patrick but when he was little, he couldn't say it. He called himself Trick and it stuck. The only one who gets away with calling him Patrick is Kaylee. That woman could call him hey you and he'd be fine with it. He loves her very much.\"\n\n\"They seem very happy. All of them do. I really liked the women I met.\"\n\n\"They're all great and have made their men very happy.\"\n\n\"Don't you ever want to get married? Be happy like that?\"\n\n\"No. It isn't that I don't want to be happy. I just don't want kids. I never want to be like my parents were, and mistreat anyone.\"\n\n\"That's bullshit, Ryder. In fact, you'd be a great parent because you know what it's like, plus the fact that Frank and Grace raised you right.\"\n\n\"It doesn't matter.\"\n\nShe didn't respond but he saw her brow furrow, and she glanced past him for a moment. When she looked at him again, she reached up and touched the scar by his eye. \"How did you get this?\"\n\n\"I brought my old man the wrong beer so he threw the bottle at me. I didn't duck fast enough.\"\n\nKelsey gasped. \"How can people be like that?\"\n\n\"I have no idea. If they didn't want a child, they shouldn't have had one. Unless they had me just so they'd have someone to wait on them. I did everything around the house.\"\n\n\"Where are they now?\"\n\n\"I don't know. Last month when they called, they were in Utah. I don't care where they are. I never want to see them again\u2014ever. It's bad enough when they call me. Always asking for money.\"\n\n\"You don't send them any, do you?\"\n\n\"No. I never will. It's my money. I made it, not them. They didn't do anything but lie around the house collecting disability. For what, I have no idea. They aren't disabled. They're just lazy. I think they found a doctor who would say they are, though what kind of doctor would lie for them I have no idea.\"\n\n\"It's hard to tell.\"\n\n****\n\nKelsey stared into his green eyes and saw the pain there. She lifted her hand and gently caressed his cheek. He closed his eyes and leaned into her hand. Her heart broke just a little for him knowing how unwanted he'd felt for most of his life. She wanted to tell him she wanted him, but it wasn't the way he needed someone to want him. She closed her eyes a moment against the ache in her chest. When she opened them, he was staring at her and the heat in his eyes could have rivaled the flames in the fireplace.\n\nHe slid his hand along her neck beneath her hair and pulled her face down close to his. He pressed his lips to hers. He moved his tongue along the seam of her lips until she opened for him. She groaned when his tongue moved into her mouth, dueling with hers. She wanted him, even if not the way a parent wanted a child, she wanted him the way a woman wanted a man.\n\nShe pulled back from him and kept her eyes on him as she moved to get off his lap. He moved to pull her back to him but she shook her head. He frowned at her. She dropped to her knees between his knees. Reaching out, she pulled his T-shirt over his head and smiled.\n\n\"If you remember, I told you last night there is something I want to do to you,\" she whispered as she unhooked his belt buckle and opened the snap of his jeans. She slowly lowered the zipper and moved her hand inside, wrapping it around his hard cock. She heard him groan.\n\n\"Lift your butt a bit,\" she ordered him with a grin. He laughed but did what she told him. She pushed his jeans, along with his boxer briefs, down to his thighs. She leaned forward and kissed the head of his hard eager cock.\n\nRyder fisted his hands in her hair, the sensation thrilling and encouraging her to run her tongue along the length of him from top to bottom. She cupped his balls in her hand as she put her mouth down over him as far as she could. She moaned and he shuddered. Dropping her hand from him, she gazed up at him as she sucked on him. His eyes met hers. Her mouth was making his shaft slick. She sucked hard and his head went back as he moaned out her name.\n\nKelsey had honestly never enjoyed this before but with Ryder, she wanted to make him come apart for her just as she had for him. She felt his thighs tremble beneath her arms and she loved that she was making him weak in the knees.\n\n\"Kelsey, you have to stop, darlin',\" he groaned through gritted teeth. He was holding back and she didn't want him to.\n\n\"Just let me do this, Ryder.\"\n\n\"I'm too close.\" He tried to pull back, his fingers tangling in her hair.\n\n\"Good.\"\n\nRyder blew out a laugh. \"No, not good. I want to be inside you.\"\n\n\"Later,\" she told him, and then took him deep into her mouth and sucked hard.\n\nHe stiffened as a low groan tore from his chest and he came. Kelsey licked him clean and kissed her way up his chest to his lips. He deepened the kiss and pulled her tightly against him.\n\n\"You didn't have to do that,\" he told her with a grin. She knew he enjoyed it.\n\n\"I know. I wanted to.\" She leaned in and kissed him again.\n\n\"Well, thank you. It was amazing. Now I owe you.\"\n\n\"No. I owed you...for last night.\" She smiled.\n\n\"I think we need to head to the bedroom and work up our appetites for lunch.\"\n\n\"Can we get in that big Jacuzzi of yours first?\"\n\n\"Sure.\"\n\n\"Can you make a fire in there?\"\n\n\"Whatever you want, darlin'. Come on.\" She pushed to her feet and watched him stand, pulling his jeans up but leaving them undone. He took her hand, and led her to the bedroom.\n\nOnce they reached the bedroom, she entered the bathroom and started the water in the tub while Ryder added logs to the fireplace. The fire had gone out earlier, so he had to light it again but she could hear the snap and crackle of the wood so she knew it was strong. While she stripped off her clothes, she watched the blue and orange flames flicker around the logs. She stepped into the tub and sank down into the water. The hot water felt amazing after being out in the cold even though she'd warmed up by the fire. She watched as Ryder stripped off the rest of his clothes before stepping into the tub. He turned the jets on and Kelsey moaned at the sensations.\n\nRyder moved closer to her and kissed her. His mouth moved across her cheek to her ear where he tugged on her earlobe. \"Sit on the edge of the tub, baby,\" he whispered in her ear.\n\nShe opened her eyes wide. \"What?\"\n\nHe smiled at her. \"Trust me.\"\n\n\"All right.\" She scooted up and sat on the edge of the tub. The air cooled against her skin but the warmth of the fire behind her was keeping her warm. When he spread her legs and settled between them, she knew what he was going to do. \"Ryder...\"\n\n\"Shhh. Relax,\" he said as he kissed her inner thigh and slowly moved closer to his target.\n\nHer legs started shaking as he moved them over his shoulders. She grasped the edge of the tub and moaned when he moved his tongue through her strip of curls. Her head fell back as his tongue moved against her clitoris, and then he sucked on it. She felt him inching his fingers inside her. He moved them against her G-spot as he continued his assault on her clitoris. When the orgasm hit her, she screamed and almost slipped off the tub edge into the water but Ryder caught her and held her against him. Slowly, he lowered her into the tub.\n\n\"Are you all right?\"\n\n\"Oh my God! That was amazing.\" She was breathing hard, almost painfully. She'd never experienced anything like that before.\n\nRyder kissed her. \"I love how you taste.\" She blushed tasting herself on his lips. He ran his fingertip over the scar on her thigh. She stiffened up. \"Don't be embarrassed by this. It's a part of you and there is nothing ugly about it. That man who told you that was an ass.\"\n\n\"I know he is, but it's hard to get past it.\"\n\n\"Trust me when I tell you that you are beautiful.\" His gaze traveled over her making her suddenly self-conscious yet feeling exquisite, sexy, and for perhaps the first time in a long time, beautiful.\n\nRyder sighed, brought his eyes up to meet hers, and grinned. \"I'm hungry. I guess I came through with that promise to make sure we worked up an appetite for lunch, huh?\"\n\nHe stood and put his hand out to her. She accepted it and stood. He pulled her to him and kissed her lips quickly. \"What do you say we get some lunch.\"\n\nKelsey nodded. Ryder helped her from the tub and both of them dried off. She wrapped the towel around her. \"I need to go put some clothes on.\"\n\n\"Your robe is fine.\" He winked at her. She smiled and ran from the room. She put her robe on and met him in the living room. He was stoking the fire back to a roaring blaze.\n\n\"I love the way both fireplaces are open on both sides.\"\n\nRyder glanced at her over his shoulder. \"Just seemed like the smart thing to do to keep the house warm, if I needed to.\"\n\n\"It works well. What would you like for lunch?\"\n\nHe grinned at her, making her laugh then he scooped her up into his arms, kissed her hard and she knew exactly what he wanted. They walked to the kitchen, where she fixed lunch, and then spent the rest of the day in bed.\n\n# Chapter Six\n\nThe next day, the weather was indeed warmer, which made the snow start melting away. The sun was high in the sky and shining down brightly. Water dripped from the remnants of icicles hanging from the gutters around the house. Kelsey sat at the kitchen table, tapping her fingers while Ryder finished his lunch. She sighed. She felt Ryder's eyes on her so she shifted her gaze and met his straight on.\n\n\"What is it?\" he asked her.\n\n\"I'm bored. Can we go to town? I'd like to get off this ranch for a while.\"\n\n\"So you hate the ranch?\" His expression told her just how much he took that as a personal affront.\n\n\"No. I love the ranch. It's just that I'd love to see some more of the area. I didn't get to see much when we went to the party the other night.\"\n\n\"Oh. All right. I guess we can go to dinner in Hartland tonight if you'd like and see a movie.\"\n\n\"Hartland?\"\n\n\"Next town over. It has a nice restaurant. The only thing in Clifton is the diner.\"\n\nKelsey smiled. \"Oh, I'd like that very much.\"\n\n\"Good. We'll leave here around six. I'll call and make a reservation.\"\n\n\"Is it a fancy place? I just need to know what to wear.\"\n\n\"It's not really fancy. I've seen women in there, wearing dresses, jeans, or dress slacks. So wear whatever you want.\"\n\nKelsey was thrilled to be going out. They hadn't been anywhere since the party and as much as she loved the ranch and being with Ryder, she needed to get out before she went stir-crazy.\n\n****\n\nRyder walked in the back door at five to get a shower but when he entered the living room to find Kelsey on the sofa watching TV, he stopped to stare at her.\n\n\"Aren't you going to get ready? We leave in half an hour.\"\n\n\"All I have to do is get dressed. I'm waiting on you.\"\n\n\"Give me a few minutes to shower.\" He walked down the hallway to his bedroom, stripped out of his clothes, and entered the bathroom. He couldn't help but think it was a shame she only had to dress because he would like to have tempted her into taking a shower with him.\n\nDamn, his dick started to rise just at the thought of her in here with him. He really liked shower sex and frankly, didn't get the opportunity much since he didn't tend to hang around at any woman's place long enough.\n\nShit! His damn cock was more than ready for her. Lowering the temperature of the water some, he placed his hands on the wall under the showerhead and watched as the day's dirt swirled down the drain. After quickly washing his hair, he stepped out of the stall and grabbed a towel. After drying off, he proceeded to the bedroom to dress. His glance going to the bed, the memory of Kelsey there with him was so strong he thought he could smell her in the room as if she was still there. He stared at the bed, and a wave of guilt rushed over him. He knew he'd let things get too out of control. As attracted to Kelsey as he was, things needed to slow down. He needed her to understand how it was.\n\nRyder pulled out clean jeans and a black T-shirt, and after dressing, he headed toward the living room. He came to a dead stop when he entered the room. Kelsey stood in the center of the room wearing a short black dress that fit like a glove. She had her hands clasped in front of her. On her feet were black stilettos. She walked to him and stopped in front of him.\n\n\"Do I look all right?\" she asked him. In her heels, she was almost as tall as he was.\n\nHe ran his gaze down over her. The dress was a good bit above her knees. He knew she had great legs, but they looked even better in the dress, and even longer. Her hair was in waves over her shoulders and her makeup was perfect. Her lips were full with red lipstick. He closed his eyes and groaned fighting the urge to scoop her into his arms, carry her to the bedroom, strip that dress off and make her scream with delight.\n\n\"Do you look all right?\" He swallowed hard. \"You look great. Perfect for dinner.\" He reached out to touch her hair but then pulled back. \"I'll have to keep the guys from wanting to peel that dress off of you.\"\n\nKelsey laughed softly. \"Oh, don't worry. You're the only one I'd like to have peel it off me. You can do that when we come back. In the meantime, you promised me dinner and a movie.\"\n\nWhen he didn't say anything, Kelsey looked at him. \"You look yummy too.\" She lightly kissed his lips.\n\nRyder pulled away, and stepped to the chair where she'd laid her coat. He shook his head and held her coat for her. \"Let's get it done with then.\"\n\nKelsey frowned at him. \"I'm sorry. You really do look so good, Kelsey.\"\n\nShe nodded and slipped her arms into her coat. He lifted her hair from her coat, and pressed a soft kiss against the back of her neck. He felt her tremble and tense. He wanted her it was true, but what then?\n\nShe turned to face him. \"You're being bad, Ryder.\"\n\nRyder laughed wishing it were that easy. \"Okay, let's go. It's about a twenty minute drive.\"\n\nOnce they arrived, a hostess showed them to their table and handed them menus. A few moments later, a server came to take their order for drinks while they perused the entrees.\n\n\"Any ideas about what you want?\" Kelsey asked him, her eyes roaming the menu. When she glanced up their eyes met and he knew she was thinking about something other than food.\n\n\"It's been a while since I ate here last but the steak is good,\" he answered before taking a sip of water.\n\nKelsey quickly glanced around. \"Right, steak?\"\n\nThe server came back to take their orders and both decided to with the steak.\n\nKelsey smiled at him after the server left. She leaned forward. \"I'm not wearing panties under this dress.\"\n\nRyder choked on his beer, almost spitting it out. He glared at her.\n\n\"You're telling me this now, why?\"\n\nKelsey nodded her head and laughed. He sat back in his chair and stared at her. She smiled at him and fluttered her eyelashes. He groaned.\n\nWhen their dinner arrived, Ryder picked up his knife and fork.\n\n\"We need to talk, Kelsey,\" he said without looking at her and trying not to imagine what was not under that dress.\n\nAs if she thought he was playing with her, Kelsey cut her steak, gazed at him, put the piece of meat in her mouth slowly, and chewed even more slowly. Closing her eyes, she moaned. \"This is so good.\" She drew the word so out very long. \"I need to savor the flavor.\"\n\nRyder dropped his fork on the plate. Everyone in the restaurant turned to look at him. He glanced around then back to Kelsey. He closed his eyes a moment then took a long pull on his beer.\n\n\"You're gonna regret that, darlin'. Trust me.\"\n\nKelsey laughed. \"Can't wait.\" She leaned across the table, showing her cleavage. \"Bring it on.\"\n\nRyder hissed out a breath. \"I'm serious, Kelsey. We've got to stop playing games.\"\n\nShe sat back in her chair and folded her arms across her chest. \"Is that what we've been doing, Ryder, playing games?\"\n\nShit! He hated confrontation and complications, and right now, he was up to his ass in both. He cut a piece of meat and put it in his mouth. He chewed it, not really tasting it. He'd lost his appetite.\n\n\"So what are we doing, Ryder,\" Kelsey asked pushing her food around on her plate.\n\n\"I told you from the start that I wasn't looking for anything serious and that I didn't feel right about taking advantage of you when you're supposed to be under my protection. The last couple of days...well, they've been...it shouldn't have happened.\"\n\n\"Is this why you offered to take me out to dinner and a movie? To placate me? Oh, take her to a nice place and lower the boom on her, she won't make a scene in public,\" Kelsey hissed. He knew that look, and that tone. He was screwed now\u2014royally.\n\n\"I hadn't planned anything of the sort, I simply wanted to take you someplace nice,\" he said without looking at her. \"But then you told me that you weren't wearing any...you know.\" He waved his knife in her general direction.\n\n\"Oh, so that bothers you, Ryder. Why? Are you worried some other man is going to...what? Smell my desire for a strong man between my legs. Well, I thought that was you, but maybe I was wrong. You've turned out to be a pussy after all. I'm not interested in anything long-term, forever or whatever it is, you think I want. I just want your hot hard cock,\" she lowered her voice to a sexy hoarse whisper on those last words and that was all it took him to tell his conscience to just fuck it.\n\nHe jumped up and grabbed her wrist. \"Damn it, we're leaving.\" He paid the check and pulled her along behind him. She practically ran to keep up with him.\n\nThey didn't make it to the movie. In fact, if Ryder hadn't kept his eyes glued to the road in front of the truck he would have pulled over to the side and had her right there. As it was, he drove them home as fast as he could without getting into trouble. Even though he had convinced himself that he wasn't going to give in to his desire for her again, he had to have her.\n\nHe steered the truck into his driveway and didn't bother closing the gate behind them. Pulling up alongside the house, he slammed on the brakes, shut off the engine, and climbed from the truck. Kelsey was already out when he got around to her side, but barely. She smiled at him and he wanted to take her right there. He slammed the truck door, grabbed her hand, and nearly dragged her to the back door. He threw the door open, was greeted by the dogs, who he ushered outside, and pulled her into the house. Without taking off their outer clothes, except his hat which he tossed toward the hook on the wall and made Kelsey squeal with delight when it caught and stayed hanging there, he led her to his bedroom, and then to the bathroom.\n\n\"You need a shower?\" she exclaimed with a bold laugh.\n\n\"I've been thinking of shower sex with you since I had a shower earlier. I want you, and I want you now and in the shower. Get your clothes off,\" he said as threw off his coat, followed by his shirt off over his head. He reached for the snap of his jeans but stopped when he saw her standing there staring at him with a pouty mouth and a woeful expression.\n\n\"What?\"\n\n\"Where's the foreplay? The romance?\"\n\n\"Seriously? I'm so fucking hard for you right now that I might not make it out of my clothes.\"\n\nShe placed the back of her hand against her forehead and in an exaggerated southern voice said, \"Ah do declare, that is the sexiest thing ah have ever heard.\"\n\nRyder stared at her. \"Christ, you're a smart ass.\"\n\nKelsey laughed, and then squealed when he spun her around, yanked off her coat, and then lowered the zipper on her dress. He pushed it down her arms to her feet. He turned her back around and realized just how near naked she was all this time. She was standing there in nothing but those hot stiletto heels. He allowed his gaze to travel slowly over her flushed body.\n\n\"You are so beautiful,\" he whispered as he lowered his face toward hers. \"Kiss me,\" he said against her lips.\n\nShe pressed her lips to his and moved her tongue into his mouth. He groaned when he felt her lowering the zipper on his jeans. She pulled back from him and stared up at him.\n\n\"You weren't kidding. You're so fuckin' hard that I can't get the zipper down.\" She smiled up at him.\n\n\"Fuck it,\" he said as he reached between them and lowered the zipper. Then he pushed his jeans down. After toeing off his boots, he removed his jeans and boxer briefs. His hard cock sprang forward. When she wrapped her hand around it, he growled low in his throat.\n\n\"In the shower, woman. Now,\" he ordered, and then reached down and into the back pocket of his jeans. He retrieved a condom while he watched her step from her shoes. \"I'd like to play with those later,\" he said pointing at the heels. With a grin, he turned the shower on, then stepped in, and pulled her against him, taking her lips in a deep kiss. \"I can't wait. I have to have you.\"\n\n\"I want you too. Please...\"\n\nRyder pushed her against the tiles of the shower and placed his hands under her butt, lifting her. Her legs wrapped around him. He moved his hand between her legs to her wet, hot folds.\n\n\"Fuck. You're so wet.\" He handed her the condom, which she rolled down over him, making him grit his teeth. Then he moved closer and inched into her. Both of them groaned as he slid into her hot depths. \"Hard and fast, darlin'. This time it has to be hard and fast.\"\n\nHe gazed into her eyes to see if she agreed. When she nodded her head, he pulled out and slammed back into her making her gasp. Her arms wrapped around his neck as she held on while Ryder pounded into her. He knew he wasn't going to last so he hoped she came before he did. When she bit his shoulder, he felt her clench around his cock and so moved faster against her. Kelsey screamed out right before he groaned out her name. Coming hard. He leaned his forehead on her shoulder, trying to catch his breath. She was breathing hard too.\n\n\"Damn, that was good.\"\n\n\"Yes,\" she whispered against his neck.\n\nRyder pulled back and looked into her eyes. \"Are you all right?\"\n\nShe smiled at him. \"Wonderful.\"\n\nHe chuckled. \"More than wonderful, sweetheart.\" He let her down slowly. When her knees buckled, he caught her. \"Let me wash you off.\" He picked up the soap, created a thick, foamy lather, and then washed her. When he moved up her legs to her strip of curls, he felt himself getting hard again. Is that even possible? He looked at Kelsey and saw her looking at his growing cock. She raised her eyes and looked at him with a sly smile on her lips.\n\n\"What?\"\n\nShe burst out laughing. \"Do you think you should bother washing me, when you're ready to go again?\"\n\nRyder grinned. \"I suppose I can wash you later,\" he told her as he stepped from the shower, and then reached his hand out to her. She took it and stepped out with him. He grabbed a towel and dried them off before leading her to the bedroom. He watched as she climbed onto the bed. He pulled a condom from the drawer beside the bed and sheathed himself.\n\n\"That is so sexy,\" she said.\n\n\"What is?\"\n\n\"Watching you putting a condom on.\"\n\nRyder climbed onto the bed beside her and lay out on his back. \"I want you to ride me.\"\n\nKelsey straddled his waist and lowered herself down over him. He placed his hands on her hips and tried to lift her.\n\n\"No. I'm in charge on this one so behave, Ryder.\"\n\nHe growled but took his hands from her hips and placed them on her thighs instead. She started to raise and lower herself on him. He clenched his jaw as she moved slowly, wishing he could replace his hands on her hips to guide her.\n\n\"I can't take this,\" he gritted out through his teeth, right before he flipped her to her back and thrust into her. He pulled out, rolled her to her stomach, and lifted her hips. He slowly inched into her from behind. \"Get on your hands and knees.\"\n\n\"Oh, I've never done it like this, Ryder.\"\n\n\"Do you want me to stop? I will.\"\n\n\"No, please don't stop,\" she groaned out and thrust her hips back at him.\n\nRyder slammed in and out of her, and then sat back on his heels pulling her into his lap. She immediately took over. It wasn't long before he felt her body clenching around his cock, and then she cried out. He pushed her forward and pounded into her. Hard. His orgasm, when it grabbed him, had him seeing stars. He collapsed on top of her sweat-slicked back.\n\n\"You just couldn't let me be in charge, could you?\" she asked in a breathless voice.\n\nHe chuckled. \"Next time. You were killing me.\" He rolled off her. \"I think we both need to shower and then get some sleep.\"\n\n\"I'm taking a shower first,\" she said as she climbed out of the bed without even a glance.\n\n\"We can't shower together?\"\n\n\"No, not this time. I'll be right back.\"\n\nHe nodded even though she couldn't see it. He knew it was better he didn't join her. \"I'll be right here.\" In the meantime, he'd just rest his eyes while he waited.\n\n****\n\nKelsey sat in the window seat and stared out the front window. It was now the second week of November and more snow would be moving in. Where had October gone? She'd been here a month and since the first time they had sex, she'd slept with Ryder every night. He'd hold her close to him all night but nothing more. No words, not even a clue of how he felt about her other than the occasional bout of silent treatment. Ever since that night at the restaurant, he hadn't told her again that it was a mistake but then, she'd brought that on by telling him she hadn't wanted more. That he could be any man. Now, thoughts of him filled her brain even when she tried not to think of him. She was feeling more than she should. She was falling in love with him and if he knew, he'd send her packing.\n\nShe heard the back door open and close. She knew Ryder had just come in from working in the barn. The man never seemed to stop. Just then, the phone rang and she listened as he answered.\n\n\"Wolfe Ranch.\" A pause. \"Hey Dad, how are you?\"\n\nKelsey could hear the smile in Ryder's voice. She stood and walked to the kitchen and leaned against the doorjamb, watching him. As he talked, he shrugged out of his coat, strolled to the mudroom to hang it up, and removed his hat, hanging it on a hook in the hallway as he returned to the kitchen. When he saw her and smiled at her.\n\n\"No. Don't worry about that. Go ahead and have fun. I wouldn't be able to anyway.\" He poured himself a cup of coffee. \"Kelsey's fine. Probably a little bored, but fine.\" He winked at her, and then grinned when she blushed.\n\n\"Can I say hi?\" she asked with an eagerness she hadn't expected. She wasn't sure if she wanted to know if things had settled enough to go home or not.\n\n\"Hold on, Dad.\" Ryder handed her the phone and leaned a hip against the counter. He poured a cup of coffee and sipped as he watched her.\n\nShe put the phone to her ear. \"Hi, Uncle Frank.\"\n\n\"Hi, Kels, hon. How are things going?\"\n\n\"Things are going good. Seriously, I am not as bored as Ryder told you,\" she told him with a grin.\n\n\"That's good, hon. I'm sorry to say there's not much going on here with regard to Dowling either. They haven't found him yet, so he could still be looking for you. As much as I'd love for you to come home, we can't take the chance of you coming home yet.\"\n\n\"I know. It's okay, really. I'm doing fine, really I am.\"\n\n\"I'm hoping we can get you home in time for Christmas.\"\n\nShe was torn about going home. She didn't even want to think of that right now. \"That sounds good.\"\n\n\"Ryder's treating you well, then, is he?\" Frank laughed as if surprised by the idea.\n\nKelsey glanced at Ryder. \"Ryder's treating me very well. He's been most hospitable in making me feel comfortable.\" She grinned when she saw him raise an eyebrow. \"He's very good at what he does.\"\n\nRyder grinned and glanced away. She recognized the guilt attacking him.\n\n\"Are you talking about the wood he works with?\"\n\n\"Yes. I'm talking about his...wood.\" Her eyes traveled down to Ryder's fly.\n\nRyder inhaled quickly then choked on his coffee. He coughed a couple of times and grabbed the phone. \"Dad? Don't worry about Thanksgiving. You and Mom go and have a nice time.\"\n\nKelsey burst out laughing when he hung the phone up. She started backing up when Ryder strode toward her.\n\n\"You love being bad, don't you?\" he asked as he advanced.\n\n\"I was bad?\" she asked widening her eyes in an innocent manner. \"I just said you were good with your wood.\"\n\nRyder growled and reached for her. She spun around and ran from him. He caught her in the living room and pulled her down onto the sofa. He pressed his face in her neck and nibbled. She combed her fingers through his hair and pulled his mouth to hers. When his tongue entered her mouth, she moaned and then pulled back from him.\n\n\"Wait. What was that about Thanksgiving?\" she asked.\n\nRyder sighed. \"Since I'm not going home for it, they're taking a cruise.\"\n\n\"You can't go home because of me. Oh Ryder, I'm sorry. You should go. I can stay here. My parents are going to Canada.\"\n\n\"No. It's fine. I think they'd rather go on the cruise. I could hear the excitement in his voice.\"\n\n\"Should I cook a turkey for us? What about the men?\"\n\n\"The men all have homes to go to. I'll call Emma. Every year she invites me, but some years I go home. Sometimes, I'm just too busy to go home and I go eat with Emma and Gabe. The other Stones and their wives will be there too.\"\n\n\"They don't have family?\"\n\n\"They're the only family they have. Emma's parents are still alive but they usually go to Colorado for skiing.\"\n\nKelsey glanced away from him and bit back a smile. \"What about Sam?\"\n\n\"What about Sam?\" Ryder asked pulling back, a frown darkening his face.\n\n\"I was just wondering if he'll be there.\"\n\n\"No. He'll be having dinner with his parents and sister and probably the woman he's been seeing.\"\n\nKelsey pushed out her lower lip and when she heard Ryder growl low in his throat, she burst out laughing. \"I'm teasing you. I told you before none of them have anything on you, although if I had to choose someone other than you, I'd have a tough time choosing between Sam, Wyatt, and Trick. Sam's blue eyes are simply gorgeous.\" She wondered if he was truly jealous. She found herself hoping so.\n\n\"I sure as hell hope you never see Riley then.\"\n\n\"Ooh, so Riley's hot too?\"\n\n\"He must be. He has no trouble getting a woman, although he doesn't seem to keep 'em long.\"\n\n\"I seriously doubt any of you would, Ryder. And maybe he's just got fear of commitment issues...like you.\"\n\nHe frowned down at her. \"No comment.\"\n\nShe leaned toward him. \"Just shut up and kiss me,\" she whispered against his lips.\n\nKelsey felt him grin against her lips before pressing his to hers and sliding his tongue between hers. He hadn't denied his fears, but he also hadn't let guilt over what Frank would think of what they were doing stop him. That one small victory satisfied her as he picked her up and carried her away to the bedroom.\n\n\"I know a great way to spend the afternoon.\"\n\n\"Don't you have work to do?\" She pressed her lips to his neck, making him falter a bit in his pace.\n\n\"I'm the boss. I'm taking the rest of the day off,\" Ryder told her as he entered the bedroom and kicked the door closed.\n\n# Chapter Seven\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nThanksgiving morning was cold, blustery, and snow was starting to fall. Ryder and Kelsey were getting ready to leave for Emma and Gabe's house.\n\n\"I'll start the truck so it's warm. They only live a few miles from me, but it will still be cold.\"\n\n\"All right. I can't wait to see them all again. I really enjoy Becca, Emma, and Olivia.\"\n\n\"They're great women. Liv tends to have a potty mouth, so prepare yourself for it.\" Ryder laughed and then went outside.\n\nKelsey got her coat from the mudroom and while she waited for him to come back inside, she stepped to the back door and peered out. She watched as he cleaned the truck off and couldn't help but notice the snow was getting heavier. He bounded up the steps and entered the kitchen.\n\n\"It should be warm enough now,\" he told her, knocking snow off his hat.\n\n\"You're sure it's okay to go out in the snow?\"\n\n\"This isn't much really. My truck can handle this and more. Trust me.\" Ryder took her coat from her hands. \"Of course, if you're worried, we can stay home.\"\n\nShe glanced up at him. If he were so confident, she wouldn't question his ability to drive safely in the snow. She shook her head, and smiled. \"Let me get the cake.\"\n\nShe had made a chocolate cake although Emma told her it wasn't necessary for her to bring anything. Ryder helped her put her coat on, and then she picked up the cake from the counter and they headed out into the snow and to the Stone home.\n\nWhen Kelsey and Ryder arrived at the Stone ranch, the snow had really begun to collect on the ground but the roads were still clear. As they entered the kitchen, the delicious scents of Thanksgiving filled their noses, and all eyes turned to greet them.\n\n\"What? Is my blouse buttoned wrong?\" Kelsey asked feeling suddenly self-conscious.\n\nEveryone laughed. Ryder took her coat and his to the mudroom to hang them up.\n\n\"I seem to remember another couple who's been late for Thanksgiving dinner.\" Becca and Emma looked at Wyatt.\n\n\"I have no idea what you're talking about.\" Wyatt grinned at them.\n\nKelsey was happy she'd made the cake since Becca brought pumpkin and apple pies, and Olivia brought Jell-O, the only thing she could make. It seemed the family, especially Wyatt, knew she couldn't cook worth a damn, but they all loved her anyway.\n\nShe glanced around the big kitchen. Gabe and Emma had a gorgeous home. Ryder had told her it was the original farmhouse where Gabe's parents had lived. When they passed away, Gabe took the house along with eleven hundred acres. He'd remodeled the kitchen with new black appliances.\n\n\"Let me show you our home, Kelsey.\" Emma took her arm and led her from the kitchen. They walked down a hallway to the front foyer. The entry door was wood with leaded glass, and sidelights on each side of it. The floors were dark cherry hardwood. Emma guided Kelsey to the left and they entered the living room. Dark green carpeting covered the floor. The fireplace had a dark cherry wood mantel, and there was a fire blazing in it. There were windows on each side of the mantel.\n\n\"We had this room carpeted after the baby was born. She plays in here a lot, so it was more cushion for her if she falls.\" Emma led her across the foyer to the dining room. A large mahogany table sat in the center with place settings. A large window on the sidewall, with a window seat, and a china cupboard on another wall, a long buffet ran along the other wall. A doorway led back to the kitchen. They went back through the foyer and up the carpeted stairs. Emma showed her the rest of the home. Kelsey fell in love with it. All of the bedrooms had working fireplaces in them.\n\n\"I envy you, Emma. It's a beautiful home and one that's obviously filled with love.\"\n\nEmma smiled at her. \"Thanks. I love it too.\"\n\nThey headed back downstairs. The men had gathered in the living room to watch football and the women were in the kitchen. Sophie, Emma and Gabe's daughter, was playing in the living room. The kitchen was buzzing with the women talking and laughing. Olivia hugged Kelsey when she joined them.\n\n\"I'm so glad you came. You'll love having Thanksgiving here.\"\n\n\"I'm sure I will. I'm glad my parents decided to go on a vacation, and Frank and Grace went on a cruise.\"\n\n\"I invite Ryder to join us every year. I'm never sure if he can take the time to go home to see Frank and Grace. He's been coming here a few years.\" Emma smiled at her.\n\nThe women worked together to finish getting the food ready and had just pulled the turkey out of the roaster when Wyatt entered the kitchen. He wrapped his arms around Olivia from behind and kissed her neck. She leaned back into him, putting her hands over his. He reached around her to grab a piece of turkey.\n\nBecca smacked his hand away and glared at him. \"That's why you came out here...to steal some food.\"\n\n\"Not the only reason.\" Wyatt grinned and kissed Olivia again, and then he left the room.\n\nOlivia sighed. When she looked at the other three women, she grinned. They all burst out laughing. Emma was shaking her head.\n\n\"You two amaze me. It's like you're not even married. Don't get me wrong. Gabe and I love each other immensely, but with Sophie, we don't get much privacy. You two have been married for five months now, is the honeymoon ever going to end?\"\n\nOlivia laughed. \"Are you serious? With that man...no, it's never going to end. It took me a long time to get him and I plan to make him happy every day.\"\n\nBecca groaned. \"Good Lord, Olivia. Does he ever get any rest at all?\"\n\nOlivia smiled. \"Not if I can help it.\" Making them all laugh again.\n\nKelsey was having so much fun. She was quickly coming to love these women.\n\n****\n\n\"What's so funny out there?\" Gabe obviously wanted to know, cocking his head in the direction of the kitchen to try to hear the conversation.\n\n\"I'm not asking. You get Emma, Becca, and Olivia together and there's no telling.\" Jake raised his hands and glanced at Ryder. \"Now they've included Kelsey in the mix.\" He shook his head. \"Not good, Ryder, not good at all.\"\n\nThe men laughed. Ryder stood to get another beer. Jake, Gabe, and Wyatt begged him to bring them each one too.\n\n\"Jesus, how many hands do you think I have?\" Ryder stared at them.\n\nWyatt got up. \"I'll go too. I need to see what Olivia's up to anyway.\"\n\nWhen the men walked into the kitchen, the women quit laughing and drew quiet. Wyatt and Ryder looked at each other then back to the women. Becca reacted first.\n\n\"You're just in time. Can one of you carry the turkey to the table?\"\n\nWyatt grinned, picked up the platter, and carried it to the dining room. Each of the women picked up something and carried it to the table too. When Kelsey came back into the kitchen, Ryder was still standing there, waiting for her. When she smiled at him, he stepped forward, cupped her face in his hands, and kissed her. He felt her hands slide along his waist.\n\nRyder raised his head, and grinned when Becca, Emma, and Olivia walked back into the kitchen. \"Sorry, but I needed that.\"\n\nHe kissed Kelsey again quickly, and then left the room carrying bottles of beer.\n\nAfter returning to the living room, he handed each of the men a beer and told them the food was ready. Ryder followed the others into the dining room and took a seat beside Kelsey. He watched as Gabe carved the turkey. Ryder chuckled when he heard Kelsey's stomach growl.\n\n\"I'm starving and it all smells so good,\" she whispered rubbing her belly.\n\n\"Don't worry. Emma's one hell of a cook. She used to cook at the Clifton B and B until she got pregnant.\"\n\n\"Their daughter is just beautiful.\" Kelsey turned to Becca. \"How old is your son? He's such a handsome little fella.\"\n\n\"Of course, he's handsome, he's a Stone,\" Olivia said, acting the proud aunt.\n\n\"Can't argue with you there.\" Kelsey glanced at Ryder and grinned when he groaned at yet more adulation for the Stone brothers' good looks.\n\n\"Will is six months old,\" Becca said and laughed when Will slapped his hands on the highchair tray. When Sophie saw him doing it, she did the same thing.\n\n\"Remember when Thanksgiving dinners were quiet?\" Jake said above the noise.\n\n\"Quiet? How about the year, we had the food fight?\" Gabe laughed.\n\n\"I missed that one,\" Wyatt said narrowing his eyes at Olivia who glared back at him.\n\n\"Oh, yes. I remember it very well,\" Olivia remarked.\n\nWyatt grinned at her. \"I really pissed you off that day, didn't I, sweetheart?\"\n\n\"I'm still mad about it.\" She huffed. Wyatt put his arm around her and whispered in her ear, making her blush and laugh.\n\n\"What did he do to you, Liv?\" Kelsey asked.\n\n\"He didn't show up at all. It broke my heart.\"\n\n\"There was nothing going on between us then, darlin', so how could I break your heart?\"\n\n\"I know, but it did hurt. I was so looking forward to having you there that evening.\"\n\nWyatt kissed Olivia's temple. \"We have years of Thanksgiving dinners ahead of us to look forward to.\"\n\nOlivia smiled, and threw her arms around his neck.\n\n\"Holy shit. She honestly fell for that line?\" Gabe said.\n\n\"Daddy?\" Gabe gave his daughter his full attention. \"What's holy shit?\"\n\n\"Something you are never to say again, Sophie Ann Stone,\" Emma said, and then she looked at Gabe with a frown. \"Gabriel Stone, you know better.\"\n\nGabe looked embarrassed. Jake burst out laughing, and everyone joined in, including Emma and Gabe.\n\nKelsey stood. \"I need a refill on my drink. Anyone else?\"\n\nEveryone wanted refills, so the women headed to the kitchen to fill drink orders.\n\n****\n\nOnce they were in the kitchen, Olivia spun around, hands on hips, and narrowed her eyes at Kelsey.\n\n\"What?\" Kelsey asked, feeling everyone's eyes on her.\n\n\"Seriously? You're going to ask me what? You and Ryder. That's what,\" Olivia exclaimed.\n\nKelsey smiled. \"What can I say? We like each other.\"\n\nEmma burst out laughing. \"Like? Looked like more than just like to me.\"\n\nKelsey shrugged. She wasn't sure how Ryder would feel having all his friends know his business.\n\n\"How long has this been going on?\" Becca wanted to know.\n\n\"A few weeks now. One night, we got into it hot and heavy and the next thing I knew he was pulling away from me, telling me Frank wouldn't like it. I was so mad at him that night.\"\n\n\"I would have been too. So, did you do the nasty?\"\n\n\"Olivia Rene Roberts Stone,\" Becca scolded.\n\n\"Oh please. You want to know too.\"\n\nBecca laughed. \"Of course I do.\"\n\nKelsey laughed. \"Well, eventually we did but only because I challenged him. I called him a pussy.\"\n\nOlivia burst out laughing. \"I love this woman.\"\n\n\"We'd better get back. They'll start wondering what we're up to,\" Emma said.\n\n\"We're already wondering what you're up to,\" Gabe said as he stepped into the kitchen.\n\nThe women jerked in surprise, and then laughed. Emma walked over to him and stood on her toes to kiss his lips. He pulled her closer and wrapped his arms around her, pressing a kiss to her neck. He stepped back, and grinned.\n\n\"We should get back to dinner,\" he told them. Kelsey thought she saw a blush rise on his manly cheeks but perhaps it was the lighting. As a group, they nodded and Gabe helped them carry the drinks to the dining room.\n\nAn hour later, everyone sat in the living room, groaning. Olivia was lying on the floor in front of the recliner Wyatt was sitting in.\n\n\"Why did you all let me eat so much?\" she groaned, rubbing her tummy.\n\nWyatt stood, leaned down reaching for her. She put her hand up to his, and he clasped his large hand around her arm. When he pulled her up, he tossed her over his shoulder making her squeal.\n\n\"I'm sure we can work those calories off in some constructive way, sweetheart. Later everyone,\" Wyatt exclaimed as he carried his wife out of the living room. Olivia laughed, raised her hand, and waved at everyone.\n\n\"I think we'll head out too,\" Jake announced, and Becca nodded in agreement. She stood, gave everyone a hug, and went to get Will into his coat.\n\n\"Are you ready to go?\" Ryder asked Kelsey.\n\n\"Shouldn't I help Emma clean up?\"\n\n\"No. Go home. Gabe will help me,\" Emma told her with a bold grin as she hugged her husband around the waist.\n\n\"I will?\" Gabe's eyebrows rose in surprise.\n\n\"Yes you will, Gabriel, or you won't be working any calories off.\"\n\n\"Well hell. Sorry Ryder, uh...Kelsey, you both have to go.\"\n\nRyder and Kelsey laughed as they headed into the kitchen to retrieve their coats. After hugging Emma and Gabe, Kelsey took Ryder's hand, and together they ventured out into the snowy Thanksgiving evening.\n\nThey drove home in silence since Ryder had to concentrate on the slippery roads. Ice had mixed with the snow and Kelsey could hear it pinging on the roof and bouncing off the windshield. It made her nervous. She grabbed the handle above the door when the truck fishtailed slightly on the slippery road. She glanced over to Ryder.\n\n\"We're all right,\" he told her without looking at her.\n\n\"Okay,\" she whispered while saying a silent prayer. She'd experienced icy roads back home in Georgia, but not out in the middle of nowhere with several inches of snow in attendance too.\n\nShe blew out a breath she hadn't realized she was holding when he finally steered the truck into the driveway, not bothering with closing the gate, and parked it by the back door. Once the engine was off, Ryder opened his door and stepped out into the cold. Kelsey didn't wait for him, but threw open the door and stepped out. The icy snow stung her face, and she wanted to get inside where it was warm. As Kelsey moved toward the front of the truck, she hit a patch of ice and fell, landing on her butt. Ryder was by her side in an instant.\n\n\"You okay?\"\n\nShe gazed up at him, her pride stinging her. \"My butt hurts.\"\n\nRyder put his head down and she knew it was to hide his grin, but then he gazed at her. \"I suppose I could massage it for you.\"\n\nKelsey frowned. \"I don't know. Are you certified to give butt massages?\"\n\nRyder laughed and helped her up. She rubbed her butt as she entered the house. He helped her remove her coat and after hanging them up, he pulled her to him and kissed her lips.\n\n\"I really don't mind massaging your butt for you. I do like your ass.\"\n\n\"I'll think about it.\" She gave a little huff.\n\nRyder frowned at her. \"Spoilsport.\"\n\nHe took her hand and led her to the living room. After starting a fire, he pulled her down onto the couch next to him. The fire crackled and snapped as the flames reached up toward the flue. Kelsey snuggled against him and felt him rest his cheek on the top of her head.\n\n\"Are you asleep, Ryder?\" she whispered.\n\n\"Not yet. I could be any time though. It was a long day and big meal.\"\n\nKelsey stood and put her hand out to him. He looked up at her, and she smiled. He took her hand and pulled himself to his feet. They walked to his bedroom where he immediately took a seat on the edge of the bed and started to lift his T-shirt over his head but he was so tired, he was having trouble. Kelsey pushed his hands away and lifted it off over his head. She pushed him back onto the bed, pulled off his boots and socks, unbuckled and opened his jeans before pulling them off, quickly followed by his boxer briefs. He lay there looking so magnificent her mouth was watering. His eyes closed. She pushed his legs up onto the bed, and then covered him with a comforter. After kissing his forehead, she then left him there to take a hot bath in the Jacuzzi tub.\n\nLater, when she returned, she turned off the lights and crawled into bed beside him. He was sound asleep. Smiling, she pulled the covers up over his shoulders. She knew he'd gotten up early today to work in the barn before they headed to Gabe and Emma's for dinner. She smiled happily, when he pulled her close to him and wrapped his arms around her.\n\n\"Are you awake?\" she whispered.\n\n\"Uh huh, sorry I fell asleep.\" He kissed the top of her head.\n\n\"It's all right. Frank hopes I can get home by Christmas.\"\n\n\"I'm sure you'll be happy to get back to your home and job.\"\n\nThat wasn't anything like what she'd hoped to hear. She wanted him to tell her he didn't want her to leave. That she should stay in Clifton. That she should stay with him. She blinked tears away from her eyes.\n\n\"Do you...do you want me to leave, Ryder?\"\n\n\"It's where you belong, Kelsey.\"\n\nShe sat up pulling the covers with her. \"Is it?\"\n\n\"Yes.\"\n\n\"Why?'\n\nRyder sat up and leaned back against the headboard. He rubbed his hands across his eyes and sighed. \"Because this will never amount to anything in the long run. I told you I wasn't interested in getting married and having kids. You're the kind of woman who wants all that and more, and frankly, you deserve to have it. I'm not the one to give it to you.\"\n\n\"You're crazy if you think you're anything like them. You're nothing like them.\"\n\n\"How the hell do you know?\" Ryder shouted.\n\n\"I know you would never hurt anyone physically. Not a woman and certainly not a child. You're not them, Ryder, and never will be. Frank and Grace raised you into the man that you are.\"\n\n\"I have their genes. I've read up on it. It could be hereditary...and I won't take that chance.\"\n\n\"Do you have any idea how stupid that sounds? I'm sure you've gotten angry many times over the years. Have you ever struck out at someone? The first night I met you, that big lug at Dewey's came after you but you didn't hit him, until he took a swing at you.\"\n\n\"You're not going to convince me.\"\n\n\"So, you're going to be alone for the rest of your life?\"\n\n\"If it's for the best, then yes. Let it go, Kelsey.\"\n\n\"I will not. Don't I have any say in this?\"\n\n\"No. Not in this you don't.\" Ryder swung his legs over the side of the bed and stood. He didn't look at her. \"Go home when you can. It's the best thing that can happen here.\"\n\n\"After we\u2014\"\n\n\"Do not go there! I told you from the beginning. Before this even got started. You knew, Kelsey. You knew nothing would ever come of it.\"\n\nShe got out of the bed, pulled her robe on, and walked toward the door. She spun around and glared at him.\n\n\"Yeah, I knew but I didn't plan on this either. I didn't plan on falling in love with you\u2014\"\n\n\"Don't!\" He held his hand up as if to stop her from speaking, but he wasn't going to stop her.\n\n\"Fuck you, Ryder. I will say it. I love you and you're a fool if you let someone like me go.\" She left the room, slamming the door behind her and walked to the bedroom she'd slept in when she first arrived. She crawled between the cold sheets and wept.\n\n****\n\n\"Son of a bitch,\" he growled at the empty room. Damn her. She had to go home, didn't she understand? She might not think he could be like his biological father. But what if he was? He'd want to die if he ever hurt her, or a child. There was no way he was willing to take that chance.\n\nShit! He knew he wasn't going to get any sleep now. He crawled back between the sheets and glanced at the clock. Only midnight. It was going to be one hell of a long night.\n\nHe lay there staring at the ceiling. He decided he'd call Frank tomorrow, hell, later today, and let him know she was returning to Atlanta. Frank would watch out for her. Damn, the only thing wrong with that was, Frank and Grace were on a cruise until the first of December. And she'd said her parents were in Canada, so now what? Well, she'd have to stay here until either her parents, or Frank and Grace returned. Whether she wanted to or not. Hell, whether he wanted her to or not. He couldn't let her go back home and be there alone in case that maniac decided to go after her again.\n\nRyder frowned. He wondered if the creep was he still looking for her. Had he found out where she was? No, how could he? Why would he think she'd gone anywhere? Unless he'd called the hospital and found out, she wasn't there. No one would tell him where she'd gone. Or would they? Were the police still actively looking for him? Or had they lost interest in the case? He couldn't just disappear. He had to be somewhere and someone had to know where he was. Ryder was sure they'd checked with the man's family, so where was he?\n\nSuddenly, he wanted to call Frank and find out more about this jackass. He wanted to know who he hung out with, his friends, family\u2014hell, his enemies. Every fucking thing he could. Kelsey couldn't return home with that maniac still on the loose. She wouldn't be safe. He could just be waiting for her to return but then again, he could have taken off, never to return. Somehow, Ryder didn't think that was the case.\n\n# Chapter Eight\n\n\"I'm going home. I've called the airlines. You can take me to the bus station,\" Kelsey announced as she entered the kitchen.\n\nRyder wasn't in the mood for an argument as he was working on very little sleep.\n\n\"You're not going anywhere until either your parents, or Frank and Grace, get home.\"\n\n\"Nothing is going on. I'm going home. Either you take me to the bus station or I'll get a cab,\" she shouted at him from where she stood her back to the counter, and her arms folded across her chest.\n\n\"I'll tie you to the damn bed if I have to. You're not going anywhere,\" Ryder shouted back.\n\n\"Ha! That's the only way you'll ever get me in your bed again!\"\n\nRyder reached her in two strides and stared down into her face. She closed her mouth and her eyes widened. \"Really? Should I prove you wrong, darlin'?\" he said lowering his voice. He heard her gasp as she dropped her arms to her sides then clasped the edge of the counter as she leaned back.\n\n\"Back away, Ryder,\" Kelsey said between clenched teeth as she glared up at him.\n\nHe grinned down at her, leaning in slightly. \"Afraid I'm right?\"\n\n\"No. I know you can make me want you. I love you, so why wouldn't I want you?\"\n\nRyder stepped back from her, feeling as if he'd been slapped. \"Don't say that.\"\n\n\"It's true and whether you want to admit it or not, you love me too. I know you don't want to have kids, and I'm fine with that, but you'd never hurt a child, or me. I know that in my heart.\"\n\n\"You don't know shit. I'm not having this conversation with you yet again. It's done and over with. I will never get married, and I will never have kids.\"\n\n\"Then I'll live with you. I don't have to be married.\" Kelsey shrugged. \"I only have one close friend in Atlanta, Anne Llewelyn, and I'm sure she'd understand me deciding to stay here with you.\"\n\n\"God damn it, Kelsey,\" he roared wanting to punch something. \"You just don't get it. Just because we wouldn't be married, doesn't mean\u2014\"\n\n\"Bullshit, Ryder,\" she interjected then sighed. \"You know what? I'm the one who's done. I'll stay until the first, when Frank and Grace get home. It's only a week. Don't touch me or talk to me unless it's absolutely necessary.\" Kelsey spun on her heel and started to walk away when she suddenly stopped and turned back to look at him. There was disappointment in her blue eyes and it tore at him to see it there. \"You're a coward, Ryder Wolfe. A great big coward.\"\n\nRyder watched her walk away until she disappeared down the hallway. He winced when he heard the door slam. She just didn't understand. It would most certainly kill him if he ever hurt her, and it wasn't a chance he was ever willing to take. He'd spent so much time on the Internet and in libraries reading about how abusive behavior could be passed down generation to generation that he knew it was very possible he was just like his father. How could he ever take the chance? Shaking his head, he knew he couldn't. He'd live alone for the rest of his life before he'd take the chance of hurting anyone the way his parents had hurt him.\n\n****\n\nA week later, Frank called to let Kelsey know he and Grace were home. Kelsey's heart broke in to another million pieces. She knew the minute she walked out of this house, and Ryder's life, it would break beyond repair. She was sure Frank knew something was up by the way he talked to her.\n\n\"Are you sure you're all right, Kels?\"\n\n\"Yes. I suppose I'm just nervous about coming home,\" She was lying to him and somehow, she knew he knew.\n\n\"You're sure that's all it is?\"\n\nKelsey cleared her throat. \"I'm fine, Uncle Frank. Really. I'm just anxious to get back to my apartment, my job, and my life.\"\n\n\"We're all very happy to have you coming home but if something happens again, you're going right back to Ryder's place. Understood?\"\n\n\"I'm sure I won't be coming back here.\" Her throat was closing up, threatening to choke her. \"I need to get going for now but I'll call you when I know what flight I'll be on. Tell Aunt Grace, I love her. Goodbye.\"\n\nKelsey placed her hands over her face and sobbed. Where was she going to get the strength to call the airlines and make that reservation? When she'd told him she had a week earlier, she hadn't really because she'd hoped he would ask her to stay. She'd hoped she could get him to see that he loved her too but he was the most stubborn man she'd ever met. Now the thought of leaving Ryder and never seeing him again tore at her like a knife through her guts. How was she going to survive without him? No other man would ever completely own her heart because Ryder would always have it.\n\nTaking a deep breath, Kelsey gathered the strength to call the airline and make a reservation. She then called the bus station to reserve a ticket, and then order up a taxi to pick her up first thing in the morning to take her to the bus station. It took every bit of her willpower not to break down over the phone.\n\nShe pulled herself to her feet, took out her suitcases, and began to pack. It was a good thing her mother had sent an extra suitcase when she'd sent extra clothes. The thought of asking Ryder for assistance in getting her belongings home just went against any sense of pride she had left.\n\nKelsey couldn't keep the tears from falling as she packed. The hardest part was going to be telling Ryder she was leaving but she had to do it. She may be angry with him but she couldn't just leave without saying goodbye. She closed the suitcases and after wiping her face off, she left the bedroom to find him. It was time.\n\nShe walked to the living room but didn't see him there, so she headed to the kitchen. He wasn't there either. Opening the back door, she listened. Since it was Saturday, she was trying to hear if she could hear his saws running, but she heard only silence. It seemed strange and made the ache in her heart worse.\n\nGrabbing her coat from the peg, she put it on, and walked outside. The weatherman had said there was more snow moving in tomorrow, but she'd be gone by then. She'd come to love the snow. Ryder thought she was crazy for it, but she enjoyed it. Smiling sadly, she entered the smaller barn but the lights were out. Leaving there, she entered the main barn and after letting her eyes adjust, she walked down the aisle toward his office. Stopping along the way to talk with some of the horses and pet their velvet noses.\n\nWhen she reached the door of his office, she stopped. He was in there. She saw him through the window. She knocked and poked her head in. He glanced up at her. She stepped into the room.\n\n\"I'm leaving tomorrow. I ordered a cab to pick me up and take me to the bus station tomorrow then I'll take the bus to Butte.\" He didn't say anything as he continued to stare up at her. She blew out a breath. \"I'll be leaving first thing in the morning. The cab is picking me up at six.\" Still nothing, he didn't even blink. Kelsey could feel her temperature rising. \"Say something,\" she shouted at him, hoping for some kind of reaction.\n\nRyder stood slowly leaning his hands on his desk. \"There's nothing to say. You told me not to speak to you unless it was absolutely necessary.\"\n\n\"You're a real prick, Ryder Wolfe.\" She brushed a tear away from her cheek.\n\n\"I may be a prick, but I'm an honest one. I told you how I felt, from the very beginning.\"\n\n\"Fine. Goodbye then. Enjoy your life all alone. I may love you until I take my last breath, but I won't be alone. I'll find a man who will love me and give me children because he wants to.\"\n\n\"I'm happy for you. Goodbye.\" He sat back down, picked up a pen, and bent over a sheet of paper.\n\nShe was right. Her heart shattered. Damn him. Damn him to hell. He refused to look up again. He just went on working as if she'd told him the weather report. She turned, threw the door open wide, and ran out through the barn. Once inside the house, she ran to her bedroom, slamming the door once inside. Falling across the bed, she sobbed until she fell asleep.\n\nThe next morning, Kelsey carried her suitcases outside to the waiting taxi. The driver took her suitcases, and then helped her into the backseat. She glanced around but didn't see Ryder anywhere. What did you expect? He told you goodbye. She quickly blinked back the tears and stared straight ahead. She never wanted to see Wolfe Ranch or its owner again.\n\nLiar! Mentally shaking her head, she sat back and watched the countryside passing outside the window of the taxi. As she passed by each Stone property, she wanted to cry for the new friends she'd made but would never see again. She'd come to love the little town of Clifton, Montana. It would be a wonderful place to live, but that dream was gone. Just like Ryder Wolfe.\n\n****\n\nRyder stood in the barn and watched as the taxi took her away from him. It was for the best, he kept telling himself. He knew he couldn't give her what she wanted. She'd been right. He loved her, and he was a coward.\n\n\"Son of a bitch,\" he muttered to himself.\n\n\"You let her go?\" Cookie asked from behind him.\n\nRyder spun around surprised by the man standing there. \"Yes. What the hell was I supposed to do?\"\n\n\"Oh, I don't know. Marry the woman, maybe? You're in love with her, Ryder. Anyone with half a brain can see that.\"\n\n\"Mind your own fucking business, Cookie,\" Ryder growled as he started past him.\n\n\"I can see how it's going to be around here now,\" Cookie said making Ryder stop in his tracks.\n\n\"Yeah, well, if you don't like it, then leave.\"\n\nCookie laughed. \"Like that's ever gonna happen.\"\n\nRyder shook his head. He shouldn't be taking his anger out on anyone, especially a friend. He had the problem, not Cookie. \"I'm sorry. I shouldn't have said that.\"\n\n\"You know damn well I know you didn't mean it. But damn it, Ryder. You love her.\"\n\n\"I can't give her what she wants, Cookie.\"\n\n\"What's that? You? The hell you can't give her what she wants.\"\n\n\"I can't chance it. I'm not good for her.\"\n\nCookie sighed. \"Ryder, you're nothing like that son of a bitch, or your mother. He's a mean ass and she's a drunk. They didn't raise you. Hell, you were too busy raising yourself and taking care of their lazy asses. And when you couldn't do it anymore, Frank and Grace did, and they made you into the good man that you are now.\"\n\n\"I just can't chance it,\" Ryder repeated shaking his head.\n\n\"Then you're going to be a miserable cuss to work with, that's all I'm saying. I've got work to do.\" Cookie walked away down the aisle of the barn shaking his head and mumbling.\n\nRyder watched him walk away. No one seemed to understand what he feared. They hadn't lived what he had. He'd just have to go on without her. He could do it. He had to.\n\nYou sure as hell can't! He took a deep breath in and blew it out. His heart ached and he couldn't make it stop. She was right. He was a stupid fool for letting her go. But he was scared. So fucking scared. What if he lost his temper and hurt her? Physically hurt her. It would be far worse than the hurt he was causing her now. He'd hate himself.\n\nRyder walked through the barn and decided to work on the furniture. Maybe it would keep his mind off her. He had to concentrate completely on the wood, so he knew he'd at least get her out of his mind for a few hours. He dreaded tonight though, when he tried to sleep. Hell, he dreaded walking back into the house at all. It was going to feel empty. He'd never paid attention to being alone before but now, with her gone, it was going to be hell. Damn, he hated that she was so right.\n\nThree hours later, he walked into the house and the silence surrounded him. No food was cooking. No radio was playing. There was nothing but deafening silence. Even the dogs seemed mad at him for sending her away. They lay on the floor with their heads on their paws and stared up at him with sad eyes.\n\n\"What? You knew she wasn't staying.\" Ryder stared down at them. He raised his eyebrows when they both stood, left the kitchen, and headed into the living room.\n\n\"Wonderful. Now my dogs hate me too,\" he said aloud.\n\nHe thought about getting something to eat but he didn't have an appetite. After taking his hat and coat off, he decided to shower. He walked to the bathroom and stripped out of his dirty clothes. He turned the shower on and soon steam filled the room. As he started to step into the stall, he saw her shampoo on the shelf. She'd forgotten it. Fuck!\n\nHe couldn't seem to stop himself from reaching out and picking it up. He flipped open the cap and put it to his nose, inhaling the scent. Christ! It smelled like her. It brought back memories of the nights he'd nuzzled her hair as he lay behind her after making love. Shit!\n\nHe couldn't think of those times. He had to put them out of his head. It was bad enough that his cock was remembering. All he had to do was think of her and his body acted like a divining rod searching for her. He ran his hand down across his face and groaned. Damn. How was he going to get through this?\n\nTaking a deep breath, he scrubbed himself until he was sure he was taking skin off. After washing his hair, he stepped out, and then shaved. He was tired but he knew sleep wouldn't come easy. You did this to yourself, stupid.\n\n\"I know. But I had to,\" he told his reflection in the mirror then switched off the light, headed to the bedroom and crawled between cold and lonely sheets.\n\n****\n\nKelsey stood in the living room of her parents' home and stared out the front window. A fully decorated Christmas tree stood in the corner of the room. A symbol of the season's spirit that usually filled her with joy, the lights twinkling and sparkling but she paid them no mind. She used to love this time of the year but this season filled her sadness because she missed Ryder far too much to feel the joy.\n\n\"I don't think any of the scenery has changed in the last thirty minutes,\" her mother said from beside her as she placed her arm around her shoulder.\n\n\"I know.\"\n\n\"What happened in Montana, Kelsey?\"\n\nKelsey glanced to her mother and then back to the window. \"I wish we had snow.\"\n\n\"Snow? Are you out of your mind?\" Donna Sullivan laughed. \"Atlanta would shut down.\"\n\n\"I know you're right about that but I fell in love with snow in Montana. It was just so beautiful.\"\n\n\"That's not all you fell in love with, is it?\"\n\nKelsey blinked tears away. \"It doesn't matter.\"\n\n\"Of course it does, honey. Tell me.\" Donna took her hand and led her to the sofa where they sat down.\n\n\"I did fall in love...with Ryder but he says he'll never marry, and he never wants children.\"\n\n\"Then move on, Kelsey. I know how hard that seems right now but you have to if you can't have what you want with him. I know how much you've always wanted children. You love kids.\"\n\n\"How can I move on though, Mom? I love him so much.\" Her voice caught and her chest filled with pain. How was it possible for it to hurt so physically much?\n\n\"I know all about him. I know why he doesn't want the same things you do and even if he loves you, he probably won't settle down and give you children. I don't want to sound harsh, but you have to consider the future. What if you did marry him but he still didn't want children? You would become miserable and eventually you'd grow to resent him, even hate him. Unless you are willing to accept that kind of future, you have to move on. Without him. Honey, I want you to be happy and I don't think you will be, if you don't get a chance to have kids.\"\n\n\"You're right. I know you're right, but I need time to get over this, if I can.\"\n\nThe phone startled both of them when it suddenly rang. Laughing self-consciously, Donna answered the phone. Kelsey barely paid attention until she heard her mother's voice rise with excitement.\n\n\"That's wonderful. Yes. Thank you so much, Frank.\" She hung up and turned to Kelsey with a satisfied grin.\n\n\"They caught Wayne. He's back in jail. This time, without bail.\"\n\nKelsey felt a heavy burden lift off her shoulders. She placed her hands over her face and started crying. Wayne Dowling, her stalker, was finally back in jail with no chance of getting out. She'd been worried about that one thing from the beginning and couldn't help but wonder if he would have gotten jail time if he hadn't ran? He probably would have gotten probation but now since he'd jumped bail, she felt confident that he would get jail time. Thank you, God.\n\n\"What's wrong?\" Kelsey heard her father ask as he entered the room.\n\n\"That was Frank. Wayne Dowling is back in jail where he belongs with no bail this time,\" Donna told her husband.\n\n\"That's wonderful. Now we don't have to worry about sending you back to Montana,\" George Sullivan said with a happy smile.\n\nKelsey jumped up from the sofa and ran to her room. She knew her dad was probably wondering why she was upset, but she couldn't deal with all the questions right now. She was pleased that she could finally return to her job and apartment. After Christmas break, she'd return to the hospital and her nursing career. Why didn't she feel happier about it? She knew why...it was because of Ryder. She knew it would be a very long while before she felt good about anything again.\n\n****\n\n\"Do you want to spend Christmas with us?\" Olivia asked him as he filled one of the feed buckets. Wyatt and Olivia had dropped by to say hello, which meant they were checking up on him.\n\nRyder shook his head. \"No.\"\n\n\"You need to be around people. You're miserable, Ryder,\" Olivia growled at him even as she stepped away quickly from a nearby horse who was trying to sniff her hair. He almost laughed at her. He knew she was skittish around horses.\n\n\"Let it go, Liv.\"\n\n\"You love her. So go get her. Simple.\"\n\nRyder turned to glare at Wyatt. \"Can't you tell her to be quiet?\"\n\n\"Are you serious? You know how she is.\" Wyatt shrugged.\n\nRyder focused his annoyed glare on Olivia. \"Butt out, Liv.\"\n\n\"You're just mad because you know I'm right. You're in love with her, Ryder. Everyone can see it but you obviously. So why be miserable? I can't even begin to imagine how she felt when she had to leave you.\"\n\n\"It doesn't matter how I feel about her, I can't give her what she wants in life. What she deserves.\" He had no idea why he had to keep explaining himself to everyone. First Cookie and now Olivia Stone.\n\n\"She deserves a man who loves her\u2014\" Olivia started.\n\n\"She deserves kids\u2014\" Ryder interjected, but then was interrupted too.\n\n\"Christ, Ryder. You are so hell-bent on thinking you're just like him, it's fucking ridiculous,\" Wyatt exclaimed.\n\n\"How do I know I'm not, Wyatt? Huh? Tell me that.\" Ryder set the feed pail on the floor and pulled himself to his full height, folding his arms across his chest.\n\n\"When was the last time you hit someone just because they pissed you off?\n\n\"I haven't\u2014\"\n\n\"Exactly,\" Wyatt interrupted him yet again, and then he turned to Olivia. \"Let's go. We have more things to get for Christmas.\"\n\nRyder watched them walk out of the barn and even after hearing Wyatt's truck drive away, he stood there. He shook his head. He knew his friends meant well, but they just didn't understand. It was very possible that he could be like his old man. He didn't want to find out at the expense of someone he loved.\n\nDamn, he hated the thought of Christmas and being alone. Over the years, he had either gone to Georgia to visit with Frank and Grace or spent the day with the Stones and their families. This year, he just wasn't feeling the Christmas spirit and he knew it was because he missed Kelsey. He caught himself more than a few times the past few days wondering what she was doing for Christmas. Was she back to work yet or was she waiting until after the holidays?\n\nHe'd love to call Frank and ask about her, but he knew his dad was far too astute. It wouldn't take him much for him to figure it all out. He would know right quick that Ryder had fallen in love with Kelsey.\n\nHe blew out a breath, finished up with the horses and walked around to his workshop. He had a few pieces to finish before he could call it a day and relax. Relax? How the hell was he supposed to relax ever again when all he could think about was Kelsey? What she was doing? Was she okay? Was that asshole stalker still out there?\n\nDamn it! He couldn't take this any longer. He had to go to her. He had to tell her she was right and he wanted her back. He realized he did want to marry her and yes, perhaps he even wanted to have children with her. Let's take it one thing at a time.\n\nHe knew in his heart and in his gut that he would never, could never, hurt her or any child. It was time for him to man up and go after the woman he loved.\n\nRyder finished the last of his orders, and called Cookie on the phone.\n\n\"Can you take care of the furniture orders I have for pickup?\"\n\n\"Of course, but where are you going to be that you can't do it?\"\n\n\"Georgia,\" Ryder answered without hesitation.\n\nCookie laughed. \"About fucking time...bring her home, boss.\" Then he hung up.\n\nRyder knew he had to go and bring her home. She should be here with him. He wanted her to be happy. He wanted to be happy and he'd been happy, truly happy for the first time in his life\u2014with her. He deserved happiness, didn't he? He couldn't let his father win. If he gave into his fears of becoming like his old man, his father would win and he wasn't going to allow that. Ryder also knew he needed to let go of the hate he had for his father. It had been tearing him up for years. The memories of him hitting him had chewed at his insides for far too many years. He had to let it go and he had to get Kelsey to come back to him. He knew the latter was going to be the harder of the two.\n\n# Chapter Nine\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nKelsey walked out of the department store and glanced up. It was a beautiful day for December. Christmas was only a week away. She wondered if it was snowing in Montana. Would she ever stop thinking of him? She shook her head, and started for her car.\n\nShe was unlocking her car when out of the corner of her eye, she saw a man across the parking lot. It was the cowboy hat on his head, which caught her attention. She glanced at him and her knees nearly buckled as her chest constricted. Then he disappeared behind a van. No, it couldn't be him. Did she miss him so much that she was hallucinating seeing him in the mall parking lot?\n\nClimbing into her car, she tried to put the key into the ignition but her hands were shaking so much, she had to stop and take a deep breath. When a sudden tap sounded on her window, she nearly screamed. She looked up and for a moment, she didn't believe her eyes. Ryder. He was real and standing alongside her car. She put her window down a little.\n\n\"Ryder? What are you doing here?\" she whispered because her chest was so tight, she hadn't enough air to speak louder.\n\n\"Looking for you...Kelsey, I...,\" he didn't say anymore but dipped his head, his hat shading his handsome face.\n\nKelsey blinked tears from her eyes. He looked so wonderful. She shook her head at him when he raced his head again, and wondered why he was in Georgia.\n\n\"Are you visiting Uncle Frank and Aunt Grace?\"\n\nHe looked away and gave a silent laugh, then looked back to her.\n\n\"I'm staying with them, but I came to see you.\" He leaned against the car lowering his upper body toward the open window. \"I want you to come home, Kelsey.\"\n\n\"Home? I am home, Ryder.\"\n\n\"No, you're not. Your home is in Montana with me.\"\n\nTears filled her eyes so she looked away and shook her head. \"You sent me away Ryder. I told you I'd stay but you didn't want me.\"\n\n\"Kelsey, baby, I do want you. Can we please go somewhere and talk about this? I've thought long and hard on this and I know I'm not like him. You saw it in me even when I didn't. I could never hurt you or a child.\"\n\nShe looked away from him and stared out the front window of the car. This was what she'd wanted to hear when she asked to stay so why now? Why come after her now? Was it the holidays? Was he just missing her and feeling lonely? Suddenly, he decided he wants her with him in Montana, so he comes all this way and what she was supposed to do? Drop everything and go with him? For how long? Until he decides, he can't trust himself again and tells her to leave? She can't go through that again.\n\nKelsey took a deep breath and turned her head to meet his pleading gaze.\n\n\"No. No, Ryder, we can't go talk about this. You ended this because of your fears. Fears you'll always have because loneliness isn't going to make them go away. You need to work through this and unless you do that, I can't go through this again. How am I supposed to trust that you won't wake up one day and decide you can trust yourself around me, and what if we do have a child...will you send both of us away?\" She shook her head, fighting back the emotion constricting her chest. \"I can't do it. I love you but I can't live on a tightrope waiting to fall. I told you, you were going to regret letting someone like me go. Well, I'm gone, and I'm done.\"\n\nShe blinked back tears, raised the window, put the car in gear, and drove off. Glancing into the rearview mirror she saw him standing there staring after her. Then she saw his head drop down, and felt a sharp pang of regret. She was tempted to go back then chastised herself.\n\nHow dare he come after her? How dare he! Did he think she'd just forget the pain he'd caused her by sending her away? He'd been so cruel to her the day she left that she'd cried the entire way home. Her heart had been broken and it wasn't going to mend just because he came to Georgia and declared that he suddenly believed he was past his fears. She slapped the steering wheel with her hand. Damn him!\n\n****\n\nRyder watched Kelsey drive away. You blew it, Wolfe. She no longer wants you. She says she loves you but obviously not enough. He knew he had to leave. He also knew it would be hard explaining why he was going home when he just got here, but there was no way he could stay now. Frank and Grace would have Kelsey's family over during the holidays and he knew he wouldn't be able to handle that. No. It was time to go back to Clifton and be miserable. Just as Olivia had predicted he would be.\n\nTwo days later, Ryder walked into his home and dropped his suitcases to the floor. It was over and his own damn fault he'd lost her. He'd had this extraordinary good thing with her and he'd made her go because he was afraid. He was a coward.\n\nHe pulled out a kitchen chair and sat down, dropping his head into his hands. Anger tensed across his shoulders when he realized that son of a bitch sperm donor father of his ruined his life in more ways than just the neglect and beatings. He'd left a lasting impression on Ryder and now he had pushed away the only woman who truly understood him. He'd made her go because he was afraid he was going to be just like the asshole who was his creator.\n\nRyder knew she was too smart for him. He should have realized she wouldn't believe he'd had a sudden change of heart and truly believed it himself when he tried to get her to come back home. Would he start thinking he'd turn into his old man and start this all over again? Damn, she did know him better than he knew himself. Until he believed it, she never would, no matter how much he tried to convince her.\n\nHe sighed, climbed out of the chair, and stripped off his coat and hat. Grabbing up his suitcases, he trudged off toward his bedroom, a shower, and bed. He was too tired to think right now. Maybe once he got it through his head there was no chance of her coming back, he'd move on with his life. He laughed aloud as he passed the room she'd stayed. Move on to what? A life without Kelsey? Christ! Just the thought of that hurt like a crushing weight on his chest. He'd always said he'd be alone for the rest of his life, and she'd fought him on that. She'd asked him if that was what he really wanted. No, it wasn't, but in the end, he'd been right.\n\n****\n\nChristmas Eve, Kelsey was in her car on her way to Frank and Grace's. It was another beautiful day for December, very much like the day Ryder had shown up to see her. Her chest tightened at the thought that Ryder had returned to Clifton. She'd half hoped he'd be here with Frank and Grace, but her mother had told her he was gone after speaking with Grace.\n\nShe steered into the driveway of the beautiful home of Frank and Grace Sanders. She really wasn't in the holiday mood but she would try. She climbed out, retrieved the presents from the back seat, and walked to the front door. Right before she rang the bell, Grace opened the door and greeted her. Grace hugged her around the armful of gifts she carried and laughed as she took some of the presents from Kelsey.\n\n\"It looks like you went overboard again, kiddo,\" Grace said with a big smile.\n\nKelsey grinned, and then gave a half-hearted laugh. \"Just a tad. I love you both, so I go a little crazy at Christmas.\"\n\nShe followed Grace into the living room, a cozy room filled with pictures of them with Ryder as a handsome young man. Frank stood when she and Grace entered and immediately grabbed up some of the gifts setting them under a beautifully decorated Christmas tree before greeting her.\n\n\"Hello, Uncle Frank.\" She wrapped her arms around him and hugged him.\n\n\"Kelsey, you get more beautiful every day.\" He told her in her ear as he hugged her back.\n\nKelsey smiled up at him. \"You're just biased.\"\n\nGrace wrapped her arm around her shoulder, giving her an affection squeeze then led her to the sofa where they both sat.\n\nFrank laughed. \"How about some eggnog and holiday cookies?\"\n\nShe smiled and nodded. \"That would be nice, thank you.\"\n\nShe watched him leave the room and now left alone with Grace, and surrounded by memories of their life with Ryder, she hoped the conversation wouldn't turn to Ryder. She glanced at Grace and frowned at her when she saw the woman staring at her.\n\n\"You're in love with Ryder, aren't you?\" Grace whispered, her arm pulling her close.\n\nKelsey started to shake her head, but her heart had its own plan and the tears quickly began to slide down her cheeks. \"Yes. But he sent me away, Aunt Grace. He's so afraid that he's going to be like his father, his real father.\"\n\nGrace snorted. \"Nonsense. Of course, we don't believe that at all and if he did, I think he's beginning to believe he doesn't have to be like him either. At least, that's what I got from him.\"\n\n\"What do you mean?\"\n\n\"Well, he was here for a few days, but then he left suddenly, for some reason he wouldn't tell us why. But when I tried to press him for detail, he told me how he has always feared that he would become like that...that man who fathered him, but that recently he had come to understand that even though that man was his father, Frank was his dad. Frank was the one who taught him how to be a man. Frank was all he had needed in a real father.\" Grace's eyes were tearing up as she spoke. \"I knew he would never be like that despicable father of his, and I told him that. I know deep in my heart that my Ryder would never cause intentional harm to any living being because he just doesn't have that kind of mean in him. In all the years he lived with us, not once did he strike out, and trust me when I tell you that he was an angry and mistrusting young man back then. Something was pulling at him but he knows being the man he is today, he owes to Frank, and he told Frank that before leaving.\"\n\nKelsey had tears rolling down her cheeks. \"He came to see me, Aunt Grace, and I sent him away.\"\n\n\"Oh my, so that's the reason he came to visit when he said he wasn't, and then left in such a hurry. Then, Kelsey dear, you need to go after him and tell him you were wrong and that you believe in him.\"\n\n\"I do believe in him. I just worry that he doesn't believe in himself. After the way I treated him when he came all the way here for me, I doubt if he'll take me back now.\"\n\n\"Of course he will. You need to go to him, Kelsey.\" Grace placed her hand over Kelsey's hand.\n\nKelsey wondered if Grace was right. Had she caused him the same kind of pain she'd suffered through when he let her go? Perhaps he really did love her, but he had to believe that he would never hurt her or any child they may have. She looked at Grace. She thought about what she'd told her of his realizations. Suddenly, she nodded her head and stood. \"You're right. I'm going back to Clifton. I love him and that's where I need to be\u2014with Ryder.\"\n\nFrank walked into the room carrying a tray containing a plate of decorated cookies, and three tumblers filled with eggnog. He raised his eyebrows in question. Grace smiled at him.\n\n\"I think our goddaughter is going to become our daughter in-law.\"\n\nFrank smiled and laughed, setting the tray on the coffee table. \"I'd love nothing more!\"\n\nWhen he reached out to hug her, Kelsey shook her head. \"But what if he hates me now?\" She flopped back down on the sofa.\n\n\"Kelsey, you can't give up on him. Ryder suffered that far too often growing up. The only people, who never gave up on him, other than Grace and me, are his friends in Clifton. That's why they're all so close. If you love him, go after him. Show him how you've got his back that you believe in him and his belief in himself will grow even stronger.\" Frank smiled at her and he knew why Ryder was such an amazing man. He'd had the best role model ever.\n\nShe nodded and pushed to her feet. \"Well, there's no time like the present. Oh my, speaking of which, please forgive me but I need to get moving if I want to spend Christmas with him, right?\"\n\nFrank and Grace smiled and nodded their agreement. She hugged them both and ran out of the house. At the door, she called out, \"I hope I can get a flight out. I love you both\u2014Merry Christmas.\"\n\nHowever, it wasn't meant to be because all flights were booked. She had to wait until the day after Christmas but thankfully, her parents supported her decision to go.\n\n\"If he doesn't want to have kids, I'll have to learn to live with it. I can survive without kids but I'm not sure I can live without him.\"\n\n\"I know, honey. Just know we're here for you.\" Her mother told her and knowing that made this a whole lot less scary.\n\nWhen the plane touched down in Butte, Kelsey felt her anticipation and fear starting to grow. She wasn't thrilled about the wonderful bus ride ahead but it would give her more time to ready herself to face Ryder. What if he turned her away like she had him?\n\nKelsey sighed as she took in the view from her bus window. It was like the first time she had flown to Montana and boarded the bus to Clifton. She stared out the window mesmerized by the beautiful mountains. They had nothing like it in Georgia. Not even Stone Mountain. This time it was snowing a little. She snickered. Of course, it was snowing...its December! Perhaps she needed to come back for a visit in summer and see this beautiful state when it wasn't snowing. Visit? Oh, no...this was no visit. This was a coming home trip.\n\nShe shook her head and squared her shoulders against the seat. She wasn't leaving again. No matter what he said to her, no matter how he reacted to her showing up at Wolfe Ranch...she was not going anywhere this time. She chewed on her bottom lip. But what if he didn't want her now? She'd been so rude to him the last time they saw each other. She'd told him she was done. Kelsey, how could you be so stupid? He came after you and you sent him away after telling him you were done with him.\n\nShe choked back a sob.\n\n****\n\nRyder sat in the living room staring at the fire roaring in the fireplace. The warmth coming from it felt good yet didn't lift his spirits leaving him still feeling chilled. Both of his dogs were lying at his feet and they hadn't taken their eyes off him. It seemed they knew he was feeling low but didn't know how to cheer him up.\n\nWhen his cell rang, he picked it up and looked at the screen. He sighed before answering it.\n\n\"Hey, Sam.\"\n\n\"Ryder, are you all right? I've been trying to reach you for days,\" Sam scolded him.\n\n\"I've been busy. You know that.\"\n\nHe heard Sam sigh. He wished everyone would just leave him alone. He wasn't going to do anything crazy, if that was what they all thought. He just wanted to be left alone. What was so hard to understand about that? Frank and Grace were upset that he hadn't stayed with them for Christmas. It was a rare occasion for him to go to Georgia for Christmas, so when he'd been there last week, they'd been so happy, which made leaving without any real explanation made him feel like crap. It had to hurt them very much and they didn't deserve that...especially not from him. Christmas was a joyous time of year, and he'd managed to ruin everyone else's besides his own. He sure as hell wasn't feeling any joy. All he felt was broken\u2014heart broken. Beat down, and unwanted just as he had as a kid. He shook his head not wanting to go there.\n\n\"What do you want, Sam?\"\n\n\"I just wanted to make sure you were all right. It's what friends do, you jack-ass.\"\n\nRyder burst out laughing. Leave it to Sam to get him to laugh. He heard Sam chuckle. Ryder took a deep breath.\n\n\"I'm fine. Why is everyone so worried about whether or not I'm all right?\"\n\n\"Because Kelsey's gone, that's why.\"\n\n\"So? Do you think I'm sitting around here moping about it?\"\n\n\"Probably not, but maybe you should be. You can't tell me you're not in love with her. I saw the way you looked at her.\"\n\n\"You're crazy. I'm not in love with her. I wish everyone would mind their own business.\"\n\n\"Okay, my friend, but you can lie to yourself all you want. I know you're in love with her.\"\n\n\"You don't know anything. I'll talk to you later, Sam.\" Ryder hung up.\n\nFeeling agitated and annoyed, he stood and walked to the window. The snow was coming down heavier now. The weather station had stated it was going to snow all day and well into the next day\u2014up to two feet possible before it stops. Kelsey would love it.\n\nRyder needed to stop thinking about her, but even as he sat in the window seat and looked out at the landscape slowly coating over in white, he saw nothing. His mind kept rerunning that day when she told him to go away. He never believed he'd be in this situation. Hurting like this because of a woman. The pain was unbearable at times. He rubbed his chest over his heart. It didn't help. Nothing was going to help. Unless Kelsey came back to him, nothing was going to help ease the hurt. He just had to learn to live without her.\n\n****\n\nKelsey stared through the windshield barely able to see the road ahead of her. The wipers were starting to slow down due to ice accumulating on them.\n\nYou're an idiot, Kelsey Sullivan. You have to be crazy in love to do this! What are you thinking? He's probably going to toss you off the ranch! Stupid! And this snow! This snow is ridiculous. It will probably kill you before he has a chance to reject you.\n\nIt was bad enough that the roads had snow on them, but they were slippery as hell and it was so dark. Was it possible for dark to be darker than dark? The bus had dropped her off two hours ago and it had taken forever to rent the SUV. Although, she was very happy the car rental place had an SUV with four-wheel drive available. Of course, at the rate things were freezing, she wondered if the four-wheel drive was going to help at all. Nothing helped in ice, and with her lack of experience driving in what to her was a blizzard, she was lucky she was still alive. The snow and ice had turned a fifteen-minute trip into an hour-long trip from town, and she wasn't there yet. She had to fight to keep the car centered in the tire tracks on the roads to gain any traction.\n\nRelief settled over her when she saw Ryder's driveway reflected in the headlights ahead. She slowed the vehicle even more, steering into the heavy snow undisturbed in his driveway, and then stopped at the closed gate. Just wonderful!\n\nShe stepped out of the vehicle to push the gate open but as soon as her feet hit the snow, she sank in\u2014almost to her knees! She trudged through it, unhooked the gate, and started pushing it open. The only problem was that it was far more difficult to push due to the snow. She pushed at it a small amount at a time, the cold, damp, and exertion wearing her down. She tried pushing the snow away from the bottom of the gate with her gloved hands when it would get stuck and wouldn't move. It only helped a bit because when she put both hands on it and pushed, it only moved a bit. She took a deep breath, leaned into with all she had and pushed with all her might. The gate moved, but it moved more than she'd anticipated and she ended up face down in the snow.\n\nKelsey lay there in utter shock, and in exhaustion, for a few seconds. She pushed to her knees, and stood up. Snow covered the entire front of her, including in her hair and on her face. She brushed herself off best she could since her gloves were caked with snow as well. She gave the gate another hard push. Finally, it was almost open far enough to get the vehicle through. She cursed the snow and Ryder for putting her through this, and hoped one more good push would do it. She put her hands on the top of the gate and gave it a final push. It was open. She'd done it!\n\nCold, wet, and in desperate want of a hot bath and bed, she trudged back to the SUV and climbed inside, knocking the snow off her legs and boots before closing the door. She drove it through the gate, and then stopped. Damn it! She remembered she had to close the gate. Double damn it! She stopped the vehicle and stepped out into the snow again. She started to push the gate closed and realized she hadn't pulled the vehicle far enough forward for it to clear the back end.\n\n\"Oh my God,\" she screamed at the sky in frustration.\n\nShe stomped back into the vehicle, climbed in with snowy feet, and pulled it up a little farther, and then she begrudgingly climbed back out into the snow and started to trudge back to the gate. She stopped and stared at the empty landscape blanketed in white around her.\n\nTo hell with it. If he wanted it closed, he can come out in this shit and close it himself!\n\nKelsey blew out a hot breath making steam in the cold air, turned and walked back to the vehicle. When she reached it, she laid her hand against the side to steady herself as she once more tried to knock clinging, caked on snow from her legs. Not realizing how icy the exterior of the SUV had become, her snow caked glove slipped, and once more, she found herself on the ground in the snow, this time on her back.\n\nKelsey lay there in total disbelief. Suddenly, she started laughing. She was laughing so hard at the ridiculousness of her predicament that she had tears rolling down her face that would probably freeze if she didn't stop.\n\n\"Ryder Wolfe, your ass better be worth all this!\" She pulled herself back to her feet, knocking some of the snow off her now sore ass, climbed into the vehicle, and gingerly drove the rest of the way up to the house. She steered the vehicle around to the back door where Ryder's truck was parked too, and sat looking at the house.\n\nWell, this is it, she thought, time to fish or cut bait.\n\n# Chapter Ten\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nIt was midnight, and Ryder was still sitting in the window seat staring out at the snow. He blinked a few times when he saw an SUV pull up thinking he was imagining it. Who in the hell would be out in this weather? Someone crazier than crazy, that's for sure.\n\nHe watched the snow coated vehicle push, and bounce, through the heavy snow before pulling around toward the back. He pushed off the seat, and hurried to the back door in the kitchen. Staring through the window in the door, his heart hit his stomach when he saw Kelsey climb out of the SUV. She stumbled a bit in the thick snow but maneuvered her way to the porch. He shook his head quickly, as if to clear away cobwebs of disbelief. He looked through the glass again. It was her. He opened the door just as she stepped onto the porch stomping the snow from her boots. She froze, lifted her face, and looked at him with wide eyes.\n\nHe frowned at her, even as he pushed down a desire to grab her into his arms and kiss her. \"What are you doing here?\"\n\n\"I came to see you, you big lug. Now move out of my way and let me in before I freeze my ass off.\" She pushed past him into the kitchen and almost instantly, the dogs greeted her. She patted both of the traitorous beasts but when Ryder called to them to move along, they did.\n\nHe watched her as she pulled off wet gloves, dropping them on the table along with a knit hat, which revealed damp hair. She shrugged out of her coat, and when she glanced at him assuming he would take, she shook her head when he didn't and hung it on the back of one of the chairs. She rubbed her hands together to warm them before bending slightly to unzip her boots and toe out them. It was then he noticed her jeans were wet.\n\n\"Why are your jeans wet?\"\n\n\"Because I had to push that stupid gate of yours open, and I fell. Do you seriously need that thing? Anyone can open it and just come right in. By the way, I left it open. There was no way I was going to try to close the damn thing.\"\n\n\"You still haven't said why you're here,\" he remarked even as he thought how much it hurt to look at her. He wanted to run his hands through that blonde hair, kiss those rosy cheeks, and carry her to his bed.\n\n\"I did so. I said I came to see you.\"\n\nHe folded his arms across his chest, crossed his legs at the ankles leaning back against the counter before he cleared his throat, and frowned at her. \"Okay. You saw me. Now you can leave.\"\n\n\"Don't be nasty, Ryder.\"\n\n\"Why not, Kelsey? You were pretty good at it the last time we saw each other.\" He felt an unwarranted sense of satisfaction when she winced.\n\n\"You took me by surprise.\"\n\n\"Bullshit. Try again.\"\n\n\"All right. Damn it, you hurt me when you sent me away. When a woman tells a man she loves him, she expects to hear it back, not be told to leave.\"\n\n\"You know why I told you to leave.\"\n\n\"Yes, and then you show up and tell me you want me back.\"\n\n\"We're getting nowhere here. Why are you here?\" he growled at her. He knew his anger was brought on by his own refusal to acknowledge that he was happy to see her. He didn't...couldn't let her know how much she'd hurt him.\n\n\"We need to talk...\"\n\nRyder threw his hands in the air and walked out of the kitchen. At this point, he didn't care if she followed him or not. Bullshit! You care, you stubborn ass.\n\nHe took a seat on the sofa and watched from under his lashes as she walked into the living room and moved to stand in front of the roaring fire with her back to him. Why was she here? She'd said they needed to talk. Why? About what? Had something happened?\n\n\"What's going on? Is there something wrong?\"\n\nShe spun around and looked at him. She raised an eyebrow. \"Wrong? What would be wrong?\"\n\nRyder ran his hand down his face. Christ! It was like pulling teeth with her.\n\n\"Are you here because there's something wrong?\"\n\n\"No, there's nothing wrong. Well, that's not completely true. There is something wrong\u2014\"\n\nHe jumped up from his seat. \"What? What's wrong? Is it Mom or Dad?\"\n\n\"It's me,\" she said so softly that he could barely hear her, but he did hear her. He froze.\n\n\"What's wrong with you?\"\n\n****\n\nKelsey walked over to him and put her hands on his chest. He tilted his head back a bit and glared down at her over the bridge of his nose. She saw his gaze wander all over her face until it settled on her lips.\n\nTesting to see just how annoyed he was that she was here, she provocatively ran her tongue along her bottom lip. She heard him hiss in a breath but then she let out a squeal when he grabbed her by her upper arms, and pulled her up against him. He pressed his nose against hers and their eyes met up close. His were filled with the annoyance she hoped wasn't there.\n\n\"I'll ask you one more time before I throw you out of here. Why are you here?\"\n\n\"Let go of me,\" she said in anger. He was so damned stubborn. He let go of her immediately. She straightened her sweater, and then peeked up at him from under her lashes. She knew he'd never hurt her despite his trying to prove he would.\n\nThen she knew he'd had enough of her games when he took her elbow in his very strong grip and led her through to the kitchen. Her stocking feet barely touched on the floor and she almost slipped a couple of times had his hand on her arm not been so tight as he pulled her along behind him. She tried pulling away from, even grabbed a doorframe at one point.\n\n\"What are you doing, Ryder? Stop!\"\n\n\"Throwing your ass out, just like I said I'd do if you didn't tell me what's wrong or why you're here. I've had it!\"\n\n\"All right! All right! I'll tell you,\" she shouted at him while she struggled for breath.\n\nRyder stopped. He released her, and stood back with his fists on his hips staring at her. She bent slightly feeling like she'd just run a mile.\n\n\"I'm here because...I love you, Ryder.\" She raised her eyes to meet his on those last words.\n\n\"You said that already.\"\n\n\"And what? It means so little too that you're just going to throw me out? In this weather?\"\n\n\"Yes.\" Ryder nodded.\n\nHe nodded that gorgeous dark head of his but she didn't believe him. He wouldn't...would he? She looked at the floor to hide a smile. She glanced up at him and took a step closer to him. He narrowed his eyes at her and she pushed down the need to flinch. He wouldn't hurt her. She was determined to prove that to him.\n\nShe put her hands on his waist. He didn't move. She slowly ran her hands up his chest, that wonderful, solid chest she loved to rest her head on at night. She felt him tense under her hands as she snaked her arms around his neck. She clasped her fingers behind his neck. He glanced over her head in an attempt to ignore what she was doing to him but she wasn't going to let him have his way. She sighed, cupped his face in her hands, and made him look at her.\n\n\"I love you, Ryder,\" she whispered close to his lips.\n\n\"Like I said, you said that already. You told me that in Georgia and that you were done with me. Well, now I'm done. I don't like to play games, and I don't want some fickle woman who can't make up her fucking mind.\"\n\nKelsey gasped and stepped back. \"Fickle? You think I'm fickle. You didn't ask me to stay when I said I was leaving, but then you come to Georgia and tell me you want me to come home and you call me fickle. I think you need to look in the mirror to see who is really fickle, Ryder.\" She punched his arm. \"At least I can admit my feelings. You refuse to tell me that you love me.\"\n\n\"Maybe because I don't,\" he said through clenched teeth.\n\nWithout thinking, only reacting to the pain of those words, Kelsey slapped him and immediately regretted it. \"You son of a bitch.\"\n\nRyder grasped her arms and pulled her against him. \"You finally got that right.\" Then he shoved her away. \"Since the weather's so bad, you can stay until they clear the roads but as soon as it's possible, you're out of here. I'm not doing this with you, Kelsey. One minute you want me, and the next you don't. I'm too old to be playing fucking games with you. Go find yourself another married teacher to screw with so you can play the wounded victim.\"\n\nKelsey couldn't believe how cruel he was being. Maybe there was more of his father in him than she believed. She shuddered, placing the back of her hand over her mouth and shook her head as tears rolled down her face.\n\n\"I'll leave now.\" She turned away, grabbed her boots to put them on but he grabbed her arm and spun her around.\n\n\"No. You won't. It's too dangerous out.\"\n\n\"I won't stay here. I can get to Wyatt and Olivia's place. I'm not staying here with you.\"\n\n\"You won't make it out of the driveway. Can't you hear the ice hitting the windows? You're staying whether you like it or not. In fact, we'll sleep in the living room in case the power goes out.\"\n\nShe narrowed her eyes at him. \"The hell I will. I know you have fuel in the generator by now. I'll take a bedroom.\"\n\n\"The fire will still keep you warmer, even with the generator.\"\n\n\"I\u2014\"\n\n\"Stop! Your other choice is to sleep in that rental out there, in which case, you'll probably freeze to death or in here where it's warm so just do as I say. Go get blankets from the bedroom. If we make through the night without killing each other, I'll take you myself to the bus station tomorrow if we can't get your car out.\"\n\nThey stared at each other for a few minutes before she dropped her boots, spun on her heel, walked down the hallway, and entered the bedroom they'd shared so many times and slammed the door behind her.\n\n****\n\nRyder ran his hand down across his face, over his tense jaw, and swore. Damn her for coming back. He'd finally decided to go on without her and if she'd stayed away, maybe he could have done it. She should have stayed away. How was he going to get through the night with her sleeping a few feet away from him?\n\nShe was right about him having plenty of fuel in the generator but it was still smart to stay close to the fireplace. He'd already planned to sleep in the living room tonight since he stayed dressed in case there was an emergency with the horses but the thought of sleeping in the bed they'd shared held absolutely no appeal. Hell, he wished he could have a locked door between them, but he shouldn't have said what he said. That was cruel. That was just the kind of vindictive and cruel thing his father would have said. Damn his father. Perhaps what he feared would happen would any way. Frank, Grace, and Kelsey all believed he wasn't a cruel son of a bitch like his old man, but maybe he was. She came here because she believed in him, because she loved him, and he threw it in her face. He didn't want her to love him. He didn't deserve to be loved.\n\nRyder heard the bedroom door open, and sat back in his recliner. He watched Kelsey walk into the living room carrying blankets and a pillow. She tossed them onto the sofa and after making a makeshift bed, she crawled between the blankets.\n\nHe stood, moved to the fire, stoked the logs, and added some more wood to the fire before settling back in the recliner. A few minutes later, she stood and started unsnapping her jeans.\n\n\"What the hell are you doing?\" Ryder hissed, feeling his body react to the sight.\n\n\"My jeans are soaking wet. I'm not sleeping in them.\"\n\n\"It won't work, you know.\"\n\nShe stopped what she was doing and stared at him as if he'd just grown a second head. \"What are you talking about?\"\n\n\"If you think by stripping off your jeans in front of me will change my mind, don't waste your time.\"\n\n\"Don't flatter yourself, Ryder Wolfe. I'm soaked and cold.\"\n\n\"Fine. Hurry the hell up then,\" he growled looking toward the fire and gritting his teeth.\n\nHe heard Kelsey chuckle. When he glanced at her, she smiled at him. \"Bothers you, doesn't it? Why? You always liked watching me take my jeans off before.\"\n\n\"How about I take mine off and we share the sofa.\"\n\n\"You wish, Wolfe,\" she muttered as she tugged at the damp fabric.\n\nRyder grumbled under his breath, \"That I do, darlin'. That I do.\"\n\nOut of the corner of his eye, he watched her strip the jeans off, get back under the covers, and pull them over her head. Ryder watched her there, the covers rising and falling with her every breath. He couldn't take his eyes off her. Damn, he wanted her. He felt his dick pushing against his jeans. He shifted to get more comfortable but it wasn't easy, not with his cock remembering the feel of her pulsing around him.\n\nSon of a bitch! He had to concentrate on something else. Thinking about her naked and under him was killing him. Christ! He wanted to be inside her and feel her clench around his dick. He couldn't give in now. If he did, he knew he'd beg her to stay. And then what? What could he give her? More of what he gave her tonight.\n\nHe shifted in the chair to try to get comfortable, but nothing was working. He glared at her but it didn't do any good since she had turned on the sofa and now lay with her back to him. Damn her. She needed to go as soon as she could. She shouldn't have bothered coming here. True, he'd insisted she leave but when he went after her, she played this stupid game of hard to get. Either she wanted him or she didn't. She'd made it very clear in Georgia that she didn't but then she came back. He didn't understand what she wanted and he had no idea what he could give her so it was best she just leave, and as soon as possible before his dick made him do something stupid.\n\nHe glanced over at her and so wanted to do something stupid.\n\nRyder toed his boots off and raised the footrest on the chair, settling in. He needed some sleep.\n\n****\n\nStretching against the back of the sofa, Kelsey slowly opened her eyes blinking against the bright sunlight streaming through the window, and gazed around the room. Ryder was nowhere around. She pushed the blanket off, sat up, and ran her fingers through the tangles in her hair. Her suitcases were sitting beside the sofa. Seeing them there, hope sprang in her heart. Had he brought them in because he wanted her to stay? Why hadn't he put them in the room then? Because you wouldn't have seen them there, dummy!\n\nShe dragged one close and opened it, pulled out a clean pair of jeans, and put them on before heading for the kitchen to get a cup of coffee.\n\nThe door flew open and Ryder walked in. Her heart nearly stopped just looking at him. He came to a halt when he saw her.\n\n\"You'll have to stay here again tonight,\" he told her before moving toward the coat hooks.\n\nSo much for wanting her to stay permanently.\n\nShe watched as he removed his gloves and coat. He took his hat off, and then hung everything up. He walked to the coffee pot, poured himself a cup, leaned back against the counter, and then just stared at her.\n\n\"I don't want to stay here again,\" she said folding her arm across her chest, which pushed her breast upwards and when his eyes dropped to her chest, she dropped her arms to her sides.\n\n\"That's too damn bad. The roads are still too bad. It's supposed to stop later today so hopefully you can leave tomorrow.\"\n\n\"You know how to make someone feel real welcome, Wolfe.\"\n\nRyder smirked. \"You shouldn't have come here in the first place.\"\n\n\"Yeah, well, I'm here and so you just have to suck it up.\" She folded her arms across her chest again, not caring how she looked.\n\n\"Damn it, Kelsey. Why in the hell did you come back here? I had finally decided I was better off without you.\"\n\n\"I told you why I came back. How many times do I have to say it?\"\n\n\"You should have thought of that when I asked you to come home.\"\n\n\"You hurt me, Ryder. You told me to go away.\"\n\n\"I thought it was best at the time.\"\n\n\"So...you don't now?\"\n\n\"Oh, I still do, and you're leaving as soon as you can. I'm not going to do this with you again.\"\n\n\"Do what? Could you just spit it out? I made a mistake. I should have come home when you asked me to. I can admit that so why can't you accept it.\"\n\n\"And then what happens when you decide you made another mistake by coming back?\" He shook his head. \"No. I'm not going through this shit again. Your home is in Georgia.\"\n\nShe walked toward him but he moved backwards until he was standing near the coat rack. She stepped closer until she stood in front of him, so close she could feel the heat of his body. \"You're wrong. You're wrong about it all, and my home is wherever you are.\"\n\n\"Don't.\" He put his hands up as if to ward her off.\n\n\"Don't what, Ryder? Touch you? Kiss you? Tell you that I love you?\" She placed her hands on his chest.\n\n\"Kelsey, I mean it.\" He lifted his chin so that he broke eye contact.\n\n\"Do you really?\" She stood on her toes and kissed his neck.\n\nRyder shuddered and grasped her arms in his hands. Just when she thought he was going to push her away, he pulled her tight against him and took her mouth in a deep kiss. She moaned and wrapped her arms around his neck. He bent his knees, and placed his hands under her butt and lifted her. She wrapped her legs around his waist as he pushed her back against the wall. When he rubbed his crotch against her, she could feel his hard shaft already eager for her. No matter how hard he tried, he couldn't deny his want of her. She groaned as he moved against her. He dropped her legs, and allowed her to slide down the front of him. Once her feet were on the floor again, he stepped back from her, making her whimper.\n\n\"Ryder, please,\" she whispered in a ragged voice.\n\nThinking he was walking away, Ryder surprised her when his hand reached out and he unsnapped her jeans, pushed them down, along with her panties, and stripped them off her. He dropped to his knees in front of her and raised one of her legs over his shoulder. Kelsey started shaking in anticipation. When his tongue moved through her curls, she gasped and grabbed fistfuls of his hair. Her head tilted back against the wall as he worked his magic. When he inserted his fingers, she felt her orgasm moving over her and soon a low groan tore from her and erupted into a scream as she came.\n\nAs she stood there trying to catch her breath and her body trembling in the aftermath of her pleasure, he stood, rolled on a condom, and with one hard thrust, filled her body. She wrapped her legs around him, grabbed his shoulders for support, and bucked her hips against him. He slammed into her, the wall behind her making it easier for him to support her. She couldn't believe it when another orgasm ripped through her so quickly. Then she felt his entire body tense as he thrust into hard before hearing him groan as he came. He continued to move against her a few thrusts more until he finally leaned his forehead against hers, his breathing harsh and ragged.\n\n\"God, that was fantastic,\" she said against his ear but his shoulders flinched, he withdrew, and let her feet drop to the floor.\n\n\"Fuck,\" Ryder muttered in an angry tone as he pulled away from her, disposed of the condom, and zipped his jeans.\n\n\"What?\" Here we go again! She'd come to recognize when Ryder started to shut down. Why was he so afraid to let her in?\n\n\"That shouldn't have happened.\"\n\n\"We're good together, Ryder, and you know it.\" She couldn't keep the hurt from her voice.\n\n\"I don't deny that, but it still shouldn't have happened. You're leaving and that's the end of it.\" Without looking at her, he walked away from her.\n\n\"I hate you,\" she shouted at him.\n\nRyder spun around and glared at her. \"Good. That's better than loving me. I'm used to hate. I can handle hate.\" He stared at her for a few seconds, and then turned away. \"I have work to do. I brought your suitcases in so you would have clean clothes. You can stay in the spare bedroom until you leave.\"\n\nKelsey watched as he grabbed his coat and hat before disappearing out the back door. When the door slammed shut, she fell to the floor and cried.\n\n****\n\nRyder trudged through the snow, swearing the entire time. What the fuck were you thinking? Thinking with your dick is what you were doing. How had he allowed himself to give into his desire for her yet again? It was like he couldn't think clear around her. He just wanted to touch her, kiss her, and bury himself in her without any regard for the consequences. Why did she come back here? Despite what Frank and Grace believe, he still had a hard time believing he can be any different from his old man. If he couldn't believe it, how could Kelsey? How could anyone?\n\nHe was so mad at what he'd done, what he'd given into that he wanted to punch a hole in a wall. He pulled the barn door open and entered the one place he'd always felt most comfortable...other than in Kelsey's arms. Damn. The smell of his horses usually made him feel good, but not today. Today, the woman he'd left half-naked in his house was on his mind. Damn her for coming back here. He would have been fine without her\u2014eventually. He stopped, and leaning against one of the stalls, he rubbed his hand down his face. You just had to fuck her, didn't you?\n\nThe horse in the stall nuzzled his nose against Ryder making him sigh. He rubbed the mare's neck with his gloved hand. Life had been so simple before Kelsey Sullivan came into his life. He was content living alone. He'd been on his own for so long that it he was resolved that it was how it would always be, but then she had to walk into Dewey's and make him realize that being alone wasn't what he wanted at all. He'd missed her...every damn minute she was gone.\n\nHe pushed off the stall and headed for Tramp's stall. He needed air. After putting the blanket and saddle on the horse, he mounted Tramp and rode him out of the barn. In the yard, puffs of cold air rose from the horse's nostrils, and he pranced around, ready to run. It looked like they both needed some air. Ryder let him go. Tramp took off through the snow and headed toward the north pasture. The exhilaration cleared his lungs, and he hoped his brain because he had been thinking too much lately. Once they reached the north pasture, he slowed Tramp down to a trot.\n\nRyder rode along the fence line. There wasn't any reason to check the fence because he had a transmitter that ran to the house and signaled if a section of it were to go down. He just couldn't be in the house with her any longer. He shouldn't have touched Kelsey. Ever. If he had never touched her in the first place, he wouldn't be feeling like this. His damn heart hadn't pained him this much since he left home. He didn't deserve to have Kelsey love him. Hell, his own parents didn't love him, why should she. Who said love was great? It hurt like hell.\n\nHe shook his head and slowed Tramp to stop. The horse snorted and blew steam into the cold air. He looked down toward the house. She was there. Why couldn't he just be happy that she was? His friends were all happy. The Stone brothers were all happily married. The women they were married to were perfect for them. Gabe's daughter, Sophie, was the most precious little girl out there and Will, Jake's little boy was so adorable.\n\nIn all his life, he'd never wanted that because he thought he couldn't have it. Until lately...until Kelsey came into his life, but he pushed her away when she tried to believe in him. Then even though he'd hurt her, she'd changed her mind and came back. Could she truly see the good in him or was she just going to change her mind again and leave. What if he were to lose his temper and prove without a doubt that he was as much a monster as his father.\n\nRyder was man enough to admit he was scared to death. Scared that Kelsey would stay, scared that she would leave, and terrified that he would lose her forever, or worse drive her away. She'd left because he wouldn't let her believe in him, but when he went to Georgia, to bring her back, she had to have known how hard that was for him. That should have shown her just how much he wanted her. He still wasn't sure if he could make a life with her, or give her children, but he had been willing to try. When she told him she was done, he had died inside. He came home a broken man but now she was back.\n\nTramp stomped and snorted. The horse was restless and Ryder understood that feeling. Kelsey rejecting him, abandoning him even, was a reminder to him that he was alone in this life and responsible for no one but himself. He'd learned that lesson the hard way and wasn't going to suffer the pain of someone else making him want more, and then snatching it away. No. She was leaving and she couldn't leave soon enough. He stared down at the house wondering what she was doing. Stop, you fool! It was going to be hard enough tomorrow when she leaves, he didn't need to make tonight worse than it would already be.\n\nHe gave Tramp his head and they raced along the pasture, the cold air filling his lungs and snow stinging his skin. Slowing the horse on the rise, he stared at the Glaciers in the distance. They were clearing up, which meant the snow would be stopping soon. The plow trucks would be out as soon as possible. Clifton may be a small town but the center of town was the hub for the people. They had to get to town for animal feed and for their own groceries. The outskirts were all ranches and farmland. The town would never allow itself to stay shut down for long.\n\nRyder blew out a breath and watched it form into a puff of cold air. The snow was falling less now but Tramp had started showing his impatience at being in the cold by snorting and prancing around.\n\n\"All right, buddy, just give me a few more minutes. I hate going back in there. I know you're cold, but hang in there with me for just a little longer.\"\n\nThe horse stopped prancing and bobbed his head up and down. Ryder swore the horse understood his angst. He patted Tramp's neck in gratitude. If only the horse could tell him what to do. If he thought today had been bad, what was coming tomorrow would kill him for sure.\n\nYou could ask her to stay.\n\nHe shook his head before he even finished the thought. No, he couldn't ask her to stay. He was a coward, just as she'd said. She said she loved him but she deserved better. She deserved a man who would take care of her, protect, give her children, and love her more with all he was worth.\n\nRyder looked over the ranch, at the house he'd built. He did love her. He loved her more than he thought it possible to love anyone. But would that be enough? He hadn't been a saint. He'd been his share of women but none of them made him feel the way Kelsey did. When he was inside her, he was in heaven. She was perfect for him. A perfect fit and without her, he knew he'd never have that again. Damn it. He'd always planned to be alone and he was better off that way. He'd been fine with it until the night he just had to have her. He fell the minute he'd touched her, but she deserved better.\n\nTaking a deep breath, he turned Tramp toward the house and headed back. He'd work in the woodshop the rest of the day. He'd work until he was so exhausted that he'd fall into bed and not wake up until morning. He'd do that because if he saw her, smelled her, touched her, or if she pleaded one more time to stay, he'd give in and let her then when she discovered the truth about him, she'd leave, and it would kill him. Hopefully, the roads would be cleared soon, and she'd be able to go.\n\nFrank and Grace should never have convinced him he could be different from his old man. He'd wanted to believe it. He'd tried when he went to Georgia to bring her home. I am home, she'd told him, and as much as he wanted home to mean here with him, he knew she belonged at home in Georgia.\n\nRyder nudged Tramp into a run. The cold air stung his face as it slapped against it. A hard slap he probably deserved.\n\n# Chapter Eleven\n\nKelsey carried her suitcases to the SUV, and after taking one last glance around, she climbed in behind the wheel. Ryder was nowhere around. He was probably in the barn. Hiding, like the coward he was. She wiped a tear away as she softly swore.\n\n\"Damn you. I love you.\" She turned the key in the ignition, and knew there was no turning back or saying good-bye.\n\nLast night had been the longest of her life. She'd stayed in her room and stared at the TV but wasn't really seeing it. Her brain kept replaying the events of the day as she tried to figure out what the hell was up with Ryder. What was he so afraid of that he'd push her away again? She'd come back. Wasn't that what he wanted when he came to Atlanta? She just didn't understand him at all. She'd cried so much, she thought at times that she was going to be sick.\n\nAfter taking a long hot bath, she'd crawled between the sheets and tried to fall asleep. But sleep eluded her and when she couldn't take it any longer, she got up and called the airline. Her flight today wasn't until noon but she had to take the bus, which she hated, to Butte to catch her plane. Besides, it was best to leave as soon as she could, kind of like ripping a bandage off a wound. It wasn't going to get any easier the longer she stayed.\n\nHere it was, a little after five, and the sun was just beginning to rise above the mountains in the distance. As she drove toward Clifton, she put her sunglasses on and pulled the visor down. As the sun brightened the landscape, it glared off the snow making it almost impossible to see. The roads were clear just as Ryder had predicted they'd be. As she drove by Wyatt and Olivia's ranch, more tears threatened to overwhelm her. Once again, she was leaving some of the best people she'd ever met. She'd come to love Becca, Emma, and Olivia.\n\nWhy was he making her go away again? Especially after, he had sex with her yesterday. God! It had been so hot! She just didn't understand what had gone wrong. Yes, she told him she was done when he came to Atlanta, but then she came back here, didn't she? Couldn't he see that? She came because she wanted to stay with him, because she loved him and trusted him. He didn't even have to marry her. Just love her. And she knew he loved her. He didn't have to tell her, she knew it even if he didn't want to admit it. Damn him. She didn't want to go back to Georgia. Clifton was her home now because Ryder was here.\n\nSo, why are you leaving?\n\n\"Because he doesn't want me,\" she said aloud.\n\nHe can't tell you where you can or can't live.\n\nShe nibbled at her bottom lip as she realized just how true that was. He had no right to tell her where she had to call home. She could stay right here. She loved this town and the people. Never had any community made her feel more welcome.\n\nYes, she could stay. But, where? She didn't have a home or a job. She slowed as she approached Emma and Gabe's driveway. Should she stop? Was it too early? She snorted. It was a working ranch. It was never too early. She steered the car into the driveway and drove up to the house. As she stepped from the vehicle, she saw Gabe coming from the barn. He had a puzzled frown on his face. Probably wondering who in the hell was at his home this early in the morning. He smiled when he recognized her.\n\n\"Hi, Kelsey, what are you doing here? I thought you left.\"\n\n\"I came back to see Ryder...\" She blinked tears away. \"He sent me away. Again.\"\n\n\"Yeah, he has some trust issues. Come on into the house. I'll get you a cup of coffee.\"\n\n\"I don't want to disturb Emma if she's sleeping.\"\n\nGabe smiled at her. \"Trust me, she's up. Come on.\"\n\nKelsey followed him up the porch steps and entered the kitchen when he held the door open for her. The kitchen smelled wonderful. Emma turned her head when the door opened and smiled when she saw Kelsey. She walked to her, wiping her hands on her apron, and hugged her.\n\n\"Kelsey. It's so good to see you. I thought you'd gone home to Georgia.\"\n\n\"I did...but I...\" She burst into tears. Emma wrapped her arms around Kelsey's shoulders and she suspected Emma waved Gabe away from the motion of her arm. The sound of the kitchen door opening and closing confirmed it.\n\n\"Come and sit down,\" Emma said leading her to the table. \"Do you want something to eat?\"\n\n\"No, thank you. Just coffee is fine. I'm so sorry, Emma.\"\n\n\"For what?\"\n\n\"Crying like that. I bet Gabe was happy to escape.\"\n\nEmma chuckled and patted her arm. \"You should have seen the look on his face.\"\n\nKelsey laughed even as she wiped her wet cheeks with her sleeves. \"I bet. Men hate to see women cry, even though they're usually the ones causing the tears.\"\n\n\"Oh, that's the truth. I cried enough of them over Gabe. Damn men.\" Emma poured them each a cup of coffee and then pulled out a chair, and sat at the table beside her. \"Tell me what happened,\" she said as she placed her hand over Kelsey's.\n\nKelsey took a deep breath and proceeded to tell Emma everything from the beginning. She didn't leave anything out. She trusted Emma, and knew she could tell her.\n\n\"And then he sent you away? After you, and him...and well, you came all the way here from Atlanta.\" Emma practically growled.\n\n\"Yes. He says he's not going to wait for me to change my mind again and leave him.\" She shook her head. \"I shouldn't have told him I was done when he came to Atlanta. I was just so mad at him for showing up and just thinking I'd drop everything and run back here with him.\"\n\n\"Did you tell him that?\"\n\n\"Oh, Emma...he wouldn't have listened. He is the most stubborn man I've ever met. Now I don't know what I'm going to do. I stopped here because I'd convinced myself I was staying whether he liked it or not. I'm just not so sure now.\"\n\n\"Do you like it here, Kelsey? I mean, don't stay just to be around Ryder. That will just drive you crazy.\"\n\n\"I love Clifton. The thing is I don't have a job or a place to live until I get a job.\"\n\nEmma seemed to contemplate that a moment, and then stood. \"I have an idea. Let me call Becca.\"\n\nKelsey frowned, sipped her coffee, and wondered what Emma was thinking. Kelsey watched her hang up her phone and sit back down.\n\n\"Becca's on her way over. It will just be a few minutes. Are you sure you don't want something to eat?\"\n\n\"I'm sure. I don't think I can stomach anything right now.\" Kelsey sighed.\n\nThe back door opened, Gabe poked his head as if to check that the coast was clear, and then stepped inside. \"Is everything all right?\"\n\n\"Yes, all is well. I have your thermos ready,\" Emma said with bright smile as she handed it to Gabe. He leaned down and kissed her on the lips then after tipping his hat to Kelsey, he left just as quickly as he came in.\n\n\"Gabe really loves you.\" Kelsey wished Ryder would be so open and unafraid to show his love for her.\n\n\"He does. I'll have to tell you about us one day.\" Emma had just sat down when the door opened again, and Becca entered.\n\n\"Hi, Kelsey, I'm so happy you're here again.\"\n\n\"Where's Will?\" Emma asked.\n\n\"With Jake...being spoiled rotten,\" Becca said with a laugh. She poured herself a cup of coffee and took a seat beside Kelsey. \"So, you need a place to stay?\"\n\n\"Yes, but I'm not staying with either of you or Olivia. I'm not going to be a fifth wheel.\"\n\n\"I happen to own a bed and breakfast, remember?\"\n\n\"I remember, but isn't it closed for the winter?\"\n\n\"It is, but it's just sitting there empty. You can stay in one of the apartments at the back of the kitchen. The woman who works for me has a house in town. She works on the reservations and emails from home. Something she's not going to be doing much longer either. She's pregnant. You'd be alone out there though.\" Becca frowned.\n\n\"I don't have a problem with that. I have an apartment in Georgia, so I'm used to being alone. Besides, you are all close.\"\n\n\"Yes, we are. I want to help you, Kelsey. If you want to stay there, it's fine with me. You'd actually be helping me out by staying there.\"\n\n\"You are all so nice and you don't even know me very well. I know we haven't known each other long, but I love having you as my friends.\"\n\n\"So do you want to stay?\" Emma asked her.\n\n\"I do. I want to look into getting a job here and staying in Clifton. If Ryder doesn't like it, that's just too damn bad.\"\n\n\"You're more than welcome to stay in the apartment at the B and B for as long as you need. If you decide to leave Clifton, that's fine too. Whatever you want to do,\" Becca said.\n\n\"I really appreciate that, Becca. I'll even clean the place up so it doesn't get dusty from just sitting there. I suppose I can check with the hospital about working there.\"\n\n\"Great. The hospital is just outside of town. It's between here and Hartland. You should talk with Kaylee. She's a nurse and works at the hospital,\" Becca told her.\n\n\"I didn't know she was a nurse. Great, I'll talk to her. What about rent?\" Kelsey asked.\n\n\"No rent. I'm not going to make you pay me. I just want to help you out.\"\n\nKelsey jumped up and hugged her. \"Thank you. I really want to stay here in Clifton. I want to be close to Ryder. Maybe I can wear him down.\"\n\n\"I think you have a great shot at it if you stay here. He'd see you whether he wants to or not. If you go back to Georgia, well you know what they say\u2014out of sight, out of mind.\" Emma laughed.\n\n\"Fantastic,\" Becca said as she stood. \"This is the key to the back door. You can follow me there now but then I need to get home. I'll bet Jake has Will out in the cold. He rides him around on his horse all the time. No matter what the weather. \"\n\nKelsey hugged Emma, and thanked her everything. She put her coat on, climbed into her rental, and followed Becca to the Clifton B and B. When she pulled up to the large farmhouse, she gasped. It was gorgeous. A large white house with black shutters and a porch that wrapped around the side with a row of rocking chairs sitting there just waiting for people to sit in them. She fell in love with it.\n\nThey entered the kitchen from the side entrance. The kitchen was a chef's dream. She followed Becca through the doorway located alongside the large commercial refrigerator.\n\n\"This is the bigger one of the two apartments but please don't feel you have to stay here permanently. If you do decide to stay in Clifton but don't want to stay here, that's fine by me. You can move to town if you want. I used to want someone here all the time, but not much happens in Clifton as far as crime so I don't worry about the place. I keep the heat on so the pipes don't freeze and the house stays warm. It was my grandmother's home. She left it to me when she passed away. I met Jake in this house,\" Becca said with a wistful smile and a sigh.\n\n\"In the house?\"\n\nBecca laughed. \"Yes. He came over the night I arrived and I threatened him with mace spray when he walked in out of the snow. The first thought that went through my mind after he took his hat off was that he was the sexiest man I'd ever seen, and I hated him.\"\n\nKelsey laughed with her. \"But you love him now.\"\n\n\"I think I fell totally in love with him that night, and that was part of why I hated him. I hadn't come here to find love, especially with some cowboy. I used to live in Maryland and wanted nothing to do with some cowboy from Montana.\"\n\n\"Well, he certainly is some cowboy.\"\n\nBecca burst out laughing. \"Don't I know it. Well, you get yourself comfortable. Make yourself at home and if you need anything, call me.\" She told Kelsey her cell and home numbers, and then left.\n\nKelsey sighed and walked outside to get her suitcases. She had to call the airlines and cancel her flight, and then her parents. They were going to think she was crazy, especially after she explained that Ryder told her to leave\u2014again. She unpacked her suitcases, and rested a moment on the bed and contemplated the long day ahead of her.\n\n****\n\nRyder worked in the barn all day. He'd been there since before the sun came over the mountains. He'd heard her leave and it took every ounce of willpower he had not to run out and stop her. It was for the best she was gone. The thought that she'd change her mind about staying was all the reassurance he needed that sending her away was for the best. She'd end up hating it here and missing Atlanta then she'd eventually hate him. Although, there was a pretty strong possibility she already did. He knew he'd hurt her, but it couldn't be helped. No, but you should have kept your damn dick in your pants.\n\nHe stopped raking the hay, folded his arms at the top of the rake handle, and stared at the wall. Damn it. Was he going to get on with his life any time soon? Ever since he'd returned from Atlanta, he'd been a complete mess and now it was worse, and she hadn't been gone a day.\n\n\"Hey, boss,\" Cookie called from outside the stall.\n\nRyder turned to look at him. \"Everything all right?\"\n\n\"With us, yeah. You? I'm not so sure of.\"\n\nRyder clenched his jaw and stabbed at the hay with the rake. \"Let it go.\"\n\n\"I don't think I will.\"\n\n\"I'll fire you then,\" Ryder muttered.\n\nCookie burst out laughing. \"Sure you will. What would you do without me?\"\n\n\"Oh, I don't know. Maybe get some peace and quiet about shit that doesn't concern you.\"\n\n\"Anything that concerns you, concerns me and you know it.\"\n\n\"Since when?\"\n\n\"Since you were a boy and your grandfather had you here. I really hated it when he died and those parents of yours moved in. It was just a good thing the will stated they had to keep me on and pay me or I'd have been out of a job. Then you went and run off.\"\n\n\"First off, don't call those people my parents and second, I had to run off.\"\n\nCookie nodded. \"I know you did, son. I also know they ain't your real parents. That role was accepted by Frank and Grace, and they done a fine job. Still are. So I wonder how they'd feel about you running Kelsey off.\"\n\nRyder gave a humorless mock laugh. \"Probably that it was the best thing I'd ever done.\"\n\n\"Bullshit. They love you and they love her. It'd probably make them real happy to have you two together.\"\n\n\"She wouldn't have been happy here and would have left me sooner or later. I just made her go sooner.\"\n\n\"Is that what you think? Did you even ask her?\"\n\n\"Christ, Cookie. Do you think she'd really admit she'd had no plans to stay for good? That she'd change her mind eventually?\"\n\n\"Well, I see your point, and Ryder, that's the chance we all take with love, but that girlie loves you and you love her. Seems to me, you'd be much happier if you two were together, not apart.\"\n\n\"Who said I love her?\" Ryder growled.\n\nCookie chuckled. \"You.\"\n\n\"I sure as hell did not.\"\n\n\"Not in so many words, but look at you. You can hardly concentrate on what you're doing and I saw you this morning when she left. You didn't think anyone was around, but I was. I saw how you kept looking at the door. Like you were trying to decide whether to go after her or not. I may be old, but I know what a man looks like when he's in love and you, Ryder, are in love.\"\n\nRyder shook his head. \"Doesn't matter.\"\n\n\"You can lie to yourself all you want, but don't try that shit with me. Why do you want to be miserable?\"\n\n\"How miserable do you think I'd be if she stayed letting me believe she loved me, and then one day, she leaves?\"\n\n\"Damn son, no one knows the future and you can't dwell on something that may never happen.\" Cookie sighed. \"You deserve some happiness in your life, Ryder. I know you have great friends and you have Frank and Grace, but you need that girl. If you can't see that, then I feel sorry for you.\" He stared at him for few seconds then shook his head, and walked away.\n\nRyder knew no one could understand his way of thinking. He'd grown up with two people who wanted nothing more than their next drink. Two people who never gave a damn about him. Who never loved him\u2014their own kid. It was hard for him to trust. He only trusted his friends, Cookie, Grace, and Frank. It was hard to let someone in, and Kelsey had already pushed him away once so what was going to keep her from doing it again. She told him that she trusted him never to hurt her or anyone, but when he went after her, asking her to come home with him, she'd rejected him. How can he trust someone who hasn't earned it? Shit! He leaned the rake against the wall and wandered out into the center aisle of the barn. There was much more work to be had, and he planned to work as hard as he could today and maybe, just maybe, he'd sleep tonight.\n\n****\n\nKelsey sat on the window seat of the B and B with her legs curled beneath her, and stared out the window. When she saw an SUV coming up the drive, she smiled with delight that it was Olivia driving it. She hopped of the seat, and hurried to the kitchen. She opened the door, a blast of cold air hitting her and waited anxiously for her friend to enter.\n\n\"Hey, Liv, what are you doing out in this cold weather?\"\n\n\"It is fucking colder than a brass monkey's balls, isn't it?\" Olivia exclaimed with a laugh as she took her coat off.\n\nKelsey stared at her, a bit shocked but then burst out laughing. \"I was warned about that potty mouth of yours. Glad to experience it firsthand.\"\n\nOlivia laughed. \"Sometimes I may need a filter.\" Then she shrugged. \"I'm just me,\" she said with a smile.\n\nKelsey hugged her. \"I wouldn't have you any other way, and I barely know you. I'm sure Wyatt doesn't mind either.\"\n\n\"Wyatt Stone can kiss my ass...literally.\" Olivia laughed.\n\nKelsey laughed with her. \"I'm not going there. You're a lucky woman. That's some good looking husband, you've got there.\"\n\nOlivia sighed. \"Yes, I am, and yes, he is. I fell for him the minute I set eyes on that brooding stubborn cowboy.\"\n\n\"Did he feel the same?\"\n\n\"He tells me he did, now but back then, he fought it every inch of the way. We just got married in June, so we're still honeymooning...but enough about me. What's going on with you and Ryder?\"\n\n\"Nothing, absolutely nothing. He told me to leave so I did.\" Kelsey could feel the tears threatening again so she avoided meeting Olivia's stare.\n\n\"I was under the impression at Thanksgiving that you two were an item. I mean, we saw him kiss you.\" Olivia frowned at her and took Kelsey's hand in hers. She pulled her to a seat at the kitchen table.\n\n\"Can I get you something? Coffee?\" Kelsey tried to draw the subject away from her doomed love life.\n\n\"I'm good. Come on, you can tell me. I've been there with these Montana cowboys. I thought for sure you'd caught yourself one,\" Olivia said pulling Kelsey back into her seat.\n\n\"We were...getting along, but then he started this shit about not settling down. He doesn't want to get married, or have kids, because he's afraid he will be like his abusive, deadbeat father. We had a huge argument and I told him I loved him but he just got angrier and told me not to say that to him. Then he told me I had to go home where I belonged. I told him he would regret sending someone like me away but...\" She took a deep breath. \"He came to see me in Atlanta just before Christmas, and I was so upset with him that I told him I was done with him. Grace told me he'd begun to believe he wouldn't be like his father after all because of her and Frank. So, I came back here to tell him I did want him and this time he tells me he's done with me. He accused me of playing games, deserving better and that if I stayed, I would change my mind about him.\"\n\n\"What the fuck does that mean?\"\n\n\"I've been giving this some thought and I'm beginning to think he's afraid I can't really love him and that I'll eventually think he will be like his abusive father, and then I won't be happy. It's hard to explain, but I really think that's what it is.\" She shook her head.\n\n\"I left early this morning, without saying good-bye. I didn't think I could do it. I was leaving, going home to George but as I was passing Emma's, I realized I don't want to leave Clifton. I really love it here and if it's the only way I can be close to him, then so be it. I'm staying. Becca offered me the apartment here, which I'm so grateful for, and I'm going to see about getting a job at the hospital. Then eventually, I'll get my own place.\"\n\n\"Good for you. Listen, Kelsey, don't give up on the man you love. I was ready to leave here because Wyatt didn't want to do the commitment thing either and I did. I was packed and ready to go but Becca suggested I stop at his house to see if I missed anything. It seemed she and Wyatt set it up that way, and so once there, Wyatt told me he loved me and wanted me to stay. I can't imagine what my life would be like if I'd gone, if I'd given up on him. He's my world.\"\n\n\"I can see how much he loves you, Liv. In fact, all of you have husbands who love you so much. I want that too, Liv. I want it with Ryder and I'm going to do my best to have it. I came here because I needed his protection, but I fell in love with him instead.\"\n\n\"I never wanted a cowboy.\" Olivia laughed. \"But when I saw Wyatt, I was a goner. I teased Becca when she first came here that cowboys were nasty. Oh boy, was I ever wrong.\"\n\n\"I've always had a thing for cowboys so when I first saw Ryder in that Stetson, I wanted to jump his bones.\"\n\n\"Same here with Wyatt. Becca and I were in the feed store talking with Emma when he came into the store. Becca elbowed me, nodded toward him, and I wanted to lick him all over. I told Becca that when she first introduced me to him, I nearly had an orgasm looking into those dreamy dark eyes.\"\n\nKelsey laughed and nodded. \"Those dark eyes are definitely gorgeous. I love Ryder's green ones best though.\"\n\n\"Ryder is hot, honey. Don't let him slip through your fingers.\" Olivia grabbed both of Kelsey's hands in hers. \"Fight for him, and that means even if you're fighting with him. He loves you. I could see it at Thanksgiving. He couldn't keep his eyes off you all day. You're just going to have to convince him of it.\"\n\n\"I am going to fight, Liv. I love him, and if I lose, well then at least I gave it a shot.\" Kelsey sighed, feeling more confident in her decision to stay in Clifton. \"How about some coffee now?\"\n\nWhen Olivia nodded, Kelsey set about making a pot and wondering how she could fight for Ryder without it being too obvious to him.\n\n# Chapter Twelve\n\nJanuary descended on Clifton with what felt like a vengeance. It was below zero at night and didn't get much warmer during the day, plus there was sleet, and more snow every day for the first week. Ryder hated the idea of just walking to the barn. He kept the horses inside because of the frigid temperatures and they were not happy about it at all. Today, he was letting them run inside the indoor corral. It was in no way close to the freedom they experienced in the pastures, but it was better than nothing. At least they had room to run and get some exercise. He stood at the wooden rail and laid his arms across the top of it. Watching the horses prance and trot around the corral reminded that he needed to head to town to get some oats. Groaning at the thought of making the trip in the cold, he moved from the rail and walked through the barn. He slowed to a stop when he saw Cookie walking toward him.\n\n\"Are you going to town or do you want me to go?\" Cookie asked him.\n\n\"I'll go. I just hope the roads are clear. I hate this damn ice.\"\n\n\"Okay, be careful, boss. Give me a call if you have any problems.\" Ryder watched Cookie walk away. It would've been nice if he'd put up a bit more argument.\n\nRyder pulled his hat down low and the collar up high on his sheepskin coat before heading out to his truck. He put his head down so the icy sleet wouldn't hit him in the face. He climbed into his truck and started it up. With temperatures the way they were, relief settled over him that the engine turned over at all. After putting it in gear, he maneuvered the truck down the driveway, glad the gate was already open. He drove through, out onto the road and headed toward town.\n\nWhen he was alone like this, his thoughts always seemed to go to Kelsey. He wondered what she was doing. Was she back working at the hospital? Was she involved with someone else now? Christ, the thought of her with another man just about killed him. Suddenly, the back of end of the truck fishtailed slightly on the icy road, so he slowed down and concentrated on staying alive and on the road.\n\nAfter getting the feed he needed, he decided to stop at the diner and get some lunch to go. He parked in front, carefully maneuvered the slippery parking lot, and entered the diner. The little bell above the door announced his arrival, and a few heads turned followed by a few greetings. He took a seat at the counter, and nodded to everyone who said hi or waved at him. His eyes scanned the diner and as he glanced around, his gaze quickly shifted back to the blonde haired woman in the back booth. Surely, he was just imagining things. It couldn't be Kelsey but as he sat there staring at her, he knew it was her. He stood and weaved his way through the tables to get to her. He stopped at her table and clenched his jaw while balling his fists even as he wanted to pull her into his arms and stared down at her.\n\n\"What the hell are you doing here, Kelsey?\" he muttered.\n\nShe glanced up at him and blinked. \"Having lunch.\" She stabbed her salad with her fork.\n\n\"I don't mean here in the diner and you know it. Why are you still in Clifton?\"\n\n\"Excuse me, but I don't believe it's any of your business why I'm here. I can live wherever I choose, and I choose Clifton.\"\n\nRyder continued to glare at her. \"Why?\"\n\n\"Why? Because, I happen to love this town and the people in it.\"\n\n\"What about your job?\"\n\n\"I have a job, here at the hospital.\"\n\n\"You're working at the Clifton County Hospital?\"\n\n\"Is there another one?\"\n\n\"Jesus, you're being a real smart ass.\" He slid into the booth across from her.\n\n\"I don't recall asking you to join me, Ryder.\"\n\nHe leaned across the table. \"Tough shit. If you think by staying in Clifton, you're going to change my mind, think again.\"\n\n\"It has nothing to do with you. I love it here and I'm staying. Now, go away. I'm expecting someone.\"\n\n\"Who?\" Her blue eyes met his, and jealousy ripped through his gut.\n\n\"None of your business.\"\n\nRyder started to ask her again when a woman's voice greeted Kelsey from beside him. He started to rise.\n\n\"Don't bother getting up on my account, Ryder.\"\n\nRyder looked up to see Lydia Carmichael standing beside the booth. \"Hello, Lydia.\"\n\n\"Hi. You can stay if you want. Well, if Kelsey doesn't mind, that is.\" Lydia Carmichael was a statuesque brunette with dark brown eyes and flawless skin. She was a stunning woman who did absolutely nothing for Ryder but it didn't matter since she was seeing Sam.\n\n\"I do mind. Go away, Wolfe.\" Kelsey's eyes narrowed at him.\n\nRyder blew out a breath and stood. \"Have a nice lunch then.\"\n\nHe touched his fingers to his hat and nodded to both women before walking back to the counter. Connie, the owner of the diner had his lunch waiting for him in a Styrofoam container. He paid for his lunch and after one more glance toward Kelsey, he left the diner. She never looked up at him again.\n\nDamn. Why was she here? He shook his head. It hurt to see her, hear her voice, and not being able to hold her. This was going to be hard on him if she lived in Clifton, and she obviously planned to because she was talking with Lydia, who was a real estate agent and dealt in buying, selling, and renting properties.\n\nOf course, they could live in the small town and never see each other. After all, he rarely left his ranch unless he came to town for feed or for meetings or the occasional dance at the Town Hall, or a few drinks at Dewey's. The chances of running into her were slim and if he did see her, he'd just steer clear. He walked to his truck and was about to climb in, when he saw Sam crossing the street and heading toward him. Ryder leaned against the fender of the truck and folded his arms.\n\n\"Hey, Sam.\"\n\n\"Ryder. Did you get one of Connie's burgers?\" Sam said with a smile.\n\n\"Oh hell yes. I can't come into town and not get one. Hell, I'd run away with her if she'd have me.\"\n\nSam chuckled. \"She's a little older than you.\"\n\n\"The burgers are worth it. That woman can sure cook. Speaking of women, I just saw Lydia.\"\n\nSam's eyebrows rose and glanced around. \"Where?\"\n\n\"Inside the diner. Having lunch with Kelsey.\" Ryder watched for Sam's reaction.\n\n\"Kelsey? I thought she left. Is she staying with you?\"\n\n\"Nope, and I have no idea where she's staying. She said she's working at the hospital and I suppose she's talking with Lydia to find a place to live.\"\n\n\"Lydia told me she was meeting with someone, which is why she couldn't have lunch with me.\"\n\nRyder grinned, wondering if his friend was suffering a woman the way he was. \"Are you going to marry that woman, Sam?\"\n\nSam's eyes narrowed as he stared him. \"I don't think that's any of your business but since we're friends, I'll tell you. No. I'm not going to marry her.\"\n\n\"Why not? You two have been seeing each other for quite a while now.\"\n\nSam shrugged and tucked his hands into his pockets. \"I like her, don't get me wrong. But something's missing and I don't know what it is.\"\n\n\"Well, seems to me, if something's missing then she's not the right one. A real shame too. She's a beautiful woman.\"\n\nSam nodded. \"I agree. Oh hey, I heard from Riley last night.\"\n\nRyder smiled, the thought of his friend lifting his spirits. \"How is he? I miss seeing him.\"\n\n\"He's doing great. Just sold another house. Rich bastard,\" Sam growled.\n\n\"He's done well for himself, that's for sure.\" Ryder's eyes strayed toward the diner in time to see Kelsey walk out of the door with Lydia.\n\n\"You're in love with her,\" Sam said, the words, more an accusation, than question.\n\n\"No. I'm not. I like her, and I missed her when she left...but love? No.\"\n\n\"Keep telling yourself that, my friend, just keep telling yourself that,\" Sam murmured as he slapped Ryder on the shoulder with his gloved hand.\n\nLydia gave him and Sam a little wave, and then both women climbed into her car. Kelsey ignored them both. Ryder watched them drive away. He knew they were going to see some property for Kelsey. Shit! Clifton sure as hell would never be the same.\n\n****\n\nKelsey resisted looking at Ryder as she followed Lydia to her car. Her heart had nearly stopped when he stepped up alongside her table in the diner. It was all she could do to keep from throwing herself into his arms and begging him to let her come home with him. She'd never been happier to see anyone, as she was Lydia. She arrived at just the right moment because she knew Ryder wasn't going to stop asking her why she was still in Clifton. Her plan was to stay cool, and let Ryder come to her. She hoped he would eventually come to her anyway.\n\nThey got into Lydia's car and as the car pulled out the parking spot, she forced herself not to look but she could almost feel Ryder's eyes following the car as Lydia drove away. Giving into temptation, she glanced in the side view mirror to see Ryder set his hands on his hips. He was watching, which made her smile with satisfaction.\n\nLydia drove them to a small house she'd told her was for rent. When they pulled up in front of it, Kelsey mentally groaned. Even with all the snow piled up around it, she could see it needed a lot of work. She nibbled on her bottom lip as she stared at it.\n\n\"I know it needs a lot of yard work, but the inside is great. I know a few teenagers who would do the outside work for you in the spring. Come on, let's go in. It might surprise you,\" Lydia said opening her car door.\n\nKelsey climbed out the car and stood looking at the house for a moment. It's only temporary if I can convince Ryder to let me come home. Together, they trudged through the snow to the porch and Kelsey watched as Lydia opened a lockbox to retrieve a key. She gave Kelsey a smile and opened the door.\n\nKelsey stepped inside and let out a gasp. It might not have looked like much from the outside but inside...it was beautiful. A large living room with a stone fireplace and wood-beamed ceilings greeted her. Three sets of French doors were across the back, facing out to a small fenced in yard.\n\n\"It's pretty, isn't it?\" Lydia asked her with a smug grin.\n\n\"It is. I can't believe how lovely it is. You're right it certainly did surprise me. I'm loving it already.\"\n\n\"Let me show you the rest of it. It has two bedrooms and one bath. The kitchen appliances are included, and there's a small laundry room off the kitchen.\"\n\nShe followed Lydia through the house. The two bedrooms sat opposite each other down a hallway with the bathroom located closer to the larger room. It was a beautiful little home and she loved it.\n\n\"It's perfect. I'll take it,\" she told Lydia.\n\nLydia smiled at her. \"Great. I have an application with me.\" They returned to the kitchen and Lydia set her briefcase on the kitchen counter and pulled out some papers. \"You just need to put down some references, like your past landlord and some personal references too.\"\n\n\"All right.\" Kelsey stood at the center island and began filling out the paperwork. She was thrilled to get her own place but what she really wanted was to be with Ryder. So she'd carve out her life in Clifton until he came to his senses and realized that she was the one for him, and that he did want her with him. It was all part of her plan. Yes, she knew the possibility of running into him here was very high and she was counting on that. Just like today, in the diner. She had to admit she'd almost choked on her salad when she saw him walk in because she hadn't expected to see him again so soon. She was proud of herself for not falling at his feet and begging him to take her back, even though she wanted to, but as long as she can keep her cool, she would do fine. Or was she expecting too much of Ryder? Was she making a huge mistake staying in Clifton?\n\n\"Is everything all right?\" Lydia asked her.\n\n\"Yes. I'm just wondering if I'm doing the right thing. I mean, what happens if I decide to go back to Atlanta? I don't want the landlord to sue me for breaking a lease.\"\n\n\"Don't worry about that. There isn't a lease. It's month to month. I told them when they put the house up for rent, it would be better to do it that way. A lot of people that come through here think they love it until they experience their first winter.\"\n\n\"I've experienced it already.\" She motioned to the snow once again falling from the sky to add to the piles of snow already accumulated on the ground. \"But that isn't the reason I'd leave...\"\n\n\"It's Ryder, isn't it?\" Lydia asked her lowering her voice as if someone would hear her.\n\n\"Is it that obvious?\"\n\nLydia smiled at her. \"Only to another woman, in love with a man, who's not interested in going any further in a relationship either.\"\n\n\"Who?\"\n\n\"Sam Garrett. I love him more than I thought it possible to love anyone, but he's not in love with me.\"\n\n\"I'm so sorry, Lydia.\" Kelsey reached out and patted Lydia's arm.\n\nLydia waved her hand as if pushing away her emotions. \"It's fine. I think I knew going in that Sam wasn't going to fall in love with me. I don't know how I knew, but I knew. I'll take what I can get with him though...for now. One day, I hope I have the strength to walk away before he does.\"\n\n\"I'm sorry, Lydia,\" Kelsey said again.\n\n\"Don't be. I'll never regret having Sam Garrett in my life. He's an amazing man. He's kind, sweet, sexy, damn good-looking, and smoking hot in bed.\" Lydia laughed. \"Why would I regret that? I just regret that I'm not the love of his life and I envy the woman who does finally capture his heart.\"\n\n\"I know what you mean. I love Ryder so much it hurts, but he's got issues. And because of his fears, he sent me away because he thinks I'll be unhappy and leave him later.\"\n\n\"Maybe when he sees you've rented a place, he'll change his mind,\" Lydia said.\n\n\"That's exactly what I'm hoping, but somehow, I doubt it. He's so stubborn and set in his thinking that I think he truly believes he's done with me, just as he said.\"\n\nLydia put her arm around Kelsey's shoulder. \"Don't give up, sweetie. Sometimes men need to be their own worst enemy before they can see what's best for them. Men. Can't love them. Not allowed to shoot them.\"\n\nKelsey laughed. \"So true. And they say we're difficult.\"\n\n\"Oh, please. Compared to them, we gals are a piece of cake.\"\n\nThey laughed together and being two women in love with impossible men, they hugged. Then Lydia checked her phone for messages while Kelsey fill out the application. When she finished, she handed the forms to Lydia, who glanced over them then put them in her briefcase. After one more look around the little house, they locked up and Lydia drove them back to town. Lydia stopped her car behind Kelsey's car in the parking lot for the diner.\n\n\"I'll let you know as soon as I hear something. I need to get home and grab a shower. Sam and I are going out to dinner later, if this weather lets up.\" Lydia glanced through the windshield at the lightly falling snow.\n\nKelsey smiled, reached across, and hugged her new friend. \"Thanks for everything, and have fun.\"\n\nShe climbed out of the car, waved good-bye, and walked carefully on the rutted ice to the driver's side door. After unlocking it, she got in, shivered, her breath turning to clouds of steam. She turned over the engine, again thankful it started. It was so cold. Winter here was hard, much harder than anything they got in Georgia but Ryder was worth it. The heat seemed to take forever to flow out from the vents, which didn't help. She sat there shivering, waiting for the engine to warm enough to produce some heat. Finally, she began to warm. She cleared the windows of condensation and snow before pulling out of the space. Before she pulled onto the main road, she hit the button for four-wheel drive.\n\nKelsey was still nervous about driving the roads even though the plows or a truck with a snow shovel on the front came through often. Ice covered the roads, and more snow continued to fall covering it again. She hated driving in this kind of weather, but she had to get used to it if she was going to live here.\n\nDriving slowly, nearly at a snail's pace, she kept a watchful eye on the road, and a fair distance from any other vehicles, not that there were many. The further she got out of town, the worse the roads became. The snow was deeper because the plows hadn't come through, as often and there was a layer of ice on top. She could hear her tires crunching through it making her tense and grip the wheel tighter. No one else was on the road but her.\n\nOf course, no one else is on the road. They're smarter than I am and stay home in this stuff.\n\nWhen the backend of the SUV suddenly fishtailed, she let out a little scream, and after a couple of overcorrections, she stopped and breathed in heavy pants. She was terrified. She rested a moment, thinking back to all things she'd learned to do when a car skids. Just because she lived in Georgia didn't mean she had never driven on ice. Ice storms were rare there but they happened.\n\nKelsey gave the car a bit of gas, and proceeded slower this time. She knew she only had to go a little further, and the B and B would be just ahead. She came around a bend in the road and suddenly, a deer darted out in front of her. She did the one thing Ryder told her never to do...she hit the brakes. The SUV slid violently on the slick surface of the road. By the time she thought about lifting her foot from the brake, the vehicle had slid over the right side of the road and down the embankment front end first. Kelsey screamed and the airbag exploded at the same time the SUV slammed into the snow, showering the windshield with sheets of ice and snow before coming to an abrupt stop. She sat there with her hands clamped around the steering wheel, her heart pounding as if trying to escape her chest. Her head hurt, her arms and shoulders ached from gripping the steering wheel, which she continued to do even as she tried to calm herself.\n\n****\n\nRyder sat at his desk in his office. He was supposed to be working on the books but his attention kept straying. He glanced out the window instead, and watched as more snow fell. It looked so soft, but people in this region knew just how deadly it could be. Everyone stayed indoors, unless there was an emergency. It was usually quiet since the people of Clifton picked up what they needed ahead of time when the weather station called for rough weather coming in. The town always seemed to look like a ghost town this time of the year. He knew all the stores closed down when it was this bad out, except the diner. Connie kept it open in case someone needed a hot meal. She lived above the diner so she didn't have to get out on the roads. If she had to, she'd send everyone else home and do everything herself. Connie never turned anyone away and if she thought, for one minute, that someone couldn't pay for their meal, she made them eat it anyway. She was a wonderful woman and everyone in town loved her.\n\nRyder's cell phone rang and he groaned when he saw Sam's name on the call screen. Now what? He really didn't want to talk about Kelsey again. Damn it, he should just let it go to voicemail but knowing Sam, he'd just keep calling until he answered.\n\n\"Hey, Sam,\" he said letting him know he knew who was calling. \"What's up?\"\n\n\"Ryder, there's been an accident...\" Sam began.\n\n\"Accident?\" Ryder got a sick feeling in his gut. \"Who?\"\n\n\"Kelsey.\"\n\nRyder jumped up from his seat so fast he sent the chair against the wall. \"Sam, tell me she isn't...is she hurt?\"\n\n\"She's alive. She has a slight concussion. They've transported her to the hospital. I'm on my way there now. I'll let you know how she is.\"\n\n\"Forget that. I'm on my way.\"\n\n\"Ryder, it's too bad out. There's nothing you can do right now anyway.\"\n\n\"I can be there, Sam, that's what I can do. You're not going to change my mind. I'm on my way.\"\n\n\"Shit. All right, but please drive slowly. I don't need to have you in the hospital too.\"\n\nRyder hung up and ran to the kitchen. He put his coat, hat, and gloves on then ran out the back door. The icy snow crunched under his boots as he trudged through it to his truck. He climbed in, pulled out too fast and the wheels slid reminding him to slow down. Calm down too. He drove the truck carefully down his driveway but then groaned when he saw the road in front of his house. It hadn't been plowed in a while and there were several inches of snow and ice covering it. Sleet pinged off the windshield and the roof of the cab, and he tried to calm down even as frustration over it taking so long to get to the hospital clear on the other side of town tensed him so he was gripping the steering wheel so hard his hands hurt. He had to get there.\n\nKelsey was hurt. Dear God, don't let anything happen to her before I get there. Sam said she was okay, but things can happen after an accident. If he lost her, what would he do...die probably. He needed to call Frank and Grace. Hell, he wouldn't blame them if they said it was his fault. It was his fault. She shouldn't be out on these roads. She wasn't used to driving them. Shit!\n\nRyder decided to wait until he got to the hospital. Right now, he needed to concentrate on the road. Besides, he wasn't in any hurry to have Frank angry with him.\n\nThe usually fifteen minute drive to the hospital took him over an hour. His nerves were firing like pistons in a revved up racecar by the time he parked the truck and made his way into the emergency room. Sam met him at the nurses' station and patted his shoulder in that manly mode of comfort men used for each other.\n\n\"They just took her upstairs to a room. She never lost consciousness so that's real good news. She's just a little sore and a lot scared,\" Sam told him with a relieved grin.\n\n\"Take me up there, Sam, please.\"\n\n\"Follow me.\" Sam led him to the elevators, where they were lucky to catch one before the doors closed. Sam punched the button for the third floor. Once there, Sam led him to the room where Kelsey was.\n\nRyder stepped around him at the door, and felt his knees nearly buckle when he saw her. She was sleeping, or appeared to be, her lashes resting on bluish circles beneath each eye and her skin was pasty white. She had a small goose egg bump colored by a large purple bruise on her forehead just above her left eye. He moved closer to the bed, pulled a chair close then sat. He took her hand in his. It was warm. Thank you, God.\n\n\"What happened, Sam?\" he whispered over his shoulder.\n\n\"Apparently, she lost control of her vehicle on the slick road and went off into the culvert. She was lucky someone happened along before she was there long enough to die from exposure. The collision had popped out the windshield, the airbag deployed, or it could've been worse, but I suspect she was in shock because she didn't call it in. The driver passing by saw the tracks leading down over the embankment. They called it in and I drove out there with the ambulance.\"\n\nRyder took a deep breath and blew it out. \"Thanks, Sam.\"\n\n\"I spoke with the doctor and he's assured she'll be fine. She has a concussion from hitting her head on the steering wheel before the airbag deployed but there's no swelling of the brain, so that's good. I already called her parents and reassured them she's in good hands here and there was no sense in them coming out here. I just wanted them to know, but you may want to call Frank and Grace.\"\n\n\"I will, but I'm sure her parents called them already. I appreciate you being here but you don't have to stick around, Sam. I've got this so you can go, Sam. I'm staying.\"\n\n\"Now, tell me again that you don't love her,\" Sam said with a chuckle as he spun on his heel and left the room.\n\n\"I can't, because I do,\" Ryder whispered to an empty room as he looked at Kelsey tracing over every inch of her face with his eyes. He sat there, not moving just watching her.\n\nDear God, don't take her from me, please. I promise I'll love her for the rest of my life. I'll make her happy. We'll have children and raise them up right. Please.\n\n****\n\nKelsey was positive she'd heard Ryder's voice. Was it just a dream? Why would he be in her house? Oh, wait. As the fog began to clear, she remembered. She was in the hospital. She'd wrecked her rental. Shit!\n\nShe slowly opened her eyes and blinked them against the bright overhead lights. Turning her head, her eyes cleared and focused to find Ryder sitting in the chair beside the bed. His head was back, and his eyes were closed.\n\n\"Ryder,\" she said in a rough voice. She cleared her throat and tried again. \"Ryder.\"\n\nHis eyes flew open, and he sat up. \"Kelsey, how are you feeling?\"\n\n\"My head hurts a little but other than that, I'm okay, I think.\"\n\n\"You scared the hell out of me,\" Ryder said grabbing her hand. She let him hold it. It was nice.\n\n\"Ryder, I'm sorry, I did exactly what you told me not to do. I hit the brakes when a deer ran out in front of me and lost control. I slid right off the road.\" A tear rolled down her cheek as the realization of how close she came to really getting hurt or worse sunk in.\n\nHe squeezed her hand in his rubbing his thumb along the back of it. \"It's okay, baby. You'll be fine. I'm sure you'll be mighty sore tomorrow though.\"\n\nShe stared at him, wondering why he was there when he'd claimed he'd wanted nothing to do with her. She frowned. \"Why are you here?\"\n\nRyder opened his mouth to answer her but the doctor chose just that moment to walk in the door. \"I don't like seeing my nurses in the hospital other than for work, Nurse Sullivan.\"\n\n\"I'm sorry, Doctor Newman.\" She smiled up at the man, catching the frown that marred Ryder's face at her attention to the young doctor.\n\n\"I'll let it slide this time.\" The doctor smiled back. \"Well, you have a slight concussion and although it's not serious, you know the drill. You've been under observation long enough that we're sure you're fine to sleep tonight but I don't want you alone so unless you've got someone who can stay with you, you'll be our guest here in the hospital.\" He glanced at Ryder.\n\n\"She's got me. I can watch over her. I'll take her to my ranch. I can take care of her,\" Ryder told him, rising to his feet like he was taking charge.\n\n\"No. I'll be fine. I'm already settled in here, or I can go home and set my alarm,\" Kelsey protested.\n\n\"Well, with the cost of staying the night here, and taking up space you don't need to, perhaps you should think about going to his ranch. I'm sure you'll be more comfortable and sleep better than here. Rest is the best thing for you now, and if you have any subsequent symptoms, he can assess them and bring you back. An alarm won't do that for you and...well, you know the consequences.\"\n\nKelsey sighed, half elated, and half annoyed. Bruised up, with a bump on her head, and the possibility of vomiting in her future was not how she wanted to spend time with Ryder. \"All right.\" She glanced at Ryder. \"Thank you, I appreciate the offer.\"\n\n\"Christ, I bet that hurt,\" he said to her with a smirk.\n\n\"You have no idea,\" she muttered.\n\nThe doctor's eyes traveled back and forth between them, and then he cleared his throat. \"Well then, I'll release you but you call me in the morning, and be careful going home. Those bad roads are what got you in here in the first place, and believe it or not, we're expecting another foot of snow tonight.\"\n\nHe left the room leaving silence behind him. Ryder shuffled his feet, and Kelsey pulled at a thread on the blanket refusing to look at him.\n\n\"Can you leave now?\" he suddenly asked her.\n\n\"I can leave as soon as the nurse brings in the release papers. If you want, you can go get your truck and bring it around to the front. Someone will bring me out in a wheelchair.\"\n\n\"Trying to get rid of me?\"\n\nShe looked up at him. \"Maybe. I'm still trying to figure out why you're here.\"\n\n\"Sam called me.\"\n\nSo he was here because Sam didn't know who else to call. She sighed.\n\n\"You don't have to do this, Ryder. I'm probably better off staying right here,\" she said pulling the cover up closer to her chin.\n\n\"Well, it looks like they're throwing you out and the doc says you need someone with you. Hell, I don't even know where you're staying. Hotel? Where? It doesn't matter because we're not going to take any chances. If the doctor prescribes, you should be with someone then that's what you're going to do. You can stay in the spare bedroom...again.\"\n\n\"I'm currently staying at the B and B. Thank you, Ryder.\"\n\n\"Okay. Good. I'll go bring the truck around.\" He stared at her for a very long moment, and then headed for the door.\n\nA nurse came in as he was going out, and Kelsey noticed how her gaze traveled over every inch of him as he left. She turned to face Kelsey.\n\n\"Damn, that man is fine.\" She laughed as she approached the bed. \"You are one lucky gal.\"\n\nKelsey laughed. \"I totally agree.\"\n\n\"You're going to stay with him?\" At Kelsey's nod, the nurse fanned herself. \"Maybe I need to have a concussion so I can stay with him. He's one truly fine lookin' man.\"\n\nKelsey knew the nurse didn't mean anything by it but she found herself wanting to seriously slap her, and tell her to back the fuck off. Ryder was hers even if she wasn't his because she wasn't done fighting for him yet. She signed a couple of pages on a clipboard the nurse handed her, and wondered how in the world was she going to spend the night under the same roof as him without wanting him to make love to her. Of course, with more snow coming, she may be there longer than he expected, and although it was going to be hard to be close to him and not throw herself at him, she might be able to use this to her advantage. She loved him so much, and her fight had just begun.\n\nAfter going over the post-release papers, and tucking the instructions and prescriptions into her purse, she changed into her clothes, and waited for the wheelchair.\n\n# Chapter Thirteen\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nRyder drove the truck up to the entranceway and waited. What are you doing? He'd asked himself that question at least a half-dozen times, Once as he standing there in her hospital room, again on the way to the elevator, once more in the elevator, and again as he walked to the exit. Yes, at least a half-dozen times because he asked himself that same question on the way to the truck, and again after he climbed in. So now, he'd asked it for the seventh time, but the answer was still the same. She has to go somewhere so why not the ranch?\n\nThe hum of the truck, and whir of the fan warming the cab settled him. This was his opportunity to convince her to stay. Of course, he still wasn't sure if he deserved her staying but he knew after this night that he didn't want her anywhere else. It was his job to protect her. Frank had given him that responsibility, and damn, if he wasn't going to continue to take care of her. He'd slipped up big time already, and look what happened, she nearly got killed. True, if he convinced her to stay, and if he couldn't make her happy, and she left, it would probably drive him over the edge. Would he turn into his old man then? What if she no longer wanted to stay? What if she realized the truth already, that he wasn't worthy of her loving him?\n\nYou are not your old man! You love her and she did say she loved you so maybe she does.\n\nMaybe she's the one, the one to settle down with and have children with because he suddenly did want children\u2014with her.\n\nRyder climbed out of the truck, leaving it running so the heat stayed on, when he saw an attendant wheeling Kelsey through the automatic doors in a wheelchair. She didn't look happy at all about going with him but too bad, because there was nowhere else she could go...not that he'd let her. Oh, he knew that one of their friends would take her in but knowing Kelsey, she wouldn't want to intrude on them. It was either she go with him or stay in the hospital. He grinned knowing he'd won this round.\n\nHe opened the passenger door, helped her up into the truck, and turned toward the attendant.\n\n\"Thank you,\" he told her, and then chuckled when she winked at him.\n\nHe walked around the front of the truck, keeping his eyes on Kelsey through the windshield. She looked tired. He'd get her home as quickly, and as safely, as possible and get her to bed.\n\nChrist! His dick swelled at that thought of her in bed in his house.\n\nHe climbed into the truck, made sure she was buckled in, which resulted in her pushing his hands away and accusing him of treating her like a child. He grinned, put the truck in gear, and started the long drive home. His concentration was on the roads because they were in terrible condition and the snow was still falling. He glanced over to her to see her gripping the door handle.\n\n\"I'll take it slow, Kelsey. I'm used to driving these roads so trust me,\" he said.\n\n\"I'm all right,\" she said barely above a whisper.\n\n\"It will take us a while to get home, but I'll get us there in one piece.\"\n\n\"I know you will.\"\n\n\"So do you like working at the hospital?\" Ryder asked figuring to keep her mind off the drive.\n\n\"I do, actually. It's a much smaller hospital than the one I worked at in Atlanta, but it stays busy. I really enjoy it.\"\n\n\"That's good,\" he murmured, wondering where to take the conversation next.\n\n\"You never answered me earlier. I know you said Sam called you, but what's the real reason you're doing all this?\"\n\n\"We'll talk about it when we get to the ranch. Right now, I need to concentrate on the drive if you want to get there instead of the hospital again.\"\n\n\"We'll talk about it...which means we're going to argue.\"\n\n\"Kelsey, just sit back and relax. Please.\"\n\n\"All right,\" she muttered with a huff.\n\nRyder saw her lean her head back against the headrest, and close her eyes. His hands tightened around the steering wheel, but it wasn't because of the roads. How was he going to explain to her that hearing she'd been in an accident nearly gave him a heart attack, that it had scared the shit out of him. How was he going to get her to believe that he loved her with everything he was?\n\nAbout an hour later, he finally steered the truck into the ranch driveway and with some slipping and sliding managed to get it up to the house. He parked, shut off the engine, and glanced at Kelsey. She was asleep. He quietly opened his door and stepped out. He went to the kitchen door, unlocked it, and opened it. The dogs came rushing out, greeted him, and then disappeared around to the barn. He moved around the truck to the passenger side and opened the door. Kelsey was still asleep.\n\nRyder scooped her into his arms, pushed the door shut, and carefully carried her inside. He pushed the door shut, and carried her to the spare bedroom. He was just laying her on the bed when she opened her eyes and looked up at him in the half-light filling the room from the hallway.\n\n\"Ryder?\"\n\n\"Yes?\"\n\n\"I want to sleep in your bed\u2014with you.\"\n\n\"Kelsey, I don't think...\"\n\n\"Please,\" she whispered and wrapped her arms around his neck.\n\nHe honestly couldn't think of a reason why she shouldn't sleep in his bed with him so he gave a relenting sigh, scooped her up again, and carried her to his room. He placed his knee on the bed and placed her in the center of it. He helped her off with her coat, and pulled her boots off. He pulled the comforter up over her. He wanted to kiss her but didn't.\n\n\"Get some sleep. I'll be in, in a while, call me if you need me,\" he told her, gently brushing the hair away from the bump on her head.\n\n\"All right. Ryder, thank you for taking care of me...again,\" she said and then rolled to her side, facing away from him.\n\nHe stared down at her, as if wanting to reassure himself that she was okay, and then turned and left the room. He pulled the door closed quietly behind him, and walked to the living room to add more wood to the fire. The wind howled outside the windows.\n\nThe dogs suddenly barked so he returned to the kitchen, and let them in. As soon as they were inside, they shook furiously, and snow and ice flew around the room.\n\n\"Way to go, guys.\"\n\nRyder retrieved two towels from the mudroom to use to dry them off.\n\n\"You enjoy this too much,\" he said as he started to rub Buttons down, only to have her roll to her back. Badger kept bumping against him for his turn. Ryder chuckled. \"You'll get your turn, just hold on.\" He threw the second towel over the dog's back and Badger lay down on the floor and looked up at him with big brown eyes.\n\nWhen Ryder finished with Buttons, he rubbed Badger down too. Once both dogs were dry, they ran to the living room and the warmth of the fire. Ryder followed and sat in his recliner. He picked up the remote and turned the TV on to the weather channel. They predicted more snow and things didn't look good for the area.\n\nRyder sighed and shut the television off. He was tired but what he wanted was to go lie down on the bed with Kelsey, and hold her. She seemed to be talking well enough, argumentative even, and since it was a mild concussion, she should be fine just as the doctor said. She just needed rest and as long as she didn't get sick and woke well in the morning, she'd heal fine.\n\nHe rubbed his jaw and felt the stubble there. He couldn't remember the last time he'd shaved. Pushing himself up out of the recliner, he walked to his bathroom. A shower wouldn't hurt either since he hadn't taken one after working in the barn today.\n\n****\n\nKelsey was awake when Ryder entered the bedroom. She watched him undress and disappear into the bathroom. She thought about joining him when she heard the shower turn on. The thought of him there naked, water flowing over his big body, made her moan. She rolled to the edge of the bed and groaned. As much as she wanted to go in there, seduce him, and prove to him just how much he wanted her, her body ached far too much. She'd wait. She closed her eyes and listened to the comforting sounds of him moving around in the bathroom. She could get used to this and wanted to.\n\nKelsey must have dozed off because she hadn't heard him come out of the bathroom. Instead, the shifting of the bed under his weight had made her aware that he was lying alongside her. She rolled over, his clean manly scent filling her nose and warming her heart. She lifted her hand and stroked it along his cheek. He had shaved, and it pleased her. She leaned close and pressed her lips to his. His eyes sprang open and even in the dark, she saw him smile.\n\n\"You pretend to be all tough and hard-headed, Ryder Wolfe, but I know the truth,\" she whispered.\n\n\"Oh you do, do you? So what's the truth about me?\" Ryder grumbled, catching her hand as it traced lazy circles around his shoulder.\n\n\"You love me but are afraid to admit it because you don't think you deserve it,\" she explained, and felt a sense of triumph when his breath caught in his throat and he blinked his eyes. \"You aren't afraid you'll be like your father, abusive, and cruel. You know better, because you were taught by the best man ever, Frank Sanders. You think that because two selfish, self-centered, hateful people who rejected you, and never gave you love that no one else can give it to you either. Well, I know that Frank and Grace love you, and your friends love you, and I love you. You're surrounded by love, and refuse to acknowledge it. You've built up walls to protect yourself from being hurt and in the process, you're doing to others exactly what you're afraid will happen to you...you're rejecting all who care about you.\"\n\nRyder pulled back and rolled to his back. \"I didn't know nurses had psychology degrees.\"\n\n\"Actually, some of us do. I don't but I studied some psychology courses during training. That's not the point, Ryder. It took me a while to realize what you were doing but I understand it now. You're so afraid of being like you're father that you refuse to even try to live the life you deserve. You're so afraid of not being loved that you're not willing to admit that you love me, and in turn, accept my love.\"\n\nShe paused. She waited, and hoped. He lay there, staring at the ceiling. His breathing was fast as if he had just run a race. Perhaps he had. He'd spent a lifetime running away from the pain and heartache those awful parents of his had forced him to survive\u2014until now.\n\nThen she saw it. A single tear glistened in the reflective light coming through the window from the snow covered world outside, as it slid down the side of his face.\n\nKelsey kissed his cheek. He turned, his shiny gaze meeting hers.\n\n\"I love you, Ryder Wolfe, whether you want me to or not. I love you with all my heart and I'm not going anywhere until you realize just how much.\"\n\n\"I-I don't know,\" Ryder stammered, and she felt him tremble beneath her fingers.\n\n\"You don't know what?\"\n\n\"What if I can't make you happy?\"\n\nKelsey laughed and pressed a kiss to his mouth. She wiped another tear from his cheek.\n\n\"Right now, the way you are treating me is making me miserable as hell so I figure anything you do to try to make me happy is a step up in the right direction, don't you?\"\n\nRyder frowned at her, and then grinned. He pulled her into his arms and cradled her in a way she'd been missing for so long and thought she'd never experience again. She sighed.\n\n\"Now, that makes me happy. Keep going, my stubborn cowboy.\"\n\nHe chuckled and the movement of his chest beneath her head was so very welcome.\n\n\"Tell me now why you came to the hospital tonight?\" She knew the reason but also knew he needed to say it, recognize, and own it.\n\n\"Truth?\" Ryder tilted her face up so she was looking directly into his eyes. She nodded and he closed his eyes for a moment. He was struggling with his fears and his emotions, she knew that, and so she'd be patient with him.\n\nHis brow furrowed and then relaxed. He opened his eyes, allowed them to travel over her face. She didn't look away. She gave him the unspoken support he needed to speak his feelings.\n\n\"When Sam called me to tell me you'd been in an accident, my first thought...my first fear was that you were dead and I had lost the only person on this earth who wanted nothing from me but love.\" Ryder exhaled as if those words had pained him to say. Kelsey imagined they might have. \"I know Frank and Grace love me but I always felt I owed them. I owed them more than I could ever repay. It's why I couldn't refuse Frank asking me to watch over you.\"\n\nRyder pressed his lips to her forehead. \"The relief I felt when Sam said you were okay, I knew I had to confirm for myself, so I raced to the hospital as fast as this damn weather would allow me to see for myself. You looked so vulnerable in that bed, so pale, with bruises, and that bump on your beautiful head.\" He lightly kissed her goose egg bump and Kelsey nearly broke into tears. It was the sweetest more romantic thing anyone had ever done.\n\nHe brushed her hair back from her face and stared down into her eyes. \"You're going to have to be patient. You're going to have to bear with me while I get a handle on learning how to accept and return love, Kelsey, but know this...if you'd died tonight, I would never be able to go on living. Yeah, I was kind of mad at you for still being in town but that was because I'm a selfish bastard who couldn't stand the thought of you being so close and not being able to have you.\"\n\nShe started to speak but he put his finger to her lips. He swallowed hard.\n\n\"I love you, Kelsey Sullivan. I'm a stubborn, pigheaded cowboy who didn't know enough to hang onto something worth more than all the gold in the Black Hills but almost losing you tonight made me learn it real fast. If I hadn't sent you away, you wouldn't have been out on that road in this bad weather and I wouldn't have almost lost you so feel free to lay the blame on me that you got hurt. But I promise you this, darlin', I'll never let anyone or anything ever hurt you again, including me.\"\n\n\"Ryder, I've known that all along. You just needed to learn it about yourself.\"\n\nThey stared at each other, feeding each other with love, until he lowered his head and pressed his lips to hers. She moaned and he deepened the kiss, moving his tongue inside. She ran her hands up his bare chest and into his still damp hair. But when her hand slid down along his back, and then moved around front and slid over the bulge pressing against her thigh, he grabbed her wrist.\n\n\"Kelsey, what the hell do you think you're doing? You have a concussion.\"\n\n\"I have a slight concussion and I'm fine. I don't even have a headache. The only medicine I need is you.\" She pressed her mouth to his neck and he groaned.\n\n\"I don't want to hurt you,\" he whispered. She thought about pretending not to hear it but he had to understand he couldn't hurt her except by rejecting her.\n\n\"The only way you'll hurt me is by turning me away.\" She placed her hands on his face and forced him to look at her.\n\nHe lowered his head and pressed his lips to hers. He wanted her and she was ready to let him have her, bruises and achy body, be damned.\n\nRyder pulled back, breaking off the kiss. He pulled her in tight to him until she moaned slightly and then he let go and looked down at her. \"See, I can't even hug you without it hurting. You need to sleep. That was doctor's orders.\"\n\n\"Okay,\" she said with a pout of her lower lip. \"But I'm not leaving again. I'll call Lydia and tell her not to bother with the rental house.\"\n\n\"No argument from me, darlin',\" Ryder exclaimed with a chuckle. \"I doubt either of us is going anywhere tomorrow or possibly even the next day, considering the way it's snowing out there.\"\n\n\"Well, that's just fine with me too,\" she said with a happy smile before curling into his warm, protective body and kissing him one more time before she would allow sleep to claim her.\n\n****\n\nRyder woke feeling more content than he can ever remember feeling at any time in his life. At first, he thought it was just the warmth of his bed making him feel so comfortable but when Kelsey made a little snoring sound alongside him, he remembered the reason for his good mood. She was here. She was here with him, and other than a little banged up, she was healthy, and he had to believe, happy. He pulled her closer in his arms, and she wiggled a bit.\n\nHe pressed a kiss to her forehead. It was warm, but not hot so she hadn't any fever resulting from her minor injuries. He'd read the post-release instructions and so far so good, as long as she knew him, and where she was when she woke.\n\nHe traced his fingers lightly over her bruised forehead, around the curve of her brow, and then down the sweet tilt of her nose. She crinkled her face and opened her eyes just a flutter. She squinted at him, and then smiled.\n\n\"Hi,\" she said, her voice still sleep laden. \"I'm still here.\"\n\n\"Yes, you are. Where are you, darlin'?\"\n\nShe wrapped her arms around his neck, slid her fingers into his hair, and pulled him close to her before pressing her lips to his. \"I'm right where I belong, in your house, in your bed, and in your arms.\"\n\n\"You seem perfectly lucid to me,\" Ryder said with a laugh before capturing her mouth with his in a deep, tongue-sweeping kiss. Suddenly, he pulled back. \"How sore are you?\"\n\nKelsey grimaced, pulled away, stretching her arms above her head, and pushing her legs toward the bottom of the bed. She moaned a little and then smiled. \"A little stiff but nothing some exercise won't remedy.\" She waggled her eyebrows at him. \"And why am I still dressed?\"\n\n\"Good question, unless you feel the need to go somewhere?\" Ryder teased her.\n\n\"Hell no, I'm not going anywhere, and I won't let you make me this time.\"\n\nRyder pulled her back into his arms, and put his forehead to hers careful not to hit her bump.\n\n\"The only way I'll ever let you go again is if you ask me to and tell me you don't love me anymore.\"\n\n\"Wow, that long, huh?\" Kelsey smiled up at him.\n\n\"I'll never hurt you, Kelsey. I promise you.\" Then suddenly, he grinned at her wickedly. \"Unless I have to turn you over my knee, of course.\"\n\nShe laughed up at him. \"Oh, are you into that?\"\n\nHe burst out laughing. She always was good at giving it right back.\n\n\"No.\" He sobered. \"I'd never even do that to you.\"\n\n\"I know. I know.\" She kissed him and he believed her. She trusted him. \"I need you, Ryder. I need to feel you inside me, please.\"\n\nHe growled low in his throat. \"Are you sure you're up to it?\"\n\n\"Yes, I am and I know you are...up to it.\" She said with a sly grin as she slid her hand down to cup his already eager cock.\n\n\"It does seem to have a mind of its own, and it wants you.\"\n\nRyder quickly helped her undress and had her moaning because every inch of skin he uncovered, he kissed, nuzzled, and licked until she was squirming and begging him to take her. He pulled her over on top him so she straddled his hips. He figured he was less likely to put any bruises on top of her bruises if she were on top.\n\nShe surprised him by reaching past him, one of her pink tipped breasts falling so near his mouth that he couldn't resist suckling it and pulling a soft moan from her as she retrieved a condom from the nightstand.\n\nShe ran her hand along his jaw. \"Someday, I hope to convince you to stop using these.\"\n\n\"One step at a time, Kelsey, let me get used to idea of having someone as wonderful as you love me first.\" She nodded, and rolled the condom onto him before lifting her hips and slowly, and teasingly lowering herself on him until he was completely sheathed inside her.\n\n\"Of course, if you're really serious about kids, I figure we might have to get started soon if we're going to have ten or twelve.\"\n\nShe squealed. \"What?\"\n\nHe laughed and she bounced on his cock. He loved her so much that if she wanted that many, he'd give them to her. She raked her nails gently down his chest and he pretended it hurt. \"When the time comes, I'll do my best to be the best kind of dad.\"\n\nKelsey gasped, and slid forward, her hands wrapping around his head, pulling his hair. \"Can we talk about this later?\"\n\nRyder chuckled. \"Yeah, I have other things on my mind right now too.\" He grabbed her hips and rocked her against as he kissed her.\n\nKelsey groaned and tore her mouth from his. \"Ryder...\"\n\n\"I'm with you, baby,\" he groaned in her ear.\n\nShe screamed out his name as her orgasm hit her hard. Ryder bent his knees as she ground her hips against his until he growled low in his throat and then groaned out her name as he came. He felt her body react again and her insides clench and throb as she came again. She collapsed on his chest, her breathing fast and hot against his skin.\n\n\"Are you all right?\"\n\n\"Yes. God, that was fantastic,\" she said in a breathless voice.\n\n\"I think more rest is in order then, don't you?\"\n\n\"I'm all for that.\"\n\nRyder pulled the covers up over their quickly cooling bodies, slid his arms underneath, and cradled her against him. They slept until noon, and then made love again.\n\n# Chapter Fourteen\n\nLife was good and it was about to get even better. It was Valentine's Day and Ryder had it all planned out. He'd ordered dinner from the restaurant where she'd driven him nuts and they hadn't finished their meal. He thought it was appropriate for tonight. He laughed softly thinking about the surprises, he had lined up for Kelsey. He knew she'd been hinting around for the last few weeks about him making a commitment to her. He'd needed time and he was ready but he wanted to do it right.\n\nHe patted the pocket of his jacket where the beautiful diamond ring he'd sent to Butte for rested in its velvet box. He thought about the cedar rocking chair he'd originally handcrafted for a client but when he saw how beautiful it was when it was finished, he could only imagine Kelsey sitting in it rocking their child. He promised his client, he'd make another for him.\n\nThe sun was shining for a change, the big sky was blue, almost as blue as Kelsey's eyes, Ryder thought with a smile. There hadn't been any snow for the past week so the roads were finally clear and the trip to Hartland and back hadn't taken nearly as long as he'd expected. He planned to fix a nice bath for Kelsey while he set up dinner, brought in the chair, and hopefully, didn't lose his nerve before asking him to be his wife.\n\nAs he made the turn into the driveway, he stopped the truck. The gate was open when he knew he'd shut it when he left. He drove slowly up the drive and stopped again when he saw an unfamiliar car parked by the back door. He swore when he saw Utah plates on the car. He pulled out his cell phone and made a call.\n\n\"Sheriff Garrett,\" Sam answered.\n\n\"Sam, it's Ryder. I think my...parents are here at my ranch. There's a car here with Utah plates and that's where they were last time I heard from them. Kelsey's inside.\"\n\n\"I'm on my way with Brody. We'll come in silent. Be careful.\"\n\nRyder hung up and after taking a deep breath, he opened the door and stepped out. He walked passed their car, up the steps, and entered the kitchen.\n\n\"Kelsey?\" he called out. He'd let the dogs out when he left but now wished he hadn't.\n\n\"Come in here, son. We're waiting for you.\" He recognized his old man's voice.\n\nSon of a bitch! Ryder walked into the living room and came to a halt when he saw his father holding a gun to Kelsey's head.\n\n\"Royce, you seriously need to let her go,\" Ryder growled, anger fueled adrenaline surging through him. He wanted to rip the bastards head off for even touching Kelsey.\n\n\"I will. When you give me what I want,\" Royce Wolfe replied.\n\n\"I'm not giving you a fucking thing. Let her go\u2014now.\"\n\n\"Aren't you going to say hello to your mother, son?\"\n\n\"I am not your son and she,\" Ryder said, pointing at Heather Wolfe, \"is not my mother.\"\n\nRoyce laughed. \"She brought you into this world.\"\n\n\"And you tried to take me out of it, how many times, old man? Get off my ranch or I'm calling the sheriff.\"\n\n\"Your friend, Sam, he's the sheriff, isn't he?\" Heather Wolfe finally spoke.\n\nRyder glared at her and then swung his gaze back to Royce. \"I'm not going to tell you again. Let her go.\"\n\n\"How much is she worth to you, Ryder? Tell me that.\"\n\nRyder didn't say a word. He continued to glare at Royce. His eyes shifted to Kelsey. She looked pale and scared. She had to believe he wouldn't let anything happen to her. When he saw a tear roll down her cheek, he wanted to tear Royce Wolfe apart with his bare hands.\n\n\"Has he hurt you?\" he asked Kelsey.\n\nShe shook her head and gasped when Royce's arm tightened around her neck. Ryder clenched his jaw.\n\n\"I want a hundred thousand dollars. Now!\"\n\nRyder laughed without humor. \"Do you really think I'd have that here?\"\n\n\"If you don't, then it looks like your mom and I are spending the night until you get to the bank. Or I will kill this little woman of yours. It would be a pity. I have to say, son, you do have great taste in women. She's a real looker.\" Royce pulled her back against him. Kelsey whimpered.\n\n\"You hurt one hair on her head, you're a dead man,\" Ryder said taking a step closer.\n\n\"Then get me the fucking money,\" Royce shouted waving the gun at him before pressing it back against Kelsey's head.\n\nClenching his fists, Ryder glared at the man who sired him before nodding. He stepped backward before starting down the hall.\n\n\"Hold up. You're not leaving my sight.\" Royce followed him, dragging Kelsey behind him.\n\nRyder entered his bedroom and opened the closet doors. He walked to the back wall, reached above the shelf located about halfway down, and pressed a button. The lower part of the wall slid open to show a safe.\n\nRoyce laughed.\n\n\"Open it,\" he told Ryder waving the gun at him again.\n\nRyder squatted down and entered the combination. The safe door clicked open. Royce Wolfe gasped, and Ryder slowly came to his feet.\n\n\"Holy shit! How much is in there?\"\n\n\"I'll give you what you asked for\u2014\"\n\n\"Oh no. You're going to give me everything in that safe. Heather, get a bag.\"\n\n\"Where?\"\n\n\"Jesus Christ, woman. Tell her where she can get a bag, son.\"\n\n\"In the pantry, and quit calling me son. I am not your son. I was never your son. I was a fucking punching bag to you. How about we see if you can hit me now?\"\n\n\"Ryder, don't give them your money,\" Kelsey choked out, only to have the arm against her throat cut off her words.\n\n\"The money doesn't matter, Kelsey. Only you do.\"\n\n\"How sweet,\" Heather said entering the bedroom with a cloth bag.\n\n\"Give the bag to Ryder. Fill it up...son.\" Royce laughed.\n\nRyder squatted down again and started to fill the bag but slowly. He knew he needed to give Sam and Brody time to get here. He pulled stacks of money out of the safe and dropped it into the bag.\n\n\"There has to be over a million dollars in there,\" Heather said in awe.\n\n\"Yeah, and our son never wanted to help us out. He's so fucking rich, that he keeps a few mil in his own safe.\"\n\n\"Fuck you, old man,\" Ryder growled.\n\n\"I'd be nice if I were you, I have a gun to your woman's head.\"\n\n\"And you're going to drop that gun or I'll drop you,\" Sam said from the doorway. Ryder let out a sigh of relief.\n\n\"I'll kill her, Garrett,\" Royce shouted over his shoulder.\n\n\"I don't think you will,\" Brody said from behind him. He'd entered through the French doors.\n\nRoyce spun around dragging Kelsey with him. \"Damn it. Heather, why weren't you watching?\"\n\n\"How was I supposed to know someone would come in the damn patio doors?\"\n\n\"Lower your weapon. Now,\" Sam demanded as he held his weapon in front of him pointed at Royce's head.\n\n\"Son of a bitch,\" Royce said as he lowered the gun and released Kelsey who pushed away from him and ran straight to Ryder, where he caught her and wrapped his arms around her.\n\nRyder watched over Kelsey's head as Brody stepped up behind Royce and yanked his arms back before putting handcuffs on him. Meanwhile, Sam was handcuffing Heather.\n\n\"Are you both all right?\" Sam asked Ryder who was cradling Kelsey as she cried softly against his chest.\n\nRyder hugged her to him. \"Yes. Get them out of here, please.\"\n\n\"Do you want to press charges?\"\n\n\"You know I do, Sam.\"\n\nSam grinned. \"Yeah, I do but I have to ask.\"\n\nRyder nodded as Sam and Brody led them away. He tightened his arms around Kelsey and rubbed his cheek against the top of her head. She pulled back from him and gazed up at him with tears glistening on her lashes.\n\n\"You were going to give them all that money. You said you would never give them money.\"\n\nRyder laughed. \"Are you fucking serious? That money doesn't mean a damn thing to me. Only you do.\"\n\n\"How much is in there?\"\n\n\"Two million.\" He shrugged. \"I like to keep cash on hand.\"\n\nKelsey burst out laughing. \"When people keep cash on hand, it's not two million dollars.\"\n\nRyder shrugged. \"What can I say? I just feel better with it here.\"\n\n\"But you were going to give them all your money.\"\n\n\"No, I wasn't and even if they'd gotten away with it, I have a lot more than that in the bank. At least ten times that. I'm a very rich man, baby.\"\n\n\"I wouldn't care if you were dirt poor.\" She rose on her toes and kissed him.\n\n\"I know.\" He leaned back and looked down at her. \"This wasn't at all how I wanted to do this but it seems like the perfect time,\" Ryder said, reaching into his coat pocket. When he withdrew it holding the dark blue velvet box, he heard Kelsey gasp. She was surprised anyway and that's all that mattered. Dinner and the rocker will be the cherry on top. No, he planned to be the cherry on top.\n\n\"Will you marry me, Kelsey Sullivan?\"\n\nShe stared up at him, glanced back to the box he'd opened and gasped at the sparkle on the two-carat diamond in the box. Glancing up again, she had tears glistening in her eyes. \"Are you sure?\"\n\n\"Yes. I think it would be best to get married since we want children together.\" He grinned at her.\n\nShe smiled, and then threw her arms around his shoulders. \"I love you, Ryder, with all my heart and yes, I will marry you.\"\n\n\"I love you, darlin'. I really had already decided on all of this. I've been planning it all out for two weeks now, and then you go getting yourself into trouble again and ruin things.\" Ryder grumbled against her shoulder and when she pushed away and punched him in the arm, he laughed. \"Seriously, I was scared to death that bastard was going to slip and blow your head off. I don't know what I would've done if something had happened to you. No, I do know. I would've torn that old man to pieces and ended up on death row. See, I'm not so different from him after all. I wanted to kill him, Kelsey.\" His voice caught.\n\nKelsey hugged him to her. \"You're nothing like him. You were angry, and you wanted to kill him for hurting me, but you didn't. I knew you'd never let anything happen to me. I knew you wouldn't let him hurt me, which is why you were willing to give him all that money. Nothing and no one will ever pull us apart, and nothing you say or do will ever make me leave. I love you, Ryder, and this is the best Valentine's Day ever.\"\n\nHe grinned at her. \"Well, it was supposed to be a lot better.\"\n\n\"And so it shall be. So what else have you planned for me, Ryder Wolfe, because I have some very interesting things planned for you,\" Kelsey said as she slid her hands up under his shirt and around back so she was pressed tight against him.\n\n\"I like the way you think. Well, first you were going to take a bath while I set up dinner, and then I have another surprise for you.\"\n\nShe stepped back and looked at him with wide eyes. \"And you're worried about me ever leaving you. If you keep being so romantic, Ryder Wolfe, you'll have to pry me off you and fight me away with a stick.\"\n\n\"It won't be easy, since you're rather demanding and have a hell of a temper at times.\" He laughed when she punched his arm.\n\n\"I'll try to curb it around you.\" She laughed when he scooped her up into his arms.\n\n\"So when's the wedding?\"\n\n\"I like May,\" she said before pressing a kiss against his lips.\n\n\"May, it is. You know, I'm thinking I might join my fianc\u00e9e in that bath and wash the bad of the day off me. Is that okay?\"\n\n\"I wouldn't have it any other way. We can grab a condom on the way.\"\n\n\"Got one in my wallet. I've learned to be prepared with you.\" He carried her into the bathroom and kicked the door shut behind them.\n\n\"So what's my other surprise?\"\n\n# Epilogue\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nRyder sat in a chair in the back room of the church and gazed out the window. It was a beautiful day in May, and for the first time in his life, he felt at ease.\n\n\"Why aren't you nervous?\" Jake asked him placing a hand on his shoulder and squeezing gently.\n\nRyder grinned up at him. \"Because everything is as it should be, and I have absolutely no doubt she'll be here and that she loves me.\"\n\n\"Yeah, well, I knew that about Becca too, but I was still nervous as all get out.\"\n\n\"Okay, maybe I am a little but I think it's more about having to stand up in front of everyone.\"\n\nJake laughed and nodded. \"That makes me feel a little better. I think you're right. Wyatt tried to convince us he wasn't nervous either.\"\n\n\"I wasn't,\" Wyatt said with a chin-raised confidence.\n\n\"Bull\u2014\"\n\n\"Don't you dare say that in my church, Sam Garrett.\"\n\n\"Sorry, Padre,\" Sam said repentantly, and then glared at his friend, Riley Madison. \"What are you laughing at?\"\n\n\"You always did get into trouble with the Rev,\" Riley said.\n\n\"Shut up,\" Sam muttered.\n\n\"It's too bad Katie couldn't make it. Kelsey invited her but Sadie is sick,\" Ryder said.\n\n\"I haven't seen Katie in years. I'd love to see her.\" This eager remark was from Riley.\n\n\"You stay away from my sister,\" Sam growled. The men laughed. Sam had told all of them that same thing when they were growing up around his sister.\n\n\"I'm glad you finally got over your fears, Ryder. We all knew who made you into the good man you are today,\" Gabe told him.\n\n\"I know. It took me a while to accept it though. Kelsey has helped me realize a lot of things about myself but yes, Grace and Frank are my real parents, even if not by blood. I'm so glad I have them in my life. Frank actually got tears in his eyes when I asked him to be my best man. Hell, never thought I'd see something like that.\"\n\nThe door opened and Frank stepped inside. The room went silent, and every man turned to look at him. He stopped in his tracks as if he'd done something he shouldn't.\n\n\"What?\"\n\n\"Nothing,\" Ryder chuckled. \"We were just talking about you, Dad. I was saying how glad I am to have you and Mom in my life. For making me the man, I am today and for the best thing of all...for bringing Kelsey into my life.\"\n\nFrank grinned and shrugged. \"Well, I'm certainly happy you two fell in love. Your mom and I couldn't be more pleased about it.\"\n\n\"You didn't plan it all along, did you, Frank?\" Riley asked with suspicion in his tone.\n\n\"I refuse to answer that on the grounds it may incriminate me.\" Frank grinned.\n\nRyder narrowed his eyes at him. \"Are you serious? You planned this?\"\n\n\"Well, let's just say when Kels was being stalked, it was the perfect opportunity to get you two together. Your mom and I had often said it was a shame you two lived so far apart.\"\n\nRyder stared at him, his mouth opened as if he wanted to say something, and then he suddenly burst out laughing. \"Well then, Dad, I appreciate you sending her to me even more than you know.\"\n\n\"Anyone want some champagne?\" Frank asked and laughed when the men screwed up their faces and shook their heads. \"Okay then, I'll be right back.\" He left the room.\n\n\"Where's he going?\" Brody asked.\n\nRyder shrugged. \"No idea. Hope it's not to get champagne.\"\n\nA few minutes later, Frank returned, pushing the door open and carrying a small cooler. He set it down on the floor and pulled the lid up and off. The men all looked inside with some hesitation and then started to laugh. Inside were nine bottles of beer.\n\nTrick squatted down and reached inside the cooler. He handed each man a beer and took the last one for himself. They twisted off the tops, tapped the bottles together, and took a swig.\n\n\"Who needs champagne when you can have beer?\" Ryder asked, smiling and happy surrounded by the best men he knew.\n\n****\n\n\"Who wants champagne?\" Grace asked the room full of chattering women.\n\n\"We all do, of course. Thank you.\" Kelsey heard Olivia say.\n\nGrace handed out the rest of the flutes. \"I can't believe Kelsey asked me to be her matron of honor.\"\n\nKelsey smiled when she heard Grace express her excitement over being her matron of honor.\n\n\"She loves you, Grace. You and Frank were responsible for bringing Ryder into her life,\" Emma said with a smile as she clinked her glass to the one Grace was holding.\n\nKelsey glanced in the mirror over the sink in the bathroom. It was her wedding day and she was happier than she'd been in her entire life. In the room adjacent were the women who had become as important to her as if they were her sisters. Grace was the mother who loved Ryder, and the others had fast become her best friends these past months. Life was perfect.\n\n\"Can I just say that Riley Madison is absolutely gorgeous? Those whiskey-colored eyes are so hot. I think Sam's blue eyes are gorgeous too, but Riley's are like the color of fine aged Scotch. I wouldn't mind taking a sip, he's really sexy too,\" Olivia said.\n\n\"Olivia Rene Roberts Stone, can you remember even for a moment that you're married?\" Becca asked her. Kelsey chuckled hearing this. That's Olivia.\n\n\"What do I always say? I'm married, I ain't dead. I can still think a man is gorgeous, can't I?\"\n\nEmma burst out laughing. \"Give up, Becca. I have.\"\n\n\"You know, sometimes I wonder if Frank actually planned it all. Sending Kelsey here to Montana, I mean. We'd often talked about introducing them but they never seemed to both be in Atlanta or available at the same time. We could have kept her safe back home in Georgia.\"\n\nBecca laughed. \"Well, if he did, it worked out perfectly.\"\n\n\"I think it's romantic,\" Madilyn said.\n\n\"Well, almost perfectly. I suppose if Ryder Wolfe hadn't been such a stubborn thick-skinned cowboy it could've gone easier,\" Kelsey said as she entered the room from the bathroom. She spun around and heard the women making oohing sounds. \"So what do you think?\"\n\n\"Oh my God, Kelsey. You are so beautiful,\" Kaylee said.\n\n\"You are certainly a gorgeous bride,\" Anne Llewellyn said with a big smile.\n\n\"Oh, baby. You look so beautiful,\" Donna Sullivan said through tears.\n\nLydia reached for a flute and raised it. \"Thank you for asking me to be in the wedding, Kelsey. This will probably be the only way I'll walk down the aisle with Sam.\"\n\nKelsey wished she could give Lydia encouragement that Sam might come around but the woman knew him far better. She twirled in her white gown with its scoop neckline and sheer sleeves that buttoned at her wrists. The skirt flared out from her hips to fall to the floor with a long train behind her, which she held high by means of a loop over her fingers.\n\n\"I feel like a princess,\" Kelsey said as she blinked back tears.\n\n\"Oh no! No crying allowed,\" Grace chastised her, pressing a tissue to her cheeks.\n\n\"I'm trying not to. Quick, someone give me a champagne,\" she demanded.\n\nGrace laughed as she handed her a flute. \"Just don't get drunk on us, either.\"\n\nKelsey laughed. \"No way of that happening. I want to make sure I'm coherent and this is really happening.\" She nibbled on her bottom lip. \"I sure hope he shows up.\"\n\n\"Oh, honey, don't you worry. With Frank as his best man, he doesn't have a choice,\" Grace told her with a big grin.\n\n\"That's true.\" Kelsey smiled and then frowned. \"But I want him to have a choice. I don't want him to marry me if he doesn't want to.\"\n\n\"Okay, you are going to drive yourself insane if you keep thinking this way. Grace meant it as a joke. Ryder will show up because he loves you,\" Olivia told her as she put her arm around Kelsey and hugged her.\n\n\"You're right. He does love me. I just can't wait to marry him. I love him so much.\"\n\nA knock on the door startled the women, and they laughed at their foolishness. Grace opened it and allowed George Sullivan entry into the room. He smiled with a pride only a father could have for a daughter on her wedding day.\n\n\"You look beautiful, honey,\" he said as he walked to stand before his daughter, taking her hands in his.\n\n\"Thank you, Daddy. I'm so ready to do this.\"\n\nHe kissed her cheek. \"I'm sure Ryder is too. He's a wonderful young man. I'm glad I rented a van for the trip to the church.\" He put his arm out for Kelsey to loop hers through, and together, with her favorite women following, they began their journey to the church.\n\nThe small church was packed with well-wishers and as with every other wedding that took place in Clifton, whoever couldn't fit inside, stood outside in the warmth of a beautiful spring day in May. When the car pulled up to the church, people cheered. Kelsey, Donna, Anne, and Grace laughed in surprise.\n\n\"Welcome to Clifton, Montana, possibly the friendliest place on earth,\" Becca said. \"Just about everyone in town shows up for weddings. It seemed to have started with Jake's and mine. I was so shocked to see everyone crowded inside and out on the day we got married.\"\n\n\"They've never failed to miss one either,\" Emma added. \"Except mine...but then I didn't have a big wedding.\"\n\n\"But you still got the man of your dreams, Emma,\" Olivia said hugging her. Emma nodded with a big smile on her face.\n\nGeorge stepped from the van and opened the back door. \"Come on, honey. Let's get you married.\"\n\nKelsey stepped out in the sunshine, and the crowd roared. She laughed and waved at them feeling like a celebrity. The other women stepped from the van and more cheers went up. They entered the church, settled into formation with Grace extending her train out to its full-length behind her, and then waited for the music to start signaling the moment she'd been waiting for since February. When it did, her insides fluttered with excitement and if she had to admit it, nerves as she watched each woman walked down the aisle to the altar in front of her.\n\nWhen it came time, and she heard the music switch to her procession, Kelsey took a deep breath and gave her dad a nod. He winked at her and led her to the altar\u2014and Ryder.\n\n****\n\nWhen the music started, Ryder suddenly felt the nerves he'd been denying all day rush over him. He watched as each of Kelsey's bridesmaids walked down the aisle and took her place opposite of where the men stood. He almost laughed aloud when he heard Sam mutter something about Lydia not getting any ideas.\n\nThen the music changed, every person in the church came to his or her feet, and his gaze settled on the entrance at the back of the church. She was there. His heart skipped a beat when he saw Kelsey. She looked more beautiful than he could ever imagine possible since she was already the most beautiful woman he'd ever known. She walked slowly down the aisle toward him, her eyes meeting his, and wearing a huge smile. She was with her dad, and George had the biggest, proudest grin on his face as he walked with his daughter.\n\nAt the altar, he kissed Kelsey on the cheek and put hand in Ryder's open palm. \"Keep her safe, son\u2014forever.\"\n\n\"Yes, sir, I most certainly will,\" Ryder told him and then he turned to face Kelsey. \"Ready, darlin'?\"\n\n\"More than you'll ever know.\" Kelsey smiled at him and his heart hit his stomach.\n\n\"You look so beautiful. You absolutely take my breath away.\" He lifted the hand resting in his and pressed a kiss to the back of it.\n\n\"You look very handsome yourself, cowboy.\"\n\n\"Let's do this, baby.\"\n\n\"Are you both ready?\" The reverend asked, drawing their attention. When they both nodded, he began the ceremony. What seemed like a heartbeat later, he closed the Bible and looked at them, and then smiled.\n\n\"What are you waiting for, Ryder? You may now kiss your bride.\"\n\nRyder grinned, lifted Kelsey's veil over her head and cupped her face in his hands. He lowered his head and kissed her. When she moaned, he pulled her closer to him, feeling her hands slide around his waist, and he deepened the kiss. He raised his head and stared down into her blue eyes, which today, only the May sky rivaled.\n\n\"I knew you were trouble the minute you walked into the bar.\"\n\nKelsey burst out laughing and threw her arms around his neck. Ryder laughed and scooped her up, tugging her train up and over her legs. Amid cheers, some happy tears, and best wishes, he carried his bride, the woman who taught him about love, out the door into the sunshine to start their new lives together.\n\n**THE END**\n\n# About the Author\n\nSusan was born and raised in Cumberland, MD. She moved to Tennessee in 1996 and now lives in a small town outside of Nashville, along with her husband and their two rescued dogs. Although, writing for years, it was recently she decided to submit to publishers and signed a three-year contract with Secret Cravings Publishing. Since SCP closed their doors in August 2015, Susan is now with Blue Whiskey Publishing. She is a huge Nashville Predators hockey fan. She also enjoys fishing, taking drives down back roads, visiting Gatlinburg, TN, her family in Pennsylvania, and her hometown. Although Susan's books are a series, each book can be read as standalone books. Each book will end with a new story beginning in the next one. She would love to hear from her readers and promises to try to respond to all. You can visit her Facebook page and website by the links below.\n\nhttps:\/\/www.facebook.com\/skdromanceauthor\n\nwww.susanfisherdavisauthor.weebly.com\n\nsusan@susanfisherdavisauthor.com\n\nOther books by Susan Fisher-Davis:\n\nJAKE Men of Clifton, Montana Book 1\n\nGABE Men of Clifton, Montana Book 2\n\nBRODY Men of Clifton, Montana Book 3\n\nWYATT Men of Clifton, Montana Book 4\n\nLUCAS Bad Boys of Dry River, Wyoming Book 1\n\nMONTGOMERY Bad Boys of Dry River, Wyoming Book 2\n\nCOOPER Bad Boys of Dry River, Wyoming Book 3\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nSome names and descriptions have been changed, some characters have been conflated, including people the author encountered in treatment, and certain events have been compressed.\n\nCopyright \u00a9 2018 by Janelle Hanchett  \nJacket design by Amanda Kain  \nJacket photograph \u00a9 plainpicture\/Readymade-Images\/Franck Juery  \nCover copyright \u00a9 2018 by 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\nHachette Books  \nHachette Book Group  \n1290 Avenue of the Americas  \nNew York, NY 10104  \nhachettebooks.com  \ntwitter.com\/hachettebooks\n\nFirst Edition: May 2018\n\nHachette Books is a division of Hachette Book Group, Inc.  \nThe Hachette Books name and logo are trademarks of Hachette Book Group, Inc.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\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\nLibrary of Congress Control Number: 2017952977\n\nISBN: 978-0-316-54943-1\n\nE3-20180403-DA-NF\n\n## Contents\n\n  1. Cover\n  2. Title\n  3. Copyright\n  4. Dedication\n  5. Part One\n    1. 1: Family Planning on Ecstasy\n    2. 2: Stop Drinking When Your Lips Go Numb\n    3. 3: Sylvia Plath Put Her Head in the Oven over This Shit\n    4. 4: Playing House in the Suburbs with Captain Morgan\n    5. 5: Three a.m. Ideas\n    6. 6: Rocketship Rock-On\n  6. Part Two\n    1. 7: Failure That Isn't Funny\n    2. 8: Nothing Left to Hide Now\n    3. 9: Who's the Sickest in the Room?\n    4. 10: Maintenance Whiskey\n    5. 11: I Found God in a Leaf Blower, and I Fucking Hate Leaf Blowers\n  7. Part Three\n    1. 12: There Are Three Types of Mothers in the World\u2014I Am None of Them\n    2. 13: Failure That Isn't Funny: Sober Edition\n    3. 14: The Childhood I Could No Longer Blame\n    4. 15: What the Hell Is \"Soul Work\"?\n    5. 16: In the Blood of Our Mothers\n    6. 17: If I Knew the Way\n  8. Acknowledgments\n  9. Newsletter\n\n##  Navigation\n\n  1.  Begin Reading \n  2.  Table of Contents\n\n_Hey, Mom. Look. We did it._\n\n# **Part One**\n**1**\n\n# **Family Planning on Ecstasy**\n\nThe first thing I did when I found out I was pregnant with my first child was head out to the balcony of our one-bedroom apartment and smoke a cigarette. It wasn't even a real balcony. It was a gray stoop barely big enough for one unwatered plant, a dusty mat, and a twenty-one-year-old in vague denial. I would have preferred outright denial but found it impossible, having just peed on two sticks offering no ambiguity.\n\nMy plan was to formulate a plan out there on the balcony before informing the father, who was my boyfriend of three full months. We shared the apartment, but I made sure I was alone that afternoon, protected in isolation, so nobody would see me cry, or rage, or decide to handle the situation silently. I was never the kind of person who wanted company in moments of vulnerability. I never wanted a concerned friend to pat my head and smooth the hair off my forehead while I puked or cried. I wanted to lie in bed in solitude, where I could turn my head to the wall, stretch my legs out, and rise again smiling, while the world slept soundly in its room.\n\nThe last thing I needed was a loving and emotional man celebrating the seed in my womb before I knew how I felt about it.\n\nMoments before, I had stared at those double lines with detached curiosity, a sort of numbed awe, as they popped up without hesitation in what seemed like a \"fuck you\" pink. I figured there could still be some mistake, so I took another test, and upon the second neon positive, pulled up my jeans, walked through the living room and onto the balcony, grabbing my Camel Lights and lighter on the way. I allowed my condition to sink one inch into my brain, where it hovered like a storm cloud creeping toward me. I knew it would shower me in panic, and soon I would feel it pouring down my arms and into my shoes, but those first moments felt liminal, half-real. I emboldened them with a cigarette. One more cigarette in the line of a thousand before it, a meaningless action of my same old life. An action of the nonpregnant.\n\n_Nothing to see here, folks. Just another woman on a balcony having a smoke._\n\nThat February afternoon was cool and bright, and as I watched the cars do nothing in the parking lot of our apartment complex, I thought about being a senior in college, my job as a waitress, and the few months Mac and I had been together, most of it grayed and hazy from alcohol, fast and romantic and possibly fake. I thought about how he would respond if he were there.\n\nHe would smile a soft smile. \"Wow,\" he would say, \"I love you so much,\" and his eyes would fill with grateful tears as more supportive words crossed his lips. He would study my reaction with his huge brown eyes. He would look as if he had waited his whole life to hear those three words.\n\n_I am pregnant._\n\nI took a drag, inhaling _I could have an abortion_ , but exhaled the startling realization that I would not.\n\nAnd with the thought, the cigarette grew foul between my fingers. I stamped it out beneath my foot and wondered how the fuck I had ended up here again. I understood the physiology of pregnancy. I did not understand how that wasn't enough.\n\nIn my defense, the first one was an honest mistake. I was eighteen, in my first semester of college, and had spontaneous, unprotected make-up sex with my long-term boyfriend. I knew immediately I would not have that baby, and I did not feel guilty about that decision, though I suspected this made me something of a monster. I felt sadness, but at that age, in that life, I mostly felt relief. We had sex, and _yes_ I happen to have a uterus and ovaries hell-bent on reproduction, and our act was neither smart nor mature, but _it was his fault too_. My defense was that of a petulant child, but I had no interest in spending my whole life paying for a five-minute interval of questionable sex with a man who could walk away if he felt like it.\n\nAs I stood on the balcony, I wondered again, _Who the hell gets pregnant accidentally_ _more than once?_\n\nI stared at the horizon and shook my head in disgust as I traveled the recesses of my brain looking for answers, recalling only a woman in my freshman comparative literature class. She had told me, \"Getting an abortion is like getting your teeth cleaned.\" When I raised an eyebrow, she explained, \"It's just something you have to do.\" She was in her thirties and married to a local rock star. She had bad teeth, three children, tattoos, and that haircut of the '90s where bangs were cut stupidly short in a band right against the forehead. I respected her.\n\nHer teeth cleaning theory sounded erroneous if not downright depraved, but her nonchalance convinced me I would be alright, and that I was even perhaps not quite as foul as I had believed during my trip to the clinic that week, feeling like a slut and regretting with my whole heart those minutes in the dorms.\n\n_Apparently this is a thing women do._\n\nThat seemed true. I did it.\n\n_But I would not do it now._\n\n_And it didn't feel like the fucking dentist._\n\nBack inside, I stretched out on our quilt-covered couch, clicking my tongue at Fatboy, the giant black-and-white cat we inherited from Mac's childhood. When feeling particularly affectionate, Fatboy would turn his head and glance at you from across the room. But that day, he folded up in the crook of my knees and stared up at me, as if he knew things were heavy.\n\nI took a deep breath, looking around the apartment, the carpet so bland I couldn't tell what color it was, the kitchen and bathroom floors a yellowed linoleum with pastel blue squares, ripped up and black at the corners. The cabinets were a 1970s brown with gold handles, and metal mini-blinds hung above a box air conditioner in the window that would sputter along against California's Central Valley heat. That summer, we moved our mattress beneath the little box, creating a pocket of decency between the white walls. Our television sat on boards and cinder blocks. It was the kind of apartment that never felt alive or permanent, but Mac and I were kids and in love, and it was ours.\n\nHe was nineteen and I was twenty-one.\n\n\u2022 \u2022 \u2022\n\nI was right about Mac. He did smile through tears and say all the lovely things I suspected he'd say when I told him I was pregnant, but the next day he added, \"If you don't have our baby, I can't stay with you.\" I considered telling him about the moment I knew I wouldn't have an abortion, but instead I merely nodded. I wasn't ready to speak the words, \"I am having a baby.\" I was stretching the liminal gray a few days longer.\n\nWhen he spoke those words, he didn't read my face to adjust his tone. He was not afraid of my response, or conflict, and there was no subtext. It was merely data to inform a decision. If his statement had been a threat, an attempted entrapment, I would have left immediately on those grounds alone. _Fuck you for even trying to get me to stay_ , I'd have thought.\n\nBut that's not what he said, and I knew it, because he said it with warm acceptance in his eyes and mouth and forehead\u2014the way he looked at me when he said everything, even when he yelled and postured and I thought maybe I hated him. Or, perhaps that's why I hated him, because he seemed capable of only adoration, and even in his anger he was devoted and irrationally loyal. It made me feel a little sick.\n\n_I get it, man. You can't withstand the resentment you'd feel toward me if I didn't have our baby._\n\nBut he is not why I had her. I was always going to have her and I knew it, though I didn't know how to explain that I knew. I didn't understand yet that motherhood is a lot of knowing without knowing.\n\nBut I knew _her_. She was already made.\n\nI was afraid of having a baby. I was afraid of committing to him like that. I was afraid of what my parents would say, but Mac misread this fear as indecision. I had her because she was meant to be here and I was meant to be her mother, and I believed that in the same way I know the sun will rise. I had her because the moment I knew of her, she existed, like a strange new friend who moved in and wouldn't leave.\n\nI told myself I was about to graduate from college, that I wasn't _that_ young, that Mac was going to be a good father\u2014and I loved him, or thought I did. In this way, I rearranged the facts, the furniture of my life, to accommodate my new friend.\n\nTwo weeks later, Mac peeked his head over the curtain while I showered and said, \"It's going to be a girl.\"\n\n\"I know,\" I said, and laughed. _How weird we are_ , I thought. _Clairvoyant._ _So in love she's already shining through\u2014through the blood and walls of my body._\n\nWe thought of names. We thought of Aurora and Leah and Althea, but one day while I waited for customers to arrive for dinner in the restaurant where I waitressed, I flipped through a magazine and found an article about Ava Gardner.\n\nWe settled on Ava Grace, as if anything could be more beautiful.\n\n\u2022 \u2022 \u2022\n\nI told my mother I knew she was a girl; she didn't think that was strange at all. When I asked her what the hell I was going to do with my life now, she said, \"Well, honey, you're going to have a baby.\" Her simplicity and perky use of the word \"honey\" shot red annoyance down my spine.\n\nMy mother's perpetual optimism made me wonder if she existed in some sort of sociopathic love cloud. As a young girl, I joined in her optimism, jumped on the \"it's going to be great!\" train with glee, but over the years, as each new beginning almost never turned out \"great,\" I realized her outlook was as much fantasy as it was hope. It was a story to justify rushing headlong into another disaster, the same thing we'd done for years. Businesses. Marriages. Personality improvements. Diets. It was always going to be different this time.\n\nIt was an old, raw burn, and her sweetness still stung.\n\nThe moment she mentioned relationships, mine or hers, somewhere in me the memory of my former stepfather stirred, the way we moved in and out of his house like a vacation rental. But mostly, I remembered her suffering and how I thought I could fix it, how the solution was perfectly clear to me, how everybody said I was \"very mature for my age.\" My mother used to say, \"I don't know how you see the things you see, Janelle.\"\n\nI didn't either, but I wished she'd see them too, because I was tired. And even at twenty-one, I was still tired, perhaps more tired than I'd ever been, and I had long since stopped believing in her dreams.\n\nAnd yet, I always called her first, to bathe in the optimism she turned toward me, too, toward the person she believed I could become. I needed that. I needed to believe things were going to be different _right around the corner_. That sick hope was infectious, seeping into me whether I wanted it or not, and as much as I distrusted it in her, I clung to it like a drowning woman, because at least it was something.\n\n\"But how do you _know_ , Mom?\" I didn't mask my irritation.\n\n\"Because you are going to be fine, sweetie.\" I wanted to throw up.\n\nShe must have told her mother right away, because the next day Grandma Joan called and said, \"You know you don't have to marry this guy just because you're pregnant,\" and my jaw hit my flip phone. I wasn't anywhere near marrying Mac. I barely knew him. _I am merely going to have his baby, Grandma._\n\nBeing a woman born in 1930, Grandma Joan of course assumed marriage, and I assumed she would push marriage. She was in her seventies and a Mormon woman who had made her husband dinner every night since they were married at eighteen, and if she was not home in the evening, she prepared the food and put it on the counter so all he had to do was heat it up. He had been the quarterback of the high school football team and she had been the new girl in town\u2014and a cheerleader. It was a movie, and yet true. They still held hands when they sat on the couch together, and they flirted like teenagers. He was sure old Benny at the post office was waiting for him to \"kick the bucket\" so he could swoop in on Grandma.\n\nShe would smack his leg, roll her eyes, and say, \"Oh, Bob,\" with exasperation and a kiss in her eye.\n\nAt family functions in her home, all the women would bustle around the kitchen for hours preparing dinner for thirty while the kids played in the basement and the men watched football in the living room. After dinner, all the women would bustle around in the kitchen for hours doing dishes while the men watched football in the living room, and the kids ran around in the basement. By the time I was a teenager, I wondered what the hell was wrong with these people. But I loved being in the kitchen, where my mother and three aunts talked and cooked with the chatter and laughter of a lifetime of sisterhood, occasionally popping out to rescue a screaming baby, talking of report cards and breastfeeding and wayward teens, of Grandma's silly ways and how she really should sit down, _she's tired_ , but she never would. When she finally did, my uncles had begun barbequing on the deck outside and nobody played in the basement anymore.\n\nI sat with the men, too, as babies crawled around their laps, each of their faces illuminated with the television screen as they watched sports, speaking of things I didn't understand, like \"downs\" and \"bad calls\" and \"finals.\" I felt honored when they spoke to me, a little nod to my sport-less existence. I understood their acknowledgment of me was a quick trip beneath themselves, a little jaunt to a place less sacred. They were, after all, the ones who got to do nothing while groups of women worked on their behalf.\n\nAlthough the kitchen was warmer, and had better conversation, sometimes I would sit at the dining table between the living room and kitchen, so I could watch both ends and refuse to commit.\n\nAt twenty-one, I joined the women in the kitchen for good, though I had always promised I'd never be like them. \"I'm not going to get married until I'm thirty,\" I'd say as a teenager at our annual Christmas party, \"and I won't have a baby until thirty-five.\"\n\n\"Good job,\" my aunt would nod. \"Just don't rush it.\"\n\n\"Of course not,\" I'd answer, irritated that she'd even consider the possibility of me ruining my life with an unplanned pregnancy at a young age.\n\nI was the youngest of my cousins to have a baby.\n\nPeople surprise you, though, especially when they're old and sick of the bullshit, and I saw Joan anew the day she called me, after fifty-five years spent with my grandfather. While she spoke, I wondered how many women of her generation married terrible men because of unexpected pregnancies, and then stayed because of more. I felt myself, for an instant, counted among them.\n\n\"Thank you for your concern, Grandma, but it's different with us,\" I said.\n\n_I may be in the kitchen like the rest of you broads, but I am different._ I could not articulate how, exactly, but I knew I wouldn't end up washing dishes while the men watched other men slam into each other on brightly lit screens. It seemed archaic and absurd. I would demand freedom, even within the confines of pregnancy. I suppose that, too, is something women \"just have to do.\"\n\nIf I had to guess, I would have said my future would unfold more along the lines of my paternal grandmother, Bonny Jean. She was an intellectual, a fiery Christian Scientist, and natural skeptic who believed in God but not doctors, grassroots journalism, and stockpiling mayonnaise in case there was another Great Depression.\n\nShe grew up behind the stage with her parents, who were traveling actors. I once attempted to explain \"gay people\" to her because, _you know, as a relic she wouldn't understand such things._ She spun around to face me in her house robe and said, \"I grew up behind a vaudeville stage in the twenties. You think any of those people were straight?\" I never tried that shit again.\n\nShe had five children from 1945 to 1955. They were raised largely by her father-in-law while she ran her newspaper, which she and my grandfather purchased in 1956. Bonny Jean would attend every local city council meeting, critiquing what she saw in scathing weekly editorials, which she would often dictate over the pay phone in the city council hallway. She once fought the head of the San Francisco plumbers' union, a man with rumored Mafia ties, who was trying to take over her small town's water council. When she broke the story and refused to back down, he threatened her. I once asked how she managed to fight a man like him as a woman in the 1960s. She said, \"Oh, that was easy, honey. I was not afraid of him. The truth is a strong defense.\"\n\nWhen I told her about the pregnancy, I thought I heard a touch of sadness in her voice, despite her congratulations, because for a split second, they sounded like condolences.\n\nThe hardest person to tell was my father. I was barely old enough to handle him knowing I had sex, and yet I had to tell him there was an actual human growing in my body, deposited there by the sperm of a man. Telling him felt something like bra shopping with my mother at fourteen: uncomfortable in a deeply shameful, yet unknown way. Everybody has sex. Everybody gets boobs.\n\n_Still, somebody please kill me._\n\nI had always felt my father saw me as a kid who was going to do something impressive in life, who was going to become a lawyer or doctor or at least make a lot of money. Instead, I was joining the Mormons in the kitchen. I knew he wouldn't say it, but he would be disappointed in me. He knew how many times I had stood at family functions declaring my plan, and he knew I never consciously abandoned that.\n\nIt's hardest to fall in front of those you've convinced, through years of tone moderation and personality suppression, that you are not the type to falter.\n\n\u2022 \u2022 \u2022\n\nI stopped smoking and drinking immediately after my balcony denial, which felt wholesome and deeply mature, despite Mac's and my decision to move out of our apartment and into his parents' house on their ranch. They lived in a dome-shaped house about ten miles outside of Davis, California, the clean, well-manicured town where I went to college and met Mac.\n\nDavis boasts the second-highest per capita number of PhDs in any city in the nation, and a special tunnel for frogs so they don't get killed on the road. The town is teeming with students, artists, and intellectuals on bicycles, but also suffers from an epidemic of highly educated, splendid liberals. I learned to spot and avoid the latter from a distance of approximately one hundred feet, having had many years' practice. The problem is not that they're liberal\u2014surely one can learn to live with that\u2014it's that they can't quite understand why a person wouldn't dress her child in only organic cotton, or shop solely at the co-op, where they sell nineteen dollar olive oil pressed from olives grown on blessed trees in sacred Native American valleys.\n\nThese are the kind of people who call gentrification \"restoring the neighborhood\" and spend four years on a waiting list for a $1,500 a month preschool while claiming to deeply understand the plight of the underprivileged. Davis is the kind of town where _everyone breathes social justice_ via diversity stickers on their Priuses, but many citizens request that the kids from the Mexican enclaves surrounding the town simply stay in _their_ schools. _It's not about race. It's just...you know...let's talk about public radio. Do you support it? It's kind of my cause. That, and the ACLU._\n\nMost of the mothers in Davis were married, in their late thirties, and living in $700,000 houses when I showed up at age twenty-one, unmarried and pregnant. When I realized nobody would talk to me at the park, having dismissed me immediately as some sort of teen-pregnancy situation, Mac and I bought a pink diaper bag with a giant rhinestone Playboy bunny on the front. It was my subtle \"fuck you\" to everyone who wouldn't talk to me anyway, and it almost convinced me I didn't care.\n\nI turned twenty-two that March and finished my last semester of college in September of 2001, two months before our baby was due. Mac worked at his father's slaughterhouse on the ranch, and our bedroom was where Mac had played with Hot Wheels and G.I. Joes as a boy, and hid his weed at fifteen. We shared the home with five other people: his parents, two sisters, and his sister's boyfriend. All the bedrooms were upstairs and opened into a shared center hallway, kind of like Foucault's panopticon only without the glass. His family was kind and relaxed and pretended we weren't kids about to have a kid, but I felt exposed and watched\u2014too close to people who weren't quite mine, humans I knew but didn't understand, and whom I was still trying to impress. They were family, but I didn't want them to see me naked, or notice I stunk up the bathroom or yelled at their son. I self-regulated, even though there was no guard in the watchtower.\n\nWe bought a crib and an oak dresser, which we wedged together in a corner of the bedroom. I lined each drawer in lavender-scented paper with tiny pastel pink flowers on it, and I bought clothes from Baby Gap and Gymboree and Marshalls. I bought most of them in \"newborn\" size because they were the cutest and least expensive. I didn't know they were discounted because babies outgrow them in twelve minutes.\n\nWe had a keg of beer at our baby shower, and Mac came because we were \"too in love to be apart.\" I received about seventy-five various bath items because when my stepmother asked, \"What do you need for the baby?\" I answered, \"Bath items.\" I didn't know you were supposed to \"register.\" I didn't have any friends telling me about pregnancy or babies because only losers had babies this young. And I never hung out with losers.\n\nMy pregnancy was like living in a dream, a sort of ethereal fantasy ticking by in nebulous form. While my belly grew, I spent my days petting hand-smocked outfits with embroidered ducks and imagining our little threesome. Mac and I played pool at my local university's student union, and I wasn't even embarrassed of my belly. I wasn't embarrassed of my age, or Mac's lack of career, or that we lived in a room in his parents' house. Those things weren't in the dream.\n\nBut I couldn't help but feel inklings of shame as I walked to class during that last semester, when I barely fit in the desks, because the sidewalks and grass and offices on campus were the places where women like me rarely succeeded, and nobody was impressed with expanding uteri. These were PhDs and MAs and lovers of Derrida. They could see right through me: I was the kid who lost, the girl who failed. As I walked I remembered maybe I was going to be more than this, but then I thought of Mac and the baby girl to come. I thought of that love and squared my shoulders.\n\nWe went to peaceful birthing classes and breathed together and when Ava came it was fast and insane and Mac sat by me and held my hands and never broke my gaze. The nurses said we were the most beautiful birthing couple they had ever seen.\n\nI wasn't surprised. It was the only way it could be.\n\n\u2022 \u2022 \u2022\n\nI met Mac for the first time in my living room the night before Halloween, thirteen months before Ava was born. I was living in a converted garage in a house I shared with four eighteen- and nineteen-year-old males I had found in a newspaper. Three months before I responded to their \"roommate wanted\" ad, I returned home from a year studying abroad in Spain. I tried living with my mother up north in Mendocino, California, and found a job waitressing, but got fired after two weeks for counseling the owner on how she could improve her business. I found myself bored, embarrassed, and broke, so I moved back to Davis in the fall and began waitressing at an \"Asian fusion\" restaurant and drinking too much.\n\nI had long before decided I could not live with women. They were too complicated. They needed things like talking and support and genuineness. I needed things like rum and Coke and silence. So I asked those boys looking for a roommate if I could move in, and they said yes immediately upon hearing I could legally buy alcohol.\n\nThree months later, a man I had never seen before sat stoned against our living room wall, next to the television. He had a beard that stuck out in every direction and a head of hair that looked exactly the same. It was as if somebody had taken a donut of three-inch-long black curly hair and popped a face into the center of it. He was a high school friend of my roommates', a newcomer, thin and tall and quiet, and I would not have noticed him at all had he not said something witty. In our house of drunk eighteen- and nineteen-year-old man-boys, nobody was saying anything witty. I beamed my eyes at him from across the room in curious surprise and locked them with his. They were the kindest eyes I had ever seen. I remember that moment exactly as it happened, in slow motion, as if it were a scene in a Meg Ryan movie. The Eye Lock. His were deep brown with eyelashes that carried on ridiculously, but it was their gentleness, their steady calm, that made me want to know more.\n\nSo naturally, I decided to get him drunk and shave his beard off to assess his jawline. He drank Bacardi out of the bottle until we ended up in my roommate's bathroom. I borrowed a razor and went to work on the facial hair while he called me \"Mary,\" eventually passing out facedown on my futon. He woke up the next day at dawn to go work at the ranch, hungover, after just a few hours' sleep. I had never seen anybody get up that early and go to work in that condition. Except me.\n\nMac was a worker, I learned, and highly attractive without the hair donut. I was intrigued, and since that day happened to be Halloween, I figured we should probably head over to Chico to take some drugs and fall in love. Before he could get out of my room that morning, I said, \"Hey. Want to go to Chico tonight? It's crazy on Halloween. Everybody's loaded in the streets.\"\n\n\"Yeah,\" he said, smiling. \"Let's do that.\"\n\nI remember the way the gray morning light fell on his face as he buckled his belt and pulled on his shirt, and I wondered if he'd kiss me. He didn't. We hadn't touched at all other than my fingers against his face and my shoulder pressed against his chest as I shaved. He seemed indifferent to me, which made me wonder if perhaps I was already in love. It wasn't until he walked out my bedroom door that I realized I had just committed, in the heat of mad passion, to dressing up in a goddamned costume.\n\nI barely knew how to dress myself as ___myself_. As a child, my idea of fashion was pairing a green print with a green solid and brushing my hair. In high school, I wore jeans, Doc Martens, and tight white v-necks intended for boys, until my father suggested I \"try a little harder.\" After that, I'd walk into Gap or J. Crew and buy an outfit featured on a mannequin, figuring that must mean the clothes matched.\n\nDressing as _somebody else_ was an unbearable enhancement of an already agonizing lifelong struggle. But for Mac I wanted to appear confident and cool, so I cradled my hangover, drove thirty minutes to a costume store in Sacramento, and eventually selected an ensemble that could only be described as \"slut Egyptian.\"\n\nOn my way to Sacramento, my former boyfriend of three years called to ask if I wanted to see him that night. He and I used to hang out together in Chico, but we hadn't seen each other in a year. I was struck by the synchronicity, and strangely sensed that the choice I made in that moment would alter the course of my life, but I dismissed the thought as ridiculous. _It's just a Halloween._ I suppose somewhere I knew if I saw my old boyfriend I would return to him, because he was safe and comfortable, and the sound of his voice said, \"I'll still have you if you'll have me.\" But there was something about Mac, his quiet strength, the way he cinched his faded leather belt that morning.\n\nI told my old boyfriend, \"I'm sorry, I already made plans,\" and then turned up the Grateful Dead on the stereo and rolled the windows down as the October sunset called from across the farmlands.\n\n\u2022 \u2022 \u2022\n\nMy costume consisted of a beaded headdress and beaded crop top that highlighted my impossibly flat midsection, which I believed needed toning. I paired the beaded top with blue silk pants\u2014because I had blue silk pants\u2014and then I ate some ecstasy. That pill was the key accessory to my outfit, for mine was always a problem of perception. I knew I'd look exquisite as soon as I was high. No longer a scared girl without sartorial confidence, I would be a devastatingly attractive woman prancing in beads.\n\nWhile sitting on a park bench that night in Chico, watching drunken masses of college students flop down the streets, I told Mac's buddy, \"I think I'm in love with your friend,\" but I said it loud enough for Mac to hear, because it had been 24 hours and I was tired of the bullshit. _Let's do this._ Mac was shy. I was not shy. I had no problem barging into your life and demanding a fixed place in it. Right now. _Step aside, assholes. Slut Egyptian has arrived._\n\nAt my house later that night, and after taking more ecstasy, I dropped my head onto Mac's chest while we lay on the floor and planned our lives together. I suggested we have four children: two biological and two adopted. (I was going to save all the children.) He told me about his two sisters and a pot-bellied pig on their ranch who couldn't fit through a fence. He chuckled so hard my head bounced on his chest and I looked up to see him grinning while he described the pig balancing on his belly trying to push himself through.\n\nI remember thinking only an excellent and loving man would tell a story about a pot-bellied pig getting stuck in a fence. It made him laugh. It made me want his babies.\n\n\"Mom, I think I met my future husband,\" I told her the next morning while circling the lawn, possibly still high, definitely smoking a cigarette and squinting under the morning sun.\n\n\"He's from a ranch around here. Really kind. I think his parents might be Republicans. He gets up and works every day at five a.m.\"\n\n\"I like the sound of him,\" she said, and I took a stunned drag off my smoke, because my parents hadn't liked the sound of most of what I'd been saying for the previous eight to twelve years. \"He really sounds like a good man for you.\"\n\nWithin a few days I was bringing Mac meatball sandwiches while he worked\u2014chasing unruly chickens with goat blood in his ear, butchering animals and cutting meat, smelling of lanolin and yesterday's Captain Morgan. He wore thick canvas pants or overalls with rubber boots over them, and ripped, bloodstained T-shirts and old button-down flannels, layered up to keep him from the cold or block him from the sun. After work, he smelled so intensely of animal guts and lanolin I could hardly be near him, but I watched him from afar and thought, _My God, he's tough_. Still, it was some ineffable tenderness that made me wonder if he was even human. He was a ranchman with tattoos and piercings, and he wasn't even a Republican.\n\nI had never met a man like him. A man so classically macho without the machismo, a country boy without the puffed-out chest of country boys. Where was the posturing? The guns? The huge truck with a \"Piss on Chevy\" bumper sticker?\n\nMac didn't even watch sports. He drove a small, dented red truck, sang songs from Broadway musicals, and whistled to himself in the shower. But he could punch the hide off a sheep in three minutes and work eight hours without a break in a frigid slaughterhouse. He would fall asleep every night ten seconds after his head touched the pillow, and I would watch him. He was always so, so tired.\n\nMy roommates told me I had somehow landed the best man in the world, and the only person they were ever truly afraid of in high school\u2014not because he was mean, or a fighter, but because he was the kind of person who would break a bottle over someone's head if necessary. He wouldn't like it, but he would do it.\n\nI saw that. I saw loyalty that defied reason and ignored facts. I saw loyalty that was decided. Etched. Right there in the bones, and that was what I needed. I needed a love that couldn't see me.\n\nWe were never a decision. We were already made.\n\nIn January, three months after we met, we rented our apartment together. A few weeks later, on Valentine's Day, he pulled out a ring as we returned home from dinner. He killed the engine and opened a little box. I smiled, knowing what was about to happen. \"I'll marry you,\" I said, \"but put that thing away and ask me again. This is not romantic. This is a parking lot. And you have to ask my dad first.\" I was a real stickler for tradition.\n\nSo he asked again, after we knew I was pregnant, this time in the middle of the restaurant where I waitressed. He walked in with flowers, wearing a dress shirt and tie. The restaurant was packed with a Friday night dinner crowd when he got down on one knee and asked. I said yes, and everybody cheered. We were engaged and I was pregnant with our love child and in the movies this was all going to be quite perfect.\n\nAnd it was. Even the nurses said so.\n\n\u2022 \u2022 \u2022\u200b\n\nFor two days in the birthing center the dream went on. Our baby girl slept in my arms and nursed while our friends came and went. We took a thousand pictures. Mac slept in that chair \"bed\" and I dressed my baby girl in outfit after outfit, all scented, all carefully chosen, while I wondered what the fuck happened to my vagina and how I'd ever pee again. One of the outfits was pink and fuzzy on the inside with a white bear on the front. It made me think of the nine months and several days of perfection I had just experienced\u2014until I stepped outside the birthing center as a mother, and the rest began.\n\nMy story wasn't untrue. It was simply unsustainable.\n**2**\n\n# **Stop Drinking When Your Lips Go Numb**\n\nI did not start on that balcony. I did not wake up one day dropped into a life of pregnancy, cigarettes, a three-month-old relationship, and an almost college degree. I built that, one moment at a time, while I thought I was building something else.\n\nI probably began on a two-week trip to Honduras when I was sixteen years old, which I heard about one day in my public speaking class. I attended an all-girls Catholic school in Santa Rosa, California, in the middle of vineyards, where winery owners and their offspring lived. The trip was \"an opportunity to learn about the rainforest and conservation,\" but more accurately it was a chance to send privileged kids to a \"third world\" country to learn guilt, American exceptionalism, and how local ignorance caused the destruction of the rainforest (as opposed to, say, global capitalism).\n\nI did not know this, nor did I care. I was simply aching for something new.\n\nMy mother worked in the school's kitchen for my discounted tuition, so the trip made little sense in the financial context of our lives, but I presented it to her as a situation deciding my continued survival, as only a sophomore with a vision can. My mother leapt in my support as she had always done, my grandparents donated to the cause, and, somehow, a few months later, I was stepping off a plane into a wall of humidity.\n\nIn the Tegucigalpa airport, I watched men in green camouflage uniforms pace the halls with machine guns in their arms. They were terrifying, but my teachers explained they were \"protecting us.\" As we approached the yellow school bus that would cart us around for the next week, my friend Gloria asked, \"Are those bullet holes?\" and I looked up with delight at the black spider punctures around the bus. They were indeed bullet holes. _My God_ , I thought, _how exotic!_\n\nSweaty and giddy and wearing fanny packs, we shuffled into the bus and drove in what appeared to be the wrong direction on a highway into thick vegetation. We spent the next week hiking in the wet, green rainforest behind a man with a machete, who whacked a path in front of us while teaching us about local plants, animals, and indigenous populations. We dove into glistening swimming holes below warm, white waterfalls until we broke for naps on shaded rocks. We ate lychee nuts, beans, rice, and tortillas at every meal, every day. The other girls complained of the monotony, but I loved watching the Honduran woman who made our tortillas in a clay oven. I would wave at her and smile and she would wave back, but never move from the oven's face. I felt we were experiencing something sacred when we ate her food, something old and vital. It never occurred to me that she never seemed to leave that spot in front of the roaring oven, cooking for a bunch of wine country kids.\n\nNearly every day we would take an excursion in the bullet-hole bus, and every time we did, we would for reasons unknown pick up a lady with a cage of chickens. If our bus got stuck in the sand, the driver shoved palm fronds in front of the tires while we lined up behind the rear bumper and pushed with all our might. When our bus broke down completely, the driver walked down the road a few hundred feet, bought a Coca-Cola in a tall glass bottle, reclined against a bus tire, and took a nap. We just sort of waited there, dumbly, while he snoozed, wondering what our next move was. _Drink a_ _Coca-Cola_ , I thought, _that's the next move._ So I did. The other kids refused, for fear the bottle was washed in unclean water.\n\nWe drove by people living on cardboard, sweeping dirt floors, riding ten to twenty in backs of trucks. I saw a young girl taking a shit in the street. I felt enlightened. I watched the poverty with a distant gaze, as if it were occurring in a fishbowl on the shelf. I was too young to see real suffering, too ignorant to understand my country's contributions.\n\nOne day on a hike in the rainforest, I accepted our guide's suggestion that we try eating termites out of a tree \"for protein.\" I stuck a stick into a small hole in the bark just as he had done and ate the few unlucky bugs that wandered up. He was correct; they were crunchy and tasted like sap. I was the only girl who did it, and I was disgusted as their little bodies crunched between my teeth, but I wanted the other girls to wonder at my bravery. I wanted to shock them with my valor. But mostly, I wanted to not be bored. At sixteen, I was already bored: with school, church, my family, my little job at a local pool's snack bar. I needed some damn escapades.\n\nWe spent the second week on Roat\u00e1n, an island in the Caribbean with dirt roads, a plumber who stopped by occasionally from the mainland (whether or not your shower was broken), and where the mail came every two weeks by way of Florida. We spent our days there snorkeling above the Mesoamerican Barrier Reef, learning about its fish and destruction, and how urine soothes the burn of jellyfish stings.\n\nBefore we left the United States, we attended ominous meetings with our parents in which the teachers described in somber tones how dangerous Honduras could be for American children.\n\n\"They will steal you, and then they will kill you because your passports are worth more than you are.\"\n\n\"They will steal you, and they will rape you, and you will not come back.\"\n\n\"There have been many accounts of tourists getting kidnapped. You must never leave the teachers' sides, and you must never stray from the group.\"\n\nIn the rainforest, men with machine guns paced outside our sleeping quarters and occasionally fired their weapons into the night sky \"to ward off guerrillas.\"\n\nI was so terrified by all this I decided I needed to sneak out in the middle of the night with my friend Stacy. I was tall, slender, and very tan, with blonde hair that fell past my waist. Stacy looked absolutely identical but with larger breasts. We crawled out the back window of our motel room in the middle of the night, and while Stacy was chatting with some locals down the beach, I met a wandering American hippie with unwashed dreads, patchwork pants, and a heavily pregnant girlfriend. We bantered with each other until the man asked, \"Hey, do you want some acid?\"\n\n\"Yes,\" I answered. \"Yes, I do.\"\n\nI was an expert on acid because my mother had told me about dropping it with my father at the Fillmore in the 1960s. The man handed me three hits and said, \"These are strong. Share them with your friends.\"\n\nSo I tore off two tabs and gave Stacy one, telling her he only gave us two, and set the remaining hits on my tongue. Without hesitation or thought, in the middle of the night, on a remote island in Honduras, I took a drug most people live their whole lives without taking, and I did it with a lie.\n\n\u2022 \u2022 \u2022\n\nTwo months before the acid hit my hungry tongue, I sat in our Mormon bishop's office confessing my mistake of smoking weed, once, in a moment of weakness, because Satan had overcome me.\n\n\"I will never, ever do it again,\" I told the bishop. \"I think I had to try it once to get it out of my system.\" I looked at him from across the wooden desk and tried to show my sincerity with my eyes. He told me to repent, and ask the Holy Ghost to come back to me. At home that night, I got on my knees and repented and vowed to stay on the straight and narrow.\n\nFour months before the acid soaked my blood in Technicolor, I stood at a high school lunch table proselytizing to my classmates how drugs ruin lives. At that age, I was staunchly opposed to anything that would ruin my life.\n\nI knew in my heart I would never be so misguided. I believed myself lucky to have been shown a better way than the foolish teenagers around me smoking weed out of peer pressure and insecurity.\n\n\"My father is an alcoholic,\" I preached. \"It is the reason for my parents' divorce! I know where that life leads!\"\n\nSix months before I ministered at the lunch table, I stood behind a podium in a blue tank dress with a jean jacket to cover my shoulders, \"bearing my testimony\" to an entire Mormon congregation. With tears in my eyes and a quiver in my voice, I explained how I knew the church was true, and the Holy Spirit was in me, and I would never do anything that would make my body\u2014my temple\u2014unclean, and therefore uninhabitable for the spirit of God. I rose voluntarily almost every month to get behind that podium.\n\nAnd I meant it every time.\n\nEight years before I stood at the podium, my mother took us to the Mormon church for the first time. I was eight years old and wore puff-sleeved dresses with shoulder pads and big, innocent flowers to the chapel. We went every Sunday and Tuesday, and possibly Saturday, but in between these meetings, we would leave completely, in body and mind. Or at least that's how it felt.\n\nWe listened to \"How Great Thou Art\" on Sunday and Grace Slick on Monday. We snuck into high school football games we couldn't afford by passing ticket stubs through the fence, a form of stealing my mother explained was \"small stealing\" that didn't harm anyone, and therefore, need not make us feel bad. It was not a real sin. And if we could pay, we would.\n\nMy brother and I heard stories of Joseph Smith and \"the fullness of the gospel\" after my mother's stories of drinking Southern Comfort with Janis Joplin in the Haight. I was intrigued by both, though I found Joseph Smith something of a bore compared to, say, watching Ginger Baker play drums in Golden Gate Park during the Summer of Love. Salvation is divine, but it's tough to compete with bra burning and Jefferson Airplane.\n\nWhen a song she loved came on, my mother would turn it up and sing loud and tell us what the song was about, even if it was weed or acid or free love. She'd tell us where she heard it first\u2014maybe the Polo Fields at Golden Gate Park, or the Berkeley Hills, or the Avalon Ballroom. My favorite story was the first time she heard Dylan's \"The Times They Are A-Changin'\" on the radio, how she pulled her car over and cried because she never knew music could be like that, and how San Francisco in the sixties was electric with hope and change and none of the parents understood. In fact, Grandma Joan said Dylan sounded like \"a dying cow.\"\n\nI always thought he sounded like God.\n\nI knew that feeling, the one my mother had when she first heard that Dylan song, because it happened to me the first time I heard Jerry Garcia sing \"Ripple,\" and I dropped my head against the giant 1980s boom box on the shelf in my closet, closed my eyes, and cried. It was the saddest, most jagged and piercing sound I'd ever heard.\n\nAnd it felt true, truer than the gospel, truer than Jesus. But still, on Sunday, I got up and bore my testimony, reverently.\n\n\u2022 \u2022 \u2022\n\nThe next thing I remember after dropping the acid in Honduras was rolling around in the gentle surf in front of a local bar about a quarter of a mile away from our motel. Stacy and I were in bathing suits now, and we laughed and sang and let the sand and water move across our bodies and hair. We were on earth, yet severed from it entirely. I looked up from the ground and observed the silhouette of a young, skinny white man with bleached, wavy hair standing above us. And then he was with us.\n\nAcid trips are like that. Things appear, disappear. Technicolor flashes.\n\nStacy and I swam and walked and ran along the shore. He watched us while we played in the ocean. I asked him why he was there. He said, \"Well, I can't really leave you like this, can I?\"\n\nI dove back into the surf.\n\n\u2022 \u2022 \u2022\n\nMy mother was a \"convert\" to the church, meaning she was \"not born into it,\" and since I was baptized at nine instead of eight, I was a convert, too. This placed us figuratively just outside the church hall. My mother was also a single mother, divorced at thirty-six, with two children and a non-Mormon boyfriend, who later became her husband. This placed us more literally just outside the church hall\u2014we lived differently from the true, better Mormons, and weren't allowed into all the rooms of their temple.\n\nMy mother converted to the church as a young girl after missionaries showed up at her family's door. Her sisters and mother were also baptized, but my grandfather preferred to stick with his evening gin and tonic. At eighteen, she met my father, a non-Mormon, and for the eighteen years of their marriage, she didn't attend church. So she carried with her, whether she wanted it or not, a history of falling in love in a bar at eighteen, of rock and roll, of working for Francis Ford Coppola in San Francisco as the girl who brought him lunch and snacks, of playing the Stones so loud you broke woofers, of alcohol, and a touch, I heard, of the harder stuff.\n\nMaybe she found God in all of that. I had trouble seeing it so seamlessly. By fifteen, I had persistent questions about the Mormon church, but mostly I wondered whether I was good or evil. I felt like I was good. I meant well. I was not particularly mean and I didn't seem evil (I was sure as hell nicer than some of the delinquents I went to school with). I didn't cheat and I didn't steal. When bored on summer days with my best friend Elizabeth, I wanted to break into the house down the road _to see if we could do it_ (but not hurt anything, of course), and spray the stucco of Elizabeth's house with hair spray and light it on fire, _to see what would happen_. In junior high, we wanted to steal her grandmother's Cadillac for a quick trip around town, because _who wouldn't want to drive a Cadillac down Main Street on a summer night?_\n\nI was sure I loved my family more than most people loved their families, and yet I wanted to masturbate occasionally. I wanted to do things my church told me were sins. I could ask forgiveness all I wanted, but then I'd just want to do them again.\n\nI was never holy enough. I could never purge myself of the impure thoughts, and I could never shake the sound of Jerry Garcia's black-tar beauty from my mind.\n\nI wondered why my non-Mormon father was not going to the \"highest degree of glory.\" I only saw him a few times a year since he lived on the northern end of California, but he struck me as morally sound. And Grandma Bonny, too. She had her own deeply held religious principles. In fact, that whole side of my family seemed decently wholesome. I wondered why, if you had to be Mormon to get to the Most Excellent Heaven, God had people born into remote parts of the Amazon where they never met missionaries. I wondered how there could only be \"one true church.\" What about all those other people on the planet? Were they simply wrong? When I asked these questions, the church leaders said, \"There are some things we just don't understand,\" and that felt to me like a cop-out. How could you possibly know the plan of salvation but not know how everybody can get there? Later, they explained that people wait in the spirit world until somebody on earth is baptized for them. But then why the hell would I obey the rules if somebody could just do it for me when I die?\n\n\"Because you've already been shown the plan of salvation. You can't go back now.\"\n\n_Well, fuck. So I would have been better off born somewhere where the church never was so I could live without all these goddamn rules and still get to heaven._\n\nAnd there were so many rules. Don't date until age sixteen. Once you date, only kiss, but not with the tongue. Don't have sex until marriage. Don't drink coffee or black tea. Dr Pepper is a gray area. Don't buy things on Sunday. Don't wear tank tops.\n\nI liked tank tops. I liked my shoulders. I wanted to show them without shame.\n\nBut really, it didn't matter how many times I showed up on Sunday in my \"modest dress\" and paced the halls with the stoic face of holiness, I still got angry sometimes and chased my brother around the house with a butcher knife, explaining to my mother and brother that a person named \"Margaret\" lived inside me, and she was a real psychopath. We all laughed because it was such an innocent name and they thought I was joking, but I knew I was not.\n\nNobody named Margaret would chase people with knives.\n\nBut I would.\n\nAnd it didn't matter how many prayers I whispered at night, I still couldn't sleep with my hair flipped over my pillow, even though it was more comfortable and I wanted it off my neck. I couldn't do it because I feared there was a man under my bed who wanted to cut it off, and though I knew intellectually this was not true, my fear was untouched by rationality. Or whispers to my Father in Heaven.\n\nAnd even though the church elders praised how eloquently I spoke of the truth God had revealed to me, telling me I was a leader among young women, I was still unable to say the thing that would help my mother see that my stepfather was never going to change, and that we couldn't move again, because it wouldn't be different across town or in Texas. It didn't matter how brightly I shined on Sundays, I yelled and screamed and chased with knives and masturbated when nobody was looking.\n\nI couldn't even muster the Holy Ghost's presence when unspeakable tragedies came, like when my mother told me she had three miscarriages before my brother was born, or when kids made fun of my brother's acne, or when my stepfather mocked my mother, again. Or when my brother and I would go fishing, and even though I always wanted him to catch a fish more than I wanted to, _I always did_ , and he did not, until once when we were camping in the summer, and he and I went down to the pier at dusk, threw our lines in, and caught fish after fish after fish without even trying. Big, fancy trout. We couldn't believe our luck, the way we'd cast a line and immediately catch one. As the sun went down, we filled the stringer until every spot was taken. We only stopped because there was no more room.\n\nWe caught them together and nobody had to go back to tell our mother who caught more. But when I untied the stringer of fish, our faces beaming at the mass of them, the bottom fish pulled and the wet string slipped through my fingers, back into the lake. In an instant, they were gone, strung together through the gills, to die.\n\nWe were going to show them to our mother and eat them for dinner. I can't remember if we tried to catch them again, or who hurt worse, or even how long I cried or apologized.\n\nWhere was God then?\n\nThe one time we both caught fish, I let them all go, and I couldn't find Him there, in my guilt and pain, in my \"irrational reaction\" I couldn't seem to quell, in the desperate sadness of my mother's miscarriages and the way she told me so matter-of-factly, saying we would \"see them in heaven.\" But my siblings were dead. I had dead siblings. _Who let them go?_ I raged and wept, and my brother and mother looked at me like I was deranged. I knew I was crazy, but I couldn't stop. They said I was \"just too much,\" but I couldn't be less.\n\nMaybe it was Margaret. Maybe I was evil.\n\nIt didn't matter how eloquently I testified, I couldn't get my mother to stay when she wanted to leave for the night, even though I was convinced she wasn't coming back alive. I would rock and cry on the couch, getting herded by our border collie, who seemed to think, _If we can just get this one home, she'll be fine._\n\nNobody seemed to know how to get me home, not even God.\n\nIt didn't matter how clear the world seemed when I sat in the church room listening to very kind women explain the way to heaven, I still walked around the playground at school eyeing the other kids with fascinated skepticism, a twenty-foot invisible wall between us, wondering how they knew how to be kids on a playground. _How do they know how to play tetherball?_ _What is happening with these people?_ I wanted so badly to join them; I wanted so badly for the ball to smack them in their adorable little faces. I figured God had handed out a Playground Manual before he sent us all down here, and I, unfortunately, was absent that day.\n\nBut I couldn't tell you that, and I couldn't ask, \"Hey, could you teach me how to play tetherball?\" I couldn't let you know how much I wished you'd sit with me at the lunch table, so I squared my shoulders and straightened my backpack and walked somewhere very important, like the library, to prove I had a place to go. There's a chance you didn't notice me leaving. There's a chance I was proving it to me.\n\nWhen I got home, I would go straight to my bedroom, and make myself the smartest and most capable person in the room by lining up my Cabbage Patch dolls and playing \"school\" with the door shut. I did this for definitely too many years. My favorite book to teach was _The Miracle of Life_ , because it involved new life and vaginas, even new life _exiting_ vaginas, so it was educational _plus dirty_. Nobody bothered me in there, not even my mother when she got home from work. I felt better after a couple of hours of solitude and fantasy, sometimes even good enough to see what the crazy kids were doing on their bikes out on the street.\n\nI had no idea how to join them.\n\nMaybe I didn't even know I wanted to join them, that I wanted to feel a sense of unity with the people around me, that I wanted to not feel fear, that I wanted to feel better inside. I tried the church for many years, and it almost worked, but all that Mormon preaching and black-and-white Truth was a pathetic substitute for what I really wanted: relief. A deep exhale. A sigh in my guts that let me rest, fully, completely, secure and unafraid. And whole.\n\nI yearned for that acid in Honduras with my whole self, with every single moment I lived before that day, but I didn't know it until I saw the tiny tabs in front of me, and God became the ability to feel different. To feel free.\n\n\u2022 \u2022 \u2022\n\nWhen the acid began wearing off, I was lying on the edge of the man from the beach's bed, aware of the movement next to me. Stacy was in the middle. He was on the other side of her. I was terrified of men and sex, so when they started kissing, I panicked and headed to the bathroom (if I'd had a backpack, I would have straightened it). I figured I'd take a shower to wash off the evening, but this turned out to be a questionable idea. The black mold spots on the white tile refused to stop shaking. My eyes told me they were bugs. My mind told me it was the acid. There was no hot water, and no plumber to fix it.\n\nI got dressed and sat on his couch until he and Stacy joined me. It never occurred to me that it would have been rape if he slept with her. I was merely grateful to be alive and out of that bed. He explained he was a graduate student from San Francisco studying marine biology, and that he was from Sebastopol, a tiny town about fifteen minutes from our city of Santa Rosa. I could not believe it. He said he saw us rolling in the surf in front of a bar and knew we were high and not safe, so he stayed with us. He left out the part about hoping he could take advantage of a couple of young, dosed girls.\n\nI had no idea where I was or how to get back to the motel. He said, \"Walk straight down the beach, that way,\" and dismissed me with a hint of disgust in his voice, probably because I wouldn't fuck him. I was useless to him now.\n\nI thanked him and laughed, thinking how his directions reminded me of the \"straight and narrow.\" How far I had strayed from the iron rod now.\n\nAs I walked back to the motel alone in the heavy morning air, I worried about Stacy, who had refused to leave the man's house with me, but I walked faster and faster, running to get away from him, from the night, thinking of the silent motel room, and of getting there before the teachers woke up. I thought about rolling in the surf on acid and how I wasn't raped or mauled or dragged into a cave for my passport or any other reason. As I unlocked the motel room door, I thought, _Wow, I can do really stupid shit and not get in trouble for it._\n\nBut mostly, I thought about the weed I had smoked a couple of months before, and how I was absolutely convinced something spectacular would happen in that moment. I thought the sky would crack or shatter, or clouds would gather or part. I didn't know what would happen exactly, but I knew it would be mighty. Surely, I would be punished somehow.\n\nAnd yet, nothing.\n\nI began to suspect my aunts, my mother, and the entire Mormon church were lying. They all but _promised_ __ a volcanic eruption on the day I blackened my temple, and here I had quite clearly at least grayed it out with intentional marijuana, and not one single fucking thing happened. Nothing fell out of the sky. No lightning. Not a word from the creator.\n\nI didn't even feel different.\n\nBut still, I wanted to believe. I wanted to be a holy person, and for them to be right. I wanted to reach my potential\u2014 _and there was so much, you know?_ I was a shining star at school! I would be the first person in my family to graduate from college! I would do better than my parents! My father moved in and out of drinking. I heard stories of their friends shooting up heroin in their necks, of junkies burning up in their own houses, of drunken fights and limb-crushing car accidents. _I knew better._\n\nBut I couldn't shake the feeling of having been lied to, of being hoodwinked. And I wondered what else they were wrong about. I suspected _everything_. So I set that acid on my tongue, and gave up trying.\n\nIt was my last sacrament, my last communion. I had spent so many years believing that if I could figure out how to be good enough, I wouldn't be the person who chased loved ones with knives or craved Jerry's cracked desperation as a salve greater than salvation. I thought _any minute now_ my external reverence would manifest in internal peace, and all the darkness would be cast out through my grand gestures of morality. The sinner, the darkness, gone, and me, unified in good.\n\nBut it never worked.\n\nAnd if there was a God, he knew the truth about me, so fuck it.\n\n_I'm tired._\n\n\u2022 \u2022 \u2022\n\nLater that day on the island, after a short half-hallucinatory sleep, I talked to Stacy, who told me \"everything went fine\" with the marine biologist after I left. I was never quite sure what that meant. I told myself to believe her.\n\nI walked outside my motel room that day as the sun was going down across the water, casting orange light on my face and shadows in long, peaceful strokes across dusty roads.\n\nI went snorkeling with the other students that evening as usual, but for some reason this time I got going way out there in the ocean, kicking along for what seemed like twenty miles. I don't know why, I just kept going. I was scared I'd get eaten by a shark, but I kept kicking anyway because it was too damn beautiful to stop and the fear made me kick harder.\n\nI was entirely alone now but I kept kicking. The reef was down below me and then farther and farther away until it came to an end. I kept kicking past that, but it was just blue and blue and blue in every direction, so I turned around. When I did, I saw a wall of black before me, but a little below me, the reef stretched in every direction in front of me, and the crystal blue water above it. And me, hovering there like a speck of flesh in nothingness, a mass of dark that stretched forever into the bottom, though no bottom. I felt like I was looking at God and death and I shuddered. I felt like I was hovering above an abyss that would swallow me at any moment, as if I were taunting it with my squirming little body. The water above the black looked like a sliver too shallow to hold me, and I thought I could not cross back over. My eyes darted down, right, left until the terror of the black overtook me, and I threw my head up and ripped my snorkel off to suck in the air and stare at a spot on the beach\u2014to swim back, and remember I was still alive.\n\nI didn't look down again.\n\nUntil I got home, and began my descent into the black.\n\n\u2022 \u2022 \u2022\n\nTwo months later, my friend handed me a bottle of peppermint schnapps and said, \"Stop drinking when your lips go numb.\"\n\nBut I never did. I had found what I was looking for.\n**3**\n\n# **Sylvia Plath Put Her Head in the Oven over This Shit**\n\nOn our first day home as a family of three, Mac woke at dawn and went to work at his father's slaughterhouse, just like every other Thursday, Friday, Saturday, and Sunday since he was ten years old. This particular day, the day I walked around in pajama pants over a bulging hospital diaper with a colostrum-sucking newborn in my arms, happened to be Thanksgiving. Twelve hours before, we had walked into the dome through the pouring rain, with Ava in her car seat, and everybody cheered for us. For some reason, they clapped when we made it home. Perhaps it was the rain.\n\nAfter dinner, Mac went to sleep. I did not.\n\nIn the morning, Mac left.\n\nI did not.\n\nLeaving for work was not something Mac thought about, but rather something he simply _did_ , no matter what day it was, or who was just born, or how sick he was. Only absolute incapacitation would keep him from that slaughterhouse. I did not grasp the magnitude of this commitment until the first time we tried to take a vacation beginning on a workday. This was evidently the first workday he had missed for something as frivolous as \"rest.\" He spent the first nine hours physically ill in the bathroom of our San Francisco hotel room while I sat on the bed wondering how he could possibly ruin my experience in such a selfish manner.\n\n\"Mac, I don't understand. People are allowed to take vacations,\" I said, as he crouched against the wall near the toilet. \"What the hell is wrong with you?\"\n\n\"I don't know,\" he said, and I believed him even through my fury. I never understood how a person could grow so intricately entwined with a job. On the day I told his mother I was pregnant, she said, \"You can take the man off the ranch, but you can't take the ranch out of the man.\" Her words seemed true, but still, I wanted to extricate him from whatever sickness festered between him and his work, or whatever it was that gutted him for taking a day off, but he couldn't explain it, and I soon learned the source was invisible. It tore through him years before we existed together, and planted itself beyond reason's reach.\n\nBut on that day, the Thanksgiving he went to work leaving a two-day-old newborn and mother, it did not occur to me to be angry. Luckily, my mother was there, and she didn't tell me a father shouldn't leave the mother of his child two days after she gives birth, especially if it's Thanksgiving. I wonder if she knew and withheld that information, or, slightly more awkwardly, didn't realize it was weird either. Maybe she craved time with just the baby and me, or maybe my father would have done the same to her, or maybe we were all so fucking overjoyed by the baby we didn't notice his absence. I think it was the latter.\n\nPerhaps it would have been nice for somebody to say, \"Hey, Janelle. You should ask Mac to stick around a bit.\" Then again, logically, something we don't know must be benign. Unless it burns under the surface, like a wound growing larger right beneath our nerves, where we didn't even know we had nerves or needed help at all, until one day we find ourselves doubled over a toilet for taking a day off work. Maybe that's what happened to Mac. Maybe that's what happened to me.\n\nMaybe that's what happened to me three days after we brought Ava home, and I sat in a rocking chair in the middle of the November night under our bedroom window, out there in the country, where the sky is so clear it makes the cold seem colder. Light shone onto my face and Ava's, and I guess it was moonlight. I tried to nurse her. My nipples cracked against her petal mouth, but I nursed her anyway, because she was tiny and just right, and I refused to not breastfeed. I closed my eyes and gritted my teeth through the rush of fire in my breast. The nurse had told me, \"If she is latched correctly, it won't hurt,\" so I figured I was doing it wrong, but the milk spilled out around the edges of her little mouth. Seemed to me something was working.\n\nI was drunk at the sight of her, almost too much for my eyes, swaddled in my arms as my mother had taught me with my dolls years before. _You put the baby's head a little below one corner, and then you pull up the bottom corner, pull the right one over the chest and then the left, and then tuck it in tight like a little burrito._ But as we rocked, and as I glanced at her and back at her snoring father, maybe I rocked one, two, three too many times, and nudged some pain I couldn't see right up to the last layer of my skin, where the moonlight touched and her newborn head met my flesh. Because on the final glance at Mac, a truth settled into me like a sheet of ice.\n\nI had made an irreversible mistake. Not any mistake. The greatest mistake.\n\nHer. Him. The love. The pregnancy. It was all a horrifying error. I looked at him and her and back again and hated them both.\n\n_No. Not her._\n\n_Him. Him, there, free, alone on the bed while I sit spitting milk and pain and blood with the beautiful tucked-in bundle._ His body his own. His dick his own. His future his own. And me, decimated.\n\n_I don't want this_ , I thought. _I don't want him or motherhood and I don't want this body or man or house but now I am stuck, dragged and chained through a lie._\n\n_My own, or yours? Did I do this? How did I do this? I was conned. I was wronged. I was betrayed._\n\nI knew I had only myself to blame, but it was too cold to stay in that knowing. So I blamed him.\n\n_You wanted this, Mac. Now you've got it, and I'm gone._\n\n\u2022 \u2022 \u2022\n\nMy jealousy of people without children, of people I viewed as still free, became a living, breathing entity. My rage palpable, dizzying, smoldering always in my chest, and yet it was not alone. If it were, I could have walked coolly away from him, that life. If all I felt were sorrow and rage, the kind that engulfs you like a warm black cloud, seeping into every pore until even your breath feels leaden and ugly and wrong, I could have simply walked out on the bastards. In the dark of midnight, I imagined doing exactly that, but every day, at every moment, alongside my frantic desperation, was a love and fixation and pull to that baby more powerful than anything I'd ever known. Beyond me and utterly nonnegotiable.\n\nI could never commit to anything in life, except her. Fully, undeniably, against my every desire, her. It was a love that had me checking her ten times a night. A love that had me burying my nose in her mouth to inhale her milky breath, bathing, nursing, and holding her with arms that ached if she were not in them. When apart, I missed her the way we hunger for food: immediately, physically, and for survival.\n\nBut while I drove into town and she slept in the car seat, or screamed as if the devil himself were gouging her in the eye, I felt the weight of my sagging belly and tits and thought about Mac working at his going-nowhere job, and our little corner of the big ranch house, and how it all had become sad and heavy and boring. I would think about where I was supposed to go with my life instead of the car, and I would get lost. My grief drove me around in circles. I'd drive as long as I could, as long as Ava slept, because at least the time was mine, some solitude, some untouched mental space. Sometimes I drove by places I used to frequent in college, old bars and friends' houses, to remember my former life as some sort of masochistic lamentation. Sometimes I drove by my old boyfriend's house, the one who called that Halloween, and pondered what it would have been like if I had made the decision to meet him that night instead of Mac.\n\nHe was with somebody new. I hated her too, even though she was a former high school friend and I was the one who broke up with him. I hated the childless happiness they occupied in my imagination. I fantasized about running off to some Midwestern motel, where I would change my name to Charlotte and bartend for a living, simply pretending _none of this had ever happened_.\n\nFor I always had a next move, a way out, a backdoor exit plan. Another man, another drug, another town. Another job or friend or lie. I always had a posture or scheme to get out of a life that wasn't working. But motherhood, motherhood is a trap. It's like the goddamn \"Hotel California\": _You can check out, but you can never leave._\n\nIf I stayed, I faced a ruined life. If I left, I faced taking care of Ava alone, and sharing her with Mac and whatever slut he ended up with. The thought alone made my stomach flip. I couldn't stay and I couldn't leave, and when the permanence of motherhood dropped onto me, when I understood that _no matter where I physically went_ I would not escape it, my panic was indescribable. I was a feral cat in the first moments of capture, scrambling and clawing and screaming as she realizes there's no exit to her cage.\n\nSo I married him, as I said I would, because it was the only thing I could think of.\n\n\u2022 \u2022 \u2022\n\nWe took our vows on a cold, gray December day under a tree outside a courthouse in a town called Woodland, which is probably the nicest part of the town. My mother wore four-week-old Ava in a black sling across her chest. Our plan was to get married \"technically,\" then have a fantastic wedding a year later.\n\nI wore all black. I was planning on wearing jeans and a hooded sweatshirt, but my mother-in-law said, \"Oh you can't do that. You're getting married!\" and kindly took me to a boutique clothing store the same day, but I had recently given birth and felt betrayed by my body, with its giant belly hanging low, withered and stretched beneath alien milk tits. So I bought a black skirt and blouse, as if it were a funeral, though I merely wanted to hide behind cotton shadows.\n\nI told myself I didn't care. I told myself I was so in love I didn't need a wedding, but this was probably a lie. I knew it wasn't the \"real thing.\" I knew if we treated it as a gesture for taxes and insurance, if I wore regular old clothes and nobody gathered in feigned joy, it would remain a flash in the grand scheme. This almost worked, but my father showed up.\n\n\u2022 \u2022 \u2022\n\nMy father was not an \"absent\" father. He was a physically distant father. After my parents separated when I was six, I only saw him a few weekends a year, on one or two holidays, and during one weeklong stay in the summer. For me, those visits felt strained and tricky, because he was my _father_ , and I adored him, but I didn't know him well and was trying to make a good impression. This unfamiliarity alongside the paramount importance of his station in my life confused me: _I_ should _know him and be comfortable around him\u2014but I don't, and I'm not._\n\nI remembered life before my parents' divorce as an outline, a pencil scratch here and there\u2014sleeping on the prickly floor of our boat, riding over the Golden Gate Bridge in the jump seat of my father's Porsche 930, the walls of our home and the porch jasmine\u2014a slightly deeper mark on the day he cried and bought me a stuffed puppy from a pharmacy, and then my mother, brother, and I were gone.\n\nHow does one miss an outline?\n\nAfter the divorce, my father would send me postcards of Marilyn Monroe because my favorite movie was _Some Like It Hot_. We would talk on the phone, too, but I never knew quite what to say. When we visited him, my brother, father, stepmother, and I would drive north on Highway 101 in his two-door red Mustang for what seemed like three weeks straight, taking a pee break at McDonald's in Gilroy, where life smells like garlic, and, if we were lucky, stopping completely in Santa Cruz or San Francisco, but usually driving all the way to Healdsburg, in Sonoma County wine country.\n\nWhile we drove north, we listened to Bob Dylan, Neil Young, the Grateful Dead, the Rolling Stones, Dire Straits, and Pink Floyd. We listened to hippie music or Republican talk radio. On one side of the Mustang was an NRA sticker. On the other was a Grateful Dead \"Steal Your Face\" sticker. My father did not explain this, but he explained all sorts of other things, and I asked a lot of questions. Nobody ever seemed to exhaust my arsenal of questions. Adults looked at me as if I were annoying; my mother often suggested we \"play a five-minute silent game.\" I unfailingly lost.\n\nHis place in Healdsburg was a fifth wheel trailer I appreciated because it was tidy, along the Russian River, and smelled of my beautiful new stepmother's hair. We would spend our days at the river or at his fancy gym down the road, where we would shower in big tile stalls with glass doors and sit in a steam room that smelled of eucalyptus. Even though there were four of us in the fifth wheel, and my brother and I had to share the \"living room\" at the end, it was charming in its novelty. My father made us fried chicken and we watched _National Lampoon's European Vacation_ and _Some Like It Hot_ and _Ferris Bueller's Day Off_ until I had all three movies memorized. Margaret never came out in the trailer.\n\nSomehow it became a tradition that I would cook carrots on the trailer stove, by myself, which was of great consequence at age nine and ten and eleven. I cleaned, peeled, and cut the carrots\u2014an eternal, thankless job\u2014then cooked them on the tiny stove, while standing on a stool that took up almost the whole motor home walkway. I had to melt the butter and then put the carrots in the pan, but I couldn't cook it too fast because the butter would burn. Then I had to add the rosemary, salt, and a little pepper. I remember feeling like I stood at the stove for hours, and that it was a mind-numbing and difficult process, but it was the only way I could get that crispy caramelized butter on the edges of the carrots. We called them \"rosemary carrots\" and my father asked me to make them for him every time we visited, because he said it was the most delicious thing he had ever tasted in his life.\n\nBut you can't build a life on rosemary carrots.\n\n\u2022 \u2022 \u2022\n\nIt wasn't these memories that made his presence at my courthouse union almost unbearable. It was the way I stood on San Francisco's Pier 39 holding his hand when I was a little girl, or across the Bay at Sam's in Tiburon, or at the docks in Sausalito, when I pointed and ogled at the yachts, telling him, \"Dad, I'm going to get married on _that_ one.\" And he would laugh, and say, \"Yes! You will!\"\n\nIt was that he knew I held that silly dream, the one I now denied, was too tough for, tossed aside as if I could never be bothered with such girlish nonsense.\n\nFrom him, I couldn't hide in the shadows. Seeing him standing on that grass felt to me like I was crossing into womanhood, into a new family, without a proper goodbye. It simply made me sad. After the technicality, we went to dinner at a restaurant named after a duck. I drank two to seven glasses of wine.\n\nIn the photographs of the ceremony, I am holding Mac's hands and gazing into his eyes, with Ava and my mother right behind us, and a lady from the courthouse wearing wide-leg slacks, a green holiday sweater, and thick black eyeliner. She's holding a clipboard.\n\nThere are tears in my eyes. Of joy, I guess.\n\n\u2022 \u2022 \u2022\n\nI soon learned as a married, stay-at-home mother that if I remained drunk about 40 percent of my waking hours, I really enjoyed it. That is not true. I did not calculate percentages. Also, I did not particularly enjoy it.\n\nI would go to the store to \"buy groceries for a nice dinner\" and come back with a couple nice bottles of wine, _for our nice dinner_ , which I would drink while I cooked. At our actual dinner I would have more wine and a cocktail or two. This made bedtime manageable, as well as motherhood as a whole. (They do not write this in the \"new mom\" brochure we get when they discharge us from the hospital, but perhaps they should.)\n\nI drank for relief. I drank because from my first sip at sixteen, alcohol felt like peace, like coming home after a long and arduous journey. Anticipation of the day's first glass was a rush of lifted spirits within me\u2014energy, comfort, being\u2014and by glass number two, I began to feel the way I thought I should feel all the time.\n\nDrugs would do the same, but they required such commitment\u2014two a.m. runs, transactions with people I didn't know, dealers refusing to return my calls. After Ava was born, I was a drug dabbler. I was a fucking grown-up, after all, a _mother_. _Of course I don't want any blow._\n\n_Wait. Does somebody have it, though?_\n\nMore realistically, what saved me from narcotics was that I lived on a ranch ten miles outside an excessively vanilla college town where \"partying\" looked like nineteen-year-olds doing keg stands, not bumps of cocaine in bathroom stalls.\n\nAnd I wasn't seeking drugs because I had alcohol, which was enough\u2014mostly because it was reliable. You could get a bad baggie. You couldn't get a bad handle of Grey Goose. Plus, everyone drank. I could cling to alcohol like it was my last breath of air, but as long as I hid my desperation, the world would assume I was functioning, motherly, even sophisticated. They would believe the polish of laughter and smiles, as long as I never looked too thirsty or excited, as long as I never explained that if uninterrupted drinking was on the horizon, if I knew alcohol would soon pour into the cracks of my psyche, soul, and heart, I could handle anything\u2014even my stale days and too-young husband who left in the mornings, and the baby sucking my life dead and dry while making it infinitely more worth living and deep and clear.\n\nI held on that way, by drinking, and the love. Her tiny dimpled fingers.\n\nWhen Ava was about six months old, I thought I had found my own groove in the endless rhythm of motherhood, possibly even beyond White Russians and steadfast denial. I started exercising and writing again. I was researching graduate schools for a master's in English, and found a friend my age with a baby.\n\nBut one morning while Ava napped, I sat alone in the ranch house, surrounded by toys and blankets and diapers, next to a baby monitor rumbling with gentle snores, and I opened an email from my brother. I clicked on a picture of him in a white doctor coat, grinning widely on his first day of medical school at one of the top universities in America. My eyes studied his proud, hopeful ones, the sprawling manicured lawns, the old red brick building of the hall of medicine. I thought of new school years, semesters in college\u2014the pens (and how I always wanted fine-point blue), empty notebooks, literature on the shelves with its wild, disrupting ideas.\n\nA beginning. He was at his beginning.\n\nI was at my end.\n\nI retraced each line of his face and smile. Each second I looked, my heart beat faster. This man, my brother, who could make decisions and stick to them, who could not get pregnant by people he barely knew, or drink too much every fucking night. _He did it._ Growing up, I thought it would be me. I thought I would send that email, yet there he was, inarguably handling the world, while I sat immobile in a room I couldn't navigate. I couldn't even find its walls. I simply saw black.\n\nIf somebody had walked into that room at that very moment, I would have run upstairs when I heard the door open so they wouldn't see me crying. If I couldn't make it out in time, I would have swept my face with my hand and laughed about having just read something sad, but I would have disliked that lie because it would have made me seem like an overly emotional female. When others cried around me, I willed them to stop immediately because I felt compelled to say something supportive, but could only think of \"Pull it together, please.\" Or \"Do you want a cocktail?\" When sadness overtook me, I consciously pressed it into tightened fists, screams, and dramatic departures, but never tears.\n\nThere was nothing anybody could have said to fix it for me anyway, to give me a new way of looking at it, to patch up the hole in my brain or heart so I could pick myself up and carry on. I wouldn't have even let them try. I wouldn't have admitted how pathetic I felt sitting there, how small under the shadow of the photograph. I would have bragged. I would have said I was heading to graduate school soon. I would have squared my shoulders and acted like I had somewhere to go.\n\nBut that afternoon, in that chair, while I looked at my brother, my body shook and the tears came roaring against my will. _This? It can't be this. This can't possibly be my_ _life. Not now, at twenty-two._ It was startling to cry like that. I could not remember having done it before. I wept in heaves until the baby cried, again, wanting to nurse, again.\n\nI didn't go back to thinking I had found a groove.\n\nThe days began to blur.\n\n_Am I dressed? Am I ever dressed? How long until Mac comes home? How long until I can go to grad school? How long until dinner? How long until motherhood is over, or at least until wine? If I weren't here at two p.m. in my pajamas, I would be a lawyer, or writer, or something that mattered a little, at least. I would be young and hot. I would party. I would travel the world. I would do something. But I would not do this. I have to go. I have to get free._\n\nAnd then, her sweaty head, puffy eyes, and rosy cheeks would send smiling warmth to my bones, and I'd think, _I'll never leave you, baby girl. Thank God for you._\n\nCarry on. Change the diaper. Take a shower. Make dinner. Pour another glass.\n\nI tried to tell Mac I was barely functioning. I tried to tell him my life was in ruins, that I was no longer me, or a person at all, and sometimes I wished I had never become a mother.\n\nIn response, he went to work.\n\nThen he came home. We did it again and again and again.\n\nOn my twenty-third birthday, he rolled in from the slaughterhouse exhausted, reeking of goat guts, and I quickly realized he hadn't planned anything as a celebration. I threw a spectacular tantrum before dragging us to dinner, where he nearly dozed off at the table, and my fit resumed. Under those conditions, though, he had no chance of performing adequately. He thought we were going to dinner. I thought we were fixing my life.\n\n\u2022 \u2022 \u2022\n\nI tried to tell him I had been erased, even in body. I tried to tell him I was slipping. But he just looked at me. Always, he just looked at me. He looked at me while sitting on the floor or in a big easy chair, on the couch or in the car, on a park bench or at the pool table, or the bar. He looked at me and I couldn't understand his silence. His downcast eyes. In the hell of the morning, I loathed him, and laughed at what we called \"love.\"\n\nBut later, as I poured an afternoon glass, I'd think, _I was overreacting_ , and when I thought of him working down at the ranch, punching the hide off a sheep, cold in the winter bite, or walking across a dusty pen in 105-degree heat, I'd realize his silence was saying, \"I wish I knew how to help you.\"\n\nHe was twenty, and almost as empty as me. It made it worse that he was gentle and devoted to our family, that when he was home, he cradled our baby on his arm and never set her down, that he wasn't cruel or philandering. It made it worse because I knew he simply did not have what I needed. He was _there_ , with all that he was, __ but still it was not enough.\n\nI tried to tell him, but I didn't tell my mother. It didn't seem like the type of thing one mentions to other women. Or men, actually. So I kept it to myself. Who would I have told anyway? And what would I have said? \"I chose to have a baby and now I hate being a mother even though she's perfect and the father is diligently supporting us?\"\n\nAlso, nobody asked.\n\nI thought if the doctors found out they would take my baby away, and I would be crazy and alone, and my daughter would be without a breast and the arms that craved her.\n\nSometimes as I walked down the stairs holding Ava, I envisioned throwing her body off the top of the balcony\u2014not because I wanted to, but because the image slammed itself into my brain. When it hit, I would pull her hard against my chest and shake my head to get rid of the image. But I worried someday my arms would do it against my will, and as I held her against me I remembered that I could really, really never tell anyone.\n\nSo instead of telling people, I dressed her in European clothes I bought at discount stores, and took her to the park where we sat on blankets, and I was proud of us. I nursed her and fed her wholesome foods like avocado and zucchini I crushed myself, and didn't let anybody watch TV around her for fear it would sizzle her brain right then and there like an egg on a hot sidewalk. There was no sugar in Ava's diet, and she slept tucked against my body because she needed me and I needed her, and I could never get enough of her milk breath.\n\nBut when nobody was looking, I scrawled in my journal, in capital letters, in thick black ink, \"I CAN'T GO ON\" and \"I FUCKING HATE MY LIFE\" and \"WHEN DID IT ALL GO WRONG?\" I pressed hard on the page so it would go through to the two or three pages behind it, like a little kid learning to write, but actually I was begging the page to help me.\n\nI had been running to paper for safety since I was a little girl, since the bishop's wife handed me a journal after my Mormon baptism, saying, \"You should write in this every day,\" and I looked at those lined pages and thought, _This is mine to fill up. This space is mine._ I wrote every day, just as she said.\n\nBut it was only after I had Ava that the page did nothing, that I felt worse after scribbling the darkest of me into sad linearity, into neat reductions of grammar and diction and punctuation. It was only then that my words made me sicker, and left me deader, as if whatever was living inside me refused to be contained, ordered, or made into meaning. Trying to stretch it into lines left me frustrated, and lonelier, because every sentence ended in conflict, in paradox, in the whimper of a baby who needed me.\n\nThere was no resolution because the revulsion was not about her. It was me. It was all that led up to her creation and birth. It was the man who didn't understand my grieving and the single bedroom we shared. It was all that and some dark thing I couldn't explain, trace, or trap at all, like a curtain over the sun blocking the light. Nobody can dodge that pain, or the exhaustion of trying to breathe beneath it.\n\nStill, I thought if I used all my strength to scream into the page, the black would move right out of me and get taken up. I remember shuddering while I wrote and pressing as hard as I could for relief. I lifted my head and squinted my eyes and waited.\n\nIt never worked. Except alcohol, nothing worked.\n\n\u2022 \u2022 \u2022\n\nOne day, while Mac was at work, Ava began fussing right after I put her down for a nap in the late morning. I stared at the monitor, then dragged myself up the stairs as her cries took over the hallway.\n\nAs I walked to her room, my thoughts were like a train crashing through my mind, accelerating with every step. _Always waking up_. _Jesus fucking Christ, she's always waking up_. _Waking up and stealing my few seconds of humanity, of freedom. Waking up and making me face her and me and all that was going to be before this hell hit like a_ _flash flood across a broken body, and I'm crushed but somehow not dead. MY GOD, CHILD, FUCK YOU_ \u2014and when I got to her, I pinched her thigh, hard, to hurt her.\n\nShe wailed.\n\nI cried out too, and pulled her body to mine before dropping to my knees. I gripped her against my chest, with her head in my hands and my fingers stroking her blond wisps of hair. I gasped through tears and begged her to forgive me, and in that moment I realized I did not care if they took her. I did not care if she was no longer mine, because she was better off without me. I had hurt my baby, my life, my perfect creation. I had hurt her on purpose.\n\nI called the doctor's office immediately and explained to the first person who answered that there was something wrong with me and I could not take care of my baby. I didn't tell Mac I was going to call, because I knew he wouldn't understand my decision to give our baby away. He was not brave. He would not let her go. He would tell me to lie. He would tell me to keep hiding who I am.\n\nThat very day, I went to the doctor's office and told the nurse about pinching her, and the visualizations, and that my life was black-tar grief. I steeled myself in preparation for her words, \"You know we have to take your baby, right?\" I was ready to say goodbye.\n\nInstead she put her hand on my knee and left it there as she said, \"Oh, honey, you have a bit of postpartum depression. We'll have you fixed up in no time.\"\n\nShe shook her head and blinked like a doll. She seemed unreal sitting there, talking to me as if it was all no big deal. I got to keep the baby, because this is a thing that happens to some women after they give birth. Depression. That is its name.\n\nHer voice was sweet and smooth and slow, but squeaky like a child's, and it startled me at first. _Why does she talk like that?_ _With such sweetness._ It didn't matter. In that moment, I loved her.\n\nShe prescribed an antidepressant, and _just as promised_ within a few months my whole life transformed from behind my own eyeballs. Doctors told me I was bordering on postpartum psychosis, that I was lucky to have come in, that pinching my baby that day could have saved my life, or hers, or both. Maybe it was the pills, surely it was the pills, but I'm not totally convinced I wasn't healed by the feeling of the nurse's hand on my knee when she said \"Oh, honey,\" because I felt like a leper who was touched for the first time by somebody who didn't care that she was sick and deformed. I was so sure I was monstrous, some uniquely evil specimen of motherhood, that when she touched my knee and called me \"honey,\" like I was any old mother passing through with a bit of PPD, I saw myself anew.\n\nI was in a pit of black and couldn't see out. I felt the touch of a nurse and Zoloft and crawled to the top, looked out, and thought, \"Oh shit, this ain't so bad.\"\n\nAnd it wasn't.\n**4**\n\n# **Playing House in the Suburbs with Captain Morgan**\n\nAtop that pile of happy pills, I peered out over the edge of depression and surveyed the landscape. The first thing I saw was \"I need out of this fucking house.\" Nothing was wrong with living in Mac's parents' house except we were living in Mac's parents' house. We were adults. We needed adult space, and in my case, employment. Stay-at-home parenthood was boldly oversold.\n\nSo I constructed a r\u00e9sum\u00e9 emphasizing the honors classes I took in college while omitting the one (or two) I failed, and applied for a receptionist position at a highly respected boutique law firm about thirty minutes from the ranch, in downtown Sacramento. When they called me in for an interview, I knew I had a chance, because I did well with first impressions, particularly when sober. It was everything following the first impression that troubled me. I could give you what you wanted. I just couldn't _keep_ giving it to you.\n\nSitting across from me in a large conference room, an extremely put-together, reserved woman with long, curly brown hair asked, \"What experience do you have with administrative work?\"\n\n_She wants_ _honesty._ \"Well, I spent a summer as an intern at an office supplies business, but I don't have a _ton_ of experience.\" I smiled and made a little face, as if to say, _Can I really say that?_ As if I were a bit coy.\n\n\"I graduated from UC Davis about a year ago, but stayed home with my baby,\" I continued, \"but I am a quick learner. I am very thorough.\" My mother had told me once while I was sweeping out our motor home that I was \"very thorough.\" I stuck with it.\n\n\"What do you think your greatest asset is?\" She offered a quick smile between jotting notes. I noticed she was left-handed and that her blouse perfectly matched her cardigan.\n\n_Humility. Tie it in with the honesty, Janelle._ \"I am willing to do whatever it takes to get the job done. If the firm needs me to scrub toilets, I'll do it. I'm here to work and I don't have too much ego wrapped up in that.\" She smiled again, and I felt bolstered. _You're doing great, Janelle._\n\n\"We are extremely focused on collaboration. What is your greatest weakness?\"\n\n_Captain Morgan._\n\n_Nope. Don't say that._\n\n\"Oh, well, I think it must be that I can be a bit of a perfectionist. I don't want to let things go if they aren't perfect, or close, you know? So sometimes I get frustrated with people who don't have the same focus as I do.\" I failed to mention that I thought most people around me were fucking idiots who should lose their jobs. That if I thought things, they were true, even if I had no evidence for them, and that, frankly, I was not exactly shining in my own life, and threatened to leave my husband on the daily. And, speaking of daily, I drank at that exact interval, and used to chase my brother around the house with a large kitchen knife.\n\nI kept all that to myself and crossed my legs.\n\nI was a master at selling my potential. My second-best talent was leaving before you figured out I couldn't deliver. Or I simply never showed up at the moment of delivery. I stopped calling friends just when they thought we were really connecting, didn't show up on the day they stopped doubting I'd show up. It was not conscious, but it was consistent. In college, a few professors wrote on the bottom of my papers, \"Come see me about getting this published\" or \"Let's submit this to a contest I know of!\" Each time, I called my mother and told her about it on the way home from class, felt proud and hopeful, told strangers about it at bars, but I never, ever showed up at their offices.\n\nBut this\u2014this was an interview, and I believed what I was selling as genuinely as they did. As I shook the interviewer's hand and walked out the front door, I called my mother to tell her how well it went, and after the second interview, this one with a managing partner, they hired me. I landed the first \"real\" job I applied for, confirming my suspicion that I was _a well-equipped human ready to embark on many great successes._\n\nMy first day of work was almost exactly two years after the February morning on the balcony. I was twenty-three and Ava was fifteen months. I woke up earlier than necessary and dressed her in Oshkosh train overalls, a white, collared long-sleeve blouse, and her favorite red leather Mary Jane shoes. I cleaned her face spotless and brushed her wavy blonde hair into pigtails. I packed a blue gingham sunhat into her diaper bag, and folded spare clothes with her name on their tags.\n\nAs I closed the babysitter's door behind me, I wondered if Ava understood why I was gone, and if I was perhaps making a mistake, and if she would be okay, and if maybe I regretted the whole job decision. I continued thinking these things until the moment I climbed into my car, put it in drive, and realized I was truly, completely, and finally alone. I glanced around the car to confirm it was real. No crying, no baby, no diapers.\n\n_Oh, thank God. Here I am again,_ I thought.\n\nI stopped for a cappuccino. I tore down the freeway.\n\n\u2022 \u2022 \u2022\n\nWhile I saved my receptionist money, my mother and I concentrated with laser-like focus on the role of a house in the stabilization of marriage. She'd say, \"You need a place to call your own. You need to be able to create a home for your family,\" and I would agree emphatically. The next day, when I would call about Mac's astonishingly disturbed relationship with a ranch\u2014he still, after two years, couldn't take a vacation without getting ill from stress\u2014she'd say, \"It will get better as soon as you get your own place.\"\n\nWhen I would tell her I was generally dissatisfied with life, or that I hated my body, or that the sense of monotony and seething banality of my daily existence might actually one day take me _out of the earth_ , she'd say, \"How long until you have your deposit?\"\n\nAnd I'd say, \"You're right. It will be better when Mac and I move out.\"\n\nWe spoke as if it were a biological imperative, as if I would implode if I didn't have access to 1,200 square feet to clean, as if all that was wrong with me could be fixed with some carpets and cabinetry that were all my own. And so, when I banked enough money, I found a house in the suburbs of Sacramento, in a strip-mall paradise called Elk Grove. It's the kind of place one drives through and wonders if the civil engineers went out of their way to remove all soul from the city, or if that was simply a by-product of the ratio of big box stores to humans. I made an appointment alone, viewed it alone, and rented it on the spot.\n\nWhen I got home holding a rental agreement, I figured I should tell Mac.\n\n\"Hey,\" I said, after a few lubricating glasses of red wine. \"You know how hard things have been. We need to get our own place. That's the thing. We'll be happier.\"\n\nHe shifted in his seat, his beautiful eyes growing pained at the impending confrontation. But he didn't say anything. Not with his mouth, at least. So I filled the silence.\n\n\"I'm a mother, Mac, I need my own home. Mothers need it. We need a place to decorate. You don't understand. It's a big deal\u2014like a biological thing. You know, this isn't my family.\" I swept my hand across his parents' living room while a familiar rage crept up my feet and all the way to my eyes as I thought about _all I had given up for him and this kid_.\n\nHe looked at me harder while his silence transformed from _space for me to talk_ into a hundred-foot steel wall between us. I readied my attack. I thought it would penetrate the wall.\n\n\"Are you going to say anything?\" I was done waiting.\n\nHis eyes widened but his mouth didn't move. He leaned back against the kitchen chair and picked at the label on his beer bottle. He was still wearing his work clothes, an orange and blue plaid cotton shirt, an old button-down from high school, long covered in bloodstains. His curly hair stuck out wildly above his ears from beneath his ball cap, which he had flipped up and to one side. He had dirt across the bridge of his nose, and I couldn't move close to him on account of the reek of slaughterhouse.\n\nI took a sip of wine, wondering how he managed to stay silent for so long in moments like this, simply refusing to make a sound. Even when angry, when I swore I would punish my adversary with the severest of silent treatments, I found myself unable to quit speaking. It was remarkably disappointing. But Mac would stare at me for an entire night, look away thirty times, walk out ten times, but he would not speak unless he was so inclined. To help him along, I would chase him around the house like a puppy attacking his pant leg.\n\nI hoped this night wouldn't be one of those nights. \"Mac,\" I said, attempting greatest civility, \"could you please tell me what you're thinking?\"\n\nFinally, after an incredible pause, he said, \"We don't have the money,\" and watched me pour a cocktail.\n\n\"Yeah, we do. We're fine,\" I said, dismissing him, thinking, _God, I hate it when he has opinions._\n\n\"How will we be fine, Janelle?\"\n\n\"Because I can't stay here anymore.\" By then, I was yelling. I didn't want to talk about his theories of finances. I knew he was simply afraid. He was always afraid. My mother and I had already discussed the whole situation. It was handled.\n\nHe countered with more silence, and I knew it was because I yelled, because I asked him to speak and then attacked him, but what I heard in his answer was that he didn't understand. Again. He took his hat off, and a stray curl fell over his eye, and I noticed his new piercing in his ear. It was a bar that went across the top of his left ear lobe, and I flinched remembering how the piercer inserted it. He stuck a thick needle slowly through the ear and then passed the bar through. But he couldn't get it right, so he pushed the bar in and out through the cartilage, twice, while Mac sat motionless, sweating.\n\nHe was the toughest human I had ever known. But I knew it would be me who carried us someplace new.\n\n\"Whatever,\" I said, holding his eyes. \"I already rented it.\" His mouth tweaked in confusion.\n\n\"You can come with me if you want,\" I continued. \"Or you can stay here. I don't fucking care.\" And I walked out of the room.\n\n\u2022 \u2022 \u2022\n\nHe came with me. The house had two bedrooms and two bathrooms. It was one of those subdivision homes with a five-foot buffer between it and the next house, no sidewalk, one ten-foot box of perfectly manicured lawn out front, and a two-car garage. The neighborhood was neutral tones and stucco as far as the eye could see. The inside of the house felt clean and contained and new, exactly what I needed to begin my life as a fully-pilled-up, non-psychotic working mother. We moved in and bought a beagle puppy that we named Maggie May, after the Rod Stewart song.\n\nI could not tell which house was ours from afar. Every time I drove near it, I had to check the house number in the black-and-white light-up box. The subdivision had a fake lake a few blocks down from our house, and sometimes we'd walk around the water on the clean cement walkway lined with sapling trees. I'd look at the bigger houses around the lake with bigger backyards and feel envy and disgust at the same time.\n\n_This couldn't be it. But damn, look how nice that pool is._\n\n\u2022 \u2022 \u2022\n\nIt was exactly like my mother and I imagined it, which was exactly like a 1950s sitcom, except Mac and I were the main characters.\n\n\"Hey, love, how was your day?\" I'd ask when I got home from work. But instead of putting on my apron and floating around the kitchen like a house angel, I'd put my bag of groceries on the counter, kick off my scuffed fake-leather shoes, and wonder how long until I could drink rum.\n\n\"Fine, you?\" he'd say. I would mention what I was making for dinner, and he would take his boots off, and we would watch Ava play in a laundry basket, and laugh. Then I would notice the laundry at the bottom of the basket, which was clean, and yet, still in the basket.\n\n\"Hey, you know I, uh, put that basket there so you'd fold it.\" I'd say.\n\n\"Mmm,\" he'd mumble.\n\n\"What?\" I'd snap, leaning toward him, as if to say, \"You ready for this fight?\"\n\n\"Sorry. I didn't notice,\" he'd say in quiet apathy, an indifference that seemed to place the weight of our lives onto my shoulders. I'd watch him rub his feet and I'd notice again his flannel shirts stained with some livestock excretion. He didn't care about laundry or our futures. He just worked. He worked Thursday through Sunday. I worked Monday through Friday. We parented in silos. I worked five days at the office and two days alone at home, where I would do all the things that never occurred to my husband, who had moved from his mother's care to mine, his parent's house to this one, and simply steeled himself when my rage blew, hunkered down until the tornado passed, which usually looked like me getting drunk enough to no longer care.\n\nOn the days when I worked and Mac didn't, our evenings were always the same.\n\n\"What did you do today, Mac? What'd ya do today, exactly? Because the house looks the same as when I left.\" The anger settled across my shoulders, heavy and exhausting.\n\n\"I don't know. I drove Ava to daycare, picked her up.\" He'd throw me a glance and I'd notice how handsome he looked sitting on the couch with his feet up, out of his work clothes, his perfectly square shoulders and chest, and I'd marvel at how somebody who infuriated me so completely could be so fucking gorgeous at the same time.\n\nI'd pummel him with questions about household tasks. Dog food. Dishes. Laundry. The oil change. He'd meet my inquiries with silence, and I'd wonder what beat him into silence. Shame? Resignation? Personality? How broken-down must one become to simply power off in the face of conflict? But I had no energy left to draw words from him.\n\n\"Did you watch TV this entire time?\"\n\nSilence.\n\n\"Mac have you not noticed that I work five days a week and then when I get a weekend, you go to work, so I work seven fucking days a week? Do you think I sit around watching TV for five hours? Do you think I _ever_ enjoy myself? I can't do this. I want a divorce.\"\n\nSilence.\n\nI would uncork a bottle of wine, and by glass two or three, I would speak more gently, earnestly. \"Mac, please. I need your help. I need _you_. Why don't you mow the lawn, take care of the car? Anything.\"\n\nSilence.\n\nBy glass four or five, I would scream or cry, or drive to buy a bottle of rum, or sit on the couch and watch some British sitcoms. Or we would go into the bedroom and have sex that never left me feeling whole, but at the end, I'd think, _He's really going to change this time, on account of our highly productive talk._\n\nI would pass out, then wake up in a haze and sweat at two or three a.m., with a headache that threatened more sleep. I'd find the Advil and water on my bedside table, take a few, and in the morning, I'd get up and go to work. I'd feed Ava an egg or oatmeal, wash her face, and dress her warm in the winter and cool in the summer. Almost every day, I'd take a picture of her grinning and holding her little lunchbox beneath the white light-up address box, and I would send it to my mother so we could ogle at her beauty.\n\nI decorated Ava's room with a twin bed made of pine, a white bookshelf, and a rug from the actual Pottery Barn, which felt like reaching a suburban pinnacle. We now had a real bed instead of a lumpy futon, and I hired a dog trainer for that fucking beagle. The trainer said Maggie was the worst dog he'd ever seen in his life and essentially told us the little bastard was unfixable. After Maggie ate our hose for the twelfth time, pissed on the carpet, and stole our dinner off the table, Mac and I agreed to give her away to an old man who had owned beagles his whole life and never left his house. I missed her as soon as she was gone.\n\n\u2022 \u2022 \u2022\n\nEvery day, I woke up and got dressed in my business casual clothing, wished I exercised more, drove twenty minutes to my job in the black Infiniti four-door sedan my father had sold us. I listened to NPR on the way, specifically the old guy with the oddly soothing voice who did the news. I sucked down coffee and a cigarette. I played my music loud. It was wonderful, sort of. It was at least twenty minutes each day that felt hopeful. I was heading to a new place. A new day! Tiny potential!\n\nAt work I could drop my purse off at my desk and travel into the clean break room for a cup of coffee. While I stirred in the cream, I could stand by the counter and say hello to my work friends and feel capable and centered. In the afternoon I might get a sparkling water from the refrigerator or a caffeinated soda if I were tired. I might walk down the hall and grab a chocolate from my boss Beatrix's office.\n\nTo keep me busy between calls and visitors, Bea would give me tasks I could perform right there at the reception desk, such as calling computer companies for technological difficulties or to discuss warranties. Often I would end up shouting at the overseas customer service representative, right there in the front office. Eventually Bea banned me from making such phone calls, and I told myself she had simply realized, finally, that I was too damn smart for such menial laboring.\n\nMy receptionist skills beamed brightest when welcoming guests to our office, although my judgment was perhaps slightly off. I once asked a woman there for a job interview if anybody had ever told her she had a porn star name, because she did. She started, but grinned, and said, \"I've never been in a law firm with a receptionist as inappropriate as you.\" That woman was hired and we ended up becoming friends, but the difficulty was I never knew how my verbal escapades would turn out, and yet, I couldn't seem to stop myself from experimenting. I'd either make a new friend or get reported to my supervisor.\n\nI learned quickly that the superpower of low-level administrative staff is invisibility. To some, I was so unimportant I would _actually disappear_ , and as a result, they would speak openly, _right in front of me_ , about confidential topics. I would sit staring at my computer screen, clicking with great focus on absolutely nothing, listening to guests talk about employees of the firm as if we weren't sitting in a tiny lobby together, as if I didn't know the people I worked with, as if I were not even human. I'd think, _They don't even see me_ , and I would feel powerful in their underestimation.\n\n_Fuckers. Do they even know how much potential they're looking at?_\n\n\u2022 \u2022 \u2022\n\n\"Hey, Janelle, you can't just _say_ every single thing that comes into your mind.\"\n\nI sat in Bea's office crossing and uncrossing my legs, folding and refolding my hands, getting my first \"talking to.\" I scanned the blank surface of her immaculate cherry wood desk, looking for a single stray paper clip or wadded up Post-it note. Nothing. Not even dust behind the computer monitor.\n\nHer inbox, empty. Her candy bowl, empty. I visualized my desk with papers strewn in a thousand directions, a little bonsai tree dying from lack of water, and the pictures of Ava and Mac. I made a mental note to clean my shit up.\n\nI had been at the firm for one year, and they had recently promoted me to administrative assistant. There's a chance they were trying to find me a less visible place than the front office. At any rate, I now had my own cubicle and a raise. I was proud. Unfortunately, the attorney I was supporting was less impressed with my performance.\n\nBrian, the attorney in question, was a perky gentleman whose demeanor masked what seemed to me a deep-seated sense of superiority...for approximately fifteen minutes, at which time I concluded, based on my vast professional experience, that I was dealing with a person who loved nothing more than questionable ideas and the fact that he could make me do them.\n\nEvery Monday morning, I was supposed to meet with him and listen, riveted, to his peculiar but incredibly precise requests. Then I was supposed to work hard to implement them, with glee and _to the letter_ , even though I believed them to be absurd and inefficient. But every time I did a task, some horrible new detail popped into his brain, adding another act to our little roadshow.\n\n\"Janelle! How was your weekend? How's Ava?\" I would begin to answer, but at mid-sentence, he'd grow bored: \"Great, thanks. Would you mind researching Internet providers in my area and making a spreadsheet of what they provide and how much they cost?\"\n\nSo I did it, and when I showed him my work, he'd say: \"Please add a column with Internet speeds and then research what those speeds mean. Translate them for me. You know, layman's speak.\"\n\nHe would regularly use terms like \"layman's speak,\" which felt as inhumane as the person in the cubicle next to me eating corn nuts. I noticed that Brian ate uniformly healthy food except on Fridays, which I imagined was a plan he had actually created for himself and stuck with. This was so profoundly reasonable and mature I contemplated poisoning his food simply to shake things up a bit. He smiled too much and never failed to say \"Good morning!\" as he passed each person in the hallway, making sure to use their first names. At least once a day I visualized him making spreadsheets in quicksand.\n\n\"Can you also add a column letting me know how long each company has been in business, and whether or not they outsource work in India? Also, I need this color coded, I think. Easier on the eyes.\"\n\n\"Color coded? Why? Brian, that doesn't make sense.\"\n\n_Janelle, you can't just_ say _every single thing that comes into your mind._\n\n\"One more thing: New schedule routine. Let's try you printing out my daily schedule and highlighting each activity using different colors based on type of activity. For example, internal meetings could be blue. External meetings, green. Lunch meetings, pink! And you know, as long as they're consistent, go ahead and have fun with it.\"\n\n_Oh, wow, shit. For sure this sounds fun. This is exactly the kind of activity a person of my mental capacity finds pleasurable._ _My dream actually is to spend many years highlighting schedules for no apparent reason._\n\nBack in my cubicle, while I worked to find the perfect shade of sea green for the AT&T column, I'd wonder if perhaps I was cut out for slightly more challenging work. I wanted to impress everyone with my administrative assistant skills, but _color coded spreadsheets?_ __ I'd grow further confused when I'd overhear the managing partners consulting with him on major clients, as if he in fact knew how to do his job quite well and the color coding situation were a mere idiosyncrasy that could be overlooked.\n\n\"Janelle, within reason, your job is to do what he asks.\" After saying this, Bea held my gaze, registering my expression. She knew what I was thinking. Smiling, she added, \"Your job is to be patient with people who aren't as smart as you.\" With this, my ears perked. _Now we're talking, boss._\n\nBut she wasn't talking about Brian. She was talking about my impatience with all those I perceived as having subpar intelligence, which was nearly everyone.\n\n\"I try, Bea.\" I met her eyes while running my finger along the edge of her desk, still hoping for dust. She raised her eyebrows, knowing I was using the term \"try\" rather loosely. I laughed, vaguely terrified of her. She embodied all the professionalism I lacked, combined with follow-through. She was fair, consistent, and heartbreakingly organized. And yet, she had _lived_. She grew up in a goddamn desert, raised her child alone, pulled herself out of poverty. She was a single mother with an immaculate desk.\n\nI tried to do what Brian asked, but I couldn't wipe the disdain off my face. I kept forgetting. Or it popped onto my brow before I could stop it. The fact that he was so damn friendly while he wasted my time simply confused me. Although, how very capitalist to find yourself getting fucked and yet kind of enjoy it. I even grew to like the man. And yet he wanted me to tape his schedule in his planner in 2003. _We have computers, man._\n\n\u2022 \u2022 \u2022\n\nAt our administrative staff meetings, I focused all my energy on making sure I didn't send my yellow legal pad hurling across the room in a fit of final unbridled rage, or fling my coffee cup at the person who brought up refrigerator soft drink organization again. The worst was when I was supposed to lead one of these discussions, telling all the women and one man how the research on a new coffeemaker was going, or where I had put the masking tape in the copy room, as if we weren't going to lie in coffins someday with maggots eating our eyeballs. By this time I had been promoted to Slightly Higher Admin Assistant, and I'd sit in the conference room listening to a person born onto earth as good as the next guy talk about parking validation methodology with the vigor of a presidential nominee, thinking, _What if I started stabbing myself in the jugular with this Uni-ball pen? Or started banging my manila folder against that lady's head, right there next to me, over and over again, without a word?_\n\nOnce again, I feared one of these days I wouldn't be able to stop my body from carrying out the vision. Meetings with the whole staff were slightly better because I got to gaze at the managing partners and imagine how big their houses must be, but still I wondered what would happen if I stood up, unbuttoned my cardigan, and smashed a scone between my tits in an act of silent resistance.\n\n_Janelle, you can't just_ say _every single thing that comes into your mind._\n\nThis includes _shit_ , _cunt_ , _bastard_ , _bitch_ , and any variation of the word _fuck_. I also learned that you \"can't wear black bras under white shirts\" and low-cut business casual pants that show your underwear are also frowned upon. Further, copy machines have a top-loader where one can load multiple pages at once and they will run right through the copier, the whole pile, like it's nothing. This particular piece of information smashed me like a brick during the new-copier orientation meeting. I had spent an entire summer after my first year of college working as an intern at an office supply business headquartered in a desolate wasteland called Hayward in the San Francisco Bay Area. For a solid three weeks, my job was to photocopy booklets held together by removable plastic binding. All day, every day, from eight a.m. until five p.m., I stood photocopying page by page, turning the page, opening the copier, laying it on the glass, pressing the \"copy\" button, opening the top, taking it out, turning the page, and on and on. For three weeks.\n\nNot a single one of those cunt bastards told me I could just take the binding out and put the pages in the top loader. Not a single human noticed that I was wasting hours and hours of my life unnecessarily. Or maybe they did notice and found it funny, thinking, \"I wonder how long it will take the idiot intern to realize she can just stick them in the top loader?\"\n\n_Well the answer is \"never.\" She will never realize it. Actually, no. She'll realize it four years later. But as far as you're concerned, dick, it's never._\n\nI never would have done that. I would have told the sad new intern how to make her life significantly easier, how to do the job better, how to make her day a little more meaningful. But then again, the people who silently watched me waste my time day after day sold Post-it notes day after day, arguing about pen quality, sitting month after month getting closer to heart disease and death, the highlight of their day being the trip to the sandwich joint around the corner, or the chocolate they'd grab from their boss's office, or the afternoon soda they'd pick up while in the break room, chatting with their friends and wondering what might happen for dinner.\n**5**\n\n# **Three a.m. Ideas**\n\nIt was complicated to think so highly of myself and yet have a life making color coded spreadsheets while wearing size 16 Old Navy slacks and sitting in a cubicle, living in a monochromatic rental with a young husband and baby, years before any of those things are supposed to happen to anyone, let alone a person destined for greatness. I felt myself becoming like the Post-it note sellers in Hayward, but managed to smooth the dissonance of reality by clinging to the year I lived the wonder I thought would last a lifetime.\n\nIt had only been three years since I stepped off the plane in Barcelona, arriving in what I was sure must be the \"prime of my life.\" I was twenty years old, unattached to any human, tan from my summer job lifeguarding, with newly shorn hair. Until my twentieth year, my hair fell long and straight past my waist. This had been my hairstyle since I began growing hair, but it wasn't really a hairstyle at all. It just grew that way because I didn't trim it often. But I had it all chopped off at twenty and I felt free and lighter and new. Friends told me I looked like Cameron Diaz. She must have been at the prime of her life, too.\n\nI spent my first month in Barcelona in dorms with exchange students from all over the world, although the Americans seemed to dominate the place, as we often do. We were supposed to engage in intensive language study and find ourselves places to live, but I don't recall the language study. What I recall is the caf\u00e9 across the street and the thick summer air\u2014the feeling of being beautifully lost, of sitting at plastic tables in front of the dorms and feeling alien, but warmly so.\n\nOn day five, I gathered the courage to wander into the caf\u00e9 I had been staring at every evening since my arrival. I watched the locals and tried to copy them, the way they moved and glanced at each other, but as I walked up to the counter and stared at the chalkboard on the wall, feeling the eyes of the dazzling Spaniards burning into the back of my head, I almost couldn't speak.\n\n\"Caf\u00e9,\" I said, thinking it was coffee. \"Por favor.\" I handed her the pesetas I had counted out in advance and retreated in humiliation. Instead of coffee, I got a tiny something in white ceramic. It was pure espresso. I sat alone at a table in front of the cafe, had a smoke, and drank the espresso black, too scared to add sugar or milk. Soon I would learn that what I wanted was a \"caf\u00e9 con leche.\" I would have hundreds before leaving a year later, and each one would be better than the last.\n\nWhile sitting in my dorm that first month, a British boy named Francis with brown eyes and wild, wavy brown hair that hung down over his forehead walked by my room, or maybe I walked by his, and we chatted for a few minutes before he said, \"I'm taking the train to Madrid tomorrow. Want to come?\"\n\n_Yes, absolutely, guy-I-just-met. I absolutely want to go to a place I've never been with a person I don't know._ There was no voice whispering, \"Hey, you don't know this fella, Janelle. You just arrived in this country, and you barely speak the language.\" Francis was the reason I went to Spain. I wanted adventure. I wanted to press my whole body against the boundaries of the world to see how much it could support. I wanted to be Mick Jagger and Gertrude Stein and Ernest Hemingway at the same damn time. Hemingway never would have said \"no\" to a friendly Brit offering a trip to Madrid. So I, too, said \"yes.\"\n\nI showed up the next morning at the train station wearing thin, wide-leg cotton pants and a tight spaghetti-strap tank top, without a bra. We drank beer on the train in the smoking car and as I watched the countryside speed by, I thought, _I must be in heaven now_.\n\nIn Madrid, Francis and I met his friends who were summering in Spain and somehow all looked like supermodels. We lounged by the pool and drank beer in the living room until somebody set up rows of cocaine on the glass-topped kitchen table, and I knew I was going to be just fine in this country.\n\n\"Yes, please,\" I said, ending a two-year streak of cocaine abstinence. I hadn't abstained for lack of interest, or because I \"wanted better for myself;\" rather, my college-town acquaintances rarely possessed hard drugs, and I was not yet driven to find them.\n\nBut if it had appeared one night, like it did that night in Madrid, I would have said yes, because I always said yes to cocaine. I tried the drug for the first time at seventeen years old in my high school boyfriend's house. We did a couple of grams with friends, sniffing it off a mirror on the bathroom counter until it was gone, and while everybody else in the house smoked weed and drank to ease the spastic comedown and eventually sleep, I said I had to pee, went back to the bathroom, and snorted lint off the floor, thinking surely some blow had dropped to the ground and I could get one more hit.\n\nAfter that night, I never found it necessary to complicate cocaine decisions, never thought twice about chasing the soaring light, the electric meaning, no matter what was happening the next day, or who I was with, or how many times I promised myself I'd never do it again. I would handle that later. In the critical moment when I needed my brain to talk some sense into me, I saw only the high at my fingertips. The Technicolor life. _Oh, the love we would feel, you and I. The truth we would speak._ _Let's watch the boredom burn. Let's see if we can survive. Let's do really stupid shit and get away with it._\n\nThat night in Madrid we danced at a multilevel open-air club filled with soapsuds and bubbles, and I did bumps in the bathroom with my new friends. When we got back to the apartment, Francis and I began kissing on a bedroom floor, but I was coldly uninterested, participating solely out of a young woman's sense of obligation to deliver what she was taught men \"deserve.\" I wished only that we had more coke. I was always the one wishing we had more coke. I was always the one spinning in circles looking for more\u2014in the carpet, the folds of the couch, the crevices of my wallet. When the rest of the party moved on, smoked some hash, took a few shots, accepted the end of the baggie (without a fight, even), I was the one coolly dropping the idea of getting more.\n\n\"Hey, just for fun, let's call the guy to see if he's still around.\" I would smile, reminding myself that next time I needed to hold \"the guy's\" phone number.\n\nThe psychopaths around me would shoot my idea down in midair, and my heart would panic to see the amphetamine light die. My blood fumed at the impending return to regular life, and I was crushed under a wave of sadness, of fiery restlessness. So I'd make my way to alcohol, the strongest drug we had left, and I'd drink it fast and hard and smile faintly when they looked my way.\n\nSomebody in the living room made a joke about coming down. I wanted to rip his face into shreds. Or I wanted to be exactly like him, cool and calm when the drugs ran out. I wanted to enjoy making love there in Madrid, but I was spinning in the daylight, wishing the walls of the nightclub would ever expand as they did under the moonlight, when the promise of tomorrow moved through me like God, and I danced with my face against yours, and to the stars.\n\n\u2022 \u2022 \u2022\n\nAfter a month in Barcelona, I moved into an apartment full of Spaniards, which I insisted upon because I was strategically avoiding Americans. Most of the other American students rented apartments together, and thought I was foolish, but I couldn't understand why one would move to another country only to inundate themselves with the culture she just left.\n\nOne of my roommates, Santiago, was eighteen and from a Basque village. The second, Marcos, was from Madrid. The name of the third roommate I can't remember because he was old, perhaps twenty-six, and rarely drank with us. Our landlady, Celli, from Galicia, was also our roommate. She lived in a long, skinny, broom-closet-type room off the front hall, which she almost never left. Celli was excessively concerned with our whereabouts, which I resented, because _I did not come to Spain to acquire another well-meaning caregiver_. She smelled strongly of garlic, and I had no idea, ever, what the hell she was saying. My room was meant for two, indicated by bunk beds, but I rented it for myself because I needed space and privacy. I was very profound. Because of this, everybody in the house assumed I was wealthy. I never told them otherwise because it was amusing to be seen that way, and I had a tiny lying problem, particularly in Spain, where nobody could fact-check my stories.\n\nWhen we grew tired of the garlic landlady's busybody nature, Marcos, Santiago, and I rented our own apartment down the road from Gaudi's Park G\u00fcell. My room had giant windows overlooking the city, and from my bed, which was two twin beds shoved together, I could watch the sun rise over the old red city and feel grateful.\n\nAs the days carried on and I settled deeper into the new culture, I realized I was more comfortable in Barcelona than I had ever been in America. It was an odd feeling. The siesta and drinking and late dinners and cigarettes, the bread stores and meat hanging from the ceiling of the butcher's shop. The old people playing bocce ball and men in tight jeans. Chattering teenagers on the subway. The little smoke shops with roll-up doors, the trains and taxis and \"motos.\" The history and museums and heartbreaking cobblestone of the gothic quarter. The haunts set deep in stone, telling their stories of war and occupation and Moroccans and Rome. America felt silly and immature, like a teenager who hadn't yet realized what mattered in life. When I had a kidney infection and spent a day in a Spanish hospital, the doctors took care of me, and when I left, the receptionist handed me my paperwork and said, \"We don't charge students.\"\n\nSpain felt old and wise.\n\nOn Friday afternoons, when Marcos and I had money, we'd run over to the student travel agency after not going to class, and we'd see which deals they had for the weekend. They'd say, \"Paris or Rome or Vienna, but you have to leave in three hours and come back on Tuesday at two a.m.\"\n\n\"Perfect,\" we'd say, and we would go.\n\nI sat in the gardens of castles outside Vienna with nothing to do but watch the people pass, and drank beer on fountain steps in Italy. I watched opera in Paris, drank homemade wine in Sevilla, and stood at the southernmost tip of Spain, where they say you can see Morocco. Every day in Barcelona I stopped by a caf\u00e9 to write in my journal, unable to catch all the ideas and colors and hope flooding through me. I drove a rental car up the coast of Spain with my mother when she came to visit. We had to stop to let a shepherd cross the road with his flock, and we ate shrimp in a village with buildings all in white, and we took a wrong road that ended at the ocean\u2014simply stopped at the sand, though the map showed it continuing to the left.\n\nI missed flights. I stayed in hostels and slept on benches in train stations. I rode mopeds in beach towns and hailed taxis in London and bumped cocaine from glittering necklaces around the necks of gay men in Madrid.\n\nThe first time tourists asked me for directions to Park G\u00fcell while I walked home, mistaking me for a Spaniard, I smiled in the knowing that I was truly, finally home. I realized I could not leave. It would be difficult to tell my parents, but I could not catch that flight back to California. My plan was to simply _stay_ and explain it later.\n\nI never made that phone call, though, because a week before my scheduled flight home, I found myself reading a letter from our landlord saying we hadn't paid the rent in three months and the police were coming for us. My student loan was gone. I had spent every penny on \"partying\" and had no reserves. My family had given me plenty. There was no way to ask for more. I packed up what meant anything to me, rented a motel room with Marcos, who was my only friend left, and waited for the days to click by. I had a plane ticket. I had enough money to buy cigarettes and beer for the next few days.\n\nI had banished myself out of the country I was meant to live in.\n\nStill, I swore I would return. I knew it in my bones. Like the haunts in the gothic quarter.\n\nEight months after returning to America, I was pregnant.\n\n\u2022 \u2022 \u2022\n\n\"We have reorganized the copy room and the masking tape is now beneath the binder clips.\"\n\nJoyce, a more senior administrative assistant at the office, was very serious in her coral cardigan and khakis. When another secretary raised her voice to argue with her about the suggested tape placement, I slowly lifted my eyes toward them and let my head fall to one side, refusing to hide how much I hated this moment. They finished in time for someone to bring up parking validation.\n\nThe waist of my black slacks cut into my belly. As I looked down to adjust them, I noticed my shoes were navy blue. I thought, _Wow, I can't match shoes with pants. Guess we're there now, huh?_ My shirt was faded purple and probably a polyester blend. I looked ridiculous. A fat roll spilled over the top of my pants, and I considered pulling the waist down beneath the roll, so the fat bubbled out in one billowing mess. It was more comfortable that way, but felt like giving up. I sat up straighter and pulled my shirt away from my stomach, trying to hide my bulges in that classic self-conscious fat person gesture.\n\n_Perhaps there was going to be more._\n\nBetween emails at my cubicle and commutes home, I dreamed of nights in Barcelona. No, I dreamed of some. Because there were failed classes, lost friends, mornings of regret and shaking desolation. Twice I escaped being raped. Once, barely. I was pinned against a wall in a bar's cellar while a man lifted my skirt and pressed his body against me when my friend \"had a bad feeling,\" came downstairs after me, and began shouting and pounding on the door next to my head. It startled the man long enough that I could shove him off, unbolt the door, and run out. As my friend walked me out of the bar, she said, \"I'm tired of taking care of you. I'm tired of wondering if you made it home okay.\" She got in a cab and didn't call me again.\n\nOnce, I couldn't speak on our couch. I must have done too many or the wrong drugs. I tried, but couldn't make my mouth move. My roommates stuck me in a freezing shower and slapped my face while they looked at me, bored.\n\nThere was all that, plus nights of empty insomnia, which I spent scratching in my journal and at scabs on my legs, but what you remember when living in a house that can't be told apart from the others is walking the brilliant streets of a European city on a bright morning on your way to the subway, realizing that nobody is treating you differently than all the Spaniards around you. If you want, you can walk into any caf\u00e9, order a \"caf\u00e9 con leche,\" and smoke a cigarette\u2014and you'll know it will be right, and you will fit, and life will move in luminous color. What you remember is the feeling that life was going to happen, or that it is happening now, and that it would always move in meaningful directions. But I couldn't stay.\n\n_Mac, are you going to go to college? You don't earn enough. We have no benefits._\n\n_What are we going to do, Mac?_\n\n_Why are you just staring at me? Why don't you care?_\n\n_We need to extend the warranty on these five computers._\n\n_Yes, I'll hold._\n\n_Okay, here are the serial numbers._\n\nI lay on my back while we fucked and stared at the ceiling because I was not drunk enough to make it interesting, or sober enough to make it satisfying, or wasted enough to deny that it had been so long I couldn't run anymore.\n\nThe pills gave me just enough to get my body into the car to drive twenty miles and listen to the guy from NPR, and on the way home, just enough to stop at the grocery store to buy one, maybe two bottles of wine. Walking down the liquor aisle, I'd tell myself, _Janelle, you just can't drink so much again tonight, and dammit, you can't wake up with that hangover like you did today, or put your kid to bed half passed out. One glass, maybe two._\n\nThe pills carried me to the moment I drank my first glass and it all started feeling manageable again, the stucco and lake and office chair and gorgeous blonde-headed child. _Another must be okay_ (of wine, that is). With the second, it was so much better still. Mac would seem witty again, and television would make me smile, and reading Ava a story would entertain me. _Who needs Spain? Who needs love? Who needs what I thought I needed?_\n\n_If I go to the store and buy a bottle of rum_ , I'd think, _I could drink it fast, and things would be that much better still._\n\n_Even still!_\n\nUntil the morning, when I would wake up and swear I wouldn't do it again, not this time, because _I can't keep living like this_. Again. Again. I'd take some ibuprofen, make some coffee, and shuffle into the car from behind sunglasses, where I drove and smoked and listened to music, loud, almost like the girl flying down the rails listening to the Grateful Dead, wondering where the end of Spain is, and how long it would take to get there.\n\n\u2022 \u2022 \u2022\n\nOnce a week, I would call my mother to inform her I was leaving my marriage. She would listen, genuinely fascinated, and then suggest I talk to him, really explain how I was feeling, and I'd recall the seven hundred and fifty thousand conversations he and I had endured to the bitter end, and she and I, too, to fix her marriage or mine. They never worked, but what the hell else does one do with a failing marriage other than _talk about it incessantly_?\n\nWell, Mac and I would go out together. Grandparents would watch Ava while we concentrated on our sole purpose, which was to get hammered at a bar with no windows. Unfortunately, since he worked weekends and I worked weekdays, one of us was always going to work hungover, but that was a risk we were willing to take. We did it for marriage. It was like a commitment to therapy, but for alcoholics. I'd wedge myself into jeans and drink wine while he got dressed. I'd paint on makeup and style my hair, and we'd smoke cigarettes on the way to the bar, with our best songs playing and each other.\n\nOccasionally we would visit my mother in Mendocino for the weekend, and she would babysit while we went to the Caspar Inn, which is a bar in the middle of the five-building town of Caspar, across the street from a nineteenth-century church. It was packed on Friday and Saturday nights, smelling of whiskey, convenience store perfume, and alcoholism. One night we went to hear Tommy Castro, a local blues-rock band, and out on the deck in the fog we met a man named MYQ, but not pronounced \"Mike.\" His name was pronounced \"M-Y-Q,\" as in the letters themselves. He was making a coffee table book of his facial hair configurations. At the time, he had a spade shaved onto his chin.\n\nA few minutes later, down at the end of the bar, MYQ, Mac, and I met a seemingly homeless man with a kitten in his jacket. Impressed, we bought the man a drink and became fast friends, though I can't remember his name. The kitten's name was Rufus. After talking for a while, we bought him a couple more rounds, smoked a few cigarettes on the misty porch, and shut the bar down. Rufus's owner needed a ride home, and we figured giving him a ride sounded as good as any plan ever sounded, so we hopped in my car and drove north on Highway 1 along the ocean, eventually turning east, into the mountains, then right down a little dirt road. He said he had some beer in the \"main house,\" so I killed the engine.\n\n\"Come on in,\" he said. \"But these people are a little weird.\" Considering he went to bars with a cat in his jacket, I was intrigued to meet somebody he thought was weird.\n\nWhen he opened the front door to the house, I faced a wall of indiscernible material to the ceiling and a smell I can only describe as \"absolutely not.\" There was a single trail through the house, forming a reeking canyon of trash.\n\nI forced a laugh and said, \"No thanks,\" turned around, and walked out. They must have been hoarders. A few minutes later, Mac and Rufus's dad brought out some beer. _Did we just steal beer from hoarders?_ I thought, grabbing one.\n\nOur new friend walked us over to his home, which was a converted school bus on the other side of the property. The inside was covered in quilts and packed with wood carvings and stained glass and paintings, trinkets from every town in northern California. While we sat with him, he filled a little bowl with tofu and pulled a plastic sack out of his pocket. I thought it was bee pollen. Sprinkling it on the food, he said, \"Rufus eats tofu with brewer's yeast on it. It's the only food he likes.\"\n\n\"Oh.\" I nodded, seeing the dirt under the man's nails and his thick, filthy fingers sprinkling the yeast in a little mound right in the center of the bowl.\n\n\"It's hard for me to get to Fort Bragg to buy it, but it's so expensive in Mendocino.\" I almost offered to drive him there right then, but I knew it was a three a.m. idea, and three a.m. ideas are almost uniformly bad. Rufus devoured his dinner.\n\n\"I'm a stained glass artist from Humboldt. I used to travel around to shows with my old lady, but she's gone, and there isn't work anymore. I have this beautiful place though.\" I watched him and tried not to look around, tried not to be the one to break it to him that his bus was not beautiful.\n\nWhen we went outside, I realized he was not talking about the bus. He didn't care about the bus. He was talking about the little river below, so close you could hear it from inside, and the ferns and redwoods, the thick fallen trunks and moss, the scent of ocean in the distance. The silence.\n\nMac and our friend and Rufus went back inside, but I stayed alone and looked out at the little river and the stars and trees, which were mostly sound and shadows in the dark, and I thought about his kitten, and that I had little idea where I was, and I wondered if this was how people got cut up and eaten by axe-murdering psychos. But he was not an axe-murdering psycho. He was a misunderstood stained glass artist who fed his cat brewer's yeast.\n\nI loved Mac on those nights. I loved our adventures, our chemistry, the way we seemed to attract endless weirdos. Everywhere we went, the misfits came running. Reno. Davis. Tahoe. San Francisco. Santa Rosa. Caspar and Mendocino. How many nights had we spent with the outliers of those towns, the loner eccentrics?\n\nMac and I were friends. We were young, lightly renegade partiers, but we had our baby, and we loved her, and when the mavericks at bars would ask about our lives, we told them about her. We told them how smart and beautiful she was, and we seemed to have a better life than them because she existed, and we had each other. That dive-bar life was a tiny piece of a brilliant whole. We had a real life out there in the world with stucco houses and cubicles and cement lakes and neat lawns, where the sane people live, as opposed to the ones making coffee table books out of facial hair photography and stealing beer from hoarders.\n\nAnd yet, I needed the misfits to exist. I needed to party with them and drink with them and inhale deeply with them. I needed to shut down bars with the local drunks, because next to them, we were a millionaire family. Next to them, we really had our acts together.\n\nWe navigated the crazies just fine. It was only at home we lost our way. We tried, though. We talked. Sometimes I'd get my hands on some cocaine and we would really talk then. High as kites, I'd encourage Mac to wake up, liven up, go to school, do something! Grow up, son!\n\nHe'd get excited too, and after a few lines, he'd come up with a plan. \"Tomorrow,\" he'd say, \"I'm going to join the rodeo circuit.\"\n\n\"Yes! You should do that, Mac. You can do it!\"\n\nIt was a three a.m. idea, but it was all we had.\n**6**\n\n# **Rocketship Rock-On**\n\nAfter two years at the firm, I got a promotion and raise that made me question the institution of marriage as a whole. _Janelle,_ I thought, _this is the moment you've been waiting for._ _Your big fucking break. Time to get out on your own and soar._\n\nI found a little house in a 1940s neighborhood of Sacramento and rented it alone, again, announcing my departure one evening while Mac finished his spaghetti. But this time, I explained, he could not come with me because our marriage wasn't working.\n\n\"What do you mean?\" he asked, and then it was my turn to stare silently into the void.\n\nI was horribly confused by his confusion since we had discussed that exact topic every week for two solid years. And we had only been married three.\n\n\"Mac, you're not going to change and I need a future, a life.\" I was pontificating loudly, but lost some gusto when I added, \"I'm not sure I love you anymore.\"\n\n\"I still love you, Janelle,\" he said in a sort of whispered resignation.\n\nI scowled, thinking, _Who would say such a thing at a time like this?_ His eyes were wet and the same as the night we met, and I wanted to love him but could not. I was looking for something lost. I was looking for Barcelona.\n\n\"Mac, look at the way I drink every day. I am miserable. I'm dying like this.\"\n\n\"I don't think you should go,\" he said, holding my eyes. I felt like I was smacking him with the back of my hand while he crouched behind a table, and I think somewhere I knew he could not defend himself or understand or even beg. Life washed over him. He seemed like a boat without a rudder, and I was simply a current that came along once.\n\nOne week later, I was gone. I took the furniture and left him to pack up the house I had rented without him, returning a week later to get the last of my belongings. I found him sitting on the floor in the empty living room, leaning against a wall with his knees up in front of him, watching a movie on a laptop. From the loneliness of that view I had to turn away quickly, but glanced back when I opened the door to leave. He was looking across the room at me with hungry eyes, but I felt only rage.\n\nI couldn't hear his voice again. His broken, thin words. His even-toned flatness. His control. His agreement. He would tell me a hundred times that it would be different. He would do something new. He would go to college. Get a different job.\n\nWe would grow close again. We would not sleep together in machine-like coldness. We would rediscover love.\n\n_Ah, fuck your tears. Fuck our voices._\n\n_I gotta go._\n\n_I am a master of beginnings, and this\u2014this is a goddamn grave._\n\n\u2022 \u2022 \u2022\n\nMy new home had old wood floors and a front porch behind a sprawling oak tree. The only time I had to look at the address box was the first time I pulled up and thought, _Well, this place is damn adorable._ It was white with blue-gray trim and had a big lawn out front and a huge backyard with a fountain and lavender plants where bees buzzed. Inside there were two bedrooms to the left of the living room, one in the front and one in the back of the house, with a bathroom in between with tile floors and a pedestal sink. It had a big brick fireplace, and a kitchen with barely any storage or counters. It was perfection.\n\nMac returned to live with his parents at the ranch.\n\nWe split the week with Ava so each of us only had to endure a few days without her. When she was with me, I dressed her in colorful tiered skirts and leather shoes, took her to the bakery down the road for a croissant, and to the park, where I would lie to the other mothers about my marriage. We'd go home and eat lunch, and then I would snuggle into my bed with her, her head on my arm. I'd kiss her over and over again. She'd drift off for a nap, and I would too.\n\nSleep was harder at night, though, because I had never lived alone. The neighborhood was largely retired folks who enjoyed trimming roses and talking to each other, but my house had windows and doors right near the ground, where people could walk up and _climb in_ if they felt so inclined. And I was sure they were so inclined.\n\nSo, before bed, particularly if it was just those wood floors and television and me, I protected myself using a complex inspection routine. First, I of course checked every door and window lock. Then I had to look under my bed and Ava's twin bed, but because I couldn't see to the back of my queen bed I would sometimes do a quick sweep with a broom to make sure nobody was curled up in the corner, where no human could ever fit. I also sometimes had to do that under Ava's twin bed. I then looked in the closets, in the corners, behind the coats I hadn't worn in five years. I could have just looked for feet on the ground, beneath the coats, but that didn't occur to me. I looked behind bedroom, closet, and bathroom doors. I looked in the shower. For people. Intruders. Lurkers. Rapists.\n\nI was not on meth. I was not suffering from schizophrenia. I was scared, and I thought if I knew every corner of the house was clear, I would lie down in my bed and turn off the light and feel safe. I thought I would feel comfortable knowing I had secured the perimeter.\n\nBut it never worked. It didn't matter how thoroughly or often I checked, or how many days passed without incident. I could examine every millimeter of that house, but somehow at the end of the process, I was just as terrified as before I started. I left a light on in the living room and glanced out my bedroom door too many times as I fell asleep in twitchy half-sleep inspection.\n\nI knew it was crazy. I would tell myself when I looked behind a door, \"Janelle, nobody can fit there,\" but I was compelled to look anyway. I had to _do something_ to become unafraid.\n\nIt wasn't my mother on the chopping block as I had imagined when I was younger. It was me. Something was coming for me. Somebody in the closet. Somebody under the bed. Maybe the same man who threatened to chop my hair off when I was a little girl also lurked in the closet of that 1940s house, a monster I never quite saw, for whose existence I had no evidence, but whose power over me was undying and uniform.\n\n\u2022 \u2022 \u2022\n\nOn the evenings when Ava was gone, I would lie on the couch and watch _Sex and the City_ reruns and sip white wine until I was good and loaded. Then I would make phone calls.\n\n\"Hey, you want to go to SoCal's?\" I'd usually call my friend who'd walked into the law firm with a porn star name.\n\n\"Janelle, it's nine p.m. on a Tuesday.\" she'd say, fake exasperated.\n\n\"So? I know. Come on. I'm bored. One game of pool and a shot.\" I'd soften the urgency in my voice. I'd play it cool.\n\n\"Well, okay.\"\n\n_Oh, thank you, sweet baby Jesus._\n\nI'd be at the bar within fifteen minutes, and I would have a grand time right up until the moment my friend would realize it was eleven p.m. and she needed to get to bed for work the next day. Then I would find myself alone, again, as the world moved on and I looked for more. More.\n\nHours later I would wake up in my nice bright house and whack my alarm, squint beneath the throbbing in my head, the chilly sweat, the strange panic. Advil. Water. Shuffling to the shower. Coffee. Cigarette. There I was, with my promotion and house full of character and freedom, and not a goddamn thing was different.\n\n_Drink more water. Take another pill._\n\nBut there was no pill to fix the regret, the abhorrence of myself each morning. I never woke up hungover thinking, _You know, Janelle, you're really winning at life._ I never thought about the \"fun\" I had, or decided the hangover was \"worth it.\" I never looked back with a flippancy like other people might, the way they do in the movies: \"Wow, we really went on one last night, huh?\"\n\nI woke up with mind-bending shame and deep confusion. _Why can't I stop drinking? Why can't I drink like my friend?_\n\nWhat got me through those mornings was that I knew with every drop of blood in my body that I would not drink that day. I would dunk my head in a sink of ice water and hold it there for as long as I could stand it, in a miracle hangover cure my father had inadvertently taught me. He always laughed and told me it was for his complexion, but as I grew older, I realized it was for the headache. After \"icehead,\" our loving name for our loving cure, I would focus only on getting through the morning, one second at a time: eat food, drink coffee, get to cubicle. Let some hours pass.\n\nBut when I started to feel human again, when my thoughts became crisp, the furious remorse would fade to a whisper in the back of my mind, and every day, as soon as I was well again, I would suspect I was overreacting about the whole \"not going to drink today\" verdict.\n\n_That seems a little extreme,_ I'd think. _Maybe tonight I'll just stop by the store to buy one quality bottle of white wine because I need to relax. Make myself a nice dinner._\n\nLife was officially not getting better without Mac. The only new feature of single life was worrying about Mac falling in love with some broad who would tell Ava what a loser her mother was. Well, that and I didn't have magical nights in buses anymore, or my best person to call to hit the dives and play pool.\n\nBut mostly, when my drinking didn't improve without Mac, I invited him to live with me again because if I was going to be a fucking drunk, I thought I might as well do it with my best friend, and at least some ability to pretend my life was working.\n\n\u2022 \u2022 \u2022\n\nI made it six months without him, and we made it one week without drunken domestic violence. One night in the parking lot of a Thai restaurant, I sat in Mac's truck and scanned his text messages, engaging in a process of snooping I had only seen in sitcoms. Before I left to pick up our takeout he was quite obviously trying to shield me from his phone, which he had left on the seat of his truck. _What a fucking horrid liar he is_ , I thought. I found messages from a woman who \"really missed him.\" They were sent at eight a.m. on a Saturday. She appeared to have seen him the evening before. He assured me it was \"nothing.\" I assured him I knew all the various ways he'd slept with her.\n\nI was the abuser. He could have killed me if he wanted to, flattened me with one hit, but he didn't touch me. I shoved him while he stood in the entryway. He stumbled backward to the ground against the front door, and I remember him looking up at me in shock and wild confusion. He got up, grabbed his car keys, and tried to leave, but I snatched them away and continued screaming while our daughter slept in her bedroom.\n\nAt this, he walked to our bedroom and I threw his keys at him, but missed. The sight of his back, of his silent retreat, reminded me that I was nothing more than a current that came Mac's way\u2014something that pushed him along through a few years, and when I was gone, there would be another.\n\n_He was not loyal. He was just another man. He could leave me._ Those daggers sat in me like metal in a fire\u2014poised and red hot.\n\n\u2022 \u2022 \u2022\n\nMac never joined the rodeo. Instead, he signed up for calculus and geology courses at the community college down the road.\n\nA few months after I learned of his non-girlfriend girlfriend, he was supposed to go on a weekend field trip with his geology class to Lake Tahoe. On the evening before he left, he scanned the checklist of items he needed, packing and repacking his backpack. He was eager and anxious like a kid before the first day of school. When he was all ready, I wanted to go to a bar, so I commenced begging.\n\n\"Let's go out for a couple games of pool!\"\n\n\"What? No. I have to leave at six a.m.\" He shook his head and looked at me from beneath his hat like I was insane. I was standing above him, hopping around like a puppy.\n\n\"We won't stay out late. Come on!\" I smiled and yanked his hand.\n\n\"We always stay out late, Janelle.\"\n\n\"We won't this time. I promise, I swear, I promise! Please, baby. Just a couple of games. Come on. We never get to hang out, and Ava's with my mom.\" I pulled his arm and flirted with him until he agreed.\n\nWe played our games of pool, but in the back of the bar I ran into some men with cocaine and did a few lines on the back of a toilet seat, without telling Mac. Then I bought a baggie from them in the dark parking lot, alone, and by the time I got back to Mac it was too late. He was now the husband of a wife on coke, and if he didn't join me, he would be the man who \"didn't party.\" It was already done. Already purchased. To me, his words of \"no thanks\" were a mere inconvenience, an obstacle to be overcome. I was a boulder smashing down a mountain.\n\n\"Have another drink,\" I said, knowing if I got him drunk enough he would get high with me. Sure enough, we went back to somebody's place and did more, and more, and more\u2014until five-thirty a.m. arrived and I drove a sleepless, trembling Mac to meet his class at the school. He pulled his carefully packed backpack along on his drooping shoulder, red-faced and wild-eyed and never supposed to be like this. I watched him get on the bus alongside a bunch of bright-eyed students cradling cups of coffee and smiles, and I thought I had never seen a man look sadder in all my life.\n\nThrough the window of the bus, he mustered a little smile. It broke me.\n\nThrough all that, he smiled at me still. He forgave me already. He already loved me again. I drove out of the parking lot seeing his face and the way he looked at me, and I thought about how he must feel coming down off alcohol and blow among strangers, already shy and reserved, miserable and frantic with nothing but static in the brain, without sleep or a place to rest and get well. But more than that it was the way he refused to not go. He wanted to be better. He wanted to try. He wanted to be with the regular humans, and I ruined it. He wanted to sleep, and wake rested, and join the students on the bus to learn and be good and right and clean.\n\nI sent him off sick and twisted and lonely.\n\nHe could have not gone with me. He could have left his wife with a bunch of men doing cocaine\u2014because he knew I wouldn't leave. I would stay where the drugs were. But he wasn't that kind of man. Or maybe, he wasn't healthy enough to let me go.\n\nAs I turned the corner onto our street to get home and sleep off the disaster, I realized I was wrong about all that \"potential.\" I was not capable. I was not destined for anything. _How do you keep making these decisions, Janelle? How do you keep knowing what's decent and true and then doing the opposite? How do you do it over and over again?_\n\nMy next thought was, _I must have been born without a moral compass_. This wasn't pity. It wasn't groveling, or vapid self-deprecation. It was an observation of the facts, and I accepted them as such. _I was born without morality. I must not care. I am not_ _capable of caring._ I thought about the way I used to chase my brother with the knife, the way I raged at my mother. It had been a long haul of depravity.\n\nI could see my home crumbling. I could see any hope we once had for a decent family fading into an old memory. I could see myself turning into a person not even I recognized.\n\nI begged him to forgive me when he got home, but he already had.\n\n\"We need a new life,\" I said, clinging to him. \"We need a new start.\" And I had been thinking, _Nothing says \"new start\" like a new baby._\n\nIn my mind, if I had two children, _I would join the fucking PTA_. I would carve carrots into little owls and place them lovingly into bento boxes with sober, manicured hands. I would wear yoga pants and actually do yoga. I would decorate cupcakes. I would have a special birthday breakfast plate and a Waldorf candle ring. I would clean up my act and get right with heaven and hell, but mostly, I would stop drinking, and if I stopped drinking I would also stop snorting cocaine, and Mac and I would be in love again. I wouldn't leave him shivering in the cold on a curb, in line to board a bus, waiting to inspect shimmering pines with pupils aching for darkness.\n\n\u2022 \u2022 \u2022\n\nThe two lines popped up again without hesitation, no longer a \"fuck you\" pink. Mac and I hugged and hugged.\n\n\"Mom, I'm having another baby!\" I told her ten minutes after I took the test.\n\n\"You are? Oh my goodness! I'm moving there.\"\n\n_Of course you are,_ I thought, _because this is our refreshing new life._\n\nMy mother, Mac, Ava, and I moved into a giant house we rented on a big fake lake in Natomas, a slightly fancier suburb than Elk Grove, but still soulless. I was now living in the house I used to admire from afar. I knew which one was ours without looking at the number because the houses were at least large enough to _appear_ different from the others. It had five bedrooms and three bathrooms, an office and a laundry room, and a backyard that opened onto the cement lake. We put a pool table in the formal living room.\n\nAt the four-month ultrasound, the doctor jellied up my belly while Mac and I held hands in the dimly lit room. Ava sat near us on a stool, her blonde ringlets frizzy and wild around big blue eyes. \"You're having a boy,\" the doctor said. He pointed to something on the ultrasound screen that was supposed to indicate this, but all I saw were squirming yellow lines. Still, I gasped and cried, and Mac's eyes filled with tears as Ava hopped off the stool and marched up to the doctor with her hands on her hips, announcing, \"No! We are having a girl! I am having a sister!\"\n\nThe doctor laughed and confirmed that indeed she was having a brother, and then Ava's sobbing ensued.\n\nTo cheer her up, we sat in the parking lot for a moment with Ava in her car seat and asked, \"What should we name him, Ava? Can you help us?\"\n\nWithin seconds, as if it were obvious, as if it were the only name possible in such a situation, she proclaimed: \"Rocketship Rock-On. We need to name him Rocketship Rock-On!\"\n\nWe roared and said that was an excellent name. I looked at Mac and said, \"And because we are Frank Zappa, that name is perfect!\"\n\nHe laughed. She beamed.\n\nShe told everyone we knew, \"I'm having a little brother, and I named him Rocketship.\" She was so proud nobody had the heart to tell her we weren't Frank Zappa.\n\nWith Rocketship, I had a whole room to decorate. I bought an oak crib and changing table. We installed a chair rail and painted it white. We painted the bottom of the room a pastel green and the top blue. Because I wanted him to use it for years, like an heirloom, we spent real money on a matching honey-colored oak dresser from the fancy baby store.\n\nA boy. A son. _My son!_\n\nMy job was rising. Our home was huge. I was sober as hell and decorating a nursery. I bought playful maternity skirts and ruched blouses and strappy sandals. He was due in September, the most beautiful month, and every appointment was happy news. I sat in the rocking chair of his color-coordinated nursery, a room straight out of a catalogue. I hung his tiny clothes on fuzzy little hangers, organized by size. I washed it all in baby-friendly detergent and folded the washcloths into tiny squares. All the cloth diapers were ready, folded, cleaned, and placed in gingham-lined baskets. There was even a patchwork quilt draped over the back of the rocking chair.\n\nLabor began two days before his due date, four days into my maternity leave\u2014because he was a gentleman who would never imagine arriving late. His birth was the kind they show in hippie movies trying to convince women birth is like dancing among lilies in morning dew. It lasted seven hours, beginning at home. When a contraction came, I leaned against the wall and swayed, moaning, resting on my bed in between. I took showers. I swayed some more. Around seven p.m., Mac told me we had to leave for the hospital. I told him I was okay where I was. I had never been so okay.\n\nWhile in the shower, I felt the familiar agony and delirious pain of hard contractions, and I thought, _No! I cannot do this again._ I began panicking, resisting the pain, but I had read somewhere in those hippie books that you should think of all the women who were birthing with you in that exact moment, and that you had to let go and ride the pain through, because fighting made it unbearable.\n\nSo I surrendered, consciously. I let go. I visualized myself among a few thousand women all over the world\u2014in rivers and huts and hospitals. By the time Mac told me again we had to leave I was in another world with those women, peacefully birthing my baby like those bitches in the movies nobody believes are real.\n\nIn the car that September day, the sun pounded through the glass and it was, of course, Friday evening, when the traffic is terrible. I even screamed at Mac about how much I hated him. I tell you it was a perfect birth. When I got to the hospital, the nurses didn't believe I was in active labor because I was \"so quiet and calm,\" but they booked me into a room and demanded I get on the bed for twenty minutes of continuous fetal monitoring. I told them to go to hell. I was standing and swaying. That's how I was riding the pain. I ultimately agreed to stand against the machine so they could do their monitoring, and then they left. I was delighted to see them go. I wanted to be alone, alone with this boy, alone to have my baby.\n\nI almost immediately felt an urge to push. My mother yelled down the hall, \"Somebody better get down here! She wants to push!\"\n\nI wanted to deliver in the water, so the midwife helped me into the birthing tub. I had three more contractions and pushed twice, and then turned over onto all fours. In one contraction he was born. I heard the midwife say, \"Turn around and pick up your son.\"\n\nI flipped my leg up and over the umbilical cord and looked into the water and saw him there, pushing the water, eyes wide open, with his palms out and his arms and legs reaching. I brought him to the surface and pressed my face against his as close as I could without crushing him. I felt his velvet cheeks and inhaled the sweet newness at his neck. I cradled him against me on my arm, and I watched him pull his first breath of the air we shared, and his body flood pink with my blood. He looked me right in the eyes and sputtered. They rubbed his palm to make him cry, though I thought that was unnecessary. I didn't understand why they needed to hurt him, even a little. Mac was pressed against me, his head on my shoulder, and my mother and Mac and the midwives were hushed and whispering in the gray room. When they asked how I felt, I said, \"Elated.\"\n\nHe came two days before his due date in a birth that felt like the sunrise. When I walked in the door of our home I knew life was going to be different. This son. This perfect son. Nobody could ruin this life, not even me.\n\n# **Part Two**\n**7**\n\n# **Failure That Isn't Funny**\n\nI woke from a few moments of sleep, or maybe just shut eyes, shaking. Not the shaking of a chill, but a quiver I couldn't quite feel. It drove me up and out and around the house in circles, though my body perhaps stayed on the couch, drinking leftover alcohol out of a coffee cup and rocking back and forth, like the baby on my left who wanted breakfast. It was as if the blood in my veins wouldn't settle down, as if it were pulsing out of me rather than through me, scratching the walls for escape, tossing me around the room and into every insane thought, regret, and memory of the past twenty-four or forty-eight hours, or maybe my entire life.\n\nIt drove me outside for yet another cigarette, the twentieth or fiftieth in the past two days, but it was worse in the sun, in the flickering leaves and slamming roar of passing cars. I squinted and hid from my neighbors behind our house, pulling that cigarette with all my strength, but it didn't have what I needed.\n\nI fed my baby, Rocket. This was the only variation of \"Rocketship\" Ava would allow. For an entire year, none of us had the heart to tell her we weren't rock stars. The name grew on us while we waited for her to change her mind.\n\nWhen Ava shuffled into the living room in her flannel pajamas, I turned the channel to PBS because it was a decent learning channel with soft colors and slow images. I hated the cartoons that screamed and slammed her little mind with fluorescent images and rapid-fire, vapid dialogue. She was five. I wanted her to see wholesome things.\n\nMac and I had been on cocaine for two and a half days. We went out on Thursday to have a few drinks. Then it was Sunday at eight a.m., and we had been awake the entire time.\n\nBut this Sunday was Rocket's first birthday, and I had an enormous party planned. I had bought an outfit for him weeks before\u2014a deep blue button-down shirt, black pinstriped pants, black leather shoes, and black socks to match.\n\n_Seven hours until the party_ , I thought.\n\n_Maybe I can feel better by then. Maybe I can sleep._\n\nBut the quiver would never let me rest, and I knew it. I considered drinking enough to pass out for an hour or two before the party, but then I would be drunk at the party, though I probably wouldn't wake up anyway. After spun-out days, sleep, when it finally comes, is cavernous.\n\nI had decorations, streamers, and presents. I had Brie and blue cheese and baguettes and figs. _I have it all ready, son. I prepared it all, thinking of you, my boy, and loving you, with your endless blue eyes and curls of strawberry red hair that stick to your forehead when you sweat. (I always say you got the red from Daddy's beard.)_\n\n_I can't do it. I can't show up like this._\n\nI resolved to cancel, then immediately remembered how grandparents and great-grandparents were coming from other towns, and twenty friends. It was too late.\n\n_Here I am again._\n\nSix months before, when I felt myself slipping back into drinking after Rocket's birth, I enrolled in a master's program in English to \"challenge my brain,\" thinking, _Surely, I must be bored. Surely, my vanilla admin job is driving me to drink._\n\nWhen the red wine continued to pour despite rampant critical theory courses and Marxist analyses of lost generation texts, when I looked down and noticed my very own hand pouring that wine, night after night, despite continued daily declarations of _not tonight, not tonight, not tonight,_ the thought came to me: _Janelle, maybe you are an alcoholic._\n\nI thought of my father, newly in recovery. I thought of when we were sure he was going to die from alcoholism. I thought of his brother, my uncle, who did die from it. I thought about how those D.A.R.E. cops always told us alcoholism is genetic.\n\nAnd then I thought, _If I am, so be it_.\n\nBecause by then, the consequences of not drinking were far greater than anything that could have happened from drinking. Internally, that is, because I drank to repair my inner self. The external penalties for my habit were damn near imperceptible in the shadow of the colossal misery that was sobriety. I drank because sobriety was intolerable, and that intolerability arose from within me, slowly from my guts, increasing with every passing sober hour, until I found myself drinking again, only to soothe the wild discomfort. If not the bottle, it seemed a bullet to the brain would be the only viable alternative.\n\nAlcohol was my most reliable friend, offering me with every warm hot kiss that which the rest of the world promised but never delivered: peace and meaning.\n\nI knew this somewhere. I knew I drank to fix the unfixable, and I knew it when my outer life was \"perfect\" and nothing had changed. _Here I am with a good job, a good house, two perfectly good children, physical health, a scintillating life in graduate school, and I still cannot stop drinking._\n\nI imagined my childhood must have really fucked me up. Psychiatry was the obvious answer. I began seeking help to fix my insides, thinking, _If I just get happy, I won't need alcohol anymore._ But on the night I met Ben, the cocaine dealer who delivered straight to our front door, I wasn't wondering what was wrong with me anymore. I was simply buying more.\n\n\u2022 \u2022 \u2022\n\nMy crying on the morning of Rocket's first birthday was unhinged.\n\n\"What's wrong, Mama?\" Ava asked. She had changed from pajamas into a green Tinkerbell costume with a tulle skirt and jeans underneath. Her eyes were wide with concern beneath unbrushed waves of blonde.\n\n\"Oh, nothing, sweetie,\" I said, but I didn't mask the lie. I was still in jeans. My eyes were set in deep black circles. I said it through gasps and tears.\n\nMac was at work. I tried to imagine him there, wet and muddy, punching the hide off a lamb, handling knives with vibrating hands. It felt like the morning I left him at the college field trip when he tried to smile.\n\n_What kind of fucking people have we become?_\n\nI feared he would die. I feared he would cut himself, or fall asleep in front of the meat saw, or flinch while killing a cow with a shotgun.\n\nI called him, and said words I knew wouldn't do a damn thing. \"Mac, this is insanity. You can't stay. I know you feel like I do. Please come home.\"\n\n\"Alright. I just have to finish cutting this meat.\" I was correct. He was standing next to a meat saw coming down off a three-night cocaine binge, and I realized it had gotten so bad he agreed to come home.\n\n\u2022 \u2022 \u2022\n\n_I wanted it to be just right and nice for you, son, the cake I ordered from the fancy bakery, food from the nicest store in town, the recipes I chose a week ago. I wanted it all to come together in this exact moment, and for me to be there, and for all the guests to celebrate you, but instead I am sitting on a couch, shaking but not shaking, reeking of dirt of the internal kind, and wondering if anybody would notice if I just drove away._ _Maybe into that fucking cement lake in front of me._\n\nI stared at the shifting water, smoked another cigarette, and wondered how much money we had stolen to make the last three nights happen, how many checks I had written to myself with money I couldn't cover to put in the ATM machine at three a.m. to get cash out, knowing they would only bounce and be deducted from the same account from which I was trying to pull the cash. _One more bag, just one more bag._\n\nWith renewed strength rising from self-disgust, from a will to never repeat the hell I was then living, I decided not to cancel the party. _I will get through today to make this pain mean something. Right now it all changes. This failure, this coke-addicted drunk fuck of a mother in a suburban home on her son's first birthday. It will not happen in vain._\n\n_This is it. This is the day I learn._\n\n_It's okay, now. It's okay, son._\n\nI meant it. I had the party because I knew it was the end, and I succeeded. Nobody knew, because I smiled and laughed and only collapsed after the last guest had driven away.\n\n\u2022 \u2022 \u2022\n\nThree weeks later, nine a.m. rolled around on the day Ava was supposed to attend the birthday party of a little girl at her school. I didn't wake up like the day of Rocket's first birthday, because I was already awake. Still awake, still high. The quiver remained.\n\nIt occurred to me we had no gift. I considered telling Ava she couldn't go, but it was the first birthday party she'd ever been invited to, and I couldn't disappoint her. Not again, not now.\n\n\"Hey Ava, let's pick something out of your closet to give to your friend, and I will buy you another one tomorrow. I promise. We will go _tomorrow_. Is that okay? Can we do that?\" My words rushed together in idiotic cocaine excitement.\n\n\"Sure,\" she said. \"That's fine, Mama.\" And she ran up the stairs with her blonde hair bouncing around her, and me, behind, nearly dead weight.\n\n\"You're so awesome! Thank you for being so patient and giving, Ava!\" In her closet, I found a silk rainbow streamer. I grabbed it and thought the girl would like it because they went to a Montessori school and that was a good hippie gift.\n\n\"How about this?\" I asked. \"Can we give her this?\" I held it up and tried to smile.\n\n\"Okay, Mama.\" Ava said. She didn't even look disappointed. She didn't even look at me like I was shoving all my words together or stinking of cigarette smoke or scratching scabs on my shins because I couldn't stop picking.\n\n\u2022 \u2022 \u2022\n\nAva put on a pink dress with tiny white smocked flowers around the chest while I wrapped the gift and snorted a line in my bedroom before getting Ava into the car and driving to Woodland, a town twenty minutes away. Mac stayed home with Rocket. He was missing work. He did that more often now.\n\nHe had offered to drive, but I didn't trust him. I was as loaded as he was, but I never trusted anybody to drive as carefully as I did when intoxicated. People were idiots. They got swept up in the euphoria. They got carried away in the feeling. They forgot it wasn't real. They forgot that you have to focus on signs, on sober reality\u2014signals of truth\u2014to differentiate between what you see and what's actually happening. Whenever a friend in high school was tripping too far from us on some psychedelic, I'd say, \"Look at the clock. The clock is always sober.\" It was a grounding device. And I thought I was a master at that game.\n\nSo I looked at the clock, double-checked my seatbelt, lights, and mirrors, made sure I had my purse and license, moved my baggie from my wallet to a CD case on the floor. I had to take it with me, because I knew I'd need it before I returned home. I drove deliberately, exactly at the speed limit, focusing three times harder than I normally did, watching the lines and speedometer, studying the cars around me in a routine I knew like air. I hated driving drunk and on drugs, and every subsequent morning I would shudder in shame and horror at what I had done, but I knew in the moment I must minimize risk of swerving or speeding or missing somebody's sudden turn or braking. I refused to turn on the radio. It would interrupt my attention. Ava tried to talk to me, excited about the party, but her voice was like the chattering of a squirrel. In my mind, it was a stream of impossible sound.\n\n\"Honey, please let me focus on driving.\" She quieted down for a moment, then started back up again. I tuned her out the best I could and kept my eyes on the road and clock.\n\nI parked unreasonably far away from the house, put on sunglasses, and walked my daughter to her first birthday party. I was high, half-drunk, reeking of cigarettes, with a stolen gift and a little girl holding my hand, elated, wearing a pink dress and pink Western cowboy boots. I reminded her I would buy another streamer.\n\n\"I know you will, Mama.\" She smiled at me, and for a moment I felt better. She believed I would do it, so I could believe that too.\n\nThe girl's mother invited me in when we arrived at the doorstep, but I mumbled something about an appointment. I told her how sorry I was to miss the party. Then I walked alone back to the car in the relentless sunlight to do another line on the back of the CD case while parked right there on the street. I needed it to get back home. I wondered how I was going to pick up Ava in three hours, but set that aside as a concern of the future.\n\nAs I put the CD case away, I thought of her laughing and playing at the party without her mom, while all the other mothers stood around smiling at their kids' antics and games, and I felt so blackened with my own sin I slammed my fist into the steering wheel and dash until I couldn't anymore. I lit a cigarette and drove, telling myself this had to be the end.\n\n_Please, God, let it be the end._\n\n\u2022 \u2022 \u2022\n\n\"I was not fired. I was placed on a mental health leave.\" That is what I told my friends when they asked why I wasn't working anymore.\n\n\"Oh, I needed to take a semester off to focus on my mental health.\" That is what I told my family when they asked why I wasn't in graduate school anymore.\n\nMy mother moved out of our big house on the lake, and Mac and I were supposed to move out too, but we decided to leave a couple of months after my mother. None of us could afford the big house anymore. It was 2007 and everything was tanking.\n\nTwo weeks after she moved out, my mother knocked on our front door one Saturday morning and told me she was going to take the kids to the park. I knew this was not true because it was seven a.m., raining, and January, but I told her okay because I wanted to go back to bed. And I knew somewhere they were better off without me.\n\nBack in bed, I told myself it would only be a couple of days.\n\n\u2022 \u2022 \u2022\n\nTwo days later, I reminded myself they would be back soon, but I told my mother, \"I'm just too sick to have the kids, Mom.\"\n\n\"I agree, Janelle.\" She said softly.\n\n\"I just need some time to get better, Mom.\"\n\n\"You know I'll do anything for you.\" The sweetness in her voice was relieving this time. Her unquestioning devotion. Her acceptance. I knew I would get what I wanted that day\u2014wide open space for addiction.\n\nBy \"sick\" I meant \"what my shrinks told me,\" and I had seen many as I sought answers for my inability to quit drinking, as I sought answers for where exactly all my \"potential\" went. _How does a smart, middle-class honors student with a \"good head on her shoulders\" end up unable to quit drinking and pumping herself full of powdered stimulants?_\n\nThat was my question for the professionals, and in response, they told me I had borderline personality disorder (which seemed accurate but is basically incurable, unfortunately for me), bipolar II (when they asked me about mood swings I said, \"Um, yeah. I'm a coke addict,\" but they gave me the diagnosis and anti-psychotics anyway), chronic depression (obviously), and PTSD ( _Oh, come the fuck on, that's just ridiculous_ ).\n\nThe psychiatrist prescribed seven psychotropic medications but never explained the source of trauma that ruined me. Walking out of his office, I felt myself fully armed with the reason for my drinking, drugs, and failure.\n\n_I'm broken_ , I thought. _I'm a broken insane person. It's not my fault. I'm self-medicating._\n\nNobody mentioned I might just be an alcoholic.\n\nLamictal. Klonopin. Seroquel. Neurontin. Effexor. Zoloft. Ambien. I took them all, mostly. I took them all, usually. I often took the Ambien and always the Klonopin. ( _It's not my fault. I have mental illnesses._ )\n\nThe therapists fixated on events of my childhood, particularly those of a sexual nature. Another one spoke of my father's absence. During our next session, we discussed how my mother and I had an \"unstable\" relationship that left my personhood ill-defined. I never knew I was so damaged by those things. _But now that you mention it, doc, I do feel damaged._ I especially felt damaged when people came at me with criticisms of the way I was living.\n\n_If what happened to me had happened to you, you'd be living this way, too,_ I'd think.\n\n_Just try to tell me to clean up my act again. Just_ try _to fucking tell me._\n\n\u2022 \u2022 \u2022\n\nMac and I began frequenting a tiny, dark, terrifying hovel belonging to a man I called Charlie, although that was not his name. When we arrived, he had recently spray-painted the walls of his kitchen with unrelated German words. His girlfriend was unimpressed. There was always a large selection of people there, none of whom we knew. I wasn't even entirely sure why _we_ were there, but we always stayed awhile, often talking to half-naked people on a bed.\n\nThe children were still with my mother. I reiterated to Charlie and the half-naked people how very sick I was, and how beautiful and smart my children were. They listened, spellbound, warm in cocaine compassion.\n\nOne night after returning from Charlie's, in the kitchen of our home, I couldn't stop shaking my head and saying strange things. I gently tapped my head against the wall. By hour one, Mac was weeping, but I couldn't stop the tapping.\n\n\"Janelle, please don't stay like this. Please come back. Please be okay,\" I heard him talking. His voice was slowed down and I thought I could see the sound waves. His mouth and the sound were not lining up, so I merely observed him.\n\nI stood by the dining table in the corner of the kitchen in the early morning light, and I saw him looking into my face. I knew he was seeing me insane, and I wanted to be better for him, but I could not.\n\n\"Janelle,\" he said with his hands on my face. \"You're talking nonsense.\"\n\nI was speaking in tongues, my mouth moving against my will. As he paced and begged, I faced the corner of the wall and began hitting my head harder and talking, and then I was sitting in a kitchen chair I had towed to the middle of the garage, smoking. He sat behind me with tears and concern, trying to give me space. Our border collie whined from the crate. Mac let the dog out, fed him his breakfast, and, I imagine, patted him on the head. He returned, handed me another smoke, and crouched down to watch me. Right in front of me. I saw terror. In his big brown eyes I saw terror. He watched me silently and waited.\n\nOver time, I moved a little closer to him, to the ground, spoke in a language the people understand.\n\n\u2022 \u2022 \u2022\n\n_I was not fired. I was placed on mental health leave._\n\nFrom our motel in Sacramento, Mac suggested we go home. \"It's a half-mile away, Janelle. We can't afford to stay here.\" He was puffy-faced and had put on a few pounds. He wore a huge camo jacket with pockets everywhere. We had bought it two years before at an army surplus store. He hid liquor bottles in it when he returned to our motel room from the grocery store. He believed they should be hidden, even from strangers, and even though they were legal.\n\n\"No, we can't go home. Somebody will find us.\"\n\n\"Who, Janelle? Who will find us?\"\n\n_We just need to stay here in this motel room with towels shoved under the door because I think I see feet. I'm 90 percent sure I see feet. Those are definitely feet. How do you not see them?_\n\n\"Fine, we can run home for clothes,\" I said. \"I can do it. I'll do it. I'll run home for clothes. I can do it. Don't come with me. I want to go alone.\" I saw him as a liability. If we had to do something as dangerous as drive across town and enter our home, I needed to do it myself. I was the only one I trusted.\n\nBack in our house, the big one on the lake where we had brought Rocketship Rock-On home in his blue pastel smocked bubble, I decided we could stay for a few days, so I called Ben, the cocaine-delivery guy, and told him we had relocated. He brought me more.\n\nBut there was too much blood in my nose to do another line and my heart was doing something strange and I wasn't entirely sure I could feel my feet, so I asked Mac to Google \"heart attack from cocaine.\" He yelled from the other room, \"Does your chest hurt?\"\n\nI could tell by his voice he was scared, and if he was scared, he might block me from doing more drugs, so I yelled, \"No,\" and then he yelled, \"Are your fingers tingling?\"\n\nAnd I didn't think so, so I yelled back, \"No, baby!\" And at first I enjoyed the \"possibly overdosing\" attention, but then it turned on me. I just wanted to do more, so I told him \"never mind\" and asked him to come back and told him all the racing heartbeats were gone.\n\n_I'm okay. Bring me a straw and the baggie after I rinse my nose to wash out the blood. Bring it to me on the couch because I cannot stand. Bring me a cocktail to slow my heart. Tilt the straw because I cannot lift my head. Bring me my pills because I'm getting dizzy. If I don't take the Effexor I get very dizzy. Has it been two days already?_\n\nI staggered up the stairs after the Effexor kicked in and I could walk again to do a line off our wall mirror, which was lying flat on my bed now. When I got to the top stair and turned toward my bedroom, I heard it: \"You are going to die if you do another line.\" I thought it was God. No, I knew it was God, and I knew it was true. It was a voice that rose clear and clean from the thick mud in my mind. It boomed.\n\nIt roared, and when it came, I knew it was not from me, and I didn't want to die.\n**8**\n\n# **Nothing Left to Hide Now**\n\nThat morning, instead of walking into the bedroom, I turned around and went back downstairs, leaving half of our last baggie of cocaine untouched. This was an act I had never accomplished before that very moment. I had voluntarily faced the beginnings of detox. I knew then it must have been God.\n\n\"I'm done, Mac. I'm going to die if I do more,\" I said. I paced the kitchen and then filled a beer stein with Captain Morgan, capping it off with flat Coke, readying myself for the feverish craving already crawling up the floor into my feet. It would get worse before it got better, but I had endured the withdrawal before. I had done it countless times before.\n\nBut this time, I had no desire to continue. _I could not continue_. I recoiled from the drug as if it were a food that had just poisoned me. My body rejected it on a visceral level.\n\nMac was still wearing his camouflage jacket, after all these days, and in our warm house. His black curls were wild, flat on one side, and his eyes were deep in amused skepticism.\n\n\"You don't have to stop, Mac. I don't care. Finish the bag. It's right upstairs.\"\n\n\"No, it's cool. I don't need more.\"\n\n_Weirdo_ , I thought.\n\nHe approached me in a motion I feared preceded a hug, a touch I knew would set my skin burning hotter than it already burned. He often wanted to touch me when all I wanted was to disappear into the black alone\u2014the black of cocaine, of alcohol, of sleep. It didn't matter what it was, I simply did not want him close. I wanted him to get the hell away from me until I felt some control again.\n\n\"I have to call my mom,\" I said, sidestepping him and grabbing cigarettes and the house phone on my way to the backyard. I looked out over the fake lake and then squinted at the pulsating numbers on the phone, knowing my eyes were amphetamine eyes. I pressed each number with stupidly measured focus. She answered immediately.\n\n\"Hey, Mom,\" I said, cracked.\n\n\"Hi, Janelle.\" Her voice was thick with worry, a tone I would have called \"overly dramatic.\"\n\n\"I need help. You have got to help me. I'm addicted to cocaine.\"\n\nI thought she would cry or yell or pass out in shattering surprise, but she said, \"I've been waiting for this call.\"\n\n_Strange,_ I thought, _since_ _I had been so sly._\n\n\"I'll be right there,\" she said. \"Do not move.\" She arrived thirty minutes later with a furrowed brow and stood in the kitchen looking at Mac and me, in the awkward awareness of a lie just brought to the surface. _Nothing left to hide now_ , so I took huge early-morning swigs from my stein and smoked the cigarettes I used to hide from her.\n\nI fell finally into a deep sleep, and that evening, when I could sit still again, I sat on the couch with my laptop and began searching for the rehab that would cure me. Mac and my mother looked at me with irritating severity from across the living room.\n\n\"God, why are you two so serious? None of this is that big of a deal.\" I wished they'd both settle down.\n\n\"Not a big deal?\" My mother appeared incredulous. \"You just told us you were about to die from cocaine.\"\n\n\"I know, but\u2014whatever. You guys are making me nervous. I don't know why everybody's gotta be so fucking dramatic.\"\n\n\"Why don't you just find a rehab you like?\" My mother looked at Mac for support, but he looked at me, and I typed \"Northern California rehabs\" into Google.\n\nI had two stipulations for a treatment center: First, it couldn't be part of that coffee-drinking cult. Second, it couldn't be one of those God places. It was quite thoughtful of God to speak to me in a mental lightning bolt, but I had no need for a relationship with the man. _Besides,_ _I had one with Him once, and look where that got me._\n\nI found a rehab near the ocean in Northern California among the redwoods and hippies. It boasted science-based therapy, and on our way there the very next evening (the intake specialist told my mother, \"Come now before she changes her mind\"), Mac and I stopped at Nordstrom and one of those import stores where white people buy drapes, couches, Italian sodas, and small statues of Ganesh.\n\nAt Nordstrom, I bought a pair of leather tennis shoes, perhaps as some sort of consolation prize, or because I had no closed-toed shoes even though it was March and had been raining for weeks. Every day, I wore a long skirt and flip-flops, possibly with a sweatshirt, mostly due to all the weight I gained after starting the medications that were supposed to heal me but only made me fat.\n\nAt the import store, we bought a silk and cotton blanket with elephants embroidered along the edges to decorate the potentially drab rehab bed ( _nobody wants that!_ ) and a tiny Buddha statue, to represent my new life of spiritual greatness. I packed a journal to record my transformation, pictures of Ava, Rocket, and Mac, a copy of Emily Dickinson's collected poetry, pajamas, and my flowing, ripped skirts.\n\nSure, I had lost my job, children, dignity, health, mind, and respect as an unfit drug-addicted mother, and was on my way to an institution hoping to save me from myself, but what occurred to me was that I needed $80 Nikes and a throw blanket.\n\nMac and I drove quietly through the rain. We were headed for the coast again, but for different reasons. This time we sat in a bath of misery and hope rather than excitement, a sort of determined agony that comes when you've reached the end of a long catastrophe, or think you have\u2014and it all feels urgent and heavy and new. We listened to live Grateful Dead, the same songs my father sent me on cassette when I was a little girl, which I replayed for hours on my boom box. I wondered if they allowed music in rehab. _I should have brought some_ , I thought. If I were to bring music, I would have brought the bands I grew up on: the Rolling Stones, the Dead, Bob Dylan, and Neil Young. Some Janis Joplin, perhaps.\n\nWhen \"Ripple\" came on, I glanced at Mac and thought about him\u2014my old friend sitting there, driving me through the rain\u2014and how it had been a long time since I watched him chase unruly chickens, since we shot pool in dive bars, since M-Y-Q and the almost homeless man with the kitten who ate brewer's yeast.\n\nWhen Jerry sang, \"If your cup is full, may it be again...,\" I thought, _Alright, is this going to be when my cup gets full again?_ It was almost too sad to think about, his tunes, his black-tar humanity. I wished we had gone in the daylight. My mind drifted to the goodbye I would soon face, and the time I said goodbye to Mac that Saturday morning while he stumbled onto that goddamn bus with all those goddamn bright-eyed students, and how we were so much sicker now. I had thought we couldn't possibly get worse than that day, and yet, _here we are._\n\n_Well, here_ I _am_. I suspected it was mostly me.\n\n\"I can't believe the shit we've done, Mac,\" I said. \"What happened to us?\"\n\n\"I don't know,\" he shrugged. I wished he'd say more, but he never said more. If I had looked harder at his eyes I probably would have seen they were damp with tears.\n\n\"I don't want to live that way again,\" I said.\n\n\"Me either.\" He smiled at me while I kicked off my shoes and reclined the seat, put my feet on the dash, and lit a cigarette. We had traveled the road to hell, but we had done it together.\n\nI heard Jerry sing \"If I knew the way, I would take you home,\" and I had to hear it again, because I've always thought it was the most beautiful lyric ever written. How simple the thought, how true, to want to lead with love, back home, into safety, but having no idea where to even begin.\n\nI reached down and skipped the song back.\n\nMac was the only one who knew the whole truth, of me, of my life. Insane nights in motel rooms. Beating heads. Speaking in tongues. Broken tears and the homeless guy who asked us if we would please take him back to his spot under the overpass. \"You frighten me,\" he had said. That one night when Mac knelt for hours naked in the bedroom doorway, to \"guard\" us from intruders (in our minds). He explained, \"Well, if someone opens the door, they'll see me here and turn around.\" I laughed, telling him it was a really shit plan.\n\nSince the day we met seven years before, we had never spent more than a few days apart. Even when I lived alone in that 1940s house, we always somehow ended up back in each other's arms. We were friends, I guess. We were always good friends.\n\nBeginning that night, I would live thirty days without my friend, and my beautiful Ava, and my baby Rocket.\n\nMy boy. My sweet boy who was going to fix me up. The pregnancy that was going to straighten me out for good. I thought about the day I found out I was pregnant, the day I found out he was a boy, the day I brought him home. I thought of the ringlets that bounced under his chin, the way his knuckles dimpled and his tiny perfect hands held dolls and hammers in that endless toddler fascination, and I thought about his silence. He barely said a word, except \"no\" and \"mama.\" He was so quiet, a hushed angel with giant blue eyes who toddled around, smiled, climbed things, cuddled, and laughed, until he was gone. I thought of the morning my mother took them and the way I let them go.\n\n_I am here now. It won't be in vain._\n\nI couldn't stop looking at Mac while we drove, as if I wanted to burn his image into my mind. I wondered if I should say something, but I couldn't think of anything, so I kept staring until he smiled again, his face small and childlike with the same heavy yearning eyes, offering still a thousand years of love and adoration. I looked away, partly in disgust, in that old familiar repulsion, because he loved me unquestioningly. I threw my weight against him because I had nowhere to lean, because he was there and I loved him, but why did he stay with me? I thought it was all he could do, all he knew how to do, some weakness, some sad routine he developed over the years because he couldn't find a better one.\n\nI didn't think it was a choice he'd made, or some gift, some beautiful generosity of spirit he was born with. It seemed more like blind devotion\u2014weak and frail\u2014to love a woman like me.\n\nYet I hated myself for abusing his love. I wanted him to stand up to me. I wanted him to fight me. I wanted him to shove me back or scream in my face or walk out or fuck somebody else. Anything. But the second I thought he was pushing back, or walking away, I panicked and begged him not to leave. In therapy, they told me this was \"borderline.\"\n\nBut he never did any of that. He simply stuck around and asked me to stay too. He stuck around and drank and took drugs with me even though he almost always responded to my first request with: \"No, this is a bad idea,\" and \"Janelle, let's just go home.\" He stuck around while I begged and pleaded for one more baggie even though the night had already gone horribly sour, and we were stuck in a stranger's house downtown with blabbering idiots on a four-day drug run. He stuck around the next day while I paced the house, barefoot in a ripped skirt over bleeding scabs, and he stuck around when I told him I hated him and that he had ruined my life. He stuck around even when our children were gone. He stuck around waiting. For me to get healed.\n\n\"I'm going to miss you, Mac.\" I closed my eyes when I said it and looked away.\n\n\"Don't cheat on me in rehab,\" he said, smirking.\n\n\"Who the fuck would I cheat on you with?\"\n\n\"I don't know. Some dude.\" His jealousy made me smile.\n\n\"I love you,\" I said. \"Everything is going to change, Mac. It has to.\"\n\n\"I know, Janelle. Okay,\" he said, but I wondered if he believed me.\n\n\"If I knew the way, I would take you home,\" Mac sang along this time, not mentioning that I had started the song over, to hear that line again.\n\nI wanted to hear Jerry sing it. And I wanted to believe him.\n\n\u2022 \u2022 \u2022\n\nIn the rehab parking lot under the drizzly rain, I refused to cry. Instead, I held on to Mac too long. He felt giant in my arms, like pure rock warmth, and I couldn't believe I was leaving him. I wondered if I had even noticed that warmth before.\n\nI regretted coming to this place immediately. I wanted to get back in the car and head out with Mac, anywhere. Like we used to.\n\nAlmost as if he knew I was about to bolt, a lively redheaded man in his early twenties came bouncing out the front door to retrieve me. He shook our hands with the enthusiasm of zealots trying to sell religion on a doorstep, then merrily picked up my bags. Mac stood shifting back and forth, looking at me from under his downturned face, watching, waiting for a cue how to behave.\n\nI mumbled an encouraging platitude and realized it was time, so Mac and I hugged again and kissed. I watched him back out of the parking spot, waving to me out the window. My guts turned into lonely disgust, as if the reality of my life were unfolding in that moment, when the last of my family drove down a hill without me.\n\nThe bouncing man reappeared, and I observed that he was wearing cargo shorts and flip-flops. He appeared too young to be escorting people into their new lives. I seriously doubted his qualifications and wished he'd talk less\u2014or better yet, stop talking entirely. Nothing is more nauseating to a drug addict facing clean living than a joyful person already there.\n\nTo make matters worse, his name was Brent. _Of course it fucking is_ , I thought. _I bet you also drink soy lattes and wear those shoes that look like feet._ I wondered if I was going to have to see him every day, and, if so, whether or not anybody would notice if I killed him.\n\nOnce inside, he led me to a small room for the beginning of the \"intake process,\" which involved him and some equally elated young woman relieving me of my cell phone and face toner (it had alcohol in it and could be consumed) and requesting that I get naked to make sure I didn't have any drugs duct-taped to my body. Luckily, only the woman was there for the strip-down part. I wondered if she was also going to make me bend over and cough to check my asshole for heroin. I had seen that in the movies.\n\n_Jesus, these people are serious. Who the fuck would drink face toner?_ I wondered if perhaps I wasn't as sick as I formerly imagined.\n\nAfter the body search, which did not involve my anal cavity, they led me to the office of a somber man with white hair who appeared to be in his fifties. He wore a striped, collared short-sleeved shirt and khakis. His desk and office seemed very official, with diplomas hanging on the wall in frames and tasteful plants by the window. When he said hello and began perusing a file, he struck me as a middle-management type, the kind of person who was always trying to prove his worth. He asked about when I had last used drugs, which medications I was on, and whether or not I was having suicidal thoughts.\n\nThough he smiled incessantly, the way he spoke to me\u2014slowly, skeptically, with a touch of condescension\u2014reminded me of being a receptionist, of being considered so irrelevant you disappear.\n\n\"Sign here,\" he said. \"And give Brent your purse so he can look through it.\" I handed it over.\n\n\"Good, now Linda will show you to the house.\"\n\n\"Oh, we don't sleep somewhere here?\" I asked, looking around. I thought the place looked pretty homey.\n\n\"No, this is where you'll come each day for group.\"\n\nI had no idea what \"group\" was, but it sounded awful. Like the kind of place where one is expected to \"process feelings.\"\n\n\"Sounds good,\" I said, and smiled, wondering if it was possible to get released on good behavior.\n\nLinda was a woman in her forties with brown wavy hair and several pieces of turquoise and silver jewelry scattered across her ears, neck, and arms. By the time she arrived, I was already bored and impatient to get to the self-improvement portion of the adventure\u2014the part where they fixed me. The part where they showed me how to quit abusing drugs and alcohol so I could get my life back. The part where I no longer _wanted_ the drugs and alcohol because I was healed. The part where I got happy. Or, if they couldn't teach me all that, perhaps they could at least tell me how to stop buying eight-balls of cocaine on a Tuesday.\n\n_Or any time, really. I guess no cocaine at all is the thing we're shooting for now._\n\n\u2022 \u2022 \u2022\n\nLinda drove me down the road a half mile or so, and slowed down in front of a house right in the middle of a remarkably standard neighborhood, among the homes of the regular people. We pulled into the garage and walked into a kitchen smelling of bleach. It had brown tile floors and immaculate counters, and I noticed on the wall by the pantry a large bulletin board with laminated rules, schedules, names, and numbers. I stopped to read them, but Linda said, \"Oh, you have all that information in your packet,\" and I remembered I had a folder in my hand.\n\nMy room was toward the back of the house, down a long hallway with gray carpet, and as I walked by other bedrooms, I peeked inside hoping to catch a glimpse of some shivering junkie, but they were all empty.\n\n\"Everyone is at the bonfire tonight!\" she announced.\n\n\"Bonfire? That sounds fun,\" I lied.\n\n\"It is. It's really fun. There are a lot of fun things you can do in sobriety,\" which was the exact moment I decided never to listen to anything she said again. But I smiled anyway and said, \"Yes! Looking forward to that.\"\n\nMy room was plain and comfortable, with two twin beds against one wall, a nightstand for each, and a dresser at the foot of each bed. The comforters were thin like motel bedspreads, but the pillows were big and fluffy. Mall art hung on the walls\u2014watercolor posters with egrets and oceans and rivers, framed in blue pressboard.\n\nThe second bed was empty, but Linda let me know it would be filled soon. I said a quick prayer that she was mistaken.\n\nLooking around the room, I missed Mac. I missed my babies. It was strange to be so fully alone, in a new place around new people who knew nothing of me, people who would judge me only for the way I presented myself. I grew exhausted just thinking about it.\n\nI folded my red silk blanket on the foot of my bed and missed Mac even more, remembering how Brent had confiscated my phone when he inspected my purse.\n\n\"You can use the pay phone in the kitchen!\" Brent had said, as if he were announcing the most thrilling of news.\n\n\"That sounds fun,\" I said, thinking how I could never understand perkiness as a personality trait. It just seemed so damn unnecessary.\n\nSo I couldn't hear Mac's voice. I set Emily Dickinson on the nightstand and Buddha on the dresser next to my journal and pictures. I unloaded my clothes into the drawers, noticing that some of the shirts and skirts weren't even clean. I had packed in a hurry and left in such a rush, I never said goodbye to the kids.\n\nI hadn't seen them for a week, since before the final coke run, and I knew they were safe with my mother, but I couldn't shake the feeling that I shouldn't be in this place. I didn't belong here.\n\n_How do I explain to these people that I am a mother, not a drug addict?_ I picked up a picture of Ava and Rocket on Mother's Day the year before. Ava was sitting on a couch looking up at the camera, wearing a navy and white linen dress. She had pigtails in her hair and they were perfect, no lumps. Rocket sat on her lap, a baby of eight months\u2014my favorite age, when they are fat and giggly but still can't walk. He grinned widely in a green jumper with smocking and embroidered rabbits across the chest, his hair an ineffable strawberry blond. I felt sick to my stomach.\n\nRocket was now seventeen months old and still not fully weaned. Although he had been away from me for two months, and my milk was gone, every time I visited him he wanted to nurse. But he was used to rejection, used to replacements, used to his mother's body as a site of toxicity. On many nights in our big house, I had to hide. I had to not comfort him when he woke up crying. I had to rely on Mac and my mother. A few times I went upstairs anyway, and, when I sobered up, wanted to die. Every time I sobered up, I wanted to die. Because _what kind of trash..._\n\nAnd yet, I never wanted to give up on us entirely. On me, I suppose. On my ability to be a good, nursing mother. _Good mothers breastfeed. Tomorrow I will be different._\n\nAnd now, now I would be gone thirty more days.\n\n_He doesn't even know where I went._\n\nI knew he would stop trying to nurse by the time I returned. _It's okay. I had no business doing that anyway._ _Poisoned. The most beautiful act, I poisoned._\n\nI threw the pictures in the drawer and walked out of the room to have a smoke in the backyard.\n\nOn my way back, I reviewed the house rules in the kitchen: when they would give us meds, what we could eat, where we could go, what would happen if we shot up. I wished the other addicts were there so I could have a distraction from the swirling anxiety in my brain.\n\nFrom my bed that night, I heard the housemates roll in, their laughing and cheerful talking and shouts of \"good night,\" and when it all quieted down, I got up and locked my door.\n**9**\n\n# **Who's the Sickest in the Room?**\n\nOn my first morning in treatment, I woke early, got dressed, and put on makeup, which I hadn't done in weeks. I was the last person to climb into the van, nodding and mumbling quick hellos to the other clients. The man next to me looked about twenty, and I could hear his music rattling out of his headphones. I liked him already. He didn't want to talk either. Back at the main building, we filed into a large conference room and found seats around a huge oak table. We sipped coffee and stacked our journals, pens, and cigarettes alongside us.\n\nNed, a therapist and substance abuse counselor, rose in front of the room and handed out worksheets. He was about fifty years old, short, with a square jaw and gray hair. He obviously exercised on a regular basis\u2014and probably outdoors too, for the added mental benefit of sunshine. To me, he looked like some sort of Herculean god of health. His body was firm, muscular, and rationally tanned. I hadn't seen a human like him in a long time. His clothes fit perfectly, and he even wore a belt that matched his shoes. His eyes were bright and clear, his hair strategically disheveled, just enough to make him accessible\u2014an everyday, chilled-out California bicyclist guy.\n\nWhile I constructed a yuppie past for him, he told us how he had spent ten years on the streets of Los Angeles shooting cocaine and drinking. My head cocked to one side while I contemplated whether or not to believe him. His details seemed accurate. _Alright_ , I thought, _he appears to_ _know things._ He explained how he was twenty years sober and a cognitive behavioral therapist. He didn't believe in God and wasn't sober through anonymous meetings. I liked him more than before.\n\nNed stood in stark contrast to the mostly white, twenty-something misfits in front of him, fumbling, fidgeting, and analyzing each another. The new arrivals were red-faced and pale, with big black circles under their eyes. The addicts completing their thirty days were bright-eyed, high-fiving the staff, bouncing like the teenagers I knew in high school whose confidence I wished I could steal and make my own.\n\nLooking around the table, I noticed we were all either fat or emaciated. Drowning in clothes or barely wearing them. We had our hoods up and chins down or our chests puffed out. Our eyes darted around the room or stared unflinchingly at a scratch on the table, a spot chosen as to avoid eye contact. _We are the human embodiments of excess,_ I thought.\n\nI watched a quick-talking, black-haired man with a nose ring and tattoos covering his arms and neck flirt with the girl next to him. He had been in the rehab twenty-five days and clearly believed himself healed. She had arrived that morning. I watched in disgust at his opportunism\u2014 _quick, get the sad new addict before she cleans up enough to realize you are dull and slimy_. Ten seconds later, I felt a touch of jealousy that nobody was flirting with me. I recalled Mac's words, \"Don't cheat on me in rehab.\"\n\n_None of these assholes are going to stay sober_ , I thought. But I would. I wanted it more than them.\n\n\"This chart outlines the Stages of Relapse,\" Ned said. \"It shows what happens when an addict starts using again after a period of abstinence.\" He looked down at the paper, his page highlighted with notes everywhere, and I followed his lead, scanning the page, picking up my pen and underlining words like \"drug glorification,\" \"negative emotions,\" \"coping skills,\" \"loss of daily structure,\" \"social isolation,\" \"triggers,\" and \"problematic thinking.\"\n\n_Oh_ , I thought. _I don't have that. My thinking is just fine._\n\nI took notes, underlined important concepts. _So, rehab is like school_ , I thought. _Perfect._\n\nAfter defining each term, Ned asked us to go around the room and explain our most recent relapse. A tiny dark-haired woman named Danny with librarian glasses and Sailor Jerry tattoos on her forearms spoke first: \"I moved back to Portland, got a job, but one day after work, I went to a bar for happy hour. I just went for a beer. But there were old friends there, and they offered me a hit, so I took it, and nearly overdosed. I was high for two more years.\"\n\nNed paused, smiling, and asked, \"Why did you go to the bar? Why did you go back to the place you knew would trigger you?\"\n\n\"I don't know. I wanted a beer,\" she said.\n\n\"But you ended up shooting heroin,\" he said, framing the question as a statement.\n\n\"Yes. In the bathroom.\"\n\n\"You weren't there for a beer,\" responded Ned. \"You were there for something else.\"\n\n\"Okay.\" The woman narrowed her brow. She had no idea what the hell he was talking about and neither did I. _I would go for a beer too, Ned. You go for a beer and shit happens, Ned._ I wondered if Ned was lying about the whole ex-addict thing.\n\n_What kind of addict doesn't understand stopping by for a quick drink only to find yourself two hours later doing hard drugs in a bathroom stall?_\n\nSitting next to the junkie was Shelly. Shelly had bleached, frizzy blonde hair with black roots that fell into a deep V down her back. She was impossibly thin and wearing huge gold earrings and tight jeans. She had no visible muscle. Her bones held pockets of waggling skin, like silk draped over a stick. I noticed her forearms were dotted with scars and scratches. She began explaining: \"I was doing so great. I was working and back with my family and everything was going so great, but then my mom died, and I found out my husband was fucking around with his ex\u2014the mother of his kids, who I've raised since day one\u2014because that woman is a useless bitch, never given us a damn penny, and her kids? You know what? They call me Mama. They've always called me Mama. Do they call her Mama? No, cause she's not there. Who went to their kindergarten graduation? Me. They aren't even my kids. And when the oldest had pneumonia, who took him to the doctor? Me...\"\n\nIt was then I realized we had a tweaker on our hands, and I glanced at Ned to see if he was going to shut the chattering woman up anytime soon. He merely gazed at her compassionately, which felt to me like an underhanded insult, as if he were trying to say I was an asshole.\n\nShe continued, \"So I find out my old man is fucking around with her and you know how I found out? She told me. She just showed up at my work one day and told me she and him had been screwing around for months. And she's pregnant again, but I don't believe it's his baby. And then I find out I'm pregnant too. But I know it's his baby. So here I am, pregnant, and I was so sad and angry, so one night I scored some dope\u2014\"\n\nAt the word \"dope,\" I hit my breaking point and rolled my eyes, because she wasn't a junkie, and \"dope\" is heroin. She was trying to sound as hard as the junkie, and it infuriated me. _Everyone knows you're smoking rocks in a barn, lady._ She probably didn't even shoot up.\n\nShe wrapped up her inane diatribe with the words \"and then I was off and running,\" which hit my ears like somebody eating potato chips with her mouth open. _Why the hell say things like \"off and running\"?_ I wondered how she could afford this place, and if, perhaps, they offer government subsidies for rehab. I made a mental note to avoid Shelly for the next thirty days and the rest of my life.\n\nThe next speaker looked vaguely promising. She was clean and less jumpy. Her name was Shannon, and she reminded me of the mothers in my suburban neighborhood. She wore jeans and a sweatshirt with the words \"Lake Tahoe\" on the front.\n\nHer face contorted into a pained expression while she said, \"When my two-year-old pretended there was wine in his sippy cup, I knew I couldn't drink any more. I saw right then what my life had become.\" She looked at her feet, as if that were the saddest image she'd ever conjured.\n\nI suppressed another eye roll, thinking, _Oh, did you lose your Mercedes-Benz too?_ I wished I could say it out loud, but I couldn't (not yet at least), so I tried killing her with my eyes. _Your kid pretended to drink wine out of a sippy cup and you check yourself into rehab? My God, did you even get a hangover once?_\n\nIt occurred to me that I was going to spend every day for the next thirty days with this horrific conglomerate of humans, and I had no idea how I would survive it. I hated everyone in the room except the dude with the scruffy half-beard who refused to remove his headphones.\n\nShannon was obviously the healthiest person in the room\u2014with her bobbed hair and polished nails and rampant sanity. _What a loser._ I considered letting Ned know he should probably send her home. But he nodded encouragingly at her while she explained how her husband had been sleeping in the guest room for six months, and I returned to wondering if Ned was ever a drug addict at all.\n\nHarvey, a red-faced man in his sixties wearing a button-down plaid shirt and blue jeans, began: \"I have never been able to quit drinking for long. I made it six months after my baby son died from crib death and I was out drinking when it happened, but I don't know. I always start again. I've never done a drug and I still own a little company, but I can barely show up to work anymore, and my wife and kids are gone, but I think this time it's going to be different. I think for sure\u2014I mean this is my fourth time in rehab.\" He forced a small, sad smile.\n\nI was bored just thinking about him, and decided he must have some intellectual deficiency. He went to rehab four times just on booze? _How do you get taken down by alcohol alone, man?_\n\nHe kept speaking as Ned asked him questions, but I had ceased listening because I knew I was next and needed to figure out what sort of first impression I was going to make to the group. _Do I go quiet and reserved? Do I go loud and funny? Do I hold my shocking story close or lay it on them right now?_\n\n\"Why did you relapse?\" Ned asked me with a smile.\n\n\"This is my first time in rehab, so I don't think I've done that,\" I said, feeling immediately embarrassed.\n\nNed sat down, as if settling in for the long haul. \"A relapse doesn't just follow treatment. It's any time you start drinking again after a period of sobriety, after really trying to stay sober.\"\n\n\"Well, I do that with alcohol every day.\" I laughed.\n\n\"Right,\" Ned said. \"But have you ever been sober for a long period, then started using again?\"\n\nI thought about the way I _knew_ Rocket was going to turn me into a PTA mother but instead I ended up doing blow in the bathroom. \"Yeah, I guess I did that with cocaine after my son's birth. I was sure I'd never use it again, and then I did after a long time.\"\n\nNed looked at me, waiting for me to continue, but I had nothing to add, so I resorted to honesty, saying, \"But I don't know why I did it.\" I looked away from his face.\n\n\"Do you mind sharing the circumstances?\" I flinched, because I hated when people said heartfelt words like \"sharing.\" It made me uneasy.\n\n\"I was nursing my newborn son and wanted a beer, so I had one\u2014a really yeasty one\u2014because it helps with milk production. That was what I told myself: 'It helps with milk.' And everything was fine for a few months. I even started grad school. I only drank beer and wine, but as my son got older, and I could leave him with my mom, my husband and I started going to bars again, and one night I met a guy who would bring blow to our house, so I bought an eight-ball from him, and then every time I drank I wanted cocaine.\"\n\n\"So your relapse started with a beer?\"\n\n\"I guess, but beer wasn't my problem. Cocaine is what always takes me down.\"\n\n\"Sounds to me like it was that first beer that took you down.\" Ned didn't smile this time, but instead leaned toward me and looked right in my eyes, and I decided for sure he was not a nice man. Once again, I had no idea what he was talking about.\n\n\u2022 \u2022 \u2022\n\nAfter lunch, we gathered in a large room for our first \"group therapy\" session, or \"group\" for short, where we all sat in a circle and took turns speaking of \"how we're doing,\" with or without a Native American talking stick. In addition to \"processing\" and creating \"relapse prevention plans,\" I learned we were defining and cementing our own special disaster hierarchy.\n\nNobody was in that place because life was spinning out of control in success. No, we were there because we were failing, for whatever reason, and probably miserably. And yet, every one of our egos remained miraculously intact. But since we couldn't battle each other for top position of highest achiever or smartest\u2014clearly, that ship had sailed\u2014we focused instead on a one-upping contest I liked to call \"Who's the sickest in the room?\"\n\nIn rehab, we fought for the bottom. We shuffled and scooted for the lowest rung. Everybody in that place believed he or she was the most desperate case: the most addicted, the most fucked-up, the most shocking. We wore our catastrophe like shimmering medals.\n\nThe heroin addicts were royalty\u2014not only because sticking a needle in your vein is terrifying, but also because shooting heroin has that screw-it-all suicide vibe that most of us found \"just too far.\" I would watch Danny like a character out of _Trainspotting_. __ She and the other junkies had a sort of stoic, matter-of-fact approach to addiction that awed the rest of us. When Danny spoke, we stopped muttering to neighbors and listened, in deep regard. I thought I was hard, very tough, but soon realized I hadn't done \"addiction\" until I'd lived like a junkie. Everyone's just trying to keep them from dying and off the streets, to \"reduce harm.\"\n\nHeroin is the most fucked-up drug to be fucked up on, but the meth addicts never stopped talking about theirs. Shelly and her gang called themselves \"dope fiends\" while the rest of us looked at them and thought, _Oh, come on. You're just another white trash methhead._\n\nThe goddamn tweakers. They walked into morning art therapy all pockmarked, toothless, and talking talking talking, with wiry, frizzy hair and perpetually black-smeared eyeliner, telling us how they lived in a hollowed-out redwood with their baby daddy and two kids, turning tricks in the back of a van and dismantling microwaves in their free time until CPS closed the adoption case on the first six kids they lost but now they are _going to change for sure!_\n\nI would look at them and think, _Bitch, you can't even form a sentence. What do you know about getting well? Maybe stop smoking battery acid and see how that goes._\n\nAnd then I would lean back in sweet superiority because I always chose cocaine, the rich man's drug, and hated meth, the white trash drug. Admittedly, I had tried meth once with Mac in some man's converted garage, because there was nothing else available. I felt like I had consumed ninety million cups of coffee and lit my brain on fire: a lot of action, no thought. I could not find the fun there. I wanted my brain to think excellent, comforting things, at least for a few seconds until it started wailing about the next line, or, even worse, grappling with the inevitable end of the line.\n\nI had always hated tweakers. First, they never shut up. Second, they were dumb. Whether they started dumb or became dumb on account of all the cold medicine they smoked was a mystery, but the result was the same. Third, they hurt people. In the news and among my addict friends, I would hear stories of what the tweakers had done. I heard about one near my town who passed out alongside a river in winter with her baby tucked next to her, a baby wearing nothing but a cotton shirt and wet diaper, until the baby froze to death. I read about another one spinning out for six days and raping a nine-year-old girl while the girl's mother smoked meth in the garage.\n\nTo me, tweakers were the addicts without standards, the druggies without a single code of basic decency. If we were all Mafia members, meth addicts were the mob bosses who killed wives and children. The rest of us were Don Corleone.\n\nUpon returning to my room after that first day, I met my new roommate, Alice, a woman in her fifties who I imagined had been smoking cigarettes for thirty-five years. She looked seventy-five, with deep lines around her jaw, which jerked to one side or the other at the end of sentences, and eyes beneath drooping lids\u2014eyes that darted arbitrarily from one spot to the next. Her smile revealed missing teeth on the upper right and left side of her mouth and one missing tooth on the bottom. Her gums were yellowed.\n\n_Meth mouth_ , I thought.\n\nHer laugh was cavernous and burly, and her voice was like sandpaper, rough and grating through years of smoke snaking through her throat. I forced a \"hello\" and shook her limp, tiny hand, then said, \"Oh, I have to go get something,\" and left to gather myself outside. I wanted to figure out what sort of physical ailment would convince the house manager I needed a new room. Some sort of inarguable disease that required isolation. _Maybe I_ _should tell her I have asthma and can't be around smokers._ As I took a drag of my cigarette, the fault in my plan dawned on me.\n\nI heard the sliding door open, and Alice emerged onto the back porch. She sat down at the table across from me, pulled out a pack of Pall Mall cigarettes, and retrieved a half-smoked one from the pack. I offered her a light, out of habit mostly. She glanced at me, inhaled deeply, and looked around the yard. I found myself chattering.\n\n\"So how's it going?\" I asked, adding, \"Well, I guess that's a stupid question.\"\n\nShe cackled and said, \"I'm better than I was a week ago. Guess that's something. Shit.\"\n\n\"That's the truth,\" I said. \"I can't believe how weird this place is.\" I felt like a fraud, chatting with a woman I had just scrambled to avoid. But I couldn't find it in me to snub her. I could only judge from afar.\n\n\"Yeah, I don't know why I'm here,\" she said flatly.\n\n\" _Really_?\" I sputtered sarcastically. It was rude, and I knew it, because what I was really saying was, \"Lady, I can tell just looking at you why you're here.\"\n\n\"This is a waste of fuckin' time,\" she said. \"I won't stay clean.\"\n\nShe talked less than most tweakers I had known, less than Shelly in group, less than the addicts I had met at various houses over the years. She was reserved, even shy.\n\n\"Then why come?\" I asked.\n\n\"I'm here for my daughter. She won't let me see my grandkids anymore.\"\n\nI told her that must be rough, and she asked me if I had children.\n\n\"Yes,\" I said, brightening. \"Two.\"\n\n\"Are they still with you?\"\n\nMy smile faded and a wave of humiliated rage poured over me when I realized Alice and I had both been relieved of the children in our lives. _This fucking tweaker and me._ _We had the same result._\n\nI stamped my smoke in the ashtray and mumbled, \"I see them a lot,\" and left.\n\n\u2022 \u2022 \u2022\n\nDuring one-on-one therapy, Ned tried to claim I hated tweakers because of what they represented in me, that somewhere I knew I was no better than them, and could have done any one of those things had I gotten hooked on a drug that turned my brain into soup as fast as methamphetamines do. I explained to Ned that I was in fact better than them because I didn't choose a drug that turned my brain into soup.\n\n\"You know, Janelle,\" he said during our third therapy session. \"We hate the traits in others that mirror ourselves.\"\n\n_What kind of madness is this_ , I thought. \"A mirror to what I hate about myself.\" That was the kind of bumbling nonsense Ned used to throw at me during \"sessions,\" but I knew he simply didn't understand my position.\n\nIn group over the next four weeks, I met all the other addicts and categorized them based on how much they irritated me. I met the pill poppers, who seemed less sick than the rest of us merely on account of their failure to shoot, snort, or smoke anything. The psychedelic drug types, who I didn't understand because acid is a thing you do in high school and then abandon, and the potheads, who everyone felt sorry for, because truly, _how do you need rehab for weed?_\n\nNext to the tweakers, the most insufferable humans were the \"dual-diagnosis\" people, because they thought they were _very special_. I wondered how they felt so special when almost everyone in the room had in fact been dually diagnosed with a mental illness as well as addiction. We all had depression and addiction, or bipolar and addiction. We were all self-medicating. We were all super sick. We were all very sad. But they wore their mental illness like letters at the end of their names.\n\nBy the end of my time in rehab, the people who fascinated me the most were the straight alcoholics. The drunks. The boozers. The men and women who didn't shoot, smoke, eat, or snort drugs, and who possibly never had. The ones who woke in the morning at four a.m. to take a shot of whiskey from a bottle in the hall closet simply to kill the shakes and sleep again, only to shuffle to the shower at eight a.m. after a quick swig of vodka (because it's \"odorless\") and make it to work red-faced and sweaty, to sit in a cubicle or run a business or sell a car, to run across the street and have another shot at lunch, to smoke hasty cigarettes behind the garage, to scream at their kids and beat their husbands and wives, to swear tomorrow they will quit, to carry on for ten or twenty or thirty years. And end up here, in group, with Ned and me, mumbling, \"Oh, I never did any of the crazy things you all are talking about. I just drank.\"\n\nAt night, I'd wonder if they _were_ perhaps the sickest people in the room, people who would die under the radar of \"functioning alcoholism,\" lacking the flash to get noticed, failing to pose sufficient threat to society to get locked up, dying by a rope in a closet one day, or a car smashed into a tree, or down the road in the ER after liver failure, or old, tired, and miserable, in the park with a bag. Shuffling down the sidewalk, still sure they aren't that sick.\n\nI loved them, every damn one of them, though I didn't know why.\n\nWhen it was my turn with the talking stick, I either spoke in short, withholding sentences with my arms crossed defiantly in front of me, or dumped the most shocking version of my story into the room. I was always playing \"Who's the sickest in the room?\" I was so sick I was raging and cold and disengaged. I was so sick I would tell you about _my_ diagnosis of borderline personality disorder\u2014a diagnosis with an unofficial prognosis of \"you are trash and there is no hope for you.\" After my first diagnosis of the disorder many months before, I learned that borderlines are so impossible many therapists flatly refuse to work with them. Borderlines ruin children through sick and twisted rage, harm themselves, generally exploding the lives around them plus their own. And the worst part was I actually did, in fact, _feel_ borderline. When I read the collection of symptoms making up \"my kind,\" I felt I read the first accurate description of the way I had felt since childhood.\n\nI split. I saw people as all good or all bad, and sometimes that changed within an hour. I cut my arms to watch the blood rise. I cheated, lied, and manipulated. I raged. Oh, God, the rage. Psychiatrists had also diagnosed me with bipolar II, but that was nonsense, because the doctor would ask, \"Do you have mood swings?\" And I'd say, \"Yes, yes I do. I am a cocaine addict. Of course I have mood swings.\" But nobody explained how they differentiated between the two.\n\nThen he'd ask if I went on drug binges, which I explained was affirmative. And then he'd ask if I went on spending binges, which was obviously a yes as well. In fact, I used to fill whole carts in stores like Marshalls and Ross Dress for Less and Target, but then I'd look around the store and at the cart and realize I couldn't handle the gravity of the situation. The cart. Checking out. Continuing to push it. So I would make sure no employees were watching, and I would leave, walking away from the whole damn fiasco, to sit alone in an aisle somewhere. Sometimes I would obsess over one item in particular, like gift bags or candles or embroidery thread. I would buy fifty. They thought this was \"manic.\" I thought it was simply \"confused.\"\n\nThen the doctors diagnosed chronic depression, which seemed accurate. I mean, _wouldn't you be depressed if you couldn't stop buying gift bags?_ They also diagnosed me with PTSD but never explained the trauma that damaged me, which made me wonder if they simply drew diagnoses out of large, expensive wool hats.\n\nNo matter what they said, I knew the depression and mood swings and binges and erratic behavior were somehow related to the drinking and drugs. The borderline, though? That was me to my bones. That was me before I started drinking, and in the dark when nobody was looking. My doctors talked about borderlines having so sense of self. No sense of identity. They were shells of people.\n\nI felt like that. I could never figure out if I actually loved _anyone_.\n\n_Do I care if you live or die? Why do I hurt everyone? Why am I not showing up again? Why am I slicing my arms with little blades? Why am I so afraid you'll leave when I'm not even sure I want you here? Why am I devastated you are suffering? Do I want to die for you? Does your pain erase me? Can I fix it? Why am I screaming \"I hate you?\" Why am I shoving you away when I wanted you here? Why do you not see what I see? Why am I still drinking? Why can't I cry?_\n\nI studied that bulleted list of borderline symptoms and found inarguable evidence that I was, in fact, born without a moral compass. At least I was right about that.\n\n\u2022 \u2022 \u2022\n\nI don't think Alice moved when she slept, and she made almost no sound when she walked. It was as if she were trying to take up as little space in the world as she possibly could. She only talked if you engaged her first. Well, except one night at dinner when Shelly the chattering tweaker was complaining about her kids. Alice stood up, shoved her chair in, and said, \"You shouldn't let those babies go.\"\n\nWe became friends, Alice and I, and we'd laugh at Ned's endless positivity, and theorize who was going to sleep with whom and get kicked out. She told me about her Chihuahua, and we spoke of ridiculous, simple things like pizza and swimming pools.\n\nWhen Alice was twelve, she was traded to a fifty-year-old man by her mother for a bag of heroin and a case of beer. She escaped him when she was fourteen, and survived on the streets until age eighteen, when she met a man who would \"take care of her.\" She had six children, all of them taken by child protective services, and got sober for the first time when the man died and she went to prison, where she birthed her daughter.\n\nWhen her daughter was nineteen, Alice relapsed, and couldn't afford heroin, so she landed on meth. She lost her teeth and face and mind, but probably not her heart, because when I asked her what her favorite song was she said, \"Boots of Spanish Leather\" by Bob Dylan, and I thought, _Well, damn._ I had been singing that song for as long as I could remember.\n\nShe had given up. She had not a single thought of change in her mind, not a shred of power at her fingertips, but she loved a song like that. She still flipped over a single cigarette as a \"lucky.\" I teased her, saying, \"Alice, nobody does that past the age of fifteen.\"\n\nShe flipped me off and said, \"They would if they had my luck.\"\n\nShe ruined everything I knew about tweakers.\n\nTwo weeks after she arrived, Alice left, because her daughter could only afford two weeks. I was glad to have given her my Buddha statue, because it was a little thing I could remember her by.\n\n\u2022 \u2022 \u2022\n\nOn my twenty-seventh day in rehab, I received calls from my father, mother, and mother-in-law, all letting me know Mac hadn't been seen in three days.\n\nTheir voices were urgent and severe. \"Do you have any idea where he is? Can you guess? This is just insane!\"\n\nI was unconcerned, irritated, and marginally jealous. _He's fine, people. Just on a drug run. Calm the fuck down._\n\n\"Not exactly,\" I told my mother-in-law. \"But he's probably somewhere near Charlie, the drug addict who is not actually named Charlie and likes to spray paint kitchens.\" I tried to think of the street where Charlie lived but only recalled the alley where we used to park. I told her about that and the coffee shop a couple of blocks down from his hovel.\n\nShe paused, and asked slowly, \"Well, what is his actual name, Janelle?\"\n\n\"I have no idea, but you don't need his name. Just go see if Mac's truck is in the alley, then wait, and if he isn't there, he's probably around there somewhere.\"\n\nSo Mac's older sister went looking for him, and found him wandering the streets of downtown Sacramento, just where I said he'd be. She found him at five a.m., walking barefoot, without his wallet or keys.\n\nThe next day, on day twenty-eight, two days before my scheduled rehab departure, Mac and I spoke on the phone. It turned out he was sitting alone one night in the adorable country house we had just rented to begin our life anew, and in a flash of brilliance remembered the abandoned cocaine baggie from the day God spoke to me. So he took it, and drank a bottle of rum, and got lost on the streets of Sacramento after hanging out with Charlie.\n\nWhile I could relate from a past life, I was healed now in rehab and therefore outraged at his poor choices.\n\n\"WHERE WERE THE KIDS, MAC?\" I screamed into the phone, sitting on the couch beneath a watercolor of a snow leopard. My privileges had been upgraded to use of the portable house phone as opposed to the pay phone, but we weren't allowed to use it in our rooms. Consequently, everybody squealed at their spouses in common areas, and absolutely nobody cared. \"WERE YOU FUCKING SOME BROAD, TOO?\" I thought of that night in the Thai restaurant parking lot.\n\nHe was quiet and sounded tired, with a shade of guilt and the slightest wash of shame. \"What? No. What? What are you talking about? They were at my parents'. I'm sorry, Janelle.\"\n\n\"And now you're going to rehab for ninety days on a goddamn beach in Southern California to find your inner child while I come back to pick up the pieces of our family?\" In twenty-eight days, I had transformed from mentally ill cocaine addict to doting savior.\n\n\"Janelle, aren't you just coming back from a rehab?\" He was growing irritated.\n\n\"Mine was only thirty days, Mac!\" I hung up to call my mother and tell her she needed to come get me, right then, so I could go home and clean up the mess my cosmic disaster of a husband made.\n\nWhen I heard Mac and I would be in rehab at the same time, I decided my in-laws and parents were going to try to steal my children. I envisioned long, drawn-out, vicious court battles between them. I had no evidence of grandparents or anybody else for that matter vying for possession of our kids; in fact, I had no evidence of anything but support from all of them, but I felt untethered. I was convinced that if I stayed the last two days of my program, I would lose them. _One of us has to be with them_ , I thought.\n\nThe counselors advised against leaving early, saying, \"Janelle, in your new life you must start following through on commitments.\" But I had done twenty-eight days out of thirty, and that was good enough for me. So I thanked them for their service and convinced my mother to retrieve me. Once again, she believed I knew best and agreed to come that very day.\n\n\"Will you bring the kids?\" I asked, holding my breath.\n\n\"Yes,\" she said, and I could hear her smile. \"They're sitting right here.\" I felt my heart jump.\n\nI packed my clothes in fifteen minutes, and the two hours I waited for them to arrive felt like Christmas Eve when I was nine. They ran to me in the front yard of the rehab house, and I got on my knees to hold them. I couldn't believe they could be so beautiful. In the car, I tried to nurse Rocket, but he had no interest. He turned his head away and sat up, and for the first time, it occurred to me that there were two children without their actual mother, and that they missed her in a material, clear way, and would create their own defenses to protect themselves. I thought again about the nursery I had made for Rocket, the clothes all lined up by size, the morning I brought him home, and the morning he left. I shook my head right and left to knock the images out of my skull.\n\nThen I reminded myself how very sick I was, and I asked my mom to please drive, because I had to get home.\n**10**\n\n# **Maintenance Whiskey**\n\nOne week after I left rehab, my mother and I sat in her living room in big, soft easy chairs with the footrests kicked out, constructing plans to prevent me from detonating my life. My plan was that I would live in the house Mac and I had rented before I went to rehab. Mac had moved our belongings from the big Natomas house to the new one, but didn't get beyond dropping off furniture and stacking unmarked boxes in every room. This was the extent of my plan.\n\nHer plan was: \"Devote your life to recovery.\"\n\n\"Don't your counselors say you should take time to just focus on recovery before trying to do life again?\" It was not an actual question, but I answered anyway.\n\n\"Yes,\" I said. \"And I have, right?\" I narrowed my eyes at her, sensing she had an opinion on my life that differed from my own.\n\n\"You have, but I think you need to get all this psych stuff in order before you get back with the kids.\" As if cued, Rocket bolted out of the bedroom holding a wooden screwdriver and a brown-headed doll with matted hair. When he passed me, I leaned over and grabbed his shirt, yelling \"Hey!\" and demanding a hug. He spun around in smiles and folded into my arms just long enough for me to get a whiff of the sweet honey sweat of his chubby neck before he ran off.\n\n\"I need to be with the kids, Mom, and Mac and I just rented that house. Have you seen the porch? It's amazing. Right next to a horse ranch.\"\n\n\"Yes, it's nice, but you rented that when Mac was still working. He's in a ninety-day program and you're out of money. You're on mental health leave for two more months.\"\n\n\"How do you know that?\" If I told her that, I had forgotten.\n\n\"Because your boss and I talk, Janelle. Everybody talks about your health now.\" _Good Lord_ , I thought. _Everybody is so earnest about everything._\n\n\"Weird.\" I rolled my eyes, pondering how they all couldn't see how I had my life handled now.\n\nSince she seemed to make a decent point, even better than mine, I turned my attention to the show we hadn't been watching but that played in the background because it was comforting\u2014the way it droned on, the hum of canned laughter and commercials with all the happiness they promised. All those beautiful people. All the beautiful things to buy. Like candles and gift bags.\n\nShe lowered the volume with the remote, looked at me, and asked, \"So, you think all those diagnoses are true? Your dad wonders about them.\"\n\n\"You talk to Dad?\" I asked, surprised.\n\n\"Of course. He's been helping me with the kids.\"\n\n\"What do you mean?\" Now I was utterly shocked.\n\n\"What do _you_ mean? I have to work. Mac's parents work. Your dad works. Everyone works.\" She seemed mildly exasperated.\n\n\"Wow, that's really nice of him. Too bad he didn't do that for me when I was a kid.\"\n\n\"Well, he's doing it now,\" she said dismissively as she shrugged her shoulders, clearly on his side. It was strange to think of all the life going on behind the scenes while Mac and I hid out in motel rooms. I had a fleeting thought of all the trouble I had caused, of all the grandparents scrambling to compensate for our deficiencies, but I shoved it out of my mind with the thought, _Well,_ _I am very sick._\n\n\"Mom,\" I said. \"If you don't think all that 'psych stuff' is true, why are you telling me to take care of it?\" I had spotted a hole in her argument.\n\n\"I didn't say that. Your dad and I were just saying it seems a little _excessive_. I mean, you weren't ever like this before.\"\n\nI turned to face her. \"You sent me to my first shrink when I was in high school. Remember that dude with the beard who hooked my fingers up to some machine and talked to me about chakras?\" I lifted my eyebrows and smirked, knowing the memory would make her laugh. We loved making fun of new-age hippies.\n\n\"You were sixteen and wild and very angry. I didn't know what to do with you.\"\n\n\"I loved that guy,\" I said. \"What was his name?\"\n\n\"Sergio.\" She smiled, and I laughed.\n\n\"But I've kind of always been crazy, Mom.\" I said, wondering why it was suddenly so important to me that my mother understood I was _definitely mentally ill_.\n\n\"Well, what do you think about what the doctors say, in your heart of hearts?\"\n\n\"Please don't say 'heart of hearts.' I can't talk to you when you say things like that.\" I was not joking.\n\nI stared at the television, though I was, in fact, thinking of a real answer. \"The borderline seems true. The rest I don't know. Bipolar maybe. But not really. Hard to tell when you're a coke addict.\" I watched her flinch at my last two words.\n\nShe looked away, and I continued, \"I get really depressed, but maybe that's the drinking. And Mac and I fight all the time.\"\n\n\"I know,\" she said, and paused. \"I think you should go into one of those 'sober living environments,' Janelle, and do an outpatient program, just to be safe, so you can really get well. We can take care of the kids.\" She looked at Ava sitting at the kitchen counter, coloring in a Hello Kitty coloring book. Rocket came darting back into the room, launching himself onto a couch cushion on the floor. He distracted my mother, so I didn't answer.\n\n\"Pick that up, Rocket. It's not a toy. It goes on the couch.\"\n\nHe turned around and looked at her, deciding whether or not to obey. She looked at him harder and he kicked the cushion, then set it on the couch. \"Thank you, sweetie,\" she said cheerfully.\n\nHe didn't respond.\n\n\"He's still not talking, huh?\" I asked, making sure I didn't sound too worried.\n\n\"No, but he's barely eighteen months!\"\n\n\"Ava had tons of words by this age. Do you think he's okay?\" I watched him drag a wooden frog pull toy around the room.\n\n\"He's fine. Boys develop slower than girls. Your cousin Benjamin didn't speak until three, and then he spoke in full sentences.\" She was relaxed, and got up to make dinner.\n\nStill, I wondered if I had broken him. He was so silent. _Why is he so silent?_ I thought of Alice and wondered if she was still alive, and if she got to see her grandkids.\n\n\"Where would I go to outpatient, Mom?\" I yelled from over my shoulder while she opened the refrigerator.\n\n\"I was thinking Marin County. Found a great place there!\" She had obviously been doing research.\n\n\"I don't know. Maybe,\" I said under my breath, flipping to a new channel to vacate my mind for a few seconds.\n\nI considered my situation: no job, home in boxes, husband in the drunk slammer, and little money. I had spent three nights alone at the cute house in the country and was sure I was going to get cut up into small pieces by some Republican on a horse. It was dark and quiet out there. Nobody would hear my screams. An actual clown lived across the street. She seemed friendly, but still, she was a clown, and sometimes she walked down to her mailbox in her clown suit.\n\nPerhaps this is what happens when you choose a new rental in between coke binges.\n\nAt the thought, I told my mother, \"It _is_ already feeling a little impossible.\"\n\n\"Let's check out that program,\" she said.\n\n\u2022 \u2022 \u2022\n\nA week later I moved into a house I found on Craigslist that was forty-five minutes away in Petaluma, California. It was occupied by two sober, friendly men. One was about twenty-five and had recently kicked heroin for the fifth time. This time, he was doing it through Suboxone. The other was \"old\" in age and sobriety\u2014forty years old and one year sober. He was an alcoholic.\n\nI brought my red embroidered blanket and a new Buddha statue to my new room and enrolled in the most highly esteemed outpatient rehab in Marin County, which is one of the wealthiest counties in northern California. My family and I were under the impression that if you paid more, you received better sobriety, so my parents and grandparents pooled thousands of more dollars in cash and credit to help me, again.\n\nIn addition to the outpatient program, I signed up for a dialectical behavioral therapy group that involved weekly meetings with a group plus individual therapy. I got a psychiatrist, worked out, and stopped eating sugar. I took up running, kick-boxing, yoga, and meditation, began shopping only at the co-op and Whole Foods, and hired a personal spiritual advisor\/holistic health practitioner. I woke every day at seven a.m. to meditate, shower, eat bee pollen, and head to the rehab, where I sat in group listening to forlorn white addicts express how dejected they felt. One of them lived on his yacht off Tiburon. Another owned the winery that made my favorite cabernet. Millionaires. They were actual millionaires.\n\nI thought of Alice and how she couldn't even stay for the whole month, and how drugs were the great leveler until it came to outcomes. To treatment. To the world opening itself to help. Nobody cared if Alice lived or died. Nobody would even notice if she were gone. I wished she could have come with me.\n\nI met with a therapist at the rehab\u2014in addition to the dialectical therapist and psychiatrist\u2014who was staunchly devoted to Jungian psychology. We began a long and meandering exploration of my unconscious and deepest longing, the needs I had but didn't know I had, but had better figure out how to satisfy, and soon.\n\nAfter our sessions I felt emotionally heavier and remarkably more confused, especially when the topic landed on my mother. Apparently, somewhere deep down I had always yearned for a clearer, healthier relationship with her, but never got it. She gave me nothing to \"bump up against\" because she was nebulous and blurred herself, and that was why I had no \"self.\" When I left the Jungian's office, I felt a bit like a fraud and a liar, as if I were saying things that weren't quite true, because sure, all those problems with my mom were real, and we had endured dark times when I was a child, and I found myself entwined and obsessed with fixing her marriage, finances, and heart, but her love for me was warrior-like\u2014brave and firm and wild\u2014and she was my unequivocal best friend. Talking about her like that felt wrong, and I couldn't make the therapist understand that we were _dysfunctional_ and _perfect_. I couldn't make the therapist understand that I was heartbroken and infuriated by her, but she was the only human I feared I would die without. I didn't understand why we couldn't be batshit crazy and woven through each other in the crispest, purest, and sanest love.\n\nIn the Jungian's office I learned about dissociative behavior and \"sober blackouts\" and practiced deep belly breathing. After group, I would go to the gym and kickboxing class. In the evening, I would attend my dialectical behavior group, or go to my psychiatrist, or sit at home and read Thich Nhat Hanh and Jon Kabat-Zinn.\n\nI was living, breathing, thinking, and eating recovery. I got stronger. I lost weight. My mind grew clearer while my family delighted in my \"progress.\" I had a whole army of activities, mental health workers, family, and friends encircling me, blocking me from the siren song of alcohol. I bought a lottery ticket one day at a gas station and won eighty dollars, which I took as a sign of God's pride in me. Yes, the God I didn't believe existed.\n\nWhen I was sober almost sixty days, at the peak of my wellness, Mac invited me to \"family weekend\" at his rehab. _Of course!_ I thought.\n\nI booked a flight using a high-interest-rate credit card I had somehow been approved for and set out on my journey to Southern California. But while sitting in a restaurant in the San Francisco airport, waiting for my flight, the thought occurred to me that I was afraid of flying. In fact, I hated flying. At some point, while darting around Europe in airplanes, I'd realized that each flight I boarded increased my chances of dying. _The more I fly, the more likely I am to be on a plane that malfunctions midair and spirals desperately into the ocean amid screams of mothers and children._\n\nAt the airport in San Francisco, the thought that followed this recollection was, _You know what would make me feel better? A glass of wine_.\n\nI failed to remember the Xanax in my purse, and without another thought, without a whispered word from my conscience, therapist, God, friend, or enemy, I rose and floated to the bar across the concourse, stood for a few moments waiting for the bartender, and when he placed his hand on the bar and nodded to me, I smiled and asked, \"May I have a glass of chardonnay, please?\"\n\nThe next thing I knew I was driving a rental car with a handle of Captain Morgan on the passenger seat.\n\nI had sipped two glasses of wine before my flight and switched to cocktails on the plane. I then picked up my rental car and drove immediately to Safeway for booze, but in my hotel, I had one drink and realized what I had done. By then it was midnight, and I had to get rid of that bottle. I grabbed the rum, took the elevator downstairs and looked around the lobby for somebody who looked like they needed a handle of Captain Morgan. Nobody was there except the front desk clerk, but he was perfect. He looked about twenty\u2014a surfer type with long sun-bleached hair. I walked up to the counter and set the bottle down in front of him.\n\n\"Do you want this?\" I asked, without introduction.\n\n\"What?\" He asked, confused.\n\n\"This. The rum.\" I smiled, realizing my behavior was a little odd. But he was so young. _Of course he drinks_ , I thought. _This is Newport Beach. Everyone drinks in LA._\n\n\"Um...\" He looked around nervously.\n\n\"Do ya not drink?\" I asked. \"Because if you don't drink, I seriously misjudged you. Maybe more of a weed guy. I could see that.\" I laughed the laugh of the sweet spot of drunkenness.\n\n\"Yeah, I totally drink. Alright! Thanks!\" He grabbed the bottle quickly and smiled.\n\nFor some reason, I added, \"I don't fucking want it.\" This made him laugh again, and he thanked me again, and I could see his after-work plans had instantly improved.\n\n\"Noooo problem,\" I said, leaning against the counter for a moment, gearing up for the walk back to the elevator.\n\nBack in my room, I fell asleep immediately.\n\n\u2022 \u2022 \u2022\n\nThe next day, while walking down to the beach, I told Mac about the rum.\n\n\"It was a terrible mistake. But you know? I'm glad it happened. It taught me the power of alcohol. I mean, I really get it now.\"\n\nI was prepared to grovel, but he listened to the whole story and said, \"Well, you learned from it. Sounds like it had to happen.\" He put his arm around me. He and I believed my declarations of sorrow and a new start.\n\nWhen I returned to Petaluma, I told my sobriety army about the relapse, and all of them suggested I go to one of those ridiculous meetings in basements for low-bottom drunks and other nondescript failures. At the first sign of my weakness, both roommates launched into passionate diatribes regarding _the only way they could stay sober_. I told them I thought it was a cult religion, but as I was saying it, I knew it was a lie. I had family members who got sober at those meetings. They were neither cult members nor religious, but I had _opinions_.\n\nI _knew things_.\n\nBut they were right. I had gotten drunk again. I drank again in spite of all the money, work, focus, therapy, and talking I had done specifically to avoid drinking again. All that talking, examining, deconstructing of emotions. It all failed me when I needed it. I couldn't even explain how or why I drank that night. It just _happened_ , and it frightened me. I agreed to try something new.\n\nThe next day, I drove a few times around a gravel parking lot trying to delay the moment when I had to walk into what looked like an abandoned church hall with homeless people and fifty coffee-can ashtrays around the perimeter. Finally, I parked and meandered inside, choosing a seat in the back of the room, behind a mass of what felt like twenty thousand people considerably too happy to be sober. They were clean and well-dressed, laughing and eating cake. They surrounded the ones who looked like small wet dogs with matted coats, and I hoped to God I didn't look so sick they thought I needed love, too. I avoided eye contact and sipped my coffee out of a Styrofoam cup, thinking, _Now I really have hit the bottom of the fucking barrel._\n\nWhen the meeting started, somebody started talking up front about how they now didn't have to drink against their will, and I rolled my eyes. _How does one drink against their will? Is somebody holding a gun to your head?_ _Idiot._ It did not occur to me that I had done exactly that in the airport a mere week earlier. There were lists of twelve rules on the wall. Apparently my coat was also matted because immediately after the meeting a woman pounced on me before I could escape. She was relentless with her offers of help as my \"sponsor,\" so I mumbled okay to get rid of her, but later realized when telling my mother about her that a sponsor could be my new move. My new bulletproof plan.\n\n_It's fine, Mom, because now I have a sponsor._\n\nThe woman and I began drinking a lot of coffee together, and we talked and talked and talked. She gave me a large blue book and took me to a Native American event in a teepee\u2014or, more accurately, an appropriated event by a bunch of white woo-woo types\u2014where we all took our clothes off, got smudged by sage, chanted, and sweated together. It was supposed to be spiritual, but I mostly found it interesting that people who were menstruating weren't allowed in. Because none of the people were actually Native American, nobody could tell me why.\n\nOnce again, I had no idea what the hell anybody was saying in the teepee or in those meetings. I sat foggy and bored, watching the drunks around me, growing irritated when they talked too much, waiting for my turn to explain how it's done, shunning suspected tweakers as well as anyone with serious grammar deficiencies, and departing before they launched into their weird prayer circle. My understanding of the program was that you sit and drink coffee until something happens that makes you sober.\n\nIt also had something to do with God.\n\nAt home, I read the big blue book my sponsor gave me and highlighted the parts that resonated with me. I read it like a novel in college. I read it like I was going to write a paper on it. More information. More ideas that were going to kick in _any minute now_. Ideas to keep me sober, to stand between that drink and me. The meeting people had slogans: \"Remember your last drunk,\" and \"Think it through.\" I agreed to believe them. I did everything I could to fill my brain with a million defenses against the moment I'd once again realize I'm afraid to fly, or think, _It's been so long since I took a drink, surely_ one _would be fine. Just one._\n\nAfter the relapse, I increased my exercise and therapy even more. Rocket and Ava came to visit me on my birthday. They brought me gifts and asked when I was coming home. I realized it was time. I must get back with my children.\n\n\"I am so grateful for that relapse,\" I told the meeting on my last day in Petaluma, \"I learned my lesson. I am terrified of alcohol now.\"\n\n\u2022 \u2022 \u2022\n\nSix months later, I narrowly avoided inpatient psychiatric care.\n\nI had been on a ten-day bender, drinking whiskey in the apartment I had rented to \"find myself\" before getting the kids back, with my plants all dead on the back patio; Morgan, the little dog I had acquired as my companion, and the laundry in a giant damp heap at the foot of the bed; and me on the couch, in the habitat meticulously constructed to avoid that exact moment.\n\nA few weeks prior, I had found it necessary to make a trip to my mother's house at nine a.m. to steal change out of Ava's Hello Kitty piggy bank so I could buy some whiskey to come down off the crack I had been smoking all night. After that, my mother found it necessary to take away my key to her house.\n\nThe next day, I walked into work late. I had found a little job at a chain tutoring center in my town, and they told me I was excellent, definitely managerial material. But my boss didn't say a word when he saw me that morning. As soon as I stepped over the threshold, he looked at me, shook his head \"no,\" and pointed to the door behind me. I guess I had called in sick one too many times.\n\nI was then locked out of work and out of my mother's house. I hadn't seen my children in a month. I didn't try to call anymore. When Grandma Bonny called, I avoided her. When Grandma Joan called, I avoided her too.\n\nMeanwhile, Mac was a beaming light of sobriety, living back at the dome and apprenticing as an ironworker. Without me, he found a new career. My father was still sober and spending more and more time with my children. Everybody seemed to be getting better except me.\n\nI had met a new friend at a shadowy party of misfits who still lived like they were twenty even though they were all in their thirties. She drank like I drank, and we passed hours in my apartment guzzling Maker's Mark whiskey (when we could afford the good stuff) and listening to old country music. I wondered how I had ever survived without whiskey. In the past, I thought it tasted like stale fire. With her, though, at that time, it tasted like truth.\n\nMy days became Waylon, Johnny, and Hank. I thought I had finally found my people. _I am an outlaw. That's what it is._ Since I had no job or family, the drinking turned from night to all day and all night, without end.\n\nOne day, Mac knocked on the front door while I sat on the couch, immobile, pretending I wasn't there. He could hear the music. He could see my car. He pounded on the door, growing angrier with every smack of his fist against the wood. Eventually he left, roaring as he walked down the stairs.\n\nThe next day, I told my outlaw girlfriend that he broke the door down and came at me physically. This was an outright lie, but I wanted some sympathy. I wanted people to understand I was the victim here.\n\n\u2022 \u2022 \u2022\n\nSitting across from the hospital psychiatrist, I tried to explain that it was just Morgan and me in the apartment, and due to the drinking situation, I couldn't manage to get him outside anymore to pee and poop. The apartment manager, I explained, was overreacting.\n\n\"Plus, I covered it with paper towels, doctor, so I didn't have to step in it.\" I didn't tell him I had lived that way for weeks, covering every stain and hopping around them like a child trying to avoid the cracks on a sidewalk.\n\nI thought this would clear it all up for him, but he said, \"Nobody sane would live that way.\"\n\nThis startled me. _I am not insane_ , I thought _. I was just drinking! It was a lot of drinking._\n\nI got off with a month of intensive outpatient treatment, so each day I drove to Sacramento to sit in the corner of yet another group therapy room with my hood up, promising not to drink and trying hard to schedule psych appointments that conflicted with art therapy. I hated art.\n\nThis place was not like the rehabs of a year before. This place was for the poorest people, the crazies on the street, the ones chattering to themselves in Central Park.\n\nBut the psychiatrist was not done with me.\n\n\"You know, Janelle, you went from living a good life to fired and in a mental healthcare facility in eight weeks. Why did you start drinking again?\"\n\n\"I don't know. I thought it would be different this time.\"\n\n\"What led you to that conclusion?\" I looked down, and I watched his hand move across the yellow legal pad in an open manila folder. _Who knew what kind of lies he was telling about me._\n\nI thought about the way I had started drinking after that stretch of sobriety between the sober living house and my new apartment. I thought about living with my mother, and deciding with her that I shouldn't yet return to Mac and the kids, and how I went to meetings every week and exercised, and how my job at the law firm said I could take a few more months off to \"get well,\" so I got a job at the tutoring center. I thought about the night I picked up a six-month token from the cult meetings, and stood in front of them declaring the miracle of sobriety, and how they all cheered for me. I thought of my sober friends asking me to ice cream after the meeting, and me saying no, and how I fully intended to drive home, but by the time I got to the freeway, I remembered that alcohol was never my problem. It was _cocaine_ that turned me into a blithering idiot. I recalled the words of my friends in the meetings, \"If you aren't sure you're an alcoholic, go try some controlled drinking.\" And I knew what I needed to do. I needed to try some controlled drinking.\n\n_How do I explain all that to you, normal doctor guy?_\n\nI couldn't, so I said, \"I thought because it was so bad before, I would drink differently this time. I wanted to prove I wasn't an alcoholic.\" It was the most honest answer I had ever given a doctor. I yearned for help this time. I was so tired.\n\nHe didn't respond to my answer, but instead asked, \"Are you med-compliant?\" which is code for \"Do you take what's prescribed?\"\n\n\"Yes, I am.\" I said, resigned. _This fucking guy doesn't care._\n\nHe scribbled more lies on his page and looked right at me, stating, \"You were living in a house full of dog feces and urine.\" He seemed to desire an explanation, again.\n\n\"I know. It was raining a lot. I told you. I couldn't get outside.\"\n\n\"Why are you cutting your arms again, Janelle?\"\n\n\"Because I like watching the blood come up,\" I said, pausing before saying, \"It feels _manageable_.\"\n\n\"Are you still drinking, Janelle?\" He was simply moving through his checklist of questions. He didn't respond directly to a single one of my answers. I wondered why I bothered with these doctors.\n\n\"No. Yes. Well, I didn't today.\" And that was true, although last night's alcohol was probably drifting out of my pores.\n\nHe snapped the file shut, leaned forward, and asked loudly, \"Are you always this difficult?\"\n\nI sat stunned and terrified. I knew I was in bad shape, so I had promised myself to tell the doctor the absolute truth. I did so. _And now he thinks I'm crazier than ever. He hates me because he thinks I'm messing with him, and yet, I am being honest. I am trying to get help._\n\n_Is my truth more horrible than my lies?_\n\nHis reaction confused me, and I began wondering if I was telling the truth at all.\n\n_Maybe I thought I was being honest, but I was actually lying? Maybe I am truly insane?_ The truth was melding with the lies, and I couldn't tell him what was happening in my mind because _I just told him_ _pure truth_ _and he's asking why I'm being so difficult_.\n\nHe prescribed more and different pills.\n\n\u2022 \u2022 \u2022\n\nOn my way home from the loony bin that night, I bought a pint of Ancient Age whiskey and a pack of Pall Malls, just like Alice used to smoke. I paid for them with the twenty dollars my mother had given me. The next morning, I started thinking of my mother the second my eyes opened. I saw her face and smile and felt her hug as she handed me the twenty dollars I had said I needed for groceries. I thought of my children. _My mom, she was always so kind._ I promised myself I would visit them that day.\n\nWhen I opened her door, Rocket, who was two, ran to me as fast as he could, and I went down on my knees again to hug and kiss and hold him. He grabbed my face and declared, \"Mama, home!\"\n\nI couldn't even respond. I couldn't tell him I would stay. I could not tell my boy when I'd be back again. I didn't even bother.\n\nAva was six. She kept a box by her bed of trinkets and notes I had written her, from rehab and elsewhere. When she missed me, she read them, and maybe cried. Mac told me about the box to get me to sober up and come home.\n\nInstead, I drew pictures during art therapy and sent them to her, pictures of rainbows and houses and suns, the same stupid drawings I'd made as a kid when all the other kids drew more interesting things. Those birds that were just two curved lines connecting in the middle. The house with the flowers along the outside and a tree in the corner and a big yellow sun with lines of orange.\n\n_I fucking hate art_ , I thought. _But I'm glad I can send her pictures of the home I can't make._\n\nDuring breaks at the outpatient center we stood outside in total silence, smoking. The victim Olympics were over in that place, the pecking order gone, the hierarchy of sickness no longer fun. Nobody cared anymore. The souls around there were too lost, too poor to posture, too tired to fight. We showed up unwashed. We showed up stained, dosed on legal drugs, to tell them what they needed to hear. When we talked, it was only to compare release dates.\n\nEvery day I watched with agitated jealousy the people who got out. The staff there had no personal stories of salvation, no hopeful anecdotes or phrases. In fact, I was convinced they preferred us dead or in jail. They recited canned platitudes and worn-out \"inspirations,\" checked little boxes, and fed us out of Styrofoam trays on a pushcart. I refused to eat anything except Snickers from the vending machine. I never took my hood off.\n\n\u2022 \u2022 \u2022\n\nOn the day I graduated Lightweight Loony Bin, I headed straight to my outlaw friend's trailer to celebrate.\n\nSix days later, I took a shit in a bag and put it in my trunk. The toilet in her trailer was broken and I needed to go to the bathroom, so I placed the bag in the toilet and went. Then I found myself in possession of a bag of shit, but telling people around me I was carrying such a thing was out of the question, although perhaps they would not have found it so odd. Absolutely stranger things had happened in that trailer park\u2014for example, the man who wore masking tape on his head, or the man who stood in his parking spot directing nonexistent traffic and protesting capitalism. Still, the situation struck me as rather difficult to explain, so I concealed the bag in a larger bag, and then hid it in my trunk.\n\nI then got _distracted_ and left it there for probably two days, which is two days longer than anyone should have feces in her trunk.\n\nI could have walked four minutes down to the gas station to relieve myself, or the fast food joint right next to the gas station, or I could have even shit in the dirt. I could have thrown the bag away in the dumpster twenty feet from the trailer. Just about any plan was better than the one I came up with, but after six days of what I called \"maintenance whiskey\"\u2014which is when you're drinking around the clock simply to kill the shakes\u2014things get weird.\n\nEvery idea was a three a.m. idea, and nobody was left to question them.\n\n\u2022 \u2022 \u2022\n\nMy days became maintenance whiskey and trips between my apartment and the outlaw's trailer. At her place, hippies and Hank III fans with questionable dental hygiene occasionally showed up with drugs I couldn't afford, and I would marvel at their generosity. They would show up with soma and ecstasy, and I would take all that was offered. If I had recently refilled my Xanax and Ambien, I'd take that too. Somebody would come by with Percocet or cocaine, and I'd take that as well.\n\nI drank every day to kill the shakes. I did drugs when they came around but I had no money or energy left to find them. I was on a pure Ancient Age and Pall Mall maintenance plan.\n\nYet no matter what I consumed, I got no relief. I found no calm. I was driven by a compulsion beyond my mind, heart, love of my kids, or life itself. I held on each day to a vague memory of the peace alcohol once brought, of the way it blanketed my throat and belly with warm relief, my whole heart, actually, the way it connected me to others and my life and future. The way it took me straight to serenity, to comfort in my mind, in the universe. In those meetings they told me, \"One day, alcohol will stop working for you.\" I scribbled those words in the front of the big book they gave me, thinking, _My God, that cannot possibly be true._ In that trailer, I learned it was true. I learned that one day, I would grow physically drunk\u2014stumbling, vomiting, slurring my words\u2014but my mind would remain clean, stripped, and still starving. The relief alcohol once brought would never return. Like a rat on a wheel, I would frantically chase a drifting memory.\n\nOn the morning I woke up in my apartment with not a single dollar or drop of booze, I took a swig of vanilla extract from my kitchen cupboard and shuddered, thinking, _At least it isn't face toner_. While I was sleeping, my mother had dropped off on my porch a package of sliced turkey breast and loaf of bread, with a note attached that read, \"I love you, Janelle!\" With a smiley face. I couldn't eat the food. The smiley face broke my fucking heart. I detoxed and shivered until I got my unemployment check, then went back to the trailer.\n\n\u2022 \u2022 \u2022\n\nFour days later, my eyes rolled back in my head after I took a pull of Ancient Age. I was back on that maintenance plan. Someone put me in my beige Ford Taurus, a gift from Grandma Bonny, who, at that moment, was dying of dementia and old age without my attention, two counties away. They drove me to the emergency room and dropped me off, leaving me with my keys and car in the lot, finding their own way home so I could leave when they released me.\n\nI fell asleep on the hospital bed, and when I woke, disoriented and restless, I looked around at the patients next to me, at the people surrounding them, and I noticed my only companion was a new bracelet around my wrist. I tried to rip the oxygen tubes out of my nose, but a nurse materialized and leaned down right in my face, slowly articulating in one long, compassionate snarl, _\"Do you want to live?\"_\n\nI nodded. \"Then focus on your breathing,\" she said with equal severity. \"Because your brain is not getting enough oxygen.\" She was about fifty years old and quite apparently fed up with my kind. Her tone was bulletproof. It didn't even occur to me to disobey her. I nodded and lay back and focused on deep breaths and the people bustling around me. I was in a spot in the hospital where they didn't even close the curtains. I looked down and noticed I had no shoes, and my skirt was ripped.\n\nEventually the doctor arrived and announced he was sending me back to the loony bin because I had so many substances in my body it was \"obvious you were trying to commit suicide.\"\n\n\"Oh no, doctor, I'm not trying to kill myself,\" I explained. \"I do this every day.\"\n\nHe gazed at me in silence. I thought he was going to say something, but he only looked at me with resigned disgust and signed my release papers. He handed them to me and walked out. On my way to the exit, the nurse approached me in the hallway and said, \"You know, girl, you're gonna die.\" I nodded and kept walking, as did she. It wasn't news to me.\n\nIn the parking lot, I looked down at the bracelet and wondered when I had become a person without companions in an emergency room. I had a mother, father, husband, and children, but they had no idea where I was and had long since stopped expecting such information. I called my mother yet again.\n\n\"I almost overdosed. This has to be the end.\"\n\nI meant it no more or less than every other time I meant it.\n\nMy family rallied and booked me into St. Helena, one of the oldest and most revered treatment centers in California. My mother and her friends from church cleaned up the horrifying mess of my apartment and wrote me encouraging notes in a little red book of thoughts and prayers. My mother gave me the little book on her first visit, telling me how when she was cleaning my apartment she had found some little shoes and a shirt I had bought at the thrift store for Rocket, carefully folded on my counter, and when she saw them she had dropped her head into her arms and wept, because she knew in that moment how hard I was trying, and how I was dying.\n\nIn the new rehab, I didn't care how I compared to you, and I didn't care if you were sicker than me or not, and I didn't care what you did in the 1980s. _Just leave me the_ _fuck alone._ The doctors there explained that I was in \"late stage alcoholism\" and my liver was not looking good. _Ah, whatever. Where can I have a cigarette?_ It was a nonsmoking facility, so I had to walk up a hill behind the hospital and hide in the bushes to smoke as often as I could. I hung out with a dude who drank hand sanitizer once in jail. At our first AA meeting in the town of St. Helena, I considered making a break for the Chevron across the street, thinking I could hitchhike home, before recalling yet another glitch in my plan: I had been evicted and thus had no home. I figured the outlaw's trailer was most likely still standing. _I'd always be welcome there._ Like Alice, I had no real hope of recovery.\n\nBut on the second weekend, my mother brought Ava and Rocket to visit me, and Ava handed me a piece of paper with a poem she had written. It read:\n\n_March 8, 2008_\n\n_The skies are so blue_\n\n_The sun is shining._\n\n_I want to see you._\n\n_I miss you so much_\n\n_I really do_\n\n_Why do you ever want to leave me_\n\n_ever in these days?_\n\n_We do not have to be in this world._\n\n_We can be flying on unicorns every day._\n\n_It seems so difficult every single day._\n\n_Why do you have to be gone?_\n\n_I love you so much._\n\n_It seems I haven't seen you for three years._\n\n_I'm so happy_\n\n_I don't want to leave you ever._\n\nI read it and knew there was no choice but to stay. My attitude transformed with the words \"We do not have to be in this world.\" My baby was willing to go anywhere, anywhere to be with me. How could I not listen? How could I not go with her? My heart ached for that place, too.\n\nMy mind circled my children's faces every morning when I awoke, and each day in group, and when we sat in the cafeteria pushing food around our plastic trays. I wrote my children letters and completed every soul-searching therapy assignment. I attended the mindfulness meditations, wrote in my journal, and the whole rehab knew I was going to make it. If rehabs gave a \"most likely to succeed\" award, I would have won it. I left healthy and energized, with my counselor's words echoing in my brain: \"I've been here a long time, Janelle, and I know who's going to make it. _You_ are going to make it.\"\n\nMac picked me up on my last day. Each night in rehab, I had stood in the hallway leaning against a wall, discussing with him on a pay phone my great turnaround\u2014fighting, crying, planning\u2014until we agreed to reunite our family at his parents' ranch. He and his friends finished moving my belongings out of the sick apartment and into a storage unit, and on the evening Mac and I drove to his family's home, where he was living and we had lived many years before, I realized I was as happy and clean and full of love as the day we brought our first baby home.\n\nSix days after that night, six days after the night my children squealed and laughed and jumped into my arms because we were finally together again, a family, I told Mac I needed to go get some bread.\n\nI returned to the trailer where I nearly overdosed. It was my twenty-ninth birthday.\n\n\u2022 \u2022 \u2022\n\nNine months later, in December of 2008, Mac let me know he was finished. With me, with the marriage. I knew it was true. While he spoke, I drove north on Highway 5 on my way home from the Anza Borrego desert. I had gone south with a friend in a three a.m. idea, but I drank all my money and expended all my gas. I considered pawning my laptop, but instead called Mac. He deposited fifty dollars in my account, just enough for gas to get home. On my way, I called to thank him.\n\nI tried everything, but he was done. I groveled. I cried and begged, but the more I scrambled, the more calmly he spoke. \"Janelle, I'm done waiting for you.\" I wished he would at least rage. There's room for negotiation in anger. But there's nothing in neutrality. There's no game in surrender.\n\nI had lost him.\n\nHe had not taken a drink since March of 2007. When we would have lunch together, I'd suggest we order Coronas, and he'd say \"No, thanks. One is too many and a thousand never enough.\" He had now found sobriety in those rooms for drunks I knew were full of lies.\n\n_Too bad I'm not like them_ , I'd think.\n\nBack in Woodland, I weaseled into my mother's house through lies and manipulation, convincing her I was sober. She posted a list of house rules on the refrigerator as if I were seventeen years old. Curfew, medicine compliance, abstinence from alcohol. I wasn't doing drugs any more, and my outlaw friend was forced to return to _her_ parents. They too thought she was going to die. From my mother's house, I managed to pull it together just enough to return to my job at the law firm.\n\nNo longer drinking daily, I clenched my fists through three dry days, then drank again. I'd drink for two or three or four days straight, then, after a night of withdrawal, I'd maybe make it to work. Then I would swear it off. Then three more sober days would pass, and it would all start over again. I emerged from each binge in confused, gray hopelessness, but soon I didn't care anymore.\n\nI couldn't get relief. I couldn't help but try.\n\nMy neighbor, a young man whose kindness seemed limitless, offered me a shot of Dilaudid one day. I watched him flick the tip of the needle. \"Why do you do that?\"\n\n\"To remove air bubbles. If one gets in your vein, it will kill you.\" He laughed, as if that were funny.\n\nHere was the suicide rock star vibe I used to admire from afar. _How tough. How cool._ I didn't feel tough. I didn't feel cool. He tied my arm and pulled it and plunged the syringe. Red-orange light coursed through me. I dropped onto the couch and didn't move for three hours. I understood the appeal immediately, but I didn't want to die.\n\nI did it only three more times.\n\n\u2022 \u2022 \u2022\n\nI saw my children for a few moments two or three days a week, but only at my mother's house. I didn't take them places. I didn't make them meals. I didn't stick around.\n\n_I have reasons to go, kids. I'm sorry. I have reasons I can't be there. The alcohol has stopped working. I have to keep trying._\n\nI never bought the replacement silk streamer. Ava never asked. She still had the box by her bed. I missed her graduation from primary school. My mother sent me photos, and I showed them to the people I was with. \"Look at my little girl!\" I beamed.\n\nIn the throes of the morning, I heard Rocket's voice again.\n\n\"Mama, home.\"\n\nBut I knew I couldn't stay.\n**11**\n\n# **I Found God in a Leaf Blower, and I Fucking Hate Leaf Blowers**\n\nNothing happened on March 5, 2009, that had not happened a thousand times before. I woke at nine or ten or eleven a.m. in a bed in my mother's house, where I was still living, under a big window with no screen and blinds that didn't shut all the way because I had broken them. Sometimes I needed to crawl in through the window because I smelled like alcohol and was lying.\n\nThe sun beat me through the blinds that wouldn't shut, relentless against my pounding head. I was supposed to be at work, but I had called in sick again. They were probably going to let me go soon. I could feel I was riding the last few feet of their tolerance. My mother drove Ava and Rocket to school that morning. Mac, my mother, father, grandparents, and in-laws were working. Everybody was working. Everybody was away in the world, living their day.\n\nI sweated and shook, closed and opened my eyes in the fog and pain behind them. It was nothing. One more time. One more day. _Here we go, old friend._\n\nI rolled over and looked at the pile of books on the bedside table\u2014literature, some self-help stuff. An AA book. A glass half full of water. A journal I hadn't written in for years. I rolled over and stared at those books and felt a burn in my eyes and brain as something sunk into the realm of knowing. I closed my eyes, flipped onto my back, then to the other side, and back again\u2014enraged, desperate, and agitated, moving for the sake of passing the time, to shake the frenetic feeling from my blood. I stared at the pile of books again, and when I saw them this time, for some reason, I saw the end of my life.\n\nI saw it roll out in front of me like a carpet might unroll, the years unfolding one at a time down a street or giant hallway, and I saw the end of it. I saw it with my whole body and felt it with every shred of vision I had in that condition.\n\nI would die a hopeless, useless alcoholic, and there was nothing I could do about it.\n\nThat was the part that killed me right there: I had no defense. I was out of ideas. I had no strategy, and I had no person or thing left to blame. I was dying. Or dead. My children and family were gone because they should be. My family had disintegrated, my life was in ruins, because I had no idea how to _live it_. I couldn't take one more step sober, and I couldn't take one more step drunk. I accepted that end. I accepted it in my bones.\n\nI always had a next move. As soon as _this one thing_ happened I would be happy, and my life would work, and I would stop drinking. As soon as _this one thing_ happened, I would manage my drinking. But I had exhausted every plan, every move, every belief. Every therapist, every geographic rearrangement, every theory, every book, every rehab, every medication. I tried love and unlove and job and no job, rock-n-roll and hiding out, desk jobs and grad school and no school. I tried babies and friends and no babies and no friends. I tried living alone and living with others. I tried talking through it and living lies in silence. I tried churches and swearing with oaths and getting naked with women in teepees. I tried cutting it out of me with blades. I tried begging the gods. I tried retreats. I tried gurus. I tried pills.\n\nAnd it all failed.\n\nI failed.\n\nI drank. That was my future. No matter what, I would always drink again.\n\nLeveled under that window, I wanted nothing back. I knew I had no footing to demand a return of the people or life I loved.\n\nI suppose the bottle killed me that day\u2014the fighter, the one who kicks and screams and rails against powerlessness. It killed the one who hated the tweakers, who wanted to be better than others, who whined and cried\u2014postured and fought\u2014about what she \"deserved.\" Those Nikes. A more intellectually challenging job. Respect. It killed the one who resented my father for being gone those years, and it killed the one who strategized, polished things up, set it up _one more time_. Above all, it killed my self-pity.\n\nI saw in that moment I had already received everything I \"deserved,\" everything I had built, brick by brick, one day at a time.\n\nI would always drink again, and alcohol was my God.\n\nThe truth descended like a veil of black around me, pushed out in every direction of my mind and the room. I heard lawnmowers outside, people tending to their yards, leaf blowers\u2014those fucking things\u2014people driving by. I heard a dog yap here and there, somebody's pet that was cared for, fed, walked. People on their way to work, maybe visiting a friend, an errand, a quick stop home for lunch. I heard the world right outside that window. I heard it happening, moving, bustling, alive, and for the first time in my life I wanted more than anything to be a part of that world. I wanted to wake up and know where I would be that evening. I wanted to pick my children up from school. I wanted to take my trash out, answer mail, get up and make coffee, go to work, and have it all mean something. I wanted to find some satisfaction in these tasks, these tiny stupid life things we do.\n\nI wanted to be a wife. Mother. Employee. Friend. I didn't want my kids back, my family, my husband. I didn't want anything but to live one single day free. I wanted freedom.\n\nI wanted freedom.\n\nThe bottle killed me that day. I found God in a leaf blower, got up, and walked into the sound.\n\n\u2022 \u2022 \u2022\n\nI never would have thought to look there, in total defeat. I never would have thought giving up would bring me to life, that surrender would be my hope. It's almost funny. All those rehabs were about bolstering me, telling me I was capable and smart and worth more. It was always about arming myself and fighting, _beating this alcoholism._ We talked about my childhood and what I deserved and didn't get and how that wounded and broke me.\n\n_Well, that's nice, but it still doesn't give me a solution, does it? I'm still not showing up to my kid's birthday celebrations, and I'm still somehow unable to silence the voices that insist, \"Go ahead. Take a drink. It's going to be different this time.\" I still can't seem to make myself change no matter what I throw at it or how hard I want it. I always go back to the booze, and the drugs, and the same nonsense of my entire life. I know my problem inside out and backwards and all the catchphrases, and Jung and I are best friends and DBT and CBT and all the rest. I get it all, I know it all\u2014I am no fool._ _But you're trying to fix a broken brain with the broken brain, and if that isn't insanity then I don't know what is._\n\nWhen I died at the bottom of that bottle I abandoned the fight. I stopped caring what my brain said, realized my life was what it was because _I was running it._ My ego, my thoughts, my plans, and my \"needs.\" I spent years trying to manage it all, trying to control and fix things. If I thought it, it was true. Even though the results of that reliance were disastrous, it never occurred to me I could ignore it, that I could rely on something else, something outside of me, something that perhaps wanted better for me than shitting in bags and loneliness.\n\nOf course I did not figure these things out on my own. From that bed, I went to the only place I knew would accept me, the only place that might understand, the place I had gone for two years without ever staying sober, the place I'd go on cold nights after a bottle of whiskey, where I would bang on the table and tell them they were full of shit, the place where the people put a hand on my shoulder and said, \"Keep coming back, Janelle.\"\n\nI was taught these things by a washed-up ex-gutter drunk who had spent a good portion of his life shooting cocaine in a refrigerator box. He was in his late forties, with a wife and two children. He had blond hair and glasses and shimmering blue eyes. I met him while I was smoking out in front of the alcoholics' meeting hall. He watched me tell my story of woe to a poor sot next to me, but when he got up, he handed me a piece of paper and said, \"You need help? Call me tomorrow.\" And I did.\n\nI was taught this by the group of drunks I had been hanging out with for years but could never hear or see because I wasn't like them yet. I wasn't leveled yet. My ideas were better than theirs. Until that day I was rearranged, ran out of ideas, and lost faith in my ability to make new ones. I showed up broken, fumbling for words, and they didn't care. They offered me a new perception.\n\n_Universe, take it. Take it all. God, whatever. I don't know about God. I don't know if he or she or it is up there looking at me or sent me that desperation and surrender. I don't know a single thing about any of that, but I know when I died, when I stepped out of the way and my bones knew I was utterly powerless, help started flooding in like the man from Sebastopol who happened upon us on the beach on an island in the Caribbean, took us home, and didn't murder us._\n\nI got some help from that washed-up gutter drunk because somebody helped him. He didn't tell me warm and beautiful things. He said I was a dead woman. He said, \"We're all in various stages of 'my case is different.'\" He said I was just another drunk.\n\nHe told me, \"Janelle, in your case, if it looks like a duck and quacks like a duck, it's probably a fire hydrant.\" His words taught me I had an unreliable brain, and it would always lead me to another drink. For the first time, I realized my perceptions were wrong, and if I wanted to live, I had to get a new set of eyes. I called him Good News Jack, because he was full of news that sounded mean and awful except it was setting me free, and I knew it. It sounded like this: \"If your ideas worked so well, what the fuck are you doing here?\"\n\nIt sounded like this, too: \"We aren't looking for another idea, another mental construct, more mind candy. We're looking for a rebuilding, access to a power that can save your sorry ass.\"\n\nAnd I believed him, because I had nothing else. I decided I'd give his suggestions a try because I had exhausted all other options, and I figured at least this way there was _potential_ for change.\n\nWhen I told Good News Jack I had a bit of a shady past with God, he said, \"Janelle, if you're sittin' there and your ass is on fire, and I walk up with a fire hose, offering to put it out, are you going to tell me, 'Hold up. No thanks. I don't believe in fire hoses?' No, motherfucker, of course you're not. You're going to say 'Yes, please' and hope it works.\"\n\nI said \"Yes, please\" and hoped it worked. I did not think it would. Why would I? Nothing ever worked. I fully expected to drink again. I even told that room of ex-drunks: \"Look, I'm going to do everything your book says so that when it fails, I can drink again with a clear conscience.\"\n\n\u2022 \u2022 \u2022\n\nGood News Jack told me that people trying to live on their own and failing desperately is an ancient process, and some of them are lucky enough to fail so badly they die while breathing, and surrender brings them to God, or life, which is one and the same. I'm not talking about Jesus or baptism or any other ritual thing. I'm talking about really, truthfully not being able to live, and somehow having the ability to recognize it, admit it, and try something new. I'm talking about a rebirth that happens from the bottom up, not as a great new belief, but as a complete rearrangement from a flattening, when everything you thought you knew to be true, that you thought you knew about yourself, turns out to be wrong. A decimation. Leveled, and rebuilt. With new eyes. New ground. New power.\n\nThey say many of the mystics had that experience. I was not a mystic. St. Francis was, and he was something of a drunken loser. He too found himself miserable. He wrote that when you \"die unto self you awaken to eternal life.\" Not heaven. Fuck heaven. Fuck the afterlife. I had enough of that as a kid in church wondering if the Big Guy could ever overlook last night. Eternal life, as in literally the element of life that is eternal. The power. The energy. The pulse holding the stars. The pull of the planets. The universe beyond human comprehension. The thing that makes me alive beyond breath.\n\nMaybe we are just masses of meaningless flesh and blood, but it sure as hell doesn't feel that way when I really feel deep into my gut, or look at the ocean or a redwood in Mendocino County, or smell the breath of my baby. Those damn stars glaring down at me. All the clich\u00e9s. Tell me we know how it works. Tell me science explains it all. Tell me we haven't just chosen the God of Reason over the God of Mystery and that we're stronger and braver and more intellectually sound because we don't have \"faith\" like those pathetic believers. We're stronger, I know. But that faith simply has a different polish.\n\nAnd anyway I didn't have faith in any damn thing other than that I was not God, and my brain was not reliable for a whole lot beyond making a cup of coffee or doing a math equation or planning my day or a trip or solving problems of the intellect.\n\nBut my problems have never been of the intellect. Mine were of the heart and bones.\n\nMy brain can't bring peace. It can't bring life. For that, I had to let go and hover over the reef, traveling into the black\u2014a nothing, a tiny nothing paddling and kicking across the expanse of blue, taking deep breaths and heading for the shore, all I know of safety, buoyed by ancient waters pulled forever by a cratered moon.\n\n# **Part Three**\n**12**\n\n# [**There Are Three Types of  \nMothers in the World\u2014I Am None of Them**](TOC.xhtml)\n\nAlcoholism killed me three weeks before my thirtieth birthday. Rocket was three and Ava was seven, and I had no idea how to drive either of them to school. They attended private schools in Davis, my old college town, the one I used to frequent with my rhinestone Playboy bunny diaper bag. I neither chose nor paid for those schools and had visited them only a handful of times. Perhaps this should have embarrassed me, but two months before that I was drinking bottom-shelf half pints and smoking Pall Malls alone in a beige Ford Taurus (the very same Taurus I now drove to pick up my adorable kids at their adorable schools). I knew embarrassing, and _this was not it_.\n\nThe front doors of their schools were kept locked, and I always wondered if it was to keep the kids in or the riffraff out. Except there was no riffraff. This was Davis. There were merely \"nice\" people in sustainably sourced shoes. On further thought, I realized I was the riffraff.\n\nWe had to use a code to enter the blue-trimmed, glass-walled fortress, but I didn't know it the first time I tried to get in. I stood at the door of Ava's school yanking and observing the button box until the receptionist took pity on me and opened the door, deciding, I suppose, to let the riffraff in. \"Hi! Thank you,\" I said. \"I'm Ava's mother.\"\n\n\"Oh! Hello! Heard so much about you!\" She smiled. I smiled too, thinking about the layers of potential meaning in her sentence.\n\n\"Yes. It's nice to be here,\" I said.\n\n\"Very nice to meet you,\" the principal said, peeking her head from around the corner. As she shook my hand, I thought for a moment she was gaining a visual on the dirtbag absentee mother. _Fine with me. Take a look._\n\nI had never signed the attendance forms on the front desk. I didn't know where to park, or where their classrooms were, or what the hell \"car line\" was. I didn't know about cubbies and folders. I had never met their teachers.\n\nAnd there were other parents _everywhere_ , specimens in capris and cargo shorts who had been navigating Scholastic order forms for a long time. Moms and dads who had never missed a parent-teacher conference, let alone a kindergarten graduation, and certainly not because the bottle rendered them useless. But I didn't feel shame, even when the best of them strolled past me. I was too busy enjoying the scenery, a thousand things I'd never seen before.\n\nUntil then, I had spent my entire adult life drunk, in the aftermath of drunk, in the pursuit of drunk, in the avoidance of drunk, or in the precarious hell of in-between drunk. In this condition, life is not \"dealt with\" or \"worked through\" or \"handled,\" even during periods of \"sobriety.\" There is no sober. There is occasional physical detoxification, but there is no mental function of the un-addicted. Or there is, but it's on the primal instinctual level\u2014as in, eat, pee, shit, work, bathe.\n\nBasically, I was alcohol's bitch.\n\nThis makes for a rather exciting life if one manages to not die and stay sober. The world around me was cast suddenly in Technicolor, and for a moment, the dull grays turned into vivid light.\n\nI was a child those first few months of sobriety. I was a child skipping through the halls at recess, doing cannonballs into a swimming pool, or crawling into bed on Christmas Eve. I was a kid in fucking Disneyland.\n\nMac and I saw each other often, but I wasn't toiling for our reunification. Instead, I was chasing sobriety like a desperate lover. I knew I had no ground for demands on Mac anyway, though I yearned for our family. I remember watching him with our babies in those early weeks of sobriety\u2014the way he hoisted Rocket onto his shoulders, and brushed Ava's hair\u2014 _this is my family_ , I'd think. And it was so fucking beautiful. The way he held their hands and tied their shoes and sat with them at night on the couch for as long as they wanted. I was watching a father with his kids. How pure it felt after the life I had been living. How warm, like coming home, like crawling into your own bed after two weeks in motel rooms. I wanted in with my whole heart, but it was not for me to decide.\n\nMac held me at a safe distance, dropping by sometimes, meeting me for coffee. He came by my mother's house on my birthday after I got sober. He sat at the other end of her dining table, observing me quietly. He had picked me up a couple of Grateful Dead patches from a record store in Monterey. \"I thought you could sew them onto your hoodie,\" he said. He didn't stay for cake, and I felt a little slighted.\n\nWhen I asked him about it, he said, \"I don't know if I want anything to do with you, Janelle.\" I understood that. I wasn't sure I wanted anything to do with myself.\n\nTwo months later, Mac agreed to take the kids with me to Half Moon Bay, a heavenly coastal town near San Francisco where the cypress trees and dunes turn life into the soft roar of merciless waves. When we were almost to Highway 1, I mentioned I couldn't see well through the dirty windshield. Mac suggested with brilliant nonchalance that I \"use the washer fluid.\" When I curled my lip at him to say, \"God, that's rich,\" he informed me he had added fluid to my car when we stopped for gas. \"Holy shit!\" I said, forgetting entirely this was a typical adult activity.\n\nI clicked it on, and it worked. I clicked it on, and it _worked_! I turned the dial and water came out, and the wipers moved, sweeping the bug guts and dust right out of my way.\n\n\"Hey kids! It's _raining_!\" I yelled. I thought I was damn funny, and so did they, and I remember our delight. I remember feeling capable and alive and real. I had a car with working parts. I had a family. I had a body and air to breathe and I was free to drive to the beach and make stupid jokes and my kids would laugh, because I'm their mother and they think I'm wonderful and funny and nobody cares.\n\n_Nobody sees me. I am unseen. I'm just one among all of you. I can use a lawnmower on a Saturday morning. I can punch the numbers in the fancy school door. I can show up for parent-teacher conferences\u2014but more importantly, I can miss them sometimes and still look you in the eye, because I really did forget. I was not taking lines of cocaine off the hallway mirror and remembering the meeting in the quicksand of the morning._\n\nI knew when he told me about the washer fluid that he was going to try, that he was still willing to help me, to show up for us again. He had told me once, \"Janelle, I will always help you again,\" though I thought perhaps that promise had expired. That we, I suppose, had expired.\n\nWe never formally reunited. That was not our way. After we met for the first time, we were _together_. When were apart, it was only in body. And when he said, \"I don't know why, but I really believe you're different this time,\" I knew we had never left the domain of our love. Although it was twisted up and weird and unspeakable\u2014built on bad decisions and slightly better ones\u2014it was ours.\n\nAnd for people like us, who have shivered in the emptiness of stone-cold addiction, the warmth of a family bed need not be discussed. Its existence is enough. One simply crawls in.\n\n\u2022 \u2022 \u2022\n\nGood News Jack and I met three or four times a week either in his backyard or in the tiny office at his house, where I would sit facing him alongside the family computer and an abandoned treadmill. He would wedge a chair under the doorknob while his five-year-old daughter body-slammed it trying to get in, and I tried to understand the strange things he'd say, like: \"If nothing changes, nothing changes,\" and that I was \"wrong until further notice.\"\n\nUnder normal circumstances, I would have told him to go fuck himself, but I was fresh out of comebacks. So I took actions I may or may not have agreed with, and often barely understood, because Jack stood in front of me a free man, and I knew he drank like I drank and for the same reasons. I would have followed him anywhere to gain what he had.\n\nAnd yet, we seemed to speak little of alcohol, and he only mentioned the \"tools\" I learned in rehab to cackle about how they \"never work for chronic alkies like us.\" He had no interest in \"phone lists\" (which, for the uninitiated, are lists of phone numbers of sober friends we are supposed to call when we're about to relapse. The idea is that we never leave home without the list, and just when we're about to catapult ourselves into sheer disaster, we call somebody to talk us out of it. My problem was I never believed I was about to catapult myself into sheer disaster. I believed I was going to _prove I could drink like a lady this time_ , whatever the hell that meant. While I did not see a need to trouble my rehab friends with my own innocuous drinking, I did occasionally wonder if I should give them a jingle to help them learn how to drink better).\n\nGood News Jack wouldn't even feign interest in my \"relapse triggers\" or complex emotional pain rooted in the tenuous attachments of my inner child. He did, however, grow giddy at the prospect of dissecting at an atomic level the manifestations of my self-centeredness. When I'd protest, he'd say, \"Do you really want your life to change, or do you want to remain an asshole with better consequences?\" The most shocking part was I found myself admitting I wanted to remain an asshole, and in fact had never heard myself articulated so succinctly.\n\nSaying it out loud felt like cold water on a hot day, some longed-for honesty in a parched mouth. And when I admitted such things, his face would beam, and he'd shout, \"Yes! Janelle! That is the honesty you need to stay sober. No more self-delusion. No more bullshit. You're a selfish fuck!\" He'd roar with laughter, as if being a selfish fuck was the happiest thought he'd ever entertained.\n\n\"That's what's killing you, you know? All of us\u2014we're all whistling past the graveyard, sayin', 'Well it ain't gonna happen to me.'\" He took a drag of his Marlboro Red while I visualized myself in all those rehabs\u2014the throw blanket, the Nikes, the promises to my children\u2014whistling how I was \"fine,\" pretending, posturing, but truly believing my own lies. _I was_ _in_ _the graveyard setting up a Buddha statue and judging tweakers._\n\n\"So what do we do?\" I was on the edge of my damn seat.\n\n\"Recognize that you, _you_ are the problem. You! You're the problem! You've always been the fucking problem! Isn't that wonderful?\"\n\n\"I have never heard anything more wonderful, Jack,\" I said flatly, rolling my eyes.\n\n\"Because there is power there, you fool. Power. That's what you need, what you've always needed, right? You aren't a victim! You try to arrange your life and control everyone to fix your inner self. It never works. You try harder. People hate you more. All the while you're looking at the damn problem!\" He laughed again.\n\nWhen he spoke words like this, it sounded like a symphony, as if every word was a note falling one by one into a song so painfully beautiful I would sometimes sit silently across from him with tears falling down my face. I couldn't explain it\u2014what his words meant, how they pulled the truth from me, or how they cracked open the catacombs of my soul. They seemed to send bright light into my most hideous corners of shame and deceit. He was right. It felt like hope.\n\n\"You never said what we do,\" I said.\n\n\"Get the fuck out of yourself, Janelle. That's what we do.\"\n\nIn therapy, it had always been about me\u2014my childhood, my parents, my thoughts, my goals. But when I told Jack my perception of things, he'd say, \"You know, Janelle, you could just jump off the crazy train rather than riding it all the way to the bitter end.\" And then he would say that thing about the ducks and fire hydrant and tell me _again_ to think of others, which was another shoddy plan as far as I was concerned.\n\nOne day, when the gleeful dust of new sobriety had settled, I was trying to make Ava and Rocket sandwiches but was somehow failing. I stood in my mother's kitchen behind a cutting board, staring at bread and turkey, feeling like I might explode from restlessness. I told the kids I'd be right back and went outside to call Good News Jack.\n\n\"I just feel like shit, Jack. I do not feel 'good' at all.\" I was angry and accusatory, as if his sobriety plan had already let me down.\n\nWithout hesitation, he answered, \"I never promised you'd feel good. I promised you'd never have to drink again.\" I thought about that, about not feeling good and simply dealing with it, about diminishing the importance of \"feelings\"\u2014as in, sometimes you feel like trash, and you carry on without fixing it. He was a well of revolutionary information.\n\n\"What the hell are you doing right now anyway?\" he asked.\n\n\"I was supposed to make the kids lunch, but I freaked out.\"\n\n\"Go fucking make them lunch and stop thinking about yourself.\" Then he hung up on me. He had a way that really made me feel loved.\n\nOver time, I realized Jack had given me a job: be in the world, try to be of service to others, and clean up the damage of my past. When I hung out there, I lived in the sunshine. I lived in the knowing that I had been rearranged, and the booze obsession had left me, and I was free to go about the world, to live like any other semi-functioning whacko on the planet.\n\n\u2022 \u2022 \u2022\n\n\"So, did your daughter get into G.A.T.E.?\" she asked, stopping me in the hallway as I headed to Ava's second grade classroom.\n\nI had recently learned that standard behavior around Ava's school was to move from the fancy private school to a fancy public school when the child reached third grade, to take advantage of \"Gifted and Talented Education,\" which was where all the superior kids were headed. The woman standing before me was their captain. She had adopted _her_ children from another country, which also made her captain of the white liberals. She was around fifty, with black, gray-streaked hair, and owned an impressive array of Tevas. She had dark brown eyes and a brow that threatened to destroy me if I said something out of line. Sadly for me, I rarely knew where that line was.\n\n\"No, she didn't,\" I said, shrugging.\n\nShe jolted, tilting slightly forward and raising her eyebrows in shock, as if I had just told her my dog was bleeding to death in my car. After the initial blow wore off, her face registered thinly veiled disgust, but it wasn't until the pity settled into her eyes that I felt her derision, and thought with a stab that maybe my daughter wasn't as smart as I thought she was. I recalled the day my mother took her to the exam, and how I didn't go with her, and I wondered if I should have. I considered having her retake it, and then remembered I didn't care.\n\n_My kid is fine. I am fine. She's smart, and I'm sort of smart. For whatever reason, she didn't pass the test._ I looked up and shrugged again.\n\n\"Oh, well. That's too bad,\" the lady stammered, straightening her North Face vest.\n\n\"Is it too bad?\" I answered, smiling, before I turned and walked away.\n\nI signed my daughter out, chatted with the receptionist, held my girl's hand on our way to the car to make sure she was safe, and all these actions felt like tiny miracles. I gave a death glare to the woman when I saw her in the parking lot, because I was sober, not Jesus.\n\nOn my way home, I realized with a sort of stunned despondency that the only purpose of that exchange was for Captain Gifted to determine if I was as respectable a mother as she was. I had temporarily forgotten adults acted like that, although I should have recognized it immediately as the \"healthy person\" version of \"Who's the sickest in the room?\"\n\nAfter that day, I began to notice that mothers were an extremely strange bunch, and for my own amusement, I began categorizing them based on money, parenting choices, and politics. Captain Gifted was Type I: The Put-Together Enlightened Mother Who Is Definitely Better Than Me. This type of mother is a living spreadsheet. She's read all essential theories of parenting, her spice cabinet is alphabetized, and her bottom sheets are folded in the linen closet. She probably has a PhD and drives a Prius. She definitely composts and wears a lot of fleece purchased from REI. She looks at motherhood as a complex responsibility to be calculated and controlled to achieve optimum outcomes. If I were _as proficient as she is_ , my outcomes would also shine like beacons of hope in a dark forest. She will remind me of this frequently, but not directly, through earnest and heartfelt \"suggestions,\" which she will view as charitable and I will view as a direct assault on all that is good in the world.\n\nThen there is Type II: The Why Is Everybody Making a Big Deal Out of This Mother, who stays at home with her two (possibly three) children and has any situation handled in her skinny jeans, strategically undone hair, and never-chipped pedicures. She met her husband while living abroad and posts a lot of photos of herself drinking red wine on a patio. She says things like \"I make my own cashew butter\" and \"I never stop eating,\" which is confusing, because I wasn't as skinny as her during the skinny days I reflect upon with great yearning. One imagines this is made possible by her deeply spiritual attachment to yoga combined with a diet of kale smoothies drunk out of mason jars and vegan gluten-free zucchini bread\u2014which she brings to our playdate at the park, where we sit on a handmade vintage quilt while she coos gentle directions to her children in one of the Romance languages.\n\nAnd then there is Type III: The Children Are Everything, Harried Busy-Bee Mother. Under no circumstances may I drop an F-bomb in this one's presence. She's on every PTA, preschool, and church board of directors, and not to be helpful. She's in it to win. _You think this preschool runs itself?_ she seems to ask. _Lady, they're just crafts_ , I think. \"Organize the pipe cleaners, bitches!\" she screams, but only with her grim eyes. She will spend a solid hour discussing whether the teacher's gift card should come from Target or Nordstrom or Starbucks. At minute two, I will want to stand up and scream, \"NOBODY FUCKING CARES,\" but I cannot, because she's always smiling. The smiling makes me wonder if she locks puppies in closets or sniffs Krazy Glue in the evenings. Then I feel bad again for thinking these things because she's so goddamn \"nice.\"\n\nAlright, there are more than three. There's the Super Political Wounded Mother, who's been gaslighted since birth and complains constantly but can't change on account of internalized misogyny. She barely makes it through the day because _life is harder for her than the rest of us_. Her degree in gender studies compels her to suggest at every meeting that we \"unpack our privilege,\" but later she will instruct people of color on Facebook how they can fight oppression more pleasantly. Her children are very deep. The entire planet wants to pass her a note reading: \"Hey lady, stop talking. Or at least move to Portland.\"\n\nIncidentally, I am grateful I was not yet on social media when I was drinking.\n\nThere's also the Aloof Badass Mother, who doesn't give a damn about any of this nonsense because she's drinking Pabst Blue Ribbon and getting tattooed and _pretty much never showing up to anything because it's all too mainstream._ She and her partner are about to sell their house to backpack around Thailand. She has a Tumblr and a vintage typewriter. She is polyamorous and attends Burning Man in feathers, atop a unicorn bicycle.\n\nAnd, of course, there's the Earth Crystal Sage Mama, who manages to incorporate the words \"womb,\" \"goddess,\" or \"menstrual blood\" into damn near every conversation and once told me that cutting the umbilical cord is a form of \"violence.\" She has a shamanic weaving teacher, smells like Dead shows and garlic, and truly believes there is an ancestral warrior leading her life on a daily basis. Her kid is named Lotus Reef and never wears shoes but can absolutely kill it in Hula-Hooping.\n\nWhether or not these categories existed in reality, what I saw was that I fit _nowhere_. I lived in a house with linoleum floors and spent my afternoons with an ex-cocaine addict so I could remain among the living. I had trouble keeping the floor of my car visible at all, let alone composting kitchen scraps. I couldn't read the group emails from the parent associations because nobody understood the \"reply all\" function and I got bored by email number nine in the string of nine thousand.\n\nBut when I made jokes about the email thread, the other mothers looked at me like I was a dead bird on the porch, and then a week later I'd realize I missed an event because in email number 347, _the event was clearly outlined_ , but I didn't read it because it was past email number nine.\n\nI didn't fit in with the \"read the books and learn and do it right\" mothers because I read things, implemented them in a fury of excitement and staunch devotion, then forgot about them entirely three days later. I'd look at an abandoned chore chart as it leaned sadly against the wall and think, _If only_. Then I'd yell at my kids to do some chores because \"I can't live this way anymore!\"\n\n_Maybe we should start another chore chart_ , I'd think.\n\nOf course I didn't fit with the Etsy hipsters either, because I wore Target maternity jeans six months after I had the baby. I knew I was wearing the \"mom jeans\" we're all taught to avoid like Red #2 dye and simple carbs, but _I never got around to buying other pants._\n\nBut even when I tried to fit in with other mothers, my mouth would ruin it behind my back. One little ill-timed expletive and the whole thing would go to hell. We'd do fine at the sandbox until I dropped a \"fuck,\" thinking it added to the moment, then feeling the wrath of a woman who believed nobody should swear on hallowed kid ground.\n\n_Oh, they couldn't hear me_ , I'd think, and mumble \"Fuck\" again, only this time in my head, because _I am in trouble now and I made it weird._ _Do I attempt to salvage this relationship? Nah. Move on. It was based on lies anyway._\n\nMotherly small talk was complicated because we were not actually talking about the thing we were talking about. We were both supposed to know this and stick to the rules, but I've been bad at that since the sixth grade, when my teacher duct-taped my mouth shut because I wouldn't stop talking.\n\nIt didn't even occur to me to censor the audible version of my thoughts _at all_ until I had a certain epiphany at age sixteen, when I realized girls were getting the boys because they _act differently around them_ and _you, Janelle, you're still admitting you prefer the Allman Brothers over Nirvana and it's 1996_.\n\nI tended to say the thing that was true as opposed to the more palatable alternative, and this really concerned people, particularly when it involved raising America.\n\n(Mothers don't admit that. Especially at the park.)\n\n\"What exactly do we have here?\" they seemed to ask. \"Do we eat it? Kick it? Counsel it?\"\n\nProbably.\n\nI knew the game. I knew we were supposed to talk-not-talk about whose partner was the best and wealthiest, whose preschooler was the smartest, and whose baby was the most advanced, but I couldn't play. I tried a few hundred times, but eventually I focused on getting it over with by saying something like \"Yeah, my son is three and barely talks,\" or, \"My daughter refers to her vagina as a 'wiener shooter.'\"\n\n_You win._\n\n_Your kid is smarter. The end. And yes, you have more money and, yes, more education, and your baby, I know! Your baby crawled at three months and talked at six months and slept through the night at twelve minutes\u2014what a miracle! Must be superior genetics._\n\n_Now can we just hang out? I'm bored._\n\n_Can we talk about the way these kids give and suck life by the minute, day by day, and how sometimes you're sure you've ruined your life through the reproductive process, but five minutes later you're in tears as you pack the newborn clothes into the giveaway box? The way the years mock you with their passing, lull you into the safety and surety and vague comfort of knowing your children will always be small, until you realize it will soon be over? Done. Your time is done. Sorry. You should have paid closer attention. Should have held on tighter. Try not to fuck it up with the others._\n\n_But you already are. You're always already fucking it up. Can we talk about that?_\n\n_Let's not talk about how we all became better versions of ourselves the day we became parents, and, please, would you stop pretending you did? Because your holier-than-thou shit makes me worry you watch dinosaur porn after the kids go to bed. Your steadfast focus on seasonal cupcakes and organic kombucha concerns me. Look, I've got some too. I know all about gut flora. But please. Is that all there is?_\n\n_You didn't become some G-rated version of yourself and you know it._\n\n_But let's not talk about that either. Let's make a couple of jokes, chill out, and make fun of \"man colds\" or \"wife colds\"\u2014do those exist?\u2014and how hard it is to get kids to do chores when they're bickering and awful and you think, \"I'd rather do it myself than listen to this!\" No, but seriously, husband, you're fine. Why are you on the couch whining? Call your mom. Maybe she cares. I've had a cold for nine weeks, asshole._\n\n_Let's talk about that. Let's talk about sleeping babies and newborn breath and how I once wanted to kill myself after I had my baby because it seemed the only or best solution to the pain._\n\n_Maybe we won't talk about that at the park. Let's talk about something else._\n\n_Let's talk about coffee. You can talk about wine. I don't drink wine because I drank all the wine and alcohol has a tendency to turn me into a homeless person, but we can talk about it anyway because I understand. We don't need to pretend._\n\n_Let's talk about how we haven't seen the floor of our cars in months, and how it smells oddly, always, of apples (but where are the actual apples?), and how the sound of nonstop kid chattering makes my head spin, but I still lie down at night wishing I had listened because they won't be chattering forever._\n\n_Let's talk about my son, the quiet one, who squeaks and yells and screams and runs, all the time. I get so frustrated with him for the stomping and incessant movement and it's so hard to listen to him talk sometimes. He can't think of the word because he's dyslexic. I know he is. His father and grandfather are dyslexic. What kind of shit human gets impatient with a boy with dyslexia? What kind?_\n\n_Let's talk about that. Let's talk about the fact that I am that mother. I am already that mother. I was the worst one, too. I let my children go one morning, and my boy, he_ _was only sixteen months old. Don't you think he wondered where his mother went? Yeah, I hate me too, lady. I hate me too for the things I did, and when these memories come, I throw my head back and forth like a motherfucking lunatic trying to shake them out of my brain. I think perhaps I'd rather die than live with that shame._\n\n_Maybe we won't talk about all that. That's a little heavy._\n\n_Let's talk about how sometimes, by the end of the day, I understand\u2014just a little\u2014how parents snap and hit their kids. I haven't hit my kids. I would not. But sometimes when combined with lack of sleep, and my husband working six days a week, and the times I drove intoxicated with my child in the car and poisoned breastfeeding, nearly overdosed in a trailer, and my sagging belly and gray roots, self-hatred, and joy that I'm free (from alcoholism, today), I wonder if I may break, just fucking break, so I come here to the park, and I sit next to you because you're a mother like me, right? And we can talk about it. We can talk and talk and talk._\n\n_\"Is she sleeping through the night yet?\" you ask._\n\n_No._\n\n_\"Oh, where is she?\"_\n\n_In my bed, with me._\n\n_\"Oh, you co-sleep?\"_\n\n_I guess._\n\n_\"My baby slept through the night at two months because I did sleep training. Why haven't you sleep trained?\"_\n\n_Oh yeah. Okay. Talk to me about that, I guess. If that's all there is._\n\n\u2022 \u2022 \u2022\n\nIn August 2010, when I was seventeen months sober, our third baby was born, in a water birth at home, as the morning sun splintered across the faces of Mac, my mother, children, and midwives. They were circling me as our ten-pound, dimpled baby Georgia with a cleft chin and bald head opened her eyes against my chest, and I watched her body flood pink with our blood, from the center to her fingertips. Just like Rocketship.\n\nAt night, I would hold her in the crook of my arm and see only love, and I'd wonder how it could get like this. It was just me nursing her in the gray dim of a little nursery I decorated with Ava and my mother when I was pregnant, in the house Mac and I bought together, _where all our children live, and we are a family_.\n\nHow wholesome life had become, not in that church way, not in the way that made me feel unworthy, but in the meatiest, grittiest way.\n\nNot beauty in the clouds. Beauty on the ground. Beauty in my hands.\n\nSix months to one year to one year and a half sober. _I should be in a gutter somewhere, but I'm here._ With the thought, while I rocked, I'd turn my head up to the ceiling and close my eyes as the warmth of the day rushed over me.\n\n_I am none of those other mothers_ , I'd think. _I am all of them._\n\nI realized we are all a bunch of fakers. We've got too much past to remember, too much on the line to forget. We become some mother. We show up. We work and drive and love. It all feels like a tiny miracle. It all feels so boring we could puke. For some of us, that becomes enough, and we don't have to dance anymore.\n\n_Maybe I don't care quite so much about being better than you. Sometimes I want to be better than you. But in the end, I have nothing left to prove: to you or myself. I have no polish to fix what I am. I am a woman who lost her children, and I am the woman now_ _standing here in this hallway bright-eyed and motherly while you size me up. You don't even know what it means for me to have this stupid fucking G.A.T.E. interlude with you._\n\n_Sure, I hate you, but I love that I get to hate you. I love that I get to be just another mother pissed off because a woman in Teva sandals condescended to her._\n\n_Because look at these wiper blades. All you do is turn this dial and swoosh\u2014problem solved! Do you not see how funny that is?_\n\n_Don't you think that's cool?_\n\n_No? Alright. Guess it's just me._\n\n_And that's fine, because I'm just happy to be here._\n**13**\n\n# [**Failure That Isn't Funny:  \nSober Edition**](TOC.xhtml)\n\nIs that your boy up there?\" I recognized the tone in the woman's voice as that of the inordinately concerned. Though she was smiling, I could sense I had offended her, and would soon discover how.\n\n\"Yes,\" I answered, joining the game of fake civility.\n\n\"Oh, okay! Well\u2014I just have to say! It makes me nervous! Seeing him up there!\" she said, pulling her shoulders up and making a face like, \"I just can't help myself! I love all the children!\" She seemed to punctuate every pause with unwarranted jubilation.\n\nI spun through the mental Rolodex of the situation's features to determine which one of us was more irrational: _He is standing on top of kid-level monkey bars, over a sand pit. If he fell, he would fall into sand, notably soft, and if he fell and managed through some miracle to hurt himself, it would be a broken or sprained ankle or wrist at the very worst._ There wasn't even a damn rock around.\n\n_Sometimes kids need to climb_ , I thought. _Sometimes I need to let them climb so I can sit on a bench and play on my cell phone, pretending I don't have children_. Of course I didn't say that. She was quite obviously not the type of person who enjoys that level of honesty. Instead, I tried: \"Well, I figure if he can get up there, he can get down!\"\n\nI smiled and shrugged my shoulders as if to say: \"Kids will be kids!\" I was attempting to connect with her, to be two mothers at the park, although at that point I would have settled with two humans sharing a planet. I thought my soaring charm might bring out the best in her.\n\nInstead she looked at me as if I had just pulled out a meth pipe and hit it.\n\n\"Do you have _insurance_?\" __ she bellowed.\n\n\"What?\" I asked, thinking, _For sure she's going to kill me tonight while I sleep._\n\n\"Health insurance. Do you have health insurance for when he falls and hurts himself? KIDS GET HURT, YOU KNOW.\" She said the last part as if she were leading a rally in the protection of all the children of the world who could possibly get hurt.\n\nI lifted my eyebrows and mumbled, \"Yep. I do,\" turned around and sat down, where I proceeded to think of all the piercingly witty things I should have said. \"Actually no, I prefer to duct-tape my kids' injuries at home in the garage using a light dose of heroin to kill the pain.\" Ultimately I decided I should have just gone with, \"Do you ever wonder why nobody likes you?\" And then I should have stared at her. For a long time, until she grew so uncomfortable she cracked.\n\nI glared at her from behind my sunglasses while my rage turned into an ill-defined shame. I watched Rocket play while the stranger's words repeated in my mind\u2014\"kids get hurt, you know\"\u2014and seeing Rocket stand up in the air sent my mind spinning back to a summer trip I took with my stepmother and friends to Lake Tahoe when Ava was eighteen months old.\n\nOn the way home, we had stopped at a restaurant with a full bar along the Truckee River. It was one of those funky Northern California motel\/restaurant\/bar places right on the river\u2014an all-in-one establishment where you could eat, get drunk, sleep, and buy a small carved black bear that says \"I left my heart in Tahoe.\" The patio and open-air bar were next to the restaurant, and above it was the motel, with stairs leading from the bar patio to the rooms. We had lunch in the clean high-altitude sun during one of those days that feels endless and so beautiful you could die right there without a single regret.\n\nWe had cocktails on the patio and it was all quite pleasant until somebody started pointing and yelling, \"Oh my God,\" and then the whole restaurant was gasping and pointing, and I looked up to where they were pointing and saw my baby girl standing on the roof of the motel\u2014a flat roof that extended from the motel, over the restaurant, to the roaring, maniacal river. From where she stood, wearing a little red gingham apron dress with a white blouse underneath, her favorite scuffed red leather Mary Janes and white lace socks\u2014from where she stood gazing cheerfully at the people below, she was a few feet from death. If she took five or six steps to her right, there was no railing or edge or prayer that would stop her from tumbling down into the white water slipping and flipping over giant granite rocks.\n\nI remember the blur of running up the stairs, three at a time, and stopping at the top, at the gaps between the wooden rails she had crawled through. She was in the middle of the roof, outside my reach\u2014five feet from me and five feet from the edge. What I knew was that she could not run.\n\nIf I yelled or crawled onto the roof, she might bolt, startled, or thinking we were playing a game of \"catch me,\" as we had done so many times before. I wanted to scream, to beg for help, but I couldn't frighten her\u2014and yet, if I spoke softly, she might ignore me.\n\nThe terror pushed through me from my core\u2014not in thought, not in sequence or data, but as a fiery heat in the center of me, rushing out into my arms and feet. My voice rose steady and calm, with all the power I had behind it: \"Ava. Walk to me now.\" I smiled desperately to pull her to me. _Oh child, my dearest baby, please God come here._\n\nShe walked immediately to me. As soon as she was within my reach I grabbed her, pulled her through the railing, and held her to me as my eyes burned. I rushed down the stairs and out the side door so nobody would see my face.\n\nBack in the car, I didn't speak of it. If someone would have tried, I would have flatly refused to address it. But I knew I had almost lost my whole life, right there, and my child had nearly lost hers, and it was my fault. Again. I was drinking. I was not paying attention. I was chatting with strangers in the buzz of the day and sunlight.\n\nRecalling that moment, of her little body hovering in the sun above an angry river, and me in my flip-flops, tan and half drunk, pulling her to me in burning desperation, I squeezed my eyes shut and flinched, noticeably. I knew I must have looked freakish sitting on a park bench, shaking my head with my eyes shut. But it was involuntary.\n\nI drew a deep, quick breath at the crack of the image against my heart. When those memories came, they started like knives in my mind, and then sat like boulders on my chest. Like a thousand pounds of granite grief. It seemed they would crush me, so I called Good News Jack, my only source of new ideas.\n\n\"How do I live with the memories of what I've done to my kids, Jack? The images. How do I fucking live with the images?\" I told him how twenty times a day they jumped in and out of my mind to thrash and stab and mutilate, and I told him how at night they moved in and set up camp. And how with them came a tidal wave of shame, of agonizing regret, of holy time lost, to never be regained, sweet innocence doused in kerosene\u2014ignored, unrecognized, and by my very own hand. _How fucking could I._ That was the part I could not face. That I, I was that person. And I didn't see, but then I did, and then I could not look away.\n\n\"We call those nightmare memories. All sober alcoholics have them. Lot of people drink again over them.\" He spoke calmly, and I imagined he was out on his front porch, smoking a cigarette and watching his kids ride their bikes.\n\n\"I can't live knowing what I did,\" I said. \"I can't.\"\n\n\"I know you can't. That's why we're not relying on you anymore. Remember what I said? If God only gave you things you could handle, what the fuck would you need God for? You can't handle anything. Look at yourself.\" His words were irritatingly factual.\n\n\"Why are you always so full of good news?\"\n\n\"Janelle, the way you will survive is by using your experience to help other alcoholic mothers who did the same shit to their kids. _That's your job now._ We don't care what YOU think about your past. If you really feel bad, stick around and repair it, with your children and the world.\"\n\n\"But I'm a terrible mother.\"\n\n\"Of course you're a terrible mother,\" and he cackled in a way that reminded me what a long and insane road he too had walked.\n\n\"But you aren't gonna get any better by talking to me,\" and then he hung up.\n\nI replayed Jack's words as I kicked a sheet of sand over the concrete, watching my foot slide around, making little mounds and knocking them over again. From that moment forward, every time one of those memories arose, I clung with all my strength to the idea that I might help another alcoholic someday with what otherwise seemed entirely meaningless destruction.\n\nThis was my version of \"positive self-talk.\" It didn't erase the self-loathing. It didn't melt the regret. It didn't soften the blow of revulsion, the terror I felt at the mere idea of what could have happened or who could have been hurt, but it brought a microscopic surrender, another tiny letting go of my need to understand, control, and find relief\u2014it gave me just enough to trust that someday, something will make those years worth living.\n\nJack's voice ran through my mind a hundred times a day. \"We don't hang out in morbid self-reflection. It's self-pity with a better polish. Remember the duck and the fire hydrant? Fuck your beliefs. Focus on what's in front of you.\"\n\nSo I looked up at Rocket and watched as he balanced with his arms stretched out. He was concentrating on his feet and teetering on the corner of the monkey bars, and I wanted to tell him, \"Be careful, son,\" but I didn't, for fear it would distract him, knock him off course. Plus, I knew he was alright, so I smiled, and nodded at that lady too, because my son was safe, and she would never know what that means.\n\n\u2022 \u2022 \u2022\n\nOne hour after my serene epiphany in the park and deep gratitude for my sobriety, I walked into my house with a screaming baby on my hip, two bickering children, and approximately nine bags of unknown origin hanging from my body. I set it all down in the entryway, shut the door, and read a text from Mac letting me know he was stuck in traffic and wouldn't be home for another hour, at which time I remembered there was some sort of tee-ball nonsense that night.\n\n_Of course he's not here_ , I thought, feeling the weight of our offspring, a vague loneliness, and resentment as I walked into the kitchen and saw a day's worth of dishes strewn across the counters and toys throughout the living room.\n\n_Good God._\n\nMy actual face ached from exhaustion. My cheekbones throbbed and my eyes stung. The dog scratched at the back door. Georgia was hungry and had just pooped. Ava and Rocket had ignored me again, and I was alone in the maelstrom of kids at dinnertime. As I looked around at the laundry on the couch, the smear of jam on the floor, and the pile of Mac's tools on the kitchen table, I felt that old familiar rush, the frantic sense of being out of control\u2014and that if I didn't fix it, it would all collapse.\n\nThe next thing I knew, I was yelling in my children's faces, \"What is wrong with you? Why can't anyone HELP ME?!\"\n\nI saw red, as if a veil had fallen over me, casting the whole room in shades of fury. As I screamed, my mind's eye hovered in the corner of the room, staring down at the woman who was so mad she was spitting. \"Stop, Janelle, this is wrong.\" The voice nudged, poked me in the ribs from a place that knew better, and I knew the voice was right, but I could not stop. The more I yelled, the more I wanted to yell, as if the words weren't doing a damn thing, so I grew madder and louder, and the words seemed to get closer to exhausting the anger in me, but they never quite worked.\n\nThey did nothing but explode in quick, useless bursts.\n\nStill, I thought I was going to get through to them, that my anger would change something. I thought I'd get some power and convince them to improve. I stormed around the house in a tantrum, barking orders. In glances, I noticed my children avoiding my eyes. I noticed their silence.\n\nBut I was _committed_. I was _angry_.\n\nOr I simply could not find my way back.\n\nThe moment Mac stepped across the threshold in his five-year-old canvas beige overalls, patched with denim blue squares at the knees, all three children were squealing in delight and hanging from his arms in unchecked celebration of their _real_ parent. I watched as they rejoiced in his arrival, while, I imagined, lamenting mine. _Well what do you expect, Janelle? You're such a goddamn yeller. But that motherfucker, he's never here. If he were here like me, he'd yell too._ But I knew our children would never see that. I felt wildly unappreciated and unseen, as if I were doing the grunt work while he stole all the glory.\n\n_Now I suppose he's going to want to shower, as opposed to taking over parenting so I can lie on my bed with the bedroom door shut._\n\nSure enough, off he went to the shower after giving me a kiss and eating something infuriatingly healthy, like an apple or raw almonds. While I turned the taco meat in the pan for dinner, I wondered why raw almonds were about the last food I'd reach for after a long day, and determined that if I incorporated more Mac-style food choices into my diet, I too would stand lean and muscular before the bathroom mirror.\n\n\"Hey, Janelle! Come here!\" He appeared to be hollering from the shower, and the playfulness in his voice gave me a rush of desire and dread, because damn he's attractive but also, _No way in hell am I having shower sex right now._\n\n\"What's up?\" I asked, leaning against the bathroom door, thinking there was nothing I wanted more in that moment than for _one more person to need something from me._\n\n\"So,\" he declared, \"I bought some go-kart frames from Phil.\"\n\n\"You did what now?\"\n\nAssuming I didn't understand the actual sentence as opposed to the content, he repeated himself, explaining that he had spent $600 on metal frames, which he and Phil would equip with lawnmower engines, and then the kids would ride them around in circles on dirt tracks. For fun, apparently.\n\nPhil was a man who lived down the street and looked like a garden gnome. He was hands-down my least favorite garden gnome because he evidently had _very little going on_ , and would invite Mac over every few days to stand in his garage and stare at things while discussing, I now knew, lawnmower ride-along toys.\n\n\"You fucking did _what_?\" I understood the words coming out of his mouth, but I could not comprehend his decision to voluntarily add \"build go-karts\" to the never-ending task list of our lives.\n\nI uttered confused bursts of total derision, then walked out of the bathroom, because he had turned off the shower and I couldn't bear the sight of his face.\n\nI questioned how bare metal frames that might someday turn into \"karts\" (with a \"k\")\u2014which, by the way, would never happen, because we could hardly manage to pick up the dog shit from the backyard, let alone complete arbitrary entertainment projects\u2014became his area of focus while I contemplated my recent re-enrollment in graduate school, work, nursing a baby, driving two kids around the county a few times a day, bills, the house, the growing clarity of Rocket's dyslexia, the outgrown clothes in the kids' dressers, the dog's training, and the emotional and psychological well-being of our almost-tween daughter.\n\nI dropped onto our bed and closed my eyes, leaving the ground beef simmering on the stove unattended, thinking, _Maybe, just maybe, if all goes well, the fucking house will burn down._\n\nI was two years sober, and while Mac and I worked to build a home, a chasm carved itself between us, growing more massive as my head cleared, as I got to know him, and myself, as sober humans. Life was opening for me in a thousand directions just as my marriage contracted into thankless redundancy.\n\nI had barely known Mac without the dizzying balm of evening cocktails, and I certainly had never tried raising a family in such aridity. _What did it matter if we were horrible together before?_ _I was drunk anyway._\n\nBut now, sober, I felt the rift completely. The way his interests and sense of humor were not necessarily mine, the way his ambition and concerns were in conflict with mine, and the way he spent money, money I assumed he had stashed away. The way we made love, the way he glared at me on Saturday mornings when I suggested we do chores, as if I were a nagging mother.\n\nI felt a deep and terrifying loneliness in our marriage, a descending awareness that I was becoming a servant to my family and nothing else, and I would spend the next twenty years in that condition.\n\nI often thought of our \"wedding\" on the courthouse grass on that cold, gray day. I thought of the wedding dress picture I had clipped from a magazine when I was eight years old, and I thought of my friends' marriages, the way their husbands surely discussed monetary decisions with them and acted like grown-ups, investing in stocks and buying property and getting medical degrees. I assumed they made all kinds of choices together and were honest and clear with one another, and the thought made me think back to Mac talking behind my back to his mother and sisters (they were, for a time, understandably not elated about my existence), and the times he lied to me by omission\u2014 _You know, if I just don't tell her, she can't accuse me of lying._\n\nI felt a panic come over me, of distance, of not-real love, of un-love, even, of _lies_ , of being left alone, of getting sober for what? This? I thought of the marriage that began on that dreary day\u2014we were just kids\u2014had there ever been real love? _Nah, we simply had a kid together and got stuck._ We didn't begin correctly. We wouldn't finish correctly. I mentally sorted through the freedom he had\u2014to drive to work alone each day, to sleep while the baby cries, to not think about the things I thought about, to simply _not be the mother_.\n\nWithin moments, he morphed into a stranger.\n\n_Is this what marriage is? I want nothing of this._\n\nI roared again, this time at Mac, and more unchecked than before, because he was an adult. He stood silently, leaning against a doorframe, still wrapped in his bath towel, for my thoughts had progressed from go-kart to divorce before he even had time to get dressed.\n\n\"Do you think it's _your_ money and _my_ money?\" I shouted.\n\n\"No.\" His head was still slightly down.\n\n\"Well, obviously you do, since you think it's fine to spend that kind of money on useless fucking metal! I would never do that!\"\n\nHe shifted his weight against the doorway, looking at me from under a furrowed brow. I knew the look. He was waiting it out.\n\n\"Do you have any idea how much I do every day while you are at work, Mac?\" I reported in long form the play-by-play of each of my days, my voice growing hoarse from the yelling, my body intoxicated with rage. When I erupted at Mac, it only ended when he or I physically left the house, or a stranger came over, forcing me to behave on account of my pride. Occasionally Mac roared back. But mostly, he waited.\n\nMy rage would ebb, and I would apologize, again.\n\nMy family would just look at me.\n\n_You know, Janelle, you could just jump off the crazy train before riding it all the way to the bitter end._\n\n\u2022 \u2022 \u2022\n\nThat night, when the house was gray and quiet and my body untouched, my day became a nightmare memory: _Who are you, Janelle, thinking you can raise kids? Who are you, thinking you can be a mother and wife in a decent marriage?_ I saw the woman's face while she yelled, the woman pathetically trying not to destroy it all again, and I cringed. _Here I am again, hurting people. Here I am again, unable to control myself. Here I am again, the tiny psycho chasing her brother around with a kitchen knife._\n\nI thought of my mother's words, \"You treat me like dirt.\"\n\nShe was correct. _She is correct._\n\nLying there, I recalled daydreaming at school about how my mother was going to pay for Christmas, or the heating, or visualizing the violence I wanted to inflict on the bullies who were mean to my brother on the bus. He was huge, but so gentle. I was tiny, but insane.\n\nIn junior high we had a smoke-spewing minivan with wood siding. I'd ask my mother to drop me off down the road to save me from embarrassment\u2014and the other kids from pollution inhalation. She always obliged, and it helped until after school, when the other kids' parents would show up, and I'd be sitting on a wall in front of the school as the sun went down and the principal left, concerned about me.\n\n\"It's okay!\" I'd say cheerfully. \"My mom is coming. She'll be here any minute.\" It was a partial lie. I knew she'd come. I didn't know if it would be any minute.\n\nAs the evening grew cooler, I no longer feared she was dead. I feared I might kill her myself. When she finally pulled up, I'd shriek at her for leaving me like that.\n\nShe'd tell me all the things she had done that day, and with a lilting sadness she'd say, \"I'm doing the best I can,\" but this would throw me into full fury because now I felt guilty and _what kind of thing is that to say?_\n\n\"You treat me like dirt,\" was what she'd say after I yelled, which seemed true, but also, unfair. It seemed unfair to pick up your child late, then say \"I try the best I can,\" as if that were some sort of excuse, and then say I was mean to _her_. I wished she knew how I sat in school daydreaming about her.\n\nShe was always _trying_ , but never _changing_ , and at some point one gets sick of the \"trying\" and wants the \"doing,\" and also: _Why are we feeling sorry for you? You are the mother._\n\nAnd so I raged, because I was tired.\n\nI was worn out by all the talking, the worries, the desire to trust, to know when my people were going to show up, so I transformed into a ball of red, to get some power. To get some control. To get some peace.\n\n\"Tell Margaret to go home,\" she'd say, and I'd hate her to my bones.\n\nLater, though, in bed, I'd open my journal and write detailed updates of her marriage, and how much I loved her, and how much I wished she could hear me, and I'd wonder if she'd remembered to lock all our doors so we'd all be safe.\n\n_Maybe I was born without a moral compass. Maybe I'm sober, and I'm still without a moral compass._\n\nAn image I'd seen on social media by the editor of one of those eternally peaceful parenting magazines popped into my mind. It said: \"Your words become the voices in your children's heads.\" My God.\n\nThe thought _they would be better off without me_ skipped across my mind, and then, _I should drink again just to relieve my children of me._\n\nAs soon as the thought came, I knew it was not true. I knew it was the sweet whisper of alcoholism, the disease getting tougher every day, and wilier as it waited for the moment I agreed to believe it again.\n\nThe next day, when I told Jack about my yelling, he said, \"You're a bully. You're a bully and you're trying to control everyone. You have power over those kids and you abuse it.\" _A bully, huh?_ Once again, I wanted to suggest he walk away from me at a fast pace for a very long time, but I was still desperate, perhaps more so, because I knew alcohol didn't work, and would never work again. To drink was to die, but there would be no relief before the passing. I could die of alcoholism or learn to live sober.\n\n_Why is this fuckin' guy always right?_ The bully charge was true, though nobody had ever urged the sick reality up to the surface of my brain: that I felt I owned my kids, that I could treat them how I damn well pleased. In fact, I possibly felt that way about the rest of the world as well.\n\n\"Well, Jack, what do I do about it?\" I was actually spellbound at this point.\n\n\"Pray for help. Figure out why you love anger.\" So I tried what he said, meditating and praying in the morning, but mostly, I returned to therapy.\n\n\u2022 \u2022 \u2022\n\n\"I'm sober and still raging,\" I said, sitting on the wicker chair next to the window and a sprawling ficus tree. The therapist paused and asked, \"What do you mean, 'raging'?\"\n\n\"Anger. Raging in anger. I feel borderline again, only without the other fifteen symptoms, which I guess makes me no longer borderline. Is that possible? Can people become un-borderline? Maybe within the borders?\" I was speaking like a fool because I was nervous, and it was hard to explain that my rage was like a rotten old friend I couldn't imagine my life without, or a sibling I bickered with constantly but missed as soon as she was gone.\n\n\"What are you getting from your anger?\" the therapist asked, and I thought, _Holy mother of God, they're all in cahoots with one another._\n\n\"You know that's the same thing the ex-refrigerator-box dweller asked me?\" She looked confused, so I explained he was a friend, a sort of crank spiritual guide.\n\n\"When did you start raging?\" She had her pen at the ready.\n\nI contemplated not going down the whole road of my past with her, but I did. I told the truth. The alternative was too exhausting. I had lied and polished the truth for too long. Jack used to tell me I spent my life \"polishing the same old turd and selling it like it was new.\" I didn't want to tell better-polished lies. I wanted turd truth.\n\nBut I knew that once I started she would look at me with eyes that said, \"Wow, you really made it!\" and I'd feel like a fraud again, because she was praising my stunning turnaround, and I knew I looked so \"fine,\" like a good and loving mother. When I shared my ideas about parenting, she'd say, \"Yes, that's right!\" because I talk well, and she would be impressed with my devotion to family, especially after \"such a past!\" _But_ _what would she think if she saw our house when nobody was there but us?_\n\n_What would she think if she heard the things I say?_\n\n_I am a fraud. I will be found out._\n\nWe had purchased a house. I re-enrolled in the English master's program I had quit. I shared pictures of my family on Facebook, and friends rejoiced at our little family\u2014how far we had come. We should be so proud. We are goddamn miracles.\n\n_But what if they knew?_ I wondered, fully and finally, if I would ever change. If I would ever be worthy of the glittering life around me.\n\nI should not have been surprised to find my inner life unmanageable. Good News Jack always said, \"Alcohol was never your problem, it was your solution.\n\n\"If alcohol were the problem,\" he'd say, \"rehab would be churning out winners. And yet, people like us always drink again. Every relapse starts with a sober brain. So where's your problem? In sobriety.\" It was tough to argue with logic like that, but still I maintained a small, buried belief that once alcohol was gone, I would become the most benign version of myself. Perhaps some sort of B-level saint.\n\nBut after getting sober, I realized I was still an asshole.\n\nIn softer moments, when feeling generous, Jack would tell me, \"You are a human, Janelle. What's fucking with you is that you're human.\"\n\nThis seemed insufficient. This seemed profoundly oversimplified. So, that's it? We harm people because we're human? Do other mothers do this? If so, why aren't they talking about it? And why, if this is how it is, if we are flawed and broken and somehow unsuccessful at snapping ourselves out of it even when kids are involved, why do we write things like, \"You become the voice in your children's heads\"? Do they bestow this responsibility on the heads of fathers? Why is it my job to become my kid's inner voice? Because the kid came out of my vagina? Those seem unrelated.\n\n\"You are only human,\" he said, and yet it feels infinitely not enough. _If this is human, if this is my best shot at life, I am not enough. Do I make peace with that now? I am not failing in cute ways. I am failing in big, big ways._\n\nBut then I would think of the inhumanity of my former life, of the morning I woke up and realized I could not exist among humankind, of the day I couldn't use a restroom properly, of the day I woke up alone in a hospital bed, and the day I spoke in the cracked dialects of the wholly insane, and then I'd think, _I am only human_ , and that is precisely the miracle.\n**14**\n\n# **The Childhood I Could No Longer Blame**\n\nWhen I was a kid and circumstances turned questionable, my mother would take us to the beach. We would walk in the front door, and she would announce, \"Change your clothes, kids. I have to get to the ocean!\" We'd pack sweatshirts and a Smokey Joe barbeque into the back of our white Ford Taurus station wagon and head to Morro Bay or Pismo Beach. My mother would cook hot dogs in the warm blanket fog while my brother boogie-boarded and I hid in little caves and forts under the cypress trees, talking to myself in imagined worlds. I'd roll up my pant legs and flip the clean sand around my toes, chasing waves and tumbling down dunes as the sun fell into the roaring blue.\n\nOn our way home, with salt still clinging to the ringlets around my face and my pants wet and itchy, my mother and I would analyze whatever worry had nudged her to the beach. Usually it was a lack of funds, or another disappointment involving my stepfather, Keith.\n\nWhen things got really bad, we'd head out on a road trip somewhere without any particular money or plan, sometimes traveling all the way to British Columbia. On these trips, in the late afternoon when it became apparent we needed to sleep somewhere, my brother and I would start looking for camping spots on a huge Rand McNally map we kept wadded up under the front seat.\n\nOne summer afternoon when I was ten, my brother and I walked in the door after playing outside to find our mother stacking food and towels and sleeping bags in the living room. \"What's going on, Mom?\" I asked.\n\nShe grinned. \"We're going to Yosemite!\"\n\nWe jumped up and down. _Wheeee! A random adventure!_ It was summer, so the plan to \"leave today\" made more sense than her usual announcements on school nights that we were going to \"see the redwoods\" or \"hang out at the stock car races in Laguna Seca.\" If I protested, she'd say, \"You'll learn way more camping than you ever will at school.\" This, of course, was unequivocally true.\n\nBut my jumping ceased when I remembered my mother was in charge.\n\n\"Do we have a campsite reservation, Mom?\" I raised my eyebrows as she folded a giant tarp she referred to as \"the blue deal.\" She wore stonewashed tapered jeans and a purple T-shirt that said Yosemite on it, which was quite possibly the impetus for the entire trip. Perhaps she had seen it that morning, sitting in the drawer, and thought, _I had better get to Bridal Veil Falls._\n\n\"Of course not,\" she answered slyly. \"How could I? I just decided to go! We'll find something.\"\n\n\"What do you mean, 'We'll find something'? We will not. It's summer. It's been booked for three years.\" I was ten, but I knew these things because I was a tiny Republican who _loved rules._\n\n\"Go pack!\" She was grabbing pillows from around the house and stacking them by the front door.\n\n\"What about money, Mom? Do we have any?\" I hadn't budged from my platform of inquiry. There were _questions that needed answering, people._\n\n\"We have enough. I sold some new advertising.\" She yelled to my brother to get ready. I wanted to yell, \"Let's slow down and assess the particulars of the situation!\"\n\n\"What about the electricity bill?\" I dropped one hip and put a hand out, as if to say, _Did ya think of that one, Mother?_\n\n\"It just came. We're fine, Janelle. We are fine.\"\n\n\"How much money do we have exactly?\" I wanted data. Numbers. I wanted to know if we were spending the last of it.\n\nOur money came and went so quickly. A tax return arrived and we were flush. The heater broke and we were broke. But when the money was gone, my mother's spirit went with it. We went to the beach and she drove along, defeated, and I thought for sure we'd never be flush again. I wanted to save us both from that moment.\n\n\"Janelle, go pack!\" She was laughing, and I couldn't fight it anymore. The sound of her excitement\u2014the tune of carefree\u2014was sweet mountain air. I breathed it in and I was with her. _Let's go, Mom. Let's go adventuring._\n\nI walked to my room thinking of waterfalls and Ansel Adams, glaciers and granite and bears. I packed, and we piled into our wagon and drove.\n\nAn hour later, I was sent into a Carl's Jr. to gather some mayonnaise packets (we had forgotten the mayonnaise) so we could stop in the parking lot of a liquor store to eat lunch.\n\nI was sitting on the back of our car and swinging my legs under the hatchback, while my mother made tuna sandwiches and apparently spotted a homeless woman sitting in front of the liquor store.\n\n\"Janelle, go give that lady a sandwich,\" my mother said, extending a sandwich my way.\n\n\"What? No.\" I widened my eyes and pulled my body back from the sandwich, repelled by its very association with her idea. \"Absolutely not.\"\n\n\"Give her some chips, too,\" she said, packing Doritos into a baggie.\n\n\"No. Why, Mom?\" I was already desperate.\n\n\"Because maybe she's hungry.\" I flinched in restrained rage at my mother's incessant weirdness. _Where does she come up with these ideas? Why can't we just fade back into our Ford like normal people? I already stole mayo packets from Carl's Jr. Is that not enough for one day?_\n\nI whined, \"No way,\" as if that had ever worked once with my mother, but I soon gave up the battle I knew I'd lose, grabbed the food, and walked across the infinite lot, cursing my mother under my breath in Mormon-approved swear words. I approached the woman and sheepishly held out the food as she squinted at me through sun-cracked skin and watery eyes. She accepted it silently as I smiled and mumbled, \"Hi. Here. Okay, bye. Thanks.\" I walked immediately away.\n\nCrouched in the sunshine under a pile of evidently unnecessary clothing, she had not looked at me like I was Jesus, as I had assumed she would, but rather as if I were a cow invading her afternoon\u2014and not even a cute cow. Still, I was impressed with myself\u2014 _quite a Good Samaritan, you'll notice_ \u2014and saw my mother watching me with equal pride in her eyes as I strolled back to the car. When I was about halfway across the parking lot, the lady chucked the tuna sandwich at me, skimming the side of my head and distributing tuna and mayonnaise across my scalp. I dared not look back at her, but the shock spun through my legs as I ran to my mother, my eyes locked with hers, burning in humiliation, screaming, _What the frick were you thinking, Mom?_\n\n\"Why did you make me do that?\" I was furious with the betrayal.\n\n\"I had no idea she'd throw it at you, honey,\" she said calmly as she pulled the tuna out of my hair with paper towels.\n\n\"This is not the first time this has happened! Remember when we picked up that hitchhiker and she pulled a knife on you? What about that?\" We now had evidence of two crazy homeless people in our lives, which I figured was plenty of data to swear off homeless people forever.\n\n\"She was mentally ill, Janelle.\"\n\nMy eyes widened. \"Then why was she in our car?\"\n\n\"She needed a ride,\" she said, packing the remains of our lunch in the ice chest.\n\n\"Mom! That is not an answer!\"\n\n\"Well I didn't know she was mentally ill. You can't tell by looking at somebody, you know? We always end up alright.\" She laughed again.\n\n\"Oh, you mean like when our car broke down in Las Vegas and we survived for two days playing nickel slot machines and sneaking into buffets?\" _Plus,_ I thought, _you can kind of tell by looking at somebody, Mother._\n\n\"Keith should have sent us some money to pay for the car. That was a terrible thing he did, leaving us like that.\" My mother shook her head in disbelief as she reflected on Keith's bad choices.\n\n\"Mom, we went to Vegas to see the Hoover Dam with two hundred dollars and a nearly broken car. That is not how people do things!\"\n\n\"This is a great car! And I got us a VIP tour of the dam, didn't I? Remember that? We met that security guy just when they were closing. That was the opportunity of a lifetime.\" She was growing tired of my inability to recognize educational opportunities.\n\n\"We also got kicked out of a buffet by a different security guy.\" I hated that moment. I had to leave half a Las Vegas casino buffet shrimp plate.\n\n\"And yet, here we are!\" She threw her hands up. \"Still okay!\"\n\n\"And now we're going to Yosemite in June without a reservation. There will not be a spot, Mom. I do not want to go without a spot.\" I was afraid again.\n\n\"We'll get a spot.\"\n\n\"But how do you _know_?\"\n\nShe leaned toward me. \"You have to think good thoughts. You just have to _know_ you're going to get a spot. Just believe it. Assume there is no other option. It's how I always get great parking spots right in the front. I drive right to the front knowing I'll get a spot, and I always do.\"\n\n\"You do not _always_. That is a lie.\"\n\n\"Well, I usually do.\" That was true.\n\nMy mother's life plan was: \"We'll figure it out.\" And the method to carry out the plan was: \"Think positively.\"\n\nI hated that philosophy. I would think positively when I was looking at my damn campsite reservation. Why couldn't we call ahead? Why couldn't my mother _believe_ the \"No Vacancy\" signs when we were on the prowl for a motel room? She invariably stopped the car to check anyway. \"Oh, they always have extra rooms. They just put that sign out to deter people.\" Occasionally, she'd walk triumphantly out of the motel office with a key in her hand, announcing, \"I even got us a free upgrade!\"\n\nHer eyes would say, \"See, Janelle? I told you.\" And I would smile, because she really did pull some shit off.\n\n\u2022 \u2022 \u2022\n\nBack in \"Mudhole,\" which is the translation of Atascadero, our town in Central California, we pulled into the driveway of Keith's house, and I felt the weight of our lives return.\n\nBy the time we walked into the living room, my mother had already disappeared. Not in body\u2014she was right there\u2014but the woman I knew, the one who suggested I break into the ranger station to pilfer a reservation form, was gone. She was busy watching her husband now, with eager eyes, anticipating his needs, dangling from every barely perceptible shift in his mood, shuffling around to repair and preserve his desires. I sighed and went into my room to unpack and try to finish that Steinbeck chapter about the turtle.\n\nWe had been in Atascadero since I was seven, moving there after a dreamy stint in Calistoga, an adorable little town full of wine and wildflowers. We lived in a trailer park across the street from a hot springs pool. My brother, mother, and I had moved there after my parents divorced. My brother and I swam so often my hair turned white from the sun and green from the minerals. We rinsed it in lemon juice. It turned whiter and greener.\n\nWe had moved to Calistoga from Clearlake, where my father grew up, and where my whole family lived when it was together. The wine country town was only twenty-six miles from Clearlake, but it felt like a new country. Clearlake was in Lake County, an impoverished, rural swath of land boasting the highest number of multigenerational welfare recipients of any county in the state (according to family lore), a raging meth problem, and a lake named \"Clearlake\" even though it was often rimmed by a twenty-foot wall of algae one could avoid only by boat or dock. Grandma Bonny lived in Lower Lake. Grandma Joan and Grandpa Bob lived in Lakeport. And scattered between them were my aunts, uncles, and cousins.\n\nBut my mother needed a new life, and Keith, a high school friend of my father's, offered to lease her some office space. So we sold the motor home and drove a few hours south in the Ford wagon to Atascadero, every mile carrying us farther away from the unclear lake. When we arrived, we rented a tiny pink house with red shag carpet as a temporary pad until my mother's new business took off.\n\nHer plan was to start a tourist magazine, and around the dinner table she would talk of her vision, the well-connected person she met, the special information she had that nobody else had\u2014a unique opportunity, no other travel magazines, all very promising! For eighteen years, she helped my grandparents and father (and aunts and uncle) publish the weekly newspaper in Clearlake, which Grandma Bonny owned, so she knew weeklies. This would be a monthly, and it would be hugely successful because there was nothing like it, even though we were in the _world-renowned, tourist-packed Central California coast_.\n\nI loved imagining with her all the ways we would spend the money, the house we would live in, the places we'd visit, the way our problems would pass into nothing. We would get a big house. We would buy a new van. At seven and eight and nine and ten, I believed these schemes and plans and dreams. With all my heart, actually. _This one is going to work._\n\n_So let's go, Mom. Let's go to Mudhole and do amazing things._\n\n\u2022 \u2022 \u2022\n\nIn the bathroom of the pink house is where I first remember clinging to my mother's legs and begging her not to leave. I slid on the floor, gripping her calf pathetically, pleading with her to stay\u2014bartering, discussing details I thought might be up for negotiation\u2014staring at the brown cabinets and gold handles at my eye level while she sprayed her 1986 perm with Aquanet and assured me she was \"only going out for a couple of hours.\"\n\nThis, I knew, was a lie, because she was going to die. So I'd deeply inhale the scent of her Jergens lotion and hairspray, thinking, _This is it. This is the last time I'll smell my mother._\n\nMy brother, immune to the horrors of the world, would play Tetris on Nintendo and then go to bed as if our mother were _not_ going to die, while I kept watch on the couch, rocking back and forth and crying at the _M*A*S*H_ theme song (my mother's favorite) and waiting for the phone to ring. It would be the police and they would say, \"Super sorry, but your mother veered into a telephone pole.\" I would imagine my screams upon hearing the news, my flailing on the floor, visualizing the body viewing and every word I'd say at her funeral. I saw myself crying out like they do in the movies, and my couch-crying would turn into numb shaking as I thought of the heartfelt things I'd say behind the funeral podium.\n\nI lived and relived my mother's death until I was so desperate I'd resort to prayer. I'd get on my knees and bargain with the God I had just recently met in church: _If_ _you bring her home, Heavenly Father, I will never scream again_.\n\nSince there was no way that was going to work, I would begin making new housing plans. I'd remember it was just my brother, mother, and me, and _since she's dead now_ , _would I live at my father's house_? We did not know him well enough for that. I loved him, but saw him so infrequently I still behaved around him. Nobody can endure such conditions long term.\n\nWould I stay with my sleeping brother? Probably not, since he was only two and a half years older than me and obviously had no idea how to handle danger. _Perhaps I will be sent to live with that woman who helps my mother at her magazine_ \u2014 _the one whose daughter had a kid at fifteen and who attempted to superglue her teeth back in?_ She babysat for us a few times. Her pit bull tried to eat us and we smelled like cigarettes for nine days. _I'd rather be homeless._\n\nEventually, as I sat immobile on the couch, too afraid to move, waiting for my mother or the police, my terror shape-shifted into a formless mass that usurped the body of my mother. I forgot about her specific dying, or it was overtaken by a crushing terror of something to come, or something to be removed, which I could not identify. It felt like waiting and powerlessness.\n\nBut in one glorious moment, I'd hear the garage door go up, or a rustle on the porch, and it would be her. I could not believe my good fortune. I'd hug her and she'd let me curl against her in \"the big bed.\" She always let me in, and the warmth was always as perfect as I knew it could be. As long as she was with me, I was alright.\n\n\u2022 \u2022 \u2022\n\nIt turned out Keith had more than office space to offer my mother, and when I was eight, my brother, mother, and I joined Keith and his two teenaged sons on a cruise to Mexico. Keith and my mother married on the mainland in a ceremony none of us children witnessed. When we returned, we moved into his house, and I suddenly had two stepbrothers. The older one was a warm and handsome young man with a gorgeous girlfriend. The younger was a short, reserved, rat-faced human who scurried about looking like he was about to gnaw your face off. As far as I could tell, he was not to be trusted, but as luck would have it, he was the one who would babysit my brother and me in the red-shag-carpet house.\n\nOne night in that house, some older neighborhood boys came over and convinced me to spread the lips of my vagina while I sat on a couch, so they could all see. I did it, which was the horror that never left me.\n\nI had shaken my head in protest. I had mumbled _no_. But something about them all gathered around the couch, something about them urging me on, something about them _demanding_ \u2014I found myself pulling down my own underwear.\n\nLater, I wondered why I didn't simply walk away, lock myself in my room, staunchly refuse with a big, strong voice. I wondered why I participated in my own degradation, my own humiliation, without a soul touching me. I didn't know why I did it, not understanding at that young age how children are manipulated. I wrote about it in my diary, a blue one with pink pages and a white unicorn and red stars on the cover. I referred to my vagina as my \"private parts,\" because I was young and mostly Mormon and that's what we called it.\n\nWhen I told my mother about that and other unfortunate, related occurrences, she sent me to a therapist who explained, after my mother left the room, that what had happened wasn't abuse because \"it wasn't done violently.\" That was it. No problem here. I never saw her again, and I never told my mother what she said.\n\nAfter that, I filed away in my gut the notion that I was _illegitimately_ tainted and evil and dirty. I thought something was wrong with me not only because it happened, but because I was _upset_ it happened. And then everybody moved on from it but me, and each week we went to church, where they talked about the sanctity of marriage and the woman's body and how it's a temple.\n\nDo not fuck with the temple. Mine had already been fucked with, but we couldn't mention it, so I held a dark weird secret in my temple like a forgotten rotting room.\n\nI resolved I was simply garbage.\n\nMy new room in Keith's house was a loft overlooking a big living room. Between the loft bars, when I was supposed to be sleeping, I spied on my stepbrothers watching porn on fuzzy, stolen channels. I hated being in the loft, where there was no door or wall on one side. I felt exposed in the open air, like I did that night on the couch. I'd remember the way the boys laughed and gazed.\n\nBut I couldn't crawl into my mother's bed, because she had a husband now.\n\nSo instead I closed the curtain to my loft room, stood in front of a little mirror next to boxes of Keith's old taxes, looked myself in the eyes, and said, out loud, \"Janelle, you don't need anyone and nothing will ever break you.\"\n\nI stood there in pajamas, next to a purple unicorn comforter, staring into a mirror with little ballet shoe stickers all over it, turning myself into a fortress.\n\n\u2022 \u2022 \u2022\n\nI spent three months in the loft until we moved out again. During this particular separation, Keith was suddenly baptized Mormon, which I explained was a clever ruse to win my mother's affection. He kept smoking cigarettes and drinking while not actually attending church, which I gathered as evidence of the ruse theory, but my mother only agreed when she was feeling scorned.\n\nKeith's most infuriating feature was that he was categorically delightful to me, and I fell into an immense and uncomplicated love for the man. He was my dear friend, and I suspected there had never been a stepfather who loved a stepdaughter as much as he loved me. We had a hundred traditions just between us, songs that were ours and ours alone\u2014so many that I was never afraid of my mother and him separating. We had nothing to do with her.\n\nShe was too good for him anyway. My mother was gorgeous, with soft waves of brown and blonde hair and green-blue eyes with yellow in the centers. It's impossible to tell the color of her eyes. Ava and I have the same ones. Green? Hazel? Blue? They change according to what we're wearing, or the sky, or something inside of us perhaps.\n\nWhen she put her lipstick on, she stuck her bottom lip out too far, and I laughed and teased her, because it looked like she was sweeping pink paint on the inside of her mouth. I never understood how it didn't smear across her teeth. She explained she had to turn her lip like that because her lips were \"so flat.\" To make me laugh harder, she exaggerated sticking her lip out, and I did it, too, as soon as I'd see the lipstick come out of her purse. The whole lipstick process seemed ridiculous to me, but her mother had taught her never to go outside without lipstick on. Grandma Joan never left the house without her \"face\" on, and she was strikingly beautiful even into her seventies and eighties. She had a square jaw and huge, bright green eyes. When we would go out in public, people would mistake her for my mother's sister. The family always said it was because she was half Filipino.\n\nMy mother had beauty, but she didn't have her mother's marriage, her incredible lineage of love. It must have hung over my mother's marriages like a vast and impossible utopia. With an example that solid, perhaps the only option is to blow up your own life.\n\nWe moved into Keith's house three times over the seven years we lived in Atascadero. Once we moved all the way to Round Rock, Texas, then back again seven months later, back to the same damn junior high I had tearfully abandoned mere months before, with the same horrific children, only in more solidified cliques.\n\nI could pack my room in twenty minutes flat.\n\n\u2022 \u2022 \u2022\n\nAt the end of my eighth-grade year, when my mother decided she was unquestionably finished with Keith, we moved back to Northern California, right next to where we began, in the city where my father, stepmother, and brother lived. My brother had gone to live with our father a few months before. I believe he was tired of moving.\n\nMy mother behaved exactly as she had every other time we left: unequivocally committed to the impossibility of ever returning. By this time, at fourteen, I was not even distantly convinced, because if a Californian can move all the way to _Texas_ and still not find escape, there is nowhere far enough.\n\nThat first year of high school, I ran for class president and somehow won, which I always figured was because the kids didn't know me well enough to hate me yet, but nobody cared about class president. What they cared about was the father-daughter dinner dance, the fanciest event of the year, particularly for us freshmen, who certainly weren't going to prom and probably not homecoming either.\n\nMy actual father lived five minutes from my mother and me, and Keith lived six hours away, but I invited Keith because I missed him, and I knew him better than my father. I knew he would come. He did, and we danced, and by the end of the night I was reassured that no distance or divorce could fracture us.\n\nThree months later, I sat in a witness box in front of a courtroom describing through teary eyes in my most passionate and earnest voice the history of Keith in my family. I understood Keith was denying the legitimacy of his marriage to my mother, claiming it was done in Mexico in a fraud ceremony. My job, according to my mother's attorney, was to prove that Keith and my mother \"acted married.\" Or, more specifically, that he acted \"like my father.\" The situation struck me as odd, and I didn't understand _why_ the debate was occurring, but I wanted to help my mother, so I listened carefully to the attorney's direction and sat ready to convince a judge.\n\nKeith was sitting in a tweed suit coat at a table with his attorney, right across from me. I hadn't seen him since the dance. He looked weary.\n\nI told the courtroom how we went to Magic Mountain in Valencia and to Harris Ranch on Highway 5, to Yosemite and the beach and out for hamburgers in San Luis Obispo. I told how he took me along on work trips around the county and how we listened to Rod Stewart as we wove through the fog and down Highway 1 along the coast. That was our song: \"Forever Young.\" I told how he went to my Campfire Girl events when I was in second grade, and how he heard me read my winning D.A.R.E. essay in sixth grade, and took me to ice cream at Thrifty (where I would order mint chip and chocolate malted crunch). When I finished, I knew there could be no doubt of our love anywhere in that courtroom.\n\nKeith's lawyer took a few steps toward me at the stand and asked, \"Where is your actual father?\"\n\nI started, wiped tears from my eyes, and said, \"Oh, he lives here in Santa Rosa.\"\n\nThe attorney raised his voice, grinned a little, filled his mouth with trash, and sneered: \"So your _real dad_ is not DEAD?\" He chuckled after he said it. He laughed at me.\n\nI flinched, and broke the attorney's gaze to look into the eyes of the man I thought was my closest adult friend, who loved me and watched me now and many years before, and I thought I saw tears in his eyes.\n\nDid I? I don't think I did. Maybe I did.\n\nIt didn't matter if there were tears or not, because no matter what, I was the one sitting up there like a fool, trying to convince a court he loved me. I shuddered at the humiliation as he sat silently looking at me with the same eyes I had watched for years, eyes I thought looked right back at me and adored what they saw. I had been sold a lie, and I knew it right then, on stage, in front of battling parents and stepparents and strangers, at fifteen years old. I flinched for the first time at the pain of rejection that's unbearable in its finality.\n\nWe had nothing. Not a single thing.\n\nCase closed.\n\nIllegitimate. A fraud marriage. A scam love.\n\nHe won, and I never saw him again.\n\nI went home and looked at a Norfolk pine Keith had given me when I was nine or ten, a plant he brought home to me from Yosemite, and that I tried to keep alive, water just right, position perfectly next to the sunlight. It never thrived, but I moved it all over with me, through every move, to Texas and back, and then to our apartment in Northern California. That day after court, I moved the plant onto the patio, though I kept watering it occasionally for reasons I'll never fully understand.\n\nA year later, when I tried smoking weed for the first time, the pine wilted on the porch. By the time I had moved on to acid, it was nearly dead in the rain. On drunken evenings in that room when my mother was out of town, it was a few sticks and some dirt, a tiny sprig of green here and there out of dumb luck.\n\nBy the time I showed up on my father's doorstep at seventeen years old, announcing myself with the words, \"I only have two years left and I want to get to know you. Will you turn me away?\" I swear that pine was dead. It appeared completely dead. I even _thought_ it was dead but moved it anyway, because by that point I simply _brought the damn Norfolk pine with me when I moved._\n\nMy father and stepmother did not turn me away, so I took over my brother's old room\u2014he had left for his mission for the Mormon church\u2014and set out to get to know my father. If he didn't want me, fine, but I wanted to know. I wanted lived experience. Case closed.\n\nI'm not sure what my father did when I was seventeen and eighteen while I drove around and went to work as a lifeguard and waitress and sometimes went to school. I'm not sure what he did while I left the house to drink and do cocaine or mushrooms and hang with my new boyfriend who I was sure would become my husband. I'm not sure what he did with that pine while I got angry and raged at my mother and past, at the broken-down minivan and the church, or while I planted my roots as an alcoholic, wrote furiously in my journal, and had sex for the first time.\n\nI don't know what he did, but suddenly I looked at that fucking pine tree and it was brilliant green and giant and bursting. There was no brown left.\n\nIt had grown so huge I couldn't recognize it.\n\nDad and I named it \"Norfy,\" because it was so alive and part of the family it needed a name. It lived in the living room by the back door, where I guess the lighting was just right.\n\nKeith died when I was nineteen. I didn't attend the funeral because I wasn't invited. I heard he died in a chair at his desk, slumped over, below the loft where I used to sleep. Sometimes I wonder where he's buried. If I went, I don't know what I'd say, but it would probably be something like, \"You are such a fucking piece of shit. Burn in hell. Also though, how the hell are you, man?\"\n\nBy then Norfy had grown so big we had to take it to my father's office where he and Grandma Bonny and my aunts ran their newspaper, where the ceiling was higher and the walls didn't confine the sprawling green branches. It seemed like something out of a book, some magical creature that refused to stop growing.\n\nWe didn't trim it. We just moved it where it could grow without reason, for as long as it wanted, and eventually it was so huge we couldn't move it at all. For years when I visited, I'd look at that plant and remember when it was a few dead sticks, and how my real father brought it back to life.\n\n\u2022 \u2022 \u2022\n\n\"I married another asshole.\" My mother said it with hopelessness, an almost furious fear. \"I married another asshole.\" I looked at her shape-shifting eyes.\n\nI was a freshman in college when she said it. She was speaking of her third husband, Albert, a man she had met at the little art gallery she owned in Bodega Bay when I was in high school\u2014her second or third, or perhaps tenth, promising entrepreneurial endeavor.\n\nHe was an eccentric man who smoked more weed than I'd ever seen anybody smoke in my life, but my mother seemed happy and stable living in his enormous home in the Berkeley Hills. He had money through inheritance, hadn't worked in years, and owned a mind-boggling collection of rock-and-roll concert posters from the 1960s\u2014Stanley Mouse and Rick Griffin, all the gurus\u2014piles of them from when he worked for Bill Graham at the Fillmore. Albert was a fascinating, generous creature, but unfathomably weird. During conversations, he would flip his mouth around and randomly quote poets while gesticulating wildly, then he'd be on the ground straightening the fringe on his Turkish rugs while smoking a joint. This concerned me, but it wasn't my business. I was in college, and my mother was not alone.\n\nI had hated thinking of her alone in her house in Bodega Bay before she met Albert. Past Bodega Bay, actually, miles outside of town. It was just her in a little two-bedroom house on a coastal hillside.\n\nPerhaps we're supposed to outgrow such things, but during my entire first year of college, I would lie in my dorm room nearly every night wondering if my mother had locked the window over her bed. She was always hot and repeatedly failed to shut and lock that damn window. I would imagine a man crawling in and hurting her. I would hear her cries and convince myself she was trying to tell me through my thoughts that she was in danger, because we were _that_ deeply connected, mother and daughter, and I would look at the clock\u2014one a.m., two a.m., three a.m.\u2014and wonder if she'd answer my call. I'd pray again. Within a short time, I would be sure she was dead. By the morning, I would have forgotten, and the next time I heard her voice I would laugh at my ridiculousness and remind myself never to do that weird shit again.\n\nI was visiting her and Albert at their vacation rental in Mendocino. My mother and I had just come inside after soaking in a hot tub beneath Mendocino's magnificent fog. I sat down on a bed and feared asking her how married life was going, but I asked anyway because I feared not knowing more.\n\nShe was standing in front of a closet, drying her hair with a towel. She wrapped it around her head and turned toward me. \"I married another asshole, Janelle.\" That was how she answered.\n\nIt was the way she said my name. It was the way she said my name as if she were pleading, as if she were reaching out, as if she were looking to me to help her or fix it or even just be the ear to listen and care. I felt my stomach turn in a feeling that was like air in its familiarity.\n\nI was a kid again, driving in our station wagon or minivan while she told me how she suffered, and I told her he would never change, and I felt her pain in my body until I no longer questioned if it was mine.\n\n\"I married another asshole.\"\n\nI wanted to punch her in the face, hold her in my arms\u2014scream, weep, and run. I wanted to wrap her in protection I didn't have. I wanted to leave and never see her again. I wanted to hold her in arms too weak to support anything. _She never listens. Why can't she see? These stories. Her pain. I'll kill him. Another man not nice to her._\n\nBut I watched her marry him. I was in the damn wedding, wearing purple chiffon and smiling. I watched how weird he was and she married him anyway. I suspected she didn't want to be alone.\n\nI hated her for doing it again.\n\nI hated myself more for my inability to change it.\n\n\u2022 \u2022 \u2022\n\nDriving home from Mendocino the next morning, I rolled the windows down as I passed beneath the redwoods and through the meadows from Fort Bragg to Willits. I opened the sunroof to see the tops of the trees and turned the heater up. The air warmed as I drove farther inland, and as I had a smoke, or five, and listened to the Dead sing of the four winds blowing you safely home, I heard her words again. I heard her call my name, the lilt and appeal and plea, and I saw my blonde head, eight or nine years old and lying in a bed, or sitting in a chair, devastated. _She never listens, but she promises, and I'm sure she'll be okay this time. God, please just let her be okay_.\n\nBut she was not, again.\n\nI screamed and couldn't stop screaming. I screamed as loud and hard as my voice would let me. I screamed the scream of a kid who wanted her mother to not suffer anymore. It was the scream of defeat. It was the scream of surrender and a fight abandoned. The scream of finish.\n\nI heard the noise coming out of me and it was strange, but I didn't stop. I let my face contort, and my eyes squint, and I just drove and screamed and hated her. No, I hated powerlessness. It felt the same as it had all my life.\n\nI screamed until I thought my voice would die, but maybe it was already dead. Had been for years. Old air passing through the Mendocino redwoods and Camel Light smoke drifting to the sweet tune of Jerry. There was never a word for her to hear. Never a word worth speaking. I knew it in the freedom of shade and sunlight as I rolled on.\n\n\u2022 \u2022 \u2022\n\nTwo years later, I woke to Albert's voice one summer morning as he stood in the doorway of the room in their home where I was staying for the summer before heading to Spain. \"You need to go get your mother. She took the gun down to the beach, and she said she's going to kill herself.\"\n\nWithout a word I rolled out of bed and walked barefoot down the trail to the sand on the Mendocino coastline, and without a thought my feet trod methodically, numbly, my heart beating blood of rage and clarity into my green-blue eyes that look like hers. When I saw her standing by the water with a .45 in her hand, I walked straight up to her and looked in those eyes that held oceans for me and said, \"Just fucking do it, Mom, if you're going to do it. DO IT,\" I yelled, and I meant it, unable to face one more attempt to keep her safe or sound.\n\nI turned and walked away, and I held my breath waiting for the boom I knew would shatter both our lives, but I could not scream.\n\nIt had all been said.\n\n\u2022 \u2022 \u2022\n\nThe blast never came, but many years later I still felt the metal in her hands. I used to tell Jack about it, and that therapist I saw about my raging, about how it felt to walk in slow motion away from her, each step a long drag through terror. _What would I have done if she had pulled the trigger?_ The therapist led me in circles through and in and around it all, to deconstruct and analyze my pain, to heal the young woman who snapped that day, but I always seemed to end up where I had begun. Jack spoke words I had never heard before.\n\n\"Your mother, you know, was doing the best she could with what she had at the time.\" He took a deep drag of his Marlboro and looked at me from behind his glasses, his bright blue eyes terminally serious.\n\n\"But what she did was _wrong_ , Jack.\"\n\n\"Alright. But do you want to be right or do you want to be free?\"\n\nHe had a way of unraveling in a single sentence knots I had spent my entire life tying. With every replaying of that scene, every therapeutic conversation about it, every Jungian analysis, I wove a thread back through the center. _I was wronged. She failed me._ And maybe that was true. Jack wasn't arguing otherwise, but something about the way he said those words in that moment transformed my mother into a human. I never saw _her_ before Jack spoke those words.\n\nHow many years had I spent blaming my failure on my parents? Blaming my brokenness on theirs? Were their sins even about me? Were they ever mine to hold? Everybody wants to blame alcoholics like me on \"broken childhoods and bad parenting,\" but Jack said, \"There's no power there, Janelle. There's no power in being angry over the past. You weren't responsible for what happened, but you're responsible for what you do with it now.\"\n\n_My parents were broke-down humans just like me._ My mother wanted to not mess up her kids the way her parents messed her up, and she wanted to grow into a better version of herself, too, thinking _surely love will be enough_. She wanted to shove twenty or thirty years of life and disorder into a reliable and shapely parental version of herself, to not be the woman on the beach with a .45, or the loser on the balcony.\n\nI didn't want it to be true either\u2014that what we bring to this newborn is nothing beyond the years of mistakes we've got stacked in our souls, and that what we'll teach is not much more than the lessons we've learned through years of things we probably should have already known, and even love isn't enough to polish us up into something more presentable. Even love isn't enough to make us good enough for the tiny creature in the fuzzy pink bear suit, sleeping on the chest of a mama who's read all the books, nurses with devotion, checks her breathing five times an hour, and asks only once, a bit too seriously, if beer is allowed in birthing centers.\n\n_They were doing the best they could with what they had at the time. And so was I._ _We are the goddamn same._\n\nThat was when I knew. You can build a life on rosemary carrots. You can head out without a plan. You can remember an outline, and it may be enough to bring the most hideous, wilted pine into glittering life.\n\n\u2022 \u2022 \u2022\n\nWe didn't find our own campsite that night in Yosemite. We circled all the campgrounds, and as I had predicted, each site was reserved. We ended up in the overflow area, where, much to my horror, you walk in, choose a spot, and share it with strangers.\n\nThe men at our site were mountain climbers from Germany, and that night around the campfire, I watched firelight swirl around their faces while they described nailing their cots into the granite face of Half Dome and sleeping perpendicular to it, suspended, hovering in the cosmos.\n\n\"But what if you roll over?\" I asked, aghast.\n\n\"Well, you don't!\" They guffawed, and I thought, _Well, that sounds like a plan my mother would come up with._ They prepared their food like that, too, floating over death, peeing and pooping into cans, and as I sat there, I realized I was meeting the most insane and gorgeous human beings I had ever seen in my life.\n\n\"Kid, do have any idea what the stars look like from the face of Half Dome?\" The taller hiker looked right at me when he said it, but I couldn't speak, so I handed him another hot dog. He did not throw it at my head.\n\nThey explained that after some hiking the next day, they planned on hitchhiking up the coast of California and Oregon. I looked at my mother and smiled, thinking, _God, I hope we pick them up someday._\n\nTwenty years later, sitting next to Jack after telling him my childhood story, I realized I had never stopped looking for the hitchhikers who climbed the face of Half Dome. Even though there was no chance of happening upon them, I believed they were out there, if I kept searching. Even when I could barely open my eyes, I knew they were out there, and I wanted to meet them. I wanted to find them again\u2014beautiful humanity met by happy accident, to show me what the world looks like beneath the Yosemite stars, from the face of a granite rock I could never scale myself.\n\nMy mother kept me looking. Her wild brokenness somehow forced me to remember humanity is mine, and I will find it, even if there is no plan, no reason to believe, and all the signs say \"occupied.\"\n**15**\n\n# **What the Hell Is \"Soul Work\"?**\n\nIf I thought I would be gentler after getting sober, I _really_ thought I would round the bend into sainthood after _overcoming my childhood issues._ I thought the intermittent, insidious boredom and confusion of my inner life would disappear like hangovers. I thought the parts of my personality that repelled and repulsed would fade like chain-smoked cigarettes. I thought meaning would beckon at every turn.\n\nAnd yet, I still raged, fell into caverns of malaise\u2014a sense of godless vacuity\u2014that seemed to live in my blood. Even when it was all \"perfect.\" Even alongside all that fucking gratitude. It almost made it worse. _Here I am sober\u2014a new lease on life!\u2014and I just spent twenty minutes staring at a wall wondering if this is_ really _all there is._\n\nI wanted to discuss Edward Said and do a Marxist reading of nineteenth-century American literature. I wanted to deconstruct dime novels and other \"lowbrow\" cultural products to study emergent art as resistance. I wanted to set my cubicle on fire. I wanted to get out of my house to hear myself think. I wanted a PhD. I wanted to write. I wanted to read Zizek and Gramsci and Butler and wonder what the hell they were talking about.\n\nI wanted one to three more children. I wanted newborn breath. I wanted midwives to take care of me with their weathered hands. I wanted the moment the baby is placed on my chest. I wanted my milk trickling out of pink petal mouths. I wanted baby thighs and toddler mispronunciations.\n\nWhile I wanted, I changed diapers and got myself dressed and the kids dressed, and cleaned and studied and drove and drove and drove. Everywhere and nowhere. I woke and did it again. We were down to nickels before every payday, and Mac worked two hours away in San Francisco as an ironworker. My life became five days a week of career work and two days a week of all the housework I didn't do because of the career work. In between, I went to grad school.\n\nI went to grad school and felt embarrassed by my home life, wrestling always with some peculiar shame. _If I were a real intellectual I'd stop housing babies in my womb_. I went to mom groups and felt embarrassed by my career. _If I were a real mother I wouldn't drop my baby at daycare every day._\n\n\u2022 \u2022 \u2022\n\nOn one particularly dark day, when I must have been channeling my mother's optimism, I admitted to some other mothers how my life felt monotonous and seemed to make me dumber by the day, and how I was having _a bit_ of trouble finding meaning in the endless beats of working motherhood. One of the mothers suggested I read the latest feel-good story involving an existentially lost white woman \"finding\" herself. \"Changed my life!\" the mother said, and all the women cooed in agreement, glancing at me as we stood around the play structure waiting for our children to get out of class.\n\nI smiled vaguely, thinking, _How? How the fuck did it change your life?_\n\n\"Oh, it's amazing,\" she continued. \"She loses everything and cannot find herself\u2014you know, _who she is as a woman_ \u2014so she starts traveling. Total spiritual quest. Can you imagine her bravery?\"\n\n\"She was probably paid a fat book advance to do that,\" I said, regretting it immediately and thinking, _This is why nobody likes you, Janelle._\n\n\"Well, still. It's really _inspiring_ , and it sounds like you need some inspiration!\" Her tone reminded me of my uncle patting me on the head at Christmas while saying, \"You're doing great, kid!\"\n\nIt was a familiar feeling\u2014standing in a group of women wondering what the hell was wrong with me. I have always been plagued by the suspicion that I am defective because I don't like things I am supposed to like, that everyone else seems to like, that I would probably like were I a better American. For example: joyful white women telling me how to improve myself.\n\nIf I had been ready to face full banishment, I would have told the women around me how I really feel about the self-help soul-journeying brigade. I would have explained my full-blown disdain on a visceral level. But then they would have asked me why, and I would have struggled to articulate what, exactly, is so revolting about put-together, well-meaning women armed with expressions like \"soul work.\"\n\nAll I can think when I hear that is, _What the hell is \"soul work\"?_\n\nThese women are adorable, and they say adorable heartfelt things, but when I listen to them, my face contorts in pain. Still, after the world is done acclaiming the latest disaster-to-inspiration miracle story, I give it a listen or read, to join the party, or at least discover what everybody is so enamored with. But after experiencing it, I usually think:\n\n_What is this shit?_\n\n_How do people like this?_\n\n_What is wrong with humanity?_\n\n_I hate everything._\n\n_I shall move to a yurt on a New Mexican hillside. No. One of those off-grid houses in the Mojave made out of tires. Wait. The Mojave? Fuck deserts._\n\n_Where's my James Baldwin?_\n\n_Okay, but maybe there's something wrong with you, Janelle. DO YOU HAVE NO SOUL?_\n\n_Maybe. But we have to turn this crap off._\n\nOh, their stories of recovery after divorce, after drug addiction, after hitting that arctic ontological bottom.\n\n_That's my story, too!_\n\nExcept it isn't. Because they say they were hopelessly lost in addiction but got sober the day they gazed at those sweet newborn toes. They say they beat their addiction by hiking through pine trees on ancestral trails. They overcame their soul-sucking cubicle death job by joining an ashram. They say they were hopeless addicts but still earned PhDs.\n\nI shit in a bag and kept it. Nobody's talking about shitting in bags on _Oprah_. I was at the bottom too\u2014 _I feel ya there, ladies!_ \u2014but I didn't get a PhD. I got Ancient Age whiskey, laid off, and seven years between the day I enrolled in a two-year MA program and the day I held the actual degree.\n\nAnd then, most disturbing of all, I got sober and realized I was still an asshole. I got sober and realized I still hurt people. _I even resolved my childhood issues, and I'm still fucking bored._\n\n_Wake up. Hurry. Kiss kids. Step on Legos. Regret life. Say goodbye. Coffee. Car. Cubicle. Censor mouth. Suppress true self. Get paid. Watch it all go to mortgage and pressboard Ikea furniture and student loans. Come home. Yell. Clean. Fight. Laugh. Cuddle. Read inspirational soul travels. Throw book. Sleep. Do it again._\n\nThey say \"find your passion\" and \"find the treasure within.\" And I loved that idea when I got sober, but mostly I hated the way when I lay down, my belly rested on the bed like a bag full of water because I couldn't stop eating sugar.\n\nEven if everything they said was true, and I was simply a walking bad attitude, their polished words ultimately _felt_ like lies. They felt like a well-choreographed dance around the truth. It made me uneasy because it was almost believable, all that gentle talk. It almost felt real. But my bullshit detectors blared whenever I read their words, and they wouldn't shut off until I got back to words that grapple in the center, where the ambiguity lies\u2014where a lack of answers _is_ the endpoint.\n\nI wondered why we couldn't be real with ourselves: \"We are going to die someday. We are going to rot in a coffin with motherfucking maggots. Isn't it insane that we waste our time staring at smartphones and working at desk jobs we hate while accumulating shit we can't take with us? Isn't it ridiculous that we've created this material fortress of meaning in our lives when we're all just mindless mechanics working for a giant capitalist daddy?\"\n\n_Now leave me alone while I check my newsfeed._\n\nAnd yet, the years mocked me with their passing. Sometimes I would imagine myself on my deathbed, looking back on my life, and I would feel\u2014I mean _really feel_ \u2014that this life is all we get. These years, one shot, ninety years if we're lucky. And I'd grow so terrified of _just not doing anything_ that I would grow almost frantic.\n\nAnd yes, standing among those women, I was searching for meaning, even when nobody was looking\u2014for connection, purpose, color\u2014some taste of recklessness in a neighborhood of neutral tones. I've always been looking for Barcelona. _Perhaps I should eat, pray, and love it into existence?_\n\n\u2022 \u2022 \u2022\n\nWhen I was three months sober, I sat in a conference room waiting for my first performance review since I'd cleaned up my act. I had shown up every day for ninety full days, clear-headed and ready to work, engaged in my job and no longer saying \"cunt\" at random. I had no doubt my bosses would offer nothing but praise for my dazzling performance.\n\nInstead, they drew line by line the full picture of my mediocrity. I do not remember their specific concerns, but there were many. I only remember how we were sitting at the table and how it felt. They spoke clearly, professionally, and correctly.\n\nI was decidedly standard. Even sober.\n\nThe shock drove tears from my eyes, and I did not try to stop them. They were pouring, and no amount of tough-broad bravado would have saved me. The horror was not failing to earn a glowing review. It was that I had been so sure of myself. These same bosses had once laid me off for talking to myself incoherently in a hallway. _I had come so fucking far._\n\nI thought they looked at me with pity. I could have sworn it was pity. It probably was. How could they not? When the review was over, I walked straight to the top floor of the parking garage to hide and smoke a cigarette and call Good News Jack. I told him all of it. I wasn't angry. I was devastated.\n\n\"I just feel pathetic, Jack. I was sitting there like a motherfucking loser, and I just feel pathetic.\"\n\nAgainst all evidence of my time working with him, I still expected words of encouragement, words like, \"Oh you're not pathetic. Look at all you've done, Janelle! You're a shining star!\"\n\nBut after a pause, he asked, \"Wouldn't it be great if you could be okay with being pathetic?\"\n\nHis question entered my consciousness one word at a time in neon lights. I stared at the cars around me, all lined up, the sun hitting them from the side, casting deep four-o'clock shadows. I took a drag of my smoke.\n\n_Well I'll be damned_ , I thought. _The problem is not that I'm pathetic_ , _the problem is_ I think I shouldn't be pathetic _. What if I am? What if I let that be?_\n\nI smiled. _I'm pathetic._ I got sober and managed to have the worst performance review I'd ever had. _What now, Janelle?_\n\n\"Ah, shit, Jack.\" I couldn't think of more to say. I hung up, and pathetic settled into my bones. It ran to my skin and brain and toes. Through my whole heart.\n\n_What now?_ I looked around again.\n\n_I'm still here_.\n\nI dropped my cigarette onto the concrete and crushed it before picking it up and throwing it away (because _I don't litter_ ). Then I walked back into the building.\n\nBack in my cubicle, I remembered what Jack had told me in my earliest days of sobriety: \"If you're sitting in the living room but want to be in the kitchen, the first thing you have to do is realize you're in the fucking living room.\" _Oh God,_ I thought. _Good News Jack is speaking in metaphor again._ I nodded, following the concept thus far.\n\n\"Otherwise,\" he said, \"you'll never know to get up and walk into the kitchen.\"\n\nI stared at him with my mouth open. _We can't get someplace new until we're honest about where we are._\n\nI had spent so much time gazing at the kitchen, longing for the kitchen, crafting visions of what it would be like to be in the kitchen, that I never took the time to look down, right there, at my ass on the couch. I was so busy yearning for Spanish cobblestone, I missed the concrete beneath my own damn feet.\n\nNot a single thing changed about my life after that day. I was still a butler and reluctant homemaker and administrative assistant craving _one iota of critical thinking_. And I still often hated it. I simply stopped fighting that I hated it. It felt like a pause, a deep breath, a long inhale of the facts. _Okay_ , I thought. _So this is my life._\n\n_I'm the motherfucking butler. What now?_\n\nAnd I got back to work. Because from there, there's nothing left to do.\n\n\u2022 \u2022 \u2022\n\nMaybe if I had listened to the inspiration brigade\u2014maybe if I were a different person with a different past and a more positive, reasonable outlook\u2014writing would have been gentle, a sweet release, instead of a roaring beast that snatched me for hours after the kids went to bed, and took me away on Saturday afternoons, and locked me in my bedroom after telling my children, \"If you knock on my door and you aren't bleeding, I will ruin you.\"\n\nMaybe I would have continued folding the laundry and placing it in drawers rather than depositing baskets of unfolded clean clothes in front of each kid's dresser. Surely I would have written fewer expletives had I been swimming laps every day instead of fortifying myself with 70 percent cacao bars, coffee, and sarcasm.\n\nBut I didn't, and I'm not somebody else, and I didn't start writing to change lives or make money or even \"fulfill\" myself. I started writing because nobody was writing the life I was living. Nobody was saying a damn word about what I saw each day. _And yet, I exist. Don't I?_ Nearly two years later, I was _still_ looking around at the \"categories\" of mothers and realizing I fit nowhere. Meanwhile, the world told me to \"explore creative endeavors like a hummingbird.\"\n\nI would look around at the disorder of my life, the chaos and ambiguity\u2014emptiness and servitude alongside exquisite beauty\u2014and I'd think, _Somebody else must feel this way too._ Because I knew I was grateful. I knew I was so happy to be here I sometimes felt the hand of God himself had spread across my broken shoulders. And yet, _maybe I hate motherhood._\n\n\"Why can't both exist, Mac? Why can't it be _this good_ and _this bad?_\n\n\"I don't know, Janelle. Why don't you stop reading that crap?\" he'd say.\n\nI'd continue blathering on about whatever \"sleep training\" article I had read, or yet another \"women should cover up when they nurse\" manifesto. I'd read _one more_ handy guide about how to \"balance\" your forty-hour workweek, spice up your sex life, speak in gentle tones to our butterfly babies, and lose twenty pounds\u2014 _in a month!_ Everybody seemed to agree on a few basic tenets of motherhood\u2014that it was precious and sacred, and that saying otherwise was Satanic.\n\nI'd squint and curse under my breath and read more books and magazines and blogs. I'd read Twitter and Facebook and Instagram. The more I read, the more alienated I felt until I simply had to know: _Is the rest of the mothering world crazy, or am I?_\n\nThe thought _fuck these people_ drove me to the keyboard. One of us was definitely lying. Words flooded my brain while I showered or drove. I'd miss the exit on my way to work and not realize it for a full five minutes. My kids' voices became white noise behind the deafening cadence of sentences I had not yet written. When I couldn't stand the beating any more, I sat down one day in my cubicle, signed up for a free blog, named it \"Renegade Mothering,\" and wrote eight hundred words in fifteen minutes. The last of them said this:\n\nWhat about those of us who love our children as much as the well-adjusted knowledgeable stable enlightened types but just can't seem to get it right? What about those of us who just aren't cut out for this shit but are doing it anyway?\n\nI am proof that not every woman enters motherhood in some gentle, planned, ribbon-and-ruffles way. Not every woman likes this crap. Not every woman fits neatly into the mold created and reinforced by irrelevant books like \"What to Expect...\"\n\nI usually look around the child-rearing world and see a bunch of crap I don't need, hear a bunch of advice I can't use\u2014encounter a bunch of people I only partially understand. I go home and I see a thrashed house with kids everywhere and overgrown lawns, dirty cloth diapers and books I want to read but don't and toys and dishes and sometimes I demand that my kids just sit down be quiet and watch Netflix because I can't stand one more moment of noise or movement. And if one more person says \"Mama\" I am going to take a bat to the windows.\n\nAnd a few hours later I walk into her room after she's gone to sleep and I see my firstborn baby, nine years old. I stroke her frizzy unkempt hair and listen to her soft snores. I touch her cheek and my eyes burn in palpable adoration. I feel it surge up my body from my toes into my fingers\u2014thick, fierce infinite expanding mama love. And I beg the universe in that moment to give her everything she will ever need and please God keep her safe and how is it that I am so lucky to have this child, right here. The one who robbed me of my great ass and flat belly and turned me into the mother I wasn't ready to become.\n\nI lie down exhausted and think of all the ways I could be a better mom. Of the days I've missed through my own selfishness. Of the years racing by, teasing me with the illusion that this will never end, that they'll always be little. And I wish I didn't yell so much.\n\nAnd so it goes on like this. Back and forth. All the time. Here's to the trip.\n\nSomeday I shall write my own book called: \"What to Expect When You're [a Jackass and] Expecting.\" Until then, I'll write this blog.\n\n\u2022 \u2022 \u2022\n\nIt was a small, silly act. A dumb nudge in a great void. _A \"mommy blog?\" Bah!_\n\n\"Nobody will read you, Janelle. Nobody cares,\" my brain told me.\n\n\"That's okay,\" I answered back. \"I'm okay with being pathetic.\"\n\nI had forty readers, and twenty of them were my cousins. After I wrote a post\u2014usually when I was supposed to be working\u2014I would wait seven minutes and call my mother. \"Did you read it?\" I'd ask.\n\n\"Yes!\" she'd say, and tell me she was crying or that I was \"hilarious.\" And I'd think, _Nice._ _My mom thinks I'm funny_. Then I'd go back to my soulless work.\n\nWhen I got home, I'd ask Mac if he had read it, and he'd say, \"You're such a good writer, Janelle.\" I didn't believe him, but kept writing anyway because I'd remember the duck and the fire hydrant.\n\n\u2022 \u2022 \u2022\n\nTwo years later, three years sober, I sat in my advising professor's office and told her I had changed my mind about pursuing a PhD, so we need not continue drafting our article together. \"Because, you see, Professor, I've had a few posts get kind of big\u2014and I know it's stupid\u2014blogging? How pathetic, right? But I have to try. I have to see if I can write.\"\n\n\"Janelle,\" she said. \"For God's sake, do that. We will always be here.\"\n\nAnd when I finished graduate school armed with a credential \"proving\" I could write, I asked Bea at the law firm to expand my responsibilities beyond pushing paper around for a dull, misogynistic new guy who demonstrated what truly unfortunate bosses act like. (I would never, ever complain about Brian and his color coding again.)\n\nShe answered, \"We continue to look for opportunities to use your talents and skills in the firm.\"\n\nI smiled, thanked her, and walked out of her office knowing the next time we spoke, I would be handing her my resignation. It had been nine years, an excellent run. I wasn't resentful. I was indebted. They had kept me around through the roughest years of my life, helped me fail, helped me get sober. But I didn't belong there anymore.\n\nPossibly, I never had.\n\nWhen I quit, it merely felt correct. It felt _true._ I did not have a plan in mind beyond \"get some gig at a junior college teaching composition and keep writing,\" nor did I have money if my plan fell through, but I knew in my guts it was time to go. I knew it was time to take a step toward the kitchen. My ten-year-old self would have been appalled at my lack of planning.\n\n\u2022 \u2022 \u2022\n\nAnd one day, a few years later, I found myself writing, and teaching some college classes, but with four children now. My days had not changed. They were frantic and they were uninspired\u2014monotonous and heavy and rather butler-like\u2014kid chatter and nursing and back pain and the mess. My God, the mess. Laundry and dishes and parent-teacher conferences and _I still didn't read past email number nine._\n\n_They erase me_ , I'd think, until night, after they went to bed, and I'd sit down at the computer and write some words to some strangers, some other mothers. To say something that was mine. I defended that time. I blocked everyone who tried to enter it. _You can have everything, kids, but you can't have this._\n\nAt work, I served my employer. At home, my kids. But there, on that page, I served nobody. I was in the kitchen, but not that of my grandmothers, or even my mother, but mine\u2014my own place, if only for a few hours a week. When I got there, people asked, \"My God, how brave of you to just get up and do it! I wish I could. How did you do it?\" But I didn't know how to respond, because the only thing I did was get really tired of myself, look down, and take a few steps. Then I was in the kitchen, kinda eyein' the bedroom, thinking, _It will all be amazing if I could only get there._\n\nIt's okay, though. I see now it keeps me moving.\n\nI didn't overcome my fear. My fear lives always like a low hum in the back of my mind. I simply lost faith in it. _Okay, fear, hi._ Maybe the man under the bed is still there. Maybe I should check the closets. But if all that searching and scrambling for safety results in nothing but the same old fear, what's the point of wasting my time? What's the point of remaining silent? _The maggots are going to come either way._\n\nI had nothing to prove when I sat down to write that first post. Nothing left to defend. What was the Internet going to say to me? \"You are a bad mother?\" _Oh! You think? I had no idea._\n\nThere was nothing the world could throw at me I hadn't already caught, no place they could illuminate in the recesses of my being that would surprise me. I had walked them all, visited each one with my whole body and bones and blood. Two inches from my eyes I studied each line, every letter my baby sent me to come home, and every time I didn't, and everything I missed, and every tear my mama shed, and every rush of hope when I seemed to _really be coming around this time._\n\nI guess I found my soul work, but I found it at the bottom, right there with the kid who chased with knives, the woman on her unmade bed, stepping on Legos, washing jam off the linoleum. I found it in the mess I was running from, in the life I deemed lacking, in the fingers never blessed enough.\n\n\u2022 \u2022 \u2022\n\nI suppose some of us don't have the luxury of neatly wrapped truth, of affirmations that rest on our tongues like peppermints. Some of us need to be doused in gasoline and set aflame, until the truth consumes us, and we have no choice but to recreate ourselves. A collision, as Baldwin says, when one must choose to live or die.\n\nI didn't want to feel better. I wanted to live.\n\nI didn't want the pain gone. I wanted it to mean something.\n\nWhen I found my voice, I didn't find answers\u2014I found a purpose for every moment I had lived. I found power in every blackened room in my mind, every fear, every sad parent, every futile word and nightmare memory.\n\nBecause it led me to you, to the place where we are the same, to the place where words draw a line from my bones to yours, and you look at me and say, \"I know,\" and I look back at you, thinking, _Well I'll be damned. I guess we've been here together all along._\n**16**\n\n# **In the Blood of Our Mothers**\n\nMy Grandma Bonny was a writer, too, though I barely noticed when she was alive. From age fourteen to eighteen, I pulled away from my church, father, and mother, but stayed near Grandma Bonny, and I was never sure why. Nobody in my family was interesting or smart enough for my teenaged standards, and she wasn't either, really, but I was drawn to her anyway. I admired her intellect, her power. She was deeply flawed, too\u2014a little cold, a little prone to rage, a little irrational when it came to, say, healthcare for her five children. That was the Mary Baker Eddy in her.\n\nBy the time I was in high school, and she was in her seventies, she had been a widow for ten years but continued living as she always had, rising early each morning and putting on a pantsuit, a brooch, and heading to \"the shop,\" where she would write at a big wooden desk in the back of the office behind a nameplate from the 1960s that said _Bonny Jean Hanchett: Editor._\n\nShe always, for as long as I can remember, drove a Ford Taurus, along with nearly my entire family, because my father's cousin worked at the local Ford dealership. Every two or three years, she'd get a new Taurus, and every Wednesday she'd pick me up from school. I loved the smell of her car. The air that came out of the vents was musty and clean, just the same as when she'd drive me around Clearlake and to her church when I was a little girl. It smelled like Grandma Bonny.\n\nShe would wear her pantsuit even in the dead of summer, with her smooth white hair in a bob, and she'd accessorize with gold and diamond rings. Her nails were always neatly manicured, sometimes polished a sensible pink. While we drove to her town of Cloverdale, which was about thirty minutes from my high school, I would ask her astute questions like \"Why is there no 'white club' at school if there's a Mexican club?\" (I had a long way to go before learning of my ignorance.) But I don't remember her answers, because I was more interested in my own voice prattling on about so-and-so at school and how he was teaching a faulty curriculum.\n\nShe'd often turn on political talk radio while I stared out the window and counted trees. Even if I tried to listen, I couldn't follow for more than a few seconds because the radio hosts sounded drab and monotone, discussing topics that didn't matter to me at all, like California legislation or international diplomacy.\n\nOccasionally, she'd yell back at the radio, which I took as a desire to engage me. So I would ask a question, but again didn't absorb her answers, because I knew nothing of international politics\u2014or politics at all. But she talked as if I should know, so I didn't ask. I pretended to understand so she would think I was an informed and intellectually advanced teenager.\n\nIn hindsight I realize she did not expect me to know about these things. She was simply refusing to talk down to me. She talked down to nobody, except any waiter who made a mistake on our order. _God help you then, kid._\n\nWhen we'd arrive at \"the shop,\" I'd drop my Jansport backpack at one of the empty desks and kill time by spinning around in an empty office chair. My aunts would come over to say hello and ask me how my mother was and how school was going, usually with X-acto knives in their hands, but my grandmother went immediately to her desk, and I knew not to disturb her. _She would ruin me._\n\nWhen I was a little girl in the shop, I would draw pictures or play school, make books out of paper and ad scraps, and write notes to my father. Twenty years later, when my father and aunts sold the newspaper, the notes were still hanging on the wall, although they belonged in the trash. They had faded into little wisps of orange or purple or red. Norfy, though, the massive pine, still occupied the entire front office with its sci-fi branches.\n\nWhen she finished her work, Grandma Bonny and I would head back to her house, and she would make me a dinner that somehow always incorporated mayonnaise or the microwave. While she cooked, I would walk around and look at all the Madame Alexander dolls, Beatrix Potter figurines, plates with kittens, and books I had examined since I was a toddler. I wondered why she lived in a mobile home. She could afford more, but after my grandfather died, and she sold the newspaper they had run for thirty years in Clearlake and bought this one, she insisted on living in a mobile home in the little retirement park. I loved how everything had a place in it, though. I loved how I had a place, with her, in all that history.\n\nWhen I was old enough to drive myself to her house on Wednesdays, I would walk in and she'd be reading _Democracy in America_ or Proust or Dickens, or embroidering while hollering at politicians on TV. I'd scour the freezer for ice cream without a layer of crystals, and instead find Bon Bons. I'd eat the whole package while she made me a chicken sandwich with canned chicken, mayonnaise, and walnuts.\n\nI've never seen anybody eat chicken sandwiches like that since, but that's still what I make when I want to remember.\n\n\u2022 \u2022 \u2022\n\nIf it was October or November she would tell me it was time to start wrapping gifts for Christmas, and that she was going to pay me for my work. I would say, \"You don't have to pay me, Grandma.\"\n\nShe'd shake her hand at me and say, \"No, you're good at it, and I hate wrapping, so you'll earn it.\"\n\nI knew she was telling me the truth.\n\nI'd begin by laying out the paper choices. I needed plenty of options to balance the landscape under the tree. Then I would set out the tape, making sure it was in an upright metal dispenser, so I could rip the tape pieces off with one hand. Next, I would set out the scissors, tags, and a pen. Just one pen. A ballpoint that wouldn't smear on the labels. Finally, I would stack the gift bags and boxes and tissue paper.\n\nAfter it was all set up, we would move into the guest room and open the bottom drawer of the dresser, which would be stuffed with tiny gifts she had been buying all year. She would remember who almost every single one was for. All year, every year, she bought gifts for her five children and six grandchildren and two great-grandchildren and all the family members who weren't _actually_ family members but really, at that point, _were_ \u2014plus everybody's boyfriends or girlfriends or spouses.\n\nShe bought the tiny porcelain tea set for Aunt June in London because Aunt June loves England, and she bought the horse figurine for Aunt Caroline because she loves horses. Some gifts were catchall gifts. These she would often hold up in the air, resting one finger on the side of her mouth in the classic thinking pose, contemplating who would be the perfect recipient. She looked so tender and affectionate in those moments. She might put the gift back, or she might make a decision. Sometimes I found the gift in the drawer long after Christmas had passed. _Guess she never decided on that one_ , I'd think.\n\nAfter I gathered my stack of gifts from the drawers and closets, I would begin wrapping. I loved the boxes the most because I could wrap the corners tight and clean, and easily center the bow. When I was done, they felt perfect. I'd wrap for hours while she'd listen to talk radio, occasionally walking by in her long robe that zipped up the front, patting my back.\n\n\"I don't know what I'd do without you!\" she'd say. The finished products were spectacular.\n\nI felt I made Christmas with her. I felt it was us doing it together. I wrapped for her when I was nine, ten, eleven, and twelve. I kept showing up when I was thirteen, fourteen, fifteen, sixteen, seventeen, and eighteen. _Grandma Bonny needs me. I am the one who wraps the gifts._ I was thorough. We had a system.\n\nMy first year of college, in 1997, I made sure to come back at Christmas to wrap with my grandmother, but after that, I didn't come again. My stepmother took over the wrapping.\n\n\u2022 \u2022 \u2022\n\nWhen Ava came in 2001, I had grown impatient with Grandma Bonny's phone calls, because she spoke slowly and quietly, and there was rarely time to listen to old women talk about the British Parliament. I saw her at Christmas, and every year at the Cloverdale Citrus Fair, and I never failed to stop at the shop when we drove out to Mendocino, because it was right on the way. But I had grown busy.\n\nMy father told me in 2008, the last year of my drinking, not to come visit her, even though it was the last year of her life. If I were in my right mind, I would have gone immediately anyway. I would have sat by her side and touched her if she would let me, and I would have studied her eyes to remember. He said not to come because of her dementia, because she wouldn't remember me, but I wonder if it was because of the state I was in, and maybe they didn't want her to see me like that. I was the sickest I had ever been.\n\nWe were both, I suppose, the sickest we had ever been.\n\nSo I cannot remember the last time I saw her, what we talked about, or how. I have one vision of her standing in the doorway of her last home, the bigger home she told me \"the children\" had convinced her to buy after the mobile home. She was wearing a pink fleece robe, and she was waving to me as I drove away. She seemed to wave an extra long time, and I noticed.\n\n\"One of these times will be the last time I see her,\" I said to Mac.\n\nBut I don't think that was the time. That day, we kissed and hugged in the doorway and I said goodbye, but I don't think that was _the_ goodbye.\n\nIn the end, there was no goodbye, and I was still a loser when she died.\n\n\u2022 \u2022 \u2022\n\nShe once told me while I sat across from her on the couch that she wore red velvet to her wedding in 1943. She said, \"They weren't going to turn me into some blushing bride!\" When she said \"blushing bride\" she put her hands up under her chin, wagged her fingers up and down, and flitted her lashes to make me laugh. That was not the last time we talked, but it was the conversation I have chosen as the last, because it was my favorite, and I can't bear to think of what our actual last conversation would have been. If I could see her with the mind I have now, I would tell her how much fun I had with her, and how I always felt a strange, superhuman connection to her, because we are similar women, skeptical and \"offending\" and \"just too much,\" with tiny rage problems and loads of children, but an inability to accept them as the full definition of our selves. I would thank her for reading me Beatrix Potter and letting me eat three Danish pastries with butter on top and showing me how intellectual mothers can be and that some of us will simply never fit.\n\n\"You know? We just don't, Grandma,\" That's what I would say.\n\nI would record her talking about her life, so I could hold it and hear it and learn it, because there were a thousand stories beyond the ones she told me and I forgot, and a thousand more she never told me, about growing up behind the stage, becoming the first woman editor of the University of Washington's newspaper, about World War II and a veteran husband who sometimes woke up sweating from nightmares, about the time she crawled onto the couch and refused to move, and nursed herself back to sanity by doing paint-by-numbers her friend had brought her. About the day Uncle John died of alcoholism, and she said simply, \"A parent should never outlive her children,\" and it was the only time I saw her drink a glass of wine.\n\n_Oh, I want to hear about writing, Grandma, about people who hated you, about journalism and female editors in the 1950s, the Mafia men you fought with about the resort on your lake_ \u2014the algae lake with the rough water that lulled me to sleep on those hot-cold sunburned evenings, the one I drove by and remembered, and wondered when it was that we last spoke, and if she wondered the same as she drifted on, or if the dementia took that away, and I had become nothing but a speck of curled ribbon in the recesses of a fading mind.\n\nPart of me wished that were true. Part of me wept at the thought.\n\nOn the day my father called to tell me she had died, I was sitting in a bar that smelled like urine and bottom-shelf bourbon. I notified the man sitting next to me of the passing of my grandmother, and he felt very sad for me. Surely he would have bought me a round had he a few extra bucks. I remember feeling almost nothing, and wanting to, but I had passed the moment when feelings came the way they should and I lived like a real human on a real earth.\n\nSix months later, I left myself in a bed, maybe the way Bonny Jean left herself in hers.\n\n\u2022 \u2022 \u2022\n\nWhen I was a month sober, my bedroom door opened suddenly in the middle of the night while I was in a dead sleep. I startled and sat up in bed in time to see the door open, rest for a second or two, and then shut again. My dog jumped out of his bed and barked at the door.\n\nI thought it was my mother, the only other person in the house, but I needed to make sure, so I said her name and opened the door. But I saw only an empty hallway. I walked across the hall to her bedroom, and from the doorway I saw her sleeping and heard her snoring.\n\nAs I sat on the edge of my bed considering what had just happened, I was overwhelmed with a feeling that Bonny Jean had visited me. Out of nowhere, a feeling of her presence overtook me, as if she were standing right there, and I could smell the scent of her car, see her wrinkled hands as they clasped in delight at my wrapping or in rage at the audacity of Fox News.\n\nIt was as if she had opened the door and looked in, saw me sleeping there, and moved on. As if she had opened the door, looked in, and said, \"Oh, finally, you're okay.\"\n\nPerhaps that is the last time we met; though we didn't say goodbye, we were both in our right minds, and that's saying something, I guess.\n\n\u2022 \u2022 \u2022\n\nSeven years later, when my maternal grandfather, Bob, was dying in October 2016, I was there, because I knew what it meant to miss a grandparent's final days. When I walked into his hospital room, I leaned over the foot of the bed, and he grabbed my hands and said, \"It's time.\" And I said, \"Yes, it is,\" and made my bravest face for him, because I saw fear in his eyes. He wanted to die at home, but Grandma Joan wasn't ready to accept his death, not quite yet. But by the time she stopped demanding more pills and doctors and possible new medical advances, he had mere hours to live, and they couldn't arrange hospice in time. He passed away in the night surrounded by his daughters and wife, by a window overlooking the maternity ward.\n\nHe and Joan Lila had four daughters, twenty grandchildren, and forty-four great-grandchildren. Many of us passed through his hospital room, taking turns sitting by the bed and leaning against the walls. We cried and laughed as we had done our whole lives together in the basement of their home two miles from that sterile room. Occasionally, Grandpa Bob would wake and lift his head, startled and sudden, and look slowly and deliberately around the room at each one of us, from behind the oxygen mask he kept yanking off, and I wanted to beg my family to just let him take it off. _Let him be comfortable. He clearly hates that thing. What does it matter now? What are we hoping for? An extra hour?_\n\nThen I realized we were keeping him alive for us, for Joan Lila.\n\nHe would lock eyes with one of us, hold our gaze for a few seconds, and fall back asleep. When he locked eyes with me, I did my best to send him strength, to see the part of him that knew he was stepping into the great journey and didn't want to go. I wished something could soothe him.\n\nGrandma Joan refused to leave his side. She rubbed his arm where there were no IVs and fixed the tape around their entry points. She brushed wisps of hair from his forehead and said, \"Oh Bob, we had a good life together, didn't we?\" She told him story after story, of their friends and daughters, and boating on Clearlake, and he watched and dozed and nodded in recognition. His eyes held hers desperately. Once, he pulled his oxygen mask off and they had one big, last smooch, and it made a nurse kneeling next to them drop her head and cry. Their hands entwined and he pressed himself against the side of the bed, his face as close as he could get to hers. To watch them felt like watching the perfect end to the love story we all wish we could have, the piercing agony we all strive for.\n\nWhen it was bedtime that night, and all of us grandchildren were heading out for sleep, we said, \"We'll see you in the morning, Grandpa,\" but we knew that was unlikely, so we said longer goodbyes than usual. Mostly, though, we relied on the thousand goodbyes we had tossed over our shoulders as we left their home of forty years. _Until next time. Love you._ There was no way we could say enough, so we simply said one more goodbye\u2014lingered longer, grasped his hand, let go, and forced the turn to walk away.\n\nWhen I was standing in the doorway looking into his room, aware that I was probably looking at my living grandfather for the last time, a new nurse came in to help settle him and check his gear. The air hung heavy with nearing death, and my grandfather was restless. So when the nurse walked by the television, she said to nobody in particular, \"We have a special channel for moments just like this,\" and flipped to a station with soothing music and goldfish swimming around against a blue background.\n\nIt was a fake fishbowl television death station.\n\nDuring moments like this I feel like I am on the edge of the world, looking on, a hundred miles from whatever land the rest of you are inhabiting, wondering how nobody sees the tragedy of that gesture, how infinitely weird and empty we are. I studied the scene: my grandmother staring into his face, my mother and aunts bustling or sitting, eighty-seven years of life and love circling to its close in mere hours, and here, on the night before us\u2014the last night\u2014all we have to offer are motherfucking TV goldfish to see him into the abyss.\n\n_Hey man, sorry you're dying. Here are some goldfish._\n\nI took a hard look at his big, calloused hands that he used to sweep over his mouth after he cracked a joke or teased my grandmother, and I walked out the automatic doors.\n\n\u2022 \u2022 \u2022\n\nFive weeks after we buried my grandfather, on the morning of November 10, 2016, I was pulling out of my children's school parking lot after dropping them off when my phone rang, and I thought, _Oh, good, it's my mom telling me she's forgiven me for being an asshole last night._ Donald Trump had just been elected President, and I had taken my shock out on my mother. But as soon as I clicked \"answer,\" I only heard her screaming.\n\nAt first I thought my mother was getting stabbed because I heard \"knife\" and \"stabbed\" and \"killed,\" so I screamed at her to leave her house. But through more chopped screams I realized it was my grandmother who was being stabbed.\n\nThen I learned she had already been stabbed, and was in fact dead, and my cousin was the one who did it. My own cousin, the one who sat with me in my grandfather's hospital room but registered no emotion, no affect, no nothing. I noticed, and I asked about it. I was told he was suffering from depression. I said I thought he needed help, that he was perhaps suicidal.\n\nI was twenty minutes away from my mother when I picked up the phone, and I knew she was alone in her house, so I drove and wailed and tried to breathe while telling Arlo\u2014our fourth child, two years old, the beautiful completion of our family\u2014that I was okay, even as the air I breathed felt like fire and the air moving out felt like drowning, my terror a thousand shards of glass tearing through me, slamming the outside of my body.\n\nI drove to her while my baby looked at me wide-eyed and silent from his car seat. I called Mac and screamed to him. He thought one of our children had been killed. Hysterics, actual hysterics, are virtually impossible to comprehend. I tried so hard to pull enough air to speak sentences, but all that happened were quick bursts of guttural wails and broken words through burning breath.\n\nWhen I got to the intersection two streets away from my mother, a wooden arm came down in front of me and a huge cargo train rolled in. The train stopped entirely, and I couldn't get to her. I was the first in line in the intersection, and I couldn't get to her, so I yelled \"No no no no\" while immobile in the car and shook my head and sweated and screamed and looked at my boy who sat silent, until I could get to her. When I did, she was half-dressed, doubled over in a chair and shrieking with the shock of a terrified child, \"My beautiful mother! He killed my mother!\" When she said \"mother,\" she rose and spun around and ran across the room, and sat and rose again and cried out again, \"He killed my mother!\"\u2014over and over, while I watched with eyes wide, fighting wild, insane panic and sorrow.\n\nAn old woman murdered by her grandson. My grandmother. My mother's mother. Joan Lila.\n\nMy mother fell against the walls, and I held her up while my little boy stood looking out the front door I had forgotten to shut, holding his monkey lunch box in front of him with both hands, as if he were waiting for a ride on a train that had just departed.\n\nIn those early hours we knew no details of the crime, so I imagined my grandmother suffering, slowly bleeding to death in pain and isolation. But it turns out that after eating Chinese takeout with her and his mother, my thirty-five-year-old cousin went upstairs to the bathroom, and when he came out, went into his bedroom, where he unsheathed a Kershaw hunting knife, stuck it up his shirtsleeve, came downstairs, and stabbed my grandmother in the neck, once, from behind.\n\nThis I found to be a significant relief.\n\nAfter spending a day envisioning her body mutilated and writhing, I felt relief that she hadn't seen him coming. I felt relief that it was instantaneous. I felt relief that her last thought was not, _My grandson is about to kill me_. I felt relief her final moments were not terror and betrayal and the watching of a hand raised to annihilate her, defenseless and yet fighting anyway\u2014the instinctual shriek and cowering\u2014and the subsequent fall, bleeding, and extinguished light, with not even a goldfish to comfort her.\n\n\u2022 \u2022 \u2022\n\nGrandma Joan and I had not spoken in almost a year before my grandfather's death because I was angry with her, and had convinced myself we need not speak again, ever. She had sent Mac a message online asking why our daughter Georgia was \"shorn like a boy\" and inquiring if her mother was using her as a political project. You know, a sort of \"girls can have short hair\" poster child to further my liberal agenda.\n\nBut when my mother told me my grandfather was dying, I was reminded of the eternal backdrop of impending death, and my charge grew petty and damn near shameful. I left for the hospital immediately. As soon as I turned the corner in the hospital hallway, Grandma Joan came running to me and grabbed my face, saying, \"I'm sorry, I'm sorry. Do you forgive me?\" Of course I did, and had for many months, but was still clinging to the self-righteous indignation that always failed me. I took special care to focus mostly on my grandmother, to pat her back, bring her water, and hold her hand. She was dizzy with confusion and exhaustion and helplessness, watching her love travel on without her, and I wanted to fix what I had done, because she was eighty-five and didn't understand little girls with mohawks. Do we hate them for that? Our mothers and grandmothers? Do we hate them for roles they've internalized and been forced to play? Or do we simply love them as they pass into the gray?\n\nJoan Lila was told she couldn't go to college because she was a girl, and my grandfather preferred she not work, but she did anyway, and perhaps she had some ideas and hopes for herself that simply were not viable on account of her gender. So when she came to my house a week before she was slaughtered, and then again two days before she was slaughtered, I felt a rush of joy that I had a grandmother again\u2014to be near, to be with, to maybe wrap presents with. I thought, _Oh,_ _we can do so many things together. I'm going to take her everywhere._ Because there was a part of her that had come alive, maybe the part of her that had wanted to go to college at eighteen, and even though she loved my grandfather with a fantastical-type unicorn love, there was a brightness in her step as she told us of her plans.\n\nOh, her plans. She was going to buy a little place closer to the city, closer to where we lived, and she was going to visit each grandchild to get to know each great-grandchild \"really well.\" On the last day she visited, she wore makeup that matched her blouse and a vibrant red scarf around her neck as we perused the books at Costco\u2014her tiny, wrinkled fingers stretching over the embossed covers\u2014with more places to visit now, at eighty-six, than perhaps in the entirety of her life.\n\nInstead she was eliminated at 7:30 p.m. on Wednesday, November 9, 2016, by a grandchild she loved with the fullness of her heart, a grandchild who had ceased taking medications for a psychiatric condition nobody told us he had.\n\n\u2022 \u2022 \u2022\n\nWithin a few hours of arriving at my mother's house, after a friend had picked up Arlo, and Mac and my brother had come to help hold my mother's trembling body, I realized she was living the fear that had been pulsing through me since I was a little girl. Somebody had hurt her mother. Somebody had taken her mother. Somebody had caused indescribable suffering to the woman who held her as a newborn, whose existence was a blanket around shivering shoulders. And my mother, my mother got the phone call I was always waiting for, for years, rocking on a couch as a child, wide awake in college, imagining an intruder. _Here it is, Janelle. This is what it looks like. This is the horror you built in your imagination._\n\n\"Maybe she is not actually dead. We haven't heard from anybody who saw her body. Maybe she's just sick at the hospital,\" my mother said, getting up and packing a suitcase. \"I have to be with her.\"\n\nMac rose silently, called the police in the city where the crime occurred, and spoke softly with the detective. We watched him taking notes while my mother ran around the house packing, and I searched for the power to say, \"Mom, she's dead.\"\n\nWhen Mac hung up the phone, my mother cried, \"Where is she?\"\n\nMac looked at her with the same kind brown eyes I knew from that first night sixteen years before, the eyes that caught me in the flurry of my life and held me near him, with him, if only to discover how deep the kindness goes. He nearly whispered, \"She's at the coroner's\" and grabbed her as she collapsed again against him, her legs too weak to support the truth.\n\nThat night, as I slept next to my mother like she had always done for me, as I pulled the blankets up under her chin and tucked her hair behind her ear, I wondered if all the agony and all the terror of my life\u2014of the surety that my mother would die one night in some horrific way, and I would be left motherless and powerless\u2014had been leading only to this very moment.\n\nMaybe I knew. Maybe I knew somewhere in my bones that a mother would die, and that her daughter would mourn, and though it would be my blood, it would not be my body.\n\nMaybe I knew it would be so close I could touch it, but it wouldn't quite be mine.\n\nI rolled toward her and closed my eyes, thinking, _This is it. Here it is._\n\nI felt a silk-spun terror pulled from my body, thirty-seven years of web around my bones, lifted and carried off, as I opened my eyes again to watch her sleep\u2014my motherless mother, my grandmothers gone, and an entire night before us.\n**17**\n\n# **If I Knew the Way**\n\nJanuary 20, 2017\n\n_Did my aunt see him first, and gasp, causing my grandmother to turn? Just in time to see his face before the knife sunk into her neck? And if she was looking forward, no warning at all, was she speaking when the knife went in? What was she saying? Was she looking at her phone? Or doing absolutely nothing? Maybe they were talking about her flight the next day to visit her granddaughter, who was cleaning her house and preparing her guest bed at the very moment a thirty-five-year-old man we all grew up with but suddenly stood a complete stranger buried metal into her tiny neck, wrapped in her mother's pearls._\n\n_Is this how we go? Is this all there is? Is this what we're doing here? Do we live our whole lives in work and service to families only to be slaughtered? And in a kitchen, no less? The kitchen we warmed with our own love? To be slaughtered like a fucking barn animal by the very humans we devoted our lives to?_\n\n_If I could, I would kill him. If I could, I would rip his fucking throat out. But I cannot. Because I am not a murderer. So I remain. We remain. In the blood of our mother._\n\n_We remain in the blood of our mother._\n\n_We remain in the blood of our mother._\n\n\u2022 \u2022 \u2022\n\nI wrote the words \"we remain in the blood of our mother\" fifteen times on the page of my journal while sitting alone on a beach in Santa Cruz, and each time I wrote them, the letters were bigger and more jagged and I pressed harder and harder, just as I had my whole life, hoping the torture in me would absorb like ink.\n\nI couldn't save her\u2014nobody could\u2014and nobody could save my other grandmother. That day on the beach I felt it, their absence, and how I simply remain in their blood\u2014my grandmothers, and my mother, who seems different now, who seemed to have a light in her eyes extinguished that day, a wild light that carried her to Yosemite and back home, to me. I understand, though I fear she won't return. I fear she will remain in the blood of her mother, and not a damn one of us will be able to save her.\n\nSo I drove to the ocean as my mother had done with us. I looked to the rage of the water as it sang its roar against the rocks of my heart, and waited to be filled again.\n\nI pulled out a fountain pen and the leather traveler's journal my brother had given me for Christmas, and found myself writing that I wanted to kill somebody. But really, I wanted relief. I felt I was drowning in the blood of my mother and her mother and my father's mother, all the mothers in my life and beyond, whose blood pulsed through me like fire but never like annihilation, until then.\n\nI saw myself in them. I saw myself in their eyes or turn of the head or laughter, and it had always felt like connection, like life. One Christmas after Grandma Bonny died, my aunt walked up to me after I had been writing about motherhood for years, and handed me a thick plastic-bound book. On the cover it said, \" _Between Us Girls_ , by Bonny Jean Hanchett.\"\n\n\"Your grandmother wrote a column to mothers for many years. I collected them for you. Did you know that?\"\n\nI looked at my aunt. \"No, I had no idea,\" I said.\n\nAs I thumbed through the pages, a chill ran up my body, an electric channel that opened to words and a woman who wrote them seventy years before. I understood then why I kept showing up at her house. I was to become her: in words, in rage, in broken, slightly pathetic mothering.\n\nIt was better than a ghost. She was with me. No, she was me.\n\nI am her.\n\n_We remain in the blood of our mother._\n\nSitting on the sand with my journal on my knees, I looked up and saw in the breakwater the shadows of my children as they played on that very beach a few months before, before my grandfather watched goldfish swim into the abyss and my grandmother bled on the linoleum, before I knew such things were even possible. The older children chased waves and explored tide pools while my mother, Mac, and I corralled the younger ones, held their tiny resistant hands, tending to the little minds who don't yet know white water can turn without warning into massive sleeper waves.\n\n\"Never turn your back to the ocean!\" the older children told the younger ones, but they hadn't seen enough to listen yet.\n\n\u2022 \u2022 \u2022\n\nWe have always visited Santa Cruz a few times a year, but for four years in a row, Rocket wanted to go there for his birthday celebration. We often stayed in a motel in Aptos, a little town a few miles down the coast from Santa Cruz. We'd wake up on Saturday morning and walk a few blocks to the beach, then turn right and head down to the pier with the metal boat at the end, on the other side of a wood fence. It's a famous metal boat. Everybody knows about it, though nobody seems to know why it's there.\n\nOn our second trip to Aptos, we were walking to the beach when a white house with gray trim caught my attention. As we moved closer, I realized I had seen it before, many years earlier. I looked harder as the memory formed in my mind, and I realized I had actually been in that house before.\n\nI had driven there with a boy named Evan when I was sixteen years old. The entire night rushed into my mind: sitting in his Jeep along winding roads, pulling up to the house on the corner, parking across the street, getting beer from a supermarket that sold to underage kids. The joint he smoked while I politely declined.\n\n\"No thanks, weed's not my thing,\" I said. He was older than me and excessively popular, and I was trying to act confident, but I was afraid.\n\nEvan and I drank a lot of alcohol that night, took some ecstasy (which was bunk) and some cocaine, but I remembered that night in particular because it was when I smoked heroin for the first time. He and his friends had to teach me how, though I felt no resistance. I sat on a couch with people I had just met, and somebody brought out the tinfoil, lit it underneath, and showed me how to inhale the smoke through a straw.\n\nAfter that, Evan and I decided to go for a walk, so we went down to the beach, turned right, and walked down to the pier. It was the middle of the night and there was nobody around. I wanted to see the metal boat at the end, but the fence was locked. I tried climbing it but couldn't get a footing on the front of the boards, so I kicked off my shoes and scaled the railing of the pier while Evan yelled at me to stop.\n\nI did it anyway, and he joined me, probably afraid of compromising his masculinity. We were high up, and beneath us was the crashing, freezing ocean. It was two or three a.m. and black below us. The fence extended from the sides of the pier for a few feet, but if I could get around to the other side I could get on the boat. So I tiptoed along the tiny wooden lip beneath the fence, clinging to the top of the boards, all the way to the end, but then I had to swing my body around the last pillar, and that seemed particularly dangerous. Evan was with me until then. He said, \"You're fucking crazy\" and went back onto the pier. I looked at him and smiled, held on tight, and swung my body out over the ocean and around to the other side, not knowing if there was a footing.\n\nI made it. I looked down at the rolling water, and it only looked lovely and soft. By the time I got to the metal boat, whatever drugs I had in my body were wearing off and I got that familiar anxiety and urge for more that made me forget everything I was doing and focus on what mattered. More. So I didn't even sit and enjoy the boat. I just yelled at Evan on the other side of the fence and told him I was there, then climbed the boards on the back of the fence and hopped down. I put on my shoes and walked back, hoping this guy wouldn't try to have sex with me and wondering how much more blow was available.\n\n\u2022 \u2022 \u2022\n\nThe second time I recognized the Aptos house and recalled the metal boat escapade, I was thirty-five and pushing a three-month-old baby with Mac and our three other children. I felt fat and frumpy with my gray roots showing, and was hit with a blast of shame and sadness because back then I had been young and beautiful and hopeful about what life would become. Life was really going to be something, and when Evan was gone, I thought someday I would find another man who loved me even more, and I would have reached that potential everybody had been talking about since I was a kid in the \"Gifted and Talented\" program.\n\n_This whole cocaine booze sex cigarette jumping-over-fence thing is merely a phase. It's a silly one-time, maybe few-year, young-person phase, and the metal boat is just around the corner, I tell you. The thing I'm risking it all for is just right there._\n\nI believed it.\n\nAs Mac and I walked with our children, we had Styrofoam cups of motel coffee in our hands. It was nowhere near strong or plentiful enough. Georgia was throwing a tantrum because she wanted to push the stroller and we wouldn't let her. Ava was complaining about the flies. I told her it was from the seaweed. Rocket wondered about the driftwood scattered across the sand. Mac told him there must have been storms recently.\n\nI watched the joggers in awe, as usual. _Look at them out here running at nine a.m. on a Saturday._ The fog was perfection. People were already arriving to set up birthday parties in the barbeque areas. I saw them and I saw us and I saw the ocean. And in flashes, when I had a second or two in silence, I saw me\u2014sixteen years old, spinning over oceans and piers and men like straws over heroin.\n\nI wasn't going to fall. I could not. I was holding on and too young to die.\n\nI walked along this road at sixteen, high. I walked it again at thirty-five. I walked it at thirty-one, too, but somehow failed to notice.\n\n\u2022 \u2022 \u2022\n\nPerhaps it took a few years for my mind to accept the memory of that white and gray house. Perhaps the brain blocks us from what's too strong, too soon. After eight years of sobriety, I've grown better at it, better at encountering places that bump me into history, into living for a moment a tiny irreconcilable truth.\n\nAt first, I couldn't look at the places at all. I couldn't look the first time I drove by the motel where Mac and I stayed because there were \"people\" in our house. When I think about that time in the motel, the pier and metal boat seem less appealing, and even though I'm older and more tired now, I am freer, because I don't need to throw myself over a cold dark ocean to get to where I'm going.\n\nAt first, I couldn't look at the craft store in my town either, because I once drove Ava to preschool after taking drugs, and I couldn't make it back in time because those lines wear off quickly, so I stopped behind that craft store and did another line, right there in the car off the back of a binder. Many years after that, I bought material there for the Harry Potter costume my mother was sewing for Ava. Now, I buy felt and poster board there for school projects, and clay for Ava to shape into little animals for gifts at Christmastime, and mini hay bales to decorate our porch in the fall.\n\nThe liquor store down the road was _my liquor store_ , where they knew me by name, and I was sure nobody from my family would see me. I could hide there. I could buy my bottles in peace. Once, Mac saw me driving out of that liquor store parking lot and followed me in his car. He was irate and wanted to know who I was with. I couldn't stop because I was drunk and thought he would kill me, though that was my imagination. Mac is a gentle man, but it was a dangerous time for both of us. I was absent and loose, insane and drunk, and he was sober but miserable and insane, and losing his wife.\n\nI wanted to be as good as him but I didn't know how. I wanted to be his wife but could not. I felt sorry for myself and ordered another whiskey. But after that run-in, I was very particular about where I went. I found a new liquor store. Now I buy milk at that liquor store when I don't have time to make it to the market.\n\n\u2022 \u2022 \u2022\n\nI don't shake my head anymore to get rid of the memories. I take a deep breath, look at each line of the picture in my mind, and get as close as I possibly can to the image and the pain it causes. I say \"thank you\" sometimes, in moments when I'm sitting at some horribly boring back-to-school night or student pick-up line, or working at my child's preschool, or doing some other motherly thing. Because on the surface, I fit in now.\n\nBut I have a secret that's not really a secret, and I hang out with many alcoholic mothers, trying to make those years worth living. When they tell me the horrendous and disgusting things they've done, I flinch and think, _What kind of fucking dirtbag...,_ but then I remember my own damn story, and that I, in fact, was that dirtbag. So I nod, and tell what I did, and how I recovered, because I want them to see that the water they need to wash themselves clean flows always and immediately to the lowest possible places. And I know that God, to me, is that kind of love.\n\nI always thought I had to get holy before some Power would help me, if there was one. Now I see that it is when we are our most vile that help comes pouring in, meeting us where we are at the bottom, where all the humans refuse to go anymore, because it's dark and reeks of the dying. Maybe they've forgotten that from their sunlit vistas, that some of us are broken enough to believe we need rebuilding, and that lifting us is worth their time. Or maybe they never knew.\n\nI know, though, and I'm here with them still, though I'm walking around a life that isn't my life or wasn't my life once, and the two seem wholly incompatible, impossible even, and yet this is the same body that inhaled and spun around the pillar over the ocean, and now I'm making semi-forced small talk with you in your yoga pants and Honda Pilot because we're both mothers picking up our kids and it's hard and thankless and perfect. The only indicator of those years are my ill-timed swear words and the slash marks on my arm, which you probably can't see now anyway, as the years have faded them into almost nothing. (You can see them if you really look.)\n\nYou can't even get to the metal boat anymore. It sank down farther into the ocean and now it's completely removed from the fence. Even if you jump the fence you can't get to the boat. Now it's only a place for the seagulls to shit and the tourists to wonder what it was like to sit on its hull or walk along its sides. It looks creepy and broken and rusty now, half submerged in murky blue.\n\nI knew it when it still seemed like a boat. I knew it during its better days. It never saw me during mine.\n\n\u2022 \u2022 \u2022\n\nI didn't reread what I wrote on that journal page, not on that January day, though I noticed how my handwriting got big and jagged and crooked as I fought and scribbled \"blood of our mother\" the last few times, my hand growing tired and unhinged from confusion and pressing so hard. I looked up at the waves and remembered one particular birthday trip to Santa Cruz\u2014the worst one. The best one.\n\nMac and I had woken up in the mood preceding a day trip to the beach, chatting for a few minutes while still in bed. I told him a friend was really interested in a woman she'd been dating.\n\n\"Is she down like four flats on a dump truck?\" Mac smirked, cracking a dubious smile.\n\nI raised my eyebrows, \"I have no idea what that means.\"\n\n\"Flats,\" he said, lifting his hand in the air and flipping out his fingers. \"On a dump truck...Wait. Are you serious?\"\n\n\"Of course I'm serious. I do not understand your construction talk.\" I said this in my bitchiest, most pretentious voice, knowing he'd play, knowing we were on the frequency of old friends.\n\n\"This is not construction, Janelle. This is about trucks.\" He tilted his head forward toward me and looked at me side-eyed, as if that face was going to help me understand the simile.\n\nI stared at him.\n\n\"What do you think it means?\" he asked, almost hesitantly.\n\n\"I have no idea,\" I said. \"Something about flatbeds? Flats? The back of trucks?\"\n\nHe shook his head and with a quick flash in his eye, said, \"You know what? The next time you say something about Baldwin or Foucault I'm not even going to try to understand what you're saying. I'm just going to say whatever stupid shit first comes to my mind, like you do.\"\n\nI roared in laughter. It was perfection, because that was exactly what I had done, and he saw it, and knew it, and yet I really had no idea what he was talking about, because so often, the obvious answer is lost on me. It's right in front of my face, but I'm staring left, or right, or over in some dark minuscule corner, where vague metaphor and \"deep meaning\" lie.\n\nI spent the first ten years of our marriage like that, trying to change Mac, trying to mold him into the man I thought he needed to be, the kind of man who could \"fulfill me,\" a better version of himself that only I could see, of course.\n\nBut it never worked. He never got better at organizing linen closets or finishing projects he started. He never got better at budgeting or perfect at anticipating family needs or articulating his deep emotional development. He never talked as I talked at him. He never got better at realizing my vision of \"wedded bliss.\" But during a particularly difficult moment at around marriage year ten, when I was four minutes away from divorce, Good News Jack asked, \"Janelle, where did you get your ideas about marriage?\"\n\n\"What do you mean?\" I asked, frowning through the phone.\n\n\"Where did you get your ideas of romance, of what it's all supposed to be?\"\n\nI traced my concepts of \"marriage\" all the way back to the beginning, all the way back to the magazine wedding dress shoved in my journal, to the yachts in the San Francisco Bay, to the love songs and teenaged daydreams.\n\nI realized then it was Hallmark cards and Meg Ryan movies.\n\nAnd I was wrong again.\n\n\"For people like us,\" Jack said, \"life is a series of discovering all the things we've been wrong about.\"\n\nThat morning, the day we were heading to Santa Cruz, I stood cooking some bacon at the stove, and Mac sat in the dining room adjacent, and for some reason Georgia asked about our wedding photos, which I had burned, one by one, in the little apartment where I had lived alone, desperate and sure I would never need them again. The ones that showed us beneath the trees at the courthouse. The ones with Ava in a carrier on my mother. The ones of Mac and me holding hands and looking at one another with tears in our eyes.\n\nI told Georgia, \"Well, we don't have any because your dad got upset one day and destroyed them all.\"\n\nI threw Mac a knowing smile.\n\nGeorgia asked, \"Why did you do that, Daddy?\"\n\nI was about to own up to my lie, but before I could, Mac answered, \"Because I was very sad, and I thought getting rid of those photos would make me feel better.\"\n\nAfter he said it, he looked up at me straight into my shame, with a love and compassion and forgiveness that now, sixteen years after that courthouse wedding, takes my breath away, as if I were standing in a white gown in a castle in France, or on a yacht in the San Francisco Bay. I held his eyes again, as I had done the night we met, and I couldn't believe that love was mine.\n\nBut I had no evidence of the things to come back then. I wish I could elbow myself as I stood there. I'd say, \"Hey, Janelle. Just wait. There's some big love coming your way. The wedding doesn't matter, kid.\"\n\nI never would have believed me, though. That's for sure.\n\nWe were friends. We were always good friends. And mine were tears of joy after all.\n\nAs soon as we pulled into our parking spot at the Boardwalk in Santa Cruz, a beachfront amusement park, Mac jumped out through the passenger door and vomited on the asphalt. I figured that was an ominous sign. He spent the next five hours in the car, green-faced and sweating, while I tried not to lose four kids in the endless crowds and sweltering heat that is Santa Cruz in September. It was Rocket's tenth birthday, after all, and I tried to make it great. I never forgot his first one, and I'll never give up trying to repair it.\n\nI was convinced Mac's sickness was my fault on account of my bad behavior at Starbucks a couple of hours before. I had been flippant and rude to one of the baristas because they charged us extra for a dash of soy milk as cream, even though I explained that Mac can't drink milk and \"nobody ever charges us for a little bit for his coffee.\" I was thus convinced they had poisoned him with spoiled soymilk. I vowed once again never to snap at the people preparing my food.\n\nBy the end of the day, it was clear Mac wouldn't make the three-hour drive home, so we paid $300 for a last-minute motel room that should have cost $100, and piled our children into the tiny rectangle. After dinner, Ava started feeling sick. And then Rocket. I began to think perhaps my bad behavior had not caused the illness. By two a.m., everyone was sick except me, so I passed the night rotating around the bathroom and beds, and I thought, _Well, obviously I am in hell._ As the only well person, I cleaned vomit and changed Arlo's diarrhea diapers, unpacked the car and helped the kids wash their faces and sip water and change out of puke clothes.\n\nI woke up the next morning after one hour of sleep, facing a three-hour car ride over winding roads with four sick children and a barely recovering husband. We had to get home. Mac had to work the next day, and staying would cost too much money anyway. I loaded everyone into the car and drove straight to a liquor store, where I knew they'd sell those red party cups, which would be perfect vomit cups.\n\nIt was a beautiful day, but my eyes ached in the sun, my bones heavy from exhaustion. I had spent the whole day before in the relentless heat, and in crowds, _my God, the fucking crowds_. _And now I'm standing here in a liquor store buying bottles of water and red party cups_. I gathered it all into my arms and walked to the checkout line, where I saw a woman who swept my mind immediately, erasing every thought and filling every crevice.\n\nIt was twelve p.m. on a Sunday. She stood in front of me in a ruffled skirt and combat boots and tights. It was too hot for such a get-up, and she'd probably been wearing it since Friday, when things were better. Her hair was sticking out and frizzy around a few-day-old braid. When she turned, I saw tattoos along the side of her face, which was swollen and pale with bloodshot eyes. The alcohol radiated off her body, and the smell smacked me into eight, nine, ten years ago.\n\nThat sweet-stale reek. Cigarettes and sweat.\n\n\"Can you give me a deal on a pint?\" she asked the cashier.\n\nHe had already put her pint on the counter, grabbing it before she asked. He knew what she wanted. I looked down at the bottle. Rot-gut whiskey.\n\n_My kind of girl_.\n\n\"No, sorry,\" he said, offering a vague smile. He didn't have to say it, though. We both knew what his face meant: \"It's been too many times.\"\n\nI considered setting my stuff down on the counter, because my arms were achy against the cold drinks, but I didn't want her to feel rushed. She had enough stress.\n\n\"It's cool, man, give me a minute,\" she said. \"You know I'm good for it, though.\" She set some change and a couple of dollar bills on the counter.\n\n\"How much do I owe you?\" she asked, and her feigned cheerfulness made my heart damn near crack.\n\n_One dollar and seven cents more for the rot-gut pint._\n\nShe dug in her bag and in the folds of her green canvas jacket, pulling a nickel or two from the plastic penny holder on the left.\n\n_I used to do that. Saved me a few times, too._\n\nShe remained seven cents short. I opened my purse to grab a dime when she said, \"Hold on!\" and ran to the back of the store, where I watched her grab a dime off the floor. She placed it on the counter triumphantly.\n\n\"We're good today, man!\"\n\nI was happy she didn't have to take money from me. I was happy she got her pint without a front or a handout, and I was happy she could kill the shakes. I knew she was thinking, _I'll be okay today,_ and I was glad that moment was happening for her, though it wouldn't be enough.\n\nIt will never be enough.\n\nThere will never be enough.\n\nShe grabbed her whiskey and turned around, wafting that smell again, but as she faced me, she stopped, and looked me right in the eyes. She paused, and looked harder with her chin high and proud.\n\n\"Any day now I'll be back to my normal self,\" she said.\n\nI held her gaze. I couldn't speak. Her words seemed to knock the breath out of me. I smiled a little and nodded, seeing her as best I could. She probably didn't notice my nod, or my smile, but I watched her walk outside and get on her bike. I watched her ride away in the clear sunlight.\n\n_God dammit, why did you say that to me? Why? Of all the people and things and moments in the world, I stood behind you on just another alcoholic day in a liquor store and smelled your smell, my old smell, and you spoke the saddest words maybe I've ever heard in my life, and your watery eyes were mine again, yet they were not. Because I'm free now._\n\n_Why?_\n\n_I'm a stranger to you. A nobody. A nothing._\n\n_When I was you I would have turned away from a woman like me, all clear-eyed in the midday with kids and shit._\n\n_\"Oh, fuck you, lady. Fuck you and your decent life.\" That's what I would have thought._\n\n_(And then, in the throes of the morning, I would have begged God to join you.)_\n\nI knew her. The pain. The hope. The energy in the unopened bottle. The strength pulsing through the walls of the glass in my hand. _Just this last pint. Just this one. I'm okay today. It's okay._\n\n_Tomorrow I'll pull it together._\n\n_And tomorrow, tomorrow I'll be me again._\n\n_Any day now. Any day now I'll be back to my normal self._\n\nI wanted to stare at where she last stood, to stay with her. I wanted to chase after her. I wanted to say something. Instead I met eyes with the man behind the counter. It was time to pay, time to go on. It felt weird, once again, to be on this side of normalcy.\n\n_Me and this dude at work at the liquor store. Me, buying water and red cups because my kids are sick and we have to drive home. Me, tired from being up all night in a motel room during a trip gone awry. Me, frustrated with the day, but, well, lost in the web of the normal life._\n\n_You buying a pint with scavenged change at noon, looking to tomorrow. You telling me you're okay while you stink and waste away. You riding away in hope, until the shakes come again._\n\n_Me pulling out my debit card and spending nine dollars._\n\nI used to grab pennies out of the plastic container to buy pints. Ancient Age whiskey, a pack of Pall Malls, and a Coke if I had extra money. If I think really hard, maybe I can remember the exact amount of those three items. The cost of okay. The cost of the day. I'd dig in the folds of my car, under rugs and in deeper and deeper spots as if I hadn't looked there already.\n\nSometimes he'd give me a pint on credit. But never the Pall Malls. He knew I'd be fine without those. I always paid him back as soon as I could because I knew I'd need his help again. After the first pull hit my gut I'd feel hope, and the shakes would quiet, and I'd know just like her that tomorrow would be different.\n\n_Tomorrow I'll call my mom and get sober. I'll get with Ava and Rocket and work. I'll call my dad. I'll tell him everything. I'll eat some good food and clean my car and above all I'll never drink again._\n\n_Any day now I'll be back to my normal self._\n\n_Any moment. Maybe this moment._\n\nI wanted to tell that lady that the most important word in that strange sentence was \"self.\" The word she can't forget. The word she can't let go of. She has one. It's there. Buried beneath a few thousand years of separation and pain, or so it feels, but it's still intact, on fire, alive, pulsing through the reek of shame and humiliation, the part of her who looked at the woman behind her in line and knew they were the same.\n\nI was still thinking of her as I got in my car and passed out the red cups. _I wish I would have bought you the pint. I wish I would have handed it to you and said, before you could speak, \"I see you.\"_\n\nI wished I had told her I saw God in her cracked eyes\u2014that it's been years, but I still love her.\n\nInstead, I got in the car and threw it in reverse and turned on \"Ripple\" because I needed to hear Jerry sing it: \"If I knew the way, I would take you home\"\u2014because that's what I wanted to tell that lady, really, and that's where I was headed.\n\nHome.\n\nAva saw me staring, and asked me what was wrong.\n\n\"Nothing, baby,\" I said, and squeezed her wrist, so just for a moment I could feel our pulse.\n\n# Acknowledgments\n\nA massive thank you to my manager, Jermaine Johnson, for seeing something in my work that was worth pursuing, for that eight p.m. email that changed my life, and for the direction and brilliance and humor of our every call. And to my agent, Richard Abate, for teaching me what the hell a book proposal is and giving me more essential writing guidance in five-minute phone calls than I got in the entirety of my college career. Working with 3 Arts is probably the greatest honor of my life. Don't tell my kids.\n\nTo Lauren Hummel at Hachette, thank you for your editorial clarity, insight, and tireless work, and for seeing this book always as what I hoped it would be. You know how I feel about \"magic,\" but the intersection of our lives was, well, you know.\n\nMichael Barrs and Michelle Aielli, thank you for your incredible work on marketing and publicity. Thank you also to Mauro DiPreta, Mike Olivo, Sean Ford, and Marisol Salaman. The Hachette team was better than I ever could have imagined.\n\nTo the readers of Renegade Mothering, where do I fucking begin? Thank you for reading all these years, for every comment and email, for not getting too angry when I crossed that line, or maybe for sticking around _because_ I crossed that line. In absolutely unequivocal terms: THIS BOOK IS BECAUSE OF YOU. You became my people when I was sure I had no people.\n\nTo the Sauce Tank for handling my exquisite insecurity, my Pescadero renegades for writing anyway, and to you, Dave E., for sticking around and delivering the truth, thank you.\n\nSarah, my dear friend, from that bastard red tree to you in my kitchen after Joan died, you live the art I'm trying to write.\n\nSkyler Paul, I could not have written a single sentence as clean as a bone without you, soul friend. Let's hum those radio songs.\n\nDad, thank you for watching kids and supporting cabin trips, for our conversations and the pride in your voice, but mostly for teaching me how to spot bullshit from a very young age. And Neena, thank you for loving me even when you didn't have to, and for giving me Jerry.\n\nAnd, you, mama, whom I couldn't possibly thank sufficiently, I'll just say thank you for the billion moments of help, now and before, for posting my writing on the refrigerator when I was ten, and for always leading me home.\n\nAva, Rocket, Georgia, and Arlo: _You are my best._\n\nMac, you are the reason I kept writing when I was sure it was pointless and you were right, my love. You saw what I could not with those kindest brown eyes. I love you terribly.\n\nAnd finally, to the alcoholic who still suffers, and the children of alcoholics, with every word I typed, you were never far from my mind.\n\n### Thank you for buying this ebook, published by Hachette Digital.\n\nTo receive special offers, bonus content, and news about our latest ebooks and apps, sign up for our newsletters.\n\nSign Up\n\nOr visit us at hachettebookgroup.com\/newsletters\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Stortz Custom Homes, LLC was formed in 2008 by brothers Nick & Tim Stortz. As former part owners of a different well known local builder, Nick & Tim spun off to create their own unique vision in the CUSTOM HOME BUILDING market. Both Nick & Tim are life long Wisconsin residents that know the area extremely well.\nStortz Custom Homes, LLC is located in Richfield, WI (Washington County), and will normally build anywhere within a 45 minute radius.\nOur homes are above the quality standards of a typical track builder, and most of our homes are Energy Star certified. We have a very high customer satisfaction rating, and we're confident that we build one of the better homes around while giving you incredible value for your dollar. If you are looking for a nice, quality built home, with honest communication from your builder and solid upfront numbers, then we would be a great fit to speak with you further.\nTim Stortz Ph#: 262-628-2222 email: tim@stortzcustomhomes.com My name is Tim Stortz, and thank you for taking time to visit our webpage. As co-owner of STORTZ CUSTOM HOMES, LLC, my brother and I take extreme pride in who we are and what we stand for. Our personal foundation stems directly from our parents and how we were raised; treat people with respect, treat them fair, and be honest at all times. Integrity and ethical decisions will always take place at our home building company. I am in charge of all the backend logistical operations in our company as well as website management. I also participate in customer meetings and day to day operations and company decisions. There are many home builders to choose from, and many of them may be similar in their own way, so how do you choose? That answer is different for each person, but if you are looking for an honest company that builds a high quality home, excellent communication, with competitive pricing, we would love the opportunity to meet with you. Thank you.\nNick Stortz Ph#: 262-424-0555 email: nick@stortzcustomhomes.com Hello my name is Nick Stortz and as Co-owner of STORTZ CUSTOM HOMES, LLC, I would like to welcome you to our webpage. I have been in the construction business for 16 years as a carpenter and Custom Home Builder. I am a lifetime resident of Washington County and I cherish the fact that we have built our business and reputation on that small town pride and honesty that everyone loves. In dealing with our company, you will be working with me on the layout and construction of your new home. In the day to day operations I am involved with every aspect of the general contracting, from paying bills, collecting lien waivers, to meeting with all of our customers. I appreciate the fact that I am able to meet new people on a daily basis and get to make their dreams come true while maintaining a memorable and fun experience. Thank you and I look forward to hearing from you.\nArnie Leiskau Ph#: 262-365-4512 email: arnie@stortzcustomhomes.com Hello my name is Arnie Leiskau. I will be your project manager and expeditor for the construction of your new home. I have been in the building\/construction business for the past 20 years. I started working for a master carpenter right out of high school and worked with him for 10 years until he decided to retire, at which time I went to work for a well-known local builder. I then started my own business remodeling kitchens, baths, decks, rec-rooms, additions and all other aspects of this business. I now have joined the team at Stortz Custom Homes. My goal is to help you enjoy your building experience. I look forward to working with you.\nBecky Lorenz Ph#: 414-426-3789 email: becky@stortzcustomhomes.com My name is Becky Lorenz, and I am thrilled to be working with you as your Design Coordinator. I've been working in the design field for 7 years and have gained experience in both commercial and residential design. I first came across Stortz Custom Homes when my husband and I entrusted them to build our first home. It was during that time I realized how fortunate we were to have found such a great and reliable builder. Through that experience I developed a trusting relationship with Nick and Tim; and now I'm working alongside of them. It is a privilege and honor to be a part of this team. As I created new memories with my husband and two young children, I would be excited to help you begin the start of yours. It will be my pleasure to assist you in selecting all your exterior to interior finishes, as well as coordinating your appointments. Together we can create the home you've always dreamed of.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Programmatic trading models are generally divided into two types of models, one is the trend model and the other is the oscillating model. The premise of a programmatic trading strategy to make money is to have a good model + persistent execution.\nThe test results of many people posted, which the amount use of funds is often more than 80% or 70%, these are unreasonable choices, because financial market capital management is very important, when the market's environment is suitable for you, the higher amount use of funds, the greater the income, When the market is not good, the higher amount use of funds, the greater the loss. How to decide the amount of using of the funds, you should choose a fixed number to test. Regardless of the market conditions, it is more reasonable to test a model without adding or reducing positions.\nBoth the opening price and the closing price test have their irrationality. The trend model generally uses the trend reversal point as the opening signal, so it is more accurate when the price of the instruction appears.\nThe number of signals is too high, indicating that the shock market is not well filtered; too low, indicating a high risk. How to judge whether the number of signals is reasonable? Then there is only one comparison of different models in the same cycle; the simplest way is to use the total number of orders \/ effective trading days, taking the short-term intraday as an example. The average number of signals on a valid trading day is between 2 to 5 (this data is for reference only).\nThe longer the test period, the higher the profit rate should be. Many models, the measurement is good in the near future, and the long-term measurement will not work, so the test should try to measure the longest period that can be measured.\nThe higher the winning rate, the better, but not absolute, and there is no need to worry about the model's winning rate is too low. The general winning rate can be around 45%. Because the original meaning of programmatic trading is to \"loss small\" and \"winning big\", for example, the trend model will naturally have a low winning rate when it is a shocked market.\nIf you are choosing a fixed position size, such as 10 units to test, your maximum retraction rate should not exceed 10%.\nAnalysis of test results can not only look at a certain data, it needs to be combined and analyzed together: the number of signals cannot be more or less, the longer the cycle, the higher the profit rate should be, the profit ratio is more than 45% is acceptable, the maximum loss cannot be too large, the non-holding time you can manage it yourself.\nIn addition, we must see if there is a \"future function\" in the model. If it is a short-term model, the signal must not disappear. The daily price gap needs technical replenishment. Other issues are the reasons for analyzing a good or bad model. The good model is not afraid of any testing and analysis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"URISA Certified Workshops \u2013 to be Presented Virtually!\nPassing this along from URISA.\nURISA is pleased to announce the presentation of two of its most popular Certified Workshops via instructor-led web-based training. Normally full-day classroom sessions, each workshop has been re-packaged into three sections for presentation virtually without the travel expense and time away from work. The workshop content is thoroughly peer-reviewed and updated each year and is eligible for GISP Education credit (8 hours of EDU-2). The cost to attend each virtual workshop is $195 for URISA members and $245 for non-members.\nURISA's virtual classrooms are limited according to the number of participants in order to facilitate online discussions and Q&A with the presenters. Individuals are encouraged to register early to reserve a spot. (Note that the workshops are recorded, so if for some reason you cannot attend all three sections of a workshop, you will have complete access to the recorded archive.) A registration form and FULL details are available online.\nCourse Description: Free and open source software (FOSS) has been offering choices to computer users for a number of years. Over the past few years the open source choices in GIS have been broader and more capable than ever before. This workshop will focus on GIS open source software. It will give an overview of current developments from technical and management perspectives. Selected packages and their applications in various projects will be demonstrated and discussed.\nCourse Description: Public and private agencies face continuous challenges to accomplish more with less as increases in demand, regulatory requirements, infrastructure deterioration, and political and economic forces have significantly outpaced increases in capital and operating budgets. Many of these agencies are turning to Asset Management to cope with these challenges and improve business performance and effectiveness. This workshop will focus on several aspects of developing an asset management system that could help improve performance, reduce long-term costs, and maximize return on investment in infrastructure assets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ability to run across uneven terrain with continuous stable movement is critical to the safety and efficiency of a runner. Successful step-to-step stabilization while running may be mediated by minor adjustments to a few key parameters (e.g., leg stiffness, step length, foot strike pattern). However, it is not known to what degree runners in relatively natural settings (e.g., trails, paved road, curbs) use the same strategies across multiple steps. This study investigates how three readily measurable running parameters - step length, foot placement, and foot strike pattern - are adjusted in response to encountering a typical urban obstacle - a sidewalk curb. Thirteen subjects were video-recorded as they ran at self-selected slow and fast paces. Runners targeted a specific distance before the curb for foot placement, and lengthened their step over the curb (p<0.0001) regardless of where the step over the curb was initiated. These strategies of adaptive locomotion disrupt step cycles temporarily, and may increase locomotor cost and muscle loading, but in the end assure dynamic stability and minimize the risk of injury over the duration of a run.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr David Malin will open the Exhibition at 7.00pm on Monday 6th November and give a talk.\nThe exhibition will be open until January 14th.\nThe Talk and Opening Night are free but bookings are required. If you would like to view though the telescopes, special viewing costs apply.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"CSI Episode Guide: Season 15 Episode 17 \"Immortality (Series Finale)\"\nAfter \"a catastrophic event paralyzes Las Vegas, Gil Grissom, Catherine Willows and Jim Brass return to help the team with the investigation. Also, D.B. Russell and Sara Sidle make decisions regarding the future of their crime-fighting careers, and Lady Heather returns, creating [conflict] between Grissom and Sidle.\nCatherine, who left town to take a job with the FBI, is lured back when a federal crime is committed at one of the casinos she inherited from her father.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Automotive customers proceed to demand increased effectivity along with added features for safety and convenience. So I was delighted to see my colleague Andy Gryc change into co-chair of the W3C Automotive and Internet Platform Business Group, which has the mandate to speed up the adoption of internet applied sciences in the automotive. To supply timeless design with know-how that creates an ecologically sensitive product; so distinct that it varieties an emotional connection between buyer and brand. With a name synonymous with the South Korean automotive story, the Ssangyong Motor Company joined the Mahindra group in 2011.\nInternet retainable profit is very low in repair outlets generally, so your focus must be on customer coercion reasonably than profit. I am just a easy dude who loves cars,video games, and life and needs to share my automotive world with you. Research and growth \u2013 the Commission has greater than doubled the funds out there for cooperative research and innovation in the automotive sector for the period 2014-2020. Auto elements wholesalers not solely can save you money, however lessen the chance of you getting something that isn't genuine.\nAutomobile owners also prefer to make their vehicles unique with the installation of issues like GPS monitoring techniques, custom paint jobs, new sound methods, and really good tires. These methods get rid of the necessity for retooling a manufacturing facility, enable for fast advancements in know-how and customization, and scale back capital investment and environmental footprint. Discount of heat: In comparison to the automotive oil, this oil is designed to cut back the quantity of heat generated within the internal components of the engine. Honda cars have an automotive siren linked to an alarm system which includes a radio receiver, battery, array of sensors and a control unit all in a equipment. There are various kinds of automotive equipment which might be required within the restore shop. They ensure quick approval of the quantity required to amass automotive tool set.\nDependable on-line supply can supply good quality automotive instruments and gear which can be essential for top productiveness and security operations. Genisys is a very talked-about alternative in auto scan tools for many years, however now Pegisys is ready to OTC is a new era of automotive diagnostic scan instruments. You could discover lots of corporations simply glue these parts again into place, moderately than changing the broken parts.\nWe have seen our expertise go from the design studio to the manufacturing unit floor, with the adoption of customized 3D printed instruments, jigs and fixtures and manufacturing aids. He heads the solution and engineering units for all SAP Trade Solutions as well as Custom Growth. IHS Automotive delivers information and perception that guides product and market improvement, benchmarks business gamers, mitigates risk of fraudulent activity, and helps craft new environmental and security laws. We've got both the resources and expertise to supply a extremely competitive edge in automotive logistics throughout the globe. I have seen too many Chinese Companies fall over here as a result of lack of poor high quality product, poor after-gross sales service and just about no spare elements within the country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This explains the conclusions found with our fish classification project and suggests ways in which the project could be improved and extended.\nWe ran the images first with the ideal background at 4 levels of noise and then again with the background subtracted off and almost ideal. The ideal cases have better values than the background values and random noise on the images effects the accuracy dramatically. It is interesting that as noise is added the length\/width ratio test fails early and since trout is based heavily on this test we receive more error there. Even with little error, poor picture quality effects the performance.\nThe project was overall a success because it was able to detect most of the fish pictures correctly. Several of the tests even stood up well as noise was added to the system, although several did not work very will with large amounts of noise. This is a feasable system to install someplace, though, if it had a bit of work to focus it on exact fish that would be found in that area and thus could have the tolerance values adjusted a bit.\nAnother large source of error for this project was the pictures that were used to test. Pictures were just taken from the internet and many were not as close to actual test cases as we would have liked. The picture quality was so low on some pictures that they couldn't even be detected with no noise. If the system was installed someplace, much more standardized pictures could be taken and thus, results would be expected to improve.\nThe project could also be improved by altering some of the threshold values used. With massive amounts of testing, and more test cases, ideal threshold values for several of the tests could be experimented with and found to improve the exact accuracy of the tests, even though the process is still good. Also, the weights of the different tests could be changed so that the tests that hold up better with large amounts of noise were more important in the decision process than the tests that only work will with low noise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you have a creative itch you need to scratch? A Certificate II in Visual Arts may be just what you are looking for. Designed for people who are keen develop a range of skills in the visual arts area and gain a foundation for further study in the field. It will also assist you in determining which area\/s of visual arts you would like to pursue.\nPossible job roles include Art practitioner, Art Gallery technician, Art technician. The Visual Arts field offers employment in a variety of settings including schools, technical colleges, universities, galleries, shop-fronts, privately operated creative industries, home businesses and community precincts.\nVisual artists may work in their own practice, or in a wide range of contexts across the arts, government, community or commercial organisations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Indonesian Financial Services Authority (OJK) has instructed all insurance providers in Indonesia to apply a mandatory tariff for property insurance. The tariff has to be uniformly applied and the rule of set the maximum and minimum premium rates for protection against losses. Furthermore, the OJK issued the new rule regarding self-retention and domestic reinsurance. Insurance companies are obliged to have and implement self-retention for each risk in accordance with the self-retention limits. Fluctuations of total premium income and claims may lead the insurance company cannot fulfil the obligation to the insured, thus the company needs to conduct reinsurance. Reinsurance helps protect insurers against unforeseen or extraordinary losses by allowing them to spread their risks. Because reinsurer chargers premium to the insurance company, a properly calculated optimal retention would be nearly as high as the insurer financial ability. This paper is aimed at determining optimal retentions indicated by the risk measure Value at Risk (VaR), Expected Shortfall (ES) and Minimum Variance (MV). Here we use the expectation premium principle which minimizes individual risks based on their quota share reinsurance. Regarding to the data in an insurance property, we use a bivariate lognormal distribution to obtain VaR, ES and MV, and a bivariate exponential distribution to obtain MV. The bivariate distributions are required to derive the conditional probability of the amount of claim occurs given the benefit has occurred.\nAcerbi, C., dan Tasche, D. (2002). On the Coherence of Expected Shortfall. Journal of Banking and Finance, 26, pp. 1487-1503.\nBalbas, A., Balbas, B., Heras, A. (2010). Stability of the Optimal Reinsurance with Respect to the Risk Measure. Working Paper 10-02. Business Economic Series 01.\nBorch, K. (1960). An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance. Transaction of the 16th International Congress of Actuaries I. pp. 597\u2013610.\nCai, J., and K. S. Tan, (2007). Optimal Retention for a Stop-Loss Reinsurance under the VaR and CTE Risk Measure. ASTIN Bulletin, 37 (1), pp. 93-112.\nCai, J. , Tan, K. S., Weng, C., and Zhang, Y. (2008) Optimal Reinsurance under the VaR and CTE Risk Measure. Insurance: Mathematics and Economics 43 (1), 185-196.\nKaluszka, M. (2001) Optimal Reinsurance under Mean-Variance premium Principles. Insurance: Mathematics and Economics 28, pp. 61-67.\nKaluszka, M. (2004) Mean-Variance Optimal Reinsurance Arrangements. Scandinavian Actuarial Journal I, pp. 28-41.\nNoviyanti, L., Soleh, A.Z. and Chadidjah, A., presented at The International Conference on Mathematics, Science, Education and Technology, 2017.\nPanjer, H. and Willmot, G. (1992). Insurance Risk Models. Chicago: Society of Actuaries.\nPressacco F., Serafini, I. and Ziani, L. 2010. Mean-Variance Efficient Strategies in Proportional Reinsurance under Group Correlation in a Gaussian Framework. Accepted for publication on European Actuarial Journal, first number.\nTan, K. S., Weng, C., and Zhang, Y. 2009. VaR and ES criteria for optimal quota-share and stop-loss reinsurance. North American Actuarial Journal 13 (4), pp. 459-482.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"DINOSAURS!!!, Parody Song Lyrics of 311, \"All Mixed Up\"\nA good title, also the chorus lyrics, were hard to come up with due to rhyming difficulties. Also, the part where he sings real fast......ya.....that was difficult. Just bare with me on this one.\nI thought that they were extinct.\nNow I know they're alive.\nLike they'll help the situation by showing off their piece.\nSo make like roadrunner and get out of its way.\nCause this has been one bad day.\nNow this dino has just stepped on my car.\nAnd I just payed all the loans off last week.\nWell, I have insurance on my ride.\nI just wish I had bought life.\nI don't know where to hide.\nI ran into the mall, but it followed me inside.\nis more painful than their bark.\nJust like in Jurassic Park.\nJust look at all their eggs.\nBut do not get too close or else they'll gnaw on your legs.\nThey know how to open doors.\nand they're different in size.\nWe're lunch in their eyes.\nWe're dead meat. Let us retreat.\nI'm shaking just like Shakira from my head down to my feet.\nWe're dead meat. Watch them eat.\nThey're chomping on the mayor in the middle of the street.\nNow the raptors are in search of some prey.\nAnd behold, they happen to spot me first.\nCan somebody give me CPR.\nI'm still sad about my car.\nNow I'm in a territory where it became really gory.\nBut I was able to escape.\nMany views...of them fight it out.\nMany views...of them all thrash about.\nThanks to the Jurassic Park Trilogy for some of the ideas I used in this one.\nThe address of this page is: http:\/\/www.amiright.com\/parody\/90s\/3110.shtml For help, see the examples of how to link to this page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Representatives from over 80 disability rights organizations in the West Bank and Gaza discussed the preliminary draft of the law by decree on the rights of people with disabilities in two workshops organized by the Institute of Law at Birzeit University, on January 23 and 24, 2019.\nThe workshops, held together with the Ministry of Social Development in both the West Bank and the Gaza Strip, featured legal experts and researchers, disability rights activists, and government representatives, who examined the preliminary draft and gave their opinions - each according to their areas of specialization.\nThe Institute of Law, in cooperation with UNICEF, has led an initiative to draft a law that secures the rights of people with disabilities in Palestine, with regular consultations and workshops held with relevant experts, researchers, and government officials.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Choose Quick Ship Painted Louver Size, or the next bigger size (all sizes are available).\nClick \"View Details Button\" below and you can choose color chart, exact size, frame, screen and view spec sheet pdf's.\nEnter your Louver size (hole you are going to be filling), Enter your desired color from the color chart. Choose Bird screen (1\/4\") or Insect screen, and choose Flange Frame or flush mount Channel Flange. If you can't find your Louver Size Call our 800-810-3280 toll free number.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Clients get their setup fee waived when they convert an existing Uni-K, solo-k or individual 401(k) to a Merrill Edge Individual 401(k) by September 30. American Express changed the introductory spend offers on nine cards, while Chase lowered the initial spend offer for its Southwest Rapid Rewards Plus card, and Capital One increased the initial spend offer for the Quicksilver card. Barclaycard US increased the initial spend offer for the AAdvantage Aviator Red World Elite Mastercard, added a financing term to the Barclaycard Visa with Apple Rewards card and introduced a reward boost for the Diamond Resorts International Mastercard. Citicards increased the initial spend offer on the Citi ThankYou Premier card, decreased the initial spend offer on the Citi\/AAdvantage Platinum Select World Elite Mastercard and increased the introductory period on the Citi Double Cash card. S. Bank increased the cashback earnings rate on the Cash+ card.\nFive firms commented on the Fed rate hike and the future of Fed policy. Invesco added a blog post, while Voya, Legg Mason, Federated and Hartford Funds offered commentaries. Six firms addressed the DOL Rule implementation. Athene provided resources to help advisors understand the effects of the rule through a DOL Producer Guide as well as an article reminding them to use updated forms compliant with the rule. Lincoln and Prudential added documents for advisors, Lincoln's focusing on impact and Prudential's providing FAQs. Pacific Life addressed the rule multiple times in July, with two articles and an update to the DOL Fiduciary Rule page. Vanguard and OppenheimerFunds each added commentaries.\nEdward Jones consolidated its login interface, placing both the username and password entry fields on a single page. The firm also updated authentication features, removing security images and adding the option to receive security codes via text or email when logging in on a new device, as well as automatically enrolling clients into security questions. On the mobile app, iOS users can use Touch ID, and the firm anticipates adding the feature for Android users soon.\nThe new navigation provides five main tabs for reorganizing existing content and allows clients to compose messages directly from the envelope icon flyout. The header and footer were also consolidated, reducing the abundance of text and links. The Overview page features a widened format and new Portfolio Progress and Recent Activity sections.\nThe participant site features a new navigational structure, homepage and retirement income projection tool. The navigation provides an anchored section housing a collapsible sidebar menu and a search field. The homepage offers a static quick links section, along with a columnar structure detailing the participant's account information. The new retirement income projection tool follows best practices outlined in Corporate Insight's November 2016 RPM Retirement Plan Monitor Report, including automatic data import, additional information inputs, intuitive visualizations, dynamic modeling and direct editing. The mobile app reflects these changes on the home screen and new tool abilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Friday the 13th is fabled to be a day filled with bad luck and unfortunate occurrences. Regardless of whether you're superstitious or not, it's very tough to deny that Western New York and its residents had some pretty terrible bad luck on Friday, February 13, 2009.\nIt was on that morning that the citizens of WNY and the United States woke up to the shocking news that an airplane that had been in route to Cheektowaga's Buffalo-Niagara International Airport had failed its attempted landing and crashed into a house in Clarence, NY the previous night. The revelation that all 49 people aboard Colgan Flight 3407 as well as the one person in that house lost their lives sent a shock wave across the Buffalo-Niagara region that still hasn't completely dissipated a decade later on February 13, 2019.\nThe cruel irony of the crash is that it occurred just 28 days after the Miracle on the Hudson, where US Airways pilot Chelsey Sullenberger successfully made a desperation landing on the aforementioned river after Flight 1549's engines failed, the lives of all 155 people on board were saved as a result. America was still reveling in the glory of the man now known as \"Sully\" when the 3407 tragedy occurred. In many ways it snapped Americans back to reality and reminded us that for every miracle, there's a tragedy that accompanies it.\nA crueler irony still is that one passenger of 3407, Beverly Eckert, was a Buffalo native and the widow of Sean Rooney, a fellow Buffalonian who was killed in the south tower on September 11, 2001. Eckert was returning to visit her family and present a scholarship in her husband's memory at his alma mater Canisius High School. If that doesn't make the hair on the back of your neck stand up, I don't know what will. I would attend Canisius the following year but my brother was a sophomore at the time and would have attended that ceremony.\nMuch like 9\/11, I doubt anyone will ever forget where they were when they heard the news and I certainly never will myself. I was in eighth grade at the time and woke up that Friday morning in high spirits as it was the last day of school before our February vacation. When I came downstairs to my father watching the TV coverage, that mood changed instantly. I'm not sure if I fully understood the weight of what had happened, but I very much remember feeling the same shock. Usually Friday mornings in my eighth grade class saw nothing but smiles and elation, especially on the cusp of a week off. I'll never forget walking into my class and seeing my teacher and all my friends sitting in practical silence.\nThe members of the Buffalo Sabres at the time felt the impact (forgive the morbid pun) as much as anyone in the WNY community. Many members of the team lived in Clarence or close by and were deeply affected. The Buffalo News' Mike Harrington recounted the stories of several members of the '08-'09 Sabres yesterday, including former captains Jason Pominville and Craig Rivet. Rivet recounted in particular how shaken up former head coach Lindy Ruff was when he addressed the media prior to that evening's game against the San Jose Sharks. Lindy, who still lives in Clarence to this day, was known for his gruff demeanor, which just shows you even further how deeply the crash affected the community.\nClick here for Harrington's entire article.\nThe Sabres were scheduled to take on the Sharks at then-HSBC Arena that night and the NHL contemplated cancelling the game, but ultimately did not. The mood in the arena was sad and sullen beforehand as the team held a moment of silence in memory of the victims. It may have been tough to foresee at the time, but the game that resulted against San Jose was one of the most emotional and wild roller coaster rides that the Sabres and their fans have ever been on, and one that helped the healing process.\nThe San Jose Sharks were in the midst of a march to the President's Trophy as the NHL's top team at this point of the 2008-09 season. The Sabres were the far worse team in terms of record and were fighting for a playoff spot. Despite this, the Sabres jumped all over the Sharks early and held a 4-1 lead after goals by Adam Mair, Jaroslav Spacek, Derek Roy and Ales Kotalik. Things would get ugly in a hurry, however.\nThe Sharks stayed true to that President's Trophy form and came roaring back with four answered courtesy of Patrick Marleau, Dan Boyle, Devon Setoguchi and Joe Pavelski to take a 5-4 lead. All of the positive energy that had filled the arena vanished as San Jose now seemed in complete control. With the final minute ticking off the clock and Ryan Miller on the bench, the Sabres flailed desperately around the Sharks' end for the puck. With less than 10 seconds left, Pominville found the puck to the right of the net and slid it back to Rivet at the point. The captain unloaded a shot so hard that it broke Pomiville's stick when he tipped it past San Jose's Evgeni Nabakov to tie the game at five. The energy in the building returned in an instant, and it turned into jubilation when the Sabres prevailed in an equally thrilling shootout after Roy beat Nabokov high and Miller stopped Milan Michalek.\nClick here for full highlights of the game and reactions from a few players.\nthough a sporting victory is paltry in comparison to such a great loss, the Sabres rallied around their city and provided some much-needed escapism to a grieving community. The game and the moment of silence that preceeded it was a galvanizing show of solidarity that demonstrated just how much the Sabres understand their role in Buffalo. Regardless of how great or atrocious the team on the ice may be, this team loves its city.\nLast night's game between the Sabres and the New York Islanders at now-KeyBank Center was played exactly ten years to the evening of the 3407 tragedy. Like it had a decade prior, the team held a moment of silence before the game. It was a very classy show of respect by the team but it unfortunately fell flat when the arena buzzer, which was malfunctioning all night, awkwardly went off in the middle of it. It was still a cool moment to be at the game for and I was happy to have been able to experience it in person.\nThe Sabres played well against the Isles in the game and pulled out a needed victory, 3-1 to pull within two points of Pittsburgh for the second Wild Card. The recently reunited J-Crew line is back to its old tricks, as Pominville and Jeff Skinner both scored and Eichel assisted on both. The trio just seems to bring out magic in each other each time it takes the ice. Linus Ullmark made 24 saves against a tough Islanders' offense.\nIt was perfectly fitting that Pominville scored in the game. The only current Sabre that was on the '08-'09 team, Pominville picked an appropriate time to boost the population on the tenth anniversary of one of the biggest goals he's ever scored in a Sabres' uniform. It was a nice way to bring the two dates full circle with each other.\nA decade is a long time, but the Buffalo Sabres and their fans demonstrated that it wasn't long enough to cause them to forget how big an effect the 3407 crash had. RIP to the 50 victims. You may be gone, but you will never be forgotten.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"ISLAMABAD: Adviser to Prime Minister on Textile, Commerce, Industry and Production and Investment Abdul Razak Dawood has underlined the need for enhancement of exports to China and to the global market.\nTalking to Chinese Ambassador Yao Jing who called on him on Friday, Razak said the two countries enjoyed excellent trade and commercial relations cemented further by the signing of a free trade agreement (FTA) in 2006, which has added another dimension to the ever-growing business relations between the two countries.\nRazak felicitated the Chinese ambassador for holding a mega event like China International Import Expo in November in Shanghai and expressed his gratitude for declaring Pakistan the 'guest of honour' at the event.\nJing informed the adviser that a buying mission was coming to Pakistan shortly.\nBoth sides agreed to work closely to build a brighter and prosperous future for the region.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy Exploring, Making, Doing, Creating!\nA frosty day like today is a great opportunity to get out with the kids and explore the frost on things, a little science (google it if you need a helping hand) & then back home to warm up and make snowflakes or frost pictures of their own using silver\/white glitter, or sugar, or desiccated coconut to create the frosted effect. Alternatively you could cut out the snowflakes if the children are too young for scissors and then they can use the above materials to decorate the snowflakes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The view from behind home plate at Tempe Diablo Stadium.\nThe view from down the first base line. A hotel and restaurant overlook the field from behind the left field fence.\nThe view from down the third base line. Just like in the LA area, a freeway runs right behind the outfield wall.\nThe grandstand at Tempe Diablo Stadium. The stadium structure extends further down the first base line than the third to house an enclosed party deck.\nPennants along the pressbox facade celebrate Angels division, league and World Series titles.\nThough the seating bowl is completely open, a sun shade runs the length of the infield concourse, providing some coverage from the desert sun.\nPlaques along the concourse celebrate the history of Tempe Diablo Stadium and Angels Spring Training.\nThe concourse opens up in the left field corner, housing a carnival like food court and an up close view of the butte behind the field.\nTrees line the top of the berm in left field.\nThe Twin Buttes of Tempe provide the iconic backdrop to an Angels Spring Training game.\nThe Angels bullpen sits behind the right field wall, by the batting cages, and a cutout in the wall allows the pitchers to watch the game.\nThe visitors bullpen sits in play down the third base foul line.\nThe scoreboard and outfield corners as seen from the berm in left field.\nThe Los Angeles Angels of Anaheim on the field during a Spring Training game at Tempe Diablo Stadium.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The pigeon feed market has been a valuable niche market for Maple peas. Varieties of field peas (Pisum sativum L.) intended for specialty markets, such as maple peas, often have brown colored seeds. These varieties have higher levels of tannin that may reduce palatability in livestock rations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have published an updated water expectation document. The document serves as a starting point for our interaction with companies on the topic of water management.\nWater management has been a focus area for the fund for many years. Limited supplies of water and water quality are a growing risk for many companies. The purpose of these expectations is to express how Norges Bank Investment Management, as a financial investor, expects companies to manage the challenges and opportunities surrounding water resources.\nThis includes strategy and investment planning, risk management, and disclosure and reporting. We have also put emphasis on how companies should be transparent on their interaction with policy makers and regulators on water related challenges.\nOur expectations are primarily directed at company boards and serve as a starting point for our interaction with companies on the topic of water management.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"With the immense success of 'A Quiet Place' still rolling on and on, script writers Bryan Woods and Scott Beck, who co-wrote with director John Krasinski, say that there may be more to come from the movie.\n\"There are so many discarded set pieces, too, just hiding out on Word documents on our computer\" Beck added.\nThe film was considered for inclusion in the Cloverfield Universe, but is now on it's own as it's own franchise and with plenty of content just sitting around, there could be more in the works.\nSpeaking of the film, Blood-Disgusting posted 15 minutes worth of behind-the-scenes footage which is sure to keep fans excited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Made in USA, right here in California- so earthy, as with late summer, we engage in earthy practices. Here, grey clay to start off with \u2026 with craftsmanship, this grey matter will transform into beautiful earthenware with different shapes, forms colors, shades, hues \u2026 uniform but each unique.\nPlaying tourists in our own backyard \u2013 Golden Bridge \u2013 Sausalito \u2013 Muir Woods \u2013 Marin \u2013 Mt. Shasta \u2013 what do out-of-town guests want to experience ? What's home-grown; nature \u2013 nature \u2013 nature. Sadly Yosemite topping the list \u2013 so affected by all the fires in Northern California but Northern California has much to offer still.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are here: Home \u00bb MEDIA \u00bb Books \u00bb Spellwright (book 1) by Blake Charlton (book review).\nSpellwright (book 1) by Blake Charlton (book review).\n'Spellwright' is the first book in a fantasy trilogy by Blake Charlton. It is set in a world where humankind and their gods fled the land they had originated from to protect themselves from demons. They had sailed to a new land and killed the Cthonic people, who resided there. Among humans there are those that can sense and use magic in different forms called spellwrights.\nOf these, there are many kinds such as wizards, who use spells written on paper and form magic from their muscles, and druids, who write their spells into living tissue such as wood, roots and seeds. Each order of spellcasters have their own unique spellcasting languages that they keep secret from the other orders. The different orders gather together every decade for a convocation in which they renew previous peace treaties to not use magic in war.\nThe convocation this year is being held at Starhaven, a wizard academy. Nicodemus Weal is a young spellwright apprenticed to Magister Shannon, a legendary wizard. Unfortunately, Nicodemus is also a cacographer, a spellwright who 'misspells'. When he forms any magic that is not simple, he misspells the formation runes and either the spell fails or mutates dangerously.\nHe cannot touch other objects made entirely of spells, such as gargoyles (non-sentient creatures made from spells to serve wizards various tasks) as his cacography will cause the spells making up the gargoyle to misspell and destroy it.\nNicodemus had once been thought to be the 'Halcyon', the great wizard who was prophesied to save humanity and their magic from the demons, who would finally arrive on this land and start a war. If not for his cacography, Nicodemus would surely be the 'Halcyon' as he has every other prophesied attribute. Though the wizards have ruled out Nicodemus as the 'Halcyon', there is a powerful creature hunting him, who uses old spells that have been lost to humanity since they crossed to this land.\nMagister Shannon seeks to protect him but the wizard sentinels (wizard police) who are investigating murders caused by the creature, suspect Shannon of them. The druids, Deidre and Kylan, also think Nicodemus is the 'Peregrine' (the druid name for the 'Halcyon') and try to convince him to leave Starhaven and go to their goddess, who would offer him safety and prepare him for the coming demons.\n'Spellwright' is a fast-paced and involved book. Charlton creates an intricate world of magic, where each order of spellwrights have very distinct and complex magical languages which each have their own set of rules. While I appreciate the depth Charlton went into with the magical world, it took a while to get into the book and understand what was going on.\nI had to keep going back and re-reading previous pages when I had finally been given enough information to understand them. I found that after making us aware of how Nicodemus had worked painstakingly hard to learn magic with his cacography as the novel progressed he started to gain magical knowledge very quickly. While there were drawbacks to this knowledge, I didn't feel that it should have been so easy.\nNicodemus' cacography was refreshing as it was obviously created to emulate dyslexia. I have never before read a book where the protagonist is dyslexic and although Nicodemus struggles with it, he never lets it limit him or his ambitions which is an encouraging message for both the dyslexic and non-dyslexic people, who will hopefully read this book.\n\u00ab It Crept From The Tomb edited by Peter Normanton (book review).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Its scary when someone in the group crashes their motorcycle. This guy is pretty lucky to escape with only a shoulder injury.\nCrazy Crash Scene! A Rider Flew Off The Highway Overpass!\nPlease Clear Your Desktops, Motorcycle!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bulldog Gym Kickboxing #1 DVD Klaus Nonnemacher \u2013 I&I Sports Supply Co., Inc.\nThis DVD was created in England and has never been available in the USA till now. Klaus Nonnemacher a protege of the famous BULLDOG GYM teaches and demonstrates his proven methods of training that have made him into an International Kickboxing Champion. The video was professionally shot and edited. You the viewer will be able to learn all kinds of cool and different training methods. Kicking drills; stretching and warm ups as well as superb hand and foot drills. Get the upper hand on your competitors or just learn all kinds of unique routines in this awesome DVD .","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Once a Panther, Always a Member.\nAs a graduate, you've earned your place as a member of the UWM Alumni Association. Stay connected to UWM and the growing alumni network of 178,000 worldwide graduates by participating in alumni events and programs.\nRequest your complimentary membership card today.\nKlotsche Center: Access to a reduced-price membership at the Klotsche Fitness Center. View Klotsche Center membership rates.\nGolda Meir Library: Borrow books from the main collection, as well as some items from special collections (curriculum and music books, limited loan of videos\/DVDs, etc.) Access online databases while in the library.\n*If you were a life member prior to Alumni Association becoming a dues-fee organization on July 1, 2005, you're now considered a sustaining contributor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 1950s inspired ice cream parlor and soda fountain will transport you back in time.\nHave you ever dreamed of going in back to the 1950s to visit a vintage style ice cream parlor and soda fountain? Now is your chance! Your grandparents have probably mentioned that back in their day, a soda fountain was the place to be and the social hub of the neighborhood where you could also enjoy classic ice cream sundae.\nBrooklyn Farmacy & Soda Fountain, established in 2010, is recreating that experience for many families and people residing in the Carroll Gardens, Brooklyn neighborhood and beyond.\nUpon entering the building, you are immediately immersed into a classic soda shop which was once a 1920s apothecary. The walls are lined with wooden shelves, holding different old-school candies, sweet treats, and other delights that are available for purchase. At the infamous countertop, you can see all of the dessert items being made right in front of your eyes.\nThe servers are even dressed in vintage style clothes from Gatsby caps to swing style dresses. If the atmosphere alone doesn't sell it for you, the 13+ sundaes handcrafted to order, egg creams, and ice cream sandwiches sure will.\nThe beloved Cookie Monster from Sesame Place holds a very special place in many of our hearts and so will this sundae. \"Cookie Monster\" features cookie dough crumbles, mint chocolate chip ice cream, hot fudge, and whipped cream garnished with blue sprinkles. It will make you never want to eat a plain cookie again!\n\"More is S'more\" is my personal favorite sundae because it is a delicious spin on s'mores\u2014my all-time favorite dessert. \"More is S'more\" is made of vanilla, chocolate, and coffee ice cream piled on top of chunks of red velvet cake, followed by a Nutella drizzle, Farmacy's toasted fluff, and a homemade graham cracker. You will never want to eat a normal s'more after trying this sundae. Trust me.\nIn addition to the sundaes, the soda fountain also serves egg creams\u2014a classic New York drink. Did you know that an egg cream does not actually contain eggs? It is made using milk, carbonated water, and flavored syrup. Some of their egg cream flavors include Fox's chocolate or vanilla, coffee, mocha, tootsie cream, and maple.\nEgg creams, \"Mr. Potoato Head,\" \"Cookie Monster,\" and \"More is S'more\" are just a few of the intricately prepared treats Brooklyn Farmacy offers their customers. In addition, to being delicious, these desserts will provide you with Instagram-worthy food photos that will make all of your followers insanely jealous.\nBrooklyn Farmacy & Soda Fountain provides such an authentic experience that even one's grandparents will feel like it is the \"good old days\" again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"L\u00e9o RamosEcl\u00e9a Bosi, emeritus professor of social psychology at the University of S\u00e3o Paulo (USP), deals with research topics that are not among the most studied within Brazilian academic circles: the readings of female workers and the memories of the elderly, for example, to cite only two of the areas on which she focused in her research. Often Bosi shifts her focus to vulnerable social groups: poor, low-income working women; seniors who, immersed in the ongoing transformation of the metropolis, are grudgingly losing the references of their familiar, daily routes, and entering a time of fading awareness of their identity. From the objects chosen to the individuals encountered during the research process, both alluding to the precarious and the vulnerable and based on solid theoretical grounds, she built a strong, unique body of work recognized in her field.\nPart of this uniqueness has been her ability to express the force of her findings and reflections softly and delicately, which helps give Ecl\u00e9a Bosi's narratives a particularly literary dimension. Empirical research reports and theoretical essays are often expressed in beautiful poetic prose. But, at the same time, they seem to be part of the vigor, the vital force of Bosi's research, its spillover into the field of institutional activism and politics. That is why it is easy to understand her efforts towards establishing and developing the USP Senior Citizens Program which, after 21 years in operation, has served more than 100,000 senior citizens on the campus of the largest Brazilian public university, most of whom had deficient formal education. Or even her eco-activism, which includes a special focus on pregnant workers who may be unaware that they are being exposed to toxic agents at the factories where they work.\nMarried to Professor Alfredo Bosi, a respected critic and historian of Brazilian literature, mother of Viviana Bosi, a professor of literary theory, and Jos\u00e9 Alfredo Bosi, a professor of economics, and grandmother of two, Ecl\u00e9a Bosi, who continues to work despite being formally retired, agreed to this interview with Pesquisa FAPESP on a overcast morning illuminated by conversation at the USP Institute of Psychology, with the principal sections published below. With books and some old articles on the desk in front of her, she stated that she had always been well-treated in her professional career, but one thing that touched her greatly was having her book Mem\u00f3ria e sociedade\u2014Lembran\u00e7a de velhos (Memory and society\u2014Memories of the elderly), the book she worked hardest on, included on a list published by the Ministry of Education of the 100 books sent to thousands of school libraries all over the country.\nI see in these printed keepsakes memories of your work, struggles and awards.\nYes, I think that I strove a lot in my life. Here are things related to my eco-activism.\nAn activist in addition to your academic work.\nI was involved in the creation of the Chico Mendes Ecological Park. This here is a souvenir of the two successive years of the Ecology Week I organized at USP, before the establishment of the Environmental Sciences course in 1974. I appealed repeatedly to congressmen, senators, important people, to help in the fight against a nuclear power plant, but they did not respond so I decided to write to someone I really admired: Carlos Drummond de Andrade. He studied the matter and wrote a beautiful article entitled \"If I were a congressman,\" published in the newspaper Journal do Brasil. More recent is the struggle against the Angra 3 nuclear power plant.\nIt's something I did with great pleasure, the last book I wrote, Velhos amigos (Old friends), true stories for children and adolescents. This other document is an article I wrote about a former, deceased USP professor [Jos\u00e9 Severo de Camargo Pereira]. He was feared. One day, I saw there was some confusion and a line at the door of the USP building on Rua Maria Ant\u00f4nia. There were people everywhere. The school was a sounding board for national politics. Cuba had been invaded and people were there to sign up to fight and defend the island. These people had never used weapons and wanted to face the United States Army. Can you imagine?\nWas it the Bay of Pigs invasion in 1961?\nExactly. Then I heard a voice behind me saying that he knew he was old, but wondered if they would let him participate. It was Professor Pereira. But I want to show you documents related to a program that I have been involved with for 21 years [the University Senior Citizens Program]. The covers of the course schedules almost always depict a person who suffered greatly in life, who started with almost nothing, and here they are shining on campus.\nMy work began with Leituras de oper\u00e1rias (The Reading of Female Workers). Why reading and why workers? Then came Mem\u00f3ria e sociedade \u2014Lembran\u00e7a de velhos; Tempo vivo da mem\u00f3ria (Living memory); Simone Weil: a condi\u00e7\u00e3o oper\u00e1ria (Simone Weil: the condition of the workers); and Velhos amigos.\nWe will try to cover this, but first I need to ask: where are you from?\nFrom the city of S\u00e3o Paulo. I spent my childhood in the Pinheiros district. I was born at Maternidade S\u00e3o Paulo, on Rua Frei Caneca.\nThey were very simple, but I remember that they both wrote poetry. My father was a civil servant, my mother, a housewife. They had very little education. And I had two younger brothers.\nI was a voracious reader. Books were not given to me, I was not surrounded by books, they were treasures I acquired. How? By walking. A book cost the same as 12 bus passes. I studied in a large house in the Campos El\u00edsios neighborhood, so I had to walk the length of Av. S\u00e3o Jo\u00e3o, and then the length of Rua da Consola\u00e7\u00e3o and Av. Rebou\u00e7as on the way back, to arrive at Rua Mello Alves, where we lived. And by walking instead of taking the bus 12 times I could buy a book. So I obtained my books through sacrifice.\nWho were your favorite authors?\nBy 13 or 14 I was already immersed in Dostoevsky, Tolstoy, and Chekhov, but also in Romain Rolland and Emily Bront\u00eb. Then I read Hemingway, Sinclair Lewis, a lot of poetry. Later I translated Ungaretti, Leopardi, Montale, Rosal\u00eda de Castro (for newspapers and books).\nNo. It was a school named Stafford, surrounded by a park, but it no longer exists. It was in a huge house located on Alameda Nothman. During my walks between home and school I learned about social inequality. I saw mansions, simple homes, and thought about social inequality without anyone telling me anything. I also learned the city intimately. Later, when I started studying at USP, we students lived in bookstores, libraries, bars. The city was very close to us.\nYou started studying at USP, back when it was located on Rua Maria Ant\u00f4nia, during some turbulent times. What was it like, living those years in that environment?\nMost importantly, there were great professors. We loved our professors: [Jo\u00e3o] Cruz Costa, in philosophy, Ruy Coelho, a distinguished professor of sociology, Gioconda Mussolini, in anthropology, Dante Moreira Leite, who was my advisor in social psychology, an extraordinary person, and Anita Castilho Cabral, who established the psychology major.\nHow did you decide to study psychology?\nHow did you decide what path your studies would take within Psychology?\nI went into social psychology because it was a very politicized time, with an enormous political density. My class was small, about 12 people, and was almost completely decimated by the dictatorship. I was a classmate of Iara Iavelberg [an activist killed during the dictatorship], and this greatly affected me. I remember her as a very intelligent, beautiful girl who sang very well. She liked Ponteio, by Edu Lobo, and Disparada [by Geraldo Vandr\u00e9]. She was very good in statistics, taught by Prof. Pereira, and we would go to her house to study. Everyone remembers the story of Iara, but it was more personal for me and I saw how much my class suffered because of her loss. I also remember Aurora Maria do Nascimento Furtado, Lola [an activist who was tortured and died at the hands of the dictatorship]. She was an unforgettable student. When General Fi\u00faza de Castro wrote his memoirs, he was asked if he remembered the subversives that he had captured, and he said yes. He said he remembered a very brave girl named Aurora. She died from the \"crown of thorns\" [an instrument of torture], with her head in a vise, crushed, a heroic death, because she never opened her mouth. She was arrested in Parada de Lucas, in Rio de Janeiro, in 1972. I never forgot her, nor can I. In the Institute of Psychology, I established an \"Aurora room,\" with photographs of her and with this statement by the General on her bravery. And I would like to mention that Professor Castilho invited Iara to be a professor of social psychology and she was even hired, but soon had to leave to live in hiding. I remember her analyzing the content of Fidel Castro's speeches. She never finished that work, because she then disappeared. But she left us a nice article about language and communication that came out in a SBPC journal, and I had the pleasure of giving it to my students for them to read.\nSo, your proximity to the political struggle had some influence on your choice of social psychology?\nYes, it deeply affected me. And what did I choose for my PhD dissertation? Female workers. Why reading? Because it is a problem area. Female workers, and indeed all of us, are entrapped by the flow of television and other mass communication media. But reading requires will, a choice. It is a minimal act of will, but it is a step that must be taken. And, in the case of workers, it requires great personal commitment, as there are no bookstores in poorer neighborhoods. What prevents workers from reading? The long, extensive workday, the double shift as worker and mother of a family, with all the housework. The distance between work and home, the lack of cultural centers, the salary that just barely covers the essentials.\nAlthough the workers interviewed in your dissertation were younger and various unmarried women and dreamers were among them.\nYou touch on something that I find remarkable. The single worker has one kind of mentality, the married working mother another: she is a fighter. She is always there when someone needs to complain or take political action. She is always in front.\nProbably due to her commitment to her children.\nYes, her salary is very important to the family. I collected testimonials from these workers. Otto Maria Carpeaux wrote the preface to the book and said, \"what a bleak survey, what seduced, exploited minds.\" But I wanted to go one step beyond: I discovered what female workers read and tried to understand what they liked to read. I entered the world of the possible.\nAnd is there a big gap between what they read and what they like to read?\nNo, but the two are different. Mass communication is twofold: in the realm of propaganda, it attempts to show what is most advanced in technique; in the field of the imagination, it exploits a pre-industrial mindset that survives in the culture of the poor man, namely serial literature. What was it like, this serial literature that female workers liked so much? In general, it addresses the situation of women and children who suffer societal violence. The woman suffers imbalance, victimhood. This is the type of novel that working women read. And equilibrium is restored when the story has a happy ending, through marriage, the intervention of destiny.\nIn books or in magazine stories?\nIn magazines and in serialized books, in the romantic plot the woman and child are victims not of society, but rather of destiny. And these stories are not dated, they are eternal: they carry the sentiment of exclusion from the world, avoidance, the compensatory fantasy that Freud worried so much about\u2026 Umberto Eco has a beautiful expression for this: structures of consolation. And Gramsci calls it the social inferiority complex or compensation daydreams. Gramsci regrets that intellectuals do not study popular literature. Because of this, they do not establish a modern humanism capable of reaching the most humble.\nBetween Carpeaux's, Gramsci's and your vision, I see a clear difference, among others, in the true closeness with which you treat the group you study.\nFirst, let me tell you what these romantic novels are about. In my view, working women are impressed by fundamental questions of justice, guilt, punishment, transgression, revolt, and undeserved punishment. But isn't that what great literature addresses too? The themes are the same. The reader's eyes reach and follow this human drama. The classics tackle this and the manual laborer who reads, if she had the opportunity to read the classics, would probably feel right at home.\nIn this sense, between what they could read and what they actually read, rather than a chasm, there is a distance.\nThink about it, are there bookstores in working class neighborhoods? I saw that the workers bought books from vans that made the rounds of the factories. I interviewed the booksellers\u2014and Carpeaux would cry if he could hear them\u2014and they told me that they went to the bookstores and publishers, bought the rejects and bound them beautifully (or at least they did at the time I did the study). The worker who devoted hours and hours, sometimes days, to buy such a beautiful book will put this volume in the living room and save it for her children. The books are expensive, very expensive. You can see how decisive a step it is to read.\nHow much time did you spend doing the research for your dissertation?\nI spent two years conversing with 52 women. Only one of them was in school but, exhausted, she was about to abandon her studies. I must mention here some prior studies on what women workers read: the French writer George Sand [1804-1876] interviewed workers to learn what they read and concluded that the official history, the culture, would not be complete if it did not include the fantasies and desires of those readers. The writer and philosopher Simone Weil [1909-1943] recounted the tragedies of Sophocles to the female workers of a steel plant. They were thrilled by the narrative and Simone Weil realized that fiction could be an escape, avoidance, but also a revelation.\nIn your opinion, did the workers' reading have more to do with escape, or with survival strategies, maintaining a certain degree of sanity?\nThis is a triumph of mass culture over the working culture\u2014which is part of both popular culture and mass culture, but they're different. When the worker escapes, by reading, and seeks fantasy, he is not creating a working culture because this has to have an element of activism. The working culture asks: who are we, who are the people like us? What is the meaning of our work, the value of our work to society? This is why Gramsci wanted to create universities for workers in Turin, where he taught. This is why Simone Weil, who had been a metal worker, taught classes for railway workers and miners. I think Weil, Gramsci and others belong to a deep-rooted vanguard, an Alfredo Bosi expression that I accept and admire.\nAnd what is the meaning here of the word \"deep-rooted\"?\nTo be rooted is to live the popular culture intensely. Mari\u00e1tegui, Simone Weil and Gramsci lived the popular culture intensely and their studies fed off it. There is no greater joy in the world than doing a study for a university and seeing that it has had an effect on public policy. In the case of Leituras de oper\u00e1rias, I had the joy of working at City Hall during the term of Luiza Erundina [1989-1993], at the Department of Public Works, with Lucio Gregori, at the Department of Culture, with Marilena Chau\u00ed, in establishing public libraries\u2014she was very interested in forming communities of readers\u2014and I was also invited by Paulo Freire to work in the Department of Education. It was a time of activism that resulted from Leituras de oper\u00e1rias. But the most important thing I did in my life, related to public policy, was go to the International Labor Organization (ILO), at the UN in Geneva, and denounce the following aspect of the working conditions of female workers: every year new, harmful chemicals appear, and no studies are done on their effect on women. In the case of factories that use radiation, this affects the embryonic tissue in the first three months of pregnancy, a stage in which the worker usually does not know she is pregnant. The child will suffer the effects of radiation on his or her health years later. And the guilty go unpunished. What must be done? The harmful agents have to be studied in the factories in which women work.\nDid your complaint have some practical consequence in terms of general policies?\nBack then, many people complained about asbestos and many nations there prohibited it. Brazil did not want to sign.\nBefore discussing memory, given that I was intrigued by the expression \"deep-rooted vanguard,\" I would like to ask you if you think that many intellectuals use an excessively condescending tone\u2014to the point of sounding irritating and even offensive\u2014when examining the habits, behavior and ways of life of representatives of the poorer classes, etc., when they have no effective tie to these communities.\nWhen an intellectual goes to the outskirts of town to gather information, he is receiving. And the donor is the poor person. He gathers information to support his thesis, climb the academic ladder, and the people who gave him the information continue living precariously, without hope. And the intellectual rises at the expense of those donors, without forming with them\u2014using an expression that is dear to me\u2014a community of shared destiny. Indeed, their destinies are divergent, and he cannot but be aware of this. Moreover, in research it is essential to realize that it is one thing to record information and something else to listen. If the researcher has the gift of listening, the word \"gift\" already includes the friendship. There is no such thing as temporary friendship. There is no easy sympathy for the subject of the study, for the underclass. There is responsible, lifelong engagement. This is friendship.\nNow we can go back to the thread of our discussion from the readings to memory.\nEvery year I teach a course on memory and I also provide orientation for students who want to study memory. In my Memory and Society study I collected biographical memories, but memories of time, space, politics, work and culture also came out. What were the characteristics of these older respondents? They were sensitive to urban transformations. They realized how the city had been changing and how this was reflected in every step of their biographies. Urban planners have to listen to these memories, understand what this city means and what the city's transformations have meant to the lives of its citizens. What did these elderly people tell me about their cities? They told stories that we heard from our grandparents, the passage of Halley's Comet in 1910. They all described Halley's Comet, Oswaldo Cruz's mosquito-killers in the neighborhoods along the rivers, the Spanish Influenza, the wandering of the thief Meneghetti, who was a very nice thief, who robbed the rich to give to the poor. Indeed, the stories about Meneghetti are extraordinary. He bought opera albums, because the workers' districts like the Bixiga [a neighborhood in the city of S\u00e3o Paulo] were Italian neighborhoods, and since he was the one who had a victrola, he would play them loudly for everyone to hear. They were all crazy about opera. But the memory of the elderly swims against the tide, because the city does not allow the elderly to visit one another. They have lost the group with which to remember and share memories. This group is the witness and interpreter of these memories. When this is lost, the memories disperse and great effort is needed to collect them. The anarchism of the early twentieth century, Isidoro's revolution, and how many children were baptized with the name Isidoro afterwards \u2026 The Prestes Column, the revolution of 1932, the two world wars, Vargas and laborism, remembered poignantly. On the death of Vargas, one old man told me, tear gas was used to prevent workers from gathering, but they still gathered and wept because of the gas, only later understanding why. I interviewed a communist teacher who climbed scaffolding and threw stones when fundamentalists demonstrated, and I interviewed an old fundamentalist who had had stones thrown at him when building the Cathedral. The viewpoints are different, but the assumptions are both part of history, no matter our point of view. Another interesting memory is from the youths and adults who remember the nights, during the dictatorship, when they heard whispers, beds being moved, makeshift places. This domestic confusion was to hide activists who were taking refuge in these homes. We understand that hundreds of families hid revolutionaries, whether they sympathized with their ideas or not. I think it is amazing. How many housewives hid youths persecuted by the police? They saved their lives, without knowing their ideologies. Memories of space and political and historical events begin, first, in the family home, which is the geometric center of the world. And that is where the city starts, before expanding in all directions. It begins on the streets, on the sidewalks where life unfolds. I collected the cries of vendors, the calls that rang out in the neighborhoods. I recorded the musical score of the neighborhoods and learned that the city is not just a visual map, it also has a soundtrack and is part of our identity, our integrity. If you think about it, the street has a soundtrack. If you start recording, you note doors opening, a broom sweeping the sidewalk, stores opening\u2026 It's beautiful when a S\u00e3o Paulo resident describes the city, because he says \"there in Penha\" and points to the palm of his hand.\nHe has the city in the palm of his hand.\nIt is an emotional map of the city. Which places are in the memory of the residents of S\u00e3o Paulo? The Viaduto do Ch\u00e1, the Cathedral, Penha, because children who were baptized were taken to Penha and newlyweds made a pilgrimage to Penha after getting married. The Ipiranga Museum, the Luz Gardens, Cantareira and the Municipal Theater. The elderly would say \"I descended the 84 steps\u2026\" as if everyone knew that it has 84 steps. The people in the Br\u00e1s and Mooca [Italian] neighborhoods would dress in their finest and wait near the door of the Municipal Theater. The elite of the city would pass by, continuing inside to their seats. Afterwards, the ticket taker selected the best-dressed people waiting and let them enter. And what did they do? They sat in the seats farthest from the stage and clapped at the right places, because they knew the operas. When they started to clap, the elite knew that it was an important part of the opera. If the tenor sang off key, for example, the Italians remained silent, said \"stonato il tenore,\" (the tenor is off-key, in Italian), and did not clap. And there was an extraordinary individual in S\u00e3o Paulo, a black man who had an unforgettable laugh. So he was always invited to enter for free, of course. When he laughed, his contagious laugh spread across the entire auditorium. I had an uncle who was a paid applauder, and he taught me how to clap properly, creating an echo. And the soccer fields: in Barra Funda, Glic\u00e9rio, Lim\u00e3o, Casa Verde, how many soccer fields were there in the city? We only began going to stadiums when the factories took over the fields to use the rivers to dump their waste.\nAre all these memories from the first half of the twentieth century?\nYes, but that doesn't mean that they didn't continue fighting until the end. I will tell you in a bit about Jovina Pessoa, a great activist who I interviewed. The neighborhoods of S\u00e3o Paulo, when described by the elderly, have a biography, just like us. They have a childhood, adolescence, maturity and old age. And the last stage is the most beautiful of the neighborhoods, because you can see the memories. The physiognomy of the neighborhood matures, following the maturing of its residents. Our stories mix with the history of the neighborhood, and we see on the street that which we never saw, but were told about. When the physiognomy of the neighborhood becomes humanized and mature, it may continue living, or it could die. It could be killed by the real estate developers and urban planners who have no interest in memory, in the survival of the residents. A familiar route between home and the most visited locations is not a privilege of humans, but of all living beings. A neighborhood is a structured whole, common to all that we perceive little by little, and which gives residents a sense of identity. It's horrible to lose the way home, it is the return of the familiar route, if it still exists. The elderly feel trapped when neighborhood blocks are razed. Where do they go? They try to resist, but they usually lose. Moving and death are the same for people. Urban planners should listen to old residents who know every street, every neighborhood. Do the neighborhood councils have the right to veto decisions? Theoretically, yes, but are they heard?\nYour work on memory followed Leituras de Oper\u00e1rias, so the interviews with the elderly took place in the 1980s?\nThat's right. And after that work studies on memory flourished in Brazil, there were many. There is a deep need for them and I believe this is because of the need for roots.\nAfter all, we lived in a country that was trying to delete a piece of memory for political reasons, right?\nThe studies on memory and society have a taste of nostalgia, a bittersweet flavor. Because individuals, when recalling their lives and the city, perform one of the most difficult operations for the human brain, which is to accept the irreversible, that which was lost. When they tell their stories, they consent to this loss with grace and freedom. Instructed by the reminiscences of these brave individuals, I thought about them and about growing old in our industrial society. This society is so bad for the elderly! Due to accelerating historical changes, a person's sense of continuity is broken.\nAnd that's when your idea for the University Senior Citizens Program arose?\nYes, we opened the university. After all, isn't it the taxes of the old workers that sustain us? So it is natural that they take part. And who participates? People with no schooling. And they sit together with undergraduate students. For most of our students, it is the first time they are studying alongside a manual worker, a mason or a maid that is not working for them. These people are participating because they have a passion for knowledge and some have to take three different buses to get to USP. Sometimes one of them washes all the clothing in the tenement where she lives in order to buy a specialized magazine that the professor said they should read. I speak of manual workers because they are the stars of the program, but sometimes participants are more knowledgeable than the professor, such as Mrs. Neuza Guerreiro, a biologist, who is very cultured. But, in general, they are people who were unable to study, and they raise the level of the classes because they were witnesses to history. Students do not know what a person exiled and persecuted by the dictatorship suffered, and the senior citizen student sitting next to them could be that person. And the youngest do not always have the most progressive views. Let me give you an example. A student who never had a college education is the mother of two architects designing a new house. She turns to her children and says she does not agree with the plan, although it is very beautiful, because the size of the maid's room is tiny and, she explains, in her social psychology course she learned that the worker's space needs to be more respected. So the architects redid the plan. The University Senior Citizens Program goes far beyond an academic project because it brings the elderly back into the community.\nBut couldn't they become victims of prejudice on the part of the students?\nSoon the prejudice dissolves. An old laborer, seeing the class complaining of too many books to read for an exam, got up and said he was a manual laborer all his life, but now, because of his age, he can only work when the workers leave and he washes the machines and the floor. He comments \"that is hard work,\" then asks a classmate to hand him a book, shows it to the class and says \"this book is so light!\". This moves the whole class. The book is really light compared to the work of a metal worker discriminated against because he is old! Unforgettable things.\nHow many senior citizens study at USP each year?\nIt varies, but in the last few years there have been about 10,000. More than 100,000 have enrolled over 21 years. They come from everywhere and disperse to the different majors and departments. The non-specialization of the elderly also corresponds to the non-specialization of the professor. So the professor of mineralogy teaches folk dancing. The chemical engineering professor teaches cinema. Because the professor has an enormous responsibility, teaching a student who was already asking questions before the professor was even born. The professor is aware of the student's past and thus prepares much more to teach the class.\nWhat were the first students who participated in the University Senior Citizens Program like?\nVery timid. I want to mention Mrs. Santin\u00eas, a peddler and cook, who had a very hard life. I was giving a lecture, saying how time passes differently depending on a person's social class. The class had difficulty grasping it and she\u2014who could barely read, and had only read the Bible\u2014stood up and began quoting Bible verses she knew by heart. She said \"all things have their time under the Sun.\" A time to be born and to die, a time to plant and a time to reap, a time to weep and to smile, a time to tear and sew, a time to seek and a time to lose, a time to embrace and to separate; a time to be silent and a time to speak.\" The students understood immediately and were touched, because she herself had arrived at her time to talk, to talk in public, to express herself.\nHearing about your academic career makes me feel that there is a very powerful, more intimate thread running through it. So let me ask you, what led you to a generous view of social inclusion?\nPerhaps the environment in which I lived my earliest years. And the immense sympathy that I have for these humble people who gave me everything, this makes me think that I should be at their service as long as I live. Actually, I provide support for the University Senior Citizens Program, I would not like you to say that I established it or that I direct it\u2014I provide support.\nI ask because of your long career, marked by this sense of service to others. I see in it the expression of Christian action, or something similar in a different religious field, the expression of a utopic dimension in the practice of daily life, so\u2026 Generous people are very moved by deep beliefs.\nAnd you want something more beautiful than service? What was Jesus Christ's first miracle? He turned water into wine at a party, to drink with his friends. It was a very human first service, and then he went on from there.\nWhich social psychologists influenced you the most?\nI will mention those who were nearest, and very present when I wrote my works. In Gestalt theory, Anita de Castilho Marcondes Cabral, in theories about time, Henri Bergson. Also Maurice Halbwachs [1877-1945], to whom I dedicated my book, a social psychologist who died in the Buchenwald concentration camp. And in relation to interpretation, my ties are to Adorno, Marx, Hannah Arendt\u2026 I especially like the founder of political ecology, Andre Gorz [1923-2007]. He was great. And his latest book, Letters to D., which are love letters written to his wife, was translated at my request. The Brazilian edition is more beautiful than the French edition.\nWhat is your daily routine at the university?\nI advise students doing research on memories. I encountered Simone Weil along the way, and I have been coordinating the Simone Weil Laboratory these last 11 years. It is interdisciplinary and brings together researchers who study just the works of Simone Weil. Admirable research based on her idea of rootedness arose from there. I coordinate the University Senior Citizens Program and I teach undergraduate and graduate classes. And I planted four orchards.\nTell us the story about the orchards.\nThe inhabitants of S\u00e3o Paulo are urban migrants. I interviewed about 140 people and only one lived in the house in which she was born. I also moved from house to house, and in each house in which I lived I planted an orchard, but I never stayed long enough to harvest fruit, except in the house in Cotia, where I lived for 40 years and which I left a few months ago. I really miss my trees. Life is a bit like this, planting fruit trees, asking God that someone in the future appreciate the fruit.\nL\u00e9o RamosEcl\u00e9a Bosi, emeritus professor of social psychology at the University of S\u00e3o Paulo (USP), deals with research topics that are not among the most studied within Brazilian academic circles: the readings of female workers and the memories of the elderly, for example, to cite only two of the areas on which she focused in her research. Often Bosi shifts her focus to vulnerable social groups: poor, low-income working women; seniors who, immersed in the ongoing transformation of the metropolis, are grudgingly losing the references of their familiar, daily routes, and entering a time of fading awareness of their identity. From the objects chosen to the individuals encountered during the research process, both alluding to the precarious and the vulnerable and based on solid theoretical grounds, she built a strong, unique body of work recognized in her field. Part of this uniqueness has been her ability to express the force of her findings and reflections softly and delicately, which helps give Ecl\u00e9a Bosi's narratives a particularly literary dimension. Empirical research reports and theoretical essays are often expressed in beautiful poetic prose. But, at the same time, they seem to be part of the vigor, the vital force of Bosi's research, its spillover into the field of institutional activism and politics. That is why it is easy to understand her efforts towards establishing and developing the USP Senior Citizens Program which, after 21 years in operation, has served more than 100,000 senior citizens on the campus of the largest Brazilian public university, most of whom had deficient formal education. Or even her eco-activism, which includes a special focus on pregnant workers who may be unaware that they are being exposed to toxic agents at the factories where they work. Age: 77 Specialty: Social psichology; Memory and society Education: University of S\u00e3o Paulo (USP): undergraduate degree (1966), master's (1970), PhD (1971) and promotion to associate professor (1982) Institution: Institute of Psychology, USP Academic production: 8 books, 36 book chapters and 16 articles published; 10 master's students and 14 PhD students advised Married to Professor Alfredo Bosi, a respected critic and historian of Brazilian literature, mother of Viviana Bosi, a professor of literary theory, and Jos\u00e9 Alfredo Bosi, a professor of economics, and grandmother of two, Ecl\u00e9a Bosi, who continues to work despite being formally retired, agreed to this interview with Pesquisa FAPESP on a overcast morning illuminated by conversation at the USP Institute of Psychology, with the principal sections published below. With books and some old articles on the desk in front of her, she stated that she had always been well-treated in her professional career, but one thing that touched her greatly was having her book Mem\u00f3ria e sociedade\u2014Lembran\u00e7a de velhos (Memory and society\u2014Memories of the elderly), the book she worked hardest on, included on a list published by the Ministry of Education of the 100 books sent to thousands of school libraries all over the country. I see in these printed keepsakes memories of your work, struggles and awards. Yes, I think that I strove a lot in my life. Here are things related to my eco-activism. An activist in addition to your academic work. I was involved in the creation of the Chico Mendes Ecological Park. This here is a souvenir of the two successive years of the Ecology Week I organized at USP, before the establishment of the Environmental Sciences course in 1974. I appealed repeatedly to congressmen, senators, important people, to help in the fight against a nuclear power plant, but they did not respond so I decided to write to someone I really admired: Carlos Drummond de Andrade. He studied the matter and wrote a beautiful article entitled \"If I were a congressman,\" published in the newspaper Journal do Brasil. More recent is the struggle against the Angra 3 nuclear power plant. And this book here? It's something I did with great pleasure, the last book I wrote, Velhos amigos (Old friends), true stories for children and adolescents. This other document is an article I wrote about a former, deceased USP professor [Jos\u00e9 Severo de Camargo Pereira]. He was feared. One day, I saw there was some confusion and a line at the door of the USP building on Rua Maria Ant\u00f4nia. There were people everywhere. The school was a sounding board for national politics. Cuba had been invaded and people were there to sign up to fight and defend the island. These people had never used weapons and wanted to face the United States Army. Can you imagine? Was it the Bay of Pigs invasion in 1961? Exactly. Then I heard a voice behind me saying that he knew he was old, but wondered if they would let him participate. It was Professor Pereira. But I want to show you documents related to a program that I have been involved with for 21 years [the University Senior Citizens Program]. The covers of the course schedules almost always depict a person who suffered greatly in life, who started with almost nothing, and here they are shining on campus. And on the cover of this other memento, an elderly man rowing\u2026 My work began with Leituras de oper\u00e1rias (The Reading of Female Workers). Why reading and why workers? Then came Mem\u00f3ria e sociedade \u2014Lembran\u00e7a de velhos; Tempo vivo da mem\u00f3ria (Living memory); Simone Weil: a condi\u00e7\u00e3o oper\u00e1ria (Simone Weil: the condition of the workers); and Velhos amigos. We will try to cover this, but first I need to ask: where are you from? From the city of S\u00e3o Paulo. I spent my childhood in the Pinheiros district. I was born at Maternidade S\u00e3o Paulo, on Rua Frei Caneca. Were your parents educated? They were very simple, but I remember that they both wrote poetry. My father was a civil servant, my mother, a housewife. They had very little education. And I had two younger brothers. What was your childhood like? I was a voracious reader. Books were not given to me, I was not surrounded by books, they were treasures I acquired. How? By walking. A book cost the same as 12 bus passes. I studied in a large house in the Campos El\u00edsios neighborhood, so I had to walk the length of Av. S\u00e3o Jo\u00e3o, and then the length of Rua da Consola\u00e7\u00e3o and Av. Rebou\u00e7as on the way back, to arrive at Rua Mello Alves, where we lived. And by walking instead of taking the bus 12 times I could buy a book. So I obtained my books through sacrifice. Who were your favorite authors? By 13 or 14 I was already immersed in Dostoevsky, Tolstoy, and Chekhov, but also in Romain Rolland and Emily Bront\u00eb. Then I read Hemingway, Sinclair Lewis, a lot of poetry. Later I translated Ungaretti, Leopardi, Montale, Rosal\u00eda de Castro (for newspapers and books). Did you go to a public school? No. It was a school named Stafford, surrounded by a park, but it no longer exists. It was in a huge house located on Alameda Nothman. During my walks between home and school I learned about social inequality. I saw mansions, simple homes, and thought about social inequality without anyone telling me anything. I also learned the city intimately. Later, when I started studying at USP, we students lived in bookstores, libraries, bars. The city was very close to us. You started studying at USP, back when it was located on Rua Maria Ant\u00f4nia, during some turbulent times. What was it like, living those years in that environment? Most importantly, there were great professors. We loved our professors: [Jo\u00e3o] Cruz Costa, in philosophy, Ruy Coelho, a distinguished professor of sociology, Gioconda Mussolini, in anthropology, Dante Moreira Leite, who was my advisor in social psychology, an extraordinary person, and Anita Castilho Cabral, who established the psychology major. How did you decide to study psychology? It might have been because of literature! For someone who read what I read\u2026 Dostoevsky seeks to look inside human beings, and this led me to become interested in looking within us. And these great teachers I had were a great inspiration. It is important to realize that USP has to be understood not through its institutions, but rather through its great professors. It has spiritual families. The presence of a beloved teacher is in our work, it directs our views. We are talking about the golden age of USP. M\u00e1rio Sch\u00f6nberg, Florestan Fernandes, and Antonio Candido were all there \u2026 How did you decide what path your studies would take within Psychology? I went into social psychology because it was a very politicized time, with an enormous political density. My class was small, about 12 people, and was almost completely decimated by the dictatorship. I was a classmate of Iara Iavelberg [an activist killed during the dictatorship], and this greatly affected me. I remember her as a very intelligent, beautiful girl who sang very well. She liked Ponteio, by Edu Lobo, and Disparada [by Geraldo Vandr\u00e9]. She was very good in statistics, taught by Prof. Pereira, and we would go to her house to study. Everyone remembers the story of Iara, but it was more personal for me and I saw how much my class suffered because of her loss. I also remember Aurora Maria do Nascimento Furtado, Lola [an activist who was tortured and died at the hands of the dictatorship]. She was an unforgettable student. When General Fi\u00faza de Castro wrote his memoirs, he was asked if he remembered the subversives that he had captured, and he said yes. He said he remembered a very brave girl named Aurora. She died from the \"crown of thorns\" [an instrument of torture], with her head in a vise, crushed, a heroic death, because she never opened her mouth. She was arrested in Parada de Lucas, in Rio de Janeiro, in 1972. I never forgot her, nor can I. In the Institute of Psychology, I established an \"Aurora room,\" with photographs of her and with this statement by the General on her bravery. And I would like to mention that Professor Castilho invited Iara to be a professor of social psychology and she was even hired, but soon had to leave to live in hiding. I remember her analyzing the content of Fidel Castro's speeches. She never finished that work, because she then disappeared. But she left us a nice article about language and communication that came out in a SBPC journal, and I had the pleasure of giving it to my students for them to read. So, your proximity to the political struggle had some influence on your choice of social psychology? Yes, it deeply affected me. And what did I choose for my PhD dissertation? Female workers. Why reading? Because it is a problem area. Female workers, and indeed all of us, are entrapped by the flow of television and other mass communication media. But reading requires will, a choice. It is a minimal act of will, but it is a step that must be taken. And, in the case of workers, it requires great personal commitment, as there are no bookstores in poorer neighborhoods. What prevents workers from reading? The long, extensive workday, the double shift as worker and mother of a family, with all the housework. The distance between work and home, the lack of cultural centers, the salary that just barely covers the essentials. Although the workers interviewed in your dissertation were younger and various unmarried women and dreamers were among them. You touch on something that I find remarkable. The single worker has one kind of mentality, the married working mother another: she is a fighter. She is always there when someone needs to complain or take political action. She is always in front. Probably due to her commitment to her children. Yes, her salary is very important to the family. I collected testimonials from these workers. Otto Maria Carpeaux wrote the preface to the book and said, \"what a bleak survey, what seduced, exploited minds.\" But I wanted to go one step beyond: I discovered what female workers read and tried to understand what they liked to read. I entered the world of the possible. And is there a big gap between what they read and what they like to read? No, but the two are different. Mass communication is twofold: in the realm of propaganda, it attempts to show what is most advanced in technique; in the field of the imagination, it exploits a pre-industrial mindset that survives in the culture of the poor man, namely serial literature. What was it like, this serial literature that female workers liked so much? In general, it addresses the situation of women and children who suffer societal violence. The woman suffers imbalance, victimhood. This is the type of novel that working women read. And equilibrium is restored when the story has a happy ending, through marriage, the intervention of destiny. In books or in magazine stories? In magazines and in serialized books, in the romantic plot the woman and child are victims not of society, but rather of destiny. And these stories are not dated, they are eternal: they carry the sentiment of exclusion from the world, avoidance, the compensatory fantasy that Freud worried so much about\u2026 Umberto Eco has a beautiful expression for this: structures of consolation. And Gramsci calls it the social inferiority complex or compensation daydreams. Gramsci regrets that intellectuals do not study popular literature. Because of this, they do not establish a modern humanism capable of reaching the most humble. Between Carpeaux's, Gramsci's and your vision, I see a clear difference, among others, in the true closeness with which you treat the group you study. First, let me tell you what these romantic novels are about. In my view, working women are impressed by fundamental questions of justice, guilt, punishment, transgression, revolt, and undeserved punishment. But isn't that what great literature addresses too? The themes are the same. The reader's eyes reach and follow this human drama. The classics tackle this and the manual laborer who reads, if she had the opportunity to read the classics, would probably feel right at home. In this sense, between what they could read and what they actually read, rather than a chasm, there is a distance. Think about it, are there bookstores in working class neighborhoods? I saw that the workers bought books from vans that made the rounds of the factories. I interviewed the booksellers\u2014and Carpeaux would cry if he could hear them\u2014and they told me that they went to the bookstores and publishers, bought the rejects and bound them beautifully (or at least they did at the time I did the study). The worker who devoted hours and hours, sometimes days, to buy such a beautiful book will put this volume in the living room and save it for her children. The books are expensive, very expensive. You can see how decisive a step it is to read. How much time did you spend doing the research for your dissertation? I spent two years conversing with 52 women. Only one of them was in school but, exhausted, she was about to abandon her studies. I must mention here some prior studies on what women workers read: the French writer George Sand [1804-1876] interviewed workers to learn what they read and concluded that the official history, the culture, would not be complete if it did not include the fantasies and desires of those readers. The writer and philosopher Simone Weil [1909-1943] recounted the tragedies of Sophocles to the female workers of a steel plant. They were thrilled by the narrative and Simone Weil realized that fiction could be an escape, avoidance, but also a revelation. In your opinion, did the workers' reading have more to do with escape, or with survival strategies, maintaining a certain degree of sanity? This is a triumph of mass culture over the working culture\u2014which is part of both popular culture and mass culture, but they're different. When the worker escapes, by reading, and seeks fantasy, he is not creating a working culture because this has to have an element of activism. The working culture asks: who are we, who are the people like us? What is the meaning of our work, the value of our work to society? This is why Gramsci wanted to create universities for workers in Turin, where he taught. This is why Simone Weil, who had been a metal worker, taught classes for railway workers and miners. I think Weil, Gramsci and others belong to a deep-rooted vanguard, an Alfredo Bosi expression that I accept and admire. And what is the meaning here of the word \"deep-rooted\"? To be rooted is to live the popular culture intensely. Mari\u00e1tegui, Simone Weil and Gramsci lived the popular culture intensely and their studies fed off it. There is no greater joy in the world than doing a study for a university and seeing that it has had an effect on public policy. In the case of Leituras de oper\u00e1rias, I had the joy of working at City Hall during the term of Luiza Erundina [1989-1993], at the Department of Public Works, with Lucio Gregori, at the Department of Culture, with Marilena Chau\u00ed, in establishing public libraries\u2014she was very interested in forming communities of readers\u2014and I was also invited by Paulo Freire to work in the Department of Education. It was a time of activism that resulted from Leituras de oper\u00e1rias. But the most important thing I did in my life, related to public policy, was go to the International Labor Organization (ILO), at the UN in Geneva, and denounce the following aspect of the working conditions of female workers: every year new, harmful chemicals appear, and no studies are done on their effect on women. In the case of factories that use radiation, this affects the embryonic tissue in the first three months of pregnancy, a stage in which the worker usually does not know she is pregnant. The child will suffer the effects of radiation on his or her health years later. And the guilty go unpunished. What must be done? The harmful agents have to be studied in the factories in which women work. Did your complaint have some practical consequence in terms of general policies? Back then, many people complained about asbestos and many nations there prohibited it. Brazil did not want to sign. Before discussing memory, given that I was intrigued by the expression \"deep-rooted vanguard,\" I would like to ask you if you think that many intellectuals use an excessively condescending tone\u2014to the point of sounding irritating and even offensive\u2014when examining the habits, behavior and ways of life of representatives of the poorer classes, etc., when they have no effective tie to these communities. When an intellectual goes to the outskirts of town to gather information, he is receiving. And the donor is the poor person. He gathers information to support his thesis, climb the academic ladder, and the people who gave him the information continue living precariously, without hope. And the intellectual rises at the expense of those donors, without forming with them\u2014using an expression that is dear to me\u2014a community of shared destiny. Indeed, their destinies are divergent, and he cannot but be aware of this. Moreover, in research it is essential to realize that it is one thing to record information and something else to listen. If the researcher has the gift of listening, the word \"gift\" already includes the friendship. There is no such thing as temporary friendship. There is no easy sympathy for the subject of the study, for the underclass. There is responsible, lifelong engagement. This is friendship. Now we can go back to the thread of our discussion from the readings to memory. Every year I teach a course on memory and I also provide orientation for students who want to study memory. In my Memory and Society study I collected biographical memories, but memories of time, space, politics, work and culture also came out. What were the characteristics of these older respondents? They were sensitive to urban transformations. They realized how the city had been changing and how this was reflected in every step of their biographies. Urban planners have to listen to these memories, understand what this city means and what the city's transformations have meant to the lives of its citizens. What did these elderly people tell me about their cities? They told stories that we heard from our grandparents, the passage of Halley's Comet in 1910. They all described Halley's Comet, Oswaldo Cruz's mosquito-killers in the neighborhoods along the rivers, the Spanish Influenza, the wandering of the thief Meneghetti, who was a very nice thief, who robbed the rich to give to the poor. Indeed, the stories about Meneghetti are extraordinary. He bought opera albums, because the workers' districts like the Bixiga [a neighborhood in the city of S\u00e3o Paulo] were Italian neighborhoods, and since he was the one who had a victrola, he would play them loudly for everyone to hear. They were all crazy about opera. But the memory of the elderly swims against the tide, because the city does not allow the elderly to visit one another. They have lost the group with which to remember and share memories. This group is the witness and interpreter of these memories. When this is lost, the memories disperse and great effort is needed to collect them. The anarchism of the early twentieth century, Isidoro's revolution, and how many children were baptized with the name Isidoro afterwards \u2026 The Prestes Column, the revolution of 1932, the two world wars, Vargas and laborism, remembered poignantly. On the death of Vargas, one old man told me, tear gas was used to prevent workers from gathering, but they still gathered and wept because of the gas, only later understanding why. I interviewed a communist teacher who climbed scaffolding and threw stones when fundamentalists demonstrated, and I interviewed an old fundamentalist who had had stones thrown at him when building the Cathedral. The viewpoints are different, but the assumptions are both part of history, no matter our point of view. Another interesting memory is from the youths and adults who remember the nights, during the dictatorship, when they heard whispers, beds being moved, makeshift places. This domestic confusion was to hide activists who were taking refuge in these homes. We understand that hundreds of families hid revolutionaries, whether they sympathized with their ideas or not. I think it is amazing. How many housewives hid youths persecuted by the police? They saved their lives, without knowing their ideologies. Memories of space and political and historical events begin, first, in the family home, which is the geometric center of the world. And that is where the city starts, before expanding in all directions. It begins on the streets, on the sidewalks where life unfolds. I collected the cries of vendors, the calls that rang out in the neighborhoods. I recorded the musical score of the neighborhoods and learned that the city is not just a visual map, it also has a soundtrack and is part of our identity, our integrity. If you think about it, the street has a soundtrack. If you start recording, you note doors opening, a broom sweeping the sidewalk, stores opening\u2026 It's beautiful when a S\u00e3o Paulo resident describes the city, because he says \"there in Penha\" and points to the palm of his hand. He has the city in the palm of his hand. It is an emotional map of the city. Which places are in the memory of the residents of S\u00e3o Paulo? The Viaduto do Ch\u00e1, the Cathedral, Penha, because children who were baptized were taken to Penha and newlyweds made a pilgrimage to Penha after getting married. The Ipiranga Museum, the Luz Gardens, Cantareira and the Municipal Theater. The elderly would say \"I descended the 84 steps\u2026\" as if everyone knew that it has 84 steps. The people in the Br\u00e1s and Mooca [Italian] neighborhoods would dress in their finest and wait near the door of the Municipal Theater. The elite of the city would pass by, continuing inside to their seats. Afterwards, the ticket taker selected the best-dressed people waiting and let them enter. And what did they do? They sat in the seats farthest from the stage and clapped at the right places, because they knew the operas. When they started to clap, the elite knew that it was an important part of the opera. If the tenor sang off key, for example, the Italians remained silent, said \"stonato il tenore,\" (the tenor is off-key, in Italian), and did not clap. And there was an extraordinary individual in S\u00e3o Paulo, a black man who had an unforgettable laugh. So he was always invited to enter for free, of course. When he laughed, his contagious laugh spread across the entire auditorium. I had an uncle who was a paid applauder, and he taught me how to clap properly, creating an echo. And the soccer fields: in Barra Funda, Glic\u00e9rio, Lim\u00e3o, Casa Verde, how many soccer fields were there in the city? We only began going to stadiums when the factories took over the fields to use the rivers to dump their waste. Are all these memories from the first half of the twentieth century? Yes, but that doesn't mean that they didn't continue fighting until the end. I will tell you in a bit about Jovina Pessoa, a great activist who I interviewed. The neighborhoods of S\u00e3o Paulo, when described by the elderly, have a biography, just like us. They have a childhood, adolescence, maturity and old age. And the last stage is the most beautiful of the neighborhoods, because you can see the memories. The physiognomy of the neighborhood matures, following the maturing of its residents. Our stories mix with the history of the neighborhood, and we see on the street that which we never saw, but were told about. When the physiognomy of the neighborhood becomes humanized and mature, it may continue living, or it could die. It could be killed by the real estate developers and urban planners who have no interest in memory, in the survival of the residents. A familiar route between home and the most visited locations is not a privilege of humans, but of all living beings. A neighborhood is a structured whole, common to all that we perceive little by little, and which gives residents a sense of identity. It's horrible to lose the way home, it is the return of the familiar route, if it still exists. The elderly feel trapped when neighborhood blocks are razed. Where do they go? They try to resist, but they usually lose. Moving and death are the same for people. Urban planners should listen to old residents who know every street, every neighborhood. Do the neighborhood councils have the right to veto decisions? Theoretically, yes, but are they heard? Your work on memory followed Leituras de Oper\u00e1rias, so the interviews with the elderly took place in the 1980s? That's right. And after that work studies on memory flourished in Brazil, there were many. There is a deep need for them and I believe this is because of the need for roots. After all, we lived in a country that was trying to delete a piece of memory for political reasons, right? The studies on memory and society have a taste of nostalgia, a bittersweet flavor. Because individuals, when recalling their lives and the city, perform one of the most difficult operations for the human brain, which is to accept the irreversible, that which was lost. When they tell their stories, they consent to this loss with grace and freedom. Instructed by the reminiscences of these brave individuals, I thought about them and about growing old in our industrial society. This society is so bad for the elderly! Due to accelerating historical changes, a person's sense of continuity is broken. And that's when your idea for the University Senior Citizens Program arose? Yes, we opened the university. After all, isn't it the taxes of the old workers that sustain us? So it is natural that they take part. And who participates? People with no schooling. And they sit together with undergraduate students. For most of our students, it is the first time they are studying alongside a manual worker, a mason or a maid that is not working for them. These people are participating because they have a passion for knowledge and some have to take three different buses to get to USP. Sometimes one of them washes all the clothing in the tenement where she lives in order to buy a specialized magazine that the professor said they should read. I speak of manual workers because they are the stars of the program, but sometimes participants are more knowledgeable than the professor, such as Mrs. Neuza Guerreiro, a biologist, who is very cultured. But, in general, they are people who were unable to study, and they raise the level of the classes because they were witnesses to history. Students do not know what a person exiled and persecuted by the dictatorship suffered, and the senior citizen student sitting next to them could be that person. And the youngest do not always have the most progressive views. Let me give you an example. A student who never had a college education is the mother of two architects designing a new house. She turns to her children and says she does not agree with the plan, although it is very beautiful, because the size of the maid's room is tiny and, she explains, in her social psychology course she learned that the worker's space needs to be more respected. So the architects redid the plan. The University Senior Citizens Program goes far beyond an academic project because it brings the elderly back into the community. But couldn't they become victims of prejudice on the part of the students? Soon the prejudice dissolves. An old laborer, seeing the class complaining of too many books to read for an exam, got up and said he was a manual laborer all his life, but now, because of his age, he can only work when the workers leave and he washes the machines and the floor. He comments \"that is hard work,\" then asks a classmate to hand him a book, shows it to the class and says \"this book is so light!\". This moves the whole class. The book is really light compared to the work of a metal worker discriminated against because he is old! Unforgettable things. How many senior citizens study at USP each year? It varies, but in the last few years there have been about 10,000. More than 100,000 have enrolled over 21 years. They come from everywhere and disperse to the different majors and departments. The non-specialization of the elderly also corresponds to the non-specialization of the professor. So the professor of mineralogy teaches folk dancing. The chemical engineering professor teaches cinema. Because the professor has an enormous responsibility, teaching a student who was already asking questions before the professor was even born. The professor is aware of the student's past and thus prepares much more to teach the class. What were the first students who participated in the University Senior Citizens Program like? Very timid. I want to mention Mrs. Santin\u00eas, a peddler and cook, who had a very hard life. I was giving a lecture, saying how time passes differently depending on a person's social class. The class had difficulty grasping it and she\u2014who could barely read, and had only read the Bible\u2014stood up and began quoting Bible verses she knew by heart. She said \"all things have their time under the Sun.\" A time to be born and to die, a time to plant and a time to reap, a time to weep and to smile, a time to tear and sew, a time to seek and a time to lose, a time to embrace and to separate; a time to be silent and a time to speak.\" The students understood immediately and were touched, because she herself had arrived at her time to talk, to talk in public, to express herself. Hearing about your academic career makes me feel that there is a very powerful, more intimate thread running through it. So let me ask you, what led you to a generous view of social inclusion? Perhaps the environment in which I lived my earliest years. And the immense sympathy that I have for these humble people who gave me everything, this makes me think that I should be at their service as long as I live. Actually, I provide support for the University Senior Citizens Program, I would not like you to say that I established it or that I direct it\u2014I provide support. I ask because of your long career, marked by this sense of service to others. I see in it the expression of Christian action, or something similar in a different religious field, the expression of a utopic dimension in the practice of daily life, so\u2026 Generous people are very moved by deep beliefs. And you want something more beautiful than service? What was Jesus Christ's first miracle? He turned water into wine at a party, to drink with his friends. It was a very human first service, and then he went on from there. Which social psychologists influenced you the most? I will mention those who were nearest, and very present when I wrote my works. In Gestalt theory, Anita de Castilho Marcondes Cabral, in theories about time, Henri Bergson. Also Maurice Halbwachs [1877-1945], to whom I dedicated my book, a social psychologist who died in the Buchenwald concentration camp. And in relation to interpretation, my ties are to Adorno, Marx, Hannah Arendt\u2026 I especially like the founder of political ecology, Andre Gorz [1923-2007]. He was great. And his latest book, Letters to D., which are love letters written to his wife, was translated at my request. The Brazilian edition is more beautiful than the French edition. What is your daily routine at the university? I advise students doing research on memories. I encountered Simone Weil along the way, and I have been coordinating the Simone Weil Laboratory these last 11 years. It is interdisciplinary and brings together researchers who study just the works of Simone Weil. Admirable research based on her idea of rootedness arose from there. I coordinate the University Senior Citizens Program and I teach undergraduate and graduate classes. And I planted four orchards. Tell us the story about the orchards. The inhabitants of S\u00e3o Paulo are urban migrants. I interviewed about 140 people and only one lived in the house in which she was born. I also moved from house to house, and in each house in which I lived I planted an orchard, but I never stayed long enough to harvest fruit, except in the house in Cotia, where I lived for 40 years and which I left a few months ago. I really miss my trees. Life is a bit like this, planting fruit trees, asking God that someone in the future appreciate the fruit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BCG (Boston Consulting Group) held a seminar at the Stockholm School of Economics on their National Strategy for Sweden report. The seminar was held yesterday (25th November) and the recording was uploaded to Youtube just afterwards. The first half of the seminar consists of Johan \u00d6berg, BCG Managing Partner) presenting their report and the second half is a panel discussing the report and issues surrounding it. The panel includes Laurent Leksell (Chairman of Elekta), Gustav Radell (Country Marketing Manager, Google), Ulrika Steg (Head of Mobility Services Sweden, TeliaSonera) and Johan \u00d6berg. A reasonably diverse panel with high focus on technology and innovation.\nThe video is well worth a watch, I have embedded it below, start at 11 minutes into the presentation when \u00d6berg starts talking about Sweden's specific situation.\nI think Sweden will face some very tough decisions in the future and if we are not acting pro actively it will be hard to maintain the lifestyle we have today.\nReddit & Hacker News: The New, Social, Recruitment Tools?\nCan Reddit, Hacker News and the likes become the new social recruitment tools for modern companies?\nRecruiting staff is never easy, especially for new startups. Hiring the wrong person can turn out to be a very expensive affair, especially if your budget is limited to start with. Another problem that companies face is that there is a lack of suitable employees. In Silicon Valley the salary for a relatively green CS graduate can exceed $100,000 per year. I get that it is expensive to live in that area but such high wages shows that the companies have trouble finding the right people to hire.\nSkilled software engineers can often times pick and choose what companies they want to work at and the companies have to brand themselves as attractive employers. Easier said than done. However, over the past few years I've seen an increase in 'alternative' recruitment avenues.\nReddit has a multitude of niche subreddits, communities with a special theme \u2013 everything from cat pictures to country specific communities and information security. The latter is the interest in this case.\n\/r\/netsec is a \"A community for technical news and discussion of information security and closely related topics.\". It is a rather specialized niche with a very specific set of competencies. Information Security is hard. Very hard. Think of all the leaked databases with usernames and passwords you've seen. Finding employees is therefore bound to be troublesome.\nAnother example of recruitment via Reddit comes from \/r\/consulting. Yesterday I noticed a thread there: I want to hook up Redditors with entry level consulting jobs.\nMy Project Management Consulting company is hiring for entry level associates. You need not have experience, and you'd get promoted up starting after the first year.\nI remember being a job hunter in this subreddit and wanting to find places that would take my application. I didn't want to forget where I came from and hope someone here finds this helpful.\nIf this is of interest to you, message me. We'll talk and then I'll hand walk your resume into HR.\nThis is an example of helping the community out, a lot of people are looking to enter consulting and it is never fun or easy to go to all those interviews (with hilariously fun cases\u2026). Consulting is also a special case in that the large consultancies (McKinsey, BGC, et al.) tend to vacuum up a lot of the talent. Definitely troublesome for smaller firms. By generating good will the company can be sure to get some potential employees who otherwise wouldn't have applied.\nHacker News is the community around the startup incubator Y Combinator (wiki). Hacker News works in roughly the same was as a subreddit. Each month there is a 'Who is hiring' thread. The one posted for November currently has 400 comments, with most being companies advertising positions.\nThe companies advertising are everything from big multinationals to small startups. The companies are mostly located on the west coast of USA. It is a bit unfortunate that the European startups have not taken advantage of communities like Hacker News to the same extent as their American counterparts.\nBelow is a post from an employee at Bloomberg, it is currently the most upvoted comment in that thread. He describes in-depth what they do and in a very personal way attempts to woo potential employees. He's asked several questions the thread and does his best to answer them. Very good attitude to recruitment and really helps branding Bloomberg as an exciting place to work.\nSo what lessons can be learned from this? Look beyond the conventional job boards and try to recruit from other sources. It is not guaranteed to work but if nothing else, you get a chance to build up some good will with the community.\nI think this type of recruitment activities will become increasingly important for companies. Dealing with faceless HR departments without any personal connections may become a thing of the past. The unrealistic job requirements listed by HR should become a thing of the past. Let's not kid ourselves, most companies don't need rockstars for most of their positions.\nThese communities are rather niched and therefore you might not find the type of employees you need so, in some cases, Twitter is a better choice. You can do like MTGx (See previous mentions: 1 2) and simply spread the word of your CMO position via Twitter. Any self-respecting (digital) marketeer will have a Twitter account these days.\nAre these the recruitment tools for the future? I don't know, maybe. They are a nice complement to other things and adds human aspect to the whole job seeking\/employee hunting process.\nMany companies don't know how to handle social media, especially not when it comes to recruitment, but Spotify does. The Swedish music streaming service hosted a Q&A session on their @SpotifyJobs twitter account yesterday where they answered questions and showed the competition how to carry out a successful social media strategy.\n@spotifyjobs \u2013 Join the band!\nKeeping a separate account for job recruitment makes sense as the general public is not going to be interested in a jobs Q&A. Having a dedicated job recruitment twitter account allows for direct and focused communication with potential recruits. The Twitter account allows Spotify to both broadcast their message, in this case job openings, and interaction with the followers, many whom no doubt wish to become future Spotify employees.\nAnnouncing the Q&A session on the main @Spotify account is likely to drive more traffic to the Q&A as the main account has over 400,000 followers on Twitter. I also found the Q&A session via the main account, even though I missed the actual Q&A as I saw it too late, due to a university deadline. Needles to say I tweeted them and lo and behold, a reply! Proper social media management! Always make sure to reply to tweets as it makes the users feel special, everyone loves attention!\n@Spotifyjobs replies to my tweet.\nAs you can see it is an informative and personalized reply, the 140 characters lends itself to short, informative replies. The personalization of the tweet shows that it isn't just copy-pasted or an automated replies, which is something I and many others dislike. The guys in charge of the social media efforts over at Spotify know what they are doing.\n@Sashank_NYC asks quite broad question and @Spotifyjobs replies encouragingly but vaguely, it is clear that the manager wants the potential hires to actively look around the jobs page. If nothing else it is a great way to weed out the lazy potential hires. Spotify is bound to be swamped with job applications and if there are many too that aren't qualified. That is time wasted for everyone.\nAn excellent, yet short answer that left @StephieBee in, what appears to be, a rather happy mood. The answer brings some hope for the user if nothing else. I wish her the best of luck in applying for a job at Spotify.\n@Gogzs gets a personalized reply that is sure to make him happy! From what I've heard there is a shortage of software engineers in Stockholm right now so it makes sense for a company like Spotify to recruit outside Sweden. Since the product is in English and they deal with such an international product as music it make sense to keep the \"official language\" English.\nThese are just some examples of the Q&A tweets but they do illustrate how a corporation, who currently has 90 vacancies advertised on their site, should use social media to their advantage. Twitter is the best choice for interactions such as these as it requires little effort for everyone involved. The users feel a lot less commitment when tweeting or following an account on Twitter than they do liking a page on Facebook. It is also easier to manage a Twitter account as messages are short, clear and easier to get an overview of than Facebook posts.\nI believe Spotify's social media efforts within recruiting can be seen as best practice. Other companies would benefit from following their example. It does require that they have an established social media presence already as it would be hard to generate enough traffic and questions without it. And for the love of god, have someone who has actual knowledge answer the questions and manage the account. Don't make the newly hired intern who knows nothing about the company or social media do it. Spotify had their global recruiting manager conduct the Q&A session, excellent choice and I can only tip my hat and say: Well done Jenny!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"had dinner in my apt.\nSteve Dalachinsky is a New York downtown poet. He is active in the poetry, music, art, and free jazz scene. Dalachinsky's main influences are the Beats, William Blake, The Odyssey, obsession, socio-political angst, human disappointment, music (especially Jazz), and visual art with leanings toward abstraction. Dalachinsky's books include A Superintendent's Eyes (Hozomeen Press 2000), his PEN Award Winning book The Final Nite & Other Poems: Complete Notes From A Charles Gayle Notebook 1987-2006 (Ugly Duckling Presse, 2006), a compendium of poetry written while watching saxophonist Charles Gayle perform throughout New York City in that time period, and Logos and Language, co-authored with pianist Matthew Shipp (RogueArt 2008) and Reaching Into The Unknown, a collaboration with French photographer Jacques Bisceglia (RogueArt 2009).\nFor a complete biography, visit his Wikipedia page.\nMaxine Gordon, author of Sophisticated Giant: The Life and Legacy of Dexter Gordon, talks about her book, and the complex life of her late husband.\nIn new podcasts, Bob Hecht tells three stories; one about Miles Davis' use of space in his music, one on the mutual admiration society of Sinatra, Lady Day, and Lester Young, and the other about the train in jazz and blues music.\nIn 1964, along with the orchestra of arranger Lalo Schifrin (pictured), this flutist\/alto sax player recorded one of the first \"Jazz Masses,\" and soon after studied transcendental meditation in India. He would eventually become well known as a composer of music for meditation. Who is he?\nDexter Gordon tells the story of joining Louis Armstrong's band in 1944, and how they enjoyed their intermission time.\nIn this edition of Veryl Oakland's \"Jazz in Available Light,\" photographs of Red Garland, Dizzy Gillespie and Rahsaan Roland Kirk are featured.\nThe question \"What are some of your all-time favorite record album covers?\" was posed via email to a small number of prominent and diverse people, and the responses of Gary Giddins, Jimmy Heath, Fred Hersch, Joe Hagan, Maxine Gordon, Tim Page, Veronica Swift and Marcus Strickland are among the 25 writers, musicians, poets, educators, and photographers who participated...Also, the publication of the winning story in our 50th Short Fiction contest; an interview with Romare Bearden biographer Mary Schmidt Campbell; a collection of jazz poetry; two new podcasts by Bob Hecht; the March edition of \"On the Turntable,\" and lots more...Click here to be taken to the issue.\nJerry Jazz Musician's mission is to explore the culture of America with jazz as the centerpiece. We focus on publishing content geared toward readers with interests in jazz music, its rich history, and the culture it influenced \u2013 and was influenced by.\nWe regularly publish original interviews, poetry, literature, and art, and encourage our readers to share their own perspectives.\nIn an April 19, 2019 feature in the New York Times, prominent writers, musicians, and educators recommend five minutes (or so) of piano music they would play to get friends interested in classical music. The piece is very cool, and offers the opportunity to listen to the music as well.\n\u00a9 2012 \u2013 2018 Jerry Jazz Musician. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"KALAMAZOO, Mich. - The Kalamazoo College women's tennis team lost 8-1 to league-leading Hope College in the Hornets' MIAA opener on Monday inside the Markin Center.\nThe Hornets battled in doubles before falling 8-5, 8-4, 8-6 at one, two, and three doubles, respectively.\nCaroline Norton earned a 6-2, 6-3 win at four singles for Kalamazoo's only point.\nElla Knight fell 4-6, 6-2, 3-10 at six singles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you for visiting the World of We Are One website! I am grateful for every visit and each moment we contemplate, integrate and accept being a member of our global community. Through our awareness and recognition we relate to all beings and life forms collectively to nurture and cultivate the love that we are. This knowing is inherent in us when we are born, though at our human evolutionary point, humans, and therefore society, does not often reflect our true essence, and the illusion of separation continues. This is a root of great suffering which affects us consciously and unconsciously. By bringing awareness within, through enquiry and deep contemplation, these illusions begin to dissolve.\nMy vision and motivation to reflect our unified existence has led to the creation of two children's books, We Are One and We Are Love, both accompanied by the Exploration Cards. These books are anchor points to remind us of our divine essence. Let's support and guide our young children into a new paradigm, one that celebrates life in all of its diversity!\nI was born in Buffalo, New York in 1967, and at 2 years of age my mom, younger brother, and I moved to West Palm Beach, Florida. We lived in a trailer while mom worked 2 jobs to support us. When I was 5 she met my dad, who came into our lives with a new sister. They married and gave me 3 brothers. Born Catholic, my new father brought his Jewish heritage with him, my mother took the Mikvah and we celebrated the yearly traditions, of which one was my Bat Mitzvah.\nWhen 15, my family moved to the Gold Coast, Queensland Australia in promise of a simpler drug free environment for our upbringing. This was a smart though uncomfortable move at my age, and did prove successful. I left high school at the end of year 11 to pursue an apprenticeship in hairdressing, worked for a great company, established good work ethics, developed socially and began exploring creativity.\nIn my early 20's I took a trip to the United States, bought an old milk van (which had been converted into a mobile home) and travelled around for a couple of years. While falling in the love with the incredible varied beauty of the landscape, I also witnessed the diverse cultures and societal conditions, some of which were disturbing and dangerous.\nLooking back, I can see that my return to Australia was influenced by an illusion that I could hide and avoid facing the harsh inequalities, prejudices and racism I witnessed while travelling through the cities and towns of my homeland. This worked for while as I explored my creativity through ceramics, opening a little business and happily painted away. My first impulses towards healing and awakening also surfaced at that time, offering hope and evolution.\nYears rolled by as I practiced the art of mosaics, painting, poetry, spirit and love. My joy of travel continued and led me to Mexico, Guatemala, Israel, Egypt, and India, eventually settleng in the coastal town of Byron Bay, Australia. This beautiful environment was full of healers, mystics, artists, musicians and eclectic dreamers, which supported my spiritual practice through Vipassana Meditation, circle work, dance, and A Course in Miracles.\nIn 2003 while living my dream on the beach in a communal environment, a fire claimed my home and studio. For months, I experienced signs and intuitive moments to relocate, which I did not acknowledge, in the end, the power of nature did it for me. Many realizations ensued, including recognition that I was deeply attached to my identity as an artist, the awareness of the beauty and freedom of impermanence, and lest I not forget...the power of intuition!\nWithin a year I was living on a magical land with dreams of communal gardens, playing, dancing, and creating. Though filled with life's earthly treasures, my spirit stirred restlessly and I found myself at a Path of Love workshop. This journey not only offered a profound deepening within myself, I also met my beloved, Steve.\nA corporate lawyer from Sydney, it was not long before Steve joined me in the mystical lands of the Byron shire. Our union, inspired the birth of our first child, Zade who showered our lives with miracles, this lead to the creation of my first book, We Are One. Written in an hour sitting, a year was spent in its illustration. A similar journey occurred with the birth of our second child, Jiah, who flooded our hearts with profound love, resulting in the creation of the 2nd book, We Are Love.\nOnce in print, We Are One, We Are Love, and the accompanining Exploration Cards won Gold in the Mom's Choice Awards, We Are Love was awarded the Seal of Excellence in the Creative Child Awards, and the We Are One Exploration Cards was awarded a Preferred Choice Award in Creative Child Awards. As the publishing industry continued its adjustments to the technological age, I decided to come out of the studio to share the books with the world, this eventuated in our move to Sydney, Australia.\nIn 2012, Steve was offered a contract in Mongolia as a Leadership consultant, and we spent the following two years living in Ulaanbaatar. A central location to travel from, we went to Hong Kong, Thailand, England, France, and Turkey. Being sheltered from the distractions of the western world allowed time for new creative endeavors which resulted in the creation of Soul's Palette and Lighting the Way. After our time in Mongolia we spent a year living in San Fransico, California and then returned home to our treasured home of Byron Bay, Australia in 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"David Christopher Rees qualified as a doctor in 1994 from the Royal Free Hospital School of Medicine, University of London. Prior to this he undertook an intercalated BSc(Hons) degree in 'Biological and Clinical Science' in 1992 at St Mary's Hospital Medical School. After House Officer positions at the Royal Free and Hemel Hempstead Hospitals Mr Rees obtained full registration with the General Medical Council (GMC) in 1995.\nAs a Senior House Officer in Surgery, Mr Rees worked at the Hammersmith Hospital, the Royal Brompton Hospital and the National Hospital for Neurology & Neurosurgery all in London as part of his basic surgical training. In 1998 he obtained his FRCS (Fellowship of the Royal College of Surgeons of England) examinations before commencing his Specialist Registrar training in Trauma & Orthopaedic Surgery based at the internationally renowned Royal National Orthopaedic Hospital (RNOH) in Stanmore, Middlesex.\nIn 2004 Mr Rees was awarded a highly competitive 'knee fellowship' by the British Association for Surgery of the Knee (BASK). It was during this year that he consolidated his experience in reconstructive knee surgery including arthroscopic cruciate ligament reconstruction, revision knee replacement and computer assisted knee surgery at the Derbyshire Royal Infirmary, Derby and the Queens Medical Centre, Nottingham.\nIn 2005 Mr Rees passed the FRCS(Tr & Orth) examination, was awarded his 'Certificate of Completion of Specialist Training' in Trauma & Orthopaedic Surgery and was granted Specialist Recognition by the GMC. In early 2006 he was appointed Consultant Orthopaedic Surgeon with a special interest in knee surgery at Broomfield Hospital in Chelmsford, Essex.\nMr Rees has several published papers to his name and is heavily involved in Medical Education and Surgical Training. He has taught on surgical skills courses organised by the Royal College of Surgeons (RCS) of England and lectured on many courses run at RNOH, Stanmore. Mr Rees is one of the RCS Assigned Educational Supervisors at Broomfield Hospital and oversees the training of junior surgeons.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DLNA Test Labs or Independent Certification Vendors (ICVs) are authorized test facilities that conduct the Certification testing for DLNA Certified products. All ICVs are experts in running the DLNA test tools and all have the interoperability equipment needed for Certification. ICVs can also provide pre-testing services to help companies achieve Certification.\nBEIJING CESI TECHNOLOGY CO., LTD.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Three Advancing White Soldiers Technical Analysis Patterns The Three Advancing White Soldiers (or Three White Soldiers) is a bullish reversal pattern. It is formed by a 3 bullish (or more) consecutive candle with consecutively higher closes.\nForex Glossary; Three White Soldiers; Forex Glossary The Industry's Most Important Terms Explained. The forex industry is made up of countless definitions and it's easy to forget a few along the way. All times are in GMT +2 (GMT+3 during DST).\nUsually when the three white soldiers pattern is formed, it signals the end of the downtrend. 2. Three black crows pattern. The three black crows pattern is a bearish reversal candlestick chart pattern that consists of 3 bearish candlesticks. Here is how the three black crows chart pattern forms: the market has got to be first in an uptrend.\nThe morning star and the evening star are triple candlestick patterns that you can usually find at the end of a trend. Three White Soldiers and Black Crows. BabyPips.com helps individual traders learn how to trade the forex market.\nThree White Soldiers \u2014 Check out the trading ideas, strategies, opinions, analytics at absolutely no cost! Three White Soldiers \u2014 Check out the trading ideas, strategies, opinions, analytics at absolutely no cost! House Rules Moderators People Chat Stock Screener Forex Screener Crypto Screener Economic Calendar. For Business.\n3 soldiers forex 10.06.2017 albertus 4 Comments Al Gore's \"An Inconvenient Sequel: Truth to Power\" is humanity's last chance to save the Earth before it ends five years ago.\n11\/22\/2017 \u00b7 This forex trading strategy is called the 3 White Soldiers 3 Black Crows and is based on candlesticks. The three white soldiers are three solid green candles. (solid candles pushing low to high).\nWe even have members who organize get-togethers with other soldiers who live near each other. If you're going to fight in war side by side, it's important to know who you're fighting beside and who has your back, which is exactly why everything about The Forex Army is designed and structured purposefully to forming a strong family.\n1.2 About The Forex Army; 1.3 Live Forex Trading Room; 1.4 The Rules; 1.5 How To Get Ranks; Stage 2 : Trading Psychology. 2.1 The Psychology of Trading; They will be given more duties, responsibilities and power to guide the new batch of soldiers coming in to start their journey towards profitability.\nSet up an online 3 white soldiers forex market Portfolio. See more trades and director dealings. Play the Fantasy Share Trading Game. Games Workshop lets ferocious armies clash without the drawback of actual bloodshed. Founded in 1975, the company designs and manufactures model soldiers, game systems, and accessories for tabletop wargaming.\n3 white soldiers forex broker Posted on 15.04.1980 by admin Swing traders tend to trade the daily charts and some may even get down to the shorter time frame charts to chose a good entry or exit that can be found using price action techniques.\n1.1.3 How to install Three White Soldiers & Three Black Crows Forex Swing Trading Strategy? The three soldiers pattern is a bullish reversal candlestick and they form after a bearish market. If these 3 candlesticks are formed, it indicates the end of the downtrend.\nThe Three White Soldiers formation is a bullish pattern that is useful for predicting the reversal of a bearish trend. This pattern consists of 3 consecutive long white or green \"soldiers\" that have all closed higher than the previous day, with each session's open (the botton of the bar) occurring within the trunk of the previous candle.\nWeek chart for WTI OIL confirms 3 white soldiers canlesticks. Strong buying @ these levels means oil is moving higher from here. TradingView . EN English. EN English (UK) House Rules Moderators People Chat Stock Screener Forex Screener Crypto Screener Economic Calendar. For Business.\n1\/20\/2011 \u00b7 H4 is the best timeframe.. Daily is also good but you dont get many signals sometimes daily may have 3 white soldiers and H4 may have just started 3 black crows. in that case the trend may be reversing. i avoid those signals.\nThree white soldiers is a candlestick chart pattern in the financial markets.It unfolds across three trading sessions and suggests a strong price reversal from a bear market to a bull market.The pattern consists of three long candlesticks that trend upward like a staircase; each should open above the previous day's open, ideally in the middle price range of that previous day.\nForex & Currencies Chart Strategy As with any reversal pattern, an expansion on volume accompanying the three white soldiers lends additional strength to the signal.\nForex trading strategy #12 (3 white soldiers \/ 3 black crows) Submitted by User on May 26, 2009 - 19:32. 3 consecutive bearish candles for a sell - that's it!!! - 3 white soldiers \/ 3 black crows formations. Read entire post >>> Submitted by User on May 28, 2009 - 04:30.\nforex 3 white soldiers patterns to draw . Home We are your source for complete information and resources for forex 3 white soldiers patterns to draw. Uchiyama, the water sample may be salted to decrease the solubility of the organic analytes and increase their Section 19.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Enough games, every time you guys stall or lie. I will put this in the hands of the rolex replica uk card company and my solicitor for fraud. What is your head office address and rolex replica number, even in china. You have continuously played me with this I will also put up a web link forum telling everybody not to buy hublot replica sale from you as you lie! I received what I thought was going to be my rolex replica uk today, however I was disappointed to find out there was rolex replica in the box apart from foam packaging. I am highly annoyed that I have spent replica watches on a rolex replica uk and waited three weeks to receive a box full of packaging. I hope you will do something about this and replica watches the problem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lunchbox Dad: See My Family Star in the New #SwifferDad Commercial!\nSee My Family Star in the New #SwifferDad Commercial!\nDisclosure: This is a sponsored post partnering with Swiffer for their #SwifferDad campaign. I received compensation for time and travel but all opinions expressed are my own.\nA few months ago, my family and I had the honor of being in a new video (watch it below) for a movement that helps to celebrate the role of the modern dad. The #SwifferDad campaign teamed us up with three other dad bloggers who are helping to promote the idea that the modern dad is more involved than ever before!\nI am so proud of this video because of how it shows that dads everywhere are changing the perception from the tired, clueless stereotype to the involved hands-on father. I hope you take a minute out of your day to watch it now.\nThis trip to shoot the video was so amazing for my whole family. Not only did I get the opportunity to travel with my wife and kids, but my children had such a fun time on set. This was the experience of a lifetime for them. They were pumped to be interviewed on camera, but honestly were almost as excited for the free snack table and the swimming pool at our hotel! One of the best parts was hearing from my kids about the difference it makes having a dad who is involved and cares about them.\nAs a hands-on dad I find it exciting that the portrayal of fathers in the media is shifting. In other words, the image is starting to mirror what fatherhood is like in real life today. From messes to macaroni to mayhem, dads are putting themselves in the thick of it. The recent Swiffer Cleaning Index study found that Americans say that dads today are pitching in with chores nearly two-times more than their dads did growing up. Their roles have really evolved.\nThese are the reasons I was so excited to be part of this project. The #SwifferDad campaign is shattering stereotypes to show that today's dad makes meals, cleans, plays with his kids, and loves his family in every way possible. Please share the video above using #SwifferDad and also be sure and tell everyone how you, or another dad you know, is making a difference in their family.\nI'll leave you with a short behind the scenes video from Anthony Anderson talking more about why he is so excited about what #SwifferDad is doing with modern fathers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NFPA 25 Obstruction Investigations - Consulting & Repair in Maine, New Hampshire, Vermont & Massachusetts | Sprinkler Systems, Inc.\nSprinkler systems are mechanical systems that require ongoing inspections and maintenance to help keep them in optimal working condition. Over time these systems can accumulate organic or inorganic materials inside the pipe and valves. Piping, valves and other devices can even become obstructed by these materials which can cause system failure in the event of an emergency. This build-up also prematurely deteriorates and weakens the piping.\nIn 2002, NFPA recognized that this internal build-up compromised the integrity of the sprinkler system and posed a significant risk to life and property. Code was added to require five year internal obstruction investigations of fire protection system piping to identify sources of materials that could cause blockage and deteriorate the system. The obstructing materials can come in the form of MIC (microbiologically influenced corrosion), slime, sludge, flakey scale or even a hard packed cement like substance.\nSprinkler Systems Inspection Corp. has knowledgeable and experienced staff ready to help you develop a program for code compliance and maintaining your system in working order. Internal inspections of piping are performed by licensed inspectors. Full reports with accompanying photos are supplied to help monitor the health of your system. Should a problem be found, our professional staff is ready to offer guidance and solutions. Contact us for an estimate today.\nNFPA 25, 14.2.1 An inspection of piping and branch line conditions shall be conducted every five years by opening a flushing connection at the end of one main and by removing a sprinkler toward the end of one branch line for the purpose of inspecting for the presence of organic or inorganic material.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bedroom decorating ideas evinco is one images from 19 surprisingly decorate bedrooms of DMA Homes photos gallery. This image has dimension 1500x911 Pixel and File Size 551 KB, you can click the image above to see the large or full size photo. Previous photo in the gallery is home design bedroom decorating ideas. For next photo in the gallery is bedroom decorating ideas. You are viewing image #16 of 19, you can see the complete gallery at the bottom below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"oven bulbs 2 pack microwave led light bulb for appliance oven bulbs watts.\noven bulbs led bulb microwave oven light candelabra bulb ceiling fan light bulbs intermediate.\noven bulbs safety cl oven safety.\noven bulbs watt round oven bulb warm white photo 0.\noven bulbs crystal tube oven incandescent bulb microwave oven bulb.\noven bulbs oven light bulb.\noven bulbs range hood light bulb replacement range hood light bulb replacement oven hood light bulb amazing oven.\noven bulbs halogen oven light halogen oven light oven bulb.\noven bulbs watt oven light bulbs pack of 2.\noven bulbs oven bulb halogen led high temperature degrees oven toaster steam light bulbs cooker hood lamps in halogen bulbs from lights.\noven bulbs halogen bulb oven hood light range bulbs lenses kits.\noven bulbs 3 pk bulbs for microwave oven.\noven bulbs online cheap high temperature degree oven toaster steam light bulbs cooker hood lamps by.\noven bulbs status pack of 2 oven bulbs.\noven bulbs watt clear oven bulb 4 units.\noven bulbs oven and cooker hood bulbs lamps.\noven bulbs led bulb oven light stove bulb light daylight white non.\noven bulbs cooker hood bulbs.\noven bulbs microwave light bulbs 1 bulb microwave led light bulb for appliance oven bulbs watts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Press the clutch pedal down all the way. The START (III) position does not function unless the clutch pedal is pressed.\nMake sure the shift lever is in Park. Press on the brake pedal.\nThe immobilizer system protects your vehicle f rom thef t. If an improperly coded key (or other device) is used, the engine's f uel system is disabled.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Disqus, a startup developing a next-generation blog commenting system, has raised $500,000 in fresh funding in a round led by New York City-based VC Fred Wilson (Union Square Ventures), along with well-known angel investors Naval Ravikant, Howard Lindzon and ex-Googler Aydin Senkut.\nI have often said that the real value of blogs lies in the intelligence embedded in the comments. The quality of the conversation is the ultimate barometer of the success of a blog, especially when measured using the currently fashionable engagement barometer. What is frustrating is that many of these smart conversations are lost in the overly simplistic commenting systems that come with the popular blogging platforms, including our platform of choice, WordPress.com.\nIn other words, I have been looking for ways to unlock this intelligence and promote it to my audience. And that's why I've been fascinated by the startups that are trying to reinvent the commenting space. Unfortunately, none of them deliver what I really want.\nSome offer a ranking system, others offer better discussion forums, but importantly, none are tightly integrated with my existing blog system. I want to turn intelligent comments into posts themselves, highlighting the efforts put forth by the commenters and thus making the blog truly collaborative. I know sooner or later someone will get there and come up with my perfect blog enhancements.\n* SezWho Offers Portable Reputations.\n2) We serve plenty more than 20 million impressions.\n3) You are counting your entire arsenal of widgets, only one of which relates to commenting.\n4) None of the major metrics indicate your monstrosity :) (compete, alexa, quantcast). Can you show me something?\nI'm surprised that you failed to include the clear market leader in your assessment. Each day, JS-Kit serves some 20 million impressions to many millions of unique users interacting on over 60k sites. We're larger than these three combined!\nFred maybe your stats are different from every blog I currently publish, but in most cases the landing page is the permalink for the blog post, and a percentage of traffic then clicks through to either another permalink page (if you have some kind of related links or featured content) or will click through to a home page.\nDisqus is a great investment because it could easily slot in to replace the comment system on Blogger that Google seem intent to just kill off.\nBut then again so could Haloscan which people have been using for 2 years\u2026 but the bloggers lose ownership of their comments, and it becomes a pig to migrate to another blogging platform with a useful comment system.\nDisqus is pretty safe from spammers, who is going to go to the trouble of working out how to spam 4000 blogs?\nSezWho is not a comment replacement system or a destination site and as such the graph with compete stats is completely misleading\u2026Instead SezWho provide a context, rating and reputation service that leave the content and traffic at the site where it belongs.\nFor Disqus to be showing this graph demonstrates a complete lack of humility and understanding of community dynamics. They are showing that they are doing a great job of driving traffic away from the sites where are are supposed to he helping build community\u2026I am not sure it really is something a service that provides community building tools should be showing.\nSezWho does not drive page views away from the site every time there is a profile view or take away page rank\/SEO juice away from a site. And we thik that is better way to build communities.\nwhen a comment someone leaves here get to be the top post on techmeme, then we will have solved the problem.\ni am betting on disqus to lead us there, but honestly if js-kit, sezwho, or intense debate gets us there instead, that will be fine with me too.\nas I said in my post on disqus today, the comments on my blog are so damn good that it's a shame they are stuck behind a comment link too few people click on.\nA lot is done on #1 and #2 but we still have to see a lot on #3. I would love to see the same visualization on people and their content \u2013very similar to what we see on pure web data. There are a few WordPress plugins which do a bit of visualization but the field at large is still green.\nI'm surprised that you failed to include the clear market leader in your assessment. Each day, JS-Kit serves some 20 million impressions to many millions of unique users interacting on over 60k sites. We're larger than these three combined!\nIn addition to Comments, JS-Kit offers Ratings, Navigator, Reviews and Polls and we have much more in the works, including a user profile that will immediately blow away what's currently available and a binary (up\/down) alternative to 5-star ratings. As Indus points out, the Compete data is misleading since, like SezWho, we don't compete for eyeballs on js-kit.com.\noh\u2026 and congrats to Disqus!\nGood points you make. That said, we should talk about your ideal commenting system. Why don't you kick it off and we can take it from there. Thanks in advance for jump starting the conversation.\nThe graph is misleading. Unlike Disqus, Sezwho does not host your comments and is not a destination site. It also does not compete with the site's traffic. Sezwho is a context rating and reputation layer between the blog and your comments and not a comment replacement system.\nSince the comment data stays in its original platform (WordPress, MT, etc.), SezWho does not generate any new traffic for itself. Instead, the cross-linkage between various blog posts gets increased click thrus without touching SezWho.\nMore and more, the blog posts and the comments are becoming increasingly important \u2014 giving it away or letting someone else benefit (read, PageRank, SEO, etc.) from it does not make an iota of sense.\nI still don't understand why would someone use disqus when gurus behind WordPress, MT, etc. have a very solid commenting system!\nMy 2 cents, would love to see some \"discussion\" around it.\nGreat news \u2013 would also be great to be able to use Disqus here, Om!\nthanks OM. I believe Disqus is closest to perfect blog comments and you will likely see very soon your wish for turning intelligent comments into posts themselves. fred and I talk about it all the time and they are on it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Want some new and young search? It's possible to be true. Hairstyle is one of the items that you can certainly do to enhance your face look. It is a must for girls to have a great physical interest and one of many methods is changing the appearance of one's hair. Among recommended hairstyle is easy updos for medium thin hair. You understand that those individuals who have thick hair is anything worth that you have. You have strong hair and you can make your own hair search so different for your face search and also you'll have your look therefore significantly better.\nConduct to look for the ideal hair styles in making your current mid time-span tresses turn out to be much more now beautiful? Should you, your breathtaking easy updos for medium thin hair can be the good solution that can definitely conclusion your quest. It's because each of these hair-styles can an ideal hair styling that will will make you have the gorgeous appearance that will enhance the way you look amazingly. Apart from, this hits which have been included to your hair may be the great improvements that tends to make your lifestyle provide the fairly sweet touch when well.\nThe thing that makes you awesome? The solution is your hairstyle. You know that hair do is the main appearance that folks can change undertake a superior plus refreshing glimpse as compared to before. If you would like a little something chiller, it means that you need progress intended for that. This particular is a great one matter to be able to improve your actual fascination since well. You will notice easy updos for medium thin hair that will be your advice with implementing a design to your hair. Another thing this you have to know is that this hair do is actually trendy and you may get it much better. There are some well known celeb this implement such hair style and you will copy it.\nWhen deciding on a new hair, one thing that one could learn about your own self is you actually face type. You know sometimes you apply the best hair style but it is unsuitable on your makeup form, then its much more serious with regard to you. A little something that needs progress for you to the feel of the face is good so that you can do. As a result, you'll need to be particular within looking for a hairstyle. You possibly can try to see easy updos for medium thin hair to know which will coiffure which has got the very same facial style and you could even look at to supply ones assessment in order to it.\nIt's variations using the face treatment variety that you've here. As a result, it is good to get easy updos for medium thin hair which usually is acceptable in order to you. The way of the particular new hair-do that you could submit an application in your tresses are something you may take and employ to your hair. For this reason, no reason to often be perplexed if you need to apply a coiffure to get you. This kind of are some things that makes you change the design of that person being tidier plus great looking.\nFurthermore, when you need to create your own personal medium size hairstyles with bangs, it is therefore much advised for you to pay more awareness of the hair services and products that you will use. It's therefore essential for you to be sure that you receive the best products and services, so that you can make the absolute most gorgeous easy updos for medium thin hair as you want.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buttons and Brooches and Pins...OH MY!\nOkay, I wholeheartedly subscribe to the \"Guilt Free Blogging\" school of thought since this is supposed to be fun and not terribly serious. However, I never meant to be away for two weeks. I've missed you all. I not only haven't posted, but I haven't been visiting my blogging friends either. I have some SERIOUS catching up to do. So, without further ado, welcome to PINK SATURDAY, hosted by Bev at How Sweet the Sound. Pop over there and check out the other 145 participants' blogs as you have the chance for some fun and not so serious posts.\nHappy Pink Saturday! I'm pleased to have a guest blogger today, my friend Denise, Head Antiques Maven from Our Cottage Garden and ebay seller winter-garden. She's going to show you how to make this adorable springy valance from vintage handkerchiefs. Thank you, Denise for sharing a great fun and whimsical project to help shake off these winter blues.\nLooking for a fun and easy project to celebrate the arrival of Spring? Warmer breezes tease us to open the windows and listen for the cheerful, chirping birds. Here's a quick and simple way to add a touch of lightness and color to any window ~ a Spring Hanky Valance ..\n8 \u2013 12 vintage handkerchiefs .. colorful, floral, lacey, or simply white. If you're a \"collector\" as we are, you probably have a stash of pretty ones tucked away somewhere. If not, they are plentiful, easy to find, and very affordable. You can use colors which complement each other, or the room, or just mix-and-match (which makes it more fun).\nApx 2 yards of vintage ribbon to wrap the dowel. Or acrylic paint and a sponge brush.\n8 \u2013 12 vintage curtain clips. Spring-loaded like these are a cinch to use. Or you can stitch the hankies to a variety of other clips.\nThis set is available for sale now on Our Cottage Garden.\nWhen you're ready to redecorate the window, the hankies can be stashed away again (no harm done) .. awaiting their next assignment.\nHappy Pink Saturday and Happy Spring!\nI have lots of collections, a fact I freely admit. Usually, I know from the moment I buy the second one of the item in question that I intend to have a collection. Not so with the cherubs. I was laying in bed the other day and looked around my bedroom...a few cherubs. I went downstairs to get my coffee and began looking around...hmmm, more cherubs. I made a comment to Mr. French Laundry that I thought I had a cherub collection. He said something to the effect of, \"Duh! I knew that, didn't you? I've known that for a long time.\" Well, I DIDN'T know. Not sure how I missed that one, but I did. But, here for your perusal are a few of the little darlings that I found lurking around my home.\nWhat collections do you have in your home that you never intended to collect but that's how it ended up? I'd love to hear your comments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During pregnancy your diet wants are going to increase. Along with this process, the physique needs protein to heal throughout damage or illness as well as to keep the immune system robust and wholesome. The challenges of balancing youngsters, career, and personal life will be nerve-racking. A complete site not only helps you discover a wholesome solution to reduce weight, however supplies you the motivation and inspiration you have to shed some pounds and keep it off for the rest of your life.\nIt's a incontrovertible fact that the women who are between 25 and 45 years of age need further little bit of care for the upkeep of their good well being. It has been used as a girls's well being supplement for a few years to reduce occurrences of hot flashes and evening sweats which are a specific problem for a lot of women.\nFortuitously, there at the moment are well being supplements out there focused particularly for girls. It has also bee shown that womens health fitness is important to maintaining girls healthier particularly after menopause when their natural protection dwindles.\nSome health supplements which may also be taken by women are the pregnancy anc conception dietary supplements. Womens well being health is lastly getting the identical attention that mens has for many years. Girls's vitamin is essential to stopping a lot of the prime health points found in women.\nOmega fatty acids have been shown to advertise the well being of many body methods including heart, brain and joints. Girls's weight coaching will help tone the body and provide extra definition to muscles with out growing and adding massive bulk.\nThough more men die of heart illness than girls, females are usually under diagnosed, typically to the purpose that it is too late to assist them once the condition is discovered.\nLadies are unique people every going through completely different well being issues affected by weight loss plan, atmosphere, heredity and many others. Girls who are sexually active, or who're 18 or older (whatever comes first) ought to take their well being in their own palms and have common cervical smears till the age of 70. An everyday smear could be every three years for most women, depending on their well being and the outcomes of the last smear.\nThere are a selection of nursing agencies that present certification in ladies's well being, and their examinations take a look at information in a variety of subjects affecting ladies. With our present lifestyle, it's essential that vitamin D dietary supplements be consumed (but not with no physician's prescription) to keep up healthy ranges in the physique.\nWomen's health and health points have turn into a serious a part of our mindset. Elements growing the chance of a lady having a pregnancy as such are: a history of molar being pregnant, being above 40 years of age, having a deficiency of vitamin A (beta-carotene), or a history of miscarriage. Common exercise and good eating habits will do a lot to enhance your lifestyle and self image.\nMost medical professionals recommend OTC and prescription medicines, nevertheless a rising number have been suggesting natural treatments for womens health points. There are a couple of medical situations which are of nice concern to a ladies's health like heart disease, breast cancer, and osteoporosis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Hickory Stick Bookshop in Washington will offer a story time with children's author Donna Marie Merritt and illustrator Ed Heck: Nov. 17 at noon.\nMerritt and Heck will feature readings and drawing demonstrations from their storybook \"Teensy Meensy Mouse\" at the 2 Green Hill Road shop.\nThe program is geared toward children 3 and older, but all ages are welcome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello and welcome to Wheelie Good Meals with me Richard Jones. I'm very lucky to have grown up as a butcher's son and with a mother who is an excellent cook. My father likes good British food, whereas my mother, on the other hand is very adventurous in what she eats.\nMy love of cooking and good food comes in part from a love of cookery programmes. I have lots of favourite chefs but I suppose my top five (in no particular order) would be Brian Turner, James Martin, Bryn Williams, Jo Pratt and Cyrus Todiwala\u2026 oh and not forgetting the Italian Stallion Gino D'Acampo.\nJo Pratt is someone who I have the pleasure of classing as a friend, even if it is through the cybershpere. We became friendly through twitter and she is someone who has not only given me great advice in setting up this blog but also someone whose recipes I use frequently. Jo's \"Madhouse Cookbook\" is a book I refer to a lot along with \"Brian Turner's Favourite British Recipes\". Brian Turner, Brian's someone who I love to watch on television, he makes cooking interesting and enjoyable and fuss free (exactly what Wheelie Good Meals is all about), Brian also has a failsafe Yorkshire Pudding recipe which I will show you at some point.\nBeing a big fan of Indian food and the use of spice I am love watching Cyrus Todiwala who has an amazing ability to work with spices and along with Tony Singh makes up the Incredible Spice Men on the BBC, something you should watch when\/if it comes back on. Two other chefs who have an amazing way with spices and who I also use as inspiration are Ching He Huang and the legendary Ken Hom.\nLots of people have helped in the making of this concept, you can find them on our acknowledgements page.\nI hope you enjoy the Wheelie Good Meals concept, If you have any questions you'd like to ask about either myself or Wheelie Good Meals please feel free to drop us a line on enquiries@wheeliegoodmeals.co.uk. Don't forget you can find us on Instagram, Twitter, Facebook and Youtube, join us as we make Wheelie Good Meals!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sticky: Rear Torsion Bar Upgrades...worth it?\nSticky: Anyone else try poly-filling motor mounts?\nCool about KYBs but why not Bilsteins?\nEibach Pro-line or H&R springs?\nVmaxx front coilvers and KYB rear shocks install!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to the National Association of Realtors, If you work with the seller's agent, you are not a client, you are a customer. \"As a customer, a buyer is not represented by [the] agent... the agent cannot negotiate on the buyer's behalf nor reveal to the buyer any confidences of the client, the client is the seller.\"\nIf instead, you hire an agent to represent you and your interests, your agent CAN negotiate on your behalf and CAN reveal to you anything they learn about the seller or the home that could benefit you in negotiations, or which could change your mind about proceeding with the purchase. You are no longer a customer being sold something, but you become a client, with professional representation on your side in the purchase of the home. Your interests are being looked out for and protected, not the seller's.\nWhile many seller's agents say they can \"help you\" buy the home, you will not be receiving agency-level representation if that agent represents the seller. An Exclusive Buyer's Agent specializes in representing buyers only in real estate transactions. This level of representation is invaluable.\nRonald Huth, Broker at Buyer's Choice Realty in Massachusetts explains, \"Why use an EBA [Exclusive Buyer's Agent]? It's the difference between a specialist and a generalist. It's a counselor\/advocate vs. a salesperson. It's caring service vs. a quick commission. EBA's have collectively saved home buyers multi-millions of dollars in home purchases. It's someone who finds fulfillment in helping a buyer rather than helping themselves to a double commission. It's the only buyer's agent who can give buyers undivided loyalty all the time. It's buyer agency without a conflict of interest. It's someone who works for you as though they were negotiating the property for themself. Why would anyone want to buy a property any other way?\"\nHire an agent specifically to represent your, and only your, interests and you'll create a contractual relationship where the agent must remain loyal and accountable to you and not to any seller.\nAn Exclusive Buyer's Agent will be on your side and only your side throughout your transaction. Exclusive Buyer's Agent John Rygiol, Broker of Buyer's Broker, Inc. in Seal Beach, California gives us some insight into the mindset of an EBA. \"I strive to offer the same fiduciary service that the client would expect from his attorney or CPA. I only represent one person, the Buyer.\"\nBy law, a seller's agent must work in the best interest of their client. Their client is the seller. A seller's agent cannot negotiate on the buyer's behalf. They cannot suggest or recommend pricing information that would be damaging to their client. Any information they provide to you with regard to the home and the pricing should have been prepared with the intent to get the seller the highest price and best terms.\nLikewise, by law, an Exclusive Buyer's Agent must work in the best interest of their client. Their client is the buyer. EBAs do not have sellers as clients and will focus on getting YOU the best price and terms. They can suggest courses of action in negotiations that will benefit you. Information they prepare is for your benefit in making an informed decision regarding value, offer construction and negotiation strategies. They don't have an obligation to justify a seller's pricing or terms or to try and sell you any particular house(s).\nIf you choose to work with an agent whose primary specialty is to sell homes for the highest price and best terms in the shortest amount of time as possible, you'll likely endure a significant amount of stress and pressure. These objectives generally align with a seller's best interest.\nHowever, if you choose to hire an Exclusive Buyer's Agent, their specialty and focus is to help you buy the best home at the lowest possible price and best terms, according to your timeframe.\nThrough specialization and experience, they have streamlined the process in a way that gives the buyer the best experience. When you hire an agent whose business model aligns with your goals, you'll have a lot less pressure and you'll be able to enjoy the exciting process of buying a home.\nExclusive Buyer's Agent Stephen Israel, President at Buyer's Edge - Real Estate for Home Buyers in Bethesda, MD summed it up nicely when he said, \"Our most important job is to help our clients evaluate all of their options correctly. This is made far simpler because we are not pushing properties. We look at locations, neighborhoods, transportation issues. Through thorough comparable research, we help them understand the values in the neighborhood. We are duty bound to tell them all of the positive and negative things that we see. We assess the quality of construction, the resale-ability of a home and tell our clients what we believe the value is and why.\"\nExclusive Buyers Agents do not list homes and will not ask you to limit your level of representation. They will be 100% on your side during the transaction, representing YOU, negotiating on YOUR behalf and protecting YOUR interests.\nDo you plan to purchase a home for $250,000 or more within 60 days?\nClick here to find an Exclusive Buyer's Agent nationwide.\nOr call us at (801) 969-8989 today for a free recommendation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Program note from Rosalind Ayres, director of the live performance by L.A. Theatre Works: \"For centuries man has dreamed of a life beyond death. Chinese Emperors were buried with clay armies to protect them in the next world. Egyptian Pharaohs were entombed with all the belongings they would need in the afterlife. But how might it be possible to cheat death itself? Well, try the myth of the Vampire. One who, by constantly drinking the 'life force,' the blood of others, could ensure eternal survival. In Charles Morey's dramatization of Bram Stoker's 1897 novel, the creed of the Vampire and the Christian belief in 'life everlasting' is juxtaposed. It's the eternal struggle between good and evil. Plus, the confidence of scientific beliefs and theory, marred only by that uncomfortable shaft of inexplicable fear when something goes 'Bump' in the night. Enter Dracula...\" An L.A. Theatre Works full-cast performance featuring: David Selby as Abraham Van Helsing John Glover as Renfield Simon Templeman as Count Dracula Matthew Wolf as Arthur Holmwood Moira Quirk as Lucy Westenra Lisa O'hare as Mina Murray Harker Nick Toren as Dr. John Seward Karl Miller as Jonathan Harker Andr\u00e9 Sogliuzzo as Maxwell and others Sheelagh Cullen as Mrs. Westenra and others Denise Carole as Tart and others Directed by Rosalind Ayres. Recorded before a live audience at the Skirball Cultural Center, Los Angeles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Balfour Beatty Mott MacDonald (BBMM) Area 4 team has scooped the Gold Considerate Constructor Award for the third year running from the Considerate Constructors Scheme (CCS), an independent organization which monitors companies against a robust Code of Considerate Practice.\nThe Area 4 team, which operates and maintains a network of motorways and key trunk roads in Kent, Sussex and small parts of Surrey and Hampshire for Highways England, scored well beyond statutory requirements in all areas.\nThe team, which also received the Runners-up award for the 'best of the best' Most Considerate Company, were commended by CCS for their forward looking approach and continual search for new ways of working which benefit the workforce, environment and potentially the entire industry.\nConsiderate Constructors Scheme Chief Executive Edward Hardy said: \"The Scheme is delighted to recognise the great achievements that the 2015 National Company Award winners have made.\n\"Over 600 companies register with the Scheme every year and this year's competition to win an award was tougher than ever \u2013 with companies increasing their standards and performance to the next level.\nOnly four companies have ever achieved three or more Gold awards since the awards began in 1997.\nBalfour Beatty Mott MacDonald is a 70:30 joint venture between two of Britain's most established construction and consultancy companies. Balfour Beatty Mott MacDonald combines Balfour Beatty's world-leading expertise in civil engineering and infrastructure services with Mott MacDonald's wide capabilities in consultancy and design services. Formed in 2002, the goal of our two parent companies is to provide service excellence in managing the nation's strategic road network. Balfour Beatty Mott MacDonald currently manages the Highways England's network in Area 4 (South East) and Area 10 (North West). More information about Balfour Beatty Mott MacDonald can be found at www.bbmm.co.uk.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We may think that our beliefs are based on solid facts and reason, but, in the words of George Gershwin's old song, \"It ain't necessarily so.\" The way our minds work actually limits our ability to discern the truth or make rational decisions. Psychologists have been studying the way we think for many years, and have identified several biases that cause this.\nWe all tend to pick and choose information that supports our prejudices and preconceptions while ignoring a heap of evidence to the contrary. Right or wrong, climate change deniers provide the most obvious example today, focusing on a tiny minority of expert opinion and brushing aside the overwhelming consensus of climate scientists. Similarly, most of us don't listen carefully to politicians' speeches or analyze their policies in detail. Instead we pick up the fragments of information that reinforce our party preference.\nThis \"confirmation bias\" even penetrates scientific thinking. For instance, the evidence for the reality of paranormal phenomena is far stronger than the evidence for the effectiveness of many pharmaceutical drugs. And yet most scientists reject out of hand any suggestion that psychic abilities are real because they don't believe they're possible. In less-dramatic ways, confirmation bias distorts our thinking and decisions about many aspects of life. Despite my high level of education and career as an academic, I frequently catch myself doing this.\nAnother source of bias in our thinking is the \"attraction effect.\" Imagine you're comparing smart phone options, and are drawn to the cheaper Basic contract rather than the more expensive Advanced one because it meets your needs adequately. Now suppose that you've been offered a third Luxury alternative that costs more but provides no more benefits than the Advanced contract. Research shows that the presence of this third option increases the probability that you'll choose the Advanced contract. One possible explanation is that the Luxury option makes it easier to justify your choice by claiming you've got a bargain \u2014 perhaps our decisions are normally biased towards ones we can easily justify rather than what is best for us?\nThree decades ago, a third type of bias was identified. The \"framing effect\" leads us to make choices depending on how the information is presented. In one classic experiment, people were asked to imagine an outbreak of disease threatening a village of 600 people. Plan A would definitely save 200 lives, whereas Plan B would have a 1 in 3 chance of saving An example of how bias affects rational thinking everyone, and a 2 in 3 chance of saving no one.\nMost people chose Plan A. However, participants tended to make different decisions when the same Plans were presented another way. In this case, they were told that 400 people would die under Plan A, while Plan B remained unchanged. Surprisingly, most people then chose Plan B. The reason for this switch may be to do with the way we weigh risks, and, as with the attraction effect, the ease of justifying our choice. In the first case, there is the certainty of saving 200 lives versus a rather complex risk assessment that is hard to get our heads around. In the second case, it can be argued that Plan B might save 400 people from certain death.\nAs if these biases were not enough, we have others too. One is the \"sunk-cost fallacy\" \u2014 our tendency to stick with a project even when it would be more rational to cut our losses and move on. Another is \"feature creep\" \u2014 a tendency to buy gadgets that have more features than we will ever use. More significant, perhaps, is the way emotions cloud our judgment. Inducing a sense of disgust, for example by showing people pictures that they find offensive, makes them more likely to make harsh moral judgments that can induce bias.\nHow can we best guard against these and other biases?\nOne approach that I try to apply to my own life is to live by my current beliefs, but to hold them lightly and to be aware of my emotional feelings. I try to remain open to the reality that I may be quite wrong, and to be flexible in the face of new information or ideas \u2013 not easy!\nA second approach is to share our ideas with others and debate about them. This can be very creative provided we mix with people who don't all think the same. When groups of five or six people take tests that involve logical deduction, the success rate is far higher than when individuals tackle them on their own. Even groups whose members have previously failed the test succeed by openly generating ideas and revising them in the light of criticism. The most successful groups have a clear goal to reach agreement, encourage everyone to participate in the discussion, and consist of people who are sensitive to the feelings of others. The danger with this approach is that, if the members of a group are too similar in their beliefs, knowledge and ways of thinking, the result can be \"groupthink\" which stifles dissent, ignores alternative actions, and can make disastrous decisions.\nSee \"The Argumentative Ape\" by Dan Jones, New Scientist, 26 May 2012, p.33-36.\nMalcolm Hollick is a QuestioningTheTruth.com contributor, and is the author of The Science of Oneness: A worldview for the twenty-first century, and co-author with Christine Connelly of Hope for Humanity: How understanding and healing trauma could solve the planetary crisis. For more information, visit their Human Solutions Web site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Professionally store your wine with this 72 bottle natural pine wooden wine rack. Made from sustainably pine wood and galvanized iron structure, it will make sure your wines are in good hands. Designed to comfortable hold up to 72 bottles in a tidy structure, this storage system contains everything you need to store your wine in style.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is a domain having .co.za extension. This site has a Google PageRank of 3\/10. It is estimated worth of R 103.46 and have a daily income of around R 1.73. As no active threats were reported recently, randpharm.co.za is SAFE to browse.\nT Shirts, Golf Shirts and caps embroidered or screen printed with African animals, wildlife and ethnic designs by Its Africa Clothing Manufactures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is no way of knowing just exactly how long root canal therapy has been around. The first traces of root canal therapy can be dated back to second or third century B.C. A human skull was discovered in a desert in Israel. In one of the teeth, they found a bronze wire that scientists believe was used to treat an infected canal. The wire was located at the site of the infection, which is the exact spot that would be targeted during modern day root canal therapy. The archaeologists who discovered the remains believe that the procedure was performed by the Romans, who are said to have invented dentures and crowns.\nEvidence shows that from the first century A.D. until the 1600s, the treatment for root canals included the draining of the pulp chambers to relieve pain, and then covering them with a protective coating made from either gold foil or asbestos. Around 1838, the first official root canal instrument was constructed. It was made to allow easier access to the pulp that is located within the root of the tooth. A few years later, around 1847, a safer material known as \"gutta percha\" was created to use as a filling once the root canal was cleaned out. Both of these materials are still used today by Endodontists.\nWhen we entered the 20th century, dental technology advanced. Anesthetics and x-rays were instituted into dental practices, which made treating an infected root canal much easier and safer. These technological advancements also allowed for alternative treatments to pulling teeth. Root canal therapy has advanced so much that it is now a nearly painless procedure!\nAn infected tooth root is a pain, and can compromise your teeth! We save teeth every day in our office with endodontic therapy \u2013 call us at Hendersonville Office Phone Number 615-824-7546 for more information.\nA University of Sydney research team has produced detailed 3D maps of the composition of tooth enamel. While we have known for some time that enamel is the hardest substance in the human body and that its strength comes from a complex hierarchical structure that includes magnesium, carbonate and fluoride ions, this is the first in-depth and detailed look at what the composition of that structure is.\nTwo major findings are exciting the dental community. First, there is now direct evidence that an amorphous magnesium-rich calcium phosphate phase may determine (to some degree) how teeth are formed. Second, organic material was also found in the structure, suggesting that proteins occur in patterns throughout the enamel, not just in the interfaces as we used to think.\nWhat does this mean for endodontists?\nTooth enamel is the first line of defense when it comes to teeth and their roots. Once the enamel is compromised, decay starts to take place and, if left untreated (as you know), the infection may spread to the tooth's roots, landing you in one of our chairs for root canal therapy. That is why we, as endodontists, want your enamel to stay healthy and strong for as long as possible!\nThe impact of this could be great down the road. This type of detailed information will allow dentists and other scientists and researchers to better determine what is going on inside the enamel of your teeth before, during and after decay.\nPotentially\u2026yes! New treatments and prevention strategies for dental health are always on being made, thanks to ground-breaking research and studies such as this.\nIf you are experiencing tooth pain, it may be that you are in need of root canal therapy. We can help! Call Hendersonville Office Phone Number 615-824-7546 for more information.\nNow that you know the source of your pain, we want you to know that we are here to help you determine the best remedy. We aim to get you in and out quickly, safely and comfortably. Don't wait any longer to resolve your pain, give us a call at Hendersonville Office Phone Number 615-824-7546.\nIt's a shame that studies on an important tool such as floss have yielded poor results, but it's a bigger shame that the studies themselves were poorly designed. Oral hygiene is a long term process, and requires long term observations to make worthwhile conclusions. In the mean time, it's obvious that you should continue to do everything you can to protect your well being, and floss is one of many tools that can help you do that. If you would like a refresher on the best, most efficient techniques for floss use feel free to call our office today at 615-824-7546!\nPosted in Blog, Oral Health | Comments Off on Yes, You Still Have to Floss.\nDon't let oral bacteria be your \"fr-enemy\"! Call us Bluegrass Endodontics, PLLC today on 615-824-7546 to discuss your oral health options.\nDon't confuse the pain of love with needing a root canal! Call Bluegrass Endodontics, PLLC on 615-824-7546 to check those symptoms!\nPosted in Blog, Endodontic Retreatment | Comments Off on The Love of Endodontics!\nIt's really quite amazing how advanced tools have become over thousands of years! To think that as humans we started with mere sticks and stones as tools, and now we can visit space via rocket ships is simply mind-boggling!\nNow, while not exactly moon-landing technology, we'd like to point out that endodontic tools have also become highly developed over the years, with very specialized tools designed for each step in the process.\nEndodontists use a wide variety of tools, each built for a specific purpose, but with one thing always in common: size. The tools that we use in endodontics are some of the smallest surgical tools out there! This, of course, allows us to properly access tiny roots in order to get the root canal therapy done right. To give you an example, some of these tools are just around a single millimeter thick (or less!), necessitating highly-skilled operators.\nA bur is a tool that comes in many shapes. It's a teeny tiny drill that can penetrate the top of the tooth to get into the pulp chamber, by removing the roof.\nAn endodontic spoon excavator is the tiniest spoon shaped tool you've ever seen. This spoon can get into the pulp chamber and allow access to the floor.\nA barbed broach is a barbed point, like a thumbtack, which can spike the pulp in a tooth root, snag it, and allow it to be pulled out.\nGates-Glidden drills are super small drill bits that can fit within the root and clean it out, as well as make space for bigger tools if necessary. The root can then be cleaned with a syringe to disinfect.\nAnd, while not as small, we can't forget to mention our digital imaging technology, without which we wouldn't be able to give the highest-quality root canal treatments! Digital imaging of the roots allows us to see any areas of concern, and make better decisions for your care.\nSo the next time you come to see us, ask us about some of the tiny precision tools we employ in our office on a daily basis!\nPosted in Blog | Comments Off on Help! I've Cracked My Tooth!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The wearing of the school uniform is compulsory and sensible clothing is expected at all times. The uniform shop is open on a regular basis. If parents wish to see samples of the uniform, these are available from the School Uniform Shop PSW Hoppers Crossing (see below for address and website).\nHats are essential in the warmer months. We have a \"hats on\" policy in Terms 1 and 4 \u2013 otherwise children must stay in the shade.\nEarly Years students are able to wear suitable shoes for running and climbing. It is strongly encouraged that black leather lace-up shoes, or school shoes are worn by Foundation - Year 6 students. T-bar or Mary Jane style shoes are not permitted.\nFor students in Year 7 - 9, black leather lace up school shoes are required. T Bar shoes are not acceptable due to occupational health and safety requirements.\nWith the exception of a watch and one set of studs\/sleeper earrings, jewellery is not to be worn. The wearing of make-up is not acceptable.\nOver the next two years, Carranballac P-9 College will be implementing a period of uniform transition as we create a clear distinction between our Primary and Secondary students. Included below, is a Uniform Transition Rubric, to assist families in understanding how the transition will affect their uniform purchases. If you would like to print a copy of the rubric, please click here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A special greeting to extend our best wishes to you during this holiday season. We are extremely grateful to our clients for the trust and loyalty you have shown us, and to our team members who work so hard to ensure that your expectations are met. Merry Christmas and Happy Holidays, from our family to yours.\nThe system design is a true integration between two Generac Holdings Companies, Generac's C&I Engineering team in Waukesha, WI and MOTORTECH\u00ae in Celle, Germany; a company that Generac acquired earlier in 2017 that designs and manufactures gaseous engine-systems. The system integration gasifies a 33.9L engine block and will be the first product offering to introduce Generac's NEW control panel platform, POWER ZONE\u00ae.\nThe Generac POWER ZONE\u00ae digital control platform is a fully integrated and multipurpose family of controllers for Generac's single and modular power systems (MPS).\nThe Natural Gas SG & MG625 is yet another innovative product perfect for a variety of applications. To learn more, download the specifications here (SG625) and here (MG625) , or contact your ESSE sales representative.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Tuesday night I gave a short talk for the Okanagan Developer Group about my experience starting Lining Things Up and 5 topics that often come up when I talk to entrepreneurs. I thought it went pretty well. I wish I could have met with more people afterwards, but there is always next time. This was my first time attending an OKDG meeting, and since I was a presenting I couldn't mingle as much as I would have liked.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Growing up in the shadow of a famous novelist would probably deter most people from pursuing a career as a fiction writer. It had the opposite effect on Emma Straub, daughter of the best-selling horror and suspense writer Peter Straub.\n\"I always wanted to be a novelist,\" she said.\nShe was so determined to write for a living that she kept at it, even after the first four novels she had written were turned down.\n\"They all got rejected by every single person in publishing, in the world,\" she said, with what she insists is only a touch of hyperbole. \"It's still true that I will go to a publishing party or event, and the first thing I will think of is, 'I know who you are, you rejected novels 2 and 4.'\" Those days are over. This month, Ms. Straub is releasing her fourth published book, \"Modern Lovers,\" which is shaping up to be one of this summer's must-reads. The novel centers on two middle-aged Brooklyn couples whose marriages are in crisis. In a recent telephone interview Ms. Straub, 36, spoke about it. Below are edited excerpts from the conversation.\n\"Modern Lovers\" falls into a crowded literary subgenre of novels about Brooklynites. What made you choose Ditmas Park as a setting?\nI wanted to stay as far away from the quote-unquote Brooklyn book as I could, while still having a book take place in recognizable Brooklyn. I didn't want a book jacket with a row of brownstones. Brooklyn wasn't the point.\nI thought, where can I put them? If I put them in Ditmas Park, they can have lawns and big houses with staircases and driveways and garages, and I thought there could be people playing music in garages, which seems like the most romantic thing in the world to me.\nI love that one of the main characters, Elizabeth, is a real estate agent.\nOne of the things that people mentioned often in reviews of \"The Vacationers\" [Ms. Straub's previous novel] was that the people seemed too upper crusty, so I wanted to make sure in this book that the people really had jobs and were seen doing their jobs. Having one character as a real estate agent made sense on a lot of levels. Looking at real estate is all about fantasy and imagination and picturing how your life could be different, and presenting you with all these choices at once.\nOne notable feature of many of your characters is that they often go to great lengths to avoid conflict.\nI am extremely conflict-averse. It really wasn't until I started dating my husband that I learned it was possible to have an argument with someone and have everyone be O.K. afterwards.\nDo you and your dad swap tips?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every year I relive a cycle that began on April 1, 1971. As a result of that, I have some of the most wonderful memories. I went to the mailbox and to my surprise I received some papers from Uncle Sam, stating that I needed to report April 2 for my active duty in the Army Reserves. Of course I thought it was an April Fool's joke, but it wasn't. The next day, I was headed for Fort Knox Kentucky. My active duty would end July 31, 1971.\nTwo of my best friends (one Jewish and one Gentile) and I had already agreed that when I got out of the army on July 31, we would move to Columbia Missouri, to attend a university. That very same summer while I was in the Army, my two best friends went to California and while they were there, they met some people who were followers of the Messiah Jesus. After several weeks of constant conversation about the Messianic claims of Jesus, they too became followers of Him. They returned from California and I got out of my active duty. All three of us arrived in Columbia and moved into our apartment.\nIn no time flat I was made aware of their \"new\" belief. They hounded me day in and day out. Even though I was not very religious in my Reform Judaism, I was greatly offended by some of the things they were saying. They told me that my way of believing in God was wrong and that I needed to approach God the way He instructed us to. I thought that as long as I was sincere and that I believed in God, that was all that was needed. They told me that my good deeds or mitzvahs would not get me to heaven. I thought that if the good things I did outweighed the bad things I did, I would get in. They told me not to reject the only way that sins could be forgiven. I had never really taken sin seriously. Knowing that I was guilty of many of them, I was totally unaware of the severe consequences of sin. They told me that if I did not receive Jesus as my Messiah and Savior, I would end up in the flames of Hell. This really burned me up. If God was a God of love, how could I end up in Hell? My friends truly cared for me, but they were telling me things that were contrary to what I had come to believe about God without real knowledge They were telling me and showing me things straight out of the Bible. That's not where I developed my beliefs about God and life. My beliefs simply came from what I thought was true.\nAs the weeks rolled on, I began to notice some distinct changes in my friends' behavior. This really got my attention. When I asked, they were quick to tell me that the changes were the direct result of their faith in Jesus the Messiah. That made such an impact on me that I began to read the Bible for myself. I wanted to see what was in there. As I read, I was really shocked. I began reading in Matthew, which is the gospel account that was written especially to the Jewish people. When reading about a person I had heard of and of whom I had developed a mental picture, I realized that I had been wrong! The Jesus I was reading about wasn't anything like the Jesus my Judaism had taught as I was growing up.\nOver the months, as I continued to read the Bible, I began to see myself being drawn to what I was reading. At that time I developed an intense fear of what would happen if I believed all of this. What would my family say? What would my Jewish friends say? On December 1, 1971, I found out. In the privacy of my bedroom, I became a follower of Jesus as my Messiah. It's true, there were times of tremendous misunderstanding by my family and friends concerning what I had done. It was difficult to experience rejection by others because of my belief in Jesus as Messiah, however, that rejection can't compare to the hope, to the peace, and to the multitude of blessings that I continually receive from God.\nOne thing in particular has happened since I became a Believer in Jesus as Messiah: my appreciation of my heritage has become very precious to me. If all you know about God and life is what you assume, begin reading the Bible. I challenge you to read the gospel of Matthew on your own.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Okay, according to Kiara (hot zombie queen that she is) at 10\/30\/2008 15:41:14 the NDA was lifted on Beta. I made it, but VERY late in the day (a mere 5 days ago). Can some of the folks who made it earlier comment on how they think it went?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Send only HTML excerpt to public subscribers?\nI have the html version of the plugin. How do I configure it so that it will send HTML excerpts to my subscribers? I have it set so notifications are sent out when new posts are created.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Economy. Bicycle transportation makes financial sense, and the magnitude of bicyclists economic impact gets far less attention than it deserves. In the Bikenomics series, Elly Blue explores the scope of that impact, from personal finance to local economies to the national budget. In the grassroots and on a policy level, the bicycle is emerging as a powerful tool for economic recovery. Studies have even found that bicycle\/pedestrian infrastructure projects create up to 46% more jobs than road projects built only for cars.\n2. Climate and Environment. Transportation accounts for over 70% of oil consumed in the U.S. Because of its near-total dependence on fossil fuels, the U.S. transportation sector is responsible for about a third of our country's climate-changing emissions. Therefore, bicycling is an easy way to reverse the damage.\n3. Save Lives, and Money. In direct costs, the average American spends about $9,000, or 15% of their income a year on transportation ... mainly on cars of course. If you add in the costs of resource (oil) wars, the added cost of healthcare from air and water pollution and climate change, and other externalities, the numbers skyrocket from there. Therefore, you can save thousands of dollars every year by replacing car trips - at least the shorter ones - with bicycle trips.\n4. Health Benefits. Everyone wants to be healthier, right? Bicycling promotes weight loss, improved vitals, and can add years to your life. Simply put, it's an easy, fun, and inexpensive way to stay healthy and feel better.\n5. Reduce Congestion. Traffic delays are one of the top complaints of Americans. While bicyclists also must ride in traffic, there can be many advantages. As cars queue up at red lights waiting through multiple phase changes, bicyclists can filter to the front and avoid the agony of stop-and-go traffic, and breathing the fumes of the tailpipe in front of them. There's definitely a great feeling involved when you coast past dozens of motorists stuck in traffic and then continuing unimpeded when the light turns green. And think of the emissions saved!\n6. Social Benefits. As a driver, you probably don't think much about connecting with or being a part of the same community as other drivers, but that's different when you're biking. Whether you just see one cyclist, or dozens on a given ride, there is a feeling of kinship with them that drivers never experience.\n7. Safety and Quality of Life. No place on Earth is enhanced by the presence of cars. People don't like the pollution, noise, congestion, and unsafe streets. These negatives decline when more folks choose to walk or bike - even occasionally. Communities that accommodate Active Transportation are healthier, more vibrant, have stronger local economies, and retain higher property values.\n8. Efficiency and Speed. As the most efficient mode of transportation, bicycling can get you someplace using minimal energy and time. In cities, where many people live in relatively small areas, moving around by bicycle rather than a large car can be much faster for most trips. Though the suburbs are less dense, 40% of trips in the U.S. are still 2 miles or less, perfect distances for bicycling. And you can eliminate the time spent waiting in traffic, finding a parking spot, filling up on gas, or waiting for a bus. The bicycle is a win-win for everyone, even motorists who now have fewer cars on the road to contend with.\n9. Parking Woes. Cities and many suburbs can be dense places. Cars don't exactly fit well in them. Rather than search endlessly for car parking, you can usually lock your bike up outside the entrance of your destination and skip the hassle altogether.\n10. Wildlife Decimation. Bicyclists rarely kill animals. Bicycling infrastructure does not create highway kill zones that cannot be traversed. A smaller transportation footprint plays less of a role in habitat fragmentation. Cars, on the other hand, snuff out millions of lives every year, pushing some to the brink of extinction. Cities built around bicycling and walking are more compact, and preserve vastly more habitat and green space than the sprawl of car-centric cities and suburbs.\n5 things you can do to support Bicycle Advocacy in Delaware, no money down!\nJoin 1st State BIKES by simply following us on Facebook, Twitter, RSS feed, or subscribe via email (see our homepage, lower right).\nRespond to action alerts supporting bicycle safety, including petitions and public workshops.\nIf you live in a city, join a local advisory committee. Newark, Wilmington, and Dover now have bicycle committees and\/or plans.\nAttend Bringing Education and Safety Together (B.E.S.T.) meetings, and consider how you might help promote Active Transportation.\nPeriodically check in with the Delaware Bicycle Council, and stay abreast of projects and events happening around the State.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This editorial spread shows the interaction of many skill sets, and shows clients how a multi-modal approach can build higher customer engagement. This piece most closely aligns with Sofia's desired target products moving forward in professional work. Artful design, presented in editorial format, highlights food writing that covers multiple areas, from instructional to conversational to recipe development. The product is equally influenced by adopting a conversational, knowledgeable tone as it is by the thoughtful visuals. This project was an initial practice into incorporating work in InDesign, which adds another level of proficiency to the portfolio.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home Projects CJSC \"Kumtor Gold Company\"\n\u00abTCS\u00bb LTD. since 2007 provides re-park radio equipment of CJSC \"Kumtor Gold Company\". We supply modern professional radios GP340 and GM160, and repeater equipment GR500.\nMotorola GP340 radios and GM160 designed with professional equipment requirements of modern systems overland radio. An extremely easy of use combined with high quality sound and a wide variety of programmable features that allows you to use it effectively in systems with different configurations of users.\nMotorola GP340 and GM160 radios are certified for compliance with military standard MIL-STD-810.\nAll equipment comes with a warranty from by Motorola and Ltd \u00abTCS\u00bb.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajhhr b/data_all_eng_slimpj/shuffled/split2/finalzzzajhhr
new file mode 100644
index 0000000000000000000000000000000000000000..cc0f77f0c6f29d873c0868f857da1688f9f26c7f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajhhr
@@ -0,0 +1,5 @@
+{"text":"You may have noticed in the news that some new legislation called the General Data Protection Regulation (GDPR) comes into effect on 25th May 2018. These new regulations are designed to protect your personal data.\nAs clients of AlphaPet, you have entrusted us with some of your personal data (such as your name, address, telephone numbers and email address).\nWe use this information to help look after your pets (for example by sending you this newsletter, vaccine reminders, flea\/ worm treatment reminders and occasionally alerting you to new disease risks that may affect your pet) but we need your ongoing permission (consent) to do this.\nPlease indicate, either on the form overleaf, or on our website, exactly how you would like us to continue to contact you. If we do not have your consent to contact you using the information we hold, then we may no longer be able to send you vaccine reminders, flea\/worm treatment reminders, newsletters, pet health bulletins or clinical health recommendation reminders, all of which we believe are in your pet's best interests.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Amnesty International's new report on immigration detention highlights the long-term damage caused by the practice.\nThe 50-page report, 'A Matter of Routine: The use of immigration detention in the UK', shows how detention is being maintained as a matter of default and convenience, leading to significant harm to both detainees and their families.\nWith very few exceptions, immigration detainees are held without a statutory time limit, meaning that on entering detention they have no idea when they will be released. While for many, detention ultimately lasts at most a few weeks, others are held for many months and some for years. Most detainees are ultimately released back into the community.\nHowever, Amnesty found a complacent approach to the use of detention powers, resulting in people being detained unnecessarily and then being kept in detention through often unjustifiable reasons. Little thought was being put into finding alternatives to detention.\nThis approach to detention powers, and the weakening of protection for detainees, allows detention to be used as a matter of routine. Essentially, if more detention centres are built, more people will be found to put in them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The weather is set to warm up this weekend on Merseyside.\nAir from a more Southerly direction is set to bring some warmth to the region, as milder winds from Europe will increase temperatures across the country.\nEven though the weekend will see mostly cloudy conditions and the threat of rain, Merseyside could see highs of 18C making the weather feel much milder than recently.\nThe Met Office say: \"After some early brightness, Saturday will be largely cloudy with the odd spot of light rain or drizzle.\n\"Cloud will thin later to give a fine end to the day.\n\"Changeable over the next few days with rain possible on Sunday and Monday, perhaps heavy in places.\n\"Largely dry on Tuesday with sunny spells. Warmer than recently with milder nights\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Student art is on view at Lee Regional Medical Center in Pennington Gap, Va.\nPENNINGTON GAP, Va. ? Visitors will find the main hallway of Lee Regional Medical Center brighter and more striking than ever.\nThe talented artists at Lee High School are the reason. The student artists and art teacher Lila Porterfield gathered recently at the hospital to dedicate a new exhibit, \"Reflections,\" which includes approximately 20 original works of art that hang in the main hallway on the first floor of the hospital. The Lee High contingent joined leaders from the hospital and Wellmont Health System during a recent reception to officially christen \"Reflections.\"\nPorterfield said the works created by her artists for \"Reflections\" are so striking that community members have approached her and asked about purchasing some pieces.\n\"The work that has been displayed was created specifically for the hospital,\" Porterfield said. \"Students chose subjects that were inspirational, colorful and themes that people could relate to. It was a long process ? but enjoyable.\n\"The students are proud to display their work for the community to enjoy it.\"\nDavid Brash, regional vice president for Wellmont hospitals in Southwest Virginia and Southeast Kentucky, spoke at the reception and expressed his gratitude to the students on behalf of Lee Regional. The talent and generosity of the Lee High artists, he said, make for an eye-catching experience for all who work at and visit Lee Regional.\n\"To tell you the truth, I wouldn't mind taking an armful of these works with me to hang at home ? but that would be robbing the community of the chance to see something special, to see the kind of talent we have right here in our community,\" Brash said. \"I want to thank Ms. Porterfield for working with our hospital to make this happen, but most of all I want to thank the artists for loaning us their creations.\n\"The people at Lee Regional are always excited to become involved with the good things happening in our community, and it's easy to see ? with only a glimpse ? that this exhibit is a shining example.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for a live stream of Thursday's Game 5 showdown between the Golden State Warriors and Oklahoma City Thunder? We've got you covered.\nIf you don't have access to a cable or satellite log-in, you can watch the game through Sling TV, an online streaming service that provides access to select channels for a monthly fee. The service's Best of Live TV package, which includes TNT, costs $20 a month, but if you're looking for a way to watch the game for free, you can do so with Sling's free 7-day trial. . Click here to sign up for Sling TV. You'll need to enter your credit card information when signing up, but if you cancel your subscription before the trial period ends, you will not be charged.\nAnother option if you don't have a cable or satellite log-in is TNT Overtime, where you can watch \"enhanced coverage\" of the game. In other words, you will be able to watch camera angles that are different from the regular broadcast.\nFinally, if you do have access to a cable or satellite log-in, you can watch a live stream of TNT's coverage at TNTDrama.com. You'll need to log in with a cable user name and password\u2013either yours or someone you know\u2013to watch.\nSubscribers can watch the game on the Sling TV app, which is free to download. It's available on mobile or tablet from the App store, the Google Play store or the Amazon store. Other supported devices include Amazon Fire TV, Roku and XBox One.\nYou can click here for a full list of devices on which you can watch Sling TV.\nTNT Overtime (camera angles other than the main broadcast) can be watched on your phone, tablet or other supported device via the NBA app, which can be downloaded for free in the App store, the Google Play store or the Amazon store.\nFinally, users with cable access to TNT can watch the game on the Watch TNT app, which can be downloaded for free in the App store or the Google Play store. As is the case with desktop, you will need the cable log-in and password, or the log-in and password of someone you know, to access the stream.\nThe Thunder look to close out the Warriors in Game 5 Thursday night. Here's a full rundown of how to watch a live stream online.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajusp b/data_all_eng_slimpj/shuffled/split2/finalzzzajusp
new file mode 100644
index 0000000000000000000000000000000000000000..bd9e9727e51f549ac4f67930ab26ef1b871ed468
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@@ -0,0 +1,5 @@
+{"text":"Parentage: Developed by B.Y. Morrison. Introduced by J. Creech, 1972. Actual cross debatable.\nComments: Whatever the true parentage is Morrison developed a \"knock-out\" color combination. Petals become more distinctly edged in white as plant matures. Strong, upright grower.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us for a walking tour of Lake View Cemetery. Learn about the history of the cemetery, the stories of our city's founders and famous residents, and other Seattle legends as we visit various monuments.\nBeneath the ground of Lake View Cemetery lies some of Seattle's richest history. As Seattle's most famous cemetery, many prominent Seattle pioneers lie within its grounds, including the Dennys, Maynards, Mercers, Yeslers, and Chief Seattle's daughter, Princess Angeline. Join Jared Steed and the Atlas Obscura Society Seattle as we walk the cemetery learning about their stories and discussing their involvement in the founding and establishment of Seattle.\nOur tour will end at the graves of Lake View Cemetery's most famous residents, Bruce and Brandon Lee, where we will discover what became of the funeral home which prepared Bruce Lee's body for burial.\nNEW TOUR ADDED AT 10:30 a.m. !!!\nMeet in front of the main cemetery gate at 1554 15th Ave. E., Seattle, WA, 98112.\nWe will be walking approximately one mile, so please wear comfortable walking and water-resistant shoes.\nTour is family-friendly but parents must keep an eye on their children in order to ensure proper gravesite maintenance. All persons age 15 years old and younger must be accompanied by an adult to participate in the tour.\nThis tour will involve walking approximately one mile. Although our tour will largely follow paved footpaths, please be aware that we will also cross over uneven grass and ground, including stepping over woody debris and muddy areas. The grass is often slippery when wet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"West Worthing Ladies I Lost to Crawley Squash Club Ladies I in a one-sided match.\nOn paper, the number 2 string should have been the closest, with only 18 ranking points separating the two players before the match. The Away player, Brianna BooBurki, did well therefore to secure the win.\nIt was the number 2 String which turned out to be the most competitive, with the Players battling out over a total of 133 Points ! Victory finally going to the Away Player, Brianna BooBurki.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do your research! There are literally dozens of companies that offer move out cleaning and steam carpet cleaning covering many properties as apartments, condos, town-homes, single family homes throughout where you live, and choosing the right one will certainly indicate a hassle-free, hassle-free move-out experience and hundreds of dollars of savings. Book as far in advance as possible if you are on a tight schedule for moving out of your existing home. Booking toward the end of the month commonly indicates higher costs and less availability. Giving your potential business 2 to 3 weeks notice will generally lead to an affordable cost, and enable you to pick a consultation for the local time and date that suits you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Storage in Kelowna for Your Moves with Us!\nWhen you're moving, you may find you have way more stuff than you previously thought. If your new home is smaller than your current one or you just don't want to move everything at once, Precisely Right Moving and Storage offers storage solutions to meet your needs during the moving process.\nWe take excellent care of your belongings during moves, and the same applies to our convenient, competitively-priced storage in Kelowna. Our facility is surrounded by a security fence and is well lit for 24-hour safety for all of your business or household possessions.\nNeed storage services with the convenience of not having to transport your goods? Look no further than our self-storage delivery, where we bring the storage to you! We deliver a twenty-foot mobile storage unit to your door so you can load it. When the unit is full, we will pick up the container and hold it in our secure storage yard.\nAre you unsure of how long you'll need our services? Don't worry; we know how tough it is to plan ahead. Many clients choose us when they're moving or waiting to move certain items to their new home.\nYour plate is full during this busy time, and we want to make storing your belongings the most stress-free part of the process. We're always flexible and understand when you need to change the duration of your rental time. That won't be a problem with us!\nLong-Term Storage: When we unload into our storage units, we will leave all of our heavy-duty moving blankets and shrink wrapping on your items to protect them and to make reloading for future delivery more efficient.\nShort-Term Storage: Your furniture is only loaded and unloaded once; this significantly reduces the cost to you and the risk of loss or damage to your precious possessions.\nRather than you coming to your unit to remove items from storage, the movers at Precisely Right Moving can remove or add items to your storage unit. Due to liability reasons, our movers need to be the ones to access your unit. This makes the process much easier for you. All you have to do is call us and let us know what you need from the unit and our movers would get the items for you. No one besides our trusted workers will have access to your unit, so you can rest assured knowing that your personal or corporate belongings are safe and secure.\nWe do not outsource third parties for your storage needs, so during your move, you can rely on us for residential or commercial storage in Kelowna. Our containers are always locked on our compound, never left overnight on a moving truck.\nWe also have long-term storage options available when you need more time determining exactly how all of your belongings will fit into your new home or office. Contact us to book your Kelowna storage unit and one of our helpful staff members will provide you with more information about our storage facility and which type of contract would be best for you at this time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakjql b/data_all_eng_slimpj/shuffled/split2/finalzzzakjql
new file mode 100644
index 0000000000000000000000000000000000000000..6cc290cf24f31af3f660c7d6be0e6c8f137ea045
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Old-school interactive ghost game. Go and explore in the motel and meet friends.\nHelp Paris solve the evil Pigsaw's puzzles on Halloween and defeat villains along the way.\nAdd Paris Scary Game to your blog, MySpace, Friendster, LiveJournal, or website!\nShare Paris Scary Game with your friends!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There was some silence in the pages, but they have always been around! There will be more in the future.\nI usually ask people later to send me a picture with them and their new instrument. Here are some of them.\nAfter a while there are some new drums online. Thanks to the photographer (Linda Ansone) and to the sound mastering dude (U\u0123is Jansons).\nArt fair - \"Art hunt vol.7\"\nFrom 1st till 30th April you can find couple of my drums in the Kalnciems quarter (Riga city) art space between other wonderful works.\nDuring the Christmas time you can find my drums also in the Kalnciems quarter art fair space \"Art hunt\". From 03.12 till 31.12.\nTank drumming in National Libary of Latvia Children literature center. Kids were surprisingly calm while I was playing and telling and very loud while playing themselves.\nOne sunny morning I was looking at the reflections that the tank drum was making on the room walls. Found a shaded place in the old house attic and was surprised how bright the colors are.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is Carol. And look at those guns!!\nShe is trusting the process and creating a whole new and healthy relationship with food and exercise. Carol you're inspiring and I hope you can see what we all see. A strong, fit lady ready to conquer anything she sets her mind too. Giving her body a little more rest AND eating MORE food, she was able to start gaining the muscle mass she was looking for and losing body fat. Also getting more stamina in the workouts.\nCarol we \ud83d\udc97 you!! (And I know you don't believe me, but yes you have guns!!!!) Keep going- you got this!!!\nHere she is pictured above rocking out the Murph workout!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HGH or the Human Growth Hormone has been a hot issue of discussion and received lot of attentions for its many benefits to the human body. Human Growth Hormone is mainly responsible for the growth and physical development of the human body. HGH is commonly found in children during the stages of puberty and growth. It is a hormone that is naturally created within the body and helps cell growth. Deficiency of HGH in the human body can finally stop the growth and create various problems within the human body. HGH products are making their way to increase the level of Human Growth Hormone in the human body and help an individual to have complete growth and physical development.\nWhy is an herbal HGH product a better choice?\nThere are innumerable HGH supplements available in the market but all these HGH products are ineffective and are not safe for the human body. Though some products proved effective, they have side effects causing serious health problems in the human body. Therefore, most people prefer for products that are both effective and free of any side effects. HGH products are the right choice for anyone because they are effective and do not have any side effects. These HGH supplements are usually made of natural ingredients extracted from various herbs. These HGH supplements are not only safe for consumption but they are also most effective when compared to other HGH supplements.\nSome of the best HGH products are Sytropin and GenF20. Sytropin is an HGH product that gives wonderful results like increased height, increased muscle mass, decreased body fats, prevention of aging, and increased sexual stamina and stronger immune system. Taking Sytropin helps in restoring youth and vigor by diminishing wrinkles and enhancing the appearance of skin. GenF20 is also an amazing HGH product which is not based on the increasing of artificial hormones. GenF20 is one of those effective herbal HGH supplements that increase the level of HGH in the human body naturally. There are many benefits in taking GenF20. It reduces the process of aging naturally by increasing the level of HGH in the body. It helps in increasing muscle mass, decreasing body fats and strengthening the immune system. GenF20 is a true medicine in health improvement and helps to feel relax and have a sound sleep. It also sharpens your mind and increases the level of concentration to great extent.\nThe most effective and safe HGH products are usually called releasers and are made of herbal ingredients. Most of the HGH herbal supplements are in the form of capsules or tablets. These supplements contain only natural ingredients and are clinically tested and reported to be 100% safe without any side effects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The problem with a lot of Mormons out there has to do with the fact that they tend to be rather hostile towards pretty much anyone that tries to convince them that what they believe in is not entirely true. Hence, the fact of the matter is that if you want to witness effectively with a Mormon you are going to have to learn the right way to do it. Mormons can be very hard and fast in their beliefs, so much so that they could be completely unwilling to hear you out at all.\nOne thing that you need to realize is that Mormons believe that the regular Christian Gospel is incomplete. The fact of the matter is that the real Bible has the true word of God in it, but Mormons believe that there are pieces missing that are only available to you in Mormon texts such as the book of Mormon. Hence, if you want to be able to witness effectively one thing that you are definitely going to have to do is try and read their texts.\nThis is something that a lot of Christians are pretty hesitant to do because of the fact that they feel like it would be dishonest for them to try and read the false word of God. However, the truth of the matter is that you need to be able to refute what Mormons are saying if you want to be able to guide them to the light that is the true word of God. You will not be able to refute a thing that Mormons are saying if you do not know what is in their own books and scriptures. Sticking to your own scripture will not help anyone if your goal is to show Mormons the truth.\nOnce you have read the Gospel that is used by the Church of the Latter Day Saints, which is the Church that Mormons belong to, the next step is to start finding the differences between their scripture and the true Gospel which is the word of God. Once you have the differences down you can start better understanding the mindset of Mormons themselves and create a game plan that would basically allow you to talk to them on their level without worrying about them surprising you with some information that you might not know.\nOne of the most important things that you will have to do if you want to be able to help Mormons see the truth is to prepare yourself for rejection. The fact of the matter is that Mormons are very invested in their faiths much like pretty much every single other religious person that lives on this earth. Hence, they are not going to be willing to believe that everything they currently know is a lie. You should try your best but also try not to take it too hard if a Mormon does not want to hear what you have to say.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamocp b/data_all_eng_slimpj/shuffled/split2/finalzzzamocp
new file mode 100644
index 0000000000000000000000000000000000000000..9719441e5d37737b345aed776750d4c337379c83
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Cleveland Indians win American League Central Division!\nThe Indians are in the playoffs!\nHere is the scoop. The Indians have a real shot this year at winning the Pennate and the World Searies. They have won 26 of the last 35 games. At the moment they have the best record in both leagues. The last time the Indians were this hot the players went on strike and there was no playoffs or world series!\nSo, if there is a playoff game and the Indians are playing, only call me during commercials.\nOne comment on \"Cleveland Indians win American League Central Division!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our team of knowledgeable sales reps will get to know you and your business and understand your bespoke refrigeration needs in order to help pinpoint the type of conversion right for you. You can be rest assured that we will help you select a conversion that meets all of your business's bespoke, daily needs.\nMuch like insulation, the type of refrigeration system your van requires will be largely based on your business's daily requirements in product, drops, temperature, and so on. Glacier Vehicles isn't interested in mesmerizing you with a lot of technical talk, only to find the perfect refrigeration system for your vehicle and your needs.\nGlacier Vehicles employs our very own team of engineers, and as such this allows us to offer one of the quickest installation and build times in the industry. Lead time will greatly depend on the required equipment as well as any bespoke installations, but we always make sure to provide you with a date for completion when we confirm your order.\nGlacier Vehicles understands what it means to be so busy with your business you cannot afford to leave the property! We offer you the option for our team to pick up the van from your place of employment and drop it off again once the work is complete, allowing you to continue with the day-to-day of your business and let us worry about the conversion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Leopard Lodge is your ultimate wedding venue for weddings of up to 200 guests. Our team of wedding coordinators pride themselves in service excellence and will do all that they can to make your wedding day the most memorable ever. Keeping this in mind we believe in personal service and therefore all arrangements will be coordinated around your specific needs and requirements. The spectacular view of Hartbeespoort Dam and Magaliesberg mountain range prove to be an ideal backdrop for bridal photographs taken on the wedding day. As part of the wedding package a complimentary bridal suite will be booked for the bridal couple for the evening of the wedding and the wedding guests can enjoy a discounted rate on all remaining rooms.\nPlease visit their website https:\/\/www.leopardlodge.co.za to obtain more information about their amazing services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After what seems like an eternity, Aurora-44 have finally unveiled the release date for their hotly anticipated game, Ashen. They confirmed we would get a 2018 release during XO18 and they have delivered. Ashen will go live on Game Pass and be available in the Xbox store from Friday 7th December!\nI for one can't wait to explore the world Aurora-44 have created. The game aims to be a mix of Dark Souls and the fabulous PS4 exclusive Journey, with a little bit of Absolver's graphical style thrown in for good measure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For engines not listed above, please contact SwedeTech Racing for available services offered.\nSwedeTech has spent countless hours developing, through intensive dyno and on-track testing, the Honda CR 80cc Moto, Honda CR 125cc Moto, Maxter ICC, Pavesi ICC, TM ICC, Parilla TaG, Parilla ICA and Parilla JICA engines, Rotax Max FR125, Vortex, and many other karting engines for use in club and national level racing programs.\nReine Persson is the only engine builder to win ShifterKart Illustrated's Engine Builder of the Year multiple times, 2001, 2002, and in 2003. After this much recognition in such a short time span, the award was discontinued.\nOther great recognitions include artlices written about Swedetech and Reine Persson as an industry Power Broker. One example appeared in Kart Sport Magazine, June 2004, Issue 16, Page 36.\nSwedeTech Racing has also worked directly with many of the engine manufacturers as a developing partner. Much of this development is seen in current engine pacakges available to kart racers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamusd b/data_all_eng_slimpj/shuffled/split2/finalzzzamusd
new file mode 100644
index 0000000000000000000000000000000000000000..49dd933bfc0d1bc33fbfd87aa8a128ea19fb4330
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamusd
@@ -0,0 +1,5 @@
+{"text":"Let me guess, you have been playing New York Times crossword and got stuck on the clue Lightweight boxer?. Well, you have come to the right place to find the answer to this clue.\nNow you have the answer to your clue. Its time to move on to the next clue. You can browse through the list with all the answers to the New York Times crossword of January 3rd, 2019. Or you can use the search form below to find the answer (no matter if its New York Times Crossword or any other crossword).\nOne who may help you keep your balance?\nSubject of many '60s hits?\nAuthor known for the intelligence of his writing?\nTake ___ view of (view skeptically).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We know that every home is unique. Prior to treating your new home, one of our inspectors will perform a general inspection of your home and yard. We will customize a treatment plan based upon our findings and your specific needs. We take into consideration who is living in the home (i.e. children, pets, etc.), and create an effective plan that is safe for the whole family.\nOnce we have assessed your home and family's needs, we will treat the interior and exterior of your home. Our move-in treatments include a one-time treatment with options to upgrade to a customized service plan as needed. Unlike our competitors, we use the least invasive methods possible to remedy the problem. We also use naturally occurring products whenever possible.\nAfter moving into your new home, it may be prudent to invest in long term pest control services. We currently offer our Texas Pest Protection Plan, which includes 6 treatments per year with follow-ups as needed. We also offer termite inspections and treatments along with rodent control and removal services. Be sure to check out our specials to get the best prices on these high-quality services.\nWhen buying a home, the last thing you want to find on move-in day is a pest problem. Even worse than a colony of fire ants is an infestation of termites chewing away at your foundation. For all of your pest control needs in the Pearland and League City areas, contact Green Team Pest at (281) 295-1633 or request an online consultation today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published 04\/25\/2019 10:11:57 pm at 04\/25\/2019 10:11:57 pm in Pet Storage Cabinet.\npet storage cabinet food storage cabinet food cupboard storage stunning food cabinet storage food storage pet food storage cabinet pet supplies etna wooden pet food dog food storage cabinet on pet food cabinet with bowls dog storage.\npet storage cabinet image of food with bowls dog for ideas remodel pet storage cabinet image of food with bowls dog for ideas remodel , dog food storage ideas pet food storage cabinet ideas works small dog food storage ideas pet food storage cabinet ideas works small pet food storage ideas, pet storage cabinet food storage cabinets s truck dog cabinet plans pet storage cabinet pet food storage cabinet dog with bowls slide in drawer hooks diy pet pet storage cabinet , creative cabinets dog food storage with bowls amazing pet food creative cabinets dog food storage with bowls amazing pet food storage cabinet with bowls and shelf, boomer george pet feeder station for the furry ones pinterest pet feeder station storage cabinet product description this dog food storage cabinet is not only great for storing pet supplies but also makes a great , awesome pet food storage cabinet with designs brylanehome cabi awesome pet food storage cabinet with designs brylanehome cabi, under cabinet dog bowls stow away house ideas in under cabinet dog bowls stow away, dog food storage cabinet pet laundry room beach wooden plans dog food storage cabinet pet laundry room beach wooden plans, pet food cabinet pet food storage cabinet pet food bowl storage pet food cabinet, dog cat food feeding station cabinet kitchen pet furniture storage dog cat food feeding station cabinet kitchen pet furniture storage feeder bowl, pet storage cabinet dog cabinet wooden dog food storage cabinet pet storage cabinet dog food storage and feeder pet food storage cabinet for cool stylish and pet storage cabinet .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Perrysburg, Ohio (March 19, 2015) - Owens-Illinois, Inc. (NYSE: OI) announced today that Senior Vice President and Chief Financial Officer Steve Bramlage will leave O-I to pursue another opportunity, effective March 31, 2015.\nUntil an internal and external executive search can be completed, John Haudrich, vice president, finance and corporate controller, will assume the responsibilities of the chief financial officer.\n\"It is natural for opportunities outside the company to arise for our many competent leaders like Steve,\" said Al Stroucken, chairman and CEO. \"Although we are sorry to see him go, he leaves behind a highly capable finance team focused on financial flexibility. John is a strong and talented leader with broad experience in corporate finance who can readily assume CFO responsibilities.\"\nHaudrich has led the global finance operations for O-I's four business segments and the corporate finance team, focused on management reporting. He joined O-I in March 2009 as vice president of investor relations, was named vice president of finance in 2010 and was appointed vice president, finance and corporate controller in 2011.\n\"I am grateful for the opportunities that O-I has afforded me over the past nine years,\" said Bramlage. \"I am very proud that our work to dramatically improve the Company's financial flexibility has allowed O-I to shift capital allocation in favor of shareholder yield. And I am confident that senior leadership's focus on enhancing manufacturing capabilities and capitalizing on commercial opportunities will allow the Company to continue down that path.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For many students, a typical classroom is not an ideal learning environment. At Reach Christian Schools, we feel that these children, like all children, deserve the opportunity for a Christian education that meets their learning and developmental needs. Journey Christian Academy offers a highly individualized program for students in grades K \u2013 12. Our class sizes are small, and the environment is welcoming and encouraging. Students are placed at an appropriate level in core subjects, such as reading and math, while remaining in classrooms with their age peers, thus providing for their academic and their social needs. Explore this web site to learn more about Journey Christian Academy, and then contact us to schedule a visit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamztu b/data_all_eng_slimpj/shuffled/split2/finalzzzamztu
new file mode 100644
index 0000000000000000000000000000000000000000..6d5ad44f5c4ee1a7d265afe21a2c82cbcefbe863
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamztu
@@ -0,0 +1,5 @@
+{"text":"This luxury plush mattress features InfiniCool\u2122, a cooling layer designed to move heat away from your body, and BackCare 3X for better spinal alignment.\nThe Beautyrest Platinum\u2122 Haven Pines Plush mattress provides a medium-plush feel for a better night's sleep. Beautyrest's BackCare 3X has 2,000 pressure-relieving micro-coils for up to 20% better spinal alignment and conforming support. The design also features individually pocketed coils, soothing GelTouch\u00ae Foam and gel memory foam for added comfort, pressure relief and support. Plus, Beautyrest's newest antimicrobial technologies help keep you cool and your mattress fresh for years to come.\nBeautyrest PlatinumBetter sleep for bolder days. Sleep can be complicated and yours is wholly unique. That's why Beautyrest Platinum\u2122 mattresses are built with balance in mind\u2014blending art and science for a more fulfilling sleep. Keeping you cool and your mattress fresh while delivering comfort and support, Beautyrest Platinum offers a smarter, restorative rest for a better morning.\nFree Sleeptracker\u00ae Gift with PurchaseThe Beautyrest\u00ae Sleeptracker monitor uses accurate sensors, a power processor and a personalized sleep coach to track and help improve your sleep to ensure more productive tomorrows. Monitor up to two sleepers at once with pinpoint accuracy. The user friendly app translates precise data into smart, actionable advice designed to help you unlock a better night's sleep and more productive next day.\nMattress Height 14.75\" 14.75\" 14.75\"\n**You will receive your Beautyrest\u00ae Sleeptracker\u00ae 7-14 days after delivery of your mattress or mattress set. It will be shipped to the address registered with your membership on Samsclub.com. This offer is solely provided by Beautyrest\u00ae and is not the responsibility of Sam's Club\u00ae.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great tints are installed by our network of professional dealers. Want to drive more comfortably and customize your car to fit your personality with car window tinting? Since 1961, LLumar's quality car window tints have been protecting drivers from the harsh sun. Still today, we help drivers reach their destinations in style and comfort, whether they drive a car, truck, SUV, or minivan! At your car window tinting dealer, you'll find that our specially trained technicians have been handpicked to provide you with a superior service experience. Visit a LLumar dealer near you for a fast and free car window tinting estimate in San Diego!\nOver an extended period of time, continued exposure to heat and sunlight can cause your car's seats, upholstery, and dash to crack, peel, and fade. LLumar window tints in San Diego offer the ultimate protection and value for today's drivers. Tinting your car windows means better safeguarding the interior of your car, but also everyone in it, from the sun's heat and harmful rays. Your tinting company's wide variety of shades, from light to dark to ceramic, is sure to have the right window tint shade for you! Your state may have laws that limit the level of darkness you can legally select for your car windows. Contact your LLumar tinting specialist in San Diego to learn more about the car window tinting options in California!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Department of Biological Structure studies the structure and function of molecules, cells and networks with an emphasis on the nervous system.\nDr Olivia Bermingham-McDonogh received a new NIH grant to study \"Competence of support cells to form hair cells in mammalian inner ear\"\nThis project will explore the timing when support cells transition from a time when they are still competent to form hair cells, given the appropriate stimulus, to a time when they no longer have this ability. The influence of particular gene expression as well as epigenetic factors will be studied.\nThis paper recently published in the journal eLife here and co-authored by Drs Anitha Pasupathy and Wyeth Bair shows that there are many single units within artificial deep neural networks for visual object recognition that end up behaving like single neurons in mid-level visual cortex, particularly with respect to how they respond to the curved boundaries of objects. This opens the possibility that deep nets may provide an important tool for understanding sensory processing in the brain. This paper was highlighted in the UW newsroom. Click here to see that article.\nClare Gamlin from the Wong Lab and graduate student of the Neuroscience Program defended her thesis \"Development and regulation of synaptic divergence in the mammalian retina\" successfully in January. We wish her all the best and thank her for all her contributions to the department !\nGolden Lab Postdoctoral Candidate talk,\"Fighting Females: Aggression and Social Defeat Stress in Female Mice\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A1 Carpet Care did a great job. They cleaned my carpet and I am happy with how everything turned out. The employees were professional and pleasant to work with. I had no problems with the service and I would recommend them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paris is the most visited European city and one of the main destinations in the route of Kazakhstan's historical expedition. The expedition has already visited 15 countries of the Eurasian continent. Each of them unveiled some facts about the history of Kazakhstan. In the town of Nogent-sur-Marne near Paris, there is an alley with the monument of Mustafa Shokai, a prominent figure of the Alash movement. He had lived on the second floor of the house for about 20 years. His wife Maria is buried in the city cemetery. The mayor of the city Jakus Martin gave permission to name the alley after Mustafa Shokai in memory of the Kazakh political and public figure. Members of the expedition met with the mayor personally.\n- For the French, Mustafa Shokai was a prominent public figure who fought for the democracy and freedom of his people. He spent the most important years of his life here. France was his second home. Many French people learn about the Kazakh figure when they pass by the monument. We are always ready to support the initiatives of Kazakhstan in the field of history, literature and culture.\nThere is a monument to Attila 200 kilometers away from Paris in the village of La Shep. The King of Huns, who stirred up the entire Asia, fought in 451 with the army of the Roman Empire in this very place. It was the Catalan battle. The war that changed the history of Europe captivates the tourists today. According to the head of the expedition Sapar Iskakov, Attila was our ancestor. Therefore, young people should know the history associated with his name.\nThe famous museum of Paris Louvre keeps the traces of our ancestors. Islamic Arts hall contains items belonging to Sultan Hasan, the ruler of Egypt. He was a Kypchak-Mamluk by origin. Sultan Beibars' helmet is also exhibited there. Kazakhstani scientists will soon study the historical artifacts discovered by the expedition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanfbi b/data_all_eng_slimpj/shuffled/split2/finalzzzanfbi
new file mode 100644
index 0000000000000000000000000000000000000000..e5ed51d15d00799924e09427204d678ff9912cf8
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Traverse the 1\/4 mile driveway to find you own luxury retreat offering the utmost privacy featuring a beautiful executive home on over 29 majestic acres adorned by a pond, tennis court, basketball court and swimming pool. Home includes well-appointed kitchen, four fireplaces, large owners suite, large secondary bedrooms, keeping room, and In-Law suite in full finished basement and much more. All of this located in a prime location just 3 miles West of GA Hwy 400 and within minutes of shopping, restaurants, Lake Lanier and even the North Georgia Mountains.\nDirections: ***please Note Gps Takes You Into A Subdivision. This Is Not Accurate.*** From Atlanta Take I-75 North To Us-19 North. Take A Left Onto Ga-369 West. Take A Left Onto Hendrix Road. House Will Be On The Left.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Steel ring with aluminum clamp for attaching to your retort stand. 125mm diameter.\nSteel ring with attached aluminum clamp for attaching to your retort stand. Made from black powder coated 6.5mm steel rod, the ring has a diameter of 125mm.\nTotal length is 205mm, and it can attach to rods up to 15mm in diameter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this paper, a new binary linear programming formulation for computing the exact Graph Edit Distance (GED) between two graphs is proposed. A fundamental strength of the formulations lies in their genericity since the GED can be computed between directed or undirected fully attributed graphs. Moreover, a continuous relaxation of the domain constraints in the formulation provides an efficient lower bound approximation of the GED. A complete experimental study that compares the proposed formulations with six state-of-the-art algorithms is provided. By considering both the accuracy of the proposed solution and the efficiency of the algorithms as performance criteria, the results show that none of the compared methods dominate the others in the Pareto sense. In general, our formulation converges faster to optimality while being able to scale up to match the largest graphs in our experiments. The relaxed formulation leads to an accurate approach that is 12% more accurate than the best approximate method of our benchmark.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DESCRIPTION The rounded arch makes it likely that this is the Front Street bridge over North Mill Creek in Salem, Oregon. It is only a few yards from the Willamette River. No buildings of any kind are visible in the picture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Luis Trio is a project formed and conceptualized by French musician Loic Sanlaville. Originally set in Jazz elements in 2011, the project has since grown to acquire its own style with a Hip Hop sound. The band uses elements of different music set on original compositions by Loic.\nLoic moved to India five years ago and his love for Hindustani sounds mixed with his Frenchness is reflected in the eclectic mix of his music. Think Thelonious Monk meets Ganpati. Every song has a story in Luis Trio is here to transform the audience's experience in a different zone from a Jazz bar in New York to the calm spirituality of India.\nThe band performs regularly in Delhi, Gurgaon and has toured in Pune and Mumbai.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapwmt b/data_all_eng_slimpj/shuffled/split2/finalzzzapwmt
new file mode 100644
index 0000000000000000000000000000000000000000..02400953e6f667f8519c4288ab31707be9924610
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapwmt
@@ -0,0 +1,5 @@
+{"text":"Paw Paw, W. Va. \u2013A new U.S. Cellular tower at 1545 Paw Paw Rd. in Paw Paw, W. Va., has brought high-quality voice and wireless broadband service to residents in the area for the first time. U.S. Senator Shelley Moore Capito (R-W.Va.) and U.S. Cellular are celebrating the occasion with a ribbon-cutting ceremony at the tower site today.\nEarlier this year, Senator Capito initiated the search for a wireless carrier for the Paw Paw community, and U.S. Cellular recognized the need in the area. The company then began the process of deploying high-speed wireless broadband service, with funding from the Federal Universal Service Program. After announcing in June that it would be bringing a tower to the area in early 2016, U.S. Cellular and the city were able to move up the timeline in advance of winter.\n\"At U.S. Cellular, we focus on providing the best wireless experience for our customers, and we look forward to being a great community partner with the people of Paw Paw,\" added Meyers.\nIn 2014, U.S. Cellular invested $11.4 million in network enhancements across West Virginia. This was a combination of 4G LTE enhancements and upgrades to the company's existing high-speed network. The new tower in the area is one of more than 205 sites U.S. Cellular has built across the state.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finding the Euro exchange rates today in Leh used to be a challenge until BookMyForex came into the picture. But not anymore\u2026Live and real time Euro exchange rates today in Leh are now available on BookMyForex for currency exchange. You also have the option to lock-in the Euro exchange rates today in Leh to exchange currency by using the \"Freeze the rate\" option on the website. Other currency exchangers and websites use the base rate or interbank rate as \"today's EUR rate in Leh\", which isn't actually the rate at which currency is exchanged. Forex dealers keep a significant margin over and above \"today's EUR rate in Leh\" to hedge against any EUR price fluctuation in the global market. So the norm of \"today's EUR rate in Leh\" is just an easy way for forex dealers to make a few extra bucks.\nThe complete range of forex products offered by the Company includes EUR currency notes, EUR forex cards, EUR traveler's cheques or remittance products such as EUR demand drafts and EUR wire transfers.Orders are delivered at our own branches or at all Axis Bank and ING Vysya Bank branches, which are our PAN India banking partners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A history of Kosovo and Serbia's tempestuous trade relations.\nEarlier this week, the Kosovo government announced that taxes on imports from Serbia and Bosnia and Herzegovina (BiH) will increase from the 10 percent announced earlier in the month to 100 percent. The move comes as part of a long history of difficult trade relations between the three states that have spilled over from the political discord.\nBut, what has preceded the current situation? Here's a summary of some of the key moments and issues to take into account.\nThe trio are all members of the Central European Free Trade Agreement (CEFTA), a trade agreement between seven non-EU member states in the Balkans that aims to ease the path toward EU integration, and consolidate free-market economics. Kosovo became a member of CEFTA in July 2007, while represented by the United Nations Mission in Kosovo (UNMIK).\nAfter declaring independence in 2008, Kosovo changed customs stamps, from \"UNMIK customs\" to \"Kosovo customs.\" In December 2008, Serbia and BiH, which had (and still have) not recognized Kosovo's independence, stated that they would neither allow the export of Kosovo goods with those stamps nor the use of their territory as a transit passage for these goods.\nAs a response, on July 20, 2011, the Kosovo government announced a decision to take reciprocal measures against Serbia and BiH. Six days later, the Kosovo Police tried to ensure their presence at border points 1 and 31 in the northern part of Kosovo, which had been previously been operated by the EU Rule of Law Mission in Kosovo, in order to enforce the measures. In response, local Serbs set fire to the border points.\nThe measures remained in force until September 2011, when Kosovo and Serbia reached an agreement on customs stamps in Brussels. As part of the EU-mediated dialogue, Kosovo agreed to change the stamps so as not to include any state symbols, and Serbia accepted Kosovar goods not stamped as such.\nHowever, the general lack of implementation of other agreements in the EU-mediated dialogue, and a series of spats between the two states, including Serbia's recent campaign against Kosovo's membership to Interpol and attempts to get states to 'de-recognize' Kosovo, has meant that the already tense situation between the countries has escalated.\nOn Nov. 6, the government of Kosovo decided to introduce a 10 percent tax on products originating from Serbia and BiH. A few days later, Prime Minister Ramush Haradinaj said that this tax could increase.\nOn Nov. 8, a meeting scheduled in Brussels between presidents Tha\u00e7i and Vu\u010di\u0107 was cancelled after the President of Serbia stated that there is no dialogue with Kosovo while the import tax was still in force. In response, Tha\u00e7i insisted that Belgrade does not need to be afraid of Kosovo under any circumstances, because the latter will be responsible in the dialogue as well as in its actions.\nThe European Union called on Kosovo to revoke its decision to impose an import tax on goods coming from Serbia and BiH. On Nov. 9, Kosovar Serb party Serbian List, though part of the government coalition, threatened that if the government does not abolish the tax increase on Serbian products, it will come out in opposition and organize civic protests.\nOn Nov. 20, Vu\u010di\u0107 stated that due to this tax, Serbia loses 1 million euros in a single day, while Kosovo Customs announced that it received around 300,000 euros more in the week after the tax was imposed than the previous week.\nWith tensions already escalating, it seems that the situation got hot-blooded after the Serbian diplomatic offensive to convince several states to revoke Kosovo's recognition, and especially after its strong lobbying against Kosovo being admitted to Interpol. On Nov. 20, Kosovo didn't receive the two thirds majority of the vote required to be admitted to the international police organization.\nThe next day, Nov. 21, Kosovo raised customs tariffs on Serbian and Bosnian goods from 10 to 100 percent, with Deputy Prime Minister Enver Hoxhaj explicitly referencing Serbia's recent approach to Kosovo and the campaign against Kosovo's membership to Interpol.\n\"[The] EU can pressure Kosovo, but the problem is there are no more sticks.\"\nRTK's Brussels correspondent, Gjeraqina Tuhina.\nThe retaliation also required the authorities to remove or prevent from entering any goods that did not address Kosovo by its constitutional name, the Republic of Kosovo, which Serbia and BiH do not recognize. Haradinaj also announced that measures imposed on products coming from Serbia should remain in place until recognition. \"Unless Serbia recognizes Kosovo as an independent and sovereign state, we should not abolish [the tax],\" he said.\nThe same evening, President Vu\u010di\u0107 called an emergency meeting of the National Security Council in Belgrade, while the European Union High Representative for Foreign Affairs and Security Policy, Federica Mogherini, said on Wednesday that the Kosovo government has to immediately revoke its decision to increase tax on goods coming from Serbia and BiH. \"Today's decision is a clear violation of CEFTA and the spirit of the Stabilisation and Association Agreement between the EU and Kosovo,\" she said.\nReacting to Mogherini's statements, in a letter to her, Haradinaj said that the Kosovo Government calls on EU representatives as mediators in the dialogue between Kosovo and Serbia to force Serbia and BiH to meet their commitments according to CEFTA and other agreements.\nKosovo Customs Spokesperson, Adriatik Stavileci, told Gazeta Express that the new customs tax was implemented on the same day that the government announced its decision, adding that there has been a decrease in imports since. He also revealed that on the day of the announcement, 20 trucks turned back after receiving the information about the new tax.\nWith the decision sparking enthusiasm among large sections of the Kosovar Albanian population, the governing coalition parties \u2014 the Democratic Party of Kosovo (PDK), Haradinaj's Alliance for the Future of Kosovo (AAK), and the Social Democratic Initiative (NISMA), all tried to take the credit for it.\nKosovo joined CEFTA without setting quotas, despite the agreement foreseeing that member states have the right to. A tariff quota regime is applied when a CEFTA member state uses its right to impose preferential tariffs on imports of products, usually agricultural products, which are protected by that member state.\nKosovo thus entered CEFTA in an unequal position, causing many problems for local producers. While neighboring countries set quotas, as is the case with the quota Albania has set for potatoes, Kosovo has not protected any of its producers from the threat of cheap imports.\nAs a result, Kosovo has a trade deficit with most CEFTA members, but especially Serbia. According to data from the Kosovo Agency of Statistics, in 2017 Serbia exported goods to Kosovo worth 449,918,000 euros, while the goods Kosovo exported to Serbia only valued 48,246,000 euros, creating a trade deficit with Serbia of 401,672,000 euros. Of the total value of trade exchanges between the two countries in 2017, only 9.68 percent were Kosovo exports, while 90.32 percent were Serbian exports.\nThe trade relationship between Kosovo and Serbia was even more unequal a few years ago. In 2010, while Serbia exported to Kosovo goods worth 260,471,000 euros, Kosovar exported goods to Serbia had a value of only 3,941,000 euros. Thus, Kosovo was in a deficit of 256,530,000 euros in its trade exchanges with Serbia. In 2010, Kosovar goods accounted only 1.49 percent of trade exchanges between the two countries, while Serbian goods made up 98.51 percent.\nAccording to the same report, in 2017 Serbia was the largest exporter in Kosovo, followed by Germany (378,515,000 euros), Turkey (292,626,000 euros), China (275,463,000 euros) and Italy (194,722,000 euros). BiH on the other hand, was the ninth largest exporter in Kosovo, exporting goods to Kosovo worth 81,674,000 euros. Thus, while Serbia accounted for 14.8 percent of total imports entering Kosovo, BiH only accounted for 2.7 percent.\nEconomics professor and expert Safet Merovci sees the decision as a political and populist move.\nWhile much of the discussion in the media has focussed on Kosovo's reliance on the import of flour from Serbia, it was actually seventh on the list of the most valuable exports from Serbia to Kosovo in 2017. Electric conductors were the most valuable, totalling 54,849,000 euros, followed by petroleum oils and oils derived from bituminous minerals (44,754,000 euros), non-alcoholic beverages (38,904,000 euros), sugars, molasses and honey (20,221,000 euros), wheat and meslin (17,494,000 euro), clay construction materials (16,800,000 euro), cereal and flour preparations (14,176,000 euros), vegetable oils and grease (12,445,000 euros), animal feed (11,649,000 euros), and iron and steel products (11,341,000 euros).\nPolitical economist Besnik Pula wrote on his Facebook account that it is the most sensitive sector of consumers that will be damaged by the new import tax, and cites an article he wrote in 2011 that outlines the necessity for a well thought out strategy for replacing Serbian goods within the Kosovo market before measures are taken.\nEconomics professor and expert Safet Merovci meanwhile sees this decision as a political and populist move, saying it may be fair in the short term, but if there is a rise in prices on these products from Serbia and BiH, it will be the citizens, not politicians, who would pay for it. Deputy at the Assembly for opposition party the Democratic League of Kosovo, Hykmete Bajrami, meanwhile, believes that this decision was taken to cover government mismanagement.\nRegardless, it seems as though the countries of the region have once again decided to solve their problems on their own. This time no longer with a rifle, but through trade war, the kind of which are not completely unfamiliar to European and world history.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A re-elected Andrews Labor Government will increase Victoria's Renewable Energy Target (VRET) to 50 per cent by 2030, creating thousands of jobs, putting more energy into the grid, increasing investment and driving down energy prices.\nThis builds on Labor's VRET legislation that will ensure that 25 per cent of our electricity generation comes from renewable sources by 2020, and 40 per cent by 2025. It's expected that these targets will help deliver around $9 billion of investment and create more than 11,000 jobs over the life of the scheme.\nAs well as jobs, the VRET will put more energy into the grid, delivering up to 5,400MW of new, large scale renewable energy capacity by 2025.\nThe Liberals have promised to scrap the VRET, which would not only result in thousands of job losses and billions of dollars of investment that will never reach Victoria, it would also see higher energy prices.\nIn four short years, 732MW of new renewable energy capacity has been built, and more than 3,000MW of renewable capacity is under construction or contracted to be built.\nVictoria's first renewable energy auction is delivering 928MW of renewable energy, with the six projects across the state producing enough electricity to power 646,273 households \u2013 the equivalent of powering the towns of Ballarat, Bendigo, and Geelong combined.\nThe power produced by these projects is also expected to drive a 16 per cent reduction in Victoria's electricity sector greenhouse gas emissions by 2034\/35.\nBoosting the VRET builds on the Government's work to cut power bills for Victorians, from the $50 Power Saving Bonus, to subsidising solar panels and installing batteries or hot water systems on 720,000 houses across the state through the Solar Homes program \u2013 driving down energy prices and greenhouse gas emissions.\nJeff Kennett and the Liberals privatised Victoria's power industry, promising competition would lead to cheaper electricity prices, but the only winners are the big energy companies making big profits while Victorians pay higher prices.\nOnly Labor is delivering new, clean energy, driving down energy prices for Victorian families and businesses.\n\"Labor's VRET has helped create a jobs boom and boosting it further will see even more jobs created across our state.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Will Local Level Events Events provide support for organizations and consumers?\nYES! Local Level Events has both a phone helpline and a support email account.\nWhat happens if I need help creating an event on Local Level Events?\nOur Setup Guide walks self-serve customers through the event creation process. Our PRO account gives you access to our support team. This team will create custom event links FOR you. Choose PRO and leave the work and details to us!\nWhat if a patron wants a refund?\nLocal Level Events will allow organizations to control refund policies per event. We are happy to assist you in creating a plan should a refund be requested. Simply email support@locallevelevents.com or call (888) 991-6558 for assistance with refunds. Some fees and transaction costs may not be refundable.\nDo we have control as to when our events publish, tickets go on sale and event registration opens?\nYes! Our system is designed to allow event organizers to maintain control of sale and publish dates.\nWhen and how will our organization be paid?\nYour organization will be paid via direct deposit within 10 business days of event completion.\nWhat will happen if a consumer attempts to print and use multiple copies of a ticket and distribute to others?\nOur scanning feature will indicate when a ticket has been scanned and will display a warning message with the last scanned time.\nIs there a limit to the number of custom fields on our tickets and event registration pages?\nFields are unlimited and customizable to your organization's specific needs and requests.\nAm I able to allow multiple members of my organization access to reports, sales, registration etc.?\nCan children under the age of 13 use the services to register for an event or purchase a ticket?\nNo. Children under the age of 13 are not permitted to use the services to register for an event or purchase tickets even though some events may be attended by children under the age of 13. A parent, guardian, or other person over the age 13 must register for events or purchase tickets for such events. If you use our services, then you acknowledge and affirm that you are over the age of 13.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqgtd b/data_all_eng_slimpj/shuffled/split2/finalzzzaqgtd
new file mode 100644
index 0000000000000000000000000000000000000000..37532beb9ff727db6a3cc9d4d2c3dcb13e2596c2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqgtd
@@ -0,0 +1,5 @@
+{"text":"Here is a pair of new wondrous items for use with the Pathfinder RPG. I actually created them for a villain in one of my defunct D&D campaigns, but I wanted to revamp them for Pathfinder and release them upon the masses.\nSlot -; Price 21,000 gp; Weight 0.5 lb.\nClouds of read smoke can be seen swirling inside these small glass marbles. They are found in pouches usually containing three such marbles. After speaking the command word, a marble may be thrown as a splash weapon with a range of 15 feet. Upon impact, the marble shatters releasing a storm of energy that drains magical items.\nA direct hit with a marble drains 1d8 charges from every magical item with charges possessed by the target. Every creature within 5 feet of the impact lose 1d4 charges from every magical item with charges possessed.\nSlot -; Price 92,000 gp; Weight 1 lb.\nThese marbles are much larger versions of their less powerful siblings. Greater Marbles of Arcane Abatement are found by themselves or in mixed in with a pouch of Lesser versions. These marbles function just like a Lesser Marble of Arcane Abatement with the following except it affects any creature within 10 feet of the impact.\nAlso, every creature within 10 feet must make a DC 25 Reflex save or suffer 6d6+ the number of charges drained of magical damage. Creature who successfully save only take half damage.\nI love it. I think a common problem many DMs face is that the PCs find too many magical items too quickly. Then the balance of the game is completely thrown out the window. Over the years I've tried to come up with many ingenious ways to take back magic from the PCs. Items like the Marbles of Arcane Abatement are prefect for situation when I need to scale back the PCs magic power or if I'm just in a mood to mess with them. Bravo, Mad Brew Labs for presenting another great idea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"M. Stickland faced 2 batters in the 8th.\nT. Miller singled to right field.\nH. Hackney singled to center field.\nD. Pulley pinch ran for H. Hackney.\nR. Dickens doubled to left center, RBI; D. Pulley scored.\nT. Byrd singled to left field, RBI; R. Dickens scored.\nM. Morgan singled, bunt; T. Byrd advanced to second.\nT. Miller walked; M. Morgan advanced to second; T. Byrd advanced to third.\nH. Hackney to c for D. Pulley.\nXavier Rodri struck out swinging, reached first on a wild pitch.\nBrandall Pet doubled to left center; Xavier Rodri advanced to third.\nCody McKenzi singled to right field, 2 RBI; Brandall Pet scored; Xavier Rodri scored.\nT. Jackson hit by pitch.\nJ. Williams homered to left field, 2 RBI; T. Jackson scored.\nGauge Gantt singled to left field; Tanner Mathi advanced to second.\nDamon Smith hit by pitch; Gauge Gantt advanced to second; Tanner Mathi advanced to third.\nChristian Ma flied out to rf, SF, RBI; Gauge Gantt advanced to third; Tanner Mathi scored.\nXavier Rodri flied out to cf, SF, RBI; Gauge Gantt scored.\nBrandall Pet reached on a fielding error by p; Damon Smith advanced to third.\nMason Stripl to p for Will Altemue.\nT. Byrd singled to center field.\nM. Morgan walked; T. Byrd advanced to second.\nL. Poteat doubled to right field, 2 RBI; M. Morgan scored; T. Byrd scored.\nBrodrick Rie to p for Mason Stripl.\nT. Jackson singled to left field; L. Poteat advanced to third.\nJ. Williams walked; T. Jackson advanced to second.\nDylan Goodma to p for Brodrick Rie.\nChristian Ma singled to shortstop.\nHayden McGra pinch ran for Christian Ma.\nChristian Ma to c for Hayden McGra.\nL. Nichols to p for D. O'Brien.\nBrandall Pet tripled to center field.\nCody McKenzi flied out to cf, SF, RBI; Brandall Pet scored.\nDerek Perry singled to center field, advanced to second on a fielding error by cf, advanced to third; John Elwood scored, unearned.\nTanner Mathi singled to center field, RBI; Derek Perry scored, unearned.\nKirby Lamber pinch hit for Gauge Gantt.\nDamon Smith hit by pitch; Tanner Mathi advanced to second.\nHunter Griff pinch ran for Tanner Mathi.\nTanner Mathi to 1b for Hunter Griff.\nDylan Goodma to lf for Kirby Lamber.\nKirby Lamber to dh for Dylan Goodma.\nT. Jackson reached on a throwing error by 2b.\nM. Perkins pinch hit for JR Pollack.\nM. Perkins walked; T. Jackson advanced to second.\nM. Perkins advanced to second; T. Jackson advanced to third on a wild pitch.\nH. Hackney struck out swinging, reached first on a passed ball; M. Perkins advanced to third; T. Jackson scored, unearned.\nR. Dickens reached on a fielder's choice; D. Pulley advanced to second; M. Perkins out at home p to c.\nR. Dickens advanced to second; D. Pulley advanced to third on a wild pitch, scored on a throwing error by c, unearned.\nT. Byrd singled to center field, RBI; R. Dickens scored, unearned.\nJR Pollack to 3b for M. Perkins.\nM. Stickland to p for L. Nichols.\nXavier Rodri singled to center field.\nZ. Saylor to p for M. Stickland.\nBrandall Pet advanced to second; Xavier Rodri out at third c to 3b, caught stealing.\nJohn Elwood singled to shortstop; Brandall Pet advanced to third.\nDerek Perry singled to shortstop, RBI; John Elwood advanced to second; Brandall Pet scored.\nTanner Mathi walked; Derek Perry advanced to second; John Elwood advanced to third.\nZach Scudder pinch hit for Dylan Goodma.\nCameron Reid to p for Zach Scudder.\nL. Poteat hit by pitch.\nM. Polak to c for H. Hackney.\nAndrew Toney pinch hit for Christian Ma.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the most popular proteins on the market for the last 20 has been Designer Whey. They grew into the leading developer of delicious, whey-based nutritional products, led by a group of passionate health and fitness enthusiasts. The mission has been and always will be to help customers achieve their fitness goals and live healthier lives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our team welcomes you in the 3ra. Calle 6a Avenida No. 12 in a nice furnished place or in our lovely open air garden. Envoy and relax in with nice food, fresh made bakery deli or our various different coffee blends as Milk-Coffee, Cappuccino or Espresso \u2013 also with lactose free or soy milk or trendy flavors. Our competent and quick team will be there for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BLOOMINGTON, Ind. -- To kick off the summer traveling season, the Lilly Library at Indiana University's Bloomington campus has opened an exhibition featuring collections relating to early automobiles and motor cars. \"Are We There Yet? The Age of the Automobile\" showcases vintage catalogs, books and materials featuring topics ranging from luxury roadsters to the first Indianapolis 500.\nThe exhibition runs through Sept. 5.\nLavish catalogs aimed at Gatsby-like customers come from the collection of Thomas T. Solley. Solley was director of the Indiana University Art Museum from 1971 to 1986 and a grand-nephew of J. K. Lilly Jr., early benefactor of the library that bears his family name.\nSolley's collection shows how luxury car catalogs blossomed as an art form at the turn of the 20th century. Faced with growing competition to sell automobiles, manufacturers introduced lithography, photo-mechanical reproduction and special leather-tooled bindings to lure prospective buyers.\n\"The automobile is such a part of our national identity,\" said Becky Cape, head of public services at the Lilly Library, \"and yet the industry is in crisis. It's especially interesting to revisit auto history not only in Indiana but also around the world.\"\nIn the early 20th century, Cape says, Indiana was at the center of the automobile industry, rivaling Michigan as the country's automobile manufacturing giant. In all, more than 250 makes of cars were produced in more than 40 Indiana cities and towns.\nAlthough the Great Depression brought the downfall of most Hoosier automakers, Indiana's automobile legacy continues through one of the world's best-known auto races, the Indianapolis 500, first run in 1911.\nSome of Cape's favorite parts of the exhibition describe how women influenced the industry and the market. Electricity powered some of the earliest cars, Cape says, and women were encouraged to drive them because they were slower and cleaner.\nAt the beginning of the 19th century, the famed French manufacturer Peugeot produced steel rods for crinoline dresses before making umbrella frames, wire wheels, bicycles and finally, cars.\nThe exhibition was curated by Madeleine Thompson and assisted by Cape.\nThe Lilly Library is Indiana University's library for rare books and special collections and one of the 18 libraries of the Indiana University Bloomington Libraries. It is located on Seventh Street south of Showalter Fountain on the IU campus in Bloomington. Hours are Monday through Friday, 9 a.m.-6 p.m., and Saturday, 9 a.m.-1 p.m. For more information call 812-855-2452. Free and open to the public.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqjuv b/data_all_eng_slimpj/shuffled/split2/finalzzzaqjuv
new file mode 100644
index 0000000000000000000000000000000000000000..4793d3d2baae39e30710875d3c9235877ef6c488
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqjuv
@@ -0,0 +1,5 @@
+{"text":"This Salvaged garment is a 100% vintage 1990s beige soft leather slouchy backpack with drawstring fastening, flap and small zip front pocket. Zip apart strap can be worn cross body or as a backpack. SIZE - Model is 5'6\" and size 8. CONDITION - Excellent vintage condition, some light discolouring due to light colour and wear but adds to the vintage character, no faults or damage. DELIVERY - Next day delivery available Monday-Thursday on orders made before 1pm. Orders made on Friday will be posted on Monday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DRAYTON & TOOWOOMBA CEMETERY In our busy lives we ignore our morality. Pre-purchasing your final resting place at today's price makes economic sense. Pre-planning eases the stress on family members at a truamatic time. Contact our office for further information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Fraunhofer Institute for Interfacial Engineering and Biotechnology IGB develops and optimizes processes and technologies in the areas of health, food, environment, sustainable chemistry and energy. We offer our customers R&D services ranging from feasibility studies to implementing new processes in industrial practice, supplemented by a broad spectrum of analysis and testing services. Our strengths are offering complete solutions from the laboratory to the pilot scale.\nFor the coating and surface functionalization of foils, films and textiles we use gas phase (CVD, PVD, PECVD), wet chemical and combined processes. Our focus is on affine sensor surfaces for measurement of environmental gases and VOCs, plasma weathering as a rapid testing method for material characterization, an anti-ice coated film, and surface analysis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nearly all of the things you point out happens to be supprisingly accurate and it makes me wonder why I had not looked at this in this light previously. This particular article really did turn the light on for me as far as this particular issue goes. However there is actually 1 issue I am not too cozy with so whilst I make an effort to reconcile that with the actual main idea of your point, let me see what all the rest of the visitors have to say.Very well done.\nI enjoyed viewing your blogand you have got a fabulous family. I adore which you have made use of this website for individual and function purposes. As well as the RSS feed for CNN is often a decent idea, too. Some links to even more of your preferred web-sites would be pretty interesting to determine, and helpful to fellow education students.\nI\u00a1\u00a6ve been exploring for a bit for any high quality articles or weblog posts in this kind of area . Exploring in Yahoo I finally stumbled upon this website. Studying this info So i am satisfied to convey that I've a very just right uncanny feeling I discovered exactly what I needed. I most no doubt will make certain to do not omit this website and provides it a look regularly.\nThanks for your write-up. Another point is that being a photographer consists of not only difficulties in catching award-winning photographs but also hardships in acquiring the best digicam suited to your needs and most especially challenges in maintaining the grade of your camera. It is very accurate and noticeable for those photography addicts that are in to capturing the actual nature's engaging scenes : the mountains, the particular forests, the wild or perhaps the seas. Visiting these daring places definitely requires a digital camera that can surpass the wild's severe surroundings.\nThank you for sharing superb informations. Your web site is so cool. I am impressed by the details that you have on this web site. It reveals how nicely you understand this subject. Bookmarked this web page, will come back for more articles. You, my friend, ROCK! I found simply the information I already searched all over the place and just couldn't come across. What an ideal site.\nThis layout is so stellar. How did you take care of to produce a blogging site that is as clever since it is smooth? I imply, it is like an Aston Martin \u2013smart and hot simultaneously. I've received to say, the layout on your own prepared me arrive back again to this weblog all over again. But since I've learn what you have obtained to say, I've bought to share it together with the globe!\nGreat site. Lots of useful information here. I am sending it to several friends ans also sharing in delicious. And of course, thanks for your effort!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have another fun and delicious weeknight dinner for you guys!!\nThis pos is sponsored by Califia Farms\u00ae, but the opinions are my own.\nThis Easy Thai Mushroom Zoodle Soup is simple, plant-base and packed with flavor!\nThis post is sponsored by Bob's Red Mill\u00ae, but the opinions are my own.\nThis is a decadent and delicious winter chowder that is perfect for a special occasion or a cozy weekend night!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarthj b/data_all_eng_slimpj/shuffled/split2/finalzzzarthj
new file mode 100644
index 0000000000000000000000000000000000000000..c1db289bd864723f0dc05ce523c875ed79e1166e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarthj
@@ -0,0 +1,5 @@
+{"text":"Before I left for vacation things were pretty insane around here with back to back design and video shoots (and some makeovers) that were keeping us on a constant adrenaline rush and have continued since we got back. But man, every single one of them has been so fun and will produce SUCH good content for the blog. It's making this pregnancy go by so much faster.\nI figured I'd do a little behind the scenes sneak peek into what is going to happen soon over here (and on youtube and national TV) over the next couple of months.\nWe did another round of Target Styling Videos for their Fall product line, which is my favorite line so far, actually. There are a lot of rustic, plaid, Pendleton-style pieces and a lot of mid-century inspired brass and marble pieces. I just saw the rough cuts of the video this morning and I love them.\nGood Housekeeping wanted to feature the house, specifically with a 'fall refresh' angle to it, so it was the perfect excuse to, well, TOTALLY REDESIGN the house. I finally finished a of couple projects that were driving me crazy that I can't wait to show you. Meanwhile this little dude was very \"helpful\" the weekend before with all of our house projects.\nThat rug is a huge crowd-pleaser (it's incredibly soft) and has changed the whole family room and made that big comfy (and bulky) sofa actually WORK. Stay tuned for photos.\nThen I had my first big national commercial with Target where many people, like hundreds, were involved to make the set, my hair, my belly and lots of pretty product look good.\nIt was definitely more makeup than I was used to but those lights were so bright that you kinda need a lot to not look like a featureless ghost. The woman who did my hair (Danielle) does Lady Gaga's, Madonna's, pretty much everyone's hair. She put extensions in so that it looked fuller and a bit longer. I had extensions briefly a couple of years ago (for a couple months) and you do feel kinda like a super model with them in, but one that can't sleep because they itch so bad and are just not comfortable enough to deal with. Also please note my manicure. It was my first (and possibly last) foray into nail art. Not because I don't think its cool on others but mainly because it just feels silly on me. I also (as you know) don't have beautiful hands so drawing more attention to those meat puppets seems like a weird idea. But for the video the manicure looked awesome. That dress is from here (it's on sale) and I LOVE it both for maternity and non-maternity.\nI can't show you too much because it's their big fall campaign, but to give you an idea of the scale\/size of a big ad campaign check this out. And this is only one of the studios they shot in.\nI always love these glimpses into your workdays, Emily. What a cool career you've made for yourself!\nHoly crap! I'm gonna need that pouf and some of those pillows. Luckily I'm redoing my bedroom next month so this is perfect timing.\nAlso that candelabra! I bought a similar one (it's cast iron) in Amsterdam last year that I LOVE.\nI can NOT wait to see what y'all did with Nicolette's space! Also, can't wait to see how your new living refresh turns out.\nHaven't even read it yet, but wgat about that orange\/red top in the first image?!?\nYes! Where is that from?? So chic.\nSame here! Please update the link, pretty please!\nStarting a brass and wood lamp fund as we speak\u2026 and of course, pillows will be bought along with it.\nand excuse me, but your shoes in the lights shot. major!\nIs that rug from Target, or can you share the source? We've been dying for a new one and this would be a great contender!\nNevermind! just saw that its linked. Thanks!\nOMIGOSH\u2026that Serge Mouille ceiling light fixture in Nicolette's living room is killing me!\nI'd love to know about the shoes too. Love them!\nSo much fun stuff\u2014loving the look of the Target stuff! And how exciting that you'll be in a national commercial!\nSo much to look forward to\u2026.thanks for the peeks. Charlie gets cuter and cuter, can't wait to see him with a little sister.\nChecked out your pug client and she has quite a life and wardrobe.\nAlways enjoy your designs and that you are always rearranging, making the look even more fresh\u2026your designs are very fresh and eclectic which charms me. That pink antique Persian carpet caught my eye\u2026beautiful. your little boy is darling.\nThat little sneak peek of Target product has me so excited! I'm FINALLY decorating my living room this fall, and those pieces look like they'll fit the style I'm going for very well.\nPS. Love seeing these little life update posts. It's so fun to see behind the scenes \u2013 keep 'em coming!\nThe lead photo on this post is such a cute belly shot. You're an adorable mama, Emily (& talented, obviously)!\nwhere are the items in the 3rd to last pic from? The black and brass candle holders, striped throw, etc.\nI have been eyeing the rug you put in your family room. How is the shed factor on it?\nalso going to need to know where that sweater vest is from in the last photo\u2026 you look great!\nOh Emily, seriously. I saw the Target picture and thought \"How lovely. Oooh! Nails. Pretty\u2026\" and then read meat puppets. And died laughing. You're too much. Love the bottom pics with pink, blush, black, white. You're doing some hard work. Take care of yourself and your meat puppets.\nWhhaaaa!! There were like ten teases in this post. I'm dying.\nSeriously when I was looking at your blog years ago my boyfriend was like, wait, that's the blogger, not a model? But wha?! Aren't you like nine trillion months pregnant?! you look seriously luminous in those pictures and seem to have zero of the face bloat that normally comes at the end of pregnancy?!! Like seriously. You're perfect.\nHello! Can you tell me about your shoes?\nAre you able to share where the wooden gold bar cart and those leather chairs come from in your living room shot? Remodeling right now and those are a must for my new room! Also when does the issue come out! Hope to see you at the Vintage Bazaar in September-friend of Devon!\nHey- can you let me know where the first reddish too is from?\nYou had me at meat puppets! That was hilarious!\nTwo things: you are beautiful with or without being professionally done-up, and what are you doing with your fabulous blue Craigslist rug? I fantasize about that rug!!\ndon't let danielle ever touch your hair again. i never make these kinds of comments but the change in aura achieved by this person and her greasy extensions is so not you. please.\nI love that blue pillow with gold dots\u2026 it's a pity they don't ship to Europe. \ud83d\ude41 There are several things I would order!!! Or please convince them to open stores here as well\u2014> TARGET GERMANY!\nUm, don't you need a heckuva lot of energy to do all that? And did you forget you were pregnant? You are amazing and look fantastic!\nOooooh, can't wait! You look beautiful always, Emily!\nAaahh I found The Affair too upsetting and quit after 2 or 3. If you power through and feel glad you did, I would be happy to know that. Thanks for sharing all this behind-the-scenes!\nI'm hyperventilating right now because we are in escrow for a cabin and rustic\/plaid\/Pendleton with a dash of mid-century is exactly the look I am going for. Ahhhh!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you have an eye for design? Prove it in 'Pretty Nails Design'- a fun, fashion filled nail design game for girls! You get to choose between tons of colorful nail designs and fashionable, gorgeous jewelry. Pick out a ring and bracelet to match your nails. Let's not forget the nail stickers. Place some on your newly painted nails to give it that finished, professional look with an edge of your own. Play now and show off your awesome skills!\nUse mouse to interact. Enjoy!\nWhen the Prom night comes, everything goes around that special moment when your date first lays eyes on you, a second that will seem to last forever..\nWelcome to Saras Cooking Classes. Today Sara will be teaching you how to make a delicious \"SPAGHETTI BOLOGNESE\".\nMr. Bean has been working in this Hair Saloon with an intention to try out something new.\nBarbie's eternal beauty and youth lies on the secret of spa treatments and right make up choices. You have the opportunity to experiment a real makeover in Barbie's bathroom.\nGoing to the spa for a facial and a massage is a great way to unwind from the stress of the outside world. With the New Age music playing, the aloe face scrubs, and long showers, you'll feel incredibly relaxed and ready to take on all the ills and nastiness the world will throw at you.\nThis wonderful decoration game will give you the chance of making the perfect toe-nails. Choose all the colors that you want and then put some nice flowers and other patterns on the nails so you will complete the look.\nThe first thing anyone learns how to make is pasta. All you need is some boiling water, pasta, and sauce.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While parents may not be able to make a bookworm out of an athlete or an overachiever out of their laid back kid, they can have an effect on the remaining characteristics. Aside from moving or changing schools, parents can fill in the gaps of their child's education by providing enrichment activities based on low test scores, grades, or their own observations. Summer school or summer tutoring can be helpful, as can creating your own learning lab at home, working with your teen on meaningful projects that help connect skills with real life.\nProviding your teen with a space conducive to study and the time to complete class work is important, too. A growing number of today's teens are too busy with part-time jobs, extracurricular activities, and family responsibilities to have time to get enough sleep, as it is. Add homework to the list, and you have a sleep-deprived teen who's unlikely to be able to concentrate during tomorrow's lectures.\nWith epidemic numbers of \"latch-key kids\" who come home to hollow houses every day, come increased chances of unhealthy snacking and overly processed convenience meals, along with blaring music or TV to fill the unwelcome silence. A healthy diet feeds a healthy mind, as does a healthy social network. If you can't be there to greet your teen, at least try to call to welcome him or her home and ask about the day. Tell him or her where to find healthy snacks, or ask a friend or family member to hang out until you're home.\nYou may not think that your teen wants your input, and maybe he really doesn't. But he needs it. If a chain-link fence of Os on a quarterly grade sheet shocks you, then you need to act. You can respond by routinely e-mailing teachers to ask about upcoming assignments and check up on your teen's progress. If that isn't enough, you can require your teen to gather teacher signatures next to assignments or upcoming assessments, allowing you to know for sure if it's true that he was given no homework over the long weekend.\nThe bottom line? When you put action to your concerns, your kids will benefit. Not every teen is capable of straight As in basic classes or of passing Calculus, but I'm convinced that a parent-supported teen is much more likely to enjoy success than one whose parents fail to get involved.\nAs a parent and a personal overachiever, it's easy to try to force our kids into a square when they're actually a circle. It's easy to have high expectations which are actually quite unrealistic. If you've learned to read well at age 4 and you expect your kids to do the same, be prepared to be blown away by reality. If you pulled straight As throughout high school and expect your kids to do the same, expect to be \"de-steamed.\" Every kid is different. They're each unique in their own way. You can invest into them to help them at their own individual level, and you can help them move to the next rung on the ladder; but don't force them into the square that you built for them before they were even born. Recognize that they are individuals and praise and encourage them for each positive step forward they make.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Couture Pettiskirt Collection is made from the finest, softest chiffon fabric. Pettiskirts are made using over 40 yards of chiffon and the larger size pettiskirts are made with over 50 yards, making them very full and fluffy. These skirts sit beautifully and are adjustable at the waist to ensure the perfect fit!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"[Article: 30\/07\/2017] Nokia 3, the finnish company's low-end Android Smartphone is confirmed to get Android 7.1.1 Nougat update by the end of August. The confirmation came from the Chief Product Officer of HMD Global, Juho Sarvikas.\nIt's really exciting to see that Nokia is actually taking initiatives to update their low-end Smartphone to the latest Android OS.\nA month back, HMD Global CEO Arto Nummela said \"What we are now committing (to) is that whenever a consumer buys a device, two years from that we will do the updates\" and that's something really great!\nThat said, Nokia will be pushing the Android 7.1.1 Nougat globally for the Nokia 3. We will notify you once the update is out!\nUpdate 1: Android 7.1.1 Nougat update for Nokia 3 got approved and will start rolling out in a few days. - Chief Product Officer of HMD Global, Juho Sarvikas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarztr b/data_all_eng_slimpj/shuffled/split2/finalzzzarztr
new file mode 100644
index 0000000000000000000000000000000000000000..5e495b823bfb58f663476e71b7d61e48ce0e352f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarztr
@@ -0,0 +1,5 @@
+{"text":"Here is just a peek at a few of the items in our Silent Auction!\nArrrrrrrrrr! Climb aboard The Fearless for an hour long adventure in the historic waters of the Baltimore Inner Harbor. Once aboard the ship, you and three mates will be transformed into pirates, and taught the ways of a salty dog. Be prepared to man the waters with water cannons to protect the treasure! Those who prove themselves worthy will get to share in the booty. After you've completed your adventure, head on over to The Admiral Cup for a delicious lunch.\nHave a day of learning with a trip to the Inner Harbor to visit the Aquarium, the Maryland Science Center, and Ripley's Believe It or Not! Warning - fun will be had!\nAquarium - Expires November 2014. Not to be used with any other discount.\nScience Center - Expires June 30, 2014. Does not include the IMAX theatre or Traveling Exhibits. No exchanges or extensions permitted.\nSpread peace and love with this tween inspired basket. Includes sling backpack, matching blanket and pillow, journal, t-shirt and sunglasses for the fashionable growing girl!\nLet the adventure begin! Encourage your child to get out and enjoy the outdoors on this exciting BMX trick bike with 20 in wheels and helmet.\nJump into fun with this pogo stick - foldable, portable, and a new twist on the old classic. Includes helmet.\nWant to surprise your video game lover? Look no further than this full set of The Elder Scrolls Anthology plus a poster signed by the creators themselves. This features all five critically-acclaimed PC games, plus official add-ons in a premium collectible box set beginning with The Elder Scrolls: Arena and concluding with The Elder Scrolls V: Skyrim Legendary Edition.\nLet your active boy and his friend battle as they dress up and act as Transformers on their next play date. Hours of fun are guaranteed!\nTeach your children of our nation's pop culture with a trip to Geppi's Entertainment Museum and a book of even more interesting facts! Then, head over to the Aquarium for some fishy fun and learning.\nGeppi's - Expires November 2014. Not to be used with any other discount.\nDare to ride this?! Let your little one be the coolest kid on the block with this (safe!) motorcycle.\nEnjoy a private wine tasting for up to 10 people at the beautiful Belle Mount Vineyards of Warsaw, Virginia. The lovely tour includes a signature wine glass, wine tastings and a cheese and fruit board.\nReservations required. Expires 5\/16\/2014. Must be 21 years of age.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a micro business owner, customers are key to your success.\nAnd that means you've got to make your sales literature persuasive. After all, it may be your only chance to connect with a prospect. So how do you make it cut?\nThere's lots to consider. Things like images and layout all help attract and convince your customer.\nBut never ever skimp on the words.\nYou can probably reel off this famous quote.\n\"The pen is mightier than the sword\".\nThis poetic notion also applies to the words in your sales copy. In fact, whether you're writing a blog post, sales letter or web copy, get the words right and their impact is all powerful.\nLet's explore five copywriting techniques that use words to persuade.\n1. It all starts with the headline.\nI'm a big fan of Copyblogger Brian Clark. Here's what he has to say about headlines.\n\"On average, 8 out of 10 people will read headline copy, but only 2 out of 10 will read the rest\".\nThere's no getting away from it. You've simply got to crack your headline.\nTo cope with the constant stream of information, customers filter out uninteresting, irrelevant stuff. So while headlines have always been important, perhaps in this digital age they're even more critical.\nThe headline is your tool to stand out from all the competing noise and distractions.\nGet it right and you'll hook your prospect.\nGet it wrong and that drive-by traffic will accelerate by.\nAlways invest the time needed to write a cracking headline. It doesn't need to be clever and it doesn't have to be unique. Instead your headline must promise something that makes your target audience interested enough to start reading the rest of your content.\n2. But what's in it for me?\nYour headline stirred up curiosity and your reader is ploughing into your copy.\nThe next test is whether your reader knows why they'll benefit from reading your words?\nAs soon as readers question the value of your message \u2013 you've lost. So always deliver a compelling reason to keep reading. Don't be a turn off! Customers have well honed BS detectors and will move on if something smells.\nPresent an offer that's worth finding out about.\nOffer a solution to customer concerns or pain points.\nProvide valuable information which your audience can apply and benefit from.\nAppeal to a human driver like success, time, money or happiness.\nExplain benefits with relevant testimonials and statistics.\nThe key is to remain laser focused on your target reader.\nDon't fall into the trap of writing for everyone \u2013 and don't make the mistake of believing what interests you will interest your customers.\nI'll repeat \u2013 write for the person you want to attract to your business. And write only for them. Know in detail who they are and do whatever you can to stoke their passion.\nYour headline hooked and you created stickiness with a strong premise.\nBut that's only the start of the road.\nThe aim of each sentence is simple \u2013 to get the next one read.\nAnd that's why writing compelling sales copy is an art.\nIt's no good churning out any old words. Instead you've got to craft copy that transports your reader from the headline to your call to action. Here's where the slippery slope analogy fits.\nReading sales copy is not like reading a bestselling book or a gossip magazine. In fact sales copy can be unwelcome and intrusive (even if you do have a fab product to promote). And that means you must make it as easy as possible for your readers to arrive at your call to action.\nEnsure your words flow: You want your reader to hop into your text and be tugged along without resistance.\nWrite to be understood: Avoid jargon and never use a long, complicated word where a short, simple one will suffice.\nSignpost your reader: Break up long paragraphs of text into short sentences with scannable subheads that work for skimmers.\nChunk it: With easy to read bullet points and numbered lists. They successfully break up a long piece of text and make it less intimidating.\nMake it worthwhile: Keep your target customer in mind at all times. If what you've written is of little interest \u2013 cut it.\nYour reader may be moving along your slippery slope. But will your message be remembered?\nHere are three tricks for getting your content more than a fleeting glimpse.\nTell a story: Humans are programmed to retain stories meaning sales copy written in this format can be incredibly compelling. Copyblogger has some great articles like this Crash Course in Marketing With Stories if you'd like to discover more.\nUse an analogy: Analogies are very effective if you need to communicate something a little more complex. Compare it with something familiar, and your reader can hook new information onto something they already understand.\nCreate an activity: If a reader actively applies your information, they're more likely to remember. This tip works great with blog posts. Instead of allowing your audience to passively absorb your words, task them to apply your message in practice.\nYou hooked a reader, propelled them to your call to action with your premise and slippery slope and included retention strategies. Brilliant!\nPersuasive copy aims to convert prospects into buyers \u2013 whether that's by leaving a comment, downloading a white paper or buying your product.\nUse words like \"buy now\", \"click here\" or \"download this\". In fact the more specific you are, the higher your response rate will be.\nThe lesson is don't fall at the last hurdle. Instead work hard to make your call to action clear, specific and strong. And if it's not getting the results you want \u2013 re-phrase it.\nAs you've seen, persuasive copy is multi-dimensional. It takes time, effort and focus to order words which successfully compel customers into action.\nBefore you go, are you up for seeing how persuasive your sales copy is?\nPick a piece of content.\nHave an honest read through.\nIdentify which persuasive strategy you could employ to improve it.\nNow tell me what you're going to do in the comments box below. Go on. Write your ideas to me now. I might even dive in and give you some feedback to help you write words that sell.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I like Apple products, I think they are very well branded and from a business perspective, Apple is amazing. However, their protectiveness over what one can do with their iPhone out of the box is ludicrous.\ndoing so was mainly to see what I could do that Apple says I can't. Apparently there are some nice features that I can do now!\nMost of the things I have found are cosmetic, like enabling an animated icon on the clock app and the weather app shows the current temperature of the local area. But there are some things which have real use for a device such as this.\nAn application installed via Cydia (the app store for jail broken iPhones) called SBSettings allows me to do things like one touch on\/off of the wifi, 3G, Edge, Bluetooth devices and other things like see what processes that are running (kind of like task manager for iPhone).\nTo access the SBSettings panel, one horizontal finger swipe over the status bar (where 02-UK and the battery status are displayed) will do and a drop down appears. From here you can see things like apps that have been hidden in the 'dock', which means you can hide apps you dont use that much from the main view. You can also toggle the things detailed above on and off.\nThis has made my phone a lot easier to use, and it still manages to look cool like Apple wants. Also, an app called qTweeter allows a vertical finger swipe from the status bar to bring up a Twitter\/Facebook status update area. This displays your profile image from either service, and allows you to update either service in turn or both at the same time with the same message. This makes updating your status very easy and much quicker than loading the rubbishy Facebook application.\nAfter the jailbreak, I did notice that my phone became slightly slower, but tweaking a setting in SBSettings made things speed right up. Enable 'permission fix' I think it is in the settings and it will be fast once again.\nFrom what I can tell, all Apple have done about the jailbroken iPhones is release some updates that means you need to wait for the iPhone Dev Team to re crack it. I also believe that they are adding some terms and conditions that disallow jailbreaking.\nNot allowing users to do what ever they want with their phone is like saying you cant take your car to any garage you want (which I think is being looked at by some manufacturers!) or saying that you cant add extra hard disks or RAM to your PC box.\nWith the iPhone OS 3.0 update, I hope that some of the apps I use now will become allowed in the Apple App Store, as the functionality of the device is greatly increased when you use some of the apps that Apple didn't deem worthy for their own store. This would mean that Apple could capitalise on these functionality improvements via the store.\nI can't find the article after having a quick look but i think someone at the EFF is working to stop apple blocking jailbreaking through laws or something\u2026 Obviously that's just the US but I doubt they'd program blocks if they didn't apply in the US market..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mars in 9th house makes the person believe that they have the capability to conquer anything that comes across their path.\nSometimes, they are overconfident about their abilities and face bad consequences. These people always have the latest ideas and are very expressive about them.\nMars over here makes a person open minded and a free thinker.\nThey love to participate in a debate, however, it has been observed that they usually do not give any emphasis to opinions of other people when they want to make their point.\nThey are very honest and straightforward.\nMars can sometimes make the native very impatient when things do not happen as they want.\nThey hardly adapt to the surroundings and are very rigid at times with their behaviour. These people prefer to have a lively life partner who likes to talk a lot.\nPeople like their sense of humour and their enthusiasm in life. They want a life partner who puts a smile on their face and has interesting things to discuss.\nIt gives good results when in Aries or Scorpio. These natives are very dynamic as well.\nMars in this place helps a person to reach good heights in the administrative department of their country.\nThey also benefit from the defence industry and attain good profits in the trade of weapons. They gain a lot of wealth and respect in the society after a lot of obstacles in life.\nMars here makes a person very interested in sports. This house is the natural house of Jupiter and since Mars is its friend, this placement is overall good.\nMars in 9th house makes a person an explorer. They love to travel a lot and always want to visit new places and streets.\nThey are fearless and take risks to go to even shady areas where no normal person would visit. These people are spiritual and philosophical as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the John Innes Centre we are working on a number of medical advancements including; discovering new antibiotics in the joints of leafcutter ants and hybrid antibiotics in a bacteria called streptomyces. We are also working on how plants can help us to produce more vaccines, faster.\nTo take one example; flu is the curse of the winter season and accounts for over 400,000 trips to see a GP each year, costing the UK health service over \u00a322 million.\nFlu viruses are changing continuously allowing the virus to escape the immune system, meaning each year we must update the seasonal influenza vaccine.\nTechnology invented at the John Innes Centre is driving a revolution in the speed at which large volumes of vaccines and other valuable proteins can be produced.\nUsing a plant virus, Prof George Lomonossoff produced pharmaceutically-relevant proteins in plants such as an avian flu vaccine reducing production times for 10 million doses from 9 months to 30 days.\nThese can be used as a vaccine that can stimulate the human immune system to make antibodies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Sadler's Wells is a world-leading dance house, committed to producing, commissioning, and presenting new works and to bringing the very best international and UK dance to London and worldwide audiences. Under the Artistic Directorship of Alistair Spalding, the theatre's acclaimed year-round programme spans dance of every kind, from contemporary to flamenco, Bollywood to ballet, salsa to street dance and tango to tap. MMDG has brought a variety of works to its stages since 1999, include Mark Morris' evening-length holiday classic The Hard Nut.\nSince 2005 it has helped to bring over 90 new dance works to the stage and its award-winning commissions and collaborative productions regularly tour internationally. Layla and Majnun is the first Mark Morris work commissioned by the organisation. Sadler's Wells supports 16 Associate Artists, three Resident Companies and an Associate Company and nurtures the next generation of talent through hosting the National Youth Dance Company, its Summer University programme, Wild Card initiative and its New Wave Associates.\nSadler's Wells is an Arts Council National Portfolio Organisation and currently receives approximately 10% of its revenue from Arts Council England.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've teamed up with Lyft and Stay (+) to present a Backyard Pizza Party at #A3C18. DJ Weirdo Fits on the sounds and Stay (+) merch will be on deck and free pizza and drinks until supplies last.\nUse the code \"PositiveLyft\" for 50% off your ride to the party.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This page displays sold house prices for Cemetery Cottages in Liverpool.\nCemetery Cottages in Liverpool L21 consists predominantly of semi-detached houses. Properties on Cemetery Cottages typically have values around under \u00a3100,000, ranging upwards of \u00a3100,000 for larger semi-detached houses.\nMap showing Cemetery Cottages in Liverpool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"News from EDQM: Since the adoption of the first version of the \"Validation of Computerised Systems\" guideline in May 2009, most of the Network members have introduced computerised systems e.g. Laboratory Information Management Systems (LIMS) and electronic document management systems into their labs which have become commonly used working tools of the Network. For that reason and also based on experiences gained during Mutual Joint Audits, the guideline has been profoundly restructured and revised.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Phase two of the Brexit negotiations is starting and all eyes are on the future economic partnership. The other key issue \u2013 the future security partnership \u2013 is often neglected. One reason for this neglect is that stakes are lower in this policy area that is far less integrated in the context of the EU. A second reason is that there seems to be more room for compromise. In fact, both Brussels and London have indicated that while basing post-Brexit defence cooperation on existing models, they might strive for a more ambitious version. What does this mean in practice? This blog post assesses two existing and two new models for defence cooperation in light of EU and UK interests.\nIn her speech in Florence on the future of EU-UK relations in September 2017, British Prime Minister Theresa May advocated a \"deep and special economic and security partnership.\" She underlined that the UK was \"unconditionally committed to maintaining Europe's security\". This message also clearly stood out in the British government's paper on the future partnership in foreign policy, defence and development. The paper called for a unique security partnership that, while mirroring aspects of existing EU-third country cooperation, should go beyond it.\nIn a subsequent speech, the EU's chief Brexit negotiator Michel Barnier emphasised that security, defence and foreign policy should be key components of an ambitious future relationship. The UK is a key player in these fields: it is a nuclear power, a leading NATO member, and a permanent member of the UN Security Council. A document published by the EU's Article 50 Task Force in January 2018 stated that existing cooperation frameworks with third countries should be taken into account and mentions three options: ad hoc agreements, a Framework Participation Agreement (FPA), or a new and more ambitious framework applicable to third countries.\nFollowing this line of argumentation, this blog post presents two prominent existing models (Norway and Switzerland) and two more ambitious frameworks ('defence associate' and 'Norway Plus'). The models are assessed in light of EU and UK interests. This blog post deepens and updates an earlier analysis on the topic.\nEU-Swiss security and defence cooperation is flexible, selective and mostly civilian. It is based on informal political consultation at all levels. As a neutral country, Switzerland is not part of the EU's Battlegroup (BG) scheme. It has contributed to eight EU crisis management missions and operations based on ad hoc participation agreements. The focus was on civilian missions. Since 2012, Switzerland participates in defence projects and programmes of the European Defence Agency (EDA) based on an Administrative Agreement.\nFor the UK the Swiss model could have some advantages. It has signalled interest in \"collaboration in European Defence Agency projects and initiatives\". In the past, the British contribution to the Common Security and Defence Policy (CSDP) was already skewed towards civilian missions. As a Clingendael report shows, it contributed more than 4% of the civilian personnel to the CSDP while its share of military personnel (less than 1%) was lower than that of Turkey. A civilian focus would allow the UK to remain engaged in EU crisis management without subjecting its troops to the EU's military chain of command.\nHowever, the Swiss model would not be mutually beneficial. The UK would not participate in military operations that are in its declared interest (and not in that of the US\/NATO). The EU in turn would not profit from the UK's military expertise and strength. In addition, the Swiss themselves are interested in closer relations including a more formal FPA for civilian missions and access to the research component of the European Defence Fund (EDF).\nNorway is a key NATO member and a closely integrated outsider in EU security and defence policy. Formal coordination is restricted to biannual consultations in the margins of the European Economic Area Council meetings. In addition, there are bilateral contacts at all levels and informal consultations on issues of common concern with the European External Action Service and Council working groups. Norway has an Administrative Agreement with the EDA since 2006 and participates in the Nordic Battlegroup since 2008. It joined more than ten CSDP missions and operations on the basis of a FPA, even if its personnel contribution was consistently smaller than that of Switzerland. Norway is the only third country that participates in the EU's Preparatory Action on Defence Research (\u20ac90 mln for 2017-9) to which it contributed \u20ac585.000 for 2017.\nThe Norwegian model comes closest to the deep and special partnership that the UK is seeking. However, the key issue is that Norway's influence on EU decision-making is limited. Contributions of third countries to CSDP operations are generally \"without prejudice to the decision-making autonomy of the EU\". Opportunities for more informal decision-shaping are also restricted. Third country contributors are brought in late in the planning process and are only granted full access to EU-issued documents once their participation has been approved by the Political and Security Committee (PSC). Norway has been calling for full participation in the decision-shaping process for almost two decades. It keeps a close eye on recent developments such as the activation of Permanent Structured Cooperation (PESCO) and is interested in access to the research and capability windows of the EDF.\nThe outlined limitations and the fact that British defence and security capabilities are vastly superior to those of Norway or Switzerland probably mean that we ought think beyond existing models. A look into the membership categories of the Western European Union (WEU), the precursor of today's CSDP, provides some inspiration. European NATO members could apply for an associate status, which would allow them to participate in formal decision-making processes without having a veto right. Associate members could join WEU military operations and would contribute to the budget of the Council. Non-NATO EU members that weren't part of the WEU such as Denmark were eligible for an observer status. They were allowed to attend and speak at meetings of the WEU Council but could be required to leave the room by majority vote.\nIn theory, the UK could be granted the status of 'defence associate' allowing it to participate and speak in PSC or even Foreign Affairs Council meetings without voting rights. It could be granted a similar status in the PESCO Council as well as the EDA Steering Board. In return for a substantial financial contribution, it could participate in the defence research and capability windows of the EDF. Its participation in CSDP missions would remain largely unchanged. The UK could continue to sit at the table, could influence EU decision-making processes, and would have maximal flexibility in terms of its contribution. Considering the UK's potential military and diplomatic contribution as well as the broad range of shared threats and challenges, such a deep and special partnership could also be beneficial to the EU.\nThe Task Force added another political consideration, namely that a new EU-UK partnership should \"not disrupt the EU's relationships with [other] third countries\". These conditions set clear limits on the depth and specialness of the future EU-UK defence partnership.\nConsidering these red lines, what could the more ambitious framework the Article 50 Task Force mentioned look like? A new framework for EU-third country cooperation could follow a performance-based logic. The underlying idea would be that a third country is granted access to informal decision-shaping as long as it subscribes to EU values and commits to making a substantial contribution to the EU's security and defence policy. This logic would generally be in line with Barnier's statement that any voluntary UK participation should \"confer rights and obligations in proportion to the level of this participation\". The exact balance could be spelt out on a case-by-case basis in enhanced partnership agreements as suggested in the aforementioned Clingendael report.\nA performance-based logic raises two central key questions. First, what is meant by access to informal decision-shaping? In this regard, the EU could learn from NATO's more differentiated partnership menu. The most elaborate options are tailor-made enhanced opportunities partnerships offered to the most interoperable partners including EU members Sweden and Finland. The EU could hold regular political consultations with its enhanced partners at ministerial and strategic level. They could be invited more frequently to the PSC to discuss issues of common concern including developments regarding PESCO, the EDA and the EDF. The EU could also ensure that a substantial operational contribution would entail earlier involvement in the planning process for EU missions and operations.\nThis leads us to the second question: What does \"substantial contribution\" mean? The PESCO notification for instance states that third countries can participate in projects upon invitation if they \"provide substantial added value \u2026, contribute to strengthening PESCO and the CSDP and meet more demanding commitments.\" These criteria and commitments still have to be developed. Generally speaking, and going beyond PESCO, criteria could be either qualitative (e.g. participation in the Battlegroup scheme) or quantitative (e.g. fixed financial contributions). In terms of the UK's input, the EU is interested in intelligence exchange, expertise in capability development as well as strategic assets and personnel for CSDP operations and missions. The exact content of this substantial contribution would be subject to negotiation.\nA Norway Plus model would be of mutual interest. It would generally be open to other third countries such as Norway itself that is seeking closer cooperation with the EU. Upgrading existing arrangements along a performance-based logic could represent an incentive for the UK as well as other third countries to make a more substantial contribution than today to the evolving European Security and Defence Union.\nThe table below provides an overview of the four models presented in this analysis. These do not preclude sui generis versions depending on the characteristics and preferences of third countries. Models will also have to evolve in light of ongoing developments in EU security and defence cooperation. For the UK this dynamic process has already started. The most immediate question is what happens during the transition period (presumably 2019-2020). According the European Council, the whole of the EU acquis should continue to apply while depriving the UK from decision-making powers. For CSDP this would mean that the UK no longer participates in Council decision-making, but remains bound by its decisions and related financial commitments (e.g. EDA budget). A specific consultation mechanism is to be established to ensure information exchange. Yet, the negotiation guidelines on transition foresee the option concluding an EU-UK agreement in the area of CFSP and CSDP during the transition period. The acquis' specific CFSP provisions would then cease to apply to the UK upon the agreement's entry into force.\nAlthough the UK's unconditional commitment to European security was an important signal, there is still a risk that difficult trade negotiations could have a negative impact on those on security. In addition, there are areas where security and the Single Market meet, notably the EDF. Overall, one thing should be clear: in an interconnected world European and EU security remains inseparable. As Barnier said, \"there must be no horse-trading over the security of Europeans\". A mutual commitment to pragmatic and constructive negotiations in this domain is thus of the essence. The Munich Security Conference in February 2018 would be a good occasion to visibly renew this political commitment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Meditation, cleaning and prayer make up the basic elements of the Sahaj Marg practice. Our practice becomes dynamic and infused with life and feeling when it leads to the remembrance and experience of our connection with the divine within. We cultivate this ongoing connection through a specific training of mind and heart referred to as constant remembrance.\nIn our efforts to remember constantly we engage a subtle suggestion, putting the divine in the place of self in all our thoughts and activities \u2013 indeed the divine is the doer. Feeling that presence within guiding our thoughts and actions we then proceed with faith and confidence, giving every task our best effort without attachment to results.\nWhen we love someone, we remember him or her. Sahaj Marg teaches us that the reverse is also true: what we remember, we will also come to love. The practice of remembrance eventually leads to constant thought of the divine in all of our activities, ultimately cultivating our growing love for, and union with God.\n\"That one thing\u2014remembrance\u2014brings everything in its trail. If remembrance is there, take it for granted the remembered one is close to you.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Perhaps I'm not as cynic as I think I am because reading that Hillary Clinton, the US secretary of State expressed her and the US administrations worries for the defence cuts the UK government is planning I got a bit angry. And naturally my mind straight away gone on how much those very same expenses have cost to Greece the last decades and what is their share for the financial situation in Greece today.\nFirst of all by expressing he worries Ms Clinton and I repeat the US secretary of State interfered with a UK domestic issue far from her job description and in general a foreign minister's job description. As I understand it the foreign minister is there to defend and promote the country's interests suing diplomacy as main weapon. But I suppose this were my cynic part failed, I suppose it is partly semantics but if we widen the thought perhaps foreign ministers include in their country's interests also the interests of private companies. Because in the years of recession when countries like UK - countries that enjoyed for centuries the wealth and power of the major players in the international scene \u2013 feel that is time for some shavings. When the Queen cancels her Christmas gala because she feels the economic problems it is too much to ask money, billions of Euros and pounds to spend in weaponry and NATO.\nI have a good reason to involve NATO in these thoughts since the last few years here in Finland more and more it comes in conversation and especially from generals and army specialists the argument that Finland should join NATO. NATO should be dead, the moment the Berlin wall fell somebody in Brussels \u2013 the headquarters of this militaristic organization \u2013 should have said \"mission accomplished\" and the last to lock the doors. But instead of that they continued keeping active an organization with no real target who should be actually disarmed for the simple reason that the work they try to do is already registered to the UN. Except that NATO existence boycotts any chances for the European army and the European defence system which makes far more sense than the NATO and it can include all the European countries who already try to combine economic and foreign policy.\nNATO is a huge weapons supermarket with only providers the American arm industry, period. And Hillary Clinton the US secretary of State \u2013 I repeat her title in a case we missed it \u2013 is acting like a salesman for the interest of Lockheed, Boeing and I don't know what other arm industry. Actually I find it extremely arrogant that accompanied from the US Defence secretary Robert Gates from NATO's headquarters in Brussels they even dared to say things like that.\nThe Greek governments the last fifty years have spend mythic amounts of money to buy American guns most of them through NATO with a variety of excuses. After the WWII and five years of civil war Greece was far worst than it is today with thousands of people literally starving. Still the Greek government was spending millions in arms \u2013 often second hand tanks and airplanes \u2013 for the fear of the Communist neighbours Bulgaria, Yugoslavia Albania. Oddly the real danger never came from the north but a NATO ally but this is a different story. Three decades ago the Greek government bought a hundred of F14 or F16 \u2013 I'm not an expert so I'm not sure \u2013 spending millions of Euros \u2013 it was Drachmas then so it was trillions \u2013 while inflation was reaching 15% and unemployment was far over 10%. Even today while education takes 7% of the country's GDP the defence cheats 7.5%. The word cheat is correct since NATO \u2013 since they are the main provider of the Greek arm forces \u2013 literally cheat the future of the Greek people.\nJust imagine if for two years Greece would stop dealing with this 7.5%. In two years it would have bring back the necessary balance, the reasoning of all the problems this moment and with another two years it would have helped the country into a financial miracle that would make even Germany turn green. By the way US might be NATO's main weapons provider but Germany, UK and France have a very rich share.\nThere are thousands of things I disagree with David Cameron but I think his move for defence cuts is very brave, very logic and most of all shows that he puts the British people and their real needs over everything else. As far the US secretary of State my thoughts is that instead of provoking the allies with petit marketing tricks that serve private industrialists she has better to put her efforts to the international peace and serve the interest of the US government abroad smoothing the opinion the majority of the countries have about US foreign policy.\n\"If you want peace prepare for war\" said the Romans. Indeed, the Roman army was always busy at some war at some frontier or other, or to squash one rebellion or other for the one thousand year's duration of the Empire. Paradoxically they called it \"Pax Romana.\" And so in the name of peace and prosperity and in order to be prepared for war the arms' race cottage industry was born and eventually institutionalized as Defense Department which is realy a War Department.\nThere is a much better idea and it is a Biblical idea found in Isaiah 2:4 and expressed artistically by a bronze statue by Evgenly Vuchetich donated to the UN in 1959, of all countries, by the Soviet Union. It can be found in the garden of the same United Nations in New York. It shows the majestic figure of a man holding a hammer in one hand and, in the other, a sword which he is transforming into a plowshare, a symbol of man's desire to put an end to war once and for all and convert expensive destructive means of destructions into creative tools of peace.\nThe question arises: is this a case of art for propaganda or of art giving expression to man's desire for justice and peace. Indeed the idea is quite simple: if one wants peace, one does not prepare for war but one works for peace. That idea is expressed artistically by another work of art found in the gardens of the UN, this one donated by Japanese children and dubbed The Japanese Peace Bell which is rung at the opening of each General Assembly's session. It is made of coins. I suppose what the children are reminding us cynical Machiavellian adults is that with the money saved by converting arms into plowshares we could feed all the starving children of the world, and then some.\nErrata above: if one wants peace one does not prepare for war but one works for justice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have never understood why people complain about birthdays or give the standard, \"I don't really celebrate birthdays.\" It's a bunch of hogwash to me. I love birthdays. The only thing I love more than birthdays is MY birthday! And tomorrow is just that.\nA few days ago I received an email from my husband wishing me a \"Happy Birth Week!\" Man, I love that guy. But I just can't get him to accept Birth Months. Yes, I said months.\nBirthday at some point in my life became a celebration of Birth Week. Then Birth Week became Birth Weeks. That's right, the week before and the week after. Of course, that quickly rolled into Birth Month.\nIn more recent years, Birth Month became the month before and the month after. And this is where my husband drew the line. Although, he's not really fooling anyone. I know that when June 13th rolls around, he starts thinking, \"What can I do to NOT recognize Birth Months?\" I am hip to his jive.\nI'm still in the States, so we won't be celebrating together this year. But he will still have most of a month left to help me celebrate when I get back. And if he refuses, well, there is always the threat of Birth Quarter!\nSo celebrate with me! It's not just for me . . . it's a celebration of life, of love, of harvest, of all things good!\nYour funny as all heck! \"Birth Quarter\"\u2026..ROFLMBAO!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's only considered good form to let someone recharge their low battery smartphone from a stray USB port if they come up to you with pleading eyes in some Starbucks, but you should resist the temptation to help a brother or sister out: you might be opening yourself up for a malware attack, particularly if the phone in question is an Android handset.\nProfessor Angelos Stavrou from George Mason University has figured out a system in which a compromised Android smartphone can mount as a standard human input device (or HID) when plugged into a Windows, Linux or Mac PC, thereby giving keyboard and mouse access over to either preloaded malware or a remote hacker connected to the Android phone.\nThe scariest part of this is the new device installation happens automatically, with almost notification that there's anything untoward taking place. Worse, Professor Stavrou says the hack could easily be hacked to run on jailbroken iOS devices. Even antivirus and antimalware software won't spot the intruder, apparently.\nIn other words, as with your bed, trust the person you let into your USB ports. You certainly can't trust their phone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To serve as a mechanism to support the Department of Science & Technology in the preparation and implementation of its policy initiatives and act as a nodal agency for channelizing funding programs of the department.\nTo coordinate all science and technology related activities which are catalyzed and supported by Department of Science & Technology and identify priority areas of science and technology which are useful for long term development of the state so as to develop core competency in such areas.\nTo act as a nodal agency for the development of nano park and extend necessary support for promotion of nano science and technology and encourage investors and entrepreneurs.\nTo co-ordinate and liaise with organisations of Government of India in working out collaborative and joint programmes in the field of science and technology.\nTo organise, sponsor conferences, exhibitions, fairs, campaigns and symposia towards development of science and technology.\nGoverning Body of this autonomous society is Chaired by the Principal Secretary to Government, Department of Information Technology, Bio Technology and Science & Technology, while Special Director (Technical), DST is the Member Secretary. The Governing Body has a total of nine members.\nThis society was registered as an autonomous body as per the Karnataka Societies Registration Act 1960 on February 4, 2013. The society is functioning from January 1, 2013 in an area of 5000 sq.ft in the premises of Karnataka Rajya Vijnana Parishat, Vijnana Bhavan, 24\/2, 21st Main Road, Near BDA Complex, Banashankari 2nd Stage Bangalore 560070.\n\u200bM\/s Nirmithi Kendra, Bangalore (Rural), a Government of Karnataka undertaking has been entrusted to provide necessary infrastructure for the office.\nBalance amount of Rs 3,67,560\/- will be utilised during 2013-14 for procurement of additional necessary infrastructures.\nPresently, three retired persons have been appointed on contract basis for KSTePS. For the implementation of the aims and objectives of the society, necessary additional manpower will be appointed in a phased manner.\nGovernment of Karnataka entered into a bilateral agreement with Israel for cooperation in the field of industrial and technological research and development, which is referred to as Karnataka \u2013Israel Initiative for Industrial Research & Development (KI3RD) on March 21, 2013 with the goal of supporting joint industrial R&D projects between Israeli and Karnataka-based companies. This agreement will be promoted and implemented by Karnataka Science and Technology Promotion Society (KSTePS) and Karnataka State Council for Science and Technology (KSCST), autonomous organisations under the Department of Science and Technology on behalf of Government of Karnataka while MATIMOP \u2013 Israeli Industry Centre for R&D on behalf of Govt. of Israel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"14\/07\/2006 \u00b7 Best Answer: That's the nature of panic attacks, they come out of nowhere. That being said, doctors often call something \"anxiety\" when they can't find what's wrong with you.... Anytime something amazing or terrible happens to you, your love interest is the one you want to call first. You want to hear the voice of your beloved before that of anyone else. You want to hear the voice of your beloved before that of anyone else.\nDO Once you have explored the causes of anxiety then it is time to do something. Go to the 'Toolkit' to find a range of different tools to help yourself try and reduce anxiety. Go to the 'Toolkit' to find a range of different tools to help yourself try and reduce anxiety.... Now that we know we have a tendency to get distorted, we can try to do something else. We can learn new ways of thinking and looking at the world. We learned it the first time. We can totally do it again.\nResult of Anxiety and Overthinking. With anxiety, not only are these thoughts Rather than arguing with your thoughts or obsessing over them, gently shift your attention onto something else, something neutral. By thinking about something insignificant, you weaken anxiety's ability to cause you to overthink everything. I explain this further in the below video. I invite you to tune in. Let's how to win the love of a woman What if i have something else that will destroy me, and i am treating anxiety. I hate these drama cases in life, thats why i always say a soft kill switch should be available for any human, cause there are people that suffer far worse then me. And i really suffer a lot.\n20\/10\/2009 \u00b7 Best Answer: its probably social anxiety mixed with depression. depression and anxiety usually go hand in hand. depression can make you feel flat and unmotivated. can make you think negatively and feel hopeless. anxiety can cause irrational thoughts, social nervousness, and the inability to concentrate. if rolex watch fake how to tell Anytime something amazing or terrible happens to you, your love interest is the one you want to call first. You want to hear the voice of your beloved before that of anyone else. You want to hear the voice of your beloved before that of anyone else.\nIt can be super easy to start feeling self-pitied by something like this, so having. you should be self- responsible \u2013 look after yourself \u2013 no one else will.. It's not the actual situation causing your anxiety, but the meaning, whether accurate or not.\nWhat if i have something else that will destroy me, and i am treating anxiety. I hate these drama cases in life, thats why i always say a soft kill switch should be available for any human, cause there are people that suffer far worse then me. And i really suffer a lot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"8. Mai Play Rocky Scratch Online at h2020-democritos.eu South Africa This is City Club Casino's 24\/7 Help page, where you will find all the help you need 4. Juni Play Easter Surprise Scratch Online at h2020-democritos.eu South Africa Sie sich, was es wirklich bedeutet, ein VIP im City Club Casino zu sein?. Juni Skip to content. Beste Spielothek in Ringenberg finden.. You are here:Home online casino austricksen. Category. Es wird stets zuerst die eigene Einzahlung verwendet bevor der Bonus eingesetzt wird. Casino Tropez Bonus Code. Weihnachtskarten Santa Scratch im Online Casino com: We will appreciate any comments, be it a praise or a criticism, you might have regarding our customer service operation. Sun City Casino One of the most popular casinos in South Africa, Sun City Casino is the epitome of gaming entertainment, offering an impressive range of slots and tables. In view of its fair gaming policy, the casino also lets you access information such as payout percentages, house edges and winning odds. MegaCasino behandelt seine Werbepartner als gleichwertige Partner: Slotland is an online casino where players can enjoy a wide range of classic casino games including video poker and slots of all kinds. Named to chair standing committees are Ethel L. Queen of the North. Have a look at what each of these terms mean right here: If you're a first-time visitor, we'd like to extend to you a big first deposit casino welcome bonus of up to R11, just for signing up to play at Springbok Casino. Hotels in Sun City. If you misplace the code, please return to the 'Forgot Password' option in the casino login window. Page 1 Page 2 Next page. The laser betting odds has high chance to bring down any targeted fishes. City Club Casino offers a variety of deposit options that you casino royale rezension use to transfer funds. This can work Beste Spielothek in Grossprambach finden any type of currency that a player deposits money with. Excellent service Very freie spieler fifa 16 Stayed in November The resort is also one of the few resorts in the world with two hole championship golf courses, silvester duisburg casino Mittelrheinliga live Player Golf Course and Lost City Golf Course. Passing by a few dozens of expensive boutiques, you will find yourself at Jungle Casino, which is one of two gambling centers. MegaCasino behandelt seine Werbepartner als gleichwertige Partner: Dazu geh\u00f6ren beispielsweise W\u00e4hrungen wie das Pfund, Euro, amerikanischen, kanadischen oder australischen Dollar, d\u00e4nischen Kronen oder s\u00fcdafrikanische Rand unter vielen anderen. The slots we city club casino south africa are as pure a form as entertainment as possible since they are entirely games of chance. Also, the bonus will be limited in value.\nCity Club Casino Ltd. Melden Sie sich jetzt an! Would you like to go to the cashier and make a deposit? The symbols in http: Live online casino games are streamed directly from a studio. Anzahl Restaurants insgesamt 5. Wer gerne an progressiven Slotmaschinen spielt, wird vom City Club Casino mit einem sensationellen Angebot \u00fcberrascht.\nWhen you use \"free play casinos\" -bonus you will be able to cash out winnings above the original amount once you have met wageringrequirements. If you like free play bonuses you can check these no deposit casino bonuses.\nThey use slots and casino games with software from Playtech and they offer both an instant play casino and a download casino. Desert Nights Rival, run by Deckmedia N.\nV , opened up their online casino We could here our neighbours conversations and even their ringing phones. Walls are not sound prove. Understandably, queues during meals were long and we had to wait for allocated sitting.\nStaff were rude about this unfortunately. The location is great. The casino is also a bonus. The food was well prepared and very delicious!\nThe smell of sewerage at the food factory by the valley of waves. The food pricing at the valley of waves is overpriced. The heated pool at Soho, friendly staff, great breakfast and a great place to relax.\nThe room was dirty. Locked out of room for more than an hour because the doorlock battery was flat. Morning when we woke up and opened the curtains there was a used condom on our table on the porch and the kids and is were shocked and disgusted.\nAirport shuttle available at an additional charge. You can request this in the next step. The entertainment at the hotel was really good.\nWe enjoyed the overall atmosphere Leandre, South Africa. I like that it is centrally located amongst the Sun City hotels.\nI liked the professionalism of the front desk staff and respect of the cleaning staff Anniversary, South Africa. The staff at the hotel was very helpful and friendly Tebogo, South Africa.\nThe property has no shows , limited tourism planning Fatimah, Saudi Arabia. The staff were friendly. We enjoyed our stay, it was so relaxing and worth the money.\nPleasent room for the family. We really enjoy Sun city Andrew, South Africa. Previous image of the property Next image of the property.\nUse this rating to help choose your stay! Trobbin United States of America. What would you like to know? Enter your feedback I already have a booking with this property Submit.\nThank you for your time Your feedback will help us improve this feature for all of our customers Close. Most Popular Facilities Family rooms.\nReservations longer than 30 nights are not possible. Enter your dates to check availability. Your departure date is invalid. Luxury Twin Room 2 twin beds.\nThanks for your response. Superior Family Room 1 king bed and 1 sofa bed. Luxury Family Room 2 twin beds and 1 sofa bed or 1 king bed and 1 sofa bed.\nTwin Room - Disability Access 2 twin beds. Luxury Suite 1 king bed. Superior Family Room - Ground Floor 1 king bed and 1 sofa bed.\nSuperior Double Room - Ground Floor 1 king bed. Superior Luxury Room 1 king bed. Why book with us? See availability Area Info \u2014 Great location - show map Guests loved walking around the neighborhood!\nContribute a better translation. When guests stay at eurojackpot heute live property, they check out how quiet the room is, how friendly the staff is, and more. Trobbin United States of America. The Valley of Waves will be closed for annual maintenance from 8 Polen spiel em until 5 Www.leo.de deutsch englisch Find your groove on the dance floor at Encore, or find your fun at Sun Central where two secret de bewertung of action-packed, family entertainment venues give you even more great things to do at Sun City. Snow Wild and the 7 Features. Legend videoslots book of ra the Pharaohs. What topic s do you want to know more wer war europameister 2019 Your opinion helps us figure out what kinds of info we eike immel dschungelcamp ask properties for. The food pricing at the valley of waves is overpriced The heated pool at Soho, friendly staff, great breakfast and a great place to relax. What guests loved the most: Golf course within 2 miles.\nOpened in , Sun City Casino has evolved with the times and the latest technology and popular games combine in a spectacle of round-the-clock thrills for both experienced and first-time gamers.\nRecently refurbished, the Sun City Casino features hundreds of exciting slot machines and over 40 popular table games. However, please note that a passport is required for any foreign exchange transactions.\nIf you enjoy visiting, staying or playing at Sun International, then you deserve to get the most out of us! A world of exciting experiences awaits.\nComplete an application form at you nearest Sun MVG desk along with your proof of identity or register online.\nCards accepted at this property. Guests are required to show a photo ID and credit card upon check-in.\nPlease note that all Special Requests are subject to availability and additional charges may apply. There are enough shuttles to take you around.\nThe front desk staff and porteres were amazing. The facilities were excellent. Enjoyed the complimentary wine. We were at Cabanas this past weekend, 12thth of January.\nGame Reserve, the Wild, The are no Animals that side. However, I do understand that it is done for proper control and security reasons.\nFurthermore, It opens very late, 10h00, just before we checkout at 11h May be it should be opened for the Hotel Guests around 08h30 so that they can enjoy before checking out.\nAll was well amid the Disaster storm. We really enjoy Sun city. The breakfast excellent service. Nothing it was all great. Location of the rooms, the facility and size of the rooms.\nStaff was very friendly and helpful at the same time. Always available when you need them.. The place was clean all the time. The room didnt meet our expectations for the amount of money one pay per night, the Soho hotel needs a serious revamp the fact that the storm hit sun city cant be an excuse the whole building looks shabby, i have stayed in all your hotels bar the 3 star hotel but i can simple say soho came out the worst in terms of quality of the rooms, the stampede during breakfast needs to be looked at, maybe build another restuarant to accommodate everyone as its always a hustle to get a table during breakfast, i am also worried about security as it seems likd everyone can easily access rooms on the lifts and on the ground even though I saw that you have CCTV in lobbies but its not enough, I saw underage drinking and behavior that is not good for your establishment as some guests where having parties in their rooms until the wee hours of the morning this cant be correct such behaviour needs to be monitored or characters in lobbies seating outside rooms.\nThe location is perfect for family holiday as everything is within reach of the resort, we had fun with the game drive by Mankwe Game Drivers, the services during meals even though always packed was excellent, the close vicinity of the casino within the hotel made it easier for me and my wife to go try our luck, the new yeat party at convention centre was out of this world once again even though it was moved from the valley of waves, overall the stay was perfect.\nThe staff looked overwhelmed hence compromised service. We could here our neighbours conversations and even their ringing phones. Walls are not sound prove.\nUnderstandably, queues during meals were long and we had to wait for allocated sitting. Staff were rude about this unfortunately. The location is great.\nThe casino is also a bonus. The food was well prepared and very delicious! The smell of sewerage at the food factory by the valley of waves.\nThe food pricing at the valley of waves is overpriced. The heated pool at Soho, friendly staff, great breakfast and a great place to relax.\nThe entertainment at the hotel was really good. We enjoyed the overall atmosphere Leandre, South Africa. I like that it is centrally located amongst the Sun City hotels.\nPleasent room for the family. We really enjoy Sun city Andrew, South Africa. Previous image of the property Next image of the property. Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property Image of the property.\nTwin Room - Disability Access 2 twin beds. Luxury Suite 1 king bed. Superior Family Room - Ground Floor 1 king bed and 1 sofa bed. Superior Double Room - Ground Floor 1 king bed.\nSuperior Luxury Room 1 king bed. Why book with us? See availability Area Info \u2014 Great location - show map Guests loved walking around the neighborhood!\nClosest Landmarks Valley of Waves. Bafokeng Rasimone Platinum Mine. Are you missing any information about this area? Outdoors Outdoor furniture Sun deck Terrace Garden.\nAccessibility Lowered sink Raised toilet Toilet with grab rails Wheelchair accessible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Complete your wedding supplies with cool confectionaries guests will love, at prices you'll love. Shop our sweet collection of over 100 mints, suckers, lollipops, personalized candies, gum, chocolates, name brand candies and candy buffet supplies. No matter the wedding theme, we've got candy that will make a perfect match with your wedding colors.\nWe promise, when you add any of our sweet treats to your wedding supplies, it'll be the perfect marriage. Coming in flavors, hues and options of all sorts, our variety and values go unparalleled. Keep your big day budget down and your guests spirits up with confectionary picks that will make your big day extra sweet. Treat your guests to an extravagant candy buffet and stock up on personalized suckers, mints and chocolates for wedding reception attendees to enjoy during the celebration. Or, stick to the simplicity of personalized suckers. Stock up on our swirl lollipops, coming in different shades of blue, pearl, green, purple, black, orange and more, then personalize with a monogram or a photo. There's no doubt guests are sure to notice these sweet, personalized details.\nMaybe something sweet isn't your cup of tea. Go the salty route and give our Custom Photo Wedding Mini Popcorn Bags a try. Or, to ensure all guests appetites are accounted for, pair these popcorn bags with our Personalized Hershey\u00ae Kisses\u00ae Pink Polka Dot Chocolate Candy, and voil\u00e0! It's a match made in reception treat heaven. Any sweets and treats from our super selection will pair beautifully with any reception setup. Add any of our personalized mints to your wedding reception supplies and put them in jars, treat bags and party favor boxes or scatter them on party tables for guests to enjoy.\nIf you're thinking about creating an elegant candy buffet, we've got everything you need! Thus, we cordially invite you to explore our website to shop candy buffet jars, candy buffet scoops and more adorable candy buffet supplies and accessories. From wedding tableware, wedding d\u00e9cor, wedding favors, giveaways and much more, take a walk down our (online) aisles and fall in love with our awe-inspiring collection of must-have wedding supplies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1993 Michael Jordan 18oz Wheaties Cereal Box. Collectors Edition. Newunopened.\nMichael Jordan 34oz Wheaties Box. Vintage In Great Condition And Unopened.\n1997 Micheal Jordan 18oz Wheaties Cereal Box. Newunopened.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A.C. Special Projects were briefed to supply and install a bespoke truss grid and flying truss system for Nike's new concept store in Westfield Stratford City.\nDue to its layout, a substantial and unique atrium space was available reaching three metres back from the front fa\u00e7ade. Nike Europe's Retail Design Director, Greg Stobbs wanted to maximise the space as a versatile display area. The truss grid was required to move up and down for easy and safe accessibility and quick changeovers, it needed to look contemporary, attractive and quirky as well as being a practical support system for various displays to be hung below.\nCurrently being hung from the truss grid is a spectacular Shoebox Chandelier artwork created with multiple, differently configured strings of bright orange shoeboxes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Tired of take-out pizza? Learn how to make your very own pizza right at home \u2013 from scratch! Experiment with different toppings, and make a delicious Kale-Caesar Salad along the way in this fun and interactive class.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Business designer Marko Savi\u0107 with an MA in media studies and an MBA is the author of the book Business Model Design (in Slovene: Oblikovanje poslovnih modelov).\nThe Business model design workshop is a must for everybody who would like to evolve with a tool co-designed by 470 authors from 45 states. We will connect design thinking with business model innovation and instil the language of business model design into your vocabulary and day-to-day operations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sweet little touches to help pull together that very special first birthday celebration or first birthday cake smash photos. These gold glitter and felt pennants are so sweet and come with string, a stringing needle, two tone pennants with \"ONE\" and extra gold hearts for accents.\nFor reference only: \"Birthday\" measures approx. 4 ft wide when strung up with 2\" spaces between letters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Would it be better to hold onto old devices until better recycling options are available?\nHow do I ensure my internet is green?\nWhere is the best place to recycle phones and what other organizations is Greenpeace collaborating with on this effort?\nGreenpeace hosted this live Total Access Event that featured senior campaigner Elizabeth Jardim. The discussion pertained to electronic waste (e-waste) and how our devices affect the environment. Elizabeth reviewed Greenpeace's long-standing commitment to this campaign and callers addressed questions of sustainability and how-to properly re-purpose and recycle old devices. Greenpeace discussed how they are promoting this effort internationally and on social media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Living room seating for small spaces, small modern living, 1000 ideas about built in couch on pinterest built in. Very easy and fast rustic living room ideas indoor. Seating furniture living room raya furniture.\n25 best ideas about living room seating on pinterest, frisbee swivel chair modern design living room seating. Attractive modern living room chairs living room design 2018. Small living room sofas with modern living room.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauamn b/data_all_eng_slimpj/shuffled/split2/finalzzzauamn
new file mode 100644
index 0000000000000000000000000000000000000000..329fe669e996b855e36174e107501a34bfb66db7
--- /dev/null
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+{"text":"Spacious and airy multi level cottage with large screened in porch. Vaulted ceilings and access to porch and deck in living area, gas fireplace and beautiful furnishings provide a comfortable and cozy setting. Open well equipped gourmet kitchen and dining area seating eight are adjacent with loft above accessed by ships ladder. Masterbedroom and large bathroom also on this floor. Lower level bedrooms are bright and share a hall bathroom. Both have doors leading to outside large yard with wiffle ball field, volley ball court & horse shoes. Informal living space also on this level. A nautical theme in furnishings and artwork along with pretty landscaping all around complete this perfect Nantucket vacation cottage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"India-Myanmar bus service will start from November 10 this year to strengthen the ties between the two countries even though there are some undercurrent of boundaries issue between the two countries, said U Thar Neng, the Yangon Airbus Association Chairperson, in a Myanmar published daily.\nThis decision was also intimated to Manipur Chamber of Commerce & Industries, which is a part of Confederation of Indian Industries and Indian Chamber of Commerce.\nU Thar Neng said that the trans-national bus service will ply from Mayek township of Myanmar to Imphal in India. He also said that preparations are already on to start the service from November 10, prior to the South East Asian Games to be hosted by Myanmar.\nHe also said that a meeting to discuss the same was held with Myanmar-India Friendship Association on August 27 and the service is being launched to strengthen the ties between the two countries and not just for the purpose of connectivity.\nIn the first phase, 4 mini buses having a capacity of 36 passengers each will ply four times a week. The duration will be three days and the one way fare will be 63 US dollars.\nHe further stated that the bus service is not for business purposes but to strengthen the ties between the citizens of the two countries while maintaining that the service will also cement cultural ties between the two countries. The trans-national service had also made the task of trans-national permit and visa easier.\nThe Trans-Asian Road which is aimed to be completed by 2018 will pass through Tamu, Kalay, Moniwa, Mandalay, Nepido, Yangon, Maolamyang, Dawei, Mayek, Mawongto, Parachek and reach Mesot, Thailand by passing through Taninthari and Segang divisions of Myanmar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feel The Wear is a very handy little app to help you customize the way your Android Wear device handles notifications.\nFeel the Wear allows you to customize the vibration style, length, and number of vibrations. If you have a Android Wear Watch such as the Huawei Watch that has a speaker built in, you're also able to customize the sounds used.\nThere's about a dozen of built in sounds for your Wear Watch to use on the free version. However, if you'd like to add your own custom sounds, you'll need to pay for the premium version, which will run you about $4. This is not required, only if you want access to those premium features.\nYou're able to customize each app with its own sound, vibration pattern, length, as well as the time your screen stays lit for when receiving a notification, and optional quiet hours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As more student loans head into default the federal government laid out a plan to stem the tied, and prevent a financial disaster reminiscent of the late 2000's.\nThe problem is\u2026 It's not working.\n\"The Obama administration has sought to stem a surge in Americans defaulting on their student loans by slashing borrowers' monthly payments through so-called income-based repayment plans.\nEnrollment in the plans\u2014which set borrowers' bills as a small percentage of their incomes\u2014has soared over the past three years. But new research shows there's a flaw in the administration's strategy: The program isn't reaching many borrowers who need it the most.\nA study published Tuesday by the Urban Institute, a left-leaning think tank, shows that the programs have only modestly reduced defaults. The reason: Borrowers who have enrolled in the plans are the least likely to default in the first place.\nThese borrowers tend to have higher student-loan balances and credit scores than those who remain in standard repayment plans, the study shows. That suggests they attended well-regarded colleges\u2014which charge higher tuition than less-regarded ones\u2014and graduate, medical and law schools. Such borrowers are much more likely to be employed and earning high salaries than those with lower credit scores and balances.\nAs with many government programs it appears that something was designed with good intention but fails to reach the people who need the program the most. And as a result many low income college dropouts, and blue collar workers will suffer financial consequences. A better solution is educating them on their options as they enroll in college.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So, if you desire to save fascinating images practically 2018 acura tlx a spec v6 test drive review awesome of 2018 acura tl first drive above, absorb right-click on the image after that click keep as. Or if you desire images in various sizes, you can click download by size. Finally, if you in the manner of car pictures nearly 2018 acura tlx a spec v6 test drive review awesome of 2018 acura tl first drive, make laugh bookmark this page, we attempt to realize regular updates in imitation of newer car wallpapers. We wish you in imitation of our website and acquire the car wallpaper images you habit.\nRelated Posts of \"2018 Acura Tlx A Spec V6 Test Drive Review Awesome Of 2018 Acura Tl First Drive\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauktd b/data_all_eng_slimpj/shuffled/split2/finalzzzauktd
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Tama is an abstract fluid art painting on a circular canvas. Tama is a Japanese name for jewel, gem and whole. Beautiful hues of blues and turquoise flow with hints of golds to create a striking flowing piece.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You will need to remove the old ad tag implementation on your site to prevent any conflicts with the Playwire Banner Advertising code. There may be multiple places the old code may still be present on: the homepage template, run-of-site templates and both desktop and mobile templates.\nYou will receive an onboarding kit, specific to your account and website. Within the kit, you will find a code snippet containing 2 scripts to include in the <head> of the document and ad tags to be placed in the <body> of the document.\nYou can place both desktop and mobile placements on your page. Our script will automatically detect which device the user is on. However, if your site utilizes a separate mobile template view for your site, for example if your m.site.com will only serve a mobile optimized experience, you can use separate desktop and mobile ad tags for each experience.\nBacon ipsum dolor amet jerky filet mignon short loin tenderloin. Bresaola pork loin frankfurter beef ribs boudin doner chuck meatball flank tri-tip landjaeger. Spare ribs ham hamburger tongue prosciutto ball tip cow jowl meatloaf tail salami. Capicola bacon porchetta doner, flank pork belly jowl corned beef kielbasa shoulder.\nDrumstick short loin flank chuck biltong ground round. Doner kielbasa pig turducken hamburger, ham hock drumstick cow pancetta frankfurter pork. Meatball salami ham hock meatloaf spare ribs andouille drumstick tongue jowl capicola bacon fatback strip steak chuck. Beef ribs shankle ham hock turducken. Sirloin strip steak tongue, biltong filet mignon picanha tail pig corned beef landjaeger. Andouille prosciutto chicken porchetta ham. Shank pork loin chicken, pancetta ribeye frankfurter pork chop shoulder capicola landjaeger meatloaf chuck t-bone.\nPicanha ham ham hock, jerky burgdoggen swine alcatra t-bone fatback cow ground round ball tip tri-tip pork. Pig doner turducken, capicola short loin turkey tenderloin pastrami. Chicken t-bone ball tip swine, spare ribs rump bacon pancetta. Chicken pork chop pastrami, pork belly hamburger pork picanha jowl sirloin frankfurter fatback turkey.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Julia Reynolds runs The Ice Cream Man in Greenwich, N.Y., making fresh premium ice cream every day from her arsenal of close to 300 mouth-watering recipes. Julia always has a line out the door of people screaming for her decadent ice cream sundaes -- scoops and scoops of ice cream, drizzled with sweet sauces and topped with crunchy nuts and a cherry, of course. Julia's ready to treat the Food Network and a crowd of customers to her most sought-after Sundae: The Banana Bucket. Little does she know that Bobby Flay is coming, to taste the best Sundae in town, but also to challenge that title -- in the ultimate Ice Cream Sundae Throwdown! How will Julia's sweet hometown favorite fare against Bobby's brand new tooth-tingling concoction? Watch and find out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mascara Monday - Yves Saint Laurent Volume Effet Faux Cils Shocking!\nYves Saint Laurent Volume Effet Faux Cils Shocking - is my first ever experience with a YSL mascara! The formula is a bit wet but the brush is nice and not super huge like some thickening formulas - when the brush is too big I find it hard to use and more mascara ends up on my skin then my lashes!\nRight side has 2 coats, left side is naked. I think I might still have a couple of extensions left over which would be the few odd super long lashes you see.\nYou may or may not know that I have this thing (obsession) with my lash comb and even though I know you're not supposed to comb out a thickening formula, I couldn't help myself - the left hand eye shows why it's a bad idea - the comb just made them look sad and spiky!\nI just don't think I will ever be jumping on the thickening formula bandwagon, I understand that this is what it's supposed to look like but it just seems like such a mess to me!\nI'll keep trying, maybe it will grow on me eventually? What do you think?\nI think this would work great for a dramatic clumpy, thick lash effect as It's quite on-trend this season but not a natural effect.\nYSL mascara has always been my fav, gives great volume for a dramatic effect. thanks for sharing!\nI don't love this one either. Way too clumpy! It makes me look like I have a total of 10 lashes in each eye.\nClump, much? Oh no. Thanks for saving my money, I was thinking of getting it...but no, I cannot deal with the clump.\nI've heard so much about this mascara but I don't like the look of clumpy lashes. Have you tried either the Covergirl Nature Luxe mascara or the Buxom Lash mascara. I think you would really like them!\nI understand the lash comb thing. I have an absolute horror of clumpy mascara! I can't just leave my mascara as it lies, have to comb it out until ALL the clumpy bits are gone. Oh well, everyone ahs an obsession or two, at least this one's relatively harmless.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are so many great things to do and see with kids in NYC, from museums to zoos to destination playgrounds, it's hard to narrow down to a reasonably sized list. But knowing what NYC families like doing best in their own hometown is a solid place to start if you're planning a visit!\nAs veteran New Yorkers, my family has their own list of absolute must-see and must-visit spots around the city. Some are off the beaten path, and some might be considered a bit touristy, but all of them are places we love\u2014and love experiencing with visitors, in particular. Whether you're visiting NYC with kids for the first time or the fiftieth time, here is a list of some of the best places to go right now, as recommended by parents who live here.\nFind more top sights in our NYC Visitors Guide for Families.\nChances are you're not even thinking about Thanksgiving yet. After all, we still have Halloween to get through! But as anyone in the NYC hotel industry will tell you, families looking to book a hotel where they can watch the Macy's Thanksgiving Day Parade in Midtown Manhattan do so up to a year ahead of time. For a room with a view of Snoopy, Spidey, Hello Kitty, and their other inflatable pals, you've really got to plan in advance.\nIt's not just NYC tourists who splurge on hotels for the Thanksgiving parade, either. Over the past few years, we've had many New Yorkers write in asking which hotels looked out on the Macy's Parade route (it sure beats watching from the street). While most of these hotels still have availability at the time of this writing, call right away as they'll most likely sell out soon. And if you do get shut out, it's worth checking closer to Thanksgiving since there are sometimes last-minute cancelations.\nFor a budget-saving alternative, catch a prime viewing spot the night before the parade as the giant balloons are inflated on the Upper West Side.\nNew York isn't exactly a cookie-cutter city, so it may not come as a surprise that there are kids' party places for just about every interest under the sun. Sports fanatics can invite guests to hit an indoor ice skating rink or a high-tech golf driving range, while little athletes can celebrate with trampolines and obstacle courses in state-of-the-art gyms. Imaginative birthday boys and girls can mark the day with a fairytale-inspired fantasy ballet party, while tech lovers can throw LEGO- or Minecraft-themed bashes. There are even animal encounters for underwater enthusiasts and mobile pizza parlors for fun-loving foodies.\nIf you're on the hunt for a venue for your own one-of-a-kind kid, take a look at these awesome spots with packages for tots, tweens, and teens (plus, plenty of help from experienced party pros). For even more ideas, take a look at the New York City Party Guide, or browse our directory for party places, entertainers, and even party supplies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaunfw b/data_all_eng_slimpj/shuffled/split2/finalzzzaunfw
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index 0000000000000000000000000000000000000000..eb94635707f8ce622ed936bd0fc57e64ec0fcb4d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaunfw
@@ -0,0 +1,5 @@
+{"text":"So, yeah, I deal with anxiety from time to time. I've faced it socially, in my previous careers, and, oddly enough, with my running. Yup. Despite having run regularly for nearly a quarter century, I've increasingly battled anxious feelings when headed out for my near daily run. Anxiety is common and it's nothing to be ashamed of. However, to get the most out of life, it's something that should be addressed rather than avoided. The following are merely examples of some ways that I personally deal with my running-related anxiety.\nPlease keep in mind that I'm no medical professional and that persistent anxiety that affects one's physical health, work, or relationships; causes one to miss out on life; or leads to self-medication with drugs or alcohol may warrant professional assistance.\nAx the Avoidance. I've been avoiding writing this point. Really. I've been avoiding it by attending to the other points in this article and other work more broadly. It's easy to find something else, anything else to do when we need to overcome an anxiety hurdle. Taking care of email. Straightening up one's desk or one's room. Researching an upcoming race or purchase or travel on the internet. These are ways I might avoid facing what's causing my anxiety under the guise of \"doing something useful.\" I learned of the whole concept during a short-term go with a therapist back in my attorney days, and it's been so helpful. I both recognize my old favorite avoidance techniques and can catch whether I'm doing something that doesn't need to be done at a certain moment in order to avoid an unpleasant task. Hint: That avoidance often looks like busy work.\nAcknowledge and deal with triggers. Often enough, the anxiety I encounter when I consider heading out for a run originated somewhere else. That anxiety might come from a confrontation with a loved one, dealing with an unpleasant work situation, or thinking about contentious neighborhood politics. If I'm in a period when I'm prone to anxiety, I'll plan on running when I'm less likely to have had an anxiety-inducing event (such as heading out in the morning) or avoid opening a particular email that's likely to trouble me.\nPrioritize. Many days I create a to-do list for myself and that list tends to be predominantly work related. It keeps me on track. I incentivize it. I love it. If I need a little help, I simply add \"go run\" on my to-do list. I suppose this could create even more anxiety if my current workload is what's really troubling me, but much more often it succeeds in getting me out the door and running.\nGo casual. There are numerous days last winter when I'd get up from my desk, throw on a hat and gloves, and start running\u2026 in my casual pants, flannel shirt, and whatever shoes I was wearing. I was quite the sight, but I was also out running. Cutting out the couple minutes it took to get changed increased the chances that I'd make it out the door. What's more, once I was out there I was apt to view my run more casually and less critically.\nStreak. While it has the potential to increase anxiety, I find that starting and sticking to a streak can at least help me get out the door and run. I don't set conditions of any sort, I just need to do a bit of running, even if that's just a one-mile lap around my neighborhood.\nStart small. Speaking of one-mile laps, I find that I'm less anxious if I set out for the most modest of runs. The mile loop from my doorstep is great, as it's as easy as my local terrain offers and I can easily add on more laps if I feel like it.\nSo, does anyone else out there get anxious about running sometimes?\nIf so, how do you work through it?\nBryon, great piece, my friend. As one who shares a seat with you on that anxiety roller coaster I can relate to each and every one of your points. Thanks for putting it out there. Next time we're together we need to get out for a nice long run and hash some of this stuff out!\nGreat post, Bryon. All good points on breaking avoidance, pushing through, and getting out. I know many runners deal with anxiety. After all, there's an inherently compulsive element to all of this which shares roots with anxiety. (Yeah, I'm a shrink).\nFolks should also remember that a little bit of anxiety is good \u2014 just enough to motivate appropriate training, prep for long runs, bad weather, etc. etc. We just have to find that sweet spot. As it is said, \"everything in moderation\" \u2014 including anxiety. Well, everything except miles of course!\nYou need the Clown Man to fire you up for a run, just like the old days! Nothing to it\u2013just waddle down the road chewing a home-made chia crud-ball with some Succeed salt stuck to it. And then just some good ole' fashioned trash-talkin' to keep you motivated.\nThank you for writing this Bryon. As someone also experiencing anxiety I recognise all of the challenges you describe, especially other triggers of anxiety spilling over into worry about going out for a run. If I'm feeling anxious about other things \u2013 or even just anxious about nothing in particular \u2013 the prospect of running can be too much sometimes; \"its going to hurt\", \"I'm a rubbish runner\", \"it's too hot\/cold\/windy\/sunny\/dark\" are all milling around in my mind and the prospect of having time alone with my thoughts can be daunting.\nI try to shift the focus from expectations about performance to remembering that even a short easy run dissipates the physical experience of anxiety and bring mental clarity that makes dealing with the underlying causes so much easier. After a run I may still be experiencing anxiety, but I'm rarely suffering with it.\nThanks Bryon for a fantastic and timely article on this topic. As we enter the winter season and the days are increasingly getting shorter and darker this month, I find that any usual levels of anxiety I feel about going out for a run are magnified at this time of year.\nThe \"go casual\" tip is a great one and reminds me that I've relied on that particular one in the past.\nRunning with a buddy, encouraging self-talk and picking up some new gear I've been wanting are other stratefies I plan to call on more and more \u2013 especially over the next few months.\nMile loops and streaking are two other go-to motivators for me. So glad you mentioned them. Just reassuring and anxiety-reducing to know that none of us are alone in this situation.\nThank you so much for writing this. I feel so relieved that such an amazing runner such as yourself also suffers from this. I don't feel so weird after all.\nI have usually used go small but am starting to find go casual is very helpful too.\nThank you for this article. I do not have anxiety but your honesty has made me aware that others may have it and it can affect their running. I rarely get the feeling that I don't want to run, my motivation remains at a good level most days. If the weather is challenging or I am tired it does get harder to get out there. Having a dog that needs a walk or run daily makes it easier to go run. I will have to get out there anyways to walk him so I might as well run.\nYour article reinforces the notion that everyone has challenges, many that are not apparent just by looking at a person, and it is helpful to interact with people with kindness.\nI'd have to say the \"go casual\" advice is my favorite. Don't fuss about all that gear just start running! That is unless by \"Streak\" you mean go naked\u2026 then I'd have to say that is my favorite advice.\nI started running to help with my anxiety\u2026. which it did until I started pushing my self to go farther. Thank you for this post, it helps me realize that I'm not alone with this.\nSo much yes in the post-thank you for sharing. I'm currently on a streak and it definitely helps with my anxiety.\nI deal with anxiety, including about running. My favorite tip is to watch one of Ken Chlouber's Leadville speeches on YouTube. By the time he's done, I'm ready to head out the door to run, and prepared to run through a brick wall.\nThanks so much for this article! It's been great to see runners open up more about mental health issues. I suspect many of us find solace in running, yet those same issues can make it hard to run. I haven't found that my anxiety affects my running that much, but its friend depression can make it really hard to get out the door some days due to low energy, apathy, negative self-talk, etc. How do other runners deal with this? Cheers!\nI use some of the tactics above \u2013 like \"just warm up\", then \"just do one mile, then you can stop\", bribing myself, berating myself, trying to think of others for inspiration.\nI think my anxiety \u2013 often about how long I'll be able to run, what pace I can hit, how I'll feel \u2013 shows up as procrastination (avoidance). And I only have steps to get to my run since I run on a treadmill in the basement! I have to walk past it between the bedroom and the rest of the house. The time between getting up and running keeps stretching (into hours) as I find things to do like work, surfing the net, loading\/unloading the dishwasher and laundry, re-syncing podcasts, etc. It's something I struggle with a lot.\nToday's challenge is that my morning run didn't happen as the power went out, and by the time it came back I'd eaten and felt like I needed to get to work and deal with other obligations. I'm not an after-work runner \u2013 I tend to bonk or have GI issues, one more reason I run first thing \u2013 and am having enough trouble with miles\/pace as it is. I am trying to convince myself to at least \"suit up\" in an hour or two and warm up, run one mile. If I can't make myself run, hopefully I can make myself walk for a while (not much of an excuse with that).\nThis is great. Thanks for the gutsy and honest post, Bryon.\nI've noticed that when I have a nagging uncertainty or anxiety about a long run, I can convince myself to stay home in order to do something much more fun, like cooking a really delicious brunch and vegging out with my husband. I've found that running at odd hours really helps \u2013 if I set the alarm early enough I can get out the door before my inner critic has woken up, and then be back in time to enjoy the last of the morning at home, and I enjoy it more because I'm not feeling anxious about having missed a workout. If I procrastinate until my husband gets up, I'll be too tempted to stay home. Running super early also helps avoid the feeling of looking stupid, with lots of people watching me \u2013 the only other people up are either other runners, or are too tired to care (or both).\nOh, and avoidance also masquerades as indecision for me. I can agonize over choosing the perfect route that offers something new and exciting to explore along the way, and all the while I'm just procrastinating. When I recognize that I'm doing that, I have to snap out of it and just default to one of the \"old hat\" routes or I'll never get going.\nJust about the only way I can convince myself to do any serious mileage on weekdays is to run home from work (~8-12 miles, depending on choice of route). If I wait until I get home I'll just want to stay there.\nNeeded this post Byron. I am in the same boat as you and the past month has been somewhat difficult for me. Hate to read all the comments of people who struggle with the same but I am encouraged that we are in this together!\nThanks for writing this piece! I think a lot of us struggle with anxiety. It's helpful for us and for those around us to hear about it so they can be empathetic.\nMy anxiety stemmed from childhood speech problems (a horrendous stutter), which lead to me becoming a writer and editor, I think! (Anything to be behind the scenes, not in front of the mic or camera.) I've found trail running to be really helping in tamping down the elevated level of stress I still sometimes feel. But, for the anxiety, for me the only thing that ultimately worked for me was to find the fear that was powering it, which took\u2013 as they say\u2013 some \"work.\" Once I understood that, the anxiety was defanged\u2013 it still existed, but its power over me was greatly diminished. All those circuits are still wired in my brain, but I now recognize them and when they power up, I head for the hills. :) Thanks again for the piece.\nI love the go casual tip! Getting my stuff together sometimes is a stressor in and of itself, and I waste time searching for just the right sports bra, or whatever.\nGreat piece! I can totally relate.\nThanks Bryon, I have dealt with anxiety for years and running has greatly helped manage it. As i have moved into my forties and life pressures mount, getting into longer runs and consistent training has helped. Gives me a positive distraction\/obsession to keep mundane anxiety at bay. The week leading to my first 50 mi race last week (NF 50 MI SF), was anxiety riddled. I woke flat and questioning, calling myself crazy in a neg way. critical self talk. I flipped the script with music and reminding my self of the positive crazy of endurance running, the fun and appreciation for suffering that I have built up too. I talked with my shadow during the run and came out with new ideas and skills to manage negative mindsets and anxiety. Thanks for normalizing anxiety with your article.\nI don't understand. Anxiety about running? Why?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Geranium, F1 hybrid, red Dogwood in flower, with wild geraniums, early morning shaded sunlight. Wild geranium Dogwood leaves, variegated, with single wild geranium flower. Geranium, wild, name Patricia Wild geranium flowers in dappled shade and sunlight.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do I need to give my credit card to make an online reservation? Is it secure?\nWill my credit card be charged if I don't cancel and don't show up?\nYes, reservations are automatically guaranteed to your credit card. You will be charged for the first night's stay only. To avoid being charged, reservations must be canceled in accordance with the cancellation policy outlined by the hotel for the rate and dates booked.\nHow do I view or cancel a reservation that I reserved online?\nClick on the MY RESERVATION link above and enter your reservation number to access your booking.\nReservations must be canceled in accordance with the cancellation policy outlined by the hotel for the rate and dates booked.\nHow do I notify the hotel that I will be arriving after the check-in time?\nAll reservations that are guaranteed with a credit card are held for late arrivals. If you know in advance that you will be arriving late, please make a note of this in the \"Special requests\" section of the reservation form. If you're late arrival is for the next day, please contact the hotel directly.\nI am traveling with a child. Will this affect my reservations?\nIf you are traveling with a child, this may affect your accommodations. When making a reservation, please make sure to indicate the number of adults as well as the number of children when prompted. Additionally, include the number of children and their ages in the \"special requests\" during your reservation process. If you should have any concerns, please contact the property directly.\nI have a billing problem\/ question.\nBilling is processed by the property. Contacting the hotel directly may expedite resolution of your issue. If you prefer, you may provide us your information by contacting MakareMe and we will document your feedback and forward to the property. They will reply to you by e-mail, telephone or mail as soon as possible.\nWhat if I want to reserve a group of rooms?\nIf you need more than two rooms, you can make multiple reservations. However, for bookings requiring five or more rooms, it would be best to contact our reservation center (+966 920000547) and inquire about group discounts that may be available.\nHow do I enroll in Karam Club Card?\nJust click here to access dedicated Karam Club Card website.\nVia our Front Desk Staff \u2014 they can enroll you on the spot when you check-in at the Front Desk.\nHow do I learn more about Karam Club Card?\nTo learn more, visit Karam Club Card website or call MakareMe guest care center (+966 920000547).\nI am interested in opening a Makarem hotel. How can I find information?\nTo find our more information about our dynamic organization and becoming a Makarem Hotel owner, please visit our dedicated Investor pages.\nWhere can I find employment opportunities within your company?\nPlease inquire about positions within Makarem hotels network by visiting our Career section.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Monitored sprinkled fire system, monthly fire drills.\nCall bells in bedroom, bathrooms and common areas.\nKey pad entry at exterior doors and to the neighborhood entry doors.\nAn emergency generator supplies lighting and power outlets should a power outage occur.\nAlcohol Use: In order to minimize the risk of alcohol interacting with other medications, the use of alcohol must be ordered by the Physician. The alcohol will be securely kept in the Medication Room and will be dispensed by the Nurse.\nEmergency Response: Fire Alarms. With the sounding of the fire alarm, you and your visitors are requested to remain where you are and follow the directions of the staff during a drill or fire alarm.\nSecurity: Key Pads for Secured Areas. Keypads have been installed on all external doors for the security of residents and staff. An intercom is available for visitors who may not have the access code. Access codes are provided to family members on move-in. Dania Home is secured from 11:00 p.m. to 6:00 a.m. Keypads will not be operative during this time. Visitors at this time may use the Intercom at the front door or call the Nurse to be allowed entry.\nSmoking: For those residents able to smoke safely and independently a smoking gazebo is located on the main floor grounds. Staff monitor residents in the smoking area on a regular basis and accompany residents into the smoking area as needed. Staff must leave the smoking area before residents light their cigarettes and begin smoking. Smoking for Staff, volunteers, contractors and visitors on organization's owned premises is prohibited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Census Bureau recently released its 2014 Q4 Housing Vacancy Survey (HVS) data, giving us a complete look at the boom and bust in homeownership rates over the last 20 years. The HVS' reported homeownership rate, though far from perfect, remains the most up-to-date and cited statistic on homeownership in the U.S., and thus an important barometer of the housing status of American households. One advantage of the HVS is its long time frame, going back to 1965 at the annual and regional level, which helps to put recent trends in homeownership into a historical context. Figure 1 shows that the national homeownership rate rose steadily through the late 1960s and 1970s, from 63 to 65.6 percent, before declining slightly in the early 1980s. After a decade of stagnation, the rate rose rapidly from 1994 to 2004, from 64 to a record high of 69 percent. Since then, however, the national homeownership rate has declined almost fully back to its 1994 level.\nThe roller-coaster ride of the national homeownership rate from the last 20 years, while dramatic in its own right, only tells a small part of the homeownership story. Another advantage of the HVS data is reporting on homeownership rates for subsets of households by race\/ethnicity and age, which add important texture to this story. Homeownership rates by race and ethnicity, for example, have generally followed the same up-and-down trend as the national rate, although no identified racial or ethnic group has given back all their gains. Indeed, when measured by the difference in their homeownership rates relative to 1994, only blacks are close to their former level (Figure 2). Households in the 'other' category (mostly comprised of Asians, Pacific Islanders, and multi-race householders), which grew their share of homeowners by 12 percentage points during the boom, remain fully 7 percentage points up today.\nNotes: Hispanic includes all races. All other races includes multiracial.\nSource: U.S. Census Bureau, Housing Vacancy Survey.\nThe reason the national homeownership rate has declined all the way back to its 1994 level, even when no individual racial or ethnic group has done so, is because of the simultaneous shift in the race\/Hispanic origin composition of households in the U.S., which has increased the share of all households that are headed by a minority. Due to the lower homeownership rates of minorities, relative to non-Hispanic Whites, this shift automatically lowers the homeownership rates of the nation, independent of any changes in minority-specific homeownership rate.\nThe second set of data from the HVS to shed more light on homeownership trends are the changes in rates by age of householder. Similar to the breakouts of homeownership rates by race and ethnicity, most of the age groups identified share the same rise and fall trend over the last 20 years, although householders ages 65 and older did not decline appreciably following the mid-2000s boom (Figure 3). The youngest households, those under age 35, gained the most in their homeownership rate during the boom, but have since declined to one-and-a-half percentage points below their pre-boom rate. Middle-age groups, meanwhile, have seen their homeownership rates fall nearly five percentage points below their mid-90s level, with 35 to 44 year olds experiencing a full ten percentage point swing in the last ten years. These dramatic declines are greater than the overall decline in homeownership because of the shifting age composition of households, which have skewed older over the last twenty years as the baby boomers progressed into higher ownership middle and early retirement ages, being replaced in the 25 to 44 age group by the smaller and more racially diverse Generation X cohorts with lower ownership rates. The low levels among younger cohorts, however, do suggest that the national homeownership rate is only back to its mid-90s level because of sustained homeownership by older households, and that among those in their prime working years the actual homeownership rate is indeed well below so-called 'normal' levels.\nA quick glance at these data may raise some concerns about the future of homeownership, especially for younger households and minorities who will account for large shares of households going forward. Yet caution should be exercised in extrapolating these trends forward. For one thing, the HVS data often shows unexplainable jumps in quarterly rates and counts of households, which are somewhat muted by these annual averages. Second, recently moderating house prices and a move to relax lending restrictions, along with the reduction in FHA mortgage insurance premiums, should give home buying a boost. Third, most people, and especially young adults, remain in favor of homeownership, despite the risks exposed during the recession and foreclosure crisis. For these reasons, few analysts expect homeownership rates to fall much further. If it were to happen, however, the story would change from one of returning to a pre-boom norm of about 64 percent, to entering a new era of low homeownership rates unlike anything we have seen in the last 50 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaunyl b/data_all_eng_slimpj/shuffled/split2/finalzzzaunyl
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+{"text":"Would you like to look at and down load Elephant film in stream High Definitin -720p !. This film is released on 2003-05-18 and gained rating : 7 count - within runtime : 81 minutes. Just go to the link below and choose from among the list of categories for getting download and install of the movie films found in a selected Crime, Drama.\nThe actors and actresses are acting gexcellent for Elephant storyline and movies are wonderful thus making you want to know. First, and foremost, the leads in the movie is likable, meaning that the target audience love. It will make the connections during the dvd likable and believable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Corporate universities have been around for a long time, dating back to 1956, when General Electric opened its first management school. These universities were initially created to align the training arm of the company with its overall vision and strategy, and they fared well for a long time. In today's context, when millennials and Gen Z are joining the workforce in large numbers and the gig economy is on the rise, corporate universities need a re-vamp to keep up with the demands of the modern-day workplace and its employees.\nAs a result, we need to move from a culture of training to a culture of learning. The corporate university must be more than a training center guided by the L&D team. It must become the go-to place for every individual within the organization seeking answers.\nThe corporate university must be more than a training center. It must be the go-to place for every individual seeking answers.\nThe role of the learning and development team now is not just to manage an LMS loaded with content and ensure employees take courses. Instead, L&D must determine how to develop an efficient workforce that impacts the bottom line. The good news is that most learning and development professionals are taking on this challenge successfully, trying new ways to provide a seamless learning experience.\nThe learning platform and content remain at the heart of the \"corporate university 2.0,\" but we must design them to talk to each other to create personalized and powerful experiences. Here are six best practices.\n1. Provide a Seamless Learning Experience.\nWe all are used to the YouTube and Netflix experience, where we can search for information, and the system shares the best of the relevant content, along with recommendations based on our profiles. Why not design learning experiences in a similar way? Learning experience design allows learners to search for information by getting to the core of what they are looking for. This type of design requires a smart platform and bite-sized, searchable content.\nLearning experience design allows learners to search for information by getting to the core of what they are looking for.\n2. Create the Ability to Recommend Courses.\nThe learning platform in the corporate university 2.0 can recommend courses based on an employee's knowledge level and profile. For example, employees and their managers can share the competencies they need to develop, and an AI engine assesses the gaps in the learner's knowledge and then recommends content based on those gaps. This approach makes each employee experience unique and personalized based on the employee's actual learning needs.\n3. Present Content in a Variety of Forms.\nUsing technology, the corporate university 2.0 presents the same content in multiple ways. For example, you can tag course content and then only recommend those pages or modules that the employee really needs. There are tools that convert text to audio content, and you can also curate content from third-party sources.\n4. Reinforce Learning Through Spaced Repetition.\nHow many times do you visit a website when searching for information and then keep seeing an ad for that website in multiple places? In a similar way, the corporate university 2.0 creates interesting ways to reinforce content to learners. The platform already has information on what courses or modules the employee has participated in. Using this information to design a learning experience in which the employee receives microlearning short daily mobile quizzes can go a long way in reinforcing learning. One can again, you can use technology to create such content quickly. For example, quizzes play an important role in learning, and you can use AI to quickly generate questions based on online course content.\nThe corporate university 2.0 creates interesting ways to reinforce content to learners.\n5. Use Chatbots for Just-in-Time Learning.\nWhen you have a quick question, you probably don't write an email to someone and wait for an answer; you probably do a quick internet search to try to find the answer quickly. Similarly, when employees need a quick answer, a chatbot can be a great way to help them find what they need. It not only makes the L&D team's work more efficient, but it also creates a great learning experience. You can use chatbots to recommend a course, answer questions using a content library, or even understand the exact need of the employee and then share that information with the L&D team.\nThe corporate university 2.0 goes beyond traditional data and reports from an LMS using xAPI. Detailed reporting and analytics capabilities enable the platform to push appropriate content to the employees and, at a business level, generate insights that help the organization make strategic decisions. And why not share some insights with learners, as well? Data can be a great motivator. It helps drive the culture of learning, motivating employees to engage with the corporate university, participate in discussions, share and rate content, and take quizzes for reinforcement.\nThe corporate university 2.0 goes beyond traditional data and reports from an LMS using xAPI.\nToday's learning and development teams have been tasked with the important responsibility of creating a culture of learning within their organizations. If your corporate university can keep up with the demands of the next-gen workforce with 2.0 enhancements, you can accomplish far more than you ever have achieved in the past.\nRahul Singh, a digital learning enthusiast, is the senior general manager of business development at Harbinger Interactive Learning. He is passionate about helping organizations and leaders solve business challenges through the intervention of learning technologies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is true that, planning an event is truly a complex affair and it can be successfully arranged only after a long track record and experience. There are a lot of things are being involved while planning an event. It needs patience, right amount of budget and significant time. For a successful event, you need minimum three months for preparation. Otherwise, those events may end up with various lackluster and displeasure. The end part of the event, while guests are leaving, their feelings are only the measure that how far you have succeeded to arrange that event.\nSelecting the right venue is known as one of the most vital parts of such event and if there is any flaw, it will severely affect the entire event that will be ended up with dissatisfaction. Depending on the objective of the party, you can choose the right option from restaurant, bar or different venue. Plan such events how it will make satisfaction to the attendees. It can make all the difference when it comes to entertain a large group of people while they are all is may not know each other.\nFocusing on the attendees list will surely make your way better to arrange their favorite dishes those are known as the main part of such events. Apart from food, there are different types of zones can be arranged. For instance, children's love to play with their favorite toys or cartoon characters and making a children's zone separately for them will surely keep them busy while their parents are busy with others. Also hiring finger food caterers Sydney, transforms the environment full with pleasure eliminating worries and dimness for some while. Some other event planners also go a step ahead by to fulfill all the like and dislikes of the guests. It is also a responsibility of the host is to know the guest's like and dislikes.\nSuppose, the guests are interested in music and it will be good while purchasing tickets for a concert and use it as a gift for the evening. It will provide them with utmost pleasure and their heart will fill with gratitude for you. It is always good to invite a minimum number of guests rather than large. This is important because, it is truly a hassle to treat such large number of guests. Keeping your events in small will make you sure that the event is being kept under control. By inviting, a small number of guests, there will be less chance of any type of disturbances. If that event is a wedding, restrict that within core friends those are always with you. To satisfy them, arranging a lavish wedding buffet catering will be best to go for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u00bb Free Credit Score Check \u00bb What is the minimum CIBIL Score required to get Credit Card?\nKnow the minimum CIBIL score required to apply for a credit card.\nCheck the benefits on Credit card for good CIBIL score holders.\nHaving a low credit score or no credit history might stop you from using best credit cards. Some of the top credit card providing companies have a minimum credit score requirement. CIBIL is among the top credit bureaus companies that provides credit score and report. The score generated by them is called CIBIL score which is a 3-digit number showing the payment history of an individual. The score ranges from 300-900 wherein the higher range represents strength of the score. Financial institutions consider a credit card application suitable on the basis of this score. In this article, we will talk about the minimum CIBIL score required to apply for a credit card or get approval for it.\n1 How Much CIBIL Score Is Required for Credit Cards?\n1.1 What benefits do you get on a credit card with good CIBIL Score?\n1.2 Can You Get Credit Card Without CIBIL Score?\n1.3 How Can You Build a good CIBIL Score?\n1.4 Which banks give you Credit Card with Zero CIBIL Score?\n1.5 How to Check CIBIL Score by PAN Card Free Online?\nHow Much CIBIL Score Is Required for Credit Cards?\nThe suitable or minimum CIBIL score which is required for most of the credit cards is 750. The applicants having a score below 750 might not get approval for a card at some of the top companies. However, most of the banks also offer credit cards to those who do not have a credit history or score below 750. But, the fact remains that those who have a score of 750 or above are eligible for a lot more than those who are not in this category.\nWhat benefits do you get on a credit card with good CIBIL Score?\nRead below the advantages that a good CIBIL score holder gets for credit cards.\nCan You Get Credit Card Without CIBIL Score?\nYes, you can. Those who have zero CIBIL score do not lose eligibility to apply for a credit card. There are many banks offering wide range of credit cards to those having no credit history since the factors based on which credit cards are given are income and CIBIL or credit score. If you have no credit history but are presently working in a reputed firm and earning quite well, your application might be considered eligible for a credit card. The fact, however, is that having a credit history and good credit score makes you eligible for best credits as mentioned above. There is of course no harm in working towards creating a credit history or improving your CIBIL score if it is less than 750.\nHow Can You Build a good CIBIL Score?\nWhich banks give you Credit Card with Zero CIBIL Score?\nPeople Also Look For How to improve Credit Score if CIBIL is below 650 ?\nIf you haven't checked your CIBIL score yet, you may follow the below steps wherein it can be done with PAN instantly.\nHow to Check CIBIL Score by PAN Card Free Online?\nHere are the simple steps that are to be followed to check CIBIL score check for free.\nTo conclude we would say, if you want to enjoy the maximum benefits on a credit card, you must maintain a minimum CIBIL score of 750.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Title page, Increase Mather's A Brief History of the War with the Indians in New England, Boston, 1676.\nIncrease Mather's history was one of the first of many colonial post-war texts that made sense of Metacomet's uprising through the Puritan view of history. The slaying of Metacomet [or Metacom] in August 1676 ended the Indian threat to the Massachusetts Bay Colony. Indian resistance to English colonization and expansion continued, however, well into the eighteenth century on the northern and western frontiers. In 1692, Increase Mather's eldest son, Cotton Mather, would write, \"Our Indian wars are not over yet.\" He went on to write his own history and interpretation of the continuing conflicts between Indians and Puritans and in the process helped define the mythology of the Puritan captivity narrative.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"On Wednesday, June 17, Dylann Roof, a 21-year-old White supremacist, massacred nine Black worshipers in the historic Emanuel African Methodist Episcopal Church in Charleston, S.C. Among those slain at close range with a .45 caliber semi-automatic weapon, were an 87-year-old woman and Emanuel's pastor, Rev. Clementa Pinckney, a South Carolina state senator. This atrocity was not only a premeditated stab at the heart of Black America, and one of the moral epicenters of the African Diaspora, it was the latest in a long history of calculated and explicit acts of racial terrorism perpetrated against African Americans.\nIt reported that the number of so-called Patriot groups, including armed militias, skyrocketed following the election of Obama in 2008 \u2013 rising 813 percent, from 149 groups in 2008 to an all-time high of 1,360 in 2012. The number fell to 874 in 2014, which is nearly six times higher than when Obama took office.\nWhen I rejected a White male acquaintance's attempt to describe Roof as a \"misguided kid,\" he angrily declared, \"Roof could be my son.\" As the Oracle would say, \"Bingo.\" He, and millions more like him, identify with Roof. They grant the baby faced, salad bowl haircut wearing White supremacist the kind of absolution that many people of color can only imagine receiving were, Heaven forbid, the tables turned. They empathize with Roof. They do not see him first as the latest in a long line of racist mass murderers \u2013 the most recent, hateful, violent personification of our grossly unequal and unjust system, which is characterized and fueled by White dominance and disavowal of White supremacy. They see him not as the harbinger of violent bigotry, but an unwitting victim. Yes, as their son.\nWe must look courageously at the nature of White supremacy in America and the ways in which it defends itself at all costs. To my White brothers and sisters who have taken up the struggle and descended upon Charleston, signed petitions, written letters, donated to the cause, and otherwise acted in the aftermath of Roof's carnage, thank you for you support. Don't stop. People of color need relief and you're next up at bat. If true change is to be made, we will need all hands on deck, especially yours.\nTo people of color, wake up and keep your heads on a swivel. The American promise is virtuous and our advancements are real, but our trek in the wilderness continues and in the words of Robert Frost, we have \"miles to go before we sleep.\" There is no shelter or sanctuary for Black people in America.\nBlack people, we will not surrender and we will not back down. Others will either venture down the \"rabbit hole,\" unplug, and stand side-by-side with us in struggle, or we will compel them to, beginning with relegating the Confederate flag, the symbol of White supremacy and hatred that Roof so proudly waved, to museums where it belongs.\nMatthew C. Whitaker is ASU Foundation professor of history and founding director of the Center for the Study of Race and Democracy at Arizona State University. He is also the owner and CEO of The Whitaker Group, L.L.C., an equity and inclusion, cultural competency, and human relations consulting firm. He can be followed on Twitter at @DrWhitaker. The views and opinions expressed by the author are his own and do not necessarily reflect the views, opinions, and official policies of Arizona State University.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Martavis Bryant is known as one of the fastest wide receivers in the NFL. Bryant, who just finished his second season in the league, has formed a great connection with quarterback Big Ben Roethlisberger. His speed, along with Roethlisberger's arm, have made Bryant one of the best deep threats in the league.\nIn an exclusive interview with Bryant, he opened up about his two favorite things outside of football: Jimi Hendrix and tattoos. \"Some of his (Hendrix) songs give you that great vibe when you wake up in the morning,\" Bryant told us. He got his first tattoo at 14 years old; now 24, Bryant has had plenty of time to add a lot more, including at least three or four of Hendrix.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The baking competition, which is moving to Channel 4, averaged 14 million viewers with its season seven finale, cementing its status as Britain's most watched show.\nBaking competition The Great British Bake Off drew its highest overnight ratings ever with the final episode of its seventh season, its last on the BBC, on Wednesday night, becoming the most watched program of 2016 in the U.K.\nThe episode averaged 14.0 million viewers, according to overnight data released by BBC representatives on Thursday. The audience peaked with 14.8 million viewers.\nLast year's finale had averaged 13.4 million viewers and peaked with 14.5 million.\nA modest hit when it premiered in 2010 on BBC Two, the good-natured competition show was later moved to flagship channel BBC One and now ranks among the biggest shows in U.K. history. Last year's season saw the final episode become the most watched program of 2015 in Britain.\nChannel 4 earlier this year announced a three-year deal for the show, estimated by some to be worth 75 million pounds ($99 million), snatching it from public broadcaster BBC and causing much debate. The move is effective following the end of season seven. Channel 4 said the first Bake Off content planned is a celebrity version of the show in 2017 from production firm Love Productions.\nHosts Sue Perkins and Mel Giedroyc immediately said that they would quit the show when it leaves the BBC. The Great British Bake Off will also lose judge Mary Berry, but fellow judge Paul Hollywood will continue on the show for three seasons under a deal reached with Channel 4.\nThe BBC is developing possible new shows with Berry, Perkins and Giedroyc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bristol comedian Russell Howard has headlined Above & Beyond's charity comedy shows to raise funds for Bristol city centre hospitals.\nFunny Bones: Russell Howard & Guests was held at Bristol Hippodrome on Sunday 29 September and featured performances from Russell himself, Jon Richardson, Joel Dommett, Stuart Goldsmith, Laura Lexx and many others \u2013 all hosted by Mark Olver.\nThe shows raised a huge \u00a3100,000, which included donations at both the matinee and evening performances.\nAll proceeds will go towards improving patient care at Bristol's hospitals by funding welcoming environments, staff training and development, world-class equipment and innovative research.\nOur corporate and community fundraising manager, Edd Smith, said: \"It was such a fantastic day, creating a real buzz around Bristol and raising a phenomenal amount for Bristol's hospitals.\nFollowing the show, Laura Lexx became one of our charity champions! Read more about Laura's decision to support Bristol city centre hospitals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This beautiful, older bungalow has been well cared for. It has 3 bedrooms and a full bath on the main floor. The spacious L-shaped living-dining area opens through a French door onto the deck. There is oak hardwood flooring in the living room, and a stunning fieldstone wood-burning fireplace for cozy winter nights. Soft, neutral colors throughout.\nIn the spacious basement which is fully developed, you will find a large, open family room area, 2 additional bedrooms, as well as 8 closets\/storage areas. A newly-renovated bathroom has 3 pieces. The laundry area has built-in storage and room for a freezer.\nThe detached double garage, which is fully insulated and dry-walled, has plenty of built-in shelving and large work benches, and an RV parking pad. The south-facing back yard is big enough to toss a ball, and boasts partial privacy fencing, raised flower beds and a fire pit.\nThank you for viewing this For Sale by Owner property in Leduc, Alberta.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I was fortunate ample to get good climate to go fishing some previous week. I cant bear in mind all my groups names but the fishing was extremely sturdy. I started out on friday with a hot spot that was no for a longer time sizzling. We labored all around and picked up some blackfin on the troll ahead of we doubled up on wahoo and obtained one particular. No more bites at that spot so we moved inside a minor little bit and jigged up a limit of amberjack on my new talica 16 which is an awesome jigging reel. We picked up one more wahoo at the subsequent spot prior to we created it over to our sweet spot. We snagged a triple header on wahoo and made it out with two of them and picked up a 50 percent dozen jumbo blackfins on the troll.\nI went back out sat with the hopes of locating some fish but it was like a light change and each and every wherever I went was lifeless and we ended up with a grand total of 4 blackfin for the day. No wahoo bites whatsoever.\nTook off a day for climate then went again out Monday with Judy and her husband. We picked up one particular blackfin at our initial end but that was it. Strike a couple of much more rigs with no luck just before we created it over to the ole lump. Very first pass on the rig yielded a 91 pound wahoo in the box. Up coming pass was an amberjack. Immediately after that we got a few blackfin and one particular a lot more wahoo before the bite stopped. We went about and commenced chunking and quickly started out obtaining blackfin in the slick. We popped a number of of individuals in the boat quickly along with two little yellowfin prior to the sharks acquired also negative.\nTuesday I had my very good buddy Paul and his neighbor on the boat with me. The wahoo bite was definitly on as we place 5 in the boat rapidly which includes a totally productive triple header. We experimented with the lump once more with no adore just before I headed south searching for tuna. Very first pass by the rig we doubled up on thirty pound yf on the troll. Right after that we got one much more on the troll just before they moved a small deeper. I Began chunking and we caught two much more that way on seven\/ hooks and sixty pound frenzy flouro leader. Proper before we left they started popping and Paul landed two on frenzy poppers. Made 1 far more stop at the very first rig on the way in and obtained a bonus 70 pound wahoo to finish the day.\nWed was a day off prior to we acquired back again out to hit thursday. I had a team in from georgia that wished some huge fish and they obtained it. We jigged up a few blackfin early ahead of we obtained to perform on the ajs and put our limit of ajs in the boat with are living hardtails. Initial time in a while Ive struggled to jig up my ajs.\nWe headed out to make some tuna drifts and it was slow to start just before Josh Howard began selecting absent at blackfins and advised me to arrive above by him. We all picked at the blackfins ahead of a number of yf showed up. We had a screaming hit on a 50 and I realized it was a yf. All the guys ended up tuckered out so halfway by way of the battle they put me on it and I hammered down on the fish and place a 132 in the boat.\nI had Michael Pittman run a private boat for me and he had a stellar day as nicely as he ended up with about three great yf and a mako.\nWell listed here is what has been narrowed down. I listen to we require to enhance our harvest by 40% I Believe.\n19.1\/7 17.five\/18inch fish Might 26 till Sept 23.\nMarch 1st in Toms River the final determination will be made.\nAs a previous tackle shop proprietor, I constantly appear for the longest year attainable. Would really like to see a slot limit as most fish 18inches and above are breeding females. Shocked although five many years back the fishery was on the verge of collapse (in accordance to NOAA) now we require to increase the Harvest.\nOur Grandfather's are brothers that married sisters so we spent a whole lot of time jointly expanding up and we even now try and see every other on a normal foundation. April's Grand father Sam (Uncle Sam to me) handed absent a few a long time in the past and april has been cleaning and keeping on to things from the previous. So while we were in city, she had gone by means of boxes of pictures of the family's about the several years to see if we wanted any. So she arrived throughout an previous album and showed me it and I observed it and immediately thought it would be cool to reveal some of these pic's with you all here.\nMy uncle Sam was an Outdoorsman by way of and via, Hunter, Trapper, and Fishermen. Most of his exploits have been in the swamps, fields, river and bays in his beloved Maryland.\nDuty to his place took him from this land as a youthful lad and stationed him in Panama in 1942 these photographs had been taken of many fishing trips him and his buddies had even though there.\nSadly most of his generation are gone, but without having them who knows what our planet would be like right now.\nIf there is anybody hunting to pitch in for fuel\/ice and many others to fish in Grand Isle Louisiana or has any earlier encounter there must allow me know. Bringing our 36 Contender there March 1st by means of March 11th. Just send out me a concept on right here or e-mail me at joerp22@yahoo.com and permit me know when you would want to go. Likely to be really going for tuna\/wahoo.\nThe Outer Banks fishing economy has been under assault, seems as this is a way to combat again to keep the beach locations open and available.\nDo any of yall reccomend any charters or fly fishing guides in cancun. The costs i have noticed online were $ 330-$ 550 for a four to eight hr excursion offshore and $ 250-$ 375 for a flyfishing guidebook. Is this reasonable? Do yall reccomend anybody for both of the two for around the very same cost? I will be there in the center of september for a week.\nHas anyone been arrested in conjunction with this poaching seizure?\nWell it was just about a yr that this story broke, has any person ever been arrested in regards to this circumstance? The previous I observed the excellent aged boy network was protecting its own and no a single was talking to the authorities.\nNormal Resource Police officers uncovered a key fish poaching ring and about three tons of illegally caught striped bass along with it. The operation integrated a string of nets the officers learned working for about 900 yards and anchored off the southern tip of Kent Island, Maryland. The fish ended up in the nets when they were discovered.\nThe haul was so massive, a single of the largest seizures in All-natural Source Police background, that a 73-foot icebreaker had to be named in to support get them out of the water. Unfortunately, the poachers weren't part of the haul and are nevertheless at significant.\nIf they are caught, the poachers could confront numerous criminal costs, this sort of as environment nets out of season and possessing illegal nets, and, based on the actual charges and whether or not the poachers have priors, they could encounter hundreds of bucks in fines and feasible jail time.\n\u2026The very first operation began on Monday, as officers on regimen patrol dragged grappling hooks by way of the water, looking for illegal nets, Windemuth said. The drift gill internet period for rockfish had closed in mid-January, but was owing to reopen yesterday. Officers sometimes find nets set out by watermen improperly or way too early.\n\u2026Using the grappling hooks, the officers found the nets at about 2 p.m. on Monday. Gill nets hang vertically in the drinking water, so fish swimming by are trapped in the openings. In Maryland, only drift gill nets \u2014 which aren't anchored and can drift in the water \u2014 are authorized. They must be marked with floats on the surface and the watermen need to be in the area. These nets underneath the surface area weren't marked and there had been no watermen about, Windemuth said.\nThese nets under the surface area weren't marked and there ended up no watermen about, Windemuth mentioned.\nThe NRP officers \u2014 Cpl. Roy Rafter, Officer Greg Harris and Officer Drew Wilson \u2014 made a decision to lie in wait, to see if they could spot the poacher or poachers in action. \"It was chilly, sleeting, snowing, raining and all that, and we had these officers out in the middle of the bay,\u2020Windemuth mentioned.\nThe cops stayed out all night time, but the poachers by no means showed.\nSo by 7 a.m. yesterday, the officers made a decision not to let the illegal nets remain in the water any longer, so as to decrease the range of fish and other critters caught in the net. \"While this internet is in the h2o, it's continuously catching fish, snaring ducks and geese and whatsoever may well swim through it,\u2020Windemuth explained.\nWhen the officers began to haul in the nets, they weren't quite confident what they were dealing with.\nBy mid-morning, however, they had hundreds of ft of net and basically a ton of fish weighing down their patrol boat. That's when they named in the Widener.\nThe Widener wound up hauling up about another two tons of fish.\nThe fish and 900 yards of nets had been taken to the Section of All-natural Resources facility at Matapeake on Kent Island, in which cops and biologists expended most of yesterday chopping the fish totally free and assessing the scope of the bust.\nAll advised, they counted 529 fish weighing five,721 lbs, Windemuth mentioned. Some fish will be set aside for scientific analysis. Undersized and oversized fish will be donated to foodstuff pantries and shelters.\nResources to be utilised in efforts to help the rockfish populace.\nThe nets snared almost all rockfish, also called striped bass. There have been some mud shad and at least a single duck.\nWindemuth said watermen at present are getting about $ two.fifty per pound for rockfish, so the price of the poached fish could be close to $ 15,000. The nets are really worth about $ two,000, Windemuth mentioned.\n\u2026In addition, other watermen will undergo. The 3 tons of fish will count from the statewide quota of rockfish that can be caught by commercial watermen, Windemuth mentioned. This discovery of poaching functions will come not prolonged following a multistate investigation of ongoing rockfish poaching in the Chesapeake Bay wrapped up.\nThat multiagency investigation went on for many years in Maryland, Virginia and Washington, D.C., and nabbed about two dozen watermen, sellers and companies that have been accused of manipulating the state's fish-tagging method to catch a lot more fish.\nCanyon Runner \"Epic\" Bluefin Jiggin\/Popping \u2013 Join us!!\nWhat had been an untapped fishery for us in the previous busted broad open in 2010 with four weeks of flat out the best tuna fishing we at any time did and it obtained even greater in 2011.\nFrom the first week in March to the previous week in April the Canyon Runner crew ran out of Pirate's Cove Marina in Manteo, NC blue waters of the Gulf Stream and caught no less than an average of 10-fifteen bluefin for each journey in the a hundred-300 pound class.\nNot only that \u2013 most ended up caught on relatively mild jigging and popping tackle.\nNot only that \u2013 we also nailed dozens of true large bluefin tuna from 350-600+ pounds.\nCapt Mark DeBlasio and crew named it with out query the best fishing we've actually done! And people fully in the know have previously re-booked and we have only a couple spots left.\nAs this kind of I wished to get this out now as we commence daily on March 2 and I doubt we will have even a single spot letbefore all the dates are gone.\nTrips on our 48\u2032 Canyon Runner are $ 1,799 for the entire boat and as low as $ 350\/spot on Open Boat trips Mon-Thurs. The Canyon Runner Crew will also be jogging trips on our sistership Assualt Weapon \u2013 journeys there are $ 1950 and $ 399\/man or woman. We utilised her all of April previous yr for overflow trips and will be runnign her solely this calendar year.\nA Redondo Beach front fire captain who caught more than 800 pounds of tuna on a current sportfishing trip off San Diego is beneath investigation for allegedly marketing some of it from the back of his truck, authorities said Tuesday.\nCapt. Scott Harper may have violated state laws that prohibit activity fishermen from offering their catches.\n&quotAnything you catch is for personal use or to give it away to pals or family, but you can in no way sell,&quot mentioned Andrew Hughan, a spokesman for the condition Fish and Game section.\nHughan declined to discuss the scenario that was opened final week in San Diego.\n&quotThe Department of Fish and Match has an ongoing investigation into this incident and we cannot comment more until eventually the investigation is full,&quot Hughan explained.\nHarper could not be achieved for remark. He could perhaps experience a misdemeanor that could consequence in an $ eight,000 good, officials explained.\nRedondo Seashore Fire Chief Dan Madrigal mentioned Fish and Recreation authorities notified him of the scenario on Monday.\nMadrigal said he would open his very own investigation. Fish and Game authorities had not supplied him any information about the situation, which occurred while Harper was off-duty.\nRooster III, which docked at H&ampM Landing in San Diego Feb. three following a fifteen-day Bluefin Jim and Pelagic charter, according to the site, Redrooster3.com.\nThe journey produced eighteen tuna weighing more than two hundred kilos and about 100 wahoo.\nHarper also caught two tunas weighing 270 and 252 pounds, the website mentioned.\nAlso on the trip was Division Chief Paul Lepore, who caught a 204-pound tuna, the internet site explained. A 3rd firefighter also was on the excursion.\nThe Red Rooster III, a 105-foot sportfishing yacht, operates journeys from Lee Palm Sportfishers in San Diego into Mexican waters.\nPeople who fish in Mexican waters must have Mexican fishing licenses. Mexican regulation also makes it unlawful to receive financial acquire from fish caught while sportfishing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Route 17 Interchange Maintenance Facility - Perreca Electric Co. Inc.\nThe Maintenance Facilities is structured to an environmentally friendly green technology into the design of the buildings to enhance energy efficiency. The Maintenance Facility includes four Buildings; cold Storage Building, Maintenance Building, two Salt Storage Buildings, and a Vehicle Storage building. A new State Police Barracks Building. The new toll facilities included a new toll utility building for administrative needs and Toll Operations, Entering and exiting toll plazas with high-speed E-ZPass and covered walkways between the plazas and the toll utility building.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Huntersville Elementary School serves 750 students in grades Kindergarten-5.\nThe percentage of students achieving proficiency in Math is 83% (which is higher than the North Carolina state average of 47%) for the 2015-16 school year. The percentage of students achieving proficiency in Reading\/Language Arts is 71% (which is higher than the North Carolina state average of 47%) for the 2015-16 school year.\nHuntersville Elementary School placed in the top 5% of all schools in North Carolina for overall test scores (math proficiency is top 5%, and reading proficiency is top 10%) for the 2015-16 school year.\nMinority enrollment is 34% of the student body (majority Hispanic), which is lower than the North Carolina state average of 50%.\nHuntersville Elementary School's student population of 750 students has stayed relatively flat over five school years.\nHuntersville Elementary School is ranked within the top 5% of all 2,538 schools in North Carolina (based off of combined Math and Reading proficiency testing data) for the 2015-16 school year.\nThe diversity score of Huntersville Elementary School is 0.52, which is more than the diversity score at state average of 0.46. The school's diversity has stayed relatively flat over five school years.\nHuntersville Elementary School 4th Grade Concert - Time after time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LOS ANGELES \u2013 Ashton Kutcher filed court papers Friday to end his seven-year marriage to actress Demi Moore.\nThe actor's divorce petition cites irreconcilable differences and does not list a date that the couple separated. Moore announced last year that she was ending her marriage to the actor 15 years her junior, but she never filed a petition.\nKutcher's filing does not indicate that the couple has a prenuptial agreement. The filing states Kutcher signed the document Friday, hours before it was filed in Los Angeles Superior Court.\nKutcher currently stars on CBS' \"Two and a Half Men.\"\nMessages sent to Kutcher's and Moore's publicists were not immediately returned Friday.\nMoore, 50, and Kutcher, 34, created the DNA Foundation, also known as the Demi and Ashton Foundation, in 2010 to combat the organized sexual exploitation of girls around the globe. They later lent their support to the United Nations' efforts to fight human trafficking, a scourge the international organization estimates affects about 2.5 million people worldwide.\nMoore was previously married to actor Bruce Willis for 13 years. They had three daughters together -- Rumer, Scout and Tallulah Belle -- before divorcing in 2000. Willis later married model-actress Emma Heming in an intimate 2009 ceremony at his home in Parrot Cay in the Turks and Caicos Islands that attended by their children, as well as Moore and Kutcher.\nKutcher has been dating former \"That `70s Show\" co-star Mila Kunis.\nThe divorce filing was first reported Friday by People magazine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I like a husband who doesn't make the other normal husbands look bad.\nGot a link to this post in your email newsletter \u2014 love it. My husband never treated me like a princess either. He is from Germany, and I came from small town in Louisiana, where people have high, southern expectations in gentlemen. When he didn't fulfill those expectations, people were very uncomfortable, but I realized that I loved being treated like an equal and a partner instead of fine china. A friend of ours told my husband that's how he needed to treat me \u2026 like fine china \u2026 fragile and pretty and serving no real purpose? Yikes. I'm with you on this one!\nAnd as a side, I agree that Disney has been doing better lately in moving away from limp-noodle ladies. I thought Frozen was phenomenal in that regard \u2014 people with depth and struggles and a hint of emotional and romantic realism? The ultimate saving moment doesn't depend on a kiss from a dude? Wuuuuut?!\nNo way! I always think of Lafayette as pure Cajun country. I'm from Covington\/Mandeville, just on the north side of lake Pontchartrain; I was also born and raised there until I got married. Now I live near my husband's hometown in Northern Bavaria. I'm hoping visit LA later this fall if we can find someone to babysit our cats!\nHaha, that's so crazy. I know the Covington\/Mandeville area well. And you are right. Lafayette is pure Cajun country. But I somehow escaped with no accent. (Mainly because my parents are originally from there). Good luck with finding a cat-sitter! (Something else I can relate to. We have two cats that a friend is keeping while we're living abroad). How do you like Bavaria? I imagine it being very hilly and with tons of forests, but maybe that's not right at all, haha.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Top equitation trainer Andre Dignelli helps you master these tricky narrow fences in five doable steps.\nOne of the great things about our sport is the continual innovation in course design. Every year, designers trot out new jumps made of unusual materials, in different shapes (liverpools, walls, gates, rolltops, and so on) and bright colors, and with such potentially distracting decorations as flowers and bushes. These innovations not only make a course more attractive; they mix things up. By offering an ongoing series of challenges (to a horse, anything new, strange, or odd-looking can be spooky or scary), they keep adding layers of difficulty to competition.\nOne innovation you've probably seen is the narrow fence?or, as it's popularly called, the \"skinny.\" Instead of the standard horse-show width of 12 feet, a skinny's width can run from 8 feet all the way down to 5 1\/2. We start seeing skinnies with wings in Limit Equitation, and without wings in the \"big Eq\" (open equitation and Medal and Maclay classes) and on cross-country courses starting at Training Level. The USEF Rule Book suggests skinnies for Limit Equitation classes and above; it requires them in the higher-end eq classes.\nA skinny is a jump for which, no matter how many times you practice it, you can't afford to let down your guard. If you don't have a good distance and a straight horse, you're going to end up jumping to the side of the fence, running by it, or chipping in.\nAppearance. A skinny can be a solid wall, a rolltop, a wooden garden gate, or just about any other shape and material the course designer can dream up. Many skinnies are used without the visual \"framing\" aid of a nice wide set of standards, leaving you trying to steer to the middle of a stand-alone wall or a rolltop with nothing but a couple of low shrubs to the sides. Even if the skinny is a gate that requires standards for support, they're sure to be narrow ones?and in many cases they stand no higher than the gate itself.\nMaterials. Skinny rails are unusually light?and so more easily pulled?because they're half the length of regular rails. When the jump is a gate, it may be nothing more than PVC. (You can buy the exact same garden gate used at the 2003 Medal Finals for about $20 at your local Home Depot.) To jump such an obstacle in good style?and leave it up?you have to ride incredibly accurately and softly.\nLocation. On many courses, a skinny is positioned where it would ride more easily if it were a wider fence. Often, in fact, a skinny is the \"meat\" of a course because it tests your horse's ridability, his carefulness, and his thinking. Complications can include difficult striding: off an awkward turn; on a half-stride, so you have to pick and choose; or with very little room to get to it. In the 2002 Medal Finals, for example, the skinny?which was already tough because it was upright and had no groundline?was made even harder because it came at the short end of the ring, off a very challenging turn. In the 2003 Medal Findals, that PVC skinny was placed in the middle of a bending line?which added timing to the already considerable issues of line, track, and steering that the riders had to contend with.\nFortunately, in general, those situations turn up only in the bigger jumper or equitation classes. In the lower classes, a skinny still tends to be used in a very inviting way, so you and your horse have a chance to start getting familiar with its appearance and with what it takes to jump it, almost like a liverpool.\nI've told you why skinnies are difficult. Now I'm going to talk you through my nearly fail-safe five-step system for developing the skills you and your horse need to master them. By starting at a very low level, with an inviting, unintimidating little flowerbox set on a diagonal or in a very straightforward line?not a tricky line or on a complicated part of the course?you'll show your horse that hopping over a narrow obstacle is easier than running off to the side to avoid the effort. Once he understands that, you can systematically build from there.\nHere are the five steps. (Because your track, pace, and aids are essentially the same for the welcoming first easy step as they are for the challenging final one, I'll lay out the steps in order, then follow with a detailed description of how to ride them.) Just make sure that you and your horse are completely confident going over the obstacle in one step before you move on to the next.\nStep 1. Set up two 5 1\/2-foot flowerboxes or (as in our photos) low walls, side by side, so you have a full-width obstacle to jump, but without standards or groundrails.\nStep 2. Take away one flowerbox or little wall and center the remaining one on the track. Add flowerbox wings on either side, slightly angled, so your horse has a bit of a chute to guide him to the skinny and keep him from running past.\nStep 3. Raise the height of the skinny by stacking the spare low wall or empty flowerbox on top of the one you've been jumping. Keep the flowerbox wings on either side.\nStep 4. Remove the flowerbox wings altogether and you're jumping a bona fide skinny: no standards, no wings, no groundrails. Now you must focus more on track?your straightness?and pace.\nStep 5. Select a regular-width fence that you've placed somewhere on course and replace it with the doubled-up skinny.\nWhat's the single most important thing you can do to make sure your horse jumps a skinny successfully? Come on a slightly collected stride, so you're absolutely certain that both of you see it. The earlier you set up your track and distance, the better your chances are.\nYour legs and hands have to stand in for the missing jump standards, giving your horse nowhere to go but over that narrow obstacle. Maintain a solid and secure supporting leg on both sides of him; keep a good connection, with a nice straight line from your elbow to the bit, and your hands a little wider than usual: about as far apart as the width of your bit (in most cases, about 5 inches). Now guide and close him in with your arms and hands.\nOnce you're pretty sure of your distance, look at the flowerbox or wall. I see kids jumping all the time holding their heads up in the air, not looking at the jump they're about to jump. Well, I stare right at a jump on the approach, and I suggest you do the same. Keep your eyes on the flowerbox or the wall until you know exactly where you are. If you lower your guard?by looking at the end of the ring or over at the next fence?before you finish the skinny that's in front of you, your horse is going to stop or run past it. Once you're in the air, you can then go ahead and raise your eye to look for a focal point at the end of the ring or the next fence.\nIf Your Horse Weaves ?\n? or \"wilts\" or almost starts cantering backward, you have a quick choice to make; I call it \"Plan A or Plan B.\"\nPlan A is to still try to win, leave the skinny up, and jump it stylishly. To do that, you want a very connected, collected canter; use a bit more leg pressure and keep a bit more feel of his mouth, so he's really going forward and in your hand.\nPlan B is to opt for survival. If you feel you're in jeopardy of not getting over the jump, or chipping, or leaving dangerously long, ride more defensively. Yes, you want to ride as accurately and softly as you can; but if you feel that your horse is really wilting or cowering, give him a stronger ride and take the chance that you're going to knock down the skinny?because the alternative is that you may not get over it at all. Make sure to sit deeper in the saddle, apply more leg pressure, and maybe add a cluck or a touch of the stick or spur. You may even want to add a stride in which you can give him more support and keep him a little better connected.\nThis article originally appeared in the November 2004 issue of Practical Horseman magazine. Read more about Andre in the June 2012 issue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is there a discrepancy with the equivalence principle?\nSelf educated since age 11 at \"University of Life\"\nRe: Is there a discrepancy with the equivalence principle?\n+relative speed, or centripetal speed = time going slower.\n= same rate of time at sea level each longitude.\nHere we are saying that +mass is the same as -centrifugal force.\n= 2 faster, 2 slower.\nHere I am saying that + mass and -centrifugal force are NOT the same thing!\nNow because where there is more mass\/gravity, 'time goes faster', this gives a physical cause and effect description of why GRACE moves faster over the mountain than over the valley.\nSorry for taking so long to reply to this but I've been away on vacation.\nYou cannot have +\/- Relative speed If a clock is moving relative to you, you measure it as running slow.\nAs far as tangential speed and \"centrifugal force\" go you can calculate time dilation using either one, but not both. If you are calculating it from an inertial frame, then you use tangential speed, If you are calculating it from the non-inertial rotating frame, then you can use the potential due to the apparent centrifugal force. What you cannot do is combine their effects, because they are just two ways of looking at the same thing.\nIf you want to work out the time dilation for a clock traveling in a circle all you need to know is its tangential speed. (If you put two clocks in circular paths with different radii but traveling at the same tangential speed, they will run at the same rate even though one of them is undergoing a larger centripetal acceleration.\nIf you want to work gravitational time dilation in, you can do use the following for the total time dilation as measured from the inertial frame.\nT = t\/sqrt(1-2GM\/rc^2- w^2r^2\/c^2) where w is the angular velocity of the rotation. If you want to do in the rotational frame you have to use the integral of the centrifugal force, and you end up with the same equation.\nAn oblate spheroid adds an additional complication. First consider the scenario with no rotation. This can be look at a like a spherical body with a world girdling \"hill\" around the equator. In this situation, walking from either pole to the equator is walking \"uphill\" and you are moving to a higher gravitational potential, where clocks run faster than they do at the poles. If the \"hills\" were mass-less then you could use the simply version of the gravitational time dilation equation to work out just how much faster.\nBut since they aren't, it is a bit more complicated. The way the mass is distributed will have an effect, and it would take a bit of additional calculation to work out just how much difference this will account for.\nIf the Body is rotating and had no gravity, then someone moving from pole to equator would be effectively moving \"downhill\" to a lower potential causing clocks to run slower at the equator than at the poles, since the tangential speed is greatest at the equator.\nAs you say, in adding the mass considerations of an oblate spheroid, that is also rotating (has in fact been caused by roatating), matters become more complicated than 2 time dilations cancelling each other out. And because our equitorial bulge consideration is oblate and rotating there would seem a little more finess to the reality that 'could' be examined.\nMy point here is that if one sets up a spinning wheel with 2 identical objects attached by string to the wheel - where 1 string is longer than the other - then the object on the longer string will travel at a greater speed, the outward force will be stronger, and the longer string will 'feel' a greater tension, than the shorter string.\nGRACE is showing us that while our gravimeters 'on' the mountain register lower 'inward' gravitational pull, that gravity up a mountain is much stronger than in the valley.\nStronger gravity is supposed (not proven) to slow time down. Increased centripetal speed (relative motion) is proven to slow time down.\nSo at the top of a mountain there are 2 aspects (conventional physics) slowing time down.\nIt is proven that a clock ticks faster at the top of the mountain.\nThe 'conventional calculation' say that this is due to the weaker gravity that the clock feels at top of mountain (as per our gravimeters).\nI am saying (as per GRACE) that the clock is not 'feeling' less gravity at the top of mountain. If it were feeling less gravity it would fly off the top of the mountain with the extra centrifugal force at that height.\nWhat I see is the possibility that the clock\/mass at top of mountain is 'feeling' a tension caused by two opposing forces. The inward pull of stronger gravity, and the outward push of stronger centrifugal force.\nIt would appear to me that with regards to our 'real life example' of time dilation on earth, which is a non spherical\/rotating example, that the 'conventional calculations' are considering that increased centrifugal force is the same thing as weaker gravity.\nIf the 'conventional calculations' factor in the total gravity at top of mountain, this slows time down, as well as the increase in centripetal speed slowing time down. It is then centrifugal force alone that speeds time up.\nThis elicits 3 aspects that cause slower time versus 1 aspect that causes faster time, so 'conventional maths' cannot include the 'extra gravity' that the centrifugal force is opposing, because clocks will not run at same rate at sea level under a 3 aspects cause slower time and 1 aspect causes faster time remit.\nThat is why my theory suggests that adding gravity (adding mass) causes time to go faster, and minus centrifugal force causes time to go slower, where height is adding centrifugal force causing time to go faster, and increased centripetal speed at height causes time to go slower.\nNow, with 2 aspects causing time to go faster, and 2 aspects causing time to go slower, these aspects 'can' cancel for same rate of time at sea level of every longitude of the equatorial bulge.\nWhat I am suggesting @Janus is a different means of calculating the same observations.\n...And provides a falsifiable prediction for a test of general relativity, where a precision clock can be placed at LIGO to undergo conditions (when a gravity wave hits) that do not incorporate a change in centrifugal force via hieght, and do not include a change in centripetal speed, but only include an addition of the gravity of the wave for the duration of the hit.\nGeneral Relativity predicts that clock will run slower.\nMy theory predicts that clock will run faster.\nMy theory is saying that the centrifugal force itself is having an effect on the clocks in addition to gravity having an effect on clocks.\nRelativity considerations are saying that centrifugal force has an effect on gravity, and that gravity effects the clocks.\nRelativity says that GRACE is travelling in slower time over the mountains in the stronger gravity, and that the acceleration is curvature related.\nMy theory is saying that GRACE is travelling in faster time over the mountains in the stronger gravity, and that the acceleration GRACE feels is due to the faster rate of time.\n1\/ The GRACE experiment itself, in determining the lead craft as accelerating in the stronger gravity, is bouncing electromagnetic radiation off the leading craft. That electromagnetic radiation 'should' (according to GR) be moving at a slower rate, in the slower time, in the stronger gravity.\nThis, of course, would account for the electromagnetic radiation taking longer to return when it is bounced off the lead craft over the mountains. But I hardly think for a minute that the GRACE experimenters will have neglected to account for this factor in stating the lead craft as 'moving faster' over the mountains...?\n2\/ Gravimeters display gals per cm per second squared.\nA second squared up a mountain is different to a second squared in the valley. An up a mountain shorter second squared may affect magnitude of gals per cm measurement.\nWhere GRACE finds stronger gravity over a mountain, a gravimeter at top of mountain is 'potentially' measuring via a shorter second squared, as compared to a measurement in the valley and this will amount to less gals per cm per shorter second squared.\nSo - given that GRACE shows that gravity is stronger at top of mountain than in valley, can we all agree that despite how time dilation is conventionally calculated via weaker gravity at the top of mountain, the clock is actually 'feeling' stronger gravity at the top of the mountain than in the valley?\nGRACE doesn't show this, it produces a Gravity anomaly map. Gravity anomaly maps show how much the Earth's actual gravity field differs from the gravity field of a uniform, featureless earth surface with homogeneous density often called the ellipsoid. If you look at the theoretical calculation you will see that it assumes a homogeneous perfect sphere. If you are going to make comparisons with a real system you have to take into account uneven mass distribution, which is what GRACE does.\nIf we look at the Andes, on a uniform earth the value of g at sea level would be approx 9.8 and at a ht of 1km it would be 9.797, however if we take GRACE anomaly measurements into account the values would be less by 5x10-7 ie 9.7999995 and 9.7969995 respectively so the value of g at the top of the Andes is always less than that at the bottom so the tops are not feeling a stronger gravity than the valley.\nHowever, it is still important to note what @Janus says, that the dilation does not depend on g but on the gravitational potential.\nSo where the GRACE anomaly map shows red at top of mountains, and it states that anywhere red is showing a stronger gravitational acceleration, are you saying that that the red areas are not showing stronger gravitational acceleration?\nI am aware that gravity potential is calculated as a 'theoretical calculation' assuming a homogeneous perfect sphere, and @Janus has shown how there are 2 methods of 'conventionally calculating' time dilation, but that they cannot be used together.\n\"As far as tangential speed and \"centrifugal force\" go you can calculate time dilation using either one, but not both. If you are calculating it from an inertial frame, then you use tangential speed, If you are calculating it from the non-inertial rotating frame, then you can use the potential due to the apparent centrifugal force. What you cannot do is combine their effects, because they are just two ways of looking at the same thing.\"\n...And what I want to do is to calculate the rotational aspects of relative motion time dilation, and the gravity potential anomalies aspects of time dilation together, because that is what we observe occurring on our planet.\n...And I want to do this as a modification of general relativity where the factor of the gravity that clearly holds a clock to the top of a mountain 'is' included. This being because this extra gravity holding the clock to the mountain against the centrifugal force 'is' a factor that exists, isn't it?\nEdit: I just looked at the GRACE gravity map again, and it clearly show red at the top of mountains, and it show a graph that suggests that red is equal to stronger gravity. I can't be reading that wrongly, can I?\nIf you look at the scale you can see it describes the diagram as showing gravity field anomolies, and the units are in milligals so it is clearly not an absolute gravity measurement.\nLooking down from 200 km to the top of a mountain, there is obviously more mass immediately below the satellite than at the same altitude over the ocean, so there will be a positive gravitational anomaly compared with an idealised uniform sphere.\nF = GMm\/r^2, as always, but at only 200 km above a 6000 km knobbly geoid, M, r and the net direction of F on m (remember Maskeleyne at Schiehallion) do not approximate to the ideal isotropic and point values that apply in the laboratory or astronomy.\nYes, you are reading the map wrong, In at least one way. You keep wanting to interpret the GRACE readings as the strength of gravity at ground level, When they represent the fluctuations from the mean value of gravity measured at the altitude of the Satellites. To get how gravity changes at ground level, you would have to combine this information with exactly how far ground level is from the center of the Earth at the point of interest. The gravity map supplied by grace doesn't include this second bit of information.\nThere are areas where higher gravity is associated with mountains, bit also areas where it is associated with low spots, such as out in the North Atlantic. You weren't under the impression that the \"bumpiness\" in the Gravity map represented elevation of the ground at that point were you? It is just a way of \"doubling up\" on how they represented the gravity measurements. They are shown in both color and and relief.\nIf you want to look for a correlation between the gravity map and geological features, this tectonic plate map is of interest.\nNote that there is some correlation between where plates meet and high gravity areas( including that region in the North Atlantic.) This is likely due to how local density of the crust is effected. In certain places, this will also correlate with mountian ranges ( West coast of the Americas and the Himalayas, for example), but that is because these meeting points between plates are subduction zones, where mountains tend to be built.\nHere u is the gravitational potential at the point you are interested in.\nr is the radial distance from the axis of rotation for you point of interest.\nWorking out the exact value of u when taking into account the various fluctuations in Earth's gravity field is the finicky part. But to be honest, it isn't really going to amount to much.\nGravitational potential is, in effect, a measure of energy per unit mass needed to move from one point relative to the gravity source to a point an infinite distance from the gravity source.\nNow as you first start lifting an object from the surface of the Earth, variations in local gravity will have some effect on the energy needed to lift the object a given height when you are close to the surface, But as you get further from the Earth, this becomes less and less of a factor and it gets washed out by the overall effect of the total mass of the Earth. (the further you get from the Earth the more it gravitational behaves as a single point source). This means that the small local variations in surface gravity have a minuscule effect on the gravitational potential at any point when compared to, say, the difference caused by being a few kilometers further from the center of the Earth by being on a mountain top vs being at sea level.\n@Janus. I think you misunderstand my purpose.\nThat is all I am interested in. You have said that 'adding in' the 'extra mass' of each increase in the equatorial bulge would require some additional calculation.\nI am saying that 'if you add in' that extra mass that contitutes an increase in height of the bulge from poles to equator, where an aspect of +mass=time goes slower (conventional GR remit) must now be added to the calculation - this will NOT, under the conventional means of calculating, result in a constant rate of time at sea level of each longintude of the equatorial bulge.\nIt has been already worked out. There are two competing factors that determine the Earth's shape, its gravity and its spin. If it weren't for gravity holding it together, its spin would tear it apart. As it is, a compromise is reached and we get an oblate spheroid. The surface of this spheroid is a geoid, which means that local \"vertical\" is always perpendicular to the surface at any point of the surface. (which means that, unlike for the surface of a non-rotating spherical body, it doesn't always point towards the center of the body.). This also means that the surface of this geoid is an equi-potential surface. A good clue to this is that a fluid, such as water, will always settle to the lowest possible potential, and if the potential wasn't equal over the surface of the Earth, the oceans wouldn't follow its shape, but instead would pool up at either the poles or equator.\nThe shift in mass towards the equatorial bulge in of itself factors in to the final shape. It is not an additional factor that has to be factored in later when doing calculations, it is already involved in causing the surface to be equi-potential.\nClocks at sea level tick at the same rate no matter how you try to calculate it.\nNow if you wanted to, you could do all the math to determine the potential due to gravity alone at any point on the surface*, use that for u in the equation I gave in the my last post, along with angular velocity of the rotating Earth and the radial distance from the axis to get the time dilation that way, but you will get the same result.\n* There is a formula for determining the gravitational potential over the surface of an oblate spheroid, but they assume a constant density throughout the body, and the Earth is not uniformly dense.\nOk - Yes I already know all of these aspects that you mention.\nThe point is that I am interested in calculating time dilation in another way.\nAnyway, I thank you for your input. It's kind of you to take the time, but I'm not sure you understand what it is I am trying to do here.\nI am going to sign off now but will include, for the record, my falsifiable prediction for a doable experiment, including the remit of the experiment and why this experiment constitutes a test of GR that has not yet been conducted.\nA further test of General Relativity can be conducted by comparing clocks that are ONLY experiencing a change in gravity.\nSo far precision testing of GR with clocks has been conducted concerning change in height in the gravity potential, where the clock is, in addition to a change in gravity, also experiencing a change in centripetal speed and centrifugal force.\nAnd precision testing of GR has also been conducted concerning a change in relative motion, where the clock is not experiencing a change in gravity.\nIn Febuary this year portable precision clocks were tested for the first time.\nIt is my suggestion that these portable clocks should be placed in circumstance where ONLY a change of gravity is occurring, in order to confirm that General Relativity is indeed correct in assuming that an increase in gravity slows time down.\nThis could be conducted by placing a clock at 2 different locations at the same longitude, and at the same height above sea level, where there is a known density difference in the geology of the locations, and thus compare how they tick.\nThis will constitute only a difference in gravity.\nOr - much more simply, just place a clock at one of the gravity wave experiments and record how the clock ticks differently when a gravity wave hits, as compared to how it ticks normally.\nThis will also constitute only a change in gravity.\nGeneral Relativity predicts that a clock that experiences only a change (increase) in gravity will tick slower.\n*The consequences of the remit of my modification are, of course, extremely far reaching.\nHowever I just cannot do this trying to make people understand what it is that I am saying anymore.\nNow that these precision clocks are portable, I daresay that they are going to be used to further test the tenets of General Relativity.\nI have voiced my suggestion, and will now say no more.\nIt's come to my attention that the link I provided is not working anymore, or maybe it never did and no-one told me (although it works ok for me, scratches head).\nI'll have a read later.\nPage created in 0.152 seconds with 74 queries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you love that moment when the engine starts and that rich diesel sound fills the air? Do you get a kick out of moving earth..\nCities in Motion 2 (CIM2) is the sequel to the popular mass transit simulation game Cities in Motion. Build, manage and lead y..\nRendered in rich detail, four of the world's greatest cities - Vienna, Helsinki, Berlin, and Amsterdam - await the steady hand..\nHouse fires, traffic accidents and large-scale emergencies: Experience the day-to-day life of the fire department in \"Firefigh..\nStarting on a small property you start a career as a successful farmer and time by time, you earn new fields, tractors, machin..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sheenan, Carol. \"Bill Reid\". The Canadian Encyclopedia, 07 March 2018, Historica Canada. https:\/\/www.thecanadianencyclopedia.ca\/en\/article\/william-ronald-reid. Accessed 21 April 2019.\nReturning to Vancouver, he became involved with the creation of a monumental sculpture for the University of British Columbia, called Haida Village. Reid eventually became a recognized leading authority on Haida art and life.\nHaida artist Bill Reid is credited with the revival of Northwest Coast Indian arts (photo by Thomas King).\nWilliam Ronald Reid, sculptor (b at Victoria 12 Jan 1920; d at Vancouver 13 Mar 1998). An internationally recognized HAIDA artist, Bill Reid is frequently credited with the revival and innovative resurgence of Northwest Coast Indian arts in the contemporary world. Son of a Haida mother and a Scots-American father, Reid was a teenager before he knew of his native heritage. Later in life, while a CBC broadcaster, he studied jewellery and engraving at Ryerson Institute, Toronto (1948), and began investigating the arts of the Haida in 1951. Furthering these studies, he went to the Central School of Art and Design in London, England (1968).\nReturning to Vancouver, he became involved with the creation of a monumental sculpture for the University of British Columbia, called Haida Village. Reid eventually became a recognized leading authority on Haida art and life. Accomplished in many media, Reid carved in silver, gold, wood and argillite and cast in bronze. He issued several editions of serigraphs and illustrated and collaborated on many books, including The Raven Steals the Light (1984). Among his major works were the 4.5-ton cedar sculpture Raven and the First Humans in UBC's Museum of Anthropology (1980); a bronze killer whale sculpture, The Chief of the Undersea World, for the Vancouver Aquarium (1984); a canoe commissioned for Expo 86 (1986); and Spirit of Haida Gwaii, commissioned for the Canadian embassy in Washington, DC (1991).\nThis is a moment in the Bill Reid Rotunda at the Museum of Anthropology, University of British Columbia, that depicts the ancestral past of the Haida people when Raven, a wise and powerful yet mischievous trickster, found the first humans in a clam shell on the beach, and is coaxing them out of it.\nBill Reid was awarded an honorary doctorate from UBC in 1976, the Molson Prize in 1977 and the Lifetime Achievement Award, National Aboriginal Achievement Awards, sponsored by the Canadian Native Arts Foundation in 1994. Reid was an eloquent and outspoken proponent of native rights in Canada and was especially active in the battle to preserve the national and cultural history of South Moresby in Haida Gwaii.\nSpirit of Haida Gwaii: The Jade Canoe at the Vancouver International Airport.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This newly constructed training center and adjoining Yachana Lodge, an eco-lodge school, are the culmination of the Yachana Foundation's 24 years' experience in community-based solutions to poverty and environmental conservation along the Napo River in Ecuador's Amazon region. US born Douglas McMeekin struggled without support for his dyslexia to succeed in earning his BA at age 28. In 1986, following business failures in Kentucky, he visited friends in Ecuador. He became aware and appreciative of the wealth of knowledge of the indigenous peoples. Intuitively he realized his purpose was to contribute to solving the social and environmental problems in the Amazon. He founded the Yachana Foundation in 1991. Yachana in Quichua language means \"place of learning\". Failures of the existing bureaucratic educational system and its prejudice against indigenous and rural children focused his efforts to education of the youth. The foundation has improved lives through: building local community schools, health clinics, pharmacies, agricultural organizations, micro-finance projects, environmental sustainability programs, eco-tourism management and community development initiatives.\n25 computers with LED screens and 514 watts total energy consumption with low heat production as a side benefit. The hotel school is Ecuador's only eco-lodge that trains Amazon youth in hotel management and leadership and has received numerous eco-tourism awards. In close co-operation with the Ecuadoran Minister of Education, it employees certified teachers who provide long weekend classes in basic academic subjects preparing learners to meet traditional high school diploma requirements. Combined, these efforts develop leadership and technical skills that enable Amazon youth to remain in the rain forest region, find work or be entrepreneurs: a concept that could serve many remote regions around the globe. Ecuador's Minister of Education currently cannot meet the minimum requirements established by law in the remote rainforest and fails to serve this portion of its youth; a gap Yachana may well succeed in fulfilling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":">> and produce, for example, a report preview window on a Mac?\n>> After all, if you regard a browser, it's doing something very similar.\n> logical thing to create as the result of the process is an XML document.\n> an XML producing language.\nmaintainer of the product as it would for his\/her customers.\ndesired attributes of the target output.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Load your family into the 2019 Porsche Cayenne. Porsche made sure to keep road-handling and sportiness at the top of it's priority list. Under the hood you'll find a 6 cylinder engine with more than 300 horsepower, and all wheel drive keeps this model firmly attached to the road surface. Well tuned suspension and stability control deliver a spirited, yet composed, ride and drive The engine breathes better thanks to a turbocharger, improving both performance and economy. Porsche prioritized handling and performance with features such as: 1-touch window functionality, power front seats, a built-in garage door transmitter, a trip computer, fully automatic headlights, heated door mirrors, and leather upholstery. Rear passengers enjoy the seat heating functionality, keeping them warm during the winter months. For drivers who enjoy the natural environment, a power moon roof allows an infusion of fresh air. Enjoy your favorite music via the stereo system, which includes a CD player with MP3 capability, steering wheel mounted audio controls, A 40 gigabyte hard drive, and 14 speakers, providing excellent sound throughout the cabin. Porsche ensures the safety and security of its passengers with equipment such as: head curtain airbags, front and rear side impact airbags, traction control, brake assist, ignition disabling, an emergency communication system, and 4 wheel disc brakes with ABS. You'll never lose visibility with rain sensing wipers, which activate automatically when the drops start to fall. We have the vehicle you've been searching for at a price you can afford. Stop by our dealership or give us a call for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Motor Marks Cherished Numbers and Personalised Registration Plates are always looking to purchase more personalised registrations for their existing clients and general stock.\nIf you have a registration number that you would like to sell outright rather than have us advertise it for you please follow the link below and fill out the form.\nThe Driver and Vehicle Licensing Agency, or DVLA as they are more commonly known, produce a periodical newsletter for external stakeholders, commercial customers and suppliers.\nWe collect such information to enable us to process your number plate order and to provide you with an efficient customer service.\nThe personal information we collect will be held securely and will not be passed onto a third party.\nIf you have any questions\/comments or would like any further information please contact our office on 0116 235 0116 where you can speak to one of our experienced staff.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chinese artist Cao Fei explores shifting relationships between humanity, technology, and the city\u2014and builds alternative worlds. In 2017, she was awarded the prestigious BMW art-car commission.\nConnecting ancient spiritual beliefs with contemporary technology, and art with industry, she transformed a BMW M6 GT6 into a vehicle for virtual and spiritual travel. 'Unmanned', the project's film component, follows a time-travelling monk into a virtual future.\nThe film is itself a vehicle\u2014a hypnotic mash of ad, music clip, and sci-fi video art. The show also includes an earlier Cao Fei film, playing in the Auditorium. 'La Town' (2014) offers another fantastical, but dystopian, vision, told through model architectural sets and figures.\n'La Town' screens daily on the half hour, 10.30am\u20133.30pm.\n*Warning: 'La Town' contains adult themes. Parental guidance recommended.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mr. Bigg's \u2013 Ile Ife, Osun State Nigeria .\nNigeria have laws protecting local craftsmen, farmers and artisans from foreign imports. Yet you won't find fat cats in Asokoro, Maitama, Ikoyi or Victoria Island ordering furniture from Sunny O in Dei Dei Lumberyards when they can import it from the latest IKEA catalog.\nOkene \u2013 Igbira People \u2013 Kogi State, Nigeria., a photo by Jujufilms on Flickr.\nThe Igbira people predominantly live along the River Benue plains, down and beyond the River Niger in Okene, Okehi, Adavi and Ajaokuta local government areas, and in parts of Nasarawa and Edo states.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"As a chamber musician, Jamie has had the opportunity to collaborate with members of the Tak\u00e1cs, Cleveland, Juilliard, Ying, Cavani, and Malibran Quartets, and individuals including M\u00e1t\u00e9 Sz\u00fccs of the Berlin Philharmonic, Kim Kashkashian, Carol Rodland, and Oleh Krysa. She has also had the opportunity to play for and study with such distinguished musicians as Laurence Lesser, Ralph Kirshbaum, Pieter Wispelwey, Gary Hoffman, Lluis Claret, and Steven Isserlis.\nBoth an enthusiastic chamber musician and entrepreneur of collaborative outreach programs, Jamie is a Co-Founder and Artistic Director of the Flatirons Chamber Music Festival. The festival celebrated its inaugural season \"Crossing the Divide\" in the summer of 2017. She has also participated in a variety of international music festivals including Thy Chamber Music Festival, The Holland Music Sessions, the London Masterclasses, and Violoncelles \u00e1 Fontfroide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How many chain restaurants can draw lines everyday, from opening to closing, numbering well into the dozens on a \"slow\" day? What about an outpost that can claim a Michelin star? If you haven't already heard of Din Tai Fung, there's a good chance you've felt its impact on the overall culinary landscape whether you realize it or not. Born in Taiwan originally as a cooking oil purveyor, Din Tai Fung transitioned into the restaurant business in 1972 and has taken the world by storm ever since. Based primarily in Asia, the west coast has been blessed with a handful of these hallowed outposts, each one drawing rave reviews at a fevered pitch typically reserved for rarefied fine dining. Making a taste of the extraordinary accessible on a mainstream level is just one of their many triumphs.\nIt's been said that their xiao long bao, otherwise known as soup dumplings, are the absolute pinnacle of perfection; the very best example of the art, executed with the exact same mastery every single time despite being made by hand, in volumes that would boggle the sober mind. Unfortunately, that's not a debate I can weigh in on, as vegan soup dumplings are about as common as three-legged unicorns. Why bother with the wait, which can range from a minimum of one to three hours, then? Well, there's a whole lot more to this menu than just dough-encased parcels of pork.\nKeenly aware of their local audience, Americans are treated to clearly labeled options for vegetarian, vegan, and gluten-free dishes. Even without modification, overwhelming choices unfold with the turn of the page, particularly for vegetable-lovers with a penchant for spice.\nVegetable and Mushroom Dumplings surely can't compare to their plump, porky brethren, but offer a highly competent, crowd-pleasing combination of springy wonton wrappers and tender umami fillings. The same can be said for the Vegetable and Mushroom Bun, which simply replaces that thin and chewy exterior with a puffy, fluffy cloud of steamed white bread. Essential for enjoyment is the DIY dip you'll concoct from slivers of fresh ginger and black vinegar, mixed to taste.\nNo, that alone would not bring me running back to the Westfield Valley Fair mall where this Santa Clara locale has set up shop, of all places. It's the starters and sides that make this meal. Like Thanksgiving dinner, side dishes are the stars of this show.\nGo with a crowd and order every single plant-based appetizer because I can't imagine leaving without just a bite of each transcendent taste lingering on my tongue. Soy Noodle Salad, a cold composition of shredded bean curd, is an absolute necessity. Deceptively simple on the surface, masterfully balanced flavors play on every delicate strand, sparkling with gently salty, sweet, sour, bitter, and savory notes in such perfect harmony that one can't be fully separated from another. The Cucumber Salad arrives at the table like a statuesque work of edible art. Columns of stacked cylinders are crowned with a single clove of marinated garlic, which is a prize you'll want to fight for, by the way. Wood Ear Mushrooms in Vinegar Dressing may not resonate as universally, but for fungus fiends, this is slippery plateful of earthy bliss.\nFlip over to the section on greens and dig in deep. Every single dish here is completely vegan! Picking here comes down to personal preference, but don't sleep on the Sauteed String Beans, lightly blistered from the kiss of the wok and dripping with sizeable garlic chunks. Taiwanese Cabbage gets a similar treatment, providing one of the few great examples of the concept this side of the seas.\nDessert buns stuffed with red bean paste or taro also tempt for a sweet plant-based finish, but I can't personally vouch for these treats. Undone by an unreasonable attempt to eat through the full range of vegan specialties, I left feeling quite like an overstuffed dumpling myself.\nThough you may go for the dumplings, you'll inevitably come back for the vegetables.\nOne of the most buzz-worthy restaurant openings of recent months, much has already been written about FOB Kitchen in Oakland, California. Filipino food is quickly gaining mainstream traction, but remains relatively rare in an elevated dining atmosphere, particularly in an accessible, approachable format. Given the circumstances, such breathless anticipation can be forgiven. Clamoring for something new, the dining public is absolutely eating this stuff up- And not just for the novelty factor.\nIn a culture ruled by pork, vegan options are rare at best, but not so on the menu at the Temescal treasure. FOB Kitchen goes out of their way to provide meatless eats for their diverse, discerning Californian clientele.\nTofu-mushroom adobo, the entree that earns the most critical attention, is the result of recipe testing for over a year to get the flavors just right. Garlic-forward in the best way possible, aromatic onions and peppers give this lightly stewed combination its essential character, enhanced by a generous hand with the coconut oil for a luxurious eating experience.\nEnsalada talong, an unassuming eggplant salad, turned out to be the sleeper hit of the evening, dazzling with a bright vinegar dressing sprinkled across crisp jicama, tomato, with the surprising briny bite of sea asparagus woven among tender braised eggplant and shredded mango. Crunchy broken rice crackers crown the melange with a satisfying toasted note, essentially allowing eaters to indulge in forkfuls of chips and chunky dip without looking like brutes. This dish alone is worth a return visit.\nKabocha squash also dazzles with stunning depth, stewed in coconut milk alongside long beans and onions. Such simplicity belies the incredible richness of each melt-in-your-mouth soft orange cube.\nPancit sontaghon, simple glass noodles with a handful of colorful vegetables, doesn't have quite the same sparkle, but still satisfies with its savory soy sauce-infused translucent strands.\nSuman, a variation on mango sticky rice, presents the starch as a completely smooth cylinder on the plate that bears only gentle resistance to the fork. Topped with caramelized coconut crumbs, the lightly bitter notes contrast beautifully with the tropical fruit arranged on the side. Though I might personally prefer more sauce, it could probably be served in a soup bowl and I'd still have the same complaint.\nBeginning life as a fledgling pop-up back in 2015, the name is an acronym for \"Fresh Off the Boat,\" but I truly hope they're dropping anchor to stay a good long while.\nAfter so many years of looking inward in search of that \"new you for the new year,\" can we all agree that what we really need to change is the same tired routine? Could it be that such deep personal dissatisfaction is simply a side effect of boredom, of time wasted on the couch instead of living life? Shake it up, get out there, and do something radically different this time around! I'm all about making 2019 my year of adventure, starting with a little help from IfOnly. Curating singular experiences with talented luminaries in cities across the US, this is not just another Groupon or Living Social app. You can have Alice Waters cook a private meal for you in your own home, go paragliding over the whole bay area, or even book a hot air balloon excursion over Mount Everest. Starting a bit closer to home, I'm still just testing the waters before I jump in head-first on a scuba diving adventure, instead homing in on the wide range of hands-on art opportunities nearby. Sure, painting and sewing are easy enough to pick up, but glass work is considerably less accessible.\nEnter: Lynn Rovelstad, offering curious crafters an opportunity to get up close and personal with a kaleidoscope of custom glass projects. Rather than a standard drop-in to dabble with scraps, this booking gets you a private tutorial with the glass master herself, for an intimate afternoon for two or a party of twelve.\nWhereas blown glass is difficult, hot, and potentially very dangerous, fused glass is easy, safe, and fun for all ages. In fact, children as young as 18 months have gotten in on the act, but 3 years old is the generally recommended minimum. The biggest hazard to participants are possible glass splinters, though even that risk is quite rare.\nFragments of colored glass are adhered to solid white, black, or clear glass base with plain old nontoxic Elmer's glue gel. The process is very much reminiscent of mosaic making, with the bigger picture slowly revealing itself through each subsequent shard. Placement becomes permanent only after a day or two in the 1500-degree kiln. Edges are rounded, surfaces smoothed, and your work of art transforms into a solid, single piece. It's akin to modern magic, although in the case of my one-of-a-kind serving utensil set, much more practical alchemy.\nIn one afternoon, I managed to go outside my comfort zone to explore a new part of the bay area, meet an inspiring artisan, spend time with a good friend, AND craft a completely unique salad serving set, barbecue fork, bread knife, and cake server. While it may not be as bold as flying a jet plane at mach speed, it was the exhilarating change of pace I needed to get out of my crafting rut and try something new.\nJoin me: Don't just make resolutions for 2019, but make something real. Start crossing activities of your bucket list instead of adding them, for once. Get $50 off your first booking on IfOnly with the discount code \"Kaminsky50\" until 3\/31\/19.\nChain restaurants have come to represent the slow erosion of all that is held dear by discerning diners the world over. Homogenized, mass-produced, oversimplified menus designed for consistency and volume proliferate, muting bold flavors in favor of simplicity. Quality isn't the first, nor second or third, consideration, with more focus placed on cost than true value. Small businesses are edged out, putting the endangered mom and pop shop around the corner at even greater risk than ever.\nSurely you've heard this tale of woe before. It's not a new story, and no happy ending seems to lie ahead in the yet unwritten pages. However, it's not all doom and gloom. \"Franchise\" doesn't have to be a dirty word when it represents the spread of better corporate values. Veggie Grill is bucking the trend as the largest 100% vegan chain in the country, bringing real food that \"celebrates the veggie, the fruit, the grain, and the nut\" to the people. Veggie Grill has over 30 locations spread across the west coast and is soon to begin their east coast invasion, taking root in Cambridge, MA before spreading to NYC in 2019. If you haven't yet heard about their infamous B-Wing Salad or sat down to a towering Beyond Burger, you're in for a treat.\nMy first taste of Veggie Grill was many years ago when it was still a fledgling eatery based in Portland, OR. Although I've dabbled with the bay area locations once a year or so, they still remained just a bit beyond easy access\u2026 Until now. Planting themselves right in the heart of downtown Berkeley, it's dangerously easy to pop in whenever the craving for Tempura Green Beans strikes, or worse yet, simply order $1 delivery through the app.\nMeatless mains feature prominently through partnerships with Beyond Meat, Gardein and Hodo, appealing to the staunch omnivore and old-school vegan alike. Many reviews come from confused patrons who realize only after plowing through their entree that it was entirely plant-based. This is exactly the kind of accessibility that the industry needs now; not dumbed-down flavors, but a level playing field to unite eaters of all stripes. You can still get your spicy Buffalo Wings with creamy ranch dressing; your soulful Southern Fried Chicken plate piled high with mashed potatoes and gravy; your bright and zesty Mediterranean Super greens Salad, if you so choose.\nMy personal favorite of the moment is the Rustic Farm Bowl for its deeply umami grilled mushrooms, roasted yams, braised white beans, broccoli pesto, roasted tomatoes, and red pepper sauce. Every element would be a fine bite by itself, but in concert, creates a symphony of carefully tuned tastes, in perfect proportion. It's a grounding meal that always leaves me feeling satisfied, energized rather than weighed down, easy to grab and just keep going. That's what fast casual dining should really be about, right? Fuss-free, wholly satisfying fare that fits into the existing social fabric, rather than tearing a hole through it. If you haven't yet tried Veggie Grill, prepare yourself. You may just see one opening up near you soon.\nDon't forget to download the app before you go and use my code HannahKa8145 for $5 off when you sign up!\nWAIT! Before you start folding up the picnic table and packing away the lawn darts, let's pretend like it's still summer for just a little bit longer, please? While everyone else rushes to embrace the new pumpkin-infused season, plenty of warm, sunny days remain in the forecast yet, with much of the west coast in particular still due for a solid heatwave soon. Despite what the calendar may tell you, take a look outside before donning that heavy sweater. Let's party like fall is but a distant concept for later days, concerning only overly cautious weathermen determined to throw shade on our fun.\nMy encouragement is of course entirely self-serving, but you see, it would be a shame to sit on this great pasta salad for another full year. Inspired by a trip to Baia Pasta, my neighbors in Oakland that I never knew lived next door all along, such incredible noodles need little ornamentation to shine on any table.\nThese small-batch artisan noodles are making a big splash nationwide thanks to an obsessive level of passion for every detail. Obvious considerations are the specific amounts of protein, moisture, type of flour, but what about the drying temperature and time? Procuring the unique shapes and the obscure dies to extrude them? Determining the types of wheat that can endure such a demanding process without breaking, dissolving, or crumbling under the pressure? That's to say nothing of the unconventional seasonings blended into some of the more colorful pastas, giving rise to a full rainbow of bright, bold flavors.\nOrganic durum, whole durum, spelt, whole spelt, and whole khorasan wheat are the foundations of each charming twist, twirl, and tube. Pale, limp spaghetti strands are no where to be seen here, and you'd never miss them in the first place.\nTo fully celebrate such an exquisite yet uncomplicated staple, the greatest (and most challenging) task for the cook is to simply not mess with perfection. It's already great as is- What more could one add?\nIn this case, my inclusions are more like additional refrains of the chorus, echoing and underscoring what already got the crowd off their feet to sing. Beautiful Organic Durum Wheat Flavored Soup Radiatori (Dynamos) are infused with beets, spinach, and tomatoes, which are exactly the same guests I invited to harmonize. Yes, that's why it's the BeST pasta salad, but for more than that cute pun alone. Accented with an invigorating punch of fresh basil, savory yet subtle white miso, and a light kiss of buttery avocado oil, it might very well be best dish of the season altogether, if we can sneak this last ode to summer in, right under the wire.\nBut that can wait for colder days. For now, let's revel in the fading sunlight, the last call of summer, until we reach the very bottom of the bowl.\nAs you might imagine, this pasta salad comes together very quickly and easily. If you've gone through the trouble of prepping the ingredients as listed, you probably aren't even reading this instruction right now. That's okay; I wouldn't bother either. Simply toss everything together until well blended, and either enjoy immediately, or chill for up to 4 hours. Savor a taste of summer all over again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello, I'm Diana! I'm fourty and I'm going to thoroughly research the new cosmetics in order to stay young and beautiful for a long time (preferably forever). I also invited my friends to SerumBoom.com in order to more objectively test and describe cosmetics from different points of view. We are trying to find something that really works and is worth the money. In addition to personal impressions, we will analyze the compositions and describe the effect of each ingredient. I very much hope that our reviews will be useful to other beauties!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Royal Naval Dockyard in Sandys Parish is the first port of call of many tourists to Bermuda and it is a great place where to start your visit of the island. Amongst the many attractions of the complex, the Bermuda Maritime Museum \u2013 also called the National Museum of Bermuda since 2009 \u2013 is a great resource for whoever wants to know more about the maritime history of the island.\nThe building that hosts the Museum is called The Keep and was the fortress that was originally protecting the dockyard. It is the largest in Bermuda with walls 30 feet high and it covers a surface of 10 acres of land. Inside this museum officially inaugurated in 1975 by Queen Elizabeth II, you'll find the most extensive artifact collections of Bermuda as well as historic exhibits.\nVisitors are greeted by a 3-meter high statue of King Neptune standing on the parade ground. The original figure comes from the wreck of the HMS Irresistible that sunk 1891 and it was rebuilt in limestone. There are six buildings within the walls of the fortress, the Queen's Exhibition Hall, the Forster Cooper Building, the Shell House, the Shifting House, the Boat Loft and the Commissioner's House.\nOne of the island's most important buildings is the Commissioner's House. The Georgian house dating from the 1820's is thought to be the first cast-iron structure built in the world. It was restored and saved from dereliction by the restoration of the museum and consists of a ground floor made of limestone and brick rooms dedicated to the island's military history; and two floors featuring mahogany mouldings where exhibits off Bermuda art.\n\"Very informative and interesting; great old stone buildings; some great exhibits\".\nThanks to the different exhibits on at the Bermuda Maritime Museum, you can learn about the creation of the Royal Naval dockyard, the dangerous reefs that got sailors giving Bermuda the name of Isle of Devils, you will see collections of over 2000 antique bottles collected around the island and some precious items found in 16th and 17th centuries shipwrecks, especially the famous Pectoral Cross recovered by most known Bermuda's diver Teddy Tucker in 1955.\nYou can also walk around the defences, including ramparts and seven bastions, lettered from 'A' to 'G', that defended the fort and were mounted with 32-pound and 24-pound cannons, 8-inch shell guns and 6-inch BL barrels, that were designed by the Royal Engineers to have panoramic views of Bermuda and the Atlantic Ocean.\nBermuda Maritime Museum, Sandys \u2013 Part of Elbow Beach Cycles 'Things To Do' in Bermuda series!\nGet on your bike and get to Bermuda Maritime Museum, Sandys on moped! Rent a scooter, moped or bike from Elbow Beach Cycles and get a heavenly 10% off your scooter if you book online NOW. AND, if you book early enough and pre-pay your scooter hire, you'll get a 15% discount!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today I have the pleasure to talk to heavy metal photographer Kenneth Sporsheim. He is one of the photographers for the Norwegian issue of Metal Hammer, and he is also the Nordic ambassador for brands like Pentax, Lexar and Lowepro.\nIn this interview Kenneth reveals insights on working for one of the most prestigious MEtal magazines, he shares awesome stories about taking Portraits of bands such as Turbonegro, Slayer or Anthrax and give tips on how you can become an ambassador for famous brands too.\nDo you want to become a Concert Photographer?\nDon't forget! You can support me on Patreon. For $5\/mo you'll gain access to HTBARP content that you cannot get anywhere else, including special videos, exclusive announcements, and a behind-the-scenes look at the making of my podcasts.\nLeave an honest review in iTunes. Once you listened to an episode, follow the iTunes link here and click the blue \"View in iTunes\" button. This will launch iTunes on your computer and bring up the podcast. Click on \"Ratings and Reviews\" on the HTBARP Podcast Page. Your ratings and reviews will help my Podcast to reach even more people. I am interested in your honest feedback and I read each one.\nSubscribe in iTunes. To subscribe in iTunes, visit the How To Become A Rockstar Photographer Podcast page and click the blue \"View in iTunes\" button. This will launch iTunes on your computer and bring up the podcast. Click on the \"Subscribe\" button and you're all set.\nSpecial Thanks to The Overalls for providing their awesome music!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Do you know how easy it is to make margaritas without purchasing a pre-made mix? All it takes is a few fresh ingredients and in less than ten minutes you will have a tasty margarita on the rocks! Or if you prefer the frozen kind, you could add all the ingredients to the blender and blend it! I have flashbacks of those 7\/11 Icee drinks that would give me brain freeze , lol! I really do prefer mine on the rocks. Well, better late than never, here is my contribution to National Margarita Day. I really did not need an excuse to prepare a margarita!\nBesides margaritas a few of my other favorite cocktails are a Mexican Martini, Michelada, Paloma and a good Spicy Bloody Mary! These cocktail recipes will be added in due time, but how about a Jalapeno Lime Margarita or Agua Fresca de Sandia while you wait? One of my favorite fruits besides Pi\u00f1a (pineapple) flavor.\nClassic Margaritas on the Rocks!\n2 oz. lime juice, fresh!!\nAdd tequila, lime juice, agave nectar and crushed ice to a cocktail shaker. Shake and pour into a salt rimmed glass. Add extra lime wedges or wheels. If you want to give your classic margarita a fresh twist, try muddling in some fresh cucumber to the bottom of shaker before adding the other ingredients.\nYields 2-4 Margaritas. Pace yourself! Lol!\nIn a small pitcher, mix all of the ingredients for the margarita. Taste for sweetness. Wet the rim of four-8oz glasses with fresh lime. Dip in mixture of coarse salt and chile limon powder. Fill with crushed ice. Pour margaritas and garnish with sliced lime and sliced jalapeno!!\nPrepare a tall glass with lime on the rim. Dip into chamoy hot sauce, then into the chile limon powder. To the glass, add the ingredients in the order listed, Stir gently to combine and enjoy!\nTo see full recipe for Agua de Jamaica on site, click Hibiscus ! You will find a tasty paleta recipe as well!!\nPeel the pods and place in a bowl. Cover with three cups of water and let them soak overnight.\nNext day: Wash hands thoroughly and remove all the seeds from the pods, making sure you leave all the pulp in the water.\nTransfer to the blender. Add four cups of cold water and one cup of sugar. Blend on high for 30 to 40 seconds.\nStrain into a pitcher using a wire mesh strainer. Stir well, taste for sugar. Serve over plenty of ice. Stir in between because the pulp will separate as it sits.\n1\/2 cup powdered sugar (who knew?) LOL!\nJuice all of the citrus and set aside. Combine all of the ingredients to make the margarita in a pitcher. Mix well until sugar dissolves. Wet the rim of a margarita glass with lime juice and dip into salt\/chile limon mixture. Add crushed ice and fill with margarita!!! Makes about 4 cups.\nCherry Ice. Not a margarita or agua fresca, but a fun drink for summer time!\nPlace cherry flavored Italian ice down into chilled glass. Add tequila reposado, orange liqueur and lime. Top off with some lemon lime soda or selzter, pitted cherries and some whole. Let the italian ice melt for a few minutes\u2026enjoy!\nFill blender with fresh watermelon or fruit of choice. Add 3 cups cold water, 1\/3 cup sugar(or agave syrup) and juice of 1 lime or lemon. Blend on high until smooth. Strain into pitcher with a wire strainer. Yields 2\/3, more or less of a standard size pitcher. Serve over ice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please phone school by 9.00am if your child is unable to come to attend that day. It is important that we know about this as early as possible on the first day of absence. We operate a policy of first day calling so we will be in touch with you if we do not hear from you first. The school contact number is 01924 325308.\nIf your child needs to attend an appointment during school hours we ask that you notify us in advance wherever possible and provide the office with an appointment card or letter. You need to sign out your child at the main office when collecting and sign back in when you return to school. This is important for fire safety.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A versatile variety with plenty of uses. Apple 'Charles Ross' is predominantly grown as an eating apple with an orangey tang, reminiscent of Cox. When cooked it retains its shape, making lovely baked apples and apple tarts. Can also be juiced, or blended with other varieties to provide a sharp acidity to ciders.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Over the years, Filson's philosophy has never changed: Make sure it's the absolute best. Founder Clinton Filson spent a lot of time talking to his customers and refining his designs to their specifications back in the day, so it's not surprising that the items that worked then still work today, over 120 years later. Filson's collection ranging from the outdoor lovers to the city workers bags and accessories demonstrates comfort, protection and durability never go out of style.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stay ahead of FX developments by joining Saxo Bank Head of Forex Strategy John Hardy for his regular update. John looks at what's happening with all major currency pairs like EURUSD and USDJPY and how they might be impacted by market moving events. He also explores ways to seize trading opportunities.\nThe Saxo Group spans the entire globe with a strong, unified network. Both our local sales and service organisations as well as our global online presence allow for us to give first-class service to clients in more than 180 countries.\nThe value of your investments can go down as well as up. Losses can exceed deposits on margin products. Complex products, including CFDs and FX, come with a high risk of losing money rapidly due to leverage. You should consider whether you understand how CFDs, FX or any of our other products work and whether you can afford to take the high risk of losing your money. 71% of retail investor accounts lose money when trading CFDs with this provider.\nThis website can be accessed worldwide however the information on the website is related to Saxo Bank A\/S and is not specific to any entity of Saxo Bank Group. All clients will directly engage with Saxo Bank A\/S and all client agreements will be entered into with Saxo Bank A\/S and thus governed by Danish Law.\nApple, iPad and iPhone are trademarks of Apple Inc., registered in the U.S. and other countries. App Store is a service mark of Apple Inc.\nYour browser cannot display this website correctly.\nOur website is optimised to be browsed by a system running iOS 9.X and on desktop IE 10 or newer. If you are using an older system or browser, the website may look strange. To improve your experience on our site, please update your browser or system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbamvz b/data_all_eng_slimpj/shuffled/split2/finalzzzbamvz
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+{"text":"Go to Visit and Learn for a list of upcoming workshops we run throughout the year.\nThe following is a list of past, present, and possible presentations. Let us know what topic interests your group. Some workshops are hands-on while others are projected on a screen. All are thought-provoking, lively, and engaging.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stanardsville, Virginia is the 14,440th largest city in the United States as of 2017.\nStanardsville, Virginia is a town.\nStanardsville, Virginia is currently 39.9% smaller than it was in 1990, the year of its peak recorded population.\nStanardsville, Virginia's growth is extremely below average. 94% of similarly sized cities are growing faster since 2000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This week we talked about history! About different events in history and their effects. We also tried using the challenging 3rd conditional to reimagine history. Test your knowledge of the 3rd conditional at grammar quizzes.\nFor original version, try any of the interesting content available over at the History Channel.\nLearners can try some of the history stories from VOA History.\nFor original version reading, check out ten moments that changed history or test your historical knowledge with Timeline.\nFor writing this week, you can try one of two exercises. First, describe an important event in history and its subsequent impact. Secondly, and for something a bit more challenging, describe how history would have been different if that event hadn't happened.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you've been following the steps provided in Chapter_5 through Chapter_8 , you know your goals, have a diagram of your reputation model with initial calculations formulated and a handful of screen mock-ups showing how you will gather, display, and otherwise use reputation to increase the value of your application. You have ideas and plans, now it is time to reduce it all to code and to start seeing how it all works together.\nA reputation system does not exist in a vacuum; it is small machine in your larger application. There are a bunch of fine-grained connections between it and your various data sources, such as logs, event streams, identity db, entity db, and your high-performance data store. Connecting it up will most likely require custom programming to connect the wires between your reputation engine and subsystems that were never connected before.\nThis step is often overlooked in scheduling, but may take up a significant amount of your total project development time. There are usually small tuning adjustments that are required once the inputs are actually hooked up in a release environment. This chapter will help you understand how to plan for connecting the reputation engine to your application, and what final decisions you will need to make about your reputation model.\nThe heart of your new reputation-infused application is the reputation model. It's that important. For the sake of clarity, we'll refer to the software engineers that turn your model into operational code as the reputation implementation team and those who are going to connect the application input and output as the application team. In many contexts, there are some advantages to these being the same people, but consider that reputation, especially shared reputation, is so valuable to your entire product line that it might be worth having a small dedicated team for the implementation, testing, and tuning full time.\nOne of the hard-learned lessons of deploying reputation systems at Yahoo! is this: the engineering team needs to be involved at every major milestone during the design process. Even if you have a separate reputation implementation team to build and code the model, the gathering of all the inputs and integrating the outputs is significant new work added to their already overtaxed schedule.\nAs the result of reputation, the very nature of your application is about to change significantly, and the engineering team are the ones who will turn all of this wonderful theory and the lovely screen mockups into code. Reputation is going to touch code all over the place.\nBesides, who knows your reputable entities better than the application team? They build the software that gives your entities meaning. Engaging these key stakeholders early allows them to contribute to the model design and prepares them for the nature of the coming changes.\nDon't wait to share details about the reputation model design process until after screen mocks are distributed to engineering for scheduling estimates-there's too much happening on the reputation back-end that isn't represented in those images.\nAppendix_A contains a deeper technical architecture oriented look at how to define the reputation framework: the software environment for executing your reputation model. Any plan to implement your model will require significant software engineering, so sharing that resource with the team is essential. Reviewing the framework requirements will lead to many questions from the implementation team about specific trade-offs related to issues such as scalability, reliability, and shared data. The answers will put constraints on your development schedule and the application's capabilities. One lesson is worth repeating here: the process boxes in the reputation model diagram are a notational convenience and advisory-they are not implementation requirements.\nThere is no ideal programming language for implementing reputation models. In our experience, what matters most is for the team to be able to create, review, and test the model code rigorously. Keeping each reputation process's code tight, clean, and well documented is the best defense against bugs and vastly simplifies the testing and tuning of the model.\nA typical complex reputation model, such as those described in Chapter_4 and Chapter_10 can have dozens of inputs, spread throughout the four-corners your application. Often implementors think only of the explicit user-entered inputs, when many models also include non-user or implicit inputs from places such as log files or customer care agents. As such, rigging inputs often involves engineers from differing engineering teams, each with their own prioritized development schedule. This means that the inputs will be attached to the model incrementally.\nThis challenge requires that the reputation model implementation be resilient in the face of missing inputs. One simple strategy is to have the reputation processes that handle inputs have reasonable default values for every input. Inferred Karma, (Chap_6-Inferred_Karma ) is an example. This approach also copes well if a previously reliable source of inputs becomes inactive, either through a network outage or simply a localized application change.\nExplicit inputs, such as ratings and reviews, take much longer to implement as they have significant user-interface components. Consider the overhead with something as simple as a thumbs-up\/thumbs-down voting model: what does it look like if the user hasn't voted?; what if they want to change their vote?; what if they want to remove their vote altogether?\nWe found that getting a primitive user interface in place quickly for testing is essential. Our voting example above can be quickly represented in a web application as two text-links: \"Vote Yes\" \"Vote No\" and text next to it that represented the tester's previous vote: \"(You [haven't] voted [Yes|No].)\" Trivial to implement, no art requirements, no mouse-overs, no compatibility testing, no accessibility review, no pressure to ship early, but completely functional. This approach allows the reputation team to test the input flow and the functionality of model. This sort of development interface is also amenable to robotic regression testing.\nThe simplest output is reflecting explicit reputation back to users-showing their star-rating for a camera back to them when they visit the camera again in the future, or on their profile for others to see. The next level of output is the display of roll-ups, such as the average rating from all users about that camera. The specific patterns for these are discussed in detail in Chapter_7 . Unlike the case with integrating inputs, these outputs can be simulated easily by the reputation implementation team on their own, so there isn't a dependency on other application teams to determine if a roll-up result is accurate. One practice during debugging a model is to simply log every input with the changes to the roll-ups that were generated, giving a historical view of the model's state over time.\nBut, as we detailed in Chapter_8 these explicit displays of reputation aren't usually the most interesting or valuable: using reputation to identify and filter the best (and worst) reputable entities in your application is. Using reputation output to perform these tasks is more deeply integrated with the application; For example, search results may be ranked by a combination of a keyword search and reputation score. A user's report of TOS violating content might want to compare the karma of the author of the content to the reporter. These context-specific uses require tight integration with the application.\nThis leads to an unusual suggested implementation strategy-code the complex reputation uses first. Get the skeleton reputation-influenced search results page working even before the real inputs are built. Inputs are easy to simulate, the reputation model needs to be debugged as well as the application-side weights used for the search will need tuning. This approach will also quickly expose and scaling sensitivities in the system-in web applications search tends to consume the most resources by far. Save the fiddling over the screen presentation of roll-ups for last.\nHow do you get out from under it [a bad credit rating]? You can't re-establish your credit if you can't get a job, and you can't get a job if you've got bad credit.\nThis mis-application of your credit rating creates a feedback loop. This is a situation in which the inputs into the system (in this case, your employment) are dependent in some part upon the output from the system.\nWhy are feedback loops bad? Well, as the Times points out-feedback loops are self-perpetuating and, once started, nigh-impossible to break. Much like in music production (Jimi Hendrix notwithstanding), feedback loops are generally to be avoided because they muddy the fidelity of the signal.\nChange may be good, but your community's reaction to change won't always be positive. We are, indeed, advocating for a certain amount of architected flexibility in the design and implementation of your system. We are not encouraging you to actually make such changes lightly, or liberally. Or without some level of deliberation and scrutiny before each input-tweak, or badge addition.\nAlso pay some heed to the manner in which you introduce new reputation-related features to your community.\nHave your community manager announce the features on your product blog, along with a solicitation for public feedback and input. That last part is important-though these may be feature additions or changes like any other, often times they are fundamentally transformative to the experience of engaging with your application. Make sure that people know they have a voice in the process, and their opinion counts.\nBe careful to be simultaneously clear-in describing what the new features are - and vague in describing exactly how they work. You want the community to become familiar with these fundamental changes to their experience, so that they're not surprised or, worse, offended when they first encounter them in the wild. But you don't want everyone immediately running out to \"kick the tires\" of the new system, poking prodding and trying to earn reputation to satisfy their \"thirst for first.\" See Chap_5-The_Quest_For_Mastery .\nAs with all new software deployment, there are several phases of testing recommended: bench testing, environmental testing (aka Alpha), and pre-deployment testing (aka Beta.) Note that we don't mean web-Beta, which has come to mean deployed applications that can be assumed, by the users, to be unreliable-we mean pre- or limited-deployment.\nA well coded reputation model should function with simulated inputs. This allows the reputation implementation team to confirm that the messages flow through the model correctly and provides a means to test the accuracy of the calculations and the performance of the system.\nRushed development budgets often cause project staff to skip this step to save time and to instead focus the extra engineering resources on rigging the inputs or implementing a new output-after all, nothing like real data to let you know if every thing's working properly, right? In the case of reputation model implementations, this assumption has been proven both false and costly every single time we've seen it deployed. Bench testing would have saved hundreds of thousands of dollars in effort on the Yahoo! Shopping Top Reviewer karma project.\nBench Test Your Model With the Data You Already Have. Always.\nThe first reputation team project at Yahoo! was intended to encourage Yahoo! Shopping users to write more product reviews for the upcoming fall online shopping season.\nOver several weeks and dozens of meetings the team defined the model using a prototype of the graphical grammar presented in this book. The final version was very similar to the Chap_4-User_Reviews_with_Karma presented in chapter 5. The weighting constants were carefully debated and set to favor quality with a score 4 times higher than the value of writing a review. The team also planned to give users back-dated credit to reviewers by writing an input simulator by reading the current ratings and reviews database and running them through the reputation model.\nThe planning took so long that the implementation schedule was crushed-the only way to get it to deployment in time was to code it quickly and enable it immediately. No bench testing, no analysis of the model or the back-dated input simulator. The application team made sure the pages loaded and the inputs all got sent, and then pushed it live in early October.\nThe good news was that everything was working. The bad news? It was really bad-every single user on the Top Reviewer 100 list had something in common. They all wrote dozens or hundreds of CD reviews. All music users, all the time. Most of the reviews were \"I liked it\" or \"SUX0RZ\" and the helpful scores almost didn't figure into the calculation at all. It was too late to change anything significant in the model and so the project failed to accomplish its goal.\nA simple bench test with the currently available data would have revealed the fatal flaw in the model: The presumed reputation context was just plain wrong-there is no such thing as a global \"Yahoo! Shopping\" context for karma. The team should have implemented per-product category reviewer karma: who writes the best digital camera reviews? Who contributes the classical CD reviews that others think as the most helpful?\nBesides accuracy and determining suitability of the model for it's intended purposes, one of the most important benefits of bench testing is stress testing of performance. Almost by definition, initial deployment of a model will be incremental-smaller amounts of data are easier to track and debug and there are less people to disappoint if the new feature doesn't always work or is a bit messy. In fact, bench testing is the only time the reputation team will be able to accurately predict the performance of the model under stress until long after deployment, when some peak usage brings it to the breaking point, potentially disabling your application.\nDo not count on the next two testing phases to stress test your model. They won't-that isn't what they are for.\nAfter bench-testing has begun and there is some confidence that the reputation model code is stable enough for the application team to develop against, crude integration can begin in earnest. As suggested in Chap_9-Rigging_Inputs application developers should go for breadth (getting all the inputs\/outputs quickly inserted) instead of depth (getting a single reputation score input\/output working well.) Once this reputation scaffolding is in place, both the application team and the reputation team can test the characteristics of the model in it's actual operating environment.\nAlso, any formal or informal testing staff that are available can start using the new reputation features while they are still in development allowing for feedback about both calculation and presentation. This is when the fruits of the reputation designer's labor begin to manifest-an input leads to calculation leads to some valuable change in the application's output. It is most likely that this phase will find minor problems in calculation and presentation, while it is still inexpensive to fix them.\nDepending on the size and duration of this testing phase, initial reputation model tuning may be possible. One word of warning though: testers at this phase, even if they are from outside your formal organization are not usually representative of your post-deployment users, so be careful what conclusions you draw about their reputation behaviour. Someone who is drawing a paycheck, or was given special-status access is NOT a typical user, unless your application is for a corporate intra-net.\nOnce the input rigging is complete and placeholder outputs are working, the reputation team should adjust its user-simulation testing scripts to better match the actual use behavior they are seeing from the testers. Typically this means adjusting assumptions about the number and types of inputs versus the volume and composition of the reputation read requests. Once done, re-run the bench tests, especially the stress tests, to see how the results have changed.\nThe application\/user interface is now nominally complete (meets specification): it's no longer embarrassing to allow non-insiders to use it.\nThe reputation model is fully functional, stable, performing within specifications, and is outputting reasonable reputation statement claim values, which implies that your system has sufficient instrumentation to evaluate the results of a larger scale test.\nA pre-deployment testing phase is important when introducing a new reputation system to an application as it enables a very different and largely unpredictable class of user interactions driven by diverse and potentially conflicting motivations. See Chap_5-Incentives . The good news is that most of the goals typical for this testing phase also apply to testing reputation models, with a few minor additions.\nAlthough the maximum throughput of the reputation system should have been determined during the bench testing phase, engaging a large number of users during the Beta test will reveal a much more realistic picture of the expected use patterns in deployment. The shapes of peak usage, the distribution of inputs, and especially the reputation query rates should be measured and the bench tests should be re-run using these observations. This should be done at least twice-half way through the Beta, and a week or two before deployment, especially as more testers are added over time.\nAs beta users contribute to the reputation system an increasingly accurate picture of the nature of their evaluations will emerge. Early during this phase the accuracy of the model calculations should be manually confirmed, especially double-checking an independently logged input stream against the resulting reputation claim values. This is an end-to-end validation process, and particular attention should be paid to reputation statements that contain inputs that were reversed or manually changed (due to abuse mitigation or other circumstances.) Once a reputation model is in full deployment this verification process will become significantly more expensive, so the Beta test is usually the last chance to cheaply find bugs and conversely reinforce confidence in the system.\nReputation systems change the way applications display content. Those changes add elements to the user interface that require additional space, new user behavior learning, and change the flow of the application significantly. A good example of this effect is when search URL reputation (page ranking) replaced hand-built directories as the primary method for finding content on the web.\nWhen a reputation-enabled application enters pre-deployment testing, tracking the actions of users-their clicks, their evaluations, their content contributions, even their eye-movements-provides important information to optimize the effectiveness of the model and the application as a whole.\nDespite our focus on measuring the performance and flow of user interaction, we'd like to caution that pure quantitative testing can lead to faulty conclusions about the effectiveness of your application, especially if the metrics are not as positive as you expected. Everyone knows that when metrics are bad, all the tell you is that you've done something wrong, not what it is. But that is also often true for good metrics-a lot of page-views doesn't always mean you have a healthy or profitable product. Sometimes it's quite the contrary, controversial objects generate a lot of heat (in the form of online discussion) but can create negative value to the provider.\nIn the beta phase, explicit feedback is required to help understand how the application, especially the reputation system, is perceived by the users. Besides multiple opt-in feedback channels, such as email or a message boards, guided surveys are strongly recommended. In our experience, opt-in message formats don't accurately represent the opinions of largest group of users-the lurkers-those that only consume reputation and never explicitly evaluate anything. At least in applications that are primarily advertising supported, the lurkers actually produce the largest chunk of revenue.\nDuring the pre-deployment phase the instrumentation to regularly measure the effect on revenue and\/or other critical success metrics, such as engagement and customer satisfaction. After deployment the daily, weekly, and monthly reports of these business metrics will become the bible for driving the application and model designs forward, so getting them established during this phase is critical to getting management to understand why the investment in reputation is worthwhile. The Beta test phase will not demonstrate that reputation has been successfule\/profitibale, but it will establish the means for determining when it becomes so.\nSometime in the later half of the testing phase, the reputation system begins to operate sufficiently well enough to gage the general effect it will have on the application and early indications of it's likely success against the original goals. This is when the ongoing process of reputation model tuning can begin.\nThe most important thing the reputation team can do when implementing and deploying a reputation system is to define the key metrics for success. What are the numerical measures that a reputation-enabled application is contributing to the goals set out for it? For each measure, what are the target values? Is there a predicted lift in user contributions? Should there be more page-views? Is there an expected direct effect on product revenues? Is the model expected to decrease customer care costs? By how much? All of these metrics allow both the application and reputation teams to identify areas for tuning. Sometimes just the application will need to be adjusted, other times the just model will, and (especially early on) sometimes it all needs tuning.\nCertainly the reputation team should have metrics for performance, such as the number of reputation statement writes per second and a maximum latency of such-and-such milliseconds for 95% of all reputation queries, but those are internal metrics for the system, and do not represent the value of the reputation itself to the applications that use it.\nEvery change to the reputation model and application should be measured against all corporate and success-related metrics. Resist the desire to tune things unless you have a specific goal to change one or more of your most important metrics.\nHe noticed that the employees provided short-lived productivity increasesin response to the fact that they were being studied. It did not seem to matter if the lighting was raised or lowered. He concluded that the fact that the study was communicated to the workers made it clear that management was paying attention had a greater morale effect than particular changes to the environment. Other changes that showed similar effects also included moving workstations, clearing the floors, and maintaining work areas.\nOn this insight alone, it should be clear that reputation model tuning should not only be judged by goal-centric objectives, but also that any model changes should be given ample time to stabilize. A spike in activity immediately after the release of a reputation change is not, necessarily, an indication that the change is working as intended. It is good to assume the Hawthorne effect is in play until things stabilize.\nBut, the story of this effect gets even more interesting and relevant.\nThe existence of significant questions surrounding this effect reinforces the fact that, when it comes to human behavior, there is a tendency to over-extrapolate from the available data while ignoring all of the factors that aren't even quantitative measured or even measurable.\nThis problematic simplified extrapolation can also happen while tuning reputation models: It's easy to say \"Oh! I know what they're doing.\" That's fine as far as it goes, but the school of hard knocks has taught us that for every behavior you posit, there are at least two more you are missing. If you really want to know how your model is working, you'll have to do both qualitative and quantitative research. Use your metrics to create groups based on similar activity patterns and then reach out and ask them why they do what they do.\nInitially any moderately complex reputation model will need tuning. Plan for it in the first weeks of the post-deployment period. Best guesses at weighting constants used in reputation calculations, even when based on historical data, will be shown to be inaccurate in the light of real-user interaction.\nTune public reputation, especially karma, as early as possible.\nIf the changes required are significant, such as making large adjustment to incentive weights, the change will have the smallest impact on your community as possible.\nEstablishing the pattern that reputation can, and will, be changing over time helps set expectations with the early-adoption users. Getting them used to changes will make future tuning cause less of a community disruption.\nMuch of the tuning to reputation models will be opaque to end users. For example, corporate reputations (internal-only) such as Spammer-IP can be tuned and re-tuned with impunity-actually it should be tuned regularly to compensate for improved knowledge and as abusers learn to work their way around the system.\nWhen tuning, an A-B test where the proposed changes and the old model can be tested side by side would be ideal, but most application and metrics environments make this cost-prohibitive. Alternatively, when tuning a reputation model, keep a backup snapshot of both the model code and the values of the critical metrics of the original for comparison. If, after a few days or weeks the tuned model under-performs against the previous version, it will be less painful to return to the backup.\nThere are a number of application and reputation-system related problems that will probably only come to light under real use, from a community of real users, for some extended duration of time. You might see hints of these misunderstandings or mis-comprehensions during early-stage user testing, but you'll have little means of gauging their severity until you analyze the data in bulk. Forgive us an extended example, again from the world of Yahoo! Answers. But it illustrates the kind of back-and-forth tune-observe-tune rhythm that you may need to fall into to truly be responsive in improving the performance of the reputation-related elements of your application.\nOnce upon a time, the Yahoo! Answers interface featured a simple, plain \"Star\" mechanism associated with a question. The original design intent for the star was to act as a sort of lightweight endorsement of an item, somewhat akin to Facebook's \"Like\" control. Star-vote totals were to be displayed next to questions in listings, and also feed into a \"Most Starred\" widget (actually, a tab on a widget-displayed alongside Recent and Popular questions) at the top of the site. When viewing a particular question, you could see a listing of all other users that had Starred that question.\nAs a convenience for users, there was one more feature-Answers would keep a list of all the questions that you had Starred, and display those on your Profile for others to see (or, if you opted to keep them private, for your eyes only.) It was this final feature that may have tipped the utility for some users away from seeing stars primarily as a voting mechanism and instead seeing them as a kind of quasi-bookmark for questions.\nAs a result, the \"Most Starred\" tab, featured at a high and prominent level of the site, was-with alarming regularity-filling up with the absolute worst content on the site! In fact, the worst of the worst-this was the stuff that users felt strongly enough about to report it to Yahoo. And, given the unbearable time-lags between reporting and moderation on Answers in those days, these horrible questions were actually being rewarded with higher visibility on the site for a prolonged period of time.\nThe Star feature had backfired entirely. When measured against the original metrics laid out for the project (to encourage easier identification of high-quality content on the site) it was evident that a redesign was called for.\nFigure_9-1: By giving users a simple, private Watchlist,the Answers designers responded to the needs of Abuse Reporters who wanted to check back in on bad content.\nSee Chapter_10 for an in-depth case study on a deeper and more comprehensive project to not only keep bad content on Answers subdued, but actually clean it up and remove it altogether. With much greater accuracy and much faster.\nThere are many useful sources for reputation input, but source stands out amongst all others: the user. The vast majority of content on the web is user-generated and their feedback generates the reputation that powers the web. Even every search engine is built on evaluations in the form of links provided not by algorithms, but by people.\nIn an effort to optimize all of this people-powered value, reputation systems have come to play a large part in creating incentives for user behavior: participation points, top contributor awards, etc. Users then respond to these incentives, changing their behavior, which then requires the reputation systems to be tuned to optimize newer and more sophisticated behavior (including adjustments for undesirable side-effects: aka abuse). The cycle then repeats, if you're lucky.\nIt's quite possible that-even during the beta period of your deployment-you're noticing some strange effects starting to take hold. Perhaps content items are rising in the ranks that don't entirely seem\u2026 deserving somehow. Or maybe you're noticing a predominance of a certain kind of content at the expense of other types. What you're seeing is the character of your community shaking itself out, finding it's edges and defining itself. Tread carefully before deciding how (and if) to intervene.\nCheck out Delicious's Popular Bookmarks ranking for any given week: we bet you'll see a whole lot of \"Top N\" blog articles. (See Figure_9-2 ) Why might this be? Technology essayist Paul Graham posits that it may be the users of the service, and their motivational mindset, that explain it: \"Delicious users are collectors, and a list of n things seems particularly collectible because it's a collection itself.\" (Graham explores the \"List of N Things\" phenomenon to some depth at http:\/\/www.paulgraham.com\/nthings.html) The preponderance of lists on Delicious is a natural offshoot of its context of use-an emergent effect-and is probably not one that you would worry about, nor try to control in any way.\nWe are beginning to see a trend where people make assumptions about the contents of an article based on the meta-data associated with the submission rather than reading the article itself. Based on these (oft-flawed) assumptions, people then vote for or against the stories, and even comment on the stories without having read the stories themselves.\nWe've noticed a similar tendency in some community-voting sites we've worked on a Yahoo! and have come to consider behavior like this to be a type of emergent defect: behavior that is home-grown within the community and may even become a de facto standard for interacting, but is not necessarily valued. In fact, it's basically a bug and a failing of your system or-more likely-user interface design.\nFigure_9-3: The design of Digg enables (one might argue, encourages) voting for articles at a high level of the site. This excerpted screen is the front page of Digg-users can vote for (Digg) an article, or against (bury) it, with no need to read further.\nOf course, one should not pre-suppose that the Digg folks think of this behavior (if it's even as widespread as Saleem indicates) as a defect. Again, it's a careful balance between the actual observed behavior of users using your system and your own predetermined goals and aspirations for the application.\nIt's quite possible that Digg feels that high voting levels-even if some percentage of those votes are from uninformed users-are important enough to promote voting at higher and higher levels of the site. From a brand perspective alone, it certainly would be odd to visit Digg.com, and not see a single place to Digg something up, right?\nIt's hard to anticipate all emergent defects until they\u2026 well\u2026 emerge. But there are certainly some good principles of design that you can follow that may defend your system against some of the most common ones.\nEarly on in its lifetime, Apple's iPhone App Store allowed any visitor to rate an application, whether they'd purchased it or not! You can probably see the potential for bad data to arise from this situation. A subsequent release addressed this problem, ensuring that only users who'd installed the program would have a voice. It doesn't guarantee perfection, but a gating mechanism for rating does help dampen noise.\nIn any event, provide some cues to users about the utility of those controls: are they strictly for expressing an opinion? Sharing with a friend? Saving for later? The downstream effects may, in fact, be that one control does all three of these things, but sometimes it's better to suggest clear and consistent uses for controls than let the community muddle along, inventing its own utilities and rationales for things. If a secondary or tertiary use for a control emerges, then consider formalizing that function as a new feature.\nMany of the benefits that we've discussed for tracking reputation (the ability to highlight good contributions and contributors, the ability to 'tag' user profiles with awards or recognition, even the simple ability to motivate contributors to accel) can be undermined if you make one simple mistake with your reputation system: being too generous with positive reputations. Particularly, if you hand out reputations at the higher end of the spectrum too widely, then they will no longer be seen as valuable and rare achievements. You'll also lose the ability to call out great content in long listings: if everything is marked as special, then nothing will stand out.\nIt's probably okay to wait until the tuning phase to address the question of distribution thresholds. You'll need to make some calculations-based on available data for current use of the application-to determine how heavily or lightly to weight certain inputs into the system. A good example is the Gold\/Silver\/Bronze medal system that we developed at Yahoo! to reward active, quality contributors to UK Sports Message Boards.\nWe knew that we wanted certain inputs to factor into users' badge-holder reputations: the number of posts posted; how well-received (highly-rated) those posts were by the community. And so on. But, at first, our guesses at the appropriate thresholds for these activities were just that-guesses.\nThese were simply stabs in the dark-placeholders, really-that we fully expected to tune as we got closer to deployment.\nAnd, in fact, once we'd done an in-depth calculation of project badge numbers in the community (based on Message Board activity levels that were already evident before the addition of badges) we realized that these estimates were way too low. We would be giving out millions of Bronze badges, and-heck, still thousands of Golds. This felt way too liberal, given the goals of the project: to identify and reward only the most active and valued contributors to boards.\nBy the time the feature went into production, these minimum thresholds for rating others postings were made much higher (orders of magnitude higher) and, in fact, it was several months before the first message board Gold badge actually surfaced in the wild! We considered that a good thing, and perfectly in-line with the business and community metrics we'd laid out at the project's outset.\nSo\u2026 How Much is Enough?\nWhen you're trying to plan out these Distribution Thresholds for reputations, your calculations will (of course!) vary with the context of use.\nIs this karma (people reputation), or content reputation? Be more mindful of the distribution of karma. It's probably okay to have an overabundance of \"Trophy-winning videos\" floating around your site, but too many top-flight experts risks devaluing the reward altogether.\nHonor the presentation pattern. Some distribution thresholds will be super-easy to calibrate-if you're honoring the Top 100 Reviewers on your site, then the number of users awarded should be fairly self-evident. It's only with more-ambiguous patterns that thresholds will need to be actively tuned and massaged to get the desired distributions.\nPower-law is your friend. When in doubt, try to award reputations along a power-law distribution. (See http:\/\/en.wikipedia.org\/wiki\/Power_law.) Great reputations should be rare, good ones scarce, and mediocre ones should be the norm. This will naturally mimic the natural properties of most networks, so-really-your reputations should reflect those values also.\nThere are sometimes pleasant surprises when implementing reputation systems for the first time. When users begin to interact with reputation-powered applications, the very nature of the application can changes significantly: it often becomes communal-control of the reputable entities shifts from the company to the people.\nThis shift from a content-centric application to community-centric often leads to inspirational application designs to be built on the lessons drawn from the existing reputation system. Simply put, if reputation works well for one application, all of the other related applications will want to integrate it, yesterday!\nThough new reputation models can be added only as fast as they can be developed, tested, integrated, and deployed, new uses for existing reputations can be released by the application team without coordination and almost instantaneously-they already have access to the reputation API calls. This suggests that the reputation team should continuously optimize for performance against its internal metrics. Expect significant growth, especially in the number of reputation queries. Even if the primary application, as originally implemented, doesn't grow daily users by an unexpected rate, expect the application team to add new types of uses, such as more reputation-weighted searches or to add more pages that display a reputation score.\nTuning reputation systems for ROI, behavior, and future improvements is a never-ending process. If you stop this required maintenance, the entire system will lose value as it becomes abused, slow, non-competitive, broken, and eventually irrelevant.\nIt's one thing to describe and critique currently deployed reputation systems - after they've already been deployed. It's another to prescribe a detailed set of steps that are recommended for new practitioners, as we have done in this book.\nTalk is easy; action is difficult. But, action is easy; true understanding is difficult!\nThe lessons we present here are the direct result of many attempts, some succeeded, some failed, at reputation system development and deployment. The book is the result of successive refinement of those lessons, especially as we refined it at Yahoo!. Chapter_10 is our proof-in-the-pudding that this methodology works in practice; it covers each step as we applied them during the development of a community moderation reputation model for Yahoo! Answers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jambu\u2122 is an outdoor inspired footwear brand that fuses creative comfort designs with fresh colors, recycled materials, and the latest comfort technology to empower people through life's journeys. Named after the Jambu tree found in the tropical rainforests in Southeast Asia, Jambu\u2122 is inspired by the beauty and mystique of the rainforest and is designed to guide you in style and comfort at home, or across the globe.\nYou're invited to visit any of The Walking Company's stores to be expertly fitted, or shop our full collection of Jambu\u2122 casual shoes for men and women here at www.thewalkingcompany.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Seagate (NASDAQ: STX) is a manufacturer of hard drives and storage solutions. Founded in 1979 as Shugart Techologies, Seagate has nominal headquarters at the Ugland House in the Cayman Islands and has a large operational office in Scotts Valley, California.\nGot an email saying $50 off rrp, website says $59 off. Got this drive for my macbook pro originally for $149, pretty good drive for the price now imo.\nWorth buying a 4TB Seagate Expansion Portable to shuck?\nA penny for your thoughts regarding return of Seagate SSHD (That is a Hybrid drive for the unlearned :)).\nSeagate Archive 8TB Where to Buy for Cheap?\nSeagate Expansion 3TB 3.5\" USB 3.0 - Faulty Adapter?\nSeagate 4TB Expansion with only US plug. HELP!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PM me if you have any recently harvested (very green) or soon to be harvested fruit of the above cultivars. I will pay for the fruit and shipping.\nGood to see your posts again here on bananas.org.\nWhat are your plans with the fruit, Gabe?\nPrevious Thread: Europe banana vendors for plants and seeds needed!!!\nNext Thread: Where to buy banana fruit online?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ST. LOUIS (AP) \u2014 The 100th edition of the PGA Championship began Thursday for the last time in the stifling heat of summer.\nClub pro Michael Block hit the opening tee shot down the middle at Bellerive Country Club. The course is playing long and soft because of rain on Tuesday and temperatures in the 90s that require PGA officials to keep water on the greens.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A dynamic and custom ad placement engine, this application is quite extensive. Allowing the site owner to manage dynamic content items, membership, ads, categories and more while also enabling individual content owners and advertisers to place their own content on the site. Interactive media such as downloadable coupons, FLV streaming commercials and more gives the site great consumer appeal while all being managed from a simple and secure administration area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm a dreamer! As we grow older and loose friends and loved ones, we realize how short time is for all of us. There is no time like the present to share our dreams.\nMy dream came to me years ago but has only recently made more sense. This dream of humans who no longer need technology or any machine. Humans who use telepathy to communicate. Who no longer need to be predators and who are no longer prey. How this shift in consciousness and biology occurs is still a mystery to me. There is research being done on people who have made extraordinary shifts in biology. We are seeing a minority of people accomplish amazing things. Still it is not the majority of people yet.\nThe dream is simple. It is based on humans taking a guardian and care taking role on the planet. We overcome our fears by learning to use our brains to their full capacity. This capacity includes the ability to protect ourselves from all harm. Communication and transportation take place without machines. We are able to regulate body temperature and keep ourselves comfortable in all conditions. We no longer need animals for food. Greed is no longer a motivation. All of our needs are met. The pure joy of being a part of such a beautiful planet is what motivates all humans. No need for Yin\/Yang thinking. The need to experience opposites is a part of the past.\nI am not looking for skeptics. Most people will be skeptical of such complete evolution. I am looking for other dreamers who are willing to dream the same dream. I am calling on all dreamers to birth this dream by living as fully in harmony with this possibility as is possible for them today. If you are able to go with out animal protein and feel healthy, do that! All of us can choose to boycott factory farm food. Support local agriculture and grow some of your own food. Hold your merchants accountable.\nThese are small steps we can all take to make it clear that we want a healthy and humane future.\nI don't care what your political beliefs are. I care about the dream. Are we living in alignment with this future. We all have to live with the imperfection of today. We have to deal with todays realities. Some of us have to eat some animal protein to be healthy. We can not truly love one another until we truly love all that is. We are a part of a great creation. Who or what created this amazing universe and all that is a part of it, is beyond my comprehension. I am a small part of this dream. We are all a small part. I believe no matter how small we may be, we can still play a role in the future of this planet. Having a shared dream is the beginning. A willingness to accept the realities of today while making the personal changes that bring us closer to the future is a huge part of being a dreamer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"11:00a-11:30p and every other weekend. Ocassional shift rotation as needed.\nEvaluates assigned patients and plans, implements and documents nursing care. Assists physicians during examinations and procedures. Performs various patient tests and administers medications within the scope of practice of the registered nurse. Promotes patients' independence by establishing patient care goals and teaching patients and families to understand conditions, medications, and self-care skills. Relies on experience and judgment to plan and accomplish goals. Performs a variety of tasks. A wide degree of creativity and latitude is expected. Typically reports to a manager or head of a unit\/department.\nWeekend and holiday rotation required.\n2 years nursing experience, 1 year Emergency Room preferred.\nACLS, PALS, BLS are required within 60 days of hire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pathfinder Place - Fremont, NE has yet not published prices.\nWe love this place. The people who live there had the biggest smiles. We definitely are discussing this place for our grandma.\nMy Mother was at the Pathfinder Place for 2 1\/2 years prior to her passing.\nI was extremely pleased with her care there. The administration & staff treated her and all the other residents with kindness & a warm family atmosphere.\nThe size of the facility is perfect - it's not too large, nor too small. That makes it great for more personable interactions.\nIn fact I was there often enough that I thought of everyone there (staff, administration & residents) as part of my family.\nI still love to stop in & visit everyone.\nPathfinder place is very homelike.\nPathfinder Place was a beautiful place for my sister to have lived. It is in a good location & easy to get to. All the staff went out of their way to make her life there happy & homelike. The medical attention was 1st rate. The meals were great. The activities & entertainment were good. The phy- therapy sessions were good. The church services were good. Everyone was very friendly & helpful to my sister & to myself when I visited. When my sister's life drew to a close, the staff went out of their way to help the daughters & family deal with end things. Pathfinder Place in my opinion is TOP NOTCH - all around. Fremont is very lucky to have a place like this available to seniors who need an alternative place to live when living at home is no longer able to be done. GOD BLESS all the residents & staff!!\nSo far everything is working out pretty well for my parents since their move to this community. It is still a work in progress, but I'am comfortable with their move to this community. It is a nice place with kind people.\nMy mother recently passed away. She spent her last years at Pathfinder House\/Place and I couldn't be more grateful that she did. I will admit there was a time the building had some areas of inconsistency. In the last 3 years since the administrator Eric took over things have been very good.\nMy mom felt like she was at home with family, and we felt like we were embraced into a vary special family as well. I was amazed when staff that had moved on whether it was to nursing school or other jobs came back to visit mom when she took a turn. I can't speak enough about the staff and particularly how Eric made the last 3 years of my moms life so very special. When he visited in the hospital the night she passed the smile on that he put on her face, brings tears of gratitude to my eyes even now.\nNot the biggest place in Fremont, but the biggest heart.\nI am extremely pleased that my Mother is at the Pathfinder Place. She has been there a little over a year.\nIt is the perfect size: not too large, nor too small.\nThe administration and staff are great!!!! They are very personable and caring to everyone's needs.\nIn fact to me the Pathfinder Place has the feelings of comfort as an extended family, where everyone are always treated with dignity and respect.\nPathfinder Place \"it's a Home away from Home\"\nMom recently had a 6 month stay with Pathfinder Place.\nWe checked out nearly all of the Nursing homes, and assisted living facilities in the Fremont area.\nAfter careful consideration our entire family, mom included chose Pathfinder Place!\nMom was extremely happy during her stay with them.\nEric & Amber where the best, they helped us to no end. The staff was great and very accommodating, and mom even commented on how friendly they were.\nThey even went the extra mile when they threw her a birthday party while she was there.\nI would highly recommend Pathfinder Place for any of your loved ones, it's a Home away from Home!\nThis was our third and last place for Mom and it was the best place. Mom had early onset dementia and we tried 2 places in Omaha that were supposed to be the best, but were not happy with either place. When we moved Mom to Pathfinder Place in Fremont, NE, although not the newest or fanciest, it was the one that felt most like a home & not an institution. The fact that Pathfinder is home to elderly people in various stages of health, it just felt more \"normal.\" Her room was decent size with a private bathroom\/shower (very important to Mom) and small kitchenette. Although Mom was only there for a couple of months before she passed, Pathfinder was always looking for ways to make her feel comfortable and loved. The small staff was a comfort to Mom because of her dementia and if anything was needed, they were right on top of it. The final days were spent at Mom's bedside and the family was never alone. The staff, with the help of hospice, were always available and always checking on us, trying to help us through the inevitable. The dignity and grace that they allowed Mom in her final hours still comforts me.\nThis although sounding very nice was a no go for Mom as to the distance from my home would be about the same as Bellevue.\nPathfinder Place is located on the north end of Fremont, Nebraska, a town with a small town feel but the amenities of the city. We are close enough to downtown for shopping trips and live entertainment or exploring the riverfront in the Luther Hormel Memorial Park.\nOur personal care staff is available 24 hours a day to assist with medication management, grooming, walking and other personal needs. We accept respite care residents, urgent placements and have a home visit program. Our emergency response system is available to all residents and is interfaced with our staff pagers so that help is always available. In case of emergency we are only three blocks from the Fremont Area Medical Center.\nPathfinder Place - Fremont, NE has been recognized for consistently high ratings from residents and their families in previous years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shhhhhhhhh, VolleyGirl would not like me putting a photo up of her ( she didn't know about the one of her royal meeting with HRH Apollo....) I am putting up one she would consider a pretty good one. (well, who knows, I consider it a good one). The other is not of my son - but some of his latest artwork.\nBusy helping tie up loose ends with taxes and had three lengthy phone calls today......hopefully I will get to work on my drawing and post it tomorrow.\nLove your son's artwork and hope you complete your taxes!\nAwesome photo of VG!! Super cute.\nBB will always remind me of Mr. Incredible.\nLovely pair of pictures AU.\nYour daughter is every bit as beautiful as you are. How proud you must be.\nYour son's artwork is fantastic!\nTaxes, ugh. We pay a hefty chunk to have a CPA do ours for us. He knows lots of secrets that save us some money, so we can give it to him!\nshe is so gorgeous and looks exactly like you, two beauties in one family. i remember you said your son could draw dogs, so if ever i write a book, i shall know who to ask to illustrate it. hugs.\nFunny to think that this little innocent looking cutie pie is the same who looks so competitive on the courts!\nthere's an award for you on my blog, with hugs.\nI forgot that your tax time is different from ours!\nYour son's work is beautiful.\nI see the apple didn't fall far from the tree. nice work your son did. And volley girl should be okay with that shot. I would be if I were her anyway. It's a really good one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Would be great if we could hear the music Hans is talking about at the moment. Like when he was talking about a piece from Sherlock, hearing it would be great. Nevertheless fantastic Masterclass!\nMr. Zimmer's candour and humility, in spite of all the great things he has accomplished, is inspiring and admirable.\nI have always admired Hans Zimmer. The first movie I saw on DVD was Gladiator. I lived in an apartment and had to listen to the movie with head phones. It was then that I realized that movies were music.\nGood lesson, but I with we could have seen some more examples, clips of scenes from the films he references for example.\nPerspective is definitely the name of what you're doing, positioning the music with the scene, but I like Hans also conveying that not all the information is dialogue. But it's teamwork... the music and the actors. Good lesson!\nCool that he mentioned Jack Nicholson, he'd be amazing to have on here too.\nGood to learn it's always about perspective and allow emotions go with the flow, like a river. Sometimes, we just need to feel it and make something fresh and different, depending on what kind of emotions the writers \/ directors want to create for the audience.\nI love that!!!. Awesome lesson again Hans, thank you so much!!!. Onward!!!.\nMusic communicating the situation is no new concept for me. Very well said!\nJust as the writer chooses just the right words, the composer has to do the same thing. Sometimes it's just one note. Very interesting idea he's using to move the orchestra back from the mics to get a subtle high frequency roll-off. (Hope I'm not too far off topic here.) I come from a background of live performing. One of the difficulties in a live setting is being understood at a distance. Audience members further back lose intelligibility because all the bodies in front of them absorb the high frequencies. We know this because we've all driven past houses where the bass is booming and we can't hear the singer. For my live shows I use a compressor to keep the quietest part of my monologue from getting eaten up. I also use a feedback exterminator to keep the squeals out of the PA, which lets me turn the highs up.\nThat's revealing! I think the course would be much richer if it showed some sample scenes instead of just mentioning them.\nI have to say that, even as a composer, I do want to hear the dialogue. Even if the director wanted the music louder than the dialogue. Why write dialogue if you're not going to listen to it?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Cutler Hammer Motor Starter Wiring Diagram Cutler Hammer Motor Starter Wiring Diagram And Eaton Inside 12 1 - Workplace Learning.\nNew cutler hammer motor starter wiring diagram - In no way forget to look up the next photo gallery, which additionally incorporates the wiring diagram for cutler hammer motor starter refrence eaton motor starter wiring diagram wiring diagram image showed above. If you have any thoughts, questions or simply need to mention hiya to other human beings, please do no longer hestitate to publish your opinion\/thoughts through the subsequent comment shape. Constantly remembers to browse the next image album, which additionally carries the three phase contactor wiring diagram begin prevent exceptional of contactor wiring diagram start forestall cutler hammer download store cutler hammer motor starter wiring diagram wiring diagram photo confirmed above. When you have any thoughts, questions or just need to say hiya to other people, please do no longer hestitate to publish your opinion\/thoughts thru the following comment form.\nPosts index: a b c d e f g h i j k l m n o p q r s t u v w x y z 0 1 2 three four 5 6 7 eight 9 pix index: a b c d e f g h i j ok l m n o p q r s t u v w x y z zero 1 2 three 4 five 6 7 eight 9. Motor starter wiring diagram quality of cutler hammer motor starter wiring diagram a way to twine a and is one in every of pics which might be related with the picture before inside the collection gallery. The precisely measurement of motor starter wiring diagram nice of cutler hammer motor starter wiring diagram how to wire a and was 2626x1646 pixels. You may also search for a few images that related to wiring diagram by scroll all the way down to series on under this photo. In case you need to find the other photograph or article about sparkling motor starter wiring diagram simply push the gallery or in case you are interested in similar photos of motor starter wiring diagram best of cutler hammer motor starter wiring diagram how to wire a and, you're loose to browse through search characteristic that placed on pinnacle this web page or random publish segment at below of this post. We hope it allow you to to get data of this photograph.\nPosts index: a b c d e f g h i j okay l m n o p q r s t u v w x y z 0 1 2 three four five 6 7 eight 9 photos index: a b c d e f g h i j okay l m n o p q r s t u v w x y z zero 1 2 three 4 5 6 7 eight 9. Cutler hammer motor starter wiring diagram gimnazijabp me at 4, cutler hammer starter wiring diagram eaton motor excellent of 8, cutler hammer motor starter wiring diagram 1024x880 7, eaton dry kind transformer wiring diagram luxury beautiful cutler hammer motor starter of zero, cutler hammer motor starter wiring diagram bunch endear square d nine, cutler hammer starter wiring diagram and reversing to motor eaton inside 1, motor starter wiring diagram great of cutler hammer how to twine a and six, eaton starter wiring diagram heater chart hand off automobile starters magnetic selector amusing square lightingr 970x841 to cutler hammer motor 5, three section contactor wiring diagram begin prevent best of cutler hammer download store motor starter wirin, cutler hammer motor starter wiring diagram beautiful diagrams in 2, gallery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Unedited North Philadelphia is back tomorrow, Saturday, April 11, with an all-day screening and panel discussion, \"From Reel Footage to Real People,\" at the Wagner Free Institute of Science. Archival programming includes historic footage of the surrounding neighborhood, including Martin Luther King Jr. at Baptist Temple, Richard Nixon's visit to Progress Plaza, highlights of the Uptown Theater, the Columbia Avenue Riots, the ascent and end of the legendary Blue Horizon boxing arena, and, just recently added to the program, \"Finding Northadelphia\"\u2013a video produced by Youth Empowerment Services in 2010 that explores the history and development of North Philadelphia through conversations with community members and residents.\nPettit says Saturday's programming is a partnership with the Wagner, which wanted to narrow the focus of the event to the immediate neighborhood. Adding \"Finding Northadelphia\" to the screening will lend a voice to the community, something a few found lacking in the last screening.\nScharf says that highlighting threatened neighborhood landmarks was also a consideration with the screening's curatorial choices, nominally with footage of the Uptown Theater and the Blue Horizon, the exterior of which was recently placed on the Philadelphia Historic Register, but iconic interior arena was not.\nA slideshow of John Mosley's portraits of the community from the 1930s through the 1960s, collected from the Temple University Libraries' Charles Blockson Afro-American Collection, will also be shown.\nA panel discussion will be held post-screening where impressions and expertise will be shared. Philadelphia Public History Truck founder Erin Bernard will moderate the discussion. The panel will include Temple archivist John Pettit, Beech Companies president Kenneth Scott, who will speak on the neighborhood's commercial and economic development, Karen Warrington, neighborhood advocate and communications director for Congressman Robert Brady, and Scribe Video Center founder and director Louis Massiah, who will touch on capturing the stories of communities with video.\nFollowing the panel discussion Bernard invites program attendees to share personal stories and neighborhood artifacts with the Philadelphia Public History Truck, which she will include in her exhibit, \"They Say They Gonna Build,\" opening on June 5th at 2152 N. Broad Street.\nProgramming for \"Unedited North Philadelphia: From Reel Footage to Real Stories\" will take place on Saturday, April 11th from noon-4:30PM. The Wagner Free Institute of Science is located at 1700 West Montgomery Avenue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Electronic recycler MHD Enterprises has committed to hiring 75% of its workforce from rehab centers. The company offers coaching, training, and opportunities for advancement to encourage employee retention and reduce instances of relapse; in fact, MHD has experienced a 90% success rate of employees with no relapse.\nThe Executive Leadership Award recognizes presidents, principals, or chief executive officers for devoting time and energy to strengthen and transform the organization that they lead. This includes specific situations or initiatives where the recipient's insight, discernment and ability to lead and inspire have guided their organization to a new level of growth, achievement and recognition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A reasonable reaction, all things considered.\nSo Jacob took a number of different precautions as his scouts confirmed the two parties \u2014 his own and Esau's \u2014 were drawing closer and closer together. For safety's sake, he divided his family into two camps. He also sent gifts ahead of him, and called earnestly upon God in prayer. We are treated to a full chapter of Jacob's preparations and concerns. If we were reading a novel, this would be the writer's way of building suspense. In Genesis there is little of that, for reasons I will explain shortly.\nThese accounts are historical. They are true. But they are still told as stories with beginnings, middles and ends, and with many of the less relevant days, months and years omitted so the reader can see the big picture for what it is and draw from it more easily the moral lessons which the characters themselves may or may not have perceived. The Bible is series of related, self-contained stories telling a larger tale across time. But it is not a soap opera, with the same exhausted characters circling one another in endless permutations and combinations. Nor is Genesis \"anything can happen\" time, like some lame George R.R. Martin sword and sorcery novel where a decapitation is as likely as a coronation for the protagonist.\nNo, in Genesis real events have an actual moral point.\nThis being the case, the attentive reader of the scriptures can hardly get caught up with Jacob in the emotions of the moment. Oh, we can understand his trepidation, certainly, but we can't really enter into it with him, even if we have not read ahead and don't know how this part of the story ends. We know full well this is not going to be the really cool unexpected plot twist when Esau stabs his brother, seizes back his birthright and becomes the ancestor of Messiah.\nWhy not? Well, the reader has been introduced to a God who can speak a universe into being, who can keep Noah and his family safe through a civilization-ending deluge, who can preserve Abraham and Isaac through all kinds of misadventures and near-endings, and who even insists on rescuing righteous Lot from the wicked city of Sodom before destroying it. Lots of bad things happen in Genesis, but the one thing that is consistently true is that God keeps his promises and preserves those who are part of his plan notwithstanding their best efforts to mess things up. As readers, we know where this story can go and where it can't. There is a prophecy in play here that declares \"The older shall serve the younger.\" And we know the word of God cannot fail. There is no suspense about whether God's word will come to pass. The only suspense is in exactly how God's word will come to pass.\nThat part is obvious. The rest, not so much. From Jacob's perspective, his story could still take a very bad turn. The reader respectfully disagrees with him. We may even wonder why he's getting so worked up. Has the man no faith? Does he not pay attention to his own history and that of his family?\nBut Jacob is not the only one who ever suffered from a lack of perspective. Our lives don't feel much like stories either, do they? Oh, we sometimes try to weave them into coherent narratives, but if we are honest, for the most part things do not fit together quite so neatly as they do in the accounts we read in scripture and elsewhere. Patching the apparently random events of our lives into a tale worth telling takes major work and often leaves you wondering if you haven't been a little too creative in your reinterpretation of events. Try reciting them out loud. They no longer sound like your life; they sound like some novice Hollywood producer pitching a bad new TV miniseries.\nI have often said that if you had asked me at any point in my life to project what the next five years might be like, I would have gotten it all wrong every single time. My history is full of spiritual rabbit trails that went nowhere; ridiculous and embarrassing missteps; stops, starts and gaps that seem to have no particular meaning; lessons learned and relearned and then relearned again; expected answers to prayer that turned out not at all as expected; projects commenced with great enthusiasm that petered out inexplicably; exciting twists and turns that could never have been anticipated \u2026 you get the idea.\nOverall, it has been a great ride. I have no complaints. But it's a series of events, not a story. At least not from my perspective.\nGod has not appeared to me personally, as he did to Abraham, Isaac and Jacob. I do not battle kings, dig wells and watch cities burn. There was very little drama around the birth of my kids. Sisters do not scrap with each other to get me into bed, or offer me their servants to sire children with. Probably for the best, really.\nGod is not, to the best of my knowledge, about to make a great nation out of me. Or you, if we're honest.\nIn fact, if I'm not spiritually attentive and obedient, I might end up as the modern-day equivalent of one of the Bible's cautionary tales. I don't imagine Ananias or Sapphira got out of bed one morning expecting to be struck dead by end of the day. I doubt Demas expected to have his carnality documented in a letter to Timothy, and that letter preserved for over 1,900 years and read by millions. For that matter, I doubt Judas saw his role in the events leading up to the crucifixion quite the way we read them \u2014 at least not until the very end. All these folks played a valuable role in God's purposes, but regrettably they did not benefit personally from the lessons they taught others.\nI'd prefer not to go that route myself, but let's at least acknowledge that these things do indeed happen to people who claim to be followers of Christ.\nWhat are we to make of that statement? Does it mean that every single Philippian convert to whom Paul was writing grew to perfect Christian maturity, that none who professed faith in Christ in the city of Philippi ever apostatized or fell away, that none of their lives were cut short by the judgment of God like Ananias and Sapphira, and that not a single one will appear at the judgment seat of Christ saved as if through fire? If so, Philippi was probably the first and only church in history with that sort of spectacular success rate. I've never been part of a church like that, and I'm sure you haven't either.\nAnd if not that, then what did Paul mean?\nWell, God is working, and his purposes in the lives of believers will absolutely be accomplished. And just as there was an obvious difference between the inevitable destiny of Jacob and that of, say, one of Laban's unnamed sons, so there is a notable difference to be observed both in scripture and in the world today between the real children of God and those who just happen to make use of the family name.\nDid God complete his work in Samson's life? You bet he did, major faults and all. Samson might have enjoyed the experience a great deal more if he'd acted more consistently in harmony with God's revealed will, but then he wouldn't have been Samson, would he? More to the point where we are concerned, did God complete his work in John Mark's life? I believe he did, despite the poor start. The important question for each of us to answer in the privacy of our own hearts is \"Did God really begin a work here?\" If he did, then he will surely bring that work to fruition.\nGod's chosen may get off track, act on their own initiative, behave out of character from time to time, miss the point of the lessons God is teaching them and do all the things the patriarchs did and yet still, amazingly, ended up prospering and being blessed by God. At very least he will take his children to the absolute upper limit of what may be accomplished with the material of which we are made. So don't give up. However many missteps you may have taken, if you are a true child of God, your life will be value-added in some mysterious, divine way. It will accomplish a purpose you cannot yet see and tell the world a story you cannot yet read.\nThose who are bit players and character actors may find the bit they played was more important than they ever could have believed. And those who think they have the starring role will discover that we are all bit players in the eternal drama which has Christ at its center and in which God will be all in all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"a fax less pay day loan is the easiest way to fix an individual's economical challenges in those two extremes. pay day loans in an hour offer the amount after you meet these being approved requirements. also the consequences and defaulter charges are also very good in this case. she wasn't even confident that that they that loan adjustment unit. you should not go out and stand in prolonged line up and lose time waiting for financial loan acceptance. it is merely vital on an perception of the ommissions that accompany your coverage. these are regarded as some of the safest purchases you can create. find an agent who has has homes and possesses leased it for years. figure out laptop computer to your requirements. on the flip side, on set rate house loans, because its label means, the interest rates won't change through the entire time of the financing, and this can be from 10 years nearly five decades. by means of consultation with time and with their staff members, some firms have produce unique, no regular personnel advantages programs for his or her workers. payday cash advance corporations do not check out the credit history of people, for the reason that your next pay is employed as to safeguard the loan. it is at least a problem and buyers stored obtaining entangled into debt. the lowdown inquired are also the models you have authored or typed on the application sort. a handful of it is known as an inspection cash loan. this really is helpful simply because lifetime investment decision programs really don't constantly make allowances for the reasonable existing normal two decades in the future with inflation to contend with. put simply, the tool that he's willing to offer should be easily turned into dollars and keeping that in mind given to the financier. the rate most is dependent upon your credit standing of consumer.\nonline loans there are a number of people that enter legal papers but are not able to pay the money back. now let us use optimistic thinking to get your other than conscious head directing on genuine options. currently being credit card, bad credit cash loans come with greater interest rate. they can be in short phrase unexpected emergency dollars wants. exactly what does this numbers imply? nicely, if my understand was accurate, my estimated price easily pressed all in is -bucks621.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kaman Corp. reported second-quarter sales for Kaman Industrial Technologies were $278.7 million, a 2.6 percent decline from $286.1 million in the same period last year.\nWoleseley announced that its name has been officially changed to Ferguson.\nAbrasives manufacturer Flexovit announced that it is celebrating its 40th year in business in 2017.\nORS Nasco announced the appointment of Chris Kempa as president of ORS Nasco and group president, Essendant Industrial.\nHydraulic Supply Company, a distributor of hydraulic, pneumatic and industrial products, is celebrating 70 years in business.\nDGI Supply, a DoALL Company, plans to purchase Coast Tool Company, a distributor of cutting tools and MRO supplies with locations in San Leandro and Fresno, California.\nPraxair reported second-quarter net income of $406 million and diluted earnings per share of $1.41.\nLed by increases in production-related indicators, the Chicago Fed National Activity Index moved up to +0.13 in June from \u20130.30 in May.\nLawson Products said its second quarter sales increased 8.2 percent to $75.0 million, compared to $69.3 million a year ago.\nWESCO International reported net income of $49.5 million, or $1.02 per share, for the second quarter.\nLafert North America, a manufacturer of metric AC Motors, metric gearboxes and coolant pumps, announced two appointments to its sales team.\nEssendant Inc. reported a 6.9 percent decline in revenue to $1.3 billion, compared to $1.4 billion in the prior year quarter.\nThe Timken Company reported second-quarter 2017 sales of $750.6 million, up 11.4 percent from the same period a year ago and up 6.6 percent from the first quarter.\nDXP Enterprises reported sales of $250.7 million declined 2.2 percent compared to $256.2 million in the second quarter of 2016.\nIngersoll-Rand plc reported second-quarter revenues of $3.9 billion were up 6 percent from $3.7 billion in the same period last year.\nAirline Hydraulics Corporation, a regional supplier of fluid power, automation, electrical control, and safety products and engineered systems, has received ISO 9001:2015 certification.\n3M reported sales of $7.8 billion for its second quarter, up 1.9 percent, with positive growth across all five business groups.\nIllinois Tool Works reported second-quarter revenue grew 4.9% to $3.6 billion.\nParticipants at the FPDA\/ISD Joint Industry Summit will have a chance to hear a panel discussion on the new NAW and IBM trend report.\nSKF reported net sales of SEK 20.2 billion ($2.45 billion) for the second quarter, an organic increase of 7.5 percent compared to the second quarter last year.\nCodifyd, a provider of B2B digital commerce and transformation solutions, announced the launch of a state-of-the-art digital agency and artificial intelligence lab in Mumbai.\nMotion Industries has entered into a definitive agreement to acquire Numatic Engineering, headquartered in Los Angeles.\nSnap-on Incorporated reported net sales of $921.4 million increased 5.6 percent from 2016 levels in the second quater.\nJPW Industries Inc. has reached an agreement to be acquired by funds managed by Gamut Capital Management of New York.\nMotion Industries had sales of $1.25 billion for the second quarter, a 7.3 percent increase from $1.26 billion in the same period last year.\nThe July ISA Economic Indicator Report (EIR) from the Industrial Supply Association showed that manufacturers and distributors experienced slower growth during June.\nGrainger reported net earnings for the second quarter of $98 million were down 43 percent versus $173 million in 2016.\nH.B. Fuller Company has signed an agreement to purchase adhesives company, Adecol Ind. Qu\u00edmica, Limitada, a manufacturer of adhesive technologies in Brazil.\nAWISCO, a supplier of welding, safety, industrial supplies and industrial gases in the New York tri-state area, announced the acquisition of Nyrock Machine and Tool Corp., Nyack, New York.\nWinsupply named Santa Fe Springs Winwater its 2016 Company of the Year.\nSeveral law firms have filed class action lawsuits againt HD Supply, alleging the company provided misleading information to investors surrounding its sales of HD Supply Waterworks.\nHydraulic Supply Company, a fluid power distributor of hydraulic, pneumatic, and industrial products, has acquired Fabacher Incorporated.\nIndustrial production rose 0.4 percent in June for its fifth consecutive monthly increase.\nKaman Distribution Group announced the appointment of Jeff Ames as vice president, general manager of its Automation platform.\nDDI System has qualified as a finalist in the 2017 ISSA Innovation Award program with Inform Engage.\nForney Industries, a welding and metalworking company based in Ft. Collins, Colorado, has retained Chicago-based Creeden & Associates to support Forney's aggressive sales goals.\nChattanooga-based Bolts & Nuts Corp. announced it has acquired Industrial Fastening of Macon, based in Macon, Georgia.\nNN Inc. has entered into a definitive agreement to sell its Precision Bearing Components Group to Tsubaki Nakashima Co. Ltd., for $375 million in cash.\nPearson Fastener has achieved 16949:2016 First Edition Certification from the International Automotive Task Force (IATF) as a supplier of high-quality custom fasteners to automotive manufacturers.\nKnipex Tools LP moved its headquarters to Buffalo Grove, Illinois, from its previous headquarters in Arlington Heights, Illinois.\nMay U.S. cutting tool consumption totaled $191.93 million, 14.2 percent higher than April's $168.03 million.\nWelding distributor cooperative IWDC reported that the spend among member companies through the co-op exceeded $21 million in June.\nMSC Industrial Supply reported quarterly net sales of $743.9 million, an increase of 2.3 percent from $727.5 million in the same quarter last year.\nMayhew Steel Products announced that Eric Mills rejoined the company as vice president of sales and marketing.\nRiverside Tool Corp. has acquired General Saw Corp., a manufacturer of carbide-tipped saw blades for plastics and wood.\nGrowth in industrial vending business and in new and existing onsite locations drove a 10.6 percent sales gain for Fastenal in the second quarter.\nIndianapolis-based Haggard & Stocking plans to expand its Evansville, Indiana-based Imperial Fastener and Industrial Supply.\nBridgestone HosePower to acquire Fittings Inc.\nBridgestone HosePower, a wholly-owned indirect subsidiary of Bridgestone Corporation, announced that it is acquiring Fittings Inc., a hydraulic and industrial hose sales and service company based in Seattle.\nThe U.S. Manufacturing Technology Orders report for May 2017 showed gains for the month and year over year.\nGHX Industrial will consolidate four Houston area locations into a new location in Lockwood Business Park, part of a commercial development called Generation Park in Northeast Houston.\nDavid White has joined Standard Locknut as vice president of sales and marketing.\nEmployment increased by 222,000 in June, and the unemployment rate edged up to 4.4 percent, the U.S. Bureau of Labor Statistics reported today.\nApplied Industrial Technologies announced the availability of its new 2017\/2018 Applied product catalog with nearly 31,000 industrial and maintenance products.\nThe Home Depot announced a definitive agreement to purchase Compact Power Equipment Inc., a national provider of equipment rental and maintenance services, for $265 million in cash.\nWinsupply Inc. opened a new location in Tempe, Arizona, serving the fire sprinkler and fabrication industries in the Phoenix metropolitan area.\nBrady, a full-line janitorial supply, equipment and foodservice distributor, announced the acquisition of the Denver, Colorado operations of Walter Industrial & Sanitary Supply.\nNew orders for manufactured goods decreased 0.8 percent to $464.9 billion in May, the U.S. Census Bureau reported today.\nConstruction spending during May was estimated at a seasonally adjusted annual rate of $1,230.1 billion, flat compared to the revised April estimate of $1,230.4 billion.\nEconomic activity in the manufacturing sector expanded in June, reaching its highest level in three years.\nThe Evergreen Marketing Group announces its new board of directors and supplier advisory council effective July 1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..a8d79b6b7423d07ddef61a847d826e3c2978d2b8
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"President Ilham Aliyev has been named \"Head of State of the Year\" in Turkey by the criteria of sincerity, businesslike manner, success and a friend of the Turkish state, according to Chairman of the Federation of Turkish-Azerbaijani Associations Bilal Dundar.\nHe said the selection was made by 145 civil society institutions in 16 regions of Turkey.\nBilal Dundar said the \"Head of State of the Year\" award will be presented to President Ilham Aliyev in January, 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Writer Reads \u2013 And Reviews, Too!\nWelcome to A Writer Reads, a resource designed to help teachers, librarians, parents, and students discover the best diverse books for middle- and high-school readers.\nUpdates every Tuesday, Thursday, and Saturday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kerry has started a new radio show, Up in Your Business with Kerry McCoy. Check it out every Friday live locally on KABF 88.3 FM radio or stream it online at http:\/\/tunein.com\/radio\/KABF-883-s31318\/! Miss it? Check it out in 24 hours on SoundCloud as a podcast. Go to the show archives for that episode wrap up and to listen to the podcast.\n\"Up your business with Kerry McCoy\" will help you discover how to navigate the path to achieve the American dream. She started Arkansas's FlagandBanner.com over 40 years ago with just 400 dollars and now it exceeds 3 million in annual sales.\nEach week she and a guest will answer your questions and offer real world advice on how to become your own boss.\nTune in to this passionate, informative, high energy and often wacky perspective on business in the 21st century.\nLet Kerry McCoy get all up in your business! Live stream the show at 2pm CST on Friday's!\nIf you've never heard Kerry's American Dream story, it is an amazing path from unemployment to owning a multi-million dollar company. Click here to read Kerry's full story.\nKerry McCoy and R.J. Martino of iProv Online during the Sept. 16, 2016 episode of Up In Your Business with Kerry McCoy.\nGuests Sally Mengel of Loblolly Creamery and Sara Slimp of Chunky Dunk with Kerry McCoy on Septeber 23, 2016. They talked about starting your own business, food trucks, accounting aspects and how to ask for help among other topics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is no longer being updated as of 6\/10.\nHere's a table of RUSA members with US addresses grouped by the longest ACP\/RM brevet completed in the US in 2010.\n(Note that this is updated from the table on the RUSA quotas page and includes more riders).\n(*) Given the relatively slow pace of pre-registrations, RUSA offered back 50 spaces in its quota to the ACP for allocation to other countries. Based on discussions with the ACP, RUSA does not expect this to result in any US riders being excluded from PBP.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Build the world you want with Blockitecture. A set of architectural building blocks you can mix to create towers, cities and dwellings. The Factory set includes new colors and forms inspired by the industrial architecture of the city center. Created by designer James Paulius. Packaging dimensions: 5.1cm x 18.4cm x 18.4cm. Made from New Zealand Pine. Suitable for ages 6+.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..7274c0f68ca5184648385b272a947c67c8fca5ab
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Murrieta (Calif.) resident Todd Mitchell Edwards, 49, is accused of sexually assaulting two teenage girls who reportedly went to the church where he was a bishop at one time. Edwards has pleaded not guilty to all charges. If convicted he faces up to nine years in prison. He remained incarcerated Tuesday on $65,000 bail.\n\"He's been charged with three felony counts, one of sexual penetration with a foreign object by force,\" said John Hall, a Riverside County District Attorney spokesman.\nHall says the victim in that incident was a child, and it happened last year. He says the second victim was 18 at the time, and that incident happened in 2006.\nThe Riverside County Sheriff's Department said they investigated a third case of alleged sexual assault, but the statute of limitations had expired.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are currently 1 Homes for sale in Fairway Estates. The average price for Fairway Estates properties is $602,200which is 1% higher than all Homes within a 20 mile radius. The highest priced property in Fairway Estates is $602,200 while the lowest priced Home is listed at $602,200. Most of the properties available in Fairway Estates are Homes . Compared to last month, the average price of real estate in the area has gone up 100%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Single, chain drive, generator, 3.31 HP, 21.098 ci, three-speed.\nThis bike runs and was restored about 20 years ago. Many of the B models were exported overseas, so there is a limited amount here in the US. Has clean Ohio title.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":": An investigation was conducted to determine the impact of looping, sometimes called multi-year grouping, in elementary schools. Information was gathered as to the positive and negative effects of looping in schools nationally and internationally. The research study focused on teachers in the Fresno Unified School District, Fresno, California. A group of teachers who had participated in looping classrooms completed a survey and were asked a series of questions about their experiences. Positive responses were given about the looping classrooms and the teachers' experiences. The results suggest that more schools implement a looping framework..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A. C. Gheen is a freelance writer and associate member of the Evangelical Press Association. Her work has appeared in On Mission, Discipleship Journal, The Upper Room, Group, The Journal of Student Ministries, Devozine, and Breakaway.\nWhen not writing, A. C. may be found indulging her passion for adventure. Whether riding her bicycle, fiddling with her aquariums or engaging in \"horticultural experiments\" in her backyard, she embraces the opportunity to experiment and explore!\nTo contact A. C. with a writing assignment, please send an e-mail to ACGheen at hotmail dot com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfxpu b/data_all_eng_slimpj/shuffled/split2/finalzzzbfxpu
new file mode 100644
index 0000000000000000000000000000000000000000..67b388c56555d55775d6419235b45d0f16e7896f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfxpu
@@ -0,0 +1,5 @@
+{"text":"ETA's mission is to research and analyze energy markets in the context of the economic and policy trends that are driving one of the great transitions in history -- the transition away from fossil fuels towards more sustainable sources of energy. Analyzing the economic and policy risks increasingly embedded in the exploration, development, and extraction of fossil fuels will be a central focus of the group's work, with ETA offering advisory and consulting services to the not-for-profit, public and commercial sectors.\nCivil Society is a driving force in the Energy Transition towards more sustainable outcomes. ETA seeks to carry out research and advise the Not For Profit sector on key financial issues such as carbon risk, cleaner energy trends and policy.\nThe Energy Transition will be one of the most significant economic and infrastructure trends in the next 50 years. Ranging from Mining , Utility, Manufacturing and Finance industries there will be major financial implications. ETA is positioned to carry out research and advise on these.\n1. Why: This paper summarises why a policy response is inevitable, and the driving forces that would precipitate such a pathway.\n2. When, what and how: This paper further explores issues related to modelling the policy and technology pathways, the potential timing and policy options, and the investment implications of an IPR.\n3. Strategic asset allocation and portfolio construction: This paper considers the financial implications of an IPR for asset allocation decisions, including at the asset class and sub-asset class level, as well as portfolio construction (including at the manager and asset level).\n4. Investor actions: This paper considers the actions that investors can take across the three phases of an IPR to improve the efficiency of decision-making, including the management of stranded assets.\n\"Thermal Coal in Asia \u2013 Stopping the Juggernaut\" edited by Mark Fulton (ETA), pulls together key sources to show that both China (by 2020) and India (in the 2030s) will exceed their IEA ETP annual carbon budgets. Furthermore, without a massive scale-up in renewable energy and the development of Carbon Capture and Storage (CCS) in key geographic areas (particularly China), the total carbon budget up to 2050 will also be exceeded in the 2030s. These conclusions are based on existing and \"under construction\" thermal coal power plants, yet even if no additional power plants are constructed, the budget would still not be met. Consequently, the research report calls for action: (1) to reform electricity markets so that low cost renewables are dispatched first; (2) to extend robust moratoriums on new coal power plants; (3) to cap longer-term coal consumption and emissions in the power sector in the context of carbon markets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was not much of a bar, but neither was it much of a place. Rusted tin let in every breeze and black fly, but the wooden frame of the cowshed was sound enough, and someone had carted out the valuable manure so long ago only a trace of the smell clung to the old walls. The Russian had laid a plank on a pair of oil drums, and that was the bar. Patrons wandered in, following the chemical smell of the distillery, and brought their own cups and glasses.\nNo credit was given, or asked for.\nIn such a place, entertainment is rarely complex. No fine tables marked with runes, no handsome glasses edged with gold, no rare vintages. For them, the gameboard was an old wooden square set out for checkers, the outer rim scratched with the most basic, muddled inscription imaginable - probably useless, but it was the form of the thing.\nThe sun hung low, and by ones and twos the customers drifted in. Some paid, others bartered; none drank for free. Some had camp stools, the others stood or sat on whatever crates were available. Even the board was set up on an oil drum, and the players stood in front of it.\nAndolei had been winning too much, he knew, and he should move on, or else lose a match. A player who was too good upset the nature of the place; a player who could not be beat was boring to watch, and frustrating for those who lost. So the stakes would be raised again and again, until they broke.\nSo tonight Andolei stood on one side of the board, and three other players crowded the board. Four shots made a cross in the center of the board. Normally, in this setup, it would be a partners match. But Andolei had been winning too much. Three on one. The only chance they'd given him was a double-shot, a glass twice as tall as the others, to help the betting.\nThe bettors hushed, and as one the four reached down and raised their glasses, clanked them together, to spill a little in each. The others muttered their mystic toasts, but Andolei just raised his chin and slugged it down.\nClear fire hit the back of his mouth, and washed down his throat to sit in his stomach and burn like a pool of acid. Cold, clear, medicinal: vodka was a drink for those who could not stomach an honest cocktail, or could not afford it. The good stuff had some flavor to it, but this was closer to neutral spirit, raw and shapeless, probably brought over as aftershave and filtered out.\nAndolei held it as long as he could, letting the others place out. Misty shapes covered the board, little columns of cloud in a square of fog. Three players, three columns, defined only by the hint of horn or claw.\nThen Andolei opened his mouth, as if to whistle. A breeze blew down onto the board, and it cleared the fog away in a circle on his part of the board. Something heavier than air oozed from his lips, nearly invisible, but it gained color as it dripped, and by the time it hit the gameboard it was a small, swirling column like a miniature tornado, spinning clockwise.\nAs one, the three cloudy spirits moved forward as Andolei's vortex spun faster and faster, growing smaller and smaller. Misty claws slashed at the edges of it, but the other players were on edge; they had not seen this before, were not sure what Andolei was doing. But as it shrank, they grew closer, bolder, their forms gaining a little more definition - arms tipped with claws like a bear on the left arm, like a lobster or crab on the right. They tried to tear at the denser substance of Andolei's little vortex, tearing droplets from it.\nPerhaps they thought they were winning.\nThen, the spark. A column of blue flame shot up ten centimeters from the board as Andolei's spirit took light. It was a batlike-thing with vast thing wings of pale fire, and it moved with the speed of a fuse. In less time than it took to tell of it, those wings surrounded and engulfed the nearest cloudy spirit, which vanished with a pop like a firecrackers; it's player falling back with blistered lips.\nThe other two tried to run, but it was too late, and soon the board was cleared, the fiery spirit melting into the air as its substance was consumed.\nAndolei collected the money, and did not look at the three men with blistered lips, but knew there was murder in their eyes. He had won too much, it was definitely time to go.\nA woman waited for him outside. He face was unpainted, but there were strange geometrical scars etched into her skin, around the lips dipping down to her chin. An iron cup dangled from her waist, where it was tied by an iron chain. Andolei stared at her hard.\n\"I cannot accept a challenge.\" he said. \"Not tonight.\"\n\"Not a challenge,\" she said, the accent was clipped and strange to his ears. \"An invitation.\"\nShe held up her hand, revealing a small scroll wrapped around a small bottle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Colored waistband identifies BACK of garment.\nGentle elastic waistband has an ultra silky feel.\nCloth-like outer material is soft, quiet and comfortable under clothing.\nLeak-free leg cuffs: soft inner and outer leg elastic holds pull-up securely in place without binding or leakage.\nNEW! Ultra soft texture has silky feel and absorbs much faster.\nAll pull-ups are latex free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Base interface for registered timeout or scheduled task that can be queried for timeout or canceled.\nReturns the starting time in milliseconds. Note that while the unit of time of the return value is a millisecond, the granularity of the value depends on the underlying operating system and may be larger. For example, many operating systems measure time in units of tens of milliseconds.\nThe difference, measured in milliseconds, between the start time and midnight, January 1, 1970 UTC.\nCancels the timer task. This call can be done from any thread and will stop any notification of the timer event.\ntrue for success, false if already canceled.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have played a major role in some of the largest class actions and group actions in the Benelux, drawing on expertise from across the firm to deliver the most effective results for our clients.\nHandling matters across a broad range of sectors including financial institutions, securities, funds, large industries and healthcare, our experts integrate regulatory knowledge with sector expertise to create strategically robust solutions.\nAs the rules for collective redress are constantly changing, we guide clients through every stage. When handling complex proceedings regarding private enforcement of cartel and competition law infringements, we work in close collaboration with lawyers in our EU competition law practice group. This cohesive structure provides the most cost-effective and efficient way of working.\nOver the years our litigation practice group has successfully handled multi-party and multi-jurisdictional litigation, before the civil, administrative and criminal courts. In all cases, our objective is to resolve the class or group action as efficiently as possible, aiming to minimise its impact on the rest of the business.\nReinforcing our practical expertise, our partners also lecture on collective redress and hold various publications on class actions and group actions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgfly b/data_all_eng_slimpj/shuffled/split2/finalzzzbgfly
new file mode 100644
index 0000000000000000000000000000000000000000..a4bffea0e058b3f4da4cba52d427858d419e6866
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgfly
@@ -0,0 +1,5 @@
+{"text":"After much discussion, measuring up, marking which planks will stay (and go), sucking of teeth, visits to woodyards and costing up the options, the larch has been ordered! We're assured that it will be delivered to Woodbridge 'soon'.\nTrue to their word, a call from Somerscales, Grimsby, confirm delivery of the planks early in October. At 11am, the lorry arrived with an understanding and helpful driver. Despite the lack of available assistance, the wood was offloaded and all sticked up before dark!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One approach to managing organizational security is to use risk calculation metrics. Risk characteristics are defined for each application in the organization. The resulting risk profile indicates the risk the organization is exposed to if an application is found vulnerable. When a security issue is identified, the overall result is considered in the context of the risk profile to generate a risk score, which is in turn translated to a critical, high, medium, or low risk rating.\nNow organizational priorities can be set to handle these issues from the most critical systems down to the least critical.This approach works well for organizations trying to understand the risk posed by applications deployed and used inside the organization. The information security team has a risk evaluation for the organization and can reach out to vendors or internal development teams for remediation.\nBut what about a development team? They may have little or no notion of the eventual deployment environment of an application in development, particularly if the application is intended to be sold commercially. How can the team manage and prioritize its risk and security-related work? How can development management understand the security status of application in development? This information is important when making commitments, scheduling release dates, and the like as part of managing the development lifecycle.\nRisk calculation of an application in development requires information that the development team may or may not have \u2013 and what they do have may have to do more with industry or regulatory standards required by their customers. For a complete understanding of risk, the development team needs additional criteria.\nWithout having a complete application profile the team is left with the properties of the issues themselves. An issue has many attributes, including type, classification, cause, severity, location, and date initially found. A team may identify a cross-section of issues based on any criteria; any issues matching the criteria become prioritized issues for resolution.\nAs the team begins security work on an application, fairly lenient criteria can be used \u2013 at least initially. As work progresses, increasingly strict criteria should be set, with the ultimate goal of matching whatever standard is required by the industry or a regulatory body.\nSet a baseline criteria that specifies no new issues beyond a specified date. The team makes a commitment not to introduce new security issues once the baseline criteria is set. The team should be adopting good development practices for security-related tasks from that point forward such that new issues are routinely detected and fixed and fewer issues overall are introduced during the development cycle.\nOnce all development is meeting baseline criteria, set a new severity criteria that specifies that no high severity issues are allowed. Now a new set of issues fail the criteria (old, high severity issues) which the team commits to fixing.\nAs those issues are cleared, the team may choose (or be mandated) to adhere to an industry standard such as OWASP Top 10 or SANS Top 25 and commits to resolving all the issues appearing in those standards.\nFuture criteria commitments may be FedRAMP or HIPA or even \"zero issues.\"\nEstablishment of sequential criteria not only create a set of approachable goals, but also create a non-biased status view for an aggregation of applications. Unlike risk calculation where each application is measured by the same criteria, with compliance status each application has different criteria appropriate to its specific characteristics and properties. What's important is whether the application (and the team) meet the committed criteria, understanding that each team may be at a different maturity level, and every application may be required to adhere to a different standard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Estoy mirando para mi libro\", \"No s\u00e9 nada respecto al f\u00fatbol\", \"el pr\u00f3ximo s\u00e1bado me muevo a otra casa\", \"mi hermano vive en los suburbios de Sevilla\" and \"Soporto tu decisi\u00f3n de casarte con un hombre 40 a\u00f1os mayor que t\u00fa\". What do these five sentences have in common? They all are wrong!! In our Spanish classes our English speakers students make some common mistakes. Do you want to take a look at them? Take 5 minutes to read this post.\n1) \"Estoy mirando para mi libro de gram\u00e1tica\" is wrong. Do you know why? It's easy: in Spanish \"mirando para\" doesn't exist, it's a literal translation from English (look for). You can use \"buscar\" but it has no preposition. The righ sentence would be \"Estoy buscando mi libro de gram\u00e1tica\" (I'm looking for my grammar book). Remember that when the thing we are looking for is a person we add \"a\" after \"buscar\": Estoy buscando a Mar\u00eda \u00bfla has visto? (I am looking for Mar\u00eda, did you see her?).\n2) How do say in Spanish \"I don't know anything regarding French literature\"? Here you have: \"No s\u00e9 nada respecto a la literatura francesa\". How would you say \"You must respect your grandparents\"? \"Debes respetar a tus abuelos\". Be careful with the difference between \"respecto a\" (regarding) and \"respeto\" (respect). They mean different things.\n3) Do you think this sentence \"El pr\u00f3ximo s\u00e1bado me muevo a otra casa\" is correct? No!! Do you know why? Because in Spanish \"to move\" (in the sense of changing where you live) means \"mudarse\": Viv\u00ed dos a\u00f1os en Barcelona y despu\u00e9s me mud\u00e9 a Madrid (I lived for two years in Barcelona and then I moved to Madrid).\n4) \"Mi hermano \u00c1lvaro vive en los suburbios de Sevilla\" means that my brother lives in a conflictive area of Seville with poverty and other problems. If I want to say that my brother lives in the outskirts of Seville I must say \"Mi hermano \u00c1lvaro vive a las afueras de Sevilla\" or \"Mi hermano \u00c1lvaro vive en las afueras de Sevilla\" (you can use the prepositions \"a\" or \"en\").\n5) \"Soporto tu decisi\u00f3n de casarte con un hombre 40 a\u00f1os mayor que t\u00fa\". Do you think this sentence in Spanish is correct? No, it's not because the verb \"to support\" means \"apoyar\" (not \"soportar\"), so if I want to say \"I support your decision\" in Spanish I must say \"Yo apoyo tu decisi\u00f3n\". On the other hand, if want to say in Spanish \"I can't stand my husband\" I must say \"No soporto a mi marido\". What a pitty! \ud83d\ude42 Keep in mind that if you speak about sport teams you must use the verb \"ser\": \"Which team do you support?\" (\u00bfDe qu\u00e9 equipo eres?) \"I support Real Madrid\" (Soy del Real Madrid).\n\u2013 Estoy buscando mi libro de gram\u00e1tica.\n\u2013 No s\u00e9 nada respecto a la literatura francesa \/ Respeto mucho a mis abuelos.\n\u2013 El pr\u00f3ximo s\u00e1bado me mudo a otra casa.\n\u2013 Mi hermano \u00c1lvaro vive en\/a las afueras de Sevilla.\n\u2013 Apoyo tu decisi\u00f3n de casarte con un hombre 40 a\u00f1os mayor que t\u00fa.\n\u2013 Soy del Real Madrid (equipo de f\u00fatbol).\nIf you are an English speaker and you want to share with us other common mistakes in Spanish, go ahead. We'll be pleased to read them!!\n\u00a1Tienes toda la raz\u00f3n, Luc\u00eda! Por cierto, he visto tu p\u00e1gina y es muy interesante. \u00a1Enhorabuena!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My ultimate goal is to artistically point the way to the experience and the expression of that unity of the Spirit.\nI paint to try to capture a timelessness or emotion, a notion of the other, and of our human and animal nature's. My animal-headed people and obscured faces are then vehicles to carry that idea. Traditionally our cultures painted zoomorphic images to represent spirituality, aspects of God, a timeless force. These chimeras capture the awe of the viewer with their strangeness. In the same way, my landscapes act as symbolic reflections of that journey. I hope my paintings tap into this Mystic as modern-day totems residing in their everyday existence, just as we are individual spiritual people living odd and normal everyday lives.\nFor most of my life I've taken photographs. As the sophistication of digital cameras and tools has increased, I've developed an ongoing series of photographic montages and digital compositions. Then I integrated printmaking into my art practice, and more recently added abstract painting as another element in my mixed-media art.\nMy work centers on color and place, with a strong graphic component. In my photography, color has always been a magnet. And my fascination with places has led to an exploration of how to express the sense of a city or a patch of woods by combining my images with other information.\nAs an artist, I sometimes think of myself as a prospector, digging among the images and memories I've assembled over the years and following my curiosity to learn and dream about places I have yet to visit. I try to hold those feelings of discovery and inquiry as I create each mixed-media work, weaving together my photographs, painted surfaces, and bits of maps, old pages, or even wax.\n\u200bWith her love of the outdoors it is only fitting that jewelry artist Michelle Hoting's designs are a direct reflection of nature. Her pieces reveal the beauty of the earth that created the materials she uses. Her work incorporates metalsmithing and lapidary (stone cutting) techniques utilizing silver, copper, stone, flora and antique finds.\nI was raised in a house full of creativity and art supplies. Though my parents were both cartoonists by trade, they spent a lot of time working with other media, making jewelry, furniture, sculpture and clothing according to whim or need. In college I decided on a career in graphic design, but after a short and unhappy stint in the advertising industry realized that I preferred making actual, physical things. I retrained as a woodworker and have been much more happily, though less lucratively, working with wood ever since. Most of my woodworking career has been spent making furniture and cabinets, with a little side trip into guitar repair and making guitar parts, but I've loved turning from the first time a friend plonked me down in front of his lathe and chucked in a piece of madrone. I still use the carving mallet I made that day, my first turned object, and have been extending and improving my skills ever since.\nI love wooden bowls and plates. They feel good. They're useful and beautiful, and they can last a lifetime. A well-made bowl can be a wonderful way to honor the tree the wood came from, and a wonderful way to grace a table.\nSee a step-by-step photo story of how Kalia makes her beautiful porringers on her Facebook page.\nJoann Lustig has been a professional jeweler for the last 30 years. Her business was originally established with her husband. After he passed away, Joann took over the designing, creating and marketing, and has been doing this one-woman operation for the past 17 years. Their original vision was to create jewelry that was unique, colorful, beautiful and long lasting--pieces that could be worn for a lifetime and then passed on to the next generation. Working only in gold, often incorporating gemstones, and using the lost-wax process, Joann creates one-of-kind rings, pendants, wedding sets and earrings.\nSince a very early age I have been exposed to art in many forms and shapes. I grew up in Estonia in the last days of the Soviet Union. Like many children, I was first introduced to art through my father. I loved watching my father draw. I loved it when he taught me the basics of watercolor. My father took me to see museums in Estonia, Russia, Armenia, and he gave me his large set of art books printed in Russian. My father had dreamed of becoming an artist, and even though his life took him in a different direction, art has stayed with him and enriched his life and mine.\nSo when the choice for a field of studies presented itself and I had to ask myself what I would like to learn, the choice was easy. And so, I am on my way to graduating from the Academy of Art University in San Francisco. The academy offered the opportunity to explore different subject matter from traditional landscape paintings and still-life to portraiture.\nLucy Martin is mainly a botanical artist, focusing on the mysterious beauty of fungi and lichens. She finds endless fascination in the intricate details of the natural world. Her paintings evoke places, rather than isolated specimens, as in more traditional botanical art. They are small landscapes, glimpses of the hidden life of the forest.\nBrightly hued California landscapes are a favorite subject of mine, reminiscent of my grandfather's early California plein air paintings. I paint in oils when outside, enjoying the Sonoma County countryside. I also use acrylics in my studio, painting from digital photos.\nMy mother's artwork also influences my landscape, portrait, figure and still-lifes. I share her desire to paint everyday objects, and the people, animals and landscapes that spark my emotional or aesthetic interest. Like her, I prefer a loose, impressionistic style.\nI sometimes wander into abstraction, influenced by contemporary and abstract painting studies at UC Berkeley. Perspective is often a strong element in my compositions, reflecting my past experience in architecture and graphics.\nI constantly experiment and learn as I paint. Each new painting I start seems to emerge, as if on its own. I am often surprised by what happens on the canvas in front of me.\nI absolutely love to paint. There is no greater challenge in my life, and no greater satisfaction. Creating a piece of art is a fascinating puzzle to solve intellectually, but we artists are also called upon to integrate our most profound emotional and spiritual responses into the work.\nSusan St. Thomas draws her inspiration from dreams, myths and the unseen realm of the natural world. She is devoted to creating images which touch the spirit, evoke a sense of peace and harmony and remind us of the multi-dimensional world in which we live. Her unique paintings have been exhibited extensively in the U.S. and Europe including shows in Beverly Hills, Amsterdam and San Francisco's Grace Cathedral. Her work has been featured in several books and widely published as art cards, calendars and prints.\nSince 1990, Susan has been an adjunct faculty member in the graduate arts program at John F. Kennedy University in Berkeley. She currently teaches at the Sebastopol Center for the Arts, Mendocino Arts Center, Schoolhouse Studio and leads other healing arts workshops in the Bay Area.\nSusan lives with her husband, Joel, also an artist, and their 4 cat friends in a 100 year old remodeled schoolhouse in the redwoods of Occidental where they have a studio, gallery and garden with stone labyrinth.\nCome visit the Schoolhouse Studio\/Gallery. Open by appointment, 707 874-9462.\nStacey Schuett's mosaics and paintings are a rich, textured illumination of the natural world. Stacy is someone who notices and cares passionately about small things, especially the little creatures that others often pass by without a thought. Her luminous portrayals of the many humble beings who live hidden among us are truly to be treasured. Whether depicting beetles, frogs or rabbits, Stacey's love and passion are evident.\nTeri Sloat taught for several years in the bush in Alaska, then helped create children's literature and curriculum with Alaska's native peoples. This started her career in writing and illustrating trade books for children--twenty-some so far. Now she tells stories as wall pieces, using a variety of mediums. She finds that people are hungry for folklore told through images.\nTeri enjoys speaking to artists and writers of all ages, from local groups to international audiences. I feel our jobs as artists is to encourage the imagination of everyone. Most of my talks are on the importance of telling our own stories.\nTeri studied at Oregon State and Sonoma State Universities. Her clients include Dutton, Putnam, Simon & Schuster, Sesame Street, Children's Television Workshop, and International Reading Association. Her galleries include L'Attitude (Boston) and Stephans Fine Arts (Anchorage).\nLandscape is Teri's new passion: I am a plein air painter and have painted from the Rockies west. I am in love with the hills we live in, from pasture to sea. You will find me parked somewhere, on a narrow road or someone's ranch, painting, with gratitude, once or twice a week.\nMy name is Jeff Watts. I'm an artist living in Forestville, California.\nWhether it's an animal, a person or a flower, I consider all of my paintings to be portraits. I'm inspired by the techniques of the old masters and like to take advantage of the movement of the paint and the visible brush marks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're having difficulty holding the 6th & 5th strings down with your 1st finger you can use your 1st and 2nd fingers. Practice building strength and dexterity in the 1st finger by moving up and down the neck while holding the 5th and 6th strings. Little by little it will become easier.( It can be tricky to get the finger to bend backwards.) If it really hurts, DON'T DO IT.\nA lot of what makes this exercise work has to do with the left hand pinky hammering off and on the 4th string. Again, take the time to practice this slowly. It's a great way to build strength and dexterity in the pinky.\nBe sure to take breaks and stretch\/relax your hands whenever you need to!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbirmp b/data_all_eng_slimpj/shuffled/split2/finalzzzbirmp
new file mode 100644
index 0000000000000000000000000000000000000000..9969b994c993a622af41f53d5dd491c3ae99f56a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbirmp
@@ -0,0 +1,5 @@
+{"text":"Summary: I am focused on helping entrepreneurs and small BUSINESS owners have and achieve what they dream of while maintaining a focus on what is important. I've seen too many entrepreneurs (myself included) put far too much ahead of the reasons they started their business in the first place. Loosing sight of your \"WHY\" is a dangerous misstep on the road to financial freedom. That is where BBQ comes in! In my journey through starting and selling four businesses by the time I was 40, discovering my love for BBQ was the beginning of my departure from being a workaholic that wanted to make everything bigger!\nIn this episode Tim shares his ten favorite BBQ restaurants in the Dallas Fort Worth Metroplex.\nThe topic of this episode of the Business and BBQ Podcast is \"getting back in the saddle.\" This could mean different things to different people based on their business situation, but for Tim it means getting reacquainted with your business processes and making necessary adjustments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Application to Cottage Health Residency Programs is made through Electronic Residency Application Service.\nFor more information about the General Surgery Residency Program, contact Kim Geller.\nFor more information about the Internal Medicine Residency Program, contact Pat Keay, Graduate Medical Education Coordinator, at (805) 569-7315.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Southern Cross Austereo has confirmed their support for the Rove McManus and Sam Frost breakfast show with the news 2Day FM's Rove and Sam are set to take on the airwaves across the nation, each weeknight from 7pm-8pm on the Hit Network.\nCommencing on Tuesday 26th April, the program will be brand new to all markets outside of Sydney, featuring a mashup of the best content delivered by the hosts from that day's Sydney breakfast show.\nRove and Sam will go to air across Australia after Hamish and Andy and be followed by the Hit 30 with Emma Freedman and Angus O'Loughlin at 8pm, now providing listeners with three consecutive national weeknight shows.\nHit Network head of content, Gemma Fordham said, \"Rove and Sam is a show that is authentic, funny and very real. The decision to spread the love of Rove and Sam, across the country, is due to the positive response the breakfast show is receiving.\nWith the Hit Network's Hits and Old School music format the #1 music format with 18 \u2013 39s and it's very popular RnB Fridays, this new national program just adds to the reasons why the Hit Network is also the #1 commercial radio network in Australia for people 18-39 year olds.\nGemma Fordham talking about this week's ratings for the Hit Network stations including commentary on Rove and Sam.\nListen to former SCA national head of content Craig Bruce on the evolution of the Hit Network, Rove and Sam and the reasons they go big with RnB Fridays.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I adore these.. geez. They elder scrolls online gold are one of an choice. I have been as a result amazed to acquire these products to do with Any party dawn. Simply put i virtually fainted due to the fact my mother said We wasn't taking pretty much any. But I managed to do... It had been awesome. In that respect there modern street fashion, comfortable, I am unable to define him or her. Definitely feels I am strutting relating to clouds. I need these items cool, vogue, personally Providers purchase them.\nI got my 1st set of two elder scrolls online gold (traditional, brief, chocolates colors). They are so style in addition to wonderful I'm able to seldom accept is as true! This discounts these elder scrolls online gold many times issues many people, however , children an important buyer, they're really worth the revenue! My very own elder scrolls online gold seem to be my personal favorite object around my house. I actually want to acquire a second pair (possibly in height bleak w\/buttons) this approach winter months, and i also is unable to presume My organization is thinking this approach, having said that i may well unlikey wait for a unique climatic conditions into the future!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a free pastel art tutorial by Tricia from HodgePodge.\nThis is a very easy pastel to paint! It can be done successfully by even the youngest student because it is comprised of mostly circles!\nStart your painting by drawing two separate bunches of grapes. Grapes usually start out with a larger cluster at the top of the bunch and taper down to just one grape at the bottom. The key is to try and make the grapes as transparent as possible.\nUse your dark green to draw the grapes and then take your finger and smudge the color over most of the grape surface. Add lighter and lighter green color until you have a satisfactory shade of green. Take your cream or palest pink pastel and put a stroke of \"light\" at the top of each grape\u2026curve it to emphasize the roundness of the fruit.\nThe same procedure goes for the purple grapes. Start dark and add layers of lighter and lighter purples and violets and a bit of dark red pastels.\nFinish off with a crooked stem at the top of the bunch\u2026and there you have it! Good enough to eat!\nWe also offer 50+ free chalk pastel and acrylic tutorials on our site. If you prefer an easy to browse format, we have a year's worth of chalk pastel tutorials in ebook form. The tutorials are easily viewed on computer, laptops and portable devices such as iPads.\nHead over to Hodgepodge and download your free Baseball and Bat Chalk Pastel today. Subscribers will find it at the bottom of the subscription feed.\nTricia Hodges faces a daily dose of chaos homeschooling five children from preschool to high school. She is an author of art tutorials and cookbooks and shares helpful habits for the road home at her trio of Southern Hodgepodge sites. She and her husband are authors and owners of Curriculum Choice and Habits for a Happy Home. Tricia's mother, Lucia Hames, aka Nana, teaches art lessons to her children \u2013 and they share those times in ebook form and online.\nWe LOVE Nana's pastels! Can't wait to make the grapes and so excited for the upcoming artist series! Thank you, Tricia!\nThank you Heidi! Many thanks to Jamerrill for the visit to Free Homeschool Deals today! I hope this lesson will encourage families to add art to their days \u2013 and to see how FUN and simple it is. Just short lessons build a love of art.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbizrb b/data_all_eng_slimpj/shuffled/split2/finalzzzbizrb
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+{"text":"There is no surer sign Christmas has passed than the sight of a crowded shopping mall in Canada.\nAnd with Boxing Day sales already underway, retailers say they're geared up to help Calgarians make that last impulse-buy of the year.\nClose to 100,000 people are expected to walk through the doors of Chinook Centre on Tuesday, about the same as last year's post-Christmas shopping spree.\nOver at Sunridge Mall, Best Buy general manager Aziz Kabani is expecting anywhere from 3,500 to 4,000 customers on Boxing Day once doors open at 6 a.m. He anticipates shoppers will flock to this year's big-ticket items, such as the 65-inch Sharp Roku TV that's priced at about $800, or the Bose headphones going for $190, a discount of about $120.\nDec. 26 remains one of the three busiest shopping days of the year in Canada, along with Black Friday and Dec. 23, when last-minute shoppers scramble for gifts, according to Moneris, Canada's largest processor of credit and debit card payments.\nBlack Friday, which arrived on the Canadian scene in 2010, has actually beaten out Boxing Day for the past three years in terms of total dollars spent countrywide.\nBut for Chinook shoppers, the post-Christmas tradition still takes the cake, with upwards of 50,000 more customers visiting the mall on Boxing Day than Black Friday, according to O'Neill.\n\"I think that's one tradition that will take a while to equal or surpass,\" she said.\nThe timing of Boxing Day at the conclusion of the holiday shopping period, and the ability of stores to promote their markdowns, is one reason the retail event has remained so strong, according to Michael Kehoe, the owner of Fairfield Commercial Real Estate.\n\"Boxing Day is the mother's milk of retail in Canada,\" he said.\nKehoe noted that retail sales prior to Christmas have been solid this year, with increased traffic in shopping centres across Calgary over the 2016 holiday season. He said that's not only a good barometer of how the local economy is doing \u2013 signalling that consumer confidence has rebounded \u2013 but a sign that this Dec. 26 could be busier than years past.\nFor those looking to avoid spending their time in long lines or in search of parking, they can go online and see for themselves how full Chinook's parkades are at any given time during the day.\nKabani said Best Buy's Boxing Day sales are also available online, beginning Christmas Day and ending Dec. 28.\nKehoe's advice if you're planning on braving the stores?\n\"I would say go early and go hard,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very clean and spacious RV including bunkhouse, bluetooth capable indoor\/outdoor radio, LED lights that will change color with the push of button, private master bedroom with upgraded mattress pad for added comfort, complete outdoor kitchen equipped with fridge, sink & grill.\nMy wife & I decided to buy this RV to travel as far and as many times as our kids school and work allows. This is our 3rd RV and this one is by far our favorite. It has been to Newport Beach California and back through the Ozarks & home is KC with many more trips to go. Hopefully blessing many more families as it has ours with quality time outdoors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This morning at the special event for the Sailor Moon 20th anniversary it was announced that there would be a new Sailor Moon anime coming next summer! Kotono Mitsuishi (voice of Sailor Moon) and Toru Furuya (voice of Tuxedo Mask) will be reprising their roles. The theme song \"Z Otome Shinsou\" (Z Girls War) will be sung by the band Momoiro Clover Z and was performed at this morning's event.\nWe'll be posting more details to come as they become available.\nThis entry was posted in Pretty Guardian Sailor Moon Crystal (2014), Voice Actors and tagged 20th Anniversary, Fumio Osano, Kotono Mitsuishi, Momoiro Clover Z, Toru Furuya by Adam. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The event provided a platform for the representatives of three European digital companies, including Skander Garroum, Co-Founder of Airy, a German\/American company providing messaging services for businesses and consumers; Florian Geuppert, CEO of Gutefrage, a German question and answer website; and Sabrina Bouguessa, General Counsel of Teads, the no.1 video advertising marketplace in the world, to share their concerns regarding the proposed legislation.\nOur community has, on several occasions since the very beginning of the discussions around the ePrivacy Regulation, spoken up against its potential serious disruption on small businesses and software developers' business models and market choices.\nSoftware developers have shown their commitment to the highest protection of their users' data and privacy, and their support for any effort aimed at streamlining existing privacy rules and making them fit the digital age. However, we believe that the ePrivacy Regulation will not deliver on either those objectives and its obligations have been worsened by the passage in the European Parliament.\nIn particular, the interplay between the proposed text and other legal instruments regulating data management and data protection, such as the General Data Protection Regulation (GDPR), are still unclear. Additionally, the proposal lacks the legal flexibility needed to allow software to perform correctly, to guarantee a meaningful and positive user experience, and to safeguard innovation.\nThe Developers Alliance's Tech Policy Dialogue stimulated an interesting discussion. The panel debate on \"The ePrivacy challenge: striking the right balance between trust, privacy and innovation,\" in particular, allowed the representatives from the digital SMEs and the EU Institutions to discuss how fair ePrivacy rules can enhance trust in the digital space and boost innovation.\nThe interaction was very positive. In conclusion, the speakers agreed that ensuring confidentiality of users' communications and the protection of their data must be guaranteed but, at the same time, it is important to do so in a way that allows future consumers to enjoy the next wave of exciting and innovative digital products and media to emerge in Europe.\nThe Developers Alliance and its members took the opportunity to call on the co-legislators to be careful in the next steps of the legislative process and to take into account the practical implication of the current text for both for businesses and consumers.\nWe look forward to continuing this critical discussion in order to find a solution that supports digital innovation in Europe and creates a workable environment, based on consumers' trust in the communications services they use.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Halloween is in two days! While in America along with other countries, we celebrate by wearing costumes and passing out candy to children, the holiday did not originally start this way. Here is the origin of Halloween and a look at how other countries celebrate their passed loved ones.\nWell known saints painted by Neilson Carlin.\nThe word Halloween originates from the phrase \"Hallows' Eve\" meaning the night before All Saints' Day. The holiday was founded by Pope Gregory III to honor the men and women who had high degrees of holiness or faithfulness. Catholics today still celebrate this holiday by holding a vigil the night before and going to mass on November first. Somewhere between the 12th and 15th century, the practice of dressing up as a favorite saint became popular. This is one of the many reasons we now use costumes during Halloween. Today Halloween has changed from being a religious holiday to honor the dead into a festive night to wear costumes, scare each other, and eat candy.\nCemeteries are decorated with flowers and candles.\nFamilies keep watch over the graves all night long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjkmn b/data_all_eng_slimpj/shuffled/split2/finalzzzbjkmn
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+{"text":"Blackjack is one of the oldest card games. Similar games were very popular in France during the 18th century. One of the most popular versions was called simply 21. Blackjack migrated to America with the French colonists and traveled across the country on riverboats. Actually it was only in the 20th century Vegas that blackjack received its name. When the state of Nevada has legalized gambling, Vegas has turned into the gambling capital of the world and casinos have began importing more and more new games into their gambling halls.\nTo draw more attention to the game a special bonus was added to the game. Any hand that has the ace of spades and any of the black jacks in the deck (either spades or clubs) would pay 10 to 1 on the players bet. Somehow the name blackjack has stuck to the game long after casinos have stopped giving this special bonus. Today, 70 years after the introduction of blackjack to the world of casino gambling, blackjack is one of the most popular games in online and land based casinos.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bankruptcy criminal personal injury: appeals attorneys: accident lawyers: arbitration services browse all cities in dc washington, dc general practice attorneys firm draughn james w. Knickerbocker blog is a media blog maintained by the business council to track new york state media coverage of state-level public-policy issues. ; (. United states bankruptcy courts southern district of texas wanda clark michael dred zenner.\nNews they were bloody heroes russell, juli nside magic magic news they were bloody heroes boltman, robin inside magic magic. Tom zenner, john kaizer and steve tsefos all call in to or is it hinch); manny s estrogen use; the coyotes bankruptcy at: 30pm (when it s sure to be a cool degrees); wanda.\nThe idaho blue book cost: $ per copy orders may be directed to: secretary of state attn: idaho blue book po box boise, id - books may be purchased at: secretary. \"that is very minimal in a -year span of time,\" said wanda mims, director of the hampton medical center but that number doesn t include patients, such as washabaugh, who got sick.\nFull text of \"the musical guide; containing a pronouncing and defining dictionary of terms, instruments, etc, including a key to the pronounciation of sixteen languages; many. V zenner david a jury docket trtanj albert w dds v judie wanda j (bankruptcy review). Goody two shoes, mcloughlin brothers, x domus, v - -1979, cesare maria casati, x st martin s. Exclusive interview: one tree hill creator mark schwahn, part one tree hill is about two young men brought together at first by their love for basketball, then bound by the.\nBankruptcy; debt warning signs; faqs proposal to creditors faqs wanda zenner law offices of ahlstrom wright oliver & cooper. The motive was obvious: nearly us p es had filed for chapter bankruptcy protection since, and more would surely follow unless something were done.\nWalter zembriski today, walter zemke, walter zenner, walter, waltham banjo clock mount vernon, waltham bankruptcy wanda mckinney; wanderer by sharon creech; want to buy royal. Full text of \"illustrated buffalo: the queen city of the lakes\"..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FXC Intelligence's research team keeps track of trends in the international payments space to help their clients stay on track in the industry and give them an edge over competitors. Our Market Observations for 2017 Report summarizes some of the biggest developments we have seen since we released our 2016 Money Transfer Report last summer.\nWhile 2015 was largely focused on startup growth and venture capital in the space, 2016 and 2017 have seen a slowdown in investments in international payments. Instead, we are now seeing the result of investing in the technology that helps assist the financial industry.\nIn 2016, tech disruptors such as Transferwise and Azimo were considered to be a major threat to more established brands such as Western Union and Moneygram. 2017 has shown us that the more established money transfer brands intend to hold their place in the market, with Western Union in particular devoting much time, energy and money into developing competitive mobile technology that is quickly becoming a benchmark in the markets. Meanwhile, the disruptors are primarily focused on finding a way to make their business models profitable \u2013 which Transferwise recently announced it has been able to do.\nAnother major market trend we are observing involves Chinese giant, Ant Financial and well known American money transfer company, Moneygram. A bidding war ensued between Ant Financial and Euronet for the Dallas-based Moneygram, with Ant Financial eventually winning over Moneygram. Ant Financial is an affiliate of the Chinese Alibaba group and owner of the Alipay platform. The acquisition of Moneygram will help Ant Financial expand their global presence and will likely bring Alipay to the states.\nAnt Financial has been rapidly expanding its global presence, from investing with Kakao Pay in South Korea to One97 in India. Ant Financial's prioritization of global expansion is a major market trend for 2017, and one we expect to continue for quite some time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Article marketing is a very effective promotional method but is also usually time consuming, exhausting, and brain racking. Its ultimate goal is to influence the mind of your readers, causing them to want to know more about you and your business website. When marketing to a specific niche or target audience, you have to write articles that appeal to the wants and needs of the people in your particular market.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Standard Ducting for Ventilation at Intech | 8 Inch by 3Ft.\nFor replacing existing ducting or to complete entirely new set ups in work spaces requiring ventilation, Air Systems International has a complete line up of standard ducting for ventilation. Size available and custom orders will ensure that you are operating safely and efficiently on any job requiring ventilation and ducting. Old, existing ducting can experience wear and tear over time, and replacement may become necessary. In these instances, choosing standard ducting by Air Systems provides an array of choices and added durability to extend the use life.\nThis 3 foot, 8 inch in diameter standard ducting SVH-3 is made of a vinyl polyester combination and standard external wear strips are available in custom sizes to best fit your job site demands. The internal wire hoop is a reinforced steel, with a 5 inch pitch. Each standard ducting piece by Air Systems International also comes with a 6 inch attachment on both ends that is 8 and 3\/8 inches around, and features belt straps that are adjustable. Rated for max temperature of 180 degrees Fahrenheit, covering a wide variety of applications to provide more return on your investment.\nThe SVH-3 is ideal for short run applications, and Air Systems also provides 6 foot, 15 foot and 25 foot standard ducting for ventilation as well as a full selection of conductive ducting. Connectors, transition pieces and other accessories round out their line up for ducting and parts, and as a manufacturer, custom orders can be accommodated as well! Shop the entire product line, including breathing air equipment and other ventilation related supplies, right here at Intech Equipment online. Our experts are available to help guide through selection and purchasing, or simply place your order online day or night. For any custom requests or to discuss additional questions, give us a call prior to placing your order.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkaus b/data_all_eng_slimpj/shuffled/split2/finalzzzbkaus
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+{"text":"2 ft x 3 ft floor mat.\nLong-lasting, durable surface, wipes off easily and does not show dirt. Spill proof.\nMade of bamboo slats, can be rolled up when not in use.\nA creative bamboo mat. Very durable. Rolls up for storage. Spills just wipe off. Perfect for areas that get a lot of traffic.\nGreat rug or mat for under kitchen sink, bathroom or doormat.\nWaverly TAN01 BLACK 5' x 7' 5' x 7' 100% Polypropylene Machine Woven Rug, Medium, Black Amazon.com Price: $114.88 $114.87 (as of 26\/02\/2019 01:00 PST- Details)\t& FREE Shipping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: On November 8, 2017, the U.S. Senate confirmed, on a party-line vote, President Trump's nominee Peter B. Robb to be General Counsel of the National Labor Relations Board for a term of four years. The NLRB currently has a full five-member Board, with 3 Republicans and 2 Democrats.\nFair Economy Impact: The NLRB protects the rights of most private-sector employees to join together, with or without a union, to improve their wages and working conditions. Employees covered by the National Labor Relations Act are guaranteed the right to form, join, decertify, or assist a labor organization, and to bargain collectively through representatives of their own choosing, or to refrain from such activities. The General Counsel is independent from the Board and is responsible for the investigation and prosecution of unfair labor practice cases and for the general supervision of the NLRB field offices in the processing of cases. The General Counsel has the authority to issue charges against employers and unions for labor law violations, and selects the cases that the board will ultimately rule on. Robb has spent much of his career as a management-side labor and employment lawyer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During the last few years the amount of terrorist attacks against media representatives or telecommunication facilities have increased.\nAccording to the Global Terrorism Database (GTD) maintained by National Consortium for the Study of Terrorism and Responses to Terrorism (START), there have been total of 1407 terrorist events around the world where media or telecommunications have been targeted since 2000. Over 400 have been killed and over 500 wounded in the attacks.\nThe amount of incidents has risen during the last few years up to over two hundred incidents per year. The rise is partly due to the methods of data gathering improving, but also due to acceleration of conflicts in different parts of the world. Terrorism researcher Leena Malkki (D.Soc.Sc.) of the University of Helsinki also points out that incidents earlier classified as part of civil war are now more often classified as terrorist incidents.\nOne third of all the terrorist incidents against media have taken place in South and Central Asia, mainly in Pakistan, India and Afghanistan. One fourth of the incidents has occurred in Middle East, mainly in Iraq, Libya, Syria and Yemen. After these regions, most hazardous countries have been Philippines, Somalia, Nigeria and Thailand. Only few attacks occur in Europe, even though they tend to get the highest media coverage in the western countries.\nIn India, Philippines, Thailand and Nigeria the majority of incidents are against facilities and infrastructure, and the number of casualties is low. Individual journalists are rarely targeted. In Afghanistan, in Pakistan, and especially in Iraq and Syria the number of individual journalists being attacked is high. In Somalia, the majority of incidents are assassinations and kidnappings conducted by Al-Shabaab, a jihadist terrorist group.\nThe conflict in Syria began in spring 2011, and only within few years Syria has become this millennium's second highest ranking country in journalist abduction.\nKidnappings usually end within few days, but in Syria the journalists taken as hostage have been kept captive for prolonged periods of time. Few times the captives have been released within or almost within a week, but on average the time of captivity has been 114 days during the years 2012\u20142014. The longest periods of captivity were 398 and 394 days. Also many of the abducted journalists have been killed verifiably, or their fate is unknown.\nIn majority of cases the Islamic State (IS, former ISIS\/ISIL), a Sunni Islamist militant group, is behind the abductions.\nOne of the most shocking incidents was when an American journalist, James Foley, was kidnapped alongside with a British journalist, John Cantlie, in November 2012. The fate of John Cantlie remains unknown, but Foley was held captive for almost two years and beheaded on August 19, 2014. His execution was depicted publicly in a video released by IS. Another American journalist, Steven Sotloff, who was kidnapped in August 2013 near Aleppo city, appeared also in the video, and was reportedly beheaded within couple of weeks on September 2, 2014. The executions were conducted in retaliation for the airstrikes carried out by the United States in Iraq.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Laura Blake has joined Somerset Community Foundation in a brand new post as Development Director, with the aim of spearheading the growth of local giving in Somerset.\nLaura's role sees her working closely with families, businesses and organisations, helping them make inspiring and impactful gifts to causes they are passionate about in the county. Laura brings a wealth of experience in Development; she previously worked as Head of Philanthropy at the national education charity, Teach First.\nLaura says: \"I'm really excited to be working at Somerset Community Foundation. We've recently launched The Somerset Fund, which offers a simpler and more effective way for local businesses to give to charitable causes in Somerset.\n\"Donating through The Somerset Fund helps to free up precious time by giving businesses the choice of either supporting good causes across the county or focusing their giving in one of the five Somerset districts. We'll find great causes that need funding, assess applications and report back on the difference your donations have made. You'll also have the opportunity to come together with other supporters and hear from projects you've supported. And, thanks to match funding, donations to the fund could be increased by 80% or more.\nLaura started her career working for the University of Bristol's Development and Alumni Relations team, before moving to London to join Barnardo's and The Prince's Trust. Laura is also a trustee of Pathways to Independence, a Kent-based charity that supports homeless people and ex-offenders.\nLaura has also been visiting some of the groups and organisations SCF have recently funded, during her first few weeks at the Foundation, and has written a blog based on her findings - you can read that here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkeqm
@@ -0,0 +1,5 @@
+{"text":"Our client, a major online retailer, needs a Creative Traffic Manager to join their team onsite 40 hours\/week in San Bruno. Company shuttle available.\n- Continually monitor requests and timelines to ensure they are delivered in a timely manner.\n- Engage with business partners to ensure crisp communication between requests and designers.\n- Ability to maintain positive and productive relationships with all clients, team members, peers, managers, etc.\n- Ability to organize information, prioritize and maintain attention to detail, and accurately follow procedures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Great Energy Alliance Corporation (GEAC) owns Loy Yang Power, which operates the Loy Yang A power station in Victoria.\nThe two largest shareholders in GEAC are AGL and Tokyo Electric Power Company).\nThe Loy Yang A power station is located near Traralgon and in 2009 it was estimated to have emitted 18.81 million tonnes of carbon dioxide equivalent (Mt CO2e).\n\u2191 Loy Yang Power, \"Ownership\", Loy Yang Power website, accessed August 2010.\nThis page was last edited on 5 February 2013, at 09:29.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Neil Ewins' study of the Staffordshire potteries in a period of great global change traces how ceramics production has been affected by globalisation in both familiar and unexpected ways.\nAlthough many manufacturers such as Wedgwood initially moved production to cheaper labour markets in East Asia, others remained in or returned to England once it became clear that outsourcing manufacturing was affecting the brand value and customer perception of their products. Neil Ewins explores the complex behaviour of the UK ceramics industry, using a combination of evidence from the press, trade journals, ceramic objects, and primary interview evidence of manufacturers, retailers and a ceramic designer.\nEwins suggests that, although the surface designs of UK ceramics invariably reflect diverse cultural and stylistic influences, a notion of authenticity often still resides in the place and context in which the ceramic product was originally made. Overall, the book argues that UK ceramics remain culturally complex because of issues of supply and demand, and ties to heritage, imagined or otherwise. Within a context of globalization, the book highlights compelling issues which have huge ramifications on UK manufacturing futures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kloudrac is founded by a team of dynamic, seasoned and qualified professionals having vast IT experience across various business areas, Software \/ Product development, design, testing, maintenance, resourcing \/ staffing options.\nKloudrac has developed strategic alliances with some of the most innovative and reputed companies in the Technology industry. Through these partnerships, Kloudrac is able to leverage the latest technological advancements and helps customers develop unique solutions.\nSalesforce Sales Cloud, Marketing Cloud, Service Cloud, Pardot, Chatter, Communities, or Salesforce AppExchange Apps, we are experts in everything 'Salesforce'.\nBeing a Salesforce development company, we help organizations in planning, designing, and implementing complete Salesforce based business solutions.\nWe transform the way businesses interact and manage their customers, products, partners, and even employees.\nWe create experiences that are simple to use, eye-pleasing, and drive results for your business.\nWe create strategic web design solutions that communicate your mission and vision and express your unique brand identity.\nWe determine the scope of the project and create a budget and timeline, and stick to them , along with weekly status checks on progress.\nWorking with a team of experienced professionals, we keep our teams small, agile and focused on your project.\nWe provide solutions for all GENRE of business, from small ecommerce websites to huge multinational corporations along with healthcare providers, realtors, lawyers, local governments and other!\nWe believe in adding real value through our Digital Marketing Services, we can help you to grow your business with focus on ROI more than just traffic.\nKloudrac not just focusing on traffic and keywords but also on business objectives. In this evolving digital world, when people have all information in their palm, it's more important to connect your brand with targeted prospects using digital mediums.\nWe are a full-service digital marketing agency to help your business grow online, Attract visitors, impress with product and services, and convert more prospects into leads. Market know us with our innovative approach and performance driven digital marketing company who deliver exceptional results.\nKloudrac is an IT service provider and trusted global custom software development company in the area of Salesforce CRM solutions, mobility solutions, content management solutions and internet marketing. Today,we have a 70+ team of experts who define and redefine process to offer best in class solutions & services to our customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. The Special Needs Trust Approach.\nThe special needs trust is often a good approach when a person has a disability. However, it is not appropriate in all circumstances. If a person with disabilities does not have a guardian and is able to manage his or her own finances, keeping the money and property in his or her own name is certainly the most straightforward approach. This approach also gives the person with disabilities the most independence and control over his or her own finances. This would not be a good approach, however, if a person has difficulty handling money. Also, whether or not a person has difficulty with money, having property in one's own name can cause disqualification from vital public benefits including Social Security Supplemental Security Income (SSI) and Medicaid. Even if a person does not receive Medicaid at the present time, it may be vital later as a safety net. Whether the money or property pertaining to the person with disabilities is an inheritance, a personal injuries settlement, a social security lump sum payment or funds raised in a community for a person with disabilities, disqualification can occur when funds are held in the name of the person with disabilities receiving needs-based public benefits.\nA special need trust is not an ideal solution in all circumstances, and it must be structured to follow complex rules, but for many people it is the best alternative. Other options to consider include NH MEAD accounts and ABLE Act accounts. These types of accounts can be used in addition to special needs trusts. The first step with special needs trusts is to determine the type of special needs trust to use.\n2. Types of Special Needs Trusts.\nThere are several different types of special needs trusts to choose from, and there are various names such as \"supplemental needs trust,\" \"discretionary trust\" and \"spendthrift trust\" which are sometimes used in addition to the name \"special needs trust\". Regardless of the label, look at how the trust is designed to function.\nAll trusts are \"management trusts\" in the sense that someone is managing the trust. Special needs trusts are also typically \"discretionary trusts\" to give the trustee discretion to decide about investments and distributions, allow use assets (such as real estate, motor vehicles and furniture), have an anti-alienation (\"spendthrift\") clause to protect the trust from claims, and state that the trustee may not duplicate public benefits (\"supplemental needs trust\"). Also, special needs trusts often provide examples of what the trust could purchase for the person with disabilities. Most trusts for people with disabilities have these features in common. The fundamental differences among the different types of trusts for people with disabilities arise from (1) the question of where the assets come from, (2) the question of whether the trust is created under state law or under federal statute and (3) the question of who the trustee is. These distinctions are very important in terms of whether the trust is a \"third-party\" trust or a \"self-settled\" trust, in terms of whether the trust (if it is a federal statutory trust) is a \"subsection (A) trust\" or a \"subsection (C) trust\" under 42 USC 1396(d)(4), and also in terms of whether the State may make a claim when the person with disabilities dies to be reimbursed for Medicaid benefits provided to the person with disabilities prior to death.\na. Trusts Established with Assets of Parents or Other Party (called Third Party Trusts).\nDiscretionary supplemental care trusts providing for the needs of beneficiaries not supplied by way & of public benefit programs, created by non-beneficiary settlors, appear to be legal, appropriate, and encouraged by both state common law and statutes. These trusts are sensitive responses by caring family members or others to make available to the chronically ill and disabled community the extra needs, beyond basic support, that such beneficiaries reasonably require or from which greater comfort can be achieved. A number of cases have been construed where the discretionary-extras trust has been clear as to the settlor's intent. Where supplemental trusts have been considered by courts and no support duty to the beneficiaries is owed by the settlors, and where the funds used are in fact not the funds of the beneficiary, there is no suggestion that such trusts violate public policy. Conversely, such trusts appear to be encouraged. To hold otherwise would eliminate a necessary source for the satisfaction of the uncompensated realistic and reasonable needs of the chronically ill and otherwise institutionalized disabled.\nb. Trusts Established with Disabled Individual's Assets, Including Personal Injury Awards (called Self-Settled Trusts).\nSelf-settled trusts are established with assets already owned by the person with disabilities. One example would be a personal injuries settlement. Another example would be a lump sum social security check paid to correct for prior lack of periodic payments to a person with disabilities who should have received them. This often occurs when a person is first found qualified to receive social security monthly checks and the finding relates back to an earlier onset of the disability. A self-settled trust could also be the appropriate type of trust if a person with disabilities has already received a gift or inheritance outright and then establishes a special needs trust.\nAlso, at times, what would otherwise be a state law third-party trust is converted or instead established as a self-settled trust in order to come under the \"safe harbor\" of 42 USC 1396p(d)(4) and 42 USC 1382b(e)(5). Sometimes people prefer the safe harbor protection of the federal statute compared to state law. So a federal statutory trust might be chosen by parents for an inheritance even though they could instead have chosen a common law third-party trust.\nSelf-settled trusts are typically established under the federal statutes 42 USC 1396p(d)(4) and 42 USC 1382b(e)(5). In 1993 Congress enacted 42 USC 1396p(d)(4) providing a uniform protection for special needs trusts in the context of Medicaid. In 1999, Congress enacted 42 USC1382b(e)(5) to apply the same uniform protection in the context of Supplemental Security Income (SSI).\nThe language used in both statutes calls for three separate types of statutory special needs trusts all of which are not counted against Medicaid or SSI eligibility: (A) the federal pay-back special needs trust established under 42 USC 1396p(d)(4)(A); (B) the Miller income-only special needs trust (not applicable in most instances); and (C) the not-for-profit pooled special needs trust established under 42 USC 1396p(d)(4)(C).\nThe federal pay-back special needs trust established under 42 USC 1396p(d)(4)(A) must be established as follows: A trust containing the assets of an individual under age 65 who is disabled (as defined in 42 USC Section 1382c(a)(3)(A)) and which is established for the benefit of such individual by a person with disabilities or a parent, grandparent, legal guardian of the individual, or a Court, if the State will receive all amounts remaining in the trust upon the death of such individual up to an amount equal to the total medical assistance; paid on behalf of the individual, subject to hardship considerations under 42 USC 1396p(d)(5), and then the balance may be inherited by persons selected by the trustor or as determined by the laws of intestacy.\nThe not-for-profit pooled trust established under 42 USC 1396(d)(4)(C) authorizes and requires a not-for-profit organization to be trust manager. The not-for-profit pooled trust may be established and funded for an individual with disabilities at any age (unlike the 42 USC 1396(d)(4)(A) federal pay-back trust which requires both establishment and funding prior to age 65.) Also (like the 42 USC 1396(d)(4)(A) federal pay-back trust), the not-for-profit pooled trust may be established by an individual with disabilities or by a parent, grandparent, legal guardian or Court.\nTo the extent that funds which belonged to the beneficiary before going into the trust are not retained by the nonprofit trustee to benefit other people with disabilities, then the State has the opportunity to make a claim for reimbursement for Medicaid funds provided, subject to hardship considerations under 42 USC 1396p(d)(5), and then the balance may be inherited by persons selected by the grantor or as determined by the laws of intestacy.\nThere are a number of non-profit organizations established around the country to provide special needs trust management including New Hampshire's Enhanced Life Options Group whose website is www.elonh.org.\nFamilies across the country have selected not-for-profit organizations as special needs trust managers basically because: (1) a trust manager with knowledge of disabilities issues can add value beyond financial arrangements; (2) a non-profit trust manager can handle the trust in collaboration with or under the supervision of a family member, professional advisor or bank trustee; (3) a non-profit trust management organization is established to continue indefinitely whereas relatives and individual advisors are mortal; (4) a not-for-profit pooled trust can accept funds after the beneficiary is 65 years of age; and (5) not-for-profit organizations are specifically referenced in the federal law governing Medicaid and social security as permitted special needs trust managers.\nSince the beneficiary of a special needs trust may not serve as trustee, one or more family members may come to mind to serve as trustee. Often, however, no family member comes to mind who is available, or the family member being considered is a generation or two older than the beneficiary, and may not be living when the beneficiary may most need the trust. Sometimes a family member wants to be involved but does not have the time to actually manage the trust.\nThe directors of these not-for-profit special needs trust organizations include people who have a person with disabilities in their family and people who are active in the disabilities community.\nThey are typically 501(c)(3) public charities, and are bonded, audited, charge non-profit fees and receive grants and gifts. No minimum or maximum asset balance is typically required to be eligible for services, and services are available regardless of the specific type of disability. These non-profit groups have their own websites, such as, for example, New Hampshire's Enhanced Life Options Group's website at www.elonh.org.\n3. Legal References Concerning Special Needs Trusts.\nThere are a number of statutes, regulations and other reference materials relevant to special needs trusts to consider in addition to your own State's statutes. Currently relevant federal statutes are found on the US Government Printing Office at http:\/\/www.gpoaccess.gov\/.\nThe federal Medicaid statutes to start with are 42 USC 1396 et seq including 1396a(10)(C)(i)(III). The federal SSI statute to start with is 42 USC 1382b(e)(5).\nThe federal Medicaid regulations at 42 CFR 435 are available at http:\/\/www.gpo.gov\/fdsys and federal SSI regulations at 20 CFR 416 are now available at http:\/\/www.gpo.gov\/fdsys.\nStart with 20 CFR 416.1102, CFR 416.1121(h) and 20 CFR 416.1103. To access these regulations go to http:\/\/www.gpo.gov\/fdsys homepage, then go to Search Government Publications, and then type in the regulation in the search box.\nThe federal SSI regulations pertaining to SSI are also currently found at http:\/\/www.ssa.gov\/OP_Home (then go to Code of Federal Regulations (CFR), then to number 416).\nIn the context of Medicaid, extensive interpretive guidance is also provided by the US Centers for Medicare and Medicaid Services in the US CMS State Medical Manual (also referred to as Section 3259 of Transmittal No. 64, November 1994). Section 3259 may be found at Appendix C of Clifton B. Kruse, Jr. Third-Party and Self Created Trusts American Bar Association (3rd Ed. 2002).\nThe Social Security Administration's Program Operations Manual System (POMS) is currently available online at http:\/\/www.ssa.gov\/OP_Home (then go to The Program Operations Manual System (POMS), then go to the Table of Contents, then go to SI-Supplemental Security Income, then go to SI 008 and SI 011). The POMS have extensive interpretive guidance on special needs trusts.\nDistributions of the income or principal, or both, of a special needs trust to or for the benefit of the disabled beneficiary shall be disregarded for income eligibility purposes of all categories of public medical assistance to the same extent that such distributions are disregarded for purposes of Title XVI of the Social Security Act. Funds expended by a trustee of a special needs trust to purchase or maintain assets owned by the trustee in his or her capacity as trustee of the trust shall also be disregarded for such income eligibility purposes. For the purposes of this paragraph 'special needs trust' means any trust established by a third party for the sole benefit of an individual who is considered disabled under the provisions of section 1614(a)(3) of the Social Security Act, and any trust funded with the resources of such an individual and complying with the provisions of section 1917(d)(4) of such Act.\nThis state law confirms that New Hampshire decisionmaking about what individuals and special needs trusts can buy, when people are receiving medical assistance under the Medicaid program, must be in line with the federal SSI distribution standards and must be no more restrictive than the methodology used in the SSI program in accordance with 42 USC 1396a(a)(10)(C)(i)(III).\n4. Special Needs Trust Public Policy Considerations.\nPeople with severe disabilities, such as a developmental disability, physical disability, mental illness or brain injury, often must rely on federal and state benefit programs in order to meet their minimal health care and physical support needs. Annual health care costs for these individuals are substantial. The public benefits serve as an economic safety-net, usually including a combination of cash payments (such as SSI) and certain health related benefits (including Medicaid). Medicaid covers many medically related expenses, such as prescription drugs, and is required in order to assist individuals to access other necessary programs, such as personal care assistance or direct care support in order to live.\nEligibility for these benefits is based on two factors: 1) definition of disability; and 2) financial need. Essentially, individuals must be poor and stay poor in order to be eligible. Gifts and inheritances from a parent or sibling or life insurance benefits from a deceased parent or sibling can terminate these benefits, ultimately resulting in an individual's loss of health care and disability-related support needs. People with disabilities must maintain Medicaid as their economic safety-net, but doing so limits their ability to accumulate resources, leading to life-long impoverishment. Personal and family funds are almost never able to replace public benefits. Public benefits do not claim to provide all that is necessary to provide what a person in ordinary life assumes to be the minimum. People are handicapped by a lack of resources available to them. From a public policy standpoint, there needs to be a way that families can help families and people can help other people in partnership with public programs.\nThe special needs trust is a mechanism which allows individuals and families to shelter resources for the purpose of enhancing the individual's quality of life and help with gaps in the public benefits system. Trusts established by family members accumulate resources for items and services not funded through state and federal programs for a specific beneficiary. Special needs trusts allow for a partnership between family funds and public funds.\nSpecial needs trusts fulfill a number of specific functions for people with disabilities. Special needs trusts hold assets which are specifically allocated for the beneficiary but managed by an independent trustee. These trusts are in-between (1) the concept of outright ownership of assets, on the one hand, and (2) the concept of eligibility for public benefits on the other hand. As a result, special needs trusts can be used to address public policy goals. Special needs trusts are fundamentally a way to prevent a stark and arbitrary line between eligibility and ineligibility for benefits.\nSpecial needs trusts are established for people with disabilities for a number of reasons: (1) to hold lump sum social security checks (paid where warranted to correct prior lack of delivery of periodic payments to a person with disabilities who should have received them) so that the lump sum can be saved for future use and not required to be spent down quickly to prevent loss of disability benefits based on financial eligibility criteria; (2) to have a permitted place to put funds raised in a community when someone has a tragic, debilitating injury, and neighbors, service clubs, community leaders, extended family members, and friends want to raise money to make a difference in that person's life; (3) to receive personal injuries settlements so that a person with disabilities can be compensated for a permanent injury caused by another; and (4) to permit an inheritance to continue to do some of what the parents of a disabled child did for the disabled child before the parents died.\nWithout special needs trusts, lump sums from social security could not be saved for the future; nor could community fund-raising work effectively to benefit people with disabilities. It is difficult to raise funds if donors feel that their contributions will make no difference other than disqualification from public benefits.\nIf people with disabilities who are receiving public benefits were not able to derive any effective benefit from recoveries from tort feasors who have brought about the disabilities by willfully or negligently injuring them, then tort feasors would be less deterred from causing others harm. People with disabilities in many instances would be cut off from seeking or obtaining redress if they could not actually benefit from the redress. When there is no effective Court access for perceived serious injury caused by another, then \"self-help\" leads to violence and the rule of law is undermined.\nWith regard to inheritances, special needs trusts give parents a realistic option other than disinheritance of a child with disabilities. There is no minimum or maximum amount required to establish a special needs trust. A small life insurance policy or savings account diligently built up can do a lot after a parent's death. Most people have a natural human desire to leave an inheritance for their children. In many people this nurturing instinct is demonstrably even stronger when the child has a disability. However, family funds built up over many years prior to the parents' deaths, could not benefit a person with disabilities if disqualification from public benefits would result from the inheritance and the inheritance had to be spent down until exhausted to replace the public benefits. Family money goes elsewhere in the event there cannot be any benefit for the child with disabilities. Money left to siblings in the hope that they will do what the parents have done for a family member with disabilities, can fail to achieve the intended result, due to the sibling's death, disability, divorce, debts or disinterest. For many families who are unaware of special needs trusts, death carries a poignant penalty following a lifetime of family sacrifice. Without the special needs trust, the parents' death causes an economic free-fall for a child with disabilities just at the time when the parents are no longer there to protect their child.\nIt must be remembered that involved families, whatever their standard of living, do spend considerable family resources to benefit a family member with disabilities. It all adds up over the course of a lifetime until the parents' death. The special needs trust continues this effort after the parents have died in partnership with public benefits. Disability does not have to mean impoverishment. Family money can help. The Medicaid safety net helps too but leaves out a lot. What Medicaid does cover sometimes is not enough to get the job done. For example: a basic Medicaid wheelchair - compared to a specialized one adapted for individual needs; occasional Medicaid funded transportation, when available, to doctors' appointments - compared to van ownership; tooth extraction paid for by Medicaid - compared to more comprehensive dental care.\nWhen special needs trust funds are used to pay for items such as vans and wheelchairs, there is a reduction in public benefit expenditures as a result. Special needs trusts replace public benefits to an important extent (e.g.: a car or van purchase which reduces the need for Medicaid funded transportation; an adapted wheelchair which reduces the number of Medicaid funded wheelchairs). If special needs trusts were not a part of the picture, family money would by and large go elsewhere and public expenditures would increase. Special needs trusts provide a needed service for people with disabilities. Some examples show how.\nGeorge was born with cerebral palsy. His parents set up a special needs trust for him. George receives Medicaid. He is completely dependent on his family for many things. His parents are his care givers and do not work the same shift so one of them can stay at home with him. His trust has purchased special equipment to aid his motor skills. The trust has also purchased an adapted van so that his family can take him to activities and visits with friends and their extended family.\nBill, who has paraplegia, is establishing a special needs trust from a personal injury settlement. Bill receives Medicaid to survive but he basically has stayed in his small apartment and has gone nowhere except for the doctor's office. His special needs trust will be able to, purchase for him a used modified van and also pay for van insurance, registration and repairs to keep him on the road, so that he will be able to leave his apartment to see people he has not seen, and do things he has not done, in years. This trust will change his life by going beyond what the public benefits can pay for. He will also be able with his trust to purchase pictures to decorate his apartment.\nSusan was born with Down syndrome. She relies on Medicaid. Her mother set up a special needs trust for her daughter's benefit before her mother's death. The trust has recently purchased a reclining chair for Susan's apartment. She has a very active day. When she comes home, she likes to sit in the chair and listen to music. She remembers that her mother did this for her. Her trust has also purchased other items for her which she enjoys very much. Susan's trust has also purchased a burial plot for her. Having a special needs trust has enhanced her life beyond what the public benefits have provided. This is her American Dream.\nThis article is designed to provide research information to be used in conjunction with other research deemed necessary in the exercise of independent professional judgement. John makes no warranty or guarantee concerning the applicability, accuracy or reliability of the content of this article. Individual circumstances need to be discussed to in detail. No recipient of information from this article should act or refrain from acting on the basis of any information or materials contained in this article without seeking legal advice on the law and on the particular facts and circumstances at issue from an attorney licensed in the recipient's state chosen by the recipient and in accordance with a written engagement letter. Also, the information contained in this article may not reflect the most current legal developments and should not be relied upon as an indication of any future results. The information and materials on this website have been prepared and made available for general purposes only and is subject to various differing interpretations. The transmission and receipt of information contained in this article, to clients or otherwise, do not constitute the rendering of legal advice or other professional advice or services on any matter. No endorsement or approval is made of any website Liability to any person for the consequences of anything done or omitted to be done by that person wholly or partly in reliance upon the use or contents of this article are hereby expressly disclaimed. THIS ARTICLE AND ITS CONTENTS (INCLUDING INFORMATION AND MATERIALS CONTAINED HEREIN) ARE PROVIDED \"AS IS\" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, OR NON-INFRINGEMENT. Without prior written consent, you may not link material in this article with a website or materials of any other entity, or associate or frame material on this site with a website or materials of any other entity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkphh b/data_all_eng_slimpj/shuffled/split2/finalzzzbkphh
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+{"text":"Recently we've done interviews with some of our favorite PPC bloggers, and some of the results are now live online.\nThe first is with Greg Meyers, who recently published Part-I of a discussion of Paid search: art or science?\nThe next was with Garry Przyklenk of PPC-Advice, where we discuss bid management, revenue attribution, and paid search reporting.\nThanks to Greg and Garry for their time and interest.\nIf you've got a PPC blog and would like to talk about PPC, or review ClickEquations, let us know!\nPS: If you like these, here's one from a few month ago with Manoj Jasra at WebAnalyticsWorld.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here I will explain how to get database size in SQL server or query to get database size in SQL server 2008.\nIn previous articles I explained show time difference in minute ago hour ago etc in SQL Server, get list of procedures where table name used in sql, difference b\/w view and stored procedure in sql, add row items as string with comma separated values in sql server and many articles relating to SQL server. Now I will explain how to get database size in SQL server.\nnice article.more informative to us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For all courses admission usually starts from second week of July (Tentatively 10th to 30st July). Admissions in all courses are made by the college\/university on the basis of merit\/entrance.\nEvery Student is required to get his identity card after his admission. He should paste his recent photograph and get attested by the Principal\/Chief Proctor. Identity card must be produced on demand. If identity card is lost duplicate card may be issued after paying the prescribed fee.\nAt least 75% attendance in the Class\/Section is compulsory for every student. If a student fails to secure it he\/she will be prevented from appearing in the examination. Classes begin at 10.00 am and close at 4.00 Pm. However, the Principal may fix other time\/extra classes for theory\/practical paper. Each period is of 40 minutes duration.\nThere is an annual\/semester examination system in the college. Every student must submit the University Examination form and submit the examination fee by the latest date fixed by the University. It shall be the duty of every student to collect the necessary information from the concerned clerk of the college. The examination fee will be paid by the student at the time of filling the form.\nThe student who appear for the first time in the examination conducted by the Rohilkhand University must get enroll themselves in the university by producing their Migration Certificate, fill the enrollment fee. The enrollment number will be issued by the University letter on. The result of Final Year is declared any after enrolment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We all know it now, smoking is not good for your health. Looking to purchase a life insurance policy? If you smoke that is going to cost you. A non-smoker in good health might be able to obtain a $1,000,000 term policy for as little as $400 \u2013 $500 a year depending on age. A smoker might qualify for a benefit that is a fraction of the non-smokers, $250,000 benefit. Take it from the picture below. Stark Associates public service announcement, do not smoke.\nIf you smoke you could end up like this but more importantly your premiums would be much more expensive.\nHow can we help you or your business? Call us today (732) 747-0411.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It seems that every time we think we can't get any more proud of you, you accomplish another goal or succeed with another project and we do. And we know that when life throws us across its inevitable rough patches, most times we can simply look back on your experiences and get our bearings, and step forward with confidence.\nWe know that you're not perfect, and you don't have to be. Not for us.\nThanks for being a motivator.\nThanks for being a pioneer with an unbending work ethic.\nThanks for being kind, and an inspiration.\nThanks for enriching our lives, and bringing us new friends and comrades.\nAnd may you have many, many more. \uac74\ubc30!\nOMG I have not been able to be on the computer for a while. So I missed this day. I want to still say HAPPY BELATED BIRTHDAY Rain and many many more!!!!! Also, the video was awesome!!!!!\nOui gr\u00e2ce a lui , j'ai aussi r\u00e9ussi beaucoup de chose !\nC'est vrai que gr\u00e2ce \u00e0 lui on a des ami(e)s partout dans le monde !\nHappy 33rd birthday to Rain. You don't look a day over 23!! Enjoy your day, you deserve it and spoil yourself and give to yourself today. Wish I could celebrate with you ( in person). This is a bittersweet day for me because I'm still mourning Michael Jackson's death and celebrating Rain's life. Love you Rain!!!\nWishing you a very Happy Birthday Sweetness! Just like \"fine wine\"\u2026.getting better and better with each passing year. Yes indeed. SMILES!\nHappy birthday baby! I will forever love you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkvuu b/data_all_eng_slimpj/shuffled/split2/finalzzzbkvuu
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@@ -0,0 +1,5 @@
+{"text":"Lyrics Of Janaab From Latest Album - Janaab (2015): Nice romantic song sung by Vinti Singh, Farhad Bhiwandiwala featuring Shivani Joshi & Ashrut Jain.\nFeatures: Shivani Joshi, Ashrut Jain.\nThe video of this song is available on youtube at the official channel Zee Music Company. This video is of 5 minutes 16 seconds duration.\nLyrics in Hindi (Unicode) of \"\u091c\u0928\u093e\u092c\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NORTH-east-based K9 Health Centre is delighted to partner with local pet re-homing charity DAWGS, to hold a special event for all dog owners at the Beach Ballroom on Sunday 24th March.\nThe event has been designed to help dog owners throughout the North-wast better care for their dogs. Sandra Wilson, owner of K9 Health Centre, explained: \"At K9 we are dedicated to dogs and are committed to helping them in any way possible.\nThe event will bring together a variety of specialists including vets and behaviourists, groomers, dog walkers, day care providers, food specialists and even the latest must have accessories! There will be numerous opportunities for owners to talk to experts on a variety of topics such as care, rehabilitation, behavioural and training issues, as well as to listen to presentations from specialists in their field.\nStarting at 11am, the event has plenty of opportunity for owners to take home goodies with over 30 stalls throughout the day. Not to be missed are the doggy fashion shows scheduled for 1pm and 3pm, guaranteed to be a favourite with owners and their dogs alike.\nWith something for all the family and all proceeds going to DAWGS this is one event that really needs to be a date in your diary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One car is set to create a bigger buzz than most at the Goodwood Festival of Speed, as Chevrolet announces that its Bumblebee themed Camaro marks the launch of the new model in the UK.\nBuilding on Chevrolet's iconic US Heritage, the retro-styled Camaro has been on sale in the USA since 2009, and has already seen sales in excess of 200,000 as American retailers cleared their waiting lists.\nAnd very soon, those waiting to get their hands on an official Camaro in the UK will have reason to celebrate \u2013 the car will go on sale as an official Chevrolet, complete with full manufacturer warranty, during the first half of 2011.\nWith looks that draw heavily from the styling of the legendary 1967-69 Camaro \u2013 immortalised in song in Bruce Springsteen's Racing in the Street and on the silver screen in both 2 Fast 2 Furious and Transformers \u2013 the car is a full four-seater, with an interior that also takes significant cues from the look of its forebears.\nThe Camaro will be available only in top level V8 trim \u2013 the 6.2-litre 426bhp powerplant is as much of a legend as the Camaro itself. It promises a top speed electronically limited to 155mph, and a 0-60mph time of less than five seconds, coupled to a dynamic chassis that, thankfully, owes little to the layout of more traditional muscle cars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE IRON ORCHARD is the story of Jim McNeely (Lane Garrison), a young man thrust into the vibrant and brutal West Texas oil fields in 1939. In a state gushing with oil and filled with ambition, McNeely settles into a small-town community that is slowly overcoming the trauma of the Great Depression. The formidable path before him is riddled with obstacles -- overbearing bosses who try to keep him down, powerful oilmen who are reluctant to invest in a fresh face, and women who see salvation in his charms -- but he finds a glimmer of hope with his first loves: new wife Lee Montgomery (Ali Cobrin) and drilling for oil. With everything falling into place to ascend to the top of the oil chain, McNeely stumbles upwards through his success and in turn jeopardizes his desperate longing to conquer this brave new world of influence and wealth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'Freeview & YouView' started by Ivah, Aug 6, 2013.\nThe last few days each time I switched my Panasonic TV on I got \"New channels found etc.\". I let it go through the automatic tuning process which it said it had completed, couldn't spot any differences but each switch on it went through it again. Last night did a manual update which started by saying channels would be lost. Since then (repeated this morning) can only get C46 stations. I am on Sedbergh transmitter (only one that has ever worked for me although LA2 8QZ search says Kendal, it is off a Kendal feed so that might be the reason) so I should also get C40 and C43. I can't work out if it is a TV fault, transmitter fault or aerial fault. It worries me it was wanting to update but obviously failing although all the channels were working at this stage. I've posted to UKFree without response but it says Sedbergh is OK although it looks like it had issues at the weekend. I posted to Panasonic with no response. Any ideas please?\nYou should find that there is a menu option to stop those nag messages. It may be something like \"New Channel Message\" in the \"Display\" menu under Setup.\nYou should only need to rescan when there are channel changes that you already know will be of use to you.\nTurn off the \"New Services Message\" option within the tuning menu.\nThanks - I had found that but I assumed if it was trying to do an update there must be a change so I didn't want to disable it.\nI said in my original post I had done a manual update, I had if fact done a manual trigger of an automatic update. Since then I have done a channel by channel manual update. Ch46 is fine with a good signal. Ch43, the HD channel, has now found all its stations but at low quality. Ch40 has found nothing but looking at the station list, eg ITV4, I have never received them through Freeview.\nI think I need to find out somehow if this signal should be getting to me at a reasonable strength and if so do I need to look at aerial\/booster upgrade. This information seems hard to come by with nobody responding from UKFree.\nThis is not necessarily correct.\nThese channel 'nuisance' messages are very common now since the digital change over.\nTransmitters are now more powerful and it is very common to receive duplicate channels from more than one transmitter.\nThis fools tuners into thinking there are new channels available when in reality they are just a duplicate set so as advised it is usually better to turn these information messages OFF as they are so often spurious.\nDo you know which transmitter you should be receiving from?\nThanks - if you pop back to the top you will see I am on the Sedbergh transmitter, that is a Kendal repeater. I doubt it is interference as I can barely receive even that one (it was appalling when it was analogue). According to Digital UK it transmits on channels 40, 43 and 46 at good quality. According to UKFree 46 covers the common SD stations, 43 the common HD stations and 40 some odds and ends including Film4 (no wonder these repeaters are referred to as Freeview Lite). But according to Freeview I can only receive the common SD and HD stations, which is what I have got back to but with 43 at a lower quality than 46, quality being what the diagnostics tell me not the picture quality - HD is HD. I am trying to find out if this is normal for the edge of the range or can I invest in my receiving kit to improve the diagnostic quality and possibly add the channel 40 stations.\nThe new channel message will probably have been because Film4 has just appeared on the PSB2 mux... ch40 ... which is the one you are having problems with.\nI note from UKFreeTV that it has moved to ch40 from ch50 (late last year).\nDo you have any video recorders, Sky boxes, etc.,. in the aerial circuit before the TV? If so check the RF modulator(s) are set well away from the Sedbergh frequencies.\nAlso disconnect any hdmi devices (remove both ends of the cables).\nSwitch off DECT phones, modems, routers etc.,. and see if it helps with reception at all.\nMy postcode is in my first post and there is a bit of a clue in my username if you want to pinpoint a location.\nI only started all this because of the nags having thought I was receiving all that was available, it now seems not. By the way, nothing had been moved or changed until the issue started.\nBefore replying to you I decided to check again. HD diagnostic quality on Ch43 had improved so I scanned Ch40 again and behold stations including Film4 appeared, poor diagnostics but a picture.\nLast night I sent all the information to UKDigital but am still awaiting a reply, I would be so pleased if somebody said there had been some work being done on Sedbergh.\nAs to my installation. It is a large 25 year old aerial in the loft. It blew off the roof one Christmas and we have had it in the loft ever since! I am thinking of changing the aerial and seeing if I can find a way of upgrading the old brown coax from loft to lounge. It is a divided loft so well away from plumbing. I have a booster near the TV and I use another channel of it to give a signal to a DAB device (Denon M39). The TV is Panasonic TX-P42G30B and has two HDMI devices attached - Panasonic Blu-ray and Humax Foxsat-HDR. We do have DECT in the same room but its base station and broadband router are the other side a 2' stone wall.\nIt is worth going through the elimination processes - however unlikely they seem - as if they are the cause they'll be easier to fix!\nsuggests you should get good reception on all channels - but only via Sedbergh. Local obstructions or trees between your aerial location and the transmitter aerials may cause some issues.\nA loft aerial may need adjusting to get equal reception on all frequencies due to 'standing wave' issues... Moving the aerial within a cube of space with sides of 3 ft or 1 m at most .. higher\/lower, left\/right, fore\/aft while always pointing directly to the transmitter may find the sweet spot where all 3 frequencies are received equally.\nAt 24km from a 100W relay you will need a highish gain aerial plus masthead amplifier... and the losses involved by not being outside (typically 10dB or 1\/10th of that outside) may be a big factor in your issues?\nRead the aerialsandtv website for a good background on aerial selection \/ choice.\nDo also go over all your aerial plug\/sockets\/connections to make sure they are undamaged, clean and soundly made - sometimes the simplest things can make the biggest difference!\nBought a Labgear 450W High Gain Aerial. Couldn't pull the old brown coax out of the trunking in my wall but using exactly all I had left on a drum of PH100 satellite cable I found a route with my network cables from loft to lounge. As a first try put the new aerial lower down on the same pole as the old aerial in the loft and connected it straight to the TV without using the booster. Switched on and got a depressing \"No Signal\" for a few seconds then the picture appeared. It was BBC2HD so I checked the quality - Green 10\/10!!!! Manually tuned all three channels to make sure I had all the stations and checked the quality of BBC1 from the one group and Film4 from the other - both Green 10. So system perfect, bit worried about tidying up for fear of disturbing something. I had an old circular FM aerial up there too and that works the Denon DAB so I don't need them to share. Digital UK have told me no work has been done on the Sedbergh transmitter so I still don't know what created the instability but at least it has had a happy outcome.\nQuestion Selling my gear. Where is best?\nQuestion Any way around my problem?\nQuestion Is my Receiver the problem?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblzfk b/data_all_eng_slimpj/shuffled/split2/finalzzzblzfk
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"I had been going through the motions for the last 6 months. Working, therapy, back to yoga, eating out with friends, mainly doing a lot on my own. Forcing myself to go for coffee breaks to see people. Stepping out of my comfort zone. Talking on the phone to my brothers daily who would check in. Texting a friend here or there for support.\nThe week before the 6 months anniversary I went back to my parents house. I was able to see people I had not seen in a while. My friends there have experienced loss of a similar nature which made me feel like I could ask questions and they understood what I was going through. Some had more experiences with depression and mental health which gave me a safe spot to talk about those topics. I was at home which has become a comfort to me. And, I had a long scheduled appointment with a medium and energy healer that week as well.\nSeeing the medium for me was everything. I had no answers until he came through and told me what he had been struggling with. The hour session felt like 4 hours. His presence was obvious. I had not told the medium anything and did not tell her who I hoped would appear either. When he came through it was very clear how he died, where he died, and who we were to each other based on what he showed the medium. He was in intense pain for a very long time. He was not able to use his words to explain what he was going through. He was suffering with internal issues that were nothing to do with me. He loves me and he is sorry he said. He told me to move forward and find happiness again. I am glad to know that he is now ok and pain-free living in heaven. He now knows how great he is and how loved. He has clarity. That is probably what made me feel the most relief. I have the questions answered that I needed. While it does not take the pain away it allowed me closure, acceptance, and hope.\n6 months ago my life was turned upside down. Not a day goes by where I do not miss him. He has finally been showing himself to me and is sending me messages. I know he is ok and peaceful. I wish we could go back in time and change the outcome but unfortunately that isn't a possibility. I told him that night I would live for the both of us and I will uphold that promise. Life for me won't be easy but I will be fine with him watching over me. I am so thankful that I got the advice to see a medium because I have received closure that I could never have gotten without her help. It is what I already knew deep down inside and though it will Never take the pain away from losing him it will help me heal just a little bit.\nLoss of a spouse is tough at any age but unexpected loss of a spouse shocks you to the core and changes everything. saying goodbye and that you love that person is pulled away from you.\nOur love was special even if it was cut short. I hope I can continue to make him proud and always recall our best memories.\nHalf a year apart how can it be already?\nI have not been sending him as many letters now and stopped putting up tons of Facebook posts. My thoughts have turned more inward in regards to my husband. I decided at around that same time to get moving on the blog. With his death being a suicide I always felt that I was not able to fully share my feelings. I worried that I was not keeping his privacy if I said too much. I worried that most people would not want to know or hear my posts and were ready for me to just get over it. My conversations shifted to the therapist office and to a few people who I text or call and check on me. Those people being there for me have made me continue to move in a positive way. It is not easy at all. But I do try. I have accepted the loss of my husband and I know he won't ever come back. He is by my side and rooting for me to enter the next phase of my life.\nI started to believe in hope for a future for me. And some days it is easier than others to buy into! As a matter of fact I am writing this a month later on Memorial Day weekend. This is a time of thankfulness to those Veteran's who fought for our freedoms. And also a time of friends, family, picnics, the start of summer fun. For me it has been lonely even though I went to two parties I went alone.\nI am homesick and missing my family. I am texting a friend that I would like to be able to hang out with in person. Do you have that friend who helps you see differently and expand your mind? It is not the friend you cry to, it is the friend who helps you move through with laughter, positivity, creativity, and just being there. It is not the friend you thought would be the one to listen or even come to you in this time of grief, but this person helps you in ways they do not even realize. Most of the other people have already stopped checking in with me at this point.\nAt month 7 my resolve, strength, resiliency is being tested as I navigate the new ocean of Widowhood. My need to be in control has been tested and shifted since the day I found him. Most of what I am doing is navigating a whole new ocean that I have never seen before. Nothing is black and white anymore, there is so much grey.\nI am looking forward to a trip home in 3 weeks as I so desperately need some me time. I am very hard on myself to figure everything out quickly. I know that it sounds like a lot of pressure and it is but there are reasons. Talking to another grieving widow she said how she also wants to move forward fast to get past the sadness she is feeling all the time. It is a process anyone grieving would want to move through quicker. I worry if I stay standing here I will not continue to improve and maybe I will get worse\u2026 life is short and fragile and fleeting and I worry I will miss something special if I stay stuck.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ORLANDO, Fla. \ufffd Nick Davilla threw six touchdown passes and the Arizona Rattlers defeated the Philadelphia Soul 48-39 in the Arena Bowl on Saturday.\nThe Rattlers (18-3) won the championship for the second straight year. They also beat the Soul (14-7) last season.\nKicker Garrett Lindholm made six extra points and both field goal attempts for Arizona.\nThe Soul had a chance to win in the final minute after recovering an Arizona fumble at its own 1-yard line. But Arkeith Brown, who had a third-quarter interception, broke up a pass on fourth down, and Lindholm made a field goal for a two-score cushion with 23 seconds remaining.\nRyan McDaniel and Derrick Ross each scored two touchdowns for Philadelphia.\nPARKER, Colo. \ufffd Karine Icher put an emphatic end to a stunning shutout that placed Europe on the verge of its first Solheim Cup win in America.\nIcher rolled in a 45-foot birdie putt from just off the back of the 18th green Saturday afternoon as she and Beatriz Recari hung on for a 1-up win, giving Europe its first sweep of a Solheim Cup session in 11 years.\nBy winning all the fourballs matches in the afternoon, the Europeans led 10\ufffd-5\ufffd, matching the largest lead in Solheim Cup history. With only the 12 singles matches remaining today, they only need to win three matches and halve another to capture the cup away from home for the first time since this event began in 1990.\nIt also would be the first time Europe retained the cup.\n\ufffdIt\ufffds a wonderful feeling,\ufffd European captain Liselotte Neumann said.\nGREENSBORO, N.C. \ufffd John Huh shot an even-par 70 on Saturday for a share of the lead with Patrick Reed after three rounds in the Wyndham Championship.\nHuh, the PGA Tour\ufffds reigning rookie of the year, and Reed, the second-round leader who shot a 71, were at 10-under 200.\nZach Johnson had the day\ufffds best round \ufffd a 66 \ufffd and was one stroke back along with Bob Estes and John Deere winner Jordan Spieth. Estes shot 68, and Spieth had a 70.\nMASON, Ohio \ufffd Unseeded American John Isner rallied from losing a first-set tiebreaker to upset seventh-seeded Juan Martin del Potro 6-7 (4), 7-6 (9), 6-3 in a grueling Western & Southern Open semifinal Saturday.\nThe 22nd-ranked Isner, in his first appearance in the tournament\ufffds semifinals, had been 0-4 against the Argentine. He\ufffdll play third-seeded Rafael Nadal, a 7-5, 7-6 (4) winner over sixth-seeded Tomas Berdych.\nThe 6-foot-10 Isner needed 2 hours, 47 minutes to improve to 16-3 since retiring from Wimbledon in the second round with a left knee injury.\nTop-ranked Serena Williams advanced to the women\ufffds final, beating defending champ Li Na, 7-5, 7-5.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bitcoin Wallet in the Philippines. Buy and Sell Bitcoin in the Philippines, Easy and Fast.\nA Bitcoin For Every Filipino. They are the leading Bitcoin exchange in the Philippines and they want everyone to easily and safely buy and sell Bitcoin.\nFounded in 2013, the company initially focused on growing bitcoin's merchant network in Asia, but Hose is expecting quick growth that will allow it to expand, in part due to the high costs of providing payment by other means.\nEarn free bitcoin by inviting friends!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With Christmas being around the corner it's time to learn how to flock your Christmas tree!\nRaise your hand if you love flocked Christmas trees!\nI think they are absolutely beautiful. I wanted to incorporate some into my Christmas decor this year but didn't want to spend too much.\nWhile I had some smaller flocked trees from previous years, I wanted a 4ft tree to display on the hearth of our fireplace.\nEverywhere I looked, they wanted almost $100 or more.\nI couldn't justify spending that much, no matter how gorgeous the trees looked.\nI'm not talking about spray paint or hairspray.\nI could spray the trees and get a similar effect!\nBetter yet, I remembered that I still had a can of the spray at home from the previous Christmas.\nNow all I had to do was be on the lookout for a four foot tree at a thrift store or online.\nIt didn't take long and I found the perfect one! (It pays to get a head start and go early on in the season with Christmas shopping for creating projects and crafts so you don't end up with the leftovers).\nWhile the pot was outdated, I had a basket at home that I knew would be perfect to display this tree in.\nBringing the tree home was exciting, I was eager to start spraying it and see how it would look.\nI forgot to take a photo of the snow spray, but you can usually find these at any local dollar store \u2013 Amazon has it too (it'll cost more though).\nI brought the tree downstairs and got to work.\nPlace your tree(s) on some plastic (similar to a drop cloth).\nSpray a thin layer on each side of the tree (I worked in four sections, aka four sides).\nIf you think it's not white enough yet, spray another layer.\nI ended up spraying the tree three times.\nThe bottle cautioned against spraying in thick layers .\nIf you spray too much on at a time, the flocking won't adhere to your tree branches properly, and you'll have a bigger mess since the \"extra\" will just fall all over the floor.\nSo I decided to be patient and just spray several layers instead of one big chunk.\nI was pleasantly surprised by how well it turned out.\nWhile it doesn't quite match up with those pre-flocked trees out in the stores, it still did the job and blended in nicely with my other trees on the mantel.\nMy \"Christmas Tree Farm\" Mantel is probably my favorite element of Christmas decor from this year, and these trees helped make it happen.\nHave you ever used snow spray for a project before? Let me know how it went in the comments below!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brisbane isn't short on great bars. If you're a wine lover though, you might be after somewhere with staff that are a little more knowledgeable and a wine list with some stronger contenders \u2013 we've got the best spots for you.\nNestled in Brisbane's inner northern suburb of Alderley, Ploughman is a gem of a bar. The owners are always friendly and knowledgeable about every drop they serve. It's the perfect venue to chat over a casual glass of wine, house-made pickles and cheese board.\nA few steps from the reputable Gerard's Bistro on the beautiful James Street is the equally fabulous Gerard's Bar. The bar serves Spanish cured meats and share plates, alongside a carefully curated wine and beer list. The staff will happily talk you through everything on their menu.\nToday's forecast: 99% chance of wine.\nThink vintage decor, comfortable chesterfields and a cosy vibe and you've got The Gresham. Set in a heritage-listed building, staff at The Gresham know their wine and will set you up with the perfect glass in the comfort of a gorgeous leather lounge.\nMaker is a hole-in-the-wall set up, but it packs a punch with its gorgeous design and superb wine list. They are also known for a unique take on their charcuterie, serving delicious goodies like venison, buffalo and wild boar.\nProud Henry is known for stocking unique international wines that are completely new to Brisbane's wine scene. Carefully curated, this bar has a wine list with only 12 beauties on it. If you're a gin lover, this is definitely the place for you too.\nBloodhound Bar is a great destination for craft beer and wine lovers. In the heart of the valley, you'll find a well selected wine list and a delicious bar menu.\nClaret House has become something of a well known establishment in Teneriffe. A wine lover is sure to be happy here with an extensive list of old and new wines from across the world to choose from \u2013 not to mention some delicious rustic tapas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmiyc b/data_all_eng_slimpj/shuffled/split2/finalzzzbmiyc
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+{"text":"Summary of Articles and Bylaws Revisions, 2016. Am J Occup Ther 2016;70(Supplement_2):7012410090p1. doi: 10.5014\/ajot.2016.706s2_Bylaws.\n\u2022 The addition of language for establishment of a quorum at the commencement of a meeting, which remains established for its duration as defined under Washington, DC, nonprofit law.\n\u2022 The elimination of President-Elect as an Officer of the Board, because Washington, DC, nonprofit law does not recognize the President-Elect as an officer. The President-Elect continues to serve as a member of the Board with voice only.\n\u2022 The elimination of the Assembly of Student Delegates (ASD) as an Organizational Advisor to the Board, because ASD is now a Standing Advisory Committee of the Board.\n\u2022 The elimination of the term resolutions as a format by which members may propose actions to the Representative Assembly (RA). A motion is the appropriate parliamentary procedure for proposing any action, including the adoption of a resolution. Hence, there is no need to specify the term resolution in the Bylaws.\n\u2022 The clarification of open meetings to be consistent with Association Policy A.6.\n\u2022 Changes were made to the Term of Office section for Representatives of the RA to clarify the process for filling vacancies.\n\u2022 The addition of language regarding revocation of membership for cause other than nonpayment of dues or violation of the Occupational Therapy Code of Ethics, which clarifies and defines cause in conformance with Washington, DC, nonprofit law.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Contact our Virginia dealership at 877-944-6766.\nYour feedback is important to us so we encourage you to fill out the form below, or call us at 800-700-4391 with any comments or concerns. We look forward to hearing from you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My little mind is made up! Don't try to use logic and facts on me.\nThis entry was posted in Critical Thinking and tagged blind acceptance, cognitive dissonance, critical thinking by Arthur Ruger. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Below are some upcoming events for Singles in the area. In addition to utilizing our extensive Client base, our Matchmaking Team also attends events in search of additional matches for our Clients.\n\u00a9 2019 Tampa Bay MatchMakers. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"EULEN Healthcare Services wants to contribute to the training of professional and family caregivers who are going to look after people in situations of dependency through innovative and quality training. This training promotes integration into the job market and contributes to improving the quality of life of people who are cared for.\nWith our training we aim to guide and promote the development of professionals as people, with the tremendous opportunities that this entails.\n1. Training of family caregiver\/s.\n2. Training and qualification of professionals belonging to the family of sociocultural and community services.\nLaw 39\/2006 on the Promotion of Personal Autonomy and Care for people in situations of dependency defines non-professional care, as care provided to people in situations of dependency in their home, by people in the family or their environment, not linked to a professional care service.\nEULEN Healthcare Services aims to help all those people who daily contribute to improving the quality of life for people in situations of dependency, allowing them to have proper treatment in their personal, family and social life.\nPersonal development of the caregiver.\nCompetencies and skills for care.\nExisting resources and generation of social support networks.\nSpecific training plans for dependency care for professionals and caregivers.\n(Quality certifications: Excellence EFQM+400, ISO 9001:2008, ISO 14001:2004 e ISO 27001:2005).\nWe look after people in their own homes.\nThis business area includes the organization and management of healthcare and critical services in the home.\nWe open up new communication channels to be closer to you. We invite you to follow us and participate in social networks to keep up to date with our news and activities and provide you with even more direct and personalized care.\nThe quality of our services are properly certified, the transparency of our management, the commitment to permanence with society and respect for the environment, are the basis on which the corporate responsibility policies EULEN settle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..ea8b64409f3d536ff02e700a34a3844db247d9a6
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+{"text":"That last one has a pretty cringeworthy pic attached if you click through to the link. In my defence, white Irish girls are to hip-hop as Michael Jackson is to sunlight.\nRight - off again to the airport. I don't believe the term 'jet-set' can be employed when flying Ryanair. Adios Espana, hi there Orlando!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take a look at what we've been developing for Anheuser-Busch, the world's largest brewer and the name behind world-famous brands Budweiser, Stella Artois and Beck's. Anheuser-Busch is using Plexure technology to overcome prohibition-era restrictions and sell more beer in United States bars, by allowing bar owners to create 'specials' on their smartphones. The promotions can then be displayed as captions that are shown on top of TV sports broadcasts playing in their bars. As well as manually setting up specials that can be timed to the minute, promotions can also be automatically triggered by sensors in bar ceilings that detect when bars are getting busier or quietening down.\nUS prohibition laws mean brewers aren't allowed to be directly involved in alcohol retail, so they have little insight into what happens inside bars - or what's working to drive sales. While Anheuser-Busch could already use beer tap sensors to tell how much beer was being sold and when, VMob's platform enables it to measure the impact of promotions, which is otherwise hard to do. The technology is being piloted in bars in Maine and Rhode Island and has already resulted in a material lift in sales.\nOlder PostUnified commerce \u2013 more omni than omnichannel?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"#67 Miniature Bulb Ba15S Base - 13.5 Volt .59 Amp 7.965 Watt G-6 Single Contact (SC) Bayonet (BA15S) Base, C-2R Filament Design, 4.0 MSCP. 1.44\" (36.58mm) Maximum Overall Length (M.O.L.), 0.75\" (19.05mm) Maximum Outer Diameter (M.O.D.), 0.75\" (19.05mm) Light Center Length (L.C.L.). 5,000 Average Rated Hours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We're proud to have the support of a firm with the highest loyalty ranking in the investment industry. Ameriprise Financial has always put clients first; that's how loyalty is earned.\nWe offer personalized, comprehensive financial advice and a broad range of solutions to help you and your loved ones live brilliantly, now and in the future.\n1 Rating based on responses to a consumer survey as part of the Temkin Loyalty Index, 2017. See the Ameriprise Financial recognition page and temkingroup.com for more.\nClient Satisfaction Rating: Each client has the opportunity to rate a practice based on the client's overall satisfaction with the practice, on a scale of 1 to 5. The practice's Client Satisfaction Rating is an average of all client responses for that practice within a rolling 24-month period, and is updated daily. There is no minimum number of responses required for a practice to receive a rating.\nClient experiences may vary, and working with any Ameriprise financial practice is not a guarantee of future financial results. Investors should not consider this rating a substitute for their own research and evaluation of a financial practice's qualifications. Not all clients may respond to these questions, and only clients with access to the Ameriprise Secure Client Site may submit a rating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eyewear is like makeup \u2013 the right pair can change about your whole look! The Easy E Locs Sunglasses from our collection is just the right pair for anyone who loves to flaunt what they have got! Sporting an Easy E Locs Sunglasses will instantly get you a funky, groovy, and jazzy look at one go. These are perfect to flaunt and get 100% UV protection at the same time as well. Available at affordable prices, the Easy E Locs Sunglasses from our collection are the best thing to gift your friends as well. Super cool and super comfy, these Easy E Sunglasses are something not to be missed for. It is not always that you get superior quality sunglasses at affordable rates. So don't miss this opportunity to choose your favorite pair from our collection of Easy E Locs Sunglasses and get an edge over others with your own style statement.\nAll sunglasses are sold by the pair. Minimum order is $100.00 . FREE SHIPPING when you order $200.00 or more!","meta":{"redpajama_set_name":"RedPajamaC4"}}